(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Logarithmic and trigonometric tables : five place and four place"




UC-NRLF 

III I Illl 

SB 35 



IN MEMORIAM 
FLOR1AN CAJORI 




PHILLIPS-LOOMIS MATHEMATICAL SERIES 



ELEMENTS OF TRIGONOMETRY 



PLANE AND SPHERICAL 



BY 

ANDREW W. PHILLIPS, PH.D. 

AM> 

WENDELL M. STRONG, PH.D. 

VALK UN1VKRSITV 




NEW YORK AND LONDON 

HARPER & BROTHERS PUBLISHERS 
1899 



THE PHILLIPS-LOOMIS MATHEMATICAL SERIES. 



ELEMENTS OF TRIGONOMETRY, Plane and Spherical. By 
ANDREW W. PHILLIPS, Ph.D., and WENDELL M. STRONG, Ph.D., Yale 
University. Crown 8vo, Half Leather. 

ELEMENTS OF GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., 
and IRVING FISHER, Ph.D., Professors in Yale University. Crown 
8vo, Half Leather, $1 75. [By mail, $1 92.] 

ABRIDGED GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., and 
IRVING FISHER, Ph.D. Crown 8vo, Half Leather, $1 25. [By 
mail, $1 40.] 

PLANE GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., and IRVING 
FISHEK, Ph.D. Crown 8vo, Cloth, 80 cents. [By mail, 90 cents.] 

LOGARITHMIC AND TRIGONOMETRIC TABLES. Five- Place 
and Four-Place. By ANDREW W. PHILLIPS, Ph.D., and WKNDKI.I. 
M. STRONG, Ph.D., Yale University. Crown 8vo. 

LOGARITHMS OF NUMBERS. Five- Figure Table to Accompany 
the "Elements of Geometry," by ANDREW W. PHILLIPS, Ph.D., and 
IKVING FISHER, Ph.D., Professors in Yale University. Crown 8vo, 
Cloth, 30 cents. [By mail, 35 cents.] 



NEW YORK AND LONDON : 
HARPER & BROTHERS, PUBLISHERS. 



Copyright, 1898, by HARPER & BROTHERS. 



All rights reserved. 



PREFACE 



IN this work the trigonometric functions are defined as 
ratios, but their representation by lines is also introduced at 
the beginning, because certain parts of the subject can be 
treated more simply by the line method, or by a combination 
of the two methods, than by the ratio method alone. 

Attention is called to the following features of the book : 

The simplicity and directness of the treatment of both 
the Plane and Spherical Trigonometry. 

The emphasis given to the formulas essential to the solu- 
tion of triangles. 

The large number of exercises. 

The graphical representation of the trigonometric, inverse 
trigonometric, and hyperbolic functions. 

The use of photo-engravings of models in the Spherical 
Trigonometry. 

The recognition of the rigorous ideas of modern math- 
ematics in dealing with the fundamental series of trigo- 
nometry. 

The natural treatment of the complex number and the 
hyperbolic functions. 

The graphical solution of spherical triangles. 

Our grateful acknowledgments are due to our colleague, 
Professor James Pierpont, for valuable suggestions regard- 
ing the construction of Chapter VI. 

We are also indebted to Dr. George T. Sellew for making 
the collection of miscellaneous exercises. 

ANDREW W. PHILLIPS, 
WENDELL M. STRONG. 

YALE UNIVERSITY, December, 1808. 



TABLE OF CONTENTS 



PLANE TRIGONOMETRY 
CHAPTER I 

THE TRIGONOMETRIC FUNCTIONS 

PAGE 

Angles i 

Definitions of the Trigonometric Functions 4 

Signs of the Trigonometric Functions .8 

Relations of the Functions 10 

Functions of an Acute Angle of a Right Triangle 13 

Functions of Complementary Angles 14 

Functions of o, 90, 1 80, 270, 360 15 

Functions of the Supplement of an Angle 16 

Functions of 45, 30, 60 17 

Functions of ( .r), (180 .r), (i8o+.r), (360 .r) 18 

Functions of (90 y), (90 +j), (270 y\ (270+^) 20 

CHAPTER II 

THE RIGHT TRIANGLE 

Solution of Right Triangles 22 

Solution of Oblique Triangles by the Aid of Right Triangles . . 28 

CHAPTER III 

TRIGONOMETRIC ANALYSIS 

Proof of Fundamental Formulas (i i)- (14) 32 

Tangent of the Sum and Difference of Two Angles 36 

Functions of Twice an Angle 36 

Functions of Half an Angle 36 

Formulas for the Sums and Differences of Functions 37 

The Inverse Trigonometric Functions 39 



vi TABLE OF CONTENTS 

CHAPTER IV 

THE OBLIQUE TRIANGLE 

PAGE 

Derivation of Formulas 41 

Formulas for the Area of a Triangle 44 

The Ambiguous Case 45 

The Solution of a Triangle : 

(i.) Given a Side and Two Angles 46 

(2.) Given Two Sides and the Angle Opposite One of Them . 46 

(3.) Given Two Sides and the Included Angle 48 

(4.) Given the Three Sides 49 

Exercises 50 

CHAPTER V 

CIRCULAR MEASURE GRAPHICAL REPRESENTATION 

Circular Measure 55 

Periodicity of the Trigonometric Functions 57 

Graphical Representation 58 

CHAPTER VI 

COMPUTATION OF LOGARITHMS AND OF THE TRIGONOMETRIC FUNC- 
TIONS DE MOIVRE'S THEOREM HYPERBOLIC FUNCTIONS 

Fundamental Series 63 

Computation of Logarithms 64 

Computation of Trigonometric Functions 68 

De Moivre's Theorem 70 

The Roots of Unity 72 

The Hyperbolic Functions 73 

CHAPTER VII 

MISCELLANEOUS EXERCISES 

Relations of Functions . . 7 

Right Triangles 80 

Isosceles Triangles and Regular Polygons 83 

Trigonometric Identities and Equations 84 

Oblique Triangles 88 



TABLE OF CONTENTS vii 



SPHERICAL TRIGONOMETRY 
CHAPTER VIII 

RIGHT AND QUADRANTAL TRIANGLES 

PAGE 

Derivation of Formulas for Right Triangles 93 

Napier's Rules 94 

Ambiguous Case 97 

Quadrantal Triangles 98 

CHAPTER IX 

OBLIQUE-ANGLED TRIANGLES 

Derivation of Formulas 100 

Formulas for Logarithmic Computation 101 

The Six Cases and Examples 104 

Ambiguous Cases 106 

Area of the Spherical Triangle 108 

CHAPTER X 

APPLICATIONS TO THE CELESTIAL AND TERRESTRIAL SPHERES 

Astronomical Problems no 

Geographical Problems 113 

CHAPTER XI 

GRAPHICAL SOLUTION OF A SPHERICAL TRIANGLE 115 

CHAPTER XII 

RECAPITULATION OF FORMULAS 119 

APPENDIX 

RELATION OF THE PLANE, SPHERICAL, AND PSEUDO-SPHERICAL 

TRIGONOMETRIES 125 



ANSWERS TO EXERCISES 129 



PLANE TRIGONOMETRY 



CHAPTER I 
THE TRIGONOMETRIC FUNCTIONS 

ANGLES 

1. In Trigonometry the size of an angle is measured by 
the amount one side of the angle has revolved from the 
position of the other side to reach its final position. 

Thus, if the hand of a clock makes one-fourth of a rev- 
olution, the angle through which it turns is one right angle ; 
if it makes one-half a revolution, the angle is two right an- 
gles; if one revolution, the angle is four right angles; if one 
and one-half revolutions, the angle is six right angles, etc. 




O' 



B 

FIG. 2 




FIG. 3 



The amount the side OB has rotated from OA to reach its final position 
may or may not be equal to the inclination of the lines. In Fig. i it is equal 
to this inclination ; in Fig. 4 it is not. 

Two angles may have the same sides and yet be different. In Fig. 2 

I 



PLANE TRIGONOME TR 1 ' 



and Fig. 4 the positions of the sides of the angles are the same ; yet in 
Fig. 2 the angle is two right angles, in Fig. 4 it is six right angles. The 
addition of any number of complete revolutions to an angle does not change 
the position of its sides. 

Question. Through how many right angles does the hour-hand 
of a clock revolve in 6^ hours? the minute-hand ? 

Question. If the fly-wheel of an engine makes 100 revolutions per 
minute, through how many right angles does it revolve in i second ? 




Initial line 




RIGHT ANGLES 



Initial line 



5} RIGHT ANGLES 



Def. The first side of the angle that is, the side from 
which the revolution is measured is the initial line; the 
second side is the terminal line. 

Def. If the direction of the revolution is opposite to that 
of the hands of a clock, the angle is positive; if the same 
as that of the hands of a clock, the angle is negative. 

Initial line 




Initial Line 

POSITIVE ANGLE 




NEGATIVE ANGLE 



The angles we have employed as illustrations those described 

by the hands of a clock are all negative angles. 

2. Angles are usually measured in degrees, minutes, and 
seconds. A degree is one-ninetieth of a right angle, a min- 
ute is one-sixtieth of a degree, a second is one-sixtieth of a 
minute. 



THE TRIGONOMETRIC FUNCTIONS 



The symbols indicating degrees, minutes, and seconds are ' "; 
thus, twenty-six degrees, forty-three minutes, and ten seconds is 
written 26 43' 10". 

3. The plane about the vertex of an angle is divided into 
four quadrants, as shown in the figure; the first quadrant 
begins at the initial line. 



in 




IV 




THE FOUR QUADRANTS 



II 



III 



ANGLE IN 1ST QUADRANT 



ANGLE IN 2D QUADRANT 





ANGLE IN 3D QUADRANT 



ANGLE IN 4TH QUADRANT 



An angle is said to be in a certain quadrant if its terminali 
line is in that quadrant. 

EXERCISES 

4. (i.) Express 2\ right angles in degrees, minutes, and seconds^ 
In what quadrant is the angle? 

(2.) What angle less than 360 has the same initial and terminal 
lines as an angle of 745? 

(3.) What positive angles less than 720 have the same sides as am 
angle of 73? 

(4.) In what quadrant is an angle of 890? 



4 PLANE TRIGONOMETRY 

DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS 

5. The trigonometric functions are numbers, and are de- 
fined as the ratios of lines. 

Let the angle A OP be so placed that the initial line is 
horizontal, and from P y any point of the terminal line, draw 
PS perpendicular to the initial line. 




S A 



ANGLE IN THE 1ST QUADRANT 





ANGLE IN THE 2D QUADRANT 




ANGLH IN THE 30 QUADRANT 



Denote the angle A OP by x. 
SP 



ANGLE IN THE 4TH QUADRANT 



- = sine of x (written sin^r). 



= cosine of x (written cos^r). 



THE TRIGONOMETRIC FUNCTIONS 



SP 

-y^ tangent of x (written tan x). 

)= cotangent of x (written cot x). 
OP 




OP 



= secant of x (written sec^r). 



= cosecant of x (written csc^r). 

To the above may be added the versed sine (written versin) and coversed 
sine (written coversin), which are defined as follows : 

versiii x = i - cos a?; coversiii x = i sin x. 

The values of the sine, cosine, etc., do not depend upon 
what point of the terminal line is taken as P, but upon the 
angle. 





S S' 



S'S 



x 



For the triangles OSP and OS'P' being similar, the ratio of any 
two sides of OS'P' is equal to the ratio of the corresponding sides 
of OSP. 

Def. The sine, cosine, tangent, cotangent, secant, and 
cosecant of an angle are the trigonometric functions 
of the angle, and depend for their value on the angle 
alone. 

0. A line may by its length and direction represent a 
number; the magnitude of the number is expressed by the 
Icngtli of the line ; the number is positive or negative ac- 
cording to the direction of the line. 



PLANE TRIGONOMETRY 



7. In 5, if the denominators of the several ratios be 
taken equal to unity, the trigonometric functions will be rep- 
resented by lines. 

SP SP 
Thus, sin^r= -~p= - = SP the number represented by 

the line, that is, the ratio of the line to its unit of length. 

Hence SP may represent the sine of x. 

In a similar manner the other trigonometric functions 
may be represented by lines. 

In the following figures a circle of unit radius is described 
about the vertex O of the angle A OP, this angle being de- 
noted by x. Then from 5 it follows that 





FIG. a 



Cot 




C Cot B 




FIG. 3 



FIG 4 



THE TRIGONOMETRIC FUNCTIONS 7 

SP represents the sine of x. 
OS represents the coiiie of x. 
A T represents the tangent of x. 
BC represents the cotangent of x. 
0/^represents the secant of x. 
OC represents the coecant of x. 

For the sake of brevity, the lines SP, OS, etc., of the preceding figures are 
often spoken of as the sine, cosine, etc. 

Hence, we may also define the trigonometric functions 
in general terms as follows : 

If a circle of unit radius is described about the vertex of 
an angle, 

(r.) The iine of the angle is represented by the perpendicular 
upon the initial line from the intersection of the terminal line with 
the circumference. 

(2.) The cosine of the angle is represented by the segment of the 
initial line extending from the vertex to the sine. 

(3.) The tangent of the angle is represented by a line tangent to 
the circle at the beginning of the first quadrant, and extending from 
the point of tangency to the terminal line. 

(4.) The cotangent of the angle is represented by a line tangent 
to the circle at the beginning of the second quadrant, and extending 
from the point of tangency to the terminal line. 

(5.) The secant of the angle is represented by the segment of the 
terminal line extending from the vertex to the tangent. 

(6.) The cosecant of the angle is represented by the segment of 
the terminal line extending from the vertex to the cotangent. 

The definitions in 5 are called the ratio definitions of the trigonometric- 
functions, and those in 7 the line definitions. The introduction of two 
definitions for the same thing should not embarrass the student. We have 
shown that they are equivalent. In some cases it is convenient to use the 
first definition, and in other cases the second, as the student will observe 
in the course of this study. It is therefore important that he should be- 
come familiar with the use of both. 



8 



PLANE TRIGONOMETRY ' 



SIGNS OF THE TRIGONOMETRIC FUNCTIONS 

8. Lines are regarded as positive or negative according 
to their directions. Thus, in the figures of 5, OS is posi- 
tive if it extends to the rigJit of O along the initial line, 
negative if it extends to the left ; SP \s positive if it extends 
upward from OA, negative if it extends downward. OP, the 
terminal line, is always positive. 

The above determines, from 5, the signs of the trigono- 
metric functions, since it shows the signs of the two terms 
of each ratio. 

By the line definitions the signs may be determined di- 
rectly. The sine and tangent are positive if measured up- 
ward from OA, and negative if measured downward. 

The cosine and cotangent are positive if measured to the 
right from OB, and negative if measured to the left. 

B Cot + Cot- B 





TIG. 3 



FIG. 4 



THE TRIGONOMETRIC FUNCTIONS 



The secant and cosecant are positive if measured in the 
same direction as the terminal line, OP\ negative if measured 
in the opposite direction. 

The signs of the functions of angles in the different quadrants are as follows : 



Quadrant 


I 


II 


Ill 


IV 


Sine and cosecant 


+ 
+ 


+ 


- 


- 


Cosine and secant 


- 


- 


+ 


Tangent and cotangent 


+ 


- 


+ 


- 



f>. It is evident that the values of the functions of an 
angle depend only upon the position of the sides of the 
angle. If two angles differ by 360, or any multiple of 360, 
the position of the sides is the same, hence the values of 
the functions are the same. 



C cot B 





Thus in Fig. i the angle is 120, in Fig. 2 the angle is 840, yet 
the lines which represent the functions are the same for both angles. 

EXERCISE 

Determine, by drawing the necessary figures, the sign of tan 1000; 
cos 810; sin 760; cot 70; cos 550; tan 560; sec 300; cot 
1560; sin 130; cos 260; tan 310. 



10 



PLANE TRIGONOMETRY 



RELATIONS OF THE FUNCTIONS 
10. By 5, whatever may be the length of OP, we have 



SP 



OS 



OP smjr ' OP * x ' n ^' 



SP 




OS 



OP 





We have, then, from Figs. 2 and 3, 



SP _ _ sinx 

OS- - 



OS COS X 

- f*Ol JC - 

SP~ " 



or 



Multiplying (i) by (2), 

tana? 

i 

cot x 

Again, from Figs. 2 and 3, 
OP 



tan x 



_ 

co*x* 

OP 1 

= CC X = - . 

SP sin a? 

From Figs. 2 and 3, OS 2 -f SP*= OP\ 
or 
and 

Also, O 
or I iKtan 2 o?=r sec 2 .x; 



OP 



B Cot 




sin' 2 jf = i cos 2 jr ; cos 9 x= i sin 2 jr. 

*=OT\ and 



FIG. 3 



(0 

(2) 

(3) 



(4) 

(5) 

(6) 



(7) 
(8) 



THE TRIGONOMETRIC FUNCTIONS u 

The angle x has been taken in the first quadrant ; the 
results are, however, true for any angle. The proof is the 
same for angles in other quadrants, except that SP be- 
comes negative in the third and fourth quadrants, and OS 
in the second and third. 

EXERCISES 

11. (i.) Prove cos-r sec.i-= i. 
(2.) Prove sin^r cscx = i. 
(3.) Prove tan x cos x sin x. 

(4.) Prove sin x \ f \ cos'- x = i cos 2 .r. 

(5.) Prove tan x + cot .r - - l - -- 
sin.r cos.r 

(6.) Prove sin 4 .r Cos 4 .r = i 2 cos'.r. 

(7.) Prove = sin.r. 

cot-r sec.r 

(8.) Prove tan x sin .r -h cosx = sec;tr. 

12. The formulas (i)-(8) of 10 are algebraic equations 
connecting the different functions of the same angle. If 
the value of one of the functions of an angle is given, we 
can substitute this value in one of the equations and solve 
to find another of the functions. Repeating the process, we 
find a third function, etc. 

In solving equation (6), (7), or (8) a square root is extracted; 
unless something is given which determines whether to choose the 
positive or negative square root, we get two values for some of 
the functions. The reason for this is that there are two angles 
less than 360 for which a function has a given value. 

EXERCISES 

13. (i.) Given x less than 90 and sin.r = ^; find all the other 
functions of x. 

Solution. 



=r rb \/ i 1 
Since x is less than 90, we kno\v that cosx is positive. 



12 PLANE TRIGONOMETRY 

Hence cosx= + 



$3 

---- 

2 



(2.) Given tan.r = and .r in quadrant IV; find sin x and cos jr. 

Solution, 



_ _ , 
COS.T ~ 
hence 3 sin .*=: COSJF, 

sin 2 .* + cos 2 .* = i ; 
hence 10 sin 2 jr = i ; 



(3.) Given sin( 30) = \\ find the other functions of 30. 

(4.) Given x in quadrant III and sin^r = ; find all the other 
functions of x. 

(5.) Given y in quadrant IV and sin/=z |, find all the other 
functions of y. 

(6.) Given cos 6o = ; find all the other functions of 60. 

(7.) Given sin o=:o; find coso and tano. 

(8.) Given tans'rrjand z in quadrant I; find the other functions 
of JT. 

(9.) Given cot45= i ; find all the other functions of 45. 

(10.) Given tan/=\/5 and cos^> negative; find all the other 
functions of y. 

(ii.) Given cot 30= \/3 i fid tne other functions of 30. 

(12.) Given 2 sin-r=i cos^r and x in quadrant II; find sin^r 
and cos jr. 

(13.) Given tan.r4-cot;r = 3 and x in quadrant I ; find sin jr. i 



THE TRIGONOMETRIC FUNCTIONS 13 

FUNCTIONS OF AN ACUTE ANGLE OF A RIGHT TRIANGLE 

14. The functions of an acute angle of a right triangle 
can be expressed as ratios of the sides of the triangle. 




Remark. Triangles are usually lettered, as in Fig. 2, the capital 
letters denoting the angles, the corresponding small letters the sides 
opposite. 

In the right triangle ABC, by 5, 

BC a 



15. From 14, for an acute angle of a right triangle, we have 

side opposite angle 
sme = -- s-s- - ^ ; 
hypotenuse 

side adjacent to angle 
cosine = - ^ - ', 
hypotenuse 

tang ent = . sjde opposite angle 
side adjacent to angle 

side adjacent to angle 
cotangent = , - ' 

side opposite angle 



PLANE TRIGONOMETRY 



FUNCTIONS OF COMPLEMENTARY ANGLES 

16* From 14, we have 
sin ^L = cos^ 



(o) 
tan A = cot B = cot (9O - A) 5 

cot A = tan B = tan (9O A). 

Because of this relation the sine and cosine are called co-func- 
tions of each other, and the tangent and cotangent are called co- 
functions of each other. 

The results of this article may be stated thus: 
A function of an acute angle is equal to the co-function of 
its complementary angle. 

Ths values of the functions of the different angles are given in " Trigo- 
nometric Tables." By the use of the principle just proved, each function 
of an angle between 45 and 90 can be found as a function of an angle less 
than 45. Consequently, the tables need to be constructed for angles up to 
45 only. The tables are so arranged that a number in them can be read 
either as a function of an angle less than 45 or as the co-function of the 
complement of this angle. 

EXERCISES 

17. (i.) Express as functions of an angle less than 45: 

sin 70 ; cos 89 30' ; tan 63 ; 

cos66; cot47; sin7239'. 

(2.) cosjr = sin ix ; find x. 
(3.) tan x = cot 3-r ; find x. 
(4.) sin 2^r = cos3-r ; find x. 
(5.) cot(3O x) = tan(3o-J- $x) ; find x. 
(6.) A, B, and C are the angles of a triangle; prove that 



Hint. A+B + C=\ So . 






THE TRIGONOMETRIC FUNCTIONS 



FUNCTIONS OF O, 90, l8o, 270, AND 360 

18. As the angle x decreases towards o (Fig. i), sin* de- 
creases and cos* increases. When OP comes into coincidence 
with OA, SP becomes o, and OS becomes OA( = i). 
Ik-nee sino = o. coso=i. 




FIG. 3 



FIG. 4 



As the angle x increases towards 90 (Fig. 2), sin* increases 
and cos* decreases. When OP comes into coincidence with OB, 
SP becomes OB(\) and OS becomes o. 
Hence sinQO ^!, cosgo^o. 

As the angle x decreases towards o (Fig. 3), tan* decreases 
and cot* increases. When OP comes into coincidence with OA, 
^/"becomes o and BC has increased without limit. 
Hence tanor=o, coto=:oo. 

As the angle x increases towards 90 (Fig. 4), tan* increases 
and cot* decreases. When OP comes into coincidence with OB, 
AT has increased without limit, and BCo. 
Hence 



Remark. liy coto=co we mean that as the angle approaches indefinitely 
near to o its cotangent increases so as to become greater than any finite quan- 
tity we may choose. The symbol co does not denote a definite number, but 
simply that the number is indefinitely great. 



i6 



PLANE TRIGONOMETRY 



Tn every case where a trigonometric function becomes indefinitely 
great it is in a positive sense if the angle approaches the limiting 
value from one side, in a negative sense if the angle approaches the 
limiting value from the other side. Thus cot o = -|- oo if the angle 
decreases to o, but cot o= oo if the angle increases from a nega- 
tive angle to o. We shall not often need to distinguish between 
H-oo and oo, and shall in general denote either by the symbol oo. 

By a similar method the functions of 180, 270, and 360 may be 
deduced. The results of this article are shown in the following table : 



Angle 





90 


1 80 


270 


360 


sin 





I 





I 





cos 


i 





-I 


O 


I 


tan 


o 


oo 


o 


00 


o 


cot 


00 


O 


CO 





00 



19. It may now be stated that, as an angle varies, its sine and cosine 
can take on "values from / to -\- r only, its tangent and cotangent all 
values from oo to -\- oo , its secant and cosecant all values from oo 
to -j- oo , except those between / and -f- /. 



FUNCTIONS OF THE SUPPLEMENT OF AN ANGLE 
20. Suppose the triangle OPS (Fig. i) equal to the tri- 
angle OP'S ' (Fig. 2), then SP=S'P f and OS=OS', and the 
angle AOP' (Fig. 2) is equal to the supplement of A OP 
(Fig. i). Also, in the triangle AOP' (Fig. 3), angle AOP' 
= angle AOP' (Fig. 2). 

V 






8' O 

FIG. 2 



FIG. 3 



THE TRIGONOMETRIC FUNCTIONS 



It follows from 5 and 8 that 

in(lO x) = in 

C0 (1O 05) = C , 

tan (10 - x) = - tan x ; ' 
cot (1O a?) = cot x. 

The results of this article may be stated thus : 

The sine of an angle is equal to the sine of its supplement, 

and the cosine, tangent, and cotangent are each equal to minus 

the same functions of its supplement. 

The principle just proved is of great importance in the solution of tri- 
angles which contain an obtuse angle. 

FUNCTIONS OF 45, 30, AND 60 

21. In the right triangle OSP (Fig. i) angle O = angle P = 4$* 
and OP i. 

Hence OS = SP = i \/2. 

Therefore sin 45 = 00545 = 1/2; 14,16 

tan 45 = cot 45= i. 

P 





8 
FIG. I 



c 

O 
FIG. 2 



In equilateral triangle 0/^4 (Fig. 2) the sides are of unit length 
PS bisects angle OP A, is perpendicular to OA, and bisects OA. 
Hence, in the right triangle OPS, OS = %, SP = ^\/^. 
Therefore sin 30 = cos 60 = i; 14 

cos 30 = sin 60 = i \/3 I 
tan 30 = cot 60 = J y'3 ; 
cot 30 = tan 60 = \/3- 
2 



1 8 PLANE TRIGONOMETRY 

22. The following values should be remembered : 



Angle 





30 


45 


60 


90 


sin 





i 


iv/2 


iVi 


I 


cos 


i 


4V5 


4^2 


k 






EXERCISES 
Prove that if x = 30, 

(i.) sin 2x = 2 sin^r COS.T; 

(2.) cos 3* = 4 cos 3 .r 3 cos x ; 

(3.) cos 2;r =r cos' 2 ;r sin 2 .r; 

(4.) sin 3-r= 3 sin x cos 2 -r sin 3 .r; 

2 tan JT 

(5.) tan2;r = . 
i tan-jir 

(6.) Prove that the equations of exercises i and 3 are cor- 
rect if .r = 45 . 

(7) Prove that the equations of exercises (2) and (4) are cor- 
rect if .r=i20 . 



The following three articles, 23-25, are inserted for 
completeness. They include the functions of (90 ;r) and 
(180 x\ which, on account of their great importance, were 
treated separately in 16 and 20. 



FUNCTIONS OF (x), (l8o x\ (l8o + ;r), (360 -*) 

23. The line representing any function as sine, cosine, etc. 
of each of these angles has the same length as the line repre- 
senting the same function of x. 

Thus in Figs. 2 and 3, triangle OS'P' triangle OSP, hence SP-^S'P', 



THE TRIGONOMETRIC FUNCTIONS 





FIG. 3 



FIG. 4 



In Figs, i and 4, triangle OSP' triangle OSP, hence SP'=SF. 

In Figs, i, 2, and 4, triangle OA T = triangle OA T, hence A T ' = A T. 

In Figs, i, 2, and 4, triangle 0/?C' = triangle OBC. hence BC'=BC. 

Therefore any function of each of the angles ( x\ (180 x), 
(i8o-f *), (360 x\ is equal in numerical value to the same function 
of x. Its sign, however, depends on the direction of the line repre- 
senting it. 

Putting in the correct sign, we obtain the following table: 



sin ( x] sin x 
cos ( x) = cos* 
tan( *) = tan* 
cot ( x) cot x 

sin (180 + *)= sin* 
cos (180 + *)= cos* 
tanCi8o-f*) = tan* 
cot (180 + *) = cot* 



sin ( 1 80 *) = sin * 
cos (180 *) cos* 
tan(i8o *) tan* 
cot (180 - *) = - cot* 

sin (360 *) = sin * 
cos (360 *) cos* 
tan (360 *) = tan * 
cot (360 *) = - cot * 



20 



PLANE TRIGONOMETRY 



FUNCTIONS OF ( 9 O -7), ( 9 O +j), (270 -y\ (270 +J/) 
4. The line representing the sine of each of these angles is 
of the same length as the line representing the' cosine of y; the 
cosine, tangent, or cotangent, respectively, are of the same length 
as the sine, cotangent, and tangent of y. 





FIG. 3 



For 




Triangle OS'P' = triangle OSP, hence S'P' = OS, and OS' = SP. 
Triangle OA T' triangle OBC, hence A T' = BC. 
Triangle OBC triangle OA T, hence BC AT. 

Therefore any function of. each of the angles (90 y\ (90 -\-y\ 
(270^), (270+^), is equal in numerical value to the co-function 



THE TRIGONOMETRIC FUNCTIONS 21 

of y. Its sign, however, depends on Ihe direction of the line repre- 
senting it. 

Putting in the correct sign, we .obtain the following table : 
sin (90 y) = cos v sin (90 -f y) = cosy 

cos (90 y) sin r cos (90 + y) = sinjv 

tan (90 r) = cot v tan (90 +}')= cot v 

cot (90 y) - tan y cot (90 + y) = tan y 

sin (270 }')= cosy sin (270 +/) = cosy 

cos (270 y) sin y cos (270 +^) = siny 

tan (270 .r) = cot y tan (270 +y) = coty 

cot (270 - y) = tan r cot (270 + r) = tan v 

25. Either of the two preceding articles enables us directly to- 
express the functions of any angle, positive or negative, in terms- 
of the functions of a positive angle less than 90. 

Thus, sin 21 2 sin (180+ 32)= sin 32; 

cos 260 = cos (270 10) = sin 10. 



EXERCISES 

(i.) What angles less than 360 have the sine equal to %\/2? the- 
tangent equal to \/3 ? 

(2.) For what values of .r less than 720 is sin.r = .Jy^? 

(3.) Find the sine and cosine of 30; 765; 120; 210. 

(4.) Find the functions of 405; 600; 1125; 45; 225. 

(5.) Find the functions of 120; 225; 420; 3270. 

(6.) Express as functions of an angle less than 45 the functions of 
233: -197: 894. 

(7.) Express as functions of an angle between 45 and 90, sin 267;. 
tan ( 254); cos 950. 

(8.) Given cos 164 = .96, rind sin 196. 

(9.) Simplify cos (90 + ,r) cos (2 70 .r) sin(i8o jr)sin(36o x).. 

(10 .) S im plify sin( ' 8o --'' ) tan(9o + .r)+ . . '. . 
y sm (270 x} sin 2 (270 x) 

(u.) Express the functions of (.r 90) in terms of functions of x. 



CHAPTER II 
THE RIGHT TRIANGLE 

27. To solve a triangle is to find the parts not given. 

A triangle can be solved if three parts, at least one of 
which is a side, are given. A right triangle has one angle, 
the right angle, always given ; hence a right triangle can 
be solved if two sides, or one side and an acute angle, are 
also given. 

The parts of the right triangle not given are found by 
the use of the following formulas: 

opposite side adjacent side 

(i) sine =-~ ; (2) cosine =r ; 14 

hypotenuse hypotenuse 



(3) tangent^ 



opposite side 



(4) cotangent = 



_ adjacent side ^ 



16 



adjacent side ' opposite side ' 

To solve, select a formula in which two given parts enter; substituting 
in this the given values, a third part is found. Continue this method till 
all the parts are found. 

In a given problem there are several ways of solving the triangle ; choose 
the shortest. 

EXAMPLE 

The hypotenuse of a right triangle is 47.653, a side is 
21.34; find the remaining parts and the area. 



THE RIGHT TRIANGLE 



SOLUTION WITHOUT LOGARITHMS 

The functions of angles are given 



in the table of 



Natural Functions." 
21-34 



sin A =-= 

f 47.653 

47.653)21.3400^4478 
190612 

227880 
190612 
372680 
333571 
391090 
381224 
9866 

sin A = .4478 

.4 = 26 36' 
bc cos A 
=47- 653 x -8942 

47.653 
.8942 
95306 
190612 
428877 
381224 
42.6113126 
* =42.61 f 

= (90 -26 36' 1 = 63 24 



. 34x42.61 



21-34 
42.61 

2134 
12804 



SOLUTION EMPLOYING LOGARITHMS 

It is usually better to solve triangles 
by the use of logarithms. 

The logarithms of the functions are 
given in the tables of " Logarithms of 
Functions." * 



* 

sin A = - 
c 



log sin A = log a log c 
log 21. 34 =1.32919 

log 47. 653 = 1.67809 

--- sub. 
log sin .4=9.65110 10 

A = 26 36' 14" 



cos A=- 

log b = log c + log cos A 
log 47- 653 = 1-67809 
log cos 26 36' 14" =9.95140 10 
log =1.62949 

=42.608 



J =( 9 o-26 36' I4")=63 23' 46" 

area = \ab 
1 og area = log 4 + log a + log b 

log ^ = 9.69897 -10 
Iog2i. 34=1. 32919 
log 42 608 = i . 62949 
log area=2. 65765 



2)909.2974 
454.6487 

area =4 54- 6 

* In this solution the five-place table of the " Logarithms of Functions" is 
used. 

t No more decimal places are retained, because the figures in them are not 
accurate ; this is due to the fact that the table of " Natural Functions" is only 
four- place. 



PLANE TRIGONOMETRY 

CHECK ON THE CORRECTNESS OF THE WORK 



= 90.263 x 5.043 
90.263 
5-Q43 



270789 
361052 
45I3I5Q 
a* = 455. 196309 

Extracting the square root, a = 
21.34, which proves the solution cor- 
rect. 



a = c- -l>i = (c + b)(c - l>) 
= 90.261 x 5.045 

log 90.261 = 1.95550 

log 5.045 = 0.70286 

2)2.65836 

log 21. 34 = 1.32918 
a = 21.34, which proves the solu- 
tion correct. 



Remark. The results obtained in the solution of the preceding 
exercise without logarithms are less accurate than those obtained in 
the solution by the use of logarithms ; the cause of this is that four- 
place tables have been used in the former method, five place in the 
latter. 

EXERCISES 

28. (i.) In a right triangle = 96.42, c= 114.81 ; find a and A. 

(2.) The hypotenuse of a right triangle is 28.453, a side is 18.197; 
find the remaining parts. 

(3.) Given the hypotenuse of a right triangle = 747.24, an acute 
angle =23 45' ; find the remaining parts. 

(4.) Given a side of a right triangle = 37.234, the angle opposite 
= 54 27'; find the remaining parts and the area. 

. .(5.) Given a side of a right triangle = 1.1293, the angle adjacent 
= 74 13' 27"; find the remaining parts and the area. 

(6.) In a right triangle A = 1 5 22' 1 1 ", c .01 793 ; find b. 

(7.) In a right triangle = 71 34' 53", = 896.33; find a. 

(8.) In a right triangle c = 3729.4, = 2869.1 ; find A. 

(9.) In a right triangle a 1247, b 1988 ; find c. 

(lo.) In a right triangle (7 = 8.6432, = 4.7815; find B. 

The angle of elevation or depression of an object is the 
angle a line from the point of observation to the object 
makes with the horizontal. 






THE RIGHT TRIANGLE 




Thus angle x (Fig. i) is the angle of elevation of P if O is the point of 

observation ; angle y (Fig. 2) is the angle of depression of P if O is the 

point of observation. 

(n.) At a horizontal distance of 253 ft. from the base of a tower the 
angle of elevation of the top is 60 20' ; find the height of the tower. 

(12.) From the top of a vertical cliff 85 ft. high the angle of depres- 
sion of a buoy is 24 31' 22"; find the distance of the buoy from the 
foot of the cliff. 

(13.) A vertical pole 31 f t. h igh casts a horizontal shadow 45 ft. long ; 
find the angle of elevation of the sun above the horizon. 

(14.) From the top of a tower 115 ft. high the angle of depression 
of an object on a level road leading away from the tower is 22 13' 44"; 
find the distance of the object from the top of the tower. 

(15.) A rope 324 ft. long is attached to the top of a building, and 
the inclination of the rope to the horizontal, when taut, is observed 
to be 47 21' 17"; find the height of the building. 

(16.) A light- house is 150 ft. high. How far is an object on the 
surface of the water visible from the top? 

[Take the radius of the earth as 3960 miles.] 

(17.) Three buoys are at the vertices of a right triangle; one side 
of the triangle is 17,894 ft., the angle adjacent to it is 57 23' 46". 
Find the length of a course around the three buoys. 

(i 8.) The angle of elevation of the top of a tower observed from a 
point at a horizontal distance of 897.3 ft. from the base is 10 27' 42" ; 
find the height of the tower. 

(19.) A ladder 42^ ft. long leans against the side of a building; its 
foot is 25! ft. from the building. What angle does it make with the 
ground ? 

(20.) Two buildings are on opposite sides of a street 120 ft. broad. 



S'.Z'b 



i 



26 



PLANE TRIGONOMETRY 



The height of the first is 55 ft. ; the angle of elevation of the top of 

the second, observed from the edge of the roof of the first, is 26 37'. 

Find the height of the second building. 

A -(21.) A mark on a flag-pole is known to be 53 ft. 7 in. above the 

j ground. This mark is observed from a certain point, and its angle. 

of elevation is found to be 25 34'. The angle of elevation of the top 

of the pole is then measured, and found to be 34 17'. Find the 

height of the pole. 

(22.) The equal sides of an isosceles triangle are each 7 in. long ; the 

base is 9 in. long. Find the angles of the triangle. 




b = 9 



Hint. Draw the perpendicular BD. BD bisects the base, and also the 
angle ABC. 

In the right triangle ABD, AB-] in., AD\\ in., hence ABD can 
be solved. 

Angle C= angle A, angle ABC 2 angle ABD. 

(23.) Given the equal sides of an isosceles triangle each 13.44 in., 
and the equal angles are each 63 21' 42"; find the remaining parts 
and the area. 

(24.) The equal sides of an isosceles triangle are each 377.22 in., 
the angle between them is 19 55' 32". Find the base and the area 
of the triangle. 

JL (25.) If a chord of a circle is 1 8 ft. long, and it subtends at the centre 

an angle of 45 31' 10" , find the radius of the circle. 

(26.) The base of a wedge is 3.92 in., and its sides are each 13.25 in. 
long; find the angle at its vertex. 



THE RIGHT TRIANGLE 



27 



(27.) The angle between the legs of a pair of dividers is 64 45', the 
legs are 5 in. long; find the distance between the points. 

(28.) A field is in the form of an isosceles triangle, the base of the 
triangle is 1793.2 ft. ; the angles adjacent to the base are each 53 27' 
^49". Find the area of the field. 

6 (29.) A house has a gable roof. The width of the house is 30 ft., 
the height to the eaves 25! ft., the height to the ridge-pole 33! ft. 
Find the length of the rafters and the area of an end of the house. 
^ (30.) The length of one side of a regular pentagon is 29.25 in. ; find 
the radius, the apothem, and the area of the pentagon. 






b 




Hint. The pentagon is divided into 5 equal isosceles triangles by its radii. 
Let AOB be one of these triangles. ^#=29.25 in.; angle AOB=\ of 
36o = 72. Find, by the methods previously given, OA, OD, and the area 
of the triangle A OB. 

These are the radius of the pentagon, the. apothem of the pentagon, and 
\ the area of the pentagon respectively. 

(31.) The apothem of a regular dodecagon is 2 ; find the perimeter. 
0(32.) A tower is octagonal ; the perimeter of the octagon is 153.7 ft. 
Find the area of the base of the tower. 

(33.) A fence extends about a field which is in the form of a regular 
polygon of 7 sides; the radius of the polygon is 6283.4 ft. Find the 
length of the fence. 

(34.) The length of a side of a regular hexagon inscribed in a circle 
is 3.27 ft. ; find the perimeter of a regular decagon inscribed in the 
same circle. 

(35.) The area of a field in the form of a regular polygon of 9 sides 
is 483930 sq. ft. ; find the length of the fence about it. 



28 



PLANE TRIGONOMETRY 



SOLUTION OF OBLIQUE TRIANGLES BY THE AID OF 

RIGHT TRIANGLES 

29. Oblique triangles can always be solved by the aid of 
right triangles without the use of special formulas ; the 
method is frequently, however, quite awkward ; hence, in a 
later chapter, formulas are deduced which render the solu- 
tion more simple. 

The following exercises illustrate the solution by means 
of right triangles : 

(i.) In an oblique triangle ^ = 3.72, ^ = 47 52', '=109 10'; find 
the remaining parts. 

The given parts are a side and two angles. 

C 




Hint. A = [i&o-(B+C)], 

Draw the perpendicular CD. 

Solve the right triangle BCD. 

Having thus found CD, solve the right triangle A CD, 

(2.) In an oblique triangle a = 89.7, c 125.3, B= 39 8'; find the 
remaining parts. 

The given parts are two sides and the included angle. 




125.3 



THE RIGHT TRIANGLE 



29 



'ii/. Draw the perpendicular CD. 
Solve the right triangle CBD. 
Having thus found CD and AD(=c-DB), solve the right triangle ACD. 

(3.) In an oblique triangle a = 3.67, b 5.81, A = 27 23'; find the 
remaining parts. 

The given parts are two sides and an anglt opposite one of 
them. 

C 




B' 



B 



Either of the triangles ACB, ACB' contains the given parts, and 
is a solution. 

There are two solutions when the side opposite the given angle is 
less than the other given side and greater than the perpendicular, 
CD, from the extremity of that side to the base.* 

Hint. Solve the right triangle ACD. 

Having thus found CD, solve the right triangle CDB (or CDB'\ 

(4.) The sides of an oblique triangle are = 34.2, = 38 A ^- = 55. 12; 
find the angles. 

The given parts are the three sides. 




c =55.12 

* A discussion of this case is contained in a later chapter on the solution 
of oblique triangles. 



Hint. 
Hence 



PLANE TRIGONOMETRY 



a* - ^ = CF? - # -(c- X}* 



In each of the right triangles A CD and BCD the hypotenuse and a side 
are now known ; hence these triangles can be solved. 

"jsi (5-) Two trees, A and B, are on opposite sides of a pond. The 
distance of A from a point C is 297.6 ft., the distance of B from C is 
8644 ft., the angle ACS is 87 43' 12". Find the distance AB. 

(6.) To determine the distance of a ship A from a point B on 
shore, a line, EC, 800 ft. long, is measured on shore ; the angles, ABC 
and ACS, are found to be 67 43' and 74 21' 16" respectively. What 
is the distance of the ship from the point B? 

j^ (7.) A light-house 92 ft. high stands on top of a hill; the distance 
from its base to a point at the water's edge is 297.25 ft. ; observed 
from this point the angle of elevation of the top is 46 33' 15". Find 
the length of a line from the top of the light-house to the point. 

(8.) The sides of a triangular field are 534 ft., 679.47 ft., 474.5 ft. 
What are the angles and the area of the field ? 

(9.) A certain point is at a horizontal distance of 117^ ft. from a 
river, and is u ft. above the river; observed from this point the angle 
of depression of the farther bank is i 12'. What is the width of the river? 

(10.) In a quadrilateral ABCD,AB= 1.41, BC 1.05, CD = 1.76, DA 
= 1.93, angle ^=75 21'; find the other angles of the quadrilateral. 







\ 



n Q . i o 






THE RIGHT TRIANGLE 31 

Hint. Draw the diagonal DB. 

In the triangle ABD two sides and an included angle are given, hence the 
triangle can be solved. 

The solution of triangle ABD gives DB. 

I laving found DB, there are three sides of the triangle DBC known, hence 
the triangle can be solved. 

(ii.) In a quadrilateral ABCD, AB=\2.i, AD = 9.7, angle A 
47 18', angle 71 = 64 49' angle D= 100; find the remaining sides 

Hint. Solve triangle ABD to find BD. 



*h 



CHAPTER III 
TRIGONOMETRIC ANALYSIS 

30. In this chapter we shall prove the following funda- 
mental formulas, and shall derive other important formulas 
from them : 



in (x + y) = in x co y + cos x in y, 
in(a?-2/) = in coi/ - cosx siiii/, 

cos (x + y] = cos a? cos?/ -sin as iny, 

cos (a? y) = 



(12) 
(13) 
(14) 



PROOF OF FORMULAS (l l)-(l4) 

31. Let angle AOQ= AT, angle QOP=y; then angle 



The angles * and j are each acute and positive, and in Fig. i 
('*'+ y) i l ess tn ^n 90, in Fig. 2 (.r-f-j) is greater than 90. 





In both figures the circle is a unit circle, and SP is perpendicular to 
OA ; hence SP= sin (x +y), OS= cos (x + y). 









TRIGONOMETRIC ANALYSIS 33 

Draw DP perpendicular to OQ ; 
then DP=siny, OD = cosy, 

angle SPD = angle AOQ = x. 

(Their sides being perpendicular.) 

Draw DE perpendicular to OA, DH perpendicular to SP. 
Sin (x +/) = SP= ED + PIP. 



cos/. 

(For OED being a right triangle, -- = sin.r.) 



HP '= (cos x) x DPCQsx sin/. 
HP 

(For HPD being a right triangle, - = cos jr.) 



Therefore, in(a? + 2/) = in.r co?y 4- cosx siny. (u) 

Cos O +/) = (95 = OE - HD. * 
= (cos x) x (9Z> = cos x cos/. 



(For OED being a right triangle. ~- = cos jr.) 



(For PHD being a right triangle, = sin x.) 



Therefore, cos (x + y] = cox eos?/-in^ in//. (13) 

5^. The preceding formulas have been proved only for 
the case when x and y are each acute and positive. The 
proof can, however, readily be extended to include all values 
of x and y. 

Let/ be acute, and let x be an angle in the second quad- 
lant ; then x = (90 4- x r ) where x' is acute. 
sin (x -f-/) = sin (90 + x' +y) 

= cos(X+/) 24 

= cosx' cos/ sin x' sin/ 

= sin (90 + x') cos/ + cos (90 4- x') sin / 24 
=smx cos/4- cos x sin/. 

* If (x +,r) is greater than 90, OS is negative. 



34 PLANE TRIGONOMETRY 

Thus the formula has been extended to the case where 
one of the angles is obtuse and less than 180. In a 
similar way the formula for cos(x+y) is extended to this 
case. 

By continuing this method both formulas are proved to 
be true for all positive values of x and y. 

Any negative angle y is equal to a positive angle y' , minus 
some multiple of 360. The functions of y are equal to 
those of y', and the functions of (x-\-y) are equal to those 
of 



Therefore, the formulas being true for \x -\-y'), are true for 



A repetition of this reasoning shows that the formulas are 
true when both angles, x and y, are negative. 

33. Substituting the angle y for y in formula (11), it 
becomes 

s\n(xy) s'mx cos( 7) + cos;r sin (y). 
But cos( y) = cosy, and sin( y) s\ny. 23. 

Therefore, sin (a? ?/) = in.x COST/ cosx *m //. (12) 

Substituting (y) for y in formula (13), it becomes 
cos (xy) cos x cos ( y) sin x sin (y), 



Therefore, cos (a? - y) = cos x cosy + siia? siny.* (14) 

EXERCISES 

34. (i.) Prove geometrically where .r and j are acute and positive : 

cos^y cos^r sinj/, 
n,r sinj. 



* Formulas (12) and (14) are proved geometrically in 34. The geometric 
proof is complicated by the fact that OD and DP are functions of y, while 
the functions of y are what we use. 



TRIGONOMETRIC ANALYSIS 

.Q 

0,4 -x ,H 



35 




Hint. Angle AOQ-x, angle POQ=y, and angle A OP (x-y). 

Draw /'/) perpendicular to 6>(). 

Then DP= sin (;) = sin r ; but Z>/* is negative, therefore PD taken 
as positive is equal to sin y: 

OD=cof,( ji')=cos y, 

Angle HTD angle AOQ=Jc. their sides being perpendicular. 
Draw DI1 perpendicular to SP, DE perpendicular to OA. 

sin(x-y)=SP=D-P//. 

From right triangle OED, D.=(^'mx)x OJ r )=sinx cos jr. 
From right triangle DHP, P//=(cosx)x PZ)=cosx sin y. 
Therefore, . sin (.*;')= sin x cos;/ cos x sin; . 



From right triangle 0&D, OE = (cos x)x 0>=cosx cosjj'. 
From right triangle DHP, SJ//=(sin x) x PD = ^\\\x sin;'. 
Therefore, cos(.r _j')=cos x cosj' + sin.r sin;'. 

(2.) Find the sine and cosine of (45+, r), (30 x\ (6o-f-,r), in terms- 
of sin^r and cos.r. 

(3.) Given sin.r=$, sin/ = ^, ,\- and y acute; find sin(,r+j) and 
sin(.r y). 

(4.) Find the sine and cosine of 75 from the functions of 30 and 45. 
Hint. 75=(45 + 30). 

(5.) Find the sine and cosine of 15 from the functions of 30 and 45. 

(6.) Given x and y, each in the second quadrant, sin x = $, siny = ^ ; 
find sin (x-\-y) and cos(.r y}. 

(7.) By means of the above formulas express the sine and cosine of. 
(180 .r), (i8o-|-,r), (270 .r), (270+ *), in terms of sin^r and cos-r, 

(8.) Prove sin (6o-f 45) + cos (60 + 45) = cos 45. 

(9.) Given sin 45 = ^^/I, cos 45 \/2 ; find sin 90 and cos 90. 

(10.) Prove that sin (60 -f- x) sin (60 .r) = 




36 PLANE TRIGONOMETRY 

TANGENT OF THE SUM AND DIFFERENCE OF TWO ANGLES 

_ sin(^r-f-j) sin;r cosj^-f-cos^r sinj/ 
~~ cos(jr+7)~cos^ cosj sin x si ny 
Dividing each term of both numerator and denominator 
of the right-hand side of this equation by COSJT cosj, and 

remembering that --- = tan, we have 
cos 

tan x + tan y 



In a similar way, dividing formula (12) by formula (14), we 

.obtain 

tana? - tan?/ 
> = !+ tan* 



FUNCTIONS OF TWICE AN ANGLE 
36. An important special case of formulas (n), (13), and 

(15) is when y~x\ we then obtain the functions of 2x in 

terms of the functions of x. 

From (n), sin (*?+.*)= sin* cos^-hcos^r sin jr. 
Hence in 2a? = 2 in x co x. (17) 

From (13), co2x = co 2 a?-in 2 x. (18) 

Since cos 2 ^r= i sin 8 JIT* and sin a jr= I cos 2 .r, 

we derive from equation (18), 

cos 2^-=: I 2sin 2 ^r, (19) 

and cos2;r = 2 cos 2 ^r I. (20) 

From (15), tanto = > , l f.. 



FUNCTIONS OF HALF AN ANGLE 

57. Equations (19) and (20) are true for any angle; there- 
fore for the angle \x. 

From(i9), cos;r= I 2 sin 2 ^; 



TRIGONOMETRIC ANALYSIS 37 

I COS.T 



or 



therefore, sin * = \~ (22) 

From (20), cos.r = 2 cos 2 4* I ; 

i -f cos^r 
or cos -.*::= --- ; 

therefore, cos-^^rfcy *j~-. (23) 

Dividing (22) by (23), we obtain 

(24) 



cos a? 



FORMULAS FOR SUMS AND DIFFERENCES OF FUNCTIONS 

38. From formulas (u)-(i4), we obtain 

sin (x + /)-!- sin (x j / ) = 2sin^r cos}' ; 
sin (irH-^) .sin (^ j / ) = 2cos;tr sinj' ; 
cos (x 4-7) + cos (*}') = 2 COS.T cosj ; 
cos (x +y) cos (x y) 2sin^r sinj/. 
Let n - (x +/) and v (x y] ; 

then x = %(2i + ii), y %(u v). 

Substituting in the above equations, we obtain 

sin ?f + sinr = 2 sin-|(e +v)cos^(u v); (25) 

siiifr - sin r = 2cos-|(/e + t')sin^(M r); (26) 

cos M + c*o r= 2 co*lr(u+v) cos|-(? f) ; (27) 

cos!/-cost'=-28iii-J(w + v) in-J(i-f). (28) 
Dividing (25) by (26), 

in t 4- sinv 



sin M- sin v 



, , 



EXERCISES 

,'i,9. Express in terms of functions of x, by means of the formulas 
of this chapter, 



38 PLANE TRIGONOMETRY 

(i.) Tan(i8o .r); tan ( 1 80 -f x\ 

(2.) The functions of (x 180). 

(3.) Sin (.r 90) and cos (^ 90). 

(4.) Sin (.r 270), and cos (x 270). 

(5.) The sine and cosine of (45 x); of (45+.*). 

(6.) Given tan 45= i, tan 30 ^ -^3; find tan 75; tan 15. 

cot./- cot*/-l 

(7.) Prove cot (05-1- y) - - . (30) 



Hint. Divide formula (13) by formula (n). 
COt.X COt 91 + 1 

(8.) Prove cot (x-y) = - - . (31) 

cot y- cot a? 

(9.) Prove cos (30 +y) cos (30 y) = sin y. 
(10.) ProVe sin ^x = 3 sin. r 4 sin 3 .*-. 

/#/. Sin 3-r=sin (x+2x). 
(n.) Prove cos 3* = 4 cosfc 3 cos x. 
(12.) If x and y are acute and tan-r = , tanj/ = J, prove that 



(i 3.) Prove that tan (.r-|~45) = 

i tan x 

(14.) Given siny= and y acute; find sin \y, cos^y, and tan \y. 

(15.) Given cos^r=: | and x in quadrant II; find sin 2x and 
cos 2.r. 

(16.) Given cos 45 i \/2 ; find the functions of 22!. 

(17.) Given tan.r = 2 and .r acute ; find tan \x. 

(i 8.) Given cos 30 = | -^3 ; find the functions of 15. 

(19.) Given cos9o = o; find the functions of 45. 
*> (20.) Find sin x in terms of sin x. 

(21.) Find COS5-T in terms of COS.T. 

(22.) Prove sin(.r-j- y -j-2-) sin x cosy cos 5- -(-cos ,r sin v coss'-J-cos.r 
cosy sin z sin x sin/ sin^. 

Hint. Sin (x+y + z) sin (x+y) co$s + cos(.*+j') sin 2. 

(23.) Given tan 2^ = 3 tan.r; find x. 
^ (24.) Prove sin 32 -|- sin 28 = cos 2. 

(25.) Prove tan x -\- cot x 2 esc 2x. 

(26.) Prove (sin!.r-|-cos..r) a =: i -)-sin^r. 

(27.) Prove (sin \x - cos l.r)- = i - sin x. 



TRIGONOMETRIC ANALYSIS 39 

^(28.) Prove cos 2.v = cos*.r sin 4 .r. 
(29.) Prove tan (45 -f x} -f tan (45 .r) = 2 sec 2x. 

2 tan -r 
< (30.) Prove sin2.r = 



-f tan 2 *' 

i tan'.r 
(31.) Prove cos 2.1- = 

i -h tan-. r 

I 4- sin 2x /tan .r-f- i\' 
(32^ Prove - ) 

i sin 2.x Vtan.r i/ 

cin r 

(33.) Prove tan|.r = 



-|- cos .r 

sin JT 
, (34.) Prove co ti .r =I -_ c . 

cos x cosy 

(35.) Express as a product .* 

cos x -f- cos_y 

COSJT cos r _ 2 sin !(*+_;') sin^(jr 

COS a + COS/ 2 COS \ (x + 1') COS ^ (x 

= -tan^(j:+;') tani(jr-.v). 

tan ,r -f tan y 

/ (36.) Express as a product : . 

cot x + cot/ 

cos (.1- 4- y) 

(37.) Prove i tan x tan y =: - . 

' - 



Till: TXVKRSE TRIGONOMETRIC FUNCTIONS 

40. Dcf. The expressions sin ^, cos-^tan-'tf, etc., de- 
note respectively an angle whose sine is a, an angle whose 
cosine is a, an angle whose tangent is <7, etc. They are 
called the inverse sine of a, the inverse cosine of <?, the 
inverse tangent of a, etc., and are the inverse trigono- 
metric functions. 

Sin-'tf is an angle whose sine is equal to a, and hence de- 
notes, not a single definite angle, but each and every angle 
whose sine is a. 

* Since quantities cannot be added or subtracted by the ordinary operations 
with logarithms, an expression must be reduced to a form in which no addition 
or subtraction is required, to be convenient for logarithmic computation. 



40 PLANE TRIGONOMETRY 



Thus, if sin*=, jr=3o , 150, (30 + 360), etc., 

and sin- 4=30, 150, (3O + 36o), etc. 

Remark. The sine or cosine of an angle cannot be less than i 
or greater than -|- i; hence sin" 1 ^ and cos~'# have no meaning unless 
a is between i and -f i. In a similar manner we see that sec-'tf 
and csc~ l a have no meaning if a is between i and -j- * 

EXERCISES 

41. (i.) Find the following angles in degrees: 

sin~ I |-v/2, tan~ T ( *) sin~ T ( ). 

cos- 1 !, cos- 1 !, 

(2.) If x cot-^, find tan x. 

(3.) If x = sin- x f , find cos x and tan x. 

(4.) Find sin (tan-'i \/3)- 
(5.) Find sin(cos- T ). 
(6.) Find cot (tan- 1 yS). 

(7.) Given sin-'fl = 2 cos- 1 *?, and both angles acute ; find a. 
(8.) Given sin" 1 ^ = cos -I ^ ; find the values of sin" 1 ^ less than 360. 
(9.) Given tan~ x i =-}tan- f o, and both angles less than 360; find 
the angles. 

(lo.) Given sin" 1 ^ cos-V? and sin-^ + cos- 1 ^ = 450; find sin~V?. 
(n.) Prove sin (cos~Vz) = \/ia*. 

Hint. Let jc=cos- 1 ^ ; then a = cos jc, 

sin x= y I COS' J JT = y I </ 2 . 

(12.) Prove tan^an-^r+tan- 1 ^)^: 

r.-b 
(13.) Prove tan(tan- r rt tan- l ^)= T- 

(14.) Prove cos(2 cos- I ^) = 2^ 2 i. 
(15.) Prove sin (2 cos 1 rt) = 2a y/i a". 

2(7 

(16.) Prove tan (2 tan- 1 a)= --- 

* 
(17.) Prove cos(2tan- I <7)= 






(i 8.) Prove sin(sin~ I rt + cos- 1 ^) = ab.y/(\ a*}(\ 



CHAPTER IV 

THE OBLIQUE TRIANGLE 
DERIVATION OF FORMULAS 

42. The formulas derived in this and the succeeding 
articles reduce the solution of the oblique triangle to its 
simplest form. 

c C C 




FIG. 3 



Draw the perpendicular CD. Let CD=/i, 

Then - sin ^4; 

o 



and 



(In Fig. 2 -=sin(i8o-X)=--sin^) 

h 

- sin B. 



(In Fig. 3 -= 



(32) 



By division we obtain, 

a _ sin A 
b ~ silTI* " 

Remark. This formula expresses the fact that the ratio of two sides of an 
oblique triangle is equal to the ratio of the sines of the angles opposite, and 
does not in any respect depend upon which side has been taken as the base. 
Hence if the letters are advanced one step, as shown in the figure, we obtain, 
as another form of the same formula, 




42 PLANE TRIGONOMETRY 

b _ sin/? 

Repeating the process, \ve obtain 
c sin C 

;, = ^A' & 

The same procedure may be applied to all the formulas for the solution of 
oblique triangles. Henceforth only one expression of each formula will lie given. 

Formula (32) is used for the solution of triangles in which 
a side and two angles, or two sides and an angle, opposite one 
of them are given. 

43. We obtain from formula (32) by division and compo- 
sition, a b sin^ sin/? 



_ 

a + b ~ sin A + sin B ' 

By formula (29), denoting the angles by A and B, in- 
stead of u and z/, 

sin A -sin.#tan-J(^ B) 



Therefore, ~~ - ~-| -^ ' ^ (33) 

This formula is used for the solution of triangles in which 
two sides and the included angle are given. 

44. Whether A is acute or obtuse, we have ^ 
C C 




FIG. 




(If A is acute (Fig. i),AD = bco*A, DH AB - AD c - b cos,-/, CD 
bs\nA. UA is obtuse (Fig. 2), AD ^cos (180-^) = - ^cos^, DBAB 
, CD b sin(i8o A)- b sin^.) 



THE OBLIQUE TRIANGLE 43 



r 2 be cosA+b* (cos* A +sinM). 

Therefore, a 2 6 2 +c 2 - 2bc co A. (34) 

This formula is used in deriving formula (37). 

// is also used in the solution without logarithms of tri- 
angles of which two sides and the included angle or three 
sides are given. 

45. From formula (34), cos^ = 7 
From formula (22), 37, 

2 sin 2 4/2 = i cos^4 = I c ~ a . 

2 be 

Hence 2 sin 2 A = 



2 be 

(b-_c 
~~2bc 



2bc 
Let J== ?|.f, then (a-6 + c) = 2(s-d), and (a + b-c) 



Substituting, 2 s\n*A = - 



Hence sin^^j = .A^)( S - *\* (35) 

V be 

From formula (23), 37, 

2 COS' = 



bc 

* In extracting the root the plus sign is chosen because it is known that 
sin A ,/ is positive. 



44 PLANE TRIGONOMETRY 

Hence cos^A 

Dividing (35) by (36), we obtain 

tan I A =. \/ (s ~~ b ^~ 

* \/ \ 

s {s a) 



(36) 

(37) 



Let 






tan i ^ = - ~ 
sa 



(38) 



Formulas (37) and (38) are used to find the angles of a tri- 
angle when tJie three sides are given. 

FORMULAS FOR THE AREA OF A TRIANGLE 
40. Denote the area by S. 
C 




D 

FIG. I 




(In Fig. i, CD=as,\\\B\ in Fig. 2, CD - sin (180-^) = asm.) 

In Figs. I and 2, S=$c.CD. 

Hence /S'^-JacsinB. (39) 

From formula (17), 

sinZ? = 2 sin $ cosZ?. 



THE OBLIQUE TRIANGLE 45 

Substituting for sinj/? and cos^fi the values found in 
formulas (35) and (36), we obtain 

sin = \ /s(s-a)(s-b}(s-e). 
ac* 

Therefore, S=*/s(8 a)(sb)(s c). (40) 

This formula may also be written, 

S=sK. (41) 

Formula (39) is used to find the area of a triangle when 
two sides and the included angle are knoivn; formula (40) or 
formula (41), when the three sides are known. 

THE AMBIGUOUS CASE 

47 The given parts are two sides, and the angle opposite 
one of them. 

Let these parts be denoted by a, b, A. 

C 




If a is less than b and greater than the perpendicular CD 
(Fig. i), there are the two triangles ACB and ACS', which 
contain the given parts, or, in other words, there are two 
solutions. 

If a is greater than b (Fig. 2), there is one solution. 

If a is equal to the perpendicular CD, there is one solu- 
tion, the right triangle A CD. 






46 PLANE TRIGONOMETRY 

If the given value of a is less than CD, evidently there 
can be no triangle containing the given parts. 

Since CD=t>sinA, there is no solution when < bs\\\A ; there is one 
solution, the right triangle A CD when a=bs\\\A; there are two solutions 
when a <! b and > fisinA. 

48. CASE I. Given a side and two angles. 

EXAMPLE 
Given a = 36.738, A = 36 55' 54", B = 72 5' 56", 

C=i8o (A + )= 180 109 i' 50" = 70 58' 10". 

To find c. 

c sin C 

a sin A 

= 1.56512 

log sin =9.97559 10 
colog sin A =0.22 1 23 
log r= i. 76194 
^=57.80 



To find b. 



a sin /4 
log rt = i. 56512 
log sin .#=9.97845 
colog sin A =o. 221 23 
log 0=1.76480 
^ = 58.184 



Determine b from r, C, and B by the formula 



This check is long, but is quite certain to reveal an error. A check which is 
shorter, but less sure, is 

b _ sin B 

c sin C 

Solve the following triangles : 

(i.) Given a 567.25, A \\ 15', ^ 47 12'. 

O (2.) Given ^ = 783.29, A = Si 52', ^ 42 27'. 

. (3.) Given c= 1125.2, A = 79 15', ^=55 n'. 

(4.) Given ^=15.346, B=it 51', Cr=58 10'. 

(5.) Given a = 5301. 5, ^4 =69 44', C=4\ 18'. 

(6.) Given =1002.1, ^=48 59', = 76 3'. 

t/f>. CASE II. Given two sides of a triangle and tlie angle 
opposite one of them. 



THE OBLIQUE TRIANGLE 



47 



EXAMPLE 
Given a = 23.203, b 35.121, A = 36 8' 10". 

C 




To find B and B '. 



sin A a 
log <$=i. 5.1556 
log sin ,4=9.7706410 
colog rz = 8. 6344 5 10 
log sin #=9.95065 10 
=63 12' 



To find C and' C'. 
C = i8o-(/f +)=So 39' 50' 
') = 27 3- 50" 



To find c and c . 
c sin C 



log 11=1.36555 
log sin (7=9.99421 10 
colog sin A =0.22936 
log f=i. 58912 
^=38.825 

log a = i. 36555 
log sin C' =9.65800 10 
colog sin A =0.22936 
log c' = i. 25291 
^' = 17.902 



Check. 

Determine b from c, C, and B by the formula 
b a tan4(/?- 



tan 



This check is long, but is quite certain to reveal an error. A check which is 
shorter, but less sure, is 



c sin 6" 

(i.) How many solutions are there in each of the following? 
(i.) A = $o, a = i$, 6 = 20-, 
(2.) A = 30, a = TO, = 20; 
(3.) ^ = 30, a =8, 6 = 20; 
(4.) = 37 23', a = 9.1, 6 = 7.$. 



v\ 



48 



PLANE TRIGONOMETRY 



Solve the following triangles, finding all possible solutions : 
7X2.) Given A = 147 12', a = 0.63735, = 0.34312. 
(3.) Given A= 24 31', a = 1.7424, = 0.96245. 
(4.) Given A= 21 21', a = 45.693, = 56.723. 
(5.)Giveny? = 61 16', # = 9.5124, = 12.752. 
(6.) Given C= 22 32', a =0.78727, = 0.47311. 

0. CASE III. Given two sides and the included angle. 



Given ^ = 41.003, 
parts and the area. 



EXAMPLE 

' = 48.718, C 68 33' 58"; find the remaining 



To find A and B. 

tan|(Z? A) _b a 
~~ 



b-a = 7.715 
b + rt = 89.721 



log (-) = 0.88734 
colog (b + a) = 8.0471010 
log tan( + ^) = o. 16639 



log tan %(B A ) = 9. 10083 10 
-^^ 7 ii' 20" 




^>2 Q 54' 21" 
=48 31' 41" 



r _ sin C 
a sin A 
loga= 1.61281 
log sin C= 9. 96888 -10 
colog sin A = o. 12535 
log<r= 1.70704 
c- 50.938 

To find the area. 
S = \ab sin C 
= 9-69897 -10 
= 1.61281 
= 1.68769 
log sin C= 9. 96888 10 
log S= 2.96835 
S= 929.72 



Check. 
sin C c 
siii~5 ~ 7 
log sin B = 9.94951 10 

log c = 1.70704 
coiog b 8.31231 10 
log sin C 9.96886 10 



THE OBLIQUE TRIANGLE 49 

Solve the following triangles, and also find their areas : 
/(7J Given A= 41 15', =0.14726, f =0.10971. Q) 
\^2.) Given C= 58 47', =11.726, #=16.147. 
(3.) Given = 49 50', # = 103 74, ^=99.975. 
1 (4.) Given A= 33 31', =0.32041, ^=0.9203. 
(5.) Given C=i28 7', =17.738, #=60.571. 

51. CASE IV. Given the three sides. 

EXAMPLE 
Given # = 32.456, = 41.724, ^ = 53.987 ; find the angles and area. 

^ = 64.084 
(. </) = 3i.628 
(s ^ = 22.360 

log A'= i. 02349 



-r)- 10.097 



*- / (J - </JU - 

A \ 



- u)(s - <>)(* - 



log (.f rt) = 1.50007 
lug (-f ^) =1.34947 

log (s r)=i. 00419 
colog j = 8. 19325 10 

2)2.0461*8 
log A'=i.o2349 

To find A. 
K 



s a 
log A'= i. 02 349 

log (s-a) = 1.50007 

sub. 

log tan|^=g. 52342 -10 

A^ = i8 27' 23" 
^=36 54 4"" 



log (/-/;) = 1.34947 

sub. 

log tan ^ .#=9.67402 - i o 
^=25 16' 16" 
^=50 32' 32" 



C* 



log A'= i. 02349 

log (j-<-)= 1.00419 

sub. 

log tanC=o.oi93O 

|(7=46 16' 22" 
C=92 32' 44" 



Find the angles and areas of the following triangles: 

(i.) Given # = 38.516, =44.873, ^=14.517. 
(2.) Given # = 2.1158, =3.5854, ^=3.5679. 

* C could be found from (A + J 9)=(i8o- C), but for the sake of the check it 
is worked out independently. 

4 



Check. 









50 PLANE TRIGONOMETRY 

(3.) Given ^=82.818. =99.871, ^=36.363. 

(4.) Given ^ = 36.789, =i 1.698, ^=33.328. 

(5.) Given a i 13.03, =131.17, c \ 14.29. 

(6.) Given a= .9763, =1.2489, < = 1.6543. 

EXERCISES 

52. (i.) A tree, A, is observed from two points, B and C, 1863 ft. 
apart on a straight road. The angle BCA is 36 43', and the angle 
CBA is 57 21'. Find the distance of the tree from the nearer 
point. 

(2.) Two houses, A and B, are 3876 yards apart. How far is a third 
house, C, from A, if the angles ABC and J5AC are 49 17' and 58 18' 
respectively ? 

(3.) A triangular lot has one side 285.4 ft. long. The angles adja- 
cent to this side are 41 22' and 31 19'. Find the length of a fence 
around it, and its area. 

(4.) The two diagonals of a parallelogram are 8 and 10, and the 
angle between them is 53 8' ; find the sides of the parallelogram. 

(5.) Two mountains, A and B, are 9 and 13 miles from a town, C; 
the angle ACB is 71 36' 37". Find the distance between the moun- 
tains. 

(6.) Two buoys are 2789 ft. apart, and a boat is 4325 ft. from the 
nearer buoy. The angle between the lines from the buoys to the 
boat is 1 6 13'. How far is the boat from the farther buoy? Are 
there two solutions ? 

(7.) Given a = 64.256, r= 19.278, C=i6 19' 11"; find the differ- 
ence in the areas of the two triangles which have these parts. 

(8.) A prop 13 ft. long is placed 6 ft. from the base of an embank- 
ment, and reaches 8 ft. up its face; find the slope of the embank- 
ment. 

(9.) The bounding lines of a township form a triangle of which the 
sides are 8.943 miles, 7.2415 miles, and 10.817 miles; find the area 
of the township. 

(10.) Prove that the diameter of a circle circumscribed about a 
triangle is equal to any side of the triangle divided by the sine of the 
angle opposite. 

' - i -J / - "' Srf! 



' *~^t b 

THE OBLIQUE TRIANGLE 




Hint. By Geometry, angle A OB =2C. 

Draw OD perpendicular to AB. 
Angle DOB=%AOB=C. 
DB=r sin DOB=r sin C. 
Hence c=2rs'mC, 

c 
or 2r=~ ;. 

sine 

(ii.) The distances AB, BC, and AC, between three cities, A, B t 
and Care i$ miles, 14 miles, and 17 miles respectively. Straight rail- 
roads run from A to B and C. What angle do they make ? 

(12.) A balloon is directly over a straight road, and between two 
points on the road from which it is observed. The points are 15847 
ft. apart, and the angles of elevation are found to be 49 12' and 
53 29' respectively. Find the distance of the balloon from each of 
the points. 

(13.) To find the distance from a point A to a point B on the op- 
posite side of a river, a line, AC, and the angles CAB and ACB were 
measured and found to be 315.32 ft., 58 43', and 57 13' respectively. 
Find the distance AB. 

(14.) A building 50 ft. high is situated on the slope of a hill. From 
a point 200 ft. away the building subtends an angle of 12 13'. Find 
the distance from this point to the top of the building. 

(15.) Prove that the area of a quadrilateral is equal to one-half 
the product of the diagonals by the sine of the angle between 
them. 

(16.) From points A and B, at the bow and stern of a ship respec- 
tively, the foremast, C, of another ship is observed. The points A 
and B are 300 ft. apart; the angles ABC and BAC are found to be 

tftr) 

' 



52 PLANE TRIGONOMETRY 

65 31' and 1 10 46' respectively. What is the distance between the 
points A and C of the two ships ? 

(17.) Two steamers leave the same port at the same time ; one sails, 
directly northwest, 12 miles an hour; the other 17 miles an hour, in 
a direction 67 south of west. How far apart will they be at the end 
of three hours ? 

(i 8.) Two stakes, A and />, are on opposite sides of a stream; a 
third stake, C, is set 92 ft. from A ; the angles ACB and CAB are 
found to be 50 3' 5" and 61 18' 20" respectively. How long is a 
rope connecting A and Z?? 

(19.) To find the distance between two inaccessible mountain-tops, 
A and B, of practically the same height, two points, C and D, are 
taken one mile apart. The angle CDA is found to be 88 34', the 
angle DC A is 63 8', the angle CDB is 64 27', the angle DCB is 87 9'. 
What is the distance? 

(20.) Two islands, B and C, are distant 5 and V 3 miles respectively 
from a light-house, A, and the angle BAC is 33" / >f f ; find the dis- 
tance between the islands. 

(21.) Two points, A and B, are visible from a third point C, but 
not from each other; the distances AC, BC, and the angle ACB were 
measured, and found to be 1321 ft., 1287 ft., and 61 22' respectively. 
Find the distance AB. \ ^ 1 ^ 

(22.) Of three mountains, A, B, and C, B is directly north of C 5 
miles, A is 8 miles from C and 1 1 from B. How far is A south of B ? 

(23.) From a position 215.75 ft. from one end of a building and 
198.25 ft. from the other end, the building subtends an angle of 
53 37' 28"; find its length. 

(24.) If the sides of a triangle are 372.15, 427.82, and 404.17 ; find 
the cosine of the smallest angle. 

(25.) From a point 3 miles from one end of an island and 7 miles 
from the other end, the island subtends an angle of 33 55' 15'''; find 
the length of the island. 

(26.) A point is 13581 in. from one end of a wall 12342 in. long, and 
10025 in - from tne other end. What angle does the wall subtend at 
this point? 

(27.) A straight road ascends a hill a distance of 213.2 ft., and is in- 



THE OBLIQUE TRIANGLE 53 

clined 12 2' to the horizontal; a tree at the bottom of the hill 
subtends at the top an angle of 10 5' 16". Find the height of the 
tree. 

(28.) Two straight roads cross at an angle of 37 50' at the point A ; 
^ miles distant on one road is the town B, and 5 miles distant on the 
other is the town C. How far are B and C apart ? 

(29.) Two stations, A and B, on opposite sides of a mountain, are 
both visible from a third station, C\ AC =11.5 miles, BC = 9.4 miles, 
and the angle ACB^cp $1'. Find the distance from A to B. 

(30.) To obtain the distance of a battery, A, from a point, B, of the 
enemy's lines, a point, C, 372.7 yards distant from A is taken ; the an- 
gles ACB and CAB are measured and found to be j$ 53' and 74 35' 
respectively. What is the distance ABt 

(31.) A town, B, is 14 miles due west of another town, A. A third 
town, C, is 19 miles from A and 17 miles from B. How far is C west 
of A? 

(32.) Two towns, A and B, are on opposite sides of a lake. A is 
18 miles from a third town, C, and B is 13 miles from C; the angle 
ACB is 13 17'. Find the distance between the towns A and B. 

(33.) At a point in a level plane the angle of elevation of the top 
of a hill is 39 51', and at a point in the same direct line from the hill, 
but 217.2 feet farther away, the angle of elevation is 2&J 53'. Find 
the height of the hill above the plane. 

(34.) It is required to find the distance between two inaccessi- 
ble points, A and B. Two stations, C and D, 2547 ft. apart, are 
chosen and the angles are measured ; they are ACB=2j 21', BCD 
=33 14', DA = i8 17', and ADC=$i 23'. Find the distance from. 
A to B. 

(35-) Two trains leave the same station at the same time on straight 
tracks inclined to each other 21 12'. If their average speeds are 40 
and -^Smiles an hour, how far apart will they be at the end of the first 
fifteen minutes? 

(36.) A ship, A, is seen from a light-house, B\ to determine its dis- 
tance a point, C, 300 ft. from the light-house is taken and the angles 
BCA and CBA measured. If CA = io8 34' and CBA=6$ Q 27', what- 
is the distance of the ship from the light-house? 



54 



PLANE TRIGONOMETRY 



(37.) Prove that the radius of the inscribed circle of a triangle is 
equal to a sin-|Z>' sin^Csec^. 




Hint. Draw OB, OC, and the perpendicular OD. 
OB and OC bisect the angles B and C respectively, and ODr. 



sm 



Hence 



sin \ /J sin -^ C sin ^ A' sin \ C 
sin ^ v9 sin 



- 
cos-i/4 



= a sin 



sm i c sec 



CHAPTER V 

CIRCULAR MEASURE GRAPHICAL REPRESENTATION 
CIRCULAR MEASURE 

53. The length of the semicircumference of a circle is 
irR (77 = 3.14159 + ); the angle the semicircumference sub- 
tends at the centre of the circle is 180. Hence an arc 
whose length is equal to the radius will subtend the angle 

1 80 

; this angle is the unit angle of circular measure, 

and is called a radian. 

7T R 





If the radius of the circle is unity, an arc of unit length 
subtends a radian ; hence in the unit circle the length of an 
arc represents the circular measure of the angle it subtends. 

Thus, if the length of an arc is , it subtends the angle - radians. 



Since one radian = 



1 80 



we have 






00 radians, 

2 

= 7r radians, 



56 PLANE TRIGONOMETRY 

270= - radians, 

360 2?r radians/etc. 

The value of a radian in degrees and of a degree in radians are : 
i radian = 57.29578, 

= 57 17' 45". 
1=:. 0174533 radian. 
In the use of the circular measure it is customary to omit the word radian ; 

thus we write - , TT, etc., denoting - radians, TT radians, etc. On the other 

hand, the symbols are always printed if an angle is measured in degrees, 
minutes, and seconds ; hence there is no confusion between tlie systems. 

EXERCISES 

(i.) Express in circular measure 30, 45, 60, 120, 135, 720, 990. 
(Take ^=3.1416.) 

(2.) Express in degrees, minutes, and seconds the angles -^, , - ,-. 

8 10 2 4 

(3.) What is the circular measure of the angle subtended by an arc 
of length 2.7 in., if the radius of the circle is 2 in.? if the radius is 

5 in. ? 

<T4. The following important relations exist between the 
circular measure x of an angle and the sine and tangent of 
the angle. 

(i .) If x is less than , sin x < x < tan x. 




O S 

Draw a circle of unit radius. 
By Geometry, SP<arcAP<AT. 
Hence sin x <x < tan^r. 



CIRCULAR MEASURE 57 

sin x tan x 
(2.) As x approaches the limit o, and approach 

& Jv 

the limit i. 

Dividing sin x < x < tan x by sin x, we obtain 



, 
I <- - < 



sinjr cos^r 

sin 4: cos JIT 
Inverting, i>~ > 

*-v 1 

As x approaches the limit o, COS.T approaches the length 
of the radius, that is, i, as a limit. 

Therefore, - - approaches the limit I. 

sin x 
Dividing i > - > cos^r by cos;r, we obtain 

Jv 

i tan x 



COS X X 

As x approaches the limit o, cos;r approaches the limit I ; 

hence approaches the limit I. 

cos-r 

Therefore, - approaches the limit I. 




PERIODICITY OF THE TRIGONOMETRIC FUNCTIONS 

o& The sine of an angle x is the same as the sine of 
(^+360), (x + 720), etc. that is, of (>+2;/7r), where n is 
any integer. 

The sine is therefore said to be a periodic* function, hav- 
ing the period 360, or 2?r. 

The same is true of the cosine, secant, and cosecant. 

* If a function, denoted by /(.*)> of a variable jc, is such that f(x + k}=f(x} 
for every value of x, k being a constant, the function f(x) is periodic; if k is 
the least constant which possesses this property, k is the period of /(.r). 



58 PLANE TRIGONOMETRY 

The tangent of an angle x is the same as the tangent of 
(x+ 180), (^4-360), etc. that is, of (x + ntr\ where n is any 
integer. 

The tangent is therefore a periodic function, having the 
period 1 80, or TT. 

The same is true of the cotangent. 

GRAPHICAL REPRESENTATION 

36. On the line OX lay off the distance OA(=x) to rep- 
resent the circular measure of the angle x. At the point A 
erect a perpendicular equal to sin x. If perpendiculars are 
thus erected for each value of x> the curve passing through 
their extremities is called the sine curve. 

If sin* is negative, the perpendicular is drawn downward. 



In a similar manner the cosine, tangent, cotangent, secant, 
and cosecant curves can be constructed. 




Sine Curve 




-1 



Cosine Curve 



GRAPHICAL REPRESENTA TION 



59 




TangentCurve 




.0 



Cotangent Curve 



6o 



PLANE TRIGONOMETRY 








3 / 2 7T 




SECANT CURVE 



If the distances on OX are measured from O' instead of 
O, we obtain from the secant curve the cosecant curve. 

In the construction of the inverse curves the number is 
represented by the distance to the right or left from O\ 
the circular measure of the angle by the length of the per- 
pendicular erected. 

All of the preceding curves, except the tangent and co- 
tangent curves, have a period of 2?r along the line OX; that 
is, the curve extended in either direction is of the same 
form in each case between 2?r and 477% 4?r and 6?r, 27r and 
o, etc., as between o and 2?r, while the corresponding inverse 
curves repeat along the vertical line in the same period. 
The period of the tangent and cotangent curves is TT. 



GRAPHICAL REPRESENTATION 



61 




-i o +1 

INVERSE SINE CURVE 




-1 



INVERSE COSINE CURVE 



J L 



-3 -2 -1 +1 + 2 +3 

INVERSE TANGENT CURVE 



62 



PLANE TRIGONOMETRY 




87T 



-3 



+ 3 



INVERSE SHCANT 



CHAPTER VI 

COMPUTATION OF LOGARITHMS AND OF THE TRIG- 

ONOMETRIC FUNCTIONS -DE MOIVRE'S THEOREM 

HYPERBOLIC FUNCTIONS 

*7f. A convenient method of calculating logarithms and 
the trigonometric functions is to use infinite series. In 
work? on the Differential Calculus it is shown that 



= x-~2+i-j-+ d) 

nf*$ OT^ ZM*^ 

= x -_+__-+...* (2) 

/y2 / y^ /y6 

l- 4j j + il -^ ! + . (3) 

Another development which we shall use later is 
or or^ or^ Gt*^ 

e * =:1 + l! + 2! + 3! + 4! + - ( 4) 

where c 2. 7 18281 8 ... is the base of the Naperian system 
of logarithms. 

58. The series (i) converges only for values of x which satisfy the 
inequality I<JT^I. The series (2), (3), and (4) converge for all 
finite values of .r. 

It is to be noted that the logarithm in (i) is the Naperian, and the 
angle x in (2) and (3) is expressed in circular measure. 

* 3! denotes 1x2x3; 4] denotes 1x2x3x4, etc. 



64 PLANE TRIGONOMETRY 

COMPUTATION OF LOGARITHMS 

59. We first recall from Algebra the definition and some 
of the principal theorems of logarithms. 

The logarithm to the base a of the number m is the number .r 
\vhich satisfies the equation, 



This is written x = 

The logarithm of the product of two numbers is equal to the sum 
of the logarithms of the numbers. 

T h us 1 og^ m 11 = \og a m +'1 og a n. 

The logarithm of the quotient of two numbers is equal to the log- 
arithm of the dividend minus the logarithm of the divisor. 

in 

Thus l<z 1R ;;/ lgtf ;z - 

n 

The logarithm of the power of a number is equal to the logarithm 
of the number multiplied by the exponent. 

Thus log^ m*=p \og a m. 

To obtain the logarithm of a number to any base a from its Na- 
perian logarithm, we have 

log in 

log* m = = M a log, m, 

tog, a 

where M rt = ; M a is called the modulus of the system. 

6*0. We proceed now to the computation of logarithms. 
The series (i) enables us to compute directly the Naperian 
logarithms of positive numbers not greater than 2. 

Example. To compute log*- to five places of decimals. 
Substitute - for x in (i): 



2/22 2 3 2' 4 2 

If the result is to be correct to five places of decimals, we must take enough 
terms so that the remainder shall not affect the fifth decimal place. Now we 



COMPUTATION OF LOGARITHMS 



know by Algebra that in a series of which the terms are each less in numerical 
value than the preceding, and are also alternately positive and negative, the re- 
mainder is less in numerical value than its first term. Hence we need to take 
enough terms to know that the first term neglected would not affect the fifth 
place. 

Positive terms 





2 


=0.5000000 


I 


1 


= .0416667 


3 


2 




j 


I 








= .0062500 


5 


2 




I 


1 


=. .OOIIl6l 


7 


2 1 




i 


1 









= .0002170 


9 


2 




i 


I 








= .OOOO444 


1 1 


- 




i 


I 








= .0000094 


13 


2 





.5493036 



Negative terms 


i 


I 




2 


2* ~~ 


3.1250000 


I 

4 


I 
' 2 1 


.0156250 


I 
6 


I 


.0026042 


i 
8 


I 


.0004883 


i 


T 






= 


.0000977 


10 


2 




I 


I 




12 


*2" = 


.OOOO2O3 


I 


I 




14 


'2" = 


.0000044 



.1438399 



Subtracting the sum of the negative from the sum of the positive terms, we 
obtain 



lo &^= -4054637- 
Denote the sum of the remaining terms of the series by. R. Then, by Alge- 



bra, 



< .0000021. 

The error caused by retaining no more decimal places in the computation is 
less than .0000006. Hence the total error is less than .0000027. Therefore 
the result is correct to five decimal places. 

61. As remarked, the series (i) does not enable us to 
calculate directly the logarithms of numbers greater than 2, 
but it can be readily transformed into a series which gives 
us the logarithm of any positive number. 

Replacing x by x in (i), we obtain 
5 



66 PLANE TRIGONOMETRY 

x* x 3 x* 
log, (i -*)=-* --_-_- 

This series converges for i <*<* 

Subtracting this from (i), we obtain 



*v\ 

.,j-H =2 (^f + I + 7 V "-)' (s) 

4 f I 

/ which converges for i < x < i . 

Putting y=[- -), we see that j passes from o to oo as x 
\i x/ 

passes from i to -f-i ; hence, if we make this substitution in 
(c), we get a 



which converges for all positive values of j, and therefore enables 
us to compute the Naperian logarithm of any number. 

From (5) we can get another series which is useful : put 

i i4-x y+i 

x - ; then, as -=-=- - , equation (5) gives us 



which converges for all positive values of y. Hence, 






This series gives us log,(jy-f-i), when log,^ is known. It con- 
verges more rapidly than (6), when y is greater than 2, and hence 
should be used under these circumstances. 

62. To construct a table we need to compute directly 
only the logarithms of prime numbers, since the others can 
be obtained by the relation 

log ,ry = \og ;r + log y. 

i- 



i 






COMPUTATION OF LOGARITHMS 



67 



Thus, to obtain the logarithms of the integers up to 10, 
we need to compute by series only the logarithms of the 
numbers 2, 3, 5, and 7. 

(For 4=2 2 , 6=2 . 3, 8 = 2*, 9=3 2 , 10=2 . 5, and log 1=0.) 
In this case we are computing the logarithms of successive integers, and 
should therefore use (7). 

6Yf. Example. Compute the Naperian logarithms of 2, 3, 4, and 5. 



. j 

3 3 3 3 5 



^.<..l + i.l 9 +. \ 

3 s 7 3 7 9 3 9 / 



-=-3333333 



.=. 0008230 
l -. ^=. 0000653 

1.1^.0000056 
.3465729 

2 



Denote the sum of the remaining 
terms of this series by A'. 
Then, by Algebra, 



or A" < .000000573. 

The error caused by not retaining 
more places of decimals in the pre- 
ceding column is less than .0000005. 

Hence, the total error is less than 
.00000165. 



log, 2 = .693 1458 

Remark. We should get the same series if we were to use (6). 



- = .2000000 
\ ' ^ = .0026667 



6 
- =.0000018 

7 5' 



.2027325 

2 



.4054650 

Add log, 2= .6931458 
log, 3 = 1.0986108 



AX- 



or A* < .00000006. 

Noting the errors in the pre- 
ceding column and in log, 2, we 
see that the total error is less than 

;OOOOO2I7. 



68 PLANE TRIGONOMETRY 

Remark. If we were to use (6) to compute log, 3, we should have 



This series converges much more slowly than the above, since its 
terms are multiples of powers of \, while the terms of the above are 
the same multiples of powers of \. Thus, we should be obliged to 
use eight instead of four terms to have the result correct to five 
places. 

log, 4 = 2 log, 2 = 1.3862916. 



or log, 5 = 1.60944. 

64. Proceeding in like manner, we may calculate any number of 
logarithms. 

The following table gives the Naperian logarithms of the first ten 
integers : 



log* J -ooooo 
log, 2= .69315 
log, 3 = 1.09861 
log* 4 =1.38629 
= 1.60944 



log, 6= 1.79176 

log, 7 = I-9459 1 
log, 8 = 2.07944 
log, 9 = 2. 19722 
log, 10 = 2.30259 

The common logarithm of any number may be found by multiply- 
ing its Naperian logarithm by M 10 =. 43429448. 59 
Thus log lo 5 = log, 5 X 43429448 = .69897. 

65. Remark. If a table of logarithms were to be computed, the 
theory of interpolation and other special devices would be employed. 

COMPUTATION OF TRIGONOMETRIC FUNCTIONS 

c . sin^r cos^r 

f>6. Since tan;tr= , cot;r= , etc., the computa- 

COS.T sin x 

tion of all the trigonometric functions depends upon that of 
the sine and cosine ; thus the developments (2) and (3) suf- 
fice for all the trigonometric functions. Further, since the 

^ 



COMPUTATION OF SINES AND COSINES 69 

sine or cosine of any angle is a sine or cosine of an angle 

p-, it is never necessary to take x greater than - in the 
^4 4 

series (2) and (3). 16 

Since - =0.785398 .,.<, these series converge rapidly; in fact, 
4 10 

= .000003 does not affect the fifth decimal place, and the 
9! 11! 

seventh. 

67. Remark. In the systematic computation of tables we should 
not calculate the functions of each angle from the series independent- 
ly. We should rather make use of the formulas (25) and (27) of 38, 
thus obtaining 

sinw.r = 2 cos.r sin (n i)^r sin (n 2) x, 
cos nx = 2 cos x cos (n i ) x cos (;/ 2) x. 

If our tables are to be at intervals of i', we should calculate the 
sine and cosine of i' by the series. The above expressions then en- 
able us to find successively the sine and cosine of 2', 3', 4', etc., till we 
have the sine and cosine of all angles up to 30 at intervals of i'. 

To obtain the sine and cosine of angles from 30 to 45 we should 
make use of these results by means of the formulas 
sin (30+ y) =cosy sin (30 y), 
cos (3o-f-_y) = cos (30^) sin y. 

68. To employ series (2) and (3) in computing the sine 
and cosine we must first convert the angle into circular 
measure. 

To do this we recall that 

i = . 017453293, i ' = .0002908882, i " = . 000004848 1 37. 
Example. To compute the sine and cosine of 12 15' 39". 

1 2= .209439516 
15' =.004363323 
39" = .000189076 
12 15' 39" = .213991915 in circular measure. 



PLANE TRIGONOMETRY 



x-. 2139919 



( 



= . 0000037 



.2139956 

X* 

subtract = .0016332 



Correct to five decimal places. 



cos*=i H 

2! 4! 

1 = 1.0000000 



= .0000874 
4 ! 

1.0000874 
subtract -= .0228963 

cos.r= .9771911 
Correct to five decimal places. 



DE MOIVRE'S THEOREM 
00. In Algebra we learn that the complex number 



(8) 



may be represented graphically thus : 
Y 




Take two lines, OX and OY, at right angles to each other. 
To the number a will correspond the point A, whose dis- 
tances from the two lines of reference are ft and a re- 
spectively. 

This geometrical representation shows at once that we 
can also write a in the form 

a=r (cos $+4 sin 3). (9) 

7O. From Algebra we recall the definition of the sum of the 
complex numbers a a + //3 and b=y + il-, namely 



Subtraction is defined as the inverse of addition, so that 
a b^a y-M'()3 c). 



DE MOIVRE'S THEOREM 71 

Multiplication is most conveniently defined when a and b are 
written in form (9). If 

a r (cos-$-M sin ) and d~s (cos^-f / sin^), 
their product is defined by the equation 

ab rs [cos (3 + 0)-|-/ sin (-r-0)]- ( IO ) 

Division is defined as the inverse of multiplication, so that 



Finally, we recall that in an equation between complex numbers, 

+173=7+13, 
we have =y, /3 = S. (n) 

*7 71. Consider the different powers of the complex number 

# = cos $+*' sin $. 
By (10) we have 

* a = (cos $ + / sin S) (cos $+/ sin $), 

= cos 2-S + / sin 2-$. 
jc 3 =A: 2 . ^ = (cos 2^+1 sin 2$) (cos $+*' sin 3), 

=cos3^-|-/ sin 3^. 
And, in general, for any integer n, 

*=:(cos 3+t sin ^)"=cos n$+i sin n. 

From this equation we have De Moivre's Theorem, which 
is expressed by the formula 

(12) 



72. An interesting application of De Moivre's Theorem 
is the expansion of sin nx and cos nx in terms of sin x and 
COS.T. Expanding the left-hand side of (12) by the bino- 
mial theorem, and substituting x for 3-, we have 

cosnx-\-i sin nx=cos n x+n cos*" 1 x (i sin x) -f - j cos w ~ 2 ^ 
( sin*)' + f*=!l=!> co8*(isin*)' + . . . 



7 2 PLANE TRIGONOMETRY 

or 

cosnx + i sinnx=(cos n x j - cos w ~ 2 ^ sin 2 *-}- . . .) 

[// (n i) (n 2) 
n cos"" 1 x smx cos w 3 x sin x+ .... 

Equating real and imaginary parts, as in (11), we have 
cosnx=cos n x cos n ~ 2 x sin' 2 #-f- ; . . (13) 



sin//* ncos n l xs'mx - cos*" 3 * sin 3 x+. . . (14) 

Example. n = 5. 

cos 5-r = cos 6 x 10 cos 3 .*- sin 2 ^r-f-5 cos^r sin*,r. 
sin 5_r= 5 cos*,r sin or 10 cos 2 x sin 3 x + sin* x. 

THE ROOTS OF UNITY 

73. We find another application of De Moivre's Theorem 
in obtaining the roots of unity. The # th roots of unity are 
by definition the roots of the equation 

x n \. 

Every equation has n roots and no more ; hence, if we 
can find n distinct numbers which satisfy this equation we 
shall have all the # th . roots of unity. 
Consider the // numbers 

2?rr 2irr 

x r = cos \-t sin , 

n n 

r=o, i, 2, ... n i. 

Geometrically these numbers are represented by the n 
vertices of a regular polygon. They are, therefore, all dif- 
ferent. We shall see now that they are precisely the ;/ th 
roots of unity. 

In fact, we have by (12), 



* ( cos \-i sin ) , 

\ ;/ n ) 



THE ROOTS OF UNITY 



(2Ttr\ . . . / 2:rr\ 
n . - )-f-j sin [. -), 
/ V / 



73 



sin 27T/-, 
= 1+*'. = 1. 
Therefore x r is one of the roots of unity. 

Thus the cube roots of unity are represented by the points A, P, 
and Q of the following figure. In the figure OA = i, angle AOP = 

= i2o, angle AOQ = = 240; that is, the circumference is di- 
vided into three equal parts by the points A, P, and Q. Then OD =4, 
and DP DQ = ^^/^. Hence we see from the method of represent- 
ing a complex number given above that A represents -\-i,P represents 
;, Q represents /' 

P. 




EXERCISES 

74. (i.) Express sin 4* and cos 4* in terms of sin x and cos*. 
(2.) Express sin 6.r and cos 6* in terms of sin x and cos*. 
(3.) Find the six 6 th roots of unity. 

(4.) Find the five 5 th roots of unity. 

THE HYPERBOLIC FUNCTIONS 

75. The hyperbolic functions are defined by the equations 

inha?= ~ , (15) 



(16) 



cosh x = 






in which sinh* and cosh* denote the hyperbolic sine and 



74 PLANE TRIGONOMETRY 

hyperbolic cosine of x respectively. These functions are 
called the hyperbolic sine and cosine on account of their 
relation to the hyperbola analogous to the relation of the 
sine and cosine to the circle. A natural and convenient 
way to arrive at the hyperbolic functions and to study their 
properties is by using complex numbers in the following 
manner. The series (2), (3), and (4) give the value of sin x, 
cos^r, and e* for every real value of x. These series also 
serve to define sin^r, cos^r, and e x for complex values of x. 
In the more advanced parts of Algebra it is shown that 
the following fundamental formulas which we have proved 
only for a real variable, 

sin (x+y) = s\nx cosjy-}- cos x sin y, (17) 

cos (x+y) CQSx cos^ sin* sinj, (18) 

e*+*=e*e", (19) 

hold unchanged when the variable is complex. 

This fact enables us to calculate with ease sin^r, COS.T, and 
e x for any complex value of the variable. 

In so doing we are led directly to the hyperbolic func- 
tions. At the same time a relation between the trigono- 
metric and hyperbolic functions is established by means of 
which the formulas of Chapter III. can be converted into 
corresponding formulas for the hyperbolic functions. 

Taking x and y real and replacing y in (17), (18), and (19) by 

*>, we get 

sin (aF+/y) = sina: cos/y-f cos* sin iy, 

cos (x+iy)=cos x cos iy sin x sin iy, 



Thus the calculation of these functions when the variable 
is complex is made to depend upon the case where the vari- 
able is a pure imaginary. 



HYPERBOLIC FUNCTIONS 75 

If we replace x by ix in series (4) we obtain 



V 



3! 5! 7! 

A comparison with series (2) and (3) shows that these two 
series are cos^r and sin^r respectively; hence the important 
formula due to Euler 

This enables us to calculate ?* from sin^r and cos;r when 
ix is a pure imaginary ; that is, when x is real. 

To find sin ix and cosix replace x in (20) by ix\ we obtain 

e*=cosix+t sin ix. (21) 

Again replacing x by ix in (20), we obtain 

e* = cos ix i sin ix. (22) 

The sum and difference of (21) and (22) give 

cos ix = == cosh a?, (23 ) 

(24) 



If we compute the value of e* by the aid of series (4) for 
a succession of values of x, we find that sinh^r and cosher 
are represented by the curves on page 76. 

The system of formulas belonging to the hyperbolic func- 
tions is obtained from those of the trigonometric functions 
by using (23) and (24). This shows that for every formula 
in analytic trigonometry there exists a corresponding for- 
mula in hyperbolic trigonometry which we get by this sub- 



7 6 



PLANE TRIGONOMETRY 



stitution. In the examples which follow, this method is 
used to obtain important formulas in hyperbolic trigonome- 
try. 

Replacing x by ix in (23) and (24), we get 

- (25) 

- (26) 



which are formulas frequently used. 



Example. sinh (jr -}- j) = / sin / ( 

= i [sin ix cos /p + cos ix sin /y], 

= / [/' sinh .r cosh^/ + / cosh x sinhj], 

= sinh x cosh_y + cosh x sinh y. 

Example. sinh x -\- sinh_y = /(sin ix -\- sin iy), 

i 2 sin i(x -\-y} cos ^ i (x y\ 
= 2 sinh (jr +7) cosh ^ (xy). 




sinh 




HYPERBOLIC FUNCTIONS 77 

EXERCISES 

70. (i.) Prove sinho=o, cosho=i. 
(2.) Prove sinh TT/ = /, cosh^7r/=o. 
(3.) Prove sinh TT/ = O, cosh7i7 = i. 

Prove that 

(4.) si n (* /,r) = sin ix. 

( 5 .) cos ( ix) = cos /Jr. 

(6.) sinh( x) = sinh x. 

(7.) cosh( x} = cosher. 

Remark. The hyperbolic tangent, cotangent, secant, and cosecant 
are defined by 

sinhjr cosher 

tanh-r = 



cosher smh^r 

sech-r = \ , eschar = 



t \^*3\-,LM. ** ^ . . 

cosher sinner 

Prove that 

(8.) tan (tx) i tanh x. 

(9.) coth ( x} = coth x. 

(10.) sech ( .r) = sech x. 

(n.) cosh a jr sinh 3 jr=:i. 

(12.) sech a .r-f- tan h a .r= i. 

(13.) coth'.r csch'jr = i . 

(14.) sinh(.i- y) = sinh^r cosh y cosher sinh^. 

(15.) cosh(.i- _y) = cosher coshj sinhjr 



(,6.) coshi.r =v /l 

(17.) sinhw sinhr/ = 2 cosh \(u + v) sinh \ (u v). 

(i 8.) cosh u -\- cosh v = 2 cosh %(u-\-v) cosh \(u^v). 

(19.) cosh u cosh v = 2 sin h^(#-f-?/) sinh %(u z/). 



CHAPTER VII 

MISCELLANEOUS EXERCISES 
RELATION OF FUNCTIONS 

77. Prove the following : 

(i.) cos^r = sin^r cot-r. 

(2.) CSC.T tan x = sec x. 

(3.) (tan x -|- cot x) s\nx cos^r=:i. 

(4.) (sec/ tan /) (sec y -f tan /) = i . 

(5.) (CSC 2 COt 2) (CSC 2 -f- COt z) = I . 

(6.) cos 2 / + (tan / cot/) sin/ cos y = sin 2 /. 
(7.) cos 4 .r sin 4 * -{-1=2 cos 2 ,r. 
.) (sinj/ cos/) 2 = 1 2 sin j cos/. 

sin^r COSJT). 



. cot x-\- tan y 

(10.) - - - = cot. r tan y. 
tan ^- + cot/ 

\-<n.) cos 2 / sin 2 / = 2 cos 2 / i. 
(12.) i tan 4 ^r = 2 sec 2 ^r sec*;r. 



(i 3-) - - 5 = tan jr. 
sm^r cot 2 ^: 

(14.) sec 2 / esc 2 / = tan 2 / -f- cot 2 / +2. 

.) cot/ esc/ sec/ (i 2 sin 2 /) = tan/. 



, / I \ 2 I COS.? 

(16.) H cot 2^) = 

Vsin / i 4- cos z 



I +cos/ sin 3 / 

(i 8.) i+ 



(19.) - -- sin 3 .r = (cos-r sin x) (i-|-sin.r cos,r). 
(20.) (sin,r cos/-f-cos.r sin/) 2 -j-(cos.r cos/ sin .r si 



MISCELLANEOUS EXERCISES 79 

(21.) (a cos.r b sin .r) 2 -(-(,* sin x + b cos.r) 2 = &*+&*. 

-> ' ^ 4tan 2 j 

sin 2 jf ~ (i tan 2 ;/) 2 ' 



Find an angle not greater than 90 which satisfies each of the fol- 
lowing equations: 
(23.) 4 cos x = 3 sec .r. 
(24.) sin^^cscj f. 
(25.) \/ 2 sin.r tan.r = o. 
(26.) 2 cos.r \/3 cot.r = o. 
(27.) tan j + cot j 2 = 0. 
(28.) 2 sin'-/ 2 = \/2 cosj. 
(29.) 3 tan 2 .r i =4 sin 2 .r. 
(30.) cos 2 .r-|-2 sin a .r -| sin.r = o. 
(31.) csc.r = tan.r. 
(32.) sec .r -(- tan .r \/3- 
(33.) tan .r + 2 v/3 cos.r = o. 
(34.) 3 sin.r 2 cos 2 .r=o. 

Express the following in terms of the functions of angles less 
than 45: 
(35.) sin 92. 
(36.) cos 1 27. 
(37.) tan 320. 
(38.) cot 350. 
(39.) sin 2f>5 . 
(40.) tan 171. 

(41.) Given sin .r = $ and x in quadrant II; find all the other 
functions of x. 

(42.) Given cos.n= f and x in quadrant III; find all the other 
functions of x. 

(43.) Given tan.r = | and x in quadrant III; find all the other 
functions of x. 

.'44.) Given cot,r = and x in quadrant IV; find all the other 
functions of x. 



8o PLANE TRIGONOMETRY 

In what quadrants must the angles lie which satisfy each of the 
following equations : 
(45.) sin^r cos.*' i\/3. 
(46.) sec.r tan .* = 2-^/3. 
(47.) tanjj/ + -y/20 cos}> = o. - 
(48.) cos x cot x = . 

Find all the values of y less than 360 which will satisfy the fol- 
lowing equations : 

(49.) tanj-f-2 sinj = o. 

(50.) (i -f- tan x) (i 2 sin x] = o. 

(51.) sin^r COS.T (i + 2 cos-r)=o. 



Prove the following: 
(52.) cos 780 = . 

(53.) sin 1485 = i\/2. 
(54.) cos 2550 = ^ -y/3- 
(55.) sin ( 3000) = cos 30. 
(56.) cos 1 300 = cos 40** 

(57.) Find the value of a sin 90 + b tano-\-a cos 180. 
(58.) Find the value of a sin 30 + ^ tan 45 -\-a cos 60 -\-b tan 135. 
(59.) Find the value of (a b) tan 225 + cos 180 a sin 270. 
(60.) Find the value of (a sin 45+^ cos 45) (a sin 135 + ^ sin 225). 

RIGHT TRIANGLES 

7. In the following problems the planes on which distances are measured 
are understood to be horizontal unless otherwise stated. 

(i.) The angle of elevation of the top of the tower from a point 
1 121 ft. from its base is observed to be 15 17'; find the height of 
the tower. 

(2.) A tree, 77 ft. high, stands on the bank of a river ; at a point on 
the other bank just opposite the tree the angle of elevation of the 
top of the tree is found to be 5 if 37". Find the breadth of the 



MISCELLANEOUS EXERCISES 81 

(3.) What angle will a ladder 42 ft. long make with the ground if its 
foot is 25 ft. from the base of the building against which it is placed ? 

(4.) When the altitude of the sun is 33 22', what is the height of a 
tree which casts a shadow 75 ft. ? 

(5.) Two towns are 3 miles apart. The angle of depression of one, 
from a balloon directly above the other, is observed to be 8 15'. 
How high is the balloon ? 

(6.) From a point 197 ft. from the base of a tower the angle of ele- 
vation was found to be 46 45' 54" ; find the height of the tower. 

(7.) A man 5 ft. 10 in. high stands at a distance of 4 ft. 7 in. from 
a lamp-post, and casts a shadow 18 ft. long; find the height of the 
lamp-post. 

(8.) The shadow of a building 101.3 ft- high is found to be 131.5 
ft. long; find the elevation of the sun at that time. 

(9.) A rope 112 ft. long is attached to the top of a building and 
reaches the ground, making an angle of 77 20' with the ground ; 
find the height of the building. 

(10.) A house is 130 ft. above the water, on the banks of a river; 
from a point just opposite on the other eank the angle of elevation 
of the house is 14 30' 21". Find the width of the river. 

(u.) From the top of a headland, 1217.8 ft. above the level of the 
sea, the angle of depression of a dock was observed to be 10 9' 13" ; 
find the distance from the foot of the headland to the dock. 

(12.) 1121.5 ft. from the base of a tower its angle of elevation is 
found to be n 3' 5 "; find the height of the tower. 

(13.) One bank of a river is 94.73 ft. vertically above the water, and 
subtends an angle of 10 54' 13" from a point directly opposite at the 
water's edge; find the width of the river. 

(14.) The shadow of a vertical cliff 113 ft. high just reaches a boat 
on the sea 93 ft. from its base ; find the altitude of the sun. 

(15.) A rope, 38 ft. long, just reached the ground when fastened to 
the top of a tree 29 ft. high. What angle does it make with the 
ground ? 

(16:) A tree is broken by the wind. Its top strikes the ground 15 
ft. from the foot of the tree, and makes an angle of 42 28' with the 

ground. Find the height of the tree before it was broken. 
6 



82 PLANE TRIGONOMETRY 

(17.) The pole of a circular tent is 18 ft. high, and the ropes reach- 
ing from its top to stakes in the ground are 37 ft. long; find the 
distance from the foot of the pole to one of the stakes, and the angle 
between the ground and the ropes. 

(i 8.) A ship is sailing southwest at the rate of 8 miles an hour. 
At what rate is it moving south ? 

(19.) A building is 121 ft. high. From a point directly across the 
street its angle of elevation is 65 3'. Find the width of the street. 

(20.) From the top of a building 52 ft. high the angle of elevation 
of another building 112 ft. high is 30 12'. How far are the buildings 
apart ? 

(21.) A window in a house is 24 ft. from the ground. What is the 
inclination of a ladder placed 8 ft. from the side of the building and 
reaching the window ? 

(22.) Given that the sun's distance from the earth is 92,000,000 
miles, and its apparent semidiameter is 16' 2" ; find its diameter. 

(23.) Given that the radius of the earth is 3963 miles, and that it 
subtends an angle of 57' 2" at the moon; find the distance of the 
moon from the earth. 

(24.) Given that when the moon's distance from the earth is 238885 
miles, its apparent semidiameter is 15' 34"; find its diameter in miles. 

(25.) Given that the radius of the earth is 3963 miles, and that it 
subtends an angle of 9" at the sun ; find the distance of the sun 
from the earth. 

(26.) A light-house is 57 ft. high ; the angles of elevation of the top 
and bottom of it, as seen from a ship, are 5 3' 20" and 4 28' 8". Find 
the distance of its base above the sea-level. 

(27.) At a certain point the angle of elevation of a tower was ob- 
served to be 53 51' 1 6", and at a point 302 ft. farther away in the 
same straight line it was 9 52' 10"; find the height of the tower. 

(28.) A tree stands at a distance from a straight road and between 
two mile-stones. At one mile-stone the line to the tree is observed 
to make an angle of 25 15' with the road, and at the other an angle 
of 45 17'. Find the distance of the tree from the road. 

(29.) From the top of a light-house, 225 ft. above the level of the 
sea, the angle of depression of two ships are 17 21' 50" and 13 50' 22", 



MISCELLANEOUS EXERCISES 83 

and the line joining the ships passes directly beneath the light-house ; 
find the distance between the two ships. 

ISOSCELES TRIANGLES AND REGULAR POLYGONS 

S^- 
79. (i.) The area of a regular dodecagon is 37.52 ft.; find its 

apothem. 

(2.) The perimeter of a regular polygon of 1 1 sides is 23.47 ft. ; find- 
the radius of the circumscribing circle. 

(3.) A regular decagon is circumscribed about a circle whose radius 
is 3.147 ft. ; find its perimeter. 

(4.) The side of a regular decagon is 23.41 ft. ; find the radius of 
the inscribed circle. 

(5.) The perimeter of an equilateral triangle is 17.2 ft.; find the 
area of the inscribed circle. 

(6.) The area of a regular octagon is 2478 sq. in. ; find its pe- 
rimeter. 

(7.) The area of a regular pentagon is 32.57 sq. ft. ; find the radius 
of the inscribed circle. 

(8.) The angle between the legs of a pair of dividers is 43, and the 
legs are 7 in. long ; find the distance between the points. 

(9.) A building is 37.54 ft. wide, and the slope of the roof is 43 36' ; 
find the length of the rafters. 

(10.) The radius of a circle is 12732, and the length of a chord is 
18321 ; find the angle the chord subtends at the centre. 

(u.) If the radius of a circle is taken as unity, what is the length 
of a chord which subtends an angle of 77 17' 40"? 

(12.) What angle at the centre of a circle does a chord which is ^ 
of the radius subtend ? 

(13.) What is the radius of a circle if a chord 11223 ft. subtends an 
angle of 59 50' 52"? 

(14.) Two light-houses at the mouth of a harbor are each 2 miles 
from the wharf. A person on the wharf finds the angle between the 
lines to the light-houses to be 17 32'. Find the distance between the 
two light-houses. 

(15.) The side of a regular pentagon is 2; find the radius of the 
inscribed circle. 



84 PLANE TRIGONOMETRY 

(i 6.) The perimeter of a regular heptagon inscribed in a circle is 
12 ; find the radius of the circle. 

(17.) The radius of a circle inscribed in an octagon is 3; find the 
perimeter of the octagon. 

(18.) A regular polygon of 9 sides is inscribed in a circle of unit 
radius; find the radius of the inscribed circle. 

(19.) Find the perimeter of a regular decagon circumscribed about 
a unit circle. 

(20.) Find the area of a regular hexagon circumscribed about a 
unit circle. 

r (21.) Find the perimeter of a polygon of 11 sides inscribed in a 
\ unit circle. 

(22.) The perimeter of a dodecagon is 30; find its area. 

(23.) The area of a regular polygon of 11 sides is 18; find its pe- 
rimeter. 

TRIGONOMETRIC IDENTITIES AND EQUATIONS 

8O. Prove the following : 
(i.) sin 



sin 2x -f- si 



(4.) cos 2 / tan 2 / -f sin 2 / cot 8 / = I. 

(5.) -. = cot x cot y cot z cot x cot y cot z. 

sin* sin/ sin z 

(6.) cos 2 (x y) sin 2 (x +/) = cos 2 x cos 2/. 

. sin jr-4-siny 

(7.) - = cot i (x y). 

cos x cos y 

. cosx sec^r 



sin 2x 

(o.) cot;r = 

i cos 2x 

I COS 2/ 

(10.) tan y = 
i + cos 2/ 

(11.) cot x tan .r = 2 cot 2x. v , 



/ 



MISCELLANEOUS EXERCISES 85 



(12.) tan^.r + 2 sin 2 ^,r cot.r 

tan x tan / 

(n.) - = dbsm^r sec.r tan/. 

cot .r cot/ 

(14.) sin .r 2 sin 3 ,r = sin x cos 2.r. 

(15.) 4 sin/ sin (60 /) sin (60 -(-/) = sin 3/. 

suy(,-tan^)/ _1_ _ - ' = si 

sec 2 / Vcos/ sin/ cos/+sin/ 



(17.) i + tan/ tan /= 

(1 8.) sin 4^r = 4 sin.r cos 3 ^r 4 cos^r si 



(20.) tan 50 4- cot 50 2 sec 10. 
(21.) cos (x + 45) 4- sin (x 45) = o. 
tan^r 



(22.) 



i cot 2x tan x 



(23.) ( i tan 2 x) sin x cos .r = cos 2.r yj 



cos 2x 

-f COS 2X ' 



- = ec 

' cos/ sin/ 



(25.) sin (*+/) cos,r cos(.r+/) sin^r = si 
(26.) cos (.r /) sin/ + sin (x /) cos/ = sin JJT. 
. sin(jr /) sin(/ g) , sin (g JT) _ 

(27 ) - _ _i_ - -- --4- u. 

COSX COS/ COS/ COS.? COS 2 COSX 

sin +sir L 2 = co 
cos .r cos 2.r 

(29.) 2 sin 2 ,r sin a /-h 2 cos 2 x cos 2 / = i + cos 2x cos 2/. 
(30.) sin 60 4- sin 30 = 2 sin 45 cos 15. 

tan (,r-/) + tan/ 



u ' ; i tan (.r,/) tan/ 



sin/ tan i/ 

(33.) sin^r+sin 2.r = 2 sin 
sin x 4- sin/ _ 



cos .* cos/ sn/ sn x 



(36.) 2 tan 2/ = tan(45+/) tan (45 /). 



86 PLANE TRIGONOMETRY 

tan2.r4-tan.r _sin3.r 
tan 2 x tan x ~ sin^r 

(38.) 



i 3 tan 2 / 

(39.) sin 60 4- sin 20 = 2 sin 40 cos 20. 
(40.) sin 40 sin 10 = 2 cos 2 5 sin 15. 
(41.) cos ix cos4-r = 2 sin yc sin x. 
(42.) tan 15 = 2 v/3- 
(43.) (\/i 4sin.r \/i sin.r) 2 =4 sin 2 
(44.) "V/i H-sin^r--'i sin^) 2 = 4 cos 



, ... N siii4.r 

(46.) . * = 2 COS 2.T. 



(47.) sin 50 sin7o-f-sinio = o. 

f .Q\ I? IT ^TT . IT 

(40-) cos -- cos- = 2 sin sin 

3 J2 12 12 



sin 750 - sin 1 5 /f 
y cos754-cosi5 V ' 



(51.) tan 8 .r(i+cot a .r) 3 = -- 



(52.) tan 7 5 = 2 

(53.) sin 3^- -f- sin 5.1- = 2 sin 4^- cos^r. 

(54.) cos 5-r -f- cos 9-r = 2 cos 7-r cos 2^r. 

~ 1 



(55.) sin 1 5 = 



\/2 



(56.) - - =tan;r. 

cos 3-r + cos ^* 

(57.) sin 5j = 5 sinj 20 sin 3 /-}- 16 sin 5 /. 
(58.) cos5/ = 5 cosj 20 cos 3 / + 16 cos 5 /. 



(60.) 

(6 1 .) cos 3^- -f- cos 5a- -}- cos 7* + cos 1 5^- = 4 cos 4^ cos 5 ^r cos >x 



(62.) sin 2 \x (cot \. i- i) 2 = i sin.r. 
sin \r 



MISCELLANEOUS EXERCISES 87 

sin 2 \x (c 
(63.) 

(64.) 

4 

(65.) 

cos .r sin.r 2 

(66.) cosj + cos I 1 2O y} + cos ( 1 2 

x si 113.1- 

(67.) . =2 cos2.r+ 1. 
sin.r 

(68) 



. 

cos 3-r 4- 3 cos x 

(64.) sin.r(i-|-tan.r)4-cos.r(i 4-cot.r) = esc ^ + sec ^r. 
cos 3 .r sin 3 .r 



(sin y sinj)(cos4_y cos6y) 
i cos.r 



4cos.r 



, , 

(70.) - - 2- = 2. 

sin.r cos.r 

s i H- sin a- 4- cos. r 
(71.) - =cotfr. 

i +sm.r cos.r 






cos (4-r 27) 4- cos (4.T 
Sin.r + sin3.r + sin 5 .r4-sin7.r = ^ 
cos .r -f- cos 3.r -|- cos $.r 4- cos 7.1- 

If A, B, and C are the angles of a triangle, prove the following 
(74.) sin2^4-|-sin2/>'4-sin2C = 4 sin A s\n sinC 
(75.) sin 2^f + sin 27)' sin 2(7 = 4 cos^4 cos# sinC. 
(76.) sinM4-sin 2 /'-f-sin 2 C=2 + 2 cos A cos B cos C 
(77.) tan ^ -f tan B 4- tan C = tan A tan ^ tan C. 

Solve the following equations for values of x less than 360. 

(78.) cos 2.r 4- cos x = i . 

(79.) sin.r4-sin7.r 

(80.) cos^" sin2.r 

(8 1.) cos.r sin3^r cos2.i- = o. 

(82.) sin^a- 2 sin2,r r=o. 

(83.) sin 2.r cos 2.r sin x + cos x = o. 

(84.) sin (60 .r) - sin (60 + x) = + i y 

(85.) sin (30 4- -r) cos (60 + x) = | ^ 



88 PLANE TRIGONOMETRY 

(86.) esc x = i + cot x. 

(87.) cos T.X = cos 2 .r. 

(88.) 2 sin_y=sin 2y. 

(89.) sin $y -\- sin 2y -\- s'my = o. 

(90.) sin a .r + 5 cos'.r = 3. 

(91.) tan(45 



OBLIQUE TRIANGLES 

81. (i.) It is required to find the distance between two points, A 
and JB.on opposite sides of a river. A line, AC, and the angles BAG 
and ACB are measured and found to be 2483 ft., 61 25', and 52 17' 
respectively. 

(2.) A straight road leads from a town A to a town B, 12 miles 
distant ; another road, making an angle of 77 with the first, goes from 
A to a town C, 7 miles distant. How far are the towns B and C apart ? 
In order to determine the distance of a fort, A, from a battery, 
B, a line, BC, one-half mile long, is measured, and the angles ABC 
and ACB are observed to be 75 18' and 78 21' respectively. Find 
the distance AB. 

(4.) Two houses, A and B, are 1728 ft. apart. Find the distance of 
a third house, C, from A if BAC=tf 51 'and ABC= 57 23'. 

(5.) In order to determine the distance of a bluff, A, from 'a house, 
B, in a plane, a line, BC, was measured and found to be 1281 yards, 
also the angles ABC and BCA 65 31' and 70 2' respectively. Find 
the distance AB. 

(6.) Two towns, 3 miles apart, are on opposite sides of a balloon. 
The angles of elevation of the balloon are found to be 13 19' and 
20 3'. Find the distance of the balloon from the nearer town. 

(7.) It is required to find the distance between two posts, A and B, 
which are separated by a swamp. A point C is 1272.5 ft. from A, and 
2012.4 ft- from B. The angle ACB is 41 9' 1 i". 

(8.) Two stakes, A and B, are on opposite sides of a stream ; a 
third point, C, is so situated that the distances AC and BC can be 
found, and are 431.27 yards and 601.72 yards respectively. The angle 
ACB is 39 53' 13". Find the distance between the stakes A and B. 



MISCELLANEOUS EXERCISES 89 

(9.) Two light-houses, A and B, are 11 miles apart. A ship, C, is 
observed from them to make the angles BAC =31 13' 31" and ABC 
= 21 46' 8". Find the distance of the ship from A. 

(10.) Two islands, A and B, are 6103 ft. apart. Find the distance 
from A to a ship, C, if the angle ABC is 37 25' and BAC is 40 32'. 

(u.) In ascending a cliff towards a light-house at its summit, the 
light-house subtends at one point an angle of 21 22'. At a point 
55 ft. farther up it subtends an angle of 40 27'. If the light-house 
is 58 ft. high, how far is this last point from its foot? 

(12.) The distances of two islands from a buoy are 3 and 4 miles 
respectively. The islands are 2 miles apart. Find the angle sub- 
tended by the islands at the buoy. 

^(13.) The sides of a triangle are 151.45, 191.32, and 250.91. Find 
the length of the perpendicular from the largest angle upon the 
opposite side. 

** (14.) A tree stands on a hill, and the angle between the slope of the 
hill and the tree is 110 23'. At a point 85.6 ft. down the hill the 
tree subtends an angle of 22 22'. Find the height of the tree. 
! (15.; A light-house 54 ft. high is built upon a rock. From the top 
of the light-house the angle of depression of a boat is 19 10', and 
from its base the angle of depression of the boat is 12 22'. Find the 
height of the rock on which the light-house stands. 

(16.) Three towns, A, B, and C, are connected by straight roads. 
A = 4 miles, BC= 5 miles, and AC= 7 miles. Find the angle made 
by the roads AB and BC. 

(17.) Two buoys, A and B, are one-half mile apart. Find the dis- 
tance from A to a point C on the shore if the angles ABC and BAC 
are 77 7' and 67 17' respectively. 

(i 8.) The top of a tower is 175 ft. above the level of a bay. From 
its top the angles of depression of the shores of the bay in a certain 
direction are 57 16' and 15 2'. Find the distance across the bay. 

(19.) The lengths of two sides of a triangle are \/2 and -v/3- The 
angle between them is 45. Find the remaining side. 

(20.) The sides of a parallelogram are 172.43 and 101.31, and the 
angle included by them is 61 16'. Find the two diagonals. 

(21.) A tree 41 ft. high stands at the top of a hill which slopes 



X 

X 



90 PLANE TRIGONOMETRY 

10 12' to the horizontal. At a certain point down the hill the tree 
subtends an angle of 28 29'. Find the distance from this point to 
the toot of the tree. 

(22.) A plane is inclined to the horizontal at an angle of 7 33'. At 
a certain point on the plane a flag-pole subtends an angle 20 3', and at 
a point 50 ft. nearer the pole an angle of 40 35'. Find the height of 
the pole. 

(23.) The angle of elevation of an inaccessible tower, situated in a 
plane, is 53 19'. At a point 227 ft. farther from the tower the angle 
of elevation is 22 41'. Find the height of the tower. 

(24.) A house stands on a hill which slopes 12 1 8' to the horizontal. 
75 ft. from the house down the hill the house subtends an angle of 
32 5'. Find the height of the house. 

(25.) From one bank of a river the angle of elevation of a tree on 
the opposite bank is 28 31'. From a point 139.4 ft. farther away in a 
direct line its angle of elevation is 19 10'. Find the width of the river. 

(26.) From the foot of a hill in a plane" the angle of elevation of 
the top of the hill is 21 7'. After going directly away 211 ft. farther, 
the angle of elevation is 18 37'. Find the height of the hill. 

(27.) A monument at the top of a hill is 153.2 ft. high. At a point 
321.4 ft. down the hill the monument subtends an angle of 11 13'. 
Find the distance from this point to the top of the monument. 

(28.) A building is situated on the top of a hill which is inclined 
10 12' to the horizontal. At a certain distance up the hill the angle 
of elevation of the top of the building is 20 55', and 115.3 ft- farther 
down the hill the angle of elevation is 15 10'. Find the height of 
.the building. 

(29.) A cloud, C, is observed from two points, A and J3, 2874 ft. 
apart, the line AB being directly beneath the cloud. At A, the angle 
of elevation of the cloud is 77 19', and the angle CAB is 51 18'. 
The angle ABC is found to be 60 45'. Find the height of the cloud 
above A. 

(30.) Two observers, A and B, are on a straight road, 675.4 ft. apart, 
directly beneath a balloon, C. The angles ABC and BAC are 34 42' 
and 41 15' respectively. Find the distance of the balloon from the 
first observer. 



MISCELLANEOUS EXERCISES 91 

(31.) A man on the opposite side of a river from two objects, A 
and B, wishes to obtain their distance apart. He measures the dis- 
tance CD = 357 ft., and the angles ACB=2^> 33', BCD = 38 52', ADB 
= 54 10', and ADC =34 n'. Find the distance AB. 
i (32.) A cliff is 327 ft. above the sea-level. From the top of the 
cliff the angles of depression of two ships are 15 11' and 13 13'. 
From the bottom of the cliff the angle subtended by the ships are 
122 39'. How far are the ships apart ? 

(33.) A man standing on an inclined plane 112 ft. from the bottom 
observed the angle subtended by a building at the bottom to be 33 
52'. The inclination of the plane to the horizontal is 18 51'. Find 
the height of the building. 

(34) Two boats, A and B, are 451.35 ft. apart. The angle of ele- 
vation of the top of a light-house, as observed from A, is 33 if. 
The base of the light-house, C, is level with the water; the angles 
ABC and CAB are 12 31' and 137 22' respectively. Find the height 
of the light-house. 

(35.) From a window directly opposite the bottom of a steeple the 
angle of elevation of the top of the steeple is 29 21'. From another 
window, 20 ft. vertically below the first, the angle of elevation is 39 3'. 
Find the height of the steeple. 

(36.) A dock is i mile from one end of a breakwater, and i miles 
from the other end. At the dock the breakwater subtends an angle 
of 31 n'. Find the length of the breakwater in feet. 

(37.) A straight road ascending a hill is 1022 ft. long. The hill 
rises i ft. in every 4. A tower at the top of the hill subtends an 
angle of 7 19' at the bottom. Find the height of the tower. 

(38.) A tower, 192 ft. high, rises vertically from one corner of a 
triangular yard. From its top the angles of depression of the other 
corners are 58 4' and 17 49'. The side opposite the tower subtends 
from the top of the tower an angle of 75 15'. Find the length of 
this side. 

(39.) There are two columns left standing upright in a certain ruins ; 
the one is 66 ft. above the plain, and the other 48. In a straight line., 
between them stands an ancient statue, the head of which is 100' ft. 
from the summit of the higher, and 84 ft. from the top of the lower 



92 PLANE TRIGONOMETRY 

column, the base of which measures just 74 ft. to the centre of the 
figure's base. Required the distance between the tops of the two 
columns. 

(40.) Two sides of a triangle are in the ratio of 1 1 to 9, and the 
opposite angles have the ratio of 3 to i. What are these angles ? 

(41.) The diagonals of a parallelogram are 12432 and 8413, and the 
angle between them is 78 44' ; find its area. 

(42.) One side of a triangle is 1012.6 and two angles are 52 21' and 
57 32' ; find its area. 

(43.) Two sides of a triangle are 218.12 and 123.72, and the included 
angle is 59 10' ; find its area. 

(44.) Two angles of a triangle are 35 15' and 47 18', and one side 
is 2104.7 I find its area. 

(45.) The three sides of a triangle are 1.2371, 1.4713, and 2.0721; 
find the area. 

(46.) Two sides of a triangle are 168.12 and 179.21, and the included 
angle is 41 14' ; find its area. 

(47.) The three sides of a triangle are 51 ft., 48.12 ft., and 32.2 ft. ; 
find the area. 

(48.) Two sides of a triangle are 1 1 1 . 1 8 and 121.21, and the included 
angle is 27 50' ; find its area. 

(49.) The diagonals of a parallelogram are 37 and 51, and they form 
an angle of 65 ; find its area. 

(50.) If the diagonals of a quadrilateral are 34 and 56, and if they 
intersect at an angle of 67, what is the area ? 



SPHERICAL TRIGONOMETRY 



CHAPTER VIII 

RIGHT AND QUADRANTAL TRIANGLES 
RIGHT TRIANGLES 

82. Let O be the centre of a sphere of unit radius, and 
ABC a right spherical triangle, right angled at A, formed by 
the intersection of the three planes A OC, AOB, and BOC 




with the surface of the sphere. Suppose the planes DAC" 
and BEC passed through the points A and B respectively, 
and perpendicular to the line OC. The plane angles DC" A 
and BC'E each measure the angle C of the spherical tri- 
angle, and the sides of the spherical triangle a, b, c have the 
same numerical measure as BOC, AOC, and AOB respec- 



94 SPHERICAL TRIGONOMETRY 



tively, then, AD = ta.nc, BE s\\\c, BC 1 = sma, 

cos6, OE = cosc, AC" = sin b. 
In the two similar triangles OEC' and OAC" ', 



OA i . cos b ' ' 


- LOS 6* LOS 6. 


V) 


In the triangle BC' E, 
^. n BE Qr ^. n 


sin^r v 


(2) 


^C?" 
In the triangle DAC" , 


sin a 


DA or 


tan ^ 


(3) 
(4) 


Combining formulas (2) and (3) with 
., tan $ 


(i). 

rrb^ 


tan a 



Again, if AB were made the base of the right spherical 
triangle ABC, we should have 

sinj5= ^- (5) 

4-i, > A 

(6) 

r (7) 

From the foregoing equations we may also obtain by 
combinations, 

cos/?=sin C cos^. (8) 

cosC sin B cose. (9) 

cos # = cot B cot 7. (10) 

NAPIER'S RULES OF CIRCULAR PARTS 
S3. The above ten formulas are sufficient to solve all 
cases of right spherical triangles. They may, however, be 



RIGHT AND QUADRANTAL TRIANGLES 



95 



expressed as two simple rules, called, after their inventor, 
Napier's rules. 

The two sides adjacent to the right angle, the complement 
of the hypotenuse, and the complements of the oblique an- 
gles are called the circular parts. 

The right angle is not one of the circular parts. 



comp B 



comp 



comp C 




Thus there are fire circular parts namely, />, c, comprt, comp/?, compC 
Any one of the five parts may be called the middle part, then the two parts next 
to it are called adjacent parts, and the remaining two parts are called the oppo- 
site parts. 

Thus if c is taken for the middle part, comp/? and b are adjacent parts, and 
comptf and comp C are opposite parts. 

The ten formulas may be written and grouped as follows : 



ist Group. 

sin comp C = tan comprt tan b. 
sin comp /?= tan compi? tan c. 
MII comp a = tan comp/? tan comp C. 
sin c = tan comp B tan b. 

sin 



b =tan comp C tan r. 



zd Group. 

sin comp rt=r cos l> cos c. 
sin b = cos comp a cos comp B. 

sin r=cos comprt cos comp C. 

sin comp j9=cos comp C cos . 
sin comp f=cos comp/? cose-. 



Napier's rules may be stated : 

I. The sine of the middle part is equal to the product of 
the tangents of- the adjacent parts. 

II. The sine of the middle part is equal to the product of 
the cosines of the opposite parts. 



9 6 



SPHERICAL TRIGONOMETRY 



84. In the right spherical triangles considered in this work, each 
side is taken less than a semicircumference, and each angle less than 
two right angles. 

In the solution of the triangles, it is to be observed, 

(i.) If the two sides about the right angle are both less or both 
greater than 90, the hypotenuse is less than 90; if one side is less 
and the other greater than 90, the hypotenuse is greater than 90. 

(2.) An angle and the side opposite are either both less or both 
greater than 90. 



EXAMPLE 



85. Given # = 63 56', = 40 o', to find c, B, and C. 



To find c. 

cotnp a is the middle part. 
c and b are the opposite parts, 
sin comp a=cos b cos c, 
cos 0=cos b cos c. 

cos a 

cos c = - 

cos b 

log cos 0=9.64288 
colog cos =0.11575 
log cos ^-=9.75863 
'=54 59 47" 



To find C. 

comp C is the middle part. 

comp a, and b are adjacent parts. 

sin comp C=tan comp a tan, 

cos C= cot a tan. 

log cot a= 9. 68946 
log tan =9 92381 

9-61327 
C=6 5 45' 58" 



To find B. 

l> is the middle part. 

comp a and comp B are the opposite 

parts. 

sin =cos comp a cos comp B, 
or sin =sin a sin B. 

sin b 



log sin =9.80807 
colog sin rt=o. 04659 
log sin =9. 85466 
^ = 45 41 '28" 

Check. 
Use the three parts originally required. 

comp C is the middle part. 
comp.# and c are opposite parts. 

sin comp C=cosc cos comp B, 
or cos C=cos c sin B, 

log cos -=9.75863 
log sin B=<). 85466 
log cos (7=9.61329 

C=6$ 45' 54" 



RIGHT AND QUADRANTAL TRIANGLES 97 

AMBIGUOUS CASE 

86. When a side about the right angle and the angle opposite 
this side are given, there are two solutions, as illustrated by the fol- 
lowing figure. Since the solution gives the values of each part in 
terms of the sine, the results are not only the values of a, b, B, but 
1 8o rt, 180 b, 1 8o #. 




Given c = 26 4'. 



To find a, a', b, b' and B, B', using Napier's rules. 



To find B and B '. 

sin comp C= cos comp B cose, 
cos C=sin B cos c, 

cos C 
~~ cos c 

log cos C=g. 90796 

colog cos -=0.04659 

log sin ^ = 9.95455 

B= 64 14' 30" 
r = i8o-=ii5 45' 30" 

To find b and b' . 
sin =tan c tan comp C, 
sin =tan c cot C 
log tan ^-=9.68946 
log cot 67=0.13874 
log sin =9.82820 

b 42 19' 17" 
=137 40' 43" 



To find a and a'. 
sin -=cos comp a cos Comp C, 
sin c=sin a sin (7, 
sin c 



or 
or 

log sin - = 9.64288 

colog sin (7=0.23078 

log sin 0=9.87366 

a= 48 22' 55"- 
fl' = i8o-a=i3i37' 5" + 
'Discrepancy due to omitted decimals.) 

Check. 

sin =cos comp a cos comp /?, 
or sin =sin a sin ^. 

log sin a or a'=g. 87366 

log sin # or .#'=9.95455 

log sin =9.82821 

= 42 19' 21" 
39" 



98 SPHERICAL TRIGONOMETRY 

QUADRANTAL TRIANGLES 

87. Def. A quadrantal triangle is a spherical triangle 
one side of which is a quadrant. 

A quadrantal triangle may be solved by Napier's rules for 
right spherical triangles as follows : 

By making use of the polar triangle where 



C=i8o ^ <r=i8o C' 

we see that the polar triangle of the quadrantal triangle is 
a right triangle which can be solved by Napier's rules. 
Whence we may at once derive the required parts of the 
quadrantal triangle. 

EXAMPLE 

Given A = 1 36 4'. B = 1 40 o'. a 90 o'. 
The corresponding parts of the polar triangle are 

a' =^3 56', V = 40 o', A' = 90. 
By Napier's rules we find 

B' = 45 41 ' 28", C' = 65 45' 58", c - 54 59' 47" ; 

whence, by applying to these parts the rule of polar triangles, we 
obtain 

b 134 18' 32", c= 114 14' 2", C=i25o' 13". 

EXERCISES 

88. (i.) In the right-angled spherical triangle ABC, the side a= 
63 56', and the side = 40. Required the other side, c, and the 
angles B and C. 

(2.) In a right-angled triangle ABC, the hypotenuse a = 91 42', and 
the angle ^ = 95 6'. Required the remaining parts. 

(3.) In the right-angled triangle ABC, the side b = 2.6 4', and the 
angle ^ = 36. Required the remaining parts. 

- (4.) In the right-angled spherical triangle ABC, the side c = 54 30', 
and the angle = 44 50'. Required the remaining parts. 

Why is not the result ambiguous in this case? 



RIGHT AND QUADRANTAL TRIANGLES 99 

(5.) In the right-angled spherical triangle ABC, the side = 55 28', 
and the side ^ = 63 15'. Required the remaining parts. 

(6.) In the right-angled spherical triangle ABC, the angle B = 69 
20', and the angle C = 58 16'. Required the remaining parts. 

(7.) In the spherical triangle ABC, the side # = 90, the angle C= 
42 10', and the angle ^ = 115 20'. Required the remaining parts. 
Hint. The angle A of the polar triangle is a right angle. 

(8.) In the spherical triangle ABC, the side = 90, the angle C= 
69 13' 46", and the angle A = 72 12' 4". Required the remaining 
parts. 

(9.) In the right-angled spherical triangle ABC, the angle C=23 
27' 42", and the side b 10 39' 40". Required the angle B and the 
sides a and c. 

(10.) In the right spherical triangle ABC, the angle = 47 54' 20", 
and the angle C=6i 50' 29". Required the sides. 



CHAPTER IX 
OBLIQUE-ANGLED TRIANGLES 

89. Let O be the centre of a sphere of unit radius, and 
ABC an oblique-angled spherical triangle formed by the 
three planes AOB, BOC, and AOC. Suppose the plane 




AED passed through the point A perpendicular to AO, in- 
tersecting the planes A OB, BOC, and AOC, in AE, ED, 
and AD respectively. Then AD=tan b, AE-tan c, OD 



In the triangle EOD, 

ED' 2 = sec a + secV 2 sec b sec c cos a. 
In the triangle AED, 

ED'* = tan 2 ^ -f tanV 2 tan b tan c cos A. 
Subtracting these two equations and remembering that 

sec 2 ^ tan 2 =i, we have 
= 2 2 sec seer cos#-|-2 tan tanr cos A. 
Reducing, we have 

coc+in& 



(i) 



OBLIQUE-ANGLED TRIANGLES 101 

If we make b and c in turn the base of the triangle, we obtain in a 
similar way, 

cos = cos cos#-|-sin<: s i n a cos B, 
and cos<r = cosd! cos/^ + sin<7 s\nb cosC. 



Remark. In this group of formulas the second may be obtained 
from the first, and the third from the second, by advancing one letter 
in the cycle as shown in the figure ; thus, writing b for 
a, c for b, a for c, B for A, C for B, and A for C. The 
same principle will apply in all the formulas of Oblique- 
Angled Spherical Triangles, and only the first one of 
each group will be given in the text. 

90. By making use of the polar triangle where 




we may obtain a second group of formulas. 

Substituting these values of a, b, c, and A in (i), and remembering 
that cos ( 1 80 A) cos A and sin (i So A) r= sin A, we have 

cos^4' = cos^'cosC'-fsin^' sin C' cosa'. 

Since this is true for any triangle, we may omit the accents and 
write, 

cos A = - cos B cos C + sin B sin C cos a. (2) 



FORMULAS FOR LOGARITHMIC COMPUTATION 
. Formula (i), cos a = cos b cose + sin & sin c cos A, 
cos a cos^ cos*: 



gives cos A 



sne 



By 36, cos^ = i 2 sin 2 -J^ 

cos a cos^ 



Whence i 

or sin 2 A 



sin b sin c 

' sin ccosa 



2 sin b sine 



102 SPHERICAL TRIGONOMETRY 

c] cos a 



2 sin b sin c 
sin sin 



sin b sin c (38) 

Putting 

c a + bc , a b-\-c 

-=s, then - =s c, and - =^ b, 



we 



/sm(s 

have sin-J^f=\/- < . 

V si 



. . . 
sin b sin c 

Since, also, cos A i+ 2 cos*$A, 
we have, similarly, 

/sin s s\n(s a] 
= V - i A 
v - 



sin b sine 

/Ii 
Hence 



By a like process, formula (2) reduces to 



% , cosScos(S-A) , 

tania^W- ^- ( TI ) 



. If, in formula I, we advance one letter, we have 



/sin (s c) sin (sa) 
=\/- f L\ 

v situ sin (s -b) 

And dividing tan^A by tan^^, and reducing, we obtain 

tan^A sin(s b) 

tan \B~ sin (j tf) ' 
By composition and division, 

tan %A-\- tan \B sin (j ^) + sin(^ a) 



tan ^ A tan ^^ ~~ sin (j b) sin (j )' 
30 38, this becomes I? 1 



in ^(A B)~~ tan ^ (a b)' 






OBLIQUE-ANGLED TRIANGLES 103 

Multiplying tanf A by tan#, and reducing, we obtain 



tan \A tan -J B sin (s c) 
i sin s 

By division and composition, and by 30, 38, this be- 

comes 

tanjc 



co-^(A B) tan -J- (a + b) ' 
Proceeding in a similar way with formula II, we obtain 

s!n-J-(a + 6)_ cot-J-C ,_,, 

*in%(a b)~ tan%(A &)' 

cow 4- ( -f 6) cot -J C 

And ~- -.- = ^-A ~. (VI) 



99. In the spherical triangle yi ^{7, suppose C7? drawn per- 
pendicularly to AB t then, by the formulas for right spher- 
ical triangles, 




In triangle A CD, sin / = sin b sin A. 

In triangle BCD, sin p~ sin # sin B. 

Whence sin a sin /?=sin b sin ^4, 

sin a in 6 



Remark. If (A + B)>i8o, then 
1 80, then (a 



, and if (A-hB)< 



104 



SPHERICAL TRIGONOMETRY 



94. All cases of oblique-angled triangles may be solved 
by applying one or more of the formulas I, II, III, IV, V, 
VI, VII, as shown in the following cases. 

CASES 

(i.) Given three sides, to find the angles. 

Apply formula I. Check : apply V or VI. 

(2.) Given three angles, to find the sides. 

Apply formula II. Check : apply III or I V. 

(3.) Given two sides and the included angle. 

Apply V and Vl\ and VII. Check : apply III or I V. 

(4.) Given two angles and included side. 

Apply III and I V, and VI L Check : apply V or VI. 

(5.) Given two angles and an opposite side. 
Apply VII, V, and III. Check : apply IV. 

(6.) Given two sides and an opposite angle. 
Apply VII, V, and IV. Check : apply III. 



EXAMPLE CASE (l) 
95. Given a = 81 10' b = 60 20' 

To find A, B, and C. 

a 81 10' 
b 60 20' 

C =:II2 25' 



<r=ii225' 



j = i26 57' 30" 
s-a=45 47' 30" 
^-^=66 37' 30" 
j-^^i432' 30" 
log sin .f =9. 90259 
log sin(.r ^0=9.85540 
log sin (s )=9. 96281 
log sin (j ^=9.39982 



To find 



sin s sin(s <i) 
log sin (s ^=9.9628 1 
log sin(j-<r)=9.39982 

colog sin s=o. 14460 

colog sin (s a)=o.ogi4i 

2) I Q .60464 

log tan .4:= 9.80232 

^=32 23' 19" 

^ = 64 46' 38" 

ur 



OBLIQUE-ANGLED TRIANGLES 



105 



To find B. 



tan A B= * /? 
V 



b] 



log sin(j a)= 

log sin (s ^=9.39982 

colog sin .r= 0.0974 1 

colog sin(j b) =0.03719 

2)19.38982 
logtan#= 9.69491 



To find C. 



/ 
V 



_ sin (j 0) 



sin s sin(j c) 
log sin (sa)=g. 85540 
log sin (s ^=9.9628 1 
colog sin j-=o. 09741 
colog sin (s r) =0.600 1 8 

2)20.51580 

log tan (7= 10.25790 
K= 6l 5' 32" 

(7=122 II' 4" 



, 
Formula V, cot A C= 



sin a 
A =64 46' 38" 
.= 52 42' 12" 



fl=8i 10' 
b =60 20' 

=141 30' ; (a+)=70 45' 
a b 20 50'; \(a ^)=io 25' 



A-B=i2 4' 26" 
4B) 6 2' 13" 

log tan ^(A ^=9.02430 
log sin 4r(rt-f-/>)=9 97501 
colog sin $(a ^=0.74279 
cot (7=9.74210 
C- 61 5' 32" 

(7=122 II' 4" 



EXAMPLE CASE (3) 

96. Given a = 78 15' = 56 20' C=I2O 

To find ^4, B, and r. 

log sin (a+ ^=9.96498 
log cos (a + ^=9.58663 
log si n^( ^=9.27897 
log cos (a ^=9.99201 
log cot (7= 9. 76144 



+ )=6 7 17' 30 
-3)= 10 57' 30 



To fin 

Formula V/may be written 

cosA(r7 ^) cot 



COS (7< 
log COS^(fl! ^)= 9.99201 

log cot C= 9. 76 1 44 
colog cos ( + b) 0.41337 
log tan $(A + 5) = 10. 16682 



(A-B)= 6 47' 4" 
A =62 31' 40" 
^=48 57' 32"- 



To find$(A-B\ 
Formula V^ may be written 
sjnj>-^ 



log sin 1(^ ^=9.27897 

log cot (7=9.76144 

colog sin(rt +^)=o. 03502 



io6 



SPHERICAL TRIGONOMETRY 



To find c. 
From Formula VII, sin c= 



sin b sin C 
sin B 



log sin =9.92027 

log sin =9.93753' 

colog sin 2? =0.12249 

log sin ^=9. 98029 
<r=io78' 



Check. 
Formula III may be written 

_ sin % (A + B} tan (a - b} 



log sin %(A + B) = 9.91725 

log tan \ (a b) = 9.28696 

colog sin %(AB) 0.92762 

log tan r== 10. 1 3 1 83 

\c= 53 33' 56"- 
^=107 7' 51" 
(Discrepancy due to omitted decimals ) 



AMBIGUOUS CASES 

97. (i.) Two sides and an angle opposite one of them are the 
given parts. 

If the side opposite the given angle differs from po more than the 
other given side, the given angle and the side opposite being either both 
less or both greater than 90, there are two solutions. 




(2.) Two angles anc} a side opposite one of them are the given parts. 

If the angle opposite the given side differs from 90 more than the 
other given angle, the given side and the angle opposite being either 
both less or both greater than 90, there are two solutions. 

Remark. There is no solution if, in either of the formulas. 



sin B= 



sin A sin b 



sin b sin A 



sin a sin B 

the numerator of the fraction is greater than the denominator. 






OBLIQUE-ANGLED TRIANGLES 



107 



cos /-. 



Formula V may be written 
cot A C- s 



EXAMPLE CASE (6) 

98. Given #=40 16' =4744' ^=52 30' 

To find B, B', C, C, and c t c'. 

To find B and B'. 
Formula VII may be written 

sin^4 sin/' 
sm B= : . 

sin a 

log sin ^=9. 89947 

log sin ^=9. 86924 

colog sin fl=o. 1 8953 

log sin jB=g. 95824 

B= 65 16' 30" 
B' = 114 43' 30" 

To find c. 
Formula IV may be written 

tanr= 

log CO! 

log tan ( + />) =9. 98484 

colog cos $(AB)= 0.002 70 

log tan i'=g. 70080 

<r=26 39' 42" 
'=53 19' 24" 
To find c. 

log tan |(^ + /')=9. 98484 

colog cos (A B') =0.06745 

log tan ' =9.09860 

*<'= 7 9' 9" 



sin(rt b) 
^)= 9.84177 

log tan(.4 -5)= 9.04901 n 
colog sin (rt <)= 1.1863311 
log cot $C= 10.0771 1 

^C=3 9 o 5 6'2 4 " 
C=79 52' 48" 

To find C, 

logsin(rt + )= 9.84177 
log tai4 (.4 -.#')= 9.7815311 
colog sin ^ (a ti\ 1. 18633 n 
log cot \ C = 10. 80963 

lc= 8 48' 41" 



Check. 
Formula III may be written 

sin B sin c 
sin b= - . _ 
smC 

log sin =9. 95824 
log sine =9. 904 1 8 
colog sin C*=o. 00682 
log sin ^=9.86924 
^=47 44' 



<r' = i4 18' 18" 

EXERCISES 

99. (i.) In the spherical triangle ABC, the side ^ = 124 53', the 
side b = 31 19', and the angle A = 16 26'. Find the other parts. 

(2.) In the oblique-angled spherical triangle ABC, angle A = 128 
45', angie C= 30 35', and the angle ,5 = 68 50'. Find the other parts. 



* The letter " n" indicates that these quantities are negative. 



lo8 SPHERICAL TRIGONOMETRY 

(3.) In the spherical triangle ABC, the side ^ = 78 15', =56 20', 
and A = 120. Required the other parts. 

(4.) In the spherical triangle ABC, the angle ,4 = 125 20', the an- 
gle (7 = 48 30', and the side ^ = 83 13'. Required the remaining 
parts. 

(5.) In the spherical triangle ABC, the side ^ = 40 35', = 39 10', 
and a = 71 15'. Required the angles. 

(6.) In the spherical triangle ABC, the angle A = 109 55', B \ 16 
38', and C= 120 43'. Required the sides. 

(7.) In the spherical triangle ABC, the angle ^ = 130 5' 22", the 
angle C= 36 45' 28", and the side = 44 13' 45". Required the re- 
maining parts. 

(8.) In the spherical triangle ABC, the angle ^ = 33 15' 7", B = 
3 l0 34' 38", and C= 161 25' 17". Required the sides. 

(9.) In the spherical triangle ABC, the side <r=ii2 22' 58", = 
52 39' 4", and a = 89 16' 53". Required the angles. 

(10.) In the spherical triangle ABC, the side ^ = 76 35' 36", b = 
50 10' 30", and the angle ^ = 34 15' 3". Required the remaining 
parts. 

AREA OF THE SPHERICAL TRIANGLE 

100. It is proved in geometry that the area of a spherical 
triangle is equal to its spherical excess, that is, 
area = (A + B + C 2 rt. angles) X area of the tri-rectangular triangle, 
where A, B, and C are the angles of the spherical triangle. 
Hence 

area _A+-\-C 180 

surface of sphere "~ 720 

The surface of the sphere is 477^, therefore 

A + B+ C-180\ 



The following formula, called Lhuilier's theorem, simpli- 
fies the derivation of (A +jB+Ci8o) where the three 



OBLIQUE-ANGLED TRIANGLES 109 

sides of the spherical triangle are given ; in it a, b, and c 
denote the sides of the triangle, and 2s= 



tan /- _ y'tan i s tan i (s-a) tan i(s-6) tan i (s-cj. 



EXERCISES 

(i.) The angles of a spherical triangle are, ^=63, =84 21', 
C=79; the radius of the sphere is 10 in. What is the area of the 
triangle ? 

(2.) The sides of a spherical triangle are, a = 6.47 in., = 8.39 in., 
^ = 9.43 in.; the radius of the sphere is 25 in. What is the area of 
the triangle ? 

(3.) In a spherical triangle, ^ = 75 16', ^ = 39 20', c = 26 in.; the 
radius of the sphere is 14 in. Find the area of the triangle. 

(4.) In a spherical triangle, a = 441 miles, ^ = 287 miles, C = 38 21'; 
the radius of the sphere is 3960 miles. Find the area of the triangle. 



CHAPTER X 

APPLICATIONS TO THE CELESTIAL AND TERRES- 
TRIAL SPHERES 

ASTRONOMICAL PROBLEMS 

101. An observer at any place on the earth's surface 
finds himself seemingly at the centre of a sphere, one-half 
of which is the sky above him. This sphere is called the 
celestial sphere, and upon its surface appear all the heavenly 
bodies. The entire sphere seems to turn completely around 
once in 23 hours and 56 minutes, as on an axis. The im- 
aginary axis is the axis of the earth indefinitely produced. 
The points in which it pierces the celestial sphere appear 
stationary, and are called the north and south poles of the 
heavens. The North Star (Polaris) marks very nearly (with- 
in i 16') the position of the north pole. As the observer 
travels towards the north he finds that the north pole of the 
heavens appears higher and higher up in the sky, and that 
its height above the horizon, measured in degrees, corre- 
sponds to the latitude of the place of observation. 

The fixed stars and nebulae preserve the same relative 
positions to each other. The sun, moon, planets, and com- 
ets change their positions with respect to the fixed stars 
continually, the sun appearing to move eastward among 
the stars about a degree a day, and the moon about thir- 
teen times as far. 



AP PLICA TIONS 1 1 1 

The zenith is the point on the celestial sphere directly 
overhead. 

The horizon is the great circle everywhere 90 from the 
zenith. 

The celestial equator is the great circle in which the 
plane of the earth's equator if extended would cut the ce- 
lestial sphere. 

The ecliptic is the path on the celestial sphere described 
by the sun in its apparent eastward motion among the stars. 
The ecliptic is a great circle inclined to the plane of the 
equator at an angle of approximately 23^. 

The poles of the equator are the points where the axis 
of the earth if produced would pierce the celestial sphere, 
and are each 90 from the equator. 

The poles of the ecliptic are each 90 from the ecliptic. 

The equinoxes are the points where the celestial equa- 
tor and ecliptic intersect ; that which the sun crosses when 
coming north being called the vernal equinox, and that 
which it crosses when going south the autumnal equinox. 

The declination of a heavenly body is its distance, meas- 
ured in degrees, north or south of the celestial equator. 

The right ascension of a heavenly body is the distance, 
measured in degrees eastward on the celestial equator, from 
the vernal equinox to the great circle passing through the 
poles of the equator and this body. 

The celestial latitude of a heavenly body is the dis- 
tance from the ecliptic measured in degrees on the great 
circle passing through the pole of the ecliptic and the 
body. 

The celestial longitude of a heavenly body is the dis- 
tance, measured in degrees eastward on the ecliptic, from 



112 SPHERICAL TRIGONOMETRY 

the vernal equinox to the great circle passing through the 
pole of the ecliptic and the body. 

EXERCISES 

(i.) The right ascension of a given star is 25 35', and its decima- 
tion is -f-( n orth) 63 26'. Assuming the angle between the celestial 
equator and the ecliptic to be 23 27', find the celestial latitude and 
celestial longitude. 




In this figure AB is the celestial equator, AC the ecliptic, P the pole of 
the equator, P' the pole of the ecliptic. ' S is the position of the star, and 
the lines SB and SC are drawn through P and P' perpendicular to AB and 
AC. AB is the right ascension and BS the declination of the star, while 
AC is the longitude and SC the latitude of the star. 

In the spherical triangle P'PS, it will be seen that P'S is the comple- 
ment of the celestial latitude, PS the complement of the declination, and 
P'PS is 90 plus the right ascension. It is to be noted that A is the ver- 
nal equinox. 

(2.) The declination of the sun on December 2ist is (south) 
23 27'. At what time will the sun rise as seen from a place whose 
latitude is 41 18' north ? 

The arc ZS which is the distance from the zenith to the centre of the sun 
when the sun's upper rim is on the horizon is 90 50'. The 50' is made up 
of the sun's semi-diameter of 16', plus the correction for refraction of 34'. 



AP PLICA TIONS 1 1 3 

(3.) The declination of the sun on December 2ist is (south) 
23 27'. At what time would the sun set as seen from a place in lati- 
tude 50 35' north ? 




SUNRISE SUNSET 

In these figures P is the pole of the equator, Z the zenith, EQ the celes- 
tial equator. ASh the declination of the sun, ZS=qcP 50', PSgoP + dec- 
lination, PZ= 90 -latitude. The problem is to find the angle SPZ. An 
angle of 15 at the pole corresponds to I hour of time. 

GEOGRAPHICAL PROBLEMS 

102. The meridian of a place is the great circle passing 
through the place and the poles of the earth. 

The latitude of a place is the arc of the meridian of the 
place extending from the equator to the place. 

Latitude is measured north and south of the equator from o to 90. 

The longitude of a place is the arc of the equator extend- 
ing from the zero meridian to the meridian of the place. 
The meridian of the Greenwich Observatory is usually taken 
as the zero meridian. 

Longitude is measured east or west from o to 180. 
The longitude of a place is also the angle between the zero meridian and 
the meridian of the place. 



ii4 SPHERICAL TRIGONOMETRY 

In the following problems one minute is taken equal to one geo- 
graphical mile. 

(i.) Required the distance in geographical miles between two 
places, D and E, on the earth's surface. The longitude of D is 60 
15' E., and the latitude 20 10' N. The longitude of E is 115 20' E., 
and the latitude 37 20' N. 







In this figure A C represents the equator of the earth, P the north pole, 
and A the intersection of the meridian of Greenwich with the equator. PB 
and PC represent meridians drawn through D and E respectively. Then 
AB is the longitude and BD the latitude of D ; AC the longitude and CE 
the latitude of E. 

(2.) Required the distance from New York, latitude 40 43' N., 
longitude 74 o' W., to San Francisco, latitude 37 48' N., longitude 
122 28' W., on the shortest route. 

(3.) Required the distance from Sandy Hook, latitude 40 28' N., 
longitude 74 i' W., to Madeira, in latitude 32 28' N., longitude 16 55, 
W., on the shortest route. 

(4.) Required the distance from San Francisco, latitude 37 48' 
N., longitude 122 28' W., to Batavia in Java, latitude 6 9' S., longi- 
tude 1 06 53' E., on the shortest route. 

(5.) Required the distance from San Francisco, latitude 37 48' 
N., longitude 122 28' W., to Valparaiso, latitude 33 2' S., longitude 
71 41' W., on the shortest route. 



CHAPTER XI 

GRAPHICAL SOLUTION OF A SPHERICAL TRIANGLE 

J.03. The given parts of a spherical triangle may be laid 
off, and then the required parts may be measured, by making 
use of a globe fitted to a hemispherical cup. 

The sides of the spherical triangle are arcs of great circles, 
and may be drawn on the globe with a pencil, using the 
rim of the cup, which is a great circle, as a ruler. The rim 
of the cup is graduated from o to 1 80 in both directions. 

The angle of a spherical triangle may be measured on a 
great circle drawn on the sphere at a distance of 90 from 
the vertex of the angle.* 

CASE I. Given the sides a, b, and c of a spherical triangle, 
to determine the angles A , B, and C. 

Place the globe in the cup, and draw upon it a line equal 
to the number of degrees in the side c, using the rim of the 
cup as a ruler. Mark the extremities of this line A and B. 
With A and B as centres, and b and a respectively as radii, 
draw with the dividers two arcs intersecting at C (Fig. i). 
Then, placing the globe in the cup so that the points A and 
C shall rest on the rim, draw the line AC=b, and in the 
same way draw BC=a. 

To measure the angle A place the arc AB in coincidence 

* Slated globes, three inches in diameter, made of papier-mache, and held 
in metal hemispherical cups, are manufactured for the use of students of 
spherical trigonometry at a small cost. 



Ii6 SPHERICAL TRIGONOMETRY 

with the rim of the cup, and make AE equal to 90. Also 
make AF in AC produced equal to 90. Then place the 
globe in the cup so that E and F shall be in the rim, and 
note the measure of the arc EF. This is the measure of the 
angle A. In the same way the angles B and C can be de- 
termined. 




CASE II. Given the angles A, B, and C, to find the sides 
a, b, and c. 

Subtract A, B, and C each from 180, to obtain the sides 
a 1 ', b' , and c' of the polar triangle. Construct this polar tri- 
angle according to the method employed in Case I. Mark 
its vertices A', B' , and C '. With each of these vertices as 
a centre, and a radius equal to 90, describe arcs with the di- 
viders. The points of intersection of these arcs will be the 
vertices A, , and C of the given triangle. The sides of 
this triangle a, b, and c can then be measured on the rim 
of the cup. 



GRAPHICAL SOLUTION 



117 



CASE III. Given two sides, b and c, and the included angle 
A, to find B, C, and a. 

Lay off (Fig. 3) the line AB equal to c, and mark the 
point D in AB produced, so that AD equals 90. With the 
dividers mark another point, F 3 at a distance of 90 from A. 
Turn the globe in the cup till D and Fare both in the rim, 
and make DE equal to the number of degrees in the angle A. 
With A and E in the rim of the cup, draw the line AC equal 
to' the number of degrees in the side b. Join C and B. The 
required parts of the triangle can then be measured. 




FIG. 3 



FIG. 4 



CASE IV. Given the angles A and B and the included side 
c, to find a, b, and C. 

Lay off the line AB equal to c. Then construct the given 
angles at A and B, as in Case III., and extend their sides to 
intersect at C. 

CASE V. Given the sides b, a, and the angle A opposite one 
of these sides, to find c, B, and C. (Ambiguous case.) 



Ii8 SPHERICAL TRIGONOMETRY 

Lay off (Fig. 4) AC equal to b, and construct the angle A 
as in Case III. Take c in the dividers as a radius, and with 
C as a centre describe arcs cutting the other side of the tri- 
angle in B and B' , and measure the remaining parts of the 
two triangles. 

If the arc described with C as a centre does not cut the other side of the 
triangle, there is no solution. If tangent, there is one solution. 

CASE VI. Given the angles A, B, and the side a opposite 
one of the angles. 

Construct the polar triangle of the given triangle by 
Case V. ; then construct the original triangle as in Case II., 
and measure the parts required. 

The constructions given above include all cases of right and quadrantal 
triangles. 



CHAPTER XII 
RECAPITULATION OF FORMULAS 

ELEMENTARY RELATIONS ( IO) 

sin x * COSJT 

tan x = - , cot x = - , 

COSJT sinx 

i i 

sec x = -- , esc x = 

cos.r 

tan x cot x = i , 
sin 3 x -\- cos 2 x=. i, 



i + cot 2 x = csc a x. 
RIGHT TRIANGLES ( 14 AND 2 7) 



cos A = - , cos B = - , 

|, tan # = -, 



cot A = - , cot B = -, , 

a b 



where c=. hypotenuse, a and b sides about the right angle; A and B 
the acute angles opposite a and . 

FUNCTIONS OF TWO ANGLES ( 30-34) 

sin (x-\-y)=.s\nx cos^-j-cos^r sinj, 
sin (x y) = sinx cos y cos x siny, 
cos (x -\-y) = cos x cosy sin x siny, 
cos (.r j) = cos.r cosj-j-sin^r si 



120 RECAPITULATION OF FORMULAS 



tan (.r-h)/)=- 

i tan^r 

tan x tan y 

tan (.r y) = 

i-f tan^r tanj/ 

cot^r cot/ i 



cot^r cot y+ i 
cot (.r /)= . 

cot/ cot x 

FUNCTIONS OF TWICE AN ANGLE ( 36) 
sin 2x = 2 sin^r cos,r, 



= 2 cos 2 .r i, 
2 tan .r 



tan 2x = 
cot 2;r = 



i tan x 
cot'.r i 



2 cot x 
FUNCTIONS OF HALF AN ANGLE ( 37) 



cos 




i cos x 



SUMS AND DIFFERENCES OF FUNCTIONS ( 38) 

sin // -f~ s i n ^ = 2 sin ^ ( -+- v) cos ^ ( T/), 
sin u sin7/ = 2 cos^( + z/) sin \(u v\ 

COS U -f- COS W = 2 COS ( + 7/) COS ( 2/), 

cos u cos ?/ = 2 sin $ (w + v) sin (w v). 
sin u -f sin y _ tan \ (u -f ^) 
sin u sin v ~~ tan \ (u v] ' 



RECAPITULATION OF FORMULAS 121 

OBLIQUE TRIANGLES ( 42-45) 

a_sin A a__s'mA b sin B 

6~sinL" ~s'mC' <r~sinC' 

a ^tan|(^~ ) 



tanC' = 

'.'*) 

a+fi+c ' 
where s= 



where AT= 



AREA OF A TRIANGLE ( 46) 

=|rtr sin ^. 5"=^ sin C 5=^ sin ^4. 



LOGARITHMIC, COSINE, SINE, AND EXPONENTIAL SERIES 

(58) 



=*- + ~ +> etc> 



122 RECAPITULATION OF FORMULAS 



,. 
r 2 r 3 x* 

^=i+.r+- + _+-+,etc. 

DE MOIVRE'S THEOREM ( 71) 



- ~ * - cos w ~ 3 ^ sin 3 ^+, etc. 

( i ) 
cos .r rr >z cos .r -- - ? - CO8* jr sin 2 ^-f, etc. 



HYPERBOLIC FUNCTIONS ( 75) 

e x -e~ x 




. 
cos /.r = =cosh jr. 

2 



SPHERICAL TRIANGLES 
RIGHT AND QUADRANTAL TRIANGLES ( 83, 87) 

Use Napier's rules. 

OBLIQUE TRIANGLES ( 89-93) 

cos a =. cos b cos c -j- sin b sin c cos ^4. 
cos A = cos # cos C-\- sin ^ sin C cos #. 



sin s sin (.$ a) 



RECAPITULATION OF FORMULAS 123 

v; 



tan | a -cos cos - 



cos (S) cos(S-C) 
tan c 



sn ^ 



cos (y ) tan 
sin $(a-\-&) cot ^ 



sini( b) tani(A 
cos$(a-\-t>)_ cotj C 



cos (# ^)~"tan (A-\-)' 
s\na __ sin If 
sin A ! ~sin B' 

AREA OF SPHERICAL TRIANGLES ( lOl) 



tan (*' 

\ 4 




APPENDIX 

RELATIONS OF THE PLANE, SPHERICAL, AND PSEUDO- 
SPHERICAL TRIGONOMETRIES 

We have up to the present considered the trigonometries 
which deal with figures on a plane or spherical surface. A 
characteristic feature of these two surfaces is that the curv- 
ature of the plane is zero, while that of the sphere is a posi- 
tive constant p. If the radius of the sphere is increased in- 
definitely, its surface approaches the plane as a limit while 
its curvature p approaches o. 

-In works on absolute geometry it is shown that there ex- 
ists a surface which has a constant negative curvature : it is 
called a pseudo-sphere, and the trigonometry upon it pseudo- 
spherical trigonometry. 

We observe that as p passes continuously from positive 
to negative values, we pass from the sphere through the 
plane to the pseudo-sphere. Thus the formulas of plane 
trigonometry are the limiting cases of those of either of the 
two other trigonometries. 

In the treatment of spherical trigonometry the radius of 
the sphere has been taken as unity. If, however, the radius 
of the sphere is r, and a, b, and c denote the lengths of the 
sides of the spherical triangle, the formulas are changed, in 

that a is replaced by -, b by -, and c by - ; thus, 



126 



APPENDIX 



becomes 



. sin^r 
sin C= 

sin a 



. c 

sin- 

r 

. a 

sm- 

r 



The formulas for pseudo-spherical trigonometry are the 
same as the formulas of spherical trigonometry, except that 

the hyperbolic functions of -, -, and - are substituted for 

the trigonometric. 

Thus, corresponding to the above formula of spherical 
trigonometry, is the formula 



sin C = 



of pseudo-spherical trigonometry. 





The pseudo-sphere is generated by revolving the curve whose equation is 
r-\- vr't x 1 



y=r log 

*A> 

about its y axis. The radius of the base of the pseudo-sphere is r. 



APPENDIX 127 

Hence the formulas of plane trigonometry can be derived 
from the formulas of either spherical or pseudo- spherical 
trigonometry by expressing the functions in series and al- 
lowing r to increase without limit. 

Example. Show that if r is increased indefinitely the following 
corresponding formulas for the spherical and pseudo-spherical right 

triangle 

a be 

cos = cos - cos - (i) 

, a i b 1 c 

cosh - = cosh- cosh-, (2) 

r r r 

reduce to the corresponding formula for a plane right triangle; that 
is, to 

a>=F+c\ (3) 

Substituting the series cos -, etc., in equation (i), we obtain 



I rt ! , 1 a 4 , I If I S i A 4 

'-i! 7< +4-, S + ' ' = '- T\ ?" 7 + r, ? + ' ' ' ^ 

Substituting in equation (2) the series for cosh - , etc. , which we obtain from 



cosh x = - , we have 



Cancelling i in equations (4) and (5), multiplying by r 2 , and, finally, allowing 
r to increase without limit, we get from either equation 



EXERCISES 

Derive each of the following formulas of plane trigonometry from 
the corresponding formula of spherical trigonometry, and also from 
the corresponding formula of pseudo-spherical trigonometry. 



123 APPENDIX 

Right triangles ; A = right angle. 
(i.) Plane, sin C=^ 

sin c 

Spherical, sin C = - 

sin a 

Pseudo-spherical, sin C= . h ' 

Oblique Triangles. 
(2.) Plane, # 2 = 2 + <r 2 2 be cos A 

Spherical, cos a = cos cos c-\- sin sin <r cos A 

Pseudo-spherical, cosh # = cosh b cosh <: -J- sinh b sinh ^ cos A. 



(3.) Plane, S=Vs (s-a) (j- J) (j-f). 

Spherical, 
(^ + 3+ C- .800) = i ^ tan (J -.^ ^ ^a-g ^ ^^-^ 

4 * r r y y 

Pseudo-spherical, 



(s-a) , (s-6) 

i_2 tan h ^ 1 -i tanh | L - 



4 (page 3). 

(i.) 192 51' 25f . 

Quadrant III. 

(2.) 2 5 . 

(3-) 287, 647^. 
(4.) Quadrant III. 

9 (page 9). 

tan 1000 is negative, 
cos 8 10 is o. 
sin 760 is positive, 
cot 70 is negative, 
cos 550 is negative, 
tan 560 is negative, 
sec 300 is positive, 
cot 1560 is negative, 
sin 130 is positive, 
cos 260 is negative, 
tan 310 is negative. 

13 (page n). 
(3.) cos 30 = ^ v/3- 
tan -3o = -iy'3. 

COt 30 = v/3, 

sec-3op=yg; 

CSC 30= 2. 

(4.) cos.r= - v/2, 
tan x = i v/2, 

COt X = 2 V/2, 

sec.r = I v/2, 
esc .r = 3. 
9 



ANSWERS TO EXERCISES 

(5-) 



cot y = , sec y = J, 



(6.) sin 60 = v/3. 

tan 60 = v/3, 

cot 60 = i ^3. 

sec 60 = 2, 

esc 60 = f v/3. 
(7-) cos o = i , tan o = o. 
(8.) sin 2 = |, cos ,3- = !, 



esc ^ = f . 

(9.) sin 45 = cos 45 = | -/2, 
tan 45= i, 

sec 45 = esc 45 = v/2. 
(10.) sin^ = v/5, cosj/ = f, 
cot_>/ = f v/5, sec j = |, 



(11.) sin3o = i co 
tan 30 = i- v/3, 
sec 30 = | v/3, 

CSC 30 = 2. 

(12.) 



= f. 



17 (page 14). 

(i .) sin 70 = cos 20, 
cos 60 = sin 30, 
cos 89 31'= sin 29', 
cot 47= tan 43, 



'3 



ANSWERS TO EXERCISES 



tan 63= cot 27, 
sin 72 39'= cos 17 21'. 
(2.) .r = 30 . 

( 3 .) * = 2230'. 

(4.).r=i8. 
(5.) jr=is. 

25 (page 21). 

(i.) 225 and 315, 

60 and 240. 
(2.) 60, 120, 420, 480. 
(3.) sin- 3 o=-i 

cos 30=^ -v/3, 

sin 765= cos 765 = -v/2, 

sin 1 20= -v/3, 

cos 1 20 = |, 

sin 210= ^, 

cos 2io= -y/3- 

(4.) The functions of 405 are 

equal to the functions of 45. 

sin 6oo= |- \/3 
cos6oo= i 
tan 600 = -Y/3. 
cot 6oo= -v/3, 
sec 6oo= 2, 
esc 6oo= f -v/3. 

The functions of 1125 are 
equal to the functions of 45 
sin 45 = ItV*. 
cos- 45= i -v/2, 
tan 45= cot 45= i , 
sec 45=-v/2, 
csc 45= -v/2. 
sin 225= cos 225= V 2 
tan 225= cot. 225= i, 
sec 225= esc 225= v/2. 
(5.) The functions of 120 are 



the same as those of 600 

given in (4). 

sin 225 = 1/2, 

cos 225 = -v/2, 

tan 225= cot 225= i, 

sec 225= "v/2, 

esc 225= -v/2, 

sin 420 = \/3, 

cos 420 =\, 

tan 420 = y^ 

cot 420= iVi 

sec 420 = 2, 

csc-42o = -t-v/3^ 

The functions of 3270 are 

equal to the functions of 30. 
(6.) sin 233 = cos 37, 

cos 233 sin 37, 

tan 233 cot 37, 

cot 233 = tan 37, 

sec 233 = esc 37, 

esc 233 = sec 37. 

sin 1 97 = sin 17, 

cos 1 97 = cos 1 7, 

tan 197 = tan 17, 

cot 197 = cot 17, 

sec 1 97 = sec 17, 

esc 1 97 = esc 17. 

sin 894 = sin 6, 

cos 894 = cos 6, 

tan 894 = tan 6, 

cot 894 = cot 6, 

sec 894 = sec 6, 

esc 894 = esc 6. 
(7.) sin 267=: sin 87, 

tan 254 = tan 74, 

cos 950 = cos 50. 
(8.) 0.28. 



ANSWERS TO EXERCISES 



(9.) 2 sin 2 x. 

(10.) i -f-sec a x. 

(ii.) sin (* 90)= cos. r, 
cos(.r oo c ) = sin-r, 
tan (JT 90) = cot x, 
cot (JT 90) = tan x, 
sec (x 90) = esc x, 
esc (.r 90) = sec x. 

$ 28 (page 24). 

(I.) a =62.324, 

^ = 32 52' 40". 
(2.) = 21.874, 

^ = 39 45' 28", 
#=50 14' 32". 
(3.) <* = 300.95. 
= 683.96, 
= 66 15'. 
(4.) = 26.608, 
* = 45-763. 
^ = 35 33'- 
area = 495. 34. 

(5.) = 3-9973- 
? = 4.1537, 

^ = 1 5 46' 33". 
area = 2. 257. 
(6.) = 0.01729. 
(7.) <* = 298.5. 
(8.) ^ = 39 42' 24". 
(9.) ^- = 2346.7. 
(10.) # = 28 57' 8". 
(if.) 444.16 ft. 

(12.) 186.32 ft. 

(i 3.) 34 33' 44". 

114.) 303.99 ft. 

(15.) 238.33 ft. 

(16.) 15 miles (about). 

(17.) 79,079 ft. 

(18.) 165.68 ft. 



(I9-) 53 33'- 
(20.) 115.136 ft. 

(21.) 76.355 ft. 
(22.) = 80 3 2", 

A = C = 49 59' 44". 
(23.) #=53i6' 3 6", 
= 12.0518 in., 

area = 72. 392 sq. in. 
(24.) = 130.52 in., 

area = 24246 sq. in. 
(25.) 23.263 ft. 
(26.) 1 7 48". 
(27.) 5.3546 in. 
(28.) 1084950 sq. ft. 
(29.) 17 ft., 885 sq. ft. 
(30.) radius = 24.882 in., 

apothem = 20. 1 3 in., 

area= 1472 sq. in. 
(31.) 12.861. 
(32.) 1782.3 sq. ft. 
(33.) 38168 ft. 

(34.) 20.21 ft. 

(35.) 2518.2 ft. 

29 (page 28). 

(I.) ^ = 22 58', 
= 7.07, 

c = 9.0046. 
(2.) = 79-435. 
A = 45 27' 14", 
C = 95 24' 46". 
(3.) ^^ = 7.6745, 
^#' = 2.6435, 
^ = 46 43' 50", 
2? ' = 133 16' 10", 
ACB^ 105 53' 10", 
ACB' = 19 20' 50". 

(4.) ^ = 37 53'- 
# = 43 52' 25", 



132 



ANSWERS TO EXERCISES 



C = 98-14' 35"- 
(5.) 902.94. 
(6.) 1253.2 ft. 
(7.) 357-224 ft. 

(8.) ^ = 44 2' 9". 
^ = 51 28' ii", 
C = 84 29' 40", 
area = 126100 sq. ft. 
(9.) 407.89 ft. 
(io.) B=i2\ 7' 16", 
C = 92 20' 38", 
D = J\ n' 6". 
(11.) #^ = 6.6885, 



v 3V/5 + : 
cos (* y) 2JL 5 



39 (P a g e 37). 
(5.) sin (45-*) = 

1/2 (cos * sin *), 
cos (45*)=: 

1 1/2 (cos * + sin*), 
sin (45+*) = 

1/2 (cos* + sin*), 



34 (page 34). 

(2.) sin (45+ x) = 

l/ 2 (cos * + sin*), 
cos (45+ f) = 

4 I/ 2 (cos .r sin*), 
sin (30*) = 

(cos* 1/3 sin*), 
cos (30 *)_ = 

i (V X 3 cos * + sin*), | 
sin (6o+*)_= 

$ (y'3 cos* -f- sin*), 
cos (60+*) = 

(cos* -y/3 sin*). 
(3.) sin(*+y)=ff, 
sin * ) = - 



( 4 .) sin 75 = 




cos 7 5 = 



(.5-) sin 15= 



4 
1/6+ 1/2" 



(15.) sin 2* = ff, 
cos 2* = T/ 5 . 

(i 6.) sin 22| = 2 -1/2. 



(170 




cos i5 = 



ANSWERS TO EXERCISES 



133 



tan i5 = 2 v/3, 
cot 15 = 2-f- V^. 
sec 1 5 = 2^/2 \/3, 

CSC I5 = 2*/2 + -v/3. 

(20.) sin 5.r = 

5 sin .r 20 sin 3 .r 

+ 1 6 sin 5 .r. 
(21.) cos 5_r = 

5 cos x 20 cos 3 x 

4- 1 6 cos 5 ,r. 

(23.) The values of .r <3oo are 

o, 30, 1 50, 1 80, 210, 330. 

(36.) tan,v tan^/. 

41 (page 40). 

(i.) sin-i ^2=45. i 

45+ 360, etc., 
cos- i = 60, 300, etc., 
tan-' (0= 1 35. 3 1 5. etc., 
cos 1 i =0, 360, etc., 
sin -i ( |) = 210, 330, etc. 

(2.) tan,r = 3- 

(3.) cos.r = d 

(5.) sin (cos- 1 $) = jj. 
(6.) cot (tan- 1 3V) =17. 

(7.) = i\/3. 

(8.) 45, 225. 

(9.) ,r = 45,_y =180. 

(10.) sin I rt = 225. 

48 (page 46). 

(i.) C=i2i33', 

^ = 2133.5, 

c = 2477.8. 
(2.) C=554i', 

^ = 534.05, 



^- = 653.52. 

(3.) C=4534 / , 

a= 1548.1, 

(4.) ^ = 105 59', 
a =54.018, 

^ = 47.738. 
(5.) ^ = 68 58', 
^ = 5274.9, 
= 3730. 

(6.) ^ = 54 58'. 
a = 923.4, 

c= 1 187.7- 

49 (page 47). 

(i.) (I.) Two solutions. 

(2.) One solution, a right tri- 
angle. 

(3.) One solution.. 

(4.) Two solutions. 
(2.) Z?=i657'2i", 

C= 1 5 50' 39"'. 

=1:0.32122. 
(3.) - = 2.5719, 

B=IT 15' i", 

C=i 4 2 1 3' 59". 
(4.) - = 93-59. c' = 54-069, 
B = 26 52' 7", B'= 1 33 7' 53", 
C = i3i46 / 53",C / =253i / 7' / . 
(5.) No solution. 
(6.) = 1.0916, ^'=0.36276, 

B = 1 1 7 50' 44", B'= 1 7 5' 1 6". 

50 (page 48). 

(i.) a = 0.0971, 
#=90 35' 36", 

5 = 0.0053261. 



134 



ANSIVERS TO EXERCISES 



(2.) C 14.211, 


^ = 48 44' 32", 


A = 76 20' 5", 


C= 95 1 5' 56", 


= 44 52' 55" 


5 = 0.60709. 


5 = 80.962. 




(3.) = 85.892, 


52 (page 50). 


A = 67 2l'42", 


(i.) 1116.6 ft. 


C = 62 4 8' 18", 
5=3962.8. 
(4.) = 0.6767, 


(2.) 308 1. 8 yards. 
(3-) 638.34 ft., 
14653 sq. ft. 


5= 15 9' 2l", 


(4.) 4.1 and 8.1. 


C= 131 19' 39", 


(5.) 13.27 miles. 


5=0.08141. 


(6.) 6667 ft. One solution. 


(5.) <: = 72.87, 


(7.) 121.97. 


^ = 40 50' 32", 


(8.) 44 2' 56". 


^=11 2' 28", 
5 = 422.65. 


(9.) 32.151 sq. miles. 
(11.) 54 29' 12". 




(12.) a = 12296 ft., 


51 (page 49). 


r= 13055 ft. 


(i.) ,4 = 55 20' 42", 
^=106 35' 36", 
C=i8 3 ' 4 2", 
5=267.92. 
(2.) A = 34 24' 26", 
B = 73 H' 56", 

C=7220' 36", 

5=3.6143. 


(13.) 294.77 ft. 

(14.) 222.1 ft. 
(16.) 42Q2J^ft r 4- ' 

(17.) 72.613 miles, 
(i 8.) 50.977 ft. 
(19.) 0.85872 miles. 
(20.) 2.98 miles. 
(21.) 1393.9 ^. 
(22.) 8.2 miles. 


(3.) /* = 52 20' 24", 

B= 107 19' 14", 


(23.) 187.39 ft. 
(24.) 0.6011. 


C = 20 20' 24", 






(25.) 4.8112 miles. 

k 


5= 1437.5. 




(4.) A =97 48', 


(26.) 60 51' 8". 


#=l82I 48", 


(27.) 37.365 ft. 


C=63 50' 12", 


(28.) 3.2103 miles. 


5=193.13. 


(29.) 10.532 miles. 


(5.) A = 54 20' 16", 


(30.) 851.22 yards. 


B = 70 27' 46", 


(31.) 9.5722 miles. 


C= 5 472', 


(32.) 6.1271 miles. 


5 = 6090. 


(33.) 280.47 ft. 


(6.) A = 35 59' 30", 


(34-) 123.33 ft. 



ANSWERS TO EXERCISES 



135 



(35-) 4-8ii2 miles. 
(36.) 2666.1 ft. 

* 53 (page 56). 

(i.) 30 = 0.5236, 
45 = 0.7854, 
60 = 1.0472, 

I 20 = 2.0944, 
135= 2.3562, 
720= 12.5664, 
990 =17.2788. 

<2.) I = 22 30', 
1O 

| = 28^ 38' 53", 
} = 100 16' 4". 
(3.) 1.35,0.54. 

74 (page 73). 

(i.) sin 4,1- = 4 cos 3 . i~ sin x 

4 cos.r sin 3 ^-, 
cos 4,1- = cos 4 x 

6 cos 2 x sin 2 x -j- sin 4 x. 
(2.) sin 6x = 6 cos 5 .r sin .r 

20 cos 3 a- sin 3 .r 
-|-6 cos.r sin 5 ^-, 
cos 6x = cos 6 .r 

15 cos*.r sin 2 ^- 
-|- 1 5 cos 1 ' x sin 4 x sin 6 x. 

1 1 \ i- v 1 I V 3 

(3-)- r o ' - r i = \ -r z > 



(4.) .r =i, ^ = o. 3090+ / 0.951 1, 
-f a = 0.8090 -|- /' 0.5878, 
.r 3 = 0.8090 / 0.5878. 
.r 4 = o. 3090 / 0.951 1. 

77 (page 78). 

(23.) .r = 3 o. 
(24.) ^ = 30. 
(25.) x = o or 45. 
(26.) A- = 6o. 
(27.)_y = 45. 
(28.) j = 45- 
(29.) -r = 45. 
(30.) .r = 3 o. 
(31.) ^- = 60. 
(32.) ^- = 30. 
(33.) No angle < 90. 
(34.) jr = 3o-. 
(35.) sin 92 = cos 2. 
(36.) cos 127 = sin 37. 
(37.) tan 320 = tan 40. 
(38.) cot 350 = cot io j . 
(39.) sin 265 = cos 5. 
(40.) tan 171= -tan 9. 
(41.) cos.r= - 
tan. r ^ 



esc -r 



(42.) 



(43.) sin.r = 
cos x-~ 
cot .1- = f , sec .r = 



136 



ANSWERS TO EXERCISES 



(44.) sin,r = 7^- -v/74. 



= f, sec jt- = 



(45.) Quadrant II or IV. 
(46.) Quadrant I or II. 
(47.) Quadrant III or IV. 
(48.) Quadrant I or II. 
(49.) .r = o, 120, 1 80, 240. 
(50.) .r = 3 o, 135, 150, 315. 
(51.) ,r = o, 90, 120, 1 80, 240 

270. 
(57-) o. 
(58.) a. 
(59.) 2 (<*). 
(60.) i(fl'-J 9 ). 

78 (page 80). 

(i.) 306.32 ft. 

(2.) 831.06 ft. 

(3.) 53 28' 14". 
(4.) 49.39 ft. 
(5.) 0.43498 mile. 
(6.) 209.53 ft. 
(7.) 7-3188 ft- 
(8.) 37 36' 30". 
(9.) 109,28 ft. 
(10.) 502.46 ft. 
(u.) 6799.8ft. 

(12.) 219.05 ft. 

(13.) 49i.76ft. 

(14.) 50 32' 44". 

(15.) 49 44' 38". 

(i 6.) 34-063 ft. 

(17.) 32.326 ft., 29 6' 35". 

(18.) 5.6569 miles an hour. 

(19.) 56.295 ft. 

(20.) 103.09 ft. 



(21.) 71 33' 54". 

(22.) 858,160 miles. 

(23.) 238,850 miles. 

(24.) 2163.4 miles. 

(25.) 90,824,000 miles. 

(26.) 432.08 ft. 

(27.) 60.191 ft. 

(28.) 0.32149 mile. 

(29.) 193.77 ft. 

79 (page 83). 
(i.) 3416 ft. 

(2.) 3.7865 ft. 
(3.) 20.45 ft- 

(4.) 36.024^. 
(5.) 8.6058 sq. ft. 
(6.) 181.23 in. 

(70 2-9943 ft. 
(8.) 5.1311 in. 
(9.) 25.92 ft. 
(io.) 92 i' 24", 
(11.) 1.2491. 
(12.) 33 12' 4". 
(13.) 11248 ft. 
(14.) 0.60965 miles. 
(15.) 1.3764. 
(16.) 1.9755- 
(17.) 19.882. 
(i 8.) 0.9397. 
(19.) 6.4984. 
(20.) 3.4641. 

(21.) 6.1981. 
(22.) 6.9978. 
(2 3 .) 15.25. 

80 (page 84). 

(78.) X OO, I 20, 240 , 270. 

(79.) ,r = o, 20, 45, 90, 100, 

I35 C , 140, 1 80, 220, 

225, 260, 270, 315, 

340. 



ANSWERS TO EXERCISES 



137 



(8o.) ^- = 0, 30, 90, 150, 1 80, 


(24.) 55.74 ft. 


270. 


(25.) 247.52 ft. 


(8 1.) _r = o, 45, 120, 240, 225, 


(26.) 556.34 ft. 


270. 


(27.) 465.72 ft. 


(82.) ;r = o , 90, 1 80, 270. 


(28.) 109.22 ft. 


(83.) .r = 0, 90, 210, 330. 


(29.) 2639.4 ft. 


(84.) ^- = 240, 300. 


(30.) 396- 54 ft. 


(85.) x = 2ioP, 330- 


(31.) 287.75 ft. 


(86.) x = o, 90. 


(32.) 2280.6 ft. 


.(87.) ;r = o, 1 80. 


(33.) 64.62 ft. 


(88.) * = o, 1 80. 


(34.) 127.98 ft. 


(89.) ^ = 0, 90, 120, 1 80, 240, (35.) 45- l8 3 ft- 


270. 


(36.) 4365-2 ft. 


(90.) ^- = 450,1350,2250,3150. 


(37.) 140.17 ft. 


(91.) ^ = 30, 150, 210, 330. 


(38.) 610.45 ft- 




(39.) 1 56.66 ft. 


81 (page 88). 


(40.) 41 48' 39" and 125 25 


(i.) 2145.1 ft. 


(41.) 51,288,000. 


(2.) 12.458 miles. 


(42.) 366680. 


(3.) 1.1033 miles. 


(43-) U586. 


(4.) 1508.4 ft. 


(44.) 947460. 


(5-) I7I9-3 yards. 


(45.) 0.89782. 


(6.) 1.2564 miles. 


(46.) 9929-3- 


(7.) 1346.3^. 


(47.) 7 5 1. 62 sq.ft. 


(8.) 387.1 yards. 


(48.) 3H5-9- 


(9.) 5.1083 miles. 


(49.) 855.1. 


(10.) 379 1 - 8 ft. 


(50.) 876.34. 


(u.) 4.4152 ft- 
(12.) 28 57' 20". 


88 (page 98). 


(13.) 115.27. 


(i.) ^=54 59' 47", 


(14.) 44.358 ft. 


^ = 45 41 '28", 


(15.) 92.258 ft. 


s~* /* o p> ' rR'' 


(16.) 101 32' 16". 


(2.) C= 7 i 36' 47". 


(17.) 0.83732 mile. 


^ = 95 22'; 


(18.) 539.1 ft. 


c = 7 i 32' 14", 


(19.) 1.239. 


(3.) C=6 4 1 4' 30", 


(20.) 152.31 and 238.3. 


C' 115 45' 3 . 


(21.) 68.673 ft- 


.00 ^^' rr" 

a m 4^ 22 55 > 


(22.) 32.071 ft. 


*' = i3i37' 5". 


(23.) 13778ft. 


^=42 19' 17". 



57' 



1 38 



ANSWERS TO EXERCfSES 



c = 137 40 43" 
(4.) C- 65 49' 54" 
a = 63 10' 6", 

= 38 59' 12". 
(5.) a = 7S 13' i", 
^=58 25' 46", 

(6.) a = 76 30' 37", 
= 65 28' 58," 
<r = 55 47' 44". 



3' 
(8.) 

(9-) 



, = 64 36' 39", 
= 47 57' 45" 
'-96 1 3' 23", 
= 73 1 7' 29", 
= 70 8' 38". 
= 66 58', 



(10.) tf = 6i4' 55", 
b = 40 30' 22", 
<r=5o 30' 32". 



99 (page 107). 




(2.) a 
b 

(3-) a 
(4.) B 



131 36' 36", 
116 36' 38", 
29 1 1' 42". 

107 7' 45", 
48 57' 29", 
62 31 '40". 
62 54' 43", 
114 30' 26", 
= 56 39' 10". 



a= 



(5.) ^ = 130 35' 56", 
^ = 30 25' 34", 
C = 3i 26' 32", 

(6.) ^=98 21' 22", 

b-=. 109 50' 8", 

*= 115 13' 4". 
(7.) B = y. 26' 9'', 

a= 84 14' 32", 

^ = 51 6' 12". 
(8.) tf-8o 5' 8", 

70 10' 36", 

r = i455'2". 
(9.) .4=70 39' 4", 

# = 48 36' 2", 

C=ii9 15' 2". 
(10.) a = 40 o' 12", 

^ = 42 15' ii", 
C=i2i 36' 19". 

100 (page 109). 

(i.) 80.895 s q- in- 
(2.) 26.869 sq. in. 
(3,) 158.41 sq. in. 
(4.) 39990 sq. miles. 

101 (page 112). 
(i.) 5C = 48 2' 43", 

^=52 53' 9". 
(2.) 7 : 24 A.M. 

(3.) 4 P.M. 

102 (page 114). 
(I.) 3029^ miles. 
(2.) 2229.8 miles. 
(3.) 2748.5 miles. 
(4.) 7516.3 miles. 
(5.) 5108.9 miles. 



THE END 



LOGARITHMIC 

AND 

TRIGONOMETRIC TABLES 

FIVE-PLACE AND FOUR-PLACE 



PHILLIPS-LOOMIS MATHEMATICAL SERIES 



LOGARITHMIC 

AND 

TRIGONOMETRIC TABLES 



FIVE-PLACE AND FOUR-PLACE 



BY 

ANDREW W. PHILLIPS, PH.D. 

AM) 

WENDELL M. STRONG, PH.D. 

YALE UNIVERSITY 




NEW YORK AND LONDON 

HARPER & BROTHERS PUBLISHERS 
1899 



THE PHILLIPS-LOOMIS MATHEMATICAL SERIES. 



ELEMENTS OF TRIGONOMETRY, Plane and Spherical. By 

ANDREW W. PHILLIPS, Ph.D., and WENDELL M. STRONG, Ph.D., Yale 

University. Crown 8vo, Cloth. 
ELEMENTS OF GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., 

and IRVING FISHER, Ph.D., Professors in Yale University. Crown 

8vo, Half Leather, $1 75. [By mail, $1 92.] 

ABRIDGED GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., and 
IRVING FISHER, Ph.D. Crown 8vo, Half Leather, $1 25. [By 
mail, $1 40.] 

PLANE GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., and IRVING 
FISHER, Ph.D. Crown 8vo, Cloth, 80 cents. {By mail, 90 cents.] 

LOGARITHMIC AND TRIGONOMETRIC TABLES. Five Place 
and Four- Place. By ANDREW W. PHILLIPS, Ph.D., and WENDELL 
M. STRONG, Ph.D., Yale University. Crown 8vo. 

LOGARITHMS OF NUMBERS. Five-Figure Table to Accompany 
the "Elements of Geometry," by ANDREW W. PHILLIPS, Ph.D., and 
IRVING FISHER, Ph.D., Professors in Yale University. Crown 8vo, 
Cloth, 30 cents. [By mail, 35 cents.] 

NEW YORK AND LONDON : 
HARPER & BROTHERS, PUBLISHERS. 



Copyright, 1898, by HARPER & BROTHERS. 



All rights reserved. 



CONTENTS 



TABLE p AGE 

INTRODUCTION TO THE TABLES v 

I. FIVE-PLACE LOGARITHMS OF NUMBERS i 

II. FIVE -PLACE LOGARITHMS OF THE TRIGONOMETRIC 

FUNCTIONS TO EVERY MINUTE 29 

III. FIVE-PLACE LOGARITHMS OF THE SINE AND TANGENT 

OF SMALL ANGLES 121 

IV. FOUR-PLACE NAPERIAN LOGARITHMS 131 

V. FOUR-PLACE LOGARITHMS OF NUMBERS 135 

VI. FOUR -PLACE LOGARITHMS OF THE TRIGONOMETRIC 

FUNCTIONS TO EVERY TEN MINUTES 139 

VII. FOUR -PLACE NATURAL TRIGONOMETRIC FUNCTIONS 

TO EVERY TEN MINUTES 149 

VIII. SQUARES AND SQUARE ROOTS OF NUMBERS 159 

IX. THE HYPERBOLIC AND EXPONENTIAL FUNCTIONS OF 

NUMBERS FROM o TO 2.5 AT INTERVALS OF .1 . . 160 
X. CONSTANTS MEASURES AND WEIGHTS AND OTHER 

CONSTANTS . 161 



INTRODUCTION TO THE TABLES 

COMMON LOGARITHMS. 

1. The common logarithm of a number is the index of 
the power to which 10 must be raised to give the number. 

Thus, log IOG = 2, because 100 = io 2 

log i =o, " i =10 

log .1 = i, .1 = io -I 

log 3 =47712, " 3 =io- 47m 

In general, log m x if ;;/ = io*. 

2. To multiply two numbers, add their logarithms. The 
result is the logarithm of the product. 

Proof. Ifaw = io* so that log m = x, 

and n = io> " " log n =y, 

then mn = io*+*" " log mn = x+y. 

Hence log mn = \ogrn -f log n. 

3. To divide one number by another, subtract the loga- 
rithm of the divisor from the logarithm of the dividend. 
The result is the logarithm of the quotient. 

Proof. = 

Hence log ^~ =x ~? 

4. To raise a number to a power, multiply the logarithm 
of the number by the index of the power. The result is the 
logarithm of the power. 



vi INTRODUCTION TO THE TABLES. 

Proof. m a = ( i o x ) a = i o- ax ; 

Hence \ogm a = ax = a logm. 

5. To extract a root of a number, divide the logarithm of 
the number by the index of the root. The result is the loga- 
ritJini of the root. 



Proof. "Im =-. * Ao* = 10*. 



* v ^>s A. / "" ~ 7 

b b 
6*. Restatement of laws : 

log nin = log in + log n ; 

log = logm logn ; 
log m a = a log m ; 



7. Most numbers are not integral powers of 10; hence 
most logarithms are of decimal form. 

Thus, log 2. 2 .34242, Iog4 =1.60206. 

S. If a logarithm is negative, it is expressed for conven- 
ience as a negative integer plus a positive decimal. 

The logarithm of a number less than I is negative. 

The negative integer is usually expressed in the form 
910, 8 10, etc. 

Thus, Iog.2i544 = i -f .33333, written 9-33333 - 10 : 
Iog.o2i544 = 2 -h. 33333, " 8.3333310; 
log .0021 544 = 3 + .33333, " 7-33333 ~ 10. 

Remark. In some books the negative integer is written i, 2, etc., 
instead of 9 10, 8 10, etc. 

The integral part of a logarithm is the characteristic; 
the decimal part is the mantissa. 

Thus, log 2 1 5.44 = 2. 33333 ; the characteristic is -f- 2 ; the mantissa 



COMMON LOGARITHMS. vii 

is +-33333: log .021544=8.33333 10 ; the characteristic is 8 10 
= 2 ; the mantissa is -f .33333. 

9. It is evident that the larger a number the larger its logarithm. 
Hence the logarithm of any number 

between i and 10 is o -f- a mantissa, 
10 " 100 " i+" " 
.1 " i "-i+" 
.01 " .1 " 2-f" " etc. 
We have, then, the following rule for obtaining the characteristic : 

10. Count the number of places the first left-hand digit of 
the number is removed from the unit's place. 

If this digit is' to the left of the unit's place, the result is the 
required characteristic. 

If this digit is to the rig/it of the unit's place, the result 
taken with a minus sign is the required characteristic. 

If this digit is in the unit' s place, t/ic characteristic is zero. 
Thus the characteristic of the logarithm of 21550 is 4 

" " " ' " 21.55 i 

2.155 " o 

' -2155 "i 

" " " " " " .02155 " -2 

11. The logarithms of numbers which differ only in the 
position of the decimal point have the same mantissa. 

For to change the position of the decimal point is to multiply or 
divide by an integral power of 10; that is, an integer is added to or 
subtracted from the logarithm, and consequently only the character- 
istic is changed. 

Thus, log 2 1 544 =3-33333 

log 2.1544 =0.33333 
log .21544 =9.33333-10 
log .021544 = 8.33333-10 

Therefore, in finding the mantissa of the logarithm of a 
number the decimal point may be disregarded. The man- 
tissa is found from the tables of logarithms. 



viii INTRODUCTION TO THE TABLES. 

USE OF THE TABLE OF LOGARITHMS OF NUMBERS. 

(TABLE i.) 

12. To find the logarithm of a number. 

Look in the column at the head of which is " N " for the 
first three figures of the number, and in the line with "N" for 
the fourth figure. In the line opposite the first three figures 
and in the column under the fourth is the desired mantissa. 

Only the last three figures of the mantissa are found thus; the 
first two must be taken from the first column ; they are found either 
in the same line or in the first line above which gives the whole man- 
tissa, except when a * occurs. If a * precedes the last three figures of 
the mantissa the first two are found in the following line : 

The characteristic is obtained by 10. 

Example. To find the logarithm of 105400. 

The characteristic = 5. 10 

The mantissa = .02284 (opposite 103 and under 4 in the tables) ; 

Hence log 105400 = 5.02284. 

13. If there are five or more figures in a number the 
figures beyond the fourth are treated as a decimal. The 
corresponding mantissa is between two successive mantissas 
of the tables. 

Example. To find the logarithm of 10543. 

The characteristic = 4. 10 

The mantissa is not in the tables, but is between the mantissa of 

1055 = .02325 
and the mantissa of 1054 = .02284 

Their difference = 41 

Hence an increase of one in the fourth figure of the number pro- 
duces an increase of 41 in the mantissa. Then an increase of .3 must 
produce an increase of 41 X .3 in the mantissa. 

41 X. 3 = 12.3 = 12 nearly. 

Hence the mantissa of 10543=1.02284-}- 12 = .02296. 

Therefore log 10543= 4.02296. 



LOGARITHMS OF NUMBERS. ix 

An easy method of multiplying 41 by .3 is to use the table of pro- 
portional parts at the bottom of the page in the tables. 
Under 41 and opposite 3 is 12. 3 (=41 X-3). 

14. Figures beyond the fifth are usually omitted in the 
use of a five -place table, as their retention does not add 
much to the accuracy of the result. For the fifth figure, 
however, we choose the one which gives most nearly the 
true value of the number. 

Thus, if the number is 157.032, we use 157.03; 

" 157.036, " " 157.04; 

" " " " 157.035. " " 157-04. 

13. To find a number from its logarithm. 
The process is the reverse of finding the logarithm from 
the number; it is illustrated by the following examples: 
Find the number of which 9.12872 10 is the logarithm. 
Since the characteristic = i, the decimal point will be before the 
first figure of the number. 

.12872 is opposite 134 and under 5 in the tables. 
Hence .12872 = the mantissa of 1345, 

and 9.12872 10 = log. 1 345. 

Find the number of which 9.12895 10 is the logarithm. 
The mantissa .12895 is not i" the tables, but is 
between .12905 = mantissa of 1346 

and .12872= " " 1345. 

.00033 = tne difference. 
.12895 mantissa given, 
.12872 = mantissa of 1345, the smaller number, 

23 = the difference. 

Change into a decimal. The first figure of this decimal will be 
the figure in the fifth place of the number. 

f = .7 nearly. 
Hence 9.12895 10 log. 13457. 



x INTRODUCTION TO THE TABLES. 

An easy method of changing into a decimal is to use the table 
of proportional parts. 

Under 33 is found 23.1 (= 23 nearly), which is opposite 7. 

Hence H = -7 nearly. 

The process we have employed in finding the logarithm 
of a number of more than four figures, or the number corre- 
sponding to a mantissa not given in the table, is called in- 
terpolation. 

EXAMPLES FOR THE USE OF LOGARITHMS. 

16. Multiply 5789.2 by .018315. 

tog 5789.2 = 3.76262 

log .018315 =8.26281 10 

2.02543 = log 106.03 
Multiply 9.8764 by .10013. 

log 9.8764 = 0.99460 
log. 10013 = 9.00056 10 

9.99516 10 = log .98892 
Find the value of 3.1416 X 7638.6 x .017829. 
log 3.1416=0.49715 
log 7638.6 = 3.88302 
log .017829 = 8.251 13 10 

2.631 30 = log 427. 86 
Divide 81.321 by 3.1416. 

Iog8i.3i2 = 1.91021 
log 3. 141 6 = 0.497 1 5 

1.41306 =log 25.886 
Find the value of (2.1345)'. 

log 2. 1 345 =0.32930 

5 

1.64650 = log 44.310 
Find the value ofy / .oio2i. 

log .01021 = 8.00903 10 
= 28.00903 30 

28.00003 30 

^ ^-=9.33634 -10 = log. 21694 



LOGARITHMS OF TRIGONOMETRIC FUNCTIONS, xi 

17. The logarithm of is called the cologarithm of m, 

and is obtained by subtracting logm from zero. 

Thus, if log m = 9.76423 10, cologw =0.23577. 

It is frequently shorter to add cologm than to subtract 
logm when we wish to divide by a number m. 

The following example illustrates this : 

Find the value of 57 f X42 ' 24 . 
644.32 

log 57. 98= 1.76328 
log 42. 24= 1.62572 
colog 644.32 = 7.19090 10 

0.57990 = log 3.801 

USE OF THE TABLE OF LOGARITHMS OF TRIGONOMETRIC 

FUNCTIONS. (TABLE n.) 

18. For an angle less than 45, the degrees are at the 
head of the page, the minutes in the column at the left, and 
"L. Sin.," "L. Tang.," etc., at the head of the correspond- 
ing columns. For angles between 45 and 90, the degrees 
are at the foot of the page, the minutes in the column at 
the right, and " L. Sin.," " L. Tang.," etc., at the foot of the 
corresponding columns. 

The characteristic is printed 10 too large where it would 
otherwise be negative. Hence, in using this table, 10 is 
to be supplied, except for the cotangent of angles less than 
45 and the tangent of angles from 45 to 90. 

EXAMPLES. 

log sin 15 25' = 9.42461 10. 
log tan 28 1 7' = 9.73084 10. 
log cos 62 14' = 9.66827 10. 
log cot 25 34' = 0.3 2020. 



xii INTRODUCTION TO THE TABLES. 

19. If the given angle contains seconds, we may reduce 
the seconds to a decimal of a minute and proceed as in 
finding the logarithms of numbers. It must be remem- 
bered, however, that log cos and log cot decrease as the 
angle increases. 

In practice we remember that 6" is one-tenth of a minute, and di- 
vide the number of seconds by 6", then use the table of proportional 
parts at the bottom of the page. 

EXAMPLES. 
Find log sin 28 14' 36" (=log sin 28 14.6'). 

log sin 28 1 5' log sin 28 14' = 23 (found in column "d.") 
log sin 28 14' = 9.67492 10 
23 X. 6 = 13.8= 14 nearly 

log sin 28 14' 36" = 9.67506 10 

Find log cos 39 17' 22" (=log cos 39 17.3!'). 
log cos 39 1 7' = 9.8887 5 10 



log cos 39 17' 22" = 9.88871 10 

Find log tan 51 27' 44" (=log tan 51 27.7^'). 
log tan 51 27' = .09862 



log tan 51 27' 44" = .0988 1 

Find log cot 67 i8'46". 

log cot 67 1 8' =9.62150 10 
6X.= 28 



Hence log cot 67 18' 46" = 9.62122 10 

20. The 'process of finding an angle, if its logarithmic 
sine or tangent, etc., is given, is the reverse of the pre- 
ceding. 



EXPLANATION OF THE TABLES. xiii 

EXAMPLES. 
Given log sin x = 9.67433 10 ; find x, 

log sin 28 ii' = 9.6742 1 10 
log sin ,r log sin 28 ii' = 12 
and log sin 28 12' log sin 28 H' = 24 
Hence .r = 28 1 1' 30" ( of i f being 30"). 

Find the angle whose log 005 = 9.88231 10. 
log cos 40 1 8' = 9.88234 10. 

60" x ft= 16". 
Hence log cos 40 18' 1 6" = 9.88231 10. 

Find the angle whose log tan =0.17844. 
log tan 56 27 =0.17839. 

6o"x&=u". 
Hence ' log tan 56 27' ii" = 0.17844. 

Find the angle whose log cot = 9.87432 10. 
log cot 53 10' = 9.87448 10. 

6o"xi=37"- 
Hence log cot 53 10' 37" = 9.87432 10. 

EXPLANATION OF THE TABLES. 

21. A dash above the terminal 5 of a mantissa, as 5, de- 
notes that the true value is less than 5. 

Thus, log 389 = 2.5899496 to seven places, but to five places 
log 389 = 2.58995. 

Tables I and II have already been explained. 

TABLE III. 

22. The logarithmic sine and tangent cannot be obtained 
very accurately from Table II if the angle contains seconds 
and is less than 2. 

Table III is to be used when greater accuracy in the sine 
or tangent of a small angle is desired than can be obtained 



xivr INTRODUCTION TO THE TABLES. 

by the use of Table II. It is to be noted that the first page 
of Table III gives the sine and tangent to every second for 
angles less than 8'. 

TABLE IV. 

23. Naperian or "natural" logarithms are logarithms to 
the base e ( = 2.71828 + ). The whole logarithm is given, 
since the integral part cannot be supplied by inspection, as 
with common logarithms. 

TABLES V AND VI. 

24. Four-place logarithms and logarithmic functions are 
used instead of five-place if the results are sufficiently ac- 
curate for the purpose in view. 

In Table VI both the degrees and minutes are in the col- 
umns at the sides of the page, otherwise this table does not 
differ in form from Table II. 

TABLE VII. 

23. This table is identical with Table VI in form, but 
gives the trigonometric functions themselves, instead of 
their logarithms. 

TABLES VIII, IX, X. 

26. These tables require no explanation. 



TABLE I 

FIVE-PLACE LOGARITHMS 
OF NUMBERS 



100-130 



N 


O 


1 


2 


3 


4 


5 


O 


7 


8 


9 


100 


oo ooo 


o43 


o8 7 


i3o 


i 7 3 


2I 7 


260 


3o3 


346 


389 


IOI 




432 


4 7 5 


5i8 


56i 


6o4 


64 7 


689 


7 32 


775 


817 


102 




860 


9o3 


945 


988 


*o3o 


*0 7 2 


*ii5 


*i5 7 






io3 


01 


284 


3 2 6 


368 


4io 


452 


494 


536 


5 7 8 


620 


662 


104 




7 o3 


745 


787 


828 


8 7 o 


912 


9 53 


995- 


*o36 


*o 7 8 


io5 


02 


119 


1 60 


202 


243 


284 


325 


366 


407. 


449 


490 


106 




53i 


5 7 2 


612 


653 


6 9 4 


735 


776 


816 


85 7 


898 


107 




9 38 


979 




'9 


*o6o 


*IOO 


*i4i 


*i8i 




*222 


*262 


*302 


108 


o3 


342 


383 


423 


463 


5o3 


543 


583 


623 


663 


7 o3 


109 




743 


7 82 


822 


862 


902 


94 1 


9 8i 




*O2I 


*o6o 


*IOO 


110 


o4 1 3g 


i 79 


218 


2 58 


297 


336 


3 7 6 


415 


454 


493 


1 1 1 




532 


5 7 i 


610 


650 


689 


7 2 7 


7 66 


8o5 


844 


883 


I 12 




922 


961 


999 


*o38, 


*77 


*n5 


*i54 


*I92 


*23l 


*26 9 


n3 


o5 


3o8 


346 


385 


423 


46 1 


500 


538 


5 7 6 


6i4 


652 


ii 


4 




690 


729 


767 


805 


843 


881 


9 i8 


956 


994 


*032 


ii 


5 


06 070 


1 08 


i45 


i83 


221 


258 


2 9 6 


333 


3 7 i 


4o8 


116 




446 


483 


521 


558 


5 9 5 


633 


6 7 o 


77 


744 


7 8i 


117 




819 


856 


8 9 3 


93o 


9 6 7 


*oo4 


*o4i 




*o 7 8 


*ii5 


*i5i 


1x8 


07 


188 


225 


262 


298 


335 


3 7 2 


4o8 


445 


482 


5i8 


119 




555 


5 9 i 


628 


664 


7 oo 


7 3 7 


773 


809 


846 


882 


120 


918 


954 


99 


*02 7 


*o63 


*99 


*i35 


*i 7 i 


*2O 7 


*a43 


121 


08 


279 


3i4 


35o 


386 


422 


458 


493 


529 


565 


600 


122 




636 


6 7 2 


707 


743 


77 8 


8i4 


849 


884 


92O 


^955 


123 




991 


*O26 


*o6i 


"096 


*l32 


*i6 7 . 


*2O2 


*23 7 


*2 7 2 




124 


09 


342 


3 77 


4l2 


447 


482 


617 


552 


58 7 


621 


656 


125 




691 


7 26 


760 


796 


83o 


864 


8 99 


934 


9 68 


*oo3 


126 


10 


o3 7 


O 7 2 


1 06 


i4o 


175 


209 


243 


278 


3l2 


346 


I2 7 




38o 


4i5 


449 


483 


5i 7 


55i 


585 


619 


653 


68 7 


128 




721 


755 


789 


8 2 3 


85 7 


890 


924 


958 


992 


*025 


I2 9 


ii 


o5g 


093 


126 


1 60 


i 9 3 


22 7 


261 




294 


32 7 


36i 


130 


3 9 4 


428 


46 1 


494 


528 


56i 


594 


628 


661 


6 9 4 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


PP 


44 


43 42 


41 40 39 


38 37 36 


i 


4.4 


4.3 4.2 


i 


4.i 


4.o 


3. 9 


i 


3.8 


3.7 


3.6 


2 


8.8 


8.6 8.4 


2 


8.2 


8.0 


7-8 


2 


7.6 


7-4 


7-2 


3 


13.2 


12.9 12.6 


3 


12.3 


I2.O 


n. 7 


3 


n.4 


ii. i 


10.8 


4 


17.6 


1-7.2 16.8 


4 


16.4 


16.0 


i5.6 


4 


15.2 


1 4.8 


i4.4 


5 


22. 


21.5 21. 


5 


20. 5 


20. o 


i 9 .5 


5 


19.0 


i8.5 


18.0 


6 


26.4 


25.8 25.2 


6 


24.6 


24.0 


23.4 


6 


22.8 


22.2 


21.6 


7 


3o.8 


3o.i 29.4 


7 


28.7 


28.0 


2 7 .3 


7 


26.6 


25.9 


25.2 


8 


35.2 


34.4 33.6 


8 


32.8 


32.0 


3l.2 


8 


3o.4 


29.6 


28.8 


9 


3 9 .6 




9 


36. 9 


36.o 


35.i 


9 


34.2 




32.4 



13O 16O 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


130 


1 1 


394 


428 


46i 


494 


5 2 8 


56i 


5 9 4 


628 


661 


694 


i3 


i 




727 


7 6o 


793 


826 


860 


8 9 3 


926 


9 5 9 


99 2 


*O24 


I 32 


12 057 


090 


123 


i56 


189 


222 


254 


287 


320 


352 


i33 




385 


4i8 


45o 


483 


5i6 


548 


58 1 


6i3 


646 


6 7 8 


1 34 




710 


7 43 


77 5 


808 


840 


872 


95 


9 3 7 


9 6 9 


*OOI 


i35 


i3 


o33 


066 


098 


i3o 


162 


i 9 4 


226 


2 58 


290 


322 


1 36 




354 


386 


4i8 


45o 


48 1 


5i3 


545 


5 77 


6o 9 


64o 


i3 7 




672 


704 


7 35 


767 


799 


83o 


862 


8 9 3 


9 2 5 


9 56 


1 38 




988 


*c 


'9 


*o5i 


*082 


*n4 


*i4g 


*i 7 6 


*208 




*2 7 


139 


i4 


3oi 


333 


364 


3 9 5 


426 


45 7 


48 9 


52O 


55i 


582 


140 


6i3 


644 


675 


7 o6 


7 3 7 


768 


799 


82 9 


860 


8 9 i 


U 


i 




922 


9 53 


983 


*oi4 


+045 


*o 7 6 


*io6 


*i3 7 


*i68 


*i 9 8 


142 


i5 


229 


25 9 


290 


320 


35i 


38i 


412 


442 


4 7 3 


5o3 


M 


3 




534 


564 


5 9 4 


625 


655 


685 


7 i5 


7 46 


77 6 


806 


i44 




836 


866 


897 


9 2 7 


9 5 7 


987 


*OI 7 


*o4 7 


*0 77 


*IO 7 


i45 


16 


i3 7 


167 


i 97 


22 7 


2 56 


286 


3i6 


346 


3 7 6 


4o6 


i46 




435 


465 


495 


524 


554 


584 


6i3 


643 


6 7 3 


702 


i4 7 




7 32 


761 


7< 


pi 


820 


850 


879 


909 


9 38 


9 6 7 


997 


i48 


i 7 


026 


o56 


085 


n4 


i43 


i 7 3 


202 


23l 


260 


289 


149 




3i 9 


348 


377 


4o6 


435 


464 


493 


522 


55i 


58o 


150 


609 


638 


66 7 


696 


725 


7 54 


782 


811 


84o 


869 


i5 


i 




898 


9 26 


9 55 


984 


*oi3 


*o4i 


"070 


*o 99 


*I2 7 


*i56 


i5 2 


18 


1 84 


2l3 


241 


270 


298 


32 7 


355 


384 


4l2 


44 1 


i53 




46 9 


4 9 8 


526 


554 


583 


611 


63 9 


6t> 7 


696 


724 


1 54 




7 52 


780 


808 


83 7 


865 


893 


921 


949 


977 


*oo5 


i55 


19 o33 


06 1 


o8 9 


117 


i45 


i 7 3 


201 


22 9 


25 7 


285 


i56 




312 


34o 


368 


3 9 6 


424 


45i 


479 


5o 7 


535 


562 


i5 7 




590 


618 


645 


6 7 3 


700 


728 


7 56 


7 83 


811 


838 


1 58 




866 


8 9 3 


921 


948 


9-76 


*oo3 


*o3o 


*o58 


*o85 


*II2 


i5 9 


20 


i4o 


167 


i 9 4 


222 


249 


276 


3o3 


33o 


358 


385 


160 


412 


43 9 


466 


493 


520 


548 


575 


602 


629 


656 


N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


PP 


35 


34 


33 


32 31 30 29 28 27 


i 


3.5 


3.4 


3.3 


i 


3.c 


3.1 


3.o i 2.9 


2.8 


2.7 


2 


7.0 


6.8 


6.6 


2 


6.4 


6^2 


6.0 2 5.8 


5.6 


5.4 


3 


io.5 


IO.2 


9-9 


3 


9.6 


9-3 


9.0 3 8.7 


8.4 


8.1 


4 


i4.o 


i3.6 


13.2 


4 


12.8 


12.4 


12. o 4 ii. 6 


I 1. 2 


10.8 


5 


i 7 .5 


I 7 .0 


i6.5 


5 


16.0 


i5.5 


i5.o $ i4.5 


i4.o 


i3.5 


6 


21. 


20.4 


19.8 


6 


19.2 


18.6 


18.0 6 17.4 


16.8 


16.2 


7 


24.5 


.23.8 


23.1 


7 


22.4 


21,7 


21. o 7 20. 3 


19.6 


18.9 


8 


28.0 


2 7 .2 


26.4 


8 


25.6 


24.8 


24.0 8 23.2 


22.4 


21.6 





3i.5 




29.7 


9 


s8.8 


57.9 


27.0 9 26.1 


25.2 


24.3 



16O-190 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


160 


20 4l2 


43 9 


466 


493 


520 


548 


575 


602 


629 


656 


161 




683 


710 


7 3 7 


7 63 


79 


817 


844 


871 


898 


9 2 5 


162 




952 


978 


*oo5 


*032 


*o59 


*o85 


*II2 


*:39 


*i65 


*I92 


i63 


21 


219 


245 


272 


299 


325 


352 


3 7 8 


405 


43i 


458 


1 64 




484 


5u 


53 7 


564 


5 9 o 


617 


643 


669 


696 


722 


i65 




748 


775 


801 


827 


854 


880 


906 


932 


9 58 


985 


166 


22 


on 


037 


o63 


089 


u5 


i4i 


167 


i $4 


220 


246 


167 




272 


298 


324 


350 


3 7 6 


4oi 


427 


453 


479 


5o5 


168 




53i 


55 7 


583 


608 


634 


660 


686 


712 


7 3 7 


7 63 


169 




7% 


8i4 


84o 


866 


891 


917 


943 


968 


994 


*oi9 


170 


23 045 


070 


096 


121 


i4 7 


172 


198 


223 


249 


2 7 4 


171 




3oo 


325 


35o 


3 7 6 


4oi 


426 


452 


477 


502 


5a8 


172 




553 


5 7 8 


6o3 


629 


654 


679 


704 


729 


7 54 


779 


i 7 3 




805 


83o 


855 


880 


95 


93o 


955 


980 


*oo5 


*o3o 


1 74 


24 


055 


080 


105 


i3o 


155 


180 


204 


229 


254 


379 


i 7 5 




3o4 


329 


353 


3 7 8 


4o3 


428 


45 2 


477 


502 


5 37 


176 




55i 


576 


601 


6 2 5 


650 


6 7 4 


699 


7 2 4, 


,748 


773 


177 




797 


822 


846 


871 


895 


920 


944 


969 


993" 


*oi8 


178 


25 


042 


066 


091 


n5 


i3 9 


1 64 


188 


212 


2 3 7 


261 


179 




285 


3io 


334 


358 


382 


4o6 


43i 




455 


479 


5o3 


180 


52 7 


55i 


5 7 5 


600 


624 


648 


672 


696 


720 


744 


181 




768 


792 


816 


84o 


864 


888 


912 


9 35 


9 5 9 


9 83 


182 


26 


007 


o3i 


055 


079 


1 02 


126 


i5o 


1 74 


198 


221 


i83 




245 


269 


293 


3i6 


34o 


364 


38 7 


4n 


435 


458 


1 84 




482 


5o5 


529 


553 


5 7 6 


600 


23 


647 


670 


6 9 4 


i85 




717 


7 4i 


764 


788 


811 


834 


858 


881 


95 


928 


186 




9 5i 


975 


998 


*02I 


*o45 


*o68 


*09i 




*u4 


*i38 


*i6i 


187 


27 


1 84 


207 


23l 


2 54 


277 


3oo 


323 


346 


3 7 o 


3 9 3 


1 88 




4i6 


43 9 


462 


485 


5o8 


53i 


554 


577 


600 


623 


189 




646 


669 


692 


7 i5 


7 38 


761 


784 


807 


83o 


85a 


190 


8 7 5 


898 


921 


944 


967 


989 


*OI2 


*o35 


*o58 


*o8i 


N 


O 


1 


2 


3 


4 


5 


6 


7 


8 9 


PP 27 


26 


25 


24 23 22 


21 20 19 


i 


2-7 


2.6 


2.5 


: 


2.4 


2.3 


2.2 


i 


2.1 


2.O 


1.9 


2 


5.4 


5.2 


5.o 


2 


4.8 


4.6 


4.4 


2 


4.2 


4.o 


3.8 


3 


8.1 


7.8 


7-5 


3 


7.2 


6.9 


6.6 


3 


6.3 


6.0 


5-7 


4 


10.8 


10.4 


IO.O 


4 


9.6 


9.2 


8.8 


4 


8.4 


8.0 


7.6 


5 


i3.5 


i3.o 


12.5 


5 


12.0 


ii.5 


n.o 


5 


io.5 


IO.O 


9-5 


6 


16.2 


i5.6 


i5.o 


6 


i44 


i3.8 


13.2 


6 


12.6 


I2*.0 


n.4 


7 


18.9 


18.2 


i 7 .5 


7 


16.8 


16.1 


i5.4 


7 


i4. 7 


i4.o 


i3.3 


8 


21.6 


20.8 


20. o 


8 


19.2 


18.4 


17.6 


8 


16.8 


16.0 


15.2 


9 


24.3 


23.4 


22.5 


9 


21.6 


20.7 


19.8 


9 


18.9 


18.0 17.1 



190-230 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


190 


27876 


898 


921 


944 


967 


989 


*OI2 


*o35 


*o58 


*o8i 


191 


28 io3 


126 


149 


171 


194 


2I 7 


240 


262 


285 


3o 7 


192 


33o 


353 


3 7 5 


398 


421 


443 


466 


488 


5n 


533 


i 9 3 


556 


5 7 8 


601 


623 


646 


668 


691 


7 i3 


7 35 


7 58 


194 


780 


8o3 


825 


84 7 


8 7 o 


892 


914 


9 3 7 


9 5 9 


9 8i 


i 9 5 


29 oo3 


026 


o48 


O 7 O 


092 


11 5 


i3 7 


i5 9 


181 


203 


196 


226 


248 


270 


292 


3i4 


336 


358 


38o 


4o3 


425 


197 


447 


46 9 


491 


5i3 


53s 


55 7 


5 79 


601 


623 


645 


198 


667 


688 


710 


7 32 


7 54 


776 


798 


820 


842 


863 


199 


885 


97 


929 


9 5i 


97 3 


994 


*oi6 


*o38 


*o6o 


*o8i 


200 


3o io3 


125 


i46 


1 68 


190 


211 


233 


255 


2 7 6 


2 9 8 


201 


32O 


34 1 


363 


384 


4o6 


428 


449 


4 7 i 


492 


5i4 


202 


535 


55 7 


5 7 8 


600 


621 


643 


664 


685 


707 


7 28 


203 


750 


771 


792 


8i4 


835 


856 


878 


899 


920 


942 


2O4 


9 63 


9 84 


*oo6 


*027 


*o48 


*o69 


*O9i 


*II2 


*i33 


*i54 


205 


3i i 7 5 


197 


218 


23 9 


260 


281 


302 


323 


345 


366 


206 


38 7 


4o8 


429 


450 


4 7 i 


492 


5i3 


534 


555 


5 7 6 


207 


5 97 


618 


639 


660 


681 


702 


7 23 


744 


765 


7 85 


208 


806 


827 


848 


86 9 


890 


911 


9 3i 


962 


973 


994 


209 


32 015 


o35 


o56 


077 


098 


118 


i3 9 


160 


181 


2OI 


210 


222 


243 


263 


284 


305 


3 2 5 


346 


366 


38 7 


4o8 


211 


428 


449 


46 9 


4 9 o 


5io 


53i 


55 2 


5 7 2 


5 9 3 


6i3 


212 


634 


654 


675 


695 


7 i5 


7 36 


7 56 


777 


797 


818 


2l3 


838 


858 


879 


899 


919 


94o 


9 6o 


980 


*OOI 


*O2I 


2l4 


33o4i 


062 


082 


102 


122 


i43 


i63 


i83 


203 


224 


2l5 


244 


264 


284 


3o4 


325 


345 


36s 


385 


4o5 


425 


216 


445 


465 


486 


5o6 


5 2 .6 


546 


566 


586 


606 


626 


217 


646 


666 


686 


706 


726 


746 


766 


7 86 


806 


826 


218 


846 


866 


885 


906 


925 


945 


9 65 


985 


*oo5 


*O25 


219 


34o44 


o64 


o84 


io4 


124 


i43 


i63 


i83 


203 


223 


220 


242 


262 


282 


3oi 


321 


34i 


36i 


38o 


4oo 


420 


221 


43 9 


45 9 


4?9 


498 


5i8 


53 7 


55 7 


5 77 


5 9 6 


616 


222 


635 


655 


6 7 4 


6 9 4 


7 i3 


7 33 


7 53 


77 2 


79 2 


811 


223 


83o 


850 


869 


889 


908 


928 


947 


967 


986 


*oo5 


224 


35 025 


o44 


o64 


o83 


102 


122 


i4i 


1 60 


1 80 


I 99 


225 


218 


238 


25 7 


276 


295 


315 


334 


353 


3 7 2 


3 9 2 


226 


4n 


43o 


449 


468 


488 


5o 7 


526 


545 


564 


583 


227 


6o3 


622 


64i 


660 


679 


698 


717 


7 36 


7 55 


774 


228 


79 3 


8i3 


83 2 


85i 


8 7 o 


889 


908 


9 2 7 


9 46 


965 


229 


9 84 


*oo3 


*O2I 


*o4o 


*o5 9 


*o 7 8 


*09 7 


*n6 


*i35 


*i54 


230 


36 i 7 3 


192 


21 I 


229 


248 


26-7 


286 


305 


324 


342 


X 


O 


1 2 


3 


4 


5 


6 


7 


8 9 



23O 26O 



N 





1 


2 


3 


4 


5 





7 


8 





230 


36 173 


192 


211 


22 9 


248 


267 


286 


3o5 


324 


342 


s3i 


36i 


38o 


399 


4i8 


436 


455 


474 


493 


5n 


53o 


232 


549 


568 




586 


605 


624 


642 


661 


680 


698 


717 


233 


7 36 


7 54 




773 


79 I 


810 


829 


847 


866 


884 


9 o3 


234 


922 


94o 




9 5 9 


977 


996 


*oi4 


*o33 


*o5i 


"070 


*o88 


235 


3 7 io 7 


125 




1 44 


162 


181 


199 


218 


236 


254 


2 7 3 


236 


291 


3io 


328 


346 


365 


383 


4oi 


420 


438 


45 7 


23 7 


475 


493 




5n 


53o 


548 


566 


585 


6o3 


621 


639 


238 


658 


676 


6 9 4 


7 I2 


7 3i 


749 


767 


7 85 


8o3 


822 


239 


84o 


858 


876 


8 9 4 


912 


9 3i 


949 


9 6 7 


985 


*oo3 


240 


38 021 


039 


057 


o 7 5 


093 


112 


i3o 


1 48 


166 


1 84 


241 


202 


220 


238 


256 


2 7 4 


292 


3io 


328 


346 


364 


242 


382 


3 99 


4i 7 


435 


453 


4 7 i 


48 9 


507 


525 


543 


243 


56i 


5 7 8 


5 9 6 


6i4 


632 


650 


668 


686 


7 o3 


7 2I 


244 


7 3 9 


7 5 7 


775 


7 9 2 


810 


828 


846 


863 


881 


899 


245 


9 i 7 


934 


9 52 


970 


987 


*oo5 


*023 


*o4i 


*o58 


*o 7 6 


246 


3 9 o 9 4 


in 




I2 9 


i46 


1 64 


182 


199 


217 


235 


252 


247 


2 7 


28 7 


3o5 


322 


34o 


358 


3 7 5 


3 9 3 


4io 


428 


248 


445 


463 


48o 


498 


5i5 


533 


55o 


568 


585 


602 


249 


620 


63 7 


655 


672 


690 


707 


724 


742 


7 5 9 


777 


250 


794 


811 


820. 


846 


863 


881 


898 


9 i5 


9 33 


9 5o 


Si 


967 


985 


*002 


*oi 9 


*o37 


*o54 


*o 


7i 


*o88 


*io6 


*I23 


252 


4o i^ 





i5 7 


175 


I 9 2 


209 


226 


243 


261 


278 


295 


253 


3l2 


329 


346 


364 


38i 


398 


4i5 


432 


449 


466 


254 


483 


5oo 


5i8 


535 


552 


56 9 


586 


6o3 


620 


63 7 


255 


654 


6 7 i 




688 


705 


722 


7 3 9 


7 56 


77 3 


79 


807 


256 


824 


84i 




858 


875 


892 


909 


926 


943 


960 


976 


257 


993 


*OIO 


*027 


*o44 


*o6i 


*078 


*95 


*ii 


i 


*I28 


*i45 


258 


4i 162 


179 


i 9 6 


212 


229 


246 


263 


280 


296 


3i3 


269 


33o 


34 7 


363 


38o 


397 


4i4 


43o 


447 


464 


48 1 


260 


497 


5i4 


53i 


54 7 


564 


58i 


5 97 


6i4 


63i 


647 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


PP 


19 


18 17 16 


15 


14 


i 


T^ 


1.8 


1.7 i 1.6 


7s" 


i.4 


2 


3'.8 


3.6 


3.4 2 3.2 


3'.o 


2.8 


3 


5-7 


5.4 


5.i 3 4.8 


4.5 


4.2 


4 


7.6 


7.2 


6.8 4 6.4 


6.0 


5.6 


5 


9 .5 


9.0 


8.5 5 8.0 


7-5 


7.0 


6 


1 1. 4 


10.8 


10.2 6 9 .6 


9.0 


8.4 


7 


i3.3 


12.6 


11.9 7 ii. 2 


io.5 


9.8 


8 


15.2 


i4.4 


i3.6 8 12.8 


12.0 


I 1. 2 


9 


17.1 


16.2 i5.3 9 r4.4 


i3.5 


12.6 



260-300 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


260 


4i 497 


5i4 


53i 


547 


564 


58i 


5 97 


6i4 


63i 


64 7 


261 


664 


681 


697 


714 


7 3i 


747 


764 


780 


797 


8i4 


262 


83o 


847 


863 


880 


8 9 6 


9 i3 


9 2 9 


946 


9 63 


979 


263 


996 


*OI2 


*02 9 


*o45 


*o62 


*078 


*95 


*in 


*I2 7 


^i44 


a64 


42 1 60 


177 


i 9 3 


210 


226 


243 


2 5 9 


27 5 


2 9 2 


3o8 


265 


325 


34 1 


35 7 


3 7 4 


3 9 o 


4o6 


423 


43 9 


455 


472 


266 


488 


5o4 


521 


53 7 


553 


-570 


586 


602 


6i 9 


635 


267 


65i 


667 


684 


700 


716 


7 32 


?49 


765 


781 


797 


268 


8i3 


83o 


846 


862 


878 


8 9 4 


9 u 


9 2 7 


9 43 


9 5 9 


269 


97 5 


991 


*oo8 


*024 


*o4o 


*o56 


*O72 


*o88 


*io4 


*I20 


270 


43 i36 


152 


169 


185 


201 


217 


233 


249 


265 


28l 


271 


297 


3i3 


329 


345 


36i 


3 77 


3 9 3 


4o 9 


425 


44 1 


272 


45 7 


4 7 3 


48 9 


505 


521 


53 7 


553 


56 9 


584 


600 


27 3 


616 


632 


648 


664 


680 


6 9 6 


712 


727 


743 


7 5 9 


274 


77 5 


791 


807 


8 2 3 


838 


854 


870 


886 


9 O2 


9 i 7 


2 7 5 


9 33 


949 


965 


981 


99 6 


*OI2 


*028 


*o44 


*o5 9 


*o 7 5 


276 


44091 


107 


122 


i38 


1 54 


I 7 


i85 


2OI 


217 


232 


277 


248 


264 


279 


295 


3n 


326 


342 


358 


3 7 3 


38 9 


278 


4o4 


420 


436 


45i 


46 7 


483 


4 9 8 


5i4 


52 9 


545 


279 


56o 


5 7 6 


5 9 2 


607 


623 


638 


654 


66 9 


685 


7 oo 


280 


716 


7 3i 


747 


762 


778 


793 


8o 9 


824 


84o 


855 


281 


871 


886 


902 


9 i 7 


9 32 


948 


9 63 


979 


994 


*OIO 


282 


46025 


o4o 


o56 


071 


086 


IO2 


117 


i33 


i48 


i63 


283 


179 


I 9 4 


209 


225 


240 


2 55 


271 


286 


3oi 


3i7 


284 


332 


34 7 


362 


3 7 8 


3 9 3 


4o8 


423 


43 9 


454 


46 9 


285 


484 


500 


5'5 


53o 


545 


56i 


5 7 6 


5 9 i 


606 


621 


286 


63 7 


652 


667 


682 


6 97 


712 


728 


743 


7 58 


773 


287 


788 


8o3 


818 


834 


84 9 


864 


879 


8 9 4 


99 


9 24 


288 


9 3 9 


954 


969 


9 84 


*ooo 


*oi5 


*o3o 


*o45 


*o6o 


*075 


289 


46 090 


105 


I2O 


i3s 


150 


165 


1 80 


195 


2IO 


225 


290 


240 


255 


270 


285 


3oo 


315 


33o 


345 


35 9 


3 7 4 


291 


38 9 


4o4 


419 


434 


44 9 


464 


479 


494 


5o 9 


523 


292 


538 


553 


568 


583 


5 9 8 


6i3 


627 


642 


65 7 


6 7 2 


293 


687 


702 


716 


7 3i 


746 


761 


776 


79 


8o5 


820 


294 


83,5 


850 


864 


879 


8 9 4 


99 


923 


9 38 


9 53 


9 6 7 


296 


982 


997 


*OI2 


*O26 


*o4i 


*o56 


*O70 


*o85 


*IOO 


*n4 


296 


47 129 


1 44 


1*9 


I 7 3 


188 


202 


217 


232 


246 


261 


297 


276 


290 


305 


3i 9 


334 


34 9 


363 


3 7 8 


3 9 2 


4o 7 


298 


422 


436 


45i 


465 


48o 


4 9 4 


5o 9 


524 


538 


553 


299 


56 7 


582 


596 


611 


6 2 5 


64o 


654 


66 9 


683 


6 9 8 


300 


712 


727 


74i, 


^ 7 56 


770 


784 


799 


8i3 


828 


842 


N 





1 


a 


3 


4 


5 


6 


7 


8 


9 



3OO 33O 



N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


800 


47712 


727 


7 4i 


7 56 


77 


784 


799 


8i3 


828 


842 


3oi 


85 7 


871 


885 


9 oo 


914 


929 


943 


9 58 


972 


986 


302 


48 ooi 


oi5 


029 


o44 


o58 


073 


087 


101 


116 


i3o 


3o3 


i44 


i5 9 


I 7 3 


187 


202 


216 


230 


244 


259 


2 7 3 


3o4 


287 


302 


3i6 


33o 


344 


35 9 


3 7 3 


38 7 


4oi 


4i6 


3o5 


43o 


444 


458 


4 7 3 


48 7 


5oi 


5i5 


53o 


544 


558 


3o6 


5 7 2 


586 


601 


615 


629 


643 


65 7 


671 


686 


7 oo 


3o 7 


7 i4 


728 


742 


756 


770 


785 


799 


8i3 


827 


84i 


3o8 


855 


86 9 


883 


897 


911 


926 




954 


968 


982 


309 


996 


*OIO 


*024 


*o38 


*052 


*o66 


*o8o 


*o 9 4 


*io8 


*I22 


310 


49 1 36 


i5o 


1 64 


178 


192 


206 


220 


234 


248 


262 


3u 


276 


290 


3o4 


3i8 


332 


346 


36o 


3 7 4 


388 


4O2 


3l2 


4i5 


429 


443 


45 7 


471 


485 


499 


5i3 


527 


54i 


3i3 


554 


568 


582 


5 9 6 


610 


624 


638 


65i 


665 


6 7 9 


3i4 


693 


707 


721 


7 34 


748 


762 


776 


79 


8o3 


817 


3i5 


83i 


845 


85 9 


872 


886 


900 


914 


927 


941 


^955 


3i6 


969 


9 82 


99 6 


*OIO 


*024 


*o37 


*o5i 


*c 


,g? 


*o 79 




3i 7 


5o 106 


I2O 


i33 


i4 7 


161 


1 7 4 


188 


202 


2l5 


229 


3i8 


243 


256 


270 


284 


297 


3n 


325 


338 


352 


365 


3i 9 


3 79 


3 9 3 


4o6 


420 


433 


447 


46i 


474 


488 


5oi 


320 


5^5 


520 


542 


556 


569 


583 


5 9 6 


610 


623 


63 7 


321 


65i 


664 


678 


691 


705 


718 


7 32 


745 


7 5 9 


772 


322 


786 


799 


8i3 


826 


84o 


853 


866 


880 


8 9 3 


97 


323 


920 


934 


947 


961 


974 


987 


*OOI 


*oi4 


*028 


*o4i 


324 


5i 055 


068 


08 1 


095 


1 08 


121 


138 


i48 


162 


'75 


325 


1 88 


202 


2l5 


228 


242 


255 


268 


282 


2 9 5 


3o8 


326 


322 


335 


348 


362 


3 7 5 


388 


402 


4i5 


428 


44 1 


32 7 


455 


468 


48 1 


495 


5o8 


521 


534 


548 


56i 


5 7 4 


328 


58 7 


601 


6i4 


627 


64o 


654 


667 


680 


6 9 3 


7 o6 


329 


720 


7 33 


746 


7 5 9 


772 


786 


799. 


812 


8 2 5 


838 


330 


85i 


865 


878 


891 


904 


917 


93o 


9 43 


9 5 7 


97 


N 





1 


2 


3 


4 


5 





7 


8 t> 


PP 15 14 13 12 


11 


! 


i.5 


i.4 


i.3 


I 1.2 


i.i 


2 


S'.Q 


2!8 


*.6 


2 2.4 


2.2 


3 


4.5 


4.2 


3. 9 


3 3.6 


3.3 


4 


6.0 


5.6 


5.2 


4 4.8 


4.4 


5 


7-5 


7.0 


6.5 


5 6.0 


5.5 


6 


9.0 


8.4 


7.8 


6 7.2 


6.6 


7 


io.5 


9-8 


9.1 


7 8.4 


7-7 


8 


12. 


II. 2 


10.4 


8 9.6 


8.8 


9 


i3.5 


12.6 


11.7 


9 10.8 


9-9 



330-370 



Jg 





1 


2 


3 


4 


5 


6 


7 


8 


9 


330 


5i 85i 


865 


878 


891 


904 


917 


93o 


943 


9 5 7 


97 


33i 


9 83 


996 


*oo9 


*022 


*o35 


*o48 


*o6i 


*75 


*o88 


*IOI 


33 2 


52 u4 


127 


i4o 


i53 


166 


179 


192 


2O5 


218 


23l 


333 


244 


257 


270 


284 


297 


3io 


323 


336 


34 9 


362 


334 


375 


388 


4oi 


4i4 


427 


44o 


453 


466 


479 


492 


335 


5o4 


5i 7 


53o 


543 


556 


569 


58 2 


5 9 5 


608 


621 


336 


634 


64? 


660 


6 7 3 


686 


699 


711 


724 


737 


7 5o 


33 7 


7 63 


776 


789 


802 


815 


827 


84o 


853 


866 


879 


338 


892 


95 


917 


930 


943 


956 


969 


9 82 


994 


*OO 7 


33 9 


53 020 


o33 


o46 


o58 


071 


o84 


097 


I IO 


122 


i35 


340 


i48 


161 


i 7 3 


186 


199 


212 


224 


2 3 7 


250 


263 


34i 


275 


288 


3oi 


3i4 


326 


33 9 


352 


364 


3 77 


3 9 o 


342 


4o3 


4i5 


428 


44 1 


453 


466 


479 


4 9 i 


5o4 


5i 7 


343 


52 9 


542 


555 


56 7 


58o 


5 9 3 


6o5 


618 


63i 


643 


344 


656 


668 


681 


6 9 4 


706 


719 


7 3 2 


744 


7 5 7 


7 6 9 


345 


782 


794 


807 


820 


832 


845 


85 7 


870 


882 


8 9 5 


346 


908 


920 


9 33 


945 


9 58 


97 


9 83 


99 5 


*oo8 


*O2O 


34 7 


54o33 


o45 


o58 


070 


o83 


o 9 5 


1 08 


120 


i33 


i45 


348 


1 58 


170 


i83 


i 9 5 


208 


220 


233 


245 


258 


2 7 


349 


283 


295 


3o 7 


320 


332 


345 


35 7 


370 


382 


3 9 4 


350 


407 


419 


43 2 


444 


456 


46 9 


48 1 


4 9 4 


5o6 


5i8 


35i 


53i 


543 


555 


568 


58o 


5 9 3 


605 


617 


63o 


642 


352 


654 


667 


679 


691 


7 o4 


716 


728 


7 4 1 


7 53 


7 65 


353 


111 


79 


802 


8i4 


82 7 


83 9 


85i 


864 


8 7 6 


888 


354 


900 


9 i3 


9 2 5 


937 


949 


9 62 


974 


9 86 


998 


*OII 


355 


55 023 


o35 


047 


060 


O 7 2 


o84 


096 


108 


121 


i33 


356 


i45 


i5 7 


169 


182 


194 


206 


218 


230 


242 


255 


35 7 


267 


279 


291 


3o3 


3i5 


328 


34o 


352 


364 


3 7 6 


358 


388 


4oo 


4i3 


425 


43 7 


44 9 


46i 


473 


485 


497 


35 9 


5og 


522 


534 


546 


558 


5 7 o 


582 


5 9 4 


606 


618 


360 


63o 


642 


654 


666 


6 7 8 


691 


7 o3 


7i5 


727 


7 3 9 


36i 


7 5i 


7 63 


775 


787 


799 


8n 


823 


835 


847 


85 9 


362 


871 


883 


895 


907 


919 


9 3i 


943 


955 


9 6 7 


979 


363 


991 


*oo3 


*oi5 


*027 


*o38 


*o5o 


*062 


*o 7 4 


*o86 


*o 9 8 


364 


56 no 


122 


1 34 


i46 


i58 


170 


182 


194 


2O5 


217 


365 


229 


24 1 


253 


265 


277 


289 


3oi 


3l2 


324 


336 


366 


348 


36o 


3 7 2 


384 


3 9 6 


407 


419 


43i 


443 


455 


367 


46 7 


478 


490 


502 


5i4 


5 2 6 


538 


549 


56i 


5 7 3 


368 


585 


5 97 


608 


620 


632 


644 


656 


667 


6 79 


691 


36 9 


7o3 


7i4 


7 26 


738 


750 


761 


773 


785 


797 


808 


370 


820 


832 


844 


855 


867 


879 


891 


902 


9 i4 


926 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



37O-4OO 



N 





1 


2 


3 


4 


5 





7 


8 


9 


370 


56 820 


832 


844 


855 


867 


879 


891 


902 


914 


926 


3 7 i 


9 3 7 


9 4 9 


961 


972 


984 


996 


*oo8 


*oi 9 


*o3i 


*o43 


3 7 2 


5 7 o54 


066 


078 


089 


101 


u3 


124 


1 36 


1 48 


i5 9 


3 7 3 


171 


i83 


1 9 4 


206 


217 


229 


241 


252 


264 


276 


3 7 4 


287 


2 99 


3io 


322 


334 


345 


357 


368 


38o 


392 


3 7 5 


4o3 




426 438 


44 9 


46 1 


4 7 3 


484 


496 


5o 7 


3 7 6 


5i 9 


53o 


542 


553 


565 


5 7 6 


588 


600 


611 


623 


377 


634 


646 


65 7 669 


680 


692 


7 o3 


715 


726 


7 38 


3 7 8 
379 


74 9 
864 


7 6i 

8 7 5 


887 


784 
898 


79 5 
910 


807 
921 


818 
933 


83o 
944 


84i 
9 55 


852 
967 


380 


97 8 


99 


*OOI 


*oi3 


*024 


*o35 


*o4 7 


*o58 


*O 7 O 


*o8i 


38i 


58 o 9 2 


io4 


n5 


127 


1 38 


149 


161 


172 


1 84 


J 95 


382 


206 


218 


229 


240 


252 


263 


274 


286 


2 97 


309 


383 


320 


33i 


343 


354 


365 


377 


388 


3 99 


4io 


422 


384 


433 


444 


456 


46 7 


478 


490 


5oi 


5l2 


524 


535 


385 


546 


55 7 


56 9 


58o 


5 9 i 


602 


6i4 


625 


636 


647 


386 


65 9 


6 7 o 


681 


692 


704 


715 


726 


737 


749 


760 


38 7 


77 i 


782 


794 


805 


816 


827 


838 


850 


861 


872 


388 


883 


8 9 4 


906 


917 


928 


9 3 9 


95o 


961 


973 


984 


38 9 


995 


*oo6 


*oi7 


*028 


*o4o 


*o5i 


*062 


*7 3 


*o84 


*095 


390 


5 9 1 06 


118 


129 


i4o 


i5i 


162 


i 7 3 


1 84 


i 9 5 


207 


3 9 i 


218 


229 


240 


25l 


262 


2 7 3 


284 


295 


3o6 


3i8 


392 


32 9 


34o 


35i 


362 


3 7 3 


384 


3 9 5 


4o6 


4i 7 


428 


3 9 3 


43 9 


45o 


46i 


472 


483 


494 


5o6 


5i 7 


5 2 8 


53 9 


3g4 


550 


56r 


5 7 2 


583 


594 


605 


616 


62 7 


638 


64g 


3 9 5 


660 


671 


682 


693 


7 64 


7i| 


> 


726 


7 3 7 


748 


759 


396 


77 


780 


791 


802, 


8i3 


824 


835 


846 


85 7 


868 


397 


879 


890 


901 


912 


923 


934 


945 


9 56 


966 


977 


398 


9 88 


999 


*OIO 


*O2I 


*032 


*o43 


*o54 


"065 


*o 7 6 


*o86 


399 


60 o 97 


1 08 


119 


i3o 


i4i 


l52 


i63 


I 7 3 


1 84 


i 9 5 


400 


206 


217 


228 


239 


249 


260 


271 


282 


293 


3o4 


N 





1 


2 


3 


4 


5 





7 


8 


9 


PP 12 11 


10 9 


I 


1.2 1,1 I 


i.o 0.9 


2 


2.4 2.2 2 


2,0 1.8 


3 


3.6 3.3 3 


3.o 2.7 


4 


4.8 4.4 4 


4.o 3.6 


5 


6.0 5.5 5 


5.o 4.5 


6 


7.2 6.6 6 


6.0 5.4 


7 


8.4 7-7 7 


7.0 6.3 


8 


9.6 8.8 8 


8.0 7.2 


9 


10.8 9.9 9 


9.0 8.1 



10 



400-440 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


400 


60 206 


217 


228 


239 


249 


260 


271 


282 


293 


3o4 


4oi 


3i4 


3 2 5 


336 


34 7 


358 


36 9 


379 


390 


4oi 


412 


402 


423 


433 


444 


455 


466 


477 


48 7 


498 


509 


520 


4o3 


53i 


54i 


55 2 


563 


5 7 4 


584 


5 9 5 


606 


617 


62 7 


4o4 


638 


649 


660 


670 


681 


6 9 2 


7o3 


7 i3 


7 24 


735 


4o5 


746 


7 56 


767 


778 


788 


799 


810 


821 


83i 


842 


4o6 


853 


863 


874 


885 


895 


906 


917 


927 


9 38 


949 


4oy 


9 5 9 


970 


981 


991 


*OO2 


*oi3 


*023 


*o34 


*o45 


*o55 


4o8 


6 1 066 


077 


087 


098 


I0 9 


119 


i3o 


i4o 


i5i 


162 


409 


172 


i83 


194 


204 


215 


225 


236 


247 


25 7 


268 


410 


278 


289 


3oo 


3io 


321 


33i 


342 


352 


363 


3 7 4 


4n 


384 


395 


4o5 


4i6 


426 


43 7 


448 


458 


46 9 


4 7 9 


412 


4go 


5oo 


5n 


621 


532 


542 


553 


563 


5 7 4 


584 


4i3 


5,5 


606 


616 


627 


63 7 


648 


658 


669 


6 79 


690 


4i4 


7OO 


711 


721 


7 3i 


742 


752 


7 63 


77 3 


7 84 


794 


4i5 


805 


8i5 


826 


836 


847 


85 7 


868 


8 7 8 


888 


899 


4i6 


90 9 


920 


93o 


94i 


9 5i 


962 


972 


982 


993 


*oo3 


4i 7 


62 oi4 


024 


o34 


045 


o5'5 


066 


076 


086 


097 


I0 7 


4i8 


118 


128 


i38 


149 


i5 9 


170 


1 80 


190 


2OI 


211 


419 


221 


232 


242 


252 


263 


2 7 3 


284 


294 


3o4 


315 


420 


325 


335 


346 


356 


366 


377 


387 


397 


4o8 


4i8 


421 


428 


43 9 


449 


45 9 


46 9 


48o 


490 


5oo 


5n 


521 


422 


53i 


542 


55 2 


56 2 


572 


583 


5 9 3 


6o3 


6i3 


624 


423 


634 


644 


655 


665 


6 7 5 


685 


696 


7 o6 


7 i6 


7 26 


424 


?3? 


747 


7 5 7 


767 


778 


788 


798 


808 


818 


829 


426 


83 9 


849 


85 9 


870 


880 


890 


900 


910 


921 


9 3i 


426 


941 


9 5i 


961 


972 


9 82 


992 


*OO2 


*OI2 


*O22 


*o33 


427 


63 o43 


o53 


o63 


073 


o83 


094 


io4 


u4 


124 


1 34 


428 


i44 


155 


165 


175 


185 


i 9 5 


205 


2l5 


225 


236 


429 


246 


256 


266 


276 


286 


2 9 6 


3o6 


3i 7 


32 7 


33 7 


430 


34 7 


35 7 


36 7 


3 77 


38 7 


397 


407 


4i 7 


428 


438 


43i 


448 


458 


468 


478 


488 


498 


5o8 


5i8 


528 


538 


432 


548 


558 


568 


5 79 


58 9 


5 99 


669 


619 


629 


63 9 


433 


649 


65 9 


669 


679 


68 9 


699 


709 


7i9 


729 


73 9 


434 


749 


7 5 9 


769 


779 


7 8 9 


799 


809 


819 


829 


839 


435 


849 


859 


869 


879 


88 9 


899 


909 


919 


9 2 9 


9 3 9 


436 


949 


9 5 9 


969 


979 


9 88 


998 


*oo8 


*oi8 


*028 


*o38 


43 7 


64o48 


o58 


068 


078 


088 


098 


108 


118 


128 


i3 7 


438 


i4 7 


5 7 


167 


177 


187 


197 


207 


217 


22 7 


2 3 7 


43 9 


246 


256 


266 


276 


286 


296 


3o6 


3x6 


326 


335 


440 


345 


355 


365 


375 


385 


3 9 5 


4o4 


4i4 


424 


434 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



440-470 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


440 


64345 


355 


365 


375 


385 


395 


4o4 


4i4 


424 


434 


44 1 


444 


454 


464 


4 7 3 


483 


493 


5o3 


5i3 


5 2 3 


53 2 


442 


542 


552 


562 


5 7 2 


582 


5oi 


601 


611 


621 


63i 


443 


64o 


65o 


660 


670 


680 


689 


699 


79 


719 


729 


444 


738 


748 


7 58 


768 


777 


787 


797 


8o 7 


816 


826 


445 


836 


846 


856 


865 


8 7 5 


885 


895 


904 


914 


924 


446 


9 33 


943 


953 


9 63 


972 


982 


99 2 


*002 


*OII 


*02I 


44? 


65o3i 


o4o 


o5o 


060 


070 


079 


089 


099 


108 


118 


448 


128 


i3 7 


i4 7 


i5 7 


167 


176 


186 


196 


2O5 


215 


449 


225 


234 


244 


254 


263 


2 7 3 


283 


292 


302 


3l2 


450 


321 


33i 


34 1 


35o 


36o 


36 9 


3 79 


38 9 


3 9 8 


4o8 


45i 


4i8 


427 


43 7 


447 


456 


466 


4 7 5 


485 


495 


5o4 


45 2 


5i4 


523 


533 


543 


552 


562 


5 7 i 


58i 


5 9 i 


600 


453 


610 


619 


629" 


63 9 


648 


658 


66 7 


677 


686 


696 


454 


.706 


7 i5 


725 


734 


744 


7 53 


7 63 


772 


782 


792 


455 


801 


811 


820 


83o 


839 


849 


858 


868 


877 


887 


456 


896 


906 


916 


925 


935 


944 


954 


9 63 


973 


982 


457 


992 


*OOI 


*OII 


*O2O 


*o3o 


*o3 9 


*o49 


*o58 


*o68 


*77 


458 


66087 


096 


106 


"5 


124 


1 34 


i43 


i53 


162 


172 


45 9 


181 


191 


200 


2IO 


219 


229 


2 38 


247 


25 7 


266 


460 


276 


285 


295 


3o4 


3i4 


323 


332 


342 


35i 


36i 


46 1 


3 7 o 


38o 


389 


398 


4o8 


4i 7 


4a 7 


436 


445 


455 


462 


464 


474 


483 


492 


5O2 


5u 


621 


53o 


53 9 


549 


463 


558 


567 


577 


586 


5 9 6 


605 


6i4 


624 


633 


642 


464 


65 2 


661 


671 


680 


689 


699 


708 


717 


7 2 7 


7 36 


465 


745 


755 


764 


773 


7 83 


792 


801 


811 


820 


829 


466 


839 


848 


85 7 


867 


876 


885 


8 9 4 


904 


9 i3 


922 


46 7 


932 


9 4i 


95o 


960 


969 


978 


987 


997 


*oo6 


*oi5 


468 


67025 


o34 


o43 


o52 


062 


071 


080 


089 


099 


1 08 


46 9 


117 


127 


i36 


i45 


1 54 


1 64 


I 7 3 


182 


191 


201 


470 


2IO 


219 


228 


23 7 


247 


256 


265 


2 7 4 


284 


293 


N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


PP 10 9 8 


i 


I.O 


i 


0.9 


i 0.8 


2 


2.0 


2 


1.8 


2 1.6 


3 


3.o 


3 


2.7 


3 2.4 


4 


4.o 


4 


3.6 


4 3.2 


5 


5.o 


5 


4.5 


5 4.0 


6 


6.0 


6 


5.4 


6 4.8 


7 


7.0 


7 


6.3 


7 5.6 


8 


8.0 


8 


7.2 


8 6.4 


9 


9.0 


9 


8.1 


9 7-2 



47O-51O 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


470 


67 2IO 


219 


228 


"^ 


*47 


256 


265 


2 7 4 


284 


2 9 3 


4?i 


302 


3ii 


321 


33o 


33 9 


348 


35 7 


36 7 


3 7 6 


385 


472 


394 


4o3 


4i3 


422 


43i 


44o 


449 


45 9 


468 


477 


4 7 3 


486 


495 


5o4 


fri4 


5 2 3 


532 


54i 


55o 


56o 


56 9 


4?4 


5 7 8 


58 7 


5 9 6 


6o5 


6i4 


624 


633 


642 


65i 


660 


4 7 5 
4?6 


669 
761 


679 
77 


688 
7 7 9 


788 


706 
797 


7 i5 
806 


724 
8i5 


7 33 
825 


742 
834 


752 
843 


477 


85a 


861 


8 7 o 


8 79 


888 


8 97 


906 


916 


925 


934 


478 


943 


952 


961 


9 7 o 


979 


988 


997 


*oo6 


*oi5 


*024 


479 


68o34 


o43 


052 


06 1 


070 


o 7 9 


088 


o 97 


1 06 


n5 


480 


124 


i33 


142 


i5i 


1 60 


169 


178 


187 


196 


205 


48 1 


215 


224 


233 


242 


25l 


260 


269 


278 


287 


296 


482 


3o5 


3i4 


3 2 3 


332 


34i 


350 


35 9 


368 


3 77 


386 


483 


395 


4o4 


4i3 


422 


43i 


44o 


449 


458 


46 7 


476 


484 


485 


494 


502 


5u 


520 


529 


538 


54 7 


556 


565 


485 


5 7 4 


583 


592 


601 


610 


619 


628 


63 7 


646 


655 


486 


664 


6 7 3 


68 1 


690 


699 


708 


7 i 7 


726 


7 35 


744 


48 7 


7 53 


762 


77 i 


7 8o 


789 


7 9 7 


806 


8i5 


824 


833 


488 


842 


85i 


860 


869 


8 7 8 


886 


895 


904 


9 i3 


922 


48 9 


93i 


940 


949 


958 


966 


975 


984 


993 


*002 


*on 


490 


69 020 


028 


o3 7 


o46 


o55 


o64 


o 7 3 


082 


090 


099 


491 


1 08 


117 


126 


i3fi 


1 44 


l52 


161 


I 7 


I 79 


188 


492 


197 


2O5 


2l4 


223 


232 


241 


249 


258 


267 


276 


493 


285 


294 


302 


3u 


320 


329 


338 


346 


355 


364 


494 


3 7 3 


38i 


390 


3 99 


4o8 


417 


425 


434 


443 


452 


4g5 


46 1 


46 9 


478 


48 7 


496 


5o4 


5i3 


522 


53i 


53 9 


496 


548 


55 7 


566 


5 7 4 


583 


592 


601 


609 


618 


627 


497 


636 


644 


653 


662 


671 


679 


688 


697 


7o5 


7 i4 


498 


7 23 


7 32 


740 


7 49 


7 58 


767 


77 5 


784 


793 


801 


499 


810 


819 


827 


836 


845 


854 


862 


8 7 I 


880 


888 


500 


897 


906 


9i4 


923 


932 


94o 


949 


9 58 


966 


97 5 


5oi 


984 


992 


*OOI 


*OIO 


*oi8 


*O27 


*o36 


*o44 


*o53 


*062 


5O2 


70 070 


079 


088 


096 


105 


n4 


122 


i3i 


i4o 


i48 


5o3 


i5 7 


i65 


174 


i83 


191 


200 


209 


217 


226 


234 


5o4 


243 


252 


260 


269 


278 


286 


295 


3o3 


3l2 


321 


5o5 


329 


338 


346 


355 


364 


3 7 2 


38i 


38 9 


398 


4o6 


5o6 


4i5 


424 


432 


44 1 


449 


458 


46 7 


4 7 5 


484 


4 9 2 


607 


5oi 


Sog 


5i8 


5 2 6 


535 


544 


552 


56i 


56 9 


5 7 8 


5o8 


586 


595 


6o3 


612 


621 


629 


638 


646 


655 


663 


609 


672 


680 


689 


697 


706 


7 i4 


72 3 


7 3i 


74o 


74 9 


510 


7 5 7 


766 


774 


7 83 


791 


800 


808 


817 


8 2 5 


834 


N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 



i3 



510-540 



N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


510 


70767 


766 


774 


7 83 


791 


800 


8o 


817 


825 


34 


5n 


842 


85i 


85 9 


868 


876 


885 


8 9 3 


9 O2 


9 io 


9 i 9 


5l2 


9 2 7 


935 


944 


9 52 


961 


969 


97 8 


9 86 


995 


*oo3 


5x3 


71 012 


020 


029 


037 


o46 


o54 


o63 


071 


079 


088 


5x4 


o 9 6 


105 


n3 


122 


i3o 


i3 9 


i4 7 


i55 


1 64 


172 


5x5 


181 


189 


198 


2O6 


2l4 


223 


23l 


240 


248 


2 5 7 


5i6 


265 


2 7 3. 


282 


2 9 O 


299 


3o 7 


3i5 


324 


332 


34i 


5x 7 


34 9 


35 7 


366 


3 7 4 


383 


3 9 i 


3 99 


4o8 


4i6 


425 


5x8 


433 


44 1 


450 


458 


466 


475 


483 


4 9 2 


500 


5o8 


5i 9 


5x7 


525 


533 


542 


55o 


55 9 


56 7 


5 7 5 


584 


5 9 2 


520 


600 


609 


617 


625 


634 


642 


65o 


65 9 


667 


6 7 5 


521 


684 


692 


700 


7<>9 


717 


7 25 


7 34 


7 42 


7 5o 


7 5 9 


522 


767 


77 5 


784 


79 2 


800 


8o 9 


817 


8 2 5 


834 


842 


523 


85o 


858 


867 


8 7 5 


883 


8 9 2 


9 oo 


9 o8 


917 


9 2 5 


524 


9 33 


941 


950 


9 58 


966 


975 


9 83 


99 i 


999 


*oo8 


5 2 5 


72 016 


024 


032 


o4i 


049 


067 


066 


o 7 4 


082 


o 9 o 


526 


99 


107 


ix5 


123 


132 


i4o 


1 48 


i56 


165 


i 7 3 


52 7 


181 


189 


198 


206 


214 


222 


230 


2 3 9 


247 


255 


5 2 8 


263 


272 


280 


288 


296 


3o4 


3i3 


321 


329 


33 7 


52 9 


346 


354 


362 


370 


3 7 8 


38 7 


395 


4o3 


4u 


4i 9 


580 


428 


436 


444 


452 


46o 


46 9 


477 


485 


493 


5oi 


53i 


5o 9 


5i8 


526 


534 


542 


55o 


558 


56 7 


575 


583 


532 


5 9 i 


5 99 


607 


616 


624 


632 


64o 


648 


656 


665 


533 


673 


681 


689 


697 


7 o5 


7 i3 


722 


7 3o 


7 38 


746 


534 


7 54 


762 


770 


779 


787 


795 


8o3 


811 


819 


827 


535 


835 


843 


85 2 


860 


868 


8 7 6 


884 


8 9 2 


900 


9 o8 


536 


916 


925 


9 33 


94 1 


949 


9 5 7 


9 65 


973 


981 


989 


53 7 


997 


*oo6 


*oi4 


*O22 


*o3o 


*o38 


*o46 


*o54 


*062 


*070 


538 


7 3o 7 8 


086 


o 9 4 


IO2 


in 


u 9 


127 


135 


i43 


i5i 


53 9 


i5 9 


167 


IT! 


r 

> 


i83 


191 


i 99 


207 


2l5 


223 


23l 


540 


239 


247 


255 


263 


272 


280 


288 


296 


3o4 


3l2 


N 


O 


1 


2 


3 


4 


5 


O 


7 


8 


9 


PP 9 


8 7 


i 


0.9 


j 


0.8 


i 


0.7 


2 


1.8 


2 


1.6 


2 


i .4 


3 


2-7 


3 


2.4 


3 


2.1 


4 


3.6 


4 


3.2 


4 


2.8 


5 


4.5 


5 


4.o 


5 


3.5 


6 


5.4 


6 


4.8 


6 


4.2 


7 


6.3 


7 


5.6 


7 


4.9 


8 


7-2 


8 


6.4 


8 


5.6 


9 


8.1 


9 


7.2 


9 


6.3 



i4 



54O 58O 



N 


1) 


1 


15 


3 


4 


5 


6 


7 


8 


9 


540 


7 3 2 3 9 


24 7 


255 


263 


2 7 2 


280 


288 


296 


3o4 


3l2 


54i 


32O 


3 2 8 


336 


344 


352 


36o 


368 


3 7 6 


384 


3 9 2 


542 


4oo 


4o8 


4i6 


424 


432 


44o 


448 


456 


464 


472 


543 


48o 


488 


496 


5o4 


5l2 


520 


528 


536 


544 


552 


544 


56o 


568 


5 7 6 


584 


592 


600 


608 


616 


624 


632 


545 


64o 


648 


656 


664 


6 7 2 


679 


687 


6 9 5 


7 o3 


711 


546 


719 


727 


7 35 


743 


7 5i 


7 5 9 


767 


775 


7 83 


79 i 


54 7 


799 


807 


815 


823 


83o 


838 


846 


854 


862 


870 


548 


8 7 8 


886 


8 9 4 


902 


910 


918 


926 


9 33 


9 4 1 


9 4 9 


549 


9 5 7 


9 65 


973 


981 


989 


997 


*oo5 


*oi3 


*O2O 


*028 


550 


7 4o36 


o44 


o5 2 


060 


068 


o 7 6 


o84 


092 


99 


107 


55i 


n5 


123 


i3i 


i3 9 


i4 7 


155 


162 


170 


I 7 8 


186 


55a 


194 


202 


2IO 


218 


225 


233 


24 1 


249 


257 


265 


553 


27 3 


280 


288 


296 


3o4 


3l2 


320 


32 7 


335 


343 


554 


35i 


35 9 


36 7 


3 7 4 


382 


390 


3 9 8 


4o6 


4i4 


421 


555 


429 


43 7 


445 


453 


46i 


468 


476 


484 


492 


500 


556 


5o 7 


5i5 


523 


53i 


53 9 


54 7 


554 


562 


5 7 o 


5 7 8 


55 7 


586 


593 


601 


609 


617 


624 


632 


64o 


648 


656 


558 


663 


671 


679 


68 7 


695 


7 02 


710 


7 i8> 


726 


7 33 


55 9 


7 4i 


749 


7 5 7 


7 64 


772 


7 80 


788 


79 6 


8o3 


811 


560 


819 


827 


834 


842 


850 


85.8 


865 


8 7 3 


881 


88 9 


56i 


896 


904 


912 


920 


927 


935 


9 43 


9 5o 


9 58 


966 


56 2 


9 7 4 


981 


989 


997 


*oc>5 


*OI2 


*O2O 


*028 


*o35 


*o43 


563 


7 5 o5i 


oSg 


066 


o 7 4 


082 


089 


97 


105 


u3 


120 


564 


128 


1 36 


i43 


i5i 


i5 9 


166 


I 7 4 


182 


i8 9 


I 97 


565 


205 


2l3 


220 


228 


236 


243 


25l 


269 


266 


2 7 4 


566 


282 


289 


297 


3o5 


3l2 


320 


328 


335 


343 


35i 


56 7 


358 


366 


3 7 4 


38i 


38 9 


397 


4o4 


412 


420 


42 7 


568 


435 


442 


45d 


458 


465 


4 7 3 


48 1 


488 


4 9 6 


5o4 


569 


5n 


5i 9 


526. 

-* 


534 


542 


549 


55 7 


565 


5 7 2 


58o 


570 


58 7 


5 9 5 


6o3 


610 


618 


626 


633 


64 1 


648 


656 


5 7 i 


664 


671 


679 


686 


694 


702 


709 


717 


724 


7 32 


5 7 2 


7 4o 


747 


755 


762 


77<> 


778 


7 85 


793 


800 


808 


5 7 3 


8i5 


823 


83i 


838 


846 


853 


861 


868 


876 


884 


5 7 4 


891 


899 


906 


914 


921 


929 


9 3 7 


944 


9 52 


9 5 9 


5 7 5 


96-7 


974 


982 


989 


997 


*oo5 


*OI2 


*O2O 


*O2 7 


*o35 


5 7 6 


7 6 042 


050 


057 


065 


7 2 


080 


08 7 


095 


io3 


no 


5 77 


118 


125 


i33 


i4o 


i48 


i55 


i63 


170 


178 


i85 


5 7 8 


193 


200 


208 


2l5 


223 


230 


2 38 


245 


253 


260 


579 


268 


275 


2 83 


290 


298 


3o5 


3i3 


320 


328 


335 


580 


343 


35o 


358 


365 


3 7 3 


38o 


388 


3 9 5 


4o3 


4io 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



i5 



580-61O 



N 





1 


a 


3 


4 


5 


6 


7 


8 


9 


580 


7 6343 


35o 


358 


365 


3 7 3 


38o 


388 


3 9 5 


4o3 


4io 


58i 


4i8 


425 


433 


44o 


448 


455 


462 


4 7 o 


4 77 


485 


582 


4 9 2 


500 


5o 7 


5^5 


522 


53o 


53 7 


545 


55 2 


55 9 


583 


56 7 


5 7 4 


582 


58 9 


5 97 


6o4 


612 


6i 9 


626 


634 


584 


64 1 


64 9 


656 


664 


671 


678 


686 


693 


7 OI 


7 o8 


585 


7 i6 


7 23 


73( 


> 


7 38 


745 


7 53 


7 6o 


768 


775 


782 


586 


79 


797 


805 


812 


8i 9 


827 


834 


842 


84 9 


856 


58 7 


864 


871 


879 


886 


8 9 3 


9 oi 


9 o8 


916 


9 23 


9 3o 


588 


9 38 


945 


9 53 


9 6o 


9 6 7 


975 


9 82 


9 8 9 


997 


*oo4 


58 9 


77 OI2 


019 


026 


o34 


o4i 


o48 


o56 


o63 


O 7 O 


078 


590 


o85 


o 9 3 


IOO 


107 


"5 


122 


I2 9 


i3 7 


i44 


i5i 


5 9 i 


i5 9 


166 


I 7 3 


181 


188 


i 9 5 


203 


2IO 


2I 7 


225 


602 


232 


240 


247 


254 


262 


26 9 


276 


283 


2 9 I 


2 9 8 


5 9 3 


3o5 


3i3 


320 


327 


335 


342 


34 9 


35 7 


364 


3 7 i 


5 9 4 


3 79 


386 


3 9 3 


4oi 


4o8 


4i5 


422 


43o 


43 7 


444 


5 9 5 


452 


45 9 


466 


474 


48i 


488 


4 9 5 


5o3 


5io 


5i 7 


5 9 6 


525 


532 


53 9 


546 


554 


56i 


568 


5 7 6 


583 


5 9 o 


5 97 


5 97 


605 


612 


6i 9 


627 


634 


64 1 


648 


656 


663 


5 9 8 


670 


677 


685 


6 9 2 


6 99 


7 o6 


7 i4 


721 


7 28 


7 35 


5 99 


743 


750 


7 5 7 


764 


772 


779 


786 


793 


801 


808 


600 


8i5 


822 


83o 


83 7 


844 


85i 


85 9 


866 


8 7 3 


880 


601 


887 


895 


902 


99 


9 i6 


924 


9 3i 


9 38 


9 45 


9 52 


602 


9 6o 


9 6 7 


974 


98 1 


9 88 


99 6 


*oo3 


*OIO 


*OI 7 


*025 


6o3 


78 o32 


o3 9 


o46 


o53 


06 1 


068 


075 


082 


089 


97 


6o4 


io4 


in 


118 


125 


132 


i4o 


i4 7 


1 54 


161 


168 


6o5 


176 


i83 


190 


i 97 


204 


211 


219 


226 


233 


240 


606 


24 7 


254 


262 


269 


276 


283 


290 


2 97 


305 


3l2 


6o 7 


3i 9 


326 


333 


34o 


34 7 


355 


362 


36 9 


3 7 6 


383 


608 


3 9 o 


3 9 8 


4os 




412 


4u; 


426 


433 


44o 


44 7 


455 


6o 9 


462 


46 9 


4 7 e 


i 


483 


4 9 o 


497 


5o4 


5l2 


5i 9 


5 2 6 


610 


533 


54o 


54 7 


554 


56i 


56 9 


5 7 6 


583 


5 9 o 


5 97 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


PP 8 


7 6 


i 


0.8 


i 


0.7 


i 


0.6 


2 


1.6 


2 


i.4 


2 


1.2 


3 


2.4 


3 


2.1 


3 


1.8 


4 


3.2 


4 


2.8 


4 


2.4 


5 


4.o 


5 


3.5 


5 


3.o 


6 


4.8 


6 


4.2 


6 


3.6 


7 


5.6 


7 


4-9 


7 


4.2 


8 


6.4 


8 


5.6 


8 


4.8 


9 


7-2 


9 


6.3 


9 


5.4 



16 



010-650 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


610 


78 533 


54o 


54? 


554 


56i 


56 9 


5 7 6 


583 


5 9 o 


5 97 


611 


6o4 


611 


618 


625 


633 


64o 


647 


654 


661 


668 


612 


6 7 5 


682 


689 


696 


704 


711 


718 


7 2 5 


7 32 


73 9 


6i3 


746 


7 53 


760 


767 


774 


781 


789 


796 


8o3 


810 


6i4 


817 


824 


83i 


838 


845 


852 


85 9 


866 


873 


880 


6i5 


888 


895 


902 


909 


916 


923 


93o 


937 


944 


9 5i 


616 


9 58 


9 65 


972 


979 


986 


99 3 


*ooo 


*oo 7 


*oi4 


*O2I 


617 


79029 


o36 


o43 


050 


057 


o64 


071 


078 


085 


O 9 2 


618 


099 


1 06 


n3 


I2O 


127 


1 34 


i4i 


1 48 


i55 


l62 


619 


169 


176 


i83 


190 


197 


204 


211 


218 


225 


232 


620 


239 


246 


253 


260 


267 


274 


28l 


288 


295 


302 


621 


309 


3i6 


323 


33o 


33 7 


344 


35i 


358 


365 


3 7 2 


622 


379 


386 


3 9 3 


4oo 


407 


4i4 


421 


428 


43s 


442 


623 


449 


456 


463 


470 


477 


484 


491 


498 


505 


5u 


624 


5i8 


5 2 5 


532 


539 


546 


553 


56o 


56 7 


5 7 4 


58i 


626 


588 


595 


602 


609 


616 


6 2 3 


63o 


63 7 


644 


65o 


626 


65 7 


664 


671 


678 


685 


692 


699 


706 


7 i3 


720 


627 


727 


7 34 


74i 


748 


7 54 


761 


768 


77 5 


782 


7 8 9 


628 


796 


8o3 


810 


817 


824 


83i 


83 7 


844 


85i 


858 


629 


865 


872 


879 


886 


893 


900 


906 


9 i3 


920 


9 2 7 


630 


934 


941 


948 


955 


962 


969 


975 


982 


989 


996 


63i 


8ooo3 


OIO 


017 


024 


o3o 


o3 7 


o44 


o5i 


o58 


065 


632 


072 


079 


o85 


092 


099 


1 06 


n3 


120 


127 


1 34 


633 


i4o 


i4 7 


1 54 


161 


168 


175 


182 


1 88 


i 9 5 


202 


634 


209 


216 


223 


229 


236 


243 


25o 


25 7 


264 


271 


635 


277 


284 


291 


298 


3o5 


312 


3i8 


325 


332 


33 9 


636 


346 


353 


35 9 


366 


3 7 3 


38o 


38 7 


3 9 3 


4oo 


407 


63 7 


4i4 


421 


428 


434 


44 1 


448 


455 


462 


468 


4 7 5 


638 


482 


48 9 


496 


5O2 


509 


5i6 


523 


53o 


536 


543 


639 


55o 


55 7 


564 


5 7 o 


5 77 


584 


591 


5 9 8 


6o4 


611 


640 


618 


625 


632 


638 


645 


65 2 


65 9 


665 


672 


6 79 


64 1 


686 


6 9 3 


699 


706 


7 i3 


720 


726 


7 33 


740 


74 7 


642 


7 54 


760 


767 


774 


781 


787 


794 


801 


808 


8i4 


643 


821 


828 


835 


84i 


848 


855 


862 


868 


8 7 5 


882 


644 


889 


8 9 5 


902 


909 


916 


922 


929 


9 36 


9 43 


949 


645 


9 56 


g63 


969 


976 


983 


990 


996 


*oo3 


*OIO 


*OI 7 


646 


81 023 


o3o 


o3 7 


o43 


o5o 


057 


o64 


070 


077 


o84 


647 


090 


097 


io4 


1 1 1 


117 


124 


i3i 


i3 7 


1 44 


i5i 


648 


i58 


1 64 


171 


178 


i84 


191 


198 


204 


211 


218 


649 


224 


23l 


238 


245 


25l 


258 


265 


271 


2 7 8 


285 


650 


291 


298 


3o5 


3u 


3i8 


325 


33i 


338 


345 


35i 


N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 



650-680 



N 





1 


ii 


3 


4 


5 


a 


7 


8 !> 


650 


81 291 


298 


305 


3u 


3i8 


325 


33i 


338 


345 


35i 


65i 


358 


365 


3 7 i 


3 7 8 


385 


3 9 i 


3 9 8 


405 


4n 


4i8 


662 


425 


43i 


438 


445 


45i 


458 


465 


4 7 i 


478 


485 


653 


491 


498 


505 


5n 


5i8 


525 


53i 


538 


544 


55i 


654 


558 


564 


5 7 i 


5 7 8 


584 


5 9 i 


5 9 8 


6o4 


6n 


6i 7 


655 


624 


63i 


63 7 


644 


65i 


65 7 


664 


671 


677 


684 


656 


690 


697 


7 o4 


7 IO 


7 i 7 


7 23 


7 3o 


7 3 7 


743 


750 


65 7 


7 5 7 


7 63 


77 


776 


7 83 


79 


796 


8o3 


809 


816 


658 


823 


829 


836 


842 


849 


856 


862 


869 


8 7 5 


882 


659 


889 


8 9 5 


902 


908 


9i5 


921 


928 


935 


94 1 


948 


660 


954 


961 


968 


974 


981 


987 


994 


*ooo 


*oo7 


*oi4 


661 


82 020 


027 


o33 


o4o 


o46 


o53 


060 


066 


o 7 3 


79 


662 


086 


092 


099 


io5 


112 


119 


125 


132 


i38 


i45 


663 


i5i 


1 58 


1 64 


171 


I 7 8 


1 84 


191 


197 


2O4 


210 


664 


217 


223 


2 3o 


2 36 


243 


249 


256 


263 


269 


2 7 6 


665 


282 


289 


2 9 5 


302 


3o8 


3i5 


321 


3 2 8 


334 


34i 


666 


34 7 


354 


36o 


36 7 


3 7 3 


38o 


38 7 


3 9 3 


4oo 


4o6 


667 


4i3 


419 


426 


432 


43 9 


445 


452 


458 


465 


4 7 i 


668 


478 


484 


491 


497 


5o4 


5io 


5i 7 


5 2 3 


53o 


536 


669 


543 


549 


556 


562 


569 


5 7 5 


582 


588 


595 


601 


670 


607 


6i4 


620 


627 


633 


64o 


646 


653 


65 9 


666 


671 


672 


679 


685 


692 


698 


75 


711 


718 


724 


7 3o 


672 


7 3 7 


743 


750 


7 56 


7 63 


769 


77 6 


782 


789 


79 5 


673 


802 


808 


8i4 


821 


827 


834 


84o 


847 


853 


860 


6 7 4 


866 


872 


879 


885 


892 


898 


95 


911 


918 


924 


6 7 5 


93o 


9 3 7 


943 


950 


9 56 


9 63 


969 


97 5 


982 


988 


676 


995 


*OOI 


*oo8 


*oi4 


*O2O 


*O27 


*o33 


*o4o 


*o46 


*052 


677 


83o5 9 


o65 


O 7 2 


078 


085 


091 


097 


io4 


no 


II 7 


678 


123 


129 


1 36 


i4a 


1 49 


155 


161 


1 68 


i 7 4 


181 


679 


187 


198 


200 


206 


2l3 


219 


225 


232 


238 


245 


680 


a5i 


25 7 


264 


270 


2 7 6 


283 


289 


296 


302 


3o8 


N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


PP 7 


6 


i 0.7 i 


0.6 


2 1.4 2 


1.2 


3 2.1 3 


1.8 


4 2.8 4 


2.4 


5 3.5 5 


3.o 


6 4.2 6 


3.6 


7 4.9 7 


4.2 


8 5.6 8 


4.8 


Q 6.3 9 


5.4 



18 



68O 72O 



N 





1 


2 


3 


4 


5 


6 


7 8 


9 


680 


83 261 


25 7 


264 


270 


276 


283 


28 9 


296 


302 


3o8 


68 1 


315 


321 


32 7 


334 


34o 


34 7 


353 


35 9 


366 


3 7 2 


682 


3 7 8 


385 


391 


3 9 8 


4o4 


4io 


4i 7 


423 


429 


436 


683 


442 


448 


455 


46i 


46 7 


4 7 4 


48o 


48 7 


493 


499 


684 


5o6 


5l2 


5i8 


525 


53i 


53 7 


544 


55o 


556 


563 


685 


56 9 


5 7 5 


582 


588 


5 9 4 


601 


6o 7 


6i3 


620 


626 


686 


632 


63 9 


645 


65i 


658 


664 


6 7 o 


677 


683 


689 


687 


696 


7 02 


708 


7i5 


721 


727 


7 34 


7 4o 


7 46 


7 53 


688 


7 5 9 


7 65 


77 i 


778 


784 


79 


797 


8o3 


809 


816 


689 


822 


828 


835 


84 1 


84 7 


853 


860 


866 


872 


8 79 


690 


885 


891 


897 


904 


910 


916 


9 23 


929 


9 35 


9 42 


691 


948 


9^4 


960 


967 


973 


979 


9 85 


992 


998 


*oo4 


692 


84 on 


017 


023 


029 


o36 


042 


o48 


055 


061 


o6 7 


6 9 3 


o 7 3 


080 


086 


092 


098 


105 


in 


n 7 


123 


i3o 


694 


1 36 


1 4a 


1 48 


155 


161 


i6 7 


I 7 3 


180 


1 86 


I 9 2 


6 9 5 


198 


205 


211 


2I 7 


223 


230 


236 


242 


248 


255 


696 


261 


26 7 


2 7 3 


280 


286 292 


298 


3os 


3u 


3i 7 


697 


323 


33o 


336 


342 


348 354 


36i 


36 7 


3 7 3 


379 


698 


386 


392 


3 9 8 


4o4 


4io 


4i 7 


423 


429 


435 


442 


699 


448 


454 


46o 


466 


4 7 3 


479 


485 


491 


497 


5o4 


700 


5io 


5i6 


522 


5 2 8 


535 


54i 


54 7 


553 


55 9 


566 


701 


5 7 2 


5 7 8 


584 


590 


5 97 


6o3 


609 


6i5 


621 


628 


702 


634 


64o 


646 


652 


658 


665 


6 7 i 


6 77 


683 


68 9 


7 o3 


696 


702 


708 


7 i4 


720 


726 


7 33 


7 3 9 


745 


7 5i 


704 


7 5 7 


7 63 


770 


776 


782 


7 88 


794 


800 


807 


8i3 


7 o5 


819 


8 2 5 


83i 


83 7 


844 


850 


856 


862 


868 


8 7 4 


706 


880 


88 7 


893 


899 


9 o5 


911 


9 i 7 


924 


93o 


9 36 


707 


942 


948 


954 


960 


967 


973 


979 


9 8 5 


991 


997 


708 


85oo3 


009 


016 


022 


028 


o34 


o4o 


o40 


052 


o58 


709 


065 


7 I 


077 


o83 


089 


o 9 5 


IOI 


I0 7 


n4 


1 20 


710 


126 


I 32 


i38 


i44 


i5o 


1 56 


i63 


169 


175 


181 


711 


i8 7 


io3 


199 


205 


211 


2I 7 


224 


23o 


236 


242 


712 


248 


254 


260 


266 


2 7 2 


278 


285 


291 


297 


3o3 


7 i3 


309 


3i5 


321 


3 27 


333 


33 9 


345 


35 2 


358 


364 


7 i4 


3 7 o 


3 7 6 


382 


388 


394 


4oo 


4o6 


412 


4i8 


425 


7 i5 


43i 


43 7 


443 


449 


455 


46 1 


46 7 


4 7 3 


479 


485 


716 


491 


497 


5o3 


509 


5i6 


522 


528 


534 


54o 


546 


717 


552 


558 


564 


5 7 o 


5 7 6 


582 


588 


5 9 4 


600 


606 


718 


612 


618 


625 


63i 


63 7 


643 


649 


655 


661 


66 7 


719 


6 7 3 


679 


685 


691 


69-7 


7 o3 


79 


7 i5 


7 2I 


727 


720 


7 33 


7 3 9 


745 


7 5i 


7 5 7 


7 63 


769 


77 5 


781 


7 88 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



72O-750 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


720 


85 7 33 


7 3 9 


7 45 


7 5i 


7 5 7 


7 63 


-769 


77 5 


7 8i 


7 88 


7 2I 


794 


800 


806 


812 


818 


824 


83o 


836 


842 


848 


7 22 


854 


860 


866 


8 7 2 


878 


884 


890 


896 


902 


908 


7 23 


9 i4 


920 


9 26 


932 


938 


944 


950 


9 56 


962 


968 


724 


974 


980 


9 86 


992 


998 


*oo4 


*OIO 


*oi6 


*O22 


*028 


725 


86o34 


o4o 


o46 


Of)2 


o58 


064 


7 


o 7 6 


082 


088 


726 


094 


IOO 


1 06 


112 


118 


124 


i3o 


i?6 


lAi 


i4 7 


727 


i53 


i5 9 


i65 


171 


177 


i83 


189 


i 9 5 


201 


207 


728 


2l3 


219 


225 


23l 


2 3 7 


243 


2 


49 


255 


26l 




729 


2 7 3 


2 79 


285 


291 


2 97 


3o3 


3o8 


3i4 


320 


326 


730 


332 


338 


344 


35o 


356 


362 


368 


3 7 4 


38o 


386 


7 3i 


392 


3 9 8 




4o4 


4io 


4i5 


421 


427 


433 


439 


445 


7 32 


45i 


45 7 


463 


46 9 


475 


48 1 


48 7 


493 


499 


5o4 


7 33 


5io 


5i6 


522 


5 2 8 


534 


54o 


546 


552 


558 


564 


734 


5 7 o 


5 7 6 


58i 


58 7 


5 9 3 


599 


6o5 


611 


617 


6 2 3 


7 35 


629 


635 


64 1 


646 


652 


658 


664 


6 7 o 


676 


682 


7 36 


688 


6 9 4 


7 oo 


7 o5 


711 


7 i 7 


7 23 


729 


735 


74 1 


7 3 7 


747 


7 53 


7 5 9 


7 64 


770 


776 


782 


7 88 


794 


800 


7 38 


806 


812 


817 


823 


829 


835 


84i 


847 


853 


85 9 


7 3 9 


864 


8 7 o 


8 7 6 


882 


888 


8 9 4 


900 


906 


911 


917 


740 


923 


9 2 9 


935 


94 1 


947 


953 


9 58 


964 


97 


976 


7 4i 


982 


9 88 




994 


999 


*oo5 


*OII 


*o 


17 


*023 


*029 


*o35 


7 42 


8 7 o4o 


o46 


052 


o58 


o64 


O 7 O 


o 7 5 


08 1 


087 


093 


7 43 


099 


105 


III 


116 


122 


128 


1 34 


i4o 


i46 


i5i 


7 44 


i5 7 


i63 


1 69, 


175 


181 


186 


192 


198 


2O4 


210 


7 45 


216 


221 




22 7 


233 


239 


245 


25l 


256 


262 


268 


7 46 


274 


280 


286 


291 


2 97 


3o3 


3o 9 


315 


320 


326 


74 7 


33 2 


338 


344 


349 


355 


36i 


36 7 


3 7 3 


379 


384 


7 48 


390 


3 9 6 


402 


4o8 


4i3 


419 


425 


43i 


43 7 


442 


74 9 


448 


454 




46o 


466 


4 7 i 


477 


483 


48 9 


495 


5oo 


750 


5o6 


5l2 


5i8 


523 


5 29 


535 


54 1 


54 7 


55 2 


558 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


PP 


6 


5 


i 


0.6 . i 


o.5 


2 


1.2 2 


I.O 


3 


1.8 3 


i.5 


4 


2.4 4 


2.O 


5 


3.o 5 


2.5 


6 


3.6 6 


3.o 


7 


4-2 7 


3.5 


8 


4.8 8 


4-0 


9 


5.4 9 


4.5 



750-79O 



N 


O 


1 


2 


3 


4 


5 


6 


7 8 


9 


750 


87 5o6 


5l2 


5x8 


523 


52 9 


535 


54i 


547 


552 


558 


761 


564 


5 7 o 


5 7 6 


58 1 


58 7 


5 9 3 


5 99 


6o4 


610 


616 


7 52 


622 


628 


633 


63 9 


645 


65i 


656 


662 


668 


6 7 4 


7 53 


679 


685 


691 


6 97 


7 o3 


7 o8 


7 i4 


720 


726 


7 3i 


7 54 


7 3 7 


7 43 


749 


7 54 


7 6o 


7 66 


77 2 


111 


7 83 


789 


7 55 


795 


800 


806 


812 


818 


823 


82 9 


835 


84 1 


846 


7 56 


852 


858 


864 


86 9 


8 7 5 


881 


88 7 


8 9 2 


8 9 8 


9 o4 


767 


910 


9 i5 


921 


9 2 7 


9 33 


9 38 


9 44 


950 


9 55 


9 6i 


7 58 


967 


973 


978 


9 84 


99 


99 6 


*OOI 


*oo 7 


*oi3 


*oi8 


7 5 9 


88 024 


o3o 


o36 


o4i 


o4 7 


o53 


o58 


o64 


O 7 O 


076 


760 


08 1 


087 


o 9 3 


o 9 8 


io4 


I IO 


116 


121 


I2 7 


i33 


761 


1 38 


1 44 


150 


i56 


161 


167 


i 7 3 


I 7 8 


1 84 


I9O, 


762 


i 9 5 


2OI 


207 


2l3 


218 


224 


230 


235 


241 


247 


7 63 


262 


2 58 


264 


2 7 


2 7 5 


281* 


287 


2 9 2 


2 9 8 


3o4 


7 64 


309 


3i5 


321 


326 


33 2 


338 


343 


34 9 


355 


36o 


7 65 


366 


3 7 2 


377 


383 


38 9 


395 


4oo 


4o6 


412 


4i 7 


7 66 


423 


429 


434 


44o 


446 


45i 


45 7 


463 


468 


4 7 4 


767 


48o 


485 


491 


4 97 


502 


5o8 


5i3 


5io 


525 


53o 


768 


536 


542 


54 7 


553 


55 9 


564 


5 7 o 


5 7 6 


58i 


58 7 


769 


5 9 3 


5 9 8 


6o4 


610 


6i5 


621 


62 7 


632 


638 


643 


770 


649 


655 


660 


666 


6 7 2 


677 


683 


68 9 


6 9 4 


7 oo 


771 


7 o5 


7 n 


717 


7 22 


728 


7 34 


7 3 9 


745 


7 5o 


7 56 


772 


762 


767 


773 


779 


784 


79 


79 5 


801 


8o 7 


812 


77 3 


818 


824 


829 


835 


84o 


846 


852 


85 7 


863 


868 


77 4 


8 7 4 


880 


885 


891 


897 


9 02 


9 o8 


9 i3 


9 i 9 


9 2 5 


77 5 


93o 


9 36 


94i 


947 


9 53 


9 58 


9 64 


9 6 9 


975 


981 


776 


986 


992 


997 


*oo3 


*oo 9 


*oi4 


*020 


*025 


*o3i 


*o3 7 


777 


89 042 


o48 


o53 


oSg 


o64 


070 


o 7 6 


081 


o8 7 


092 


778 


098 


io4 


109 


"5 


120 


126 


i3r 


i3 7 


i43 


i48 


779 


y i54 


i5 9 


165 


170 


176 


182 


i8 7 


i 9 3 


i 9 8 


2O4 


780 


209 


2l5 


221 


226 


232 


23 7 


243 


248 


254 


260 


781 


265 


271 


2 7 6 


282 


28 7 


2 9 3 


2 9 8 


3o4 


3io 


3i5 


782 


321 


326 


33 2 


33 7 


343 


348 


354 


36o 


365 


3 7 i 


7 83 


3 7 6 


382 


38 7 


3 9 3 


3 9 8 


4o4 


4o 9 


4i5 


421 


426 


7 84 


432 


43 7 


443 


448 


454 


45 9 


465 


4 7 o 


4 7 6 


48 1 


7 85 


48 7 


4 9 2 


498 


5o4 


5o 9 


5i5 


520 


526 


53i 


53 7 


786 


542 


548 


553 


55 9 


564 


5 7 o 


5 7 5 


58i 


586 


592 


787 


5 97 


6o3 


609 


6i4 


620 


6 2 5 


63i 


636 


642 


64 7 


788 


653 


658 


664 


66 9 


675 


680 


686 


6 9 i 


697 


7 O2 


789 


708 


7 i3 


7i9 


724 


7 3o 


7 35 


7 4i 


746 


7 52 


7 5 7 


790 


7 63 


7 68 


774 


779 


785 


79 


796 


801 


807 


812 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



79O 82O 



N 





1 


2 


3 


4 


5 


G 


7 


8 


9 


790 


89 7 63 


768 


774 


779 


785 


79 


79 6 


801 


807 


812 


79 1 


818 


823 


829 


834 


84o 


845 


85i 


856 


862 


86 7 


79 2 


8 7 3 


878 


883 


889 


8 9 4 


900 


9 


o5 


911 


9 i6 


922 


793 


9 2 7 


9 33 


9 38 


944 


949 


955 


960 


966 


97i 


977 


794 


982 


988 


1 


993 


998 


*oo4 


*oo 9 


*o 


jr 


*O2O 


*O26 


*o3i 


79 5 


90 o3 7 


o4s 


t 


o48 


o53 


o5 9 


064 


069 


075 


080 


086 


79 6 


091 


097 


102 


1 08 


n3 


119 


124 


129 


135 


i4o 


797 


1 46 


i5i 




i5 7 


162 


1 68 


i 7 3 


1-79 


1 84 


i8 9 


^95 


798 


200 


206 


211 


2I 7 


222 


22 7 


233 


238 


244 


249 


799 


255 


260 


266 


2 7 I 


2 7 6 


282 


287 


293 


2 9 8 


3o4 


800 


309 


3i4 


320 


325 


33i 


336 


342 


347 


35 2 


358 


801 


363 


36 9 


3 7 4 


38o 


385 


390 


3 9 6 


4oi 


407 


4l2 


802 


4i 7 


423 


428 


434 


43 9 


445 


450 


455 


46 1 


466 


8o3 


4 7 2 


477 


482 


488 


493 


499 


5o4 


509 


5 I 5 


620 


8o4 


526 


53i 




536 


542 


54 7 


553 


558 


563 


56 9 


5 7 4 


8o5 


58o 


585 


590 


5 9 6 


601 


6o 7 


612 


617 


623 


628 


806 


634 


63 9 


644 


650 


655 


660 


666 


671 


677 


682 


8o 7 


68 7 


6 9 3 


698 


7 o3 


79 


7 i4 


720 


725 


7 3o 


7 36 


808 


7 4 1 


747 


7 52 


7 5 7 


7 63 


7 68 


77 3 


779 


784 


789 


809 


795 


800 


806 


811 


816 


822 


827 


83 2 


838 


843 


810 


849 


854 


85 9 


865 


8 7 o 


8 7 5 


881 


886 


891 


897 


811 


902 


97 


9 i3 


9 i8 


924 


929 


934 


940 


945 


9 5o 


812 


956 


961 




966 


97 2 


977 


982 


988 


99 3 


998 


*oo4 


8i3 


91 009 


oi4 


020 


O25 


o3o 


o36 


o4i 


o46 


OD2 


o5 7 


8i4 


062 


068 


o 7 3 


o 7 8 


o84 


089 


094 


IOO 


io5 


I IO 


8i5 


116 


121 




126 


I 32 


i3 7 


142 


i48 


i53 


i58 


1 64 


816 


169 


i 7 4 


180 


185 


190 


196 


201 


206 


212 


2I 7 


817 


222 


228 


233 


238 


243 


249 


254 


25 9 


265 


2 7 


818 


275 


281 




286 


291 


2 97 


302 


307 


312 


3i8 


323 


819 


3 2 8 


334 


33 9 


344 


350 


355 


36o 


365 


3 7 i 


3 7 6 


820 


38i 


38 7 


3 9 2 


397 


4o3 


4o8 


4i3 


4i8 


424 


42 9 


N 


O 


1 


2 


3 


4 


5 


G 


7 


8 


9 


PP 


6 


5 


i 


0.6 i 


o.5 


2 


1.2 2 


I.O 


3 


1.8 3 


i.5 


4 


2.4 4 


2.O 


5 


3.o 5 


2.5 


6 


3.6 6 


3.o 


7 


4.2 7 


3.5 


8 


4.8 8 


4.o 


9 


5.4 9 


4.5 



82O-860 



N 





1 


a 


3 


4 


5 





7 


8 


9 


820 


91 38 1 


387 


3 9 2 


3 97 


4o3 


4o8 


4i3 


4i8 


424 


429 


821 


434 


44o 


445 


45o 


455 


46i 


466 


471 


477 


482 


822 


48 7 


492 


498 


5o3 


5o8 


5i4 


5i 9 


524 


529 


535 


828 


54o 


545 


55i 


556 


56i 


566 


5 7 2 


5 77 


582 


58 7 


824 


5 9 3 


5 9 8 


6o3 


609 


6i4 


6i 9 


624 


63o 


63s 


64o 


8 2 5 


645 


65i 


656 


661 


666 


6 7 2 


6 77 


682 


687 


6 9 3 


826 


698 


7 o3 


79 


7 i4 


719 


7 24 


7 3o 


735 


7 4o 


745 


827 


7 5i 


7 56 


761 


7 66 


772 


777 


782 


787 


79 3 


79 8 


828 


8o3 


808 


8i4 


819 


824 


82 9 


834 


84o 


845 


85o 


829 


855 


861 


866 


871 


8 7 6 


882 


887 


892 


897 


9 o3 


830 


908 


9 i3 


918 


924 


9 2 9 


9 34 


9 3 9 


944 


950 


955 


83i 


960 


965 


97 i 


9-76 


9 8i 


9 86 


991 


997 


*002 


*oo7 


83a 


92 OI2 


018 


023 


028 


o33 


o38 


o44 


049 


o54 


o5 9 


833 


065 


070 


075 


080 


o85 


091 


096 


101 


1 06 


in 


834 


117 


122 


1*7 


132 


i3 7 


i43 


1 48 


i53 


i58 


i63 


835 


169 


1 7 4 


i 79 


i84 


i8 9 


195 


200 


205 


2IO 


2l5 


836 


221 


226 


23l 


236 


241 


24 7 


252 


25 ? 


262 


267 


83 7 


2 7 3 


278 


283 


288 


2 9 3 


298 


3o4 


309 


3i4 


3i 9 


838 


324 


33o 


335 


34o 


345 


35o 


355 


36i 


366 


3 7 i 


83 9 


376 


38i 


38 7 


392 


3 97 


402 


407 


4l2 


4i8 


423 


840 


428 


433 


438 


443 


44 9 


454 


45 9 


464 


46 9 


4?4 


84i 


48o 


485 


490 


4 9 5 


5oo 


5o5 


5n 


5i6 


521 


526 


842 


53i 


536 


542 


54 7 


552 


55 7 


56 2 


56 7 


572 


5 7 8 


843 


583 


588 


5 9 3 


5 9 8 


6o3 


609 


6i4 


619 


624 


62 9 


844 


634 


63 9 


645 


650 


655 


660 


665 


670 


6 7 5 


681 


845 


686 


691 


696 


7 OI 


7 o6 


711 


716 


722 


727 


7 32 


846 


7 3 7 


7 42 


747 


7 52 


7 58 


7 63 


768 


77 3 


778 


7 83 


84? 


788 


79 3 


799 


8o4 


8o 9 


8i4 


819 


824 


829 


834 


848 


84o 


845 


850 


855 


860 


865 


870 


8 7 5 


881 


886 


849 


891 


896 


901 


9 o6 


9 n 


916 


921 


927 


9 32 


9 37 


850 


942 


94? 


952 


9 5 7 


9 62 


96-7 


973 


978 


983 


9 88 


85i 


99 3 


998 


*oo3 


*oo8 


*oi3 


*oi8 


*024 


*029 


*o34 


*o3 9 


85 2 


93o44 


o4 9 


o54 


o5 9 


o64 


069 


75 


080 


085 


o 9 o 


853 


095 


IOO 


io5 


no 


u5 


120 


125 


i3i 


i36 


i4i 


854 


i46 


i5i 


i56 


161 


166 


I 7 I 


176 


181 


186 


192 


855 


197 


202 


20-7 


212 


2I 7 


222 


227 


282 


23 7 


242 


856 


247 


252 


258 


263 


268 


2 7 3 


278 


283 


288 


293 


85 7 


298 


3o3 


3o8 


3i3 


3i8 


323 


328 


334 


33 9 


344 


858 


349 


354 


35 9 


364 


36 9 


3 7 4 


3 79 


384 


38 9 


394 


859 


3 99 


4o4 


409 


4i4 


420 


425 


43o 


435 


44o 


445 


860 


450 


455 


46o 


465 


4 7 o 


4 7 5 


48o 


485 


490 


495 


N 


o 


1 


2 


3 


4 


5 


6 


7 


8 


9 



23 



860-890 



N 





1 


2 


3 


4 


5 


6 


7 


8 9 


860 


93450 


455 


46o 


465 


470 


4 7 5 


48o 


485 


490 


495 


861 


5oo 


5o5 


5xo 


5x5 


52O 


526 


53i 


536 


54 1 


546 


862 


55i 


556 


56i 


566 


5 7 i 


5 7 6 


58i 


586 


5 9 i 


5 9 6 


863 


60 1 


606 


611 


616 


621 


626 


63i 


636 


64i 


646 


864 


65i 


656 


661 


666 


671 


6 7 6 


682 


687 


6 9 2 


697 


865 


702 


707 


712 


717 


722 


7 2 7 


732 


737 


7 42 


747 


866 


752 




762 


767 


772 


777 


782 


787 


792 


797 


86 7 


802 


807 


812 


817 


822 


827 


83 2 


83 7 


842 


847 


868 


852 


85 7 


862 


867 


872 


877 


882 


887 


892 


897 


869 


902 


907 


9 I2 


917 


922 


927 


932 


937 


942 


947 


870 


952 


9 5 7 


962 


967 


972 


977 


982 


987 


992 


997 


871 


94 002 


007 


OI2 


017 


022 


02 7 


032 


o3 7 


042 


047 


872 


052 


o5 7 


062 


067 


O 7 2 


77 


082 


086 


091 


096 


8 7 3 


IOI 


1 06 


III 


116 


121 


126 


I 


3i 


1 36 


i4i 


1 46 


874 


i5i 


1 56 


161 


1 66 


I 7 I 


176 


181 


186 


191 


196 


8 7 5 


2OI 


206 


21 I 


216 


221 


226 


23l 


236 


240 


245 


876 


25o 


255 


260 


265 


2 7 O 


2 7 5 


280 


285 


290 


295 


877 


3oo 


305 i 3io 


3x5 


320 


325 


33o 


335 


34o 


345 


878 


349 


354 


35 9 


364 


36 9 


3 7 4 


3 79 


384 


389 


3 9 4 


879 


399 


4o4 


409 


4x4 


419 


424 


429 


433 


438 


443 


880 


448 


453 


458 


463 


468 


4 7 3 


478 


483 


488 


493 


881 


498 


5o3 5o 7 


5l2 


5i 7 


522 


52 7 


532 


53 7 


542 


882 


54 7 


552 55 7 


562 


56 7 


5 7 i 


576 


58i 586 


691 


883 


596 


601 


606 


6x1 


616 


621 


626 


63o 


635 


64o 


884 


645 


65o 


655 


660 


665 


6 7 o 


675 


680 


685 


689 


885 


6 9 4 


699 


7 o4 79 


7 i4 


719 


724 


729 


7 34 


7 38 


886 


743 


748 


7 53 


7 58 


7 63 


7 68 


77 3 


778 


7 83 


787 


887 


792 


797 


802 


807 


812 


817 


822 


827 


832 


836 


888 


84i 


846 


85i 


856 


861 


866 


871 


876 


880 


885 


889 


890 


8 9 5 


900 


95 


910 


9*5 


919 


924 


929 


934 


890 


9 3 9 


9 U 


9 4 9 


954 


9 5 9 


9 63 


968 


973 


978 


9 83 


N 


O 


1 


2 


3 


4 


5 


a 


7 


8 


9 


PP 5 


4 


i o.5 i 


o.4 


2 I.O 2 


0.8 


3 i.5 3 


I .2 


4 2.0 4 


1.6 


5 2.5 5 


2.O 


6 3.o 6 


2.4 


7 3.5 7 


2.8 


8 4.o 8 


3.2 


9 4.5 9 


3.6 



24 



89O 93O 



N 


O 


1 


a 


3 


4 


5 


6 


7 


8 


9 


890 


94939 


9 44 


949 


9 54 


9 5 9 


9 63 


968 


973 


978 


9 83 


8 9 i 


988 


993 


99 8 


*OO2 


*oo7 


*OI2 


*OI 7 


*022 


*02 7 


*032 


8 9 2 


9 5o36 


o4i 


o46 


o5i 


o56 


061 


066 


7 I 


o 7 5 


080 


8 9 3 


o85 


o 9 o 


095 


100 


105 


109 


n4 




124 


129 


8 9 4 


1 34 


i3 9 


i43 


1 48 


i53 


i58 


i63 


168 


i 7 3 


177 


8 9 5 


182 


187 


192 


197 


202 


20 7 


211 


216 


221 


226 


8 9 6 


a3i 


236 


240 


245 


25o 


255 


260 


265 


2 7 


2 7 4 


897 


279 


284 


289 


294 


2 99 


3o3 


3o8 


3i3 


3i8 


3 2 3 


8 9 8 


3 2 8 


332 


33 7 


342 


34 7 


352 


35 7 


36i 


366 


3 7 i 


899 


3 7 6 


38i 


386 


390 


3 9 5 


4oo 


405 


4io 


4i5 


419 


900 


424 


42 9 


434 


439 


444 


448 


453 


458 


463 


468 


901 


472 


477 


48a 


487 


4 9 2 


497 


5oi 


5o6 


5u 


5i6 


9 O2 


521 


525 


53o 


535 


54o 


545 


55 


554 


55 9 


564 


9 o3 


56 9 


574 


5 7 8 


583 


588 


5 9 3 


5 9 8 


602 


607 


612 


904 


617 


622 


626 


63i 


636 


64 1 


646 


65o 


655 


660 


95 


665 


670 


6 7 4 


679 


684 


689 


6 9 4 


6 9 8 


7 o3 


7 o8 


906 


7 i3 


718 


7 22 


727 


7 32 


7 3 7 


742 


746 


761 


7 56 


907 


761 


766 


770 


77 5 


7 8o 


785 


789 


794 


799 


8o4 


908 


809 


8i3 


8l8 


8 2 3 


828 


832 


83 7 


842 


847 


852 


909 


856 


861 


866 


871 


8 7 5 


880 


885 


8 9 o 


895 


899 


910 


9 o4 


99 


914 


918 


923 


928 


9 33 


9 38 


942 


9 4 7 


911 


9 52 


9 5 7 


961 


966 


* 9?I 


97 6 


9 8o 


985 


99 


* 995 


912 


999 


*oo4 


*oo 9 


*oi4 




*023 


*028 


*o33 


*o38 




9 i3 


9 6 047 


052 


o5 7 


06 1 


066 


7 I 


o 7 6 


080 


o85 


090 


914 


95 


99 


io4 


109 


n4 


118 


123 


128 


i33 


i3 7 


9 i5 


142 


i4 7 


I 52 


i56 


161 


166 


171 


i 7 5 


1 80 


185 


916 


190 


i 9 4 


199 


204 


209 


2l3 


218 


223 


227 


232 


917 


2 3 7 


242 


246 


a5i 


2 56 


261 


265 


2 7 O 


275 


280 


918 


284 


28 9 


294 


298 


3o3 


3o8 


3i3 


3i 7 


322 


32 7 


919 


332 


336 


34i 


346 


35o 


355 


36o 


365 


36 9 


3 7 4 


920 


379 


384 


388 


3 9 3 


398 


402 


4o 7 


4l2 


4i 7 


421 


921 


426 


43i 


435 


44o 


445 


450 


454 


45 9 


'464 


468 


922 


4 7 3 


478 


483 


487 


4 9 2 


497 


5oi 


5o6 


5n 


5x5 


923 


52O 


5 2 5 


53o 


534 


53 9 


544 


548 


553 


558 


562 


924 


56 7 


5 7 2 


5 77 


58i 


586 


5 9 i 


5 9 5 


600 


605 


609 


925 


6i4 


6i 9 


624 


628 


633 


638 


642 


647 


652 


656 


926 


661 


666 


6 7 o 


6 7 5 


680 


685 


68 9 


6 9 4 


6 99 


7 o3 


927 


7 o8 


7 i3 


717 


722 


727 


7 3i 


7 36 


7 4i 


7 45 


7 5o 


928 


755 


7 5 9 


7 64 


7 6 9 


774 


778 


7 83 


7 88 


792 


797 


9 2 9 


802 


806 


811 


816 


820 


825 


83o 


834 


83 9 


844 


930 


848 


853 


858 


862 


867 


8 7 2 


8 7 6 


881 


886 


890 


N 


O 


1 2 


3 


4 


5 


6 


7 


8 


9 



93O-96O 



N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


980 


9 6848 


853 


858 


862 


867 


872 


8 7 6 


881 


886 


890 


9 3i 


895 


900 


9 o4 


99 


9 i4 


918 


923 


9 28 


932 


9 3 7 


9 3 2 


942 


946 


9 5i 


9 56 


960 


965 


97 


974 


979 


9 84 


9 33 


988 


993 


997 


*OO2 


*oo 7 


*OI I 


*oi6 


*O2I 


*025 


*o3o 


9 34 


97035 


o3 9 


o44 


o4 9 


o53 


o58 


o63 


067 


O 7 2 


077 


9 35 


08 1 


086 


o 9 o 


o 9 5 


IOO 


104 


109 


n4 


118 


123 


9 36 


128 


132 


i3 7 


142 


i46 


i5i 


i55 


160 


165 


i6 9 


9 3 7 


1 7 4 


*79 


i83 


188 


192 


19-7 


202 


206 


211 


216 


9 38 


220 


225 


230 


234 


23 9 


243 


248 


2 53 


25 7 


262 


9 3 9 


267 


271 




276 


280 


285 


290 


294 


299 


3o4 


3o8 


940 


3i3 


3i 7 


322 


32 7 


33i 


336 


34o 


345 


350 


354 


9 4i 


35 9 


364 


368 


3 7 3 


377 


382 


38 7 


391 


3 9 6 


4oo 


942 


4o5 


4io 


4i4 


4i 9 


424 


428 


433 


43 7 


442 


447 


943 


45 1 


456 


46o 


465 


470 


4 7 4 


479 


483 


488 


4 9 3 


944 
945 


497 
543 


5O2 

548 


5o6 
55 2 


55 7 


5i6 
562 


520 

566 


5^5 
5 7 i 


5 29 

5 7 5 


534 
58o 


53 9 

585 


946 


58 9 


5 9 4 




5 9 8 


6o3 


607 


612 


617 


621 


626 


63o 


947 


635 


64o 


644 


64 9 


653 


658 


663 


66 7 


672 


676 


948 


681 


685 


6 9 o 


695 


699 


7 o4 


708 


7 i3 


717 


722 


949 


727 


7 3i 




7 36 


74o 


745 


749 


7 54 


759 


7 63 


768 


950 


772 


777 


782 


786 


791 


79 5 


800 


8o4 


809 


8i3 


9 5i 


818 


823 


827 


832 


836 


84 1 


845 


85o 


855 


85 9 


952 


864 


868 


873 


877 


882 


886 


891 


896 


9 oo 


95 


9 53 


909 


9 i4 


9 i8 


9 23 


928 


932 


9 3 7 


9 4i 


9 46 


9 5o 


954 


955 


9 5 9 


9 64 


9 68 


973 


978 


9 82 


987 


99 i 


996 


955 


98 ooo 


005 


oo 9 


oi4 


019 


023 


028 


o3a 


o3 7 


o4i 


9 56 


o46 


o5o 


055 


o5 9 


o64 


068 


073 


o 7 8 


082 


o8 7 


957 


091 


o 9 6 


IOO 


105 


109 


n4 


118 


123 


127 


I 32 


9 58 


i3 7 


i4i 




i46 


i5o 


155 


i5 9 


1 64 


168 


i 7 3 


177 


9 5 9 


182 


186 


I 9 i 


i 9 5 


200 


2O4 


20 9 


2l4 


218 


223 


960 


227 


232 


236 


241 


245 


250 


254 


259 


263 


268 


N 


O 


1 


2 


3 


4 


5 


O 


7 


8 


9 


PP 


5 


4 


i 


o.5 i 


o.4 


2 


I.O 2 


0.8 


3 


i.5 3 


1.2 


4 


2.0 4 


1.6 


5 


2.5 5 


2.O 


6 


3.o 6 


2.4 


7 


3.5 7 


2.8 


8 


4.o 8 


3.2 


9 


4.5 9 


3.6 



26 



960-1OOO 



N 





1 


ti 


3 


4 


5 


6 


7 


8 9 


960 


9 822 7 


232 


236 


24 1 


245 


250 


254 


2 5 9 


263 


268 


961 

962 
963 

964 
965 
966 

967 
968 
969 

970 

971 
972 
973 

974 
975 
976 

977 
978 
979 
980 


272 
3i8 
363 

4o8 
453 
4 9 8 

543 
588 
632 


2 77 
322 
367 

412 
457 
5O2 

54 7 

5 9 2 

63 7 


281 
3 27 

3 7 2 

4i 7 
462 
507 

55b 
597 
64 1 


286 
33i 
3 7 6 

421 
466 
5n 

556 
601 

646 


2 9 O 

336 
38i 

426 

4 7 i 
5i6 

56i 
6o5 
65o 


2 95 
34o 
385 

43o 

4 7 5 

520 

565 
610 
655 


299 

345 
390 

435 
48o 
525 

5 7 o 
6i4 
65 9 


3o4 
34 9 
3 9 4 

43 9 

484 
529 

5 7 4 
619 
664 


3o8 
354 
3 99 

444 
48 9 
534 

5-79 
623 
668 


3i3 
358 
4o3 

448 
4 9 3 
538 

583 
628 
6 7 3 


677 


682 


686 


691 


6 9 5 


700 


704 


709 


7 i3 


717 


722 
76.7 

in 

856 
900 
945 
989 
99 o34 
078 


726 
771 
816 

860 

95 
9 4 9 

994 
o38 
o83 


7 3i 
776 
820 

865 
909 
954 

998 
o43 
087 


7 35 
780 
825 

869 
914 
958 

*oo3 
047 
092 


74o 
784 
829 

874 
918 
9 63 

*oo 7 

052 

096 


744 
789 
834 

878 
923 
967 

*OI2 

o56 

IOO 


749 
793 

838 

883 
927 
972 

*oi6 
06 1 
105 


7 53 
798 

843 

887 
932 
976 

*02I 
065 
109 


7 58 
802 
84 7 

892 
9 36 
981 

*025 

069 
n4 


762 
807 
85 1 

8 9 6 
9 4i 

9 85 

*02 9 
074 

118 


123 


127 


i3i 


i36 


i4<> 


i45 


1 49 


1 54 


i58 


162 


981 
982 
983 

984 
9 85 
986 

987 
988 
989 

990 

991 

992 
99 3 

994 
99 5 
99 6 

997 

99 8 

999 
1000 


167 

21 I 

255 

3oo 
344 

388 

432 
476 

52O 


171 
216 
260 

3o4 
348 
392 

436 

48o 
524 


176 
220 
264 

3o8 
352 

396 

44 1 
484 
5 2 8 


180 
224 
269 

3i3 
35 7 
4oi 

44s 
48 9 
533 


185 

22 9 
2 7 3 

3i 7 
36i 

4o5 

44 9 
4 9 3 
53 7 


189 
233 
277 

322 

366 
4io 

454 
498 
542 


I 9 3 

238 
282 

326 
3 7 o 

4i4 

458 

502 

546 


198 
242 
286 

33o 

3 7 4 
419 

463 
5o6 
55o 


202 

247 
291 

335 
3 79 

423 

46 7 
5n 
555 


207 

25l 
2 9 5 

33 9 
383 
427 

4 7 i 
5i5 
55 9 


564 


568 


572 


5 77 


58i 


585 


590 


5 9 4 


5 99 


6o3 


607 
65i 
695 

7 3 9 

782 
826 

870 
9 i3 
9 5 7 


612 
656 
699 

7 43 
787 
83o 

8 7 4 
917 
961 


616 
660 
704 

747 
791 
835 

878 
922 
9 65 


621 

664 
708 

752 
7 9 5 
839 

883 
926 
97 


625 
669 
712 

756 
800 

843 

887 
93o 

974 


629 
6 7 3 
717 

760 

8o4 
848 

891 

935 
978 


634 
677 
721 

765 

808 
852 

896 

9 3 9 

9 83 


638 
682 
726 

7 6 9 
8:3 
856 

9 oo 

9 44 
9 8 7 


642 
686 
73o 

774 
817 
861 

9 o4 
9 48 
99 i 


647 
691 

7 34 

778 
822 
865 

909 

952 
996 


00 OOO 


oo4 


009 


oi3 


017 


022 


026 


o3o 


o3 


o3 9 


X 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 



27 



TABLE II 

FIVE -PL ACE LOGARITHMS 

OF THE 

TRIGONOMETRIC FUNCTIONS 

TO EVERY MINUTE 



0. 





L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 
























. OO OOO 


60 


I 


6.46 373 


30103 


6.46 3 7 3 






3.53627 


O.OO OOO 


5 9 


2 

3 


6.94085 


17609 


6.76476 
6.94085 


3 010 3 
17609 


3.23 524 
3.o5 9 i5 


O.OO OOO 
0.00 000 


58 
5 7 


4 


7.06679 


12494 
9691 


7.06 579 


12494 

nfinr 


2.9 


3421 


O.OO OOO 


56 


5 


7.16 270 


7018 


7.16 270 


909 




2.8 


3 7 3o 


O.OO OOO 


55 


6 


7.24 188 


7910 


7.24 188 


791? 




2.75 812 


O.OO OOO 


54 








6694 






6694 
















7 


7.30882 




7.30882 


c8oo 


2.6 9 1 1 8 


O.OO OOO 


53 


8 


7 .36682 




7 .36682 






2.633:8 


O.OO OOO 


52 


9 


7-4i 797 


5"5 


7-4i 797 


5"5 


2.58 2o3 


O.OO OOO 


5i 


10 


7 .463 7 3 




7.46 3 7 3 


457 


2.53 627 


O.OO OOO 


50 


1 1 


7.5o 5i2 


4 J 39 


7.5o 5i2 


4139 


2.49488 


O.OO OOO 


49 


12 

i3 


7.54 291 
7.57 767 


3476 


7.54 291 
7.57767 


3476 


2.45 709 
2.42 233 


O.OO OOO 
O.OO OOO 


48 
47 


i4 
i5 


7.60985 
7.63 982 


3218 
2997 


7.60 986 
7.63 982 


3219 
2996 


2. Sg Ol4 

2.36 018 


0.00 000 
O.OO OOO 


46 

45 


16 


7.66784 




7.66785 


2803 


2.332:5 


O.OO OOO 


44 








2633 






263 


1 














17 


7-69417 


2483 


7.69418 


2482 


2.3o 582 


9-99999 


43 


18 


7.71 900 

7-74248 


2348 


7.71 900 

7.74248 


2348 


2.28 :oo 

2.25 752 


9-99999 
9-99999 


42 

4i 


20 


7.76475 


2227 


7.76476 




2.23 524 


9-99999 


40 


21 


7-7* 


J 5 9 4 




7 . 7 85 9 5 




2.2: 405 


9-99999 


3 9 


22 


7.80615 




7.80 6i5 






2.: 9 385 


9-99999 


38 


23 


7.82545 


1930 


7 .82546 


*93 




2.:7 454 


9-99999 


37 








1848 






184 


1 














24 


7 .843 9 3 




7-843 9 4 






2.: 5 606 


9 -.99 999 


36 


25 


7 .8( 


5 166 




7.86167 






2 .:3833 


9-99999 


35 


26 


7.87870 


1704 


7.87871 


1704 


2.:2 :29 


9-99999 


34 








1639 






i63( 


1 














27 
28 


78 9 509 
7.91 088 


1579 


7.89 5io 
7.91 089 


1579 


2. :o 4 9 o 
2.08 9 : i 


9-99999 
9-99999 


33 

32 


29 


7.92 612 


1524 


7.92 6i3 


152- 


\ 


2.07 387 


9-99 99 s 


3: 


30 


7.94 o84 




7.94 086 




1 


2.o5 9 :4 


9-99998 


30 




L. Cos. 


d. 


L. Cotg. d. 


L. Tang. 


L. Sin. 


' 


89 3O . 


PP 


9691 


4576 


2997 




2483 


2119 


I8 4 8 




1704 


1579 


1472 


.1 


969 


458 


300 


. x 


248 


212 


185 


.! 


170 


158 


*47 


.2 


1938 






.2 


497" 


424 


37 


.2 


341 


316 


294 


3 


2907 


1372 


899 


3 


745 


636 


554 


3 


5" 


474 


442 


4 


3876 


1830 


"99 


4 


993 


8 4 8 


739 


4 


682 


632 


589 


5 


4846 


2288 


1498 


5 


1242 


1060 


924 


5 


8.S2 


789 


736 


.6 




2646 


1798 


.6 


i 49 p 


I2 7 I 


1109 


.6 


IO22 


947 


883 


7 


6784 


3203 


2098 


7 


1738 


M83 


1294 


. 7 


"93 


1105 


1030 


.8 


7753 


3661 


2398 


.8 


1986 


l6 95 


1478 


.8 




1263 


1178 






4118 2697 






1663 




1421 


1325 



O 3O . 





L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 






30 


7 . 94 o84 




7.94 086 






2.o5 914 


9.99998 


30 


3i 

32 


7.95 5o8 
7.96 887 


1424 
1379 


7.96 5io 
7.96 889 


1424 

1379 


2.04 490 

2.O3 III 


9.99998 
9.99998 


29 

28 


33 


7.98 223 


133 


7-9* 


1 225 






2.01 775 


9.99998 


27 








1297 






129; 
















34 


7.99 520 




7.99 522 


1259 


2.OO 478 


9.99998 


26 


35 
36 


8.00 779 
8. 02 002 


1223 
1190 


8.00 781 

8.02 004 


1223 
1190 


I.992I 9 
1.97996 


9.99998 
9.99998 


25 
24 


37 


8.o3 192 


1158 


8.o3 194 


1159 


I .96 806 


9-99 997 


23 


38 


8.o4 35o 


1128 


8.04353 




i .96 647 


9-99997 


22 


3 9 


8.o5 


478 




8.o548i 




i . 94 5 1 9 


9-99997 


21 


40 


8.06 5 7 8 




8.06 58i 




1.93419 


9-99997 


20 


4i 


8.07 650 


1046 


8.07 653 


1047 


1.92 347 


9-99997 


I9 


42 


8.08 696 




8.08 700 




i .91 3oo 


9-99997 


18 


43 


8.09 718 




8.09 722 


1022 


i .90 278 


9-99997 


17 








999 






998 
















44 


8. 10 717 


976 


8. 10 720 




i .89 280 


9.99996 


16 


45 


8. ii 


6 9 3 




8. ii 


696 




1.88 3o4 


9.99996 


i5 


46 


8.12 


647 


954 


8.12 65 1 


955 


1.87 349 


9.99996 


i4 


47 


8.i3 58i 


934 


8.i3 585 


934 


1.86415 


9.99996 


i3 


48 


8.i4 


495 




8. 1 4 500 




i.85 5oo 


9.99996 


12 


49 


8.16391 


896 


8.i53 9 5 


895 

0_0 


i.84 605 


9.99996 


II 


50 


8.16268 


877 


8.16273 




i. 83 727 


9.99995 


10 


5i 


8.17 128 


840 


8.17 i33 


843 


i .82 867 


9.99996 


9 


52 


8.17 971 




8.17976 


828 


i .82 024 


9-99 99 5 


8 


53 


8.18 798 


27 
812 


8.18 8o4 


812 


i .81 196 


9-99995 


7 


54 
55 


' 8.19 610 
8. 20 407 


797 


8.19 616 
8.2o4i3 


797 

_0_ 


i. 80 384 
1.79687 


9-99995 
9.99994 


6 
5 


56 


8.21 


189 


782 


8.21 196 


702 


1.78 805 


9.99994 


4 


57 


8.21 


968 


769 


8.21 964 


769 

7 rA 


1.78036 


9.99994 


3 


58 


8.22 713 


755 


8.22 720 






1.77 280 


9.99994 


2 


5 9 


8.23456 


743 


8.23462 


742 


1.76 538 


9-99 994 


I 


60 


8.24 186 


730 


8.24 192 


73 


1.75808 


9.99993 







L. Cos. d. 


L. Cotg. 


d. 


L, Tang. 


L. 


Sin. 




' 


89. 








PP 1379 


1223 


IIOO 




999 


914 


860 




812 


769 


730 


r 


138 


122 


no 


.1 


IOO 


9* 


86 


i 


81 


77 


73 


.2 


276 


245 


220 


.2 


200 


183 


172 


2 


162 




146 


3 


414 


367 


330 


3 


300 


274 


258 


3 


244 


231 


219 


4 


552 


489 


440 


4 


400 


3 66 


344 


4 


325 


308 


292 




690 


612 






500 


457 


430 


5 


406 


tf-5 


365 


.6 


827 


734 


660 


.6 


599 


548 




6 


487 


461 


438 


7 


96s 


856 


770 


7 


699 


640 


602 


7 


568 


538 


5" 


.8 


1103 


978 


880 


.8 


799 


73i 


688 


8 


650 


615 


584 




uoi 990 


9 


899 823 


774 


9 73 1 


692 


657 



3 1 



1. 



, 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 









8.24 186 




8.24 192 






i. 7 5 808 


9.99993 


60 


I 


8.24903 


717 
706 


8.24 910 


710 
706 




i .75 090 


9.99 993 


5 9 


2 


8.25 609 




8.25 616 






1.74384 


9.99 993 


58 


3 


8.26 3o4 


095 

68 4 


8.26 3i2 


2J 




1.73688 


9.99993 


5? 


4 


8.26 988 


673 


8.26 996 


671 




1.73 oo4 


9.99992 


56 


5 


8.27 661 




8.27 669 






i .72 33i 


9.99 992 


55 


6 


8.28 324 


663 


8.28 332 


66 3 




1.71 668 


9.99992 


54 


7 


8.28977 


653 


8.28 986 


64? 




1.71 oi4 


9.99992 


53 


8 


8 .29 621 




8 .29 629 






1.70 3 7 i 


9.99992 


52 


9 


8.3o 255 


34 


8.3o 263 


6 34 




i.6 97 3 7 


9-99991 


5i 


10 


8,3o 879 


024 
616 


8.3o888 


625 




i .6 9 112 


9-99 99 1 


50 


n 


8.3i 


495 


608 


8.3i 5o5 


1 7 
607 




i.684 9 5 


9-99 99 1 


49 


12 


8.32 io3 




8.32 112 






1.67888 


9.99990 


48 


:3 


8.32 702 


599 


8.32 711 


599 




1.67 289 


9-9999 


47 


14 


8.33 292 


59 


8.33 3o2 


^ 




1.66698 


9.99990 


46 


i5 


8.33875 




8.33 886 


5 4 




i .66 1 14 


9.99990 


45 


16 


8.3445o 


568 


8.3446r 


575 

568 




i.6553 9 


9.99989 


44 


17 


8.35oi8 


560 


8.35 029 






i .64 971 


9.99989 


43 


18 


8.35 578 




8.35 590 


5 




i .64 4io 


9.99989 


42 


19 


8.36 i3i 


553 


8.36 i43 


553 




i.6385 7 


9-99 9 8 9 


4i 


20 


8. 36 678 




8. 36 689 


M 




i.63 3n 


9 . 99 9 88 


40 


21 


8.37 217 


533 


8.37 229 


54 


i .62 771 


9-999 88 


3 9 


22 


8.37750 


526 


8.37 762 






1.62 238 


9.99988 


38 


23 


8.38 276 


520 


8.38 289 


5 2 7 
520 


i .61 711 


9.99987 


3? 


24 


8.38 796 


514 


8.38 809 




i .61 191 


9.99987 


36 


25 


8.39 3io 




8.3 9 323 






i .60 677 


9.99987 


35 


26 


8. 3 9 8i8 


SOB 


8.3 9 83 2 


s9 


i. 60 168 


9.99986 


34 


27 


8.4o 320 


502 
406 


8.4o 334 


502 

406 


1.59666 


9 . 999 86 


33 


28 


8.4o 816 




8.4o83o 




i .5 


9 170 


9 . 99 9 86 


32 


2 9 


8.4i 307 


491 


8.4i 32i 


491 


i.5 


8679 


9 . 99 9 85 


3i 


30 


8.4i 792 


485 


8.4i 807 


486 


i.58 i 9 3 


9 . 999 85 


30 




L. Cos. 


d. 


L. Cotg. d. 


L. Tang. 


L. 


Sin. 




' 


88 30 . 


PP 


706 


663 


634 




599 


575 


553 




533 


SM 


496 


.! 


70.6 


66.3 


63.4 


.1 


59-9 


57-5 


55-3 


.! 


53-3 


51-4 


49- 6 


.2 


141.2 


132.6 


126.8 


.2 


119.8 


115.0 


1 10. 6 


.2 


106.6 


102.8 


99.2 


3 


211. 8 


198.9 


190.2 


3 


179.7 


172-5 


165.9 


3 


159-9 


I 54- 2 


148.8 


4 


282.4 


265.2 


253-6 


4 


239.6 


230.0 


221.2 


4 


213.2 


205.6 


198.4 


5 


353- 


33 I -5 


3 1 7- 


t e 


299-5 


287.5 


276.5 


5 


266.5 


257.0 


248.0 


.6 


423.6 


397-8 


380.4 


6 


359-4 


345-0 


33^.8 


.6 


319.8 


308.4 


297.6 


7 


494.2 


464.1 


443-8 


7 


4 r 9-3 


402.5 


387.1 


. 7 


373-1 


359-8 


347-2 


.8 


564.8 


53- 4 


507.2 


.8 


479-2 


460.0 


442-4 


.8 


426.4 


411.2 


396.8 


.9 635.4 


596.7 ' 570.6 




539.1 5 T 7-5 


497 -J 


9 479-7 




446.4 



32 



> 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 






30 


8.4i 7 9 2 




8.4i 807 






i .5s i 9 3 


9.99985 


30 


3i 

32 


8.42 272 
8.42 746 


480 

474 


8.42 287 
8.42 762 


475 


i.5 77 i3 
1.57 238 


9.99985 
9.99 984 


2 9 

28 


33 


8.43 216 


470 
464 


8.43 232 


470 
464 


i.56 768 


9.99984 


2 7 


34 
35 


8.43 680 
8.44 i3 9 


459 


8. 43 696 
8.44 i56 


460 


i.56 3o4 
i.55 844 


9.99984 
9.99 983 


26 
25 


36 


8 .44 5 9 4 


455 


8.446u 


455 


i.5538 9 


9.99983 


24 


37 

38 


8.45 o44 
8.45489 


45<> 
445 


8.45o6i 
8.45 507 


450 
446 


1.54939 
1.54493 


9.99983 
9.99982 


23 
22 


3 9 


8.45 930 


441 


8.45 948 


44' 




i.54o52 


9.99982 


21 


40 


8.46 366 


430 


8.46385 


437 




i.536i5 


9.99982 


20 


4i 


8.46 799 


433 


8.46817 


43 2 

428 


i.53 i83 


9.99 981 


19 


42 


8.47 226 




8.47 245 




1.52 755 


9.99981 


18 


43 


8.47650 


424 


8.47 669 


424 




i.52 33i 


9.99981 


'7 








419 






4* 














44 
45 


8.48 069 
8.48485 


416 


8.48 089 
8.485o5 


416 


i . 5i 911 
i.5i 495 


9.99980 
9.99980 


16 
i5 


46 


8.48896 


411 

408 


8.4* 


J 917 


412 

408 




j.5i o83 


9.99979 


i4 


4 7 


8.49 3o4 




8.49 325 






i 5o 675 


9.99979 


i3 


48 


8.49 708 


404 


8.49 729 






i .5o 271 


9.99979 


12 


49 


8.5o 1 08 


400 


8.5o i3o 


401 




i .49 870 


9.99 978 


I I 


50 


8.5o 5o4 


39 


8.5o 527 


397 




i .49 473 


9.99978 


10 


5i 


8.5o 897 


393 


8.5o 920 


393 


i .49 080 


9.99977 


9 


52 


8.5i 


287 


-Of: 


8.5i 3io 


-Of. 


i .48 690 


9.99977 


8 


53 


8.5i 


6 7 3 


330 

382 


8.5i 696 


300 
383 


i.48 3o4 


9 99977 


7 


54 


8.52 o55 




8.52 079 


180 


i .47 921 


9.99976 


6 


55 


8.52434 




8.52459 




i.4754i 


9.99976 


5 


56 


8.52 810 


376 


8.52 835 


37 6 


1.47 i65 


9.99975 


4 








373 






37: 














57 


8.53 i83 


060 


8.53 208 




i .46 792 


9-99975 


3 


58 


8.53 552 




8.53 578 




i .46 422 


9.99974 


2 


5 9 


8.53 919 


367 


8. 53 9 45 


367 

,6, 


i.46o55 


9.99974 


I 


60 


8.54 282 




8.54 3o8 




i .45 692 


9-99 974 







L. Cos. d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 




! 




88. 




PP 


470 


455 


441 




424 


4 o8 


396 


386 


376 


367 


.! 


47.0 


45-5 


44.1 


.1 


42.4 


40.8 


39-6 -i 


3 8.6 


376 


36.7 


. 2 


94.0 


91 o 


88.2 


.2 


8 4 .8 


81.6 


79.2 .2 


77-2 


752 


73-4 


3 


141.0 


136-5 


132-3 


3 


127.2 


122.4 


118.8 .3 




1128 


1 10. 1 


4 


188.0 


182.0 


176.4 


4 


169.6 


163.2 


158.4 .4 


154-4 


1504 


146.8 


5 


235-0 


227.5 


220.5 


5 


2I2.O 


204.0 


198.0 .5 


193.0 


1880 


183-5 


6 


282.0 


273.0 


264.6 


.6 


254-4 


244.8 


237-6 -6 


231.6 


225.6 


220.2 


.7 


329.0 


318.5 


308.7 


7 


296.8 


285.6 


277.2 .7 


270.2 


263.2 


256.9 


.8 


376.0 


364.0 


352.8 


.8 


339- 2 


326.4 


316.8 .8 


308.8 


300.8 


293-6 


9 4 2 3- 


409. 5 396. 9 




381.6 367.2 


356-4 -9 347-4 


338.4 


33-3 



33 



2. 



/ 


L. 


Sin. 




d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 









8.54282 


760 


8.54 3o8 






i .45 692 


9.99974 


60 


I 


8.54642 


357 


8.54669 


358 


i.45 33i 


9.99 97 3 


5 9 


2 


8.54 999 




8.55 027 




i.44 9 7 3 


9.99973 


58 


3 


8.55 354 


355 


8.55 382 


355 


i.446i8 


9.99 972 


5 7 


4 


8.55 7 o5 




8.55 7 34 


352 


1.44266 


9.99 972 


56 


5 


8.56o54 




8.56o83 


349 


i .43 917 


9.99 971 




55 


6 


8.56 4oo 


34 6 


8.56429 


346 


i.43 5 7 i 


9.99971 


54 


7 


8.56 743 


343 


8.56 773 


344 


i.43 227 


9.99 9 7 o 


53 


8 


8.67 o84 




8.57 n4 




1.42 886 


9.99 9-70 


52 


9 


8.57 421 


337 


8.57452 


338 


1.42 548 


9-99 9 6 9 


5i 


10 


8.5 77 5 7 


33 


8.5 77 88 


33 6 


I .42 212 


9-99 9 6 9 


50 


1 1 


8.58 089 




8.58 121 


333 
33 


I .4l 879 


9.99968 


49 


12 

i3 


8.584i9 
8.58 7 4 7 


328 


8.5845i 
8.58 779 


328 


i .41 549 

I .4l 221 


9.99968 
9.99 9 6 7 


48 
47 


i4 
i5 


8.59 072 
8.5 9 3 9 5 


325 
323 


8.5 9 io5 
8.69428 


326 
323 


i.4o 895 
1 .40 672 


9.9996-7 
9.9996-7 


46 

45 


16 


8.69 716 


318 


8.59 7 49 


321 


i .4o 261 


9.99966 


44 


17 
18 


8.6oo33 
8.60 349 


3 i6 


8.60068 
8.60 384 


316 


1.39 932 
1.39 616 


9.99966 
9.99965 


43 

42 


'9 


8.60 662 


3*3 


8.60 698 


3i 


| 


1.39 3o2 


9.99 964 


4i 


20 


8.60 973 


39 


8. 6 r 009 


3" 


i.38 991 


9.99 9 64 


40 


21 


8.61 282 


307 


8.61 3i 9 


310 


i. 3868i 


9.99963 


3 9 


22 


8.61 58 9 




8.61 626 




i.38 3 7 4 


9.99963 


38 


23 


8.61 894 




8.61 931 


35 


i.38 069 


9.99962 


37 










302 








3 


1 












24 


8.62 196 


301 


8.62234 




i. 3 77 66 


9.99962 


36 


25 


8.62 497 




8.62 535 




1.37465 


9.99961 


35 


26 


8.62795 


298 


8.62 834 


299 


1.37 166 


9.99961 


34 


27 
28 


8.63 091 
8.63 385 


296 
294 


8.63 i3i 
8.63426 


297 
295 


i.36 869 
i.365 7 4 


9.99960 
9.99960 


33 

32 


29 


8. 63 678 


293 


8.63 718 


292 


1.36282 


9.99 9 5 9 


3i 


30 


8. 63 968 


290 


8.64 009 


291 


i.35 991 


9.99959 


30 




L. 


Cos. 




d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 




' 


87 3O'. 


PP 


360 


350 


34 




33 


320 


310 


300 


290 


285 


.1 


36 


35 


34 


.1 


33 


32 


31 .1 


30 


2 9 


28.5 


.2 


72 


7 


68 


.2 


66 


64 


62 .2 


60 


58 


57- 


3 


1 08 


105 


1 02 


3 


99 


9 6 


93 -3 


90 


87 


85-5 


4 


144 


140 


136 


4 


132 


128 


124 .4 


120 


116 


114.0 




180 




170 




65 


160 




ISO 


145 


M2.5 


.6 


216 


2IO 


204 


.6 


198 


192 


1 86 .6 


1 80 




171.0 


7 


252 


245 


238 


7 


231 


224 


217 .7 


2IO 


203 


199-5 


.8 


288 


280 


272 


.8 


264 


256 


248 


240 


232 


228.0 


.0 


324 


315 


306 




2Q 7 


288 


279 .9 270 


161 


256.5 



2 3O . 



/ 


L. 


Sin. 




d. 


L. Tang. 


d. 


L. 


Cotg. 


L. 


Cos. 






30 


8.63 9 68 




8 .64 009 




i .35 991 


9.99 9 5 9 


30 


3i 

32 


8.64256 
8.64543 


287 


8.64 298 
8.64 585 


289 
287 


i .35 702 
i.35 415 


9 . 999 58 


2 9 

28 


33 


8.6 


4827 




284 
283 


8.64 870 


285 
284 


i.35 i3o 


9.99957 


2 7 


34 


8.65 no 




8.65 i54 




1.34846 


9.99 9 56 


26 


35 


8.653 9 i 




8.65435 




1.34565 


9 . 999 56 


25 


36 


8. 65 670 


279 


8. 65 715 


280 


1.34285 


9.99955 




24 


37 
38 


8.65 9 47 
8.66223 


277 
276 


8.65 99 3 
8.66 269 


278 
276 


i .34 007 
i.33 7 3i 


9.99955 
9.99954 


23 
22 


3 9 


8.66 497 


274 


8.66543 


274 


1.33457 


9-999 5 4 


21 


40 


8.66 769 


272 


8.66816 


273 


i.33 1 84 


9-99 9 53 


20 


4i 

42 


8.67 039 
8.6 7 3o8 


270 
269 


8.67 087 
8. 67 356 


271 
269 


i.32 913 
i.32 644 


9 . 999 5 2 
9 . 999 52 


9 

18 


43 


8.6 7 5 7 5 


267 
266 


8.67624 


266 


i.3 2 3 7 6 


9.99961 


7 


44 
45 


8.67841 
8.68 io4 


263 


8.67 890 
8.68 1 54 


264 


i .32 no 
i.3i 846 


9.99961 
9.99950 


16 

i5 


46 


8.68 367 


263 
260 


8.684i 7 


263 
261 


i.3i 583 


9.99949 


i4 


47 


8.68627 




8.68678 


260 


i.: 


I 322 


9.99949 


i3 


48 


8.68886 


259 


8.68 938 




1.2 


(I 062 


9.99948 


12 


49 


8.69 1 44 


258 


8.69 196 


258 


i.3o 8o4 


9.99948 


I I 


50 


8.69 4oo 


256 


8.69453 


257 


i.3o54 7 


9.99 947 


10 


5i 


8. 69 654 


254 


8.69 708 


255 


i .3o 292 


9.99946 


9 


52 


8.69 907 




8.69 962 




i.3oo38 


Q. 99946 


8 


63 


8.70 i5 9 


252 


8.70 214 


252 


1.29 786 


9.99945 


7 










250 






25 
















54 


8.70 409 




8.70465 




i. 29 535 


9.99944 


6 


55 


8.70658 




8.70 714 




.29 286 


9.99944 


5 


56 


8.70 905 


247 

246 


8.70 962 


240 
246 


.29 o38 


9.99943 


4 


57 


8.71 i5i 




8.71 208 




.28 792 


9.99942 


3 


58 


8.71 3 9 5 




8. 7 i453 




.28547 


9.99942 


2 


5 9 


8.71 638 


243 


8. 7 i 697 


244 


.28 3o3 


9.99941 


I 


60 


8.71 880 




8.71 940 




i .28 060 


9-99 9 4o 







L. 


Cos. 




d. 


L. Cotg. 


d. L. Tang. 


L. 


Sin. 




f 








87. 










PP 


280 


275 


270 




265 


260 


255 




250 


245 


240 


.1 


28.0 


27-5 


27.0 


.1 


26.5 


26.0 


25-5 


.1 


25.0 


24-5 


24.0 


.2 


56.0 


55-0 


54-o 


.2 


53-o 


52.0 


51.0 


.2 


50.0 


49- 


48.0 


3 


84.0 


82.5 


81.0 


3 


79-5 


7 8.o 


76.5 


3 


75-0 


73-5 


72.0 


4 


112. 


1 10.0 


108.0 


4 


106.0 


104.0 


102.0 


4 


100.0 


98.0 


96.0 


t e 


140.0 


137-5 


i35-o 


5 


132-5 


130.0 


I2 7-5 


5 


125.0 


122.5 


12O.O 


.6 


168.0 


165.0 


162.0 


.6 


159.0 


156.0 


i53-o 


.6 


150.0 


147.0 


144.0 


.7 


196.0 


192.5 


189.0 


7 


185-5 


182.0 


178-5 


7 


175-0 


i7i-5 


168.0 


.8 


224.0 


22O.O 


216.0 


.8 


2I2.O 2O8.O 


204.0 


.8 


200.0 


196.0 


192.0 


9 




247-5 


243.0 


9 


238.5 234.0 




.9 225.0 




216.0 



35 



( 


L. 


Sin. 




d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 









8.71 880 


340 
239 
238 
237 
235 
234 
232 
232 
230 
229 

228 
226 
226 
224 
223 

222 
2 2O 
2 2O 
2I 9 
2I 7 

216 
216 

2I 4 
213 
212 
211 
210 
209 
208 
208 




8.71 940 


241 
239 
239 
237 

236 

234 
234 
232 
231 
229 

229 
227 
226 
225 
224 

222 
222 
220 
2I 9 
2I 9 

217 

216 

215 
214 

213 

211 
211 

210 
209 
208 


i .28 060 


9 .99 940 


60 


I 

2 

3 

4 
5 
6 

8 
9 


8.72 I2O 

8. 7 235 9 

8.72 597 

8.72 834 
8.73069 
8. 7 3 3o3 

8.73 535 
8.73 767 
8. 7 3 997 




8.72 181 
8.72 420 
8.72 659 

8.72 896 
8. 7 3i3 2 
8. 7 3366 

8.73 600 
8. 7 383 2 
8 . 74 06-3 


i .27 819 
1.27 58o 
1.27 34 i 

1.27 io4 
1.26868 
1.26634 

i .26 4oo 
1.26 168 
i .25 937 


9.99 940 
9.99939 
9.99938 

9.99938 
9.99937 
9.99936 

9.99936 
9.99935 
9.99934 


5 9 
58 
5 7 

56 
55 
54 
53 

52 

5i 


10 


8. 7 4 226 




8.74 292 


i.25 708 


9.99934 


50 


ii 

12 

i3 

i4 
i5 
16 

17 

18 

'9 


8.74454 
8.74 680 
8.74 906 

8. 7 5 i3o 
8. 7 5353 
8.75575 

8. 7 5 79 5 
8. 7 6oi5 
8. 7 6234 




8.74 52i 
8. 7 4748 
8.74974 

8.75 199 
8.75 423 
8.75 645 

8. 7 586 7 
8.76087 
8.76 3o6 


i.25 479 

I .25 252 
I .25 026 

i .24 801 
1.24 577 

1.24355 

1.24 i33 
i.23 913 
i.23 6 9 4 


9.99933 
9.99932 
9.99932 

9.99931 
9.99930 
9.99929 

9.99929 
9.99928 
9.99 927 


4 9 
48 

47 
46 
45 
44 

43 

42 

4r 


20 


8. 7 645i 




8.76 525 


I .23 475 


9.99926 


40 


21 
22 
23 

24 
25 

26 

27 
28 
29 


8.76667 
8. 76 883 
8.77097 

8.77 3io 

8.77 522 
8.77943 

8.78 i52 
8.78 36o 




8.76 742 
8. 7 6 9 58 
8.77173 

8. 77 38 7 
8.77 600 
8.77 811 

8.78 022 
8. 7 8232 

8. 7 844i 


i.23 258 

I .23 O42 
I .22 827 

1.22 6l3 
I .22 400 
I .22 189 

I .21 978 
I. 21 7 68 
I .21 559 


9.99926 
9.99925 
9.99924 

9.99923 
9.99923 
9.99922 

9.99921 
9.99920 
9.99 920 


39 

38 

37 
36 
35 
34 
33 

32 

3i 


30 


8.78 568 




8.78 649 


i. 2 1 35 1 


9.99 919 


30 




L. 


Cos. 




d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 




' 






80 


3O 


. 






PP 


238 


234 


229 




.2 

3 
4 

: 7 8 

9 


225 


220 


216 


2X2 


208 


204 


.2 

3 

4 

9 


23.8 
47.6 
71.4 

95-2 
119.0 
142.8 

166.6 
190.4 


23-4 
4 6.8 
70.2 

93-6 
117.0 

140.4 

163.8 

187-2 


22.9 

a; 

91.6 

"4-5 
137-4 

160.3 
183.2 
206. i 




22.5 

45-0 
67-5 

90.0 
112.5 
135-0 

157-5 
180.0 

52-5 


22. 
44.0 

66.0 

88.0 

IIO.O 

132.0 

176.0 


21.6 .1 

43-2 ' .2 

64.8 .3 
86.4 1 .4 

108.0 .5 
129.6 ! .6 

151-2 .7 

172.8 .8 

194.4 .9 


21.2 

42.4 
63.6 

84.8 

ro6.o 
127.2 

148.4 
[69.6 

90.8 


20.8 

41.6 
62.4 

83.2 
104.0 
124.8 

145.6 
166.4 

187.2 


20.4 
40.8 
61.2 

81.6 

IO2.O 
122-4 

142.8 
163.2 
183.6 



36 



3 3O'. 



/ 


L. Sin. 


d. 


L. Tang. d. 


L. Cotg. 


L. 


Cos. 




30 


8.78 568 




8.78 64 9 






i .21 35i 


9-99 9'9 


30 


3i 


8.78 774 




8.78 855 


206 
206 


1.21 145 


9.99918 


2 9 


32 


8.78979 




8.70. 061 






i .20 939 


9.99917 


28 


33 


8.79 i83 


204 


8.79266 


205 


I .20 734 


9.99917 


27 








203 






20< 














34 
35 


8.79 386 
8.79 588 


202 


8.79 4?o 
8.79673 


20 3 


i .20 53o 

i .20 327 


9.99916 
9.99915 


26 
25 


36 


8.79789 




8.79875 




I .20 125 


9.99 9 i4 


24 














20J 














37 


8.79990 




8.80076 






1.19 924 


9.99913 


23 


38 


8.80 189 


199 


8.80 277 




1 .19 723 


9.99913 


22 


3 9 


8. 80 388 


199 


8.80476 


199 


i . 19 524 


9.99912 


21 


40 


8. 80 585 


197 


8.8o6 7 4 


198 


1.19 326 


9.99911 


20 


4i 


8.80782 


197 
1 06 


8.80 872 


198 
1 06 


1.19 128 


9.99910 


i 9 


42 

43 


8.80 978 
8.81 173 


'95 


8.81 068 
8.81 264 


I 9 6 


1. 18 932 
i.i8 7 36 


9.99909 
9.99909 


1 8 

17 








194 






X 9S 














44 


8.81 36 7 




8.81 459 






1. 18 54i 


9.99908 


16 


45 


8.81 56o 


Z 93 


8.81 653 


X 94 


i.i8347 


9.99907 


i5 


46 


8.81 752 


192 


8.81 846 


X 9-. 




1. 18 i54 


9.99906 


i4 


47 


8.81 944 


192 


8.82o38 


192 


i . 17 962 


9.99905 


i3 


48 


8.82 1 34 




8.82 23o 




1.17 770 


9.99904 


12 


49 


8.82 324 


190 


8.82 420 


190 


1.17 58o 


9.99904 


I I 


50 


8.82 5i3 




8.82 610 




i.i 7 3 9 o 


9 . 999 o3 


10 


5i 


8.82 701 




8.82 799 


1 88 


I.I7 201 


9.99902 


9 


r 


8.82888 




8.820,87 


18 


5 


1.17 oi3 


9.99901 


8 


53 


8.83075 


lOJ 

186 


8.83i 7 5 


1 81 


i 


1.16825 


9.99900 


7 


54 


8.83 261 


18 


8.8336i 


i8C 


( 


1.16 63 9 


9.99899 


6 


55 


8.83446 




8.83547 




1. 16453 


9.99898 


5 


56 


8.8363o 


104 

183 


8.83 7 3 2 


105 
184 


1.16268 


9.99898 


4 


57 


8.83 8i3 


181 


8.83 9 i6 


1 8 A 


1. 1 6 o84 


9-99 8 97 


3 


58 


8.83 996 




8.84 ioo 





i . 1 5 900 


9 . 99 8 9 6 


2 


5 9 


8.84 177 


181 


8.8 


4282 


182 


i . 


[5 718 


9-99 8 95 


I 


60 


8.84 358 




8.84464 




i.i5536 


9.99 8 9 4 







L. 


Cos. d. 


L. Cotg. d 


. L. Tang. 


L. 


Sin. 


' 






86. 








PP 


201 


198 


195 




192 


I8 9 


187 




185 


183 


181 


.1 


20.1 


19.8 


19-5 


.1 


19.2 


,8.q 


i8. 7 


. x 


18.5 


18.3 


18.1 


.2 


40.2 




39.0 


.2 


38.4 


37-8 


37-4 


.2 


37- 


3 6.6 


36-2 


3 


60.3 


59-4 


58-5 


3 


57-6 


56-7 


56.1 


3 


55-5 


54-9 


54-3 


4 


80.4 


79-2 


78.0 


4 


76.8 


75-6 


74-8 


4 


74.0 


73-2 


72.4 




100-5 


So 


97-5 


.5 


96.0 


Q4-5 


93-5 


5 


92-5 


91.5 


90-5 


.6 


1 20. 6 


8 


117.0 


.6 


115-2 


"3-4 


II2.2 


.6 


III.O 


109.8 


co8.6 




140.7 


138.6 


136-5 


7 


134-4 


132.3 


130.9 


7 


129.5 


128.1 


126.7 




160.8 


158.4 


156.0 


.8 




151.2 


149.6 


.8 


148.0 


146.4 


r 44 .8 


9 


180.9 


178.2 175.5 








168.3 




164.7 '62.9 



4. 



/ 


L. 


Sin. 




d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 







8.84358 


_0 


8.84464 




1. 15536 


9.99 894 


60 


I 


8. 8453 9 


179 


8.84646 


1 80 


.16 354 


9.99893 


r 


2 


8.8 


4718 






8.84 826 




.i5 174 


9.99 892 


Do 


3 


8.8 


48 97 




179 


8.85 006 


1 80 


.14994 


9.99891 


5? 


4 


8.85 075 


178 


8.85 185 


179 

T ,o 


.i48i5 


9.99 891 


56 


5 


8.85 252 




8.85363 




.i463 7 


9.99890 


55 


6 


8.85429 


177 


8.85 54o 


177 


. i4 46o 


9.99889 


54 


<7 


8.85605 


176 


8.85 717 


177 

176 


i.i4283 


9-^ 


> 9 888 


53 


8 


8.85 780 




8.858 9 3 




i . i4 107 


9.99887 


52 


9 


8.85 955 


175 


8.86069 


176 


i.i3 9 3i 


9.99 886 


5i 


10 


8.86 128 


173 


8.86 243 


174 


i.!3 7 5 7 


9.99 885 


50 


1 1 


8.86 3oi 




8.86417 




1. 13583 


9.99884 


49 


12 


8.86474 




8.865 9 i 




1. 1 3 409 


9.99 883 


48 


i3 


8.86645 


171 


8.86 7 63 


172 


i.i323 7 


9.99 882 


4? 


i4 


8.86816 


171 


8.86935 


172 


i.i3 065 


9.99 881 


46 


i5 


8.86987 


1 60 


8.87 106. 




I .12 894 


9.99 880 


45 


16 


8.87 i56 


169 


8.87277 


171 

170 


I . 12 723 


9.998-79 


44 


18 


8. 87 325 
8.87494 


169 


8. 87 447 
8.87 616 


169 


1. 12 553 
1. 12 384 


9-99 8 79 
9-99 8 78 


43 

42 


'9 


8.87661 


107 
168 


8.87785 


169 


I .12 2l5 


9.99 8 77 


4i 


20 


8.87 829 


1 66 


8.8 79 53 


..- 


1. 12 047 


9.99 876 


40 


21 


8.87995 


1 66 


8.88 120 




1. 1 1 880 


9.99 875 


3 9 


22 


8.88 161 




8.88 287 


1 7 


i. ii 7i3 


9.998-74 


38 


23 


8.8 


83 2 6 


164 


8.8 


8453 






i. ii 547 


9.99873 


3? 


24 


8.88 490 


164 


8.88 618 


.fie 


i. ii 382 


9.99872 


36 


25 


8.8 


8654 


if>i 


8.8 


8 7 83 






i. ii 217 


9.99871 


35 


26 


8.88817 


103 

163 


8.8 


8 9 48 




163 


I. 1 1 052 


9.99870 


34 


27 


8.88 980 


162 


8.89 in 


163 


1.10889 


9 . 99 86 9 


33 


28 


8.8 


9 142 






8.89 274 




i . 10 726 


9.99868 


32 


2 9 


8.89 3o4 


162 


8.8 9 43 7 


163 


i.io563 


9.99867 


3i 


30 


8.89464 




8.89 598 




I.I0402 


9.99 866 


30 




L. 


Cos. 




d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


9 


85 30 . 


PP 


181 


179 


177 




175 


173 


171 


1 68 


166 164 


.1 


18.1 


17.9 


17.7 


.! 


17-5 


7-3 


17.1 ; i 


16.8 


16.6 16.4 


.2 


36.2 


SS-8 


35-4 


.2 


35-o 


34-6 


34-2 2 


33.6 


33-2 32.8 


3 


54-3 


53-7 




3 


52-5 




51.3 3 


50.4 


49.8 49.2 


4 


72.4 


71.6 


70.8 


4 


70.0 


6q.2 


68.4 4 


67.2 


66.4 65.6 




9-5 




88.5 




87- S 


86. S 


85-5 ' 5 


84.0 


83.0 82.0 


.6 


108.6 


107.4 


106.3 


.6 


105.0 


103.8 


102.6 | 6 


100.8 


99.6 98.4 


7 


126.7 


125-3 


123.9 


7 


122.5 


121. 1 


119.7 .7 


117.6 


116.2 114.8 


.8 


144-8 


143.2 


141.6 


.8 


140.0 


138.4 


136.8 : .8 


134-4 


132.8 131.2 


9,, 


162.9 


161.1 


*59-3 


9 


'57-5 


155-7 


153.9 -9 iS 1 - 2 


149.4 r 47-6 



38 



4 30 . 



; 


L. Sin. 


d. 


L. Tang. 


d. L. 


Cotg. 


L 


Cos. 






30 


8.8 9 464 




8.89 5 9 8 






I . 


10 402 


9.99 866 


30 


3i 

32 


8.C 
8.C 


*9 625 

19784 


159 


8.8 9 760 
8.8 9 9 2o 


162 

160 


I . 
I . 


IO 240 

10 080 


9.99 865 
9.99864 


2 9 
28 


33 


8.8 99 43 


159 

159 


8. 9 o 080 


160 
160 


I . 


09 920 


9.99 863 


27 


34 


8. 9 o 1 02 


I eg 


8. 9 o 240 




I .09 760 


9.99 862 


26 


35 


8. 9 o 260 




8. 9 o 3 99 




I . 


09 601 


9.99 861 


25 


36 


8. 9 o 417 


157 


8. 9 o 55 7 


158 


1.09 443 


9.99 860 


24 










J 57 






15 

















37 
38 


8. 9 o 674 
8. 9 o 730 


156 


8.90715 
8.90872 


157 


1.09 285 
i .09 128 


9.99859 
9 . 99 858 


23 
22 


39 


8. 9 o885 


155 


8.91 029 


157 


i .08 971 


9-99857 


21 


40 


8.9; o4o 


155 


8.91 185 


150 


1.08 8 1 5 


9 . 99 856 


20 


4i 

42 


8. 9 i i 9 5 
8. 9 i 34 9 


155 
154 


8.91 34o 
8.91 495 


155 


i .08 660 
i. 08 505 


9.99855 
9.99 854 


i a 


43 


8. 9 i 5o2 


I 53 


8.91 650 


X 55 


i. 08 35o 


9.99853 


17 


44 
45 


8. 9 i 655 
8. 9 i 807 


153 
152 


8.91 8o3 
8.91 9 5 7 


153 
154 


i .08 197 
i. 08 o43 


9.99 852 
9.99 85i 


16 

i5 


46 


8. 9 i 9 5 9 


152 


8. 9 2 110 


J 53 


i .07 890 


9.99850 


i4 


47 


8. 9 2 IIO 


151 


8. 9 2 262 


152 


1.07 738 


9.99848 


i3 


48 


8. 9 2 261 


151 


8. 9 24i4 




i. 07 586 


9.99847 


12 


49 


8.92 4i i 


150 


8. 9 2 565 


*5 




1.07435 


9.99 846 


II 


50 


8.92 56 1 


15 


8. 9 2 716 


15 




i .07 284 


9.99845 


10 


5i 


8.92 710 


149 


8. 9 2866 


S" 


1.07 i34 


9.99 844 


9 


52 


8.92859 




8. 9 3 016 




i .06 984 


9.99843 


8 


53 


8. 9 3 007 




148 

M7 


8. 9 3 165 


149 

148 


i.o6835 


9.99 842 


7 


54 
55 


8.93 i54 
8.93 3oi 


*47 


8. 9 33i3 
8.93462 


149 


i. 06 687 
i.o6538 


9.99 84i 
9.99 84o 


6 
5 


56 


8.93448 


X 47 
146 


8.93 609 


J 4 
*4 


/ 

7 


i .06 391 


9-99 83 9 


4 


57 


8. 9 3 5 9 4 


1 46 


8. 9 3 7 56 




i .06 244 


9 . 99 838 


3 


58 


8. 9 3 7 4o 




8.93 903 




i .06 097 


9 .( 


?9 83- 


1 


2 


5 9 


8.93885 


145 


8.94 049 


140 
., 


i.o5 951 


9 . 99 836 


I 


60 


8.94 o3o 




8.94 195 






i.o5 805 


9 . 99 834 







L. 


Cos. 




d. 


L. Cotg. 


d. 


L. Tang. 


L 


Sin. 




' 








85. 






PP 


162 


160 


159 




157 


155 


153 




151 


149 


147 


.1 


16.2 


16.0 


15-9 


.! 


J5-7 


i-vS 


15-3 


I 


iS-i 




14.7 


.2 


32-4 


32.0 


31-8 


.2 




31.0 


30.6 


2 


30.2 


2 J.8 


29.4 


3 


48.6 


48.0 


47-7 


3 


47.1 


4b-5 


45-9 


3 


45-3 


44-7 


44.1 


4 


64.8 


64.0 


63.6 


4 


62.8 


62.0 


61.2 


4 


60.4 


59-6 


58.8 


5 


81.0 


80.0 


79-5 


5 


78.5 


77-5 


76.5 


5 


75-5 


74-5 


73- S 


.6 


97.2 


96.0 


95-4 


.6 


94.2 


93.0 


91.8 


6 


90.6 


89.4 


88.2 


7 


"3-4 


112. 


111.3 


7 


109.9 


08.5 


107.1 


7 


i5-7 


104.3 


102.9 


.8 


129.6 


128.0 


127.2 


.8 


125.6 


124 o 


122.4 


8 


120.8 


119.2 


117.6 


9 145-8 


144.0 


143- 1 




I4J-3 139-5 


137-7 


9 -135-9 


J34-i 





3 9 



5. 



f 


L. 


Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L 


Cos. 







8 .94 o3o 




8.94 195 






i .o5 805 


9.99 834 


60 


I 


8 ,g4 174 


144 




4340 


MS 


i .o5 660 


9.99 833 


5 9 


2 


8. 9 4 3i 7 




8. 9 4485 




i.o5 515 


9.99 832 


58 


3 


8.94461 




8.94 63o 


M 


b 


i .o5 370 


9.99 83i 


5 7 










142 








i^ 


^ 










I 


4 


8.94 6o3 




8.5 


4 773 








i .o5 227 


9.99 83o 


56 


5 


8.94746 




8.5 


4 917 






i.o5 o83 


9.99 829 


55 


6 


8.94 887 


141 


8.95 060 


X 4 


j 


i ,o4 940 


9.9982 


9 


54 










142 








i/ 


7 












7 


8.95 029 




8.95 202 




i .04 798 


9.99827 


53 


8 


8.95 170 




8. 9 5 344 




i.o4656 


9.99 825 


52 


9 


8.95 3io 


140 


8.95486 


142 


i .04 5i4 


9.99 824 


5i 


10 


8. 9 5 450 




8. 9 5 627 


141 


i.o4 3 7 3 


9.99 823 


50 


ii 


8. 9 5 58 9 


I 39 


8. 9 5 7 6 7 


140 


i.o4233 


9.99 822 


49 


12 


8.95 728 




8.95 908 




i .o4 092 


9.99 821 


48 


i3 


8. 9 5 867 


Z 39 
138 


8.96 047 


140 


i.o3 953 


9.99 820 


47 


i4 
i5 


8.96 oo5 
8.96 i43 


138 


8.96 187 
8.96 325 


138 


i.o3 8r3 
i.o3 675 


9.99819 
9.99 817 


46 
45 


16 


8.96 280 


J 37 


8.96464 


138 


i.o3 536 


9.99 816 


44 


17 


8.96417 


i 6 


8.96 602 






i.o3 398 


9.99 8i5 


43 


18 


8. 96 553 


13 


8.96 739 


*3 




i ,o3 261 


9.99 8i4 


42 


'9 


8.96 689 


136 


8.96 877 




i.o3 123 


9.99 8i3 


4i 


20 


8.96 825 


I* 


8.97 oi3 


130 


i .02 987 


9.99 812 


40 


21 


8 . 96 960 


135 


8.97150 


135 


i .02 85o 


9.99 810 


3 9 


22 


8.97 095 




8.97 285 


1-36 


i .02 715 


9.99809 


38 


23 


8.97 229 




8.97 421 




i .02 579 


9.99 808 


3 7 


24 


8. 97 363 


133 


8.97 556 




1.02 444 


9.99807 


36 


25 


8.97496 




8.97 691 




i .02 3og 


9.99 806 


35 


26 


8.97 629 




8.97825 


X 34 


i .02 175 


9.99 8o4 


34 


















X 3 


4 












27 


8.97762 




8.97 959 




I .02 O4l 


9.99 8o3 


33 


28 


8.97 8 9 4 




8.98 092 




I .OI 908 


9.99 802 


32 


29 


8.c 


)8 026 


132 


8.98 225 


133 


i. oi 775 


9.99 801 


3i 


30 


8.1 


>8i5 7 




8.98 358 




i .01 642 


9.99 800 


30 




L. 


Cos. 




d. 


L. Cotg. 


d. L. Tang. 


L. 


Sin. 




' 


84 3O . 


PP 


145 


143 


141 




139 


138 


136 


135 


133 


131 


.2 


14-5 

2Q.O 


28.6 


14.1 
28.2 


.1 

.2 


13-9 
27.8 


13.8 
27.6 


13-6 -I 
2 7 .2 .2 


13-5 
27.0 


11:1 


2&2 


3 


43-5 


42.9 


42.3 


3 


41-7 


41.4 


40-8 .3 


40-5 


39-9 


39-3 


4 


S 8.o 


57-2 


56.4 


4 


55-6 


55-2 


54-4 -4 


54- 


53.2 


52.4 


.5 


72.5 


7 z -5 


70.5 


. e 


60 5 


69.0 


68.0 .5 


07- s 


66.5 


6s* 5 


.6 


87.0 


85.8 


8 4 .6 


.6 


83-4 


82.8 


81.6 .6 


81.0 


79-8 


78.6 


7 


101.5 


100. 1 


9 8. 7 


7 


97-3 


96.6 


95-2 .7 


94-5 


93-1 


91.7 


9 


1 30. s 


114.4 

128.7 


126.9 




25-1 


124.2 


122.4 -9 ] 




119.7 


117.9 



5 30 . 





L. 


Sin. 




d. 


L. Tang. 


d. 


L. 


Cotg. 


L. 


Cos. 






30 


8.98 157 


8.98 358 






I .01 642 


9 . 99 8ot 




30 


3 1 


8.98 288 




8.9 


8 4 9 o 




132 


i .01 5io 


9-99 79 8 


2 9 


32 


8.98419 




8.9 


8 622 






i. 01 378 


9.99 797 


28 


33 


8.98 549 


130 


8.9 


8 753 




13 




i .01 247 


9.99 796 


27 


































34 


8.98 679 




8.9 


8884 








i .01 116 


9-99 795 


26 


35 
36 


8. 9 8 808 
8. 9 8 9 37 


129 


8.99 015 
8.99 i45 


130 


i .00 9 85 
i. oo 855 


9 v -99 79 3 
9.99 792 


25 

24 


37 


8. 99 066 


129 
128 


8.99275 


130 


i .00 725 


9.99791 


23 


38 


8. 99 i 9 4 




8.99405 




i .00 5 9 5 


9.99790 


22 


3 9 


8. 99 322 


128 


8.99 534 


129 

T _o 


i .00 466 


9.99 788 


21 


40 


8. 99 450 




8.99 662 




i. oo 338 


9.99 787 


20 


4i 


8. 99 5 77 


127 
127 


8.99791 


129 
128 


I .OO 2O 9 


9.99 786 


'9 


42 


8.99 704 


126 


8.99919 






I .00 08 I 


9-99 7 8 5 


18 


43 


8.99 83o 


126 


9 . oo o46 


I2/ 
128 


o. 99 9 54 


9.99783 


17 


44 


8.99 g56 


126 


9.00 174 




0.99 826 


9.99 782 


16 


45 


9.00 082 




9.00 3oi 




0.99699 


9.99 781 


i5 


46 


9.00 207 


12 5 
125 


9.00 427 


126 


0.99 573 


9.99 780 


i4 


47 

48 


9.00 332 
9.00 456 


124 


9.00 553 
9.00 679 


126 


0.99 447 
0.99 32i 


9.99778 
9.99 777 


i3 

12 


49 


9.00 58 1 


125 


9.00 805 


126 


o. 99 195 


9.99 776 


I I 


50 


9.00 704 


123 


9.00 gSo 




0.99 070 


9-99775 


10 


5i 


9.00 828 


124 
123 


9.01 o55 


124 


0.98 945 


9-99 77 3 


9 


52 


9.00 95 1 




9.01 179 




0.98 821 


9.99 772 


8 


53 


9.01 074 


123 


9.01 3o3 




O ' 


? 86 97 


9.99771 


7 




















4 














54 


9.01 196 




9.01 427 




0.98 573 


9-99 7 6 9 


6 


55 


9.01 3i8 




9.01 55o 




O . ' 


?845o 


9-99 7 68 


5 


56 


9.01 44o 




9.01 673 


123 


0.98 327 


9.99 767 


4 










121 










1 














57 


9.01 56i 




9.01 796 




o. 9 8 204 


9.99 7 65 


3 


58 


9.01 682 




9.01 918 




0. 


58 082 


9.99 764 


2 


5 9 


9.01 8o3 




9.02 o4o 




o. 9 7 960 


9-99 7 63 


I 


60 


9.01 923 




9.02 162 




0.97838 


9-99 76i 







L. 


Cos. 


d. 


L. Cotg. 


d 


. 


L. 


Tang. 


L. 


Sin. 




' 


84. 


PP 


13 


129 


128 




126 


5 


123 




122 


121 


I2O 


.1 


13-0 


12.9 


12.8 


.1 


12.6 


12.5 


12.3 


.! 


12.2 


12. I 


12.0 


.2 


20. o 


25.8 


25.6 


.2 


25.2 


25.0 


24.6 


.2 


24.4 


24.2 


24.0 


3 


39-o 


38.7 


38.4 


3 


37-8 


37-5 


3 6 -9 


3 


36.6 


36.3 


36.0 


4 


52.0 


51.6 


51.2 


4 


5 o. 4 


50.0 


49.2 


4 


48.8 


48.4 


48.0 


5 


65.0 


64- S 


64.0 




63.0 


62. S 


61.5 




61.0 


60.5 


6o.O 


.6 


78.0 


77-4 


76.8 


.6 


75-6 


75-o 


73-8 


.6 


73-2 


72.6 


72.0 


7 


91.0 


9-3 


89.6 


. 7 


88.2 


87-5 


86.1 


7 


8.S-4 


84.7 


84.0 


.8 


104.0 
117.0 


103.2 

16. i 


102.4 

IIS- 2 




100.8 
"3-4 


loo.o 

112.5 


98-4 
110.7 


.8 
9 


97.6 
109.8 


108.9 


96.0 
I08.0 



6. 



/ 


L. 


Sin. 




d. 


L. Tang. 


d, 


L. 


Cotg. 


L. 


Cos. 









9.01 923 




9.02 162 




0.97 838 


9-99 7 bl 


60 


I 


9.02 o43 


120 


9.02 283 


121 


0.97 717 


9.99760 


5 9 


2 


9.02 i63 




9.02 4o4 






0.97 596 


9.99 7 5 9 


58 


3 


9.02 283 




9.02 525 






-97475 


9.99 757 




57 










119 








12 


> 














4 


9.O2 4O2 


118 


9.02 645 




0.97 355 


9.99 7 56 


56 


5 


9.O2 52O 




9.02 766 


121 


0.97 234 


9.99 755 




55 


6 


9.02 63g 


119 


9.02 885 


II 9 


0.97 115 


9.99 7 53 


54 










118 








12 


j 














7 


9 .02 7 5 7 




9.o3 005 




0.96 995 


9.99752 


53 


8 


9.02 874 




9.o3 124 




0.96 876 


9.99 751 




52 


9 


9.02 992 




9.0 


3 242 




118 


0.96 758 


9-99 749 


5i 


10 


9o3 109 


117 


9-o3 36i 


II 


) 
3 


0.96 63g 


9-99 748 


50 


ii 


g.oS 226 


116 


9.o3 479 




a 


0.96 52i 


9.99 747 


49 


12 


9.o3 342 




9.o3 597 






0.96 4o3 


9.99 745 


48 


1 3 


9. o3458 




9.0 


3 7 i4 




117 


0.96 286 


9.99 744 


4? 










116 








II 


* 














i4 


9.o3 574 


116 


9.03 832 






0.96 168 


9.99 742 


46 


i5 


9.o3 690 




9.o3 948 






0.96 o52 


9.99 74 1 




45 


16 


9 .o3 805 


"5 


9.04 o65 


117 

116 


0.95 935 


9.99 740 


44 


'7 


9.o3 920 


114 


9.04 181 


116 


o.< 


? 58i 9 


9.99738 


43 


18 


9.04 o34 




9.04 297 




0. 


P 5 7 o3 


9.99737 


42 


'9 


9.04 1 49 




9.04 4i3 




0.95 587 


9.99 7 36 


4i 


20 


9.04 262 


114 


9.04 528 


"5 


0.95 472 


9.99 734 


40 


21 


9.04 376 


114 


9.04 643 


"5 


0.95 357 


9.99 7 33 


3 9 


22 


9.04 490 




9.04 ?58 




0.95 242 


9.99 7 3i 


38 


23 


9.04 6o3 


112 


9.04873 


"5 
114 


0.96 127 


9.99 730 


3 7 


24 


9 . o4 7 1 5 


"3 


9.04 987 




0.95 oi3 


9.99728 


36 


25 


9.04828 




9.o5 101 




0.94 899 


9.99 727 


36 


26 


9.04 940 




9.0 


5 214 




"3 


0.94 786 


9.99726 


34 


27 


9-o5 o52 


112 


9.o5 328 


114 


0.94 672 


9.99 724 


33 


28 


g.oS 1 64 




9.o5 44i 


" 


3 


0.94 559 


9.99 723 


32 


2 9 


9 .o5 275 




9. o5553 


112 


0.94 447 


9.99 721 


3i 


30 


9-o5 386 




9-o5 666 


"3 


0.94334 


9.99 720 


30 




L. 


Cos. 




d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. 




83? 30 . 


PP 


121 


120 


"9 




118 


117 


116 




"5 


114 


"3 


.1 


12. 1 


I2.O 


11.9 


.1 


ii. 8 


ii. 7 


ii. 6 


.1 


"5 


11.4 


"3 


.2 


24.2 


24.0 


23-8 


2 


23.6 


23-4 


23.2 


.2 


23.0 


22.8 


22.6 


3 


36.3 


36.0 


35-7 


3 


35-4 


35-i 


34-8 


3 


34-5 


34-2 


33-9 


4 


48.4 


48.0 


47-6 


4 


47-2 


46.8 


46.4 


4 


46.0 


4S.6 


45-2 


5 


60-5 


60.0 


59-5 


5 


59-o 


S8.S 


58.0 


5 


57-5 




56-5 


.6 


7 2.6 


72.0 


71.4 


.6 


70.8 


70.2 


69.6 


.6 


69.0 


68.4 


67.8 


7 


84.7 


84.0 


83-3 


7 


82.6 


81.9 


81.2 


7 


80. S 


79.8 


79.1 


8 


96.8 


96.0 


95-2 


.8 


94-4 


93-6 


92.8 


.8 


92.0 


91.2 


90.4 


.9 108.9 


I08.0 


107.1 .9 


106.2 


105.1 


104.4 


9 I0 3-5 


IO2.6 


101.7 



6 3O . 



; 


L. 


Sin. 




d. 


L v Tang. 


d. 


L. 


Cotg. 


L. 


Cos. 






30 


9 .o5 36 




9.06 666 






0.94 334 


9-99 7 20 


30 


3i 

32 


9-o5 497 
9o5 607 


IIO 


9-o5 778 
9.o5 890 


112 
112 


0.94 222 

0.94 no 


9.99 718 
9.99 717 


29 

28 


33 


9 .o5 7 i 7 


no 


9.06 002 


112 


0.93 998 


9.99 716 


27 


34 
35 


g.oS 827 
9 .o5 9 3 7 


IIO 


9.06 1 13 
9.06 224 


III 


o'. 


9 388 7 
3 3 776 


9.99714 
9.99 713 


26 

25 


36 


9.06 o46 


109 


9.06 335 


III 


o. 9 3 665 


9.99 711 


24 


37 
38 


9.06 i55 
9.06 264 


109 
109 


9.o6445 
9 .o6556 


IIO 
III 


0.93555 
0.93444 


9.99 710 
9.99 708 


23 
22 


3 9 


9.06 372 


108 


9.06 666 


IIO 


o. 9 3 334 


9.99707 


21 


40 


9.06 48i 


109 


9.06 77 5 


109 


0.93 225 


9.99 705 


20 


4i 

42 


9.06 589 
9.06 696 


107 


9 .o6 885 
9.06 994 


109 


0.93 n5 
0.93 006 


9.99 704 
9.99 702 


18 


43 


9.06 8o4 


107 


9.07 io3 


109 

108 


0.92 897 


9.99 701 


17 


44 
45 
46 


9.06 911 
9.07 018 
9.07 124 


107 
106 
107 


9.07 211 

9.07 32O 

9.07 428 


109 
108 
108 


0.92 789 
0.92 680 
0.92 572 


9.99699 
9.99698 
9.99696 


16 
i5 

i4 


47 

48 


9.07 23i 
9.07337 


106 


9.07 536 
9.07 643 


107 


0.92 464 
0.92 357 


9.99695 
9.99 6 9 3 


i3 

12 


49 


9.07 442 


105 

infi 


9.07 7 5i 




0.92 249 


9.99 692 


I I 


50 


9.07 548 




9.07 858 


107 


0.92 142 


9.99690 


10 


5i 


9.07 653 


105 


9.07 964 




0.92 o36 


9.99689 


9 


62 


9.07 768 




9.08 071 


ir/i 


0.91 929 


9.99687 


8 


53 


9.07863 


105 
105 


9.08 177 


106 


0.91 823 


9.99 686 


7 


54 


9.07 968 




9.08 283 


106 


0.91 717 


9.99 684 


6 


56 


9.08 072 




9.08 389 




0.91 611 


9.99 683 


5 


56 


9.08 176 




9.08495 




0.91 5o5 


9.99 681 


4 










104 








10 


i 














57 


9.08 280 




9.08 600 






0.91 4oo 


9.99 680 


3 


58 


9.08 383 




9.08 705 




3 


0.91 295 


9.99678 


2 


59 


9.08 486 


103 


9.08 810 


105 


0.91 190 


9.99677 


I 


60 


9.08 589 




9.08 914 




0.91 086 


9.99 676 







L. 


Cos. 




d. 


L. Cotg. 


d. 


L. Tang. 


L 


Sin. 






83. 


PP 


112 


in 


IIO 




109 


108 


107 




1 06 


105 


104 


.1 


II. 2 


ii.i 


II. 


.1 


10.9 


10.8 


10-7 


: i 


10.6 


10.5 


10.4 


.2 


22-4 


22.2 


22.0 


.2 


21.8 


21.6 


21-4 


.2 


21.2 


21.0 


20.8 


3 


33-6 


33-3 


33-o 


3 


32.7 


32.4 


32.1 


3 


31.8 


31-5 


31.2 


4 


44.8 


44-4 


44.0 


4 


43.6 


43-2 


42.8 


4 


42.4 


42.0 


41.6 


:J 


56.0 
67.2 


55-5 
66.6 


55- 
66.0 


:I 


54-5 


54-o 
64.8 


5 


5 
.6 


53- 
63.6 


52-5 
63.0 


52.0 
62.4 


7 


78.4 


77-7 


77.0 


7 


76-3 


75-6 


74-9 


7 


74-2 


73-5 


72.8 


8 


89.6 


88.8 


88.0 


.8 


87.2 


86.4 


85.6 


.8 


84.8 


84.0 


83-2 









Q 


98.1 


97.2 


96-3 


9 95-4 


94-5 


93- 6 



43 



1 


L. 


Sin. 




d. 


L. Tang. 


d. 


L. 


Cotg. 


L. 


Cos. 







9.08 589 


103 
103 

IO2 
102 
102 
101 
IO2 
1OI 
101 
1OO 

101 
IOO 
100 

99 

IOO 

. 99 
99 
98 

99 
98 

98 
98 
98 
97 
97 
97 
97 
96 

97 
96 


9.08 914 


is 
104 
104 
103 
104 
103 
103 

102 
103 
102 

102 
101 
102 
IOI 
IOI 
IOI 
IOI 
IOO 
IOO 
IOO 

IOO 

99 
99 
99 
99 
99 
98 

98 
98 
98 


0.91 086 


9.99675 


60 


I 

2 

3 

4 
5 
6 

8 
9 


9.08 692 
9.08 795 
9.08 897 

9.08 999 
9.09 101 
9.09 202 

9.09 3o4 
9.09405 
9.09 5o6 


9.09 019 
9.09 123 
9.09 227 

9.09 33o 
9.09434 
9 .o 9 53 7 

9.09 64o 
9.09 742 
9.09 84s 


0.90 981 
0.90 877 
0.90 773 

0.90 670 
0.90 566 
0.90 463 

0.90 36o 
0.90 258 
0.90 i55 


9.99 6 7 4 
9.99 672 
9.99670 

9.99 669 
9.99 667 
9.99 666 

9.99 664 
9.99 663 
9.99 661 


5 9 
58 
5 7 
56 
55 
54 
53 

5l 


10 


9.09 606 


9.09947 


0.90 o53 


9.99659 


50 


1 1 

12 

i3 

i4 
i5 
16 

7 

18 

J 9 


9.09 707 
9,09 807 
9.09907 

9. TO OO6 
9.IO 1 06 

9. 10 205 
9.10 3o4 

9-10 4O2 

9.10 5oi 


9.10 049 
9.10 i5o 

9. IO 252 

9.10353 

9.10 454 

9.10555 

9.10 656 
9. 10 756 
9.10 856 


0.89 g5i 
0.89 850 
0.89 748 

0.89 647 
0.89 546 
0.89445 

0.89 344 
0.89 244 
0.89 i44 


9.99 658 
9.99 656 
9.99655 

9.99 653 
9.99 65i 
9.99 650 

9.99 648 
9.99647 
9-99 645 


49 
48 

47 
46 
45 
44 

43 

42 

4i 


20 


9.10 599 


9.10 g56 


0.89 o44 


9.99 643 


40 


21 
22 
23 

24 
25 
26 

2 7 
28 
2 9 


9.10 697 
9. 10 795 
9. 10 893 

9. 10 990 
9.11 087 
9.11 i 84 

9.11 281 

9.11 377 

9.11 474 


9.11 o56 
9. ii i55 
9.11 254 

9.1 353 
9. 452 
9. 55i 

9. 649 

9- ?4? 
9. 845 


0.88 944 
0.88 845 
0.88 746 

0.88 647 
0.88 548 
0.88 449 

0.88 35i 
0.88 253 
0.88 165 


9.99 642 
9.99 64o 
9.99 638 

9.99 637 
9 .99 635 
9.99 633 

9.99 632 
9.99 63o 
9.99 629 


39 

38' 

37 

36 

35 
34 

33 

32 

3i 


30 


9.11 570 


9.1 


I 943 




0.88 057 


9.99627 


30 




L. 


Cos. 




d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


' 


82 3O . 


PP 


105 


104 


103 




102 


IOI 


IOO 




99 


98 97 


.1 

.2 

3 

4 
5 
.6 

:i 

i&i 


10.5 

21.0 

31-5 

42.0 

52.5 
63.0 

73-5 

84.0 

04. 5 


10.4 

20.8 

31.2 

41.6 
52.0 
62.4 

72.8 
83.2 


10.3 

20.6 

30-9 
41.2 

I,':i 

72.1 

82.4 

^2^ 








.0 

.0 

.0 

.0 

.0 

.0 
.0 

.0 

.0 


. i 

.2 

3 
4 

:! 
:l 


9-9 
19.8 
29.7 

39- 6 
49-5 
59-4 

69-3 
79.2 

SQ.I 


9.8 9.7 
19.6 19.4 
29.4 29.1 

39.2 38-8 
49.0 48.5 
58.8 58.2 

68.6 67.9 
78.4 77-6 
88.2 87.3 


.2 

3 

4 
5 
.6 

:1 

9 


20.4 
30.6 

4 0.8 
51.0 
6l.2 

71.4 
8l.6 

QT.8 


20.2 
3-3 

40.4 

50-5 
60.6 

70.7 
80.8 


2C 

3 C 

4 c 

5c 
6c 

70 

8c 

DC 



7 30 . 



: 


L. 


Sin. 




d. 


L. Tang. 


d. 


L. Cotg. 


L, 


Cos. 






30 


9.1 


i 570 




9.11 943 




0.88 067 


9-99 62 7 


30 


3i 


9.11 666 


90 

95 


9. 12 C 


>4o 


97 
08 


0.87 960 


9 . 9 9 625 


29 


32 

33 


9.11 761 
9.11 857 


96 


9.12 i 38 

9. 12 235 


97 


0.87 862 
0.87 765 


9.99 624 
9.99 622 


28 
27 


34 


9.11 952 


95 
95 


9.12 332 


97 
06 


0.87 668 


9.99 620 


26 


35 


9.12 047 




9. 12 428 




0.87 572 


9.99 618 


25 


3b 


9.12 142 


95 
94 


9. 12 525 


97 
96 


0.87 475 


9.99617 


24 


38 


9.12 236 
9.12 33i 


95 


9.12 621 
9.12 717 


96 


o. 8 7 3 79 
0.87 283 


9.99 6i5 
9.99 6i3 


23 
22 


3 9 


9.12 425 


94 


9.12 8i3 


96 


0.87 187 


9.99 612 


21 


40 


9.12 519 


94 


9 . 12 909 


96 


0.87 091 


9.99 610 


20 


4i 


9.12 612 


93 
94 


9 . 1 3 oo4 


95 
qe 


0.86 996 


9.99 608 


'9 


42 


9.12 706 




9. 1 3 099 




0.86 901 


9.99607 


18 


43 


9.12 799 


93 


9 . 1 3 1 94 


95 


0.86 806 


9.99605 


17 










93 






95 














44 


9.12 892 




9-i3 289 




0.86 711 


9.99 6o3 


16 


45 


9. 12 985 




9. 1 3 384 




0.86 616 


9.99 601 


i5 


46 


9-i3 078 


93 


9.13478 


94 


0.86 522 


9.99 600 


i4 










93 






95 














47 
48 


9.i3 171 
9-i3 263 


92 


9 .i35 7 3, 
9.13667 


94 


0.86 427 
0.86 333 


9.99 598 
9.99 5 9 6 


i3 

12 


49 


9 .i3 355 


92 


9. 1 3 761" 


94 


0.86 239 


9.99595 


I I 


50 


9-i3 447 


92 


9.i3 854 




0.86 i46 


9.99 5 9 3 


10 


5i 


9 .i3 53 9 


92 


9.i3 948 


94 
93 


o.86oi 


>2 


9 . 99 5 9 i 


9 


52 


9. 1 3 63o 




9. 1 4 o4i 




o.85 959 


9 . 99 589 


8 


53 


9. i3 722 


92 


9 .i4 i 


34 


93 
93 


o.85 866 


9.99 588 


7 


54 


9.i38i3 




9 . i4 227 


93 


o.85 77 3 


9.99 586 


6 


55 


9. 1 3 904 




9 .i43 


20 




o.85 680 


9.99 584 


5 


56 


9-i3 994 


9 


9.14^ 


12 


9 2 


o.85 588 


9.99 582 


4 










91 






9 2 














57 


9 .i4 o85 




9. i4 5o4 




o.85 496 


9.99 58i 


3 


58 


9.14 175 




9-i45 9 7 




o.854o3 


9-99 5 79 


2 


5 9 


9. i4 266 


9 1 


9 .i4 688 


9 1 


o.85 3i2 


9. 99 5 77 


I 


60 


9.14 356 




9. i4 780 




o.85 220 


9 . 99 575 







L. 


Cos. 


d. 


L. Cotg. d. 


L. Tang. 


L. 


Sin. 




V 








82. 








PP 


97 


96 


95 




94 


93 


92 




91 


90 


.j 


9-7 


9-6 


9-5 


. i 


9-4 


9-3 


9.2 


.1 


9-i 


9.0 


.2 


19.4 


19.2 


19.0 


.2 


18.8 


18.6 


i8. 4 


.2 


18.2 


18.0 


3 


29.1 


28.8 


28.5 


3 


28.2 


27.9 


27.6 


3 


27.3 


27.0 


4 


,,8.8 


38.4 


38.0 


4 


37-6 


37-2 


3 6.8 


-4 


36-4 


36.0 


5 


48- s 


48.0 


47-5 


5 


47.0 


46-5 


46.0 


5 


45-5 


45-o 


6 


58.2 


57-6 


57-o 


.6 


56.4 


55-8 


55-2 


.6 


54-6 


54-o 


I 


67.9 
77.6 


67.2 
76.8 


66.5 
76.0 


'I 


65-8 
75-2 


65.1 
74-4 


64.4 
73-6 


i 


63.7 
72.8 


63.0 
72.0 


9 


8 7 . s 


86.4 


85-5 


9 


84.6 


83-7 


82.8 


9 


81.9 


81.0 



45 



, 


L. Sin. 


d. 


L. Tang. d. 


L. Cotg. 


L. Cos. 







9.i4356 


0_ 




9.14 780 




o.85 220 


9 


. 99 5 7 5 




60 


I 


9-14445 


09 

9 




9.14 872 


9 2 


o.85 128 


9 


.99 5 7 4 




5 9 


2 


9.14535 






9.14 963 




o.85 037 


9 


.99 572 




58 


3 


9.14624 


89 




9. 1 5 o54 


9 1 


o.84 946 


9 


.99 5 7 o 


5 7 


4 


9.14 71 


4 


90 
80 




9 .i5 i45 


9i 


o.84 85s 


9 


.99 568 


56 


5 


9.i48o3 






9.i5 236 


9* 


o.84 764 


9 


.99 566 


55 


6 


9.14 891 






9.i5 327 


91 


0.84673 


9 


.99565 


54 


7 


9. i4 980 


89 

80 




9.i5 417 


90 


0.84583 


9 


.99 563 


53 


8 


9. 1 5 069 






9-i5 5o8 


9 1 


o.84 492 


9 


.99 56i 




52 


9 


9oi5 157 






9..i5 598 


90 


o.84 i 


{02 


9 


.99 55 9 


5i 


10 


9. 1 5 245 


88 




9 .i5688 


90 


o.84 3i2 


9 


99 55 7 


50 


ii 


9 .i5333 


88 




9 .i5 777 


59 


0.84 223 


9 


.99 556 


49 


12 


9 . 1 5 421 






9 .i5 867 




o.84 i33 


9 


. 99 554 


48 


i3 


9 .i5 5o8 


87 




9. i5 956 


89 


o . 84 o44 


9 


.99 552 


47 


i4 


9 . i5 5 9 6 


88 
87 




9.1 


6o46 


90 
80 


o.83 


,54 


9 


.99 55o 


46 


i5 


9 .i5 683 


Q_ 




9.16 135 




o.83* 


365 


9 


.99 548 


45 


16 


9 .i5 770 


87 




9. 16 224 


89 
88 


o.83 776 


9 


.99 546 


44 


'7 


9 .i5 857 


87 




9. 16 3i2 


80 


0.83688 


9 


.99545 


43 


18 


9. 1 5 944 


86 




9.16 4oi 




o.83 599 


9 


.99 543 


42 


'9 


9. 16 o3o 


86 




9. 16 489 




o.83 


Sn 


9 


.99 54i 




4i 


20 


9.16 1 16 


87 




9.16 5 77 


88 

00 


o.83 423 


9 


.99 53 9 


40 


21 


9. 16 2o3 


86 




9.16 665 


88 


o.83 335 


9 


.99537 


39 


22 


9.16 289 


g. 




9.16 753 




o.83 247 


9 


.99 535 


38 


23 


9.16 374 


86 




9.16 84i 


87 


o.83 159 


9 


.99533 


37 


24 


9.16 46o 


j 5 




9. 16 928 


88 


o.83 072 


9 


.99 532 


36 


25 


9 .i6545 


86 




9. 17 016 




0.82 984 


9 


.99 53o 


35 


26 


9.16 63i 


8s 




9. 17 io3 


87 
87 


0.82 897 


9 


.99 528 


34 


27 


9.16 716 






9.17 190 


87 


0.82 810 


9 


.99 526 


33 


28 


9.16 801 


J 




9.17 277 




0.82 723 


9 


.99 524 


32 


2 9 


9.16 886 


85 




9.17 363 


86 


0.82 637 


9 


.99 522 


3i 


30 


9. 16 970 


84 




9.17450 


87 


0.82 55o 


9 


.99 52O 


30 




L. Cos. 


d. 


L. 


Cotg. 


d. 


L. Tang. 


L. Sin. 


' 


813O. 


PP 92 


9 


9 




89 


88 




87 


86 


.1 9.2 


9.1 


9.0 


.1 


8. 9 


8.8 


.1 


8.7 


8.6 


.2 18.4 


1 8. 2 


18.0 


.2 


17.8 


17.6 


.2 


17.4 


17.2 


3 27.6 


27-3 


27.0 


3 


26.7 


26.4 


3 


26.1 


25.8 


4 36.8 


36.4 


36.0 


4 


35-6 


35-2 


4 


34-8 


34-4 


5 46-0 


45-5 


45-o 


.5 


44-5 


44.0 




43-5 


43- 


6 55-2 


54- 6 


54-o 


.6 


53-4 


52-8 


.6 


52.2 


51.6 


7 6 4-4 


63-7 


63.0 


7 


62.3 


61.6 


7 


60.9 


60.2 


.8 73.6 


72.8 


72.0 


.8 


71.2 


70.4 


.8 


69.6 


68.8 


.9 82.8 


81.9 


81.0 




^ _ 


78-3 


77-4 



46 



8 3D . 



; 




L. Sin. 


d. 




L. Tang. 


d. 


L. Cotg. 


L. Cos. 




30 


9 


. 16 970 






9.17 450 




0.82 55o 


9 


>99 52O 


30 


3i 


9 


.17055 


85 
8-1 




9.17 536 


86 
86 


0.82 464 


9 


99 5i8 


29 


32 


9 


.17 i3 9 






9.17 622 




0.82 378 


9 


99 5i7 




28 


33 


9 


. 17 223 


84 
84 




9.17 708 


86 
86 


0.82 292 


9 


99515 




27 


34 


9 


.I 7 3o 7 


. 




9.17 794 


Rfi 


0.82 206 


9 


99 5i3 




26 


35 


9 


.i 7 3 9 i 






9.17 880 




O.82 120 


9 


99 5i i 




25 


36 


9 


.17474 


83 
84 




9.17 965 


85 
86 


0.82 c 


3 5 


9 


995o9 


24 


37 


9 


.17 558 






9.18 o5i 




0.8 1 949 


9 


99 507 




23 


38 


9.17 64i 


3 




9.18 i36 


5 


0.81 864 


9 


.99 5o5 




22 


3 9 


9 


.17 7 2 4 






9.18 221 


85 


0.81 779 


9 


.99 5o3 




21 


40 


9.17807 


8 3 




9.18 3o6 


5 


0.8 1 694 


9 


. 99 5oi 




20 


4i 


9 


.17 890 


83 
8? 




9.18 391 


85 


0.81 609 


9 


99499 




19 


42 


9 


.17973 






9.18475 




0.81 525 


9 


99497 




1 8 


43 


9 


.i8o55 


82 




9.18 56o 


8 5 
84 


0.8 1 44o 


9 


.99495 




17 


44 


9 


.18 i3 7 


0, 




9.18 644 


o . 


0.81 356 


9 


99494 


16 


45 


9 


. l8 220 


3 




9.18 728 




0.81 272 


9 


.99492 


i5 


46 


9.18 3o2 


81 




9.18 812 


84 
84 


0.81 1 88 


9 


.99 490 


i4 


47 


9.18 383 






9.18 896 


o.. 


0.81 io4 


9 


.99488 


i3 


48 


9.18465 






9.18 979 




0.81 021 


9 


.99486 


12 


49 


9.18 547 


82 




9. 19 o63 


84 

0_ 


0.80 937 


9 


99 484 




I I 


50 


9.18 628 






9.19 i 46 


3 
Ro 


0.80 854 


9 


. 99 482 


10 


5i 


9.18 709 


81 




9.19 229 


3 
83 


0.80 771 


9 


.99 48o 


9 


52 


9. 18 790 






9. 19 3i2 




0.80688 


9 


.99478 


8 


53 


9.18 871 


8r 




9 .i 9 3 9 5 


8 3 
83 


0.80 605 


9 


.99476 


7 


54 


9. 18 952 


81 




9.19478 


83 


O.8o 522 


9 


.99474 


6 


55 


9. 19 o33 






9. 19 56i 


P_ 


0.80439 


9 


.99472 


5 


56 


9.19 1 13 


80 




9.19 643 


82 


0.80 357 


9 


.99470 


4 


57 


9. 19 193 


80 




9.19 725 


82 


O.8o 275 


9 


.99 468 


3 


58 


9.19 273 






9.19 807 




O.8o ] 


9 3 


9.99 466 


2 


5 9 


9.19353 


80 




9. 19 889 


82 


0.8o 1 


1 1 


9 


.99 464 


I 


60 


9.19433 






9.19 971 




o. 80 029 


9 


. 99 462 




" 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. Sin. 










81. 








PP 


86 


85 


8 4 




83 


82 




81 


80 




8.6 


8-5 


8. 4 


.1 


8.3 


8.2 


>z 


8.1 


8.0 


.2 


17.2 


17.0 


16.8 


.2 


16.6 


,6.4 


.2 


16.2 


16.0 


3 


25-8 


25-5 


25.2 


3 


24.9 


24.6 


3 


24-3 


24.0 


4 


34-4 


34- 


33-6 


4 


33-2 


32.8 


4 


32-4 


32.0 


5 


43- o 


4 2 -5 


42.0 


5 


41- s 


41.0 


5 


40.5 


40.0 


.6 


51-6 


51-0 


50.4 


.6 


49.8 


49.2 


.6 


48.6 


48.0 


.7 


60.2 


59-5 


58.8 


. 7 


58.1 


57-4 


. 7 


56-7 


56.0 


.8 


68.8 


68.0 


67.2 


.8 


66.4 


65.6 


.8 


64.8 


64.0 


9 


7*-4 






9 74-7 73-8 


9 


72.9 


72.0 



47 



9. 



I 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 







9.19 433 






9. 19 971 




0.80 029 


9 


.99 462 


60 


I 


9,19 5i3 






9.20 o53 


82 

81 


0.79 947 


9 


.99 46o 


5 9 


2 


9.19 5 9 2 






9.20 1 34 




0.79 866 


9 


.99458 


58 


3 


9. 19 672 


79 




9.20 216 


81 


0.79 784 


9 


.99 456 


57 


4 


9.19 761 






9.20 297 


81 


0.79 703 


9 


99 454 


56 


b 


9.19 83o 






9.20 378 




0.79 622 


9 


.99 452 


55 


6 


9.19909 


79 




9.20 459 




0.79 54i 


9 


.99450 


54 








79 








81 














7 


9.1998 


8 






9.20 54o 




0.79 46o 


9 


99448 


53 


8 


9.20 067 






9.20 621 




0.79 3 79 


9 


.99 446 


52 


9 


9.20 i45 


7 

_Q 




9.20 701 





-79 2 99 


9 


99 444 


5i 


10 


9.20 223 






9.20 782 




0.79 218 


9 


.99 442 


50 


1 1 


9.20 3o2 


79 
78 




9.20 862 


80 


0.79 i38 


9 


.99 44o 


49 


12 


9.20 38o 






9.20 942 




o. 79 o58 


9 


.99 438 


48 


i3 


9.20458 


7 
77 




9.21 022 


80 


0.78 978 


9 


.99 436 


47 


i4 


9.20 535 


78 




9.21 102 


80 


0.78 898 


9 


.99434 


46 


i5 


9.20 6i3 






9.21 l82 




0.78 818 


9 


.99432 


45 


16 


9.20 691 


JO 

77 




9.21 26l 


79 
80 


0.78739 


9 


.99 429 


44 


17 


9.20 768 






9.21 34i 




0.78 659 


9 


.99427 


43 


18 


9.20 845 






9.21 420 




0.78 58o 


9 


.99 425 


42 


'9 


9.20 922 


77 




9.21 499 


79 


0.78 5oi 


9 


. 99 423 




4i 


20 


9.20999 






9.21 578 


79 


o. 78 422 


9 


.99 421 




40 


21 


9.21 076 


77 




9.21 657 


79 


o.78'343 


9 


.99419 




39 


22 


9.21 i53 


76 




9.21 736 


_0 


0.78 264 


9 


.99417 


38 


23 


9.21 229 


77 




9.21 8i4 


7 
79 


0.78 186 


9 


.99415 


37 


24 


9.21 3o6 


76 




9.21 893 


78 


0.78 107 


9 


.99 4i3 




36 


25 


9.21 382 






9.21 971 




0.78 029 


9 


.99411 




35 


26 


9.21 458 


76 




9 . 22 o4g 


7 
78 


o. 779 5i 


9 


.99 409 


34 


27 


9.21 534 


76 




9-22 127 


78 


0.77873 


9 


.99407 


33 


28 


9,21 610 






9-22 2O5 


_Q 


-77 795 


9 


.99 4o4 


32 


29 


9.21 685 


76 




9.22 283 


78 


0.77 717 


9 


.99 402 


3i 


30 


9.21 76 


i 






9.22 36i 




0.77 63 9 


9 


. 99 4oo 


30 




L. Cos. 


d. 


L. Cotg. d. 


L. Tang. 


L. Sin. 










8O 3O . 






PP 82 


81 


so 




79 


78 




77 


76 


.1 8.2 


8.1 


8.0 


.1 


7-9 


7.8 


. i 


7-7 


7.6 


.2 16.4 


16.2 


16.0 


.2 


15-8 


15.6 


.2 


15-4 


15.2 


3 24.6 


24-3 


24-0 


3 


23-7 


23-4 


3 


23.1 


22.8 


4 32-8 


32.4 


32.0 


4 


31-6 


31.2 


4 


30.8 


30-4 


5 4i-o 


4-5 


40.0 


5 


39-5 


39- 


5 


38.5 


38.0 


.6 49.2 


48.6 


48.0 


.6 


47-4 


46.8 


.6 


46.2 


45-6 


-7 57-4 


56.7 


56.0 


7 


55-3 


54-6 


7 


53-9 


S3- 2 


.8 65.6 


64.8 


64.0 


.8 


63-2 


62.4 


.8 


61.6 


60.8 


9 73-8 


72.9 


72.0 


9 ?i-i 


70.2 


9 


69-3 


68.4 



48 



9 SO. 



, 


L. Sin. 


d. 




L. Tang. d. 


L. Cotg. 


L. Cos. 




30 


9.21 761 


9.22 36i 




0.77 63 9 


9 


.99 4oo 


30 


3i 


9.21 836 


75 
76 




9.22 438 


77 
78 


0.77 562 


9 


. 99 3 9 8 




29 


32 


9.21 912 






9.22 5i6 




o. 77 484 


9 


.99 3 9 6 


28 


33 


9.21 987 


75 




9.22 5g3 


77 


0.77 407 


9 


99 3 94 




27 


34 


9.22 062 


75 




9.22 670 


77 


0.77 33o 


9 


.99392 




26 


35 


9.22 187 






9.22 747 




0.77 253 


9 


.99 390 


25 


3b 


9.22 211 


74 




9.22 824 


77 


0.77 176 


9 


.99 388 


24 








75 








77 














3? 


9.22 286 






9.22 901 




0.77099 


9 


.99 385 




23 


38 


9.22 36i 






9.22 977 


7 


0.77 O23 


9 


.99 383 


22 


3 9 


9.22435 


74 




9.23 o54 


77 


0.76 946 


9 


.99 38i 




21 


40 


9.22 5og 






9.23 i3o 




0.76 870 


9 


99 3 79 


20 


4i 


9.22 583 


74 
74 




9.23 206 


70 


0.76 794 


9 


99 3 77 


19 


42 


9.22 657 






9.23 283 




0.76 717 


9 


.99375 


18 


43 


9.22 731 


74 
74 




9.23 SSg 


7 
76 


0.76641 


9 


99 3 7 2 


17 


44 


9.22 805 






9.23435 




0.76 565 


9 


.99 370 


16 


45 


9.22 878 






9.23 5io 




0.76 490 


9 


.99 368 


i5 


46 


9.22 952 


74 




9.23 586 


7 


0.76^ 


n4 


9 


.99 366 


i4 


47 


9.23 O25 


73 




9.23 661 


75 
76 


0.76 339 


9 


.99 364 


i3 


48 


9.23 098 






9.23 737 




0.76 263 


9 


.99 362 


12 


49 


9.23 171 


73 




9.23 812 


75 


0.76 188 


9 


. 99 35 9 


I I 


50 


9.23 244 






9.23 887 




0.76 1 13 


9 


.99 357 


10 


5i 


9.33317 


73 




9.23 962 


75 


0.76 o38 


9 


.99 355 


9 


52 


9.23 390 






9. 24 037 




0.75 963 


9 


.99353 


8 


53 


9.23462 


72 
73 




9.24 112 


74 


0.75888 


9 


. 99 35i 




7 


54 


9.23535 






9.24 186 




0.75 8i4 


9 


.99348 


6 


55 


9.23 607 






9.24 261 




0.75 739 


9 


.99 346 


5 


56 


9.23 679 


72 




9.24335 


74 


0.75 665 


9 


.99 344 


4 








73 








75 














57 


9.23 752 






9.24 4io 




0.75 590 


9 


.99 342 


3 


58 


9.23 823 






9.24484 




0.75 5i6 


9 


. 99 34x 





2 


5 9 


9.23 895 


72 




9.24558 


74 


0.75 442 


9 


. 99 33 7 


I 


60 


9.23 967 






9.24 632 




o. 7 5368 


9 


.99335 







L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. Sin. 


' 








80. 








PP 77 


76 


75 




74 


73 




72 


71 


.1 7.7 

2 15-4 


7 .6 
15-2 


7-5 
15-0 


.1 

.2 


a 


,5:1 


.1 

.2 


7-2 

14.4 


14.2 


3 23.1 


22.8 


22-5 


3 


22.2 


21.9 


3 


21.6 


21.3 


4 30-8 


3-4 


30.0 


4 


29.6 


29.2 


4 


28.8 


28.4 


5 38-5 


38-0 


37-5 


5 


37- 


36.5 


5 


36.0 


35-5 


.6 46.2 


45-6 


45-o 


.6 


44-4 


43-8 


.6 


43-2 


42.6 


7 53-9 
.8 61.6 


g; 


60.0 


7 
.8 


51-8 
59-2 


5 1 -! 

58.4 


:i 


50.4 
57-6 


8:1 


9 6 9-3 


68.4 


67-5 


.9 66.6 


65-7 


9 


64.8 


63.9 



I 


L. Sin. 


d. 


L. 


Tang. d. 


L. Cotg. 


L. Cos. 


d. 







9.23 967 




9- 


24 632 


74 
73 
74 
73 
74 
73 
73 
73 
73 
73 
72 
73 
72 
73 
72 
72 
72 
72 
72 
7 1 
72 
7 1 
72 

7 1 
7 


0.75 368 


9-99 


335 




60 


I 

2 

3 

4 
5 
6 

7 
8 

9 


9.24 no 
9.24 181 

9.24253 
9.24 324 
9.24395 

9.24466 
9.24536 
9.24 607 


71 

7 1 
72 

7 1 

7 1 
70 

7 1 


9- 
9- 
9- 

9- 
9- 
9- 

9- 
9- 
9- 


24 706 

24 779 
24853 

24 026 
25 ooo 
25 073 

25^219 
2*5 292- 


0.75 294 

O.75 221 

0.75 147 

0.75 074 
0.75 ooo 

0.74927 

o. 7 4854 
0.74 781 
0.74 708 


ooo ooo ooo 

ooo ooo ooo 
ooo ooo ooo 


333 
33i 
328 

3 2 6 

324 

322 

3i 9 
3i 7 
3i5 


2 

3 

2 
2 
2 

3 

2 
2 
2 

3 

2 
2 
2 

3 
2 

2 

3 

2 
2 

2 

3 

2 
2 

3 

2 
2 

3 

2 
2 


5 9 

58 
5? 

56 

55 
54 
53 

52 

5i 


10 


9.24677 


7 


9- 


25365 


o. 7 4635 


9.99 


3i3 


50 


II 

12 

i3 

U 

i5 
16 

17 

18 


9.24 748 
9.24818 
9.24888 

9.24 g58 
9.25 028 
9.25 098 

9.25 168 
9.25 237 
9.25 307 


70 
70 
70 
70 
70 
70 
69 
70 


000 000 000 


2 543 7 
25 5io 
25 582 

25655 
25 727 
2 5 799 

25 871 

25 943 

26 015 


0.74563 
0.74 490 
0.74418 

0.74345 
0.74273 

0.74 201 
O.74 129 

0.74 057 
0.73 985 


ooo ooo ooo 

ooo ooo ooo 
ooo ooo ooo 


3io 
3o8 
3o6 

3o4 
3oi 
299 

297 
294 
292 


49 

48 

47 
46 
45 
44 

43 

42 

4i 


20 


9.25 376 


09 


9- 


26 086 


0.73 914 


9.99 


290 


40 


21 
22 
23 

24 
25 

26 

27 
28 
29 


9 .25445 
9.25 5i4 
9.25 583 

9-25652 
9.25 721 
9.25 790 

9.25 858 
9.25927 
9.25 995 


9 

68 
69 
68 
68 


ooo ooo ooo 


26 i58 
26 229 
26 3oi 

26 372 
26443 
2 65i4 

26 585 
26655 
26 726 


0.73 842 
o. 7 3 77 i 
0.73 699 

0.73 628 
o. 7 355 7 
0.73 486 

o! 7 3345 
o. 7 3 274 


9.99 
9.99 
9.99 

9-99 
9.99 

9.99 

9.99 
9.99 
9-99 


288 
285 
283 

281 
278 
276 

274 
271 
269 


3 9 
38 
37 

36 
35 

34 
33 

32 

3i 


30 


9. 26 o63 


9- 


26 797 


0.73 2o3 


9.99 


267 


30 




L. Cos. 


d. 


L. 


Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 




79 30'. 


PP 

.2 

3 
4 

i 

.9 


74 


73 72 


.1 

.2 

3 
4 


71 


70 


69 


.2 

3 

4 
5 
.6 


68 3 


22.2 
29.6 

44-4 


7-3 7-2 
14.6 14.4 
21.9 21.6 

29.2 28.8 
36.5 36.0 
43-8 43-2 

Si. i 5-4 
58.4 57.6 
65.7 64.8 


14.2 
21.3 

28.4 
35-5 
42.6 


7.0 

21. 
28.0 

35-o 
42.0 

49-o 
56.0 


6.9 
13-8 
20.7 

27.6 
34-5 
41.4 

48-3 
SS- 2 
62.1 


6.8 0.3 
13.6 0.6 
20.4 0.9 

27.2 1.2 

34.0 1.5 

40.8 1.8 

47.6 2.1 

54-4 2.4 
61.2 2.7 



5o 



3O. 



/ 


L. Sin. 


d. 


L. Tang. 


d. 


L. 


Cotg. 


L. Cos. 


d. 




30 


9.26 o63 




9.26 


797 




o. 


73 2o3 


9-99 


267 




30 


3i 


9.26 i3i 


68 


9.26 


867 


70 


o. 


7 3 i33 


9.99 


264 


2 


29 


32 


9.26 199 




9.26 937 




0. 


7 3o63 


9.99 


262 




28 


33 


9.26 267 




9.27 


008 


71 


o. 


7 2 992 


9.99 


260 




2 7 


34 


9-26335 


68 
68 


9.27 


078 


7 o 


0.72 922 


9.99 


25 7 


3 

2 


26 


36 


9.26 4o3 




9.27 


i48 


7 


0.72 852 


9.99 


255 




25 


36 


9.26 470 


67 


9.27 


218 


70 


0.72 782 


9.99 


252 




24 


37 


9.26 538 


68 

67 


9.27 


288 


70 
60 


0.72 712 


9.99 


25o 


2 


23 


38 


9.26 6o5 




9.27 


35 7 




o. 


7 2643 


9 . 99 248 




22 


3 9 


9.26 672 


07 
fit 


9.27 427 


70 


o. 


72 5 7 3 


9.99 


245 




21 


40 


9 


.26 739 


07 


9.27 496 


69 


0. 


7 2 5o4 


9.99 


243 




20 


4i 


9 


.26 806 


6? 


9.27 


566 


7 
60 


o. 


7 2434 


9.99 


241 


3 


19 


42 


9 


.26873 




9.27 


635 




0.72 365 


9.99 


238 




1 8 


43 


9.26 940 


07 

67 


9.27 


704 


69 
69 


0. 


7 2 296 


9-99 


236 


3 


17 


44 


9 


.27007 


66 


9.27 


773 


60 


o. 


7 2 227 


9.99 


2 33 


2 


16 


45 


9.27073 




9.27 


842 




0. 


72 i58 


9.99 


23l 




i5 


46 


9.27 i4o 


67 
66 


9.27 


911 


69 
69 


o. 


72 089 


9.99 


229 




i4 


47 


9.27 206 


67 


9.27 


980 


60 


0. 


72 020 


9.99 


226 


2 


i3 


48 


9.27 2 7 3 




9.28 


049 




o. 


71 9 5i 


9.99 


224 




12 


49 


9.27 33 9 




9.28 


117 




o. 


71 883 


9.99 


221 




I I 


50 


9 


.27 405 




9.28 


186 


69 


0. 


71 8i4 


9.99 


219 




10 


5i 


9.2 7 4 7 i 


66 


9.28 


254 


60 


0. 


71 746 


9.99 


2I 7 


3 


9 


52 


9.2-7 53 7 




9.28 


323 




o. 


71 677 


9.99 


2l4 




8 


53 


9.2 7 602 


65 


9.28 


3 9 i 




o. 


71 6o 9 


9.99 


212 




7 










66 






68 










3 




54 


9.27 668 


66 


9 .2845 9 


68 


o. 


71 54i 


9.99209 


2 


6 


55 


9.27734 




9 .28 


52 7 




0. 


7 i4 7 3 


9.99 


20 7 




5 


56 


9.27799 


65 


9 .28 


5 9 5 




0. 


71 4o5 


9.99 


204 




4 










65 






67 














57 


9.27 864 


66 


9 .28 


662 


68 


o. 


71 338 


9.99 


202 


2 


3 


58 


9 . 27 g3o 




9.28 


7 3o 




0. 


71 2-70 


9.99 


2OO 




2 


5 9 


9.27 995 




9.28 798 


67 


o. 


7 i 202 


9.99 


197 




I 


60 


9.28 060 


S 


9.28 865 




o. 


71 135 


9.99 


195 









L. Cos. | d. 


L. Cotg. d. 


L. 


Tang. 


L. Sin. d. 


' 








79. 




PP 


70 


69 


68 


67 


66 




65 3 


.1 


7.0 


6.9 


6.8 


.1 6. 7 


6.6 


.1 


6.5 0.3 


.2 


14.0 


13-8 


13-6 


.2 13.4 


13.2 


.2 


13.0 0.6 


3 


21.0 


20.7 


20.4 


, 3 20. i 


19.8 


3 


19.5 0.9 


4 


28.0 


27.6 


27.2 


.4 26.8 


26.4 


4 


26.O 1.2 


5 


35- 


34-5 


34-o 


5 33-5 


33-o 


5 


32-5 1-5 


.6 


42.0 


41.4 


40.8 


.6 40.2 


39- 6 


.6 


39.0 1.8 


7 


49.0 


48-3 


47.6 


7 46-9 


46.2 


7 


45-5 2.1 


.8 


56.0 


55-2 


54-4 


8 53-6 


52.8 


.8 


52.0 2.4 


9 


63.0 


62.1 




9 6o -3 


59-4 




58-5 2.7 



5i 



11. 



' 


L. Sin. 


d. 


L. 


Tang. d. 


L. Cotg. 


L. Cos. i d. 







9.28 060 


65 
65 
6 4 
65 
65 
6 4 
6 4 

65 
6 4 


9 


28 865 


68 
67 
67 
67 
67 
67 
67 
67 
66 
67 

66 
67 
66 
66 
66 
66 
66 
66 
66 
65 
66 
65 
65 
66 

6 5 
65 
65 
65 
64 


0.71 135 


9.99 195 


3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 


60 


J 

2 

3 

4 
5 
6 

8 
9 


9.28 125 
9 .28 190 
9.28 254 

9.28 319 
9.28 384 
9.28448 

9.28 5i2 
9 .285 77 
9.28 64 1 


9 
9 
9 

9 
9 
9 

9- 
9- 
9- 


28 933 
29 ooo 
29 067 

29 1 34 

29 201 
29 268 

2 9 335 
29 4O2 

29 468 


o. 71 067 
0.71 ooo 
0.70 9 33 

0.70866 
0.70 799 
0.70 732 

0.70 665 
0.70 598 
0.70 532 


OOO OOO OOO 

ooo ooo ooo 


192 
I9O 

I8 7 

185 
182 

180 
177 

172 


5 9 
58 
5? 
56 
55 
54 
53 

52 

Si 


10 


9.28 705 




9.29535 


0.70 465 


9.99 


170 


50 


1 1 

12 

i3 

i4 
i5 
16 

18 
'9 


9.28 769 
9.28 833 
9.28 896 

9.28 960 
9.29 024 
9.29 087 

9.29 i5o 
9.29 214 
9.29 277 


04 
6 4 

63 
6 4 
6 4 
63 
63 
6 4 

63 


9 
9- 
9- 

9- 
9- 
9- 

9- 
9- 
9 


29 601 
29 668 
29 734 

29 800 
29 866 
2 99 3 2 

2 999 8 
3oo64 
3o i3o 


0.70 399 
0.70 332 
0.70 266 

0.70 200 
0.70 1 34 
0.70068 

0.70 002 
0.69 936 
0.69 870 


9.99 
9.99 
9.99 

9.99 
9.99 
9-99 

9-99 
9-99 
9.99 


167 
165 
162 

1 60 
157 

162 
150 

i47 


49 

48 

4? 
46 
45 
44 

43 

42 

4i 


20 


9.29 34o 


03 


9- 


3o i 9 5 


0.69 805 


9.99 


145 


3 

2 

3 

2 

3 

2 

3 
3 

2 


40 


21 
22 
23 

24 
25 

26 

27 
28 
29 


9.29 4o3 
9.29 466 
9.29 529 

9.29 591 
9 .29 654 
9.29 716 

9- 2 9779 
9 .2 9 84i 
9.29 903 


63 
63 
63 
62 

63 
62 
63 
62 
62 


9 
9- 
9- 

9- 
9- 
9- 

9- 
9- 
9- 


3o 261 
3o326 
3o 391 

30 522 

3o58 7 

3o652 
3o 7 i 7 
3o 782 


0.69 739 
0.69 674 
0.69 609 

o,6 9 543 
0.69 478 
0.69 4i3 

0.69 348 
o.6 9 283 
0.69 218 


ooo ooo ooo 

ooo ooo ooo 
ooo ooo ooo 


142 
i4o 
i3 7 

132 

i3o 

127 
124 

122 


39 

38 

37 
36 
35 
34 

33 

32 

3i 


30 


9.29 966 


6 3 


9- 


3o846 


0.69 1 54 


9.99 


119 




30 




L. Cos. 


d. 


L 


,Cotg 


. 


d. 


L. Tang. 


L. Sin. 


d. 


' 


78 30 . 


PP 

t. .2 

3 
4 

.u 


68 


67 66 


.2 

3 

4 
5 
.6 

.7 
.8 


65 


6 4 


63 




62 3 


6.8 
13-6 
20.4 

27.2 
34-o 
40.8 

47.6 
54-4 


6.7 6.6 
13.4 13.2 
20. i 19.8 

26.8 26.4 
33-5 33-o 
40.2 39.6 

46.9 46.2 
53-6 52-8 
60.3 59.4 


6-5 
13-0 

19-5 

26.0 
32-5 

45-5 
52.0 

58. s 


6. 4 

12.8 

19.2 

256 

32.0 

38.4 
44.8 


63 

12.6 

18.9 

25.2 

37-8 

44.1 
50-4 


.2 

3 

4 

mti 


6.2 .3 
12.4 .6 
18.6 .9 

24.8 1.2 
31.0 1.5 
37.2 1.8 

43-4 2.1 
49.6 .2.4 
5S.8 2.7 



1 1 3D . 





L. Sin. d. 


L. 


Tang. 


d. 


L. Cotg. 


L. Cos. d. 




30 


9.29 966 


9 


3o 846 






0.69 1 54 


9.99 119 




30 


3i 

32 


9-3o 028 
93o 090 


62 


9 
9- 


3o 91 1 
3o 975 


6 4 




0.69 089 
0.69 025 


9.99 
9-99 


117 

u4 


3 


29 

28 


33 


9.3o 


i5i 




9- 


3 1 o4o 


65 




0.68 960 


9.99 


I 12 


2 


27 


34 


9.3o 2i3 


62 
62 


9- 


3i io4 






0.68 896 


9.99 


109 


3 


26 


35 


9.30275 




9- 


3i 168 






0.68 832 


9.99 


1O6 


3 


25 


36 


9.3o 


336 




9- 


3i 233 


65 




0.68 767 


9.99 


104 


2 


24 


37 


9-3o 398 


62 
61 


9- 


3i 297 


6 4 




0.68 703 


9.99 


1OI 


3 


23 


38 


93o 459 




9- 


3i 36i 


4 




0.68 639 


9.99 


099 




22 


3 9 


9 . 3o 


521 


61 


9- 


3i 425 


6 4 




0.68 575 


9-99 


096 


3 


21 


40 


9.3o 582 


61 


9- 


3i 48 9 


04 

6? 




0.68 5n 


9.99 


093 


3 


20 


4i 


9 .3o643 


61 


9- 


3i 552 


61 




0.68448 


9.99 


091 




I9 


42 


9.3o 704 


61 


9- 


3i 616 


fio 




0.68 384 


9-99 


088 




18 


43 


9. 3o 765 


61 


9- 


3i 679 


3 




0.68 32i 


9.99 


086 


3 


17 


44 


9-3o 826 


61 


9- 


3i 7 43 


6-j 




0.68 257 


9.99 


o83 




16 


45 


9.30887 




9- 


3 1 806 






0.68 194 


9.99 


080 




i5 


46 


9.30947 


61 


9- 


3i 870 


64 




0.68 i3o 


9.99 


078 


3 


i4 


47 


9. 3 1 008 


60 


9- 


3i 9 33 






0.68 067 


9.99 


076 




i3 


48 


9 .3i 068 




9- 


3 1 996 


3 




0-68 oo4 


9.99072 




12 


49 


9 .3i 


129 


60 


9- 


32 o5c 


1 


3 




0.67 941 


9.99070 




1 I 


50 


9. 3i 


189 


61 


9- 


32 122 


fio 




0.67 878 


9-99 


067 


3 


10 


5i 


9.3i 250 


60 


9- 


32 i85 


3 
63 




0.67815 


9.99 o64 




9 


52 


9-3i 3io 


60 


9- 


32 248 






0.67 752 


9-99 


062 




8 


53 


9.3i 370 


60 


9- 


32 3u 




63 
62 




0.67 689 


9.99 


059 


3 


7 


54 


9. 3 1 43o 


60 


9- 


3 2 3 7 3 






0.67 62 7 


9.99 


066 




6 


55 


9. 3 1 490 




9- 


32436 


3 




0.67 564 


9.99 


o54 




5 


56 


9.3i 549 


59 
60 


9- 


32498 


62 




0.67 5o2 


9.99 


o5i 


3 


4 


57 


9. 3 1 609 


60 


9- 


32 56i 


62 




0.67 439 


9.99 


o48 


2 


3 


58 


9. 3 1 669 




9 


32 623 






o.6 7 3 77 


9.99 


o46 




2 


5 9 


9-3i 


728 


59 


9- 


32685 


62 




o.6 7 3i5 


9.99 


o43 


3 


I 


60 


9 .3i 


788 




9 


32 7 4 7 






o.6 7 253 


9,99 


o4o 









L. Cos. ! d. 


L, 


Cotg 


. d. 




L. Tang. 


L. Sin. 


d. 


' 






78. 






PP 


65 


64 63 




6a 


61 


60 




59 3 


.1 


6.5 


6.4 6.3 


.! 


6.2 


6.1 


6.0 


. i 


5-9 0.3 


.2 


13.0 


12.8 12.6 


.2 


12.4 


12.2 


I2.O 


.2 


ii. 8 0.6 


3 




19.2 18.9 


3 


18.6 


l8-3 


18.0 


3 


17.7 0.9 


4 


26.0 


25.6 25.2 


-4 


24.8 


24.4 


24.0 


4 


23.6 1.2 




32.5 


32.0 31-5 


.5 


31.0 


3-5 


30.0 


.5 


29-5 1-5 


.6 




38.4 37-8 


.6 


37-2 


36.6 


36.0 


.6 


35-4 1-8 


.7 


45-5 


44.8 44.1 


7 


43-4 


42.7 


42.0 


7 


41-3 2.1 


.8 


52.0 


51.2 50.4 


.8 


49-6 


48.8 


48.0 


.8 


47.2 2.4 


9 58-5 








54-9 


54- -9 


S3- 1 2.7 



53 



12 C 



, 


L. Sin. 


d. 


L. 


Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 







9 .3i 788 




9- 


32 747 


61 




0.67 253 


9.99040 




60 


I 


9 .3i 847 


60 


9- 


32 810 


62 




0.67 190 


9.99 


o38 


3 


5 9 


2 


9.3i 907 




9- 


32 872 






0.67 128 


9.99 


o35 




58 


3 


9-3i 966 


59 


9- 


32 9 33 


62 




0.67 067 


9.99 


032 


3 

2 


$7 


4 


9.32 025 


59 


9- 


3 2 99 5 


62 




0.67 005 


9.99 


o3o 




56 


5 


9.32 o84 




9- 


33o5 7 






0.66 943 


9.99 


027 




55 


6 


9.32 i43 


59 


9- 


33 119 


61 




0.66 881 


9.99 


024 


3 

2 


54 


7 


9.32 202 


59 


9- 


33 180 


62 




0.6 


6 820 


9.99 


022 




53 


8 


g.32 261 


_0 


9- 


33242 






0.66 7 58 


9.99 


019 




52 


9 


9.32 319 




9- 


33 3o3 






0.66 697 


9.99 


016 


3 


5i 


10 


9.32 378 


59 


9- 


33 365 






0.66635 


9-99 


oi3 




50 


ii 


9.32437 


5 8 


9- 


33426 


61 




0.66 5 7 4 


9.99 


on 




49 


12 


9.32 L 


fo5 




9- 


3348 7 






o.665i3 


9.99 


008 




48 


i3 


9.32 553 


5 8 


9- 


33548 


61 




0.66 452 


9.99 


oo5 


3 


47 








59 








61 












3 




i4 


9 .32 612 




9- 


33609 


61 




o.663 9 i 


9.99 


002 




46 


i5 


9 .32 670 




9- 


33 670 






0.66 33o 


9.99 


ooo 




45 


16 


9.32 728 


5 8 


9- 


33 7 3i 


61 




0.66 269 


9.98 


997 


3 


44 








58 








61 












3 




17 


9.32 786 




9- 


33 792 


61 




0.66 208 


9.98 


994 


3 


43 


18 


9.32 844 




9- 


33853 






0.66 i47 


9.98 


991 




42 


r 9 


9.32 902 


S 8 


9- 


33 9 i3 


60 




0.66 087 


9.98 


989 




4i 


20 


9.32 960 


S 8 


9- 


33 97 4 


61 




0.66 026 


9.98 


986 




40 


21 


9.33 018 


5 8 


9- 


34o34 


61 




o.65 966 


9.98 


9 83 


3 


3 9 


22 


9.33 075 




9 


34 095 






o.65 905 


9 . 9 8 


9 8o 




38 


23 


9 .33 i33 


5 8 


9- 


34 1 55 






o.65 845 


9.98 


97 8 




37 








57 








60 












3 




24 


9-33 190 


eg 


9- 


342i5 


61 




o.65 785 


9.98 


975 




36 


25 


9.33 248 




9- 


34276 






o.65 724 


9.98 


97 2 




35 


26 


9.33 3o5 


57 


9- 


34336 






o.65 664 


9.98 


9 6 9 


3 


34 








57 








60 












2 




27 


9.33 362 


,0 


9- 


343 9 6 


60 




o.65 6o4 


9.98 


967 




33 


28 


9.33 420 




9- 


34456 






0.65544 


9.98 


9 64 




32 


29 


9-33477 


57 


9 .345i6 






0.65484 


9.98 


9 6i 


3 


3i 


30 


9.33 534 


57 


9- 


345 7 6 






o.65 424 


9.98 


9 58 




30 




L. Cos. 


d. 


L. 


Cotg. 


d. 


L. Tang. 


L. Sin. d. 




77 3O. 


PP 


63 


62 61 




60 


59 


58 




57 3 


., 


6.3 


6.2 6.1 


.! 


6.0 


5-9 


5-8 


.! 


5-7 0.3 


.2 


12.6 


12-4 12.2 


2 


12.0 


ii. 8 


n.6 


2 


11.4 0.6 


3 


18.9 


18.6 18.3 


3 


18.0 


17.7 


17.4 


3 


17.1 0.9 


4 


25.2 


24.8 24.4 


4 


24.0 


23.6 


23.2 


4 


22.8 1.2 


5 


31.5 


31.0 30.5 


5 


30.0 


29-5 


29.0 


5 


28.5 1.5 


.6 


37-8 


37.2 36.6 


6 


36.0 


35-4 


34- 8 


6 


34.2 1.8 


.7 


44.1 


43-4 42-7 


7 


42.0 


4i-3 


40.6 


7 


39.9 2. I 


.8 


50.4 


49.6 48.8 


8 


48.0 


47.2 


46.4 


.8 


45-6 2.4 


9 56-7 


55- 8 54-9 


9 


54- 


53- ! 


52.2 


.9 


5 r -3 2.7 



54 



12 30 '. 



; 


L. Sin. 


d. 


L. Tang. 


d. 


L 


, Cotg. 


L. Cos. 


d. 




30 


9.33 534 


57 


9.34 


5 7 6 




o.65 424 


9.98 958 




30 


3i 


9.33 591 


S6 


9-34 


635 


60 


o. 


65 365 


9.98955 


3 


29 


32 


9.33647 




9-34 


695 


60 


0. 


653o5 


9.98 


953 




28 


33 


9-33 704 


57 


9-34 


755 


59 


o. 


65245 


9.98 


950 


3 


27 


34 


9.33 761 


57 


9.34 


8i4 


60 


o. 


65 186 


9.98 947 




26 


35 


9.338i8 


-< 


9.34 


8 7 4 




0. 


65 126 


9.98 


944 




25 


36 


9 .338 7 4 


5 


9-34 


9 33 


59 


o. 


65 067 


9.98 


94 1 


3 


24 


37 
38 


9 .33 9 3i 
9.33 987 


57 
56 

efi 


9.34 
9 .35 


992 
o5i 


59 
59 
60 


o. 
o. 


65 008 
64 949 


9.98 938 
9.98 936 


3 

2 


23 
22 


3 9 


9.34o43 


5 


9 .35 


in 




0. 


64 889 


9.98 


933 


3 


21- 


40 


9.34 100 


cfi 


9.36 


170 




o. 


6483o 


9.98 930 




20 


4i 


9 .34i56 


5 


9 .35 


229 


59 


o. 


64 771 


9.98 


927 


3 


19 


42 


9.34 212 


5 


9.35 


288 




0. 


64 712 


9.98 


924 




18 


43 


9.34268 


5 6 
56 


9 .35 


34 7 


59 
58 


o. 


64653 


9.98 


921 


3 

2 


i? 


44 
45 


9.34324 
9.34380 


56 


9 .35 
9.35 


4o5 

464 


59 


o.64 595 
o.64 536 


9.98 
9.98 


919 
916 


3 


16 
i5 


46 


9.34436 


56 


9-35 


523 


59 


o. 


64477 


9.98 913 


3 


i4 








55 






58 










3 




47 
48 


9.34491 
9.34547 


56 


9 .35 
9 .35 


58i 
64o 


59 


o. 
o. 


644i9 
64 36o 


9.98 910 
9.98 907 


3 


i3 

12 


49 


9.34 602 


55 


9 .35 


698 


58 


o. 


64 3o2 


9.98 


904 


3 


I I 


50 


9.34658 


5 


9 .35 


7 5 7 


59 

eg 


0. 


64243 


9.98 


901 


3 


10 


5i 


9.34 71 


1 


56 


9 .35 


815 


58 


o. 


64 1 85 


9.98 


898 


2 


9 


52 


9.34 769 




9.35 


8 7 3 




o. 


64 127 


9.98 


896 




8 


53 


9.34 824 


55 
55 


9 .35 




58 
58 


o. 


64 069 


9.98 


8 9 3 


3 


7 


54 


9.34879 


55 


9.35 


989 


eg 


o. 


64 01 1 


9.98 


890 


3 


6 


55 


9.34934 




9.36 047 




o. 


63 9 53 


9.98 


887 




5 


56 


9.34989 


55 


9.36 


io5 


58 


0. 


638 9 5 


9.98 


884 




4 


57 


9.35 o44 


55 


9.36 


i63 


58 

eg 


0. 


6383 7 


9.98 


881 


3 
3 


3 


58 


9-35 099 




9.36 


221 




o. 


63 779 


9.98 


878 




2 


5 9 


9 .35 1 54 


55 


9. 36 


279 


58 


o. 


63 721 


9.98 


8 7 5 




I 


60 


9.35 209 


55 


9. 36 


336 


57 


0. 


63 664 


9.98 


872 









L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. 


d. 


' 




77V 


. 




PP 60 


59 


58 


57 


56 


? 


55 3 


.1 6.0 


5-9 


5-8 


i 5-7 


5-6 


j 


5-5 0.3 


.2 I2.O 


n.8 


11.6 


.2 II-4 


II. 2 


2 


ii. o 0.6 


.3 18.0 


17.7 


17.4 


3 i7-i 


16.8 


3 


16.5 0.9 


4 24.0 


23.6 


23-2 


.4 22.8 


22-4 


4 


22.0 1.2 


5 3- 


29-5 


29.0 


5 28.5 


28.0 


5 


27-5 i-5 


.6 36.0 


35-4 


34-8 


.6 34.2 


33-6 


6 


33-o 1.8 


.7 42-0 


4 x -3 


40.6 


7 39-9 


39-2 


7 


38-5 2.1 


.8 48.0 


47-2 


46.4 


.8 45-6 


44-8 


8 


44.0 2.4 


9 54- 


S3- 1 5 2 - 2 


9 1 5i-3 


50.4 9 


49-5 2 -7 



55 



' 


L. Sin. 


d. 


L. Tang-. 


d. 


L. 


Cotg. 


L. Cos. d. 







9. 35 209 




9 .36 


336 




0. 


63 664 


9.98 


8 7 2 




60 


I 


9 .35 2 63 


55 


9. 36 3 9 4 


5 
c8 


o. 


63 606 


9.98 86 9 




5 9 


2 


9.35 3i8 




9. 36 


452 




0. 


63 548 


9 . 9 8 


86 7 




58 


3 


9 .353 7 3 




9. 36 


509 


57 


o. 


63491 


9 . 9 8 


864 


3 


57 








54 






57 










- 






4 


9.35427 


54 


9. 36 


566 


rg 


o. 


63434 


9 . 9 8 


861 




56 


5 


9-35481 




9.36 


624 




o . 


63 3 7 6 


9.98 


858 




55 


6 


9.35536 


55 


9. 36 


681 


57 


0. 


63 3i 9 


9.98 


855 


: 




54 


7 


9.35 590 


54 


9.36 


7 38 


57 


0. 


63 262 


9.98 


852 


3 


53 


8 


9.35 644 




9-36 


79 5 




0. 


63 205 


9.98 


84 9 




52 


9 


9.35 698 


54 


9 .36 


85 2 


57 


0. 


63 i48 


9.98 


846 


3 


5i 


10 


9.35 7 5 2 




9 .36 


99 


57 


o. 


63 091 


9.98 


843 


" 




50 


ii 


9.35 806 


54 


9 .36 


9 66 


57 


0. 


63o34 


9.98 


84o 


3 


49 


12 


9.35 860 




9-3 7 


023 




o. 


62 977 


9.98 


83 7 




48 


i3 


9.35 914 


54 


9 .3 7 


080 


57 


o. 


62 920 


9 . 9 8 


834 


- 


> 


47 


i4 


9. 35 968 


54 


9 .3 7 


i3 7 


57 
cfi 


0. 


62863 


9 . 9 8 


83i 


3 


46 


i5 


9-36 022 




9 .3 7 


I 9 3 


5 


0. 


62 8o 7 


9 . 9 8828 






45 


16 


9.36 075 


53 


9 .3 7 


250 


57 


o. 


62 7 5o 


9 . 9 8825 





i 


44 


17 


9.36 129 


54 


9 .3 7 


3o6 


5t> 


o. 


62 6g4 


9 . 9 8 


822 


3 


43 


18 


9.36 182 




9 .3 7 


363 


57 


0. 


62 63 7 


9.98 


8i 9 






42 


1 9 


9. 36 236 


54 


9 .3 7 


4i 9 


56 


o. 


62 58i 


9.98816 


3 


4i 


20 


9.36 289 


53 


9 .3 7 


4 7 6 


57 


o. 


62 524 


9.98 


8i3 





> 


40 


21 


9. 36 342 


53 


9 .3 7 


532 


56 


o. 


62468 


9.98 


810 




3 9 


22 


9 .363 9 5 


53 


9 .3 7 


588 


5 6 


o. 


62 412 


9.98 


8o 7 




38 


23 


g.36 449 


54 


9.37 


644 


56 


o. 


62 356 


9.98 


8o4 


3 


37 


24 


9-36 5o2 


53 


9 .3 7 


700 


56 


0. 


62 3oo 


9-98 


801 


3 


36 


25 


9.36555 


53 


9 .3 7 


7 56 


56 


o. 


62 244 


9 . 9 8 


798 




35 


26 


9.36 608 


53 


9 .3 7 


812 


50 


o. 


62 188 


9 . 9 8 


79 5 







34 


27 
28 


~9?3\66o 
9.36 7 i3 


52 
53 


9 .3 7 
9 .3 7 


868 
924 


56 
56 


o. 

0. 


62 1 32 
62 o 7 6 


9 . 9 8 
9 . 9 8 


79 2 

789 


3 

3 


33 

32 


2 9 


9. 36 766 


53 


9 .3 7 


980 


56 


0. 


62 020 


9 . 9 8 


786 




3i 


30 


9.36 819 


53 


9. 38 


o35 


55 


0. 


61 965 


9 . 9 8 


7 83 




30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. 


d. 


- 






76 3O. 






PP 58 


57 


56 


55 


54 




53 


3 


5-8 


5-7 


5.6 


i 5-5 


5-4 


.1 


5-3 


3 


.2 II. 6 


n-4 . 


II. 2 


.2 II. 


10.8 


.2 


10.6 


.6 


3 17-4 


17.1 


1 6. 8 


3 16.5 


16.2 


3 


159 


9 


4 23 2 


22.8 


22.4 


.4 22.O 


21.6 


4 


21.2 


1.2 


5 29.0 


28.5 


28.0 


5 27.5 


27.0 


.5 


26.5 


i-5 


.6 34-8 


34- 2 


33-6 


6 33.0 


32-4 


.6 


3 1.8 


1.8 


.7 4- 6 


39-9 


392 


7 38.5 


37-8 


7 


37- * 


2.1 


.8 46.4 


45-6 


44-8 


.8 44.0 


43-2 


.8 


42.4 


2-4 


9 5 2 - 2 


51.3 ! 50.4 


9 49-5 


48.6 




47-7 


2.7 



56 



133O 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9.36 


819 




9 


.38 o35 






0.61 965 


9.98 


7 83 




30 


3i 

32 


9-36 
9-36 


8 7 I 
924 


52 

53 


9 
9 


.38 091 
.38 147 


56 




0.61 909 
0.61 853 


9.98 
9.98 


780 
777 


3 
3 


2 9 

28 


33 


9.36 


976 


5 2 


9 


.38 202 


55 




0.61 798 


9.98 


774 


3 


27 


34 


9 .3 7 


028 


52 


9 


.38 2 5 7 


55 
eft 




0.61 743 


9.98 


771 


3 


26 


35 


9 .3 7 


08 1 




9 


.383i3 


5 




0.61 687 


9.98 


768 


3 


25 


36 


9 .3 7 


i33 


5 2 


9 


.38368 


55 




0.61 632 


9.98 


765 


3 


24 


37 


9 .3 7 


i85 


52 


9 


.38423 


55 




0.61 577 


9.98 


762 


3 


23 


38 


9-37 


2 3 7 




9 


.38479 


J 




0.61 52i 


9.98 


7 5 9 


3 


22 


3 9 


9 .3 7 


289 


5 2 


9 


.38 534 


55 




0.61 466 


9 . 9 8 


7 56 


3 


21 


40 


9 .3 7 


34i 




9 


.38 58 9 






0.61 4n 


9.98 


7 53 


3 


20 


4i 


9 .3 7 


3 9 3 


5 2 


9 


38 644 


55 


0.61 356 


9.98 


75o 


3 


'9 


42 


9 .3 7 


445 




9 


38 699 






0.61 3oi 


9 . 9 8 


7 46 




18 


43 


9 .3 7 


497 




9 


38 7 54 






0.61 246 


9.98 


743 


3 


ij 








52 








54 












3 




44 


9 .3 7 


54 9 


51 


9 


38 808 






0.61 192 


9.98 


74o 




16 


45 


9-37 


600 




9 


38863 






0.61 137 


9.98 


737 




i5 


46 


9.3 7 652 




9 


38 918 


55 




0.61 082 


9 . 9 8 


7 34 


J 


i4 


47 


9-3 7 


7 o3 


51 
52 


9 


38 9 7 


2 


54 




0.61 028 


9.98 


7 3i 


3 


i3 


48 


9 .3 7 


755 




9 


39 027 






0.60 973 


9.98 


728 




12 


4 9 


9.3 7 806 


5 1 


9 


39 082 


55 




0.60 9 i8 


9.98 


725 


3 


II 


50 


9.3 7 858 




9 


3 9 1 36 


54 




0.60 864 


9.98 


722 


3 


10 


5i 


9.3-7909 


51 


9 


39 190 






0.60 810 


9.98 


719 




9 


52 


9.3-7 960 




9 


3 9 24 


5 






0.60 755 


9.98 


7 i5 




8 


53 


9-38 on 




9 


3 9 2 99 


54 




0.60 701 


9 . 9 8 


712 


3 


7 








51 








54 












3 




54 


9.38 062 




9 


3 9 353 






0.60 647 


9 . 9 8 


79 




6 


55 


9. 38 


n3 




9 


3 9 4o 


7 






0.60 5 9 3 


9.98 


7 o6 




5 


56 


9.38 


1 64 


5 1 


9 


3 9 46i 


54 




0.60 53 9 


9 . 9 8 


7 o3 


3 


4 








51 








54 












3 




57 


9 .38 


2l5 




9 


3 9 5i5 






0.60485 


9 . 9 8 


7 oo 




3 


58 


9 .38 266 




9 


3 9 56 9 






o.6o43i 


9 . 9 8 


6 97 




2 


5 9 


9. 38 


3i 7 


5 1 


9 


3 9 623 


54 




0.60 377 


9 . 9 8 


6 9 4 


3 


I 


60 


9. 38 


368 


5 1 


9 


3 9 6 77 


54 




0.60 323 


9 . 9 8 


6 9 o 


4 







L. Cos. d. 


L. Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 


' 








76. 








PP 


56 


55 


54 




53 


52 


51 




4 3 


! 


5.6 


5-5 


5-4 


.1 


5-3 


5-2 


5.1 


.1 


0.4 0.3 


2 


II. 2 


II. 


10.8 


.2 


10.6 


10.4 


IO.2 


.2 


0.8 0.6 


3 


16.8 


16.5 


16.2 


3 


15-9 


15.6 


15-3 


3 


1.2 0.9 


4 


22.4 


22. 


21.6 


4 


21.2 


20.8 


20.4 


4 


1.6 1.2 


5 


28.0 


27.5 


27.0 


5 


26.5 


26.0 


25-5 


5 


2.O 1.5 


6 


33-6 


33-0 


32.4 


.6 


31-8 


31.2 


30.6 


.6 


2.4 1.8 


7 


39-2 


38. s 


37-8 


7 


37-1 


*4 


35-7 


7 


2.8 2.1 


8 


44-8 


44.0 


43-2 


.8 


42.4 


41.6 


40.8 


.8 


3.2 2.4 


9 5-4 


49.5 48.6 




47-7 


46.8 


45-9 -9 


3.6 2.7 



14. 



f 


L. Sin. 


d. 


L. 


Tang. 


d. 


L. Cotg. 


L. Cos. d. 







9. 38 368 




9- 


39677 


54 
54 
53 
54 
53 
54 
53 
54 

53 

54 
53 
53 
53 
53 
53 
53 
52 
53 

53 
52 

52 
53 
52 
52 
S 2 
53 
52 


0.60 323 


9.98 


690 




60 


2 

3 

4 
5 
6 

7 
8 

9 


9 .384i8 
9.38469 
9. 38 5i 9 

9.38 570 
9. 38 620 
9. 38 670 

9.38 721 
9. 38 771 
9-38 821 


51 
50 

51 

5 
5 
Si 
50 
So 


9- 
9- 
9- 

9- 
9- 
9- 

9- 
9- 
9- 


3 97 3i 
39785 
3 9 838 

39 892 
3 99 45 
3 9999 

4o o52 
4o 1 06 
4o i5 9 


0.60 269 
0.60 2i5 
0.60 162 

0.60 1 08 
0.60 055 
0.60 ooi 

0.59 948 
0.59 84i 


9.98 687 
9.98 684 
9.98 681 

9 . 9 8 678 
9.98675 
9.98 671 

9.98668 
9.98 665 
9.98 662 


3 
3 
3 
3 
4 
3 
3 
3 


5 9 

58 

56 
55 
54 
53 

52 

5i 


10 


9-38 871 




9- 


4o 212 


0.59 788 


9.98 


65 9 




50 


1 1 

12 

i3 

i4 
i5 
16 

7 

18 

'9 


9-38 921 
9.38 971 
9.39 02 1 

9.39 071 

9.39 121 
9.39 170 

g.Sg 22O 
9.39270 

9 .3 9 3i 9 


So 
So 
50 
50 
49 
So 
5 
49 


9- 
9- 
9- 

9- 
9- 
9- 

9- 
9- 
9- 


4o 266 
4o 3i 9 
40372 

40425 
40478 
4o53i 

4o584 
4o636 
40689 


o.5 9 7 34 
o.5 9 68i 
0.59 628 

o.Sg 575 

O.Sg 522 
O.Sg 469 

0.59 4i6 
0.59 364 
o.Sg 3i i 


9.98 
9.98 
9.98 

9.98 
9.98 
9.98 

9.98 
9.98 
9.98 


656 
652 
649 

646 
643 
64o 

636 
633 
63o 


4 
3 
3 
3 
3 
4 
3 

3 


49 

48 

4? 
46 

44 
43 

42 

4i 


20 


9.39 369 


5 


9- 


4o 742 


0.59 258 


9.98 


627 


40 


21 
22 
23 

24 
25 

26 

27 
28 
29 


9.39 4i8 
9.39467 
9.39 5i 7 

9.39 566 
9.39615 
9.39 664 

9 .3 97 i3 
9.39 762 
9.39 8n 


49 
49 
5 
49 
49 
49 
49 
49 
49 
49 


9- 
9- 
9- 

9- 
9- 
9- 

9- 

9- 
9- 


40795 
40847 
4o 900 

4o 952 
4i 005 
4i 057 

4i 109 
4i 161 
4i 214 


o.Sg 2o5 
0.59 i53 
0.59 100 

0.59048 
o.58 995 
0.58943 

o.58 891 
o.58 83 9 
o.58 786 


9.98 
9.98 
9.98 

9.98 
9.98 
9.98 

9.98 
9.98 
9.98 


623 
620 
617 

6i4 
610 
607 

6o4 
601 

5 97 


3 
3 
3 
4 
3 
3 
3 
4 


3 9 
38 
3? 

36 
35 

34 
33 

32 

3i 


30 


9.39 860 


9- 


4i 266 


o.58 734 


9.98 


594 


3 


30 




L. Cos. 


d. 


L, 


Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 


' 


75 30 . 


PP 

.2 

3 
4 

:I 

7 

-8 

q 


54 


53 53 


.1 

.2 

3 
4 

:I 
:l 

~ 


51 


50 


49 




4 | 3 


,11 

16.2 

21.6 

27.0 
32-4 

37-8 
|f 


53 5-2 
10.6 10.4 
15.9 15.6 

21.2 20.8 

26. 5 26.0 

31.8 31.2 

37-1 36-4 
42.4 41.6 

47-7 46.8 


IO.2 

15-3 
20.4 

25 I 
30.6 

35-7 
40.8 

45 9 


5-o 

IO.O 

15-0 

20.0 
25.0 
30.0 

35-o 
40.0 


4.9 
9.8 
14.7 

19.6 
24-5 
29-4 

34-3 
39-2 
44.1 


.2 

3 
4 

1 

:i 

9 


:J :i 

1.2 .9 

1.6 1.2 
2.0 I. 5 

2.4 1.8 

2.8 2.1 

3.2 2.4 

3.6 2.7 



58 



143O 



> 


L. Sin. 


d. 


L. 


Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9.39 860 


49 


9.41 266 




o.58 734 


9.98 594 




30 


3i 


9.39909 


49 


9.41 3i8 


52 


o.58 682 


9.98 591 




29 


32 


9.39 958 


48 


9.41 370 




o.5863o 


9.98 588 




28 


33 


9.40 006 


49 


9.41 422 


52 


o.58 578 


9.98 584 


^ 


27 


34 


9.40 c 


55 


48 


9.41 4?4 


s 2 


o.58 526 


9.98 58 r 


3 


26 


35 


9.40 io3 




9- 


4i 526 




0.58474 


9.98 


5 7 8 




25 


36 


9.40 i 


52 




9.41 5 7 8 




0.58422 


9 . 9 85 7 4 




24 








48 






5* 










3 




37 


9.40 200 


49 


9- 


4i 629 


s 2 


o.58 3 7 i 


9.98 


5 7 i 


3 


23 


38 


9.40 249 




9- 


4r 681 




o.58 319 


9.98 568 




22 


39 


9.40 297 


4 


9- 


4i 733 


52 


o.58 267 


9.98 


565 


3 


21 


40 


9.40 346 


.0 


9- 


4i 784 




o.58 216 


9.98 


56i 




20 


4i 


9 .4o 3 


94 


48 


9- 


4i 836 




o.58 i64 


9.98 


558 


3 


19 


42 


9.40 442 




9- 


4i 887 




o.58 n3 


9.98 


55,5 




18 


43 


9.40490 


48 


9- 


4i 939 


52 
51 


o.58 061 


9.98 


55i 




l l 


44 


9.40 538 




9- 


4i 990 




o.58 oio 


9.98 


548 


3 


16 


45 


9.40 586 




9- 


42 o4i 




0.57 959 


9.98 


545 




i5 


46 


9.40 634 


48 


9- 


42 093 


52 


0.57 907 


9.98 


54i 




i4 








48 






51 










3 




47 


9.40 682 


4.8 


9- 


42 i44 




o.5 7 856 


9.98 


538 


3 


i3 


48 


9.40 730 




9- 


42 195 




0.57 805 


9.98 


535 




12 


4 9 


9.40778 


48 


9- 


42 246 


51 


0.57 754 


9.98 


53i 


3 


I I 


50 


9.40 825 


47 

.0 


9- 


42 297 


5 1 


0.57 703 


9.98 


5 2 8 




10 


5i 


9.40 873 


48 


9- 


42 348 


5 1 


0.57 652 


9.98 


525 


4 


9 


52 


9.40 921 




9- 


42 3 99 




0.57 601 


9.98 


521 




8 


53 


9.40968 


47 
48 


9- 


42 45o 


Si 
51 


0.57 550 


9.98 


5i8 


3 


7 


54 


9.41 016 




9? 


42 5oi 




0.57 499 


9.98 


515 


4 


6 


55 


9.41 o63 




9- 


42 552 




o.5 7 448 


9.98 


5n 




5 


56 


9.41 in 


4 


9- 


426o3 


51 


o.5 7 3 97 


9.98 


5o8 




4 


57 


9.41 i58 


47 


9- 


42653 


So 


o.5 7 347 


9.98 


505 


3 
4 


3 


58 


9.41 2o5 




9- 


42 704 




0.57 296 


9.98 


5oi 




2 


5 9 


9-4l 252 


47 


9- 


42755 


bl 


0.57 245 


9.98 


498 




I 


60 


9.41 3oo 


40 


9- 


42 8o5 


5 


o.5 7 195 


9.98 


494 






" 




L. Cos. 


d. 


L. 


Cotg. 


d. L. Tang. 


L. Sin. 


d. 








75. 






PR 


52 


51 50 


49 


4 8 


47 




4 


3 


.! 


52 


5.1 5.0 


.1 4.9 


4 .8 


4-7 


.1 


0.4 


-3 


2 


10.4 


10.2 10. 


.2 9.8 


9 .6 


9.4 


.2 


0.8 


0.6 


3 


15-6 


15-3 15 


3 14-7 


14.4 


14.1 


3 


1.2 


0.9 


4 


20.8 


20. 4 20. o 


.4 19.6 


19.2 


18.8 


4 


1.6 


1.2 


I 


26.0 

31 2 


25.5 25.0 
30.6 30.0 


5 24.5 
.6 29.4 


24.0 
28.8 


23-5 
28.2 


5 
.6 


2.O 
2. 4 


1^8 


7 


36.4 


35-7 35-o 


7 34-3 


33-6 


32-9 


7 


2.8 


2.1 


8 


41.6 


40.8 40.0 


.8 39.2 


38-4 


37-6 


.8 


3-2 


2-4 


.9 46.8 


459 45' 


.9 44.1 43.2 


42.3 .9 





59 



15. 






L. Sin. d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. d. 







9-4i 


3oo 


9.42 bo5 


51 
50 
51 
50 
50 
51 
50 
50 
50 
5 

5 

5 
50 
50 
50 
49 
So 
5 
49 
50 

49 
5o 
49 
5 
49 
49 
49 
5 
49 
49 


o.5 7 195 


9.98 494 




60 


2 

3 

4 
5 
6 

7 

8 

9 


9-4i 
9>4i 
9.41 

9,41 
9.41 
9-4i 

9-4i 
9.41 
9.41 


34? 
394 
44 1 

488 
535 
582 

628 
670 
722 


47 
47 
47 
47 
47 
46 

47 
47 


9.42 856 
9.42 906 
9.42 957 

9.43 007 
9.43o5 7 
9.43 108 

9 .43i58 
9.43 208 
9.43258 


0.57 1 44 
0.57 094 
0.57 043 

0.56993 
o.56 943 
o.56 892 

0.56842 
o.56 792 
o. 56 742 


9.98 491 
9.98 488 
9.98484 
9.98 481 

9.98477 
9.98 474 

9.98 471 
9.98 467 
9. 98 464 


3 
4 
3 
4 
3 
3 
4 
3 


5 9 

58 
5? 
56 
55 
54 

53 

52 

5i 


10 


9.41 


768 


47 
46 

47 
46 

47 
46 
46 
47 
46 


9.43 3o8 


o.56 692 


9 


. 98 460 




50 


1 1 

12 

i3 

i4 
i5 
16 

17 
18 

19 


9.41 
9.41 

9-4i 

9.41 
9.42 

9.42 

9.42 
9.42 
9.42 


815 
861 
908 

954 

OOI 

o47 

ogS 

i4o 

186 


9.43358 
9.434o8 
9-43458 

9 .435o8 
9.43558 
9.43 607 

9 .4365 7 
9.43 707 
9 .43 7 56 


0.56642 
o.56 592 
o., 56 542 

o.56 492 
0.56442 
0.56393 

0.56343 
o.56 293 
o.56 244 


9 
9 
9 

9 
9 
9 

9 
9 
9 


.98457 
.98453 
.98450 

.98447 
.98443 
.98440 

.98436 
.98433 

.98 429 


4 
3 

4 
3 
4 
3 

4 


49 

48 

4? 

46 
45 
44 

43 

42 

4i 


20 


9.42 


232 


46 


9.43 806 


o.56 194 


9 


.98426 


4 
3 
4 
3 
3 
4 
3 
4 
3 
4 


40 


21 
22 
23 

24 
25 

26 

27 
28 
29 


9.42 
9.42 
9.42 

9.42 
9.42 
9.42 

9.42 
9.42 
9.42 


278 
324 
3 7 o 

4i6 
46 1 
507 

553 
5 99 

644 


46 
46 
46 
46 
45 
46 
46 
46 
45 
46 


9.43855 
9.43 905 
9 .43 9 54 

9.44 oo4 
9.44 o53 

9.44 102 

9.44 i5i 
9.44 20 1 
9-44 250 


o.56 i4 
o.56 095 
o.56 o46 

o.55 996 
o.55 947 
o.55 898 

0.55849 
o.55 799 
0.55 7 5o 


9 
9 
9 

9 
9 
9 

9 

9 
9 


.98 422 

.98419 
.98415 

.98412 
.98 409 
.98 4o5 

.98 402 
.98 398 
.98395 


3 9 
38 
3 7 
36 
35 
34 
33 

32 

3i 


30 


9.42 


690 


9.44 299 


o.55 701 


9 


.98 391 


30 




L. Cos. 


d. 


L. Cotgr. d. 


L. Tang. 


L. Sin. 


d. 








74 3D '. 






PP 

.1 

.2 

3 
4 

:l 

:i 


51 


50 


49 




48 47 46 


.1 

.2 

3 

4 

5 
.6 

:l 

9 


45 


4 3 


5-i 

10.2 
15-3 

20-4 
25.5 
30.6 

35-7 
40.8 

45 9 


5-o 

10.0 

15-0 

20. o 

25-0 
30.0 

35-o 
40.0 

45.0 


49 
9.8 
14.7 

19.6 
24-5 
29.4 

34-3 
39- 2 
44.1 


.2 

-3 

4 
5 
.6 

3 


4.8 4-7 4-6 
9.6 9.4 9.2 
14.4 14.1 13.8 

19.2 18.8 18.4 
24.0 23.5 23.0 
28.8 28.2 27.6 

33-6 32.9 32.2 
38.4 37.6 36.8 
43.2 423 41 4 


4-5 
9.0 
i3-5 

18.0 
22.5 
27.0 

3'-5 
36.0 

4-5 


0.4 0.3 
0.8 0.6 
1.2 0.9 

1.6 1.2 
2.0 1.5 

2.4 1.8 

2.8 2.1 

3.2 2.4 

3.6 2.7 



60 



15 3O 





L. Sin. d. 


L 


. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9.42 


690 




9 


.44 299 






o.55 701 


9.98 


3 9 i 






30 


3i 


9.42 


7 35 


46 


9 


.44348 


49 




0.55652 


9.98 


388 


3 


29 


32 


9.42 781 




9 


.44397 






o.556o3 


9.98 


384 




28 


33 


9.42 


826 


46 


9 


.44446 






0.55554 


9.98 


38i 


3 
4 


27 


34 


9.42 872 


45 


9 


.44495 


49 




o.555o5 


9.98 


377 




26 


35 


9.42 


917 




9 


.44544 


Q 




0.55456 


9.98 


3 7 3 




25 


36 


9.42 


962 


45 


9 


.44 592 






o. 554o8 


9.98 


370 


3 


24 








46 








49 


















37 


9.43 008 




9 


,4464i 






o.5535 9 


9.98 


366 




23 


38 


9-43o53 




9 


44 690 


.0 




o.55 3io 


9.98 


363 




22 


39 


9-43 


098 


45 


9 


44738 






o.55 262 


9.98 


35 9 


4 


21 


40 


9.43 


i43 


45 


9 


.44 787 






o.55 2i3 


9.98 


356 




20 


4i 


9-43 


188 


45 


9 


.44836 






o.55 1 64 


9.98 


352 


4 


'9 


42 


9.43 


233 




9 


4488 


4 


1 




o.55 116 


9.98 


349 




18 


43 


9.43 


278 


45 


9 


44933 


49 




o.55 067 


9.98 


345 


4 


17 


44 


9-43 


323 


45 


9 


.44981 


48 




o.55 019 


9.98 


342 


3 


16 


45 


9.43 


36 7 




9 


45 029 


1 




0.54 971 


9.98 


338 




i5 


46 


9.43412 


45 


9 


45 078 


49 




o.54 922 


9.98 


334 


4 


i4 


4 7 


9.43457 


45 


9 


.45 126 


4 H 




0.54874 


9.98 


33i 


3 


i3 


48 


9.43 


502 




9 


45 174 






0.54826 


9.98 


32 7 




12 


49 


9-43 


546 


44 


9 


.45 222 


48 




o.54 778 


9.98 


324 


3 


I I 


50 


9.43 


5 9 i 


45 


9 


.45 271 


49 




o.54 729 


9.98 


320 




10 


Si 


9.43635 


44 


9 


.453l 9 


48 




o.5468i 


9.98 


3i 7 


3 


9 


52 


9.43 680 


45 


9 


.4536 7 


4 




0.54633 


9.98 


3i3 




8 


53 


9-43 


724 


44 




.4541 


s 


48 




0.54585 


9.98 


309 


4 


7 


54 


9-43 


769 


45 


9 


.45463 


48 

.0 




0.54537 


9.98 


3o6 


3 


6 


55 


9.43 


8i3 


44 


9 


.455n 






0.54489 


9.98 


302 




5 


56 


9.43 


85 7 


44 


9 


.4555 9 


48 




o.5444i 


9.98299 




i 


4 


5 7 


9.43 


901 


44 


9 


.456o6 


47 

.Q 




o.543 9 4 


9.98 295 


4 


3 


58 


9-43 


946 


45 


9 


.45654 






0.54346 


9.98 


291 




2 


5 9 


9.43 


990 


44 


9 


.45 702 






o.54 298 


9.98 288 


3 


I 


60 


9-44 


o34 


44 


9 


.45750 


4 




0.54 25o 


9.98 


284 









L. Cos. 


d. 


L 


. Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 


' 








74 


. 








PP 


49 


48 47 




46 


45 


44 




4 


3 


.1 


4-9 


4-8 4-7 


,i 


4 .6 


4-5 


4-4 


.1 


0.4 


'3 


.2 


9.8 


9.6 9.4 


.2 


9-2 


9.0 


8.8 


.2 


0.8 


0.6 


3 


14.7 


14.4 14.1 


3 


13-8 


13-5 


13-2 


3 


1.2 


0.9 


4 


19.6 


19.2 18.8 


4 


18.4 


18.0 


17.6 


4 


1.6 


1.2 


f c 


24-5 


24.0 23.5 


. tj 


23.0 


22.5 


22.0 


5 


2.0 


1.5 


.6 


29.4 


28.8 28.2 


.6 


27.6 


27.0 


26.4 


.6 


2. 4 


1.8 




34-3 


33- 6 32-9 


7 


32.2 


3i-5 


30.8 


7 


2.8 


2.1 




39 2 


38-4 37-6 


.8 


36.8 


36.0 


35-2 


.8 


3-2 


2. 4 


9 44- x 


43 .2 4^ 




41.4 


40-5 


39-6 -9 


3-6 


2-7 



61 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 







9.44 o34 




9.45 750 




o.54 25o 


9 


.98 284 




60 


I 


9.44078 




9.45 797 


47 


0.54 203 


9 


.98 281 




5 9 


2 


9.44 122 




9.45845 


4 


o.54i55 


9 


.98277 




58 


3 


9.44 1 66 


44 


9.45 892 


47 


o.54 108 


9 


.98273 




57 








44 






48 












3 




4 


9 . 44 2 1 o 




9.45 940 




o.54 060 


9 


.98 270 




56 


5 


9.44253 




9.45987 




o.54oi3 


9 


.98 266 




55 


6 


9.44 297 


44 


9.46 035 


48 


o.53 9 65 


9 


.98262 




54 


7 


9.4434i 


44 


9.46 082 


47 
j.8 


o.53 918 


9 


.98 259 


3 


53 


8 


9.44385 




9.46 i3o 




o.53 870 


9 


.98 255 




52 


9 


9.44428 


43 


9-46 177 


47 


0.53823 


9 


.98 261 




5i 


10 


9.44 472 




9.46 224 


47 


o.53 776 


9 


.98 248 




50 


ii 


9.44 5i6 




9.46 271 


47 

.0 


o.53 729 


9 


.98244 




49 


12 


9.4455 9 




9.46 319 




0.5368T 


9 


.98 240 




48 


i3 


9-44 602 


43 


9.46 366 


47 


0.53634 


9 


. 9 823 7 


3 


47 


i4 


9-44646 


44 


9-464i3 


47 


o.5358 7 


9 


.98 233 


4 


46 


i5 


9.44689 




9-46 46o 


47 


o.53 54o 


9 


.98 229 




45 


16 


9.44733 


44 


9-46 507 


47 


0.53493 


9 


.98 226 


3 


44 


1 7 


9-44 776 


43 


9.46554 


47 


0.53446 


9 


.98 222 


4 
4 


43 


18 


9.44 819 




9.46 601 


47 


o.53 399 


9 


.98 218 




42 


'9 


9.44 862 


43 


9.46648 


47 


o.53 352 


9 


.98215 


3 


4i 


20 


9.44 go5 




9.46 694 


46 


o.533o6 


9 


.98 211 


4 


40 


21 


9.44948 




9.46 741 


47 


o.53 259 


9 


.98 207 


3 


3 9 


22 


9.44 992 




9.46 788 


47 


O.53 212 


9 


.98 204 




38 


23 


9.45 035 


43 


9.46835 


47 


o.53i65 


9 


.98 2OO 


4 


37 


24 


9.45077 


42 


9.46 881 


46 


o.53 119 


9 


.98 196 


4 
4 


36 


25 


9.45 120 


43 


9.46 928 


47 


o.53 072 


9.98 192 




35 


26 


9.45 i63 


43 


9.46975 


47 


o.53 025 


9 


.98 189 




34 


27 


9.45 206 


43 


9-47 021 


46 


o.52 979 


9 


.98 185 


4 


33 


28 


9.45 249 




9.47 068 


47 


o.52 932 


9 


.98 181 




32 


29 


9.45 292 


43 


9-47 


i4 


46 


o.52 886 


9 


.98 177 




3i 


30 


9 .45334 


42 


9.47 160 


46 


o.52 84o 


9 


.98 174 




30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L.Sin. Id. 


' 






73 3D . 








PP 


48 


47 


46 




45 


44 


43 




42 


4 3 


,! 


4 .8 


4-7, 


4.6 


.1 


4-5 


4-4 


4-3 


.i 


4.2 


0.4 0.3 


.2 


q.6 


9-4\ 


9.2 


.2 


q.o 


8.8 


8.6 


.2 


8. 4 


0.8 0.6 


3 


14.4 


14.1 


13-8 


3 


13-5 


13.2 


12.9 


3 


12.6 


1.2 0.9 


4 


19.2 


18.8 


18.4 


4 


18.0 


17.6 


17.2 


4 


16.8 


1.6 1.2 


.6 


24.0 
28.8 


S5 


23-0 
27.6 


:! 


22.5 
27.0 


22.0 
26. 4 


21-5 

25-8 


5 
.6 


21. 
25.2 


2.0 J-5 

2.4 1.8 


7 


336 


32.9 


32.2 


. 7 


*.5 


30.8 


30.1 


7 


29.4 


2. 8 2. I 


.8 


38.4 


37-6 


36.8 


.8 


36.0 


35-2 


tf-4 


.8 


33-6 


3 .2 2. 4 


9 43.2 


42.3 41-4 




40. s 3Q.6 38.7 




37.8 


3.6 2.7 



16 30 '. 





L. Sin. 


d. 


L. Tang. d. 


L. Cotg. 


L. Cos. 


d. 




30 


9.45 344 




9.47 160 




o.52 84o 


9 


.98 174 




30 


3i 

32 


9.45 419 


43 
42 


9.47 207 
9.47 253 


47 
46 


o.52 793 
o.52 747 


9 
9 


.98 170 
.98 1 66 


4 
4 


29 

28 


33 


9-45 


462 


43 


9.47299 


46 


o.52 701 


9 


.98 162 


4 


27 


34 
35 


9-45 
9.45 


5o4 
54 7 


4 2 
43 


9.47346 
9-47392 


47 
46 


o.52 654 

0.52 608 


9 
9 


.98 159 
.98 i5 


3 
4 


26 

25 


36 


9.45 


589 




9-47438 


46 


o.52 562 


9 


.98 i 


5i 


4 


24 


37 


9-45 


632 


43 


9-47484 


46 
46 


o.52 5i6 


9 


.98147 


4 


23 


38 


9-45 6 7 4 




9.47 53o 




o.52 470 


9 


.98 1 44 


3 


22 


3 9 


9.45 


716 


4 2 


9.47 576 


4 6 


O.52 424 


9 


.98 i4o 


4 


21 


40 


9.45 


7 58 




9.47 622 


4 


o.5 2 3 7 8 


9 


.98 1 36 


4 


20 


4i 


9-45 


801 


4 2 


9.47668 


46 


o.52 332 


9 


.98 i32 


4 


19 


42 


9-45 


843 




9-47 7*4 




0.52 286 


9 


.98 129 




18 


43 


9-45 


885 


42 


9.47 760 


46 


0.52 24O 


9 


.98125 


4 


17 


44 


9.45 


9 2 7 


42 


9.47 806 


.f. 


o.52 194 


9 


.98 121 




16 


45 


9-45 


969 




9.47852 




0.52 li 


18 


9 


.98 117 




i5 


46 


9-46 


OI I 




9.47897 


45 


o.52 io3 


9 


.98 ii3 


4 


i4 


47 


9.46o53 


42 


9-47943 


46 
.f. 


O.52 of 


>7 


9 


.98 no 


3 


i3 


48 


9.46 095 




9.47989 


4 


O.52 Oil 


9 


.98 106 




12 


49 


9-46 


i36 


4 1 


9.48 035 


4'' 


o.5i 965 


9 


.98 102 


4 


I I 


50 


9-46 


178 




9.48 080 


45 


o.5i 920 


9 


.98 098 


4 


10 


5i 


9-46 


220 


4 2 


9.48 126 




o.5i 8 7 4 


9 


.98 094 


4 


9 


52 


9-46 


262 




9.48 171 




o.5i 829 


9 


.98 090 




8 


53 


9-46 


3o3 




9.48 217 


40 


o.5i 783 


9 


.98 087 


3 


7 


54 


9-46 


34.5 


4 2 


9.48 262 


45 


o.5i 7 38 


9 


.98083 


4 


6 


55 


9 .46 386 




9.48 307 




o.5i 693 


9 


.98 079 




5 


56 


9-46 


428 


42 


9.48 353 


4 6 


o.5i 64 7 


9 


.98 075 


4 


4 








4> 






45 












4 




57 


9.4646 9 




9.48 398 




o.D i 602 


9 


.98 071 




3 


58 


9-46 


5ii 




9-48443 




o.5i 55 7 


9 


.98 067 




2 


5 9 


9.46 


552 


4 1 


9.48489 


46 


o.5i 5ii 


9 


.98 o63 


4 


I 


60 


9-46 


594 




9 .48 534 


45 


o.5i 466 


9 


.98 060 









L. Cos. 


d. 


L. Cotg. 


d. 


L, Tang. 


L. Sin. 


d. 


' 




73. 






PP 


47 


4 6 


45 




44 


43 


42 




41 


4 3 


.2 


4-7 
9-4 


4-6 
9.2 


4-5 
9.0 


.2 


a 


tl 


t: 


.1 

2 


4.1 

8.2 


0.4 0.3 
0.8 0.6 


3 


14.1 


i3- 8 


13-5 


3 


13-2 


12.9 


12.6 


3 


12.3 


1.2 0.9 


4 


18.8 


18.4 


18.0 


4 


17.6 


17.2 


16.8 


4 


16.4 


1.6 1.2 


5 
.6 


23-5 
28.2 


23.0 
27.6 


22.5 
27.0 


.6 


22.O 
26.4 


21.5 
25.8 


21.0 
25.2 


5 
.6 


20.5 

24.6 


2.0 1.5 

2.4 1.8 


7 


32.9 


32.2 


3i 5 


7 


30.8 


30.1 


29.4 


7 


28.7 


2.8 2.1 


.8 


37-6 


36.8 


36.0 


.8 


35-2 


34-4 


33-6 


.8 


32.8 


3.2 2.4 


9 42.3 


41.4 40.5 




39-6 38.7 37.8 









63 



17 



f 


L. Sin. 


d. 


L. 


Tang 


. 


d. 


L. Cotg. 


L. Cos. 


d. 







9 .46 5 9 4 




9- 


48 534 


45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
44 
45 
44 
45 
44 
45 

44 
45 
44 
44 
45 
44 
44 
44 
44 
44 


o.5i 466 


9 . 98 060 




60 


I 

2 

3 

4 
5 
6 

8 
9 


9-46635 
9.46 676 
9.46717 

9 .46 758 
g.46 800 
9.46 84i 

9.46 882 
9.46 923 
9.46 964 


4 1 
4 1 

42 
4 1 
4 r 

40 
4 1 
4 1 
4 1 

40 
40 

40 

4 1 

40 

4* 

40 

4 
4 1 

40 
40 
40 


9.48 579 
-48 624 
9.48 669 

9.48 714 
9-48 759 
9.48 8o4 

9.48849 
9 .48 8 9 4 
9.48 939 


o.5i 421 
o.5i 376 
o.5i 33i 

o.5i 286 
o.5i 241 
o.5i 196 

o.5i i5i 
o.5 1 1 06 
o.5i 06 1 


9.98 o56 
9 . 9 8o52 
9.98 o48 

9.98 o44 
9.98 o4o 
9.98 o36 

9.98 o32 
9.98 029 
9.98 025 


4 
4 
4 
4 
4 
4 
3 
4 


5 9 

58 
5 7 

56 
55 
54 
53 

52 

5i 


10 


9.47 005 


9- 


48 9 84 


o.5i 016 


9.98 


O2I 




50 


1 1 

12 

i3 

i5 
16 

17 
18 

'9 


9.47 o45 
9.47 086 
9.47 127 

9.47 168 
9.47 209 
9.47 249 

9.47 290 
9.47371 


9.49 029 
9.49 073 
9.49 118 

9.49 i63 
9.49 207 

9.49 252 

9.49 296 
9.49 34i 
9.49 385 


o.5o 971 
o.5o 927 
o.5o882 

o.5o 837 
o.5o 793 

0.50748 

o.5o 704 
o.5o 65g 
o.5o 615 


9.98 017 
9.98 oi3 
9.98 009 

9.98 oo5 
9.98 ooi 
9.97 997 

9.97993 
9.97989 
9.97986 


4 
4 
4 
4 
4 
4 
4 
3 
4 

4 
4 
4 
4 
4 
4 
4 
4 
4 
4 


49 
48 

47 
46 
45 
44 

43 

42 

4i 


20 


9.47 4i i 


9- 


49 43o 


o.5o 570 


9-97 


9 82 


40 


21 
22 
23 

24 
25 

26 

27 
28 
29 


9.47 452 
9.47 492 
9.47533 

9.47 6i3 
9.47654 

9.47 694 
9.47 734 
9.47 774 


9.49 474 
9 .49 5ig 
9.49 563 

9.49 607 
9.49 652 
9.49 696 

9.49 74o 
9.49 784 
9.49 828 


o.5o 526 
o.5o48i 
o.5o 437 

o.5o 393 
o.5o348 
o.5o 3o4 

o.5o 260 
o.5o 216 
o.5o 172 


9.97978 
9.97 974 
9.97970 

9.97966 
9.97962 
9.97 958 

9.97954 
9.97950 
9.97 946 


3 9 
38 
37 
36 
35 
34 

33 

32 

3i 


30 


9.47 8i4 


40 


9- 


49872 


o.5o 128 


9.97 942 


30 




L. Cos. 


d. 


L. 


Cotg. d. 


L. Tang. 


L. Sin. 


d. 


' 


72 3O . 


PP 

.1 

.2 

-3 

4 

1 

9 


45 


44 43 


.2 
3 

4 


42 


41 


40 




4 3 


4-5 
9.0 

13-5 

18.0 
22.5 
27.0 

3 r -5 
36.0 

40.5 


tl *i 

13.2 12.9 
17.6 17.2 

22.0 21-5 
26.4 25.8 

30.8 30.1 

35-2 34-4 














8.4 

12.6 

16.8 

21. 
25-2 

29.4 

33-6 


8.2 

12.3 

16.4 

20.5 

24.6 

28.7 

32.8 


8.0 

12. 

16.0 
20. o 

24.0 
28.0 

32.0 

36.0 


.2 

3 

4 
5 
.6 

7 
.8 


0.8 0.6 
1.2 0.9 

1.6 1.2 
2.O 1-5 

2.4 1.8 

2.8 2.1 
3-2 2.4 

3-6 2.7 



64 



17 30 



J 


L. Sin. 


d. 


L 


. Tang. d. 


L. Cotg. 


L. Cos. 


d. 




30 


9.47 8i4 




9 


.49 872 






o.5o 128 


9-97 


942 




30 


3i 


9.47 


854 




9 


.49 916 


44 




o.5o o84 


9-97 


9 38 




29 


32 


9.47 


8 9 4 




9 


.49 960 






o.5o o4o 


9-97 


934 




28 


33 


9-47 


934 


40 


9 


. 5o oo4 


44 




0.49 996 


9-97 




4 


27 


34 
35 
36 


9.47 
9-48 
9-48 


974 
014 
o54 


40 
40 


9 
9 
9 


.5oo48 
. 5o 092 
.5o 1 36 


44 
44 
44 




0.49 952 
0.49 908 
0.49 864 


9.97926 
9.97922 
9.97918 


4 
4 
4 


26 

25 
24 


37 
38 
3 9 


9-48 
9-48 
9.48 


094 
i33 
i 7 3 


40 

39 
40 


9 
9 
9 


.5o 180 

.50 223 

. 5o 267 


44 
43 
44 




0.49 820 
0.49 777 
0.49 733 


9-97 
9-97 
9-97 


910 
906 


4 
4 

4 


23 
22 
21 


40 


9-48 


2l3 




9 


.5o3n 






0.49 689 


9-97 


902 




20 


4i 


9.48 


252 


39 


9 


.5o355 


44 




0.49 645 


9-97 


898 




19 


42 


9-48 


292 




9 


.5o3 9 8 






0.49 602 


9-97 


8o4 




18 


43 


9.48 


332 


4 


9 


. 5o 442 


44 




0.49558 


9-97 


890 




'7 


44 


9-48 


3 7 i 


39 


9 


.5o485 


43 




0.49 5i*j 


9-97 


886 


4 


16 


45 


9-48 


4u 




9 


.5o 529 






0.49 471 


9-97 


882 




i5 


46 


9-48 


45o 


39 


9 


.5o 572 


43 




0.49 428 


9-97 


878 




i4 








4 








44 












4 




47 


9.48 490 


39 


9 


.5o6i6 






0.49 384 


9.97874 


4 


i3 


48 


9.48 


529 




9 


.5o65 9 






0.49 34i 


9-97 


870 




12 


49 


9-48 


568 


39 


9 


.5o 703 


44 




0.49 297 


9-97 


866 




I I 


50 


9.48 


607 


39 


9 


.50746 


43 




0.49 254 


9-97 


861 




10 


5i 


9-48 


647 




9 


.5o 789 


43 




0.49 211 


9-97 


85 7 




9 


52 


9-48 


686 




9 


5o833 






O.49 167 


9-97 


853 




8 


53 


9.48 


7 25 


39 


9 


.50876 


43 




O.49 124 


9-97 


849 




7 


54 


9-48 


764 


39 


9 


.5o 919 


43 




0.49 08 I 


9-97 


845 


4 


6 


55 


9-48 


8o3 




9 


. 5o 962 






0.49 o38 


9-97 


84 1 




5 


56 


9-48 


842 


39 


9 


. 5 1 oo5 


43 




0.48 995 


9-97 


83 7 




4 


5 7 


9-48 


881 


39 


9 


.5i o48 


43 




0.48952 


9-97 


833 


4 
4 


3 


58 


9-48 


920 




9 


.5l 9 2 






o.48 908 


9-97 


820 




2 


5 9 


9-48 


9 5 9 


39 


9 


.5i 135 


43 




0.48865 


9-97 


825 




I 


60 


g.48 


998 


39 


9 


.5i 178 


43 




o.48 822 


9-97 


821 









L. Cos. d. 


L 


. Cotg. d. L. Tang. 


L. Sin. d. 


f 






72 








PP 


44 


43 


42 




* 


40 


39 




5 4 


, 


4-4 


4-3 


4.2 


.1 


4.1 


4.0 


3-9 


.1 


0.5 0.4 


.2 


8.8 


8.6 


8.4 


.2 


8.2 


8.0 


7.8 


.2 


i.o 0.8 


3 


13.2 


12.9 


12.6 


3 


12.3 


I2.O 


11.7 


3 


1-5 1-2 


4 


17.6 


17.2 


16.8 


4 


,6.4 


16.0 


15-6 


4 


2.0 1.6 


5 


22. 


21-5 


21. 


5 


20.5 


20. o 


*9* 5 


5 


2.5 2.0 


.6 


26.4 


25-8 


25-2 


.6 


24.6 


24.0 


23-4 


.6 


3.0 2.4 


7 


30.8 


30.1 


29.4 


7 


28.7 


28.0 


27-3 


7 


3-5 2.8 


.8 


35-2 


34-4 


33-6 


.8 


32-8 


32.0 


31.2 


.8 


4.0 3.2 


9 39-6 


3 8 -7 37- 8 


.9 




36.0 


35- x 




4-5 3- 6 



65 



18 C 





L. Sin. 


! d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 







9.48 998 






9-5i 178 


40 


0.48 822 


9 


97821 




60 


I 


9.49 037 




39 


9-51 221 


43 


0.48 779 


9 


97817 




5 9 


2 


9.49 076 






9.61 264 




o.48 736 


9 


97812 




58 


3 


9.49 115 




39 
38 


9. 5 1 3o6 


43 


o.48 694 


9 


97 808 


4 


57 


4 


9.49 i53 






9.5i 349 


43 


o.4865i 


9 


97 8o4 




56 


5 


9.49 192 






9.5i 392 




o.48 608 


9 


97 800 




55 


6 


9.49 23i 




39 


9 .5i4 


*5 




o.48 565 


9 


97796 


4 


54 








3 8 






43 










4 




7 


9.49 269 






9.51478 


42 


0.48 522 


9 


97792 




53 


8 


9.49 3o8 






9.5i 5- 


20 




o.4848o 


9 


97788 




52 


9 


9.49 347 




39 


9 .5i 563 


43 


0.48437 


9 


97784 


4 


5i 


10 


9.49 385 


JC 


9. 5 1 606 




o.48 3g4 


9 


97779 




50 


1 1 


9.49 424 




39 


9.61 648 




0.48352 


9 


97775 




49 


12 


9.49 462 




3 


9.5i 691 




o.48 309 


9 


97 771 




48 


i3 


9.49 5oo 




38 


9 .5i 7 34 


43 


o.48 266 


9 


97 767 


4 


47 


i4 


9.49 539 




39 

,0 


9.61 776 


42 


0.48 224 


9 


97763 


4 


46 


i5 


9.49 577 




3 


9-5i 819 




0.48 181 


9 


97 759 




45 


16 


9.49 6 1 5 




38 


9-5i 861 


42 


o.48 139 


9 


97754 


5 


44 


17 


9.49654 




39 

-0 


9-5i 903 


42 


0.48 097 


9 


9 7 7 5o 


4 


43 


id 


9.49692 






9. 5 1 946 




o.48o54 


9 


97 746 




42 


1 9 


9.49 7 3 o 




38 


9 .5i 9! 


48 


4 2 


O.48 012 


9 


97 742 




4i 


20 


9.49 768 


38 


9.52 o3i 


43 


0.47 969 


9 


977 38 




40 


21 


9.49806 




38 

-0 


9.52 073 


42 


0.47 927 


9 


977 34 


4 


39 


22 


9.49844 




3 


9.52 1 15 




0.47885 


9 


97 729 




38 


23 


9.49882 




38 


9.52 157 


4 2 


0.47843 


9 


97725 


4 


37 


24 


9.49920 




38 

_0 


9.52 200 


43 
42 


O.47 8OO 


9 


97 721 


4 


36 


25 


9.49958 




3 


9.52 242 




0.47758 


9 


97 7*7 




35 


26 


9.49996 




38 


9.52 284 


42 


0.47 716 


9 


97 7 i3 


4 


34 


2 7 


9.5o o3^ 




38 


9.52 326 


42 

42 


0.47 674 


9 


97708 


5 


33 


28 


9.5o 072 




3 


9.52 368 




0.47 632 


9 


97 74 




32 


2 9 


9.5o 1 10 




38 


9 .524 


10 


4 2 


0.47 590 


9 


97 700 


4 


3i 


30 


9.5o i48 


38 


9 .5 2 452 




0.47548 


9 


97696 


4 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 






71 3D. 






PP 


43 


42 




39 


38 




5 


4 


.1 


4-3 


4.2 


. T 


3-9 


3-8 


.1 


o-5 


0.4 


.2 


8.6 


8. 4 


.2 


7.8 


7.6 


.2 


I.O 


o.tf 


3 


12.9 


12.6 


3 


11.7 


11.4 


3 


i-5 


1.2 


4 


17.2 


16.8 


4 


15-6 


15-2 


4 


2.0 


1.6 


5 


21.5 


21.0 


5 


IQ- 5 


19.0 


. 5 


2-5 


2.0 


.6 


25-8 


25.2 


.6 


23-4 


22.8 


.6 




2-4 


7 


30-1 


29.4 


7 


27-3 


26.6 


7 


3-5 


2.8 


.8 


34-4 


33-6 


.8 


31.2 


30.4 


.8 


4.0 


3-2 


9 


38.7 


37-8 




35- l 34-2 


9 


4-5 3- 6 



66 



18 3O 





L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9. 5o i48 




9.52 452 




o. 


47548 


9.97696 






30 


3i 


9.5o i85 


38 


9.52 


494 


42 


o. 


47506 


9-97 


6 9 I 




29 


32 


9. 5o 223 


,g 


9.52 


536 




0. 


4 7 464 


9.97 687 




28 


33 


9-5o 261 




9.52 


5 7 8 




o. 


47422 


9-97 


683 


4 


27 








37 






42 










- 






34 


95o 298 


38 


9.52 


620 


41 


o. 


4738o 


9-97 


679 






26 


35 


9 .5o336 


18 


9.52 


661 




o. 


4 7 33 9 


9.97674 






25 


36 


9-5o 374 




9.52 


7o3 




o. 


47297 


9-97 


670 


4 


24 








37 






42 
















37 


9.5o4n 


3 8 


9.52 


745 


42 


0. 


47255 


9-97 


666 




23 


38 


9-5o 44 


9 




9.52 


787 




0. 




9-97 


662 




22 


3 9 


9-5o486 


37 


9.52 


829 


42 


o. 


47 171 


9-97 


65 7 


b 


21 


40 


9.5o 523 


37 


9 .5 2 


870 




0. 


47 i3o 


9-97 


653 




20 


4i 


9.5o 56i 


38 


9.52 


912 




0.47 088 


9-97 


649 




19 


42 


9.5o 598 




9.52 


9 53 




o. 


47 047 


9-97 


645 




18 


43 


9 .5o635 


37 


9.52 995 


42 


0.47 005 


9-97 


64o 


5 


'7 


44 


9.5o 673 


38 


9.53 037 


42 


o.46 9 63 


9.97636 


4 


16 


45 


9-5o 710 




9.53078 




o 46 922 


9-97 


63 2 




i5 


46 


9-5o 747 


37 


9 .53 


I2O 


42 


o. 


4688o 


9-97 


628 




i4 


47 


9.5o 784 


37 


9 .53 


161 


41 


o. 


46839 


9-97 


623 


5 


i3 


48 


95o 821 


37 


9 .53 


2O2 




o. 


46798 


9.97619 


1 


12 


49 


9-5o858 


37 


9 .53 


244 


42 


o. 


46756 


9-97 


6,5 


4 


II 


50 


9.50896 


38 


9.53 


285 


4 1 


o.46 715 


9-97 


610 




10 


5i 


9.5o 9 33 


37 


9 .53 


32 7 


4 2 


o. 


46673 


9-97 


606 


1 


9 


52 


9 .5o 970 


37 


9 .53 


368 




o. 


46632 


9-97 


602 




8 


53 


9-5i 007 


37 


9.53 409 


4 1 


0. 


465gi 


9-97 


597 


5 


7 


54 


9. 5 1 o43 


36 


9.5345o 


4 1 


0. 


46550 


9-97 


5 9 3 


4 


6 


55 


9. 5 1 080 


37 


9.53 


492 




o. 


465o8 


9-97 


589 




5 


56 


9 .5i 117 


37 


9 .53 


533 


4 1 


o. 


46 46 7 


9-97 


584 







4 


57 


9 .5i i54 


37 


9.53 


5 7 4 


4 1 


o. 


46426 


9-97 


58o 


4 


3 


58 


9-5i 191 


37 


9 .53 


6i5 




o. 


46385 


9-97 


5 7 6 




2 


5 9 


9-5i 227 


36 


9 .53 


656 


4i 


0. 


46 344 


9-97 


671 




I 


60 


9. 5 1 264 


37 


9 .53 


697 


4i 


o. 


46 3o3 


9-97 


56 7 









L. Cos. 


d. 


L. Cotg. 


d. j L. 


Tang. 


L. Sin. 


d. 








71. 








PP 42 


4i 


38 


37 


36 




5 


4 


.1 4.2 


4.1 


3-8 


i 3-7 


3-6 


.! 


0.5 


04 


8.4 


8. a 


7.6 


.2 7.4 


7-2 


.2 


l.O 


0.8 


3 I2 -6 


12.3 


11.4 


3 "- 1 


10.8 


3 


i-5 


1.2 


.4 16.8 


16.4 


15-2 


.4 14.8 


14.4 


4 


2.O 


1.6 


5 21.0 


20.5 


19.0 


5 18.5 


18.0 




2-5 


2.0 


.6 25.2 


24.6 


22.8 


.6 22.2 


21.6 


.6 


3- 


2.4 


7 29.4 


28.7 


26.6 


7 25.9 


25.2 


7 


3-5 


2.8 


.8 33-6 


32.8 


30.4 


.8 29.6 


28.8 


.8 


4.0 


3- 2 




36.9 34.2 




3 2 -4 




4-5 





67 



19. 



, 


L. Sin. 


d. 


L. Tang. 


d. 


L 


. Cotg. 


L. Cos. d. 







9 .5i 264 




9.53 697 







46 3o3 


9-97 


56 7 




60 


I 


9 .5i 3oi 


37 


9 .53 7 38 




o 


46 262 


9-97 


563 


5 


5 9 


2 


9 .5i 338 


,6 


9.53779 




O.46 221 


9-97 


558 




58 


3 


9 .5i 3 7 4 


37 


9 . 53 820 


4 1 





46 180 


9-97 


554 


4 
4 


57 


4 


9.5i 4i 


I 


36 


9.53 861 




o 


46 i3 9 


9-97 


550 




56 


5 


9. 5 1 44 7 




9.53 902 




o 


46 098 


9-97 


545 




55 


6 


9 .5i 484 


36 


9 .53 9 43 


4 1 
41 


o.46 057 


9-97 


54 1 


4 

5 


54 


7 


9 .5i 52O 


37 


9 .53 9 84 




o. 


46 016 


9-97 


536 




53 


8 


9 .5i 55 7 




9 .54 025 




o 


45 97 5 


9-97 


532 




52 


9 


9 .5i 5 9 3 


Jj 


9 .54 


o65 


40 


0. 


45 935 


9-97 


528 


4 


5i 


10 


9 .5i 629 




9.54 1 06 




0. 


458 9 4 


9-97 


523 




50 


ii 


9.5i 666 


*6 


9.54 147 


4 1 


o. 


45853 


9-97 


5i 9 


4 


49 


12 


9 .5i 702 




9-54 


187 




o. 


458i3 


9-97 


515 




48 


i3 


9.5i 738 


30 
36 


9-54 


228 


41 


0. 


45 772 


9-97 


Sio 


b 
4 


47 


i4 


9 .5i 774 


37 


9-54 


269 




0. 


45 7 3i 


9-97 


5o6 




46 


i5 


9 .5i 81 


I 




9-54 


309 




0. 


45 691 


9-97 


5oi 




45 


16 


9.5i 847 


3 6 


9-54 


350 


41 


0. 


45 65o 


9-97 


497 


4 


44 


'7 


9.5i 883 


,6 


9-54 


390 


40 


o. 


456io 


9-97 


492 


b 


43 


18 


9.5i 91 


q 




9-54 


43i 




0. 


45 569 


9-97 


488 




42 


J 9 


9 .5i 9 5 


5 


36 


9-54 


471 


4 


0. 


45 529 


9-97 


484 


4 


4i 


20 


9 .5i 99 i 


3 


9-54 


5l2 


4 1 


0. 


45488 


9-97479 




40 


21 


9 .52 027 


3 6 


9-54 


552 


4 


o. 


45448 


9-97475 


4 


3 9 


22 


9.52 o63 




9-54 


5 9 3 




0. 


45407 


9-97 


4 7 o 




38 


23 


9.52 099 


3 


9.54633 


40 


0. 


45 367 


9-97 


466 


4 


37 








3 6 






4 










5 




24 


9.52 i3 


5 


16 


9-54 


6 7 3 




0. 


45 327 


9-97 


46i 




36 


25 


9.52 171 




9-54 


7 i4 




0. 


45 286 


9-97 


45 7 




35 


26 


9.52 207 


3 6 


9-54 




4 


o. 


45 246 


9-97 


453 


4 


34 


27 


9.52 242 


35 
16 


9-54 


794 


40 


0. 


45 206 


9-97 


448 


5 


33 


28 


9.52 278 




9-54 


83,5 




0. 


45 i65 


9-97 


444 




32 


2 9 


9.52 3i4 


3 6 


9 .54 


875 


40 


o. 


45 125 


9-97 


43 9 


b 


3i 


30 


9.52 350 


3 6 


9 .54 


915 


4 


0. 


45 o85 


9-97 


435 




30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. 


d. 








7O 3D . 




PF 


> 41 


40 


37 


36 


35 




5 4 


.1 


4-1 


4.0 


3-7 


3-6 


3-5 


.1 


0.5 0.4 


.2 


8.2 


8.0 


7-4 


.2 7 .2 


7.0 


.2 


i.o 0.8 


3 


12.3 


12.0 


n. i 


.3 10.8 


10.5 


3 


1-5 !- 2 


4 


16.4 


16.0 


14.8 


4 14-4 


14.0 


4 


2.0 1.6 


5 


20.5 


20.0 


18.5 


.5 18.0 


17-5 


5 


2.5 2.0 


.6 


24.6 


24.0 


22.2 


.6 21.6 


21. 


- .6 


3.0 2.4 


7 


28.7 


28.0 


25.9 


7 25.2 


24-5 


7 


3-5 2.8 


.8 


32.8 


32.0 


20.6 


.8 28.8 


28.0 


.8 


4-0 3-2 


9 36-9 


36.0 33.3 


9 3 2 -4 


31-5 


9 


4-5 3- 6 



68 



19 3D . 



t 


L. Sin. 


d. 


L. Tang. 


d. 


L 


, Cotg. 


L. Cos. 


d. 




30 


9.52 350 


35 
36 
35 
36 
35 
36 
35 
36 
35 
36 

35 
35 
36 
35 
35 
35 
35 
35 
35 


9-54 


9*5 


40 
4 o 

40 
40 
4 o 
40 
40 
40 
4 o 
40 

40 
40 

39 
40 
40 
40 

39 
40 
40 
39 
40 

39 
40 

39 
40 

39 
40 

39 
39 
40 


o. 


45o85 


9.97435 


5 
4 
5 
4 
5 
4 
5 
4 
5 


30 


3i 

32 

33 

34 
35 
36 

37 

38 

39 


9.52 385 
9.52 421 
9.52456 

9.52 492 
9.52 527 
9.52 563 

9.52 598 
9.52634 
9.52 669 


9.54905 
9.54 995 
9.55 o35 

9-55 075 
9.55 n5 
9.55 i55 

9.55 i 9 5 
9. 55 2 35 
9.55 275 


0. 

o. 
o. 

0. 

o. 
o. 

0. 

o. 
o. 


45 045 
45 005 
44 965 

44925 
44885 
44845 

44805 
44765 
44 725 


9.97430 
9.97 426 
9.97421 

9.97417 
9.97 412 
9.97 4o8 

9.97 4o3 
9.97 399 
9.97 3 9 4 


29 
28 
27 

26 

25 
24 

23 
22 
21 


40 


9.52 705 


9 .55 


315 


o. 


44685 


9-97 


390 




20 


4i 

42 

43 

44 

45 
46 

47 
48 

49 


9.52 740 
9.52 775 
9.52 811 

9.52 846 
9.52 881 
9.52 916 

9.52 gSi 
9.52 986 

9.53 021 


9-55355 
9 .553 9 5 
9.55434 

9.55474 
9 .555i4 
9.55554 

9 .555 9 3 
9.55633 
9 .556 7 3 


0.44645 
o.44 6o5 
0.44566 

o.44 526 
0.44486 
0.44446 

o.44 407 
o.44 367 
o.44 327 


9-97 
9-97 
9-97 

9-97 
9-97 
9-97 

9-97 

9-97 
9-97 


385 
38i 
3 7 6 

3 72 
36 7 
363 

358 
353 
349 


4 
5 
4 
5 
4 
5 
5 
4 


9 
1 8 

17 

16 
i5 
i4 
i3 

12 
I I 


50 


9-53 o56 


35 
-e. 


9 .55 


712 


o. 


44288 


9-97 


344 




10 


Si 

52 

53 

54 
55 
56 

57 
58 
5 9 


9.53 092 
9.53 126 
9.53 161 

9-53 196 
9.53 23i 
9-53 266 

9.53 3oi 
9.53 336 
9 .53 3 7 o 


3 
34 
35 
35 
35 
35 
35 
35 
34 


9.55 752 
9-55 791 
9. 5583i 

9-55 870 
9. 55 910 
9-55 949 

9.55 989 
9.56 028 
9.56 067 


o.44 248 
o.44 209 
0.44 169 

o.44 i3o 
o.44 090 
o.44 o5 1 

0.44 on 
o.43 972 
o.43 933 


9-97 
9-97 
9-97 

9-97 
9-97 
9-97 

9-97 
9-97 
9-97 


34o 
335 
33i 

326 

322 

3i 7 

3l2 

3o8 
3o3 


5 
4 

5 
4 

5 
5 
4 

5 


9 

8 

7 
6 
5 
4 
3 

2 


60 


9.534o5 


35 


9.56 


107 


o.438 9 3 


9-97 


2 99 









L.Cos. d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. 


d. 


/ 




70. 






PP 40 




39 


36 


35 


34 


.1 

.2 

3 
4 

:I 

:l 

9 


5 4 


.1 4.0 

.2 8.0 

3 12.0 

.4 16.0 

5 20.0 

.6 24.0 

.7 28.0 
.8 32.0 
.g 36.0 


1:1 
11.7 

15-6 
19-5 
23-4 

27-3 
31.2 
35. i 


3-6 
7.2 
10.8 

14.4 
18.0 

21.6 

25.2 
28.8 

32.4 


i 3-5 
.2 7.0 
3 i-5 

.4 14.0 
5 17-5 

.6 21.0 

7 24.5 
.8 28.0 
9 3 T -5 


u 

10.2 

13.6 
I 7 .0 
20.4 

23-8 
27.2 
30.6 


0.5 0.4 
i.o 0.8 
1.5 i- 2 * 

2.0 1.6 
2-5 2.0 
3-0 2.4 

3-5 2.8 
4-0 3-2 

4-5 3-6 



69 



2O. 





L. Sin. 


d. 


L. Tang. d. 


L. 


Cotg. 


L. Cos. d. 







9.534o5 




9.56 


I0 7 




0. 


43 893 


9-97 


2 99 






60 


I 


9.53440 


35 


9.56 


i46 


39 


o. 


43854 


9.97 294 


5 


5 9 


2 


9-53475 




9.56 


i8b 




0. 


438i5 


9-97 


289 




58 


3 


9.53 509 


34 


9.56 


224 


39 


o. 


43 77 6 


9-97 


285 


4 




5 7 








35 






4 
















4 


9 .53544 




9.56 


264 


39 


o. 


43 7 36 


9-97 


280 




56 


5 


9.53578 




9.56 


3o3 




o. 


436 97 


9-97 


276 




55 


6 


9 .536i3 


35 


9.56 


342 


39 


o. 


43658 


9-97 


271 


b 


54 


7 


9.53 647 


34 


9.56 


38i 


39 
39 


0.43 619 


9-97 


266 


5 


53 


8 


9.53682 




9.56 


420 




o. 


4358o 


9-97 


262 




52 


9 


9.53 716 


34 


9.56 


459 


39 


o. 


4354i 


9-97 


25 ? 


5 


5i 


10 


9 .53 7 5i 


35 


9.56 


498 


39 


0. 


43 5o2 


9-97 


252 




50 


1 1 


9.53 7 85 




9 .56 


53 7 


39 


o. 


43463 


9-97 


248 


4 


49 


12 


9.53819 




9.56 


5 7 6 




o. 


43424 


9-97 


243 


' 




48 


i3 


9.53854 


35 


9.56 


6i5 




o. 


43385 


9-97 


238. 


b 


47 








34 






39 
















i4 


9.53888 


34 


9.56 


654 


39 


o. 


43 346 


9-97 


234 




46 


i5 


9-53 922 




9.56 


693 




o. 


433o 7 


9-97 


229 


' 




45 


16 


9 .53 9 5 7 


35 


9.56 


7 32 




0. 


43268 


9-97 


224 


- 




44 








34 






39 
















7 


9.53991 


34 


9.56771 


39 


o. 


43 229 


9-97 


220 




43 


18 


9.54 025 




9.56 


810 




o. 


43 190 


9-97 


2l5 






42 


1 9 


9.54 059 


34 


9.56 


849 


39 
,a 


o. 


43 i5i 


9-97 


210 


- 




4i 


20 


9.54093 


34 


9.56 


887 




o . 


43 n3 


9-97 


2O6 


4 


40 


21 


9.54 127 


34 


9.56 


926 


39 


o. 


43o 7 4 


9-97 


2OI 


5 


3 9 


22 


9.54 161 




9.56 


965 




o. 


43o35 


9-97 


196 




38 


23 


9.54 195 


34 
34 


9 .5 7 


oo4 


18 


0. 


42996 


9-97 


192 


4 
5 


37 


24 


9.54 229 




9 .5 7 


042 


39 


o. 


42 9 58 


9-97 


l8 7 




36 


25 


9.54263 




9 .5 7 


08 1 




o. 


42 919 


9-97 


182 






35 


26 


9.54297 


34 


9 .5 7 


I2O 


39 


o. 


42 880 


9-97 


1 7 8 


4 


34 


27 


9 .5433i 


34 


9 .5 7 


i58 


3 
39 


o. 


42 842 


9-97 


I 7 3 




) 


33 


38 


9 .54365 




9 .5 7 


197 


_Q 


o. 


42803 


9-97 


1 68 






32 


29 


9.54399 


34 


9 .5 7 


235 


3 


o. 


42765 


9-97 


i63 


b 


3i 


30 


9 .54433 


34 


9 .5 7 


274 




o. 


42 726 


9-97 


i5 9 






30 




L. Cos. 


d. 


L. Cotg. d. 


L. Tang. 


L.Sin. 


d. 


' 






69 3O . 




PP 40 


39 


38 


35 


34 




5 


4 


I 4.0 


3-9 


3-8 


i 3-5 


3-4 


., 


0.5 


0.4 


2 8.0 


7.8 


7.6 


.2 7.0 


6.8 


.2 


I.O 


0.8 


3 12.0 


11.7 


11.4 


3 JQ-S 


IO. 2 


3 


i-5 


1.2 


4 16.0 


15-6 


1-5-2 


.4 14.0 


I 3 .6 


4 


2.0 


1.6 


5 20. o 


19-5 


19.0 


5 17-5 


I 7 .0 


5 


2-5 


2.0 


6 24.0 


23-4 


22.8 


.6 21. 


20-4 


.6 


3-o 


2-4 


7 28.0 


27-3 


26.6 


7 24.5 


238 


7 


3-5 


2.8 


8 32.0 


31.2 


3-4 


.8 28.0 


27.2 


.8 


4.0 


3-2 


Q 36-0 




34.2 


9 V-S 






4-5 


3-6 



70 



2O 3O 



' 


L. Sin. d. 


L. Tang. d. 


L. 


Cotg. 


L. Cos. 


d. 




30 


9.54433 




9 .5 7 


274 38 


0. 


42 726 


9-97 


i5 9 




30 


3i 


9.54466 


34 


9 .5 7 


3i2 M 


o. 


42688 


9-97 


1 54 


5 


29 


32 


9.54 5oo 




9 .5 7 


35i 




o. 


42 64 9 


9-97 


i4g 




28 


33 


9.54534 


34 


9.5 7 


389 


3 8 


0.42 611 


8. 97 


145 


4 


27 


34 


9 .5456 7 


33 

34 


9.57428 


39 
38 


o. 


42 572 


9-97 


i4o 


5 


26 


35 


9.54601 




9.57466 




0. 


42 534 


9-97 


i35 




25 


36 


9.54635 


34 


9 .5 7 


5o4 


38 


o. 


424 9 6 


9-97 


i3o 


b 


24 


3? 


9-54668 


33 


9 .5 7 543 


39 
-,% 


0.42457 


9-97 


126 


4 


23 


38 


9. 54 702 




9.57 


58i 




o. 


42 4i 9 


9-97 


121 




22 


3 9 


9 .54 7 35 


33 


9.5 7 


619 


38 


0. 


42 38 1 


9-97 


116 


s 


21 


40 


9-54 769 


34 


9 .5 7 


658 


08 


0.42 342 


9-97 


in 




20 


4i 


9.54802 


33 


9 .5 7 


696 


3 

og 


o. 


42 3o4 


9-97 


107 


4 


i9 


42 


9-54836 




9 .5 7 


7 34 




0. 


42266 


9-97 


102 




1 8- 


43 


9.54869 


33 


9 .5 7 


772 


3 8 


0. 


42 228 


9-97 


97 


b 


17 


44 

45 


9.54903 
9.54936 


34 
33 


9.57 810 
9-5 7 849 


38 
39 


0. 

o. 


42 I 9 

42 i5i 


9-9709 2 
9-97 8 7 


5 

5 


16 
i5 


46 


9.54969 


33 


9 .5 7 


887 


38 


o. 


42 n3 


9 . 9 7o83 


4 


i4 


4? 


9-55 oo3 


34 


9.57925 


38 
18 


o. 


42 075 


9-97 


7 8 


5 


i3 


48 


9-55o36 




9.5 79 63 




o. 


42o3 7 


9-97 


o 7 3 




12 


4 9 


9.55 069 


33 


9-58 ooi 


38 


o. 


4i 999 


9-97 


068 


b 


I I 


50 


9-55 102 


33 


9.58 039 


3 8 


o. 


4i 9 6i 


9-97 


o63 


5 


10 


5i 


9.55 i36 


34 


9.58077 


38 
18 


0. 


4i 9 23 


9-97 


o5 9 




9 


52 


9.55 169 




9.58 


"J 




o.4i 885 


9-97 


o54 




8 


53 


9-55 202 


33 


9.58 


i53 


38 


0. 


41847 


9 . 97 o4 9 


S 


7 


54 


9-55235 


33 


9. 58 


191 


38 
18 


o. 


4 1 809 


9-97 


o44 


5 


6 


55 


9.55 268 




9.58 


229 




0. 


4i 77 1 


9-97 


o3 9 




5 


56 


9.55 3oi 


33 


9-58 


267 


38 


o. 


4i 7 33 


9-97035 


4 


4 


5? 


9.55334 


33 


9-58 


3o4 


37 


0. 


4 1 6 9 6 


9-97 


o3o 


5 


3 


58 


9.55 367 


33 


9.58 


342 


3 


o. 


4i 658 


9-97 


2 5 




2 


5 9 


9.55 4oo 


33 


9.58 


38o 


38 


0. 


4i 620 


9-97 


020 


5 


I 


60 


9.55433 


33 


9.584i8 


S 8 


o. 


4i 582 


9-97 


oi5 









L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L.Sin. 


d. 


' 






69. 




PP 39 


38 


37 


34 


33 




5 4 


i 3-9 


3-8 


3-7 


.! 3 . 4 


3-3 


.1 


0.5 0.4 . 


7-8 


7.6 


7-4 


.2 6.8 


6.6 


.2 


i.o 0.8 


3 "-7 


11.4 


u. i 


3 IO. 2 


9-9 


3 


1-5 1-2 


4 5-6 


15-2 


14.8 


4 13-6 


13.2 


4 


2.0 1.6 


5 19-5 


10. 


18.5 


5 i7- 


16.5 


5 


2.5 2.O 


.6 23.4 


22.8 


22.2 


.6 20.4 


19.8 


6 


3-0 2.4 


7 27-3 


26.6 


25-9 


7 23-8 


23.1 


7 


3-5 2.8 


.8 31.2 


3-4 


29. 6 


.8 27.2 


26.4 


.8 


4.0 3.2 


9 35- * 


34-2 33-3 


9 30-6 


29.7 


9 


4-5 3-6 



7 1 



21. 



1 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 







9 .55433 


33 
33 
33 
32 
33 
33 
33 
32 
33 


9.584i8 


37 
38 
38 
38 
37 
38 
37 
38 
38 
37 
38 
37 
38 
37 
37 
38 
37 
38 
37 
37 

37 

38 
37 
37 
37 
37 
38 
37 
37 
37 


o.4i 582 


9.97 oi5 


5 
5 
4 
5 
5 
5 
5 
5 
5 
5 

4 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
4 
5 
5 
5 
5 
5 
5 
5 


60 


I 

2 

3 

4 
5 
6 

7 

8 

9 


9.55466 
9.55499 
9.55532 

9. 55 564 
9.555 9 7 
9.55 63o 

9 .55663 
9 .556 9 5 
9.55 728 


9.58 455 
9.58493 
9. 5853i 

9.58 569 
9.58 606 
9.58 644 

9.58 681 
9 58 719 
9 .58 7 5 7 


o.4i 545 
o.4i 507 
o.4i 469 

o.4i 43i 
o.4i 394 
o.4r 356 

o.4i 319 
o.4i 281 

o.4i 243 


9.97 oio 
9.97 oo5 
9.97 ooi 

9.96996 
9.96991 
9.96 986 

9.96 981 
9.96 976 
9.96971 


5 9 

58 

56 
55 
54 
53 

52 

5i 


10 


9-55 761 




9-58 79 4 


0.4 1 206 


9.96 966 


50 


ii 

12 

i3 

i4 

i5 

16 

17 

18 

'9 


9-55 793 
9.55826 
9.55858 

9-55 891 
9.55 923 
9.55956 

9.55988 
9.56 02 1 
9 .56o53 


33 
32 
33 
32 
33 
32 
33 
32 
32 

33 
32 
32 
33 

3 2 
32 
32 
32 
32 
33 


9.58832 
9.58869 
9-58 907 

9.58944 
9.58 981 
9.59 019 

9.59 o56 
9.59 094 
9.59 i3i 


o.4i 168 
o.4i i3i 
o.4i 093 

o.4i o56 
o.4i 019 
o.4o 981 

o.4o 944 
o.4o 906 
o.4o 869 


9.96 962 
9.96957 
9.96 952 

9.96947 
9.96 942 
9.96937 

9.96 932 
9.96927 
9 . 96 922 


49 

48 

47 

46 
45 
44 

43 

42 

4i 


20 


9-56o85 


9.59 168 


o.4o 832 


9.96917 


40 


21 
22 
23 

24 
25 
26 

27 
28 
2 9 


9,56 118 
9.56 iScf 
9-56 182 

9.56 215 
9.56 247 
9.56 279 

9. 563u 
9. 56 343 
9.56 375 


9.59 2o5 
9.59 243 
9.59 280 

9-5 9 317 
9 .5 9 354 
9.5 9 391 

9.59 429 
9.59 466 
9.59 5o3 


o.4o 795 
o.4o 757 
o.4o 720 

o.4o683 
o.4o646 
o.4o 609 

o.4o 571 
o.4o534 
o.4o 497 


9.96 912 
9.96907 
9.96 903 

9.96 898 
9.96 893 
9.96 888 

9.96 883 
9.96 878 
9.96 873 


3 9 
38 
37 

36 
35 
34 
33 

32 

3i 


30 


9.56 4o8 


9.59^540 


o.4o 46o 


9.96 868 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 








68 30'. 




PP 

.1 

.2 

3 

4 

7 
.8 




38 


37 




33 


32 




5 


4 


11.4 
15-2 

19 o 

22.8 

26.6 
3-4 

34-2 


3-7 
7-4 
n. i 

14.8 
18.5 

22.2 

25-9 
29.6 


. i 

.2 

3 

4 
5 
.6 

' 


U 

9.9 

13.2 
16.5 
19.8 

23.1 
26.4 


3-2 

6.4 
9.6 

12.8 

16.0 
19.2 

22.4 

25.6 


.1 0.5 

.2 1.0 

3 i-5 

.4 2.0 

5 2.5 

7 3-5 
.8 4.0 

9 4-S 


0.4 
0.8 

1.2 

1.6 

2.0 
2.4 

2.8 



72 



21 3O 





L. Sin. 


d. 


L. Tang. d. 


L. Cotg. 


L. Cos. 


d. 




30 


9.564o8 


32 
32 

32 
32 
32 

3 1 
32 
32 
32 


9. 59 54o 


37 
37 
37 
37 
37 
37 
37 
36 
37 
37 
37 
37 
36 
37 
37 
37 
36 
37 
37 
36 

37 
36 

37 
36 
37 
36 
37 
36 
37 
36 


o.4o 46o 


9 


96 868 


5 
5 

5 
5 
5 
5 
5 
5 
5 
5 

5 

5 
5 
5 
5 
5 
5 
5 
6 

5 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


30 


3i 

32 

33 

34 
35 
36 

38 
3 9 


9-56 44o 
9.56 472 
9.565o4 

9.56536 
9.56 568 
9. 56 599 

9. 5663i 
9.56663 
9-56 695 


9.59 5 77 
9.59 6i4 
9 .5 9 65i 

9.59 688 

9- 5 9 7 2 5 
9.59 762 

9.59799 
9.59835 
9.59 872 


0.40 423 
o.4o 386 
o.4o 34 9 

o.4o 3i2 
o.4o 275 
o.4o238 

O.4o 2OI 

o.4o 165 
o.4o 128 


9.96 863 
9.96 858 
9.96853 

9.96848 
9.96 843 
9.96 838 

9.96833 
9.96 828 
9.96 823 


29 

28 
27 

26 

25 

24 

23 
22 
21 


40 


9.56 727 




9.59909 


o.4o 091 


9 


96 818 


20 


4i 

42 

43 

44 
45 
46 

47 
48 

4 9 


9.56 759 
9-56 790 
9.56822 

9.56854 
9.56886 
9.56917 

9-j>6 949 
9.56 980 

9.57 012 


S 2 
32 
32 

32 
32 


9.5 9 946 
9.59 983 
9 .60 019 

9-6oo56 

9.60 i3o 

9.60 166 
9.60 2o3 
9.60 240 


o.4o o54 
o.4o 017 
0.39 981 

0.39 9 44 
0.39 907 
0.39 870 

0.39834 
o. 3 9797 
0.39 760 


9.96 8i3 
9.96 808 
9.96 8o3 

9.96 798 
9.96793 
9.96 788 

9.96 783 
9.96 778 
9.96772 


'9 

18 

17 
16 
i5 
i4 
i3 

12 
I I 


50 


9.57 o44 




9.60 276 


0.39 724 


9- 


96 767 


10 


5i 

52 

53 

54 
55 
56 

57 

58 

59 


9.57075 
9.57 107 
9.57 i38 

9.57 169 

9.57 2OI 
9.57 232 

9,57 264 
9 .5 7 326 


32 

32 
3' 
32 

3' 


9.60 3 1 3 
9.60 349 
9.60 386 

9.60 422 
9.60 459 
9.60 495 

9.60 532 
9.60 568 
9.60 605 


0.39 687 
0.39 65i 
0.39 6i4 

0.39 578 
0.39 54 1 
0.39 505 

0.39468 
0.39 432 

0.39 395 


000 OOO 000 


96 762 
96 757 
96 752 

96747 
96 742 
9 6 7 3 7 

96 732 

96 727 
96 722 


9 

8 

7 
6 
5 
4 
3 

2 
I 


60 


9 .5 7 358 




9 .60 64 1 


0.39 359 


9- 


96 717 







L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L.Sin. d. 








68. 




PP 

.1 

.2 

3 
4 

9 


37 


36 




32 


31 


2 

3 
4 

I 

9 


6 


5 


3-7 
7-4 
n. i 

14.8 
18.5 

22.2 

25-9 
29.6 

33-3 


3-6 
,o'.8 

21 6 

25 2 
288 
32.4 


.1 

.2 

3 
4 

: 7 8 


12.8 

16.0 
19.2 

22.4 
25.6 


K 

9-3 

12.4 
15-5 
18.6 

21.7 
24.8 

27.9 


06 

I 2 

i 8 
24 

t! 

5-4 


0.5 

I.O 

2.0 
2-5 

3-5 
4.0 

4-5 



/ 


L. Sin. 


d. 


L. 


Tang. 


d. 


L. Cotg. 


L. Cos. d. 







9 .5 7 


358 




9. 60 64 1 




o. 3g 


359 


9.96 




60 


I 


9 .5 7 38 9 


31 


9 .6o 677 


37 


0.39 


323 


9.96 


711 




5 9 


2 


9 .5 7 420 




9.60 714 


,6 


0.39 


286 


9.96 


706 




58 


3 


9 .5 7 45i 


31 


9.60 750 


36 


o.Sg 


25o 


9-9 6 


7 oi 


5 

S 


57 


4 


9 .5 7 482 


32 


9.60 786 


37 


0.39 


214 


9 . 9 6 


696 




56 


5 


9- 5 7 


5i4 




9- 


60823 


,5 


o. 39 


177 


9.96 


691 




55 


6 


9 5 7 


545 


3 1 
3 1 


9.60 85g 


36 


o. 39 


i4i 


9.96686 


5 
5 


54 


7 


9.57 


5 7 6 




9- 


60 895 


36 


o. 3g 


105 


9.96 


681 




53 


8 


9 . 57 607 




9- 


60931 


16 


0.39 


069 


9.96 


676 




52 


9 


9 .5 7 638 


3 1 


9- 


60 967 


3 


0.39 


o33 


9.96 


670 




5i 


10 


9 .5 7 669 


3 r 


9- 


6 1 oo4 


36 


o.38 


996 


9.96 


665 




50 


1 1 


9- 5 7 


7 oo 


31 


9- 


6 1 o4o 


36 


o.38 


960 


9.96 


660 




49 


12 


9- 5 7 


7 3i 




9- 


61 076 




o.38 


924 


9.96 


655 




48 


i3 


9.57 


762 


3 1 


9- 


61112 


36 


o,38 


888 


9.96 


650 


5 
5 


47 


i4 


9 .5 7 


79 3 


31 


9- 


61 1 48 


36 


o.38 


852 


9.96 


645 




46 


i5 


9.57 


824 




9- 


61 1 84 


,6 


o.38 


816 


9.96 


64o 




45 


16 


9 5 7 


855 


3 1 

3 


9- 


6l 220 


36 


o.38 


780 


9.96 


634 


S 


44 


17 


9 . 57 885 


31 


9- 


6r 256 


36 


o.38 


744 


9.96 


629 




43 


18 


9 .5 7 


916 




9- 


61 292 


16 


o.38 


708 


9.96 


624 




42 


r 9 


9 .5 79 47 




9- 


61 328 


06 


o.38 


672 


9.96 


619 




4i 


20 


9 .5 7 


978 




9- 


61 364 


16 


0.38 


636 


9.96 


6i4 




40 


21 


9 .58 008 


3 


9- 


6 1 4oo 


36 


o.38 


600 


9.96 608 




3 9 


22 


9 .58 


o3 9 




9- 


61 436 




o.38 


564 


9.96 


6o3 




38 


23 


9 .58 


070 


31 


9.61 472 


3 
36 


o.38 


5 2 8 


9.96 


5 9 8 


b 
5 


37 


24 


9 .58 


IOI 




9.61 5o8 


16 


o.38 


492 


9.96 


5 9 3 




36 


25 


9 .58 


i3i 




9.61 544 




o.38 


456 


9.96 


588 




35 


26 


9 .58 


162 


3 1 


9,61 579 


35 


o.38 


421 


9.96 


582 




34 


2 7 


9 .58 


192 


30 


9.61 6i5 


36 


o.38 


385 


9.96 


5 77 


5 


33 


28 


9 .58 


223 




9.61 65i 




o.38 


34 9 


9.96 


5 7 2 




32 


2 9 


9 .58 


253 


30 


9.61 687 


36 


o.38 


3i3 


9,96 


56 7 


b 


3r 


30 


9 .58 284 


3 1 


9.61 722 


35 


o.38 


278 


9.96 


562 


5 


30 




L. Cos. 


d. 


L. 


Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 










67 3O . 




PP 


37 


36 35 




32 


31 


3 




6 5 


.1 


3-7 


3-6 3 5 


, 


3-2 


I' 1 


3 


I 


o. 6 o. 5 




7-4 


7-2 70 


.2 


6.4 


6.2 


6.0 


2 


12 1.0 


3 


u. i 


10 8 10,5 


3 


9.6 


9-3 


9.0 


3 


1.8 1.5 


4 


14.8 


14-4 14-0 


4 


12.8 


12.4 


12. 


4 


2 4 2.0 


:i 


18.5 

22.2 


18.0 17.5 

21.6 21.0 


5 
6 


16.0 
19.2 


III 


15.0 

18.0 


.1 


3.0 2.5 

36 3-0 


7 


25-9 


25.2 24 5 


, 7 


22 4 


21 7 


21. O 


7 


42 3-5 


.8 


29.6 


28.8 280 


,8 


25-6 


248 


2 4 .0 


.8 


4.8 4.0 


9 33 3 


32.4 31 5 


9 


28.8 27.9 




9 


5-4 4-5 



74 



22 3O 



, 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9- 


58 284 






9.61 722 




o.38 278 


9.96 562 


6 


30 


3i 


9 .583i4 


3 


9.61 758 


3 
76 


o.38 242 


9. 96 556 


s 


2 9 


32 


9 


58 345 






9.61 ' 


r 9 4 




o.38 206 


9.96 55i 




28 


33 


9 


58 375 




3 


9.61 83o 


3 6 


o.38 170 


9.96 546 




2 7 


34 


9 


58 4o6 




3' 


9.61 865 


35 
_< 


o.38 135 


9.96 54 1 


5 
6 


26 


35 


9 


58436 






9 61 901 


3 




o. 38 099 


9 . 9 6535 




25 


36 


9 


58467 




3' 


9.61 936 


35 


o.38 o64 


9>9 6 53o 




24 


37 


9 


584 9 7 




30 


9,61 972 


36 

3* 


o.38 028 


9 . 9 6525 




23 


38 


9 


58 527 






9.62 008 






0.37992 


9 . 9 6 52O 


6 


22 


3 9 


9 


5855 7 




3 


9.62 o43 


35 


0.37 957 


9 . 9 6 5i4 


5 


21 


40 


9 


58588 




3 1 


9.62 079 


3 


0.37 921 


9 . 9 6 5o 9 


5 


20 


4i 


9 


.586i8 




3 


9.62 u4 


35 
06 


o.3 7 886 


9 . 9 6 5o4 


6 


19 


42 


9 


58648 






9 ,62 


50 




0.37 85o 


9 . 9 6 4 9 8 




18 


43 


9 


.58 678 




3 


9 .62 i85 


35 


0.37815 


9 . 9 6 4 9 3 




17 


44 


9 


.58 70 9 


3' 


9 .62 221 


36 


0.37 779 


9.96488 


5 

5 


16 


45 


9 


.58 7 3 9 


30 


9 .62 256 




0.37 744 


9.96483 


6 


i5 


46 


9 


.58 76 9 


30 


9.62 292 


36 


0.37 708 


9.96477 




i4 


47 


9 


.58 799 


30 


9.62 327 


35 


o.3 7 6 7 3 


9.96472 




i3 


48 


9 


.58 829 




9.62 362 




o.3 7 638 


9.96 467 


fi 


12 


49 


9 


.5885 9 


3 


9.62 398 


3 6 


0.37 602 


9.96 46 1 


5 


I I 


50 


9 


.58 88 9 


3 


9.62 433 




o.3 7 56 7 


9.96 456 




10 


5i 


9 


. 58 9 i 9 


3 


9.62 468 


35 
16 


0.37 532 


9.96 


45i 


6 


9 


52 


9 


.58 9 4 9 


3 


9. 62 5o4 




0.37 4 9 6 


9.96 445 




8 


53 


9 


.58 979 


30 


9.62 


53 9 


35 


0.37461 


9.96 44o 




7 


54 


g 


. 5 9 oo 9 


30 


9.62 


5 7 4 


35 


0.37426 


9.96 


435 


6 


6 


55 


g 


.5 9 o3^ 


I 


3 


9.62 609 




0.37 391 


9 . 9 6 


42 9 




5 


56 


9 


.5 9 o6 9 


3 


9.62 


645 


36 


o.3 7 355 


9-9 6 


424 




4 


57 

58 


9 
9 


.5 9 o 9 8 
.5o 128 


29 
30 


9.62 
9.62 


680 


35 
35 


0.37 320 

o.3 7 285 


9 . 9 6 4i 9 
9 . 9 6 4i3 


6. 


3 

2 


5 9 


9 .5 9 i 58 


3 


9.62 


7 5o 


35 


0.37 250 


9 . 9 6 4o8 


5 


I 


60 


9 


.59 188 


3 


9.62 


7 85 


35 


o. 


37215 


9 . 9 6 4o3 









L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. 


d. 


' 


67. 


PP 


36 


35 


31 




30 


29 




6 


5 


.2 


3 .6 
7.2 


3-5 
7.0 


K 


.1 

2 


3- 
6.0 


!; 


.1 
.2 


0.6 

1.2 


o-S 

I.O 


3 


10.8 


10.5 


9-3 


3 


9.0 


8. 7 


3 


1.8 


15 


4 


14.4 


14.0 


12.4 


4 


12. 


it 6 


4 


2.4 


2.0 




18.0 

21.6 


21 


SI 


1 


15-0 

18.0 


M-5 
17.4 


:1 


3- 
3-6 


25 


.7 


25.2 


24-5 


21.7 


7 


21.0 


20.3 


7 


4-2 


3-5 


.8 


28.8 


28.0 


24.8 


.8 
g 


24.0 
27.0 





.8 

9 


4.8 


40 






23. 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. d. 







9.69 i8 


3 
29 

3 
3 
29 

30 
3 
29 
30 
2 9 

3<> 
29 
3 
29 
30 
2 9 
29 
3 
29 
29 
30 
29 
29 

2 9 
29 

3 
29 
29 

29 
29 


9.62 


7 85 


35 
35 
35 
36 
35 
35 
35 
35 
35 
34 

35 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
34 
35 
35 
34 
35 
34 
35 
35 
34 


0. 


37215 


9 . 9 6 4o3 


6 
5 
5 
6 

5 
6 

5 
5 
6 
5 
6 
5 
5 
6 

5 
6 
5 
6 
5 
6 

5 
5 
6 
5 
6 
5 
6 

5 
6 
5 


60 


I 

2 

3 

4 
5 
6 

7 
8 

9 


9.69 218 
9.69 247 
9.69277 

9.59 807 
9.59 336 
9.5 9 366 

9.59 396 

9.59 425 
9.5 9 455 


9.62 820 
9.62 855 
9.62 890 

9.62 926 
9.62 961 
9.62 996 

9.63 o3i 
9.63 066 
9-63 101 


o. 
o. 

0. 
0. 

o. 

0. 

o. 
o. 

0. 


37 180 
3 7 145. 
37 1 10 

37074 

3 7 o3 9 
37 oo4 

36 9 6 9 
36 9 34 
36899 


9.96 

9 . 9 6 
9 . 9 6 

9 . 9 6 
9 . 9 6 
9 . 9 6 

9 . 9 6 
9 . 9 6 
9 . 9 6 


3 97 
3 9 2 
38 7 

38i 
3 7 6 
3 7 o 

365 
36o 

354 


5 9 
58 
57 
56 
55 
54 

53 

52 

5i 


10 


9.59 484 


9 .63 


i35 


0. 


36865 


9 . 9 6 


34 9 


50 


1 1 

12 

i3 

i4 
i5 
16 

'7 

18 

'9 


9.59 5i4 
9 .5 9 543 
9 .5 9 5 7 3 

9.59 602 
9.59 632 
9.59661 

9.59 690 
9.59 720 
9.59749 


9-63 170 
9.63 2o5 
9.63 24o 

9.63 275 
963 3io 
9. 63 345 
9 .633 79 
9 .634i4 
9 .63 44 9 


0. 

o. 
o. 

o. 
o. 

0. 

o. 
o. 

0. 


3683o 

36 7 95 
36 760 

36725 
36 6 9 o 
36655 

36621 
36586 
36 55i 


9 . 9 6 
9 . 9 6 
9.96 

9.96 
9.96 
9.96 

9.96 
9.96 
9.96 


343 
338 
333 

32 7 
322 

3i6 

3ii 
3o5 
3oo 


49 
48 

47 

46 

45 
44 

43 

42 

4i 


20 


9.59778 


9 .63 


484 


0. 


365i6 


9.96 


294 


40 


21 
22 
23 

24 
25 

26 

27 
28 
29 


9.59808 
9 .5 9 83 7 
9.59866 

9.59895 
9.59924 
9.59 954 

9.59 983 
9.60 012 
9.60 o4i 


9 .63 
9 .63 
9 .63 

9 .63 
9 .63 
9 .63 

9 .63 
9 .63 
9 .63 


5i 9 
553 

588 

623 
65 7 
6 9 2 

726- 
761 

79 6 


o. 
o. 
o. 

o. 
o. 
o. 

o. 
o. 

0. 


3648i 
36447 
364i2 

363 77 
36 343 
363o8 

362 7 4 
3623 9 
362o4 


9.96 289 
9 . 9 6 284 
9 . 9 6 278 

9.96 273 
9. 96 267 
9.96 262 

9.96 256 
9.96 25i 
9.96 245 


39 

38 

37 
36 
35 
34 

33 

32 

3i 


30 


9. 60 070 


9 .63 


83o 


0. 


36 170 


9 . 9 6 


240 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. d. 


' 




66 30 . 






PP 36 


35 


34 




3 


29 


.1 

.2 

3 
4 

! 

:l 




6 5 


i 3-6 

.2 7 .2 

.3 10.8 

4 M-4 
.5 18.0 
.6 21.6 

7 25-2 
.8 28.8 

.q ^2.4 


3-5 
7.0 
10.5 

14.0 
I7-S 

2I.O 

24-5 
28.0 

31.5 


3-4 
6.8 

IO.2 

13-6 
17.0 
20.4 

2 3 .8 
2 7 .2 

' 


3.0 

2 6.0 
3 9- 

.4 12.0 

s 15-0 

.6 18.0 

.7 21.0 

,8 24.0 
9 2 7- 


2.9 
5-8 
8. 7 

n.6 
M-5 
17.4 

20.3 

a: 


0.6 0.5 

1.2 1.0 

1.8 1.5 

2.4 2.O 
3.0 2-5 
3.6 3.0 

4-2 3-5 
4.8 4.0 

5- 4 4-5 



7 6 



23 3O 



' 


L. Sin. 


d. 


L. Tang. d. 


L. Cotg. 


L. 


Cos. 


d. 




30 


9 


.60 070 




9. 6383o 


35 
34 
35 
34 
35 
34 
35 
34 
34 
35 

34 
34 
35 
34 
34 
35 
34 
34 
34 
34 

35 
34 
34 
34 
34 
34 
34 
34 
34 
34 


o.36 170 


9.96 240 




30 


3i 

32 

33 

34 
35 
36 

3? 

38 
3 9 


9 
9 
9 

9 
9 

9 

9 
9 
9 


.60 o 99 
.60 128 
.60 157 

.60 186 
.60 215 
.60 244 

.60273 
.60 3o2 
.6o33i 


29 

29 
29 

29 

29 

29 
29 
29 


9. 63 865 
9.63899 
9.63 934 

9. 63 968 
9.64oo3 
9.64 037 

9.64 072 
9.64 1 06 
9.64 i4o 


o.36i35 
o.36 101 
o.36 066 

o.36o32 
o.35 997 
o.35 9 63 

o.35 928 
o.358 9 4 
o.35 860 


9.96 234 
9.96 229 

9.96 223 

9.96 2 1 8 

9.96 212 
9.96 207 

9.96 2O1 
9.96 196 
9.96 190 


5 
6 
5 
6 

5 
6 
5 
6 
5 


29 
28 
27 

26 

25 
24 

23 
22 
21 


40 


9 .6o35 9 




9. 64 175 


0.35825 


9-96 185 


20 


4i 

42 

43 

44 
45 
46 

47 
48 

49 


9 
9 
9 

9 
9 
9 

9 
9 
9 


.60 388 
.60 417 
.60 446 

.60474 
.6o5o3 
.60 532 

,6o56i 
.6o58 9 
60618 


29 

29 
29 
28 
29 
29 

29 
28 
29 


9.64 209 
9.64243 
9.64 278 

9.64 3i2 
9.64346 
9 .6438i 

9.64415 
9-64449 
9.64483 


o.35 791 
0.35757 
o.35 722 

0.35688 
0.35654 
o.35 619 

0.35585 
o.3555i 
o.355i 7 


9.96179 
9.96 174 

9.96 168 

9.96 162 
9.96 i5 7 
9 . 9 6i5i 

9.96 1 46 
9.96 i4o 
9.96135 


5 
6 
6 
5 
6 
5 
6 

5 
g 


19 

17 

16 
i5 
i4 

i3 

12 
I I 


50 


9 


60 646 


29 
29 
28 
29 
28 
29 
. 28 
. 29 
28 


9 .64 5 1 7 


0.35483 


9 . 9 6 i2 9 




10 


5r 

52 

54 
55 
56 

5 7 ' 
58 
5 9 


9 
9 
9 

9 
9 
9 

9 

9 

9 


6o6 7 5 
60 704 
.60 732 

60 761 
60 78 9 
60818 

.6o846 
.60875 


9.64 552 
9.64586 
9.64 620 

9.64654 
9.64688 
9.64 722 

9.64756 
9.64 790 
9.64 824 


0.35448 
o.354i4 
o.3538o 

0.35346 
o.35 3i2 
o.35 278 

0.35244 
o.35 210 
o.35 176 


9 . 9 6 123 
9 . 9 6 118 
9 . 9 6 112 

9 . 9 6 107 
9.96 101 
9.96 095 

9.96 090 
9.96 o84 
9.96079 


5 
6 

5 
6 
6 
5 
6 
5 


1 

7 
6 
5 
4 

3 

2 

I 


60 


9 


.60 9 3i 


28 


9 . 64 858 


o.35 142 


9.96073 









L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 




66. 






PP 

.2 

3 
4 

J 

>9 


35 


34 




n 


28 


.2 

3 
4 

9 


6 


5 


3-5 

7.0 
10.5 

14.0 
J 7 5 

21.0 

24-5 
28.0 

31.5 


U 

IO. 2 

13-6 
17.0 
20.4 

23-8 
27.2 

3O.O 


,1 

.2 

3 

4 

Jt 

9 


tl 

n.6 
17.4 

20.3 
23.2 

26.1 


2.8 

5-6 
8.4 

11.2 
14.0 

16.8 

19.6 
22.4 


0.6 

1.2 

1.8 
2.4 

tJ 

5-4 


0-5 

I.O 

2.O 
2-5 

3-5 
4.0 



77 



24 C 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. 


Cotg. 


L. Cos. 


d. 







9.60 931 




9 .64 


858 




0. 


35 142 


9 . 9 6 


o 7 3 




60 


I 


9.60 960 


28 


9-64 


892 


34 


o. 


35 108 


9 . 9 6 


067 




5 9 


2 


9.60 988 


28 


9-64 


926 




o. 


35o 7 4 


9 . 9 6 


062 




58 


3 


9.61 016 




9-64 


960 


34 


o. 


35 o4o 


9 . 9 6 


o56 




57 








29 






34 










6 




4 


9.61 045 


28 


9-64 


994 




0. 


35 006 


9.96 


o5o 




56 


5 


9.61 o 7 3 


_0 


9. 65 


028 




o. 


34972 


9.96 


045 




55 


6 


9.61 10 


i 




9.65 


062 


34 


o. 


34938 


9.96 


o3 9 




54 


7 


9.61 129 


29 


9. 65 


096 


34 


o. 


34 904 


9.96 


o34 


5 

6 


53 


8 


9.61 i58 




9 .65 


i3o 




o. 


348 7 o 


9.96 


028 




52 


9 


9.61 186 


28 


9 .65 


1 64 


34 


0. 


34836 


9.96 


022 




5i 


10 


9.61 214 




9. 65 


197 


33 


o. 


348o3 


9.96 


OI 7 




50 


ii 


9.61 242 


28 


9.65 


23l 


34 


0. 


34 7 69 


9.96 


Ol I 


6 


49 


12 


9.61 270 




9.65 


265 




o. 


34 7 3<) 


9.96 


oo5 




48 


i3 


9.61 298 


28 


9.65 


2 99 


34 


o. 


34 701 


9.96 


ooo 


5 


47 


i4 


9.61 326 


28 

28 


9 .65 


333 


34 


0. 


3466 7 


9 . 9 5 


994 


fi 


46 


i5 


9.61 354 




9. 65 


366 


33 


o. 


34634 


9-9 5 


988 




45 


1 6 


9.61 382 


28 


9. 65 


4oo 


34 


o. 


34 600 


9 . 9 5 


982 




44 


I? 


9.61 4i 


i 


29 


9. 65 


434 


34 


0. 


34 566 


9 . 9 5 


977 


6 


43 


18 


9.61 438 




9. 65 


46 7 


33 


o. 


34533 


9 . 9 5 


971 




42 


'9 


9.61 466 




9 .65 


5oi 


34 


o. 


34499 


9 . 9 5 


9 65 




4i 


20 


9.61 4 9 4 




9 .65 


535 


34 


o. 


34465 


9 . 9 5 


960 


5 
g 


40 


21 


9.61 522 


28 


9 .65 


568 


33 


0. 


34432 


9.96 


954 


6 


39 


22 


9.61 55o 




9-65 


602 


34 


o. 


343 9 8 




948 




38 


23 


9.61 5 7 8 


28 


9. 65 


636 


34 


o. 


34364 


9 . 9 5 


942 




37 


24 


9.61 606 


28 

28 


9.65 


669 


33 


o. 


3433i 


9 . 9 5 


9 3 7 


b 
6 


36 


25 


9.61 634 




9.65 


7 o3 


34 


o. 


34 29 7 


9 . 9 5 


9 3i 


5 


35 


26 


9.61 662 


28 


9 .65 


7 36 


33 


o. 


34 264 


9 . 9 5 


925 




34 


27 


9.61 689 


27 

28 


9.65 


770 


34 


o. 


3423o 


9 . 9 5 


920 


b 
6 


33 


28 


9.61 717 




9-65 


8o3 


33 


0. 


34 197 


9 . 9 5 


914 


g 


32 


2 9 


9.61 745 


28 


9.65 


83 7 


34 


o. 


34 i63 


9 . 9 5 


908 


6 


3i 


30 


9.61 773 




9.65 


8 7 o 


33 


o . 


34 i3o 


). 9 5 


902 




30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. |d. 








65 30 . 






PP 34 


33 


29 


28 


27 




6 5 


i 3-4 


3-3 


2.9 


.1 2.8 


2.7 


.1 


0.6 0.5 


.2 6.8 


66 


5-8 


2 5-6 


54 


.2 


1.2 1.0 


3 10.2 


9-9 


8.7 


3 8-4 


8.1 


3 


1.8 1.5 


4 13-6 


13.2 


11.6 


. 4 II. 2 


10.8 


4 


2.4 2.0 


.5 17.0 


16.5 ' 


14-5 


.5 14.0 


J 3-5 


5 


3.0 2.5 


.6 20.4 


19.8 


17.4 


.6 16.8 


16.2 


.6 


3-6- 3-o 


7 238 


23.1 


20.3 


.7 19.6 


18.9 


7 


42 3-5 


.8 27.2 


26.4 


23.2 


.8 22.4 


21.6 


.8 


4.8 4.0 


.9 30.6 


29.7 26.1 


9 25.2 


24-3 




5-4 4-5 



7 8 



24 3D' 



> 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9 


61 773 


27 


9 .65 870 


34 


o. 34 i3o 


9.95 902 




30 


3i 


9 


61 800 


28 


9 .65 90^ 




33 


o.34 o 9 6 


9 . 9 5 8 97 


6 


29 


32 


9 


61 828 


28 


9 .65 9 3 7 






o.34 o63 


9. 95 891 




28 


33 


9 


61 856 


27 


9 .65 97 i 




33 


O.34 O2 9 


9.95885 


6 


27 


34 


9 .6i 883 


28 


9 .66 oo4 


34 


0.33 99 6 


9.95 879 




26 


35 


9 


61 9 n 


28 


9 .66 o38 




o.33 9 62 


9 . 9 58 7 3 




25 


36 


9 


61 9 3 9 


27 


9 .66 071 




33 


o.33 9 2 9 


9.95 868 


6 


24 


37 


9 


61 9 66 


28 


9 .66 10^ 




34 


o.33 896 


9.95 862 


6 


23 


38 


9 


61 99 4 




9. 66 1 38 






o.33 862 


9 . 9 5 856 




22 


3 9 


9 


62 021 


27 

28 


9 .66 171 




33 


o.33 829 


9.95 85o 


5 


21 


40 


9 


62 o4 9 




9 .66 204 




o.33 796 


9.95 844 




20 


4i 


9 


62 076 


28 


9 . 66 238 




34 


o.33 762 


9.95 839 


6 


, 9 


42 


9 


62 io4 




9 .66 271 






o.33 729 


9 . 9 5 833 




18 


43 


9 


62 i3i 


28 


9 .66 3o4 


33 
33 


o.33 696 


9.95 827 


6 


17 


44 


9 


62 i5 9 


27 


9 .66 337 




0.33663 


9.95 821 


g 


16 


45 


9 


62 186 


28 


9 .66 371 






o.33 629 


9. 9 5 8i5 




i5 


46 


9 


62 214 




9 .66 4o4 




33 


o.33 5 9 6 


9.95 810 


5 


i4 








27 






33 














47 


9 


62 241 


27 


9 . 66 437 




o.33 563 


9.95 8o4 


f, 


i3 


48 


9 


62 268 




9 .66 470 


33 


o.33 53o 


9 . 9 5 798 




12 


49 


9 


62 2 9 6 




9 .665o3 


33 


0.33497 


9.95792 


6 


II 


50 


9 


62 323 




9 .66 53 7 


34 


0.33463 


9.95 786 


6 


10 


5i 


9 


62 35o 


27 


9 .66 570 


33 
33 


o.3343o 


9.95 780 


s 


9 


52 


9 


6 2 3 77 




9 .66 6o3 




o.33 397 


9-9 5 775 


g 


8 


53 


9 


62 405 




9. 66 636 


33 


o.33 364 


9.95 769 




7 








27 






33 














54 


9 


62432 




9 .66 66 9 




o.33 33i 


9 . 9 5 7 63 


6 


6 


55 


9 


62 45 9 




9 .66 702 




o.33 298 


9 . 9 5 757 


6 


5 


56 


9 


62486 


27 


9 .66 735 


33 


o.33 265 


9 . 9 5 7 5i 




4 


57 


9 


62 5i3 


27 
28 


9 .66 768 


33 


o.33 232 


9.95 745 


6 


3 


58 


9 


62 54 1 




9 .66 801 






o.33 199 


9-9 


5 739 


5 


2 


5 9 


9 


62 568 


27 


9. 66 834 


33 


o.33 166 


9 . 9 5 7 33 




I 


60 


9 


62 5 9 5 


27 


9 .66 867 


33 


o.33 1 33 


9 . 9 5 728 









L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 








65. 






PP 


34 


33 


28 


27 




6 5 


, 


3-4 


3-3 -i 


2.8 


2-7 


., 


0.6 0.5 


.2 


6.8 


6.6 .2 


5.6 


5-4 


.2 


1-2 I.O 


3 


IO. 2 


9-9 3 


8.4 




3 


1.8 1.5 


-4 


I 3 .6 


13-2 -4 


II 2 


0.8 


4 


2-4 2.0 


-5 


17.0 


16.5 5 


14.0 


3-5 


. tj 


3 .0 2.5 


.6 


20-4 


19.8 .6 


16.8 


6.2 


6 


3- 6 3- 


7 


23-8 


23-1 -7 


19,6 


8.9 


.7 


4-2 3-5 


.8 


27.2 


26.4 8 


22.4 


1.6 


8 


4.8 4.0 






25.2 4.3 


9 


5-4 4-5 



79 



25. 



/ 


L. Sin. 


d. 


L. Tang. 


d. 


L.Cotg. 


L. 


Cos. d. 







9 


.62695 




9.66 867 




o.33 i33 


9.96 728 


g 


60 


I 


9 


.62 622 


27 


9.66 900 


33 


o.33 100 


9.96 722 


6 


5 9 


2 


9 


.62 649 




9.66933 




0.33067 


9.96 716 




58 


3 


9 


.62676 


27 


9.66 966 


33 


o.33 o34 


9.96 710 


6 


57 


4 


9 


.62 703 


27 


9.66999 


33 


o.33 ooi 


9.96 704 


6 


56 


5 


9 


.62 730 




9.67 o32 




0.32 968 


9.96 698 




55 


6 


9 


.62767 




9.67 065 




o.32 9 35 


9.96 692 




54 








27 






33 










6 




7 


9 


.62 784 




9.67 098 


33 


o.32 902 


9.96 686 


6 


53 


8 


9 


.62 811 




9 .6 7 i3 


[ 




0.32 869 


9.96 680 




52 


9 


9 


.62838 


27 


9.67 i63 


3 2 


o.3 2 83 7 


9.96 674 




5i 


10 


9 


.62 865 




9.67 196 


33 


o.32 8o4 


9.96 668 




50 


1 1 


9 


.62 892 


26 


9.67 229 


33 


o.32 771 


9.96 663 


6 


49 


12 


9 


.62 918 




9.67 262 




o.32 738 


9.96 667 




48 


i3 


9 


.62 945 


27 


9.67295 


32 


o.32 706 


9.96 65i 


6 


47 


i4 


9 


.62 972 


27 


9.67327 


33 


o.32 673 


9-95645 


6 


46 


16 


9 


.62 999 




9.67 36o 




o.32 64o 


9.96 639 




45 


16 


9 


.63 026 


27 
26 


9.6739 


3 


33 


o.32 607 


9.96 633 


6 


44 


17 


9 


.63 062 


27 


9.67 42< 


5 


3 2 


o.32 674 


9.96 627 


fi 


43 


18 


9 


.63 079 




9.67458 




0.32 542 


9.96 621 




42 


'9 


9 


.63 106 


27 


9.67 491 


33 


o.32 609 


9.96615 




4i 


20 


9 


.63 i33 


/: 


9.67 624 




o.32 476 


9.96 609 




40 


21 


9 


.63 169 


27 


9.67 556 


33 


o.32 444 


9.96 6o3 


6 


3 9 


22 


9 


.63 186 




9.67 689 




o.32 4 


[ I 


9.96 697 




38 


23 


9 


.63 2i3 


27 
26 


9.67 622 


33 
3 2 


o.32 378 


9.96691 


6 


37 


24 


9 


.63 23 9 




9.67 654 


33 


o.32 346 


9.96685 




36 


26 


9 


.63266 




9.67 687 




o.32 3i3 


9.96 679 




35 


26 


9 


.63 292 




9.67 719 


32 


0.32 28l 


9.96673 




34 








27 






33 










6 




27 


9 


.633i 9 


26 


9.67 762 




0.32 248 


9.96667 


6 


33 


28 


9 


.63345 




9.67785 




0.32 2l5 


9.96 56i 




32 


29 


9 


.63 3 7 2 


27 


9.67 817 


32 


o.32 i83 


9. 9 5555 




3i 


30 


9 


.63 3 9 8 




9.67 850 


33 


o.32 i5o 


9.96 549 




30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. d. 




64 30 . 


PP 


33 


32 




37 


26 




6 


5 


i 


3-3 


3-2 


. .1 


2-7 


2.6 


, I 


0.6 


o-5 


2 


6.6 


6.4 


.2 


5-4 


5-2 


.2 


1.2 


I.O 


3 


9-9 


9.6 


3 


8.1 


7.8 


3 


1.8 


i-5 


4 


13.2 


12.8 


4 


10.8 


10.4 


4 


2-4 


2.O 


5 


16.5 


16.0 


5 


13-5 


13.0 


5 


3- 


2-5 


6 


19.8 


19.2 


6 


16.2 


15.6 


.6 


3-6 


3- 


7 


23-1 


22.4 


. 7 


18.9 


18.2 


7 


4.2 


3-5 


8 


26.4 


25-6 


.8 


21.6 


20.8 


.8 


4.8 


4.0 


Q 






24.3 23.4 




S-4 4-5 



80 



25 3D 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9 


63 398 




9.67 850 




o.32 i5o 


9.95 549 




30 


3i 


9 


63425 


26 


9.67 882 


32 


o.32 118 


9.95543 




29 


32 


9 


6345i 




9.67 915 


33 


o.32 o85 


9.95537 




28 


33 


9 


63478 


27 


9.67947 


32 


o.32 o53 


9. 9 553i 


6 


27 


34 
35 


9 
9 


63 53i 


26 
27 


9.67-980 

9.68 012 


33 
32 


O.32 O2O 

o.3i 988 


9.95525 
9.95 5i 9 


6 
6 


26 

25 


36 


9 


63 557 




9.68044 


32 


o.3i 956 




3 5i3 


6 


24 


3? 


9 


63583 


26 


9.68 077 


33 


o.3i 923 


9.95 507 


6 


23 


38 


9 


636io 




9.68 109 




o.3i 891 


9.95 5oo 




22 


3 9 


9 


63636 


26 


9.68 142 


33 


o.3i 858 


9.95 494 


6 


21 


40 


9 


63 662 




9.68 174 


3 2 


o.3i 826 


9.95488 




20 


4i 


9 


63 689 


26 


9.68 206 


33 


o.3 1 794 


9.95 482 


6 


I9 


42 


9 


63 715 


26 


9.68 239 




o.3i 761 


9.95 476 




18 


43 


9 


63 7 4i 




9.68 271 






o.3i 729 


9.95470 




17 








26 






32 








6 




44 


9 


63 767 


27 


9.68 3o3 




o.3i 697 


9.95464 


6 


16 


45 


9 


63 79 4 




9.68 336 




o.3i 664 


9.95458 




i5 


46 


9 


63820 


26 


9.68 368 


32 


o.3i 632 


9.95 452 


6 


i4 


47 


9 


63846 


26 


9.68 4oo 




o.3i 600 


9.95 446 


6 


i3 


48 


9 


63872 




9.68 432 


3 2 


o.3i 568 


9.95 44o 




12 


4 9 


9 


63898 




9.68 465 


33 


o.3i 535 


9.95434 




II 


50 


9 


63 924 


26 


9.68 497 


32 


o.3i 5o3 


9.95427 




10 


5i 


9 


63 9 5o 


26 


9.68 529 




o.3i 471 


9.95 421 


6 


9 


52 


9. 63 976 


26 


9. 6856i 




o.3i 439 


9.95 4i5 




8 


53 


9.64 002 


26 


9 .685 9 3 


3 2 
33 


o.3i 407 


9.95 4oQ 


6 


7 


54 


9 


64 028 


26 


9.68 626 




o.3i 3 7 4 


9.95 4o3 


6 


6 


55 


9 


64o54 




9.68 658 




o.3i 342 


9 . 9 5 3 97 




5 


56 


9 


.64 080 




9.68 690 


32 


o.3i 3io 


9.95 3 9 i 




4 


5? 


9 


.64 1 06 


26 
26 


9.68 722 


32 


o.3i 278 


9.95 384 


7 
6 


3 


58 


9 


.64 1 32 




9.68 7 54 




o.3i 246 


9 . 9 53 7 8 




2 


5 9 


9 


.64 i58 




9.68 786 


32 


o.3i 214 


9.95372 


g 


I 


60 


9 


.64 1 84 




9.68 818 


32 


o.3i 182 


9 . 9 5 366 









L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 


64. 




PP 


33 


32 




27 


26 


7 


6 


.1 


3-3 


3-2 


., 


2.7 


2.6 .1 


0.7 


0.6 


.2 


6.6 


6.4 


.2 


5-4 


5-2 -2 


M 


1.2 


3 


9.9 


9.6 


3 


8.1 


7-8 -3 


2.1 


1.8 


4 


13.2 


12.8 


4 


10.8 


10.4 .4 


2.8 


2.4 


.5 


16.5 


16.0 




!3-5 


i3- 5 


3-5 


3.0 


.6 


19.8 


19.2 


.6 


16.2 


15.6 .6 


4.2 


3-6 


.7 


23.1 


22.4 


7 


18.9 


18.2 .7 


4-9 


4.2 


.8 


26.4 


25.6 


.8 


21.6 


20.8 .8 


5-6 


4.8 


9 


29.7 28.8 


9 


24-3 2 3-4 -9 


6.3 5-4 



81 



26. 



/ 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d. 







9.64 1 84 


26 


9.68 818 




o.3i 182 


9.95 366 




60 


I 


9.64 210 


26 


9.68 85o 


3 2 


o.3i 150 


9.95 36o 


6 


5 9 


2 


9. 64 236 


26 


9.68 882 




o.3i 118 


9. 9 5354 




58 


3 


9.64 262 


26 


9.68 914 


32 

3 2 


o.3i 086 


9.95 348 


7 


5 7 


4 


9.64 288 


25 


9.68 946 




o.3i o54 


9.95 34i 


6 


56 


5 


9.64 3i3 


26 


9.68 978 




o.3i 022 


9.95335 




55 


6 


9.64 33 9 


26 


9.69 oio 


3 2 

32 


o.3o 990 


9.95 329 


6 


54 


7 


9 


.64365 


26 


9.69 042 




0.30958 


9.95 323 


6 


53 


8 


9 


.643 9 i 


26 


9.69074 




o.3o 926 


9.95 3i 7 




52 


9 


9 


.64417 




9.69 106 


32 


o.3o 894 


9.95 3io 


7 


5i 


10 


9 


.64442 


26 


9.69 1 38 


32 


o. 3o862 


9.95 3o4 




50 


ii 


9 


.64468 


26 


9.69 170 


32 


o.3o 83o 


9.95 298 


6 


49 


12 


9 


.64494 




9.69 202 




o.3o 798 


9.95 292 




48 


i3 


9 


.645i 9 


25 


9.69 234 


32 


o.3o 766 


9.95 286 




47 


i4 


9 


.64545 


26 
26 


9.69 266 


32 


o.3o 734 


9.95 279 


7 

6 


46 


i5 


9 


.645 7 i 




9.69 298 




o.3o 702 


9.95 273 




45 


16 


9 


.64 5 9 6 


25 


9.69 329 


31 


o.3o 671 


9.95 267 




44 


17 


9 


.64622 


26 


9.69 36i 


32 


o.3o 6 


3 9 


9.95 261 


b 


43 


18 


9 


.64647 




9 .6 9 3 9 


3 


3 2 


o.3o 607 


9.95 254 




42 


'9 


9 


.646 7 3 


26 


9.69425 


3 2 


o.3o 575 


9.95 248 


6 


4i 


20 


9 


.64 698 


25 


9.69 457 


32 


o.3o 543 


9.95 242 




40 


21 


9 


.64 724 




9.69 488 


3 1 


o.3o 5i2 


9.95 236 




3 9 


22 


9 


.64 749 




9.69 52O 




o.3o48o 


9.95 229 




38 


23 


9 


64775 




9.69 552 


3 2 


o.3o448 


9.95 223 




37 








25 






3 2 










h 




24 


9 


.64 800 


26 


9.69 584 




o.3o4i6 


9.95 2I 7 




36 


25 


9 


.64826 




9.69 6i5 




o.3o385 


9.95 211 




35 


26 


9 


.6485i 


25 

26 


9.69 647 


32 


o.3o 353 


9.95 2O4 


7 
f, 


34 


27 


9 


.648 77 




9.69679 




o.3o 32i 


9.95 198 


6 


33 


28 


9 


.64 902 




9.69 710 




o.3o 290 


9.95 192 




32 


2 9 


9 


.64 927 


25 


9.69 742 


32 


o.3o 258 


9.95 i85 


7 


3i 


30 


9 


.64953 




9.69 77 4 


32 


o.3o 226 


9.95 179 




30 




L. Cos. 


d. 


L. Cotg. d. 


L. Tang. 


L. 


Sin. 


d. 


' 


63 3D'. 


PP 


32 


31 




26 


25 




7 


6 


i 


3-2 


i' 1 


! 


2.6 


2-5 


.! 


0.7 


0.6 


2 


6.4 


6.2 


2 


5-2 


5- 


2 


1.4 


1.2 


3 


9.6 


9-3 


3 


7.8 


7-5 


3 


2.1 


1.8 


4 


12.8 


12.4 


4 


10.4 


10. 


4 


2.8 


2.4 


5 


16.0 




5 


13-0 


12.5 




3.5 


3- 


6 


19.2 


18.6 


.6 


15-6 


15.0 


6 


4.2 


3-6 


7 


22.4 


21.7 


7 


18.2 


17-5 


7 


4-9 


4.2 


8 


'5-6 


24.8 


.8 


20.8 


20.0 


.8 


5-6 


4.8 


9 






23.4 22.5 




6.3 5-4 



82 



26 3O . 





L. Sin. d. 


L. Tang. 


d. 


L. 


Cotg. 


L. Cos. 


d. 




30 


9.64 953 




9 .6 9 


774 




o. 


3o 226 


9 . 9 5 


179 


g 


30 


3i 


9.64 978 
9.65 oo3 


25 


9 .6 9 
9 .6 9 


8o5 
83 7 


3 1 
32 


o. 

0. 


3o 195 
3o i63 


9 . 9 5 
9 . 9 5 


178 

167 


6 


2 9 

28 


33 


9-65 029 




9.69 


868 


3i 


o. 


3o 1 32 


9 . 9 5 


1 60 


7 


27 


34 


9.65o54 


25 


9.69 


32 
9 


o. 


3o 100 


9 . 9 5 


1 54 


f> 


26 


35 


9.65 079 




9.69 


9 32 




0. 


3o 068 


9 . 9 5 


i48 




25 


36 


9-65 io4 


25 
26 


9.69 


9 63 


3 1 
3 2 


o. 


3oo37 


9 . 9 5 


i4i 


6 


24 


37 


9.65 i3o 


2 5 


9.699^5 




0. 


3o oo5 


9 . 9 5 


1 35 


6 


23 


38 


9 .65i55 




9.70 026 




o. 


29974 


9 . 9 5 


I2 9 




22 


39 


9.65 i8o 


2 5 


9.70 o58 


3 2 


0. 


29 942 


9 . 9 5 


122 


7 
g 


2* 


40 


9-65 2o5 


2 5 


9.70 089 


3 1 


o. 


29 91 1 


9 . 9 5 


116 


g 


20 


4i 

42 


9.65 23o 
9. 65 2 55 


2 5 
25 


9.70 
9.70 


121 

i5a 


3 2 
3 1 


o. 
o. 


29 879 
29 848 


9 . 9 5 
9 . 9 5 


I IO 

io3 


7 


19 

18 


43 


9-65 281 




9.70 


1 84 


32 


o. 


29 816 


9 . 9 5 


97 




17 


44 


9 .653o6 


25 


9.70 


2l5 




o. 


29785 


9 . 9 5 


o 9 o 


7 

6 


16 


45 


9.65 33i 




9.70247 




o. 


29 753 


9 . 9 5 


084 




i5 


46 


9-65 356 


25 


9.70 


278 


3* 


0. 


29 722 


9 . 9 5 


078 




i4 


47 


9.65 38i 


25 


9.70 


3o 9 




0. 


29691 


9 . 9 5 


071 


6 


i3 


48 


9. 65 4o6 




9.70 


34i 




o. 


29 659 


9 . 9 5 


065 




12 


49 


9 .6543i 


2 5 


9.70 


3 7 2 


3 1 


0. 


29 628 


9 . 9 5 


o5 9 




I I 


50 


9. 65 456 




9.70 4o4 




0. 


29 596 


9 . 9 5 


052 




10 


5i 


9 .6548i 


2 5 


9.70 


435 


3 1 


0. 


29 565 


9 . 9 5 


o46 


7 


9 


52 


9 .65 5o6 




9. 70 466 




o. 


29 534 


9 . 9 5 


o3 9 


g 


8 


53 


9. 6553i 


25 


9.70498 


32 


o. 


29 5o2 


9 . 9 5 


o33 




7 


54 


9 .65 556 


25 


9.70 


52 9 




o. 


29471 


9 . 9 5 


027 


7 


6 


55 


9 .65 58o 




9.70 


56o 




o. 


29440 


9 . 9 5 


020 


g 


5 


56 


9 .65 6o5 


25 


9.70 


5 9 2 


32 


0. 


29 4o8 


9 . 9 5 


oi4 




4 


57 


9 .65 63o 


25 


9.70 623 




o. 


29 377 


9 . 9 5 


007 


7 

6 


3 


58 


9 .65 655 




9.70 654 




0. 


29 346 


9 . 9 5 


OOI 




2 


5 9 


9 .65 680 


25 


9.70 685 


3 1 


0. 


29315 


9 . 9 4 


995 




I 


60 


9 .65 705 


2 5 


9.70 717 


3 2 


o. 


29 283 


9 . 9 4 9 88 









L. Cos. 


d. 


L. Cotg. d. 


L. 


Tang. 


L. Sin. 


d. 


f 




63. 




PP 


32 


31 


26 


25 


24 




7 6 


x 


3.2 


3- 1 


2.6 


.1 2.5 


2.4 


.1 


o. 7 o. 6 


. 2 


6. 4 


6.2 


5-2 


2 5-0 


4 .8 


.2 


1.4 1.2 


3 


9 .6 


9-3 


7.8 


3 7-5 


7-2 


3 


2.1 1.8 


4 


12.8 


12.4 


10.4 


.4 10.0 


9.6 


4 


2.8 2.4 


.5 


16.0 




13.0 


r 12. 5 


12.0 


5 


3-5 3-o 


.6 


19.2 


18.6 


15.6 


.6 15.0 


14.4 


.6 


4.2 3.6 


-? 


22.4 


21.7 


18.2 


7 17-5 


16.8 


7 


4-9 4- 2 


.8 


25-6 


24.8 


20.8 


8 20.0 


19.2 


.8 


5.6 4.8 


9 




27.9 23.4 








6 -3 5-4 



83 



27. 



/ 


L. Sin. 


d. 


L. Tang. 


d. 


L 


. Cotg. 


L. Cos. 


d. 







9.66 705 




9.70717 




o 


29283 


9 . 9 4 


988 




60 


I 


9.65 729 


25 


9.70 748 


31 


o 


29 252 


9.94 


982 




5 9 


2 


9 .65 7 54 




9.70779 




o 


29 221 


9.94 


Q7 5 




58 


3 


9.66779 




9.70 810 


3 1 


o 


29 190 


9.94 


969 




5 7 








25 






3 1 










7 




4 


9.65 8o4 


24 


9.70 84i 


3 2 


o 


29 1 59 


9.94 


962 


6 


56 


5 


9.66828 




9.70873 




o 


29 127 


9.94 


g56 




55 


6 


9.65853 


25 


9.70904 


3 1 


o 


29 096 


9.94949 


7 


54 








2 5 






31 










6 




7 


9.60878 


24 


9.70935 


31 


o 


29065 


9.94943 




53 


8 


9.65 902 




9.70 966 




o 


29 o34 


9.94 936 




52 


9 


9.65 927 




9.70997 


3 1 


o 


29 oo3 


9.94930 




5i 


10 


9.65 962 


24 


9.71 


028 


31 


o 


28 972 


9-94 


92 3 


g 


50 


1 1 


9.65 976 


2 5 


9.71 


o5g 


31 


o 


28941 


9.94917 


6 


49 


12 


9.66 ooi 




9.71 


090 




o 


28 910 


9.94 911 




48 


i3 


9.66026 


2 5 


9.71 


121 


32 


o. 


28 879 


9.94904 


7 

6 


47 


i4 


9.66 o5o 


25 


9.71 


i53 


31 


o. 


28847 


9.94 


898 




46 


i5 


9.66075 




9.71 


1 84 




o, 


28816 


9.94 


891 




45 


16 


9.66 099 




9.71 


215 




o. 


28 785 


9.94 


885 




44 








2 5 






3 










7 




J 7 


9.66 124 


24 


9.71 


246 


31 


o. 


28754 


9-94 


878 




43 


18 


9.66 i48 




9.71 


277 




o. 


28 723 


9-94 


871 




42 


'9 


9.66 173 




9.71 


3o8 




o. 


28 692 


9 . 9 4 


865 




4i 


20 


9.66 197 




9.71 


33 9 




o. 


28661 


9.94 858 




40 


21 


9.66 221 


25 


9.71 


3 7 o 


31 


o. 


2863o 


9.94852 




39 


22 


9.66 246 




9.71 


4oi 




0. 


28 599 


9.94 


845 




38 


23 


9.66 270 




9.71 


43i 




0. 


2 856 9 


9.94 


83 9 




37 








2 5 






3 1 










7 




24 


9.66 295 


24 


9.71 


462 


3 1 


o. 


28538 


9-94 


832 


6 


36 


25 


9.66 319 




9.71 


49^ 




o. 


28 507 


9.94 


826 




35 


26 


9.66343 


2 4 


9.71 


5 2 4 




o. 


28476 


9.94 


819 


7 


34 








2 5 






3 1 










6 




27 


9.66368 




9.71 


555 


31 


o. 


28445 


9.94 


8i3 




33 


28 


9.66 392 




9.71 


686 




o. 


284i4 


9-94 


806 




32 


29 


9.66416 


24 


9.71 


617 


3 1 


o. 


28 383 


9 . 9 4 


799 


7 


3i 


30 


9.66 44i 




9.71 


648 




o. 


28352 


9 . 9 4 


79 3 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. ! d. 








62 3O . 






PP 


32 


31 


30 


25 


24 




7 6 


.! 


3-* 


3-i 


3-o 


i 2.5 


2.4 


. x 


o. 7 o. 6 


.2 


6.4 


6.2 


6.0 


.2 5.0 


4 .8 


2 


1-4 1.2 


3 


9.6 


9-3 


9.0 


3 7-5 


7.2 


3 


2.1 1.8 


4 


12.8 


12.4 


12.0 


.4 xo.o 


9 .6 


4 


2.8 2.4 


5 


16.0 


iS-5 


15.0 


5 12.5 


12.0 




3-5 3-o 


.6 


19.2 


18.6 


18.0 


.6 15.0 


14.4 


6 


4.2 3.6 


7 


22.4 


21.7 


21. 


7 17-5 


16.8 


7 


4.9 4.2 


.8 


25-6 


24.8 


24.0 


.8 20.0 


19.2 


.8 


5-6 4-8 


9 




27.0 27.0 


.9 22. S 


2 r.6 




6.3 5-4 



84 



27 30 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. 


Cotg. 


L. Cos. 


d. 




30 


9.66441 


24 
24 
24 
24 
25 
24 
24 
24 
24 
24 

24 
25 
24 
24 
24 
24 
24 
24 
24 
23 

24 
24 
24 
24 
24 
24 
24 
23 
24 
24 


9.71 


648 


31 

30 
31 
31 
31 

31 
30 
31 
31 
30 

31 
31 

3* 
3 

3 
3 1 
30 

3 
3 
30 
30 
3 


0.28 352 


9.94 


79 3 


7 

6 

7 
6 
7 
7 
6 
7 
6 
7 
7 

7 
7 

t 

3 

d 
3 

(. 

3 

(. 

3 
3 

e 

3 
3 
'i 




30 


3i 

32 

33 

34 
35 
36 

37 
38 
3 9 


9.66465 
9.66489 
9.665i3 

9 .6653 7 
9.66 562 
9.66586 

9.66 610 
9.66634 
9.66658 


\O vo "O VO vO VO vo VO vO 


679 
709 
740 

771 
802 
833 

863 
894 
925 


0.28 32i 
0.28 291 
0.28 260 

0.28 229 
0.28 198 
0.28 167 

0.28 137 
0.28 106 
0.28 075 


9.94 786 
9.94 780 
9.94773 

9.94767 
9.94 760 
9.94753 

9.94747 
9.94 740 
9.94734 




29 

28 
27 
26 

25 

24 

23 
22 
21 


40 


9.66682 


9-71 


9 55 


o. 


28045 


9 . 9 4 


727 




20 


4i 

42 

43 

44 
45 
46 

47 

48 

49 


9.66 706 
9.66 7 3i 
9.66 755 

9.66779 
9.66 8o3 
9.66 827 

9.6685r 
9.66 875 
9.66 899 


9-71 
9-7 2 
9-7 2 

9.72 
9.72 
9.72 

9.72 
9.72 
9-72 


986 
017 

o48 

078 
109 
i4o 

170 

2OI 

23l 


0. 

o. 
o. 

o. 
o. 

0. 

o. 
o. 

0. 


28014 
27983 

27 952 
27 922 

27891 
27 860 

27830 

27799 
27769 


9.94 720 
9.94 714 
9-94707 

9.94 700 
9.94694 
9.94687 

9.94680 
9.94674 
9.94 667 


'9 
1 8 

17 

16 
i5 
i4 

i3 

12 
I I 


50 


9.66 922 


9-72 


262 


o. 


2 77 38 


9.94 


660 


10 


5i 

52 

53 

54 
55 
56 

57 
58 

59 


9.66 946 
9.66 970 
9.66994 

9.67 018 
9.67 042 
9.67 066 

9.67 090 
9.6 7 1 13 
9 .6 7 i3 7 


9.72 293 
9.72 323 
9.72 354 

9. 72 384 
9.72 415 
9.72 445 

9.72476 
9.72 5o6 
9.72 537 


o. 

0. 

o. 

o. 
o. 
o. 

o. 
o. 
o. 


27 707 
27677 

27 646 

27 616 
27585 
27555 

27 524 
27494 
27463 


9-94654 
9.94647 
9.94640 

9.94 634 
9.94 627 
9.94 620 

9.94614 
9.94607 
9.94 600 


9 

8 

7 

6 
5 
4 
3 

2 

I 


60 


9.6-7 161 


9.72 


56 7 


o. 


2 7 433 


9 . 9 4 


5 9 3 







L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. 


d. 


' 








62. 








PP 

.1 

2 

3 
4 


31 




30 


35 




34 


23 




7 


6 


3- 1 

6.2 

9-3 
12.4 
i8!6 

21.7 
24.8 














0.6 

1.2 

1.8 

2-4 

3- 
3-6 

4-2 

4.8 

5-4 


6.0 
9.0 

12.0 

I 5 .0 

18.0 

21.0 
24.0 

27.0 


5-o 
7-5 

10. 

12.5 
15-0 

17-5 

20.0 

22. =; 


.3 4.8 

3 7-2 

.4 9.6 

5 I2.O 

.6 14.4 

.7 16.8 
.8 19.2 


4-6 
6.9 

9-2 
"5 
13-8 

16.1 
18.4 


.2 

3 
4 


1.4 

2.1 

2.8 

3-5 
4.2 

4-9 
5-6 

6-3 



85 



28. 



' 


L.Sin. 


d. 


L. Tang 


. d. 


L. Cotg. 


L. Cos. d. 







9 


67 161 




9.72 567 


3i 
30 
3i 
30 
3i 
30 
30 
3i 
3 
3i 

30 
30 
3i 
30 
3 
3i 
30 
3 
30 
3i 

30 
3 
30 
30 
3i 
3 
3 
3 
3 
3 


0.27 433 


9.94 5 9 3 


6 


60 


I 

2 

3 

4 
5 
6 

7 

8 

9 


9.67 185 
9.67 208 

9.67 232 

9.67 256 
9.67 280 
9.67 3o3 

9.67327 
9.67 35o 
9-67 374 


23 
24 
24 
24 
23 
24 
23 
24 


9.72 598 
9.72 628 
9.72 65g 

9.72 689 
9.72 720 
9.72 750 

9.72 780 
9.72 811 
9.72 84i 


0.27 402 

0.27 372 
0.27 34i 

0.27 3n 
0.27 280 
0.27 250 

O.27 220 
O.27 189 

0.27 iSg 


9-94 58 7 
9.94 58o 
9.94573 

9.94 567 
9.94 56o 
9 .94553 

9.94 546 
9.94 54o 
9 . 9 4533 


7 
7 
6 

7 
7 
7 
6 

7 

7 

7 
6 
7 
7 
7 
7 
6 
7 
7 
7 

7 

6 
7 
7 
7 
7 
7 
6 
7 
7 


5 9 
58 
5 7 
56 
55 
54 

53 

52 

5i 


10 


9 


67 3 9 8 




9.72 872 


0.27 128 


9.94 526 


50 


1 1 

12 

i3 

i4 
i5 
16 

*7 

18 

[9 


9.67 4ai 
9.67 445 
9.67468 

9.67 492 
9 .6 7 5i5 
9.67539 

9.67 562 
9.67 586 
9.67 609 


24 
23 
24 

23 
24 

23 
24 

23 


9.72 902 
9.72 932 
9.72 963 

9.72 993 

9.73 023 

9.73054 

9.73 o84 

9.73 n4 
9.73 i44 


0.27 098 
0.27 068 
0.27 037 

0.27 007 
0.26 977 
0.26 946 

0.26 916 
0.26 886 
0.26 856 


9.94 5 19 
9.94 5i3 
9.94 5o6 

9.94499 
9.94 492 
9 . 9 4485 

9.94479 
9.94 472 
9.94 465 


49 

48 

47 
46 
45 
44 

43 

42 

4i 


20 


9 


.67 633 




9 . 7 3 175 


0.26 825 


9-94458 


40 


21 
22 
23 

24 
25 

26 

27 
28 
29 


9 
9 

9 

9 
9 
9 

9 
9 
9 


.67656 
.67 680 
.6 77 o3 

.67 726 
.67 750 
.67773 

.67 796 
.67 820 
.6 7 843 


24 
23 
23 
24 
23 
23 
24 
23 


9.73 205 
9.73 235 
9.73 265 

9.73 2 9 5 
9.73 326 
9. 7 3356 

9 . 7 3 386 
9.73416 
9.73446 


0.26 795 
0.26 765 
0.26735 

0.26 705 
0.26 674 
0.26 644 

0.26 6i4 
0.26 584 
0.26 554 


9.94 45 1 

9.94445 
9. 9 4438 

9-94431 
9.94 424 
9.94417 

9.94 4 10 
9.94 4o4 
9 . 9 4 3 97 


39 
38 

37 

36 
35 
34 
33 

32 

3i 


30 


9 


.67866 




9.73476 


o. 26 524 


9.94 390 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. d. 


' 


61 30 . 


PP 

.1 

.2 

3 

4 
5 
.6 

3 

9 


3* 


30 




24 


23 




.1 

.2 

3 
4 

:i 

9 


7 


6 


11 

9-3 

12.4 

I5-S 
18.6 

21.7 
24.8 

27.9 


3- 
6.0 
9.0 

12.0 
15.0 

18.0 

21. 
24.0 


.2 

-3 
4 

:! 

7 
.8 

9 


:i 

7-2 

9 .6 

I2.O 
14.4 

16.8 
19.2 


a 

6. 9 
9.2 
i 3 .8 

16.1 
18.4 




0.7 
1.4 

2. I 

2.8 

3-5 

4-2 

4-9 
5-6 
6.3 


0.6 

1.2 

1.8 

2-4 

30 

3.6 

4.2 

4.8 

5-4 



86 



28 3O 





L. Sin. d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9.67866 




9.73476 


3 
30 
30 

3 
3 
3 
30 
3<> 
30 
30 

3 
30 
3 
3 
30 
30 
30 
3 
3 
3 

30 
30 
29 
30 
3 
3 
3 
3> 
2 9 
3 


0.26 


524 


9 . 9 4 


3 9 o 


7 

7 
7 
7 
7 
6 

7 

7 
7 
7 

7 
7 
7 
7 
7 
7 
6 

7 
7 
7 

7 
7 
7 
7 
7 
7 
7 
7 
7 
7 


30 


3i 

32 

33 

34 
35 
36 

3? 
38 
3 9 


9.67 890 
9.67913 
9.67 936 

9.67959 
9.67 982 
9.68 006 

9.68 029 
9.68 o52 

9.68075 


23 
23 
23 
23 
24 
23 
23 
23 


9 . 7 35o 7 
9 . 7 3537 
9 . 7 356 7 

9-73597 
9.73627 
9 . 7 365 7 

9 . 7 3 687 
9.73 717 
9.73 747 


0.26 493 
0.26 463 
0.26433 

o.264o3 
0.26 373 
0.26343 

0.26 3i3 
0.26283 
0.26 253 


9.94383 
9.94376 
9.94 369 

9.94 362 
9.94355 
9 . 94 349 

9.94342 
9.94335 
9.94 328 


2 9 

28 
27 

26 

25 

24 

23 
22 
21 


40 


9.68098 


23 


9.73777 


0.26 


223 


9 . 9 4 


321 


20 


4i 

42 

43 

44 
45 
46 

4? 
48 

49 


9.68 121 

9.68 i44 
9.68 167 

9.68 190 
9.68 21 3 

9.68 237 

9.68 260 

9.68283 

9.68 3o5 


23 

23 
23 
23 
23 
24 
23 
23 

22 


9.73807 
9 . 7 383 7 
9 . 7 3 867 

9-73897 
9.73927 
9.73957 

9.73987 
9-74oi7 
9.74 047 


0.26 193 
0.26 i63 
0.26 i33 

0.26 io3 
0.26073 
0.26 o43 

0. 26013 

0.25983 

0.25 953 


9.94 3i4 
9.94 3o7 
9.94 3oo 

9.94 293 
9.94 286 
9.94 279 

9.94 273 
9.94 266 
9.94 259 


*9 

18 

'7 
16 
i5 
i4 

i3 

12 
II 


50 


9.68 328 


2 3 

23 
23 
23 
23 
2 3 
23 
23 
23 
22 
23 


9.74077 


0.25 


923 


9.94 


252 


10 


5i 

52 

53 

54 
55 
56 

5? 
58 
5 9 


9.68 35i 
9.68374 
9.68397 

9.68 420 
9.68443 
9. 68 466 

9.68489 
9.68 5i2 
9.68 534 


9.74 107 
9 . 7 4 i3 7 
9.74 166 

9.74 196 
9.74 226 
9.74 256 

9.74 286 
9.74 3i6 
9-74345 


0.26 893 
0.25863 
0.25834 

o.25 8o4 
o.25 774 
o.25 744 

o.25 714 
o.25 684 
0.25655 


9 . 9 4 
9 . 9 4 
9.94 

9 . 9 4 
9-94 
9 . 9 4 

9 . 9 4 
9-94 
9,94 


245 

238 

23l 

224 
2I 7 
2IO 

203 

i 9 6 
i8 9 


9 

8 

7 
6 
5 
4 
3 

2 
I 


60 


9.68 55 7 


9 . 7 43 7 5 


0.25 


625 


9 . 9 4 


182 







L. Cos. 


d. 


L. 


Cotg. 


d. L. Tang. 


L. Sin. d. 


' 






61. 






PP 

.2 

3 
4 

:! 

9 


31 


30 29 


.1 

.2 

3 

4 

:! 
:i 

9 


24 


33 


22 




7 


6 


1:1 

9-3 
12.4 

;i:i 

21.7 
24.8 

27.4 


3-o 2.9 
o.o 5.8 
9.0 8.7 

12.0 II. 6 

15.0 14.5 

18.0 17.4 

21. 20.3 

24.0 23.2 
27.0 26.1 


2.4 

4 .8 

7-2 
9.6 

12.0 
14.4 

16.8 
19.2 


::i 

6. 9 

9.2 
11.5 

13.8 

16.1 
18.4 

20.7 


2.2 

4-4 
6.6 

8.8 

II. 

13-2 

5* 


.2 

3 
4 

:I 
:I 

__ 


0.7 
1.4 

2.1 
2.8 

3-5 
4.2 

4.9 
5-6 
6-3 


0.6 

1.2 

1.8 

2.4 

rs 

ti 

5-4 



87 



- 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. d. 







9 


.68 557 




9.74375 


3 
3 
3 
29 

3 
3 
29 

3 
3 
30 
29 
3<> 
30 
29 
30 
30 
29 

30 
29 
30 
29 
30 
3 
29 

3 
29 
3 
29 
3 
29 


o.25 625 


9.94 182 


7 
7 
7 
7 
7 
7 
7 
7 
7 
7 

7 
7 
8 

7 

7 
7 
7 
7 
7 


60 


I 

2 

3 

4 
5 
6 

8 
9 


9 

9 
9 

9 
9 
9 

9 

9 
9 


.68 58o 
.686o3 
.68625 

.68648 
.68671 
.68694 

.68 716 
.68 739 
.68 762 


23 

22 

23 
2 3 
23 
22 

23 
23 


9.74405 
9.74435 
9.74465 

9.74 494 
9.74 524 
9-74554 

9. 7 4 583 
9.74 6 1 3 
9-74643 


o.25 5g5 
o.25 565 
0.26 535 

o.25 5o6 
o.25 476 
o.25 446 

o.25 417 
o.25 387 

0.25 357 


9.94 175 

9.94 1 68 
9.94 161 

9.94 1 54 
9.94 147 
9.94 i4o 

9.94 i33 
9.94 126 
9.94 119 


5 9 
58 
5 7 
56 
55 
54 
53 

52 

5i 


10 


9 


.68 7 84 




9.74673 


0.25 327 


9.94 112 


50 


1 1 

12 

i3 

i4 
i5 
16 

17 

id 

'9 


9 
9 
9 

9 
9 
9 

9 
9 
9 


.68 807 
.68 829 
.68852 

.68875 
.68 897 
.68 920 

.68 942 
.68 965 

.68 987 


22 
23 
23 
22 
23 
22 

23 
22 


9.74 702 
9.74732 
9.74 762 

9.74 79 1 
9.74 821 
9.7485i 

9.74880 
9.74 910 
9.74939 


o.25 298 

0.25 268 

o.25 238 

o.25 209 
o.25 179 

0.2D l49 
O.25 I2O 

o.25 090 
o.25 061 


9.94 105 
9.94 098 
9.94 090 

9.94083 
9.94 076 
9.94 069 

9.94 062 
9.94 o55 
9.94 o48 


49 
48 

47 

46 

45 
44 

43 

42 

4i 


20 


9 


.69 oio 




9.74969 


o.25 o3i 


9.94 o4i 


7 


40 


21 
22 
23 

24 
25 

26 

27 
28 
29 


9.69 o32 
9.69 o55 
9.69077 

9.69 100 

9.69 122 

9.69 i44 

9.69 167 
9.69 189 
9.69212 


23 
22 

23 
22 
22 

23 
22 

23 


9.74998 
9.75 028 
9.75 o58 

9.76087 
9.75 117 
9.75 i46 

9 . 7 5 176 
9.75 2o5 

9.75235 


O.25 OO2 

0.24 972 

0.24 942 

0.24 9i3 
0.24883 
0.24854 

0.24824 
0.24795 

0.24 765 


9.94 o34 
9.94 027 
9.94 020 

9.94 OI2 

9.94 oo5 

9.93998 
9.93991 

9.93 984 
9.93 977 


7 
7 
8 
7 
7 
7 
7 
7 
7 


39 

38 

37 

36 
35 

34 

33 

32 

3i 


30 


9 


.69234 




9.75 264 


0.24 736 


9-9 3 970 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 


60 30 . 


PP 

.2 

3 
4 

:i 

Q 


30 


39 




23 


22 


.1 

.2 

3 
4 

:I 

J 


8 


7 


3- 
6.0 
9.0 

12.0 
15-0 

18.0 

21.0 
24.0 


11 

8. 7 

ii. 6 

i4-5 
17.4 

20.3 
23-2 


. i 

.2 

3 

4 

5 
.6 

7 
.8 
q 


:i 

6. 9 

9-2 
"5 
13.8 

16.1 
18.4 
20.7 


2.2 

4-4 
6.6 

8.8 

II. O 

13-2 

15-4 

17.6 

iq. 8 


0.8 
1.6 
2.4 

3-2 

4.0 
4.8 

5-6 
6.4 


0.7 
1.4 

2.1 
2.8 

3-5 
4-2 

4.9 
*' 



88 



29 30 . 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9 


69 234 


22 

23 
22 
22 
22 
2 3 
22 
22 
22 


9.76 264 


30 
29 

3 
29 
29 

3 
29 
3 
29 
29 

3 
29 
3 
29 
29 
30 
29 
29 
29 
3 
29 
29 
29 
3 
29 
29 

29 
30 
2 9 
29 


0.24 736 


9.93970 




30 


3i 

32 

33 

34 
35 
36 

37 
38 
3 9 


9 
9 
9 

9 
9 
9 

9 
9 
9 


69 256 
69 279 
69 3oi 

69 323 
69 345 
69 368 

69 Sgo 
69 412 
6 9 434 


9.75 294 

9 . 7 5 323 
9.75 353 

9 . 7 5382 
9.75 4n 
9.75441 

9.75470 
9 . 7 5 500 
9. 7 5 52 9 


0.24 706 
0.24 677 
0.24 647 

0.24 618 
0.24 589 
0.24 559 

0.24 53o 
0.24 5oo 
0.24 471 


9.93 963 
9 . 9 3 9 55 
9.93 948 

9.93 941 
9.93 934 
9 . 9 3 9 2 7 

9.93 920 
9 . 9 3 9 i2 
9 . 9 3 9 o5 


8 
7 
7 
7 
7 
7 
8 

7 

7 

7 
7 
8 

7 
7 
7 
8 
7 
7 
7 

7 

8 

7 
7 
8 
7 
7 
7 
8 
7 


29 

28 
27 

26 

25 
24 

23 
22 
21 


40 


9 


69456 




9 . 7 5558 


0.24 442 


9.93 898 


20 


4i 

42 

43 

44 
45 
46 

47 
48 

4 9 


9 
9 
9 

9 
9 
9 

9 
9 
9 


69479 
69 5oi 
69 5 2 3 

69 545 
69 567 
69 589 

69 61 1 
69 633 
69 655 


22 
22 
22 
22 
22 
22 
22 
22 


9 . 7 5 588 
9.75617 
9.75 647 

9.75676 
9.75 705 
9-75735 

9.75 7 6 4 
9 <7 5 793 
9.75 822 


0.24 4i2 
0.24383 
0.24353 

0.24 324 
0.24 295 
0.24 265 

0.24 236 
0.24 207 
0.24 178 


9.93 891 
9.93 884 
9.93 876 

9.93 869 
9.93 862 
9. 9 3855 

9-93847 
9.93 84o 
9 . 9 3 833 


'9 

18 

17 
16 
i5 
i4 
i3 

12 
II 


50 


9 


69677 




9.75 852 


0.24 i48 


9.93 826 


10 


5i 

52 

53 

54 
55 
56 

5 7 

58 

59 


9.69699 
9.69721 
9.69 743 

9.69 765 
9.69787 
9.69 809 

9.69 83i 
9.69853 
9.69875 


22 
22 
22 
22 
22 
22 
22 
22 


9.75 881 
9.75 910 
9 . 7 5 939 

9.75969 
9.75998 
9.76027 

9.76 o56 
9. 76 086 
9.76 115 


0.24 119 
0.24 090 
0.24 061 

0.24 o3i 
0.24 002 
o.23 973 

0.23 944 

o.23 914 
0.23885 


9.93 819 
9.93 81 1 
9.93 8o4 

9.93 797 
9 . 9 3 789 
9 . 9 3 782 

9-9 3 775 
9 . 9 3 768 
9.93 760 


9 

8 

7 
6 
5 
4 
3 

2 
I 


60 


9 


.69897 




9.76 i44 


0.23856 


9.93 753 







L. Cos. d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. d. 


' 




6O. 






PP 

2 

3 

4 

:I 


30 


29 




23 


22 


.i 

.2 

3 
4 

:I 

7 

.8 


8 


7 


3- 
6.0 
9.0 

I2.O 
15-0 

18.0 

21.0 
24.0 


I* 
8. 7 

ii. 6 
M-5 
17-4 

20.3 


.1 

.2 

3 
4 

:i 
:1 

9 


ii 

6. 9 

9 .2 

SI 

16.1 
18.4 

20.7 


2.2 

4-4 
6.6 

8.8 

II. O 

13.2 

J5-4 
17.6 
19.8 


0.8 
1.6 

a -4 

3-2 

4 'R 
4.8 

5-6 
6-4 

7.2 


0.7 
i-4 

2,1 

2.8 

3-5 

4-2 

4-9 

I' 6 

6-3 



89 



3O. 



' 


L.Sin 




d. 


L. Tang. 


d. 


L 


Cotg. 


L.Cos. |d. 







9.69 897 




9.76 


1 44 




o. 


23856 


9 . 9 3 


7 53 




60 


I 


9.69919 


22 


9.76 


178 


29 


o. 


23 82 7 


9 . 9 3 


7 46 


a 


5 9 


2 


9.69 941 




9.76 


202 




0. 


2 3 79 8 


9-9 3 


7 38 




58 


3 


9.69 963 


21 


9-76 


23l 


30 


0. 


2 3 7 6 9 


9 . 9 3 


7 3i 


7 
7 


57 


4 


9.69 984 


22 


9-76 


261 


29 


o. 


2 3 7 3 9 


9 . 9 3 


724 


7 


56 


5 


9.70 006 




9 . 7 6 


2 9 O 




0. 


23 7 IO 


9 . 9 3 


7 i 7 




55 


6 


9.70 028 




9.76 


3i 9 




o. 


2368i 


9 . 9 3 


79 




54 








22 






29 










7 




7 


9. 70 050 




9.76 


348 


29 


o. 


23652 


9 . 9 3 


702 




53 


8 


9.70 072 




9.76 


3 77 




o. 


23623 


9 . 9 3 


6 9 5 




52 


9 


9. 7 o 093 




9.76 4o6 


29 


o. 


235 9 4 


9 . 9 3 


68 7 




5i 


10 


9. 7 o n5 




9.76435 




0. 


23 565 


9 . 9 3 


680 




50 


ii 


9. 7 o 1 3 


7 




9-76464 


29 


o. 


2 3 536 


9.93 


6 7 3 


8 


49 


12 


9.70 i5g 




9.76 


493 




0. 


23 507 


9 . 9 3 665 




48 


i3 


9.70 180 




9.76 


522 


29 


o. 


2 34 7 8 


9 . 9 3 658 


7 


47 


i4 


9.70 202 


22 


9.76 


55i 


29 


0. 


23449 


9.93 65o 


8 


46 


i5 


9 .70 224 


22 


9.76 


58o 




o. 


23 42O 


9 . 9 3 643 




45 


16 


9.70 245 


21 


9.76 


609 


29 


o. 


23 Sgi 


9 . 9 3636 


7 


44 


17 


9.70 267 


22 


9.76 639 


3 


o. 


2 336i 


9 . 9 3 


628 




43 


18 


9.70 288 




9.76 


668 




o. 


23332 


9 . 9 3 


621 




42 


X 9 


9.70 3io 


22 


9.76697 


29 


0. 


2 3 3o3 


9-9 3 


6i4 


7 

g 


4i 


20 


9.70 332 


22 


9.76 


7 25 




0. 


23275 


9 . 9 3 606 




40 


21 


9.70 353 




9.76 


7 54 


29 


o. 


23 246 


9 . 9 3 


5 99 


7 

8 


39 


22 


9.70 3 7 5 




9-76 


7 83 




0. 


23 217 


9 . 9 3 


5oi 




38 


23 


9.70 396 




9-76 


812 


29 


o. 


23 188 


9-93 


584 


7 


37 








22 






29 










7 




24 


9.70 4i8 




9-76 


84 1 


29 


o. 


23 i5g 


9-93 


5 77 


8 


36 


25 


9 .7o43 9 




9-76 


8 7 o 




o. 


23 i3o 


9-93 


56 9 




35 


26 


9. 7 o 46i 




9.76899 




o. 


23 IOI 


9 . 9 3 


562 


7 


34 








21 






29 










K 




27 


9.70 482 




9 <7 6 928 


29 


o. 


23 072 


9 . 9 3 


554 




33 


28 


9. 7 o 5o4 




9.76 


9 5 7 




o. 


23o43 


9 . 9 3 


54 7 




32 


29 


9 . 7 o 525 




9.76 9 86 


29 


o. 


23 Ol4 


9 . 9 3 


53 9 




3i 


30 


9 . 7 o 54 7 




.9-77 


015 




0. 


22 985 


9 . 9 3 


53 2 


7 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. 


d. 


' 






59 3O . 




PP 30 


29 


28 


22 


21 




8 7 


i 3-0 


2.9 


2.8 


.1 2.2 


2.1 


.1 


0. 8 0. 7 


.2 6.0 


5-8 


5-6 


.2 4.4 


4.2 


.2 


1.6 1.4 


3 9-o 


8.7 


8.4 


.3 6.6 


6.3 


3 


2. 4 2.1 


4 12.0 


n.6 


II. 2 


.4 8.8 


8.4 


4 


3.2 2.8 


5 15-0 


14-5 


14.0 


5 ii-o 


10.5 


5 


4- 3-5 


.6 18.0 


17.4 


16.8 


.6 13.2 


12.6 


.6 


4.8 4.2 


.7 210 


20.3 


19.6 


7 15-4 


14-7 


7 


5.6 4.9 


.8 24.0 


23.2 


22.4 


.8 17.6 


16.8 


.8 


6.4 5.6 


9 27.0 




.9 19.8 


18.9 


.9 





9 



30 30 



t 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 

3i 

32 

33 

34 
35 
36 

37 

38 

39 


9- 


70547 




9-77 015 


29 
29 
28 
29 
29 
29 
29 
29 
28 
29 

29 
29 
29 
28 
29 
29 
29 
28 
29 
29 
28 
29 
29 
29 
28 
29 
28 
29 
29 
28 


0.22 985 


9.93 532 




30 


9- 
9- 

9- 

9- 

9- 
9- 

9- 
9- 
9- 


70 568 
70 5go 
70 61 1 

70 633 
70654 
70 675 

70697 
70 718 
70 7 3 9 


22 
21 
22 
21 
21 
22 
21 
21 


9.77 o44 
9.77073 
9.77 101 

9.77 i3o 
9.77 i5 9 
9.77 1 88 

9.77217 
9.77 246 
9-77 274 


O. 22 9 56 
0.22 927 
O.22 899 

0.22 870 
O.22 84l 
O.22 8l2 

0.22 783 
0.22 754 
O.22 726 


9.93525 
9.93517 
9.93 5io 

9.93 5o2 

9.93495 
9.93487 

9.93480 
9.93 472 
9.93 465 


8 

7 
8 

7 
8 

7 
8 

7 


29 

28 
27 

26 

25 

24 

23 
22 
21 


40 


9- 


70 761 




9 . 77 3o3 


O.22 697 


9.93457 


7 
8 
7 
8 

7 
8 

7 
8 

7 
8 

7 
8 

7 
8 
8 
7 
8 

7 
8 

7 


20 


4i 

42 

43 

44 
45 
46 

47 
48 

49 


9- 
9- 
9 

9- 

9 
9 

9 

9 
9 


70 782 
70 8o3 
70824 

70 846 
70 867 
70 888 

70909 
70 9 3i 
70 952 


21 
21 
22 
21 
21 
21 
22 
21 


9. 77 332 
9.77 36i 
9.77 390 

9.77418 
9.77447 
9.77476 

9.77 505 
9.77533 
9.77 562 


O.22 668 
O.22 639 
0.22 6lO 

O.22 582 

0.22 553 

0.22 524 

O.22 4g5 
0.22 467 
0.22 438 


9.93450 
9.93 442 
9.93435 

9.93427 
9.93 420 
9.93 412 

9.93405 
9.93 397 

9.93 390 


1 8 

16 
i5 
i4 
i3 

12 
II 


50 


9 


70973 


21 

21 
21 
22 
21 
21 
21 
21 
21 
21 


9-77 59i 


0.22 409 


9.93 382 


10 


5i 

52 

53 

54 
55 
56 

57 
58 
5 9 


9 
9 
9 

9 

9 

9 
9 
9 


70994 
71 oi5 
71 o36 

71 o58 
71 079 
71 100 

71 121 

71 142 
71 i63 


9-77 619 
9.77 648 
9.77 677 

9.77 706 
9.77 7 34 
9.77763 

9.77 791 
9.77820 
9.77 849 


O.22 38l 
0.22 352 
0.22 323 

O.22 294 
O.22 266 
0.22 237 

O.22 209 
O.22 l8o 
O.22 l5l 


9.93375 
9 . 9 336 7 
9.93 36o 

9.93 352 
9.93344 
9.93337 

9. 9 332 9 

9.93 322 

9 . 9 3 3i4 


9 

8 

7 
6 
5 
4 

3 

2 

I 


60 


9 


.71 1 84 


9.77 877 


0.22 123 


9- 9 3 3o 7 







L. Cos. 


d. 


L. Cotg. d. 


L. Tang. 


L. 


Sin. 


d. 








59. 




PP 

.1 

2 

3 

4 

7 
.8 
9 


39 


28 


22 


21 


.1 

2 

3 
4 

7 
.8 

9 


8 7 


2.9 

5-8 

8.7 

ii. 6 
"4-5 
17.4 

20.3 
23.2 
26.1 


2.8 .1 

5-6 
8.4 .3 

II. 2 4 

14.0 .5 
16.8 .6 

19.6 .7 

22.4 

25.2 .9 


2.2 

4-4 
6.6 

8.8 

II. O 

13-2 

\7.6 

19.8 


2. I 
4-2 
6-3 

8.4 
10.5 

12.6 

14.7 
16.8 
18.9 


o. 8 o. 7 
1.6 1.4 

2 4 2.1 

3.2 2.8 
4-o 3-5 
4.8 4.2 

5-6 4-9 
6.4 5.6 



31. 



/ 


L. Sin. d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 







9.71 1 84 


9.77877 




O.22 123 


9.93 307 




60 


I 




.71 205 


9.77906 


29 


0.22 094 


9.93 299 


g 


5p 


2 


9.71 226 




9-77 9 3 5 


28 


0.22 o65 


9.93 291 




58 


3 


9.71 247 




9.77963 




0.22 037 


9.93 284 


' 


57 














29 










K 




4 


9.71 268 


21 


9.77992 


28 


0.22 008 


9.93 276 




56 


5 


9.71 289 




9.78 020 




O.2I 980 


9.93 269 




55 


6 


9.71 3io 


21 


9.78 049 


2 8 


O.2I 95 I 


9.93 261 


8 


54 


7 


9.71 33i 


21 


9.78077 




0.21 923 


9.93 253 




53 


8 


9.71 352 




9.78 106 




0.21 894 


9.93 246 




52 


9 


9.71 3 7 3 




9.78135 


29 
28 


0.21 865 


9.93 238 




5i 


10 


9 


.71 3 9 3 




9.78 i63 


20 


0.21 837 


9.93 23o 




50 


ii 


9 


.71 4i4 


21 


9.78 192 


28 


0.21 808 


9.93 223 


7 
g 


49 


12 


9 


.7i435 




9.78 220 




O.2I 780 


9.93 21 5 




48 


i3 


9 


.71 456 


21 


9.78 249 


28 


0.21 751 


9.93 207 


8 


47 


i4 


9 


.71 477 


21 


9.78277 


29 


O.2I 723 


9.93 2OO 


7 
g 


46 


i5 


9 


.71498 




9.78 3o6 


28 


0.21 694 


9.93 192 




45 


16 


9 


.71 5i 9 


2O 


9.78334 


29 


0.21 666 


9.93 184 




44 


17 


9 


.71 53 9 


21 


9.78 363 


28 


0.21 637 


9.93 177 


g 


43 


18 


9 


.71 56o 




9 . 7 83 9 i 


?8 


0.21 609 


9.93 169 




42 


'9 


9 


.71 58i 




9.78419 




0.21 58i 


9.93 161 




4i 


20 


9 


.71 602 




9.78448 


28 


O.2I 552 


9.93 i54 


7 


40 


- 21 


9.71 622 


21 


9.78476 


29 


O.2I 524 


9.93 i46 


g 


39 


22 


9 


.71 643 




9-78505 


28 


O.2I 495 


9.93 i38 




38 


23 


9 


.71 664 




9.78533 




O.2I 467 


9.93 i3i 


7 


37 








21 






29 










X 




24 


9 


.71685 




9.78 562 


28 


0.21 438 


9.93 123 


g 


36 


25 


9 


.71 705 




9.78 590 


28 


0.21 4 


to 


9.93 n5 




35 


26 


9 


.71 726 




9.78 618 




0.21 382 


9.93 108 


7 


34 








21 






29 










H 




27 


9 


.71 747 




9.78 647 


28 


0.21 353 


9.93 100 


g 


33 


28 


9.71 767 




9.78675 




O.2I 325 


9 . 93 092 




32 


29 


9.71 788 




9.78 704 


28 


O.2I 296 


9.93084 




3r 


30 


9 


.71 809 




9.78732 




O.2I 268 


9.93077 


7 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 


58 30 . 


PP 


29 


28 




21 


20 




8 


7 


, 


2.9 


2.8 


j 


2.1 


2.0 


.1 


0.8 


0.7 


2 


5.1 


5-6 


.2 


4.2 


4.0 


.2 


1.6 


1.4 


3 


8-7 


8.4 


3 


6.3 


6.0 


3 


2.4 


2. 1 


4 


ii. 6 


II. 2 


4 


8.4 


8.0 


4 


3 2 


2.8 


5 


*4* 5 


14.0 


5 


10.5 


10. 


5 


4.0 


3-5 


.6 


17-4 


1 6. 8 


.6 


12.6 


12.0 


.6 


4.8 


4.2 


7 


20.3 


19.6 


. 7 


14.7 


14.0 


.7 


5-6 


4.9 


.8 


23.2 


22.4 


.8 


16.8 


16.0 


.8 


6.4 


5-6 


9 






18.9 18.0 


9 


7.2 6.3 



92 



31 3O . 



, 


L. Sin. d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d. 




30 


9 


.71 809 




9.78732 


28 
29 
28 
28 
29 
28 
28 
29 
28 
28 

28 
29 
28 
28 
28 
29 
28 
28 
28 
28 

29 
28 
28 
28 
28 
28 
29 
28 
28 
28 


O.2I 268 


9.93077 


8 
8 
8 

7 
8 
8 
8 
8 

7 

8 

8 
8 

7 
8 
8 
8 
8 
8 

7 
g 


30 


3i 

32 

33 

34 
35 
36 

37 
38 

39 


ooo ooo ooo 


.71 829 
.71850 
.71 870 

.71 891 
.71 9 n 
.71932 

.71 952 
.71 97 3 

.71 99 4 


21 
20 
21 
2O 
21 
20 
21 
21 


9.78 760 
9.78789 
9.78817 

9.78845 
9.78874 
9.78 902 

9.78 930 
9.78959 
9.78987 


O.2I 24O 
O.2I 211 
0.21 183 

0.21 155 

0.21 126 
O.2I 098 

0.21 070 
O.2I o4l 
0.21 Ol3 


9.93 069 
9.93 061 
9.93 o53 

9.93 o46 

9.93 o3o 

9.93 022 
9.93 oi4 
9.93 007 


29 
28 
27 
26 

25 
24 

23 
22 
21 


40 


9 


.72 014 




9.79015 


O.2O 985 


9.92999 


20 


4i 

42 

43 

44 
45 
46 

4 7 
48 

49 


ooo ooo ooo 


.72 o34 
.72055 
.72 o 7 5 

.72 096 
.72 1 16 

. 7 2l3 7 

.72 157 
.72 177 
.72 198 


21 
20 
21 
2O 
21 
20 
20 
21 


9.79 o43 
9.79072 
9.79 100 

9.79 128 

9.79 i56 
9.79185 

9.79 2i3 
9.79 241 

9.79269 


O.2O 957 
O.2O 928 
O.2O 900 

O.2O 872 
O.2O 844 
O.2O 8l5 

0.20 787 
O.2O 759 
O.2O 781 


9.92 991 
9.92 983 
9.92976 

9.92 968 
9.92 960 
9.92 952 

9.92 944 
9.92 936 
9.92929 


18 
17 

i5 
i4 
i3 

12 
II 


50 


9 


.72 218 




9.79297 


0.20 703 


9.92 921 


8 
8 
8 
8 
8 

7 
8 
8 
8 
g 


10 

9 

8 

7 

6 

5 

4 

3 

2 
I 


5i 

52 

53 

54 
55 
56 

57 
58 
5 9 


9 

9 

9 

9 
9 
9 

9 

9 
9 


.72 238 
.72 259 
.72 279 

.72 299 
.72 32o 
.72 34o 

.72 36o 
.72 38i 
.72 4oi 


21 

20 
20 
21 
20 
20 
21 
20 


9.79 326 

9.79354 
9.79382 

9.79410 
9.79438 

9.79 466 

9.79495 
9.79 523 
9.79 55i 


O.2O 674 

0.20 646 
0.20 618 

o 20 5oo 

0.20 562 

0.20 534 
0.20 5o5 

O.2O 477 
O.2O 449 


9.92 913 
9.92 905 
9.92897 

9.92 889 
9.92 881 
9.92 8 7 4 

9.92 866 
9.92 858 
9.92 850 


60 


9 


.72 421 




9-79 5 79 


0.2O 421 


9.92 842 









L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


t 






58. 






PP 

.1 

.2 

3 

4 
5 
.6 

9 


29 


28 




21 


20 


2 

3 
4 

1 

9 


8 


7 


ii. 6 
14.5 
17.4 

20.3 
23.2 

26.1 


2.8 

8.4 

II. 2 

14.0 

16.8 

19.6 
22.4 
25.2 


.1 

.2 

3 
4 

J 

7 
.8 
9 


2.1 
4-2 
6-3 

8. 4 
10-5 
12.6 

14.7 

16.8 
18.9 


2.0 

4.0 

6.0 
8.0 

IO.O 
12.0 

14.0 
16.0 
18.0 


0.8 
1.6 
2.4 

3-2 

ts 


0.7 
1.4 

2.1 

2.8 

3-5 
4.2 

4.9 

y 



93 



32. 



, 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. d. 







9. 7 2 421 




9.79 5 79 


28 


O.2O 


421 


9-9 2 


842 

D 




60 


I 


9.72441 


20 


9-79 6 7 


28 


0.20 


3 9 3 


9.92 


834 


8 




5 9 


2 


9.72 46i 




9. 7 9635 




0.20 


365 


9.92 


826 


r> 




58 


3 


9.72 482 


2O 


9 . 79 663 


28 


0.2O 


33 7 


9.92 


818 


8 




5 7 


4 


9.72 602 


20 


9.79691 




O.2O 


3o 9 


9.92 


810 


7 




56 


5 


9.72 522 




9.79 719 


28 


0.20 28l 


9.92 


8o3 


| 




55 


6 


9.72 542 


20 


9.79 747 


29 


O.2O 


253 


9.92 


795 


8 




54 


7 


9.72 562 


2O 


9.79776 


28 


O.2O 


224 


9.92 


787 


8 




53 


8 


9.72 582 




9.79 8o4 




O.2O 


196 


9.92 


779 







52 


9 


9.72 602 




9.79 832 




0.20 


168 


9.92 


771 


8 




5i 


10 


9.72 622 




9.79 860 




O.2O l4o 


9.92 


7 63 







50 


1 1 


9.72 643 


2O 


9.79888 


28 


0.20 


112 


9.92 


7*5 


t 




49 


12 


9.72 663 




9.79916 




0.20 o84 


9.92 


747 






48 


i3 


9.72 683 




9.79944 


28 


O.2O 


o56 


9.92 








47 








20 








28 
















i4 


9.72 703 




9.79972 


28 


O.2O 


028 


9.92 


7 3i 


I 




46 


i5 


9.72 723 




9.80 ooo 




O.2O 


ooo 


9.92 


7 23 






45 


16 


9.72 743 




9.80 028 


28 


0.19 972 


9.92 


7 i5 






44 


17 


9.72763 


2O 


9.80 o56 


28 
28 


o. 19 


944 


9 . 9 2 


77 


I 




43 


18 


9.72783 




9.80084 




o. 19 


916 


9.92 


699 


f 




42 


'9 


9.72 8o3 




9.80 112 


28 


o. 19 


888 


9.92 


691 


5 




4i 


20 


9.72 823 




9.80 i4o 




o. 19 


860 


9.92 


683 






40 


21 


9.72843 




9.80 1 68 




o. 19 


832 


9.92 


6 7 5 


| 




3 9 


22 


9.72 863 




9.80 195 




o. 19 


805 


9.92 


66 7 






38 


23 


9.72 883 


19 


9.80 223 


28 


0.19 


777 


9.92 


65 9 


1 




37 


24 


9.72 902 




9.80 25i 


28 


o. 19 


74 9 


9.92 


65i 


| 




36 


25 


9.72 922 




9.80 279 




o. 19 


721 


9.92 


643 






35 


26 


9.72 942 


20 


9.80 307 


28 


0.19 693 


9.92 


635 


1 




34 


27 


9.72 962 




9. 80 335 


28 


o. 19 


665 


9.92 


62 7 


| 




33 


28 


9.72 982 




9. 80 363 




o. 19 


63 7 


9.92 


6i 9 






32 


2 9 


9.73 002 


20 


9.80 391 


28 


o. 19 


6o 9 


9.92 


611 






3i 


30 


9.73 022 




9.80 419 




o. 19 


58i 


9.92 


6o3 






30 




L. Cos. 


d. 


L. 


Cotg. d. 


L. Tang. 


L. Sin. 


d. 




57 30. 


PP 


29 


28 27 




21 


20 


19 




8 


7 


.1 


2.9 


2.8 2.7 


.1 


2.1 


2.0 


1.9 


.1 


0.8 


0.7 


.2 


5-8 


5-6 5-4 


.2 


4-2 


4.0 


3 


.2 


1.6 


1.4 


3 


8.7 


8.4 8.1 


3 


6.3 


6.0 


5-7 


3 


2.4 


2. I 


4 


n.6 


II. 2 10.8 


4 


8.4 ' 8.0 


7.6 


4 


3-2 


2.8 


5 


14.5 


14.0 13.5 


5 


10.5 10.0 


9-5 


. 5 


4.0 


3-5 


.6 


i7-4 


16.8 16.2 


.6 


12.6 


12.0 


11.4 


; -6 


4.8 


4-2 


7 


20.3 


19.6 18.9 


7 


14.7 


14.0 


3*3 


! . 7 


5-6 


4-9 


.8 




22.4 21.6 


.8 


16.8 


16.0 


15.2 


.8 


6.4 




.9 26.1 


25.2 24.3 




18.9 18.0 


17.1 


9 


7.2 





94 



32 3D 



- 


L. Sin. d. 


L. Tang. 


d. L. Cotg. 


L. Cos. 


d. 




30 


9 


73 022 


20 

20 
2O 
20 

9 
20 
2O 
20 

'9 


9.80419 


28 
27 
28 
28 
28 
28 
28 
28 
27 
28 

28 
28 
28 
27 
28 
28 
28 
27 
28 
28 

28 
27 
28 
28 
27 
28 
28 
27 
28 
28 


0.19 58i 


9 . 9 2 6o3 




30 


3i 

32 

33 

34 
35 
36 

37 
38 
3 9 


9 

9 
9 

9 
9 
9 

9 
9 
9 


73o4i 
73 061 
73 081 

73 101 

73 121 

7 3 i4o 

73 160 
73 180 
73 200 


9.80447 
9.80 4?4 
9.80 5o2 

9.8o53o 
9. 80 558 
9.80 586 

9.80 6i4 
9.80642 
9.80 669 


0.19553 
o. 19 526 
0.19498 

0.19470 
0.19 442 
o. 19 4i4 

0.19386 
0.19358 
0.19331 


9.92595 
9 . 9 2 58 7 
9-9 25 79 
9.92571 
9.92 563 
9.92555 

9.92 546 
9.92 538 
9.92 53o 


8 
8 
8 
8 
8 

9 
8 
8 
g 


29 

28 
27 

26 

25 
24 

23 
22 
21 


40 


9 


73 219 


9.80 697 


0.19303 


9.92 522 


8 

8 
8 
8 
8 

9 
8 
8 
8 
8 

8 
8 

9 
8 
8 
8 

8 
8 

9 
8 


20 


4i 

42 

43 

44 
45 
46 

47 
48 

49 


ON ON ON ON ON ON ON ON ON 


7 3 2 3 9 
7 3 2 5 9 

7 32 7 8 

7 32 9 8 
73 3i8 
7 333 7 

7 335 7 
7 33 77 
7 33 9 6 


20 

'9 
20 
20 

19 

20 

20 

9 

2O 

'9 

20 

20 

'9 
20 

2O 

19 
2O 


9.80725 
9.80 753 
9.80 781 

9.80808 
9.80 836 
9.80864 

9.80 892 
9.80 919 
9.80 947 


o. 19 275 
o. 19 247 
o. 19 219 

0.19 192 
0.19 1 64 
0.19 i 36 

0.19 108 
0.19081 
o. 19 o53 


9.92 5i4 
9.92 5o6 
9.92498 

9.92 490 
9.92 482 
9.92473 

9.92 465 
9.92457 
9.92 449 


18 

16 
i5 
i4 
i3 

12 
I I 


50 


9 


7 34i6 


9.80975 


O. 19 O25 


9.92441 


10 


5i 

52 

53 

54 
55 
56 

57 
58 
5 9 


9 
9 
9 

9 
9 
9 

9 
9 
9 


7 3435 
73455 
7 34 7 4 

73494 
7 35i3 
7 3533 

. 7 355 2 
. 7 35 7 2 
.73 591 


9.81 oo3 
9.81 o3o 
9.81 o58 

9.81 086 
9.81 ii3 
9.81 i4i 

9.81 169 
9.81 196 
9.81 224 


0.18997 
o. 1 8 970 
o.i 8 942 

0.18 9 i4 
0.18 887 
0.18 859 

0.18 83i 
o.i 8 8o4 
o. 18 776 


9.92433 
9.92425 
9.92 4i6 

9.92 4o8 
9.92 4oo 
9.92 392 

9.92 384 
9.92 376 
9.92 367 


9 

8 

7 

6 
5 
4 
3 

2 
I 


60 


9 


.73611 


9.81 252 


o.i 8 748 


9.92 359 







L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 


57. 






PP 

.2 

3 

4 
5 
.6 

9 


28 


27 




20 


19 


.2 

3 
4 

7 
.8 


9 8 


2.8 

5-6 

8.4 

II 2 
I 4 .0 

16.8 

19.6 
22.4 


2.7 .1 

5-4 -2 
8.1 .3 

10.8 .4 
ll'.l '.I 

18.9 .7 
21.6 .8 
2 4-3 -9 


2.O 
4-0 

6.0 
8.0 

10.0 
12. 

14.0 

16.0 

18.0 


1.9 

3-8 

5-7 

7-6 
9-5 
11.4 

5-2 
17.1 


o. o o. 8 
1.8 1.6 
2.7 2.4 

3.6 3.2 
4.5 4.0 

5-4 4-8 

6.3 5-6 
7.2 6.4 
8. i 7. 2 



33. 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d. 







9.73 611 




9.81 252 


27 


0.18 748 


9.92359 




60 


I 


9.73 63o 


20 


9.81 279 


28 


o. 18 721 


9.92 35i 


8 


5 9 


2 


9.73 650 




9.81 307 


28 


0.18 693 


9.92 343 




58 


3 


9.73 669 


20 


9.81 335 


27 


o.i8665 


9.92335 


9 


57 


4 


9.73 689 


19 


9.81 362 


28 


0.18 638 


9.92 326 


8 


56 


5 


9.73708 




9.81 390 


28 


o. 18 610 


9.92 3i8 




55 


6 


9.73 727 


I 9 


9.81 4i8 




0.18 582 


9.92 3io 




54 








20 






27 










8 




7 


9 


.73747 




9.81 445 


28 


0.18 555 


9.92 3o2 




53 


8 


9 


. 7 3 7 66 




9 81 473 




0.18 527 


9.92 293 




52 


9 


9 


. 7 3 7 85 


19 


9.81 5oo 


27 

28 


o. 18 500 


9.92 285 


g 


5i 


10 


9 


.73805 




9.81 628 


28 


o. 18 472 


9.92 277 


g 


50 


1 1 


9 


. 7 3824 


19 


9 .8i 556 


27 


o.i 8 444 


9.92 269 


Q 


49 


12 


9 


. 7 3 843 




9 .8i 583 




0.18 417 


9.92 260 




48 


i3 


9 


. 7 3863 


19 


9 .8i 61 


I 


27 


0.18 389 


9.92 252 


8 


47 


i4 


9 


.73882 


19 


9 .8i 638 


28 


0.18 362 


9.92 244 


Q 


46 


i5 


9 


.73901 




9 .8i 666 




0.18 334 


9.92 235 




45 


16 


9 


.73921 




9 .8i 6 9 3 


27 
28 


0.18 307 


9.92 227 


8 


44 


17 


9 


. 7 3 9 4o 


1 9 


9.81 721 


27 


o. 18 279 


9.92 219 


8 


43 


18 


9 


. 7 3 9 5 9 




9.81 748 


_Q 


O. l8 252 


9.92 211 




42 


19 


9 


. 7 3 97 8 




9.81 776 




o. 18 224 


9.92 2O2 


9 


4i 


20 


9 


-7 3 99 7 




9.81 8o3 




0.18 197 


9.92 194 


g 


40 


21 


9 


.74017 




9.81 83i 




o. 18 169 


9.92 186 


9 


3 9 


22 


9 


.74o36 




9.81 858 




0.18 142 


9.92 177 




38 


23 


9 


.74o55 


19 


9.81 886 




0.18 u4 


9.92 169 




3 7 


24 


9 


. 74 074 


19 


9.81 913 


27 
28 


0.18 087 


9.92 161 




36 


25 


9 


.74 o 9 3 




9.81 g4 


I 




o. 18 o5 9 


9.92 i52 




35 


26 


9 


. 7 4n3 




9.81 968 


27 

28 


0.18 o32 


9.92 1 44 


8 


34 


27 


9 


.74 1 32 


19 


9.81 996 




o. 1 8 oo4 


9.92 i36 


9 


33 


28 


9 


. 7 4i5i 




9.82 023 




0.17 977 


9.92 127 


g 


32 


29 


9 


74 17 


19 


9.82 o5i 




0.17 949 


9.92 119 


8 


3i 


30 


9 


74 189 




9.82 078 


27 


o. 17 922 


9.92 in 




30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 




56 30 . 


PP 


28 


27 




2O 


19 




9 


8 


.1 


2.8 


2.7 


.1 


2.0 


1.9 


i 


0.9 


0.8 


.2 


5-6 


5-4 


.2 


4.0 


3-8 


2 


1.8 


1.6 


3 


8.4 


8.1 


3 


6.0 


5-7 


3 


2.7 


2.4 


4 


II.2 


10.8 


4 


8.0 


7-6 


4 


3-6 


3.2 


5 


14.0 


*35 


5 


10.0 


9-5 


5 


4-5 


4.0 


.6 


16.8 


16.2 


.6 


12. 


KM 


6 


5.4 


4-8 


7 


19.6 


18.9 


7 


I 4 .0 


13-3 


7 


6.3 


5-6 


.8 


22.4 


21.6 


.8 


10. 


15-2 


8 


7-2 


6.4 


9 


25.2 24.3 


9 


1 8.0 17. I 


9 


8.1 7.2 



96 



33 3O . 





L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d. 




30 


9 


74 189 




9.82078 


28 


0.17 922 


9.92 in 






30 


3i 


9 


.74 208 




9.82 106 


27 


0.17 894 


9.92 1 02 




9 
8 


29 


32 


9 


.74227 




9.82 i33 


28 


0.17 867 


9.92 094 






28 


33 


9 


.74 246 


19 


9.82 161 


27 


0.17 839 


9.92 086 




9 


27 


34 


9 


.74265 


19 


9.82 188 


27 


0.17 812 


9.92077 




s 


26 


35 


9 


.74284 




9.82 2i5 


28 


0.17785 


9.92 069 






25 


36 


9 


. 7 43o3 


X 9 


9.82243 




0.17 7 5 7 


9.92 060 




9 


24 








19 






27 












8 




37 


9 


. 7 4322 


19 


9.82 270 


28 


0.17 780 


9.92 o52 




8 


23 


38 


9 


.74341 




9.82 298 




o. 17 702 


9.92 o44 






22 


3 9 


9 


7 436o 


19 


9.82 325 




0.17675 


9.92 o35 




9 

8 


21 


40 


9 


. 7 43 79 




9.82 352 


28 


0.17 648 


9.92 027 






20 


4i 


9 


. 7 43 9 8 




9.82 38o 




o. 17 620 


9.92 018 




8 


'9 


42 


9 


.744i7 




9.82 407 


_0 


0.175^ 


3 


9.92 oio 






18 


43 


9 


. 7 4436 


*9 


9.82435 




o. 17 565 


9.92 002 






'7 








jg 






27 












<) 




44 


9 


.74455 




9.82462 




0.17 538 


9.91 99 3 




8 


16 


45 


9 


.74474 




9.82489 


O o 


0.17 5u 


9.91 985 






i5 


46 


9.74493 


'9 


9.82 517 


27 


0.17483 


9.91 976 




9 


i4 


47 


9 


74 5i2 




9.82 544 




o. 17 456 


9.91 968 






i3 


48 


9 


7 453i 




9.82 671 






o. 17 429 


9.91 959 




(i 


12 


49 


9 


74549 




9.82 599 




o. 17 4oi 


9.91 951 






I I 


50 


9 


74568 


19 


9.82 626 




0.17 3 7 4 


9.91 942 






10 


5i 


9 


7458 7 


19 


9.82 653 


27 
28 


0.17 347 


9.91 934 






9 


52 


9 


74 606 


*9 


9.82 681 




0.17 319 


9.91 9 25 






8 


53 


9 


74625 


'9 


9.82 708 


27 


0.17 292 


9.91 917 






7 


54 


9-74644 


'9 


9.82 735 


2 7 


0.17 265 


9.91 908 




9 
8 


6 


55 


9 


74662 




9.82 762 




0.17 238 


9.91 900 






b 


56 


9 


.74681 


19 


9.82 790 




O.I7 2IO 


9.91 891 




9 


4 


57 


9 


.74 700 


i9 


9.82 817 


27 


0.17 i83 


9.91 883 







3 


58 


9 


74 7*9 


J 9 


9.82 844 




o. 17 i56 


9.91 874 






2 


5 9 


9 


.74737 


18 


9.82 871 


27 


o. 17 129 


9.91 866 






I 


60 


9 


. 7 4 7 56 


19 


9.82 899 




0.17 101 


9.91 85 7 




9 







L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. d. 


' 






56. 








PP 


28 


27 




19 


18 




9 


8 


.1 


2.8 


2.7 


.1 


1.9 


1.8 


.1 


0.9 


0.8 


.2 


5.6 


5-4 


.2 


3.8 


3-6 


.2 


1.8 


1.6 


3 


8.4 


8.1 


3 


5-7 


5-4 


3 


2.7 


2.4 


4 


II. 2 


10.8 


4 


7.6 


7.2 


4 


3-6 


3- 2 


, e 


14.0 


t3-5 


5 


9-5 


9.0 


,c 


4-5 


4.0 


.6 


16.8 


16.2 


.6 


11.4 


10.8 


.6 


5-4 


4.8 


7 


19.6 


18.9 


7 


13-3 


12.6 


7 


6-3 


5-6 


.8 


22.4 


21.6 


.8 


15-2 


14.4 


.8 


7.2 


6.4 




25.2 24.3 




17.1 16.2 


9 


8.1 


7.2 



97 



, 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d. 







9 


74756 




9.82 899 




o. 17 101 


9.91 85 7 


g 


60 


! 


9 


74775 


19 


9.82 926 


27 


0.17 074 


9.91 849 




5 9 


2 


9 


.74794 


18 


9.82 953 




0.17 047 


9.91 84o 


g 


58 


3 


9 


.74812 


19 


9.82 980 


28 


0.17 020 


9.91 832 


q 


^>7 


4 


9 


. 7 483i 


19 


9.83 008 


27 


o. 16 992 


9.91 823 


8 


56 


5 


9 


.74850 


18 


9 .83o3< 






o. 16 965 


9.91 815 




55 


6 


9 


, 7 4868 




9.83 062 




0.16 938 


9.91 806 


9 


54 








19 






27 










8 




7 


9 


7 488 7 


19 


9.83 089 


28 


0.16 911 


9.91 798 




53 


8 


9 


.74 906 


18 


9-83 117 




o.i6883 


9.91 789 


g 


52 


9 


9 


74 924 




9.83 1 44 


27 


o.i6856 


9.91 781 




5i 


10 


9 


.74943 


18 


9.83 171 




0.16 829 


9.91 772 




50 


1 1 


9 


.74 961 




9.83 198 


27 


o. 16 802 


9.91 763 


8 


49 


12 


9 


.74 980 




9.83 225 




0.16 775 


9.91 755 




48 


i3 


9 


.74999 


18 


9. 83 252 


28 


0.16 748 


9.91 746 


9 

8 


47 


i4 


9 


. 7 5 017 




9-83 280 


27 


o. 16 720 


9.91 738 




46 


i5 


9 


. 7 5o36 


18 


9.83 307 




0.16 693 


9.91 729 




45 


16 


9 


. 7 5o54 




9.83334 


27 


o. 16 666 


9.91 720 


9 
8 


44 


l l 


9 


. 7 5 o 7 3 


18 


9.83 36i 


27 


0.16^39 


9.91 712 




43 


18 


9 


.75 091 




9. 83 388 




o. 16 612 


9.91 703 




42 


J 9 


9 


.75 1 10 


18 


9 .834i5 




0.16 585 


9.91 695 




4i 


20 


9.75 128 




9-83 442 


28 


o.i6558 


9.91 686 


9 


40 


21 


9 


.75 i47 


18 


9.83470 




0.16 53o 


9.91 677 


g 


39 


22 


9 


.75 i65 




9-83497 




0.16 5o3 


9.91 669 




38 


23 


9 


. 7 5 1 84 


r 9 

18 


9.83 524 


27 
27 


0.16476 


9.91 660 


y 

9 


37 


24 


9 


.75 202 




9 .8355i 






o. 1 6 449 


9.91 65i 


g 


36 


25 


9 


.75 221 




9 .835 7 8 




0.16 422 


9.91 643 




35 


26 


9 


.75 23g 




9.836o5 


27 


0.16395 


9.91 634 


y 


34 


27 




. 7 5 258 


'9 
18 


9. 83 632 


27 


o.i6368 


9.91 625 


9 

g 


33 


28 


9 


.75 276 




9 .8365 9 




0.16 34i 


9.91 617 




32 


29 


9 


.75 294 




9. 83 686 


27 


0.16 3i4 


9.91 608 


y 


3i 


30 


9 


.76 3i3 


I 9 


9.83 713 


27 


0.16 287 


9.91 5 99 


y 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 


55 30 . 


PP 


23 


27 




19 


18 




9 


8 


.1 


2.8 


2.7 


., 


1.0 


1.8 


., 


0.9 


0.8 


.2 


5-6 


5-4 


.2 


3-8 


3-6 


.2 


1.8 


1.6 


3 


8.4 


8.1 


3 


5-7 


5-4 


3 


2.7 


2.4 


4 


II. 2 


10.8 


4 


7-6 


7.2 


4 


3-6 


3-2 


1 


14.0 

16.8 


'I' 5 

16.2 


5 
.6 


9-5 
11.4 


9.0 

10.8 


:i 


4-5 
5-4 


4.0 
4.8 


7 


19.6 


i8.q 


7 


13-3 


12.6 


. 7 


6-3 


5-6 


.8 


22.4 


21.6 


.8 


15-2 


14.4 


. 8 


7.2 


6.4 





25.2 24.3 




17.1 16.2 




8.1 7.2 



98 



34 30' 



! 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9 


.75 3i3 


18 


9 .83 


7 i3 


27 
28 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
27 

27 

26 
27 
27 
27 

27 
27 
27 
27 
27 
27 


0. 


i628 7 


9.91 


5 99 




30 


3i 

32 

33 

34 
35 
36 

37 
38 
3 9 


9 
9 

9 

9 
9 

9 

9 
9 
9 


. 7 533i 
. 7 535o 
. 7 5 368 

. 7 5386 

. 7 544i 
. 7 545 9 


18 

18 

'9 
18 

18 
18 

19 

18 

18 

if 

18 
if 
if 

18 
18 
if 

18 
if 

if 

18 
if 
f 

18 
if 

18 


9 .83 740 
9 .83 768 
9. 83 795 

9-83 822 
9.83 849 
9.83 876 

9.83 903 
9.83 930 
9 .83 9 5 7 


0. 
0. 

o. 

o. 
o. 

0. 

o. 
o. 
o. 


16 260 

16 232 

16 2o5 

16 178 
16 i5i 
1 6 124 

16 097 
1 6 070 
i6o43 


9.91 
9.91 
9.91 

9.91 
9.91 
9.91 

9.91 
9.91 
9.91 


5 9 i 

582 
5 7 3 

565 
556 
54 7 
538 
53o 

521 


9 
9 
8 

9 
9 
9 
8 

9 


2 9 

28 
27 

26 

25 

24 

23 
22 
21 


40 


9.75496 


9 .83 


984 


o. 


16 016 


9.91 


5l2 


8 
9 
9 
9 
8 

9 

9 
'9 
9 
8 

9 
9 
9 
9 
8 

9 
9 
9 
9 

9 


20 


4i 

42 

43 

44 
45 
46 

47 

48 

49 


9 . 7 55i4 
9 . 7 5533 
9.7555i 

9. 75 569 
9 . 7 558 7 
9.75 6o5 

9.75 624 
9.75 642 
9.75 660 


9.84 01 1 
9 .84o38 
9.84 065 

9.84 092 
9.84 119 
9.84 i46 

9.84 i 7 3 
9.84 200 

9.84 22 7 


o. 

0. 
0. 

o. 

0. 
0. 

0. 
0. 

o. 


i5 989 
1 5 962 
i5 9 35 

1 5 908 
i588i 
1 5 854 

i5 827 
1 5 800 
i5 77 3 


9.91 
9 . 9 i 
9.91 

9.91 
9.91 
9.91 

9.91 
9.91 
9.91 


5o4 

495 

486 

477 
46 9 
46o 

45i 
442 
433 


19 
18 

17 

16 
i5 
i4 
i3 

12 
I I 


50 


9 


. 7 5 678 


9-84 


254 


o. 


i5 7 46 


9.91 


425 


10 


5i 

52 

53 

54 
55 
56 

5 7 
58 
5 9 


9.70 696 
9.75 714 
9 . 7 5 7 33 

9 . 7 5 7 5i 
9.75 769 
9.75787 

9.7^805 
9.75 823 
9.75 84 1 


9.84 280 
9.84 3o 7 
9.84334 

9.8436i 
9.84388 
9-844i 5 

9.84442 
9.84469 
9.84496 


0. 
0. 
0. 

0. 
0. 
0. 

o. 
o. 

0. 


1 5 720 
i56 9 3 
1 5 666 

i563 9 
i5 612 

i5558 
i553i 
i55o4 


9.91 
9.91 
9.91 

9.91 
9.91 
9.91 

9.91 
9.91 
9.91 


4i6 
407 
3 9 8 

38 9 
38i 

3 7 2 

363 
354 
345 


9 

8 

7 

6 
5 

4 
3 

2 

I 


60 


9 . 7 585 9 


9-84 


523 


0. 


i5 477 


9.91 


336 







L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. 


d. 








55. 






PP 

.2 

3 

4 

9 


28 


27 


26 


19 


if 


.2 

3 

4 


9 8 


2.8 

5-6 
8.4 

II. 2 
14.0 

1 6. 8 
19.6 

22-4 


2.7 

10.8 

II; 2 

18.9 

21.6 

24-3 


2.6 

10.4 
,3.0 

15-6 

18.2 

20.8 

2 3-4 


.1 1.9 

.2 3-8 
3 5-7 

4 7-6 
5 9-5 
.6 11.4 

7 J 3-3 
.8 15-2 

.q 17.1 


1.8 
3-6 

5-4 

7.2 
9.0 
10.8 

12.6 

14.4 

16.2 


0.9 0.8 
1.8 1.6 
2-7 2.4 

3-6 3-2 
4-5 4-o 
5-4 4-8 

6-3 5.6 
7.2 6.4 

8.1 7.2 



99 



35. 



/ 


L. Sin. d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d* 







9 


. 7 585 9 


18 


9.84 5-23 




o.i5 477 


9.91 336 




60 


I 


9 


758 77 


18 


9.84 550 


26 


o. i5 45o 


9.91 328 




5 9 


2 


9 


. 7 58 9 5 


18 


9.845 7 6 




o. i5 424 


9.91 3i 9 




58 


3 


9.75913 




9.846o3 




o.i 5 397 


9.91 3io 


9 


57 














27 










9 




4 


9 


. 7 5 9 3i 


18 


9.8463o 


27 


o. i5 3 7 o 


9.91 3oi 




56 


5 


9 


.75949 


18 


9.8465 7 




o.i5343 


9.91 292 




55 


6 


9 


7 5 9 6 7 


18 


9.84684 


27 


o.i5 3i6 


9.91 283 


y 

9 


54 


7 


9 


. 7 5 9 85 


18 


9.84 7 n 


27 


o.i5 289 


9.91 274 


8 


53 


8 


9 


.76 oo3 




9.84 7 38 




o.i5 262 


9.91 266 




52 


9 


9 


.76 O2I 




9-84 7 64 




o.i5 236 


9.91 2 5 7 


9 


5i 


10 


9 


.76039 


18 


9.84 7 9i 


27 


o.i 5 209 


9.91 248 


50 


ii 


9 


.76057 


18 


9.84818 


27 


o.i5 182 


9.91 239 


g 


49 


12 


9 


.76075 


18 


9.84845 




o.i5 i55 


9.91 23o 




48 


i3 


9 


.76 o 9 3 


18 


9.84872 


27 


o.i5 128 


9.91 221 


Q 


47 


i4 


9 


.76 in 


18 


9.84899 


26 


o. i5 101 


9.91 212 


9 


46 


i5 


9 


.76 I2 9 




9.84 925 




o.i5 o r 


75 


9.91 2o3 




45 


16 


9 


. 7 6l46 


18 


9.84 952 


27 


o . 1 5 o48 


9-91 194 


9 


44 


'7 


9 


.76164 


18 


9.84979 


27 


o. 1 5 02 1 


9.91 i85 


9 


43 


18 


9 


.76 l82 


18 


9.85 006 




0.14994 


9.91 176 




42 


19 


9 


.76 200 




9 .85o33 


26 


o. 1 4 967 


9.91 167 




4i 


20 


9 


.76 218 




9.85 059 




o . 1 4 94 1 


9.91 i58 




40 


21 


9 


.76 236 


17 


9.85 086 


27 


o. i4 914 


9.91 149 


8 


39 


22 


9 


.76253 




9 .85 n3 




0.14887 


9.91 i4i 




38 


23 


9 


.76271 


18 


9.85 i4o 


26 


o. i4 860 


9.91 i32 


9 


37 


24 


9 


.76 28 9 


18 


9.85 166 


27 


o.i4834 


9.91 123 


9 


36 


25 


9 


. 7 63o 7 




9.85 193 




o. i4 807 


9.91 n4 




35 


26 


9 


. 7 6 324 


18 


9.85 220 


27 


o. 14 780 


9.91 105 


9 


34 


27 


9 


. 7 6342 


18 


9.85 247 


26 


o.i4 7^ 


>3 


9.91 096 


9 


33 


28 


9 


. 7 636o 




9.85 273 




o.i4 727 


9.91 087 




32 


29 


9 


. 7 6 3 7 8 




9.85 3oo 




o. i4 700 


9.91 078 


9 


3i 


30 


9 


. 7 63 9 5 


I 7 


9. 85 32 7 




o. i4 673 


9.91 069 




30 




L. Cos. d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 


54 3O . 


PP 


27 


26 




18 


17 




9 8 


.1 


2-7 


2.6 


.1 


1.8 


r -7 


.1 


0.9 0.8 


.2 


5-4 


5-2 


.2 


3-6 


3-4 


.2 


1.8 1.6 


3 


8.1 


7.8 


3 


5-4 


5- 1 


3 


2.7 2.4 


4 


10.8 


10.4 


4 


7.2 


6.8 


4 


3-6 3-2 




[11 


13.0 

15-6 




9.0 
10.8 


8-5 

10.2 


5 
.6 


4-5 4- 
5-4 4-8 


7 


18.9 


18.2 


-7 


12.6 


ii. 9 


7 


6.3 5-6 


.8 


21.6 


20.8 


.8 


14.4 


13-6 


.8 


7.2 6.4 


-9 


24--? 2}. 4 


9 


16.2 15.3 





8.1 7.2 



35 3O 



r 


L. Sin. d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d. 




30 


9 


.76 3 9 5 




9.85 327 


27 
26 
27 
27 
26 
27 
27 
26 
27 
27 

26 
27 
27 
26 
27 
27 
26 
27 
27 
26 

27 
26 
27 
27 
26 
27 
26 
27 
27 
26 


o. 14 673 


9.91 069 


9 
9 
9 
9 
ro 

9 
9 
9 
9 


30 


3i 

32 

33 

34 
35 
36 

3? 
38 
3 9 


9 
9 
9 

9 
9 
9 

9 
9 
9 


. 7 64i3 
. 7 643i 
. 7 6448 

.76466 
.76 484 
.76 5oi 

.76 519 
. 7 653 7 
. 7 6554 


18 
17 

18 
18 

'7 
18 
18 
17 


9. 85 354 
9. 85 38o 
9.85407 

9.85434 
9.85 46o 
9. 85 48 7 

9.85 5i4 
9.85 54o 
9 .8556 7 


o. i4 646 
o. i4 620 
o. i4 5g3 

o.i4566 
o. i4 54o 
o.i4 5i3 

o.i4486 
0.14 46o 
o.i4433 


9.91 060 
9.91 o5i 

9.91 042 

9.91 o33 
9.91 oa3 
9.91 oi4 

9.91 oo5 
9.90996 
9.90987 


29 

28 
27 

26 

25 

24 

23 
22 
21 


40 


9 


. 7 65 7 2 




9.85 5 9 4 


o.i4 4o6 


9.90978 


9 
9 
9 
9 
9 
9 
9 
9 

10 

9 

9 
9 
9 
9 
9 

10 

9 
9 
9 
9 


20 


4i 

42 

43 

44 
45 
46 

47 
48 

49 


9.76 590 
9.76 607 
9.76 625 

9.76 642 
9.76 660 
9.76677 

9.76695 
9.76 712 
9.76 7 3o 


17 

18 

'7 
18 

17 

18 

7 

18 


9.85 620 
9.85 647 
9.85 674 

9-85 700 
9.85 727 
9 .85 7 54 

9-85 780 
9.85 807 
9 .85 834 


o.i4 38o 
o.i4353 
o.i4 326 

o. 1 4 3oo 
o.i4 273 
o.i4 246 

0. l4 220 

o.i4 193 
0.14 166 


9.90969 
9.90 960 
9. 9 o 9 5i 

9.90 942 
9.90 933 
9.90 924 

9.90915 
9.90 906 
9.90 896 


J 9 

18 

'7 

16 
i5 

i4 

i3 

12 
II 


50 


9 


.76 747 




9-85 860 


o. 1 4 i4o 


9.90 887 


10 


5i 

52 

53 

54 
55 
56 

5? 
58 
5 9 


9 
9 
9 

9 
9 
9 

9 
9 
9 


76765 
76 782 
76 800 

.76817 
.76835 
.76852 

.76 870 

76 887 
.76 904 


'7 

18 

17 

18 

17 

18 
'7 
17 




9.85 887 
9.85 913 
9-85 940 

9.85 967 
9-85 993 
9.86 020 

9.86 o46 
9.86 o 7 3 
9.86 joo 


o.i4 n3 
o. i4 087 
o. i4 060 

o. i4 o33 
o. i4 007 
o.i 3 980 

o.i3 954 
o. i3 927 
o. 1 3 900 


9.90 878 
9.90 869 
9.90 860 

9.90 85i 
9.90 842 
9.90 832 

9.90 823 
9.90 8i4 
9.90 805 


9 

8 

7 

6 

5 

4 

3 

2 
I 

~o~ 


60 


9 


.76 922 




9.86 126 


o.i 3 874 


9.90796 




L. Cos. 


d. 


L. Cotg. d. 


L. Tang. 


L. Sin. 


d. 


t 






54. 






PP 

.2 

3 
4 

:5 


27 


26 




18 


17 


.2 

3 
4 

:! 
:J 

-.9 


10 


9 


2.7 

5-4 
8.1 

10.8 

'3-5 
16.2 

18.9 

21.6 

a* T, 


2.6 
5-2 

7-8 

10.4 
13.0 
15-6 

18.2 

20.8 

23.4 


.2 

3 
4 

:i 
:i 

9 


1.8 
3-6 
5-4 

7.2 
9.0 
10.8 

12.6 

sy 


i-7 
3-4 
5-i 

6.8 
8-5 

IO.2 

II-9 
I 3 .6 

jia 


I.O 

2.0 

3-o 

4.0 

|- 
6.0 

7.0 
8.0 

9.0 


ti 

2.7 

3-6 
4-5 
5-4 

6-3 
tj 



101 



36 C 



1 


L.Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. i d. 







9 


.76922 




9.86 126 


27 
26 
27 
26 
27 
26 
27 
26 
27 
27 

26 
27 
26 
27 
26 
27 
26 
26 
27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
26 
27 


o.i3 874 


9.90 796 


9 

10 

9 
9 
9 
9 

10 

9 
9 
9 

10 

9 
9 
9 
ro 

9 
9 
9 

10 

9 


60 


I 

2 

3 

4 
5 
6 

8 
9 


9 
9 
9 

9 
9 
9 

9 
9 
9 


. 7 6 9 3 9 
.76 957 
.76974 

.76991 
.77009 
.77026 

77043 
.77 061 

.77078 


18 
7 

17 
18 

17 
*7 
18 

17 


9.86 i53 
9.86 179 
9.86 206 

9.86 232 
9.86 259 
9.86285 

9.86 3i2 
9.86338 
9.86 365 


o. i3 847 
o.i3 821 
o.i 3 794 

o.i3 768 
o.i3 7 4i 
o.i3 715 

o.i3688 
o.i3 662 
o,i3 635 


9.90787 
9.90777 

9 .90 768 

9.90759 
9.90750 
9.90 741 

9.90731 
9.90 722 
9.90713 


5 9 

58 
5 7 
56 
55 
54 

53 

52 

5i 


10 


9 


.77095 


J 7 

18 

*7 
7 
7 
18 

*7 
17 
*7 


9.86 392 


o.i 3 608 


9.90 704 


50 


ii 

12 

i3 

i4 
i5 
16 

'7 
18 

'9 


9 

9 
9 

9 
9 
9 

9 
9 

9 


77 112 

. 77 i3o 
.77147 

77 i64 
.77181 

77 199 
.77216 
.77233 
.77 25o 


9.86418 

9-86445 
9.86471 

9.86498 
9. 86 5 2 4 
9 .8655i 

9. 865 77 
9.86 6o3 
9.86 63o 


o.i3582 
o.i3555 
o.i3 529 

o.i3 5o2 
0.13476 
o . 1 3 449 

o.i3423 
o.i3 397 

o.i3 370 


9.90 694 
9.90 685 
9.90 676 

9.90 667 
9.90 657 
9.90 648 

9.90 639 
9.90 63o 
9.90 620 


49 

48 

47 
46 
45 
44 

43 

42 

4r 


20 


9 


.77 268 




9.86656 


o.i3 344 


9.90 61 1 


40 


21 
22 
23 

24 
25 

26 

27 
28 
29 


9 
9 
9 

9 
9 
9 

9 
9 
9 


.77285 

.77 3o2 

.77319 

.77336 
. 77 353 
. 77 3 7 o 

7738 7 
77405 

.77 422 


*7 
*7 
17 
*7 
*7 
V 
18 

7 
*7 


9 .86683 
9.86 709 
9.86 736 

9.86 762 
9.86 789 
9.86 8i5 

9.86842 
9.86868 
9.86 894 


o.i3 317 
o.i3 291 
o.i 3 264 

o.i3 2 38 

O.l3 211 

o.i3 185 

o.i3i58 
o.i3 i32 
o. 1 3 1 06 


9.90 602 
9.90 592 
9.90 583 

9.90 574 
9. 90 565 
9.90 555 

9.90 546 
9.90 537 
9.90 527 


10 
9 
9 
9 
10 

9 
9 
10 


3 9 
38 
37 
36 
35 
34 

33 

32 

3i 


30 


9 


7743 9 


9.86 921 


o. 1 3 079 


9.90 5i8 


y 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 




53 3O . 


PP 

.2 

3 
4 

:I 
:i 

9 


27 


26 




18 


17 


.1 

2 

3 
4 

:l 

7 
.8 

9 


10 


9 


2.7 

I:J 

10.8 

111 

18.9 

21.6 

24.3 


2.6 

5- 2 
7.8 

10.4 
13.0 
15.6 

18.2 

20.8 

23.4 


. i 

.2 

3 

4 
5 
.6 

i 

9 


1.8 
3-6 
5-4 

7.2 
9.0 
10.8 

12.6 

XJ 


i-7 
3-4 
5- 1 

6.8 
8-5 

10.2 

II.Q 

I 3 .6 
15.3 


I.O 
2.0 

3- 

4.0 

5- 
6.0 

7.0 
8.0 
9.0 


0.9 
1.8 
2.7 

3-6 
4-5 
5-4 

6-3 

g 



102 



30 30 . 



! 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9 


77 43 9 




9.86 921 


26 


o. 1 3 079 


9.90 5i8 




30 


3i 


9 


77456 


17 . 


9.86 947 


27 


o.i3o53 


9.90 509 




29' 


32 


9 


77 4 7 3 




9.86 974 


26 


o.i 3 026 


9.9 


0499 




28 


33 


9 


77490 


17 


9.87 ooo 




o. 1 3 ooo 


9.90490 


9 


27 








17 






27 










IO 




34 


9 


77 5o7 




9.87027 


26 


0.12 973 


9.90 48o 




26 


35 


9 


77 5 24 




9.87 o53 


26 


0.12 947 


9.90471 




25 


36 


9 


7754i 


17 


9.87079 




O. 12 921 


9.90 462 


9 


24 


37 


9 


77 558 




9.87 106 


27. 


O. 12 894 


9.90 452 


IO 


23- 


38 


9 


775 7 5 




9.87 i32 




26 


0,12 868 


9.90443 




22 


3 9 


9 


77 592 


17 


9.87 i58 




0.12 842 


9.90434 


9 


21 


40 


9 


77609 




9.87185 


26 


0.12 8i5 


9.90424 




20 


4i 

42 


9 
9 


77626 
77643 


17 
17 


9.87 211 

9.87 238 


27 


0.12 789 
0.12 762 


9.90415 
9.90 4o5 


y 

10 


'9 

18 


43 


9 


77 660 


^ 


9.87 264 


26 


o. 12 736 


9.90 396 


9 

10 


17 


44 


9 


.77677 




9.87 290 




0.12 710 


9.90 386 




16 


45 


9 


77694 


7 


9 .8 7 3i 7 




0.12 683 


9.90377 




i5 


46 


9 


.77711 


*7 


9.8 7 343 


26 


o. 12 657 


9.90 368 


y 

10 


i4 


47 


9 


77728 


16 


9.87 369 




o. 12 63i 


9-90358 




i3 


48 


9 


77744 




9.87 396 




o. 12 6o4 


9.90 349 




12 


49 


9 


77 761 


*7 


9.87 422 




O. 12 578 


9.90 339 




II 


50 


9 


77778 


17 


9. 87 448 




0. 12 552 


9.90 33o 


9 


10 


5i 


9 


77 79 5 


'7 


9.87475 


27 
26 


O. 12 525 


9 . 9 320 




9 


52 


9 


.77812 


17 


9.87 5oi 






O. 12 499 


9.90 3 1 1 




8 


53 


9 


.77829 


17 


9.87527 


26 


0.12473 


9.90 3oi 


10 


7 


54 


9 


. 77 846 


17 


9.87554 


27 
26 


0.12 446 


9.90 292 


9 


6 


55 


9 


.77862 




9.87 58o 




O. 12 42O 


9.90 282 




5 


56 


9 


.77879 


17 


9.87 606 




0.12 394 


9.90 273 


y 


4 


57 


9 


.77896 


17 


9.87633 


27 
26 


O. 12 367 


9.90 263 


10 


3 


58 


9 


. 779 i3 


*7 


9.87 659 




0.12 34l 


9.90 254 




2 


5 9 


9 


. 779 3o 


17 


9.87685 




0.12 315 


9.90 244 




I 


60 


9 


77946 


16 


9.87 711 




0. 12 289 


9.90 285 


9 







L. Cos. 


d. 


L. Cotg. 


d. L. Tang. 


L. 


Sin. 


d. 


' 




53. 






PP 


27 


26 


17 


,6 




IO 


9 


.1 


2-7 


2.6 .1 


i. 7 


1.6 


., 


1.0 


O.Q 


2 


5-4 


5-2 .2 


3-4 


3- 2 


.2 


2.0 


1.8 


3 


8.1 


7-8 -3 




4.8 


3 


3-o 


2.7 


4 


10.8 


10.4 .4 


6.8 


6.4 


4 


4.0 


3-6 


5 


13-5 


13.0 .5 


8-5 


8.0 






4-5 


6 


16.2 


15.6 .6 


IO.2 


9.6 


.6 


6.0 


5-4 


7 


18.9 


18.2 .7 


ii 9 


II. 2 


-7 


7.0 


6-3 ' 


.8 


21.6 


20.8 .8 


136 


12.8 


.8 


8.0 


7-2 


9 


24-3 23.4 .9 


15.3 14.4 


9 


9.0 





io3 



37 



, 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 







9 


77946 




9.87 711 




0.12 28 9 


9 . 9 0235 




60 


I 


9 


77963 


17 


9 .8 77 38 


26 


O. 12 262 


9 . 9 O 225 


9 


5 9 


2 


9 


77980 




9.87 764 




O. 12 236 


9 . 9 o 216 




58 


3 


9 


77997 


16 


9.87 790 


27 


0.12 210 


9 . 9 o 206 


9 


$7 


4 


9 


.78 oi3 


17 


9.87817 


26 


0.12 l83 


9 . 9 oi 97 


IO 


56 


5 


9 


.78 o3o 




9.87843 




O. 12 l5 7 


9.90 187 




55 


6 


9 


.78047 


1 7 

16 


9.87 869 


26 


0.12 l2 


tl 


9.90 178 


9 

10 


54 


7 


9 


.78 o63 


17 


9.87 895 




O. 12 105 


9.90 168 


9 


53 


8 
9 


9 
9 


.78 080 
.78097 


7 


9.87 922 
9 .87 9 48 


26 

26 


0.12 078 
O. 12 o52 


9 . 9 oi5 9 
9.90 i4 9 


10 


52 

5i 


10 


9 


.78 n3 




9-87 9 ?4 




O. 12 026 


9 . 9 o i3 9 




50 


1 1 


9 


.78 i3o 




9 .88 ooo 




O. 12 OOO 


9.90 i3o 


9 

IO 


4 9 


12 


9 


.78147 


16 


9 .88 027 




o.ii 9 73 


9.90 120 




48 


i3 


9 .78 i63 




9.88 o53 




o.ii 9.47 


9.90 in 


y 


47 














26 










IO 




i4 


9 


.78 180 


17 


9.88 079 


26 


o.ii 921 


9.90 101 




46 


i5 


9 


.78 197 


16 


9.88 io5 




o. 1 1 895 


9.90 091 




45 


16 


9 


.78213 




9.88 i3i 




26 


o.ii 869 


9.90 082 


y 


44 


"7 


9 


.78 23o 


16 


9.88 i58 


27 
26 


o.ii 842 


9.90 072 


10 


43 


18 


9 


.78246 




9.88 184 




o.ii 816 


9.90 o63 




42 


'9 


9 


.78263 




9.88 210 


26 


o.ii 790 


9.90 o53 




4i 


20 


9 


.78 280 


16 


9.88 236 




o.ii 764 


9.90 o43 




40 


21 


9 


.78 296 


17 


9.88 262 




o.ii 738 


9.90 o34 


y 


3 9 


22 


9 


.78 3i3 


16 


9.88 289 




o.ii 711 


9.90 024 




38 


23 


9 


. 7 832 9 




9.88 315 




o.ii 685 


9.90 oi4 




37 








17 






26 










9 




24 


9 


.78 346 


16 


9. 8834i 


26 


o.ii 6v 


>9 


9.90005 




36 


25 


9 


.78 362 




9.88 367 




o.ii 633 


9.89995 




35 


26 


9 


.78 379 




9.88 393 




o.ii 607 


9.89 985 




34 


27 


9 


. 7 83 9 5 


17 


9.88420 


27 
26 


o.ii 58o 


9.89976 


9 


33 


28 


9 


.78412 


16 


9.88446 




o.ii 554 


9.89 966 




32 


29 


9 


.78428 




9.88 472 




O.II 528 


9.8 


99 56 




3i 


30 


9 


.78445 




9.88 4 9 > 




O.II 502 


9.8 


994? 


y 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. d. 


' 


52 30 . 


PP 


27 


26 




17 


16 




IO 


9 


.! 


2.7 


2.6 


! 


i-7 


1.6 


.1 


1.0 


0.9 


2 


5-4 


5-2 


2 


3-4 


3-2 


.2 


2.O 


1.8 


3 


8.1 


7.8 


3 




4-8 


3 


3-o 


2-7 


4 


10.8 


10.4 


4 


6.8 


6.4 


4 


4.0 


3-6 


5 


3-5 


13.0 


5 


8-5 


8.0 


, e 


5-O 


45 


.6 


1 6. 2 


15.6 


6 


IO.2 


9.6 


.6 


6.0 


5-4 


7 


18.9 


18.2 


7 


II-9 


II. 2 


. 7 


7.0 


6-3 


.8 


21.6 


20.8 


8 


13-6 


12.8 


.8 


8.0 


z- 2 


9 


24-3 23.4 


9 


15.} 14.4 


9 


9.0 8.1 



104 



37 30 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9.73445 


16 


9.88 4 9 8 


26 


O. I I 5O2 


9.89947 




30 


3i 


9 


78461 




9.88 524 




26 


o.u 476 


9 .8 99 3 7 




29 


32 


9 


78478 




9.88 55o 




O. I I 45 


9.89927 




28 


33 


9 


78494 


16 


9 .885 77 




27 
26 


o. 1 1 423 


9.89 918 


y 

IO 


27 


34 


9 


78 5io 




9.88 6o3 




26 


o.u 397 


9.89 908 




26 


35 




78 5 27 




9.88 629 




o.u 371 


9 .8< 


? 898 




25 


36 


Q 


7 8543 




9.88655 






o.u 345 


9.8 


?888 


10 


24 


37 


9 


78 56o 


7 
16 


9.88 681 




26 
26 


o.u 3i 


9 


9.89879 


9 


23 


38 


9 


7 85 7 6 




9.88 707 




o.u 293 


9.89 869 




22 


3 9 


9 . 7 85 9 2 




9.88 7 33 


26 


o. 1 1 267 


9.89 859 


10 


21 


40 


9 


78 609 


16 


9 .88 75 9 




O.U 24 I 


9.89849 




20 


4i 


9 


.78625 


17 


9 .88 786 


26 


O.U 2l4 


9.89 84o 




'9 


42 


9 


.78 642 


16 


9 .88 812 


26 


o.u 188 


9.89 83o 




18 


43 


9 


78 658 


16 


9 . 88 838 


26 


o.u 162 


9.89 820 


IO 


17 


44 


9 


.78674 




9 . 88 864 


26 


o.u i36 


9.89 810 




16 


45 


9 


.78691 


16 


9 .888 9 o 




O.U 110 


9.89 801 




i5 


46 


9 


.78 707 




9 .88 9 i6 




o.u 084 


9.89791 




i4 








16 






26 










IO 




47 


9 


.78723 


16 


9 .88 9 42 


26 


o.u o58 


9.89 781 




i3 


48 


9 


.78739 




9 .88 9 68 




O.U 032 


9.89 771 




12 


4 9 


9 


.78 7 56 




9 .88 99 4 


26 
26 


o.u 006 


9.89 761 




I I 


50 


9 


.78772 




9 .8 9 020 


26 


o. 10 980 


9.89 752- 


y 


10 


5i 


9 


.78 788 


17 


9 .8 9 o46 


27 


o.io 954 


9.89 742 




9 


52 


9 


.78 805 




9 .8 9 073 


26 


o.io 927 


9.89732 




8 


53 


9 


.78821 


16 


9 .8 9 o 99 


26 


o. 10 901 


9.89 722 


10 


7 


54 


9 


. 7 883 7 


16 


9.89 125 


26 


o.io 875 


9.89 712 




6 


55 


9 


. 7 8853 




9.89 i5i 




26 


o.io 849 


9.89 702 




5 


56 


9.78869 


17 


9.89 177 


26 


0. 10 823 


9.8 


9 6 9 3 


9 

IO 


4 


57 


9.78 886 


16 


9.89 2o3 


26 


0.10797 


9.89 683 




3 


58 


9 


.78 902 




9.89 229 


26 


o.io 771 


9.89 673 




2 


5 9 


9 


.78 918 




9.89255 


26 


o. 10 745 


9.89 663 




I 


60 


9 


,78934 




9.89 281 




o.io 719 


9.8 


9 653 









L. Cos. d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 




52. 






PP 


27 


26 




17 


16 




10 


9 


.1 


2.7 


2.6 


.1 


i. 7 


1.6 


.1 


I 


0.9 


.2 


5-4 


5-2 


.2 


3-4 


3.2 


.2 


2 O 


1.8 


3 


8.1 


7.8 


3 




4.8 


3 


3 


2.7 


4 


10.8 


10.4 


4 


6.8 


6.4 


4 


40 


3-6 




13-5 


13.0 


5 


85 


8.0 


5 


5 


4-5 


.6 


16.2 


15-6 


.6 


10 2 


9.6 


.6 


6 o 


5-4 


.7 


18.9 


18.2 


7 


ii 9 


II. 2 


7 


7.0 


6-3 


.8 


21.6 


20.8 


.8 


136 


12.8 


.8 


8.0 


7.2 




24-3 2 3-4 -9 


*5 3 *4-4 


9 





io5 



38. 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 







9.78 934 


16 


9.89 


28l 


26 


0. 


10719 


9. 89 653 




60 


I 


9.78 <)5o 


17 


9.89 


3o 7 


26 


o. 


10 6 9 3 


9.89 


643 


5 9 


2 


9.78967 


16 


9.89 


333 


26 


o. 


IO 66 7 


9.89633 


58 


3 


9.78 983 


16 


9.89 


35 9 


26 


o. 


10 64 1 


9.89 624 


IO 


$7 


4 


9.78999 


16 


9.89 


385 


26 


o. 


10 615 


9.89 


6i4 


IO 


56 


5 


9.79015 


16 


9 .8 9 


4n 


26 


o. 


10 58 9 


9.89 


6o4 




55 


6 


9.79 o3i 


16 


9.89 


43 7 




o. 


10 563 


9.89 


5 9 4 


IO 


54 


7 


9.79047 


16 


9 .8 9 


463 


26 


0. 


io53 7 


9.89 


584 


x 





53 


8 


9.79 o63 




9 .8 9 


48 9 


26 


o. 


10 5n 


9.89 


5 7 4 




52 


9 


9.79079 


16 


9.89 


5i5 


26 


o. 


10485 


9.89 


564 




bi 


10 


9.79095 


16 


9.89 


54i 


26 


o. 


10 459 


9.89 


554 




50 


1 1 


9-79 " 


i 


17 


9.89 


56 7 


26 


o. 


io433 


9.89 


544 


IO 


49 


12 


9.79 128 


16 


9.89 


5 9 3 


26 


o. 


10 4o 7 


9.89 


534 




48 


i3 


9.79 i44 


16 


9.89 


619 


26 


0. 


10 38i 


9.89 


524 


IO 


47 


i4 


9.79 160 


16 


9.89 645 


26 


0. 


10 355 


9.89 


5i4 


IO 


46 


i5 


9-79 176 


16 


9.89 


671 




0. 


10 329 


9.89 


5o4 






45 


16 


9-79 *9 2 


16 


9.89 


697 


26 


0. 


10 3o3 


9.89 


495 


y 

10 


44 


"7 


9.79 208 


16 


9.89 


?2 3 


26 


o. 


I02 77 


9.89485 






43 


18 


9.79 224 


16 


9.89 


74 9 




o. 


10 25l 


9.89 


475 




42 


1 9 


9.79 240 


16 


9.89 


77 5 




o. 


10 225 


9.89 


465 




4i 


20 


9.79 256 


16 


9.89 


801 




o. 


10 199 


9.89 


45s 




40 


21 


9.79272 


16 


9.89 


827 


26 


o. 


10 i 7 3 


9.89 


445 


IO 


3 9 


22 


9.79 288 




9.89 


853 




o. 


10 147 


9.89435 




38 


23 


9.79 3o4 


15 


9.89 


879 


26 


o. 


IO 121 


9.89425 


IO 


37 


24 


9-79 3i 


9 


16 


9.89 


95 


26 


o. 


10 095 


9.89 


415 


IO 


36 


25 


9.79 335 




9.89931 




o. 


10 069 


9.89405 




35 


26 


9.79 35i 


16 


9.89957 


26 


o. 


10043 


9.89 


3 9 5 


10 


34 


27 


9-79367 


16 


9.89 983 


26 


o. 


10 017 


9.89 


385 


10 


33 


28 
29 


9. 79 383 
9-79 3 99 


16 
16 


9.90 009 
9.90 035 


26 


o. 
o. 


09991 

09 965 


9.89 
9 .8 9 


375 
364 


ii 


32 

3i 


30 


9.79415 




9.90 06 1 




0. 


09 9 3 9 


9 .8 9 


354 




30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. 


Tang. 


L. Sin. 


d. 


' 






51 30. 








pp 26 


17 


16 


15 


ii 




10 


9 


.1 2.6 


*-7 


1.6 


i i-5 


i.i 


.j 


I.O 


0.9 


.2 5 .2 


3-4 


3.2 


2 3.0 


2.2 


.2 


2.0 


1.8 


3 7-8 




4.8 


3 4-5 


3-3 


3 


3- 


2.7 


.4 10.4 


6.8 


6.4 


4 6.0 


4.4 


4 


4.0 


3-6 


5 13-0 
.6 15.6 


8.5 

10.2 


8.0 
9.6 


5 7-5 
6 9.0 


11 


.1 


5- 
6.0 


4-5 
5-4 


.7 18.2 


II. 9 


II. 2 


7 10.5 


7-7 


7 


7.0 


6-3 


.8 20.8 


13-6 


12.8 


.8 12.0 


8.8 


.8 


8.0 


7.2 


9 23-4 


15.3 14.4 


9 J 3-5 




9 


9.0 


8.1 



1 06 



38 3O' 



/ 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9-79415 


16 


9.90 061 




0.09 939 


9.89 354 




30 


3i 


9 


7943i 


16 


9.90 086 


26 


0.09 914 


9.8 


9344 




29 


32 


9 


.79447 


16 


9.90 112 


26 


0.09888 


9.89 334 




28 


33 


9 


.79 463 


15 


9.90 1 38 


26 


0.09 862 


9.89 324 


IO 


27 


34 


9 


.79478 


16 


9.90 164 


26 


0.09836 


9.8 


93i4 




26 


35 


9 


.79494 


16 


9.90 190 


26 


o . 09 8 1 o 


9.89 3o4 




25 


36 


9 


.79510 


16 


9.90 216 


26 


0.09 784 


9.89 294 


IO 


24 


37 


9 


.79 526 


16 


9.90 242 


26 


0.09 758 


9.89 284 




23 


38 


9 


.79 542 




9.90 268 


26 


0.09 732 


9.89 274 




22 


3 9 


9 


.79 558 




9.90 294 


26 


0.09 706 


9.89 264 




21 


40 


9 


. 79 5 7 3 


16 


9 . 9 320 


26 


0.09 680 


9.89 254 




20 


4i 


9.79589 


16 


9.90 346 


25 


0.09 654 


9.8 


9 244 




19 


42 


9 


.79605 


16 


9.90 371 




_ 


0.09 629 


9.89 233 




1 8 


43 


9 


.79 621 


15 


9.90397 


26 


0.09 6o3 


9.89 223 




17 


44 


9 


.79 636 


16 


9.90423 


26 


0.09577 


9.89 2i3 




16 


45 


9 


.79652 


16 


9.90449 




0.09 55 1 


9.89 2o3 




i5 


46 


9 


.79 668 


16 


9.90475 


26 


0.09 525 


9.89 193 


10 


i4 


47 


9 


.79684 


15 


9.90 5oi 


26 


o . 09 499 


9.89 i83 




i3 


48 


9 


.79699 




9.90 527 




0.09 473 


9.89173 




12 


49 


9 


.79 716 




9.90 553 




0.09447 


9.89 162 




I I 


50 


9 


79 7 3 ' 




9.90 578 


2 5 
26 


0.09422 


9.89 i52 




10 


5i 


9 


79 746 


16 


9.90 6o4 


26 


o . 09 396 


9.89 142 




9 


52 


9 


.79 762 




9.90 63o 


26 


0.09 370 


9.89 i3 2 




8 


53 


9 


79 77 s 


15 


9.90 656 


26 


0.09344 


9.89 122 


10 


7 


54 


9 


. 7979 3 


16 


9.90 682 


26 


0.09318 


9.89 112 


II 


6 


55 


9 


.79809 




9.90 708 


26 


0.09 292 


9.89 101 




5 


56 


9 


.79825 




9.90 734 




0.09 266 


9.89 091 




4 


57 


9 


.79 84o 


5 
16 


9-90759 


25 
26 


0.09 24 1 


9.89 081 


IO 


3 


58 


9 


.79856 




9.90785 




o . 09 2 1 5 


9.89 071 




2 


5 9 


9 


.79872 




9.90 811 




0.09 189 


9.89 060 




I 


60 


9 


.79887 




9.90 837 




0.09 1 63 


9 .8 


9 o5o 









L. Cos. 


d. 


L. Cotg 


. d. 


L. Tang. 


L. 


Sin. 


d. 








51. 






PP 


26 


25 




16 


'5 




ii 


IO 


.1 


2.6 


2-5 


.1 


1.6 


i. s 


., 


i.i 


1.0 


.2 


5-2 


5-o 


.2 


3-2 


3- 


.2 


2.2 


2.O 


3 


7.8 


7-5 


3 


4.8 


4-5 


3 


3-3 


3- 


4 


10.4 


IO.O 


4 


6.4 


6.0 


4 


4.4 


4.0 


j 


13.0 
15-6 


12.5 
15-0 


1 


8.0 
9.6 


7-5 
9.0 


:J 


Ii 


C 


7 


18.2 


17-5 


7 


II. 2 


10.5 


7 


7-7 


7.0 


.8 


20.8 


20. o 


.8 


12.8 


12. 


.8 


8.8 


8.0 




23.4 22.5 .9 


14.4 13.5 


9 


9.9 9.0 



107 



39. 



! 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. d. 







9 


.79887 


16 


9.90 837 


26 


0.09 i63 


9 . 89 o5o 




60 


i 


9 


.79903 


15 


9.90 863 


26 


0.09 1 37 


9.89 o4o 


IO 


5 9 


2 


9 


799' 8 


16 


9.90 889 




0.09 in 


9.89 o3o 




58 


3 


9 


.79934 


16 


9.90 914 


26 


0.09 086 


9.89 020 


ii 


D 7 


4 


9 


.79950 


15 


9.90 940 


26 


0.09 060 


9.89 009 


JO 


56 


5 


9 


.79965 


16 


9.90 96^ 


) 


26 


0.09 o34 


0.88999 




55 


6 


9 


.79981 


15 


9.90992 


26 


o . 09 008 


9.88989 


II 


54 


7 


9 


.79996 


16 


9.91 018 


25 


0.08 982 


9.8 


8978 


10 


53 


8 


9 


.80 012 




9.91 o4- 




26 


0.08 957 


9.88 968 




52 


9 


9 


.80 O27 


16 


9.91 069 


26 


0.08 9: 


h 


9.88 958 




5i 


10 


9 


.80043 


15 


9.91 ogS 


26 


0.08 905 


9.88948 




50 


ii 


9 


.8oo58 


16 


9.91 121 




26 


0.08 879 


9.88 937 




49 


12 


9 


.80 074 




9.91 147 




0.08 853 


9.88 927 




48 


i3 


9 


.80089 


16 


9.91 172 


26 


0.08 828 


9.88917 


II 


47 


i4 


9 


.80 105 


15 


9.91 198 


26 


0.08 802 


9.88 906 




46 


i5 


9 


.80 I2O 


16 


9.91 224 


26 


0.08 776 


9.88 896 




45 


16 


9 


.80 i36 


15 


9.91 250 


26 


0.08 750 


9.8 


8 886 


II 


44 


17 


9 


.80 i5i 


15 


9.91 276 


25 


0.08 724 


9.8 


8 875 




43 


18 


9 


.80 166 


16 


9.91 3oi 


26 


0.08 699 


9.88 865 




42 


19 


9 


.80 182 




9.91 32^ 


r 


26 


0.08 673 


9.88 85s 




4i 


20 


9 


.80 197 


16 


9.91 353 


26 


0.08 647 


9.88844 




40 


21 


9 


.80 2i3 


15 


9.91 379 


25 


0.08 621 


9.8 


8834 




3 9 


22 


9.80228 


16 


9.91 4o^ 


i 


26 


0.08 596 


9.88 824 




38 


23 


9 


.80 244 


15 


9.91 43o 


26 


0.08 570 


9.88 8i3 


IO 


37 


24 


9 


.80259 


15 


9.91 456 


26 


0.08 544 


9.88 8o3 




36 


25 


9 


.80274 




9.91 482 




0.08 5i8 


9.88 793 




35 


26 


9 


.80 290 




9.91 507 


25 


0.08 493 


9.88 782 




34 








15 






26 










IO 




27 


9 


.80 3o5 


15 


9.91 533 


26 


0.08 467 


9.88 772 




33 


28 


9 


.80 320 




9.91 55 9 




0.08 44 1 


9.88 761 




32 


2 9 


9 


.80 336 




9.91 58s 




0.08 4i5 


9.8 


8 7 5i 




3 1 


30 


9 


.8o35i 




9.91 610 


25 


0.08 390 


9.8 


74i 




30 




L. Cos. 


d. 


L. Cotg. | d. 


L. Tang. 


L. 


Sin. 


d. 




50 30 . 


PP 


26 


25 




16 


15 




ii 


10 


.1 


2.6 


2.5 


.1 


1.6 


'5 


.1 


i.i 


I.O 


>2 


5-2 


5.0 


.2 


3-2 


3- 


.2 


2.2 


2.O 


3 


7.8 


7-5 


3 


4.8 


4-5 


3 


3-3 


3- 


4 


10.4 


IO.O 


-4 


6.4 


6.0 


4 


4-4 


4.0 


.5 


13.0 


12.5 


5 


8.0 


7' 5 




5-5 


5o 


.6 


15.6 


15.0 


.6 


9.6 


9.0 


.6 


6.6 


6.0 


I 


18.2 

20.8 


17-5 

20. 


:l 


II. 2 
12.8 


10.5 

12.0 


:! 


11 


7.0 
8.0 


Q 


23.4 22.5 


9 


14.4 


13.5 







1 08 



39 3O . 






L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9 


.8o35i 




9.91 610 


26 
26 
26 
25 
26 
26 
26 
25 
26 
26 

25 
26 
26 
26 
25 
26 
26 
25 
26 
26 

25 
26 
26 

25 
26 
26 
25 
26 
26 
25 


0.08 390 


9.88 741 




30 


3i 

32 

33 

34 
35 
36 

3? 
38 
3 9 


9 
9 
9 

9 
9 
9 

9 
9 
9 


.80 366 
.80 382 
.80 397 

.80412 
.80428 
.8o443 

.80 458 
.8o473 
.80489 


16 
i5 
15 
16 

15 
i5 
15 
16 

15 


9.91 636 
9.91 662 
9.91 688 

9.91 713 
9.91 739 
9.91 765 

9.91 791 
9.91 816 
9.91 842 


0.08 364 
0.08 338 
0.08 3i2 

0.08 287 
0.08 261 
0.08 235 

0.08 209 
0.08 i84 
0.08 i58 


9.88 730 
9.88 720 
9.88 709 

9.88 699 
9.88688 
9.88678 

9.88668 
9 .8865 7 
9.88 647 


10 

II 

IO 

II 

10 
IO 

II 

10 


29 

28 
27 

26 

25 
24 

23 
22 
21 


40 


9 


80 5o4 


9.91 868 


0.08 i32 


9.88636 




20 


4i 

42 

43 

44 
45 
46 

4? 
48 

49 


9 
9 
9 

9 
9 
9 

9 
9 
9 


.80 519 
.80 534 
.80 550 

.80565 
.80 58o 
.80595 

.80 610 
.80625 
.8o64i 


15 
i5 

16 

15 

15 
15 
15 
IS 
16 
15 

i5 

15 

15 

5 
5 
15 

16 

15 
15 

IS 


9.91 893 
9.91919 
9.91 945 

9.91971 
9.91 996 
9.92 022 

9.92 o48 
9.92 073 
9.92099 


0.08 107 
0.08 081 
o.o8o55 

0.08 029 
o . 08 oo4 
0.07 978 

0.07 952 
0.07 927 
0.07 901 


9.88 626 
9.886i5 
9.88 605 

9.88 594 
9.88 584 
9.88673 

9.88563 
9.88 552 
9.88 542 


II 

10 

II 

IO 

II 

IO 

11 

10 

II 

IO 

II 
II 

IO 

II 

IO 

II 

10 

II 
II 


i9 

18 

!? 

16 
i5 
i4 
i3 

12 
II 


50 


9 


.80 656 


9.92 125 


0.07 875 


9.88 53i 


10 

9 

8 

7 

6 
5 

4 
3 

2 

I 


5i 

52 

53 

54 
55 
56 

5 7 
58 
5 9 


9.80671 
9.80686 
9.80 701 

9.80 716 
9.80 731 
9.80 746 

9.80 762 
9.80777 
9.80 792 


9.92 i5o 
9.92 176 
9.92 202 

9.92 227 
9.92 253 
9.92279 

9.92 3o4 
9.92 33o 
9.92 356 


0.07 850 
0.07 824 

0.07 798 

0.07 773 
0.07747 

0.07 721 

0.07 696 
0.07 670 
0.07 644 


9.88 521 

9.88 5io 
9.88499 

9.88489 
9.88478 
9.88468 

9.88457 
9-88447 
9.88436 


60 





.80807 


9.92 38i 


0.07 619 


9.88425 







L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 






5O. 






PP 

.2 

3 

4 
5 
.6 

7 
.8 


26 


25 


16 


15 


.1 

.2 

3 
4 

9 


II 10 


2.6 

5-2 

7.8 

10.4 

13.0 
156 

18.2 

20.8 

2 V4 


2.5 .1 

5.0 .2 

7-5 -3 

10.0 .4 
12-5 .5 
15.0 .6 

17-5 -7 

20.0 .8 

22.5 .9 


1.6 

4 3 :s 
6. 4 

8.0 
9.6 

II. 2 

12.8 

M-4 


1 5 
3- 
45 

6.0 
7-5 
9.0 

10.5 

12. 

13-5 


I.I 1.0 
2.2 2.0 

3-3 3-o 

4-4 4- 
5-5 5- 
6.6 6.0 

7-7 7-o 
8.8 8.0 

9.9 9.0 



109 



40< 



/ 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. d. 







9 


.80807 


15 
15 
15 
15 
15 
15 
15 
15 
15 


9.92 38i 


26 
26 

25 
26 
26 
25 
26 
26 
25 
26 

25 
26 
26 

25 
26 
26 
25 
26 
25 
26 

26 
25 
26 
25 
26 
26 

25 
26 

25 
26 


0.07 619 


9.88425 


10 
ii 

10 

II 
II 

10 

II 
II 

10 

II 
II 

10 

II 
II 

10 

II 
II 

10 

II 


60 


I 

2 

3 

4 
5 
6 

8 
9 


9 

9 
9 

9 
9 
9 

9 
9 
9 


.80 822 
.8o83 7 
.80 852 

.80 867 
.80882 
.80897 

.80 912 
.80 927 
.80 942 


9.92 407 
9.92433 
9.92458 

9.92484 
9.92 5io 
9.92 535 

9.92 56i 
9.92 587 
9.92 612 


0.07 5g3 
0.07 567 
0.07 542 

0.07 5i6 
0.07 490 
0.07465 

0.07 439 
0.07 4i3 
0.07 388 


9.88415 
9.88404 
9.88 3 9 4 

9.88 383 
9.88 372 
9.88 362 

9.8835i 
9.88 34o 
9 .8833o 


5 9 
58 
5 7 
56 
55 
54 
53 

52 

5i 


10 


9 


.80957 


J 5 


9.92 638 


0.07 362 


9.8 


8 3i 9 


50 


1 1 

12 

i3 

i4 
i5 
16 

i? 

18 

J 9 


9 
9 
9 

9 

9 
9 

9 
9 
9 


.80 972 
.80 987 

.8l 002 

.8l 017 
.8l 032 

.81 047 

.81 061 
.81 076 
.81 091 


15 
i5 
15 
'5 
15 
*4 
15 
15 
15 

15 
i5 
15 
i5 
14 
i5 
15 
i5 
15 
'4 


9.92 663 
9.92 689 
9.92715 

9.92 740 
9.92 766 
9.92792 

9.92817 
9.92 843 
9.92 868 


0.07 337 
0.07 3i i 
0.07 285 

0.07 260 
0.07 234 
0.07 208 

0.07 i83 
0.07 157 
0.07 i32 


9.88 3o8 
9.88298 
9.88 287 

9.88 276 
9.88266 
9.88 255 

9.88 244 
9.88 234 

9.88 223 


49 

48 

47 

46 
45 
44 

43 

42 

4i 


20 


9 


.81 1 06 


9.92 89^ 


1 


0.07 106 


9.88 212 




40 


21 
22 
23 

24 
25 
26 

27 
28 
29 


9 
9 
9 

9 
9 
9 

9 

9 
9 


.8l 121 

.81 i36 

.81 i5i 

.81 166 
.81 1 80 
.81 i 9 5 

.8l 210 

.81 225 
.81 240 


9.92 920 
9.92 945 
9.92971 

9.92996 

9.93 022 
9.93048 

9 . 9 3o 7 3 
9.93099 
9.93 124 


0.07 080 
0.07 o55 
0.07 029 

0.07 oo4 
0.06 978 
0.06 952 

0.06 927 
0.06 901 
0.06 876 


9.88 201 
9.88 191 
9.88 l8o 

9.88 169 

9.88 i58 
9.88 i48 

9.88 137 
9.88 126 
9.88 u5 


IO 

II 
II 
II 

10 

II 
II 
II 


3 9 
38 
37 
36 
35 
34 
33 

32 

3i 


30 


9 


.81 254 


9.93 150 


0.06 85o 


9.8 


8 105 




30 




L. Cos. d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. }d. 


r 


49 30 . 


PP 

.1 

.2 

3 
4 

:l 

7 
.8 

9 


26 


25 




15 


M 


.1 
.2 

3 

4 
5 
.6 

7 

.8 


ii 


IO 


2.6 
5-2 

7.8 

10.4 
13.0 

15.6 

18.2 

20.8 

23-4 


2.5 
5-0 
7-5 

10.0 

12.5 
15-0 

'7-5 

20.0 

22.5 


.1 

.2 

3 
4 

7 
.8 


r.S 

3-o 
4-5 

6.0 
7-5 
9.0 

10.5 

12.0 

*3-5 


M 

2.8 
4-2 

5.6 

7.0 

8.4 
9-8 

II. 2 


i.i 

2.2 

3-3 

4-4 
5-5 
6.6 

7-7 
8.8 

9-9 


1.0 
2.O 

3-0 

4.0 

5- 
6.0 

7.0 
8.0 
9.0 



40 30 ' 



/ 


L. Sin. d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9.81 254 




9 . 9 3 150 


25 
26 
26 
25 
26 
25 
26 
25 
26 
26 

25 
26 
25 
26 

25 
26 

25 

26 
26 


. 0.06 85o 


9.88 105 


ii 
ii 
ii 
ii 

IO 

II 
II 
II 
II 
II 

II 

10 

II 
II 
II 
II 
II 
II 
II 
II 


30 


3i 

32 

33 

34 
35 
36 

37 
38 
3 9 


9.81 269 
9.81 284 
9.81 299 

9.81 3i4 
9.81 328 
9.81 343 

9.81 358 
9.81 372 
9.81 387 


15 

15 

15 
15 
M 

15 
15 
*4 

15 


9.93 i 7 5 

9.93 2OI 
9.93 227 

9.93 252 
9.93278 

9.93 3o3 

9.93 329 
9. 9 3354 
9.93 38o 


0.06 825 
0.06 799 
0.06 773 

0.06 748 
0.06 722 
0.06 697 

0.06 671 
0.06 646 
0.06 620 


9.88 094 
9 .88o83 
9.88 072 

9.88 061 
9.88 o5i 
9.88 o4o 

9.88 029 
9.88 018 
9.88 007 


29 

28 
27 

26 

25 
24 

23 
22 
21 


40 


9 


.81 402 




9.93 4o6 


0.06 5g4 


9.87 996 


20 


4i 

42 
43 

44 
45 
46 

4? 
48 

49 


9 
9 
9 

9 
9 
9 

9 
9 
9 


.81 417 

.81 43i 
.81 446 

.81 46 1 
.8i4?5 
.81 490 

.81 505 

.81 519 
81 534 


M 
15 
15 
M 
15 
15 
'4 
15 


9 . 9 343i 
9.9 3 457 
9.93 482 

9.93 5o8 
9.93 533 
9.9355 9 

9.93 584 
9.93 610 
9.93636 


0.06 569 
o.o6543 
0.06 5i8 

0.06 492 
0.06 467 
0.06 44 1 

0.06 4i6 
0.06 390 
0.06 364 


9.87 9 85 

9- 8 7975 
9.87 964 

9.87953 
9.87 942 
9 .8 79 3i 

9.87 920 
9.87909 
9.87 898 


J 9 

18 

!7 

16 
i5 

i4 

i3 

12 
II 


50 


9.81 549 




9.93 661 


26 

25 
26 
25 
26 
25 
26 
25 
26 
25 


0.06 339 


9.87 887 


10 


5i 

52 

53 

54 
55 
56 

5? 
58 
5 9 


9 

9 
9 

9 
9 
9 

9 

9 
9 


81 563 

81 578 
81 592 

81 607 
81 622 
.81 636 

.81 65i 
81 665 
.81 680 


15 
14 
IS 
IS 
14 
5 
14 
15 


9.93 687 
9. 9 3 712 
9 . 9 3 7 38 

9 . 9 3 7 63 
9.93789 
9.93 8i4 

9.93 84o 
9.93865 
9.93 891 


0.06 3 1 3 
0.06 288 
0.06 262 

0.06 237 

. 06 2 I I 

0.06 186 

0.06 1 60 
0.06 135 
0.06 109 


9.87877 
9.87 866 
9.87855 

9.87844 
9.87833 
9.87 822 

9.87 811 
9.87 800 
9.87789 


II 

II 
II 
II 
II 
II 
II 
II 
II 


9 

8 

7 

6 
5 
4 
3 

2 

I 


60 




.81 694 




9.93 916 


0.06 o84 


9.87778 







L. Cos. d. 


L. Cotg 


d. L. Tang. 


L. 


Sin. 


d. 


f 






49. 






PP 

.1 

.2 

3 
4 

:J 


26 


5 




15 


14 


.2 

3 

4 
5 
.6 

:5 


ii 


10 


2.6 

5. 2 

7.8 

10.4 
13.0 
'5.6 

18.2 

20.8 

23-4 


2-5 

5-o 
7-5 

IO.O 

12.5 
15-0 

17-5 

20. o 


.1 

.2 

3 

4 
5 
.6 

:i 


i-5 
3-o 
4-5 

6.0 
7-5 
9.0 

10.5 

12.0 

n-5 


1.4 

2.8 

4.2 

5-6 
7.0 
8.4 

9.8 

II. 2 


1. 1 

2.2 

3-3 

4-4 
5-5 
6.6 

7-7 
8.8 

9.9 


J.O 
2.0 

3-o 

4.0 

5-o 
6.0 

7.0 
8.0 



41. 



> 


L. Sin. d. 


L. Tang. d. 


L. Cotg. 


L. Cos. 


d. 







9.81 694 




9.93 916 


26 


0.06 084 


9 .8 777 8 




60 


I 


9.81 709 


14 


9.93 942 


25 


0.06 o58 


9.87767 




5Q 


2 


9.81 723 




9.93967 


26 


0.06 o33 


9.8 77 56 




58 


3 


9.81 738 




9.93993 


25 


0.06 007 


9.87745 




5 7 


4 


9.81 7 52 


15 


9 . 94 o 1 8 


26 


o.o5 982 


9.87734 




56 


5 


9.81 767 




9.94 o44 




0. 05956 


9.87723 




55 


6 


9.81 781 


15 


9.94069 


26 


o.o5 931 


9.87712 




54 


7 


9.81796 


14 


9.94095 


25 


o.o5 905 


9.87 701 




53 


8 


9.81 810 




9.94 I2O 


26 


o.o5 880 


9.87 690 




52 


9 


9.81 825 




9.94 i46 




o.o5854 


9.87679 




5i 


10 


9.81 839 


15 


9-94 171 


2 5 
26 


o.o5 829 


9.87668 




50 


ii 


9.81 854 




9.94197 


25 


o.o5 8o3 


9 .8 7 65 7 




49 


12 


9.81 868 




9.94 222 


26 


o.o5 778 


9.87 646 




48 


i3 


9.81 882 


15 


9.94 248 


25 


o.o5 752 


9.87635 


M 


47 


i4 


9 


.81 897 


14 


9.94 273 


26 


o.o5 727 


9.87624 




46 


i5 


9.81 911 




9-94299 




o.o5 701 


9.87613 




45 


16 


9 


.81 926 




9.94324 


26 


o.o5 676 


9.87 601 




44 


17 


9 


.81 940 


15 


9.94350 


25 


o.o5 65o 


9.87590 


TI 


43 


18 


9 


.81955 




9.94375 


26 


o.o5 625 


9-87579 




42 


'9 


9 


.81 969 




9 . 94 4o i 




o.o5 599 


9.87 568 




4i 


20 


9 


.81 983 




9.94426 


26 


o.o5 574 


9 .8 7 55 7 




40 


21 


9.81998 


14 


9.94 452 


2 5 


o.o5 548 


9.87 546 




39 


22 


9 


.82 OI2 




9.94477 


26 


o.o5 523 


9.87535 




38 


23 


9 


.82 026 




9.94 5o3 




o.o5 497 


9.87 524 




3 7 








15 






25 














24 


9 


.82 o4i 




9.94 528 


26 


o.o5 472 


9.87613 


12 


36 


25 


9 


.82 065 




9.94554 




o.o5 446 


9.87 5oi 




35 


26 


9 


.82 069 


I 4 


9.94579 




o.o5 421 


9.87490 




34 








15 






25 














27 


9 


.82 o84 




9 . 94 6o4 


26 


o.o5 396 


9.87479 


II 


33 


28 


9 


.82 098 




9.94 63o 




o.o5 370 


9.87468 




32 


29 


9 


.82 112 


14 


9.94655 


26 


o.o5 34s 


9-87457 




3i 


30 


9 


.82 126 




9.94 681 




o.o5 319 


9.87446 




30 




L. Cos. d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 


48 30 . 


PP 


26 


35 




15 


M 




12 


ii 


. x 


2.6 


25 


.! 


1.5 


1.4 


.1 


1.2 


i.i 


.2 


5-2 


5-o 


.2 


3-o 


2.8 


.2 


2.4 


2.2 


3 


7.8 


7-5 


3 


45 


4-2 


3 


3-6 


3-3 


4 


10.4 


IO.O 


4 


6.0 


5-6 


4 


4.8 


4-4 




13.0 


12.5 




7-5 


7.0 


.5 


6.0 


5-5 


.6 


15.6 


15-0 


.6 


9.0 


8.4 


.6 


7.2 


6-. 6 


7 


18.2 


I 7-5 


7 


10.5 


9.8 


7 


8.4 


7-7 


.8 


20.8 


20. o 


.8 


12.0 


II. 2 


.8 


9.6 


8-8 





23.4 22.5 


9 


13.5 12.6 


9 


10.8 9.9 



41 3D . 



' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9.82 126 




9.94 68 1 




o.o5 319 


9- 87 446 




30 


3i 


9.82 i4i 




9.94 706 


25 
26 


o.o5 294 


9.87434 


12 


29 


32 


9.82 155 




9.94732 




o.o5 268 


9.87423 




28 


33 


9.82 169 


15 


9.94757 


25 
26 


o.o5 243 


9.87412 




27 


34 


9.82 1 84 




9.94 783 




o.o5 217 


9.87 4oi 




26 


35 


9.82 198 




9.94 808 




o.o5 192 


9.87 390 




25 


36 


9.82 212' 


14 


9. 9 4834 




o.o5 1 66 


9.87 378 


12 


24 


3 7 


9.82 226 


14 


9-9485 


9 


25 


o.o5 i 


4i 


9 .8 7 36 7 


II 


23 


38 


9.82 240 




9.94884 




o.o5 116 


9.87 356 




22 


39 


9.82 255 


15 


9.94 910 




o.o5 090 


9.87345 


II 


21 


40 


9.82 269 




9.94935 


26 


o.o5 065 


9.87334 


II 


20 


4i 


9.82 283 


14 


9.94961 


25 


o.o5 039 


9.87 322 




'9 


42 


9.82 297 




9.94 986 


26 


o.o5 oi4 


9.87 3n 




18 


43 


9 


.8 2 3n 




9.95 OI2 




o.o4 9 


88 


9.87 3oo 




l l 








15 






25 










12 




44 


9 


.82 326 




9.95 037 




o.o4 963 


9.87 288 




16 


45 


9 


.82 34o 




9.95 062 




o.o4 938 


9.87 277 




i5 


46 


9 


.82 354 




9.95 088 




o.o4 912 


9.87 266 




i4 














25 










II 




47 


9 


.82 368 




9.95 n3 


26 


0.04887 


9.87255 




i3 


48 


9 


.82 382 




9.95 i3 


) 




o.o4 861 


9.87 243 




12 


49 


9 


.82 3 9 6 


14 


9.95 1 64 


25 

26 


o.o4836 


9.87 232 


II 


I I 


50 


9 


.82410 




9.95 190 




o.o4 810 


9.87 221 




10 


5i 


9 


.82424 


15 


9.95 2i5 


25 


o.o4 785 


9.87 209 




9 


52 


9 


.82439 




9-95 24< 


i 


26 


o.o4 760 


9.87 I 9 8 




8 


53 


9 


.82453 


14 


9 .95 266 




o.o4 734 


9.87187 


11 


7 








*4 






25 










12 




54 


9 


.82467 




9.95 291 


26 


o.o4 709 


9.8 


7 J 75 




6 


55 


9 


.82 48 1 




9. 9 53i 7 




o.o4 683 


9.87 1 64 




5 


56 


9 


.82495 




9.95 342 


26 


o.o4658 


9.87 i53 


12 


4 


57 


9 


.82 509 




9.95 368 


25 


o.o4 632 


9.87 i4i 




3 


58 


9 


.82 523 




9.95 393 




o.o4 607 


9.87 i3o 




2 


5 9 


9 


.82 53 7 


4 


9.95 4x8 


25 

26 


o.o4 582 


9.87 119 




I 


60 


9 


.8 2 55i 


14 


9.95 444 




o.o4556 


9.87 107 









L. Cos. 


d. 


L. Cotg. 


d. 


L, Tang. 


L. 


Sin. 


d. 


' 






48. 




PP 


26 


35 




'5 


M 




12 


ii 


.1 


2.6 


2.5 


i 


i. s 


1.4 


.i 


1.2 


i.i 


.2 


5-2 


S-O 


2 


3-O 


2.8 


.2 


2. 4 


2.2 


3 


7.8 


7-5 


3 


4-5 


4-2 


3 


3-6 


3-3 


4 


10.4 


IO.O 


4 


6.0 


5.6 


4 


4.8 


4-4 


5 


13-0 


12.5 


^ 


7-5 


7.0 


-5 


6.0 


5-5 


.6 


15-6 


15-0 


6 


9.0 


8.4 


.6 


7.2 


6.6 


7 


18.2 


17-5 


7 


10.5 


9.8 


. 7 


8.4 


7-7 


.8 


20.8 


20. o 


8 


I2.O 


II. 2 


.8 


9.6 


8.8 




23.4 22.5 


9 


13.5 12.6 


9 


10 8 





n3 



42. 



! 


L. Sin. 


d. 


L. Tang 1 , d. 


L. Cotg. 


L. Cos. 


d. 







9.82 55i 


14 

*4 
*4 
M 

J 4 

*4 
J 4 
14 
14 


9.95 444 


25 
26 
25 
25 
26 

25 
26 
25 
25 
26 

25 
25 
26 

25 
26 
25 
25 
26 
25 
26 

25 
25 
26 
25 
25 
26 
25 
26 
25 
25 


0.04 556 


9.87107 


ii 
ii 

12 
II 

12 
II 
II 
12 
II 


60 


i 

2 

3 

4 
5 
6 

8 
9 


9.82 565 
9.82679 
9.82 593 

9.82 607 
9.82 621 
9.82635 

9.82 649 
9.82663 
9.82 677 


9.95 469 
9.95495 

9.95 52O 

9.95545 

9 . 9 55 7 i 
9.95 596 

9.95 622 
9-95 647 
9.95 672 


o.o4 53i 
o.o4 5o5 
o.o4 48o 

0.04455 
o.o4 429 
o.o4 4o4 

o.o4 378 
o.o4353 
o.o4 328 


9.87 096 
9.87 085 
9.87073 

9.87062 
9.87 o5o 
9.87 039 

9.87028 
9.87 016 
9.87 005 


5 9 
58 
57 

56 
55 
54 

53 

52 

5i 


10 


9 


.82 691 


J 4 
14 
'4 
M 
J 4 
1 4 
13 
M 
X 4 


9.95 698 


o.o4 3o2 


9.86993 




50 


1 1 

12 

i3 

i4 
i5 
16 

r ? 

18 

1 9 


9 

9 
9 

9 
9 
9 

9 
9 
9 


.82705 
.82719 
.82733 

.82747 
.82 761 
.82775 

.82 788 
.82 802 
.82 816 


9.95 723 
9.95 748 
9 <9 5 774 

9.95 799 
9.95825 
9.95 85o 

9 . 9 58 7 5 
9.95 901 
9.95 926 


o.o4 277 

0.04 252 

o.o4 226 

O.O4 2OI 

o.o4 175 
o.o4 150 

o.o4 125 
o.o4 099 
o.o4 074 


9.86 982 
9.86 970 
9.86959 

9.86 947 
9.86936 
9.86 924 

9.86 913 
9.86 902 
9.86 890 


12 
II 
12 
II 
12 
II 
II 
12 


49 

48 

47 
46 
45 
44 

43 

42 

4i 


20 


9 


.82 83o 




9-9 5 9 5 2 


o.o4 o48 


9.86 879 




40 


21 
22 
23 

24 
25 
26 

2 7 
28 
29 


9 
9 
9 

9 
9 

9 

9 
9 
9 


.82 844 
.82858 
.82872 

.82885 
.82 899 
.82 913 

.82 927 
.82941 
.82955 


14 
*4 
13 
4 
4 
J 4 
14 
M 


9-9 5 977 
9.96 002 
9.96 028 

9.96 o53 
9.96 078 
9.96 io4 

9.96 129 
9.96155 
9.96 180 


o.o4 023 
o.o3 998 
o.o3 972 

o.o3 947 
o.o3 922 
o.o3 896 

o.o3 871 
o.o3845 
o.o3 820 


9.86 867 
9 .86855 
9.86844 

9.86832 
9.86 821 
9.86 809 

9.86 798 
9.86 786 
9.86775 


12 

12 
II 
12 
II 
12 
II 


3 9 
38 
37 
36 
35 
34 

33 

32 

3i 


30 


9 


.82 968 




9.96 2o5 


o.o3 795 


9.86 7 63 




30 




L. COS. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


- 


47 3D . 


PP 

.2 

3 

4 
5 
.6 

:J 


26 


25 




M 


13 


.2 

3 

4 

5 
.6 

7 
.8 

9 


12 




ii 


2.6 

Si 

10.4 

13.0 

15.6 

18.2 

20.8 

23.4 


2-5 

50 

75 

10.0 

I2 -5 
15-0 

i7-5 

20.0 


2 

3 

4 
-5 
.6 

1 

9 


3 

4.2 

5-6 
7.0 
8.4 

9.8 

II. 2 


"3 

2.6 

3.9 

5-2 

6 -5 
7.8 

9.1 
10.4 
11.7 


1.2 

2-4 

3-6 

4.8 
6.0 
7.2 

b 


i.i 

2. 2 

3-3 
4-4 

1:1 

U 



u4 



42 3O 





L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d. 




30 


9 


.82 968 


14 
14 
14 
13 
14 

*4 

*4 

*3 

J 4 
M 

M 

13 
14 

*4 
3 
*4 
14 
3 
14 


9.96 2o5 


26 
25 
25 
26 

25 
25 
26 
25 
25 
26 

25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 

26 


o.o3 79 5 


9.86 7 63 




30 


3i 

32 

33 

34 
35 
36 

3? 

38 
3 9 


9 
9 
9 

9 
9 
9 

9 
9 

9 


.82 982 
.82 996 
.83oio 

.83023 
.83o3 7 
.83o5i 

.83065 
.83078 
.83 092 


9.96 23i 
9.96 256 
9.96 281 

9.96 307 
9.96 332 
9 . 9 635 7 

9 . 9 6383 
9.96408 
9.96 433 


o.o3 769 
o.o3 744 
o.o3 719 

o.o3 693 
o.o3 668 
o.o3643 

o.o3 617 
o.o3 592 
o.o3 567 


9.86 762 
9.86 740 
9.86 728 

9.86 717 
9.86 705 
9.86694 

9.86 682 
9.86 670 
9.86 659 


12 
12 
II 
12 
II 
12 
12 
II 
12 


2 9 

28 
27 

26 

25 
24 

23 
22 
21 


40 


9 


.83 106 


9.96459 


o.o3 54i 


9.86 647 


20 


4i 

42 

43 

44 
45 
46 

4? 

48 

49 


9 
9 
9 

9 

9 
9 

9 
9 
9 


.83 120 
.83 i33 
.83 147 

.83 161 

.83 i 7 4 
.83 188 

.83 202 

.832i5 
.83 229 


9.96484 
9.96 5io 
9.96535 

9.96 56o 
9.96 586 
9.96611 

9.96 636 
9.96662 
9.96 687 


o.o3 5i6 
o.o3 490 
o.o3465 

o.o3 44o 
o.o34i4 
o.o3 38 9 

o.o3 364 
o.o3338 
o.o3 3i3 


9.86635 
9.86624 
9.86 612 

9.86 600 
9 .8658 9 
9 .865 77 

9 .86565 
9. 86 554 
9 . 86 542 


II 
12 
12 
II 
12 
12 
II 
12 


i 9 

'7 
16 
i5 
i4 

i3 

12 
II 


50 


9 


.83 242 




9.96 712 


o.o3 288 


9 .86 53o 




10 


5i 

52 

53 

54 
55 
56 

5? 
58 
5 9 


9. 83 2 56 
9.83 270 
9.83283 

9. 83 297 
9.83 3io 
9.83324 

9.83338 
9 .8335i 
9.83 365 


H 
i3 
14 
3 
H 
M 
13 
*4 
3 


9.96 738 
9.96 763 
9.96 788 

9.96 8i4 
9.96 839 
9.96 864 

9.96 890 
9.96915 
9.96 940 


o.o3 262 
o.o3 237 

O.o3 212 

o.o3 186 
o.o3 161 
o.o3 i36 

o.o3 1 10 
o.o3o85 
o.o3 060 


9 .865i8 
9.86 507 
9.86 4 9 5 

9 . 86 483 
9 . 86 472 
9 .8646o 

9 . 86 448 
9 . 86 436 
9 . 86 425 


II 

12 
12 
II 
12 
12 
12 
II 


9 

8 

7 

6 
5 

4 

3 

2 


60 


9 


.833 7 8 


9.96 966 


o.o3 o34 


9 .864i3 









L. Cos. d. 


L. Cotg. d. 


L. Tang. 


L. 


Sin. 


d. 


' 




47. 






PP 

.1 

2 

3 

4 

:l 

.7 

.8 
9 


26 


25 


14 


13 


.1 

.2 

<3 
4 

:i 


12 II 


-2.6 

5-2 

7.8 

10.4 
13.0 
15-6 

18.2 

20.8 

2:5.4 


2.5 .1 

5-O .2 

7-5 -3 

10.0 .4 

12.5 .5 

15.0 .6 
17-5 -7 

20.0 8 


LI 

4.2 

5-6 
7-o 
8.4 

9.8 

II. 2 

12.6 


1:1 

3-9 

Is 

7.8 

9.1 
10.4 

11.7 


1.2 I.I 
2.4 2.2 

3-6 3-3 
4-8 4.4 

6 ' I'i 

7.2 6.6 

8.4 7.7 
9.6 8.8 

10.8 9.9 



i5 



43 C 



/ 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 







9 


,833 7 8 




9.96 966 


25 
25 
26 
25 
25 
26 

25 
25 
25 
26 

25 
25 
26 

25 

25 
26 
25 
25 
26 

25 

25 
26 
25 
25 
25 
26 
25 
25 
26 

25 


o.o3 o34 


9 .864i3 




60 


2 

3 

4 
5 
6 

8 
9 


9.83 392 
9 .834o5 
9.83419 

9.83432 
9.83446 
9.83459 

9.83473 
9.83486 
9 .83 500 


*3 
*4 
i3 
14 
13 
H 
13 
J 4 


9.96991 
9.97 016 
9.97 o4a 

9.97067 
9.97 092 
9.97 118 

9-97 '43 
9.97 168 
9.97 i 9 3 


o.o3 009 
0.02 9 84 
0.02 958 

O.O2 933 
O.O2 908 
0.02 882 

0.02 857 

0.02 832 
O.O2 807 


9.86 4oi 
9.86 389 
9 .863 77 

9 .86 366 
9.86 354 
9.86 342 

9.86 33o 
9 .86 3i8 
9 .86 3o6 


12 
12 
II 
12 
12 
12 
12 
12 


5 9 
58 
5 7 
56 



53 

52 

5i 


10 


9 


.83 5i3 


J 3 


9.97219 


O.O2 781 


9 .86 295 




50 


1 1 

12 

i3 

i4 
i5 

16 

17 
18 

9 


9.83 527 
9.83 54o 
9 .83554 

9 .8356 7 
9. 8358i 
9 .835 9 4 

9.83 608 
9.83 621 
9.83634 


i3 
N 

13 
H 
*3 
H 
1 3 
13 


9.97 244 
9.97 269 
9.97295 

9.97 320 

9.97345 
9.97371 

9.97 396 

9-97 42i 
9,97447 


0.02 756 
0.02 731 
0.02 705 

O.O2 680 
O.O2 65g 
O.O2 629 

0.02 6o4 

0.02 579 

0.02 553 


9. 86 283 
9 .86 271 
9.86 259 

9.86 247 
9 .86235 
9.86223 

9.86 211 
9 .86 2OO 

9 .86 188 


12 
12 
12 
12 
12 
12 
II 
12 
12 


4 9 

48 

47 
46 
45 
44 

43 

42 

4,i 


20 


9 


.83648 




9.97472 


O.02 528 


9 .86 176 


40 


21 
22 
23 

a4 

25 

26 

27 
28 
29 


9 

9 
9 

9 
9 
9 

9 
9 
9 


.8366i 
.83674 
.83688 

.83 701 
.83 715 

.83 728 

.83 7 4i 
.83 755 
.83 768 


3 
*4 
3 
r 4 
*3 
3 
H 

*3 

13 


9.97497 
9.97 523 
9-97548 

9.97573 
9.97 5 9 8 
9.97 624 

9.97649 
9.97 674 
9-97 7<>o 


0.02 5o3 

O.O2 477 
O.O2 452 

O.O2 427 
O.O2 402 
O.O2 376 

0.02 35i 

O.O2 326 

0.02 3oo 


9 .86 1 64 
9 .86 162 
9.86 i4o 

9.86 128 
9.86 116 
9.86 io4 

9.86 092 
9.86 080 
9.86 068 


12 
12 
12 
12 
12 
12 
12 
12 
12 


39 
38 

37 
36 

35 
34 
33 

32 

3r 


30 


9 


.83 781 


9-97 725 


O.02 275 


9 .86 o56 


30 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. 


Sin. 


d. 


' 


46 30 . 


PP 

i 

2 

3 

4 
5 
6 

7 
.8 

9 


26 


25 




14 


13 


.1 

.2 

3 

4 
5 
.6 

.7 

.8 


12 


ii 


2.6 

% 

10.4 

13.0 

15-6 

18.2 

20.8 

23.4 


2-5 

5-o 
7-5 

IO.O 

12.5 
15-0 

17-5 

20. o 


.2 

3 

4 

:I 

7 
.8 

9 


a 

4.2 

5.6 
7.0 
8. 4 

9 .8 

II. 2 

12.6 


1.3 

2.6 

3-9 

5-2 

6.5 
7.8 

9.1 
10.4 

11.7 


1.2 

2-4 

3-6 

4.8 
6.0 
7-2 

8.4 
9.6 
10.8 


1. 1 

2.2 

3-3 

4-4 

II 

7-7 
8.8 

Q.Q 



n6 



43 3O 



I 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9 


83 781 




9-97725 


25 
26 
25 
25 
25 
26 

25 
25 
26 

25 

25 
26 
25 
25 
25 
26 

25 
25 
26 
25 

25 
25 
26 

25 
25 
26 
25 
25 
25 
26 


O.O2 275 


9- 


86o56 




30 


3i 

32 

33 

34 
35 
36 

37 
38 
3 9 


9. 83 795 
9. 83 808 
9.83 821 

9.83834 
9.83 848 
9-83 861 

9.83 874 
9.83 887 
9-83 901 


13 

13 

13 
13 

13 


9.97750 
9.97 776 
9.97 801 

9.97 826 
9.97 85i 
9.97877 

9.97902 
9.97927 
9.97953 


0.02 250 
O.O2 224 
O.O2 199 

O.O2 174 
O.O2 iJcj 
O.O2 123 

O.O2 098 
O.O2 O73 

0.02 o47 


000 000 OOO 


86o44 
86o32 
86 020 

86 008 
85 996 
85 9 84 

85 972 
85 960 

85 9 48 


12 
12 
12 
12 
12 
12 
12 
12 


29 

28 
27 

26 

25 

24 

23 
22 
21 


40 


9 


83 9 i4 


13 


9.97 978 


0.02 022 


9 


85 9 36 




20 


4i 

42 

43 

44 
45 
46 

47 
48 

49 


9 
9 

9 

9 
9 
9 

9 
9 
9 


83 927 
83 940 
83 954 

83 967 
83 980 
83 993 

84 006 

84 020 
84o33 


13 
14 

13 
13 


9.98 oo3 
9.98 029 
9.98 o54 

9.98079 
9.98 io4 
9.98 i3o 

9.98 i55 
9.98 180 
9.98 206 


o.oi 997 
o.oi 971 
o.oi 946 

o.oi 921 
o.oi 896 
o.oi 870 

o.oi 845 
o.oi 820 
o.oi 794 


ooo ooo ooo 


85 924 
85 912 
85 900 

85888 
85 876 
85 864 

8585i 
8583 9 
8582 7 


12 
12 
12 
12 

12 

'3 
12 
12 
12 

12 
12 
12 
13 
12 
12 
12 
12 
12 
13 


'9 

18 
17 

16 
i5 
i4 
i3 

12 
I I 


50 


9 


84 o46 




9.98 23i 


o.oi 769 


9 


85 8i5 


10 

9 

8 

7 

6 
5 
4 
3 

2 
I 


5i 

52 

53 

54 
55 
56 

57 
58 
5 9 


9.84059 
9.84 072 
9.84 o85 

9.84 098 
9 .84 112 
9.84 125 

9.84 1 38 
9.84 1 5 1 
9 .84 1 64 


13 
'3 

'3 

'3 
'3 


9.98 256 
9.98 281 
9.98 307 

9.98 332 
9 . 9 835 7 
9. 98 383 

9.98408 
9 .98433 
9.98 458 


o.oi 744 
o.oi 719 
o.oi 693 

o.oi 668 
o.oi 643 
o.oi 617 

o.oi 592 
o.oi 567 

O.OI 542 


9 
9 
9 

9 
9 

9 
9 
9 


858o3 
85 79 i 
85779 
85 766 
85 7 54 
85 742 

85 7 3o 

85 718 
85 706 


60 


9 


.84177 


9 . 98 484 


o.oi 5i6 


9 


856 9 3 







L. Cos. d. 


L. Cotg. d. 


L. Tang. 


L. Sin. 


d. 


f 






46. 






PP 

.1 

.2 

3 

4 
5 
.6 

7 
.8 


26 


25 


.2 

3 
4 


14 


13 




12 


2.6 

S- 2 
7.8 

10.4 
13.0 

15-6 

18.2 

20.8 

23-4 


2 -5 
5-o 
7-5 

10.0 

12.5 
15.0 

17-5 

20. o 


1.4 

2.8 
4-2 

5.6 
7.0 
8.4 

9.8 

II. 2 


1.3 

2.6 

3-9 

5-2 
6-5 
7.8 

9.1 
10.4 

11.7 


.1 

.2 

3 
4 

7 
.8 


1.2 

2-4 

3-6 

4.8 
6.0 

7-2 

8.4 
9.6 
10.8 



117 



44. 



/ 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. d. 







9 


84i?7 


13 


9.98 484 




o.oi 5i6 


9- 


85 693 




60 


! 


9 


84 190 


13 


9.9 


8 509 


25 


o.oi 491 


9 


8568i 


12 


5 9 


2 


9 


842o3 




9.98534 




o.oi 466 


9 


85 669 




58 


3 


9 


84216 




9.98 56o 




o.oi 44o 


9 


85 65 7 




5 7 








J 3 






25 














4 


9 


84 229 


13 


9.9 


858s 


25 


o.oi 4i5 


9 


85 645 


13 


56 


5 


9 


84242 




9.98 610 




o.oi 390 


9 


85 632 




55 


6 


9 


.84255 


M 


9. 98 635 


25 
26 


o.oi 365 


9 


85 620 


12 


54 


7 


9.84 269 


1-3 


9.98 661 


25 


O.OI 339 


9 


85 608 


12 


53 


8 


9 


.84282 




9.98 686 




o.oi 3i4 


9 


85 5 9 6 




52 


9 


9 


.84295 


J 3 


9.98 711 


25 

_fC 


o.oi 289 


9 


85583 


J 3 


5i 


10 


9 


.843o8 




9.98 7 3 7 




o.oi 263 


9 


855 7 i 




50 


ii 


9 


.84 3ai 


13 


9.98 762 


25 


o.oi 238 


9 


85 55 9 




49 


12 


9 


.84334 




9.98787 




O.OI 2l3 


9 


85 547 




48 


i3 


9 


.84347 


13 


9.98 812 


25 
26 


o.oi 188 


9 


85534 


I 3 

12 


47 


i4 


9 


.8436o 


13 


9.98838 




o.oi 162 


9 


85 522 




46 


i5 


9 


.843 7 3 




9.9 


8 863 


25 


o.oi 137 


9 


85 5io 




45 


16 


9 


.84 385 




9.98 888 


25 


O.OI 112 


9 


.85497 


*3 


44 


i? 


9 


.843 9 8 


*3 
13 


9.98 913 


25 
26 


o.oi 087 


9 


.85485 


12 


43 


18 


9 


.844ii 




9.98939 




o.oi 06 i 


9 


.85473 




42 


'9 


9 


.84424 




9.98 964 


25 


o.oi o36 


9 


.85 46o 


J 3 


4i 


20 


9 


.8443 7 




9.98989 


25 
/: 


O.OI OI I 


9 


85448 




40 


21 


9 


.8445o 


I 3 


9.99015 


2 5 


o.oo 985 


9 


.85436 




3 9 


22 


9 


.84463 




9.99 o4o 




o.oo 960 


9 


.85423 




38 


23 


9 


.84476 




9.99 o65 


2 5 


o.oo 935 


9 


.854ii 




37 








X 3 






2 5 










12 




24 
25 


9 
9 


.84489 
.84 5o2 


13 


9.99090 
9.99 1 1 6 


26 


o . oo 910 

0.00884 


9 
9 


.85 399 
.85386 


13 


36 
35 


26 


9 


.845i5 




9.99 i4i 


25 


o.oo 859 


9 


.853 7 4 




34 


27 


9 


.84528 


*3 


9.99 1 66 


25 


o.oo 834 


9 


.8536i 


*3 


33 


28 


9 


.8454o 




9.99 191 




o.oo 809 


9 


.85349 




32 


2 9 


9 


.84553 


'3 


9.99 217 




o.oo 783 


9 


.85 33 7 


12 


3f 


30 


9 


.84566 




9.99 242 


2 5 


o.oo 758 


9 


.85 3 2 4 


J 3 


30 




L. Cos. d. 


L. Cotgr. 


d. 


L. Tang. 


L. Sin. d. 


/ 




45 30 . 






PP 


26 


5 




M 


13 




12 


.1 


2.6 


2-5 


.1 


1-4 


1.3 


.1 


1.2 


.2 


5-2 


5-o 


.2 


2.8 


2.6 


.2 


2.4 


3 


7.8 


7-5 


3 


4.2 


3-9 


3 


3-6 


4 


10.4 


10.0 


4 


5-6 


5-2 


4 


4-8 


5 


13.0 


12.5 


5 


7.0 


6.5 


.5 


6.0 


.6 


15-6 


15.0 


.6 


8.4 


7.8 


.6 


7.2 


7 


18.2 


17.5 


7 


9-8 


9- 1 


7 


8.4 


.8 


20.8 


20.0 


.8 


II. 2 


10.4 


.8 


9.6 


9 


23.4 22.5 


.9 12.6 


"7 


.9 10.8 



118 



44 3O 



/ 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




30 


9 


.84 566 


13 

*3 
i3 
13 

12 
13 
3 
13 

r 3 

12 

3 
13 

'3 

12 
13 
3 
3 
f2 

*3 

'3 
'3 

12 
13 

*3 

12 

'3 

13 

12 

'3 


9.99 242 


25 

26 

25 
25 
25 
26 

25 
25 

25 
26 

25 
25 
25 
26 

25 
25 
26 
25 
25 
25 
26 
25 
35 
25 
26 
25 
25 
25 
26 
25 


o.oo 758 


9 


.85 324 




30 


3 1 

32 

33 

34 
35 
36 

3? 
38 
3 9 


9 
9 
9 

9 
9 
9 

9 

9 
9 


.845 79 
.84 592 
.84 605 

.846i8 
.8463o 
.84643 

.84656 
.8466 9 
.84682 


9.99 267 
9 99 2 9 3 
9.99 3i8 

9.99 343 
9.99 368 
9.99394 

9.99419 
9.99 444 
9.99469 


o.oo 733 
o.oo 707 
o.oo 682 

o.oo 657 
o.oo 632 
o.oo 606 

o.oo 58 1 
o.oo 556 
o.oo 53i 


9 
9 
9 

9 
9 

9 

9 
9 
9 


.853i2 
.85 299 
.85 287 

.85 27 4 
.85 262 
.85 250 

.85 2 3 7 - 
.85 225' 

.85 212 


*3 
12 

13 
12 
12 
3 
-12 
3 


29 

28 
27 

26 

25 
24 

23 
22 
21 


40 


9 


.84 694 


9.99495 


o.oo 5o5 


9 


.85 200 




20 


4i 

42 

43 

44 
45 
46 

4? 
48 

4 9 


9.84 707 
9.84 720 
9 .84733 

9-84745 
9. 84 758 
9-84 77 1 

9.84784 
9.84 796 
9.84 809 


9.99 52O 

9.99 545 

9.99 570 

9.99 5 9 6 
9.99 621 
9.99 646 

9.99672 
9.99697 
9.99722 


o.oo 48o 
o.oo 455 
o.oo 43o 

o . oo 4o4 
o.oo 379 
o.oo 354 

O.OO 328 

o.oo 3o3 
o.oo 278 


9 
9 
9 

9 
9 
9 

9 
9 
9 


.85 187 
.85175 
.85 162 

.85 150 
.85 i3 7 

.85 125 

.85 112 
.85 100 
.85087 


13 

12 
13 
12 

13 
12 
13 
12 

*3 


; 8 9 

17 

16 
i5 

i4 

i3 

12 
I I 


50 


9 


.84 822 


9.99 747 


o.oo 253 


9 


.85 074 




10 


5i 

52 

53 

54 
55 
56 

5? 

58 

59 


9 
9 
9 

9 
9 

9 

9 
9 

9 


.84835 

.84847 
.8486o 

.84873 
.84885 
.848 9 8 

.84911 
.84923 
.84936 


9-99773 
9.99798 
9.99 823 

9.99848 
9.99874 
9.99899 

9.999 2 4 
9.99949 
9-99975 


o.oo 227 

O.OO 2O2 

o.oo 177 

O.OO I 52 

o.oo 126 

O.OO IOI 

o.oo 076 
o.oo o5i 

O.OO O25 


9 
9 
9 

9 
9 
9 

9 

9 
9 


.85 062 
.85 049 
.85o3 7 

.85o24 
.85 012 
.84 999 

.84986 
.84974 
.84961 


3 

12 

13 
12 

3 
13 
12 

13 


9 

8 

7 
6 
5 

4 

3 

2 
I 


60 


9 


.84949 


r 3 


0.00 OOO 


0.00 OOO 


9 


.84949 









L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 


' 






45. 




PP 

.2 

3 
4 

:! 
:l 

9 


26 


25 


. i 

.2 

-3 
4 

:I 

9 


14 


13 


.1 

.2 

3 

4 
5 
.6 

7 
.8 

9 


12 


2.6 

5-2 

7.8 

10.4 

13.0 

15-6 

18.2 

20.8 

23-4 


2.5 

5-o 
7-5 

IO.O 

12.5 
15.0 

17-5 
20. o 

22-5 


a 

4 .2 

5.6 
7.0 
8. 4 

9 .8 

II. 2 

12.6 


i-3 

2.6 

3-9 

5-2 

6-5 

7.8 

9.1 
10.4 
11.7 


1.2 
2-4 

3-6 

4 .8 
6.0 
7-2 

8.4 
9.6 

10.8 



119 



TABLE III 
FIVE-PLACE LOGARITHMS 

OF THE 

SINE AND TANGENT OF 
SMALL ANGLES 

THE SINE AND TANGENT TO EVERY SECOND FROM O TO 8' J TO EVERY 
TEN SECONDS FROM O TO 2. 

THE COSINE AND COTANGENT TO EVERY SECOND FROM 90 TO 89 
52' ; TO EVERY TEN SECONDS FROM 90 TO 88. 



0. 



FUNCTIONS OF SMALL ANGLES 

LOGARITHMIC SINE AND TANGENT. 





0" 


1" 


2" 


3" 


4" 


5" 


6" 


7" 


8" 


9" 


10" 




o 

10 
20 


5- 68557 
98660 


68557 
72697 

*779 


98660 
76476 
$02800 


#16270 
79952 

*0473Q 


,28763 
83170 
1*06579 


#38454 
86167 
#08351 


#46373 
88969 

# IO 55 


*53o7 
91602 
,11694 


#58866 
94085 
#13273 


#63982 
96433 
**4797 


*68 557 
98660 
^16270 


50 
40 
30 


30 

40 

5 


6. 16270 
28763 
38454 


29836 
39315 


19072 
30882 
40158 


20409 

3i9 4 
40985 


21705 
32903 
41797 


22964 
33879 
42594 


24188 
34833 
43376 


25378 
357 6 7 
44145 


26536 
36682 
44900 


27664 

37577 
45 6 43 


28763 
38454 
46373 


20 
IO 

o 59 


t o 

IO 

20 


6.46373 
6. 5 3067 
8866 


7090 
3683 
9406 


7797 
4291 

9939 


8492 
4890 
#0465 


975 
548i 
#0985 


ft 9 
6064 

**499 


ss 

*200 7 


*"6s 
7207 
*2509 


*'8f 
7767 
*3o6 


*2442 

8320 
*3496 


*367 
8866 
*39 g 2 


50 
40 
3 


30 
40 

50 


6.63982 

* 8 F 
6.72697 


4462 
8990 
3090 


4936 
9418 
3479 


5406 
9841 
3865 


5870 
#0261 
4248 


633 
^0676 
4627 


6 78! 

*io88 
5003 


7235 
*i49 6 
5376 


7680 
*i90Q 
5746 


8121 
*23o 
6112 


8557 
#2697 
6476 


20 

10 

o 58 


2 o 

10 
20 


6 47 6 
9952 
6.83170 


6836 

*028 

3479 


793 

*o6i 
3786 


7548 
*943 
4091 


7900 
*i268 
4394 


8248 

*i59' 
4694 


8595 
*i9" 

4993 


8938 

*22 3 
5289 


9278 

*2545 
5584 


9616 
+2859 
5876 


9952 
*3 I 7 
6167 


50 
40 
30 


30 

4 

So 


6167 
8 9 6g 
6.9 1602 


6 455 
9240 
1857 


6742 
9509 

2110 


7027 
9776 
2362 


73io 

#0042 
2612 


759i 

#0306 
2861 


7870 
*o 5 68 
3109 


8l 47 
*082 9 

3355 


8423 
*io88 

3599 


8697 
**34 6 
3843 


8969 
^1602 
4085 


20 
IO 

o 57 


3 o 

10 
20 


4085 

6433 
8660 


4325 
6661 
8877 


4565 

6888 
993 


4803 
7113 
937 


5039 
7338 
9520 


5275 
7561 

9733 


559 
7783 
9944 


5742 
8004 
*i55 


5973 
8224 
#0364 


6204 
8443 
*0572 


6433 
8660 

*0779 


50 
40 
30 


3 
4 
50 


7.00779 
2800 
4730 


0986 
2997 
49'9 


1191 

3193 
5106 


1395 
3388 
5293 


1599 
3582 
5479 


1801 
3776 
5664 


2003 
3968 
5849 


2203 
4160 
6032 


2403 
4351 
6215 


2602 
454i 
6397 


2800 
473 
6579 


20 

IO 

o 58 


4 o 

10 

20 


6579 
8351 
7. 1 0055 


6759 
8525 

O222 


s 


7118 
8870 
0553 


7296 
9041 
0718 


7474 
9211 
0882 


7651 
9381 
1046 


7827 
955i 
1209 


8003 
9719 
137 1 


8177 
9887 
1533 


8351 
*55 
1694 


50 
4 
30 


3 
40 
5 


1694 
3273 

4797 


1854 
3428 

4947 


2014 
3582 
5096 


2174 
3736 
5244 


2333 
3889 

539 2 


2491 
4042 
5540 


2648 
4194 
5687 


2805 
4346 
5833 


2962 
4497 
5979 


38 
4647 
6125 


3273 
4797 
6270 


20 
IO 

o 55 


5 o 

IO 
20 


7.16270 
7694 
9072 


6414 

7834 
9208 


6558 

7973 
9343 


6702 
8112 
9478 


6845 
8250 
9612 


6987 
8389 
9746 


7130 
8526 
9879 


7271 

8663 

# OOI2 


7413 
8800 
*oi45 


7553 
8937 
#0277 


7694 
9072 
#0409 


50 
40 

30 


30 

4 

5" 


7.20409 
1705 
2964 


0540 
1833 
3088 


67i 
1960 
3212 


0802 
2087 
3335 


0932 
2213 
3458 


1062 
2339 
358o 


1191 
2465 
3702 


I 3 20 
2590 
3824 


1449 
2715 
3946 


1577 
2840 
4067 


i7 5 
2964 
4188 


20 
IO 

o 54 


6 " 

10 

20 


4188 
5378 
6536 


4308 

5495 
6650 


4428 
5612 
6764 


4548 
5728 
6877 


4668 
6991 


4787 
596i 
7104 


4906 
6076 
7216 


5024 
6192 
7329 


SH 2 
6307 

744 > 


5260 
6421 
755 2 


5378 
6 536 
7664 


50 
40 
3 


3 
40 
So 

7 o 

IO 

20 


i 


7775 
8872 
9942 


7886 
8980 
*47 


7997 
9088 
#0152 


8107 
9196 
*0257 


8217 
9303 
*036 2 


8327 
9410 

+0467 


8437 
9517 

*57 1 


8546 
9623 
*o6 7 5 


8655 
9730 
*779 


8763 
9836 
#0882 


20 
IO 

o 53 


7.30882 
1904 
2903 


0986 
2005 
3001 


1089 
2106 
3100 


1191 
2206 
3*9 8 


1294 
2306 
3296 


1396 
2406 
3393 


1498 
2506 

349 


1600 
2606 
3588 


1702 
2705 
3685 


1803 
2804 
3782 


1904 
2903 
3879 


50 
40 
30 


3 
40 

50 

M 


3879 
4833 
5767 

M^^M 

10" 


3975 




4071 
5022 
5952 


4167 
5"6 
6044 

m^mi 


4263 
5209 
6i35 

^ 


4359 
5303 
6227 


4454 
5396 
6318 

__ 


4549 
5489 
6409 


4644 
5582 
6500 


4739 
5675 
659 1 




4833 
5767 
6682 




20 
IO 

o 52 

^ 


9" 


8" 


7" 


6" 


5" 


4" 


3" 


2" 


1" 


0" 





LOGARITHMIC COSINE AND COTANGENT. 



89 C 



FUNCTIONS OF SMALL ANGLES. 



' 


L. Sin. L. Tang. 






L. Sin. 


L. Tang. 




o 

IO 

20 

3o 
4o 
5o 


5.6855 7 
5.98 660 

6. 16 270 
6.28 763 
6.38454 


5.68557 
5.98 660 
6. 16 270 
6.28763 

6.38454 


o 60 

5o 
4o 
3o 
20 

10 


73o 
4o 
5o 


7. 33 879 
7.34833 
7.35767 


7. 33 879 
7.34833 
7.35 7 67 


3o 
20 

IO 


8 o 

IO 
20 

3o 
4o 
5o 


7.36 682 
7.37577 
7.38454 

7 .3 9 3i4 
7.40 1 58 
7.40985 


7.36682 
7 .3 7 5 77 
7-38455 

7-3 9 3i5 
7.40 i58 
7.40985 


o 52 

5o 
4o 
3o 
20 

IO 


1 o 

IO 

20 
3o 
4o 
5o 


6.46 3 7 3 
6.53o67 
6.58866 
6. 63 982 
6.6855 7 
6.72 697 


6.46 373 
6.53o67 
6.58866 
6. 63 982 
6.6855 7 
6.72 697 


o 59 

5o 
4o 
3o 
20 

IO 


9 o 

10 

20 
3o 
4o 
5o 


7-4i 797 
7.42 594 
7.43376 

7.44 i4s 
7.44 900 
7.45643 


7-4i 797 
7.42 594 
7.43376 

7.44i45 
7-44 900 
7.45643 


o 51 

5o 
4o 
3o 
20 

IO 


2 o 

IO 
20 

3o 
4o 
5o 


6.76476 
6.79952 
6.83 170 
6.86 167 
6.88 969 
6.91 602 


6.76476 
6.79 952 
6.83 170 

6.86 167 
6.88 969 
6.91 602 


o 58 

5o 
4o 
3o 
20 

10 


10 o 

IO 

20 
3o 
4o 
5o 


7 .463 7 3 
7.47 090 
7.47797 
7.48491 
7.49 175 
7.49849 


7 .463 7 3 
7.47091 
7.47797 
7.48492 
7.49 176 
7.49 849 


o 50 

5o 
4o 
3o 

20 
10 


3 o 

10 
20 

3o 

4o 
5o 


6.94085 
6.96433 
6.98 660 
7.00779 
7.02 800 
7.04 73o 


6.94085 
6.96433 
6.98 660 

7.00779 
7.02 800 
7.04 73o 


o 57 

5o 
4o 
3o 

20 
10 


11 o 

IO 

20 
3o 
4o 
5o 


7-5o 5i2 
7-5i 165 
7.5i 808 

7.52 442 
7.53067 
7.53683 


75o 5i2 
7.5i i65 
7. 5 1 809 

7.52443 
7.53 067 
7.53683 


o 49 

5o 

4o 
3o 

20 
10 


4 o 

10 

20 
3o 
4o 

5o 


7.o65 79 
7 .o835i 
7.10 o5s 

7.11 694 
7 .i3 27 3 
7.14 797 


7.06 579 
7. 08 35 2 
7.10 o55 

7.11 694 
7 .i32 7 3 
7.14797 


o 56 

5o 
4o 
3o 

20 
10 


12 o 

10 
20 

3o 

4o 
5o 


7.54 291 
7.54890 
7.5548i 

7.56 o64 
7 .5663 9 
7.57 206 


7.54 291 
7.54 890 
7.5548i 

7.56o64 
7 .5663 9 
7.57207 


o 48 

5o 
4o 
3o 
20 

IO 


5 o 

IO 
20 

3o 
4o 
5o 


7. 16 270 

7- 1? 6 94 
7. 19 072 

7.20 409 
7.21 7 o5 
7.22 964 


7. 16 270 
7.17694 
7.19073 

7.20 409 
7.21 705 
7.22 964 


o 55 

5o 
4o 
3o 

20 
10 


13 o 

IO 

20 
3o 
4o 
5o 


7.57767 
7.58 320 
7-58866 
7.59 4o6 
7.5 99 3 9 
7.60 465 


7.57767 
7.58 320 
7.58867 
7.59 4o6 
7.59939 
7.60466 


o 47 

5o 
4o 
3o 
20 

10 


6 o 

10 

20 
3o 
4o 
5o 


7.24 188 
7.25 3 7 8 
7.26 536 

7.27 664 
7.28763 
7.29 836 


7.24 188 
7.25 3 7 8 
7.26536 
7.27 664 
7.28764 
7.29 836 


o 54 

5o 

4o 
3o 

20 
10 


14 o 

10 
20 

3o 
4o 
5o 


7.60985 
7.61 499 
7.62 007 

7.62 5og 
7.63 006 
7.63496 


7.60 986 
7.61 500 
7.62 008 
7.62 5io 
7. 63 006 
7-63 497 


o 46 

5o 
4o 
3o 
20 

IO 


7 o 

IO 

20 
3o 


7.3o 882 
7.3i 904 
7.32 go3 

7. 33 879 


7.30882 
7. 3 1 904 
7 . 32 go3 

7. 33 879 


o 53 

5o 

4o 
3o52 


15 o 


7.63 982 


7.63 982 


o 45 




L. Cos. 


L. Cotg. 


a i 




L. Cos. 


L. Cotg. 


n , 



123 



89. 



FUNCTIONS OF SMALL ANGLES. 
0. 



r rt 


L. Sin. 


L. Tang. 




, 


L. Sin. 


L. Tang. 




15 o 

10 

20 
3o 
4o 
5o 


7.63 92 
7-64461 
7.64936 
7.654o6 
7.65 870 
7 .6633o 


7.63 982 
7.64462 
7.64937 
7.65 4o6 
7.65871 
7 .6633o 


o 45 

5o 
4o 
3o 
20 

10 


22 3o 
4o 
5o 


7 .81 591 
7.81 911 
7.82 229 


7.81 5 9 i 
7.81 912 
7.82 23o 


3o 

20 
IO 


23 o 

IO 

20 
3o 
4o 
5o 


7.82 545 
7.82 859 
7. 83 i 7 o 

7 .834 7 9 
7 .83 7 86 
7.84 091 


7.82 546 
7.82860 
7.83 171 

7.8348o 
7. 83 787 
7 .84 092 


o 37 

5o 
4o 
3o 

20 
10 


16 o 

10 
20 

3o 
4o 
5o 


7.66 784 
7.67235 
7.67 680 

7.68 121 

7 .6855 7 
7.68 989 


7.66785 
7.67235 
7.67680 

7.68 121 

7.68558 
7.68 990 


o 44 

5o 
4o 
3o 
20 

IO 


24 o 

IO 

20 
3o 
4o 
5o 


7.84 393 
7-846 9 4 
7.84 992 
7.85 289 
7 .85583 
7 .858 7 6 


7-843 9 4 
7.846 9 5 
7.84993 
7.85 290 
7.85 584 
7. 85 877 


o 36 

5o 
4o 
3o 
20 

IO 


17 o 

IO 
20 

3o 

4o 
5o 


7-69417 
7.69 84i 
7. 70 261 

7.70676 
7.71 088 
7-7 1 496 


7.69418 
7.69 842 
7.70 261 
7.70677 
7.71 088 
7-7 1 496 


o 43 

5o 
4o 
3o 
20 

10 


25 o 

IO 

20 
3o 
4o 
5o 


7 .86 166 
7 . 86 455 
7 86 741 

7.87 026 
7.87 309 
7.87590 


7.86 167 
7.86456 
7.86743 
7.87027 
7.87310 
7 .8 7 5 9 i 


o 35 

5o 
4o 
3o 

20 
IO 


18 o 

10 
20 

3o 
4o 
5o 


7.71 900 
7.72 3oo 
7.72697 
7.73090 

7.73479 
7-73865 


7.71 900 
7,72 3oi 
7.72697 
7.73 090 
7 . 7 348o 
7 . 7 3866 


o 42 

5o 

4o 
3o 

20 
10 


26 o 

10 

20 
3o 

4o 
5o 


7.87870 
7.88 147 
7-88423 

7.88697 
7.88 969 

7.89 24o 


7 .8 7 8 7 i 
7 .88 i48 
7.88424 
7.88 698 
7.88 970 
7.89 241 


o 34 

5o 
4o 
3o 
20 

IO 


19 o 

10 

20 

3o 

4o 
5o 


7.74248 
7-74627 
7.75 oo3 
7 . 7 53 7 6 
7.75745 

7.76 112 


7.74248 
7.74628 
7.75 oo4 
7.75377 
7.75 7 46 
7.76 n3 


o 41 

5o 
4o 
3o 

20 
10 


27 o 

IO 

20 

3o 
4o 
5o 


7.89 509 

7.89776 
7.90 o4 1 
7.90 3o5 
7.90 568 
7.90 829 


7.89 5io 
7.89777 
7.90043 
7.90307 
7.90 569 
7.90 83o 


o 33 

5o 
4o 
3o 

20 
10 


20 o 

IO 
20 

3o 
4o 
5o 


7.76475 

7. 76 836 
7-77 193 
7-77548 
7.77899 
7.78248 


7. 7 64 7 6 
7 . 7 683 7 

777 i94 
7.77549 
7.77900 
7.78 249 


o 40 

5o 
4o 
3o 

20 
IO 


28 o 

IO 
20 

3o 
4o 
5o 


7.91 088 
7.91 346 
7.91 602 
7 . 9 i85 7 
7.92 1 10 
7.92 362 


7.91 o8 9 

7.91347 
7.9,1 6o3 

7.91 858 
7.92 in 
7.92 363 


o 32 

5o 
4o 
3o 
20 

10 


21 o 

10 

20 
3o 
4o 
5o 


7 . 7 85 9 4 
7 . 7 8 9 38 
7.79278 
7.79616 
7.79952 
7.80284 


7 . 7 85 9 5 
7 . 7 8 9 38 
7.79279 

7.79617 
7.79952 
7.80285 


o 39 

5o 
4o 
3o 
20 

10 


29 o 

IO 
20 

3o 
4o 
5o 


7.92 612 
7.92 861 
7.93 108 

7.93354 
7.93599 
7 . 9 3 842 


7.92 6i3 
7.92 862 
7.93 no 

7 . 9 3 356 
7.93 601 
7-93844 


o 31 

5o 
4o 
3o 

20 
10 


22 o 

IO 
20 

3o 


7.80615 

7.80 942 
7.81 268 
.81 591 


7.8o6i5 
7.80943 
7.81 269 
7.81 591 


o 38 

5o 

4o 

3o37 


30 o 


7.94 o84 


7.94 086 


o 30 




L. Cos. L. Cotg. 


" ' 




L. Cos. i L. Cotg. 


// / 



89. 



FUNCTIONS OP SMALL ANGLES. 
0. 



, " 


L. Sin. 


L. Tang. 




, 


L. Sin. 


L. Tang. 




30 o 

10 
20 


7.94 o84 
7.94325 
7.94564 


7.94086 
7.94 326 
7.94 566 


o 30 

5o 
4o 


37 3o 
4o 
5o 


8.o3 77 5 
8.o3 967 
8.o4 1 59 


8.o3 777 
8.o3 9 7 o 
8.o4 162 


3o 
20 

10 


3o 
4o 
5o 


7.94 802 
7.96 o3 9 
7.96 274 


7.94804 
7.96 o4o 
7.95 276 


3o 
20 
10 


38 o 

IO 

20 


8.o435o 
8.o4 54o 
8.o4 729 


8.o4353 
8.o4543 
8.o4732 


o 22 

5o 
4o 


31 o 

10 
20 


7.96 5o8 
7.9574i 
7.95 97 3 


7.95 5io 
7.95743 
7.95974 


o 29 

5o 
4o 


3o 
4o 
5o 


8.o4 918 
8.o5 io5 
8.o5 292 


8.o4 921 
8.o5 108 
8.o5 295 


3o 
20 

IO 


3o 
4o 
5o 


7.96 2o3 
7.96 432 
7.96 660 


7.96 205 
7.96434 
7.96 662 


3o 

20 
10 


39 o 

IO 
20 


8.05478 
8.o5663 
8.o5848 


8.o548i 
8.o5666 
8.o585i 


o 21 

5o 

4o 


32 o 

10 

20 


7.96887 
7.97 n3 
7.97337 


7.96889 

7-97 n4 
7.97339 


o 28 

5o 

4o 


3o 
4o 
5o 


8.o6o3i 
8.06214 
8.06 396 


8.o6o34 
8.06 217 
3.o6 399 


3o 
20 

10 


3o 
4o 
5o 


7.97660 

7-97 7 82 
7.98 oo3 


7.97662 

7.97 7 84 
7.98 oo5 


3o 

20 
10 


40 o 

IO 

20 


8.06678 
8.o6 7 58 
8.o6 9 38 


8.o658i 
8.06 761 
8.06 94 1 


o 20 

5o 
4o 


33 o 

10 

20 


7.98 223 
7.98442 
7.98 660 


7.98 225 

7.98444 
7.98 662 


o 27 

5o 
4o 


3o 
4o 
5o 


8.07 117 
8.07 296 
8.07473 


8.07 I2O 
8.07 298 

8.o 7 4 7 6 


3o 

20 
IO 


3o 
4o 
5o 


7.98 876 
7.99 092 

7.99 3o6 


7.98 878 
7.99 094 

7.99 3o8 


3o 
20 

IO 


41 o 

IO 
20 


8.07 650 
8.07 826 
8.08 002 


8. 07 653 
8.07 829 
8.08 005 


o 19 

5o 
4o 


34 o 

10 
20 


7.99 620 
7.99 732 
7.99943 


7.99 522 
7.99 734 

7.99 946 


o 26 

5o 
4o 


3o 
4o 
5o 


8.08 176 
8.o835o 
8.08 624 


8.08 1 80 
8.08 354 
8.08 627 


3o 
20 

IO 


3o 
4o 
5o 


8.00 1 54 
8.00 363 
8.00 571 


8.00 1 56 
8.oo365 
8.00 574 


3o 
20 

10 


42 o 

IO 

20 


8.08696 
8.08868 
8.09 o4o 


8 .08 700 
8.08 872 
8.09043 


o 18 

5o 
4o 


35 o 

10 
20 


8.00 779 
8.00 985 
8.01 190 


8.00 781 
8.00987 
8.01 193 


o 25 

5o 
4o 


3o 
4o 
5o 


8.09 2IO 

8.09 38o 
8.09 650 


8.09 214 
8.09 384 
8. 09 553 


3o 
20 
10 


3o 
4o 
5o 


8.01 395 
8.01 598 
8.01 801 


8.01 397 
8.01 600 
8.01 8o3 


3o 

20 
10 


43 o 

IO 

20 


8.09 718 
8.09 886 
8.ioo54 


8.09 722 
8.09 890 
8. 10 067 


o 17 

5o 
4o 


36 o 

10 

20 


8.02 OO2 
8.02 203 

8. 02 402 


8. 02 oo4 
8. 02 2o5 
8.02 405 


o 24 

5o 
4o 


3o 
4o 
5o 


8. 10 220 
8.io386 
8.10 552 


8. 10 224 
8.10 390 
8.io555 


3o 

20 
IO 


3o 
4o 
5o 


8. 02 601 
8. 02 799 
8.02 996 


8.02 6o4 
8.02 801 
8.02 998 


3o 

20 
10 


44 o 

10 
20 


8.10 717 
8.10 881 
8 . 1 1 o44 


8. 10 720 
8.io884 
8. 1 1 o48 


o 16 

5o 

4o 


37 o 

10 

20 


8.o3 192 
8.o3 387 
8.o3 58i 


8.o3 194 
8.o33 9 o 
8.o3 584 


o 23 

5o 
4o 


3o 

4o 
5o 


8. 1 1 207 
8. 1 1 370 
8. ii 53i 


8. I I 211 

8. ii 3 7 3 
8. ii 535 


3o 

20 
IO 


3o 


8.03775 


8.o3 777 


3o22 


45 o 


8. 1 1 693 


8. ii 696 


3 15 




L. Cos. 


L. Cotg. 


" ' 




L. Cos. 


L. Cotg. 


" ' 



125 



89. 



FUNCTIONS OP SMALL ANGLES. 



/ // 


L. Sin. 


L. Tang. 




/ tr 


L. Sin. L.Tang. 




45 o 

10 

20 
3o 
4o 
5o 


8.11 693 
8. ii 853 
8.i2oi3 
8. 12 172 
8.i233i 
8.12489 


8. n 696 
8. ii 85 7 
8.12 017 
8.12 176 
8.12335 
8.12493 


o 15 

5o 
4o 
3o 

20 
10 


52 3o 
4o 
5o 


8.18 387 
8.18 524 
8.18662 


8.18 392 
8.18 53o 
8.18667 


3o 

20 
IO 


53 o 

10 
20 

3o 
4o 
5o 


8.18 798 
8.18 935 
8.19 071 
8.19 206 
8.i 9 34i 
8.19476 


8.18 8o4 
8.18 940 
8.19076 

8. 19 212 

8.1 9 347 
8.19481 


o 7 

5o 
4o 
3o 

20 
IO 


46 o 
10 

20 
3o 
4o 
5o 


8.12 647 
8.12 8o4 
8.12 961 

8.i3 117 
8.i3 272 
8.13427 


8.12 65i 
8.12 808 
8.12 965 

8.i3 121 
8.i32 7 6 
8.i343i 


o 14 

5o 
4o 
3o 
20 

IO 


54 o 

10 
20 

3o 
4o 
5o 


8.19610 

8.19744 
8.19877 

8.20 OIO 

8.20 i43 
8. 20 275 


8.19616 
8.19 749 

8.19 883 
8. 20 016 
8. 20 149 
8.20 281 


o 6 

5o 
4o 
3o 

20 
IO 


47 o 

10 

20 
3o 
4o 
5o 


8.i3 58i 
8.i3 7 35 
8.13888 
8.i4o4i 
8.i4 193 
8.i4344 


8.i3 585 
8.i3 739 
8.i3 892 
8.14045 
8.i4 197 
8.i4348 


o 13 

5o 
4o 
3o 
20 

IO 


55 o 

IO 
20 

3o 
4o 
5o 


8.20 407 
8.20538 
8.20 669 

8.20 800 
8.20 930 
8.21 060 


8.20 4i3 
8.20 544 
8.20675 

8.20806 
8.20936 
8.21 066 


o 5 

5o 
4o 
3o 

20 
10 


48 o 

10 

20 
3o 
4o 
5o 


8.14496 
8.i4646 
8.i4 796 
8.i4945 
8. 1 5 094 
8.i5 2 43 


8.i4 500 
8.i465o 
8.i48oo 

8. i4 950 
8. 1 5 099 
8.i5 247 


o 12 

5o 
4o 
3o 

20 
IO 


56 o 

IO 
20 

3o 
4o 
5o 


8.21 189 
8.21 319 
8.21 447 
8.21 5 7 6 
8.21 7 o3 
8.21 83i 


8.21 195 
8.21 324 
8.21 453 

8.21 58i 
8.21 709 
8.21 837 


o 4 

5o 
4o 
3o 
20 

IO 


49 o 

10 
20 
3o 
4o 
5o 


8.i5 3 9 i 
8.15538 
8.15685 
8.i5832 
8.i5 97 8 

8.l6 123 


8.i5 3 9 5 
8.15543 
8.i5 690 

8.i5836 
8.i5 982 
8.16 128 


o 11 

5o 
4o 
3o 

20 
IO 


57 o 

IO 
20 

3o 
4o 
5o 


8.21 958 
8.22 085 

8.22 211 
8.22 337 

8.22463 
8.22 588 


8.21 9 64 

8.22 091 
8.22 217 
8.22 343 
8.22 469 
8.22 595 


o 3 

5o 
4o 
3o 

20 
IO 


50 o 

10 
20 

3o 
4o 
5o 


8.16 268 
8.i64i3 
8.i655 7 

8.16 700 
8.i6843 
8.16986 


8.16 273 
8.16417 
8.i656i 
8.16 705 
8.i6848 
8. 16 991 


o 10 

5o 
4o 
3o 

20 
10 


58 o 

IO 
20 

3o 
4o 
5o 


8.22 713 
8.22838 

8.22 962 

8. 23 086 

8.23 210 

8.23 333 


8.22 720 
8.22844 
8.22968 

8.23 092 
8.23 2 i6 
8.23 33 9 


o 2 

5o 
4o 
3o 
20 

IO 


51 o 

10 
20 

3o 
4o 
5o 


8.17 128 
8.17 270 
8. 17 4i i 
8.17 552 
8. 17 692 
8.17882 


8.17 i33 
8.17275 
8.17416 
8. 17 55 7 
8.17697 
8.i 7 83 7 


o 9 

5o 
4o 
3o 
20 

IO 


59 o 

10 

20 
3o 

4o 
5o 


8.23456 
8.23578 
8.23 700 
8.23 822 
8.23 9 44 
8.24065 


8.23462 
8.23 585 
8.23 707 
8.23 829 
8.23 950 
8.24 071 


o 1 

5o 
4o 
3o 
20 

10 


52 o 

10 

20 

3o 


8.17971 
8.18 no 
8.18 249 

8. T* 3s 7 


8.17976 
8.18 ii5 
8.18254 
8.18 392 


o 8 

5o 
4o 
3o 7 


60 o 


8.24 186 


8.24 192 


o 




L. Cos. L. Cotg 1 . 


" ' 




L. Cos. 


L. Cotg". 


" ' 



126 



89. 



FUNCTIONS OF SMALL ANGLES. 
1. 



/ II 


L. Sin. 


L.Tang. 




r ti 


L. Sin. 


L. Tang. 




o 

10 
20 


8.24 186 
8.243o6 
8.24426 


8.24 192 
8.243i3 
8.24433 


o 60 
5o 

4o 


7 Jo 

4o 
5o 


8.29 3oo 
8.29 407 
8.29 5i4 


8.29 309 
8.29416 
8.29 523 


3o 

20 
IO 


3o 
4o 
5o 


8.24546 
8.24665 

8.24785 


8.24553 

8.24 672 
8.24 791 


3o 

20 

10 


8 o 

IO 
20 


8.29 621 
8.29727 
8. 29 833 


8.29 629 
8.29 7 36 
8.29842 


o 52 

5o 
4o 


1 o 

10 
20 


8.24903 
8.25 022 
8.25 i4o 


8.24 910 
8.25 029 
8.25 i4 7 


o 59 

5o 
4o 


3o 
4o 
5o 


8.29939 
8-.3oo44 
8.3o 150 


8.29 947 
8.3oo53 
8.3o i58 


3o 

20 
10 


3o 

4o 
5o 


8.25258 
8.25 375 
8.25493 


8.25265 
8.25 382 
8.25 500 


3o 
20 
10 


9 o 

IO 
20 


8.30255 
8.3o35 9 
8.3o464 


8.3o263 
8.3o368 
8.3o473 


o 51 

5o 

4o 


2 o 

10 

20 


8.25 609 
8.25 726 
8.25842 


8.25 616 
8.25 7 33 
8.25849 


o 58 

5o 
4o 


3o 

4o 
5o 


8.3o568 
8.30672 
8.30776 


8.3o577 
8.3o68i 
8.30785 


3o 

20 
IO 


3o 
4o 
5o 


8.25958 
8.26074 
8.26 189 


8.25 9 65 
8.26081 
8.26 196 


3o 

20 
10 


10 o 

IO 

20 


8.30879 
8.3o 9 83 
8.3i 086 


8.3o888 
8.3o 992 
8.3i 095 


o 50 

5o 

4o 


3 o 

10 
20 


8.263o4 
8.26419 
8.26533 


8.26 3i2 
8.26426 
8.2654i 


57 

5o 

4o 


3o 
4o 
5o 


8.3i 188 
8.3i 291 
8.3i 3 9 3 


8. 3 1 198 
8. 3 1 3oo 

8. 3 1 4o3 


3o 

20 
10 


3o 

4o 
5o 


8.26648 
8.26 761 
8.26875 


8.26655 
8.26 769 
8.26882 


3o 

20 
10 


11 o 

10 
20 


8.3i 4g5 
8.3i 5 97 
8.3i 699 


8.3i 505 
8.3i 606 
8.3i 708 


o 49 

5o 

4o 


4 o 

10 
20 


8.26988 
8.27 101 
8.27 214 


8.26 996 
8.27 109 

8.2 7 221 


o 56 

5o 

4o 


3o 
4o 
5o 


8.3i 800 
8.3i 901 

8.32 002 


8. 3 1 809 
8.3i 911 

8.32 012 


3o 
20 

IO 


3o 
4o 
5o 


8.27 326 

8. 27 438 
8.27 550 


8.2 7 334 
8.2 7 446 
8.27558 


3o 

20 
IO 


12 o 

10 

20 


8.32 io3 
8.32 2o3 
8.32 3o3 


8.32 112 

8.32213 
8.32 3i3 


o 48 

5o 

4o 


5 o 

10 

20 


8.27 661 
8.2 777 3 
8.27883 


8.27 669 
8.27 780 
8.27 891 


o 55 

5o 
4o 


3o 
4o 
5o 


8.324o3 
8.3 2 5o3 
8.32 602 


8.324i3 
8.32 5i3 
8.32612 


3o 

20 
1C 


3o 
4o 
5o 


8.27 994 
8.28 104 
8.28 215 


8.28002 
8.28 112 

8.28 223 


3o 

20 
IO 


13 o 

10 

20 


8.32 702 
8.32 801 
8.32 899 


8.32 711 
8.32811 
8.32 909 


o 47 

5o 
4o 


6 o 

10 
20 


8.28324 
8.28434 
8.28 543 


8.28 332 

8.28442 

8. 2 855i 


o 54 

5o 
4o 


3o 
4o 
5o 


8.32 998 
8. 33 096 
8.33 195 


8.33 008 
8.33 106 
8.33205 


3o 

20 
IO 


3o 
4o 
5o 


8.28652 
8.28 761 
8.28 869 


8.28660 
8.28 769 
8.28 877 


3o 
20 

10 


14 o 

10 

20 


8.33292 
8. 333 9 o 

8.33488 


8.33 3o2 
8.334oo 
8.33498 


o 46 

5o 
4o 


7 o 

10 

20 


8.28977 
8.29085 
8.29 193 


8.28 986 
8.29 094 

8.29 2OI 


o 53 

5o 
4o 


3o 
4o 
5o 


8.33 585 
8.33682 
8.33779 


8.33 5 9 5 
8. 33 692 
8.33789 


3o 
20 

10 


3o 


8 . 29 3oo 


8.29 3og 


3o52 


15 o 


338 7 5 


8.33886 


o 45 




L. Cos. 


L. cotg. 


" ' 




L. Cos. 


L. Cotg. 





FUNCTIONS OP SMALL ANGLES 
1. 



/ tr 


L. Sin. 


L. Tang. 




, 


L. Sin. 


L. Tang. 




15 o 

10 
20 


8.33876 
8.33972 
8.34068 


8.33 886 
8.33982 
8.34078 


o 45 

5o 

4o 


22 3o 
4o 
5o 


8.38 oi4 
8.38 101 
8.38 189 


8.38 026 
8.38 n4 
8.38 202 


3o 

20 
10 


3o 
4o 
5o 


8.34 i64 
8.34260 
8.34355 


8.34174 
8.34270 
8.34366 


3o 
20 

10 


23 o 

10 

20 


8.38 276 
8.38363 
8.3845o 


8.38289 
8.383 7 6 
8.38463 


o 37 

5o 
4o 


16 o 

10 

eo 


8.3445o 
8.34546 
8.3464o 


8.3446i 
8.34556 
8.3465i 


o 44 

5o 
4o 


3o 

4o 
5o 


8.38 53 7 
8.38 624 
8.38 710 


8.38 550 
8.38636 
8.38 723 


3o 

20 
10 


3o 
4o 
5o 


8.34735 
8.3483o 
8.34924 


8.34746 
8.3484o 
8.34 9 35 


3o 
20 
10 


24 o 

10 

20 


8.38 796 
8.38882 
8.38,968 


8.38 809 
8.388 9 5 
8.38981 


o 36 

5o 
4o 


17 o 

I O 

20 


8.35oi8 
8.35 112 
8.35206 


8.35 029 
8.35 123 
8.35217 


o 43 

5o 
4o 


3o 

4o 
5o 


8.39 o54 
8.39 1 39 

8.39 225 


8.39067 
8.3 9 i53 
8.3 9 238 


3o 
20 

IO 


3o 
4o 
5o 


8.35 299 
8. 353 9 2 
8.35485 


8.353io 
8.354o3 
8.35497 


3o 
20 
10 


25 o 

10 

20 


8.3 9 3io 
8.3 9 3 9 5 
8.39480 


8.3 9 323 
8.39408 
8.39493 


o 35 

5o 
4o 


18 o 

10 

20 


8.35578 
8. 35 671 
8.35 7 64 


8.35 590 
8.35682 
8.35775 


o 42 

5o 

4o 


3o 
4o 
5o 


8.3 9 565 
8.39 649 
8.3 97 34 


8.3 9 5 7 8 
8.3 9 663 
8.3 97 47 


3o 
20 

IO 


3o 
4o 
5o 


8.35856 
8.35 9 48 
8.36 o4o 


8.35867 
8.35 9 5 9 
8.36o5i 


3o 
20 
10 


26 o 

IO 

20 


8.39818 
8.3g 902 
8.39986 


8.3 9 832 
8.39 916 
8.4o ooo 


o 34 

5o 
4o 


19 o 

10 
20 


8.36 i3i 
8.36223 
8.363i4 


8.36 i43 
8.3623s 
8.36326 


o 41 

5o 
4o 


3o 
4o 
5o 


8.4o 070 
8.4o i53 
8.40237 


8.4oo83 
8.4o i6 7 
8.4o25i 


3o 
20 

IO 


3o 
4o 
5o 


8.364o5 
8.36496 
8.36587 


8.36417 
8.365o8 
8.36 599 


3o 

20 
IO 


27 o 

10 

20 


8.4o 320 
8.4o4o3 
8.4o486 


8.4o334 
8.4o4i 7 
8.4o 500 


o 33 

5o 
4o 


20 o 

10 

20 


8.36678 
8.36768 
8.36858 


8.36 689 
8.36 780 
8.36870 


o 40 

5o 
4o 


3o 
4o 
5o 


8.40669 
8.4o65i 
8.40734 


8.4o583 
8.4o665 
8.4o 7 48 


3o 
20 

IO 


3o 
4o 
5o 


8.36948 
8.37038 
8.37 128 


8.36 960 
8.37050 
8.3 7 i4o 


3o 
20 

IO 


28 o 

IO 
20 


8.40816 
8.40898 
8.40980 


8.4o83o 
8.40913 
8.4o 995 


o 32 

5o 
4o 


21 o 

10 

20 


8.37217 
8.37 3o6 
8.37395 


8.37 229 
8.3 7 3i8 
8.3 7 4o8 


o 39 

5o 
4o 


3o 

4o 
5o 


8.4i 062 
8.4i 1 44 
8.4i 225 


8.4i o 77 
8.4i i58 
8.4i 240 


3o 

20 
10 


3o 
4o 
5o 


8.37484 
8.37 573 
8.37662 


8. 3 7 497 
8.3 7 585 
8.37674 


3o 
20 

10 


29 o 

IO 

20 


8.4i 307 
8.4i 388 
8.4i 469 


8.4i 32i 
8.4i4o3 

8.4i484 


o 31 

5o 
4o 


22 o 

10 

20 


8.37750 
8.3 7 838 
8.37 926 


8.37762 
8.3 7 85o 
8.37938 


o 38 

5o 
4o 


3o 
4o 
5o 


8.4i 55o 
8.4i 63i 
8.4i 711 


8.4i 565 
8.4i 646 
8.4i 726 


3o 

20 
IO 


3o 


8.38oi4 


8.38026 


3o37 


30 o 


Mi 79 2 


Mi 8o 7 


o'30 




L. Cos. 


L. Cotg. 


n , 




L. Cos. 


L. Cotg. 


' " 



128 



88. 



FUNCTIONS OF SMALL ANGLES. 

1. 



/ " 


L. Sin. L.Tang. 




/ ft 


L. Sin. L.Tang. 




30 o 

10 

20 
3o 
4o 
5o 


8.4i 79- 
8.4i 872 
8.4i 962 
8.42 o32 

8.42 112 
8.42 192 


8.4i 807 
8.4i 887 
8.4i 967 
8.42 o48 
8.42 127 
8.42 207 


o 30 

5o 
4o 
3o 
20 
10 


37 3o 
4o 
5o 


8.45 267 
8.4534i 
8.454i5 


8.45 285 
8.45 35 9 
8.45433 


3o 
20 

IO 


38 o 

IO 

20 
3o 
4o 
5o 


8.45489 
8.45563 
8.45637 
8. 45 710 
8. 45 7 84 
8.4585 7 


8. 455o 7 
8.4558i 
8.45655 

8.45728 
8.45 802 
8.45875 


o 22 

5o 
4o 
3o 

20 
IO 


31 o 

10 
20 
3o 
4o 
5o 


8.42 272 

8.4235i 
8.4243o 

8.42 5io 
8.42689 
8.42667 


8.42 287 
8.42 366 
8.42446 
8.42625 
8.42 6o4 
8.42683 


o 29 

5o 
4o 
3o 
20 

10 


39 o 

IO 
20 

3o 
4o 
5o 


8.45 930 
8.46oo3 
8.46076 
8.46 149 
8.46 222 
8.46 294 


8.45948 
8.46 021 
8.46094 

8.46 167 
8.46 240 
8.46 3i2 


oSl 

5o 
4o 
3o 

20 
10 


32 o 

10 
20 

3o 
4o 
5o 


8.42 746 
8.42825 
8.42903 
8.42982 
8.43 060 
8.43 i38 


8.42 762 
8.42 84o 
8.42 919 

8.42 997 
8.43075 
8.43x54 


o 28 

5o 
4o 
3o 
20 
10 


40 o 

10 

20 
3o 
4o 
5o 


8.46 366 
8.4643 9 
8.465n 

8.46583 
8.4665s 
8.46727 


8.46 385 
8.46457 
8.46529 

8.46602 
8.466 7 4 
8. 46 7 45 


o 20 

5o 
4o 
3o 

20 
10 


33 o 

10 
20 
3o 
4o 
5o 


8.43 2ltJ 

8.432 9 3 
8. 433 7 i 

8.43448 
8.43526 
8.436o3 


8.43232 
8.43 3o 9 
8. 4338 7 

8.43464 
8.43542 
8.436i 9 


o 27 

5o 
4o 
3o 
20 

10 


41 o 

IO 

20 
3o 
4o 
5o 


8.46 799 
8.46870 
8.46942 
8.47oi3 

8.47084 
8.47 i55 


8.46817 
8.46889 
8.46960 

8.47o3a 
8.47 io3 
8.47 i?4 


o 19 

5o 
4o 
3o 

20 
10 


34 o 

10 

20 
3o 
4o 
5o 


8.4^680 
8.43 7 5 7 
8.43834 
8.43910 
8.43987 
8.44o63 


8.43 696 
8.43773 
8.4385o 

8.43 927 
8.44oo3 
8. 44 080 


o 26 

5o 
4o 
3o 

20 
10 


42 o 

IO 

20 
3o 
4o 
5o 


8.47226 
8.47297 
8.47368 

8.47439 
8.47 509 
8.47 58o 


8.47245 
8.473i6 
8.47387 
8.47458 
8.47528 
8.47 599 


o 18 

5o 
4o 
3o 
20 

10 


35 o 

10 

20 

3o 
4o 
5o 


8.44 139 
8.44216 
8.44292 
8.4436 7 
8.44443 
8.445i 9 


8.44 1 56 
8.44232 
8.443o8 

8.44384 
8.4446o 
8.44536 


o 25 

5o 
4o 
3o 

20 
IO 


43 o 

IO 

20 
3o 

4o 

5o 


8.47 650 
8.47 720 
8.47 790 
8.47 860 
8.47 930 
8.48 ooo 


8.47 669 
8.47 74o 
8.47810 

8.47 880 

8.4795 
8.48 020 


o 17 

5o 
4o 
3o 
20 

10 


36 o 

10 
20 
3o 
4o 
5o 


8.44594 
8.44669 

8.44745 
8.44820 
8.44895 
8.44969 


8.446n 
8.44686 
8.44762 
8.44837 
8.44912 
8.44987 


o 24 

5o 
4o 
3o 

20 
IO 


44 o 

IO 
20 

3o 
4o 
5o 


8.48 069 
8.48 i3 9 
8.48 208 

8.48 278 
8.48 347 
8.484i6 


8.48090 
8.48 i5 9 
8.48 228 

8.48 298 
8.48 367 
8.48436 


o 16 

5o 
4o 
3o 
20 

10 


37 o 

10 

20 

3o 


8.45o44 
8.45 119 
8.45 193 

8.45 -j-;- 


8.45 061 
8.45 i36 
8.45 210 
8.45 285 


o 23 

5o 

4o 

3o22 


45 o 


J. 48 485 


M8 5o5 


o 15 




L. Cos. L. Cotg. 


" ' 




L. Cos. 


L. Cotg. 


" ' 



88. 



129 



FUNCTIONS OF SMALL ANGLES 
1. 



, 


L. Sin. 


L. Tang. 




, 


L.Sin. 


L. Tang. 




45 o 

10 

20 


8.48 485 
8.48 554 
8.48622 


8.48 5o5 
8.48574 
8.48643 


o 15 

5o 
4o 


52 3d 

4o 
5o 


8.5i 48o 
8.5i 544 
8. 5 1 609 


8.5i 5o3 
8.5i 568 
8.5i 63 2 


3o 

20 
IO 


3o 
4o 
5o 


8.48 691 
8.48 760 
8.48828 


8.48 711 
8.48 780 
8.48849 


3o 

20 
IO 


53 o 

10 

20 


8.5i 6 7 3 
8.5i 7 3 7 
8.5i 801 


8.5i 696 
8.5i 760 

8;5l 824 


o 7 

5o 
4o 


46 o 

10 
20 


8.. 48 896 
8.48 965 
8.49033 


8.48917 
8.48985 
8.49.053 


o 14 

5o 

4o 


3o 
4o 
5o 


8.5; 864 
8.5i 928 
8.5i 992 


8.5i 888 
8.5i 952 
8.52 oi5 


3o 
20 

IO 


3o 

4o 
5o 


8.49 101 
8.49 169 
8.49236 


8.49 121 
8.49 189 
8.49257 


3o 

20 
10 


54 o 

IO 

20 


8.52o55 
8.52 119 
8.52 182 


8 . 52 079 
8.52 i43 
8.52 206 


o 6 

5o 
4o 


47 o 

10 

20 


8.49 3o4 
8.49372 
8.49439 


8.49325 

8.49 3 9 3 
8.49460 


o 13 

5o 

4o 


3o 
4o 
5o 


8.52245 
8.5 2 3o8 
8.5 2 3 7 i 


8.52 269 
8.52 33 2 
8.52 3 9 6 


3o 
20 

10 


3o 
4o 
5o 


8.49 5o6 
8.49574 
8.49641 


8.49528 
8.49 5 9 5 
8.49 662 


3o 

20 
10 


55 o 

IO 

20 


8.5 2 434 
8.52 497 
8.52 56o 


8.52 459 

8.52 522 

8.52 584 


o 5 

5o 

4o 


48 o 

10 
20 


8.49 708 

8.49775 
8.49842 


8.49 729 
8.49 796 
8.49863 


o 12 

5o 
4o 


3o 
4o 
5o 


8.52623 
8.52685 
8.5 27 48 


8.52647 
8.52 710 

8.52 772 


3o 

20 
10 


3o 
4o 
5o 


8.49908 
8.49975 
8.5oo42 


8.49930 

8.49997 
8.5oo63 


3o 

20 
10 


56 o 

IO 

20 


8.52 810 
8.52872 
8.52935 


8.52 835 
8.52897 
8.52 960 


o 4 

5o 
4o 


49 o 

10 

20 


8.5o 108 
8.5o i 7 4 
8.5o24i 


8.5o i3o 
8.5o 196 
8.50263 


o 11 

5o 
4o 


3o 
4o 
5o 


8.52 997 
8.53o5 9 
8.53 121 


8.53022 
8.53o84 
8.53 i46 


3o 
20 

10 


3o 
4o 
5o 


8.5o 307 
8.5o373 
8.5o439 


8.5o329 
8.50395 
8.5o46i 


3o 
20 

IO 


57 o 

IO 
20 


8.53 i83 
8.53245 
8.533o6 


8.53 208 
8.53 270 
8.53 332 


o 3 

5o 

4o 


50 o 

10 
20 


8.5o 5o4 
8.5o 570 
8.5o636 


8.5o527 
8.5o5 9 3 
8.5o658 


o 10 

5o 

4o 


3o 
4o 
5o 


8.53368 
8.53429 
8.53491 


8.53 3 9 3 
8.53455 
8.535i6 


3o 

20 
IO 


3o 
4o 
5o 


8.5o 701 
8.5o 767 
8.5o83 2 


8.5o 724 
8.5o 789 
8.5o855 


3o 
20 

10 


58 o 

IO 
20 


8.53552 
8.536i4 
8. 53 675 


8.535 7 8 
8.5363 9 
8.53 700 


o 2 

5o 
4o 


51 o 

10 
20 


8.50897 
8.5o 9 63 
8.5i 028 


8.5o 920 
8.5o 9 85 
8.5i o5o 


o 9 

5o 

4o 


3o 
4o 
5o 


8.53 7 36 
8.53797 
8.53858 


8.53762 
8.53823 
8.53884 


3o 

20 
10 


3o 
4o 
5o 


8.5i 092 
8.5i i5 7 

8.5l 222 


8.5i u5 
8.5i 180 
8.5i 245 


3o 

20 
10 


59 o 

10 
20 


8.53919 
8. 53 979 
8 . 54 o4o 


8.53 9 45 
8.54oo5 
8. 54 066 


o 1 

5o 

4o 


52 o 

10 

20 


8.5i 287 
8.5i 35i 
8.5i 4i6 


8.5i 3io 
8.5i 3 7 4 
8.5i 439 


o 8 

5o 

4o 


3o 
4o 
5o 


8.54 ioi 
8.54 161 
8.54 222 


8.54 127 
8.54 187 
8. 54 s48 


3o 
20 

10 


3o 


8.5i 48o 


.5i 5o3 


3o 7 


60 o 


8.542* 


8.54 3o8 


> 




L. Cos. 


L. Cotg. 


,, * 




L.Cos. 


L. Cotg. 


' 



130 



88. 



TABLE IV 

FOUR-PLACE 
NAPERIAN LOGARITHMS 



NAPERIAN LOGARITHMS. 

LOGARITHMS OF POWERS OF 10. 



Num. 


Log. 




Num. 


Log. 


10 


2.3O26 




. i 


3~.6 97 4 


IOO 


4.6o52 




.01 


5.3 9 48 


1000 


6.9078 




.001 


7.0922 


IOOOO 


9.2103 




.0001 


^.7897 


IOOOOO 


1 1 .5129 




.00001 


72.4871 


IOOOOOO 


i3.8i55 




.00000 I 


a. 1845 


IOOOOOOO 


16.1181 




.000000 I 


17.8819 


IOOOOOOOO 


18.4207 




.00000001 


19.5793 


I OOOOOOOOO 


20.7233 




.00000000 1 


21 .2767 


Num. 


Log. 




Num. 


Log. 



LOGARITHMS OF NUMBERS FROM i TO 10. 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


1.0 


o.oooo 


OIOO 


0198 


0296 


0392 


o488 


o583 


o6 77 


0770 


0862 


.1 

.2 

.3 


0.0953 

0.1823 

0.2624 


io44 
1906 
2700 


n33 

1989 
2776 


1222 
2070 
2852 


i3io 

2l5l 

2927 


1398 

223l 

3ooi 


i484 

23ll 

3o 7 5 


i5 7 o 
2390 
3i48 


i655 
2469 

3221 


i 7 4o 
2546 
32 9 3 


.4 
.5 
.6 


0.3365 
o.4o55 
0.4700 


3436 

4l2I 

4762 


35o7 

4i8 7 
4824 


35 77 
4253 
4886 


3646 
43i8 
4947 


3 7 i6 

4383 
5oo8 


3784 
444 7 
5o68 


3853 
45n 

5i 2 8 


392O 

45 7 4 
5i88 


3 9 88 
463 7 
524 7 


7 

.8 

9 


o.53o6 

o.58 7 8 
0.6419 


5365 
5 9 33 
6471 


5423 
5988 
6523 


548i 
6o43 
6575 


5539 
6098 
6627 


55 9 6 
6i52 
66 7 8 


5653 
6206 
6-729 


5 7 io 
6259 
6 7 8o 


5 7 66 
63i3 
683i 


5822 

6366 
6881 


2.0 


0.6931 


6981 


7o3i 


7080 


7129 


7178 


722-7 


7275 


7 3 2 4 


7 3 7 2 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



NAPERIAN LOGARITHMS. 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


2.0 

2.1 

2.2 
2.3 


0.6931 


6981 


7o3i 


7080 


7129 


7178 


7 22 7 


7275 


7324 


7 3 7 2 


0.7419 
o. 7 885 
0.8329 


7 46 7 

79 3 
83 7 2 


75i4 
797 5 
84i6 


7 56i 
8020 
845 9 


7608 
8o65 
85o2 


7 655 
8109 

8544 


77 OI 

8i54 
858 7 


7747 
8198 
8629 


779 3 
8242 
8671 


783 9 
8286 
8 7 i3 


2.4 
2.5 
2.6 


o.8 7 55 
0.9163 
o. 9 555 


8796 
9203 
9^94 


8838 
9243 
9632 


88 79 

9 282 

9 6 7 o 


8920 
9822 
9708 


8961 
936i 
9746 


9002 
9400 
9783 


9042 
9 43 9 

9 82I 


9 o83 
9 478 
9 858 


9 I23 

9 5i 7 
9 8 9 5 


2.7 

2.8 

2.9 

3.0 

3.i 

3.2 

3.3 


0.9933 
1.0296 
1.0647 


9969 
o332 
0682 


6006 
0367 
0716 


6o43 
o4o3 
0750 


6080 
o438 
0784 


61 16 
o473 
0818 


Ol52 

o5o8 
o852 


018,8 
o543 
0886 


0225 

o5 7 8 
o 9 i 9 


6260 
o6i3 
o 9 53 


1.0986 


1019 


io53 


1086 


1119 


n5i 


n84 


1217 


I24 9 


1282 


I.i3i4 
i.i632 
i.i 9 3 9 


1 346 
i663 
1969 


i3 7 8 
1694 

2000 


i4io 
1725 
2o3o 


1 442 
i 7 56 
2060 


i4 7 4 
i 7 8 7 
2090 


i5o6 
181-7 

21 19 


i53 7 

1 848 

2l4 9 


i56 9 

1878 

21 79 


1600 
I 9 o 9 
2208 


3.4 
3.5 
3.6 


1.2238 
1.2528 

1.2809 


2267 
2556 
2 83 7 


2296 

2585 
2865 


2326 

26i3 

28 9 2 


2355 
2641 
2920 


2384 
2669 
2 9 4 7 


2 4i3 
2698 
29-75 


2442 
2726 

3002 


2470 
2754 

3o2 9 


24 99 

2782 
3o56 


3.7 
3.8 
3. 9 

4.0 

4.i 

4.2 

4.3 


i.3o83 
i.335o 
i.36io 


3uo 
3376 
3635 


3i3 7 
34o3 
366i 


3i64 
342 9 
3686 


3i 9 i 

3455 
3712 


3218 
348 1 
3 7 3 7 


3244 
35o 7 
3 7 62 


3271 

3533 

3 7 88 


3 297 
3558 
38i3 


3324 

3584 
3838 


1.3863 


3888 


3913 


3 9 38 


3962 


3 9 8 7 


4012 


4o36 


4o6i 


4o85 


i.4no 
i.435i 
1.4586 


4i34 
43 7 5 
4609 


4i5 9 
43 9 8 
4633 


4i83 

4422 

4656 


4207 
4446 
4679 


423i 
446 9 
4702 


4255 
4493 

4 7 2D 


42 79 
45i6 

4748 


43o3 

454o 
477<> 


432 7 
4563 
4 79 3 


4.4 
4.5 
4.6 


i.48i6 
i.5o4i 
1.5261 


483 9 
5o63 
5282 


486i 
5o85 
53o4 


4884 
5107 
5326 


4907 
5129 

534 7 


4929 
5i5i 
536 9 


49^1 

5i 7 3 
5390 


4 9 74 
5i 9 5 
54i2 


4 99 6 
5217 
5433 


5oi 9 
523 9 
5454 


4-7 
4.8 

4-9 
5.0 

5.i 

5.2 

5.3 


1.5476 
1.5686 
1.5892 


54 97 
5 7 o 7 
SgiS 


55i8 
5728 
5 9 33 


553 9 

5 7 48 
5 9 53 


556o 
5 7 6 9 
5 97 4 


558i 
5 79 o 
5 9 94 


56o2 
58io 
6oi4 


56 2 3 
583i 
6o34 


5644 
585i 
6o54 


5665 
58 7 2 
6o 7 4 


1.609^ 


6n4 


6i34 


6i54 


6i 7 4 


6194 


6214 


6233 


6 2 53 


62 7 3 


1.6292 
i.648 7 
1.6677 


63i2 
65o6 
6696 


6332 
65 2 5 
6 7 i5 


635i 
6544 
6 7 34 


63 7 i 
6563 
6752 


63go 
6582 
6771 


6409 
6601 
6-790 


642 9 
6620 

6808 


6448 
663 9 
6827 


646 7 
6658 
6845 


5.4 
5.5 
5.6 


1.6864 
1.7047 
1.7228 


6882 
7066 
7246 


6901 
7 o84 
7 263 


6919 
7102 
7281 


6938 
7120 
7299 


6 9 56 
7 i38 
7 3i 7 


69 7 4 
7 i56 
7 334 


6 99 3 
7 i 7 4 
7 352 


7011 

7 I 9 2 

7 3 7 o 


7029 

7 21O 

7 38 7 


5-7 
5.8 
5. 9 

6.0 


i. 7 4o5 
1.7579 
i.775o 


7422 

7 5 9 6 
7766 


744o 
7 6i3 

7783 


745 7 
763o 
7800 


7 4 7 5 
7 64 7 
7817 


7 4g2 
7 664 
7 834 


7 5o 9 
7 68i 
7 85i 


7 52 7 

7699 

7867 


7 544 
77 i6 

7 884 


7 56i 
7733 
79 oi 


1.7918 


79 34 


7951 


7967 


7984 


8001 


8017 


8o34 


8o5o 


8066 







1 


2 3 


4 


5 


6 


7 


8 


9 



i33 



NAPERIAN LOGARITHMS. 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


6.0 


1.7918 


79 34 


79 5i 


79 6 7 


794 


8001 


8017 


8o34 


8o5o 


8066 


6.1 

6.2 

6.3 


i.8o83 
.8245 
.84o5 


8099 
8262 
8421 


8116 

8278 
843 7 


8i32 
82 9 4 
8453 


8i48 
83io 
8469 


8i65 
8326 

8485 


8181 
8342 
85oo 


8i 97 
8358 
85i6 


8 2 i3 
83 7 4 
8532 


822 9 

83 9 o 

854 7 


6.4 
6.5 
6.6 


.8563 

.8718 
.8871 


85 79 
8733 
8886 


85 9 4 
8 7 4 9 
8 9 oi 


8610 
8 7 64 
8 9 i6 


8625 
8779 
8931 


864i 
8 79 5 
8 9 46 


8656 
8810 
8 9 6i 


86 7 2 

8825 
8 97 6 


868 7 
884o 
8 99 i 


8 7 o3 
8856 
9 oo6 


6.7 
6.8 
6.9 


.9021 
.9169 
.93i5 


9o36 
9184 
933o 


9 o5i 
9 i 99 
9 344 


9 o66 

9 2l3 

9 35 9 


9081 
9228 
9 3 7 3 


9 o 9 5 
9242 
9 38 7 


9 i 10 

9 25 7 
9 402 


9 I25 
9 2 7 2 

9 4i6 


9 i4o 
9 286 
9 43o 


9 i55 
9 3oi 
9 445 


7.0 


.9459 


9473 


9 488 


9 5o2 


95i6 


9 53o 


9 544 


9 55 9 


9 5 7 3 


9 58 7 


7-i 
7.2 
7 .3 


.9601 
. 97 4i 
.9879 


9 6i5 
97 55 
9892 


9 62 9 

9769 
99 o6 


9 643 
97 8a 

99 20 


9 65 7 
9796 
99 33 


9 6 7i 
9 8io 

9947 


9 685 
9 824 
99 6i 


9 6 99 

9 838 
9974 


97 i3 
9 85i 
99 88 


97 2 7 
9 865 

OOOI 


7-4 
7 .5 

7-6 


2. 001 5 

2.0149 
2.0281 


0028 
0162 
0295 


0042 
0176 
o3o8 


oo55 

oi8 9 

0321 


0069 

O2O2 

o334 


0082 
02 1 5 
o347 


oo 9 6 

O22 9 

o36o 


oio 9 
0242 
o3 7 3 


OI22 
0255 

o386 


oi36 
0268 
o3 99 


7-7 
7.8 

7-9 


2.0412 

2.o54i 

2.0669 


o425 
o554 
0681 


o438 
0567 
o6 9 4 


o45i 
o58o 
0707 


o464 
0592 
0719 


o477 
o6o5 
0732 


o4 9 o 

0618 
o 7 44 


o5o3 
o63i 

o 7 5 7 


o5i6 
o643 
o 7 6 9 


o528 
o656 

0-782 


8.0 


2.0794 


0807 


o8i 9 


o83 2 


o844 


o85 7 


o86 9 


0882 


o8 9 4 


o 9 o6 


8.1 

8.2 

8.3 


2 .0919 
2.I04I 

2.u63 


ogSi 
io54 
1175 


o 9 43 
1066 

1187 


o 9 56 
1078 
1199 


0968 
1090 

I2II 


o 9 8o 
1 1 02 

1223 


99 2 

1 1 14 
1235 


ioo5 
1126 

I24 7 


IOI 7 

n38 
1258 


IO2 9 

i i5o 

I2 7 O 


8.4 
8.5 
8.6 


2.1282 

2.l4oi 

2.i5i8 


1294 
1412 
1529 


i3o6 

1424 
i54i 


i3i8 
i436 
i55 2 


i33o 

1 448 
1 564 


1 342 

i45 9 
1576 


i353 
i4 7 i 
i58 7 


i365 

i483 
i5 99 


i3 77 
i4 9 4 
1610 


i38 9 
i5o6 
1622 


8.7 
8.8 
8.9 


2.i633 

2.1748 
2.1861 


1 645 
1759 

1872 


i656 
1770 

i883 


1668 
1782 
1894 


1679 
i 79 3 
I 9 o5 


i6 9 i 
1804 
I 9 i7 


1702 
i8i5 

I 9 28 


I 7 i3 
182-7 
i 9 3 9 


I 7 25 

i838 
i 9 5o 


i 7 36 
1849 
1961 


9.0 


2. 1972 


i 9 83 


i 99 4 


2006 


2017 


2028 


2o3 9 


2o5o 


2061 


2072 


9.1 
9.2 
9 .3 


2.2083 
2.2192 
2.23OO 


2094 

22O3 
23ll 


2IO5 
22l4 
2322 


21 16 

2225 
2332 


2127 

2235 

2343 


2i38 

2246 

2354 


2148 

225 7 

2364 


2l5 9 

2268 

23 7 5 


2I 7 O 
227 9 

2386 


2181 
2289 
2396 


9 .4 
9 .5 
9.6 


2.24O7 
2.25l3 

2.2618 


2418 

2523 

2628 


2428 

2 534 
2638 


243 9 

2544 
2649 


245o 
2555 
265 9 


2460 
2565 
2670 


24 7 r 
25 7 6 
2680 


2481 
2586 

26 9 O 


24 9 2 

25 9 7 
2701 


2502 
2607 
2711 


9-7 
9.8 
9.9 


2.2721 
2.2824 
2.2925 


2732 
2834 
2 9 35 


2742 
2844 
2 9 46 


2752 
2854 
2g56 


2762 
2865 
2 9 66 


2 77 3 

2875 

2 97 6 


2 7 83 

2885 
2 9 86 


2?9 3 
2 8 9 5 
2 99 6 


28o3 
2 9 o5 
3oo6 


28l4 
29l5 

3oi6 


10.0 


2.3026 


3i26 


3224 


3322 


34i8 


35i4 


36o 9 


3 7 o3 


3 79 6 


3888 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



1 34 



TABLE V 

FOUR-PLACE LOGARITHMS 
OF NUMBERS 



FOUR-PLACE LOGARITHMS. 



N 





1 


2 


3 


4 


5 




6 


7 


8 


9 


"IT 


oooo 


o43 


086 


128 


170 


212 


B^^^^B 

253 


BBBMH 

2 9 4 


334 


~ 


1 1 


4i4 


453 


4 9 2 


53i 


56 9 


607 


645 


682 


719 


7 55 


12 


792 


828 


864 


8 99 


9 34 


969 


100^ 


ro38 


1072 


1 106 


i3 


ii3 9 


I 7 3 


206 


23 9 


271 


3o3 


335 


36 7 


3 99 


43o 


i4 


46i 


492 


5 2 3 


553 


584 


6i4 


644 


6 7 3 


7 o3 


7 32 


i5 


1761 


79 


818 


84 7 


8 7 5 


903 


9 3i 


9 5 9 


9 8 7 


2OI, 


16 


20 


4i 


068 


o 9 5 


122 


1 48 


i 7 5 


2OI 


227 


253 


2 79 


l l 


3o4 


33o 


355 


38o 


4o5 


43o 


455 


48o 


5o4 


52 9 


18 


553 


577 


60 1 


625 


648 


672 


6 9 5 


718 


7 42 


7 65 


'9 


788 


810 


833 


856 


878 


900 


9 23 


9 45 


9 6 7 


989 


20 


3oio 


032 


o54 


o 7 5 


o 9 6 


118 


i3 9 


1 60 


181 


201 


21 


222 


243 


263 


284 


3o4 


324 


345 


365 


385 


4o4 


22 


424 


444 


464 


483 


5o2 


522 


54 1 


56o 


5 79 


5 9 8 


23 


617 


636 


655 


6 7 4 


6 9 2 


7' 


I 


729 


747 


766 


7 84 


24 


802 


820 


838 


856 


874 


8 9 2 


99 


927 


9 45 


9 62 


25 


3 979 


997 


4or4 


4o3i 


4o48 


4o65 


4082 


4o 99 


4n6 


4i33 


26 


4i5o 


1 66 


i83 


200 


216 


232 


24 9 


265 


281 


2 9 8 


27 


3i4 


33o 


346 


362 


3 7 8 


3 9 3 


4o 9 


425 


44o 


456 


28 


4 7 2 


48 7 


502 


5i8 


533 


548 


564 


5 79 


5 9 4 


6o 9 


2 9 


624 


63 9 


654 


66 9 


683 


698 


7 i3 


728 


742 


7 5 7 


30 


477i 


786 


800 


8i4 


829 


843 


85 7 


87, 


886 


9 oo 


3i 


9 i4 


928 


942 


9 55 


969 


9 83 


997 


5oi i 


5o24 


5o38 


32 


5o5i 


o65 


o 79 


9 2 


io5 


n 9 


i3a 


i45 


i5 9 


I 7 2 


33 


1 85 


198 


21 I 


224 


2 3 7 


25o 


263 


276 


28 9 


302 


34 


3i5 


3 2 8 


34o 


353 


366 


3 7 8 


3 9 i 


4o3 


4i6 


428 


35 


544 1 


453 


465 


4 7 8 


490 


502 


5i4 


527 


53 9 


55i 


36 


563 


5 7 5 


58 7 


5 99 


611 


623 


635 


647 


658 


6 7 o 


3 7 


682 


6 9 4 


7 o5 


717 


729 


74o 


7 52 


7 63 


77 5 


7 86 


38 


798 


8o 9 


821 


83 2 


843 


855 


866 


877 


888 


8 99 


3 9 


911 


9 22 


933 


9 44 


9 55 


9 66 


9 77 


9 88 


999 


6010 


40 

i 


6021 

MM^^^M 


o3i 




042 

HM^M 


o53 

Mi 


o64 


o 7 5 


o85 


o 9 6 


IO 7 


117 


N 







1 


2 


3 


4 


5 




6 


7 


8 


9 


PP 


38 


32 


28 


35 




22 


21 


19 




18 


17 


16 


.1 

.2 


7-6 


1! 






















> 5-6 


5-o 


.2 


4-4 


4-2 


3-8 


.2 


3-6 


3-4 


3- 2 


3 


11.4 


9 .e 


8.4 


7-5 


3 


6.6 


6-3 


5-7 


3 


5-4 




4.8 


4 


5-2 


I2.i 


II.2 


IO.O 


4 


8.8 


8.4 


7.6 


4 


7.2 


6.8 


6.4 


5 


19.0 


j6.c 


> 14.0 


12.5 


5 


I.O 


10.5 


9-5 


5 


9.0 


8-5 


8.0 


6 


22.8 


19.2 


16.8 


15.0 


.6 


3-2 


12.6 


11.4 


.6 


10.8 


10.2 


9.6 


7 


26.6 


22.4 


19.6 


J 7-5 


7 


5-4 


'4-7 


'3-3 


7 


12.6 


ii. 9 


II. 2 


.8 


30-4 


25.6 


22.4 


20.0 


.8 


7.6 


1 6.' 8 


'5-2 


.8 


14.4 


13-6 


12.8 


9 34-2 


28.8 25.2 22.5 


9 


9.8 18.9 








J5- 3 '4-4 



1 36 



FOUR-PLACE LOGARITHMS. 



N 





1 


2 


3 


4 




5 


6 




7 


8 


9 


40 


6021 


o3i 


042 


o53 


064 


M^M 

07 


HIM 

5 


o85 


^ 
096 


I0 7 


117 


4i 


128 


1 38 


i4 9 


160 


I 7 


180 


191 


201 


212 


222 


42 


232 


243 


253 


263 


2 7 4 


284 


294 


3o4 


3i4 


325 


43 


335 


345 


355 


365 


3 7 5 


385 


3 9 5 


4o5 


4i5 


425 


44 


435 


444 


454 


464 


4 7 4 


484 


4 9 3 


5o3 


5x3 


522 


45 


653 2 


542 


55i 


56i 


5 7 i 




58o 


5 9 o 


5 99 


6o 9 


618 


46 


628 


63 7 


646 


656 


665 


6 7 5 


684 


6 9 3 


702 


712 


47 


721 


7 3o 


7 3 9 


749 


7 58 


767 


776 


7 85 


794 


8o3 


48 


812 


821 


83o 


83 9 


848 


85 7 


866 


8 7 5 


884 


8 9 3 


49 


902 


911 


9 2O 


928 


937 


9 46 


9 55 


9 64 


972 


981 


50 


6990 


998 


7 oo 7 


7016 


7 024 


7o33 


7 042 


7 o5o 


75 9 


7 o6 7 


5i 


7076 


o84 


o 9 3 


IOI 


I IO 


118 


126 


i35 


i43 


152 


52 


1 60 


168 


I 77 


i85 


i 9 3 




202 


210 


218 


226 


235 


53 


243 


25l 


269 


267 


275 




284 


2 9 2 


3oo 


3o8 


3i6 


54 


324 


332 


34o 


348 


356 




364 


3 7 2 


38o 


388 


3 9 6 


55 


7 4o4 


4l2 


4i 9 


427 


435 




443 


45i 


45 9 


466 


4 7 4 


56 


482 


490 


497 


5o5 


5i3 




52O 


5 2 8 


536 


543 


55i 


5 7 


55 9 


566 


5 7 4 


582 


58 9 




597 


6o4 


612 


6i 9 


62 7 


58 


634 


642 


649 


65 7 


664 




672 


679 


686 


6 9 4 


701 


5 9 


709 


716 


7 23 


7 3i 


7 38 




745 


752 


760 


767 


774 


60 


7782 


789 


796 


8o3 


810 


818 


825 


832 


83 9 


846 


61 


853 


860 


868 


8 7 5 


882 




88 


9 


8 9 6 


9 o3 


9 io 


917 


62 


924 


9 3i 


9 38 


945 


952 




9 5 9 


9 66 


973 


9 8o 


987 


63 


99 3 


8000 


8oo 7 


8oi4 


802 1 




8028 


8o35 


8o4i 


8o48 


8o55 


64 


8062 


o6 9 


o 7 5 


082 


089 




o 9 6 


102 


io 9 


116 


122 


65 


8l2 9 


i36 




149 


i56 




162 


i6 9 


176 


182 


189 


66 


i 9 5 


202 


209 


2l5 


222 




228 


235 


2 


4r 


248 


254 


67 


261 


267 


2 7 4 


280 


28 7 




2 9 3 


2 99 


3o6 


3l2 


319 


68 


3 2 5 


33i 


338 


344 


35i 




35 7 


363 


3 7 o 


3 7 6 


382 


69 


388 


3 9 5 


4oi 


4o 7 


4i4 




420 


426 


432 


43 9 


445 


70 


45i 


45 7 


463 


4?o 


476 




482 


488 

^v 


4 9 4 

i^^M 1 


5oo 

^^V 


5o6 


N 









2 


3 


4 




5 




6 


7 


8 


9 


PP 


15 


14 13 


12 






IO 


9 




8 


7 


6 


., 


1.5 


1.4 1.3 


1.2 


.! 


i.i 


I.O 


0.9 


.1 


0.8 


0.7 


0.6 


.2 


3-o 


2.8 2.6 


2.4 


.2 


2.2 


2.0 


1.8 


.2 


1.6 


1.4 


1.2 


3 


4-5 


4-2 3-9 


3-6 


3 


3-3 


3- 


2.7 


3 


2.4 


2,1 


1.8 


4 


6.0 


5-6 5-2 


4-8 


4 


4-4 


4.0 


3-6 


4 


3-2 


2.8 


2.4 


5 


7-5 


7.0 6.5 


6.0 


5 




5-o 


4-5 


5 


4.0 


3-5 


3-o 


.6 


9.0 


8.4 7-8 


7.2 


.6 


6.6 


6.0 


5-4 


.6 


4.8 


4.2 


3-6 


7 


10.5 


9.8 9.1 


8.4 


7 


7-7 


7.0 


6-3 


7 


5-6 


4-9 


4.2 


.8 


I2.O 


1 1. 2 10.4 


9.6 


.8 


8.8 


8.0 


7.2 


8 


6.4 


5-6 


4.8 


9 i3-5 


126 11.7 10.8 .9 9.9 











i3 7 



FOUR-PLACE LOGARITHMS. 



N 







1 


2 


3 


4 


5 


6 


7 


8 


9 


70 


845i 


~ 


463 


470 


476 


482 


488 


4 9 4 


5oo 


5o6 


7 1 


5i3 


5i 9 


5 2 5 


53i 


53 7 


543 


54 9 


555 


56i 


56 7 


72 


5 7 3 


5 79 


585 


5oi 


5 97 


6o3 


6o 9 


6i5 


621 


62 7 


73 


633 


63 9 


645 


65i 


65 7 


663 


66 9 


6 7 5 


681 


686 


74 


692 


6 9 8 


704 


710 


716 


7 22 


727 


7 33 


7 3 9 


745 


75 


875! 




7 56 


762 


768 


774 


779 


7 85 


7 9 i 


797 


802 


76 


808 


8i4 


820 


8 2 5 


83i 


83 7 


842 


848 


854 


85 9 


77 


865 


871 


876 


882 


887 


8 9 3 


8 99 


9 o4 


9 io 


9 i5 


78 


9 2I 


9 2 7 


9 32 


9 38 


9 43 


949 


9 54 


9 6o 


9 65 


971 


79 


97 6 


9 82 


9 8 7 


99 3 


99 8 


9 oo4 


9 oo 9 


9 oi5 


9 O2O 


9025 


80 


9 o3i 


o36 


042 


047 


o53 


o58 


o63 


o6 9 


o 7 4 


079 


81 


o85 


o 9 o 


o 9 6 


IOI 


106 


112 


117 


122 


128 


i33 


82 


1 38 


i43 


i4 9 


1 54 


i5 9 


i65 


170 


i 7 5 


180 


186 


83 


| 9 I 


i 9 6 


2OI 


206 


212 


2I 7 


222 


227 


232 


2 38 


84 


243 


248 


253 


258 


263 


26 9 


2 7 4 


2 79 


284 


289 


85 


9 2 9 4 


2 99 


3o4 


3o 9 


3i5 


320 


3 2 5 


33o 


335 


34o 


86 


345 


35o 


355 


36o 


365 


3 7 o 


3 7 5 


38o 


385 


390 


87 


3 9 ! 


j 


4oo 


4o5 


4io 


4i5 


420 


425 


43o 


435 


44o 


88 


445 


45o 


455 


46o 


465 


46 9 


474 


479 


484 


48 9 


89 


4 9 4 


499 


5o4 


5o 9 


5i3 


5i8 


5 2 3 


5 2 8 


533 


538 


90 


9 542 


54 7 


55 2 


55 7 


562 


566 


5 7 i 


5 7 6 


58i 


586 


9 1 


5 9 ( 


) 


5 9 5 


600 


6o5 


6o 9 


6i4 


6i 9 


624 


628 


633 


92 


63( 


J 


643 


647 


652 


65 7 


661 


666 


671 


6 7 5 


680 


93 


685 


68 9 


6 9 4 


6 99 


7 o3 


7 o8 


7 i3 


717 


7 22 


727 


94 


7 3i 


7 36 


7 4 1 


745 


7 5o 


7 54 


7 5 9 


7 63 


7 68 


773 


9 5 


9777 


782 


786 


7 9 i 


7 9 5 


800 


8o5 


8o 9 


8i4 


818 


9 6 


823 


827 


832 


836 


84 1 


845 


85o 


854 


85 9 


863 


97 


868 


872 


877 


881 


886 


8 9 o 


8 9 4 


8 99 


- 9 o3 


9 o8 


9 8 


9 I2 


9 i7 


081 


9 26 


9 


3o 


9 34 


9 3 9 


9 43 


9 48 


9 52 


99 


9 56 


9 6i 


9 65 


9 6 9 


974 


97 8 


9 83 


9 8 7 


99 i 


99 6 


100 


oooo 


oo4 


oo 9 


oi3 


OI 7 


022 


026 


o3o 


o35 


o4o 


N 







1 


2 


3 




4 


5 


6 


7 


8 


9 


PP 


7 


6 


5 


4 


.1 


0.7 


0.6 


.1 0.5 


0.4 


.2 


M 


12 


.2 1.0 


0.8 


3 


2.1 


i 8 


3 *-5 


1.2 


.4 


2.8 


2.4 


.4 2.0 


1.6 


5 


3-5 


3 o 


3 2.5 


2.O 


.6 


4.2 


3-6 


.6 3.0 


2.4 


7 


4-9 


4.2 


7 3-5 


2.8 


.8 


5-6 


4.8 


.8 4-0 


3-2 


-9 




5-4 


9 4-5 3- 6 



i38 



TABLE VI 

FOUR-PLACE LOGARITHMS 

OF THE 

TRIGONOMETRIC FUNCTIONS 

TO EVERY TEN MINUTES 



POUR-PLACE LOGARITHMIC FUNCTIONS. 



O ' 


L. Sin. 


d. 


L.Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




o 

10 
20 

3o 
4o 
5o 


7 

7 

8 
8 


.463 7 
. 7 648 

.9408 
.o658 
.1627 


son 
1760 
1250 

969 
792 
66 9 
580 

5" 

458 
4i3 

578 




3011 
1761 
i2 49 
9 6 9 
792 
670 
580 
5" 

457 
4i5 
378 
348 
322 
300 
281 
263 
249 
235 
223 

213 

202 
194 
l8 5 
I 7 8 
I 7 I 
l6 5 
158 
54 
I 4 8 


2.5363 

2.2352 

2.0591 
1.9342 
i.83 7 3 


o.oooo 

. OOOO 

o.oooo 

o.oooo 
o.oooo 
o.oooo 




o 
o 





I 




o 



I 



I 



o 

I 



I 
I 
I 


o 90 

5o 

4o 

3o 
20 

10 


7 .463 7 
7 . 7 648 

7-9409 

8.o658 
8.1627 


1 o 

10 

20 

3o 
4o 
5o 


8 
8 
8 

8 
8 
8 


.2419 
.3o88 
.3668 

.4179 
.463 7 
.5o5o 


8.2419 
8.3089 
8.3669 

8.4i8i 

8.4638 
8.5o53 


i.758i 
i .691 i 
i. 633 i 

1.5819 
1.5362 

i .4947 


9-9999 
0.9999 

9.9999 

9.9999 
9.9998 
9.9998 


o 89 

5o 
4o 

3o 

20 
IO 


2 o 

10 
20 

3o 
4o 
5o 


8 
8 
8 

8 
8 
8 


.5428 
.5 77 6 
.6097 

.63 97 
.6677 
.6940 


348 
321 
300 
280 
263 
248 
235 

222 
212 
2O2 
I 9 2 


8.543i 
8.5 779 
8.6101 

8.64oi 
8.6682 
8.6945 


.456 9 
.4221 
.38 99 

.35 99 
.33i8 
.3o55 


9-9997 
9-9997 
9.9996 

9.9996 
9 . 999 5 
9-999^ 


o 88 

5o 
4o 

3o 
20 

IO 


3 o 

10 

20 

3o 
4o 
5o 


8 
8 
8 

8 
8 
8 


.7188 
. 7 423 
. 7 645 

.7867 
.8069 
.8261 


8.7194 
8.7429 
8.7652 

8. 7 865 
8.8067 
8.8261 


.2806 
.2571 
.2348 

.2i35 
.i 9 33 
.1739 


9-9994 
9.9993 
9.9993 

9.9992 
9.9991 
9-999 


o 87 

5o 
4o 

3o 
20 

IO 


4 o 

IO 
20 

3o 
4o 
5o 


8 
8 
8 

8 
8 
8 


.8436 
.86i3 

.8 7 83 

.8946 
.9104 
.9266 


185 
I 77 
170 
l6 3 
158 
IS 2 
M7 


8.8446 
8.8624 
8.8 79 5 

8.8960 
8.9118 
8.92-72 


.i554 
.1376 

.1205 

. io4o 

.0882 
.0728 


9.9989 
9.9989 
9.9988 

9.9987 
9.9986 
9 . 99 85 




I 


o 86 

5o 
4o 

3o 

20 
10 


5 





8.9403 


8.9420 


i.o58o 


9 . 99 83 




o 85 






L 


. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L.Sin. 


d. 


' O 


PP 

.2 

3 
4 

:I 
1 


348 


300 


263 


2 

3 
4 

I 
I 


235 


213 


185 


.i 

.2 

3 
4 

J 

:! 


171 


158 147 


S3 
104.4 

139.2 
174.0 
208.8 

j 


g 

9 o 

120 
150 
1 80 

210 
240 


26.3 

52-6 
78.9 

105.2 
i3i-5 
157-8 

184., 
210.4 




23-5 
47.0 
70.5 

94.0 
117.5 
141.0 

164.5 
i88.c 


21.3 

42. c 

63-9 

85.2 
106.5 

127.1 

149.1 
170.4 


18.5 

37-o 
55-5 

74.0 
92-5 

III.O 

129.5 
148.0 
166.5 


17.1 
34-2 
5i-3 

68.4 
85-5 

102.6 

119.7 
136.8 


15.8 14.7 

31.6 29.4 

47-4 44-i 

63.2 58.8 
79-o 73-5 
94.8 88.2 

no. 6 102.9 
126.4 117-6 

142.2 i3 2 -3 



POUR-PLACE LOGARITHMIC FUNCTIONS. 



O ' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




5 o 

IO 

20 

3o 

4o 
5o 


8.9403 
8. 9 545 
8.9682 

8.9816 
8.9945 
9.0070 


142 

137 
'34 
129 
125 

122 

"5 
"3 
109 
107 
104 

102 

99 
97 
95 
93 

89 
87 
85 
84 
82 
80 

79 
78 
76 
75 
73 
73 


8.9420 
8. 9 563 
8.9701 

8.9836 
8.9966 
9.0093 


143 
138 
135 
130 
127 
123 

120 

"4 
in 

1 08 
105 
104 

101 

98 

97 
94 
93 

89 
87 
86 

84 
82 
81 
80 
78 
77 
76 
74 


i.o58o 
1.0437 
1.0299 

i.oi64 
i.oo34 
0.9907 


9 . 99 83. 
9 . 99 82 
9 . 99 8i 

9 . 99 8o 
9-9979 
9-9977 


2 

I 
I 
2 
I 
I 
2 
I 
2 
2 
I 
2 
2 


o 85 

5o 

3o 
20 

IO 


6 o 

IO 

20 

3o 
4o 
5o 


9.0192 
9.081 i 
9.0426 

9.0539 
9.0648 
9.0755 


9.0216 
9.o336 
9.o453 

9.0667 
9.0678 
9.0786 


0.9784 
0.9664 
0.9547 

0.9433 
0.9322 
0.9214 


9-9976 
9-997 5 
9-997 3 
9.9972 
9.9971 
9.9969 


o 84 

5o 
4o 

3o 

20 
10 


7 o 

IO 

20 

3o 
4o 
5o 


9.0859 
9.0961 
9. 1060 

9 .n5 7 

9-1252 

9 .i345 


9.0891 
9 .o 99 5 
9. 1096 

9.1194 
9.1291 
9. i 385 


0.9109 
0.9005 
0.8904 

0.8806 
0.8709 
o.86i5 


9.9968 
9.9966 
9.9964 

9.9963 
9.9961 
9.9959 


o 83 

5o 

4o 

3o 
20 

IO 


8 o 

IO 

20 

3o 

4o 
5o 


9 .i436 

9. 1612 

9.1697 
9.1781 
9 .i863 


9.1478 
9.1569 
9.i658 

9.1745 
9.i83i 
9 .i 9 i5 


0.8522 

o.843i 
0.8342 

0.8255 
0.8169 
o.8o85 


9 . 99 58 
9 . 99 56 
9.9954 

9.9952 
9 . 99 5o 
9.9948 


I 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


o 82 

5o 
4o 

3o 

20 
10 


9 o 

IO 

20 

3o 
4o 
5o 


9.1943 
9.2022 
9.2100 

9.2176 
9.2324 


9-1997 
9 .2O78 
9 .2i58 

9.2236 
9.23i3 

9.2889 


o.8oo3 
0.7922 
0.7842 

0.7764 
0.7687 
0.7611 


9.9946 
9.9944 
9.9942 

9.9940 
9 . 99 38 
9 . 9936 


o 81 

5o 

4o 

20 
IO 


10 o 


9 .23 97 


9.2463 


o. 7 53 7 


9.9934 


o 80 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 


' o 


PP 138 


"5 


"7 


.1 

.2 

3 
4 

9 


104 97 


89 


.1 

.2 

3 
4 

I 


84 


78 73 


' I3 'J 
27. o 

3 4'-4 

4 55-2 
5 69.0 
.6 82.8 

.7 96.6 
.8 110.4 
.9 124.2 


12.5 
25.0 
37-5 

50.0 
62.5 
75- 

87-5 

1OO.O 

112.5 


11.7 
23-4 

46.8 
58.5 
70.2 

8..g 
93.6 

105-3 


10.4 9.7 
20.8 19.4 
31.2 29.1 

41.6 38.8 
52-0 48.5 
62.4 58.2 

72.8 67.9 
83.2 77.6 

93.6 87.3 


8. 9 
17.8 
26.7 

35-6 
44-5 
53-4 

62.3 
71.2 

80. i 


25-2 

33-6 
42.0 
50.4 

58.8 
67.2 


7-8 7-3 
15.0 14.6 
23.4 21.9 

31.2 29.2 
39-o 36.5 
46.8 43.8 

54-6 51.1 
62.4 58.4 
70. 2 65. 7 



i4r 



FOUR-PLACE LOGARITHMIC FUNCTIONS. 



O ' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. Cos. 


d. 




10 o 

10 

20 

3o 
4o 
5o 


9 
9 

9 

9 
9 
9 


. 2 3 97 
.2468 
.2538 

.2606 
.2674 
.2740 


71 
7 o 

68 

68 
66 


9. 2463 
9 . 2 536 
9 .2609, 

9.2680 
9.2750 
9.2819 


73 
73 
7i 
70 
69 
68 
66 
67 
6 5 
64 
63 
63 
61 
61 
61 
59 
59 
58 
57 
57 
56 
55 
55 
54 
53 
53 
53 
5 
52 
5i 


0.7537 
0.7464 
o. 7 3 9 i 

0.7320 
0.7250 
0.7181 


9 . 99 34 
9 . 99 3i 
9.9929 

9.9927 
9.9924 
9.9922 


3 

2 
2 

3 

2 


o 80 

5o 
4o 

3o 

20 
10 


11 o 

IO 
20 

3o 
4o 
5o 


9 

9 
9 

9 
9 
9 


.2806 
.2870 
.2 9 34 

.2997 
.3o58 
.3119 


64 
64 
63 
61 
61 


9.2887 
9.2953 
9 .3o2o 

9.3o85 
9 .3i49 

9 .32I2 


0.7113 
0.7047 
0.6980 

o.6 9 i5 

o.685i 

0.6788 


9.9919 
9.9917 
9.9914 

9.9912 
9.9909 
9.9907 


3 

2 

3 

2 

3 

2 


o 79 

5o 
4o 

3o 
20 

IO 


12 o 

10 
20 

3o 

4o 
5o 


9 
9 
9 

9 
9 
9 


3i79 
.3238 
.3296 

.3353 
.34io 
.3466 


59 
58 
57 
57 
56 


9 .3275 
9 .3336 
9 .33 97 

9 .3458 
9 .35 i 7 
9 .35 7 6 


0.6725 
0.6664 
o.66o3 

O.6542 
0.6483 
0.6424 


9.9904 
9.9901 
9.9899 

9.9896 
9.9893 
9.9890 


3 
3 

2 

3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
4 
3 
3 
4 


o 78 

5o 

4o 

3o 
20 

10 


13 o 

10 

20 

3o 
4o 
5o 


9 .352i 
9 .35 7 5 
9.3629 

9.3682 
9 .3 7 34 
9.3786 


55 
54 
54 
53 
5 2 
5 2 
5i 
5 
50 
49 
49 
48 

47 


9 .3634 
9 .36 9 i 
9 .3 7 48 

9 .38o4 
9 .385 9 
9 .3 9 i4 


0.6366 
o.63o 9 
0.6252 

o.6i 9 6 
o.6i4i 
0.6086 


9.9887 
9.9884 
9.9881 

9.9878 
9 . 9 875 
9.9872 


o 77 

5o 
4o 

3o 

20 
10 


14 o 

10 

20 

3o 
4o 
5o 


9 .383 7 
9 .388 7 
9 .3 9 3 7 

9 .3 9 86 
9 .4o35 
9 .4o83 


9 .3 9 68 
9.4021 
9 .4o74 

9.4127 
9.4178 
9.4a3o 


o.6o32 
o.5 9 7 9 
o.5 9 26 

0.5873 

0.5822 

0.5770 


9 . 9 86 9 
9 . 9 866 
9 . 9 863 

9 . 9 85 9 
9.9856 
9.9853 


o 76 

5o 

4o 

3o 
20 

IO 


15 





9 .4i3o 


9.4281 


o.5 7 i 9 


9 . 9 84 9 


o 75 






L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 


' o 


PP 

.2 

3 
4 

:l 


7i 


68 


66 


.1 

.2 

3 

4 

:! 
:i 

9 


64 61 58 


.1 

.2 

3 
4 

:! 

:i 


55 


53 5i 


7- 1 
14.2 
21.3 

28.4 

33 

Si 


6.8 
13-6 
20.4 

27.2 
34-o 
40.8 

47.6 
54-4 
61.2 


6.6 

!|! 

26.4 
33-o 
39-6 

46.2 
52.8 
59-4 


6.4 6.1 5.8 

12.8 12.2 II. 6 
19.2 l8. 3 17.4 

25.6 24.4 23.2 
32.0 30.5 29.0 
38.4 3 6.6 34 .8 

44.8 42.7 40.6 
5 1.2 48.8 46.4 

57- 6 54-9 52-2 


5-5 

II. 

16.5 

22.0 
27-5 

33-o 

38.5 

44.0 

49-5 


5-3 5-i 

10.6 10.2 

iS-9 '5-3 

21.2 20.4 
26.5 25.5 
3 1.8 30.6 

37- * 35^7 
42.4 40.8 

47-7 45-9 



142 



FOUR-PLACE LOGARITHMIC FUNCTIONS. 



O ' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d. 




15 o 

10 

20 

3o 

4o 
5o 


9 .4i3o 

9.4177 
9.4223 

9.4269' 
9 .43i4 
9.4359 


47 
46 
46 
45 
45 
44 
44 
44 

42 

43 
4 2 
4i 
4 1 
4i 
40 
40 
40 
39 
39 
38 
38 
37 
38 
36 
37 
36 
36 
35 
36 
35 


9.4281 
9 .433i 

9 .438i 

9 .443o 

9.4479 
9.4527 


50 
50 

49 
49 
48 
48 
47 
47 
47 
46 
46 
45 
45 
' 45 
44 
44 
44 
43 
43 
42 
42 
42 
42 
4i 
4* 
40 
40 
40 
40 
40 


o.5 7 i 9 
0.5669 
0.5619 

0.5570 
0.5521 

0.5473 


9.9849 
9.9846 
9.9843 

9 . 9 83 9 
9.9836 
9.9832 


3 
3 
4 
3 
4 


o 75 

5o 
4o 

3o 
20 

IO 


16 o 

10 
20 

3o 
4o 
5o 


9.44o3 
9-4447 
9.4491 

9.4533 
9 .45 7 6 
9.4618 


9 .45 7 5 
9.4622 
9.4669 

9.4716 
9.4762 

9.4808 


o.5425 
0.5378 
o.533i 

0.5284 
0.5238 
0.5192 


9.9828 
9.9825 
9.9821 

9.9817 
9.9814 
9.9810 


3 
4 
4 
3 
4 


o 74 

5o 

4o 

3o 

20 
IO 


17 o 

IO 

20 

3o 
4o 
5o 


9-465 9 
9.4700 
9.4741 

9.4781 
9.4821 
9.4861 


9.4853 
9.4898 
9.4943 

9.4987 
9-5o3i 
9.5075 


o.5i47 

O. 5 I 02 

o.5o57 

o.5oi3 
0.4969 
0.4925 


9.9806 
9.9802 
9.9798 

9.9794 
9.9790 
9.9786 


4 
4 
4 
4 
4 
4 


o 73 

5o 

4o 

3o 
20 

IO 


18 o 

10 
20 

3o 

4o 
5o 


9.4900 
9.4939 
9.4977 
9. 5oi5 
9-5o52 
9.5090 


9.5u8 
95i6i 
9.52o3 

9.5245 
9.5287 
9.5329 


0.4882 
0.4839 
0.4797 
o.4755 
o.47i3 
0.4671 


9.9782 
9.9778 
9-9774 

9-977 
9 . 9 765 

9-97 6 * 


4 
4 
4 
4 
5 
4 


o 72 

5o 
4o 

3o 

20 
IO 


19 o 

IO 

20 

3o 
4o 
5o 


9.5126 
9.5i63 
9.5199 

9.5235 
9.5270 
9.53o6 


9.5370 
9.54n 
9 .545i 

9.5491 
9.553i 

9.5571 , 


o.463o 
0.4589 
0.4549 

0.4509 
0.4469 
0.4429 


9.9757 
9.9752 
9-9748 

9.9743 
9.9739 
9.9734 


5 
4 
5 
4 
5 


o 71 

5o 

4o 

3o 
20 

10 


20 





9 .534i 


9.56n 


0.4389 


9. 97 3o 




o 70 




L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 


' O 


PP 

2 

3 

4 
5 
.6 

7 
.8 

9 


49 47 


45 


.1 

.2 
3 

4 

5 
.6 


44 


43 


41 




.1 

.2 

3 

4 
5 
6 

! 


40 


38 


36 


4-9 4-7 
9.8 9.4 
14.7 14.1 

19.6 18.8 
24-5 23.5 
29.4 28.2 

34-3 32.9 
39-2 37-6 
44- 1 42.3 


4-5 
9.0 
13-5 

18.0 
22.5 
27.0 

3'- 5 
30.0 

4-5 


ti 

13-2 
17.6 

22. 
26.4 

30.8 
35-2 


tl 

12.9 

17.2 
21.5 
25.8 

30.1 

34-4 

38.7 


t: 

12.3 

i6. 4 
20.5 
24.6 

28.7 

32.8 




4.0 

8.0 

12.0 

16.0 

20.0 
24.0 

28.0 
32.0 
36.0 


3-8 

7-6 
11.4 

15-2 
19.0 

22.8 

26.6 
30-4 

34-2 


3-6 

7-2 

10.8 

14.4 
18.0 

21.6 

25.2 
28.8 
32-4 



i43 



POUR-PLACE LOGARITHMIC FUNCTIONS. 



O ' 


L. Sin. 


d. 


L. Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d. 




20 o 

10 

20 

3o 
4o 
5o 


9 .534i 
9 .53 7 5 
9.5409 

9 .5443 
9.5477 
9.55io 


34 
34 
34 
34 
33 
33 
33 
33 
32 
32 
3 
32 
3i 
3 
3<> 
3* 
3 
3 
29 
3 
29 
29 

29 
28 
28 
28 
28 
28 
27 
27 


956i i 
9 .565o 
9. 5689 

9.5727 
9 .5 7 66 
9.58o4 


39 
39 
38 
39 
38 
38 
37 
38 
37 
37 
37 
36 
36 
36 
36 
36 
35 
36 
35 
34 
35 
34 
35 
34 
34 
33 
34 
33 

34 


33 


o.438 9 
o.435o 
o.43n 

0.4273 
0.4234 
o.4i 9 6 


9. 97 3o 
9.9725 
9.9721 

9.9716 
9.9711 
9.9706 


5 
4 
5 
5 
5 


o 70 

5o 
4o 

3o 

20 
10 


21 o 

10 
20 

3o 

4o 
5o 


9 .5543 
9.55 7 6 
9.5609 

9 .564i 
9 .56 7 3 
9.5704 


9-5842 
9 .58 79 
9.5917 

9 .5 9 54 
9 .5 99 i 
9.6028 


o.4i58 

0.4 I 21 

o.4o83 

o.4o46 
o.4oo 9 
o.3 9 72 


9 . 9 702 

9.9697 
9.9692 

9.9687 

9.9682 

9.9677 


5 

5 
5 
5 
5 


o 69 

5o 
4o 

3o 

20 
IO 


22 o 

10 
20 

3o 
4o 
5o 


9 .5 7 36 
9.5767 
9 .5 79 8 

9.5828 
9.5859 
9.5889 


9.6064 
9.6100 
9 .6i36 

9.6172 
9.6208 
9 .6243 


o.3 9 36 
o.3 9 oo 
0.3864 

0.3828 
o.37 9 2 
o.3 7 5 7 


9.9672 
9.9667 

9.9661 
9.9656 

9.9651 
9.9646 


5 
6 

5 
5 
5 


o 68 

5o 
4o 

3o 
20 

IO 


23 o 

10 

20 

3o 
4o 
5o 


9.5919 
9 .5 9 48 
9.5978 

9.6007 
9.6o36 
9.6o65 


9 .62 79 
9 .63i4 
9.6348 

9. 6383 
9.6417 
9. 6452 


0.3721 
0.3686 
0.3652 

0.3617 
0.3583 
0.3548 


9.9640 

9.9635 
9.9629 

9.9624 

9.9618 

9.9613 


5 
6 
5 
6 
5 


o 67 

5o 
4o 

3o 

20 
10 


24 o 

IO 

20 

3o 

4o 
5o 


9.6093 
9.6121 
9.6149 

9.6177 
9.6205 
9.6232 


9.6486 
9.6520 
9. 6553 

9 .658 7 
9 .662o 
9 .6654 


o.35i4 
o.348o 
0.3447 

o.34i3 
o.338o 
0.3346 


9.9607 

9.9602 

9.9596 

9.9590 
9 . 9 584 
9 . 9 5 79 


5 
6 
6 
6 
5 


o 66 

5o 
4o 

3o 

20 
10 


25 





9.6259 


9 .6687 


o.33i3 


9 . 9 573 




o 65 






L. Cos. 


d. 


L.Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 


' 


PP 

. i 

.2 

3 
4 

7 
.8 

9 


39 


37 


35 


,i 

.2 

3 

4 
5 
.6 

1 

9 


34 


33 


32 


2 

3 
4 

1 


31 


3 


29 


1:1 

11.7 

15-6 
'9-5 
23-4 

27-3 
3' ? 
35-i 


3-7 
7-4 
n. i 

14.8 
18.5 

22.2 

25-9 
29.6 


3-5 
7.0 
10.5 

14.0 
17-5 

21.0 

24-5 
28.0 

3'-5 


U 

10.2 

I 3 .6 
17.0 
20. 4 

2 3 .8 
2 7 .2 

30.6 


U 

9-9 

III 

19.8 

23.1 
26.4 

20. 7 


E 

9 .6 

12.8 

16.0 
19.2 

22.4 

25.6 


n 

9-3 
12.4 

;i:i 

J 

27.9 


3-o 
6.0 
9.0 

12.0 
15-0 

18.0 

21.0 
24.0 


I 1 

8.7 

ii. 6 
i4-5 
17-4 

20.3 
23.2 
26.1 



1 44 



FOUR-PLACE LOGARITHMIC FUNCTIONS. 



' 


L. Sin. 


d. 


L.Tang. 


d. 


L. Cotg. 


L. 


Cos. 


d. 




25 o 

10 

20 

3o 

4o 
5o 


9 
9 
9 

9 
9 
9 


.6269 

.6286 
.63i3 

.634o 
.6366 
.6392 


27 
27 
27 

26 
26 
26 
26 
26 
25 
26 

25 


9.6687 
9.6720 
9.6762 

9.6786 
9.6817 
9.6860 


33 
32 
33 
32 
33 
32 
32 
32 
31 
32 

32 
3 1 

3 

30 

30 
3* 
30 

30 
30 
29 
30 
29 
30 
29 
29 


o.33i3 
0.3280 
0.3248 

o.32i5 
o.3i83 
o.3i5o 


9.9673 
9.9667 
9.9661 

9.9666 
9 . 9 54 9 
9.9643 


6 
6 
6 

6 
6 


o 65 

5o 
4o 

3o 
20 
10 


26 o 

10 
20 

3o 
4o 
5o 


9 
9 
9 

9 
9 
9 


.64i8 
.6444 
.6470 

.6496 
.6621 
.6546 


9.6882 
9.6914 
9.6946 

9.6977 
9.7009 
9.7040 


o.3n8 
o.3o86 
o.3o54 

o.3o23 
0.2991 
0.2960 


9 . 9 53 7 
9 . 9 53o 
9 . 9 524 

9.9618 
9.9612 
9.9606 


7 
6 
6 
6 
7 


o 64 

5o 
4o 

3o 

20 
10 


27 o 

10 
20 

3o 
4o 
5o 


9 
9 
9 

9 
9 
9 


.6670 
.6696 
.6620 

.6644 
.6668 
.6692 


24 
25 
25 
24 
24 
24 


9.7072 
9.7103 
9 . 7 i 34 

9.7166 
9 . 7 i 9 6 

9 . 7 226 


0.2928 
0.2897 
0.2866 

0.2835 
0.2804 

0.2774 


9.9499 
9.9492 
9.9486 

9.9479 
9.9473 
9.9466 


7 
6 

7 
6 

7 


o 63 

5o 

4 
3o 
20 
10 


28 o 

10 
20 

3o 
4o 
5o 


9 
9 
9 

9 
9 
9 


.6716 
.6740 
.6 7 63 

.6787 
.6810 
.6833 


24 
23 
24 
23 
23 


9-7 25 7 
9.7287 
9.7317 

9 . 7 348 
9.7408 


o.2 7 43 
0.2713 
0.2683 

0.2662 
0.2622 
0.2692 


9.9459 
9.9453 
9.9446 

9.9439 
9.9432 
9.9426 


6 
7 
7 
7 
7 


o 62 

5o 
4o 

3o 

20 
10 


29 o 

10 

20 

3o 
4o 
5o 


9 
9 
9 

9 
9 
9 


.6856 
.6878 
.6901 

.6923 
.6946 
.6968 


22 

23 
22 

23 
22 


9 . 7 438 
9.7467 
9.7497 

9.7626 
9.7666 
9 . 7 585 


0.2662 
0.2533 
o.25o3 

0.2444 
0.24 i 5 


9.9418 
9.9411 
9 . 9 4o4 

9.9397 
9.9390 
9 . 9 383 


7 
7 
7 

7 
7 


o 61 

5o 
4o 

3o 
20 
10 


30 





9 


.6990 




9.7614 


0.2386 


9 . 9 3 7 5 




o 60 




L 


. Cos. 


d. 


L. Cotg. 


d. 


L.Tang. 


L. Sin. 


d. 


' O 


PP 

.2 

3 
4 

:J 

9 


28 


27 


26 


.1 

.2 

3 
4 

:i 


25 


24 23 


.1 


22 


7 


6 


2.8 

5-6 
8.4 

II. 2 
14.0 

16.8 

19.6 
22.4 


2-7 

ti 

10.8 
13-5 
16.2 

18.9 

21.6 

24-3 


2.6 

10.4 

13.0 
15.6 

18.2 

20.8 

23-4 


2-5 

7-5 

10.0 

12.5 
15-0 

17-5 

20.0 

22-5 


2.4 2.3 

4 .8 4 .6 
7.2 6.9 

9.6 9.2 

12.0 II.5 
14.4 13.8 : 

16.8 16.1 i 
19.2 18.4 
21.6 20.7 


2.2 

n 

8.8 

II.O 

13-2 

19.8 


0.7 
1.4 

2.1 

2.8 

3-5 
4.2 

4.9 
5-6 
6-3 


0.6 

1.2 

1.8 

2-4 

5-4 



i45 



FOUR-PLACE LOGARITHMIC FUNCTIONS 



o 


r 


L. Sin. 


d. 


L. Tang. 


d. 


L.Cotg. 


L. Cos. 


d. 




30 o 

10 
20 

3o 
4o 
5o 


9.6990 

9 . 7 OI2 
9.7088 

9.7O55 
9.7076 
9.7097 


22 
21 
22 
21 
21 
21 
21 
21 
21 
20 
21 
20 
20 
20 
20 
20 
20 
19 
19 
20 

19 

18 

18 

18 
18 


9- 7 6i4 
9.7644 
9-7673 

9-77i 
9-77 3 o 
9.7759 


30 
29 
28 

29 
29 


0.2886 
0.2356 
0.2827 

0.2299 
0.2270 

O.224l 


9 . 9 3 7 5 
9.9868 
9.9861 

9.9353 
9.9346 
9-9338 


7 
7 

8 

7 
8 


o 60 

5o 
4o 

3o 

20 
10 


31 o 

10 
20 

3o 
4o 
5o 


9.7II8 

9.7160 

9.7181 
9.7201 
9. 7222 


9.7788 
9.7816 
9. 7 845 

9.7878 
9.7902 
9.7980 


29 
28 

29 
28 
29 
28 
28 
28 
28 
28 
28 
27 
28 
28 
27 
28 

27 
28 
27 
27 
27 
27 
27 
27 


O.22I2 
0.2184 

o.2i55 

0.2127 
0.2098 
0.2070 


9 . 9 33i 
9 . 9 323 
9.9815 

9.9808 
9.9800 
9.9292 


7 
8 
8 
7 
8 
8 


o 59 

5o 
4o 

3o 

20 

10 


32 o 

10 

20 

3o 

4o 

5o 


9.7242 
9.7262 
9.7282 

9.7802 
9.7822 
9.7342 


9 . 79 58 
9.7986 
9.8014 

9.8042 
9.8070 
9.8097 


O.2O42 
O.2OI4 
0.1986 

0.1958 

o. 1980 
o. 1908 


9.9284 

9.9268 

9.9260 
9.9252 
9.9244 


8 
8 
8 
8 
8 


o 58 

5o 
4o 

3o 
20 

10 


33 o 

10 

20 

3o 
4o 

5o 


9 . 7 36i 
9. 7 38o 
9 . 7 4oo 

9.7419 
9. 7 438 
9.7457 


9.8125 
9 .8i53 
9.8180 

9.8208 
9.8235 
9.8268 


0.1875 

0.1847 
o. 1820 

0.1792 
o. 1765 

o.i 7 3 7 


9.9286 
9.9228 
9.9219 

9.9211 
9.9208 
9.9194 


8 

9 
8 
8 
9 


o 57 

5o 

4o 

3o 
20 
10 


34 o 

10 
20 

3o 

4o 
5o 


9.7476 
9.7494 
9 . 7 5i3 

9 . 7 53i 
9 . 7 55o 
9 . 7 568 


9.8290 
9 .83i 7 
9. 8344 

9 .83 7 i 
9.8898 
9.8425 


0. I 7 IO 

0.1688 

o.i656 

o. 1629 
o. 1602 
o. i5 7 5 


9.9186 
9.91-77 
9.9169 

9.9160 
9. 9 i5i 
9.9142 


9 
8 

9 
9 
9 
8 


o 56 

5o 
4o 

3o 

20 
IO 


35 





9 . 7 586 


9.8452 


27 


o.i548 


9.9184 


o 55 






L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L.Sin. 


d. 


' O 


PP 

2 

3 

4 

j 


29 28 


27 


.2 

3 

4 
5 
.6 

9 


22 


21 


20 


.2 

3 
4 


19 


8 7 


2.9 2.8 
5-8 5-6 
8.7 8.4 

II. 6 II. 2 

14-5 14-0 
17.4 16.8 

20.3 19.6 
23.2 22.4 
26.1 25.2 


2.7 

10.8 
16.2 
18.9 

21.6 

24-3 


2.2 

a 

8.8 

II. 

13-2 

\7'.6 

IQ.8 


2.1 

8-4 
10.5 

12.6 

i6!8 
18.9 


2.0 
4-0 

6.0 

8.0 

10.0 
12. 

14.0 

16.0 

18.0 


'9 

3-8 
5-7 

7.6 
9-5 
11.4 

13-3 
15-2 


0.8 0.7 
1.6 1.4 
2.4 2.1 

3-2 2.8 

4- 3-5 
4.8 4.2 

5-6 4-9 
6.4 5.6 



i46 



POUR PLACE LOGARITHMIC FUNCTIONS. 



O I 


L. Sin. 


d. 


L.Tang. 


d. 


L. Cotg 


L. Cos. 


d. 




35 o 

10 
20 

3o 

4o 
5o 


9.7686 
9.7604 
9.7622 

9.7640 
9.7667 
9.7676 


18 
18 
18 

18 
18 
'7 


9.8462 

9.8479 
9.8606 

9.8533 
9.8669 
9.8686 


27 
27 
27 
26 

27 
27 
26 
27 
26 

27 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

25 
26 
26 
25 


o.i548 
o. 1621 
0.1494 

o. 1467 
o. i44i 
o. i4i4 


ON ON ON ON ON ON 


9134 
9126 
9116 

9107 
9098 
9089 


9 
9 

9 
9 
9 
9 

IO 

9 
9 

10 

9 


o 55 

5o 

4o 

3o 

20 
10 


36 o 

10 

20 

3o 

4o 
5o 


9 

9 
9 

9 
9 
9 


.7692 
.7710 
.7727 

7744 
.7761 

.7778 


9 .86i3 
9 .863 9 
9.8666 

9.8692 

9.8718 
9.8746 


0.1387 
o. i36i 
o.i334 

o.i3o8 
0.1282 
0.1266 


9.9080 
9.9070 
9.9061 

9.9062 
9.9042 
9.9033 


o 54 

5o 
4o 

3o 

20 
10 


37 o 

IO 

20 

3o 
4o 
5o 


9.7796 
9.7811 
9.7828 

9.7844 
9.7861 

9.7877 


I 7 
16 

16 
'7 
16 


9.8771 
9.8797 
9.8824 

9.8860 
9.8876 
9.8902 


o. 1229 

O. I2O3 

o. 1176 
o. i 160 

O. I 124 

0.1098 


9.9023 
9.9014 
9.9004 

9.8996 
9.8986 
9.8976 


9 

10 

9 

10 
10 


o 53 

5o 
4o 

3o 

20 
IO 


38 o 

10 

20 

3o 
4o 

5o 


000 000 


.7893 
.7910 
.7926 

.7941 
.7967 
.7973 


16 

15 
16 
16 


9.8928 
9.8964 
9.8980 

9.9006 
9.9032 
9.9068 


o. 1072 
o . i o46 

O. IO2O 

0.0994 

0.0968 
0.0942 


9.8966 
9.8966 

9.8946 

9.8935 
9.8926 
9.8916 


10 
10 
10 
10 
10 


o 52 

5o 

4o 

3o 

20 
IO 


39 o 

10 

20 

3o 
4o 
5o 


000 000 


.7989 
.8oo4 
.8020 

.8o35 
.8o5o 
.8066 


15 
16 

15 
15 
16 


9.9084 
9.9110 
9 . 9 i35 

9.9161 
9.9187 
9.9212 


0.0916 
0.0890 

0.0866 
0.0839 

o.o8i3 
0.0788 


9.8906 
9.8896 

9.8884 

9.8874 
9.8864 
9-8853 


10 

II 

10 
10 

II 


o 51 

5o 
4o 

3o 
20 

IO 


40 





9 


.8081 




15 


9.9238 


26 


0.0762 


9.8843 




o 50 




L 


.Cos. 


d. 


L. Cotg. 


d. 


L.Tang. 


L. Sin. 


d. 


' 


PP 

.1 

.2 

3 

4 
5 
.6 

7 
.8 

9 


36 


as 


18 


.2 

3 
4 

7 

.8 


I? 


if 


15 


.1 

.2 

3 

4 

5 
.6 

7 

.8 

9 


ii 


IO 


9 


2.6 
5-2 

7.8 

10.4 

13.0 

15-6 

18.2 

20.8 

2 3-4 


2-5 

5-o 
7-5 

IO.O 

12.5 
15.0 

17-5 

20. o 

22.5 


1.8 
3-6 
5-4 

7.2 
9.0 
10.8 

12.6 

14.4 

' 


3-4 

6.8 
8-5 

10.2 

II-9 
13-6 

'5-3 


1.6 

tf 

6.4 
8.0 
9.6 

II. 2 

12.8 

14.4 


3-o 
4-5 

6.0 
7-5 
9.0 

10.5 
12:0 


i.i 

2.2 

3-3 
4-4 

1:1 
1.1 


1.0 

2.O 

4.0 
6.0 

7.0 
8.0 

9.0 


0.9 
1.8 
2.7 

3-6 
4-5 
5-4 

6-3 
7.9 
8.1 



FOUR-PLACE LOGARITHMIC FUNCTIONS. 



O ' 


L. Sin. 


d. 


L.Tang. 


d. 


L. Cotgr. 


L. Cos. 


d. 




40 o 

IO 
20 

3o 

4o 
5o 


9.8081 
9.8096 
9.8111 

9.8126 
9.8140 
9 .8i55 


15 
15 
14 
15 
15 


9.9238 
9.9264 
9.9289 

9 . 9 3i5 
9.9341 
9.9366 


26 
25 
26 
26 

25 

26 

25 
26 

25 

26 

25 

25 
26 

25 
26 

25 

25 
26 

25 
25 
25 
25 
25 
25 
26 

25 
25 
25 
26 
25 


0.0762 
0.0736 
0.071 i 

o.o685 
0.0669 
o.o634 


9.8843 
9.8832 
9.8821 

9.8810 
9.8800 
9.8789 


ii 
ii 

10 

II 


o 50 

5o 
4o 

3o 
20 

IO 


41 o 

IO 

20 

3o 
4o 
5o 


9.8169 
9.8184 
9.8198 

9.8213 

9.8227 

9.8241 


15 
14 
15 
14 
14 


9.9392 
9.9417 
9.9443 

9.9468 
9.9494 
9 . 9 5i 9 


0.0608 
o.o583 
o.o557 

o.o532 
o.o5o6 
o.o48i 


9 

9 
9 

9 
9 
9 


.8778 
.8767 
.8 7 56 

.8 7 45 
.8 7 33 
.8722 


II 
II 
II 

12 
II 


o 49 

5o 
4o 

3o 

20 
IO 


42 o 

IO 

20 

3o 
4o 
5o 


9.8255 
9.8269 
9.8283 

9.8297 
9-83ii 
9 .8324 


4 
H 
J 4 
14 
3 


9.9544 
9.9570 
9.9595 

9.9621 
9.9646 
9.9671 


o.o456 
o.o43o 
o.o4o5 

0.0379 
o.o354 
0.0329 


9.8711 
9.8699 

9.8688 

9.8676 
9.8665 
9.8653 


12 
II 
12 
II 
12 


o 48 

5o 
4o 

3o 
20 

IO 


43 o 

IO 

20 

3o 
4o 
5o 


9.8338 
9.835i 
9.8365 

9.8378 
9.8391 
9-84o5 


'3 
'4 
3 
13 
4 


9.9697 
9.9722 
9.9747 

9.9772 
9.9798 
9.9823 


o.o3o3 
0.0278 
0.0253 

0.0228 

O.O2O2 
0.0177 


9 

9 
9 

9 
9 
9 


.864i 
.8629 
.8618 

.8606 
.85 9 4 

.8582 


12 
II 
12 
12 
12 


o 47 

5o 

4o 

3o 

20 
10 


44 o 

IO 
20 

3o 
4o 
5o 


9.8418 
9-843i 
9-8444 

9 .845 7 
9.8469 
9.8482 


3 
3 
13 

12 

3 


9.9848 
9.9874 
9.9899 

9 . 9924 
9.9949 
9.9975 


0.0152 

0.0126 

O.OIOI 

0.0076 
o.oo5i 

O.OO25 


9 
9 
9 

9 
9 
9 


.8569 

.855 7 
.8545 

.8532 
.8520 
.85o 7 


12 
12 
3 
12 

13 


o 46 

5o 

4o 

3o 
20 

IO 


45 o 


9.8495 




0,0000 


o.oooo 


9 


.8495 




o 45 






L. Cos. 


d. 


L. Cotg. 


d. 


L. Tang. 


L. Sin. 


d. 


' O 


PP 

.1 

.2 

3 

4 

:i 

.1 

9 


26 


5 


15 




M 


13 


12 


i 

.2 

3 

4 

:! 
:l 

9 


IX 


IO 


2.6 
5-2 

78 

10.4 

130 

15-6 
182 

20.8 

23 4 


2-5 

5-o 
7-5 

10.0 

12 5 

150 

17 5 

20.0 


* 5 
3-o 

4-5 

60 

75 

9.0 

10.5 

I2.O 

1 3-S 


.2 

3 
4 

:! 
;J 

9 


a 

4-2 

5-6 
7-0 
8.4 

9.8 

H.2 

12.6 


a 

3-9 

5-2 

6-5 

7.8 

9.1 
104 
11.7 


1.2 

y 

4 .8 

6.0 
7.2 

8-4 
9.6 

10.8 


I.I 

2.2 

3-3 
4.4 

1:1 

7-7 
8.8 

9-9 


1.0 
2.O 
3-0 

4-0 

5- 
6.0 

7.0 
8.0 
9.0 



i48 



TABLE VII 

FOUR-PLACE 

NATURAL TRIGONOMETRIC 
FUNCTIONS 

TO EVERY TEN MINUTES 



FOUR-PLACE NATURAL FUNCTIONS. 



O ' 


Sin. 


d. 


Tang. 


d. 


Cotg. 


d. 


Cos. 


d. 




o 

10 

20 

3o 
4o 
5o 


o.oooo 
0.0029 
0.0068 

0.0087 
o.oi 16 
o.oi45 


29 
29 
29 
29 
29 
3 
29 
29 
29 
29 
29 


o 
o 

o 

o 



o 


.0000 
.0029 

.0068 

.0087 
.01 16 

.0145 


29 
29 
29 
29 
29 

3 
29 
29 
29 
29 
29 
29 
29 
29 

3 
29 
29 
29 
29 
29 

3 
29 
29 
29 
30 
29 
29 
29 
30 
29 


infinit. 

343.773 7 
171.8864 

114.6887 
86.9398 
68.7601 


it* 

818 
613 
477 
382 
312 
2 6c 

22C 

i8fi 
163 
H3 
126 

112 

IOC 

9 c 
81 
7A 
6i 
6-. 
5y 
55 
4C 
4! 
4- 
3< 




i 

i 





.0000 
.0000 
.0000 

.0000 
9999 
9999 


o 
o 

o 
I 
o 


o 90 

5o 

4o 

3o 
20 

10 


1 o 

IO 
20 

3o 

4o 
5o 


0.0176 
0.0204 
0.0233 

0.0262 
0.0291 

O.O320 




o 

o 

o 



o 


.0176 
. oo4 
.0233 

.0262 
.0291 

032O 


67.2900 
49. 1039 
42.9641 

38.i885 
34.3678 
3i.24i6 


61 
98 
56 
07 
62 


o 
o 
o 






.9998 
.9998 
9997 

9997 
.9996 
.9996 




i 
o 

I 

I 
I 

2 

I 
I 


o 89 

5o 

4o 

3o 
20 

10 


2 o 

10 

20 

3o 
4o 
5o 


o.o349 
0.0378 

0.0407 

o.o436 
o.o465 
o.o494 


29 
29 
29 
29 
29 


o 



o 






.o349 
.o3 7 8 
.0407 

.o43 7 
.o466 
.0496 


28.6363 
26.43i6 
24.54i8 

22.9038 
21.4704 
20.2066 


bj 

47 
98 
80 

34 

48 


o 






o 
o 


.9994 
. 999 3 
.9992 

999 
.9989 

.9988 


o 88 

5o 

4o 

3o 

20 
IO 


3 o 

10 

20 

3o 
4o 
5o 


0.0623 
0.0662 
o.o58i 

0.0610 
o .o64o 
0.0669 


29 
29 
29 

30 
29 




o 



o 
o 




.0624 
.o553 
.0682 

.0612 
.o64i 
.0670 


19.0811 
18.0760 
17.1693 

16.3499 
i5.6o48 
14.9244 


4b 
61 

57 
94 
5i 
04 

37 
40 
98 
07 
57 
43 
,61 


o 



o 






.9986 
.9986 
. 99 83 

.9981 
.9980 
.9978 


I 
2 
2 
I 
2 


o 87 

5o 
4o 

3o 
20 

10 


4 o 

10 
20 

3o 

4o 
5o" 


0.0698 
0.0727 
0.0766 

0.0786 
0.08 i 4 
o.o843 


29 
29 
29 
29 
29 
29 











.0699 
.0729 
.0768 

.0787 
.0816 
.o846 


i4.3oo7 
13.7267 
13.1969 

12. 7062 
I2.25o5 
11.8262 









o 


.9976 
9974 
.9971 

.9969 
.9967 
.9964 


2 

3 

2 
2 

3 


o 86 

5o 

4o 

3o 

20 
10 


5 o 


o. 


0872 





.0876 


ii.43oi 





.9962 




o 85 




Cos. 


d. 


Cotg. 


d. 


Tang. 


d. 


Sin. 


d. 


/ 


PP 

.2 

3 
4 

J 

:i 

9 


26053 


16380 


i "45 


.i 

.2 

3 

4 
5 
.6 

7 
.8 

9 


8194 


6237 


4907 


.1 

.2 

3 

4 

i 

:I 

9 


3961 


30 


29 


2605 
5211 
7816 

10421 
13027 
15632 

18237 
20842 

23448 


1638 
3276 
4914 

6552 
8190 
9828 

11466 
13104 

14742 


1125 
2249 
3374 

4498 
5 62 3 
6747 

7872 
8996 

IOI2I 


819.4 
1638.8 
2458-2 

3277-6 
4097.0 
4916.4 

5735-8 
6555.2 
r 374-6 


623.7 
1247.4 
1871.1 

2494.8 

3"8.s 
3742.2 

4365-9 
4989.6 


490.7 
981.4 
1472.1 

1962.8 
2453-5 
2944.2 

3434-9 
3925.6 
4416. 


396-1 
792.2 
1188.3 

1584-4 
1980.5 
2376.6 

2772.7 
3168.8 
$564-9 


3-0 
6.0 
9.0 

12.0 
15-0 

18.0 

21.0 
24.0 


2.9 

5-8 

8-7 

ii. 6 
14-5 
17.4 

20.3 
23.2 
26.1 



160 



POUR-PLACE NATURAL FUNCTIONS. 



' 


Sin. 


d. 


Tang. 


d. 


Cotg. 


d. 


Cos. 


d. 




5 o 

10 

2O 


0.0872 
0.0901 
0.0929 


29 
28 


o.o8 7 5 
0.0904 
0.0934 


29 

3 


n.43oi 
u.o5 9 4 
io. 7 n 9 


3707 

3475 


o 





. 99 62 

99 5 9 
99 5 7 


3 

2 


o 85 

5o 

4o 


3o 


0.0968 


29 
29 


0.0963 


^9 
29 


10.3854 


32^ 


5 


o 


.9954 


3 

3 


3o 


4o 


0.0987 




0.0992 




io.o 7 8o 









.9951 




20 


5o 


o. 1016 


29 


O. IO22 


3 


9 . 7 882 







9948 


3 


10 


6 o 

10 


o. io45 
o. 1074 


29 


o. io5i 
o. 1080 


2 9 


9 .5i44 
9 .2553 


2738 
2591 







9945 
.9942 


3 


o 84 

5o 


20 


0. 


no3 


29 


O.IIIO 


3 


9 .oo 9 8 


2455 


o 


99 3 9 


3 


4o 


3o 
4o 


o. 
o. 


I I 32 

1161 


29 
29 


o. i 189 

o. 1 169 


*y 

30 


8. 77 6 9 
8.5555 


2329 
2214 




o 


.9936 
.9932 


3 
4 


3o 
20 


5o 


o. 


1190 


'y 


0.1198 


^9 


8.345o 







.9929 


3 


IO 


7 o 

IO 

20 


o. 
o. 
o. 


1219 
1248 
1276 


29 
28 


o. 1228 
o. 1267 
0.1287 


29 
30 


8.i443 
7 . 9 53o 
7.7704 


1913 
1826 








.9926 
.9922 
.9918 


3 
4 


o 83 

5o 
4o 


3o 


o. 


i3o5 


29 


o.i3i7 


30 
2 9 


7 .5 9 58 







.9914 


4 


3o 


4o 


o. 


i334 




o.i346 




7 .428 7 







.9911 




20 


5o 


o. 


i363 


^9 


0.1376 




7 . 2 68 7 







.9907 


4 


IO 


8 o 

IO 


o. 

0. 


1392 
1421 


29 
29 


o.i4o5 
o.i435 


30 


7 .ii54 

6.9682 


J 533 
1472 






.9903 
.9899 


4 


o 82 

5o 


20 


o. 


1449 




o.i465 


3 


6.826 9 


I 4 I 3 





.9894 


5 


4o 


3o 


0. 


1478 


29 


o.i4 9 


3 


29 


6.6 9 i2 


'357 


o 


.9890 


4 


3o 


4o 
5o 


o. 

0. 


1607 
i536 


29 


o. 1624 
o.i554 


30 


6.56o6 
6.4348 


1258 







.9886 
.9881 


5 


20 

IO 


9 





o. 


1 564 




o.i584 


3 


6.3i38 


,,68 





.9877 




o 81 


IO 
20 


0. 
0. 


1693 
1622 


29 


o.i6i4 
o. i644 


3 


6.i 97 o 

6.o844 


1126 






.9872 
.9868 


4 


5o 

4o 










?K 






29 






10 


K 






5 




3o 


o. 


i65o 




0.1673 


3 


5. 97 58 







. 9 863 




3o 


4o 


o. 


i6 7 g 




o. 1703 




5.8 7 o8 


1050 





.9868 




20 


5o 


0. 


1708 


29 


o.i 7 33 


3 


5. 7 6 9 4 


1014 





. 9 853 


5 


IO 


10 


O 


o. 


i 7 36 




o.i 7 63 




5.6713 


901 





.9848 




o 80 




Cos. 


d. 


Cotg. 


d. 


Tang. 


d. 




Sin. 


d. 


f O 


PP 


2738 


1533 


981 


30 


29 


28 




5 4 


3 


tl 


273.8 


153-3 


98.1 .1 


3.0 


2.9 


2.8 


.! 


0.5 0.4 


o-3 


2 


547-6 


306.6 


196.2 .2 


6.0 


5-8 


5.6 


.2 


i.o 0.8 


0.6 


3 


821.4 


459-9 


294-3 -3 


9.0 


8-7 


8.4 


3 


1 5 i 


1.2 


0.9 


4 


1095.2 


6'3-2 


392.4 .4 


12. 


n.6 


II. 2 


4 


2.0 


1.6 


1.2 


5 


1369.0 


766.5 


490.5 .5 


15-0 


14-5 


14.0 


.5 


2-5 


2.O 


-5 


.6 


1642.8 


919.8 


588.6 .6 


18.0 


J 7-4 


16.8 


.6 


3-o 


2.4 


1.8 


. 7 


1916.6 


1073.1 


686.7 -7 


21.0 


20.3 


19.6 


. 7 


3-5 


2.8 


2.1 


.8 


2190.4 


1226.4 


784.8 .8 


24.0 


23.2 


22.4 


.8 


4.0 


3-2 


2.4 




2464.2 


T.7Q.7 882.0 .9 


27.0 26.1 




9 


4.5 3.6 2.7 



POUR-PLACE NATURAL FUNCTIONS. 



' 


Sin. 


d. 


Tang. 


d. 


Cotg. 


d. 


Cos. 


d. 




10 o 

IO 

20 

3o 
4o 
5o 


o. 

0. 

o. 

o. 
o. 
o. 


1735 

i 7 65 

1794 
1822 

1861 

1880 


29 

29 
28 
29 

29 
28 
29 
28 
29 
28 
29 
28 

29 
28 
a 8 

29 
28 

29 
28 
28 
28 
29 
28 
28 
28 

29 
28 
28 
28 


0.1763 

o.i 79 3 
0.1823 

o.i853 
o.i883 
o. 1914 


3 
3 
3 
3 
3 1 
30 
30 
3 
3i 
3 
3 
3i 
3 
30 
3 1 
3 
3i 


5.6713 
5.5764 
5.4845 

5.3 9 55 
5.3o 9 3 
5 .2257 


94 
9' 
8c 
8( 
8; 
81 
7* 
7< 
74 

r< 

7< 

6 
6f 
6< 
62 
61 
5c 
5* 
5< 
5! 
54 
52 
5i 
5C 
45 
4* 
4 
45 
44 
43 


9 
9 

)0 

>2 

6 


0.9848 
0.9843 
0.9838 

o. 9 833 
0.9827 
0.9822 


5 
5 
5 
6 
5 


o 80 

5o 
4o 

3o 
20 

10 


11 o 

10 

20 

3o 
4o 
5o 


o. 
o. 

0. 

o. 
o. 
o. 


1908 
i 9 3 7 
1966 

i 99 4 

2O22 

2o5i 


0.1944 
o. 1974 

0.2OO4 

o.2o35 
o.2o65 
0.2095 


5.i446 
5.o658 
4.9894 

4-9i52 
4.843o 
4.7729 


8 
4 

2 
2 
)j 

3 
4 
6 
9 
3 
7 

n 


0.9816 
0.9811 
0.9805 

0.9799 
0.9793 
0.9787 


5 
6 
6 
6 
6 


o 79 

5o 
4o 

3o 
20 

IO 


12 o 

IO 
20 

3o 
4o 
5o 


o. 
o. 
o. 

o. 
o. 
o. 


2079 
2IO8 

2i36 

2164 
2193 

2221 


0.2126 
o.2i56 
0.2186 

0.2217 
0.2247 
0.2278 


4.7046 
4.6382 
4.5 7 36 

4.5i07 
4.4494 
4.38 97 


0.9781 
0.9775 
0.9769 

0.9763 

0.9757 
0.9750 


6 
6 
6 
6 
7 


o 78 

5o 
4o 

3o 
20 

10 


13 o 

IO 

20 

3o 
4o 
5o 


o. 

0. 
0. 

o. 
o. 
o. 


225O 

2278 

23o6 

2334 
2363 
2391 


0.2309 
0.2339 
0.2370 

0.2401 
0.2432 
0.2462 


30 
3 1 
3i 
3i 
30 


4.33i5 

4.2747 
4.2193 

4.i653 
4. 1126 
4.0611 


8 

4 



7 

5 
3 
i 
i 
9 
9 
8 

9 


0.9744 
0.9737 
0.9730 

0.9724 
0.9717 
0.9710 


7 
7 
6 

7 
7 
7 
7 
7 
8 

7 

7 
g 


o 77 

5o 
4o 

3o 
20 

IO 


14 o 

IO 

20 

3o 
4o 
5o 


o. 
o. 
o. 

o. 

o. 

0. 


2419 

244? 
2476 

2604 

2532 

256o 


0.2493 
0.2524 

0.2555 

0.2586 
0.2617 
0.2648 


3* 
3i 
3i 
3 
3 


4.0108 
3.9617 
3. 9 i36 

3.8667 
3.8208 
3.7760 


0.9703 
0.9696 
0.9689 

0.9681 
0.9674 
0.9667 


o 76 

5o 
4o 

3o 

20 
IO 


15 


O 


o. 


2588 







.2679 




3. 7 32i 





. 9 65 9 






o 75 






Cos. 


d. 


Cotg. 


d. 


Tang. 


d. 




Sin. 




d. 


' o 


PP 

i 

2 

3 
4 

I 

I 


742 


448 


31 


.1 

.2 

3 

4 
-5 
.6 

:i 


30 


29 


28 


.1 

.2 

3 

4 

5 
.6 

7 
.8 


7 


6 


5 


74.2 
148.4 

222.6 

296.8 
371-0 

445-2 

S'9-4 
593-6 


44-8 
89.6 
134-4 

179.2 
224.0 
268.8 

313-6 
358.4 
403.2 


t: 

9-3 
12.4 

;i:I 

21.7 
24.8 


3-o 
6.0 
9.0 

12.0 
15-0 
18.0 

21.0 

24.0 


* 

8. 7 

ii. 6 
M-5 
17.4 

20.3 


2.8 

5-6 
8.4 

II. 2 
14.0 

16.8 

19.6 
22.4 


0.7 
1.4 

2.1 
2.8 

3-5 

4-2 

4.9 

r 


0.6 

1.2 

1.8 

2-4 

H 

ti 

5-4 


o-5 

I.O 

i-5 

2.0 
2-5 

3-o 

3-5 
4.0 

4-5 



152 



POUR-PLACE NATURAL FUNCTIONS. 



O ' 


Sin. 


d. 


Tang. 


d. 


Cotg. 


d. 


Cos. 


d. 




15 o 

10 

20 

3o 
4o 
5o 


o. 

0. 
0. 

o. 
o. 

0. 


2588 
2616 
2644 

2672 
2700 
2728 


28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
27 
28 
28 
27 
28 
28 

27 
28 
28 
27 


0.2679 
0.271 i 
0.2742 

0.2773 
o.28o5 
0.2836 


32 
3 

32 

32 
32 
3* 
32 
32 
3 
32 
32 
32 
32 
32 


3.7321 
3.6891 
3.6470 

3. 6059 
3.5656 
3.5 2 6i 


4 
4 
4 
4 
3 
3 
3 
3 

3 
3 
3 
3 
3 
3 
3 
3 
3< 
3 

2 
2 
2 
2 
2 
2 
2< 
2< 
2. 
2. 
2. 


3 

:i 
ii 
33 
M 

3 7 

79 
71 
^5 
37 
50 


0.9659 
0.9652 
0.9644 

0.9636 
0.9628 
0.9621 


7 

8 
8 
8 
7 


o 75 

5o 
4o 

3o 
20 

IO 


16 o 

IO 

20 

3o 

4o 
5o 


o. 
o. 
o. 

o. 
o. 
o. 


2 7 56 
2784 
2812 

2840 
2868 
2896 


0.2867 
0.2899 
0.2931 

0.2962 
0.2994 
o.3o26 


3.48 7 4 
3.4495 
3.4i24 

3.3 7 5 9 
3.34o2 
3.3o5 2 


0.9613 
0.9605 
0.9596 

0.9588 
0.9580 

0.9572 


8 

9 
8 
8 
8 


o 74 

5o 

4o 

3o 
20 

IO 


17 o 

IO 

20 

3o 

4o 
5o 


0.2924 
0.2952 

0.2979 

0.3007 
o.3o35 
o.3o62 


0.3089 

0. 3l2I 

o.3i53 

o.3i85 
0.3217 


3.2709 
3.23 7 i 
3.2o4i 

3.1716 
3.i3 97 

3.io84 


13 
38 
5 

25 
9 
3 


0.9563 
o. 9 555 
o. 9 546 

o. 9 537 
o. 9 528 

O. 9 52O 


8 
9 
9 
9 

8 


o 73 

5o 
4o 

3o 
20 

IO 


18 o 

IO 
20 

3o 
4o 
5o 


o. 
o. 
o. 

o. 
o. 
o. 


3090 
3n8 
3i45 

3i 7 3 

3201 
3228 


0.3249 
o.328i 
o.33i4 

0.3346 
o.33 7 8 
o.34n 


32 

33 
32 
32 
33 


3.0777 
3.o475 
3.0178 

2.9887 
2.9600 
2 . 9 3i 9 


J l 

m 

17 

H 

37 
h 


o. 9 5n 
o. 9 5o2 
o. 9 4 9 2 

o. 9 483 
o. 9 474 
o. 9 465 


9 

IO 

9 
9 
9 


o 72 

5o 
4o 

3o 
20 

IO 


19 o 

IO 
20 

3o 
4o 
5o 


o. 
o. 
o. 

o. 
o. 
o. 


3256 
3283 
33u 

3338 
3365 
33 9 3 


27 
28 
27 

27 
28 


0.3443 
0.3476 
o.35o8 

o.354i 
0.3574 
0.3607 


33 

33 
33 
33 


2.9042 
2.8 77 o 

2.8502 

2.8239 

2. -7980 
2. 77 25 


1 

2 

,8 
>3 
9 
5 


o. 9 455 
o. 9 446 
o. 9 436 

o. 9 426 
o. 9 4i 7 
o. 9 4o 7 


9 

10 
10 

9 

IO 


o 71 

5o 
4o 

3o 
20 

IO 


20 





o. 


3420 


27 


o.364o 




2.7475 





o. 9 3 97 




o 70 






Cos. 


d. 


Cotg 




d. 


Tang. 


d 





Sin. 




d. 


' O 


PP 

.1 

.2 

3 
4 

:J 

9 


255 


33 


32 


31 


28 


27 




10 


9 


8 


25-5 
51.0 
76.5 

102.0 

127.5 
i53-o 

178-5 
204.0 


1:1 

9-9 

13.2 
16.5 
19.8 

23.1 
26.4 
29-7 


3.2 

6.4 .2 

9.6 .3 

12.8 .4 

16.0 .5 
19.2 .6 

22.4 .7 
25.6 .8 


11 

9-3 

12.4 

S3 
i 


2.8 

tf 

II. 2 
14-0 

16.8 

19.6 
22.4 


2-7 

K 

10.8 

'3-5 
16.2 

18.9 

21.6 

24-3 


.1 I.O 
.2 2.0 

3 3- 

.4 4.0 
5 5' 
.6 6.0 

7 7- 
.8 8.0 
.9 9.0 


a 

2-7 

3-6 
4-5 
5-4 

6-3 

K 


0.8 
1.6 
2.4 

3-2 

4.0 
48 

5-6 
6.4 

7-2 



i53 



FOUR-PLACE NATURAL FUNCTIONS. 



O ' 


Sin. 


d. 


Tang. 


d. 


Cotg. 


d. 


Cos. 


d. 




20 o 

IO 
20 

3o 
4o 
5o 


o. 
o. 

0. 

o. 

0. 

o. 


3420 
3448 
3475 

35o2 
352 9 
355 7 


28 
27 
27 
27 
28 
27 
27 
27 
27 
27 
27 


o.364o 
0.3673 
0.3706 

0.3739 

0.3 77 2 

o.38o5 


33 

33 
33 
33 
33 


2.7475 
2.7228 
2 .6 9 85 

2.6746 
2.65n 
2.6279 


247 
243 
239 
235 
232 

228 


o. 9 3 97 
0.9387 
o. 9 3 77 

o. 9 36 7 
o. 9 356 
o. 9 346 


IO 
IO 
10 

II 

IO 


o 70 

5o 

4o 

3o 
20 

IO 


21 o 

IO 
20 

3o 
4o 
5o 


o. 
o. 

0. 

o. 

0. 
0. 


3584 
36n 
3638 

3665 
3692 
3 7 i 9 











.383 9 
.38 7 2 
.3906 

.3 9 3 9 
.3 97 3 
.4oo6 


33 
34 
33 
34 
33 


2.6o5i 
2.5826 
2.56o5 

2.5386 
2.5172 
2.4960 


225 

221 
2I 9 
214 
212 


o. 9 336 
o. 9 325 

o. 9 3o4 
o. 9 2 9 3 
0.9283 


II 

IO 

II 
II 

IO 


o 69 

5o 
4o 

3o 
20 

IO 


22 o 

IO 

20 

3o 

4o 

5o 


o. 
o. 
o. 

o. 
o. 
o. 


3 7 46 
3 77 3 
38oo 

382 7 
3854 
388i 


27 
27 
27 

27 
27 











.4o4o 
4o 7 4 
.4108 

'4i 7 6 
.4210 


34 
34 
34 
34 
34 


2.4545 
2.4342 

2.3 9 45 
2.3750 


206 
20 3 
2OO 
I 97 
195 


0.9261 
0.9250 

0.9239 
0.9228 
0.9216 


II 
II 
II 
II 

12 


o 68 

5o 
4o 

3o 
20 

IO 


23 o 

10 
20 

3o 
4o 
5o 


o.3 9 o 7 
o.3 9 34 
o.3 9 6i 

o.3 9 8 7 
o.4oi4 
o.4o4i 


27 
27 
26 
27 
27 
26 
27 
26 
27 
26 
27 











.4245 

42 79 

.43i4 

.4348 
.4383 
.44i7 


35 
34 
35 
34 
35 
34 
35 
35 
35 
35 
35 
36 
35 


2.3559 
2.336 9 
2 .3i83 

2.2998 
2.2817 
2.2637 


I 9 

1 86 
185 
181 
1 80 
177 
174 
i73 
170 
168 
1 66 
i6<i 


0.9205 
o . 9 i 94 
0.9182 

o. 9 i 7 i 
0.9159 

o. 9 i4 7 


II 
12 

II 
12 
12 


o 67 

5o 
4o 

3o 
20 

IO 


24 o 

10 
20 

3o 
4o 
5o 


o. 
o. 
o. 

0. 

o. 
o. 


4o6 7 
4094 
4120 

4i47 
4173 
4200 









o 


.4452 
.4487 

.4522 

.455 7 
.45 9 2 
.4628 


2.2460 
2.2286 

2.2II3 

2.i 9 43 

2.1 77 5 
2. 1609 


0.9135 
0.9124 
0.9112 

o. 9 ioo 

o. 9 o88 
o. 9 o 7 5 


II 
12 

12 
12 

13 
12 


o 66 

5o 
4o 

3o 
20 

IO 


25 





o. 


4226 







.4663 


2.i445 







. 9 o63 


o 65 






Cos. 


d. 


Cotg. 


d. 


Tang. 


d. 


Sin. 


d. 


' O 


PP 

.1 

.2 

3 

4 
5 
.6 


177 


35 


34 


.2 

3 

4 
5 
.6 

7 
.8 

9 


33 


27 


26 


.1 

.2 

3 

4 
5 
.6 


12 


ii 


IO 


17.7 
35-4 

70.8 
88.5 
106.2 

123.9 
141.6 


3-5 
7.0 
10.5 

14.0 

21.0 

24-5 
28.0 
3 z -5 


U 

10.2 

13-6 
I 7 .0 
20. 4 

2 3 .8 
2 7 .2 


1:1 

9-9 

13.2 
16.5 
19.8 

III 


2.7 

5-4 
8.1 

10.8 
'3-5 
16.2 

18.9 

21.6 

2 4-3 


2.6 

10.4 

13.0 

15-6 

18.2 

20.8 

23-4 


1.2 

2.4 

3-6 

6.0 
7.2 

8.4 
9.6 
10.8 


2.2 

3-3 

4-4 

ii 

1.1 


1.0 
2.O 

3-o 

4.0 
5- 
6.0 

7.0 
8.0 



1 54 



FOUR-PLACE NATURAL FUNCTIONS. 



O ' 


Sin. 


d. 


Tang. 


d. 


Cotg. 


d. 


Cos. 


d. 




25 





o. 


4226 




0.4663 


06 


2.i445 


162 


0.9063 




o 65 


IO 

20 


o. 
o. 


4253 

4279 


26 


0.4699 
0.4734 


35 
36 


2.1283 
2.II23 


160 
158 


0.9051 
0.9038 


13 

12 


5o 
4o 


3o 


o. 


43o5 


26 


0.4770 


06 


2.0965 


1*6 


0.9026 




3o 


4o 


o. 


433i 




o.48o6 




2.0809 




0.9013 




20 


5o 


0. 


4358 




o.484i 


35 


2 o655 


J 54 


0.9001 




IO 


26 





0. 


4384 


26 


0.4877 


36 


2.o5o3 




0.8988 




o 64 




10 


o. 


44io 




0.4913 




2.o353 






0.8975 




5o 


20 


0. 


4436 


26 


0.495 





37 


2.O2O4 


149 


0.8962 


IJ 


4o 










fh 






3 6 






i 


7 








JO 




3o 


o. 


4462 


26 


0.4986 


16 


2.0057 




0.8949 




3o 


4o 


o. 


4488 




O.5O22 




.9912 




0.8936 




20 


5o 


o. 


45i4 


26 


o.5o5 


9 


37 


.9768 


144 


0.8923 


M 


IO 


27 





o. 


454o 


26 


o.Sog 


5 




.9626 




0.8910 




o 63 


10 


o. 


4566 




o.5i32 




. 9 486 




0.8897 




5o 


20 


o. 


4592 


26 


0.5169 


37 


. 9 34 7 


1 


59 


0.8884 


1J 


4o 


3o 

4o 


0. 

o. 


46i 7 
4643 


25 

26 


0.5206 
0.5243 


37 
37 


.9210 
.9074 


136 


0.8870 
0.8857 


13 


3o 

20 


5o 


0.4669 


26 


0.5280 


37 


.8940 


134 


0.8843 


M 


IO 


28 





o. 


46 9 5 




o.53i7 




.8807 


X 33 


0.8829 




o 62 


10 


o. 


4720 




0.5354 




.8676 




0.8816 




5o 


20 


o. 


4746 


20 


0.5392 


3^ 


.8546 


130 


0.8802 


14 


4o 


3o 


0. 


4772 


26 


o.543o 


38 


.84i8 


128 


0.8788 


14 


3o 


4o 
5o 


o. 
o. 


4797 
48 2 3 


36 


o. 5467 
o.55o5 


38 


.8291 
.8i65 


126 


0.8774 
0.8760 




20 
IO 


29 





o. 


4848 




0.5543 


3 8 


i.8o4o 


I2 5 


0.8746 




o 61 


IO 


o. 


48 7 4 




o.558i 


S 8 


1.7917 


123 


0.8732 


14 


5o 


20 


o. 


4899 


25 


0.5619 


3 


1.7796 


121 


0.8718 


M 


4o 


3o 

4o 


o. 
o. 


4924 


25 
26 


0.5658 
0.5696 


39 
38 


1.7675 
1.7556 


121 


0.8704 
0.8689 


14 
15 


3o 
20 


5o 


o. 


4 97 5 


3 


0.5735 


39 


i. 7 43 7 




o.86 7 5 


'4 


IO 


30 





o. 


5ooo 




0.5774 




1.7321 




0.8660 




o 60 






Cos. 


d. 


Cotg. 


d. 


Tang. 


d. 




Sin. 




d. 


> o 


PP 


149 


I3i 


39 


38 


37 


36 




25 


M 


13 


. I 


14.9 

2Q 8 


26 2 


3-9 -i 

78 2 


3-8 


3-7 


3-6 


.1 


2.5 


LI 


26 


3 


44-7 


39-3 


"7 -3 


11.4 


II. I 


10.8 


3 


7-5 


4-2 


3-9 


4 


59-6 


52.4 


15.6 .4 


15.2 


14-8 


14.4 


4 


10. 


5-6 


5-2 


.5 


74-5 


65-5 


9-5 -5 


19.0 


i8.S 


18.0 


5 


12.5 


7.0 


6-5 


.6 


89.4 


78.6 


23-4 -6 


22.8 


22.2 


21.6 


.6 


15-0 


8.4 


7.8 


7 


104.3 


91.7 


27-3 -7 


26.6 


25.9 


25-2 


.7 


I 7-5 


9.8 


9.1 


.8 


119.2 


104.8 


31.2 .8 


30.4 


29.6 


28.8 


.8 


20.0 


II. 2 


10.4 






117.9 35-i -9 




32-4 




22.5 


12.6 11.7 



i55 



FOUR-PLACE NATURAL FUNCTIONS. 



o 


' 


s 


in. 


d. 


1 


rang 


. 


d. 


Cot 


r. 


d 


. 




Cos. 


d. 




30 

i 



i 
| 








^o 




o. 
o. 
o. 

o. 
o. 
o. 


5ooo 
5o 2 5 
5o5o 

5o 7 5 
5ioo 
5i25 


25 
25 
25 
25 
25 












.577 

.58i 
.585 

.58 9 
.5 9 3 
.5 9 6 


i 
2 
I 

D 
3 
I 


38 

39 
39 
40 

39 


. 7 3 

I. 7 2< 
. 7 0( 

.6 9 r 

.68( 
.67, 


21 

D5 
? 

77 
34 
)3 


ii 

ii 
U 

ii 
ii 


6 

5 
3 
3 


C 







o 


.8660 

.8646 
.863i 

.8616 
.8601 

.858 7 


14 
15 
15 
15 

J 4 


o 60 

5o 
4o 

3o 

20 
IO 


31 

i 



i 
\ 








o 




o. 
o. 
o. 

o. 
o. 
o. 


5i5o 
5i75 

52OO 
5225 

525o 
5 27 5 


25 
25 
25 
25 
'25 





o 

o 

o 




.600 
.6o4i 
.608 

.612! 
.616! 
.620! 


) 

3 

3 
3 
3 


39 
40 
40 
40 
40 


.66. 

.65: 

.645 

.63 
.62: 
.6ic 


i3 
M 

26 

9 

2 

>7 


1C 
1C 
10 

10 
10 


9 

8 

7 
7 

5 





o 

o 
o 




.85 7 2 
.8557 
.8542 

.8526 
.85n 
.84 9 6 


i5 
15 
16 
15 

15 
16 


o 59 

5o 
4o 

3o 
SO 

IX) 


32 

i 


c 

/! 
i 





o 







o. 
o. 
o. 

o. 
o. 
o. 


52 99 
5324 
5348 

5373 
53 9 8 
5422 


25 
24 
25 
25 
24 








o 




.6241 
.628* 
.633* 

.63 7 
.64i! 

.645 


? 

? 

5 

[ 
2 

3 


40 
4 1 
4 1 
4i 
4i 


.6o( 
.5 9 c 

.5 7 c 

.56c 
.55c 
.54c 


>3 

)0 

>8 

>7 

>7 
^7 


10 
10 
1C 
10 
IO 


3 

2 

I 

O 


o 
o 
o 

o 
o 
o 


.848o 
.8465 
.845o 

.8434 
.84i8 
.84o3 


15 
IS 

16 
16 

15 
ifi 


o 58 

5o 

4o 

3o 

20 
10 


33 

i 



4 

c 


C) 








o. 
o. 
o. 

o. 
o. 
o. 


5446 
5471 
54 9 5 

55i 9 

5544 
5568 


25 
24 
24 
25 
24 











.64 9 ^ 
.6531 
.657 

.66i< 
.666 
.670 


i 

7 

? 

[ 
5 


42 
4i 
42 
4 2 
4 2 


.53c 
.53c 

.52( 

.5i( 
,5oi 

4 9 


>9 

>i 

>4 

>8 
3 

9 


y 

9 
9 
9 
9 
9 


8 

7 

6 

5 

4 


o 
o 



o 
o 




.8387 
.83 7 i 
.8355 

.833 9 
.8323 
.83o 7 


16 
16 
16 
16 
16 


o 57 

5o 

4o 

3o 
20 

10 


34 

i 



C 

2 
f. 





o 



o 




o. 
o. 
o. 

o. 
o. 
o. 


55 9 2 
56i6 
564o 

5664 
5688 
5712 


24 
24 
24 
24 
24 
24 











.6 7 4 
.678 
.683< 

.687 
.6 9 i( 
.6 9 5 


7 

5 

5 

I 


42 
43 
43 
43 
43 


.48i 

.4 7 : 
.4& 

.45J 
44( 
.43- 


56 

*3 
il 

)0 
)0 
JO 


y 

9 
9 
9 
9 
9 


3 

2 
I 

O 


o 
o 
o 

o 
o 




.82 9 o 

.8274 

.8258 

.8241 
.8225 
.8208 


J 7 
16 
16 
*7 
16 

17 
16 


o 56 

5o 
4o 

3o 
20 

IO 


35 





o. 


5 7 36 


24 





.700 


I 


43 


1.42* 


h 









.8l 9 2 




o 55 






c 


OS. 


d. 


( 


:otg. 




d. 


Tanj 


? 


d 







Sin. 


d. 


' 


PP 




13 


42 


41 








40 


25 


2 


t 






17 


16 


15 


.1 

.2 

3 
4 

:l 

:i 


I 

2 
2 

3 

3 


tl 

2.9 
7.2 

;:i 

0.1 

ti 


t; 

12.6 

16.8 

21. 
25-2 

29.4 

33-6 


t 

12. 

16. 

20. 
24. 

28. 


I 

2 

3 
4 

! 


.1 

.2 

3 
4 

' 


i 
i 

i 


4.0 

8.0 

2.O 

6.0 

0.0 

4.0 

8.0 

J2.0 

(6.0 


2.5 

5-o 
7-5 

IO.O 

12.5 
15-0 

17-5 
20. o 


2 

4 
7 

9 

12 
H 

16 
"9 

21 


5 

2 

6 
o 

4 

8 

.2 

.6 




I 

2 

3 

4 
5 
6 

7 8 


*-7 
3-4 
5-i 

6.8 
8-5 

10.2 

II.9 
I 3 .6 


1.6 

4 3 :I 

6. 4 
8.0 
9.6 

II. 2 
12.8 
14.4 


1.5 

3- 
4-5 

6.0 
75 
9.0 

10.5 

12.0 
13-5 



1 56 



POUR PLACE NATURAL FUNCTIONS. 



o / 


Sin. 


d. 


Tang. 


d. 


Cotg. 


d. 


Cos. 


d. 




35 o 

10 

20 

3o 
4o 
5o 


o 
o 
o 

0. 

o. 
o 


5 7 36 
6760 
5 7 83 

58oy 
583i 

5854 


24 
23 
24 
24 
23 


0. 7 002 

o. 7 o46 
o. 7 o8 9 

o. 7 i33 

o. 7 i 77 

O. 7 22I 


44 
43 
44 
44 
44 
44 
45 
45 
45 
45 
45 
46 
45 
46 
46 
47 
46 


.4281 

.4i 9 3 
4io6 

.4oi 9 
.3 9 34 

,3848 


88 
87 
87 
85 
86 
84 
84 
83 
83 
82 
Si 


0.8l 9 2 

o.8i 7 5 
o.8i58 

o.8i4i 
0.8124 
o.8io 7 


*7 
i7 
17 
J 7 
17 


o 55 

5o 

4o 

3o 

20 
IO 


36 o 

10 

20 

3o 

4o 
5o 


o. 

0. 

o. 

o. 
o. 
o. 


58 7 8 
5901 
5 92 5 

5 9 48 
6972 
5 99 5 


23 
24 
23 
24 
23 


O. 7 265 

o. 7 3io 
o. 7 355 

o. 7 4oo 
o. 7 445 
o. 7 4 9 o 


.3 7 64 
.368o 
.35 97 

.35i4 
.3432 
.335i 


o.8o 9 o 
o.8o 7 3 
o.8o56 

o.8o3 9 
0.8021 
o.8oo4 


*7 
17 
*7 
18 

17 


o 54 

5o 

4o 

3o 
20 

10 


37 o 

10 

20 

3o 

4o 
5o 


o. 
o. 
o. 

o. 
o. 
o. 


6018 
6o4i 
6o65 

6088 
61 1 1 
6i34 


23 
24 
23 
23 
23 


o. 7 536 
o. 7 58i 

O. 7 62 7 

o. 7 6 7 3 

O. 77 2O 

o. 77 66 


.32 7 

.3i 9 o 
.3iu 

.3o32 
.2 9 54 
. 2 8 7 6 


80 
79 
79 
78 
78 


o. 79 86 
o. 79 6 9 
o. 79 5i 

o. 79 34 
o. 79 i6 

o. 7 8 9 8 


17 
18 

17 

.8 
18 


o 53 

5o 

4o 

3o 
20 

10 


38 o 

10 

20 

3o 

4o 
5o 


o. 
o. 
o. 

o. 
o. 

0. 


6i5 7 
6180 
6202 

6 22 5 
6248 

62 7 I 


23 

22 

23 
23 
23 


o. 7 8i3 
o. 7 86o 
o. 79 o 7 

o. 79 54 
0.8002 
o.8o5o 


47 
47 
47 
47 
48 
48 


.2 799 

.2 7 23 

.264 7 

.25 7 2 
24 97 
.2423 


77 
76 
76 
75 
75 
74 


o. 7 88o 
o. 7 862 
o. 7 844 

o. 7 826 
o. 7 8o8 
o. 779 o 


18 
18 
18 
18 
18 


o 52 

5o 
4o 

3o 

20 
IO 


39 o 

10 

20 

3o 
4o 
5o 


o. 

0. 
0. 

o. 

0. 
0. 


62 9 3 
63i6 
6338 

636i 
6383 
64o6 


23 
22 

23 
22 

23 


o.8o 9 8 
o.8i46 
o.8i 9 5 

0.8243 

O.82 9 2 

0.8342 


4* 
48 

49 
48 

49 
So 


.234 9 

.22 7 6 
.2203 

.2l3l 

.2o5 9 

.1988 


74 

73 
73 
72 
72 
7i 


o. 777 i 
o. 77 53 
o. 77 35 

o. 77I 6 
o. 7 6 9 8 
o. 7 6 79 


18 
18 

*9 
18 

19 


o 51 

5o 
4o 

3o 

20 
IO 


40 





0. 


6428 




o.83 9 i 


49 


i.i 9 i8 


70 


o. 7 66o 


'9 


o 50 






Cos. 


d. 


Cotg. 




d. 


Tang. 


d. 




Sin. 


d. 


t O 


PP 

.1 

.2 

3 

4 

5 
.6 

:l 


48 


47 


46 


45 


44 


23 


.1 

.2 

3 

4 

5 
.6 

:! 


22 


19 


18 


4 .8 

9 .6 
14.4 

19.2 
24.0 
28.8 

tf 

43-2 


4-7 
9.4 
14.1 

18.8 

2 2 3 8: 2 
32.9 

37-6 
42.3 


4 .6 

9 .2 .2 

13-8 -3 

18.4 .4 

2 3- -5 
27.0 .6 

32-2 .7 
36.8 .8 
41.4 .9 


4-5 
9.0 
13-5 

18.0 
22.5 
27.0 

3' -5 
36.0 
40.5 


44 
8.8 
3-2 

17.6 

22.0 

26.4 

30.8 

35-2 


2.3 
4.6 
6.9 

9.2 
"5 
13-8 

16.1 
18.4 


2.2 

4-4 
6.6 

8.8 

II. 

13.2 

3 

19.8 


i. 9 
3-8 
5.7 

7-6 
9-5 
11.4 

13-3 
15-2 

17.1 


1.8 
3-6 
5-4 

7- 2 
9.0 
10.8 

12.6 

14.4 

16.2 



i5 7 



POUR-PLACE NATURAL FUNCTIONS. 



O ' 


Sin. 


d. 


Tang. 


d. 


Cotg. 


d. 


Cos. 


d. 




40 o 

10 

20 

3o 

4o 
5o 


0.6428 
o.645o 
0.6472 

0.6494 
0.6617 
o.653 9 


22 
22 
22 

23 

22 


0.8391 

o.844i 
0.8491 

o.854i 
0.8691 
0.8642 


50 

5 
5 
So 
51 
Si 
51 
52 
51 
52 
53 
52 
53 
53 
53 
54 
54 
54 
55 
55 
55 
55 
56 
c.6 


1.1918 

1.1847 
1.1778 

1.1708 
i.i64o 
i . 1671 


7* 
6 9 
70 

68 
6 9 


o. 7660 
0.7642 

0.7623 
0.7604 

0.7686 
0. 7 566 


18 
*9 
9 
19 
J 9 


o 50 

5o 
4o 

3o 

20 
10 


41 o 

10 
20 

3o 
4o 
5o 


o. 
o. 
o. 

o. 
o. 
o. 


656i 
6583 
66o4 

6626 

6648 
6670 


22 
21 
22 
22 
22 


0.8693 

o.8 7 44 
0.8796 

0.8847 
0.8899 
0.8962 


i.i5o4 
i.i436 
1.1369 

i.i3o3 
1.1237 
1.1171 


67 
68 
67 
66 
66 
66 


0.7647 
0.7628 

0.7609 

0.7490 
0.7470 
0.7461 


9 

19 

'9 
19 

20 
19 


o 49 

5o 
4o 

3o 
20 

IO 


42 o 

10 

20 

3o 
4o 
5o 


o. 
o. 

0. 

o. 

o. 
o. 


6691 
6 7 i3 
6 7 34 

6 7 56 
6777 
6799 


22 
21 
22 
21 
22 


0.9004 
0.9067 
0.9110 

0.9163 
0.9217 
0.9271 


i . i 106 
i .io4i 
1.0977 

i .0913 
i.o85o 
1.0786 


65 
65 
64 
64 
63 
64 


o. 7 43i 
0.7412 
0.7392 

o. 7 3 7 3 
o. 7 353 
o. 7 333 


9 

20 

19 
2O 
20 


o 48 

5o 

4o 

3o 

20 
10 


43 o 

10 
20 

3o 
4o 
5o 


o. 
o. 
o. 

o. 
o. 
o. 


6820 
684i 
6862 

6884 
6906 
6926 


21 
21 
22 
21 
21 


0.9326 
0.9380 
0.9435 

0.9490 
0.9645 
0.9601 


i .0724 
i .0661 
i .0699 

i.o538 
1.0477 
i . o4 i 6 


63 
62 
61 
61 
61 


o. 7 3i4 
o. 7 294 

0. 7 2 7 4 

o. 7 254 
0.7234 
0.7214 


'9 

20 
20 
2O 
20 
2O 


o 47 

5o 
4o 

3o 

20 
IO 


44 o 

IO 
20 

3o 
4o 

5o 


0.6947 
0.6967 

0.6988 

0.7009 
0.7030 
0.7060 


20 
21 
21 
ZI 
20 
21 


0.9667 
0.9713 
0.9770 

0.9827 
0.9884 
0.9942 


56 
57 
57 
57 
58 


i.o355 
1.0296 
1.0235 

i .0176 
1.0117 
i. 0068 


60 
60 
59 
59 
59 


0.7193 
o. 7 i 7 3 
o. 7 i53 

o. 7 i33 

O. 7 II2 
O. 7 O92 


20 
20 
20 
21 
20 


o 46 

5o 
4o 

3o 

20 
10 


45 





o. 


7071 


i .0000 




i .0000 




O. 7 O 7 I 




o 45 






Cos. 


d. 


Cotg. 


d. 


Tang. 


d. 


Sin. 




d. 


' o 


PP 

.1 

.2 

3 

4 

j 

:i 

9 


57 


55 


54 


53 


51 


22 21 


20 


19 


5-7 
11.4 
17.1 

22.8 

28.5 

34-2 

39-9 
45.6 

5 1 .1 


5-5 

II. O 

16.5 

22.0 
27-5 

33-o 

38.5 
44.0 
49-5 


5-4 ' 

10.8 .2 

16.2 .3 

21.6 .4 
27.0 .5 
32.4 -6 

37-8 -7 
43-2 .8 
48.6 .9 


5-3 

10. 

iS-9 

21.2 

26.5 
31.8 

37-i 
42.4 
47-7 


5-i 

10.2 
5-3 

20-4 
25-5 
30-6 

35-7 
40.8 

45-9 


2.2 .1 2.1 
4.4 .2 4.2 

6.6 .3 6.3 

8.8 .4 8.4 
ii. o .5 10.5 
13.2 .6 12.6 

*5-4 -7 14-7 
17.6 .8 16.8 
19.8 .9 18.9 


2.0 
4-0 

6.0 
8.0 

IO.O 
12. 

14.0 
IO.O 

18.0 


3 

5-7 

7-6 
9-5 
11.4 

i3-3 
15.2 
17.1 



i58 



TABLE VIII. 
SQUARES AND SQUARE ROOTS OF NUMBERS. 

SQUARES OF INTEGERS FROM 10 TO 100. 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


mmmumum 

IO 


MMHMM^ 

100 


^^^^M 

121 


1 44 


mmmmm^^ 

169 


196 


226 


2 56 


289 


324 


36i 


20 


4oo 


44 1 


484 


629 


5 7 6 


625 


676 


729 


784 


84 1 


3o 


900 


961 


1024 


1089 


n56 


1225 


1296 


1 369 


1 444 


l52I 


4o 


1600 


1681 


1764 


1849 


i 9 36 


2025 


2116 


2209 


2 3o4 


2401 


5o 


2600 


2601 


2704 


2809 


2916 


3o25 


3i36 


3249 


3364 


348 1 


60 


36oo 


8721 


3844 


3 9 6 9 


4096 


4225 


4356 


448 9 


4624 


4761 


70 


4900 


5o4i 


5i84 


532 9 


5476 


56a5 


5 77 6 


5 9 2 9 


6o84 


6241 


80 


64oo 


656i 


6724 


6889 


7066 


7225 


?3 9 6 


7 56 9 


7744 


7921 


90 


8100 


8281 


8464 


8649 


8836 


9025 


9216 


9409 


9604 


9801 



SQUARE ROOTS OF NUMBERS FROM TO 10, AT INTERVALS OF .1. 



N 

i 




.0 

^^M 




.1 

.3i6 


.2 

.447 


.3 

.548 


.4 

.632 


.5 

.707 


.6 

"775 


.7 

~ 


.8 

.8 9 4 


.9 

.949 


i 


I.OOO 


i. 049 


1.095 


i.i4o 


i.i83 


1.225 


1.265 


i.3o4 


1.342 


1.378 


2 


i.4i4 


1.449 


1.483 


i.5i 7 


1.549 


i.58i 


1.612 


1.643 


1.673 


1.703 


3 


1.732 


1.761 


1.789 


1.817 


1.844 


1.871 


1.897 


1.924 


1.949 


i. 97 5 


4 


2.000 


2.025 


2.049 


2.074 


2.098 


2. 121 


2.i45 


2.168 


2.191 


2.214 


5 


2.236 


2.258 


2.280 


2.302 


2.324 


2.345 


2.366 


2.38 7 


2.408 


2.429 


6 


2.449 


2.470 


2.490 


2.5lO 


2.53o 


2.55o 


2.569 


2.588 


2.608 


2.627 


7 


2.646 


2.665 


2.683 


2.702 


2.720 


2.739 


2.7^7 


2.775 


2.793 


2.811 


8 


2.828 


2.846 


2.864 


2.881 


2.898 


2.915 


2. 9 33 


2.950 


2.966 


2. 9 83 


9 


3.000 


3.017 


3.o33 


3.o5o 


3.o66 


3.082 


3.098 


3.n4 


3.i3o 


3.i46 



SQUARE ROOTS OF INTEGERS FROM 10 TO 100. 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


m^mmmm* 

IO 


3.162 


I^T" 


3.464 


3.6o6 


3. 7 42 


3.8 7 3 


4.000 


4.123 


4.243 


4.35 9 


20 


4.472 


4.583 


4.690 


4. -796 


4.899 


S.ooo 


5.099 


5.196 


5.292 


5.385 


3o 


5.477 


5.568 


5.65 7 


5.745 


5.83i 


5.916 


6.000 


6.o83 


6.i64 


6.245 


4o 


6.325 


6.4o3 


6.48i 


6.55 7 


6.633 


6.708 


6.782 


6.856 


6.928 


7.000 


5o 


7.071 


7-Ui 


7.21 1 


7.280 


7 .348 


7-4i6 


7-483 


7 .55o 


7.616 


7.681 


60 


7-746 


7.810 


7.874 


7-9^7 


8.000 


8.062 


8.124 


8.i85 


8.246 


8.3o 7 


70 


8.367 


8.426 


8.485 


8.544 


8.602 


8.660 


8.718 


8.775 


8.832 


8.888 


80 


8.944 


9.000 


9 .o55 


9.1 10 


9.i65 


9.220 


9.274 


9.327 


9 .38i 


9-434 


90 


9-48 7 


9 .53 9 


9.592 


9-644 


9.6 9 5 


9-747 


9.798 


9.849 


9.899 


9.950 



z5 9 



TABLE IX. 



THE HYPERBOLIC AND EXPONENTIAL FUNCTIONS OF 
NUMBERS FROM TO 2.5, AT INTERVALS OF .1. 



X 


cosh a? 


sinha? 


tanh a? 


e* 


e * 





i .000 


o 


o 


i .000 


I .000 


. i 


i .oo5 


.100 


.100 


i.io5 


.905 


.2 


i .020 


.201 


.197 


I .221 


.819 


.3 


i.o45 


.3o5 


.291 


i.35o 


. 7 4i 


.4 


1.081 


.4n 


.38o 


1 .492 


.670 


.5 


1.128 


.521 


.462 


1.649 


.607 


.6 


i.tSS 


.63 7 


.53 7 


1.822 


.549 


7 


1.255 


. 7 5 9 


.6o4 


2.014 


497 


.8 


i.33 7 


.888 


.664 


2.226 


.449 


9 


1.433 


1.027 


.716 


2 .46o 


.407 


1.0 


i.543 


1.176 


.762 


2.718 


.368 


. i 


i .669 


1.336 


.801 


3.oo4 


.333 


.2 


1.811 


i .509 


.834 


3.320 


.3oi 


.3 


1.971 


1.698 


.862 


3.669 


.2 7 3 


.4 


2.l5l 


i .904 


.885 


4.o55 


.247 


.5 


2.352 


2.129 


.905 


4.482 


.223 


.6 


2.577 


2 .3 7 6 


.922 


4. 9 53 


,202 


7 


2.828 


2.646 


. 9 35 


5.474 


.183 


.8 


3.107 


2.942 


947 


6.o5o 


.i65 


9 


3.4i8 


3.268 


956 


6.686 


.i5o 


2.0 


3.762 


3.627 


.964 


7 .38 9 


.i35 


2. I 


4.i44 


4.022 


.970 


8.166 


. 122 


2.2 


4.568 


4.457 


.976 


9.025 


.III 


2.3 


5.o3 7 


4.937 


.980 


9.974 


. IOO 


2.4 


5.55 7 


5.466 


.984 


II .023 


.091 


2.5 


6.i32 


6.o5o 


.987 


12.182 


.082 



1 60 



TABLE X 

CONSTANTS 

MEASURES AND WEIGHTS 
AND OTHER CONSTANTS 







, .... 




"* '' *' 

.* 



/*' " " - - - 

,. /x , , .,. :.. xx.' ., ., 

" " 



' ' ' ' '" ' ..... ' ' 



- 



( 
rw^rf ^r&rW^r 

,-'>< ' :< 







>.. . '. 

' ',' ' '' 



' 









<v v 









B' 

Si 

w P- 



rt P- 



IT 

f 



, Elements 
plans and 



ottBtry 



Y 9 194 



May'47Pf 




M3O6247 



THE UNIVERSITY OF CALIFORNIA LIBRARY 






UNIVERSITY OF CALIFORNIA LIBRARY 
BERKELEY 

Return to desk from which borrowed. 
This book is DUE on the last date stamped below. 



REC'D LD 

APR 9 1961 



30cf 



LD 

SEP 2 1962 



LD 21-100m-ll,'49(B7146sl6)476 



5 










V 

^~ 



' 3 ^ ul L^, 



rr / 



k -l>,ks-c,. ru-y*- frf 0* 

, 

Oa ju~r ^*. 



t X 






A f /MA4^e 

; e,^ ot U. T -,H*" 

*fjj^cti4 * */* 
v^- J^^XA-^> j Wt^ 

UA_