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Xtbrari? 

of tl>e 

'University of Mlsconsin 



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I 



• 



Digitized by VjjOOQIC 



Digitized by VjjOOQIC 



Digitized by VjjOOQIC 



THE 

RAILEOAD TAPER 

I 

I 

THE THEOEY AND APPLICATION OF A 
COMPOUND TRANSITION CURVE 
BASED UPON THIRTY- 
FOOT CHORDS 



BY 
LEE PERKINS 

FoBicBBLT Assistant Enginbbb, Southdbn Pacific Linbs in 

Arizona and Mbxico 

Mbmbbr Ambbican Railway Enginbbbing Association 



FIRST EDITION 
ViBST THOUSAND 



NEW YORK 
JOHN WILEY & SONS, Inc. 

London: CHAPMAN & HALL, Luiitbd 
1915 



Google 



COPTEIOHT. 1915 
BY 

LEE PERKINS 



Btaxfbopt pttSB 

F. H. GILSON COMPANY 
BOSTON, U.S.A. 



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204129 

JUN -9 i9IS 
oSFL 



C^cP^^^H 



PREFACE 



A VARIETY of transition curves are used on the different 
railroads, most of them fully explained by published hand- 
books. A number of years ago Mr. Wm. Hood, Chief 
Engineer of the Southern Pacific, devised a compound 
transition curve of unusual simplicity, and developed 
several tables for this curve. For a number of years this 
taper has been used on many roads with somewhat in- 
complete data, usually in the form of blue prints. 

This handbook has been prepared as an extension and 
elaboration of Mr. Hood's earlier work, in an effort to 
simplify the use of the taper, and to lessen the work of the 
man in the field.. With this in view, to a brief explanation 
of the taper has been added the solution of all the usual 
curve problems as applied to tapered curves. Since the 
various transition curves differ only in their method of 
development and their numerical values, differing not at 
all in the method of inserting them between tangent and 
curve, or between curves, these curve problems for 
tapered curves are available for use with the tables of any 
transition curve. 

The transition curve herein developed is, theoretically, a 
series of compound curves, each branch of the curve having 
a thirty-foot chord (in the metric system a ten-meter 
chord), the last branch compounding with the central 
circular curve, the degree of curvature of the various 
branches and ending with the main curve increasing in 
direct arithmetical progression. The various tables are 
to enable one to run in this series of compound curves with 

iii Digitized by Google 



IV PREFACE 

the ease of running in a simple curve, and from a single 
setting of the transit. 

The primary taper tables give the various taper func- 
tions for every thirty feet of taper. That all the tapers 
may be available for use with all degrees and fractional 
degrees of main curve, avoiding the use of inappropriate 
taper lengths, additional tables have been furnished giving 
the taper functions for main curves of intermediate degree, 
the values having been trigonometrically interpolated. 
If further interpolation be necessary it may be linear 
between the values given in these tables of intermediate 
values. In addition, fairly simple equations are given for 
all the angular functions, frequently useful in setting 
intermediate points on the taper. 

The radius of main curve is assimaed based on a fifty- 
foot chord throughout, mainly because the roads that have 
always • used this taper have always used that radius. 
However, this usage is in accordance with the present 
tendency, obviating the necessity of using long chain 
lengths in setting fifty-foot points, which are set much 
more frequently than not. 

All the functions of the taper for both field and drawing- 
room work are given for eleven different tapers of increas- 
ing taper increment, which with their forty to eighty 
lengths comprise some seven hundred tapers. Three of 
these eleven, however, will be found to answer most needs 
except in mountainous country. 

The taper numbers used are those which have long been 
employed for these curves, and are retained for the benefit of 
present records, though a more logical system of number- 
ing, such as the degree of curvature increment per thirty 
feet of taper, might seem desirable, and is suggested for 
use where this taper is hereafter adopted. 

The problem of tapering old circular curves is essentially 
a different one from locating new lines, and is handled 
separately. The insertion of tapers between the branches 

. Digitized by VjOOQli 



PREFACE V 

of compound curves should receive more attention than 
it has generally had. A fairly simple solution of this 
problem is given. 

The tapers have been calculated for metric curves for 
the use of the constantly increasing number of American 
engineers in Mexico and South America, where this taper 
is already extensively used. 

Acknowledgments are due to the following, for the use 
of tables reprinted from their respective books: 

Professor Daniel Carhart, Logarithms of Numbers and 
Trigonometric and Miscellaneous Formulas; Professor J. 
C. Nagle, Logarithmic Sines and Cosines, Logarithmic 
Tangents and Cotangents, Functions of a One Degree 
Curve, Natural Secants and Cosecants, and Metric Con- 
version Table; Mr. Walter Loring Webb, Logarithmic 
Versed Sines and External Secants, Natiu'al Sines, Cosines, 
Tangents and Cotangents, and Natural Versed Sines 
and External Secants; and the American Railway En- 
gineering Association, for the Table of Theoretical Super- 
elevation of Rails, prepared by its Committee on Track 
and pubhshed in the Manual of that Association. 

LEE PERKINS. 
Sn.VER City, New Mexico, 
Marchy 1916. 



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y Google 



CONTENTS 



CHAPTER I 

INTRODUCTION 

Paqk 

1. Purpose of the Taper 1 

DEVELOPMENT OF THE TAPER 

2. Series of Compound Curves 1 

3. Central Angle, o 3 

4. Co-ordinates x and y , 3 

5. Deflection Angles 3 

6. Long Chords 3 

7. t and d, giving location of B.C 6 

CHAPTER II 
LOCATION PROBLEMS 

Simple Curves 

8. Semi-Tangent, T 7 

9. External, E 11 

10. Tangent Offset and Distance along the Tangent to any 

Point on a Curve 11 

11. To Pass a Curve through a Given Point, uniting two Given 

Tangents 11 

12. To Pass a Curve of Given Radius through a Given Point, 

and uniting with a Given Tangent 15 

13. To Pass a Curve through two Given Points and uniting 

with a Given Tangent 16 

14. To fix upon a Tangent at the End of a Curve 18 

15. To Locate a Tangent to two Located Curves 20 

16. To Relocate a Curve making the E.C. fall at a Given Point 

in a Given Parallel Tangent 24 

17. The Intersection of a Line with a Curve 26 

Compound Curves 

18. Partial Taper between the Branches of the Compound 

Curve .^ 27 

ylJ Digitized by CjOOQ IC 



VIU CONTENTS 

Paob 

19. Taper Functions of the Partial Taper 28 

20. To find the Semi-Tangents, both Radii and both Central 

Angles given 31 

21. To Compound a Simple Curve so the E.C. shall fall in a 

Given Parallel Tangent, both Radii given 33 

22. To Compound a Simple Curve so the E.C. shall fall in a 

new Located Tangent, both Radii, or both Central Angles 
given 34 

23. To find the Second Semi-Tangent and the Central Angles, 

one Semi-Tangent, the Intersection Angle and both Radii 
given 36 

Curves Beyond the Frog 

24. Taper beginning at the Heel of Frog 37 

CHAPTER III 
FIELD WORK . 

25. Selection of the Taper 40 

26. Laying out a Tapered Curve 41 

OBSTRUCTIONS ON THE TAPER 

27. Curve run from Main Curve Produced 42 

28. Curve run from a Point beyond the C.C 44 

29. Taper Points set by Offset 45 

30. Taper Points set from Point on Semi-Tangent 45 

31. Curve Notes for the Field Book 46 

CHAPTER IV 
OFFICE WORK 

32. Plotting a Tapered Curve 47 

33. Paper Location with Tapers 47 

34. Curve Compensation with Tapers 47 

35. Right of Way Descriptions with Tapers 48 

CHAPTER V 
TAPERING OLD TRACK 

36. Condition of Track 49 

37. Operating Conditions Qicuti?qfii)^C7QQgii 60 



CONTENTS IX 

METHODS OF INSERTING TAPERS 

Simple Curves 

' Pagb 

3a. Shifting the Curve 50 

39. Shifting the Tangents 52 

40. Shifting Curve and Tangents 53 

41. Increasing Degree of Curve 53 

42. Increasing Degree and Shifting Curve 54 

43. Maintaining Length of Line Unchanged 55 

44. Increasing Degree and Shifting Curve and Tangents 57 

45. Combination of Methods 57 

46. Increasing Degree of Portion of Curve , 59 

Compound Curves 

47. Similarity of Procedure 59 

48. Shifting the Curves 59 

49. Shifting Curves and Tangents 60 

50. Shifting Lighter Branch and Increasing Degree of Sharper 

Branch 60 

61. Field Work with Compound Curves 61 

TABLES 

Radii of Curves and their Logarithms 63 

Notation used in the Taper Tables 69 

Important Equations abstracted from the Text 69 

Taper i, A.= i 71 

1 ,A- i 74 

1§,A= i 78 

2 , A = 1 82 

2i, A = 11 86 

2i, A = li 90 

2i, A = 2 94 

3 ,A :;=2i 97 

3i, A = 3 100 

3i, A = 4 103 

3,i A = 5 105 

Comparative Values of i, d, and o 108 

Maintaining Length of Line Unchanged 115 

Logarithms of Numbers 117 

Logarithmic Sines and Cosines 135 

Logarithmic Tangents and Cotangents 150 

Logarithmic Versed Sines and External Secants. . (^f^/Ntr|r • • ^^ 



X CONTENTS 

Paob 

Futictions of a One Degree Curve 210 

Natural Sines, Cosines, Tangents and Cotangents 240 

Natural Secants and Cosecants 263 

Natural Versed Sines and External Secants 276 

METRIC TABLES 

Radii of Curves and their Logarithms 299 

Notation used in the Taper Tables 305 

Important Equations abstracted from the Text 305 

Taper i, A = J 307 

1 ,A = i 310 

li, A = i 313 

2 , A = 1 317 

2i, A = li 321 

2i, A = li 325 

2i, A = 2 329 

3 ,A -2i 333 

3i, A = 3 337 

3i, A = 4. . . . . . .'. 340 

3i, A = 5 342 

Comparative Values of Z, d, and o 344 

Mutual Conversion of Feet and Inches into Meters and Centi- 
meters 351 

Trigonometric and Miscellaneous Formulas 352 

Super-elevation of Rails on Curves 355 



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THE RAILROAD TAPER 



CHAPTER I 
INTRODUCTION 

PURPOS^J OF THE TAPER 

1. Tapered curves have gained such wide usage that 
it is needless to dwell at length on their advantages. 
Where circular curves alone are used, the transition from 
tangent to curve is sudden, giving a decided lurch to the 
cars, unpleasant to passengers and damaging alike to 
rolling stock and track. Running the super-elevation 
back on the tangent does not obviate this, iand it can be 
remedied only by increasing the curvature gradually, in- 
creasing the super-elevation of rails at the same time. 

This the tapered curve accomplishes in a satisfactory 
and practical manner, much as it has always been done by 
track foremen, who lined in an approximate taper by eye. 
The proper use of the taper, however, insures a uniform 
curve beyond the taper, a result not easily obtained by the 
trackman unaided. 

Development of the Taper 

2. The railroad taper consists, theoretically, of a series 
of short curves, each of thirty-foot chord length, and of 
uniformly increasing curvature, compounding at the end 
with the central main curve. The various tapers differ 
merely in the curvature increment, and, therefore, in the 
length needed to lead up to the curvature of the main 
curve. Fig. 1 illustrates the first one hundred and twenty 

Digitized by VjjOOQ IC 



THE RAILROAD TAPER 



zl> 




Fig. 2. 



Google 



DBVBLOPMENT OF THE TAPER 8 

feet of Taper 2, compounding with the main five-degree 
curve. 

3. The central angle (o) of the taper is, of course, the 
summation of the central angles of its parts. 

4. In Fig. 2 is shown the same curve represented by 
its thirty-foot chords, showing the method of obtaining 
the X and y co-ordinates of the chord-points referred to 
the initial tangent and the B.C. (beginning of curve) of the 
taper. The angle k is the azimuth of the various chords, 
referred to the initial tangent, and the x and y co-ordinates 
are the departures and latitudes of the chord-points. 
These co-ordinates are given in the tables. 

6. From x and y were calculated for the tables the deflec- 
tion angles (/) from the initial tangent at the B.C. to the 
various chord-points, by the equation 

tan/ = ?. (1) 

The backsight deflection (6) from tangent at any chord- 
point to the B.C. equals the central angle of the taper 
up to that point minus the deflection at the B.C. to that 
chord-point, and is given in the tables. 

As shown in Fig. 3, the deflection from tangent at any 
chord-point to any other chord-point equals the azimuth 
of the long chord between the chord-points, minus the 
central angle of the taper up to the first chord-point. 

x** — x' 
This azimuth is the angle whose tangent is -jj-^ — y 

As shown in Fig. 4, the backsight deflection from tangent 
at any chord-point to any other chord-point is 

and is given in the tables. 

6. The long chord {}£) from the B,C. to any chord- 
point, given in the tables, is 

l<i = ^i (2) 



^ Digitized by Google 



THE RAILROAD TAPER 




Fig. 3. 






Initial Tangent"^ 




FiQ.4. 



Google 




Fig. 6. 



Digitized by VjOOQIC ^5) 



6 THE RAILROAD TAPER 

Similarly, as in Fig. 6, the long chord between any other 
chord-points is 

^ - cos (e' +/'-")■ ^^^ 

7. In order to make room for the taper, the simple curve 
may be considered as ofifset from its natural position a 
distance from the tangent that will be called d, the taper 
beginning at a point back of this ofifset position of the 
B,C, of the simple curve a distance along the tangent that 
will be called t 

This tangent distance, t, is very nearly half the length 
of the taper, its exact value, from Fig. 6, being 
t=^BD-IE 
= y — R&inOj (4) 

where R is the radius of the main curve, and o the central 
angle of the taper. 
The value of the circular curve ofifset, d, from Fig. 6, is 
d = DE-HI 
= X — R vers o. (5) 

The values of t and d are given in the tables. 

These two equations, 4 and 5, are the fundamental 
equations of the taper, locating the center of curvature 
of the main curve with reference to the tangents and the 
beginning and end of curve. The equations are identical 
in form for all transition curves, and similar values under 
dififerent letter designations are invariably given in their 
tables. On these two distances, t and d, are based all the 
solutions of curve problems in the succeeding chapters. 



y Google 



CHAPTER II 
LOCATION PROBLEMS 

8. To Find the Semi-tangents of a Tapered Curve. 
Given R and / to find T. 




Fig. 7. 

Case I. Tapered at both ends with the same taper. 
From Fig. 7, 

T = AD + AB 

= t + OAtB,n^I 

= t+{R + d)UniI. (6) 

7 Digitized by VjjOOQ IC 



8 



THE RAILROAD TAPBR 



In the tables is given a value D = R + d,m the above 
may be written 

T = t + DtAniL (7) 

Example, Find the semi-tangent of a 5° 00' curve, 
tapered with Tap. 2, total angle of curve 35° 04'. 

ByEqn. 7 logZ)60Tap.2 3.05949 

log tan 17° 32' 9.49960 
2.55909 
362.32 
t = 59.99 
T= 422.31 
Case II. Tapered at one end only. 
It may occasionally be necessary to taper one end only 
of a curve, or, considering the parallel ofifset tangent, to 
treat the curve as tapered at one end only. 




Fig. 8. 

In this case the two semi-tangents will be of different 
lengths, the one for the tapered end being the longer, when 
/ is less than 90°, by an amount less than t. 

From Fig. 8. 

MN = MK = dcscI, 
Tt = t + OAts^niI^MN 

= < + DtanH-dcsc/, (8) 

Tn = /? tan I / + d CSC 7.^,,,^^, ,,Google 0) 



LOCATION PBOBLBMS 9 

Theee equations do not change for angles greater than 
90°, the algebraic sign of the last term remaining un- 
changed. 

Example. Find the semi-tangents of a 7" 00' curve, 
tapered at the B.C. with Tap. 2^, no taper at the E.C., 
total angle of curve 64" 22'. 

By Eqn. 8 esc 64° 22' 1 . 23037 

d 0.95 

615185 
1 007333 



1.0688515 



logZ)7«Ta|).2i 2.91360 

log tan 27° 11' 9.71059 
2.62419 

420.91 

+ t 54.98 

476.89 

1.07 



B.C. to P.I. 474.82 

By Eqn. 9 log Rr 2.91309 

log tan 27° 11' 9.71059 

2.62368 

420.42 
+ 1.07 

P.I. to B.C. 421.49 

Case III. Tapered at both ends, wUh different tapers. 
From Fig. 9, 

MN = MK'=dicscI, 
BM = di esc /, 
BJ = BN = MN - BM 

= (di — dj) CSC I, 
Ti = ti + Ditanil -(di-dt)cacl. (10) 
Tt='tt + Data.n^I -ldt-di)cacl. (11) 
These equations do not change for angles greater than 
90°, the algebraic sign of the last term remaining un- 
changed. • og„„e, by Google 



10 



THB RAILBOAD TAPBB 



Example. Find the semi-tangents of a 10'*00' curve, 
tapered at the B.C. with Tap. 2^, and at the E.G. with 
Tap. 3, total angle of curve 113" 42'. 




By Eqn. 10 



Fig. 9. 



esc 1 13' 42' (esc 66" 18') 1 . 0921 1 
di - tfe = 2.84 - 0.98 = 1.86 

655266 
873688 
1 09211 



By Eqn. 11 



log DiQo Tap. 2J 

log tan 56° 51' 



+ h 



B.C. to P.I. 

log Dio° Tap. 3 

log tan 56° 51' 



2.03 

2.76041 
0.18500 
2.94541 

881.88 

84.88 

966.76 

2.03 

964.73 

2.75900 
0.18500 
2.94400 

Digitized by 



Google 



LOCATION PROBLEMS 11 





879.02 


+ u 


44.97 




923.99 


+ 


2.03 


P.I. to E.C. 


926.02 



9. Given i^ and JT, to find the External, E. 

Case I. Tapered at both ends vxith the same taper. 
From Fig. 7, 

E = OB-R 

= Dsec|/-i2, (12) 

or E = {R + d) sec^ I - R - d + d 

= (ft + d)(secK- l)+d 

= D exsec ^I + d, (13) 

Example. Find the external of a 5° 00' curve, Tap. 2, 
total angle 27° 12'. 

By Eqn. 12 or 13, E = 33.88. 

Case II. Tapered at one end only, or at both ends, with 
different tapers. 
From Fig. 10, 

e = Vo^Ta^j-r 

= VZ)2 + (T - 0' - fi, (14) 

or E =^D see tan-^ — ^ - R, (15) 

T — t . T ^ t 

where tan~^ j^ is the angle whose tangent is — yr—* 

Example. Find the external to the curve given in the 
example under equations 8 and 9. 
By Eqn. 14 or 15, E = 102.24. 

10. To Find the Tangent Offset and Tangent Distance 
to any Point on the Curve. 

From Fig. 11, 

PD = AC + d 

= RyersI + d, (16) 

ED = BD + t 

= 72 sin / + ^.^.g,^^, by Google (17) 



12 



THE RAILROAD TAPBR 




Fig. 10. 




Fig. 11. 



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LOCATION PROBLEMS 



13 



Or from the Table of Functions of a 1° curve, AC is the 
mid-ordinate for a central angle equal to 2 /, and BD is 
half the long chord for a similar central angle. 

11. Given a Point, JP, referred to the P.J., to find 
the Radius of a Curve passing through JP uniting the 
Given Tangents. 

Since the solution of this problem depends upon the 
taper chosen and its value of d, and the choice of taper 
depends largely upon the degree of curve, a trial solution 
for an untapered curve is first made. 




Fig. 12. 



From Fig. 12, 



OS . . 

sin 2/ =gpsma 

— -R sec ^ / 
" R 
__ sing 
"" cos^/' 
R=PO 
sin a 



sma 



PB. 

sin 3C Digitized by 



Google 



(18) 
(19) 



14 



THE RAILROAD TAPER 



Since this problem occurs mainly in location, and it is 
necessary merely to have the curve pass close to the 
determining point without using a curve of fractional 
degree, to allow room for the taper a curve of next higher 
even d^ree is chosen, and when desirable it may be 
determined as follows how close the chosen curve will pass 
to the given point: 

In Fig. 13, P is the point through which the curve should 
pass, P' a point on the chosen curve radially opposite. 
PP'=O P-R 

-^ VOD^ + DP^-R 

= V(OA - APy + {AB - CBY - R 

= V(Z>-FBsm6)2+(Z>tanJ7-f'5cos6)2-/2. (20) 



U c 

\ 

D ~^^^v5^ 



Fig. 13. 
Example. In Fig. 13, let PB = 194.3, 6 = 42^ 24', 
I = 54° 22'. 



Then a = 90° - ^V^ - 42^24' 



= 20° 25'. 

By Eqn. 18, y = 23° 05' 

x = y^a= 2° 40', 

By Eqn. 19, R = 1456.8. 

3° 56' curve. 



y Google 



LOCATION PROBLEMS 



15 



Deciding now upon a 4° 00' curve with Tap. 1, to 
determine how close this curve will pass to point P; 
By Eqn. 20, {D-PB sin fe)2 = 1698100 
(D tan i I-PB cos hy= 351675 
2049775 

\/2049775 = 1431.7 
- R 1432.5 

PP' = - 0.8 

The negative result shows that P is inside the curve 
0.8 ft., which on most location is sufficiently close. 

12. Find the B.C. of a Curve of Given Radius that will 
pass through a Given Point referred to a Given Tangent. 




Fig. 14. 



From Fig. 14, 



AB = R sin vers~^ 



BPj-d 
D 



(21) 



Ir times the sine of the angle whose versine is — j^ — j j 



or 



W^(P- BPy + A&, 
AB = v^-(i>-BP)^,.,,,,Google(22) 



16 



THE BAILROAD TAPES 



Example. Given BP = 82 ft., M = 6** 00', Tap. 2, 
then D = 956.41, R = 955.04. 



By Eqn. 21, vers 



,, BP-d _ 



R 



= 22°45i'. 



AB = 369.45. 



13. Given Two Points referred to a Tangent^ to find 
the Radius and B.C. of a Curve passing through them 
and joining the Given Tangent with a Given Taper. 

Since the solution of this problem depends upon the 
taper chosen and its value of d, the choice of which depends 
largely upon the degree of curve, a trial solution for an 
untapered curve is first made. 




Fia. 15. 



In Fig. 15, given points P and S, referred to the tangent 
by lengths DP and DS, and the angle a, to find the radius 
OA and distance AD. 



By geometry 



AD^ = DPX DS, 
AD = VDP X DS. 



(23) 



DB and BP may be obtained from a solution of the tri- 



angle BDP. 



Digitized by VjOOQIC 



LOCATION PBOBLBM8 



17 



J, AB^ + BP* 

K = T>T^ > 



or 



fi = 



2BP 
AB 



• o* iBP 
sin 2 tan~^ -j^ 



(24) 
(25) 



A curve of next higher even degree is now chosen with 
an appropriate taper, passed through iS as in paragraph 
12, and the distance of P from the curve determined as 
follows. 




Fia. 16. 
From Fig. 16, 

A'C = Rsin vers"' — j; — > 



R 



or 



A'C = Vie* - (D - CS)*; 
PP' = O P-R 

= VOEP + EP*-R 



or 



PP' = 



= V(D - fiF)« + A'B' - R, 
D-BP 



cos tan" 



A'B 
D-BP 



R. 



(26) 
(27) 

{2S) 
(29) 



y Google 



18 THE RAILBOAD TAPER 

Example. DP = 94 ft., PS = 127 ft., angle a = 16" 12'. 
ByEqn. 23, AD = lU.l. 

From A BDP, BP = 26.2 

and Z>B= 90.3; 

then AB = AD + DB = 2MA. 



By Eqn. 25, 


tan--^-6»23', 


id 


5 = 1060.7 




5° 24' curve. 



Deciding now upon a 5° 30' curve. Tap. 2, through S: 
From A DSC, C/S = 61.66 . 

-d 1.05 

CS-d = 60.61 
and DC = 212.2 

-DB 90.3 



BC = 121.9. 
By Eqn. 26, vers-^ "^^ = 19° 38' 

and A'C = 350.06 

-5C 121.9 



A'B = 228.16. 

By Eqn. 29, tan-i^4!§p = 1^° 39', 

and PP' = 0.14. 

From the positive value of PP\ P lies outside the curve 
assumed. If the curve had not passed sufficiently close 
to Pj a slightly sharper curve would pass further out at 
P', or if the curve had passed outside of P, a slightly 
lighter curve would pass closer to P. 

14. To Fix upon a Tangent at the End of a Curve. 

In the previous problem of passing the curve through 
two determining points and tangent to the back-located 
tangent, it is not practical to first locate- the tangent 



LOCATION PROBLEMS 



19 



ahead and make an intersection. The curve must be 
run in beyond the determining points, then a meander line 
run to the next controlling point on the tangent ahead 
and the tangent located as follows: 

When the curve has reached a point A, Fig. 17, from 
which a view may be had of the tangent desired, or of 
some controlling point, B, on it, through which it is 
desired to run the tangent because of properly supporting 
ground, a flag may be sent to C, opposite B, (the uncer- 



„/ \ 




L 


B 


J ^ — 


S 




/ ^^n[ 


F ^--s^ 


«s. 


, 

^i-^ 


"^^^^ 


/v\*^ 





■> 


r::^ 




""^^^ 


^ 


/^ 


.^ 





Fig. 17. 

tainty of right angle direction will aflfect this very little), 
and a distance therefrom equal to d of the taper. With 
improvised targets of some nature, or double points and 
angle between, a fairly accurate stadia shot may be 
obtained, and with the angle from tangent at A to C, the 
distance BL and the central angle of the rest of the curve 
may readily be calculated. 
From Fig. 17, 

AG = 2i2sina, 
OK = R cos a, 
CF = y/CG X AC, 
OK 



tann = 



KC 



ficosa 
AC — Rsina 



y Google 



(30) 



20 THE RAILROAD TAPER 

R 

R 



tanm-^^ 



- / > (31) 

V(AC -2R&ma)AC 

6 = a + m - n. (32) 

Given by AD and DL may be calculated by equations 
9 and 8. Then 

LB = CF-J^ 

= VlAC - 2Rsina)AC - t. (33) 

Example. Assume a 5® 30' curve, Tap. 2, to have been 
run to point A, Distance AC by stadia is 610 ft., and 
angle a is 6® 10'. 

By Eqn. 30, n = 64^ 19'. 

ByEqn. 31, w = 66^01'. 

By Eqn. 32, a 6^ 10' 

+ m 66^01' 
-n 64° 19' 
6= 7^62'. 
ByEqn. 33, LB = 417.9. 

16. To Connect two Located Curves by a Tangent. 

In the preceding problem, if the next controlling point 
had been on a curve, instead of a tangent, it would often 
be necessary to locate this next curve first and then con- 
nect the two curves by a tangent. 

Case I. Curves of contrary flexure. 

Let AB and CD, Fig. 18, be the two located curves, to 
locate tangent LM fitting the two curves with given 
tapers, the curves being ofiFset from the tangent di and cfe, 
respectively, to provide room for the tapers. 

From some point E on tangent at B, run one or more 
courses, EF, to some point F on tangent at D. Calculate 
the latitude and departures of N and with reference to 
tangent at S as base and radius BN as meridian. The 
diflference of the latitudes equals OG, the difference of the 
departures equals NG. 

Digitized by VjOOQIC 



LOCATION PROBLEMS 



Then, 




21 



tane = ^, 



(34) 
(35) 



cos c = ^t + <^i + fe + rf. 
NO 

NO ' 
a = 90'' - e - c, 
f - g = a-b, 

f> = <^-f+g, 
LM = ST- it, + 1,) 

= NOdn C — (fi -j- ^)3».zedbyG00glggv 



(36) 
(37) 

(38) 



22 



THE RAILROAD TAPER 



Example. Given a 5*^ 30' curve AB and a 4® 00' curve 
CD, to locate tangent connecting them, using Taper 2 
on the 5"^ 30' curve and Taper U on the 4° 00' curve. 

At B turn tangent to curve and set point E, and at D 
turn tangent and set F. Assume BE = 100 ft., angle 
/ = 9"^ 11', EF = 384.0 ft., angle ^ = 6° 38', FD = 100 ft. 

Then a traverse from iV to is as follows: 



Course. 


Length. 


Asimuth. 


Lat. 


Etop. 


NB 

BE 

EF 

FD 

DO 


1041.8 
100.0 
384.0 
100.0 

1432.5 


180° 00' 
90° 00' 
80° 49' 
87° 27' 

177° 27' 


-1041.8 


+ 61.3 
+ 4.4 
-1431.1 



+100.0 
+379.1 
+ 99.9 
+ 63.7 


NO 


2491.5 


165° 03' 


-2407.2 


+642.7 



From this traverse 





iVO = 2491.5, 




NG = 642.7, 




GO = 2407.2, 




e = n" 03'. 


By Eqn. 36, 


c = 6° 23'. 


By Eqn. 37, 


a = 8° 34'. 


By Eqn. 38, 


6 = 6" 01'. 


By Eqn. 39, 


LM = 144.5. 


By Eqn. 9, 


BH = 85.08. 


By Eqn. 8, 


HL = 138.54. 


By Eqn. 8, 


MK = 133.54. 


By Eqn. 9, 


KD= 82.05. 



If in this solution LM had been too short, or even 
negative, one of the curves would have to be shifted or 
sharpened, or the tapers shortened. 



Case II . Curves of similar flexure. 
Determine NG and GO as in Case I. 



y Google 



LOCATION PROBLEMS 



23 




Fig. 19. 



Then from Fig. 19, 



NO 


= VOG^ + NG^, 

OG 
-NG' 


(34) 


tane 


(35) 


cose 


A- A 

• NO ' 


(36a) 


a 


= e + c - 90°, 


(37a) 


b 


^f + g-a, 


(38a) 


LM 


Digitized by V!jOOQ 


1, (39) 

J i 



24 



THE RAILROAD TAPER 



Example. Assume the same conditions as in the 
example under Case I, angle g merely being in the opposite 
direction. 

Then a traverse from iV to is as follows: 



Courae. 


Lencth. 


Asimuth. 


Lat. 


Dep. 


NB 

BE 

EF 

FD 

DO 


1041.8 
100.0 
384.0 
100.0 

1432.5 


180° 00' 
90° 00' 
80° 49' 
74° 11' 

344° 11' 


-1041.8 


-1- 61.3 
+ 27.3 
-1-1378.3 




-hioo.o 

4-379.1 
4- 96.2 
-390.4 


NO 


485.4 


23° 30' 


-h 425.1 


-hl84.9 



From this traverse 



By Eqn. 36a, 
By Eqn. 37a, 
By Eqn. 38a, 
By Eqn. 39, 



iVO = 485.4, 

e = 90^^-23° 30' 
= 66^^30'. 

c = 36°29'. 

a = 12^59'. 

6= 2^50'. 
LM = 156.1. 



16. To Relocate a Curve so the B.C. shall fall at a 
Given Point in a Given Parallel Tangent. 

In Fig. 20 let ABC be the located curve, tapers omitted 
for clearness, and DEF the required curve ending at F. 
CG and GF are known. 

By a solution of triangle KMIf KF becomes known. 
The value of R2 now depends solely upon the taper chosen 
and its value of fe and tfe, and may best be determined by 
trial, using the tables of functions of a one-degree curve, 
assuming at first a value of fe somewhat larger, or smaller 
as the case may be, than ti. It will usually be sufficient 
for the semi-tangent of the new curve approximately to 
equal KF. .^ , 

Digitized by VjOOQlC 



LOCATION PROBLEMS 



25 



Example. Assume curve ABC to be a 6° 00' curve, 
Taper 2, total angle 36° 14'. CG = 7.1, GF = 42.4. 

D K I 




N*-'" 



FiQ. 20. 



From A KMI 


KM 


= 9.7 






and 


KI = 12.0. 




By Eqn. 7, 


IC = 387.88 
+ KM 9.7 
+ GF 42.4 

KF = 440.0. 




From Tables, ] 


[" curve 

Tsa-M' = 1874.7. 




If the new curve were untapered, 






^ 440.0 *•''*' • 




If Taper 2 be used on the new curve, 


then 






M 


1847.7 
M 


\i 


Approx. T. 




/ 


4° 45' 
50' 
55' 

5° 00' 


394.7 

.387.9 

381.3 

374.9 


56.2 
57.5 

58.7 
60.0 

il 


450.9 
445.4 
440.0 
434.9 

i.PrlhvC^OOgl' 


> 



26 



THE RAILROAD TAPER 



Assuming a 4° 65' curve, Taper 2, then 
by Eqn. 7, Tt = 440.29. 

In this particular case it is unnecessary to interpolate 
to obtain a more exact value for the semi-tangents, 0.29 ft. 
being sufficiently close. Linear interpolation between the 
intermediate values given in the tables would be quite 
accurate when necessary. 

17. Intersection of a Line with a Curve. 
In Fig. 21, given R, d, AB and angle 6, to find BP and 
angle a. 



^^r^^^s:^ 




^'/\^ 


B 


/ y^^" 




jL^-y\ 




^l^ T I 1 R (/ 


A 


D-R+d 


^r 


B.O. 


j; 







Fig. 21. 



D 



,ann 


= AB' 




KW) 


c 


= 180" - n 


-h, 


(41) 


d 


= n + 6' - 


180', 


(41a) 


OB 


D ' 
Sinn 






sine 


OB . 




Digitized by GoOQIc 



LOCATION PROBLBUS 27 

(42) 



Dsinc 
Rsian 



g^^ gsin(e-c) ^ (43) 

Sin c 

= 90° - n + e - c, (44) 

= 90° - n + c' - 6'. (44a) 

In equations 42 and 43, c' may be substituted for c with 
no other changes. 

Example. Given a 4° 00' curve, Taper Ij, ilB = 195.9 
ft., 6 = 67° 07'. 

By Eqn. 40, n = 82° 13'. 

By Eqn. 41, c = 30° 40'. 

By Eqn. 42, e = 31° 00'. 

By Eqn. 43, SP = 16.34. 

By Eqn. 44, a = 8° 07'. 



Compound Curves 

18. A Taper between the Branches of a Compound 
Curve is often more necessary than between curves and 
tangents, as the change in curvature is often greater. 
Only that portion of the complete taper required to gradu- 
ally increase the curvature as desired, is used. 

Example. Taper 2 from tangent to a 6° 00' curve is 
built up from 30 ft. of 1°, 30 ft. of 2°, 30 ft. of 3°, 30 ft. of 
4° and 30 ft. of 5° curves, but in compounding from a 2° 
to a 6° curve with Taper 2, the first 60 ft. is dropped oflF, 
the deflection tables being used as if the transit were at 
Z = 60, the length of taper to fit a curve of M° = Mi + A, 
where Mi is the degree of the lighter branch, and A the 
curvature increment in degrees for the taper used. 



28 THE RAILROAD TAPER 

19. Taper Functions of the Partial Taper. 

From the method of developing the taper, the length of 
any taper from tangent is, 

Z = ^(M-A). *(45) 

Then the entire taper from tangent to fit the branch of 
the compound curve of greater curvature is 

fe = ^(M8-A), *(46) 

and the length replaced by the branch of lesser curvature 
is that needed to fit a curve of M° = Af i + A degree, or 

ii = f (Mi + A-A), 

Then the partial taper retained between the branches 
of the compound has a length of 

Ic == h — 'i 

= ^(M2-Mi-A). *(48) 

In Fig. 22, Ml is the curve of lesser degree, M2 the curve 
of greater degree, and ABC the taper joining the two 
curves, the tapers at the beginning and end of the com- 
pound curve being omitted for clearness of diagram. 

The two curves are offset from each other to make room 
for the taper an amount KF = dr, radially to both curves, 



* Equations 45, 46, 47, 48, 50, 51a, 56 and 57 are empirical 
equations derived from the tables, and cannot be used with other 
transition curves. When the taper functions of Mi and M2 are 
given in the tables these equations are not needed. When needed 
for other transition curves, somewhat similar equatiqns, but with 
other constants, must be derived. Digitized by Google 



LOCATION PROBLEMS 



29 



the curve of greater curvature lying within the one of 
lesser curvature. 

d. = d for M = Ma - Mi (49) 

to be taken from the tables. 
The central angle of the partial taper is 

9 (M2 + Ml) (M2 - Ml - A) 



Oc = 



*(50) 



or Oc may be taken from the tables, being the difference 
between for h and for Zi. 




Fig. 22. 

The common radial line, OF, divides this angle, Oc, 
proportionally to the degrees of curve of the two branches 
of the compound curve. 



* See footnote, page 28.jitized by Google 



30 THE RAILROAD TAPER 



Ml /ci\ 



9 Ml (Mi- Ml - A) 



* 



(51a) 



A 

Let DA be tangent to the curve of lesser degree at the 
beginning of the partial taper, and let the curve of greater 
degree be backed up through angle Ocj so that its tangent, 
ET, shall be parallel to DA, then 

tc = (fii — R2 — dr) sin a, (52) 

dc = Ri — R2 — tc cot a, (53) 

or = (fii — R2) vers a + dr cos a. (53a) 

From Eqn. 4, yc = tc + R2 sin Oc. (54) 

From Eqn. 5, Xc = dc + R2 vers Oc. (55) 

Using the method of second differences with the tables, 
the deflection to the end of the taper, from tangent at the 
beginning of the partial taper, is 

. (3^1 + ^0 + 45);. + 450 ^,.^, 

^'^ 300 . ^^^^ 



-IP 



A 
Mi + 2M2 + A)(M2 - Ml - A) 



+ A . *(57) 



Equation 56 may be used for obtaining all the deflec- 
tions from tangent at^, the beginning of the partial taper, 
to points on the taper, using for Ic the distance of the point 
beyond the beginning of the partial taper. A, k remaining 
constant from equation 46. 

Where Afi is a multiple of A, the deflections to points 
on the partial taper may be taken directly from the tables, 
transit at I = Zi, deflection to I = h + k, using for k the 
distance of the point beyond the beginning of the partial 
taper, A. 

* See footnote, page 28. jtizedbyCiOOgle 



LOCATION PROBLEMS 31 

The angle of backsight to the beginning of the partial 
taper, from tangent at the end of the taper, is 

bs = Oc-U (58) 
The long chord of the partial taper is, from Eqn. 2, 

Icc = Vc sec/c. (59) 
20. The Semi-tangents of the Compound Curve. 

From Fig. 22 /i = c - a, • (60) 

h = g + a, (61) 

By Eqn. 8, SD = ti + Di tan | /i - di esc h. (62) 

By Eqn. 9, DA = fti tan ^h + di csc.7i. (63) 
By Eqn. 10 

AG. = tc + *Dc tan ^h- {dc - (h) esc h. (64) 
By Eqn. 11, 

GH = t2 + Di tan ^h- {dz - dc) esc h, (65) 

DG = DA+AG, (66) 

Solving triangle DNG gives values for DN and NG. 

Then SN = SD + DN, (67) 

NH = NG + GH. (68) 

Where it is desirable to obtain the semi-tangents of the 
compound curve without first finding the taper functions of 
the partial taper, DG may be drawn tangent to Mi at F 
and equations 62 to 68 become as follows: 

SD = ti + Di tan J c - di esc c, (62a) 

DF = fti tan ^ c + di esc c, (63a) 

FG = ^Dr tan I gr - {dr - d^) esc g, (64a) 

GH = ^ + D2 tan \g — {fk — dr) esc g, (65a) 
DG = DF + FG, (66a) 

SN = SD + DN, (67) 

NH = NG + GH. (68) 

Example. Given / = 114° 24', c = 34° 40', g = 79°44', 
Ml = 3° 00', M2 = 8° 00'. Use Taper 1 between tangent 

* Dc = ^2 -h do. 

t Dr = Ri + dr. Digitized by Google 



32 THE RAILROAD TAPER 

and 3° 00' curve, Taper 2 between 3° 00' and 8° 00' curves, 
and Taper 2J between the 8° 00' curve and tangent. 
Find the semi-tangents. 



By Eqn. 48, 


Z« = ^(8-3-1) 




= 120. 


By Eqn. 49, and the tables, 




dr = 0.79. 


By Eqn. 60, 


9 (8 + 3) (8 - 3 - 1) 
1 




= 396', 




= 6° 36'. 


By Eqn. 51a, 


9 X 3 (8 - 3 - 1) 
1 




= 108' 




= 1°48'. 


By Eqn. 52, 


<„ = * 37.47. 


By Eqn. 53, 


de= 1.38. 


By Eqn. 54, 


yc= 119.80. 


By Eqn. 55, 


x^= 6.13. 


By Eqn. 57, 


/, = 175^ = 2° 55J'. 


By Eqn. 58, 


6s = 3°40j'. 


By Eqn. 59, 


fcc= 119.96. 


By Eqn. 60, 


/, = 32°52'. 


By Eqn. 61, 


h = 81° 32'. 


By Eqn. 62, 


SD= 637.26. 


By Eqn. 63, 


DA = 564.59. 


By Eqn. 64, 


AG= 656.32. 


By Eqn. 65, 


GH= 683.74. 


By Eqn. 66, 


DG = 1220.91. 


From A DGN, 


NG= 727.57, 


id 


DN= 1326.06. 


By Eqn. 67, 


SN = 1963.32. 


By Eqn. 68, 


NH = mi.zi. o,„,e,.,C 



Google 



LOCATION PROBLEMS 



33 



21. Given a Located Curve, to Compound with Curve 
of Given Radius so as to End on a Given Parallel Tangent, 
using Three Tapers. 

In Fig. 23, given located curve SAFV of radius fii, 
parallel tangent NH distant p from the original tangent 
WV, R%y dij dky dry ti Siiid tt, to find /i, h and NH. 




Fig. 23. 



OE = Di-p- D2, 

OP = Ri-dr- ft2, 
OE 



cosg = 



OP 

Ri — R2 + di — dz — p 



*(69) 
(70) 



Ri — Ri — dr 
c = I -g. 

* Where the new tangent is beyond the original tangent, requir- 
ing i^ to be greater than Ru p and dr in equations 69, 71 and 72 
change sign. For dr see Eqn. 49. Digitized by vjOOglC 



84 THB RAILROAD TAPBR 

The new semi-tangents may now be determined by the 
jnethods of the preceding problem. If the semi-tangents 
of the two parts of the curve, very useful in running in the 
curve, are not needed, the semi-tangents of the entire 
curve may be obtained more easily as follows: 

SN = (Ri + di) tanH - pcscJ + tu *(71) 

NH = (fti + di) tanH - (ft - ^ - dr) sing 

+ pcotI + t2. *t(72) 

Example, Given a located 4^30' curve, Taper I5, in- 
tersection angle 82® 14', to compound onto a parallel 
tangent 71 ft. beyond the original tangent, using a 2° 45' 
curve. Taper 1, with Taper 1 between the two curves. 

By Eqn. 69, cos g = ZTgioJ^' 

g = 24° 14'. 
By Eqn. 70, c = 58°00'. 

By Eqn. 71, SN = 1258.99. 

By Eqn. 72, NH = 1502.68. 

22. Given a Located Curve, to Compound onto a new 
Located Tangent, Point of Compound Approximately 
Fixed, full Tapered. 

In Fig. 24, AN* is the located curve, to be compounded 
at about iV' so as to unite with located tangent UF. 

Run tangent to located curve at iV', intersecting located 
tangent. From Functions of a One-degree Curve deter- 
mine degree of curve, Af' , with which to compound if 
tapers were neglected. Then use such a degree of curve 
that 

R, = R'^ ^-±i', (73) 

and the resulting g will be very nearly g\ 

* See footnote, page 33. 

t Care should be taken to use a negative value for cot / when 
/ is greater than 90°. Digitized by Google 



LOCATION PROBLEMS 



85 



*(74) 



OE-OB + BF- FE 

= i2, coay + iV'C sin g' - (fi, + A), 
OP = i2, - iJ, - dr, 

cos sf = ^ 

_ fii cosg' + iV'Csing' - {Rt + dj) 
R\ — Ri — dr 

Example. Given located 4° 00' curve AiV', to com- 
pound at about iV' (iV' to be about the center of the taper 
between parts of the compound), so as to end on located 
tangent CF, N'C = 340.5 ft., g' = 42° 12', both meas- 
ured on the ground. 

- A c u. 




Fia. 24. 



From Functions of a 1" Curve: 
r42''12' = 2211.0, 

R' = 5730 X 



= 882.4, 
M' = 6°29'+. 



340.5 
2211 



* For (2r see Eqn. 49. 



y Google 



sd 



THE RAILBOAD TAPEB 



By Eqn. 73, dj = 1 . 76 for 6* 30' Tap. 2, 

dr = 0.09. 

A j> ooo A 1-75 + 0.09 
Approx. Ri = 882.4 ^ 

= 881.52, 
Rt'w = 881.59. 

Use 6° 30' curve, Tap. 2. 

Ty T^ nA 406.53 

By Eqn. 74, 008^=^5^^ 

fir = 42° 26'. 

23. Given one of the Semi-tangents of a Compound 
Curve, the Intersection Angle, and both Radii, to find 
the two Central Angles and the other Semi-tangent. 

In Fig. 25, given SN, I, Bi and Ri, to find c, g, and NH. 




Fra. 25. 

OE = OW + WU- UE 

= {Ri + di) COS/ + TN^nI -{Rt + dt), 
OP = Ri-Ri-dr, 
OE 



coag = 



OP 



y Google 



LOCATION PROBLEMS 37 

_ (fii + di) cos/ + TN sin/ -(R^ + d^) 

NH = LU -LN -UV + k 

= (ft + di) sin / - TNcosI 

- {Ri - ft - *dr) sin g + fe. t(76) 

Example. Given semi-tangent SiV = 1963.32, Mi = 
3° 00' Tap. 1, Ma = 8° 00' Tap. 2J, and Tap. 2 between 
ilfi and M2, / = 114° 24', to find c, f/ and NH. 

212 62 ' 
By Eqn. 75, cos gf = 339278' 

flf = 79° 44'. 
By Eqn. 76, NH = 1411.32. 

Tapers Beyond the Frog 

24. When the Track Beyond the Turnout curves 
back parallel to the main line, there is seldom room and 
little need for a taper. When the track beyond the turn- 
out curves almost immediately in the same direction as 
the turnout, the taper is of much use, both as to looks and 
from practical considerations. 

It has been the practice on many roads to introduce a 
short tangent, of twenty to thirty feet, beyond the frog, 
before beginning the curve. This answers the double 
purpose of giving the trackman room for an approximate 
taper, and by removing the B.C. from the heel of the frog, 
prevents the kink that otherwise is very likely to occur 
there due to the rail joint on the outside rail. However, 
this presents the very bad appearance and gives the bad 
riding of the carefully avoided "broken-backed curve." 
If the curve begin at the heel of the frog with a taper of 
moderate length, both appearance and service are much 
improved. 

* For dr see Eqn. 49. 

t When R2 is greater than i^i, dr becomes negative. When / is 
greater than 90^ cos /becomes negative. ....^..^Google 



38 



THE RAILROAD TAPER 



Considerable ease in location will often be obtained if 
the curve beyond the taper may be handled as a simple 
curve tangent to a parallel with the main line, instead of 
with the tangent through the frog. 

In Fig. 26, / is the distance from frog angle point to 
actual point of frog, usually given by standard drawings. 
It is 

f=i{g + v)coi^F, (77) 

where g = gauge, u = thickness of frog point, usually 
one-half inch, and F = frog angle, m = length of frog 
from actual point to heel. 




Fig. 26. 



Then 



p = HE + KO-OD 

= (J + m + 1) sinF + (R + d) cosF ' 
w = CH -SH -CP 



fi, (78) 



= (; + ^ + 0cosF-(i2 + d)^ng^^^(79) 



LOCATION PROBLBHS 39 

From Eqn. 10, 
AP = w + (R+p) tan i (F+I) - (p-d) esc (f +/). (80) 

From Eqn. 11, 
AN =t+ (R+d) tan § {F+I) - (d-p) esc (F+I), (81) 
^^^(AP+f)^F^ - (82) 

Sin/ 

The resulting calculations for the entire problem may 
be longer than if the semi-tangent BT be calculated by 
Eqn. 7, / + m added, and triangle ACB solved. However, 
p and w are very useful in a preliminary consideration of 
the location of the turnout. 

Example. Given a No. 9 turnout from a side track to 
a branch line, with a 6° 00' curve, Tap. 2J, intersection 
angle 72'' 48', beginning at the heel of the frog, g = 4'-8^", 
V = J", m = S'-IO", F = 6° 22'. 

By Eqn. 77, / = 42.70. 

By Eqn. 78, p = 10.70 + 949.72 - 955.04 

= 5.38. 

By Eqn. 79, w = 95.92 - 105.97 - 42.70 

= -52.75. 

The negative value of w means that S lies between P 
and C. 

By Eqn. 80, AP ^ - 52.75 + 794.08 - 4.88 
= 736.45. 

By Eqn. 81, AN = 44.99 + 790. 10 + 4.88 
= 839.97. 

By Eqn. 82, AB = 90.45. 



y Google 



CHAPTER III 

FIELD WORK 

Selection of the Taper 

25. The Selection of the Proper Taper for use on 

each curve is an important function of the engineer. As 
suggested elsewhere, the proper length of taper to use is 
one on which the super-elevation of rails given the main 
curve may be run oflF on the taper at from one-half to one 
inch per rail length. This will suggest the use of Taper 1, 
1| or 2, for curves that do not limit the speed of trains, 
with a maximum length of about 360 ft., and thence 
gradually decreasing in length as velocity and super- 
elevation decrease, to a minimum of 90 ft. Although 
some very short tapers are given in the tables, they are 
for exceptional conditions. The Table of Comparative 
Values will aid in the selection of the taper. 

A table of the theoretical elevation needed to counter- 
act the centrifugal forces is included with the other tables, 
copied from the Proceedings of the American Railway 
Engineering Association^ but as the super-elevation de- 
pends more upon the velocity of trains than upon the 
degree of curve, and as this velocity varies with trains, 
seasons and traffic, an arbitrary rule, usually a compromise, 
is in force on most roads. 

The beginning and end of every taper should be clearly 
marked for the track foreman, so that he may know 
between what points he is to give the rails a gradually 
increasing super-elevation. The proper maintenance of 
this gradual increase, however, is no easy task, even when 
the points are well marked, and if the distance over which 

Digitized by VjOOQIC 



FIELD WORK 41 

it occurs be too great, and the increase in super-elevation 
be too gradual, it is very likely to become and remain very 
irregular, totally destroying the value of the taper. 
Therefore, though theoretically a curve tapered through- 
out might seem perfection, as maintained practically it 
would be a very poorly riding curve. Too long a taper 
is as bad as none. 

The somewhat prevalent practice of fitting a circular 
curve to the ground, and tangents at some undetermined 
offset therefrom, then connecting the two by the neces- 
sary taper to suit the offset, has the support of a few 
eminent engineers, but should be condemned as slipshod 
work. It is rarely the case that more careful thought 
and investigation will fail to place some other curve or 
tangents in a different location, with a suitable taper 
which should be chosen for totally different reasons. 
The weight and velocity of trains, the weight and char- 
acter of the engines and cars whose direction is to be 
changed, and the amount and quickness of the change, 
should alone govern the choice of taper. 

If any additional cost of construction be involved in 
the proper choice of taper, which is doubtful with good 
choice of location, it would be justified by the resulting 
lessened operating costs. Most certainly the higher 
quality of locating ability required to properly place a 
line to suit economic conditions is well able to solve this- 
lesser problem. 

Laying out a Tapered Curve 

26. After the location of tangents and main curve has 
been decided upon and choice of taper made, the B.C. 
and E,C. should be set by measurement from the PJ. 
The taper at each end should then be run from the B.C. 
and B.C., the C.C, to the main curve being set preferably 
by long chord from the B.C. and J?.C., and then the main 
curve should be run from C,C. to C.C. of tapers, or fror 



42 THE RAILROAD TAPER 

each CC. to the middle of the curve if it is unusually 
long. 

It is usual, especially for final track centers, to stake the 
taper every thirty feet from B.C. and S.C, closing to the 
C,C. with a short chord where the length of taper, Z, is 
not a multiple of 30 ft.; but on location, and especially for 
slope staking on construction, it is often desirable to 
stake the station and plus fifty points, which may be done 
by obtaining the deflections from Eqn. 107. 

If the main curve is of moderate length, it is usual, 
with the transit at the C.C, to turn the transit at the 
B.C. with the backsight deflection set on the reverse side of 
the vernier, thus bringing the instrument into tangent 
at the C,C, with the vernier at zero, and thence run in 
the main curve with foresight on the other C,C. If the 
main curve is unusually long, and the other C.C. not 
visible, this involves running a comparatively long curve 
from a short backsight, frequently resulting in poor 
closing. In this case a flag may be set at an offset from 
the initial tangent equal to x of the taper, preferably 
opposite the P./., but in any event as far from the C.C. 
as the curve to be run from it. With the vernier set at 
the total central angle of the taper, o, and the instrument 
directed at this flag, bringing it into a direction parallel 
to the initial tangent, the instrument will then be tangent 
to the taper at the C.C. when the vernier is brought to 
zero. 

Passing Obstructions on the Taper 

27. Curve run from Main Curve Produced. 

It frequently happens that either the B.C. or C.C. of 
the taper is inaccessible, when it is often necessary to run 
the main curve from the offset B.C. of a simple curve, or 
from a point on the simple curve produced backward as 
far as necessary beyond the B.C. of the simple curve, such 
point being set by offset from the initial tangegLj^ 



FIELD WORK 



43 



In Fig. 27, point C is the B,C. of the simple curve, D 
the B.C. and B the C.C, of the taper. The central angle 
of the portion of the simple curve from C to B is the 
central angle of the taper, o. Point A, a distance back 
on the simple curve equal to CB, and about opposite the 
B.C. of the taper, is a point frequently chosen from which 




Fig. 27. 

to run the main curve, because of the ease with which its 
position may be determined. Its offset from tangent is 
X of the taper, and its distance along the tangent from the 
B.C. of the taper is 

n = y — 2Rsmo. 

From Ekin. 4, Rsino = y — t, 

then n = 2t-y. (83) 

Any other point, J?, may be chosen, its offset from 
tangent being, 

r = d + 22vers/', (84) 

where 7' is the central angle of the curve back of the \B.C. 
of the simple curve. 

The distance along the tangent from the B.C. of the 
taper is 

s = t- Rversr. (85) 

This is merely equivalent to suggesting that the main 
curve may be run in beyond the C.C. from any point back 
of the C.C. by considering it a simple curve from a parallel 
*^angent. ^ Digitized by Google 



44 THE RAILROAD TAPER 

28. Curve run from a Point Beyond the CC. 

When on location the C,C. is in a bad position to occupy 
with the transit, a point may be set on the main curve 
beyond the C,C, from a point on the semi-tangent, by 
working the problem under paragraph 17 backwards. 
The plus of the point to be set is assumed, and from that 
angle a, Fig. 28, is known. D and AB are known. Then 

tann = ^, (86) 

OB=^^^y (87) 

cos n 

g = a- (90^ - n) 

= a + n - 90°. (88) 

Solving triangle OBP, two sides and the included angle 
given, 

tan(c + |i/)=g|±^tan|(7, (89) 

6 = 180*^ - n - c, (90) 

Sp^R^. . (91) 

sm c 

For point P', equations 88 and 90 become 

g^ = 90° - (a' + n), (88a) 

6' = 180° -n + c'y (90a) 

the other equations remaining the same. 

Example. Given a 4° 00' curve. Taper Ij, C.C. in a 
poor location for a transit set up, the second station 
beyond, 137.9 ft. from the C.C.y being a particularly 
good transit point. On the same ridge with this second 
station is a point on the semi-tangent, PoST, 260.89 ft. 
from the B.C. Determine the angle and distance from 
PoST to PoC. 

a = 4 X 0.6' X 137.9 + 2° 36' 
= 8° 07'. 
By Eqn. 86, n = 82° 13'. 

'ByEqn.87, 05 = 1446.50. ,,^_,^GoogIe 



FIELD WORK 



45 



By Eqn. 88, 
By Eqn. 89, 

By Eqn. 90, 
By Eqn. 91, 



fir = 0° 20'. 
c + iff = 30°50', 
c = 30°40'. 
6 = 67° 07'. 
BP = 16.34. 



"^^^Sn^^ 




^'x^w 


B 


/ y^^ 




Q^^ I 1 R L 


A 


D - R+d 

B.O. 










Fig. 28. 



Fig. 29. 



29. Taper Points set by Offset. 

When both the B.C. and the C.C, are inconvenient to 
occupy with the transit, the taper points may be set by 
offset from the tangent, using x and y, or they may be 
set by offset from the tangent on the first half and from 
the 'curve on the second half. In the latter case the offset 
from the curve is the same as from tangent at equal 
distances from B.C. and C.C. 

30. Taper Points set from Point on Semi-tangent. 

A few or all of the taper points are occasionally set by 
measurement and angle from a point on the semi-tangent, 
PoSTy by a solution of the triangle in Fig» 29. 

X 



coso 



(92) 



,Googl( (93) 



46 THE RAILROAD TAPER 

31. Curve Notes for the Field Book. 

Assuming the B.C. at Sta. 489 + 15.2, the transit 
notes for a tapered curve would be as follows: 

497 

O +62.2 4* 41 J' 2-37i' B.C. 

+32.2 4* 14 J' 

496 +02.2 SMIJ' 

+72.2 3*021' 

+42.2 2*17J' 

495 +12.2 r26J' 

+82.2 0**291' 2-195 

O +67.2 13* 231' 13* 23J' C.C. Tap. 2 o - 7* ISf 

494 10* 62' \ 

+50 8* 59J' TA - 41* 24' 

493 7*07' r- 387.16 

+60 6*14J' Bxi, - 55.64 

492 3*^22' R = 764.08 

+60 1*29J' L^ - 357.0 

O 491 +10.2 2--37J' 4*4U' C.C. T* dO' Ft A - 26* 46J' 

+96.2 2*16J' 

+66.2 1*39' t - 97.41 

+35.2 1*07J' d - 2.71 

490 +06.2 0*42' x - 8.93 

+76.2 0*22J' fc - 194.86 

+46.2 0*09' I - 195 

O +15.2 0*00' B.C. Tap. 2 o = 7* 18}' 

489 

In the above, the deflections for the taper at the E.C, 
are given from tangent at the CC, calculated from the 
equation 

., a + 2g + 45)(Z-s)+450 ,^., 

^ ^ 300 ' ^^^^ 

A 

an empirical equation derived from the tables by the 
method of second differences, in which I is the distance 
from B.C. or E.C. to the point for which deflection is 
wanted, and s is the distance of the transit from B.C. or 
E.C. The equation gives the deflection in minutes, and 
is from tangent at the transit. The deflections from 
tangent at the C.C. given in the notes above are merely 
to make the notes complete. Usually both tapers are 
run from B.C. and E.C. towards the main curve, using 
same set of deflections. ^,^,,,^^^ by Google 



CHAPTER IV 
OFFICE WORK 

32. Plotting a Tapered Curve. 

In the plotting of tapered curves, it is customary, after 
plotting the tangents and locating the B.C. and E.C., to 
draw the simple curve in its offset position, then draw a 
curve of radius twice that of the main curve extending 
from B.C. and E.C. a distance equal to Z, the length of 
the taper, coinciding with the main curve at the C.C. 
The difference between such a curve and the exact taper 
is noticeable only if the drawing be on a very large scale. 

33. Paper Location with Tapers. 

The usual method of paper location with tapers, as 
indeed frequently in location on the ground, is to offset 
the simple curve from its tangents an amount equal to 
d for the taper chosen. If curves predominate, they will 
be located first, and with parallel curves near their prob- 
able ends of radius greater by d, the tangents may be 
drawn in their offset position. If tangents predominate, 
they will be drawn first, with short parallel tangents near 
their probable ends, and the curves drawn tangent to 
these offset tangents. 

After the location of the simple offset curves, and the 
tangents, the tapers may be drawn or merely indicated as 
desired, depending upon the scale of map used. 

34. Curve Compensation with Tapers. 

It is a practice on some roads to use a compensation on 
the taper of half that on the main curve. An exactly 
equivalent result, with a simpler and more satisfactory 
grade line, is obtained if the change of grade from tangent 

Digitized by VjOOQIC 



48 THE RAILROAD TAPER 

to curve be made at the point opposite the offset position 
of the B.C. of the simple curve, which is at the middle of 
the taper, and a suitable vertical curve inserted. 

This brings correct theoretical and practical results, 
increasing the compensation in exact ratio with the curva- 
ture, and is well suited to location in rough country, where 
final consideration of the taper must be postponed until 
the location of curves and tangents. 

35. Right of Way Descriptions with Tapered Curves. 

A suitable description of right of way when the taper 
is used would be: ". . . , the center line of said rail- 
road extending thence along a tangent N 1° 14' W to 
station 4937 + 48.2, a distance of 465 ft., thence along 
a taper to the right of curvature increasing one degree 
every thirty feet to station 4938 + 98.2, a distance of 
150 ft., thence along a 6° 00' curve to the right (radius 
955.04) to station 4945 + 42.8, a distance of 644.6 ft., 
thence along a taper to the right of curvature decreasing 
one degree every thirty feet to station 4946 + 92.8, a 
distance of 150 ft., thence along a tangent N 46° 26^' E to 
station 5052 + 47.1, a distance of 554.3 ft., . . . ." 



y Google 



CHAPTER V 
TAPERING OLD TRACK 

36. The Tapering of Old Track, located and con- 
structed with circular curves only, calls for most careful 
analysis of the conditions encountered, and generally 
repays in results obtained a great deal of otherwise need- 
less transit work. Although after long experience in this 
particular work one may decide almost offhand upon the 
taper to use and the changes to make in tangents and 
curves, the beginner must usually make more than one 
trial before arriving at the best results obtainable, which 
usually combines with the most satisfactory curve for 
operation the lowest cost in making the change in aUgn- 
ment. Such work is frequently compUcated by bridges 
on the curves or tangents near the curves, limiting the 
amount track may be thrown, or by the slow, though 
frequently very decided, sliding of embankment on side 
hill work, distorting the original circular curve. 

It will also frequently be found that the trackman has 
thrown the B-C. in towards the center of curvature, 
lining in an approximate taper, and if the curve be short, 
may have taken care of the additional curvature by throw- 
ing the middle of the curve out. If the curve be long, it 
will probably be found that a short portion beyond this 
approximate taper is of unduly heavy curvature. On the 
other hand, the passage of heavy engines of rigid wheel 
base has frequently thrown the curve out at the B.C. and 
E.C.J with which is coupled unduly heavy curvature just 
beyond. When allowed sufficient time, the engineer will 
rerun the original simple curve to determine just where 
the track is, before attempting to insert tapers. The 

Digitized by VjOOQIC 



50 THE RAILROAD TAPER 

resulting transit notes, with measured offsets to the track 
as it is found, together with notes showing all bridges and 
openings, narrow embankments and cuts, and points 
where embankments and cuts may be cheaply widened 
and where not, will give the engineer the data for a care- 
ful office study of the conditions to be encountered. 
With an understanding of the various methods suggested 
below of inserting tapers, a tentative plan may be decided 
upon for each curve, subject to revision when running in 
the curves. Since the method involving the lowest maxi- 
mum shift of track is the shifting of both curve and tan- 
gents, which will then influence the next curve in each 
direction, the entire series of curves and tangents must be 
studied as a whole. 

The insertion of tapers in old circular track will gener- 
ally be accompanied by a resurfacing out of face of the 
entire portion of the line involved, so that usually there 
is no need to confine the shifting of track to the curves, 
if the shifting of tangents also will improve results. If 
the track may be ballasted at the same time the cost of 
inserting tapers is practically eliminated. 

37. Although, unfortunately, in the insertion of tapers 
in old circular curve alignment, the physical conditions 
encountered must very largely influence the choice of 
taper, frequently necessitating the use of short tapers, 
the operating conditions due to amount and direction of 
grade, velocity, weight and character of trains, amount 
of traffic, direction of heaviest tonnage, etc., must not be 
lost sight of, but must, as largely as possible, influence the 
choice of taper. 

Methods of Inserting Tapers in Circular 
Track. Simple Curves 

38. Shifting the Curve. The simplest, though usually 
the least desirable, method of making room for the taper 
is by shifting the curve radially inward, ^e degree of 

Digitized by VjOOQli 



TAPERING OLD TRACK 



51 



curve reijiaining unchanged. The amount track will be 
shifted will vary from zero at the new B.C. and E.C.j to 
MN, Fig. 30, at the middle of the curve (with equal tapers 




at the two ends), varying approximately with the distance 
from B.C. or E.C. 

MN = 00' = AB = dseoh I. (95) 

As this piethod will shorten the line somewhat, by an 
amount equal to 

S = 2t + 2dta,nil -l, (96) 

necessitating the cutting of rails under usual conditions^ 
it is to be avoided if possible except at summits, where 
the rail joints are likely to be quite open, or at sags, 
where the cutting of rails would be desirable to relieve 
joints already too tight. 

This method is limited to small values of d and 7, to 
avoid throwing the inner rail onto or close to fresh em- 
bankment, accentuating the tendency of the inner rail to 
settle, especially on single and double track roads, where 
the super-elevation must fit the speed of the fastest trains, 
throwing a large part of the weight of slower trains upon 
the inner rail. Digitized by Google 



52 



THE RAILROAD TAPER 



39. Shifting the Tangents. Where consecutive curves 
are of contrary flexure, and in a lesser degree where of 
similar flexure, shifting a short tangent, Fig. 31, forms a 
desirable method of making room for the taper. The 
tangent is shifted at its ends amounts approximately equal 
to d of the tapers adopted. 

I 




Fig. 31. 



With this method it is usual first to fit the main portion 
of each curve as closely as desired by trial, not over- 
looking provision for inserting a proper taper at the other 
end of each curve, then connecting the two curves by a 
meander Une, preferably the old tangent, and proceeding as 
in paragraph 15. Due to irregularities found in the track 
it is not Ukely that these curves, fitted to the track as 
found, will be exactly tangent to the old tangent, but this 
is immaterial. 

With this method the length of track is slightly in- 
"^eased, but seldom enough to be obiectiona^^^^l^ 



TAPERING OLD TRACK 



53 



40. Shifting Curve and Tangents. This method, 
Fig. 32, usually gives the most desirable rearrangement of 
alignment, involving the lowest maximum throw of track, 
and maintaining the length of line practically unchanged. 
To obtain this latter result the tangents should be thrown 
out about two-thirds of d of the taper opposite the old 
B.C. and E,C,, then run to an intersection and the curve 
run in. The new curve will cross the old one on the 




Fig. 32. 

second half of the taper, the entire central portion of the 
new curve lying inside the old curve, about one-third d 
from it at the middle of the curve. These amounts will 
vary with the length of the tangents and their resulting 
angular shift. 

41. Increasing the Degree of Curve. Where the pre- 
ceding methods involve too great a throw of track with 
the taper desired, the curve may be sharpened, as in Fig. 
33, enough to make room for the taper. 

d= (E-fi')vers|7. (97) 

As the value of d depends upon both R' and the taper 
chosen, which latter itself is influenced by R\ trial solu- 
tions of the -second half of this equation miist be made 

Digitized by VjOOQI^ 



54 



THE RAILROAD TAPER 



until a desirable value is obtained for d. For a desirable 
simplification of this point see the solution in the next 
problem. 




With this method of inserting tapers, track will be 
thrown in toward the center of curvature throughout the 
curve, the throw being a maximum about at the C.C. of 
the tapers. This will be 

A = (E - K') vers (H - o). (98) 

42. Increasing Degree and Shifting Curve. Curves 
of odd minutes curvature are to be avoided as much as 
possible, as, the trackman being unfamiliar with them, 
they are difficult to maintain. If the curve of the pre- 
ceding problem be sharpened so that (R — R') vers § I 
will give a value greater than d, and the curve shifted away 
from the center of curvature to bring it tangent to the 
original tangents, Fig. 34, a desirable realignment is 
obtained. 

At the middle of the curve track will now be thrown out 
an amount 



or 



A = [(ft - ft') vers J7 - d]secH, (99) 

A = (ft - DO exsec ^I -d. r-^^^i (99a) 

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TAPERING OLD TRACK 



66 



For preliminary study, where / is not large, the factor 
sec 1 1 may be disregarded as insignificant. The new 
curve will now lie within the old throughout the tapers 




and a portion of the main curve, and outside the old curve 
on the middle portion of the main curve, a very desirable 
rearrangement. The best balance is obtained by making 
h equal about one-third d, when the maximum throw of 
track at the three points of greatest divergence will be 
about equal. 

43. Maintaining Length of Line Unchanged. With 
the proper value of A, Elqn. 99, R' may be assumed such as 
to leave the length of line unchanged. 

The length of line from A \x> B, Fig. 34, by the new 
alignment is 

^' = ^ + T* ^^^^ 

The length of line from 4 to B by the old alignment is 
1007 



L = 2AC + 



M 



= 2(-2(E-Z)')tan§7 + 



100/ 



M .. , 

Digitized by VjOOQlC 



(101) 



56 THE RAILROAD TAPER 

The table on page 115 gives the values of M', I and h for 
a large range of M and / with the more important tapers. 
Where curves are sharper than ten-degree, tangents will 
be short, and paragraphs 39 and 40 will give better results. 
Where / is greater than given in the tables, paragraph 46 
will usually give better results unless curves of fractional 
minutes curvature are used. 

When the value of Af or 7 Ues between those given in 
the table, M' may be interpolated and then checked by 
equations 100 and 101, recalculating when necessary. If 
on the first trial L' — L is too great, decrease M\ The 
irregularity of h in the table is due to using the nearest 
minute for Af' . 

Equations 100 and 101 are based upon fifty-foot chord 
measurement of both old and new curves, but any dis- 
crepancy over arc measurement introduced is small and 
may usually be neglected. With both curves based upon 
a fifty-foot chord, increasing a 5° 00' curve to a 5** 15' 
curve introduces a discrepancy due to arc excess of only 
0.0002 ft. per hundred feet. With the old curve based 
upon a hundred-foot chord and the new curve based upon 
a fifty-foot chord the difference in arc excess is quite large, 
especially with the higher degrees of curve, but this is 
almost exactly balanced by the decreased value of AC^ 
equation 101, over that assumed for the table, due to the 
increased value of R. 

For arc measurement equations 100 and 101 become 

L'(arc)=Z + f^. (100a) 

L (arc) = 2< - 2 (E - jD') tan*/ + ^. (101a) 

Using these equations, if a 10° 00' curve, 7 = 50° 00', 
be sharpened to a 10° 17' curve, Tap. 3, as given in the 
table, the new alignment would be 0.04 ft. longer than the 
old. The curve would have to be sharpened to a 10° 16' 
curve, Tap. 3, to maintain the length^unch^g^,^ 



TAPERING OLD TRACK 



57 



The length of line may be maintained unchanged with 
many of the other problems in this chapter by arranging 
by the manner of shifting curves and tangents that 
approximately half the new alignment he within and half 
without the old. 

44. Increasing Degree and Shifting Curve and Tan- 
gents. Where I is large the smallest increase of curvature 
avoiding odd minutes will frequently give too large a value 




Fig. 35. 



for h of equation 99, when to avoid throwing the curve out 
too much at the middle the tangents may be thrown in 
at the B,C, and C,C. of the old curve, as in Fig. 35. 

c + A = (72 - EO vers \I -d (approx.). (102) 

For the best balance of track h should about equal c. 
The new tangents may be run to an intersection, new 
external calculated and h checked on the ground, and if 
satisfactory the curve run in. 

45. Any of the preceding methods may be combined 
in any one curve, by shifting the tangent at one end only 



58 



THE RAILROAD TAPBR 



//A. 




\ .Jlr 






/ 




Fia. 37. 





— ? 

/ 

/ 
/ 

/ 


***«^«*._ 


/ 

/ 




Fia. 38. 



Digitized by VjjOOQIC 



TAPERING OLD TRACK 59 

either the full or partial amount of d, and the curve at the 
other end, or the tangents may be shifted different amounts 
at the two ends of the curve, thus taking advantage of 
different widths of roadbed available at different points. 
On heavy grades a shorter taper may be used at the 
lower end of the curve when room is limited, due to the 
lower speed of ascending trains. 

46. Increasing Degree of Portion of Curve. Where I 
is too great to sharpen the entire curve, and the tangents 
too long to be shifted advantageously, a portion only of 
the curve may be sharpened to make room for the taper, 
as in Fig. 36. The difference between the degree of the 
old curve and the portion sharpened should preferably not 
be greater than the curvature increment of the taper to 
be used, and may often be less. 

vers/' = -^4^. (103) 

r must of course be larger than the central angle of the 
taper, or the increase in curvature to provide the offset 
has been too great. /' should not be less than twice o, 
and may usually be as much as three times o. 

Compound Curves 

47. Any combination of the preceding problems of 
this chapter may be used in inserting tapers at the ends 
or between the branches of old compound curves. A few 
of the possible combinations follow. 

48. Shifting the Curves. In Fig. 37 the relative values 
of the various d's are exaggerated, increasing the angular 
change in the position of the point of compounding. How- 
ever, this arrangement disturbs the branch of sharper 
curvature disproportionately, which may be modified as in 
Fig. 38 by shifting the branch of lighter curvature for the 
entire amount of dr. Digitized by Google 



60 



THE RAILROAD TAPER 



49. Shifting Curves and Tangents. Where both the 
curves and tangents may be shifted, as in Fig. 39, the 




Fig. 39. 

maximum throw of track may be much reduced. This 
aids particularly with the branch of sharper curvature. 

50. Shifting the Lighter Branch, and Increasing the 
Curvature of the Sharper Branch. This change of align- 

L 




Fig. 40. 



ment is shown in Fig. 40. With this may of course 
be combined the shifting of the end of the sharper branch 
to avoid curvature of odd minutes, as in paragraph 42, or 
also the shifting of one or both tangents as ii^g^gf^ph 44. 



TAPERING OLD TRACK 



61 



Field Work 

61. The principal objection to compound curve align- 
ment is the difficulty trackmen find in maintaining it to 
proper curvature. It follows that in inserting tapers in 
compoimd curves it is seldom profitable to make any- 
lengthy calculations based upon the theoretical location 
of the curves. They will be somewhere else. 

The best method of procedure is to pick out portions of 
the curves of apparently fairly uniform curvature, which 
can be tested with the mid-ordinate- to a hundred-foot 
tape held against the rail at the ends of the tape, set up the 
transit on these portions, extend them as far in each direc- 
tion as they fit the track fairly well, then turn tangents and 
intersect as in Fig. 41. It will usually be found that these 




Fig. 41. 



curves, fitted to the track, are not tangent to each other nor 
to the original tangents, the offsets from which may or may 
not be in a direction to facilitate the insertion of tapers. 



62 THE RAILROAD TAPER 

At the tangent ends, the existing offset from tangent is, 
from Fig. 41, 

c = AB sin a — fi vers a. (104) 

Between the two branches of the compound curve, the 
existing offset along the common radial line may be found 
from Fig. ^1. 

A solution of right triangles OPK and O'PaS will give the 
sides O'P and OP and the included angle / of triangle 
O'PO, from the solution of which 00' and angle g may be 
obtained, from which angle n may be obtained. Then 

m = b-n, (105) 

e = R' -R- OU (106) 

With these data the shifting of curves and tangents 
most desirable may be decided upon. Then to insure 
perfect fitting of the curves to each other and to their 
tangents, the tangents may be located in their new posi- 
tions, the longer of the two branches located in its new 
position by the problem in paragraph 12, and then the 
' shorter of the two branches located by the problem in 
paragraph 22. 



y Google 



TABLES 



63 



B^ 



RADII 
25 



—M 











sin- 


4 


- 






Deg. 


Radius. 


Losrad. 


Deg. 


Radius. 


Log. r»d. 


Deg. 


Radius. 


Log. rad. 


0"00' 


Infinite 




I'OO' 


5729.60 


3.75812 


2" 00' 


2864.82 


3.45710 


1 


343776.76 


5.53627 


1 


5636.67 


3.75095 


1 


2841.15 


3.45349 


2 


171887.41 


5.23524 


2 


5544.77 


3.74388 


2 


2817.86 


3.44992 


3 


114591.60 


5.05915 


3 


5456.76 


3.73693 


3 


2794.95 


3.44637 


4 


85943.67 


4.93421 


4 


5371.50 


3.73010 


4 


2772.41 


3.44286 


6 


68754.94 


4.83730 


5 


5288.86 


3.72336 


5 


2750.23 


3.43937 


6 


67295.78 


4.75812 


6 


5208.73 


3.71673 


6 


2728.41 


3.43591 


7 


49110.67 


4.69118 


7 


5130.99 


3.71020 


7 


2706.93 


3.43248 


8 


42971.83 


4.63318 


8 


5055.53 


3.70377 


8 


2685.78 


3.42907 


9 


37197.19 


4.58203 


9 


4982.26 


3.69743 


9 


2664.96 


3.42569 


10 


34377.47 


4.53627 


10 


4911.09 


3.69118 


10 


2644.46 


3.42234 


11 


31252.24 


4.49488 


11 


4841.92 


3.68502 


11 


2624.27 


3.41901 


12 


28647.89 


4.45709 


12 


4774.67 


3.67895 


12 


2604.39 


3.41571 


13 


26444.80 


4.42234 


13 


4709.26 


3.67295 


13 


2584.81 


3.41243 


14 


24555.34 


4.39016 


14 


4645.63 


3.66704 


14 


2565.52 


3.40918 


15 


22918.30 


4.36018 


15 


4583.69 


3.66121 


15 


2546.52 


3.40595 


16 


21485.92 


4.33215 


16 


4523.37 


3.65546 


16 


2527.79 


3.40274 


17 


20222.04 


4.30583 


17 


4464.63 


3.64979 


17 


2509.35 


3.39956 


18 


19098.60 


4.28100 


18 


4407.39 


3.64418 


18 


2491.16 


3.39640 


19 


18093.80 


4.25753 


19 


4351.60 


3.63865 


19 


2473.24 


3.39327 


20 


17188.74 


4.23524 


20 


4297.21 


3.63319 


20 


2455.57 


3.39015 


21 


16370.20 


4.21405 


21 


4244.16 


3.62779 


21 


2438.16 


3.38706 


22 


15626.13 


4.19385 


22 


4192.40 


3.62246 


22 


2420.99 


3.38399 


23 


14946.70 


4.17555 


23 


4141.87 


3.61720 


23 


2404.06 


3.38095 


24 


14323.89 


4.15606 


24 


4092.58 


3.61200 


24 


2387.36 


3.37792 


25 


13750.99 


4.13833 


25 


4044.43 


3.60686 


25 


2370.90 


3.37491 


28 


13222.11 


4.12130 


26 


3997.39 


3.60177 


26 


2354.67 


3.37193 


27 


12732.40 


4.10491 


27 


3951.46 


3.59676 


27 


2338.65 


3.36896 


28 


12277.67 


4.08912 


28 


3906.56 


3.59179 


28 


2322.85 


3.36602 


29 


11854.34 


4.07388 


29 


3862.66 


3.58689 


29 


2307.26 


3.36310 


30 


11459.17 


4.05915 


30 


3819.75 


3.58203 


30 


2291.88 


3.36019 


31 


11089.52 


4.04491 


31 


3777.77 


3.57724 


31 


2276.70 


3.35731 


32 


10742.97 


4.03112 


32 


3736.71 


3.57249 


32 


2261.72 


3.35444 


33 


10417.66 


4.01777 


33 


3696.53 


3.56779 


33 


2246.94 


3.35159 


34 


10111.03 


4.00480 


34 


3657.21 


3.56315 


34 


2232.35 


3.34876 


35 


9822.16 


3.99221 


35 


3618.71 


3.55855 


35 


2217.95 


3.34595 


36 


9549.31 


3.97997 


36 


3581.02 


3.55401 


36 


2203.73 


3.34316 


37 


. 9291.66 


3.96809 


37 


3544.10 


3.54951 


37 


2189.70 


3.34038 


> 38 


9046.71 


3.95649 


38 


3507.93 


3.54505 


38 


2175.84 


3.33763 


39 


8814.75 


3.94521 


39 


3472.50 


3.54064 


39 


2162.15 


3.33489 


.40 


8594.38 


3.93421 


40 


3437.78 


3.53628 


40 


2148.64 


3.33216 


41 


8384.76 


3.92341 


41 


3403.74 


3.53196 


41 


2135.29 


3.32946 


42 


8185.12 


3.91303 


42 


3370.37 


3.52768 


42 


2122.12 


3.32677 


43 


7994.77 


3.90281 


43 


3337.65 


3.52344 


43 


2109.09 


3.32410 


44 


7813.13 


3.89282 


44 


3305.56 


3.51924 


44 


2096.24 


3.32144 


45 


7639.45 


3.88306 


45 


3274.08 


3.51509 


45 


2083.53 


3.31880 


46 


7473.38 


3.87352 


46 


3243.19 


3.51097 


46 


2070.98 


3.31618 


47 


7314.37 


3.86418 


47 


3212.88 


3.50689 


47 


2058.56 


3.31357 


48 


7161.99 


3.85503 


48 


3183.13 


3.50285 


48 


2046.33 


3.31098 


49 


7015.82 


3.84608 


49 


3153.93 


3.49885 


49 


2034.22 


3.30840 


50 


6875.51 


3.83730 


50 


3125.26 


3.49489 


50 


2022.26 


3.30584 


51 


6740.70 


3.82870 


51 


3097.10 


3.49096 


51 


2010.43 


3.30329 


52 


6611.09 


3.82027 


52 


3069.45 


3.48706 


52 


1998.74 


3.30076 


53 


6486.33 


3.81200 


53 


3042.29 


3.48320 


53 


1987.19 


3.29824 


54 


6366.22 


3.80388 


54 


3015.60 


3.47937 


54 


1975.77 


3.29574 


55 


6250.46 


3.79691 


55 


2989.38 


3.47558 


56 


1964.48 


3.29325 


56 


6138.85 


3.78809 


56 


2963.61 


3.47182 


56 


1953.32 


3.29077 


57 


6031.15 


3.78640 


67 


2938.28 


3.46809 


57 


1942.28 


3.28831 


58 


5927.17 


3.77285 


68 


2913.38 


3.46440 


58 


1931.37 


3.28587 


69 


5826.71 


3.76542 


59 


2888.89 


3.46073 


59 


1920.58 


3.28343 



64 



THE RAILROAD TAPER 



RADII — (Continued) 



Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


S^'OO' 


1909.91 


3.28101 


4-00' 


1432.47 


3.15608 


5«»00' 


1146.01 


3.05919 


1 


1899.36 


3.27861 


1 


1426.52 


3.16430 


1 


1142.20 


3.06774 


2 


1888.93 


3.27622 


2 


1420.63 


3.15248 


2 


1138.42 


3.05630 


3 


1878.61 


3.27384 


3 


1414.78 


3.15069 


3 


1134.66 


3.05487 


4 


1868.40 


3.27147 


4 


1408.99 


3.14891 


4 


1130.93 


3.06344 


5 


1868.30 


3.26912 


6 


1403.24 


3.14713 


6 


1127.20 


3.05201 


6 


1848.31 


3.26677 


6 


1397.63 


3.14636 


6 


1123.54 


3.05059 


7 


1838.42 


3.26446 


7 


1391.88 


3.14360 


7 


1119.88 


3.04917 


8 


1828.65 


3.26213 


8 


1386.26 


3.14185 


8 


1116.24 


3.04776 


9 


1818.97 


3.25983 


9 


1380.70 


3.14010 


9 


1112.63 


3.04636 


10 


1809.40 


3.25753 


10 


1375.17 


3.13836 


10 


1109.04 


3.04495 


11 


1799.93 


3.25526 


11 


1369.70 


3.13662 


11 


1105.48 


3.04365 


12 


1790.56 


3.26299 


12 


1364.26 


3.13490 


12 


1101.94 


3.04216 


13 


1781.27 


3.26073 


13 


1358.87 


3.13318 


13 


1098.42 


3.04077 


14 


1772.09 


3.24849 


14 


1363.53 


3.13146 


14 


1094.92 


3.03938 


16 


1763.01 


3.24626 


15 


1348.21 


3.12976 


15 


1091.44 


3.03800 


16 


1754.01 


3.24403 


16 


1342.95 


3.12806 


16 


1087.99 


3.03662 


17 


1745.11 


3.24182 


17 


1337.72 


3.12637 


17 


1084.66 


3.a^'>26 


18 


1736.30 


3.23962 


18 


1332.64 


3.12468 


18 


1081.15 


3.U3389 


19 


1727.67 


3.23744 


19 


1327.39 


8.12300 


19 


1077.76 


3 03252 


20 


1718.93 


3.23526 


20 


1322.29 


3.12133 


20 


1074.39 


3.03116 


21 


1710.38 


3.23309 


21 


1317.22 


3.11966 


21 


1071.05 


3.02981 


22 


1701.92 


3.23094 


22 


1312.19 


3.11800 


22 


1067.72 


3.02846 


23 


1693.53 


3.22879 


23 


1307.21 


3.11634 


23 


1064.42 


3.02711 


24 


1685.23 


3.22666 


24 


1302.26 


3.11470 


24 


1061.13 


3.02577 


25 


1677.01 


3.22464 


25 


1297.34 


3.11305 


25 


1067.87 


3.02443 


26 


1668.87 


3.22242 


26 


1292.47 


3.11142 


26 


1054.62 


3.02310 


27 


1660.81 


3.22032 


27 


1287.63 


3.10979 


27 


1061.40 


8.02177 


28 


1652.83 


3.21823 


28 


1282.82 


3.10817 


28 


1048.19 


3.02044 


23 


1644.92 


3 21614 


29 


1278.05 


3.10655 


29 


1045.01 


3.01912 


30 


1637.09 


3.21407 


30 


1273.32 


3.10494 


30 


1041.84 


3.01780 


31 


1629.33 


3.21201 


31 


1268.62 


3.10333 


31 


1038.69 


3.01649 


32 


1621.64 


3.20996 


32 


1263.96 


3.10173 


32 


1036.67 


3.01618 


33 


1614.03 


3.20791 


33 


1259.33 


3.10014 


33 


1032.46 


3.01387 


34 


1606.49 


3.20588 


34 


1254.73 


3.09855 


34 


1029.37 


3.01257 


35 


1599.02 


3.20385 


35 


1260.17 


3.09697 


36 


1026.29 


3.01127 


36 


1591.62 


3.20184 


36 


1245.64 


3.09539 


36 


1023.24 


3.00998 


37 


1584.28 


3.19983 


37 


1241.16 


3.09382 


37 


1020.20 


3.00869 


38 


1577.01 


3.19784 


38 


1236.68 


' 3.09226 


38 


1017.19 


3.00740 


39 


1569.81 


3.19686 


39 


1232.26 


3.09070 


39 


1014.19 


3.00612 


40 


1662.68 


3.19387 


40 


1227.86 


3.08915 


40 


1011.20 


3.00484 


41 


1655.61 


3.19190 


41 


1223.48 


3.08760 


41 


1008.24 


3.00356 


42 


1548.60 


3.18994 


42 


1219.14 


3.08606 


42 


1006.29 


3.00229 


43 


1541.66 


3.18799 


43 


1214.84 


3.08452 


43 


1002.36 


3.00102 


44 


1534.78 


3.18606 


44 


1210.66 


3.08299 


44 


999.46 


2.99976 


45 


1527.96 


3.18411 


45 


1206.31 


3.08146 


45 


996.56 


2.99850 


46 


1521.19 


3.18218 


46 


1202.10 


3.07994 


46 


993.67 


2.99724 


47 


1514.49 


3.18027 


47 


1197.91 


3.07842 


47 


990.81 


2.99699 


48 


1607.85 


3.17736 


48 


1193.75 


3.07691 


48 


987.96 


2.99474 


49 


1601.27 


3.17646 


49 


1189.62 


3.07541 


49 


986.13 


2.99350 


60 


1494.74 


3.17457 


60 


1185.61 


3.07391 


50 


982.32 


2.99225 


51 


1488.27 


3.17268 


61 


1181.44 


3.07241 


51 


979.62 


2.99101 


52 


1481.86 


3.17081 


62 


1177.40 


3.07092 


62 


976.74 


2.98978 


53 


1475.50 


3.16894 


63 


1173.38 


3.06944 


53 


973.97 


2.98855 


54 


1469.19 


3.16708 


54 


1169.39 


3.06796 


54 


971.22 


2.98732 


55 


1462.94 


3.16523 


56 


1165.43 


3.06649 


65 


968.48 


2.98609 


56 


1456.75 


3.16338 


66 


1161.49 


3.06602 


66 


965.76 


2.98487 


57 


1460.60 


3.16155 


67 


1157.58 


3.06356 


57 


963.06 


2.98865 


58 


1444.60 


3.15972 


68 


1153.70 


3.06209 


68 


960.37 


2.98244 


59 


1438.46 


3.15790 


59 


1149.84 


3.06064 


59 


957.70 


2.98123 



Digitized by VjOOQIC 



TABLES 



65 



RADII — {Continued) 



rH«. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


6" 00' 


955.04 


2.98002 


7«»00' 


818.64 


2.01309 


8" 00' 


716.34 


2.85612 


1 


952.39 


2.97882 


1 


816.09 


2.91206 


1 


714.85 


2.86422 


2 


949.76 


2.97762 


2 


814.76 


2.91103 


2 


713.37 


2.86332 


3 


947.15 


2.97642 


3 


812.83 


2.91000 


3 


711.90 


2.85242 


4 


944.55 


2.97522 


4 


810.92 


2.90898 


4 


710.43 


2.85152 


5 


941.96 


2.97403 


5 


809.01 


2.90796 


6 


708.96 


2.85062 


6 


939.30 


2.97284 


6 


807.11 


2.90693 


6 


707.50 


2.84973 


7 


936.83 


2.97166 


7 


805.22 


2.90692 


7 


706.05 


2.84884 


8 


934.28 


2.97048 


8 


803.34 


2.90490 


8 


704.60 


2.84795 





931.76 


2.96930 


9 


801.47 


2.90389 


9 


703.16 


2.84706 


10 


929.23 


2.96812 


10 


799.61 


2.90288 


10 


701.73 


2.84617 


11 


926.73 


2.96695 


11 


797.76 


2.90187 


11 


700.30 


2.84528 


12 


924.24 


2.96578 


12 


795.91 


2.90086 


12 


698.88 


2.84440 


13 


921.76 


2.96462 


13 


794.07 


2.89986 


13 


697.46 


2.84352 


14 


919.29 


2.96^6 


14 


792.24 


2.89886 


14 


696.06 


2.84264 


16 


916.84 


2.96230 


15 


790.42 


2.89786 


15 


694.64 


2.84176 


16 


914.41 


2.96114 


16 


788.61 


2.89686 


16 


693.24 


2.84089 


17 


911.98 


2.95999 


17 


786.80 


2.89587 


17 


691.86 


2.84001 


18 


909.67 


2.95884 


18 


785.01 


2.89487 


18 


690.46 


2.83914 


19 


907.17 


2.95769 


19 


783.22 


2.89388 


19 


689.08 


2.83827 


20 


904.78 


2.96656 


20 


781.44 


2.89290 


20 


687.70 


2.83740 


21 


902.41 


2.95640 


21 


779.67 


2.89191 


21 


686.33 


2.83653 


22 


900.05 


2.96427 


22- 


777.91 


2.89093 


22 


684.96 


2.83567 


23 


897.70 


2.96313 


23 


776.15 


2.88995 


23 


683.60 


2.83480 


24 


895.36 


2.95200 


24 


774.40 


2.88897 


24 


682.26 


2.83394 


25 


893.04 


2.95087 


26 


772.66 


2.88799 


25 


680.90 


2.83308 


26 


890.72 


2.94974 


26 


770.93 


2.88702 


26 


679.56 


2.83222 


27 


888.42 


2.94862 


27 


769.21 


2.88604 


27 


678.21 


2.83136 


28 


886.13 


2.94750 


28 


767.49 


2.88607 


28 


676.88 


2.83051 


29 


883.86 


2.94638 


29 


765.78 


2.88410 


29 


676.55 


2.82966 


30 


881.59 


2.94627 


30 


764.08 


2.88314 


30 


674.22 


2.82880 


31 


879.34 


2.94416 


31 


•762.39 


2.88218 


31 


672.90 


2.82795 


32 


877.09 


2.94305 


32 


760.70 


2.88121 


32 


671.59 


2.82710 


33 


874.86 


2.94194 


33 


769.02 


2.88025 


33 


670.28 


2.82626 


34 


872.64 


2.94084 


34 


757.35 


2.87930 


34 


668.98 


2.82541 


35 


870.44 


2.93974 


36 


755.69 


2.87834 


35 


667.68 


2.82467 


36 


868.24 


2.93864 


36 


764.03 


2.87739 


36 


666.39 


2.82373 


37 


866.05 


2.93754 


37 


752.38 


2.87644 


37 


665.10 


2.82289 


38 


863.88 


2.93645 


38 


750.74 


2.87549 


38 


663.81 


2.82205 


39 


861.71 


2.93536 


39 


749.10 


2.87464 


39 


662.54 


2.82121 


40 


869 66 


2.93428 


40 


747.48 


2.87360 


40 


661.26 


2.82037 


41 


857.42 


2.93319 


41 


745.85 


2.87266 


41 


659.99 


2.81954 


42 


866.28 


2.93211 


42 


744.24 


2.87171 


42 


658.73 


2.81871 


43 


863.16 


2.93103 


43 


742.63 


2.87077 


43 


657.47 


2.81788 


44 


851.05 


2.92996 


44 


741.03 


2.86984 


44 


656.22 


2.81705 


45 


848.95 


2.92888 


46 


739.44 


2.86890 


46 


664.97 


2.81622 


46 


846.86 


2.92781 


46 


737.86 


2.86797 


46 


653.72 


2.81639 


47 


844.78 


2.92674 


47 


736.28 


2.86704 


47 


662.48 


2.81457 


48 


842.71 


2.92568 


48 


734.70 


2.86611 


48 


661.25 


2.81375 


49 


840.66 


2.92461 


49 


733.14 


2.86519 


49 


650.02 


2.81293 


50 


838.60 


2.92355 


60 


731.58 


2.86426 


50 


648.79 


2.81211 


51 


836.56 


2.92250 


51 


730.03 


2.86334 


51 


647.57 


2.81129 


62 


834.63 


2.92144 


62 


728.48 


2.86242 


52 


646.36 


2.81047 


53 


832.61 


2.92039 


63 


726.94 


2.86150 


63 


646.14 


2.80966 


54 


830.50 


2.91934 


64 


725.41 


2.86058 


64 


643.93 


2.80884 


55 


828.60 


2.91829 


65 


723.88 


2.85967 


65 


642.73 


2.80803 


66 


826.51 


2.91726 


56 


722.36 


2.86875 


56 


641.53 


2.80722 


57 


824.53 


2.91620 


57 


720.85 


2.85784 


57 


640.34 


2.80641 


58 


822.56 


2.91516 


58 


719.34 


2.85693 


58 


639.15 


2.80560 


50 


820.59 


2.91413 


59 


717.84 


2.85603 


69 


637.96 


2.80480 



Digitized by VjOOQIC 



66 



THE RAILROAD TAPER 



RADII — {Continued) 



Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


9* 00' 


636.78 


2.80399 


10*00' 


573.14 


2.76826 


IP 00' 


621.07 


2.71690 


1 


635.60 


2.80319 


1 


572.19 


2.75764 


1 


620.28 


2.71624 


2 


634.43 


2.80239 


2 


571.24 


2.75682 


2 


519.50 


2.71558 


3 


633.27 


2.80159 


3 


670.29 


2.75610 


3 


618.71 


2.71493 


4 


632.10 


2.80080 


4 


669.35 


2.76638 


4 


617.93 


2.71427 


5 


630.94 


2.79999 


5 


568.41 


2.75466 


5 


617.16 


2.71362 


6 


629.79 


2.79920 


6 


667.47 


2.75394 


6 


616.38 


2.71297 


7 


628.64 


2.79840 


7 


666.63 


2.76323 


7 


615.61 


2.71232 


8 


627.49 


2.79761 


8 


565.60 


2.75251 


8 


514.84 


2.71167 


9 


626.35 


2.79682 


9 


564.67 


2.75180 


9 


614.07 


2.71102 


10 


625.21 


2.79603 


10 


663.75 


2.76109 


10 


513.30 


2.71037 


11 


624.08 


2.79524 


11 


662.83 


2.75038 


11 


512.54 


2.70972 


12 


622.95 


2.79445 


12 


561 91 


2.74967 


12 


611.77 


2.70908 


13 


621.82 


2.79367 


13 


560.99 


2.74896 


13 


511.02 


2.70843 


14 


620.70 


2.79288 


14 


560.08 


2.74826 


14 


510.26 


2.70779 


15 


619.58 


2.79210 


15 


569.17 


2.74764 


16 


509,60 


2.70714 


16 


618.47 


2.79132 


16 


568.26 


2.74684 


16 


508.76 


2.70660 


17 


617.36 


2.79054 


17 


567.36 


2.74613 


17 


508.00 


2.70586 


18 


616.25 


2.78976 


18 


666.46 


2.74543 


18 


507.25 


2.70522 


19 


615.15 


2.78898 


19 


665.66 


2.74473 


19 


506.60 


2.70468 


20 


614.05 


2.78821 


20 


554.66 


2.74403 


20 


505.76 


2.70394 


21 


612.96 


2.78743 


21 


563.77 


2.74333 


21 


506.02 


2.70330 


22 


611.87 


2.78666 


22 


552.88 


2.74263 


22 


604.28 


2.70267 


23 


610.78 


2.78589 


23 


561.99 


2.74193 


23 


503.64 


2.70203 


24 


609.70 


2.78612 


24 


561.11 


2.74124 


24 


502.80 


2.70140 


25 


608.62 


2.78435 


26 


560.23 


2.74054 


26 


502.06 


2.70076 


26 


607.55 


2.78368 


26 


549.36 


2.73985 


26 


601.33 


2.70013 


27 


606.48 


2.78281 


27 


648.47 


2.73916 


27 


600.60 


2.69960 


28 


605.41 


2.78206 


28 


647.60 


2.73847 


28 


499.88 


2.69887 


29 


604.35 


2.78129 


29 


546.73 


2.73777 


29 


499.16 


2.69824 


30 


603.29 


2.78052 


30 


546.87 


2.73709 


30 


498.43 


2.69761 


31 


602.23 


2.77976 


31 


545.00 


■ 2.73640 


31 


497.72 


2.69698 


32 


601.18 


2.77900 


32 


644.14 


2.73571 


32 


497.00 


2.69635 


33 


600.13 


2.77824 


33 


643.28 


2.73602 


33 


496.28 


2.69672 


34 


599.08 


2.77749 


34 


642.42 


2.73434 


34 


496.56 


2.69610 


35 


598.04 


2.77673 


36 


541.57 


2.73365 


36 


494.85 


2.69447 


36 


597.00 


2.77598 


36 


640.72 


2.73297 


36 


494.14 


2.69385 


37 


695.97 


2.77623 


37 


639.87 


2.73229 


37 


493.43 


2.69323 


38 


594.93 


2.77447 


88 


639.03 


2.73161 


38 


492.73 


2.69261 


39 


593.91 


2.77372 


39 


538.18 


2.73093 


39 


492.02 


2.69198 


40 


592.89 


2.77297 


40 


637.34 


2.73025 


40 


491.32 


2.69136 


41 


691.87 


2.77223 


41 


636.51 


2.72957 


41 


490.62 


2.69074 


42 


590.85 


2.77148 


42 


635.67 


2.72890 


42 


489.92 


2.69013 


43 


589.84 


2.77074 


43 


534.84 


2.72822 


43 


489.22 


2.68951 


44 


588.83 


2.76997 


44 


534.01 


2.72765 


44 


488.53 


2.68889 


45 


587.83 


2.76925 


46 


633.18 


2.72687 


46 


487.84 


2.68828 


46 


586.82 


2.76851 


46 


532.35 


2.72620 


46 


487.15 


2.68766 


47 


585.82 


2.76777 


47 


631.53 


2.72563 


47 


486.46 


2.68705 


48 


584.83 


2.76703 


48 


530.71 


2.72486 


48 


486.77 


2.68643 


49 


583.84 


2.76629 


49 


629.90 


2.72419 


49 


486.09 


2.68682 


50 


582.85 


2.76656 


50 


629.08 


2.72352 


60 


484.40 


2.68521 


51 


581.86 


2.76482 


61 


528.27 


2.72286 


61 


483.72 


2.68460 


52 


580.88 


2.76409 


62 


527.46 


2.72219 


62 


483.04 


2.68400 


53 


579.90 


2.76335 


63 


626.65 


2.72162 


63 


482.37 


2.68338 


54 


578.93 


2.76262 


54 


625.86 


2.72086 


54 


481.69 


2.68277 


55 


577.95 


2.76189 


66 


625.06 


2.72020 


56 


481.02 


2.68216 


56 


576.98 


2.76116 


.66 


524.25 


2.n953 


66 


480.35 


2.68156 


57 


576.02 


2.76044 


57 


623.46 


2.71887 


67 


479.68 


2.68096 


68 


575.05 


2.76971 


68 


622.66 


2.71821 


58 


479.01 


2.68035 


59 


574.10 


2.76899 


69 


621.86 


2.71756 


69 


478.35 


2.67974 



Digitized by VjOOQIC 



TABLES 



67 



RADII — (Continued) 



Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


12*00' 


477.68 


2.67914 


13*00' 


440.97 


2.64441 


14*00' 


409.51 


2.61226 


1 


477.02 


2.67854 


1 


440.41 


2.64386 


1 


409.02 


2.61175 


2 


476.36 


2.67794 


2 


439.85 


2.64330 


2 


408.54 


2.61123 


3 


475.70 


2.67734 


3 


439.29 


2.64276 


3 


408.05 


2.61072 


4 


475.05 


2.67674 


4 


438.73 


2.64219 


4 


407.57 


2.61020 


5 


474.39 


2.67614 


5 


438.17 


2.64164 


5 


407.09 


2.60969 


6 


478.74 


2.67554 


6 


437.61 


2.64109 


6 


406.61 


2.60918 


7 


473.09 


2.67494 


7 


437.05 


2.64054 


7 


406.13 


2.60667 


8 


472.44 


2.67435 


8 


436.50 


2.63999 1 


8 


405.66 


2.60616 


9 


471.79 


2.67375 


9 


435.95 


2.63944 


9 


405.18 


2.60764 


10 


471.16 


2.67316 


10 


435.40 


2.63889 


10 


404.70 


2.60713 


11 


470.50 


2.67256 


11 


434.85 


2.63834 


11 


404.22 


2.60662 


12 


469.86 


2.67197 


12 


434.30 


2.63779 


12 


403.75 


2.60611 


13 


469.22 


2.67138 


13 


433.75 


2.63724 


13 


403.28 


2.60560 


14 


468.58 


2.67078 


14 


433.21 


2.63669 


14 


402.81 


2.60510 


15 


467.94 


2.67019 


15 


432.66 


2.63615 


15 


402.33 


2.60459 


16 


467.31 


2.66960 


16 


432.12 


2.63560 


16 


401.87 


2.60408 


17 


466.68 


2.66901 


17 


431.58 


2.63506 


17 


401.40 


2.60357 


18 


466.04 


2.66843 


18 


431.04 


2.63451 


18 


400.93 


2.60307 


19 


465.41 


2.66784 


19 


430.50 


2.63397 


19 


400.46 


2.60256 


20 


464.79 


2.66725 


20 


429.96- 


2.63343 


20 


400.00 


2.60206 


21 


464.16 


2.66667 


21 


429.42 


2.63289 


21 


399.53 


2 60155 


22 


463.53 


2.66608 


22 


428.89 


2.63235 


22 


399.07 


2.60105 


23 


462.91 


2.66550 


23 


428.36 


2.63181 


23 


398.61 


2.60055 


24 


462.29 


2.66491 


24 


427.82 


2.63127 


24 


398.15 


2.60005 


25 


461.67 


2.66433 


25 


427.29 


2.63073 


25 


397.69 


2.59954 


26 


461.05 


2.66375 


26 


426.76 


2.63019 


26 


397.23 


2.59904 


27 


460.43 


2.66317 


27 


426.23 


2.62965 


27 


396.77 


2.59854 


28 


459.82 


2.66259 


28 


425.71 


2.62911 


28 


396.32 


2.69804 


29 


459.20 


2.66201 


29 


425.18 


2.62858 


29 


395.86 


2.59754 


30 


458.59 


2.66143 


30 


424.66 


2.62804 


30 


395.41 


2.59704 


31 


457 98 


2.66085 


31 


424.14 


2.62750 


31 


394.95 


2.5C655 


32 


457.37 


2.66027 


32 


423.62 


2.62697 


32 


394.50 


2.56605 


33 


456.77 


2.65970 


33 


.423.09 


2.62644 


33 


394.05 


2 59555 


34 


456.16 


2.65912 


34 


422.57 


2.62590 


34 


393.60 


2.59505 


36 


455.56 


2.65855 


35 


422.06 


2.62537 


35 


393.15 


2 59456 


36 


454.96 


2.65797 


36 


421.54 


2.62484 


36 


3S2.70 


2.59406 


37 


454.36 


2.65740 


37 


421.02 


2.62431 


37 


392.56 


2.59357 


38 


453.76 


2.65682 


38 


420.51 


2.62378 


38 


391.81 


2.59307 


39 


453.16 


2.65625 


39 


420.00 


2.62325 


39 


391 36 


2.59258 


40 


452.57 


2.65568 


40 


419.49 


2.62272 


40 


390.91 


2.59209 


41 


451.97 


2.65511 


41 


418.98 


2.62219 


41 


390.48 


2.59160 


42 


451.38 


2.65454 


42 


418.47 


2.62166 


42 


390.04 


2.59110 


43 


450.79 


2.65397 


43 


417.96 


2.62133 


43 


389.60 


2.59061 


44 


450.20 


2.65340 


44 


417.45 


2.62061 


44 


389.16 


2.59012 


45 


449.61 


2.65284 


45 


416.95 


2.62008 


45 


388.72 


2.58963 


46 


449.02 


2.65227 


46 


416.44 


2.61955 


46 


388.28 


2.58914 


47 


448.44 


2.65170 


47 


415.94 


2.61903 


47 


387.84 


2.58865 


48 


447.86 


2.65114 


48 


415.44 


2.61851 


48 


387.40 


2.58816 


49 


447.28 


2.65057 


49 


414.94 


2.61798 


49 


386.97 


2.58767 


60 


446.70 


2.65001 


50 


414.44 


2.61746 


50 


386.53 


2.58719 


61 


446.12 


2.64945 


51 


413.94 


2.61694 


51 


386.10 


2.68670 


62 


445.54 


2.64889 


52 


413.44 


2.61642 


52 


385.67 


2.58621 


63 


444.96 


2.64832 


53 


412.95 


2.61589 


53 


385.24 


2.58573 


54 


444.39 


2.64776 


54 


412.45 


2.6153? 


54 


384.81 


2 58524 


65 


443.82 


2.64720 


55 


411.96 


2.61485 


55 


384.38 


2 58476 


66 


443.25 


2.64664 


56 


411.47 


2.61434 


56 


383.95 


2 58423 


67 


442.67 


2.64608 


57 


410.98 


2.61382 


57 


383.52 


2.58379 


58 


442.11 


2.64553 


58 


410.49 


2.61330 


58 


383.09 


2 58331 


69 


441.54 


2.64497 


59 


410.00 


2.61278 


59 


382.67 


2 58282 



Googi 



68 



THE RAILROAD TAPER 



RADII — (Continued) 



Deg. 


Radius. 


Log.rad. 


Deg. 


Radius. 


Log.rad. 


Deg. 


Radius. 


Loc.rad. 


IS" 00' 


382.25 


2 58234 


19" 00' 


301.91 


2.47988 


23*00' 


249.53 


2.39712 


05 


380.14 


2.57994 


05 


300.59 


2.47797 


15 


246.86 


2.39245 


10 


378.05 


2.57755 


10 


299.28 


2.47608 


30 


244.24 


2.38782 


15 


375.99 


2.57518 


15 


297.99 


2.47420 


45 


241.68 


2.38324 


20 


373.95 


2.57281 


20 


296.71 


2.47233 








25 


371.93 


2.57046 


25 


295.44 


2.47047 


24-00' 


239.17 


2.37871 


30 


369.93 


2 56812 


30 


294.18 


2.46861 


15 


236.71 


2.37422 


35 


367.96 


2.56580 


35 


292.94 


2.46678 


30 


234.30 


2.36978 


40 


366.00 


.2.56348 


40 


291.69 


2.46492 


45 


231.95 


2.36539 


45 


364.07 


2.56118 


45 


290.46 


2.46309 








50 


362.16 


2.55890 


50 


289.25 


2.46127 


25-00' 


229.64 


2.36104 


55 


360.26 


2.55662 


55 


288.04 


2.45945 


15 
30 


227.37 
226.15 


2 35673 
2.35248 


WOO' 


358.39 


2.55436 


20^00' 


286.84 


2.45764 


46 


222.98 


2.34827 


05 


356.54 


2.55211 


05 


285.66 


2.45585 








10 


354.70 


2.54986 


10 


284.48 


2.45405 


26-00' 


220.84 


2.34408 


15 


352.88 


2.54763 


15 


283.31 


2.46226 


15 


218.75 


2.33995 


20 


351.09 


2.54542 


20 


282.15 


2.45048 


30 


216.69 


2.33584 


25 


349.31 


2.54321 


25 


281.00 


2.44871 


45 


214.68 


2.33179 


30 


347.55 


2.54101 


30 


279.86 


2.44695 








•35 


345 80 


2.53883 


35 


278.73 


2.44518 


27-00' 


212.70 


2.32776 


40 


344.08 


2.53666 


40 


277.61 


2.44344 


15 


210.78 


2.32379 


45 


342.37 


2.53450 


45 


276.50 


2.44170 


30 


208.85 


2.31983 


50 


340.68 


2.53234. 


60 


275.40 


2.43996 


45 


206.98 


2.31593 


55 


339.00 


2.53020 


55 


274.30 


2.43823 


28-00' 


205.14 


2.31205 


17«00' 


337.34 


2.62807 


21«00' 


273.22 


2.43651 


15 


203.33 


2.30820 


05 


335.70 


2.52595 


05 


272.14 


2.43479 


30 


201.56 


2.30440 


10 


334.07 


2.52384 


10 


271.07 


2.43309 


45 


199.81 


2.30062 


15 


332.46 


2.62174 


15 


270.01 


2.43138 








20 


330.87 


2.51965 


20 


268.96 


2.42969 


29-00' 


198.10 


2.29688 


25 


329.29 


2.51758 


25 


267.91 


2.42799 


15 


196.42 


2.29318 


30 


327.72 


2.51551 


30 


266.88 


2.42632 


30 


194.76 


2.28950 


35 


326.17 


2.51344 


35 


265.86 


2.42465 


45 


193.13 


2.28585 


40 


324.64 


2.51140 


40 


264.83 


2.42298 


30-00' 


191.53 


2.28224 


45 


323.12 


2.50936 


45 


263.82 


2.42131 








50 


321.61 


2.50733 


50 


262.83 


2.41966 








55 


320.12 


2.50531 


55 


261.82 


2.41800 








18- Oy 


318.64 


2.50330 


22-00' 


260.83 


2.41637 








05 


317.17 


2.50129 


05 


259.85 


2.41472 








10 


315.72 


2.49930 


10 


258.88 


2.41310 








15 


314.28 


2.49732 


15 


257.91 


2.41147 








20 


312.86 


2.49534 


20 


256.95 


2.40986 








25 


311.44 


2.49337 


25 


256.00 


2.40824 








30 


310.04 


2.49142 


30 


265.05 


2.40664 








35 


308.66 


2.48948 


35 


254.12 


2.40504 








40 


307.28 


2.48754 


40 


253.19 


2.40344 








45 


305.92 


2.48561 


45 


252.26 


2.40185 








50 


304.57 


2.48368 


50 


261.36 


2.40027 








55 


303.23 


2.48177 


55 


250.44 


2.39870 









y Google 



TABLES 



NOTATION 



A = Curvature increment of taper per 30 ft. of taper. 
M = Degree of main circular curve. 
I = Length of taper from the B.C. of the taper. 
/ = Deflection from tangent at the B.C. of the taper. 
/'» /i> hi etc. = Deflection from tangent at some point on the 

taper. 
h = Deflection from tangent at the end of the taper, C.C. of 

main curve, to the B.C. of the taper. 
o » Central angle of the taper at each end of the curve. 
d = Circular curve ofifset from tangent to provide room for 

the taper. 
t = Distance along the tangent, from opposite the offset 

position of the B.C. of the simple curve to the B.C. of 

the taper. 
D = 12 + d , where M is the radius of the main circular curve. 
X and y are co-ordinates of points on the taper, referred to the 

tangent at the B.C. of the taper as meridian, and to a 

radial line at the B.C. of the taper as base. 
Ic = Long chord of the taper, from the B.C. of the Taper to 

points on the taper. 
8 s= Distance along the taper, from the B.C. of the taper to the 

transit, when an intermediate set-up is necessary. 
T = Semi-tangent of the entire tapered curve, from P.I. to 

B.C. or E.C. 
E = External, the distance from the P.I. to a point on the 

curve radially opposite. 

EQUATIONS 

Eqn. No. 

T = < -h i> tan i /. (7) Both ends, the same 

taper, 
r^ = < -h D tan J / - d esc /. (8) Tapered end — other 

end not tapered. 
Tn^ R tan J / -|- d esc /. (9) Untapered end — other 

end tapered. 
Ti = <i -h Di tan i / - {di - di) esc /. (10) Both ends tapered 

with different tapers. 
^2 = <2 + I>2 tan i / - (d. - di) esc /. (11) Both ends tapered, 

with different tapers. 
E ^ DeeG\I - R, (12) Both ends the same 

taper. Google 



70 THE RAILROAD TAPER 

EMPIRICAL EQUATIONS DERIVED FROM THE 
TABLES 



Z = ^ (M - A). (45) 

;=(i±I^M. (107) 



^ 300 ^^^' 



A 

Z (I + 30) 
300 

A 



(108) 



y Google 



TABLES 



71 







TAPER \ 


\. INTERMEDIATE 


VALUES 




A=i 


M 


I 


f 


b 





D 


logD 


d 


t 


X 


V 


Ic 


030 


30.00 


/ 

02J 


/ 

02! 


e / 

04! 


11459.18 


4.05915 


0.01 


15.00 


0.02 


30.00 


30.00 


35 


40.00 


003i 


03! 


07 


9822.17 


3.99221 


0.02 


20.00 


0.04 


40.00 


40.00 


40 


50.00 


004i 


05! 


010 


8694 41 


3.93422 


0.03 


25.00 


0.06 


50.00 


50.00 


45 


60.00 


051 


008 


13! 


7639.49 


3.88306 


0.04 


30.00 


0.10 


60.00 


60.00 


60 


70.00 


07 


OlOi 


017! 


6875.56 


3.83731 


0.06 


35.00 


0.14 


70.00 


70.00 


55 


80.00 


0081 


013! 


22 


6250.53 


3.79592 


0.07 


40.00 


0.20 


80.00 


80.00 


100 


90.00 


0101 


016! 


27 


5729.70 


3.75813 


0.10 


45.00 


0.27 


90.00 


90.00 


05 


100.00 


012i 


020 


32! 


5288.99 


3.72337 


0.13 


60.00 


0.36 


100 00 


100.00 


10 


110.00 


014i 


024 


38! 


4911 25 


3.69119 


0.16 


65.00 


0.47 


110.00 


110.00 


15 


130.00 


016! 


28! 


45 


4583.89 


3.66123 


0.20 


60.00 


0.59 


120.00 


120.00 


'20 


130.00 


019i 


32! 


52 


4297.4J5 


3.63321 


0.24 


66.00 


0.73 


130.00 


130.00 


25 


140.00 


22 


37! 


69! 


4041.72 


3.60689 


0.29 


70.00 


0.89 


140.00 


140.00 


30 


150.00 


24f 


42! 


107! 


3820.09 


3.58207 


0.34 


75.00 


1.08 


149.99 


160.00 


35 


160.00 


27! 


48! 


116 


3619.11 


3.55860 


0.40 


80.00 


1.29 


159.99 


160.00 


40 


170.00 


30! 


54! 


125 


3438.25 


3.53634 


0.47 


85.00 


1.62 


169.99 


170.00 


45 


180.00 


34 


100! 


134! 


3274.63 


3.61516 


0.65 


90.00 


1.79 


179.98 


179.99 


60 


190.00 


37i 


107 


144! 


3125.89 


3.49497 


0.63 


94.99 


2.08 


189.98 


189.99 


55 


200.00 


41! 


113! 


a55 


2990.10 


3.47569 


0.72 


99.99 


2.40 


199.98 


199.99 


200 


210.00 


45 


121 


206 


2865.64 


3.45722 


0.82 


104.99 


2.76 


209.97 


209.99 


05 


220.00 


49 


128! 


2 17! 


2761.16 


3.43952 


0.93 


109.99 


3.13 


219.96 


219.98 


10 


230.00 


053 


136! 


2 29! 


2645.51 


3.42251 


1.05 


114.99 


3.65. 


229.95 


229.98 


15 


240.00 


57J 


144! 


2 42 


2547.70 


3.40615 


1.18 


119.99 


4.00 


239.94 


239.97 


20 


250.00 


101! 


153! 


2 55 


2456.89 


3.39039 


1.32 


124.98 


4.49 


249.93 


249.97 


25 


260.00 


lOOi 


2 02 


3 08! 


2372.36 


3.37618 


1.46 


129.98 


5.02 


259.92 


269.97 


30 


270.00 


111! 


211! 


3 22! 


2293.50 


3.36050 


1.62 


134.98 


6.69 


269.90 


269.96 


35 


280.00 


116! 


2 20! 


3 37 


2219.74 


3.34630 


1.79 


139.97 


8.20 


279.88 


279.95 


40 


290.00 


121! 


2 30! 


3 52 


2150.61 


3.33256 


1.97 


144.97 


6.86 


289.86 


289.94 


45 


300.00 


126i 


2 41 


4 07! 


2085.69 


3.31925 


2.16 


149.96 


7.66 


299.84 


299.94 


50 


310.00 


132 


2 51! 


4 23! 


2024.62 


3.30634 


2.36 


164.96 


8.30 


309.81 


309.93 


55 


320.00 


137! 


3 02! 


440 ' 


1967.06 


3.29382 


2.58 


169.96 


9.09 


319.78 


319.91 


300 


330.00 


143i 


3 13! 


4 57 


1912.72 


3.28165 


2.81 


164.94 


9.93 


329.74 


329.89 


05 


340.00 


149) 


325 


514! 


1861.35 


3.26983 


3.05 


169.93 


10.82 


339.70 


339.87 


10 


350.00 


155i 


3 37 


5 32! 


1812.70 


3.25833 


3.30 


174.93 


11.76 


349.66 


349.85 


15 


360.00 


2 01! 


3 49! 


5 51 


1766.58 


3.24713 


3.67 


179.92 


12.75 


369.61 


359.83 


20 


370.00 


2 08! 


4 01! 


610 


1722.78 


3.23623 


3.85 


184.90 


13.80 


369.65 


369.81 


25 


380.00 


2 15 


4 14! 


6 29! 


1681.16 


3.22661 


4.15 


189.89 


14.90 


379.49 


379.78 


30 


390.00 


2 21! 


4 27! 


6 49! 


1641.56 


3.21526 


4.47 


194.88 


16.07 


389.42 


389.75 


35 


400.00 


2 28! 


4 41! 


710 


1603.82 


3.20616 


4.80 


199.86 


17.29 


399.36 


399.72 


40 


410.00 


2 35! 


4 55! 


7 31 


1567.82 


3.19530 


5.14 


204.85 


18.67 


409.27 


409.69 


45 


420.00 


2 43 


5 09! 


7 52! 


1533.46 


3.18567 


5.60 


209.83 


19.91 


419.18 


419.65 


50 


430.00 


2 50i 


5 24 


8 14! 


1500.61 


3.17627 


5.87 


214.81 


21.31 


429.08 


429.61 


55 


440.00 


2 58! 


5 38! 


8 37 


1469.20 


3.16708 


6.26 


219.79 


22.78 


438.97 


439.56 


400 


450.00 


306 


554 


900 


1439.14 


3.15810 


6.67 

Di 


224.77 

litized by 


24.31 


448.85 


449.51 



72 



THE RAILROAD TAPER 



II 
< 






o 

> 

p 
o 






98 



o^ 






OS 
0?o 



dig 






OS 

6u 



dS 
d-c 



o^ 
dt, 



d§ 
dt^ 



0:2 
dt, 



d8 
d«^ 



o!S 

d°o 



d^ 
d-o 



OS 



^ W F^^ W|^ H|W WW^^ rt I w ^^ w|^ h)w Wt' 

o CO CO CO CO CO CO CO CO C^ C<l C^ ^ 1-H fH o 



n\^ co|^ H|c9 «H|n ^^ h}^ €0|^e^^w|^H|N , wM* 

^ 10 10 W5 »0 »0 Tt< CO S O 'S Ca 10 CO !co 



(M C^ C^ C^ C^ (M C^ C^ »-H ^ t-H O O -Ot-H 



o o »-H i-H 5) c5 10 '^j* csi »-H 55 e^ .Jsioeo 

C^ C^ C^ C<l <N C<l 1-1 tH T-H »-H O O -Oi-i^ 



,H tH tH T-H tH t-H »-H tH »-H O O ' O O i-t N 









• 0»-H,-iOOOOO • O O »-H fh C^ c<l CO 



rtld H-* «!■* Hn H-* wf-* 



iHl^COl^ iH|C^P9|^H(^H1C4 



0000000 ■ O O »-H »-H C<l C^ CO CO 



• 000000 •OOO'-i'-iC^C^CO'^ 



53J2^<^'~' . CO o o CO O) 00 a> Tt< ca CO 

C^(M(M.-li-l . »-H CO 10 »-H CO O CO tH 10 CO 
00000 ■ O O O »-H 1-1 C<l C<l CO CO Tt< 



coi>-Tt<g5 . »-H CO M ^ i> iS -t* ?o w »-H ca 

^^'-HO . »-H (M Tp O W »0 C5| 10 CO 1-H »o 
0000 ■ O O O 1-1 T-H tH C^ c<l CO -^ "^ 



'228 :gSS§!gS5S%2^S52 

•000 " O O O O 1-1 1-1 C^ C^ CO CO '^^t^ »o 



SS :S2gSgSS5S8SS8^S 

00 • O O O O tH 1-1 th c<l C<l CO Tj< "^ 10 



.0'-HC<ico«5»-icoococ555i-iioTj< 

• O •OOOOOfHi-hC<IC^C0C0"^tJ<iC 



- :8fe285^8S^S$2?S5SS 

• •00000»-h»-HthC^c<ICOCO'^»C»C 






TABLES 



73 



I] 


^ 


88888SSfe$SSS8KSS 




gS8|Sg|||||||2| 


1 


Ok 


30.00 
60.00 
90.00 
120.00 
149.99 
179.99 
209.97 
239.94 
269.90 
299.84 
329.74 
359.61 
389.43 
419.18 
448.86 


T 


H 


82^S8ffK8gS§{2feSS 

tH »-t »-H C^ 


1 


- 


8888888gg§S888Sfc 
SS^S{28S2^§SgS8^ 


O 

o 
O 


t» 


3S28^§g32SS5Sfe!3;Sfe 

oooooooTHTHCsic^co-^iocd 


1 

o 




TtJcocococococococococococococo 


Q 


11459.18 
7639.49 
5729.70 
4583.89 
3820.09 
3274.63 
2865.64 
2547.70 
2293.50 
2085.69 
1912.72 
1766.58 
1641.56 
1533.46 
1439.14 


o 


o O O O O th *-• C^ Cv| CO -Tj* Tj* »o CO t>- 05 




- 


^SS8Sg25S8SS8§S 




:« 


-g«8:2g:§82g«82g^8 

O O »-H tH 1-ii-H cq C<I(N CS| CO CO CO CO tP 

Digitized by VjC 



)gle 



74 



THB RAILROAD TAPER 



> 



o 

CO 

o 






P5 






08 



dt, 



d8 
d««, 



dSS 
dSs- 



dS 
di. 



dS 
dt^ 



dS 
d'^ 



d^ 
d?o 



dS 
d?^ 



dS 



OS 
ou 



d^ 
dtn 



dS 
dt. 



d8 



^4^ W|V^WH(^ Wp#HlC<^i^ WH'^C^Hf^ 

o 10 10 to 10 »0 iO »C iC Tt< Tj< CO CS| 1-H T-H • 



"^ 10 o o o lO CO i-H Tt< rH CO 35 25 .c5 

tP Tj< 10 10 10 Tt< '«i< ^ CO CO C^ fH o "i-i 



Tj< TP Tt< Tt< Tj< Tt< CO CO C^ C^ »-H O "OC^ 






o CO CO CO CO CO CO CO C^ <N J-H O 'Oi-iCO 



C^ CO CO CO C^ N N i-H »-H O -Oi-HNCO 



• C^ C^ C<l Cq N C^ 1-1 1-1 O •OfHC^COTti 



»-H C^ C^ fh ,-1 ,-1 1-1 O • O 1-1 C9 CO -Tji 10 



T-l 1-1 1— I T-l T-l O O ■ O '-• C^ CO Tt< IC o 



»-4 »-4 ,_( »-4 O O • O i-i i-i C<l CO "^ »o t>- 



00000 • O 1-1 1-1 <N CO '^ »0 CO l^ 



cS^S2 ;?53S^oioS$SoS 

0000 ; O O t-« C^ C^ CO tP W3 1^ 00 



Sf2 isis^SSS^^^ 

000 • O O 1-1 1-1 C^ CO '^ »0 CO t^ 00 



;5S 

00 



COC6QC^O'^»Ot-«c6t-«OCQ 

i-HC0OC0»-l»O'^Tt<TfllCO^ 

oo»-i'-i<Nc^co'^iooooa> 



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Hc«n<-« ^■*HN»|-* H-*H«M|^ H-«Hnc«Ni 

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fH ^H fH C^ CI W (N CO 00 CO ^ 

8QQg5t*"^ooiN.i-i»ci^'*i-Hco 
o5o^i-i"^i>.ooooo5Su5oo^ 

l-Hl-Hl-H^CS^NCOCOCO"^ 

8S8g§§2Sg?^??S3§S 

dddddi-Hi-HCsieo'^ioi^o6d 

C0^C0t^l>.Cp05'^^Ql^b-OC0 

t^25TFcoc!i(Nt-Ht-HOOOcbo5oo 

COCOCOCOCOCOCOCOCOCO&4C4<N<N 

SS?SS5S22S^2Sf2SS 

U5COC<ICV|i-lr-<,-)l-HT-li-l 



o O O O i-H C^ CO "^ >0 CO 00 05 i-H CO »c 



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76 THE RAILROAD TAPER 

TAPER 1. INTERMEDIATE VALUES 



A-J 



M 


I 


/ 


b 





D 


logD 


d 


t 


a;. 


y 


Ic 


o / 

100 


30.00 


o / 

04i 


o / 

04} 


o / 

009 


5729.62 


3.76813 


0.02 


16.00 


0.04 


30.00 


30.00 


05 


35.00 


05i 


06 


11} 


6288.89 


3-72336 


0.03 


17.60 


0.06 


35.00 


36.00 


10 


40.00 


om 


07} 


14 


4911.12 


3.69118 


0.03 


20.00 


0.08 


40.00 


40. OU 


15 


45.00 


07J 


009} 


16} 


4583.73 


3.66122 


0.04 


22.50 


0.10 


45.00 


46.00 


20 


50.00 


08} 


11} 


020 


4297.26 


'3.63319 


0.05 


25.00 


13 


50.00 


50.00 


25 


65.00 


10 


13} 


23} 


4044.49 


3.60686 


0.06 


27.60 


0.16 


65.00 


65.00 


30 


60.00 


Olli 


15} 


27 


3819.83 


3.58204 


0.08 


30.00 


0.20 


60.00 


60.00 


35 


65.00 


012} 


18 


30} 


3618.80 


3.56866 


0.09 


32.60 


0.24 


65.00 


66.00 


40 


70.00 


014} 


20} 


35 


3437.89 


3.53629 


0.11 


36.00 


0.29 


70.00 


70.00 


45 


75.00 


15} 


23} 


39} 


3274.21 


3.61511 


0.13 


37.50 


0.36 


76.00 


76.00 


50 


80.00 


om 


26} 


044 


3126 41 


3.49491 


0.15 


40.00 


0.41 


80.00 


80.00 


55 


85.00 


19} 


29} 


48} 


2989.66 


3.47561 


0.17 


42.60 


0.47 


86.00 


85.00 


200 


90.00 


21 


33 


64 


2866.02 


3.45713 


0.20 


45.00 


0^65 


90.00 


90.00 


05 


95.00 


023 


36} 


59} 


2760.46 


3.43940 


0.22 


47.50 


0.63 


96.00 


95.00 


10 


100.00 


25 


040 


106 


2644.71 


3.42238 


0.25 


50.00 


0.72 


100.00 


100.00 


15 


105.00 


27 


43} 


110} 


2546.80 


3.40599 


0.28 


62.60 


0.82 


106.00 


105.00 


20 


110.00 


29} 


47} 


1 17 


2466.89 


3.39021 


0.32 


56.00 


0.93 


109.99 


110.00 


25 


115.00 


3U 


52 


123} 


2371.26 


3.37498 


0.36 


67.60 


1.06 


114.99 


116.00 


30 


120.00 


33} 


66} 


130 


2292.27 


3.36027 


0.39 


60.00 


1.18 


119 99 


120.00 


35 


125.00 


36} 


100} 


136} 


2218.39 


3.34604 


0.44 


62 50 


1.31 


124.99 


126.00 


40 


130.00 


38} 


105} 


144 


2149.12 


3.33226 


0.48 


65.00 


1.46 


129.99 


130.00 


45 


135.00 


41} 


110} 


161} 


2084.06 


3.31891 


0.53 


67.49 


1.62 


134.98 


134 .M 


60 


140.00 


044 


115 


159 


2022.84 


3.30696 


0.68 


69.99 


1.79 


139.98 


139.99 


65 


145.00 


46} 


120 


2 06} 


1966.11 


3.29339 


0.63 


72.49 


1.97 


144.98 


144.99 


300 


150.00 


49} 


125} 


2 16 


1910 60 


3.28117 


0.69 


74.99 


2.16 


149.97 


149.99 


06 


m.OQ 


62} 


131 


2 23} 


1869.06 


3.26929 


0.76 


77.49 


2.36 


154.97 


154.99 


10 


160.00 


55} 


136} 


2 32 


1810.21 


3.26773 


0.81 


79.99 


2.57 


169.97 


159.99 


15 


166.00 


58} 


142} 


2 40} 


1763.89 


3.24647 


0.88 


82.49 


2.80 


164.96 


164.98 


20 


170.00 


101} 


14.S} 


2 60 


1719.88 


3.23550 


0.96 


84.99 


3.06 


169.96 


169.98 


26 


175.00 


105 


153} 


2 69} 


1678.03 


3.22480 


1.02 


87.49 


3.30 


174.95 


174.98 


30 


180.00 


108} 


2 00} 


309 


1638.19 


3.21436 


1.10 


89.98 


3.57 


179.94 


179.98 


35 


185.00 


111} 


2 07 


3 18} 


1600.20 


3.20417 


1.18 


92.48 


3.86 


184.93 


184.97 


40 


190.00 


116} 


2 13} 


3 29 


1663.95 


3.19422 


1.27 


94.98 


4.16 


189.92 


189.97 


45 


195.00 


118} 


2 20} 


3 39} 


1529.32 


3.18460 


1.36 


97.48 


4.47 


194.91 


194.96 


50 


200.00 


122} 


2 27} 


360 


1496.19 


3. f 7499 


1.46 


99.97 


4.80 


199.90 


199.96 


55 


205.00 


126} 


2 34} 


400} 


1464.49 


3.16569 


1.66 


102.47 


5.14 


204.89 


204.95 



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TABLES 



77 



TAPER .1. INTERMEDIATE VALUES — (Continued) A = i 



M 


I 


/ 


b 





Z> 


logi> 


i 


t 


z 


V 


U 


o / 

400 


210.00 


130 


1 
242 


4 12 


1434.12 


3.15659 


1.65 


104.97 


5.50 


209.88 


209.95 


05 


215.00 


134 


2 49} 


4 23} 


1405.00 


3.14768 


1.76 


107.47 


5.87 


214.87 


214.95 


10 


220.00 


138 


2 57 


435 


1377.04 


3.13895 


1.87 


109.96 


6.26 


219.85 


219.94 


15 


225.00 


142 


3 04} 


4 46} 


1350.19 


3.13039 


1.98 


112.46 


6.67 


224.83 


224.93 


20 


290.00 


146i 


312} 


4 59 


1324.39 


3.12201 


2.10 


114.95 


7.10 


229.82 


229.93 


25 


235.00 


150i 


3 21 


5 11} 


1299.57 


3.11380 


2.23 


117.45 


7.54 


234.80 


234.92 


30 


240.00 


154} 


3 29} 


5 24 


1275.68 


3.10574 


2:36 


119.94 


8.01 


239.77 


239.91 


35 


245.00 


im 


3 37} 


5 36} 


1252.66 


3.09783 


2.49 


122.44 


8.49 


244.75 


244.90 


40 


250.00 


2 03! 


346} 


550 


1230.48 


3.09007 


2.63 


124.93 


8.99 


249.73 


249.89 


45 


255.00 


208i 


355} 


6 03} 


1209.08 


3.08246 


2.77 


127.43 


9.51 


254.70 


254.88 


50 


280.00 


2 13 


404 


6 17 


1188.43 


3.07497 


2.92 


129.92 


10.05 


259.67 


259.87 


55 


205.00 


2 171 


4 13 


6 30} 


1168.51 


3.06763 


3.08 


132.41 


10.60 


284.64 


284.85 


500 


270.00 


2 22^ 


4 22} 


6 45 


1149.25 


3.06041 


3.24 


134.91 


11.18 


269.61 


289.84 


05 


275.00 


2 27J 


4 32 


6 59} 


1130.60 


3.05331 


3.40 


137.40 


11.78 


274.57 


274.82 


10 


280.00 


2 32^ 


4 41} 


714 


1112.61 


3.04634 


3.57 


139.89 


12.40 


279.53 


279.81 


15 


285.00 


2 37J 


450} 


7 28} 


1095.19 


3.03949 


3.75 


142.38 


13.04 


284.49 


284.80 


20 


290.00 


2 421 


5 01} 


744 


,1078.33 


3.03275 


3.94 


144.87 


13.71 


289.45 


289.78 


25 


295.00 


248 


511} 


7 59} 


1062.00 


3.02612 


4.13 


147.36 


14.39 


294.40 


294.76 


30 


300.00 


2 53i 


5 21} 


815 


1048.16 


3.01960 


4.32 


149.85 


15.10 


299.35 


299.73 


35 


305.00 


2 581 


5 32 


830} 


1030.81 


3.01318 


4.52 


152.34 


15.83 


304.30 


304.71 


40 


310.00 


304i 


5 42} 


8 47 


1015.93 


3.00686 


4.73 


154.83 


16.58 


309.24 


309.69 


45 


315.00 


309} 


5 53} 


9 03} 


1001.49 


3.00065 


4.94 


157.32 


17.36 


314.18 


314.66 


50 


320.00 


315} 


6 04} 


920 


987.48 


2.99453 


5.16 


159.81 


18.16 


319.12 


319.63 


55 


325.00 


3 2U 


615} 


9 36} 


973.86 


2.98850 


5.38 


162.30 


18.98 


324.06 


324.61 


600 


330.00 


327 


6 27 


954 


960.65 


2.98257 


5.61 


164.77 


19.83 


328.97 


329.57 


15 


345.00 


3 45 


7 01} 


1046} 


923.18 


■ 
2.96529 


6.34 


172.22 


22.53 


343.73 


344.47 


30 


360.00 


4 03} 


7 38} 


1142 


888.73 


2.94877 


7.14 


179.67 


25.46 


358.44 


359.34 


45 


375.00 


4 23} 


8 16} 


12 39} 


856.95 


2.93296 


8.00 


187.10 


28.63 


373.10 


374.20 


700 


390.00 


4 43} 


8 55} 


13 39 


827.56 


2.91780 


8.92 


194.52 


32.05 


387.71 


389.03 


15 


405.00 


5 04} 


9 36} 


14 40} 


800.33 


2.90327 


9.91 


201.93 


35.73 


402.25 


403.84 


30 


420.00 


5 26} 


1018} 


15 45 


775.05 


2.88933 


10.97 


209.33 


39.67 


416.73 


418.62 



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o l^ l^ l^ l^ l^ !>. CO CO »C rt< CO (N »-i 



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i-hCS|CO»OCO 



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o 

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Q 
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CO®00'H^|>.QMCOC6C^»O00 
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co CO CO CO CO CO c<i d c^ c4 (N c^ csi 



Q0'^Q^?"5©'^05^CSIO5C0'Tt^ 

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o OOi-HC^ICO-^COOOOC^-^l^O 

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THE RAILROAD TAPER 







TAPER li. INTERMEDIATE 


VALUES 




A = J 


M 


I 


/ 


b 


• 


D 


logZ) 


d 


t 


X 


V 


2c 


130 


30.00 


o / 

061 


o / 

06} 


o / 

a 13} 


3819.78 


3.68204 


0.03 


15.00 


0.06 


30.00 


30.00 


35 


33.33 


07} 


008 


15} 


3818.74 


3.55856 


0.03 


16.67 


0.07 


33.33 


33.33 


40 


36.67 


08} 


09} 


18} 


3437.82 


3.53628 


0.04 


18.33 


0.09 


38.67 


36.67 


45 


40.00 


09} 


011} 


21 


3274.13 


3.51509 


0.05 


20.00 


0.11 


40.00 


40.00 


50 


43.33 


010} 


13 


23} 


3125.32 


3.49489 


0.06 


21.67 


0.13 


43.33 


43.33 


65 


46.67 


11} 


15 


26} 


2989.45 


3.47559 


0.07 


23.33 


0.16 


48.67 


46.67 


200 


50.00 


13 


17 


30 


2864.90 


3.45726 


0.08 


25.00 


0.19 


50.00 


50.00 


05 


53.33 


014} 


019 


033} 


2750.32 


3.43938 


0.09 


26.67 


0.22 


53.33 


53.33 


10 


56.67 


15i 


21} 


36} 


2644.56 


3.42235 


0.10 


28.33 


0.26 


56.67 


56.67 


15 


60.00 


016} 


23} 


40} 


2546.64 


3.40597 


0.12 


30.00 


0.30 


60.00 


60.00 


20 


63.33 


18} 


26 


44} 


2455.70 


3.39018 


0.13 


31.67 


0.34 


63.33 


63.33 


25 


66.67 


19} 


28} 


48} 


2371.05 


3.37494 


0.15 


33.33 


0.38 


66.67 


66.67 


30 


70.00 


21} 


31} 


52} 


2292.04 


3.36022 


0.16 


35.00 


0.43 


70.00 


70.00 


35 


73.33 


22} 


34 


56} 


2218.13 


3.34599 


0.18 


36.67 


0.49 


73.33 


73.33 


40 


76.67 


24^ 


36} 


101} 


2148.84 


3.33220 


0.20 


38.33 


0.55 


76.66 


76.67 


45 


80.00 


26} 


39} 


106 


2083.75 


3.31885 


0.22 


40.00 


0.61 


80.00 


80.00 


50 


83.33 


28 


42} 


110} 


2022.50 


3.30589 


0.24 


41.66 


0.68 


83.33 


83.33 


55 


86.67 


29} 


46 


115} 


1964.75 


3.29331 


0.27- 


43.33 


0.75 


86.66 


86.66 


300 


90.00 


31i 


49} 


121 


1910.20 


3.28108 


0.29 


45.00 


0.82 


89.99 


89.99 


05 


03.33 


33^ 


52} 


126} 


1858.62 


3.26919 


0.32 


46.66 


0.90 


93.33 


93.33 


10 


96.67 


35} 


56} 


131} 


1809.75 


3.25762 


0.35 


48.33 


0.99 


96.66 


96.66 


15 


100.00 


37J 


100 


137} 


1763.39 


3.24635 


0.38 


50.00 


1.09 


99.99 


99.99 


20 


103.33 


39^ 


103} 


143} 


1719.34 


3.23536 


0.41 


51.66 


1.19 


103.32 


103.33 


25 


106.67 


4U 


107} 


149} 


1677.45 


3.22465 


0.44 


53.33 


1.29 


106.65 


106.66 


30 


110.00 


43} 


111} 


155} 


1637.56 


3.21420 


0.47 


54.99 


1.40 


109.99 


109.99 


35 


113.33 


46 


115} 


2 01} 


1599.53 


3.20399 


0.51 


56.66 


1.52 


113.32 


113.33 


40 


116.67 


48} 


120 


2 08} 


1563.23 


3.19402 


0.55 


58.33 


1.64 


116.65 


116.66 


45 


120.00 


50i 


124} 


215 


1528.55 


3.18428 


0.59 


59.99 


1.77 


119.98 


119.99 


50 


123.33 


53 


128} 


2 21} 


1495.37 


3.17475 


0.63 


61.66 


1.90 


123.31 


123.32 


55 


126.67 


55i 


133} 


2 28} 


1463.61 


3.16543 


0.67 


63.33 


2.04 


126.64 


126.65 


400 


130.00 


58 


138 


2 36 


1433.18 


3.15630 


0.71 


64.99 


2.19 


129.97 


129.99 


05 


133.33 


lOOJ 


1421 


2 43i 


1404.00 


3.14737 


0.78 


66.66 


2.35 


133.30 


133.32 


10 


136.67 


103} 


147} 


2 50i 


1375.98 


3.13861 


081 


68.32 


2.51 


136.63 


136.65 


15 


140.00 


105} 


152i 


2 58i 


1349.07 


3.13003 


0.86 


69.99 


2.68 


139.96 


139.99 


20 


143.33 


108^ 


157} 


3 06} 


1323.20 


3.12163 


0.91 


71.65 


2.86 


143.29 


143.32 


25 


146.67 


111} 


2 02f 


3 14i 


1298.31 


3.11338 


0.97 


73.32 


3.05 


146.62 


146.65 



y Google 



TABLES 



81 



TAPER li. INTERMEDIATE VALUES — (Continued) A = i 



M 


I 


/ 


b 


o 


D 


logD 


d 


/ 


z 


V 


Ic 


a / 

430 


150.00 


a / 

114J 


o / 

2 08} 


o / 

3 22} 


1274.35 


3.10529 


1.08 


74.98 


3.24 


149.94 


149.98 


35 


153.33 


117i 


2 12} 


330} 


1251.26 


3.09735 


1.09 


76.66 


3.44 


153.27 


163.31 


40 


156.67 


1201 


2 19 


3 39} 


1229.00 


3.08956 


1.15 


78.31 


3.65 


156.60 


166.64 


45 


leo.oo 


123i 


2 24} 


348 


1207.52 


3.08189 


1.21 


79.98 


3.87 


159.92 


159.97 


50 


163.33 


126i 


230} 


356} 


1186.79 


3.07437 


1.28 


81.64 


4.10 


163.25 


163.30 


55 


166.67 


129i 


236} 


405} 


1166.78 


3.06699 


1.35 


83.31 


4.33 


166.68 


166.63 


500 


170.00 


132i 


2 42} 


415 


1147.43 


3.06973 


1.42 


84.97 


4.67 


169.90 


169.96 


05 


173.33 


135} 


2 48} 


4 24} 


1128.69 


3.06267 


1.49 


86.64 


4.83 


173.22 


173.29 


10 


176.67 


139 


2 64} 


4 33} 


1110.61 


3.04556 


1.57 


88.30 


5.09 


176.66 


176.62 


15 


180.00 


1421 


3 01} 


4 43} 


1093.09 


3.03866 


1.65 


89.96 


6.36 


179.87 


179.95 


20 


183.33 


1461 


3 07} 


4 63} 


1076.12 


3.03186 


1.73 


91.63 


6.64 


183.19 


183.28 


25 


186.67 


149i 


3 14 


5 03} 


1069.68 


3.02517 


1.81 


93.29 


6.93 


186.51 


186.60 


30 


190.00 


162} 


319} 


6 13} 


1043.74 


3.01859 


1.90 


94.95 


6.23 


189.83 


189.93 


35 


193.33 


1561 


3 27} 


5 23} 


1028.28 


3.01211 


1.99 


96.62 


6.64 


193.15 


193.25 


40 


196.67 


200 


3 34} 


5 34} 


1013.28 


3.00673 


2.08 


98.28 


6.86 


196.47 


196.69 


45 


200.00 


2 03} 


3 41} 


5 46 


998.72 


2.99944 


2.17 


99.94 


7.19 


199.79 


199.71 


50 


203.33 


2 07i 


3 48} 


5 56} 


984.69 


2.99326 


2.27 


101.61 


7.53 


203.10 


203.24 


55 


206.67 


2 11J 


3 55} 


6 06} 


970.85 


2.98716 


2.37 


103.27 


7.88 


206.42 


206.57 


600 


210.00 


215 


403 


618 


957.61 


2.98114 


2.47 


104.93 


8.24 


209.73 


209.89 


15 


220.00 


2 261 


4 25} 


6 52} 


919.64 


2.96362 


2.80 


109.92 


9.39 


219.66 


219 86 


30 


230.00 


2 39} 


4 49} 


7 28} 


884.74 


2.94682 


3.15 


114.90 


10.64 


229.68 


229.83 


45 


240.00 


2 52} 


5 13} 


806 


862.48 


2.93068 


3.63 


119.88 


12.00 


239.49 


239.79 


700 


250.00 


3 06i 


5 39} 


8 46 


822.68 


2.91618 


3.94 


124.85 


13.47 


249.38 


249.75 


15 


260.00 


3 m 


606 


9 26} 


794.80 


2.90026 


4.38 


129.82 


16.06 


269.26 


259.70 


30 


270.00 


3 33} 


6 33} 


10 07} 


768.93 


2.88689 


4.86 


134.79 


16.76 


269.12 


269.64 


45 


280.00 


3 48} 


7 02} 


10 61 


744.80 


2.87204 


6.36 


139.76 


18.68 


278.96 


279.57 


800 


290.00 


404 


7 32 


1136 


722.24 


2.86868 


5.90 


144.72 


20.63 


288.76 


289.49 


15 


300.00 


420 


8 02} 


12 22} 


701.11 


2.84679 


6.47 


149.67 


22.61 


298.64 


299.40 


30 


310.00 


4 36} 


8 34} 


13 10} 


681.30 


2.83334 


7.08 


164.62 


24.83 


308.29 


309.29 


45 


320.00 


453 


907 


14 00 


662.70 


2.82132 


7.73 


169.66 


27.18 


318.02 


319.17 


900 


330.00 


510} 


940} 


14 61 


646.20 


2.80969 


8.42 


164.49 


29.68 


327.70 


329.04 


16 


340.00 


5 28§ 


1015 


16 43} 


628.72 


2.79846 


9.14 


169.42 


32.32 


337.34 


338.88 


30 


350.00 


5 46} 


10 50} 


16 37} 


613.19 


2.78760 


9.90 


174.34 


36.11 


346.94 


348.72 


45 


360.00 


6 06} 


1127} 


17 33 


598.63 


2.77709 


10.70 


179.26 


38.06 


358.60 


358.54 


1000 


370.00 


6 25 


12 06 


18 30 


684.69 


2.76693 


11.66 


184.15 


41.16 


366.01 


368.33 


15 


380.00 


6 46} 


12 42} 


19 28} 


671.61 


2.76710 


12.44 


189.04 


44.43 


376.47 


378.09 


30 


390.00 


7 05} 


13 23} 


20 28} 


669.24 


2.74760 


13.37 


193 92 


47.85 


384.87 


387.82 



y Google 



THE RAILROAD TAPER 



II 



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^ 



O 



(4 



On 



Cj8 



t 



'if 

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08 
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68 
6So 



68 
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98 



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68 
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68 
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68 
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68 



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O t* t>. t* t>. O CO »0 rt< CO »-* "-H 



'^Of-iOMjS^^o^ w ;coco 

• »C «0 CO CO »0 »0 -^ CO (N ♦-• -i-iCO 



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Digitized by VjjOOQ||C 



TABLES 



83 



II 



8 









^ 



O 






i-ii-Hi-lc5C^C^CNICOCv5 



O Q Oi O) C3 Od Q^ O 00 t^ ^ CO 



80iOOCS|'«*i000iQQ0'^i-H 
COr-lCOCOT-lOSO>COO'^iO 



OOOOiOSOSOOt^SThfHS 









ooooo55'^oocO'^C)cooqc^ 
"^ocvi^cdQ05i-HcJa>o6»o 

QQO5TfrHOa00t^»»O»O^Tfi 



• O O 1-1 CO •«*< «0 00 O CO CO OS CO 

1-1 1-4 t-H t-H W 



gS88SS2§SS 

1-1 rH 1-1 C^ C<l C^ C 



8SSS8888SS88 

CS|COtJ1iOCOI>-00050i-hC^CO 
1-H »-l »-< 1— ( 

Digitized by 



(jOoqIc 



84 THE RAILROAD TAPER 

TAPER 2. INTERMEDIATE VALUES 



A=l 



M 


I 


/ 


b 





D 


iogD 


d 


t 


z 


y 


Ic 


e / 

200 


30.00 


o / 

009 


o / 

009 


o / 

18 


2864.86 


3.45710 


0.04 


16.00 


0.08 


30.00 


30.00 


05 


32.50 


10 


10} 


020} 


2750.28 


3.43938 


0.05 


16.26 


0.09 


32.50 


32.50 


10 


35.00 


oioi 


12 


22} 


2644.51 


3.42234 


0.05 


17.60 


0.11 


36.00 


35.00 


15 


37.60 


Olli 


13} 


25} 


2546.58 


3.40596 


0.06 


18.75 


0.13 


37.60 


37.50 


20 


40.00 


012i 


15} 


028 


2455.64 


3.39016 


0.07 


20.00 


0.15 


40.00 


40.00 


25 


42.50 


014 


016} 


030} 


2370.98 


3.37493 


0.08 


21.25 


0.17 


42.60 


42.50 


30 


45.00 


015 


18} 


33} 


2291.97 


3.36021 


0.09 


22.50 


0.20 


45.00 


45.00 


35 


47.50 


016i 


020} 


36} 


2218.05 


3.34597 


0.10 


23.75 


0.22 


47.50 


47.50 


40 


50.00 


017i 


22} 


;0 40 


2148.76 


3.33219 


0.11 


25.00 


0.25 


50.00 


50.00 


45 


52.60 


18^ 


24} 


43} 


2083.65 


3.31882 


0.12 


26.26 


0.28 


52.50 


52.50 


50 


55.00 


191 


27 


46} 


2022.39 


3.30586 


0.13 


27.50 


0.32 


55.00 


56.00 


55 


57.50 


21^ 


029 


60} 


1964.62 


3.29326 


0.14 


28.75 


0.35 


57.60 


57.60 


300 


60.00 


22^ 


31} 


064 


1910.07 


3.28105 


0.16 


30.00 


0.39 


60.00 


60.00 


05 


62.60 


024 


33} 


57} 


1858.47 


3.26916 


0.17 


31.25 


0.43 


62.50 


62.50 


10 


65.00 


26i 


36} 


101} 


1809.59 


3.25758 


0.19 


32.50 


0.48 


65.00 


65.00 


15 


67.60 


26i 


39 


106} 


1763.21 


3.24630 


0.20 


33.76 


0.53 


67.50 


67.60 


20 


70.00 


28i 


41} 


110 


1719.15 


3.23531 


0.22 


35.00 


0.58 


7i).00 


70.00 


25 


72.60 


030 


44} 


114} 


1677.25 


3.22460 


0.24 


36.25 


0.63 


72.50 


72.50 


30 


75.00 


3H 


47} 


118} 


1637.35 


3.21414 


0.26 


37.50 


0.69 


75.00 


75.00 


35 


77.50 


33i 


050 


123} 


1599.30 


3.20393 


0.28 


38.75 


0.75 


77.60 


77.50 


40 


80.00 


34i 


53} 


128 


1562.98 


3.19395 


0.30 


40.00 


0.81 


79.99 


80.00 


45 


82.50 


36i 


66} 


132} 


1628.28 


3.18420 


0.32 


41.25 


0.88 


82.49 


82.50 


50 


85.00 


38i 


59} 


137} 


1495.08 


3.17437 


0.34 


42.60 


0.95 


84.99 


85.00 


55 


87.60 


40i 


102} 


142} 


1463.31 


3.16534 


0.37 


43.75 


1.02 


87.49 


87.60 


400 


90.00 


42 


104 


148 


1432 86 


3.15620 


0.39 


45.00 


1.10 


89.99 


90.00 


05 


92.60 


044 


109} 


153} 


1403.66 


3.14726 


0.42 


46.24 


1.18 


92.49 


92.50 


10 


95.00 


451 


113 


158} 


1375.62 


3.13860 


0.46 


47.49 


1.27 


95.99 


95.00 


15 


97.50 


47} 


116} 


2 04} 


1348.69 


3.12991 


0.48 


48.74 


1.36 


97.49 


97.50 


20 


100.00 


491 


120} 


2 10 


1322.80 


3.12149 


0.51 


49.99 


1.45 


99.98 


99.99 


25 


102.60 


52 


123} 


215} 


1297.87 


3.11323 


0.54 


51.24 


1.56 


102.48 


102.49 


30 


105.00 


054 


127} 


2 21} 


1273.89 


3.10613 


0.67 


52.49 


1.65 


104.98 


104.99 


35 


107.60 


56i 


131} 


2 27} 


1250.77 


3.09718 


0.60 


63.74 


1.76 


107.48 


107.49 


40 


110.00 


58i 


135} 


2 34 


1228.48 


3.08937 


0.63 


54.99 


1.87 


109.98 


109.99 


45 


112.60 


lOOi 


139} 


2 40} 


1206.98 


3.08170 


0.67 


56.24 


1.98 


112.47 


112.49 


50 


115.00 


102} 


144 


2 46} 


1186.22 


3.07416 


0.71 


57.49 


2.10 


114.97 


114.99 


55 


117.60 


106i 


148 


2 53} 


1166.18 


3.06677 


0.76 


58.74 


2.23 


117.47 


117.49 


500 


120.00 


107i 


152} 


300 


1146.80 


3.05949 


0.79 


59.99 


2.36 


119.96 


119.99 


05 


122.50 


110 


166} 


3 06} 


1128.03 


3.05232 


0.83 


61.24 


2.49 


122.46 


122.49 


10 


126.00 


112} 


2 0U 


3 13} 


1109.91 


3.04529 


0.87 


62.48 


2.63 


124.96 


124.99 



y Google 



TABLES 85 

TAPER 2. INTERMEDIATE VALUES— (Con/iwtted) A=l 



ill 


I 


/ 


b 





D 


log/) 


d 


t 


X 


V 


Ic 


o / 

5 15 


127.50 


114} 


O f 

206 


o / 

320} 


1092.35 


3.03836 


0.91 


63.73 


2,77 


127.45 


127.49 


20 


130.00 


117i 


2 10} 


3 28 


1075.35 


3.03165 


0.96 


64.98 


2.92 


129.95 


129.98 


25 


132.50 


120 


215} 


3 36} 


1068.87 


3.02484 


1.00 


66.23 


3.08 


132.44 


132.48 


30 


135.00 


122i 


220} 


3 42} 


1042.89 


3.01823 


1.06 


67.48 


3.24 


134.94 


134.98 


35 


137.50 


126i 


225 


3 50i 


1027.39 


3.01174 


1.10 


68.73 


3.41 


137.44 


137.48 


40 


140.00 


127} 


230} 


358 


1012.35 


3.00490 


1.15 


69.98 


3.58 


139.93 


139.98 


45 


142.50 


130J 


2 36} 


4 05} 


997.76 


2.99903 


1.21 


71.22 


3.75 


142.42 


142.48 


50 


145.00 


133} 


240i 


4 13} 


983.68 


2.99281 


1.26 


72.47 


3.93 


144.92 


144.97 


55 


147.50 


136} 


2 45i 


4 21} 


969.80 


2.98668 


1.32 


73.72 


4.12 


147.41 


147.47 


600 


150.00 


139 


2 51 


430 


956.41 


2.98064 


1.37 


74.97 


4.32 


149.90 


149.97 


15 


157.60 


147} 


3 07* 


4 55} 


918.39 


2.96303 


1.55 


78.71 


4.94 


157.37 


157.47 


30 


165.00 


157 


3 24} 


5 21} 


883.34 


2.94613 


1.75 


82.45 


5.61 


164.84 


164.96 


45 


172.50 


2 06i 


3 42} 


5 49} 


850.92 


2.92989 


1.97 


86.19 


6.34 


172.32 


172.44 


700 


180.00 


2 16i 


4 0U 


618 


820.84 


2.91426 


2.20 


89.93 


7.14 


179.77 


179.92 


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187.50 


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4 21 


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792.86 


2.89920 


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93.67 


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187.22 


187.39 


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195.00 


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766.79 


2.88468 


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194.66 


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7 50} 


742.43 


2.87066 


2.99 


101.14 


9.92 


202.10 


202.34 


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210.00 


300 


6 24 


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719.63 


2.85711 


3.29 


104.87 


10.98 


209.52 


209.81 


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217.50 


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8 58} 


698.26 


2.84402 


3.62 


108.60 


12.12 


216.93 


217.26 


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225.00 


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678.18 


2.83134 


3.96 


112.33 


13.33 


224.33 


224.72 


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232.50 


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6 33} 


1010} 


659.29 


2.81908 


4.32 


116.05 


14.62 


231.72 


232.18 


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240.00 


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6 68i 


10 48 


641.49 


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4.71 


119.78 


15.99 


239.10 


239.63 


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247.50 


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7 24 


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2.79566 


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123.49 


17.44 


246.46 


247.07 


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608.83 


2.78460 


6.64 


127.21 


18.97 


253.80 


264.61 


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262.50 


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12 47} 


593.82 


2.77366 


6.99 


130.92 


20.69 


261.12 


261.92 


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270.00 


4 46 


8 45 


13 30 


679.61 


2.76314 


6.47 


134.63 


22.30 


268.42 


269.36 


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277.50 


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9 13i 


1413} 


566.14 


2.76292 


6.97 


138.33 


24.11 


276.71 


276.76 


30 


285.00 


6 15 


9 42} 


1467} 


553.36 


2.74301 


7.49 


142.03 


26.00 


282.96 


284.16 


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292.50 


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1543} 


641.22 


2.73337 


8.04 


146.72 


27.99 


290.20 


291.54 


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300.00 


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1630 


629.69 


2.72402 


8.62 


149.41 


30.08 


297.40 


298.92 


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307.50 


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1116 


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618.73 


2.71494 


9.23 


153.09 


32.26 


304.58 


306.28 


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315.00 


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508.29 


2.70611 


9.86 


156.77 


34.55 


311.72 


313.64 


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322.60 


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498.36 


2.69754 


10.52 


160.44 


36.94 


318.84 


319.50 


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330.00 


6 54 


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488.88 


2.68920 


11.20 


164.10 


39.44 


325.91 


328.29 


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479.86 


2.68111 


11.92 


167.76 


42.04 


332.96 


335.59 


30 


345.00 


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471.26 


2.67325 


12.66 


171.41 


44.76 


339.95 


342.88 


45 


352.60 


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14 39} 


22 28} 


463.05 


2.66563 


13 44 


175.05 


47.59 


346.90 


350.16 


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465.22 


2.66822 


14.25 


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30.00 


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2291.93 


3.36020 


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30.00 


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32.00 


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2218.01 


3.34596 


0.06 


16.00 


0.11 


32.00 


32.00 


40 


34.00 


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2148.70 


3.33218 


0.06 


17.00 


0.13 


34.00 


34.00 


45 


36.00 


14 


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2083.60 


3.31881 


0.07 


18.00 


0.16 


36.00 


36.00 


50 


38.00 


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32} 


2022.34 


3.30686 


0.08 


19.00 


0.17 


38.00 


38.00 


55 


40.00 


16 


19 


35 


1964.67 


3.29327 


0.09 


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0.19 


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300 


42.00 


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1910.00 


3.28103 


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1868.40 


3.26914 


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1809.51 


3.25756 


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23.00 


0.26 


46.00 


46.00 


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1763.13 


3.24628 


0.12 


24.00 


0.29 


48.00 


48.00 


20 


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50 


1719.06 


3.23529 


0.13 


25.00 


0.32 


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23 


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1677.16 


3.22457 


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1637.25 


3.21411 


0.16 


27.00 


0.38 


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64.00 


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1599.19 


3.20390 


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261 


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1562.86 


3.19392 


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1528.16 


3.18417 


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1494.95 


3.17463 


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1375.45 


3.13844 


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69.99 


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1348.50 


3.12985 


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0.77 


71.99 


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1322.60 


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1297.67 


3.11316 


0.33 


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75.99 


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1273.67 


3.10506 


0.36 


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1206.73 


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1058.50 


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3.01156 


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TABLES 



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TAPER 2i. INTERMEDIATE YALVES— (Continued) A = li 



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2.99883 


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113 


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3 12* 


983.11 


2.99260 


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109.96 


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969.31 


2.98646 


0.83 


65.98 


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111.96 


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3 25} 


955.90 


2.98041 


0.86 


56.98 


2.67 


113.96 


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3 46 


917.82 


2.96276 


0.98 


59.98 


2.96 


119.94 


119.98 


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126.00 


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234 


4 051 


882.70 


2.94581 


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62.97 


3.36 


126.93 


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45 


132.00 


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2 48 


4 27} 


850.19 


2.92952 


1.24 


65.97 


3.81 


131.91 


131.97 


700 


138.00 


147 


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4 49} 


820.03 


2.91383 


1.39 


68.96 


4.30 


137.89 


137.96 


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3 18 


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791.97 


2.89871 


1.66 


71.96 


4.83 


143.87 


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5 37* 


765.80 


2.88412 


1.72 


74.95 


5.40 


149.84 


149.94 


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6 02} 


741.34 


2.87002 


1.90 


77.94 


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156.81 


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6 28} 


718.43 


2.85638 


2.09 


80.93 


6.67 


161.78 


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696.93 


2.84319 


2.29 


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7.37 


167.73 


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4 43 


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676.73 


2.83042 


2.51 


86.91 


8.12 


173.69 


173.88 


45 


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7 52* 


657.71 


2.81803 


2.74 


89.90 


8.92 


179.63 


179.86 


900 


186.00 


3 01 


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639.77 


2.80602 


2.99 


92.88 


9.77 


185.57 


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192.00 


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622.83 


2.79437 


3.25 


95.86 


10.68 


191.61 


191.82 


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6 02 


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606.81 


2.78305 


3.62 


98.84 


11.64 


197.43 


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9 66} 


691.64 


2.77206 


3.81 


101.82 


12.66 


203.34 


203.76 


10 00 


210.00 


3 45 


645 


10 30 


677.26 


2.76137 


4.12 


104.80 


13.71 


209.25 


209.70 


15 


216.00 


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7 07J 


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563.61 


2.75098 


4.44 


107.78 


14.84 


216.14 


215.65 


30 


222.00 


4 08} 


7 30J 


1139} 


650.64 


2.74087 


4.77 


110.75 


16.03 


221.02 


221.60 


45 


228.00 


4 2U 


754 


12 15} 


538.30 


2.73102 


5.12 


113.72 


17.27 


226.89 


227.55 


1100 


234.00 


434 


8 18} 


12 52} 


526.56 


2.72146 


6.49 


116.69 


18.58 


232.74 


233.49 


15 


240.00 


4 47 


8 43 


13 30 


615.38 


2.71213 


6.88 


119.66 


19.96 


238.59 


239.42 


30 


246.00 


6 00} 


9 081 


14 08} 


504.71 


2.70304 


6.28 


122.61 


21.40 


244.42 


245.36 


45 


252.00 


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9 34i 


14 48} 


494.64 


2.69420 


6.70 


126.57 


22.90 


250.23 


251.29 


12 00 


258.00 


5 27i 


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15 28} 


484.82 


2.68558 


7.14 


128.52 


24.47 


256.02 


257.20 


15 


264.00 


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10 28* 


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475.54 


2.67719 


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131.47 


26.11 


261.79 


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466.67 


2.66901 


8.08 


134.42 


27.82 


267.64 


268.98 


45 


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458.19 


2.66105 


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137.36 


29.61 


273.27 


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282.00 


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450.06 


2.66327 


9.09 


140.30 


31.47 


278.98 


280.75 


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19 04} 


442.29 


2.64671 


9.63 


143.23 


33.40 


284.66 


286.61 


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294.00 


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12 53* 


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434.85 


2.63834 


10.19 


146.16 


35.40 


290.32 


292.47 


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427.71 


2.63115 


10.76 


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37.49 


296.95 


298.32 


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306.00 


7 29i 


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420.87 


2.62415 


11.36 


152.00 


39.65 


301.55 


304.15 


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414.31 


2.61733 


11.98 


154.91 


41.89 


307.12 


309.96 


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408.04 


2.61070 


12.63 


157.81 


44.21 


312.66 


315.76 


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402.01 


2.60424 


13.29 


160.71 


46.61 


318.16 


321.66 


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330.00 


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396.23 


2.59795 


13.98 


163.60 


49.09 


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y Google 



90 



THE RAILROAD TAPER 



II 



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TABLES 



91 



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y Google 



92 



THE RAILROAD TAPER 







TAPER 2i. 


INTERMEDIATE 


VALUES A = li 


M 


I 


/ 


b 





D 


IogZ> 


d 


t 


X 


y 


Ic 


3 00 


30.00 


o / 


/ 

13} 


/ 

27 


1909.97 


3.28103 


0.06 


15.00 


0.12 


30.00 


30.00 


05 


31.67 


OUh 


141 


29} 


1868.37 


3.26913 


0.07 


15.83 


0.13 


31.67 


31.67 


10 


33.33 


15} 


16} 


311 


1809.47 


3.25756 


0.07 


16.67 


0.15 


33.33 


33.33 


15 


35.00 


16i 


18} 


34} 


1763.09 


3.24627 


0.08 


17.50 


0.17 


35.00 


35.00 


20 


36.67 


17i 


19} 


36} 


1719.02 


3.23528 


0.09 


18.33 


0.18 


36.67 


36.67 


25 


38.33 


18i 


21 


39} 


1677.10 


3.22466 


0.09 


19.17 


0.20 


38.33 


38.33 


30 


40.00 


19i 


22} 


42 


'1637.19 


3.21410 


0.10 


20.00 


0.22 


40.00 


40.00 


35 


41.67 


020i 


24} 


44} 


1599.13 


3.20388 


0.11 


20.83 


0.24 


41.67 


41.67 


40 


43.33 


2H 


26} 


47} 


1562.80 


3.19390 


0.12 


21.67 


0.27 


43.33 


43.33 


45 


45.00 


22^ 


28} 


60} 


1528.09 


3.18416 


0.13 


22.50 


0.29 


45.00 


45.00 


60 


46.67 


231 


30 


63} 


1494.88 


3.17461 


0.14 


23.33 


0.32 


46.67 


46.67 


55 


48.33 


24f 


32 


56} 


1463.09 


3.16627 


0.15 


24.17 


0.35 


48.33 


48.33 


400 


50.00 


26 


34 


100 


1432.63 


3.15613 


0.16 


25.00 


0.38 


60.00 


50.00 


05 


61.67 


27i 


36 


103} 


1403.41 


3.14716 


0.17 


26.83 


0.41 


51.66 


51.67 


10 


53.33 


28} 


38} 


106} 


1376.35 


3.13841 


0.18 


26.66 


0.44 


63.33 


53.33 


15 


66.00 


291 


040} 


110} 


1348.41 


3.12982 


0.20 


27.50 


0.48 


55.00 


65.00 


20 


56.67 


31 


42} 


113} 


1322.60 


3.12140 


0.21 


28.33 


0.51 


56.66 


56.67 


25 


58.33 


32i 


44} 


117} 


1297.66 


3.11313 


0.22 


29.16 


0.55 


68.33 


58.33 


35 


60.00 


33i 


47} 


121 


1273.66 


3.10502 


0.24 


30.00 


0.59 


60.00 


60.00 


30 


61.67 


35i 


49} 


124} 


1260.42 


3.09706 


0.25 


30.83 


0.63 


61.66 


61.67 


40 


63.33 


36J 


62} 


128} 


1228.12 


3.08924 


0.27 


31.66 


0.67 


63.33 


63.33 


45 


66.00 


38 


64} 


132} 


1206.69 


3.08166 


0.28 


32.50 


0.72 


64.99 


85.00 


60 


66.67 


39} 


67} 


136} 


1185.81 


3.07402 


0.30 


33.33 


0.77 


66.66 


66.67 


55 


68.33 


41 


59} 


140} 


1165.74 


3.06660 


0.31 


34.16 


0.82 


68.33 


68.33 


600 


70.00 


42i 


102} 


145 


1146.34 


3.06931 


0.33 


36.00 


0.87 


69.99 


70.00 


05 


71.67 


44 


105} 


149} 


1127.65 


3.06214 


0.35 


35.83 


0.92 


71.66 


71.66 


10 


73.33 


46} 


108 


163} 


1109.41 


3.04609 


0.37 


36.66 


0.97 


73.32 


73.33 


15 


75.00 


47i 


111 


158} 


1091.83 


3.03816 


0.39 


37.49 


1.03 


74.99 


75.00 


20 


76.67 


49 


1 13} 


2 02} 


1074.80 


3.03133 


0.41 


38.33 


1.09 


76.66 


76.66 


25 


78.33 


60} 


1 16} 


2 07} 


1068.30 


3.02461 


0.43 


39.16 


1.15 


78.32 


78.33 


30 


80.00 


52} 


119} 


2 12 


1042.29 


3.01799 


0.45 


39.99 


1.22 


79.99 


80.00 


35 


81.67 


064 


122} 


2 16} 


1026.76 


3.01147 


0.47 


40.83 


1.28 


81.66 


81.66 


40 


83.33 


56} 


126 


2 21} 


1011.69 


3.00506 


0.49 


41.66 


1.35 


83.32 


83.33 


45 


85.00 


67} 


129} 


2 26} 


997.07 


2.99873 


0.62 


42.49 


1.42 


84.98 


84.99 


60 


87.67 


69} 


132} 


2 31} 


982.86 


2.99249 


0.64 


43.32 


1.49 


86.66 


86.66 


65 


88.33 


101} 


135} 


2 36} 


969.04 


2.98634 


0.56 


44.16 


1.57 


88.31 


88.33 



y Google 



TABLES 93 

TAPER 2i. INTERMEDIATE VALVES— {Continued) A=li 



M 


/ 


/ 


b 





D 


logi> 


d 


t 


X 


y 


Ic 


600 


90.00 


103 


o t 

139 


2 42 


966.63 


2.98029 


0.59 


44.99 


1.65 


89.98 


89.99 


16 


95.00 


1081 


149} 


2 58} 


917.61 


2.96261 


0.67 


47.49 


1.90 


94.97 


94.99 


30 


100.00 


114J 


200} 


3 15 


882.35 


2.94564 


0.76 


49.98 


2.18 


99.96 


99.99 


45 


105.00 


121 


2 11} 


3 32} 


849.80 


2.92932 


0.86 


52.48 


2.48 


104.96 


104.99 


700 


110.00 


127i 


2 23} 


3 61 


819.69 


2.91360 


0.95 


54.98 


2.80 


109.96 


109.98 


15 


115.00 


134i 


2 36 


4 10} 


791.48 


2.89844 


1,06 


57.47 


3.15 


114.93 


114.98 


30 


120.00 


141i 


2 48} 


4 30 


765.26 


2.88381 


1.18 


59.97 


3.63 


119.92 


119.97 


45 


125.00 


148} 


3 02} 


4 60} 


740.76 


2.86967 


1.31 


62.46 


3.94 


124.90 


124.96 


800 


130.00 


156 


3 16 


5 12 


717.78 


2.85699 


1.44 


64.96 


4.38 


129.88 


129.96 


15 


135.00 


2 03} 


380} 


5 34} 


696.22 


2.84275 


1.58 


67.45 


4.85 


134.86 


134.96 


30 


140.00 


2 111 


3 46} 


6 57 


675.95 


^.82991 


1.73 


69.94 


6.36 


139.84 


139.94 


45 


146.00 


220 


4 00} 


6 20} 


656.86 


2.81747 


' 1.89 


72.44 


5.90 


144.81 


144.93 


900 


150.00 


2 281 


4 16} 


6 46 


638.84 


2.80639 


2.06 


74.93 


6.47 


149.77 


149.91 


15 


155.00 


2 371 


433 


7 10} 


621.82 


2.79366 


2.24 


77.42 


7.08 


154.74 


154.90 


30 


160.00 


2 46} 


4 49} 


7 36 


605.72 


2.78227 


2.43 


79.91 


7.73 


159.70 


150.88 


45 


166.00 


2 661 


5 07} 


8 02} 


590.46 


2.77119 


2.63 


82.39 


8.42 


164.66 


164.86 


10 00 


170.00 


305 


5 26 


830 


675.98 


2.76041 


2.84 


84.88 


9.14 


169.60 


169.84 


15 


175.00 


3 141 


6 43} 


8 68} 


662.23 


2.74991 


3.06 


87.37 


9.90 


174.54 


174.82 


30 


180.00 


3 24} 


6 02} 


9 27 


649.16 


2.73970 


3.29 


89.85 


10.70 


179.47 


179.79 


45 


185.00 


336 


6 21} 


9 66} 


636.71 


2.72974 


3.53 


92.33 


11.66 


184.40 


184.76 


1100 


190.00 


3 461 


6 41} 


10 27 


624.86 


2.72004 


3.79 


94.81 


12.44 


180.32 


189.73 


15 


195.00 


3 66} 


7 02 


10 68} 


613.56 


2.71069 


4.06 


97.29 


13.37 


194.24 


194.69 


30 


200.00 


4 07} 


7 22} 


1130 


502.77 


2.70137 


4.34 


99.77 


14.36 


199.14 


199.65 


45 


206.00 


4 18} 


7 44} 


12 02} 


492.47 


2.69238 


4.63 


102.24 


16.37 


204.03 


204.61 


12 00 


210.00 


4 30 


806 


12 36 


482.62 


2.68361 


4.94 


104.72 


16.44 


208.92 


209.56 


15 


215.00 


4 41} 


8 28} 


13 10} 


473.20 


2.67504 


6.26 


107.19 


17.66 


213.79 


214.51 


30 


220.00 


4 63} 


8 51} 


13 45 


464.18 


2.66669 


5.59 


109.65 


18.73 


218.65 


219.46 


45 


225.00 


606 


9 14} 


14 20} 


466.54 


2.65853 


5.93 


112.12 


19.95 


223.50 


224.40 


13 00 


230.00 


6 18} 


9 38} 


14 57 


447.26 


2.65056 


6.29 


114.58 


21.22 


228.34 


229.33 


15 


235.00 


6 31} 


10 03 


15 34} 


439.32 


2.64278 


6.66 


117.04 


22.54 


233.17 


234.25 


30 


240.00 


6 44} 


10 27} 


16 12 


431.71 


2.63519 


7.05 


119.50 


23.91 


237.98 


239.17 


45 


245.00 


6 67} 


10 53} 


16 60} 


424.40 


2.62778 


7.45 


121.95 


25.34 


242.77 


244.08 


14 00 


250.00 


611 


1119 


17 30 


417.38 


2.62063 


7.87 


124.40 


26.82 


247.66 


248.99 


15 


256.00 


6 24} 


1145} 


18 10} 


410.63 


2.61345 


8.30 


126.85 


28.36 


252.30 


253.89 


30 


260.00 


6 38} 


12 12} 


18 51 


404.15 


2.60654 


8.74 


129.29 


29.95 


257.04 


258.78 


45 


265.00 


6 63 


12 39} 


19 32} 


397.92 


2.59980 


9.20 


131.73 


31.60 


261.77 


263 67 


15 00 


270.00 


7 07} 


13 07} 


20 15 


391.93 


2.59321 


9.68 


134.17 


33.31 


266.47 


268.64 


30 


280.00 


7 37} 


14 04} 


2142 


380.62 


2.58049 


10.69 


139.03 


36.90 


275.81 


278.26 


16 00 


290.00 


808 


15 04 


23 12 


370.15 


2.56838 


11.76 


143.87 


40.74 


286.05 


287.94 


30 


300.00 


8 39} 


16 05} 


24 45 


360.45 


2.55684 


12.90 


148.68 


44.82 


294.19 


297.68 



y Google 



94 



THE KAILROAD TAPER 



TAPER 2i. CURVATURE INCREASING 2*» 00' 

EVERY 30 FT. A=2 













Deflections to 










Transit 
at 






















B.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 


, .-, 


2" 00' 


4«00' 


6«00' 


8" 00' 


10*00' 


12" 00' 


14*00' 


16*00' 


18*00' 


20*00' 


/ 





30 


60 


90 


120 


150 


180 


210 


240 


270 




e / 


e 1 


o / 


• 1 


O 1 


o / 


O 1 


e / 


e 1 


O 1 







18 


45 


124 


2 15 


3 18 


433 


600 


7 39 


930 


30 


018 




36 


121 


2 18 


327 


448 


6 21 


806 


10 03 


60 


103 


36 




054 


157 


3 12 


4 39 


6 18 


809 


1012 


90 


2 12 


139 


054 




112 


2 33 


406 


5 51 


748 


9 57 


120 


3 45 


306 


2 15 


112 




130 


309 


500 


703 


9 18 


150 


542 


4 57 


400 


2 51 


130 




148 


3 45 


554 


8 15 


180 


803 


7 12 


609 


454 


3 27 


148 




206 


4 21 


648 


210 


10 48 


9 51 


8 42 


7 21 


5 48 


403 


206 




224 


4 57 


240 


13 57 


12 54 


1139 


1012 


8 33 


642 


439 


224 




2 42 


270 


17 30 


16 21 


15 00 


13 27 


1142 


9 45 


7 36 


5 15 


242 


.... 



M 


/ 





D 


logD 


d 


t 


X 


V 


Ic 


400 


30 


o / 

36 


1432.55 


3.15611 


0.08 


15.00 


0.16 


30.00 


30.00 


600 


60 


148 


955.35 


2.98016 


0.31 


29.99 


0.79 


59.99 


60.00 


800 


90 


3 36 


717.12 


2.85559 


0.78 


44.98 


2.20 


89.96 


89.98 


10 00 


120 


600 


674.71 


2.75945 


1.57 


59.95 


4.70 


119.85 


119.95 


12 00 


150 


900 


480.42 


2.68162 


2.74 


74.87 


8.63 


149.60 


149.85 


14 00 


180 


12 36 


413.90 


2.61690 


4.39 


89.73 


14.25 


179.07 


179.64 


16 00 


210 


16 48 


364.96 


2 56225 


6.57 


104.50 


21.87 


208.08 


209.24 


18 00 


240 


2136 


328.02 


2.51590 


9.38 


119.12 


31.77 


236.44 


238.64 


20 00 


270 


27 00 


299.71 


2.47670 


12.87 


133.54 


44.13 


263.76 


267.43 



y Google 



TABLES 



95 



TAPER 2i. INTERMEDIATE VALUES A=2 



M 


I 


/ 


b 


o 


D 


logD 


d 


t 


X 


y 


fc 


e f 

400 


30.00 


018 


O 1 

018 


O 1 

36 


1432.66 


3.15611 


0.08 


15.00 


0.16 


30.00 


30.00 


05 


31.25 


019 


019} 


38} 


1403.33 


3.14716 


0.09 


15.62 


0.17 


31.25 


31.25 


10 


32.50 


019f 


21 


040} 


1375.26 


3.13838 


0.09 


16.25 


0.19 


32.50 


32.50 


15 


33.75 


0201 


22} 


43 


1348.31 


3.12979 


0.10 


16.87 


0.20 


33.75 


33.75 


20 


35.00 


21} 


23} 


45} 


1322.40 


3.12136 


0.11 


17.50 


0.22 


36.00 


36.00 


25 


36.25 


221 


25} 


048 


1297.45 


3.11309 


0.11 


18.12 


0.24 


36.25 


36.25 


30 


37.50 


23! 


27 


050} 


1273.44 


3.10498 


0.12 


18.75 


0.26 


37.60 


37.50 


35 


38.75 


24f 


28i 


53} 


1250.30 


3.09701 


0.13 


19.37 


0.28 


38.75 


38.75 


40 


40.00 


25} 


030} 


56 


1227.99 


3.08919 


0.14 


20.00 


0.30 


40.00 


40.00 


45 


41.25 


26} 


32 


58} 


1206.46 


3.08151 


0.15 


20.62 


0.32 


41.25 


41.25 


50 


42.50 


27} 


33} 


101} 


1185.66 


3.07396 


0.15 


21.25 


0.34 


42.50 


42.50 


55 


43.75 


029 


35i 


104} 


1165.59 


3.06655 


0.16 


21.87 


0.37 


43.75 


43.75 


600 


46.00 


30 


37J 


107} 


1146.18 


3.05925 


0.17 


22.50 


0.39 


45.00 


45.00 


05 


46.25 


31} 


39} 


110} 


1127.38 


3.05207 


0.18 


23.12 


0.42 


46.25 


46.25 


10 


47.50 


32} 


41} 


113} 


1109.23 


3.04502 


0.19 


23.75 


0.44 


47.50 


47.50 


15 


48.75 


33i 


43} 


116} 


1091.64 


3.03808 


0.20 


24.37 


0.47 


48.75 


48.75 


20 


50.00 


34} 


45} 


120 


1074.60 


3.03125 


0.21 


25.00 


0.50 


50.00 


50.00 


25 


51.25 


0.36 


47} 


123} 


1058.09 


3.02452 


0.22 


25.62 


0.54 


51.25 


51.25 


30 


52.50 


37 


49i 


126} 


1042.08 


3.01790 


0.24 


26.25 


0.57 


52.50 


52.50 


'35 


53.75 


om 


51i 


130 


1026.54 


3.01138 


0.25 


26.87 


0.60 


53.75 


53.75 


40 


55.00 


39} 


53} 


133} 


1011.46 


3.00495 


0.26 


27.50 


0.63 


55.00 


55.00 


45 


56.25 


041 


56 


137 


996.83 


2.99862 


0.28 


28.12 


0.67 


56.25 


56.25 


50 


57.50 


42} 


68} 


140} 


982.61 


2.99238 


0.29 


28.75 


0.71 


57.50 


57.50 


55 


68.75 


43} 


1001 


144} 


968.78 


2.98623 


0.30 


29.37 


0.75 


58.75 


58.76 


600 


60.00 


45 


103 


148 


955.35 


2.98016 


0.31 


29.99 


0.79 


59.99 


60.00 


15 


63.75 


49} 


110} 


159} 


917.20 


2.96246 


0.36 


31.87 


0.91 


63.74 


63.75 


30 


67.50 


53} 


118 


2 11} 


882.00 


2.94547 


0.41 


33.74 


1.06 


67.49 


67.50 


45 


71.25 


58} 


126 


2 24} 


849.41 


2.92912 


0.46 


35.62 


1.21 


71.24 


71.25 


700 


75.00 


103 


134} 


2 37} 


819.15 


2.91336 


0.51 


37.49 


1.38 


74.98 


75.00 


15 


78.75 


108 


143} 


2 51} 


790.99 


2.89817 


0.57 


39.36 


1.56 


78.73 


78.74 


30 


82.50 


113 


152} 


3 05} 


764.72 


2.88350 


0.64 


41.24 


1.76 


82.47 


82.49 


45 


86.25 


im 


2 02 


320} 


740.15 


2.86932 


0.71 


43.11 


1.97 


86.22 


86.24 


800 


90.00 


124 


212 


3 36 


717.12 


2.85559 


0.78 


44.98 


2.20 


89.96 


89.98 


15 


93.75 


129} 


2 22} 


3 52 


695.50 


2.84230 


0.86 


46.85 


2.45 


93.70 


93.73 


30 


97.60 


135} 


233 


4 08} 


675.17 


2.82941 


0.95 


48.73 


2.71 


97.44 


97.48 


45 


101.25 


141} 


2 44 


4 25} 


656.01 


2.81691 


1.04 


50.60 


2.99 


101.18 


101.23 



y Google 



96 THE RAILROAD TAPER 

TAPER 2|. INTERMEDIATE YAUJE8— (Continued) A =2 



M 


I 


/ 


b 





D 


logZ) 


d 


t 


X 


y 


Ic 


o t 

900 


105.00 


e / 

148 


O 1 

2 55} 


o / 

4 43} 


637.91 


2.80476 


1.13 


52.47 


3.30 


104.92 


104.97 


15 


108.75 


154J 


3 07} 


5 01f 


620.81 


2.79296 


1.23 


54.34 


3.62 


108.66 


108.71 


30 


112.50 


2 01 


3 19} 


5 20} 


604.63 


2.78149 


1.34 


56.21 


3.96 


112.39 


112.46 


45 


116.25 


208 


3 32 


5 40 


589.28 


2.77032 


1.46 


58.08 


4,32 


116.12 


116.21 


10 00 


120.00 


2 16 


3 45 


600 


574.71 


2.75945 


1.67 


69.95 


4.70 


119.85 


119.96 


15 


123.75 


2 22i 


3 58} 


620} 


660.86 


2.74885 


1.69 


61.82 


5.11 


123.58 


123.69 


30 


127.50 


2 29} 


412 


6 41} 


547.69 


2.73853 


1.82 


63.68 


6.54 


127.31 


127.43 


45 


131.26 


2 37J 


4 26 


7 03} 


536.14 


2.72847 


1.96 


66.65 


6.00 


131.03 


131.17 


1100 


135.00 


2 45 


440} 


7 26} 


623.17 


2.71864 


2.10 


87.42 


6.48 


134.75 


134.91 


15 


138.75 


2 63 


4 65} 


7 48} 


511.75 


2.70906 


2.25 


69.28 


6.98 


138.47 


138.66 


30 


142.50 


3 01 


510} 


8 11} 


500.84 


2.69970 


2.41 


71.15 


7.51 


142.18 


142.39 


45 


146.25 


3 09} 


6 26 


8 35} 


490.41 


2.69056 


2.57 


73.01 


8.06 


145.89 


146.12 


12 00 


150.00 


3 18 


5 42 


900 


480.42 


2.68162 


2.74 


74.87 


8.63 


149.60 


149.85 


15 


153.75 


3 261 


6 68} 


9 25} 


470.86 


2.67289 


2.92 


76.74 


9.23 


163.30 


153.68 


30 


157.50 


3 35} 


6 16 


9 50} 


461.69 


2.66435 


3.10 


78.60 


9.86 


157.00 


157.31 


45 


161.25 


3 44i 


6 32 


10161 


452.91 


2.65601 


3.30 


80.46 


10.52 


160.69 


161.04 


13 00 


165.00 


354 


6 49} 


10 43} 


444.47 


2.64784 


3.60 


82.32 


11.20 


164.37 


164.76 


15 


168.75 


4 03} 


7 07} 


1110} 


436.37 


2.63986 


3.71 


84.17 


11.92 


168.06 


168.48 


30 


172.50 


4 13 


7 25} 


1138} 


428.59 


2.63204 


3.93 


86.03 


12.67 


171.74 


172.20 


45 


176.25 


4 23 


7 44 


12 07 


421.10 


2.62439 


4.15 


87.89 


13.45 


176.41 


176.8(5 


14 00 


180.00 


4 33 


803 


12 36 


413.90 


2.61690 


4.39 


89.73 


14.25 


179.07 


179.64 


15 


183.75 


4 43} 


8 22} 


13 06} 


406.96 


2.60956 


4.63 


91.59 


16.09 


182.73 


183.35 


30 


187.50 


4 53} 


8 42 


13 36} 


400.29 


2.60237 


4.88 


93.44 


15.96 


186.38 


187.06 


45 


191.25 


5 04i 


9 02 


14 06} 


393.86 


2.59634 


5.14 


96.29 


16.86 


190.02 


190 76 


15 00 


195.00 


5 15 


9 32} 


14 37} 


387.66 


2.68845 


5.41 


97.14 


17.79 


193.66 


194.46 


30 


202.50 


5 37} 


10 04} 


15 41} 


375.90 


2.57507 


6.97 


100.82 


19.76 


200.89 


201.86 


16 00 


210.00 


600 


10 48 


16 48 


364.96 


2.56226 


6.57 


104.60 


21.87 


208.08 


209.24 


30 


217.50 


6 23} 


1133 


17 56} 


354.76 


2.54993 


7.21 


108.18 


24.12 


216.26 


216.60 


17 00 


225.00 


6 48 


12 19} 


19 07} 


345.23 


2.63811 


7.89 


111.86 


26.62 


222.37 


223.94 


30 


232.50 


713 


13 07} 


20 20} 


336.33 


2.52677 


8.61 


115.49 


29.07 


229.43 


231.26 


18 00 


240.00 


7 39 


13 67 


2136 


328.02 


2.61590 


9.38 


119.12 


31.77 


236.44 


238.54 


30 


247.50 


8 05} 


14 48 


22 53} 


320.22 


2.50545 


10.18 


122.76 


34.63 


243.38 


245.83 


19 00 


255.00 


8 38 


15 40} 


24 13} 


312.94 


2.49546 


11.03 


126.38 


37.64 


250.26 


253 07 


30 


262.50 


9 01 


16 34} 


25 35} 


306 10 


2.48586 


11.92 


129.98 


40.81 


257.06 


260.27 


20 00 


270.00 


9 30 


17 30 


27 00 


299.71 


2.47670 


12.87 


133.54 


44.13 


263.76 


267.43 



y Google 



TABLES 



97 



TAPER 3. CURVATURE INCREASING 2** 30' 

EVERY 30 FT. A=2i 



Transit 
at 








Deflections to 








B.C. 
2" 30' 


C.C. 

5-00' 


C.C. 

7" 30' 


C.C. 

lO'OO' 


C.C. 

12" 30' 


C.C. 
15*00' 


C.C. 

17«30' 


C.C. 

20*00' 


C.C. 

22*30' 


I 





30 


60 


90 


120 


150 


180 


210 


240 



30 
60 
90 
120 
150 
180 
210 
240 


e r 

22} 

1 18! 

2 45 
4 4U 
7 07i 

10 03! 
13 30 
17 26! 


e / 

22i 

45 

2 03! 

3 52i 
6 11! 
900 

12 18! 
16 07i 


o / 

56! 
45 

107i 
2 48! 
500 
7 41! 
10 52i 
14 33! 


145 
141! 

107i 

130 
3 33! 
6 07J 
911! 
12 45 


2 48! 
2 52i 
2 261 
130 

152h 
4 18! 
7 15 
10 41! 


4 07i 

4 18! 
400 
3 11! 
152i 

2 15 

5 03! 
8 22i 


/ 

5 41! 
600 
5 48! 
5 07i 
3 56! 
2 15 

2 37§ 
5 48} 


e / 

7 30 
7 66! 
7 52i 
7 18! 
6 15 
4 41! 
2 37i 

300 


o / 

9 33! 
10 07i 
10 11! 
9 45 
8 48! 
7 22i 
5 26! 
300 



M 


I 


o 


D 


logD 


d 


t 


X 


y 


Ic 


/ 

500 


30 


o / 

45 


1146.11 


3.05923 


0.10 


15.00 


0.20 


30.00 


30.00 


7 30 


60 


2 15 


764.47 


2.88336 


0.39 


29.99 


0.98 


59.99 


60.00 


10 00 


90 


430 


574.12 


2.75900 


0.98 


44.97 


2.75 


89.94 


89.98 


12 30 


120 


730 


460.55 


2.66328 


1.96 


59.91 


5.88 


119.77 


119.91 


15 00 


150 


11 15 


385.68 


2.58623 


3.43 


74.80 


10.77 


149.37 


149.76 


17 30 


180 


15 45 


333.21 


2.52270 


5.48 


89.59 


17.78 


178.54 


179.41 


20 00 


210 


2100 


295.04 


2.46988 


8.20 


104.22 


27.26 


207.01 


208.78 


22 30 


240 


27 00 


266.74 


2.42609 


11.69 


118.63 


39.49 


234.42 


237.72 



y Google 



98 THE RAILROAD TAPER 

TAPER 3. INTERMEDIATE VALUES 



A=2§ 



M 


I 


/ 


b 





D 


logD 


d 


t 


X 


V 


Ic 


600 


30.00 


e / 

224 


22} 


O 1 

46 


1146.11 


3.06923 


0.10 


16.00 


0.20 


30.00 


30.00 


06 


31.00 


23} 


23} 


47} 


1127.30 


3.06204 


0.10 


16.60 


0.21 


81.00 


31.00 


10 


32.00 


24i 


26} 


49} 


1109.15 


3.04499 


0.11 


16. oa 


0.22 


32.00 


32.00 


15 


33.00 


26J 


26} 


62 


1091.56 


3.03806 


0.12 


16.60 


0.24 


33.00 


33.00 


20 


34.00 


26i 


28} 


64} 


1074.61 


3.03121 


0.12 


17.00 


0.26 


34.00 


34.00 


26 


36.00 


27 


030 


67 


1068.00 


3.02449 


0.13 


17.60 


0.28 


36.00 


35.00 


30 


36.00 


028 


31} 


59} 


1041.98 


3.01786 


0.1> 


18.00 


0.29 


36.00 


36.00 


36 


37.00 


029 


033 


102 


1026.44 


3.01133 


0.16 


18.60 


0.31 


37.00 


37.00 


40 


38.00 


30 


34} 


104} 


1011.36 


3.00490 


0.16 


19.00 


0.33 


38.00 


38.00 


46 


39.00 


31 


36} 


107} 


996.71 


2.99867 


0.16 


19.60 


0.35 


39.00 


39.00 


60 


40.00 


32 


38 


110 


982.49 


2.99233 


0.17 


20.00 


0.37 


40.00 


40.00 


66 


41.00 


33i 


39} 


112} 


968.66 


2.98617 


0.18 


20.60 


0.40 


41.00 


41.00 


600 


42.00 


34i 


41} 


116} 


966.23 


2.98011 


0.19 


21.00 


0.42 


42.00 


42.00 


16 


46.00 


37i 


46} 


124} 


917.06 


2.96239 


0.21 


22.60 


0.49 


46.00 


45.00 


30 


48.00 


41 


62} 


133} 


881.83 


2.94638 


0.24 


24.00 


0.57 


48.00 


48.00 


46 


61.00 


44} 


58} 


143} 


849.23 


2.92903 


0.28 


26.60 


0.66 


51.00 


51.00 


700 


64.00 


48i 


106} 


163} 


818.96 


2.91326 


0.31 


26.99 


0.76 


53.99 


64.00 


16 


67.00 


621 


111} 


204 


790.77 


2.89806 


0.36 


28.49 


0.87 


56.99 


67.00 


30 


60.00 


66i 


118} 


2 16 


764.47 


2.88336 


0.39 


29.99 


0.98 


69.99 


60.00 


46 


63.00 


lOOi 


126 


2 26} 


739.88 


2.86916 


0.44 


31.49 


1.11 


62.99 


63.00 


800 


66.00 


104f 


133} 


2 38} 


716.82 


2.86641 


0.48 


32.99 


1.24 


66.98 


66.00 


16 


69.00 


109J 


141} 


2 60} 


696.17 


2.84209 


0.53 


34.49 


1.39 


68.98 


69.00 


30 


72.00 


114 


149} 


3 03} 


674.81 


2.82918 


0.59 


36.99 


1.65 


71.98 


71.99 


46 


75.00 


118i 


168 


3 16} 


665.61 


2.81666 


0.64 


37.48 


1.72 


74.97 


74.99 


900 


78.00 


123} 


2 06} 


3 30} 


637.48 


2.80447 


0.70 


38.98 


1.90 


77.97 


77.99 


16 


81.00 


128i 


2 16 


3 44} 


620.36 


2.79264 


0.77 


40.48 


2.09 


80.96 


80.99 


30 


84.00 


134 


2 26} 


3 59} 


604.13 


2.78113 


0.84 


41.98 


2.30 


83.96 


83.99 


46 


87.00 


139} 


2 36 


4 14} 


688.74 


2.76992 


0.91 


43.47 


2.62 


86.96 


86.98 


10 00 


90.00 


146 


2 46 


4 30 


574.12 


2.76900 


0.98 


44.97 


2.75 


89.94 


89.98 


16 


93.00 


160} 


2 66} 


4 46 


660.23 


2.74837 


1.06 


46.47 


2.99 


92.93 


92.98 


30 


96.00 


166} 


306 


5 02} 


647.01 


2.73800 


1.14 


47.96 


3.26 


96.92 


95.97 


46 


99.00 


2 02} 


3 16} 


619} 


634.41 


2.72787 


1.23 


49.46 


3.63 


98.90 


98.96 


1100 


102.00 


2 08} 


3 27} 


6 36} 


622.39 


2.71799 


1.32 


60.96 


3.82 


101.89 


101.96 


16 


106.00 


2 16 


3 39} 


6 64} 


610.92 


2.70835 


1.42 


62.45 


4.12 


104.87 


104.96 


30 


108.00 


2 21} 


3 61 


612} 


499.96 


2.69893 


1.62 


53.94 


4.44 


107.86 


107.96 


46 


111.00 


2 28 


4 03} 


6 31} 


489.46 


2.68972 


1.62 


55.44 


4.78 


110.84 


110.94 



y Google 



TABLES 99 

TAPER 3. INTERMEDIATE YAUJES— (Continued) A=2i 



Af 


/ 


/ 


b 


o 


D 


Iogi> 


d 


t 


X 


V 


le 


e » 

12 00 


114.00 


2 34} 


o / 

4 15} 


6 50} 


479.41 


2.68071 


1.73 


56.93 


5.13 


113.82 


113.93 


15 


117.00 


2 41} 


4 28} 


7 10 


469.78 


2.67189 


1.84 


58.42 


5.60 


116.80 


116.92 


30 


120.00 


2 48} 


4 41} 


730 


460.55 


2.66328 


1.96 


59.91 


5.88 


119.77 


119.91 


45 


123.00 


2 66 


4 54} 


750} 


451.69 


2.65484 


2.08 


61.41 


6.29 


122.76 


122.90 


13 00 


126.00 


3 03} 


5 08} 


811} 


443.18 


2.64658 


2.21 


62.90 


6.71 


126.72 


125.89 


15 


129.00 


3 10} 


5 22 


8 32} 


435.00 


2.63849 


2.34 


64.39 


7.15 


128.69 


128.88 


30 


132.00 


sm 


536 


7 54} 


427.14 


2.63057 


2.48 


65.88 


7.61 


131.65 


131.87 


45 


135.00 


3 26} 


5 50} 


916} 


419.58 


2.62281 


2.63 


67.37 


8.09 


134.61 


134.86 


14 00 


138.00 


3 34} 


6 05} 


9 39} 


412.29 


2.61520 


2.78 


68.86 


8.58 


137.57 


137.84 


15 


141.00 


3 42} 


620} 


10 02} 


405.26 


2.607/3 


2.93 


70.34 


9.10 


140.53 


140.82 


30 


144.00 


350i 


6 36 


10 26} 


398.50 


2.60043 


3.09 


71.83 


9.64 


143.48 


143.80 


45 


147.00 


3 59 


6 51} 


10 50} 


391.98 


2.59326 


3.26 


73.32 


10.19 


146.43 


146.78 


15 00 


150.00 


4 07i 


7 07} 


1115 


385.68 


2.58623 


3.43 


74.80 


10.77 


149.37 


149.76 


30 


156.00 


425 


7 40} 


12 05} 


373.72 


2.57265 


3.79 


77.77 


11.99 


155.25 


166.71 


16 00 


162.00 


4 43} 


8 14} 


12 57} 


362.56 


2.55938 


4.17 


80.73 


13.30 


161.11 


161.65 


30 


168.00 


5 02 


8 49} 


13 51} 


352.13 


2.54670 


4.58 


83.69 


14.70 


166.94 


167.58 


17 00 


174.00 


5 21} 


9 26} 


14 47} 


342.36 


2.53448 


5.02 


86.64 


16.19 


172.75 


173.50 


30 


180.00 


5 41} 


10 03} 


15 45 


333.20 


2.52270 


5.48 


89.58 


17.78 


178.54 


179.41 


18 00 


186.00 


6 01} 


10 42} 


16 44} 


324.61 


2.51136 


5.97 


92.62 


19.47 


184.30 


185.31 


30 


192.00 


623 


1122} 


17 45} 


316.53 


2.50041 


6.49 


95.46 


21.26 


190.03 


191.20. 


19 00 


198.00 


6 44} 


12 03} 


18 48} 


308.94 


2.48987 


7.03 


98.39 


23.15 


195.73 


197.07 


30 


204.00 


7 07 


12 46} 


19 53} 


301.78 


2.47969 


7.60 


101.31 


25.14 


201.39 


202.93 


20 00 


210.00 


7 30 


13 30 


2100 


295.04 


2.46988 


8.20 


104.22 


27.25 


207.01 


208.78 


30 


216.00 


7 53i 


14 15 


22 08} 


288.69 


2.46043 


8.83 


107.12 


29.47 


212.59 


214.62 


2100 


222.00 


817} 


15 00} 


23 18} 


282.72 


2.45136 


9.50 


110.02 


31.80 


218.13 


220.44 


30 


228.00 


8 42} 


15 48 


24 30} 


277.07 


2.44259 


10.19 


112.90 


34.25 


223.62 


226.23 


22 00 


234.00 


9 07} 


16 36} 


25 44} 


271.76 


2.43417 


10.92 


115.77 


36.81 


229.05 


231.99 


30 


240.00 


9 33} 


17 26} 


27 00 


266.74 


2.42609 


11.69 


118.63 


39.49 


234.42 


237.72 



y Google 



1 00 THE RAILROAD TAPER 

TAPER 3i. CURVATURE INCREASING 3** 00' 

EVERY 30 FT. A = 3 



Transit 
at 


Deflections to 


B.C. 

3'00' 


C.C. 

6*00' 


C.C. 

9*00' 


C.C. 

12*00' 


C.C. 

15-00' 


C.C. 

18" 00' 


C.C. 

21** 00' 


C.C. 

24*00' 


I 





30 


60 


90 


120 


150 


180 


210 



30 
60 
90 
120 
150 
180 
210 


o / 

27* 
134i 
3 18 
5 37^ 
8 33 
12 04J 
16 12 


27 

54' 
2 28i 
4 39 
7 25i 
10 48 
14 46J 


107J 
54 

'l i2i' 

3 22i 
600 
9 13i 
13 03 


206 
2 01i 
121 

'148' 

4 16i 

7 21 

llOli 


o / 

322i 
3 27 
2 551 
148 

2 is* 

5 10i 
8 42 


e / 

4 57 

5 10i 
4 48 
3 49J 
2 15 

2^" 

6 04i 


o t 

6 49J 

7 12 
6 58i 
609 
4 43i 
242 

3 09' 


o / 

900 
9 31 J 
927 
8 46i 
7 30 
5 37J 
3 09 



M 


I 


o 


D 


logD 


d 


t 


X 


y 


Ic 


e / 

600 


30 


o / 

54 


955.16 


2.98008 


0.12 


15.00 


0.24 


30.00 


30.00 


900 


60 


242 


637.25 


2.80431 


0.47 


29.99 


1.18 


59.98 


60.00 


12 00 


90 


5 24 


478.84 


2.68021 


1.18 


44.96 


3.30 


89.91 


89.97 


15 00 


120 


900 


384.60 


2.58501 


2.35 


59.88 


7.06 


119.67 


119.88 


18 00 


150 


13 30 


322.75 


2.50887 


4.11 


74.71 


12.91 


149.10 


149.65 


2100 


180 


18 54 


279.78 


2.44682 


6.56 


89.40 


21.29 


177.91 


179.17 


24 00 


210 


25 12 


248.99 


2.39618 


9.82 


103.88 


32.58 


205.71 


208.28 



y Google 



TABLES 



101 



TAPER 3i. INTERMEDIATE VALUES A = 3 



M 


I 


/ 


b 





D 


IogZ> 


d 


t 


X 


y 


Ic 


O 1 

600 


30.00 


/ 

27 


f 

27 


' 

064 


955.16 


2.98008 


0.12 


15.00 


0.24 


30.00 


30.00 


16 


32.50 


291 


31} 


101 


916.98 


2.96236 


0.14 


16.26 


0.28 


32.50 


32.50 


30 


35.00 


32i 


36J 


108} 


881.76 


2.94635 


0.16 


17.60 


0.33 


35.00 


35.00 


45 


37.60 


35} 


40} 


116 


849.13 


2.92897 


0.18 


18.75 


0.39 


37.50 


37.60 


700 


40.00 


38i 


45} 


124 


818.84 


2.91320 


0.20 


20.00 


0.45 


40.00 


40.00 


15 


42.50 


411 


50} 


132} 


790.66 


2.89798 


0.23 


21.25 


0.52 


42.60 


42.60 


30 


45.00 


46 


66} 


141} 


764.34 


2.88329 


0.26 


22.49 


0.69 


46.00 


45.00 


45 


47.50 


48} 


102 


160} 


739.73 


2.86907 


0.29 


23.74 


0.67 


47.49 


47.60 


800 


50.00 


52 


i08 


200 


716.66 


2.85531 


0.32 


24.99 


0.76 


49.99 


60.00 


15 


52.60 


661 


114} 


• 
2 10 


694.99 


2.84198 


0.35 


26.24 


0.85 


62.49 


62.50 


30 


55.00 


69} 


120} 


2 20} 


674.61 


2.82905 


0.39 


27.49 


0.96 


54.99 


55.00 


45 


57.50 


103} 


127} 


2 31 


666.40 


2.81661 


0.43 


28.74 


1.06 


57.49 


67.60 


900 


60.00 


107} 


134} 


242 


637.26 


2.80431 


0.47 


29.99 


1.18 


69.98 


60.00 


15 


62.60 


llU 


141} 


2 63} 


620.09 


2.79246 


0.61 


31.24 


1.31 


62.48 


62.50 


30 


65.00 


116 


149} 


3 06} 


603.85 


2.78093 


0.66 


32.48 


1.44 


64.98 


64.99 


45 


67.50 


120} 


167 


3 17} 


688.44 


2.76970 


0.61 


33.73 


1.68 


67.48 


67.49 


10 00 


70.00 


126 


206 


3 30 


573.80 


2.75876 


0.66 


34.98 


1.73 


69.97 


69.99 


15 


72.60 


129i 


2 13} 


3 43 


659.88 


2.74810 


0.71 


36.23 


1.89 


72.47 


72.49 


30 


75.00 


134} 


2 21} 


3 66} 


546.64 


2.73770 


0.77 


37.48 


2.06 


74.96 


74.99 


45 


77.50 


139} 


2 30} 


4 10 


534.01 


2.72755 


0.83 


38.73 


2.24 


77.45 


77.49 


1100 


80.00 


144} 


2 39} 


4 24 


521.96 


2.71764 


0.89 


39.97 


2.43 


79.95 


79.98 


15 


82.50 


149i 


2 48} 


4 38} 


610.46 


2.70796 


0.96 


41.22 


2.63 


82.44 


82.48 


30 


85.00 


156 


2 58} 


4 63} 


499.46 


2.69850 


1.03 


42.46 


2.84 


84.93 


84.98 


45 


87.60 


2 00} 


3 08 


5 08} 


488.94 


2.68926 


1.10 


43.71 


3.06 


'87.42 


87.47 


12 00 


90.00 


206 


3 18 


524 


478.86 


2.68021 


1.18 


44.96 


3.29 


89.91 


89.97 


15 


92.60 


2 Hi 


3 28} 


5 40 


469.20 


2.67136 


1.26 


46.20 


3.54 


92.40 


92.47 


30 


95.00 


2 17} 


3 38} 


5 56} 


459.93 


2.66269 


1.34 


47.45 


3.80 


94.89 


94.96 


45 


97.50 


2 23} 


3 49} 


6 13 


461.03 


2.66421 


1.42 


48.69 


4.07 


97.37 


97.46 



y Google 



102 



THE RAILROAD TAPER 



TAPER 3i. INTERMEDIATE 


VALUES 


— {Continued) 


A'=3 


M 


I 


/ 


6 





D 


logZ> 


d 


t 


X 


y 


Ic 


13 00 


100.00 


e / 

229i 


400} 


e / 

6 30 


442.48 


2.64589 


1.51 


49.94 


4.35 


99.86 


99.95 


15 


102.60 


2 35! 


4 1U 


6 47} 


434.26 


2.63776 


1.60 


61.18 


4.64 


102.35 


102.45 


30 


105.00 


2 42 


4 23} 


7 05} 


426.36 


2.62978 


1.70 


52.42 


4.94 


104.82 


104.94 


45 


107.50 


2 48} 


4 36 


7 23} 


418.75 


2.62195 


1.80 


63.67 


5.26 


107.30 


107.43 


14 00 


110.00 


2 55 


4 47 


7 42 


411.41 


2.61427 


1.90 


54.91 


6.69 


109.78 


109.92 


15 


112.50 


3 01i 


4 59} 


8 01 


404.34 


2.60675 


2.01 


56.15 


6.94 


112.26 


112.41 


30 


115.00 


3 08} 


5 11} 


8 20} 


397.53 


2.59937 


2.12 


57.39 


6.30 


114.73 


114.90 


45 


117.60 


315} 


5 24} 


840 


390.95 


2.59212 


2.23 


68.64 


6.67 


117.20 


117.39 


16 00 


120.00 


3 22} 


5 37} 


900 


384.60 


2.58601 


2.35 


59.88 


7.06 


119.67 


119.88 


30 


125.00 


3 37 


6 04} 


9 41} 


372.53 


2.57116 


2.60 


62.36 


7.88 


124.61 


124.85 


16 00 


130.00 


3 52 


6 32 


10 24 


361.26 


2.55782 


2.87 


64.84 


8.76 


129.64 


129.82 


30 


135.00 


4 07} 


7 00} 


1108} 


360.70 


2.54494 


3.15 


67.31 


9.70 


134.45 


134.79 


17 00 


140.00 


4 23} 


7 30} 


1154 


340.79 


2.53249 


3.45 


69.78 


10.70 


139.34 


139.75 


30 


145.00 


440 


8 01} 


12 41} 


331.49 


2.62047 


3.77 


72.25 


11.77 


144.22 


144.70 


18 00 


150.00 


4 57 


833 


13 30 


322.75 


2.60887 


4.11 


74.71 


12.91 


149.10 


149.65 


30 


155.00 


5 14} 


9 05} 


14 20} 


314.51 


2.49763 


4.47 


77.17 


14.12 


153.96 


164.69 


19 00 


160.00 


5 32} 


9 39} 


15 12 


306.76 


2.48678 


4.84 


79.63 


16.40 


168.78 


169.52 


30 


165.00 


5 51 


10 14} 


16 05} 


299.42 


2.47628 


5.24 


82.08 


16.76 


163.59 


164.44 


20 00 


170.00 


6 10 


10 50 


17 00 


292.60 


2.46613 


5.66 


84.52 


18.19 


168.39 


169.36 


30 


175.00 


6 29} 


1126} 


17 66} 


285.96 


2.46631 


6.10 


86.96 


19.70 


173.16 


174.27 


2100 


180.00 


6 49} 


12 04} 


18 64 


279.78 


2.44682 


6.56 


89.40 


21.29 


177.91 


179.17 


30 


185.00 710 


12 43} 


19 63} 


273.92 


2.43762 


7.04 


91.83 


22.96 


182.62 


184.06 


22 00 


190.00 7 31 


13 23 


20 64 


268.38 


2.42875 


7.56 


94.26 


24.71 


187.30 


188.93 


30 


195.00 7 62} 


14 03} 


2156} 


263.13 


2.42017 


8.08 


96.67 


26.55 


191.95 


193.79 


23 00 


200.00 8 14} 


14 45} 


23 00 


268.17 2.41191 


8.64 


99.08 


28.47 


196.59 


198.63 


24 00 


210.00; 9 00 

1 


1612 


25 12 


248.99 2.39618 


9.82 


103.88 


32.68 


205.71 


208.28 



y Google 



TABLES 



103 



TAPER 3i. CURVATURE INCREASING 4'* 00' 

EVERY 30 FT. A=4 









DeflectionB to 








Transit 
at 
















B.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 




4*'00' 


8'00' 


12- 00' 


16*00' 


20*00' 


24*00' 


28*00' 


/ 





30 


60 


90 


120 


150 


180 




e / 


/ 


o t 


o t 


e f 


o / 


1 







36 


130 


248 


430 


6 36 


906 


30 


36 




1 12 


242 


436 


654 


9 36 


60 


206 


1 12 




148 


3 54 


6 24 


9 18 


90 


424 


3 18 


148 




224 


606 


8 12 


120 


730 


6 12 


430 


224 




300 


6 18 


150 


1124 


9 54 


800 


5 42 


300 




3 36 


180 


16 06 


14 24 


12 18 


9 48 


654 


336 





AT 


I 


o 


D 


logD 


d 


t 


X 


y 


U 


O 1 

800 


30 


e 1 

112 


716.50 


2.85522 


0.16 


15.00 


0.31 


30.00 


30.00 


12 00 


60 


3 36 


478.31 


2.67971 


0.63 


29.98 


1.57 


59.97 


59.99 


16 00 


90 


7 12 


359.96 


2.55625 


1.57 


44.92 


4.39 


89.84 


89.95 


20 00 


120 


12 00 


289.97 


2.46235 


3.13 


59.78 


9.40 


119.42 


119.79 


24 00 


150 


18 00 


244.63 


2.38851 


5.46 


74.49 


17.17 


148.40 


149.39 


28 00 


180 


25 12 


213.85 


2.33011 


8.71 


88.95 


28.24 


176.29 


178.52 



y Google 



104 THE RAILROAD TAPER 

TAPER 3i. INTERMEDIATE ^VALUES A = 4 



M 


I 


/ 


h 





D 


logD 


d 


t 


X 


V 


U 


e / 

800 


30.00 


36 


e 1 

36 


1 

112 


716.50 


2.85522 


0.16 


15.00 


0.31 


30.00 


30.00 


16 


31.88 


381 


040 


1 18} 


694.82 


2.84187 0.18 


15.93 


0.36 


31.87 


31.87 


30 


33.75 


4U 


44} 


126 


674.42 


2.828931 0.20 


16.87 


0.41 


33.74 


33.75 


45 


35.62 


44^ 


49i 


133} 


655.19 


2.81637 0.22 


17.81 


0.46 


35.62 


35.62 


900 


37.50 


47i 


054 


141} 


637.02 


2.80416' 0.24 


18.74 


0.52 


37.50 


37.50 


15 


39.38 


50J 


59 


149} 


619.84 


2.79228' 0.26 


19.68 


0.58 


39.37 


39.37 


30 


41.25 


53i 


104 


157} 


603.58 


2.78073' 0.29 


20.62 


0.64 


41.24 


41.25 


45 


43.12 


56} 


109} 


2 06} 


688.16 


2.76949 


0.32 


21.55 


0.71 43.12 


43.12 


10 00 


46.00 


100 


115 


2 15 


573.49 


2.75853 


0.36 


22.49 


0.79 44.99 


45.00 


15 


46.88 


103} 


1201 


2 24} 


569.55 


2.74784 


0.38 


23.43 


1 
0.87 46.87 


46.87 


30 


48.75 


107 


126} 


2 33} 


546.28 


2.73742 0.41 


24.36 


0.95 48.74 


48.75 


45 


50.62 


llOi 


132} 


2 43} 


533.62 


2.72723 0.44 


25.30 


1.04 50.61 


50.62 


1100 


52.50 


1141 


139 


2 53} 


521.54 


2.71729 


0.47 


26.24 


1.13 52.48 


52.50 


15 


54.38 


118 


145} 


3 03} 


510.01 


2.70758 0.51 


27.17 


1.23 64.35 54.37 


30 


56.25 


122 


152 


3 14 


498.98 


2.69808' 0.56 


28.11 


1.34 56.22J 56.24 


45 


68.12 


126 


158} 


3 24} 


488.43 


2.68880 


0.59 


29.04 


1.45 58.10; 58.12 


12 00 


60.00 


130 


206 


3 36 


478.31 


2.67971 


0.63 


29.98 


1.57 


59.97 59.90 


15 


61.88 


\U\ 


2 13} 


3 47} 


468.61 


2.67081 


0.67 


30.91 


1.69 


61.84 61.87 


30 


63.75 


\m 


2 20} 


3 59 


459.31 


2.66211 


0.72 


31.85 


1.82 


63.71 63.74 


45 


65.62 


142J 


2 28} 


4 11 


450.38 


2.65358 


0.77 


32.79 


1.96 


65.59 65.62 


13 00 


67.50 


147i 


2 36 


4 23} 


441.79 


2.64522 


0.82 


33.72 


2.11 


67.46 67.49 


15 


69.38 


15U 


244 


4 36} 


433.53 


2.63702 


0.87 


34.65 


2.26 


69.33 69.36 


30 


71.25 


156i 


2 52 


4 48} 


425.68 


2.62898 


0.92 


35.59 


2.42 


71.19 71.23 


45 


73.12 


2 0U 


3 00} 


5 01} 


417.92 


2.62109 


0.97 


36.52 


2.58 


73.061 73.11 


14 00 


75.00 


206 


309 


5 15 


410.54 


2.61336 


1.03 


37.46 


2.75 


74.93 


74.98 


15 


76.88 


2 11 


3 17} 


5 28} 


403.42 


2.60576 


1.09 


38.39 


2.93 


76.79 


76.86 


30 


78.75 


216 


3 26} 


5 42} 


396.56 


2.59831 


1.16 


39.33 


3.12 


78.66 


78.72 


45 


80.62 


2 21 


3 35} 


6 56} 


389.94 


2.59100 


1.22 


40.26 


3.31 


80.53 


80.59 


15 00 


82.50 


2 26i 


3 45 


6 11} 


383.53 


2.58380 


1.28 


41.19 


3.61 


82.39 


82.46 


30 


86.25 


2 37 


404 


6"41 


371.35 


2.56978 


1.42 


43.06 


3.93 


86.12 


86.21 


16 00 


90.00 


2 48 


4 24 


7 12 


359.96 


2.55625 


1.67 


44.92 


4.39 


89.84 


89.96 


30 


93.76 


2 59} 


4 44} 


7 44 


349.28 


2.54317 


1.73 


46.79 


4.89 


93.66 


93.69 


17 00 


97.60 


3 Hi 


506 


8 17} 


339.23 


2.63049 


1.89 


48.65 


5.42 


97.27 


97.43 


30 


101.25 


3 23} 


5 28 


8 51} 


329.79 


2.51824 


2.07 


50.61 


5.98 100.98 


101.16 


18 00 


105.00 


3 36 


6 51 


9 27 


320.90 


2.50637 


2.26 


52.37 


6.59 404.68 


104.89 


30 


108.75 


3 49 


614} 


10 03} 


312.50 


2.49485 


2.46 


54.22 


7.23 |108.38 


108.82 


19 00 


112.50 


4 02i 


6 39 


10 41} 


304.58 


2.48370 


2.67 


56.08 


7.91 112.07 


112.34 


30 


116.25 


4 16 


704 


1120 


297.07 


2.47286 


2.89 


57.93 


8.63 '115.76 


116.07 


20 00 


120.00 


4 30 


7 30 


12 00 


289.97 


2.46236 


3.13 


59.78 


9.40 119.42 


119.79 


30 


123.75 


4 44} 


7 56} 


12 41 


283.24 


2.45216 


3.38 


61.63 


10.21 


123.08 


123.51 


2100 


127.50 


4 59i 


8 24 


13 23} 


276.86 


2.44226 


3.64 


63.48 


11.06 


126.74 


127.22 


30 


131.25 


6 14} 


8 52 


14 06} 


270.79 


2.43263 


3.91 


65.32 


11.96 


130.38 


130.98 



Digitized by VjOOQIC 



TABLES 105 

TAPER 3i. INTERMEDIATE VALUES ^ (Con^inwed) A = 4 



M 


I 


/ 


6 





D 


\ogD 


d 


t 


X 


V 


Ic 


22 00 


135.00 


5 30 


o / 

9 21 


14 51 


265.02 


2,42328 


4.19 


67.16 


12.90 


134.01 


134.63 


30 


138.76 


5 46 


9 50i 


15 36} 


259.53 


2.41419 


4.48 


69.00 


13.90 


137.63 


138.33 


23 00 


142.50 


6 02i 


10 21 


16 23i 


254.33 


2.40540 


4.80 


70.83 


14.94 


141.23 


142.02 


24 00 


150.00 


6 36 


1124 


18 00 


244.63 


2.38851 


5.46 


74.49 


17.17 


148.40 


149.39 


25 00 


157.50 


7 1U 


12 30 


19 41i 


235.82 


2.37258 


6.18 


78.14 


19.61 


156.50 


156.73 


26 00 


165.00 


7 48 


13 39 


2127 


227.81 


2.35757 


6.97 


81.76 


22.26 


162.52 


164.04 


27 00 


172.50 


8 26i 


14 51 


23 17i 


220.61 


2.34343 


7.81 


85.37 


25.14 


169.45 


171.31 


28 00 


180.00 


906 


16 06 


25 12 


213.85 


2.33011 


8.71 


88.95 


28.24 


176.29 


178.62 



TAPER 3f. CURVATURE INCREASING 5° 00' 

EVERY 30 FT. A = 5 





Deflections to 


at 


B.C. 
5'00' 


C.C. 

10*00' 


C.C. 

15-00' 


C.C. 

20-00' 


C.C. 

25-00' 


C.C. 

30-00' 


I 


. 


30 


60 


90 


120 


150 




30 

60 

90 

120 

150 


'045* 
2 37i 
5 30 
9 22 J 

14 15 


45 

"isd* 

4 07i 

745 

12 22i 


152i 
130 

"215 ■ 
5 37i 
10 00 


e / 

3 30 
3 22i 
2 15 

"sod* 

7 07J 


a / 

5 37J 
5 45 
4 52J 
300 

'345' 


e # 

8 15 
8 37i 
800 
6 22i 
3 45 



a 


I 





I> 


logD 


d 


t 


X 


V 


U 


10 00 


30 


/ 
130 


573.34 


2.75841 


0.20 


14.99 


0.39 


30.00 


30.00 


15 00 


60 


4 30 


383.13 


2.58323 


0.78 


29.97 


1.96 


59.96 


59.99 


20 00 


90 


900 


288.80 


2.45939 


1.96 


44.88 


5.49 


89.75 


89.92 


25 00 


120 


15 00 


233.54 


2.36836 


3.90 


59.66 


11.73 


119.09 


119.67 


30 00 


150 


22 30 


198.34 


2.29741 


6.81 


74.21 


21.39 


147.50 


149.04 



y Google 



106 THE RAILROAD TAPER 

TAPER 3i. INTERMEDIATE VALUES A=5 



M 


I 


/ 


b 





D 


logD 


d 


t 


z 


V 


U 


• 1 
10 00 


30.00 


o / 

045 


o / 

45 


a 1 

130 


573.34 


2.75841 


0.20 


14.99 


0.89 


30.00 


30.00 


15 


31.50 


47i 


49 


136} 


559.38 


2.74771 


0.21 


15.74 


0.43 


31.50 


31.50 


30 


33.00 


050i 


53} 


144 


546.10 


2.73727 


0.23 


16.49 


0.48 


33.00 


33.00 


45 


34.50 


53i 


58 


151} 


533.43 


2.72708 


0.25 


17.24 


0.53 


34.50 


34.50 


1100 


36.00 


056 


102} 


158} 


521.34 


2.71712 


0.27 


17.99 


0.59 


36.00 


36.00 


15 


37.50 


59i 


107i 


2 06} 


509.79 


2.70739 


0.29 


18.74 


0.65 


37.49 


37.50 


30 


39.00 


102 


1 12} 


2 14} 


498.74 


2.69787 


0.31 


19.49 


0.71 


38.99 


39.00 


45 


40.50 


105i 


117} 


2 22} 


488.18 


2.68858 


0.34 


20.24 


0.77 


40.49 


40.50 


12 00 


42.00 


108J 


122} 


2 31} 


478.05 


2.67947 


0.37 


20.99 


0.84 


41.99 


42.00 


15 


43.50 


llli 


128 


2 39} 


468.34 


2.67056 


0.40 


21.74 


0.91 


43.49 


43.50 


30 


45.00 


115 


133} 


2 48} 


459.02 


2.66183 


0.43 


22.49 


0.98 


44.99 


45.00 


45 


46.50 


118} 


139} 


2 57} 


450.07 


2.65328 


0.46 


23.23 


1.06 


46.49 


46.50 


13 00 


48.00 


122 


145} 


3 07} 


441.46 


2.64489 


0.49 


23.98 


1.14 


47.98 


48.00 


15 


49.50 


125i 


151} 


3 16} 


433.18 


2.63667 


0.52 


24.73 


1.23 


49.48 


49.50 


30 


51.00 


129 


157} 


3 26} 


425.21 


2.62860 


0.55 


25.48 


1.32 


50.98 


51.00 


45 


52.50 


1321 


2 03} 


3 36} 


417.53 


2.62069 


0.58 


26.23 


1.42 


52.48 


52.50 


14 00 


54.00 


136} 


2 10} 


3 46} 


410.13 


2.61292 


0.62 


26.98 


1.52 


53.97 


63.99 


15 


55.50 


140} 


2 16} 


3 57} 


402.99 


2.60529 


0.66 


27.72 


1.62 


66.47 


55.49 


30 


57,00 


144i 


2 23} 


408 


396.11 


2.59782 


0.70 


28.47 


1.73 


56.97 


56.99 


45 


58.50 


148i 


2 30} 


4 18} 


389.46 


2.59046 


0.74 


29.22 


1.84 


58.46 


58.49 


15 00 


60.00 


152} 


2 37} 


4 30 


383.03 


2.58323 


0.78 


29.^7 


1.96 


59.96 


59.99 


30 


63.00 


2 01 


2 52 


453 


370.80 


2.56914 


0.87 


31.46 


2.21 


62.95 


62.99 



y Google 



TABLES 107 

TAPER 3i. INTERMEDIATE VALUES— (Continued) A =5 



M 


I 


/ 


b 





D 


logD 


d 


t 


X 


V 


Ic 


16 00 


66.00 


e / 

2 091 


e / 

3 07 


5 16} 


359.35 


2.55552 


0.96 


32.96 


2.48 


64.44 


64.99 


30 


69.00 


2 m 


323 


5 414 


348.61 


2.54234 


1.06 


34.45 


2.77 


68.92 


68.98 


17 00 


72.00 


2 271 


3 394 


6 074 


338.51 2.52957 


1.17 


35.94 


3.09 


71.90 


71.97 


30 


75.00 


2 37^ 


3 56i 


6 33} 


329.00 2.51720 


1.28 


37.44 


3.43 


74.88 


74.97 


18 00 


78.00 


2 47i 


4 13f 


7 014 


320.04 2.50520 


1.40 


38.92 


3.79 


77.86 


77.96 


30 


81.00 


2 57J 


4 32 


7 294 


311.57 2.49356 


1.53 


40.41 


4.18 


80.84 


80.95 


19 00 


84.00 


308 


4 501 


7 58} 


303.58 


2.48227 


1.67 


41.90 


4.59 


83.81 


83.94 


30 


87.00 


3 19 


510 


829 


295.99 


2.47128 


1.81 


43.39 


5.03 


86.78 


86.93 


20 00 


90.00 


3 30 


5 30 


900 


288.80 


2.46060 


1.96 


44.88 


.5.49 


89.75 


89.92 


30 


93.00 


3 41J 


5 504 


9 32 


281.98 


2.45022 


2.12 


46.36 


6.98 


92.71 


92.91 


2100 


96.00 


3 53 


6 Hi 


10 04} 


275.50 


2.44012 


2.28 


47.85 


6.50 


95.67 


95.89 


30 


99.00 


405 


6 334 


10 384 


269.33 


2.43028 


2.45 


49.33 


7.05 


98.62 


98.87 


22 00 


102.00 


4 174 


6 551 


1113} 


263.46 


2.42071 


2.63 


50.81 


7.63 


101.56 


101.85 


30 


105.00 


430 


7 181 


1148} 


257.87 


2.41140 


2.82 


52.29 


8.23 


104.50 


104.82 


23 00 


108.00 


4 43 


7 424 


12 254 


252.55 


2.40235 


3.02 


63.76 


8.86 


107.44 


107.78 


24 00 


114.00 


5 094 


8 3U 


13 40} 


242.61 


2.38491 


3.44 


56.72 


10.23 


113.28 


113.73 


25 00 


120.00 


5 374 


9 224 


15 00 


233.54 


2.36836 


3.90 


59.66 


11.73 


119.09 


119.67 


26 00 


126.00 


6 064 


10164 


16 22}" 


225.24 


2.35265 


4.40 


62.59 


13.37 


124.87 


125.59 


27 00 


132.00 


6 37 


11124 


17 494 


217.64 


2.33774 


4.94 


65.51 


15.15 


130.61 


131.49 


28 00 


138.00 


7 084 


12 101 


19 194 


210.66 


2.32358 


5.52 


68.42 


17.07 


136.29 


137.36 


29 00 


144.00 


7 41 


13 11} 


20 52} 


204.24 


2.31014 


6.14 


71.32 


19.15 


141.93 


143.21 


80 00 


150.00 


815 


14 15 


22 30 


198.34 


2.29741 


6.81 


74.21 


21.39 


147.50 


149.04 



y Google 



108 



THE RAILROAD TAPER 



SSs- 8S55 



§8g SS^ 8855 823. 

bdo »2d^ QOo jod^ 



8S3- 8S'- 8255 88'^ SSht SSh. 

dd<=> dd^ ooo SOo QO® oo^ 

COo Oo 05o^0"^<» Quo 

O O "^ rH ^ •-! ^ 



^^ O .*^t3 O 



^^^3 O -^t3 © ^Td O 



*'« © 



S 



8 2 S? !S 

"'-' "digitized by (sOOg IC^ 



TABLES 



109 



•c 


CO 

CO 












1 





























■«♦« 














CO 








CO 












^ 


CO 












... 














e« 


sr 


, 














H? 


s? 










88'^ 


8S* 




















^°°o 


























•4« 


sy 






CO 




8S|r 




88?: 







- 


M 


8S2 


S8^ 







.0 


82s 





i«« 


>«« 


8Sg. 


8S^ 




82;j 


8^8 


8Ss 


8Ssr 
8®o" 

'-' 1-1 


^ 


- 




88^- 

2 tn 


S3 - 


83*- 

»-• ,-H 


8SS 


888^ 

1-" (M 


^*» 


H« 


ddo 


S2g 




82rr 


8«E; 


8e;s 

QCOo 
CO *^ 


< 


1 


^'TS 


^•^ 


^TS 


-^'T3 


~»r3 


-"^ti 




:^ 


8 




to 

1-H 


(N 





? 8 ^S 

digitized by t^OOgll'^*' 



110 



THE RAILROAD TAPER 



Mi 


'SS 












. . . 1 


. . . . . . . . . * * * '.'.', '.'.'. 

'.'.'. '.'.'. '.'.'. ... ... ... 




'i* 


CO 








CO 


^ 








M 


ec 








esi 


s? 






8S^g 


!22-5 


O 


o 


;r 


^ 


82g 


82^ 

O 


82g 


88^^ 


8^S 


88S?r 

1— 1 


t: 


esi 


82^ 

O 


8Spir 

1-H 


8S!s 


1-H 


8S?^^ 

t^ o 

1-H 


8^^ 

l-t 


- 


C4 


8^s? 

b- o 




8^^ 




8fe«^* 


1-H o 


-.. 


He* 




8Ss 


8f=^^ 

CO c^ 




-< re 


8c5^^ 

2 ^ 


Hn 


- 


82g 

2 CO 


8^3? 

^ CO 


8Sg 

Oi-Ho 


8§^ 

»o I-* ^ 


Q<No 


8{^| 


H. 


- 


CO o 


8S^^ 


8^8 

bdo 








<1 


1 


^Td o 


-^-^ o 


-**'« O 


^73 O 


-«^'« o 


--tJ c> 




^ 


o 
CO 


o 
CO 


8 

o 


t2 g « 

o o o 

digitized by SOOgl(f' 



TABLES 



111 



•o 


s? 




^ 


«$• 




eo 


S!? 


: i ■: ; ; ; : ; ; : ; ; s^s ss^^ 

SOo »2do 

. . . CO o CO o 
. . . . . . . . . . . ^__| 


M 


CO 


82ig 8^g SSgj SS-r 8Si«. S^^^ 

s®b ??®S5 ^°f g°? ?s°r ^®r 

O 1^ 1-1 rH 


N 


3? 


8J2h- {28^ SS^ ^SJ; 8S^ S5!«j 

^®f ^®r s°r s°°^ 8="^ fe^o" 

,-1 rH 1-H C<l 


y^ 


c< 


8??^^ S^i- SiSa 8S^ 8Sg 82^ 

ddo jod»S ddo 52dc^ odo ddo 

l^rHt^o (»c^00o o^c^Oeo 


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8^$5 8fe^ 8S^ 8S^ 8S^^ Sq^j 

ddo dd<=^ cjd^ god® '^jid^ di-Io 

OSc^OSo Oo Oo »-to C^o 

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11 


cq 


120.00 

0.79 

3° 00' 

127.50 

0.91 

3°20i' 

135.00 

1.05 

3°42i' 

142.50 

1.21 

4° 05i' 

150.00 

1.37 

4° 30' 

165.00 

1.75 

5°21|' 


i«« 


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8^2 83^ 88^ 8^^^ 8^2 8)2^- 
ci-Ho si'Hi' drt'^ d«o dtio dw*^ 


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112 



THE RAILROAD TAPER 



•o 


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30.00 

0.20 

1°30' 


■^ 


eo 


30.00 

0.16 

1M2' 

33.75 

0.20 

1°26' 

37.50 

0.24 

IMli' 

45.00 

0.35 

2° 15' 


eo 


eo 


88'^ 88h; 8?S8 8^^ 8^g 88^ 

ddo lod'* odo lad^ Qdo ddo 
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SSs- 8^s 8^'^ 8§^ S^s- 8§g 

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m3o COc^CDo l^o l^o 0>^ 
T-H <N CO CO 


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lod^ (Ndo ddo i^do io,-i"^ S'-'o 

r^o OOo OSeO^o Oo C^^o 
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Hn 


c^ 


110.00 

0.95 

3° 51' 

120.00 

1.18 

4*^30' 

130.00 

1.44 

5° 12' 

140.00 

1.73 

5° 57' 

150.00 

2.06 

6° 45' 

170.00 

2.84 
8° 30' 


>*» 


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8^S? S^S- 8Ss? 8S«^ 8§|y 8Sb 

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TABLES 



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114 



THE RAILBOAD TAPER 



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Swo ocoo go'^w i^cd^ 

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8S* 88g 8Sg 

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TABLES 



115 



ALIGNMENT REVISION 

MAINTAINING LENGTH UNCHANGED 

(See page 55, par. 43) 



M 


roo' 


1-30' 


2-00' 


A 


i 


i 


i 




4 


Taper. 


4 


4 


4 


1 


/ 


/ 


h 


M' 


I 


h 


M' 


I 


h 


M' 


I 


h 


W 


2-30' 
5 
10 

15 
20 
25 


110.00 
96.00 


0.04 
0.14 


IMO' 
1*»03' 


m.bo 

158.00 
154.00 


6!6l 
0.23 

0.35 


1-48' 
1-34' 

1-32' 


266!6o 

224.00 
218.00 
214.00 


6!42 

0.37 
0.50 
0.26 


2-25' 

2-07' 
2-04' 
2-02' 


ii6!6o 

94.00 
92.00 


6!67 

0.22 
0.19 


2' 20' 
2-04' 

2-02' 


M 


2° 30' 


3-00' 


A 


i 


4 


4 


1 


Taper. 


i 


1 


1 


14 


/ 


I 


h 


JIf' 


I 


h 


M' 


I 


h 


3f' 


I 


h 


W 


W 

15 
20 
25 

30 
40 
50 


350.00 
296.00 
286.00 

280.00 
276.00 
274.00 


0.84 
0.71 
0.87 

0.75 
1.05 
1.25 


3" 10 
2° 43 
2° 38 

2-35 
2-33' 
2*32' 


132.00 

125.00 
123.00 


0.15 

0.19 
0.27 


2-42' 

2-35' 
2-33' 


165.00 
157.00 
154.00 

153.00 


0.31 
0.32 
0.25 

0.36 


3-14' 
3-07' 
3-04' 

3-03' 


96.67 

93.33 
92.00 


0.03 

0.12 
0-17 


3-10' 

3-05' 
3-03' 


M 


4** 00' 


A 


i 


i 


* 1 


Taper. 


1 


14 


2 


/ 


I 


h 


M' 


I 


h 


JIf' 


I 


k 


w 


10* 
15 
20 
25 

ao 

40 


233.00 

224.00 
217.00 
214.00 
213.00 


*6!8i' 
0.81 

0.76 
0.69 
0.51 
0.75 


4- 49' 
4-23' 

4-14' 
4-07' 
4-04' 
4-03' 


139.33 
135.33 

133.33 
132.00 


0.34 
0.35 
0.31 

0.24 
0.35 


'4-30' 
4-14' 
4-08' 

4-05' 
4-03' 


110.00 
96.00 
93.00 
92.00 

91.50 


0.15 
0.12 
0.10 
0.15 

0.20 


4-40' 
4-12' 
4-06' 
4-04' 

4-03' 


60 
00 


,H^'i; 








































o 







116 



THE RAILROAD TAPER 



MAINTAINING 


LENGTH UN 


CHA 


NG] 


5D- 


- (Continued) 


M 


5° 00' 


A 


§ 


i 


1 


n 


Taper. 


1 


H 


2 


2i 


/ 


/ 


k 


M' 


I 


h 


M' 


I 


k 


M' 


I 


h 


M' 


20 
25 
30 
40 
50 


349.00 
296.00 
285.00 
280.00 
277.00 
275.00 
274.00 


1.64 
1.43 
1.50 
1.60 
1.53 
1.31 
1.48 


6° 19' 
6° 26' 
6M6' 
6M0' 
6° 07' 
5° 06' 
5° 04' 


223 '.33 
190.00 
182.67 
176.67 
174.00 
172.67 
172.00 


6! 76 
0.68 
O.W 
0.70 
0.66 
0.62 
0.72 


6° '20' 
6° 30' 
5° 19' 
5° 10' 
6° 06' 
5° 04' 
5° 03' 


164.00 
132.50 
127.00 
124.60 
122.50 
121.60 


0.36 
0.34 
0.32 
0.29 
0.32 
0.28 


6° 08' 
5° 26' 
6° 14' 
5° 09' 
6° 06' 
6° 03' 


102.00 

96.20 
93.20 
92.00 
91.20 


0.26 
0.17 
0.18 
0.13 
0.19 


5° 30' 
6° 13' 
6° 08' 
6° 05' 
5° 03' 


60 
70 
80 
90 








M 


6° 00' 


A 


i 


1 


li 


H 


Taper. 


H 


2 


M' 


2i 


2h 


/ 


I 


h 


M' 


I 


k 


I 


k 


M' 


I 


h 


M' 


15° 

20 

25 

30 

40 

50 

60 

70 

80 

90 


258.67 
228.67 
220.67 
217.33 
215.33 
214.00 
212.67 


1.21 
1.12 
1.09 
1.25 
1.35 
1.39 
0.72 


7° 13' 
6° 28' 
6° 16' 
6° 11' 
6° 08' 
6° 06' 
6° 04' 


m.50 

163.50 
157.00 
154.00 
153.00 
152.00 


0.58 
0.66 
0.52 
0.76 
0.58 


6° 47' 
6° 27' 
6° 14' 
6° 08' 
6° 06' 
6° 04' 


13i!60 
123.60 
120.00 
117.20 
116.00 
116.20 


0.36 
0.33 
0.35 
0.34 
0.20 


6° 44' 
6° 24' 
6° 16' 
6° 08' 
6° 06' 
6° 03' 


111.00 
98.33 
95.00 
93.33 
92.00 
91.00 


0.25 
0.22 
0.24 
0.26 
0.34 
0.14 


7° 03' 
6° 25' 
6° 16' 
6° 10' 
6° 06' 
6° 03' 


M 


8° 00' 


A 





1 




U 


U 


2 


Taper. 


2 




2i 




2h 


2} 


/ 


I 


h 


M' 


I 


h 


M' 


I 


h 


M' 


I 


h 


M' 


20° 

25 

30 

40 

50 

60 

70 

80 

90 


258! 50 
232.50 
224.00 
219.50 
217.00 
215.50 


iiei 

1.68 
1.67 
1.48 
1.50 
1.65 


■9° 37' 
8° 45' 
8° 28' 
8° 19' 
8° 14' 
8° 11' 


220.40 
179.60 
172.00 
168.40 
166.40 
165.60 
164.80 


1.04 
0.96 
1.08 
0.92 
0.81 
1.14 
1.18 


10° 26' 
8° 44' 
8° 25' 
8° 16' 
8° 11' 
8° 09' 
8° 07' 


176.67 
160.00 
139.00 
135.33 
133.67 
132.67 
132.00 
131.67 


'6'.7i 

0.67 
0.64 
0.63 
0.70 
0.75 
0.74 
0.95 


16° 20' 
9° 00' 
8° 27' 
8° 16' 
8° 11' 
8° 08' 
8° 06' 
8° 05' 


112.26 
101.00 
97.00 
93.25 
92.00 
91.60 
91.00 


0.43 
0.40 
0.43 
0.31 
0.31 
0.43 
0.33 


9° 29' 
8° 44' 
8° 28' 
8° 13' 
8° 08' 
8° 06' 
8° 04' 


M 


10° 00' 


A 


u 


U 


2 


21 


Taper. 


2i 


2J 


21 


3 


/ 


I 


A 


M' 


I 


h 


M' 


I 


k 


M' 


I 


k 


M' 


25° 
30 
40 
50 
60 
70 
80 
90 


258".6o 
234.80 
226.40 
221.60 
218.80 


i'.99 
1.90 
1.95 
1.86 
1.90 


12° 00' 
11° 02' 
10° 41' 
10° 29' 
10° 22' 


2i9!6o 

190.67 
182.67 
178.67 
176.33 
176.00 


i'.ze 

1.32 
1.33 
1.31 
1.30 
1.46 


12° 27' 
11° 02' 
10° 38' 
10° 26' 
10° 19' 
10° 15' 


156! 50 
132.50 
127.00 
124.76 
123.25 
122.50 
122.00 


*6!76 
0.69 
0.65 
0.73 
0.64 
0.71 
0.82 


i2°22' 
10°50' 
10° 28' 
10° 19' 
10° 13' 
10*10' 
10°08' 


112.00 
101.40 
95.40 
93.40 
92.20 
91.60 
91.20 
91.00 


0.45 
0.41 
0.39 
0.46 
0.89 
0.42 
0.41 
0.53 


11°50' 
10° S7' 
40° 27' 
10° 17' 

io°ir 

10°08' 

woe 

10°06' 





LOGARITHMS OF NUMBERS 11^ 


N 


0123456780 


100 


00000 00043 00087 00130 00173 00217 00200 00303 00346 00380 




0432 Q476 0618 0661 0604 0647 0689 0732 0775 0817 




0860 0903 0946 0988 1030 1072 1115 1167 1199 1242 




1284 1326 1368 1410 1452 1494 1636 1678 1620 1662 




1703 1746 1787 1828 1870 1912 1963 1995 2036 2078 


K 


2119 2160 2202 2243 2284 2326 2366 2407 2449 2490 




2631 2672 2612 2663 2694 2735 2776 2816 2867 2898 




2938 2979 3019 3060 3100 3141 3181 3222 3262 3302 




3342 3383 3423 3463 3603 3643 3683 3623 3663 3703 




3743 3782 3822 8862 3902 3941 3981 4021 4060 4100 


110 


04139 04179 04218 04268 04297 04336 04376 04415 04464 04493 




4632 4671 4610 4650 4689 4727 4766 4806 4844 4883 




4922 4961 4999 6038 6077 6116 6164 6192 6231 6269 




5308 6346 6385 6423 5461 6500 6638 6576 6614 6652 




6690 6729 6767 6805 5843 6881 6918 6966 6994 6032 




6070 6108 6146 6183 6221 6268 6296 6333 6371 6408 




6446 6483 6621 6658 6696 6633 6670 6707 6744 6781 




6819 6866 6893 6930 6967 7004 7041 7078 7115 7161 


n 


7188 7225 7262 7298 7336 7372 7408 7446 7482 7618 




7665 7691 7628 7664 7700 7737 7773 7809 7846 7882 


120 


07918 07964 07990 08027 08063 08099 08135 08171 08207 08243 




8279 8314 8360 8386 8422 8468 8493 8629 8665 8600 




8636 8672 8707 8743 8778 8814 8849 8884 8920 8966 




8991 9026 9061 9096 9132 9167 9202 9237 9272 9307 




9342 9377 9412 9447 9482 9617 9662 9687 9621 9656 




9691 9726 9760 9795 9830 9864 9899 9934 9968 10003 




10037 10072 10106 10140 10175 10209 10243 10278 10312 0346 




0380 0415 0449 0483 0617 0661 0686 0619 0663 P687 




0721 0765 0789 0823 0867 0890 0924 0968 0992 1026 




1069 1093 1126 1160 1193 1227 1261 1294 1327 1361 


180 


11394 11428 11461 11494 11528 11561 11694 11628 11661 11694 




1727 1760 1793 1826 1860 1893 1926 1959 1992 2024 


SB 


2057 2090 2123 2156 2189 2222 2264 2287 2320 2352 


_ 


2386 2418 2460 2483 2516 2648 2681 2613 2646 2678 




2710 2743 2776 2808 2840 2872 2905 2937 2969 3001 


5 


3033 3066 3098 3130 3162 3194 3226 3258 3290 3322 




3354 3386 3418 3450 3481 3513 3646 3677 3609 3640 




3672 3704 3735 3767 3799 3830 3862 8893 3925 3966 




3988 4019 4061 4082 4114 4145 4176 4208 4239 4270 




4301 4333 4364 4396 4426 4467 4489 4520 4661 4582 


140 


14613 14644 14675 14706 14737 14768 14799 14829 14860 14891 




4922 4953 4983 6014 6045 6076 6106 5137 6168 6198 




6229 6269 6290 6320 5351 5381 6412 6442 6473 6603 




5534 5664 6694 6625 6655 6686 6716 5746 6776 5806 




5836 6866 6897 6927 6967 5987 6017 6047 6077 6107 




6137 6167 6197 6227 6266 6286 6316 6346 6376 6406 




6436 6465 6495 6524 6554 6684 6613 6643 6673 6702 




6732 6761 6791 6820 6850 6879 6909 6938 6967 6997 




7026 7056 7085 7114 7143 7173 7202 7231 7260 7289 




7319 7348 7377 7406 7435 7464 7493 7622 7651 7680 


150 


17609 17638 17667 17696 17725 17754 17782 17811 17840 17869 



118 


LOGARITHMS OF NUMBERS 


N 


0123456780 


150 


17609 17638 17667 17696 17725 17764 17782 17811 17840 17869 




7898 7926 7965 7984 8013 8041 8070 8099 8127 8166 




8184 8213 8241 8270 8298 8327 8366 8384 8412 8441 




8469 8498 8626 8664 8683 8611 8639 8667 8696 8724 




8762 8780 8808 8837 8865 8893 8921 8949 8977 9006 




9033 9061 9089 9117 9146 9173 9201 9229 9267 9285 




9312 9340 9368 9396 9424 9461 9479 9507 9535 9562 




9690 9618 9646 9673 9700 9728 9766 9783 9811 9838 




9866 9893 9921 9948 9976 20003 20030 20068 20085 20112 




20140 20167 20194 20222 20249 0276 0303 0330 0358 0385 


160 


20412 20439 2046^20493 20620 20548 20675 20602 20629 20656 




0683 0710 0737 0763 0790 0817 0844 0871 0898 0925 




0962 0978 1006 1032 1069 1086 1112 1139 1165 1192 




1219 1245 1272 1299 1326 1362 1378 1405 1431 1468 




1484 1511 1537 1664 1690 1617 1643 1669 1696 1722 




1748 1775 1801 1827 1854 1880 1906 1932 1958 1985 




2011 2037 2063 2089 2115 2141 2167 2194 2220 2246 




2272 2298 2324 2350 2376 2401 2427 2463 2479 2505 




2631 2557 2583 2608 2634 2660 2686 2712 2737 2763 




2789 2814 2840 2866 2891 2917 2943 2968 2994 3019 


170 


23045 23070 23096 23121 23147 23172 23198 23223 23249 23274 




3300 3825 3360 3376 3401 3426 3452 3477 3502 3528 




3563 3678 3603 3629 3654 3679 3704 3729 3754 3779 




3805 3830 3865 3880 3905 3930 3965 3980 4005 4030 




4065 4080 4105 4130 4165 4180 4204 4229 4254 4279 




4304 4329 4363 4378 4403 4428 4462 4477 4602 4627 




4661 4576 4601 4626 4650 4674 4699 4724 4748 4773 




4797 4822 4846 4871 4896 4920 4944 4969 4993 5018 




6042 5066 5091 5116 5139 5164 5188 5212 6237 6261 




5285 5310 6334 6368 6382 6406 5431 5455 5479 5603 


180 


25627 25551 26575 25600 26624 26648 25672 25696 25720 25744 




5768 6792 5816 6840 5864 6888 5912 5935 6959 6983 




6007 6031 6055 6079 6102 6126 6160 6174 6198 6221 




6246 6269 6293 6316 6340 6364 6387 6411 6435 6468 




6482 6606 6629 6563 6576 6600 6623 6647 6670 6694 




6717 6741 6764 6788 6811 6834 6858 6881 6905 6928 




6961 6975 6998 7021 7045 7068 7091 7114 7138 7161 




7184 7207 7231 7254 7277 7300 7323 7346 7370 7393 




7416 7439 7462 7486 7608 7631 7664 7677 7600 7623 




7646 7669 7692 7716 7738 7761 7784 7807 7830 7852 


190 


27876 27898 27921 27944 27967 27989 28012 28035 28068 28081 




8103 8126-8149 8171 8194 8217 8240 8262 8285 8307 




8330 8353 8375 8398 8421 8443 8466 8488 8611 8633 




8556 8578 8601 8623 8646 8668 8691 8713 8735 8758 




8780 8803 8825 8847 8870 8892 8914 8937 8959 8981 




9003 9026 9048 9070 9092 9115 9137 9159 9181 9203 




9226 9248 9270 9292 .9314 9336 9358 9380 9403 9425 




9447 9469 9491 9513 9635 9667 9679 9601 9623 9645 




9667 9688 9710 9732 9754 9776 9798 9820 9842 9863 




9885 9907 9929 9961 9973 9994 30016 30038 30060 30081 


800 


30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 





LOGARITHMS OF NUMBERS 119 


N 


0123456789 


200 


30103 30125 30146 30168 30190 30211 30233 30265 30276 30298 


1 


0329 0341 0363 0384 0406 0428 0449 0471 0492 0514 


2 


0536 0557 0578 0600 0621 0643 0664 0686 0707 0728 


8 


0750 0771 0792 0814 0836 0856 0878 0899 0920 0942 


4 


0963 0984 1006 1027 1048 1069 1091 1112 1133 1164 


5 


1176 1197 1218 1239 1260 1281 1302 1323 1345 1366 


6 


1387 1408 1429 1450 1471 1492 1613 1634 1655 1576 


7 


1697 1618 1639 1660 1681 1702 1723 1744 1765 1786 


8 


1806 1827 1848 1869 1890 1911 1931 1952 1973 1994 


9 


2015 2036 2056 2077 2098 2118 2139 2160 2181 2201 


210 


32222 32243 32263 82284 32305 32326 32346 32366 82387 82408 


1 


2428 2449 2469 2490 2510 2631 2562 2572 2593 2613 


2 


2634 2654 2675 2695 2716 2736 2766 2777 2797 2818 


3 


2838 2858 2879 2899 2919 2940 2960 2980 3001 3021 


4 


3041 3062 3082 3102 3122 3143 3163 3183 3203 3224 


6 


3244 3264 3284 3304 3325 3345 3365 3386 3405 3425 


6 


3445 3465 3486 3506 3526 3546 3566 3586 3606 3626 


7 


3646 3666 3686 3706 3726 3746 3766 3786 3806 3826 


8 


3846 3866 3885 3905 3925 3945 3965 3985 4005 4025 


9 


4044 4064 4084 4104 4124 4143 4163 4183 4203 4223 


220 


34242 34262 34282 34301 34321 34341 34361 34380 34400 34420 


1 


4439 4459 4479 4498 4518 4537 4667 4577 4596 4616 


2 


4635 4655 4674 4694 4713 4733 4753 4772 4792 4811 


8 


4830 4850 4869 4889 4908 4928 4947 4967 4986 5005 


4 


5025 5044 5064 5083 5102 6122 6141 5160 5180 5199 


6 


6218 6238 5257 6276 5296 6315 5334 6363 6372 6392 


6 


6411 5430 6449 5468 6488 6507 5626 6545 5664 5583 


7 


5603 5622 6641 6660 6679 5698 6717 6736 6755 5774 


8 


5793 6813 5832 5851 5870 5889 5908 6927 5946 5965 


9 


6984 6003 6021 6040 6059 6078 6097 6116 6135 6154 


280 


36178 36192 36211 36229 36248 36267 36286 36305 36324 36342 


1 


6361 6380 6399 6418 6436 6456 6474 6493 6511 6630 


2 


6649 6668 6686 6605 6624 6642 6661 6680 6698 6717 


3 


6736 6754 6773 6791 6810 6829 6847 6866 6884 6903 


4 


6922 6940 6959 6977 6996 7014 7033 7051 7070 7088 


5 


7107 7125 7144 7162 7181 7199 7218 7236 7254 7273 


6 


7291 7310 7328 7346 7365 7383 7401 7420 7438 7457 


7 


7475 7493 7511 7530 7548 7566 7585 7603 7621 7639 


8 


7668 7676 7694 7712 7731 7749 7767 7785 7803 7822 


9 


7840 7858 7876 7894 7912 7931 7949 7967 7985 8003 


240 


38021 38039 38057 38075 38093 38112 38130 38148 38166 38184 


1 


8202 8220 8238 8256 8274 8292 8310 8328 8346 8364 


2 


8382 8399 8417 8435 8453 8471 8489 8507 8625 8543 


8 


8561 8578 8696 8614 8632 8650 8668 8686 8703 8721 


4 


8739 8757 8775 8792 8810 8828 8846 8863 8881 8899 


6 


8917 8934 8962 8970 8987 '9005 9023 9041 9058 9076 


6 


9094 9111 9129 9146 9164 9182 9199 9217 9235 9252 


7 


9270 9287 9305 9322 9340 9358 9375 9393 9410 9428 


8 


9445 9463 9480 9498 9515 9533 9560 9668 9585 9602 


9 


9620 9637 9655 9672 9690 9707 97^4 9742 9759 9777 


260 


39794 39811 39829 39846 39863 39881 39898 39916 39933 39960 



120 


LOGARITHMS OF NUMBERS 


N 


0123456789 


250 


39794 39811 39829 39846 39863 39881 39898 39916 39933 39950 


1 


9967 9985 40002 40019 40037 40064 40071 40088 40106 40123 


3 


40140 40167 0175 0192 0209 0226 0243 0261 0278 0295 


8 


0312 0329 0346 0364 0381 0398 0415 0432 0449 0466 


4 


0483 0500 0518 0635 0662 0669 0586 0603 0620 0637 


6 


0664 0671 0688 0705 0722 0739 0766 0773 0790 0807 


6 


0824 0841 0868 0875 0892 0909 0926 0943 0960 0976 


7 


0993 1010 1027 1044 1061 1078 1095 1111 1128 1145 


8 


1162 1179 1196 1212 1229 1246 1263 1280 1296 1313 


9 


1330 1347 1363 1380 1397 1414 1430 1447 1464 1481 


260 


41497 41514 41531 41547 41564 41581 41597 41614 41631 41647 


1 


1664 1681 1697 1714 1731 1747 1764 1780 1797 1814 


2 


1830 1847 1863 1880 1896 1913 1929 1946 1963 1979 


8 


1996 2012 2029 2045 2062 2078 2095 2111 2127 2144 


4 


2160 2177 2193 2210 2226 2243 2269 2275 2292 2308 


5 


2325 2341 2357 2374 2390 2406 2423 2439 2455 2472 


6 


2488 2604 2521 2537 2553 2670 2586 2602 2619 2635 


7 


2651 2667 2684 2700 2716 2732 2749 2765 2781 2797 


8 


2813 2830 2846 2862 2878 2894 2911 2927 2943 2959 


9 


2975 2991 3008 3024 3040 3066 3072 3088 3104 3120 


270 


43136 43162 43169 43185 43201 43217 43233 43249 43265 43281 


1 


3297 3313 3329 3345 3361 3377 3393 3409 3425 3441 


2 


3457 3473 3489 3605 3521 3537 3553 3569 3584 3600 


8 


3616 3632 3648 3664 3680 3696 3712 3727 3743 3759 


4 


3775 3791 3807 3823 3838 3854 3870 3886 3902 3917 


6 


3933 3949 3965 3981 3996 4012 4028 4044 4059 4075 


6 


4091 4107 4122 4138 4154 4170 4185 4201 4217 4232 


7 


4248 4264 4279 4295 4311 4326 4342 4358 4373 4389 


8 


4404 4420 4436 4451 4467 4483 4498 4514 4529 4545 


9 


4560 4576 4592 4607 4623 4638 4654 4669 4685 4700 


280 


44716 44731 44747 44762 44778 44793 44809 44824 44840 44865 


1 


4871 4886 4902 4917 4932 4948 4963 4979 4994 6010 


2 


5025 6040 5056 6071 5086 5102 6117 6133 6148 5163 


8 


5179 5194 5209 5225 5240 6255 6271 5286 5301 5317 


4 


5332 5347 5362 6378 5393 6408 5423 5439 5454 5469 


5 


6484 6500 5515 5530 5545 5561 5576 5691 5606 6621 


6 


5637 5652 5667 5682 5697 6712 5728 5743 5758 5773 


7 


5788 5803 5818 5834 5849 5864 5879 5894 5909 6924 


8 


5939 5954 5969 5984 6000 6015 6030 6045 6060 6075 


9 


6090 6105 6120 6135 6150 6165 6180 6195 6210 6225 


290 


46240 46255 46270 46285 46300 46315 46330 46345 46369 46374 


1 


6389 6404 6419 6434 6449 6464 6479 6494 6509 6523 


2 


6538 6553 6568 6683 6598 6613 6627 6642 6657 6672 


8 


6687 6702 6716 6731 6746 6761 6776 6790 6805 6820 


4 


6835 6850 6864 6879 6894 6909 6923 6938 6953 6967 


5 


6982 6997 7012 7026 7041 7056 7070 7085 7100 7114 


6 


7129 7144 7159 7173 7188 7202 7217 7232 7246 7261 


7 


7276 7290 7305 7319 7334 7349 7363 7378 7392 7407 


8 


7422 7436 7461 7465 7480 7494 7509 7524 7538 7563 


9 


7667 7582 7696 7611 7625 7640 7664 7669 7683 7698 


800 


47712 47727 47741 47766 47770 47784 47799 47818 47828 47842 





LOGARITHMS OF NUMBERS 121 


N 


0123456789 


800 


47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 


1 


7857 7871 7885 7900 7914 7929 7943 7958 7972 7986 


2 


8001 8015 8029 8044 8058 8073 8087 8101 8116 8130 


8 


8144 8169 8173 8187 8202 8216 8230 8244 8259 8273 


4 


8287 8302 8316 8330 8344 8359 8373 8387 8401 8416 


5 


8430 8444 8458 8473 8487 8501 8515 8530 8544 8558 


6 


8572 8586 8601 8615 8629 8643 8657 8671 8686 8700 


7 


8714 8728 8742 8756 8770 8785 8799 8813 8827 8841 


8 


8855 8869 8883 8897 8911 8926 8940 8954 8968 8982 


9 


8996 9010 9024 9038 9052 9066 9080 9094 9108 9122 


810 


49136 49150 49164 49178 49192 49206 49220 49234 49248 49262 


1 


9276 9290 9304 9318 9332 9346 9360 9374 9388 9402 


2 


9415 9429 9443 9457 9471 9485 9499 9513 9527 9641 


8 


9554 9568 9582 9596 9610 9624 9638 9651 9665 9679 


4 


9693 9707 9721 9734 9748 9762 9776 9790 9803 -9817 


5 


9831 9845 9859 9872 9886 9900 9914 9927 9941' 9955 


6 


9969 9982 9996 50010 50024 50037 6005150065 60079 50092 


7 


50106 60120 60133 0147 0161 0174 0188 0202 0215 0229 


8 


0243 0266 0270 0284 0297 0311 0325 0338 0352 0366 


9 


0379 0393 0406 0420 0433 0447 0461 0474 0488 0601 


820 


60515 60529 50642 60666 50569 60683 60596 50610 60623 60637 


1 


0651 0664 0678 0691 0705 0718 0732 0745 0759 0772 


2 


0786 0799 0813 0826 0840 0853 0866 0880 0893 0907 


8 


0920 0934 0947 0961 0974 0987 1001 1014 1028 1041 


4 


1065 1068 1081 1095 1108 1121 1135 1148 1162 1175 


6 


1188 1202 1215 1228 1242 1255 1268 1282 1296 1308 


6 


1322 1335 1348 1362 1375 1388 1402 1415 1428 1441 


7 


1455 1468 1481 1495 1508 1521 1534 1548 1561 1574 


8 


1587 « 1601 1614 1627 1640 1654 1667 1680 1693 1706 


9 


1720 1733 1746 1759 1772 1786 1799 1812 1825 1838 


330 


61861 61865 61878 61891 61904 61917 61930 61943 61957 61970 


1 


1983 1996 2009 2022 2035 2048 2061 2075 2088 2101 


2 


2114 2127 2140 2153 2166 2179 2192 2205 2218 2231 


3 


2244 2257 2270 2284 2297 2310 2323 2336 2349 2362 


4 


2375 2388 2401 2414 2427 2440 2453 2466 2479 2492 


5 


2504 2517 2530 2543 2556 2569 2582 2595 2608 2621 


6 


2634 2647 2660 2673 2686 2699 2711 2724 2737 2750 


7 


2763 2776 2789 2802 2815 2827 2840 2853 2866 2879 


8 


2892 2905 2917 2930 2943 2956 2969 2982 2994 3007 


9 


3020 3033 3046 3058 3071 3084 3097 3110 3122 3135 


340 


63148 63161 53173 53186 53199 63212 53224 53237 53250 53263 


1 


3276 3288 3301 3314 3326 3339 3362 3364 3377 3390 


2 


3403 3415 3428 3441 3453 3466 3479 3491 3504 3517 


3 


3629 3542 3555 3567 3580 3593 3605 3618 3631 3643 


4 


3656 3668 3681 3694 3706 3719 3732 3744 3757 3769 


6 


3782 3794 3807 3820 38B2 3845 3857 3870 3882 3895 


6 


3908 3920 3933 3945 3958 3970 8983 3995 4008 4020 


7 


4033 4045 4058 4070 4083 4095 4108 4120 4133 4145 


8 


4168 4170 4183 4196 4208 4220 4233 4245 4258 4270 


9 


42S3 4295 4307 4320 4832 4345 4357 4370 4382 4394 


350 


54407 64419 54432 64444 54456 54469 54481 54494 64606 54618 



122 


LOGARITHMS OF NtrMBEKS 


N 


0123456789 


850 


64407 64419 64432 54444 54466 54469 54481 64494 64506 64518 




4631 4543 4556 4568 4580 4693 460^ 4617 4630 4042 




4654 4667 4679 4691 4704 4716 4728 4741 4763 4766 




4777 4790 4802 4814 4827 4839 4851 4864 4876 4888 




4900 4913 4925 4937 4949 4962 4974 4986 4998 6011 




6023 6035 6047 6060 5072 5084 5096 5108 5121 6133 




6145 5157 5169 6182 6194 6206 6218 6230 6242 6265 




5267 6279 5291 5303 6316 5328 6340 6362 5364 6376 




6388 5400 6413 6425 6437 6449 6461 6473 5486 6497 




5609 6522 5634 6546 6558 6670 5682 5594 6606 6618 


860 


56630 56642 55664 65666 55678 56691 66703 65715 65727 55739 




5751 5763 6775 6787 5799 5811 5823 6835 5847 6869 




6871 5883 5895 6907 5919 6931 5943 5965 6967 6979 




6991 6003 6015 6027 6038 6050 6062 6074 6086 6098 




6110 6122 6134 6146 6158 6170 6182 6194 6205 6217 




6229 6241 6253 6265 6277 6289 6301 6312 6324 6336 




6348 6360 6372 6384 6396 6407 6419 6431 6443 6465 




6467 6478 6490 6502 6514 6526 6638 6649 6661 6673 




6685 6597 6608 6620 6632 6644 6666 6667 6679 6691 




6703 6714 6726 6738 6750 6761 6773 6785 6797 6808 


870 


66820 56832 66844 66866 66867 66879 66891 66902 66914 66026 




6937 6949 6961 6972 6984 6996 7008 7019 7031 7043 




7054 7066 7078 7089 -7101 7113 7124 7136 7148 716^ 




7171 7183 7194 7206 7217 7229 7241 7252 7264 7276 




7287 7299 7310 7322 7334 7345 7367 7368 7380 7392 




7403 7415 7426 7438 7449 7461 7473 7484 7496 7507 




7519 7630 7542 7663 7565 7576 7588 7600 7611 7623 




7634 7646 7667 7669 7680 7692 7703 7715 7726 7738 




7749 7761 7772 7784 7795 7807 7818 7830 7841 7852 




7864 7875 7887 7898 7910 7921 7933 7944 7956 7967 


880 


57978 67990 68001 68013 68024 68035 58047 58058 58070 58081 




8092 8104 8116 8127 8138 8149 8161 8172 8184 8195 




8206 8218 8229 8240 8262 8263 8274 8286 8297 8309 




8320 8331 8343 8354 8365 8377 8388 8399 8410 8422 




8433 8444 8466 8467 8478 8490 8601 8612 8524 8635 




8646 8657 8569 8580 8691 8602 8614 8625 B636 8647 




8659 8670 8681 8692 8704 8715 8726 8737 8749 8760 




8771 8782 8794 8805 8816 8827 8838 8850 8861 8872 




8883 8894 8906 8917 8928 8939 8950 8961 8973 8984 




8995 9006 9017 9028 9040 9061 9062 9073 9084 9095 


890 


69106 59118 69129 69140 69161 59162 59173 59184 69196 69207 




9218 9229 9240 9261 9262 9273 9284 9296 9306 9318 




9329 9340 9351 9362 9373 9384 9396 9406 9417 9428 




9439 9450 9461 9472 9483 9494 9606 9617 9528 9539 




9650 9561 9572 9583 9694 9605 9616 9627 9638 9649 




9660 9671 9682 9693 9704 9715 9726 9737 9748 9759 




9770 9780 9791 9802 9813 9824 9835 9846 9867 9868 




9879 9890 9901 9912 9923 9934 9945 9956 9966 9977 




9988 9999 60010 60021 60032 60043 60064 60065 60076 60086 




60097 60108 0119 0130 0141 0152 0163 0173 0184 0196 


400 


60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 





LOGARITHMS OF NUMBERS 123 


N 


0123456789 


400 


60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 


1 


0314 0326 0336 0347 0358 0369 0379 0300 0401 0412 


2 


0423 0433 0444 0456 0466 0477 0487 0498 0609 0620 




0631 0641 0662 0663 0674 0584 0596 0606 0617 0627 




0638 0649 0660 0670 0681 0692 0703 0713 0724 073^ 




0746 0766 0767 0778 0788 0799 0810 0821 0831 0842 




0863 0863 0874 088i 0896 0906 0917 0927 0938 0949 




0969 0970 0981 0991 1002 1013 1023 1034 1045 1065 




1066 1077 1087 1098 1109 1119 1130 1140 1161 1162 




1172 1183 1194 1204 121i 1226 1236 1247 1257 1268 


410 


61278 61289 61300 61310 61321 61331 61342 61352 61363 61374 




1384 1395 1406 1416 1426 1437 1448 1458 1469 1479 




1490 1500 1611 1621 1632 1542 1563 1663 1574 1684 




1596 1606 1616 1627 1637 1648 1668 1669 1679 1690 




1700 1711 1721 1731 1742 1752 1763 1773 1784 1794 




1805- 1815 1826 1836 1847 1867 1868 1878 1888 1899 




1909 1920 1930 1941 1961 1962 1972 1982 1993 2003 




2014 2024 2034 2045 2066 2066 2076 2086 2097 2107 




2118 2128 2138 2149 2159 2170 2180 2190 2201 2211 




2221 2232 2242 2262 2263 2273 2284 2294 2304 2315 


420 


62325 62335 62346 62366 62366 62377 62887 62397 62408 62418 




2428 2439 2449 2469 2469 2480 2490 2500 2511 2521 




2631 2542 2562 2662 2572 2583 2693 2603 2618 2624 




2634 2644 2655 2665 2675 2685 2696 2706 2716 2726 




2737 2747 2757 2767 2778 2788 2798 2808 2818 2829 




2839 2849 2869 2870 2880 2890 2900 2910 2921 2931 




2941 2951 2961 2972 2982 2992 3002 3012 3022 3033 




3043 3053 3063 3073 3083 3094 3104 3114 3124 3134 




3144 3155 3165 3175 3185 3195 3205 3216 3226 3236 




3246 3266 3266 3276 3286 3296 3306 3317 3327 3337 


430 


63347 63367 63367 63377 63387 63397 63407 63417 63428 63438 




3448 3458 3468 3478 3488 3498 3608 3618 3628 3638 




3548 8668 3568 3579 8589 8699 3609 3619 3629 3639 




8649 3659 3669 3679 8689 3699 3709 8719 3729 3739 




3749 3759 3769 3779 3789 8799 3809 8819 3829 8839 




3849 8869 3869 8879 8889 3899 3909 3919 3929 3939 




3949 3959 3969 3979 3988 8998 4008 4018 4028 4088 




4048 4058 4068 4078 4088 4098 4108 4118 4128 4187 




4147 4157 4167 4177 4187 4197 4207 4217 4227 4237 




4246 4256 4266 4276 4286 4296 4306 4316 4326 4385 


440 


64845 64365 64365 64875 64385 64395 64404 64414 64424 64434 




4444 4464 4464 4473 4483 4493 4503 4513 4623 4632 




4542 4662 4562 4572 4682 4591 4601 4611 4621 4681 




4640 4650 4660 4670 4680 4689 4699 4709 4719 4729 




4788 4748 4768 4768 4777 4787 4797 4807 4816 4826 




4836 4846 4856 4866 4875 4885 4895 4904 4914 4924 




4983 4943 4953 4963 4972 4982 4992 6002 6011 6021 




5031 6040 6060 5060 5070 6079 6089 5099 5108 6118 




5128 6137 5147 6157 6167 6176 5186 6196 6205 6215 




5225* 6284 5244 6264 6263 6273 6283 6292 6302 6312 


450 


66821 66831 65341 65350 66360 65369 65379 66389 65898 65408 



124 



LOGARITHMS OF NUMBERS 



450 


0123456789 


65321 65331 66341 65350 66360 65369 65379 66389 65398 65408 


1 


5418 5427 5437 5447 5456 5466 6475 6486 5495 5504 


2 


5514 5623 5533 5543 5552 5662 6571 6581 6591 5600 


8 


5610 5619 5629 5639 6648 6658 6667 6677 5686 5696 


4 


5706 6715 6725 6734 6744 6753 6763 5772 5782 5792 


5 


6801 5811 5820 5830 5839 5849 5858 6868 6877 5887 


6 


5896 5906 5916 6926 5935 5944 6954 6963 5973 5982 


7 


5992 6001 6011 6020 6030 6039 6049 6058 6068 6077 


3 


6087 6096 6106 6115 6124 6134 6143 6153 6162 6172 


d 


6181 6191 6200 6210 6219 6229 6238 6247 6257 6266 


460 


66276 66285 06295 66304 66314 66323 66332 66342 66351 66361 


1 


6370 6380 6389 6398 6408 6417 6427 6436 6445 6455 


2 


6464 6474 6483 6492 6502 6511 6521 6530 6539 6549 


3 


6558 6567 6577 6586 6596 6605 6614 6624 6633 6642 


4 


6652 6661 6671 6680 6689 6699 6708 6717 6727 6736 


6 


6745 6755 6764 6773 6783 6792 6801 6811 6820 6829 


6 


6839 6848 6857 6867 6876 6885 6894 6904 6913 6922 


7 


6932 6941 ,6950 6960 6969 6978 6987 6997 7006 7016 


S 


7025 7034 7043 7052 7062 7071 7080 7089 7099 7108 


9 


7117 7127 7136 7146 7154 7164 7173 7182 7191 7201 


4:70 


67210 67219 67228 67237 67247 67256 67266 67274 67284 67293 


1 


7302 7311 7321 7330 7339 7348 7357 7367 7376 7385 


2 


7394 7403 7413 7422 7431 7440 7449 7459 7468 7477 


3 


7486 7495 7504 7514 7523 7532 7541 7550 7560 7569 


4 


7578 7587 7596 7605 7614 7624 7633 7642 7651 7660 


5 


7669 7679 7688 7697 7706 7716 7724 7733 7742 7762 


6 


7761 7770 7779 7788 7797 7806 7816 7825 7834 7843 


7 


7852 7861 7870 7879 7888 7897 7906 7916 7925 7934 


8 


7943 7952 7961 7970 7979 7988 7997 8006 8015 8024 


9 


8034 8043 8052 8061 8070 8079 8088 8097 8106 8115 


4S0 


68124 68133 68142 68151 68160 68169 68178 68187 68196 68205 


1 


8215 8224 8233 8242 8251 8260 8269 8278 8287 8296 


2 


8305 8314 8323 8332 8341 8350 8359 8368 8377 8386 


3 


8395 8404 8413 8422 8431 8440 8449 8458 8467 8476 


4 


8485 8494 8502 8511 8520 8529 8538 8547 8566 8666 


6 


8574 8583 8592 8601 8610 8619 8628 8637 8646 8655 


6 


8664 8673 8681 8690 8699 8708 8717 8726 8736 8744 


7 


8753 8762 8771 8780 8789 8797 8806 8816 8824 8833 


8 


8842 8851 8860 8869 8878 8886 8896 8904 8913 8922 


9 


8931 8940 8949 8958 8966 8976 8984 8993 9002 9011 


490 


69020 69028 69037 69046 69055 69064 69073 69082 69090 69099 


1 


9108 9117 9126 9135 9144 9152 9161 9170 9179 9188 


2 


9197 9205 9214 9223 9232 9241 9249 9268 9267 9276 


3 


9285 9294 9302 9311 9320 9329 9338 9346 9366 9364 


4 


9373 9381 9390 9399 9408 9417 9425 9434 9443 9452 


5 


9461 9469 9478 9487 9496 9504 9513 9522 9631 9539 


6 


9648 9557 9566 9574 9583 9592 9601 9609 9618 9627 


7 


9636 9644 9653 9662 9671 9679 9688 9697 9706 9714 


8 


9723 9732 9740 9749 9758 9767 9775 9784 9793 9801 


9 


9810 9819 9827 9836 9845 9854 9862 9871 '9880 9888 


500 


69897 69906 69914 69923 69932 69940 69949 69968 69966 69975 



LOGARITHMS OF NUMBERS 126 



N- 


0123456789 


500 


09897 69906 69914 69923 69932 69940 69949 69958 69966 69975 


1 


9984 9992 70001 70010 70018 70027 70086 70044 70063 70062 


2 


70070 70079 0088 0096 010^ 0114 0122 0131 0140 0148 


3 


0167 0165 0174 0183 0191 0200 0209 0217 0226 0234 


4 


0243 0252 0260 0269 0278 0286 0295 0303 0312 0321 


5 


0329 0338 0346 035i 0364 0372 0381 0389 0398 0406 


6 


0416 0424 0432 0441 0449 0468 0467 0475 0484 0492 


7 


0601 0509 0518 0626 0536 0544 0662 0661 0669 0678 


8 


0686 0695 0603 0612 0621 0629 0638 0646 0665 0663 


9 


0672 0680 0689 0697 0706 0714 0723 0731 0740 0749 


510 


70767 70766 70774 70783 70791 70800 70808 70817 70826 70834 


1 


0842 0861 0859 0868 0876 0885 0893 0902 0910 0919 


2 


0927 0935 0944 0952 0961 0969 0978 0986 0995 1003 


3 


1012 1020 1029 1037 1046 1054 1063 1071 1079 1088 


4 


1096 1105 1113 1122 1130 1139 1147 1166 1164 1172 


5 


1181 1189 1198 1206 1214 1223 1231 1240 1248 1257 


6 


1265 1273 1282 1290 1299 1307 1315 1324 1332 1341 


7 


1349 1367 1366 1374 1383 1391 1399 1408 1416 1425 


8 


1433 1441 1450 1458 1466 1475 1483 1492 1500 1608 


9 


1617 1625 1633 1542 1560 1559 1567 1675 1684 1592 


520 


71600 71609 71617 71626 71634 71642 71660 71669 71667 71675 


1 


1684 1692 1700 1709 1717 1726 1734 1742 1750 1759 


2 


1767 1776 1784 1792 1800 1809 1817 1826 1834 1842 


3 


.I860 1868 1867 1876 1883 1892 1900 1908 1917 1925 


4 


1933 1941 1950 1958 1966 1975 1983 1991 1999 2008 


6 


2016 2024 2032 2041 2049 2057 2066 2074 2082 2090 


6 


2099 2107 2116 2123 2132 2140 2148 2156 2165 2173 


7 


2181 2189 2198 2206 2214 2222 2230 2239 2247 2255 


8 


2263 2272 2280 2288 2296 2304 2313 2321 2329 2337 


9 


2346 2364 2362 2370 2378 2387 2395 2403 2411 2419 


680 


72428 72436 72444 72452 72460 72469 72477 72485 72493 72501 


1 


2609 2518 2526 2534 2542 2550 2568 2667 2575 2583 


2 


2691 2699 2607 2616 2624 2632 2640 2648 2656 2665 


.3 


2673 2681 2689 2697 2705 2713 2722 2730 2738 2746 


4 


2764 2762 2770 2779 2787 '2795 2803 2811 2819 2827 


5 


2836 2843 2852 2860 2868 2876 2884 2892 2900 2908 


6 


2916 2925 2933- 2941 2949 2957 2965 2973 2981 2989 


7 


2997 3006 3014 3022 3030 3038 3046 3054 3062 3070 


^ 8 


3078 3086 3094 3102 3111 3119 3127 3135 3143 3151 


9 


3169 3167 3175 3183 3191 3199 3207 3215 3223 3231 


640 


73239 73247 73255 73263 73^72 73280 73288 73296 73304 73312 


1 


3320 3328 3336 3344 3352 3360 3368 3376 3384 3392 


2 


3400 3408 3416 3424 3432 3440 3448 3456 3464 3472 


3 


3480 3488 3496 3504 3512 3520 3528 3536 3544 3552 


4 


3560 3568 3576 3584 3592 3600 3608 3616 3624 3632 


5 


3640 3648 3656 3664 3672 3679 3687 3695 3703 3711 


6 


3719 3727 3735 3743 3751 3759 3767 3775 3783 3791 


7 


3799 3807 3815 3823 3830 3838 3846 3854 3862 3870 


8 


3878 3886 3894 3902 3910 3018 3926 3933 3941 3949 


9 


3957 3965 3973 3981 3989 3997 4005 4013 4020 4028 


550 


74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 



126 




LOGARITHMS OF NUMBESS 


N 


O 


1234567 8 9 


550 


74036 74044 74052 74060 74068 74076 74084 74092 74099 74107'| 


1 


4116 


4123 4131 4139 4147-4155 4162 4170 4178 4186 


2 


.4194 


4202 4210 4218 4225 4233 4241 4249 4267 4265 


3 


4273 


4280 4288 4296 4304 4312 4320 4327 4335 4343 


4 


4361 


4369 4367 4374 4382 4390 4398 4406 4414 4421 


5 


4429 


4437 4445 4463 4461 4468 4476 4484 4492 4500 


6 


4607 


4615 4523 4631 4639 4647 4654 4562 4570 4678 


7 


4686 


4693 4601 4609 4617 4624 4632 4640 4648 4656 


8 


4663 


4671 4679 4087 4695 4702 4710 4718 4726 4733 


« 


4741 


4749 4767 4764 4772 4780 4788 4796 4803 4811 


560 


74819 74827 74834 74842 748^0 74868 74866 74873 74881 74889 | 


1 


4896 


4904 4912 4920 4927 4935 4943 4950 4958 4966 


2 


4974 


4981 4989 4997 6005 6012 5020 6028 5035 6043 


8 


6061 


5059 6066 6074 5082 6089 6097 5105 6113 6120 


4 


6128 


6136 6143 6151 5169 6166 6174 5182 6189 6197 


6 


6205 


6213 6220 6228 6236 5243 5261 6259 6266 6274 


6 


6282 


5289 6297 6305 6312 6320 5328 5336 5343 6351 


7 


6368 


5366 5374 5381 6389 6397 6404 6412 6420 6427 


8 


5435 


6442 5460 6468 6465 6473 5481 6488 5496 5504 


9 


6611 


6619 6526 6634 6642 5649 6567 6565 6672 6580 


570 


76687 76696 75603 76610 76618 76626 76633 76641 76648 76666 | 


1 


6664 


6671 6679 6686 6694 5702 6709 5717 5724 6732 


2 


6740 


5747 6765 6762 5770 6778 5786 6793 5800 5808 


3 


5815 


5823 5831 6838 5846 6853 5861 6868 687^ 5884 


4 


5891 


6899 6906 5914 6921 6929 6937 6944 6962 5969 


6 


6967 


6974 5982 5989 6997 6005 6012 6020 6027 6035 


6 


6042 


6050 6067 6065 6072 6080 6087 6095 6103 6110 


7 


6118 


6125 6133 6140 6148 6155 6163 6170 6178 6185 


8 


6193 


6200 6208 6216 6223 6230 6238 6246 6253 6260 


9 


6268 


6275 6283 6290 6298 6306 6313 6320 6328 6335 


5S0 


76343 76360 76358 76366 76373 76380 76388 76396 76403 76410 | 


1 


6418 


6425 6433 6440 6448 6455 6462 6470 6477 6485 


2 


6492 


6500 6507 6515 6522 6530 6537 6545 6562 6559 


3 


6567 


6574 6582 6589 6597 6604 6612 6619 6626 6634 


4 


6641 


6649 6656 6664 6671 6678 6686 6693 6701 6708 


6 


6716 


6723 6730 6738 6746 6753 6760 6768 6775 6782 


6 


6790 


6797 6805 6812 6819 6827 6834 6842 6849 6856 


7 


6864 


6871 6879 6886 6893 6901 6908 6916 6923 6930 


8 


6938 


6945 6953 6960 6967 6975 6982 6989 6997 7004 


9 


7012 


7019 7026 7034 7041 7048 7066 7063 7070 7078 


590 


77086 77093 77100 77107 77115 77122 77129 77137 77144 77161 1 


1 


7169 


7166 7173 7181 7188 7196 7203 7210 7217 7225 


2 


7232 


7240 7247 7254 7262 7269 7276 7283 7291 7298 


8 


7305 


7313 7320 7327 7335 7342 7349 7367 7364 7371 


4 


7379 


7386 7393 7401 7408 7415 7422 7430 7437 7444 


5 


7462 


7459 7466 7474 7481 7488 7495 7503 7610 7517 


6 


7625 


7632 7639 7546 7554 7561 7568 7676 7683 7590 


7 


7697 


7605 7612 7619 7627 7634 7641 7648 7656 7663 


8 


7670 


7677 7685 7692 7699 7706 7714 7721 7728 7735 


9 


7748 


7750 7767 7764 7772 7779 7786 7793 7801 7808 


600 


77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 





LOGARITHMS OF NUMBERS 127 


N 


0123456789 


600 


77816 77822 77830 77837 77844 77861 77869 77866 77873 77880 




7887 7895 7902 7909 7916 7924 7931 7938 7946 7962 




7960 7967 7974 7981 7988 7996 8003 8010 8017 8025 




8032 8039 8046 8063 8061 8068 8075 8082 8089 8097 




8104 8111 8118 8126 8132 8140 8147 8164 8161 8168 




8176 8183 8190 8197 8204 8211 8219 8226 8233 8240 




8247 8264 8262 8269 8276 8283 8290 8297 8305 8312 




8319 8326 8333 8340 8347 8365 8362 8369 8376 8383 




8390 8398 8405 8412 8419 8426 8433 8440 8447 8465 




8462 8469 8476 8483 8490 8497 8604 8612 8619 8626 


610 


78633 78640 78647 78664 78661 78669 78676 78683 78690 78597 




8604 8611 8618 8626 8633 8640 8647 8664 8661 8668 




• 8676 8682 8689 8696 8704 8711 8718 8725 8732 8739 




8746 8763 8760 8767 8774 8781 87-89 8796 8803 8810 




8817 8824 8831 8838 8845 8852 8859 8866 8873 8880 




8888 8895 8902 8909 8916 8923 8930 8937 8944 8961 




8958 8966 8972 8979 8986 8993 9000 9007 9014 9021 




9029 9036 9043 9050 9067 9064 9071 9078 9085 9092 




' 9099 9106 9113 9120 9127 9134 9141 9148 9166 9162 




9169 9176 9183 9190 9197 9204 9211 9218 9226 9232 


620 


79239 79246 79253 79260 79267 79274 79281 79288 79296 79302 




9309 9316 9323 9330 9337 9344 9361 9368 9366 9372 




9379 9386 9393 9400 9407 9414 9421 9428 9435 9442 




9449 9456 9463 9470 9477 9484 9491 9498 9505 9511 




9618 9625 9532 9539 9546 9553 9660 9567 9674 9581 




9688 9695 9602 9609 9616 9623 9630 9637 9644 9650 




9667 9664 9671 9678 9685 9692 9699 9706 9713 9720 




9727 9734 9741 9748 9764 9761 9768 9776 9782 9789 




9796 9803 9810 9817 9824 9831 9837 9844 9861 9858 




9866 9872 9879 9886 9893 9900 9906 9913 9920 9927 


680 


79934 79941 79948 79955 79962 79969 79975 79982 79989 79996 




80003 80010 80017 80024 80030 80037 80044 80051 80068 80065 




0072 0079 0085 0092 0099 0106 0113 0120 0127 0134 




0140 0147 0164 0161 0168 0175 0182 0188 0195 0202 




0209 0216 0223 0229 0236 0243 0250 0257 0264 0271 




0277 0284 0291 0298 0305 0312 0318 0325 0332 0339 




0346 0353 0359 0366 0373 0380 0387 0393 0400 0407 




0414 0421 0428 0434 0441 0448 0465 0462 0468 0476 




0482 0489 0496 0602 0509 0516 0523 0530 0636 0543 




0660 0657 0564 0570 0577 0584 0591 0598 0604 0611 


640 


80618 80625 80632 8063P 80645 80652 80669 80665 80672 80679 




0686 0693 0699 0706 0713 0720 0726 0733 0740 0747 




0754 0760 0767 0774 0781 0787 0794 0801 0808 0814 




0821 0828 0835 0841 0848 0855 0862 0868 0876 0882 




0889 0895 0902 0909 0916 0922 0929 0936 0943 0949 




0966 0963 0969 0976 0983 0990 0996 1003 1010 1017 




1023 1030 1037 1043 1050 1057 1064 1070 1077 1084 




1090 1097 1104 1111 1117 1124 1131 1137 1144 1151 




1158 1164 1171 1178 1184 1191 1198 1204 1211 1218 




1224 1231 1238 1245 1251 1268 1265 1271 1278 1285 


660 


81291 81298 81305 81311 81318 81325 81331 81338 81345 81361 



L28 LOGARITHMS OF NUMBERS 



N 


0123456789 


650 


81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 




1368 1365 1371 1378 1385 1391 1398 1405 1411 1418 




1425 1431 1438 1445 1451 1458 1465 1471 1478 1485 




1491 1498 1505 1511 1518 1525 1531 1538 1544 1551 




1558 1564 1571 1578 1584 1591 1598 1604 1611 1617 




1624 1631 1637 1644 1651 1657 1664 1671 1677 1684 


6 


1690 1697 1704 1710 1717 1723 1730 1737 1743 1750 


7 


1757 1763 1770 1776 1783 1790 1796 1803 1809 1816 


8 


1823 1829 1836 1842 1849 1856 1862 1869 1875 1882 


9 


1889 1895 1902 1908 1915 1921 1928 1935 1941 1948 


660 


81954 81961 81968 81974 81981 81987 81994 82000 82007 82014 


1 


2020 2027 2033 2040 2046 2053 2060 2066 2073 2079 


2 


2086 2092 2099 2105 2112 2119 2126 2132 2138- 2145 


8 


2151 2158 2164 2171 2178 2184 2191 2197 2204 2210 


4 


2217 2223 2230 2236 2243 2249 2256 2263 2269 2276 


6 


2282 2289 2296 2302 2308 2315 2321 2328 2334 2341 


6 


2347 2354 2360 2367 2373 2380 2387 2393 2400 2406 


7 


2413 2419 2426 2432 2439 2445 2452 2458 2465 2471 


8 


2478 2484 2491 2497 2504 2510 2517 2523 2530^ 2536 


9 


2643 2549 2556 2562 2569 2575 2682 2588 2595 2601 


670 


82607 82614 82620 82627 82633 82640 82646 82653 82669 82666 


1 


2672 2679 2685 2692 2698 2705 2711 2718 2724 2730 


3 


2737 2743 2750 2756 2763 2769 2776 2782 2789 2796 


3 


2802 2808 2814 2821 2827 2834 2840 2847 2863 2860 


4 


2866 2872 2879 2886 2892 2898 2905 2911 2918 2924 


. 5 


2930 2937 2943 2950 2956 2963 2969 2975 2982 2988 


6 


2995 3001 3008 3014 3020 3027 3033 3040 3046 3052 


7 


3059 3065 3072 3078 3085 3091 3097 3104 3110 3117 


8 


3123 3129 3136 3142 3149 3155 3161 3168 8174 3181 


9 


3187 3193 3200 3206 3213 3219 3226 3232 3238 3245 


680 


83251 83257 83264 83270 83276 83283 83289 83296 83302 83308 


1 


3315 3321 3327 3334 3340 3347 3363 3359 3366 3372 


2 


3378 3385 3391 3398 3404 3410 3417 3423 3429 3436 


3 


3442 3448 3465 3461 3467 3474 3480 3487 3493 3499 


4 


3606 3612 3518 3525 3531 3537 3544 3550 3666 3563 


5 


3669 3576 3582 3588 3594 3601 3607 3613 3620 3626 


6 


3632 3639 3645 3651 3658 3664 3670 3677 3683 3689 


7 


3696 3702 3708 3715 3721 3727 3734 3740 3746 3763 


8 


3759 3766 3771 3778 3784 3790 3797 3803 3809 3816 


9 


3822 3828 3835 3841 3847 3863 3860 3866 3872 3879 


690 


83885 83891 83897 83904 83910 83916 83923 83929 83935 83942 


1 


8948 3954 3960 3967 3973 3979 3985 3992 3998 4004 


2 


4011 4017 4023 4029 4036 4042 4048 4055 4061 4067 


3 


4073 4080 4086 4092 4098 4105 4111 4117 4123 4130 


4 


4136 4142 4148 4155 4161 4167 4173 4180 4186 4192 


5 


4198 4205 4211 4217 4223 4230 4236 4242 4248 4255 


6 


4261 4267 4273 4280 4286 4292 4298 4305 4311 4317 


7 


4323 4330 4336 4342 4348 4354 4361 4367 4373 4379 


8 


4386 4392 4398 4404 4410 4417 4423 4429 4435 4442 


9 


4448 4454 4460 4466 4473 4479 4485 4491 4497 4604 


700 


84610 84516 84522 84528 84635 84541 84647 84663 84569 84560 





LOGARITHMS OF M UMBERS 129 


N 


0123456789 


700 84610 84516 84522 84528 84535 84541 84547 84553 84559 84566 | 


1 


4672 4678 4584 4590 4597 4603 4609 4616 4621 4628 


2 


4634 4640 4646 4652 4658 4665 4671 4677 4683 4689 


8 


4696 4702 4708 4714 4720 4726 4733 4739 4745 4761 


4 


4757 4763 4770 4776 4782 4788 4794 4800 4807 4813 


6 


4819 4825 4831 4837 4844 4850 4866 4862 4868 4874 


6 


4880 4887 4893 4899 4906 4911 4917 4924 4930 4936 


7 


4942 4948 4954 4960 4967 4973 4979 4985 4991 4997 


8 


6003 6009 5016 6022 6028 5034 5040 5046 5062 6058 


9 


6065 5071 5077 5083 6089 5095 5101 5107 5114 5120 


710 


86126 86132 86138 86144 86150 85166 86163 86169 85175 85181 


1 


5187 5193 5199 6206 6211 6217 5224 5230 5236 6242 


2 


6248 5254 5260 5266 5272 5278 5285 5291 6297 6303 


8 


6309 5315 6321 5327 5333 5339 5346 5352 5358 6364 


4 


6370 6376 6382 5388 5394 5400 5406 6412 5418 6425 


5 


6431 6437 5443 5449 5465 5461 M67 6473 5479 6486 


6 


6491 6497 6603 5609 5516 5522 6528 5634 5540 5546 


7 


6652 5558 6564 6670 5576 5582 6688 6694 6600 5606 


8 


6612 5618 5625 6631 6637 5643 5649 6665 6661 6667 


9 


6673 5679 5685 5691 6697 5703 6709 5715 5721 6727 


720 


85733 85739 85746 85761 86757 85763 85769 85776 86781 86788 


1 


6794 6800 6806 6812 5818 5824 6830 6836 5842 6848 


2 


6854 5860 6866 6872 5878 6884 6890 5896 5902 6908 


8 


6914 5920 6926 6932 6938 6944 5950 5966 5962 5968 


4 


6974 6980 6986 6992 5998 6004 6010 6016 6022 6028 


6 


6034 6040 6046 6052 6068 6064 6070 6076 6082 6088 


6 


6094 6100 6106 6112 6118 6124 6130 6136 6141 6147 


7 


6153 6159 6166 6171 6177 6183 6189 6196 6201 6207 


8 


6213 6219 6226 6231 6237 6243 6249 6255 6261 6267 


9 


6273 6279 6285 6291 6297 6303 6308 6314 6320 6326 


780 


86a32 86338 86344 86350 86356 86362 86368 86374 86380 86386 


1 


6392 6398 6404 6410 6416 6421 6427 6433 6439 6445 


2 


6461 6457 6463 6469 6475 6481 6487 6493 6499 6604 


8 


6610 6616 6522 6528 6634 6640 6646 6662 6668 6664 


4 


6670 6576 6681 6687 6693 6699 6606 6611 6617 6623 


5 


6629 6635 6641 6646 6662 6668 6664 6670 6676 6682 


6 


6688 6694 6700 6706 6711 6717 6723 6729 6735 6741 


7 


6747 6763 6759 6764 6770 6776 6782 6788 6794 6800 


8 


6806 6812 6817 6823 6829 6835 6841 6847 6863 6869 


9 


6864 6870 6876 6882 6888 6894 6900 6906 6911 6917 


740 


86923 86929 86935 86941 86947 86963 86958 86964 86970 86976 


1 


6982 6988 6994 6999 7005 7011 7017 7023 7029 7035 


8 


7040 7046 7062 7068 7064 7070 7075 7081 7087 7093 


8 


7099 7105 7111 7116 7122 7128 7134 7140 7146 7161 


4 


7167 7163 7169 7175 7181 7186 7192 7198 7204 7210 


5 


7216 7221 7227 7233 7239 7245 7261 7266 7262 7268 


6 


7274 7280 7286 7291 7297 7303 7809 7315 7320 7326 


7 


7332 7338 7844 7349 7855 7361 7367 7378 7379 7384 


8 


7390 7396 7402 7408 7413 7419 7425 7431 7437 7442 


9 


7448 7464 7460 7466 7471 7477 7483 7489 7495 7600 


750 


87606 87612 87618 87628 87629 87586 87541 87647 87662 87658 



130 


LOGARITHMS OF NUMBERS 


N 


123456789 


750 


87606 87612 87618 87623 87629 87635 87541 87647 87652 87668 


1 


7664 7570 7676 7581 7587 7593 7599 7604 7610 7616 


2 


7622 7628 7633 7639 764i 7651 7656 7662 7668 7674 


8 


7679 7686 7691 7697 7703 7708 7714 7720 7726 7731 


4 


7737 7743 7749 7754 7760 7766 7772 7777 7783 7789 


6 


7795 7800 7806 7812 7818 7823 7829 7835 7841 7846 


6 


7852 7858 7864 7869 7876 7881 7887 7892 7898 7904 


7 


7910 7915 7921 7927 7933 7938 7944 7950 7966 7961 


8 


7967 7973 7978 7984 7990 7996 8001 8007 8013 8018 


9 


8024 8030 8036 8041 8047 8063 8068 8064 8070 8076 


760 


88081 88087 88093 88098 88104 88110 88116 88121 88127 88133 


1 


8138 8144 8150 8156 8161 8167 8173 8178 8184 8190 


2 


8196 8201 8207 8213 8218 8224 8230 8236 8241 8247 


8 


8262 8258 8264 8270 8275 8281 8287 8292 8298 8304 


4 


8309 8316 8321 8326 8332 '8338 8343 8349 8355 8360 


6 


8366 8372 8377 8383 8389 8395 8400 8406 8412 8417 


6 


8423 8429 8434 8440 8446 8451 8457 8463 8468 8474 


7 


8480 8485 8491 8497 8502 8508 8513 8519 8525 8630 


8 


8536 8542 8647 8553 8559 8564 8570 8676 8581 8687 


9 


8693 8598 8604 8610 8616 8621 8627 8632 8638 8643 


770 


88649 88665 88660 88666 88672 88677 88683 88689 88694 88700 


1 


8705 8711 8717 8722 8728 8734 8739 8745 8760 8756 


2 


8762 8767 8773 8779 8784 8790 8796 8801 8807 8812 


3 


8818 8824 8829 8835 8840 8846 8852 8867 8863 8868 


4 


8874 8880 8886 8891 8897 8902 8908 8913 8919 8925 


5 


8930 8936 8941 8947 8963 8958 8964 8969 8975 8981 


8 


8986 8992 8997 9003 9009 9014 9020 9026 9031 9037 


7 


9042 9048 9053 9059 9064 9070 9076 9081 9087 9092 


8 


9098 9104 9109 9115 9120 9126 9131 9137 9143 9148 


9 


9164 9159 9165 9170 9176 9182 9187 9193 9198 9204 


780 


89209 89216 89221 89226 89232 89237 89243 89248 89264 89260 


1 


9265 9271 9276 9282 9287 9293 9298 9304 9310 9315 


2 


9321 9326 9332 9337 9343 9348 9354 9360 9366 9371 


8 


9376 9382 9387 9393 9398 9404 9409 9415 9421 9426 


4 


9432 9437 9443 9448 9464 9459 9465 9470 9476 9481 


5 


9487 9492 9498 9504 9509 9515 9620 9526 9631 9637 


6 


9642 9548 9563 9559 9564 9570 9576 9581 9586 9692 


7 


9597 9603 9609 9614 9620 9626 9631 9636 9642 9647 


8 


9663 9668 9664 9669 9675 9680 9686 9691 9697 9702 


9 


9708 9713 9719 9724 9730 9735 9741 9746 9762 9757 


790 


89763 89768 89774 89779 89785 89790 89796 89801 89807 89812 


1 


9818 9823 9829 9834 9840 9845 9851 9856 9862 9867 


2 


9873 9878 9883 9889 9894 9900 9906 9911 9916 9922 


8 


9927 9933 9938 9944 9949 9955 9960 9966 9971 9977 


4 


9982 9988 9993 9998 90004 90009 90015 90020 90026 90031 


5 


90037 90042 90048 90053 0059 0064 0069 0075 0080 0086 


6 


0091 0097 0102 0108 0113 0119 0124 0129 0135 0140 


7 


0146 0151 0157 0162 0168 0173 0179 0184 0189 0195 


8 


0200 0206 0211 0217 0222 0227 0233 0238 0244 0249 


9 


0265 0260 0266 0271 0276 0282 0287 0293 0298 0304 


800 


90309 90314 90320 90326 90331 90336 90342 90347 90352 90368 





LOGARITHMS OF NUMBERS 131 


N 


0123456789 


800 


90309 90314 90320 90326 90331 90336 90342 90347 90352 90368 


1 


0363 0369 0374 0380 0385 0390 0396 0401 0407 0412 


2 


0417 0423 0428 0434 0439 0445 04i0 0466 0461 0466 


3 


0472 0477 0482 0488 0493 0499 0604 0509 0615 0620 


4 


0626 0631 0636 0642 0547 0663 0558 0663 0569 0674 


5 


0680 0685 0690 0696 0601 0607 0612 0617 0623 0628 


6 


0634 0639 0644 06^0 0666 0660 0666 0671 0677 0682 


7 


0687 0693 0698 0703 0709 0714 0720 0726 0730 0736 


8 


0741 0747 0762 0767 0763 0768 0773 0779 0784 0789 


9 


P79i 0800 0806 0811 0816 0822 0827 0832 0838 0843 


810 


90849 90854 90859 90865 90870 90875 90881 90886 90891 90897 


1 


0902 0907 0913 0918 0924 0929 0934 0940 0945 0960 


2 


0966 0961 0966 0972 0977 0982 0988 0993 0998 1004 


8 


1009 1014 1020 1026 1030 1036 1041 1046 1062 1067 


4 


1062 1068 1073 1078 1084 1089 1094 1100 1106 1110 


5 


1116 1121 1126 1132 1137 1142 1148 1163 1168 1164 


6 


1169 1174 1180 1185 1190 1196 1201 1206 1212 1217 


7 


1222 1228 1233 1238 1243 1249 1264 1259 1265 1270 


8 


1276 1281 1286 1291 1297 1302 1307 1312 1318 1323 


9 


1328 1334 1339 1344 1350 1365 1360 1366 1371 1376 


820 


91381 91387 91392 91397 91403 91408 91413 91418 91424 91429 


1 


1434 1440 1445 1450 1456 1461 1466 1471 1477 1482 


2 


1487 1492 1498 1503 1608 1514 1619 1624 1529 1635 


8 


1640 1645 1561 1666 1661 1566 1672 1677 1682 1687 


4 


1693 1698 1603 1609 1614 1619 1624 1630 1635 1640 


5 


1645 1661 1666 1661 1666 1672 1677 1682 1687 1693 


6 


1698 1703 1709 1714 1719 1724 1730 1735 1740 1745 


7 


1761 1766 1761 1766 1772 1777 1782 1787 1793 1798 


8 


1803 1808 1814 1819 1824 1829 1834 1840 1845 1860 


9 


1855 1861 1866 1871 1876 1882 1887 1892 1897 1903 


880 


91908 91913 91918 91924 91929 91934 91939 91944 91950 91955 


1 


1960 1966 1971 1976 1981 1986 1991 1997 2002 2007 


2 


2012 2018 2023 2028 2033 2038 2044 2049 2(564 2069 


8 


2065 2070 2075 2080 2085 2091 2096 2101 2106 2111 


4 


2117 2122 2127 2132 2137 2143 2148 2153 2168 2163 


6 


2169 2174 2179 2184 2189 2195 2200 2206 2210 2216 


6 


2221 2226 2231 2236 2241 2247 2252 2267 2262 2267 


7 


2273 2278 2283 2288 2293 2298 2304 2309 2314 2319 


8 


2324 2330 2335 2340 2346 2360 2365 2361 2366 2371 


9 


2376 2381 2387 2392 2397 2402 2407 2412 2418 2423 


840 


92428 92433 92438 92443 92449 92464 92469 92464 92469 92474 


1 


2480 2485 2490 2496 2600 2606 2611 2516 2521 2626 


2 


2531 2636 2542 2547 2552 2567 2662 2667 2572 2578 


8 


2683 2588 2693 2698 2603 2609 2614 2619 2624 2629 


4 


2634 2639 2645 2650 2655 2660 2665 2670 2675 2681 


6 


2686 2691 2696 2701 2706 2711 2716 2722 2727 2732 


6 


2737 2742 2747 2762 2758 2763 2768 2773 2778 2783 


7 


2788 2793 2799 2804 2809 2814 2819 2824 2829 2834 


3 


2840 2845 2850 2855 2860 2866 2870 2875 2881 2886 


9 


2891 2896 2901 2906 2911 2916 2921 2927 2932 2937 


860 


92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 



132 


LOGARITHMS OF NUMBERS 


N 


0123456789 


850 


02942 92947 92952 92967 92962 92967 92973 92978 92983 92988 


1 


2993 2998 3003 3008 3013 3018 3024 3029 3034 3039 


2 


3044 3049 3064 3069 3064 3069 3075 3080 3085 3090 


8 


3095 3100 3106 3110 3116 3120 3126 3131 3136 3141 


4 


3146 3161 3166 3161 3166 3171 3176 3181 3186 3192 


6 


3197 3202 3207 3212 3217 3222 3227 3232 3237 3242 


6 


3247 3262 3268 3263 3268 3273 3278 3283 3288 3293 


7 


3298 3303 3308 3313 3318 3323 3328 3334 3339 3:^44 


8 


3349 3354 3369 3364 3369 3374 3379 3384 3389 3394 


9 


3399 3404 3409 3414 3420 342i 3430 343i 3440 344^ 


860 


934^0 93465 93460 93465 93470 93476 93480 93485 93490 03496 


1 


3600 3606 3610 3616 3620 3626 3631 3636 3541 3546 


2 


3561 3556 3661 3566 3571 3576 3581 3586 3691 3596 


8 


3601 3606 3611 3616 3621 3626 3631 3636 3641 3646 


4 


3661 3666 3661 3666 3671 3676 3682 3687 3692 3697 


6 


3702 3707 3712 3717 3722 3727 3782 3737 3742 3747 


6 


3762 3757 3762 3767 3772 3777 3782 3787 3792 3797 


7 


3802 3807 3812 3817 3822 3827 3832 3837 3842 3847 


8 


3862 3857 3862 3867 3872 3877 3882 3887 3892 3897 


9 


3902 3907 3912 3917 3922 3927 3932 3937 3942 3947 


870 


93962 93967 93962 93967 93972 93977 93982 93987 93992 93997 


1 


4002 4007 4012 4017 4022 4027 4032 4037 4042 4047 


2 


4052 4067 4062 4067 4072 4077 4082 4086 4091 4096 


8 


4101 4106 41H 4116 4121 4126 4131 4136 4141 4146 


4 


4161 4166 4161 4166 4171 4176 4181 4186 4191 4196 


5 


4201 4206 4211 4216 4221 4226 4231 4236 4240 4246 


6 


4260 4255 4260 4266 4270 4276 4280 4285 4290 4296 


7 


4300 4305 4310 4315 4320 .4325 4330 4335 4340 4345 


8 


4349 4354 4369 4364 4369 4374 4379 4384 4389 4394 


9 


4399 4404 4409 4414 4419 4424 4429 4433 4438 4443 


880 


94448 94463 94468 94463 94468 94473 94478 94483 94488 94493 


1 


4498 4603 4607 4512 4517 4622 4527 4632 4637 4642 


2 


4647 4552 4667 4662 4567 4571 4576 4681 4586 4691 


8 


4696 4601 4606 4611 4616 4621 4626 4630 4636 4640 


4 


4646 4650 4656 4660 4665 4670 4675 4680 4685 4689 


5 


4694 4699 4704 4709 4714 4719 4724 4729 4734 4738 


6 


4743 4748 4763 4768 4763 4768 4773 4778 4783 4787 


7 


4792 4797 4802 4807 4812 4817 4822 4827 4832 4836 


8 


4841 4846 4861 4856 4861 4866 4871 4876 4880 4885 


9 


4890 4895 4900 4905 4910 4915 4919 4924 4929 4934 


890 


94939 94944 94949 94964 94959 94963 94968 94973 94978 94983 


1 


4988 4993 4998 6002 6007 5012 6017 6022 6027 6032 


2 


6036 5041 5046 5061 5056 6061 6066 6071 6076 6080 


8 


6085 6090 6095 5100 5105 6109 6114 6119 5124 6129 


4 


6134 6139 6143 6148 5153 6158 5163 6168 6173 6177 


5 


6182 6187 5192 6197 5202 6207 6211 5216 6221 6226 


6 


6231 5236 6240 5246 5260 5255 5260 6265 6270 5274 


7 


6279 6284 5289 5294 6299 5303 5308 5313 5318 5323 


8 


6328 5332 6337 6342 5347 5352 6367 6361 6366 6371 


9 


5376 6381 6386 6390 5395 6400 6405 6410 6415 5419 


900 


96424 95429 96434 96439 96444 95448 95463 96468 96463 96468 





LOGARITHMS OF NUMBERS 133 


N 


0123456 7 89 


900 


95424 95429 95434 95439 95444 95448 95453 95458 96463 95468 


1 


5472 5477 5482 5487 5492 5497 5501 5506 5511 5516 


2 


5521 5525 5530 5535 5540 554^ 5650 5554 5559 5564 


3 


5569 5574 5678 5583 5588 5593 5598 5602 5607 5612 




5617 5622 5626 5631 5636 5641 5646 6650 5655 5660 




5665 5670 5674 5679 5684 5689 5694 5698 5703 5708 




5713 5718 5722 5727 5732 5737 5742 5746 5751 5756 




5761 5766 5770 5775 5780 5785 6789 5794 5799 5804 




5809 5813 5818 5823 5828 5832 6837 5842 5847 5852 




5856 5861 5866 6871 5875 5880 5885 5890 5895 5899 


910 


95904 95909 95914 96918 95923 95928 95933 96938 96942 96947 




6962 6957 5961 5966 6971 5976 5980 6986 6990 6995 




6999 6004 6009 6014 6019 6023 6028 6033 6038 6042 




6047 6062 6067 6061 6066 6071 6076 6080 6086 6090 




6095 «>^d 6104 6109 6114 6118 6123 6128 6133 6137 




6142 6147 6152 6156 6161 6166 6171 6175 6180 6185 




6190 6194 6199 6204 6209 6213 6218 6223 6227 6232 




6237 6242 6246 6261 6266 6261 6265 6270 6275 6280 




6284 6289 6294 6298 6303 6308 6313 6317 6322 6327 




6332 6336 6341 6346 6350 6355 6360 6365 6369 6374 


920 


963^9 96384 96388 96393 96398 96402 96407 96412 96417 96421 




6426 6431 6436 6440 6445 6450 6464 6469 6464 6468 




6473 6478 6483 6487 6492 6497 6501 6606 6611 6515 




6620 6625 6530 6534 6539 6544 6648 6653 6568 6562 




6567 6572 6577 6581 6586 6591 6696 6600 6605 6609 




6614 6619 6624 6628 6633 6638 6642 6647 6652 6656 




6661 eem 6670 6675 6680 6685 6689 6694 6699 6703 




6708 6713 6717 6722 6727 6731 6736 6741 6745 6760 




6765 6769 6764 6769 6774 6778 6783 6788 6792 6797 




6802 6806 6811 6816 6820 6825 6830 6834 6839 6844 


980 


96848 96863 96858 96862 96867 96872 96876 96881 96886 96890 




6895 6900 6904 6909 6914 6918 6923 6928 6932 6937 




, 6942 6946 6951 6956 6960 6965 6970 6974 6979 6984 


A i 


^ 6988 6993 6997 7002 7007 7011 7016 7021 7025 7030 




7035 7039 7044 7049 7063 7058 7063 7067 7072 7077 




7081 7086 7090 7096 7100 7104 7109 7114 7118 7123 




7128 7132 7137 7142 7146 7161 7166 7160 7165 7169 




7174 7179 7183 7188 7192 7197 7202 7206 7211 7216 




7220 7225 7230 7234 7239 7243 7248 7263 7267 7262 




7267 7271 7276 7280 7286 7290 7294 7299 7304 7308 


940 


97313 97317 97322 97327 97331 97336 97340 97346 97350 97354 




7359 7364 7368 7373 7377 7382 7387 7391 7396 7400 




7405 7410 7414 7419 7424 7428 7433 7437 7442 7447 




7451 7456 7460 7465 7470 7474 7479 7483 7488 7493 




7497 7502 7606 7611 7616 7520 7525 7529 7534 7639 




7643 7648 7652 7567 7562 7666 7571 7575 7580 7685 




7589 7694 7598 7603 7607 7612 7617 7621 7626 7630 




7635 7640 7644 7649 7663 7668 7663 7667 7672 7676 




7681 7685 7690 7695 7699 7704 7708 7713 7717 7722 




7727 7731 7736 7740 7745 7749 7754 7759 7763 7768 


950 


97772 97777 97782 97786 97791 97795 97800 97804 97809 97813 



lU 


LOGARITHMS OP NUMBER 


950 


01234567 89 


07772 97777 07782 07786 97701 07705 07800 07804 97800 07818 


1 


7818 7823 7827 7832 7836 7841 7846 7860 7855 7850 


2 


7864 7868 7873 7877 7882 7886 7891 7806 7000 70Oi 


8 


7000 7014 7018 7023 7028 7032 7037 7041 7046 7050 


4 


7055 7050 7064 7068 7073 7078 7082 7087 7001 7096 


6 


8000 8005 8000 8014 8010 8023 8028 8032 8037 8041 


6 


8046 8050 8055 8050 8064 8068 8073 8078 8082 8087 


7 


8001 8006 8100 8105 8100 8114 8118 8123 8127 8132 


8 


8137 8141 8146 8150 8155 8150 8164 8168 8173 8177 


9 


8182 8186 8101 8105 8200 8204 8200 8214 8218 8223 


960 


08227 08232 08236 08241 08245 08250 08264 08260 08263 08268 


1 


8272 8277 8281 8286 8200 8205 8200 8804 8308 8313 


2 


8318 8322 8327 8331 8336 8340 8345 8340 8354 8358 


3 


8363 8367 8372 8376 8381 8385 8300 8304 8309 8403 


4 


8408 8412 8417 8421 8426 8430 8435 8430 8444 8448 


5 


8453 8457 8462 8466 8471 8476 8480 8484 8480 8493 


6 


8408 8502 8507 8511 8516 8520 8525 8520 8534 8638 


7 


8543 8547 8552 8556 8561 8565 8570 8574 8570 8683 


8 


8688 8502 8507 8601 8606 8610 8614 8610 8623 8628 


9 


8632 8637 8641 8646 8650 8665 8660 8664 8668 8673 


970 


08677 08682 08686 08691 98696 98700 98704 98709 ^8713 98717 


I 


8722 8726 8731 8735 8740 8744 8749 8763 8768 8762 


2 


8767 8771 8776 8780 8784 8780 8703 8708 8802 8807 


3 


8811 8816 8820 8825 8820 8834 8888 8843 8847 8861 


4 


8856 8860 8865 8860 8874 8878 8883 8887 8802 8806 


5 


8000 8005 8000 8014 8018 8023 8027 8932 8936 8941 


6 


8945 8949 8954 8058 8063 8067 8072 8976 8981 8985 


7 


8989 8994 8008 0003 0007 0012 0016 9021 9025 9029 


8 


0034 0038 0043 0047 0052 0056 0061 0065 -0060 9074 


9 


0078 0083 0087 0002 9096 9100 9105 9109 9114 9118 


980 


99123 00127 90131 00136 00140 09145 00140 99154 99168 99162 


1 


0167 0171 0176 0180 0185 9189 9193 9198 9202 9207 


2 


0211 9216 0220 0224 0220 0233 9238 9242 9247^9261 


8 


0266 0260 0264 0260 0273 0277 9282 9286 9291*9296 


4 


9300 0304 0308 0313 0317 0322 0326 0380 0335 9339 


5 


9344 0348 9352 0367 0361 9366 9370 9374 9379 9383 


6 


0388 0302 0306 0401 0406 0410 9414 9419 9423 9427 


7 


9432 9436 9441 9445 9449 9454 9468 9463 9467 9471 


8 


9476 9480 9484 9489 9493 9498 9602 9506 9611 9615 


9 


9520 9524 9528 9633 9537 9642 9646 9660 9565 9659 


990 


99564 99668 99672 99577 99581 99685 99690 99694 99699 99608 


1 


0607 9612 9616 9621 9625 9629 9634 9688 9642 9647 


2 


9651 9656 0660 0664 0660 0673 0677 9682 9686 9691 


8 


9695 9699 9704 9708 9712 9717 9721 9726 9730 9784 


4 


9739 9743 9747 9762 9766 9760 9765 9769 9774 9778 


5 


9782 9787 0791 9795 9800 9804 9808 9813 9817 9822 


6 


9826 0830 0835 0830 0843 0848 0852 9866 9861 9866 


7 


9870 9874 0878 0883 0887 0891 9896 9900 9904 9909 


8 


9918 9917 9922 9926 9980 998S 9939 9944 9948 9962 


9 


9957 9961 9966 9970 9974 9978 9988 9987 9991 9996 


'HJO 


00000 00004 00009 00013 00017 00022 00026 00080 00085 00a39 





LOGARITHMIC SINES AND COSINES 


135 


t 




0« 


1 


. 


2« 




9 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





—00 


10.00000 


8.24186 


9.99993 


8.54282 


999974 


60 


1 


6.46373 


00000 


24903 


99903 


64642 


99973 


69 


s 


764T6 


00000 


25609 


99993 


54909 


99978 


58 


8 


94085 


00000 


26304 


99993 


56354 


99972 


57 


4 


7.0G579 


00000 


26988 


99992 


65705 


99972 


56 


5 


16270 


00000 


27661 


99992 


56054 


69971 


56 


6 


24188 


00000 


28324 


99992 


56400 


99971 


54 


7 


30882 


00000 


2897i 


99992 


56743 


99970 


58 


8 


86682 


■ 00000 


29621 


99992 


57084 


99970 


52 


9 


41797 


00000 


30255 


99991 


57421 


99969 


51 


10 


7.46373 


10.00000 


8.30679 


9.99991 


8.57757 


9.99969 


50 


11 


50512 


00000 


31495 


99991 


58089 


99968 


49 


12 


54291 


00000 


82103 


99990 


58419 


99968 


48 


13 


57767 


00000 


32T02 


99990 


58747 


99967 


47 


14 


60985 


.00000 


88292 


99990 


59072 ' 


99967 


46 


15 


63982 


00000 


83875 


99990 


59395 


99967 


46 


16 


66784 


00000 


34450 


99989 


59715 


99966 


44 


17 


69417 


9.99999 


35018 


99989 


60033 


99966 


43 


18 


71900 


99999 


35578 


99989 


60349 


99965 


42 


19 


74248 


99999 


36131 


99989 


60662 


99964 


41 


20 


7.76475 


9.99999 


8.36678 


9.99988 


8.60978 


9.99964 


40 


21 


78594 


99999 


37217 


99988 


61282 


99963 


39 


22 


80615 


99999 


37750 


99988 


61689 


99963 


38 


23 


82545 


99999 


38276 


99987 


61894 


99962 


37 


24 


84393 


99990 


38796 


99967 


62196 


99962 


36 


25 


86166 


99999 


39310 


99987 


68497 


99961 


36 


26 


87870 


99999 


39818 


99986 


62795 


99961 


34 


27 


89509 


99999 


40320 


99986 


63091 


99960 


38 


28 


91068 


99999 


40816 


99986 


63385 


99960 


32 


29 


92612 


AQQQQ 


41307 


99985 


68678 


99959 


31 


80 


7.94084 


9.99998 


8.41792 


9.99986 


8.68968 


9.99959 


80 


81 


95508 


99998 


42272 


9998& 


64256 


99958 


29 


32 


96887 


99998 


42746 


99984 


64543 


99958 


28 


83 


98223 


99998 


43216 


99984 


64827 


99957 


27 


34 


99520 


99998 


43680 


99964 


65110 


99956 


26 


35 


8.00779 


99998 


44139 


99983 


65391 


99956 


26 


36 


02002 


99998 


44594 


99983 


65670 


99955 


24 


37 


03192 


99997 


45044 


99963 


65947 


99955 


23 


88 


04350 


99997 


45489 


99982 


66223 


99954 


22 


39 


05478 


99997 


. 45930 


99982 


66497 


90954 


21 


40 


8.06578 


9.99997 


8.46366 


9.99982 


8.60769 


9.99958 


20 


41 


07650 


99997 


46799 


99981 


67039 


99952 


19 


42 


06696 


99997 


47226 


99961 


67808 


99952 


18 


43 


09718 


99997 


47650 


99981 


67575 


99951 


17 


44 


10717 


99996 


48069 


99980 


67841 


99951 


16 


45 


11693 


99996 


48485 


99980 


68104 


99950 


15 


46 


12647 


99996 


48896 


99979 


68367 


99949 


14 


47 


13581 


99996 


49304 


99979 


68627 


99949 


18 


48 


14495 


99996 


49708 


99979 


68886 


99948 


12 


49 


15391 


99996 


50108 


99978 


69144 


99948 


11 


50 


8.16268 


9.99995 


8.50504 


9.99978 


8.69400 


9.99947 


10 


51 


17128 


99995 


50897 


99977 


69654 


99946 


9 


52 


17971 


99995 


51287 


99977 


69907 


99046 


8 


53 


18798 


99995 


51678 


99977 


70159 


99945 


7 


54 


19610 


99995 


52055 


99976 


70409 


99944 


6 


55 


20407 


99994 


52434 


99976 


70658 


99944 


6 


56 


21189 


99994 


52810 


99975 


70905 


99943 


4 


57 


219.58 


99994 


53183 


99975 


71151 


99942 


8 


58 


22713 


99994 


53552 


99974 


71395 


99942 


2 


50 


23456 


99994 


53919 


99974 


71638 


99941 


1 


60 


24186 


99993 


54282 


99974 


71880 


99940 





9 


Cosfne 


Sine 


Cosine 


Sine 


Cosine 


Sine 


9 




89° 


88° 


87 


o 



Digitized by VjOOQIC 



136 


LOGARITHMIC SINES AND COSINES 




/ 




«• 




4» 


fi' 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 




8.71880 


9.99940 


8.84858 


9.99894 


8.94080 


9.99884 


60 




72120 


99940 


84589 


09808 


94174 


99883 


69 




72859 


999:i9 


84718 


99892 


94817 


99882 


68 




72597 


()9988 


84897 


99691 


94461 


99831 


67 




78884 


99988 


85075 


99891 


94608 


99880 


66 




78069 


99987 


86252 


99890 


94746 


99829 


66 




78303 


99986 


86429 


99889 


94887 


99828 


54 




78635 


999:J6 


86606 


99888 


95029 


99827 


63 




73767 


99935 


86780 


99887 


95170 


09826 


62 




78997 


99934 


85956 


99686 


95310 


99824 


61 




8.74226 


9.99984 


8.86128 


9.99886 


8.96450 


9.99828 


60 




74454 


99938 


86301 


99684 


96689 


99822 


40 




74680 


99982 


86474 


9988:^ 


95728 


99821 


48 




74906 


99932 


86646 


99682 


05867 


99820 


47 




75130 


99931 


86816 


99881 


96006 


99819 


46 




76868 


99930 


86987 


99680 


96143 


99817 


46 




76675 


99929 


87156 


99679 


96280 


99816 


44 




76795 


90929 


87826 


99870 


96417 


99816 


48 




76015 


99928 


87494 


998:8 


96563 


99814 


42 




76834 


99927 


87661 


99877 


96689 


99818 


41 


20 


8.76451 


9.99926 


8.87829 


9.99676 


8.96826 


9.99812 


40 


SI 


76667 


99926 


87996 


99675 


96960 


99610 


89 


22 


76888 


99926 


88161 


99874 


97096 


99809 


88 


28 


77097 


00924 


88326 


99873 


97229 


99808 


87 


24 


77810 


99928 


88490 


99872 


97868 


99807 


86 


26 


77522 


99908 


88654 


99871 


97496 


99806 


85 


28 


77733 


999-^2 


88817 


99870 


97629 


99804 


34 


27 


77948 


99921 


88980 


99869 


97768 


9fl803 


88 


28 


78162 


99920 


89142 


99868 


97894 


99602 


82 


29 


78360 


99920 


89:j04 


99867 


98026 


99801 


31 


80 


8.78568 


9.99919 


8.89464 


9.99666 


8.98167 


9.99800 


80 


81 


78774 


99918 


89625 


99665 


96288 


99798 


89 


32 


78979 


99917 


89784 


99864 


96419 


99797 


28 


83 


791S3 


99917 


89943 


99863 


98549 


99796 


27 


84 


79386 


99916 


P0102 


9986S 


98679 


99796 


26 


85 


79588 


99915 


90-^60 


99861 


98806 


99798 


26 


86 


79789 


99914 * 


90417 


99860 


98987 


99792 


^ 


87 


79990 


99918 


80574 


99859 


99066 


99791 


88 


88 


80189 


99918 


90730 


99658 


99194 


99790 


88 


89 


80388 


99912 


90685 


99857 


. 99822 


99788 


81 


40 


8.80585 


9.99911 


8.91040 


9.99656 


8.99450 


9.99787 


80 


41 


8078-i 


99910 


91195 


99865 


99577 


99786 


19 


42 


80978 


99909 


91849 


99864 


99704 


99785 


18 


48 


81173 


99909 


91502 


99853 


99830 






44 


81367 


99908 


91655 


99852 


99956 






4ft 


81560 


99907 


91807 


99851 


9.00082 


99781 




46 


81762 


99906 


91959 


99850 


00207 


99780 




47 


81044 


999a5 


92110 


99848 


00:«2 


99778 




48 


82134 


09904 


92261 


99847 


00458 






49 


82324 


99904 


92411 


99846 


00681 


99776 




60 


8.82513 


9.90903 


8.9S561 


9.99846 


9.00704 


9.90776 




51 


82701 


99902 


92710 


99844 


00828 


99778 




62 


82888 


99901 


92869 


99843 


00961 


99772 




68 


88075 


99900 


9;W07 


99842 


01074 


99771 




54 


88261 


9<.>899 


9:^154 


99841 


01196 


99769 




55 


88446 


90898 


93301 


99840 


01318 


99768 




66 


83630 


99S98 


9:^448 


99839 


01440 


99767 




57 


88818 


99897 


93694 


99838 


01561 


99766 




58 


88996 


99896 


98740 


99887 


01682 


99764 




60 


84177 


99895 


98885 


09836 


01808 


09768 




60 


84868 


99894 


94030 


99884 


01928 


99761 




» 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


\ 




SB' 




860 




84- 



Digitized by VjOOQIC 





LOGARITHMIC SINES AND COSINES 


137 


t 


6 


o 




?• 


1 


J- 


/ 


Sine 


Cosine 


Sine 


Cosine 


Bine 


Cosine 





9.01928 


9.90761 


9.08680 


9.99676 


0.14856 


0.09675 


60 


1 


02043 


99760 


08608 


99674 


14445 


99574 


60 


2 


02163 


99759 


08705 


09678 


14585 


00578 


56 


8 


02288 


99757 


08807 


99670 


14624 


00570 


57 


4 


0^402 


09766 


06999 


99669 


14714 


00668 


56 


5 


02520 


99755 


09101 


99667 


14808 


00566 


55 


6 


02689 


99758 


09202 


99666 


14801 


09666 


64 


7 


02757 


99752 


09804 


OOMtA 


14980 


90563 


58 


8 


02874 


99751 


09405 


09663 


16069 


09661 


58 


9 


02992 


99749 


09506 


99661 


15167 


90560 


51 


10 


9.03109 


9.99748 


0.09606 


O.006G0 


0.15846 


0.00567 


50 


11 


06226 


00747 


00707 


00668 


15388 




40 


12 


0:}342 


00745 


09607 


00666 


15481 


00554 


48 


13 


03458 


00744 


00007 


09665 


16608 


00S62 


47 


14 


08374 


0974S 


10006 


99658 


16606 


00550 


46 


15 


0)690 


00741 


10106 


00661 


15668 


00548 


46 


10 


03805 


99740 


10005 


09660 


15770 


00546 


44 


17 


039iO 


99788 


10304 


09648 


15867 


09645 . 


43 


18 


04034 


99787 


10408 


99647 


15944 


09543 


48 


19 


04149 


00786 


lOBOl 


90645 


16080 


09541 


41 


90 


9.04262 


9.00784 


0.10500 


0.00648 


9.16116 


0.00580 


40 


21 


04376 


00783 


10607 


09642 


16208 


00537 


SO 


23 


04490 


00781 


10705 


00640 


16889 


00585 


88 


2H 


0J603 


00780 


10608 


00688 


16874 


00633 


87 


24 


04715 




10000 


00687 


16460 


00532 


86 


25 


048-38 


00727 


11087 


00685 


16545 


09530 


35 


26 


04940 


097^ 


11164 


09638 


16681 


09588 


34 


27 


06a52 


90784 


11881 


09688 


16716 


09626 


33 


28 


05164 


00728 


11377 


99630 


16801 


09584 


88 


29 


05275 


00721 


11474 


99689 


16886 


00522 


31 


SO 


9.05886 




0.11670 


0.99627 


0.16970 


0.00520 


30 


81 


05497 


00718 


11666 


99625 


17055 


99518 


20 


32 


05607 


00717 


11761 


99684 


17189 


99517 


28 


33 


06717 


00716 


lia57 


99682 


17888 


99515 


27 


34 


05827 


09714 


11058 


99680 


17807 


99518 


26 


85 


05937 


ooris 


12047 


00618 


17801 


00611 


26 


86 


06046 


00711 


12148 


00617 


17474 


09509 


84 


37 


06155 


00710 


18286 


00615 


17568 


09607 


88 


88 


06264 


ooro8 


12381 


09618 


17641 


09605 


28 


80 


06872 


00707 


18485 


99618 


17784 


90503 


81 


40 


9.06481 


0.00706 


0.12610 


&.99610 


0.17807 


0.09501 


80 


41 


06680 


00704 


12618 


99608 


17890 


00499 


10 


42 


06696 


00703 


18706 


99607 


17978 


99497 


18 


43 


06804 


00701 


12700 


99605 


18055 


99495 


17 


44 


06911 




12802 


99608 


18137 


99494 


16 


45 


07018 


VinwO 


12086 


99601 


18220 


99492 


15 


46 


07184 


09096 


18078 


99600 


18308 


99490 


14 


47 


07231 


09605 


18171 


00598 


18383 


99488 


13 


48 


07837 


00603 


13268 


09596 


18465 


99486 


12 


40 


07442 


00602 


18355 


99595 


18647 


99484 


11 


SO 


9.0;iS48 


0.00600 


0.18447 


9.00608 


9.18688 


9.99482 


10 


51 


07653 


00680 


18580 


00591 


18709 


99480 





m 


07758 


00687 


13680 


99580 


18700 


00478 


8 


58 


07863 


00666 


18728 


00588 


18871 


09476 


7 


54 


07968 


00684 


18818 


09586 


18968 


99474 


6 


56 


06072 


90688 


18004 


99684 


19088 


99472 


6 


56 


08176 


09681 


13994 


99688 


19118 


99470 


4 


57 


08280 


00680 


14085 


99581 


19193 


99468 


8 


56 


08383 


00678 


14176 


99570 


19278 


99466 


8 


SO 


08486 


00677 


14266 


00577 


10358 


99464 


1 


60 


06589 


00675 


14S66 


00576 


10483 


00462 





t 


CkMlne 


Sine 


CkMine 


Sine- 


Cosine 


Sine 


» 




88» 




82« 




8l» 



Digitized by VjOOQIC 



138 


LOGARITHMIC SINES AND COSINES 




/ 




DO 


lOo 


ll- 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.19433 


9.99462 


9.83967 


9.99386 


9.28060 


9.99196 


60 


1 


19518 


09460 


84069 


99383 


28125 


99198 


60 


2 


19592 


99468 


84110 


99331 


88190 


99100 


58 


8 


19672 


99456 


84181 


99328 


28254 


99187 


67 


4 


19751 


99464 


84253 


99326 


28319 


99185 


56 


6 


19830 


99452 


84384 


99324 


28384 


99188 


55 


6 


19909 


99460 


84396 


09322 


88448 


99180 


64 


7 


19988 


99448 


84466 


99319 


88518 


99177 


58 


8 


80067 


99446 


84536 


99317 


28577 


99176 


52 


9 


20145 


99444 


84607 


99316 


28641 


99172 


61 


10 


9.80883 


9.99442 


9.84677 


9.99318 


9.88706 


9.99170 


50 


11 


20808 


99440 


84748 


99310 


28769 


99167 


49 


13 


80380 


99438 


84818 


99308 


28833 


99165 


48 


18 


80468 


99436 


84888 


99306 


28896 


99162 


47 


14 


80536 


99484 


84968 


99304 


88960 


99160 


46 


16 


80613 


99482 


85028 


99301 


29084 


99157 


45 


16 


80691 


99429 


85098 


99299 


89067 


99156 


44 


17 


. 80768 


99427 


85168 


99297 


29150 


99158 


48 


18 


20846 


99425 


25237 


99294 


29214 


99160 


48 


19 


80988 


99428 


25307 


99292 


89277' 


99147 


41 


20 


9.20909 


9.99421 


9.26376 


9.99290 


9.29340 


9.99145 


40 


21 


81076 


99419 


85445 


99288 


89403 


99142 


30 


22 


81163 


99417 


86514 


99285 




99140 


38 


2:^ 


21229 


99415 


25688 


99283 


89529 


99187 


37 


24 


21806 


99418 


85652 


99281 


29591 


99186 


86 


25 


21388 


99411 


85781 


99278 


29664 


99182 


36 


20 


21468 


99409 


85?90 


992T6 


29716 


99180 


34 


27 


21534 


99407 


85858 


99274 


29779 


99187 


33 


28 


21610 


99404 


26927 


99271 


29841 


99184 


88 


29 


81686 


99402 


26995 


99269 


29903 


99188 


31 


80 


9.21761 


9.99400 


9.26063 


9.99267 


9.29966 


9.99119 


30 


31 


21886 


99398 


26131 


99-264 


30028 


99117 


89 


32 


21912 


99396 


86199 


99262 


80090 


99114 


88 


83 


21987 


99394 


86867 


99260 


80151 


99112 


87 


34 


82062 


99392 


26385 


99257 


30213 


99109 


86 


35 


28187 


99390 


26408 


99255 


80275 


99106 


25 


86 


28811 


99388 


86470 


99258 


80836 


99104 


84 


37 


82386 


99385 


86638 


99250 


30398 


99001 


88 


88 


22861 


99388 


86606, 


99248 


80450 


99099 


28 


39 


22436 


99381 


86678 


99246 


80581 


99096 


81 


40 


9.22609 


9.99879 


9.86789 


9.99843 


9.80688 


9.99098 


80 


41 


2258:i 


99377 


86806 


99241 


80648 


99091 


19 


42 


22657 


99375 


86878 


99288 


80704 


99068 


18 


48 


22731 


99372 


86940 


99236 


80786 


09066 


17 


44 


88805 


99370 


87007 


99233 


80686 


99088 


16 


45 


22878 


99368 


87078 


99281 


80887* 


99080 


15 


46 


88952 


99366 


87140 


99229 


80947 


09078 


14 


47 


83025 


99364 


87206 


99226 


81006 


99075 


13 


48 


88098 


99368 


87273 


99224 


81068 


99078 


18 


49 


83171 


99859 


87839 


99221 


81180 


99070 


11 


50 


9.83844 


9.99367 


9.87405 


9.99^19 


9.81189 


9.v«067 


10 


51 


23317 


99255 


87471 


99217 


81860 


99064 


9 


58 


83890 


99363 


27587 


99214 


81810 


99068 


8 


58 


83462 


99351 


87602 


99218 


81870 


99060 


7 


54 


28535 


99348 


27668 


99209 


81480 


99066 


6 


66 


88607 


99846 


27734 


99207 


81490 


99064 


5 


56 


83679 


99844 


27799 


99204 


81649 


99061 


4 


67 


23762 


9934? 


27864 


99202 


81609 


99048 


8 


58 


83823 


99340 


27930 


99200 


•81669 


99046 


8 


69 


88895 


99337 


87996 


99197 


81788 


09048 


1 


60 


88967 


99385 


88060 


99195 


81788 


99040 





» 


CkMlne 


Sine 


Ckwine 


Sine 


Cosine 


Sine 


f 




SO" 




Ji- 




78* 



y Google 





LOGAMTflMiC SINfiS AND COSINES 


m 


/ 


120 


18° 


14» 


1 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.31788 


9.99040 


9.85209 


9.98872 


9.38368 


9.98690 


60 


1 


81847 


99038 


35263 


98869 


38418 


98687 


59 


2 


31907 


99035 


35318 


98867 


38469 


96684 


58 


3 


31966 


99082 


35373 


98664 


38519 


98681 


67 


4 


32025 


99030 


35427 


98861 


36570 


98678 


56 


6 


32084 


99027 


35481 


98856 


38620 


98675 


55 


6 


32143 


99034 


35536 


96855 


38670 


98671 


54 


7 


32202 


99022 


35590 


98852 


38721 


98668 


58 


8 


32261 


99019 


35644 


96849 


38771 


98665 


52 


9 


32319 


99016 


35698 


98846 


36821 


98662 


51 


10 


9.32378 


9.99013 


9.35762 


9.98843 


9.88871 


9.98659 


50 


11 


32437 


99011 


35806 


98840 


38921 


98656 


49 


12 


32495 


99008 


35860 


9863? 


38971 


98652 


48 


13 


39553 


99005 


15914 


98634 


39021 


96649 


47 


14 


32612 


99002 


86966 


98asi 


39071 


96646 


46 


15 


32670 


99000 


86022 


96628 


39121 


98643 


45 


16 


32728 


98997 


36075 


986:5 


39170 


98640 


44 


17 


32786 


98994 


36129 


98822 


39220 


98636 


43 


18 


32844 


98991 


86182 


. 98619 


39270 


98633 


42 


19 


32902 


98969 


36286 


98816 


39319 


98030 


41 


20 


9.32960 


9.98986 


9.36289 


9.96613 


9.39369 


9.98627 


40 


21 


38016 


98963 


36842 


98610 


39418 


96623 


39 


22 


38075 


98980 


36395 


98807 


39467 


98620 


38 


28 


33133 


98978 


36449 


98604 


39517 


98617 


37 


24 


33190 


96975 


86502 


98801 


39566 


98614 


36 


25 


33248 


98972 


36556 


96798 


39615 


98610 


35 


26 


88305 


98969 


36606 


98796 


39664 


98607 


34 


27 


83362 


98967 


36660 


98792 


39713 


98604 


33 


28 


88420 


98964 


36713 


98769 


39762 


98601 


32 


29 


83477 


98961 


86766 


98766 


39811 


96597 


31 


30 


9.33534 


9.98958 


9.36819 


9.98783 


9.39860 


9.98594 


30 


81 


33591 


98955 


36671 


98780 


39909 


98591 


29 


82 


33647 


98953 


36924 


98'/77 


39958 


98586 


28 


83 


33704 


96950 


36976 


96774 


40006 


98584 


27 


34 


33761 


98947 


37028 


98771 


40055 


96561 


26 


&5 


88818 


96944 


87081 


98768 


40108 


96578 


25 


86 


33874 


98941 


87133 


98766 


40152 


9a574 


24 


87 


83931 


98938 


37165 


96762 


40200 


98571 


28 


88 


88987 


98936 


8728'; 


98759 


40249 


98566 


22 


89 


34043 


98988 


37289 


98756 


40297 


98566 


21 


40 


9.84100 


9.96960 


9.87341 


9.98763 


9.40346 


9.98561 


20 


41 


34166 


98927 


87393 


96750 


40894 


98558 


19 


42 


84212 


98924 


37445 


96746 


40442 


98555 


18 


43 


84268 


96921 


37497 


96743 


40490 


98561 


17 


44 


84824 


98919 


87549 


98740 


40638 


98546 


16 


45 


84880 


98916 


37600 


98787 


40586 


98545 


15 


46 


84436 


98918 


87652 


98734 


40684 


98541 


14 


47 


84491 


96910 


37703 


98731 


40682 


98538 


18 


48 


84547 


98907 


37756 


96726 


40730 


98535 


12 


49 


84602 


98904 


37806 


98725 


40776 


98531 


11 


50 


9.34658 


9.98901 


9.87856 
37909 


9.96722 


9.40825 


9.'J8528 


10 


51 


34718 


98896 


98719 


40673 


96525 


9 


58 


84769 


96696 


37960 


98716 


40921 


98521 


8 


53 


84824 


98893 


38011 


98712 


40968 


98518 


7 


54 


34879 


98690 


38062 


98709 


41016 


98515 


6 


55 


84934 


98687 


88118 


98706 


41068 


96511 


5 


56 


34989 


98864 


38164 


98708 


41111 


96508 


4 


67 


35044 


96681 


38215 


98700 


41156 


98505 


8 


56 


35099 


98676 


38266 


96697 


41206 


98501 


2 


59 


35154 


96875 


38317 


98694 


41262 


98496 


1 


60 


85209 


98672 


38368 


98690 


41300 


98494 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


1 




n* 




w 




W 



y Google 



140 


LOGARITHMIC SINES AND COSINES 




f 


16» 


16' 




17« 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.41300 


9.96494 


9.44034 


9.98284 


9.46594 


9.98060 


60 


1 


41347 


98491 


44078 


96281 


466:35 


98056 


69 


2 . 


41394 


98488 


44122 


98277 


46076 


U8052 


58 


8 


41441 


98484 


44166 


98273 


46717 


98048 


57 


4 


41488 


98481 


44210 


98270 


46758 


98044 


56 


6 


41535 


98477 


4425:3 


98266 


46800 


98040 


55 


6 


41582 


9H474 


44297 


98262 


46841 


98036 


54 


7 


410-48 


98471 


44841 


98259 


46882 


98032 


58 


8 


41675 


98467 


44385 


98255 


46923 


98029 


52 


9 


41?22 


98464 


44428 


98251 


46964 


98025 


51 


10 


9.41768 


9.98460 


9.44472 


9.98248 


9.47005 


9.96021 


60 


11 


41815 


98457 


41510 


9R244 


47045 


98017 


49 


18 


41861 


98453 


44659 


96240 


47086 


98013 


48 


18 


41908 


98450 


44602 


98237 


47127 


98009 


47 


14 


41954 


98447 


44646 


98233 


47168 


98005 


46 


15 


4:2001 


98443 


44689 


98229 


47209 


98001 


45 


16 


4^047 


9S440 


44733 


98226 


47249 


97997 


44 


17 


4<J093 


98 1:^ 


44776 


98222 


47-290 


97993 


43 


18 


4-2140 


9f^:J 


44819 


98218 


47*30 


97989 


42 


19 


4:2186 


98429 


44862 


96215 


4T371 


9; 986 


41 


20 


9.J228-4 


9.98426 


9.44905 


9.98211 


9.47411 


9.97982 


40 


21 


42278 


98422 


44948 


98207 


47452 


97978 


39 


22 


42324 


98419 


44992 


98:^ 


47492 


97974 


38 


23 


42370 


98415 


45035 


98200 


47583 


97970 


87 


24 


42416 


98412 


45077 


96196 


47573 


979t6 


36 


25 


42461 


98409 


46120 


98192 


47613 


97962 


35 


26 


42507 


98405 


45163 


98189 


47654 


979ft8 


34 


27 


42553 


98402 


45-206 


98185 


47694 


97954 


3:3 


28 


42599 


98:396 


45249 


96181 


47734 


97950 


82 


29 


42644 


98395 


45292 


98177 


47774 


97946 


31 


30 


9.42690 


9.98391 


9.45834 


9.98174 


9.47814 


9.97942 


80 


81 


4-2785 


98388 


45377 


98170 


47854 


97988 


29 


82 


42781 


96884 


45419 


96166 


47894 


97984 


28 


83 


428-26 


98881 


45462 


96162 


47934 


97980 


27 


84 


42872 


98377 


45604 


98169 


479T4 


97926 


26 


35 


42917 


98873 


45547 


98155 


4J=014 


97922 


25 


36 


42962 


98370 


45589 


98151 


48054 


97918 


24 


87 


43008 


98366 


45632 


96147 


48094 


97914 


28 


88 


43053 


98363 


45674 


98144 


48188 


97910 


22 


39 


4:i098 


98359 


45716 


98140 


48173 


97906 


21 


40 


9.43143 


9.98856 


9.4.5758 


9.96136 


9.48218 


9.97908 


20 


41 


43188 


98:35-2 


45801 


96182 


48268 


97898 


19 


42 


43-231 


98349 


45843 


98129 


48298 


97894 


18 


43 


43-278 


98345 


45886 


98125 


48382 


97800 


17 


44 


433-23 


98342 


4.5927 


96121 


48871 


97886 


16 


45 


43367 


98338 


45969 


96117 


48411 


97882 


15 


46 


43412 


983:34 


46011 


9611S 


48450 


97878 


14 


47 


43457 


98331 


46053 


98110 


48490 


97874 


18 


48 


4:^502 


983-27 


46095 


98106 


48529 


97870 


12 


49 


43546 


98:324 


46186 


96108 


48568 


97866 


11 


50 


9.43591 


9.98320 


9.46178 


9.96096 


9.48607 


9.97861 


10 


51 


43635 


98317 


46220 


98094 


48647 


97857 


9 


52 


4:^80 


98813 


46268 


98090 


48686 


97868 


8 


58 


48724 


98309 


46308 


98067 


48725 


97849 


7 


64 


48769 


98306 


46345 


98088 


48764 


97645 


6 


55 


48813 


98302 


46386 


98079 


48808 


97841 


5 


66 


43857 


98299 


46428 


96075 


48842 


97887 


4 


57 


43901 


98295 


46469 


98071 


48881 


97888 


8 


58 


43946 


98291 


46511 


96067 


48920 


97829 


8 


69 


48990 


98288 


46552 


96063 


48959 


97886 


1 


60 


44084 


96284 


46594 


98060 


48998 


97881 





t 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


i 




74« 




78" 




7«« 





Digitized by VjOOQIC 





LOG.VKITHM1C SINES AND COSINES 


141 


1 


IS' 


19. 


SO' 


t 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.48998 


9.97821 


9.51264 


9.97567 


9.53405 


9.97299 


60 


1 


49037 


97817 


51801 


97568 


53440 


97294 


59 


2 


49076 


97812 


51838 


97558 


58475 


97289 


58 


8 


49115 


97808 


51874 


97554 


58509 


97286 


57 


4 


49153 


97804 


51411 


97550 


53644 


97280 


50 


5 


49192 


97800 


51447 


97545 


53578 


97276 


55 


6 


49231 


97796 


51484 


97541 


58613 


97271 


54 


7 


49269 


97792 


51520 


97586 


53647 


97266 


53 


8 


49:»8 


97788 


61557 


97532 


58682 


97262 


52 


9 


49347 


97784 


61593 


97528 


58716 


97257 


51 


10 


9 49385 


9.97779 


9.51629 


9.07S28 


9.58751 


9.97262 


50 


11 


49424 


97775 


61666 


97519 


53786 


97248 


49 


12 


49462 


97771 


51702 


97515 


53819 


97248 


48 


13 


495U0 


97767 


51788 


97510 


63854 


97288 


47 


14 


49539 


97763 


51T74 


97506 


63868 


97284 


46 


15 


49577 


97759 


51811 


97501 


53922 


97229 


45 


16 


49615 


97764 


61847 


97497 


53967 


97224 


44 


17 


49654 


97750 


51888 


97492 


58991 


97220 


43 


18 


49692 


97746 


51919 


97488 


54025 


97215 


42 


19 


49730 


97742 


51955 


97484 


F060 


97210 


41 


20 


9 497<58 


9.97788 


9.51991 


9.97479 


9.54098 


9.97206 


40 


21 


49806 


97734 


62027 


97475 


54127 


97201 


39 


22 


49341 


97729 


52063 


97470 


54161 


97196 


38 


5>8 


498S-' 


97725 


62099 


97466 


54195 


97192 


37 


24 


49920 


97721 


62135 


97461 


54229 


97187 


36 


35 


49958 


91717 


63171 


97457 


54288 


97162 


35 


26 


49996 


97713 


62207 


97458 


54297 


97178 


34 


27 


50034 


97708 


52242 


97448 


54331 


97178 


38 


28 


60072 


97704 


62278 


97444 


54365 


97168 


82 


29 


50110 


97700 


62314 


97439 


54399 


97168 


81 


SO 


9.50148 


9.97696 


9.62850 


9.97436 


9.54433 


9.97159 


80 


31 


50185 


97691 


52885 


97480 


54466 


97154 


29 


32 


50223 


97687 


62421 


97426 


54500 


97149 


26 


33 


50261 


97683 


62456 


97421 


54584 


97145 


27 


34 


50296 


97679 


52492 


97417 


54567 


97140 


26 


85 


50836 


97674 


62627 


97412 


64601 


97185 


25 


36 


50374 


97670 


52568 


97406 


546a5 


97130 


24 


37 


60411 


97666 


52696 


97408 


64668 


97126 


23 


38 


50449 


97662 


52684 


97899 


54702 


• 97121 


22 


30 


50486 


97657 


52669 


97304 


64736 


97116 


21 


40 


9.60528 


9.97658 


9.52706 


9.97890 


9.54769 


9.97111 


20 


41 


50561 


97649 


52740 


97385 


64802 


97107 


19 


42 


5059S 


97645 


52775 


97881 


54836 


97102 


16 


43 


50635 


97640 


52811 


97876 


54869 


97097 


17 


41 


50G73 


97686 


62846 


97372 


54903 


97092 


16 


45 


60710 


97032 


52881 


07867 


54936 


97067 


16 


46 


60747 


97628 


52916 


97868 


54969 


97088 


14 


47 


60784 


97628 


62951 


97358 


^5008 


97078 


18 


48 


50821 


07619 


52986 


97358 


66036 


97073 


12 


49 


50658 


97616 


58021 


97349 


66069 


97066 


11 


60 


9.50896 


9.97610 


9.58056 


9.97344 


9.66102 


9.97068 


10 


51 


500:fi) 


97606 


58092 


97340 


65136 


97059 


9 


62 


60970 


97602 


63126 


97836 


66169 


97054 


8 


53 


61007 


97597 


53161 


97881 


55202 


97049 


7 


54 


51048 


97598 


58196 


97826 


55285 


' 97044 


6 


55 


51060 


97589 


58281 


97822 


65368 


97089 


5 


56 


51117 


97684 


53266 


97317 


55801 


97085. 


4 


67 


51164 


97680 


53301 


97312 


65884 


07030 


8 


58 


51191 


97576 


53336 


97806 


65867 


97026 


2 


50 


51287 


97571 


68870 


97803 


66400 


97020 


1 


00 


51964 


97567 


58405 


07299 


55488 


97016 





f 


CoBine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




71« 




70» . 




69» 



Digitized by VjOOQIC 



142 



LOGARITHMIC SINES AND COSINES 



/ 




2V 


220 


88- 1 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Conine 


9.55433 


9.97015 


9.57358 


9.96717 


9.59188 


9.96408 


60 


1 


55466 


97010 


57389 


96711 


59218 


96897 


59 


2 


55499 


97005 


57420 


96706 


69247 


96892 


58 


3 


5553-4 


97001 


57451 


96701 


59277 


96887 


67 


4 


55564 


96996 


57482 


96696 


69307 


96381 


66 


6 


55597 


96991 


57514 


96691 


59336 


96376 


56 


6 


55680 


96986 


57545 


96686 


5936* 


96370 


54 


7 


55663 


96981 


57576 


96681 


59396 


96865 


53 


8 


55695 


96976 


57607 


96676 


59425 


96360 


68 


9 


55728 


96971 


57638 


96670 


59455 


96854 


61 


10 


9.55761 


9.96966 


9.57669 


9.96665 


9.59484 


9.96349 


50 


n 


55793 


9C962 


57700 


96660 


59514 


96343 


49 


12 


568^6 


96957 


57731 


96665 


59543 


96838 


48 


13 


55858 


96952 


5rr62 


96650 


59573 


96883 


47 


14 


55891 


96947 


57793 


96645 


59602 


96827 


46 


15 


55923 


96942 


57824 


96640 


69632 


96822 


45 


16 


55956 


96937 


57855 


96684 


59661 


96816 


44 


17 


55988 


96932 


57885 


96629 


59690 


96311 


48 


18 


66021 


96927 


57916 


96624 


59720 


96306 


42 


19 


56053 


969-22 


57947 


96619 


59749 


96800 


41 


80 


0.56085 


9.96917 


9.57978 


9.96614 


9.59778 


9.96294 


40 


21 


56118 


96912 


58003 


96608 


59808 


96289 


89 


22 


56150 


96907 


58039 


96608 


59837 


96284 


88 


28 


56182 


96903 


58070 


96596 


59866 


96278 


87 


24 


56215 


96808 


58101 


96603 


59895 


96278 


86 


25 


56-247 


96898 


58181 


96588 


59924 


96267 


86 


26 


56279 


96888 


58162 


96582 


59954 


96282 


84 


27 


56311 


96883 


5819-i 


96577 


59988 


96256 


88 


28 


56343 


96878 


58228 


96572 


60012 


96251 


88 


29 


56376 


96873 


58253 


96567 


60041 


96245 


31 


80 


9.50408 


9.96868 


9.58284 


9.96562 


9.60070 


9.96240 


80 


81 


56440 


96863 


58814 


96666 


00099 


96284 


29 


32 


.56472 


96858 


58345 


96561 


60128 


96229 


28 


83 


56504 


96863 


58375 


96546 


60157 


962$8 


27 


84 


56536 


96848 


58406 


96541 


60186 


96218 


26 


85 


56568 


96848 


58436 


96585 


60215 


96212 


25 


86 


56599 


96838 


58467 


96580 


.60244 


96-207 


24 


87 


56631 


96833 


58497 


96525 


60278 


96201 


28 


88 


56663 


96828 


58527 


96520 


60802 


96196 


28 


89 


56695 


96823 


58557 


96514 


60331 


96190 


21 


40 


9.56727 


9.96818 


9.58588 


9.96509 


9.60359- 


9.96185 


20 


41 


56759 


96818 


58618 


96504 


60388 


96179 


19 


42 


56790 


96808 


58648 


96496 


60417 


96174 


18 


43 


56822 


96803 


58678 


96493 


60446 


96168 


17 


44 


56a'>4 


96798 


58709 


96488 


60474 


96162 


16 


45 


66886 


96793 


58739 


96483 


60503 


96157 


16 


46 


136917 


96788 


58769 


96477 


60532 


96151 


14 


47 


56949 


96783 


68799 


96472 


60561 


96146 


18 


48 


56980 


96Y78 


58829 


96467 


60589 


96140 


12 


49 


67012 


96772 


58859 


96461 


60618 


96135 


11 


60 


9.57044 


9.96767 


9.58889 


9.96456 


9.60646 


0.<)6129 


10 


51 


57075 


96762 


58919 


96451 


60675 


96123 


9 


52 


57107 


96757 


58949 


96445 


60704 


96118 


8 


58 


57138 


96752 


58979 


96440 


60782 


96112 


7 


54 


57169 


96747 


69009 


96436 


60761 


96107 


6 


55 


57201 


96742 


59039 


96429 


60789 


96101 


5 


56 


57.>82 


96737 


59069 


96424 


60818 


96095 


4 


57 


57264 


96732 


59098 


96419 


60646 


96O00 


t 


68 


57295 


96727 


69128 


96418 


60875 


96084 


8 


59 


57326 


96722 


59158 


96406 


60903 


96079 


I 


60 


57858 


96717 


59188 


96403 


60981 


96078 





t 


CoBine 


Biae 


Coalne 


Sine 


Cosine 


Sine . 


# 




•8* 




%7* 




««• 












Digitized by 


GooQle 







LOGARITHMIC SINES AND COSINES 


143 


/ 


«4'» 




86« 


2QO 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.60931 


9.96073 


9.62595 


9.95?-38 


9.64184 


9.95866 


60 


1 


60960 


96067 


69629 


95722 


64210 


95360 


69 


s 


60988 


96068 


62649 


95716 


64286 


05854 


68 


8 


61016 


96066 


62676 


95710 


64262 


95348 


67 


4 


61045 


96060 


62708 


95704 


64288 


96341 


66 


6 


610r8 


96045 


69r30 


05698 


64313 


95335 


65 


6 


61101 


96030 


62757 


95692 


64839 


95329 


54 


7 


61129 


96034 


62764 


95686 


64365 


05328 


63 


8 


61158 


96098 


62611 


95680 


64391 


95317 


62 


9 


61186 


96029 


62888 


95674 


64417 


95310 


61 


10 


9.61914 


9.96017 


9.62865 


9.05668 


9.64449 


9.95304 


50 


11 


61949 


96011 


62692 


95663 


64468 


95998 


49 


18 


61970 


96005 


68918 


96657 


64194 


95298 


48 


19 


61996 


06000 


62915 


95651 


61519 


95286 


47 


14 


61396 


95004 


62972 


95645 


64545 


95279 


46 


16 


61854 


95988 


69090 


95639 


64571 


96273 


45 


T6 


61889 


95982 


68026 


95638 


64596 


95267 


44 


17 


61411 


95977 


63062 


95627 


64622 


95261 


43 


18 


61488 


95971 


63079 


95621 


64647 


95254 


42 


19 


61466 


95965 


63106 


95616 


64673 


95246 


41 


90 


9.61404 


9.95960 


9.63188 


9.95600 


9.64696 


9.95248 


40 


91 


61589 


95954 


68159 


95603 


64724 


95936 


80 


99 


61560 


95948 


68186 


95597 


64749 


05229 


38 


98 


61578 


95949 


63913 


95501 


64775 


95928 


87 


94 


61606 


95987 


63289 


05585 


64800 


95217 


86 


95 


61681 


95981 


63966 


05570 


64826 


95211 


86 


96 


61669 


95096 


68299 


05578 


64651 


95204 


34 


97 


61689 


95990 


63319 


95567 


64877 


95196 


33 


98 


61717 


95914 


63845 


95661 


64902 


95192 


89 


99 


61745 


95006 


68372 


95555 


64927 


95185 


81 


80 


9.61778 


9.95909 


0.63396 


9.95649 


9.64953 


9.95179 


80 


81 


61800 


95897 


63425 


95548 


64976 


96173 


29 


39 


61828 


95891 


63451 


96587 


65003 


95167 


28 


83 


61856 


96885 


63478 


95531 


65090 


95160 


27 


84 


61888 


95679 


68504 


95525 


65054 


95154 


26 


85 


61911 


95878 


68581 


95519 


65079 


95148 


25 


86 


61989 


95868 


63567 


95518 


65104 


95141 


24 


87 


61966 


95868 


63583 


95507 


65130 


95135 


93 


88 


61994 


95856 


63610 


95500 


65155 


95199 


22 


39 


69021 


95850 


63686 


05404 


65180 


95122 


91 


40 


9.69049 


9.06644 


0.68669 


9.05486 


9.65S05 


9.95116 


80 


41 


69076 


96839 


68680 


95469 


66280 


95110 


19 


49 


69104 


05888 


63716 


95476 


66265 


95108 


18 


48 


69181 


95897 


68741 


95470 


65281 


95097 


17 


44 


69159 


96891 


68767 


95464 


65306 


95090 


16 


46 


69186 


95816 


68794 


95458 


65381 


95064 


15 


46 


69914 


96810 


63690 


05459 


65356 


05078 


14 


47 


62941 


95804 


63646 


95446 


65381 


95071 


18 


48 


69968 


95708 


68878 


05440 


65406 


95065 


12 


40 


68296 


05798 


68896 


95484 


65431 


95059 


11 


60 


9.69328 


9.95786 


9.63984 


9.95427 


9.65456 


9.95059 


10 


51 


69850 


95780 


63960 


95421 


65481 


95046 


9 


69 


69877 


95775 


63076 


95415 


65506 


95039 


8 


68 


69405 


95769 


64008 


95409 


65531 


95033 


7 


54 


69489 


96768 


64098 


95403 


65565 


95027 


6 


65 


68459 


96767 


64064 


95397 


65580 


95090 


6 


66 


68486 


96751 


64060 


95391 


65605 


95014 


4 


67 


68618 


96745 


64106 


95384 


65630 


95007 


t 


68 


69541 


95789 


64132 


95378 


65655 


95001 


8 


60 


68568 


95788 


64156 


95379 


65G80 


04995 


1 


60 


68696 


05798 


64184 


96366 


65705 


94988 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


f 




flfi*' 




«4« 




«• 



y Google 



144 



LOGARITHMIC SINES AND COSINES 



/ 


27« 


as* 




20O 


t 




Sine 


Cosine 


Sine 


Cosine 


Sine. 


Cosine 






9.66705 


9.94968 


9.67161 


9.94598 


9.68557 


9.94182 


60 






65729 


94982 


67186 


94587 


68580 


94175 


69 






65754 


94975 


67208 


94580 


68608 


94168 


58 






65779 


94969 


67282 


94573 


68625 


94161 


67 






65804 


94962 


67256 


94567 


68648 


94154 


66 






65828 


94956 


67280 


94560 


68(i71 


94147 


55 






65853 


94949 


67803 


94563 


DOOV4 


94140 


54 






65878 


94948 


67827 


94546 


68716 


94183 


53 






6590-4 


94936 


67850 


94540 


68739 


94126 


52 






65927 


94980 


67874 


94588 


68762 


94119 


51 






9.65952 


9.94923 


9.67398 


9.94526 


9.68784 


9.94112 


50 






65976 


94917 


67421 


04&19 


68807 


94105 


49 






66001 


94911 


67445 


94618 


68829 


94098 


48 






66025 


94904 


67468 


94506 


68852 


94090 


47 






66050 


94898 


67492 


94499 


68875 


94083 


46 






66075 


94891 


67515 


94492 


68897 


94076 


45 






66099 


94885 


67589 


94485 


68920 


94069 


44 






66124 


94878 


67562 


94479 


68942 


94062 


48 






66148 


94871 


67586 


94472 


68965 


94055 


42 






661T3 


04866 


67609 


94466 


68987 


94048 


41 




20 


9.66197 


9.94868 


9.67688 


9.94458 


9.69010 


9.94041 


40 




21 


66221 


94852 


67656 


94451 


69082 


94084 


89 




22 


66246 


94845 


67680 


94445 


69055 


94027 


88 




28 


mm 


94889 


67708 


94438 


69077 


94020 


87 




24 


66295 


94882 


67726 


94431 


69100 


94012 


86 




25 


66319 


94826 


67750 


94424 


69122 


94006 


86 




26 


66343 


94819 


67778 


(M417 


69144 


98998 


84 




27 


66368 


94818 


67796 


94410 


69167 


98991 


88 




88 


66892 


94806 


67820 


94404 


69189 


98964 


82 




20 


66416 


94799 


67843 


94397 


69212 


98977 


81 




80 


9.66441 


9.94798 


9.67866 


9.94890 


9.69S84 


9.98970 


80 




81 


66465 


94786 


67890 


94388 


69256 


98968 


29 




82 


66489 


94780 


67918 


94376 


69279 


98955 


28 




83 


66513 


94778 


67986 


94369 


69801 


93948 


27 




84 


66587 


94767 


67959 


94862 


69823 


93941 


26 




35 


66562 


94760 


67982 


94855 


69346 


93984 


25 




36 


665^6 


94763 


68006 


94349 


69868 


93927 


84 




87 


66610 


94747 


t8029 


94842 


69390 


93920 


88 




88 


66684 


94740 


68052 


94836 


69412 


98912 


82 




89 


66668 


94734 


68075 


94828 


69434 


93905 


81 




40 


9.66682 


9.94727 


9.68098 


9.94821 


9.69456 


9.98896 


80 




41 


66706 


94780 


68121 


94814 


69479 


98891 


19 




42 


66781 


94714 


68144 


94807 


69501 


98884 


18 




43 


66765 


94707 


68167 


94300 


69628 


98876 


17 




44 


66779 


94700 


68190 


94298 


69545 


93869 


16 




45 


66803 


94694 


68218 


94286 


69567 


9S862 


16 




46 


66827 


94687 


68287 


94279 


69589 


98a^6 


14 




47 


66851 


94680 


68260 


94278 


69611 


98847 


It 




48 


66875 


94674 


6828:) 


94266 


69638 


93840 


12 




49 


66899 


94667 


68305 


94259 


69655 


98838 


11 




60 


9.66922 


9.94660 


9.68828 


9.94252 


9.69677 


9.98826 


10 




61 


66946 


94654 


68861 


94245 


69699 


98819. 


9 




52 


66970 


94647 


68874 


94238 


69721 


98811 


8 




58 


66994 


94640 


68897 


94231 


69748 


93804 


7 




64 


67018 


94634 


68420 


94224 


69765 


98797 


6 




65 


67042 


94627 


68443 


94217 


69787 




6 




66 


67066 


94620 


68466 


94210 


69609 




4 




67 


67090 


94614 


68489 


94203 


69881 


98775 


8 




68 


67113 


94607 


68512 


94196 


69858 


93768 


8 




59 


67187 


94600 


68534 


94189 


69875 


98760 


1 




60 


67161 


94693 


68567 


94182 


69697 


98758 







f 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


» 


«£■* 




«P 




W 














Digitized by 


GooQle 









LOGARITHMIC SINES AND COSINES 


145 


t 


80» 


SP 


82» 


/ 


sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.69897 


9.93753 


9.71184 


9.93807 


9.78421 


9.92842 


60 


1 


69919 


93746 


71205 


98299 


72441 


92884 


59 


8 


69941 


93738 


71226 


03891 


72461 


92826 


58 


8 


69963 


98731 


71247 


93284 


72482 


9«18 


57 


4 


69984 


93724 


71268 


93276 


78602 


92810 


56 . 


5 


70006 


93717 


71289 


93269 


72528 


92803 


55 


6 


70028 


98709 


71810 


93261 


72542 


92795 


54 


7 


70050 


9:^708 


71831 


93258 


72562 


92787 


58 


8 


70078 


98096 


71352 


93240 


72582 


92779 


58 


9 


70093 


93687 


71373 


93238 


7-'602 


92771 


51 


10 


9.70115 


9.93680 


9.71393 


9.982S0 


9 72622 


9.92763 


50 


11 


70137 


98678 


71414 


9:3K8 


72G48 


92155 


49 


IS 


70159 


98665 


71485 


93215 


72683 


92747 


48 


18 


70180 


98658 


71456 


93207 


72C83 


92r39 


47 


14 


70808 


98650 


71477 


a8,>oo 


72703 


92731 


46 


15 


70224 


98648 


71496 


93192 


72723 


92723 


46 


16 


70245 


96686 


71519 


93184 


72743 


92715 


44 


17 


70267 


93688 


71589 


931T7 


72703 


92707 


48 


18 


70288 


93621 


71560 


93169 


72783 


92099 


48 


19 


70310 


93614 


71581 


93161 


72803 


92691 


41 


80 


9.70382 


9.93606 


9.71608 


9.93154 


9.72828 


9.92683 


40 


81 


70358 


98599 


71688 


93146 


72818 


92675 


89 


88 


70875 


98591 


71643 


93138 


72863 


92667 


88 


28 


70896 


98584 


71664 


93181 


72888 


9-2659 


87 


84 


70418 


93577 


71685 


98123 


72902 


92651 


86 


85 


70439 


08569 


71705 


93115 


72922 


98648 


85 


86 


70461 


93568 


71726 


98108 


72942 


92635 


84 


87 


70188 


98554 


71747 


93100 


78962 


92687 


88 


88 


70504 


93547 


71767 


93098 


72982 


9-2619 


88 


89 


70585 


98539 


71788 


98064 


73002 


92611 


81 


80 


9.70647 


9.93582 


9.71809 


9.93077 


9.73022 


9.92603 


80 


81 


70568 


98585 


71889 


93069 


78041 


92595 


89 


88 


70690 


93517 


71850 


93061 


7:^061 


92587 


88 


88 


70011 


93510 


71870 


93053 


73081 


98579 


87 


84 


70688 


93508 


71891 


93046 


^8(01 


92571 


26 


85 


70654 


98495 


71911 


93088 


73121 


9-2568 


85 


86 


70675 


98487 


71938 


93030 


73140 


92555 


84 


87 


70697 


98480 


7195-3 


93022 


73160 


9-2546 


88 


88 


70718 


98478 


71973 


9:K)14 


73180 


9-2688 


88 


89 


70739 


93465 


71994 


93007 


73200 


98580 


81 


40 


9.70761 


9.93467 


9.78014 


9.92999 


9.7S819 


9.98528 


80 


41 


70r82 


93450 


72034 


92991 


78239 


'92514 


19 


43 


70603 


98148 


72055 


92988 


78259 


92506 


18 


48 


70824 


93436 


72076 


92976 


73278 


9-2498 


17 


41 


70646 


98427 


72095 


92968 


78298 


9*2490 


16 


45 


70867 


93480 


72116 


92960 


78318 


98488 


15 


46 


70868 


96418 


72137 


92958 


73337 


98473 


14 


47 


70909 


93405 


72157 


92944 


73857 


92465 


18 


48 


70931 


93897 


72177 


92986 


73377 


92457 


18 


49 


70958 


98890 


V8196 


92929 


73396 


92449 


11 


60 


9.70973 


9.93388 


9.78818 


9.02921 


9.78416 


9.98441 


10 


61 


70994 


93375 


72288 


92918 


78435 


92438 


9 


68 


71015 


93367 


72259 


92905 


78455 


92425 


8 


68 


71036 


93360 


72279 


92897 


78474 


92416 


7 


64 


71058 


93858 


78299 


92889 


73494 


92406 


6 


66 


71079 


98344 


78380 


92881 


78518 


92400 


5 


66 


71100 


93887 


78340 


92874 


78588 


92398 


4 


67 


71121 


93329 


72860 


92866 


78562 


92384 


8 


68 


71148 




72881 


92858 




98376 


8 


60 


71168 


93314 


72401 


92850 


78591 


92367 


1 


60 


71184 


93307 


72481 


92848 


78611 


92369 





f 


CoBine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




SO' 




SS' 




670 



Digitized by VjOOQIC 



146 


LOGARITHMIC SINES AND COSINES 




/ 


88«» 


84"> 


86» 1 





Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.73611 


9.92359 


9.74756 


9.91857 


9.75859 


9.91336 


60 


1 


73680 


92351 


74775 


91849 


75877 


918i^8 


59 


2 


73650 


92348 


74794 


91840 


76895 


91319 


58 


8 


73669 


92335 


74812 


91832 


75913 


91310 


67 


4 


73689 


92326 


74831 


91823 


76931 


91301 


56 


5 


73708 


92318 


74850 


91615 


75949 


91292 


56 


6 


73727 


92310 


74868 


91806 


75967 


91288 


64 


7 


73747 


92302 


74887 


91798 


75985 


91274 


58 


8 


73766 


92293 


74906 


91789 


76003 


91266 


62 


9 


73785 


92286 


74924 


91781 


76021 


91267 


61 


10 


9.73805 


9.92277 


9.74943 


9.91772 


9.7C0S9 


9.91248 


60 


11 


73824 


92269 


74961 


91763 


76067 


91289 


49 


12 


73848 


92260 


74980 


91756 


7C076 


91280 


48 


13 


73863 


92252 


74999 


91746 


76098 


91221 


47 


14 


':3882 


92244 


76017 


91738 


76111 


91212 


46 


15 


73901 


92235 


75036 


91729 


76129 


91208 


45 


16 


73921 


92227 


75054 


91720 


76146 


91194 


44 


17 


73940 


92219 


75073 


91712 


76164 


91186 


43 


18 


73959 


92211 


75091 


91708 


76182 


91176 


42 


19 


73978 


92202 


75110 


91696 


76200 


91167 


41 


20 


9.73997 


9.92194 


9.75128 


9.91C86 


9.76218 


9.91168 


40 


21 


74017 


92186 


75147 


91677 


76^86 


91149 


89 


22 


74036 


92177 


75165 


91669 


76!i53 


91141 


86 


28 


74055 


92169 


75184 


91660 


76271 


91182 


87 


24 


74074 


92161 


75-202 


91661 


76289 


91128 


86 


25 


74093 


92152 


75221 


91648 


76307 


91114 


35 


26 


74113 


92144 


75239 


916;^ 


76324 


91106 


84 


27 


74132 


92136 


76-268 


91626 


76342 


91096 


83 


28 


74151 


92127 


75276 


91617 


76860 


91087 


82 


29 


74170 


92119 


75294 


91608 


76878 


91078 


31 


30 


9.74189 


9.92111 


9.75313 


9.91599 


9.76396 


9.91069 


30 


3! 


74208 


92102 


76331 


91691 


76418 


01060 


29 


32 


74227 


92094 


75350 


91588 


76431 


91061 


28 


33 


74246 


92066 


76368 


91578 


76448 


91042 


27 


34 


•i4265 


92077 


75386 


91566 


76466 


91083 


26 


a5 


74284 


92069 


75405 


91556 


76484 


91028 


26 


36 


74303 


92060 


76423 


91547 


76501 


91014 


24 


3/ 


74322 


92052 


75441 


91538 


76619 


91006 


28 


38 


74341 


92044 


75459 


91580 


76537 


90996 


28 


39 


74360 


92036 


76478 


91521 


76564 


90967 


21 


40 


9.74379 


9.92027 


9.75496 


9.91512 


9.76672 


9.90978 


20 


41 


74898 


92018 


76514 


91504 


76590 


90969 


19 


42 


74417 


92010 


76533 


91495 


76607 


90960 


18 


43 


74436 


92002 


75.561 


91486 


76626 


90951 


17 


44 


74455 


91993 


76569 


91477 


76642 


90942 


16 


45 


74474 


91965 


75587 


91469 


76660 


90938 


15 


46 


74493 


919T6 


75605 


91460 


76677 


90924 


14 


47 


74512 


91968 


75624 


91451 


76696 


90915 


18 


48 


74531 


91959 


75642 


91442 


76712 


90906 


12 


49 


74549 


91951 


75660 


91483 


76780 


90696 


11 


50 


9.74.'568 


9.91942 


9.75678 


9.91426 


9.76747 


9.90887 


10 


51 


74587 


91934 


75696 


91416 


76765 


90878 


9 


52 


74606 


91925 


7.5714 


91407 


76788 


90869 


8 


53 


74625 


91917 


75738 


91398 


76800 


90660 


7 


54 


74644 


91908 


75761 


91.389 


76817 


90851 


6 


55 


74662 


91900 


76769 


91881 


76836 


90642 


6 


56 


74681 


91891 


76787 


91373 


76658 


90682 


4 


57 


74700 


91883 


75806 


91368 


76870 


90828 


8 


58 


74719 


91874 


76828 


91364 


76887 


90614 


S 


59 


74737 


91866 


76841 


91346 


76904 


.90806 


1 


60 


74756 


91867 


76869 


91336 


769« 


90796 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


66» 




66» 




64« 



Digitized by VjOOQIC 





LOGARITHMIC SINES AND COSINES 


147 


tf 


860 




B7» 


SS' 


/ 


Sine 


Cosiiie 


Sine 


Cosine 


Sine 


Cosine 





9.76922 


9.90796 


9.77946 


0.90286 


9.78034 


9.89663 


60 


1 


76939 


90787 


77968 


90226 


78950 


89643 


69 


2 


76957 


90777 


77980 


90216 


78967 


89633 


58 


3 


76974 


90768 


77997 


90206 


78983 


89624 


57 


4 


76991 


90759 


78013 


90197 


78999 


89614 


56 


5 


77009 


90750 


78080 


90187 


79015 


89604 


55 


6 


77026 


90741* 


78047 


90178 


79031 


89594 


54 


7 


77048 


90731 


78068 


90168 


79047 


89584 


53 


8 


77061 


90722 


78060 


901.59 


79063 


89574 


52 


9 


77078 


90718 


78097 


90149 


79079 


89564 


51 


10 


9.77096 


9.90704 


9.78118 


9.90139 


9.79095 


9.-89554 


50 


n 


77112 


90694 


78180 


90180 


79111 


89644 


49 


12 


77130 


90685 


78147 


90120 


79128 


89534 


48' 


18 


77147 


90676 


78168 


90111 


79144 


89524 


47 


14 


77164 


90667 


78180 


90101 


79160 


89514 


46 


15 


77181 


90667 


78197 


90091 


79176 


89504 


^ 


16 


77199 


00648 


78218 


90062 


79192 


89495 


44 


17 


77216 


90638 


78230 


90072 


79206 


.89485 


48 


18 


77283 


90680 


78246 


90068 


79224 


89475 


42 


19 


77260 


00620 


78263 


90058 


79240 


89466 


41 


20 


9.77268 


9.90611 


9.7-1280 


9.90048 


9.79256 


9.89455 


40 


21 


77286 


90602 


78296 


90084 


79272 


89445 


39 


22 


77302 


00692 


78813 


90024 


79288 


89435 


38 


28 


77819 


90588 


78329 


90014 


79804 


89125 


87 


24 


77886 


90674 


788 4G 


90005 


79619 


89416 


36 


26 


77853 


90666 


78362 


89995 


79335 


89405 


36 


20 


77870 


90666 


78179 


89986 


79851 


89395 


34 


27 


77387 


90646 


78395 


89976 


79367 


89386 


33 


28 


77406 


90537 


78412 


89966 


79383 


89375 


.32 


29 


77422 


90627 


78428 


80950 


79399 


89364 


31 


80 


9.77489 


9.90618 


9.78445 


9.89947 


9.79416 


9.89364 


30 


81 


77466 


90509 


78481 


89937 


79431 


89844 


29 


82 


77478 


90499 


78178 


89927 


79447 


89834 


28 


83 


77^90 


90490 


78494 


89918 


79463 


89824 


27 


34 


77507 


90480 


7a5lO 


89908 


79478 


89314 


26 


85 


77624 


00471 


78527 


89808 


79494 


89304 


25 


80 


77541 


90462 


78648 


89888 


79510 


89294 


24 


37 


77558 


90452 


78560 


89679 


79.526 


89284 


23 


88 


77575 


90443 


78576 


89869 


79542 


89274 


22 


89 


77692 


90434 


78592 


89859 


79558 


89264 


21 


40 


9.77609 


9.90424 


9.78609 


9.89849 


9.79673 


9.89264 


20 


41 


77626 


90415 


78626 


89640 


79589 


89244 


19 


42 


rr643 


90405 


78642 


88680 


79606 


89233 


18 


48 


77660 


90396 


79658 


89820 


78621 


89228 


17 


44 


77677 


90386 


78674 


89810 


796.36 


89213 


16 


46 


77694 


90377 


78691 


89801 


79652 


'89203 


16 


46 


77711 


90868 


78707 


89791 


79668 


89198 


14 


47 


77728 


90368 


78723 


89781 


79684 


89183 


13 


48 


77744 


90349 


78789 


89771 


79699 


89173 


12 


49 


77761 


90389 


78766 


89761 


79715 


89162 


11 


50 




9.90330 




9.89752 


9.79731 


9.89162 


10 


51 


77795 


90820 


78788 


89742 


79746 


89142 


9 


52 


77812 


90311 


78806 


89782 


79762 


89132 


8 


53 


77829 


90301 


78821 


89722 


79778 


89122 


7 


64 


77846 


90292 


78887 


89712 


79798 


89112 


6 


66 


77862 


90282 


78853 


89702 


79809 


89101 


6 


56 


77879 


90273 


78869 


89698 


79825 


89091 


4 


57 


77896 


90263 


78886 


89688 


79640 


89081 


3 


58 


77918 


90254 


78902 


89678 


79866 


89071 


2 


59 


77980 


90244 


78918 


89663 


79872 


89060 


1 


60 


77946 


90236 


78984 


89653 


798R7 


89050 





f 


Ckwine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


68« 




62« 




Bl' 



Digitized by VjOOQIC 



148 


LOGARITHMIC SINES AND COSINES 




f 


89* 


40* 


41* 


/ 


Sins 


Coafoe 


Sine 


Co«lne 


Stne 


OoBine 





9.79887 


9.60050 


9.80807 


9.68425 


9.81694 


9.87778 


60 


1 


79903 


89040 


80622 


88415 


81709 


87767 


59 


2 


79918 


89030 


80837 


88404 


81728 


87756 


58 


8 


79984 


89090 


80652 


88894 


81788 


87745 


67 


4 


79950 


89009 


80867 


88888 


81752 


87V84 


66 


5 


79966 


88999 


80882 


88372 


81767 


87728 


65 


6 


79981 


UQQQQ 

OOvOV 


80897 


88862 


• 81781 


87712 


54 


7 


79996 


88978 


80912 


88851 


81796 


87701 


53 


8 


80012 


88968 


80927 


88840 


81810 


87690 


! 62 


9 


80027 


88958 


80942 


88880 


81825 


87679 


51 1 


10 


9.80048 


9.88948 


9.80957 


9.86819 


9.81889 


9.87668 


60 


n 


80058 


8H987 


80972 


88808 


81864 


87657 


* 49 


12 


80074 


88927 


80987 


88296 


81868 


87646 


48 


18 


80089 


88917 


81002 


88287 


81882 


87686 


47 


14 


80105 


88906 


81017 


88276 


81897 


87624 


46 


15 


80120 


88896 


81032 


88266 


81911 


87618 


45 


16 


80186 


88886 


81047 


88256 


81926 


87601 


44 


17 


80151- 


88875 


81061 


88244 


81940 


87690 


48 


18 


80166 


88865 


81076 


88284 


81956 


87679 


42 


19 


80182 


88855 


81091 


88228 


61969 


87568 


41 


20 


9.80197 


9.88844 


9.81106 


9.88212 


9.81988 


9.87657 


40 


21 


80218 


88834 


81121 


88201 


81998 


87646 


89 


22 


80228 


88824 


81186 


88191 


82012 


87686 


86 


28 


80244 


88818 


81151 


88180 


82026 


87624 


87 


24 


80259 


8S803 


81166 


88169 


82041 


87518 


86 


25 


80274 


88793 


81180 


88168 


82056 


87601 


86 


26 


80290 


88782 


81195 


88148 


82069 


87490 


84 


27 


80305 


88772 


81210 


88187 


82064 


87479 


88 


28 


80320 


88761 


81225 


88126 


82096 


87468 


82 


29 


80336 


88761 


81240 


881J5 


82112 


87467 


81 


80 


9.80&>1 


9.88741 


9.81264 


9.88105 


9.82126 


9.87446 


80 


81 


8a366 


88790 


81269 


88094 


82141 


87434 


29 


82 


8(>882 


88720 


81284 


88088 


82166 


67428 


28 


83 


80397 


88709 


81299 


88072 


82160 


87412 


27 


84 


80412 


88699 


81314 


88061 


62184 


87401 


26 


86 


80428 


88688 


81328 


88051 


82196 


87890 


25 


36 


80443 


88678 


81343 


88040 


82212 


87878 


84 


87 


80458 


88668 


81858 


88029 


82226 


87867 


88 


88 


80473 


88667 


81872 


88018 


82240 


87366 


88 


39 


80489 


88647 


81887 


88007 


82256 


87846 


91 


40 


9.80504 


9.88686 


9.81402 


9.87996 


9.82269 


9.87884 


20 


41 


80519 


88626 


81417 


87985 


82288 


87822 


19 


42 


80534 


88615 


81431 


87976 


82297 


87811 


18 


48 


80550 


88605 


81446 


87964 


82811 


87800 


17 


44 


80565 


88594 


81461 


87953 


82826 


87288 


16 


45 


80580 
805§5 


88584 


81475 


87942 


82840 


87277 


16 


46 


8^573 


81490 


87931 


82854 


87266 


14 


47 


80610 


88563 


81505 


87920 


82868 


87265 


IS 


48 


80625 


88552 


81519 


87909 


82882 


87248 


12 


49 


80641 


88542 


81584 


87898 


82896 


87282 


11 


60 


9.80656 


9.88631 


9.81649 


9.87887 


9.82410 


9.872;i!l 


10 


61 


80671 


88521 


81663 


87677 


82424 


87209 


9 


62 


80686 


88510 


81578 


87866 


82489 


87106 


8 


58 


80701 


88499 


81592 


87855 


824.'>8 


87187 


7 


64 


80716 


88489 


81607 


87844 


82467 


87176 


6 


65 


80731 


88478 


81622 


87833 


82481 


87164 


6 


66 


80746 


88468 


81686 


87822 


82495 


87168 


4 


57 


80762 


88457 


81651 


87811 


82509 


87141 


8 


68 


80777 


88447 


81666 


87800 


8:;>623 


87180 


8 


59 


80792 


88486 


81680 


87789 


82537 


87119 


1 


• 60 


80807 


88425 


81694 


87778 


82551 


87107 





/ 


Cosine 


Sine 

50" 


Cosine 


Sine 


Cosine 


Sine 


9 


49'» 




48* 



Digitized by VjOOQIC 





LOGARITHMIC SINES AND COSINiiiS 


149 


f 


4a» 


480 


440 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.88661 


9.87107 


9.88878 


9.86418 


9.84177 


9.85693 


60 


1 


82565 


87096 


88892 


86401 


84190 


85681 


69 


8 


88579 


87085 


88406 


8C889 


84203 


85669. 


58 


8 


88598 


87078 


88419 


86877 


84216 


»)657 


57 


4 


88607 


87062 


83482 


8C866 


84829 


85645 


56 


6 


88681 


87050 


83446 


86354 


84242 


85632 


55 


6 


88685 


87039 


88469 


86842 


84255 


85620 


54 


7 


82649 


87028 


83478 


86330 


84269 


85608 


58 


8 


88663 


87016 


83486 


86318 


84282 


85506 


52 





82677 


87005 


83500 


86306 


84295 


85583 


51 


10 


9.89691 


9.66993 


9.88518 


9.86295 


9.84308 


9.85571 


60 


n 


88705 


86968 


88587 


86283 


84381 


85559 


49 


IS 


88719 


80970 


83540 


86871 


64834 


85547 


48 


18 


88738 


86959 


83554 


86859 


84847 


86534 


47 


14 


88747 


86947 


88567 


86247 


84360 


85522 


46 


15 


88r6t 


86936 


88581 


86235 


84873 


85610 


46 


16 


88776 


86924 


88594 


86828 


84885 


86497 


44 


17 


88788 


86913 


83608 


86211 


84398 


85485 


48 


18 


82803 


86902 


aS681 


86800 


84411 


86473 


48 


19 


88816 


86890 


83684 


86188 


84424 


85460 


41 


90 


9.88830 


9.86879 


9.88648 


9.86176 


9.84437 


9.85448 


40 


81 


8-2844 


86867 


88661 


86164 


84450 


85436 


89 


22 


88858 


86856 


88674 


86152 


84463 


85423 


88 


88 


88878 


86844 


83688 


86140 


84476 


85411 


87 


34 


82885 


86882 


88701 


86128 


84489 


85399 


86 


85 


82899 


86821 


83715 


86116 


84502 


a5386 


35 


86 


82918 


86809 


83788 


86104 


84515 


85874 


34 


87 


82927 


86798 


88741 


86092 


84528 


85361 


88 


88 


82941 


86786 


83755 


86080 


84540 


85349 


83 


89 


82955 


86775 


88768 


86068 


84553 


86337 


31 


80 


9.88968 


9.86763 


9.88781 


9.86066 


9.84566 


9.86384 


30 


81 


8.2982 


86768 


83795 


86044 


84579 


86312 


89 


88 


88996 


86740 


83806 


86038 


84598 


86299 


88 


88 


83010 


86728 


88821 


86080 


84605 


85887 


87 


84 


83023 


86717 


83884 


86008 


84618 


85874 


86 


85 


88087 


86705 


88848 


85996 


64680 


85262 


36 


86 


89061 


86694 


88861 


85984 


84648 


85850 


94 


37 


83065 


86682 


83874 


85978 


84666 


85887 


88 


88 


88078 


86670 


83887 


809C0 


(LiiUtQ 


86825 


28 


89 


88098 


86669 


88901 


85048 


84682 


85212 


81 


40 


9.83106 


9.86647 


9.88914 


9.85066 


9.84694 


9.85800 


80 


41 


88120 


86635 


88927 


85924 


84707 


86187 


19 


48 


88188 


86624 


88940 


85918 


84730 


86175 


18 


48 


88147 


86612 


83954 


85900 


84788 


85162 


17 


44 


88161 


86600 


83967 


85888 


84745 


85150 


16 


45 


83174 


86589 


88980 


85876 


84768 


85137 


16 


46 


88188 


86577 


83993 


85864 


84771 


86125 


14 


47 


88202 


86565 


84006 


a%861 


84784 


85112 


18 


46 


88215 


86564 


84020 


85889 


84796 


85100 


13 


49 


88829 


86548 


84038 


85827 


84809 


85087 


n 


60 


9.83848 


9.86530 


9.84046 


9.85816 


9.84888 


9.86074 


10 


51 


88856 


8651R 


84069 


85803 


84885 


86062 


9 


80 


88870 


86507 


84078 


85791 


84847 


86049 


8 


58 


88283 


86495 


84065 


86779 


84860 


86037 


7 


54 


83897 


86488 


84096 


85766 


84873 


86024 


6 


56 


88310 


86472 


84118 


85754 


84885 


85012 


6 


56 


83824 


86460 


84185 


85742 


84898 


84999 


4 


07 


88838 


86448 


84138 


85730 


84911 


84986 


8 


58 


88851 


86436 


84151 


85718 


84928 


84974 


3 


59 


88865 


86425 


84164 


85706 


84986 


84961 


1 


60 


83378 


86413 


84177 


85698 


84949 


84949 





9 


Ooeine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




470 


46* 




46* 



Digitized by VjOOQIC 



150 


LOG. TANGENTS 


AND COTANGENTS 




/ 




O* 




lo 


8 


» 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





— oo 


00 


8.24192 


11.75808 


8.54808 


11.45692 


60 


1 


6.46373 


13.53627 


24910 


75090 


54669 


45331 


50 


3 


764T6 


2:i524 


25616 


74884 


55027 


44973 


58 


8 


94085 


05915 . 


26312 


73688 


55882 


44618 


57 


4 


7.06579 


12.93421 


26996 


73004 


55734 


44266 


66 


6 


16270 


83730 


27669 


72331 


56083 


43917 


55 , 


6 


24188 


75812 


28332 


71668 


56429 


43571 


54 


7 


80882 


69118 


28986 


71014 


66773 


43227 


53 


8 


36682 


63818 


29629 


70371 


5ni4 


4*^86 


63 


9 


41797 


58203 


80263 


69737 


57452 


42548 


51 


10 


7.4637:} 


12.58627 


8.30888 


11.69112 


8.57788 


11.42212 


50 


11 


5051 2 


49488 


31505 


68495 


58121 


41879 


49 


12 


54291 


45709 


82112 


67888 


58461 


41549 


48 


13 


57767 


42233 


82711 


67289 


58779 


41221 


47 


14 


00986 


39014 


83302 


66698 


59106 


40695 


46 


15 


6:}08-2 


36018 


33886 


66114 


59428 


40572 


45 


16 


66785 


33215 


84461 


65539 


59749 


40851 


44 


17 


(59418 


30682 


35029 


64971 


60068 


39938 


43 


18 


71900 


28100 


86590 


64410 


60884 


89616 


4:i 1 


19 


74248 


25752 


86143 


63857 


60698 


89302 


41 


20 


7 76476 


12.23524 


8.36689 


11.63311 


8.61000 


11.38991 


40 


21 


78695 


21405 


37229 


62771 


61819 


38681 


m 


22 


80615 


19385 


37762 


62288 


61626 


38874 


38 


23 


82546 


17454 


38289 


61711 


61931 


88069 


37 


24 


84394 


15606 


38809 


61191 


62284 


87766 


86 


25 


86167 


13833 


39323 


60677 


62635 


87465 


35 


26 


87871 


12129 


89832 


60168 


62834 


87166 


34 


27 


89510 


10490 


40884 


69666 


63131 


86869 


as 


28 


91069 


08911 


40880 


59170 


6.3426 


36574 


as 


29 


92618 


07387 


41321 


58679 


63718 


86282 


31 


30 


7.94086 


12.05914 


8.41807 


11.58198 


8.64009 


11.36991 


30 


31 


95510 


04490 


42287 


57713 


64298 


85702 


29 


32 


96889 


03111 


42762 


57288 


64585 


35415 


88 


33 


08225 


01775 


48232 


56768 


64870 


85180 


37 


34 


99.')22 


00478 


43696 


56304 


65154 


34846 


26 


35 


8.00781 


11.99219 


44156 


55844 


65435 


84566 


25 


36 


02004 


97996 


44611 


55389 


66715 


84265 


24 


37 


a3194 


96806 


45061 


54989 


65998 


84007 


23 


38 


04353 


95647 


45507 


54493 


66269 


83781 


!S! 


39 


05481 


94519 


45948 


54052 


66643 


38457 


21 


40 


8.06581 


11.93419 


8.46885 


11.63615 


8.66816 


11.88184 


20 


41 


07653 


92347 


46817 


53183 


67087 


82918 


19 


42 


08700 


91300 


47245 


52755 


67356 


82644 


li 


43 


09722 


90278 


47669 


52331 


67624 


82376 


17 


44 


10720 


89280 


48089 


51911 


07690 


88110 


16 


45 


11696 


88304 


48505 


51495 


68154 


81846 


15 


46 


12651 


87349 


48917 


51083 


68417 


81583 


14 


47 


18585 


86415 


49326 


50675 


68678 


81828 


13 


48 


14500 


86500 


49729 


50271 


68938 


81068 


12 


49 


15395 


84605 


50130 


49870 


69196 


80604 


11 


50 


8.16273 


11.83727 


8.50527 


11.49478 


8.69458 


11.30547 


10 


51 


17133 


82867 


50920 


49080 


69706 


80292 


9 


52 


179:6 


82024 


61310 


4S690 


6996S 


80038 


8 


53 


18804 


81196 


51696 


48304 


70214 


29786 


7 


54 


19616 


80384 


52079 


47921 


70466 


29585 


6 


55 


20413 


79.587 


524.59 


47541 


70714 


29886 


5 


56 


21195 


78805 


52835 


47165 


70968 


89088 


4 


57 


21964 


78086 


53208 


46792 


71206 


88798 


8 


58 


22720 


77880 


53578 


46422 


71458 


86647 


8 


59 


23462 


76588 


53945 


46055 


71697 


86808 


1 


60 


24192 


75806 


54808 


45692 


71940 


28060 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 







8»'' 




SS* 




iJ* 



y Google 



LOG. TANGENTS AND COTANGENTS 



15 



/ 


a 


• 


4 


o 


5- 1 


/ 


T&n 


Ooton 


Tan 


Cotan 


Tan 


Cotan 







8.71040 


11.88060 


8.84464 


11.15636 


6.94195 


11.05805 


00 


1 


78181 


27819 


84646 


15854 


94340 


06660 


69 


2 


78480 


87580 


84826 


15174 


94485 


05515 


68 


8 


78659 


27841 


85006 


14994 


94680 


05370 


57 


4 


78896 


27104 


86185 


14815 


94773 


05227 


66 


6 


73138 


26868 


85868 


14637 


94917 


05083 


65 





73366 


26634 


85540 


14460 


95060 


04940 


54 


7 


73600 


26400 


85717 


14283 


95202 


04798 


63 


8 


TSSSi 


26163 


85893 


14107 


95344 


04656 


52 





74063 


25937 


86069 


18981 


95486 


04614 


51 


10 


8.74893 


11.85708 


8.86243 


11.18767 


6.96627 


11.04378 


50 


11 


745^1 


25479 


86417 


13583 


95767 


042S8 


49 


12 


74748 


85252 


86591 


18409 


95908 


04092 


48 


18 


74974 


26026 


86763 


18237 


96047 


03953 


47 


14 


75199 


84801 


86935 


13065 


96187 


03613 


46 


16 


75483 


24577 


87106 


12894 


96825 


03675 


45 


16 


75645 


24365 


87277 


12723 


96464 


03536 


44 


17 


75867 


24183 


87447 


12558 


96602 


08896 


43 


18 


76087 


23913 


87616 


12884 


96789 


03261 


42 


10 


76806 


28694 


87785 


12215 


96877 


02128 


41 


SO 


S 76525 


11.88475 


8.87963 


11.12047 


8.97018 


11.02967 


40 


Si 


76742 


23256 


88120 


11880 


97160 


02650 


89 


83 


76958 


88042 


88-387 


11713 


97266 


02715 


88 


88 


77173 


288<!7 


88453 


11547 


97421 


02579 


87 


84 


77387 


22618 


88618 


11382 


97556 


02444 


36 


85 


77600 


88400 


88783 


11217 


97691 


02809 


85 


88 


77811 


28189 


88948 


11062 


9T825 


08J76 


84 


87 


78088 


21978 


89111 


10880 


97959 


02041 


33 


88 


78838 


21768 


89274 


10786 


96092 


01906 


82 


89 


78441 


21559 


89487 


10563 


96225 


01775 


31 


80 


8.78649 


11.21351 


8.89598 


11.10402 


6.96358 


11.01642 


80 


81 


78855 


21145 


89760 


10240 


96490 


01510 


29 


89 


79061 


20939 


89920 


10060 


96622 


01378 


28 


83 


79866 


20734 


90080 


09920 


96753 


01247 


27 


84 


79470 


20630 


90240 


09760 


96664 


01116 


26 


35 


796T8 


20327 


90399 


09601 


99015 


00965 


25 


86 


79875 


20125 


90557 


09443 


99145 


00855 


24 


87 


80076 


19924 


90716 


09286 


99-^5 


00726 


28 


88 


80877 


19728 


90872 


09128 


99405 


00595 


22 


89 


80476 


19524 


91029 


08971 


99534 


00466 


21 


40 


8.80074 


11.19386 


8.91185 


11.06815 


8.99662 


11.00888 


80 


41 


8087« 


19128 


91840 


08660 


99791 


00200 


19 


48 


81068 


18932 


91495 


06505 


99919 


00061 


18 


48 


81964 


18786 


91660 


08850 


9.00046 


10.99964 


17 


44 


81460 


18541 


91803 


06197 


00174 


99626 


16 


46 


81653 


18847 


91957 


06043 


00801 


99699^ 


15 


46 


81fl46 


18154 


92110 


07890 


00427 


99573 


14 


47 


88038 


17962 


92262 


07788 


00558 


99447 


18 


48 


82830 


17770 


92414 


07586 


00679 


998>1 


12 


49 


88490 


17580 


92565 


07485 


00605 


99195 


11 


60 


8.88610 


11.17890 


8.92716 


11.07864 


9.00930 


10.99070 


10 


61 


88799 


17201 


92866 


07184 


01055 


98945 


9 


68 


88987 


17018 


99016 


06964 


01179 


96821 


8 


68 


83175 


16826 


98165 


06885 


01803 


96697 


7 


64 


83361 


16689 


98313 


06667 


01427 


9R573 


6 


65 


83547 


16458 


93468 


06638 


01660 


96450 


5 


66 


88738 


16268 


98609 


06891 


01673 


98327 


4 


57 


89916 


16084 


967S6 


06244 


01796 


96204 


8 


68 


81100 


15900 


98903 


06097 


01916 


06082 


2 


69 


84888 


15718 


94049 


05051 


02040 


07960 


1 


60 


81464 


15586 


94195 


05805 


08162 


97836 





/ 


Cotan 


Tan 


Ootan 


Tan 


Cotan 


Tan 


f 






86» 




86» 




84« 





Digitized by VjOOQIC 



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LOG. TANGENTS 


AND C< 


3TANG 


ENTS 




/ 


6 


o 


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/ 




Tan 


Cotan 


Tan 


Cotan 


Tan 


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9.02162 


10.97888 


9.08914 


10.91086 


9.14780 


10.88220 


60 


1 


02283 


97717 


09019 


90981 


14872 


85138 


69 


2 


02404 


97596 


09128 


90877 


14968 


85087 


68 


8 


02525 


97475 


09227 


90778 


15054 


84946 


67 


4 


03645 


97356 


09830 


90670 


15145 


84856 


66 


6 


02766 


97284 


09484 


90566 


15286 


84764 


66 


6 


02885 


97116 


09687 


90463 


16827 


84678 


64 


7 


08005 


96995 


09640 


90860 


15417 


84588 


68 


8 


08124 


96876 


09742 


90258 


15508 


84492 


62 


9 


08242 


96758 


09845 


90155 


15598 


84402 


61 


10 


9.08861 


10.96689 


9.09947 


10.90058 


9.16688 


10.84812 


60 


11 


08479 


96521 


10049 


89951 


15777 


84228 


49 


12 


08597 


96408 


10150 


89850 


15867 


84133 


48 


18 


03714 


96286 


10252 


89748 


16966 


84044 


47 


14 


08883 


96168 


10853 


89647 


16046 


83964 


46 


15 


08948 


96062 


10454 


89546 


16)86 


88865 


46 


16 


04065 


96985 


10665 


89445 


16224 


88776 


44 


17 


04181 


95819 


10656 


89844 


16812 


88688 


48 


18 


04297 


95708 


10756 


89244 


16401 


88599 


42 


19 


04418 


95587 


10866 


89144 


16489 


88611 


41 


20 


9.04528 


10.95472 


9.10966 


10.89044 


9.16677 


10.88428 


40 


21 


04643 


95357 


11056 


88944 


16665 


88385 


88 


22 


04758 


95242 


11155 


88845 


16758 


88247 


88 


28 


04873 


95127 


11354 


88746 


16841 


83169 


87 


24 


04987 


95018 


11858 


88647 


16928 


88072 


86 


25 


05101 


94899 


11452 


88648 


17016 


82984 


35 


26 


05214 


94786 


11551 




17108 


83897 


84 


27 


05338 


94672 


11649 


88361 


17190 


82810 


88 


28 


05441 


94569 


11747 


88268 


17277 


82723 


82 


29 


05553 


94447 


11845 


88155 


17863 


82687 


31 


SO 


9.05666 


10.94384 


9.11943 


10.88057 


9.17460 


10.82650 


90 


81 


05778 


94322 


12040 


87960 


17586 


82464 


29 


82 


05890 


94110 


12188 


87862 


17632 


82878 


28 


88 


06002 


93996 


12285 


87785 


17708 


82292 


27 


84 


06118 


98887 


12382 


87668 


17794 


82306 


26 


86 


06324 


93776 


12428 


87572 


17880 


82120 


25 


86 


06835 


98665 


12525 


87475 


17966 


82085 


94 


87 


06445 


93555 


13631 


87879 


18061 


81949 


28 


88 


06556 


93444 


12717 


87288 


18136 


81864 


22 


89 


06666 


98334 


13818 


87187 


18221 


81779 


21 


40 


9.06r75 


10.93225 


9.12909 


10.87091 


9.18806 


10.81694 


20 


41 


06885 


93115 


13004 


86996 


18891 


81609 


19 


42 


06994 


93006 


13099 


80901 


18475 


81525 


18 


48 


07103 


92897 


13194 


86806 


18560 


81440 


17 


44 


07211 


92789 


18289 


86711 


18644 


81856 


16 


45 


07880 


93680 


13884 


8C616 


18728 


81272 


15 


46 


07428 


92572 


13478 


86522 


18813 


81188 


14 


47 


\536 


93464 


13573 


86427 


18896 


81104 


18 


48 


07643 


93857 


13667 


86388 


18979 


81021 


12 


49 


07751 


93249 


13761 


86239 


19063 


80987 


11 


50 


9.07858 


10.92142 


9.13854 


10.86146 


9.19146 


10.80854 


10 


51 


07964 


92036 


13948 


86052 


19339 


80771 


9 


62 


08071 


91929 


14041 


85959 


19312 


80688 


8 


58 


08177 


91838 


14134 


85866 


19395 


80605 


7 


54 


08288 


91717 


14237 


85773 


19478 


80522 


6 


66 


08389 


91611 


14330 


85680 


19561 


80489 


6 


66 


08495 


91605 


14412 


85588 


19643 


80357 


4 


67 


08600 


91400 


14504 


85496 


19725 


80275 


8 


68 


08706 


91295 


14597 


85403 


19807 


80193 


2 


69 


08810 


91190 


14688 


85312 


19889 


80111 


1 


60 


08914 


91086 


14780 


85220 


19971 


80039 





t 


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Tan 


Cotan 


Tan 


Cotan 


Tan 


# 






88- 




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LOG 


. TANGENTS AND COTANGENTS 


153 


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10» 


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t 


Tan 


Cotan 


Tan 


Cotan 


Tan 


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9.19971 


lOSOOiO 


9.94639 


10.75368 


9.98866 


10.71186 


60 


1 


20068 


79947 


24706 


75294 


98933 


71067 


69 


9 


20184 


79%6 


94779 


75291 


29000 


71000 


58 


8 


20216 


79784 


94858 


76147 


29067 


70988 


57 


4 


20297 


79703 


24928 


75074 


29134 


70866 


50 


5 


90378 


79622 


25000 


76000 


29201 


70799 


55 


6 


20459 


79541 


26073 


74927 


89268 


70782 


64 


7 


90510 


79460 


25146 


74854 


29835 


70665 


53 


6 


20621 


79379 


25219 


74781 


29402 


70698 


62 


9 


20701 


79299 


25292 


74708 


29468 


70532 


51 


10 


9.90782 


10.79218 


9.25365 


10.74685 


9.29535 


10.70465 


60 


11 


20862 


79188 


25487 


74568 


29601 


70399 


49 


18 


20942 


79058 


25510 


74490 


29668 


70832 


48 


18 


91082 


78978 


95589 


74418 


29734 


70266 


47 


14 


91102 


78896 


25655 


74345 


29800 


70200 


46 


16 


91182 


78818 


95727 


74273 


29866 


70184 


45 


16 


91961 


78789 


25799 


74201 


29989 


70068 


44 


17 


91841 


78659 


25871 


74129 


29998 


70002 


48 


18 


21490 


78580 


95943 


74057 


30064 


69936 


42 


19 


91499 


78601 


26015 


73985 


30180 


69870 


41 


90 


9.91578 


10.78492 


9.26066 


10.73914 


9.30195 


10.69805 


40 


91 


91667 


78348 


96158 


73842 


30261 


69739 


39 


92 


21786 


78964 


26229 


78771 


80896 


69674 


88 


38 


91814 


78186 


26801 


73699 


30391 


69009 


87 


94 


91893 


78107 


26372 


73628 


80457 


69643 


36 


95 


91971 


78099 


96443 


73567 


30592 


69478 


85 


26 


92049 


77951 


96614 


73486 


30687 


69413 


84 


27 


92127 


77873 


26585 


73416 


30652 


69348 


83 


28 


92905 


77795 


26655 


73345 


80717 


69283 


82 


29 


922a3 


77717 


26726 


73274 


80782 


69218 


81 


80 


9.82361 


10.77639 


9.26797 


10.73203 


9.80846 


10.69154 


SO 


31 


29438 


77562 


26S67 


73133 


80911 


69089 


99 


32 


99316 


77484 


26987 


78063 


80975 


69025 


28 


33 


22598 


77407 


27008 


72992 


81040 


68060 


97 


34 


22670 


77330 


27078 


72922 


81104 


68896 


26 


So 


82747 


77258 


27148 


72862 


81168 


68832 


25 


36 


2',>824 


77176 


27218 


72788 


81283 


68767 


94 


87 


92901 


77099 


27288 


72712 


81297 


68703 


28 


88 


22977 


77028 


27357 


72648 


31361 


68689 


22 


39 


83054 


76946 


97427 


72578 


81425 


68575 


91 


40 


9.28130 


10.76870 


9.27496 


10.72504 


9.31489 


10.68511 


90 


41 


28206 


78794 


97566 


72484 


31559 


68448 


19 


42 


23283 


76717 


97635 


72365 


81616 


68384 


18 


48 


28859 


76641 


97704 


72296 


81679 


68321 


17 


44 


93435 


76565 


97778 


72227 


81743 


68257 


16 


46 


28510 


76490 


97842 


72168 


81806 


68194 


15 


46 


28586 


76414 


27911 


72069 


31870 


68130 


14 


47 


93661 


76339 


27980 


72020 


81933 


68067 


13 


48 


9878? 


76263 


28049 


71951 


31996 


68004 


19 


49 


98312 


76188 


28117 


71883 


32059 


67941 


11 


50 


9.23887 


10.76113 


9.28186 


10.71814 


9.82122 


10.67878 


10 


61 


98962 


76088 


28264 


71746 


32185 


67815 


9 


52 


94037 


75963 


28323 


71677 


82248 


67752 


8 


58 


94112 


75888 


98391 


71609 


82811 


67689 


7 


64 


91186 


75814 


98469 


71541 


39873 


67627 


6 


56 


94261 


75739 


98527 


71473 


89480 


67564 


5 


66 


84835 


75666 


28595 


71405 


82498 


67602 


4 


67 


94410 


75590 


98662 


71388 


89561 


67489 


3 


68 


94484 


75616 


98730 


71270 


32623 


67377 


9 


59 


84558 


75442 


98798 


71909 


82685 


67815 


1 


60 


84632 


75368 


28865 


71135 


89747 


67253 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 






80* 




W 




78» 



Digitized by VjOOQIC 



164 


LOG. TANGENTS 


AND COTANGENTS 




/ 


12» 


18* 


14» 


/ 


Tan 


Cotan 


Tftxi 


Cotan 


Tan 


Cotan 





9.82747 


10.67258 


9.36336 


10.63664 


9.89677 


10.60828 


60 


1 


82810 


67190 


86394 


63606 


89781 


60269 


60 


2 


32872 


67128 


86452 


63548 


89785 


60215 


68 


8 


82938 


67067 


86509 


68491 


89838 


60162 


67 


4 


32995 


67005 




68434 


89892 


60106 


66 


6 


83057 


66948 


86624 


68376 


89945 


60066 


65 


6 


38119 


66881 


36681 


63319 


39999 


60001 


54 


7 


83180 


66820 


36788 


63262 


40052 


69048 


53 


8 


88-242 


66758 


86795 


68205 


40106 


69894 


62 


9 


83308 


66097 


86852 


68148 


40159 


59641 


51 


10 


9.38865 


10.66636 


9.86909 


10.68091 


9.40212 


10.59788 


50 


11 


884d6 


66574 


86966 


68034 


40266 


69784 


49 


12 


88487 


66518 


87028 


62977 


40310 


59681 


48 


18 


88548 


66462 


37080 


62920 


40872 


59628 


47 


14 


88609 


66391 


37187 


628C3 


40425 


69575 


46 


15 


38670 


66380 


87198 


02807 


40478 


59522 


45 


16 


88781 


66269 


87260 


62750 


40581 


69469 


44 


17 


88792 


66208 


87306 


62G94 


40584 


69416 


48 


18 


88858 


66147 


87363 


62687 


' 40686 


59364 


42 


19 


88913 


66087 


37419 


62581 


40689 


60311 


41 


20 


9.83974 


10.66026 


9.87476 


10.62524 


9.40742 


10.69268 


40 


21 


34034 


66966 


87632 


62468 


40795 


60006 


89 


^ 


84095 


66005 


87568 


62412 


40647 


691.':i8 


88 


23 


84165 


65845 


87644 


62856 


40900 


60100 


87 


24 


84216 


65785 


87700 


6280O 


40962 


69048 


86 


25 


34276 


66724 


37766 


6:3344 


41005 


68995 


86 


26 


84836 


65664 


87812 


62188 


41057 


58943 


34 


27 


34896 


65604 


37868 


62182 


41109 


£8891 


83 


28 


34456 


65544 


87924 


62076 


41161 


58889 


88 


29 


84516 


65484 


37980 


62020 


41214 


68786 


81 


80 


9.34576 


10.65424 


9.88035 


10.61965 


9.41266 


10.58784 


80 


81 


84635 


65865 


38001 


61909 


41818 


58682 


29 


32 


84695 


66806 


38147 


61858 


41870 


68680 


28 


88 


34756 


66245 


88202 


61798 


41422 


68678 


87 


84 


34814 


65186 


38257 


61743 


41474 


68526 


86 


85 


34874 


65126 


88813 


61687 


41626 


66474 


25 


86 


34933 


65067 


88368 


61632 


41578 


68422 


24 


87 


8499S 


65008 


38423 


61577 


41629 


68871 


88 


88 


35051 


64949 


38479 


61521 


41681 


58819 


82 


89 


36111 


64889 


88534 


61466 


41788 


56267 


81 


40 


9.85170 


10.64830 


9.38589 


10.61411 


9.41784 


10.6S»6 


20 


41 


35229 


64771 


38644 


61366 


41886 


68164 


19 


42 


35288 


64712 


38699 


61801 


41887 


66118 


18 


48 


85847 


64668 


88754 


61246 


41939 


68061 


17 


44 


85405 


64595 


88808 


61192 


41990 


68010 


16 


45 


85464 


64536 


88868 


61137 


42041 


57969 


16 


46 


36523 


64477 


88918 


61082 


42093 


67907 


14 


47 


85581 


64419 


88972 


61028 


42144 


67856 


13 


48 


85640 


64860 


80027 


60978 


42195 


67806 


18 


49 


8569S 


64802 


89082 


60918 


42246 


57754 


11 


60 


9.85767 


10.64848 


9.89186 


10.60664 


9.42297 


10.57708 


10 


51 


35815 


64185 


89190 


60610 


42848 


57652 





52 


35873 


64127 


89246 


60765 


42399 


57601 


8 


58 


35931 


64069 


89299 


60701 


42450 


57660 


7 


64 


85989 


64011 


39868 


60647 


42601 


67409 


6 


55 


88047 


63958 


39407 


60508 


42652 


57448 


6 


66 


86106 


68895 


89461 


60.539 


42603 


67897 


4 


57 


36163 


63687 


89615 


604K5 


42653 


57847 


8 


58 


86221 


63779 


89569 


60431 


42704 


57296 


8 


69 


36279 


68721 


39623 


60877 


42765 


57246 


1 


60 


86886 


63664 


89677 


60:^ 


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57106 





f 


Ck>tan 


Tan 


Ck>tan 


Tan 


Cotan 


Tan 


•• 




„• 




76- 




75» 



Digitized by VjOOQIC 





LOG. 


TANGENTS AND COTANGENTS 


155 


^ 


15» 


ie« 




17* 


/ 




Tan 


Ootan 


Tan 


CSotan 


Tan 


Cotan 





9.42805 


10.57195 


9.45750 


10.54250 


9.48584 


10.61466 


60 


1 


42856 


57144 


45797 


54208 


48579 


51421 


69 


2 


42906 


57094 


45845 


54155 


48624 


51376 


58 


3 


42957 


57043 


45892 


64106 


48669 


51831 


57 


4 


43007 


56993 


46940 


54060 


48714 


61286 


56 


5 


43057 


56948 


45987 


54013 


48759 


51241 


55 


6 


43108 


56892 


46035 


53965 


48804 


61196 


54 


7 


48158 


56842 


46062 


5:^918 


48849 


61151 


&i 


8 


48206 


56792 


46180 


53870 


48894 


61106 


52 


9 


43258 


56742 


46177 


53823 


48989 


61061 


51 


10 


9.48306 


10.56692 


9.46221 


10.58776 


9.48984 


10.61016 


60 


11 


4:^58 


56642 


46271 


53729 


49029 


50971 


49 


12 


43408 


56592 


46319 


53681 


49073 


50927 


48 


18 


48458 


56542 


46366 


53634 


49118 


50882 


47 


14 


48506 


56492 


46413 


58587 


49168 


60887 


46 


15 


43558 


56442 


46460 


58540 


49207 


50793 


45 


16 


43807 


56393 


46507 


58493 


49252 


50748 


44 


17 


43657 


56343 


46564 


5844G 


49296 


50704 


43 


18 


48707 


56298 


46601 


53399 


49341 


50659 


42 


19 


43756 


56S44 


46648 


58352 


49885 


60615 


41 


20 


943806 


10.56194 


9.46694 


10.58306 


9.49480 


10.60670 


40 


21 


"^ 43856 


56145 


46741 


53259 


49474 


50526 


39 


2i 


43905 


56095 


46788 


58212 


49519 


50481 


88 


24 


48954 


56046 


46835 


53165 


49563 


50437 


37 


24 


44004 


55996 


46881 


58119 


49607 


60393 


W 


25 


44053 


55947 


46928 


58072 


49652 


50348 


B5 


26 


44102 


onovo 


46975 


58025 


49696 


50304 


34 


27 


44151 


55849 


•47021 


52979 


49740 


50260 


33 


28 


44201 


65799 


47068 


52982 


49784 


50216 


8> 


29 


44250 


55750 


47114 


52886 


49628 


60172 


31 


80 


9.44299 


10.55701 


9.47160 


10.52840 


9.49872 


10.50128 


80 


31 


41348 


55652 


47207 


52798 


49916 


60084 


29 


32 


44397 


65603 


47263 


52747 


49960 


60040 


28 


83 


44446 


56554 


47299 


52701 


50004 


49996 


27 


34 


44495 


55505 


47846 


52654 


50048 


49952 


26 


35 


44544 


55456 


47892 


62608 


50092 


49908 


25 


86 


4459J 


65408 


47438 


62562 


50136 


49664 


24 


87 


44641 


55359 


47484 


63616 


50180 


49820 


23 


88 


44690 


55810 


47680 


62470 


60323 


49777 


22 


89 


44788 


55262 


47676 


62424 


50267 


49733 


21 


40 


9.44787 


10.55213 


9.47622 


10.62878 


9.50311 


10.49689 


20 


41 


44836 


55164 


. 47668 


62832 


50355 


49645 


19 


42 


44884 


55116 


47714 


52286 


50398 


49603 


18 


43 


44938 


55067 


47760 


58240 


50442 


49558 


17 


44 


44981 


55019 


47806 


52194 


50485 


49515 


16 


45 


45029 


54971 


47852 


52148 


50529 


49471 


15 


46 


45078 


54922 


47897 


52108 


60672 


49428 


14 


47 


45126 


54874 


47943 


52057 


50616 


493«4 


18 


48 


45174 


54826 


47989 


52011 


50659 


49341 


12 


49 


45222 


54778 


48035 


51965 


5aro3 


49297 


11 


60 


9.45271 


10.54729 


9.48080 


10.51920 


9.50746 


10.49254 


10 


51 


45319 


54681 


48126 


51874 


60789 


49211 


9 


52 


45867 


54683 


48171 


51829 


50888 


49167 


8 


53 


45415 


54585 


48217 


51783 


50876 


49124 


7 


54 


45463 


54587 


48262 


51786 


50919 


49081 


6 


55 


46511 


54489 


48307 


51693 


50962 


49088 


5 


56 


45559 


54441 


48358 


61647 


61005 


48996 


4 


57 


45606 


54894 


48396 


51602 


51048 


48952 


3 


58 


46654 


54346 


48443 


51557 


51092 


48908 


2 


59 


46702 


54298 


48489 


51511 


61185 


48K65 


1 


60 


45750 


54250 


48534 


61466 


61178 


48822 





/ 


Ck>taD 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




74* 




78» 




720 



Digitized by VjOOQIC 



156 


LOG. TANGENTS 


AND COTANGENTS 




/ 


18* 


19» 




20« 


t 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.61178 


10.48822 


9.53697 


10.46303 


9.56107 


10.43893 


60 


1 


51221 


48779 


53738 


46262 


56146 


43854 


59 


2 


51264 


48736 


53779 


46221 


56185 


438.5 


58 


8 


51306 


48694 


53820 


46180 


56-^24 


43776 


67 


4 


51349 


48651 


53861 


46139 


66264 


43736 


56 


6 


51392 


48606 


63902 


46098 


66303 


43697 


56 


6 


51435 


485C5 


53943 


46057 


56342 


43658 


54 


7 


61478 


48522 


53984 


46016 


56381 


43619 


58 


8 


51520 


48480 


54025 


45975 


56420 


43580 


52 


9 


51563 


48437 


54065 


45935 


56459 


48,541 


51 


to 


9.51606 


10.48394 


9.64106 


10.45894 


9.56498 


10.4.3502 


50 


11 


51648 


48352 


54147 


45853 


56537 


43463 


49 


12 


51691 


48:^ 


54187 


45813 


56576 


43424 


48 


18 


61734 


48266 


64228 


45772 


56615 


43385 


47 


14 


61776 


48224 


54269 


45731 


56654 


4:1346 


46 


16 


51819 


48181 


54309 


45691 


56693 


43;-K)7 


45 


16 


51861 


48139 


54350 


45650 


56732 


43268 


44 


17 


51903 


48097 


54390 


45610 


56771 


4.'229 


43 


18 


51946 


48054 


64431 


45569 


56810 


43190 


42 


19 


51988 


48012 


54471 


45529 


56849 


43151 


41 


20 


9.52031 


10.47969 


9.54512 


10.45488 


9.56887 


10.43118 


40 


21 


52073 


47927 


54552 


45448 


66926 


43074 


39 


22 


52115 


47885 


54593 


45407 


5()965 


43035 


38 


28 


52157 


47843 


54633 


46367 


57004 


42996 


37 


24 


52200 


47800 


54673 


45827 


57042 


4'29f)8 


86 


25 


5224-2 


47758 


54714 


45286 


57081 


42919 


35 


26 


52284 


47716 


54754 


45246 


57120 


42880 


34 


27 


52326 


47674 


54794 


45206 


57158 


42842 


33 


28 


62368 


.47632 


54835 


45165 


67197 


42803 


82 


29 


62410 


47590 


54875 


45125 


57236 


42765 


31 


30 


9.52458 


10.47548 


9.54915 


10.45085 


9.57274 


10.42726 


90 


31 


52494 


47606 


- 54955 


45045 


57312 


42688 


29 


32 


52536 


47464 


54995 


45005 


67851 


42649 


28 


S3 


52578 


47422 


55036 


44965 


57889 


42611 


27 


34 


62620 


4';880 


55075 


44926 


57428 


42.572 


26 


35 


52661 


47339 


55115 


44885 


57466 


42534 


25 


36 


52708 


47-,'97 


56155 


44845 


57.504 


42496 


24 


37 


52W6 


47265 


55195 


44805 


67543 


42467 


23 


38 


52787 


47218 


55235 


44765 


57581 


42419 


22 


39 


52829 


47171 


65275 


447T25 


57619 


42381 


21 


40 


9.52870 


10.47130 


9.55815 


10.44685 


9.57668 


10.42842 


20 


41 


52912 


47088 


55856 


44645 


57696 


42304 


19 


42 


52953 


47047 


55895 


44605 


57734 


42266 


18 


43 


62996 


47005 


65434 


44566 


57772 


42228 


17 


44 


58037 


46963 


55474 


446-26 


57810 


42190 


16 


45 


53078 


46922 


55514 


44486 


57849 


42151 


16 


46 


53120 


46880 


65554 


44446 


57887 


42113 


14 


47 


53161 


46889 


55593 


44407 


57925 


42075 


13 


48 


53202 


46798 


55633 


44867 


67968 


42(»37 


12 


49 


53244 


46756 


55678 


44327 


5S001 


41999 


11 


50 


9.532a5 


10.46716 


9.55712 


10.44288 


9.58039 


10.41961 


10 


51 


53327 


46678 


55752 


44248 


58077 


41923 


9 


52 


53868 


46632 


55791 


44209 


58116 


41885 


8 


53 


53409 


46591 


55a31 


44169 


68153 


41847 


7 


54 


53450 


46550 


65870 


44180 


58191 


41809 


6 


55 


53492 


46506 


55910 


44090 


68229 


41771 


6 


56 


53533 


46467 


55949 


44051 


.58267 


41733 


4 


57 


58574 


46426 


65989 


44011 


58804 


41696 


8 


58 


53616 


46386 


56028 


43972 


58342 


41658 


2 


59 


53656 


46344 


56067 


43033 


68380 


41620 


1 


60 


58697 


46303 


56107 


43893 


68418 


41582 





9 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


f 




71* 




70«» 





Digitized by VjOOQIC 





LOG 


. TANGENTS 


AND COTANGENTS 


16: 


t 


«!• 


22» 


28» 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 







9.58418 


10.41582 


9.60641 


10.30859 


9.62785 


10.87815 


60 


1 


58455 


4154ft 


60677 


8982:1 


62820 


87180 


69 


s 


5A493 


41507 


60714 


89286 


68856 


87146 


68 


3 


58531 


41460 


60750 


30250 


62890 


87110 


67 


4 


68569 


41481 


60786 


80214 


62986 


87074 


56 


5 


58606 


41894 


60828 


;«177 


62961 


87039 


55 


6 


58644 


41356 


60859 


39141 


62996 


87004 


64 


7 


58681 


41310 


60895 


39105 


68031 


869U9 


53 


8 


58719 


41281 


60981 


89069 


68066 


86984 


52 


9 


58757 


41243 


60967 


39083 


63101 


86899 


61 


10 


9.68794 


10.41206 


9.61004 


10.38996 


9.631.35 


10.86865 


50 


11 


58aS2 


41168 


61040 


88960 


68170 


86680 


49 


1:8 


58869 


41181 


61076 


38024 


63806 


86795 


48 


18 


68907 


41093 


61112 


38886 


68240 




47 


14 


58944 


41066 


61148 


38858 


68876 


86725 


46 


15 


58961 


41019 


61184 


88816 


63310 


mw 


45 


16 


59019 


40981 


61220 


38780 


68.345 


36655 


44 


17 


69056 


401)41 


61266 


88744 


63879 


36021 


48 


18 


59094 


40906 


61292 


;»706 


63414 


86586 


42 


10 


59131 


40809 


61328 


88672 


63449 


86551 


41 


90 


9.59168 


10.40682 


. 9.61864 


10.38686 


9.63484 


10.86616 


40 


21 


59205 


40795 


61400 


38600 


63519 


36481 


39 


82 


59248 


40757 


61436 


8856^ 


63653 


86447 


88 


:^8 


59280 


40720 


eU72 


88528 


C3588 


86412 


87 


24 


59817 


40688 


61508 


3^492 


63683 


86377 


36 


95 


59854 


40646 


61544 


38456 


68657 


36848 


sa 


26 


59:^91 


40609 


61579 


88421 


G8692 


36806 


84 


87 


59429 


40571 


61615 


88885 


08786 


86274 


33 


28 


59466 


405:14 


61651 


88:n9 


63761 


86219 


38 


29 


59503 


40497 


61687 


88818 


63796 


86204 


81 


80 


9.59540 


10.40460 


9.61722 


10.88278 


9.63880 


10.86170 


80 


81 


595T7 


40428 


<51758 


8S242 


68865 


86135- 


89 


Zi 


59614 


40386 


61794 


38200 


68899 


86101 


28. 


88 


59651 


40849 


61880 


S8I70 


63934 


86066 


87 


31 


59688 


40812 


61865 


.S81.35 


6:^068 


86032 


86 


Ki 


597^25 


40275 


61001 


88099 


64008 


85997 


85 


86 


59762 


40238 


61986 


38064 


64087 


85963 


84 


87 


r>9799 


40201 


61972 


88028 


64072 


85926 


» 23 


88 


59885 


40165 


62008 


37998 


64106 


86894 


22 


89 


598T2 


40128 


62043 


37957 


64140 


86860 


21 


40 


9.59909 


10.40091 


9.62079 


10.37921 


9.64175 


10.86826 


20 


41 


59946 


40054 


6-2114 


87886 


642U9 


85791 


19 


42 


59983 


40017 


62150 


87860 


64843 


86767 


18 


48 


60019 


39981 


62185 


87815 


64278 


36722 


17 


44 


60056 


39944 


62221 


87779 


64812 


85668 


16 


45 


60093 


39907 


62250 


87744 


64846 


85654 


15 


46 


60130 


89870 


62292 


37708 


64381 


35619 


14 


47 


60166 


89834 


62327 


37678 


64416 


85585 


18 


48 


60208 


89797 


62862 


87688 


64449 


85651 


12 


49 


60240 


89760 


62398 


37602 


64483 


85617 


11 


50 


9.60276 


10.89724 


9.62433 


10.87567 


9.64517 


10.&5488 


10 


61 


60313 


89687 


62468 


87582 


64662 


86448 


9 


52 


60349 


89651 


62504 


37496 


64686 


36414 


8 


63 


60886 


39614 


62539 


87461 


64680 


85880 


7 


54 


• 60422 


89578 


62574 


87426 


64664 


86846 


6 


65 


60459 


39541 


62609 


87891 


64666 


85.312 


6 


66 


60495 


89505 


62645 


37855 


64728 


85278 


4 


67 


60532 


39468 


62680 


87820 


64766 


85244 


8 


68 


60568 


89482 


62715 


37285 


64790 


85210 


2 


69 


60605 


89895 


62750 


37850 


64824 


35176 


1 


00 


60641 


39359 


62785 


87215 


64858 


85148 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




88» 




e?* 




»6« 





Digitized by VjOOQIC 



158_ 


LOG 


. TANGENTS 


AND COTANGENTS 




/ 


24« 


as* 


««• 


/ 


Tan 


Coten 


Tan 


Cotan 


Tan 


Cotan 





9.64868 


10.86142 


9.66867 


10.83138 


9.68818 


10.81182 


60 


1 


64892 


86108 


66900 


88)00 


68850 


81150 


60 


2 


64926 


85074 


66938 


88067 


68882 


81118 


68 


8 


64960 


85040 


66966 


83034 


68914 


81066 


67 


4 


64994 


85006 


66999 


88001 


68946 


81064 


66 


6 


65028 


84972 


67032 


82968 


68978 


81022 


66 


6 


65062 


34938 


67066 


82935 


69010 


80990 


64 


7 


66096 


84904 


67098 


82908 


69042 


80958 


63 


8 


66180 


84870 


67131 


82869 


69074 


80926 


62 


9 


65164 


84886 


67168 


82887 


69106 


80694 


61 


10 


9.65197 


10.84808 


9.47196 


10.82804 


9.69188 


10.30862 


60 


11 


65281 


84769 


67229 


32771 


69170 


30630 


49 


12 


66265 


84786 


67262 


82738 


69202 


80798 


48 


13 


65299 


84701 


67295 


82706 


69234 


80766 


47 


14 


65883 


84667 


67827 


82673 


69266 


80734 


46 


16 


65366 


84634 


67860 


82640 


69298 


80702 


46 


16 


65400 


34600 


67398 


82607 


69829 


80671 


44 


17 


65484 


34666 


67426 


82574 


69361 


30689 


48 


18 


65467 


34633 


67468 


82542 


69893 


80607 


42 


19 


65601 


84499 


67491 


825(?9 


69425 


80576 


41 


20 


9.66635 


10.84466 


9.67624 


10.82476 


9.69457 


10.80643 


40 


21 


66668 


84482 


67656 


82444 


69488 


80512 


89 


22 


65602 


84898 


67589 


82411 


69620 


80480 


88 


28 


66686 


84364 


67628 


82378 


69662 


80448 


87 


24 


656C9 


84831 


67664 


82346 


69684 


80416 


86 


25 


66708 


84297 


67687 


82318 


69616 


30385 


86 


26 


65786 


84264 


67719 


82281 


69647 


80858 


84 


27 


65770 


84280 


67752 


82248 


69679 


30321 


88 


28 


65S08 


84197 


67786 


82215 


69710 


80290 


82 


29 


05837 


84163 


67817 


82188 


69742 


80258 


81 


80 


9.65870 


10.34130 


9.67860 


10.32160 


9.69774 


10.80226 


80 


81 


66904 


84096 


67882 


32118 


69805 


30195 


29 


32 


65937 


84063 


67916 


82065 


69887 


80163 


28 


83 


65971 


84029 


67947 


88053 


69868 


30182 


27 


84 


66004 


83996 


67960 


82020 


69900 


80100 


26 


85 


66038 


33962 


68012 


81988 


69932 


80066 


26 


M 


66071 


83929 


68044 


31956 


69063 


30037 


24 


87 


66104 


88896 


68077 


31928 


69995 


80005 


88 


88 


66138 


33862 


68109 


81891 


70026 


29974 


28 


89 


66171 


88829 


68142 


81858 


70068 


29942 


21 


40 


9.66204 


10.88796 


9.68174 


10.31826 


9.70089 


10.29911 


80 


41 


66288 


33762 


68206 


81794 


70121 


29879 


19 


42 


66271 


83729 


68289 


31761 


70152 


29846 


18 


48 


66304 


88696 


68271 


81729 


70184 


29816 


17 


44 


66887 


88668 


6&303 


31697 


70216 


29785 


16 


45 


66871 


83629 


68886 


81664 


70247 


29758 


15 


46 


66404 


83696 


68368 


81632 


70278 


29722 


14 


47 


66487 


38568 


68400 


81600 


70809 


29691 


13 


46 


66470 


a%80 


68432 


81568 


70841 


29659 


12 


49 


66603 


33497 


68465 


31585 


70372 


29628 


11 


60 


9.66637 


10.33463 


9.68497 


10.31503 


9.70404 


10.29606 


10 


61 


66670 


33480 


68529 


31471 


704&5 


29665 





62 


66603 


83.397 


68561 


31489 


70466 


29584 


8 


68 


66636 


88864 


6a')98 


81407 


70498 


29502 


7 


64 


60669 


88831 


68626 


81374 


70529 


29471 





55 


66702 


33298 


68658 


81342 


70660 


29440 


6 


66 


66785 


88265 


68690 


81810 


70692 


89406 


4 


67 


66768 


88232 


68722 


81278 


70628 


29877 


a 


68 


66801 


38199 


68754 


81246 


70664 


89846 


8 


50 


66a34 


88166 


68786 


81214 


70686 


29315 


1 


60 


66867 


83183 


68818 


31182 


70717 


29888 





» 


Oot*u 


Tan 


CoUii 


Tftri 


Cotan 


Tan 


/ 




lA*^ 




84^ 




68» 



Digitized by VjOOQIC 





LOG. 


TANGENTS AND COTANGENTS 


169 





«?• 


a8« 


19- 1 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.70717 


10.29288 


9.72667 


10:27488 


9.74875 


10.86626 


60 


1 


70748 


. 29252 


72598 


27402 


74406 


9B696 


69 


9 


70779 


29221 


72628 


27872 


74435 


QMMK 
XDOOO 


68 


3 


70810 


29190 


72659 


27841 


74466 


85585 


67 


4 


70841 


29159 


72689 


27811 


74494 


26506 


56 


5 


70873 


29127 


'iifm 


27280 


74524 


26476 


55 


6 


70904 


29096 


72750 


27250 


74554 


25446 


54 


7 


70935 


29065 


72780 


27220 


74588 


85417 


53 


8 


70966 


29034 


72811 


27189 


74618 


25887 


52 


i ^ 


70997 


29008 


72841 


27159 


74643 


• 26357 


61 


10 


9.71028 


10.28972 


9.72872 


10.27128 


9.74678 


10.95327 


60 


11 


71059 


28941 


7290:2 


2; 098 


74702 


95298 


49 


13 


71090 


28910 


72982 


27068 


74782 


25266 


48 


13 


71121 


28879 


72963 


27087 


74762 


25938 


47 


14 


71158 


28847 


72993 


27007 


74791 


95209 


*46 


15 


71184 


28816 


78023 


26977 


74821 


95179 


46 


16 


71215 


28785 


73054 


26946 


74851 


25149 


44 


17 


71246 


28754 


;3084 


26916 


74880 


25120 


43 


18 


71277 


28723 


73114 


26886 


74910 


26090 


42 


19 


71308 


28692 


73144 


26856 


74939 


25061 


41 


90 


9.71339 


10.28661 


0.78175 


10.26825 


9.74969 


10.25081 


40 


21 


71370 


28680 


73205 


26795 


74996 


95009 


39 


'22 


71401 


28599 


78235 


26765 


75026 


24972 


38 


vS 


71431 


28569 


73266 


26735 


75058 


24942 


87 


24 


71462 


28588 


73295 


26705 


75087 


94918 


36 


% 


71493 


28507 


73326 


26674 


75117 


24883 


35 


!!» 


71624 


28476 


78856 


96644 


75146 


24854 


84 


87 


71555 


28445 


73886 


26614 


75176 


94624 


83 


28 


71686 


28414 


73416 


26684 


75206 


24795 


82 


2» 


71617 


28383 


73446 


96554 


75235 


94765 


81 


SO 


9.71648 


10.28352 


9.73476 


10.26524 


9.75264 


10.94736 


80 


31 


71679 


28321 


73507 


26498 


75294 


94706 


29 


32 


71709 


28291 


73637 


26463 


76323 


94677 


28 


38 


71740 


28260 


78567 


264^3 


75353 


24647 


27 


34 


71771 


28229 


78697 


26408 


75382 


24618 


26 


85 


71802 


28198 


78627 


26373 


75411 


21589 


25 


36 


71833 


28167 


73667 


26343 


75441 


24559 


24 


37 


71863 


28137 


73687 


26813 


75470 


24530 


23 


38 


71894 


28103 


73717 


26283 


75500 


24500 


22 


39 


71925 


28075 


73747 


26253 


75529 


24471 


21 


40 


9.71955 


10.28045 


9.78777 


10.26223 


9.75558 


10.24442 


20 


41 


71986 


28014 


78807 


26198 


75588 


24412 


19 


42 


72017 


27983 


73837 


26163 


75617 


24883 


16 


44 


72048 


279.52 


73867 


26133 


75647 


2435;i 


17 


44 


72078 


27922 


73897 


2610H 


75676 


24824 


16 


46 


72109 


27891 


73927 


26073 


75705 


24295 


15 


46 


72140 


27860 


73957 


26043 


75735 


24265 


14 


47 


72170 


27^30 


78987 


26018 


75764 


24236 


18 


48 


72201 


27799 


74017 


S5983 


75793 


24207 


12 


1 49 


72231 


27769 


74047 


25953 


75822 


24178 


11 


1 50 


9.72262 


10.27788 


9.74077 


10.25923 


9.75852 


10.24148 


10 


51 


72293 


27707 


74107 


26698 


75881 


24119 


9 


52 


72823 


27677 


74137 


26868 


75910 


94090 


8 


53 


72854 


27646 


74166 


25834 


75939 


24061 


7 


54 


72384 


27616 


74196 


26804 


7.5969 


24081 


6 


65 


78415 


27585 


74226 


85774 


75996 


24002 


5 


56 


72445 


27555 


74256 


25744 


76027 


28973 


4 


57 


72476 


27524 


74286 


25714 


76056 


28944 


3 


58 


72506 


27494 


74316 


25684 


76066 


28914 


2 


59 


72537 


27463 


74345 


25655 


76115 


238a5 


1 


60 


72567 


27483 


74875 


25625 


76144 


23656 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


! 




62« 




«!• 




•O* 



Digitized by VjOOQIC 



160 


LOG 


. TANGENTS 


AND COTANGENTS 




/ 


80» 


«!• 




B2*» 


§ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.76144 


10.23866 


9.77877 


10.22123 


9.79579 


10.20421 


60 


1 


761T3 


28827 


77906 


22094 


79607 


20393 


69 


2 


70202 


28798 


77936 


22065 


79086 


20865 


58 


3 


76231 


28769 


77963 


22037 


79663 


20337 


67 


4 


76261 


28739 


77992 


22008 


79691 


20309 


56 


6 


76290 


28710 


78020 


21980 


79719 


20281 


55 


6 


76319 


88681 


78049 


21961 


79747 


20253 


64 


7 


76348 


28662 


78077 


21923 


79776 


20224 


58 


8 


76377 


23623 


78106 


21894 


79804 


20)96 


52 


9 


76406 


23594 


78185 


21865 


79682 


20168 


51 


10 


9.76485 


10.23566 


9.78163 


10.21837 


9.79860 


10.20140 


60 


11 


76464 


28536 


78192 


21808 


79888 


20112 


49 


12 


76493 


28507 


78220 


21780 


79916 


20084 


48 


13 


76522 


23478 


78249 


21751 


79944 


20056 


47 


14 


76661 


28449 


78277 


21723 


79972 


20028 


46 


15 


76680 


23420 


78306 


21694 


80000 


20000 


45 


16 


76609 


28391 


78834 


21666 


80028 


19972 


44 


17 


76639 


23361 


78863 


21637 


80056 


19944 


t 


18 


76668 


28332 


78891 


21609 


80084 


19916 


19 


76697 


23303 


78419 


21681 


80112 


19688 


41 


20 


9.76725 


10.23276 


9.78446 


10.21562 


9.80140 


10.19660 


40 


21 


76754 


28246 


78476 


81524 


80168 


19832 


89 


22 


76788 


23217 


78505 


21495 


80196 


19805 


38 


2i^ 


76812 


23168 


78633 


. 21467 


80288 


19777 


87 


24 


76841 


23159 


78662 


21438 


80251 


19749 


86 


25 


768^0 


28180 


78690 


21410 


80279 


19721 


36 


26 


16899 


28101 


78618 


21882 


80807 


19698 


34 


27 


76928 


28072 


78647 


21863 


80335 


19665 


38 


28 


76957 


23043 


78676 


21325 


80863 


19637 


82 


29 


76986 


28014 


78704 


21296 


80391 


19609 


31 


80 


9.77016 


10.22966 


9.78732 


10.21268 


9.80419 


10.19681 


30 


81 


77044 


22956 


78760 


21240 


80447 


19553 


29 


S2 


77073 


22927 • 


78789 


21211 


80474 


19526 


28 


83 


TTlOl 


22899 


78817 


21183 


80502 


19498 


27 


34 


77130 


22870 


78846 


21156 


80580 


19470 


26 


35 


77169 


22841 


78874 


21126 


80558 


19442 


25 


36 


77188 


22812 


78902 


21096 


80586 


19414 


24 


37 


77217 


22788 


78930 


21070 


60614 


19886 


28 


38 


77246 


22754 


78959 


21041 


80642 


19368 


22 


39 


77274 


22726 


78987 


21018 . 


80669 


19831 


31 


40 


9.77808 


10.22697 


9.79015 


10.20986 


9.80097 


10.19308 


90 


41 


77882 


22668 


79043 


20957 


80725 


19276 




42 


77361 


22689 


79072 


20928 


80763 


19247 




43 


77890 


22610 


79100 


20900 


80781 


19219 




44 


77418 


22582 


79128 


20872 


80806 


19192 




45 


77447 


22553 


79156 


20844 


80886 


19164 




46 


77476 


22524 


79185 


20815 


80864 


19136 




47 


77505 


22495 


79213 


20787 


80892 


19108 




48 


77533 


22467 


79241 


20759 


80919 


19081 




49 


77562 


22438 


79269 


20731 


80947 


19053 




50 


9.77691 


10.22409 


9.79297 


10.20703 


9.80976 


iO. 19026 




61 


77619 


22S81 


79326 


20674 


81003 


18997 




62 


77648 


22352 


79354 


20646 


81030 


18970 




53 


77677 


22823 


79382 


80618 


81068 


18942 




54 


77706 


22294 


79410 


20590 


81086 


18914 




56 


77784 


22266 


79488 


20562 


81118 


18887 




66 


77768 


22237 


79466 


20584 


81141 


18869 




67 


77791 


22209 


79495 


20506 


81169 


18881 




58 


77820 


22180 


79523 


20477 


81196 


18804 


2 


59 


77849 


22151 


79551 


20449 


81224 


18776 




60 




22123 


79579 


20421 


81252 


18748 




/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 

6r 


' 


59* 




68* 





Digitized by VjOOQIC 





LOG 


. TANGENTS 


AND COTANGENTS 


161 




88» 


84» 




86« 


/ 


Tan 


Ootan 


Tan 


Cotan 


Tan 


Cotan 





9.81258 


10.18748 


9.88899 


10.17101 


9.84583 


10.16477 


60 


1 

2 


81279 


18721 


88986 


17074 


81550 


16460 


69 


81807 


18698 


88968 


17047 


84676 


a6484 


88 


8 
4 
6 

7 
8 
9 


81885 


18665 


88960 


17080 


84608 


15397 


67 


81S62 


18638 


83008 


16992 


84630 


16870 


66 


81890 


18610 


88085 


16965 


84657 


16343 


65 


81418 


18588 


83068 


16968 


84684 


16316 


64 


81446 


18556 


83089 


16911 


84711 


15289 


68 


81478 


18587 


88117 


16883 


84738 


15262 


68 


81500 


18500 


83144 


16856 


84764 


15286 


61 


10 
11 
18 
18 

14 
15 
16 
17 
18 
18 


0.81688 


10.18478 


9.88171 


10.16829 


9.84791 


10.15809 


60 


81666 


18444 


83198 


16808 


84818 


15188 


40 


81588 


18417 


83286 


16775 


84845 


16166 


48 


81611 


18889 


83868 


16748 


84873 


15188 


47 


81688 


18368 


88880 


16780 


84809 


15101 


46 


81666 


18881 


83307 


16693 


84985 


16075 


45 


81693 


18807 


838S4 


16666 


84958 


15048 


44 


81781 


18279 


83361 


16638 


84979 


15081 


48 


81748 


18852 


83888 


16612 


85006 


14994 


42 


81776 


18824 


83416 


16585 


85033 


14967 


41 


20 
81 
88 
88 


9.81808 


10.18197 


9.88448 


10.16568 


9.85059 


10.14941 


40 


81881 


18169 


83470 


16530 


85086 


14914 


89 


81858 


18142 


88497 


1650!) 


861 18 


14887 


88 


81886 


18114 


88584 


16476 


85140 


14860 


87 


1 84 


81918 


18067 


83551 


16449 


85166 


14834 


86 


85 
88 


81941 


18069 


88578 


16488 


86198 


14807 


85 


81968 


18088 


83605 


16395 


86880 


14780 


84 


87 


81996 


18004 


83638 


16368 


85847 


14753 


88 


88 


82088 


17977 


83659 


16841 


85878 


14727 


82 


20 


82051 


17949 


83686 


16814 


85800 


14700 


81 


80 


9.82078 


10.17988 


9.88713 


10.16887 


9.85387 


10.14678 


80 


81 


82106 


17894 


83740 


16860 


85354 


14646 


89 


82 


82138 


17847 


83768 


16888 


• 86880 


14620 


88 


83 
84 


88161 


17839 


88795 


16205 


85407 


14693 


27 


88188 


17818 


88888 


16178 


86484 


14566 


86 


86 


88815 


17785 


83849 


16151 


86460 


14540 


25 


86 
87 
88 
80 


88843 


17757 


83876 


16124 


85487 


14513 


84 


88270 


17730 


83908 


16097 


86514 


14486 


88 


82806 


17708 


83980 


16070 


85540 


14460 


28 


82385 


17675 


83957 


16048 


85567 


14433 


81 


40 


9.88858 


10.17648 


9.88984 


10.16016 


9.86594 


10.14406 


80 


41 


82880 


17620 


84011 


15989 


86680 


14380 




48 


82407 


17593 


84088 


15968 


85647 


14363 




48 


88436 


17565 


84066 


15986 


85674 


14826 




44 

46 


82468 


17538 


84098 


15906 


86700 


14300 




88489 


17511 


84119 


16681 


85727 


14278 




46 


88617 


17483 


84146 


15864 


86764 


14846 




47 


88544 


17456 


84173 


16887 


85780 


14220 




48 
49 


88571 


17429 


84800 


15800 


85807 


14193 




82599 


17401 


84887 


15778 


85834 


14166 




60 
61 
68 
68 
64 


9.88686 


10.17874 


9.84864 


10.16746 


9.86860 


10.14140 




82653 


17847 


84880 


15780 


85887 


14113 




82681 


17819 


84807 


15698 


86913 


14087 




88708 


17898 


84884 


1.5666 


85940 


14060 




88735 


17865 


84861 


16689 
ft612 


85967 


14088 




65 
66 

57 
68 


88762 


17888 


84888 


86993 


14007 




82790 


17810 


84415 


15585 




18960 




88817 


17188 


84448 


15558 


86046 


13954 




82844 


17166 


84469 


16631 


86078 


18927 




69 
60 


82871 


17129 


84406 


15504 


86100 


13900 




82899 


17101 


84688 


15477 


86186 


18874 




/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 


I 


560 




66* 




64* 



Digitized by VjOOQIC 



62 


LOG 


. TANGENTS 


AND COTANGENTS 




/ 


S6« 


S7» 


•8- 


/ 




Tan 


Coten 


Tan 


Ootan 


Tan 


Cotan 





9.86126 


10.13874 


9.87711 


10.12289 


9.89281 


10.10719 


60 


1 


86153 


18847 


87788 


12262 


89807 


10698 


68 


2 


86179 


18821 


87764 


12286 


89833 


10667 


68 


8 


86206 


13794 


87790 


12210 


89359 


10641 


57 


4 


86232 


13768 


87817 


12183 


89886 


10615 


06 


6 


86259 


13741 


87848 


12157 


89411 


10589 


66 


6 


86285 


13715 


87869 


12131 


89437 


10563 


54 


7 


86312 


13688 


87895 


12105 


89468 


10537 


53 


8 


86338 


18662 


87922 


12078 


89489 


10511 


62 





86365 


18635 


87948 


12052 


89515 


10485 


61 


10 


9.86392 


10.18609 


9.87974 


10.12026 


9.89641 


10.10459 


60 


11 


86418 


18582 


88000 


12000 


89567 


10483 


49 


12 


86445 


13555 


88027 


11973 


89598 


10407 


48 


13 


86471 


13529 


88058 


11947 


89619 


10381 


47 


14 


86498 


13502 


88079 


11921 


89645 


10355 


46 


15 


86524 


13476 


88105 


11895 


89671 


10329 


45 


16 


86551 


18449 


88131 


11869 


89697 


10808 


44 


17 


86577 


13428 


88158 


11842 


89728 


10277 


48 


18 


86608 


13397 


88184 


11816 


89749 


10251 


42 


19 


86630 


13370 


88210 


11790 


89775 


10225 


41 


SO 


9.86656 


10.13844 


9.88236 


10.11764 


9.89801 


10.10199 


40 


21 


86683 


18817 


88262 


11738 


89827 


10178 


89 


23 


86709 


13291 


88289 


11711 


89853 


10147 


88 


23 


86786 


18264 


88315 


11685 


89879 


10121 


87 


24 


86762 


13288 


88341 


11659 


89905 


10095 


86 


25 


86789 


18211 


88867 


11683 


89931 


10069 


85 


26 


86815 


18186 


8S893 


11607 


89957 


10048 


84 


27 


86842 


18158 


88420 


11580 


89983 


10017 


88 


28 


86868 


13182 


88446 


11554 


90009 


09991 


82 


29 


86894 


18106 


88472 


11528 


90035 


09965 


81 


80 


9.86921 


10.18079 


9.88498 


10.11602 


9.90061 


10.09939 


80 


81 


86947 


18a'^8 


88524 


11476 


90086 


09914 


29 


32 


86974 


18026 


88550 


11450 


90112 


09888 


26 


83 


87000 


18000 


88677 


11423 


90188 


09862 


27 


84 


87027 


12973 


886a3 


11397 


90164 


09636 


26 


35 


87058 


12947 


88629 


11871 


90190 


09810 


85 


86 


87079 


12921 


88655 


11345 


90216 


09784 


84 


87 


87106 


12894 


88681 


11319 


90242 


09758 


88 


38 


87182 


12868 


88707 


11293 


90268 


09732 


28 


39 


87158 


12842 


88733 


11267 


90294 


09706 




40 


9.87185 


10.12815 


9.88759 


10.11241 


9.90320 


10.09680 




41 


87211 


12789 


88786 


11214 


90346 


09654 




43 


87288 


12762 


88812 


11188 


9a371 


09629 




43 


87264 


12736 


88838 


11162 


90397 


09608 




44 


87290 


12710 


88884 


11136 


90423 


09577 




45 


87817 


12683 


888JK) 


11110 


90449 


. 09561 




46 


87343 


12657 


88916 


11084 


90475 


09525 




47 


87869 


12631 


88942 


11058 


90501 


09499 




48 


87396 


12604 


88968 


11032 


90527 


09473 




49 


87422 


12578 


88994 


11006 


90568 


09447 




50 


9.87448 


10.12652 


9.89020 


10.10980 


9.90578 


10.09422 




51 


87475 


12525 


89046 


10964 


90604 


09396 




52 


87501 


12499 


89078 


10927 


90680 


09870 




58 


87527 


12473 


89099 


10901 


90656 


09844 




54 


87554 


12446 


89125 


10875 


90682 


09818 




55 


87580 


12420 


89151 


10649 


90708 


09292 




56 


87606 


12894 


801 77 


10823 


90734 


09266 




57 


87638 


12367 


89203 


10797 


90759 


09241 




58 


87659 


12341 


89229 


10771 


90785 


09215 




59 


87885 


12815 


89256 


10745 


90811 


09189 




60 


87711 


12289 


89281 


10719 


90687 


09163 







Ootan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 






68<» 




sa* 




61* 



Digitized by VjOOQ IC 





LOG 


. TANG 


ENTS AND CX)TANGEN'i'8 


163 


» 


»•• 


40* 


41» 


/ 


Tao 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.90887 


10.09163 


9.92381 


10.07619 


9.93916 


10.06064 


00 




90663 


09187 


92407 


07593 


93942 


06058 


so 




90889 


09111 


92433 


07567 


93967 


06038 


58 




90914 


09066 


92458 


07542 


93993 


. 06007 


67 




90940 


09060 


92484 


07516 


91018 


05982 


60 




90966 


09084 


92510 


07490 


94044 


05956 


56 




90992 


09008 


92585 


07465 


94069 


06931 


64 




91018 


08982 


92661 


07439 


. 94095 


05905 


58 




91043 


08957 


92587 


07413 


91120 


05880 


52 




910d9 


08981 


92612 


07388 


94146 


05854 


51 




9 91095 


10.08905 


9.92638 


10.07368 


9.94171 


10.058-29 


50 




91121 


06879 


92663 


07887 


94197 


0580:i 


49 




91147 


06853 


92689 


07811 


94222 


05778 


48 




91172 


08828 


92715 


07285 


91248 


05752 


47 




91198 


08802 


92740 


07260 


94273 


05727 


46 




9/224 


08776 


92766 


0?234 


94299 


a5701 


45 




91250 


08750 


92792 


07208 


94324 


05676 


44 




91276 


06724 


92817 


07183 


94330 


05650 


48 




91301 


06699 


.92843 


07157 


94375 


056-25 


42 


19 


91327 


08673 


92868 


07132 


94401 


05599 


41 


90 


9.9]a53 


10.06647 


9.92894 


10.07106 


9.94426 


10.05574 


40 


21 


91379 


06621 


92920 


07080 


94452 


05548 


39 


2S 


91404 


06596 


92945 


07065 


94477 


05528 


88 


28 


91430 


06670 


92971 


07029 


94603 


05497 


87 


24 


91456 


08544 


92996 


07004 


94628 


05472 


36 


25 


91484 


06518 


93022 


06978 


94554 


06446 


35 


26 


91507 


08493 


93048 


06952 


94579 


05421 


84 


27 


91533 


06467 


98078 


06927 


94604 


05396 


88 


28 


91559 


08441 


93099 


06901 


94630 


05370 


82 


29 


91685 


06415 


93124 


06876 


946.55 


05345 


81 


30 


9.91610 


10.08390 


9.98150 


10.06850 


9.94681 


iO.06819 


80 


31 


91636 


08364 


93175 


06825 


94706 


05294 


29 


32 


91662 


0a338 


asgoi 


06799 


94732 


06268 


28 


33 


9168S 


08312 


93227 


06773 


94757 


05243 


27 


34 


91713 


08287 


93252 


00748 


94783 


05217 


26 


35 


91789 


08261 


98278 


06722 


94808 


05192 


25 


36 


91765 


08235 


93303 


06697 


94834 


05166 


24 


37 


91791 


08209 


98329 


06071 


94859 


05141 


23 


38 


91816 


08184 


9JW54 


06646 


94884 


05116 


22 


39 


91842 


08158 


98380 


06620 


y4910 


05090 


21 


40 


9.91868 


10.06132 


9.93406 


10.06594 


9.94985 


10.05065 


20 


41 


91893 


08107 


93431 


06569 


94961 


05089 


19 


42 


91919 


08061 


93457 


06543 


94986 


05014 


18 


48 


91945 


06055 


93482 


06518 


95012 


04988 


17 


44 


91971 


08029 


93508 


06492 


96037 


04963 


16 


45 


91996 


08004 


98583 


06467 


?I50C2 


04938 


16 


46 


92022 


07978 


9a'i59 


06441 


95088 


04912 


14 


47 


92048 


07968 


!)8584 


06416 


95113 


04887 


18 


48 


92073 


07937 


9:^610 


00390 


96139 


04861 


12 


49 


92099 


07901 


93636 


06361 


95164 


04836 


11 


50 


9.92125 


10.07875 


9.98661 


10.06389 


9.96190 


10.04810 


10 


51 


92160 


07850 


93687 


06318 


95215 


04785 


9 


62 


92176 


07824 


9J712 


06288 


95240 


04760 


8 


68 


92202 


07798 


93738 


06262 


95266 


04734 


7 


54 


92227 


07773 


93763 


06237 


95291 


04709 


6 


55 


92253 


07747 


93789 


06211 


95817 


04688 


5 


56 


92279 


07721 


93814 


06186 


95842 


04658 


4 


57 


92304 


07696 


93840 


06160 


95368 


04632 


8 


58 


92:330 


07670 


93865 


06185 


95393 


01607 


2 


59 


92356 


07644 


93891 


06109 


95418 


04582 


1 


GO 


92381 


07619 


93916 


06084 


95444 


04556 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




50* 




490 




48» 



Digitized by VjOOQIC 



164 


LOG 


. TANGENTS 


AND COTANGENTS 




1 


42'* 


48«» 




44*» 


1 


Tan 


Ootaii 


Tan 


Cotan 


Tan 


Cotan 





9.95444 


10.04556 


9.96966 


10.08084 


9.96484 


10.01516 


60 


1 


95469 


04531 


96991 


08009 


98509 


01491 


69 


2 


95495 


04505 


97016 


02984 


98584 


01466 


58 


8 


955-20 


04480 


97042 


02968 


98560 


01440 


57 


4 


95545 


04465 


97067 


02938 


08585 


01415 


56 


6 


95571 


04429 


97092 


02906 


98610 


01890 


56 


6 


95596 


04404 


97118 


02862 


98685 


01365 


64 


7 


95622 


04878 


97143 


02857 


98661 


01339 


58 


8 


95647 


04353 


97168 


02882 


98686 


01314 


52 


9 


95672 


04328 


97193 


02807 


98711 


01280 


61 


10 


9.95698 


10.04302 


9.97219 


10.02761 


9.98787 


10.01263 


50 


11 


95723 


04277 


97244 


02766 


98762 


01238 


49 


IS 


95748 


04252 


97269 


02781 


9878"/ 


01213 


46 


13 


95774 


04226 


97296 


02705 


98812 


01188 


47 


14 


95799 


04201 


97320 


02660 


98688 


01162 


46 


15 


95825 


04175 


97845 


02665 


98868 


01187 


45 


16 


95860 


04150 


97871 


02629 


98888 


01112 


44 


17 


95875 


04125 


97396 


02604 


98913 


01087 


48 


18 


95901 


04099 


07421 


02579 


98989 


01061 


42 


19 


95926 


04074 


97447 


02553 


98964 


01066 


41 


ao 


9.95952 


10.04048 


9.97472 


10.02528 


9.98989 


10.01011 


40 


21 


95977 


04028 


97497 


02503 


99015 


00986 


89 


23 


96002 


08998 


97528 


02477 


99040 


00960 


88 


28 


96028 


03972 


97548 


02462 


99065 


00986 


87 


24 


96053 


08947 


97573 


02427 


99090 


00910 


86 


25 


96078 


08922 


97598 


02402 


99116 


00884 


35 


26 


96104 


03896 


97624 


02876 


99141 


00859 


84 


27 


96129 


03871 


97649 


02851 


99166 


00884 


88 


28 


96155 


08845 


97674 


02826 


99191 


00809 


82 


29 


96180 


08820 


97700 


02800 


99217 


00788 


31 


SO 


9.96205 


10.03795 


9.97725 


10.02276 


0.99242 


10.00768 


30 


81 


96231 


03769 


97760 


02250 


99267 


00788 


29 


82 


96256 


03744 


9^/76 


02224 


99298 


00707 


28 


83 


96281 


08719 


97801 


02199 


99818 


00662 


27 


84 


96307 


03693 


97826 


02174 


99343 


00667 


26 


85 


96882 


03668 


97851 


02149 


99366 


00682 


-85 


86 


96357 


08648 


97877 


02123 


99894 


00606 


24 


37 


96383 


03617 


97902 


02098 


99419 


0C581 


28 


88 


96406 


03592 


97927 


02073 


99444 


00556 


28 


89 


96433 


03567 


97958 


02047 


99469 


00531 


81 


40 


9.96459 


10.03541 


9.97978 


10.02022 


9.99495 


10.00505 


80 


41 


96484 


03516 


98003 


01997 


99620 


00480 


19 


42 


96510 


03490 


98029 


01971 


99545 


00465 


16 


43 


96585 


03465 


98054 


01946 


99670 


00480 


17 


44 


96560 


03440 


98079 


01921 


99596 


00404 


16 


45 


96686 


08414 


98104 


01896 


99621 


00879 


16 


46 


96611 


03389 


98130 


01870 


99646 


00864 


14 


47 


96686 


03364 


98155 


01845 


99672 


00828 


18 


48 


96662 


03338 


98180 


01880 


99697 


00808 


12 


49 


96687 


03818 


98206 


01794 


99722 


00278 


11 


50 


9.96712 


10.03288 


9.98231 


10.01769 


9.99747 


10.00258 


10 


51 


96738 


03262 


96256 


01744 


99773 


00227 


9 


68 


96763 


08287 


96261 


01719 


99798 


00202 


6 


S8 


96788 


03212 


9830r 


01693 


99828 


00177 


7 


64 


96814 


08186 


98382 


01668 


99848 


00152 


6 


66 


96889 


08161 


98357 


01648 


99874 


00126 


6 


66 


96864 


08186 


96388 


01617 


99899 


00101 


4 


57 


96890 


08110 


96406 


01592 


99924 


00076 


8 


58 


06915 


08065 


98483 


01567 


99949 


00051 


8 


59 


90940 


03060 


98458 


01 .M2 


99975 


00025 


1 


m 


96966 


uatm 


^m 


01516 


laooooo 


OWKW 





* 


CoUn 


Taa 


Ootan 


Tap 


Ootan 


Tan 


* 




47- 




^ 




ih^~ 



y Google 



LOC. VERSED SINES AND EXTERNAL SECANTS 165 



' Log. Vers. 



2 . 62642 
3-22848 
3-58066 
3-83054 



-02436 
• 18273 
.31662 
-43260 
.53499 



. 62642 
. 7O920 
.78478 
■85431 
.01868 



97860 
03466 
08732 
13696 
■18393 



-42230 
.45636 
.48915 
.52073 
.55121 



.58066 
-60911 

.63fi72 

■ MUl 

■ 71455 

.73ao2 

. 76282 
.78596 
^80851. 



5.8305J 
.85191 
.87291 
.8933!) 
.91332 



5.93284 

.95193 

.97061 

5-98890 

6 00680 



6 02435 
.0415S 
.05842 
.07496 
.09120 



6 10714 
.12279 
•13816 
.15327 
.16811 



6 18271 
Log. Vers. 



Log. Exsec 



2. 62642 
3-22848 
3-58066 
3 83054 



l-0243§ 
-18272 
.31662 
.43260 
.53491 



4-62642 
.70921 
.78478 

. .85431 
-91868 




5.42231 
.45638 
.48916 
•52075 
.55123 



5. 58068 
•60916 
.63674 
.66346 
.68940 



5.71457 
.73904 
. 7628i 
.78601 
80857 



5.8305( 
.85201 
.87295 
.89338 
.91335 



5. 93288 

.95197 

.97065 

5. 98894 

6-00685 



6-02440 
.04160 
.05847 
.07501 
.09125 



6-10719 
.12281 
.13822 
-15333 
.16818 



6 18278 
Log. Exsec, 



Log. Vers. 




6-31660 
.32892 
.34107 
-35305 
■36487 



-37653 
-38803 
-39931 
•41059 
■42165 



.43258 
-44337 
.45403 
.46455 
.47496 



-48524 
.4953S 
. 50544 
.51536 
•52518 



•53488 
• 54448 
•55397 
.56336 
.57265 




6-62639 
.63508 
.64359 
.65206 
. 66045 




6-74777 
.7552§ 
•76275 
•77013 
77 747 
6 ■>78474 
Log. Vers. 



■Z> Log. Exsec. 



25231 
26557 
27864 
29151 
30419 



3166§ 
32901 
34116 
35315 
36497_ 
37663 
38814 
3994§ 
41070 
42177 



48537 
49553 
50557 
51550 
52532 



53503 
54463 
55413 
56352 
57281 



58201 
59110 
60011 
60902 
61781 



62657 
63522 
64378 
65226 
66065 



6-78500 



Log. Ex 



1436 

1412 

1390 

1368 

1347 

1326 

1306 

1287 

1268 

1250 

1232 

1215 

1198 

1182 

1166 

1151 

1135 

1121 

1106 

1093 

107S 

1066 

1053 

1O40 

1028 

1015 

lOOi 

993 

982 

970 

960 

950 

939 

929 

919 

909 

900 

891 

882 

873 
863 
856 
848 
839 

83l 
828 
816 
808 
800 
794 
786 
779 
775 
765 
759 
755 
746 
739 
733 
727 



Digitized by VjOOQIC 



166 LOG. VERSED SINES AND EXTERNAL SECANTS 
2^* 3** 



' Log. Vers. 



Log. Exsec. 



Log. Vers. 



Log. Exsec, 



91862 
92480 
9309J 
93703 
94308 



94909 
9550G 
96099 
96688 
9727? 
97853' 
98435 
99004 
99573 
00139 
00701 
01259 
018li 
02366 



03458 
03999 
04537 
05071 
05^03 



0613S 
0665S 
07177 
07695 
_Qi21L 
08723 
0923S 
09739 
10242 
10743 



11245 
1173S 
1222? 
12716 
13203 



605 






6. 97895 

98472 

.99045 

6-99615 

7.00182 

7.00745' 

.01304 

.01860 

.02412 

02966 



7. 03505 
04047 
.04585 
.05125 
•05652 



13687 




585 



663 
559 
555 
552 
54S 

545 
541 
53S 
535 
53l 
529 
525 
522 
519 
516 

513 
5n§ 
pn? 
50S 
501 
405 

4«>3 
490 
487 
48i 



7-13687 
14168 
14645 
15122 
15595 



16066 
16534 
17000 
17463 
17923. 
18382 
18837 
19291 
19742 
20191 
2068J 
21081 
21523 
21965 
22405 




29116 
2951S 
29919 
30319 
80716 



81112 
31505 
31897 
32288 
32676 



33063 
3344g 
338?T 
34213 
3A5P3 
3497I' 
35345 
85723 
36097 
36465 



36839 
3720? 
37575 
37945 
38301 



38a 
rTVc 



61 



47S 

*7i 

473 
475 
468 
466 
463 
465 

45S 
455 
453 
451 
44S 
445 
444 
442 
440 
437 
485 
488 
481 
429 
426 

423 
426 

111 

4li 

415 
409 
,406 
405 
403 
401 
895 
397 

395 
393 
892 
396 
388 
386 
885 
885 
382 
380 

878 
377 
875 
373 
87l 
37?^ 
865 
367 
366 
364 
36S 



13745 
14228 
14707 
15183 
15657 
16129 
16598 
17064 
17528 
1798S 
1844S 
18905 
19359 
19811 
20260 



20707 
21153 
21595 
22035 
22475 



22905 
23345 
28775 
2420i 
24632 



25057 
25480 
25902 
26321 
26735 



27155 
27567 
27975 
28387 
28795 
29205 
29601 
30005 
3G405 



31201 
81595 
31985 
32875 
32768_ 
33155 
33542 
33926 
34309 
34685 
35069 
35446 
35823 
36196 
36568 



36940 
87810 
37678 
8804i 
38405 
38773 



Log. Vers. 



Log. Ex 



Log. Vers. 



Log. Exsec, 



Digitized by VjOOQIC 



LOG. VERSED SINES AND EXTERNAL SECANTS 167 
4* 5* _^^_^__ 



' Lg.Vers. O Ug.txs, » Lg.Vers. J> Ug.Exs.l » ' 




GooqIc 



168 



LOG. VERSED SINES AND EXTERNAL SECANTS 



Lg.Vm.l J> 



7. 73863 n.T 

.74104^1^ 

7. 75060^^1 

'7S534i;52 
-757701:;^^ 

•76006r^? 

■^7e24'0??| 
76475^11 

"76708 ii% 
76&4I n|* 

-7717^ JJ^ 

■7740§H? 
77636*5^; 
77867^^^ 

■ 73326 1 ■'^" 
7-78554^^1 

■ 78783 ;5^ 

'7§oio iii 

'79237!^n| 
79463 ^^Z 

7^79699'^^t 

■ 8CSG2 n.-^^ 
JO 58 6 '^'^J 

7 . 80 80S 11% 

■ 81031 ^;;f 
■81352'?; ;5; 

■^^^'^i220 

7-81914,^^2 
<82133';^S 
^82352 ^{g 
■12570 i\% 
82783 -^' 



Ug.Exs. -?> 



LgiVers. J> Ug.Exs 



o 



p.p. 



o 

1 

2 

8 

JL 

5 

6 

7 

8 

A 

lU 

11 

12 

13 

14 

15 
16 
17 
18 

19 



2iY 

31 
22 
23 
24 
25 
26 
27 
28 
29. 
30 
31 
52 
38 
Si 
35 
SB 
37 
38 
39 



^0 
41 
42 
43 
44 
45 
48 
47 
48 
49 

50 
Bl 
62 
fiS 
54 
60 
68 
67 
68 
5i 
Alt 



74101 
74343 
74585 
74826 

75066 
7^75305 
75544 
75732 
76019 
782^ 

7M92 
76723 
76963 
77197 
77431 



82783 "'' 

7.830p5|2[7 

.83438!^^! 
83553:;[^ 
83868 _ 

DjifiDa 214 



84083^}* 
84297 511 

-84510 ;!j 

■84723 i\% 

7. 85147^^? 
85359 i\\ 
855705.^^ 

85990>^" 



o a p a u 
-86199 ^g^ 
■86403 *^g 
■ 86616 f}i\a 

86824 ill 



^703 
7 a723» 



Lg. Vers. ^ 



[1 '^"Z 

20a 



7-77664 
77897 
7812S 
78360 
78590 



7-78820 
'79050 

■79279 

,79507 

^797.35 

7-79962 

80183 

804li 

80639 

BO^M 

81081 

B1312 

81535 

81758 .^^.^ 

81930 ^"^"^ 
221 



242 

241 

241 

240 

239 

239 

238 

237 

237 

236 

2S5 

235 

23^ 

233 

233 

232 

23l 

23 

23 

230 

225 

229 

228 

228 

227 

226 

226 

225 

225 

224 

224 

223 

222 



7. 87233 
87444 
.87650 
87955 
^88060 
7B8264 
88468 
88672 



206 

205 
20& 
20i 
20 i 
204 
^203 



8B875^^i^n 
89D77 "^^^ 

-- - 202 



7-82201 
82422 
8284^ 

82862 
MP. 31 



7-833001 
83518 
83735 
839n? 
64169 

7 84385 

■84600 

-84815 

35030 

-8 5245 

7^85457 

85670 

85332 

'86094 

: 86305 

7' 8651 G 
86726 
8693^ 
87146 
B7354 



7 B75fi3 



221 
220 
219 
219 
219 
2L8 
21? 
217 
216 
216 
215 
215 
214 
213 
213 
213 
213 
21I 
21! 
211' 
210 
210 
209 
20§ 
208 



7 8927§ 
■89481 
.89682 
■89882 

_. 90082 

7,90282 

.80481 

■90680 

■90878 
-91076] 



7.91273 
91470 

91667 
91B03 

92053 
92253 
92448 
9264| 
82836 
9_3020 
7^9322S 
i3415 
93607 
93799 
93990 

7-9418] 
94371 
94561 

94751 
94940 



95129 
95317 
95505 
95683 
95880 
86066 
96253 
96439 
86624 
96809 
96994 
97171 
97362 
97546 
97729 
97912 
98G94 
H8276 
9B458 
986391 
BB|2oi^®^ 



201 
201 

20a 
200 
19& 
199 
19B 
19tt 
197 
197 
197 
196 
196 
195 
195 
195 
194 
194 
193 
193 
19S 
192 
191 
191 
I9Q 
190 
190 

lay 

189 
189 
18B 
187 
188 
187 

18G 
IBS 

leg 

185 
135 
1B4 
1B4 
184 
IB'V 
183' 
183' 
182 
1B2 
182 
181 



7.87563 
►87771 
87978 
88185 
8839 1 
88597 
88803 
89008 

8921 a 

89416 
89620 
69823 
90025 
80228 
9042S 
9063C 
90831 
91032 
91231 
91431 



91630 
9i82B 
92027 
92224 
92421. 

7-9261B \ll 
^92815 ^^^ 
.930irs 
.93206 
■9340L 



7^93596 

93790 
939841 

94177 
94370 
7-94?35 
.94754 
►94946 
■95137 
^532g 



7.95519 
95709 
9S89S 



96^76 

7-96465 

96653 

96841 
97028 

97215 
97401 
97587 
97773 
97955 
9B143 



98327 
90512 
98695 
98879 
99062 
7.91244 



195 

1 

195 

195 

194 

I&4 

193 

193 

192 

192 

192 

191 

191 

190 

190 

leO 

is| 
1 

138 
188 

1 

187 

187 

!E6 

186 

I8i 

18 

18 

185 

133 

183 

IBS 

133 

182 





liiiu 


9 


» 


6 18-0 05 


0-9 


7 


21 


M 


1-fy 


8 


24 


,1-S 


1:1 


8 


27 





14 


10 


30 





1-B 


rs 


20 


60 





3-1 


3-0 


30 


90 


D 


4,7 
6.S 


4.5 


40 


120.0 


e^o 


5C 


150 


c 


7-9 


7.S 



8^ 8 



0-S 


W 


1-n 


14 


1-3 




1 ■ 4 


2-8 


2 ■ E 


4.? 


4- 


5 6 


5, 9 


7-1 


66 








1 
1 

2 
303 



0-i 
7; 
O.S 
1-0 

11 

2 

S 



0-0 
1-0 

\\ 



6 

0-B 



404. S4. 5 
50,5.fii5.4 



^ ii i 

e0.5O-5!0.i( 
70 60 60 5 ( 

roei 

?.o, 

BO 

5 1-5: 
2-S: 



8;0 

go 

100-8 

20'1-s 

30 2.? _ 
40:3 6| 3 

50I4-6I4 




Ug.Exs, i> 



Lg.Vers.| ^^ 



Ug.ExSh J> 



y Google 



LOG. 



VERSED SINES AND EXTERNAL SECANTS 



169 




y Google 



170 LOG. VERSED SINES AND EXTERNAL SECANTS 
10*' 11° 




p. p* 



10 



130 

71 15 

8 17 
g 19 

21 

43 

65 



20 
30 
10 

solioa 



120 

i-.o 

14 

la 

18 
20 
40 

Oi ttO 
^ BO 

!lioo 



1 3, 

lo 4;0'S 

S'o^I 
60-5 
SlO-l 

31.1 

ofi I 





,1 


3 


B 


Si 


a a 


7 





1 


6 


0.4 





g 


□ -4 





4 


10 


0-5 





4 


20 


uo 





9 


30 


L-5 


1 


5 


40 


2-0 


1 


3 


&0 


2.5 


3 


1 




7 
8 

10 

20 

30 

40^0 

50'0 



1 O. 

10 



p, p 



y Google 



LOG. VERSED SINES AND EXTERNAL SECANTS 
12* IS** 



171 




GooqIc 



17^ LOG. VERSED SINES AND EXTERNAL SECANTS 

14' 15' 




P.R 



103 101? 




100 

6 10-0 

8,13-3 
S,15.0 

1016-6 
20,33. S 
30,S0 C 
40 66-E 
50)83*" 




sea 





07 


06 


95 


e 


9-7 


9,6 9.S 


7 


U-JI 


n.'I IM 


8 


12. ( 
14. 


13-B 12. S 


» 


14 4 


14| 1 


10 


la. 

32. 


IfiO 




20 


32*0 


31' 


30 


48,5 


48, 


47-5 


40 


64. 6 


B4.0 




50 


80-fi 


80.0 





04 


93 


92 


fi 


SJ.4 


9-3 


9-3 


7 


10-9 


10.6 


IQ.f 


fi 


13,5 


13.4 
13-§ 


12. S 


9 


14-1 


13,8 


10 


15-$ 


15,515-5 


20 


31-5 


31-030. 6 


SO 


47-0 


46-546,0 


40 


62,6 


ei?,o'.6iJ 


&0 


78-3 


77 5 


76-6 





ot 


no 


3. 


8 


a-i 


&-G 


0-5 


7 


lO-e 

12-1 


10-5 


0-8 


8 


12-0 


0.8 


9 


13-5 


13-6, 


D.L 


10 


15- 


15-0 


0.1 


20 


30-v 


30-0 


o-I 


30 


45. S 


45.0 


O-S 


40 


fiO- > 


AO.O 


0,5 


50 


75^8 


75. 


0-4 



P.P. 



vGooQle 



gi 



LOG. VERSED SINES AND EXTERNAL SECANTS 
16* 17** 



17.^ 




P- P» 



1*3 B^4 HI 

9.31 9.21 ft.l 

I2.4!l2.2 12.1 
13.9 13 8;i3 6 



15-5 
31.0 
46.5 

77-5 

90 

9.0 



10 

12 

13 

15 

30 

45 

60^0 

75.0 



15 S 15 1 

30-030. 5 

46.0 45-5 

61 Sleo g 

76^175 B 
89 HS 

esi e 3 

10.4|10.3 
U.B 11,? 

ia-3 13-2 
14-8 U. I 
29,6 29-3 
44 5 44 
59 558. 1 

74. 1 73-1 





87 


8a 


8o 


« 


^U 


B-e 


B 


7 


10-0 


9. 


fl 


n-fi 


n.4 


11. 


9 


13. 


12,9 


12. 


10 


14,5 


14-5 


14. 


20 


29.0 


28.E 28. 


30 


43 5 


43-042- 


40 


53.0^57-£|56- 


50 


72-6J71.6I70. 





84 


S3 


83 


5 


B.4 


8.3 


8 


7 


9 8 


9 7 


9 


« 


11.2 


11. C 


10. 





12.6 


12.4 


12 


10 


11-0 


13^ 


13. 


20 


23-0 27.6 


27. 


SO 


42.0,41,5 


41- 


40 


56,0!55-S 


54. 


60 


70-0 


59-1 


63, 



81 80 79 



11 8 
.4' 9 

8 10 
.1 12 
-513 

26 
-5:40 
• 0,53-^ 
-5lGe.6 



o 

0. 

oa 

u 

3 



p. p, 



y Google 



174 LOG. VERSED SINES AND EXTERNAL SECANTS 

18° 19* 




y Google 



LOG. VERSED SINES AND EXTERNAL SECANTS 
20* 21* 



175 



Lg^Vers. X> Log.ExS, 1> Ig-Vers. -»■ log.Exs. B 



P. P, 



S8 

27 

it 
80 
Bl 
12 
1« 



4U 
41 
42 
49 

45 
46 
47 
4fl 



00^ 

n 

E2 

fit 
ft? 

GO 



79B0§ 
TeA7i 
78041 

aooia 
aoosi 

fl015§ 
a022l 

ao29& 

e043B 



SO&Di 
80&77 
a0647 
00716 
ft07Be 



B1M& 

sieu 
ai«82 

81751 



aifiss 
Big5^ 

82093 

H2229 
Lg.Vers, 



li.8073B 

soaU 

80891 
80967 
H10i3 

aiii& 

81195 

B1271 
81346 

81498 
81573 
81649 
81725 
fllJOD 
81878 
81951 
8202B 
S2102 
62177 



62252 
82321 
82402 
«t247? 
3:^552 
82627 
a270g 
82776 
82851 
.82926 
8300S 
83075 
83149 
83224 
83298 
83373 
63447 
83521 
83585 
3670 
83744 
8^818 
83892 
B396§ 
84Q39 
6411^1 
64187 
9426; 
B433' 
84406 
64481 
845^5 
846?g 
B4'T02 
84775 
84841 
B4922 
8499^ 
85088 
85141 



8.84250 
84316 
B43B3 

84441 
M^5 



a. 84582 
.B464g 

-8471' 
-847Si 

_P'1B46 



8.B4912 
8497^ 
B504^ 

^8517i 

I - 85242 

-85308 

B6373 

8543e 

■85505 

i.e55''0 

85626 

^S'^Oi 

■ Ed'^m 

.8R832 



6? 



87024 
.a7C9| 
■81167 
-B7239 
^73jB 

8738? 
.87453 

.87525 
-8759B 
..87f56B 



66 



85214 



3 8B095 
88166 
88237 

^88308 

_pm^ 

8 B844l^ 
88520 
88591 
88661 
88132 

8 8S"03 

88^44 
880U 

J e 0,8 5 
8.39155 
89226 
89295 
89366 
89438 

8-8B508 



Log. El 



XSi 





T6 


75 


6 


7^ef 7.§ 


7 


B.af 8.7 


a 


10^1 


10.0 


9 


U 4 


11.2 


10 


I2^t 


12.5 


20 


25^3 


:5 c 


30 


£8^C 


S7.5 


40,50 1 


£0 C 
62^5 



74 

7 

8 

9 

U 
12 
21 
37 
4& 
81 



71 

7-1 



73 73 

7.Jj] 7.2; 

B.E 8 4 a. 

9-5i 9 6; 8 
10.5 10 a 10 
12-1 12-0 11- 
24 324. 023- 
fi6 5i3e 35 
48 6]48 047. 

eoBlao.osft^ 




6-7 
7 
8 
10 

11 
22 
33 
44 
55 



66 

8 6 



65 

6 6 

7 

8 



10 

21 

32 

043 

0[&4 









1 
1 
1 

0.5 
0.3 

0*4 



TTT 



y Google 



176 



LOG. VERSED SINES AND EXTERNAL SECANTS 
22* 23** 



* i Lg.Vera, J> [log.Exs. Z» [ig.Vflr&.j 2» Log.Ejci> 2» 



O 

1 
2 
3 



6 
7 

a 

J. 
10 

XI 
12 
13 
U 
15 
IB 
17 
IS 
19 



31 
22 
23 
24 
25 
2% 
27 
28 
?i 
90 
31 
32 
33 

35 
36 
37 
3S 
39 



40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
/lO 
51 
62 
53 
54 

55 
5« 
57 

se 

59. 
61) 



8^86223 

.86237 
^86352 
8641? 
86482 
86547 
86812 
86676 
86741 

B6eo6 

86370 
E6934 
86999 
B7065 
87127 
87132 
87256 
B7320 
87384 
87445 



87512 
87576 
87640 
87701 
87768 
87832 
87895 
8796^ 
83023 
880 86 

88150 155 

88213 ?| 

38277 - 

88340 

88404 

88467 

88530 

83593 

88656 

88720 



B8783 
88846 
88909 
88971 
89034 

8.C909? 
89160 
8922 
8928 
89348 

a 89411 
89473 
89536 
89595 
89665 



89723 
8978! 
8984? 
89910 

89972 
8-00034 



CiT^ 



ers. 



T> 



■91244 
.91313 
^91332 
.91451 
1_915^ 
^91536 
^91657 
■91726 
*G1794 
.91863 



-91932; 
^92000 
■92068 
■92137 



8 9295§ 

93022 

93095 

93158 

93226 

a, 93291 

-93361 

.93429 

-93496 

_93564 

8 93631 

Ug/Ejts, 



90034 
90096 
9015a 
90220 
10282 
90344 
9O406 
90467 
90529 
90591 
9065^ 
90714 
90776 
90837 
90899 

9096Q 

91021, 
91083 
9U44. 
91205 



91267 
91328 
91389 
91450 

91511 
91572 
91633 
91694 
91755 
9_1S15 
91876 
91937 
91997 
9205B 
92119 
9217^ 
92249 
92300 
92361 



92485 

92542 
92602 
9286^ 

92722 

92782 

92842 
92902 
9296| 
93022 
930B2 
93142 
9320: 
9326; 
9332, 
93381 
9344^ 
93505 
93560 
9Sfil|j 
93679 



Lg^Vors. 



8-93631 
.93691 
.93766 
.93833 
■93901 



8.93968 

• 9403' 
■94102 
.94X70 
■ 94^^ 

3-94304 
^94371 
■94438 
' 94505 
- 94572 

8- 9463b 
, 94705 
■94772 
.94839 

.__S>^1905 



■94972 
■95039 
■95105 
-95172 
_. 9523^3 

.95305 
■95371 
■95437 
-95504 
-9557fj 
■9563^ 
■95703 
■95769 
^95835 
^95901 

8 95967 

96033 
96099 

96165 
96231 



8-96297 

■96362 
■9642S 

■ 93494 
-965pp 
.96e2§ 
.915691 
■96752 
-96823 
-968_8S 

8-9695 

■ 97011 
■97084 

97X4r 

■ 9721- 

■97280 
.97345 
-974ir 
.9747" 

^97541 
8 97606 



p.p. 





70 09 


8 


7-0[ »'& 


7 


a J! 8^0 


8 


9^3; 9,2 


9 


10^5 lO^b 


10 


11. b n I 


20 


23.3,23^C 


30 


35-0 34-5 


40 


46.6 46-C 


50 


e8^3fl7^B 



6S 

«.8 

1:1 

22^B 

;)4-o 

45- 1 






61 


m 


m 




6.1 


60 


5-9 




7- 


7-0 


6.| 




8^ 


8-C 


7- 




9- 


^■0 


B- 


10 


10- 


10-0 


&■ 


20 


20 ; 


20. C 


1*. 


30 


30. S 


30.0 


28. 


40 


40-6 


40-0 
50-0 


38^ 


50 


5o^a 


49 -I 





o^' 

0^ 

0. 

0. 

0. 

0. 

0-5 

0-3 

0^4 



li 



TTT 



y Google 



LOG. VERSED SINES AND EXTERNAL SECANTS 177 

34" 



I ' Lff.Ver^. 



? 
2 
3 

! -* 

1 1 
1 1 

I ^ 
! lola 

i 11 

12 
f 13 

14 
' 15 

[ 16 

■ 17 

i IB 

1& 



93670 
9373B 

03857 
93916 
"S397B 
94034 
94094 
94153 
94212 



I SO 

|. 21 
22 

f 23 
34 

[ 25' 

[ 27 

[ 2B 

[ 20 

1 30 

f 31 

i 32 

; 33 
34 

f 35 

S 36 

I 37 

Y 38 

i li 

: 40 

^ 41 

t 42 

[ 43 

^ 45 
. 46 
47 
4B 
49 
50 
51 
52 
; 63 
54 
55' 
Bfl 
57 
5S 
59 



goa 



94271 
94330 
94389 
B4448 
94506 
94S05 
94524 
945^3 
94742 
04 800 



94859 
94917 
94976 
95034 
9509 3 
95151 
95210 
95268 
95326 
B53aj 



95443 
65501 
95559 
95617 
95675 



95735 
95791 
95^49 
95907 

95965 



96023 
96083 
96138 
96196 
96255 



96311 
9e36g 
96426 
96485 
96541 
96593 
96655 
96713 
9677S 
96327 



96885 
96942 
9699^ 
9705B 
97115 
97175 



CirVers. 




LogrExs. 



y Google 



178 



LOG. VERSED SINES AND EXTERNAL SECANTS 
26** 27* 






' Lg.Vers, 



O 

1 
2 
3 
J_ 

8 
7 
t 

10 
IX 
12 
13 
ii 
15 
16 
17 
IS 

Ii 

21 

£2 
£3 
Si 
2;^ 

£C 
£7 



30 

81 

ea 
ii 

ee 

S7 
88 

a 



40 

41 
42 
43 
44 
45 
46 
47 
4« 
46 
50 
61 
S2 
G3 
B4 
65 
Ge 
G7 
GB 
5^_ 
CO! 



00520 
O057S 

ooesD 

00685 
00739 

00 7 M' 
0084fi 

00903 
00957 

OKJlT 

01066 
01120 
01175 
01229 
012B3 
01337 
01391 
01445 
0U9I 
01554 



0160S 
01302 
01711 
01769 
01833 

01B77 
01931 
01985 

0203a 
020^2 



02H6 
02191 
02253 
02307 
02360 



02414 
03467 

02521 
02574 

0262'/ 



026aj 
02734 
027B7 
02840 
02894 
0294^7 
03000 
03053 
03106 
tt3159 
03212 
03265 
03318 
0337! 
03423 
03475 
03529 
03582 
09634 
0368? 



Lg.Vefs. 



Log. Exs. 



05154 

05215 

05276 

05337 

053a8 

05456 

05519 

05580 

05640 

05701 

05762 

05B22I 

0588! 

0594: 

O6OO4: 

0606|| 

06124 

06165 

06245 

06305 



06366 
0642 g 
06486 

06546 
06606 
9 06667 
06727 
06787 
06847 
06907 

0/06967 

07027 
07037 
07146 
P_720^ 
072SG 
07326 
07386 
07445 
7505 
07565 
07324 
07B84 
07743 
OJ8O! 

07963 

07922 
07981 
08041 
08100 
9 08160 
0B21? 
09278 
0833B 
0839 
08456 
OB515. 
09574 
0B634 
09693 

ijim2 
Log* £xs, 



/J Lg.Vefs. l> Log*E?(s. 1* 



03740 
0379i 
03B45 
03898 
03950 
04002 
04055 
04107 
04160 
04212 
04264 
04317 
04369 
04421 
0447S 
04525 
04577 
04630 
04682 
04734 

047 B6 

04337 

048B& 

04&41 

04993 

O5045 

05097 

05l4i 

05200 

05252 

0530J, 

0535 

05407 

05458 

0551_0. 

05561 

05613 

05664 

05715 

057671 



05818 
05869 
05921 
05972 

P6O2J 
06074 
06125 
0617<1 
06227 
06279 
06330 
D63B^ 
06431 
0e4B2 
0(^53^ 
06581 
06635 
06681 
0673 G 
0678? 
n6S.3B 



Lg. Vers. 



0B752 
08911 
08B70 
08929 
0898B 
09047 
O9106 
09164 
09223 
09282 
09341 
09400 
0945b 
09517 
09576 
09634 
09693 
09752 
09810 
09969 



0992? 
099 8§ 
10044 

10103 
10161 
10219 
10278 
1033§ 
10394 
1045_2 
10511 
10569 
10627 
1068S 
10743 
IO8OI 

loasl 

1091? 
10975 
1103^ 



11091 

11149 
11207 
11265 
11323 
113BD 
1U35 
11496 
11551 
1161- 
11665 
11727 
1178? 
11B4 
1199 
1195? 
12015 
12072 
12129 
1218i 
9 12244 



Log. Ex 



P. P, 



ei 

1 
1 



10 

20 

30 

40,6 
50 g 



60 

6.0 

7-0 

8.0 

9 
10.0 
20 O'lS 
30.039 
40.0 30 
50. 0148 



A9 

5 9 

8- 

9. 



5H 

55 



B 7 
9 B 
10 9 
20 19 
80;29 
403B 
50]4& 



55 

7| 6. 
B 7. 
9, 8' 



1B\ 

27, i 
36' I 
45- i 





33 


6 


5, a 


7 


6 2 


8 


7.1 


& 


7.6 


10 


Si 


20 


17. g 


30 


26 ■ 5 


40 


35 3 


GO 


44.1 



57 

n 
n 

9-S 

la.O 

28'$ 
38^5 
47.9 



S-S 

n 

18^0 
27.0 
38*9 
4II.S 



,1;! 



51 

?;l 

8 5 



17 



25. B 
34 
50k2.5 



U 



oa 
0." 



0.9 

0-4 



I 



TTT 



y Google 



LOG. 



VERSED SINES AND EXTERNAL SECANTS 



179 




y Google 



180 



LOG. VERSED SINES AND EXTERNAL SECANTS 



LgpVert. i> Ug,E3cs, n Lg.Vers. i> Log.Exs 



P,P. 



50 a. 15026 



/> 



1B949 

19003 
1905S 
19112 

19m 

19221 
19275 
19329 
19334 

19438 
19492 
1954U 
19601 
19655 
19709 

1&76S 
l&ai7 
19871 
19925 
19979 



20033 
200S7 
20141 
3019S 

20249 



20 SO 3 
20357 
2041 
20465 

30518 
20572 
20626 
20680 
20733 
2078? 
20841 
20894 
2094g 
21002 
2105S 



15710 
1575S 

.15801 
15846 
15891 



16163 
,16201 

^6253 
.1029B 
,lfl345i 



^6331 
16434 
.16479 

.16525 
16568 

-16613 
16658 
■16703 
-16748 
JJW 
16838 
^6883 
,lfi9^7 
■16972 
■17017 

a7oeI 

-17106 
17151 
1719S 

■17240 



.17284 
.173: 
. 1737L^ 

17418 
■17465 
-17507 

17551 
■17596 
,17640 
■17684 
^17729' 
-17773 
■17817 
.1786! 
J 7906: 
■17950 
.17994 

1803^ 

■ laoti^ 

18l2fi 



9.22176 
.22229 
■22283 
-22335 
.22JJ8 

9-22441 
■22494 
,3264? 
■22600 
■2265S 



9 24285 
24138 
24390 
24442 
2449J 

9.24547 

24599 
34651 
24704 
24756 

9 24808 
2486n 
24915 
24964 
25011 

9^2506| 
25125 
25172 
2522^ 
25276 

9_2532fi 

loja;.Ex5H 



O 

1 
2 
3 

-A 

5 
6 
7 
8 
_9 

10 
11 

12 
13 
ii 
15 
16 
17 
18 
19^ 
30 
21 
32 
23 
1^ 
25 
26 
27 
28 
29 

31 
32 
33 
_34_ 
35 
36 
37 
33 
39, 

41 
42 
43 
_44 
45 
46 
47 
48 
49 
JIO 
51 
52 
53 
54 
55 
56 
57 
58 
59 



n 

8.3 
9 1 
18. T 
27 2 
36 S 
45 4 





B3 


5f 


6 


53 


if 


7 


6-3 
7^0 


8 


7 


9 


7-9 


7-9 


10 


e-A 


8.7 


20 


17-^ 


17.5 


30 


26^5 


26. 2 


40 
50 


35-3 
44 I 


35 
43. 7 




54 S3, 

6^3 e.S 

7,2 7.1 
^■l 80 
9 Ol 8, 1 

18.0117.1 
27-0^26,7 
3«. 035-6 
45.0[44,fl 



ti 

,•' 

17. 3 
2fl 
34 
43 



46^ 

4 g 
54 
6 2 

?:? 



47 
4 7 

6 5 

6-2 
7.5 

7 8 

15 
23 



5 23 



31 §,3I.Q 
39 list. 7 



45 

4 5 

5 

6 

6 

7 
15 
22 
30 
37 



45 

U 
?;? 

15-0 
23,5 
30-0 
37.5 



fi 


t\ 


7 
B 


U 


9 


6 7 


10 


7 4 


20 


14^j 
23-2 


30 


40 


29. ft 


50 


37,1 



44 

it 



P, P. 



y Google 



liOG. VERSED SINES AND EXTERNAL SECANTS 181 

32* 33* 



' Lg.Vers. 



10 s 

11 

12 
13 
14 

IS 

16 

17 

18 

19^ 

130 i 

21 

22 

23 

24 

27 
2A 
29 
30 1 

Si 
32 
33 
84 
Z5 

97 
88 
89 



41 

4S 

43 

44 

45 

4fl 

47 

48 

il 

50 ! 

51 

52 

S3 

§4 

BS\ 

56 

57 

58 

5fi 

§Q9 



i»2el 

19308 
19351 
19395 
19438 
19481 
19525 
19568 
19611 
19654 
1969B 
19741 
19781 
19827 
19E70 



19914 
19957 
20000 
20043 
20086 



20129 
20172 
20215 
30258 
30301; 

30343 
30386 
20429 
20475 
20515 
20558 
2080?^ 
20843 
20688 
20X2R 
20771 



J> LogiEjcs. 



0532^ 

25380 
25432 
25484 
25535 
255B6 
35640 
25692 
25743 
25795 
25847 
25899 
25950 
28005 
26054, 
26105 
26157 
28209 
3828e 
26312 



26384 
26415 
26467 
26518 
26570 
26621 
2B673 
28724 
26776 
2^8^7 
2687e 
26930 
26981 
27032 
37084 
27135 
27186 
27238 
37289 
2734fi 



37391 
27443 
27494 

27545 
27^^ 
3764? 
2769B 
277^0 
27805 
17853 
27903 
27954 
28005 
28056 
26107 
28157 
2820^ 
28259 
283lS 
28361 

?R41_3 



i> Lg>Vers> » Ug>Exs, n 



9 20771 
20814 
20856 
20399 
20942 
20984 
21027; 
21069 
21112 
31154 
2119^ 
21239 
21281 
21324 
21366 
31408 
21451 
21493 
31535 
_21577 



21620 
21663 
31704 
21746 
21788 
2183b' 
31873 
21G14 
3195(^ 
3199f 

22040' 
22081 

32124 
22166 

2330S 
22250 
23392 
32334 
22376 
3341? 



23459 
23501 
22543 
2358i 
23636 
33668 
2370§ 
23751 
22792 
23B34 
23876 
23917 
22959 
3 30 OR 
23042 
23083 
23324 
23166 
2320? 
2334S; 

23290 



3942B 
2947g 
29527 
29577 
2962? 



29678 
29721 
29779 
23829 
29875 

39930 
29980 
30030 
30081 

30131 

30181 

30231 

30282 

3033 

3038 



30431 

30483 

30533 

30583 

_3063a 

30683 

30733 

3078^ 

3083: 

30B8 

30=35 

30985 

31033 

31083 

3113^ 



31183 
31233 
31283 
31333 
31383 
31432 



P, P- 





B2 


th 


8 


5.2 


7 


6. ft 


60 


8 


6-9 


6 8 


e 


78 
SB 


7 7 


10 


8 6 


20 


17^3 


17.] 


30 


26 


25^:? 

34. 3 


40 


34. e 


50 


43-3 


42 9 



51 

n 
?i 

8-5 
17 [} 

2b■l^ 
34 E) 
42.0 






44 


^\ 


6 


4^ 
5- 


4-3 


7 


5^1 


8 


5^ 


5 B 


8 


6- 


fi-fx 


10 


7.,' 


7.2 


20 


14- 


14 5 


30 


32. 


21.7 


40 


29.3 


29 


50 


36.6 


36.5 



43 

4.3 

5 
5-7 

6 I 

14 3 

21, 
28- 
35. 



4.3| 
4.9 

8 3 
7 



7.1[ 
14 114-013 
21 5 21,0,20 

28.328-0,27 
35 4 35 0!84 



41 

4-1 
4-1 
5-I 
6.1 
8 8 
— ,13.B 
30^20.5 
40 27-3 
50 34-1 



TTF 



177^ 



LogrExs, 



Lg. Vers, 



Log.Exs 



y Google 



182 LOG. VERSED SINES AND EXTERNAL SECANTS 
34** as' 



' Lg>Vm. 



XI Ug.Exs, 



1> Lg.Vers. 



LogiExs* /> 



P.P. 



10 

11 

32 
13 
li 
15 
16 
17 
IS 
19 



2m 

31 

22 

23 

24 

2S 
26 
37 
2ft 
3_i 

SI 
82 
83 
34 
S5 

se 

37 
38 



23290 
23331 
23372 
23414 
23455 



2349r> 
23537 
23573 
23620 
23661 
23702 
23743 
23784 
23&2S 
23B6G 

23&07 
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11 
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13 
14 
15 
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25 
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29 
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31 
32 
33 
34 
35 
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41 
42 
43 
44 
45 
46 
47 
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51 

52 

53 

54 

55 

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70137 
70159 
70181 
70202 



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70572 
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7061! 
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70723 
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10. 002 19 
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10 00438 
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10. 01091 
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71218 
71239 
71261 
71282 



71304 
71325 
71345 
71368 
71385 



71411 
71432 
71455 
71475 
71495 



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10.02861 
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10.03082 
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10.03745 
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10. 03975 
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10.04193 
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21 
22 
23 
24 
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28 
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40 

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72597 
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72745 
72764 



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198 LOG. VERSED SINES AND EXTERNAL SECANTS 



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77402 
77432 
77441 
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777&0 



77809 
77128 
77847 
7786Z 
77866 

77905 
77&25 
77944 
77963 
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78002 
78021 
78045 
78059 
78078 



78098 
76117 
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7515| 
73174 
78194 
78213 
78232 
76251 
7827g 
78289 
78309 
78328 
78347 
78365 



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76404 
78423 
78442 
78462 
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16489 
16537 
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16633 
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16824 



16872 
1692& 
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17010 
17064 

17112 
17160 
17209 
17257 
17.301 



173 

1740i 
17449 
17498 

17546 



173»4 
17642 
1769C 
17739 
177,^7 
178'3S 
17684 
17932 
17980 
18029 
1807? 
1812g 
18174 
18222 
18271 



1831& 
18368 
1841fi 
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19611 
1865S 
1870E 
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18654 
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19000 



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9-78481 

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9-78766 

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9 . 78956 
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9.79051 
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9 ■79145 
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9.79239 
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9.79333 
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9.79427 
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10 19782 
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10-20767 
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15 


6 


1.5 


7 


1.8 


6 


2^0 


9 


2,3 


10 


26 


20 


5 1 


30 


7.7 


40 


10. i 


60 


12.9 



rt 



u 

o.a 

?:l 

2.9 
2. 1 
8.3 

2 5 

5.0 

7 5 

10^0 

12. a 

1:1 

2.2 

^ 

9 6 
12.1 



P.P. 

y Google 



206 



LOG. VERSED SINES AND EXTERNAL SECANTS 
82' 83* 



Lg^Vers, 



a 

1 

2 
3 
J, 
5 
6 
7 

n 

12 
IS 
14 
15 
16 
17 
18 



30 
21 
22 
23 
24 
25 
26 
2? 
28 
^9 

-^0 

31 
12 
33 
34 
35 
36 
37 
38 
39 



40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
AO 
51 
52 
53 
54 
6B 
SB 
67 
Efi 
P 



9 93i0l 
93508 
93520 
§3535 
93549 



93564 
93578 
93593 
93607 
93622 



93635 
93651 
93661 
93830 
93604 
93709 
9372S 
93738 
93755 
93787 



93731 
93796 
93Slfl 
9382^ 
93639 
93953 
93868 
93682 
93897 
93911 
93925 
93940 
93951 
93969 
93983 
93997 
94012 
9402fi 
94041 
94055 



94093 
94112 
94127 
94141 
94155 
94170 
94181 
9419B 



94213 
94227 
94241 
94258 
94170 
94284 
94299 
94313 
94327 
94341 
94358 



Lg,Ve, 



11 
14 

14 

14 
14 

14 

ll 
li 

II 

14 
14 
ll 
1| 
14 
ll 
14 
14 

14 
14 
14 
14 
14 
14 

U 

14 
14 
14, 
14 
14 
14 
14 
14 

It 

14 
14 
14 

94069 }t 
94084 ll 



Log. Exs. 



10. 79130 
.79240 
79345 
79450 
■79555 

10-79660 
79766 
79371 
,79977 
800SS 
10.80189 
. 60296 
.90402 

.ao&os 

80616 

10 80723 
80331 
.80931 
-81045 
.81154 



10 81262 
81371 
.31479 
.81588 
81697 

10-81306 
,81916 
.82023 
.82135 
^82245 

10 32'356 

. 82486 
-82577 
82688 
.82799 

10.82910 
.83022 
-B3133 

.83245 
■83358 



1? Lg. Vers, -» Log. Exs 



10.83470 
.83583 
.8369S 
.83809 
_, 83922 
10. 84035 
.84149 
,84263 
.8437? 
84492 

10 84607 
-8472! 
3i837 
.84952 
35068 

10-8516] 
.85291 
'85416 
.85532 
.B564§ 

10 8576^ 



104 
105 
104 
105 
105 
105 
105 
106 
106 
106 
106 
106 
107 
107 
107 
10? 
107 
108 
108 
108 
lOl 
105 
109 
109 
109 
109 
109 
110 
110 

no 

no 

llQ 

111 
111 

nl 
111 
nl 

iig 

112 

115 

115 

112 

lis 

113 

115 

114 
114 

114 
114 
115 
114 

115 
115 
116 
115 

lie 
ni 

116 
117 
117 



9. 94356 

94370 

94334 

94398 

94413 

9 94427 

94441 

94456 

94470 

94484 

9. 94491 

.94512 

.94527 

.9454] 

^4555 

9 94569 

.94534 

.94598 

.9461S 

.9462G 



Log, ExSi i* 



9 9464G 

94655 

94689 

.94683 

94697 

9 94711 

94726 

94740 

94754 

94765 

9.9478i 

.94795 

.94810 

.94825 

^94339 

9 . 94853 

94867 

94881 

94895 

94909 



9.9492! 

94938 

94952 

94968 

94980 

9 - 94991 

.95008 

-9502S 

.9503^ 

9505fl 

9 95064 

95078 

, 95093 

-95107 

^5m 

9. 95135 
95149 
95163 
95177 
95191 
9^95205 
Lg.Vers. 



10 



85766 
85684 
36001 
86119 
86237 
86355 
86474 
.86592 
,86711 
.86931 
. 86956 
-87070 
.8719i) 
37310 
,87431 



10 



87552 
37673 
87794 
87916 
3 80 33 



10 



10 



10 



38160 

8828S 

88405 

8352& 

68651 

88775 

8889' 

8902 

89147 

_3927 l 

8939B 
89521 
69647 
89773 



90025 
90152 
.90279 
9040* 

■ 9n{>;;! 



10 



90661 

90739 

90917 

9104 

JJ117 

91304 

91435 

.91584 

.91694 

.91825 



10 



10 



10 



91956 
92087 
9221^ 
92350 
12485 
92e"l4 
92747 
9288(1 
93014 
S3 147 
93SaI 



Log, E] 



Ki. 



11? 

117 
117' 
118 
lis 
118 
118 

lis 

llg 
11§ 

120 
120 
120 
120 
121 
121 
121 
121 
122 
122 
122 
12^ 
12^ 

12; 

121 

125 
124 
124 
124 
125 
125 
125 
126 
126, 
12^ 
120 
il27 
127 
127 
128 
12? 
128 
129 
129 
129 
130 
129 
130 
135 
131 
131 
13T 

132 

132 

I3S 
133 
IS? 
131 
134, 



p.p. 





ISO 


e 


13 p 


7 


15. 1 


8 


17,3 


9 


19. 5 


10 


21, 6 


20 


43-5 


30 


65. 


40 


86-6 
108.3 


50 



12.0 
U.O 
18. 
18-0 
20.0 
40 -0 
60.0 
80.0 
100-0 



110 

e 11 

7^12 i 

8|U.e 

916.5 
1018 3 
20 36.B 
80155 
40[73^ 
60I9I.6 



100- 
3011. 

soil 

40;2 
6OI3, 



13'lln^a 



TF 



d by Google 



LOG. VERSED SINES AND EXTERNAL SECANTS 

84" 85** 



207 




J> Ug,E)(s, jD 



11 



11 



11 



11 



11 



11 



11 



11 



11 



02010 
02161 

02327 
024S1 
026461 
02S07I 
029681 
031291 
03291 
034&3 
03616 
03780 
03944 
04108 

04438 

04604 
0477] 
04938 
0S1Q6 



0S27.1 

05443, 
05612 
05783 
05953 



oai2S 

06295 
06467 
08640 
06813 



06987 
07161 

07336 
07511 
07fiflS 
07865 
08043 
08221 
08400 
08579 



08759 
08640 
09121 

09303 
09486 
09669 
09853 
1003B 
1022S 
10409 
10595 
10783 
10971 
11160 
11349 
11539 
1173fi 
1192:' 
12114 
12307 

ij_moi 
Uf,E?£S. 



158 
159 

159 
159 
IBO 
161 
161 
161 
162 
163 
16^ 
164 
164 
165 
165 
166 
167 
167 
167 
168 
169 
169 
16i 
170 

171 
171 
172 
173 
173 
174 
174 
175 
176 
176 
177 
177 
178 
179 
179 
180 
180 
181 
1B2 
182 

IB3 

104 
185 
1B5 
1 

1 

187 

188 

189 

189 

190 

191 

19' 

192 

193 

19^ 

"77 



P.P. 





nm 


6 


19-0 


7 


33 1 


8 


25.3 


a 


28 5 

31.6 


10 


20 


63-3 


30 


05-0 


40 


1266 


50 


158.3 



6 


17-0 


7 


19 e 


8 


22-6 


9 


25.5 


10 


28-3 


20 


56.6 


30 


85 


40 


113-J 
141.1 


50 





im 


e 


15.0 


7 


17.5 


« 


20. 


9 


22-5 


10 


25-0 


30 


50-0 


30 


75-0 


40 


100 


50 


135 



180 
21.0 
24 
27.0 
30.0 

ao 

90.0 
120 
150.0 

160 

16 
18 Fi 

21^1 

24 g 

26.6 

53 3 

80 

106 5 

133. 3 

140 

14.0 
16. 1 
IB. I 
21 

23-1 
46 6 
70 
93 I 

iia.6 





130 





8 


fi 


13 p 


930.8 


7 


15-1 


1.0 


OS 


8 


17.3 


^■i 


1-5 


B 


19 5 


1 S 


1 -3 


10 


21 6 


15 


1 -3 


20 


43 S 


30 


2 fl 


30 


65-C 


4 5 


4-p 


40 


86 i 


6 


5-3 


50 


108-3 


7.5 


6^ 






^'\ 


14 


6 


1.4 




7 


17 




8 


l.S 




9 


32 




10 


2 4 




20 


4.S 




30^ 7-2 


7-0 


40 9.S 


9 A 


5012-1 


n-fl 



p.p. 



y Google 



208 LOG. VERSED SINES AND EXTERNAL SECANTS 
86' 87* 



Lg. Vers> 




11.2578^ 

26045 
26297 
2655i 
26B14 
2707i 
2733| 
27591 
278G4 
2Sin 



11 



los. Exs. I D 



11 



11 



11 



11 



U 





1 

2 

3 

„ 4 



25§ 

|2S6 

257 

253 



28398 
28668 
2S938 
29211 
29485 
29760 
30037 
30316 
3059F 

mn 



282 
263 
265 
26? 

2mTo 



31162 
3144X 
31734 
32023 
32313 



32606 

32900 
33196 
33494 
13793 
34095 
34398 
34704 
3501T 
35321 



35632 
3!J946 
36261 
36571 

36899 



26] 

27C 
272 
274 

I 27^ 
277 
27r 
2Bi 
282 

I 2a 
28 
287 

'281 
29& 

■29l 
294 
29& 
29a 
29^ 

,30] 

30; 
5301 

;30^ 
,305 

3lT 
Sll 
3lB 
313 
320 



3722T 
37546 
37872 
3B201 
38532 
38866 
3920T 
39540 
398B0 
40224 
40561 
40918 
41269 
41625 
41979 
4233B 
42695 
43064 
;I 



32^ 
323 
321 
328 

331 



4343 

43802 

m 



Ml 



333 
33| 
338 
34r 
34; 

34^ 
34 
35 
35^ 

35^ 

35 

36^ 
1364 

367 

375 
,373 



m 

21 
22 

23 
24 
2S 
26 
27 
28 
29 

au 

31 
32 
33 
31 
35 
36 
37 

41 
42 
43 
44 



P. P. 



fi 


25 


7 


20.1 


a 


33. S 


e 


37-5 


10 


41.^ 


20 


S3-i% 


an 125.0 


40 Ifl6-S 


&0i2OB.| 



fl 


23 p 


7 


2G.S 


8 


30. e 


9 


34^ 
38-3 


10 


20 


78-6 


30 


115-0 


40 


153.3 


50 


191-6 



'4^0 

24.0 
28-0 
32-0 
3B O 
40-0 

ao.0 

120 -0 

lao.o 

200^0 



33-0 

73 I 
110-0 





210 


6 


21-0 


7 


24-B 


d 


28-0 


9 


a. 5 


10 


35-0 


20 


70-0 


30 


105-0 


40 


140-0 


50 


175-0 



I8S' 

2QO 
20.0 
23 I 

26-i 
30-0 
S3 I 

100. g 

133 J 





100 


4 


» 


A 
7 


l«-0 
22-1 
25 S 


0.40-1 

0-1 oi 


8 


0.^0-1 
0-SO.I 


B 


28-5 


10 


31 5 


oJo s 

iJ L.O 


20 


63J 


30 


95^0 


2.0 


1-5 


40 


126.6 


2-fi 


2.0 


SO 


158. ^ 


3.3 


2-$ 



610-2 

7 0^3 

8P-S 

90.3 

100.3 

20O.g 

30 10 

401 J 

SOllS 

14 

1-^ 

20 
4 ' 



30 7 ( 
^Oi 9-] 

SO'U.I 



1 
0.1 
0.1 
04 
0.1 
0-1 

oj 

0-5 

S:i 

1^3 
1-B 
1-8 

I:? 

4.5 

B.^ 

&.0 

11-5 



01 

II 

O.i 

la 

1-3 



lo.i 



Lg. Vefs- D 



log. Exs. I D 



PP. 



y Google 



LOG. VERGED SINES AND EXTERNAL SECANTS 
88* 89' 



209 




y Google 



210 



THE RAILROAD TAPER 



FUNCTIONS OF A ONE-DEGREE CURVE 

The Long Chords, Mid-Ordinates, Externals, and Tangent 
Distances of this table are for a curve of 5730 feet radius. To 
find the corresponding functions of any other curve divide the 
tabular values by the degree of curve. 

For metric curves having a 20-meter chord, this is a table of 
the functions of a two-tenth degree curve. Multiply the values in 
the table by 0.2 to obtain the functions, in meters, of a one-degree 
metric curve. 





CO 


1» 


/ 


L. C. 

0.00 


M. 
0.000 


E. 
0.000 


T. 
0.00 


L. C. 

100.00 


M. E. 
0.yi8 0.218 


T. 

50.00 








2 


8.33 


000 


0.000 


1.67 


103.33 


0.233 0.283 


51. 07 


2 


4 


6.67 


001 


0.001 


3.38 


106.66 


0.248 0.248 


53.33 


4 


6 


10.00 


0.002 


0.002 


5.00 


110.00 


0.264 0.264 


55.00 


6 


8 


13.33 


0.004 


0.004 


6.67 


113.83 


0.280 0.280 


56.67 


8 


10 


16.67 


0.006 


0.006 


8.33 


116.66 


0.297 0.297 


58.33 


10 


12 


20.00 


0.009 


0.009 


10.00 


120 00 


0.314 0.314 


60.00 


12 


14 


23.38 


0.012 


0.012 


11.67 


123.33 


0.332 0.332 


61.67 


14 


16 


26.67 


0.015 


0.015 


18.33 


126.66 


0.350 0.850 


63.33 


16 


18 


80.00 


0.019 


0.019 


15.00 


130.00 


0.368 0.368 


65.00 


18 


20 


33.33 


0.024 


0.024 


16.67 


138.33 


0.388 0.388 


66.67 


20 


22 


36.67 


0.029 


0.029 


18.33 


136.66 


0.407 0.407 


68.39 


22 


24 


40.00 


0.035 


0.035 


20.00 


140.00 


0.427 . 0.427 


70.00 


24 


26 


43.33 


0.041 


0.041 


21.67 


143.38 


0.448 0.448 


71.67 


26 


28 


46.67 


0.048 


0.048 


23.33 


146.66 


0.469 0.469 


73.33 


28 


80 


50.00 


0.054 


0.054 


25.00 


150.00 


0.491 0.491 


75.00 


30 


32 


53 38 


0C2 


0.062 


26.67 


153.33 


0.513 0.513 


76.67 


3S 


34 


56.67 


0070 


070 


28.33 


156.66 


0.536 0.636 


78.33 


34 


36 


60 00 


0.079 


0.079 


30.00 


160.00 


0.559 0.559 


80.00 


36 


38 


63.33 


0.088 


0.088 


81.67 


163.33 


0.582 0.582 


81.67 


88 


40 


66.67 


0.097 


0.097 


83 33 


166 66 


0.606 0.606 


83.33 


40 


42 


70.00 


0.107 


0.107 


35.00 


170.00 


0.630 0.630 


85.00 


42 


44 


73.33 


0.117 


0.117 


36.67 


173 33 


0.655 0.655 


86.67 


44 


46 


76.67 


128 


0.128 


38.33 


176.66 


0.681 0.681 


88.33 


46 


48 


80.00 


0.140 


0.140 


40.00 


180.00 


0.706 0.706 


90.00 


48 


50 


83.33 


0.151 


0.151 


41.67 


183 33 


738 0.733 


91.67 


60 


52 


86 67 


0.164 


0.164 


43. as 


186.66 


0.760 0.760 


93.3:^ 


62 


54 


90 00 


176 


0.176 


45.00 


190.00 


0.788 0788 


95.00 


54 


56 


93 33 


0.190 


0.190 


46.67 


193.33 


0.815 0.815 


96.67 


56 


58 


96.67 


0.204 


0.204 


48 33 


196.66 


0.844 0.844 


98.33 


68 


60 


100.00 


0.218 


0.218 


50.00 


199.98 


0.873 878 


100.00 


80 



y Google 





FUNCTIONS OF A ONE-DEGREE CURVE 


211 


» 


«• 




« 


• 




/ 


L.C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





199.98 


0.878 


0873 


100.00 


299.96 


1.964 


1.964 


150.07 





2 


203.31 


0.902 


0.902 


101.67 


803.29 


8.008 


8.009 


151.74 


8 


4 


206.64 


932 


0.932 


103.34 


306.62 


2.053 


8.054 


153.41 


4 


6 


209.97 


0.9G2 


0.962 


105.01 


309 95 


. 2.098 


8.099 


155 08 


6 


8 


213.31 


0.993 


0.993 


106.68 


313.29 


2.143 


8.144 


156.75 


8 


10 


216.64 


1.024 


1.024 


108.35 


316.62 


2.188 


8.189 


168.42 


10 


12 


219.97 


1.056 


1.056 


110.02 


319.95 


2.235 


2.236 


160.09 


18 


14 


223.30 


1.088 


1.088 


111.69 


823.28 


2.282 


2.283 


161.76 


14 


16 


226.64 


1.121 


1.121 


113.36 


826.62 


8.329 


2.330 


163.43 


16 


18 


229.97 


1.154 


1.154 


115.02 


829.95 


8.876 


2.377 


165.09 


18 


80 


233.30 


1.188 


1.188 


116.60 


833.28 


r.424 


2.425 


166.76 


80 


23 


236.63 


1.222 


1.222 


118.36 


836.61 


8.473 


2.474 


168.43 


82 


24 


239.97 


1.256 


1.256 


120.a3 


a39.95 


8.523 


2.528 


170.10 


84 


26 


243.30 


1.292 


1.292 


121.70 


343.28 


8.573 


2.573 


171.77 


26 


28 


246.63 


1.328 


1.328 


123.37 


846.61 


8.628 


2.628 


173 44 


28 


ao 


249.96 


1.364 


1.364 


125.08 


849.94 


8.678 


8.673 


175.10 


80 


32 


263.29 


1.399 


1.399 


126.70 


853.27 


8.724 


2.725 


176.72 


82 


34 


256.62 


1.437 


1.437 


128.37 


856.60 


8.776 


2 777 


178.89 


84 


36 


259.96 


1.475 


1.475 


130.04 


359.94 


8.828 


2.829 


180.06 


86 


88 


263.29 


1.513 


1.513 


181.71 


863.27 


8.880 


2.881 


181.73 


88 


40 


266.62 


1.552 


1.558 


133.88 


366.60 


8933 


2.934 


183.40 


40 


42 


269.96 


1.592 


1.592 


135.05 


369.94 


2.987 


2.988 


185:07 


42 


44 


273.29 


1.632 


1.632 


136.72 


373.27 


8.042 


3 043 


186.74 


44 


46 


276.62 


1.672 


1.672 


138.38 


876.60 


8.096 


3 097 


188.40 


46 


43 


279.96 


1.712 


1.712 


140.05 


879.94 


8.151 


8.158 


190.07 


48 


SO 


288.29 


1.758 


1.752 


141.72 


883.27 


8.206 


8.207 


101 .74 


60 


62 


286.62 


1.794 


1.794 


143.89 


386.60 


3.263 


3.264 


193.41 


62 


54 


289.96 


1.836 


1.836 


145.06 


389.94 


8.320 


3.321 


195.08 


51 


56 


293.29 


1.878 


1.878 


146.73 


393.2r 


3.877 


8 378 


196.75 


56 


68 


296.62 


1.921 


1.921 


148.40 


396.60 


8.4:34 


3.435 


198.42 


58 


60 


299.96 


1.964 


1.964 


150.07 


399.94 


8.491 


3.492 


2U0 09 


60 


# 




4 








6 


» 




t 


L. C. 


U. 


E. 


T. 


L. 


M. 


E. 


T. 





399.94 


3.491 


8.492 


200.09 


499.88 


6.454 


5.459 


250.17 





2 


403.27 


8.550 


3.551 


201.76 


503.21 


6.&27 


6.038 


261.84 


8 


4 


406.60 


3.609 


8.610 


203.43 


606.54 


6.601 


6.607 


253.51 


4 


6 


409.98 


3.668 


3.670 


205.10 


509.87 


6.675 


6681 


255.18 


6 


8 


413.26 


8.727 


8.730 


206.77 


513.20 


6.749 


5.755 


266.86 


8 


10 


416.60 


3.787 


8.790 


208.44 


516 53 


6.828 


5.829 


258.52 


10 


13 


419.92 


3.848 


3.851 


210.11 


619.86 


5.899 


5.905 


260.20 


18 


14 


423.86 


8.910 


8 913 


211.77 


523.10 


6.975 


6 981 


261.86 


14 


16 


426.59 


8.972 


8.975 


213.45 


626.52 


6.052 


6.058 


263.54 


16 


18 


429.92 


4.084 


4.037 


215.11 


529.85 


6.129 


6.135 


265.20 


18 


20 


483.25 


4.096 


4.099 


216.78 


583.18 


6.206 


6.212 


266.87 


20 


22 


436.58 


4.160 


4.163 


218.45 


536.51 


6.284 


6.290 


208 54 


22 


24 


439.91 


4.224 


4.227 


220.12 


539.84 


6.362 


6.369 


270.21 


24 


26 


448.24 


4.288 


4.291 


821.79 


543.17 


6.441 


6.448 


271.88 


26 


28 


446.58 


4.358 


4.356 


283.46 


646.50 


6.520 


6.527 


273.54 


28 


30 


449.91 


4.418 


4.421 


225.13 


649.83 


6.599 


6.606 


275.21 


30 


32 


453.24 


4.484 


4.487 


226 80 


653 17 


6.680 


6.687 


276.88 


32 


34 


456.57 


4.650 


4.554 


228.47 


656.50 


6.761 


6.768 


278.55 


34 


36 


459.90 


4.617 


4.621 


230.14 


559.83 


6.842 


6.849 


280.23 


36 


38 


463.28 


4.684 


4.688 


231 81 


663.16 


6.923 


6.931 


281.90 


38 


40 


466.56 


4.751 


4.755 


233 48 


666.49 


7.005 


7013 


283.57 


40 


42 


469.89 


4.820 


4.8*24 


285.15 


669.82 


7.088 


7.096 


285.24 


48 


44 


478.23 


4.889 


4.69^ 


236.88 


673.15 


7 171 


7.180 


286 91 


44 


46 


476 56 


4.958 


4 962 


238.48 


576.48 


7255 


7.264 


288 59 


46 


48 


479.89 


5.05J7 


5.031 


240.15 


579.81 


7389 


7848 


290.26 


48 


50 


483 22 


5.096 


5 100 


241 82 


583.14 


7 428 


7.432 


291.93 


50 


52 


486.55 


5.167 


5.171 


243.49 


586.47 


7.508 


7.517 


293.60 


58 


54 


489 88 


5.238 


5.243 


245.16 


589.80 


7.593 


7.603 


295.27 


54 


56 


493.21 


5.310 


5.315 


246 88 


593.13 


7.678 


7 689 


896 95 


56 


56 


496 54 


6.388 


6.387 


248 60 


596.46 


7.764 


7.775 


298.62 


58 


60 


499.88 


5454 


6.459 


250.17 


599.80 


7.850 


7 861 


800.80 


60 



Digitized by VjOOQIC 



212 


FUNCTIONS OF A ONE-DEGREE CURVE 




/ 


e- 1 




7 


« 




/ 


L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





509.80 


7.850 


7.861 


800.30 


699.60, 


10.69 


10.71 


850.44 





2 


608.18 


7.940 


7.951 


301.97 


702.93 


10.79 


10.81 


ai2.11 


2 


4 


606 46 


8.080 


8.041 


806.64 


706.26 


10.90 


10.92 


353.79 


4 


6 


609.78 


8.120 


8.131 


305.31 


709.58 


11.00 


11.02 


855.46 


6 


8 


613.11 


8.210 


8.221 


806.98 


712.91 


11.11 


11.13 


357.13 


8 


10 


616.44 


8.300 


8.311 


306.65 


716.24 


11.21 


11.23 


858.81 


10 


12 


619.76 


8.390 


8.401 


810.32 


719.56 


11.31 


11.33 


360.48 


12 


14 


623.09 


8.480 


8.491 


311.99 


722.89 


11.42 


11.44 


362.15 


14 


16 


626.42 


8.570 


8.581 


313.66 


726.21 


11.52 


11.54 


868 83 


16 


13 


629.74 


8.660 


8.671 


315.33 


729.53 


11.68 


11.65 


365.60 


18 


20 


638.07 


8.750 


8.761 


817.00 


732.86 


11.78 


11.75 


867.17 


20 


22 


686.40 


8.844 


8.856 


818.67 


736.19 


11.84 


11.86 


868.86 


22 


24 


689.72 


8.939 


8.951 


820.34 


739.51 


11.96 


11.97 


370.62 


24 


26 


648.05 


9.0&3 


9.046 


32201 


742.84 


12.06 


12 08 


372 19 


26 


28 


646.88 


9.128 


9.141 


828.68 


746.17 


12.17 


12.19 


873.80 


28 


80 


649.70 


9.222 


9.236 


s^.sa 


749.49 


12.27 


12.30 


875.54 


30 


82 


653.03 


9.817 


9.331 


327.02 


752.82 


12.88 


12.41 


877.22 


82 


84 


656.86 


9.411 


9.426 


828.69 


756.15 


12.49 


12 52 


378.89 


34 


86 


669.69 


9.506 


9.521 


330.37 


759.47 


12.60 


12.63 


380 57 


36 


38 


663.02 


9.600 


9.616 


382 04 


762.80 


12.71 


12.74 


38-2.24 


38 


40 


666.84 


9.695 


9.712 


883.71 


766.13 


12.82 


12.86 


383.92 


40 


42 


669.67 


9794 


9.812 


835.88 


768.45 


12.98 


12.96 


385.60 


42 


44 


673.00 


9.894 


9.912 


337.05 


772.78 


18.04 


18.08 


387.27 


44 


46 


676.32 


9 993 


10.01 


838.78 


776.11 


18.15 


13.19 


388.95 


46 


48 


679.65 


10.09 


10.11 


840.40 


779.48 


18.26 


13.81 


390.62 


48 


50 


682 9d 


10.19 


10.21 


842.07 


782.76 


18.87 


18.42 


392.30 


50 


52 


686.80 


10.29 


10.31 


343.74 


786.09 


18.48 


13.53 


393.98 


52 


54 


689 68 


10.39 


10.41 


845.41 


789.41 


18.59 


18.65 


395.65 


54 


66 


692.96 


10.49 


10.51 


347.08 


792.74 


13.70 


13.76 


897.33 


56 


58 


696.28 


10.59 


10.61 


848.76 


796.07 


18.81 


18.88 


899.01 


68 


60 


699.60 


10.69 


10.71 


850.44 


799.40 


18.96 


1:^.99 


400.70 


60 


/ 




8 


o 






fi 


« 




# 


L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





799.40 


13.96 


13.99 


400.70 


899.10 


r,66 


17.71 


450.95 





S 


802.72 


14.07 


14.10 


402.37 


902.42 


17.79 


17.84 


452.63 


3 


4 


806.04 


14.19 


14.22 


404.06 


905.74 


17.92 


17.98 


454.31 


4 


6 


809.87 


14.31 


14.34 


405.72 


909.07 


18.06 


18.11 


455.98 


6 


8 


812.69 


14 43 


14.46 


407.89 


912.89 


18.19 


18.25 


457.66 


8 


10 


816.01 


14.55 


14.58 


409.06 


915.71 


18.32 


18.88 


459.34 


10 


12 


819.34 


14.66 


14.70 


410.74 


919.04 


18.46 


18.52 


461.02 


12 


14 


822.66 


14.78 


14.82 


412.41 


922.36 


18.59 


18.66 


462.70 


14 


16 


825.98 


14.90 


14.94 


414.08 


925.68 


18.72 


18.79 


464.37 


16 


18 


829.31 


15.02 


15.06 


415.75 


929.01 


18.86 


18.93 


466.05 


18 


20 


882.68 


15.14 


15.18 


417.48 


932.33 


18.99 


19.06 


467.73 


30 


22 


885.95 


16.26 


15.30 


419.10 


935.65 


19.12 


19.19 


469.41 


33 


24 


839.28 


15.38 


15.48 


420.77 


938.98 


19.26 


19.33 


471.08 


34 


26 


842.60 


15.51 


15.56 


422.45 


942.30 


19.40 


19.47 


472.76 


26 


28 


845.92 


15.63 


15.68 


424.12 


945.62 


19.54 


19.61 


471.43 


28^ 


80 


849.26 


15.75 


15.80 


425.79 


948.95 


19.68 


19.75 


476.10 


80 


82 


852.57 


15.88 


15.93 


427.47 


952.27 


19.82 


19.89 


477.78 


83 


84 


855.89 


16.00 


16.05 


429.15 


955.59 


19.96 


20.08 


479.46 


84 


86 


859.22 


16.12 


16.18 


430.82 


958.92 


20.10 


20.17 


481.14 


36 


88 


862.54 


16.26 


16.30 


432.50 


962.24 


20.24 


20.81 


482.83 


38 


40 


865.86 


16.38 


16.48 


434.18 


965.56 


20.38 


20.45 


484.51 


40 


42 


869.19 


16.50 


16.55 


435.86 


968.89 


20 52 


20.59 


486.19 


42 


44 


872.61 


16.63 


16.68 


437.54 


972.21 


20.66 


20.74 


487.87 


44 


46 


876.88 


16.76 


16.81 


439.21 


975.53 


20.80 


20.88 


489.56 


46 


48 


879.16 


16.89 


16.94 


440.89 


978.86 


20.94 


21.03 


491.24 


48 


50 


882.48 


17.02 


17.07 


442.57 


98--J.18 


21.09 


21.17 


492.92 


50 


52 


886.80 


17.14 


17.19 


444.25 


985.50 


21.23 


21.81 


494.60 


53 


64 


889.18 


17.27 


17.32 


445.93 


988.83 


21.37 


21 46 


496.28 


54 


66 


892.46 


17.40 


17.45 


447.60 


992.15 


21.51 


21.60 


497.96 


56 


58 


895.77 


17.53 


17.58 


449.28 


995.47 


81.65 


21.75 


499.65 


58 


60 


899.10 


17.66 


17.71 


450.95 


998.80 


21.80 


21.89 


501.82 


60 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEGREE CURVE 


213 




lOo 1 


11» 


/ 


t 


L C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





998.8 


21.80 


81.89 


501.32 


1098.4 


26.38 


26.50 


651.74 





2 


1002.1 


21.94 


82.03 


503.00 


1101.7 


26.54 


86.06 


568.42 


2 


4 


1006.4 


22.09 


82.18 


604.68 


1105.0 


26.70 


86.83 


655.10 


4 


6 


1008.8 


82.24 


22.33 


606.36 


1108 8 


26.86 


86.99 


656.78 


6 


8 


1012.1 


22.39 


28.48 


508.04 


1111.7 


27.02 


87.16 


658.46 


8 


10 


1015.4 


82.54 


82.63 


509.';2 


1115.0 


27.19 


87.82 


660.14 


10 


12 


1018.7 


22.68 


82.78 


511.40 


1118.3 


27,35 


27.48 


661.82 


12 


14 


10220 


32.83 


82.98 


513.08 


1121.6 


87.51 


87.65 


663.50 


14 


16 


1025.4 


22.98 


23.08 


514.76 


1124.9 


27.67 


27.81 


565.18 


16 


18 


1028.7 


23.13 


23.28 


516.44 


1128.2 


27.83 


87.98 


666.86 


18 


20 


1038.0 


83.28 


23.38 


518 12 


1131.6 


28.00 


88.14 


668.64 


20 


23 


1035.8 


83.43 


23.53 


519.80 


1134.9 


2€.17 


88.80 


670.22 


22 


24 


1038.6 


23.58 


23 66 


521.48 


1138.2 


28.34 


88.47 


671.90 


24 


26 


1042.0 


83.73 


23.84 


523.16 


1141.5 


28.50 


88 64 


673.58 


26 


28 


1045.3 


83.88 


23.99 


524.86 


1144.8 


28.67 


28.81 


675 27 


28 


80 


1048.6 


24.04 


84.14 


526.53 


1148.1 


28.84 


28.98 


676.95 


30 


32 


1051.9 


24.19 


84.30 


528.21 


1151.5 


29.00 


29.14 


678.63 


32 


34 


1055.2 


24.34 


24.45 


529.89 


1154.8 


29.17 


29.31 


580.32 


34 


36 


1058.6 


24.49 


24.60 


531.57 


1158.1 


29.34 


89.48 


582.00 


36 


38 


1061.9 


24.64 


24.76 


533 25 


1161.4 


29.50 


89.65 


583.69 


38 


40 


1065.2 


24.80 


24.91 


534.93 


1164.7 


29.67 


29.82 


685.87 


40 


42 


1068.5 


24.95 


25.06 


536.61 


1168.0 


29.84 


29.99 


587.05 


42 


<i\ 


1071.8 


23.11 


25.22 


538 29 


1171.4 


30.01 


80.17 


688.74 


44 


46 


1075.2 


25.87 


25.38 


539.97 


i 174.7 


30.18 


80.84 


590.42 


46 


48 


1078.5 


25.43 


85.54 


541.65 


1178.0 


30.35 


30.52 


592.11 


48 


50 


1081.8 


25.59 


85.70 


543.33 


1181.3 


80.53 


30 69 


693.79 


50 


52 


1085.1 


25.74 


85.86 


645.01 


1184.6 


80.70 


30.86 


595 47 


52 


54 


1088.4 


25.90 


86.02 


546.09 


1187.9 


30.87 


31.04 


597.16 


64 


56 


10U1.8 


26.06 


26.18 


548.37 


1191.3 


31.04 


31.21 


598.84 


56 


68 


1095.1 


26 22 


26.84 


550 06 


1194.6 


31.21 


31.39 


600.53 


68 


60 


1098.4 


26.38 


26.50 


561.74 


1197.9 


81.39 


81.56 


602.22 


60 


» 


12«» 1 


18» 


* 


L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





1197.9 


31.39 


31.56 


602.22 


1297.3 


36.83 


37.07 


652.87 





2 


1201 .2 


31.57 


%\AZ 


603.91 


1300.6 


87.02 


37.26 


654.56 


2 


4 


1204.5 


81.74 


31.91 


605.60 


1303.9 


37.21 


37.46 


656.25 


4 


6 


1207.8 


31.92 


32.09 


607.28 


1307.2 


37.40 


37.65 


657.93 


6 


8 


1211 1 


32.09 


32.27 


608.97 


1310.5 


37.59 


37.85 


659.62 


8 


10 


1214.5 


32.27 


32.45 


610.66 


1313.8 


37 79 


88.04 


661.31 


10 


12 


1217.8 


82.45 


32.63 


612.35 


1317.2 


37.98 


88.23 


663.ro 


12 


14 


1221.1 


32.62 


32.81 


614.04 


1H20.5 


38.17 


38.43 


664.69 


14 


16 


1224.4 


82.80 


82.99 


615.72 


1323.8 


38.86 


38.62 


666.37 


16 


18 


1227.7 


32.97 


33.17 


617.41 


1327.1 


38.55 


38.82 


668.06 


18 


20 


1231.0 


33.15 


33.35 


619.10 


1330.4 


38.75 


89.01 


669.75 


20 


22 


1281.3 


33.33 


33.53 


6J0.':9 


1333.7 


38.95 


39.2C 


671.44 


22 


24 


1237.7 


33.51 


33.72 


622 48 


1337.0 


39.15 


39.40 


673.13 


24 


26 


1241.0 


33.69 


33.90 


6-.;4.16 


1340.3 


39.35 


39.60 


674.81 


86 


28 


1244.3 


33.87 


34.09 


6-.'5.85 


1343.6 


39.54 


39.80 


676.51 


28 


30 


1247.6 


34.06 


34.27 


627.55 


1346.9 


39.74 


40.00 


678.20 


30 


32 


1250.9 


34.24 


34.45 


629.24 


1350.3 


39.94 


40.19 


679.89 


32 


84 


1254.2 


84.42 


34 64 


630.93 


1353.6 


40.13 


40.. 39 


681.58 


34 


36 


1257.5 


34.60 


34.82 


632.61 


1856.9 


40.33 


40.59 


683.26 


36 


38 


1260.8 


34.78 


35.01 


634.30 


1360.2 


40.52 


40.79 


684.95 


38 


40 


1264.2 


34 97 


35.19 


635.99 


1363.5 


40.71 


40.99 


686.64 


40 


42 


1267.5 


35.16 


a5.37 


637.68 


1366.8 


40.91 


41.19 


688 33 


42 


44 


1270.8 


35.84 


35.56 


639.37 


1370.1 


41.11 


41.40 


690.02 


44 


46 


1274.1 


35.53 


35.75 


641.05 


1373.4 


41.31 


41.60 


691.70 


46 


48 


1277.4 


35.71 


35.94 


642.74 


1376.7 


41.51 


41.81 


693.39 


48 


50 


1280.7 


35.90 


36.13 


644.43 


1380.0 


41.71 


42.01 


695.08 


50 


52 


1284.0 


36.09 


36.31 


646.12 


1383.4 


41.91 


42 21 


696.77 


62 


54 


1287.4 


36.27 


36.50 


647.81 


1386.7 


42.11 


42.42 


698.46 


54 


66 


1290.7 


36.46 


36. G9 


649.49 


1390.0 


42 31 


42.62 


700.14 


56 


68 


1294.0 


36.64 


36.88 


651.18 


1393.3 


42.51 


42.83 


701.83 


68 


60 


1297.3 


36.88 


87.07 


652.87 


1396.6 


42.71 


48.03 


703.53 


60 



Digitized by VjOOQIC 



214 


FUNCTIONS OF A ONE-DEGREE CURVE. 




/ 


14- 1 


16* 


t 


L.C. 


M. 


E. 


T. 


L.C. 


M. 


E, 


T. 





1896.6 


42.71 


48.08 


r08.53 


1495.9 


49.02 


40.44 


754.86 





s 


1899.9 


42.92 


43.28 


705.28 


1499.2 


49.24 


49.66 


766.05 


2 


4 


1403.2 


48.12 


43.44 


706.92 


1602.5 


49.46 


49.89 


757.74 


4 


6 


1406.5 


43.38 


43. G5 


708.62 


1505.8 


49.68 


60.11 


759.44 


6 


8 


1409.8 


43.53 


43.86 


710.31 


1609.1 


49.90 


60.34 


761.18 


8 


10 


1413.1 


48. T4 


44.07 


712.01 


1512.4 


60.12 


60.56 


762.88 


10 


IS 


1416.5 


48.94 


44.28 


713.71 


1515.7 


50.34 


60.78 


764.58 


12 


14 


1419.8 


44.15 


44.49 


716.40 


1519.0 


60.56 


61.01 


766.22 


14 


16 


14^3.1 


44.85 


44.70 


717.10 


1522.3 


50 78 


61.23 


767.92 


16 


18 


1426.4 


44.56 


44.91 


718.79 


1525.6 


61.00 


61.46 


769.61 


18 


20 


1429.7 


44.77 


45.12 


720.49 


1528.9 


61.22 


61.68 


771.81 


SO 


22 


1438.0 


44 98 


45.33 


722.20 


1532.2 


61.44 


61.90 


778.01 


22 


24 


1436.8 


45.19 


45.54 


728.89 


1585.5 


51.67 


52.13 


774.70 


84 


26 


1439.6 


45.40 


45.76 


726.59 


1538.8 


51.89 


62.36 


776.40 


86 


28 


1442.9 


45.61 


45.97 


727.28 


1542.1 


62.12 


62.59 


778.09 


28 


30 


1446 2 


45.82 


46.18 


728.97 


1545.4 


62.34 


62.82 


779.79 


80 


32 


1449.6 


46.03 


46.40 


730.66 


1548.7 


62.57 


53.05 


781.49 


82 


84 


1452.9 


46.24 


46.61 


732.35 


1552.0 


52.79 


53.28 


788.19 


84 


86 


1456.2 


46.45 


46.82 


734.05 


1555.8 


68.02 


63.51 


784.89 


86 


88 


14)9.5 


46.66 


47.04 


785.74 


1558.6 


63.24 


68.74 


786.59 


88 


40 


1462.8 


46.87 


47.25 


737.43 


1561.9 


63.47 


63.97 


788.20 


40 


42 


146G.1 


47.08 


.47.46 


739.12 


1565.2 


63.69 


64.20 


789.99 


48 


44 


1469.4 


47.30 


47.68 


740.81 


1568.5 


68:92 


54.44 


791.69 


44 


46 


1472.7 


47.51 


47.90 


742.51 


1571.8 


54.16 


54.67 


793.39 


46 


48 


1476.0 


47.73 


48.12 


744.20 


1575.1 


54.38 


64.91 


795.09 


48 


60 


1479.8 


47.94 


48.34 


745.89 


1573.4 


64.61 


85.14 


796.79 


50 


52 


1482.7 


48.16 


48.56 


747.58 


1581.7 


64.84 


56.37 


798.49 


68 


64 


1486.0 


48.87 


48.78 


749.27 


1585.0 


65.07 


55.61 


800.19 


54 


66 


1489.8 


48.69 


49.00 


750.97 


1588.8 


65.30 


55.84 


801.89 


66 


58 


1492.6 


48.80 


49.22 


752.66 


1591.6 


55 58 


66.08 


808.59 


68 


60^ 


1495.9 


49.02 


49.44 


754.35 


1594.9 


65.76 


66.81 


805.29 


60 




160 1 


!?• , 




/ 


L C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 


§ 





1594.9 


66.76 


56.81 


805.29 


1693.9 


62.94 


68.64 


856.36 





2 


1598.2 


65.99 


56.54 


806.99 


1697.2 


68.18 


68.89 


858.05 


8 


4 


1601.5 


56.28 


56.78 


806.64 


1700.5 


68.43 


64.16 


850.76 


4 


6 


1604.8 


56.46 


57.02 


810.89 


1708.8 


68.68 


64.40 


861.46 


6 


8 


1608.1 


56.70 


57.26 


812.09 


1707.1 


68.93 


64.66 


868.16 


8 


10 


1611.4 


56.93 


57.50 


813.79 


1710.4 


64.18 


64.91 


864.87 


10 


12 


1614.7 


57.17 


67.74 


815.49 


1713.7 


64.42 


66.16 


866.57 


18 


14 


1618.0 


57.40 


57.98 


817.19 


1716.9 


64.67 


66.42 


868.27 


14 


16 


1621.8 


57.64 


58.22 


818.89 


1720.2 


64.92 


66.67 


869.98 


16 


18 


1624.6 


67.87 


58.46 


820.59 


1728.6 


66.17 


66.98 


871.68 


18 


20 


1627.9 


68.11 


58.70 


822.29 


1726.8 


65.42 


66.18 


878.88 


80 


22 


1681.2 


58.34 


58.94 


828.99 


1780.1 


65.67 


66.43 


876.09 


22 


24 


1684.5 


68.58 


59.19 


825.69 


1738.4 


65.98 


66.69 


876.79 


24 


26 


1687.8 


58.82 


59.48 


827 39 


1736.7 


66.18 


66.95 


878.49 


86 


28 


1641.1 


59.06 


59.68 


829.09 


1740.0 


66.44 


67.21 


880.20 


88 


80 


1644.4 


59.80 


59 92 


880.79 


1748.3 


66.69 


67.47 


881.00 


80 


82 


1647.7 


59.54 


60.16 


832.49 


1746.6 


66.94 


67.72 


888.61 


82 


84 


1651.0 


59.78 


60 41 


834.20 


1749.9 


67.20 


67.98 


886.82 


34 


86 


1654.3 


60.02 


60 65 


885.90 


1758.2 


67.46 


68 24 


887.02 


86 


88 


1657.6 


60.26 


60.90 


837.61 


1756.6 


67.71 


68.50 


888.73 


88 


40 


1660.9 


60.50 


61.14 


839.81 


1759.8 


67.96 


68.76 


890.44 


40 


42 


1664.2 


60.74 


61.89 


841.01 


1768.1 


68.21 


69.08 


892.15 


42 


44 


1667.6 


60.99 


61.64 


842.72 


1766.8 


68.47 


69 29 


898.86 


44 


46 


1670.8 


61.28 


61.89 


844.42 


1769.6 


68.78 


69.56 


895.56 


46 


48 


1674.1 


61.48 


62.14 


846.18 


1772.9 


68.99 


69.82 


897.27 


48 


fiO 


1677.4 


61.72 


62.39 


847.88 


1776 2 


69.25 


70.09 


898.98 


50 


62 


1680.7 


61.96 


62.64 


849.53 


1779.5 


69.50 


70.86 


900.69 


52 


64 


1684.0 


62.21 


62.89 


851 24 


1782.8 


69.76 


70.62 


902.40 


54 


66 


1687.8 


62.45 


68.14 


852.94 


n86.1 


70.02 


22 S? 


904.10 


66 


66 


1690.6 


62.70 


68 89 


854.65 


1789.4 


70.28 


71.17 


905.81 


68 


60 


1698.9 


62.94 


68 64 


856 35 


1792.7 


70.54 


71.42 


907.58 


60 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEGREE CURVE 


215 


$ 


18' 1 


W 1 


# 


L.C. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 





17M.7 


70.54 


71.42 


907.62 


1891.5 


78.56 


79.65 


958.86 





2 


1726.0 


70.80 


71.69 


909.28 


1894.8 


78.86 


79.94 


960.67 


8 


4 


1790.8 


71.06 


71.96 


910.94 


1898.1 


79.13 


80.22 


962.80 


4 


6 


1802.6 


71.88 


72.28 


912.65 


1901.8 


79.41 


80.51 


964.00 


6 


8 


1806.9 


71.69 


72.50 


914.86 


1904.6 


79.68 


80.79 


965.72 


8 


10 


1809.2 


71.85 


72.77 


916.07 


1907.9 


79.96 


81.08 


967.43 


10 


12 


1812.6 


72.12 


78.04 


917.78 


1911.2 


80.24 


81.87 


969.15 


18 


14 


1816.7 


72.88 


78.81 


919.49 


1914.5 


80.51 


81.65 


970.86 


14 


16 


1819.0 


72.64 


78.59 


921.20 


1917.8 


80.79 


81.94 


972.58 


16 


18 


1832.8 


72 91 


78.85 


922.91 


1921.0 


81.07 


82.22 


974.29 


18 


20 


1826.6 


78.17 


74.18 


924.68 


1924.8 


81.85 


82.51 


976.01 


80 


82 


1828.9 


78.48 


74.89 


926.34 


1927.6 


81.63 


82.80 


9Tr.72 


28 


24 


1882.2 


78.70 


74.67 


928.05 


1930.9 


81.91 


83.09 


979.44 


24 


20 


1886.5 


78.97 


74.94 


929.76 


1934.2 


82.20 


88.88 


981.15 


86 


28 


1888.8 


74.24 


76.22 


931.47 


1937.6 


82.48 


88.67 


982.86 


28 


80 


1842.1 


74.51 


75.49 


938.18 


1940.7 


82.76 


88.97 


984.58 


80 


82 


1845.4 


74.77 


75.77 


984.89 


1944.0 


88.05 


84.26 


986.30 


32 


84 


1848.7 


75.04 


76.04 


986.60 


1947.3 


88.83 


84.55 


988.02 


34 


86 


1852.0 


75.31 


76.82 


938.32 


1960.6 


83.61 


84.84 


969.74 


86 


88 


1856.8 


75.68 


76.59 


940.08 


1953.9 


83.90 


85.18 


991.46 


88 


40 


1868.6 


76.86 


76.87 


041.74 


1957.2 


84.18 


85.43 


993.18 


40 


42 


1861.9 


76.12 


77.14 


948.45 


1960.4 


84.47 


85.73 


994.90 


42 


44 


1865.1 


76.89 


77.42 


945.16 


1963.7. 


84.75 


86.08 


996.68 


44 


40 


1868.4 


76.67 


77.70 


946.88 


1967.0 


85.04 


86.82 


996.84 


46 


48 


1871.7 


78.94 


77.98 


948.69 


1970.8 


85.82 


86.61 


1000.0 


48 


60 


1876.0 


77.21 


78.26 


950.30 


1978.6 


85.61 


86.91 


1001.8 


60 


62 


1878.8 


77.49 


78.68 


952.01 


1976.9 


85.90 


87.21 


1003.5 


52 


64 


1881.6 


77.76 


78.81 


953.72 


1980.1 


86.19 


87.50 


1005.2 


64 


60 


1884.9 


78.03 


79.09 


955.44 


1988.4 


86.47 


87.80 


1006.9 


66 


68 


1888.2 


78.81 


79.87 


957.15 


1986.7 


86.76 


88.09 


1006.6 


68 


60 


1891.5 


78.58 


79.65 


958.86 


1990.0 


87.06 


88.89 


1010.4 


60 


/ 


20- 1 


210 


#v 


L.C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





1990.0 


87.05 


88.89 


1010.4 


2068.5 


95.95 


97.58 


1068.0 





2 


1998.8 


87.84 


88.69 


1012.1 


2091.8 


96.26 


97.90 


1068.7 


8 


4 


1996.6 


87.68 


88.99 


1013.8 


2095.0 


96.56 


98.81 


1065.4 


4 


6 


1999.8 


87.92 


89.29 


1015.5 


2096.3 


96.87 


98.53 


1067.2 


6 


8 


2008.1 


88.21 


89.59 


1017.2 


2101.6 


97.17 


98.84 


1068.9 


8 


10 


2006.4 


88.50 


89.89 


1019.0 


2104.9 


97.48 


99.16 


1070.6 


10 


12 


2009.7 


88.79 


90.19 


1020.7 


2106.1 


97.79 


99.48 


1072.4 


18 


14 


sota.o 


88.08 


90.49 


1022.4 


2111.4 


96.09 


99.70 


1074.1 


14 


16 


2016 8 


89.87 


90.79 


1024.1 


2114.7 


98.40 


100.1 


1075.8 


16 


18 


2019.6 


89.66 


91.09 


1025.8 


2118.0 


98.70 


100.4 


1077.5 


18 


20 


8022.8 


89.96 


91.40 


1087.6 


8121.2 


99.00 


tOO.7 


1079.3 


80 


22 


2026.1 


90.25 


91.71 


1029.8 


2124.5 


99.30 


JOl.l 


1061.0 


82. 


24 


2029.4 


90.55 


92.01 


1081.0 


2187.8 


99.60 


101.4 


1082.7 


84 


26 


2082.7 


90.85 


92.82 


1082.7 


8131.0 


99.90 


101.7 


1064.4 


26 


28 


2086.0 


91.15 


98.62 


1084.4 


2134.8 


100.2 


108.0 


1066.2 


28 


80 


2080.8 


91.45 


92.98 


1086.1 


2187.6 


100.6 


102.3 


1087.9 


80 


82 


8042.6 


91.74 


93.24 


1087.9 


2140.9 


100.8 


102.7 


1069.6 


82 


84 


2045.8 


98.04 


96.54 


1089.6 


8144.1 


101.1 


103.0 


1091.3 


84 


86 


2049.1 


92.84 


93.85 


1041.8 


2147.4 


101.4 


103.8 


1093.1 


86 


88 


2068.4 


92.64 


94.15 


1043.0 


2150.7 


101.7 


108.6 


1094.8 


88 


40 


2066.7 


92.94 


94.46 


1044.8 


2154.0 


102.1 


104.0 


1096.5 


40 


42 


2068.0 


96.24 


94.78 


1046.6 


2167.2 


102.4 


104.3 


1096.8 


48 


44 


2068.2 


98.64 


95.09 


1048.2 


2160.5 


102.7 


104.6 


1100.0 


44 


46 


2066.6 


96.84 


96.40 


1049.9 


2168.8 


103.0 


104.9 


1101.7 


40 


48 


2066.8 


94.14 


95.71 


1061.7 


2167.1 


108.8 


106.8 


1103.4 


48 


50 


2072.1 


94.44 


96.08 


1058.4 


2170.8 


108.6 


105.6 


1105.2 


60 


52 


2076.4 


94.74 


96.84 


1055.1 


2178.6 


103.9 


105.9 


1106.9 


68 


64 


2078.6 


96.04 


96.65 


1066.8 


8176.9 


104.2 


106.8 


1108.6 


64 


66 


2061.9 


96.84 


96.96 


1058.6 


2180.1 


104.5 


106.6 


1110.8 


66 


68 


8068.2 


96.64 


97.27 


1060.8 


2188.4 


104.8 


106.9 


1118.1 


68 


« 


8088.6 


96.96 


97.86 


1068.0 


2186.7 


106.8 


107.8 


1118.8 


60 



Digitized by VjOOQIC 



216 


FUNCTIONS OF A ONE-DEGREE CURVE 




/ 


28- 1 


28« 


/ 


L, C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





2186.7 


105.2 


107.2 


1113.8 


2284.8 


115.0 


117.4 


1165.8 





2 


2100.0 


105.6 


107.6 


1115.5 


8288.1 


115.3 


117.7 


1167.5 


8 


4 


2193.2 


105.9 


107.9 


1117.3 


2291.8 


116.7 


118.1 


1169.8 


4 


6 


2196.6 


106.2 


108.2 


1119.0 


2294.6 


116.0 


118.4 


1171.0 


6 


6 


2199.8 


106.5 


108.6 


1120.7 


2297.8 


116.4 


118.8 


1178.7 


8 


10 


2203.0 


106.8 


108.9 


1122.4 


2301.1 


116.7 


J19.1 


1174.4 


10 


12 


2206.8 


107.1 


109.3 


1124.8 


2304.4 


117.0 


119.5 


1176.3 


12 


14 


2209.6 


107.4 


109.6 


1125.9 


2807.6 


117.4 


119.8 


1177.9 


14 


16 


2212 9 


107.7 


109.9 


1127.6 


2310.9 


1J7.7 


120.8 


1179.7 


16 


18 


2216.1 


108.0 


110.2 


1129.4 


2814 1 


118.1 


120.5 


1181.4 


18 


20 


2219.4 


106.4 


110.6 


1131.1 


8317.4 


118.4 


120.9 


1183.1 


80 


22 


2222.7 


108.7 


110.9 


1182.8 


2820.7 


118.7 


121.8 


1184.9 


88 


24 


2225.9 


109.0 


111.8 


1134.6 


2323.9 


119.1 


181,6 


1186.6 


24 


26 


2229.3 


109.4 


111.6 


1136.3 


2327.2 


119.4 


121.9 


1188.4 


26 


28 


2282.5 


109.7 


111.9 


1138.0 


2330.4 


119.8 


122.8 


1190.1 


88 


30 


2285.7 


110.0 


112.8 


1139.7 


2833.7 


120.1 


122.6 


1191.8 


30 


32 


2239.0 


110.4 


112.6 


1141.5 


2337.0 


120.4 


123.0 


1193.6 


S8 


34 


22428 


110.7 


112.9 


1143.8 


2340.8 


120.8 


188.8 


1195.3 


84 


36 


2245.6 


111.0 


118.8 


1144.9 


2343.6 


121.1 


128.7 


1197.1 


36 


88 


2248.8 


111.4 


118.6 


1146.7 


2346.7 


121.5 


184.1 


1196.8 


38 


40 


2252.1 


111.7 


113.9 


1148.4 


2350.0 


121.8 


184.4 


1200.5 


40 


42 


2255.4 


112.0 


114.8 


1150.1 


2858.3 


122.1 


184.8 


1202.3 


42 


44 


2258.6 


112.3 


114.6 


1151.9 


2356.5 


122.5 


186.1 


1804.0 


44 


46 


2261.9 


112.7 


115.0 


1153.6 


2359.8 


122.8 


125.5 


1205.8 


46 


48 


2265.2 


118.0 


115.8 


1155.4 


23(13.0 


123.8 


125.8 


1207.5 


48 


50 


2268.4 


113.8 


115.7 


1157.1 


2366.8 


183.5 


126.3 


1209.2 


50 


52 


2271.7 


113.7 


116.0 


1158.8 


2369.6 


183.8 


126.6 


1211.0 


52 


54 


2275.0 


114.0 


116.8 


1160.6 


2372.8 


124.8 


186.9 


1812.7 


54 


56 


2278.8 


114.8 


116.7 


1162.8 


2876.1 


124.5 


187.3 


1214.5 


56 


58 


2281.5 


114.7 


117.0 


1164.0 


2379.8 


124.9 


127.6 


1216.8 


58 


60 


2284.8 


115.0 


117.4 


1165.8 


2382.6 


125.8 


128.0 


1218.0 


60 


/ 


24«» 1 


260 


/ 


L. 0. 


M. 


E. 


T. 


L 0. 


M. 


E. 


T. 





2882.6 


125.2 


128.0 


1218.0 


2-180.4 


135.8 


189.1 


1270.8 





2 


2385.9 


125.5 


128.4 


1219.7 


8483.6 


-Jd6.2 


139.5 


1272.0 


8 


4 


2389.1 


125.9 


128.7 


1221.4 


2486.9 


136.5 


189.9 


1273.8 


4 


6 


2392.4 


126.2 


129.1 


1223.2 


2400.1 


136.9 


140.3 


1275.5 


6 


8 


2395.6 


126.6 


129.6 


1224.9 


2498.4 


187.2 


140.6 


1277.3 


8 


10 


2398.9 


126 9 


129.8 


1226.7 


2406.6 


137.6 


141.0 


1279.0 


10 


12 


2402.2 


127.3 


130.8 


1228.4 


2499.9 


138.0 


141.4 


1280.8 


12 


14 


2405.4 


127.6 


130.6 


1230.2 


2503.1 


138.8 


141.8 


1282.5 


14 


16 


2408.7 


128.0 


130.9 


1231.9 


2506.4 


138.7 


142.3 


1284.8 


16 


18 


2411.9 


128.3 


131.8 


1233.6 


2509.6 


139.0 


142.5 


1286.1 


18 


20 


2415.2 


128.7 


131.7 


1235.4 


2512.0 


189.4 


148.0 


1287.8 


80 


22 


2418.5 


129.0 


132.0 


1237.1 


2.516.1 


139.8 


148.8 


1289.6 


88 


24 


2421.7 


129.4 


132.4 


128S.9 


2519.4 


140.1 


143.7 


1291.3 


84 


26 


2«5.0 


129.7 


182.8 


1240.6 


2522.6 


140.5 


144.1 


1298.1 


86 


28 


2428.2 


130.1 


133 1 


1242.4 


2525.0 


140.8 


144.5 


1894.8 


88 


30 


2431.5 


130.4 


133.5 


1244.1 


2529 1 


141.8 


144.9 


1890.6 


80 


32 


2434.8 


130.8 


133.9 


1215.8 


2532^.4 


141.6 


145.3 


1298.8 


88 


34 


2439.0 


131.1 


134.3 


1247.6 


2536.6 


142.0 


145.6 


1800.1 


84 


36 


2441.3 


131.5 


134.6 


1249.3 


2538.9 


142.3 


146.0 


1801.8 


86 


88 


2444.5 


131.8 


135.0 


1251.1 


2542.1 


142 7 


146.4 


1806.6 


88 


40 


2447.8 


182.2 


135.4 


1252.8 


2545.4 


143.1 


146.8 


1306.8 


40 


42 


2451.1 


132.6 


185.7 


1254.6 


2548.6 


143.5 


147.8 


1807.1 


48 


44 


2454.3 


132.9 


136.1 


1266.8 


2651.9 


143.8 


147.6 


1808.8 


44 


46 


2457.6 


133.3 


130.5 


1258.1 


2555.1 


144.8 


148.0 


1810.6 


46 


48 


2460.6 


133.6 


136.9 


1259.8 


2558.4 


144.5 


148 4 


1818.4 


48 


50 


2464.1 


134.0 


137.2 


1261.5 


2561.6 


144.9 


148.8 


1814.1 


50 


52 


2467.4 


184.4 


137.6 


1263.3 


2564.9 


145.3 


149.8 


J815.9 


58 


54 


2470.6 


184.7 


138.0 


1265.0 


8568.1 


145.7 


149.5 


1817.6 


54 


56 


2473.9 


185.1 


138.4 


1266.8 


2571.4 


146.0 


149.9 


1819.4 


56 


58 


8477.1 


135.4 


188.7 


1268.5 


25T4.6 


146.4 


150.8 


1821.1 


68 


60 


2480.4 


136.8 


139.1 


1270.3 


2577.9 


146.8 


160.7 


1882.9 


60 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEGREE CURVE 


217 


/ 


S6« 


«?• 


i 


L. C. 


M. 


E. 


T. 


L. C. M. 


E. 


T. 





2577.9 


146.8 


160.7 


1322.9 


2675.3 158.8 


162.8 


1875.6 





2 


2581.1 


147.1 


151.1 


1324.6 


2678.5 158.6 


163.2 


1377.4 


• 


4 


2584.4 


147.6 


151.5 


1326.4 


2681.8 159.0 


163.7 


1379.2 


4 


6 


2587.6 


147.9 


151.0 


1328.1 


2686.0 159.4 


164.1 


1380.9 


e 


8 


2590.0 


148.3 


162.3 


1329.9 


2688.2 159.8 


164.5 


1882.7 


8 


10 


2594.1 


148.7 


162.7 


1331.6 


2691.5 160.2 


164.9 


1384.5 


10 


12 


2597.4 


149.1 


153.1 


1333.4 


2694.7 160.6 


165.3 


1386.2 


12 


14 


2600.6 


149.4 


153.5 


1335.2 


2698.0 161.0 


165.7 


1388.0 


14 


16 


2603.9 


149.8 


153.9 


1336.9 


2701.2 161.4 


166 1 


1389.8 


16 


18 


2607.1 


150.2 


154.3 


1338.7 


2704.4 161.8 


166.5 


1391.5 


18 


20 


2610.4 


150.6 


154.7 


1340.4 


2707.7 162.2 


167.0 


1393.3 


20 


22 


2613.6 


151.0 


155.1 


1342.2 


2710.9 162.6 


167.4 


1395.0 


28 


24 


2616.9 


151.4 


156.6 


1343.9 


2714.1 163.0 


167.8 


1896.8 


24 


26 


2620.1 


151.7 


155.9 


1345.7 


2717.4 163.4 


168.2 


1896.6 


26 


28 


2623.4 


152.1 


156.8 


1347.4 


2720.6 163.8 


168.6 


1400.3 


28 


30 


2626.6 


152.5 


156.7 


1349.2 


2723.8 164.2 


169.1 


1402.1 


SO 


82 


2629.8 


152.9 


157.1 


1351.0 


2727.1 164.6 


169.5 


1403.9 


82 


84 


2633.1 


158.3 


157.6 


1352.7 


2730.3 165.0 


169.9 


1405.6 


84 


86 


2636.3 


153.7 


157.9 


1854.5 


2733.6 166.4 


170.3 


1407.4 


36 


38 


2639.6 


154.0 


158.3 


1356.2 


2736.8 165.8 


170.8 


1409.2 


88 


40 


2642.8 


154.4 


168.7 


1358.0 


2740.0 166.2 


171.2 


1410.9 


40 


42 


2646.1 


154.8 


159.1 


1359.8 


2743.3 166.6 


171.6 


1412.7 


42 


44 


2649.3 


155.2 


159.6 


1361.5 


2746.6 167.0 


172.0 


1414.6 


44 


46 


2652.6 


155.6 


160.0 


1363.3 


2749.7 167 4 


172.5 


1416.3 


46 


48 


2655.8 


156.0 


160.4 


1365.1 


2753.0 167.8 


172.9 


1418.0 


48 


60 


2659.1 


156.3 


160.8 


1366.8 


2756.2 168.2 


173 3 


1419.8 


60 


C2 


2662.3 


156.7 


161.2 


1368.6 


2759.5 168.6 


173.7 


1421.6 


62 


54 


2666.6 


157.1 


161.6 


ITOA 


2762.7 169.0 


174.1 


1423.3 


54 


66 


2668.8 


157.5 


162.0 


13'5'2.1 


2766.9 169.4 


174.6 


1425.1 


56 


58 


2672.1 


157.9 


162.4 


1373.9 


2?69.2 169.8 


175.0 


1426.9 


68 


60 


2675.3 


158.3 


162.8 


1375.6 


2772.4 170.2 


175.4 


1428.6 


60 


/ 


28* 1 


go- 


/ 


L. C. 


M. 


E. 


T. 


L. C. M. 


E. 


rT- 





2772.4 


170.2 


175.4 


1428.6 


2869.4 182.5 


188.5 


1481.9 





2 


2775.6 


170.6 


175.8 


1430.4 


2872.6 182.9 


189.0 


1483.7 


2 


4. 


2778.9 


171.0 


176.3 


1432.2 


2875.8 183 3 


189.4 


1485.4 


4 


6 


2782.1 


171.4 


176.7 


1434.0 


2879.1 183.7 


189.9 


1487.2 


6 


8 


2785.3 


171.8 


177.1 


1436.7 


2882.3 184.2 


190.3 


1489.0 


8 


10 


2r88.6 


1?2.2 


177.6 


1487.5 


2885.5 184.6 


190.8 


1490.8 


10 


12 


2791.8 


172.6 


178.0 


1439.8 


2888.7 185.0 


191.2 


1492.6 


12 


14 


2795.0 


173.0 


178.4 


1441.1 


2892.0 185.4 


191.7 


1494.8 


14 


16 


2798.8 


173.4 


178.9 


1442.8 


2895.2 186.8 


192.1 


1496.1 


16 


18 


2801.5 


173.8 


179.3 


1444.6 


2898.4 186.3 


192.5 


1497.9 


18 


20 


2804.7 


174.3 


179.7 


1446.4 


2901.6 186.7 


193.0 


1499.7 


20 


22 


2808.0 


174.7 


180.2 


1448.2 


2904.8 187.1 


193.5 


1501.5 


22 


24 


2811.2 


175.1 


180.6 


1449.9 


2908.1 187.5 


193.9 


1503.2 


24 


26 


2814.4 


175.5 


181.0 


1451.7 


2911.8 188.0 


194.4 


1505.0 


26 


28 


2817.7 


175.9 


181.6 


1463.5 


2914.5 188.4 


194.8 


1506.8 


28 


30 


2820.9 


176.3 


181.9 


1455.2 


2917.7 188.8 


195.3 


1608.6 


30 


32 


2824.1 


176.7 


182.8 


1457.0 


2921.0 189.2 


195.7 


1510.4 


32 


84 


2827.4 


177.1 


182.8 


1458.8 


2924.2 189.7 


196.2 


1512.1 


34 


36 


2830.6 


177.5 


183.2 


1460.6 


2927.4 190.1 


196.7 


1513.9 


36 


38 


2833.8 


177.9 


183.6 


1462.3 


2930.6 190.5 


197.1 


1615.7 


38 


40 


2837.1 


178.4 


184.1 


1464.1 


2933.9 190.9 


197.6 


1517.5 


40 


42 


2840.3 


178.8 


184.5 


1465.9 


2937.1 191.4 


198.0 


1619.3 


42 


44 


28435 


179.2 


185.0 


1467.7 


2940.3 191.9 


198.5 


1521.0 


44 


46 


2846.8 


179.6 


185.4 


1469.5 


2943.5 192.4 


198.9 


1522.8 


46 


48 


2850.0 


1800 


185.9 


1471.2 


2946.8 192.8 


199.4 


1524.6 


48 


50 


2853.2 


180.4 


186.8. 


1473.0 


2950 193.2 


199.8 


1526.4 


60 


62 


2856.5 


180.8 


186.8 


1474.8 


2953.2 193.6 


200.8 


1528.2 


52 


54 


2859.7 


181.2 


187.2 


1476.6 


2956.4 194.0 


200.8 


1580.0 


54 


56 


2862.9 


181.6 


187.6 


1478.3 


2959.6 194.4 


201.2 


1531.7 


66 


68 


2866.2 


182 


188.1 


1480.1 


2962.9 194.8 


201.7 


1538.5 


68 


60 


2869.4 


182 5 


188.5 


1481 9 


2966.1 195.2 


202.1 


1585 3 


60 



Digitized by VjOOQIC 



218 


FUNCTIONS OF A ONE-DEGREE CURVE 




f 


W 


81» 


/ 


L. C. 


M. 


E. 


T. 


L. C. M. 


E. 


T. 





2966.1 


195.2 


202.1 


1535.3 


3062.6 208.4 


216.8 


1689.0 





2 


2969.3 


195.6 


202.6 


1537.1 


3065.8 208.8 


216.8 


1690.8 


2 


4 


2972.5 


196.1 


203.1 


1538.9 


3069.0 209.3 


217 2 


1592.6 


4 


6 


2975 7 


196.5 


^m.h 


1540.7 


3072.2 209.7 


217.7 


1594.4 


6 


8 


26790 


197.0 


204.0 


1542.5 


3075.4 210.2 


218.2 


1596:2 


8 


10 


2982.2 


197.4 


204.5 


1544 3 


3078.6 210.6 


218.7 


1598.0 


10 


12 


2985.4 


197.8 


204.9 


1546.0 


3081.8 211.1 


219.2 


1599.8 


12 


14 


2968.6 


198.2 


205.4 


1547.8 


3085.0 211.5 


219.6 


1601.6 


14 


16 


2991.8 


198.6 


2059 


1549.6 


8088.3 212.0 


220.1 


1603.4 


16 


18 


2995.0 


199.1 


206 3 


1551.4 


3091.5 212.4 


220.6 


1605.2 


18 


20 


2998.3 


199.5 


206.8 


1553.2 


3094.7 212.9 


221.1 


1607.0 


20 


22 


3001.5 


199.9 


207,3 


1555.0 


3097.9 213.3 


221.6 


1608.8 


22 


24 


3004.7 


200.4 


207.7 


1556.8 


3101.1 213.8 


822.1 


1610.6 


24 


26 


3007.9 


200.8 


208.2 


1558.6 


3104.3 214.2 


222.6 


1612.4 


26 


28 


3011.1 


201.3 


208.7 


1560.4 


3107.5 214.7 


223.0 


1614.2 


28 


SO 


3014.3 


201.7 


209.1 


1562.2 


3110.7 215.1 


223.5 


1616.0 


80 


32 


3017.6 


202.1 


209.6 


1564.0 


3113.9 215.6 


224.0 


1617.8 


82 


84 


3020.8 


202.6 


210.1 


1565.7 


3117.1 216.0 


224.5 


1619.6 


84 


86 


8024.0 


203.0 


210.5 


1567.5 


3120.3 216.5 


226.0 


1621.4 


86 


88 


3027.2 


203.5 


211.0 


1569.3 


3123.5 216.9 


225.6 


1623.2 


88 


40 


8030.4 


203.9 


211 5 


1571.1 


3126.7 217.4 


226.0 


1625.0 


40 


42 


8033.6 


204.3 


212.0 


1572.9 


3129.9 217.8 


226.5 


1626.8 


42 


44 


3086.9 


204.8 


212.4 


1574.7 


3133.1 218.3 


227.0 


1628.6 


44 


46 


8040.1 


205.2 


212.9 


1576.5 


3136.4 218.7 


227.5 


16.30.5 


46 


48 


3043.3 


205.7 


213.4 


1578.3 


3139.6 219.2 


228.0 


1632.3 


48 


50 


3046.5 


206.1 


213.9 


1580.1 


3142.8 219.6 


228.4 


1634.1 


60 


52 


3049.7 


206.5 


214.4 


1581.9 


8146.0 220.1 


228.9 


1635.9 


52 


54 


3052.9 


2070 


214.8 


1583.7 


3149.2 220.5 


229.4 


1637.7 


54 


56 


3056.2 


207.4 


215.3 


1585.5 


3162.4 221.0 


229.9 


1639.5 


66 


58 


8059.4 


207.9 


215.8 


1587.2 


8155.6 221.5 


230.4 


1641.3 


58 


60 


8062.6 


208.4 


216.3 


1589.0 


3158.8 222.0 


230.9 


1643.1 


60 


/ 


820 1 


88« 


t 


L. C. 


M. 


E. 


T. 


L. C M. 


E. 


T. 





3158.8 


222.0 


230.9 


1643.1 


3254. D 236.0 


246.1 


1697 3 





2 


3162.0 


2225 


231.4 


1644.9 


3258 1 236.4 


246.6 


1699.1 


2 


4 


8165.2 


222.9 


231.9 


1646.7 


8261.3 236.9 


247.1 


17009 


. 4 


6 


3168.4 


223.4 


282.4 


1648.5 


3264.5 237.4 


247.7 


1702.7 


6 


8 


3171.6 


223.8 


232.9 


1650.3 


3267 7 237.9 


248.2 


1704.5 


8 


10 


3174.8 


224.3 


233.4 


1652.1 


3270.8 238.4 


248.7 


1706.4 


10 


12 


3178.0 


224 8 


233.9 


1653.9 


3274 238.9 


249.2 


1708.2 


12 


14 


3181.2 


225.2 


234.4 


1655 7 


3277.2 239 3 


249.7 


1710.0 


14 


16 


8184.4 


225.7 


234.9 


1657.5 


3-^80 4 239.8 


250.2 


1711.8 


16 


18 


3187.6 


226.1 


235.4 


1659.3 


3283.6 240.3 


250.8 


1718.6 


18 


20 


3190.8 


226.6 


235.9 


1661 1 


3286.8 240.8 


251.3 


1716.5 


90 


22 


3194.0 


227.1 


236.4 


1662.9 


3290 241.2 


251.8 


1717.8 


22 


24 


8197 2 


227.5 


286 9 


1664.7 


3293.2 241.7 


252.3 


1719.1 


24 


26 


3200.4 


228.0 


237.4 


1666 5 


3296.4 242.2 


252.9 


1720.9 


26 


28 


3203.6 


228.4 


237.9 


1668 3 


3299.6 242.7 


253.4 


1722.7 


28 


80 


3206.8 


228 9 


238.4 


1670.1 


3302 7 243.2 


253.9 


1724.6 


ao 


32 


8210.0 


229.4 


239.0 


1671.9 


3305 9 243.6 


254.4 


1726.4 


82 


34 


8213.2 


229 8 


239.5 


1673 7 


3309 1 244.1 


255.0 


1728.2 


84 


36 


3216.6 


230.3 


240.0 


1675.5 


3312 3 244.6 


256.5 


1730.0 


86 


88 


8519.7 


230.7 


240.5 


1677.4 


3315 5 245.1 


256.0 


1731..8 


38 


40 


32229 


231 2 


241.0 


1679.2 


3318.7 245.6 


256 5 


1733.6 


40 


42 


8226 1 


281.7 


241.5 


1681.0 


3321.9 246 


257 1 


1785.5 


42 


44 


3629.3 


232.2 


242.0 


1682 8 


3325.1 246.5 


257.6 


1787.3 


44 


46 


3232.5 


232.6 


242.5 


1684.6 


3328.3 247.0 


258.1 


1789.1 


46 


48 


8235.7 


233.1 


243.0 


1686.4 


3331.5 247 5 


258 6 


1740.9 


48 


60 


8238.9 


233.5 


243.5 


1688.2 


3334.6 .248.0 


259.2 


1742.7 


60 


52 


3242.1 


234.0 


244 1 


1690.0 


33:^7.8 248.4 


259.7 


1744.6 


62 


54 


3245 3 


234.5 


244.6 


1691.8 


3341.0 248.9 


2602 


1746.4 


54 


56 


3248.5 


235.0 


245.1 


1693.7 


3344. 2 249.4 


260.8 


1748.2 


56 


58 


3251.7 


235.5 


245.6 


1695 5 


3347.4 249.9 


261.3 


1750.0 


68 


60 


8254.9 


236.0 


246.1 


1697.3 


3350.6 250.4 


261 8 


1751.8 


60 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEGREE CURVE 


219 


§ 


84« 1 


36» 


§ 


L. 0. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 





3350.6 


250.4 


261.8 


1751 .8 


3446.1 


265.2 


278.1 


1806.7 





2 


3358.8 


250.8 


262.3 


1753.7 


3449.3 


265.7 


278.6 


1808.5 


2 


4 


3357.0 


251.2 


262.9 


1755.5 


3452.5 


266.2 


279.2 


1810.3 


4 


6 


3360.1 


251.7 


263.4 


1757.3 


3455.6 


266.7 


279.7 


1812.2 


6 


8 


3363.3 


252.2 


204.0 


1759.1 


3458.8 


267.2 


280.3 


1814.0 


8 


10 


8366.5 


252.7 


264.6 


1761.0 


3462.0 


267.7 


280.8 


1815.8 


10 


12 


8369.7 


253.2 


265.0 


1762.8 


3465.2 


268.2 


281.4 


1817.7 


12 


14 


3872.9 


253.7 


265.6 


1764.6 


3468.3 


268.7 


281.9 


1819.5 


14 


16 


3876.1 


254.2 


266.1 


1766.4 


3471.5 


269.2 


282.5 


1821.3 


16 


18 


3379.2 


254.7. 


266.7 


1768.3 


3474.7 


269.7 


283.0 


1823.2 


18 


20 


3882.4 


255.2 


267.2 


1770.1 


84V7.9 


270.2 


283.6 


1825.0 


20 


22 


8385.6 


255.7 


267.7 


1771.9 


3481.0 


270.7 


284.2 


1826.8 


22 


24 


3388.8 


256.2 


268.3 


1773.7 


3484.2 


271.2 


284.7 


1828.7 


24 


26 


3392.0 


256.7 


268.8 


1775.6 


3487.4 


271.7 


285.3 


ia30.5 


26 


28 


3395.2 


257.2 


269.3 


1777.4 


3490.6 


272.2 


285.9 


1832.3 


28 


30 


3398.8 


257.7 


269.9 


1779.2 


3493.7 


272.7 


286.4 


1834.2 


30 


8-2 


3401.5 


258.2 


270.4 


1781.0 


3496.9 


273.2 


287.0 


1836.0 


32 


34 


3404.7 


258.7 


271.0 


1782.9 


8500.1 


273. V 


267.5 


1837.8 


34 


86 


3407.9 


259 2 


271.5 


1784.7 


3503.3 


274.2 


288.1 


1839 7 


36 


88 


3411.1 


259.7 


272 


1786.5 


3506.5 


274.7 


288.7 


1841.5 


88 


40 


3414.3 


260.2 


272.6 


1788.4 


3509.6 


275.2 


289.2 


1843.4 


40 


42 


3417.4 


260.7 


273.1 


1790.2 


3512.8 


275.7 


289.8 


1845.2 


42 


44 


3420.6 


261.2 


273.7 


1792.0 


8516.0 


276 2 


290,4 


1847.1 


44 


46 


3428.8 


261.7 


274.2 


1793.9 


3519.2 


276.7 


290.9 


1848.9 


46 


48 


3427.0 


263.2 


274.8 


1795.7 


3522.3 


277.2 


291.5 


1850.7 


48 


50 


3430.2 


262.7 


275.3 


1797.5 


8525.5 


277.7 


292.0 


1852.6 


50 


52 


3433.4 


263.2 


275.9 


1799.3 


8528.7 


278.2 


292.6 


1854.4 


52 


54 


3436.6 


263.7 


270.4 


1801.2 


8531.9 


278.7 


293.2 


1856.8 


54 


66 


3430.7 


264.2 


277.0 


1603.0 


3535.0 


279.2 


298.7 


1858.1 


56 


58 


3442.9 


264.7 


277.5 


1804.8 


3538.2 


279.8 


294.3 


1859.9 


58 


60 


3446.1 


265.2 


278.1 


1806.7 


8541.4 


280.4 


294.9 


1861.8 


60 


/ 


860 1 


37« 


' 


L. C. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 





8541.4 


280.4 


294.9 


1861.8 


3636.3 


296.1 


312.8 


1917.3 





2 


8544.6 


280 9 


295.4 


1863.6 


3639.5 


296.6 


312.8 


1919.1 


2 


4" 


3547.7 


281.4- 


296. 


1865.5 


3642 6 


2971 


813.4 


1921.0 


4 


6 


3550.9 


281.9 


296.6 


1867.3 


8645.8 


297.7 


814.0 


1922.8 


6 


8 


8554.0 


282.5 


297.2 


1869.2 


8648.9 


298.2 


314.6 


1924.7 


8 


10 


3557.2 


283.0 


297.7 


1871.0 


8652.1 


298.7 


315.2 


1926.5 


10 


12 


8660.4 


283.5 


298.3 


1872.9 


8655.2 


299.3 


315.8 


1928.4 


12 


14 


8563.5 


284.0 


298.9 


1874.7 


8658.4 


299.8 


316.4 


1930 2 


14 


16 


8566.7 


284.6 


299.5 


1876.5 


8661.6 


800.3 


817.0 


1932.1 


16 


18 


3569.9 


285.1 


300.0 


1878.4 


8G61.7 


800.9 


317.5 


1933.9 


18 


20 


8573.0 


285.6 


800.6 


1880.2 


3667.9 


801.4 


318.1 


1935.8 


20 


22 


3576.2 


286.1 


801.2 


1882.1 


8671.0 


801.9 


318.7 


1937.6 


22 


24 


3579.4 


286.7 


801.8 


1883.9 


8674.2 


302.5 


319.3 


1939.5 


24 


26 


8582.6 


287.2 


802.3 


1885.8 


3677.3 


803.0 


319.9 


1941.3 


26 


28 


3585.7 


287.7 


302.9 


1887.6 


8680.5 


303.6 


320.5 


1943.2 


28 


30 


8588.8 


288.2 


803.5 


1889.5 


3683.6 


304.1 


821.1 


1945.0 


80 


82 


8592.0 


288.8 


804.1 


1891.8 


8686.8 


304.6 


321.7 


1946.9 


32 


84 


8595.2 


289.8 


804.6 


1893.2 


3690.0 


305.1 


822.3 


1948.8 


34 


86 


3598.8 


2898 


805.2 


1895.0 


3693.1 


305.7 


822.9 


1950.6 


36 


88 


8601.6 


290.3 


805.8 


1896.9 


8696.3 


306.2 


323.5 


1952.5 


38 


40 


8604.7 


290.9 


306.4 


1898.7 


8699.4 


306,7 


824.2 


1954.4 


40 


42 


8607.8 


291.4 


807.0 


1900.6 


8702.6 


307.3 


324.8 


1956.2 


42 


44 


8611.0 


291.9 


807.6 


1902.4 


3705.7 


307.8 


825.4 


1958.1 


44 


46 


8614.1 


292.4 


808.1 


1904.8 


8708.9 


808.3 


326.0 


1960.0 


46 


48 


8617.8 


293.0 


808.7 


1906.1 


8712.1 


308.9 


826 6 


1961.8 


48 


50 


8620.5 


296.6 


809.3 


1908.0 


3715.2 


309.4 


8272 


1963. V 


50 


52 


8623.6 


294.0 


809.9 


1909.8 


3718.4 


809.9 


327.8 


1965.5 


52 


54 


8626.8 


294.5 


810.5 


1911.7 


3721 5 


310.5 


838.4 


1967.4 


54 


66 


8630.0 


295.1 


811.1 


1913.5 


3724.7 


811.0 


329.0 


1969.8 


56 


58 


8633.1 


295.6 


811.7 


1915.4 


3727.8 


811.6 


829.6 


1971.1 


58 


60 


3686.8 


296.1 


312.3 


1917.8 


8731.0 


812.2 


830.2 


1973.0 


60 



Digitized by VjOOQIC 



220 


FUNCTIONS OF A ONE-DEGREE CURVE 




1 



38«» 


1 SO^* 


, 


L. C. 


M. 


E. 


T. 


L. C. 


M 


E. 


T. 


3731,0 


312.2 


330.2 


1978.0 


3825 6 


8287 


848.7 


2029.1 





8 


3734.1 


312 7 


380.8 


1974.9 


8828.6 


329.2 


349 3 


2031.0 


2 


4 


3737.3 


813.8 


381.4 


1976.7 


8831.8 


329.8 


349.9 


2032 9 


4 


« 


3740.4 


318.8 


832.0 


1978.6 


8834.9 


3303 


360.6 


2034.7 


6 


8 


8743.6 


314.4 


832.6 


1980.5 


3838.0 


330.9 


351.2 


2086.6 


8 


10 


3746.7 


314.9 


383.2 


1982.8 


3841.2 


331 5 


851.8 


2038.5 


10 


12 


3749 9 


315.5 


388.8 


1984.2 


3844 3 


332.0 


352.4 


2040 4 


12 


14 


3753 


816 


334.5 


1986.1 


3847.4 


332.6 


358.1 


2042 3 


14 


16 


3756 2 


816.6 


385.1 


1987.9 


3850.6 


833.2 


853.7 


2044.1 


16 


18 


3759.3 


317.1 


335 7 


1989.8 


3a53 7 


333 7 


864.3 


2016.0 


18 


20 


3762 5 


317 7 


8368 


1991.7 


8856 8 


384.3 


364.9 


2047.9 


20 


'Zi, 


3766.6 


318 2 


336 9 


1998-6 


38600 


3;«.9 


355.6 


2049.8 


28 


24 


3768.8 


318.8 


837.6 


1995.4 


3863.1 


335.4 


856.2 


2061.7 


S4 


26 


3771 9 


819.3 


338.1 


1997.8 


3866.2 


336.0 


356.9 


2063.5 


26 


28 


3775.1 


319.9 


388.7 


1999.2 


3869.4 


336.6 


357 5 


2055.4 


28 


80 


3778.2 


8204 


339.4 


2001.0 


3872.5 


8:J7.1 


858 1 


20573 


30 


82 


3781 .4 


821.0 


840.0 


2002.9 


3875 6 


837 7 


a58.8 


2059.2 


32 


34 


3784 5 


321 .5 


340.6 


2004.8 


3878.8 


338.3 


3504 


2061.1 


34 


86 


8787.7 


322.1 


841.2 


2006.6 


3881.9 


338.8 


360.1 


2063.0 


36 


38 


37908 


822.6 


841.8 


2008.6 


3885.0 


889 4 


360.7 


2064.8 


38 


40 


3794.0 


328 2 


342.4 


2010 4 


8888.2 


340.0 


361.3 


2066.7 


40 


\l 


3797 1 


323.7 


348.1 


2012.8 


8891.8 


340.5 


862.0 


2068.6 


42 


44 


3800.3 


824.8 


343.7 


2014.1 


3894 4 


341 1 


362.6 


2070.6 


44 


46 


3803.4 


324 8 


344.3 


2016 


3897.6 


341.7 


363 3 


2072.4 


46 


48 


8806.6 


8-25.4 


344.9 


2017.9 


8900 7 


342 2 


863.9 


2074 2 


48 


50 


3809.7 


325 9 


345.6 


2019.7 


3903.8 


342 8 


364.6 


2076 1 


50 


52 


3812.9 


3265 


346 2 


2021.6 


8907.0 


343 4 


366.2 


2078.0 


52 


54 


3816.0 


3270 


346.8 


2023.5 


8910.1 


348.9 


865.8 


2079.9 


54 


56 


3819.2 


827.6 


847 4 


20254 


3913 2 


344.5 


366.5 


2081.8 


56 


68 


3822.8 


328.1 


348 1 


2027.2 


3916.4 


345 1 


367 1 


2083.7 


58 


60 


3825.5 


328.7 


848.7 


2029.1 


3919 5 


345.6 


367.7 


20866 


60 


/ 


40^ - 1 


410 




L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





8919.5 


845.6 


367.7 


2085.5 


4018.4 


862.9 


887.4 


2142.3 





2 


392-^.6 


846.1 


368.4 


2087 4 


4016.5 


363. 4 


888.1 


2144.2 


8 


4 


3925.8 


346 7 


8G9 


2089.3 


4019.6 


364.0 


888.8 


2146.1 


4 


6 


3928 9 


347.2 


369.7 


2091 2 


4022.7 


364.5 


889.4 


2148.0 


6 


8 


8932 


347 8 


370.3 


2098.1 


4025.9 


865.1 


390 1 


2149.9 


8 


10 


39:35.1 


348.4 


371.0 


2095 


4029.0 


366.6 


390.7 


2161.9 


10 


12 


3988.3 


348.9 


371 6 


2096 9 


4032.1 


366.2 


891.4 


2158.8 


12 


14 


3941.4 


349.5 


372.8 


2098.8 


4035.2 


866.8 


892.1 


2155.7 


14 


16 


3944 5 


350 1 


3T2.9 


2100.7 


4038.3 


367.4 


392.7 


2157.6 


16 


18 


8947.7 


350.7 


373.6 


2102 6 


4041.4 


368.0 


39:3.4 


2159.6 


18 


20 


3950.8 


351.3 


874.3 


2104.5 


4044. 6| 


868.6 


394.1 


8161.4 


20 


2-^ 


3953.9 


361 8 


374.9 


2106.3 


4047.7 


869.2 


894.7 


8168.3 


88 


224 


3957.1 


852.4 


375.6 


2108.2 


4050.8 


369.8 


895.4 


2165.2 


24 


26 


3960.2 


358.0 


376.2 


2110.1 


4053 9 


870.4 


896.1 


2167.1 


26 


28 


8963.3 


353 6 


376.9 


2112 


4057.0 


371.0 


896.8 


8169.0 


28 


30 


3966.4 


354.2 


8775 


2113 9 


4060.1 


371.6 


897.5 


8170 9 


80 


3.i 


3969.6 


854.7 


378.2 


2115 8 


4063.8 


372.2 


898.1 


2172.8 


88 


34 


8972.7 


855.3 


878.8 


2117 7 


4066.4 


372.8 


898.8 


2174.7 


84 


36 


3975.8 


355.9 


879.5 


2119 6 


4069.5 


378.4 


899.5 


2176.6 


86 


88 


8979.0 


356.5 


380 1 


2121.5 


4072.6 


374.0 


400.2 


2178.5 


88 


40 


8982.1 


357.1 


380.8 


2123.4 


4075.7 


874.6 


400.9 


8180.4 


40 


Al 


3985.2 


357 6 


381 4 


2125 3 


4078.8 


875.2 


401.6 


2182.4 


42 


44 


8988.4 


858.2 


382.1 


2127.2 


4082.0 


875.8 


402.2 


2184.8 


44 


46 


3991 5 


858 8 


8828 


2129.1 


4085.1 


876.4 


402.9 


2186.2 


46 


48 


3994.6 


859.4 


883.4 


2131.0 


4088.2 


877.0 


408.6 


2188.1 


48 


00 


3997 7 


360 


384.1 


2132.9 


4091.3 


877.6 


404.8 


2190.0 


60 


52 


4000 9 


360 5 


884.8 


2134.7 


4094.4 


878.2 


404.9 


2191.9 


62 


54 


4004.0 


361 1 


885.4 


8136.6 


4007.5 


378.8 


405.6 


2198.8 


54 


56 


4007 1 


361.7 


886.1 


2138.6 


4100.7 


879.4 


406.3 


8195.7 


66 


58 


4010 3 


862.8 


886 8 


8140 4 


4108.8 
4106.9 


880.0 


407.0 


8197.6 


58 


60 


4013.4 


862.9 


887.4 


8142.8 


880.6 


407.7 


8199.6 


60 



Digitized by VjOOQIC 





FUNCTIONS OF A ONiE-DEGREE CURVE 


^i 


# 


42» 1 


W 




L.C. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 







4106.9 


380.6 


407.7 


2199.6 


4200.1 


398.7 


428.6 


2257.1 





2 


4110.0 


381.2 


408.8 


2201.4 


4203.2 


809.3 


429.3 


2259.0 


8 


4 


4113.1 


381.8 


409.0 


2203.3 


4206.3 


399.9 


430.0 


2261.0 


4 


6 


4116.2 


382.4 


409.7 


2205.3 


4209.4 


400.6 


430.7 


2262.9 


6 


8 


4119.8 


383.0 


410.4 


2207.2 


4212.5 


401.1 


431.4 


2264.8 


8 


10 


4122.4 


883.6 


411.1 


2209.1 


4215.6 


401.7 


432.1 


2266.7 


10 


i2 


4125.5 


884.2 


411.8 


2211.0 


4218.7 


402.4 


432.8 


2268.7 


12 


14 


4128.6 


384.8 


412.5 


2212.9 


4221.8 


403.0 


433.5 


2270.6 


14 


16 


4131.8 


38c 4 


413.2 


2214.9 


4224.9 


403.6 


434.2 


2272.6 


16 


18 


4134.9 


386 


413.9 


2216.8 


4228.0 


404.2 


434.9 


2274.5 


18 


80 


4138.0 


3866 


414.6 


2218.7 


4231.1 


404.8 


435.6 


2276.4 


80 


22 


4141.1 


887.2 


415.8 


2220.6 


4234.2 


405.4 


480.3 


2278.3 


22 


24 


4144.2 


387.8 


416.0 


2222.5 


4237.8 


406.1 


487.0 


2280.2 


24 


26 


4147.3 


388.4 


416.6 


8224.4 


4240.4 


406.7 


437.8 


2282.2 


26 


28 


4150.4 


389.0 


417.3 


2228.4 


4243.5 


407.3 


438.5 


2284.1 


28 


30 


4158.5 


389.6 


418.0 


2228.3 


4246.6 


407.9 


439.2 


8286.0 


80 


32 


4156.6 


390.2 


418.7 


2230.2 


4249.6 


408.6 


439.9 


2288.0 


32 


84 


4159.7 


390.8 


419.4 


2232.1 


4252.7 


409.1 


440.6 


2289.9 


84 


36 


4102 8 


391.4 


420.1 


2334.0 


4255.8 


409.8 


441.4 


2291.8 


86 


38 


4165.9 


392.0 


420.8 


8236.0 


4258.9 


410.4 


442.1 


2293.8 


38 


40 


4169.0 


392.6 


421.5 


2237.9 


4262.0 


411.0 


442.8 


2295.7 


40 


42 


4172.1 


.393.2 


422.2 


2239.8 


4265.1 


411.6 


443.5 


8297.7 


42 


44 


4175.2 


893.8 


422.9 


2241.7 


4268.2 


412 2 


444.2 


2299.6 


44 


46 


4178.4 


894.4 


423.6 


2243.6 


4271.3 


412.8 


445.0 


2301.5 


46 


48 


4181.5 


895.0 


424.8 


2245.6 


4274.4 


413.5 


445.7 


8303.5 


48 


50 


4184.6 


895.6 


425.0 


2247.5 


4277.5 


414.1 


446.4 


2305.4 


60 


52 


4187.7 


896.2 


425.7 


8249.4 


4280.6 


414.7 


447.1 


8307.3 


62 


54 


4190.8 


396.8 


426.4 


2251.3 


4?283.7 


415.3 


447.8 


8309.3 


54 


56 


4193.9 


897 4 


427.1 


2258.8 


4286.8 


415.9 


448.6 


8311.2 


66 


58 


4197.0 


398 


427.8 


22552 


4289.9 


416.6 


449.8 


2313.1 


68 


60 


4>00.1 


898.7 


428.6 


2257.1 


4293.0 


417.2 


450.0 


2815.1 


60 


/ 


440 1 


is- r 




L.C. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 







4393.0 


417.2 


450.0 


2315.1 


438.-).5 


436.2 


472.1 


2373.4 





2 


4296 1 


417.8 


450.7 


2317.0 


4388.6 


436.8 


472.9 


2375.4 


2 


4 


4399.2 


418.4 


451.5 


2319.0 


4391.7 


437.6 


473.6 


2877.3 


4 


6 


4302.2 


419.1 


452.2 


2820.9 


4394.7 


438.1 


474.4 


2379.8 


6 


8 


4305.3 


419.7 


452 9 


2322.8 


4397.8 


438.8 


475.1 


2381.2 


8 


10 


4308.4 


420.3 


453.7 


2324.8 


4400.9 


439.4 


476.9 


2388.2 


10 


VI 


4811.5 


421.0 


454.4 


2826.7 


4404.0 


440.0 


476 6 


2386.2 


12 


14 


4314.6 


421.6 


4r.5.i 


2328.7 


4407.0 


440.7 


477.4 


2387.1 


14 


10 


4317.7 


422.2 


455.9 


2330.6 


4410.1 


441.3 


478.1 


2380.1 


16 


18 


4320.7 


42i.9 


456.6 


2332.6 


4413.2 


442.0 


478.9 


2891.0 


18 


20 


4823.8 


423 5 


457.3 


2:^34.5 


4416.8 


442.6 


479.6 


2393.0 


20 


22 


4326.9 


424.1 


458.1 


2336.4 


4419.8 


443.2 


480.4 


2394.9 


22 


24 


4330.0 


424 8 


458.8 


2338.4 


4422.4 


443.9 


481.1 


2396.9 


24 


26 


4333.1 


4254 


459.5 


2840 3 


4425.5 


444.5 


481.9 


2396.8 


26 


28 


4336.2 


426.0 


460.3 


2342.3 


4428.6 


445.2 


482.6 


2400.8 


28 


80 


4::)89.2 


426.7 


461 


2344.2 


4431.6 


445.8 


483.4 


2402.8 


30 


»i 


4342.3 


427 3 


461.7 


2346.1 


443*. 7 


446.4 


484.2 


2404.7 


82 


34 


4346.4 


427 9 


462.5 


2348.1 


4437.8 


447.1 


484.9 


2406.7 


34 


36 


4348.5 


428 G 


463.2 


23500 


4440.9 


447.7 


485.7 


2406.6 


36 


38 


4851.6 


429.2 


463.9 


2352.0 


4444.0 


448.3 


486.5 


2410.6 


3. 


40 


4354.7 


429.8 


464.7 


23589 


4447.0 


448.9 


487.2 


2412.6 


40 


42 


4867.7 


430.5 


485.4 


2355.9 


4450.1 


449.5 


488.0 


2414.5 


42 


44 


4360.8 


431.1 


466.2 


2a57.8 


4458.2 


450.2 


488.7 


2416.6 


44 


46 


4868.9 


431.7 


466.9 


2369.8 


4456.3 


4.50.8 


489.5 


2418.5 


46 


48 


4867.0 


432.4 


467.7 


8361.7 


4459.3 


451.5 


490. :i 


8420.4 


48 


50 


4370.1 


433.0 


468 4 


2863.7 


4462.4 


452.1 


491.0 


2422.4 


50 


52 


4373.2 


4iS».6 


469.1 


2366.6 


4465.5 


452.7 


491.8 


2424.4 


62 


54 


4376 2 


434 3 


469.9 


2367.6 


4468.6 


458 4 


492.5 


2426.3 


54 


56 


48*^. 3 


434.9 


4T0.6 


2869.5 


4471.6 


454.1 


493.3 


2428.8 


66 


58 


4882.4 


435 b 


471.4 


28^1.5 


4474.7 


414.8 


494.1 


2480.2 


68 


60 


4886.5 


486.2 


472.1 


2873.4 


4477.8 


466.5 


494.8 


2482.2 


60 



Digitized by VjOOQIC 



222 


FUNCTIONS OP A ONE-DEGREE CURVE 




/ 


46- 1 


47* 


/ 



L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





4477.8 


455.5 


494.8 


2488.2 


4569.7 


475.2 


518.8 


2491.5 


2 


4480.9 


456.1 


495.6 


2484.2 


4572.7 


475.9 


619.0 


2493.4 


2 


4 


4483.9 


456.8 


496.5 


2486.1 


4575.8 


476.5 


519.8 


8495.4 


4 


6 


4487.0 


457.4 


497.2 


2438.1 


4578.8 


477.2 


520.6 


2497.4 


6 


8. 


4490.0 


458.1 


497.9 


2440.1 


4581.9 


477.8 


521.4 


2499.4 


8 


10 


4493.1 


458.7 


496.7 


2442.1 


4584.9 


478.5 


522.2 


2501.4 


10 


13 


4496.2 


459.4 


499.5 


2444.0 


4588.0 


479.2 


528.0 


2508.4 


18 


14 


4499.2 


460.0 


500.8 


2446.0 


4591 .0 


479.8 


523.6 


2505.4 


14 


16 


4502.8 


460.7 


501.0 


2446.0 


4594.1 


480.5 


524.6 


2,')07.8 


16 


18 


4505.4 


461.8 


501.8 


2449.9 


4597.1 


481.1 


525.4 


2509.8 


18 


20 


4508.4 


462.0 


502.6 


2451.9 


4600.2 


481.7 


526.2 


8511.8 


80 


22 


4511.5 


468.7 


503.4 


2453.9 


4603.2 


488.3 


527.0 


2513.8 


28 


24 


4514.6 


463.3 


504.1 


2465.9 


4606.3 


483.0 


527.8 


2515.8 


84 


26 


4517.6 


464.0 


504.9 


2457.8 


4609.3 


483.7 


628.6 


2517.8 


86 


28 


4520.7 


464.6 


505.7 


2469.8 


4612.4 


484.8 


529.4 


2519.8 


88 


80 


4528.7 


465.8 


506.5 


2461.8 


4615.4 


485.0 


630.2 


8521.2 


80 


32 


4526.3 


466.0 


507.3 


2463.8 


4618.5 


485.7 


531.0 


8523.2 


82 


34 


4529.9 


466.6 


506.0 


2465.7 


4621.5 


486.8 


531.8 


2525.2 


84 


36 


4532.9 


467.3 


508.8 


2467.7 


4624.6 


487.0 


5:32.6 


8527.2 


86 


88 


4536.0 


467.9 


509.6 


2469.7 


4627.6 


467.7- 


533.4 


8589.8 


88 


40 


4539.1 


468.6 


510.4 


8471.7 


4680.7 


488.4 


534.8 


8681.8 


40 


42 


4542.1 


469.3 


511.1 


2473.6 


4683.7 


489.1 


535.0 


2538.2 


48 


44 


4546.2 


469.9 


511.9 


2475.6 


4636.8 


489.6 


535.8 


2535.2 


44 


46 


4548.2 


470.6 


612.7 


2477.6 


4639.8 


490.6 


536.6 


2587.2 


46 


48 


4551.8 


471.2 


513.5 


2479.6 


4642.9 


491.2 


537.4 


2589.2 


48 


50 


4554.4 


471.9 


514.3 


2481.6 


4645.9 


491.9 


5as.8 


2541.2 


50 


52 


4557.4 


472.6 


515.1 


2483.5 


4649.0 


492.6 


539.0 


2548.1 


58 


54 


4560.6 


473.2 


515.9 


2485.5 


4652.0 


493.3 


539.6 


2545.1 


54 


56 


4563.6 


473.9 


516.7 


2487.6 


4655.1 


494.0 


540.6 


2547.1 


66 


58 


4566.6 


474.5 


517.5 


2489.5 


46,58.1 


494.7 


541.4 


2549.1 


66 


60 


4569.7 


475.2 


518.3 


2491.5 


4661.2 


495.4 


542.3 


2.551.1 


60 


/ 



48« .1 


4»« 1 


r 


L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 


4661.2 


495.4 


542.8 


2551.1 


4762.8 


515.9 


567.0 


2611.3 





2 


4664.2 


496.0 


543.1 


2558.1 


4755.3 


516.5 


567.8 


8613.3 


8 


4 


4667.8 


496.7 


643.9 


2555.1 


4758.4 


517.2 


568.7 


2615.3 


4 


6 


4670.3 


497.4 


544.7 


2557.1 


4761.4 


517.9 


569.6 


2617.3 


6 


8 


4678.8 


498.1 


545.5 


2559.1 


4764.4 


518.6 


570.3 


2619.3 


8 


10 


4676.4 


498.8 


546.4 


2561.1 


4767.4 


519.3 


571.2 


2621.4 


10 


12 


4679.4 


499.4 


547.2 


8563.1 


4770.5 


520.0 


578.0 


2683.4 


12 


14 


4682.5 


600.1 


548.0 


2565.1 


4773.5 


520.7 


572.8 


2629'.4 


14 


16 


4685.5 


500.8 


548.8 


2567.1 


4776.5 


521.4 


573.7 


2627.4 


16 


18 


4688.5 


501.5 


549.6 


2569.1 


4779.6 


522.1 


574.5 


2629.4 


18 


20 


4691.6 


502.2 


550.5 


26n.l 


4782.6 


522.6 


575.3 


2631.4 


80 


22 


4694.6 


502.8 


551.8 


2573.1 


4785.6 


523.5 


576.8 


2683.5 


82 


24 


4697.6 


503.5 


552.1 


2576.1 


4788.7 


524.2 


577.0 


2685.5 


84 


26 


4700.7 


504.2 


552.9 


2677.1 


4791.7 


524.9 


577.9 


2687.5 


86 


28 


4703.7 


504.9 


553.7 


2579.1 


4794.7 


525.6 


578.7 


2689.5 


88 


80 


4706.7 


505.6 


564.6 


2581.1 


4797.7 


526.3 


579.6 


2641.5 


80 


82 


4709.8 


506.2 


555.4 


2583.1 


4800.8 


527.0 


580.4 


8648.6 


82 


84 


4712.8 


506.9 


556.2 


2585.1 


4803.8 


527.7 


581.3 


8645.6 


84 


86 


4715.9 


507.6 


557,0 


2587.2 


4806.8 


528.4 


588.1 


8647.6 


86 


88 


4718.9 


506.8 


567.8 


2589.2 


4809.9 


589.1 


568.0 


2649.6 


88 


40 


4721.9 


509.0 


658.7 


2591.2 


4812.9 


529.8 


688.6 


8651.6 


40 


42 


4725.0 


509.6 


559.5 


2593.2 


4815.9 


580.5 


584.7 


2658.7 


48 


44 


4728.0 


510.3 


560.3 


2595.2 


4819.0 


531.8 


586.5 


2655.7 


44 


46 


4731.0 


511.0 


561.2 


8597.2 


4882.0 


631.9 


586.4 


8657.7 


46 


48 


4734.1 


511.7 


562.0 


2599.2 


4825.0 


532.6 


587.8 


2659.7 


48 


50 


4T87.t 


512.4 


562.8 


2601.2 


4828.0 


588.S 


588.1 


2661.8 


50 


53 


4740.2 


613.1 


563.7 


2603.2 


4881.1 


584.0 


5b8.9 


2663.8 


58 


64 


4748.2 


513.8 


564.5 


8605.2 


4834.1 


534.7 


589.6 


8665.6 


54 


66 


4746.2 


514.5 


565.3 


2607.2 


4837.1 


685.4 


590.6 


8667.8 


66 


58 


4749.3 


515.2 


566.2 


2609.3 


4840.2 


586.1 


591.5 


8669.9 


66 


60 


4758.8 


615.9 


667.0 


2611.3 


484S.2 


686.8 


598.4 


8671.9 


60 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEGREE CURVE 


223 


• / 


60' 


SI* 


/ 


L. C. 


M. 


E. 


T. 


L.C. M. 


E. 


T. 





4843.2 


586.8 


692.4 


2671.9 


4933 6 558.2 


6i8.5 


2738.0 





2 


4846.2 


587.5 


593.2 


26789 


4936.6 558.9 


619.8 


2735.1 


2 


4 


4849.2 


588.2 


594.1 


2676.0 


4939.6 659.7 


6;20.2 


2787.1 


4 


6 


4852.2 


538.9 


594.9 


2676.0 


4942.6 560.4 


621.1 


2730.8 


6 


8 


4855.2 


539.6 


506.8 


2680.0 


4945.6 561.1 


682.0 


2741.8 


6 


10 


4858.3 


540.8 


596.7 


2682.1 


4948.6 561.8 


622.9 


2748.8 


10 


18 


4861.8 


641.0 


597.5 


2684.1 


4951.6 562.5 


628.7 


2745.8 


12 


14 


4864.3 


641.7 


596.4 


2686.1 


4954.6 563.3 


624.6 


2747.4 


14 


16 


4867.3 


542.4 


599.3 


2688.2 


4957.6 564.0 


625.5 


2749.4 


16 


18 


4870.8 


548.1 


600.1 


2690.2 


4960.6 664.7 


686.4 


2751.5 


18 


80 


4878.8 


548.9 


601.0 


2692.8 


4968.6 566.4 


627.8 


2758.5 


80 


82 


4876.8 


544.6 


601.9 


2694.8 


4966.6 666.2 


688.8 


.2755.6 


82 


24 


4879.4 


545.8 


602.7 


2696.8 


4969.6 666.9 


629.9 


2757.7 


24 


26 


4862.4 


546.0 


603.6 


2698.4 


4972.6 567.6 


630.0 


2759.7 


86 


28 


4886.4 


646.7 


604.5 


2700.4 


4975.6 668.3 


680.9 


2761.8 


86 


90 


4888.4 


547.4 


605.3 


2702.4 


4978.6 669.1 


681.6 


2768.8 


80 


32 


4891.4 


548.1 


606.2 


2704.6 


4961.6 569.8 


682.7 


2765.9 


88 


84 


4894.4 


548.8 


607.0 


2706.5 


4964.6 570.5 


638.6 


2767.9 


84 


88 


4897.4 


649.5 


607.9 


2708.6 


4987.7 671.2 


634.5 


2770.0 


86 


38 


4900.4 


550.2 


60S.8 


2710.6 


4990.7 572.0 


685.3 


2772.0 


86 


40 


4908.5 


551.0 


609.7 


2712.6 


4998.7 572.7 


636.2 


2774.1 


40 


42 


4906.6 


551.7 


610.5 


2714.7 


4996.7 573.4 


687.1 


2776.2 


48 


44 


4909.5 


552.4 


611.4 


2716.7 


4999.7 574.1 


688.0 


2778.8 


44 


46 


4912.5 


553.1 


612.3 


2718.8 


5002.7 574.9 


638.0 


2780.3 


46 


48 


4915.5 


m.s 


613.2 


2720.8 


6006.7 575.6 


639.8 


2782.3 


48 


50 


4918.5 


564.6 


614.1 


2722.8 


5008.7 676.3 


640.7 


2784.4 


60 


52 


4921.5 


555.2 


614.9 


2724.9 


5011.7 677.0 


641.6 


2786.4 


62 


54 


4924.6 


555.9 


615.8 


2726.9 


5014.7 577.8 


642.5 


2788.5 


54 


56 


4927.6 


556.6 


616.7 


2729.0 


6017.7 678.5 


643.4 


2790.6 


56 


58 


4930.6 


557.4 


617.6 


2731.0 


6020.7 679.2 


644.3 


2792.6 


58 


60 


4933.6 


558.2 


616.5 


2788.0 


5083.7 579.9 


645.8 


2794.7 


60 


/ 


620 , 


68«> 


/ 


L. 0. 


M. 


E. 


T. 


L. C. M. 


E. 


T. 





5023.7 


579.9 


645.2 


2794.7 


5118.5 608.0 


678.7 


2a56.9 





2 


5026.7 


580.6 


646.1 


2796.8 


5116.5 60:>.8 


673.7 


2856.0 


8 


4 


6029.7 


581.8 


647.0 


2798.8 


5119.4 603.5 


674.6 


2861.0 


4 


6 


6032.7 


582.1 


647.9 


2800.9 


5122.4 604.8 


675.5 


8863.1 


6 


8 


6085.7 


582.8 


648.9 


2808.0 


6126.4 605.0 


676.4 


8665.2 


8 


10 


5088.7 


583.5 


649.8 


2805.0 


5128.4 605.8 


677,4 


2867.8 


10 


12 


5041.7 


684.3 


650.7 


2807.1 


5131.3 606.5 


678.3 


2869.4 


18 


14 


6044.7 


585.0 


651.6 


2809.2 


5184.3 607.3 


679.2 


2871.5 


14 


16 


5047.7 


585.7 


652.5 


2811.2 


5187.8 608.0 


680.2 


2873.6 


16 


18 


5050.7 


586.5 


653.4 


2813.3 


5140.8 608.8 


681.1 


2875.6 


16 


20 


5068.0 


587.2 


654.3 


2815.4 


6143.2 609.5 


682.0 


2877.7 


80 


22 


5056.6 


587.9 


655.2 


2817.4 


5146.2 610.8 


688.0 


2879.8 


88 


24 


5059.6 


588.7 


656.2 


2819.5 


5149.8 611.0 


683.9 


2881.9 


24 


26 


6062.6 


589.4 


657.1 


28-,'1.6 


5152.1 611.8 


684.9 


2884.0 


26 


28 


6065.6 


590.1 


658.0 


2823.6 


5155.1 618.5 


685.8 


2886.1 


28 


80 


5068.6 


590.0 


658.9 


2825.7 


61.58.1 613.8 


686.7 


2888.1 


30 


32 


6071.6 


591.6 


659.8 


2827.8 


5161.1 614.0 


687.7 


2890.2 


82 


84 


6074.6 


592.8 


660.7 


2829.8 


5164.0 614.8 


688.6 


2892.3 


34 


36 


5077.6 


593.1 


661.6 


2831.9 


6167,0 615.5 


689.6 


2894.4 


36 


38 


6080.6 


593.8 


662.6 


2884.0 


5170.0 616.3 


690.5 


2896.5 


36 


40 


6088.6 


594.5 


663.5 


2636.1 


5173.0 617.0 


691 .5 


2698.6 


40 


42 


5086.6 


595.8 


664.4 


2838.2 


5175.9 617.8 


692.4 


2900.7 


42 


44 


5069.6 


596.0 


665.3 


2840.2 


5178.9 61S.5 


693.4 


2902.6 


44 


46 


6092.6 


596.7 


666.2 


2842.3 


51H1.9 619.3 


694.3 


2904.0 


46 


48 


5095.6 


597.5 


667.2 


2844.4 


5184.9 620.1 


695.3 


2907.0 


48 


60 


5098.6 


598.2 


668.1 


2846.5 


51b7.8 620.8 


696.2 


2909.1 


50 


62 


5101.6 


598.9 


669.0 


2848.5 


5190.8 621.5 


697.1 


2911.2 


52 


54 


5104.6 


599.7 


669.9 


2850.6 


5198.8 622.3 


698.1 


2013.3 


54 


56 


6107.6 


600.4 


670.9 


2852.7 


5196.7 623.0 


699.0 


2915.4 


56 


66 


6110.6 


601.2 


671.8 


2854.8 


5199.7 6ti3.8 


700.0 


2917.5 


58 


60 


5118.5 


609.0 


672.7 


2856.9 


,'S'.i0J.7 624.6 


700.9 


2919.5 


60 



Digitized by VjOOQIC 



224 


FUNCTIONS OF A ONE-DEGREE CURVE 






64- 


660 




t 


L.C. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 







5202.7 


624.6 


700.9 


2919.5 


5291.7 


647.4 


7-29.9 


2982.8 





2 


5206.7 


625.4 


701.9 


2921.6 


6294.6 


648.1 


730.9 


2984.9 


2 


4 


5206.6 


626.1 


702.8 


2928.8 


6297.6 


648.9 


731.9 


2987.1 


4 


6 


5211.6 


626.9 


703.8 


2925.0 


5300.5 


649.6 


732.9 


2989.2 


6 


8 


5214.6 


627.6 


704.8 


2928.0 


6803.5 


650.4 


783.8 


2991.3 


8 


10 


5217.6 


628.4 


705.7 


2930.1 


6306.4 


651.2 


734.8 


2993.4 


10 


12 


5220.6 


629.2 


706.7 


2932.2 


6309.4 


652.0 


735.8 


2996 5 


12 


14 


5228.5 


629.9 


707.7 


2934.8 


5312.8 


652.7 


736.8 


2997.7 


14 


16 


5226.4 


630.7 


708.6 


2936.4 


5315.3 


658.5 


737.8 


299U.8 


16 


18 


6229.4 


631.4 


709.6 


2938.6 


5818.2 


654.3 


738.7 


3001.9 


18 


20 


5232.4 


632.3 


710.6 


2940.6 


5321.2 


655.1 


739.7 


3004.0 


20 


22 


6235.8 


633.0 


711.5 


2942.7 


5324.1 


655.8 


740.7 


3008.2 


22 


24 


6238.8 


633.7 


712.6 


2944.8 


5327.1 


656.6 


741.7 


3008.3 


24 


26 


5241.3 


634.5 


718.4 


2946.9 


5330.0 


667.4 


742.7 


3010.4 


26 


28 


5244.2 


636.2 


714.4 


2949.0 


5333.0 


658.2 


743.7 


3012.5 


28 


30 


6247.2 


636.0 


715.3 


2951.1 


5335.9 


658.9 


744.7 


3014.7 


30 


32 


6250.2 


636.8 


716.3 


2953.2 


6338.8 


659.7 


745.7 


3016.8 


32 


34 


5253.1 


637.6 


717.8 


2955.8 


6341.8 


660.5 


746.7 


3018.9 


34 


36 


5256.1 


638.3 


718.2 


2957.5 


5844.7 


661.3 


747.7 


3021.1 


36 


38 


5259.1 


639.0 


719.2 


2959.6 


6347.7 


662.0 


748.7 


3023.2 


38 


40 


5262.0 


639.8 


720.2 


2961.7 


6850.6 


662.8 


749.7 


3025.3 


40 


42 


6265.0 


640.6 


721.1 


2963.8 


5353.6 


663 6 


750.7 


8027.5 


42 


44 


5268.0 


641.3 


722.1 


2965.9 


5356.5 


664.4 


751.7 


3029.6 


44 


46 


6270.9 


642.1 


723.1 


2968.0 


5359.5 


665.1 


752.6 


3031.7 


46 


48 


5273.9 


642.8 


724.1 


2970.1 


5362.4 


665.9 


753.6 


3033.8 


48 


50 


5276.9 


643.6 


725.0 


2972.2 


5365.4 


666.7 


754.6 


3036.0 


60 


62 


5279.8 


644.4 


726.0 


2974.4 


5368.3 


667.5 


755.6 


8038.1 


62 . 


64 


5282.8 


645.1 


727.0 


2976.5 


5371.3 


668.3 


756.6 


3040.2 


54 


66 


5285.8 


645.9 


728.0 


2978.6 


5374.2 


669.1 


757.6 


8042.4 


66 


58 


5288.7 


646.6 


729.0 


2980.7 


5377.2 


669.9 


758.6 


3044.6 


68 


60 


5291.7 


647.4 


729.9 


2982.8 


5380.1 


670.7 


759.6 


3046.6 


60 


/ 


56«> 1 




6 


7° 




/ 


L. C. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 





5380.1 


670.7 


759.6 


3046.6 


5468.2 


694.4 


790.2 


8111.1 





2 


5383.0 


671.4 


760.6 


8048.8 


6471.1 


695.2 


791.2 


3118.3 


2 


4 


5386.0 


672.2 


761.6 


3050.9 


5474.0 


696.0 


792.2 


8115.4 


4 


6 


5388.9 


672.9 


762.7 


3058.1 


5477.0 


696.8 


793.3 


3117.6 


6 


8 


5391.8 


673.7 


763.7 


8055.2 


5479.9 


697.6 


794.3 


3119.7 


8 


10 


5394.8 


674.4 


764.7 


3057.4 


5482.8 


698.4 


795.8 


3121.9 


10 


Vi 


5;i97.7 


675.2 


765.7 


3059.5 


5485.7 


699.2 


796.3 


8124.1 


12 


14 


5400.7 


676.0 


706. 7 


8061.6 


5488.7 


700.0 


797.4 


8126.2 


14 


10 


5403.6 


676.8 


7G7.7 


3063.8 


5491.6 


700.8 


798.4 


8128.4 


16 


18 


5100.5 


677.6 


7G8.7 


3065.9 


5494.5 


701.6 


799.4 


8130.6 


18 


ao 


5409.5 


678.4 


769.7 


3068.1 


6497.4 


702.4 


800.6 


8132.7 


20 


22 


5412.4 


679.2 


770.8 


3070.2 


5500.8 


703.2 


801.6 


8134.9 


22 


24 


5415.3 


680.0 


771.8 


3072.4 


6503.3 


704.0 


802.6 


8137.0 


24 


26 


5418.8 


680.8 


772.8 


8074.6 


6506.2 


704.8 


803.6 


8139.2 


26 


28 


54:21.2 


681.6 


773.8 


3076.6 


6509.1 


705.6 


804.7 


8141.4 


28 


SO 


5424.1 


682.4 


774.8 


3078.8 


5512.0 


706.4 


805.7 


8143.5 


SO 


Z'l 


6427.1 


683.2 


775.8 


8080.9 


5515.0 


707.2 


806.8 


3145.7 


82 


34 


5430.0 


684.0 


776.8 


8083.1 


6517.9 


708.0 


807.8 


3147.9 


84 


36 


5438.0 


684.8 


777.8 


3085.2 


6520.8 


708.8 


808.8 


Sl'iO.O 


36 


38 


5435.9 


685.6 


778.9 


8087.4 


6523.7 


709.6 


809.9 


3152.2 


88 


40 


6438.8 


686.4 


779.9 


8089.6 


5526.7 


710.4 


810.9 


3154.4 


40 


42 


5441.8 


687.2 


780.9 


8091.7 


5529.6 


711.8 


812.0 


3156.6 


42 


44 


5444.7 


688.0 


781.9 


8093.9 


6582.5 


712.0 


818.0 


3158.7 


44 


46 


6447.6 


688.8 


783.0 


3096.0 


6535.4 


712.8 


814.1 


3160.9 


46 


48 


6450.6 


689.6 


784.0 


3098.2 


5538.4 


713.6 


815.1 


3163.1 


48 


50 


5458.5 


690.4 


785.0 


8100.3 


5541.3 


714.4 


816.2 


8165.8 


50 


62 


5456.5 


691.2 


786.0 


8102.5 


6544.2 


715.2 


817.2 


3167.4 


62 


54 


5459.4 


692.0 


787.1 


8104.6 


5547.1 


716.0 


818.8 


3169.6 


64 


66 


6462.3 


692.8 


788.1 


3106.8 


5550.0 


716.8 


819.3 


8171.8 


56 


58 


6465.3 


693.6 


789.1 


3108.9 


5553.0 


717.6 


820.4 


8174.0 


58 


60 


6468.2 


694.4 


790.2 


8111.1 


5555.9 


718.4 


821.4 


3170.1 


60 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEGREE CURVE 


225 




680 


69* 




f 


L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 


/ 





5555.9 


718.4 


821.4 


3176.1 


6643.1 


742.8 


853.5 


3241.9. 





2 


5558.8 


719.2 


822.5 


3178.3 


6646.0 


743.6 


854.6 


8244.1 


2 


4 


5561.7 


720.0 


823.5 


3180.6 


6648.9 


744 4 


855.7 


3246.3 


4 


6 


5564.6 


720.8 


824.6 


3182.7 


5651.8 


745.3 


856.8 


8248.5 


6 


e 


6567.5 


721.6 


825.7 


3184.9 


6664.7 


746.1 


857.9 


8250.7 


8 


10 


6670.4 


722.4 


826.7 


3187.1 


6657.6 


746.9 


859.0 


3252.9 


10 


12 


5573.3 


728.2 


827.8 


3189.2 


6660.6 


747.7 


860.0 


3255.1 


12 


14 


5576.2 


724.0 


828.9 


8191.4 


6663.4 


748.6 


861.1 


3257.8 


14 


16 


6679.2 


724.8 


829.9 


3193.6 


6666.8 


749.4 


862.2 


3259.6 


16 


18 


5582.1 


726.6 


831.0 


8195.8 


6669.2 


760.2 


863.3 


3261.7 


18 


20 


6585.0 


726.5 


882.1 


8198.0 


6672.1 


751.1 


864.4 


3263.0 


20 


22 


6587.9 


727.8 


883.1 


8200.2 


5675.0 


751.9 


865.6 


3266.1 


22 


24 


6590.8 


728.1 


834.2 


3202.4 


5677.9 


752.7 


866.6 


3268 3 


24 


26 


6593.7 


728.9 


835.3 


8204.6 


5680.8 


753.6 


867.7 


3270.6 


26 


28 


6596.6 


729.7 


836.3 


3206.7 


6683.7 


764.4 


868.8 


3272.7 


28 


80 


6599.6 


730 5 


837.4 


3208.9 


5686 5 


755.2 


869.0 


8274.9 


30 


32 


6602.4 


731.8 


838.4 


3211.1 


5689.4 


756.0 


871.0 


3277.1 


83 


34 


5605.3 


732.1 


83ft. 6 


3213.8 


5692.3 


756.9 


872.1 


3279.4 


34 


36 


5608.2 


782.9 


840.6 


3215.5 


5695.2 


757.7 


873.2 


3281 6 


36 


. 88 


6611.1 


733.7 


841.6 


3217.7 


6698 1 


768.6 


874.3 


3288.8 


38 


40 


6614.0 


734.6 


8427 


8219.9 


5701.0 


7594 


876.4 


8286.0 


40 


42 


6616.9 


735.4 


843.8 


3222.1 


6703.9 


760.2 


876.5 


3288.2 


42 


44 


6619.8 


736.2 


844.9 


3224.3 


5706.8 


761 


877.6 


3290.6 


44 


46 


6622.8 


737.0 


846.0 


3226.6 


5709 7 


761.9 


878.7 


3292.7 


46 


48 


6625.7 


737.8 


847.0 


3228 7 


5712 6 


762.7 


879.8 


3294.9 


48 


50 


6628.6 


788.6 


848.1 


8230 9 


6715.5 


763.5 


880.9 


3297.1 


60 


52 


5631.5 


739.4 


849.2 


8238.1 


5718.4 


764.4 


882.0 


3299.3 


52 


• 54 


5634.4 


740;2 


&50.8 


8235.3 


6721.3 


765.2 


883.1 


3301.5 


64 


56 


6637.3 


741 


851.4 


3237.6 


6724.2 


766.0 


884.2 


8303.8 


56 


58 


6040. 2 


741.9 


85-2.5 


3239.7 


6727.1 


766.8 


885.3 


3806.0 


68 


60 


5643.1 


74-^.8 


853.5 


8241.9 


5730.0 


767.7 


886.4 


3308.2 


60 


/ 


60» 1 


6P 


/ 


L. C. 


M, 


E. 


T. 


L. C. 


M. 


E. 


T. 





6730.0 


767.7 


886.4 


8808.2 


6816.4 


792.9 


920.2 


3375.2 





2 


6732.9 


768.6 


887.5 


3310.4 


6819.3 


793.7 


921.4 


3377.4 


2 


4 


6735.8 


769.4 


888.7 


8312.7 


5822.1 


794.6 


922.5 


8379.7 


4 


6 


6738.6 


770.2 


889.8 


8314.0 


6825.0 


795.4 


923 6 


3381.9 


6 


8 


5741.5 


771.1 


890.9 


3317.1 


6827.9 


796.3 


924.8 


3384.2 


8 


10 


5744 4 


771.9 


892.0 


3819.8 


6830.7 


797.1 


925.0 


SS86.4 


10 


12 


5747.3 


772.7 


893.1 


3321.6 


5833.6 


798.0 


927.1 


3388.7 


12 


14 


6750.2 


773.6 


894.8 


3323.8 


6836.5 


798.8 


928.2 


8890.9 


14 


16 


5753.0 


774.4 


895.4 


3326.0 


5839.3 


799.7 


929.3 


3393.2 


16 


18 


6755.9 


775.3 


896.5 


8328.8 


5842.2 


800.5 


930.5 


3395.4 


18 


20 


6758.8 


776.1 


897.6 


3330.6 


6846.1 


801.4 


931.6 


3397.7 


20 


22 


6761.7 


776.9 


898.8 


8882.7 


6847.9 


802.2 


932.8 


3399.9 


22 


24 


6764.6 


777.8 


899.9 


3334.9 


6850.8 


803.1 


933.9 


3402.2 


24 


26 


5767.4 


778.6 


901.0 


8:337.2 


5853.7 


803.9 


935.1 


3404.4 


26 


28 


6770.3 


779.5 


902.1 


3839.4 


5856.6 


804.8 


938.3 


3406.7 


28 


30 


5773.2 


780.8 


903.2 


8341.6 


5859.4 


805.6 


937.4 


3408.9 


30 


32 


6776.1 


781.1 


904.4 


3343.9 


6662.3 


806.5 


938.6 


3411.2 


32 


84 


5779.0 


782.0 


905.5 


3346.1 


5865.1 


807.3 


939.7 


3413.5 


34 


36 


5781.8 


782.8 


906.6 


3348.8 


5868.0 


808.2 


940.9 


3416.7 


36 


38 


6784.7 


783.7 


907.7 


3350.6 


5870.9 


809.0 


942.1 


3418.0 


38 


40 


6787.6 


784.5 


908.8 


3852.8 


5873.7 


809.9 


943.2 


8420.8 


40 


42 


6790.6 


785.3 


910.0 


Sa'iS.O 


5876.6 


810.7 


944.4 


3422.5 


42 


44 


6793.4 


786.2 


911.1 


3357.3 


5879.5 


811.6 


946.6 


3424.8 


44 


46 


5796.2 


787.0 


912.3 


8359.5 


5882.8 


812.4 


946.7 


3427.1 


46 


48 


5799.1 


787.9 


913.4 


3361.8 


5885.2 


813.3 


947.8 


8429.3 


48 


50 


5802.0 


788.7 


914.5 


3364.0 


5888.1 


814.1 


949.0 


3481.6 


50 


52 


5804.9 


789.5 


915.7 


3366.2 


5890.9 


815.0 


950.2 


3433.9 


53 


54 


5807.8 


790.4 


916.8 


3368.5 


5893.8 


815.8 


951.3 


3436.1 


64 


56 


5810.6 


791.2 


918.0 


3370.7 


.5896.7 


816.7 


952.5 


3438.4 


56 


58 


5813.6 


792.1 


919.1 


3378.0 


5899.6 


817.5 


953.6 


3440.7 


68 


60 


6816.4 


792.9 


920.2 


3375.2 


5902.4 


818.4 


954.8 


3442.9 


60 1 



Digitized by VjOOQIC 



226 


FUNCTIONS OF A ONE-DEGREE CURVE 




i 


68* 1 


68* 


/ 


L.C. 


M^ 


E. 


T. 


L. C. 


M. 


B. T. 





5902.4 


818.4 


954.8 


8442.9 


5987.8 


844.4 


990.8 8511.8 





s 


6805.2 


819.8 


956.0 


3445.2 


5990.6 


845.8 


991.5 8513.6 


2 


4 


6906.1 


820.1 


957.2 


8447.5 


5993.5 


846.2 


992.7 8515.9 


4 


6 


5910.9 


821.0 


9583 


8449.7 


5996.8 


847.1 


993.9 8518.2 


6 


8 


5913.8 


821.8 


959.5 


3452.0 


5999.1 


847.9 


995.1 8520.5 


8 


10 


5916.6 


822.7 


960.7 


8454.8 


60020 


848.8 


996.3 8522.8 


10 


12 


5919.5 


823.6 


961.9 


8456.6 


6004.8 


849.7 


997.5 3525.1 


12 


14 


6922.8 


824.4 


963.0 


8458.8 


6007.7 


850.6 


998.7 3527.4 


14 


16 


6925.2 


825.8 


904.2 


8461.1 


6010.5 


851.4 


999.9 3529.7 


16 


18 


5928.0 


826.1 


965.4 


3468.4 


6013.3 


862.8 


1001.1 8532.0 


18 


ao 


5930.9 


827.0 


966.6 


3465.7 


6016.2 


853.2 


1002.3 8534.8 


20 


22 


5933.7 


827.9 


967.8 


3167.9 


6019.0 


864.1 


1003.5 8536.6 


22 


24 


5936.6 


828.7 


968.9 


8470.2 


6021.8 


854.9 


1004.7 3538.9 


24 


26 


5939.4 


829.6 


970.1 


3472.5 


6024.7 


855.8 


1005.9 8541.2 


26 


28 


5942.8 


830.4 


971.8 


3474.7 


6027.6 


856.7 


1007.1 8543.5 


28 


30 


5945.1 


881.3 


972.5 


3477.0 


6030.3 


857.6 


1008.4 8545.8 


30 


82 


6947.9 


832.2 


978.6 


3479.3 


6033.2 


858.4 


1009.6 8548.1 


32 


34 


5950. R 


883.0 


974.8 


3481.6 


6*36.0 


859.8 


1010.8 8550.4 


34 


36 


5953.6 


833.9 


976.0 


3483.9 


6038.9 


860.2 


1012.0 8562.7 


36 


:i8 


5956.6 


834.7 


977.2 


3486.2 


6041.7 


861.1 


1013.2 8556.0 


38 


40 


5959.3 


835.6 


978.4 


3488.5 


6044.5 


861.9 


1014.5 8557.3 


40 


42 


5962.2 


836.5 


979.6 


8490.7 


6047.4 


862.8 


1015.7 8559.6 


42 


44 


5965.0 


837.4 


980.8 


3493.0 


6050.2 


868.7 


1016.9 8562.0 


44 


46 


5967.9 


888.3 


982.0 


3495.3 


6053.0 


864.6 


1018.1 3564.8 


46 


48 


5970.7 


839.1 


983.2 


3497.6 


6065.9 


865.4 


1019.8 8566.6 


48 


50 


5978.6 


840.0 


984.4 


3499.9 


6058.7 


866.3 


1020.6 8568.9 


50 


52 


5976.4 


840.9 


985.5 


3502.2 


6061.6 


867.2 


1021.8 3571.2 


52 


54 


5979.8 


841.7 


986.7 


3504.5 


6064.4 


868.1 


1028.0 3673.5 


54 


56 


5962.1 


842.6 


987.9 


3506.8 


6067.2 


868.9 


1024.2 3575.8 


56 


58 


5985.0 


848.6 


989.1 


3609.0 


6070.1 


869.8 


1025.4 8578.1 


58 


60 


5987.8 


844.4 


990.3 


3511.3 


6072.9 


870.7 


1026.7 8580.4 


60 


/ 


64o: , 


660 


/ 


L.C. 


M. 


E. 


T. 


L. C. 


M. 


B. T. 





6072.9 


870.7 


1026.7 


8580.4 


6157.6 


897.8 


1064.0 8650.4 





2 


6075.7 


871.6 


1027.9 


3582.8 


6160.3 


898.2 


1065.2 8662.8 


8 


4 


6078.6 


872.4 


1029.2 


8585.1 


6163.1 


899.1 


1066.5 8655.1 


4 


6 


6081.4 


878.3 


1030.4 


8587.4 


6165.9 


900.0 


1067.7 8657.5 


6 


8 


6084.2 


874.2 


1031.7 


8689.7 


6168.7 


900.9 


1069.0 8659.8 


8 


10 


6087.0 


876.1 


1032.9 


8692.1 


6171.5 


901.8 


1070.2 8662.2 


10 


18 


6089.8 


875.9 


1084.1 


8594.4 


6174.3 


902.7 


1071.5 8664.5 


12 


14 


609-3.6 


876.8 


1085.4 


8596.7 


6ITr.l 


903.6 


1072.7 8666.9 


14 


16 


6095.5 


8r?.7 


1036.6 


3599.1 


6179.9 


904.5 


1074.0 8669.2 


16 


18 


6098.3 


878.6 


1037.9 


3601.4 


6182.7 


905.4 


1076.2 3671.6 


18 


20 


6101.1 


879.5 


1039.1 


3603.7 


6185.5 


906.3 


1076.6 8673.9 


20 


22 


6103.9 


880.8 


1040.3 


3606.0 


6188.3 


907.2 


1077.8 8676.2 


22 


2ft 


6106.7 


881.2 


1041.6 


3608.4 


6191.1 


908.1 


1079.1 3678.6 


24 


26 


6109.6 


882.1 


1042.8 


3610.7 


6193.9 


909.0 


1080.4 3680.9 


26 


28 


6112.4 


888.0 


1044.1 


3613.0 


6196.7 


909.9 


1081.7 8688.8 


28 


80 


6115.2 


883.9 


1045.3 


3615.8 


6109.5 


910.8 


1088.0 8685.6 


80 


32 


6118.0 


884.7 


1046.5 


3617.7 


6202.3 


911.7 


1084.2 8688.0 


38 


34 


6120.8 


885.6 


1047.8 


8620.0 


6205.1 


912.6 


1085.5 8690.4 


84 


36 


6128.7 


886.5 


1049.0 


3622.3 


6208.0 


918.6 


1066.8 8692.7 


86 


38 


6126.5 


887.4 


1050.3 


3624.7 


6210.8 


914.4 


1088.1 3695.1 


88 


40 


6129.8 


888.8 


1051.5 


3627.0 


6213.6 


915.3 


1069.4 3697.4 


40 


42 


6182.1 


889.2 


1052.7 


3629.4 


6216.4 


916.2 


1090.6 8690.8 


42 


44 


6134.9 


890.1 


1054.0 


3681.7 


6219.2 


917.1 


1091.9 3702.2 


44 


46 


6187.8 


891.0 


1055.2 


3634.0 


6222.0 


918.0 


1093.2 3704.5 


46 


48 


6140.6 


891.9 


1056.5 


8636.4 


6224.8 


918.9 


1094.5 8706.9 


48 


50 


6143.4 


892.8 


1067.7 


8688.7 


6227.6 


919.8 


1095.8 3709.8 


50 


52 


6146.2 


898.7 


1059.0 


8641.1 


6290.4 


920.7 


1097.0 3711.6 


58 


54 


6149.0 


894.6 


1060.2 


3643.4 


6238.2 


921.6 


1098.3 8714.0 


54 


66 


6151.9 


895.5 


1061.5 


3645.7 


6236.0 


982.6 


1099.6 8716.8 


56 


58 


6154.7 


896.4 


1062.7 


3648.1 


6288.8 


928.4 


1100.9 3718.7 


58 


60 


6167.6 


897.3 


1064.0 


3650.4 


6241.6 


924.3 


1102.2 8721.1 


60 



Digitized by VjOOQIC 





FUNCTIONS OF 


A ONE-DEGREE CURVE 


227 


1 


W 1 




6? 


• 


/ 


L.C. 


M. 


E. 


T. 


L. 0. 


M. 


B. T. 





6241.6 


924.8 


1102.2 


3721.1 


6325.2 


951.6 


1141.5 8798.6 





2 


6244.4 


925.2 


1103.6 


3728.4 


6328.0 


952.7 


1142.8 8795.0 


2 


4 


6247.2 


926.1 


1104.8 


37-25.6 


6330.7 


953.6 


1144.1 3797.4 


4 


6 


6250.0 


927.0 


1106.1 


3728.2 


6333.5 


954.5 


1145.4 8799.8 


6 


8 


6252.7 


927.9 


1107.4 


3730.6 


6336.3 


965.5 


1146.7 3802.2 


6 


10 


6255.6 


928.8 


1108.7 


37:i2.9 


6339.0 


956.4 


1148.1 3804.6 


10 


12 


6^58.3 


929.8 


1110.0 


8735.3 


6341.8 


957.8 


1149.4 8807.0 


12 


14 


6261.1 


930.7 


1111.3 


8V3V.7 


6344.6 


958.2 


1160.7 8809.4 


14 


16 


6263.9 


931 6 


1112.6 


3740.1 


6347.4 


959.2 


1152.0 3811.8 


16 


18 


6266.7 


932.5 


1118.9 


8742.4 


6350.1 


960.1 


1153.3 8814.2 


18 


20 


6269.5 


933.4 


1115.2 


3744.8 


6352.9 


961.0 


1154.7 8816.6 


20 


22 


6272.3 


984.3 


1116.6 


8747.2 


6365.7 


961.9 


1156.0 3819.0 


22 


24 


6275 


935.3 


1117.8 


8749.6 


6358.4 


962.9 


1157.4 8621.4 


84 


r» 


6277.8 


936.2 


1119.1 


8751.9 


6301.2 


968.8 


1158.7 3823.8 


26 


28 


6280.6 


937.1 


1120.4 


8754.3 


6364.0 


964.7 


1160.1 38-26.2 


28 


90 


6288.4 


938.0 


1121.7 


3756.7 


6^366.7 


965.6 


1161.4 3828.6 


SO 


82 


6386.2 


938.9 


1123.0 


8759.1 


6369.5 


966.6 


1162.8 3831.0 


32 


84 


6289.0 


939.8 


1124.3 


3761.6 


6372.3 


907.6 


1164.1 8833.4 


84 


36 


6291.8 


940.8 


1125.6 


8763.9 


6376.1 


968.4 


1165.5 8835.9 


36 


88 


6294.5 


941.7 


1126.9 


3766.3 


6377.8 


969.8 


1166.8 3838.8 


86 


40 


6297.3 


9426 


1128.3 


3766.7 


6380.6 


970.8 


1168.2 8840.7 


40 


42 


6300.1 


948.5 


1129.6 


8771.0 


6363.4 


971.8 


1169.6 3848.1 


42 


44 


6302.9 


944.4 


1130.9 


3773.4 


6386.1 


972.1 


1170.9 8845.6 


44 


46 


6305.7 


946.8 


1132.2 


8775.8 


6388.9 


973.0 


1172.8 8647.9 


46 


48 


6306.5 


946.3 


1183.5 


8778. i 


6391.7 


974.0 


1173.6 8850.4 


46 


60 


6311.3 


947.2 


1184.9 


8780.6 


6394.4 


974.9 


1174.9 3852.8 


60 


52 


6314.1 


948-. 1 


1136.2 


8788.0 


6397.2 


975.8 


1176.8 3865.2 


62 


6t 


6316.8 


949.0 


1137.6 


3785.4 


6400.0 


976.8 


1177.6 3657.6 


64 


56 


6319.6 


949.9 


1138.8 


8787.8 


6402.8 


977.7 


1179.0 8860.0 


66 


58 


6322.4 


950.8 


1140.1 


8790.8 


6405.5 


978.6 


1180.3 38626 


66 


60 


63-25.2 


951.8 


1141J5 


3792.6 


6408.3 


979.6 


1161.6 8864.9 


60 


' 




6g 


• 




69» 


/ 


L. C. 


M. 


E. 


T. 


L.C. 


M. 


E. T. 





6408.3 


979.6 


1181.6 


3664.9 


6491.1 


1007.7 


1282.9 8938.1 





2 


6411.1 


980.6 


1183.0 


3867.3 


6498.6 


1008.7 


1224.3 8940.6 


2 


4 


6418.8 


981.4 


1184.4 


8869.7 


6496.6 


1009.6 


1225.7 8948.0 


4 


6 


C416.6 


982.4 


1185.7 


8872.2 


6499.8 


1010.6 


1227.1 3945.5 


6 


8 


C419.8 


983.3 


1187.1 


3874.6 


6502.1 


1011.5 


1228.5 8947.9 


6 


10 


6422.1 


984.2 


1188.5 


3877.0 


6504.8 


1012.5 


1229.9 3950.4 


10 


12 


6424.9 


985.2 


1189.8 


8879.5 


6607.5 


1013.4 


1231.8 8952.9 


12 


14 


0427.6 


986.1 


1191.2 


3881.9 


6610.3 


1014.4 


1232.7 8955.8 


14 


16 


6430.4 


987.0 


1192.6 


3884.8 


6513.0 


1015.3 


1234.1 8957.8 


16 


18 


6433.1 


988.0 


1193.9 


3886.8 


6515.6 


1016.3 


1835.5 8960.2 


18 


20 


64^5.9 


988.9 


1195.3 


3889.2 


6518.5 


1017.2 


1286.9 3962.7 


80 


22 


0438.7 


989.6 


1196.7 


8891.6 


6521.2 


1018.2 


1288.3 8965.2 


22 


24 


6441,4 


990.6 


1198.0 


3894.1 


6524.0 


1019.1 


1289.7 8967.6 


24 


26 


6444.2 


991.7 


1199.4 


3896.5 


6526.7 


1020.1 


1241.1 8970.1 


26 


28 


6446.9 


992.6 


1200.8 


3898.9 


6529.5 


1021.0 


1242.5 8972.5 


28 


30 


6149.7 


991.6 


1202.1 


3901.4 


6532.2 


1022.0 


1243.9 8975.0 


30 


32 


6452.6 


994.5 


1203.5 


3903.8 


6534.9 


10-22.9 


1245.3 8977.6 


32 


34 


6455.2 


995.4 


1204.9 


3906.3 


6587.7 


1028.9 


1246.7 8980.0 


84 


36 


6458.0 


996.4 


1-200.2 


3908.7 


6540.4 


1024.8 


1248.1 8982.4 


36 


38 


6460.7 


997.3 


1207.6 


3911.2 


6543.2 


1025.6 


1249.6 3984.9 


38 


40 


6463.5 


998.2 


1209.0 


3913.6 


6545.9 


1026.7 


1250.9 3987.4 


40 


42 


6466.3 


999.2 


1210.3 


3916.1 


6548.6 


1027.7 


1262.8 3989.9 


42 


44 


6469.0 1000.1 


1-211.7 


8918.5 


6551.4 


1028.6 


1253.7 8992.3 


44 


46 


6471.8 1001.0 


1-213.1 


39-21.0 


6554.1 


1029.6 


1266.1 8994.8 


46 


48 


6474.5 1002.0 


1214.5 


3923.4 


6556.9 


1030.5 


1266.6 3997.3 


48 


50 


6477.8 1002.9 


1215.9 


3925.9 


6559.6 


1031.5 


1267.9 3999.8 


50 


52 


6460.1 


1003.8 


1217.3 


3928.3 


6562.3 


1032.4 


1269.3 4002.2 


62 


54 


6482.8 1001.8 


1218.7 


3930.8 


6565.1 


1033.4 


1260.7 4004.7 


54 ' 


56 


6486.6 100.5.7 


12-20.1 


3933.2 


6567.8 


1034.3 


1262.1 4007.2 


66 


68 


6486.3 1006.7 


1-221.5 


3985.7 


6570.6 


1035.3 


1263.5 4009.7 


68 


60 


6491.1 


1007.7 


12-22.9 


3938.1 


6573.3 


1036.3 


1265.0 4012.1 


60_, 



Digitized by VjOOQIC 



228 


FUNCTIONS OF A ONE-DEGREE CURVE 






70* 


710 






L C. 


M. 


E. 


T. 


L. C 


M. 


E. 


T. 


# 





6573.3 


1036.3 


1265.0 


4012.1 


6654.9 


1065.1 


1308.4 


4087.1 





2 


6576.0 


1037.3 


1266.4 


4014.6 


6657 6 


1066.1 


1809.9 


4089.7 


2 


4 


6578.7 


1038.2 


1267.9 


4017.1 


6660.3 


1067.0 


1311.3 


4092.2 


4 


6 


6581 .5 


1039.2 


1269.8 


4019.6 


6663.0 


1068.0 


1312.8 


4094.7 


6 


8 


6584.2 


1040.1 


1270.8 


4022.1 


6666.7 


1068.9 


1314.2 


4097.2 


8 


10 


65<^6.9 


1041.1 


1272.2 


4024.6 


6668.4 


1069.9 


1315.7 


4099.8 


10 


12 


6580.6 


1042.1 


1273.6 


4027.1 


6671.1 


1070.9 


1317.2 


4102.3 


12 


14 


6592.3 


1043.0 


1275.1 


4029.6 


6673.8 


1071.9 


1318.6 


4104.8 


14 


16 


6595.1 


1044.0 


1276.5 


4a32.1 


6676.6 


1072.9 


1320.1 


4107.3 


16 


18 


6597.8 


1044.9 


1278.0 


4034. G 


6679.3 


1073.8 


1321.5 


4109.8 


18 


ao 


6600.5 


1045.9 


1279.4 


4037.1 


6682.0 


1074.8 


1323.0 


4112.4 


ao 


22 


6608.2 


1046.0 


1280.8 


4039.6 


6684.7 


1075.8 


1324.4 


4114.9 


22 


24 


6605.9 


1047.8 


1282.3 


4042. 1 


6687 4 


1076.8 


1325.9 


4117.4 


S4 


26 


6608.7 


1048.8 


1283.7 


4044. C 


6690.1 


1077.7 


1327.4 


4119.9 


26 


28 


6611.4 


1049.7 


1285.2 


4047.1 


6692.8 


1078.7 


1328.9 


4122.4 


28 


80 


6614.1 


1050.7 


1286.6 


4049.6 


6695.5 


1079.7 


1330.4 


4125.0 


80 


32 


6616.8 


1051.7 


1288.0 


4052.1 


6698.2 


1080.7 


1331.8 


4127.5 


32 


34 


6619.5 


1052.6 


1289.5 


4a54.6 


6700.9 


1081.6 


1333.3 


4130.1 


34 


36 


6622.3 


1068.6 


1290.9 


4057.1 


6703.6 


1082.6 


1334.8 


4132.6 


36 


88 


6625.0 


1054.5 


1292.4 


4059.6 


6706.3 


1083.6 


1336.3 


4135.1 


88 


40 


6627.7 


1055.5 


1293.8 


4062.1 


6709.0 


1084.B 


1387.8 


4137.7 


40 


42 


6630.4 


1056.5 


1295.3 


4064.6 


6711.7 


1085.5 


1339.2 


4140.2 


42 


44 


6633.1 


1057.4 


1296.7 


4067.1 


6714.4 


1086.6 


1340.7 


4142.7 


44 


46 


6635.9 


1058 4 


1298.2 


4069.6 


6717.2 


1087.5 


1342.2 


4145.3 


46 


48 


6688.6 


1059.8 


1299.6 


4072.1 


6719.9 


1088.4 


1343.7 


4147.8 


48 


60 


6641.3 


1060.3 


1301.1 


4074.6 


6722.6 


1089.4 


1845.2 


4150.4 


50 


62 


6644.C 


1061.3 


1302.6 


4077.1 


6725.8 


1090.4 


1346.7 


4152.9 


£2 


54 


6646.7 


1062.2 


1304.0 


4079.6 


6728.0 


1091.3 


1348.2 


4155.4 


5t 


56 


6649.5 


1063.2 


1305.5 


4082.1 


6730.7 


1092.3 


1349.7 


4158.0 


56 


68 


6652.2 


1064.1 


1306.9 


4084.6 


6733.4 


1093.3 


1351.2 


4160.5 


58 


60 


6654.9 


1065.1 


1308.4 


4087.1 


6736.1 


1094.3 


13.52.7 


4163.1 


60 


1 




72 


o 






78 


» 




/ 


/ 


L. C. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 


o" 


6736.1 


1094.3 


1352.7 


4163.1 


6816.6 


1123.9 


1398.1 


4240.0 





2 


6738.8 


1095 2 


1354.2 


4165.6 


6819.3 


1124.8 


1399.6 


4242.6 


2 


4 


6741 .5 


1096.2 


1865.7 


4168.2 


6821.9 


1125.8 


1401.2 


4245.1 


4 


6 


6744.1 


1097.2 


1857.2 


4170.7 


6824.6 


1126.8 


1402.7 


4247.7 


6 


8 


6746.8 


1098.2 


1358.7 


4173.3 


6827.3 


1127.8 


1404.2 


4250.3 


8 


10 


6749.5 


1099.2 


1360.2 


4175.8 


6830.0 


1128.8 


1405.8 


4252.9 


10 


12 


6752 2 


1100.1 


1361.7 


4178.4 


6832.6 


1129.8 


1407.3 


4255.5 


12 


14 


6754.9 


1101.1 


1363.2 


4181.0 


6835.3 


1130.8 


1408.8 


4258.1 


14 


16 


6757 6 


1102.1 


1364.7 


4183.5 


6838.0 


1181.8 


1410.4 


4260.7 


16 


18 


6760.2 


1103.1 


1366.2 


4186.1 


6840.7 


1132.8 


1411.9 


4263.2 


18 


20 


6762.9 


1104.1 


1367.7 


4188.6 


6843.3 


1133.8 


1413.6 


42C5.8 


20 


22 


6765.6 


1105.1 


1369.2 


4191.2 


6846.0 


1134,8 


1415.1 


4268.4 


22 


24 


6768.3 


1106.0 


1370.7 


4193.7 


6848.7 


1135.8 


1416.6 


4271.0 


24 


26 


6771.0 


1107.0 


1372.2 


4196.3 


6851.3 


1136.8 


1418.2 


4373.6 


26 


28 


6773.7 


1108.0 


1373.7 


4198.8 


6854.0 


1137.8 


1419.7 


4276.2 


28 


80 


6776.3 


1109.0 


1375.2 


4201.4 


6856.7 


1138.8 


1421.3 


4278.8 


30 


32 


6779.0 


1109.9 


1376.7 


4204.0 


6859.4 


1139.8 


1422.9 


4281.4 


32 


34 


6781.7 


1110.9 


1378.2 


4206.5 


6862.0 


1140.8 


1424.4 


4284.0 


34 


36 


6784.4 


1111.9 


1379.7 


4209.1 


6864.7 


1141.8 


1426.0 


4286.6 


36 


88 


6787.1 


1112.9 


1381.2 


4211.7 


6867.4 


1142.8 


1427.5 


4289.2 


38 


40 


6789.8 


1113.9 


1382.8 


4214.3 


6870.1 


1143.8 


1429.1 


4391.8 


40 


42 


6792.4 


1114.9 


1384.3 


4216.8 


6872.7 


1144.8 


1430.7 


4294.4 


42 


44 


6795.1 


1115.9 


1385.8 


4219.4 


6875.4 


1145.8 


1432.2 


4297.0 


44 


46 


6797.8 


1116.9 


1387.4 


4222.0 


6878.1 


1146.8 


1433.8 


4299.6 


46 


48 


6800.5 


1117.9 


1388.9 


4224.5 


6880.8 


1147.8 


1435.3 


4802.2 


48 


60 


6803.2 


1118.9 


1390.4 


4227.1 


6883.4 


1148 8 


1436.9 


4304.8 


60 


62 


6805.9 


1119.9 


1392.0 


4229.7 


6886.1 


1149.8 


1438.5 


4307.4 


52 


64 


6808.5 


1120.9 


1893.5 


4232.3 


6888.8 


1150.8 


1440.0 


4310.0 


54 


56 


6811.2 


1121.9 


1395.0 


4234.8 


6891.4 


1151.8 


1441.6 


4312.6 


56 


68 


6813.9 


1122.9 


1396.6 


4237.4 


6894.1 


1152.8 


1443.1 


4315.2 


68 


60 


6816.6 


1123.9 


1398.1 


4240.0 


6896.8 


ll.'J3.8 


1444.7 


4317.8 


60 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEOREE CURVE 


^2\ 






7 


4» 




76» 




L.C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 







6896.8 


1153.8 


1444.7 


4317.8 


6976.4 


1184.1 


1492.5 


4396.7 





2 


6899.4 


1154.8 


1446.2 


43-.'0.6 


6979.0 


1186.1 


1494.1 


4399.4 


2 


4 


6902.1 


1155.8 


1447.8 


4323.1 


6981.7 


1186.1 


1495.7 


4402.1 


4 


6 


6904.8 


1156.8 


1449.4 


4325.7 


6984.3 


1187.1 


1497.3 


4404.7 


6 


8 


6907.4 


1157.8 


1451.0 


4328.3 


6988.9 


1188.1 


1499.0 


4407.4 


8 


i ^^ 


6910.1 


1158.8 


1452.6 


43JJ0.9 


6989.6 


1189.2 


1500.6 


4410.0 


10 


12 


6912.7 


1159.8 


1454.1 


433:3.6 


6992.2 


1190.2 


1502.2 


4412.7 


12 


14 


6915.4 


1160.8 


1456.7 


4336.2 


6994.9 


1191.2 


1503.8 


4415.3 


14 


16 


6918.0 


1161.8 


1457.3 


4338.8 


6997.5 


1192.2 


1505.4 


4418.0 


16 


18 


6920.7 


1162.8 


1458.9 


4341.4 


7000.1 


1193.2 


1507.0 


4420.7 


18 


20 


6923.3 


1163.9 


1460.5 


4344.0 


7002.8 


1194.3 


1508.7 


4423.3 


20 


22 


6926.0 


1164.9 


1462.0 


4346.7 


7005.4 


1195.3 


1510.3 


4426.0 


22 


24 


6928.6 


1166.9 


1463.6 


4349.3 


7008.0 


1196.3 


1512.0 


4428.6 


24 


26 


C931.3 


1166.9 


1465.2 


4351.9 


7010.7 


1197.3 


1513.6 


4431.3 


26 


28 


6933.9 


1167.9 


1466.8 


4354.5 


7013.3 


1198.3 


1515.3 


4434.0 


28 


80 


6936.6 


1168.9 


1468.4 


4357.1 


7015.9 


1199.4 


1516.9 


4436.6 


80 


32 


6939.2 


1169.9 


1469.9 


4359.8 


7018.6 


1200.4 


1518.5 


4439.3 


32 


84 


6941.9 


1170.9 


1471.5 


4362.4 


7021.2 


1201.4 


1520.2 


4442.0 


34 


8t 


6944.6 


1171.9 


1473.1 


4365.1 


7023.9 


l•^02.4 


1521.8 


4444.6 


36 


88 


6947.2 


1172.9 


1474.7 


4367.7 


7026.5 


1203.4 


1623.5 


4447.3 


33 


40 


6949.9 


1174.0 


1476.4 


4370.3 


7029.1 


1204.5 


1526.1 


4450.0 


40 


42 


6952.5 


1175.0 


1478.0 


4378.0 


7031.8 


1205.5 


1526.7 


4462.7 


42 


44 


6955.2 


1176.0 


1479.6 


4375.6 


7034.4 


1206.6 


1528.4 


4455.3 


44 


46 


6957.8 


1177.0 


1481.8 


4378.8 


7037.0 


1207.5 


15.30.0 


4458.0 


46 


48 


6960.5 


1178.0 


1482.8 


4380.9 


7039.7 


1208.5 


1531.7 


4460.7 


48 


50 


6963.1 


1179.0 


1484.4 


4388.5 


7042.3 


1209.6 


1533.3 


4463.4 


60 


62 


6965.8 


1180.0 


1486.0 


4386.2 


7045.0 


1210.6 


1584.9 


4466.0 


52 


54 


6968.4 


1181.0 


1487.7 


4388.8 


7047.6 


1211.6 


1536.6 


4468.7 tl 54 


56 


6971.1 


1182.0 


1489.3 


4391.5 


7050.2 


1212.6 


158S.2 


4471.4 


56 


58 


6973.7 


1183.0 


1490.9 


4394.1 


7052.9 


1213.6 


1539.9 


4474.1 


58 


60 


6976.4 


1184.1 


1492.5 


4396.7 


7055.5 


1214.7 


1.541.5 


4476.7 


60 


/ 


76° 




7 


70 






L. C. 


M. 


E. 


T. 


L. C. 


M 


E. 


T. 


/ 





7055.5 


1214.7 


1541.5 


4476.7 


7134.0 


1245.6 


1591.7 


4557.8 





2 


7058.1 


1215.7 


1543.2 


4479.4 


7136.6 


124C.6 


1593.4 


4560.5 


2 


4 


7060.7 


1216.7 


1544.9 


4482.1 


7139.2 


1247.7 


1595.1 


4563.3 


4 


6 


7063.3 


1217.8 


1546.5 


4484.8 


7141.8 


1248.7 


1506.8 


4566.0 


6 


8 


7066.0 


1218.8 


1548.2 


4487.5 


7144.4 


1249.8 


1598.5 


4568.7 


8 


10 


7068.6 


1219.8 


1549.9 


4490.2 


7147.0 


1250.8 


1600.2 


4571.5 


10 


12 


7071.2 


1220.9 


1551 .5 


4492.9 


7149.6 


1251.8 


1601.9 


4574 2 


12 


14 


7073.8 


1221.9 


1653.2 


4495.6 


7152.2 


1252.9 


1603.6 


4576.9 


14 


16 


7076.4 


1222.9 


1554.9 


4498.3 


7154.8 


1253 9 


1605.3 


4579.7 


16 


18 


7079.0 


1224.0 


1556.5 


4501.0 


7157.4 


1255 


1607.0 


4582.4 


18 


20 


7081.7 


1225.0 


1558.2 


4.503.7 


7160.0 


1256.0 


1608.7 


4585.1 


20 


22 


7084.3 


1226.0 


1559.9 


4506. S 


7162.6 


3257.0 


1610.4 


4587.9 


22 


24 


7086.9 


1227.1 


1561.5 


4509.0 


7165.2 


1258.1 


1612.1 


4590.6 


24 


26 


7089.5 


1228.1 


1563.2 


4511.7 


7167.8 


1259.1 


1613.8 


4593.8 


26 


28 


7092.1 


1229.1 


1564.9 


4514.4 


7170.4 


1260.2 


1615.5 


4590.0 


28 


SO 


7094.7 


1230.2 


1566.5 


4517.1 


7173.0 


1261 2 


1617.3 


4598 8 


30 


82 


7097.4 


1231.2 


1568.2 


4519.8 


7175.6 


1262 2 


1619.0 


4601 .5 


32 


81 


7100.0 


1232.2 


1569.9 


4522.5 


7178 2 


1263.3 


1620.7 


4604.3 


34 


36 


7102.6 


1233.3 


1571.5 


4625.3 


7180.8 


1264.3 


1622.4 


4607 


36 


88 


7105.2 


1284.8 


1573.2 


4528.0 


7183.4 


1265.4 


1624.1 


46098 


38 


40 


7107.8 


1235.3 


1574.8 


4530.7 


7186.0 


1266.4 


1625.9 


4612.5 


40 


42 


7110.4 


1236.4 


1576.4 


4533.4 


7188 6 


1207.4 


1627.6 


4615.3 


42 


44 


7113.1 


1237.4 


1578.1 


4536.1 


7191.2 


1268.5 


1629.3 


4618.0 


44 


46 


7115.7 


1238.4 


1579.8 


4538.8 


7193.8 


1269.5 


1631.0 


4620.8 


46 


48 


7118.3 


12395 


1581.5 


4541.5 


7196.4 


1270 6 


1632.7 


4623.5 


48 


50 


7120.9 


1240.5 


1583.2 


4544.2 


7199.0 


1271.6 


1634 5 


4626.3 


50 


53 


7123.5 


1241.6 


1584.9 


4547.0 


7201.6 


1272. 7 


1636.2 


4629.0 


52 


54 


7126.1 


1242.6 


1586.6 


4549.7 


7204.2 


1273.7 


1687.9 


4631.8 


54 


66 


7128.8 


1243.6 


1588.3 


4552.4 


7206.8 


1274.8 


1639.6 


4634.5 


56 


58 


7131.4 


1244.6 


1590.0 


4555.1 


7209.4 


1275.8 


1641.8 


4637.3 


58 


60 


7134.0 


1245.6 


1591.7 


4557.8 


7212.0 


1276.9 


1643.1 


4640.0 


60 



Digitized by VjOOQIC 



230 


FUNCTIONS OF A ONE-DEGREE CURVE 




/ 


780 


790 1 


/ 



L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





7212.0 


1276.9 


1643.1 


4640.0 


7289.5 


1308.6 


1696.0 


4728.4 


s 


7214.6 


1278.0 


1644.8 


4642.8 


7292.1 


1309.5 


1697.7 


4726.2 


2 


4 


7217.2 


1279.0 


1646.6 


4645.6 


7294.6 


1310.6 


1699.6 


4729.0 


4 


6 


7219.7 


1280.1 


1648.3 


4648.3 


7297.2 


1311.7 


1701.3 


4781.8 


6 


8 


7228.3 


1281.1 


1650.1 


4651.1 


7299.7 


1312.7 


1703.1 


4734.7 


8 


10 


7224.9 


1282.2 


1651.8 


4658.9 


7302.8 


1813.8 


1704.9 


4737.6 


10 


12 


7227.5 


1283.2 


1658.6 


4656.7 


7304.9 


1314.9 


1706.6 


4740.3 


12 


14 


7230.1 


1284.3 


1655.8 


4659.4 


7307.4 


1316.9 


1708.4 


4748.1 


14 


16 


7232.7 


1285.8 


1657.1 


4662.2 


7310.0 


1317.0 


1710.2 


4745.9 


16 


18 


7235.2 


1286.4 


1658.8 


4665.0 


7812.6 


1318.1 


1712.0 


4748.7 


18 


20 


7237.8 


1287.4 


1660.6 


4667.7 


7815.1 


1319.1 


1718.8 


4751.6 


20 


22 


7240.4 


1288.5 


1662.3 


4670.5 


7817.7 


1320.2 


1715.6 


4761.3 


22 


24 


7248.0 


1289.5 


1664.1 


4673.3 


7320.8 


1821.3 


1717.4 


4757.1 


24 


26 


7245.6 


1290.6 


1665.8 


4676.0 


7322.8 


1822.3 


1719.2 


4760.0 


26 


28 


7248.2 


1291.6 


1667.6 


4678.8 


7326.4 


1323.4 


1721.0 


4762.8 


28 


SO 


7^i50.7 


1292.7 


1C69.3 


4681.6 


7:327.9 


1324.6 


1722.8 


4766.6 


80 


32 


7253.3 


1293.7 


1671.1 


4684.4 


7330.6 


1326.5 


1724.6 


4768.4 


82 


34 


7255.9 


1294.8 


1672.8 


4687.2 


7338.1 


1326.6 


1726.4 


4771.2 


84 


36 


7258.5 


1295.8 


1674.6 


4689.9 


7335.6 


1327.7 


1728.2 


4774.1 


86 


88 


7261.1 


1296.9 


1676.8 


4692.7 


7388.2 


1328.7 


1730.0 


4776.9 


88 


40 


7263.7 


1297.9 


1678.2 


4696.5 


7340.8 


1329.8 


1731.9 


4779.7 


40 


42 


7266.2 


1299.0 


1679.9 


4698.3 


7348.3 


1330.8 


1783.7 


4782.6 


4'i 


44 


7968.8 


1300.0 


1681 .7 


4701.1 


7345.9 


1831.9 


1735.5 


4786.4 


44 


46 


7271.4 


1301.1 


1683.5 


4703.9 


734S.4 


1338.0 


1787.3 


4788.2 


46 


48 


7274.0 


1302.1 


1685.3 


4706.7 


7351.0 


1384.1 


1739.1 


4791.0 


48 


50 


7276.6 


1803.2 


1687.1 


4709.5 


7353.6 


1335.2 


1740.9 


4793.9 


50 


52 


7279.2 


1804.2 


1688.8 


4712.2 


7356.1 


1836.2 


1742.7 


4796.7 


52 


54 


7281.7 


1305.8 


1690.6 


4715.0 


7858.7 


1337.3 


1744.6 


4799.5 


64 


66 


7284.3 


1306.3 


1692.4 


4717.8 


7361.8 


1338.4 


1746.3 


4802.4 


66 


58 


7286.9 


1307.4 


1694.2 


4720.6 


7363.8 


1839.5 


1748. 1 


4805. •,> 


68 


60 


7289.5 


1808.5 


1696.0 


4728.4 


7366.4 


1340.6 


1750.0 


4808.0 


60 


/ 


80° 1 


8l» 


f 


L.C. 


M. 


B. 


T. 


L. C. 


M. 


E. 


T. 





7366.4 


1340.6 


1760.0 


4808.0 


7442.7 


1372.8 


1805.5 


4893.9 





2 


7368 9 


1341.7 


1751.8 


4810.9 


7445.2 


1378.9 


1807.3 


4896.8 


2 


4 


7371.5 


1342.7 


1758.7 


4818.7 


7447.7 


1375.0 


1809.2 


4899.7 


4 


6 


7374.0 


1343.8 


1755.5 


4816.6 


7450.3 


1376.1 


1811 . 1 


4902.6 


6 


8 


7376.6 


1344.9 


1757.4 


4819.4 


7452.8 


1377.1 


1813.0 


4906.4 


8 


10 


7379.1 


1346.0 


1759.2 


4822.3 


7455.3 


1378.2 


1814.9 


4908.8 


10 


12 


7381.7 


1847.0 


1761 .0 


4825.1 


7457.8 


1379.3 


1816.8 


4911.2 


12 


14 


7384.2 


1348.1 


1762.9 


4828.0 


7460.4 


1380.4 


1818.6 


4914.1 


14 


16 


7386.7 


1349.2 


1764.7 


4830.8 


7462.9 


1381.4 


1820.5 


4917.0 


16 


18 


7389.3 


1350.3 


1766.6 


4883.7 


7465.4 


1382.6 


1822.4 


4919.9 


18 


20 


7391.8 


1351.8 


1768.4 


4836.5. 


7467.9 


1388.6 


1824.2 


4922.8 


20 


22 


7394.4 


1352.4 


1770.2 


4839.4 


7470.4 


1384.7 


1826.1 


4925.7 


22 


24 


7396.9 


1353.6 


1772.1 


4842.2 


7473.0 


i:«5.7 


1828.0 


4928.6 


24 


26 


7899.5 


1354.6 


1773.9 


4845.1 


7476.5 


1386.8 


1829.9 


4931.5 


26 


28 


7402.0 


1355.6 


1775.8 


4847.9 


7478.0 


1387.9 


1881.8 


4984.4 


28 


30 


7404.5 


1356.7 


1777.6 


4850.8 


7480.5 


1389.0 


1833.7 


4937.2 


80 


32 


7407.1 


1357.8 


1779.4 


4853.7 


7483.1 


1390.1 


18%.6 


4940.2 


82 


84 


7409.6 


1358.9 


1781.8 


4866.6 


7485.6 


1391.2 


1887.6 


4943.1 


84 


36 


7412.2 


1359.9 


1783.1 


4859.4 


7488.1 


1392.8 


1889.4 


4946.0 


86 


38 


7414.7 


1361.0 


1785.0 


4862.3 


7490.6 


1393.4 


1841.8 


4948.9 


88 


40 


7417.8 


1362.1 


1786.8 


4866.1 


7493.2 


1394.5 


1848.2 


4951.8 


40 


42 


7419.8 


1863.2 


1788.6 


4868.0 


7495.7 


1395.6 


1846.1 


4964.7 


42 


44 


7422.3 


1364.2 


1790.5 


4870.9 


7498.2 


1396.7 


1847.0 


4967.6 


44 


46 


7424.9 


1365.8 


17^.4 


4878.8 


7500.7 


1397.8 


1848.9 


4960.6 


40 


48 


7427.4 


1366.4 


1794 3 


4876.6 


7503.3 


1396.9 


1850.8 


4968.5 


48 


50 


7430.0 


1867.5 


1796.2 


4879.5 


7505.8 


1400.0 


1862 7 


4966.4 


60 


52 


7432.5 


1368.5 


1798.0 


4882.4 


7508.3 


1401.1 


1854 6 


4960.8 


62 


54 


7485.1 


1369.6 


1799.9 


4885 3 


7510.8 


1402.2 


1866.5 


4972.2 


54 


56 


7437.6 


1370 7 


1801.8 


4888.1 


7518.3 


1403.3 


186S.4 


4976.1 


66 


58 


7440.1 


1371.8 


1803.7 


4891.0 


7515.9 


1404 4 


1860.3 


4978.0 


68 


60 


744S.7 


1372.8 


1805.5 


4898.9 


7518.4 


1405.6 


1862 8 


4961. C 


«0 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEGRKK CURVE 


231 


1 


8«- 


880 


/ 


L. C. 


M. 


E. 


T 


L. C. 


M. 


E. 


T. 





7518.4 


1405.5 


1862.8 


4981.0 


7.593.6 


1438.5 


1920.6 


5069.4 





2 


7520.9 


1406.6 


1864.2 


4983.9 


7596.1 


1439.0 


1922.6 


5072.4 


2 


4 


7628.4 


1407.7 


1866.1 


4986.8 


7598.6 


1440.7 


1924.6 


5075.4 


4 


6 


7525.9 


1408 8 


1868.1 


4989.8 


7601.1 


1441.8 


1926.5 


5078.4 


6 


8 


7528.4 


1409.9 


1870.0 


4992.7 


7603.6 


1442.9 


1928.5 


5081.4 


8 


10 


7530.9 


1411.0 


1871 9 


4995.7 


7606.0 


1444.0 


1930.5 


5084.4 


10 


12 


7588.4 


1412.1 


1873 9 


4998.6 


7608.5 


1445.1 


1982.4 


5087.3 


12 


14 


7535.9 


1413.2 


1875.8 


5001.5 


7611.0 


1446.2 


1934.4 


5090.8 


14 


16 


7538.5 


1414.3 


1877.7 


5004.5 


7613.5 


1447.3 


1036.4 


5093.3 


16 


18 


7541.0 


1415.4 


1879.7 


5007.4 


7616.0 


1448.4 


1938.4 


509G.3 


18 


20 


7548.5 


1416.5 


1881.6 


5010.8 


7618.5 


1449.6 


1940.4 


5099.8 


20 


22 


7546.0 


1417.6 


1883.5 


5013.3 


7621.0 


1450.7 


1942.4 


5102.3 


23 


24 


7548.5 


1418.7 


18S>.5 


5016.2 


7623.5 


1451.8 


1944.4 


5105.2 


24 


26 


7551.0 


1419.8 


1887.4 


5019.2 


7626.0 


1452.9 


1946.4 


6108.2 


26 


28 


7553.5 


1420.9 


1889.3 


5022.1 


7628.5 


1454.0 


1948.4 


5111.2 


28 


80 


7556.0 


1422.0 


1891.3 


5025.0 


7630.9 


1455.1 


1950.4 


5114 2 


80 


32 


7558.5 


1423.1 


1893.2 


5028.0 


7633.4 


1456.2 


1952.4 


5117.2 


83 


84 


7561.0 


1424.2 


1895.1 


5031.0 


7C35.9 


1457.3 


1954.4 


5120.2 


84 


36 


7563.5 


1425.8 


1897.1 


5033.9 


7638.4 


1458.4 


1956.4 


5123.2 


86 


88 


7566.0 


1426.4 


1899.0 


50:36.9 


7640.9 


1459.5 


1958.4 


5126.2 


88 


40 


7568.5 


1427.5 


1901.0 


6039.8 


7643.4 


1460.7 


1960.4 


6129.2 


40 


42 


7571.0 


1428.6 


1902.9 


504-2.8 


7645.9 


1461.8 


1962.4 


5182.2 


42 


44 


7573.5 


1429.7 


1904.9 


5045.8 


7C48.4 


1462.9 


1964.4 


5135.2 


44 


46 


7576.1 


1430.8 


1906.9 


5048.7 


7650.9 


1464.0 


1966.4 


6138.2 


46 


48 


7578.6 


J431.9 


1908.8 


5051.7 


7653.4 


1465.1 


1968.4 


5141.2 


48 


60 


7581.1 


1433.0 


1910.8 


5054.6 


7655.8 


1466.2 


1970.4 


5144.8 


60 


52 


7583.6 


1434.1 


1912.8 


5057.6 


7658.3 


1467.3 


1972.4 


6147.8 


62 


54 


7586.1 


1435.2 


1914.7 


5060.6 


7660.8 


1468.4 


1974.4 


5150.3 


54 


56 


7588.6 


M36.3 


1916.7 


5063.5 


7663.3 


14G9.5 


1976.4 


5153.8 


56 


68 


7591.1 


1437.4 


1918.7 


5066.5 


7665.8 


1470.6 


1978.4 


5150.3 


68 


60 


7593.6 


14:^.5 


1920.6 


5069.4 


7668.3 


1471.8 


1980.5 


5159.3 


60 


/ 


84° 


86» 


/ 


L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


1 





7668.8 


1471.8 


1980.5 


5159.8 


7742.4 


1505.4 


8041.8 


5250.6 





2 


7670.8 


1472.9 


1982.5 


5162.3 


7744.8 


1506.5 


2043.9 


5258.6 


2 


4 


7673.2 


1474.0 


1984.5 


5165.3 


7747.8 


1507.6 


2046.0 


6256.7 


4 


6 


7675.7 


1475.1 


1988.6 


5168.4 


7749.7 


1508. R 


2048.0 


5259.8 


6 


8 


7678.2 


1476.2 


1988.6 


5171.4 


7752.2 


1509.9 


2050.1 


6262.9 


8 


10 


7680.6 


1477.4 


1990.6 


5174.4 


7754.6 


1511.0 


2052.2 


5266 


10 


12 


7683.1 


1478.5 


1992.7 


5177.5 


7757.1 


1512.2 


2054.2 


5269.0 


12 


14 


7685 6 


1479.6 


1994.7 


5180.5 


7759 5 


1513.3 


2056.3 


5272.1 


14 


16 


7688.1 


1480.7 


1996.7 


5183.5 


7762.0 


1514.4 


2058.4 


6275.3 


16 


18 


7690.5 


1481.8 


1998.8 


5186.6 


7764 4 


1515.6 


2060.5 


5278.3 


18 


20 


7698.0 


1488.0 


2000.8 


5189.6 


7766.9 


1516.7 


2062 .i) 


5281.4 


20 


22 


7695.5 


1484 1 


200.J.8 


5192.6 


7769.8 


1517.8 


2064.7 


5284.4 


22 


24 


7697.9 


1485.2 


2004.9 


5195.6 


7771 .8 


1519.0 


2066.8 


6287.5 


24 


26 


7700.4 


1486.3 


2006.9 


5198.7 


7774.2 


1520.1 


2068.9 


5290.6 


26 


28 


7702.9 


1487.4 


2008.9 


5201.7 


7776.7 


1521.2 


2071.0 


5293.7 


28 


80 


7705.3 


1488 6 


2011.0 


6204.7 


7779.1 


1522.4 


2073.1 


5296.7 


30 


32 


7707.8 


1489.7 


2013.0 


5207.8 


7781.5 


1523.5 


2075 2 


5299 8 


32 


84 


7710.3 


1490.8 


2015.0 


5210 8 


7784.0 


1524.6 


2077 3 


5302.9 


84 


86 


7712.8 


1491.9 


2017.0 


5218.9 


7786.4 


3525.8 


2079.4 


5306.1 


86 


88 


7715.2 


1493.0 


2019.1 


6218.9 


7788.9 


1526.9 


2081.5 


5309.2 


38 


40 


7717.7 


1494.2 


2021.2 


5220.0 


7791.8 


1528.0 


2083.7 


5312.3 


40 


42 


7720 2 


1495.3 


2023.2 


5223.1 


7793.8 


1529 2 


2085.8 


5815.4 


42 


44 


7722 6 


1496.4 


2025.3 


5226.1 


7798 2 


1530.3 


2087.9 


5318.5 


44 


40 


7725. 1 


1497.5 


2027.4 


5229.2 


7798 7 


1531 .4 


2090.0 


5321.6 


46 


48 


7727.6 


1496.6 


a)29.4 


5232.2 


7801.1 


1532.6 


2092.1 


5324.7 


48 


50 


7780.0 


1499.8 


2031.5 


5235.3 


7803.6 


1533.7 


2094.2 


5327.8 


60 


52 


7782 6 


1500.9 


2033.6 


6238.3 


7806.0 


1534 8 


2096.3 


53809 


52 


54 


7785.0 


1502.0 


2035.6 


5^41 4 


7808.5 


1536.0 


2098.4 


5884.0 


54 


56 


7787 5 


1503.1 


2037.7 


5244 5 


7810.9 


1537.1 


2100.6 


5337.1 


56 


58 


7789 9 


1504.2 


2089.8 


5247.5 


7813 4 


1538.2 


2102.7 


5340.2 


68 


60 


7742.4 


15054 


2041.8 


5250.6 1 7815.8 


1589.3 


2104.8 


5843.3 


60 



Digitized by VjOOQIC 



232 


FUNCTIONS OF A ONE-DEGREE CURVE 




/ 


8«» 1 




8 


t** 




/ 


L. C. 


M. 


E. 


.T. 


L. C. 


M. 


E. 


T. 





7815.8 


1639.3 


2104.8 


6343.8 


7888.5 


1673.6 


2169.6 


6437.5 





2 


7818.2 


1540.4 


2106.9 


6346.4 


7890.9 


1674.8 


2171.6 


6440.7 


2 


4 


7820.6 


1541.6 


2109.1 


6349.5 


7893.3 


1676.9 


2173.8 


6443.9 


4 


6 


7823.1 


1542.7 


2111.2 


5352.7 


7895.7 


1577.1 


2176.0 


6447.1 


6 


6 


7825.5 


1543.9 


2113.4 


5355.8 


7898.1 


1678.3 


2178.2 


6450.3 


8 


10 


7827.9 


1545.0 


2115.5 


5358.9 


7900.5 


1579.4 


2180.4 


6453.4 


10 


13 


7830.3 


1546.1 


2117.6 


5862.0 


7903.0 


1580.5 


2182.6 


5466.6 


13 


14 


7832.8 


1547.3 


2119.8 


5865.2 


7905.4 


1581.7 


2184.7 


6459.8 


14 


16 


7835.2 


1548.4 


2121.9 


5868.3 


7907.8 


1582.9 


2186.9 


6463.0 


16 


18 


7837.6 


1549.6 


2124.1 


5371.4 


7910.2 


1584.0 


2189.1 


5466.2 


18 


20 


7840.0 


1550.7 


2126.3 


5374.6 


7912.6 


1585.1 


2191.3 


6469.4 


20 


22 


7812.4 


1551.8 


2128.3 


5377.7 


7915.0 


1586.3 


2193.5 


6472.6 


22 


24 


7844.9 


1553.0 


2130.5 


5?80.8 


7917.4 


1587.4 


2196.7 


6476.7 


24 


26 


7847.3 


1554.1 


2132.6 


5383.9 


7919.8 


1588.6 


2197.9 


6478.9 


26 


28 


7849.7 


1555.8 


2134.8 


5387.1 


7922.3 


1589.7 


2200.1 


6482.1 


28 


30 


7852.1 


1556.4 


2136.9 


5390.2 


7924.6 


1590.9 


2202.3 


6485.8 


30 


32 


7854.6 


1557.5 


2139.0 


5393.4 


7927.1 


1692.0 


2204.5 


6488.5 


32 


34 


7857.0 


1558.7 


2141.2 


5396.5 


7929.5 


1503.2 


2206.8 


6491.7 


34 


36 


7859.4 


1559.8 


2143.3 


5399.7 


7931.9 


1594.3 


2209.0 


6494.9 


36 


38 


7861.8 


1561.0 


2145.5 


5403.8 


7934.3 


1595.6 


2211.2 


6496.1 


38 


40 


7864.3 


1562.1 


2147.7 


5406.0 


7936.7 


1596.6 


2213.4 


6501.3 


40 


42 


7866.7 


1563.2 


2149.8 


5409.1 


7939.1 


1597.8 


3216.6 


5604.5 


42 


44 


7869.1 


1564.4 


2152.0 


.M12.3 


7941.5 


1598.9 


2217.8 


6607.7 


44 


46 


7871.5 


1565.5 


2154.2 


5415.4 


7913.9 


1600.1 


3230.0 


5610.9 


46 


48 


7874.0 


1566.7 


2156.4 


5418.6 


7946.8 


1601.2 


3222.3 


5514.1 


48 


50 


7876.4 


1567.8 


2158.6 


5421.8 


7948.7 


1602.4 


3224.6 


6517.8 


50 


52 


7878.8 


1568.9 


2160.7 


5424.9 


7951.2 


1608.5 


2226.7 


6520.6 


52 


54 


7881.2 


1570.1 


2162.9 


5428.1 


7953.6 


1604.7 


2228.9 


W23.7 


54 


56 


7883.6 


1571.2 


2165.1 


5431.2 


7966.0 


1605.8 


2281.1 


6526.9 


56 


58 


7886.1 


1572.4 


2167.3 


5434.4 


7958.4 


1607.0 


2233.3 


56S0.1 


58 


60 


7883 ._5_ 


1573.6 


2169.5 


5437.5 


7960.8 


1608.2 


2235.6 


6588.3 


60 




880 


8t»° 


f 


' 


L. C. 


M. 


E. 


T. 


L.C. 


M. 


B. 


T. 





7960.8 


1608.2 


2235.6 


5533.3 


8082.4 


1643.0 


2S03 fl 


M30.8 





2 


7963. .J 


1609.4 


2237.8 


5536.6 


8034.8 


1644.1 


2305.9 


5631.1 


2 


4 


7965.6 


1610.5 


2240.1 


5589.8 


8037.1 


1645.3 


2:iOS.2 


5637.4 


4 


6 


7968.0 


1611.7 


2242.3 


5543.1 


8039.6 


1646.5 


'2^\0.5 


6640.7 


6 


8 


7970.3 


1612.8 


3244.6 


5546.3 


8041.9 


1647.7 


2312,8 


6644 


8 


10 


7972.7 


1614.0 


2246.8 


5549.5 


8044.2 


1648.9 


2315.1 


5647.3 


10 


1? 


7975.1 


1615.2 


2249.1 


5552.8 


8046.6 


1650.0 


S317.4 


5650.^ 


12 


14 


7977.5 


1616.3 


2251.3 


5556.0 


8049.0 


1651.2 


2319.7 


5(35,1.9 


14 


16 


7979.9 


1617.5 


2253.6 


5559.2 


8051.4 


1652.4 


i^^S.O 


5057,1 


16 


18 


7982.8 


1618.6 


2255.8 


5562.5 


8053.7 


1653.6 


23S4.3 


5660.4 


18 


20 


7984.7 


1619.8 


2258.1 


5565.7 


8056.1 


1654.8 


23^. ■; 


nm.i 


30 


22 


7987.1 


1621.0 


2260.4 


5568.9 


8058.5 


1655.9 


2320.0 


5Ge7.0 


22 


24 


7989.4 


16J2.1 


2262.7 


5572.2 


8060.8 


1657.1 


Sfi31.3 


5670 3 


2^ 


26 


7991.8 


1623.3 


2264.9 


5575.4 


8063.2 


1668.3 


2333.7 


6073.0 


26 


28 


7994.2 


1624.4 


2267.2 


5578.6 


8065.6 


1659.5 


2336.0 


5070.9 


m 


30 


7996.6 


1625.6 


2269.5 


5581.9 


8067.9 


1660.7 


£>338,S 


bm.t 


30 


32 


7999.0 


1626.8 


2271.7 


5585.1 


8070.3 


1661.8 


2340.7 


56ft3.5 


33 


34 


8001.4 


1627.9 


2273.9 


5588.4 


8073.7 


1603.0 


2343.0 


6*ie«,e 


34 


36 


8003.8 


1629.1 


2276.2 


5591.7 


8075.1 


1664.2 


2345.3 


5690.2 


36 


38 


8006.1 


1630.2 


22r8.5 


5594.9 


8077.4 


1665.4 


2347.7 


5093.5 


38 


40 


8008.5 


1631.4 


2-280.8 


5598.2 


8079.8 


1666.6 


S3.W.0 


5696.8 


40 


42 


8010.9 


1632.6 


2283.0 


5601.4 


8082.2 


1667.7 


s>:^^2.3 


5700.1 


4a 


44 


8013.3 


1633.7 


2285.3 


5604.7 


8084.5 


1668.8 


235^,7 


570S.4 


44 


46 


8015.7 


1634.9 


2-87.6 


5608.0 


8086.9 


1670.0 


2357.0 


570S 8 


4fi 


48 


8018.1 


1636.0 


2289 9 


5611.2 


8089.3 


1671.3 


23.'j9.3 


5710. I 


48 


50 


8020.5 


1637.2 


2292.2 


5614.5 


8091.6 


1672.4 


2361.7 


5713.4 


50 


52 


8022.9 


1638.4 


2294.4 


5617.8 


8094.0 


1673.6 


2364.0 


6716 7 


52 1 


54 


8025.2 


1639.5 


2296.7 


5621.0 


8096.4 


1674.7 


2366.3 


S7S0 


54 


56 


8027.6 


1640.7 


2299.0 


5634.8 


8098.8 


1675.9 


2368 7 


5728 4 


56 


58 


8030.0 


1641.8 


2301.3 


6627.6 


8101.1 


1677.1 


ySTi.o 


5726.7 


58 


60 


8038.4 


1643.0 


2303.6 


6630.8 


8103.6 


1678.3 


2373,4 


BTBO.O 


60 



y Google 





FUNCTIONS OF A ONE-DEGREE CURVE 


233 


/ 


W 1 


91» 1 


/ 


L. C. 


M. 


E. T. 


L.C. 


M. 


E. 


T. 





8103.5 


1678.3 


2873.4 5730.0 


8178.9 


1713.8 


2445.1 


5830.9 





3 


8ia5.8 


1679.5 


2375.8 5733.3 


8176.2 


1715.0 


2447.5 


6834.3 


2 


4 


8108.2 


1680.6 


2878.2 5786.7 


8178.5 


1716.2 


2450.0 


6887.7 


4 


6 


8110.5 


1681.8 


2380.5 5740.0 


8180.9 


1717.4 


2452.4 


6841.1 


6 


6 


8112.9 


1683.0 


2382.9 5743.4 


8188.2 


1718.6 


2454.8 


6844.5 


8 


10 


8115.2 


1684.2 


2385.8 5746.7 


8ia5.5 


1719.7 


2457.2 


5847.9 


10 


12 


8117.6 


1685.4 


2387.6 6750.0 


8187.9 


1720.9 


2459.7 


5861.8 


12 


14 


8119.9 


-1G86.5 


2390.0 5758.4 


8190.2 


1722.1 


2462.1 


5854.7 


14 


16 


8122.8 


1687.7 


2392.4 5756.7 


8192.5 


1723.3 


2464.5 


5858.1 


16 


18 


8124.6 


1688.9 


2894.7 5760.1 


8194. 8^ 


1724.5 


2467.0 


6861.5 


18 


20 


8127.0 


1690.1 


2897.1 5763.4 


8197.2 


1725.7 


2469.1 


5864.9 


20 


fti 


81-^.3 


1691.8 


2399.5 5766.8 


8199.5 


1726.9 


2471.9 


6868.3 


22 


24 


8181.7 


1692.5 


2401.9 5770.1 


8201.8 


1728.1 


2474.3 


5871.8 


24 


26 


8134.0 


1693.6 


2404.8 5773.5 


8204.2 


1729.8 


2476.7 


5875.2 


26 


28 


8136.4 


1694.8 


2406.6 5776.9 


8206.5 


1730.5 


2479.2 


6878.6 


28 


80 


8138.7 


1696.0 


2409.0 5780.2 


8208.8 


1731.7 


2481.6 


5882.0 


80 


33 


8141.1 


1697.2 


2411.4 5788.6 


8211.1 


1782.9 


2484.1 


6885.4 


82 


34 


8143.4 


1698.4 


2413.8 5787.0 


8213.5 


1734.1 


2486.5 


5888.9 


84 


86 


8145.8 


1699.6 


2416.2 5790.8 


8215.8 


1735.3 


2489.0 


5892.3 


36 


88 


8148.1 


1700.7 


2418.6 5798.7 


8218.1 


1786.4 


2491.5 


5895.7 


88 


40 


8150.4 


1701.9 


2421.0 5797.1 


8220.4 


1737.6 


2493.9 


5899.2 


40 


42 


8152.8 


1703.1 


2423.4 5800.4 


8222.8 


1738.8 


2496.4 


6902,6 


42 


44 


8155.1 


1704.3 


2425.8 5808.8 


8225.1 


1740.0 


2498.9 


6906.0 


44 


46 


8157.5 


1705.5 


2428.2 5807.2 


8227.4 


17-41.2 


2601.3 


6909.4 


46 


48 


8159.8 


1706.7 


2430.6 5810.6 


8229.7 


1742.4 


2,508.8 


6912.9 


48 


50 


8162.2 


1707.9 


2483.0 5814.0 


8282.0 


1743.6 


2506.8 


6916.3 


60 


52 


8164.5 


1709.0 


2435.4 5817.3 


8234.3 


1744.8 


2508.7 


6919.8 


62 


54 


8166.8 


1710.2 


2437.9 5820.7 


8286.7 


1746.0 


2511.2 


6923.2 


64 


66 


8169.2 


1711.4 


2440.3 5824.1 8289.0 


1747.2 


2513.7 


6926.7 


66 


58 


8171.5 


1712.6 


2442.7 5827.5 8241.3 


1748.4 


2516.2 


6980.1 


68 


_60^ 


8173.9 


1713.8 


2445.1 5830.9 8243.6 


1749.6 


2518.7 


5933.6 


60 ; 


/ 


920 1 


98° 





L. C 


M. 


E. T. 


L.C. 


M. 


E. 


T. 





8243.6 


1749.6 


2518.7 5933.6 


8312.8 


1785.7 


2594.2 


6088.2 


2 


8245.9 


1750.8 


2521.2 5937.0 


8315.1 


1786.9 


2596.8 


6041.7 


2 


4 


8248.2 


1752.0 


2523.6 5940.5 


8317.4 


1788.2 


2599.3 


6045.2 


4 


6 


8250.6 


1753.2 


2526.1 5944.0 


8319.7 


1789.4 


2601.9 


6048.7 


6 


8 


8252.9 


1754.4 


2528.6 5947.4 


8322.0 


1790.6 


2604.4 


6052.2 


8 


10 


8256.2 


1755.6 


2531.1 5950.9 


8324.3 


1791.8 


2007.0 


6055.8 


10 


12 


8257.5 


1756.8 


2533.6 5954.4 


8326.6 


1793.0 


2609.6 


6059.3 


12 


14 


8259.8 


1758.0 


2536.1 5957.8 


8328.8 


1794.2 


2612.1 


6062.8 


14 


16 


8262.2 


1759.2 


2538.6 5901.3 


8331.1 


1795.4 


2614.7 


6066.4 


16 


18 


8264.5 


1760.4 


2541.1 5964.8 


8333.8 


1796.6 


2617.3 


6069.9 


18 


20 


8266.8 


1761.6 


2548.6 5968.2 


8335.6 


1797.8 


2619.8 


6073.4 


20 


22 


8269.1 


1762.8 


2546.1 5971.7 


8337.9 


1799.1 


2622.4 


6077.0 


22 


24 


8271.4 


1764.0 


2.548.6 5975.2 


8840.2 


1800.8 


2625.0 


6080.5 


24 


26 


8273.7 


1765.2 


2551.2 5978.7 


8342.5 


1801.5 


2627.6 


6084.1 


26 


28 


8276.0 


1766.4 


2558.7 5982.2 


8344.8 


1802.7 


2630.2 


6087.6 


28 


80 


8278.8 


1767.6 


2556.2 5985.6 


8347.1 


1808.9 


2632.7 


6091 .2 


80 


82 


8280.6 


1768.8 


2558.7 5989.1 


8319.4 


1805.1 


2635.3 


6094.7 


82 


84 


8282.9 


17i0.0 


2561.2 5992.6 


8:351.7 


1806.8 


2637.9 


6098.3 


84 


86 


8285.2 


1771.2 


2563.8 5996.1 


8354.0 


1807.6 


2640.5 


6101.8 


86 


3d 


8287.5 


1772.5 


2566.3 5999.6 


8356.3 


1808.8 


2643.1 


6105.4 


88 


40 


8289.8 


1773.7 


2568.8 6008.1 


8358.5 


1810.0 


2645.7 


6109.0 


40 


42 


8292.1 


1774.9 


2571.3 6006.6 


a360.8 


1811.2 


2648.3 


6112.5 


42 


44 


8294.4 


1776.1 


2573.9 6010.1 


8363.1 


1812.4 


2650.9 


6116 1 


44 


46 


8296.7 


1777.8 


2576.4 6013.6 


8365.4 


1813.6 


2653.5 


6119.7 


46 


48 


8299.0 


1778.5 


2578.9 6017.1 


8367.7 


1814.9 


2656.1 


6123.2 


48 


50 


8301.3 


1779.7 


2581.5 6020.0 


8369.9 


1816.1 


2658.7 


6126.8 


50 


52 


8303.6 


1780.9 


2584.0 6024.1 


8372.2 


1817.3 


2661.3 


6130.4 


52 


54 


8805.9 


1782.1 


2586.6 6027.0 


8374.5 


1818.5 


2668.9 


6133.9 


54 


56 


8308.2 


1788.8 


2589.1 6031.1 


8876.8 


1819.7 


2666.6 


6137.5 


56 


58 


8810.5 


1784.5 


2591.7 6034.6 


8379.1 


1820.9 


2669.2 


6141.1 


68 


60 


a312.8 


1785.7 


2594.2 6038.2 


8881.3 


1822.2 


2671.8 


6144.7 


60 



Digitized by VjOOQIC 



234 


FUNCTIONS OF A ONE-DEGREE CURVE 




f 


94» 


»&• 






L.C. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 


/ 





8381.3 


1822.2 


2671.8 


6144.7 


8449.2 


-1858.9 


2751.5 


6253.2 





2 


8383.6 


1828.4 


2674.4 


6148.8 


8451.5 


1860.1 


2754.2 


6256.9 


8 


4 


8385.9 


1824.6 


2677.0 


6151.9 


8453.7 


1861.3 


2756.9 


6860.5 


4 


6 


8388.1 


1825.8 


2679.7 


6155.4 


8456.0 


1862.6 


2759.6 


6264.2 


• 6 


6 


8390.4 


1827.0 


2682.3 


6159.0 


8458.2 


1863.8 


2762.3 


6267.8 


8 


10 


8392.7 


1828.3 


2684.9 


6162.6 


8460.4 


1865.0 


2T65.0 


6271.5 


10 


18 


8395.0 


1829.5 


2687.6 


6166.2 


8462.7 


1866.3 


2767.7 


6275.2 


12 


14 


8397.2 


1830.7 


2690.2 


6169.8 


8464.9 


1867.5 


2770.4 


6878.8 


14 


16 


8399.5 


1831.9 


2092.8 


6173.4 


8467.2 


1868.7 


2773.1 


6282.5 


16 


18 


8401.7 


1833.1 


2695.6 


6177.0 


8469.4 


1869.9 


2775.8 


6286.2 


18 


20 


8404.0 


1834.4 


2698.1 


6180.6 


8471.7 


1871.2 


2778.6 


6889.8 


20 


2i 


8406.3 


1885.6 


2700.8 


6184.2 


8473.9 


1872.4 


2781.2 


6293.5 


22 


24 


8408.5 


1836.8 


2703.4 


6187.8 


8476.2 


1873.6 


2784.0 


6297.2 


24 


26 


8410.8 


1888.0 


2706.1 


6191.5 


8478.4 


1874.9 


2786.7 


6300.9 


26 


28 


8413.1 


1889.3 


2708.7 


6195.1 


8480.7 


1876.1 


2789.4 


6804.6 


28 


30 


8415.3 


1840.5 


2711.4 


6198.7 


8482.9 


1877.3 


2792.1 


6308.» 


SO 


3-^ 


8417.6 


1841.7 


2714.0 


6202.3 


8485.1 


1878.6 


2794.9 


6311.9 


82 


34 


8419.9 


1842.9 


. 2716.7 


6205.9 


8487.4 


1879.8 


8797.6 


6315.6 


34 


36 


8422.1 


1844.2 


2719.3 


6209.5 


8489.6 


1881.0 


2800.8 


6319.3 


36 


38 


8424.4 


1845.4 


2722.0 


6213.2 


8491.9 


1882.3 


2803.1 


6388.0 


88 


40 


8426.6 


1846.6 


U724.7 


6216.8 


8494.1 


1883.5 


2805.8 


6386.7 


40 


42 


8428.9 


1847.8 


a727.3 


6220.4 


8496.3 


1884.8 


2808.6 


6330.4 


42 


44 


8431.2 


1849.1 


2730.0 


6224.1 


8498.6 


1886.0 


2811.8 


6334.1 


44 


46 


8433.4 


1860.3 


2732.7 


6227.7 


8500.8 


1887.8 


2814.1 


6387.8 


46 


48 


8435.7 


1851.5 


2736.4 


6231.3 


8508.0 


1888.5 


2816.8 


6341.5 


48 


50 


8487.9 


1852.7 


2738.0 


6235.0 


8505.3 


1889.7 


2819.6 


6345.8 


50 


52 


8440.2 


1854.0 


2740.7 


6238.6 


8507.5 


1890.9 


2822.3 


6849.0 


52 


54 


8442.4 


1855.2 


2743.4 


6242.3 


8509.8 


1892.2 


2825.1 


6358.7 


54 


56 


8444.7 


1856.4 


2746.1 


6245.9 


8512.0 


1893.4 


2827.8 


6356.4 


56 


58 


8447.0 


1857.6 


2748.8 


6249.6 


8514.2 


1894.6 


2830.6 


6360.1 


58 


60 


8449.2 


1858.9 


2751.5 


6253.2 


a516.4 


1895.9 


2833.4 


636:^.8 


60 


■ 


960 1 


W* 


/ 


/ 


L.C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





8516.4 


1895.9 


2833.4 


6368.8 


8583.0 


1933.2 


2017.6 


6476.6 





2 


8518.7 


1897.1 


2836.1 


6367.5 


8585.2 


1934.4 


2920.3 


6480.4 


8 


4 


8520.9 


1898.4 


2838.9 


6371.3 


8587.5 


19&5.7 


8923.2 


6484.2 


4 


6 


8523.1 


1899.6 


2841.7 


6375.0 


8589.7 


1936.9 


2926.0 


6488.0 


6 


8 


8525.4 


1900..S 


2844.5 


6378.7 


8591.9 


1988.2 


2928.9 


6491.8 


8 


10 


8527.6 


1902.1 


2847.2 


6382.5 


8594.1 


1939.4 


2931.7 


6495.6 


10 


12 


8529.8 


19a3.3 


2850.0 


6386.2 


8596.8 


1940.7 


2984.6 


6499.4 


12 


14 


a532.0 


1904.6 


2852.8 


6389.9 


859S.5 


1941.9 


2937.5 


6503.2 


14 


16 


8534.8 


1905.8 


2855.6 


6393.7 


8600.7 


1943.2 


2940.3 


6507.1 


16 


18 


8536.5 


1907.0 


2868.4 


6397.4 


8602.9 


1944.4 


2943.2 


6510.9 


18 


90 


8638.7 


1908.3 


2861.2 


6401.2 


8605.1 


1945.7 


2946.1 


6514.7 


80 


2:2 


8540.9 


1909.5 


2864.0 


6404.9 


8607.3 


1946.9 


2948.9 


6518.5 


82 


24 


8543.2 


1910.8 


2866.7 


6408.7 


8609.5 


1948.2 


2961.8 


6522.3 


84 


26 


8545.4 


1912.0 


2869.5 


6412.4 


8611.7 


1949.4 


2954.7 


6686.2 


26 


28 


8547.6 


1913.8 


2872.8 


6416.2 


8613.9 


1950.7 


2957.6 


6530.0 


28 


30 


8549.8 


1914.5 


2875.1 


6419.9 


8616.1 


1952.0 


2960.4 


6633.8 


80 


32 


85S2.0 


1915.7 


2877.9 


6428.7 


8618.3 


1953.2 


2963.3 


6587.7 


88 


31 


8554.3 


1917.© 


2880.8 


64275 


8620.5 


1954.5 


2966.2 


6541.5 


84 


36 


8556.5 


1918.2 


2883.6 


6431 2 


8622.7 


1955.7 


2969.1 


6545.3 


86 


38 


8558.7 


1919.5 


2886.4 


64350 


8624.9 


1957.0 


2972.0 


6549.2 


88 


40 


8S60.9 


1920.7 


2889.2 


6438.8 


8627.1 


1958.2 


2974.9 


6553.0 


40 


42 


8563.1 


1922.Q 


2892.0 


6442 5 


8629.3 


1959 5 


2977.8 


6566.9 


42 


44 


86653 


1923.2 


28W 8 


6446.3 


8631.5 


1960.7 


2960.7 


6560.7 


44 


46 


8567 6 


1924 5 


2897.7 


6450.1 


8633.7 


1962.0 


2983.6 


6S64.6 


46 


48 


8569.8 


1925.7 


2900.5 


6453.9 


8635.8 


1963.2 


2966.5 


6668.4 


48 


50 


8572.0 


1927.0 


2903.3 


6457.6 


8638.0 


196i.5 


2989.4 


6578.3 


50 


52 


8574.2 


1928.2 


2906.1 


6461.4 


8640.2 


1965.8 


8992.3 


6576.8 


58 


54 


8576.4 


1929 4 


2909.0 


6465.2 


8642.4 


1967.0 


2995.2 


6580.0 


54 


56 


8578.6 


1930.7 


2911.8 


6469.0 


8044.6 


1968.3 


2988.1 


6683.9 


56 


58 


8580.8 


1931.9 


2914.7 


6472.8 


8646 8 


1969.5 


9001.1 


6587.7 


58 


00 


8583.0 


19332 


2917.5 


6478.6 


86490 


1970.8 


90040 


6591.6 


» 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEGREE CURVE 


235 




»8» 


W 







L.C. 


M. 


.». T, 


L. C. 


M. 


E. 


T. 




8649.0 


1970.8 


8004.0 659t.6 


8714.3 


2008.7 


3092.9 


6709.0 





2 


8651.8 


1972.0 


3006.9 6695.5 


8716.4 


2009.9 


3095.9 


6718.9 


8 


4 


8653.8 


1973.8 


8009.8 6599.4 


8718.6 


8011.8 


3098.9 


6716.9 


4 


6 




1974.6 


3018.8 6603.2 


8720.7 


2018.5 


3101.9 


6780.8 


6 


8 


8657.7 


1975.8 


3015.7 6607.1 


8728.9 


8013.7 


3104.9 


6784.8 


8 


10 


8659.0 


1977.1 


3018.6 6611.0 


8726.1 


8015.0 


8107.9 


6728.8 


10 


18 


8668.1 


1978.3 


3081.6 6614.9 


8727.8 


2016.8 


8111.0 


6788.7 


12 


14 


8664.3 


1979.6 


8084.5 6618.8 


8729.4 


2017.5 


3114.0 


6736.7 


14 


16 


8666.4 


1980.9 


3027.5 6688.7 


8731.5 


8018.8 


3117.0 


6710.7 


16 


18 


8668.6 


1982.1 


3030.4 6G86.6 


8733.7 


2080.1 


3180.0 


6744.6 


18 


ao 


8670.8 


1988.4 


3033.3 6630.5 


8735.9 


2081.4 


3183.1 


6748.6 


80 


28 


8673.0 


1984.6 


3036.8 6634.4 


8738.0 


8088.6 


8186.1 


6758.6 


28 


84 


8675.8 


1985.9 


3039.3 6638.8 


8740.2 


2083.9 


3129.1 


6756.6 


84 


86 


8677.8 


1987.2 


3042.8 6648.8 


8742.8 


2025.8 


3133.8 


6760.6 


86 


88 


8679.5 


1988.4 


8045.8 6C46.1 


8744.5 


2026.4 


8135.8 


0764.0 


88 


SO 


8681.7 


1989.7 


8048.1 6650.0 


8746.6 


2087.7 


3138.3 


6768.6 


80 


38 


8683.9 


1991.0 


3051.1 6653.9 


8748.8 


8089.0 


3141.3 


67?2.6 


88 


34 


8686.0 


1992.2 


3054.1 6657.8 


8750.9 


8030.3 


3144.4 


6776.6 


34 


36 


8688.2 


1993.5 


3057.0 6661.7 


8753.1 


2081.5 


3147.4 


6780.6 


36 


88 


8690.4 


1994.7 


8060.0 6665.7 


8755.3 


8038.8 


3150.5 


6784.6 


38 


40 


8693.6 


1996.0 


8063.0 6669.6 


8757.4 


8034.1 


3153.5 


6788.6 


40 


48 


8694.7 


1997.3 


8066.0 6673.5 


8759.5 


8a85.4 


3156.6 


6798.6 


42 


44 


8696.9 


1998.6 


8068.9 6677.4 


87C1.7 


2036.6 


8159.7 


6796.6 


44 


46 


8699.1 


1999.8 


8071.9 6681.4 


8763.8 


2037.9 


3168.7 


6800.6 


46 


48 


8701.8 


2001.1 


3074.9 6685.3 


876G.0 


2039.8 


3165.8 


6804.6 


48 


60 


8703.4 


2008.3 


3077.9 6689.8 


8768.1 


2040.6 


3168.9 


6808.6 


50 


&8 


8705.6 


8003.6 


3080.9 6693.2 


8770.8 


8041.7 


3178.0 


6818.6 


58 


54 


8707.8 


8004.9 


3083.9 6697.1 


8772.4 


2043.0 


8175.1 


6816.7 


54 


56 


8709.9 


2006.1 


8086.9 6701.1 


8774.6 


2044.3 


3178.1 


6880.7 


56 


58 


8712.1 


2007.4 


8089.9 6705.8 


8776.7 


2045.6 


3181.2 


6884.7 


58 


60 


8714.3 


8008.7 


8092.9 6709.0 


8778.9 


2046.8 


3184.3 


6888.8 


60 


1 



100° { 


lOl* 


/ 


L C. 

8778.9 


M. 


E. T. 


L. C. 


M. 


E. 


T. 


8046.8 


3184.3 6888.8 


8842.8 


2085.3 


3278.3 


6951.0 





8 


8781.0 


8048.1 


3187.4 6838.8 


8844.9 


2066.6 


8281.5 


6955.8 


8 


4 


8783.1 


2049.4 


3190.5 6836.8 


8847.0 


2087.8 


8284.7 


6959.3 


4 


6 


8785.3 


8050.7 


3193.6 6840.9 


8849.2 


2069.1 


3287.9 


6963.4 


6 


8 


8787.4 


8051.9 


3196.7 6844.9 


8861.3 


8090.4 


8291.1 


6967.6 


8 


10 


8789.6 


8058.8 


3199.8 6849.0 


8853.4 


8091.7 


3294.3 


6971.7 


10 


18 


8791.7 


2054.5 


3208.9 6853.0 


8856.5 


8093.0 


3297.5 


6975.8 


12 


14 


8793.9 


2055.8 


3806.0 6857.1 


8857.6 


8094.3 


3300.7 


6960.0 


14 


16 


8796.0 


2057.1 


3809.1 6861.1 


8859.8 


8095.6 


3303.0 


6964.1 


16 


18 


8798.9 


2058.8 


3818.8 6865.8 


8861.9 


8096.9 


3307.1 


6988.8 


18 


80 


8800.8 


2059.6 


3815.4 6869.2 


8864.0 


8098.8 


asio.s 


6998.4 


80 


88 


8808.4 


2060.9 


3818.6 6873.3 


8866.1 


8099.4 


3313.5 


6996.6 


82 


84 


8804.5 


2068.2 


3281.6 6877.4 


8868.2 


8100.7 


3316.7 


7000.7 


24 


86 


8806.7 


8063.5 


3884.7 6881.4 


8870.3 


8108.0 


3319.9 


7004.9 


26 


88 


8808.8 


8064.7 


3887.9 6885.5 


8872.4 


8103.3 


8323.1 


7009.0 


88 


80 


8810.9 


8066.0 


8831 6889.6 


8874.5 


8104.8 


3326.4 


7013.8 


80 


88 


8813.1 


8067.3 


3834.1 6893.7 


8876.7 


8105.9 


3329.6 


7017.3 


38 


84 


8815.8 


8068.6 


38373 6897.8 


8878.8 


8107.8 


3332.8 


7081.5 


34 


86 


8817.3' 


8069.9 


8840.4 6901.8 


8880.9 


8108.5 




7085.7 


36 


38 


8819.6 


8071.1 


3848.5 6905.9 


8883.0 


8109.8 


3339i3 


7089.9 


38 


40 


8881.6 


8078.4 


8846.7 6010.0 


8886.1 


2111.1 


3348.5 


7084.0 


40 


48 


8888.7 


8073.7 


3849.8 6914.1 


8887.2 


2118.4 


3345.8 


7038.8 


48 


44 


8825.8 


8075.0 


3853.0 6918.8 


8889.8 


8113.6 


3349.0 


7048.4 


44 


46 


8888.0 


8076.3 


3856.8 6988.3 


8891.4 


8114.9 


3358.3 


7046.6 


46 


48 


8880.1 


8077.6 


3859.3 6986.4 


8893.5 


8116.8 


3355.5 


7050.8 


48 


60 


8838.2 


8078.9 


3268.5 6930.5 


8895.6 


8117.5 


3358.8 


7056.0 


50 


68 


8834.3 


8080.1 


8865.7 6934.6 


8897.7 


8118.8 


3368.0 


7059.8 


58 


54 


8836.4 


8081.4 


3868.8 6038.7 


8899.8 


8120.1 


83655 


7063 4 


54 


56 


R83R6 


8088.7 


3878.0 6948.8 


8901.9 


2121.4 


3368.7 


7067.6 


56 


68 


8840.7 


8084.0 


8875.8 6946.9 


8904.0 


8122.7 


3378.0 


7071.8 


58 


60 


8842.8 


8065.3 


3878.3 6951.0 


8906.1 


8184.0 


3375.1 


7076.0 


60 



Digitized by VjOOQIC 



^6 


PUNOTIONS OF A ONE-DEGREfi CtTRVE 




* / 


IW 1 


10»- 


/ 


L.C. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 





8906.1 


2124.0 


3:^75.1 


7076.0 


8968.7 


2163.0 


8474.6 


7208.6 





2 


8908.2 


2125.3 


3378.3 


7080.2 


8970.8 


2164.3 


8478.0 


7207.9 


2 


4 


8910.3 


2126 6 


3381.6 


7084.4 


8972.9 


2165.6 


3481.4 


7212.2 


4 


6 


8912.4 


2127.9 


8384.9 


7088.6 


8974.9 


2166.9 


3484.7 


72165 


6 


8 


8914.5 


2129.2 


33S8.2 


7092.8 


8977.0 


2168.2 


3488.1 


7220.8 


8 


10 


8916.6 


2130.5 


3391.5 


7097.1 


8979.1 


2169.5 


3491.5 


7225.1 


10 


12 


8918.7 


2131.8 


8394 7 


7101.3 


8981.1 


2170.8 


3494.9 


7229.5 


12 


14 


8920.8 


2133.1 


;«98.0 


7105.6 


8983.2 


2172.1 


3498.3 


7233.8 


14 


16 


8922.9 


2134.4 


8101.3 


7109.7 


8985.3 


2173.4 


8501.6 


7238.1 


16 


18 


8925.0 


2135.7 


8104.6 


7114.0 


8987.3 


2174.7 


3505.3 


7242 4 


18 


20 


8927.0 


2137.0 


8407.9 


7118.2 


89894 


2176.1 


3506.4 


7246.8 


20 


22 


8929.1 


2138.3 


8U1.2 


7122.4 


8991.5 


2177.4 


3511.8 


7251.1 


ais 


S4 


8981.2 


2139.6 


3114.5 


7126.7 


8998.5 


2178.7 


3515 2 


7265.4 


24 


26 


89:j3.3 


2140.9 


;3417.9 


7180.9 


6995.6 


2180.0 


3518.7 


7259.8 


26 


28 


893) 4 


2142.2 


3121.2 


7135.2 


8997.7 


2181.3 


3522.1 


7264.1 


28 


30 


8937.5 


2143.5 


3424.5 


7139.4 


8999.7 


2182.6 


3525.5 


7268.6 


80 


3^ 


8939.6 


2144 8 


3427.8 


7143.7 


9001.8 


2183.9 


3528.9 


7272.8 


32 


34 


8941.6 


2146.1 


3481.1 


7148.0 


9003.9 


2185.2 


3532.3 


7277.2 


34 


36 


8943.7 


2147.4 


3434.5 


7152.2 


9005.9 


2186.5 


3585.7 


7281.5 


36 


38 


8945.8 


2148.7 


3437.8 


7156.5 


9006.0 


8187.8 


3539.2 


7285.9 


38 


40 


8947.9 


2150.0 


8141.1 


7160.7 


9010.0 


2189.1 


3542.6 


7290.3 


40 


42 


8960.0 


2151.3 


3444.4 


7165.0 


9012.1 


2190.5 


3546.0 


7294.6 


42 


44 


8952.1 


2152.6 


3447.8 


7169.3 


9014.2 


2191.8 


3549.5 


7299.0 


44 


46 


8954.1 


2158.9 


3451.1 


7173.6 


9016.2 


2193.1 


8552.9 


7303.4 


46 


48 


8956.8 


2155.2 


34.54.5 


7177.9 


9018.3 


2194.4 


3656.3 


7807.7 


48 


50 


8958.3 


2156 5 


8457.8 


7182. 1 


9020.3 


2195 7 


3659.8 


7312.1 


50 


52 


8960.4 


2157.8 


3461.2 


7186.4 


9022.4 


2197.0 


8563.2 


7316.5 


52 


54 


8982.5 


2159.1 


3464.5 


7190.7 


9024.5 


8198.3 


8666.7 


7380.9 


54 


56 


8964.5 


2160.4 


3467.9 


7195.0 


9026.5 


2199.6 


3570.2 


7825.3 


56 


58 


H966.6 


2161.7 


3471.2 


7199.3 


9028.6 


2200.9 


3573.6 


7829.7 


58 


60 


8968.7 


2163.0 


3474.6 


7203.6 


9030.6 


2202.3 


3577.1 


7334.1 


60 


' 


104" 1 


106° 


/ 


L. C. 


M. 


E. 


T. 


L.C. 


M. 


E. 


T. 





90:50.6 


2202.3 


3577.1 


7334.1 


9091.8 


2241.8 


3682.6 


7467.5 





2 


9082 7 


2203.6 


3580.5 


7338.5 


9093.9 


2243.1 


3686.1 


7472.0 


2 


4 


90:i4.7 


2204.9 


3584.0 


7342.9 


9095.9 


2244.4 


3689.7 


V476.5 


4 


6 


9036.8 


2206.2 


3587.5 


7347.3 


9097.9 


£245.8 


3693.3 


7481.0 


6 


8 


90J38.8 


2207.5 


3591.0 


7351.7 


9099.9 


2247.1 


3696.9 


7485.5 


8 


10 


9040.9 


2208 8 


3594.4 


7356.1 


9102.0 


2248.4 


3700.4 


7490.0 


10 


12 


9042.9 


2210 2 


8597.9 


7360.5 


9104.0 


2249.7 


3704.0 


7494.5 


12 


14 


90450 


2211.5 


3601.4 


7364.9 


9106.0 


2251.1 


3707.C 


7499.1 


14 


16 


9047.0 


2212.8 


3604.9 


7869.4 


9108.0 


2852.4 


3711.2 


7503.6 


16 


18 


9049.1 


2214.1 


3606.4 


7378.8 


9110.1 


2253.7 


8714.8 


7508.1 


18 


20 


9051.1 


2215.4 


3611.9 


7378.2 


9112.1 


2255.0 


3718.4 


7512.6 


20 


22 


9053.1 


2216.7 


3615.4 


7382.0 


9114.1 


2256.4 


3722.0 


7517.2 


28 


24 


9055.2 


2218 


3618.9 


7387.1 


9116.1 


2257.7 


3726.6 


7521.7 


24 


26 


9057.2 


2219 4 


3622.4 


7391.5 


9118.1 


22ri9.0 


3729.3 


7526.3 


26 


28 


9059.3 


2220.7 


3625.9 


7396.0 


9180.2 


2260.3 


3732.9 


7630.8 


28 


30 


9061.3 


2222.0 


3029.4 


7400.4 


9122.2 


2261.7 


3730.5 


7535.3 


80 


3-2 


9063.3 


2223.3 


3633.0 


7404.8 


9124.2 


2268.0 


3740.1 


7539 9 


82 


34 


9065.4 


2224.6 


3fi36.5 


7409 3 


9126.2 


2264.3 


3748.7 


7544.4 


34 


36 


9067.4 


22260 


3640.0 


7413 8 


9128 2 


2266.7 


3747.4 


7549.0 


86 


38 


9J69.5 


2227.3 


364:^.5 


7418.2 


9130.2 


2267.0 


8751.0 


7553.6 


88 


40 


9071.5 


2228.6 


3647.1 


7422.7 


9182.3 


2268.3 


8754.6 


7568.1 


40 


42 


9073.5 


2229.9 


3650.6 


7427.1 


9134.3 


2269.6 


3758.3 


7562.7 


42 


44 


9075 6 


2231 2 


3654.1 


7431.6 


9136.3 


2271.0 


8761.9 


7667.3 


44 


46 


9077 6 


2232.6 


3657.7 


7436.1 


9138.3 


2272.3 


3766.6 


7671.8 


46 


48 


9079.6 


2233.9 


3661.2 


7440.6 


9140.3 


2273 6 


3769.2 


7576.4 


48 


60 


9081.7 


2235.2 


3664.8 


7445.0 


9142.3 


2275.0 


3772.9 


7581.0 


60 


52 


9083.7 


2236.5 


3668.3 


7449.5 


9144.3 


2276.3 


3776.5 


7586.6 


58 


54 


9085.7 


2237.8 


3671.9 


7454.0 


9146.3 


2277.6 


3780.2 


7690.2 


54 


66 


9087.8 


2239.2 


3675.4 


7458 5 


9148.3 


2278.9 


3783.9 


7594.8 


50 


58 


9089.8 


2-240 5 


3679.0 


7463.0 


9150.4 


2280.3 


8787.5 


7599.4 


^ , 


60 


9091 8 


2241.8 


3682.6 


7467 5 


91524 


2281.6 


3791 2 


7fi04 


60 ! 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEGREE CURVE 


237 


/ 


106O 1 


107» 




L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





9152.4 


2281.6 


8791.2 


7604.0 


9212.2 


2821.7 


8903.1 


7743.7 





2 


9154.4 


2282.9 


8794.9 


7608.6 


9214.2 


2323.0 


3906.9 


7748.4 


2 


4 


9156.4 


2284.3 


8798.6 


7613.2 


9216.2 


2324.4 


3910.7 


7753.1 


4 


6 


9158.4 


2285.6 


3802.8 


7617.8 


9218.1 


2325.7 


3914.5 


7757.8 


6 


8 


9160.4 


2286.9 


3805.9 


7622.4 


9220.1 


2327.0 


3918.3 


7762.5 


8 


10 


9162.4 


2288.8 


8809.6 


7627.0 


9222.1 


2328.4 


8922.1 


7767.3 


10 


12 


9164.4 


2289.6 


8813.8 


7631.7 


9224,1 


2329.7 


3925.9 


7772.0 


12 


14 


9166.4 


2290.9 


8817.0 


7636.8 


9226.1 


2331.1 


3929.7 


7776.7 


14 


10 


9168.4 


2292.3 


8820.7 


7610.9 


9228.1 


2332.4 


3933.6 


7781 .5 


16 


18 


9170.4 


2293.6 


8824.4 


7645.5 


92:30.0 


2333.7 


3937.4 


7786.2 


18 


20 


9172.4 


2294.9 


8828.1 


7650.2 


9232.0 


2335.1 


8941.2 


7791.0 


20 


22 


9174.4 


2296.3 


8881.9 


7654.8 


9234.0 


2836.4 


3045.0 


7795.7 


22 


24 


9176.4 


2297.6 


8835.6 


7659.5 


9235.9 


2337.8 


3948.9 


7800.5 


24 


26 


9178.4 


2298.9 


3839.3 


7664.1 


9237.9 


2839.1 


3952.7 


7805.2 


26 


28 


9180.4 


2800.3 


3843.0 


7668.8 


9239.9 


2340.5 


3956.5 


7810.0 


28 


80 


9182.4 


2801.6 


3846.7 


7673.4 


9241.9 


2841.8 


8960.4 


7814.7 


SO 


32 


9184.4 


2802.9 


8850.6 


7678.1 


9243.8 


2343.1 


3964 2 


7819.5 


82 


84 


9186.4 


2304.8 


3854.2 


7682.7 


9245.8 


2844.5 


3968.1 


7824.3 


34 


86 


9188.4 


2805.6 


8858.0 


7687.4 


9247.8 


2345.8 


8971.9 


7829.1 


86 


88 


9190.4 


2306.9 


8861.7 


7692.1 


9249.7 


2847.2 


3975.8 


7833.8 


38 


40 


9192.4 


2808.3 


8865.4 


7696.7 


9251.7 


2848.6 


3979.6 


7888.6 


40 


42 


9194.4 


2809.6 


3869.2 


7701 .4 


9258.7 


2849.9 


3983.6 


7848.4 


42 


44 


9196.3 


2811.0 


8873.0 


7706.1 


9255.6 


2351.2 


8987.4 


7848.2 


44 


46 


9198.3 


2812.3 


8876.7 


7710.8 


9257.6 


2352.6 


3991.8 


7853.0 


46 


48 


9200.3 


2813.6 


8880.5 


7715.5 


9259.6 


23.53.9 


3995.1 


7857.8 


48 


50 


9202.3 


2315.0 


8884.2 


7720.1 


9261.5 


2355.3 


8999.0 


7862.6 


50 


62 


9204.3 


2316.8 


8888.0 


7724.8 


9263.5 


2356.6 


4002.9 


7867.4 


52 


54 


9206.3 


2817.7 


8891.8 


7729.5 


9265.4 


2358.0 


4006.8 


7872.2 


54 


56 


9208.2 


2819.0 


3895.6 


7784.2 


9267.4 


2359.3 


4010 7 


7877.0 


56 


58 


9210.2 


2820.8 


3899.3 


7739.0 


9269.4 


2860.7 


4014.6 


7881.9 


58 


60 


9212.2 


2321.7 


3903.1 


7743.7 


9271.8 


2362.0 


4018.5 


7886.7 


60 


/ 


108° 


109« 


/ 


L.C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





9271 3 


2362.0 


4018.6 


7886.7 


9329.8 


2402.6 


4137.4 


8033.2 





2 


92; 1.3 


2868.3 


4022.4 


7891.5 


9831.7 


2403.9 


4141.4 


8038.1 


2 


4 


9275.3 


2864.7 


4026.8 


7896.8 


9333.6 


2405.3 


4145.4 


8043.1 


4 


6 


9277,2 


2366.0 


4080.2 


7901.2 


9335.6 


2406.6 


4149.5 


8048.0 


6 


8 


92Tff.2 


2367.4 


4034.1 


7906.0 


9337.5 


2408.0 


4153.6 


8053.0 


8 


10 


92;' 1. 1 


2368.7 


4038.0 


7910.8 


9339.4 


2409.4 


4157.5 


8057.9 


10 


12 


9Z^i A 


2370.1 


4042.0 


7915.7 


9341.4 


2410.7 


4161.6 


8062.9 


12 


14 


92:^5.0 


2871.4 


4045.9 


7920.5 


9343.3 


2412.1 


4165.6 


8067.9 


14 


16 


92ti:.0 


2872.8 


4049.8 


7925.4 


9345.2 


2418.4 


4169.7 


8072.8 


16 


18 


9288.9 


2874.1 


4053.8 


7930.3 


9347.2 


2414.8 


4178.8 


8077.8 


18 


20 


9290.9 


2375.5 


4057.7 


7985.1 


9349.1 


2416.2 


4177.8 


8082.8 


20 


22 


9292.8 


2876.8 


4061.6 


7940.0 


9351.0 


2417.5 


4181.9 


8087.8 


22 


24 


9294.8 


2378.2 


4065.6 


7944.8 


9852.9 


2418.9 


4186.0 


8092.8 


24 


26 


9296.7 


2879.5 


4069.5 


7949.7 


9354.9 


2420.2 


4190.0 


8097.8 


26 


28 


9298.7 


2880.9 


4073.5 


7954.6 


9356.8 


2421.6 


4193.1 


8102.8 


28 


80 


93U0.6 


2382.3 


4077.5 


7959.5 


9358.7 


2423.0 


4198.2 


8107.8 


30 


82 


9802.6 


2888.6 


4081.4 


7964.4 


9360.6 


2424.3 


4202.3 


8112.8 


32 


84 


9804.5 


2885.0 


4085.4 


7969.3 


9362.6 


2425.7 


4206.4 


8117.8 


34 


86 


9806.5 


2386.3 


4089.4 


7974.1 


9364.5 


2427.0 


4210.5 


8122.8 


86 


88 


9808. 4 


2887.7 


4093.4 


7979.0 


9366.4 


2428.4 


4214.6 


8127.8 


38 


40 


9810.4 


2889.0 


4097.3 


7983.9 


9368.3 


2430.0 


4218.7 


8182.8 


40 


42 


0312.3 


2390.4 


4101.3 


7988.8 


9870.2 


2431.1 


4222.8 


8187.9 


42 


44 


9814.2 


2391.7 


4105.3 


7993.8 


9372.2 


2482.6 


4226.9 


8142.9 


44 


46 


9816.2 


2393.1 


4109 8 


7998.7 


9374.1 


2438.9 


4231.0 


8147.9 


46 


48 


9318.1 


2394.4 


4113.8 


8003.6 


9376.0 


2436.2 


4235.1 


8168.0 


48 


50 


9320.1 


2395.8 


4117.3 


8008.5 


9377.9 


2436.6 


4239 3 


8158.0 


50 


52 


9822.0 


2397.2 


4121.8 


8013.4 


9379.8 


2438.0 


4243.4 


8163.1 


52 


54 


9323.9 


2398.5 


41.i5.8 


8018.4 


9381.7 


2439.3 


4247.5 


8168.1 


54 


56 


9325.9 


2899.9 


4129.3 


80:>3 3 


9383.7 


2440.7 


4251.7 


8173.2 


56 


58 


9327.8 


2401.2 


4133 4 


80282 


9385.6 


2442.1 


4255.8 


8178.2 


58 


60 


9329.8 


2402.6 


4137.4 


8033 2 1 93S7.5 


2443 4 


4260.0 


8188.3 


60 



Digitized by VjOOQIC 



238 


FUNCTIONS OF A ONE>-DEGREE CURVE 




/ 


no* 1 


111* 


/ 


L. C. 


M. 


E. 


T. 


L. 0. 


M. 


E. 


T. 





9387.5 


2443.4 


4260.0 


8183.3 


9444.5 


2484.5 


4386.4 


8337.2 





2 


9389.4 


2444.8 


4264.1 


8188.4 


9446.4 


2485.9 


4890.7 


8842.4 


8 


4 


9391.3 


2446.1 


4268.3 


8193.4 


9448.3 


2487.2 


4395 


8847.6 


4 


6 


9393 2 


2447.6 


4272.4 


8198.5 


9450.1 


2488.6 


4399 3 


a352.8 


6 


8 


9395.1 


2448.9 


4276.6 


8208.6 


9452.0 


2490.0 


4403.6 


8358.0 


8 


10 


9397.0 


2450.2 


4280.8 


8208.7 


9458.9 


2491.4 


4407.9 


8368.2 


10 


12 


9398.9 


2451.6 


4284.9 


8213.8 


9465.8 


2492.7 


4412.2 


8368.6 


12 


14 


9400.8 


2453.0 


4289.1 


8218.9 


9457.7 


2494.1 


4416.5 


8373.7 


14 


1« 


9402.7 


2454.8 


4293.3 


8224.0 


9459.6 


2496.5 


4420.8 


8378 9 


16 


18 


9404.7 


2455.7 


4297.6 


8229.1 


9461 .4 


2496.9 


4425.1 


8384 1 


18 


30 


9406.6 


2467.1 


4301.7 


8234.8 


9463.8 


2498.2 


4429.5 


8389.4 


20 


22 


9408.6 


2458.4 


4305.9 


8239.3 


9465.2 


2499.6 


4433.8 


8894 6 


23 


24 


9410.4 


2459 8 


4310.1 


8244.4 


9467.1 


2501.0 


4438.1 


83t9.9 


24 


26 


9412.8 


2461.2 


4314.8 


8249.5 


9469.0 


2502.4 


4442.5 


8406.1 


86 


28 


9414.2 


2462.6 


4318.6 


8254.6 


9470.8 


2608.8 


4446.8 


8410.4 


28 


80 


9416.1 


2463.9 


4322.7 


8259.8 


9472.7 


2605.1 


4461.2 


8415.6 


80 


3-J 


9418.0 


2465.3 


4326.9 


8264.9 


9474.6 


2606.5 


4455.5 


84209 


82 


34 


9419.9 


2466.7 


4331.1 


8270.0 


9476.5 


2507.9 


4459.9 


8426.2 


34 


86 


9421.8 


2468.0 


4.3:35.4 


8275.2 


9478.8 


2500.8 


4464.2 


8431.4 


86 


38 


9423.7 


2469.4 


4339.6 


8280.3 


9480.2 


2610.6 


4468.6 


8486.7 


88 


40 


9426.6 


2470.8 


4343.8 


8285.5 


9482.1 


2512.0 


4478.0 


8442.0 


40 


42 


9427.6 


2472.1 


4348.1 


8290.6 


9484.0 


2518.4 


4477.8 


8447.3 


42 


44 


9429.3 


2473.5 


4352.3 


8296.8 


9485.8 


2514.8 


4481.7 


8452.6 


44 


46 


9481.2 


2474.9 


49^.6 


8300.9 


9487.7 


2516.2 


4486.1 


8467.9 


46 


48 


9483.1 


2476.3 


4360.8 


8306.1 


9489.6 


2517.5 


4490.5 


8468.2 


48 


50 


9435.0 


2477.6 


4365.1 


8311.3 


9491.4 


2518.9 


4494.9 


8468.5 


60 


62 


9436.9 


2479.0 


4369.3 


8316.6 


9493.8 


2520.3 


4499.3 


8473.8 


52 


54 


9488.8 


2480.4 


4373.6 


a321.6 


9495.2 


2621.7 


4503.7 


8479.1 


54 


&6 


9440.7 


2481.7 


4377.9 


8326.8 


9497.0 


2523.1 


4508.1 


8484.4 


56 


58 


9442.6 


2483.1 


438-2.2 


8332.0 


9498.9 


2524.5 


4512.5 


8489.7 


58 


60 


9444.6 


2484. 5 


4386.4 


8337.2 


9500.8 


2525.8 


4516.9 


8496.1 


60 


/ 


112» 1 


118* 


/ 


L.O. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





9500.8 


2525.8 


4516.9 


8495.1 


9556.8 


2567.4 


4651.6 


8667.1 





2 


9502.6 


2527.2 


4621.4 


8500.4 


9558.2 


2568.8 


4656.2 


8662.6 


8 


4 


9504.6 


2528.6 


4525.8 


8605.8 


9560.0 


2570.2 


4660.8 


8668.0 


4 


6 


9506.4 


2530.0 


4530.2 


8511.1 


9561.8 


2571.6 


4665.4 


8678.6 


6 


8 


9508.2 


2531.4 


4534.6 


8516.4 


9563.7 


2573.0 


4669.9 


6679.0 


8 


10 


9510.1 


2532.7 


4539.1 


8.521.8 


9565.6 


2574.4 


4674.6 


8684.6 


10 


12 


9511.9 


2534.1 


4543.5 


8527.1 


9567.4 


2575.8 


4679.1 


8690.0 


18 


14 


9513.8 


2536.5 


4548.0 


8532.5 


9569.2 


257r.l 


4688.7 


8696.6 


14 


16 


9516.7 


2536.9 


4552.4 


8537.9 


9571.0 


2578.6 


4688.3 


8701.0 


16 


18 


9517.5 


2538.3 


4556.9 


8543.2 


9572.9 


2679.9 


4692.9 


8706.5 


18 


20 


9519.4 


2539.7 


4561.8 


8648.6 


9574.7 


2581.8 


4697.5 


8712.0 


20 


22 


9521.2 


2541.0 


4565.8 


8554.0 


9576.5 


2582.7 


.4702.1 


8717.6 


22 


24 


9523.1 


2542.4 


4570.8 


8559.4 


9578.4 


2584.1 


4706.8 


8728.1 


84 


26 


9524.9 


2543.8 


4574.8 


8564.8 


9580.2 


2586.5 


4711.4 


8728.6 


86 


28 


9526.8 


2545.2 


4579.8 


8570.2 


9582.0 


2686.9 


4716.0 


8784.2 


28 


80 


9628.6 


2546.6 


4683.7 


8576.6 


9583.8 


2688.3 


4720.6 


8789.7 


80 


32 


9680.5 


2548.0 


4588.2 


8581.0 


9586.7 


2589.7 


4726.8 


8746.8 


82 


34 


9532.8 


2549.4 


4592.7 


8586.4 


9587.6 


2691.1 


4729.9 


8760.8 


84 


86 


9684.2 


2550.7 


4597.2 


8591.8 


9589.3 


259^6 


4734.6 


8766.4 


86 


88 


95860 


2552.1 


4601.7 


8597.2 


9591.1 


2593.9 


4739.2 


8761.9 


88 


40 


9537.9 


2653.5 


46062 


8602.6 


9593.0 


2595.8 


4748.9 


8767.5 


40 


42 


9539.7 


2554.9 


4610.8 


8608.0 


9694.8 


2596.7 


4748.5 


8773.1 


42 


44 


9641.6 


2556.3 


4615.3 


8618.6 


9596.6 


2598.1 


4758.2 


8778.6 


44 


46 


9643.4 


2557.7 


4619.8 


8618.9 


9598.4 


2599.4 


4757.9 


8784.2 


46 


48 


9646.8 


2559.1 


4624.3 


8624.3 


9600.8 


2600.8 


4762.6 


8789.8 


48 


50 


9647.1 


2560.6 


4628.9 


8629.8 


9602.1 


2602.2 


4767.2 


8796.4 


60 


52 


9549.0 


2561 .8 


4633.4 


8635.2 


9603.9 


2608.6 


'4771.9 


8801.0 


68 


54 


9550.8 


2.^68 2 


4638.0 


8640.7 


9605.7 


2605 


4776.6 


8806.6 


64 


56 


9562 6 


2664.6 


4642.5 


8646.2 


9607.5 


2606.4 


4781.3 


8812.2 


66 


58 


9564. 5 


2566.0 


4647.1 


8651.6 


9609.4 


2607.8 


4786.0 


8817.8 


68 


«P 


9556 8 


2567.4 


4651.6 


8657.1 


9611.2 


2609.2 


4790.7 


8823.4 


60 



Digitized by VjOOQIC 





FUNCTIONS OF A ONE-DEGREE CURVE 


239 


/ 


114<» 


116» 


/ 


L.C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





9611 2 


2609.2 


4790.7 


8823.4 


0666.3 


2651.3 


4034.4 


8004.3 





2 


0018.0 


2610.6 


4795.5 


8829.1 


0667.1 


2652.7 


4030.3 


0000.1 


2 


4 


9614.8 


2612.0 


4800.2 


8834.7 


0668.8 


2654.1 


4044.2 


0005.0 


4 


6 


9616.6 


2613.4 


4804.9 


8840.3 


0670.6 


2655.5 


4049.1 


0011.6 


6 


8 


9618.4 


2614.8 


4809.6 


8846.0 


0672.4 


2656.0 


4954.0 


0017.4 


8 


10 


9620.2 


2616.2 


4814.4 


8851.6 


9674.2 


2658.8 


4958.9 


0023.2 


10 


12 


9622.0 


2617.6 


4819.1 


8857.3 


0676.0 


2659.7 


4963.8 


0029.0 


12 


14 


9623.8 


2619.0 


4828.0 


8862.9 


9677.8 


2661.1 


4968.7 


9034.8 


14 


16 


9625.7 


2620.4 


4828.6 


8868.5 


9670.6 


2662.6 


4973.6 


9040.7 


16 


18 


9627.5 


2621.8 


4838.4 


8874.2 


0681.4 


2663.0 


4978.5 


0046.5 


18 


ao 


9639.8 


2623.2 


4888.1 


8870.0 


0683.1 


2665.4 


4083.4 


0052.3 


20 


22 


9681.1 


2624.6 


4842.9 


8885.6 


0684.0 


2666.8 


4068.3 


0068.1 


22 


24 


9682.9 


2626.0 


4847.7 


8801.2 


9686.7 


2668.2 


4003.3 


0064.0 


24 


26 


9684.7 


2627.4 


4852.4 


8806.0 


9688.6 


2669.6 


4008.2 


0060.8 


26 


28 


9636.5 


2628.8 


4857.2 


8002.6 


9690.3 


2671.0 


6003.2 


0075.7 


28 


80 


9638.3 


2680.2 


4862.0 


8908.8 


9692.0 


2672.4 


5008.1 


9081.5 


30 


32 


9640.1 


2631.6 


4866.8 


8914.0 


9693.8 


2673.8 


5013.1 


9067.4 


32 


84 


9641.9 


2638.0 


4871.6 


8919.7 


9695.6 


2675.2 


5018.0 


9093.2 


84 


86 


9643.7 


2684.4 


4876.4 


8925.4 


9697.4 


2676.6 


5023.0 


9099.1 


86 


88 


9645.5 


2635.8 


4881.2 


8931.1 


9699.1 


2678.0 


5028.0 


9105.0 


38 


40 


9647.3 


2637.2 


4885.0 


8036.8 


9700.9 


2679.5 


5082.0 


9110.8 


40 


42 


9649.1 


2638.6 


4890.9 


8042.6 


9702.7 


2680.9 


,5037.0 


9116.7 


42 


44 


9650.9 


S640.0 


4895.7 


8948.3 


9704.5 


2682.3 


5042.0 


9182.6 


44 


46 


965.>.7 


2641.4 


4900.5 


8054 


9706.2 


2683.7 


5047.0 


9128.5 


46 


48 


9654.6 


2642.9 


4905.3 


8059.8 


9708.0 


2685.1 


5052.0 


9134 4 


4S 


SO 


9666.8 


2644.3 


4910.8 


8065.5 


9709.8 


2686.5 


6057.0 


9140.3 


50 


52 


9658.1 


2645.7 


4916.0 


8071.3 


9711.6 


2687.0 


5062.0 


9146.2 


b'2 


54 


9659.9 


2647.1 


4919.0 


8077.0 


0713.3 


2680.8 


5067.9 


9152.1 


54 


56 


9661.7 


2648.5 


4924.7 


8082.8 


9715.1 


2690.7 


5072.9 


9158.1 


56 


58 


9668.5 


2649.9 


4929.6 


8088.5 


9716.0 


2692.2 


5078.0 


9164.0 


58 


t «0 


9665.3 


2861 8 


4934.4 


8094.8 


0718.6 


2693.6 


5083.0 


9169.9 


60 


/ 


lift* 


117« 


/ 


L. 0. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





9718.6 


2693.6 


6083.0 


0160.0 


9771.3 


2736.1 


5236.6 


0350.5 





2 


9720.4 


2695.0 


5088.0 


9175.0 


9778.0 


2737.5 


5241.8 


0356.6 


2 


4 


9722.2 


2696.4 


5093.1 


0181.8 


9774.7 


2788.0 


5247.0 


0362.7 


4 


6 


9723.9 


2697.8 


5098.1 


0188.8 


9776.6 


2740.4 


5252.2 


0368.0 


6 


8 


9725.7 


2699.2 


5103.2 


9193.7 


9778.8 


2741.8 


5257.4 


9375.0 


8 


10 


9727.4 


2700.6 


5108.2 


9190.7 


9770.0 


2743.2 


5262.6 


9381.1 


10 


12 


9729.2 


2702.1 


5118.8 


9205.6 


0781.7 


2744.6 


5267.9 


9387.3 


12 


14 


0781.0 


2703.5 


5118.4 


9211.6 


9:83.4 


2746.0 


5273.1 


9308.4 


14 


16 


9782.7 


2704.9 


5123.4 


9217.6 


9785.2 


2747.5 


5278.4 


0300.5 


16 


18 


9784.5 


8706.8 


5128.5 


92:28.6 


9;86.0 


8748.0 


6888.6 


0405.7 


18 


20 


9736.3 


2707.7 


5133.6 


92206 


0788.6 


2750.3 


5288.9 


0411.0 


20 


22 


9738.0 


2709.1 


5138.7 


9285 5 


9700.4 


8751.7 


5294.2 


0418.0 


22 


24 


9739.8 


2710.6 


5148.8 


9241.5 


9792.1 


2753.2 


5299.5 


0424.2 


24 


26 


9741.6 


2712.0 


5148.9 


9247.6 


9798.8 


2754.6 


5304.7 


0430.4 


26 


28 


9748.8 


2713.4 


5154.0 


9253.6 


9795.6 


2756.0 


5310.0 


0486.6 


28 


80 


9745.0 


2714.8 


5160.1 


9259.6 


9797.3 


2757.4 


5315.3 


0442.8 


30 


82 


9746.8 


2716.2 


5164.2 


9265.6 


9799.0 


2758.0 


5320.6 


0440.0 


32 


84 


9748.5 


2717.6 


5169.4 


9271.6 


9800.7 


2760.3 


5325.9 


0455.2 


34 


36 


9750.3 


2719.1 


5174.5 


9277.7 


9802.5 


2761.7 


6331.2 


0461.4 


86 


88 


9752.0 


2720.5 


5179.7 


9283.7 


9804.2 


2763.1 


5336.5 


0467.6 


38 


40 


9758.8 


2721.9 


5184.8 


9289.8 


9805.0 


2764.6 


5341.8 


0473.8 


40 


42 


9755.6 


2723.8 


5190.0 


9295.8 


9807.7 


2766.0 


5347.2 


0480.0 


42 


44 


9757.8 


2724.7 


5195.1 


9301.9 


9809.4 


2767.4 


5352.5 


0486.8 


44 


46 


9759.0 


2?^. 2 


5200.3 


9307.9 


9811.1 


2768.8 


5357.9 


0408.5 


46 


48 


9760.8 


2727.6 


5205.4 


9314.0 


9812.8 


2770.8 


5863.2 


0408.7 


48 


60 


9762.6 


2729.0 


5210.6 


9320.1 


9814.5 


2771.7 


5368.6 


0506.0 


50 


63 


9764.8 


2730.4 


5215.8 


9326.1 


9816.3 


2778.1 


5378.0 


0511.8 


52 


64 


9766.0 


2781.8 


5221.0 


9332.2 


9818.0 


2774.6 


5870.3 


0517.6 


54 


66 


9767.8 


2738.3 


5226.2 


9338.3 


9819.7 


2776.0 


53S4.7 


9528.8 


56 


68 


9769.5 


2734.7 


5231.4 


9344.4 


9H21.4 


2777.4 


5390.0 


0530.0 


68 


60 


9771.8 


2786.1 


5236.6 


9350.5 


9823.1 


2778.8 


5305.4 


0636.3 


60 



Digitized by VjOOQIC 



240 NATURAL SINES, COSINES, TANGENTS, COTAN. 









0' 








!• 














Siti. 


C63, \ Tan. ] Cot. | 


Sin- 


Cos- 


Tan, 


Cot- 


t 




1 

2 
3 

5 

e 

7 
8 

ft 


^00000 
.00029 
.000&8 

-Dooa? 
.ooufl 

.00145 
.00175 
.00204 
.00333 
.00262 


Oue 
One 
One 
Oae 
One 
Une 
One 
Om 
OQe 


1.00000 
^. 0002 9 
.OOOaS 

.oooa7 

.00116 
.00145 
.00175 
.00204 
.00233 
.00262 
1.00291 
.00330 
,00349 
.003?e 
.00407 

.00436 
.00465 
.00495 

.00524 
-00553 
00583 
.00611 
.00640 
,00669 
.00698 
.00727 
.00756 
.00735 
.00615 
.0C844 

.00673 
-00902 
.00931 
.00960 
,00939 


Infinite 

3437. 7S 

1713. 37 

1145-92 

aj9.436^ 

687,549 

572, 957 

49M06 

429.718 

361 971 

343 774 

312.521 

266, 478 

364.441 

245.552 

229-133 
214. 858 
202.219 
190.964 
180 933 

171. S35 
163. 700 
156 259 
149.465 i 
J43.337_ 
137 507 
132 219 
127.321 
133.774 
118.540 
114 589 
110. 393 
107.426 
104.171 
101.107 
9S.3179 
95.4^95 
92.9085 
90- 4633 
88-1436 
85. 9398 
63 8435 
81 8470 
79. 9434 
79-1263 

76 3900 
74 7393 
73-1390 
71.B151 

,70._15.3_3. 

' €8. 7501 
67 4019 
66.1055 
fi4.85S0 
63. 6557 
62.4993 
61.3829 
60.3058 
59. 2659 
58.2612 
57-2900 
Tan. 


-01745 
-01774 
.01303 
-01833 

-01662 
.01691 

01930 
.01949 

01978 
.02007 
-02036 
-02065 
.02094 
-02133 
-02152 
.02181 
.02311 
-03340 
.02269 
-02398 


.99986 
.99984 
.99964 
.99983 

99983 
.99982 
-99963 
.99961 
.96930 
,99930 

99979 
.99979 
.99976 
.99977 
.99977^ 

99976^ 
.99973 

99975 
.99974 
.99974 


.0174ft 
.01775 
.01804 
,0ia83 
.01M3 
.01891 
.01920 
.01949 
.01973 
.02007 
.0203ft 
.02066 
.02095 
.03124 
-03153 

-02182 
.02211 
.02240 
-02269 

02296. 
.02326 

02357 
-02366 
.02415 
.02444 


67*2900 
56-3506 
56.4415 
54.6613 
53.7080 
52-6821 
62.0807 
61-3032 
50*5435 
40.6157 


59 
58 
5f 
_5« 
5ft 
54 
53 
53 
51 


10 

n 

12 
13 

u 


,00391 
.00320 
.00349 
.00378 
.00407 
,00436 
.00465 
.00495 
.00524 
,00553 


,99999 
.99999 
.99999 
.99999 
.99999 
.99999 
.99999 
.99999 
.99999 

.99998 
.99998 
.99993 
.9999S 
.9999d 
.99997 
-99997 
. 99997 
.99997 
.99996 

99996 
-99996 
.99996 
.99995 

99995 
.99995^ 
.99995 
.99994 
.99994 
.99994 


49.1030 
4a* 4121 
47.7395 
47-0853 
46.4489 


4i 
4A 

47 
46 


15 
16 
17 
18 


45*8294 
45-2261 
44.6386 
44.0661 
.43.5081 

42.9641 
42.4335 
41-9158 
41^4106 
40.9174 


46 
44 

48 
42 

-M. 


30 
91 

23 
34 


.00583 
■OOSll 
-00640 
.00669 

-00698 


-02337 .99973 
.03356 -99972 

02385 -99973 

02414 -99971 
,02443_ ^970_ 

02472 .999fi9 
-02501 .99969 
.02530 .99968 
.02560 .99967 
.02589 99966^ 

02616 '-99966 
.03647 .99965 
.02676 .99964 . 

03705 .99963 
.0373^ -.99963^ 
.03763 ,99962 
.02792 .99961 | 

02821 .99960 , 
.03350 1.99959 
.02679 .99959 
-02908 .99959 
-02938 .90957 

02967 99956 
■02996 .99955 
^3025_|.9995^ 
-03054 .99953 
.03083 .99952 
-03112 .99952 

03141 -99951 

03 170 .99950 

.03199 .99949 
.03238 .99948 
.03257 .99947 
-03236 .99946 
.Ori^ie .99945 ' 


40 

39 
38 
37 
36 


25 , 

26 

27 

28 

2P 


.00727 
.00756 
^0735 
.00814 
.00844 
.00S73 
,00902 
.00931 
.00960 
,00939 
-01018 
.01047 
-01076 
.01105 
-01134 


.02473 
.02502 
.02531 
.02560 
. 02569. 
.'02619" ' 

02648 
.03677 
.02706 

02735 
. 03764 
.02793 
.02832 
.03ES1 
.02881 
.02910 

02939 
.03968 
.02997 
.03036 
.03055 

03084 
.03114 
-03143 
.03172 


40.4358 
39.965a 
39.5059 
39.0568 
36-6177 


36 
34 

33 
33 
31 


HO 
31 
32 
83 
34 


36.1685 
37.7686 
37. 3579 
36.9560 
36. 5627 
36-1776 
35.8006 
35. 4313 
35,0695 
34 7151 
34. 3678 
34 0273 
33.6935 
33. 3662 
,33.0453 

33-7303 
33, 4313 
32.1181 
31.8205 
31.6m 


30 

29 
28 
27 
26 


35 
86 
37 
88 
39 


.010^8 

.01047 
-01076 
.01105 
-01135 


25 
24 
23 
23 
_21 


40 

41 
42 
43 
44 


.01164 
.01193 
.01222. 
.01351 

.oi2no 

.01309 
.01338 
.01367 
01396 
.0142^_ 
.01454 
.01483 
.01513 
,01542 
.01571 

.01600 
.01639 
01658 
01687 
01 715 


.99993 
.99993 
.99993 
.99992 
-99093 


-01164 
.01193 
.01233 
.01251 
.01260 

.01309 
.01338 
.01367 
-01396 
.01425 

-01455 
.01484 

.01513 
.01542 
01571 

.01600 
.01629 
.01658 
.01687 
.01716 
.01746 
Cot. 


30 

19 

1? 

18 


4J 
46 
47 
4B 
49 


99991 
.99991 
.99991 
.99990 
.99990 

.999B9 
.99989 
.99989 
.999^8 
^999&a 
.99987 
.99987 
.99906 
-9^986 
.99985 


16 
14 

la 


S2 
S3 
B4 


-03201 
.03230 
,03259 

03236 
.03317 
,03346 
.03376 
.03405 
.03434 

03463 


31.2416 
30.9599 
30.6833 
30. 41 16 
30-1446 


10 
9 
8 
7 
8 


55 

bB 

57 
58 
59 


.03345 
.03374 
■03403 
.03432 
.03461 


.99944 . 
.99943 
.99942 1 
.99941 i 
.99940 


29.8623 
29-6545 
r^ft.3711 
5^-1220 
28-8771 
J^6363 
TalL 


5 
4 

3 
2 

1 


gCL 




''^ 


. 03490 


.99939 


Cot, 


■Tfi 



89* 



Digitized by VjOOQ IC 



NATURAL SINES, COSINES, TANGENTS, COTAN. 241 







8* 






3* 






9 


Sin. 1 


Cos. 


Tan. 


t^ot. 


Wiru 1 


*;tw. 


Inn. 


Via. 


t 





-03490 


.9^939 


.03492 


28-8363 


.05234 ! 


.09363 


.05241 


19. 0811 


^ 


1 


^03519 


.99938 


^03531 


20,3994 


05263 


.99661 


.05270 


18.9755 


59 


3 


.03543 


.9si937 


.03650 


28.1664 


.05292 


.99860 


.05299 


18.8711 


58 


8 


-03577 , 


.9^936 


.03579 


27-9372 


.05321 


.99853 


.05328 


18-7678 


57 


_i^ 


tp3_6Q6_' 


9^935 


-03609 


.27-7117_ 


-0535O 


.94857 


.05357 


18-6656 


58 


fi 


.03635 


.99934 


.03638 


27 4898 


.05379 


.99855 


.05387 


18.5845 


55 


6 


-03564 


99933 


.03667 


27.2715 


-O5408 


. 99854 


.05416 


18. 4845 


54 


7 


.03693 


.99932 


.03696 


27.0566 


-05437 


.99853 


.05445 


18.3655 


53 


8 


.03723 , 


99931 


.03725 


26. 8450 


-05466 


.99851 


,05474 


18.2677 


53 


_9_ 


i037&2 


.99930 
.99929 


-03754, 
.03783 


3fi.fi387 
26. 4316 


.05495. 

.05524 


.99849 
.98847 


.05503, 
.05533 


18 1708 
18,0750 


_|i 


10 


.03781 


m 


11 


-03810 


.99927 


,03812 


26-2296 


-05553 


.99846 


.05562 


17.9802 


49 


13 


.03639 


.99926 


.03842 


25.0307 


05582 


■99644 


.05591 


17. 8863 


48 


13 


.03868 


,99935 


.03871 


25.8343 


-05611 


,99342 


.05620 


17-7934 


47 


14 


.03397 


.99924^ 


.0390QL 


25-6418 


. 05640_ 


.99841 


.05949_ 


17.7015 


46 


15 


.03926 


.99923 


.03929 


25.4517 


05669 


-99839 


.05878 


17.6106 


45 


16 


.03965 


.99922 


.03958 


25-2644 


.05698 


.99838 


.05708 


17-6205 


44 


17 


.08984 


.99921 


.03987 


25.0793 


.05737 i 


.99838 


.05737 


17. 4314 


43. 


18 


.04013 


.99919 


.04016 


24-8978 


.05756 


.99834 


.05766 


17.3432 


42 


19_ 


- 04042 


,99918 


. 04046 


24.7185 


•05785 1 


-99633 


-05795 


17.2558 


_|1 


:s(i 


.04071 


-99917 


04075 


24.5418 


-05314 


.99631 


.05824 


17-1693 


40 


31 


04100 


.99916 


,04104 


24-3675 


.05844 


.99829 


.05854 


17.0837 


39 


33 


.0412& 


-09915 


04133 


24.1957 


.05873 


99827 


-05883 


16. 9990 


38 


28 


.0415& 


.99913 


.04182 


24-0263 


.05902 


.99826 


.05912 


169150 


87 


?i^ 


._04138_ 


-99912 


04191^ 


23-8593 


-05931 


.99024 


-05941 


16-8319 


.36 


2@ 


.04217 


-99911 


04220 


33. 8945 


.05960 


. 9'9E22 


05970 


16. 7496 


35 


26 


.04246 


.99910 


,04250 


23-5321 


. 05989 


.99821 


.05999 


16-6631 


34 


37 


.04275 


.99909 


,04279 


33.3718 


.06018 


.99619 


06029 


16.5874 


33 


28 


.Q4304 


.99907 


. 04308 


23.3137 


.06047 


.99817 


.06058 


16-6076 


82 


2i_ 


.04333 


- 99906 
.99905 


-04337 
04366 


33-0577 
32-9033 


.06076 
.06105 


.99815_ 
99813 


-06087 
.06116 


_16.4333^ 
18.3499 


31 


ao 


.04382 


30 


«i 


-04391 


-99904 


.04395 


22.7519 


.06134 


.99812 


.06145 


18.2722 


29 


83 


. 04420 


.99902 


,04424 


32-6020 


.06163 


-99810 


.06175 


16.1952 


28 


83 


.04440 


-99901 


.04454 


22-4541 


.06192 


.99B0B 


06204 


16.1190 


27 


84 


.04478 


.99900 


-04483 


22-3081 


.06231. 


-99806 


06333 


1 6. 0435 


28 


85 


.04507 


. 99B98 


-04512 


22.] 640 


06250 


-99804 


-06263 


15. 9587 


aa 


86 


-04536 


-99897 


,04541 


22-0217 


.06279 


99803 


.06391 


15.8945 


24 


87 


.04565 


.99896 


,04570 


21.B813 


06309 


.99801 


-06321 


15-8211 


23 


38 


.04594 


.99894 


-04599 


21-7425 


.06337 


.99799 


06350 


15.7483 


22 


%%_ 


.04623 


-99893 


.04628 


2\.um 


,06366 


-99797 


.116379 


15-6763 


21 


40 


.04853 


-90892 


.D4f>58 


21-4704 


06395 


-99795 


-06408 


15. 6048 


;io 


41 


.04882 


-99S90 


.04687 


21-3369 


.06424 


.99793 


-06437 


15.5340 


19 


42 


,04711 


. 99889 


-04716 


21-2049 


.06453 


.99792 


06467 


15-4638 


18 


48 


.04740 


.99888 


-04745 


21.0747 


•06482 


-99790 


06496 


15.3943 


17 


44 


.04769 


-99886 


04774 


30 - 9460 


.06511_ 


-99788 


-06525 


15.9254 


_ie 


45 


.047fi8 


-99885 


04803 


20.aies 


06540 


.99786 


06554 


15.3571 


15 


46 


,04827 


-99883 


.04833 


20-6932 


.06569 


,99784 


.06584 


15-1893 


14 


47 


.04856 


,99882 


04862 


20-5691 


.06598 


99733 


-06613 


15.1322 


13 


48 


.04885 


,99881 


04891 


20,4465 


,06627 


.99780 


.06643 


15. 0557 


12 


m^ 


-04914 


^9879 _ 


. 04920 


20.3253 


06656 


-997.7L 


-06671 


14-9898 


u 


m 


-04943 


-99878 


04949 


20.3056 


06685 


.99776 


.06700 


14 9344 


10 


61 


04973 


.99378 


.04979 


20.0872 


.06714 


-99774 


.06730 


14 8596 


9 


S3 


.05001 


.99875 


05007 


19.9702 


06743 


.99772 


.06759 


14.7954 


3 


63 


,05030 


99373 


05037 


19-8546 


06773 


.99770 


.06788 


14.7317 


7 


54 


.05059 
05088 


.99872 
-99870 


05^.6i_ 
05095 


19.7403 
19.6273 


06902 


.99769 


-0681L 


14 6685 


a 


6S 


06831 


.99766 


.06847 


14.6059 


9 


50 


.05117 


.99369 


.05124 


19. 5156 


.06860 


.99764 


.06676 


14,5438 


4 


57 


05148 


99867 


■05153 


19-4051 


06889 


.99762 


.06905 


14. 4823 


3 


se 


.05175 


.99388 


05182 


1 9 2959 


.06918 


.99760 


06934 


14 4212. 


2 


fli^ 


J)5205_ 


.99364 


.05212 


19-1879 


.06947 1 


.99759 


.06963 


14.3907 


'^ 


62. 


-05234 


709863 


-05241 


19 0811 


06976 


-99758 


.0^993 


14.3007 


_s 


*^ 


Co... 1 


Hln. 


C^jt- 


Tan. 


:'u.- i 


Sin- 


Cut. 


Tiiii. 


t ' 



^T 



Wt 



y Google 



242 NATURAL SINES, 
4* 



COSINES, TANGENTS. COTAN. 
ft** 



$ 


Siii, 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan, 


Cot. 


9 




\ 
a 

8 

4 


.06976 
.07005 
.07034 
-07063 

jP7oa^ 

-07121 
.07150 
.07179 
,07208 
07237 


.99756 
.99754 
.99752 
.99750 
.99746 
.09746 
,99744 
.99742 
.95740 
.99738 


0G993 
07022 
07051 
07080 
07110 


14.3007 
14.2411 
14.1821 ' 
14. 1235 
11-0655 


.0«71fl 
,08745 
.08774 
.06803 
-08831 
-08860 
.08889 
-08918 
.08947 
-08976 


.99619 
.99617 
.99614 
.99612 
99609 


-08749 
-06778 
-08807 
-08837 
.08866 


11^4301 

I1.S019 
11.3640 
11.3163 
11.2788 


SO 

58 
53 
57 
G6 


V 

6 
.9 


07139 14-0079 
07166 1 13 9507 
.07197 13.6940 
^07227 , 13 8376 
.07256 13.7821 


.99607 
.99804 
99602 
.99599 
.99598 


■08805 
08026 
.08854 
,08983 
.09013 


11-2417 

11.2048 
11-1681 
11-1316 
11.0954 


55 

64 
63 
52 
61 


12 
13 
14 


.07266 
.07295 
.07324 
.07353 

-07382^ 

.07411 
.07440 
.07463 
.07498 

,07527 

.07556 
,07565 
.07614 
07643 
07672 


.99736 
.99734 
.99731 
.9G729 
.99727_ 
99725 
.99723 
.99721 
.99719 
.99716 

.99714 
.99712 
.S9710 
. 99708 
.99705 


.07285 
-07314 
.07344 
.07373 
.07402 

-07431 
.07461 
.07490 

-07519 
.07548 
.07578 
.07607 
,07636 
.07665 
,07695 


13.7267 
13.6719 
13-6174 
13.5634 
13 5098 
13 4566 
13.4039 
13.3515 
13. 2996 
13.34S0_ 

13.1969 
13.1461 
13.0958 
13. 0458 
12.9962 


.09006 
-09034 
.09063 

-09092 
,09121 
.09150 
.0917& 
.09308 
.09237 
. 09366 

.09295 
-09324 
.09353 
.09382 

.09411 


.99594 
99591 

.99583 
99586 

.99583 

.99580 
.99578 
.99675 
-99572 
.J9_570_ 
.89667 
99564 
.99562 
-99559 
.99556 
.99553 
.99561 
.99548 
.99545 
-99543 


-09042 

.09071 
.09101 
-09130 
.09159 

.09169 
.09213 
.09247 
.09277 
■09306 
-08335 
-09365 
.09394 
.09423 
.09453 


11-OS94 
11-0237 
10-8882 
10-8529 
10. 8178 


50 
48 

48 
47 
46 


IS 

i? 

iL 

21 
22 
23 
24 


10.8828 
10-6483 
10.8138 
10-7787 
10-7457 
10.7118 
10.6783 
10-6450 
10. 8118 
10.5789 

10 5482 
10.5138 
10-4813 
10.4481 
10 4172 
1D.3B&4 
10.3638 
10-3224 
10.2813 
10-2603 

10. 2284 
10-19B8 
10-1683 
' .0.1381 
10.1080 


45 
44 
43 
42 

40 

39 
38 
37 

38 


2h 
27 
29 


-07701 
07730 
.07759 
.07763 
.07317 


.99703 
.99701 
.99699 
.99696 
.99694 


-07724 
.07753 
.07782 
.07812 
.07841 


12.9469 
12. 8981 
12-8496 
12 8014 

12.7536 


-09440 
-09469 
.09498 
,09527 
.09558 

09585 
.09614 

-09642 
-09671 
,09700 
. 09729 
09756 
09787 
.09816 
00B45 


.09482 
.09511 
.09541 
,09570 
. 09600 
. 09629 
.09853 
09688 
.09717 
.09746 

.09776 
.09805 
-09834 
.09864 
,09893 


35 
34 
33 

32 
31 


30 

31 
32 

3a 

84 
36 
86 
87 
SB 
39 


.07349 
07675 

-07904 
-07933 
07062 

.07991 
.08030 
08049 
.08076 

-08107 


.99693 
.99689 
.99637 
.99685 
-96683 
.99680 
.99678 
.99676 
99673 
.99671 


.07870 
.07899 
-07929 

.07958 
.079S7 
.08017 
-08046 

.08075 
.08104 
.08134 


13.7063 
12-6591 

12,8124 
12.5660 
_I2.5199_ 

12-4743 
12.4286 
12-3838 

12.3i90 
13.2946 


,99540 
.99537 
.99534 
.99531 
.99528 
T99526 
-89523 
-99520 
-99517 
99514 


30 

29 
28 
27 
26 
25 
24 
23 
23 
_21 


40 
41 
42 
43 
44 


-08136 
-08165 
.08194 
.08223 
. 06252 

,"63281 
-06310 

08339 
.03368 

06397 


.99663 
.99666 
-99664 
-99861 
^9_659 
.99657 
.99654 
-99653 
.99649 
.99647 
,99644 
.99642 
-99639 
.99637 
.99^3.5, 
99632 
,99630 
.99627 
,99625 
.99622 


.08163 
.08193 
.08321 
.08351 
,06280 


12-2505 
13.2D67 
13,1632 
12-1201 
12 0772 

12. 0348 
11.99S3 
11 9504 
11. 9087 
11,8673 
'11.8262 
11.7853 
11.7448 
11-7045 
11.6B46 
11- 8248 
11.5S53 
11.5461 
11,5072 
1 ] . 4685 


.09674 

09903 
.09933 
.09961 
.09990 

10019 
.10048 

10077 
.10106 
.10135 
.10184 

10193 
.10231 

10250 
.10279 
- 10308 
.10337 
. 10368 
, 10395 
.10424 
,10463 


-99511 
99508 
-99506 
.99503 
.09500 

.99497 
.99494 
.09491 
.99488 
.99485 

.99483 
.99479 
-99476 
.99473 
.99470 
,99487 
.99484 
.99461 
,99458 
.99455 
799452 , 
.Sin. 


,09623 
.09952 
.09981 
-lOOil 
.10040 
.10089 
.10099 
.10128 
.10168 
.10187 


10-0780 
10- 0483 
10 0187 
g. 08831 

9-83101 
9-00211 
9,87358 
8-84482 
8 81641 
8. 76817 
8, 76009 
0.73217 
8-70441 
8-67660 
8.64885 
8,82205 
9.69490 
9.68781 
9.54106 


20 

19 
18 
17 


45 
48 
47 
4B 
4t 


-08309 
08339 

,08368 
-08397 
.08427^ 
.08458 
-08485 
.08514 
.08544 
.08573 
08802 
.08632 
.08661 
.08690 
.08730 


IS 

14 
13 


60 

Gl 
62 

m 

64 

56 
GB 
67 
68 
60 


.08426 
.06455 
,08484 

08513 
.08542 
.08571 
.08600 

08629 
.08658 
.06687 


10216 
.10248 
.10275 
-10305 
.10334 
.10883 
.10393 
.10422 
.10452 
.10481 
.10510 

Cot. 


10 

9 
8 
7 
6 

1 

4 
3 

2 

1 


^- 


.067]e 


.99619 

Sin. 


.08749., n.4301 


9-61438 
Tan. 


■^ 



85* 



84* 

Digitized by VjOOQ IC 



NATURAL SINES. COSINES, TANGENTS, COTAN. ^43 



r 


Bin. 


Cob* 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


r 


"o" 


a0468 


.B9452 


-10510 


9.61436 


.12187 


.99255 


.12276 


8.14435 


"55 


1 


■10483 ; 


■99449 


. ID540 


9. 48781 


.12216 


-99251 


.12308 


8, 12481 


59 


s 


.10611 


■08446 


10569 


9- 46141 


12245 


. 99248 


.12338 


8.10536 


58 


8 


.10540 


.99443 


.10599 


9-43515 


12274 


-99344 


.13367 


a. 06600 


57 


4 


.10569 
i0537 


.89440 
.39437 


.10628 
-10657 


9. 409 04 


.12302 
-13331 


.99240 
.99237 


.12397 
.13436 


8-06674 
6.04756 


56 


6 


9.38307 


55 


6 


10626 


.g9434 


10687 


9.35724 


.12360 


,99233 


.12456 


8.02848 


54 


7 


.10655 ; 


.99431 


10716 


9. 33155 


.12389 


.99330 


.12485 


8 00948 


53 


8 


10634 


.09428 


. 10746 


9-30599 


.12418 


.98226 


.12516 


7-98058 


53 


g 


.10713^ 
. 10742 


.99424 
■99421 


-10775 
.10805 


9 26053 


12447 


99222 
.99218 


-12544 


7.97176 
7. 95302 


51 


to 


8. 25530 


.12476 


.12574 


50 


11 


.10771 


.98418 


.10834 


23016 


.13504 


.89216 


.12603 


7.9343B 


49 


13 


r 10800 


.99415 


.10863 


9.20516 


.12533 


.89211 


.13633 


7.91532 


43 


la 


.10820 


.99412 


-10803 


8.18028 


.12562 


.99308 


.12663 


7.89734 


47 


H 


.10658 


,80409 


10922_ 


9.15554. 


JL2581, 


-9920^ 


.12692^ 


J -87895 


46 


15 


.10687 


. 99406 


10953 


9.13093 ' 


.12620 


.99200 


-12722 


7. 86064 


45 


le 


.10916 


.09402 


.10981 


8 10646 


.13649 


.99197 


.12761 


7.84243 


44 


17 


,10S45 


.98399 


.11011 


8.08211 


.12878 


99193 


.12781 


7.82423 


43 


18 


.1097a 


.9L396 


.11040 


9.05789 


.12706 


.99188 


.13810 


7.80633 


43 


IL^ 


.11002 


.99383 l.l]Q70 


_9^033ZL 
9.00983 


,13735 
.13764 


,89186 
-99132 


.12840. 
.12869 


J7.7JB35 
7.77035 


-11 


20 


11031 


983B0 


ai099 


40 


21 


11060 


.9938B 


.11128 


8 98598 


.12793 


.99178 


.12899 


7. 75254 


39 


S2 


1108B 


.99383 


-11158 


8 96227 


.12822 


S8175 


.12920 


7.73480 


38 


£3 


1UI8 


.98380 


.11187 


8.93367 


.12851 


.99171 


-12958 


7-71715 


37 


24 


11147 


.99377 


.11317 


8,91520 


.12880 


.99167 


.12988 


7.69957 


36 


25 


11176 


.08374 


.11246 


8.89165 


.12908 


.99163 


.13017 


7.682D8 


35 


36 


11205 


.99370 


.11376 


8 86362 


,12937 


99160 


.13047 


7.66466 


34 


E7 


11334 


.09367 


.11305 


8. 84551 


.12066 


99166 


.13076 


7-64732 


33 


38 


.11263 


.99364 


.11335 


8. 82252 


.12995 


.99153 


.13106 


7*68005 


32 


29 


-11291 


.99360 


.11364 


8. 79964 


.13024 


.99148_ 


.13136 


7.61387 


31 


30 


11320 


.99357 


.11394 


8^77689 


.13053 


.99144 


-13165 


7.59575 


^io 


81 


-11340 


.89354 


11423 


8. 75425 


.13081 


89141 


.13195 


7-57873 


29 


S2 


.11378 


.99351 


.11452 


8.73172 


,13110 


.99137 


.13224 


7^5B176 


28 


33 


.11407 


,99347 


.11482 


8.70931 


.13139 


.99133 


.13254 


7.54487 


27 


U 


.11436^ 


.89344 


.11511 


8. 63 70 L 


.13168 


.99129 


.13284 


7, 62806 


26 


35 


.11465 


§9341 


-11541 


3.6B483 


13197 


.99125 


.13313 


7.51132 


25 


3fi 


11494 


90337 


.11570 


8.64373 


.13226 


.98133 


13343 


7. 49465 


24 


S7 


.11523 


.99334 


.11600 


8. 62078 


.13254 


,99113 


.13372 


7 47306 


23 


as 


-11552 


-99331 


.11629 


8 59893 


.13283 


.99114 


.13402 


7^46154 


23 


S9_ 


.11530 


.99327 


.11659 


8. 57718 


._13312 


9mo 


■mp^ 


7. 44509 


21 


40 


.11609 


.99324 


11688 


8.55555 


.13341 


.99106 


.13461 


7-42871 


flO 


11 


11633 


.99320 


11718 


8. 53402 


.13370 


-90103 


-13491 


7 ■41240 


19 


42 


11667 


.99317 


.11747 


8. 51259 


.13399 


.99098 


.13521 


7^39616 


18 


43 


iie&a 


.89314 


11777 


8.49126 


.13427 


.99094 


.13650 


7^37999 


17 


44 


.11725 


.89310 


11806 


S. 47007 


.13456 


.09091 


■18580 


7 ■36389 


]fl 


4& 


-11754 


99307 


.11836 


8.44896 


.13485 


.89087 


.13608 


7.34"786 


15 


46 


.11788 


.98303 


.11865 


8.42795 


.13514 


.99083 


.13639 


7^33190 


14 


47 


-11813 


.08300 


.11895 


6.40705 


13543 


.99079 


.13669 


7.31600 


13 


48 


.11840 


.99287 


11924 


8.38625 


-13572 


.99075 


.13696 


7^30018 


13 


4@ 


1188ft 


.99293 


11954 


8. 36555 


.13600 


:1807L 


.13728 


7. 28442 


11 


50 


.11898 


.99290 


11983" 


8.34496 


.13629 


.99067 


.13753 


7.28873 


to 


61 


11937 


.99286 


13013 


8.32446 


.13658 


.98063 


.13787 


7^35310 


9 


62 


.11958 


99233 


.12042 8 30406 


,13687 


.99059 


.13817 


7, 23754 


8 


53 


11985 


B9279 


.12073 


8.38376 


.1371B 


■08055 


-13848 


722204 


7 


64 


12014 
12043 


.99276 
.99373 


-12101 
.12131 


8.26355 
8. 24345 


,13744 
.13773 


,99051 
99047 


.13876 
■13906 


7 20661 


8 


65 


7.19125 


6 


56 


12071 


.99269 


.12160 


8 22344 


-13302 


.99043 


.13935 


?■ 17594 


4 


57 


.12100 


.082e5 


.12190 


8.2D352 


13831 


99039 


-13965 


7.16071 


3 


A3 


.12129 


.98262 


12219 


8. 18370 


13860 


99035 


-13995 


7 ■14553 


2 


m_ 


J.2158_ 


.99253 


1224^ 1 8 16398 


.13E89 


.99031 


.14024 


7- 13042 


I 


m 


«- 


^ 


12278 a 1443B 


.13917 


99027, 


;.14054 


7,11537 


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Cot. iati. 


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r,frt. 


, Tan. 


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9W 



y Google 



244 NATURAL SINES, COSINES, TANGENTS, COTAN. 
8* 9* 



# 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


» 


? 

2 
8 
4 
5 
6 
7 
8 
9 


.13917 
.13946 
.13975 
.14004 
.14033 
.14061 
.14090 
.14119 
.14148 
.14177 


.99027 
.99028 
.99019 
.99015 
.99011 
.99006 
.99002 
.98998 
.98994 
.98990 


.14054 
.14084 
.14118 
.14143 
.14173 
.14202 
.14232 
.14262 
.14291 
.14321 


7.11537 
7.10038 
7.08546 
7.07059 
7.05579 
7.04105 
7.02637 
7.01174 
6.99718 
6.98268 


.15643 
.15672 
.15701 
.15730 
.15758 
.15787 
.15816 
.15845 
.15873 
15902 


.98769 
.98764 
.98760 
.98765 
.98761 
.98746 
.98741 
.98737 
.98782 
.98728 


.15838 
.15868 
.15898 
.15928 
.15958 
.15988 
.16017 
.16047 
.16077 
.16107 


6.81875 
6.80189 
6.29007 
6.27829 
6.26666 
6.26486 
6.24321 
6.28160 
6.22003 
6.20851 


60 

69 
68 
57 
56 
55 
64 
53 
52 
61 


10 

11 
12 
13 
14 


.14268 
.14292 
.14320 
.14349 
.14378 
.14407 
.14436 
.14464 


.98986 
.98982 
.98978 
.98973 
.98969 


.14351 
.14381 
.14410 
.14440 
.14470 


6.96823 
6.96885 
6.93952 
6. 92525 
6.91104 


.15931 
.15959 
.15988 
.16017 
.16046 


.98728 
.98718 
.98714 
.98709 
.98704 


.16187 
.16167 
.16196 
.16226 
.16256 


6.19708 
6.18659 
6.17419 
6.16283 
6.15151 


48 
47 
46 


15 
16 
17 
18 
19- 


.98965 
.98961 
.98957 
.98953 
.98948. 


.14499 
.14529 
.14559 
.14588 
.14618 


6.89688 
6. 88278 
6.86874 
6.85475 
6.84082 


.16074 
.16103 
.16132 
.16160 
.16189, 


.98700 
.98696 
.98690 
.98686 
.98681 


.16286 
.16316 
.16346 
.16376 
.16406 


6.14028 
6.12899 
6.11779 
6.10664 
6-0»S52 


46 
44 

48 

42 


SO 

21 
22 
23 
24 


.14498 
.14522 
.14551 
.14580 
.14608 


.98944 
.98940 
.98936 
.98931 
.98927 


.14648 
.14678 
.14707 
.14737 
.14767 


6.82694 
6.81312 
6.79936 
6.78564 
6.77199 


.16218 
.16246 
.16275 
.16304 
.16333 


-.98676 
.98671 
.98667 
.98662 
.98657. 


.16436 
.16466 
.16496 
.16525 
.16555 


6.08444 
6.07340 
6.06240 
6.05143 
6.04061 


40 

80 
88 
87 
86 


25 

28 
29 


.14637 
.14666 
.14695 
.14723 
.14752 


.98923 
.98919 
.98914 
.98910 
.98906 


.14796 
.14826 
.14856 
.14886 
.14915 


6.75838 
6.74483 
6.73133 
6.71789 
6.70450 


.16361 
.16390 
.16419 
.16447 
.16476 


.98652 
.98648 
.98648 
.98638 
.98688 


.16585 
.16615 
.16645 
.16674 
.16704 


6.02962 
6.01878 
6.00797 
6.99720 
5. 98646 


85 
84 
83 
82 
SI 


30 

81 
82 
83 
84 


.14781 
.14810 
.14838 
.14867 
.14896 


..98902 
.98897 
.98898 
.98889 
.98884 


.14945 
.14976 
.15005 
.15034 
.15064 


6.69116 
6.67787 
6.66463 
6.65144 
6.63831 


.16505 
.16533 
.16562 
.16591 
.16620 


.98629 
.98624 
.98619 
.98614 
.98609 


.16734 
.16764 
.16794 
.16824 
.16854 


5.97576 
6.96610 
6.95448 
6.94390 
6.93335 


30 

29 
28 
27 
26 


85 
36 
37 
88 
5L,. 


.14926 
.14954 
.14982 
.15011 
.15040 


.98880 
.98876 
.98871 
.98867 
.98863 


.15094 
.15124 
.16158 
.15183 
.15213 


6.62523 
6.61219 
6.59921 
6.58627 
6.57339 


.16648 
.16677 
.16706 
.16734 
.16763 


.98604 
.98600 
.98595 
■98590 
.98585 


.16884 
.16914 
.16944 
.16974 
.17004 


6.92283 
6.91286 
6.90191 
6.89161 
6J8114 


25 
24 
28 
22 
21 


40 

41 
42 
43 
44 


.15069 
.15097 
.15126 
.15155 
.15184 


.98858 
.98854 
.98849 
.98846 
.98841 


.15243 
.15272 
.15302 
.15332 
.15362 


6.66056 
6.64777 
6.58508 
6.62284 
6.60970 


.16792 
.16820 
.16849 
.16878 
.16906 


.98580 
.98575 
.98570 
.98565 
.98561 


.17033 
.17068 
.17098 
.17123 
.17153 


5.87080 
6.86051 
6.85024 
5.84001 
6.82982 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.15212 
.15241 
.15270 
.15299 
.15327 


.98836 
.98832 
.98827 
.98828 
.98818 


.15391 
.15421 
.15451 
.16481 
.15511 


6.49710 
6.48456 
6.47206 
6.46961 


.16935 
.16964 
.16992 
.17021 
-17050 


.98556 
.98551 
.98546 
.98541 
98536 


.17183 
.17218 
.17248 
.17278 
.17308 


6.81966 
6.80958 
6.79944 
6.78988 
6.77936 


18 
14 
13 
12 
11 


60 

61 
62 

§1 


.15356 
.15385 
.15414 
.15442 
.15471 


.98814 
.98809 
.98805 
.98800 
.98796 


.15540 
.15570 
.15600 
.15630 
.15660 


6.43484 
6.42253 
6.41026 
6.39804 
6.38587 


.17078 
-17107 
.17136 
.17164 
-17193 


.98581 
.98526 
.98521 
.98516 
.98511 


.17333 
.17363 
.17393 
.17423 
.17453 


6.76987 
6.75941 
6.74949 
6.78960 
6.72974 


lO 

9 

1 

6 


67 
58 
59 


.15500 
.15529 
.15557 
.15586 
.15615 


.98791 
.98787 
.98782 
.98778 
.98773 


.15689 
.15719 
.15749 
.15779 
.15809 


6.37374 
6-36165 
6-34961 
6.33761 
6.32566 


.17222 
.17250 
.17279 
.17308 
.17336 


.98606 
.98601 
.98496 
.98491 
.98486 


.17488 
.17518 
.17548 
.17578 
.17608 


6.71992 
6.71018 
6.70037 
6.69064 
6.68094 


5 

4 
8 
2 

1 


S9L. 


.16643 


.98769 


.15838 


6-31375 


.17865 


.98481 


.17633 


5.67128 


^ 


"V 


Cob. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


!ran. 


} 



at' 



Digitized by VjOOQIC 



NATURAL SINES, COSINES, TANGENTS, COTAN. 245 



* 




Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


f 


Y 


.mei 


.98481 


.17633 


6.67128 


.19081 


.98163 


.19488 


6.14455 


"eo 


I 


• 17898 


.98476 


.17663 


5.66165 


.19109 


.98157 


.19468 


5- 13658 


59 


2 


. 17422 


.98471 


.17693 


5.65205 


.19138 


.98152 


.19498 


6.12862 


58 


8 


.17461 


.98466 


.17723 


5.64248 


.19167 


.98146 


.19529 


6.12069 


57 


4 


.17479 


..98461 
.98466 


.17753 
.17783 


5-63295 
5.62344 


.19195 
.19224 


.98140 
.98135 


.19559 
.19689 


6-11279 


56 


6 


.17508 


6.10490 


56 


6 


.17687 


.98460 


.17813 


5.61397 


.19252 


.98129 


.19619 


6.09704 


54 


7 


.17666 


.98446 


.17843 


5.60452 


19281 


98124 


.19649 


5. 08921 


58 


8 


.17694 


.98440 


.17873 


6.59511 


.19309 


98118 


.19680 


5. 08139 


52 


9 


.17623 


.98435 


.17903 


5.58573 


.19338 


98112 


.19710 


5-07360 


61 


lO 


.17651 


.98480 


17983 


5. 57638 


.19366 


.98107 


.19740 


6-06584 


50 


11 


.17680 


.98425 


.17963 


5.56706 


.19396 


-98101 


.19770 


5-05809 


49 


12 


.17708 


.98420 


.17998 


6.65777 


.19423 


98096 


.19801 


5 05037 


48 


13 


.17737 


.98414 


.18028 


6.64851 


.19452 


98090 


.19881 


5 04267 


47 


14_ 


.17766 


.98409 


.18058 


6.53927 


.19481 


-98084 


.19861 


5. 03499 


46 


15 


.17794 


.98404 


.18088 


6.53007 


.19509 


98079 


.19891 


5-02784 


•46 


16 


.17828 


.98399 


.18118 


6.62090 


.19538 


.98073 


.19921 


5-01971 


44 


17 


.17852 


.98894 


.18143 


6.51176 


.19566 


98067 


.19952 


5-01210 


43 


18 


.17880 


.98889 


.18178 


6.60264 


.19595 


.98061 


.19982 


5-00451 


42 


12. 


.17909 
17987 


.98383 
.98878 


.18208 


5.49356 


.19623 


.98056 


.20012 


4-99695 


-il 


20 


.18233 


5.48451 


-19652 


.98050 


.20042 


4-98940 


40 


21 


.17966 


.98873 


.18263 


5.47548 


.19680 


.98044 


20078 


4-98188 


39 


22 


.17996 


.98868 


.18293 


5.46648 


.19709 


.98039 


.20103 


4. 97438 


38 


28 


.18023 


.98362 


.18323 


5.45751 


.19737 


.98033 


.20188 


4.96690 


37 


?£. 


.18052 
.18081 


.98357 
.98352 


.18358 


5.44857 


.19766 


.98027 


.20164 


4.95945 


36 


26 


.18384 


6.48966 


.19794 


.98021 


.20194 


4.95201 


35 


26 


.18109 


.98347 


.18414 


6.43077 


.19823 


.98016 


.20224 


4.94460 


34 


27 


.18188 


.98341 


.18444 


5.42192 


-19851 


.98010 


.20254 


4.98721 


33 


28 


.18166 


.98336 


.18474 


6.41809 


■19880 


-98004 


.20286 


4.92984 


32 


?9 


.18195 
.18224 


.98331 
.98325 


.18504 


6. 40429 


- 19908 


-97998 


.20315 


4- 82249 


81 


SO 


.18684 


5.b9552 


19937 


.97992 


.20345 


491516 


30 


81 


.18252 


.98320 


.18564 


6.38677 


19965 


.97987 


.20376 


4. 90785 


29 


82 


.18281 


.98315 


.18594 


6. 87805 


19994 


-97981 


.20406 


4. 90056 


28 


83 


.18809 


.98310 


.18624 


5. 36936 


20022 


.97975 


.20436 


4. 89330 


27 


84 


.18838 


.98304 


.18654 


6- 36070 


.20051 


.97969 


.20466 


4- 88605 


-26 


86 


.18867 


.98299 


.18634 


5.35206 


.20079 


.97963 


.20497 


4- 87882 


25 


86 


.18395 


.98294 


.18714 


5.34345 


.20108 


.97958 


.20527 


4.87162 


24 


87 


.18424 


.98288 


.18746 


6.33487 


.20186 


.97952 


.20557 


4. 86444 


28 


88 


.18452 


.98203 


.18775 


5.32631 


.20165 


.97946 


.20588 


4- 85727 


22 


8JL. 


.18481 


.98277 


.18806 


5-31778 


-20193 


.97940 




4 85018 


21 


40 


.18509 


.98272 


.18835 


5-30928 


-20222 


.97934 


.20648 


4- 84300 


^0 


41 


.18538 


.98267 


.18865 


5. 30080 


.20250 


.97928 


.20679 


4-83590 


19 


42 


.18567 


.98261 


.18895 


5.29235 


20279 


.97922 


.20709 


4- 82882 


18 


48 


.18596 


98256 


.18925 


5.28393 


.20307 


.97916 


.20739 


4- 82175 


17 


44_ 


.18624 


.98260 


.18955 


6.27553 


■20336 


.97910 


.20770 


4. 81471 
4-80769 


1» 


46 


.18652 


.98245 


.18986 


5.26715 


.20364 


.97905 


.20800 


18 


46 


.18681 


.98240 


.19016 


5.25880 


.20393 


.97899 


.20830 


4-80068 


14 


47 


.18710 


.98234 


.19046 


5.25048 


.20421 


.97893 


.20861 


4. 79370 


18 


48 


.18788 


.98229 


.19076 


5.24218 


.20450 


.97887 


-20891 


4.78673 


12 


49 


■18767 


.98223 


.19106 


6.23391 


.20478 


.97881 


•20921 


4.77978 


Ul 


50 


18795 


.98218 


.19136 


5.22566 


20507 


97875 


-20952 


4.77286 


10 


51 


.18824 


.98212 


.19166 


5.21744 


-20535 


.97869 


-20982 


4.76595 


9 


62 


.18852 


.98207 


.19197 


5- 20925 


.20563 


97863 


.21013 


4. 75906 


8 


68 


.18881 


.98201 


.19227 


5.20107 


. 20592 


.97857 


.21043 


4.75219 


7 


64 


.18910 


.98196 


.19257 


5. 19293 


.20620 


.97851 
.97845 


.210781 
.21104 


4.74534 


_8 


66 


.18988 


.98190 


.19287 


5.18480 


.20649 


4.73851 





66 


.18967 


.98185 


.19317 


5.17671 


-20677 


-97839 


.21134 


4.73170 


4 


67 


.18996 


.98179 


.19847 


5.16863 


.20706 


.97883 


.21164 


4.72490 


8 


68 


.19024 


.98174 


.19378 


5.16058 


-20734 


-97827 


-21195 


4.71813 


2 


M- 


.19052 


.98168 


.19408 


5. 15256 


.20763 


.97821 


.21225 


4.71137 
4-70463 


_1 


go. 


«^ 


.98163 


.19488 


5. 4455 


.20791 
Cos. 


.97815 
Sin. 


• n256 
Cot. 


-^ 


5in. 


Cot. 


Tan. 



70* 



Digitized by VjOOQIC 



246 NATTJRAL SINES, COSINES, TANGENTS, COTAN. 



/ 


3in. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 








20791 


.97815 


.21256 


4.70463 


■22495 


.97437 


23087 


4. 88148 


"eo 


1 


■20820 


.97809 


.21286 


4.69791 


.22523 


.97430 


28117 


4.32573 


59 


2 


20848 


.97803 


.21316 


4.69121 


22552 


.97424 


.23148 


4. 82001 


58 


8 


■20877 


.97797 


.21347 


4.68452 


.22580 


.97417 


.23179 


4. 81430 


57 


4 


■20905 


.97791 


.21377 
.21408 


4.67786 


.22608 


.97411 


.28209 


4.80860 


56 


5 


■20938 


.97784 


4.67121 


.22637 


.97404 


.23240 


4.30291 


55 


6 


.20962 


.97778 


.21438 


4.66458 


.22665 


.97398 


.23271 


4.29724 


64 


7 


■ 20990 


.97772 


.21469 


4.65797 


. 22698 


.97391 


.23301 


4.29159 


58 


8 


■21019 


.97766 


. 21499 


4.65138 


.22722 


.97384 


23332 


4.28595 


52 


9 


■21047 


.97760 


.21529 


4.64480 


.22750 
.22778 


.97378 


■23363 


4. 28032 


51 


10 


■21076 


.97754 


.21560 


4.63825 


.97371 


23893 


4.27471 


50 


11 


■21104 


.97748 


.21590 


4.63171 


.22807 


.97365 


■23424 


4.26911 


49 


12 


.21132 


.97742 


■21621 


4.62518 


.22885 


.97358 


23455 


4.26852 


48 


13 


.21161 


.97735 


.21651 


4 61868 


.22863 


.97351 


23485 


4.25795 


47 


14 


■21189 


.97729 


.21682 


461219 


.22892 


.97345 


■23516 


4. 25239 


46 


15 . 


.21218 


.97723 


.21712 


4 60572 


. 22920 


.97338 


.23547 


4.24685 


45 


16 


.21246 


.97717 


•21748 


4-59927 


.22948 


.97331 


.23578 


4. 24182 


44 


17 


.21275 


.97711 


.21778 


4 ■59283 


.22977 


.97325 


. 23608 


4.23580 


43 


18 


.21303 


97705 


.21804 


4. 58641 


.23005 


.97318 


.23639 


4.28030 


42 


ISL 


.21331 
.21360 


: 97698 


.21834 


4.58001 


.23033 


.97311 


■23670 


4.22481 


.41 


so 


.97692 


.21864 


4.57363 


.23062 


97304 


. 23700 


4.21983 


40 


21 


.21388 


.97686 


.21895 


4.56726 


.23090 


97298 


23731 


4.21387 


39 




.21417 


.97680 


.21925 


4. 56091 


.23118 


■97291 


■23762 


4. 20842 


88 


28 


.21445 


.97673 


.21956 


4.55458 


.23146 


■97284 


.23793 


4. 20298 


37 


24 


.21474 


.97667 


.21986 


4 .54826 


.23175 


■97278 


. 23823 


4.19756 


36 


25~ 


.21502 


.97661 


.22017 


4.54196 


.23203 


.97271 


.23854 


4.19215 


85 


26 


.21530 


.97655 


.22047 


4. 53568 


23231 


.97264 


.23885 


4.18675 


84 


27 


.21559 


.97648 


.22078 


4.52941 


.23260 


■97257 


.23916 


4.18187 


83 


28 


.21587 


.97642 


.22108 


4.52816 


23288 


.97251 


.28946 


4.17600 


82 


29 


.21616 


.97636 


.22139 


4-51693 


.23316 


.97244 


.28977 


4.17064 


81 


30 


.21644 


.97630 


.22169 


4. 51071 


.23345 


.97287 


.24008 


4.16530 


30 


81 


.21672 


.97628 


.22200 


4.50451 


■23373 


.97230 


.24039 


4.15997 


29 


82 


■21701 


•97617 


.22231 


4. 49832 


23401 


.97223 


.24069 


4.15465 


28 


88 


.21729 


.97611 


.22261 


4.49215 


23429 


.97217 


.24100 


4. 14934 


27 


84 


.21758 


.97604 


.22292 


4.48600 


■23458 


.97210 


.24131 


4 14405 


26 


85 


.21786 


.97598 


.22322 


4 47986 


23486 


.97203 


.24162 


4. 13877 


25 


86 


.21814 


.97592 


.22353 


4.47374 


■23514 


.97196 


.24193 


4.13350 


24 


37 


.21843 


.97585 


.22388 


4.46764 


■23542 


.97189 


.24223 


4.12825 


28 


38 


.21871 


.97579 


.22414 


4.46155 


■23571 


.97182 


.24254 


4.12801 


22 


39_ 


.21899 


.97573 


.22444 


4 45548 


■23599 


.97176 


■24285. 


4-11778 


-2i 


40 


.21928 


.97566 


.22475 


4.44942 


23627 


.97169 


■24316 


4-11256 


20 


41 


.21956 


.97560 


.22505 


4.44338 


■23656 


.97162 


.24347 


4. 10786 


19 


42 


.21985 


.97553 


.22536 


4. 43735 


■23684 


.97155 


.24377 


4.10216 


18 


43 


.22013 


.97547 


.22567 


4.43134 


.23712 


97148 


.24408 


4. 09699 


17 


44_ 


.22041 


.97541 


■22597 


4.42534 
4.41936 


23740 


■97141 


.24439 


4.09182 


16 


45 


.22070 


.97534 


.■ 22628 


■23769 


.97134 


.24470 


4.08666 


15 


46 


22098 


.97528 


■22658 


4.41340 


23797 


.97127 


.24501 


4.08152 


14 


47 


.22126 


.97521 


.22689 


4.40745 


.23825 


.97120 


.24532 


4. 07689 


18 


48 


.22155 


.97515 


.22719 


4.40152 


.23853 


.97113 


.24562 


4.07127 


12 


49 


.22183 


.97508 


■22750 
.22781 


4 39560 


.23882 


.97106 


.24593 


4.06616 


11 


50 


.22212 


.97502 


4. 38969 


.28910 


.97100 


.24624 


406107 


lO 


51 


. 22240 


.97496 


.22811 


4.38381 


< 23938 


.97093 


.24655 


4. 05599 


9 


52 


.22268 


.97489 


.22842 


4.37793 


.23966 


.97086 


.24686 


4. 06092 


8 


58 


.22297 


.97488 


22872 


4.37207 


.23995 


.97079 


.24717 


4. 04586 


J 


54 


.22325 


.97476 


.22908 


4.36623 


■ 24023 


.97072 


. 24747 


4.04081 


_• 


55 


.22353 


.97470 


.22934 


4.36040 


.24051 


.97065 


.24778 


4. 08578 


6 


66 


.22382 


.97468 


.22964 


4.35459 


. 24079 


.97058 


.24809 


4. 08076 


4 


67 


.22410 


.97457 


.22995 


4.34879 


.24108 


.97051 


. 24840 


4. 02574 


8 


58 


.22488 


97450 


■23026 


4. 34300 


.24136 


.97044 


.24871 


4. 02074 


2 


M_ 


.22467 


.97444 


■23056 


4.33723 


^4184_ 


■97037 
■97030 


■24902 


4. 01576 


1 


60 


«- 


4 33148 
Tan. 


■24192 
Cos. 


.24933 


4.01078 


_fi 


"7" 


Sin. 


Cot. 




^~ 



ir 



Digitized by VjOOQIC 



NATURAL SINES, COSINES, TANGENTS. COTAN. 247 

14* 16* 



r 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 







.24192 


.97030 


.24938 


4.01078 


.25882 


96593 


26795 


3. 73205 


GCI 


1 


.24220 
.24249 


.97023 


.24964 


4.00582 


.25910 


.96585 


.26826 


3.72771 


5i, 


2 


.97015 


.24995 


4. 00086 


.25933 


96578 


.26857 


3.72338 


50 


8 


.24277 


.97008 


.25026 


3. 99592 


.25966 


96570 


26888 


3.71907 


57 


4 


.24805 


.97001 


.25056 


3. 99099 


.25994 


.96562 


•26920 


3. 71476 


56 


n 


. 24833 


.96994 


.25087 


3. 98607 


.26022 


.96555 


.26951 


3. 71046 


55 


6 


24362 


.96987 


.25118 


3. 98117 


.26050 


.96547 


■26982 


3. 70616 


54 


7 


.24390 


.96980 


.25149 


3. 97627 


.26079 


■96540 


■27013 


3. 70188 


53 


8 


24418 


.96973 


.25180 


3.97139 


.26107 


96532 


•27044 


8 .69761 


52 


9 


.24446 


.96966 
96959 


.25211 


3.96651 


•28135 


.96^24 


.27076 


3. 69335 
3-68909 


51 


10 


. 24474 


.25242 


3. 96165 


•26163 


.96517 


•27107 


60 


11 


.24503 


.96952 


25278 


3. 95680 


.26191 


96509 


■27138 


3 ■68485 


49 


u 


. 24531 


. 96945 


.25304 


3. 95196 


.26219 


.96502 


.27169 


3 ■68061 


48 


18 


. 24559 


.96937 


.25385 


8. 94713 


.26247 


.96494 


27201 


3 •67638 


47 


14 


.24587 


.96930 
.96923 


.25866 
.25397 


3.94232 


.26275 


•96486 


•27232 


3. 67217 
3. 66796 


40 


15 


24615 


3. 93751 


.26303 


.96479 


.27263 


45 


IR 


24644 


.96916 


25428 


3. 93271 


.26331 


96471 


.27294 


3 66376 


44 


17 


.24672 


. 96909 


. 25459 


3- 92793 


.26359 


•96463 


.27326 


3. 65957 


43 


18 


. 24700 


96902 


.25490 


3. 92316 


.26387 


96456 


.27357 


3 ■65538 


42 


19. 


.24728 


96894 


.25521 


3.91839 


.26415 


.96448 


.27388. 


3 • 65121 


-il 


20 


.24756 


•96887 


25552 


3. 91364 


.26443 


.96440 


.27419 


3 64705 


40 


21 


24784 


96880 


.25583 


3. 90890 


.26471 


.96433 


.27451 


3 •64289 


39 


22 


24813 


96873 


.25614 


3.90417 


.26500 


.96425 


27482 


3. 63874 


38 


23 


24841 


96866 


.25645 


3. 89945 


.26528 


.96417 


.27513 


3.63461 


87 


24 


,24869. 
.24897 


96858. 
: 96851 


25676 


3.89474 


.26556 


. 96410 


.27545 


3 ■63048 


36 


26 


.25707 


3-89004 


.26584 


•96402 


.27576 


3. 62636 


35 


26 


.24925 


•96844 


.25738 


3. 88536 


.26612 


•96394 


.27607 


3.62224 


34 


27 


.24954 


•96837 


.25769 


8. 88068 


. 26640 


.96386 


27638 


3.61314 


33 


28 


24982 


96829 


.25800 


8. 87601 


26668 


96379 


.27670 


3. 61405 


32 


29 


.25010 
.25038 


96822 
96815 


.25831. 
.25862 


8.87136 


26696 


.96371 


.27701 


3. 60996 


31 


30 


3. 86671 


.26724 


•96363 


.27732 


3 ■60588 


30 


31 


.25066 


•96807 


.25893 


3. 86208 


.26752 


96355 


.27764 


8 60181 


29 


82 


.25094 


96800 


.25924 


3. 85745 


.26780 


•96347 


.27795 


3. 59775 


28 


83 


.25122 


•96793 


25955 


3. 85284 


.26808 


•96340 


.27826 


3.59370 


27 


84. 


.25151 


•96786 
96778 


.2598B 


3.84824 


.26836 


.96332 


■27858 


3. 58966 


26 


85 


.25179 


.26017 


3. 84364 


.26864 


•96324 


.27889 


3.58562 


25 


86 


25207 


96771 


.26048 


3. 83906 


.26892 


96316 


■27921 


3 •58160 


24 


87 


25235 


96764 


26079 


3. 83449 


.26920 


■96308 


•27952 


3.57758 


23 


88 


.25263 


96756 


.26110 


3.82992 


26948 


96801 


•27983 


3. 57357 


22 


89. 


•25291 
.25320 




.26141 


3 •82537. 
3 •82088 


.26976 
•27004 


.96293 


.28015 


3.56957 


21 


40 


96742 


.26172 


96285 


.28046 


3. 56557 


20 


41 


25348 


96734 


.26203 


8. 81630 


• 27032 


•96277 


.28077 


3. 56159 


19 


43 


25876 


96727 


.26235 


3. 81177 


•27060 


•96269 


.28109 


3. 55761 


18 


48 


25404 


.96719 


.26266 


3. 80726 


• 27088 


96261 


.28140 


3. 55364 


17 


M 


.25432 


.96712 


.26297 


3.80276 


.27116 


96253 


.28172 


3.54968_ 
3-54573 


16 


45 


25460 


.96705 


.26328 


3. 79827 


•27144 


.96246 


.•28203 


15 


46 


•25488 


.96697 


.26359 


3. 79378 


•27172 


96238 


.28234 


3 54179 


14 


47 


.25516 


96690 


. 26390 


8. 78931 


. 27200 


96230 


28286 


3 53785 


18 


48 


.25545 


96682 


.26421 


3.78485 


27228 


.96222 


.28297 


8. 53393 


12 


49 


.25573 
.25601 


.98675 
•96667 


.28452 


3.78040 


.27258 


.96214 


28329 
28360 


3. 53001 
3 52609 


11 


50 


.26483 


3. 77595 


.27284 


.96206 


10 


61 


.25629 


.96660 


.26515 


3. 77152 


.27312 


.96198 


28391 


3.52219 


9 


52 


25657 


.96653 


26546 


3.76709 


•27340 


.96190 


.28423 


3. 51829 


8 


Si 


.25685 


.96645 


.26577 


3.76268 


.27368 


.96182 


.28454 


3.51441 


y 


25713 


.96638 


.26608 


3. 75828 


27898 
•27424 


.96174 
■96166 


.28486 
.28517 


3-51053 
3-50666 


6 


IS 


.25741 


.96630 


.26639 


3.75388 


5 


.25769 


.96623 


.26670 


3.74950 


.27452 


.96158 


.28549 


8 50279 


4 


67 


.25798 


.96615 


.26701 


3.74512 


. 27480 


.96150 


.28580 


8. 49894 


3 


56 


.25826 


.96608 


.26733 


3.74075 


•27508 


.96142 


.28612 


8.49509 


2 


59 


.25854 


96600 


•26764 


3. 73640 


27536 


.96134 


.28643 


8. 49125 


1 


«^ 


'^- 


W- 


.28795 


3.73205 


^^ 


•96126 
Sin. 


.28675 


Tan. 





Cot. 


Tan. 





?«• 



Digitized by VjOOQIC 



248 NATURAL SINES, COSINES, TANGENTS, OOTAN. 







16 


*» 






17^ 






t 


Bin. ! 


i:m. \ 


Tan. t'rrt. 


Sin. 


Cos. 


Tan. 


Cot, 


' 


1 

2 
3 
4 


27664 

.27592 
.27620 
-27646 
.27675 


.96126 
.96118 
-96110 
.9B102 
.9a094 


28675 3 48741 
23706 1 3 48359 
28738 3 47977 
.28769 3.47596 
.28300 3 47236 


.29237 
.29265 
.29293 
.29321 
. 29348 
29376 
29404 
-39432 
-29460 ; 
.29487 1 


95630 

.95623 
95613 
.95605 
-95596 
.95586 
-95579 
.95571 
-95563 
.95554 


30573 
30606 
30637 
30669 
30700 

.30732 
30764 

.30796 
30828 

.80880 


3. 27065 
3. 26745 
3 26408 
3. 260 6 7 
3-25739 
3.25393 
3-25055 
3-34719 
3.24^63 
3-24049 


60 

59 
5B 
57 
S6 


G 
6 
7 
8 
9 


.27704 
27731 
,27759 
37787 
.27815 
27843 
.37871 
.27899 
.27927 
.27955 
.27983' 
.28011 
-28039 
.28067 
.58095 


.96088 
.98076 
.06070 
.»60fl2 
.96054 
.98046 
.96037 
.96029 
-90021 
-96013 


.28B32 
28364 
.28895 
.38927 
.28958 


3-46837 
3 46458 
3-46080 
3 45703 

3-45327 


55 
51 

51 
63 
51 


n 

13 

u 


.28990 

. 39021 
,29053 
.39084 
,29116 


3, 44951 
3-44576 

3.44202 1 
3.43829 
3^3i56_ 
3 43084 
3^43713 
3 - 42343 
3 41973 
3.41604 , 


.29515 
.39543 

,29571 
29599 

.29626 
.29354 
.29882 
. 29710 
-29737 
-29765 


.95545 
95536 
.95528 
95519 
95511 
95503 
.95493 
-95435 
.95476 
-95467 


.30891 
30923 
30955 

.30987 

J1019 
31051 
31083 
31115 

.31147 
31178, 


3-23714 
3 ^ 233S1 
3-23048 
a. 33715 
_3,223J4. 
3.23053 
3 21722 
3. 21393 
3,21063 
3 20734 


50 

49 
48 

47 
46 


IB 

ifi 

17 
IS 


.96005 
.95997 
.95989 
.95981 

.95972. 


.29147 
.29179 
29310 
. 29242 

,29274 


45 
14 
43 
42 


20 

21 
22 
23 
24 


-28123 
-28150 
28173 

-2820e 

728262 
. 28290 

28318 
-28346 
-23374 
"28402 
-23429 
.28457 
.28435 

23513 
-23541 
.28589 

28597 
.28625 
.28652 

.28380 

.26708 
.28736 
.28764 

.28792 


.95964 
.95956 

95948 
.95940 
.95931 
.95923 
-95915 

95907 
.95898 

.05882 

.95874 
,95865 
.95857 
.95849 

.95841 
.95832 
-95824 
.95816 
. 95807 


.29305 3.41236 
-29337 3 40869 
-29368 3.40502 
■39400 3.40136 
,29432 3,39771 
,39463 : 3-39408 
,29495 ' 3.39043 
-29526 3. 38879 
-29558 3 38317 
.29590_ 3.37955 
-29621 3.37594 
.29653 • 3.37234 
-29685 3.36875 
-29716 3-36516 
29748 : 3 36158 
-397S0 3.35B0O 
.29811 g. 35443 
-29843 3.35087 
.29675 3.34732 
.299n6_ 3.34377 


.29793 
-29821 
.29649 
29876 
.29904 
-29932 
.29960 
-29937 
-30015 
.30043 
-30071 
.30098 
.30126 
-30154 
-301S2 
-30200 1 
.30237 
-30365 
.30292 
-30320 


-95459 
.954^9 
,9544x 
-95433 
-95434 


31310 
31242 
.31374 
,31306 
31338 


3^20406 
3.20079 
3. 19752 
3.19426 
3. 19 100 


40 

39 
38 
37 
36 


25 
2& 
27 
28 
2& 


,95415 ' 
-95407 
.86398 
.95339 
95380 


31370 
.31402 
.31434 

31466 

31498 
.81530 
.31562 
.31594 
,31626 

31858 


3. 18775 
3. 18451 
3. 16127 
3-17304 
3 1748] 

3-17159 
3 ■16838 
3,16517 
3-18197 
315877 
3 15558 
3 15240 
3 14932 
3 14605 
3-14288 


85 
34 
33 
32 
31 


30 

31 
82 
33 
34 


.95373 
.95363 , 
-95354 ' 
.95345 
95337 
.05323 
95319 
95310 
.95301 ' 
,95293 


30 

as 

28 

37 
26 


35 
35 
37 
38 
39. . 


31890 
31723 

.31754 
31786 

.31813 


25 

24 
23 

22 
-11 


10 

41 
42 
48 
44 


-95799 
,95791 

.95782 
95774 

.95?6e_' 
'95757 • 
95749 
95740 

.95732 ; 
.95724 


-29938 
.29970 

.30001 
.30033 
.30065 

30097 

.30128 
.30160 
.30192 
.30224 


3.34023 
3 33670 
3 33317 
3. 32965 
3. 33614 
3. 33364 
3. 31914 
3. 31565 
3.31316 
3-30&B8 


-30348 

,30376 
.30403 
.30431 
.30459 
.30486 
.30514 
-30,542 
.30570 
-30597 
-30625 
.30653 
.30680 
-30708 
-30736 
.30763 
.30791 
.30819 
.30846 
.30874^ 
.30902' 
Cos. 


95284 

95275 
.95266 

95257 
.95243 

95240 
.95231 
.95222 
.95313 
.95204 
.95195 
■95186 
.95177 
.95168 
.95159^ 
.951^56 
■95143 
.95133 

95124 
.95115 
.953,06, 


.31850 
31382 
31914 
31946 
.31978 
32010 
32042 
.32074 
33106 
32139 
32171 
32203 
32235 
.32267 
-32209 
-32331 
33363 
.32396 
.32428 
-32480 
.32493, 
Cot. 


3 13973 
3. 13858 
3 13341 
3. 13027 
3 12713 


30 
19 
18 
17 
i6 


45 
46 
47 
4a 
49 


.28820 
.28847 

-28875 
.28903 
-28931 
.28959 
-28987 
.29015 
.29042 
^29070_ 
, 29098 
29128 
.29154 
.29182 
.29209 


3 12400 
3, 12087 
3^1775 
3. 11464 
a 11153 


15 
14 
13 
12 
11 


50 

&l 
52 
63 
54 


.95715 
95707 
.95G9B 

.95690 
.95e_81 
.95673 
95664 
.95658 
.95647 
.95639 


.30255 
30287 
30319 

.30351 
30382 

30414 
. 30446 

30478 
.30509 
J0541_ 
.30573 
Cot. 


3 30521 
3 30174 
3. 39829 
3-29483 
3.39139_ 
3 28795 
3^28452 
3-28109 
3.27767 
_3. 27428 
3-270B5 
Tan. 


3.1DB42 
3. 10532 
3. 10223 
309914 
3.09606 


10 

e 

8 

7 


55 
55 
67 

69 


3-09393 
3,08991 
3.03885 
3. 083 79 
3. 03073 
3.07^66 
Tan. 


5 
4 

3 

1 


filL 


-29237 


.95fl30 


^ 



1^ 



Digitize?b?t^OOgle 



NATURAL SINES, COSINES, TANGENTS, COTAN. 249 

18* 19* 



t 


Sin. 


Coii. 


Tftii. 


Cot. 1 


Sin, 1 


C(J?3. 


Tan. 


Cot. 


f 




1 

3 
S 
4 


30902 

.30929 

.30957 

.30965 

.31012 

-31040 

.31006 

31095 

.31123 

31151 

.31176 

-31206 

,31233 

31261 

.3126a 

31316 

31344 

31373 

31399 

.31454 
31482 

-31510 
-31337 
.31585 
-31593 
.31620 
-31646 
31875 
31703 


*9Q1O0 
.95097 
.95066 
.95079 
.95070 
.95061 
.95052 
.95043 
,95033 
.95024 
.95015 
.95006 
,94697 

94988 
.94979 

94970 

94901 
.94952 
.94943 
_?4933 
.94924 
,&4915 
.94900 
.94897 
- 948A8_ 
■94878 
-94869 
.94860 
-94651 
. 94J42^ 
r 9483 3 
.94823 
^94814 

94905 
.94795 
-94788 
.94777 
. 94766 
-94758 
.94749 


-32492 
.32524 
33556 
.33588 
-32621 
33653 
.32685 
-32717 
.32749 
32783 
32614 
33846 
.3287B 
-329U 
.32943_ 
-33975 
33007 
33040 
-33072 
-33104 


3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
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 
2 
2 


07768 
07464 
07160 
06857 
06554 
06253 
05950 
05649 
05349 
05049 
04749 
04450 
04152 
03854 
03556 

03260 
02963 
03667 

02372 

02077 

01783 

01489 

01198 

00903 

00611 

00319 ! 

00028 

99736 

99447 

99158 

93868 
98580 
98292 
98004 
97717 
97430 
97144 
96856 
96573 
9fl280 


.32557 
32584 
.32612 , 
32639 ' 
-33667 ■ 
.32694 
.32723 
.33749 
.32777 
32804 
32632 
32859 
. 33867 
.32914 
-33942 
-32969 
-32997 
33024 
33051 
.33079 
33106 
.33134 
.33161 
33189 
33216 
33344 
33271 
33398 
33326 
-33353 
33381 
.33408 
33436 , 
33463 
33490 
33518 
33545 
.33573 
.33600 
.33627 


.94552 
.94542 

94533 
-94523 
-94514 
-94504 

94495 
.94485 
.94476 
-94466_ 

94457 
. 94447 
-94438 

94428 

94^18 

94409 
.94399 
-94390 

94380 
.94370 


34433 
34465 
34496 
.34530 
-34663 
34596 
, 34626 
.34661 
.34693 
-34736 

34758 
. 34791 

34824 

34856 
.3488_9_ 
. 34922 

34954 
-34987 
.35030 
-35052 


2 90421 
2.90147 
2,89873 
2. 89600 
2 69327 


60 

59 
53 
57 
GS 


7 

8 

g 

10 

IL 
13 
13 

u^ 
la 

17 

le 

IS 


2. 69055 
2. 88783 

2 68511 
2. 88240 
2-87970 

3 87700 

2 87430 
2. 87161 
2. 86892 
2.66624 

3 66356 
2. 66089 
2. 653 2 2 
2-85555 
2 65289 


5& 

&4 

62 

ao 

43 

48 
47 

4S 
44 
43 
43 
41 


30 

21 
22 
23 
24 
25 
26 
27 

2a 

29 


33136 
33169 
33201 
33233 

.33266^ 
.33296^ 
33330 
33363 
33395 
■33427 
.33480 
-33493 
.33524 
.33557 
33589 


04361 
-94351 
-94342 
,94332 
-94S22 

"94313' 

,94303 

-94393 

.64284 

-94374 


-35085 

.35118 

-35150 

.35183 

.95_21ft.. 

.35243 

-35381 

.35314 

35346 
.35379 
.35412 
-35445 
.35477 
-35510 
.35543 

35576 
.36608 

35641 
.35674 
.35707 


2. 85023 
2.84756 
2-84494 
2 84229 
2. 83985 
2- 83703 
2 83439 
3. 33176 
2 62914 
2 32653 

3. 32391 

2 82130 

3 81870 
2-81610 
3. 81350 
2.81091 
2-60833 
2. 60574 
2 80316 

2 60059 1 


40 

39 
3S 

37 
36 
35 
34 
33 
32 
31 


30 

31 
33 
33 
34 


31730 
.31758 
-31786 
31813 
31841 


94364 
94254 

-94245 

.94235 
94235 
94215 

.94206 
94196 

.94186 
94176 


30 

2% 
28 
27 
,18 


35 
36 
37 
38 
3J_ 


31663 
31396 

.31923 
-31951 
31979 


33621 
33654 
.33666 

.33718 
33751 


25 
24 
23 
22 
21 


40 

41 
42 
43 
41 


32006 
32034 
-32061 
- 3206& 
-321X6 


.94740 
-94730 
-94721 
-94712 
.94702 
.94693 
-94634 
.94674 
.94685 
-94656 


.33783 
33816 
33846 

-33361 

. 33913 
33945 
33976 
34010 
34043 

.34075 


3.96004 
2 95721 
3.95437 
2.95155 
2 94872 
3.94591 
2 94309 
3.94028 

2 93746 

3 93466 


.33655 
.33682 
33710 
33737 
33764^ 
. 337'92 
.33619 
.33846 
33874 
33901 


-94167 
-94157 

94147 
.94137 

94127 
.94118 
-94108 

94098 

94038 
,94073 


35740 
-35773 

35805 
-35638 
.35871 


3 79802 
2.79545 
3 79289 
2.79033 
2-78778 


30 

19 
18 
17 
16 


45 

46 
47 
4B 
49 


-32144 
.32171 
-32199 
.33227 
-32254 
-32232 
.32309 
,32337 
.32364 
-32392 
32419 
.32447 
32474 
32502 
33529 


-35304 
-35937 
-35969 
-36002 
-36035 


3 78523 
2 78369 
2 78014 
377781 
2 77507 


IS 
14 
13 

13 
11 


BO 

&1 
52 
S3 
64 


-94846 

94637 
-94627 

94618 
.94609 

94599 
.94590 

94560 
.94571 
-945«1 
,94652 


-34108 

.34140 

.34173 
34305 
34238 
34270 
34303 

-34335 
34368 
34400 

.34433 


3. 93189 
2-92910 

3.93632 
2 92354 

2.93076 


,33929 
-33956 
.33933 
.3401 1 
•3-403_H_ 
-34066 
. 34093 
.34120 
.34347 
.343 75 


.94066 

.94056 
94049 

-94039 

-94D29 
94019 
94009 
93999 
93989 

.93979 
93969 
Sin. 


36066 
.36101 
-36134 
,36167 
-36I99_ 

36235 
.36265 

36398 
-36331 
-36364 


3.77354 
2,77002 
2.76750 
2,76498 
2.76347 
2.75996 
2.75746 
2.75496 
2^75246 
2,74997 


10 

I 


65 
57 
59 


2.91799 
2. 91533 

2-91246 
2.90971 
2 90696 
2.90421 


B 

4 

1 

1 


60 


33557 


. S4202 


.36397 
Cot. 


2.74748 
Tan. 


_ffl 


f 


C<J!i. 


Sb. 


Cot. 




Tan. 1 


t 



IX"" 



DigitizlftyGoOgLe ^ 



250 NATURAL SINES, COSINES, TANGENTS, COTAN. 



# 


Sin. 


Cos. I 


Tan. 


Cot. 


Sin. Cos. 1 


Tan. 


Cot. 


f 




1 

2 
8 

4 


.34202 
.84229 
.84257 
.84284 
.34311 


.93969 
.93959 
.93949 
.93989 
.98929 


.36397 
.36430 
.36468 
.36496 
.36529 


2.74748 
2.74499 
2.74261 
2.74004 
2.73756 


.86887 
.36864 
.86891 
.86918 
.85946 


.03868 
.98848 
.93887 
.98827 
.93816 


.88880 
.88420 
.88463 
.88487 
.88620 


2 •60509 
2.60283 
2.60057 
2.59881 
2.59606 


60 

59 
68 
67 
66 


6 
6 
7 
8 
9 


.34339 
.34366 
.34398 
.34421 
.34448 


.93919 
93909 
.93899 
.93889 
.93879 


.36562 
.36595 
.36628 
.36661 
.36694 


2. 78609 
2.78268 
2.78017 
2.72771 
2.72626 


.85978 
.86000 
.86027 
.86064 
•36081 


.98806 
.93296 
98285 
.98274 
.98264 


i 88587 
.88620 
.88654 
.88687 


2. 69881 
2. 59166 
2. 58982 
2. 58708 
2. 68484 


56 
54 

53 
52 
51 


10 

11 
12 
18 
14 


.34475 
.84508 
.34530 
.34557 
34584 


.93869 
.93859 
.93849 
.93839 
.93829 


.86727 
.36760 
.36793 
.36826 
.36859 


2. 72281 
2. 72086 
2.71792 
2. 71648 
2.71305 


•86108 
•36185 
•36162 
•86190 
•86217 


.98263 
.98243 
98282 
.98222 
•93211 


•38721 
•88754 

38787 
•38821 

88864 


2 58261 
2. 58088 
2.67815 
2.57593 
2.67871 


50 

49 
48 
47 
46 


16 
16 
17 
18 


.84612 
34689 
.84666 
.34694 
.84721 


.93819 
.93809 
.93799 
.93789 
.93779 


.86892 
.36925 
■36958 
.36991 
.37024_ 


2.71062 
2.70819 
2.70577 
2.70336 
2.70094. 


•86244 
•86271 
86298 
.86826 
'3635;^ 


93201 
93190 
.93180 
.93169 
•93169 


•38888 
•88921 
•38965 
•38988 
•?«022 
•89056 
•89089 
.89122 
•89156 
•89190 


2. 57160 
2.56928 
2.56707 
2. 66487 
2 •56266 


46 
44 
43 
42 
_41 


'^0 

21 
22 
28 
24 


84748 
.34775 
.84808 
34830 
34857 


.93769 
.93759 
.93748 
.93738 
.93728 


.37057 
.37090 
.37123 
.37157 
.37190 


2.69853 
2.69612 
2.69871 
2.69131 
2.68892 


•86379 
•86406 
.86484 
•86461 
•36488 


93148 
.98137 
.93127 
.93116 
.93106 


2.56046 
2. 56827 
2. 56608 
2.65889 
2.55170 


40 

IS 

37 
36 


25 
26 
27 
28 
29 


34884 
34912 
.34939 
.34966 
.34993 


•93718 
.93708 
.93698 
.93688 
.93677 


.37228 
.37256 
.37289 
.37322 
.37856 


2. 68668 
2.68414 
2.68176 
2.67987 
2.67700 


86515 
86542 
.86569 
.86596 
.36623 


.93096 
•93084 
.93074 
.93063 
.93052 


•89223 
89257 
.39290 
.89324 
.39857 


2. 64962 
2.54784 
2.54516 
2.54299 
2.54082 


85 
84 
83 
82 
81 


30 

81 
32 
83 
34 


.35021 
.35048 
.35075 
.35102 
35130 

35157 
35184 
.35211 
.35230 
.35266 


.93667 
.93657 
.93647 
.93637 
.93626 
.93616 
.98606 
.93596 
.93585 
.98575 


.37888 
.37422 
•37456 
37488 
37521 


2.67462 
2.67225 
2.66989 
2.66752 
2.66516 


.86650 
.36677 
•36704 
86731 
.36758 


.93042 
.93081 
93020 
93010 
.92999 


89391 
.39425 
•39458 
•89492 
•39526 


2.53865 
2.53648 
2 •53432 
2.63217 
2.58001 


30 

IS 

27 
M 


85 
36 
87 
88 
89 


37554 
87588 
37621 
•37654 
.87687 


2 .66281 
2 66046 
2. 65811 
2 •65576 
2.66842 


36786 
.36812 
86839 
86867 
•86?94 


92988 
92978 
92967 
92956 
•?2945 


•39669 
•39598 
89626 
•39660 
J96Si. 


2.62786 
2.62571 
2.52367 
2.52142 
2.51929 


25 
24 
28 
22 

-21 


40 

41 
42 
43 
44 


35293 
35320 
35347 
35375 
.35402 

35429 
.35456 
.35484 
35511 
35538 


.93565 
.93555 
.03544 
.93534 
.93524 
.98614 
.93503 
.93498 
.93483 
.93472 


.87720 
•37754 
•37787 
.37820 
.37853 

.87887 
.37920 
•37958 
•87986 
•38020 


2. 66109 
2. 64875 
2. 64642 
2 .64410 
2.64177 
2 •63945 
2 63714 
2 63483 
2 63252 
2 •63021 


36921 
•86948 
■36975 

37002 
. 37029 


•92935 
92924 
92913 
92902 

.92892 


•39727 
39761 

•39795 
39829 
39862 


2. 61716 
2.61602 
2.61289 
2.61076 
2.50864 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


37056 
.37083 
.37110 

37187 
•87164 

i 87218 
87245 
87272 

•37209 


92881 
92870 
92859 
•92849 
•92838 
•92827 
.92816 
.92805 
.92794 
•92784 


•39896 
•39980 
•39963 
•39997 
.40031 


2.50652 
2. 60440 
2.50229 
2.50018 
2.49807 


15 
14 
13 
12 
-11 


50 

61 
52 
53 
54 


.35565 
.35592 
.35619 
.35647 
.35674 


.93462 
.93452 
.93441 
.93431 
.98420 


•88058 
.88086 
•38120 
•38153 
•88186 


2 •62791 
2 62561 
2. 62332 
2-62108 
2. 61874 


40065 
40098 
40182 
.40166 
.40200 


2.49597 
2.49386 
2. 49177 
2.48967 
2-48758 


lO 

9 
8 
7 
6 


65 
56 
57 
58 
59 


.85701 
.85728 
.35755 
35782 
35810 


.93410 
.93400 
.93389 
.93879 
.93868 


.38220 
•38268 
•88286 
•38320 
•38358 


2. 61646 
2. 61418 
2 61100 
2. 60963 
2. 80736 


•37326 
37853 
87380 
37407 
87434 


•92773 
■92762 
92751 
.92740 
. 92729 


•40234 
•40267 
•40801 
•40886 
.40869 


2 48549 
2 48340 
2 •48132 
2. 47924 
2. 47716 


5 

4 
3 
2 

1 


S2. 


,3588i 
Cos. 


sin. 


«^ 


2.60509 
Tan. 


Cos. 


'W- 


«^ 


?.V509. 


-rfi 



eo** 



DigiSi^by Google 



NATURAL SINES, COSINES, TANGENTS, COTAN. 



t 


Sin. 


.92715 ; 
.92707 ' 

92697 
.92686 
.92575 1 

92664 I 
.92653 
.92642 
.92631 

92620 
.92609 
.92593 
.92587 
,82576 
.92555 


Tan. 


Cot. 


Sin. I 


Cos, ' 


Tan. 


Cot. 




1 

2 
3 
4 

G 

e 

7 

s 

9 


37461 
. 37408 

37615 

37542 
^7569 
-37595 
.37622 
-37649 
.37576 

37703 


-40403 
. 40436 
.40470 
.40504 
.40533 
.40572 

40606 
. 40640 
.40674 
.40707 
.40741 

40775 
.40309 
.40H43 

40877 


2.47509 
2.47303 
2-47095 
2 46365 
J-46e82_ 
2 . 46476 
2 46270 
2.46065 
2-45S6D 
2. 45655 


39073 
39100 
39127 
39153 
39130^ 
.39207 
-39234 
.39250 
39287 
.39314 


.92050 
.92039 
-92028 
.92015 
■92005 
"91994 

91982 
.91971 
-91959 
.91943 

91936 
.91925 
.91914 
-91902 
.91591 
.91879 
-91865 
-91656 

91845 

91833 


.42447 
42432 
42516 

.42551 
42555 

.42619 
.42554 
42688 

. 42722 
-42757 

42791 
42826 
42860 
42394 
.42929 


2-35585 
2. 35395 
2 35205 
2.35015 
2-34825 
2.34636 
2-34447 
2 34258 
2 34069 
2-33581 


10 

11 

12 

n 

14 


-37730 
37757 
37764 
37911 
37935 


2. 45451 

2.45246 
2. 45043 
3 44539 

2-44636 

2-44433 
2. 44230 
2-44027 

2-43825 
2-43623 


-39341 
-39367 

.39294 
.39421 
39448 

39474 
39501 
39528 
39555 
39581 


2-33593 
2.33505 
2-33317 
2.33130 
2-32943 


16 

16 
17 

la 

19 


37965 
37H&2 
.37919 
,37946 
.37973 
- 37&99 
39026 
38053 
38080 
.38107 
-38134 
.33151 
-35158 
35216 
,38241 
.38268 
.38295 
.38322 
.38349 
.38376 


-92554 1 
.92543 ! 
.92532 

-92521 
-92510 

.92499 t 
.92403 i 
,92477 
.92466 
.92455 
.92444 
.92432 
.92421 
.92410 
.92399 
T92385 
.92377 
.92366 
.92355 
.92343 
.92332 
.92321 
.92310 
.92299 
.92287 
.92276 
.92265 
92254 
92243 
.92231 
792220 
,92209 
1.92198 
-92155 
.92175 

.92164 
.92152 
.92141 
.92130 
,92119. 
-92107 
.92096 
.92055 
1.92073 
.92062 
92050 


40911 
-40945 
-40979 
.41013 
,41047 
.41081 
.41115 ' 
.41149 
.41153 
.41217 
"41251 
.41385 

41319 
.41353 
.41387 

.41421 
.41455 
-41490 

.41524 
.41558 


-42963 
42998 

.43032 
43067 

.43101 


2. 32758 
2 32570 
2- 32383 
2.32197 
2 32012 


30 

21 
22 
23 
24 


2.43422 
2.43220 
2 43019 
2. 4281 9 
2.42615 


39608 
39635 
39661 

.39683 
39715 
39741 

.39765 
39795 
39822 
39848 


-91822 
91810 
91799 

.9178? 

-91775 


.43136 
43170 
43205 
43239 

.43274 


2. 31326 
3. 31641 
2-31455 
2 31271 
2. 31086 


25 
26 
27 
29 
29 


2-42418 
2-42218 
2 42019 
2 41319 
2.ili20_ 

2 41421 
2. 41 223 
2. 41 025 
2. 40827 
2-40629 


.91764 
91752 
.91741 
-91729 
.91718 
'91706 
91694 
91683 
91671 
1.91660 


-43308 
.43343 
-43373 
.43412 
-43447 

43481 
.43616 

43550 
.43555 

43620 

43654 
-43689 
-43724 
-43758 
-43793 


2.30902 
2 30718 
2 30534 
2 30351 
2. 30167 


30 

31 
32 
33 
34 


39875 
39902 
.39928 

-39955 
.39982 


2- 29984 
2. 29801 
2-29819 
2-29437 
2. 29254 


35 

36 
37 
38 

39 


.38403 
.35430 
.35456 
.35433 
,35510 

.35537 
.38564 
.38591 
36517 
.38544 


.41592 
.41626 
41660 
.41594 
_4172S 
.41763 
.41797 
.41831 
.41865 
.41899 

.41933 
.41963 
.42002 
.42036 
.42070 
-42105 
.42139 
.42173 
.42207 
.42242 
,42276 
.42310 
-42345 
42379 
.42413 

42447 


2 40432 

2-40235 
2 40038 
2 39841 

2.39645 

2.39449 
2-39253 
2-39058 
2 38863 
2 38668 
2.38473 
2 38279 
2-33084 
2. 37891 
2 37697 
2-37504 
2-37311 
2 37118 
2. 36925 
2. 36733 
2 36541 
2 36349 
2 36158 
2 35967 
2 35776 


40008 

40035 
-40062 
40083 
40115 
40141 
-40168 
-40195 
-40221 
. 40248 


1-9] 645 
.91636 
-91625 
.91613 
-91601 


1 2. 29073 
2. 25891 
2. 28710 
2 28528 
2 28348 


40 

41 
42 
43 
44 


; -91590 
91578 
91566 
91555 

,-91543 
91531 
91519 
91508 
-91496 
91484 
. 91472 
91461 
. 91449 

,.91437 
91425 


43828 
.43852 

43897 
(43932 

43966 
^44001 
-44036 
-44071 
.44105 
-44140 
-44175 
.44210 
-44244 
1.44279 
.443M 


, 2 
1 2 

'i 

2 
2 

2 
2 
2 
2 


28167 
27987 
27806 
27626 
27447 


43 

43 

47 
48 
49 


.38671 
.35695 
,33725 
.35752 
-33778 
.35805 
.35832 
.38859 
.388B6 
.33912 


.40275 
40301 
40328 

-40355 
40351 
4040 S 

- 40434 

-40461 

.40438 
40514 

.40541 
40567 
40594 

-40621 
40347 


27257 
27038 
26909 
26730 
26552 


50 

61 
62 

aa 

64 


2 
2 
2 
2 
2 


26374 
26196 
26018 
25340 
25663 


65 
56 
67 
50 
6^ 


.33939 
.38966 
.38993 
.39020 
.39046 


.91414 1-44349 
91402 .44354 

.91390 -44418 
91375 .44453 

-91356 44488 

.91355 .44523 


2 
2 
2 
2 

: 2 


25466 
25309 
25132 
34955 
24780 


60 


2.35585 


40674 


2-24604_ 


T- 


Cot. 


TaD. 


Cos. 


Sin. J Cot, 


J_ 


Tnu 



er 



Digitized by VjOOQIC 



252 NATURAL SINES, COSINES, TANGENTS, COTAN. 







24» 






a6» 






• 


Sin. 


Ck)s. 


Tan. 


C3ot. 


Sin. 


Cos. 


Tan. 


Cot. 


9 


"o" 


.40674 


.91355 


.44528 


2.24604 


.42262 


.90631 


.46631 


2.14451 


"60 


1 


.40700 


.91343 


.44558 


2.24428 


.42288 


.90618 


.46666 


2.14288 


69 


2 


.40727 


.91331 


.44593 


2.24252 


.42315 


.90606 


.46702 


2.14125 


58 


3 


.40753 


.91319 


.44627 


2.24077 


.42341 


.90594 


.46737 


2.13968 


57 


4 


.40780 


.9x307 


.44662 


2.23902 


.42867 


.90082 


.46772 


2.13801 


66 


5 


.40806 


.91295 


.44697 


2.23727 


.42394 


.90569 


.46808 


2.18689 


55 


6 


.40838 


.91283 


.44732 


2.23553 


.42420 


.90557 


.46843 


2.18477 


54 


7 


.40860 


.91272 


.44767 


2.23378 


.42446 


.90545 


.46879 


2.13316 


58 


8 


.40886 


.91260 


.44802 


2.28204 


.42473 


.90532 


.46914 


2.13154 


52 





.40913 


.91248 


.44837 


2. 23030 


■42499 


.90520 


■46950 


2.12993 


51 


10 


.40939 


.91236 


.44872 


2.22857 


■42525 


.90507 


.46985 


2.12833 


60 


11 


.40966 


.91224 


.44907 


2.22683 


■42552 


.90495 


.47021 


2.12671 


49 


12 


.40992 


.91212 


■44942 


2.22510 


.42578 


.90488 


.47056 


2.12511 


48 


18 


.42019 


.91200 


44977 


2.22337 


.42604 


■90470 


.47092 


2.12850 


47 


14 


:41045 
.41072 


.91188 
.91176 


.45012 
.45047 


2.22164 
2.21992 


.42631 
.42657 


■90458 
■90446 


.47128 


2.12190 


_46 


16 


.47168 


2.12030 


45 


16 


.42098 


.91164 


.45082 


2.21819 


.42683 


.90433 


.47199 


2.11871 


44 


17 


.41125 


.91152 


.45117 


2.21647 


.42709 


90421 


.47234 


2.11711 


48 


18 


.41151 


.91140 


.45152 


2.21476 


.42736 


.90408 


.47270 


2.11552 


42 


iS. 


^iUZL 


.91128 


.45187 


2.21304^,. 


;i2Zfi2. 


J039i. 


.47305 


vwm 


^ 


90 


.41204 


.91116 


.45222 


2.21132 


.42788 


.90383 


.47341 


2.11233 


40 


21 


.41231 


.91104 


.45257 


2.20961 


.42815 


.90871 


.47377 


2.11076 


89 


22 


.41257 


.91092 


.46292 


2.20790 


.42841 


.90358 


.47412 


2.10916 


88 


28 


.41284 


.91080 


.45327 


2.20619 


.42867 


.90846 


.47448 


2.10768 


87 


ai« 


.41310 
.41337 


91068 
.91066 


.45362 
.46397 


2.20449 


■42894 


■90334 


■47483 


2.10600 


86 


25 


2.20278 


.42920 


■90321 


■47519 


2.10442 


85 


26 


.41363 


.91044 


45432 


2.20108 


.42946 


■90309 


.47555 


2.10284 


84 


27 


.41390 


.91032 


.45467 


2.19938 


.42972 


.90296 


.47590 


2.10126 


88 


28 


41416 


.91020 


.45502 


2.19769 


.42999 


.90284 


.47626 


2.09969 


82 


29 


.41443 


.91008 


.45538 


2.19599 


.43026 


.90271 


.47682 


2.09811 


81 


80 


.41469 


.90996 


^45573 


2.19430 


.43051 


.90259 


.47698 


2.09854 


30 


81 


.41496 


.90984 


.45608 


2.19261 


43077 


.90246 


. •7733 


2.09498 


39 


82 


.41622 


.90972 


.45643 


2.19092 


.43104 


.90238 


.47769 


2.09341 


2J 


88 


.41549 


.90960 


.45678 


2. 18928 


.43130 


.90221 


.47805 


2.09184 


37 


84 


.41575 
.41602 


.90948 
.90936 


.45713 


2.18755 


■43156 


.90208 


■47840 


2. 09028 


-28 


85 


.45748 


2. 18587 


.43182 


.90196 


47876 


2.08872 


25 


86 


.41628 


.90924 


.45784 


2.18419 


.43209 


.90183 


.47912 


2.08716 


24 


87 


.41656 


.90911 


.45819 


2.18251 


.43235 


.90171 


.47948 


2.08560 


28 


88 


.41681 


90899 


.45854 


2. 18084 


.43261 


.90158 


.47984 


2.08406 


23 


89. 


:4122I. 


.90887 


.45889 


2.1791§ 


.43287 


90146 


.48019 


2.P8250 


-21 


40 


41734 


..90875 


45924 


2-17749 


.43313 


.90133 


.48055 


2.08094 


90 


41 


41760 


.90863 


.45960 


2. 17682 


43340 


.90120 


.48091 


2.07939 


19 


42 


41787 


.90861 


.45995 


2.17416 


43366 


.90108 


.48127 


2.07785 


18 


48 


.41813 


.90839 


.46030 


2. 17249 


43392 


.90095 


.48163 


2.07630 


17 


44. 


.41840 


. 90826 


.46065 


2. 17083 


.43418 


.90082 


48198 


2.07476 


16 


45 


.41866 


'.90814 


.46101 


2. 16917 


43445 


.90070 


.48234 


2.07821 


15 


46 


.41892 


.90802 


.46136 


2. 16751 


.43471 


90057 


.48270 


2.07167 


14 


47 


.41919 


.90790 


.46171 


2.16585 


. 43497 


.90045 


.48306 


2. 07014 


18 


48 


41945 


.90778 


.46206 


2. 16420 


.43523 


.t)0032 


.48342 


2. 06860 


13 


4S« 


.41972 


.90766 
.90753 


,46242. 
.46277 


2. 16255 
2.16090 


.43549 
.43575 


.90019 
.90007 


■48378 
.48414 


J ■06706 
2.06558 


^1 


60 


.41998 


10 


51 


42024 


.90741 


.46312 


2.15925 


43602 


.89994 


.48450 


2. 06400 


9 


52 


•42051 


.90729 


.46348 


2. 16760 


.48628 


.89981 


.48486 


2.06247 


8 


53 


.42077 


.90717 


.46383 


2.15596 


48654 


.89968 


.48521 


2.06094 


7 


64. 


.42104 
.42130 


90704 
.90692 


.46418 
.46454 


_2J.5432_ 


48680 


89956 


48557 


2.06942 


-_e 


55 


2.15268 


.43706 


.89943 


.48593 


2.06790 


6 


56 


.42156 


.90680 


.46489 


2.15104 


.48788 


.89930 


.48629 


2. 06687 


4 


57 


.42183 


.90668 


.46525 


2.14940 


.43759 


.89918 


.48665 


3.05485 


8 


68 


.42209 


.90655 


.46560 


2. 14777 


.43785 


.89905 


.48701 


2.06888 


3 


60 


.42235 


.90643 


.46595_ 

.46631 

Cot. 


_2. 14814 

2.144?1 

Tan. 


.43811 


■89892 


.48737 

JkSZZl, 

Cot. 


2.06183 


1 


a^ 


.42262 


^^ 


-7^ 



65» 



64" 

Digitized by VjOOQIC 



NATURAL SINES, COSINES, TANGENTS, COTAN. 253 



9 


Sin. 


Cob. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


9 





43837 


89879 


.48773 


2.05080 


.45899 


.89101 


.50953 


1.96261 


"So 


1 


.43863 


.89867 


.48809 


2.04879 


.46425 


.89087 


.50989 


1.96120 


59 


2 


.43889 


.89854 


48845 


2. 04723 


.45451 


.89074 


.51026 


1.95979 


68 


3 


.43916 


.89841 


.48881 


2.04577 


.45477 


.89061 


.51068 


1.95838 


57 


4 


.43942 


.89828 


.48917 
.48953 


2. 04426 
2.04276 


.45503 


.89048 


.51099 


1.95698 


56 


6 


.43968 


.89816 


.45529 


.89035 


51136 


1.95557 


55 


6 


.43994 


.89803 


.48989 


2.04125 


.45554 


.89021 


.51173 


1.95417 


54 


7 


.44020 


.89790 


.49026 


2.03975 


.45580 


.89008 


.51209 


1.95277 


53 


8 


.44046 


.89777 


.49062 


2 03825 


.45606 


.88995 


.51246 


1.95187 


52 


9 


.44072 


.89764 


.49098 


2.08675 


.45632 


.88981 


.51283 
.51319 


1.94997 


51 


lO 


.44098 


.89752 


.49134 


2.03526 


.45658 


.88968 


1.94858 


50 


11 


.44124 


.89789 


.49170 


2.03376 


.45684 


.88955 


.51356 


1.94718 


49 


12 


.44151 


.89726 


.49206 


2.03227 


.45710 


.88942 


.51393 


1.94579 


48 


IS 


.44177 


.89713 


.49242 


2.03078 


.45736 


.88928 


.51430 


1.94440 


47 


14 


.44208 


.89700 


.49278 
.49815 


2.02929 


.45762_ 


.88915 
.88902 


.51467 


1.94301 


46 


15 


.44229 


.89687 


2.02780 


.45787 


.5; 503 


1.94162 


45 


16 


.44255 


.89674 


.49851 


2.02631 


.45813 


.88888 


.51540 


1.94023 


44 


17 


.44281 


.89662 


.49387 


2.02483 


.45839 


.88875 


.51577 


1.93885 


43 


18 


.44307 


.89649 


.49423 


2.02335 


.45865 


.88862 


.51614 


1.93746 


42 


li- 


.44333 


.89636 


.49459 


2. 02187 


•45891 


.88848 


.51651 


1.93608 


_ii 


20 


.44359 


.89623 


.49495 


2.02039 


.45917 


88835 


.51688 


1.93470 


40 


21 


.44385 


.89610 


.49532 


2.01891 


.45942 


.88822 


.51724 


1.93332 


89 


22 


.44411 


.89597 


.49568 


2.01743 


.45968 


88808 


.51761 


1.93195 


38 


28 


.44437 


.89584 


.49604 


2.01596 


■45994 


.88795 


.51798 


1.93057 


37 


24 


.44464 


.89571 
.89558 


.49640_ 
.49677 


2.01449_ 
2.01302 


.46020 


.88782 
.88768 


51835 
.51872 


1.9."920 
1.92782 


86 


25 


.44490 


.46046 


35 


26 


.44516 


.89545 


.49713 


2.01155 


.46072 


.88755 


.51909 


1.92645 


34 


27 


.44542 


.89532 


.49749 


2.01008 


.46097 


.88741 


.51946 


1.92508 


33 


28 


.44568 


.89519 


.49786 


2.00862 


.46128 


.88728 


.51983 


1.92371 


32 


29 


.44694 


.89506 
.89493 


.49822 
.49858 


2.00715 


46149 


.88715 


.52020 


1.92235 


-31 


30 


.44620 


2.00569 


.46175 


.88701 


.52057 


1.92098 


30 


SI 


.44646 


.89480 


.49894 


2. 00423 


.46201 


.88688 


. 52094 


1.91962 


29 


82 


.44672 


.89467 


.49931 


2.00277 


.46226 


.88674 


.52131 


1.91826 


28 


88 


44698 


.89454 


.49967 


2.00131 


.46252 


.88661 


.52168 


1.91690 


27 


84 


.44724 
.44750 


.89441 
.89428 


.50004 
.50040 


1.99986 


.46278 


.88647 


.52205_ 


1.91554 


26 


85 


1.99841 


.46304 


.88634 


.52242 


1.91418 


25 


86 


.44776 


.89415 


.50076 


1.99695 


.46330 


.88620 


.52279 


1.91282 


24 


87 


.44802 


.89402 


.50113 


1.99550 


.46355 


.88607 


.52316 


1.91147 


23 


38 


44828 


.89389 


.50149 


1.99406 


.46381 


.88593 


.52353 


1.91012 


22 


2L. 


.44854. 
.44880 


.89376 
.89863 


.50185 


1.99261 


.46407. 


.88580 


.52390. 


1.90876 


-21 


40 


.50222 


1.99116 


.46433 


88566 


.52427 


1.90741 


20 


41 


.44906 


.89350 


.50258 


1.98972 


46458 


.88553 


.52464 


1.90607 


19 


42 


44932 


.89337 


.50295 


1.98828 


.46484 


. 88539 


.52501 


1.90472 


18 


43 


44958 


.89324 


.50331 


1.98684 


.46510 


88526 


.52538 


1.90637 


17 


44 


.44984 


.89811 
.89298 


.50368 


1.98540 


.46536 


J8512_ 
. 88499 


.52575_ 
.52613 


1. 90203 


16 


45 


.45010 


.50404 


1.98396 


46561 


1.90069 


16 


46 


.45036 


.89285 


. 50441 


1.98253 


.46587 


.88485 


.52650 


1.89935 


14 


47 


.45062 


.89272 


.50477 


1.98110 


.46613 


.88472 


52687 


1.89801 


13 


48 


.45088 


.89259 


.50514 


1.97966 


.46639 


.88458 


. 52724 


1.89667 


12 


49 


.45114 


.89245 


.50550 


1.97823 


.46664 
.46690 


.88445 


.52761 


1.89533 


11 


50 


.45140 


.89282 


.50587 


1.97681 


.88431 


.52798 


1.89400 


~10 


51 


.45166 


.89219 


.50623 


1.97538 


.46716- 


88417 


52836 


1.89266 


9 


52 


46192 


.89206 


50660 


1.97395 


.46742 


88404 


.52873 


1.89133 


8 


53 


45218 


.89193 


.50696 


1.97253 


.46767 


88390 


.52910 


1 . 89000 


7 


54 


45243 


.89180 
.89167 


.50733 
. 50769 


1^7111. 
1.96969 


.46793 
.46819 


.88377 


.52947_ 
.52985 


1.88867 


6 


55 


.45269 


88363 


1.88734 


5 


56 


.45295 


.89153 


.50806 


1.96827 


.46844 


88349 


.53022 


1. 88602 


4 


57 


.45321 


.89140 


.50843 


1.96685 


48870 


.88336 


.53059 


1.88489 


8 


58 


.45347 


89127 


50879 


1.96544 


46896 


.88322 


.53096 


1.88337 


2 


59 


.45373 


.89114 


50916 
.50953 


1.96402 
1.96261. 


.46921 
,4S94L 


.88308 


.53134 


1.88205 


1 


§2- 


•46399 


.89101 


.88295 


J31ZL 


1.88073 


-0 


•^ 


Coe. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 






e, 


J** 




ea** 








^ 










Digitize 


JbyGO( 


3gle 





254 NATURAL SINES, COSINES, TANGENTS, COTAN. 

28** 29' 



» 


Sin. 


Cob. 


Tan. 


Cot. 


Sin. 


CoB« 


Tan. 


Cot. 


f 




I 
I 


.46947 
.46978 
.46999 
.47024 
.47050 


.88295 
.88281 
.88267 
.88254 
.88240 


.53171 
.53208 

53246 
.53283 
.53320 
.53358 
.53895 
.63482 

53470 
.53607 


1. 88078 
1.87941 
1.87809 
1.87677 
1.87646 


48481 
48506 

.48682 
48557 

.48688 


87462 
.87448 
.87434 

87420 
.87406 


66481 
.56469 
.56607 
.56646 
.65583 


1 80405 
1.80281 
1.80158 
1.80034 
1.79911 


60 

69 
58 
57 
56 


5 

? 

8 
9 


.47076 
.47101 
.47127 
.47163 
.47178 


88226 
88213 
.88199 
.88185 
88172 


1.87416 
1.87283 
1.87152 
1.87021 
1.86891 


.48608 
.48684 
.48669 
48684 
■48710 


.87891 
.87377 
.87368 
.87849 
.87836 


.55621 
.65659 
.55697 
.55786 
.55774 


1 79788 
1 70665 
1.79642 
1.79419 
1.79296 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.47204 
.47229 
.47265 
.47281 
.47306 


.88158 
.88144 
.88130 
.88117 
.88103 


.68646 
.63582 
.53620 
.53667 
.53694 


1.86760 
1.86630 
1.86499 
1.86369 
1.86239 


.48785 
48761 
.48786 
.48811 
48887 


.87821 
.87806 
.87292 
.87278 
.87264 


.55812 
.55850 
.55888 
.56926 
.55964 


1.79174 
17905] 
1.78929 
1.78807 
1. 78685 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.47882 
•47858 
.47888 
.47409 
.47434 
.47460 
.47486 
.47511 
.47537 
.47562 


.88089 
.88075 
.88062 
.88048 
.88084 
.88020 
88006 
.87993 
.87979 
.87966 


.68782 
.53769 
.53807 
.68844 
.6.8882 
.53920 
.53957 
.68995 
.54032 
.54070 


1.86109 
1.85979 
1.86850 
1.85720 
1.85591 


.48862 
.48888 
.48918 
.48938 
.48964 


.87260 
.87286 
.87221 
.87207 
.87198 


.56008 
.56041 
.56079 
.56117 
.66166 
.56194 
.56232 
.66270 
.66309 
.66347 


1. 78563 
1.78441 
1. 78319 
1.78198 
1. 78077 
1. 77955 
1. 77884 
1. 77713 
1.77592 
1.77471 


45 
44 

43 
42 


20 

21 
22 
28 
24 


1. 85462 
1.85383 
1.8G204 
1.85075 
1.84946 


.48989 
.49014 
.49040 
.49065 
.49090 


.87178 
.87164 
.87150 
.87136 
.87121 


40 

89 
38 
87 
86 


26 
26 
27 
28 
29 


.47588 
.47614 
.47689 
.47665 
.47690 


.87951 
.87987 
.87923 
.87909 
.87896 


.54107 
.64146 
. 64183 
.54220 
.54268 


1.84818 
1.84689 
1.84661 
1.84483 
1.84306 


.49116 
.49141 
.49166 
.49192 
.49217 


.87107 
87098 
.87079 
.87064 
.87050 


.56386 
.56424 
.56462 
.56501 
■56539 


1.77851 
1. 77280 
1.77110 
1.76990 
1. 76869 


85 
84 
88 
82 

31 


30 

81 
82 

11 


.47716 
.47741 
.47767 
.47793 
.47818 


.87882 
.87868 
.87854 
.87840 
.87826 


.54296 
.54333 
•64371 
.54409 
54446 


1.84177 
1.84049 
1.83922 
1.83794 
1.83667 


.49242 
.49268 
.49298 
.49318 
.49844 


.87086 
.87021 
.87007 
.86993 
.86978 


.56577 
.66616 
.56654 
.56693 
.66731 


1.76749 
1.76629 
1.76510 
1.76390 
1 ■ 76271 


30 

29 
28 
27 
26 


85 
86 
87 
88 
89 


.47844 
.47869 
.47895 
.47920 
.47946 


.87812 
.87798 
.87784 
.87770 
.87756 


.54484 
.54522 
.54560 
. 54597 
.54635 


1.83540 
1.83413 
1.83286 
1.83159 
1.83033 


.49369 
.49394 
.49419 
.49445 
.49470 


.86964 
.86949 
.86985 
.86921 


.56769 
.56808 
.56846 
.56885 
.56928 


1.76151 
1.76032 
1.75918 
1.76794 
1.76676 


25 
24 
28 
22 
2^ 


40 

41 
42 
48 
44 


.47971 
.47997 
.48022 
.48048 
.48073 
.48099 
.48124 
.48150 
.48175 
.48201 


87743 
.87729 
.87715 
.87701 
.87687 


54673 
.54711 
.64748 

54786 
.54824 


1.82906 
1.82780 
1.82654 
1.82528 
1.82402 


.49495 
.49621 
.49546 
.49671 
49596 


.86892 
.86878 
.86863 
.86849 
.86834 


.56962 
. 57000 
.57039 
.57078 
.57116 


1.76556 
1. 75487 
1.76819 
1.76200 
1.75082 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.87673 
.87659 
.87645 
.87681 
.87617 


.54862 
.54900 
.64938 
.64975 
.55018 


1.82276 
1.82150 
1.82025 
1.81899 
1.81774 


.49622 
.49647 
.49672 
.49697 
.49723 


.86820 
.86805 
.86791 
.86777 
.86762 


.57155 
.57193 
.57232 
.67271 
. 57309 


1.74964 
1.74846 
1.74728 
1.74610 
1.74492 


15 
14 
18 
12 
11 


50 

61 
52 
58 
54 


.48226 
.48252 
.48277 
.48308 
.48328 


.87603 
.87689 
.87575 
.87661 
. 87546 


.56051 
.65089 
.55127 
.65165 
65203 


1.81649 
1.81524 
1.81399 
1.81274 
1.81160 


.49748 
.49773 
.4979a 
.49824 
.49849 


.86748 
.86733 
86719 
.86704 
.86690 


.57348 
.57386 
•57425 
.57464 
.57603 


1.74876 
1..4267 
1.74140 
1.74022 
1.78905 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.48354 
.48379 
.48405 
.48430 
.48466 


.87532 
87618 
.87504 
.87490 
.87478 


.65241 
.56279 
.56317 
.55865 
.66898 


1.81026 
1.80901 
1.80777 
1.80653 
1.80629 


.49874 
49899 
49924 

.49950 
40976 


.86675 
.86661 
.86646 
.86632 
.86617 


•57541 
.57580 
.57619 
.57657 
57696 


1.78788 
1.78671 
1.73555 
1.73438 
1.73321 


6 

4 
8 
2 

1 


fiSL. 


Sin. 


«■ 


V 80496 


50000 
Cos. 


.86608 


■57735 


1.73205 





/ 


Sin. 


Cot. 


Tan. 


"^ 



61'» 



Digitized by VjOOQIC 



NATURAL SINES, COSINES, TANGENTS, COTAN. 255 







30* 






31'* 






» 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


» 


"o" 


.50000 


.86603 


.57735 


1.73205 


.51504 


.85717 


.60086 


1. 66428 


"55 


1 


.50025 


.86588 


.57774 


1.73089 


.51529 


.85702 


.60126 


1.66318 


59 


2 


.50050 


.86578 


.57813 


1.72978 


.51554 


.85687 


.60165 


1.66209 


58 


8 


.50076 


.86559 


.57851 


1.72857 


.51579 


.85672 


.60205 


1.66099 


57 


4 


.50101 


.86544 


.57890 


1.72741 


.51604 


.85657 


.60245 


1.65990 


56 


6 


.50126 


.86530 


.57929 


1.73625 


.51628 


.85642 


.60284 


1. 65881 


65 


6 


.50151 


.86515 


.57968 


1.72509 


.51658 


.85627 


60324 


1.65772 


64 


7 


50176 


.86501 


.58007 


1.72393 


51678 


.85612 


.60364 


1. 65663 


58 


8 


.60201 


.86486 


.58046 


1.72278 


.51708 


.85597 


.60403 


1-65554 


53 


» 


.60227 
50253 


.86471 


.58085 


1.72168 


.51728 


.85582 


.60443 


1.65445 


51 


10 


.86457 


•58124 


1.73047 


.51758 


•85567 


.60483 


1. 65337 


60 


11 


50277 


.86442 


.58162 


1.71933 


.51778 


.85551 


.60522 


1.65228 


49 


12 


50802 


.86427 


.58201 


1.71817 


.51808 


.85536 


.60562 


1. 65120 


48 


18 


.50327 


.86418 


.58240 


1.71703 


.51828 


85521 


.60602 


1.65011 


47 


li_ 


.50352 
50877 


.86898 


.58279 


1.71588 


.51853 


.85506 


•60642 


1.64908 


-48 


15 


.86884 


58818 


1.71473 


.51877 


85491 


.60681 


1.64795 


45 


16 


.50403 


.86369 


.58357 


1.71358 


.51902 


.85476 


.60721 


1.64687 


44 


17 


50428 


.86354 


.58896 


1.71244 


.51927 


.85461 


.60761 


1.64579 


48 


18 


50458 


.86340 


.58435 


1.71129 


•51953 


.85446 


. 60801 


1.64471 


43 


li. 


.50478., 
. 50503 


.86825 
.86310 


.58474 
.58513 


1.71015 


.51977 


.85431 


.60841 


1.64363 


-ii 


SO 


1.70901 


52002 


.85416 


.60881 


1.64256 


40 


21 


.50528 


.86295 


.58552 


1.70787 


.52026 


.85401 


.60921 


1.64148 


89 


22 


50558 


.86281 


.58591 


1.70673 


•52051 


.85385 


.60960 


1.64041 


88 


28 


.50578 


.86266 


.58631 


1.70560 


•52076 


.85370 


.61000 


1.63934 


87 


24^ 


.50608 


.86251 


.58670 


1.70446 


.52101 


.85355 


•61040 


1.63826 


86 


25 


50628 


.86237 


58709 


1.70382 


.52126 


.85340 


.61080 


1-63719 


35 


26 


60654 


.86222 


.58748 


1.70219 


52151 


.85325 


•61120 


1.63612 


34 


27 


50679 


.86207 


.58787 


1.70106 


•52175 


.85310 


.61160 


1.63505 


38 


S! 


50704 


.86192 


.58826 


1.69992 


.52200 


•85294 


•61200 


1.63398 


83 


«L_ 


.50729 


.86178 


.58865 
.58905 


1.69879 
1. 69766 


. 52225 
.52250 


•85279 


.61240 


1.63292 


31 


30 


50754 


.86163 


•85264 


.61280 


1.63185 


30 


81 


50779 


.86148 


.58944 


1.69658 


.52275 


• 85249 


.61320 


1.68079 


39 


82 


•50804 


.86133 


.58983 


1.69541 


52299 


•85234 


.61360 


1.62972 


38 


88 


50829 


.86119 


.59023 


1.69428 


.52324 


.85218 


.61400 


1.62866 


27 


84 


50854 
50879 


.86104 


.59061 


1.69316 


^2349 


.85208 


.61440 


1.62760 


26 


!! 


.86089 


.59101 


1.69203 


•52374 


.85188 


.61480 


1.62654 


35 


86 


.50904 


.86074 


.59140 


1.69091 


■52399 


85173 


•61520 


1.62548 


34 


87 


50929 


.86059 


.59179 


1.68979 


• 52428 


.85157 


.61561 


1.62442 


28 


88 


.50954 


.86045 


59218 


1.68866 


.52448 


.85142 


.61601 


1.62336 


23 


<SL 


•6P?7? 


:Mm^ 


.59258 


1.68754 


.52473 


.85127 


.61641 


1.62230 


-21 


40 


.61004 


.88015 


.59297 


1.68643 


•52498 


.85112 


.61681 


1.62125 


»0 


41 


51029 


86000 


.59836 


1.68531 


•52522 


.85096 


.61721 


1.62019 


19 


42 


.51054 


85985 


.59876 


1.68419 


.52547 


85081 


.61761 


1.61914 


18 


48 


51079 


.85970 


.59415 


1.68308 


.52572 


.85066 


.61801 


1.61808 


17 


44_ 


.51104 


.85956 


.59454 


1.68196 


.52597 


.85051 


.61842 


1.61703 


_ie 


46 


.51129 


.85941 


.59494 


1.68085 


.52621 


85085 


.61882 


1.61598 


15 


46 


.51154 


.85926 


.59533 


1.67974 


.52646 


. 85020 


•61922 


1.61498 


14 


47 


61179 


.85911 


.59573 


1.67863 


.52671 


. 85005 


■61962 


1.61388 


13 


48 


.51204 


.85896 


59612 


1.67752 


.52696 


84989 


. 62003 


1.61283 


13 


&_ 


.51229 


.85881 


69651 


1. 67641 


.52720 


•84974 


.62043 


1.61179 


11 


W 


.51254 


.85866 


59691 


1.67580 


. 52745 


• 84959 


.62083 


1.61074 


10 


51 


.61279 


.85851 


59730 


1.67419 


.52770 


•84943 


.62124 


1.60970 


9 


62 


.61804 


.85836 


.59770 


1.67809 


. 52794 


• 84928 


.62164 


1-60865 


8 


68 


.51829 


.85821 


.59809 


1-67198 


.52319 


•84918 


.62204 


1.60761 


7 


94 


.51854 


.85806 


.59849 


1.67088 


.52844 


.84897 


.62245 
.62285 


1.60657 
1.60553 


6 


65 


.51879 


.85792 


.59888 


1.66978 


.52869 


84882 


6 


66 


61404 


.85777 


.59928 


1.66867 


. 52898 


.84866 


.62325 


1.60449 


4 


67 


.61429 


.85762 


59967 


1.66757 


.52918 


.84851 


.62366 


1.60345 


8 


68 


.51454 


.85747 


60007 


1.66647 


52943 


.84836 


.63406 


1. 60241 


3 


fiS- 


:61479 


85732 


.60046 


_1.66538_ 


.52967 


.84820 


. 62446 


1.60187 
1. 60038 


1 


«^ 


«■ 


Vr- 


. 60086 
Cot. 


1.66428 


.52992 


.84805 


.62487 





Sin. 


Cot. 


Tan. 


' 



59<» 



58* 

Digitized by VjOOQIC 



256 NATURAL SINES, COSINES, TANGENTS, COTAN. 







32* 






83« 






# 


Sin, 


Cos 


Tan. 


Cot 


Sin. 


Cos. 


Tan. 


Cot. 


« 




I 
I 


.52992 
.58017 
.58041 
.58066 
.68091 


.84805 
.84780 
.84774 
.84759 
.84743 


.62487 
•62527 
.62568 
.62608 
.62649 


1.60083 
1.59980 
1.59826 
1.59728 
1.59620 


.54464 
.54488 
.54518 
.54537 
.54561 


.88867 
.88851 
.83835 
.83819 
.83804 


.64941 
.64982 
.65024 
.65066 
.65106 


1.53986 
1. 68888 
1.58791 
1. 58698 
!• 53595 


00 

57 
58 


8 
9 


.58116 
.58140 
.58164 
.58189 
.68214 


.84728 
.84712 
.84697 
.84681 
.84666 


.62689 
.62730 
.62770 
.62811 
.62862 


1.59517 
1.59414 
1.69811 
1.59208 
1.59105 


.5458S 
.64810 
.54635 
■ 54659 
54683 


.83788 
.88772 
.83756 
•83740 
.88724 


.65148 
.65189 
.65231 
.65272 
.65314 


1.53497 
1. 53400 
1.58302 
1.53206 
1.63107 


58 
54 


10 

11 


.63288 
.68268 
.53288 
.63312 
.53337 


.84660 
.84635 
.84619 
.84604 
.84588 


.62892 
.62983 
.62978 
.68014 
.63055 


1.69002 
1.68900 
1.68797 
1.58695 
1.58593 


54708 
.54732 
.54758 
.54781 
.54805 


.83708 
.83692 
.83676 
.83660 
.83645 


.65355 
.65897 
.65488 
.65480 
•65521 


1.53010 
1.62913 
1.52816 
1.52719 
1.52622 


46 


15 

i? 

18 


.53361 
.58386 
.68411 
.53435 
.53460 


.84578 
.84557 
.84542 
.84526 

.84611 


.68095 
.63136 
.63177 
.63217 
.63258 


1.58490 
1.58388 
1.68286 
1.58184 
1.58088 


.54829 
.54854 
.54878 
. 54902 
.54927 


.83629 
.83613 
.88597 
.83581 
.83565 


.65568 
.65604 
.65646 
.65688 
.65729 


1.52525 
1.52429 
1.52882 
1.52285 
1.52189 


45 

iS 


21 
23 
28 
24 


.68484 
.53509 
.53534 
.58558 
.53583 


.84495 
.84480 
.84464 
.84448 
.84438 


.63299 
.63840 
.63880 
.63421 
. 63462 


1.67981 
1.67879 
1.67778 
1.67678 
1.57575 


54951 
.54975 
.54999 
.55024 
.55048 


.83549 
.83538 
.83517 
.83501 
.83485 


.65771 
.65813 
.65854 
.65896 
.65938 


1.62043 
1.61946 
1.61860 

l!61668 


"15 


26 
26 
27 
28 
29 


.53607 
.68682 
.68656 
.63681 
.58705 


.84417 
.84402 
.84386 
.84370 
.84355 


.63508 
.63544 
.63584 
.63625 
.63666 


1.57474 
1.57372 
1.57271 
1.57170 
1.57069 


.55072 
.55097 
.55121 
.55145 
.55169 


.83460 
.83453 
.83437 
.83421 
. 83405 


.65980 
.66021 
.66063 
.66106 
.66147 


1.51662 
1.61466 
1.51370 
1.51276 
1.51179 


88 
84 
88 
82 
81 


80 

81 
82 
88 
84 


.53730 
.53754 
.53779 
.58804 
.53828 


. 84339 
.84324 
.84308 
.84292 
.84277 


.63707 
63748 
.63789 
.63830 
.63871 


1.56969 
1.66868 
1.56767 
1.56667 
1.66566 


.55194 
.55218 
.55242 
.55266 
.55291 


.83389 
.83373 
.83356 
.83340 
.83324 


.66180 
.66230 
.66272 
.66314 
.66356 


1.61084 
1.50988 
1.60898 
1.60797 
1.50702 


30 

29 

if 

26 


85 

IL 


.53858 
.63877 
.68902 
.58926 
.53951 


.84261 
.84245 
.84230 
.84214 
.84198 


.63912 
.63958 
.63994 
.64035 
.64076 


1.56466 
1.56366 
1.56265 
1.66165 
1.56065 


.55315 
.65339 
. 55368 
.55388 
.55412 


.83308 
.83292 
.83276 
.83260 
.83244 


.66898 
.66440 
.66482 
.66624 
.66566 


!• 50607 
1.50512 
1.50417 
1.50322 
1.50228 

1.50183 
1.50088 
1.49944 
1.49849 
1.49755 


25 
24 
28 
22 
-21 


40 
41 
42 
48 
44 


.53975 
.54000 
.54024 
.54049 
.54078 


.84182 
.84167 
.84151 
.84135 
.84120 


.64117 
.64158 
.64199 
.64240 
.64281 


1.55966 
1.65866 
1.55766 
1.65666 
1.55567 


.55436 
.55460 
.55484 
.55509 
.55533 


.83228 
.88212 
.83195 
.88179 
.83163 


.66608 
.66650 
.66692 
.66734 
.66776 


20 

19 
18 
17 
16 


45 


.54097 
.54122 
.54146 
.54171 
.64195 


.84104 
.84088 
.84072 
.84057 
.84041 


.64322 
.64868 
.64404 
.64446 
.64487 


1.55467 
1.55368 
1.55269 
1.55170 
1.55071 


.55557 
.55581 
.55605 
.55630 
•55654 
.55678 
•55702 
.55726 
.55750 
.55775 
.55799 
.55828 
.55847 
.55871 
.55895 


.83147 
.83131 
.83115 
.83098 
.83082 


.66818 
.66860 
.66902 
.66944 
.66986 


1.49661 
1.49566 
1.49472 
1.49378 
1.49284 
1.49190 
1.49097 
1.49008 
1.48909 
1.48816 
1.48722 
1.48629 
1.48536 
1.48442 
1. 48349 
1. 48256 
■Tan. ' 


16 
14 
IS 
12 
11 


50 

51 
6« 
68 
54 


.54220 
.64244 
.64269 
.54298 
.54317 


.84025 
.84009 
.83994 
.83978 
83962 


.64528 
.64569 
.64610 
.64662 
.64693 
.6'4734 
.64775 
.64817 
.64868 
■64899 


1.54972 
1.54873 
1.54774 
1.64675 
1. 54576 


.83066 
•83050 

83034 
•83017 

83001 

■82985 
82969 
82953 
.82936 
• 82920 


.67028 
.67071 
.67118 
.67165 
.67197 
.67289 
.67282 
.67324 
.67366 
.67409 


8 
7 
6 


65 
56 
57 
58 
59 


.54842 
•54866 
.54891 
.64416 
.54440 
.64464 


.83946 
•88980 
.83915 
.83899 
83883 
83867 


1.64478 
1. 54379 
1.64281 
1.54183 
1. 54085 


5 

4/ 

1 


(BO 


64941 


1. 53986 


.55919 


82904 


.67461 


« 


' f 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. ' 


Cot. 


^r^ 



*r 



Digitized by VjOOQIC 



NATURAL SINES, COSINES, TANGENTS, COTAN. 257 
34* 35' 





Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


# 





.55919 


82904 


67451 


1.48256 


■57358 


.81915 


.70021 


1.42815 


"55 


1 


.56943 


.82887 


.67493 


1.48163 


57381 


81899 


.70064 


1.42726 


69 


2 


.55968 


.82871 


.67536 


1.48070 


.57405 


.81882 


.70107 


1.42638 


68 


8 


.55992 


.82855 


.67578 


1.47977 


.57429 


.81865 


.70151 


1.42550 


67 


.*- 


.56016 


82839 
.82822 


■67620 
■67663 


1.47885 
1.47792 


.57453 


.81848 


.70194 


1.42462 


56 


6 


.56040 


.57477 


.81832 


.70238 


1.42374 


55 


6 


.56064 


.82806 


.67705 


1.47699 


.57501 


.81815 


.70281 


1.42286 


64 


7 


.56088 


.82790 


67748 


1.47607 


.57524 


.81798 


.70325 


1.42198 


58 


5 


.56112 


.82773 


.67790 


1.47514 


.57548 


.81782 


70368 


1.42110 


52 


« 


.56136 
.56160 


.82757 


.67832 


1.47422 


.57572 


.81765 


.70412 


1.42022 


51 


10 


.82741 


.67875 


1.47330 


.57596 


.81748 


.70455 


1.41934 


50 


11 


.56184 


.82724 


.67917 


1.47238 


.57619 


•81731 


.70499 


1.41847 


49 


12 


.56208 


.82708 


.67960 


1.47146 


.57643 


■81714 


•70542 


1.41759 


48 


13 


.56282 


.82692 


.68002 


1.47053 


.57667 


•81698 


. 70586 


1.41672 


47 


14 


.56256 
.56280 


.82675 


■68045 


1.46962 


.57691 


•81681 


■70629 
.70673 


1.41584 


46 


16 


.82659 


.68088 


1.46870 


.57715 


•81664 


1.41497 


46 


16 


.56305 


.82643 


■68130 


1.46778 


57738 


•81647 


70717 


1.41409 


44 


17 


.56329 


.82626 


■68178 


1.46686 


.57762 


.81631 


■70760 


1.41322 


48 


18 


.56353 


.82610 


.68215 


1.46595 


57786 


.81614 


.70804 


1.41235 


42 


iS- 


.56877 


82593 
.82577 


J8258 
.68301 


1.46503, 
1.46411 


■57810 


.81597 


.70848 


1.41148 


41 


2® 


.56401 


.57833 


.81580 


.70891 


1.41061 


40 


21 


.56425 


.82561 


.68343 


1.46320 


57857 


.81563 


.70985 


1.40974 


39 


22 


.56449 


.82544 


.68386 


1.46229 


•57881 


.81546 


.70979 


1. 40887 


38 


28 


.56473 


.82528 


.68429 


1.46137 


.57904 


.81530 


.71023 


1.40800 


87 


?i_ 


56497 


.82511 


.68471 


1.46046 


•57928 


.81513 


.71066 


1. 40714 


86 


26 


.56521 


.82495 


.68514 


1.45955 


57952 


.81496 


.71110 


1.40627 


36 


26 


. 56545 


.82478 


.68557 


1.45864 


.5^976 


•81479 


.71154 


1.40540 


84 


27 


.56569 


.82462 


.68600 


1.45773 


.57999 


.81462 


.71198 


1.40454 


33 


28 


.56593 


.82446 


.68642 


1.45682 


.58023 


.81445 


•71242 


1. 40367 


32 


29_ 


.56617 


.82429 


68685 


1.45592 


.58047 


.81428 


.71285 


1. 40281 


31 


30 


.56641 


.82413 


.68728 


1.45501 


.58070 


.81412 


.71329 


1.40195 


30 


81 


.56665 


.82396 


■68771 


1.45410 


.58094 


.81395 


.71373 


1.40109 


29 


82 


■56689 


.82380 


.68814 


1.45320 


.58118 


.81878 


.71417 


1.40022 


28 


88 


.56718 


.82363 


.68857 


1.45229 


.58141 


.81361 


.71461 


1.39936 


27 


84 


.56736 
.56760 


.82347 


. 68900 


1.45139 


■58165 


.81344 


.71505 


1. 39850 


26 


85 


.82330 


■ 68942 


1.45049 


■58189 


.81327 


.71549 


1.39764 


25 


86 


.56784 


.82314 


68985 


1.44958 


■58212 


.81310 


.71593 


1.39679 


24 


87 


.56803 


.82297 


.69028 


1.44868 


■58236 


.81293 


.71637 


1.395^8 


23 


88 


.56882 


.82281 


.69071 


1-44778 


.58260 


.81276 


.71681 


1.39507 


22 


89. 


. 568ft6_ 
.56880 


.82264 
.82248 


■69114 
.69157 


1.44688 


■ 58283 


■81259 


.71725 


1.39421 


-21 


40 


1^ 44598 


■58307 


.81242 


.71769 


1.39336 


20 


41 


.56904 


.82231 


.69200 


1.44508 


■58330 


.81225 


.71813 


1.39250 


19 


42 


.56928 


.82214 


.69243 


1.44418 


■58354 


.81208 


.71857 


1.39165 


18 


48 


.56952 


.82198 


•69286 


1.44329 


•58378 


•81191 


.71901 


1.39079 


17 


44 


.56976 


.82181 


69329 


1.44239 


.58401 


.81174 


.71946 


1.38994 


16 


45 


.57000 


•82165 


■69372 


1.44149 


■58425 


.81157 


■71990 


1.88909 


15 


46 


.57024 


.82148 


■69416 


1.44060 


58449 


•81140 


.72034 


1.38824 


14 


47 


.57047 


.82132 


■69459 


1^43970 


■58472 


■81123 


. 72078 


1.38738 


18 


48 


.57071 


82115 


■69502 


1^ 43881 


58496 


81106 


.72122 


1.38653 


12 


40 


.57095 


.82098 


■69545 


1^43792 


.58519_ 
. 58543 


■81089 


.72167 


1.38568 


11 


50 


.57119 


.82082 


■69588 


1.43703 


■81072 


.72211 


1.38484 


10 


51 


.57148 


.82065 


■69631 


1.43614 


.58567 


■81055 


.72255 


1.38399 


9 


52 


.57167 


.82048 


■69675 


1. 43525 


. 58590 


■81038 


.72299 


1.38314 


8 


68 


.57191 


82032 


■69718 


1. 43436 


■58614 


■81021 


.72344 


1.38229 


7 


64_ 


.57215 


.82015 


■69761 


1.43347 


■58637 


.81004 


.72388 


1.38145 


e 


66 


.57238 


.81999 


■ 69804 


1^ 43258 


■58661 


80987 


. 72432 


1.38060 


8 


66 


.57262 


■81982 


■69847 


1^ 43169 


■58684 


■80970 


■72477 


1.87976 


4 


67 


.57286 


■81965 


■69891 


1^ 43080 


•58708 


■80953 


■72521 


1.37891 


3 


68 


.57310 


81949 


■69934 


1^42992 


•58731 


■80936 


■72565 


1.37807 


2 


69 


.57334 


■81932 


■69977 


1^42903 
1 ■42815 


•58755 


.80919 


■72610 


1. 37722 


1 


eo 


■57358 


■81915 


.70021 


•58779 


■80902 


■ 72654. 


1^ 37638 


-J> 


■■^ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



M* 



Digitized by VjOOQIC 



258 NATURAL SINES, COSINES, TANGENTS, COTAN. 
Ber 87* 





Sin. 


Cos, 


Tan. 


Cot. 


Sin. 1 


Cos. Tan, 


Gat. 


' 





-58779 


.30902 


,72654 


1-37838 


.60183 


■ 79384 ■75355 t" 


1 ■33704 


"5o 


1 


■Gsaos 


.80835 


72699 


1.37554 


60205 


■ 79846 


75401 


1.82634 


69 


3 


58826 


.80867 


.72743 


1.37470 


60228 


.79329 


.75447 


1.33544 


58 


3 


.58849 ' 


.80350 


.72783 


1-37388 


.6025i 


■7S811 


.75492 


1. 32464 


57 


J_ 


.68373 


80833 


,72.e32_ 


_r,373^2_ 


_e0274^ 


-79793 


■75538 


1.32884 


_56 


5 


6839B 


soaifl 


. 72877 


1.37218 


. 60398 


■79776 


75584 


1 ■ 33304 


K5 


6 


.58920 


.80799 


.72921 


1.37134 


.60321 


.79768 


.75629 


1. 82324 


64 


7 


68943 


80783 


.72966 


1-37060 


■ 60344 , 


-79741 


.75676 


1. 83144 


63 


8 


58967 


.80765 


-73010 


1.36967 


■60367 ' 


■79723 


.76721 


1.32064 


52 


9 


53090 : 
59014 


-80748 


.73055_ 
.73100 


1.36883 


.60390 
- 60414 


.79706 


.75767 


r 31984. 
1.31904 


61 


io 


.80730 , 


1.36800 


■79688 


75812 


50 


11 


59037 


.80713 


.73144 


1-36716 


■60437 


.79671 


.75858 


1 ■31825 


49 


13 


.59061 


.30696 


73189 


i. 36633 


-60460 


■79653 


.75904 


3.31745 


46 


13 


59084 


.30679 


73234 


1.36549 


60483 


-79635 


.75950 


1.31666 


47 


1^ 


.B910S 


.80662 


^3278, 
73323 


1 36456, 
1.36383 


60506 
^60539 


79618 
. 79600 


.75996 


1.31636 


46 


15 


.SfllSl 


.80844 


.78042 


1-81507 


46 


le 


.59154 


■80627 


73388 


1.36300 


.60553 


-79583 


.76083 


1.31437 


44 


17 


-69178 


.80610 


73413 


1.36217 


-60576 


■79565 


76134 


1-31348 


43 


li 


69201 


.80593 


■73457 


1.36134 


. 60599 


79547 


.76160 


1-31269 


43 


i»_ 


.59225 


.80576 


.73502 


1. 36051 
1.35986 


.60622, 
.60645 


.79530 76228 


1-31190 


-Jli 


20 


5924B 


.80558 


.73547 


■79512 .76272 


131110 


40 


n 


59272 


80541 


73592 


1.35885 


.50863 


.79494 76318 


1.31031 


39 


22 


.59296 


.80524 


73837 


1.35802 


-60691 


■79477 .78364 


1-30953 


38 


23 ' 


-59318 


.80507 


73681 ' 


1.33719 ' 


.H0714 


.79459 1-76410 


1.30873 


37 


2*_ 


.59S42 


-80481,' 


.73726 


1.35637 


.60733 


79441 1.76468 


1 30795 


3| 


25 


,59365 


.80473 


-73771 


1.36564 


60761 


.79424 


.76502 


] ■30716 


"35 


25 


.59389 


30455 


73816 


1.35473 


.60784 


-7940fi 


.76548 


1 ■30637 


34 


37 


.69412 


.80438 


-73861 


1.35:-3B9 


■60807 


79338 


.76594 


1^305B8 


33 


28 


59436 


.30420 


.73906 


1.35307 


■60830 


79371 


.76640 


1. 80480 


32 


2fl_ 


.59469 


.80403 


-73951 


X. 35234 


-60853 
-60876 


,79853 


.76686 


1. 30401. 
1.30323 


31 


30 


-59482 


-80338 


.73996 


1.35143 


.79335 


.f6733 


30 


31 


.59508 


80368 


.74041 


1.35060 


. 60899 


.79318 


-78779 


1. 30244 


2a 


S2 


-59529 


-80351 


.74086 


1.34978 


.60922 


79300 


.78825 


1 ■30166 


28 


33 


59552 


80334 


.74131 


1.34898 


.60945 


.79282 


.76871 


1-30087 


27 


34 


59576 


.80316 


-74176 


1. 34814 


■60968 


79264 


.76918 
.76964 


1^ 30009 
1.29931 


26 


^5 


-69599 


. 80299 


74221 


1.34733 


■60991 


, 79247 


25 


3B 


.59622 


80282 


-74287 


1.34650 


.61015 


1.79229 


.77010 


1- 29853 


24 


37 


59646 


. S0264 


.74312 


1.34568 


81038 


-79211 


.77057 


1.39775 


23 


3S 


59669 


.80247 


-74357 


1.34487 


.61061 


.79193 


.77103 


1.29698 


23 


3»_ 


-59693 


- 80230 . 


.74402 


1. 3440 5 


61034 


.79176 


■77149 


1-39618 


21 


40 


-59716 


.30212 


. 74447 


1.34323 


-61107 


.79153 1.77196 


1.29541 


30 


41 


.59739 


80195 


.74493 


1.34242 


61130 


..'9 140 L 77242 


1-29463 


19 


42 


58763 


80178 


.74538 


1.34160 


81153 


.79122 


.77289 


1^39335 


18 


43 


.59738 


.80160 


.74583 


1.34079 


31178 


.79105 


.77335 


1.29307 


17 


44 


-59809 
59832 


30143 


. 7462S_ 

."74674 


1-33998 
1.33916 


61199 
■61233 


.79087 
.79069 


77382 
. 77428 


1. 39329 
1.29152 


1ft 


45 


.80125 


15 


46 


.69356 


S0108 


.74719 


1.33335 


-61245 


.79051 


.77475 


1. 28074 


14 


47 


.59879 


.80091 


.74764 


1.33754 


61268 


!. 79033 


.77521 


1 ■28997 


13 


48 


.59902 


.30073 


.74810 


1.33673 


.61291 


.79016 


.77568 


1 ■28919 


12 


49 


.59936 
.59949 


.80058 


.74855 
.74900 


1.33593 
1.33511 


.61314^ 
■61337 


.73993 


.77615 


!■ 28842 


11 


50 


.80038 


.78980 *. 77661 


1.28764 


16 


61 


,59972 


.60021 


.74946 


1.33430 


.61360 


.78963 1-77708 


1 ■28687 


9 


62 


.59995 


.80003 


.74991 


1.33349 


■61383 


.78944 


.77754 


1.28619 


a 


53 


.60019 


-79986 


.75037 


1.33268 


.61406 


.78926 


-77801 


1.28533 


7 


5^ 


.fioq42_ 


.79988 


.75082 


1.33187 


61429 


i. 73908 


.77848 


1.23466 


A 


55 


. 60065 


.79951 


."75128 


1.33107 


.61451 


.78891 


-77895 


1 ■28379 


5 


5e 


■60089 


-79934 


.75173 


1-33026 


■61474 


.78873 


.77941 


1.28S02 


4 


57 


.6011^ 


.79916 


.75219 


1.32946 


-61497 


.78355 


.77988 


!■ 28226 


3 


GS 


80135 


.79899 


.75264 


1.32865 


.61520 


.78837 


■78036 


1.28U3 


2 


59 


.60158 


■jiaiL 


. .75310 


_1JI27J5, 


61543 


78819 


■78082 


!■ 28071 


I 


§o_ 


,6018^ 
Cm, 


..79684 


.75355 


1.32704 
1 Tao. 


.61666 


.78801 


l,7812£ 
j Cot. 


1.27994 


^ 


^7^ 


Tan. 





M* 



or 

Digitized by VjOOQ IC 



NATURAL SINES, COSINES, TANGENTS, COTAN. 259 
as' 39* 



t 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 


"o" 


.61566 


.78801 


.78129 


1.27994 


62932 


.77715 


.80978 


1.23490 


'm 


1 


.61589 


.78783 


.78175 


1.27917 


.62955 


.77696 


.81027 


1.23416 


59 


2 


.61612 


.78765 


.78222 


1.27841 


.62977 


.77678 


.81075 


1.23343 


58 


3 


.61635 


.78747 


•78269 


1.27764 


•63000 


.77660 


.81123 


1.23270 


57 


4 


.61658 


.78729 


78316 


1.27688 


•63022 


.77641 


81171 


1.23196 


56 


5 


61681 


.78711 


•78363 


1.27611 


•63045 


.77628 


.81220 


1.23123 


55 


6 


• 61704 


.78694 


•78410 


1-27535 


.63068 


.77605 


.81268 


!• 23050 


54 


7 


61726 


.78676 


.78457 


1.27458 


.63090 


77586 


.81316 


1.22977 


58 


8 


61749 


•78658 


.78504 


1.27382 


.63113 


.77568 


.81364 


1.22904 


52 


9 


61772 


.78640 


•78551 


1.27306 


•63135 


.77550 


.81413 


1.22831 


51 


10 


■61795 


78622 


78598 


1.27230 


•63158 


.77531 


.81461 


1.22758 


50 


11 


61818 


.78604 


•78645 


1.27153 


63180 


.77513 


.81510 


1.22685 


49 


12 


.61841 


.78586 


-78692 


1.27077 


•63203 


.77494 


.81558 


1.22612 


48 


13 


.61864 


.78568 


.78739 


1.27001 


•63225 


.77476 


.81606 


1.22539 


47 


14 


61887 


•78550 


•78786 


1.26925 


• 63248 


.77458 


.81655 


1.22467 


46 


15 


61909 


.78532 


78834 


1.26849 


•63271 


.77439 


.81703 


1.22394 


45 


16 


•61932 


.78514 


78881 


1.26774 


•63293 


.77421 


.81752 


1.22321 


44 


17 


.61955 


.78496 


78928 


1.26698 


•63316 


.77402 


.81800 


1.22249 


43 


18 


61978 


78478 


78975 


1.26622 


63338 


.77384 


-81849 


1.22176 


42 


ii- 


.62001 


•78460 


•79022 


1.26546 


•63361 


.77366 


.81898 


1.22104 


-ii 


20 


.62024 


.78442 


•79070 


1.26471 


•63383 


.77347 


•81946 


1.22031 


40 


21 


.62046 


.78424 


.79117 


1-26395 


63406 


•77329 


.81995 


1.21959 


39 


22 


•62069 


78405 


.79164 


1.26319 


•63428 


.77310 


. 82044 


1.21886 


38 


23 


62092 


.78387 


.79212 


1.26244 


63451 


77292 


.82092 


1.21814 


37 


24 


■62115 


•78369 


79259 


1.26169 


•63473 


.77273 


.82141 


1.21742 


_^ 


25 


•62138 


.78351 


.79306 


1.26093 


•63496 


-77255 


.82190 


1.21670 


85 


26 


•62160 


.78333 


.79354 


1.26018 


.63518 


-77236 


.82238 


1.21598 


84 


27 


•62183 


.78315 


.79401 


1.25943 


.63540 


.77218 


.82287 


1.21526 


33 


28 


62206 


.78297 


.79449 


1.25867 


.63563 


.77199 


.82336 


1.21454 


32 


29 


62229 


•78279 


.79496 


1.25792 


.63585 


.77181 


.82385 


1.21382 


31 


30 


.62251 


•78261 


.79544 


1.25717 


.68608 


.77162 


-82434 


1.21310 


30 


81 


•62274 


78243 


.79591 


1.25642 


.63630 


.77144 


82483 


1.21238 


29 


82 


.62297 


•78225 


.79639 


1.25567 


63658 


.77125 


. 82531 


1^ 21166 


28 


38 


.62320 


•78206 


.79686 


1.25492 


.68676 


.77107 


82580 


1.21094 


27 


34 


.62342 


•78188 


.79734 


1.25417 


.63698 


.77088 


.82629 


1.21023 


26 


85 


.62365 


.78170 


.79781 


1.25343 


-63720 


-77070 


.82678 


1.20951 


25 


86 


62388 


•78152 


.79829 


1.25268 


63742 


.77051 


-82727 


1.20879 


24 


87 


.62411 


•78134 


.79877 


1.25193 


.63765 


-77038 


.82776 


1.20808 


23 


88 


.62433 


.78116 


.79924 


1.25118 


.63787 


•77014 


.82825 


1.20736 


22 


SL. 


• 62456 


. 78098 


.79972 


1.2S044 


•63810 


.76996 


.82874 


1.20665 


-21 


40 


.62479 


.78079 


.80020 


1.24969 


-63832 


76977 


-82923 


1.20593 


30 


41 


.62502 


.78061 


.80067 


1.24895 


.63854 


76959 


.82972 


1.20522 


19 


42 


.62524 


.78043 


.80115 


1.24820 


•63877 


. 76940 


.83022 


1. 20451 


18 


43 


62547 


.78025 


.80163 


1.24746 


63899 


•76921 


.83071 


1.20379 


17 


44 


•62570 


.78007 


.80211 


1.24672 


•63922 


76903 


.83120 


1.20308 


16 


45 


.62592 


.77988 


80258 


1.24597 


•63944 


. 76884 


.83169 


1. 2023 7 


15 


46 


.62615 


.77970 


.80306 


1.24523 


•63966 


•76866 


.83218 


1.20166 


14 


47 


.62638 


.77952 


.80854 


1.24449 


63989 


. 76847 


.83268 


1 • 20095 


13 


48 


.62660 


.77934 


.80402 


1.24375 


•64011 


76828 


83317 


1 . 20024 


12 


49 


•62683 


.77916 


.80450 


1.24301 


•64033 


.76810 


83366 


1.19953 


11 


fiO 


•62706 


•77897 


.80498 


1.24227 


•64056 


-76791 


•83415 


1. 19882 


10 


61 


62728 


77879 


.80546 


1.24153 


•64078 


.76772 


83465 


1.19811 


9 


52 


.62751 


.77861 


.80594 


1.24079 


.64100 


.76754 


.83514 


1 . 19740 


8 


63 


.62774- 


•77843 


.80642 


1.24005 


64123 


.76735 


.83564 


1.19669 


7 


54 


.62796 


.77824 


.80690 


1.23931 


64145 


.76717 


.83613 


1- 19599 


6 


55 


.62819 


77806 


.80738 


1.23858 


•64167 


.76698 


83662 


1.19528 


5 


66 


.62842 


.77788 


.80786 


1.23784 


•64190 


.76679 


•83712 


1.19457 


4 


67 


.62864 


.77769 


.80834 


1.23710 


• 64212 


.76661 


•83761 


1.19387 


3 


58 


.62887 


•77751 


.80882 


1.23637 


•64234 


.76642 


83811 


1.19316 


2 


59 


.62909 


.77733 


.80930 


1.23563 


. 64256 
.64279 


.76623 
.76604 


•83860 


1.19246 


1 


60. 


«- 


^^ 


80978 


1 • 23490 


•83910 


1.19175 


«o 


^T^ 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


^T" 



51' 



50" 

Digitized by VjOOQIC 



^m NATURAL SINES, COSINES, TANGENTS, COTAN. 







40'' 






41 


f 






/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


# . 


o 

1 

2 
8 

4 


.64279 
.64301 
.64828 
.64846 
.64868 


.76604 
.76586 
.76567 
76548 
.76530 


.83910 
83960 
.84009 
.84059 
.84108 


1.19175 
1.19105 
1.19035 
1. 18964 
1.18894 


65606 
65628 
.66650 
.65672 
.65694 


.75471 
.75452 
.75433 
.75414 
.75895 
.75375 
.75356 
.75387 
•75818 
.75299 


86929 
86980 
87081 
■87082 
■87183 


1.15087 
1.14969 
1.14902 
1.14884 
1.14767 


60 

59 
58 
57 
66 


5 
6 
7 
8 
9 


.64390 
.64412 
.64435 
.64457 
.64479 


.76611 
.76492 
76473 
.76456 
.76436 


.84168 
.84208 

! 84807 
.84357 


1.18824 
1.18754 
1.18684 
1.18614 
1.18544 
1.18474 
1.18404 
1.18334 
1.18264 
1.18194 


65716 
.65788 
.65769 
.65781 
.65808 


87184 
87236 
■87287 
■87338 
87889 


1.14699 
1.14682 
1.14565 
1.14498 
1.14480 


56 
54 
58 
62 
51 


10 

11 

12 

8 

14 


■64501 
.64524 
64546 
64568 
64590 


.76417 
.76898 
.76880 
.76861 
.76342 


.84407 
.84457 
84507 
■84556 
■84606 


.65825 
.65847 
.65869 
65891 
.65913 
■65935 
■65956 
■65978 
• 66000 
.66022 


.75280 
.75261 
.75241 
.76222 
.75203 


■87441 
87492 
.87643 
■87595 
■87646 


1. 14863 
1.14296 
1.14229 
1.14162 
1. 14095 


BO 

49 
48 
47 
46 


15 
16 
17 
18 

19 , 


.64612 
64685 
.64657 
64679 
.6479; 
.64723 
.64746 
.64768 
.64790 
.64812 


.76323 
.76304 
.76286 
.76267 
.76248 


.84656 
84706 
.84756 
•84806 
.84856 


1.18125 
1.18055 
1.17986 
1.17916 
1.17846 


76184 
76165 
76146 
■75126 
•75197 
■75088 
■75069 
■75060 
■75080 
■75011 


87698 
87749 
.87801 
■87852 
Jim. 
.87955 
.88007 
88059 
■88110 
■88162 


1.14028 
1.13961 
1.13894 
1.13828 
.J.J3761 
1.13694 
1.13627 
1.13561 
1.13494 
1.13428 


46 
44 

48 

42 


ao 

21 
22 
23 
24 


.76229 
.76210 
.76192 
.76173 
■76154 


■84906 
■84956 
■85006 
■85057 
■85107 


1.17777 
1.17708 
1.17638 
1.17569 
1.17500 


■66044 
■66066 
■ 66088 
■66109 
.66131 


40 

89 
88 


26 
26 
27 
28 
29 


.64834 
.64856 
.64878 
.64901 
. 64923 


.76135 
.76116 
.76097 
.76078 
.76059 


.85157 
■85207 
■85257 
■85308 
■85358 


1.17430 
1. 17361 
1.17292 
1.17223 
1.17154 


■66053 
■66175 
.66197 
■66218 
■66240 


.74992 
.74973 
.74953 
.74934 
■74915 


88204 
.88265 
.88317 
.88869 
.88421 


1.18361 
1.13295 
1.18228 
1.18162 
1.18096 


85 
84 
83 

82 
81 


30 

81 
82 
83 
84 


.64945 
.64967 
■64989 
65011 
■65083 


.76041 
.76022 
.76008 
.76984 
.75965 


.85408 
85458 
.85509 
■85559 
■85609 


1.17085 
1. 17016 
1.16947 
1.16878 
1.16809 


. 66262 
.66284 
.66306 
.66827 
■66349 


■74896 
■74876 
■74857 
■74838 
■74818 


88473 
88524 
.88576 
88628 
88680 


1.13029 
1.12963 
1.12897 
1.12881 
1.12765 


80 

29 
28 
27 
26 


35 
86 
87 
88 
89_ 


■65055 
■65077 
.65100 
.65122 
.65144 


■75946 
.75927 
■75908 
■76889 
■75870 


■85660 
■85710 
■85761 
.85811 
■86862 


1.16741 
1.16672 
1.16603 
1.16535 
1^ 16466 


.66371 
. 66393 
.66414 
66436 
.J6458 


.74799 
■74780 
■74760 
74741 

•7i722 


88732 
■88784 
■88836 

88888 
^8940 


1.12699 
1.12683 
1.12667 
1.12501 
_!■ 12485 


25 
24 
23 
22 
-21 


40 

41 
42 
43 
44 


■65166 
■65188 
.65210 
■65232 
■65254 


■ 75851- 
■75882 
■75818 
.75794 
• 75775 


■85912 
■85963 
.86014 
.86064 
■86115 


1.16398 
1.16329 
1.16261 
1.16192 
1.16124 


■66480 
■66501 
.66528 
.66546 
■66566 


■74703 
74683 
74664 

■74644 
74625 


88992 
.89045 
.89097 
.89149 
■89201 


1.12860 
1.12303 
1.12238 
1.12172 
1.12106 


20 


45 
46 
47 
48 
49 


■65276 
65298 
■65320 
■65342 
■65364 


.75756 
.75738 
.75719 
■75700 
■75680 


■86166 
86216 
.86267 
.86318 
■86368 


1.16056 
1^ 15987 
1.15919 
1.15851 
1. 15783 


■66588 
■66610 
■66632 
■66653 
66675 


. 74606 
. 74686 
74567 
.74548 
■74528 


■89253 
■89806 
89858 
.89410 
.89468 


1.12041 
1.11975 
1.11909 
1.11844 
1.11778 




50 

51 
52 
63 
54 


■65386 
■65408 
.65480 
.65452 
■65474 


■75661 
.75642 
.75623 
.75604 
.75585 


.86419 
.86470 
.86521 
.86572 
■86623 


1.16715 
1.15647 
1.15579 
1.15611 
1.15443 


■66697 
■66718 
■66740 
■66762 
.66783 


.74509 
74489 
.74470 
.74451 
■74431 


.89515 
.89567 
•89620 
89672 
■89725 


1.11718 
1.11648 
1.11582 
•1.11517 
1.11452 




65 
56 
57 
58 
59 


.65496 
.65518 
. 65540 
.65562 
.65584 


.75566 
.75547 
.75528 
.75509 
.75490 


.86674 
■86725 
■86776 
■86827 
86878 


1.15375 
1.15308 
1.15240 
1.15172 
1.15104 


■66805 
66827 
■66848 
■66870 
■66891 


.74412 
.74392 
.74873 
.74863 
.74334 


.89777 
.89830 
.89883 
.89935 
89988 


1.11387 
1.11821 
1.11256 
1.11191 
1.11126 




6CL 


.65606 


.75471 
Sin. 


■86929 
Cot. 


1.15037 


•?m§. 


.90040 


l.llO^l 
Tan. 


-Tfi 



40<' 



4r^ 

Digitized by VjOOQIC 



NATtmAL SINES, COSINES, TANGENTS, COTAN. 261 

48" 43* 






Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


# 




I 

2 
8 

4 
6 
6 
7 

I 


.66913 
■66935 
.66956 
.66978 
.66999 
.67021 
67043 
.67064 
.67086 
.67107 


.74314 
.74295 

74276 
.74256 
•74237 
.74217 
.74198 

74178 
.74159 
.74139 


.90040 
.90093 
.90146 
.90199 
.90251 


1.11061 
1 10996 
1.10931 
1.10867 
1.10802 


■68200 
68221 
.68242 
.68264 
.68285 


73185 
.78116 
.73096 
.73076 
•73056 


.93252 
.93306 
.93360 
•98415 
.98469 


1. 07287 
1.07174 
1. 07112 
1.07049 
1.06987 


00 

59 

56 


.90304 
.90357 
.90410 
.90463 
•90516 


1.10737' 

1.10672 

1.10607 

1.10548 

1.10478 


68306 
.68327 
.68349 
.68370 
.68891 


78086 
•73016 
.72996 
.72976 
•72957 


.98524 
.98578 
•93688 
•93688 
.98742 
•93797 
.93852 
.98906 
•93961 
.94016 


1.06925 
1. 06862 
1.06800 
1. 06788 
1.06676 


56 

1? 


10 

11 
2 
.8 
4 


.67129 
.67151 
.67172 
.67194 
.67215 


.74120 
.74100 
.74080 
.74061 
.74041 


.90569 
.90621 
.90674 
90727 
.90781 

■X 

.90940 
.90993 
.91046 

.91099 
.91158 
.91206 
.91259 
,91313 
•91366 
.91419 
•91473 
•91526 
.91580 


1.10414 
1.10349 
1.10285 
1.10220 
1.10156 


.68412 
68434 
.68455 
.68476 
.68497 


•72937 
.72917 

• 72897 
•72877 

• 72857 


1. 06618 
1. 06551 
1.06489 
1.06427 
1 06365 


60 

49 

if 

46 


15 
16 
7 
18 
19 


.67237 
.67258 
.67280 
67301 
.67323 


.74022 
.74002 
.78983 
.73963 
.73944 


1.10091 
1.10027 
1.09963 
1.09899 
,1.0J83|^ 
1.09770 
1.09706 
1.09642 
1.09578 
1.09514 


.68518 
68539 
.68561 
.68582 
.68608 
: 68624 
.68645 
.68666 
.68688 
.68709 


.72887 
.72817 
.72797 
.72777 
.72757 
.72737 
72717 
.72697 
. 72677 
.72657 


.94071 
.94125 
.94180 
.94285 
•94290 


1. 06308 
1. 06241 
1. 06179 
1^ 06117 
1.06056 


45 
44 
43 
42 


SO 

21 
22 

ii 


•67344 
.67366 

67387 
.67409 

67430 


.73924 
.78904 
.78885 
.73865 
.78846 


.94345 
.94400 
94455 
.94510 
.94565 


1.05994 
1.05982 
1. 05870 
1. 05809 
1. 05747 


40 

89 
88 
87 
86 


25 
26 
27 
28 
29 


67452 
67473 
.67495 
.67516 
.67538 


.73826 
.73806 
.73787 
.73767 
•73747 


1.09450 
1.09386 
1.09322 
1^ 09258 
1. 09195 


.68730 
.68751 
.68772 
. 68793 
.68814 


.72687 
•72617 
.72597 
.72577 
. 72557 


.94620 
.94676 
.94731 
.94786 
•94841 


1. 05685 
1-05624 
1.05562 
1.05501 
1. 05439 


35 

34 
38 
82 
81 


30 

81 
82 
83 

84 


67559 
67580 
•67602 
•67623 
• 67645 
67666 
67688 
67709 
67730 
67752 
.67773 
•67795 
67816 
67837 
.67859 


•73728 
.73''08 
.73688 
.73669 
• 7«49 


.91683 
•91687 
•91740 
91794 
.91847 


1.09181 
1.09067 
1.09003 
1. 08940 
1. 08876 


68835 
68857 
.68878 
.68899 
. 68920 


•72537 
.72517 
.72497 
.72477 
J2457_ 

.72437 
.72417 
.72397 
.72377 
.72857 


.94896 
. 94962 
•95007 
■95062 
■95118 


1.05378 
1-05317 
1.05255 
1.05194 
1. 05183 


80 

27 
29 


85 
86 
87 
38 
A? 


.73626 
.73610 
.73590 
73570 
• 73.5Ji. 
.73531 
.73511 
.73491 
.73472 
.73452 
•78432 
.73418 
•73398 
•73378 
.73358 
.73333 
•73314 
•73294 
•73274 
.73254 


.91901 
.91955 
.92008 
92062 
.92116 


1. 08818 
1. 08749 
1. 08686 
1.08622 
!• 08559 


68941 
• 68962 

68983 
•68004 
•69025 


95173 
95229 
.95284 
.96340 
.95395 


1.05072 
1-05010 
1-04949 
1. 04888 
1. 04827 


25 
24 
23 
22 
-21 


40 

41 
42 
48 
44 


92170 
.92224 
.92277 
•92331 
.92385 
.92439 

92493 
.92547 
•92601 
.92655 


1. 08496 
1.08432 
1.08369 
1. 08306 
1.08243 


69046 
.69067 

69088 
.69109 
.69130 


.72337 
.72317 
.72297 
•72277 
.72257 


.95451 
.95506 
.95562 
.95618 
.95678 
.95729 
.95785 
.95841 
.95897 
.95952 
.96008 
.96064 

96120 
.96176 
.96282 
.96288 

96844 

965] 8 


1. 04766 
1. 04705 
1.04644 
1.04583 
1-04522 

1.04461 
1.04401 
1- 04340 
1. 04279 
1.04218 
1.04158 
1.04097 
1-04036 
1.08976 
1.03915 


20 

\l 

17 
16 


45 
46 
47 
48 
49 


.67880 
67901 

.67928 
67944 

.67965 


1. 08179 
1. 08116 
1. 08053 
1.07990 
1^07927 
1.07864 
1.07801 
1. 07788 
1.07676 
1.07613 


69151 
.69172 
.69193 

69214 
.69235 
.69256 

69277 
.69298 
.69319 
•69340 


.72236 
72216 
.72196 
.72176 
.72156 


15 

it 


50 
61 
52 
58 
54_ 


.67987 
68008 
68029 
68051 

.68072 


.92709 
.92768 
.92817 
.92872 
.92926 


.72136 
.721] 6 
.72095 
•72075 
.72055 


? 


55 
56 
57 
58 
69 


68093 
•68115 

68136 
•68157 
.68179 


73234 
•73215 
.78195 
.78175 
.78155 


.92980 
•93034 
.93088 
.93148 
.93197 


1.07550 
1.07487 
1.07425 
1.07362 
1.07299 
1. 07337 


•69361 
■69382 
•69403 
■69424 
■69445 


72035 
.72015 
.71995 
.71974 
.71964 


1.08856 
1.08794 
1.08784 
1-03674 
1.03613 


8 
t 


n- 


«^ 


.73185 


.93252 


69466 


71934 


.96669 


1.03553 


^ 


£tm. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


fan. 





47* 



40" 

Digitized by VjOOQ IC 



262 NATURAIv SINES, COSINES, 

44' 



TANGENTS, COTAN. 

44'* 



* 


Sin. 


Cos. 


Tan. 1 Cot. 


t 


f 


Sin. [ Cos. 1 Tan. 


Cot. 


r 




1 

2 
8 

4 


69466 
.694S7 
69508 

.£9529 
eS549 
.69570 
.69591 
■69612 
69633 
.69654 


■71934 
■7I9L4 
71394 

.71873 
■71853 


-9S569 
- 96625 
■96681 

96738 
■96794 

96850 
■96907 
.&6963 
■97030 

97076 


1.0355S' 
1-03493 
!■ 03433 
1,03372 
1- 033 12 


60 

59 
58 
57 
66 


30 
31 

32 
33 
34 
35 
36 
37 
38 
39 


. 70091 
.70112 
■70133 
.70153 
■70174 


-71325 ,98270 
.71305 .n^2i 
■71284 ,98384 
.71264 98441 
■71243 98499 


101761 
1. 01702 
1 01642 
1. 01583 
1 01534 


30 

2^ 
2S 
27 

2B 


G 
5 
7 
8 

e 


■71833 
.713L3 
.71792 
71773 
.71762 


1^ 03352 
!■ 03192 
1,03132 
1.03072 
1-03012 


65 
54 
53 
53 
51 


■70195 

.70215 
. 70236 
^70257 

■70277 


-71323 
■71203 
71182 

. 71162 

Jim 

.71121 
.71100 
■71080 
■71059 
■71039 


■98556 
-98613 
,93671 
98738 
-93736 
98843 
98901 
.98958 
,99016 
^99073 


1.01465 
^01406 
1.01347 
1-01288 
1.01239 
1-01170 
1^01112 
1-01053 
1. 00994 
1,00935 


2S 
24 
23 
22 
31 


10 

11 

12 
13 
^4 


69675 
.69696 

69717 
.69737 

69758 
.6977e 
. 69800 
.69321 
-69342 
.69362 


^71733 
.71711 
.71691 
■71671 

71550 
.71630 
.71610 
.71590 
-71569 

71549 


97133 
.97189 
97246 

.97302 
97359 
■97416 
■97473 
■97529 
■97586 
-97643 


1 02952 

1.02893 
1. 02332 

1.02773 
1.02713 

1-02653 
1^02593 
1^02533 
1-02474 
1.02414 


50 

49 
48 
47 
46 

15 
44 
43 
42 

40 
39 
88 
37 
36 
35 
34 
33 
32 
31 

_20. 


40 

41 
42 
43 
44 


■70298 
-70319 
■ 70339 
.70360 
.70361 


20 

la 

17 

la 


15 
16 
17 
IS 
19 


45 
46 
47 
48 

_49^ 
50 
51 
52 
53 

-54^ 
65 
56 
57 
58 
59 


-70401 

■ 70422 
70443 

■70463 
■70484 

■ 70 505 
■70525 
■70546 
.70567 

■ 70587 
,70608 
■70623 
.70649 
■70670 
. 70690 
-70711 


71019 
.70993 
- 70978 
■ 70957 

70937 


■99131 
.99X89 
.99247 
■99304 
.99363 


1,00876 
1. 0081 8 
1,00759 
1. 00701 
1.00842 


15 
14 
13 

n 
11 


m 

21 

S3 
P4 


■69883 
■69904 
-69925 
69946 
69966 
699B7 
70008 
70029 
70049 
70070 


■71539 
-71508 
-71438 
.71468 
■71447 
.71427 
71407 
.71386 
■71366 
.71345 
^71325 


■ 97700 
.97756 
.97813 
-97870 
■97927 

■ 97984 
98041 

■98098 

99155 

^98213 

-93270 

Cut.. 


1-02355 
1.02295 
1-02236 
1.02] 76 
1 .02117 
1,02057 
1 01998 
1,01939 
1.01879 
1-01820 


, 70916 
70896 
70875 

.70855 
_70834 

- 70813 
■70793 
.70772 
. 70752 
-70731 


.99420 1-00583 
99478 1.00525 
99536 1.00467 

■ 99594 1. 00408 
99652 1,00350 


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1 ■00391 
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1,00118 
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NATl 


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SECANTS AND COSECANTS 


263 


9 


SECANTS. 


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20 


80 


4° 


6» 


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1.00000 


1.00016 


1.00061 


1.00187 


1.00344 


1.00383 


1.00551 


60 


1 


00000 


00016 


00062 


00189 


00346 


00385 


00564 


60 


2 


00000 


00016 


00063 


00140 


00348 


00887 


00557 


68 


3 


00000 


00017 


00064 


00143 


00350 


00390 


00560 


67 


4 


00000 


00017 


00066 


00148 


00253 


00393 


00563 


66 


5 


00000 


00018 


00066 


00145 


00354 


00395 


00566 


55 


6 


00000 


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00067 


00147 


00357 


00397 


00569 


54 


7 


00000 


00019 


00068 


00148 


00359 


00400 


00578 


53 


8 


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00030 


00069 


00160 


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53 


9 


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00363 


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51 


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00153 


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50 


11 


1.00001 


r.00031 


1.00073 


1.00165 


1.00367 


1.00411 


1.00686 


49 


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48 


13 


00001 


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45 


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44 


17 


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43 


19 


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00083 


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00384 


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41 


90 


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00037 


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40 


81 


1.00003 


1.00028 


1.00084 


1.00171 


1.00389 


1.00438 


1.00617 


39 


83 


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00086 


00173 


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00440 


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38 


33 


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86 


85 


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84 


27 


00003 


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00183 


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00454 


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33 


28 


00003 


00083 


00093 


00183 


00305 


00457 


00640 


83 


20 


00004 


00084 


00094 


00185 


00307 


00460 


00644 


81 


80 


00004 


00034 


00096 


00187. 


00309 


00463 


00647 


80 


81 


1.00004 


1.00086 


1.00097 


1.00189 


1.00813 


1.00465 


1.00650 


39 


83 


00004 


00086 


00098 


00190 


00814 


00468 


00654 


38 


83 


00005 


00087 


00099 


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37 


84 


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86 


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87 


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89 


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31 


40 


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00305 


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41 


1.00007 


1.00043 


1.00110 


1.00307 


1.00835 


1.00494 


1.00684 


19 


42 


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18 


43 


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44 


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16 


46 


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15 


46 


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60 


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61 


1.00011 


1.00053 


1.00124 


1.00336 


1.00859 


1.00534 


1.00719 


9 


63 


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63 


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64 


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6 


65 


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5 


66 


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00057 


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00539 


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4 


67 


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00058 


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3 


68 


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69 


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COSECANTS. 







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264 


NATURAL 


SECANTS AND COSECANl^ 




t 


SECANTS. 


/ 


70 


80 


9° 


100 


11« 


120 


18» 





1.00761 


1.00983 


1.01247 


1.01543 


1.01872 


1.02234 


1.02680 


60 


1 


00756 


00987 


01251 


01548 


01877 


02240 


02637 


59 


2 


00758 


00991 


01256 


01553 


018a3 


02247 


02644 


58 


8 


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00995 


01261 


01558 


01889 


02253 


02651 


57 


4 


00765 


00999 


01266 


01564 


01895 


02259 


02658 


56 


6 


00769 


01004 


01270 


01569 


01901 


02266 


02665 


55 


6 


007r3 


01008 


01275 


01574 


01906 


02272 


02672 


54 


7 


00776 


01012 


01279 


01579 


01912 


02279 


02679 


53 


8 


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01016 


01284 


01585 


01918 


02285 


02686 


52 


9 


00784 


01020 


01289 


01690 


01924 


02291 


02693 


51 


10 


00787 


01024 


01294 


01595 


01030 


02298 


02700 


50 


11 


1.00791 


1.01029 


1.01298 


1.01601 


1.01936 


1.02304 


1.02707 


49 


12 


00795 


01033 


01303 


01606 


01941 


02811 


08714 


48 


13 


00799 


01037 


01308 


01611 


01947 


02817 


02721 


47 


14 


00802 


01041 


01313 


01616 


01953 


02328 


02728. 


46 


15 


00806 


01046 


01318 


01622 


01959 


02830 


02735* 


45 


16 


00810 


01050 


01822 


01627 


01965 


02336 


02742 


44 


17 


00813 


01054 


01327 


01633 


01971 


02348 


02749 


43 


18 


00817 


01059 


01832 


01638 


01977 


02349 


02756 


42 


19 


00821 


01063 


01337 


01643 


01983 


02356 


02763 


41 


iiO 


00825 


01067 


01342 


01649 


01989 


02362 


02770 


40 


21 


1.00828 


1.01071 


1.01346 


1.01054 


1.01995 


1.02869 


1.02777 


39 


2-^ 


00882 


01076 


01351 


01659 


02001 


02875 


02784 


38 


23 


00836 


01080 


01356 


01665 


02007 


02882 


02791 


37 


24 


00840 


01084 


01361 


01670 


02013 


02888 


02799 


36 


25 


00844 


01089 


01366 


01676 


02019 


02895 


02806 


35 


26 


00848 


01093 


01371 


01681 


02025 


02402 


02818 


84 


27 


00851 


01097 


01376 


01687 


02031 


02408 


02820 


83 


28 


00655 


01102 


01381 


01692 


02037 


08416 


02827 


33 


..'9 


00859 


01106 


01386 


01698 


02043 


02421 


028:34 


31 


30 


00863 


01111 


01391 


01703 


02049 


02428 


02812 


30 


31 


1.00867 


1.01115 


1.01395 


1.01709 


1.02056 


1.02436 


1.02819 


29 


32 


00871 


01119 


01400 


01714 


02061 


03441 


0285C 


28 


3:3 


00875 


01124 


01405 


01720 


02067 


02448 


02868 


27 


34 


00878 


01128 


01410 


01725 


02073 


02454 


02870 


26 


35 


00882 


01133 


01415 


01731 


02079 


02461 


02878 


25 


36 


00886 


01137 


01420 


01736 


C2085 


02468 


02885 


24 


37 


00690 


01142 


01425 


01742 


02091 


02474 


02892 


23 


38 


00894 


01146 


01430 


01747 


02097 


02481 


02899 


22 


39 


00898 


01151 


01435 


01753 


02103 


02488 


02907 


21 


40 


00902 


01155 


01440 


01758 


02110 


02494 


02914 


20 


41 


1.00906 


1.01160 


1.01446 


1.01764 


1.02116 


1.02501 


1.02921 


19 


42 


00910 


01164 


01450 


01769 


02122 


08508 


02928 


18 


43 


00914 


01169 


01455 


01775 


02128 


02515 


02986 


17 


44 


00918 


01173 


01461 


01':81 


02184 


02521 


02943 


16 


45 


00922 


01178 


01466 


01786 


02140 


02628 


08950 


15 


46 


00926 


01182 


01471 


01792 


02146 


02585 


08968 


14 


47 


00930 


01187 


01476 


01798 


02158 


02548 


02965 


13 


48 


00934 


01191 


01481 


01803 


02159 


02548 


02978 


18 


49 


00988 


01196 


01486 


01809 


02165 


025.n5 


08980 


11 


50 


00942 


01200 


01491 


01815 


02171 


02562 


02987 


10 


51 


1.00946 


1.01205 


1.01496 


1.01880 


1.02178 


1.02569 


1.08994 


9 


52 


00950 


01209 


01601 


01826 


08184 


02576 


08002 


8 


53 


00954 


0121' 


01506 


01882 


02190 


02582 


03009 


7 


54 


00958 


01219 


01512 


01837 


02196 


02589 


08017 


6 


65 


00962 


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01517 


01843 


02203 


02696 


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5 


56 


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01228 


01622 


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02209 


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4 


57 


00970 


01233 


01527 


01854 


02215 


02610 


03039 


8 


58 


00975 


01237 


01532 


01860 


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08046 


8 


59 


00979 


01242 


01537 


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1 


60 


00983 


01247 


01543 


01872 


02234 


02680 


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1 


820 


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80» 


79" 


78« 


77" 


76" 


» 


COSECANTS. 



Digitized by VjOOQIC 





NATURAL 


SECANTS AND COSECANTS 


265 


t 


SECANTS. 


/ 


14» 


160 


16<> 


170 


18° 


190 


20° 





1.03061 


1.03528 


1.04030 


1.04569 


1.05146 


1.05762 


1.06418 


60 


1 


03069 


03536 


04039 


04578 


05166 


05773 


06429 


59 


2 


03076 


03544 


04047 


04588 


05166 


05783 


06440 


5S 


3 


03084 


03552 


04066 


04597 


05176 


05794 


06452 


67 


4 


08091 


03560 


04065 


04606 


05186 


05806 


06463 


56 


5 


03099 


03568 


04073 


04616 


05196 


05816 


06474 


55 


6 


08106 


03676 


04082 


04626 


05206 


05826 


06486 


64 


7 


08114 


03584 


04091 


04636 


05216 


05836 


06497 


58 


8 


03121 


03692 


04100 


04644 


05226 


08847 


06508 


52 


9 


03129 


03601 


01108 


04653 


06236 


05858 


06520 


51 


10 


08137 


03609 


04117 


04663 


05246 


05869 


06531 


60 


11 


1.03144 


1.03617 


1.04126 


1.04672 


1.06256 


1.06879 


1.06542 


49 


12 


03152 


08625 


04136 


04682 


06260 


06890 


06654 


48 


13 


03169 


03633 


04144 


04691 


05276 


05901 


06565 


47 


H 


03167 


03642 


04162 


04700 


06286 


05911 


06577 


46 


15 


08175 


03650 


04161 


04710 


05297 


06922 


06588 


45 


16 


03182 


08668 


04170 


04719 


05807 


06933 


06600 


44 


17 


03190 


03666 


04179 


04729 


05317 


06944 


06611 


43 


18 


08198 


03674 


04188 


04788 


05327 


05956 


06622 


42 


19 


08.?05 


08683 


04197 


04748 


05837 


05965 


06634 


41 


20 


03213 


08691 


04206 


04757 


05347 


05976 


06645 


40 


21 


1.03221 


1.03699 


1.04214 


1.01767 


1.06367 


1.05987 


1.06657 


39 


22 


03228 


03708 


04223 


04776 


05867 


05998 


06668 


38 


23 


03236 


03716 


04232 


04786 


06378 


06009 


06680 


37 


24 


03244 


03724 


04241 


04795 


05388 


06020 


06691 


36 


25 


03251 


03732 


04260 


04805 


06398 


06030 


06703 


36 


26 


08259 


03741 


04259 


04815 


06408 


06041 


06715 


84 


27 


08-'67 


03749 


04268 


04824 


06418 


06052 


06726 


33 


28 


03275 


03768 


04277 


04834 


05429 


06063 


06788 


82 


29 


0828^ 


03766 


04286 


04843 


05439 


06074 


06749 


31 


30 


03290 


03774 


04295 


04853 


06449 


06085 


06761 


30 


31 


1.03298 


1.03783 


1.04304 


1.04868 


1.06460 


1.06096 


1.06773 


29 


88 


03306 


03791 


04313 


04872 


05470 


06107 


06784 


28 


33 


08313 


03799 


04322 


04882 


05480 


06118 


06796 


27 


34 


03321 


03808 


04331 


04891 


05490 


06129 


06807 


26 


85 


03329 


03816 


04340 


04901 


06601 


06140 


06819 


25 


36 


03337 


03825 


04349 


04911 


06511 


06161 


06881 


24 


87 


03345 


03833 


04358 


04920 


05621 


06162 


06843 


23 


38 


03358 


03842 


04367 


04930 


05632 


06173 


06854 


22 


89 


03360 


03860 


04376 


04940 


06642 


06184 


06666 


21 


40 


03368 


03868 


04385 


04950 


05552 


06196 


06878 


20 


41 


1.03376 


1.03867 


1.04394 


1.04969 


1.05563 


1.06206 


1.06889 


19 


42 


03.384 


03876 


04403 


04969 


05573 


06217 


06901 


18 


43 


08392 


03884 


04413 


04979 


05584 


06228 


06913 


17 


44 


03400 


03892 


04422 


04989 


05694 


06239 


06955 


16 


45 


03408 


03901 


04481 


04998 


05604 


06250 


06930 


15 


46 


03416 


08909 


04440 


06008 


06615 


06261 


06948 


14 


47 


03424 


03918 


04449 


05018 


05626 


06272 


06960 


13 


48 


03432 


08927 


04458 


05028 


05636 


06283 


06972 


12 


49 


03439 


08935 


04468 


05038 


05646 


06295 


06984 


11 


60 


03447 


03944 


04477 


05047 


05657 


06306 


06995 


10 


61 . 


1.03455 


1.03962 


1.04486 


1.05067 


1.05667 


1.06317 


1.07007 


9 


52 


03463 


08961 


04495 


05067 


05678 


06328 


07019 


8 


68 


03471 


03969 


04504 


05077 


06688 


06339 


07031 


7 


64 


03479 


03978 


04514 


05087 


05699 


06350 


07043 


6 


65 


03487 


03987 


04523 


05097 


06709 


06362 


07055 


5 


56 


03495 


03996 


04632 


a'5107 


05790 


06373 


07067 


4 


57 


03508 


04004 


04541 


05116 


05730 


06384 


07079 


3 


58 


03512 


04013 


04551 


05126 


05741 


06395 


07091 


2 


59 


08520 


04021 


04560 


05136 


05751 


06407 


07103 


1 


60 


03528 


04030 


04569 


05146 


05762 


06418 


07115 





/ 


76» 


740 


78* 


720 


7JO 


70O 


69° 


/ 






COSECANTS. 







Digitized by VjOOQIC 



266 


NATURAL 


SECANTS AND COSECANTS 


, 


§ 


SECANTS. 


/ 


«!• 


2«- 


«8«» 


a*' 


«6» 


26« 


27" 





1.07115 


1.07858 


1.06636 


1.09464 


1.10388 


1.11260 


1.12233 


60 




07126 


07866 


06649 


09478 


10353 


11276 


12249 


50 




07138 


07879 


06663 


09492 


10368 


11292 


12266 


58 




07150 


07892 


08676 


09506 


10383 


11808 


12283 


57 




07162 


07904 


08600 


09520 


10398 


118-23 


12299 


66 




07174 


07917 


08703 


00535 


10413 


11339 


12316 


65 




07186 


07930 


06717 


09549 


10428 


11355 


12333 


54 




07199 


07948 


06780 


09563 


10443 


11371 


12349 


63 




07211 


07955 


08744 


09577 


10458 


11387 


12366 


52 




07228 


07968 


08757 


09592 


10473 


11408 


12383 


51 




07235 


07981 


08771 


09606 


10488 


11419 


12400 


50 


11 


1.07247 


1.07994 


1.06784 


1.09620 


1.10608 


1.11435 


1.12416 


49 




07259 


06006 


06798 


09685 


10518 


11451 


12433 


48 




07271 


06019 


08811 


09649 


la'wa 


11467 


12450 


47 




07283 


06032 


08625 


09663 


10649 


11488 


12467 


46 




07295 


08045 


06839 


09678 


10564 


11499 


12484 


46 




07807 


08058 


08852 


09692 


10679 


11516 


12501 


44 




07820 


08071 


08866 


09707 


10594 


11531 


12518 


48 




07832 


08064 


08880 


09721 


10609 


11647 


12584 


42 




07344 


06097 


08893 


09735 


10625 


11563 


12561 


41 


20 


07856 


06109 


06907 


09750 


10640 


11679 


12566 


40 


21 


1.07368 


1.06122 


1.06921 


1.09764 


1.10655 


1.11596 


1.12585 


89 


22 


07380 


08135 


08934 


09779 


10670 


11611 


12602 


88 


28 


07393 


08148 


06948 


09793 


10686 


11627 


12619 


87 


24 


07405 


06161 


08962 


09806 


10701 


11643 


12636 


86 


25 


07417 


08174 


06975 


09622 


10716 


11659 


12653 


85 


26 


07429 


08187 


06969 


09687 


10731 


11675 


12670 


84 


27 


07442 


08200 


09003 


09851 


10747 


11601 


12687 


SS 


28 


07454 


06213 


09017 


09866 


10762 


11706 


12704 


82 


29 


07466 


06226 


09030 


09880 


10777 


11724 


12721 


81 


80 


07479 


06239 


09044 


09895 


10793 


11740 


12738 


30 


31 


1.07491 


1.06252 


1.09058 


1.09909 


1.10608 


1.11756 


1.12755 


29 


82 


07508 


08265 


09072 


09924 


10824 


11772 


12772 


28 


38 


07516 


08278 


09066 


09939 


10889 


11789 


12780 


27 


34 


07528 


06291 


09099 


09953 


10664 


11805 


12807 


26 


85 


07540 


08805 


09113 


09968 


10870 


11821 


128e4 


25 


36 


07558 


08318 


09127 


09982 


10885 


11888 


12841 


24 


37 


07566 


08831 


09141 


09997 


10901 


11854 


12858 


28 


38 


07578 


06344 


09155 


10012 


10916 


11870 


12875 


22 


39 


07590 


08357 


09169 


10026 


10932 


11886 


12892 


21 


40 


07602 


08370 


09183 


10041 


10947 


11903 


I;i9l0 


20 


41 


1.07615 


1.08383 


1.09197 


1.10056 


1.10963 


1.11919 


1.12927 19 


42 


07627 


09897 


09211 


10071 


10078 


11036 


12944 18 


43 


07640 


08410 


09224 


10065 


10994 


11952 


12961 17 


44 


07652 


06423 


09238 


10100 


11009 


11968 


12979 16 


45 


07665 


08436 


09252 


10115 


11025 


11985 


12996 15 


46 


07677 


06449 


09266 


10130 


11041 


12001 


13013 14 


47 


07690 


084C3 


09280 


10144 


11056 


12018 


13031 


13 


48 


07702 


08476 


09294 


10159 


11072 


12034 


13048 


12 


49 


07715 


08489 


09308 


10174 


11087 


12051 


13065 


11 


50 


07727 


06503 


09323 


10189 


11103 


12067 


18083 


10 


51 


1.07740 


1.06616 


1.09387 


1.10204 


1.11119 


1.12064 


1.18100 


9 


52 


07752 


06529 


09351 


10218 


11134 


12100 


18117 


8 


53 


07765 


06542 


09365 


10233 


11160 


12117 


13185 


7 


54 


07778 


08556 


09379 


10248 


11166 


12133 


18152 


6 


55 


07790 


06569 


09393 


10263 


11181 


12150 


18170 


6 


56 


07803 


06582 


09407 


10278 


11197 


12166 


18187 


4 


67 


07816 


06596 


09421 


10293 


11213 


12188 


18205 


8 


58 


07828 


08609 


09436 


10308 


11229 


12199 


13222 


2 


59 


07841 


08623 


00449 


10323 


11244 


12216 


18240 


1 


60 


07868 


06636 


09464 


10338 


11260 


12288 


18257 





/ 


68« 


67«» 


66* 


650 


640 


680 


62- 









COSECANTS. 






' 1 



y Google 





NATURAL SECANTS AND COSECANTS 


267 


/ 






SECANTS 








! / 

1 


28" 


29» 


80« 


81* 


82" 


88« 


84" 





1.13257 


1.14385 


1.15470 


1.16663 


1.17918 


1.19236 


1. 200-29 


60 


1 


13275 


14354 


15489 


16684 


17939 


19859 


20C4: 


59 


3 


18292 


14878 


15509 


16704 


17961 


19281 


20669 


58 


8 


13310 


14391 


15528 


16725 


17988 


19804 


80693 


57 


4 


18827 


14409 


15548 


16745 


18004 


193 '7 


20717 


56 


6 


18345 


14428 


16567 


16766 


18025 


19349 


20740 


55 


6 


13362 


14446 


15587 


16786 


18047 


1937» 


20764 


54 


7 


13380 


14465 


15606 


16806 


1S068 


19394 


20788 


58 


8 


13398 


14488 


15626 


16827 


18090- 


19417 


20812 


52 


9 


13415 


14502 


15645 


16848 


18111 


19440 


80836 


51 


10 


13433 


14521 


15665 


16868 


18133 


19463 


20859 


50 


n 


1.13451 


1.14589 


1.15684 


1.16889 


1.18155 


1.19485 


1.20883 


49 


Vi 


18468 


14558 


15704 


16909 


18176 


19508 


80907 


48 


13 


18486 


14576 


15724 


16930 


18198 


18581 


80931 


47 


14 


18504 


14595 


15748 


16950 


18820 


19554 


20955 


46 


15 


18521 


14614 


15763 


16971 


18841 


19576 


20979 


45 


16 


18589 


14688 


15782 


16992 


18263 


19599 


81003 


44 


17 


13657 


14651 


15808 


17012 


18285 


19622 


81027 


43 


18 


13575 


14670 


15888 


17083 


18307 


19645 


81051 


42 


10 


18593 


14689 


15841 


17054 


18328 


19668 


21075 


41 


90 


13610 


14707 


15861 


17075 


18350 


19691 


21099 


40 


81 


1.18638 


1.14726 


1.15881 


1.17095 


1.18372 


1.19713 


1.21123 


89 


32 


18646 


14745 


15901 


17116 


18394 


19736 


21147 


38 


33 


18664 


14764 


15920 


17137 


18416 


19759 


21171 


87 


34 


18682 


14782 


15940 


17158 


18437 


19782 


81195 


86 


25 


18700 


14801 


15960 


17178 


18459 


19805 


81320 


85 


26 


18718 


14820 


15980 


17199 


18481 


19828 


81244 


84 


27 


18735 


14839 


16000 


17220 


18503 


19851 


81868 


83 


28 


13753 


14858 


16019 


17241 


18585 


19874 


81892 


82 


39 


18771 


14877 


16039 


17268 


18547 


19897 


81316 


81 


80 


18789 


14896 


16059 


17283 


18569 


19920 


81841 


80 


31 


1.18807 


1.14914 


1.16079 


1.17304 


1.18691 


\.19944 


1.81365 


89 


83 


13825 


14933 


16099 


17385 


18613 


19967 


21389 


88 


38 


18848 


14952 


16119 


17846 


18635 


19990 


81414 


87 


84 


13861 


14971 


16139 


17867 


18657 


80018 


31438 


26 


85 


18879 


14990 


16159 


17388 


18679 


20036 


81468 


85 


36 


18897 


15009 


16179 


17409 


18T01 


20059 


21487 


84 


87 


18916 


15088 


16199 


17480 


18783 


80083 


21511 


88 


38 


13984 


16047 


16819 


17451 


18745 


80106 


21535 


88 


89 


18952 


15066 


16839 


17478 


18767 


30120 


81560 


31 


40 


18970 


15085 


16859 


17493 


18790 


80153 


21584 


20 


41 


1.13968 


1.15106 


1.16279 


1.17514 


1.18812 


1.80176 


1.21609 


19 


43 


14006 


15184 


16299 


17535 


18834 


80190 


81638 


18 


43 


14034 


16143 


16319 


17556 


18856 


80883 


81668 


17 


44 


14048 


15168 


16339 


17577 


18878 


20846 


81688 


16 


45 


14061 


16181 


16359 


17698 


18901 


30869 


31707 


15 


46 


14079 


15200 


16380 


17680 


18928 


30298 


31731 


14 


47 


14097 


15819 


16400 


17641 


18945 


30316 


31756 


13 


48 


14115 


16289 


16420 


17668 


18967 


80339 


81781 


12 


49 


14184 


16258 


16440 


17683 


18990 


80363 


21805 


11 


50 


14153 


15877 


16460 


17704 


19012 


80886 


21830 


10 


51 


1.14170 


1.15896 


1.16481 


1.17786 


1.19034 


1.20410 


1.21855 


9 


52 


14188 


15315 


16501 


17747 


19057 


804as 


21879 


8 


53 


14807 


15335 


16681 


17768 


19079 


' 80457 


21904 


7 


64 


14225 


15854 


16541 


17790 


19108 


80480 


21029 


6 


65 


14243 


15878 


16568 


17811 


19184 


80504 


81958 


5 


56 


14862 


15398 


16588 


17888 


19146 


30587 


81978 


4 


67 


14880 


15418 


16608 


17854 


19169 


30661 


88003 


8 


68 


14299 


15481 


16688 


17875 


19191 


20575 


82088 


2 


69 


14817 


15451 


16643 


17896 


19814 


80598 


82053 


1 


60 


14835 


15470 


16663 


17918 


19836 


80688 


. 88077 





t 


ev 


60° 


69" 


68" 


67° 


66° 


So" 


/ 






COSECANTS. 







Digitized by VjOOQIC 



NATURAL SECANTS AND COSECANTS 



/ 


SECANTS. 


* 






860 


86° 


S?" 


88<> 


890 


iOo 


4P 







1.22077 


1.23607 


1.26214 


1.26902 


1.28G76 


1.30541 


1.32501 


60 




1 


22102 


23638 


26241 


26931 


28706 


30573 


32535 


60 




2 


22127 


23659 


25269 


26960 


28787 


80605 


32568 


58 




3 


22152 


28685 


25296 


26988 


28767 


80686 


32602 


57 




4 


22177 


23711 


25324 


27017 


28797 


30668 


82636 


66 




6 


22202 


23788 


25851 


27046 


28828 


30700 


82669 


66 




6 


22227 


23764 


25379 


27075 


28858 


30732 


32703 


64 




7 


22252 


23790 


25406 


27104" 


28889 


30764 


82737 


68 




8 


22277 


28816- 


26434 


27183 


2S919 


30796 


32770 


58 




9 


22802 


23843 


25462 


27162 


28950 


30829 


32804 


51 




10 


22327 


23869 


25489 


27191 


28980 


30861 


38838 


60 




11 


1.22352 


1.23896 


1.25517 


1.27221 


1.29011 


1.30898 


1.88878 


49 




u 


22377 


23922 


25545 


87250 


29042 


30925 


82905 


48 




13 


22402 


23948 


25572 


27279 


29072 


80957 


82939 


47 




14 


22428 


23975 


25600 


2^08 


29103 


80989 


32973 


46 




15 


22453 


24001 


25628 


27887 


29188 


81022 


83007 


46 




16 


22478 


24028 


25656 


27866 


89164 


31054 


38041 


44 




17 


22503 


24054 


25683 


27396 


29195 


81086 


88075 


48 




18 


22528 


24081 


26711 


87425 


29226 


31119 


88109 


42 




19 


22554 


24107 


25739 


27454 


29256 


31161 


88143 


41 




20 


22579 


24134 


25767 


27483 


29287 


31183 


38177 


40 




81 


1.22604 


1.24160 


1.26796 


1. '27513 


1.29318 


1.81216 


1.38211 


39 




22 


22629 


24187 


25823 


27542 


29349 


31248 


83246 


38 




23 


22666 


24213 


25861 


27572 


29380 


81281 


38279 


37 




24 


22680 


24240 


85879 


27601 


29411 


31813 


33314 


86 




25 


22706 


24267 


25907 


27630 


29442 


31846 


33348 


85 




26 


22731 


24298 


25936 


27660 


29473 


81878 


33382 


84 




27 


22756 


24320 


85963 


27689 


29504 


81411 


33416 


83 




28 


22782 


24347 


25991 


27719 


29535 


31443 


83461 


32 




29 


22807 


24373 


26019 


27748 


29566 


31476 


83486 


81 




80 


22833 


24400 


26047 


27778 


29597 


81509 


83519 


80 




31 


1.22858 


1.24427 


1.26075 


1.27807 


1.29628 


1.31541 


1.83554 


29 




32 


22884 


24454 


26104 


27837 


29659 


31f)74 


83588 


28 




33 


22909 


24481 


26132 


27867 


29690 


31607 


38628 


87 




34 


22935 


24508 


86160 


27896 


29721 


81640 


33657 


96 




85 


22960 


24534 


26188 


27926 


29752 


81672 


88691 


85 




36 


22986 


24561 


26216 


27956 


29784 


81706 


83726 


24 




37 


23012 


24588 


26245 


27985 


29815 


31788 


&S760 


88 




S8 


28087 


24615 


26273 


28015 


29846 


81771 


83795 


22 




39 


23063 


24642 


26301 


28045 


29877 


81804 


33830 


81 




40 


23089 


24669 


26330 


28075 


29909 


31837 


33864 


20 




41 


1.28114 


1.24696 


1.26358 


1.28105 


1.29940 


1.31870 


1.33899 


19 




42 


28140 


24723 


26887 


28134 


29971 


31903 


88934 


18 




43 


23166 


24750 


26415 


28164 


30003 


31936 


83968 


17 




44 


23192 


24777 


26443 


28194 


30084 


81969 


84008 


16 




45 


33217 


24804 


26472 


28224 


30066 


82002 


84088 


16 




46 


23243 


24832 


26500 


28254 


30097 


88086 


8407S 


14 




47 


23269 


21859 


26529 


28284 


30129 


38068 


84106 


18 




48 


23295 


24886 


26557 


28814 


30160 


88101 


84142 


12 




49 


23821 


24913 


26586 


28344 


30192 


32184 


84177 


11 




60 


23347 


24940 


26615 


28374 


30223 


82168 


84818 


10 




61 


1.23878 


1.24967 


1.26643 


1.28404 


1.80255 


1.82801 


1.34847 


9 




62 


23899 


24995 


26672 


28434 


80287 


32284 


84882 


8 


68 


23424 


25022 


26701 


28464 


80318 


32267 


84817 


7 1 


64 


23460 


25049 


26729 


28495 


80850 


82801 


84362 


8 1 


66 


23476 


25077 


26758 


28625 


80382 


82334 


84887 


6 


66 


28502 


25104 


26787 


2a')55 


80413 


88868 


84423 


4 


67 


28629 


25181 


26815 


28685 


80445 


88401 


84458 


8 


68 


23666 


26169 


26844 


28615 


80477 


32484 


84498 


2 


69 


28681 


25186 


26873 


28646 


80509 


38468 


84629 


1 


60 


28607 


26214 


26902 


28676 


80641 


88601 


84668 







t 


640 


680 


62» 


6P 


60<» 


49<» 


480 


/ 


COSECANTS. 1 





Digitized by VjOOQIC 





NATURAL 


SECANTS AND COSECANTS 


269 


# 


SECANTS. 


t 


48» 


48« 


44» 


46» 


46» 


47* 


48» 





1.34568 


1.36733 


1.89016 


1.41481 


1.43956 


1.46688 


1.49448 


60 


1 


34599 


86770 


39066 


41463 


43999 


46674 


49496 


69 


% 


34634 


86807 


89095 


41604 


44043 


46719 


49644 


68 


8 


34669 


36844 


89134 


41545 


44086 


46766 


49698 


67 


4 


84704 


86881 


39178 


41586 


44129 


46811 


49641 


66 


6 


34740 


86919 


39318 


41637 


44173 


46867 


49690 


56 





34776 


86956 


89361 


41669 


44817 


46908 


49738 


64 


7 


84811 


86993 


39391 


41710 


44260 


46949 


49787 


68 


8 


34846 


87080 


39330 


41752 


44304 


46996 


49885 


58 


9 


34883 


87068 


89369 


41798 


44847 


47041 


49884 


61 


10 


34917 


87105 


39409 


41835 


44391 


47087 


49988 


50 


11 


1.34958 


1.87143 


1.39448 


1.41876 


1.44435 


1.47184 


1.49961 


49 


12 


34988 


87180 


39487 


41918 


44479 


47180 


60060 


48 


13 


35034 


87318 


39637 


41959 


44528 


47286 


60079 


47 


14 


35060 


87255 


39566 


43001 


44667 


47878 


50188 


46 


15 


35095 


87393 


89606 


42042 


44610 


47819 


60177 


46 


16 


35181 


37330 


39646 


43064 


44654 


47866 


60886 


44 


17 


35167 


37368 


89685 


48126 


44698 


47411 


60876 


48 


18 


85303 


87406 


89725 


43168 


44743 


47468 


60894 


48 


10 


86338 


87448 


89764 


42210 


44787 


47604 


60878 


41 


90 


85274 


37481 


89804 


43251 


44881 


47651 


60483 


40 


21 


1 .35310 


1.87519 


1.89844 


1.42898 


1.44875 


1.47698 


1.60471 


89 


23 


35846 


37556 


89884 


43835 


44919 


47644 


60681 


88 


23 


.35388 


87594 


89984 


42877 


44968 


47691 


60670 


87 


24 


35418 


37633 


89963 


48419 


46007 


47788 


60619 


86 


25 


35454 


37670 


40003 


43461 


45053 


47784 


60669 


86 


26 


85490 


37708 


40043 


48503 


45096 


47a31 


60718 


84 


27 


85536 


87746 


40083 


42545 


45141 


47878 


60767 


88 


28 


35563 


87784 


40183 


42587 


45185 


47935 


60817 


83 


29 


85598 


87838 


40163 


42630 


45339 


47973 


50866 


81 


80 


85684 


87860 


40803 


42672 


45374 


46019 


60916 


80 


81 


1.35670 


1.37898 


1.40843 


1.43714 


1.46319 


1.48066 


1.60966 


89 


32 


3.5707 


87986 


40883 


43766 


46863 


48113 


51016 


38 


33 


86743 


87974 


40884 


43799 


46408 


48160 


51066 


37 


34 


85779 


88013 


40864 


48841 


46463 


48807 


61115 


86 


35 


86816 


88061 


40404 


42888 


46497 


48364 


61165 


86 


86 


85868 


88069 


40444 


43926 


46548 


48801 


51816 


84 


87 


86888 


88187 


40486 


43968 


45687 


48849 


61366 


83 


88 


86924 


88166 


40685 


48011 


45681 


48396 


61314 


88 


89 


35961 


88804 


40666 


43063 


45676 


48443 


61364 


31 


40 


86997 


88848 


40606 


48096 


45781 


48491 


61415 


80 


41 


1.36084 


1.88880 


1.40646 


1.43139 


1.46766 


.48638 


1.61466 


19 


42 


86070 


88819 


40687 


43181 


45811 


48586 


61615 


18 


43 


86107 


88857 


40727 


48834 


46866 


48688 


61665 


17 


44 


86148 


88896 


40768 


48867 


46901 


48681 


61615 


16 


45 


36180 


88434 


40808 


48810 


45946 


48738 


61665 


15 


46 


36317 


38478 


40849 


48353 


45993 


48776 


61716 


14 


47 


86368 


88618 


40890 


43395 


46067 


48834 


61766 


13 


48 


86390 


88660 


40930 


43438 


46063 


48871 


61817 


13 


49 


86837 


88689 


40971 


43181 


46137 


48919 


61867 


11 


50 


86868 


38688 


41018 


48584 


46176 


48967 


51918 


10 


51 


1.36400 


1.88666 


1.41068 


1.48567 


1.46218 


3.49016 


1.51968 


9 


53 


36437 


38706 


41093 


48610 


46863 


49063 


63019 


8 


53 


36474 


88744 


41134 


43653 


46809 


49111 


68069 


7 


54 


86511 


88783 


41175 


43696 


46354 


49159 


68180 


6 


55 


86548 


38882 


41916 


43789 


46400 


49207 


68171 


6 


56 


86685 


88860 


41267 


48783 


46445 


49255 


68388 


4 


57 


36633 


88899 


41398 


43836 


46491 


49303 


68373 


8 


58 


36659 


88988 


41339 


43869 


46637 


49351. 


58338 


2 


59 


86696 


88977 


41380 


48913 


46583 


49899 


68874 


1 


60 


36738 


39016 


41431 


43956 


46638 


49448 


68435 





/ 


47« 


4e«» 


460 


44* 


48° 


420 


410 


/ 


COSECANTS. 



Digitized by VjOOQIC 



270 


NATURAL 


SECANTS AND COSECAN'1'8 




/ 


SECANTS. 


/ 


4»» 


60» 


61' 


62» 


68» 


64«» 


65° 





1.52425 


1.55572 


1.68902 


1.62437 


1.66164 


1.70130 


1.74345 


60 


1 


53476 


56626 


68959 


62487 


66228 


70196 


74417 


59 


s 


52527 


65680 


59016 


62548 


66393 


70267 


74490 


68 


3 


52579 


55784 


69073 


62609 


66857 


70335 


74562 


67 


4 


52680 


V 55789 


69180 


62669 


66421 


70403 


74685 


66 


5 


52681 


56843 


59188 


62730 


66486 


70472 


74708 


55 


6 


53782 


65897 


59245 


63791 


66550 


70540 


74781 


64 


7 


62784 


55951 


69302 


62852 


66615 


70609 


74861 


58 


8 


52835 


56005 


59360 


62913 


66679 


70677 


74927 


53 


9 


52886 


56060 


59418 


62974 


66744 


70746 


75000 


51 


10 


52988 


56114 


59476 


68035 


66809 


70815 


75073 


50 


11 


1.52969 


1.56169 


1.59533 


1.68096 


1.66873 


1.70884 


1.75146 


49 . 


U 


68041 


56228 


59590 


63157 


66938 


70953 


75219 


48 


13 


58093 


66278 


69648 


63218 


67003 


71022 


75293 


47 


11 


58144 


66332 


59706 


63279 


67068 


71091 


:6366 


46 


16 


68196 


56387 


69764 


63341 


67133 


71160 


75440 


45 


16 


68247 


66442 


69822 


68402 


67199 


71229 


75513 


44 


17 


58299 


66497 


69680 


63464 


673&4 


71298 


75587 


43 


18 


58851 


56551 


59938 


63525 


67329 


71868 


75661 


42 


19 


58408 


66606 


59996 


63687 


67394 


71487 


76734 


41 


20 


58456 


66661 


60054 


63648 


67460 


71506 


75808 


40 


21 


1.58507 


1.56716 


1.60113 


1.63710 


1.67525 


1.71576 


1.75882 


39 


22 


58559 


66771 


60171 


63772 


67591 


71646 


75956 


88 


28 


58611 


66826 


60229 


63834 


67656 


71715 


76031 


87 


24 


58663 


56881 


60287 


63895 


67722 


71785 


76105 


36 


25 


53715 


66937 


60346 


63957 


67788 


71855 


76179 


35 


26 


58768 


56992 


60404 


&1019 


67853 


71925 


76253 


34 , 


27 


58820 


57047 


60463 


64081 


67919 


71996 


76828 


88 


28 


53872 


57103 


60521 


64144 


67985 


72065 


76402 


33 


29 


53924 


57158 


60580 


64206 


6S051 


72186 


76477 


31 


ao 


58977 


57218 


60639 


64268 


68117 


72205 


76552 


30 


81 


1.54029 


1.57269 


1.60698 


1.64330 


1.68183 


1.72275 


1.76C26 


29 : 


32 


54082 


67321 


60756 


64:^93 


6825a 


72846 


76701 


28 


88 


64134 


67880 


60815 


64455 


68316 


72416 


76776 


97 


84 


64187 


57436 


60874 


64518 


68882 


72487 


76851 


26 


85 


51240 


57491 


60933 


64580 


68449 


72567 


76926 


25 


36 


54292 


67547 


60992 


64643 


68515 


72688 


77001 


24 


87 


54345 


57608 


61051 


64705 


68582 


72696 


77077 


28 


38 


54398 


57669 


61111 


64768 


68648 


72769 


77162 


28 


89 


54451 


57715 


61170 


648:S1 


68715 


72840 


77337 


21 


40 


54504 


57771 


61229 


64894 


687&^ 


72911 


77303 


30 


41 


1.54557 


1.57827 


1.61288 


1.64957 


1.68848 


1.72982 


1.77378 


19 


42 


54610 


57883 


61348 


66020 


68915 


73058 


77454 


18 


48 


54668 


57989 


61407 


65068 


68962 


78134 


77580 


17 


44 


54716 


. 67995 


61467 


65146 


69049 


':8196 


77606 


16 


45 


64769 


58051 


61526 


65209 


69116 


'58267 


77881 


15 


46 


54822 


58108 


61586 


65272 


69183 


73388 


77767 


14 


47 


64876 


58164 


61646 


65336 


69250 


':8409 


77888 


13 


48 


54929 


58221 


61705 


66399 


69818 


78481 


77910 


13 


49 


54962 


58277 


61765 


65462 


69385 


78652 


77986 


11 


50 


55086 


68338 


61825 


65536 


69452 


78624 


78062 


10 


51 


1.65069 


1.66890 


1.61885 


1.65589 


1.69520 


1.78696 


1.78188 


9 


62 


56148 


68447 


61946 


65658 


69687 


78768 


78215 


8 


58 


65196 


68608 


62005 


65717 


69655 


78840 


78291 


7 


54 


65250 


68660 


62065 


66780 


69733 


73911 


78868 


6 


55 


55808 


68617 


62125 


65844 


69790 


78983 


78445 


5 


56 


^^867 


58674 


62186 


65908 


69858 


74066 


78521 


4 


57 


55411 


58781 


62246 


65972 


69926 


74128 


78598 


3 


58 


56465 


68788 


62306 


66086 


AOQOd 
UWvv 


74200 


78675 


3 


59 


56518 


68846 


62366 


66100 


70062 


74272 


78752 


1 


60 


66572 


58902 


62427 


66164 


70130 


74845 


78829 





L , 


40* 


8»« 


88«» 


87» 


86» 


86« 


84* 


r 


COSEGAKTBL 



y Google 





NATURAL 


SECANTS AND COSECANTS 


271 


/ 



SECANTS. 


/ 


56'> 


s?- 


68» 


69» 


60«» 


61» 


620 




1.78829 


1.83608 


1.88708 


1.94160 


2.00000 


2.06267 


2.13005 


00 


1 


T89C6 


83690 


88796 


94254 


00101 


06375 


18122 


59 


2 


78984 


83773 


oo8o4 


94349 


00002 


06483 


13239 


58 


3 


79061 


83855 


88972 


94443 


00303 


06692 


18356 


57 


4 


79138 


88938 


89060 


94637 


00404 


06701 


18473 


66 


5 


79216 


84020 


89148 


94632 


00505 


06809 


18590 


56 


6 


79298 


84103 


89237 


94726 


00607 


06918 


18707 


64 


7 


79371 


84186 


89325 


94821 


00708 


07027 


13825 


53 


8 


79449 


84269 


89414 


94916 


00810 


07137 


18942 


52 


9 


79627 


84352 


89503 


96011 


00912 


07246 


14060 


51 


10 


79604 


84485 


89591 


96106 


01014 


07856 


14178 


50 


11 


1.79682 


1.84518 


1.89680 


1.96201 


2.01116 


2.07465 


2.14296 


49 


U 


79761 


84601 


89769 


96296 


01218 


07575 


14414 


48 


13 


79639 


84685 


89858 


96392 


01320 


07686 


14683 


47 


14 


79917 


84768 


89948 


95487 


01422 


07796 


14651 


46 


15 


79996 


84852 


90087 


95688 


01526 


07905 


14770 


45 


16 


80074 


84935 


90126 


95678 


01628 


08016 


14889 


44 


17 


80152 


85019 


90216 


95774 


01730 


08126 


16008 


43 


18 


80381 


85103 


90305 


95870 


01883 


08286 


15127 


42 


19 


80809 


85187 


90895 


95966 


01986 


08847 


16246 


41 


20 


80388 


85271 


90485 


96062 


02039 


08468 


16866 


40 


21 


i.80467 


1.85356 


1.90575 


1.96158 


2.02143 


2 08569 


2.15486 


39 


22 


80546 


85439 


90665 


96255 


02246 


08680 


16605 


38 


23 


80625 


85523 


90755 


96351 


02349 


08791 


16726 


37 


24 


80704 


85G08 


90845 


96448 


02458 


08908 


16846 


36 


25 


8078S 


85692 


90935 


96544 


02667 


09014 


15966 


36 


26 


80862 


85777 


91026 


96641 


02661 


09126 


16066 


34 


27 


80942 


85861 


91116 


96738 


02765 


09238 


16806 


83 


28 


81021 


85946 


9120r 


96835 


02869 


09360 


16326 


32 


29 


81101 


86031 


91297 


96932 


02978 


09462 


16447 


81 


80 


81180 


86116 


91388 


97029 


03077 


00574 


16568 


30 


81 


1.81260 


1.86S01 


1.91479 


1.97127 


2.08182 


2.09686 


2.16689 


29 


82 


81340 


86286 


91570 


97224 


03286 


09799 


16810 


28 


33 


81419 


86871 


91661 


97822 


03891 


09911 


16982 


27 


34 


81499 


86457 


91752 


97420 


03496 


10024 


17063 


26 


85 


81579 


86542 


91844 


97517 


03601 


10187 


17175 


25 


86 


81659 


86627 


91935 


97615 


06706 


10250 


17297 


24 


87 


81740 


86718 


92027 


97713 


08811 


10863 


17419 


28 


88 


81820 


86799 


92118 


97811 


03916 


10477 


17541 


22 


89 


81900 


86885 


92210 


97910 


04022 


10690 


17663 


21 


40 


81981 


86990 


92302 


98006 


04128 


10704 


17786 


20 


41 


1.82061 


1.870ii« 


1.92394 


1.98107 


2.04288 


2.10817 


2.17909 


19 


42 


82142 


87142 


92486 


96205 


04839 


10931 


18031 


18 


48 


82222 


87229 


92578 


98304 


04445 


11046 


18154 


17 


44 


82303 


87316 


. 92670 


98403 


04651 


11160 


18877 


16 


45 


82884 


87401 


92762 


98502 


04658 


11274 


18401 


16 


46 


82465 


87488 


92855 


98601 


04764 


11388 


18524 


14 


47 


82546 


87574 


92947 


98700 


04870 


11608 


18648 


18 


48 


82627 


87661 


98040 


98799 


04977 


11617 


18772 


12 


49 


82709 


87748 


93133 


98699 


05084 


11782 


18896 


11 


50 


88790 


87834 


93226 


98998 


05191 


11847 


19019 


10 


61 


1.8S871 


1.87921 


1.98819 


1.99098 


2.05298 


2.11963 


2.19144 


9 


52 


82953 


88008 


93412 


99198 


05406 


12078 


19268 


8 


58 


83034 


88095 


93505 


99298 


05612 


12193 


19898 


7 


54 


83116 


88183 


93598 


99398 


05619 


12809 


19517 


6 


56 


83198 


88270 


93692 


99498 


05727 


12425 


19642 


5 


66 


83280 


88857 


93785 


99598 


06885 


12540 


19767 


4 


57 


88862 


88445 


98879 


99698 


06942 


12667 


19692 


8 


58 


83444 


88582 


98978 


09799 


.06060 


12773 


20018 


2 


59 


83526 


88620 


94066 


99899 


06158 


12889 


20148 


1 


60 


83608 


88708 


04160 


2.00000 


06267 


13005 


20269 





/ 


880 


82« 


81» 


«0«» 


290 


28° 


270 


/ 


COSECANTS. 1 





Digitized by VjOOQIC 



272 


NATURAL 


SECANTS AND C 


OSECA 


NTS 




/ 


SECANTS. 


/ 





68» 


64«» 


efio 


66° 


67« 


68«» 


69" 


2.20269 


2.28117 


8.36620 


2.45859 


2.55930 


2.66947 


2.79043 


60 


1 


20895 


28*^^ 


36768 


460i0 


56106 


67139 


79254 


59 


2 


20521 


28390 


8691G 


46181 


56282 


67332 


79466 


58 


3 


20647 


28526 


87064 


4G342 


56458 


67525 


79679 


57 


4 


20773 


28668 


87212 


46504 


56634 


67718 


79891 


56 


6 


20900 


28800 


8';361 


46665 


5681 1 


67911 


80104 


56 


6 


21026 


28937 


87509 


46827 


56988 


68105 


80318 


54 


7 


21153 


29074 


87658 


46989 


57165 


68299 


80531 


53 


8 


21280 


29211 


87808 


47152 


57342 


68494 


80746 


52 





21407 


29349 


87957 


47314 


5T5-20 


68689 


80960 


61 


10 


21535 


29487 


88107 


47477 


57698 


68884 


81175 


50 


11 


2.21662 


2.29625 


3.88266 


2.47640 


2.67876 


2.69079 


3.81890 


49 


u 


21790 


29763 


88406 


47804 


58054 


60275 


81605 


48 


13 


21918 


29901 


88566 


47967 


58333 


69471 


81821 


47 


14 


2*i045 


30040 


88767 


48131 


58412 


69667 


82037 


46 


15 


22174 


30179 


88857 


48295 


58591 


69864 


82254 


45 


16 


22302 


30318 


39008 


48459 


58771 


70061 


82471 


44 


17 


22430 


30467 


89159 


48624 


58950 


70258 


82688 


43 


18 


22559 


30596 


89311 


48789 


69130 


70455 


82906 


48 


19 


22688 


30785 


39462 


48954 


59311 


70653 


83124 


41 


20 


22817 


30875 


39614 


49119 


59491 


70851 


83342 


40 


21 


2.22946 


2.81015 


2.39766 


2.49284 


2.59672 


2.71050 


8.83661 


39 


22 


23075 


31155 


39918 


49450 


59853 


71249 


83780 


38 


23 


23205 


31295 


40070 


49616 


60086 


71448 


83999 


87 


24 


23834 


31436 


40222 


49^82 


60217 


71647 


84219 


36 


25 


23464 


31578 


40375 


49948 


60399 


71847 


84439 


35 


26 


23591 


31717 


40528 


50115 


60581 


72047 


84659 


84 


27 


237.>4 


31868 


40681 


50282 


60763 


72247 


84880 


33 


28 


28855 


81999 


40685 


50449 


60946 


72448 


85102 


32 


29 


23985 


82140 


40988 


50617 


61129 


7TW49 


85323 


81 


30 


24116 


32282 


41142 


50784 


61318 


72850 


85545 


80 


31 


2.24247 


2.32424 


2 41296 


2.50952 


2.61496 


2.7:3052 


2.85767 


89 


82 


24378 


:^66 


41450 


51120 


61680 


73254 


85990 


88 


JJ3 


24509 


32708 


41605 


51289 


61864 


73456 


86213 


27 


34 


24640 


32850 


41760 


51457 


62049 


78659 


86437 


86 


36 


24772 


32993 


41914 


51626 


62234 


73863 


86661 


85 


36 


24903 


83185 


42W0 


51795 


62419 


74065 


86885 


84 


87 


25035 


832T8 


42225 


51965 


62604 


74269 


87109 


83 


38 


25167 


3342i 


42880 


52134 


62790 


74478 


87884 


83 


39 


25300 


83565 


42536 


52304 


62976 


74677 


87560 


81 


40 


3548;2 


33708 


42692 


62474 


63162 


74881 


87785 


80 


41 


2.25565 


2.33852 


2.42848 


2.52645 


2.63348 


2.75086 


8.88011 


19 


42 


2.5697 


33996 


43005 


52815 


63585 


75292 


88288 


18 


43 


25880 


34140 


43162 


52986 


68722 


75497 


88465 


17 


44 


25963 


31284 


43318 


58157 


68909 


75703 


88692 


16 


45 


26097 


34429 


43476 


53329 


64097 


75909 


88920 


15 


46 


26230 


34573 


436:« 


53500 


64285 


76116 


89148 


14 


47 


26364 


34718 


43790 


53672 


64473 


76323 


89376 


13 


48 


26498 


34863 


43948 


53845 


64662 


76530 


89605 


12 


49 


26632 


35009 


44106 


54017 


64851 


76737 


89884 


11 


50 


26766 


35154 


44264 


54190 


65040 


76945 


90063 


10 


51 


2.26900 


2.35300 


2.44423 


2 54368 


2.65229 


2.77154 


2.90293 


9 


52 


27035 


35446 


44582 


54536 


65419 


77362 


90524 


8 


53 


27169 


3.')59«> 


41741 


54709 


65609 


77571 


90754 


7 


64 


27304 


35738 


44900 


54883 


65799 


77780 


90986 


6 


65 


27439 


35885 


45059 


65057 


65989 


77990 


91817 


5 


56 


27574 


36031 


45219 


55231 


66180 


78200 


91449 


4 


57 


27710 


36178 


45378 


55405 


66371 


78410 


91681 


8 


58 


27845 


36325 


45539 


M580 


66563 


78621 


91914 


8 


59 


27981 


36473 


45699 


65755 


66755 


78832 


92147 


1 


60 


28117 


86620 


45859 


55980 


66947 


79048 


98380 





' 


26° 


2o«» 


240 


28«» 


220 


21° 


20° 


- 






COSECANTS. 







y Google 





NATURAL 


SECANTS AND COSECANTS . 


273 


1 / 


SECANTS. 


'/ 


i 


70« 


710 


72«» 


780 


74° 


76« 


76«» 





2.92380 


3.07155 


8.83607 


8.42030 


3.62796 


8.86370 


4.13357 


60 


1 


92614 


07415 


23897 


42356 


63164 


86790 


13839 


69 


2 


92849 


07675 


24187 


42683 


63533 


87311 


14323 


58 


8 


93083 


07936 


24478 


43010 


63903 


87638 


14809 


57 


4 


93818 


08197 


24770 


43337 


64274 


88056 


15395 


56 


6 


93554 


08459 


25062 


43666 


64645 


88479 


15782 


66 


« 


93790 


08721 


25355 


43995 


65018 


88904 


16271 


54 


7 


94026 


08983 


25648 


44324 


65391 


893;)0 


16761 


58 


8 


94263 


09246 


25942 


44655 


65765 


89756 


17253 


53 


9 


94500 


09510 


26237 


44986 


66140 


90184 


17744 


51 


10 


94737 


09774 


26531 


45317 


66515 


90G13 


18338 


50 


11 


2.94975 


3.10038 


8.26837 


3.45650 


8.66893 


8.91043 


4.18738 


49 


12 


95213 


10303 


27138 


45983 


67369 


91473 


19228 


48 


13 


95452 


10568 


27420 


46316 


67647 


91904 


19725 


47 


14 


95691 


10834 


27717 


46651 


68025 


93337 


20234 


46 


15 


95931 


11101 


28015 


46986 


68405 


93770 


20r33 


45 


16 


96171 


11867 


28813 


47331 


68785 


93304 


21234 


44 


17 


96411 


11G35 


28612 


47658 


69167 


93640 


21726 


43 


18 


96652 


11903 


28912 


47995 


69549 


94076 


23239 


43 


19 


96H93 


13171 


29313 


48333 


69931 


94514 


22734 


41 


20 


97185 


12440 


29512 


48671 


70315 


94953 


28339 


40 


21 


2.97377 


8.12709 


8.29814 


8.49010 


8.70700 


8.95393 


4.23746 


89 


22 


97619 


12979 


30115 


49350 


71085 


95833 


24255 


88 


23 


97862 


13249 


30418 


49691 


71471 


96374 


24764 


87 


24 


98106 


13520 


30731 


50032 


71858 


96716 


25275 


86 


25 


98849 


13791 


31024 


50374 


72246 


97160 


25787 


86 


26 


98594 


14063 


31328 


50716 


72635 


97604 


26800 


34 


27 


98838 


14335 


81633 


51060 


73024 


98050 


26814 


83 


28 


99083 


14608 


81939 


51404 


73414 


98497 


27880 


82 


29 


99829 


14881 


33244 


51748 


73806 


98944 


27847 


81 


80 


99574 


15155 


82551 


62094 


74198 


99393 


28366 


80 


81 


2.99821 


8.15429 


8.83858 


3.52440 


8.74591 


8.99843 


4.28885 


29 


82 


8.00067 


15704 


33166 


52787 


74984 


4.00293 


29406 


28 


83 


00315 


15979 


33474 


58134 


75379 


00745 


29929 


27 


34 


00562 


16255 


83788 


63482 


75775 


01198 


30462 


26 


85 


00810 


16531 


34092 


68831 


76171 


01652 


30977 


25 


86 


01059 


16808 


84403 


64181 


76568 


02107 


81603 


24 


a*/ 


01308 


17085 


34713 


54581 


76966 


02563 


82081 


23 


88 


01557 


17368 


35025 


54883 


77365 


03020 


82560 


22 


39 


01807 


17641 


35336 


55235 


77765 


03479 


38090 


21 


40 


02057 


17920 


85649 


55587 


78166 


08938 


33622 


20 


41 


8.02306 


8.18199 


8.85963 


8.55940 


8.7&568 


4.04396 


4.84154 


19 


42 


02.%9 


18479 


36276 


56294 


78970 


04860 


84689 


18 


43 


02810 


18759 


36590 


56649 


79874 


05322 


86224 


17 


44 


03062 


19040 


86905 


67005 


79776 


05786 


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16 


45 


08315 


19322 


87221 


57861 


60183 


06351 


86299 


16 


46 


03568 


19604 


87537 


57718 


80569 


06717 


86839 


14 


47 


08821 


19886 


37854 


58076 


80996 


07184 


87380 


18 


48 


04075 


20169 


38171 


58484 


81404 


07652 


87928 


12 


49 


04829 


20458 


88489 


58794 


81813 


08121 


38466 


11 


50 


04584 


20787 


8S808 


59154 


82223 


08591 


39012 


10 


51 


3.04889 


8.21021 


8.39128 


3.59514 


8.82688 


4.09063 


4.39558 


9 


53 


05094 


21306 


89448 


59676 


88046 


09535 


40106 


8 


63 


05850 


21592 


89768 


60288 


88457 


10009 


40656 


7 


54 


0560r 


21878 


40089 


60601 


88871 


10484 


41206 


6 


55 


05864 


22165 


40411 


60965 


84286 


10960 


41769 


6 


56 


06131 


22452 


40734 


61880 


61700 


11437 


42812 


4 


67 


06879 


22740 


41057 


61695 


85116 


11915 


42867 


8 


58 


06637 


28028 


41881 


62061 


86638 


12894 


48434 


S 


59 


06896 


23817 


41705 


62428 


85961 


12875 


48983 


1 


60 


07155 


28607 


42080 


62796 


86870 


18357 


44541 





t 


19» 


18» 


!?• 


160 


16» 


14» 


18° 


/ 


COSECANTS. 



Digitized by VjOOQIC 



274 NATURAL SECANTS AND COSECANTS. 



/ 


SECANTS. 


/ 


77* 


78« 


790 


80» 


81° 


Sfi*- 


83«» 





4.44541 


4.80973 


6.24064 


5.76877 


6.39246 


7.18530 


8.20551 


60 


1 


45102 


81688 


24870 


76829 


40422 


£0020 


22500 


69 


2 


45664 


8-2294 


26656 


77784 


41602 


21517 


24457 


68 


8 


46228 


82956 


26448 


78742 


42787 


23019 


26425 


67 


4 


46796 


83621 


27241 


79703 


43977 


24529 


28402 


56 


5 


47360 


84288 


28036 


80667 


45171 


26044 


30388 


65 


6 


47928 


64956 


28833 


81635 


46369 


27566 


82384 


54 


7 


48498 


86627 


29634 


82606 


47572 


29095 


84390 


63 


8 


49069 


86299 


30436 


83681 


48779 


30630 


36405 


62 


9 


49642 


86978 


31241 


84558 


49991 


32171 


88431 


51 


10 


50216 


87649 


32049 


85539 


51208 


33719 


40466 


50 


11 


4.50791 


4.88827 


6.32859 


5.86524 


6.52429 


7.S62T4 


8.42511 


49 


12 


{^868 


89007 


33071 


87511* 


5:3655 


36836 


44566 


48 


13 


51947 


89689 


34486 


88502 


64886 


38403 


46632 


47 


14 


52527 


90373 


35304 


89497 


56121 


89978 


48707 


46 


15 


58109 


91058 


36124 


90496 


57361 


41560 


50793 


45 


16 


53692 


91746 


36947 


91496 


'58606 


43148 


52889 


44 


17 


54277 


92436 


37772 


92501 


59a55 


44743 


54996 


43 


18 


64863 


98128 


38600 


98509 


61110 


46846 


67113 


4.' 


19 


55451 


93821 


39430 


94521 


62m9 


47955 


59241 


41 


SO 


56041 


94517 


402(53 


95586 


63033 


49571 


61379 


40 


21 


4.56632 


4.95215 


6.41099 


5.9C655 


6.64902 


7.51194 


6.63528 


39 


22 


57224 


95914 


41937 


97577 


66176 


62885 


65688 


88 


28 


57819 


96616 


42778 


98603 


67454 


64462 


67859 


87 


24 


58414 


97320 


436-->2 


93683 


68738 


66107 


70041 


86 


25 


50012 


98026 


44466 


6.00666 


70027 


57759 


72234 


85 


26 


59611 


98733 


45317 


01703 


71321 


59418 


74438 


84 


27 


60211 


99443 


46169 


02748 


7-2620 


61085 


76658 


33 


28 


60813 


5.00156 


47023 


03787 


73924 


62759 


78880 


82 


29 


61417 


00869 


47881 


04834 


75233 


64441 


81118 


31 


30 


6J023 


01585 


48740 


05886 


76547 


66130 


83367 


30 


31 


4 62630 


5.02303 


5.49603 


6.06941 


6.77866 


7.67826 


8.8.5628 


29 


32 


632:« 


03024 


50468 


08000 


79191 


69530 


87901 


28 


as 


6:«49 


03746 


51337 


09062 


80521 


71242 


90186 


27 


34 


64461 


04471 


52208 


10129 


81856 


72962 


92482 


26 


35 


65074 


05197 


53081 


11199 


83196 


74689 


94791 


25 


36 


65690 


05926 


63958 


12273 


84542 


76424 


97111 


24 


37 


66307 


06657 


54837 


13350 


85893 


78167 


99444 


23 


38 


66925 


0739C 


55720 


14432 


87250 


79918 


9.U1788 


22 


39 


67546 


08125 


56605 


15517 


88612 


81677 


04146 


21 


40 


68167 


08863 


57493 


16607 


89979 


83448 


06515 


20 


41 


4.68791 


5.09602 


5.58388 


6.17700 


6.91352 


7.85218 


9.08897 


19 


43 


69417 


10344 


59277 


18797 


92731 


87001 


11292 


18 


43 


T0044 


11088 


60174 


19898 


94115 


88792 


13699 


17 


44 


70673 


11835 


61073 


21004 


95605 


90692 


16120 


16 


45 


71303 


1258? 


61976 


22113 


96900 


92400 


18558 


15 


46 


71935 


13334 


62881 


23226 


98301 


94216 


20999 


14 


47 


72569 


14087 


63790 


24343 


99708 


96040 


23459 


13 


48 


73205 


14842 


64701 


25464 


7.011.'20 


97878 


26931 


12 


49 


73843 


15599 


65616 


26590 


025:i8 


99714 


28417 


11 


50 


74482 


16359 


66583 


27719 


03962 


8.01566 


30917 


10 


51 


4.75128 


5.17121 


5.67454 


6.28853 


7.05392 


8.08428 


9.88480 


9 


52 


75766 


17886 


68377 


29991 


06828 


05291 


85957 


8 


63 


76411 


18652 


69304 


81133 


08269 


07167 


38497 


7 


54 


77057 


19421 


70234 




09717 


09052 


41052 


6 


55 


77705 


20193 


71166 


83429 


11171 


10946 


43620 


6 


56 


7RS55 


20906 


72102 


84584 


12630 


12849 


46203 


4 


57 


79007 


21742 


73041 


35743 


14096 


14760 


48800 


3 


58 


79661 


22521 


78983 


36906 


15568 


16681 


51411 


2 


59 


80316 


23301 


74929 


36078 


17046 


18612 


54a37 


1 


60 


80973 


24084 


75877 


89245 


18530 


20551 


56677 





/ 


120 


II' 


10» 


9'> 


8° 


70 


6«» 


' 


COSECANTS. 1 



Digitized by VjOOQIC 



NATURAL SECANTS AND COSECANTS 



275 



1 



/ 


SECANTS, 


' 


840 


85«» 


S6'> 


87«» 


88» 


8e«» 





9.56677 


11.47371 


14.33559 


19.10732 


28.65371 


57.29869 


60 


1 


69338 


. 51199 


39547 


21397 


89440 


.')8. 26976 


59 


2 


68008 


55058 


45586 


32182 


29.13917 


59.27431 


58 


3 


64687 


68938 


51676 


43088 


38812 


60.31411 


57 


4 


67387 


63837 


57817 


54119 


64137 


61.39105 


56 


5 


70108 


66769 


64011 


65275 


89903 


62.50716 


56 


6 


78833 


70788 


70258 


76560 


30.16120 


63.66460 


64 


7 


75579 


74714 


76558 


87976 


42802 


64.86572 


53 


8 


78341 


78787 


88918 


99524 


69960 


66.11804 


58 


9 


81119 


82768 


89383 


20.11208 


97607 


67.40987 


51 


10 


83918 


86837 


95788 


23028 


31.26758 


68.75736 


50 


11 


9.86788 


11.90934 


16.02310 


80.34989 


31.54425 


70.16047 


49 


18 


89547 


96060 


08890 


47093 


83623 


71.62207 


48 


13 


' 98389 


99314 


15527 


59841 


38.18366 


73.14583 


47 


14 


95848 


18.08397 


88883 


71737 


43671 


74.73586 


46 


15 


98183 


07610 


28979 


84388 


74554 


76.3d655 


45 


16 


10.01015 


11858 


85796 


96982 


33.06030 


78.13874 


44 


17 


03983 


16185 


42678 


21.09838 


38118 


79.94968 


43 


18 


06849 


20427 


49611 


22852 


70835 


81.85815 


43 


19 


09798 


24761 


56614 


■ 36027 


34.04199 


83.84947 


41 


80 


18763 


29125 


63679 


49368 


38838 


85.94561 


40 


81 


10.15730 


18.83521 


15.70810 


21.62876 


34.72952 


88.14924 


39 


28 


18725 


37948 


78005 


76556 


35.08380 


90.46886 


88 


23 


21739 


42408 


85268 


90409 


44539 


92.91387 


37 


24 


24770 


46900 


92597 


22.04440 


81452 


95.49471 


36 


25 


27819 


51484 


99995 


18658 


86.19141 


98.22303 


35 


86 


30887 


55988 


16.07462 


33050 


57633 


101.1 lia5 


34 


27 


38973 


60678 


14999 


4Y685 


96953 


104.17574 


33 


28 


37077 


65197 


22607 


62413 


37.37127 


107.43114 


32 


89 


40801 


69856 


30287 


77386 


78186 


110.89656 


81 


30 


43343 


74550 


38041 


92559 


38.20166 


114.59301 


30 


31 


10.46505 


18.798';8 


16.45869 


23.07986 


38.63068 


118.54440 


29 


3:3 


49685 


84048 


53772 


23520 


39.06957 


122.77808 


2H 


33 


52886 


88841 


61751 


89316 


51856 


127.32526 


27 


34 


56106 


93677 


69808 


55329 


97797 


132.22229 


26 


85 


59346 


98649 


77944 


71563 


40.44820 


137.61108 


25 


36 


68605 


13.03458 


86159 


88028 


92963 


148.24061 


24 


37 


65885 


08404 


94456 


84.04712 


41 .42266 


149.46837 


23 


88 


69186 


13388 


17.02885 


21637 


92772 


156.26288 


82 


39 


78507 


18411 


11297 


38808 


42.44525 


163.70325 


21 


40 


75849 


83478 


19848 


56818 


97671 


171.88831 


80 


41 


10.79218 


13.88578 


17.28476 


84.73873 


43.51961 


180.93496 


19 


48 


82596 


83718 


37196 


91790 


44.07746 


190.98680 


18 


43 


86001 


38891 


46005 


25.09969 


64980 


808.88132 


17 


44 


89488 


44112 


54903 


28414 


45.23780 


214.85995 


16 


45 


92877 


49378 


63893 


47134 


84026 


889.18385 


15 


46 


96348 


54676 


72975 


66132 


46.45963 


845.65402 


14 


47 


99841 


60021 


82158 


85417 


47.09596 


264.44269 


13 


48 


11.03356 


65408 


91424 


26.04994 


74997 


286.47948 


12 


49 


06894 


70838 


18.00794 


24869 


48.42241 


312.52297 


11 


50 


10455 


76318 


10868 


45051 


49.11406 


343.77516 


10 


51 


11.14089 


13.81829 


18.19830 


86.65546 


49.82576 




9 


52 


17646 


87391 


29501 


86360 


50.55840 


429.71873 


8 


53 


21277 


92999 


39274 


87.07503 


51.81290 


491.10708 


7 


54 


84938 


98651 


49153 


88981 


52.09027 


578.95809 


6 


55 


88610 


14.04350 


59139 


50804 


89156 


687.64960 


6 


66 


32313 


10096 


69233 


78978 


53.71790 


859.43689 


4 


57 


36040 


16889 


79438 


95513 


54.67046 


1145.9167 


8 


68 


89798 


21730 


89755 


88.18417 


55.45053 


1718.8785 


8 


59 


43569 


27620 


19.00185 


41700 


56.35946 


3437.7468 


1 


60 


47371 


33559 


10738 


65371 


67.29869 


00 





■ 


6o 


4« 


8» 


go 


V 


O^ 


/ 


COSECANTS. 



Digitized by VjOOQIC 



276 NATURAL VERSED SINES AND EXTERNAL SEC. 
©• 1* 2* 3** 



t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. . 


Ex. sec. 


Vers. 


Ex. sec. 


» 




1 

2 
3 

4 


■00000 
.00000 

00000 
.00000 

00000 


.00000 
.00000 
.00000 
.00000 
.00000 


00015 
•00016 
•00016 
.00017 
.00017 


.00015 
.00016 
.00016 
.00017 
.00017 


.00061 
.00062 
■00063 
00064 
00065 


.00061 
.00062 
.00063 
.00064 
.00065 
.00066 
.00067 
.00068 
.00069 
.00070 


00137 
00139 
00140 
00142 
.00143 


.00137 
.00139 
.00140 
.00142 
.00143 




] 
2 
3 

4 


5 
6 
7 
6 
9 


.00000 
.00000 
.00000 
00000 
•00000 


.00000 
.00000 
.00000 
.00000 
.00000 


.00018 
.00018 
.00019 
.00020 
•00020 


.00018 
.00018 
.00019 
.00020 
.00020 


.00066 
.00067 
00068 
.00069 
.00070 


00145 
•00146 
•00148 

00150 
■00151 


.00145 
.00147 
.00148 
.00150 
.00151 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


00000 
.00001 
.00001 
•00001 
•00001 


.00000 
.00001 
.00001 
.00001 
.00001 


.00021 
.00021 
.00022 
00028 
00023 


00021 
.00021 
.00022 
.00023 
.00023 


.00071 
.00078 
•00074 
•00075 
.00076 


.00072 
.00073 
.00074 
.00075 
.00076 


.00153 
.00154 
.00156 
.00158 
.00159 


.00153 
.00155 
.00156 
.00158 
.00159 


10 

11 
12 
IS 
14 


15 
16 
17 
18 
19 


00001 
00001 
00001 
00001 
• 00002 


.00001 
.00001 
.00001 
.00001 
•00002 


.00024 
.00024 
00025 
00026 
.00026 


.00024 
.00024 
.00025 
.00026 
.00026 


.00077 
00078 

.00079 
00081 

•00082 


.00077 
.00078 
.00079 
.00081 
00082 


.00161 
.00162 
.00164 
.00166 
.00168 


.00181 
.00163 
.00164 
.00166 
.00168 


15 
16 
17 
18 
-1» 


20 

21 
22 
23 
24 


00002 
00002 
00002 
•00002 
00002 


.00002 
.00002 
.00002 
.00002 
.00002 


00027 
00028 
00028 
.00029 
•00080 


.00027 
.00028 
.00028 
.00029 
.00030 


00083 
00084 
00085 
■00087 
•00088 


.00088 
.00084 
.00085 
•00087 
.00088 


.00169 
00171 
■00178 
.00174 
■00176 


00169 
00171 
.00173 
.00175 
00176 


20 
21 

22 
23 
24^ 


25 
26 
27 
28 
29 


00008 
00003 
•00008 
•00003 
00004 


.00008 
.00003 
.00003 
.00003 
.00004 


•00031 
•00031 
00032 
00033 
00034 


.00031 
00031 
.00032 
.00033 
.00034 


. 00089 
00090 
•00091 
.00093 
•00094 


.00089 
.00090 
.00091 
• 00093 
■00094 


.0017d 
■00179 
■00181 
•00183 
■00185 


00178 
.00180 
.00182 
.00183 
.00185 


25 
26 
27 
2S 
29 


30 

31 
32 
33 
84 


•00004 
•00004 
•00004 
•00005 
00005 


.00004 
.00004 
.00004 
.00005 
.00005 


00034 
•00035 
.00036 
•00037 
•00037 


.00034 
.00035 
•00036 
.00037 
00037 


00095 
.00096 
00098 
00099 
•00100 


.000G5 
■00097 
•00098 
.00099 
■00100 


00187 
00188 
•00190 
•00192 
•00194 


.00187 
.00189 
.00190 
00192 
00194 


30 

31 
32 
83 
84 


85 
36 
37 
88 
89 


00005 
00005 
00006 
•00006 
•00006 


.00005 
.00005 
•00006 
00006 
.00006 


00038 
00039 
•00040 
00041 
00041 


00038 
00039 
•00040 
00041 
00041 


•00102 
•00103 
.00104 
•00106 
00107 


.00102 
.00103 
.00104 
.00106 
.00107 


•00196 
.00197 
.00199 
.00201 
.00203 


.00196 
.00198 
.00200 
.00201 
00203 


85 
36 
37 
38 
89 


40 

41 
42 
43 
44 


00007 
00007 

.00007 
00008 

•00008 


00007 
•00007 
•00007 
.00008 

00008 


.00042 
•00043 
00044 
•00045 
■00046 


.00042 
00043 
.00044 
.00045 
.00046 


•00108 
.00110 
00111 
•00112 
•00114 


.00108 
.00110 
.00111 
00113 
00114 


•00205 
.00207 
•00208 
00210 
00212 


.00205 
.00207 
.00209 
00211 
00218 


40 
41 
42 
48 
44 


45 
46 
47 
47 
49 


•00009 
00009 
00009 

•00010 
00010 


00009 
.00009 
.00009 

00010 
•00010 


•00047 
00048 

•00048 
00049 

•00050 


.00047 
.00048 
•00048 
.00049 
00050 


•00115 
.00-17 
•00118 
•00119 
•00121 


.00115 
•00117 
.00118 
.00120 
•00121 


00214 
00216 
•00218 
.00220 
•00222 


00215 
•^0216 
.00218 
.00220 
.00222 


45 
46 
47 
48 
49 


50 

61 
62 
63 
64 


00011 
00011 

•00011 
00012 

•00012 


00011 
00011 

.00011 
00012 

.00012 


.00051 
00052 

•00053 
00054 

.00055 


.00051 
.00052 
.00053 
.00054 
.00055 


•00122 
•00124 
.00125 
•00127 
•00128 


.00122 
•00124 
.00125 
00127 
•00128 


■00224 
■00226 
•00228 
00230 
00232 


.00224 
.00226 
.00228 
00230 
.00232 


60 
51 
52 
53 
54 


66 
66 
67 
58 


•00013 
•00018 
00014 
•00014 
•00015 


.00013 
.00013 
00014 
.00014 
.00015 


00056 
•00057 
•00058 

00059 
■00060 


.00056 
.00057 
.00058 
.00059 
00060 


00130 

00131 
•00133 
.00134 

•P0139 . 


.00130 
.00131 
00133 
00134 
.00136 


•00234 
00236 
00238 
00240 

.00242 


.00284 
.00236 
.00238 
.00240 
00242 


55 
56 
67 
58 
69 


60 


00016 


.00015 


00061 


.00061 


00137 


00137 


00244 


.00244 


60 



Digitized by VjOOQIC 



NATURAL VERSED SINES AND EXTERNAL SEC. 
4' «• 6* 7* 



277 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vera. 


Ex. sec. 


f 


o 


00244 


.00244 


.00381 


.00382 


.00548 


•00551 


•00745 


00751 


~ 


1 


00246 


.00246 


00383 


•00385 


•00551 


.00554 


• 00749 


. 00755 


] 


2 


•00248 


.00248 


•00386 


.00387 


. 00554 


.00557 


00752 


00758 


2 


3 


.00250 


.00250 


•00388 


.00890 


00557 


•00560 


00756 


00762 


3 


4 


•00252 


.00252 


.00391 


.00392 


.00560 


•00563 


■ 00760 


00765 


4 


5 


.00254 


.00254 


.00393 


.00395 


.00563 


.00566 


•00763 


.00769 


5 


6 


.00256 


.00257 


.00396 


•00397 


.00566 


•00569 


•00767 


• 00773 


a 


7 


00258 


.00259 


.00398 


.00400 


•00569 


•00573 


00770 


00776 


7 


8 


.00260 


.00261 


.00401 


.00403 


•00572 


•00576 


• 00774 


.00780 


8 


9 


.00262 


00263 


.00404 
.00406 


•00405 


•00576 


.00579 


00778 


•00784 


9 


10 


00264 


.00265 


.00408 


00579 


•00582 


.00781 


.00787 


10 


11 


.00266 


.00267 


00409 


.00411 


00582 


.00585 


.00786 


.00791 


11 


12 


.00269 


.00269 


00412 


.00413 


00585 


•00588 


00789 


.00795 


12 


IS 


00271 


.00271 


.00114 


•00416 


00588 


.00592 


00792 


.00799 


13 


14 


00273 


.00274 
.00276 


.00417 


•00419 


.00591 


•00595 


.00796 


00802 


14 


15 


00275 


.00420 


•00421 


00594 


•00598 


. 00800 


■ 00806 


15 


16 


00277 


.00278 


.00422 


.00424 


•00598 


.00601 


.00803 


■ 00810 


16 


17 


00279 


■00280 


•00425 


•00427 


•00601 


.00604 


. 00807 


.00813 


17 


18 


00281 


•00282 


•00428 


•00429 


.00604 


•00608 


.00811 


.00817 


18 


-il. 


00284 


.00284 


•00430 


•00432 


.00607 


■00611 


.00814 


00821 


19 


20 


00286 


•00287 


•00433 


•00435 


.00610 


00614 


.00818 


00825 


20 


21 


00288 


.00289 


00436 


•00438 


.00614 


.00617 


00822 


.00828 


21 


22 


00290 


.00291 


.00438 


•00440 


.00617 


.00621 


.00825 


.00832 


22 


23 


•00293 


00293 


•00441 


•00443 


• 00620 


.00624 


00829 


.00836 


23 


Ji- 


■ 00295 


.00296 


•00444 


•00446 


■00623 


•00627 


•00833 


.00840 


24 


26 


•00297 


.00298 


•00447 


•00449 


•00626 


.00630 


.00837 


.00844 


25 


26 


00299 


.00300 


•00449 


•00451 


.00630 


00634 


00840 


00848 


26 


27 


.00301 


.00302 


•00452 


•00454 


00633 


•00637 


00844 


.00851 


27 


28 


00304 


.00305 


.00455 


•00457 


.00636 


•00640 


.00848 


.00855 


28 


2?_ 


•00306 


.00307 


00458 


•00460 


•00640 


•00644 


.00852 


. .00859 


29 


30 


.00308 


.00309 


.00460 


•00463 


•00643 


.00647 


. 00856 


.00863 


30 


31 


.00311 


.00312 


.00463 


•00465 


00646 


.00650 


00859 


•00867 


31 


32 


•00313 


.00314 


.00466 


.00468 


.00649 


. 00654 


•00863 


.00871 


32 


83 


00315 


.00316 


00469 


.00471 


.00653 


00657 


•00867 


.00875 


33 


84 


•00317 


.00318 


.00472 


•00474 


•00656 


.00660 


•00871 


00878 


34 


85 


.00320 


.00821 


.00474 


•00477 


■00659 


.00664 


•00875 


.00882 


35 


86 


•00322 


.00323 


.00477 


.00480 


. 00663 


.00667 


• 00878 


00886 


36 


87 


•00324 


.00326 


00480 


.00482 


.00666 


.00671 


•00882 


.00890 


37 


88 


■ 00327 


.00328 


•00483 


•00485 


. 00669 


.00674 


00886 


.00894 


38 


iL- 


00829 


•00330 


00486 


•00488 


■00673 


.00677 


•00890 


.00898 


-J» 


40 


00332 


.00333 


.00489 


•00491 


.00676 


.00681 


•00894 


.00902 


40 


41 


•00334 


.00335 


.00492 


•00494 


00680 


.00684 


•00898 


.00906 


41 


42 


•00336 


.00337 


.00494 


.00497 


00683 


.00688 


•00902 


.00910 


42 


43 


00339 


•00340 


00497 


•00500 


■00686 


.00691 


.00906 


.00914 


43 


44 


•00341 


•00342 


•00500 


•00503 


.00690 


•00695 


00909 


■00918 


44 


46 


.00343 


.00345 


•00503 


•00506 


■00693 


.00698 


.00913 


.00922 


45 


46 


.00346 


.00347 


•00506 


.00509 


00607 


•00701 


00917 


.00926 


46 


47 


00348 


00350 


■00509 


.00512 


. 00700 


•00705 


00921 


.00930 


47 


48 


00851 


.00352 


•00512 


•00515 


■00703 


•00708 


00925 


.00934 


48 


,^ 


00353 


00354 


•00515 


.00518 


■00707 


•00712 


.00929 


00938 


49 


50 


00356 


.00357 


.00518 


.00521 


.00710 


.00715 


.00933 


00942 


50 


61 


.00358 


.00359 


•00521 


.00524 


•00714 


.00719 


00937 


.00946 


51 


62 


00361 


.00362 


.00524 


.00527 


•00717 


.00722 


•00941 


.00950 


52 


53 


.0036d 


00364 


.00527 


•00530 


•00721 


.00726 


.00945 


•00954 


53 


64 


00365 


.00367 


00530 


00533 


.00724 


.00730 


•00949 


■00958 


54 


66 


00368 


.00369 


00533 


. 00536 


.00728 


.00733 


.00953 


•00962 


55 


5b 


.00370 


.00372 


00536 


.00639 


.00781 


.00737 


00957 


.00966 


66 


67 


.00378 


•00374 


.00539 


.00542 


.00735 


.00740 


.00961 


•00970 


67 


58 


.00376 


00377 


•00542 


.00545 


•00738 


.00744 


■ 00965 


•00975 


58 


SL 


.00878 


•00379 


00545. 


00548 


•00742, 


■00747 


.0096? 


.00979. 


-SS 


60 


.00381 


.00382 


00548 


.00551 


.00746 


.00751 


.00973 


•00983 


60 



Digitized by VjOOQIC 



278 NATURAL VERSED SINES AND EXTERNAL SEC. 

8* 9* 10* II* 



t 


Vers. I 


:x.Bec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. I 


ilx. sec. 


' 


~ 


.00973 


00983 


.01231 


.01247 


.01519 


.01543 


.01837 


01872 


"o 


1 


.00977 


00987 


.01236 


.01251 


.01524. 


.01548 


.01843 


01877 


1 


2 


.00981 


00991 


.01240 


.01256 


.01529 


.01553 


.01848 


01883 


2 


8 


.00985 


00995 


.01245 


.01261 


.01534 


.01558 


.01854 


01889 


3 


4 


.00989 


00999 


.01249 


^01265 
.01270 


.01540 


.01564 


.01860 


01895 




6 


00994 


01004 


01254 


.01545 


.01569 


.01865 


.01901 


~i 


6 


00998 


01008 


.01259 


.01275 


.01550 


.01574 


01871 


.01906 


6 


7 


.01002 


01012 


01263 


.01279 


.01555 


.01579 


.01876 


.01912 


7 


8 


.01006 


01016 


■01268 


01284 


01560 


.01585 


01382 


.01918 


8 


JL 


.01010 


01020 


■01272 


.01289 


.01565 


.01590 


01888 


.01924 


9 


10 


.01014 


01024 


.01277 


.01294 


01570 


.01595 


.01893 


.01980 


10 


11 


.01018 


01029 


.01282 


.01298 


.01575 


.01601 


01899 


.01936 


11 


12 


.01022 


01033 


.01286 


.01303 


.01580 


.01606 


.01904 


.01941 


12 


18 


.01027 


01087 


01291 


.01808 


.01586 


.01611 


.01910 


.01947 


18 


ii. 


.01031 


01041 


.01296 


.01818 


.01591 


.01616 


01916 


.01953 


14 


16 


01035 


01046 


.01300 


.01818 


.01596 


.01622 


01921 


.01959 


15 


16 


■01039 


01050 


■01305 


.01822 


01601 


.01627 


01927 


.01965 


16 


17 


01048 


01054 


01310 


.01827 


.01606 


.01633 


.01933 


.01971 


17 


18 


.01047 


01059 


•01314 


.01332 


.01612 


.01638 


.01939 


01977 


18 


i2- 


.01052 


0106?. 


01319, 


.01337 


.01617 


.01648 


.01944 . 


.01983 


-ia 


20 


•01056 


01067 


.01324 


.01342 


.01622 


.01649 


01950 


01989 


so 


21 


01060 


01071 


.01329 


.01346 


.01627 


.01654 


01956 


.01995 


21 




01064 


01076 


.01333 


.01351 


.01632 


.01659 


.01961 


.02001 


22 


28 


01069 


01080 


01338 


.01356 


.01638 


.01665 


01967 


.02007 


28 


24 


.01078 


01084 
01089 


.01343 
.01848 


.01361 
.01366 


.01643 


.01670 


.01973 


02018 


_24 


26 


.01077 


•01648 


.01676 


01979 


02019 


25 


26 


.01081 


01098 


.01352 


.01371 


01653 


.01681 


.01984 


02025 


26 


27 


.01086 


01097 


.01357 


.01376 


.01659 


.01687 


01990 


.02081 


27 


28 


■01090 


01102 


.01362 


.01381 


01664 


.01692 


•01996 


.02037 


28 


29 


.01094 - 


01106 


.01367 


01386 


.01669 


•01698 


.02002 


.02048 


29 


30 


01098 


01111 


.01371 


.01391 


•01675 


.01703 


.02008 


.02049 


30 


81 


.01103 


01116 


.01376 


.01395 


.01680 


.01709 


02018 


.02055 


81 


82 


01107 


01119 


.01381 


.01400 


01685 


.01714 


02019 


.02061 


82 


83 


.01111 


01124 


01386 


.01405 


.01690 


.01720 


02025 


.02067 


88 


8^ 


.01116 


01128 


01391 


.01410 


.01696 


.01725 


02031 


.02073 


84 


85 


.01120 


01133 


01396 


.01415 


.01701 


.01731 


■02037 


.02079 


35 


86 


.01124 


01137 


■01400 


.01420 


01706 


.01736 


02042 


.02085 


86 


87 


01129 


01142 


.01405 


.01425 


.01712 


.01742 


02048 


.02091 


87 


88 


01133 


01146 


.01410 


.01430 


01717 


.01747 


02054 


.02097 


88 


39. 


■01137 


01151,. 


.01415 


.01435 


01728 


.01753 


■02060 


02108 


-22 


40 


01142 


01155 


01420 


.01440 


■01728 


.01758 


02066 


02110 


40 


41 


.01146 


01160 


.01425 


.01445 


01738 


.01764 


02072 


02116 


41 


42 


.01151 


.01164 


01430 


01450 


.01739 


.01769 


02078 


02122 


42 


48 


01155 


.01169 


■01435 


01455 


.01744 


.01775 


.02084 


02128 


48 


44 


01159 


01178 


.01439 


.01461 


.01750 


.01781 
01780 


.02090 


02184 


44 


45 


.01164 


.01178 


.01444 


.01466 


01755 


.02095 


02140 


45 


46 


01168 


.01182 


.01449 


.01471 


.01760 


.01792 


02101 


02146 


46 


47 


01178 


.01187 


.01454 


.01476 


01766 


.01793 


02107 


02158 


47 


48 


■01177 


01191 


01459 


.01481 


.01771 


.01803 


.02118 


02159 


48 


49 


■01182 


01196 


.01464 


.01486 


01777 


.01809 
.01815 


02119 


02165 


49 


50 


01186 


01200 


■01469 


01491 


01782 


.02125 


02171 


60 


61 


01191 


01205 


.01474 


01496 


01788 


.01820 


.02181 


02178 


51 


R2 


■01195 


01209 


.01479 


.01501 


.01798 


.01826 


.02137 


02184 


52 


68 


■01200 


01214 


.01484 


■01506 


.01799 


.01832 


.02148 


02190 


58 


64_ 


■01204 


01219 


01489 


.01512 


■01804 


01837 


.02149 


02196 


54 


55 


.01209 


01223 


.01494 


.01517 


.01810 


01843 


.02155 


0220S 


55 


66 


.01218 


01228 


.01499 


.01522 


.01815 


01849 


02161 


02209 


56 


57 


01218 


01283 


.01504 


.01527 


01821 


.01864 


02167 


02215 


57 


68 


■01222 


01237 


.01509 


.01532 


.01826 


.01860 


.02178 


02221 


58 


,?». 


■01227 


01242 


J)1514 


.01537 


.01882 


.01866 


02185 


02228 
02284 


JI 


60 


01231 


01247 


.01519 


.01548 


01887 


.01872 


60 



Digitized by VjOOQIC 



NATURAL VERSED SINES AND EXTERNAL SEC. 279 





ir 


13* 


14** 


15^ 






» 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 


o 

1 

i 

4 


.02185 
02191 
02197 
.02203 
.02210 


.02234 
.02240 
.02247 
.02253 
.02259 


.02563 
02570 
.02576 
.02583 
■02589 


.02680 
.02637 
.02644 
.02651 
.02658 


.02970 
.02977 
.02985 
.02992 
.02999 


.03061 
.03069 
.08076 
.08084 
.03091 


.08407 
.03416 
.03422 
.03480 
■08438 




03528 
03586 
08544 




1 

1 


5 
6 
7 
8 
9 


.02216 
.02222 
.02228 
.02234 
02240 


.02266 
.02272 
.02279 
.02285 
.02291 


•02596 
.02602 
.02609 
.02616 
.02622 


.02665 
.02672 
.02679 
.02686 
.02693 


.03006 
.03013 
.03020 
.03027 
■03034 


.03099 
.03106 
.03114 
.03121 
.03129 


■03445 
■03458 
03460 
.03468 
■03476 




08568 
03576 
03584 
03592 
08601 


5 

1 

% 


lO 

11 
12 
18 
14 


.02246 
.02252 
02258 
02265 
.02271 


.02298 
■02804 
.02311 
.02317 
■02323 


.02629 
•02635 
.02642 
.02649 
.02655 


.02700 
.02707 
.02714 
.02721 
.02728 


.08041 
■03048 
.03055 
■03063 
■03070 


.03137 
.03144 
.03152 
.03159 
.03167 


.03483 
■03491 
•03498 
■08506 
■03514 




03609 
03617 
03625 
03633 
03642 


10 

11 

ii 

14 


16 
16 
17 

il 

21 
22 
28 
24 


02277 
■02288 

02289 
.02295 
•02302 
■02308 
.02314 

02320 
.02327 
.02333 


.02330 
.02336 
.02343 
.02349 
.02356. 
.02362 
.02369 
.02375 
02382 
.02388 


.02662 
.02669 
■02675 
02682 
,02689 
.02696 
■02702 
■02709 
.02716 
.02722 


.02735 
.02742 
.02749 
.02756 
.02763 
.02770 
.02777 
.02784 
.02791 
.02799 


■03077 
■03084 
■03091 
■03098 
■03106 

■03113 
■03120 
■03127 
■03184 
■03142 
■03149 
.03156 
.03163 
03171 
.03178 


.03175 
.03182 
.03190 
.03198 
.,03205 
03218 
03221 
.03228 
.08286 
.03244 


03521 
■03529 
■03587 
■08544 
,03552. 
■03560 

03567 
■03575 
■03588 
■03590 


- 


03650 
03658 
03666 
03674 
03688 

03691 
03699 
08708 
03716 
03724 


16 

i? 

18 

21 
22 
28 
24 


IS 

IS 

29 


.02339 
.02345 
■02352 
■02358 
■02364 


.02395 
.02402 
.02408 
.02415 
.02421 


■02729 
02736 

■02743 
02749 

■02756 


.02806 
.02813 
.02820 
.02827 
.02834 


.03251 
.03259 
.03267 
.03275 
■03282 


.03598 
03606 
.03614 
.08621 
.03629 




03732 
03741 
03749 
03758 
03766 


25 
26 

29 


80 

81 

ii 

84 


•02370 
02377 
■02383 
■02389 
02396 


.02428 
.02435 
.02441 
.02448 
02454 


■02763 
■02770 
■02777 
.02783 
02790 
■02797 
■02804 
■02811 
■02818 
■02824 


.02842 
.02849 
.02856 
.02863 
_J)2870 
.02878 
.02885 
.02892 
.02899 
.02907 


.03185 
.03193 
.03200 
.03207 
■03214 
.03222 
.03229 
.03236 
.03244 
•P?251 


■03290 
.03298 
.03306 
.03313 
.■ 03821 
.03329 
.03337 
.03345 
.03353 
03860 


03687 
.03645 
■03658 
■03660 
■03668 

03676 
■03684 
■08692 
■03699 

08707 




03774 
03783 
08791 
08799 
08808 


30 

81 
32 
38 
34 


8o 
86 
37 
88 
J9_ 


■02402 
02408 

■02415 
02421 

■02427 


.02461 
.02468 
•02474 
.02481 
02488 




08816 
08825 
08883 
03842 
03850 


35 
36 
37 
38 
-39 


40 

41 
42 
48 
44 


•02434 
■02440 
■02447 
•02453 
•02459 


02494 
.02501 
.02508 
.02515 
.02521 


■02831 
■02838 
■02845 
02852 
•02859 


.02914 
.02921 
.02928 
.02936 
.02943 


.03258 
.03266 
.03273 
.03281 
.03288 


.03368 
.08876 
.03884 
.03892 
■03400 


03715 
■03723 
■03781 
■03789 

03747 




08858 
03867 
.08875 
.08884 
.08892 


40 

41 
42 
48 
44 


45 
46 
47 
48 
49 


■02466 
■02472 
•02479 
■02485 
■02492 


.02528 
.02535 
.02542 
.02548 
.02555 


■02866 
■02873 
■02880 
■02887 
■02894 


.02950 
.02958 
.02965 
.02972 
.02980 


.03295 
.03303 
■03310 
■03318 
■03325 


■03408 
■03416 
.03424 
.03432 
.03439 


■03754 
■03762 
03770 
.03778 
•03786 




.03901 
.03909 
08918 
08927 
08935 


45 
46 
47 
48 
49 


50 

61 
52 
53 
54 


.03498- 
•02504 
•02511 
.02517 
■02524 
■02580 
02537 
■02543 
■02550 
■02556 


.02562 
.02569 
.02576 
.02582 
.02589 


■02900 
02907 
■02914 
•02921 
02928 


.02987 
.02994 
03002 
.03009 
.03017 


■03333 
■03340 
■ 03347 
.03355 
■03362 


.03447 
.03455 
.03463 
.03471 
.03479 


■08794 
■03802 
■03810 
■03818 
■08826 




03944 
03952 
03961 
03969 
03978 


50 

51 
52 


05 
08 


02596 
.02603 
.02610 
■02617 
•02624 


02935 
•02942 

02949 
•02956 
■ 02963 


.03024 
.03032 
.08039 
03046 
.08054 


■03370 
■03377 
■03385 
.03392 
■03400 


.03487 
■03495 
.08503 
.03512 
■085?p 


■03834 
03842 
.03850 
.03858 
^8866 




03987 
03996 
04004 
04018 
04021 


56 
56 
57 
68 


00 


■02563 


02680 


02970 


.08061 


03407 


.08528 


08874 




04030 


60 



Digitized by VjOOQIC 



280 NATURAL VERSED SINES AND EXTERNAL SEC. 
le* 17* 18' 19** 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vera. 


Ex. sec 


r 




1 

2 
3 
4 


03874 
•03882 
•03890 

03898 
•03906 


.04030 
.04039 
.04047 
.04056 
.04065 


04370 
•04378 

04387 
•04395 
•04404 


.04569 
.04578 
.04588 
.04597 
04606 


04894 
04903 
.04912 
.04921 
.04930 


.06146 
.05156 
.06166 
.05176 
.05186 


05448 
.05458 
.05467 
•05477 
.05486 


05762 
05773 
05783 
.05794 
■05805 




1 

2 
8 

4 


5 
6 
7 
8 
9 


•03914 
•03922 
03930 
.03938 
.03946 


■04078 
.04082 
•04091 
.04100 
.04108 


04412 
.04421 
.04429 
.04438 
.04446 


■04616 
■04625 
04635 
.04644 
.04653 


.04939 
.04948 
.04957 
.04967 
■04976 


.05196 
.05206 
.05216 
.05226 
■05236 


.05496 
.05505 
.05515 
.05524 
.05534 


■05815 
05826 
■05836 
■05847 
05858 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•03954 
•03963 
•03971 
•03979 
•03987 


.04117 
04126 
.04135 
.04144 
•04152 


•04455 
.04464 
•04472 
04481 
•04480 


.04663 
.04672 
.04682 
.04691 
.04700 


■04985 
■04994 
.05003 
.05012 
•05021 


.05246 
.05256 
.05266 
.05276 
.05286 


.05543 
.05553 
.05562 
.05572 
.05582 


■05869 
■05879 
05890 
05901 
05911 


10 

11 
12 
18 
14 


15 
16 
17 
18 
19. 


•03995 
•04003 
•04011 
•04019 
•04028 


.04161 
.04170 
.04179 
.04188 
•04197 


•04498 
•04507 
•04515 
•04524 
•04533 


.04710 
.04719 
.04729 
.04738 
.04748 


•05030 
■05039 
■05048 
■05057 
.05067 


.05297 
.05807 
.05317 
.05327 
.05337 


■05591 
.05601 
05610 
05620 
05680 


■05922 
.05933 
.05944 
.05955 
•05965 


16 
16 
17 
18 
^ 


20 

21 
22 
23 
24 


•04036 
•04044 
■04052 
04060 
•04069 


•04206 
.04214 
.04223 
.04232 
•04241 


•04541 
•04550 
•04559 
.04567 
.04576 


.04757 
.04767 
.04776 
.04786 
.04795 


05076 
05085 
05094 
•05103 
•05112 


.05347 
■05357 
.05367 
.05378 
.05388 


.05639 
.05649 
■05658 
■05668 
•05678 


05976 
.05987 
.05998 
.06009 
•06020 


20 

21 
22 
28 
24 


25 
26 
27 
28 
29 


•04077 
04085 

•04093 
04102 

•04110 


.04250 
.04259 
.04268 
.04277 
.04286 


•04585 
•04593 
•04602 
•04611 
•04620 


.04805 
.04815 
.04824 
04834 
.04843 


.05122 
.05131 
•05140 
•05149 
•05158 


.05398 
.05408 
.05418 
.05429 
.05439 


■05687 
.05697 
05707 
■05716 
.05726 
■05736 
■05746 
.05755 
.05765 
.05775 


.06030 
.06041 
.06052 
06068 
.06074 


25 
26 
27 
28 
29 


30 

31 
32 
83 
34 


•04118 
■04126 
•04135 
04143 
•04151 


.04295 
.04304 
.04313 
.04322 
•04331 


•04628 
•04637 
•04646 
04655 
•04663 


.04853 
■04863 
•04872 
.04882 
.04891 


.05168 
.05177 
.05186 
.05195 
.05205 


■05449 
.05460 
.05470 
.05480 
.05490 


.06085 
.06096 
.06107 
.06118 
•06129 


30 

31 
82 
88 
84 


35 
36 
87 
88 


•04159 
•04168 
•04176 
•04184 
•04193 


.04340 
•04349 
.04358 
.04367 
.04376 


.04672 
.04681 
•04690 
•04699 
•04707 


.04901 
.04911 
04920 
•04930 
.04940 


.05214 
.05223 
.05232 
.05242 
.05251 


.05501 
•05511 
.05521 
.05532 
.05542 


■05785 
.05794 
• 05804 
•05814 
■05824 


.06140 
.06151 
.06162 
.06178 
•06184 


86 
30 
87 
88 
-2fi 


40 

41 
42 
43 
44 


•04201 
•04209 
•04218 
•04226 
•04234 


04335 
.04394 
.04403 
.04413 
.04422 


■04716 
.04725 
•04734 
•04743 
•04752 


•04950 
•04959 
•04969 
04979 
.04989 


•05260 
■05270 
05279 
■05288 
■05298 


.05552 
■05563 
.05573 
.05584 
.05594 


05833 
■05843 
■05853 

05868 
•05873 


.06195 
.06206 
.06217 
.06228 
.06230 


40 
41 
42 
43 
44 


45 
46 
47 
48 
49 


•04243 
•04251 
04260 
•04268 
•04276 


.04431 
.04440 
.04449 
.04458 
.04468 


•04760 
•04769 
•04778 
•04787 
•04796 


■04998 
■05008 
■05018 
.05028 
.05038 


.05307 
.06316 
.05326 
.06335 
.05344 


.05604 
.06615 
.05625 
.05686 
.05646 


■05882 
.05892 
05902 
.05912 
.05922 


.06250 
06261 
06272 
06288 

.06298 


45 
46 
47 
48 

49 


50 

51 
52 
53 
54 


04285 
04293 
•04302 
•04310 
•04319 


04477 
.04486 
.04495 

04504 
.04514 


•04806 
•04814 
04323 
.04832 
•04841 


.05047 
.05057 
.05067 
.05077 
.05087 


.05354 
.05363 
.05873 
■05382 
05391 


.05657 
.05667 
•05678 
•05688 
.05699 


.05932 
.05942 
.05951 
.05961 
05971 


.06306 
.06317 
.06328 
.06339 
.06350 


ffO 
51 
52 
63 
54 


55 
66 
57 
58 
69. 


•04327 
•04336 
04344 
•04353 
.043fll 


.04523 
.04532 
.04541 
.04551 
•04560 


•04850 
•04858 
•04867 
•04876 
04885 


.05097 
■05107 
■05116 
.05126 
■05136 


•05401 
.05410 
.05420 
.05429 
.05439 


.05709 
•05720 
■05730 
■05741 
•05751 


05981 
05991 
.06001 
■06011 
.06021 


.06362 
.06378 
06384 
.06395 
.06407 


55 
66 
57 
58 
59 


60 


04370 


•04569 


•04894 


.05146 


•05448 


.05762 


.06031 


.06418 


60 



Digitized by VjOOQIC 



NATURAL VERSED SINES AND EXTERNAL SEC. 281 
20* 21* 22f* 23* 



f 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 




1 

2 
3 
4 


.06081 
.06041 
.06051 
.06061 
.06071 


.06418 
.06429 
.06440 
.064^ 
064^ 


.06642 
.06652 
06663 
.06673 
.06684 




07115 
07126 
07138 
07150 
07162 


.07282 
.07293 
.07303 
.07314 
.07325 


07853 
.07866 
.07879 
.07892 

07904 


07950 
07961 
.07972 
.07984 
.07995 


08636 
08649 
08668 
08876 
.08690 




1 
2 
8 
4 


5 
6 
7 
8 
9 


.06081 
.06091 
.06101 
.06111 
06121 


.06474 
.06486 
06497 
.06508 
.06520 


06694 
06705 
06715 
.06726 
06736 




07174 
07186 
07199 
07211 
07223 


07336 
07347 
■07358 
.07369 
07380 


.07917 
07930 
07943 
.07955 
.07968 


08006 
•08018 
•08029 
•08041 
•08052 


08703 
.08717 
.08730 
.08744 

08757 


6 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.06131 
.06141 
.06151 
.06161 
06171 


.06531 
.06542 
.06554 
.06565 
.06577 


06747 
.06757 

06768 
.06778 

06789 




07235 
07247 
07259 
07271 
07283 


.07391 
•07402 
.07413 
.07424 
.07435 


.07981 
.07994 
.08006 
.08019 
.08032 


•08064 
.08075 
08086 
08098 
■08109 


.08771 
.08784 
.08798 
.08811 
.08825 


10 

11 

12 
18 
14 


15 
16 
17 
18 
,19 


.06181 
.06191 
.06201 
.06211 
.06221 


06588 
.06600 
.06611 
.06622 
.06634 


•06799 
.06810 
06820 
.06831 
.06841, 




07295 
07307 
07320 
07332 
07344 


.07446 
.07457 
.07468 
.07479 
i.0,7_490 


.08045 
.08058 
.08071 
.08084 
.08097 


.08121 
.08182 
.08144 
.08155 
08167. 


.08839 
08852 
.08866 
.08880 
.08893 


15 
16 
17 
18 
US 


20 

21 
22 
23 
24 


.06231 
.06241 
.06252 
.06262 
.06272 


.06645 
.06657 
.06668 
.06680 
.06691 


06852 
.06868 
.06873 
.06884 
.06894 




07356 
07368 
07380 
07393 
07405 


.07501 
.07512 
.07523 
.07534 
.07545 


.08109 
.08122 
.08135 
.08148 
.08161 


.08178 
.08190 
.08201 
.08218 
.08225 


.08907 
.08921 
08984 
.08948 
.08962 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


06282 
.06292 
.06302 
.06312 
.06328 


.06703 
.06715 
.06726 
.06738 
.06749 


.06905 
.06916 
.06926 
.06937 
.06948 




07417 
07429 
.07442 
07454 
07466 


.07556 
.07568 
.07579 
.07590 
.07601 


.08174 
.08087 
.08200 
.08218 
.08226 


.08236 
08248 
.08259 
.08271 
•08282 


.08975 
.08989 
.09003 
.09017 
.09030 


25 
26 

29 


30 

81 
82 
88 
84 


.06333 
.06343 
.06358 
.06363 
.06374 


.06761 
.06773 
.06784 
.06796 
.06807 


.06958 
.06969 
.06980 
.06990 
.07001 




07479 
07491 
07503 
07516 
07528 


.07612 
07628 
.07634 
.07645 
.07657 


.08239 
.08252 
.08265 
.08278 
.08291 


.08294 
.08806 
.08817 
.08329 
■08840 


.09044 
.09058 
.09072 
.09086 
.09099 


30 

81 
82 
88 
34 


85 
86 
87 
88 
89_ 


06384 
.06394 

06404 
.06415 
.06425 


.06819 
.06831 
.06843 
.06854 
.06866 


.07012 
.07022 
.07033 
.07044 
.07055 




07540 
07553 
07565 
07578 
07590 


.07668 
.07679 
.07690 
07701 
.07713 


08305 
.08318 
.08831 
.08844 
.08357 


08852 
08364 
08875 
.08387 
.08399 


.09113 
■09127 
.09141 
.09155 
.09169 


85 
86 
87 
88 
«89 


40 

41 
42 
48 
44 


.06435 
.06445 
.06456 
.06466 
.06476 


.06878 
.06889 
06901 
.06913 
.06925 


.07065 
.07076 
07087 
07098 
.07108 




07J02 
07615 
07627 
07b40 
07652 


.07724 
.07735 
.07746 
.07757 
.07769 


.08870 
.08888 
08397 
.08410 
.08428 


.08410 
.08422 
08434 
.08445 
.08457 


.09183 
.09197 
09211 
.09224 
.09238 


40 

41 
42 
48 
44 


45 
46 
47 
48 
49 


06486 
.06497 
.06507 
.06517 
.06528 


.06936 
.06948 
.06960 
.06972 
06984 


.07119 
.07130 
.07141 
.07151 
.07162 




07665 
07677 
07690 
07702 
07715 


■07780 
.07791 
.07802 
07814 
.07825 
.07836 
07848 
07859 
.07870 
.07881 


.08486 
.08449 
.08463 
08476 
.08489 
.08508 
.08516 
.08529 
.08542 
.08556 


08469 
.08481 
.08492 
.08504 
.08510 
.08528 
.08539 
.08551 
.08568 
.08575 


.09252- 
.09266 
.09280 
.09294 

09308 
.09323 
.09337 
.09351 
.09365 

09379 


45 
46 
47 
48 
49 


50 
51 
52 
68 
64 


.06588 
.06548 
.06559 
06569 
.06580 


.06995 
.07007 
.07019 
07031 
.07048 


.07173 
.07184 
.07195 
.07206 
.07216 




07727 
07740 
07752 
07765 
07778 


50 

51 
52 
53 
54 


55 
66 
57 
68 
?9 


.06590 
.06600 
06611 
06621 

■06642 


.07055 
.07067 
.07079 
.07091 
.07108 
.07115^ 


.07227 
.07238 
.07249 
.07260 
.07271 
.07282 


- 


07790 
07808 
07816 
07828 
07841 
07853 


.07893 
.07904 
.07915 
.07927 
07938 


.08569 
.08582 
.08596 
.08069 
.08623 


.08586 
08598 
■08610 
.08622 
08634 


.09393 
.09407 
09421 
.09435 
■09449 


55 
56 
57 
58 
59 


00 


.07950 


.08636 


.08645 1 .09464 


60 



Digitized by VjOOQIC 



282 NATURAL VERSED SINES AND EXTERNAL SEC. 
24* ««• ae** 27** 



/ 


Vers. 


Ex. see. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 




1 

i 


08645 
.08657 
.08669 
.08681 
.08693 


.09464 
.09478 
.09492 
.09506 
.09520 


.09369 
.09382 
.09894 
.09406 
09418 




10838 
10353 
.10368 
10383 
10898 


10121 
.10133 
.10146 
.10159 

10172 


.11260 
.11276 
.11292 
.11308 
.11323 


.10899 
10913 
.10926 
.10989 
•10952 


.12288 
.1224b 
.12266 
12283 
12299 


O 

1 
2 
8 

4 


6 
6 
7 
8 
9 


.08705 
.08717 
08728 
.08740 
.08752 


.09535 
.09549 
.09563 
.09577 
.09592 


09431 
.09443 
.09455 

09468 
.09480 




.10413 
10428 
10443 
10458 
10473 


.10184 
.10197 
.10210 
.10223 
.10236 


.11839 
11355 
.11371 
.11387 
•11403 


10965 
.10979 

10992 
•11005 
.11019 


.12316 
.12838 
.12349 
.12366 
12888 


5 
6 
7 
8 
9 


10 

.1 
12 
.8 
14 


.08764 
.08776 
.08788 
.08800 
.08812 


.09606 
.09620 
.09635 
.09649 
.09663 


.09493 
.09505 
.09517 
.09530 
.09542 




10488 
10503 
10518 
10533 
10549 


.10248 
.10261 
.10274 
.10287 
.10300 


.11419 
.11435 
.11451 
.11467 
•11483 


.11082 
.11045 
.11058 
11072 
.11085 


.12400 
.12416 
.12438 
.12450 
.12467 


10 

11 
12 
18 
14 


15 
16 
17 
18 
10 


.08824 
.08836 
.08848 
.08860 
.08872 


.09678 
.09d92 
.09707 
.09721 
^09735 


09554 
.09567 
.09579 
.09592 

Mm 

.09617 
.09629 
.09642 
.09654 
.09666 




10564 
10579 
10594 
10609 
10625 


. 10318 
.10826 
.10838 
.10351 
.10364 


•11499 
11515 
.11531 
.11547 
.11563 


•11098 
.11112 
.11125 
.11138 
.11152 


.12484 
.12501 
.12518 
.125&4 
.12551 


15 
16 
17 
18 
-19 


20 

21 
22 
28 
24 


.08884 
.08896 
.08908 
.08920 
.08932 


.09750 
.09764 
.09779 
.09793 
.09808 




10640 
10655 
10670 
10686 
10701 


.10377 
.10890 
.10408 
.10416 
10429 


.11579 
.11595 
11611 
.11627 
.11643 


.11165 
.11178 
.11192 
.11205 
•11218 


.12568 
.12586 
.12602 
.12619 
12638 


20 

21 
22 
28 
24 


25 
26 
27 
28 
29 


08944 
.08956 
■08968 
.08980 
•08992 


.09822 
.09837 
.09851 
.09866 
.09880 


.09679 
.09691 
.09704 
.09716 
.09729 




10716 
10781 
10747 
10762 
10777 


.10442 
.10455 
.10468 
.10481 
.10494 


.11659 
.11675 
.11691 
.11708 
.11724 


.11232. 

.11245 

.11259 

.11272 

.11285 


.12653 
.12670 
.12687 
.12704 
12721 


25 
26 
27 
28 
29 


30 

81 
82 
88 
84 


.09004 
.09016 
.09028 
.09040 
.09052 


.09895 
.09909 
.09924 
.09939 
.09953 


.09741 
09754 
09767 
.09779 
.09792 




10793 
10808 
10824 
10839 
10854 


.10507 
.10520 
.10538 
.10546 
•10559 


.11740 
.11756 
.11772 
.11789 
■11805 


.11299 
•11312 
.11326 
.11839 
.11353 


.12738 
.12755 
.12772 
.12789 
.12807 


30 

81 
32 
38 
34 


85 
86 
87 
88 
89 


.09064 
.09076 
.09089 
.09101 
.09118. 


.09968 
.09982 
.09997 
.10012 
.10026 


.09804 
09817 
09829 
09842 

.09854 




10870 
10885 
10901 
10916 
10932 


.10572 
.10585 
.10598 
10611 
.10624 


.11821 
.11838 
.11854 
.11870 
.11886 


.11366 
.11380 
.11393 
.11407 
.11420 


.12824 
.12841 
.12858 
.U875 


35 
36 
87 
38 

39 


40 

41 
42 
48 
44 


.09126 
.09187 
.09149 
.09161 
09174 


.10041 
.10055 
.10071 
.10085 
.10100 


•09867 
•09880 
•09892 
09905 
•09918 




10947 
10963 
10978 
10994 
11009 


•10687 
.10650 
•10663 
.10676 
•10689 


•11903 
.11919 
.11936 
•11952 
•11968 


•11434 
.11447 
.11461 
.11474 
• 11488 


12910 
.12927 
.12944 
.12961 
.12979 


40 

41 
42 
48 
44 


45 
46 
47 
48 
49 


.09186 
09198 
09210 
.09222 
•09234 


.10115 
.10130 
.10144 
.10159 
.10174 


09930 
09943 
09955 
09968 
•09981 




11025 
.11041 

11056 
.11072 

11087 


•10702 
.10715 
•10728 
•10741 
•10755 


11985 
.12001 
.12018 
.12034 
•12051 


•11501 
•11515 
•11528 
.11542 
•11555 


•12996 
.18013 
.13031 
.18048 
18065 


45 
46 

47 


50 

61 
62 
68 
54 


09247 
.09259 
■09271 

09288 
.09296 


.10189 
.10204 
.10218 
.10238 
.10248 


.09993 
•10006 
.10019 
.10032 
•10044 




11103 
11119 
11134 
11150 
11166 


•10768 
10781 
10794 
10807 
10820 


.12067 
12084 
. 12100 
.12117 
.12138 


11569 
.11583 
11596 
11610 
•11623 


.18083 
•13100 
.18117 
13135 
13152 


60 

51 
52 
53 
54 


65 
56 
67 
68 
Si. 


.09808 
.09320 
.09882 
•09346 
•09357 


.10263 
.10278 
.10293 
.10308 
•103?3 


.10057 
.10070 
10082 
.10095 
•10108 




11181 
11197 
11213 
11229 
11244 


• 10833 
.10847 
.10860 
.10873 
.10886 


.12150 
.12166 
.12188 
.12199 
.12216 


.11637 
.11651 
.11664 
.11678 
.11692 


.13170 
.18187 
13205 
.13222 
.13240 


55 
56 
57 
58 
6? 


60 


09869 


.10388 


10121 




11260 


.10899 


.12283 


.11705 


.18257 


60 



y Google 



NATURAL VERSED SINES AND EXTERNAL SEC. 283 
as* 29* 30* 31' 



/ 


Vers. 


Ex. 860. 


Vera. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 


"o" 


.11705 


.13257 


.12538 


.14335 


.13397 


.16470 


.14283 


.16663 


~ 


1 


.11719 


.18275 


.12552 


.14354 


.18412 


.16489 


•14298 


.16684 


1 


2 


.11738 


.18292 


.12566 


.14372 


.13427 


.15509 


.14813 


.16704 


2 


8 


.11746 


.13310 


.12580 


.14891 


.13441 


.15528 


•14328 


.16725 


8 


4 


.11760 


.18327 


.12595 


.14409 


■13456 


•15548 


.14343 


.16745 


4 


5 


.11774 


.13345 


.12609 


.14428 


.18470 


.15567 


.14858 


.16766 


6 


6 


.11787 


.18362 


.12623 


.14446 


.13485 


•15587 


■14373 


.16786 


6 


7 


.11801 


.13380 


.12637 


.14465 


.13499 


.15606 


14388 


.16806 


7 


8 


.11815 


.18898 


.12651 


.14483 


.13514 


.15626 


14403 


.16827 


8 


_9L 


11828 


.18416 
. .18433 


■12665 
.12679 


_U4502 


.13629 


•15645 


■14418 


•16848 


9 


10 


.11842 


.14521 


. 13543 


■15665 


. 14433 


.16868 


10 


11 


.11856 


.13451 


.12694 


.14539 


•18558 


.15684 


.14449 


.18889 


11 


12 


.11870 


.13468 


.12708 


.14558 


• 18573 


. 15704 


. 14464 


.16909 


12 


18 


11888 


.13486 


.1272a 


.14576 


.18587 


.15724 


.14479 


.16980 


18 


14 


.11897 


.13504 


. 12736 


.14595 


.13602 


.15743 


■14494 


.16950 


14 


15 


.11911 


.18521 


.12750 


.14614 


13616 


.15763 


.14509 


.16971 


16 


16 


.11925 


.18539 


.12765 


.14632 


■13631 


. 15782 


.14524 


•16992 


16 


17 


.11988 


.13557 


.12779 


.14651 


. 18646 


.15802 


.14539 


•17012 


17 


18 


.11952 


.13575 


.12793 


.14670 


■13660 


•15822 


.14554 


•17033 


18 


19- 


.11966 




.12807 


.14689 


13675 


•15841 


.14569 


.17054 


-19 


20 


.11980 


.18610 


.12822 


.14707 


.13690 


.15861 


■14584 


.17075 


20 


21 


.11994 


.18628 


.12836 


.14726 


■ 13705 


•15881 


.14599 


.17095 


21 


22 


.12007 


.13646 


.12850 


.14745 


.18719 


•15901 


.14615 


.17116 


22 


28 


.12021 


.18664 


•12864 


.14764 


.13734 


.15920 


.14630 


.17187 


23 


?i_ 


.12085 


.18682 


.12879 


.14782 


.18749 


.15940 
.15960 


■14645 
■ 14660 


.17158 
.17178 


24 


26 


.12049 


■18700 


.12898 


.14801 


.18763 


25 


26 


.12068 


.13718 


•12907 


.14820 


.18778 


•15980 


.14675 


.17199 


26 


27 


■12077 


.18735 


.12921 


.14839 


.18793 


.16000 


.14690 


.17220 


27 


28 


.12091 


.13753 


.12986 


.14858 


. 13808 


.16019 


•14706 


.17241 


28 


29_ 


■12104 


.13771 


.12950 


.14877 


.18822 


■16089 


•14721 


.17262 


29 


30 


.12118 


.13789 


.12964 


.14896 


.13837 


.16059 


•14736 


17283 


30 


31 


.12132 


.18807 


.12979 


.14914 


.18852 


.16079 


.14751 


.17804 


31 


82 


.12146 


.18825 


.12993 


.14938 


.13867 


.16099 


.14766 


.17325 


32 


38 


.12160 


.13843 


.18007 


•14952 


.18881 


.16119 


.14782 


.17846 


33 


8JL 


12174 


.18861 


.13022 


.14971 


.13896 


.16139 


.14797 


■17367 


34 


85 


.12188 


.18879 


.13036 


.14990 


.13911 


.16159 


.14812 


17388 


35 


86 


.12202 


.18897 


•13051 


.15009 


■13926 


.16179 


.14827 


.17409 


5® 


37 


.12216 


.18916 


.18065 


.16028 


.13941 


.16199 


.14843 


.17480 


37 


38 


.12230 


.18984 


.13079 


.15047 


■18955 


•16219 


.14858 


.17451 


38 


39. 


.12244 


13952 


.13094 


.15066 


.18970 


. 16239 


.14873 


iim 


-22 


40 


•12257 


.13970 


■ 13108 


.15085 


13985 


.16259 


•14888 


.17493 


40 


41 


.12271 


.13988 


.13122 


.15105 


14000 


.16279 


.14904 


.17514 


41 


42 


.12285 


.14006 


•13137 


.15124 


■140] 5 


.16299 


.14819 


•17635 


42 


43 


.12299 


.14024 


.13151 


15143 


■14080 


.16319 


.14934 


.17556 


48 


44_ 


.12813 


.14042 


13166 


.16162 


.14044 


.16339 


.14949 


.17577 


.44 


45 


•12327 


•14061 


.18180 


.15181 


14059 


16359 


•14965 


•17598 


45 


46 


.12341 


.14079 


18195 


.15200 


14074 


.16380 


.14980 


•17620 


46 


47 


.12355 


.14097 


.13209 


.15219 


.14089 


.16400 


.14995 


.17641 


47 


48 


.12369 


.14115 


.18228 


.15289 


■14104 


.16420 


.15011 


.17662 


48 


49 


12383 


♦14134 


.13238 


.15258 


14119 


•16440 


■15026 


.17688 


-i? 


50 


.12397 


.14152 


.13252 


.15277 


■14134 


■16460 


15041 


.17704 


50 


51 


.12411 


.14170 


.13267 


.16296 


.14149 


16481 


•15057 


•17726 


51 


52 


.12425 


.14188 


.18281 


.15315 


.14164 


.16501 


15072 


.17747 


52 


58 


.12439 


.14207 


•13296 


.15335 


■14179 


•16521 


.15087 


•17768 


53 


54 


■12454 


•14225 


.13310 
.13825 


.16354 


.14194 


.16541 


■15103 


■17790 


54 


55 


.12468 


.14248 


.15373 


.14208 


.16562 


15118 


.17811 


55 


66 


.12482 


14262 


.13389 


.16393 


.14228 


.16582 


.15184 


.17882 


56 


67 


.12496 


.14280 


.13854 


.15412 


.14288 


.16602 


15149 


.17854 


67 


58 


.12510 


.14299 


.13868 


.15431 


•14253 


.16623 


•15164 


.17875 


58 


69. 


.12524 
.12588 


.14817 
.14835 


.18883 
.13397 


.15470 


.14268 
■14283 


.16648 
•16663 


.15180 


.17896 


59 


60 


.15195 


•17918 


60 



Digitized by VjOOQIC 



284 NATURAL VERSED SINES AND EXTERNAL SEC. 
3»* 83* 34* 36* 



/ 


Vers. 


Ex. sec. 


Vers. I 


Sx. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


r 


o 


.15195 


.17918 


.16138 


.19236 


.17096 


.20622 


.18085 


.22077 


"^ 


1 


.15211 


.17939 


.16149 


.19259 


.17118 


.20645 


.18101 


.22102 


1 


2 


.15226 


.17961 


.16165 


.19281 


.17129 


.20669 


.18118 


.22127 


3 


8 


.15241 


.17982 


.16181 


.19304 


.17145 


.20693 


.18135 


.22162 


8 


JL 


.15257 


•18004 


16196 


.19327 


.17161 


.20717 


.18152 


.22177 




5 


.15272 


.18025 


.16212 


.19349 


.17178 


.20740 


.18168 


.22202 


^~1 


6 


.15288 


18047 


. 16228 


.19372 


.17194 


.20764 


.18185 


.22227 


6 


7 


.15303 


18068 


.16244 


.19394 


.17210 


.20788 


.18202 


.22252 


7 


8 


.15819 


. 18090 


.16260 


.19417 


-17227 


.20812 


.18218 


.22277 


8 


9 


■15334 


•18111 


.16276 


19440 


•17243 


.20836 


.18285 


.22802 


9 


10 


.15350 


.18138 


.16292 


.19463 


.17259 


.20859 


.18262 


.22327 


ID 


11 


.15396 


.18155 


.16808 


.19485 


.17276 


.20883 


.18269 


.22362 


11 


12 


.15381 


■18176 


.16324 


.19508 


•17292 


.20907 


.18286 


.22377 


12 


13 


.16396 


.18198 


16340 


.19531 


.17308 


.20931 


.18302 


.22402 


18 


14 


.15412 


•18220 


•16355 


.19554 


.17325.. 


.20955 


.18319 


22428 


JHl 


16 


.15427 


.18241 


.16371 


.19576 


-17341 


.20979 


.18336 


.22463 


15 


16 


.15448 


.18263 


16387 


.19599 


.17357 


.21003 


.18358 


.22478 


16 


17 


.15458 


•18285 


.16403 


19622 


.17374 


.21027 


.18369 


.22608 


17 


18 


.15474 


.18307 


16419 


.19645 


.17890 


.21051 


.18386 


.22628 


18 


iL. 


15489 


19S28 


■ 16435 


19668 


•17407 


.21075 


•18403 . 


.22554 


-ii 


20 


. 15505 


.18350 


•16451 


19691 


•17423 


.21099 


.18420 


.22579 


20 


21 


15520 


18372 


•16467 


19713 


•17439 


.21123 


. 18437 


.22604 


21 


22 


.16536 


.18394 


•16483 


19736 


•17456 


.21147 


•18454 


.22629 


22 


23 


.15552 


.18416 


■16499 


19759 


.17472 


.21171 


.18470 


.22655 


28 


24 


15567 


18437 


■16515 


19782 


.17489 


.21195 


.18487 


.22680 


24 


26 


15583 


.18459 


■16531 


19805 


.17506 


.21220 


.18504 


.22706 


26 


26 


15598 


.18481 


•16547 


19828 


.17522 


.21244 


.18521 


.22781 


28 


27 


■15614 


.18503 


•16563 


19851 


.17538 


.21268 


.18538 


.22756 


27 


28 


.15630 


.18525 


•16579 


19874 


.17554 


.21292 


.18555 


.22782 


28 


29 


.15645 


.18547 


16595 


19897 


.17571 


.21316 


18572 


.22807 


29 


80 


15661 


.18569 


.16611 


19920 


•17587 


.21341 


.18588 


22838 


30 


81 


•15676 


.18591 


•16627 


19944 


•17604 


.21365 


. 18605 


.22868 


81 


82 


15692 


.18618 


■16644 


19967 


•17620 


.21889 


.18622 


.22884 


82 


88 


. 16708 


.18685 


•16660 


19990 


17637 


.21414 


18639 


.22909 


38 


84 


.15723 


.18657 


16676 


20013 


■17653 


.21438 


•18656 


.22935 


84 


85 


.15739 


.18679 


•16692 


20036 


•17670 


.21462 


•18673 


.22960 


35 


86 


15765 


.18701 


•16708 


20059 


•17686 


.21487 


•18690 


.22986 


86 


87 


.15770 


.18728 


■ 16724 


20083 


•17703 


.21511 


18707 


.28012 


87 


88 


15786 


.18745 


•16740 


20106 


•17719 


.21535 


• 18724 


.23037 


88 


89_ 


•15802 


•18767 


im9 


20129 


imsL . 


•21560 


•18741 


.28063 


_89 


40 


.15818 


.18790 


• 16772 


20152 


•17752 


.21584 


• 18758 


.23089 


40 


41 


.15833 


.18812 


•16788 


20176 


•17769 


.21609 


•18775 


.23114 


41 


42 


15849 


18834 


•16805 


20199 


•17786 


.21633 


• 18792 


.23140 


42 


48 


.15865 


18856 


.16821 


20222 


•17802 


.21658 


18809 


.23166 


48 


44 


.15880 


.18878 


•16837 


20246 


•17819 


•21682 


•18826 


.23192 


44 


45 


.15896 


•18901 


.16853 


20269 


•17835 


.21707 


■18843 


.23217 


45 


46 


.15912 


.18923 


16869 


20292 


17852 


.21731 


. 18860 


.23243 


46 


47 


15928 


.18946 


16885 


20316 


•17868 


.21756 


18877 


.23269 


47 


48 


• 15943 


.18967 


16902 


20339 


•17885 


.21781 


•18894 


23295 


48 


49 


•15959 


.18990 


.16918 


20363 


17902 


.21805 


•18911 


.23321 


48 


ffO 


15975 


.19012 


.16934 


20386 


•17918 


.21830 


■18928 


28847 


ffO 


51 


.15991 


.19034 


16950 


20410 


•17935 


.21855 


•18945 


.23378 


51 


62 


• 16006 


19067 


.16966 


20433 


17952 


.21879 


•18962 


.23399 


52 


68 


.16022 


.19079 


.16983 


20457 


17968 


.21904 


•18979 


.23424 


53 


64_ 


•18038 


.19102 


.16999 


20480 


■17985 


.21929 


.18996 . 


.23450 


_54 


65 


•16054 


.19124 


.17015 


20504 


•18001 


.21953 


•19018 


.28476 


55 


56 


• 16070 


.19146 


•17031 


20527 


18018 


.21978 


.19080 


.23502 


56 


57 


. 16085 


.19169 


•17047 


20551 


•18035 


.22008 


19047 


28529 


57 


58 


■16101 


.19191 


.17064 


20575 


18051 


. 22028 


.19064 


.23555 


58 


W- 


•mi7 


•19214 


.17080 


20598 


■18068 


.22053 


.19981 


?3581 


-Si 


60 


.16133 


.19286 


.17096 


20622 


•18085 


.22077 


19098 


.23607 


eo 



Digitized by VjjOOQIC 



NATURAL VERSED SINES AND EXTERNAL SEC. 2S5 

36'* 37** 38** 39' 



9 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. I 


]x. sec. 


Vers. 


Ex. sec. 


9 


"o" 


■19098 


.23607 


■20186 


.25214 


■21199 


26902 


. 22285 


•28676 


— 


1 


19115 


.23633 


■20154 


■25241 


■21217 


26931 


■22304 


.28706 


1 


2 


.19133 


.23659 


■20171 


■25269 


.21235 


26960 


■ 22322 


.28737 


2 


3 


•19150 


.23685 


20189 


■25296 


■21253 


26988 


■22340 


.28767 


I 


4 


.19167 


.28711 


.20207 


■25324 


21271 


27017 
27046 


■22359 


•28797 


4 


6 


.19184 


.23738 


■20224 


■25351 


.21289 


■22377 


•28828 


6 


6 


.19201 


.23764 


.20242 


.25379 


■21307 


27075 


22395 


•28858 


6 


7 


19218 


.23790 


■20259 


.25406 


■21324 


27104 


■22414 


•28889 


7 


8 


.19235 


.23816 


■20277 


.25434 


.21342 


27133 


22432 


•28919 


8 


9 


.19252 


.23843 


■ 20294 


_^25462_ 


■21360 


27162 


■22450 


•28950 


9 


lO 


.19270 


.23869 


■ 20312. 


■25489 


■21378 


27191 


■ 22469 


•28980 


10 


11 


.19287 


.23895 


■20329 


.25517 


.21396 


27221 


■22487 


.29011 


11 


12 


.19304 


.23922 


■20347 


■25545 


.21414 


27250 


■22506 


.29042 


12 


18 


.19321 


.23948 


■20365 


■25572 


.21432 


27279 


■22524 


29072 


18 


ii. 


.19338 


.23975 


■20382 


.25600 


■21450 


27308 


■22542 


.29108 


14 


16 


.19356 


.24001 


■20400 


.25628 


■21468 


27337 


■22561 


.29133 


15 


16 


.19373 


.24028 


.20417 


.25656 


■21486 


27366 


■22579 


29164 


16 


17 


.19390 


.24054 


.20435 


.25683 


■21504 


27396 


■22598 


29195 


17 


18 


.19407 


.24081 


.20453 


■25711 


■21522 


27425 


•22616 


29226 


18 


ii. 


.19424 


.24107 


.20470 


.25739 


.21540 


27454 


■22634 


■29256 


19 


20 


.19442 


.24134 


.20488 


■25767 


■21558 


27483 


■22653 


■29287 


;90 


21 


.19459 


■24160 


■20506 


■25795 


.21576 


27513 


■22671 


■29318 


21 


22 


.19476 


.24187 


■20523 


■25823 


■21595 


27542 


.22690 


29349 


22 


28 


.19493 


.24213 


■20541 


.25851 


■21613 


27572 


22708 


.29380 


28 


34l. 


.19511 


.24240 


■ 20559 


■25879 


.21631 


27601 


22727 


•29411 


24 


25 


.19528 


.24267 


■20576 


.25907 


■21649 


27630 


•22745 


•29442 


25 


26 


.19545 


.24293 


■20594 


.25935 


■21667 


27660 


■22764 


■ 29473 


26 


27 


■19562 


.24320 


.20612 


.25963 


■21685 


27689 


22782 


.29504 


27 


28 


•19580 


■ 24347 


■ 20629 


25991 


.21703 


27719 


■22801 


.29535 


28 


29 


■19597 


24373 


■20647 


.26019 


.21721 


27748 


■22819 


.29566 


29 


30 


■19614 


.24400 


20665 


.26047 


■21739 


27778 


■22838 


.29597 


30 


81 


.19632 


.24427 


■20682 


.26075 


.21757 


27807 


■22856 


.29628 


81 


82 


.19649 


.24454 


■ 20700 


.26104 


■21775 


27837 


22875 


29659 


82 


83 


.19666 


.24481 


■20718 


.26132 


■21794 


27867 


•22893 


29690 


88 


84 


.19684 


■24508 


.20736 


.26160 


■21812 


27896 


•22912 


■29721 


84 


86 


.19701 


■24534 


.20753 


.26188 


■21830 


27926 


■22930 


■29752 


85 


86 


.19718 


■24561 


.20771 


.26216 


.21848 


27956 


■22949 


■29784 


86 


87 


■19736 


.24588 


■20789 


■26245 


■21866 


27985 


■22967 


■29815 


37 


88 


.19753 


■24615 


.20807 


.26273 


•21884 


28015 


■22986 


■29846 


38 


^ 


.11^770 


■24642 


■20824 


■26301 


■21902 _ 


28045 


..23004 


■29877 


-82 


40 


.19788 


■24669 


.20842 


■26330 


•21921 


28075 


■23023 


.29909 


40 


41 


.19805 


■24696 


■20860 


■26358 


•21939 


28105 


■23041 


.29940 


41 


42 


.19822 


.24723 


■20878 


■26387 


.21957 


28134 


■23060 


.29971 


42 


48 


.19840 


■ 24750 


■20895 


.26415 


■21975 


28164 


23079 


30003 


48 


44 


■19857 


.24777 


•20913 


26443 


■21993 


28194 


23097 


.30034 


_44 


46 


■19875 


■24804 


■20981 


■26472 


■22012 


28224 


•23116 


.80066 


45 


46 


■19892 


■24832 


■20949 


■26500 


■ 22030 


28254 


■23134 


.80097 


46 


47 


.19909 


24859 


■20967 


.26529 


■22048 


28284 


23153 


.30129 


47 


48 


.19927 


■24886 


■20985 


■26557 


■22066 


28314 


■23172 


.30160 


48 


49 


■19944 


.24913 


■21002 


■26586 
.26615 


■22084 


28344 


■23190 


.30192 


49 


50 


19962 


.24940 


■21020 


■22103 


28374 


■ 23209 


.30223 


50 


61 


.19979 


.24967 


.21038 


.26643 


•22121 


28404 


■23228 


.30255 


51 


62 


19997 


■ 24995 


■21056 


.26672 


■22139 


28434 


• 23246 


30287 


52 


68 


.20014 


■25022 


.21074 


.26701 


■22157 


28464 


■23265 


■30318 


68 


64 


■20032 


■25049 
.26077 


.21092 


. 26729 


■22176 


28495 


■23283 


.30350 


64 


65 


.20049 


■21109 


.26758 


■22194 


28525 


■23302 


.30382 


55 


66 


20066 


.25104 


■21127 


.26787 


• 22212 


28555 


•23321 


■30413 


56 


67 


20084 


.25181 


■21145 


.26815 


■22231 


28585 


•23339 


■30445 


57 


68 


.20101 


■25159 


■21168 


26844 


.22249 


28615 


■23858 


.80477 


56 


ift. 


■20119 


25186 


■21181 


■26873 


■22267 


28646 


■23377 


■80509 


-52 


60 


.20186 


.25214 


.21199 


.26902 


■22285 


28676 


•28396 


.80541 


60 



Digitized by VjOOQIC 



286 NATURAL VERSED SINES AND EXTERNAL SEC. 
*©• 41* 42* 43* 



t 


Vera. 


Ex. 060. 


Vers. ] 


Sx.Bec. 


Vers. 


Ex. Bee. 


Vera. 


Ex. sec. 


9 





• 23896 


.80541 


.24529 


.82501 


.25686 


.84563 


.26866 


.86788 


"o 


1 


• 23414 


.80578 


.24548 


.82535 


.25706 


.34599 


.26884 


.86770 


1 


2 


•23488 


.80605 


.24567 


.82568 


.25724 


.34634 


.26904 


.86807 


2 


3 


■ 23452 


.80636 


.24586 


.82602 


.25744 


.34669 


.26924 


.86844 


8 


_4_ 


.23470 


.30668 


.24605 


.32636 


J6768_ 
.25783 


, .34704 


.26944 


.86881 


-i 


5 


23489 


.80700 


.24625 


.82669 


.34740 


.26964 


.86919 


6 


6 


. 23608 


.80732 


.24644 


82703 


.25802 


.84775 


.26984 


.86956 


6 


7 


•28527 


.80764 


.24668 


82737 


.25822 


.34811 


.27004 


.86998 


7 


8 


.23545 


.80796 


.24682 


82770 


.25841 


.84846 


.27024 


.87030 


8 


9 


28564 


.30829 


.24701 _ 


32804 
82838 


.25861 


.34882 


.27048 


.37068 


_« 


10 


.23588 


.30861 


.24720 


.25880 


.84917 


27063 


.37105 


10 


11 


.23602 


.80893 


.24739 


82872 


.25900 


\ 34953 


: 27083 


.37143 


11 


12 


.23620 


.30925 


.24759 


82905 


.25920 


.84988 


.27103 


.37180 


12 


13 


23689 


.30957 


.24778 


32939 


.25989 


.85024 


.27123 


.37218 


18 


14 


•23658 


.80989 


.24797 _ 


32973 


.25959 


.85060 


.27143 


.87256 


_J4 


16 


.23677 


.81022 


.24816 


33007 


.26978 


.85095 


.27163 


.87298 


16 


16 


.23696 


.31054 


.24835 


83041 


.26998 


.85131 


.27183 


.87880 


16 


17 


.23714 


.81086 


.24854 


33075 


.26017 


.85167 


.27203 


.87368 


17 


18 


.28738 


.81119 


.24874 


33109 


.26087 


.85203 


.27223 


.87406 


18 


iL. 


.23752 


.31151 


.24898 


33143.. 


.26056 


.35238 


:.2I243 


.87448 


-il 


20 


23771 


.31183 


.24912 


33177 


.26076 


.35274 


.27268 


.37481 


20 


21 


.23790 


.81216 


.24931 


83211 


.26096 


.35310 


.27283 


.87519 


21 


22 


■ 23808 


.81248 


.24950 


33245 


.26116 


.35346 


.27303 


.87556 


22 


23 


.23827 


.81281 


.24970 


33279 


.26135 


.35382 


.27328 


.87594 


28 


24 


23846 
■23865 


.31318 


.24989 


33314 


.26154 


.85418 


.27343 


.37632 


24 


25 


.81346 


.25008 


83348 


.26174 


.85454 


.27363 


.37670 


26 


26 


.23884 


.81378 


.25027 . 


33382 


.26194 


.35490 


27383 


.87708 


26 


27 


. 23903 


.81411 


.25047 


33416 


.26213 


.35526 


.27408 


.37746 


27 


28 


.23922 


.31443 


.25066 


33451 


.26233 


.35562 


.27428 


.87784 


28 


29 


23941 
.23959 


.31476 
.31509 


.25085_ 
.25104 


33485 
83519 


.28253 
.26272 


.35598 


.27448 


.37822 


29 


80 


.35634 


.27468 


.37860 


30 


81 


.23978 


.31541 


.25124 


83554 


.26292 


.35670 


.27488 


.37898 


31 


82 


.23997 


.31574 


.25148 


38588 


.26312 


.35707 


.27603 


.87936 


82 


83 


.24016 


.31607 


.25162 


88622 


.26881 


.35743 


.27523 


.37974 


83 


84 


.24035 


.31640 


25182 
.25201 


33657 
83691 


.26351 


.35779 


.27543 


.88012 


84 


85 


.24054 


.81672 


.26371 


.35815 


.27663 


.38051 


86 


86 


.24078 


.31705 


.25220 


33726 


.26390 


.35852 


27583 


.88089 


36 


87 


.24092 


.31738 


.25240 


88760 


.26410 


.35888 


.27603 


.88127 


87 


88 


.24111 


.31771 


.25259 


88795 


.26480 


.35924 


.27623 


.88165 


88 


ia. 


•24130 


.31804 


•25278 


83830 


.26449 


.85961 


.27643 


.88204 


-sa 


40 


■24149 


.31837 


.25297 


88864 


.26469 


.85997 


.27663 


.88242 


40 


41 


.24168 


.31870 


.25317 


88899 


.26489 


.36034 


.27683 


.88280 


41 


42 


.24187 


.31903 


.25336 


33934 


.26509 


.86070 


.27703 


.88319 


42 


48 


.24206 


.81936 


.25356 


83968 


.26528 


.86107 


.27723 


.88857 


48 


44 


.24225 
.24244 


.31969 


.25375 


34003 


.26548 


.36143 


.27748 


.38396 


44 


45 


.82002 


.25394 


34038 


.26568 


.36180 


.27764 


.88434 


45 


46 


.24262 


.32035 


.25414 


84073 


26588 


.86217 


.27784 


.38473 


46 


47 


.24281 


.32068 


.25433 


84108 


.26607 


.36253 


.27804 


.38612 


47 


48 


.24300 


.32101 


.25452 


84142 


.26627 


.36290 


.27824 


.38550 


48 


19 


• 24320 


.32134 


.25472 


_84177 


26647 


.36327 


.27844 


.88589_ 


49 


50 


24339 


.32168 


.25491 


84212 


26667 


.36363 


.27864 


.88628 


50 


51 


.24358 


.82201 


.25511 


.84247 


.26686 


.86400 


.27884 


.88666 


51 


62 


24377 


.32234 


.25530 


84282 


26706 


36437 


.27906 


.88705 


52 


63 


.24396 


.82267 


.25549 


84817 


.26726 


.86474 


.27925 


.88744 


53 


54 


.24415 


.32301 


.25569 


.34352 


26746 


.36511 


.27946 


.88783 


54 


55 


.24434 


.82334 


.25588 


.84887 


.26766 


.36548 


.27965 


.88822 


56 


56 


.24453 


.32368 


.25608 


.84423 


.26785 


.36585 


.27988 


.88860 


56 


67 


.24472 


.32401 


.25627 


.34458 


.26805 


.36622 


.28005 


.88899 


57 


58 


.24491 


.82434 


.25647 


.34493 


.26826 


.36659 


28026 


.88988 


58 


iL. 


.24510 


.82468 


.25666 


.34528 


.26846 


.36696 


.28046 


.88977 


_59 


60 


.24529 


.82501 


.25686 


.84568 


.26865 


.86783 


.28066 


.89016 


60 



Digitized by VjOOQIC 



NATURAL VERSED SINES AND EXTERNAL SEC. 287 
44* 45** 46^ 47* 



# 


Vm. 


Ex. &ee. 


Vers* 


Ks- fiec. 


Verg. 


Ejcfiec, 


Vers, 


Ex. sec. 


t 


1 
a 
a 

f 


.28066 

.3«oa6 

.23108 
.28127 
.28147 




39016 
39055 
39 09 5 
39134 
39173 


-29289 
-29310 
29330 
.29351 
,29372 
.29392 
.29413 
.29433 
.29454 
.29475 


.41421 
-41463 
.415U4 
.41545 
.41586 


. 30534 

.30555 

.30576 

.30587 

30618 


.43958 
.43999 
-44042 
.44088 

.44129 


.31800 -46628 

-31821 -46674 

31843 .46719 

31864 -46765 

318S5. 46811 




1 
2 
3 
4 


6 

I 

i 


.28ie7 
.28187 
.28209 
.28228 
.2824S 
.28268 
-28289 
.28309 
■38329 
il8|5IL 
.28370 
28390 
28410 
.38431 
-28451 




39212 
39251 
S9291 
39330 
39369 


-41827 
.41669 
-41710 
41762 
.41793 


.30639 
.30660 
.30681 
.30702 
.30723 


.44173 

.44217 
^44260 
44304 
-44347 


'81907 -46857 
31928 .46903 
31949 -46949 
31971 -46995 

.31992 47041 


5 
6 
7 
B 
9 


10 

11 

12 
13 
14 




39409 
39448 
39487 
39527 
39566 
39606 
39646 
39685 
33725 
39764 


.29495 
.29S16 
.29537 
-29557 
.29578 


.41835 

.41878 

.41918 

.41959 1 

.42001 

.42042 ■ 

.42084 

.42126 

.42168 

-42210 


.30744 
.30765 
.30786 
30807 
.30628 


.44391 
-44435 
-44479 
-44523 
.44567 


32013 
.32035 

.32056 
33077 

.33099 


47087 
-47134 , 

47180 
.47226 

47272 


11 

12 
13 
14 


IS 
Ifi 
17 
IB 


.29599 
.29619 
. 29640 
-29661 
.^^681 


.30849 
-30870 
. 30891 
30912 
30933 


.44610 
-44654 
-44698 
-44742 
-44787 


.32120 
.32141 
33163 
32184 

32205 


.47319 
-47365 
.47411 
47458 
47504 


15 
16 
17 
IB 
1» 


20 

21 

as 

23 
34 


.28471 
.28492 
28512 
.28532 
.28553 




39804 
39844 
3S884 
39924 
39963 


.29702 
29723 
. 29743 
.29764 

:_£9J 85 _ 
29805 
.29826 
-29847 
29868 
.29888 
.29909 
. 29930 
.29951 
.29971 
-29992 


.42251 
.42293 
.42335 
.42377 
.42419 


30954 
3097& 
-30996 
31017 
31038 


-44831 
-44875 
-44919 
-44963 
-45C07 


-33227 .47551 
32248 .47598 

-32270 .47644 
32291 -47891 

-32312 -47738 


21 
22 
23 
24 


35 

27 
38 
29 


.28573 
.28593 
-28614 
.28834 
.28855 




40003 
40043 
40083 

40123 
40163 


.42481 
-42503 

.42545 
.42587 
.42630 


31059 
-31080 

31101 
.31122 

31143. 


.45052 
-45096 
.45141 
-45185 
-45229 
.45274 
-45319 
.45363 
-45408 
.45452 


-32334 
.32355 
-32377 

32398 
-32420 
-32441 
-32462 
-32484 

32505 
-32527 


-47784 
.47831 
-47878 
-47926 

-47972 
.48019 
.48086 
-48113 
-48180 
-48207 


25 
26 
27 
28 
39 


30 

31 
32 
33 
34 


.28875 
.2869S 
.28716 
.28736 
.28757 




40203 
40243 
40283 

4032* 
40364 


.42672 
.42714 
.42756 
.42799 
.42841 


-31165 
.31186 
-31207 
-31228 

.31270 
31291 
-31313 

.31334 

.31378 
.31397 
-31418 
.31439 
31461 


30 

31 
33 
33 


35 
U 
37 
38 
33 


-28777 
28797 
.28818 
-2S838 
28859 




40404 
40444 
40485 

40525 
405f5 


.30013 
-30034 

.30054 
30075 

-30096 


.42863 
.42926 
.42968 
.43011 
.43053 
.43096 
.43138 
.43181 
.43224 
-43267 


.45497 

-45543 
.45587 
.45631 

-45676 


. 33548 

32570 

32591 

32613 

_a_2634 


48254 
-48301 
.48349 
-48396 

48443 


as 

36 
37 
38 
39 


40 
41 
42 

44 


■28879 
.28900 
. 28920 
.28941 
-28961 




40B06 
40646 
40687 
40727 
40768 


-30117 
-30138 
.30158 
.30170 
.30200 


.45721 
.45766 
.45611 
.45856 


-32656 
-32677 
32699 

-32720 
-32742 


-48491 
.48538 
-48636 
.48833 
48681 


10 

41 
42 
43 
44 


45 
46 
47 
48 
4f> 


.28981 

.29002 
.29022 
.29043 
29083 


~ 


40808 
40849 

40890 
40930 
40971 
41012 
41053 
41093 
41134 
41175 
-41216 
.41257 
.4! 29 9 
^41339 
.413R0 


-30221 
. 30342 
30263 
-302S3 
. 3 0304 


^43310 
.43352 
.43395 
.43438 
.43481 


-31482 1 .45946 

.31503 .45992 
-31524 1 -46037 
.31545 46082 
.31567 1 .4ftt27 


32763 
32785 
32806 
32828 
33849 


48728 
-48776 
.48824 , 
48871 
48919 


45 
46 
47 
48 
49 


AO 

fil 
52 
53 
54 


.29084 
29104 
29125 

.29145 
29188 


.30325 
.30346 
30367 
30388 
30409_ 
. 30430 
-30451 
.30471 
.30492 
.30613 


.43524 
-43567 
.43610 
-43653 
-43696 
-43739 
.43783 
.43836 
43669 
^3912 


31688 

.31609 
.31630 
.3165] 
.31673 


.46173 
46218 
.46263 
.46309 
-46354 


32871 
32893 
.32914 
-32936 

.3rr57_ 

. 32979 
.33001 

-33023 
.33044 


48967 
-49015 
.49063 
-49111 

49159 1 


50 
51 

53 
53 
54 


55 
56 
67 
SS 
6© 


29187 
29207 
29228 
29248 

■ 29269 


-31694 
.31715 

-31736 
-31758 
31779 
-31800 


.46400 
.46445 
.46491 
.46537 
.46582 
.48628 


-49207 

.49255 
49303 
.49351 

.49399 


55 
56 
57 
58 
59 


00 


^29289 1 


.41421 


30534 


.43958 


33087 1 


.49448 


*MI 



y Google 



288 NATURAL VERSED SINES AND EXTERNAL SEC. 





48' 


49« 


50** 


51» 




1 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 


~ 


.33087 


.49448 


.84394 


.52425 


.35721 


.55572 


.37068 


.58902 


O 


1 


.33109 


.49496 


.34416 


.52476 


.35744 


.55626 


.37091 


.58959 


1 


2 


.33130 


.49544 


.34438 


.52527 


.35766 


.55680 


.37113 


.59016 


2 


8 


.33152 


.49593 


.34460 


.52579 


.35788 


.55734 


.37136 


.59078 


8 




.33173 


.49641 


.34482 


.52630 


■35810 


.55789 


.37158 


59130 


4 


"S" 


.33195 


.49690 


.34504 


.52681 


.35833 


.55843 


.37181 


.59188 


5 


6 


.33217 


.49738 


.34526 


.52732 


.35855 


.55897 


.37204 


.59245 


6 


7 


.33238 


.49787 


.34548 


.52784 


.35877 


.55951 


.37226 


.59302 


7 


8 


.33260 


.49835 


.34570 


.52835 


.35900 


.56005 


.37249 


.59360 


8 


9 


.33282 


.49884 


.34592 


.52886 
.52938 


■35922 


.56060 


.37272 


59418 


9 


10 


.33308 


.49938 


.84614 


.35944 


.56114 


.37294 


.59475 


lO 


11 


.33325 


.49981 


.34636 


.52989 


.35967 


.56169 


.37317 


.59533 


11 


12 


.33347 


.50030 


.34658 


.53041 


.35989 


.56223 


.37340 


.59590 


12 


13 


■33368 


.50079 


.34680 


.53092 


.36011 


.56278 


.37382 


.59648 


13 


14 


.33390 


.50128 


.34702 


.53144 


.36034 


.56332 


.37385 


.59706 


14 


15 


.33412 


.50177 


.34724 


.53196 


.36056 


.56387 


.37408 


.59764 


15 


16 


.33434 


.50226 


.34746 


.53247 


.36078 


.56442 


.37430 


.59822 


16 


17 


.33455 


.50275 


.34768 


. 53299 


.36101 


.56497 


.37453 


.59880 


17 


18 


.33477 


.50324 


.34790 


.53351 


.36123 


.56551 


.37476 


.59938 


18 


IL- 


•m99 


.50373 


.84812 


.53403 


.36146 


.56606 


■37498 


.59996 


-li 


20 


.33520 


.50422 


.34834 


.53455 


.36168 


.56661 


.37521 


.60054 


20 


21 


33542 


.50471 


.34856 


.53507 


.36190 


.56716 


.37544 


.60112 


21 


22 


.33564 


.50521 


.34878 


.53559 


.36213 


.56771 


.37567 


.60171 


22 


23 


.33586 


.50570 


.34900 


.53611 


.36235 


.56826 


.37589 


.60229 


23 


24_ 


.33607 


.50619 


.34928 


.53663 


.36258 


.56881 


.37612 


.60287 


24 


25 


.33629 


.50669 


.34945 


.53715 


.36280 


.56937 


.37635 


.60346 


25 


26 


.33651 


.50718 


.34967 


.53768 


.36302 


.56992 


.37658 


.60404 


26 


27 


.33673 


.60767 


.34989 


.53820 


.36325 


.57047 


.37680 


. 60463 


27 


28 


.33694 


.50817 


.35011 


.53872 


.36347 


.57103 


.37703 


.60521 


28 


29 


.33718 


.50866 


.35033 


.53924 


.36370 


.57158 


.37726 


. 60580 


29 


30 


.33788 


.50916 


.35055 


.53977 


.36392 


.57213 


.37749 


.60639 


30 


81 


.33760 


.50966 


.35077 


.54029 


.36415 


.57269 


.37771 


.60698 


31 


82 


.33782 


.51015 


.35099 


.54082 


.36437 


.57324 


.37794 


.60756 


32 


83 


.33803 


^ 51065 


.35122 


.54134 


. 36460 


.57380 


.37817 


.60815 


33 


84 


.33825 


.51115 


.35144 


.54187 


.36482 


.57436 


.37840 


.60874 


34 


35 


33847 


.51165 


.35166 


.64240 


.36504 


.57491 


.37862 


.60933 


35 


36 


.33869 


.51215 


.85188 


.54292 


36527 


.57547 


.37885 


.60992 


36 


37 


.33891 


.51265 


.35210 


.54345 


.36549 


.57603 


.37908 


.61051 


37 


88 


.83912 


.51314 


.35232 


.54398 


.36572 


.57659 


.37931 


.61111 


38 


89- 


.33934 


.51364 


.35254 


.54451 


.36594 


.57715 


.37954 


.61170 


-2fi 


40 


.33956 


.51416 


.35277 


.54504 


.36617 


.57771 


.37976 


.61229 


40 


41 


.33978 


.51465 


.35299 


.54557 


36639 


.57827 


.37999 


.61288 


41 


42 


.34000 


.51515 


.35321 


.54610 


.36662 


.57883 


.38022 


.61348 


42 


43 


.34022 


.51565 


.35343 


.54663 


.36684 


. 57939 


. 38045 


.61407 


43 


a^ 


•34044 


.51615 


.35365 


.54716 


•36707 


.57995 


.38068 


.61467 


44 


45 


34065 


.51665 


.35388 


.54769 


■36729 


.58051 


.38091 


.61526 


45 


46 


34087 


.51716 


.35410 


.54822 


■36752 


.58108 


38113 


.61586 


46 


47 


•34109 


.51766 


.85432 


.54876 


.36775 


.58164 


.38136 


.61646 


47 


48 


.34131 


.51817 


.35454 


.54029 


.36797 


.58221 


■38159 


.61705 


48 


49 


•34153 


.51867 


.35476 


.54982 


.36820 


.58277 


.38182 


.61765 


49 


50 


.34175 


.51918 


.35499 


.55036 


.36842 


.58333 


.38205 


.61825 


50 


61 


34197 


.51^68 


.35521 


.55089 


.36865 


.58390 


.38228 


.61885 


51 


52 


.34219 


.52019 


■35543 


.55143 


.36887 


.58447 


.38251 


.61945 


ii 


53 


.34241 


.52069 


.35565 


.55196 


.36910 


.58503 


.38274 


. 62005 


54 


.34262 


.52120 


.35588 


.55250 


.36932 


.58560 


.38296 


.62065 


54 


55 


.34284 


.52171 


.35610 


.55303 


.86955 


.58617 


.38319 


.62125 


55 


56 


.34306 


.52222 


.35632 


.55357 


.36978 


.58674 


.38342 


.62185 


56 


57 


34328 


.52273 


.35654 


.55411 


.37000 


.58731 


.38365 


.62246 


57 


58 


.34350 


.52323 


.35677 


.55465 


.37023 


.58788 


.38388 


.62306 


58 


59. 


•34372 


. .52374 


.35699 


.55518 


37045 


.58845 


.38411 


.62366 


-59 


60 


.34894 


.52425 


.85721 


.55572 


.37068 


.58902 


.38434 


.62427 


60 



Digitized by VjOOQIC 



NATURAL VERSED SINES AND EXTERNAL SEC. 289 
S*** 53** 54' 55* 



* 


Vera. 


Ex.»e*L 


Vera, 


Esc sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. pec. 


I 


o 
1 

a 

4 


■38434 
-33457 
.3S4«a 
.38603 
^8526_ 

■38549 
.38571 
■38594 
■38617 
■38640 


■§2427 
.32487 
■82&48 
.32309 
82869 
.32730 
-62791 
.63852 
-32913 
. 32974 


32819 
39342 i 
39865 1 
32338 
J9911. 
30935 
39958 
30&81 
40005 
40028 
40051 
40074 
40098 
40121 
40144 


-68184 
.66228 

-862S2 
.36357 
.66421 
-66436 
.86550 
.66615 
.86879 
■66744, 
.68809 
.66873 
68938 
.67003 
.67068 

.87133 
.67199 
.67284 
.67322 
.87394 
-67460 
.67525 
.67591 
67656 
.67722 1 


.41221 
-41245 
.41269 
-41292 
.41316 

41339 
.41383 

41386 
.41410 

41433 
.414B7 
-41431 
■41504 
.41529 
_41551 , 


-70130 

.70198 

.70267 

.70335 

^^0_403. 

.70472 ' 

.70540 

.70609 

.70677 

70748 

70815 

.70384 

-70953 

.71022 

71091 


.42642 
■42868 
-42620 
■42714 
.42733 


.74345 
.74417 
.74490 
.74562 
-74635 




1 
2 
3 
4 


5 

a 

7 
8 

9 


■42762 
43735 
42309 
42333 

-42857 


■74708 
.74781 
.74864 
.74227 
.75000 


5 

6 

I 

9 


la 

11 

12 
13 
U 


.38633 
■38686 
■3870B 
■38732 
■38755 


■63035 
.83026 
.63157 
.83218 
.63279 


-42831 
■42905 
-42222 
■42953 
.42273 


.75073 
.75146 
.76219 
.75293 
.75363 


10 

11 
12 
13 
14 


13 
16 
17 
Ifl 

21 
22 
23 
24 


■33773 
.33801 
■38834 
-38347 
_38B70 
■38333 
■38818 
■3833& 
.38982 
■3fl&35 
,38009 
■39032 
39055 
-39073 
.3&101 


.63341 
. 83402 
.63464 
.63535 
.63587 
.63643 
-83710 
.63772 
.63834 
-83895 
.63957 
.64012 
.64081 
.64144 
■34208 
■64363 
.64330 
■64393 
■64455 

64518 
■64680 
.64343 
.64705 
-64763 

S4931 
■64394 
■84937 
■65020 
.65083 
,65140 
■65209 
-86272 
.66336 
-85399 
■65462 
.65526 ' 
■6G532 
.65053 
■85717 
.05780^ 
■ 85844 
■65CD8 
.66972 
-68038 

-061OO 

. 86164 


40168 
.40191 
.40214 
.40237 
J0261 
■40284 
■40307 
.40331 
.40354 
■40378 


.41575 
■41592 
.41622 
.41646 
.41670 
-41693 
.41717 
.41740 

41764 
.41788 
-41811 
.41335 
.41859 
.41832 
.41906 

41830 
.41953 
.41977 
-42001 

42024 


-71160 
■71229 
.71298 
.71388 
„- 71437 
-71506 
.71576 
-71646 
-71715 
.71785 


.43000 

,43024 
43048 
43072 

.43096 
43120 
43144 
4S168 

.43192 
43218 


.75440 
.75513 
.75587 
.76661 

-7580S 
.76882 
.75956 
.76031 
■76105 


15 
16 
17 
18 

ua 

30 

21 
22 
23 
24 


25 
26 
27 
2ft 
2« 


.40401 
■40424 
.40448 
.40471 
.40494 

40513 
.40541 

40565 
.40533 
^40611_ 
.40635 
.40858 
.40632 
.40705 
.40728 
.40752 
.40775 
.40799 
.40822 
.40048 


.07788 

.67353 

.67919 

.87985 

-68051 

.68117 ' 

.68133 

.68250 

.68316 

-60382 

.68449 

.63515 

.38532 

.68843 

.08715 

.68732 

.68348 

.68916 

.68982 

. 69049 

.69116 

.69183 

-69250 

.69313 

.69385 

.09452 
.62520 
.69537 
.69655 
^^69723- 
.39790 
.69858 
-69926 
.39994 
.70062 
.70130 


.71355 
.71925 
-71995 
.72065 
.72135. 
.72205 
-72275 
.72346 
-72418 
. 72487 


43240 
43264 
.43287 
-4331 
-43335 


.76179 
.76253 
-73328 
.73402 
.73477 


2S 
23 
27 

2a 

22 


30 

31 

a2 

33 

34 


.39124 
■39147 
.32170 
-391*3 
.32218 


.43359 
■43383 
-43407 
43431 
.43455 


.76652 
-76626 
.76701 
.78776 
76851 


30 
3i 

33 
33 
34 


35 
37 

9? 


39239 
.30262 
.39288 
■39309 
,3^9332 
■39355 
.59378 
-39401 

39424 
.39447 


.42048 
-42072 
.43096 
-42119 
-42143 


72557 
.72628 
.72693 
-72769 

72840 


-43479 
.43503 
-43527 
-43551 
-43575 


.76926 
-77001 
-77077 
.77152 
.77227 


35 
36 
37 
33 


40 
41 
42 
43 
44 


42167 

-42191 
.43214 
-42238 
.42262 
.42285 
.42309 
.42333 
.42357 
.42381 


.72911 
.72982 
.73053 
.73124 
.73195 
.73267 
.73338 
. 734D2 
.73481 
.73552 


-43599 
-48623 
■43647 
.43671 
43695 


-77303 
.77379 
-77464 
.77530 
. 77806 


40 

41 
42 
43 
44 


45 
48 
47 
48 
4A 


.39471 
39494 
■39517 
.39540 
■39583 
■39586 
.32610 
.39333 
.39858 
.39379 
.39702 
■39726 
.39740 
■32772 

,5m5_ 

.39812 


.40B6S 
.40393 
.40916 
.40939 
.40263 
40280 
41010 
41033 
.41057 
.410BO 
.41104 
.41127 
.41151 
.41174 
.41198 
.41221 


.43720 
■43744 
43768 
43792 
.43813 


.77681 
.77757 
.77833 
.77910 
■77988 


45 
43 
47 
48 
49 


50 

51 
52 
53 
^4 


.42404 
.42428 
.42452 
.42476 
42492 
-42523 
.42547 
.42571 
.42525 
-42819 
.42642 


.73624 
.73696 
.73768 
.73840 
.73911 


.43840 
-43864 
.43888 
.43212 
4393B 


■78082 
.78138 
■78215 
■78291 
■79368 
.73445 
■79521 
.78598 
.79675 
78752 
.79829 


J50 

51 
52 
53 
54 


55 
55 
57 
5S 

GO 


.73983 
-74056 
.7*123 
.74200 

. 74272 
.74345 


43960 
.43984 
44008 
44033 
.44057 
-44081 


55 
56 
57 
58 
59 
60 



y Google 



290 NATURAL VERSED SINES AND EXTERNAL SEC. 
56' 57* 58' 59* 



t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


9 




1 

2 
3 
4 


44081 
.44105 
.44129 
.44153 
.44177 


.78829 
.78906 
.78984 
.79061 
.79138 


.45536 
45560 
45585 
•45609 
■45634 


.88608 
.83690 
.83773 
.83855 
.83938 


47008 
•47033 
•47057 

47082 
.47107 




88708 
88796 
88884 
88972 
89060 


48496 
•48521 
•48546 

48571 
■48596 




.94160 
94254 
94349 
.94443 
.94537 


o 

1 
a 

8 

4 


5 
6 
7 
8 
9 


.44201 
.44225 
.44250 
•44274 
■44298 


.79216 
. 79293 
.79371 
.79449 
.79527 


■45658 
■45683 
■45707 
■45731 
•45756 


.84020 
.84103 
.84186 
. 84269 
•84352 


47131 
•47156 
■47181 
•47206 
•47230 




89148 
89237 
89325 
89414 
89503 


■48621 
■48646 
■48671 
■48696 
48721 




.94632 
.94726 
.94821 
.94916 
.95011 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.44322 
.44346 
.44370 
.44395 
•44419 


.79604 
.79682 
.79761 
.79839 
.79917 


•45780 
■45806 
•45829 
■45854 
.45878 


.84435 
.84518 
.84601 
.84685 
• 84768 


.47255 
.47280 
.47304 
.47329 
.47354 




89591 
89680 
89769 
89858 
89948 


48746 
•48771 
48796 
48821 
.48846 




95106 
.95201 
.95296 
.95392 
.95487 


10 

11 
13 
13 
14 


15 
16 
17 
18 


44443 
•44467 
•44491 
•44516 
.44540 


.79995 
.80074 
.80152 
.80231 
• 80309 


•45908 
■45927 
•45951 
■45976 
• 46000 


.84852 
.84935 
.85019 
.85103 
•85187 


47379 
.47408 
47428 
47453 
•47478 




90037 
90128 
90216 
90305 
90395 


48871 
48896 
48921 
48946 
48971 




95588 
.96678 
.96774 
.95870 
.96966 


15 
16 

iS 


SO 

21 
22 
23 

24 


44564 
44588 

■44612 
44637 

■44661 


.80388 
.80467 
.80546 
.80625 
.80704 


■46025 
■46049 
■46074 
•46098 
•46123 


.85271 
.85355 
.85439 
.85523 
•85608 


■47502 
■47527 
■47552 
.47577 
•47601 




90485 
90575 
90665 
90755 
90845 


■48996 
49021 
■49046 
■49071 
.49096 




.96062 
.96158 
.96256 
.96351 
96448 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


■44685 
■44709 
•44734 
■44758 
■44782 


.80783 
.80862 
.80942 
.81021 
.81101 


•46147 
•46172 
46196 
•46221 
■46246 


.85692 
.85777 
.85861 
.85946 
.86031 


•47626 
■47651 
•47678 
•47701 
•47725 




90935 
91026 
91116 
91207 
91297 


•49121 
49146 
49171 
•49196 
■49221 




96544 
96641 
96738 
96835 
96932 


25 
26 
27 


30 

31 
32 
33 
34 


44806 
■44831 
•44855 
■44879 
■44903 


.81180 
.81260 
.81340 
.81419 
•81499 


■46270 
■46295 
•46319 
•46344 
•46368 


.86116 
.86201 
.86286 
.86371 
•86457 


.47750 
•47775 
.47800 
47825 
.47849 




91388 
91479 
91570 
91661 
91752 


•49246 
■49271 
.49296 
49321 
.49346 




97029 
.97127 
97224 
97322 
97420 


30 

31 
82 
38 
34 


35 
36 
87 
38 
J9. 


44928 
•44952 

44976 
■45001 
■45025 


.81579 
.81659 
.81740 
.81820 
■81900 


•46393 
•46417 
■46442 
.46466 
•m91 


.86542 
.86627 
.86713 
.86799 


•47874 
•47899 
.47924 
•47949 
•47974 




91844 
91935 
92027 
92118 
92210 


49372 
•49397 

49422 
•49447 
•49472 




97617 
97615 
97713 
.97811 
97910 


35 
36 
37 
38 
-32 


40 

41 
42 
43 
44 


■45049 
■45073 
•45098 
•45122 
•45146 


■81981 
.82061 
. 82142 
.82222 
•82303 


46516 
•46540 
•46565 
•46589 
•46614 


.86970 
.87056 
.87142 
.87229 
■87315 


47998 
.48023 
•48048 
•48073 
•48098 




92302 
92394 
92486 
92578 
92670 


■49497 
■49522 
■49547 
49572 
■49597 




98008 
98107 
98205 
98304 
98403 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•45171 
45195 
■45219 
■45244 
45268 


.82384 
.82465 
.82546 
.82627 
82709 


•46639 
•46663 
■46688 
■46712 
46737 


■87401 
•87488 
■87574 
.87661 
■87748 


48123 
48148 
•48172 
•48197 
• 48222 




92762 
92855 
92947 
93040 
93133 


.49623 
.49648 
49673 
•49698 
■49723 




98502 
98601 
98700 
98799 
98899 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


4529^ 
45317 
45341 
•45365 
■45390 


.82790 
.82871 
.82953 
.83034 
.83116 
.83198 
.83280 
.83362 
.83444 
83526 


■46762 
■46786 
■46811 
■46836 
■ 468e0__ 

■46885 
•46909 
•46934 
•46959 
.46983 


.87834 
.87921 
.88008 
.88095 
■88183 


■ 48247 
■48272 

48297 
•48322 
■48347 
•48372 
■48396 
■48421 
•48446 

48471 




93226 
93319 
93412 
93505 
93598 


49748 
49773 
49799 
•49824 
49849 




98998 
99098 
99198 
99298 
99398 


50 

51 
52 
53 
54 


55 
56 
67 
58 
59. 


■45414 
■45439 

45463 
■45487 

45512 


.88270 
.88357 
.88445 
.88532 
.88620 




93692 
93785 
93879 
93973 
94066 


•49874 
■49899 
■49924 
•49950 
■49975 




99498 
99598 
99698 
99799 
99899 


55 
56 
57 
58 
59 


60 


■45536 


.83608 


.47008 


.88708 


.48496 




94160 


.50000 


1 


00000 


60 



Digitized by VjOOQIC 



NATURAL VERSED SINES AND EXTERNAL SEC. 
eo** 61* ea* 63* 



291 



9 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


f 


^ 


. 50000 


1. 00000 


51519 


1.06267 


.53053 


1.13005 


.54601 


1.20269 


~ 


1 


50025 


1.00101 


.51544 


1.06375 


.53079 


1.13122 


.54627 


1.20395 


1 


2 


.50050 


1.00202 


.51570 


1. 06483 


.53104 


1.13239 


.54653 


1. 20521 


a 


3 


50076 


1.00303 


51595 


1. 06592 


.53130 


1.13356 


.54679 


1.20647 


8 


4 


.50101 


1.00404 


.51621 


1.06701 


.53156 


1.13473 


.54705 


1. 20773 


4 


5 


.50126 


1. 00505 


51646 


1.06809 


.53181 


1.18590 


.54731 


1.20900 


6 


6 


.50151 


1. 00607 


.51672 


1.06918 


.53207 


1.13707 


.54757 


1.21026 


6 


7 


.60176 


1. 00708 


.51697 


1.07027 


.53233 


1.18825 


. 54782 


1.21153 


7 


8 


.50202 


1.00810 


.51723 


1.07137 


.53258 


1.13942 


. 54808 


1.21280 


8 


9 


.50227 


1.00912 


.51748 


1.07246 


.53284 


1.14060 


54834 


1.21407 


9 


10 


.60252 


1. 01014 


.51774 


1.07356 


53310 


1.14178 


. 54860 


1.21535 


10 


11 


. 50277 


1. 01116 


.51799 


1.07465 


. 53336 


1.14296 


.54886 


1.21662 


11 


12 


.50303 


1.01218 


.51825 


1.07575 


• 53361 


1 . 14414 


. 54912 


1.21790 


12 


13 


.50328 


1.01320 


.51850 


1.07685 


. 53387 


1.14533 


.54938 


1.21918 


13 


14 


.50853 


1.01422 


.51876 


1.07795 


• 53413 


1.14651 


.54964 


1.22045 


14 


15 


.50378 


1.01525 


.51901 


1.07905 


. 53439 


1.14770 


54990 


1.22174 


15 


16 


.50404 


1.01628 


.51927 


1.08015 


. 53464 


1.14889 


55016 


1.22302 


16 


17 


.50429 


1.01730 


.51952 


1.08126 


. 53490 


1.15008 


55042 


1.22430 


17 


18 


.50454 


1.01833 


51978 


1.08236 


.53516 


1.15127 


. 55068 


1. 22559 


18 


iL. 


.50479 


1.01936 


.52008 


1.08347 
1.08458 


.53542 




.65094 
55120 


1,22?88 
1. 22817 


-12 


20 


.50505 


1.02089 


.52029 


53567 


1.15366 


20 


21 


.50530 


1. 02148 


52054 


1. 08569 


.53593 


1.16485 


55146 


1. 22946 


21 


22 


.50555 


1.02246 


.52080 


1.08680 


.53619 


1.15605 


55172 


1. 28075 


22 


23 


.50581 


1.02349 


.52105 


1.08791 


.53645 


1.15725 


55198 


1. 23205 


23 


24 


.50606 


1.02453 


.52131 


1.08903 
1.09014 


.53670 
53696 


1.15845 
1.15965 


55224 
55250 


;. 28384 
1.23464 


24 


25 


.50631 


1.02557 


.52156 


25 


26 


. 50656 


1. 02661 


.52182 


1.09126 


.53722 


1.16085 


.55276 


1. 23594 


26 


27 


.50682 


1. 02765 


.52207 


1.09238 


.53748 


1.16206 


. 65302 


1. 23724 


27 


28 


.50707 


1. 02869 


•52233 


1.09350 


.53774 


1.16326 


55328 


1. 23855 


28 


29 


.50732 


1.02978 


. 52259 


1.09462 


.53799 


1. 16447 


.65854 


1-28985 


-29 


30 


.50758 


1.03077 


.52284 


1.09574 


.53825 


1.16568 


55380 


1. 24116 


30 


81 


.50783 


1. 03182 


.52310 


1.09686 


.53851 


1.16689 


. 55406 


1 . 24247 


31 


32 


.50808 


1.03286 


.52335 


1.09799 


.53877 


1.16810 


.55432 


1. 24378 


82 


33 


50834 


1.03891 


.52361 


1.09911 


• 53903 


1.16982 


55458 


1. 24509 


83 


34 


50859 


1. 03496 


.52386 


1.10024 


53928 


1.17053 


.55484 


1. 24640 


34 


35 


.50884 


1. 03601 


. 52412 


1.10137 


. 53954 


1. 17175 


.55510 


1.24772 


35 


36 


.60910 


1. 03706 


. 52438 


1.10250 


.53980 


1.17297 


.55536 


1.24903 


86 


87 


.50935 


1. 03811 


. 52463 


1.10868 


. 54006 


1.17419 


.55563 


1.25035 


87 


38 


60960 


1. 03916 


52489 


1.10477 


.54032 


1.17541 


55589 


1. 25167 


38 


39. 


.50986. 


1.04022 


.52514 


1.10590 


.54058 


]. 17663 




L?5300 


-J9 


40 


.51011 


1- 04128 


. 52540 


1.10704 


.54083 


1.17786 


.55641 


1.25432 


40 


41 


.51036 


1- 04233 


•52566 


1.10817 


.54109 


1.17909 


55667 


1. 25565 


41 


42 


.51062 


1.04339 


52591 


1.10931 


.54135 


1.18031 


55693 


1. 25697 


42 


43 


.51087 


1. 04445 


52617 


1.11045 


.54161 


1.18154 


55719 


1.25880 


43 


44 


.61113 


1. 04551 


.52642 
• 52668 


1.11159 
1.11274 


.54187 
.54213 


1. 18277 
1.18401 


.55745_ 
55771 


1.25963 
1. 26097 


_44 


45 


.51138 


1.04658 


45 


46 


.51163 


1.04764 


52694 


1.11388 


54238 


1. 18524 


.55797 


1. 26230 


46 


47 


51189 


1. 04870 


52719 


1.11503 


. 54264 


1.18648 


.55828 


1.26364 


47 


48 


.51214 


1. 04977 


. 52745 


1.11617 


.54290 


1.18772 


. 55849 


1. 26498 


48 


49 


.51239 


1.05084 


•52771 


1.11732 


.54316 


1.18895 
1.19019 


.55876 
.55902 


1-26632 
1.26766 


49 


60 


.51265 


1. 05191 


52796 


1.11847 


.54342 


50 


51 


.51290 


1.05298 


52822 


1.11963 


. 54368 


1.19144 


.55928 


1.26900 


51 


52 


.51316 


1. 05405 


52848 


1.12078 


.54394 


1.19268 


.55954 


1.27035 


52 


53 


.51341 


1.05512 


• 52873 


1. 12193 


. 54420 


1.19393 


.55980 


1.27169 


53 


54 


.51366 


1. 05619 


52899 


1.12309 


•54446 


1.19517 


.56006 
.56082 


1.27304 
1. 27439 


54 


55 


.51392 


1. 05727 


.52924 


1.12425 


.54471 


1.19642 


55 


66 


.51417 


1. 05835 


.52950 


1.12540 


.54497 


1. 19767 


.56058 


1.27574 


li 


67 


. 51448 


1.05942 


.52976 


1.12657 


. 54523 


1.19892 


.56084 


1.27710 


57 


58 


.51468 


1.06050 


.53001 


1.12773 


.54549 


1.20018 


.56111 


1.27845 


1 


ja. 


.51494 


1.06158 


.53027 




.54575 


1.20143 


.56187 


^. 27981 


So 


.51519 


1. 06267 


.53053 


1.18005 


.54601 


1.20269 


.56163 


1.28117 


60 



Digitized by VjOOQIC 



292 NATURAL VERSED SINES AND EXTERNAL' SEC. 
64* es'* 66* 67' 



# 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


r 


"o" 


.56168 


1.28117 


.57738 


1.86620 


59826 


1.46859 


60927 


1.55980 


""5 


1 


.56189 


1.28258 


.57765 


1.36768 


.59858 


1.46020 


.60954 


1. 66106 


1 


2 


.56215 


1.28390 


.57791 


1.S691'* 


.59379 


1.46181 


.60980 


1.56282 


2 


8 


.56241 


1.28526 


.57817 


1.37064 


.59406 


1.46342 


.61007 


1.56458 


s 


4 


■56267 
.56294 


1.28663 
1.28800 


.57844 
.57870 


1.37212 


.59433 


1.46504 
1.46665 


.61034 
.61061 


1.56684 
1.56811 




5 


1. 37361 


.59459 


■^ 


6 


•56320 


1.28937 


.57896 


1. 87509 


59486 


1.46827 


.61088 


1. 56988 


ft 


7 


■56346 


1.29074 


.57923 


1. 37658 


.59512 


1.46989 


.61114 


1. 57165 


7 


8 


■56372 


1.29211 


■57949 


1. 37808 


.59589 


1.47152 


.61141 


1. 57842 


8 


9 


■56398 


1.29349 


■57976 


1. 37957 


.59566 


1-47314 


-61166 


1.57520 


9 


10 


■ 56425 


1.29487 


■58002 


1^ 38107 


.59592 


1- 47477 


-61195 


1-57698 


10 


11 


■56451 


1.29625 


■58028 


1.38256 


■50619 


1.47640 


-61222 


1.57876 


11 


12 


.56477 


1.29763 


■58055 


1.38406 


■59645 


1.47804 


-61248 


1- 58054 


12 


18 


■56503 


1. 29901 


■58081 


1.38556 


. 59672 


1.47967 


-61275 


1.58233 


13 


14 


56529 


1. 30040 


58108 


1-58707 


•59699 


^ 48131. 
1.48295 


-61302 
-61829 


1. 58412 
1-58691 


U 


15 


56555 


1.30179 


■58134 


1-88857 


59725 


10 


16 


■ 56582 


1^ 30318 


58160 


1-39008 


.59752 


1.48459 


-61356 


1-58771 


18 


17 


■56608 


1.30457 


58187 


!■ 39159 


.59779 


1.48624 


-61888 


1-58950 


17 


18 


■56634 


1. 30596 


58218 


1.39311 


59805 


1.48789 


-61409 


1-59180 


18 


ia. 


.56660 
■ 56687 


l.?02?i 
1. 30875 


5^9 
58266 


1^ 39462 
1.39614 


•5983i^ 
■59859 


1.48954 


-61436 


1-59311 


-it 


20 


1. 49119 


.61463 


1-59491 


20 


21 


56718 


1.81015 


58293 


1.89766 


■59885 


1-49284 


.61490 


1. 59672 


21 


22 


■56739 


1.31155 


58319 


1.89918 


59912 


1-49450 


.61517 


1. 59853 


22 


28 


56765 


1.31295 


58345 


1-40070 


.59938 


1-49616 


.61544 


1. 60035 


28 


24_ 


56791 


1. 81436 


.58372 
58398 


1-40222 
1.40375 


59965 
•59992 


1-49782 
1-49948 


.61570 


1. 60217 


24 


25 


56818 


1.31576 


.61597 


1-60899 


20 


26 


56844 


1. 31717- 


•58425 


1.40528 


60018 


1.50115 


.61624 


1-60581 


28 


27 


■56870 


1^ 31858 


•58451 


1.40681 


■60045 


1.50282 
1.50449 


.61651 


1-6P763 


27 


28 


56896 


1^ 31999 


58478 


1.40835 


■60072 


.61678 


1-60946 


28 


29 


■56923 


1^32140 


•58504 


1.4P98? 
1.41142 


.60098 
■60125 


1.50617 
1.50784 


-61705 
-61782 


1-61129 


29 


30 


■56949 


!■ 82282 


•58531 


1.61813 


30 


81 


.56975 


1^32424 


58557 


1.41296 


.60152 


1.50952 


-61759 


1-61496 


81 


82 


■ 57001 


1.32566 


•58584 


1.41450 


.60178 


1.51120 


-61785 


1-61680 


32 


83 


■57028 


1. 32708 


58610 


1.41605 


■60205 


1.51289 


.61812 


1-61864 


83 


84 


■57054 


1. 32850 


•58637 
•58663 


1.41760 
1.41914 


■60232 
.60259 


1.51457 
1.51626 


-61839 


1-62049 


84 


35 


■57080 


1^ 82993 


-61866 


1-62234 


80 


86 


■57106 


1^ 33135 


58690 


1.42070 


.60285 


1.51796 


■61898 


1-6:2419 


36 


87 


■57133 


1.83278 


•58716 


1.42225 


.60312 


1.51965 


■61920 


1- 62604 


37 


38 


■57159 


1. 33422 


•58743 


1.42380 


.60339 


1.52184 


■61947 


1-62790 


38 


89. 


J57185 


1.33565 


•5876?. 


1.4253fi. 


. 60865, 




.61974 


1-62976 


uJI 


40 


■57212 


1.38708 


58796 


1.42692 


.60392 


1.52474 


.62001 


1-68162 


40 


41 


■57238 


1. 33852 


•58822 


1.42848 


■60419 


1.62645 


62027 


1- 63348 


41 


42 


■57264 


1. 83996 


58849 


1.43005 


■60445 


1.52815 


-62054 


1-68535 


42 


48 


.57291 


1. 34140 


•58875 


1.43162 


60472 


1.52986 


-62081 


1-68722 


48 


jML 


.57317 
57343 


1.84284 


.58902 


1-4331$. 


.60499. 
.60526 


1-53157 
1-53829 


-62108 
-62185 


1-63909 
1-64097 


45 


45 


1.34429 


58928 


1.43476 


46 


■57369 


1.34578 


■58955 


1.43633 


.60552 


1-53500 


■62163 


1. 64285 


46 


47 


57396 


1.34718 


■58981 


1.43790 


.60579 


1-53672 


■62189 


1. 64473 


47 


48 


57422 


1.34863 


.59008 


1.43948 


60606 


1-53845 


■62216 


1-64662 


48 


49 


57448 


1.35009 
1-35154 


^59034 
59061 


1.44106 
1.44264 


■ 60683, 
.60659 


1.54017 
1.54190 


62248 
. 62270 


1-64851 
1. 65040 


49 


50 


.57475 


50 


51 


57501 


1.35300 


59087 


1.44428 


■60686 


1- 64368 


-62297 


1^ 65229 


51 


52 


■57527 


1.85446 


.59114 


1. 44582 


-60718 


1.54536 


-62324 


1- 65419 


52 


58 


.57554 


1.35592 


.59140 


1.44741 


.60740 


1.54709 


62351 


1-65609 


58 


54 


■57580 


!■ 35738 


.59167 


1-44900 


^0_76e_ 
.60793 


1^ 64883 


62878 


1-65799 


54 


55 


.57606 


1^ 35885 


.59194 


1^ 45059 


1-55057 


■62406 


1-65989 


58 


56 


.57638 


1^ 36031 


-59220 


1.45219 


.60820 


1.55231 


■62481 


1-66180 


56 


57 


.57659 


1. 36178 


.59247 


1.45378 


-60847 


1-55405 


■62458 


1-66871 


57 


58 


■57685 


1. 36325 


. 59273 


1.46589 


-60878 


1-55580 


62485 


1-66563 


58 


6», 


•5771? 


1.3647? 


.59300 


1.45690 


J0900. 


1.55755 


mi^ 


^W55 


-S2 


60 


■57738 


1^ 36620 


■59326 


1. 46859 


60927 


1-65980 


■62539 


1- 66947 


00 



Digitized by VjOOQIC 



NATURAL VERSED SINES AND EXTERNAL SEC. 293 





68*» 


69** 


70** 


7 1'* 




f 


Vers. 


Ex. ettc. 


Ver.^. 


Kx, sj.i-. 


Wrs. 


Kx- ^cc. 


Ver?<. 


Kx, per. 


f 


o 
1 

s 
s 

4 


■33339 
32366 

■33393 
32320 

■32347 


1-66947 
1-67139 
1-67332 
1 67525 
1-67718 


64163 
64190 
64218 
64245 
-64272 


1-79043 
1-79254 
1-79466 
1-79679 
1 7989J_ 
1-30104 
1-30318 
1-80531 
1 80746 
1-60960 


65798 
66825 
65853 
65830 

J59pX 

.65935 
65962 

.65939 
66017 
66044 

-66071 

- 66099 

- 66123 
.66154 

66181 


1 92380 

1-92614 
1-92849 
1 93033 
1,93318 


67443 2. 071 55 
-67471 3-07415 
,67496 2. 07675 
-37523 12 07936 
-67553 3 0B197 




1 
2 
3 
4 


5 
6 

1 

9 


.32374 
.32701 
■32723 
.32755 
■33782 


1-67911 
1,6«105 
1 68299 
1-63494 
1-68680 


.64293 
-64326 
-64353 
-64381 
-64408 


1-93554 
1 . 93790 
1 94026 ' 

1 . 94263 
1 94500 


-67581 
67608 

-67636 
67663 

-67691 


3 06459 
2 08721 
2 03933 
2 09246 
2.09510 


5 

a 

7 
8 

9 


11 
12 
13 
14 


62809 
62836 

■62833 
63890 

-fl2&17 


1-38384 
1.39079 
1-39275 
1^ 69471 
1-69667 


-64435 
-64462 
-34489 
-64517 
-64544 

-64571 
-64598 
■64625 
-64653 
-64680. 

■64707 

. 64734 

. 64761 

64789 

64616 


1-81175 

1.81390 
1. 81 605 
1.81621 
1 82037 


1 94737 
1-94975 
1-95213 
1-95453 
1-95691 


-67718 
-67746 
-67773 
-67801 
-67829 


2 09774 
2a0038 
2^0303 
2a0568 
2 10834 


10 

11 
12 
13 
U 


IS 
IB 
17 
18 


62344 
.62971 
629B8 
63025 
63052 


1.39364 
1-70061 
1^ 70253 ! 
1-7045S 
1 ^ 70653 


1 32254 
1-82471 
1 82688 

1-82906 

1.83124 , 

1.B3342 

1.33561 

1. 33780 

1,83999 

1.84219 

1-34439 

1-84659 

1-84380 

1-S5102 

1- 35323 

1- 85545 

1-85767 

1-85990 

1-36213 

1-86437 


66208 
66236 
-66263 
66290 
66318 
66345 
66373 
66400 
66427 
66455 

6e4i2 

66510 
66537 
66564 

.66592 


1-95931 

1-96171 
1-96411 
1 96652 
1 96893 
1.97135 
1.97377 
1-97619 
1,97862 
i 93106 
1 93349 
1 98594 
1.96833 
1 99083 
1.99329 


-67856 21 1101 
.67684 3-11367 
-67911 2-11635 
-67939 2-11903 
-67963 2^13171 
-67994 '2-12440 
68021 2 12709 
68049 2 12976 
6B077 i2 13249 
-68104 l2 13520 


13 
13 
17 
IB 
_19 


241 

21 
32 
23 
24 


63079 
.63106 
63133 
63161 
63138 


1^70S51 
1-71050 
1^ 71249 
!■ 71448 
1 71647 
1 71847 
1 . 72047 
1 72247 
1. 72448 
1 72649 


21 
32 
23 
24 


2fi 
2§ 
37 
2t 
3fl 


63215 
63242 
63269 
S3296 
B3323 


64343 
64870 
64898 

-64925 
6495JL 
64979 
65007 
65034 
65061 
65083 

.65116 
65143 
65170 
65197 

. 6ft225 


-66132 
-68159 
-66137 
-68214 
-68242 


2-13791 
2 14063 
2-34335 
2 . 14608 
2^14831 


25 
33 
27 
33 
39 


au 

31 
82 
33 
34 


33350 
33377 

-63404 
63431 

-63458 


1 72S50 
1.73052 

1 73254 
1. 73456 
1 73 559 


66619 
36647 
66674 
66702 
66729 


1 99574 
1. 99821 

2 00067 
3.00315 

2-00562 


-68270 
66297 
68325 
£8352 
r8380 


2 ■15155 
2-15429 
2 15704 
2 15979 
2-16255 
2-16531 
2. 16808 

2 170B5 

3 17363 
^17641 


m 

31 
32 
33 
34 


Si, 
3« 
37 
38 


-63465 
63512 
63539 
63536 

.63594 


I. 73862 

1.74065 
1 74269 
1. 74473 
1.74B77 


1-66661 
1 66885 
1.B7109 
1 87334 
1.87560 


-66756 
56784 
66811 

.66839 
.66866 


;2-008lO 
,2-01059 
2 01308 

2^01557 
2^01807 


6B408 
68435 
-68463 
68490 
68518 


35 
36 
37 
38 
38 


40 
41 

42 
43 
44 


63621 
63648 
63675 
63702 
6372J_ 
637G6 
63783 
63810 
63838 
63865 
e3B&2 
63919 
33943 
-33973 
-34000 
64027 
64055 
640B2 
64109 
64138 


1 74881 ' 
1. 75086 

1 . 75292 
1 . 75497 
1 . 75703 


-65252 
65279 
65306 
-65334 
.65361 


1. 87785 
1 68011 
1 68238 
1.88465 
1 88692 
1. 88920 
1.89148 
1 8937B 
1 89605 
1. 89834 
1.90063 
1 90293 
1,90524 
1 90754 
1-90986 
1 91217 
1-91449 
1-B16B1 
1 91914 
1-92147 


66894 
66921 
66949 
66976 
67003 
-67031 
67058 
67066 
67113 
67141 
67168 
67196 
-67223 
.f7951 
■67278. 
-67306 
-67333 
-67361 
-673B8 
67416 


2.02057 
2 02308 
2 02559 
2 0281O 
2.03062 

2 03315 
2-03568 
2-03821 
2-04075 

3 04329 
2 04584 
2 04839 
2 . 05094 
2 05350 
2 05607 

2 05864 

3 06121 
2 06379 
2 06637 
2 06896 


68546 
68573 
63601 
68628 
_6835e 


3.17920 
2-18199 
2-18479 

2 18759 

3 19040 


40 

41 
42 
43 

44 


45 
46 
47 
48 
46 


1.75909 

1-76116 

1-76323 

1-76530 

1.76737 

i 76945 

1.77154 

1 77362 

1.77571 

1 77780_ 

i 77990 

1-78200 

1.78410 

1 78621 

l.7efl32 


65388 
.65416 
65443 
65470 
65497 
65535 
66552 
65579 
65807 
.65634 
^65661 
.65689 
65716 
65743 
.65771 


68684 2 19333 
68711 12 19604 
6B739 12 198B6 
53767 12 20169 
.38794 '3 30453 


45 
43 

47 
48 
49 


50 
51 
32 
33 
54 


63822 
63849 
63877 
63905 
68932 


2-20737 
2 21031 
2 21306 
.2-21592 
2-21378 


50 
51 

52 
S3 
S4 


55 
56 
37 
58 
53 


-68960 
-68938 
.69015 
69043 

_e90ZL 

39093 


2 23165 
2-23452 
2-22740 
2 23028 

2.23317 
2-33607 


53 
53 
57 
58 
«B0 


60 


64163 


1 79043 


65798 jl- 92360 


67443 


2-07135 


m 



y Google 



294 NATURAL VERSED SINES AND EXTERNAL SEC. 
730 73«» 74" 76** 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


f 




1 

2 

8 

^4 


■69098 
.69126 
.69154 
.69181 
.69209 


2.28607 
2.23897 
2. 24187 
2.24478 
2. 24770 


. 70768 
.70791 
•70818 
•70846 
•70874 


2. 42080 
2.42856 
2. 42688 
2. 48010 
2-43337 


.72436 
72464 
.72492 
.72520 
-72548 


2. 62796 
2.68164 
2. 68538 
2.68903 
2^64274 


74118 
-74146 
. 74174 
.74202 
. 74231 


2. 86370 
2-86790 
2-87211 
2. 87633 
2-88056 




1 
2 
8 

4 


5 
6 
7 
8 
9 


.69237 
.69264 
. 69292 
.69820 
. 69347 


2.25062 
2.25355 
2. 25648 
2-25942 
2.23287 


. 70902 
•70930 
.70958 
.70985 
•71013 


2.43666 
2. 43995 
2. 44324 
2.44655 
2.44986 


.72576 
.72604 
.72632 
.72660 
.72688 


2 64645 
2 65018 
2.65891 
2. 65765 
2.66140 


. 74259 
-74287 
-74315 
-74343 
.74871 


2-88479 
2-88904 
2-89330 
2-89756 
2.90184 


5 
6 
7 
8 
8 


10 

11 
12 
18 
14 


■69375 
69403 
■69480 
•69458 
.69486 


2.26531 
2. 26827 
2.27123 
2.27420 
227717 


.71041 
•71069 
•71097 
•71125 
•71153 


2.45317 
2.45650 
2.45988 
2.46816 
2. 46651 


.72716 
.72744 
.72772 
. 72800 
. 72828 


2-66515 
2. 66892 
2.67269 
2.67647 
2.68025 


.74399 
-74427 
.74455 
.74484 
.74512 


2-90613 
2-91042 
2-91473 
2.91904 
2-92337 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.69514 
.69541 
69569 
69597 
■69624 


2.28015 
2. 28318 
2. 28612 
2. 28912 
2.29212 


•71180 
71208 
.71286 
.71264 
.71292 


2.46986 
2.47321 
2.47658 
2. 47995 
2-48333 


.72856 
72884 
.72912 
. 72940 
-72968 


2. 68405 
2. 68785 
2. 69167 
2.69549 
2.69931 


-74540 
•74568 
-74596 
.74624 
.74652 


2-92770 
2. 98204 
2-93640 
2-94076 
2-94514 


15 
16 
17 
18 
^2 


no 

21 
22 
28 
24 


69652 
.69680 
.69708 
.69735 

69763 


2. 29512 
2.29814 
2. 30115 
2.30418 
2.30721 


.71320 
71348 
71375 
-71403 
-71431 


2- 48671 
2.49010 
2.49850 
2. 49691 
2. 50032 


.72996 
. 73024 
. 78052 
.78080 
•73108 


2-70315 
2- 70700 
2-71085 
2-71471 
2- 71858 
2.72246 
2. 72685 
2. 78024 
2. 78414 
2.78806 


74680 
. 74709 
.74737 

74765 
.74793 
.74821 
.74849 
.74878 
. 74906 
.74934 


2.94952 
2-95392 
2-95832 
2-96274 
2-96716 
2-97160 
2-97604 
2-98050 
2-98497 
2-98944 


ISO 

21 
22 
28 
24 


25 
26 
27 
28 
29 


69791 
69818 
.69846 
69874 
69902 


2.31024 
2.81328 
2.81683 
2.31939 
2.32244 
2.32551 
2-32858 
2-33166 
2. 33474 
2. 33783 


-71459 
71487 
.71515 
.71543 
.71571 
.71598 
71626 
71654 
.71682 
.71710 


2. 50374 
2. 50716 
2. 51060 
2.51404 
2.61748 
2.52094 
2. 52440 
2. 52787 
2.53134 
2. 53482 


.78136 
.73164 
.78192 
.73220 
.73248 
.73276 
. 73304 
- 73332 
.73360 
.73388 


26 
26 
27 
28 
29 


30 

81 
82 
33 
34 


■69929 
69957 
69985 
. 70013 
.70040 


2. 74198 
2. 74591 
2-74984 
2.75379 
2.75775 


-74962 
74990 
.75018 
.75047 
. 75075 


2-99393 
2. 99843 
3-00293 
3-00745 
3-01198 


30 

31 
82 
38 

34 


35 
86 
87 
88 


.70068 
.70096 
.7P124 
■ 70151 
.70179 


2. 34092 
2.34403 
2. 34718 
2. 35025 
2-35336 
2. 35649 
2.35962 
2. 36276 
2.36590 
2.86905 


71738 
.71766 

71794 
. 71822 
•71850 


2.53831 
2. 54181 
2. 54531 
2. 54883 
2.55235 


.73416 
.73444 
. 73472 
.73500 
■73529 


2.76171 
2.76568 
2-76966 
2- 77365 
2.77765 


. 75103 
-75131 
-75159 
.75187 
■75216 


3.01652 
8. 02107 
8. 02563 
3. 03020 
3-03479 
3. 08938 
3. 04898 
3. 04860 
8-05322 
3. 05786 
3. 06251 
3-06717 
3. 07184 
3-07652 
3-08121 


35 
36 
37 
88 

-21 


40 

41 
42 
43 

44 


70207 
•70235 
. 70263 
.70290 

70318 


.71877 
.71905 
.71933 
71961 
■71989 


2. 55587 
2.55940 
2.56294 
2.56649 
2.57005 


.73557 
73585 
.73618 
.73641 
.73669 
.73697 
73725 
.73758 
.73781 
. 73809 


2.78166 
2. 78568 
2. 78970 
2.79374 
2.79778 
2-80188 
2-80589 
2.80996 
2- 81404 
2.81818 


.75244 
75272 
■75300 
75328 
•75356 
•75885 
.75418 
75441 
.75469 
.75497 


40 
41 
42 
43 
-44 


45 
46 
47 
48 
49 


.70346 
.70374 
.70401 
.70429 
.70457 


2. 37221 
2.37537 
2. 37854 
2.88171 
2.38489 


.72017 
.72045 
. 72078 
.72101 
.72129 


2.57361 
2.67718 
2.58076 
2.58484 
2.58794 


45 
46 
47 
48 
49 


50 

51 
52 
63 
54 


.70485 
.70513 
.70540 
70568 
.70596 
.70624 
.70652 
.70679 
.70707 
.70735 


2.38808 
2.39128 
2.89448 
2.39768 
2.40089 
2.40411 
2.40734 
2.41057 
2.41381 

3-«170§ 


.72157 
.72185 
.72213 
.72241 
.72269 
.72296 
.72324 
.72352 
.72880 
.72408 


2.59154 

2.59514 

2.59876 

2.60238 

2.60601, 

2.60965 

2.61380 

2.61695 

2.62061 

2.62428 


.73837 
.73865 
73893 
•73921 
.73950 


2-82223 
2-82633 
2-83045 
2. 83457 
2-83871 


.75526 
.75554 

75582 
.75610 

75639 


3-08591 
3.09063 
3-09585 
3-10009 
3-10484 


50 

51 
52 
58 
54 


55 
56 
57 
58 
§9. 


.73978 
.74006 
.74034 
.74062 
.74090 
.74118 


2-84285 

2-84700 

2.85116 

2.85588 

2-85951 , 

2-86370 


75667 
-75695 
.75723 
-75751 
.75780 
.75808 


3-10960 
8-11487 
311915 
3. 12894 
8-12875 


56 
56 
57 
58 
59 


60 


.70763 


2.42080 


.72436 


2.02796 


8.18367 


60 



Digitized by VjOOQIC 



NATURAL VERSED SINES AND EXTERNAL SEC. 295 
76* 77* 78' 79* 



• 


Vers. 


Ex. sec. 


Vers. 


Ex. nee. 


Vera, 1 Ex. Fee, 


V*!i^. 


Ex, ^w. 


f 


"5" 


^75808 


3.13357 


-77505 


3.44541 


79209 '3 80973 


-80919 


4 24084 


~ 


1 


75838 


3 13839 


77533 


3. 45102 


79237 3 &163S 


-80948 


4. 24870 


1 


2 


75884 


3 . 14323 


77562 


a 45664 


79266 3. 82294 


80976 


4 25658 


2 


3 


75892 


3 14809 


77590 


3. 46328 


79294 3 82956 


81005 


4 26448 


3 


4 


-75921 


8.15295 


J761S 
77647 


3 46793 
3. 47360 


.79323 3.83621 


81033 
81063 


4^37241 
4 . 28036 


4 


5 


75949 i3 15782 


79351 3 84288 


6 


e 


75B77 , a 16271 


77675 


3-47928 


.79310 3 34956 


81090 


4-28833 


e 


7 


76005 ^3 167a 1 


777D3 3-4B4fl8 


79408 3-86627 


.81119 


4 . 29684 


7 


S ! 


760S4 


3-17252 


-77732 3 49069 


.79437 13.86299 


81148 


4 30436 


8 


9 


76062 


3 17744 


■77760 3-49642 


7S465 .3 86973 


_31176 


4 31241 


9 


10 


78090 


3, 18238 


77788 3-50216 


-79493 3. 87649 


81305 


4 32049 


10 


11 


78118 


3 18733 


77817 3 50791 


79522 3. 88327 


81233 


4 32859 


11 


12 


-78147 


3 19228 


77845 13 51368 


.79550 3 89007 


-81262 


4, 33671 


12 


15 


.78175 


3. 19725 


77874 13 51947 


.79579 3 89689 


.81290 


4-34486 


13 


1* 


-76203 


3-20224 


77902 


3 52527 


79607 13 90373 
79636 '3 91058 


_81319 
61348 


4.35304 
4.36124 


14 


15 


76231 '3 20723 


77930 


3 53109 


15 


Ifl 


76260 3 21224 


77959 


3.33692 


-79664 ]3 91746 


-81376 


4 36947 


18 


17 


76288 '3 21726 


77987 


3 54277 


-79693 Is 92436 


-81405 


4. 37772 


17 


18 


76316 3 2222& 


78015 


3 . 54863 


-79721 3 93128 


-81433 


4 38600 


IB 


iS-i 


.76344 3 23734 


78044 


3 5545! 


79750 3-93821 


81462 4 3B430 


_1» 


20 


76373 3 23239 


-78072 


3-56041 


-79778 3 94517 


81491 4 40263 


30 


21 


76401 3 23746 


-78101 


3-56632 


79807 a 95215 


-81519 4 41099 


3. 


92 


76429 3-24255 


78129 


3,57224 


-79835 3 95914 


.81548 4 41937 


3^ 


23 


76458 3 24764 


78157 


3-57819 


79864 3.96616 


-81576 4. 42778 


23 


24 


76488 3 25275 
76514 3. 25787 


:J818e^ 
78214 


3.5B414 


79892 3-97320 
-79921 3^98025 


.81605 4.43622 
81633 4-44468 


24 


25 


3 59012 


26 


26 


,76542 S. 26300 


7S242 


3 59611 


79949 3-98733 


81662 4-45317 


26 


27 


76571 I3 26814 


-78271 


3 60211 


79973 3-99443 


81691 4 46189 


27 


28 , 


76599 3 27330 


78299 


3-60813 


80006 4-00155 


.81719 4 47023 


28 


.29 


76627 3.27847 
76655 3.28366 


78328 
-78356 


3 61417 
3 62023 


,80035 
-80063 


i 00869 


81748 4 47881 


29 


1? 


,4 01585 


81776 14. 48740 


30 


81 


76684 3 28885 


78384 


3 . 62630 


80092 4 02303 


-81805 4 49603 


31 


|2 


76712 3. 28406 


78413 


3-63233 


80120 4.03024 


81834 14.50468 


32 


83 


-76740 3 29929 


78441 


3, 63849 


80149 4 03746 


.81862 14 51337 


33 


B- 


78709 3.30452 | 
76797 3. 30977 


_7_847Q_ 


3 64461 
3-65074 


_80177 
80206 


4 04471 


^1891 I4. 52308 


34 


8S 


-78498 


4-05197 


81919 


4 53081 


35 


86 


76825 3. 31503 


.78526 


3.65690 


.80234 4.05928 


81948 


4. 53956 


36 


3? 


768S4 3-32031 


78555 


3 66307 


-80263 4 06057 


,81977 


4-54837 


37 


38 


76882 3 33560 


78583 


3. 66925 


.80291 14-07390 


82005 


4-55730 


38 


3fl 


.76910 


3 33090 

3-33622 


^8.612 
78640 


3 67545 
3-68167 


.80330 i,0fl]25 
80348 4 08803 


82034 
- 82063 


4 5660^^ 
4 57493 


_33 


40 


-78938 1 


40 


il 


76967 jf -34154 


78669 


3.68791 


80377 4 09603 


82091 


4 58383 


41 


42 


-76995 '3.34S89 


-78897 , 


3-69417 , 


-80405 4-10344 


82120 


4. 59277 


42 


4S 


77023 3.35224 


78725 


3 . 70044 


80434 4. 11038 


82148 


4 60174 


43 


4^ 


-77552 3 357B1 


78754 
78782^ 


3-70673 
3 71303 


80462 |4 11835 


,82177 
82206 


4^61073 
4-81978 


JA 


4S 


77080 3-36299 


8049- 


4. 12583 


4S 


4e 


77108 3 36839 


78311 


3 71935 


80520 


4 13334 


82234 


4. 62861 


JS 


47 


77137 3-37380 


78839 


3 72569 


30548 


4.14087 


82263 


4 63790 


47 


4a 


77165 3 37923 


78888 


3 . 73305 


80577 


4-14842 


82292 


4. 64701 


48 


Ji. 


77]J3_'3 38466 
77222 3 39012 


78896 
78924 


3 73843 
3 74482 


.60605 


4-15599 


82320 
82349 


4 65616 


_4B 


50 


60634 


4 16359 


4 66S33 


50 


51 


77250 |3 39558 


78953 


3-75133 


80662 


4.17121 


82377 


4-67454 


61 


m 


77278 3 40106 


78981 


3. 75766 


.80691 4-17866 j 


82406 


4 88377 


52 


&3 


77307 I3. 40858 


79010 


3 7641] 


80719 


4-18652 


82435 


4 69304 


53 


§4 


J7335 3 41206 


79038 


3 - 77057 


80748 


±^J9421_ 


_^S24fi3 


4 70234 


54 


65 


77383 I3 41759 


79067 


J 77705 


80776 


4 30193 


83492 


4 71196 


56 


68 


77392 !3 42312 


79095 


3.78355 


80805 


4-20866 


82521 


4,72102 


58 


67 


77420 3 42867 


79123 


3,79007 


80833 


4.21742 


82549 


4-73041 


57 


58 


77448 I3 43424 


79152 


3. 79601 


80862 


4 22531 


82578 


4 73983 


58 


ii- 


77477 3 43982 


793 80 
78209 


3-80316 
3.80973 |~ 


00891 
80919 


4_23301 
4-24084 


82607 
82635 


4 74929 
4,76877 


_6B 


eo 


77505 8 44541 


00 



y Google 



296 NATURAL VERSED SINES AND EXTERNAL SEC. 
80* 81* S2r ss** 



# 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec 


Vers. 


Ex. sec. 


# 


"o" 


82635 


4.75877 


.84357 


5.39245 


86083 


6.18680 


.87818 


7.20661 


"o 


1 


.82664 


4.76829 


.84885 


5.40422 


.86112 


6.20020 


.87842 


7.22600 


1 


2 


.82692 


4.77784 


.84414 


5.41602 


.86140 


6.21517 


.87871 


7.24467 


2 


8 

4 


.82721 
•8275Q 
82778 


4.78742 
4.79703 
4.80667 




6.42787 
5.43977 


.86169 
.86198 


6.23019 
6.24529 


.87900 
•87929 


7*26425 


3 


.84471 
.84500 


7.28402 


_4 


6 


5.45171 


.86227 


6.26044 


.87957 


7.80388 


5 


8 


82807 


4.81635 


.84529 


5.46369 


.86256 


6.27566 


.87986 


7.32884 


6 


7 


82836 


4.82606 


.84558 


5.47572 


.86284 


6.29095 


.88016 


7.34390 


7 


8 


82864 


4.83581 


.84586 


5.48779 


.86313 


6.30630 


.88044 


7. 36405 


8 


9 


•82898 


4.84558 


.84615 


6.49991 


■86342 


6.32171 


■88073 


7. 38481 


9 


10 


■82922 


4.85539 


.84644 


5. 61208 


■86371 


6.33719 


.88102 


7.40466 


10 


11 


.82950 


4.86524 


84678 


5.52429 


.86400 


6.35274 


.88131 


7.42611 


11 


12 


•82979 


4.87511 


84701 


5.53655 


■86428 


6. 36835 


.88160 


7.44666 


12 


18 


88008 


4.88502 


.84730 


5. 54886 


86457 


6. 38403 


88188 


7.46632 


18 


14 


■83036 


4. 89497 


.84759 


5.56121 


86486 


6- 39978 


.88217 


7.48707 


14 


16 


•83066 


4.90496 


.84788 


5.57361 


.86515 


6.41560 


.88246 


7. 60793 


15 


16 


•83094 


4.91496 


.84816 


5.58606 


86544 


6.43148 


.88275 


7.52889 


16 


17 


33122 


4.92501 


.84845 


5.59855 


■86573 


6.44743 


■88304 


7. 64996 


17 


18 


•83151 


4.98509 


.84874 


5.61110 


■86601 


6.46346 


■88338 


7.57118 


18 


i£. 




4.94521 


JI903. 


5.62869 


■86680 


8-4795L 


.88862 


7. 59241 


^ 


20 


■83208 


4.95536 


.84931 


5.63633 


■86659 


6.49571 


.88391 


7.61379 


ZO 


21 


33237 


4.96555 


84960 


5.64902 


.86688 


6.51194 


.88420 


7.68528 


21 


22 


■83266 


4.97577 


.84989 


5.66176 


86717 


6.52825 


88448 


7. 65688 


22 


28 


•83294 


4.98608 


.85018 


5.67454 


86746 


6.54462 


■88477 


7.67859 


23 


24 


•83323 


4.99638 


.85046 


5. 68738 


■86774 


6.56107 


■88506 


7. 70041 


24 


25 


88852 


5.00666 


.85075 


6.70027 


.86803 


6.57759 


.88535 


7. 72234 


25 


26 


83380 


5.01708 


.85104 


5. 71321 


86882 


6. 59418 


.88564 


7.74438 


26 


27 


83409 


5.02748 


.85133 


5.72620 


.86861 


6.61085 


88593 


7.73663 


27 


28 


83438 


5.03787 


•86162 


5.73924 


■86890 


6.62759 


88622 


7.78880 


28 


29. 


•83467 
■83496 


5.04834 
5.05886 


.J519Q 
.85219 


5.75288 
5.76647 


•86919 
■86947 


6.64441 


•88651 
88680 


7.81118 
7.83367 


29 


30 


6.66130 


30 


81 


■83524 


5.06941 


.85248 


5. 77866 


.86976 


6.67826 


88709 


7.85628 


31 


82 


•83553 


5.08000 


.85277 


6.79191 


87005 


6.69580 


88737 


7. 87901 


32 


88 


83581 


5.09062 


.85305 


5.80521 


.87034 


6.71242 


88766 


7.90186 


33 


M_ 


• 83610. 
•83639 


5.10129. 
5.11199 


.85334 
.85363 


5.81856 


.87063 


6.72962 


88796 


7.92482 


-84 


85 


5.83196 


•87092 


6.74689 


88824 


7.94791 


85 


86 


•83667 


5.12273 


.85392 


5. 84542 


■87120 


6.76424 


•88853 


7.97111 


36 


87 


■83696 


5.13350 


.85420 


5.85893 


.87149 


6. 78167 


88882 


7.99444 


37 


88 


83725 


5.14432 


.85449 


5.87250 


87178 


6. 79918 


88911 


8.01788 


38 


iS. 


jmL 


5. 1551.7. 


;854ZS. 


5.88612 


•»7207 


fi. 81677 


.88940 


8.W46 


-sa 


40 


■83782 


5.16607 


.85507 


5.89979 


■87236 


6.83443 


■88969 


8.06515 


40 


41 


.83811 


5 17700 


.85536 


5.91352 


■87265 


6.85218 


■88998 


8.08897 


41 


42 


83840 


5.18797 


.85564 


5.92731 


■87294 


6.87001 


■89027 


8.11292 


42 


48 


88868 


5.19898 


.85598 


6.94115 


■87322 


6.88792 


■89055 


8. 18699 


48 


Ak. 


■ 8889L 


5.21004 


.85622 


5. 95505 


■87351 


6.90592 


89084 


8.16120 


.44 


45 


.88926 


5.22113 


85651 


5.96900 


. 87380 


6.92400 


89118 


8. 18553 


45 


46 


.83954 


5.28226 


■85680 


5 ■98301 


■87409 


6.94216 


.89142 


8.20999 


48 


47 


.83983 


5.24343 


85708 


5.99708 


•87438 


6.96040 


.89171 


8.23459 


47 


48 


.84012 


5.25464 


.85787 


6 01120 


■87467 


6.97873 


.89200 


8.25931 


48 


M^ 


.84041. 
.84069 


5.26590 
5.27719 


.85766 


6 02538 


•87496 


6.99714 
7.01565 


89229 
.89258 


8.28417 
8.30917 


« 


50 


.85795 


6.03962 


.87524 


50 


61 


■84098 


5.28853 


.85823 


6.05392 


87553 


7.03423 


.89287 


8.83430 


51 


52 


.84127 


5.29991 


.85852 


6.06828 


.87682 


7.05291 


.89316 


8. 35957 


52 


53 


.84155 


6.31133 


.86881 


6.08269 


87611 


7.07167 


.89345 


8. 38497 


58 


54 


.84184 


5.32279 


.85910 


6.09717 


•87640 


7.09052 


.89374 


8.41052 


54 


55 


.84213 


5.33429 


.85939 


6.11171 


.87669 


7.10946 


. 89403 


8. 43620 


56 


56 


.84242 


5.34584 


.85967 


6.12630 


87698 


7.12849 


.89481 


8.46203 


66 


57 


.84270 


5.36748 


.85996 


6.14096 


87726 


7.14760 


.89460 


8.48800 


67 


58 


.84299 


5.36906 


.86025 


6.15568 


.87755 


7.16681 


.89489 


8.51411 


58 


.ML 


.84367 


5.38073 
5.39245 


86054 
.86083 


6.17046 
6.18530 


J7Z84. 
.87818 


7.20551 


^518 
.89547 


8 54087 


LJS 


60 


8.56677 


60 



Digitized by VjOOQIC 



NATURAL VERSED SINES AND EXTERNAL SEC. 
84** 85** 8«* 



297 



» 


Vers. 


Ex. sec. 


Vers. 


Ex. 860. 


Vers. 


Ex. sec. 


t 


~ 


.89647 


8.66677 


.91284 


10.47371 


.93024 


18.33559 




1 


.89676 


8.69332 


.91818 


10.51199 


.98053 


13.39547 


1 


2 


.89606 


8.62002 


.91842 


10.55052 


.93082 


13.46586 


2 


8 


.89634 


8.64687 


.91871 


10.68932 


.93111 


18.51676 


3 


4 


.89663 


8.67387 


.91400 


10.62837 


.98140 


18.57817 


4 


6 


.89692 


8.70108 


.91429 


10.66769 


.93169 


13.64011 


5 


6 


.89721 


8.72833 


.91468 


10.70728 


.93198 


13.70268 


6 


7 


.89760 


8.76679 


.91487 


10.74714 


.93227 


13.76668 


7 


8 


.89779 


8.78841 


.91616 


10.78727 


.93257 


13.82913 


8 


9 


.89808 


8.81119 


.91545 


10.82768 


.93286 


13.89328 


9 


10 


.89836 


8.83912 


.91674 


10.86887 


.93316 


13.96788 


10 


11 


.89865 


8.86722 


.91603 


10.90934 


.93344 


14.02310 


11 


12 


.89894 


8.89547 


.91632 


10.96060 


.93373 


14.08890 


12 


IS 


.89923 


8.92389 


.91661 


10.99214 


.93402 


14.16627 


13 


14 


.89962 


8.95248 


.91690 


11.03397 


.93431 


14.22223 
14.28979 


14 


16 


.89981 


8.98123 


.91719 


1.07610 


.93460 


15 


16 


.90010 


9.01015 


.91748 


11.11862 


.93489 - 


14.85795 


16 


17 


90039 


9.03923 


.91777 


11.16125 


.98618 


14.42672 


17 


18 


.90068 


9.06849 


.91806 


11.20427 


.93547 


14.49611 


18 


-12- 


.90097 


9.09792 


.91835 


11.24761 


.93576 


14.66614 


U9 


20 


90126 


9.12762 


.91864 


11.29125 


.93605 


14.63679 


30 


21 


.90165 


9.16730 


.91893 


11.33621 


.93634 


14.70810 


21 


22 


.90184 


9.18726 


.91922 


11.37948 


.93663 


14.78005 


22 


23 


.90213 


9.21739 


.91961 


11.42408 


.93692 


14.86268 


23 


24 


.90242 


9.24770 


.91980 


11.46900 


.93721 


14.92597 


24 


25 


.90271 


9.27819 


.92009 


11.61424 


.93750 


14.99995 


25 


26 


.90300 


9.30887 


.92038 


11.65982 


.93779 


15.07462 


26 


27 


.90329 


9.83978 


.92067 


11.60672 


.93808 


15.14999 


27 


28 


.90358 


9.37077 


.92096 


11.66197 


.93837 


15.22607 


2? 


M_ 


.90386 


9.40201 


.92125 


11.69866 


.93866 


15-30287 


-29 


30 


.90415 


9.43343 


.92154 


11.74650 


.93895 


16.38041 


30 


31 


.90444 


9.46605 


.92183 


1.1.79278 


.93924 


15.46869 


31 


82 


.90473 


9.49686 


.92212 


11.84042 


.93963 


15.53772 


32 


33 


.90602 


9.62886 


.92241 


11.88841 


.93982 


15.61761 


33 


J4_ 


.90631 


9.56106 
9.69346 


.92270 


11.93677 


94011 


16.69808 
15.77944 


34 


36 


.90560 


.92299 


11.98649 


.94040 


35 


36 


.90589 


9.62605 


.92328 


12.03458 


94069 


15.86159 


36 


87 


.90618 


9.66886 


.92367 


12.08404 


.94098 


15.94456 


37 


88 


.90647 


9.69186 


.92386 


12.13388 


.94127 


16.02835 


38 


i2- 


.90676 


9.72507 


.92415 


12.18411 


.94156 




39 


40 


.90706 


9.76849 


.92444 


12.23472 


.94186 


16.19843 


40 


41 


.90784 


9.79212 


.92473 


12.28572 


.94215 


16.28476 


41 


42 


.90763 


9.82596 


.92502 


12.33712 


.94244 


16.37196 


42 


43 


.90792 


9.86001 


.92531 


12.38891 


.94278 


16.46005 


43 


44 


.90821 


9.89428 


.92660 


12.44112 


.94302 


16.54903 


44 


45 


.90850 


9.92877 


.92589 


12.49373 


.94331 


16.63893 


46 


46 


.90879 


9.96348 


.92618 


12.64676 


.94360 


16.72975 


46 


47 


.90908 


9.99841 


.92647 


12.60021 


.94389 


16.82152 


47 


48 


.90987 


10.03356 


.92676 


12.66408 


.94418 


16.91424 


48 


«_ 


.90966 


10.06894 


.92705 


12.70838 


.94447 


17.00794 


_y 


50 


.90996 


10.10465 


.92734 


M. 76312 


.94476 


17.10262 


50 


51 


.91024 


10.14089 


.92763 


12.81829 


.94505 


17.19830 


51 


62 


.91058 


10.17646 


.92792 


12.87891 


.94634 


17.29501 


52 


58 


.91082 


10.21277 


.92821 


12.92999 


.94663 


17.39274 


53 


5*. 


.91111 


10.24932 


.92850 


12.98651 


.94692 


17.49163 


54 


55 


.91140 


10.28610 


.92879 


13.04850 


.94621 


17.59139 


55 


56 


.91169 


10.82313 


.92908 


13.10096 


.94660 


17.69283 


56 


57 


.91197 


10.86040 


.92937 


13.16889 


.94679 


17.79438 


57 


58 


.91226 


10.89792 


.92966 


13.21730 


.94708 


17.89755 


58 


69_ 


.91265 


10.48669 


.92995 


13.27620 


.94787 


18.00185 


-SS 


60 


.01284 


10.47371 


.98024 


18.38569 


.94766 


18.10732 


60 



Digitized by VjOOQIC 



298 NATURAL VERSED SINES AND 
87* 88* 



EXTERNAL SEC. 

89" 



' 


Vera. 


Ex.seo. 


Vera. 


Ex.seo. 


Vera. 


Ex. seo. 


1 


'o' 


.94766 


18.10782 


.96510 


27. 65371 


.98265 


69. 29869 


~ 


1 


.94796 


18.21397 


.96539 


27.89440 


98284 


57.26976 


1 


2 


.94825 


18.32182 


.96568 


28.18917 


.98318 


58.27431 


2 


3 


.94854 


18.43088 


.96597 


28.38812 


98342 


69.31411 


3 


4 


.94883 


18.54119 


.96626 


28.64137 


.98371 


60.39105 


4 


5 


.94912 


18.65275 


.96655 


28.89903 


.98400 


61.50715 


5 


6 


.94941 


18.76560 


.96684 


29 16120 


.98429 


62.66460 


6 


7 


.94970 


18.87976 


.96714 


29.42802 


.98458 


63. 86572 


7 


8 


.94999 


18.99524 


.96743 


29.69960 


.98487 


65.11304 


8 


9 


.95028 


19.11208 


.96772 


29.97607 


.98517 


66.40927 


_9 


10 


.95057 


19.28028 


.96801 


30.25758 


.98546 


67.76736 


10 


11 


.95086 


19.84989 


.96830 


30.54425 


.98575 


69.16047 


11 


12 


.95115 


19.47093 


.96859 


30. 83623 


.98604 


70.62207 


12 


13 


.95144 


19.59341 


.96888 


31.13366 


.98688 


72.14683 


13 


14 


.95173 


19.71737 


.96917 


31.43671 


.98662 


73. 73586 


14 


15 


.95202 


19. 84283 


.96946 


31.74554 


.98691 


75.39655 


15 


16 


.95231 


19.96982 


.96975 


32.06030 


.98720 


77.13274 


16 


17 


.95260 


20.09838 


.97004 


32.38118 


.98749 


78. 94968 


17 


18 


.95289 


20.22852 


.97033 


32.70835 


.98778 


80- 85315 


18 


li. 


.95318 


20.36027 


.97062 


33.04199 


.98807 


82-84947 


19 


20 


.95347 


20.49368 


.97092 


33. 38232 


.98836 


84-94561 


20 


21 


.95377 


20.62876 


.97121 


33.72952 


.98866 


87.14924 


21 


22 


.95406 


20.76555 


.97150 


34. 08380 


.98895 


89. 46886 


22 


23 


.95435 


20.90409 


.97179 


34. 44539 


.98924 


9191387 


28 


24 


.95464 


21.04440 


.97208 


34.81452 


.98953 


94- 49471 


24 




.95498 


21.18653 


.97237 


35.19141 


.98982 


97- 22308 


25 


36 


.95522 


21.33050 


.97266 


35.57633 


.99011 


100.1119 


26 


27 


.95551 


21.47635 


.97295 


35.96953 


.99040 


103.1757 


27 


28 


.95580 


21.62413 


.97324 


36.37127 


.99069 


106.4311 


28 


29 


95609 


21. 77386 


.97353 


86.78185 


.99098 


109-8966 


29 


30 


95638 


21.92559 


.97382 


37.20155 


.99127 


113-5930 


30 


31 


.95667 


22. 07935 


.97411 


37.63068 


.99156 


117.5444 


81 


32 


.95696 


22. 23520 


.97440 


38.06957 


.99186 


121.7780 


32 


33 


.96726 


22.39316 


.97470 


38.51855 


.99215 


126.3253 


33 


34 


■95754 


22.55328 


.97499 


38. 97797 


.99244 


131-2223 


34 


35 


.95788 


22.71568 


.97528 


39.44820 


.99278 


136. 5111 


35 


36 


.95812 


22.88022 


.97557 


39.92963 


.99302 


142.2406 


36 


37 


.95842 


23.04712 


.97586 


40. 42266 


.99331 


148.4684 


37 


38 


.95871 


23. 21687 


.97615 


40.92772 


.99360 


155.2623 


38 


M. 


95900 


28. 38802 


.97644 


41.44525 


.99389 


162.7033 


-3S 


40 


.95929 


23-56212 


.97673 


41.97571 


.99418 


170-8883 


40 


41 


95958 


23.73873 


.97702 


42.51961 


.99447 


179.9350 


41 


42 


.95987 


23.81790 


.97731 


43.07746 


.99476 


189.9868 


42 


43 


.96016 


24.09969 


.97760 


43.64980 


.99505 


201-2212 


43 


44 


.96045 


24.28414 


.97789 


44.23720 


.99535 


213. 8600 


44 


45 


.96074 


24.47134 


.97819 


44.84026 


.99564 


228.1839 


45 


46 


.96103 


24.66132 


.97848 


45.45963 


.99593 


244.5540 


46 


47 


.96132 


24.85417 


.97877 


46-09596 


.99622 


263.4427 


47 


48 


.96161 


25. 04994 


.97906 


46.74997 


.99651 


285.4795 


48 


49 


.96190 


25.24869 


.97935 
.97964 


47.42241 


.99680 


311.6230 


49 


50 


.96219 


25.45051 


48.11406 


.99709 


342.7752 


50 


51 


.96248 


25.65546 


.97993 


48.82576 


.99788 


380.9723 


51 


62 


.96277 


25.86360 


.98022 


49.55840 


.99767 


428.7187 


52 


53 


.96307 


26.07503 


.98051 


50.31290 


.99796 


490.1070 


53 


64_ 


.96836 


26. 28981 


•98080 


51.09027 


.99825 


571.9581 


M 


55 


.96365 


26.50804 


.98109 


51.89156 


.99855 


686.5496 


55 


56 


.96394 


26.72978 


.98138 


52.71790 


.99884 


858.4369 


56 


57 


.96423 


26.95513 


.98168 


53.57046 


.99913 


1144.916 


57 


58 


.96452 


27.18417 


.98197 


54.45053 


.99942 


1717.874 


58 


.Sfl. 


.96481 


27.41700 


.98226 


55.35946 


.99971 


3436.747 


«59 


60 


.96510 


27.65371 


.98265 


56.29869 


1.00000 


Infinite 


60 



Digitized by VjOOQIC 



TABLES 



299 



RADII — METRIC 
i2 = 10cscy 



Deg. 


Radkis. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


0*00' 


CO 


00 


TOO' 


1145.930 


3.05916 


2" 00' 


572.987 


2.75814 


01 


68754.931 


4.83730 


01 


1127.145 


3.05198 


01 


668.252 


2.76454 


02 


34377.468 


4.53627 


02 


1108.966 


3.04492 


02 


563.595 


2.75097 


03 


22918.310 


4.36018 


03 


1091.364 


3.03797 


03 


559.013 


2.74742 


04 


17188.735 


4.23524 


04 


. 1074.311 


3.03113 


04 


564.505 


2.74391 


05 


13750.987 


4.13833 


05 


10)7.785 


3.02440 


05 


560.070 


2.74042 


06 


11459.157 


4.05912 


06 


1041.757 


3.01777 


06 


545.705 


2.73696 


07 


9822.134 


3.99221 


07 


1026.209 


3.01124 


07 


541.408 


2.73362 


08 


8594.369 


3.93421 


08 


1011.118 


3.00480 


08 


537.179 


2.73012 


09 


7639.438 


3.88306 


09 


996.465 


2.99846 


09 


533.015 


2.72674 


10 


6875.496 


3.83730 


10 


982.230 


2.99221 


10 


628.916 


2.72339 


11 


6250.451 


3.79591 


11 


968.396 


2.98605 


11 


624.879 


2.72006 


12 


5729.582 


3.75812 


12 


954.947 


2.97998 


12 


620.903 


2.71676 


13 


5288.845 


3.72336 


13 


941.866 


2.97399 


13 


616.986 


2.71348 


14 


4911.070 


3.69118 


14 


929.139 


2.96808 


14 


513.129 


2.71023 


15 


4583.666 


3.66121 


15 


916.751 


2.96225 


15 


509.328 


2.70700 


16 


4297.187 


3.63318 


16 


904.689 


2.95660 


16 


505.584 


2.70379 


17 


4044.412 


3.60686 


17 


892.940 


2.95082 


17 


601.894 


2.70061 


18 


3819.723 


3.58203 


18 


881.492 


2.94522 


18 


498.258 


2.69745 


19 


3618.685 
3437.752 


3.55855 


19 


870.335 


2.93969 


19 


494.674 


2.69432 


20 


3.53627 


20 


859.456 


2.93422 


20 


491.141 


2.69121 


21 


3274.050 


3.51509 


21 


848.846 


2.92883 


21 


487.658 


2.68812 


22 


3125.230 


3.4948a 


22 


838.496 


2.92350 


22 


484.224 


2.68605 


23 


2989.351 


3.47558 


23 


828.393 


2.91824 


23 


480.838 


2.68200 


24 


2864.795 


3.45709 


24 


818.532 


2.91304 


24 


477.500 


2.67897 


25 


2750.203 


3.43936 


25 


808.902 


2.90790 


25 


474.207 


2.67597 


26 


2644.427 


3.42233 


26 


799.497 


2.90282 


26 


470.960 


2.67298 


27 


2546.485 


3.40594 


27 


790.308 


2.89780 


27 


467.756 


2.67002 


28 


2455.540 


3.39015 


28 


781.327 


2.89283 


28 


464.596 


2.66708 


29 


2370.867 


3.37491 


29 


772.549 


2.88793 


29 


461.479 


2.66415 


30 


2291.838 


3.36018 


30 


763.966 


2.88307 


30 


468.403 


2.66125 


31 


2217.909 


3.34594 


31 


755.571 


2.87828 


31 


455.367 


2.65836 


32 


2148.600 


3.33216 


32 


747.369 


2.87363 


32 


462.372 


2.65650 


33 


2083.491 


3.31879 


33 


739.323 


2.86883 


33 


449.416 


2.66265 


34 


2022.212 


3.30583 


34 


731.468 


2.86419 


34 


446.498 


2.64982 


35 


1964.435 


3.29324 


35 


723.759 


2.85959 


36 


443.618 


2.64701 


36 


1909.868 


3.28100 


36 


716.221 


2.85505 


36 


440.775 


2.64422 


37 


1858.251 


3.26910 


37 


708.837 


2.85055 


37 


437.968 


2.64144 


38 


1809.350 


3.25752 


38 


701.605 


2.84609 


38 


435.196 


2.63868 


39 


1762.957 


3.24624 


39 


694.518 


2.84168 


39 


432.460 


2.63696 


40 


1718.883 


3.23525 


40 


687.574 


2.83732 


40 


429.767 


2.63322 


41 


1676.959 


3.22452 


41 


680.766 


2.83300 


41 


427.088 


2.63052 


42 


1637.032 


3.21406 


42 


674.093 


2.82872 


42 


424.452 


2.62783 


43 


1598.962 


3.20384 


43 


667.549 


2.82448 


43 


421.849 


2.62516 


44 


1562.623 


3.19385 


44 


661.130 


2.82029 


44 


419.277 


2.62250 


45 


1527.899 


3.18409 


45 


664.834 


2.81613 


45 


416.737 


2.61986 


46 


1494.684 


3.17455 


46 


648.667 


2.81202 


46 


414.227 


2.61724 


47 


1462.883 


3.16521 


47 


642.595 


2.80794 


47 


411.747 


2.61463 


48 


1432.406 


3.15607 


48 


636.646 


2.80390 


48 


409.296 


2.61204 


49 


1403.174 


3.14711 


49 


630.806 


2.79990 


49 


406.876 


2.60946 


50 


1375.111 


3.13834 


50 


625.072 


2.79593 


60 


404.482 


2.60690 


51 


1348.148 


3.12974 


61 


619.441 


2.79200 


51 


402.117 


2.60435 


52 


1322.223 


3.12130 


52 


613.910 


2.78810 


52 


399.780 


2.60182 


53 


1297.276 


3.11303 


63 


608.478 


2.78424 


53 


397.469 


2.69930 


54 


1273.253 


3.10491 


54 


603.141 


2.78042 


54 


395.186 


2.69680 


55 


1250.103 


3.09695 


56 


597.897 


2.77663 


56 


392.928 


2.69431 


56 


1227.780 


3.08912 


56 


592.743 


2.77287 


56 


390.696 


2.69184 


57 


1206.241 


3.08143 


57 


587.677 


2.76914 


57 


388.489 


2.58938 


58 


1185.444 


3.07388 


58 


582.698 


2.76544 


58 


386.307 


2.58693 


59 


1165.352 


3.06646 


59 


677.802 


2.76178 


69 


334.149 


2.58460 














,,, 


■ < 














jiliz^J L 




^ 



300 



THE RAILROAD TAPER 
RADII — METRIC — (Continued) 



Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


3" 00' 


382.016 


2.58208 


4''00' 


286.537 


2.45718 


5«00' 


229.256 


2.36032 


[01 


379.906 


2.57968 


•01 


285.348 


2.45537 


01 


228.495 


2.35888 


02 


377.818 


2.57728 


02 


284.170 


2.45358 


02 


227.739 


2.35744 


03 


375.754 


2.57490 


03 


283.001 


2.46179 


03 


226.987 


2.35600 


04 


373.713 


2.57254 


04 


281.842 


2.45001 


04 


226.241 


2.35457 


05 


371.693 


2.57018 


05 


280.692 


2.44823 


06 


225.500 


2.35315 


06 


369.696 


2.66784 


06 


279,551 


2.44646 


06 


224.764 


2.35173 


07 


367.719 


2.66652 


07 


278.420 


3.44470 


07 


224.032 


2.35031 


08 


365.763 


2.66320 


08 


277.298 


2.44295 


08 


223.305 


2.34890 


09 


363.828 


2.56090 


09 


276.185 


2.44120 


09 


222.583 


2.34749 


10 


361.914 


2.55861 


10 


275.080 


2.43946 


10 


221.865 


2.34609 


11 


360.020 


2.56633 


11 


273.985 


2.43773 


11 


221.152 


2.34469 


12 


358.145 


2.55406 


12 


272.898 


2.43600 


12 


220.444 


2.34330 


13 


356.290 


2.65180 


13 


271.820 


2.43428 


13 


219.740 


2.34191 


14 


354.454 


2.54956 


14 


270.750 


2.43257 


14 


219.041 


2.34053 


15 


352.637 


2.54733 


15 


269.689 


2.43086 


15 


218.346 


2.33915 


16 


350.838 


2.54511 


16 


268.636 


2.42916 


16 


217.656 


2.33777 


17 


349.058 


2.54290 


17 


267.591 


2.42747 


17 


216.969 


2.33640 


18 


347.295 


2.54070 


18 


266.656 


2.42579 


18 


216.288 


2.33503 


19 


345.550 


2.53851 


19 


265.526 


2.42411 


19 


215.610 


2.33367 


20 


343.823 


2.63633 


20 


264.505 


2.42243 


20 


214.937 


2.33231 


21 


342.113 


2.53417 


21 


263.492 


2.42077 


21 


214.268 


2.33096 


22 


340.420 


2.53202 


22 


262.487 


2.41911 


22 


213.603 


2.32961 


23 


338.743 


2.52987 


23 


261.489 


2.41745 


23 


212.942 


2.32826 


24 


337.084 


2.52774 


24 


260.499 


2.41581 


24 


212.285 


2.32692 


25 


335.440 


2.52561 


25 


259.517 


2.41417 


25 


211.632 


2.32558 


26 


333.812 


2.52350 


26 


258.542 


2.41253 


26 


210.984 


2.32425 


27 


332.200 


2.62140 


27 


257.674 


2.41090 


27 


210.339 


2.32292 


28 


330.603 


2.51931 


28 


256.613 


2.40928 


28 


209.698 


2.32159 


29 


329.022 


2.51722 


29 


255.660 


2.40766 


29 


209.061 


2.32027 


30 


327.455 


2.51515 


30 


254.713 


2.40605 


30 


208.428 


2.31896 


31 


325.904 


2.51309 


31 


253.774 


2.40446 


31 


207.799 


2.31764 


32 


324.367 


2.61104 


32 


252.841 


2.40285 


32 


207.174 


2.31634 


33 


322.845 


2.50899 


33 


251.916 


2.40126 


33 


206.552 


2.31503 


34 


321.337 


2.50696 


34 


250.997 


2.39967 


34 


205.934 


2.31373 


35 


319.842 


2.60494 


35 


260,085 


2.39809 


35 


205.320 


2.31243 


36 


318.362 


2.50292 


36 


249.179 


2.39651 


36 


204.709 


2.31114 


37 


316.896 


2.50092 


37 


248.280 


2.39494 


37 


204.102 


2.30985 


38 


315.442 


2.49892 


38 


247.387 


2.39338 


38 


203.499 


2.30856 


39 


314.003 


2.49693 


39 


246.501 


2.39182 


39 


202.899 


2.30728 


40 


312.576 


2.49496 


40 


246.621 


2.39027 


40 


202.303 


2.30600 


41 


311.162 


2.49299 


41 


244.748 


2.38872 


41 


201.710 


2.30473 


42 


309.761 


2.49103 


42 


243.880 


2.38718 


42 


201.121 


2.30346 


43 


308.372 


2.48907 


43 


243.019 


2.38564 


43 


200.535 


2.30219 


44 


306.996 


2.48713 


44 


242.164 


2.38411 


44 


199.952 


2.30093 


45 


305.632 


2.48520 


45 


241.314 


2.38268 


45 


199.373 


2.29Q67 


46 


304.280 


2.48327 


46 


240.471 


2.38106 


46 


198.798 


2.29841 


47 


302.940 


2.48136 


47 


239.634 


2.37955 


47 


198.225 


2.29716 


48 > 


301.612 


2.47945 


48 


238.802 


2.37804 


48 


197.656 


2.29591 


49 


300.295 


2.47755 


49 


237.976 


2.37653 


49 


197.090 


2.29466 


50 


298.990 


2.47566 


50 


237.156 


2.37503 


50 


196.528 


2.29342 


51 


297.696 


2.47377 


51 


236.342 


2.37354 


51 


195.968 


2.29219 


52 


296.414 


2.47190 


52 


235.533 


2.37205 


52 


195.412 


2.29095 


53 


295.142 


2.47003 


53 


234.730 


2.37057 


53 


194.859 


2.28972 


54 


293.881 


2.46817 


54 


233.932 


2.36909 


64 


194.309 


2.28849 


55 


292.631 


2.46632 


55 


233.139 


2.36761 


55 


193.762 


2.28727 


56 


291.392 


2.46448 


56 


232.352 


2.36615 


56 


193.218 


2.28605 


57 


290.163 


2.46264 


57 


231.570 


2.36468 


57 


192.677 


2.28483 


58 


288.944 


2.46081. 


58 


230.794 


2.36322 


58 


192.140 


2.28362 


59 


287.735 


2.45899 


59 


230.022 


2.36177 


59 


191.605 


2.28241 



TABLES 301 

RADII — METRIC — (Continued) 



Deg. 


Radius. 


Log.rad. 


Deg. 


Kadius. 


Log. rad. 


Dog. 


Radius. 


Log. rad. 


6*»00' 


191.073 


2.28120 


7-00' 


163.804 


2.21432 


8'»00' 


143.356 


2.16642 


01 


190.544 


2.28000 


01 


163.415 


2.21329 


01 


143.058 


2.16551 


02 


190.018 


2.27879 


02 


163.029 


2.21226 


02 


142.762 


2.15461 


03 


189.496 


2.27760 


03 


162.644 


2.21124 


03 


142.467 


2.15371 


04 


188.976 


2.27641 


04 


162.261 


2.21021 


04 


142.173 


2.15282 


05 


188.458 


2.27521 


06 


161.880 


2.20919 


05 


141.880 


2.15192 


06 


187.944 


2.27403 


06 


161.600 


2.20817 


06 


141.589 


2.15103 


07 


187.432 


2.27284 


07 


161.122 


2.20715 


07 


141.299 


2.15014 


08 


186.923 


2.27166 


08 


160.746 


2.20614 


08 


141.010 


2.14925 


00 


186.417 


2.27049 


09 


160.372 


2.20513 


09 


140.722 


2.14836 


10 


185.914 


2.26931 


10 


160.000 


2.20412 


10 


140.436 


2.14748 


11 


185.413 


2.26814 


11 


159.629 


2.20311 


11 


140.160 


2.14659 


12 


184.915 


2.26697 


12 


159.260 


2.20211 


12 


139.865 


2.14571 


13 


184.420 


2.26581 


13 


168.892 


2.20110 


13 


139.582 


2.14483 


14 


183.927 


2.26465 


14 


168.627 


2.20010 


14 


139.300 


2.14396 


15 


183.437 


2.26349 


15 


168.163 


2.19910 


16 


139.019 


2.14307 


16 


182.950 


2.26233 


16 


167.800 


2.19811 


16 


138.739 


2.14220 


17 


182.465 


2.26118 


17 


157.440 


2.19712 


17 


138.460 


2.14132 


18 


181.983 


2.26003 


18 


157.081 


2.19612 


18 


138.183 


2.14045 


ig 


181.503 


2.25888 


19 


156.724 


2.19514 


19 


137.906 


2.13958 


20 


181.026 


2.25774 


20 


156.368 


2.10415 


20 


137.631 


2.13872 


21 


180.552 


2.26660 


21 


156.014 


2.10316 


21 


137.367 


2.13785 


22 


180.079 


2.25546 


22 


165.661 


2.19218 


22 


137.084 


2.13699 


23 


179.610 


2.25433 


23 


155.310 


2.19120 


23 


136.812 


2.13612 


24 


179.142 


2.25320 


24 


164.961 


2.19022 


24 


136.541 


2.13526 


25 


178.678 


2.25207 


25 


164.613 


2.18925 


25 


136.271 


2.13440 


26 


178.215 


2.25094 


26 


164.267 


2.18827 


26 


136.002 


2.13355 


27 


177.755 


2.24982 


27 


153.923 


2.18730 


27 


136.734 


2.13269 


28 


177.298 


2.24870 


28 


153.580 


2.18633 


28 


135.468 


2.13184 


29 


176.842 


2.24759 


29 


163.238 


2.18537 


29 


135.202 


2.13098 


30 


176.389 


2.24647 


30 


152.898 


2.18440 


30 


134.037 


2.13013 


31 


175.939 


2.24536 


31 


152.559 


2.18344 


31 


134.674 


2.12028. 


32 


175.490 


2.24425 


32 


162.222 


2.18248 


32 


134.411 


2.12843 


33 


175.044 


2.24315 


33 


151.887 


2.18162 


33 


134.150 


2.12759 


34 


174.600 


2.24204 


34 


151.553 


2.18066 


34 


133.880 


2.12674 


35 


174.159 


2.24095 


36 


161.220 


2.17961 


35 


133.630 


2.12590 


36 


173.720 


2.23985 


36 


160.889 


2.17866 


36 


133.371 


2.12506 


37 


173.283 


2.23876 


37 


160.559 


2.17771 


37 


133.114 


2.12422 


38 


172.848 


2.23766 


38 


160.231 


2.17676 


38 


132.857 


2.12338 


30 


172.415 


2.23668 


39 


149.904 


2.17581 


39 


132.602 


2.12255 


40 


171.084 


2.23649 


40 


140.679 


2.17487 


40 


132.347 


2.12171 


41 


171.556 


2.23441 


41 


140,255 


2.17393 


41 


132.004 


2.12088 


42 


171.130 


2.23333 


42 


U8.932 


2.17290 


42 


131.841 


2.12005 


43 


170.705 


2.23226 


43 


148.611 


2.17205 


43 


131.580 


2.11022 


44 


170.284 


2.23117 


44 


148.291 


2.17111 


44 


131.330 


2.11830 


45 


160.863 


. 2.23010 


46 


147.973 


2.17018 


45 


131.080 


2.11757 


46 


160.446 


2.22903 


46 


147.666 


2.16925 


46 


130.840 


2.11674 


47 


169.030 


2.22796 


47 


147.340 


2.16832 


47 


130.603 


2.11592 


48 


168.616 


2.22690 


48 


147.026 


2.16739 


48 


130.346 


2.11610 


40 


168.204 


2.22684 


49 


146.713 


2.16647 


49 


130.100 


2.11428 


50 


167.794 


2.22478 


50 


146.401 


2.16564 


50 


129.855 


2.11346 


&1 


167.387 


2.22372 


61 


146.091 


2.16462 


51 


129.611 


2.11264 


52 


166.981 


2.22267 


52 


145.782 


2.16370 


62 


129.368 


2.11183 


53 


166.677 


2.22162 


63 


145.474 


2. .16278 


63 


129.125 


2.11101 


54 


166.175 


2.22057 


64 


145.168 


2.16187 


54 


128.884 


2.11020 


55 


165.775 


2.21962 


56 


144.862 


2.16095 


65 


128.644 


2.10939 


56 


165.377 


2.21848 


66 


144.669 


2.16005 


56 


128.404 


2.10858 


67 


164.981 


2.21743 


57 


144.266 


2.15913 


67 


128.166 


2.10777 


58 


164.587 


2.21640 


58 


143.955 


2.15823 


68 


127.928 


2.10607 


50 


164.195 


2.21536 


50 


143.655 


2.16732 


59 

Digitizet 


1 byXj'OO 


2.10616 

gii-_ 



302 



THE RAILROAD TAPER 
RADII — METRIC — (Continued) 



Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


9° 00' 


127.455 


2.10536 


lO'OO' 


114.737 


2.05970 


iroo' 


104.334 


2.01843 


01 


127.220 


2.10456 


01 


114.547 


2.05898 


01 


104.177 


2.01777 


02 


126.986 


2.10376 


02 


114.357 


2.05826 


02 


104.020 


2.01712 


03 


126.752 


2.10295 


03 


114.168 


2.06754 


03 


103.864 


2.01646 


04 


126.520 


2.10216 


04 


113.979 


2.05682 


04 


103.708 


2.01581 


05 


126.288 


2.10136 


05 


113.791 


2.06611 


06 


103.552 


2.01516 


06 


126.057 


2.10057 


06 


113.604 


2.06639 


06 


103.397 


2.01461 


07 


125.827 


2.09977 


07 


113.417 


2.05468 


07 


103.243 


2.01386 


08 


125.598 


2.09898 


08 


113.231 


2.05397 


08 


103.089 


2.01321 


09 


125.370 


2.09819 


09 


113.046 


2.05326 


09 


102.935 


2.01256 


10 


125.142 


2.09740 


10 


112.861 


2.06264 


10 


102.782 


2.01192 


11 


124.916 


2.09662 


11 


112.677 


2.05184 


11 


102.629 


2.01127 


12 


124.690 


2.09583 


12 


112.493 


2.06113 


12 


102.477 


2.01063 


13 


124.465 


2.09605 


13 


112.310 


2.05042 


13 


102.325 


2.00998 


14 


124.241 


2.09426 


14 


112.128 


2.04971 


14 


102.174 


2.00934 


15 


124.017 


2.09348 


15 


111.946 


2.04901 


16 


102.023 


2.00870 


16 


123.795 


2.09270 


16 


111.765 


2.04831 


16 


101.873 


2.00806 


17 


123.573 


2.09192 


.17 


111.584 


2.04760 


17 


101.723 


2.00742 


18 


123.352 


2.09116 


18 


111.404 


2.04690 


18 


101.573 


2.00678 


19 


123.132 


2.09037 


19 


111.224 


2.04620 


19 


101.424 


2.00614 


20 


122.912 


2.08969 


20 


111.046 


2.04660 


20 


101.275 


2.00550 


21 


122.694 


2.08882 


21 


110.867 


2.04480 


21 


101.127 


2.00487 


22 


122.476 


2.08806 


22 


110.689 


2.04410 


22 


100.979 


2.00423 


23 


122.259 


2.08728 


23 


110.512 


2.04341 


23 


100.832 


2.00360 


24 


122.043 


2.08651 


24 


110.336 


2.04272 


24 


100.685 


2.00296 


25 


121.827 


2.08674 


25 


110.160 


2.04202 


26 


100.538 


2.00233 


26 


121.612 


2.08498 


26 


109.984 


2.04133 


26 


100.392 


2.00170 


27 


121.398 


2.08421 


27 


109.809 


2.04064 


27 


100.247 


2.00107 


28 


121.185 


2.08346 


28 


109.636 


2.03995 


28 


100.102 


2.00044 


29 


120.973 


2.08269 


29 


109.461 


2,03926 


29 


99.967 


1.99981 


30 


120.761 


2.08193 


30 


109.288 


2.03857 


30 


99.812 


1.99918 


31 


120.550 


2.08117 


31 


109.116 


2.03788 


31 


99.668 


1.99856 


32 


120.340 


2.08041 


32 


108.943 


2.03720 


32 


99.525 


1.99793 


33 


120.130 


2.07965 


33 


108.771 


2.03651 


33 


99.382 


1.99731 


34 


119.921 


2.07890 


34 


108.600 


2.03583 


34 


99.239 


1.99668 


35 


119.713 


2.07814 


35 


108.430 


2.03615 


36 


99.097 


1.99606 


36 


119.606 


2.07739 


36 


108.260 


2.03447 


36 


98.956 


1.99544 


37 


119.299 


2.07664 


37 


108.090 


2.03379 


37 


98.813 


1.99481 


38 


119.093 


2.07689 


38 


107.921 


2.03311 


38 


98.672 


1.99419 


39 


118.888 


2.07514 


39 


107.753 


2.03243 


39 


98.532 


1.99358 


40 


118.684 


2.07439 


40 


107.685 


2.03176 


40 


98.391 


1.99296 


41 


118.480 


2.07364 


41 


107.418 


2.03108 


41 


98.251 


1.99234 


42 


118.277 


2.07290 


42 


107.251 


2.03040 


42 


98.112 


1.99172 


43 


118.074 


2.07215 


43 


107.084 


2.02972 


43 


97.973 


1.99111 


44 


117.873 


2.07141 


44 


106.919 


2.02906 


44 


97.834 


1.99049 


45 


117.672 


2.07067 


45 


106.753 


2.02838 


46 


•97.696 


1.98988 


46 


117.471 


2.06993 


46 


106.588 


2.02771 


46 


97.558 


1.98926 


47 


117.272 


2.06919 


47 


106.424 


2.02704 


47 


97.420 


1.98865 


48 


117.073 


2.06846 


48 


106.261 


2.02637 


48 


97.283 


1.98804 


49 


116.875 


2.06772 


49 


106.097 


2.02570 


49 


97.147 


1.98743 


60 


116.677 


2.06699 


60 


105.935 


2.02604 


60 


97.010 


1.98682 


51 


116.480 


2.06625 


61 


105.772 


2.02437 


61 


96.874 


1.98621 


52 


116.284 


2.06652 


62 


105.611 


2.02371 


62 


96.739 


1.98560 


53 


116.088 


2.06479 


63 


105.449 


2.02304 


63 


96.604 


1.98500 


54 


115.893 


2.06406 


64 


105.289 


2.02238 


64 


96.469 


1.98439 


55 


116.699 


2.06333 


55 


105.128 


2.02172 


65 


96.334 


1.98378 


56 


115.505 


2.06260 


56 


104.968 


2.02106 


66 


96.200 


1.98318 


57 


115.312 


2.06187 


57 


104.809 


2.02040 


67 


96.067 


1.98257 


58 


115.120 


2.06115 


68 


104.660 


2.01974 


58 


96.933 


1.98197 


59 


114.928 


2.06043 


59 


104.492 


2.01908 


ized by V 


'Jhf, 


1.98137 



TABLES 
RADII — METRIC — (Continued) 



303 



Deg. 


Radius. 


Log.rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


WOO' 


95.668 


1.98077 


13" 00' 


88.337 


1.94614 


14-00' 


82.065 


1.91410 


01 


95.536 


1.98017 


01 


88.224 


1.94559 


01 


81.958 


1.91359 


02 


95.404 


1.97957 


02 


88.112 


1.94504 


02 


81.861 


1.91308 


03 


95.272 


1.97897 


03 


88.000 


1.94448 


03 


81.765 


1.91257 


04 


95.141 


1.97837 


04 


87.888 


1.94393 


04 


81.668 


1.91205 


05 


95.010 


1.97777 


05 


87.776 


1.94338 


05 


81.572 


1.91154 


06 


94.880 


1.97717 


06 


87.665 


1.94283 


06 


81.476 


1.91103 


07 


94.760 


1.97658 


07 


87.554 


1.94228 


07 


81.380 


1.91052 


08 


94.620 


1.97598 


08 


87.444 


1.94173 


08 


81.285 


1.91001 


09 


94.491 


1.97639 


09 


87.333 


1.94118 


09 


81.190 


1.90950 


10 


94.362 


1.97480 


10 


87.223 


1.94063 


10 


81.095 


1.90899 


11 


94.233 


1.97420 


11 


87.114 


1.94009 


11 


81.000 


1.90848 


12 


94.105 


1.97361 


12 


87.004 


1.93954 


12 


80.905 


1.90798 


13 


93.977 


1.97302 


13 


86.895 


1.93899 


13 


80.811 


1.90747 


14 


93.850 


1.97243 


14 


86.786 


1.93845 


14 


80.717 


1.90696 


15 


93.723 


1.97185 


15 


86.677 


1.93790 


15 


80.623 


1.90646 


16 


93.596 


1.97126 


16 


86.569 


1.93736 


16 


80.529 


1.90595 


17 


93.469 


1.97067 


17 


86.461 


1.93682 


17 


80.436 


1.90545 


18 


93.343 


1.97008 


18 


86.353 


1.93628 


18 


80.342 


1.90494 


19 


93.217 


1.96950 


19 


86.245 


1.93573 


19 


80.249 


1.90444 


20 


93.092 


1.96891 


20 


86.138 


1.93519 


20 


80.156 


1.90394 


21 


92.967 


1.96833 


21 


86.031 


1.93465 


21 


80.064 


1.90344 


22 


92.842 


1.96774 


22 


85.924 


1.93411 


22 


79.971 


1.90293 


23 


92.717 


1.96716 


23 


85.818 


1.93358 


23 


79.879 


1.90243 


24 


92.693 


1.96658 


24 


85.711 


1.93304 


24 


79.787 


1.90193 


25 


92.469 


1.96600 


25 


85.605 


1.93250 


25 


79.696 


1.90144 


26 


92.346 


1.96542 


26 


85.500 


1.93197 


26 


79.604 


1.90093 


27 


92.223 


1.96484 


27 


85.394 


1.93143 


27 


79.513 


1.90044 


28 


92.100 


1.96426 


28 


85.289 


1.93089 


28 


79.422 


1.89994 


29 


91.977 


1.96368 


29 


85.184 


1.93036 


29 


79.331 


1.89944 


30 


91.855 


1.96310 


30 


85.079 


1.92982 


30 


79.240 


1.89894 


31 


91.733 


1.96253 


31 


84.975 


1.92929 


31 


79.149 


1.89845 


32 


91.612 


1.96195 


32 


84.871 


1.92876 


32 


79.059 


1.89795 


33 


91.491 


1.96138 


33 


84.767 


1.92823 


33 


78.969 


1.89746 


34 


91.370 


1.96080 


34 


84.663 


1.92769 


34 


78 879 


1.89696 


35 


91.249 


1.96023 


35 


84.560 


1.92716 


35 


78.790 


1.89647 


36 


91.129 


1.95966 


36 


84.457 


1.92664 


36 


78.700 


1.89597 


37 


91.009 


1.95908 


37 


84.354 


1.92611 


37 


78.611 


1.89548 


38 


90.890 


1.95852 


38 


84.251 


1.92558 


38 


78.522 


1.89499 


39 


90.770 


1.96794 


39 


84.149 


1.92505 


39 


78.433 


1.89450 


40 


90.652 


1.95738 


40 


84.047 


1.92452 


40 


78.344 


1.89401 


41 


90.533 


1.95681 


41 


83.945 


1.92399 


41 


78.256 


1.89352 


42 


90.415 


1.95624 


42 


83.843 


1.92347 


42 


78.168 


1.89303 


43 


90.297 


1.95567 


43 


83.742 


1.92294 


43 


78.080 


1.89254 


44 


90.179 


1.95511 


44 


83.640 


1.92241 


44 


77.992 


1.89205 


45 


90.061 


1.95454 


45 


83.540 


1.92189 


45 


77 ..904 


1.89156 


46 


89.944 


1.95397 


46 


83.439 


1.92137 


46 


77.817 


1.89107 


47 


89.828 


1.95341 


47 


83.339 


1.92085 


47 


77.729 


1.89058 


48 


89.711 


1.95285 


48 


83.238 


1.92032 


48 


77.642 


1.89010 


49 


89.595 


1.95228 


49 


83.138 


1.91980 


49 


77.556 


1.88962 


50 


89.479 


1.95172 


50 


83.039 


1.91928 


50 


77.469 


1.88913 


51 


89.363 


1.95116 


51 


82.939 


1.91876 


51 


77.382 


1.88864 


52 


89.248 


1.95060 


52 


82.840 


1.91824 


52 


77.296 


1.88816 


53 


89.133 


1.96004 


53 


82.741 


1.91772 


53 


77.210 


1.88767 


54 


89.019 


1.94948 


54 


82.642 


1.91720 


54 


77.124 


1.88719 


55 


88.904 


1.94892 


55 


82.544 


1.91669 


55 


77.039 


1.88671 


56 


88.790 


1.94836 


56 


82.446 


1.91617 


56 


76.953 


1.88623 


57 


88.676 


1.94781 


57 


82.348 


1.91565 


57 


76.868 


1.88575 


58 


88.663 


1.94725 


58 


82.250 


1.91514 


58 


76.783 


1.88527 


59 


88.450 


1.94670 


59 


82.152 


1.91462 


59 


76.698 


1.88478 














... , , 


£1 
















c5 





304 



THE RAILROAD TAPER 
RADII — METRIC — (Continued) 



Deg. 


Radius. 


Log. rad. 


Dog. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


16" 00' 


76.613 


1.88430 


19-00' 


60.689 


1.78239 


23-00' 


50.169 


1.70035 


05 


76.192 


1.88191 


05 


60.326 


1.78060 


15 


49.626 


1.69671 


10 


75.776 


1.87953 


10 


60.067 


1.77864 


30 


49.106 


1.69113 


15 


75.364 


1.87716 


15 


59.809 


1.77677 


46 


48.696 


1.68660 


20 


74.957 


1.87481 


20 


59.654 


1.77491 


24-00' 


48.097 


1.68212 


25 


74.554 


1.87247 


25 


39.300 


1.77306 


15 


47.609 


1.87769 


30 


74.156 


1.87015 


30 


59.060 


1.77122 


30 


47.130 


1.67330 


35 


73.762 


1.86783 


35 


68.801 


1.76938 


45 


46.662 


1.66896 


40 


73.372 


1.86563 


40 


68.554 


1.76756 


25-00' 


46.202 


1.66486 


45 


72.986 


1.86324 


45 


68.309 


1.76574 


15 


45.752 


1.86041 


50 


72.604. 


1.86096 


60 


68.067 


1.76393 


30 


45.311 


1.66620 


55 


72.227 


1.85870 


65 


57.826 


1.76212 


45 


44.878 


1.65203 


16" 00' 


71.853 


1.85644 


20-00' 


67.688 


1.76033 


26-00' 


44.454 


1.64791 


05 


71.483 


1.85420 


05 


57.351 


1.75864 


15 


44.038 


1.64383 


10 


71.117 


1.85197 


10 


67.117 


1.76677 


30 


43.630 


1.63979 


15 


70.765 


1.84976 


16 


56.884 


1.75499 


45 


43.230 


1.63579 


20 


70.396 


1.84755 


20 


66.653 


1.75322 


27-00' 


42.837 


1.63182 


25 


70.041 


1.84636 


25 


56.425 


1.75147 


16 


42.451 


1.62789 


30 


69.690 


1.84317 


30 


66.198 


1.74972 


30 


42.072 


1.62399 


35 


69.342 


1.84100 


35 


55.973 


1.74798 


46 


41.701 


1.62015 


40 


68.998 


1.83884 


40 


56.749 


1.74624 


28-00' 


41.336 


1.61633 


45 


68.667 


1.83668 


46 


65.528 


1.74451 


15 


40.977 


1.61254 


50 


68.320 


1.83466 


50 


55.308 


1.74279 


30 


40.625 


1.60879 


55 


67.986 


1.83242 


66 


56.090 


1.74107 


45 


40.279 


1.00508 


17*' 00' 


67.665 


1.83030 


21-00' 


54.874 


1.73937 


29-00' 


39.939 


1.60140 


05 


67.327 


1.82819 


06 


64.660 


1.73767 


16 


39.605 


1.69775 


10 


67.003 


1.82609 


10 


64.447 


1.73597 


30 


39.277 


1.59414 


15 


66.682 


1.82401 


16 


64.236 


1.73429 


45 


38.964 


1.59055 


20 


66.363 


1.82193 


20 


64.026 


1.73260 


30-00' 


38.637 


1.58700 


25 


66.068 


1.81986 


25 


53.819 


1.73094 








30 


66.736 


1.81780 


30 


53.612 


1.72926 








35 


65.427 


1.81576 


35 


53.408 


1.72761 








40 


65.121 


1.81372 


40 


53.205 


1.72695 








45 


64.818 


1.81170 


45 


53.003 


1.72430 








50 


64.617 


1.80967 


60 


62.804 


1.72267 








55 


64.219 


1.80766 


56 


52.605 


1.72103 








18'*00' 


63.924 


1.80566 


22-00' 


52.408 


1.71940 








05 


63.632 


1.80368 


05 


52.213 


1.71778 








10 


63.343 


1.80170 


10 


52.019 


1.71616 








15 


63.066 


1.79973 


15 


61.827 


1.71456 








20 


62.772 


1.79777 


20 


51.636 


1.71295 








25 


62.490 


1.79581 


25 


51.446 


1.71135 








30 


62.211 


1.79387 


30 


51.258 


1.70976 








35 


61.935 


1.79194 


35 


51.072 


1.70818 








40 


61.661 


1.79001 


40 


50.886 


1.70660 








45 


61.389 


1.78809 


45 


50.702 


1.70503 








60 


61.120 


1.78618 


60 


60.620 


1.70346 








55 


60.863 


1.78428 


65 


50.338 


1.70190 









y Google 



TABLES 305 

NOTATION 
A = Curvature increment of taper per 10 meters of taper. 
M = Degree of main circular curve. 
I = Length of taper from the B.C. of the taper. 
/ = Deflection from tangent at the B.C. of the taper. 
ft fly fiy etc. = Deflection from tangent at some point on the 

taper. 
b = Deflection from tangent at the end of the taper, C.C. of 

main curve, to the B.C. of the taper. 
o = Central angle of the taper at each end of the curve. 
d = Circular curve offset from tangent to provide room for 

the taper. 
t = Distance along the tangent, from opposite the offset posi- 
tion of the B.C. of the simple curve to the B.C. of the 

taper. 
D = R -{- dj where R is the radius of the main circular curve. 
X and y are co-ordinates of points on the taper, referred to the 

tangent at the B.C. of the taper as meridian, and 

to a radial line at the B.C. of the taper as base. 
Ic = Long chord of the taper, from the B.C. of the taper to 

points on the taper. 
s = Distance along the taper, from the B.C. of the taper to 

the transit, when an intermediate set-up is necessary. 
T = Semi-tangent of the entire tapered curve, from P.I. to 

B.C. or E.C. 
E = External, the distance from the P.I. to a point on the curve 

radially opposite. 

EQUATIONS 

Eqn. No. 

T = « -h Z) tan i /. (7) Both ends the same 

taper. 

Tt =^ t + Dtani I - dcQcI. (8) Tapered end — other 

end not tapered. 

Tn = /2 tan I 7 + (/ CSC /. (9) Untapered end — 

other end tapered. 

Ti = ti + Di tan n - (di - dz) csc I. (10) Both ends tapered, 

with different ta- 
pers. 

T2 = ^ + D2 tan \I — {di — di) csc I. (11) Both ends tapered, 

with different ta- 
pers. 

J^ = Dseci/-/2. (12) Both ends the same 

Dig^Fcf^y'GOOgle 



306 THE RAILROAD TAPER 

EMPIRICAL EQUATIONS DERIVED FROM THE 
METRIC TABLES 



i = ^(M-A). 




, (l + 5)(l + 10) 
•' ~ 20 
A 




„ (l + 2s+15)(l- 


-s)+50 


^ 20 
A 




3/(i + 10) 
20 • 
A 





(45m) 
(107m) 

(94m) 

(108m) 



The equations under paragraph 18 as derived from the metric 
taper tables become : 

Z = ~ (M - A). (45m) 

^2=^(M2-A). (46m) 

ii = - -- • (47m) 

ir == ^ (M2 - Ml - A). (48m) 

^ 15(M2+M.)(M2-Mi-A) ,^. 

Oc = -7 • (50m) 



15^Mi (M 2 - Ml - A) 
A 



(51am) 



^^ = 20 ^^^""^ 

A 
. 5r(4 M,+2M. + A) (M2-M, -A) . ,1 



y Google 



TABLES 



307 





OS 
o%, 


i 


o lO lO lO lO iO »0 lO lO Tfi Tfi CO CO c^ ^ o ■ 




ok 
dSo 


1 


o 1# -^ TtH Tfl "^ TtH -^^ TtH CO CO (N C^ t-H O "O 




d?c 


s 


o CO "^ TtH -^ -^ TfH CO CO CO (N (N ^ O "Oi-i 




ds 
dSo 


1 


ocOCOCOCOCOCo"cO(N(N^t-iO -Oi-HtN 




dS 


o 


<» (N CO CO CO CO W C<I(M ^ ^ O 'Oi-HtNCO 




d^§ 
d-c 


8 


o (N <N (N C^ (M (M 1-H 1-1 1-H O "O^WCO-^ 




d^^ 
d-c 


8 


o ^ (N (N (M t-Hi-H ^ 1-H O • O 1-H (N CO rj< lO 


s 

1 


dis 
d-c 


g 






dS 
d?^ 


g 


O r-l^rHT-H^OO • O ^ 1-H (N CO TtH lO CO 




ok 

dtn 


S 


o 0»-H0000 'OO^C^COCO-^^OI^ 




d^^ 

dt," 


g 


o OOOOO •00^(N(NCO'<i<iOCOI> 




d53 

dtn 


§ 


o OOOO • O O 1-H ^ (N CO Tt< Tj^ lO CD 00 




dS 

^' dtn 

d^§ 
dh 


^ 


o OOO -OO^^tNC^COTj^iOcOt^OO 




g 


o OO -OOOt-h t-ttNCOCO-^iOCDI^OS 




d^g 
di:. 


o 


o O •OOOrH^(MC^C0rJ<i0cDl>.0005 




dis 


o 


• • O O O O i-Hi-H (N C<l CO TfH lO CD l> OO Oi 


•5 
1 


. 


- 


=2Sg5SSgS§82gg5S 

rH rH T— 1 T— 1 T— 1 T-H 



308 



THE RAILROAD TAPER 



II 
< 



ft 



o 

> 



P 

H 
O 

6 

H 



H 









2SS^?'gSg§SS28^^ 









i§§i§ili5ti^gk 



^ lO Oi "^ O b» 
OOOOOOOOO^i-ifHC^COCO 









I lO T-l Oi tN. CD 40 I 






2Sw5SSi>SS8SS^SS§ 



Digitized by vjjOOQ 



TABLES 309 

TAPER i METRIC -INTERMEDIATE VALUES A=i 



M 


I 


/ 


6 





Z) 


i6gZ> 


d 


t 


2 


V 


Ic 


e / 




o # 


o # 


o t 
















45 


20.000 


09i 


13 • 


22} 


1627.921 


3.18410 


0.022 


10.000 


0.065 


20.000 


20.000 


50 


23.333 


OUi 


017* 


29} 


1376.141 


3.13835 


0.030 


11.667 


0.080 


23.333 


23.333 


55 


26.067 


014} 


22J 


36} 


1260.144 


3.09696 


0.041 


13.333 


0.113 


26.666 


26.667 


1 00 


30.000 


017i 


27i 


45 


1146.984 


3.06918 


0.064 


16.000 


0.163 


29.999 


30.000 


06 


33.333 


020f 


33} 


64} 


1057.855 


3.02443 


0.070 


16.667 


* 
0.201 


33.333 


33.333 


10 


36.667 


24i 


040 


104} 


982.318 


2. 59226 


0.088 


18.332 


0.259 


36.665 


36.666 


15 


40.000 


028 


47 


115 


916.860 


2.96230 


0.109 


19.999 


0.327 


39.998 


39.999 


20 


43.333 


32i 


54} 


126} 


859.689 


2.93429 


0.133 


21.666 


0.407 


43.331 


43.332 


25 


46.667 


36i 


102} 


139} 


809.062 


2.90798 


0.160 


23.331 


0.497 


46.662 


46.665 


30 


50.000 


41i 


111} 


152} 


764.166 


2.88318 


0.190 


24.998 


0.600 


49.994 


49.998 


35 


53.333 


46i 


120} 


2 06} 


723.984 


2.85973 


0.226 


26.664 


0.716 


53.325 


63.331 


40 


56.667 


51i 


130} 


2 21} 


687.837 


2.83749 


0.263 


28.330 


0.847 


66.656 


56.663 


45 


60.000 


67 


140} 


2 37} 


665.139 


2.81633 


0.306 


29.996 


0.993 


59.986 


69.996 


50 


63.333 


102i 


151} 


2 54} 


625.423 


2.79617 


0.361 


31.662 


1.164 


63.316 


63.326 


55 


66.667 


108i 


203 


311} 


598.299 


2.77692 


0.402 


33.327 


1.332 


66.644 


66.668 


2 00 


70.000 


115 


215 


330 


673.445 


2.75849 


0.458 


34.992 


1.527 


69.972 


69.989 


05 


73.333 


12U 


2 27} 


3 49} 


550.688 


2.74083 


0.618 


36.667 


1.740 


73.299 


73.319 


10 


76.667 


128i 


2 40} 


4 09} 


629.499 


2.72387 


0.683 


38.321 


1.972 


76.623 


76.650 


15 


80.000 


\m 


2 54} 


430 


609.982 


2.70765 


0.654 


39.986 


2.224 


79.948 


79.979 


20 


83.333 


143 


308} 


4 51} 


491.872 


2.69185 


0.731 


41.651 


2.497 


83.270 


83.307 


25 


86.667 


160f 


3 23} 


5 14} 


476.020 


2.67671 


0.813 


43.314 


2.791 


86.590 


86.635 


30 


90.000 


1681 


3 38} 


6 37} 


459.303 


2.66210 


0.900 


44.977 


3.107 


89.909 


89.962 


35 


93.333 


207 


3 54} 


6 01} 


444.612 


2.64798 


0.994 


46.639 


3.446 


93.224 


93.288 


40 


96.667 


2 15i 


4 11} 


6 26} 


430.851 


2.63433 


1.094 


48.302 


3.808 


96.538 


96.614 


45 


100.000 


2 24i 


4 28 


6 52} 


417.937 


2.62111 


1.200 


49.964 


4.196 


99.849 


99.937 


50 


103.333 


2 33i 


4 46} 


719} 


406.795 


2.60831 


1.313 


51.626 


4.608 


103.157 


103.259 


55 


106.667 


2 42} 


604 


7 46} 


394.361 


2.69589 


1.433 


63.285 


5.047 


106.461 


106.681 


300 


110.000 


2 52} 


5 22} 


8 15 


383.676 


2.68385 


1.560 


64.945 


5.512 


109.762 


109.900 


06 


113.333 


3 02} 


1)41} 


8 44} 


373.387 


2.57216 


1.694 


56.605 


6.006 


113.059 


113.217 


10 


116.667 


3 12i 


6 01} 


9 14} 


363.760 


2.56080 


1.836 


68.262 


6.526 


116.351 


116.534 


15 


120.000 


323 


6 22 


9 45 


354.622 


2.54977 


1.986 


69.920 


7.077 


119.639 


119.848 


20 


123.333 


334 


6 42} 


10 16} 


345.965 


2.53903 


2.142 


61.576 


7.658 


122.922 


123.160 


25 


126.667 


3 45 


7 04} 


10 49} 


337.747 


2.62859 


2.307 


63.231 


8.269 


126.199 


126.470 


30 


130.000 


3 56i 


7 26} 


1122} 


329.936 


2.51843 


2.480 


64.886 


8.911 


129.470 


129.776 


35 


133.333 


4 07} 


7 49 


1156} 


322.603 


2.50863 


2.661 


66.639 


9.586 


132.734 


133.078 


40 


136.667 


419} 


8 12 


12 31} 


315.427 


2.49890 


2.851 


68.189 


10.294 


135.992 


136.380 


45 


140.000 


4 32 


8 35} 


13 07} 


308.683 


2.48961 


3.051 


69.840 


11.035 


139.242 


139.678 


50 


143.000 


4 44i 


900 


13 44} 


302.249 


2.48036 


3.269 


71.489 


11.810 


142.484 


142.973 


55 


146.667 


4 57 


9 24} 


14 21} 


296.107 


2.47146 


3.476 


73.135 


12.620 


146.718 


146.265 


4 00 


150.000 


510 


960 


15 00 


290.240 


2.46276 


3.703 


74.781 


13.467 


148.942 


149.650 



Digitized by VjOOQIC 



310 



THE RAILROAD TAPER 



< 


5 

1 
1 




1 


o OS (3> OJ Oa Oi Oi Oa 00 b* t>- »0 "^ CO 1-1 * 




d8 


1 


o l^ X 00 OO 00 OO t>- 1>- CO »0 "^ CO 1-I 'i-H 


>i 


OS. 


§ 


o CO t* t>- t>- t* «0 CO lO Tj< CO C^ tH *rHCO 


o 

>-• 


OS 
o«« 


O 


o lO O CO CO CO lO »0 "^ CO W ^ "t-iCOkO 




ds 


8 


o rl< lO lO «0 -* -^ CO CO W 1-I "rHCSI-^CO 


o 

o 


OS 
ou 


8 


o CO '^ ''i^ ^ CO CO (N <N -H -rHC^TjiCOX 


O 


t 


S 


• CO CO CO CO C^ CS| 1-1 1-1 • 1-1 W CO »o t* 05 


O 


0%, 


g 


(N (N CM <M <N 1-1 O : 1-4 C^ CO »0 CO 00 O 


15 

1— 1 


d?o 


S 


o 1-H CM tH f-i tH O ; O 1-1 CO t!* CO t^ 05 1-* 


t 


s 


^^^^o : O 1-1 CM Tj< lO t* 00 O CM 




dk 
d-c. 


s 




u 

1-1 


dS 
d'„ 


s 


OOO : O i-< C^ CO "^ »0 1>« 00 O C^ Tj< 

f-4 ,.Hi-l 


dg 


R 


o OO : O O »-< C^ CO "^ CO t* 05 1-1 CO W5 

rH rH T-l 


t 


o 


O :OOi-iCMCO'^»OCOOOOCM"^CO 
rH 1-H rH rH 




a 


o 


o :OOOi-iC<ICO'^»Ot*OOOCMrt<t* 




J 


13 


- 


Digitize l5y<ri«(m>glC 



TABLES 



311 



T 

o 

1—1 

> 



O 



I 

d 



Ah 

H 






5 05 w 05 Oi O^ 00 1^ O CO '"H t* 



§Qooc^t>"<©oiC^<©g5C<icooo 
QQigiI>-^0005C00ii0OS^ 
0050>050SOOt^«OffOOiOOOOi 



iiiil 






§oc5t^W'^05cpi-HOO^oo«pg 



00'-<C^CO«DOiCOt>-CO»-HOa050 



OOOOOOOi-Hi-ic^COCOTjH^O 



COOCO»OrH«PI>b-(NWC3C^iO;iaj 

T-i»-Hd^Th»oo»oc^t>-oiC^»-4cqco 

o5ot^<o25u5'*^coco<NC5w^ 
CO w* c^ c^ c^ w c^ c^ 051 c^ w c^ c<i c^ 



0>OOOC0OHHOO00t-iC0t*0s 

iO'^*COCOC<ioQt^«dTHO"^QQOOO 

TtHCOt^KoQC5|QQ»OCOi-tOi002>»0 

^t^iO'Shwcociic^cQc^^^i-Ht-H 



o oo^c^co»ot^oaTHcooo5gg 



SSg§gSgS8S28g5 



o i-(i-H(NiNcocoT*<Tt<ioio<©<© ^*/^- 

Digitized by v^J' 



oo 



igle 



312 THE RAILBOAD TAPER 

TAPER 1 METRIC. — INTERMEDIATE VALUES A=i 



M 


\ 


/ 


h 


o 


D 


logZ) 


i 


t 


z 


y 


Ic 


9 1 

1 30 


20.000 


o / 

18! 


/ 

26} 


e $ 

45 


764.009 


2.88310 


0.043 


10.000 


0.109 


20.000 


20.000 


35 


21.667 


21 


030 


51 


723.810 


2.85962 


0.051 


10.833 


0.133 


21.666 


21.667 


40 


23.333 


23i 
26i 


34 


58 


687.634 


2.83736 


0.060 


11.666 


0.160 


23.333 


23.333 


45 


25.000 


39 


105 


654.905 


2.81618 


0.071 


12.499 


0.191 


24.999 


25.000 


50 


26.667 


029 


044 


113 


626.164 


2.79699 


0.082 


13.333 


0.226 


26.665 


26.666 


55 


28.333 


32 


49 


121 


697.991 


2.77669 


0.094 


14.166 


0.263 


28.332 


28.333 


200 


30.000 


035 


066 


130 


673.095 


2.75823 


0.108 


14.999 


0.306 


29.998 


29.999 


05 


31.666 


38} 


100} 
106} 


139 


560.193 


2.74052 


0.123 


16.832 


0.352 


31.664 


31.665 


10 


33.333 


41i 


148} 


529.055 


2.72350 


0.139 


16.666 


0.403 


33.330 


33.332 


15 


35.000 


045 


113 


158 


509.484 


2.70713 


0.166 


17.498 


0.458 


34.996 


34.998 


20 


36.667 


48J 


1191 


2 08} 


491.316 


2.69136 


0.176 


18.331 


0.618 


36.661 


36.665 


25 


38.333 


52} 


126 


219 


474.402 


2.67615 


0.196 


19.164 


0.683 


38.327 


38.331 


30 


40.000 


056} 


133 


230 


458.620 


2.66145 


0.217 


19.997 


0.664 


39.992 


39.997 


35 


41.667 


100 


141 


2 41} 


443.858 


2.64725 


0.240 


20.829 


0.730 


41.657 


41.663 


40 


43.333 


104 


148 


2 53 


430.022 


2.63347 


0.265 


21.662 


0.812 


43.322 


43.328 


45 


45.000 


108 


156 


305 


417.028 


2.62017 


0.291 


22.495 


0.900 


44.986 


44.995 


50 


46 667 


113 


205 


3 18 


404.801 


2.60724 


0.319 


23.327 


0.993 


46.660 


46.661 


55 


48.333 


117 


IMl 


3 31 


393.277 


2.69470 


0.349 


24.160 


1.093 


48.314 


48.326 


300 


50.000 


122^ 


346 


382.396 


2.68251 


0.380 


24.992 


1.200 


49.977 


49.992 


05 


51.667 


127} 


2 31} 


3 59 


372.106 


2.57067 


0.413 


25.824 


1.313 


51.640 


61.667 


10 


53.333 


132 


2 41 


4 13} 


362.363 


2.55914 


0.449 


26.657 


1.432 


53.302 


63.321 


15 


55.000 


137 


2 50* 


4 28 


353.123 


2.64793 


0.486 


27.489 


1.559 


64.964 


64.986 


20 


56.667 


142 


300 


4 43} 


344.348 


2.53700 


0.525 


28.320 


1.693 


66.626 


66.650 


25 


58.333 


148 


3 10 


4 59 


336.006 


2.52635 


0.666 


29.152 


1.836 


68.286 


68.314 


30 


60.000 


153 


3 21 


5 15 


328.064 


2.61596 


0.609 


29.984 


1.984 


59.946 


69.978 


35 


61.667 


159 


3 32 


6 31} 
5 48 


320.496 


2.50582 


0.664 


30.815 


2.141 


61.606 


61.643 


40 


63.333 


2 05 


3 43 


313.278 


2.49593 


0.702 


31.646 


2.306 


63.264 


63.306 


45 


65.000 


2 11 


3 64} 


605 


306.384 


2.48627 


0.762 


32.477 


2.480 


64.922 


64.969 


50 


66.667 


2 17 


406 


623 


299.794 


2.47682 


0.804 


33.308 


2.662 


66.579 


66.633 


55 


68.333 


2 23 


4 18 


6 41 


293.489 


2.46759 


0.868 


34.139 


2.862 


68.236 


68.294 


4 00 


70.000 


230 


4 30 


700 


287.452 


2.45857 


0.915 


34.969 


3.061 


69.889 


69.956 


05 


71.667 


2 361 


4 424 


7 19 


281.666 


2.44973 


0.974 


35.799 


3.260 


71.643 


71.618 


10 


73.333 


2 43} 


455 


7 38} 


276.115 


2.44109 


1.035 


36.629 


3.477 


73.196 


73.278 


15 


75.000 


250 


608 


758 


270.788 


2.43263 


1.099 


37.459 


3.704 


74.847 


74.939 


20 


76.667 


2 57 


6 21} 


8 18} 


266.671 


2.42434 


1.166 


38.288 


3.941 


76.497 


76.600 


25 


78.333 


304 


535 


8 39 


260.752 


2.41623 


1.235 


39.117 


4.187 


78.146 


78.259 


30 


80.000 


311} 
3 18} 


6 48} 


900 


256.020 


2.40827 


1.307 


39.946 


4.444 


79.792 


79.918 


35 


81.667 


6 03 


9 21} 


251.467 


2.40048 


1.382 


40.774 


4.710 


81.437 


81.576 


40 


83.333 


3 26 


6 17} 


943 


247.080 


2.39284 


1.459 


41.602 


4.987 


83.081 


83.232 


45 


85.000 


3 331 


6 3l| 


10 05 


242.854 


2.38536 


1.640 


42.430 


6.274 


84.723 


84.888 


50 


86.667 


3 41 


6 46} 


10 28 


238.779 


2.37800 


1.623 


43.267 


6.672 


86.363 


86.643 


55 


88.333 


3 49 


702 


10 51 


234.848 


2.37079 


1.709 


44.064 


6.882 


88.000 


88.196 


600 


90.000 


3 67 


717} 


1116 


231.054 


2.36372 


1.798 


44.911 


6.202 


89.636 


89.850 


05 


91.667 


4 05} 


7 33} 


1139 


227.390 


2.35677 


1.890 


45.736 


6.534 


91.270 


91.503 


10 


93.333 


414 


7 49 


12 03} 


223.850 


2.34996 


1.985 


46.662 


6.877 


92.901 


93.154 


15 


95.000 


4 22} 


805 


12 28 


220.429 


2.34327 


2.083 


47.387 


7.232 


94.629 


94.805 


20 


96.667 


4 31 


8 22 


12 53} 


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EVERY 10 M. A=3 



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EVERY 10 M.— {Continued) A=3 



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t 


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2.28135 


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0.131 


9.999 


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900 


20 


430 


127.716 


2.10625|0.261 


9.986 


0.654 


19.986 


19.997 


12 00 


30 


900 


96.320 


1.98372 


0.652 


14.951 


1.830 


29.916 


29.972 


15 00 


40 


15 00 


77.912 


1.89160 


1.299 


19.868 


3.910 


39.697 


39.889 


18 00 


50 


22 30 


66.186 


1.8207712.262 


24.703 


7.129 


49.166 


49.680 


2100 


60 


3130 


58.471 


1.766943.597 


29.405 


11.684 


58.076 


59.240 


24 00 


70 


42 00 


53.451 


1.72796 


5.354 


33.906 


17.709 


66.090 


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TAPER 3i METRIC. CURVATURE INCREASING 4** 00' 
EVERY 10 M. A=4 









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C.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 




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8-00' 


12*00' 


16*00' 


20*00' 


24*00' 


28*00' 


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15 10 


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200 


430 


7 40 


1130 


16 00 


20 


330 


200 




300 


630 


10 40 


16 30 


30 


720 


530 


300 




400 


830 


13 40 


40 


12 30 


10 20 


730 


400 




500 


10 30 


50 


19 00 


16 30 


13 20 


930 


500 




600 


60 


26 60 


24 00 


20 30 


16 20 


1130 


600 





TAPER 3i METRIC. CURVATURE INCREASING 4*^ 00' 
EVERY 10 M. — (Continued) A=4 



M 


I 





D 


log D 


d 


t 


X 


y 


Ic 


o » 

800 


10 


o / 

200 


143.443 


2.16668 


0.087 


4.996 


0.175 


9.999 


10.000 


12 00 


20 


600 


96.016 


1.98234 


0.348 


9.974 


0.872 


19.974 


19.993 


1600 


30 


12 00 


72.719 


1.86165 


0.866 


14.912 


2.437 


29.851 


29.951 


20 00 


40 


20 00 


59.309 


1.77312 


1.721 


19.767 


5.195 


39.463 


39.804 


24 00 


50 


30 00 


51.083 


1.70828 


2.986 


24.477 


9.432 


48.526 


49.434 


28 00 


60 


42 00 


46.067 


1.66339 


4.731 


28.957 


15.347 


56.616 


58.660 



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TABLES 



341 



TAPER 3} METKIC. INTERMEDIATE VALUES 


A=4 


M 


I 


/ 


b 





D 


log D 


d 


t 


X 


V 


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12 00 


20.000 


o / 

230 


330 


e f 

600 


96.016 


1.98234 


0.348 


9.974 


0.872 


19.974 


19.993 


16 


20.625 


2 37 


3 42 


619 


94.095 


1.97367 


.0.372 


10.284 


0.941 


20.596 


20.617 


30 


21.250 


244 


3 53J 


6 38} 


92.252 


1.96498 


0.397 


10.594 


1.013 


21.217 


21.241 


45 


21.875 


2 5U 


407 


658} 


90.484 


1.95657 


0.423 


10.904 


1.089 


21.838 


21.865 


13 00 


22.500 


2 58i 


420 


718} 


88.787 


1.94835 


0.450 


11.214 


1.169 


22.458 


22.488 


15 


23.125 


306i 


4 33} 


7 39} 


87.155 


1.94029 


0.478 


11.623 


1.252 


23.077 


23.111 


30 


23.750 


314 


4 47 


8 01 


85.587 


1.93241 


0.508 


11.833 


1.339 


23.696 


23.734 


45 


24.375 


823 


500} 


8 22} 


84.078 


1.92468 


0.638 


12.142 


1.430 


24.315 


24.357 


14 00 


26.000 


830 


5 15 


8 45 


82.625 


1.91711 


0.570 


12.450 


1.525 


24.933 


24.979 


15 


25.625 


3 38i 


5 29} 


9 07} 


81.225 


1.90969 


0.602 


12.769 


1.624 


25.550 


25.602 


80 


26.250 


3 46i 


5 441 


9 31 


79.876 


1.90242 


0.636 


13.067 


1.727 


26.167 


26.224 


45 


26.875 


355 


5 59} 


9 54} 


78.575 


1.89528 


0.671 


13.376 


1.834 


26.783 


26.846 


15 00 


27.500 


4 03i 


615 


1018} 


77.321 


1.88830 


0.708 


13.683 


1.945 


27.398 


27.467 


30 


28.750 


4 2U 


6 47 


1108} 


74.940 


1.87471 


0.784 


14.298 


2.182 


28.626 


28.710 


16 00 


30.000 


4 40 


720 


12 00 


72.719 


1.86165 


0.866 


14.912 


2,437 


29.851 


29.951 


30 


31.250 


4 59 


7 54} 


12 53} 


70.643 


1.84906 


0.963 


16.624 


2.710 


31.072 


31.190 


17 00 


32.500 


5 18} 


8 30 


13 48} 


68.700 


1.83696 


1.046 


16.135 


3.002 


32.288 


32.427 


30 


33.750 


5 39 


9 07 


14 46 


66.879 


1.82529 


1.143 


16.745 


3.315 


33.499 


33.662 


18 00 


35.000 


6 00 


945 


15 45 


65.170 


1.81405 


1.246 


17.353 


3.648 


34.704 


34.895 


30 


36.250 


6 2U 


10 24} 


16 46 


63.567 


1.80323 


1.356 


17.959 


4.002 


35.904 


36.126 


19 00 


37.500 


6 43} 


1105 


17 48} 


62.000 


1.79281 


1.471 


18.563 


4.377 


37.097 


37.355 


30 


38.750 


706i 


1147 


18 53} 


60.643 


1.78278 


1.593 


19.166 


4.775 


38.284 


38.581 


30 00 


40.000 


730 


12 30 


20 00 


59.309 


1.77312 


1.721 


19.767 


5.195 


39.463 


39.804 


30 


41.250 


764 


13 14} 


2108} 


58.053 


1.76382 


1.865 


20.365 


5.639 


40.638 


41.023 


2100 


42.500 


818} 


14 00 


22 18} 


56.870 


1.75488 


1.996 


20.961 


6.106 


41.794 


42.238 


30 


43.760 


844 


14 47 


23 31 


65.755 


1.74628 


2.143 


21.554 


6.598 


42.945 


43.449 


22 00 


45.000 


9 10 


15 35 


24 45 


64.706 


1.73804 


2.298 


22.145 


7.114 


44.086 


44.666 


30 


46.250 


9 36^ 


16 24} 


26 01 


53.717 


1.73011 


2.459 


22.733 


7.665 


45.215 


46.858 


23 00 


47.500 


10 03} 


17 15 


27 18} 


52.787 


1.72253 


2.628 


23.317 


8.222 


46.332 


47.066 


24 00 


50.000 


1100 


19 00 


30 00 


51.063 


1.70828 


2.986 


24.477 


9.432 


48.526 


49.434 


25 00 


52.500 


1158} 


20 50 


32 48} 


49.577 


1.69528 


3.375 


25.623 


10.749 


50.660 


61.787 


26 00 


55.000 


13 00 


22 45 


35 45 


48.248 


1.68348 


3.794 


26.753 


12.173 


52.726 


54.111 


27 00 


57.500 


14 03} 


24 45 


38 48} 


47.083 


1.67286 


4.246 


27.865 


13.705 


54.714 


56.403 


28 00 


60.000 


15 10 


26 50 


42 00 


46.067 


1.66339 


4.731 


28.957 


16.347 


56.616 


58.660 



y Google 



342 



THE RAILROAD TAPER 



TAPER 3f METRIC. CURVATURE INCREASING 5° 00' 
EVERY 10 M. A =5 



Transit 
at 






Deflections to 






B.C. 

6° 00' 


C.C, 

lO^OO' 


C.C. 

16" 00' 


C.C. 

20*00' 


C.C. 

25^*00' 


C.C. 

30*00' 


I 





10 


20 


30 


40 


50 



10 
20 
30 
40 
50 


o / 

i'i5"* 

4 22} 

9 10 
15 37} 
23 45 


o / 

1 15 

2 30' 
6 52} 

12 55 
20 37} 


o / 

3 07} 
230 

3 45 ' 
9 22} 
16 40 


550 
5 37} 
3 45 

'566" 
1152} 


/ 

9 22} 
9 35 

8 07} 
500 

615' 


1 

13 45 

14 22} 
13 20 
10 37} 

6 15 



TAPER 3| METRIC. CURVATURE INCREASING 5° 00' 
EVERY 10 M. — (Continued) A=5 



M 


I 





D 


log D 


A 


i 


X 


V 


Ic 


/ 
10 00 


10 


1 
2 30 


114.847 


2.06012 


0.110 


4.993 


0.218 


9.998 


10.000 


15 00 


20 


7 30 


77.048 


1.88676 


0.435 


9.960 


1.090 


19.960 


19.990 


20 00 


30 


15 00 


58.667 


1.76839 


1.079 


14.863 


3.041 


29.767 


29.922 


25 00 


40 


25 00 


48.338 


1.68429 


2.136 


19.638 


6.465 


39.164 


39.694 


30 00 


50 


37 30 


42.331 


1.62666 


3.694 


24.192 


11.675 


47.713 


49.120 



y Google 



TABLES 



343 



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344 



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THE RAILROAD TAPEB 


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THE BAJUIOAD TAPEB 



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TABIiBS 



351 



CX)NVERSION OF ENGLISH INCHES INTO CENTIMBTElttS. ( 


Ins. 





1 


9 


8 


4 


5 


6 


7 


8 







Cm. 


Cm. 


Cm. 


Cm. 


Cm. 


Cm. 


Cm. 


Cm. 


Cm. 


^^ 





0.000 


9.640 


5.060 


7.690 


10.16 


19.70 


15.94 


17.78 


90.89 


98.86 


10 


96.40 


97.94 


80.48 


88.09 


86.56 


88.10 


40.64 


48.18 


46.72 


48.96 


SM) 


60.80 


53.84 


66.88 


68.42 


60.96 


63.50 


66.04 


68.58 


71.19 


78.66 


ao 


76.80 


78.74 


81.98 


88.82 


86.36 


88.90 


91.44 


98.98 


96.52 


99.06 


40 


101.00 


104.14 


106.68 


109.29 


111.76 


114.30 


116.84 


119.38 


191.92 


194.46 


60 


197.00 


199.54 


189.08 


184.62 


187.16 


189.70 


149.94 


144.78 


147.89 


149.86 


00 


169.40 


154.94 


167.48 


160.02 


169.56 


165.10 


167.64 


170.18 


179.79 


175.« 


70 


177.80 


180.84 


189.88 


186.42 


187.96 


190.60 


198.04 


195.58 


198.12 


200.96 


80 


903.90 


906.74 


906.98 


910.82 


918.36 


915.90 


218.44 


290.96 


998.69 996.C6I 


90 


998.60 


931.14 


938.68 


986.99 


288.76 


941.80 


248.84 


246.38 


948.99 


961.46 


100 


964.00 


966.64 


959.06 


961.69 


964.16 


966.70 


269 24 


971.78 


274.82 


976.W 


CONVERSION OF CENTIMBTERS INTO ENGLISH INCHES. 


Cm. 





1 


9 


8 


4 


5 


6 


7 


8 


9 




Ins. 


Ids. 


Ins. 


Ins. 


Ids. 


Ins. 


Ids. 


Ins. 


Ins. 


Ids. 





0.000 


0.894 


0.787 


1.181 


1.576 


1.969 


9.369 


9.756 


3.150 


8.548 


10 


8.987 


4.881 


4.749 6.118 


6.512 


5.906 


6.999 


6.698 


7.087 


7.480 


80 


7.874 


8.968 


8.669 9.056 


9.449 


9.843 


10.986 


id.eao 


11.094 


11.418 


ao 


11.811 


19.905 


19.699 19.909 


18.886 


18.780 


14.173 


14.667 


14.961 


15.866 


40 


16.748 


16.149 


16.686 16.999 


17,323 


17.717 


18.11118.504 


18.898 


19.998 


60 


19.686 


90.079 


90.473 90.867 


21.260 


21.654 


22.048 29.441 


92.886 


98.990 


00 


98.699 


94.016 


94.410, 94.804 


95.197 


95.591 


25.986,96.878 


26.779 


97.106 


70 


97.660 


97.968 


98.847 98.741 


99.134 


99.628 


29.99980.316 


80.709 


31.108 


80 


81.497 


31.890 


89.984 39.678 


88.071 


33.465 


33.859,34.253 


84.646 


86.040 


90 


86.434 


86.897 


36.921 36.615 


87.009 


37.402 


87.796'88.190 


88.688 


38.977 


100 


89.370 


89.764 


40.158| 40.559 


^.945 


41.339 41.783 42.126 


42.690 


42.914 


CONVERSION OF ENGLISH FEET INTO METERS. | 


Feet. 





1 


9 


8 


4 


6 


6 


7 


8 


9 




Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 


Met. 





0.000 


0.8048 


0.6096 


0.9144 


1.9199 


1.6939 


1.8287 


9.1835 


2.4863 


2.7481 


10 


8.0479 


3.3697 


8.6676 


8.9693 


4.9671 


4.5719 


4.8767 


5.1815 


6.4868 


5.7911 


90 


6.0959 


6.4006 


6.7066 


7.0109 


7.8160 


7.6198 


7.9946 


8.9994 


8.5349 


8.8390 


80 


9.1438 


9.4486 


9.7584 


10.068 


10.363 


10.668 


10.979:11.977 


11.589 


11.887 


40 


19.199 


12.496 


12.801 


13.106 


13.411 


18.716 


14.090;i4.825 


14.680 


14.935 


60 


16.989 


15.644 


15.849 


16.154 


16.459 


16.763 


17.06817.373 


17.678 


17.988 


60 


18.987 


18.699 


18.897 


19.909 


19.507 


19.811 


20.116 20.421 


20.796 


21.081 


70 


21.336 


91.640 


21.945 


92.250 


99.555 


22.859 


23.164 23.469 


28.774 


24.079 


80 


94.383 


24.688 


94.993 


95.296 


95.609 


95.907 


26.919 26.517 


96.892 


97.196 


90 


97.431 


97.736 


98.041 


28.346 


98.65t 


98.955 


29.960 29.565 


29.870 


30.174 


100 


30.479 


30.784 


31.089 


31.394 


31.698 


89.003i39.806 82.613 


39.918 


33.999 


CONVERSION OF METERS INTO ENGLISH FEET. 


Met. 





1 


9 


3 


4 


6 


6 


7 


8 


9 




Feet. 


Feet. 


Feet. , Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 





0.000 


3.2809 


6.5618 9.8427 


13.128 


16.404 


19.685 


92.966 


26.247 


99.598 


10 


82.809 


36.090 


39.371 42.651 


45.938 


49.213 


59.494 


55.775 


59.056 


69.887 


90 


65.618 


68.899 


72.179, 75.481 


78.741 


82.022 


85.303 


88.584 


91.865 


95.146 


80 


96.497 


101.71 


104.99 108.27 


111 66 


114.83 


118.11 


121.39 


124.67 


197.96 


40 


131.94 


134.52 


137.80 141.08 


144.36 


147.64 


150.99 


154.90 


157.48 


160.76 


60 


164.04 


167.83 


170 61 178.89 


177.17 


180.45 


183.73 


187.01 


190.29 


198.67 


00 


196.85 


200.18 


203.49 906.70 


909.98 


213.26 


216.54 


219.82 


223.10 


226.88 


70 


999.66 


939.94 


236.22 239.51 


949.79 


946.07 


249.35 


259.63 


255.91 


959.19 


80 


969.47 


965.75 


269.03; 279.31 


975.60 


978.88 


282.16 


285.44 


288.72 


292.00 


90 


996.98 


996.56 


801.84 305.12 


308.40 


311.69;314.97 


318.26 


321.53 


324.81 


100 


898.09 


881.87 


384.651 337.931 341.91 


344. 49! 347. 78 


351.06 


354.84 


357.09 



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352 



THE RAILBOAD TAFEB 



TRIGONOMETRIC AND MISCELLANEOUB 
FORMULAS. 

80LUTI0N OF RIGHT TRIANGLES. 



Beqnfrod. 


Given. 


A, C,c 


a, b 


A, C, 6 


a,e 


C, b, c 


A, a 


C,a,c 


A,b 


C,a,& 


A,c 



FormnlM. 



■in^: 



= co«0==?; c = ^/(6+a)(6~«). 





tan-4=cotB=«"; 
c 

C =^ 90° — • -/4 ; c = acot-4; 6 = rtcoeecA. 

C = 9(y* — -4; a^bsinA; c = bcos\nA. 



Required. 
b 

B 

H^ + B) 
UA-B) 

A 
B 



Area 
Area 

Area 



SOLUTION OF OBLIQUE TRIANGLES. 

FormtUafl. 



Oiyen. 
A, B, a 

A^ a, b 

a,b,C 



a, &, c 



A, &f c 



6 
BinB 



flsinB 
sin A 
bain A 



i(A4-B)=ri(l80-C) 

Uni(^ - B) = 5Lzi tan J(X + J5) 
a + 



If s=ri(a4-6 + c), 8inM=^i^Z5ffiE£i 
\ 6c 



coR^A 
taniA 



\ s(8-a) 



6c 
Area = ^/a(« — a)(8 — 6)(« — c) 
Area =: 1 6c sin ii 



Area« 



e* sin A sin B 



2sm(^+B> 



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^ 



TABLES 

TRIGONOMETRIC AND MISCELLANEOUS 
FORMULAS. 

GENERAL FORMULAS. 



353 



Bln-4 = \/l — co^A = tan-4 coeA. 
saiA=2smiAconiA, 

Bin ^ = _J_ = V^l - COS2A). 
QoaecA 

C08-4 = -i~ = -w/l — 8in2^ = cot-4 sin A, 
sec -4 

cos-4 = 1—2 aiD^iA = 1 — vers A. 



COB A = V J + 1C082 A = coa^iA — sinH^ 
cos -4 



. n/1 — cosg^ ^ sin2 ii 

cos-4 1 + C0S2-4' 

* . 1 1 — C OS 2^ 

tan -A = ■ = — • 

cot A sin 2^ 

cot^ = -^ = ^^=v^co8ecM-L 
tan A sin A 

tA = sin 2^ _ 1 4- cos2^ 
^^ 1-C082^ 8in2^ 

aecA = — =— = the reciprocal of any expression for oos^, 
cos -4 

cosec A = -i- = the reciprocal of any expression for sin -4. 
sin^ 

ver8-4 = 1 — cos-4 = 2 sin^i A. 

•xsec^ = 8ec^ - 1 ^I^Il^. 
cos^ 

«i^i A /l — C06-4 / vers 4 

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364 THE BAILROAD TAPER 

TRIGONOMETRIC AND MISCELLANEOUS 
FORMULAS. 



(iOsiA 



V l + cc 



008*3 



* 1 + secA sin^ 1 + cosA 

^. . 1 + C08-4 BinA 
cot 1^ = — - 



sin -4 1 — cos-4 

sir» 2 -4 = 2 sin A cos A. 
C0H2A = co82^ — sin2-4. = 2 coa^A — L 

2tan^ 



tan2-4: 



^012 A = 



1-tanM 
cot^A — 1 



2cot-4 

sin (A± B)= sin A cos J5 ± cos-4 sinB. 

cos(^ ± 5) =co8^ cosB zf sinAsinS. 

/ /I _i_ r>\ tan A ± tan B 

tan (A ± B) = — — =- • 

^ ' liptan^taniJ 

sin^ +sinS = 2sini(^ + B)cosi(^ — B>. 

sin^— sinB = 2cosi(^ + JB)sini(^ — B). 

co&A + cosB==2cosi(^ + B) cosi(^ - 5). 

cosB — cos-4 = 2sini(^ 4- B) sini(^ — B). 

sin2^-8in2B = cos2B-cosM==5in(^ +B)Gm(^-a). 

cos2^ - sin2^ = cos (A + B) cos(^ -^ B). 

tan^±tanB = ?Hi<4A^. 

cot4±cotB-±^'»-M.m 



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TABLES 



355 



TRIGONOMETRIC AND MISCELLANEOUS 
FORMULAS. 

MISCELLANEOUS FORMULAS. 



Reqnirod. 
Area of 
Trapezoid 

Regular Polygon 

Circle 

Ellipse 

Parabola 

Surfojce of 
Cone 

Cylinder 
Sphere 

Zone 

Volume of 
Prism or cylinder 

Pyramid or cone 

Erustam of 

Pyramid or cone 



Sphere 



Given. 


Formnlafl. 


Parallel sides = m and n 
Perp. dist. bet. them =p 


|(m + n) 


Length of side = I 
Number of sides = n 


71/2 180» 

— cot 

4 n 


Radius = r 


7cr^l7r=SA4m 


Semi-axes = a and b 


Ttab 


Base == 6, height = h 


Ibh 


Radius of base = r 




Slant height = a 
Radius = r, height = h 


irrB 
2 7erh 


Radius = r 


iTtf^ 


Height = A 

Radius of iu sphere — r 


27trh 


Area of base = b 


bh 


Height = ^ 




Area of base = b 


bh 


Height = Ji 


3 


Area of bases = b and 6' 
Height = /i 


3 



Radius = r 



Jjrr* 



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356 MISOELLAKEOUS FOBHULAS 

THEORETICAL SUPER-ELEVATION IN INCHES 



•3 . 
Mo 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 



Velocity in Miles per Hour. 



10 15 20 25 



35 40 45 50 55 60 65 70 



1 
2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 



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v^ 




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