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Full text of "The railroad taper; the theory and application of a compound transition curve based upon thirty-foot chords"

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•".ipf 




/ 



T ►rt in 

LiDraiy 



2/7 
. P4 6 




THE 



RAILROAD TAPER 



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



BY 



LEE JERKINS 



FoBiiBBLT Assistant Enqinbsb. Southbbn Pacific Linbs in 

Arizona and Mexico 
Mbiibbr AiiBBiCAN Railwat Enqinbbbinq Association 



^<- 



FIR 




JOHN WIEB¥ & SOTIS; Inc. 

London : CHAPMAN & HALL, Limited 

« 

1915 



GOPTBIGHT, 1915 
BT 

LEE PERKINS 



Stanbq^e IhreM 

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



■■- ^ - "-I 



r 



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 

• • • 
HI 



../_.. i-lt 



J 



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 nimiber 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 

■ • • 

111 



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 assumed 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 
•^ .rately. The insertion of tapers between the branches 



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, Natural 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 published in the Manual of that Association. 

LEE PERKINS. 

Silver City, New Mexico, 
March, 1915. 



CONTENTS 



CHAPTER I 
INTRODUCTION 

Page 

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. / 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, imiting 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 

vu 



vm CONTENTS 

Pagb 

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 Froq 

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 

Condition of Track 49 

'^. Operating Conditions .^ 50 



CONTENTS ix 

METHODS OF INSERTING TAPERS 

Simple Curves 

Page 

38. Shifting the Curve 60 

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 

51. 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 

li, A = J 78 

2 , A = 1 82 

2i, A = li 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 /, 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 165 



X CONTENTS 

PaGI! 

Functions 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 iA= i 307 

1 , A = J 310 

li, A = i 313 

2 , A = 1 317 

2i, A = li 321 

2i, A = li 325 

2J, A = 2 329 

3 , A = 2i 333 

3i, A = 3 337 

3i, A = 4 340 

3f, 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 



THE RAILROAD TAPER 



CHAPTER I 

INTRODUCTION 

Purpose 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, and 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 

1 



THE BAILBOAD TAPER 




Taageni 
at 0.0. 



FlQ. 2. 



DEVELOPMENT OF THE TAPER 3 

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. 

5. 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/ = 5. (1) 

if 

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. 



This azimuth is the angle whose tangent is 



x — X 



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

6"-' = o" - o' - /'-", 

and is given in the tables. 

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

Ic = -^/ (2) 

cos/ 



THE RAILROAD TAPER 



o 



£ /'Initial Tangent 




Fig. 3. 






M 



Initial Tangept>^ 




Taagent 
at 0.0. 



Fig. 4. 






Vi 



-y^i :— — 



— y-Vt — 

/initial Tangent 




/Tangent 
at 2 



Fig. 5. 




Fia.6. 



v^ 



THE RAILROAD TAPER 



Similarly, as in Fig. 5, the long chord between any other 
chord-points is 



.// 



7/.'-/' __ y y (Q\ 

^ ~ cos (e' +/'-") ^' 

m 

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

This tangent distance, t, is very nearly half the length 
of the taper, its exact value, from Fig. 6, being 

t = BD-IE 
= y — RsinOj (4) 

where R is the radius of the main curve, and o the central 
angle of the taper. 
The value of the circular curve offset, 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 
different 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. 



CHAPTER II 



LOCATION PROBLEMS 

8. To Find the Semi-tangents of a Tapered Curve. 
Given fl and 7 to find T. 









. 








1 " 


7 ' 




■« 












^v' 




[ 





Case I. Taj>ered at both ends with the same taper. 
From Fig. 7, 

T = AD + AB 

= t + OA tan 1 1 

= i + {ffi + d) tan § /- (6) 



8 



THE RAILROAD TAPER 



In the tables is given a value Z> = 22 + d, so the above 

may be written 

r = < + D tan § 7. (7) 

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

By Eqn. 7 log Dy Tap. 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 offset 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 + OAta,n^I - MN 

= t + Dismal - dcsci, (8) 

Tn = Rta.n^I + dcscI. (9) 



LOCATION PROBLEMS 9 

These 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 54° 22'. 

By Eqn. 8 esc 54° 22' 1 . 23037 

d 0.95 

615185 
1 007333 

1 . 0688515 

logZ)rTap.2i 2.91360 
log tan 27° 11' 9.71059 

^- 2.62419 

420.91 
+ t 54.98 

475.89 
1.07 



B.C. to P.I , 474.82 

By Eqn. 9 log ft?- 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, with different tapers. 
From Fig. 9, 

MN = MK = dicscI, 
BM = d2 CSC I, 
BJ = BN = MN - BM 

= (di — d2) CSC /, 
Ti = ti + Di tan H - (di - ^2) esc I. (10) 
^2 = fe + Z>2 tan 1 7 - (d2 - di) esc I. (11) 
These equations do not change for angles greater than 
90°, the algebraic sign of the last term remaining un- 
changed. 



10 



THE RAILROAD TAPER 



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




By Eqn. 10 



Fig. 9. 

CSC 1 13° 42' (esc 66° 18') 1 . 0921 1 
di - dz = 2.84 - 0.98 = 1.86 





• 


655266 
873688 
1 09211 

2.03 




log Dl0» Tap. 2} 

log tan 56' 51' 


2.76041 
0.18500 
2.94541 




+ <i 


881.88 
84.88 

966.76 
2.03 




B.C. to P.I. 


964.73 


By Eqn. 11 


log Z)l0» Tap. 3 

log tan 56° 51' 


2.75900 
0.18500 



2.94400 



LOCATION PROBLEMS 11 

879.02 

+fe 44.97 

923.99 
+ 2.03 

PJ. to E£. 926.02 

9. Given B and J, to find the External, E. 

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

E = OB-R 

= Dsec^I -R, (12) 

or E = {R + d) seci I - R - d + d 

= (ft + d) (sec H - 1) + d 

= D exsec f / + 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 = VOA^ + AB^- R 

= VD2 +{T - ty - R, (14) 

T — t 
or E = D sec tan-^ — ^^ R, (15) 

T — t , T — t 

where tan~^ —jr— is the angle whose tangent is — j^ — 

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 Ofif set and Tangent Distance 

to any Point on the Curve. 

From Fig. 11, 

PD = AC + d 

= R vers 7 + d, (16) 

ED = BD + t 

= ReinI + t. (17) 



12 



THE RAILROAD TAPBR 




Fig. 10. 



E 


- t ^ 


• 

B 


5 


o 


-f 


~^^\^ 


P 






\ 



Fig. 11. 



LOCATION PROBLEBfS 



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, P, referred to the P.r., to find 
the Radius of a Curve passing through P 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, 



sini/ = 



R = 



OB . 
Qpsma 




Rsec^I 


sin a 


R 


sin a 




cos J/ 




PO 




sin a p„ 





(18) 



sinx 



(19) 



14 



THE RAILROAD TAPBR 



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 degree 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 

^y/ OD^ + DP^-R 

= y/{OA - ADY + (AB - CBy - R 

= V{D-PBsmby+{DtB,TLhI-PBcosby-R. (20) 




Fig. 13. 
Example. In Fig. 13, let PB = 194.3, 6 = 42° 24', 

7 = 54° 22'. Then a = 90° - ^^^^ - 42°24' 

= 20° 25'. 

By Eqn. 18, y = 23° 05' 

x = y^a= 2° 40', 

By Eqn. 19, R = 1456.8. 

3° 56' curve. 



LOCATION PROBLEMS 



15 



1 



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 by = 1698100 
{D tan I I-PB cos by= 351675 

2049775 

V2049775 = 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"^ 



BP-d 
D 



(21) 



R times the sine of the angle whose versine is — yi — ) 



or 



/? = (D - BPy + A&, 
AB = ViP - (D - BP)\ 



(22> 



16 



THE RAILROAD TAPER 



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



2, 



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 witii 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. 




Fig. 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^ = DP X DS, 

AD = VDP X DS. 



(23) 



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



LOCATION PBOBLBUS 



17 



IP^AB' + iR- BPy 

= 4B« + fl* - 2/2 X BP + BP^, 
AB^ + BP^ 



or 



B = 



R = 



2BP 
AB 



sin 2 tan~^ -th 
AB 



(24) 
(25) 



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



B 





E 


f4 




1 ""' 


^h^^^^^^^^'''^ i\ 




i .--^ 


.^- 



\U* 



or 



From Fig. 16, 

A'C 
pp, 



FiQ. 16. 
R sm vers^ — 5 — » 

\/m-{D- csy; 

O P-R 

VoWTep^-R 

V{D - BPy + A'& - R, 



or 



PP' = 



D-BP 



cos tan 



-1 



A'B 
D-BP 



-R. 



(26) 
(27) 



(28) 
(29) 



18 THE BAILBOAD TAPER 

Example. DP = 94 ft., PS = 127 ft., angle a = 16^ 12'. 
ByEqn. 23, i4Z) = 144.1. 

From A BDP, BP = 26 . 2 

and DB= 90.3; 

then AB = AD + DB = 234: A. 

By Eqn. 25, tan-^ f^ = 6° 23', 

and R = 1060.7 

5° 24' curve. 

Deciding now upon a 5° 30' curve, Tap. 2, through S: 

From A DSC, CS = 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 

-BC 121.9 



A'B = 228.16. 

By Eqn. 29, tan-^ D^^BP " ^^"^ ^^'' 
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 P, 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 By (the uncer- 




FiQ. 17. 

tainty of right angle direction will affect 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 ^hot 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 = 2/2 sin a, 
OK = R cos a, 

CF = VCG X AC, 
OK 



tann = 



KC 



72 cos a 



AC — R&ina 



(30) 



20 THE RAILROAD TAPER 

ta.nm = ^ 

V{AC -2Rsma)AC' 
b = a + m — n, (32) 

Given 6, AD and DL may be calculated by equations 

9 and 8. Then 

LB = CF-t 

= V{AC - 2 ft sin a) 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'. 

By Eqn. 31, m = 66° 01'. 

By Eqn. 32, a 6° 10' 

+ m 66° 01' 
- n 64° 19' 
6=7° 52'. 
By Eqn. 33, LB = 417.9. 

15. 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 offset 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 B as base and radius BN as meridian. The 
difference of the latitudes equals OG, the difference of the 
departures equals NG. 






LOCATION PROBLEMS 



Then, 




21 



Fig. 18. 



NO 

tan e = 

cos c = 



a 

f-9 
b 

LM 



= ^OCP + NCP, 

= 99. 

NG' 

Ri + di + Rt + d> 
NO 

NO ' 
= 90° - e - c, 
= a — b, 

= ST- it, + <,) 

= NO sin c - «, + tt). 



(34) 
(35) 



(36) 
(37) 

(38) 

(39) 



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 1| 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 gf = 6° 38', FD = 100 ft. 

Then a traverse from iV to is as follows: 



Ck>ur8e. 


Length. 


Azimuth. 


Lat. 


Dop. 


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 





NO = 


2491 . 5, 




NG^ 


642.7, 




G0 = 


2407.2, 




e = 


75" 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. 



LOCATION PBOBLBMS 



23 




1^ Fig. 19. 

Then from Fig. 19, 



NO = V0(? + N(?, 

OG 

tane = jy-Q. 

a = e + c - 90% 
b=f + g-a, 
L3f = iVO sine - («i + U). 



(34) 
(35) 

(36a) 

(37a) 

(38a) 

(39) 



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: 



Course. 


Length. 


Azimuth. 


». 

TiHt. 


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 


+ 61.3 
+ 27.3 
+ 1378.3 




+ 100.0 
+379 . 1 
+ 96.2 
-390.4 


NO 


485.4 


23° 30' 


+ 425.1 


+ 184.9 



From this traverse 





iVO = 485.4, 




e = 90° - 23° 30' 




= 66° 30'. 


By Eqn. 36a, 


c = 36° 29'. 


By Eqn. 37a, 


a = 12° 59'. 


By Eqn. 38a, 


b= 2° 50'. 


By Eqn. 39, 


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 DBF the required curve ending at F. 
CG and GF are known. 

By a solution of triangle KMIj KF becomes known. 
The value of 722 now depends solely upon the taper chosen 
and its value of fe and (k, 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. 



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. 




Fig. 20. 

From A KMI, KM = 9.7 

and KI = 12.0. 

ByEqn. 7, IC = 387.88 

+ KM 9.7 

+ GF 42.4 

KF = 440.0. 
From Tables, 1° curve 

^36-14' = 1874.7. 
If the new curve were untapered, 

■^ 440.0 ^'^^ • 
If Taper 2 be used on the new curve, then 



M 


1847.7 
M 


ii 


Approz. T, 


4° 45' 


394.7 


56.2 


450.9 


50' 


387.9 


57.5 


445.4 


55' 


381.3 


58.7 


440.0 


5*^00' 


374.9 


60.0 


434.9 



26 



THE RAILROAD TAPER 



Assuming a 4° 65' curve, Taper 2, then 
by Eqn. 7, 7^2 = 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 SLud 
angle a. 




Fig. 21. 



.nn-^g, 


(40) 


c = 180° - n -b, 


(41) 


c' = n + 6' - 180°, 


(41a) 


Sinn 




OB . 

iin 6 = -^ sm c 





lACATION PBOBIiElIS 27 



(42) 





Dsinc 




% 

BP 


^ R^mn 

R sin {e — c) 
sine 




a 


= f + g 






= 90" -n + e- 


c, 


a' 


= f-g' 






= 90° - n + c' - 


-e'. 



(43) 



(44) 

(44a) 

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

Example. Given a 4° 00' curve, Taper 1 J, AB = 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, BP = 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 off, 
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 

k = ^{M,- A), *(46) 

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

/i = ^(Mi + A-A), 

30 Ml 
= -^ — *(47) 

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

Ic ^^ h — h 

= J(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 Afa are 
given in the tables these equations are not needed. When needed 
for other transition curves, somewhat similar equations, but with 
other constants, must be derived. 



LOCATION PROBLEMS 



29 



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

dr = d for M = M2- Ml (49) 



to be taken from the tables. 

The central angle of the partial taper is 

9 (Ma + Ml) (Ma - Mi - A) 



Or = 



=(50) 



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




Fig. 22. 



The common radial line, OF, divides this angle, 0^ 
proportionally to the degrees of curve of the two branches 
of the compound curve. 



* See footnote, page 28. 



30 THE RAILROAD TAPER 

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 Oc, so that its tangent, 
ET, shall be parallel to DA, then 

to = {fix - R2- dr) sin a, (52) 

dc = Ri — R2 — tc cot a, (53) 

or = (Bi — Ri) 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) ^0 + 450 „ ._, 

A 
^3r (4Mi + 2M2 + A)(M2~Mi-A) ^ ^1 ^^^^^ 

Equation 56 may be used for obtaining all the deflec- 
tions from tangent at A, 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, h remaining 
constant from equation 45. 

Where Mi 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 Z = Zi ^ ic, using for Ic the 
distance of the point beyond the beginning of the partial 
taper, A. 

* See footnote, page 28. 



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 — fc (58) 

The long diord 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 ^h-di esc h, (62) 

By Eqn. 9, DA = Ri tan ih + di esc h. (63) 

By Eqn. 10 

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

GH = h + D2 tan J /2 - (^2 - 4) esc hy (65) 

DG = DA + AG, (66) 

Solving triangle DNG gives values for DN and NG. 

Then SiV = 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 \c — di esc c, (62a) 

DF = Ri tan \c + di esc c, (63a) 

FG = \Dr tan ^g — {dr — d^) esc gf, (64a) 
GH = k + D2 tan J fif — (^2 — cir) esc gf, (65a) 

DG = DF + FGy (66a) 

SN = SD + DN, (67) 

NH = NG + GH. (68) 

Example. Given 7 = 114° 24', c = 34° 40', gf = 79°44', 
Ml = 3° 00', M2 = 8° 00'. Use Taper 1 between tangent 



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, 


Zc = y (8-3-1) 


\ 


= 120. 


By Eqn. 49, and the tables. 




dr = 0.79. 


By Eqn. 50, 


_ 9 (8 + 3) (8 - 3 - i: 

Oc — - 




= 396', 




= 6° 36'. 


By Eqn. 51a, 


9 X 3 (8 - 3 - 1) 
1 




= 108' 




= 1° 48'. 


By Eqn. 52, 


U= 37.47. 


By Eqn. 53, 


dc= 1.38. 


By Eqn. 54, 


yc = 119.80. 


By Eqn. 55, 


Xc = 6. 13. 


By Eqn. 57, 


/. = 175^ = 2° 55J'. 


By Eqn. 58, 


bs = 3° 40|'. 


By Eqn. 59, 


kc = 119.96. 


By Eqn. 60, 


/i = 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, 


DC =1220.91. 


From A DGN, 


NG= 727.57, 


and 


DN = 1326.06. 


By Eqn. 67, 


SN = 1963.32. 


By Eqn. 68, 


NH = 1411.31. 



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 Ru 
parallel tangent NH distant p from the original tangent 
WV, JBa, diy d2, dr, h and fe, to find 7i, I2 and NH. 



V 

-7 — : 







Fig. 23. 



OE = Di-p- D2, 
OP = Ri-dr- /?2, 

OE 



_ Ri — R2 -\- di — d2 — p 



*(69) 
(70) 



Ri — R2 
c = I -g. 

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



34 THE RAILROAD TAPER 

The new semi-tangents may now be determined by the 
methods 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 = (fli + di) tan i / - P CSC / + t^ *(71) 

NH = (Ri + di) tan i / - (^i - R2 - dr) sin g 

+ pcot7 + fe. *t(72) 

Example, Given a located 4° 30' curve, Taper 1^, 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 = ZTgiQ^' 

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

By Eqn. 71, SiV = 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 N' so as to unite with located tangent UF. 

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

/22 = «' - --±^% (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**. 



LOCATION PROBLEMS 



35 



OE=^OB + BF - FE 

= Ri cosy + N'C sin g' - {Ri + (h)y 
OP = Ri-R2-dr, 
OE 

cos fif = ^ 

^ Ri cos g' + N'C sin g' - (^2 + (k) ^.^^. 

Rl — /?2 — dr 

Example. Given located 4° 00' curve AN\ to com- 
pound at about N' {N' 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. 

F i c u 




Fig. 24. 

From Functions of a V Curve: 
r42°12' = 2211.0, 



R' = 5730 X 

= 882.4, 
M' = 6°29'+. 



340.5 
2211 



♦ For dr see Eqn. 49. 



16 



THE RAILROAD TAPER 



By Eqn. 73, 



dg = 1 . 75 for 6° 30' Tap. 2, 
dr = 0.09. 

1.75 + 0.09 



Approx. B2 = 882 . 4 - 

= 881.52, 
Rqo^ = 881.59. 



Use 6° 30' curve, Tap. 2. 



By Eqn. 74, 



eosfif = 



406.53 



550.79 
g = 42r 26'. 

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

In Fig. 25, given SJV, /, Ri and R2, to find c, g, and NH. 




Fig. 25. 



OE = OW + WU- UE 

= {Ri + di) cos/ + TNsinl - {R2 + d^), 
■ Ri — R2 — dff 
OE 



OP 

cosg 



OP 



LOCATION PROBLEMS 37 

^ (fli + di) cos/ + TN sin/ -- (fla + (h) 

iVfl = Lf/ - LiV - f/F + fe 

= (Bi + (ii) sin / - TNcosI 

- (fii - i?2 - *rfr) sin gf + fe. t(76) 

Example, Given semi-tangent SAT = 1963.32, mI = 
3° 00' Tap. 1, Ma = 8° 00' Tap. 2|, and Tap. 2 between 
Ml and M2, / = 1 14° 24', to find c, ^ and iV/f . 

212 62 
By Eqn. 75, cos g = ^^^^ ^^ ^ 

fif = 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 Rt is greater than Ri, dr becomes negative. When / is 
greater than 90°, cos 7 becomes negative. 



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 frc^. 

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

f=i(jg + v)cot^F, (77) 

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




= HE + KO-OD 

= if + m + t)amF + (R + d)cosF - R, (78) 

= (7ff- SH - CP 

= (7-f OT + cos F- (R + d)atvP-i. VI'S^ 



LOCATION PBOBLEHS 39 

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

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

AB = (^^P+Ami. (82) 

Sin i 

The resulting calculations for the entire problem may 
be longer than if the semi-tangent BT be calculated by 
E}qn. 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 = i", 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. 



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 off 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 

40 



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 solv6 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.G. should be set by measurement from the PJ. 
The taper at each end should then be run from the B.C. 
and E.C.y the C.C. to the main curve being set preferably 
by long chord from the B.C. and E.C.j and then the main 
curve should be run from C.C. to C.C. of tapers, or froin 



42 THE RAILROAD TAPER 

each C.C, 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 E.C.y 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 tangent. 



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 5 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 = 2/ — 2 i? sin 0. 

From Eqn. 4, Rsino = y — tj 

then n = 2t-y, (83) 

-Any other point, E, may be chosen, its offset from 

tangent being, 

r = d + R vers /', (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 l)eyond the C.C. from any point back 
of the C.C. by considering it a simple curve from a parallel 
tangent. 



44 THE RAILROAD TAPER 

28. Curve run from a Point Beyond the C.C. 

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 

tan n = -j^ , (86) 

A R 

0B=-^^^ (87) 

cos n 

g = a- (90° - n) 

= a + n - 90°. • (88) 

Solving triangle OBP, two sides and the included angle 

given, 

OB 4- 7? 

tan (c + ^g) = q^ _ ^ tan J g, (89) 

b = 180° -n-c, (90) 

BP=.^^. (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 1|, C,C. in a 

poor location for a transit set up, the second station 

beyond, 137.9 ft. from the CC, 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'. 

By Eqn. 87, OB = 1446.50. 



FIELD WORK 



45 



By Eqn. 88, 
By Eqn. 89, 

By Eqn. 90, 
By Eqn. 91, 



g 

c + ig 
c 

b 

BP 



0° 20'. 
30° 50', 
30° 40'. 
67° 07'. 
16.34. 




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 2/> 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. 

^ (92) 



tan b = 



AB = 



CB-y 

CB-y 

cos 6 



(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 

+62.2 4Mli' 2°37i' E.C. 

+32.2 4M4i' 

496 +02.2 3Mli' 

+72.2 3°02i' 

+42.2 2M7i' 

495 +12.2 l*'26i' 

+82.2 0''29i' Z » 195 

O +67.2 13° 23i' 13" 23i' C.C. Tap. 2 = 7** 18J', 

494 10° 52' 

+50 8° 59i' TA = 41° 24' 

493 7° 07' r» 387.16 

+50 5°14i' Ext. = 56.64 

492 3°,22' R =764.08" 

+50 l°29i' L^ » 357.0 

491 +10.2 2°37i' 4°4U' C.C. 7° 30' ft A = 26° 46J' 

+95.2 2° 16i' 

+65.2 1°39' t =. 97.41 

+35.2 r07|' d = 2.71 

490 +05.2 0°42' x = 8.93 

+75.2 0°22i' Ic = 194.86 

+45.2 0°09' I = 195 

+15.2 0° 00' B.C. Tap. 2 o = 7° 18i' 

489 

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

^ = 300 ' ^^^^ 



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 
the same set of deflections. 



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.j 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.G. 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 

47 



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 V 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 aloiig 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., . . . ." 



CHAPTER V 

TAPERING OLD TRACK 

ft 

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 align- 
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.Cj 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 

40 



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, the degree of 



OG. VEHSED SINES AND EXTERNAI. SECANTS 199 



■2a«S8 " 

0-2350S I 

■aim I 



e.8D72E , 
—747 } 
—765 f 
-80783 , 



.2B707 i 
.2B7&E ' 



). 28061 5 
■26171 R 



t 7 8 77 
1 86 8 6 
i:U17 3 17.1 
30 26 f 267 



■2381J I 
-23Bfl8 c 
-23919 c 



-26741 I 

■ 28793 ° 

2CB4& c 

-26897 5 

^27001 5 



.2412J B 
■24175 E 
■24223 ° 



30|2a-g 
BDHI-B 



-24835 ^ 
-248Se e 
.34987 I 



_^ 27837 ° 
-2794! 5 



80 95 t 

40.12 ei: 

GO 15 -S If 



" Log.Eu. ~ 



3P 9 
40 12.5 

__5DisO 

P P. 



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. 




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 line, preferably the old tangent, and proceeding as 
in paragraph 15. Due to irregularities found in the track 
it is not likely 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- 
creased, but seldom enough to be objectionable. 



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 Une 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 = (/? - R') vers \ I. (97) 

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



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. 




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 line, preferably the old tangent, and proceeding as 
in paragraph 15. Due to irregularities found in the track 
it is not likely 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- 
creased, but seldom enough to be objectionable. 



TAPERING OLD TRACK 



66 



For preliminary study, where / is not large, the factor 
sec I / may be disregarded as insignificant. The new 
curve will now he 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 /i, Eqn. 99, i?' may be assumed such as 
to leave the length of line unchanged. 

The length of line from A to 5,* Fig. 34, by the new 
alignment is 

1 + '^- (100) 



L' 



M' 



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

100 J 



L = 2AC + 



M 



= 2t- 2{R - D')i&Vi\I + 



100/ 
M 



(101) 



56 THE RAILROAD TAPER 

The table on page 115 gives the values of ilf ', 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 ilf or 7 lies between those given in 
the table, ilf' 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 M'. 

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)=i + ^, (100a) 

L (arc) =2t-2{R-D') tan^ + ^- (101a) 

Using these equations, if a 10° 00' curve, I = 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 unchanged. 



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 aUgnment lie within and half 
without the old. 

44. Increasing Degree and Shifting Curve and Tan- 
gents. Where / 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 + h = (R- R') 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. 

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



56 THE RAILROAD TAPER 

The table on page 115 gives the values of Af', 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 / lies 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 M\ 

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 diflterence in arc excess is quite large, 
especially with the higher degrees of curve, but this is 
almost exactly balanced by the decreased value of ACj 
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)=? + ^, (100a) 

L (arc) =2t-2(R-D') tanj/ + ^- (101a) 

Using these equations, if a 10° 00' curve, / = 50° 00', 

be sharpened to a 10° 17' curve. Tap. 3, as given in the 

tabJe^ the new alignment would be 0.04 ft. longer than the 

^^d. The curve would have to be aYiaipeiied \^ t^. Y^"^ W 

^urve. Tap. 3, to maintain the length \itvc\vaxv^'&^. 



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 Umited, due to the 
lower speed of ascending trains. 

46. Increasing Degree of Portion of Curve. Where / 
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. 

vers7' = ^^. (103) 

/' 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- 



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. 

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




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 in paragraph 44. 



TAPERING OLD TRACK 



61 



Field Work 

51. 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 
compound 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. 



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. Shiftmg the Lighter Branch, and Increasing the 
Curvature of the Sharper Branch. This change of align-" 




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 
=^Iso the shifting of one or both tangents as in paragraph 44. 



TABLES 



63 



R = 



RADII 
25 



. M 

sin — 
4 



Deg. 


Radius. 


Lograd. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


O'OO' 


Infinite 




roo' 


5729.60 


3.75812 


2*00' 


2864.82 


3.46710 


1 


343775.76 


5.53627 


1 


6636.67 


3.76096 


1 


2841.15 


3.45349 


2 


171887.41 


5.23524 


2 


6544.77 


3.74388 


2 


2817.86 


3.44992 


3 


114591.60 


5.05915 


3 


6466.76 


3.73693 


3 


2794.95 


3.44637 


4 


85943.67 


4.93421 


4 


6371.50 


3.73010 


4 


2772.41 


3.44286 


5 


68754.94 


4.83730 


5 


6288.86 


3.72336 


5 


2750.23 


3.43937 


6 


57295.78 


4.75812 


6 


6208.73 


3.71673 


6 


2728.41 


3.43691 


7 


49110.67 


4.69118 


7 


6130.99 


3.71020 


7 


2706.93 


3.43248 


8 


42971.83 


4.63318 


8 


6056.53 


3.70377 


8 


2685.78 


3.42907 


9 


37197.19 


4.58203 


9 


4982.26 


3.69743 


9 


2664.96 


3.42669 


10 


34377.47 


4.63627 


10 


4911.09 


3.69118 


10 


2644.46 


3.42234 


11 


31252.24 


4.49488 


11 


4841.92 


3.68602 


. 11 


2624.27 


3.41901 


12 


28647.89 


4.45709 


12 


4774.67 


3.67896 


12 


2604.39 


3.41671 


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 


2665.52 


3.40918 


15 


22918.30 


4.36018 


16 


4683.69 


3.66121 


15 


2646.62 


3.40595 


16 


21465.92 


4.33215 


16 


4623.37 


3.65646 


16 


2627.79 


3.40274 


17 


20222.04 


4.30583 


17 


4464.63 


3.64979 


17 


2609.35 


3.39956 


18 


19098.60 


4.28100 


18 


4407.39 


3.64418 


18 


2491.16 


3.39640 


19 


18093.80 


4.25763 


19 


4361.60 


3.63865 


.19 


2473.24 


3.39327 


20 


17188.74 


4.23524 


20 


4297.21 


3.63319 


20 


2455.67 


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 


4092.68 


3.61200 


24 


2387.36 


3.37792 


25 


13750.99 


4.13833 


26 


4044.43 


3.60686 


25 


2370.90 


3.37491 


26 


13222.11 


4.12130 


26 


3997.39 


3.60177 


26 


2364.67 


3.37193 


27 


12732.40 


4.10491 


27 


3961.46 


3.69676 


27 


2338.66 


3.36896 


28 


12277.67 


4.08912 


28 


3906.66 


3.69179 


28 


2322.86 


3.36602 


29 


11854.34 


4.07388 


29 


3862.66 


3.68689 


29 


2307.26 


3.36310 


30 


11459.17 


4.06916 


30 


3819.76 


3.58203 


30 


2291.88 


3.36019 


31 


11089.52 


4.04491 


31 


3777.77 


3.57724 


31 


2276.70 


3.36731 


32 


10742.97 


4.03112 


32 


3736.71 


3.57249 


32 


2261.72 


3.35444 


33 


10417.66 


4.01777 


33 


3696.63 


3.56779 


33 


2246.94 


3.36159 


34 


10111.03 


4.00480 


34 


3667.21 


3.66316 


34 


2232.36 


3.34876 


35 


9822.15 


3.99221 


36 


3618.71 


3.66856 


35 


2217.96 


3.34595 


36 


9549.31 


3.97997 


36 


3681.02 


3.55401 


36 


2203.73 


3.34316 


37 


9291.65 


3.96809 


37 


3644.10 


3.64961 


37 


2189.70 


3.34038 


38 


9046.71 


3.96649 


38 


3607.93 


3.64506 


38 


2175.84 


3.33763 


39 


8814.75 


3.94621 


39 


3472.60 


3.64064 


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 


3.63196 


41 


2135.29 


3.32946 


42 


8185.12 


3.91303 


42 


3370.37 


3.62768 


42 


2122.12 


3.32677 


43 


7994.77 


3.90281 


43 


3337.66 


3.62344 


43 


2109.09 


3.32410 


44 


7813.13 


3.89282 


44 


3306.66 


3.51924 


44 


2096.24 


3.32144 


45 


7639.45 


3.88306 


46 


3274.08 


3.51609 


46 


2083.63 


3.31880 


46 


7473.38 


3.87362 


46 


3243.19 


3.61097 


46 


2070.98 


3.31618 


47 


7314.37 


3.86418 


47 


3212.88 


3.60689 


47 


2068.56 


3.31357 


48 


7161.99 


3.85503 


48 


3183.13 


3.60285 


48 


2046.33 


3.31098 


49 


7015.82 


3.84608 


49 


3163.93 


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 


62 


3069.45 


3.48706 


62 


1998.74 


3.30076 


53 


6486.33 


3.81200 


53 


3042.29 


3.48320 


53 


1987.19 


3.29824 


54 


6366.22 


3.80388 


64 


3015.60 


3.47937 


54 


1975.77 


3.29574 


55 


6250.46 


3.79691 


66 


2989.38 


3.47658 


55 


1964.48 


3.29325 


56 


6138.85 


3.78809 


66 


2963.61 


3.47182 


56 


1963.32 


3.29077 


57 


6031.15 


3.78640 


67 


2938.28 


3.46809 


57 


1942.28 


3.28831 


58 


5927.17 


3.77286 


68 


2913.38 


3.46440 


58 


1931.37 


3.28587 


59 


5826.71 


3.76642 


69 


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. 


3*00' 


1903.91 


3.28101 


4" 00' 


1432.47 


3.15608 


5*00' 


1146.01 


3.05919 


1 


1899.36 


3.27861 


1 


1426.52 


3.15430 


1 


1142.20 


3.05774 


2 


1888.03 


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 


3.05344 


5 


1858.30 


3.26912 


5 


1403.24 


3.14713 


5 


1127.20 


3.05201 


6 


1848.31 


3.26677 


6 


1397.53 


3.14536 


6 


1123.54 


3.05059 


7 


1838.42 


3.26445 


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 


3.04635 


10 


1809.40 


3.25753 


10 


1375.17 


3.13836 


10 


1109.04 


3.04495 


11 


1799.93 


3.25525 


11 


1369.70 


3.13662 


11 


1105.48 


3.04355 


12 


1790.56 


3.25299 


12 


1364.26 


3.13490 


12 


1101.94 


3.04216 


13 


1781.27 


3.25073 


13 


1358.87 


3.13318 


13 


1098.42 


3.04077 


14 


1772.09 


3.24849 


14 


1353.53 


3.13146 


14 


1094.92 


3.03938 


15 


1763.01 


3.24625 


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 


1084.56 


3.03525 


18 


1736.30 


3.23962 


18 


1332.54 


3.12468 


18 


1081.15 


3.03389 


19 


1727.57 


3.23744 


19 


1327.39 


3.12300 


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 


3.02677 


25 


1677.01 


3.22454 


25 


1297.34 


3.11305 


25 


1057.87 


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 


1051.40 


3.02177 


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 


1035.57 


3.01518 


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 


1250.17 


3.09697 


35 


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 


1241.15 


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 


3.19585 


39 


1232.25 


3.09070 


39 


1014.19 


3.00612 


40 


1562.68 


3.19387 


40 


1227.85 


3.08915 


40 


1011.20 


3.00484 


41 


1555.61 


3.19190 


41 


1223.48 


3.08760 


41 


1008.24 


3.00356 


42 


1548.60 


3.18994 


42 


1219.14 


.3.08606 


42 


1005.29 


3.00229 


43 


1541.66 


3.18799 


43 


1214.84 


3.08452 


43 


1002.36 


3.00102 


44 


1534.78 


3.18605 


44 


1210.56 


3.08299 


44 


999.45 


2.99976 


45 


1527.96 


3.18411 


45 


1206.31 


3.08146 


45 


996.55 


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 


2.995d9 


48 


1507.85 


3.17736 


48 


1193.75 


3.07691 


48 


987.96 


2.99474 


49 


1501.27 


3.17646 


49 


1189.62 


3.07541 


49 


985.13 


2.99350 


50 


1494.74 


3.17457 


50 


1185.51 


3.07391 


50 


982.32 


2.99225 


51 


1488.27 


3.17268 


51 


1181.44 


3.07241 


51 


979.52 


2.99101 


52 


1481.86 


3.17081 


52 


1177.40 


3.07092 


52 


976.74 


2.98978 


53 


1475.50 


3.16894 


53 


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 


55 


1165.43 


3.06649 


55 


968.48 


2.98609 


56 


1456.75 


3.16338 


56 


1161.49 


3.06502 


56 


965.76 


2.98487 


57 


1450.60 


3.16155 


57 


■ 1157.58 


3.06355 


57 


963.06 


2.98365 


58 


1444.50 


3.15972 


58 


1153.70 


3.06209 


58 


960.37 


2.98244 


59 


1438.46 


3.15790 


59 


1149.84 


i 3.06064 


59 


957.70 


2.98123 



TABLES 



65 



RADII — (Continued) 



Deg. 

6** 00' 
1 
2 
3 

4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 

42 

43 

44 

45 

46 

47. 

48 

49 

50 

51 
52 
53 
.54 
55 
56 
57 
58 
59 



Radius. 


Log. rad. 


Deg. 


. Radius. 


955.04 


2.98002 


7*00' 


818.64 


952.39 


2.97882 


1 


816.69 


949.76 


2.97762 


2 


814.76 


947.15 


2.97642 


3 


812.83 


944.55 


2.97522 


4 


810.92 


941.96 


2.97403 


5 


809.01 


939.39 


2.97284 


6 


807.11 


936.83 


2.97166 


7 


805.22 


934.28 


2.97048 


8 


803.34 


931.75 


2.96930 


9 


801.47 


929.23 


2.96812 


10 


799.61 


926.73 


2.96695 


11 


797.75 


924.24 • 


2.96578 


12 


795.91 


921.76 


2.96462 


13 


794.07 


919.29 


2.96346 


14 


792.24 


916.84 


2.96230 


15 


790.42 


914.41 


2.96114 


16 


788.61 


911.98 


2.95999 


17 


786.80 


909.57 


2.95884 


18 


785.01 


907.17 


2.95769 


19 


783.22 


904.78 


2.95655 


20 


781.44 


902.41 


2.95540 


21 


779.67 


900.05 


2.95427 


22 


777.91 


897.70 


2.95313 


23 


776.15 


895.36 


2.95200 


24 


774.40 


893.04 


2.95087 


25 


772.66 


890.72 


2.94974 


26 


770.93 


888.42 


2.94862 


27 


769.21 


886.13 


2.94750 


28 


767.49 


883.86 


2.94638 


29 


765.78 


881.59 


2.94527 


30 


764.08 


879.34 


2.94416 


31 


762.39 


877.09 


2.94305 


32 


760.70 


874.86 


2.94194 


33 


759.02 


872.64 


2.94084 


34 


757.35 


870.44 


2.93974 


35 


755.69 


868.24 


2.93864 


36 


754.03 


866.05 


2.93754 


37 


752.38 


863.88 


2.93645 


38 


750.74 


861.71 


2.93536 


39 


749.10 


859.56 


2.93428 


40 


747.48 


857.42 


2.93319 


41 


745.85 


855.28 


2.93211 


42 


744.24 


853.16 


2.93103 


43 


742.63 


851.05 


2.92996 


44 


741.03 


848.95 


2.92888 


45 


739.44 


846.86 


2.92781 


46 


737.86 


844.78 


2.92674 


47 


736.28 


842.71 


2.92568 


48 


734.70 


840.65 


2.92461 


49 


733.14 


838.60 


2.92355 


50 


731.58 


836.56 


2.92250 


51 


730.03 


834.53 


2.92144 


52 


728.48 


832.51 


2.92039 


53 


726.94 


830.50 


2.91934 


54 


725.41 


828.50 


2.91829 


55 


723.88 


826.51 


2.91725 


56 


722.36 


824.53 


2.91620 


57 


720.85 


822.55 


2.91516 


58 


719.34 


820.59 


2.91413 


59 


717.84 



Log. rad. 


Deg. 


2.91309 


8" 00' 


2.91206 


1 


2.91103 


2 


2.91000 


3 


2.90898 


4 


2.90795 


5 


2.90693 


6 


2.90592 


7 


2.90490 


8 


2.90389 


9 


2.90288 


10 


2.90187 


11 


2.90086 


12 


2.89986 


13 


2.89886 


14 


2.89786 


15 


2.89686 


16 


2.89587 


17 


2.89487 


18 


2.89388 


19 


2.89290 


20 


2.89191 


21 


2.89093 


22 


2.88995 


23 


2.88897 


24 


2.88799 


25 


2.88702 


26 


2.88604 


27 


2.88507 


28 


2.88410 


29 


2.88314 


30 


2.88218 


31 


2.88121 


32 


2.88025 


33 


2.87930 


34 


2.87834 


35 


2.87739 


36 


2.87644 


37 


2.87549 


38 


2.87454 


39 


2.87360 


40 


2.87265 


41 


2.87171 


42 


2.87077 


43 


2.86984 


44 


2.86890 


45 


2.86797 


46 


2.86704 


47 


2.86611 


48 


2.86519 


49 


2.86426 


50 


2.86334 


51 


2.86242 


52 


2.86150 


53 


2.86058 


54 


2.85967 


55 


2.85875 


56 


2.85784 


57 


2.85693 


58 


2.85603 


59 



Radius. 

716.34 
714.85 
713.37 
711.90 
710.43 
708.96 
707.50 
706.05 
704.60 
703.16 
701.73 

700.30 
698.88 
697.46 
696.05 
694.64 
693.24 
691.85 
690.46 
689.08 
687.70 

686.33 
684.96 
683.60 
682.25 
680.90 
679.55 
678.21 
676.88 
675.55 
674.22 

672.90 
671.59 
670.28 
668.98 
667.68 
666.39 
665.10 
663.81 
662.54 
661.26 

659.99 
658.73 
657.47 
656.22 
654.97 
653.72 
652.48 
651.25 
650.02 
648.79 

647.57 
646.35 
645.14 
643.93 
642.73 
641.53 
640.34 
639.15 
637.96 



Log. rad. 



85512 
85422 
85332 
85242 
85152 
85062 
84973 
84884 
2.84795 
2.84706 
2.84617 

2.84528 
2.84440 
2.84352 
2.84264 
2.84176 
2.84089 
2.84001 
2.83914 
2.83827 
2.83740 

2.83653 
2.83567 
2.83480 
2.83394 
2.83308 
2.83222 
2.83136 
2.83051 
2.82966 
2.82880 

2.82795 
2.82710 
2.82626 
2.82541 
.82457 
.82373 
.82289 
.82205 
.82121 



2. 
2. 
2. 
2. 
2. 



2.82037 

2.81954 
2.81871 
2.81788 
2.81705 
2.81622 
2.81539 
2.81457 
2.81375 
2.81293 
2.81211 

2.81129 
2.81047 
2.80966 
2.80884 
2.80803 
2.80722 
2.80641 
2.80560 
2.80480 



66 



THE RAILROAD TAPER 



RADII f- {Continued) 



Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


9»00' 


636.78 


2.80399 


10" 00' 


573.14 


2.75826 


11*00' 


621.07 


2.71090 


1 


635.60 


2.80319 


1 


572.19 


2.75754 


1 


620.28 


2.71624 


2 


634.43 


2.80239 


2 


571.24 


2.75682 


2 


519.60 


2.71558 


3 


633.27 


2.80159 


3 


570.29 


2.76610 


3 


618.71 


2.71493 


4 


632.10 


2.80080 


4 


569.35 


2.75538 


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 


567.47 


2.75394 


6 


616.38 


2.71297 


7 


628.64 


2.79840 


7 


566.53 


2.75323 


7 


616.61 


2.71232 


8 


627.49 


2.79761 


8 


565.60 


2.75251 


8 


614.84 


2.71167 


9 


626.35 


2.79682 


9 


564.67 


2.75180 


9 


514.07 


2.71102 


10 


625.21 


2.79603 


10 


563.75 


2.75109 


10 


613.30 


2.71037 


11 


624.08 


2.79524 


11 


562.83 


2.75038 


11 


612.54 


2.70972 


12 


622.95 


2.79445 


12 


561 91 


2.74967 


12 


511.77 


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 


2.74825 


14 


610.26 


2.70779 


15 


619.58 


2.79210 


15 


559.17 


2.74754 


15 


509.60 


2.70714 


16 


618.47 


2.79132 


16 


558.26 


2.74684 


16 


608.76 


2.70660 


17t 


617.36 


2.79054 


17 


557.36 


2.74613 


17 


608.00 


2.70586 


18 


616.25 


2.78976 


18 


556.46 


2.74543 


18 


607.25 


2.70522 


19 


615.15 


2.78898 


19 


555.56 


2.74473 


19 


606.60 


2.70468 


20 


614.05 


2.78821 


20 


554.66 


2.74403 


20 


606.76 


2.70394 


21 


612.96 


2.78743 


21 


553.77 


2.74333 


21 


606.02 


2.70330 


22 


611.87 


2.78666 


22 


552.88 


2.74263 


22 


604.28 


2.70267 


23 


610.78 


2.78589 


23 


551.99 


2.74193 


23 


503.64 


2.70203 


24 


609.70 


2.78512 


24 


551.11 


2.74124 


24 


502.80 


2.70140 


25 


608.62 


2.78435 


25 


550.23 


2.74054 


25 


502.06 


2.70076 


26 


607.55 


2.78358 


26 


549.35 


2.73985 


26 


601.33 


2.70013 


27 


606.48 


2.78281 


27 


548.47 


2.73916 


27 


600.60 


2.G9960 


28 


605.41 


2.78205 


28 


547.60 


2.73847 


28 


499.88 


2.60887 


29 


604.35 


2.78129 


29 


546.73 


2.73777 


29 


499.16 


2.69824 


30 


603.29 


2.78052 


30 


545.87 


2.73709 


30 


498.43 


2.60761 


31 


602.23 


2.77976 


31 


545.00 


2.73640 


31 


497.72. 


2.09698 


32 


601.18 


2.77900 


32 


544.14 


2.73571 


32 


497.00 


2.00635 


33 


600.13 


2.77824 


33 


543.28 


2.73502 


33 


496.28 


2.00572 


34 


599.08 


2.77749 


34 


542.42 


2.73434 


34 


496.56 


2.00510 


35 


598.04 


2.77673 


35 


541.57 


2.73366 


36 


494.85 


2.00447 


36 


597.00 


2.77598 


36 


540.72 


2.73297 


36 


494.14 


2.00385 


37 


595.97 


2.77523 


37 


539.87 


2.73229 


37 


493.43 


2.00823 


38 


594.93 


2.77447 


38 


539.03 


2.73161 


38 


492.73 


2.00261 


39 


593.91 


2.77372 


39 


538.18 


2.73093 


39 


492.02 


2.00198 


40 


592.89 


2.77297 


40 


537.34 


2.73025 


40 


491.32 


2.00130 


41 


591.87 


2.77223 


41 


536.51 


2.72957 


41 


490.62 


2.00074 


42 


590.85 


2.77148 


42 


535.67 


2.72890 


42 


489.92 


2.00013 


43 


589.84 


2.77074 


43 


534.84 


2.72822 


43 


489.22 


2.08051 


44 


588.83 


2.76997 


44 


534.01 


2.72755 


44 


488.63 


2.08880 


45 


587.83 


2.76925 


45 


533.18 


2.72687 


45 


487.84 


2.08828 


46 


586.82 


2.76861 


46 


532.35 


2.72620 


46 


487.16 


2.08700 


47 


585.82 


2.76777 


47 


531.53 


2.72553 


47 


486.46 


2.08705 


48 


584.83 


2.76703 


48 


530.71 


2.72486 


48 


486.77 


2.08043 


49 


583.84 


2.76629 


49 


529.90^ 


2.72419 


49 


486.09 


2.08682 


60 


582.85 


2.76556 


50 


529.08 


2.72352 


60 


484.40 


2.08521 


51 


581.86 


2.76482 


51 


528.27 


2.72286 


51 


483.72 


2.08400 


52 


580.88 


2.76409 


52 


527.46 


2.72219 


52 


483.04 


2 08400 


53 


579.90 


2.76335 


53 


526.65 


2.72152 


63 


482.37 


2.08338 


54 


578.93 


2.76262 


54 


525.85 


2.72086 


64 


481.69 


2.08277 


55 


577.95 


2.76189 


55 


525.05 


2.72020 


56 


481.02 


2.08210 


56 


576.98 


2.76116 


56 


524.25 


2.71963 


56 


480.35 


2.08150 


57 


576.02 


2.76044 


57 


523.45 


2.71887 


67 


479.68 


2.08005 


58 


575.05 


2.76971 


58 


522.65 


2.71821 


68 


479.01 


2.08035 


59 


574.10 


2.76899 


59 


521.86 


2.71756 


69 


478.35 


2.07074 



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 


2.64275 


3 


408.05 


2.61072 


4 


475.05 


2.67674 


4 


438.73 


2.64219 


4 


407.57 


2.61020 


6 


474.39 


2.67614 


5 


438.17 


2.64164 


5 


407.09 


2.60969 


6 


473.74 


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 


2.60867 


8 


472.44 


2.67435 


8 


436.50 


2.63999 


8 


405.66 


2.60816 


9 


471.79 


2.67375 


9 


435.95 


2.63944 


9 


405.18 


2.60764 


10 


471.15 


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 


2.59S54 


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 


2.59804 


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 


2.59655 


32 


457.37 


2.66027 


32 


423.62 


2.62697 


32 


394.50 


2.59605- 


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 


35 


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 


392.70 


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 


2.65626 


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 


50 


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 


2.58670 


52 


445.54 


2.64889 


52 


413.44 


2.61642 


52 


385.67 


2.58621 


53 


444.96 


2.64832 


1 53 


412.95 


2.61589 


53 


385.24 


2.58573 


54 


444 39 


2.64776 


54 


412.45 


2.61537 


54 


384.81 


2.58524 


55 


443.82 


2.64720 


55 


411.96 


2.61485 


55 


384.38 


2 58476 


56 


443.25 


2.64664 


56 


411.47 


2.61434 


56 


383.95 


2.58423 


57 


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 


59 


441.54 


2.64497 


59 


410.00 


2.61278 


59 


382.67 


2.58282 



68 



THE RAILROAD TAPER 



RADII — (Continued) 



Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


15*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 


2.46157 


25*00' 


229.64 


2.36104 


55 


360.26 


2.55662 


55 


288.04 


2.45945 


15 
30 


227.37 
225.15 


2.35673 
2.35248 


WOO' 


358.39 


2.55436 


20*00' 


286.84 


2.45764 


45 


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 


2.45226 


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 


210.76 


2 32379 


45 


342.37 


2.53450 


. 45 


276.50 


2.44170 


30 


208.85 


2.31983 


50 


340.68 


2.53234. 


50 


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 


2.52807 


21* Oy 


273.22 


2.43651 


15 


203.33 


2.30820 


05 


335.70 


2.62595 


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 


2.52174 


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*00' 


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 


267.91 


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 


255.05 


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 


251.35 


2.40027 








55 


303.23 


2.48177 


55 


250.44 


2.39870 









TABLES 69 

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, Sit 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 offset 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 — R -{- dy 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. 
8 = 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 J /. (7) Both ends, the same 

taper. 
Tt =^ t + D tan § 7 — dcac I. (8) Tapered end — other 

end not tapered. 
Tn "= R tan i I ■}■ dc8c I. (9) Un tapered end — other 

end tapered. 
Ti = h + Di tan J 7 — (rfi — (^) esc 7. (10) Both ends tapered 

with different tapers. 
Ti = tt -\- Di tan J 7 — (rf. — (/i) esc 7. (11) Both ends tapered, 

with different tapers. 
E'=DaeciI — R. (12) Both ends the same 

taper. 



70 THE RAILROAD TAPER 

EMPIRICAL EQUATIONS DERIVED FROM THE 

TABLES 

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

_ (1 + 16) (l + SO) 

^ 300 (^^^) 

A 

., (/ + 2g + 45)(Z~g)+450 ,-.. 

^ = 300 ^^^^ 

a' 

I (I + 30) 
o = ±^- (108) 

A 



TABLES 

TAPER i. INTERMEDIATE VALUES 



71 



Af 


I 


/ 


6 





D 


logD 


d 


t 


X 


V 


Ic 


030 


30.00 


e / 

02i 


e / 

02} 


e / 

04) 


11459.18 


4.05915 


0.01 


15.00 


0.02 


30.00 


30.00 


35 


40.00 


03i 


03} 


07 


9822.17 


3.99221 


0.02 


20.00 


0.04 


40.00 


40.00 


40 


50.00 


04i 


05} 


10 


8594 41 


3.93422 


0.03 


25.00 


0.06 


50.00 


50.00 


45 


60.00 


05) 


08 


13) 


7639.49 


3.88306 


0.04 


30.00 


0.10 


60.00 


60.00 


50 


70.00 


07 


10) 


17) 


6875.56 


3.83731 


0.05 


35.00 


0.14 


70.00 


70.00 


55 


80.00 


08} 


13} 


22 


6250.53 


3.79592 


0.07 


40.00 


0.20 


80.00 


80.00 


100 


90.00 


OlOJ 


016) 


27 


5729.70 


3.75813 


0.10 


45.00 


0.27 


90.00 


90.00 


05 


100.00 


12i 


020 


32) 


5288.99 


3.72337 


0.13 


50.00 


0.36 


100 00 


100.00 


10 


110.00 


014) 


024 


38) 


4911 25 


3.69119 


0.16 


55.00 


0.47 


110.00 


110.00 


15 


120.00 


16} 


28} 


45 


4583.89 


3.66123 


0.20 


60.00 


0.59 


120.00 


120.00 


20 


130.00 


019i 


32} 


52 


4297.45 


3.63321 


0.24 


65.00 


0.73 


130.00 


130.00 


25 


140.00 


22 


37) 


59) 


4044.72 


3.60689 


0.29 


70.00 


0.89 


140.00 


140.00 


30 


150.00 


24} 


42} 


107) 


3820.09 


3.58207 


0.34 


75.00 


1.08 


149.99 


150.00 


35 


leo.oo 


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 


1.52 


169.99 


170.00 


45 


180.00 


34 


100) 


134) 


3274.63 


3.51516 


0.55 


90.00 


1.79 


179.98 


179.99 


50 


190.00 


37) 


107 


144) 


3125.89 


3.49497 


0.63 


94.99 


2.08 


189.98 


189.99 


55 


200.00 


41} 


113} 


155 


2990.10 


3.47569 


0.72 


99.99 


2.40 


199.98 


199.99 


200 


210.00 


46 


121 


2 06 


2865.64 


3.45722 


0.82 


104.99 


2.75 


209.97 


209.99 


05 


220.00 


49 


128) 


2 17) 


2751.16 


3.43952 


0.93 


109.99 


3.13 


219.96 


219.98 


10 


230.00 


53 


136) 


2 29) 


2645.51 


3.42251 


1.05 


114.99 


3.55 


229.95 


229.98 


15 


240.00 


57} 


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 


106) 


2 02 


3 08) 


2372.36 


3.37518 


1.46 


129.98 


5.02 


259.92 


259.97 


30 


270.00 


111} 


2 11} 


3 22) 


2293.50 


3.36050 


1.62 


134.98 


5.59 


269.90 


269.96 


35 


280.00 


116} 


2 20} 


3 37 


2219.74 


3.34630 


1.79 


139.97 


6.20 


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 


126) 


2 41 


4 07) 


2085.69 


3.31925 


2.16 


149.96 


7.56 299.84 


299.94 


50 


310.00 


132 


2 51) 


4 23) 


2024.62 


3.30634 


2.36 


154.96 


8.30 1309.81 


309.93 


55 


320.00 


137} 


3 02} 


4 40 


1967.06 


3.29382 


2.58 


159.95 


9.09 


319.78 


319.91 


300 


330.00 


143) 


3 13) 


4 57 


1912.72 


3.28165 


2.81 


164.94 


9.93 


329.74 


329.89 


05 


340.00 


149) 


3 25 


5 14) 


1861.35 


3.26983 


3.05 


169.93 


10.82 


339.70 


339.87 


10 


350.00 


155) 


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 


3.57 


179.92 


12.75 


359.61 


359.83 


20 


370.00 


208} 


4 01} 


6 10 


1722.78 


3.23623 


3.85 


184.90 


13.80 


369.55 


369.81 


25 


380.00 


2 15 


4 14) 


6 29) 


1681.16 


3.22561 


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} 


7 10 


1603.82 


3.20516 


4.80 


199.86 


17.29 


399.35 


399.72 


40 


410.00 


2^} 


4 55} 


7 31 


1567.82 


3.19530 


5.14 


204.85 


18.57 


409.27 


409.69 


45 


420.00 


2 43 


5 09) 


7 52) 


1533.46 


3.18567 


5.50 


209.83 


19.91 


419.18 


419.65 


50 


430.00 


2 50) 


5 24 


8 14) 


1500.61 


3.17627 


5.87 


214.81 


21.31 


429.08 


429 61 


55 


440.00 


258} 


5 38} 


8 37 


1469.20 


3.16708 


6.26 


219.79 


22.78 


438.97 


439.56 


400 


450.00 3 06 


554 


900 


1439.14 


3.15810 


6.67 


224.77 


24.31 


448.85 


449.51 



THE RAILROAD TAPEH 



s 




s 


• gasasassssstsssss ; 


t 


s 


-ssssfssssississ :8 


d? 


i 




; S^ISSSRSSSSSlsI :3? 




i 

S 


|Si = Hgi3SS = S'S :SS!; 


;S9S3S5SSSSSS ;SSl6 


OS 

ds 
dj, 


1 
g 


; jsgssssssi ;sfsgs 


l=2S38gsss isssssa 


dC 

d8 
d'„ 


3 




S 




■SSft3KS2 :2SSS3?=S 


d"s 

dt. 


§ 


■SgSSSS :S3l5Sgsi8S2 


qf 


s 


■ OOOOO OOOW— INdPSPJ-n- 


s 


= OQOO -OOO— -^-HMCin^V 


dg 


• 


■ 22S ;sSss2ssS3fs2 

■ 000 0000 — rtcflwroes^io 


d^ 


• 


• Si :S2SSSSSg3SS3S 
■ 00 0000 — — ^MMW^-tin 


qS 
"■=> 


s 


- S :i'2lBS3sIsSS2SS 

■ -00000 — — N«Mm-f^in 




' 


- :SS2SSSS!t3SSSSS3 


■ 
1 


« 


- 


°SSS§SS2S||||||| 



TABLES 



73 



II 
< 






1^ 



> 

T— I 

o 

O 

O 
»— t 

»— t 
P 

O 



PL, 



Q 
!? 



^ 






'-^C^»COI>.t>-0»CiOOit>-005CO 

1-H 1-H 1— I C^ 
T-l *-H *-H ,-( ^ ,-H ,-( J5 6l 

OO»-i^C0»O00rHQ5i-i00»0'^»0<0 

"^cococooococooococooococococo 



HnHn 



*^l«lHfN 



*^C^'*iCI 



Hnhim . , ^, 

S^J^iSfeS89SfeteS5S8 

o O O O O 1-H 1-1 01 01 CO -^ T}i »0 <0 !>. 05 



1-Hi— ii-hOIC^OIcOCOCOCO"^"^ 



Q '_5 Q ^'^ Q »0 Q «C O »C Q iO O »0 O 

WTt<Oi-icorrOi-ico"^Oi-io5'^0 

OO'-ht-It-ht-iC^C^C^C^coCOCOCO'^ 



THE RAILROAD TAPER 



S5!B^S!3SS32S;9SS 



■tf loSoSS H — ■* -"SS H :3 



ssasssSsfesss ; 



sssssaassss ixss 



SSS9SS83 iSSSS 



:? :SS 



Ssg'sssss ;sBas 



g!?SS2igS iSSSSSSiS 



SSgsftS iSSSSSSES 



SBSSS :SS9SS2SS» 



£ SSSSSSSiSSS 



ssussissstass 



i323S;53SS 



3 ;8S*SSS2S3gSSSS 



iss^gasssaaasss 



=ssssgg||g|i||| 



TABLES 



75 



II 
< 









> 

o 

CO 
o 

o 
O 
»— t 

CO 

P 

O 






42 


SoOOOiOiOiSSt^ioSocO 


eJ5©05c5'«i<r»ocoo05?iicoo^ 

T-l^i-H^CSJC^WCOCOCO"^ 


at 


QQQ»tN.T*<0QtN.i-iUpi^^^CO 
OOO05050500t^C0C00i'^r*t^ 


W©O»-H'«1<tN.pC0O05C<lS00rH 


H 


^OOOOOt^Qi-iOOOCOCpiObj 


1— 1 rH 1— 1 C5 CO CO 


••a 


OOOoSoi050505Wl>'COiCCO 


^co'^«Dt^oooi-ico'^cDr*050 


•w 


888§S2S?§?!!?5S2:§g 


OOOOOrH^C^C0'^»Ct^Q0O 


M 


75813 
58204 
45713 
36027 
28117 
21436 
15659 
10574 
06041 
01960 
98257 
94877 
91780 
.88933 


cococococococococococ^c^c^c^ 




S§8SJ^SS2So52gKS8 


1 
1 


5729. 

3819. 

2865. 

2292. 

1910. 

1638. 

1434. 

1275. 

1149. 

1046 
960. 
888 
817 
775 


o 


OC^I^COi-HOi-iC^rfii-iiO-^COTti 

o O OOi-H C^ CO "^ lO CO X 05»-H CO iO 

t— ( I— 1 rH 


•>>* 


CoSSS«OQO^§l-.8ccSSc2 
»— 1»— ti— •C^C^IC^COCOCOCO'^ 


>. 


* 8m8S8w8S8S8S8S 

o T-Hr-((M(NC0C0'^'<^»OiO<UDcOl^l^ 



THE RAILROAD TAPER 







TAPER 


. INTERMEDIATE 


VALUES 




a=l 


« 


' 


/ 


" 




D 


logZ) ;■ i 


' 


-. 


» 


T 


loo 


30 00 


004i 


041 


009 


B72e.fl2 


3,75813 


0.02 


IS.OO 


0,04 


30.no 


30.W 




33 DO 


OOBi 




om 


B2SSS9 








0.06 




36 69 




4U.0O 


ooai 


071 






3.09118 


0.U3 




D.08 


40.00 


40,01 




43 00 


071 


oooi 


oiei 




3 58122 






0.10 




45 00 






OOBI 


0111 














BO, 00 


50.00 


as 


SS.DO 


010 


OlJi 


23) 


4(«4.40 


3 60O8fl 


0.06 


27.50 


0-lS 


M.OO 


55, (tl 




no 00 


0111 


15i 












0,20 




eo.u) 




B5.I10 


012J 




30J 












fie 00 


ss.ou 


40 


7UO0 


0141 


oaoi 


35 


3437 89 


3 53829 


O.ll 


36.00 




70,00 


70 0l> 




73 00 


oiai 


231 


3BI 


3274 21 






37.60 




75,00 






SO. no 


017i 


2Bi 












0,41 




80.60 


S6 


iis.oo 


Olfli 


om 


4Si 


2089.55 




17 




0.47 


8S.0O 




!00 


90 00 


21 


033 


064 


288B.02 


3.45713 


0.20 


45.00 


0.B6 


60,00 


60.61) 


OB 


95.00 


023 


3B1 


OSS) 


2TB0.4B 


3.43940 


0.22 


47. SO 


0,63 


86.00 


S5.60 




100 00 








M44,71 


3 42238 






0.72 


100,00 






lOB.OO 






\Z\ 




3 4O500 








lOB.DO 


OS 01) 




no. DO 


291 






24MS9 




0.32 


56.00 




109,89 




36 




0311 




123} 


2371. 2B 


3 37408 






l.OS 


114. 9Q 






naiti 


33i 


06fll 














119 68 




35 


125.00 


30i 


100) 


1381 


2218.30 


3 34504 








124.89 


125 06 


40 


130,00 


381 


I OBI 




2149.12 


3 33226 






148 


129.99 








41i 


1101 


ISI) 












134. OS 


134.99 












2023.84 


3 30506 








139 as 


139 99 




14B00 


OMj 




2 061 




3 21)3311 


0.63 


72 49 








300 


IBOOO 


0«) 


125) 




1910 60 


3.2SII7 


0,90 


71.99 


2,16 


149.07 


148 99 


OB 


156.1)0 


521 


,., 


aasi 


1S.59 05 


3.28929 


0.76 


77. 4D 


2.36 


tM.B7 


154 86 






OBBi 


136) 










76, M 






69. W 






OSS) 


1421 


2 401 


1783 89 


3 24847 


0.88 


82 49 




ifl*-ea 


ei.M 


M 


170.00 


ion 


14fil 






3 23550 






3,05 


iu,«a 


w« 


ZB 






153) 


2sn( 




3 22480 






3,30 




74.98 


30 




loy 


aooj 


309 














78 98 


3B 


ISBOO 


iiij 


a 07 


3 18] 


1800 20 




MB 


S2.4S 


3.86 


184,83 




40 




I15i 


2 131 


320 


!S83,95 


3 1W22 








186.82 


80.97 






USi 


2201 


3 38) 






1.36 










50 


200 00 


IM) 


2 271 






3.17489 


1.45 




480 


166.80 


69,96 


BB 


sos.oo 


1281 


2 34) 


4D0i 


I4H,49 


,.,«. 


1.BB 


102.47 


6. 11 


204.86 


204 65 



TAPER 1 


INTERMEDIATE 


VALUES - 


- (Continued) 




a = l 


M 


' 


/ 


» 


- 


D 


ksC 


- 




« 


. 


" 


ita 


210.00 


130 


2 42 


4 12 


1434.12 


3.16860 


7^ 


101.07 


6,50 


200,88 


200,95 




21S.DD 




2 49! 


4 23) 


1405. DO 


















220,00 


338 








3.13893 




109 06 
















3 041 


4 461 








112.4* 




67 




224.93 




330.00 


1461 


3121 




1324.39 






114 06 








229,93 


16 


23*00 


ISOi 


3 21 


5 11) 


1290-57 


3-11380 










334 so 


234.02 




2*0.00 


154) 


3 201 




1275.88 






110.94 




01 


330.77 


230.01 




14SO0 


IBBl 


3 371 


8 381 


1263.68 










49 




244,90 


40 


260.00 


2 03] 


HH 


S50 


1230.48 


3.09007 


2.03 


124,93 


( 


99 


240.73 


249-89 


w 


25«.no 

260.00 


3 08J 


3S5i 


B031 


1209.08 


3mo7 


j-^ 


137.43 
129.92 


,J 


05 


364,70 


354.88 


Si 


26S00 


2 171 


4 13 


3 301 




3.00763 


3.08 


132.41 


U 


60 




36186 


SCO 


270 00 


2 22i 


4 22) 




1140 25 


3.06011 


3 21 


134 91 


" 


18 


360,61 


369,84 


OS 


376,00 


2 371 


4 32 


S681 


1130.10 


3.0S331 


3.40 


137.10 


11 78 


271.57 


371.82 






2 32J 


4 4li 










130 80 


12.40 


378.63 


270.81 




2S5.DO 


2 3Ti 


4501 










143 38 






28149 


284.80 


za 


300.00 


2121 


6 011 




1078.33 


3.03275 


3.04 


144,87 












29!. m 




Bin 


7 SB) 








147,38 




30 


294 40 


294. 7B 


30 




3 531 


5211 
















209, 3S 




u 




2 681 


632 


8 301 


1030.81 


3.01318 


4.52 








304 


30 




40 




3 041 


5 421 






3.00686 










300 


24 


300M 




315.00 


3091 


5 531 






3 09U65 
















50 


320.00 


31B1 


6WJ 




087.11 


2.00453 




150 81 


18 


18 


31( 




319.83 


BOO 


330.00 


3 211 
3 27 


«m 


938) 
9 64 


060 56 


2 0B257 


5. 51 


154,77 


lo.^ 


m 


V 


321.81 


15 


348.00 


3 45 


7 011 


10 4IH 


923 18 




' ' 


-4173 


344,47 


30 


m.m 


4 031 


7 381 




SR8 73 








37S 00 


4231 


81Si 


12 39) 


»0 95 


n 10 


374,20 




390 00 


4 431 


8 65i 


13 39 






389,03 


IS 


40i.M 


S041 


BJBJ 


14 40! 


S0OT3 


2 . 1 U)2 21 


103 84 


30 


420.™ 


SMI 


10 181 


15 45 


775 06 


jaso^Y" '-{""''V "-h'"" 


WKi 



THE RAILROAD TAPER 



II 
<1 



8 

> 

o 

O 

O 

CQ 

I— I 






O 



H 



5 

S 
o 



Q 






kO 



6% 



dS 



>o 



OS 

d- 



00 



dg§ 
djL 



>o 



d?= 






d2 



d^ 
dL 



CO 



d" 



CO 



ds 



d2 
d* 



CS| 



dg 
dt. 



OJg 



CO 



CO 



o 



g 
w 



o 



o 

ft 



o 

00 



§ 



o 

<M 



o 

CO 



« i>. i>. i>. r* i>. i>. o CO io ""*! CO c^ T-i • 



HwcoH" 



V lO "Tf U5 CO 00 »-< ^ O CO CO C<l i-H . t* 

o cs CO CO c^i i-H ""i^ —t 01 CO CO c^i .e^ 

e CO CO CO CO CO CO »0 »0 rt< CO C^ 1-H ' ^^ 



o lO »0 iO lO »0 lO "^ CO CO 01 rH 






o ^ ^ ■^ ^ ^ "^ CO C^ d 1-H 



1-H CO »-H 



* C0cp05"^g5iOC0r-(O 

co^'«i<'^?iocOiOo 

o CO CO CO CO CO CO Ol rH i-H 



t>. lO fH t> 

i-hC^COIO 



H^ Hn 'H|e« HM n1'« 

- <M 23 CO lo 00 cvi 00 "^ 

i0OO»0C0t-(C0»0 
o Ol CO CO (N <N (M i-H O 



81-1 i-H QOQ 
rH C^COICCO 



«I'*^|e«iH|«i-i(«iH|'* 



» »0 CO »-H rH 01 "TM l>» 

T-H c5 oi T-H »o oi T}< 

o OlOIOIOl^i-iO 



'^OO^CO^'* 
lO »0 1-H CO O ^ 

o T-i co-^ cot^ 



- c<i 00 *<*< oi ^^ Q . i>. Tti 1-H CO o "^ ;o 

o T-H 1-H T-H T-H rH O * O »-< Ol ^ »0 |>. 00 



- rt< l>.01 l^ CQ 

1-H 1-H »-H lO CO 
o 1-H 1-H rH O O 



•^COCOCOiOCSJiO-^ 
Oi-H0lC0'^C0l>.05 



i-<|W«|<« 
V Q T-H Tt< t> 

o oooo 



«H'iH(«H«H|e«iHN nh* 

CQI^OOICOCOOIQCO 

CO»-Hl-Hi-HOlTtii-Hl25CO 

OT-Hdco-^iot^doo 



* I-H OO 
coco C^ 

o ooo 



—!■* "ilN »^lc«Hc««j^ 



OT-i^0IC0i0C0t^05T-i 



t>- CO --. - ^ - - 



o oo 



OOi-H01CO'^»Ol>.00001 



-8 :2S;.S3??5!S8S2S« 

o o ; o o 1-H 1-H oi CO ""*! CO t^ oi o oi 



»!'*H«Hc«H«ih('<* 



.^ .^ .^ -^ ,^ ^ wNiHwHctHNHH" 

8S55S;SS82;s?8§fig5 

OOOrH01C0"^»0C00005T-lC0 



SS88SS2Sg8§S8 



TABLES 



79 



ii 






CO©00^"^t^OC«5COOf^ 



_ _ ___OOOO00l^ 
^T*<tN.oc«5coo5iiooo 






H 

PZ4 



^^ 

o 

o 

o 

»— t 

o 

»— t 

> 
o 






^ 



8QO00'«1<t>-00O^'^OOt^ 
OOOOOOt^'^i-l»Ol>'»OQO 

QQOOOOOOOOOt>CO'^ 



oSoot^Sco^ot^SSoS 
ooot-Hoouiooc^'cdoioQdt^ 

»-H »-H C<l CT CO "^ 



8QQOOO<;OCOOQOr*OiQC^ 
ooooooooi>.«0'^c5o 

»oo*oO'^o^o^O'«i'oco 

i-)CO'^»Ot^OOO*-HeO'^COt^O 



Sc^ooeo»ot^eocot5:rHOi>> 
^CSI»00«0'^»000'*'*l^CO 

oooo^»-Icaco'«i^cdo6oco 



S^b-OOOQO»OTt<COOOOOQ 
qooc<icsico»-hSooi>»«ooo 
^»OT-ITj<iOOO»-HO«5iOOl^t>- 
Q0Q00C5M00C00Q:^Ot^'* 
^C^T-l^OOOOOOOOOl^t^ 

COCOC0COCOC0c4ca(NC^<N(NC^ 



00"^Q»0W3O»-H0P'^C^OC0'^ 

r»cDW»oooo»OTpoi-Hi-iioc^ 



O^O00'«*<C0t>-C^ 

ooSooSooS 

CO C< »-H T-l »-i 1— • 



Tj<COt>-C^QQ»-HiOOOO 



_ O'^0»0 
b« t>. CO LO lO 



HlwHN H«Hn H«He« 

OO^C^CO'^OOOOC^'^t^O 

>— ( T-H T-H 1-H CS 



^SoSSooi— i"^i>-Ocooo 

J1h|-h^C^(NC^C0COC0CO 



« rH O "^ 55 T-H O "^ CO r-t O -^ CO 

o i-H CV| CO CO "^ »0 «0 «0 l>. 00 o o o 



8 


6= 


THE 

i 


KAILBOAD TAPES 


US. 


i 


; ||ss|=ss js'ss :s . 


t 


a 


;2|ssss|j|'|3 ;sg 


§•" 


s 


■ sSsl'ssSsss ;ss2 


qS 


e 


; Sifjffg!! IsIs 


1)3 


1 


■Sisslsss ;8=S8S 


§j 


g 


•2SS3SSS :SSS = SSf 


t 


i 


■ f !*"2^f :**""Sgsf 


t 


s 


: !f :!i :f sllsf J! 


OS 


i 


.oooo :o — MM-f.rar-aoO 


t 


s 


■S^S :S3SSSgSS«S 


s 


s 




s? 


s 




« 




= :ooo — (NmTfuscsoocBrtm 


1 


= 


- 


°sssass2|||||| 



TABLES 81 

TAPER li. INTERMEDIATE VALUES — (Continued) A-} 



M 


I 


1_ 


b 


" 


n 


Ipgfl 


^ 


' 


' 


. 


\ 


' ' 


ISO. DO 


1141 


2 0S1 


3 22J 


1274.35 


3.1OI20 


101 


74.98 


3 24 


149 94 


149 BS 








1171 


ZlSi 


3301 


1251.26 


3.0B73S 


1.09 


78,85 




163.27 


IS3!31 




lU 


87 


ISOl 




3 3M 




3.03955 


















liai 


2 241 






3 0SLS9 


















laaj 


230) 


3 SSI 


1194.79 


3.07437 


1,28 


81. 6t 


4.10 


163,25 


183]30 


M 


16S 


S7 


139i 


2 361 


405J 




3.06899 






4.33 




186.83 


soo 


170 


00 


13!1 


2 42J 


416 


""■" 




1,42 


84 97 


4,57 


169 90 


169,91 


OB 


173 


33 


13B1 


!4S( 


4241 




3.05257 


1.49 


88.84 


4.93 


173.32 


173.20 










2 HI 


4 331 




3,04558 


1.57 


88.30 














i«ai 


3 011 


4 431 


1093!dB 


3.03966 




89.68 


6,36 


179 87 


179.96 


20 


193 


33 


1461 


3 07J 


















25 






1491 




















30 






15!i 


3 1111 


6131 


1043.74 


3.01869 






8^23 


189.83 


189.93 


SB 


103 




ISOi 


3 271 


S231 








98.62 


8.H 












200 


3 341 


5 341 


1013.28 


3.00673 


2,08 


98.28 


e,S8 












2 031 


3 411 


546 


9BE.V3 


2.W944 




90.04 




19B.7B 


199.71 


SO 


303 


33 


2 0Tt 


3 481 


sssi 






















2 111 


3 561 


6 06! 










7,88 






eoo 


210 


00 






fllS 


957.61 


2.98114 


2.47 


104.03 


8.24 


209.73 


209.89 


IS 


no 


00 


2 261 


4 26i 


8 52i 


919,84 


2 98362 


2,30 


109,92 


039 


219 66 


219 86 








2 391 


4 491 


V2Si 




2.94«82 






10.64 


229.58 


220.83 


4S 


240 


00 


2B21 


S13i 






2,93088 


3.S3 




12.00 














SflSJ 


6 391 








394 










246.71 








3 18i 


floa 


9 25J 


704.80 


2 00026 


4.38 




15,05 


259 




256.70 


30 


270 


00 


3 331 


6 331 


10 07) 






486 




18,76 






266.84 






00 


3-46) 


7 021 








5.36 








95 




800 


290 


00 






naa 


722.24 


2,8S868 


6.00 


144.72 


20.53 


288 


78 


286,49 


IB 


300 


00 


4» 


8 02) 


12 321 


701.11 


2 84576 


6 47 


149,67 


21,81 


29S 


54 


260.40 






DO 


4381 


3 341 


13 101 


681.30 


2 83334 


7.08 


154]82 


24.83 


308 


a 


309,20 


« 


320 




463 




14 00 
















316.17 


BOO 


330 




510} 


9 401 












29,68 




70 






340 


00 


5 2§1 


mis 


15 43! 




2 79848 


0.14 




32.32 


337 




338 68 


30 


3S0 




6 461 


10 501 


16 371 






B,90 






346 


84 


34S73 




30OOO 


SOS) 


11271 














356 


50 






37(n« 




12 05 


18 30 


ES4,8S 


2,76893 


11,55 










3BS 33 


IS 


3S0I» 


«4tl 


12 42! 


19 2B1 


671.81 


2.7S710 


12,44 


189]o. 


44^43 


37( 




378,09 


30 


300 00 


70SJ 


13 231 


20 261 


559.24 


2.74780 


13.37 


193 92 


4-.SS 


384 


87 


397,82 



THE RAILROAD TAPER 



II 






^ 



8 



I— I 
o 

I— I 






o 



CI 



S 

o 

Q 



08 



o 



8 



* o 



0.8 



o 



8 



^2 



08 
6% 



08 



o- 



00 



08 



98 



98 



08 



HlN .-||C4 iHfM rHiN »-l(M Hln 

* 00 00 00 00 00 00 t^ «0 40 "^ CO i-H 



CO 



* "* <r> t^ iO 01 CO Q5 Oi 00 ^ 05 
» O h* tN. tN. l>. O O iO "^ CO »-H 



00 



<N 



He« He« rH|e« 
" »0 O CO CO »0 »0 -^ CO C^ rH 



0> 1-H 

CO CO 

i-H CO 



o 



TJHO 



h|w ^Jw h|w 
00 Oi tN. Tt< 00 rH 



iOCOOC^C^<M 



s 



Tj<iO»O'*'^T}<C0C^i-i 



oco® 

CO i-hO 
1— I CO »C 



H« Hm «-i|m 



OCO'^'^rHtN.OC^ 
'^OO»0C0l0rHi-H 

«» CO '^ "^ CO CO (N 01 T-H 






o 

ft 

09 



»-•!« Hf« hn 

" 0005U50(N 
O1— I o»oco »o 

o C0C0C0<MC^i-ii-H 



"IN 



T-H CO >— ' O O 
1-HC^rt^COOO 



S 



1-H Ol »-H O CO iO 



CO 05 00 00 1— ' »0 
Oi-i"^0l0l01 

I-H oico^ci^o 



0> CO CO CO iO 

CO ^ CO ^^ ^ 

rH 1—1 1—1 1—1 O 



^ 1— I 1— I O^ CD I-H 05 
»0 O 01 »0 CO CO CO 

OOICO-^COOOO 



o 



t>- 05 00 CO 
OOU5C0 



lOCO-'tCOi-lt^COcp 
"■^■^lOi— iiOCOCO"^ 

O"— '0^^lOl^O5»-H 



8 



o 000 

I 



»h(n He* He* iH|e« 

^ J5 fe 9 3R £2 S? ;$3 22 

COOlOl^O^COCO^ 
Oi-H01C0»0C000O0l 



d8 


8 


• ^W fHjW H|W H|t9 fHjW ^IW 

00 • »H 01 CO tMC t^ 05 I-H CO 


OS 
ou 


^ 


'^(W >'4)C9 ^|W ^M rt(W 

-S :25???5^SS28SS? 

oO •OOthOICOtJhcOOOOOI'^ 

1-H iH T— I 


OS 





'^fW ^W ^^|M >4P'' ^(W fHlW 
• ^ r-^ CO O^ 1-H I-H Tf Ob »0 CO "^ CO 

:ocoo»ooooi»?5'^Tf»ib»-H 

-OOtH rH 01'^»CC000O0^»0 

rH 1-H tH 



09 



rH 1— I tH O^ C^ 0^ 



TABLES 



83 



11 
< 












s 



o 









O 






Q 

^ 



:^ 



gggOt>C5 









Q Q O) O) O O) 
CO © 00 ^ Tt< t>- ' 



01 C^?! CO CO 



OSOOC^'^OOOSOOOtHt-i 
C0rHC0C0»-H050iC0O"^»0 



OO'-HC<l'^l^Oi0MO0SO 
1-H 1— I C^ CO CO »o 



8QQOil>.C0t^00C0'-'OQ0 

rHCOTplCtN.OOOT-ICO'^COl^ 



OT-icor*co?icoi>.'^«o 



-^ 
c^ 



OOOOt-hC^CO'^OOO^'^ 



0*000 

iooou5ioob'-iu50;0c<ioo>o 
'^C<i'-Hc5oia>oooot^t^c25<o 

cococococ^c^ocic^c^c^c^c<i 



CO ^ 05 "^ 5J Q CVI 

Ti^ t^ t:- CO ^ a> 00 
oiooooot^t^ 



OOOOOOO'^OOCDTtHcOCOOOC^ 

"^oc^<p«oooi^cjcjoo»o 

00O>"^rHO500l^CDUi»jOTfir}i 



^ i-Hio^oco^^^TOW^c^i 

o OOt-iCO-^ COOOOCO COOiCO 

t— ( >-H 1-H »-H C^ 



o 

CO 




-888888888888 

o C^ CO -^ iO CO tN. 00 OS O i-H (N CO 



84 



THE RAILROAD TAPER 



TAPER 2. INTERMEDIATE VALUES 



A = l 



M 


I 


e / 




2 00 


30.00 


05 


32.50 


10 


35.00 


15 


37.50 


20 


40.00 


25 


42.50 


30 


45.00 


35 


47.50 


40 


50.00 


45 


52.50 


50 


55.00 


55 


57.50 


3 00 


60.00 


05 


62.50 


10 


65.00 


15 


67.50 


20 


70.00 


25 


72.50 


30 


75.00 


35 


77.50 


40 


80.00 


45 


82.50 


50 


85.00 


55 


87.50 


4 00 

• 


90.00 


05 


92.50 


10 


95.00 


15 


97.50 


20 


100.00 


25 


102.50 


30 


105.00 


35 


107.50 


40 


110.00 


45 


112.50 


50 


115.00 


55 


117.50 


500 


120.00 


05 


122.50 


10 


125.00 



09 
10 
OlOJ 
11} 
12} 
14 
015 
161 
17i 

om 

19} 
21} 
22i 

24 
25} 
26} 
28} 
30 
31i 
33} 
34} 
36^ 
38} 
40} 
42 

44 
45} 
47} 
49} 
52 
54 
56} 

58} 

im 

102} 
105J 
107i 

1 10 

1 12} 



009 
10} 
12 

om 

15} 
16} 
18} 
20J 
22} 
24} 
27 
029 
31J 

33} 
361 
39 
41} 
44} 
47} 
50 
53} 
56} 

59i 
102i 
104 

109} 

1 13 

1 m 

120} 

123} 

127} 

13H 

135} 

139} 

144 

148 

152i 

156} 

2 0U 



18 
20 
22 
25 
28 
30} 
33 
36 
^0 40 
43 
46 
50 
54 

67 
101 
105 
110 

1 14 

1 18 
123 
128 
132 
137 
142 
148 

153 
158 
204 

2 10 
2 15 
2 21 
2 27 
2 34 
2 40 
2 46 
253 
300 



3 06} 
3 13} 



D logD 



2864.86 
2750.28 
2644.51 
2546.58 
2455.64 
2370.98 
2291.97 
2218.05 
2148.75 
2083.65 
2022.39 
1964.52 
1910.07 

1858.47 
1809.59 
1763.21 
1719.15 
1677.25 
1637.35 
1599.30 
1562.98 
1528.28 
1495.08 
1463.31 
1432 86 

1403.66 
1375.62 
1348.69 
1322.80 
1297.87 
1273.89 
1250.77 
1228.48 
1206.98 
1186.22 
1166.18 
1146.80 

1128.03 
1109.91 



3.45710 
3.43938 
3.42234 
3.40596 
3.39016 
3.37493 
3.36021 
3.34597 
3.33219 
3.31882 
3.30586 
3.29326 
3.28105 

3.26916 
3.25758 
3.24630 
3.23531 
3.22460 
3.21414 
3.20393 
3.19395 
3.18420 
3.17437 
3.16534 
3.15620 

3.14726 
3.13850 
3.12991 
3.12149 
3.11323 
3.10513 
3.09718 
3.08937 
3.08170 
3.07416 
3.06677 
3.05949 

3.05232 
3.04529 



0.04 
0.05 
0.05 
0.06 
0.07 
0.08 
0.09 
0.10 
0.11 
0.12 
0.13 
0.14 
0.16 

0.17 
0.19 
0.20 
0.22 
0.24 
0.26 
0.28 
0.30 
0.32 
0.34 
0.37 
0.39 

0.42 
0.45 
0.48 
0.51 
0.54 
0.57 
0.60 
0.63 
0.67 
0.71 
0.75 
0.79 

0.83 
0.87 



15.00 
16.25 
17.50 
18.75 
20.00 
21.25 
22.50 
23.75 
25.00 
26.25 
27.50 
28.75 
30.00 

31.25 
32.50 
33.75 
35.00 
36.25 
37.50 
38.75 
40.00 
41.25 
42.50 
43.75 
45.00 

46.24 
47.49 
48.74 
49.99 
51.24 
52.49 
53.74 
54.99 
56.24 
57.49 
58.74 
59.99 

61.24 
62.48 



0.08 
0.09 
0.11 
0.13 
0.15 
0.17 
0.20 
0.22 
0.25 
0.28 
0.32 
0.35 
0.39 

0.43 
0.48 
0.53 
0.58 
0.63 
0.69 
0.76 
0.81 
0.88 
0.95 
1.02 
1.10 

1.18 
1.27 
1.36 
1.45 
1.55 
1.65 
1.76 
1.87 
1.98 
2.10 
2.23 
2.36 

2.49 
2.63 



30.00 
32.50 
35.00 
37.60 
40.00 
42.50 
45.00 
47.60 
50.00 
52.50 
55.00 
57.50 
60.00 

62.50 

65.00 

67.50 

70.00 

72.50 

76.00 

77.50 

79.99J 

82.49 

84.99 

87.49 

89.99 

92.49 

95.99 

97.49 

w . tfo 

102.48 

104.08 

107.48 

109.98 

112.47 

114.97 

117.47 

119.96 

122.46 
124.96 



Ic 



30.00 
32.50 
35.00 
37.60 
40.00 
42.60 
45.00 
47.60 
60.00 
62.50 
66.00 
67.60 
60.00 

62.60 
66.00 
67.50 
70.00 
72.50 
76.00 
77.50 
80.00 
82.60 
86.00 
87.50 
90.00 

92.60 

96.00 

97.50 

99.99 

102.49 

104.99 

107.49 

109.99 

112.49 

114.99 

117.49 

119.99 

122.49 
124.99 



TABLES 85 

TAPER 2. INTERMEDIATE YALVES— {Continued) A=l 



.» 


1 


/ 


b 


» 


o 


kvD 


- 


' 


^ 


. 


^ 


5 15 


irf SO 


1141 


° ' 


SMi 


1002 36 




flOl 


63. ra 


a.77 


127.45 


127 40 


30 


130 00 


117i 


2I(H 


328 


1075.3! 


3 03155 




04. H8 


2.B2 


129,05 


129.98 








ZlSi 


3 351 




3.02484 


MX) 








133,48 




135.00 


122) 


2201 


3«1 




3.01823 




67.48 






134.98 




J37.S0 


1251. 


22S 


3 601 


1027.30 


3.01174 




88,73 


3.41 


137.44 


137.48 




IM.DO 


137i 


2301 




1012.35 


3.00490 


MS 




3.58 






45 


m,50 


130i 


2351 


4 05i 


097.76 
















145.00 


1331 


2401 


4 131 




2,99281 


1 28 


72!l7 




144 92 


144.87 






1361 


245! 


4 211 






1.33 










flOO 


150.00 


l3ft 


2 51 


4 30 


063,41 


2.08064 


1.37 


74.97 


4,32 


146,90 


149.97 


15 


157.60 


I47i 


3 071 


4 561 


919 30 


296303 


155 


7S.T1 


4.04 


157 37 


157 47 


30 


185.00 


157 


3 24t 


6 211 


8B3 34 


2.94013 


l.VS 


82.45 




164 84 








21«J 


3 42i 


5 401 












172.32 


172.4* 




lao.oo 


2 161 


.4 011 


618 


S20S4 






BO 93 








IS 


187.50 


aaai 




6 471 


702.88 


2.S992D 




il3,fl7 






187,39 






2 37J 


4 4U 


7181 






5^71 








104.86 




!03.60 


2 48t 


5 021 


7501 














202.34 


800 


aio.oo 








710.B3 


2.95711 


3.20 


104 87 


10.98 


309 63 


200,81 




!I7.50 


3 111 


S46! 


8S«i 






3.62 






216,93 


217.28 






3 24 


SOU! 


9 331 


678 18 














41 


aai.so 


asii 


fl331 


10101 


660.2(1 


2.91008 


4.32 


118 06 


14.82 


331 73 


233,18 


II 00 


240,00 


a4H 


8 581 


10 4S 


641.49 


2.80710 


4.71 


110.78 


16 09 


230,10 


239.63 


IS 


W7.50 


4021 


724 


1136) 


624 69 


2 70566 


5.11 


123 49 


17 44 


346.46 


247.07 




255,00 


41S1 


7801 


1SD6I 












353 90 


264 SI 


45 


as2 6o 


4 301 


8 171 


13 471 


593 B2 








20,59 






iooo 




445 


8 45 


13 30 




3.16314 


6 47 


134.83 


22 30 


288.42 


369 3« 






4S9) 


9131 


14131 












276 71 


276.78 




285.00 




42} 


34 57] 






7 40 




28,00 


282. 0« 




4S 


202.50 


530) 


\am 


16431 


641.33 


2.73337 




146.72 


27 09 


2B0.M 


291,64 


1100 


300 00 


5 48J 


10 431 


1630 


629.09 


2,73402 


8.02 


149 41 


30,08 


297,40 


298.02 


15 


S07 50 


6 021 


1115 


17171 


618 73 


2 71494 


023 


153 00 


32.28 


304.58 


306.28 




315,00 


6im 


11471 


ism 


508 20 


2 7D(tl1 


9.86 








313,64 


45 


332.50 


63t\ 


12 301 


18561 


498 36 












810. 50 


12 00 


330.00 


S54 


12 54 


1049 


488 SS 


2.68020 


11.20 


104 10 


30,44 


325.91 


328,W 




337. U 


7 ill 


13 581 


20401 














336.60 


30 


345.00 


T» 


14 03) 


21331 






12.66 






339.05 


242.88 


4S 


m.ta 


7481 


am 


22 281 


463.01 


2.66563 


13 44 


!75,0fi 


47.51 


346,90 


350,15 


MM 


3S0.0D 


8 071 


ISlfli 


23 34 


4S5.22 


2.6S822 


14. 2S 


178.86 


50 51 


353,8 


357.40 



THE RAILROAD TAPER 



11 
<1 






o 

o 

»— t 
CQ 

H 






O 






5 

S 
.2 

« 
Q 



02 






6U 



* o 






05S 



OS. 



OS 



d8 



O' 



»o 



d!S 
d" 



CO 



dM 



O" 



c* 



ds 






CO c5 1-H T-i o C<l "^ CO (N B5 c5 
o 00 O O Od Od 00 t>> CO kC CO C4 



8 

CO 



* CO t^ r» rH Q U5 "* op 00 c^ 

o t* tN. t^ r* tN. CO »o -^ CO T-i 



c^ 



C4 



o lO CO CO CO »0 »0 '^ CO T-i 






8 



o TjH lO »C ^ "^ CO 01 »-H 



^ CO *c> 

1-H CO lO 



o 



" iO 00 CO Oi l>» O 00 

o CO CO CO CO CO 01 »-i 



^co»ot^ 



s 



o 
00 



1 Tj^COQOI^ 

o C^ CO C^ Ol i-H »-H 



T— 1 »5 ^ «5 »-H 
^c^Tt<coa> 



* CO ^ Q »OCO 
OOOCO »o 



CO »-iO »H CO 
1-H C^ Tt< CO 00 O 



S 



V '^ COCO »o 

o t— ( T— I rH O 



'4«Hc«HMHM«Hi 

COOSt^pOOOlQ 

»OOCO^T-iCOO 

O C^ CO »0 t^ 05 C<l 



s 



^^W H|W Wpii 

V <NQCO 
iO^CO 

o ooo 



OrHCO-^COOOOCO 



2 



o oo 



OrHC^c6»or*05»-i'^ 





H'* • 




* ^ 


o 


1— 1 


CO 






O ; 



CqOJCOiOQQJCOCOb-t^ 
CS|0»COeOOO»-iCOi-i 

Oi-iT-iC0'<^C000OCS|»C 



^W H|W H|W ^5?^ "i^ 



.h»«*HmHm 



co?58c?8^5S^& 

OOi-i(NCO»OCOOOOCOCO 



H 



^ 



OQ 



coSSSSSi-H^b-' 

rH »-( »-H Ol Ol C^ 



TABLES 



87 



II 
< 



IS 



I 



P5 



o 

o 

CO 






O 



H 



Q 
^ 






oocbcococsiouDOdco 









oooorHci-^iooooco 



OO^CPOQ^OI-HCPCOO 












Oi-iC^OO»Ct^OCO«OQ'^ 

1— I i-H »-H ^ 01 



1-H f— ( »-H C^ 



o 



o 

?? 



COrfOi-HeOTpO'-^W'^O 
C^C0»C«0tN.00OrHC^C0»0 



88 



THE RAILROAD TAPER 



TAPER 2i. INTERMEDIATE VALUES A-IJ 



M 


/ 


/ 


6 





D 


logZ) 


d 


t 


X 


y 


Ic 


o / 

230 


30.00 


e / 

OIU 


e / 

11} 


e / 

22} 


2291.93 


3.36020 


0.05 


15.00 


0.10 


30.00 


30.00 


35 


32.00 


12 


12i 


24} 


2218.01 


3.34596 


0.06 


16.00 


0.11 


32.00 


32.00 


40 


34.00 


13 


14i 


27} 


2148.70 


3.33218 


0.06 


17.00 


0.13 


34.00 


34.00 


45 


36.00 


14 


151 


29} 


2083.60 


3.31881 


0.07 


18.00 


0.15 


36.00 


36.00 


50 


38.00 


15 


17J 


32} 


2022.34 


3.30585 


0.08 


19.00 


0.17 


38.00 


38.00 


55 


40.00 


16 


19 


35 


1964.57 


3.29327 


0.09 


20.00 


0.19 


40.00 


40.00 


300 


42.00 


17 


20i 


37} 


1910.00 


3.28103 


0.09 


21.00 


0.21 


42.00 


42.00 


05 


44.00 


18i 


22} 


40} 


1858.40 


3.26914 


0.10 


22.00 


0.23 


44.00 


44.00 


10 


46.00 


19i 


24} 


43} 


1809.51 


3.25756 


0.11. 


23.00 


0.26 


46.00 


46.00 


15 


48.00 


20} 


26} 


46} 


1763.13 


3.24628 


0.12 


24.00 


0.29 


48.00 


48.00 


20 


50.00 


21i 


28} 


50 


1719.06 


3.23529 


0.13 


25.00 


0.32 


50.00 


50.00 


25 


52.00 


23 


30} 


53} 


1677.16 


3.22457 


0.15 


26.00 


0.35 


52.00 


52.00 


30 


54.00 


24i 


32} 


56} 


1637.25 


3.21411 


0.16 


27.00 


0.38 


54.00 


54.00 


35 


56.00 


25} 


34} 


100} 


1599.19 


3.20390 


0.17 


28.00 


0.41 


56.00 


56.00 


40 


58.00 


26i 


37 


103} 


1562.86 


3.19392 


0.18 


29.00 


0.45 


58.00 


58.00 


45 


60.00 


28 


39} 


107} 


1528.16 


3.18417 


0.20 


30.00 


0.49 


60.00 


60.00 


50 


62.00 


29} 


41i 


111} 


1494.95 


3.17463 


0.21 


31.00 


0.53 


62.00 


62.00 


55 


64.00 


31 


44} 


1 15} 


1463.17 


3.16529 


0.23 


32.00 


0.57 


64.00 


64.00 


400 


66.00 


32} 


461 


119} 


1432.71 


3.15616 


0.24 


33.00 


0.62 


66.00 


66.00 


05 


68.00 


34 


49} 


123} 


1403.50 


3.14721 


0.26 


34.00 


0.67 


68.00 


68.00 


10 


70.00 


35} 


52 


127} 


1375.45 


3.13844 


0.28 


35.00 


0.72 


69.99 


70.00 


15 


72.00 


37 


54} 


131} 


1348.50 


3.12985 


0.29 


36.00 


0.77 


71.99 


72.00 


20 


74.00 


38} 


57} 


136} 


1322.60 


3.12143 


0.31 


37.00 


0.83 


73.99 


74.00 


25 


76.00 


40i 


100} 


140} 


1297.67 


3.11316 


0.33 


38.00 


0.89 


75.99 


76.00 


30 


78.00 


411 


103} 


145} 


1273.67 


3.10506 


0.35 


39.00 


0.95 


77.99 


78.00 


35 


80.00 


43} 


106} 


150 


1250.54 


3.09710 


0.37 


40.00 


1.01 


79.99 


80.00 


40 


82.00 


45i 


109} 


154} 


1228.25 


3.08929 


0.40 


40.99 


1.08 


81.99 


82.00 


45 


84.00 


47 


112} 


159} 


1206.73 


3.08161 


0.42 


41.99 


1.15 


83.99 


84.00 


50 


86.00 


48f 


1 16 


2 04} 


1185.95 


3.07407 


0.44 


42.99 


1.22 


85.99 


86.00 


55 


88.00 


50} 


1 19} 


2 09} 


1165.90 


3.06666 


0.47 


43.99 


1.30 


87.99 


88.00 


600 


90.00 


52} 


122} 


2 15 


1146.50 


3.05937 


0.49 


44.99 


1.37 


89.98 


90.00 


05 


92.00 


54} 


125} 


2 20} 


1127.72 


3.05220 


0.52 


45.99 


1.45 


91.98 


91.99 


10 


94.00 


56i 


129} 


2 25} 


1109.58 


3.04516 


0.54 


46.99 


1.54 


93.98 


93.99 


15 


96.00 


58i 


133 


2 31} 


1092.01 


3.03823 


0.57 


47.99 


1.63 


95.98 


95.99 


20 


98.00 


lOOi 


136} 


2 36} 


1074.99 


3.03140 


0.60 


48.99 


1.72 


97.98 


97.99 


25 


100.00 


102i 


140} 


2 42} 


1058.50 


3.02469 


0.63 


49.99 


1.81 


99.98 


99.99 


30 


102.00 


104i 


144 


2 48} 


1042.50 


3.01808 


0.66 


50.99 


1.91 


101.97 


101.99 


35 


104.00 


106} 


147} 


2 54} 


1026.98 


3.01156 


0.69 


51.99 


2.01 


103.97 


103.90 


40 


106.00 


108} 


151} 


3 00} 


1011.92 


3.00515 


0.72 


52.99 


2.11 


105.97 


105.99 



TABLES 



89 



TAPER 2}. INTERMEDIATE Y ALV EH — {Continued) A = li 



M 



5 45 
50 
55 

600 
15 
30 
45 

700 

15 
30 
45 

800 
15 
30 
45 

900 

15 
30 
45 

10 00 
15 
30 
45 

1100 

15 
30 
45 

12 00 
15 
30 
45 

13 00 

15 
30 
45 

14 00 
15 
30 
45 

15 00 



/ 


/ 


b 





D 


logD 


d 




e t 


/ 


o t 








108.00 


llOJ 


155^ 


3 06} 


997.31 


2.99883 


0.76 


110.00 


113 


159} 


3 12} 


983.11 


2.99260 


0.79 


112.00 


115i 


2 m 


3 18} 


969.31 


2.98646 


0.83 


114.00 


117i 


2 07} 


3 25} 


955.90 


2.98041 


0.86 


120.00 


124i 


2 20} 


3 45 


917.82 


2.96276 


0.98 


126.00 


131} 


2 34 


4 05} 


882.70 


2.94581 


1.11 


132.00 


139i 


2 48 


4 27} 


850.19 


2.92952 


1.24 


138.00 


147 


3 02} 


4 49} 


820.03 


2.91383 


1.39 


144.00 


•155i 


3 18 


5 13} 


791.97 


2.89871 


1.55 


150.00 


2 03} 


3 33} 


5 37} 


765.80 


2.88412 


1.72 


156:00 


2 12J 


3 50} 


6 02} 


741.34 


2.87002 


1.90 


162.00 


2 2U 


4 07} 


6 28} 


718.43 


2.85638 


2.09 


168.00 


2 31 


4 24} 


6 55} 


696.93 


2.84319 


2.29 


174.00 


2 40} 


4 43 


7 23} 


676.73 


2.83042 


2.51 


180.00 


2 50} 


5 01} 


7 52} 


657.71 


2.81803 


2.74 


186.00 


3 01 


5 21} 


8 22} 


639.77 


2.80602 


2.99 


192.00 


3 1U 


5 41} 


8 52} 


622.83 


2.79437 


3.25 


198.00 


3 22} 


6 02 


9 24} 


606.81 


2.78305 


3.52 


204.00 


3 33} 


6 23} 


9 56} 


591.64 


2.77206 


3.81 


210.00 


3 45 


6 45 


10 30 


577.26 


2.76137 


4.12 


216.00 


3 56} 


7 07} 


1104} 


563.61 


2.75098 


4.44 


222.00 


4 08} 


7 30} 


1139} 


550.64 


2.74087 


4.77 


228.00 


4 21} 


754 


12 15} 


538.30 


2.73102 


5.12 


234.00 


4 34 


8 18} 


12 52} 


526.56 


2.72145 


5.49 


240.00 


4 47 


8 43 


13 30 


515.38 


2.71213 


5.88 


246.00 


500} 


9 08} 


14 08} 


504.71 


2.70304 


6.28 


252.00 


5 13} 


9 34} 


14 48} 


494.54 


2.69420 


6.70 


258.00 


5 27i 


10 01} 


15 28} 


484.82 


2.68558 


7.14 


264.00 


5 41} 


10 28} 


16 10} 


475.54 


2.67719 


7.60 


270.00 


5 56} 


10 56} 


16 52} 


466.67 


2.66901 


8.08 


276.00 


6 11 


1124} 


17 35} 


458.19 


2.66105 


8.58 


282.00 


6 26 


1153} 


18 19} 


450.06 


2.65327 


9.09 


288.00 


6 411 


12 23} 


19 04} 


442.29 


2.64571 


9.63 


294.00 


6 57} 


12 53} 


19 50} 


434.85 


2.63834 


10.19 


300.00 


7 13} 


13 24} 


20 37} 


427.71 


2.63115 


10.76 


306.00 


7 291 


13 55} 


2125} 


420.87 


2.62415 


11 36 


312.00 


7 46 


14 27} 


22 13J 


414.31 


2.61733 


11.98 


318.00 


8 02} 


15 00J 


23 a*?} 


408.04 


2.61070 


12.63 


324.00 


820 


15 23} 


23 53} 


402.01 


2.60424 


13 29 


330.00 


8 37i 


16 07J 


24 45 


306.23 


2.5979.5 


13.98 



t 


X 


53.99 


2.22 


54.98 


2.33 


55.98 


2.45 


56.98 


2.57 


59.98 


2.95 


62.97 


3.36 


6.5.97 


3.81 


68.96 


4.30 


71.96 


4.83 


74.95 


5.40 


77.94 


6.01 


80.93 


6.67 


83.92 


7.37 


86.91 


8,12 


89.90 


8.92 


92.88 


9.77 


95.86 


10.68 


98.84 


11.64 


101.82 


12.65 


104.80 


13.71 


107.78 


14.84 


110.75 


16.03 


113.72 


17.27 


116.69 

4 •• r\ ABT 


18.58 

4 r\ /\A 



119.65 
122.61 
125.57 
128.52 
131.47 
134.42 
137.36 
140.30 

143.23 

146.16 

149.08 

152.00 

154.91 

157.81 

160.71' 

163. 60 ! 



19.96 
21.40 
22.90 
24.47 
26.11 
27.82 
29.61 
31.47 

33.40 
35.40 
37.49 
39.65 
41.89 
44.21 
46.61 
49.09 



107 

109, 

111. 

113. 

119. 

125. 

131. 

137. 

143. 
149. 
155. 
161. 
167. 
173. 
179. 
185. 

191. 
197. 
203. 
209. 
215. 
221. 
226. 
232. 

238. 
244. 
250. 
256. 
261. 
267. 
273. 
278. 

284. 
290. 
295. 
301. 
307. 
312. 
318. 
323. 



Ic 



107.99 
109.99 
111.99 
113.98 
119.98 
125.97 
131.97 
137.96 

143.95 
149.94 
155.93 
161.91 
167.89 
173.88 
179.86 
185.84 

191.82 
197.79 
203.75 
209.70 
215.65 
221.60 
227.55 
233.49 

239.42 
245.36 
251.29 
257.20 
^63.09 
268.98 
274.87 
280.75 

286 61 
292.47 
298.32 
304.15 
309.96 
315.76 
55 
33 



97 
96 
96 
95 
94 
93 
91 
89 

87 
84 
81 
78 
73 
69 
63 
57 

51 
43 
34 
25 
14 
02 
89 
74 

59 
42 
23 
02 
79 
54 
27 
98 

66 
32 
95 
55 
12 
66 

16 321 
63 1 327 



THE RAILROAD TAPER 



• o 

* O 



CO 



« 00 Oi Oi Oi 00 00 1^ »o Tt< c^ 






> 



o 

CO 



P^ 

O 

P 
H 

> 
P^ 
P 






2 
.S 



* O 

0"5 



• o 

• o 

Oeo 



62 



O" 



e^i 



6% 



63 
6" 



a> 



6^ 
djL 



63 
6" 



to 



6^ 



OS 



o° 



CO 






o 



W|^»'^Mp <|# W| ^1 ^W 



o 

» t>. t^ t^ t^ «D CO »0 CO C^ 



" C0C0C^»0i-hO"^O 






o 



^ Tf TjH qp i-H t>. lo »o op 
o lO «D «D »0 »0 Tt< CO »H 



fh CO 



o 



i2 22 25 12 28 ;$! 

^ ^ Cn ^ ^ CO 
^ ^ Tfi ^ CO c^ 1^ 



rMr*'-^ 



i-HCOCO 



CON* •h|^hc««I'4) 
rjl CO C5 "^ '— ' »— I 
" (NCO^IOC^C^ 

o CO CO CO CO C^ '^ 



i-H CO »ooo 



- 0Q"5TjH'^t^ 

c<icoc^»oo 






S 



»-H CO t^ ^ 
1— I 1-H 1— I O 



t>. io i-H "^ ;b CO 

1-H C^ -^ CO 00 i-H 



s 



-§85 

o 1-H »-H0 



iOO^COCOCTt* 

oc^co»ot^oc^ 



s 



cor* 

cocq 

oo 



fH|c«iH|Hi wNH wH '* 
Q»-HOCO»-H"^iOCO 

^ -rococo "n<»-tc5 

Oi-HC0"^«D00'-HTt< 





rHlW • 


^ 


. CO • 

1-H • 




O . 



r-l|-4)H|C«n|HI HHIHWWI^ 

O'-HC^C0i0t^05C^»0 



HC«fHN) nHif-ijMHHi mNiHC«iHHi 
COt^CiOOCD^JOt^t^^O 

THTfCO'^^T-HOOC^OO 

ooi-HC^'^oooococo 



coSS8w5§^-H^^-! 

1-H 1-H tH C^ CS| W I 



TABLES 



91 



11 









> 



O 

H 
> 



PL, 

H 



g* 



:^ 






O O C!^ C) O C) 



_ _ _ _ ^ Oi 

1-H 1-H 1-H C< C^ ' 



S 






t^CO'^ 



O Q O O O) C) 

T-H 1— I 1— I C^ C^ C^ 



O0'-H'^t>-55COOC^ 



i-H f-l C^ CO Tj< 



SQOit^eO»OC^Qt^QO 



S;S528SSgS8 



OOO'-HC^CO'^t^OSC^ 



83 



•A 



1— I 05 O »—< Oi »—< 
00 CO t^ «D i-H c^ 
CO »Q O5C0 U5 CO 



ooOoooOOPOoOcocJ?>^5 



05 00 OOt^ 



lO lO 



cococ^c^c^c^c^c^c^c^ 



t>-CPC0«D'^COC^'-HC0»C 






"^ 00 CO OS o 

IQ TJ1 ^ CO CO 



" cqc^TticO'^c^co^'-^'^ 

« fh i-< C^ C^ 



* Smowo^ocoo^ 

CO "^ «D t^ 05 O C^ CO iC «D 
■ 1—1 1—1 1—1 tH 1—1 



THE RAILROAD TAPER 

TAPER 2i. IXTEKMEDIATE VALUES A = li 



,, 


. 


/ 


. 


„ 


J, 


logfl 


' 


' 




. 


' 


3oa 


3QM 


13) 


131 


27 


11)09.97 


3 28103 


o.ofl 


15.00 


0,12 


30.00 


30.90 




31 87 


om 


0141 


oaoi 


1858 37 


3 28913 










31 67 






D151 


0191 


311 


180B.47 


3.25755 


0,07 


18 87 




33 33 


33.31 




35 00 


Olfll 


181 


0S4) 




3.24827 






0,17 


34.00 


36,00 


21) 


36.87 


0171 


013} 


38( 




3.23638 










36 67 




38.33 


0181 




391 


1877.10 


3,22466 


009 






as S3 


38 32 




10 00 


Olfll 


221 












22 


40.00 


40.00 




11.87 


02(4 


24) 


Oil] 




3.20383 












40 




D2Ii 


291 


0171 


1682.80 


»,193B0 


0.12 


21 67 




43.33 




« 


15 00 


2!) 


281 


501 




3,18416 




22.50 


0.29 


46,00 


46.00 


sn 


18.87 


231 


030 


631 












46.87 








241 




5SI 


1493.00 


3.19)27 




24.17 


0.36 


48.33 


48.JJ 


100 


JO.OO 




031 




1432.83 


3.15613 


o,ia 


23,00 


0.38 


60,00 


SO.OO 


06 


51 67 


271 


038 


1031 


1403.41 


3.I47I6 


0.17 


25.93 


D.ll 


61.66 


51.67 


10 


.W.33 


2S( 


3S1 


loet 








20,88 












291 


040) 


1101 


1318.41 






27.50 














0«I 


1131 


1322.50 




o'2l 




0.61 


58,66 


56,67 


as 


53 33 


32) 


Dill 


1171 












58.33 


6S.33 






331 


171 


121 


1273.58 






30.00 




60.00 








361 


OIB) 


1241 


1250.42 


3]0970« 


0.24 


30.83 


0,63 


61.66 


81.67 


40 


03.33 


381 


52i 


128! 








31 68 


0.87 


63,33 


63,33 








Oill 


1321 


1200,69 




0.28 


32 50 




M.«9 


86 00 






39} 


571 


1381 




3"074O2 


0.30 


33.33 




66,66 


66 67 




88.33 




OSfll 


1401 


1195.74 










68.33 


68,33 


BM 




42) 


102) 




1149 34 


3,05931 


D.33 


35.00 


0.87 


69, M 


70.00 


06 


ri.B7 


044 


1051 


IMl 


1127.66 


3.00311 


0.3S 


36 83 


0,92 


71,66 


71,06 






4Si 




1631 


1109.11 


3.04*» 


0.37 


36,69 










T5.00 


471 




1581 










LOS 


74,90 


76.00 




r8.B7 


49 


im 


2 021 












76,66 


76.66 


26 




0401 


net 


2 071 


1058.30 


3.0!4«l 


0.13 


39 19 




7B.SJ 


78.33 


30 


SU.OO 


521 


iifll 




1042.20 


3,01709 


0.4S 


39.09 


1.22 


70.90 


80.00 








1221 


aiBi 












81,85 


81.66 






OSSl 




2 211 


1011,89 


300505 


49 


11 89 


1.35 


83 32 






M.OO 


67( 


121)1 


2Wli 


997.07 


2.90873 


0.82 


42.40 


1.42 


84.98 


»4-W 


M 


87. 07 


601 


132) 


2311 


982.88 










86,65 


88.«6 


M 




1011 


135) 


2 381 


999 01 


2.98631 


0.58 


14,16 


1.S7 


88.31 


8X.31 



TABLES 



93 



TAPER 21 INTERMEDIATE VALUES 


— {Continued) 


A=li 


A/ 


/ 


/ 

o / 

103 


6 





D 


logZ) 


d 
0.59 


t 


X 


V 


Ic 


e / 

600 


90.00 


/ 

139 


e / 

2 42 


955.63 


2.98029 


44.99 


1.65 


89.98 


89.99 


15 


95.00 


108! 


149} 


2 58} 


917.51 


2.96261 


0.67 


47.49 


1.90 


94.97 


94.99 


30 


100.00 


1141 


2 00} 


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 


0.85 


52.48 


2.48 


104.96 


104.99 


700 


110.00 


127i 


2 23} 


3 51 


819.59 


2.91360 


0.95 


54.98 


2.80 


109.95 


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 


14U 


2 48} 


4 30 


765.26 


2.88381 


1.18 


59.97 


3.53 


119.92 


119.97 


45 


125.00 


148} 


3 02} 


4 50} 


740.75 


2.86967 


1.31 


62.46 


3.94 


124.90 


124.96 


8 00 


130.00 


156 


3 16 


5 12 


717.78 


2.85599 


1.44 


64.96 


4.38 


129.88 


129.95 


15 


135.00 


2 03f 


330} 


5 34} 


696.22 


2.84275 


1.58 


67.45 


4.85 


134.86 


134.95 


30 


140.00 


2 11} 


3 45} 


5 57 


675.95 


2.82991 


1.73 


69.94 


5.36 


139.84 


139.94 


45 


145.00 


220 


4 00} 


6 20} 


656.86 


2.81747 


1.89 


72.44 


5.90 


144.81 


144.93 


900 


150.00 


2 28} 


4 16} 


6 45 


638.84 


2.80539 


2.06 


74.93 


6.47 


149.77 


149.91 


15 


155.00 


2 37J 


433 


7 10} 


621.82 


2.79366 


2.24 


77.42 


7.08 


154.74 


154.90 


30 


160.00 


2 46i 


4 49} 


7 36 


605.72 


2.78227 


2.43 


79.91 


7.73 


159.70 


159.88 


45 


165.00 


255} 


5 07} 


8 02} 


590.46 


2.77119 


2.63 


82.39 


8.42 


164.65 


164.86 


10 00 


170.00 


305 


5 25 


8 30 


575.98 


2.76041 


2.84 


84.88 


9.14 


169.60 


169.84 


15 


175.00 


3 14} 


5 43} 


8 58} 


562.23 


2.74991 


3.06 


87.37 


9.90 


174.54 


174.82 


30 


180.00 


3 24} 


6 02} 


9 27 


549.16 


2.73970 


3.29 


89.85 


10.70 


179.47 


179.79 


45 


185.00 


335 


6 21} 


9 56} 


536.71 


2.72974 


3.53 


92.33 


11.55 


184.40 


184.76 


1100 


190.00 


3 45} 


6 41} 


10 27 


524.86 


2.72004 


3.79 


94.81 


12.44 


189.32 


189.73 


15 


195.00 


3 56} 


7 02 


10 58} 


513.56 


2.71059 


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 


14.35 


199.14 


199.65 


45 


205.00 


4 18} 


7 44} 


12 02} 


492.47 


2.69238 


4.63 


102.24 


15.37 


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 


5.26 


107.19 


17.56 


213.79 


214.51 


30 


220.00 


4 53} 


8 51} 


13 45 


464.18 


2.66669 


5.59 


109.65 


18.73 


218.65 


219.46 


45 


225.00 


506 


9 14} 


14 20} 


455.54 


2.65853 


5.93 


112.12 


19.95 


223.50 


224.40 


13 00 


230.00 


5 18} 


9 38} 


14 57 


447.26 


2.65056 


6.29 


114.58 


21.22 


228.34 


229.33 


15 


235.00 


5 31} 


10 03 


15 34} 


439.32 


2.64278 


6.66 


117.04 


22.54 


233.17 


234.25 


30 


240.00 


5 44} 


10 27} 


16 12 


431.71 


2.63519 


7.05 


119.50 


23.91 


237.98 


239.17 


45 


245.00 


5 57} 


10 53} 


16 50} 


424.40 


2.62778 


7.45 


121.95 


25.34 


242.77 


244.08 


14 00 


250.00 


611 


11 19 


17 30 


417.38 


2.62053 


7.87 


124.40 


26.82 


247.55 


248.99 


15 


255.00 


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 


653 


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 


268.54 


30 


280.00 


7 37} 


14 04} 


21 42 


380.62 2.58049 


10.69 


139.03 


36.90 


275.81 


278.26 


16 00 


290.00 


808 


15 01 


23 12 


370.15 2.56838 


11.76 


143.87 


40.74 


285.05 


287.94 


30 


300.00 


8 39} 


16 05} 


24 45 


360.45 2.55684 

1 


12.90 


148.68 


44.82 


294.19 


297.58 



94 



THE RAILROAD TAPER 



TAPER 2i CURVATURE INCREASING 2^ 00' 

EVERY 30 FT. A =2 













Deflections to 










Transit 
at 






















B.C. 

2*00' 


C.C. 

4*00' 


C.C. 

6*00' 


C.C. 

8*00' 


C.C. 

10*00' 


C.C. 

12*00' 


C.C. 

14*00' 


C.C. 

16*00' 


C.C. 

18*00' 


C.C. 

20*00' 


/ 





30 


60 


90 


120 


150 


180 


210 


240 


270 




o / 


o / 


e / 


e / 


o t 


e / 


o / 


e / 


o / 


e / 







18 


45 


124 


2 15 


3 18 


4 33 


600 


7 39 


930 


30 


18 




36 


121 


2 18 


3 27 


4 48 


6 21 


806 


10 03 


60 


103 


36 




54 


157 


3 12 


4 39 


618 


809 


1012 


90 


2 12 


139 


054 




112 


2 33 


406 


5 51 


7 48 


9 57 


120 


3 45 


3 06 


2 15 


112 




130 


309 


500 


7 03 


9 18 


150 


5 42 


4 57 


400 


2 51 


130 




148 


3 45 


554 


8 15 


180 


8a3 


7 12 


609 


454 


3 27 


148 




206 


4 21 


6 48 


210 


10 48 


9 51 


8 42 


7 21 


5 48 


4 03 


206 


• • • • • 


224 


4 57 


240 


13 57 


12 54 


1139 


10 12 


8 33 


6 42 


4 39 


224 




2 42 


270 


17 30 


16 21 


15 00 


13 27 


1142 


9 45 


7 36 


5 15 


242 






M 


/ 





D 


logZ) 


d 


t 


z 


V 


Ic 


O 1 

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 


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 


574.71 


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 


21 36 


328.02 


2.51590 


9.38 


119.12 


31.77 


236.44 


238.54 


20 00 


270 


27 00 


299.71 


2.47670 


12.87 


133.54 


44.13 


263.76 


267.43 



TABLES 

TAPER 21 INTERMEDIATE VALUES 



M 


' 


/ 


<■ 




D 


logo 


^ 


' 


' 


» 


^ 


too 


MOO 


OlS 


OlS 


036 


i432,55 


3.1561 


008 


15. DO 


O.lfl 


30 00 


30.00 








019f 


038i 






















191 


21 


0-401 














32.50 


32.60 


IS 


33.75 


0201 


221 


043 


1348.31 


3,12976 


0.10 


18 


87 


0.20 




33,76 






oaij 


231 


461 




















30.25 


22* 


251 
















36.28 


38.25 


30 


37.60 


23J 


27 


601 


1273. « 


a. 10498 


0.12 


18 


7S 


0.26 










241 


28t 


531 


1260.30 










D.iS 




38.75 




to. 00 


261 


O30i 


050 




3,08919 




20 


00 


0.30 


40.60 


40.00 






20} 




OSSi 


iaoa]46 




O.IS 


20 


62 






41,25 




4!. 50 


271 


33! 


1011 








21 




0.34 




42.60 








OS.'il 


1041 


1165,59 


3,06656 










43.75 


43.76 


flOO 


4b!oo 


<I30 


371 


107! 


1146. IS 


3,0SI»26 


0.17 


22 


50 


0.39 


45,00 


45.00 


05 


«,M 


031i 


391 


110) 


1127.38 


3 05207 


18 


23 


12 


0,42 


46.25 


48.25 




a.m 


321 


D41J 


113) 






O.IO 


23 


76 


0.4i 


47,50 






M.75 


«m 


4S1 


1101 


1091.64 












4S.76 


48,76 


20 


50,00 


0341 


461 




1074.60 




21 






0.60 


60.00 


60,00 








47i 


1231 


1068.00 














51.26 






03! 


4IH 


I2ei 


1042.08 








26 












038) 


611 




1026.64 


3,01138 




28 




080 


63.75 


83. 7S 


40 




391 


631 


1331 


1011.46 


3,00495 


0.28 


27 


SO 


063 








U.25 


41 
















0.67 




68,26 




67,50 


42! 


06« 


140) 


982.61 


2 99238 


0,29 


28 






£7.60 


67,80 




68.78 


0«l 


100) 


1«1 


988. 7S 


3,98823 




29 


37 


0^75 


«.76 




BOO 


SOKI 


046 


103 




B5S.3B 


2.118018 


0.31 


29 


99 


0.79 


63,99 


60.00 


IS 


63.76 


^m 


uoi 


169) 


917 20 


2,96246 


0.36 


31 


8J 


0.91 


*3.74 


63 78 


SO 




AMI 




2 11} 
















67.60 


u 


71-H 


0681 


12B 


2 241 


849.4! 


2 92912 


0-40 








71,34 




700 


7S.I» 


103 


134) 


2 37) 




2.91336 




37 


49 




74.98 


76.00 






IW 


1*31 


2 611 


790. » 














78,74 


m 


K.JO 


113 


162J 


306) 


764.72 


2.88350 


0,84 






1,76 


82,47 




4S 


M.26 


1181 


302 


3 20) 


740,15 


















M.OO 




212 








0.76 








89,98 


89-98 


IS 


93.75 


I2M 


2 221 


3 52 


695,60 


2,84230 










93.70 


93.73 




B7.S0 


1361 




4 081 




2 82941 


0,05 


48 




2.71 




97,48 


K 


01.25 


1411 


244 


4 2Si 


B56.0I 


2,81891 


1,04 


BO 


00 


2,99 


101.18 


101,23 



96 THE RAILROAD TAPER 

TAPER 2i. INTERMEDIATE VALUES — (Continued) 



M 


' 


/ 


" 


- 


D 


fogfl 


i 


' 




• 


Ic 


BOO 


lOS.DO 


MB 


3 6B1 


4 431 


B37B1 


2 80476 


I 13 


62 47 


3 30 


1M03 


104 97 


IS 


108. 7S 


lS4i 


3 071 


SOI) 


020 




2.792BI 








B3 


108,86 


108.71 








3 19t 


6 20) 


601 


63 


2.78143 




68.21 






112.39 


113.46 




iie.2s 








989 


















10 00 


120.00 


2 IS 


3 4S 


BOO 






3.769*6 










110.85 


119-9* 






2 321 


368} 


B201 


660 


86 


2.748*6 










123 38 


133.69 






239) 




8 411 




















49 


131. M 


!37i 


4 21 


7 031 


539 






1,96 


65.55 






131.03 




11 00 


136.00 




4 40) 


7261 




17 












134.75 


134.91 




IBS 76 




4 661 


7 481 






2.7090B 




80 28 




98 




138,66 


30 


143 SO 


3 01 


SlOl 


SllJ 


50Q 




2.80970 


2.41 


71.15 


7 




143.18 


143 30 






SOflj 




S35) 










73.01 




06 


149.89 


148.12 


12 00 


ISO.OO 




S« 




480 


42 


2.88162 


2.74 


74. S7 


s 


63 


149,60 




IS 


m ;.s 


3 2Bi 


6SS) 


2S1 


470 


m 


1 67289 


2.92 


76.74 


823 


163.30 


153.5S 






3 361 




9 50j 










78 6(1 


8 86 


167.00 




4S 


161 26 


3 44i 


3 32 


10 ig; 


492 91 


2.BS80I 


3.30 


SO 46 


10.52 


160.89 


161.0* 


13 00 






em 


10 43! 










11'. 20 


184.37 


mn 






4 031 


7 071 


lllOi 










11 92 


168.06 




» 


172.60 


4 13 


7 2J) 


11381 


438 59 


2.B3304 


3.03 


seo3 


12.67 






41! 


I7fl.as 








421.10 


3,82439 




87 89 




176.41 


17S.93 














3.61390 












IS 


183. 7S 


4 431 


8 221 


13 061 


406 96 


2.60965 




91.59 


15.09 




183.38 


30 




4 63J 




13 3.61 


400 39 


2 60237 






15 08 


186.38 


187 06 


4E 


101.211 


6«1 




14 061 




2.69S34 




96.29 


18.88 






IS 00 


iBS.on 


Sli 


9 3!) 


14 371 


387 68 


S.68S45 


5.41 


97.14 


17.70 


193,86 


194 46 


30 


302.30 


6 371 


10 04i 


13 41] 


379.90 


2.B7S07 


6.97 


00 83 


19.76 


300.80 


301.86 


IB 00 


310,00 


600 


10 48 


!B48 


3H 




2.68235 




04.50 


2187 


208.08 


109 24 


30 


317,50 


fl331 




17 681 


364 




2,54993 




108.18 


24 13 


216.35 


116,60 




326.00 




1213) 


13 071 












36.52 


233.37 


223 .M 


30 


332.SD 


7 13 


13 071 


20 301 


3» 




3 52877 


S.81 


16 49 




239.43 


231.36 


18 00 




739 


13 S7 


21 3B 


328 


02 


2.61.990 




10.12 


31 77 


256,44 


238 6* 






SOB) 




33 631 














246.83 








15 40) 


24 131 








11.03 








30 


ita.iD 


BDl 


IB 341 


25 351 


3n» 10 


!,4B5B8 




20 9K 






20 00 


370.00 


30 


17 30 




2110.71 


3.47670 


12.S7 


133 54 


„,=«:,. 


267,43 



TABLES 



97 



TAPER 3. CURVATURE INCREASING 2° 30' 

EVERY 30 FT. A = 2J 



Transit 
at 





30 

60 

90 

120 

150 

180 

210 

240 



Deflections to 



B.C. 

2*30' 







22i 

1 18f 

2 45 
4 41i 
7 07i 

10 031 
13 30 
17 26i 



C.C. 

5° 00' 



30 



22i 

45 

2 03i 

3 52^ 

6 1U 

900 

12 181 

16 07i 



C.C. 

7° 30' 



60 



561 
45 



107f 



2 48? 



'5 

500 

7 4U 
10 52J 



C.C. 

10° 00' 



90 



145 
1 41} 

107J 



14 33? 



1 30 
3 331 
6 07J 

9 1U 
12 45 



C.C. 

12* 30' 



120 



2 481 
2 52i 
2 26i 
130 



152^ 

4 181 

7 15 

10 4U 



C.C. 

15° 00' 



150 



4 07i 

4 18} 
400 

3 11} 
152i 

2 15 

5 031 
8 22i 



C.C. 


C.C. 


17" 30' 


20*00' 


180 


210 


o / 


e / 


5 41} 


730 


600 


7 56} 


5 48f 


7 52i 


5 07i 


7 18} 


3 56} 


6 15 


2 15 


4 41} 




2 37i 


2 37i 




5 48} 


300 



C.C. 

22* 30' 



240 



9 331 
10 07i 
10 11} 
9 45 
8 48} 
7 22} 
5 26} 
300 



M 


I 


• 




D 


logZ) 


d 


t 


X 


y 


U 


o / 

500 


30 


e / 

045 


1146.11 


3.05923 


0.10 


15.00 


0.20 


30.00 


30.00 


730 


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 


7 30 


460.55 


2.66328 


1.96 


59.91 


5.88 


119.77 


119.91 


15 00 


150 


1115 


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 


27.25 


207.01 


208.78 


22 30 


240 


27 00 


266.74 


2.42609 


11.69 


118.63 


39.49 


234.42 


237.72 



THE RAILBOAD TAPER 

TAPER 3. INTERMEDIATE VALUES 



» 


' 


/ 


. 


- 


O 


IokD 


d 


' ' 


. 


Ic 


tOQ 


30.00 


22i 


0221 


OIB 


1146,11 


3 0SW3 


10 


IS. 00 


020 


SO 00 


30,00 


OS 






23( 


231 


OlJi 


1127 








16. so 


0^21 


31.00 


31,00 








341 


251 


Dlfll 










18.00 








32.00 


IS 


33 


00 


2ii 


281 


62 




59 


3,03805 




18.50 




33 




SiM 


so 




00 


fl2« 


281 


OMi 










17,00 






00 


34,00 


3i 




























3S,0D 


30 




00 


28 


0311 


691 


1011 


98 


3.01786 




18.00 


0,29 






36.00 






















18.50 




37 


00 


37,00 


40 








0311 


1011 










19,00 
















31 


381 


1071 


996 




2.99867 


0,16 


19.60 


0,35 






39.00 








32 


fl38 












20 00 


0,37 


40 




40.00 


S5 


" 


00 


331 


3B1 


1121 


908 66 


2,9B617 


O.IS 


20,50 


0.40 


" 


00 


1100 


BDO 


12 


00 


341 


fllJl 


ns( 


9S5.23 


2,08011 


0,19 


21.00 


0.42 


42 


DO 


12-00 


\i 


46 




371 


48i 


1241 










22.50 










30 






4[ 


tun 


1331 




83 


2.91938 


0.24 




0^57 


48 


09 


48,00 








0141 


0S8} 


1431 










23.60 








Sl-OO 


700. 






U181 


l«i 


153! 










26. BB 


*,76 














531 


1111 




790 




2S9806 




23.10 


0.87 


68 




67.00 


SO 


«0 




OMi 


IIB! 
















59 


S9 


60.00 


49 




1001 




2 231 










31, IB 


l.Jl 


di 




61.00 






1041 


1331 


2 38J 


7H 


82 


2.86641 


0.48 






68 


BS 


66.00 




«e.DD 


1»1 


UU 


2S0) 


99S17 








1.39 


68 


es 


69,00 


JO 




URl 


3 03) 








36.90 


J. 65 


71 




71.90 


la 


7d 


oo: 1 181 


158 


3 1Bi 


r: 


3.81889 


0.61 


37,18 


1.72 


74 


87 


74,99 


900 


78 


oo'iss) 


2 09i 


3301 


637 43 


3.90447 


0,70 


38,98 


J, 90 


J7 


97 


77, » 






UO IM( 




3111 


620 












80 


96 


so.oo 




M 


00 Ul 


2 25! 


3 501 


604 










iHb 




ts 


83 91 


IS 


87 


00 1391 




4 Hi 


588 


74 


2.76992 


0,91 


13.47 


3,62 


86 




86,98 




flO 


00 US 


l^t 








2.756O0 




M-97 


2,7a 


89 


91 


so.ts 


IS; 93 


oo 


ISO! 


2 561 




S60 


23 








2,90 




» 


92.» 


M| 98 


00 


I6«l 


306 


6 021 


M7 


01 


2.73S00 


1,11 


17.99 


3.23 




B3 


96,07 


IS M 




2 021 


3 1M 


filBi 














98 


90 


98.06 






zoai 


3 271 


5 361 




3» 








3.82 




SO 


101.06 


IS lOS 


00 


215 


3 301 


SMI 


610 


02 


3.708JB 


1,12 


52,46 


1.12 




87 




30;lO8 




2211,361 


9121 




05 








144 


107 


88 


107.05 


15 '111 


00 


228 |403i 


fl311 


480 


46 


2.68972 


1,82 


5S.11 


4 7B 


no 


84 


110,91 



TAPER 3 


INTERMEDIATE 


VALUES - 


- (ContiniiM) 




fi- 


2i 


« 


' ' ' 


• 


D 


logC 


- 


~ 


' 


' 


le 


liOO 


IH.OO 2 m 


4 151 


tm 


470.41 


3.68071 


1,73 


58 03 


5.13 


113 82 


tl3.«3 


IS 


117,00 S 411 


4 381 


710 


4M 




2.87180 








116 81 






so 


ISO MS Ml 


4 411 


730 


480 




2.88328 


1.98 


59 






119.77 


no 


01 




ua.ooliM 


4 54J 


760) 




89 












M 


13 00 


IIS 00', 3 03i 


oOSJ 


Sll) 


443 


18 


2,84658 


2.21 










U 


IM.UO 3 lOi 




8^1 


435 


00 








7.16,128 69128 


88 


30 


183 00. s m 




7S41 






2 830S7 


2 48 ■ 85 










a 


13i. 00 3 261 


SW! 


SIS) 


419 


58 




2 83 67 






134.61,134 


88 


HOO 




3 341 


051 


tm 


















84 






3 42) 


fla>| 


10021 








2.03 


70 




9.10 








30 


IMOtt 


3 50) 




10 281 




60 


2.80043 










48il43 


SO 


44 


147.00 




8 Mi 


1050) 


301 


98 


2.50328 


3.28 


73 


32 10 19 


148 


43 


146 


78 


1100 


tso.oo 


4 071 


7 07) 


1115 


388 


88 


2.S8823 


3.43 


74 


80 


10.71 


149 


37 


149 


78 


30 


IMDO 




7 40) 


12 06} 








3,79 


77 




11. SO 










1100 


IM.OO 


4 431 


SHI 


12 67) 


382 


sa 


2.55938 


4.17 




73 


13 30 


!61 






65 


30 


iss.oo 




8 4Bi 


13 SI) 






2.54870 






89 






94 


187 


6S 




174.0OJS2U 


9 261 


14 47) 








5,02 


88 








75 




60 


30 


180fl0'6 411 


loaii 


18 46 


333 


20 


2.62270 




89 


58 


17.78 










IS 00 


isa.oo, fion 


10 42i 


10 44) 




















les 


31 


30 




823 


1122) 


17 4H 




53 


2,60041 


6 40 


05 


46 


21.28 








20 


»00 


m.w 


B44I 


12 031 


18 48) 


308 


04 


2.48987 






39 




196 








K> 






12 481 


19 63) 


301 




2.4T9S0 


7.80 








201 






OS 


WOO 






13 30 


2100 






2.48988 


8.20 






27.25 


207 






78 


30 


iis.ott 


J53i 




22 08) 


288 


00 


2.48043 






12 










82 


noo 




817} 


IS 00! 


23 IS) 






















44 


30 




8 4St 


1S« 


24 30) 




07 




10.19 


112 




34. 2S 










BOD 


moo 


9 071 


18 381 


26 44) 


271 


75 


2]43417 


10.92 




77 


38.81 


220 




231 


00 


30 


240.00 


B331 


17 Mi 




288 


74 


2.42809 


11.60 


118 


83 


30,40 


234 


42 


237 


73 



100 



THE RAILROAD TAPER 



TAPER 3i. CURVATURE INCREASING 3*^00' 

EVERY 30 FT. A = 3 











Deflections to 




Transit 
at 












B.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 




3*00' 


6° 00' 


9*00' 


12° 00' 


15*00' 


18*00' 


21*00' 


24*00' 


I 





30 


60 


90 


120 


150 


180 


210 




o / 


e t 


o / 


O f 


o / 


/ 


e t 


e / 







27 


107i 


206 


3 22^ 


4 57 


6 49i 


900 


30 


di27 




54 


2 01i 


327 


5 10i 


7 12 


9 31i 


60 


134J 


54 




121 


2 55i 


4 48 


6 58i 


927 


90 


3 18 


2 28i 


121 




148 


3 49i 


609 


8 46i 


120 


5 37i 


4 39 


3 22i 


148 




2 15 


4 43i 


7 30 


150 


8 33 


7 25i 


600 


4f6§ 


2 15 


■ • • • 


242 


5 37i 


180 


12 04i 


10 48 


9 13i 


7 21 


5 10i 


242 


• • • • 


309 


210 


16 12 


14 46i 


13 03 


llOli 


8 42 


6 04J 


309 


• • a • 



M 


/ 





D 


logZ) 


i 


i 


z 


y 


2c 


e t 

600 


30 


O 1 

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 


524 


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 













TABLES 










lUl 


TAPER 3i. INTERMEDIATE 


VALUES 


A = 3 


M I 


/ 


6 





D 


logZ> 


i 


t 


X 


V 


Ic 


a / 






o 1 e / 














600 


30.00 27 


27 ' 54 


955.16 


2.98008 


0.12 


15.00 


0.24 


30.00 


30.00 


15 


32.50 


29} 


3U 


101 


916.98 


2.96236 


0.14 


16.25 


0.28 


32.50 


32.50 


30 


35.00 


32} 


35} 


108} 


881.75 


2.94535 


0.16 


17.50 


0.33 


35.00 


35.00 


45 


37.50 


35} 


40} 


116 


849.13 


2.92897 


0.18 


18.75 


0.39 


37.50 


37.50 


700 


40.00 


38} 


45} 


124 


818.84 


2.91320 


0.20 


20.00 


0.45 


40.00 


40.00 


15 


42.50 


4U 


50} 


132} 


790.65 


2.89798 


0.23 


21.25 


0.52 


42.50 


42.50 


30 


45.00 


45 


56} 


141} 


764.34 


2.88329 


0.26 


22.49 


0.59 


45.00 


45.00 


45 


47.50 


48} 


102 


150} 


739.73 


2.86907 


0.29 


23.74 


0.67 


47.49 


47.50 


800 


50.00 


52 


108 


200 


716.66 


2.85531 


0.32 


24.99 


0.76 


49.99 


50.00 


15 


52.60 


55} 


114} 


2 10 


694.99 


2.84198 


0.35 


26.24 


0.85 


52.49 


52.50 


30 


55.00 


59} 


120} 2 20} 


674.61 


2.82905 


0.39 


27.49 


0.95 


54.99 


55.00 


45 


57.50 


103} 


127} 


2 31 


655^0 


2.81651 


0.43 


28.74 


1.06 


57.49 


57.50 


900 


60.00 


107} 


134} 


242 


637.25 


2.80431 


0.47 


29.99j 1.18 


59.98 


60.00 


15 


62.50 


111} 


141} 


2 53} 


620.09 


2.79245 


0.51 


31.24 1.31 


62.48 


62.50 


30 65.00; 1 16 


149} 


3 05} 


603.85 


2.78093 


0.56 


32.48 


1.44 


64.98 


64.99 


45 


67.50; 1 20i 1 1 57 


3 17} 


588.44 


2.76970 


0.61 


33.73 


1.58 


67.48 


67.49 


10 00 


70.00' 125 


205 


330 


573.80 


2.75876 


0.66 


34.98 


1.73 


69.97 


69.99 


15 


72.50 


129} 


2 13} 


3 43 


559.88 


2.74810 


0.71 


36.23 


1.89 


72.47 


72.49 


30 


75.00 


134} 


2 21} 3 56} 


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} 


424 


521.96 


2.71764 


0.89 


39.97 


2.43 


79.95 


79.98 


15 


82.50 


149} 


2 48} 


4 38} 


510.46 


2.70796 


0.96 


41.22 


2.63 


82.44 


82.48 


30 


85.00 


155 


2 58} 


4 53} 


499.46 


2.69850 


1.03 


42.46 


2.84 


84.93 


84.98 


45 


87,50 


200} 


308 


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 


02.50 


211} 


3 28} 


540 


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 18 


451.03 


2.65421, 

1 


1.42 


48.69 


4.07 


97.37 


97.46 



102 THE RAILROAD TAPER 

TAPER 3{. INTERMEDIATE VALUES — (Cimtinued) A=3 



M 


I 


/ 


» 


- 





lojO 


- 


' 


. 


. 


"^ 


WOO 


100.00 


2 291 


4 00) 


630 


442 48 


264589 


1 ii 


49 64 


4.35 


99 86 


99,96 






60 


1351 




fl471 




28 




1,80 


51. tS 




102. K 


102 


46 


so 








4 231 


7 0Si 




38 


j! 82978 


1.70 




4H 


104 


82 


104 


94 








2 48) 




7 23) 




















a 


14 00 






2SS 










2.St437 






5.69 


109 


78 




92 


IS 


112 


» 


3 011 


4 601 


8 01 






2.80876 


2]oi 




5.94 










30 






3 08) 


Slli 


ssoi 












6,30 














3iS( 


5 24) 


840 


390 


95 






58.64 


6,67 


117 


20117 


39 


11 00 


120 


00 


3 22) 


S37) 


900 


384 


60 


2,68601 


2.36 


59.88 


7,06 


119 


67 


110 


88 


30 


I3S 


00 


3 37 


8 04) 


9 41) 


372 


53 


3.57116 


2,60 


62.38 


786 


124 


61 


124 


86 


IS 00 


130 


00 


3S2 


B32 


10 24 




28 2,65782 












12» 




30 






4 07) 


7 001 


UOSJ 












9.70 




45 




79 


17 00 






4 23} 


730) 




340 




2.53249 


3.45 


89.78 


10,70 


139 


34 


139 


76 


30 


H5 


00 


440 


a 01) 


12 411 


331 




2 62047 




73.25 


11,77 




22 






18 <M 














2.A0887 




74.71 




149 


10 




«t 


90 






5l4i 


oosi 


14 20) 




2,4B7S3 


4.47 


77.17 


14] 12 


163 


H 


164 


69 


19 00 


lao 


00 


5 32) 


9 39) 


1612 


308. IS 


2,4§878 


484 


79.63 


IS, 40 


168 


78 


159 


SI 


30 


Kt^DO 


S5l 


10 111 


ifloai 


2M.42 


a.4782B 


5.24 


92.08 


16.76 


1.3 69 


164 


a 


JOOO 


170 00[ 10 


10 so 


17 00 








s.«a 






168 39 


169 


38 






ooflsej 


11291 


17 601 








6.10 


S8,96 


19.70 






174 


27 






00 fi4fit 


12 041 






78 


2,44BS2 


6,56 


86,40 


21.29 


177 




179 


17 


30 


18S 




12 43i 


10 531 


273 




2,43782 


7,04 


91,83 


22,96 




62 




OS 


KOO 


IDO 






20 54 






2,42876 


7.6S 










188 


03 






00 7B2) 


14 031 


21681 








8.08 


66,67 


26 55 


191 


96 


193 


71 






00 8141 


14 4S) 


23 00 


m 


17 


2,4[1»1 


8.64 


99,06 












2*110 


aio 




IB 12 


25 12 


248 


Ma.3fl6l8 


6.3S 


iai.88 


32. S8 


306 


71 


208 


IS 



TABLES 



103 



TAPER 3i. CURVATURE INCREASING 4** 00' 

EVERY 30 FT. A = 4 









Deflections to 








Tnuuit 
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' 


1 





30 


60 


90 


120 


150 


180 




• / 


e / 


e / 


o / 


o / 


o / 


e / 







36 


130 


248 


4 30 


6 36 


906 


30 


36. 




1 12 


2 42 


4 36 


6 54 


9 36 


60 


206 


1 12 




148 


3 54 


6 24 


9 18 


90 


424 


•3 18 


148 




2 24 


506 


8 12 


120 


730 


612 


430 


2 24 




300 


6 18 


150 


1124 


9 54 


800 


5 42 


300 




3 36 


180 


1606 


14 24 


12 18 


9 48 


6 54 


3 36 





M 


I 





D 


logD 


d 


t 


X 


V 


Ic 


• 1 
800 


30 


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 



THE SAU^BOAD TAPES 

TAPEE 3i. INTERMEDIATE, VA^HES 



M 


I / ■ fr 


-. 


O 


-logo i 


': 




■ I 


soil 


ao.oo oas oSs 


°,j 


718 SO 


2 86632 


16 


a.* 31 


30 00 30 00 




31.SS, O3ailo40 


1181 


flwaa 


2.84187 


0.18 


«* 


0.36 


31 87* 31,87 




S3 75 04U.04fl 




074.42. 






to.n 




33,74] 33,76 






044i:0 4Bl 


1331 


fl66,» 








0.4B 




BOO 


37.eo 


0471 1 64 


1411 


637,02 


2.80416 


O.M 


WW 


0.S3 


37. so! 37,60 




31.3^ 


osinjiiao 


1491 






o.» 


19,88 




39,87 39,37 






0S3i jlM 


1571 


803.68 






ao-M 


0.64 41 24l 41:25 




*a'i2 


05B1 1 i OBJ 


2 061 




2.78949 


0.32 


21, M 


71 43 12! 43, U 


10 00 


4S.00 


100 


116 




S73.1B 


2.76893 


0.35 


12.46 


79 44 991 45 DO 


15 


Ifl.SS 


1031 


ISO] 


2 241 


666 55 


J 74784 


0.38 


22.43 


0,87 48,87 


48 87 






lOJ 


1201 


2 331 










CDS 48,74 




15 


Msa 


not 


1321 


!431 


833 


62 






as, 30 




50,81 


1100 


S2.50 


iwt 




2 531 








0.47 


,X 


113 , 6248 


52.50 








145) 


3 031 














30 


«.2S 


122 


132 


3H 


498 


08 


2.89808 




28.11 


1 34 58.22 






SS.12 




1581 


3 241 




40 


i.easso 


0.69 




08 !0 


68.12 


12 00 


mm 


130 




330 


478 


31 


J 87971 


0,63 


29.28; 1.57 


»,„ 


69.99 


IG 


81,88 


1341 


2 131 


3 47) 


488 SI 


2 87081 


0,87 


30 91 


1 69 


6184 


81 87 






13B! 


2 201 






2 86211 








63 71 




4S 


95.83 


1421 


2 281 






2.85368 


0.77 


32.79 


1.96 


85,59 


85 82 


13 OO 


87,50 


1471 












33.72 




67 46 


87.49 




BS.33 


1611 




4 351 




2 83702 








89.38 


89.38 




71, 2S 


ISBJ 


2 52 


4 481 




2.82898 


92 


35.59 


3 42 




71 23 






2 0U 


3 00! 


6 on 






0,97 


38 52 2,68 


7106 


73.11 


14 00 


76.00 


20B 


309 


613 


410,64 


2,81338 


1.03 


37.48 


"* 


74 93 


74.98 


15 


76,88 


2 11 


3 171 


5 281 


403,42 


3 80578 


1,09 


38 39 




78 79 


78 86 




78 7S 


216 


3 20) 


5 4Si 


306 58 


2.69831 








78.68 




4fi 


SO. 82 




3 351 


5 661 




1.23 


40 26 


3.31 


80.53 


80.69 






2 261 i 3 45 


fllll 


3S3 ,M 2 58380 








83,39 


S!4« 




S« 35 


237 14« 


fl"4l 


371 35 2 68978 




43,06 




88.12 




IB 00 


eooo 




712 




1.67 




130 


99 S4 


89.95 


M 




2 5IIJ I 4 Mi 












93.68 


93 B> 


17 00 


S7 10 


3111 


sn« 


8l7i 


33U23 


2.M(H9 


ISO 


48,66 


5,42 


97.27 


97.41 


30 


101,25 


3 23) 


6 28 


8 611 


3M79 


2,61824 


2,07 


50.61 


5,98 


100. B8 


101. IB 




105.00 


3 3B 


SSI 




320,60 


2 6063" 




S2,3: 




104.68 


104,89 


30 


108.75 




8141 


10 03) 


312, M 




2,48 




7^23 


108.38 


108,0 




112.50 41121 




10 411 


30t,6S 


2,48370 








112.07 


1U,H 






704 




267,07 


3.472S8 


2,89 


97.93 


8.63 


lis. 78 


118 07 


JO 00 


l»,004 3O 




li^ 


289,97 




3,13 






1U,42 


m,7B 


30 


123.75' 4 44) 


7 501 














123,08 


121 .SI 




127.60 4 5RI 


824 


13 231 


276.88 


3, 44221 




63.4! 




138.7* 


w-n 


30 


'■'"' 


614) 


8 52 


14 081 


270 


79 


3.43283 


3.91 


86 32 


11.96 


■»» 


lao.n 



A '■' 



TABLES 



105 



TAPER 3J. INTERMEDIATE VALUES — (Contwuerf) A =4 




b 





D 


O 1 


e / 




9 21 


14 51 


265.02 


95(H 


15 36§ 


259.53 


10 21 


16 23i 


254.33 


1124 


18 00 


244.63 


12 30 


19 41i 


235.82 


13 39 


2127 


227.81 


14 51 


23 m 


220.51 


16 06 


2512 


213.85 



logZ) 



42328 
41419 
40540 
38851 
37258 
35757 
34343 
33011 



4.19 
4.48 
4.80 
5.46 
6.18 
6.97 
7.81 
8.71 



67.16 
69.00 
70.83 
74.49 
78.14 
81.76 
85.37 
88.95 



12.90 
13.90 
14.94 
17.17 
19.61 
22.26 
25.14 
28.24 



134. 

137. 

141. 

148. 

155 

162 

169 

176 



01 
63 
23 
40 
50 
52 
45 
29 



Ic 



134.63 
138.33 
142.02 
149.39 
156.73 
164.04 
171.81 
178.52 



TAPER 3i. 



k — 



CURVATURE INCREASING 5° 00' . 
EVERY 30 FT. A = 5 







• 


Deflections to 






Tnuttit 












at 


B.C. 


C.C. 


C.C. 


C.C. 


C.C. 


C.C. 




6«00' 


10*00' 


15-00' 


20" 00' 


25" 00' 


30*00' 


\ 





30 


60 


90 


120 


150 




e 1 


e / 


e / 


e / 


o / 


e / 







045 


152} 


3 30 


5 37} 


8 15 


30 


045 




130 


3 22} 


5 45 


8 37} 


60 


2 371 


130 




2 15 


4 52} 


800 


90 


530 


4 07J 


2 15 




300 


6 22} 


120 


9 22} 


745 


5 37} 


300 




3 45 


150 


1415 


12 221 


10 00 


7 07} 


3 45 





U 


I 





D 


logD 


i 


i 


X 


y 


Ic 


1000 


30 


e / 

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 


2.459.39' 1.96 


44.88 


5.49 89.75 


89.92 


25 00 


120 


15 00 


233. 54i2. 36836 3.90 


59 . 66 


ll.73119.09jll9.67 


30 00 


150 


22 30 


198.3412.29741 

! 


6.81 


74.21 


21..39 147. .50 149.04 



106 THE BAILROAD TAPES 

TAPER 3|. INTERMEDIATE VALUES A=5 



« 


. 


/ 


h 


" 


D 


logo 


d 


' 


• 


- 


' 


lOflO 


30.00 


Hi 


45 


1°30 


m.M 


2.76841 


020 


14 M 


0.30 


30 00 


30.00 


IS 




471 


01» 


ISfli 




38 


2.74771 


0.21 


is! 74 


0.43 


31 SO 




50 


30 




osoi 


531 
















00 




M.M 


631 




1611 






2.72708 










1100 






I03i 


tMl 


621 


34 


2.71712 


0.27 


17. M 




30.00 


38 


00 




37.5( 


59i 


107i 


2 00) 




















30 






112i 


2 14) 


m 




2 89787 






0.71 


38,W 


39 


00 






lOSl 


1171 


2S2| 






2.8SSS8 




20 24 


77 




40 




12 00 


«,00 


105) 


122i 




478 


05 


2.87047 


0.37 


20.00 


0.84 


41 09 


42 


00 


15 


43.51 


llli 


12S 


2 301 


468 


34 


2,87058 


0.40 


21.74 


0.01 


43,40 


43 


50 


SO 


45 (» 




1 33! 1 2 *»i 














44. et 




oo 


« 


48.50 


1181 


139i 2 571 


410 


07 




0.48 


23.23 


I.0« 


48 49 




JO 


13 CO 


4S.fla 




145i 


3m 














17,98 


48 


00 




4»,S0 


[m 


ISli 


3 15) 
















40 




30 


SI. 0(1 




1571 


3M) 


12S 






6.55 


25.48 


1.3! 


50.98 




00 






1321 


2 031 


339} 










28.23 




52. 4« 




90 


UOO 


ai.oo 


136i 


a 101 


3 4U1 


410 


13 


2.eiM2 


0.82 


"■" 


1.62 


53.07 


63 


90 


It 


55 50 


I4ni 


Slfli 


3 571 


402.00 


,««, 


0.88 


27 72 


1.82 


6S.47 


SS 


49 


30 


57 00 1 44i 


a 231 




3flB 11 


!..«.. 




2...7 




56,97 


56 








2 30) 


4IM 


350 40 


,mm 


0.74 


2.22 










IS 00 


60,00 1 52i 


2 371 






2.9^323 




2.,.7 


l.W 


ie.9« 






30 


"T" 


2ia 


153 


•»■- 


■■-" 


0.87 


■■" 


2.21 


62.06 


62 


M 



TABLES 107 

INTERMEDIATE VALUES — (Coniinued) i=5 



w 


• 


/ 


b 


- 





lcj«D 


i 


. 




^ 


Ic 


UOD 


«.oo 


3001 


307 


ilflj 


350.35 


a,5BBJ2 


0.06 


32 96 


3 48 


84.44 


64,»» 


10 


w.oo 


2m 




5 41) 






IM 


31^45 




«S 


92 


OS 


aa 


17 0) 


7S.0O 


am 


3 301 


son 








35.94 














71.00 


3 871 


3M1 


6 3Ji 


329.00 


3.51720 








88 






ano 


78.00 


!«i 


1131 


7 011 






MO 


38^92 


3:79J 77 


88 


77 


08 


» 


81.00 


!SJi 


132 


7 2S1 


311 57 2*8350 






4,18 80 


81 




OS 


MOO 


H.OO 


300 


itoi 


7 5»i 




1.07 


41.90 


4.69' 83 


81 


83 


91 


w 


87.00 


Il> 


SIO 










5.03' 86 




88 




MOO 


00.00 


130 


130 


900 


2SS,80 


2.46000 


106 


44,88 


.,..:» 


75 


BS 


92 


to 


n.oD 


S*U 


6S0i 


9 32 


281.98 


1. 15022 


2.12 


46.36 


J. „ 


71 


02 


91 




BB.OO 




SUi 


10 Mi 


37S,50 






47,85 


«,so| as 






SO 


to 


W.OD 


405 


033) 


10 381 




a:i302( 






7DS M 




08 


87 


800 


in 00 


*17J 


«t6) 


iiiai 










7.»|lGl 






85 


to 




110 


718) 


11181 


257,87 
















BOD 


lOB.OO 


143 


712) 


12 251 








MJfl 


Sim 




107 


7S 


MOO 




too) 


8 31) 


1310] 








ib:i2 


lOiJlU 


2) 






itoo 


IIO.OO 


»I7) 


B39) 


15 00 


233,54 


2.36830 


3.90 


69,86 


"■"■'• 


09 


119 


67 


aoo 


lie. 00 


SOS) 


mm 


lfi22] 


225.24 


2.3S205 


1.40 


62.59 


13 37' 124 


87 


125 


f> 


noo 


112.00 


37 


urn 


17101 


!17.»1 




4.94 


85,51 








49 


soo 


mM 


708) 


13101 


19191 


210,58 








17'o7 13t 


29 


137 


30 


SOD 






Hill 


20 521 






S.tl 


71,32 










uoo 


wo.oo 


Sit 


14 IS 


23 30 


198.34 


2 ami 


8 81 


74.21 


21,39|l47 


50 


14B 


01 



110 



THE RAILROAD TAPER 



■o 



CO 



C4 



eo 



eo 



eo 



eo 



Si 

T 



> 

H 
<1 
P^ 

PUi 

§ 
O 
O 



04 



CS| 



Oo 

o 



eOOo 
CO Q 



1-^ 



CO o 

o 



'^ o 

O 



C4 



82^g 82 






12® 

'T' o 

o 



82g 8S^ 8^55 

SOo lOO*"^ QOo M5 

^ ^ o O ^ CO 



o 



04 



82^ 8S^ 



o 

O 



oo 

o 



8^^ 

o 



n\^ 



8S 

t^ o 



t^ o 



00 o 



Oi 



CS| 



CO ^ 
b* o 



c^oco 

00 o 



855^^ 



O o 






o 



»0 CO^ 

rH o 



o o ^ 

1-1 o 



8g^ 

SOo 



. CO 
CO c^ 



QO»^ 

^ o 

1-1 CS| 






o 

CO 



8S^ 



o 
CO 



82g 8^3? 8§a 



8- 



o 

CO 



1/5 1-H W 

O o 

i-H CO 



Oi-Ho 



8* 



SS 



o 



QCSlo 

25 »o 



8^^ 



CS|0 
o 
CO 



CO 5Q 






8 



QCOo 



I 



■*-"« O '-»»'^ O "-o*^ O '«o'« O '-»»'^ O «*»'tJ ^ 



8 

o 
CO 



o 
CO 



o 



1— > 
o 



o 



o 



TABLES 



111 







s? 






^ 


^ 






eo 


00 


. . . . . . . . ^__j 




s? 


CO 


S21g 82g 83*^ 82.; 82* SS* 

s®^ 1?°^ s°r §§=»? ^^o-" ^°r 


s 


e« 


04 


855*. {28^ SSi* SSfe 8Sa S5!«: 

tH 1-4 1-4 C^ 


1 

P 

< 
> 




a- 


8§?fi 8^3? 8«g 8S^ 8Sg 8^2 

pOo JOO»^ QOo i20^ OOo QOo 
1-1 C^ '-' 


•*• 


ssr 


8^S SS;^ 8S3? 8g* 8^^ 8=:ij? 


> 

i 


»< 


ea 


8^8 SS^ 83^ S53i^ 8S;g 8J2™; 

OOo i^O^ lOi-^"^ C^i-JO Q1-J0 iQi-iC^ 
WcoNo COo ^0 »O^COo 

1-" i-^co'~'co'~^'^'^ i-iio 


8 


■»• 




8^2 8S*. 88* 8J:i« 8^*2 82* 

Oi-"o Q'-<'^ Oi-4'-< QC^o OC^o OCOC^ 




•«• 


v4 


8S;^ 8K^ 8^i2 SSg- 85s S-l? 

OCOo iCCO^ Q"^o iO-^O 0»0o QI^o 
fe«o2So SOOSo g2<3^^^ 




•«• 


HN 






<I 


1 


i***^ -.a-Q ".o*^ — '^ '-^'^ '— '^ 






ai 


^ V. •. ^ ^ •■ 

2 w -^ 8 S 
000000 

10 - to ^ >0 CO CO 



112 



THE RAILROAD TAPER 





•o 














8S~S 






































CO rH 








































•^ 


CO 


. 




822 




1^0'* 

CO 


8S?2 

tCOo 




CO 


CO 




8^^ 

10 d'* 
^ 

r-t 


8"S 

ddo 


8^3? 

lodc^ 
10 


ddo 


SSg 

OOo 

t^ CO 




CM 


CO 




ddo 

(M 


^ ^ 


8S^ 

(NdO 
t>- 

CO 


060^ 
t^ 

CO 


05 -^ 




W 


ON 
CM 


lOO^O 

I> 


00 
CO 


8Sg 

OOo 
Oi CO 


t^do 

Oi 


8^^ 


8^8 


o 

W 
P 

< 
> 




CM 


OOo 
^5 CO 




CO U3 


01-H0 

2 ^ 


88~§ 

Q(No 

r- 1 


8Sg 

p(No 


>4« 


>4« 
CM 


CO 


8?2^ 

O1-4 W 

»o 


«o 


174.00 

2.51 

7° 23J' 


00 

^ 00 


'^o 


> 

l-H 

H 
< 


tH 


CM 


8S2 


195.00 
2.71 

7° 18J' 


CO 

55 00 


8S«? 

icco^o 
8 1:, 


8f='^ 

^ 


8^1 

p«Oo 

S5 2 


< 

PL, 

O 

O 


MM 
•44 


»-1 


8S'§ 

OCOo 


drt^O 
t^ 
(N 


880 

OOo 


oog' 
di^-^'^ 

r-i 
CO CO 


OOo 
CO ,-4 


8Sg 

Oi-Ho 

CO y-{ 




«-4 
He* 


8§g 

QOOo 
^ CO 

CO _4 


420.00 

10.97 

15° 45' 
























































































1 




























. 


















<1 




^~s 


-^-t: C 


-^'^ 


r-.^'^ 


^'^ 


.^•^3 






" 4 


8 







8 


00 




00 


8 




s 



s 



TABLSS 



113 



00 



CO 



0*^5 
CO o 



53 



O CO 
o 



OQOO 
^1 o 
CO 



. CO 
»0 o 



Oo 

CO 



CD o 



Oo 



. CO 






s 



o 
CO 



>0 •-<o 



Q 1-Ho Q 1-Ho O 

Oi irt O CO '-^ 



. CO 

SOo 



(Ni-H'-' 
00 o 






o 



ggl, g^Hj. 8E;£j 



8" 



o 



8gg 8S?8 Sfeg; 

Oi-io Q^o S^o 

t^ g oi ?5 o 



CI 



eo 



I 



T 

I 



e« 



e4 



C^ 1-4 CO 

O o 



^ 






O 

CD 



cbc^'-' 

C^ o 

'^ 00 



CO o 



8^8 

S(No 



>OC0 



o 

o 



oo<nco 

CO o 



. CO 

95 <N 



8^2 8!^ 



OCOo 






lO »o ^ 



OcDo 

« 2 



e« 



oi 






e^ 



8S|5 8 



CO o 

S 2 



O ^ o 

53 2 



?5 S 



8S5e 8Ss 8S5« 



So9 

. CO 



E2 S 



O C3 o 

o 



8 



o 



^ to 



883r 

00 o 

c^ 00 



oco iQ 

S'l-i^ 
CO »-t 



8 



QcOo 



8S2 

. CO 

OOo 



CO 2 



8S^5 

Qr-to 

CO _4 



Q -^ o 
CO c5 



O «-<o 



I 



.'Q O -o*^ O '-i'^ C> — '^ O --»'X3 O --i "t: C; 



o 



o 

CO 



o 



o 



o 

CD 



112 



THE RAILROAD TAPER 



M3 



ec 



OOo 
CO ,-1 



CO 



82 



<M 



OOo 

CO ^ 



CO O o 



O rt<^, 
00 o 



8 



S§2 

»C Oo 
-^ CI 



CO 



CO 






8 



»o o "^ 

■^ o 



cog 

OOo 



05^ 
CO 



too 

»0 o 



Oo 



8 



OOo 
t^ CO 



c^ 



eo 



8 



. CO 
Tf O*^ 

»0 o 



Oo 



go CO 
o 



8§^ 

t^ o 

CO 



8g?s- 8 



goo 



o 

CO 



80o 



1 

o 



CO 

W 
P 

l-H 

H 

PL, 

o 

o 



C4 



CS| 



ifi^CO 
l> o 



C^O® 
00 o 

CO 



. CO 

gOo 
CO 






05J5;* 



i>o 



s 



He* 



o 



8 CO 

1—1 _u 



8!o8 



S 



«o 



CS| 



CSI 



c< 



OOo 
!~I CO 



S-2 

. CO 
C^ ^ ro 



S^a SKe; sr^ 



O »-to 
CO 1/5 



§•" 



o 



S(No 
CO 



. CO 
0(No 



ggg, 8S*. 88"- 8Ss- 88~- 8Sg 

gOi-H"^ Q1-4CO C^C^C^ rJ<C^<N OC^^ O'^o 

COo lOo CDo t^o OOo '"^O 
'-^'^'-<iO'~^<O'~^t>-'~^00^'-t 

SSSo 8i^s« 8S'^ 8§«^ 8S'^ 8^g 

(Mo »OC^'-' OCOo lOCOW Q"^o OOo 

8^'^ 8Sh. 88^ 8S* 85!s 8Sg 

OCOo O"^*^ Q»Oo Ot^'"' 0060 O1-H0 
<M°^(NOC>1^:;COCOCOSCOS 

g» 8Sa : : : ■ : ^ ^ — : ■ ■ 

.CO . . ^ ... ... ... ... 

OOo OOo ... ... ... ... 

CO^"^^ ... ... ... ... 






^-^2 O 



-t: O — '^ C> 



'« O 



'td o 



•« o 



8 



CO 


8 


g 


8 


8 




















I>- 


l>. 


00 


00 


Oi 






TABLES 



113 



8^3 

CQ o 



53 



o 



OQOO 
^1 o 

CO 






»0 o 

CO 



Oo 



^3? 



o 



ss*^ 






»0 o 



Oo 

CO 



Soo; 



\^^ 8S^ SSh; 8&2 

^ o t* lo 00 o Oi 



o 



CO 



SSg; 822'si 83g SSg SJSg § 

O i-Ho 



Oo 



81-10 Q »-to O 

ift O CO '"' 



o 



0(No 
?3 Oi 



C^o 

O 



o 

T 



I 



CI 



CO 



8 



1-4 w 
o 



8Kh- 8^i« 8 



o 
CO 



C^ o 

'-' 00 



(MCO 
o 



SCOo 






CI 



C4 



CO o 



8?' 



S(No 



O" 

o 

O 



O "^o 
00 c< 



8^^? 8 



»oco 






OcOo 



C4 



COo 

S 2 



8 



O '^o 






8 



. CO 

r^o 



»o 



8S 

O C3 o 



O «-<o 



oi 



8 



o 






883r 

00 o 
C^ 00 



O CO ijO 

8" r-i^ 
1— 10 

CO r-t 



^^ 



8 



O CO o 



ci 



. CO 

8^?o 

CO S 



8S^5 



CO '--• 

CO ;i; 



o 

05 



Q "^ o 
CO c5 



<1 


1 




39 



TS O 



•^ O 



'^ C> r— -^ C> 



'^ O --^'^ o 



*>* 


*>* 


%.. 


**. 


•». 


8 


8 


8 


8 


8 







CO 










CO 



114 



THE BAILBOAD TAPER 



7S 



8S^ 8§8 8Ss^ 888 



l:^ o 



S-lft s 



c^ 



s 



s^s. 



Q«Do 

2 gj 



CO 



8S|; 828 {2^3^ SSh; 






SeOo 



g? 



O 



\fi o 



CO 



eo 



§"2 S 



888 






Oi o 

1-4 1-4 



c< 



CO 



^1* SSg 



o 



OOOo 

'-^ 1-4 






I 

•tea 

o 



w 



en 



C4 



04 



. CO 
Q Od o 
S 1-4 



8Sog 

p<No 



> 
> 

S 

o 
u 



99 



C< 



P4 

I 



'tt O '•o'XJ O -*'« O -o'Q O -^'^ o 



:^ 



o 

00 



o o 



o 

3 



8 

o 



ALIGNHENT REVISION 

MAINTAINING LENGTH UNCHANGED 

(See page 65, par. 43) 



u 


I'oo' 


1-30- 


2-.0' 


Tipw. 


\ 


i 


^ 


— r 




' 


' 


t 


«' 


J 


' 


«■ 


' 


A 


in- 


' 


' 


.»■ 






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LOGARITHMS OF NUMBERS 



JJ 


o 


12346 67 89| 


100 


00000 00043 00087 00130 00173 00217 00260 00303 00S46 00389 | 




0432 


0476 0518 0661 0604 0647 0689 0732 077S 0817 




0800 


0903 0916 0988 1030 1072 lllS 1167 1199 1242 




1284 


1326 1368 1410 1452 1404 1536 1678 1620 1662 




1703 


1746 1787 1828 1870 1912 1963 1995 2036 2078 




2119 


2160 2202 2243 2284 2325 2366 2407 2449 2490 




2531 


2572 2012 2653 26S4 273i 2776 2818 2857 2898 






2979 3019 3060 3100 3141 3181 3222 3262 8302 




3342 


3383 3423 3463 3503 3543 3683 3623 3663 3703 




3743 


3782 3822 3862 3902 3941 3981 4021 4060 4100 


110 


04139 041T9 04218 04258 04297 0433S 04376 044I£ 04454 04493 | 




4532 


4671 4610 4650 4689 4727 4766 4805 4844 4883 




41)22 


4961 4999 5038 5077 5115 5154 5192 5231 5260 




6308 


6346 5385 5423 5461 5500 6538 5676 5614 5662 




60eo 


6729 5767 6805 6843 5881 6018 6956 5084 6032 




6070 


6108 6145 6183 6221 6258 6296 6333 6371 6408 




6446 


6483 6521 6558 6505 6633 6070 6707 6744 6731 




0819 


6856 6893 6930 6967 7004 7041 7078 7II5 7151 




7188 


7225 7262 7298 7335 7372 7408 7445 7482 7518 




7555 


7591 7028 7664 7700 7737 7773 7809 7846 7882 


ISO 


07918 07054 07990 08027 08063 08099 08135 08171 08207 08243 | 




8279 


8314 8350 8386 8422 8468 8493 8529 8565 8600 




8636 


8672 8707 8743 8778 8814 8849 8884 8920 8955 




8001 


9026 9061 9096 9132 9167 0202 0237 9272 9;J07 




9342 


0377 0412 0447 9482 9517 9552 9587 9621 9658 




9691 


9726 9760 9796 9830 9884 9890 0034 9968 10003 




10037 10072 lOlOfl 10140 1017S 10209 10243 10278 10312 0346 | 




0380 


0415 044fi 0483 0517 0551 0586 0619 0663 0687 




0721 


0755 0789 0823 0857 08BO 0924 0858 0992 1025 




1059 


1093 1126 1160 1193 1227 1281 1294 1327 1361 


ISO 


11304 11428 11461 11494 11528 I166I 11594 11628 11861 11604 | 




1727 


1760 1703 1826 1860 1893 1926 1969 1992 2024 




2057 


2090 2123 2156 2189 2<i22 2264 2287 2320 2362 




2386 


2418 2450 2483 2516 2548 2581 2613 2646 2678 




2710 


2743 2775 2808 2840 2872 2905 2937 2969 3001 




3033 


3066 3098 3130 3162 3194 3226 3258 3200 3322 




3354 


3386 3418 3450 3481 3613 3546 3577 3809 3640 




3672 


3704 3735 3767 3799 3830 3882 3803 3925 3056 




3988 


4019 4051 4083 4114 414S 4176 4208 4239 4270 




4301 


4333 4364 4396 4426 4457 4489 4520 4551 4582 


110 


14613 


4644 14675 14T06 14737 14768 14799 14829 14860 14801 




4922 


4053 4983 5014 6045 6076 5106 6137 5168 6198 




6220 


8259 5290 5320 6351 5381 6412 5442 5473 &m 




5534 


5664 5504 5625 6855 5685 5715 6746 5776 5806 




6836 


6866 6807 5027 5967 5087 6017 6047 6077 8107 




6137 


6187 6197 6227 6256 6286 6316 6346 6378 6406 




6435 


6465 6495 6624 6664 6584 8613 6843 6673 6702 




6733 


6761 6701 8820 6850 6879 6009 6038 6067 6097 




7026 


7066 7085 7114 7143 7173 7202 7231 7280 7289 




7819 


7348 7377 7406 7435 7464 7493 7522 7561 7580 


160 


17600 1763B 17607 17606 17726 17754 17782 17811 17B40 17860 f 



LOGARITHMS OF NUMBERS 



N 


0123456789 


160^ 


17000 17638 17807 17696 1772S 17754 17782 17811 17840 17809 




7898 7026 7956 7084 8013 8041 8070 8090 8127 8168 




8184 8213 8241 8270 8298 8327 8355 8384 8412 8441 




8480 84B8 8526 8664 8583 8611 8639 8067 8906 8724 




8762 8780 8808 8837 8805 8893 8B2I 8049 8977 9005 




0033 9061 0080 9117 0146 0173 9201 0229 9257 9285 




0312 9310 9308 9306 9424 0451 9470 0507 9535 0502 




0590 9618 9645 0973 0700 9728 0756 0788 9811 0838 




0866 9803 9021 9048 S976 20003 20030 20058 20085 20112 




20140 20167 20194 20222 20249 0279 0303 0330 0358 0365 


160 


20412 20439 20466 20403 20520 20548 20575 20602 20620 20659 




0983 0710 0737 0763 0700 0817 0844 0871 OHOS 0025 




0062 0978 1005 1032 1059 1085 1112 1139 1195 1192 




1219 1246 1272 1290 1325 1352 1378 1405 1431 1458 




1484 1511 1637 1564 1590 1617 1043 1669 1090 1722 




1748 1775 1801 1827 1864: 1880 1006 1932 1968 1085 




2011 2037 2063 2089 2115 2141 2167 2194 2220 2249 




2272 2298 2321 2350 2379 2401 2427 2453 2479 2605 




2531 2667 2583 2608 2034 2660 2686 2712 2737 2763 




2789 2814 2840 2866 2891 2917 2943 2968 2904 3019 


ISO 


23045 23070 23099 23121 23147 23172 23108 23223 23240 23274 




3300 3326 3360 3379 3401 342Q 3452 3177 3602 3528 




3653 3578 3603 3629 3654 3679 3704 8729 3754 3779 




3805 3830 3855 3880 3905 3030 3055 3980 4005 4030 




4065 1080 4105 4130 4165 4180 4204 4229 4264 4279 




4304 1329 4363 4378 4103 4428 4452 4477 4502 1527 




1551 4579 4901 1625 46S0 1674 4999 4724 1718 1773 




4707 4822 4849 4871 4805 4920 4944 4909 4903 5018 




5042 5066 5091 5116 5130 6164 6188 5212 6237 5261 




6285 6310 6334 6358 5382 5106 5131 5465 6470 5503 


180 


25527 25551 25675 26600 25624 25648 S5672 25606 25720 25714 




6798 6792 6819 5810 5864 5888 6012 5935 5059 5983 




6007 0031 6065 6079 6102 6126 6160 6174 9198 6221 




6245 9269 6293 9316 6340 6364 6387 6411 6435 6458 




6482 6505 6620 6653 6576 6600 6023 6617 6670 6694 




6717 9741 6764 6788 6811 8834 6888 6881 6905 6928 




8951 0075 6998 7021 7045 7068 7091 7114 7138 7161 




7184 7207 7231 7254 7277 7:»0 7323 7316 7370 7393 




7419 7130 7182 7485 7508 7631 7564 7577 7600 7023 




7646 7669 7692 7715 7738 7761 7784 7807 7830 7862 


190 


27875 27808 27921 27044 27967 27989 28012 28036 28058 28081 




8103 8128 8149 8171 8104 8217 8240 8262 8285 8307 


2 ' 8330 8353 8375 8398 8421 8443 84(ie 8489 8511 8533 


a , 8658 8578 8601 8023 8(i46 8988 8691 8713 8736 8758 


4 ; 8780 8B03 8825 8847 8870 8892 8914 8937 8950 8081 


fi , 9003 0026 9048 0070 9092 911S 9137 0169 9181 9203 


6 1 0226 9248 0270 0292 R3U 9336 9358 0380 9403 0425 


7 9447 G4fl9 9491 0513 &m 9557 9570 9C01 0823 0645 


' */ m(f7 0688 6710 S732 0754 9776 9708 9820 0842 0863 


*/ Mtfii 0907 9929 9951 9973 ««ft4 30016 SMiaft aswi 'JWViA. 


90/st 


7ja3 3012S 30146 30168 30190 30211 3oaaa •m^'i'smt. was* 





LOGAUITHMS OF NUMBERS 119 


N 


1 2 3 4 fi 6 


7 8 9 1 


200 


30103 3012S 30146 30168 30100 S02U 30233 3025; 30270 30298 | 




0329 0341 0363 0384 0403 0428 D449 


0471 0492 0614 




06S6 0667 0578 0800 0021 0043 0664 


0685 0707 0728 




07^0 0771 0702 0814 083G 0860 0S78 


0899 0920 0942 




0963 0084 10O6 1027 1048 1060 1001 


1112 1133 1154 




1176 1107 1218 1239 1260 1281 1302 


1323 1345 1366 




1387 1408 1420 14j0 1471 1402 1613 


1634 1556 1576 




1687 1618 1630 1660 1681 1702 1723 


1744 176i 1786 




1806 1827 1849 1869 1890 1911 1931 


1952 1073 1094 




201S 2036 2056 2077 2008 2118 2139 


2160 2181 2201 


210 


S2223 32243 32263 32284 32305 32325 32346 


32366 32387 32408 




2428 2449 2460 2400 2510 2531 2552 


2572 2693 2813 




2634 2664 2675 289^ 2715 2736 2756 


2777 2707 2818 




SS38 2858 2879 2890 2019 2040 2060 


2980 3001 3021 




8041 3062 3082 3102 3122 3143 8163 


3183 3203 32-M 




8244 M64 3284 3304 3325 3345 3365 


3385 3405 3425 




3445 3466 3480 3508 8520 3540 3666 


3586 3606 3626 




8S46 3666 3686 3706 3726 3749 3766 


3786 3806 3826 




3840 3866 3885 3006 3925 3945 3965 


3985 400S 4025 




4044 4084 4084 4104 4124 4143 4163 


4183 4203 4223 


320 


S4242 34282 34283 34301 34321 34341 34361 34380 34400 34420 | 




4430 4459 4470 4498 4^.18 4537 4557 


4577 4506 4616 




4635 4655 4874 4694 4713 4733 4753 


4772 4792 4811 




4830 4850 4869 4889 4908 4928 4947 


4967 4980 5006 




6025 6044 5064 5083 5102 5122 5141 


5100 6180 5199 




6218 6238 5267 6276 6295 5315 5334 


6353 6372 5392 




5411 6430 5449 5468 5488 5507 5520 


6645 6564 6683 




6603 6622 5641 6660 5679 6698 6717 


5736 6755 6774 




6793 6813 6833 6851 5870 6889 5908 


5927 6046 5965 




6984 a003 6021 6040 0059 6078 6097 


6116 6135 0164 


2tO 


36173 36102 36311 36229 36248 30267 36286 30305 36324 30342 | 




0361 6380 6390 6418 6436 6455 6474 


0493 6511 0530 




6649 6568 6586 OOOS 6624 0642 6661 


6680 6098 6717 




6736 6754 6773 6791 6810 6829 6847 


6866 68B4 0903 




6022 6940 6059 6977 6900 7014 7033 


7061 7070 7088 




7107 7125 7144 7162 7181 71B9 7218 


7230 7264 7273 




7201 7310 7328 7340 7365 7383 7401 


7420 7438 7457 




7475 7403 7611 7530 7548 7586 7585 


7603 7621 7039 




7658 7676 7004 7712 7731 7749 7707 


7786 7803 7822 




7840 7868 7876 7804 7012 7031 7949 


7667 7985 8003 


240 


38021 38039 38067 38076 38003 38112 38130 38148 38106 38184 1 




8202 8220 8238 8250 8274 8202 8310 


8328 8340 8364 




8382 8300 8417 8435 8453 8471 8480 


8507 8525 8543 




8561 8673 8500 8611 8632 8050 8668 


8686 8703 8721 




8739 8757 8775 8792 8810 8828 8846 


8803 8881 8809 




8917 8084- 8952 8970 8987 9005 9023 


9041 9068 9076 




D0&4 9111 9129 9140 0104 9182 9100 


9217 9235 9262 




9270 9287 0305 9322 0340 9358 9375 


9303 9410 9428 




9446 9463 0480 0498 9515 9533 9550 


0608 9586 0602 




-9820 9637 9665 9672 0(190 9707 9724 


9742 0759 9777 


UO 


891H 89811 89829 39846 39863 39881 39808 39016 39933 39960 | 



120 


LOGARITHMS OF NUMBERS 


N 


0123456789 


250 


39794 39811 39829 39846 39863 39881 39898 39915 39933 39960 


1 


9967 9985 40002 40019 40037 40054 40071 40088 40106 40123 


8 


40140 40157 0175 0192 0209 0226 0243 0261 0278 029i 


8 


0312 0329 0346 0364 0381 0398 0415 0432 0449 0466 


4 


0483 0500 0518 0535 0552 0569 0586 0603 0620 0637 


6 


0654 0671 0688 0705 0722 0739 0756 0773 0790 0807 


6 


0824 0841 0858 0875 0892 0909 0926 0943 0960 0976 


7 


0993 1010 1027 1044 1061 1078 1095 HH 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 


8 


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 2259 2275 2292 2308 


5 


.2325 2341 2357 2374 2390 2406 2423 2439 2455 2472 


6 


2488 2504 2521 2537 2553 2570 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 3056 3072 3088 3104 3120 


270 


43136 43152 43169 43185 43201 43217 43233 43249 43265 43281 


1 


3297 3313 3329 3345 3361 3377 3393 3409 3425 3441 


2 


3437 3473 3489 3505 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 394^ 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 44856 


1 


4871 4886 4902 4917 4932 4948 4963 4979 4994 5010 


2 


5025 5040 5056 5071 5086 5102 6117 5133 5148 6163 


8 


5179 5194 5209 5225 5240 5255 5271 6286 5301 6317 


4 


5332 5347 5362 6378 5393 6408 6423 6439 6464 6469 


5 


5484 6500 5515 6530 5545 5561 6576 5591 6606 6621 


6 


5637 5652 5667 5682 5697 6712 6728 6743 6758 6773 


7 


5788 5803 5818 5834 6849 6864 5879 6894 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 46359 46374 


1 


6389 6404 6419 6434 6449 6464 6479 6494 6509 6623 


8 


6538 6553 6568 6583 6598 6613 6627 6642 6657 6672 


8 


6687 6702 6716 6731 6746 6761 6776 6790 6806 6820 


4 


6835 6850 6864 6879 6894 6909 6923 6938 6963 6967 


6 


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 



S/ 7422 7436 7451 7465 7480 7494 7609 7524 7638 7653 
P/ 7667 7682 7596 7611 7625 7640 1^54 IW'i l^'^^ im% 

^800 47712 47727 47741 47766 47770 477%4 477 W ^I^V^ Vl«i& V\^i 





LOGARITHMS OF NUMBERS 121 


ir 


0123456789 


•00 


47712 47727 47741 47766 47770 47784 47799 47813 47828 47842 




7867 7871 7886 7900 7914 7929 7943 7968 7972 7986 




8001 8016 8029 8044 8068 8073 8087 8101 8116 8130 




8144 8169 8173 8187 8202 8216 8230 8244 8269 8273 




8287 8302 8316 8330 8344 8369 8373 8387 8401 8416 




8430 8444 8468 8473 8487 8601 8616 8630 8644 8668 




8572 8586 8601 861^ 8629 8643 8667 8671 8686 8700 




8714 8728 8742 8766 8770 8785 8799 8813 8827 8841 




8855 8869 8883 8897 8911 8926 8940 8964 8968 8982 




8906 9010 9024 9038 9062 9066 9080 9094 9108 9122 


810 


49186 49150 49164 49178 49192 49206 49220 49234 49248 49262 




9276 9290 9304 9318 9332 9346 9360 9374 9388 9402 




9415 9429 9443 9467 9471 9486 9499 9613 9627 9641 




9554 9568 9682 9696 9610 9624 9638 9661 9666 9679 




9693 9707 9721 9734 9748 9762 9776 9790 9803 9817 




9881 .9845 9869 9872 9886 9900 9914 9927 9941 9965 




9969 9982 9996 60010 60024 60037 60051 6006i 60079 60092 




60106 50120 60133 0147 0161 0174 0188 0202 0216 0229 




0843 0266 0270 0284 0297 0311 0325 0338 0362 0366 




0379 0393 0406 0420 0433 0447 0461 0474 0488 0601 


820 


60515 50529 60642 60666 60609 60683 60696 60610 60623 60637 




0651 0664 0678 0691 0705 0718 0732 0745 0769 0772 




0786 0799 0813 0826 0840 0853 0866 0880 0893 0907 




0920 0934 0947 0961 0974 0987 1001 1014 1028 1041 




1055 1068 1081 1095 1108 1121 1135 1148 1162 1175 




1188 1202 1216 1228 1242 1256 1268 1282 1296 1308 




1822 1836 '1348 1362 1376 1388 1402 1415 1428 1441 




1465 1468 1481 1495 1608 1621 1534 1548 1561 1574 




1587 1601 1614 1627 1640 1654 1667 1680 1693 1706 




1720 1733 1746 1769 1772 1786 1799 1812 1825 1838 


880 


61851 61865 61878 61891 61904 61917 61930 61943 61967 61970 


T^ 


1983 1996 2009 2022 2036 2048 2061 2075 2088 2101 




2114 2127 2140 2153 216.0 2179 2192 2205 2218 2231 




2244 2267 2270 2284 2297 2310 2323 2336 2349 2362 




2375 2388 2401 2414 2427 2440 2453 2466 2479 2492 




2504 2617 2630 2643 2556 2569 2582 2595 2608 2621 




2634 2647 2660 2673 2680 2099 2711 2724 2737 2750 




2763 2776 2789 2802 .2815 2827 2840 2853 2866 2879 




2892 2905 2917 2930 2943 2950 2969 2982 2994 3007 




8020 8a33 3046 3068 3071 3084 3097 3110 3122 3135 


840 


53148 58161 63173 63186 63199 53212 53224 53237 532^0 53263 




8275 8288 3301 3314 3320 3339 3852 3364 3377 3390 




8403 3416 3428 3441 3453 3406 3479 3491 3504 3517 




8529 3542 3655 3507 3580 3593 3605 3618 3631 3643 




8656 8668 3681 3694 3706 3719 3732 3744 3757 3769 




8782 8794 3807 3820 3832 3845 3857 3870 3882 3895 




8908 3920 3933 3945 3958 3970 3983 3995 4008 4020 




4033 4046 4058 4070 4083 4095 4108 4120 4133 4145 




4158 4170 4183 4196 4208 4220 4233 4245 4258 4270 




4283 4295 4307 4320 4332 4345 4357 4370 4382 4394 


880 


54407 54410 64432 64444 64466 54469 54481 54494 64606 54518 



112 



THE RATLliOAD TAPER 






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30.00 

0.20 

1°30' 


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CO 


30.00 

0.16 

1°.12' 

33.75 

0.20 

1°26' 

37.50 

0.24 

l°41i' 

45.00 

0.35 

2° 15' 


CO 


-** 


40.00 

0.20 

1°24' 

45.00 

0.26 

1°41J' 

50.00 

0.32 

2° 00' 

55.00 

0.39 

2°20t' 

60.00 

0.47 

2° 42' 

70.00 

0.66 

3° 30' 




eo 


^^^ 8^^- 8^^- SS^- 8gs' 8gb 

Tt^o"' QOo 00=^ c^id® 000'=^ oOo 

lOo ?0(^COo l>-o t^o 05.^ 
i-H (N CO CO 


C^ 


sr 


SSff. SSg. 8Kg S§| 82* Sfeg 

»OOCO C30O OOo l^OO 10 1-H '^ S'-'o 
t^o OOo OieO^o Oo Cvico 
Cq CO "^ '-^ -* '-^ 






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' 


1-H 




8ggr SSs- Sg| 8S|^ 8§^- 82g 

QQi-H'^ Qi-hCO (N(NC^ "^CqC^ SSC^^ O'^O 

cOo »oo <Oo i>o OOo rio 

'-^TtH'-^»O'~'CO'~'t>»'~'00^i-H 


l-t 


esi 


180.00 

2.20 

6° 18' 

195.00 

2.71 

7° 18J' 

210.00 

3.29 

8° 24' 

225.00 

3.96 

9° 33J' 

240.00 

4.71 

10° 48' 

270.00 

6.47 

13° 30' 


MN 


He* 


8Si§ 8Sh« 880 8g^^ 8^3 8Sg 

OCOo O'^® OOo or^'"' 0X0 O1-H0 

C^^C^OC<I^C0C0C0^C0,-i 
1-H »-H 


H« 


He* 


8§'^ 8S55 ; : : ; ; : : : : : : : 

OOOo OOo ... ... ... ... 

^ T-K ^ y-K . . . . . . . . . . . . 


»♦• 


1 
a«« ••> ••• ... ... •.. 1 




'^'^ o 



tsO •"-^'XsO f-^'^O 



'^ O ^-i*^ o 



8 



o 



O 

00 

o 



o 

00 



O 

CO 

o 
QO 



o 



LOGARITHMS OF NUMBERS 



N 


Oia34S6780 


400 


60206 60217 00223 6023Q 00249 602U0 60271 60232 602S!i 60304 




0314 0325 0336 0347 03fl8 0309 0379 0390 0401 0412 




0423 0433 0444 0466 0406 0477 0487 0408 0509 0520 




0581 0541 0652 0663 0574 0584 0595 0606 0Q17 0627 




0638 064B 0660 0670 0081 0002 0703 0713 0724 0735 




0746 0766 0707 0778 0788 0709 0810 0821 0831 0842 




0863 0863 0874 088^ 0805 0906 0017 0027 0038 0849 




0B69 0970 0081 0991 1002 1013 1023 1034 104S 1055 




1060 1077 1087 1098 1100 lllB II30 lUO 1151 1162 




1172 1183 1194 1204 1215 1225 1238 1247 1257 1268 


410 


61278 61280 61300 81310 61321 61331 61342 01352 61303 61374 




1384 1395 1405 1416 1-126 1437 1448 1458 1460 1479 




1490 1500 1511 1621 1532 1542 1553 1563 1574 1584 




1596 1606 1616 1627 1637 1648 1068 1669 1070 1690 




1700 1711 1721 1731 1742 1752 1703 1773 17B4 1794 




1805 1815 1826 1830 1317 1857 1888 1378 1638 1899 




1900 1020 1930 1941 1951 1962 1072 1982 10l>3 2003 




2014 2024 2034 2015 2055 2066 2078 2036 2097 2107 




2118 2128 2138 2149 2159 2170 2180 2100 2201 2211 




2221 2232 2242 2262 2203 2273 2284 2294 2304 2315 


420 


62325 6233S 62346 02358 62360 62377 62387 G23S7 62408 82418 




2428 2439 2449 2460 2409 2480 2490 2500 2511 2521 




2631 2542 2552 2562 2572 2583 2593 2003 2013 2624 




2684 2644 206i 2665 2C75 2035 2696 2706 2716 2720 




2737 2747 2757 2767 2778 2788 2798 2808 2818 2829 




2839 2840 2859 2870 2880 2890 2900 2910 2921 2931 




3941 2951 2901 2972 2082 2002 3002 3012 3022 3033 




3043 3063 3003 3073 3083 3094 3104 3114 3124 3134 




3144 3155 8165 8175 8185 3196 3203 3215 3225 3236 




3246 3256 3266 3276 3286 3298 3306 3317 3327 3337 


480 


63347 63367 63367 63377 03387 03307 63407 83417 63428 83438 




3448 3453 3403 3478 3488 3493 35118 3.'>18 3628 3538 




3548 8568 3588 3579 3689 3590 3609 3619 3820 3630 




8640 3869 3869 3679 3080 3009 3700 3710 3720 3739 




3749 3769 3769 3779 3780 3790 3809 3310 3620 3839 




SS40 3860 38Q9 3870 3889 3890 3009 3010 3020 3930 




8940 8950 3969 3979 3088 3998 4003 4018 4023 403B 




404B 4058 4008 4078 4088 4008 4108 4118 4123 4137 




4147 4167 4167 4177 4187 4107 4207 4217 4227 4237 




4246 4256 4266 4276 4286 4296 4306 4318 4328 4336 


440 


64845 64355 84385 64375 64385 64395 64404 84414 64424 64434 




4444 4454 4404 4473 4488 4403 4503 4513 4523 4r,;J2 




4642 4562 4562 4572 4582 4591 4601 4U11 4621 4631 




4640 4650 4600 4670 4680 4080 4609 4709 4719 47^9 




4788 4748 475B 4768 4777 4787 4707 4807 4M1G 4K2il 




4836 4846 4856 4365 4875 4885 489, 4004 4014 4924 




4933 4043 4953 4963 4072 4082 4092 5002 6011 6021 




6081 6040 5050 6080 6070 6070 fiObO 5099 5103 6118 




6128 5137 5147 5167 6I'17 5178 5180 5196 5205 6215 




eS2S 6234 5244 5254 5^03 5278 5283 6292 5302 5312 


4W 


66331 65331 66341 66350 6r.:i60 O.-hWO 65^70 65380 05398 05408 



1» 


, LOGARITHMS OF NUMBEBS 


"n~ 


Oia34S«7 8» 


450 


85321 65331 65341 06350 B5390 65360 65378 65389 65398 65408 




5418 5427 6437 6«47 64se 5466 5475 6485 549; 65(M 




5614 6523 6533 6643 6662 5683 6571 5581 6691 5600 




5610 5619 6629 6630 5848 5058 5667 5677 5688 5608 




5706 5715 5725 6734 5744 5783 5703 5772 6782 6702 




5801 5811 6820 6S30 5830 5849 5858 5868 6S77 5887 




5896 5900 5916 5925 5935 6944 5964 6083 6973 5082 




5902 6001 6011 8020 6030 6039 8049 6058 6068 6077 




6087 6096 6106 6II3 6124 6134 6143 6153 6102 6172 




6181 6191 0200 G210 6219 6229 6238 6247 6257 8266 


460 


66276 86285 86295 06304 65314 66323 68332 86342 66351 66361 




0370 6380 6389 6398 6408 6417 6427 6438 6445 645S 




6484 6474 6483 6492 6502 6511 6521 6530 6539 6549 




6558 8587 8577 6586 6696 6805 6614 6824 6633 66*2 




66r>2 6661 0671 6880 6089 6699 6708 6717 6727 6736 




6746 8765 878* 6773 6783 6792, 6801 6811 6820 6820 




6S39 8848 6857 6867 6876 6885 0804 6904 6913 6922 




6932 8941 6060 6980 6909 6978 6987 6997 7006 7015 




7025 7034 7043 7052 7082 7071 7080 7089 7099 7108 




7117 7127 7136 7145 7154 7164 7173 7182 719] 7201 


«0 


67210 67210 67228 67237 67247 67256 87265 67274 67284 67293 




7302 7311 7321 7330 7339 7348 7367 7387 7376 7385 




7304 7*)3 7413 7422 7431 7440 7449 7459 7468 747T 




7488 7495 7604 7514 7623 7532 7641 7650 7560 7560 




7578 7587 7596 7605 7614 7624 7033 7642 7651 7660 




7669 7670 7888 7697 7708 7716 7724 7733 7742 7762 




7761 7770 7779 7788 7797 7806 7815 7825 7B34 7843 




7852 7861 7870 7879 7888 7897 7906 7918 7925 7934 




7943 7952 7981 7970 7979 7988 7997 8008 8015 8024 


B 


8034 8043 8052 8061 8070 8070 8088 8007 8100 8116 


480 


08124 68133 081*2 08151 68160 68160 68178 68187 68198 68206 




8215 8221 8233 8242 8261 8280 8269 8278 8287 8298 




8305 8314 8323 8332 8341" 8350 8359 8388 8377 8386 




8395 8*04 8413 8422 8431 8440 8440 8458 8467 8478 




8185 8494 8502 8511 8520 8620 8538 8647 8558 8586 




8674 8583 8502 8801 8610 8810 8628 8637 8646 8655 




8684 8673 8681 8691) 8699 8708 8717 8728 8736 8744 




8753 8762 8771 8780 8789 8797 8808 8816 .8824 8833 




8842 8851 8880 8860 8878 8886 8805 8004 8913 8922 




89:31 89*0 8949 8958 8988 8976 8984 8993 9002 9011 


490 


69020 60028 09037 69046 69055 69064 89073 69082 69000 60099 




9108 9117 9126 0135 0144 6152 9181 9170 0179 9188 




9197 0205 0214 0223 0232 9241 0249 9268 9287 9276 




9285 1304 9302 9311 0320 9329 9338 0346 9355 9364 




. 9373 9381 0390 9399 9408 9417 0425 9434 9443 9452 




9461 9469 9478 9487 9498 9604 9513 0522 9531 9639 




0548 9557 9666 9574 9583 9592 6601 9609 9618 9627 




9636 9844 9663 9062 0871 0679 9688 0697 9705 9714 


*/ 0723 0732 9740 074fl &768 %m 01T5 6784 9703 1)801 | 


V 


8SI/> 9819 9827 9830 ft845 9R&4 ^mi •XSiX «6Wi yM*^ 



/ maST 60900 eggu 69923 6»932 69S40 BftftW ftWi&ft WW** «itnfe\ 



LOGARITHMS OF NUMBERS 



125 



N 


O 1 2 


8 4 5 6 7 9 


600 


69897 69906 69914 69923 69932 69940 69949 69968 69966 69975 


1 


9984 9992 70001 70010 70018 70027 70036 70044 70063 70062 


■8 


70070 70079 0088 


0096 0105 0114 0122 0131 0140 0148 


S 


0157 0166 0174 


0183 0191 0200 0209 0217 0226 0234 


4 


0243 0252 0260 


0269 0278 0286 0295 0303 0312 0321 





0329 0338 0346 


0355 0364 0372 0381 0389 0398 0406 


6 


0415 0424 0432 


0441 0449 0458 0467 0475 0484 0492 


7 


0501 0509 0518 


0526 0535 0544 0552 0561 0569 0578 


8 


0586 059i 0603 


0612 0621 0629 0638 0646 0655 0663 


9 


0672 0680 0689 


0697 0706 0714 0723 0731 0740 0749 


610 


•70757 70766 70774 70783 70791 70800 70808 70817 70825 70834 | 


1 


0842 0851 0869 


0868 0876 0885 0803 0902 0910 0919 


i 


0927 0935 0944 


0952 0961 0969 0978 0986 0995 1003 


t 


1012 1020 1029 


1037 1046 1054 1063 1071 1079 1088 


4 


1096 1105 1113 


1122 1130 1139 1147 1155 1164 1172 


ft 


1181 1189 1198 


1206 1214 1223 1231 1240 1248 1257 


6 


1265 1273 1282 


1290 1299 1307 1315 1324 1332 1341 


7 


lft49 1867 1366 


1374 1383 1391 1399 1408 1416 1425 


8 


1433 1441 1450 


1458 1466 1475 1483 1492 I5OO 1608 


9 


1517 1526 1633 


1542 1550 1559 1567 1575 1584 1592 


620 


71600 71609 71617 71625 71634 71642 71660 71659 71667 71676 


1 


1684 1692 1700 


1709 1717 1725 1734 1742 1750 1759 


8 


1767 1775 1784 


1792 1800 1809 1817 1825 1834 1842 


ar 


1860 1858 1867 


1875 1883 1892 1900 1908 1917 1925 


4 


1933 1941 1950 


1958 1966 1975 1983 1991 1999 2008 


ft 


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 2206 2304 2313 2321 2329 2337 


9 


2346 2354 2362 


'2370 2378 2387 2395 2403 2411 2410 


680 


72428 72436 72444 72452 72400 72469 72477 72485 72403 72501 | 


1 


2809 2618 2526 


2534 2542 2550 2558 2507 2575 2583 


9 


2591 2599 2607 


2616 2024 2032 2640 2648 2656 2665 


8 


2673 2681 2689 


2697 2705 2713 2722 2730 2738 2746 


4 


2754 2762 2770 


2779 2787 2795 2803 2811 2819 2827 


ft 


2835 2843 2852 


2860 2868 2876 2884 2802 2900 2908 


8 


2916 2925 29e33 


2941 2949 2957 2905 2973 2981 2989 


7 


2997 3006 3014 


3022 3030 3038 3046 3054 3062 3070 


8 


8078 3086 3094 


3102 3111 3119 3127 3135 3143 3151 


9 


8159 3167 3175 


3183 3191 3199 3207 3215 3223 3231 


640 


73239 73247 73255 


/3263 73272 73280 73288 73200 73304 73312 


1 


3320 .3328 3336 


3344 3352 8360 3308 337(*) 3384 3392 


9 


3400 3408 3416 


3424 3432 3440 3448 345(5 3404 3472 


8 


8480 3488 3406 


3504 3512 352) 3528 3536 3544 3552 


4 


8560 3568 3r)76 


3584 3592 3(500 3(508 3(516 3624 3632 


8 


3640 3648 8656 


3(5(54 3(572 3(57i> 3(587 3(W5 3703 3711 


8 


3719 3727 3735 


3743 :)75l 3759 3767 3775 3783 3791 


7 


3799 3807 3815 


3823 :i830 3838 3«4() 3854 38(52 3870 


8 


8878 SSm 3894 


3902 3910 3018 392(5 3033 3941 3949 


9 


8957 3965 3973 


3981 3989 3997 4005 4013 4020 4028 


660 


74086 74044 74052 74060 74068 74076 74084 74092 74099 74107 



LOGARITHMS OF NUMBERS 



690 



faoo 7 



74030 74044 74052 74000 740118 

4116 4123 4131 4111!) 1117 

4101 4202 4210 42IS 4225 

4273 4280 4288 4293 1304 

4351 1369 4307 4371 43S2 

442» 4437 444£ 4463 4401 

4507 4615 4523 4631 4630 

1588 4503 4001 4009 4617 

4(103 4671 4079 41187 4605 

4711 4710 1757 4794 4772 



74078 71084 

415i 4102 

4233 4241 

4312 4320 

4390 4398 

4468 4470 

4647 4554 

4024 4632 

4702 4710 

4780 4788 



74002 74009 741071 

4170 4178 1180 

4249 4257 42flS 

4327 4335 4343 

4406 4111 4421 

4484 4492 4J00 

4563 4670 4678 

4040 4018 4650 

4719 4726 4733 

47B6 4803 4811 



71819 74827 74834 

4806 4904 4912 

4074 4981 1989 

5051 5069 6066 

6128 6136 6143 

620S 5213 5220 

5282 6280 5297 

5368 6309 6374 

643j 5442 5460 

5511 5619 6520 

75587 75595 75603 

5061 6071 6079 

6740 5747 5765 

581a 6623 6831 

6891 5899 5906 

5067 5971 5982 

6042 OOSO 0057 

one 0125 0133 



TieiO 74858 74805 74873 74B81 74889 
4027 4035 4943 1050 4958 4960 
6005 6012 5020 5028 6036 5043 
5082 6089 6097 6105 6113 6120 
6159 5160 6174 6182 5189 6167 
5236 6243 5261 6260 6200 5274 



76010 7. 

5686 I 

5702 I 

6838 I 

6914 I 

5089 I 

6065 i 

0140 I 

6215 I 



) 6397 6404 

i 6173 5181 

'. 6649 5557 

1 75020 75033 

1 6702 5709 

I 6778 5785 

\ 6853 6801 

. 5920 6037 

' 600^ 0012 

I OOBO 0087 

I 6165 6163 



. 6313 < 



6412 6420 6427 

5188 5490 5504 

656j 6672 5580 

760J1 75048 75066 

5717 5724 5732 

6793 6800 6B08 

5808 5870 5884 

5944 5062 6959 

0020 0027 6035 

0O9i 6103 0110 

0170 0173 6185 

0245 6263 6260 



10343 76350 76358 76366 76373 70380 703B8 76396 76103 76410 

6118 6126 was 0440 644B 8155 6162 6470 6477 0485 

6492 6^00 6507 OalJ 6622 6530 6537 6645 0552 8659 

6567 6674 6582 0530 6597 6604 8012 0819 0028 0634 

8611 6040 0050 0004 0071 0078 8080 0603 6701 6708 

6710 0723 6730 0738 6745 6763 6760 6768 6775 87B2 

6700 6797 6805 0812 6810 6827 6834 8842 0849 0850 

6804 8871 0879 6880 0893 6901 6008 0916 6923 11930 

6938 6045 6963 6900 0967 0975 0982 6089 6907 7001 

7012 7019 7020 7034 7041 7048 7056 7063 7070 7078 

77085 77093 77100 77107 77115 1^122 77129 77137 77144 77161 

7169 7106 7173 7181 7188 7106 7203 7210 7217 7225 

7232 7240 7217 7264 7262 7260 7276 7283 7291 7298 

7305 7313 7320 7327 7335 7342 7319 7357 7301 7371 

7379 7386 7393 7401 7108 7415 7422 7430 7437 7444 

7462 7459 7466 7474 7481 7488 7495 75a3 7510 7617 

7625 7632 7639 7646 7554 7561 7568 7576 7683 7590 

7597 7605 7012 7010 7027 7831 7611 7848 7056 7663 

' 7670 7677 788S 7693 7609 7700 7714 IT-iV 1T28 7735 

7743 77J0 7767 7764 7772 7779 778ft llft^ l*a\ 1W» 

feOO I7781B 77822 77830 77837 77844 77851 77aM 71666 nV.^ llW 



k 



LOGARITHMS OF NUMBERS 127 



N 



600 
1 

% 
8 

4 
5 
6 

7 

8 
9 

610 

1 
8 
8 
4 
6 
6 
7 
8 
9 

680 
1 

8 
8 
4 
5 
8 
7 
8 
9 



1 

8 
8 
4 
5 
6 
7 
8 
9 

640 



8 

4 
6 
S 

7 

8 
9 

6M 



8 9 



77816 77822 77830 77837 77844 77851 77869 77866 77873 77880 

7887 7895 7902 7909 7916 7924 7931 7938 7946 7962 

7960 7967 7974 7981 7988 7996 8003 8010 8017 802i 

8032 8039 8046 8063 8061 8068 8075 8082 8089 8097 

8104 8111 8118 8125 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 8*340 8347 8355 8362 8369 8376 8383 

8390 8398 8405 8412 8419 8420 8433 8440 8447 8455 

8462 8460 8476 8483 8490 8497 8504 8512 8519 8626 

78633 78640 78647 78654 78561 78569 78576 78583 78590 78697 

8604 8611 8618 8625 8633 8640 &H7 8654 8661 8668 

8676 8682 8689 8696 8704 8711 8718 8725 8732 8739 

8746 8763 8760 8767 8774 8781 8789 Sim 8803 8810 

8817 8824 8831 8838 8845 8852 8859 8866 8873 8880 

8888 8895 8902 8909 8916 8923 8930 8937 8944 8961 

8968 8966 8972 8979 8986 8993 9000 9007 9014 9021 

9029 9036 9043 9050 9057 9064 9071 9078 9085 9092 

9099 9106 9113 9120 9127 9134 9141 9148 9155 9162 

9169 9176 9183 9190 9197 9204 9211 9218 9225 9232 

70239 79246 79263 79260 79267 79274 79281 79288 79296 79302 

9309 9316 9323 9330 9337 9344 9351 9358 9365 9372 

9379 9386 9393 9400 9407 9414 9421 9428 9435 9442 

9449 9466 9463 9470 9477 9484 9491 9498 9505 9511 

9618 9626 9632 9539 9546 9553 9500 9567 9574 9581 

9688 9695 9602 9609 9616 9023 9630 9637 9644 9650 

0667 9664 9671 9678 9685 9692 9699 9706 9713 9720 

0727 9734 9741 9748 9754 9761 0708 9775 9782 9789 

0796 0803 9810 9817 9824 9831 9837 9844 9851 9858 

0866 0872 9879 9886 9893 9900 990(> 9913 9920 9927 

■70034 70941 79948 79955 79962 79969 79975 79982 79989 79996 

80003 80010 80017 80024 80030 80037 80044 80051 80068 80065 

0072 0079 0085 0092 0009 0106 0113 0120 0127 0134 

0140 0147 0154 0161 0108 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 0363 0359 0360 0373 0380 0:]«7 0303 0400 0407 

0414 0421 0428 0434 0441 0448 0455 04r>2 0468 0475 

0482 0489 0496 0502 0509 0510 052:; 05:U) 0536 0543 

0660 0667 0564 0570 0577 0584 0591 0598 0604 0611 

80618 80625 80632 80038 S0645 80052 80050 80(565 80072 80679 

0686 0693 0699 0706 0713 0720 072(5 073:J 0740 0747 

0764 0760 0767 0774 0781 0787 0794 0801 0808 0814 

0821 0828 0835 0841 0848 0855 08(52 0868 0875 0882 

0880 0896 0902 0009 091(5 0922 0929 093(5 0943 0949 

0066 0963 0J)69 0976 0983 0990 0990 1003 1010 1017 

1023 1030 1037 1043 1050 1057 10(54 1070 1077 1084 

1000 1097 1104 1111 1117 1124 1131 1137 1144 1151 

1168 1164 1171 1178 1184 1191 1198 1204 1211 1218 

1224 1231 1238 1245 1251 1258 12(55 1271 1278 1285 

81801 81298 81305 81311 81318 81325 81331 81338 81345 81361 



128 



LOGARJTHMS OF NUMBERS 





N 


01234567 89 






650 


81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 






1 


1368 1365 1371 1378 1385 1391 1398 1405 1411 1418 






2 


1425 1431 1438 1445 1461 1468 1465 1471 1478 1485 






3 


1491 1498 1605 1611 1618 1525 1631 1638 1644 1661 






4 


1668 1664 1671 1678 1684 1691 1698 1604 1611 1617 






6 


1624 1631 1637 1644 1651 1667 1664 1671 1677 1684 






6 


1690 1697 1704 1710 1717 1723 1730 1737 1743 1750 






7 


1767 1763 1770 1776 1783 1790 1796 1803 1809 1816 






8 


1823 1829 1836 1842 1849 1866 1862 1869 1876 1882 






9 


1889 1895 1902 1908 1915 1921 1928 1935 1941 1948 






660 


81964 81961 81968 81974 81981 81987 81994 82000 82007 82014 






1 


2020 2027 2033 2040 2046 2063 2060 2066 2073 2079 






2 


2086 2092 2099 2105 2112 2119 2126 2132 2138 2145 






3 


2151 2158 2164 2171 2178 2184 2191 2197 2204 2210 






4 


2217 2223 2230 2236 2243 2249 2266 2263 2269 2276 






5 


2282 2289 2296 2302 2308 2315 2321 2328 2334 2341 






6 


2347 2364 2360 2367 2373 2380 2387 2393 2400 2406 






7 


2413 2419 2426 2432 2439 2446 2452 2458 2465 2471 






8 


2478 2484 2491 2497 2604 2510 2617 2523 2530 2636 






9 


2643 2649 2666 2662 2669 2676 2682 2588 2595 2601 






670 


82607 82614 82620 82627 82633 82640 82646 82653 82669 82666 






1 


2672 2679 2686 2692 2698 2705 2711 2718 2724 2730 






2 


2737 2743 2750 2756 2763 2769 2776 2782 2789 2795 






3 


2802 2808 2814 2821 2827 2834 2840 2847 2863 2860 






4 


2866 2872 2879 2885 2892 2898 2905 2911 2918 2924 






6 


'2930 2937 2943 2950 2956 2963 2969 2976 2982 2988 






6 


2995 3001 3008 3014 3020 3027 3033 3040 3046 3062 






7 


3059 3065 3072 3078 3085 3091 3097 3104 3110 3117 






8 


3123 3129 3136 3142 3149 3155 3161 3168 3174 3181 






9 


3187 3193 3200 3206 3213 3219 3225 3232 3238 3245 






680 


83251 83257 83264 83270 83276 83283 83289 83296 83302 83308 






1 


3315 3321 3327 3334 3340 3347 3353 3359 3366 3372 






2 


3378 3385 3391 3398 3404 3410 3417 3423 3429 3436 






8 


3442 3448 3455 3461 3467 3474 3480 3487 3493 3499 






4 


3606 3612 3518 3525 3531 3537 3544 3660 3666 3663 






6 


3569 3576 3582 3588 3694 3601 3607 3613 3620 3626 






6 


3632 3639 3646 3651 3658 3664 3670 3677 3683 3689 






7 


3696 3702 3708 3715 3721 3727 3734 3740 3746 3763 






8 


3759 3765 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 


3948 3964 3960 3967 3973 3979 3985 3992 3998 4004 






2 


4011 4017 4023 4029 4036 4042 4048 4055 4061 4067 






8 


4073 4080 4086 4092 4098 4105 4111 4117 4123 4130 






4 


4136 4142 4148 4155 4161 4167 4173 4180 4186 4192 






6 


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 




/ S 4386 4392 4398 4404 4410 4417 44^S 4429 4435 4442 | 




/ P/ 4448 4464 4460 4466 4473 4479 44%^ 44^\ ^^^1 ^^^v\ 


/» 


Oj84 


^610 84616 84622 84628 84535 84541 «4647 %466a ^bb'Ji ^^^ 


^\ 





LOGAHlTHMS OF NUMBERS 129 


K 


1234567 89 


700 


84510 B4616 84522 84528 6463i 84541 8464T 84663 84560 84568 




4572 4578 4584 4600 4597 4003 4G09 4fil5 4621 4028 




4634 4840 4640 48,-,2 4858 40(lj 4671 4077 4083 4080 




4696 4702 4708 4714 4720 4726 4733 4731) 474j 4751 




4767 4763 4770 4776 4782 4788 4794 4800 4807 4813 




4819 4826 4831 4887 4844 48i0 4856 4862 4868 4874 




4880 4887 4803 4800 4905 4011 4017 40-2+ 4B30 4936 




4942 4048 4954 4060 4067 4073 4070 498i 4001 4007 




6003 5009 5016 5022 5028 5034 61M0 6046 5052 5058 




6065 6071 5077 5083 5080 5096 5101 6107 6114 5120 


710 


85126 86132 85139 85144 85160 85156 85103 85109 8517S 85181 




6187 5103 5100 6205 5211 5217 5224 5230 6236 5242 




5248 6364 6260 5266 5272 6278 6285 6201 6207 fh303 




5300 6315 5321 5327 5333 5330 5345 5362 5358 5364 




6370 6376 6382 5388 5304 6400 6406 5412 6418 5425 




5431 6437 5443 5440 645S 5461 ^467 6473 6470 5485 




5401 6497 6503 5600 6516 6622 6628 6534 6540 5546 




5562 6558 6664 5670 5570 55S2 5588 6504 5600 5606 




6612 6618 5625 5631 6037 5643 6640 6665 5661 5067 




6673 6679 6685 6(101 6697 6703 5700 6715 6721 5727 


720 


86733 86739 86746 85751 85757 85763 86709 86775 85781 85788 




6704 5800 5806 5812 5818 5824 5830 5830 5842 6848 




5854 6860 6866 5872 6878 6884 6800 6806 6002 5008 




6914 5020 6026 5032 6038 5044 5050 5050 5062 5008 




6974 6080 6086 5092 6008 6004 0010 6016 6022 6028 




6034 6040 6O40 6062 6058 6064 6070 6076 6082 8088 




6004 6100 6100 6112 6118 6124 0130 6136 6141 0147 




6163 6159 6165 6171 6177 6183 6180 0105 6201 6207 




6213 6219 6226 6231 6237 0243 0249 6255 6261 6267 




6278 6279 6285 6291 6207 8303 6308 6314 6320 6326 


780 


86332 86338 86344 80350 86366 86362 86368 86374 86380 86386 




6302 6308 6404 6410 6416 6421 6427 6433 6430 6445 




M51 6457 6463 6460 6475 0481 6487 6403 6490 6504 




6610 0516 6622 6628 6634 0640 6546 6562 6558 0664 




6670 6676 6681 0587 65B3 Um 6605 6011 6017 0623 




6620 6635 6641 0646 6652 0658 6664 6070 0676 6082 




6688 6694 6700 0705 6711 6717 0723 6729 6735 6741 




6747 6763 6750 6704 0770 0776 6782 67B8 6704 0800 




6806 6812 6817 6823 0829 08:)6 0841 0847 0853 0860 




6864 6870 6370 0882 6888 6804 0900 6000 6011 0017 


740 


86023 86929 86935 86041 86047 800r>3 86058 86064 80070 80076 




flS82 6089 6004 6090 7006 7011 7017 7023 7020 7035 




7040 7046 7052 7058 7064 7070 7075 7081 7087 7003 




7090 7105 7111 7116 7122 7128 7134 7140 7146 7161 




7157 7163 7160 7175 "l"! TIHH 7102 7198 7204 7210 




7216 7221 7227 723-i 7230 724S 7251 7256 7262 7268 




7274 7280 7286 7201 7297 7:!03 7:100 7315 7320 7320 




7832 7338 7344 7340 7355 7:i01 7;Wi7 7:173 7:170 7:JM4 




7800 7396 7402 7408 741:! 7419 7425 74:11 74:i7 7442 




7448 7454 7460 7406 7471 7477 7483 T4H0 7105 7500 


710 


BT606 87612 87618 H7523 87520 87536 87541 87547 87562 67668 



'son 



130 


LOGARITHMS OF NUMBERS 


"n~ 


12 3 4 5 


6 7 8 9 1 


J60^ 


8750« 87512 87518 87523 87629 87535 87641 87547 87652 87668 1 




7564 7570 7578 7581 7587 7603 


7690 7604 7610 7610 




7622 7628 7633 7639 7845 7661 


7656 7662 7668 7674 




7879 7685 7001 70!I7 7703 7708 


7714 7720 7726 7731 




7737 7743 7749 7754 7780 7766 


7772 7777 7783 7789 




7795 7800 7808 7812 7B18 7823 


7820 7835 7841 7840 




7862 7858 786* 7860 7875 7881 


7887 7892 7898 7904 




7010 7915 7921 7927 7833 7938 


7944 7950 7055 7961 




7987 7973 7978 7084 7000 7906 


8001 8007 8013 8019 




8024 8030 8036 80*1 8047 8053 


8053 8084 8070 8O70 


760 


88081 88087 88093 88098 88104 881 10 88118 88121 88127 88133 | 




8138 8144 8150 8166 8161 8167 


8173 8178 8184 8190 




8195 eaOl 6207 8213 8^218 8224 


8230 8235 82*1 8247 




8252 8268 8264 8270 8276 8281 


8287 8202 8208 8304 




8309 8315 8321 8328 8332 83:18 


8343 8349 8365 8360 




8366 8372 8377 8383 8380 830S 


8400 8406 8412 8417 




8423 8*29 8434 8440 8446 8151 


8457 8483 8468 8*74 




8480 8486 8491 8497 8602 8508 


8613 8510 8525 8530 




8636 85*2 8547 8663 8560 8504 


8570 8578 8581 8687 




8593 8598 8604 8610 8615 8021 


8827 8632 8638 8643 


!7a 


88849 88655 88680 88866 88672 88677 


88883 88689 8869* 88700 




8706 8711 8717 8722 8728 873* 


8739 8745 8760 8756 




8762 8767 8773 8770 8784 8700 


8705 8801 8807 8812 




8818 8824 8829 883i 8840 8846 


8862 8867 8803 8808 




8874 8880 8885 8801 8897 8902 


8008 8013 8910 8925 




8930 8036 8941 8947 8963 8058 


8084 8069 8975 8081 




8986 8992 8907 0003 0009 9014 


9020 0025 0031 0037 




0042 0048 0053 9059 0064 9070 


0078 0081 9087 0092 




9098 9104 9109 0115 0120 9126 


0131 9137 9143 0148 




0154 9159 9165 9170 9176 9182 


0187 0103 9108 0204 


780 


89209 89215 80221 89226 89232 89237 89243 89248 89264 89280 | 




9265 9271 0278 0282 0287 0293 


0298 030* 9310 0316 




9321 9328 9332 0337 93*3 0348 


9354 0360 0305 0871 




9378 9382 9397 0393 9398 0404 


0409 9415 9421 9426 




9432 9437 9443 9448 9454 0469 


0*65 9470 0470 0481 




9487 9402 0498 960* 9609 O6I5 


0520 9628 9531 9537 




0542 9548 9553 9560 9684 0570 


9676 9581 0580 9592 




9507 9603 0800 9814 9620 9025 


9031 90:t0 9842 9647 




9853 9658 9864 9869 9675 0680 


96B6 9601 0697 0702 




9708 9713 0710 9724 0730 0735 


9741 0740 0762 0757 


790 


80783 89768 89774 89779 89785 89700 80708 80801 89807 80812 | 




9818 9823 9829 0834 9840 9845 


9851 0856 9862 0867 




0873 0878 98S3 9880 0804 9GO0 


9005 0011 9910 9022 




9927 0933 9938 00*4 9940 995S 


9060 0966 9971 9977 




9982 9088 0993 0008 00004 00000 00015 90020 00026 900311 




90037 00042 90048 90053 0059 0064 


0089 0075 0080 0088 




0091 0007 0102 0109 0113 0110 


0124 0320 0135 0140 




0148 0161 0157 0162 0168 0173 


0170 0184 0189 0105 


*/ 0200 0208 0311 0217 Oflaa 0227 


OaSS 0238 0244 0249 


*/ 


02s/ 0300 0266 0271 0270 0282 


028T W4S5 QfiSfe ll^HW 





LOGARITHMS OF NUMBERS 131 


w 


0123456789 


800 


90309 90314 90820 90325 00331 90336 90342 90347 90352 90358 


1 


0363 0369 0374 0380 0385 0390 0396 0401 0407 0412 


s 


0417 0423 0428 0434 0439 044i 04^0 0465 0461 0466 


8 


0472 0477 0482 0488 0493 0499 0504 0509 051i 0520 


4 


0626 0631 0536 0542 0547 0553 0558 0563 0569 0574 


6 


0680 0685 0590 0596 0601 0607 0612 0617 0623 0628 


6 


0634 0639 0644 06^0 0655 0660 0606 0671 0677 0682 


7 


0687 0693 0698 0703 0709 0714 0720 0725 0780 0736 


8 


0741 0747 0752 0757 0763 0768 0773 0779 0784 0789 


9 


0795 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 0950 


8 


0966 0961 0966 0972 0977 0982 0988 0993 0998 1004 


8 


1009 1014 1020 1026 1030 1036 1041 1046 1052 1057 


4 


1062 1068 1073 1078 1084 1089 1094 1100 1105 1110 


6 


1116 1121 1126 1132 1137 1142 1148 1153 1158 1164 


6 


1169 1174 1180 1185 1190 1196 1201 1206 1212 1217 


7 


1222 1228 1233 1238 1243 1249 1254 1259 1265 1270 


8 


1275 1281 1286 1291 1297 1302 1307 1312 1318 1323 


9 


1328 1334 1339 1344 1350 1355 1360 1365 1371 1376 


820 


91381 91387 91392 91397 91403 91408 91413 91418 91424 91429 


1 


1434 1440 1445 1450 1455 1461 1466 1471 1477 1482 


8 


1487 1492 1498 1503 1508 1514 1519 1524 1529 1535 


8 


1540 1645 1551 1556 1561 1566 1572 1577 1582 1587 


4 


1693 1698 1603 1609 1614 1619 1624 1630 1635 1640 


5 


1645 1661 1656 1661 1666 1672 1677 1682 1687 1693 


6 


1698 1703 1709 1714 1719 1724 1730 1735 1740 1745 


7 


1751 1766 1761 1766 1772 1777 1782 1787 1793 1798 


8 


1808 1808 1814 1819 1824 1829 1834 1840 1845 1850 


9 


1866 1861 1806 1871 1876 1882 1887 1892 1897 1903 


880 


91908 91913 91918 91924 91929 91934 91939 91044 91950 91955 


1 


1960 1965 1971 1976 1981 1986 1991 1997 2002 2007 


9 


2012 2018 2023 2028 2033 2038 2044 2049 2054 2059 


8 


2065 2070 2075 2080 2085 2091 2096 2101 2106 2111 


4 


2117 2122 2127 2132 2137 2148 2148 2153 2158 2163 


6 


2169 2174 2179 2184 2189 2195 2200 2205 2210 2215 


8 


2221 2226 2231 2236 2241 2247 2252 2257 2262 2267 


7 


2273 2278 2283 2288 2298 2298 2304 2309 2314 2319 


8 


2324 2330 2335 2340 2845 2350 2355 2361 2366 2371 


9 


2376 2881 2387 2392 2397 2402 2407 2412 2418 2423 


940 


92428 92433 92438 92443 92449 92454 92459 92464 92469 92474 


1 


2480 2485 2490 2495 2500 2505 2511 2516 2521 2526 


9 


2531 2636 2542 2547 2552 2557 2562 2567 2572 2578 


8 


2683 2688 2593 2598 2608 2600 2614 2619 2624 2629 


4 


2634 2639 2645 2650 2655 2660 2665 2670 2675 2681 


5 


2686 2691 2696 2701 2706 2711 271(J 2722 2727 2782 


6 


2787 2742 2747 2752 2758 2768 2768 2773 2778 2783 


7 


2788 2793 2799 2804 2809 2814 2819 2824 2829 2884 


tf 


2840 2845 2850 2855 28(M) 2865 2870 2875 2881 2886 


9 


2891 2896 2901 290(i 2911 2916 2921 2927 2932 2987 


860 


02942 92947 92952 92957 92(M)2 92967 92973 92978 92983 92988 



LOGARITHMS OF NUMBERS 



) S2042 92047 82062 92967 92fi62 02967 82973 82878 828B;J 92988 

L 2093 2888 3001! 3008 3013 3018 3024 3029 3034 3039 

1 3044 3040 3054 3069 3D64 3068 SOTf 3080 308$ 3080 

9 309j 3100 8105 3110 3115 3120 3126 3131 3136 3141 

I 3146 3151 3166 3161 3106 3171 3170 3181 3186 3192 

i 3197 3202 3207 3212 3217 3222 3227 3232 3237 3242 

i 3247 3252 3268 3263 3268 3273 3278 3283 3288 3293 

1 3288 3303 3308 3313 8318 3323 3328 3334 3338 3344 

9 3340 3364 3369 3364 3S68 3374 3370 3384 3388 3394 

9 3388 3404 3408 3414 3420 342} 3430 343j 3440 344J 

> 034iO 8345^ 93460 9346J 93470 93475 93480 934B6 83490 83486 

1 3600 3606 3610 3516 3520 3526 3531 3536 3641 3546 

i 3551 3556 3561 3666 3671 3576 3681 3686 3591 3596 

J 3601 3606 3611 3616 3021 3626 3631 3636 3^1 3646 

I 3651 3660 3001 3600 3fl71 3676 3682 3687 3692 3687 

J 3702 3707 3712 3717 3722 3727 3732 3737 3742 3747 

i 3752 3767 3762 3767 3773 3777 3782 3787 3792 3707 

1 3802 3807 3812 8817 3822 3827 3832 3837 3842 3847 

9 3852 3867 3862 3887 3872 3877 3882 3887 3892 3897 

i 3902 3907 3012 3917 3922 3027 8932 3937 3942 3947 

) 93852 83057 03962 93867 03972 93977 93882 03987 93992 93987 

1 4002 4007 4012 4017 4022 4027 4032 4037 4042 4047 

B 4052 4067 4062 4067 4072 4077 4082 4086 4001 4096 

J 4101 4106 4111 4116 4121 4126 4131 4130 4U1 4146 

I 4161 4166 4161 4166 4171 4176 4161 4186 4101 4196 

5 4201 4200 4211 4216 4221 4226 4231 4236 4240 4245 

S 4260 4256 4260 4265 4270 4276 4280 4286 4290 4295 

7 4300 4305 4310 4315 4320 4325 4330 433S 4;W0 4345 

B 4349 4364 4359 4364 4369 4374 4370 4384 4389 4304 

9 4300 4404 4400 4414 4419 4424 4429 443;! 4438 4443 

) 04448 84453 94458 84403 84468 84473 94478 94483 94488 04493 

1 4408 4503 4507 4612 4617 4622 4527 463i 4537 4542 

E 4547 4552 4557 4562 4B67 4571 4676 4681 4588 4681 

9 4586 4601 4606 4611 4fll6 4621 4626 4630 46:35 4040 

1 4645 4660 4665 4000 466$ 4670 4675 4680 4085 4688 

J 4894 4609 4704 4709 4714 4719 4724 4729 4734 4738 

S 4743 4748 4753 4758 4703 4768 4773 4778 4783 4787 

: 4792 4797 4802 4807 4812 4817 4822 4827 4B32 4830 

I 4841 4846 4861 4856 4861 4866 4871 4876 4880 4885 

9 4890 4806 4000 4005 4810 4915 4819 4024 4929 4934 

) 94938 94944 94840 84064 94969 94963 04868 84073 94978 94983 

1 49S8 4983 4008 5002 5O07 6012 6017 6022 6027 6032 

I 8036 6041 5046 5051 6066 6061 6066 5071 5075 6080 

J 5086 6080 5095 6100 6105 6100 5114 6119 5124 5129 

I 5134 5139 5143 5148 6153 6158 5163 6168 5173 5177 

} 5182 6187 5192 5197 5202 6207 5211 6216 5221 6226 

9 5231 6236 6240 5246 6260 6266 6260 6265 6270 6274 

r 5279 6284 5288 6284 6299 6303 5308 6313 5318 6323 

/ AS3g 5HS2 5337 5342 6347 6352 &3ol &361 6368 6371 

' AUrff 5381 6388 5390 6305 6400 540S &\W M\i SA\^\ 

»«5/ S34S9 95434 96439 96444 95446 954&3 »&4Eft B&«a %b««\ 





LOGARITHMS OF NUMBERS 133 


N 


1 


2 3 


4 3 6 7 8 9 I 


»00 


96424 65429 »5434 95430 96444 95448 96463 95468 66463 95468 | 




6472 5477 


5482 5487 


6492 5497 6501 6606 6511 5516 




6621 5525 


5530 6535 


5540 5545 5550 5554 6550 5564 




6569 5674 


6578 5583 


5688 6503 6698 6002 6807 6612 




6617 5622 


5626 663] 


58;i6 6641 5646 6650 6655 5660 




666i 5670 


6674 6679 


5684 5689 6694 5608 6703 5708 




6713 5718 


6722 5727 


5732 6737 5742 6746 6761 5758 




5761 5766 


5770 5776 


5780 578S 5789 6704 5799 5804 




5809 6813 


5818 5823 


6828 6832 5837 6842 5847 5862 




6856 6661 


5866 5871 


6875 6880 5885 6890 6895 5899 


dio 


85904 95909 96S14 B6918 96923 B5928 96933 059:)8 95B42 B5947 | 




6962 6B6T 


6961 5966 


5971 5976 6980 5085 5090 6995 




5999 6004 


6009 COU 


6019 6023 6028 6033 6038 6042 




6047 6052 


6067 6061 


6066 6071 8076 6080 6085 8090 




B09S 6099 


6104 6109 


6114 6118 6123 6128 8133 8137 




8142 6147 


0162 6166 


6161 6166 6171 6176 6180 6185 




6190 61S4 


6199 6204 


6209 6213 6218 6223 6227 0232 




6237 6242 


6246 6261 


6266 6261 6265 6270 6275 62B0 




8284 6289 


6294 6298 


6303 6308 6313 8317 8322 8327 




6332 6336 


6341 6346 


6350 6355 6360 6365 6369 6374 


980 


96379 90384 


96388 98393 


96398 96402 96407 96412 96417 96421 




6426 6431 


6435 6440 


8445 6450 0454 6469 6484 6468 




6473 6478 


6483 6487 


6492 6497 6501 6506 6511' 6616 




6620 662i 


6630 6534 


6539 8544 6648 6563 6558 6562 




8667 6672 


6577 6581 


6586 6591 6695 6600 8605 6000 




8814 0619 


8624 6628 


0033 6038 6642 6647 6652 6666 




6661 6666 


6670 6675 


6680 6685 6689 0694 6699 8703 




8708 6713 


0717 6722 


6727 0731 6736 6741 6745 6750 




e76S 6769 


6764 6769 


6774 67T8 6783 6788 6792 6707 




6802 8808 


6811 6816 


6820 682S 6830 6834 6839 6844 


980 


96848 96863 96858 96862 96807 96872 96876 96881 96886 96890 | 




6895 6900 


6904 6909 


6014 8918 6923 6928 6932 6037 




6942 6946 


6961 6966 


6900 6965 6970 6974 6979 6984 




6988 6993 


6997 7002 


7007 7011 7010 7021 7025 7030 




703S 7039 


7044 7049 


7053 7068 7063 7067 7072 7077 




7081 7086 


7090 7096 


7100 7104 7109 7114 7118 7123 




7128 7132 


7137 7142 


7146 7151 7185 7100 7105 7169 




7174 7179 


7183 7188 


7192 7197 7202 7206 7211 7216 




7220 7225 


7230 7234 


7230 7243 7248 7253 7257 7262 




7267 7271 


7276 7280 


7285 7290 7204 7299 7304 7308 


940 


97813 97317 


B7322 97327 97331 97336 97340 97345 97350 97354 1 




7359 7364 


7368 7373 


7377 7382 7387 7301 7390 74(10 




7405 7410 


7414 7419 


7424 7428 7433 7437 7442 7447 




7461 7456 


7460 7465 


7470 7474 7479 7483 7488 7493 




7497 7602 


7506 7611 


7616 7620 7525 7520 7534 7539 




7643 7648 


7552 7557 


7562 7566 7671 7575 7580 7685 




7689 7694 


7608 7803 


7607 7012 7617 7021 7628 78SO 




7685 7640 


7644 7649 


7653 7656 7603 7607 7672 7676 




7681 7685 


7690 7605 


7609 7704 7708 7713 7717 7722 




7727 7781 


7736 7740 


7745 7749 7754 7759 7763 7788 


»«0 


97773 97777 97782 97786 97791 97795 97800 97804 97809 97B13 1 



134 



looarithms of numbers 



950 


0123456789 




97772 97777 97782 97786 97791 97796 97800 97804 97809 97813 




1 


7818 7823 7827 7832 7836 7841 7845 7860 7855 7869 




2 


7864 7868 7873 7877 7882 7886 7891 7896 7900 7905 




8 


7909 7914 7918 7923 7928 7932 7937 7941 7946 7950 




4 


7055 7959 7964 7968 7973 7978 7982 7987 7991 7996 




5 


8000 8005 8009 8014 8019 8023 8028 8032 8037 8041 




6 


8046 8050 8055 8059 8064 8068 8073 8078 8082 8087 




7 


8091 8096 8100 8105 8109 8114 8118 8123 8127 8132 




8 


8137 8141 8146 8150 8155 8159 8164 8168 8173 8177 




9 


8182 8186 8191 8195 8200 8204 8209 8214 8218 8223 




960 


98227 98232 98236 98241 98245 98250 98254 98259 98263 98268 




1 


8272 8277 8281 8286 8290 8295 8299 8304 8308 8318 




2 


8318 8322 8327 8331 8336 8340 8345 8349 8354 8358 




8 


8363 8367 8372 8376 8381 8385 8390 8394 8399 8403 




4 


8408 8412 8417 8421 8426 8430 8435 8439 8444 8448 




5 


8453 8457 8462 8466 8471 8475 8480 8484 8489 8493 




6 


8498 8502 8507 8511 8516 8520 8525 8529 8534 8638 




7 


8543 8547 8552 8556 8561 8565 8570 8574 8679 8683 




8 


8588 8592 8597 8601 8605 8610 8614 8619 8623 8628 




9 


8632 8637 8641 8646 8650 8655 8669 8664 8668 8673 




970 


98677 98682 98686 98691 98696 98700 98704 98709 98713 98717 




1 


8722 8726 8731 8735 8740 8744 8749 8763 8768 8762 




2 


' 8767 8771 8776 8780 8784 8789 8793 8798 8802 8807 




8 


8811 8816 8820 8825 8829 8834 8838 8843 8847 8851 




4 


8856 8860 8865 8869 8874 8878 8883 8887 8892 8896 




6 


8900 8905 8909 8914 8918 8923 8927 8932 8936 8941 




« 


8945 8949 8954 8958 8963 8967 8972 8976 8981 8986 




7 


8989 8994 8998 9003 9007 9012 9016 9021 9026 9029 




8 


9034 9038 9043 9047 9052 9056 9061 9065 9069 9074 




9 


9078 9083 9087 9092 9096 9100 9105 9109 9114 9118 




980 


99123 99127 99131 99136 99140 99145 99149 99164 99158 99162 




1 


9167 9171 9176 9180 9185 9189 9193 9198 9202 9207 




2 


9211 9216 9220 9224 9229 9233 9238 9242 9247 9261 




8 


9255 9260 9264 9269 9273 9277 9282 9286 9291 9296 




4 


9300 9304 9308 9313 9317 9322 9326 9330 9335 9339 




5 


9344 9348 9352 9357 9361 9366 9370 9374 9379 9383 




6 


9388 9392 9396 9401 9405 9410 9414 9419 9423 9427 




7 


9432 9436 9441 9445 9449 9454 9468 9463 9467 9471 




8 


9476 9480 9484 9489 9493 9498 9502 9606 9511 9516 




9 


9620 9524 9628 9533 9537 9642 9646 9650 9665 9669 




990 


99564 99568 99572 99577 99581 99686 99690 99694 99699 99603 




1 


9607 9612 9616 9621 9625 9629 9634 9638 9642 9647 




2 


9651 9656 9660 9664 9669 9673 9677 9682 9686 9691 




8 


9695 9699 9704 9708 9712 9717 9721 9726 9730 9734 




4 


9739 9743 9747 9762 9756 9760 9765 9769 9774 9778 




5 


9782 9787 9791 9795 9800 9804 9808 9813 9817 9822 




• 


9826 9830 9835 9839 9843 9848 9862 9856 9861 9866 




7 9870 9874 9878 9883 9887 9891 9896 9900 9904 9909 




^/ ffPIS 0917 9922 9926 9930 993^ W?^9 «ft44 9948 9952 


^/ 09^7 9961 9965 9970 9974 9978 W^S ^%1 Wi'^^ "^f^fi^X 


^/ooi 


^00 00004 00009 00013 00017 00022 00026 W»^ ^5^^^ ^^^*^ 


^ 



LOGARITHMIC SINES 



AND COSINES 
¥' 



11 



136 LOGARITHMIC SINES AND COSINES 



t' 6» 



II 





LOGARITHMIC SINES AND COSINES 


137 


# 


6 


• 




?• 


1 


8» 


i 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





o.oioa 


9.90761 


9.06689 


9.99675 


9.14856 


9.99575 


60 


1 


0^18 


99760 


08699 


99674 


14445 


99574 


50 


2 


0BI68 


90759 


06795 


99678 


14535 


99578 


58 


S 


09^ 


99757 


06897 


99670 


14684 


99570 


67 


4 


(ymi 


99756 


06999 


99669 


14714 


99568 


56 


ft 


08S80 


99755 


09101 


99667 


14803 


99566 


55 


6 


O8089 


99753 


09808 


99666 


14891 


99565 


54 


7 


04757 


99769 


09304 


99664 


14980 


99563 


53 


8 


a«74 


99751 


09405 


99663 


15069 


99561 


58 





0299;$ 


99749 


09506 


99661 


15167 


99559 


61 


10 


0.08109 


9.99748 


9.09606 


9.99659 


9.15845 


9.99557 


50 


11 


06386 


99747 


09707 


99658 


15333 


99556 


49 


]« 


0:^3 


99745 


09807 


99656 


15481 


99554 


48 


U 


03158 


99744 


09907 


99665 


15508 


99558 


47 


)4 


06574 


99749 


10006 


99653 


15696 


99550 


46 


15 


01690 


99741 


10106 


99651 


15683 


90548 


45 


Itf 


(m^ 


99740 


10905 


99650 


15770 


09546 


44 


17 


OWJO 


99788 


10304 


99648 


15857 


99645 


43 


18 


01034 


99787 


. 10408 


99647 


15944 


99543 


48 


19 


04149 


99736 


10501 


99645 


16030 


99541 


41 


90 


9.04908 


9.99784 


9.10599 


9.99643 


9.16116 


9.99539 


40 


91 


04876 


99783 


10697 


99648 


16203 


99537 


39 


98 


04190 


99731 


10795 


99640 


16289 


09535 


38 


94 


01603 


99780 


10893 


99688 


16374 


99533 


37 


94 


04715 


99788 


10990 


99637 


16460 


99538 


86 


95 


048-28 


99727 


11087 


99635 


16545 


99530 


35 


96 


04940 


99786 


11184 


99633 


16631 


99588 


34 


97 


06058 


99794 


11881 


99638 


16716 


99586 


33 


98 


06164 


99798 


11377 


99630 


16801 


99584 


88 


90 


06875 


99791 


11474 


99689 


16886 


99582 


31 


80 


9.05866 


9.99780 


9.11570 


9.09687 


9.16970 


9.99580 


30 


81 


05497 


99718 


11666 


99685 


17055 


99518 


89 


88 


06607 


99717 


11761 


99684 


17139 


99517 


28 


88 


06717 


99716 


11857 


9968:^ 


17883 


99515 


27 


84 


06887 


99714 


11958 


99680 


17807 


99513 


86 


86 


06087 


90718 


12047 


99618 


17391 


99511 


85 


86 


06046 


99711 


18149 


99617 


17474 


99509 


84 


87 


06165 


99710 


18-^36 


99615 


17568 


99507 


88 


8B 


00864 


99708 


18331 


99613 


17641 


99505 


23 


80 


06879 


99707 


18485 


99618 


17784 


99503 


81 


i 40 


9.06481 


9.99706 


9.18519 


&.99610 


9.17807 


9.99501 


80 


41 


06689 


99704 


18618 


99606 


17890 


99499 


19 


48 


06606 


99708 


18706 


99607 


17978 


99497 


18 


48 


06804 


99701 


18799 


99605 


18055 


99495 


17 


44 


00911 


99699 


18898 


99603 


18137 


99494 


16 


45 


07018 


99098 


18985 


99601 


18820 


99492 


15 


46 


07194 


99096 


13078 


99600 


18302 


99490 


14 


47 


07981 


99695 


13171 


99598 


18383 


99488 


13 


48 


07887 


99693 


13863 


99506 


18465 


99486 


18 


1 ^ 


07449 


99698 


18355 


99595 


18547 


99484 


11 


50 


9.07648 


9.99690 


9.13447 


9.99593 


9.18628 


9.99488 


10 


51 


07658 


99689 


13539 


99591 


18709 


99480 


9 


SO 


07768 


99687 


13630 


99589 


18790 


99478 


8 


56 


07868 


99686 


13788 


99588 


18871 


99476 


7 


64 


07968 


99684 


13813 


99586 


18952 


99474 


6 


85 


08079 


90683 


18904 


99584 


19033 


99472 


5 


66 


06176 


99681 


13994 


99588 


19113 


99470 


4 


57 


06«80 


99680 


14085 


99581 


19193 


99468 


8 


! 68 


08888 


99678 


14175 


99579 


19273 


99466 


8 


50 


06486 


99677 


14866 


99577 


19358 


99464 


1 


60 


08680 


99675 


14356 


99575 


19483 


99468 








CkMiiie 


Sine 


Cosine 


Sine 


Cosine 


Sine 


§ 




88» 




82° 




Sf 



138 LOGARITHMIC SINES AND COSINES 



"• 



II 

E 






_!E !«1_ 



LOGARITHMIC SINES AND COSINES 139 



11 




'M 



140 


LOGARITHMIC SINES AND COSINES 




' 




6° 


i 






J" 


j_ 




CobIbb 


Sine 


_C™ne_ 


Slh« 


_rQj^ 




B iiroo 


e.9»iH 


0.44034 


9.08SB4 


9.48.W) 


o.sweu 


60 






»»I9I 


Mfm 






98060 


69 






9ttJK8 




9«lt77 


4iiii:« 


99062 












98873 


46717 








iivu 


B84SI 










56 






3*177 


44l!M 


9H2«e 






66 








44ie? 


mei 


40811 


96036 














46882 


98032 


68 




4ifirs 




443SS 






98039 


69 




41TiH 




4J4At 




J69M 


g»tl!5 












9,98248 




9,9Soa;[ 


















49 














980 IB 


48 




IIBOB 


BSJW 




98337 


47127 


SHOOK 










446111 


98iB3 




oeoos 


lU 










OKSS 




S8D0I 














47St9 


07907 






4«W1 


!«i™ 


44778 


98893 




971183 






«140 




44B19 






978t» 






4«IIJ0 




448Si; 




47371 


97 980 


41 




fl.]K3i 


0.68t«l 


9.44900 


9.08311 


9.47411 


9.9788* 


40 














o;9;8 


69 
















38 




42^0 


IM115 






4^^) 




37 






iiftiia 


46077 


89190 


47B73 


9:8M1 


36 








4SIJ0 






97683 




M 






46163 
















45llDe 




476114 




33 


as 


«w» 


oe-iss 


4KJ49 


wiai 


477M 


97950 


a; 


99 




DBssa 


46aw 


98in 


47T74 


onM6 




30 


fl.4^ 


g.gfiSfli 


9.46334 






9.9:»4t 


30 




4JT3S 


98388 


4W77 


'bbito 


4:854 


97BS8 


99 




4;J79! 


gKSS4 




S8166 




97 981 




K 




983S1 


4646a 














»S37T 








9:o£e 


Sli 




JMI7 


988;3 


46547 


881 S6 


4>014 


9;934 


35 


3B 


42B6a 




46699 






B7918 




ST 


4W» 


sesas 


4IM33 


88147 






93 


38 




wsea 




98144 


4Si3S 


07910 


32 




4.Wi»8 


99SBQ 


46T1S 


88140 


48176 


07M6 




« 




9. 98896 






9.4S31S 




30 






Bsas-i 


'45801 




4»1G2 


'97B9S 






48iSl 


983)9 


45843 


98I» 


4829! 


97994 




43 






468BS 


981M 




97800 




44 






4BB37 
















46969 




4B41I 










SflitH 


46011 


98113 


48450 


91878 








983111 


46063 


98110 


4B190 


0J8T4 




48 






4B09S 




JK5S9 








4iiMe 




46136 




49668 


OTsee 








0.98330 


e.4(ii79 


9.99098 


9.4860J 


0.97861 




Bl 






46*» 


981194 


48647 












43'JO! 




49686 








4371M 








4»;SG 


9:«49 






4*^9 


9«J0« 


46945 


96083 


4K7M 


07846 




» 


43at» 


BB303 


46386 


98079 


46803 


OTSIl 




U 






4M:i8 




4884^ 


07S8T 


4 / 




43W)I 


jgpoa 


46460 






K8BS 






43M« 




46511 


99067 


48930 


979119 


* 1 




430W 








48969 




1 


flO 












sisii 


° 


' 


COftiD? 


Blue 


OMlnu 


Sloe 


QwlDe 


BliW 


3 


74. ,a« H. 





LOGARITHMIC SINES AND COSINES 


lil 




IS* 




B" 


!0> 




Sine 


Codne 


Siae 


CDRlne 


8i»e 


Cosmo 




g.tem 










B.079BB 




1 








'btsbs 


■i3440 




SB 


■ 


^KnB 


9T8IS 


BIS38 


OTttS 


M4T? 


07289 


58 


s 


4SI1S 


SraOB 


BIS74 






0T2RS 






«!» 


BTBOi 








B7!80 






41IBX 


BTSOO 






6SB™ 






a 


<«ai 


BJ7» 


GI48t 


B7S41 


a»l3 






i 


«ua 


omB 


GISSO 


B768a 


















saeei 


BT»9 






4BMr 


0IT84 




om 


S3T16 


97857 


51 


10 


R4SSU 


e.OTTTB 


9.iim 


0.076S3 






50 




40134 




Bioes 






'07348 






t94Bt 










97B43 




IS 




mm 


BiTse 


orsio 


5.1864 


97238 






4K13R 


mti 


si;t4 


1)7W6 


538SH 


97284 


48 




WST! 


97^9 








97Sv'9 






MtIS 


977M 






53967 


smat 




IT 




OmiD 






MBSl 




<s 


IS 


WW 


»774fl 


SIBIfl 


97488 










WW 


8774* 




07484 




OT-JIO 




■0 


g-4«Tss 






9.97479 


9.5(093 


9.97Wg 




91 


40806 


'87784 


'Bsoa; 








30 


W 




flTTM 


SiMSS 


9T470 






38 


a 


4118+j 




ftJOBO 






9710'.! 




u 


4M.'0 


STTSl 








071 B7 




» 


4IK1^8 


0:717 


S317I 


B74B7 


S4M3 






M 




onis 


Bssor 


B7458 




97178 








flJTOS 








97173 




ts 




S7T04 












» 


MHO 


BITOO 


SJ314 


97439 


S199B 




81 


■0 




n.DTSBS 








9.97159 


SO 


>1 






s3aa-i 






971B4 




K 


Mm 


WS87 


S«KI 




64500 








■■Met 


97688 


BM56 


974SI 






27 






BTSTB 


G£4BS 












EOSW 




SSS!7 




64601 


97185 




M 


MWT* 


B76T0 


B%M 


B7408 


5183.1 








00(11 


BTass 


BAOe 


BT40» 


B46fle 




S3 


M 


»H9 


nwi 


9Se34 


078BO 








so 


BUM 


S78GT 






MT3B 


97118 




U 


s.iwim 


B.B7663 


O.BKW 


9-traBO 


D.S47flfl 


0.07111 


90 










B7385 


B4aw 


B7)07 




iS 


HW9 


Bn4G 


son- 




B4«38 


97 101 




a 


MOSf 


BTOW 


aeaii 




S4S89 


97097 








Broad 


ft!»l46 




woai 


97002 






»T10 


B7S39 




B7M7 


B4BM 






« 


B0T4T 


B7HS 


S3B16 




54gBe 


B7US3 




« 


MIBt 


B^wa 


B«S1 










*s 




BmiB 


ram 


B73B8 


BSOW 






« 


lOffiS 






»ra49 


GfiOSO 


97088 




DO 


B.SOW 




».Bai}.'>« 






0.07093 








O^K! 


BSOB-J 




asis6 


97059 










6»(5fl 


97335 


55189 






SS 








B7SB1 










SIOIS 


B7B9e 








970M 






6T080 


B7688 


Bsasi 




BS-J68 










imu 


BMW 


B731T 


SS301 


OTreo 




ST 




STueo 




9731 S 








u 


9I1B1 




SSSM 


07306 








flS 


Mm 


07STI 


SSS70 


07103 




BTOM 




«0 


SIM 


BTIHT 


am 


07SB0 


85183 


07DIS 






OoabM 


SlH 


Cosine 


Bine 


Co«lne 


Sine 






71* 




70° 




SB> 



U)GAHITHMIC SINES AND COSINES 






LOGARITHMIC SINES AND COSINES 


143 




14' 




a- 


sa- 




fUiw 


C«lM 


sine 


Cnsinn 


Sine 


C<Hl>>e 





t.itaai 


as«978 


B.saiw 


0.35728 


9.U-1181 


0.958U0 


go 








6«h!a 








69 






flSOM 


«ffi49 




U423I1 




58 




BlOlfl 


KOGB 




B57I0 


M-Mi 






4 


ClOJS 


BBOM 


62703 


B5704 


e4U8a 


95311 




a 


SIOTH 




82730 






B.'>336 


65 


fl 


















«MV9 


eaoM 


B-'78* 












BUM 


wm 


6-jaii 


iim 


«43fll 


eeai: 


B* 


9 




900911 


oaasa 






05310 




10 




9.96017 


o.fiUBBa 


B. 05688 










'emi 


mil 


8SKB 


95003 




'85388 


48 


It 




96005 


aaais 


BB857 




B5«9 


48 


ia 




neooo 














BiToe 


ssw 




B504S 










fllWM 


M98S 


62999 


B50J9 


01571 


86273 












B5US3 




B5a«r 




17 
















18 


ti4sa 


flSSTl 


63079 












t14ll< 


93996 


Bsioe 


05016 


01673 


85:^ 


41 


a) 


8. BUM 


9.96960 












31 


*\m 


BS9M 










39 


« 


SIUO 


W9<8 


03180 


9M9; 


Ofiitt 


85iffl 


88 


SB 


«ST8 


KM2 




05591 


64775 


85*3 


37 


H 


SISM 


96937 








96ai7 


36 


W 


«IM1 


95WL 


63i»G 




64826 


S6»t 




N 


tlBOi 


939S6 


63398 


95578 


648S1 


96201 


U 




eitag 


9SBS0 












>« 
















» 


S1746 


M9oa 






84987 










9.9SW>3 














eiBOD 








' 6 1078 


'uisi73 




3t 


SIB^ 


9BS91 


83451 




65003 


96187 




3a 


aiBBfl 


wess 


63479 


96581 


85089. 




117 




«S83 


96879 


















63S3I 






81148 




M 


S1S39 


«58SS 


63667 


95513 


65101 


95111 








!am 


03583 


0MO7 


85130 




93 


sa 






66B10 






B.5129 




as 


B«Wl 


assso 


63636 






85iaa 




w 


B.S9H> 


9.as»4 


9.B3«63 


R.K4gS 


9.66205 




99 




teoTS 




S3S89 


85483 


«5sao 






« 


BS104 




6371 B 




BB36.-i 


85103 




« 


Bsin 




M741 






95097 












05184 


85306 




16 


45 


BSiSB 


B5BIS 












« 


SK14 


96810 


63890 






05078 




47 


a»ii 


96S01 


68848 






95071 


18 


4B 




B5709 


6»«7a 




BMOa 


awos 


18 


49 


««s« 




fl38!l8 












0.BS3SS 


B.asIBS 


B.03994 






D.05052 


10 


61 




9n;B0 


68950 


OSl-'l 


65481 


0.5040 


9 






BBTTB 












U 




S6TS9 














BM13 


9sm 


84018 










M 


SMBa 


BBTOi 


B4W4 


03397 


B5580 


95020 


6 


6S 






64080 












WIS 


BaT45 








05007 




B8 


ami 


96739 




OMva 




B5001 


3 


H 


•MS 


KBTSS 


641S8 


033^ -J 


OSWIil 






W 




9flTlS 




B58011 




U49SM 






OMina 


Bine 


Coainfl 


lilDe 


CosIdb 


Bine 






efi* 




■4° 




if 



144 LOGARITHMIC SINES AND COSINES 



II 





LOGABITHMIC SINES AND COSINES 


145 




(O- 








a- 


^ 


ai»e 


Coxlae 


Bine 


Co«lae 


Sine 


riw'ne 





».«)WT 


9.»37ri3 


8.71184 


B.eSBW 


B.TUJSl 


B9S84* 


00 




sniB 


OST« 


T1S06 


9S399 








a 


seui 


93738 






79)61 


9t!S» 




a 


89983 








ja4Bi 


i»l% 






60981 


IBTll 


Tiaas 


BSeTB 


TS60* 


9^810 


68 








TtM9 


»3«B 


7i«a> 


















93799 






raow 








TflSfli 




83 


B 


TOOTS 


Dsrao 


7ISS* 


93StO 


735SS 


03T7B 


63 






03887 






T.;BOi 


smi 




10 


9 7011S 


B.fiWSO 






9 l-Stii 


B.B8T03 






WIST 


98873 


71114 


9:li.3 


Tjwa 




49 






MOW 


T14M 


9:WI5 


74883 




48 


11 


TOIM 


B38BB 








9^739 




14 


70909 


9SSM) 






7K03 


93731 






Toat 


99843 


TI1B8 


mm 


TmM 




46 


IB 


TOMS 


MM 


TiB19 


93184 










TDMT 


»8S8 












IS 


T0W8 


93811 




93169 




B-JUBO 




IS 


IMIO 


93614 


71S61 


03181 


Tasoa 


8:«91 








9.B3808 


9.71808 


8.93154 


B.7S83S 






■1 


TOUS 




T1S89 










» 


7087B 


OlMl 






T-Wl 


OiflflT 


38 


» 


TOBM 


99884 


TI684 


BS131 


7SS8S 




87 


M 




9357T 


T188S 


83123 










tOU9 








Tisai 


99U8 






70«1 


B3S8a 


7IT« 




73949 


BSB3B 


M 




Tom 


BK54 


71T47 


9SI00 


7B98-J 


oasT 




19 


TOftM 


03AtT 












n 


TOMB 


«5S« 


7i;88 




73tf0:i 


BSBll 






B.TIWT 


0.9363i 


0.T1B09 


9.93077 


9.T30&J 


9.9^608 


30 


II 


Toue 


9SS»> 


71899 


93089 




92801 




M 


TOStO 


SUIT 




BW6I 




09687 




a 


lomi 


98S10 




8305.1 


73081 


S»7B 


ST 






SR0« 


71881 


»WM 
















83038 




00563 




M 


TO»TB 












24 


m 


TOMT 


98480 




BKU« 


73180 




28 


n 




93<Ta 


71973 


9MI4 










TDTtS 


S3465 








9«SiO 




40 


o-Tonii 


9.S8«7 


e.7!014 


9.93998 


9 73SI9 


S.9K99 


20 




nra 


SUM 




8«&1 




99514 




41 


Toaos 


93HS 


raB5 


1K083 


78»» 


92606 




a 


TOBM 








73 -VR 








TOBtS 


9«W 






7S*9H 




IS 


4a 


TOMT 


98(90 




9-J9W 




62489 




4S 










73337 


eU73 




47 


TOOOB 


9S4DB 






738,^7 






48 


T0U1 


93S1IT 






73,177 




IS 




TD9M 


9S390 


72198 


wmi 


73388 






00 


9.TDI>T3 


9.98889 








0.B9411 






Ta9<H 


93376 




*asi8 


'7313S 











ft338T 


TM69 


»fllOS 


7S1SS 


9'^is 


B 


' u 


Tioas 




7WT9 








T 


M 


T1058 






9S889 


73491 


92408 


« 




noT» 


99344 






73013 


82400 


B 


U 




S933T 




B'*T4 








S7 


Tllgl 






8aM8 




92384 


1 


«a 


Tim 


Bssai 


T^HItl 






92S;« 




ss 


TIIM 


93314 


7«01 




73691 




1 






sraoT 


7WS1 


B«I2 


T361I 


9-^ 







Ookliw 


Hne 


UOtlMB 


Sloe 


Coring 


Bin, 






W 




6S" 




57' 



146 


LOGARITHMIC SINES AND COSINES 






88° 


84° 


B 






Rine 


Came 


aba 


CosliiB 


Biiio 


CciD* 





a.Tsaii 


a.ftswa 


~9 7475a~ 


D,9:8,W 


9.T6SBB 


B. 91 888 










'74718 






mvts 










71791 




78898 


018IB 


58 




I8«B 


oasM 


7481S 


91fl3* 


75818 




87 






B'JSM 




918^ 




















81 «» 














78987 




84 




MT17 


esao-j 




91798 


751«« 




63 






Bssea 


;4»06 






B11MB 
















B11W7 






1)T3905 


9,92277 




S,917TS 


9.76080 


s.eiuB 


60 






Mapj 


i*»l 


91 759 




91ESB 


49 


IS 










7007B 


eisio- 












BI74S 


78098 


9isn 


47 




la'*! 






91788 


76111 




46 






e2as5 


7.1088 




70129 


gian 




19 












91194 


41 




73940 








78164 


0118G 


48 




T3S5a 






917DS 


78188 




43 






Bsaoj 


78110 


91<He 








SO 


B.Tawr 




0.781SS 




9.78818 


9.91158 








KIM 




91877 


7fiS«8 


91 H9 


SB 


ss 




Mirr 


781SS 


BISBO 












Ml fid 


I5IB1 


pieao 


78S71 


91113 




^ 










7e»« 


9iias 


38 




Tioea 




752(1 


91848 


78S07 




88 


X 


ItllS 


WH4 


782119 


018.14 






84 


i! 










^8841 


91098 




88 


T415I 








78880 


91067 


aa 


W 


T41T0 


oans 




9:008 


78379 




.11 






B.osm 


0.76SI3 




9.78305 


8.010C9 






"71208 


«103 






78413 


91080 




Si 


7^n 


BSOBt 




9 8S 


78431 




« 






flsoeo 


78388 


9Ka 








M 


'rlESS 




75as« 


9 eoe 


7D46S 


91018 






T4S81 








7B4M 


9ioes 














78501 




I 






Ba»a 


75MI 


9838 


78619 






88 


T434I 






9£ai 


7858? 


90998 




m 


T4a60 








788M 


90907 


9 


40 


0.TJST8 


s.gacr 


9.75)99 


B.B 518 


9.78871 


B.9DB7S 


99 






WOIS 






78690 


90989 


1 / 








msi 


fiwa 


78807 


BOBBO 


I 




T1J3U 






9*88 


7M1S& 


909B1 










76.18a 




reeia 


9DB4B 








01985 




9 4e> 


78880 


90988 




48 






75Ii05 


9 40) 


70877 


90SM 












9 181 


78805 


90918 


1 


W 


T4S3I 
?4Mfl 


11k! 


761100 




78730 


90900 
90890 


1 


50 






9.78078 


B,014Hi 


fl.78747 


B.908B7 


1 




















7-I6P6 




7^714 




7878* 






S3 


7Kia 




78T33 


91198 


78800 








74M4 


91908 




91389 










T4S«S 










B064S 






74881 




75787 






90B8( 




ST 




Biess 


78805 


B1:H3 




90828 




58 


74718 




7B8S3 


BIS54 


70887 


90814 




M 


71TS7 


SIBW 






7eOIH 


sons 






T4T5e 

Cosine 


9IBB7 






Taow 


99798 




Sine 
fi8° 


Cosine 


Slue 


Cortna 


Bid. 




GB° 




C4> 





LOGARITHMIC SINES AND COSINES 


147 


■ 


ae- 




7- 


le 






Sine 


Cosine 


Slue 


ttslne 


fibe 


Cosiae 


^ 


».r69ia 


9.9079* 


9.77910 


9.6I«86 


9.78934 


9.896S3 


80 












78850 


BB848 


BO 


! 


fflft^ 


90777 


7^60 














907B8 


77997 


9D3US 


78983 




67 


* 


77009 




reoia 


90197 
S01H7 


78999 


80601 


68 


6 


rroso 


9(fi-41 
















80731 


7MW3 
















78080 


901^9 




89ri-4 


69 




T7078 








79079 


89501 




10 




9.90704 


0.7Hlia 


9.90139 




9.B9&.M 








»(»91 






79111 


89514 


49 






Boess 






70128 


89534 


48 


la 


77UT 


9067B 


7tfI63 






895:14 








90667 


78180 


90101 


















79178 








T7ira 










SBIBS 


41 


IT 


JT»18 


90639 


ra«3o 








43 


IB 




90680 




90068 


79.'il 
















'.ana 




n 


10 


0,T7W8 


B.9061I 


9.7-iiSO 








40 


11 




90fl<W 


78*96 


90081 










iraoj 








79288 








77319 






90011 




8B1S5 


87 


M 


TJ33a 


90671 


78310 




79019 




38 


as 






. 78S6J 


89995 




89105 


35 










SOBSS 


793S1 


80395 


31 




7T387 


90S 10 


T.'«a5 




79387 


fiat8B 


33 


K8 




9I»S7 


781 la 










as 




B06>7 


78428 


8S9M 




B»30t 




M 


B. 77139 


9.90613 








0.89351 


so 


81 


7T4M 


90609 


'twbi 








sa 


M 






78178 


mi! 


70447 


89BSI 


m 


sa 


77190 






89918 




89IK1 


X! 


Si 


riM? 


90480 


Tsaio 


unsosi 






ss 


K 


T75»l 


90471 


7B&i7 


149898 








M 






T861S 


89888 
















89879 




88281 


93 


S8 


7T5TB 


90H3 












at 


77S9S 


00434 


IM93 


898,™ 




89U64 




40 








9.89849 


0. 9673 


B.80564 


20 




TTtua 




'ffiOSS 






80*14 


IB 


43 




OOIC© 


7B64a 


89830 


-9805 






49 


f»W) 




79SSB 


B08« 


8W1 


80-^33 




41 


TTflTT 


90886 






-TWSli 


89218 




43 


■mat 


9037T 






-985! 






4fl 




90S99 


78707 


89791 


790fiB 










B03G3 


787SS 


89781 


79eRl 






48 


ma 


9DMD 


78739 




7W109 


80173 




40 


TJTfll 


B038B 




89761 




80162 


11 






0.903.^ 


9.78rra 


s.ssmi 


0.79731 






SI 




00390 






79716 








rrsis 


B0311 






797Ga 


81132 


8 










89r'jB 




80 28 


7 


M 


77846 






S07111 


70793 








77882 


9aes^ 




89T09 


79809 


89 01 






77S79 


90973 


78880 






86091 






77B90 




7BB80 


HB088 




89081 




U 


77013 




7BB0a 


80073 










TTVK 


90244 






71B7i 






ao 


mu 


ooa«6 






7D«t7 


80050 




' 


amine 


sine 


l^lD« 


BIlIB 


Cosine 


Sine 






tt° 




sa- 


G 





148 


LOGAttlTHMIC SINES AND COSINES 




~ 


>■• 


40 




■ii" 


• 


SiDB 


CMlne 


Sine 


Cqi^bb 


Slue 


OoalDB 





g.nssr 


B8BOS0 


9.80S07 


9. §8445 


g igB, 


B.gTVTS 


00 




nmM 










81167 


BO 




T»9ia 


sgoaa 




B8101 


^Taa 






3 




890»l 


S085S 


88391 


ITS8 


87745 






IBBM 


S9009 




wass 




877S4 






;9M5 


8B»U9 






1787 


BTW8 






7B961 


X8989 




Bsaw 


1781 






J 


7BWW 


S8968 




88851 


ITBd 


t.;TOI 


S ; 


» 


soiiiir 






















B.8BS1S 


S.8)88B 


9.BTB68 


60 
















41 




SOD74 






»<1IS8 


81868 


87ftl8 


48 


IB 


SMI8B 


881117 


BIOM 


88887 


8188* 




47 






SBSOfl 


81017 






8iw4 


46 






888ea 




»i!K 






4S 






8B88H 




mK 


SlWfi 






IT 


80ISI 




simi 




B1M0 






IB 


WIW 


K8H85 
















mss 






6ie6B 
















fl-SlBB3 


9.87857 


40 


« 










81908 


B7MB 


89 






SHtm 


S1l» 






87MS 




B8 


B0844 


88813 






BMM 




2 






BSSM 


SUM 




88041 


BTfllB 


M 


■it 






81180 






BTOOl 




X 




88TH3 


81 m 










Z! 


mca 


asTTS 




MIIS7 






£• 


a 










9»98 


1^468 


?1 


s 












8T467 










9.B1«5( 


B.BSIO.'S 


B.BSm 


9,87446 


S 1 


SI 








8B0B4 


8SI41 


874S4 


so 




MiW-! 




B1!84 






BI43I 


n 








81«S9 


mtn 


811 m 










Bsrm 




88061 


Bil84 




S 1 


S5 


NU4'^H 


B8S88 




RBOSI 


8il98 


8!«(I0 


to 


36 


»>4t» 










8TB78 








B8«8 


BiaSB 




ea» 




n 1 


18 




SSffil 


613W 






mee 


» 


3> 








8800T 


saw 


87846 


n 






e.SSDH 


B.8140f 






9.STS34 


90 


41 






81417 




esaBs 














SiBTB 


smr 


87811 


IB 


« 


SOaM 


HBSOS 








87S00 








SS9D4 








S7KS8 




« 






B14T5 


6:943 


81340 


87*77 


IS 


« 








KWl 


mM 


87966 






eoaio 


BS663 




BTBW 


88368 


87*66 




48 




8SMU 






B;a8e 


STMS 




4S 






81684 




8«396 




11 




B.soGse 






fl.BT88T 


9.83410 


B.87Sai 


10 




HOST] 


'8BSS1 








emg 




U 




seMO 


8I6T8 






87198 












8785fi 


81!4.-S 


RTlSr 






80718 






(fTSM 


88497 


871TB 


I 






BB478 






8:2481 


sriM 




W 




ilns 


SICBB 






871 BB 








BB4&T 






8!sa9 


87141 


• 




8mT7 








Kisas 


87in 


■ 


ss 


m-.as 








WMT 


B7Ilfl 




so 






B1WI 


BTTTS 


BOM 


enm 


9 


Sine 


(■oeli-B 


aiue 


CiHlDe 


SlM 


, 




_ 


0" 




9" 


48* 







LOGAHITHMIC SINES AND 


COSINES 


149 




4a* 


48° 


4 








Sine 


Coiil.8 


SI lie 


(^iDO 


Sine 


Ccwins 




B.8K91 


9.81107 


8.8B378 


0.88413 


0.81177 


0.85003 


60 


1 


ftHKU 


8J0» 


8330^ 






8.3681 




a 


^79 


8T0BS 


B810S 






&W69 




3 








8(377 
H6866 


SlKfl 


Mfl45 


57 






STiea 


mtie 






H603J 




S 




87039 


SJIBfl 






85020 


54 


7 


BiSlS 




83478 


Mmo 






G8 




mm 








B4;J83 


B5B86 






aan 






88800 




B55S3 




10 






9.83513 








60 




'er.K 


' 86982 




86i8.1 


B438I 




49 




anis 


SeOTD 






WS3I 


B5SJ7 


46 


\ I) 


81793 


fma 


88854 






BM31 




1 '* 


ma 


80MT 


83587 








4B 


IS 




eeeM 




86a3,'i 








11 




86SH 


88594 


882^ 


81385 


85497 




1 „ 




sesis 








85485 




I IS 


SMOl 




8Mai 


saitoo 






48 


. 1* 


89810 


ensM 


83081 










» 




B.868T9 


9.83S4B 


9.88178 








n 




80M7 


88681 










1 a 


8a8G8 




83874 


BfllSB 




85I'J3 


38 






8U844 


8.^88 




84478 






H 




WSSd 


88701 




844BS 


H,}309 


S6 


» 


838» 




«37;iS 


BUI 18 








« 


»«I3 






80104 


84015 


h58;i 


34 


? 


uui 


887^8 






845»l 


».-.!« 1 




« 


B»HI 


8828 






84540 




33 


a> 


MKS 




83768 


BOOUH 






31 


w 


e.838AS 


8.88781 






0.84588 


O.M,334 


ao 


« 




88763 








H6313 




N 


S«<M 




83808 


8C03S 




Bir.w 


ss 


3) 


BWIO 








84805 


85387 


ST 






88717 


83834 




B4B18 


8M74 


38 


30 




86703 


BSHIB 




84880 




25 


M 




86801 


83861 


B5B81 






34 


n 


eM6.1 


MflSj 




85973 


8(858 


859^7 


33 


» 




88870 


83887 




84609 


853-S 




» 


BSQ93 


86eMI 


eSBOl 










« 


S.8310S 


0.86847 




B.Wioari 


0.81694 


0.85300 


W 






88838 












« 




emt 


esBJO 






85175 




4S 






8.3054 


85900 


84733 


85163 


17 




881 St 














S 


MJ» 


RSSSO 












4t 


8Iin 


B857T 


88993 






85135 




<T 


8WS 


88588 


84008 




81781 


BS114 


IB 


M 


SMS 


B8M1 


B4ft» 


85839 


H4T9C 


aMOn 


la 


« 


SMdl 


8^13 








85087 




5 


B-BIHS 


B.BS8.W 
a8»lH 


■ftlOSO 


'■S 


t.Mm 


0.^ 


9 


n 






K107S 


s.^79l 


WS47 




8 


w 


ssai 


88IK 






81880 


W037 




1 M 


sam 


88488 


num 










W 














8 


« 


s»u 


88l«6 


84135 


85743 








ST 


m» 


BHIB 




85730 


81911 


84986 




n 


gmi 


86498 






84933 


04071 




w 


8»U 




811M 








1 


ao 


ssrs 


88418 


84177_ 


8.W83 _ 


_B1W9_ 









{Mne 


Sine 


CobIub 


*6* 


<'.OBln9 


4fi' ^'"' 






iJ* 



150 


LOG 


TANGENTS 


AND COTANGENTS 






«■ 






i 






Tan 


Colan 


Tan 


town 


Ton 


Cotan 


~ 


em-i 


n f>w>7 


"wifo 


II.TSSOS 
7S090 


"■"mS 


11.46693 


60 


I 


'?(ii;u 




»aio 


71381 


65037 










u^9lA 






66B83 


44618 


67 








sum 






44!M9 






IflKO 




Km 








63 




S41HS 




S8833 


71668 


6M» 


43571 


64 




8aH»i 




S89tia 




6fl7TS 


49Sa7 


63 






Giiiim 


se«» 






43hBS 


63 








B0M3 


Bfl7S7 












li.'Olii! 






8,67798 


11.42iH 


60 


\l 




m»» 










4V 


flttiSI 


4ST08 




B-.m 






48 


«?ru7 






mm 


68779 


4IK!I 






niDM 






floflsa 




4069S 




IS 




36018 








40673 






33116 


84481 


06639 




tout] 






(ifl'iiK 




afiOito 


WflTl 


60068 








TIDOO 


^LOO 






60384 


39616 






74«S 


ara 








39303 










a.aaMB 


11.S8311 


8.6100B 


11.88991 


40 


Xl 


THWi 










38681 


sa 




I Bass 








8837« 


38 








MWB 












K43H 




31«OT 


8I19I 


tesM 


87766 


88 




KUJ6T 






80677 




37465 


38 


ai 




13IS9 
















10190 


403»l 


B9S60 










DIWO 




40H80 


69170 


BWM 


S66T4 




£ii 








fiSeiB 


63718 


96383 


31 








8.4:807 








30 








43887 


5771* 


64998 


Mica 


at 


f 








67SBS 


64MB 


35415 


38 




oirr.^ 






64870 


3E130 






V^i 




4ieM 








36 




B.omi 




«16« 


5*814 


664S6 










'btow 




waw 


6SJ10 


B438S 


34 


s; 




guHOC 




04139 


06998 


K1D97 


ii3 




ut»s.i 








66169 








OMm 


B15I9 




siosa 


86M3 


33467 


fl 








8.40S8S 




8.60818 


11.3SIB4 












68163 
















6'.'75S 


67S56 


83644 










47«B 


ESSSl 


676M 


88876 








SB-.tW 


480SB 












1169U 




iseos 


















moss 




81688 












50076 


686^ 


Sins 














eSBSg 


81(W3 






ia»95 


Midin 


60130 


49870 


69198 


80604 








ii.MTsr 


s.fwaj 




8.66458 


11.30B47 










D09i!n 




BS768 








179^8 


HDOt!) 


61310 


ifmm 


099a 


80938 












4«3n4 


70!a4 


S9J86 








RoSHi 






704B6 










mw 












w 






eaKB 


47IGa 




39038 








^SDM 


sa-ios 


46TW 


71!W8 


S87KS 








77W0 




4Maj 


71453 


SM47 






asM-i 


TOBM 




480» 




38303 




JO 


■urn 




!Mmi 






iSOBO 




CpUu 


Tm 


_CO«M 


Tan 




Taa 






-^ 


BB* 




88- 




87' 








LOO. 


TANGENTS 


IND CO 


TANGENTS 


151 




■ 








fi° 




Tin 


Cotao 


Tan 


Coum 


Tan 


CoUn 


^ 


B.T1M0 


l.gaMt 


B644B1 


11.16fl36 


H 9419,3 


11.03805 


60 


1 


niai 




' 8461(1 




'9434a 


osseo 


EM 


1 






MSM 




W485 




iS 




7S8M 


27841 


85000 




Moao 






* 


s 


IS 


8il85 


14637 


94BI7 


0.-1089 


M 






»ea4 




144W) 














SKI- 










9 


?s 


asoK 


860119 


i^i 


asim 


0IS14 


61 




8.7tm 


1.2S708 


8.96S4S 












■;»ii 










' 04:^38 




1 


7*7(8 


SS1I6^ 


86591 








48 






8IMJB 


Bfl7M 


laasr 


imi7 




47 


1 




swoi 


86913 












7B«S 


21577 






963^3 








pws 


aatt- 




12723 


93494 


036.16 




1 




Hiat 


87*17 


liM3 




03398 


43 


1 




a»L.) 


87616 










1 


Tens 














» 


9 TIB'S 




8.87913 








40 


a 




■«K58 


BSliO 


nem 








n 


7WM 


BUMa 








08713 




K 




U3^ 






97431 




37 


H 




nsiS 


88618 


llWl 








a 


ITHU 


«M00 


S3;ai 






l»309 


3S 


» 


77SII 


taisa 




1 lOB-i 


9:ius 


02176 




ti 


■am 






1<MW 


97939 




33 


» 


■ma 




M9j;4 










» 


»44l 


KIBSB 


8B4sr 










» 


B.7B8« 








8.983.38 




■0 










■lOS40 






29 


m 


Twei 


aosas 


SODW 




mats 




KS 


a 




J0T3I 


SOOBO 






01li47 


27 












98881 






n 






90399 










M 


79873 


20133 


90667 








ii 


m 




ie«4 




09-J85 


99.'7S 






1 M 






Doen 


09128 


99406 






W 






910^-9 


0H9TI 








M 


e.90«T4 


I.IKMO 








11.00338 


20 




80HT* 






'0BU6IJ 


"93TB 


oa'.i« 




43 


sioes 


18934 


B49S 


mtim 








a 




18786 


9650 


omo 




10.9MM 




M 


SUM 


lUtl 


om 










ts 


SIOM 


ISBiT 






OOSlI 






4S 


81K4a 




g^iio 


07890 








n 


B«MB 


JMW 


D«6'j 


07788 




»ll'l4T 




a 


s«.w 


17770 








91iS.'l 




4» 


suso 


irs89 






00WI3 






M 


e.suia 




a.9a7i6 


11.07281 




10,Wii70 




SI 


S8TM 


17201 


1WS60 






9>>"ir. 




SB 




17018 




069^4 




BWJI 




SB 




leflSS 


















933ia 


lttOB7 


01427 


9W73 




U 


BIMT 


1BIS3 


93163 






9^m 


G 


u 


sKas 


IMflS 




llfi:;til 




e^ii 




57 


s«ia 


1«M 






01799 












0^9*1 


(11W7 




9Hi83 




sg 


BWW 


1571H 












■0 


BMU 


issaa 


91195 






91838 


f 




COIBQ 


Tun 




es. -^ -— 


Cotan 


Till. 
B4* 




88* 





152 LOG. TANGENTS AND COTANGENTS 



_Ii5 e2l5S 1511 C"" 



N 




-Jg p ' "» » I?n__ 



LOG 


TANGENTS AND COTANGENTS 


1S3 




»• 


10° 


1i° 






Tm 


CotuD 


Tan 


Cotau 


Tun Cotan 


B.lUTl 


loetws 


B.B4«8a 


10.753C8 


B,K865 I0.7II3S 


~tio~ 






9170S 




SBBBS 






10184 


;Bi»B 










SMIS 




W858 


7S147 


awM7 TOBBS 




soar 


jb™ 


24»3B 


76074 


a)lS4 70888 


















78541 




740S7 






BMW 






74SS1 


assis 


ocw. 




mn 


i»a;» 




74781 






















B.TOTW 


10.79218 


s.essBs 


lo.-Am 


B.asnao lo.-ww 




!08« 


7UISB 








0399 


49 




7B058 












78978 












J88W 


iSK^ 


7«1S 


S980O 






am 


7R818 


SBVJT 




33888 -OlSl 


45 










9998S 














S99Ba 70001 




ai48o 






740S7 


sooai ewse 




Bueg 


78601 


86015 


79885 


S01S0 


9870 




».ai9TS 


io.7mai! 






B.8fl1B5 10. 




40 


amr 


783(3 


'SSIM 


"tssis 










7a2M 


aewo 


73771 


303SC 69674 














MOB 


S7 














30 




TSOM 


WM-i 




80A»II fiBlTB 










73180 


30587 


9113 












soastt 




3H 


imi 








30717 






^stm 


77717 


KTSil 


73?74 


307fiJ 


JBSIS 




0.1«WI 


10.7T639 












S!1W 


775«i 






'30911 




SB 


9KIB 


7T4S( 


iBBB7 


^8063 


»9T5 


19035 










7i99a 


31040 


18980 


ST 


saam 






789S2 




B8D6 


86 


S4717 


nB53 












a-»ii 


7T17U 


S7!I18 


7a7S( 


31!!>S 


8787 




89901 




SK88 


rjTia 








2M77 


ma 












SiOH 










8*75 




o.sma 


10.788™ 


B.e74M 


10.7aE04 














™bi 




8448 






78717 


srea.-> 


73806 


31818 


8384 




ssasB 


mil 


S7T04 


7ffl!9a 


318T9 












7aS87 








aasio 




srsia 








asw 


70414 


47011 


TWWB 




9130 




33IWI 


76JSa 


rvm 


TMSfl 


31988 






H7S7 




a904fl 










9S:tLJ 
















10.7BI19 


B.WtSC 


10,7IBH 


g.sam lo. 


7878 








assM 












7-i9«a 












Wlli 


7MS8 




71809 


Mil 


:«.S9 






75811 


asiM 


aMi 


tsars 


7887 




Si»l 








>m!K a7H4 




■««5 
















76SB0 


asssa 






MS9 








3S7*1 


'J™ 


3Mih 














iWHHS 






MBM 


75.188 






38747 B7S53 




Cown 


Tan 


Cotaii 


Tan 


0«.>n Tan 






80* 




JB- 


J8" 





154 


LOG. TANGENTS AND COTANGENTS 






11" 




8- 


1*° 




Tan 


Coian 


Tim 


to ran 


Thh 


Cotau 


~~ 


g 827*7 


lo.araw 


0.3MM 


10-03034 


O.aOOJT 


10.60323 


60 




'inaw 
















3397a 


S7I!a 










OB 


a 






30509 


63401 


39838 


60102 










30600 


63134 


89803 


















BOOSfi 


65 




33119 










00001 














40052 


ceMS 












eaw5 






02 














60841 


G1 


10 


0.3385.1 


10.S003S 


0,36909 




9.40e]B 


ln.5fl7gB 












■«ao34 


40266 


G0734 


49 




















33Ma 




HTOoa 






60388 




11 


38609 


60391 




ogsoa 


4«K 


MBIB 


40 








37193 


OiflCr? 








la 




60260 








69469 
















69410 


43 




38853 


861*7 


37J63 




40636 




li 


19 


S3313 


660S7 


37410 




40680 








9.3307* 










10.B9S68 


40 




3403* 


flsoeo 


3753i 




40705 


EOfiOO 


ao 








87588 


0S41S 






38 


i3 


















31il5 


65785 






40952 




36 




Mm 


8S7ai 


37760 


fl3i44 


4100B 




30 








3;ai2 


Mies 




5804.1 


3* 


sn 


B48D3 






62138 




£8891 




as 


3415B 










sseta 


sa 




84SI3 


05*81 


37081) 








Bl 


so 






0.38035 






10.687M 


so 




'84385 










68089 


«9 




84BBS 


05300 


38147 




41B70 












38KK 


61798 




08578 


«7 








38*57 






oessA 




ss 


34874 








41826 


EB174 


£5 


m 


34D38 


05007 


38308 




41678 


58423 


S* 




31002 




384iBi 






BSB71 


SI 


S8 












08916 




as 


3ini 


fi4BU9 




61400 


41733 


6SWT 


« 


40 












10.58216 


90 








'38044 






68164 




4S 










4iBar 


G8118 


IB 






610B3 


ai7&i 








17 








388<W 




41000 


68010 


10 


« 






388(8 




4C041 


67SB0 




4B 


WB23 


B417? 


38018 




42093 








35B81 










67850 




ffi 






330-17 


COOTS 




57806 




40 




















10.04WJ 


9.39130 


10.60854 


9.4JK07 


10.67703 














403(B 


5;«5ti 






39ST8 








*e30U 








83031 




80299 


80701 


4MM 










04011 


30303 


606(7 




67499 
















57448 






861 OG 


63805 






4iI603 






S7 


36103 


03837 


3061.'] 


Ofuna 


4SC53 


67347 




S8 






39.-^ 


60)81 




67906 










30023 


60377 


427M 






_l_ 


3633(1 




aiKir? 


003^3 


428W, 








CoUa 


Tao 


CoMn 


T'au 


UdUn 


Tan 






77" 




7lf 




7S" 





LOG. 


TANGENTS AND COTANGENTS 


165 




16' IB" 11" 






Tan 


Oown 


Tan 


Coian 


Tan 


Cotan 




's 


'li 


9.45760 


10.54250 


B.4t"53( 


10.61468 


69 








45^ 


hIm 


48669 


waat 






43007 


MWI 


4M40 


BIOIW 


48714 


S1SB6 






43057 


MBI8 


45887 




4K75B 
















48801 








4S19S 


US IS 


460S-J 




4s»4g 




63 




49208 


E6TB-i 






4NS91 


61109 








sarii 


46177 


msi 


48939 
























■4:1358 








■490«l 








mm 


MM* 


4IW1D 


636S1 


4M7S 


60927 


48 




OKS 








4anH 








asos 








«■*? 








43007 


MITO3 


46507 


63453 


40362 


M7?2 


44 




4SBB7 
















4SW 


HtI93 
















fi6£44 


46648 












ii4»oa 


lo.aeigi 


9.4aCBI 


10.53306 


9.40430 




40 








49:41 












4agi» 




467^ 












439M 


69048 


48835 


63166 


40663 








44DM 




46B81 


53113 


49607 


eosfti 


ie 
























46975 




49690 






ta 


«4iSl 


668»B 


47Q-JI 


»07» 


49740 












47oea 


6393'i 


407M4 


60218 


a: 


ai 








tlSSSO 








K 


B.41W9 


10-Bsroi 


9.471 BO 






lO.BOlSB 










i7;J0: 


BCTM 


40016 


60081 






44^r 








40960 




28 




44146 










40996 




u 


41105 








60018 


49962 






44641 




473W 


B'WOa 




49008 










47438 






40891 






44611 










49890 




% 


416W 


ttevi 




6M70 


hfl-.;j3 


49777 




99 






47578 


m-n 


5038! 


40733 


W 








9.47992 






10.49689 






■448.W 








■5036B 


496 in 










47T14 


6iM8 


60398 


49603 




































a 


4I»W 






62148 


MMI9 


49I71 






4S0TS 


SlflW 


47KBT 


53108 


H0S72 










64871 




SE057 












USi'i 












« 


isasi 




48035 


B1005 










S.45W1 


I0.54ra9 
S4«t1 


S.4S0W 


'""sic 


0.60746 
60789 


10.49254 






433«: 








60633 






5a 


4541s 


64695 


4SSI7 


61 783 


50879 








4UC3 


MMT 


48383 




rooi9 


49081 






4fiSII 


641BS 






60WK 






















» 


4.'M0« 




48'i9S 




S1M8 








4S6M 








51003 


48UOH 








Mssa 












90 


4A750 






51486 








, 


00 MD 


Tan 


Colnii 


!8° 


(■■itaii 


;o.- "— 








„o 





166 


U)G. TANGENTS AND COTANGENTS 






18' 




9' 




SO- 




Tbd Cotan 


Tan 


CoUp 


Tan 


Cotan 


~~T~ 


9.61178 10 48823 


9 53697 


10.46303 


9 56107 


]0.4«lfta 


60 




Smi 19779 


■68758 


49i5! 


"6614B 


4SSBI 














488.5 




i 


51349 JBMI 


B-iaao 


46139 


swu 




67 




B133i 1S«08 


6:^90S 


4U0(« 




4B897 














43958 




I 


Birio 4^ 


r*)35 


45K5 


5B«0 




68 




iim jsjsr 


BlOffi^ 




5B459 


43541 




10 


B-SJM6 '"J^ 


9«jW 


'"■^fla" 


9..WIM 


10.43508 


60 




MOgi 4SH09 




45SI8 


66578 


«4«l 






MTW 4Bi»a 


64SM 


4.'f77^ 




4S3((B 






BITTO 48-ei 










46 


IS 








56693 




46 


IS 


51S61 481.19 




45350 


58731! 








SIBie 4flOU7 


WSM 






4rs;9 
















4S 


IB 








5684B 






iO 


9.52081 10.47959 


o.M5ia 


10.46488 


9.B68W 


10.48118 
















sg 


ea 












58 


ss 




64033 


45367 


670(14 


4»98 


S7 


m 


BSSOO 4-fWO 


54873 


45&I7 






as 




nm-i 4-TB8 








42919 


35 


» 








57190 


42B80 


34 




flasa J-S74 


517W 




67168 


4*84* 


33 


88 


sms im 


6fH85 




67197 


48803 


B2 




S)!4J0 476M 










31 








10.45085 


9.57274 


10.4!i79C 


30 


SI 


6i4W ITBOG 


B19.U 


4WHS 


5 51* 


4)M88 


» 


S3 


53536 4T4B4 








49849 


IS 






55035 






4S811 


87 








44»» 


5-428 


4:2572 


M 




weei »7389 


55115 


44885 


6 466 


4S534 


» 


3S 


sjrns 4T-;fl7 


5SI55 


44S45 




4e(«e 
















as 


sa 










4!4I9 




sg 


BfflSS 4JI71 


m-.s 


447» 


5-6iB 


4X381 




40 




B.5SS_I5 


10.44685 


9.B7BW 


10.4!S41 


so 








44646 




4x304 


19 


«» 


Sa953 4rM7 




44605 


677W 


tana 


la 


43 


B8M5 47005 


65434 


44666 




4-£!sa 
















IB 












4»SI 




4S 






44446 


67887 


42113 




47 


6.116I ««:» 


W693 


44107 








48 










t»«r 


11 










68001 


41999 


11 


BO 


9.6.iaHS 10.48715 


8.SKI! 


I0.44il88 


S.6S089 








BSaSI 46673 












ta 


53368 4Ua-.! 
B.^WO 46fflt 


55™i 


41169 


68155 


418BJ 


8 


G4 


53450 465M 


55870 


















68SU9 






H 


5.'iB-13 46*67 




44061 




417S3 


4 


B7 


B3574 4M« 


S.^099 


44011 


tlFSM 


41«Bt 


a 


B8 


53615 46.18B 


5B0M 






41B68 












DR3S0 


4ie!o 


1 


to 


53907 46303 






6641B 


416S1 







Colan T.n 


Colan 


Toti 




Tui 
89' 




II- 




iO- 







LOG 


TANGENTS AND COTANGENTa 


1S7 




I 




li 




ss° 






Tm 


Cotan 


T>n 


C<.tai< 


Tan 


Cotan 






9.88(19 


10.41582 


0.60U41 


i>.S9SSB 


B&I7BS 


loaTSIS 


~^' 










wsei 


■ffiSSO 


87160 






1(4 ira 




60714 








SB 




smi 


















41431 


ms» 




WfflSS 








B6G0S 




e0fR8 
















60SS0 






mm 


Kl 




B898I 




eo^a 






saiM 




s 




41^1 






6W»6 












150987 


3Wa:i 




a68ti9 






S.BB7U 






U,:«B90 




10,S6S«.l 


50 




EieSSJ 


'4i"iia 




asmi 










sssn 


41181 


oioja 




oaaos 














88898 




BBTflO 


4T 




ABHI 








e)H7B 


S87S5 


4« 




SBWl 


41018 
















40M81 


ijiwo 


387B0 


















ttffl79 


mil 






S9CM 


401)0(1 








80.JH6 




IS 


09131 


40099 




3807-J 
















o.seoss 


W.034M 


lU.BMiB 






'ctm 






38B0(- 




3B4SI 


39 


et 




40787 












■A 






0117^ 














40i!!ia 




a.-4tr* 


1:9645 


30877 


38 




Msai 










30818 




H 




40608 






II36M 


36SW 






(MW 




fllBlS 




ia7ai 
















63;«i 


364 19 


Si 


M 


sesoa 




GltiST 


ssai8 




ao«M 


SL 


ao 




lO.lOIBO 


9.I11TW 






10.86170 


SO 




'm)~ 


101-J3 




»*M< 


08K83 


36I3B 






mu 


40gt» 




a8:iU« 




86101 


a 


n 




40810 










K 


ai 




40Sia 






B49aH 


SDUti 


a 




NTfl 




UI901 


awi» 


64003 


35907 


IB 




MTN 






asosi 


041187 


85903 


«4 


i; 




4*!0L 




^180-2H 






ffl 


m 


MSW 


401M 


BiODIf 










m 




■wiis 




;i;«i7 




311860 




ia 








o.sTftai 




10,S5e-« 








400M 




ai)«i) 














BJISU 






35TBT 






00019 




Ba_iM 




m-in 


aim 






DOOM 






87779 




S,'«8M 








3890T 












4S 




B0S7O 




S7708 










B0I06 




qi»J7 


37673 










ooaoa 






!17«»H 


01449 






19 


«(»10 






3700-J 






11 


SO 


o.sasTB 


lO.SflTM 


».SJ4H3 






10.3.^4X3 








39B8T 


naifjK 


3;ssa 










MB49 






3749B 


Mm 


.45414 






IKOM 








6483) 


8.^a«i 




u 




SM78 




S71i» 










aous 








iU6ni 








M49S 




OSMS 


^1!K& 


04rj3 


ssrrs 






IKISKt 




fWSBO 




64766 


ffilio 


3 


S8 


K6S6 


M(M 






IM790 








soeoa 


suss:. 


iiaf.w 


ataw 








go 


wen 




BS7'B 


»7i!ir, 




_. *-^]^t. 




f 


COUD 


T^~ 


__(^ftn_ 


'Tan 


__Co;sii_ 


_ Tail 






eg* 


"n 


■ 




66" 






1S8 


LOG 


TANGENTS 


AND nOTANGRNTfi 






9t' 


SB" 


tt' 




Tflr 


Colnn 


Tan 


Colan 


Toil 


Colan 


~o~ 


9 BiSDB 


10 361 4-i 


3 60867 


10 3.1133 


68818 


10 81183 


00 


1 


'ftlSM 


Mils 




33)00 


88850 


'ail 50 


BB 


s 




95071 


00B3S 




tm'l 


81118 


as 








eoMO 












64 901 




oeoeo 










! 


flwwa 


a4SM 




n 


68BT8 


S 


66 








B70SS 




6W4S 


80968 







kIm 


54B88 


siios 


39837 


69CI74 
B0106 


anew 

80SH 


B3 










lO.SSBOl 








11 


'<I5^I 




■b72«b 






*3UB30 


4B 


1> 




3473S 




33738 


6ei«M 


■07W 


4H 








B7a06 


S-J705 


69334 


30700 








34067 




M0T8 






46 


19 


USMG 








60393 


80703 


46 


1« 


05400 


34000 






60339 


80B71 




11 




34560 


67JB0 


8-J574 


61(381 


30«as 












3S543 








l» 


OSMl 


34409 






ena 


aosns 




stt 






0.87834 


iu.3a4;e 


fl.8W57 


10.80S4S 


40 












694S8 






sa 


BMI02 








tMiXl 


30430 


38 


sa 


esflM 




67^ 


araio 


60584 


30448 


87 






S 


670S7 


33313 


OSOIB 


ao3» 




3t 


















06770 


Sim 


87752 




00070 


ami 


S3 


% 




84197 


67785 


!asi6 


60710 


30990 


3a 










39183 


00743 


aOS68 


31 


30 


e. 06870 


10.34190 








10.30336 


ao 


81 




31090 


■8788S 


asiis 


'dokis 


00 95 




3a 








8085 


89ffl7 




aa 


ss. 










B9SD8 


30M 




84 


mm 


33»W 






09900 




» 


as 


liBosa 


S36«a 




3I98H 


m'Jus 


90D0S 


3S 


t« 






68044 


310.16 


69903 


30037 










6tio;7 




69BB5 




93 


3S 




33HflJ 








EII074 


aa 


se 




S3SU8 






10058 


Sana 




40 








10.31896 


9.TU0SB 




ao 








'othlOfi 






'3118:9 


10 


43 


ftoJri 


337« 


03339 






WB48 


18 


4a 




33090 


88l!71 


3in» 


;0184 


»HI6 








33083 


08303 


31697 


T0916 


30786 


le 


49 


Wi?! 




08380 












uioi 




0830-* 




7037B 






4? 


owa? 


.IBS&I 






:03D9 


SOOBI 




48 




S«30 


WMTIJ 


St5(M 


70341 


99060 


la 


4S 














11 


90 


a.eem 


lO.SHJoa 


0,6X497 


10,31.™ 


0. 70404 


'"ffi 


9 




eOMKi 










W684 


B 




«6t»0 


33364 










7 


M 




33331 


08096 






M471 








3ias8 


08a^8 




70660 


39440 








33-JflB 


flWIBO 




70693 


30408 
















3W7T 


3 


U 


OHBOl 








70664 


xm 




l» 




3aind 


S8TH0 




706M 


30316 


1 


DO 

















Cot«»_ 


Tan 


Cotiii, 


Tati 




T>n 


' 


84" ^ 




«<■ 







LOG. 


TANGENTS AND COTANGENTS 


159 


1 


s 


J. 


t»' 


10- 










CotBD 


Tan 






Cotan 




t.Tirm 


lonsss 


a JiMT 


10 37438 


9.7J37B 


lo.noss 


~eo~ 




ma 


!WSS 


reiB8 


KJOa 


74405 


SSMS 






TOTTK 










aS66S 


38 




Tueio 


20] so 












4 


raui 


SS159 


7!»S9 










B 


TMTa 


asm 


J«?o 


i[7aw 


74594 


S547B 














74554 


!a44S 


H 






^WHG 


7ST80 










! 


-oow 


S9031 

aiHoa 


;asii 


27ir.o 


S 


33357 


M 




e. I03S 


lO.aWi! 








10.33387 




H 


m 


.■891 1 


'TWOi 


"l;o9h 


■ '7*7*! 




49 


IS 








UTOlifl 


7473a 


9328a 




] 13. 








97087 




95338 


47 




IS8 










33:^ 






IB) 


98810 


TSK3 




74IW1 




49 


i '* 




287R5 


73064 


ten 1(1 


74^11 


alio 


44 












7iasu 




43 




















IWtt 


asfiw 


7ai« 


^tf^Qi) 


iwa 


isoi;i 




1 10 


0. i:l38 








•J.7UIW 








STO 


'ssesa 


'laais 




liKIK 


"scoiw 




a 


-Ml 


aR5» 


73^36 




aiis 




38 


■A 






I.HflB 


wra 


50B8 


BJflK 


37 




481! 


:28S3'< 


Tsain 




Si .1*7 


B1U13 




X 


m 


iSS07 










35 


1 M 




a94r6 


i^sii 


98844 


sua 


94854 


34 






|»4IK 


TSSBB 


^14 


-SlTli 




33 


', *^ 


BM 








laas 






1 W 


Sir 


S88S3 




96654 


7KIS5 






80 


0. 184B 






io.aB5i4 




10,94736 


3D 


' SI 


1ST9 
















IM 


S8»l 


73537 




753^ 






1 ■* 




»J60 


raM7 


9»H»S 


IB353 


94617 




91 










75Wi 


■J4BK 




x> 


Tieo: 


i»im 


^Bfl-i7 










M 


IB3i 






9li341 










isu 


|J8ia7 


rawtj 


aC3]H 


75171) 


9)536 


93 


{ SB 










TTfiOil 




99 


' » 


IKS 


S8U75 




«:»£ 




94471 




a 


e.TIDH 




B.T37T7 


10.9flS33 


H^!^ 


irt.ii4« 


90 


41 








aoi!i3 






13 1 


41 


TSMS 


kb:.:; 


7aMi7 


2iiisa 


Vm7 


law 


Ifl 1 
17 1 


1 ^ 


TOTS 


S-Bi-i 


73807 


90lU:l 


7.ili7« 




IB 1 


w 




2-fiBI 


;3B-J7 


illlOT!) 


75705 






' 4a 


^140 
















W170 




7floJ7 


wai3 




£-49411 




48 




a-TB9 
















awiiu 


74u4r 


9riUA3 


;.-i8.': 


HllTM 




SO 


B.T3BSII 


10.S77SH 






B.75Sr.-j 










T«OT 


=(77U7 




■9S8M 


76tWl 










"san 


H7BTT 


74197 


9fi«a 


75910 


94>»ll 






M 


AM 






astiii 






J 




M 


1E«U 






a6«it 


WMH 








S5 


Mil* 


itTBW 




SKT74 












tm 




T43ftB 


M7.H 


TtiniT 


93973 






ST 


~-U7B 




7 m 


MTll 


7B0M 


S3911 


3 




58 


SMS 


97l9t 




ISW 


Tl-flM 








9B 


ms7 


arifls 


74115 












M 




gT43a 
Tan 


7437r. 


9flfl9!> 


— ™"*_ 


93R'« 






fotai] 


Tan 








AS- 































160 


LOG 


TANGENTS 


AND COTANGENTS 






80> 


81" 




2- 




Tan 


Cotau 


Tun 


Cotan 


Tan 


Cataii 


^ 


B.TOIM 


10.S8S58 


B. 77877 


10 M1V3 


0.79579 


loswn 


80 










'>9D9I 


79607 


90393 






Tsaw 


83798 


T7B3B 


|B06S 


79085 


90365 






Toasi 


aSTBB 


;796S 




79663 


20337 


57 




T«Ml 




TTwe 


isooH 
















91980 












gaesi 


T6W9 


SI951 


70747 








IflSIS 


sscaa 




2192a 


79770 




51 




76ST7 


asoaa 








80196 














79839 








1I.:M35 


10.33565 


9.78168 






10.90110 






















TBIBS 


£8507 


TSHO 


91780 












33478 


78iM9 


B1761 


79944 


mxM 






imi 


83449 


7HK7 






20098 


46 




TKSO 


£34aa 




















21066 


eD09R 


1B978 










78363 


81837 


80066 


IBOM 


48 




70668 


33383 


78891 






19916 


4» 




761IB7 


113303 














fl.TOTSS 


10.S3a76 


B.78448 




9.80140 


10.19860 


40 






98240 








19888 


39 






«Rai7 




21 495 


89195 


19806 


38 






saiss 


786S8 


31467 


60998 


19TT7 




» 


78841 


SSIBB 


7BS83 




80951 


19749 


K 




70870 




78BM 








35 












80807 


1969S 






7flN8 




TB647 


S1363 


80336 


19666 


38 


98 


7SB57 


SB043 


rSBTS 




H»38S 


19637 


St 




jflsse 










19509 


31 




9.7T015 




e.78;3S 


10.21908 


B.80410 


10.19BSI 




II 


7T0H 


sagse 




91340 






89 


X 




83827 










98 






vases 








19498 








3S!«7a 


788(6 


91155 


W5B0 




98 


W 


7T1M 


SW*) 


78874 




B0568 








771BS 














ar 






78980 




80614 


193SB 


a 


S8 


77940 


SK54 


78959 


SlOll 


80042 


19338 




N 


77S74 


asTsfl 


78987 




80809 




» 


40 








10. 90985 




10.19303 


10 


41 




>i0«8 


79043 


mK 


*807a 


9978 




4« 


77361 


^■8839 


79072 


mat 


80758 




IS 
















17 






iana 








9)99 


It 








TBI 50 


B0B44 


80888 


9IM 




» 


77478 


xmi 


rOJBS 


20815 


60864 


gtsa 




47 


77605 


«4t5 








9106 




48 




83467 






80919 


9081 








93488 


79309 




80947 


BOU 




BO 


9.775B1 


10.SS4De 






0.80B75 


10.19085 


ID 








' 79396 






IB997 


9 


B9 






79.454 




81080 


1B970 


8 




77BT7 


Easoa 


79889 


aoBia 


81058 






H 


rrme 


32294 


79410 








i 


B5 


rnM 










laSHT 


E 


H 




23887 


79109 


aofai 


8I14I 


18869 


4 




777B1 


aaw 


79)96 


ai505 


81)69 




a 


sa 


77880 


aaiM 








18801 




r» 


77S4B 










18778 


1 


DO 


7TS77 










18748 





Cnrai, 


T»~ 


"coun 


Tan 


~ Cotsn 


Tan 


, 






SB' 




88° 




17° 





LOG. TANGENTS AND COTANGENTS 



161 





88* 


84'» 






S6« 


/ 


Tan 


Cotan 


Tan Cotan 


Tan 


Cotan 





0.81SSS 


10.16748 


9.82899 10.17101 


9.84523 


10.16477 


60 


^^ 


81279 


18721 


82926 


17074 


84550 


15450 


50 




81807 


18698 


88958 


17047 


84676 


15424 


68 




81886 


18665 


82960 


17020 


84608 


15397 


67 




8188S 


18638 


83008 


16992 


84630 


15870 


56 




81880 


18610 


83085 


16965 


84657 


15343 


65 




81418 


18588 


83062 


16988 


84684 


15316 


64 




81446 


18665 


83069 


16911 


84711 


15289 


58 




81478 


18587 


88117 


16883 


84738 


15262 


68 




81500 


18500 


83144 


16856 


84764 


15236 


51 




9.81688 


10.18478 


9.83171 10.16829 


9.84791 


10.15209 


50 




81566 


18444 


83198 


[6802 


84818 


15182 


49 




81588 


18417 


83925 ] 


16775 


84845 


15155 


48 




81611 


18889 


83262 


16748 


84872 


15128 


47 




81688 


18868 


83280 


16720 


84899 


15101 


46 




81666 


18881 


83307 


[6693 


84925 


15075 


45 




81698 


18807 


83334 


[6666 


84052 


15048 


44 




81781 


18879 


83361 


16639 


849r9 


15021 


48 




81748 


18852 


83888 


16612 


85006 


14994 


42 




81778 


18294 


83415 


16585 


85033 


14967 


41 


so 


9.81808 


10.18197 


9.88442 10.16558 


9.85059 


10.14941 


40 


SI 


81881 


18169 


83470 


16530 


85086 


14914 


89 


ts 


81868 


18148 


83497 


1650!) 


85113 


14887 


38 


S8 


81886 


18114 


88594 


16476 


85140 


14860 


37 


' S5 


81018 


18067 


83551 


16449 


85166 


14834 


36 


81941 


18059 


83578 


16422 


85193 


14807 


85 


1 ^^ 
96 


81968 


18039 


83605 


16395 


85220 


14780 


34 


S7 


81996 


18004 


83632 


16368 


86847 


14753 


33 


S8 


89088 


17977 


83659 


16841 


85273 


14727 


82 


89 


89051 


17949 


83686 


16814 


85300 


14700 


31 


80 


9.89078 


10.17982 


9.83n3 10.16887 


9.85327 


10.14673 


80 


81 


89106 


17894 


83740 


16260 


85354 


14646 


29 


88 


89188 


17867 


83768 


16232 


85880 


14690 


28 


18 


89161 


17889 


83795 


16905 


85407 


14593 


27 


84 


88188 


17812 


88822 


16178 


85434 


14566 


26 


86 


88816 


17785 


83849 


16151 


85460 


14540 


25 


86 


88S43 


17757 


83876 


16124 


85487 


14513 


24 


87 


8S970 


17780 


83903 


16097 


85514 


14486 


23 


88 


89806 


1T708 


83930 


16070 


85540 


14460 


28 


80 


88385 


17675 


83957 


16043 


85567 


14433 


81 


40 


9.88868 


10.17648 


9.88984 10.16016 


0.85594 


10.14406 


80 


41 


89880 


17680 


84011 


15989 


85620 


14380 


19 


48 


89407 


17698 


84038 


15962 


85647 


14353 


18 


48 


88436 


17565 


84065 


15935 


85674 


14326 


17 


44 


89468 


17538 


84092 


15908 


85700 


14300 


16 


46 


88489 


17511 


84119 


16881 


85727 


14273 


15 


46 


89817 


17483 


84146 


15854 


85754 


14246 


14 


47 


88544 


17456 


84173 


15827 


85780 


14220 


18 


w ■ 

48 


88571 


17429 


84200 


15800 


85807 


14193 


18 


49 


88599 


17401 


84227 


15773 


85834 


14166 


11 


60 


9.89686 


10.17874 


9.84254 10.15746 


9.85860 


10.14140 


10 


61 


88853 


17347 


84280 ] 


15720 


85887 


14113 


9 


68 


89681 


17319 


84807 ] 


5693 


85913 


14087 


8 


68 


88706 


17292 


84334 ] 


5666 


85940 


14060 


7 


54 


88785 


17265 


84361 ] 


6689 


85967 


14033 


6 


65 


88769 


17238 


84388 : 


5612 


85993 


14007 


6 


66 


88790 


17810 


84415 1 


5585 


86020 


18980 


4 


57 


88817 


17188 


84442 1 


5558 


86046 


13954 


3 


66 


89844 


17166 


84469 1 


5581 


86078 


18927 


8 


69 


89871 


17129 


8441)6 1 


5504 


86100 


13900 


1 


60 


88899 


17101 


84523 1 


[5477 


86196 


13874 1 





Cotan 


Tan 


Cotan 


ran 


Cotan 


Tan ; 


/ 


68* 


56» 






64« 1 



162 


LOG. TANGENTS 


AND COTANGENTS 






86' 


fl)- 


SB- 




Tnn C<.ta» 


Tan 


(Nilan 


Tun 


Cotan 


~~^ 


B.eaiaa io.imti 


0.87711 


10.1SS89 


0.8W81 


10.1071B 


00 










88807 


10898 


59 


a 




877U 




8»8.« 






B 


seauo 1.17M 


B7790 












aoiaa 13788 


87917 


1S1W 


BB38n 


oeiB 


86 










89411 


0880 














0568 






(esn 13BSS 






89488 




53 




masi i380;j 


879thi 


12079 


e»>e9 




81 










80615 


ioie& 






9.86392 10.13(103 






9.89641 








8011S 1858-J 


■gsooo 




895117 


'lD438 


40 






WMW7 


llO 


89598 






13 


6iI4t'| 1S!M9 














estnK ijisoi! 


aims 


llM5 


»I8T1 


1D8S9 






SBMi 18449 


88131 




898117 


10S03 


s 


IT 








897!8 


I0ST7 
















4t 




8B63U 13370 


BBZIO 






tDS% 




so 








0.80801 


lO.lOlM 


40 




eiSGH3 'l33l7 










80 






88189 








88 


sa 




88815 


1I6«1 


8087B 


101*1 


37 


&i 








89005 


10005 


8« 




86;b9 i%!1i 


88307 






10009 


38 


m 


88816 IBISB 






8SW7 




84 




8GS4a tsiss 


884a) 


11580 


89983 


10017 




JS 




BB440 






(19991 


81 


fH 


B68M isioa 








UWOS 


31 


K 


B.mai 10.18079 


0.88498 






10.00039 


80 


81 








00088 


00914 


HI 




80074 18030 








09888 


88 




MTowi isuno 


88S77 








«7 


34 


870-JT li»7M 


8800.? 




Botei 


otesa 


St 


3-t 


STOsa lew" 


8h8-i9 


11871 


90190 


09810 














09784 


84 




«Tintl liSH 










88 




S:m lltS6H 


88707 


ll^lSl 




D97ft2 


S 


3S 




887S1 


11367 


90»4 


00708 






9.871»i lO.lffilfi 








10.09680 


10 






'hbtw 






09BH 




4a 


S?JSH 1*703 


B881S 


11188 




ODwra 


18 










90307 


09808 


IT 














18 












flosai 






87843 ia8B7 


8891« 






BKia 




*T 


87309 li-Wl 




IIO.-*! 


90601 


OBiog 


IS 


4S 








90W7 




11 






88994 






09447 








9,80030 


10.10980 


9.90S78 


10.00483 


10 


SI 


'87475 'llH-JS 








oaiwo 


9 




R7M1 inns 




KfidT 


90630 


o»t7a 






87587 1W73 


80099 


lOBfll 


BOOM 


0IIS44 








BOliiS 


10S75 


90889 


oteiR 




ii 










own 


8 


M 


87SSS l!iSfl7 


89»3 


IS 


9076U 


owti 








8wao 


10771 


9078S 


09S15 


a 










«08M 


00180 


I 


60 


87711 I!ai9 








ootm 







CoU.. Tan 


Ootaii 


Tan 


Cotan 


Tan 


' 


68- 




Si° 




61' 





LOG 


TANGENTS 


AND COTANGENTS 


163 




SB" 




0" 


41" 


' 


T»a 


Cotftu 


Tau 


Uolau 


tui 


C.otaa 





a.nen 


lo.wm 


B.BSaSL 


lO.OTBlB 


B.Wflltf 


I0.OCOB4 


00 




















WS8B 


DBUI 












I 


<mj4 


OBQW 


SHX 


07fi4i 


ttlBffil 




fi; 






oaoeo 


B-JI84 






USiWJ 


M 


1 


iui)»i 


OBOUH 


vasi 


otIk 


OloilB 




M 






o«i«a 


IHMl 


UTJ3B 








s 






Bi3B7 






WSSi 






















«»I09S 


lo.osyce 


S.BMSS 


lo.oTsia 




10.058^9 
















0.-,BO:i 








<1GH6» 


BUGBB 












911W 


(BSiM 


W715 


OTAl 


BIIJS 






14 


illlH 


osww 




maw 




osTs: 




IS 
















lA 


















BI-J7S 


OSTW 


9«1T 


msi 










S19UI 




MiM3 






flSO'JB 




IS 














41 


10 


».B1MS 


io.o8e4T 


S.BfflSl 


10,07106 


y.w43« 






11 




ami 








05WS 




U 
















a 




OB6'0 










87 




gi4U 




ftJflOe 










a 












0541B 




M 








OOBW 








« 




0SI87 




dom: 








3g 


Bir^ 




iHomi 


oaeoi 


Wttso 


0S.1TU 




» 








OOHTU 








s 


■U1686 


■«t»64 


8.98150 


■«MiO 


■91TOO 


IO.0M19 

onaw 


S 




61002 














a 


ftlOSI 














SI 




asm 


93Mi 


OUTB 


917!« 




!C 




flIJTO 


osm 


BajTH 


OOTW 


aim 


061 w 




» ' 


WTB 


aB«S5 




OliBBT 












(»iOS 




OflO?) 










OlgIB 












K 


as 


BIM2 


(B1S8 


HS3K0 


ouujo 




IWKK) 




« 


B.SIMS 


lo.Miae 














SI no 














« 


sisrs 


flHOSI 






n4LIH0 








OIBIS 


09M5 






B.-.Jia 








sisii 


0S0S9 












41 


SI BBS 








!iR(K^ 


IKBDB 




« 


t-Mil 




{MTias 




aw8« 


04UR' 




47 


saws 


OTOM 


1«.-*1 


muiQ 




ojMHr 




48 


WOTS 


071K7 












4B 
















U 


S.BSIffl 




U.flSBBl 


lo-osao 


O.WJW 


lO.WSIO 






BJISD 




BMOT 
































IH738 












Bsar 


OTTTS 


9a;M 


DMXi 


95-!91 








«uu 










ww-s 












iwTmi 




inuw 






«gu 


OT(HM 


9»>J0 


oaieo 










SMSO 


crom 


ntiMji-i 


oiiias 


BMB3 


oioii: 




U 


sxtiw 








B.IUN 






u) 


SJS8I 










_J'*L''^- 


-7- 




CotBQ 


Tun 


"ci.taii 


Tbo 






BO* 




40- 




S' 



164 


LOG 


TANGENTS 


AND COTANGENTS 










_Tan_ 


Cotaii 


Tbq 


Colau 


Tail 


Cotan 


~a' 


9.95M4 


10,0455* 


9.969BS 


10.08084 


9.98454 


10,01518 


00 


1 








oaooB 


98600 


01491 




a 
















3 






3704^ 








67 




93*15 


044SS 


97067 


0*933 


W5B5 






S 






B70B3 


08008 


98610 


01380 




G 








oasfls 


086B5 


rasoB 






SSfliW 












53 




S5ti47 








geess 






S 




IMSM 


971113 


osBOi 


98711 


01E98 














9.D8737 


jo.oiaia 




11 


'B5T!ffl 
0SM8 


olSa 


arara 


OSTSl 


0878: 


01«8 


1 


IS 




DJOTl 




0S7O5 


08811! 


01188 


46 




U5H9a 


OIITS 








OllBT 






B^ 


WIM 


97371 


oxeM 


OBBM 






17 








0SM4 


08913 


01087 


43 


IB 










98089 




41 






D4074 








01 OM 




SO 










0.B69SS 


10.01011 




n 


'bbbtt 












IB 




9UIUi 


ifiiess 




02177 


99040 


ooseo 


38 






03B?J 


07548 


03463 




OOOHS 


37 


u 










99090 


00910 


36 


S5 












DGSM 


36 




ni]a4 






0^711 




00869 


81 




mua 






oasni 


B9IB6 




83 


IS 








OSBW 


1)9)91 


00009 




l» 




oaso 








007S3 


31 


so 


i.xwi 




9.97TSS 


io.o2-j:s 


g.99eie 


10.00768 


30 


ai 






97760 


O-OM 


99367 


0078S 




sx 


asise 








99398 


00707 


3S 


S3 


seasi 


08718 






99818 


00683 






l)U.<KI7 


09093 










36 






03M8 


97861 


0il4B 


99368 


00688 


. » 


Sfl 




05IH8 


979JT 




99S94 


00606 


■4 


37 










90419 








BGiue 










00666 




sg 


W«H 


OS507 






99469 


00631 


31 


40 








lO.OBOSS 


0.99490 


19.00506 


30 






'oa&ie 


'ewxB 


01iW7 


B9IK0 


00480 






wsio 












18 




SBTjBS 


034S6 








00480 


17 








K8079 


01981 


99S96 


00404 




49 










990!tl 


00ST9 




4< 










9964B 


00364 


H 




96«aa 








98673 


00828 


IS 




Bdflsa 


osase 


SfllBO 


U18K) 


89007 


oosos 




» 






osaw 




9973* 


00378 




60 


a.tmiii 


lO.OSiPH 


9.98331 






10.00363 


19 




90738 


oaaw 


98356 




08773 


00337 




m 






oasBi 


01719 


98798 


00303 




ss 


sents 








99833 








K814 


03186 






99848 


00161 




u 




03IB1 






99874 


ooise 


6 


u 


9sau 




96389 




88899 


00101 




» 


DBBSO 








99934 






u 


MSIS 










ooosi 




5S 




03oeo 






99971 


00096 








osas4 


98484 


OlSlfl 




00000 






CoMD 


Tin 


Ooian 


Tao 


CotHQ 


Tan 






47- 




«e* 




4S> 



I/)G. VERSED SINES AND EXTERNAL SECANTS 165 

if r* 



Log. Vers. 




1 
2 
t 

4 



S 

e 

t 
1. 

10 

u 

12 
18 

1*- 

15 

16 

17 

It 

1» 



90 

21 

22 

28 

31 

25 

26 

27 

28 

30 

81 

82 

88 

84_ 

85 

86 
87 
88 

IL. 

40 

41 

42 

48 

44_ 

45 
46 

47 
48 

^ 

50 

51 

52 

58 

M. 

65 
56 
57 
58 
59 

60 



62642 
22848 
5806S 
83054 



0248 

1827 

81662 

43260 

58499 



626 

7092 

7847 

85481 

91868 



20 
78 



97860 
03466 
08732 
1369S 
18893 



22848 

27086 

8112r 

3498! 

38681 



42230 
45636 
48915 
52073 
55121 



58068 
60914 
63672 
66344 
68937 



7145 

7390! 

76282 

78598 

8085f 



8305 

8519 

87291 

8933S 

91332 

93284 
95193 
97061 
98890 
00680 



02435 
04155 
05842 
07496 
09120 



10714 
12279 
13816 
15327 
16811 



6 18271 



Log. Vers. 



60206 
35218 
24987 

19382 
15835 
13389 
11598 
10230 



3861 
3697 



1952 
1909 
1868 
1829 
1790 




Log. Exsec. 



— 00 

62642 
22848 
58066 
83054 

02438" 

18273 

3166 

4326 

53491 



62642 

7092 

7847 

85431 

91868 



97861 
03468 
08733 
13697 
18393 



22849 
27087 
81127 
3498S 
3868S 



42231 
45638 
48916 
52075 
55123 



580^8 
60916 
63674 
66346 
68940 



71457 
7390$ 
7628$ 
78601 
80857 



8305 

8520 

87295 

89331 

91335- 

93288 

95197 

97065 

98894 

00685 



02440 
04160 
05847 
07501 
09125 



10719 
12284 
13822 
15333 
16818 

6 18278 

Log. Exsec. 



1} 



60206 
35218 
24987 

19383 
15836 
1338§ 
11598 
10230 

9151 
8279 
7557 
6952 
6437 

5993 
560S 
5266 
496i 
4696 

4456 
4238 
4040 
386l 
3697 

3545 
3407 
3278 
3159 
3048 

2945 
2848 
2758 
2672 
259S 

2517 
2447 
2380 
2316 
2258 

2199 
2145 
2093 
204S 
1997 

1952 
1909 
1868 
1829 
1791 

1755 
1720 
1687 
165i 
1623 

1594 
1565 
1537 
1511 
1485 

1460 



Log. Vers. 



6. 18271 
19707 
2111§ 
22509 
23877 

25223' 
26549 
27856 
29143 
30410 



6 



6 



6 



6 



6 



6 



6 



6 



6 



6 



1} 



31660 
32893 
34107 
35305 
36487_ 

37653 
38803 
39938 
41059 
42165 



43258 
44337 
45403 
4645S 
4 7496 

48524 
49539 
50544 
51536 
52518 



53488 
54448 
55397 
56336 
57265 



58184 
59093 
59993 
6088$ 
61768 



62639 
63503 
64359 
65206 
66045_ 

66876 
67700 
6851S 
69323 
70124 

70917 
71703 
72482 
73254 
7401 9_ 

74777 
755:?9 
76275 
77014 
77747 

6 78474 



Log. Vers. 



1} 



1435 
1412 
1389 
1368 

1348 
1326 
1306 
1288 
1268 

1250 
1232 
1214 
1198 
1182 

1166 
1150 
1135 
1121 
1106 

1093 
1078 
1066 
1053 
1040 



/> 



Log. Exsec. 



6 18278 
19714 
21128 
22518 
2388i 



6 



6 



6 



6 



6 



6 



6 



6 



6 



25231 
26557 
27864 
29151 
30419 

316G§ 
32901 
34118 
35315 
36497 

37663 
38814 
3994§ 
41070 
42177 



43270 
44349 
45415 
46468 
47509 



48537 
49553 
50557 
51550 
52533 



53503 
54463 
55413 
56352 
57281 



58201 
59110 
60011 
60902 
6178i 



62657 
63522 
64378 
65226 
66065 

66897 
67720 
68536 
69345 
701 4 5_ 

70939 
71725 
72505 
73277 
74043 

74802 
75553 
76300 
770''0 
77773 



6 . 785O0 
Log. Exsec. 



1> 



1436 
1413 
1390 
1368 

1347 
1326 
1308 
1287 
1268 

1250 
1232 
1215 
1198 
1182 

1166 
1151 
1135 
1121 
1108 

lC9i5 
1079 
1066 
1053 
1040 

1028 

1015 

1004 

993 

982 

970 
960 
950 
939 
929 

919 
909 
900 
891 
883 

873 
86i 
856 
848 
839 

83l 
823 
816 
808 
800 

794 
786 
779 
772 
765 

759 
753 
746 
739 
735 

727 



/> 



]m LOO. \T.RRED SINES AND EXTERNAL SECANTS 

2° 8» 



O 

1 

2 

3 

_4_ 

5 
6 
7 
8 
_9 

10 

11 
12 
IS 
14 

15 
16 
17 
18 

19 



20 

21 
22 
23 
2i_ 

25 
26 
27 
28 
29 

30 

31 
32 
33 
Si. 

35 
36 
37 
38 
89_ 

40 

41 

42 

43 

4JL 

45 

46 

A7 

48 

49 

r»o 

51 
52 
53 
54 

55 
56 
57 
58 
59 

(SO 



Log. Vers. 



6 . 78474 
79195 
7990i 
80618 
81322 



6 



6 



6 



6 



6 



6 



82019 
82711 
83398 
84079 
84755 

85425~ 

86091 

86751 

87407 

8805.7 

887U3 
89341 
89980 
90612 
91239 



91862 
92480 
93093 
93703 
94308 

94909 
95500 
96099 
96688 
9727? 

97853 
98430 
99004 
99573 
00139 

00701 
01259 
01814 
02366 
n9qi4 



03458 
03999 
04537 
0507T 
n5fi03_ 

06130 
06655 
07177 
07695 
08211 

08723 
09232 
09739 
10245 
10743 

11240 
1173?) 

JL ^j Ad Ad t 

i27in 

13203 
13687 



Log. Vers. 



n 



721 
715 
709 
703 

697 
692 
686 
681 
676 

670 
665 
660 
655 
650 

646 
641 
638 
631 
627 

622 
618 
613 
609 
605 

601 
597 
592 
589 
58i 

581 
577 
573 
569 
535 

562 
55S 
555 
551 
548 

54i 
541 
537 
53? 
5ST 

527 
525 
52T 
518 
515 

51? 

5 on 

50fi 

5nS 

500 

497 
495 
492 
489 
48g 

484 



J> 



Log. Exsec. 



6 



6 



6 



6 



6 



6 



6 



78500 
79221 
79937 
80646 
81350 

82048 
82740 
83427 
84109 
84785. 

85457 
86123 
86783 
87439 
88090 

88737 
8937S 
90015 
90647 
91275 



91898 
92516 
93131 
93741 
94346 

94948 
95545 
96139 
96728 
97313 

97895 
98472 
99046 
99616 
00182 

00745 
01304 
01860 
02412 
0?960 



03505 
04047 
04585 
05120 
05652. 

06180 
06706 
07228 
07747 
08263 

08776 
09286 
09793 
10?99 
107nR 

'lT>n7 
1179? 
12^85 
12775 
13262 

13746 



Log. Exsec. 



X> 



721 
715 
709 
703 

698 
692 
687 
682 
676 

67l 
666 
660 
656 
651 

640 
64l 
636 
632 
628 

623 
61R 
615 
610 
605 

601 
597 
593 
589 
585 

581 
577 
574 
570 
566 

563 
559 
555 
552 
545 

545 
54l 
533 
535 
531 

525 
525 
522 
519 
516 

513 
509 
507 

505 
501 

498 
405 
493 
490 
487 

48i 



7> 



Log. Vers. 



13687 
14168 
14646 
15122 
15595. 

16066 
16534 
17000 
17463 
17923 

18382 
18837 
19291 
19742 
20191 

20637 
21081 
21523 
21963 
224O0 



22836 
23269 
23700 
24129 
24555 

24980 
25402 
25823 
26241 
26658 

27072 
27485 
27895 
28304 
28711 

29116 
29515 
29919 
30319 
30716 



31112 
31505 
31897 
32288 
32676 

330P3 
33445 
338?1 
34213 
3450^ 

3497T 
35345 
35723 
Stf097 
36465 

'36839 
37207 
37574 
37940 
38305 

38667 



Log. Vers. 



n 



478 
475 
473 

470 
468 
466 
463 
460 

455 
455 
453 
45l 
445 

446 
444 
442 
440 
437 

4S5 
433 
431 
429 
426 

425 
422 
420 
415 
416 

415 
412 
410 
409 
406 

405 
402 
401 
399 
S97 

395 
393 
392 
390 
S83 

386 
385 
383 
382 
380 

378 
377 
375 
373 
37T 

3 70 
365 
367 
366 
364 

362 



Log. Exseci 



13745 
14228 
14707 
15183 
16657 

16129 
16698 
17064 
17528 
17989 

18448 
18906 
19359 
19811 
20260 

20707 
21152 
21596 
22035 
22473 



2290 

2334 

28775 

24205 

24682 



25067 
25480 
25902 
26S21 
26738 

27153 
27667 
27978 
28S87 
28795 

29200 
2960} 
80006 
80406 
8O8OI 



8120] 

8159 

3198J 

8237 

82761 

88156 
83542 
38926 
84809 
84689 

8506J 

8544 

3582J 

861B 

36561 



36940 
87810 




7 88778 
Log. Extac. 



x> 



TT 



UXi. VERSED SINES AND EXTERNAL SECANTS lfi7 




. Vl-^RSE]) SINES AND EXTEllNAL SECANTS 




liOG. VERSED SINES AND EXTERNAT. SECANTS 




170 i/)G. \']-:hhi':d rixks axd external secants 






5 OB 

00. 5 



LOG. VERSED SINES AND EXTERNAL SECANTS 171 



M7B4 
■ «nil76 



30 SD.059.BISS-0 
"-", :"U79 5178. 6 
iCilDO-OlSDaISS S 



S8-5S8.g 



11-4111-3,11-3 
13-3 13-2,13.3 
15.2rl6 Dil4.9 

niiie e;ifl a 

19 C'lS 9!l8-S 
aSOlST GISTS 
ETGISB 3 5fi Q 
7fl D75.!7i-8 
B5.b<94.llB3-9 



i-oiio.fi 

8-SilS-! 

le-SseJ 

jG 0G4-6 
73.S.72-B 



ia.fll2-S12-3 



172 LOC VERSED SINES \ND EXTERNAL SECANTS 







» 


gEx 








Log.E". 


D 




P.P. 
















8 64748 










1 




J 












99 








; 




3 












2 




















6504! 


99 






03 lOS 101 
















" 


_5514i 
55^'3 


-* 




0. 




0.1 


















^ 


S 








BB34S 
55-41 


It 


6 




6^ 




i 




8 
















6563^ 


ll 


6 


20 


7. 












t 








I 




11 
55933 


98 
88 
9! 


n 


tl 


1-1 




i 






















3 






— 









— 


t 














"seaai 


8 r 


9" 


■ B 
















56324 


g| 






11. 




j 










i 


«"! 




S642| 


9 


B 


n; 


|: 


ill 


s 

7 


1. 
















-i 


Si' 


3^032 


t 






! 


40 




"sflnj 


I 






















30 0-0 






) 
























40 B.6 


!8.0le5 


! 
























Bol a.s 




1S3 


J 


- 


- 












B 


"57201 


1 


24 

i 


?'l fl^RI t\ 


















57491 


s« 




I 




I 






















1™: 


i 








s- 






















57771 
































3 






B. 




























*: 


4. 


63 




















5806; 






0. 


0. 


79 






















9s 


"a" 




















58351 


3 










^ 


} 








Sit 




3 
3 


g ?*.. 93 . »8 


2_ 
















5B54( 




J_ 




o'. 






















r 


6873* 


1 


4 




2: 


-0 






I 










I 




tSI'^ 








s'- 


5' 


16 






5 










I 




68937 








1.3 


1. 






















































"59117 


1 




tofl2.| 


2! 


61 




















5921 


1 


50178. a 


7. 


76 






I 




44 


? 








59300 
68401 
.59495 




91 . BO . 0_ 


~a 














^ 


BSBSr 

5968? 
5B96: 




'61 




2^ 


12 


1 


1 


64 


8~537 ! 


7 


641j0 


loo 


Si"7S 


9& 




1 


64 




o'. 


13 
45 


1 




6B 








Gfl 


5''a 8 


B435( 


584E; 


OB 


















6^ 


539 ! 


Ie 


6M 






60241 


5 
















St54! 




9U 






5 






511 1 




B4B19 




687.4 






93 






^ 


8.5324: 


98 


8.54748 


98 


8.58B14 




r 




^ 






&.c^,y-»;i»5-£'=. 


77 


[iTT^ 


— 


L g.Exs 


"» 




F:T^ 



LOG. VERSED SINES AND EXTERNAL SECANTS 173 



.69083 g 
■B»IT3 ' 
B.592B2 ! 



8.69708 ! 
-69787 I 
"888 I 

m a 



-60633 B 
.607)6 t 
.60610 I 



-85133 S 
_J52lB ' 
8.66302 c 

■86396 I 

-85551 B 



3.86883 n 
•88951 E 
-87038 S 
-8712E I 



1 


»I2" .13 


J 


Cj3o!|30 


8 
7 


6l76iEi7S 



-83183 S 
.83243 " 
8.83338 S 



.63S04 ° 
8.B32fl5 I 
-633SB g 



-81289 Q 
-84376 g 



-66295 t 
-6637B J 

8 68542 p 



.87195 I 
.87277 ° 
8.87369 I 



.68944 " 

-89115 I 

■88201 J 

.6928G , 
119372 



84 £3 83 



6-3 91 8.3 
O-BlO.b 10-S 
2-1 12-0 11. S 



^EiSi 



-66807 S 
.66895 " 
8 86984 ^ 



3.635B5 % 
-6B6D0 B 
.68730 S 



-70476 
-70560 " 
-70B44 ° 
8.70728 S 
■70813 2 
.70887 , 
-70981 S 



174 lAXf. VERSED SINES AND EXTERNAL SECANTS 



8.13621 
73700 
73775 
73851 



3 8-73853 ijn 



IJOG. VERSED SINES AND EXTERNAL SECANTS 176 



■ 7903 < ' 
9-79105 I 

•79175 i 

■ 792)!, ' 
■7931; i 
■7B3B7 ' 

S^7M5B I 

■ 79528 Ij 

-79689 I 

■79733 ' 

S-79B09 ' 

■7BB7a , 



■ 80435 ° 

■80B77 t 
-80641 i 



■B2102 , 

■B2177 ' 

8^B2252 I 

-B2B27 , 

■83*02 i 

■S2477 , 

_^32552 I 

8-82B27 i 

-82702 i 

■82776 i 

■B2B51 ^ 

S-B3OD0 I 

.B3076 , 



■83670 ' 

1-63744 I 

■83818 7 

■838»a 7 

■88966 7 

-iB*Q5B ' 

I -84119 I 

■84187 ^ 

"•?8i i 



-83515 ' 
B-S35e2 I 
-83649 F 



■839BS I 
-84060 S 



847BB t 

B484B " 

9.84B12 I 

■S497S i 

-B504i l 



8 B66"0 I 



e- 87024 ; 

-8ioe! i 
■snef 7 
-STSse ^ 

_873in ' 
8.87382 , 
■8745S 7 
■87626 ; 

_J766B ' 
8-fl77"^3B I 



ij- 


9 6 9 


















deo. 


60-0 B9- 



7 8-l| 8-6' 7-fl 
9 10 510 310 a 



OflS S 

m I 

B^ 83335 ' 



_^B9438 ' 
>|Lag.Ex>. ) 



176 LOG. VERSED SINES AND EXTERNAL SECANTS 



SBfiSS 
SBflSi 

'" BIJ83 
B8B4e 

ssao; 
BSqsi 



Si 



■92*21 



aeies 

- MStJ 



^'-?- 



7 T-a 7- 

s bJ i- 



il 



LOG. VERSED SINES AND EXTERNAL SECANTS 177 



24' 



25' 



Lg. Vers. 



8.93679 
93738 
93797 
93857 
93916 



8 



8 



8 



8 



8 



8 



98975 
9403i 
94094 
94153 
94212 



94271 
94330 
94389 
94448 
94506 



94505 
94621 
94683 
94742 
94800 



94859 
94917 
94976 
9503i 
95093 



95151 

95210 

9526 

9532 

9533 



95443 
95601 
95569 
95617 
95675 



9573 

9579 

95849 

95907 

95965 



96023 
96080 
96138 
96196 
96253 



96311 
96868 
96426 
96483 
B6541 



96599 

96666 

96711 

06771 

9682' 



96885 
96942 
96999 
97056 
97118 



* 




ere. 



j> 

59 
59 
59 
59 

59 
59 
5§ 
59 
59 

59 
59 
59 
59 
5S 

59 
59 
58 
59 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
57 
58 
58 

5 

57 
58 
57 

5! 

5! 

67 

57 
57 

II 

67 
57 



Log.Exs. 



8.97606 
97671 
97736 
97801 
97865 



8 



8 



8 



8 



8 



8 



9 



9 



9 



9 



97930 
97995 
98060 
98125 
98190 



9825i 
98319 
98385 
98448 
98513 



98577 
98642 
98706 
98770 
98835 



98899 
98965 
99028 
99092 
9915g 



99220 
9928$ 
99348 
99412 
99476 



99540 
99601 
99668 
99732 
99796 



99860 
99925 
99987 
00051 
00113 



00178 
00242 
00305 
00369 
00432 



00495 
00559 
00622 
00686 
00749 

00812 
00875 
00938 
01002 
01065 



01128 
01191 
01254 
01317 
01380 



9.01443 
Log.Exs. 



J> 

65 
65 
65 
64 

63 

65 
63 
65 
65 

64 
64 
64 
65 
6i 

44 
64 
64 
64 
64 

63 
64 
63 
64 
63 

64 
64 
64 
64 
64 

64 

64 
64 
64 
63 

64 
63 
64 
65 
65 

64 
63 
63 
65 
65 

63 
63 
65 
65 
65 

65 
63 
63 
65 
63 

63 
63 
63 
63 
63 

63 

IT 



Lg.Vers. 



8.9717D 
97227 
97284 
97341 
97398 



8 



8 



8 



8 



8 



8 



8 



8 



8 



9 



9 



97455 
97511 
97568 
97625 
97681 



97738 
97795 
97851 
97908 
97963 



98020 
98077 
98135 
98190 
98246 



98302 
98355 
98413 
98470 
98527 



98583 
98639 
98695 
98750 
98808 



98802 
98918 
98974 
99030 
99085 



99141 
99197 
99252 
9930g 
99363 



99419 
99473 
9952§ 
99585 
99640 



99695 
99751 
99806 
99861 
99916 



9997T 
0002B 
00081 
00138 
00191 

00241 
00301 
00358 
0041] 
00466 



0059.r, 



Lg. Vers. 



n 

57 
58 
57 
57 

57 
56 
57 
56 
56 

56 
57 
56 
56 
56 

56 
50 
56 
56 
56 

58 
56 
56 
56 
56 

56 
56 
56 
5b 
56 

56 
56 
55 
56 
55 

55 
56 
55 
55 
55 

51 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
53 
55 

53 



1} 



Log. Exs. 



9 



9 



9 



9 



9 



9 



9 



9 



9 



01443 
01505 
01568 
01631 
01694 



01756 
01818 
01882 
01943 
02007 



0207C 

02132 

02195 

0225 

0231 



02382 
02444 
0250b 
0256S 
02631 



0269^ 

0275 

02817 

0288C 

0294 



03004 
03066 
03128 
03190 
03252 



03311 
03375 
03437 
03499 
03561 



03622 
03684 
03746 
03807 
03869 



03930 
03992 
04055 
04115 
04176 



04238 
04299 
04360 
04421 
04483 



04544 
04605 
04666 
04727 
0478r 



04850 
04911 
04972 
05033 
05095 



05153 



Log.Exs. 



62 
63 
62 
63 

62 
63 
62 
62 
63 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

6l 
62 
62 
6l 
62 

61 
61 
62 
61 
6l 

6l 

6l 

61 
61 

61 
61 
61 
61 
61 

61 
6l 
61 
61 
61 

61 
61 
61 
61 
60 

61 

77 





1 

2 

3 

„4 

5 
6 
7 
8 

9 



10 

11 
12 
13 
14 



15 
16 
17 
18 

21 
20 

21 
22 
23 
24 



25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
_39 

40 

41 
42 
43 
44 



45 
46 
47 
48 
49 

50 

51 

52 

53 

-54 

55 
56 
57 
58 
59 

f?0 



P.P. 





65 


64 


6 


6.5' 641 


7 


7.6 


73 


8 


8.0 


8.5 


9 


9.V 


96 


10 


10.8 10.6 


20 


21.621.5 


3G 


32.5,32.0 


4C 


43.5142.6 


5C 


54.1 


53.5 





62 


61 


6 


6.2 


6. 


7 


7.2 


7] 


8 


82 


8. 


9 


9.3 


9-; 


IC 


10.5 


10. 


2C 


20.6 


20.5 


8C 


31.0 


30.5 


4C 


4].^ 


40.6 


50 


51.6 


50. & 





59 


58 


6 


5.9 


5.8 


7 


69 


6.7 
7-7 


8 


7.fi 


9 


Bl 


8.7 


10 


9.§ 


9.6 


2C 


19.6 


19.5 


3C 


29.5 


29. C 


40 


39.5 


38.6 


50 


49.1 


48.5 





56 


55 


6 


5.6 


5.5 


7 


6.5 


6-4 


8 


7.4 


7.5 


9 


84 


8.2 


10 


9.5 


9.1 


20 


18.6 


18.3 


SO 


28.0 


27.5 


40 


37.5 


36. f 


60 


46.6 


45-1 



63 

6.3 

7.3 

84 

9.4 

10.5 

21.0 

31. 5 

42.0 

52.5 



60 

60 

70 

8.0 

90 

10.0 

20.0 

30.0 

40.0 
50.0 



57 

5-2 

66 

7.6 

8.5 

95 

19.0 

28.5 

38.0 

47.5 



54 

5-4 
63 
7-2 
81 
9.0 



18 
27. 
36. 
45 



6 

7 

8 

9 

IC 

20 

3C 

40 

50 



P. P. 



i:n 



i/)r:. VEP.SFD SIXES \xn faterxal secants 

27" 



26" 



Lg.Vers.i i> 



O 

1 
2 
8 

_4 

5 
6 
7 
8 

10 

11 
12 
13 

ii 
15 
16 
17 
18 
29 

io] 

21 
22 
23 
24 



9 



9 



9 



9 



9 



2J 
2C 
27 

2C 

80 

81 

88 

a 

85 

86 

87 

88 

£91 

40 

41 

42 

43 

44 

45 
46 
47 
48 
49 

50 

51 
62 
53 
64 

55 
66 
57 
58 
59 



9 



9 



9 



9 



9 



9 



9 



9 



00520 
00575 
00630 
00684 
00739 

00794 
00843 
00903 
00957 
01011 

01066 
0112Q 
01174 
01229 

01283 

01337 
01391 
0144S 
01493 
^554' 

01608 
01662! 
01715: 
0176ii 
01823* 

dl8'77 

01931 

01985' 

020381 

02092 1 

O2149I 
02199' 
02253 
02307 

0236^1 

02414 
02467 
02521 
02574 
02627 

02681 
02734 
02787 
02840 
02894 

02947 
03000 
03053 
03106 
03159 



03212 
03265 
03318 
03371 
03425 

03476 
03529 
03582 
03634 
03687 

_ 03740 

" JLg.Vers.^ 



5 
5 
5 

5 

5 

5 

53 

53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
52 

53 
53 
52 
52 
53 

!>2 



/> 



Log.Exs. ii 



9 



9 



9 



9 



9 



9 



9 



9 



9 



9 



0515^1 
05215. 
052761 
053371 

0539t{i 

0545o 
05519 
05580 
05640 
05701 

05762 
05822 
05883 
05943 

06004, 

0606Ji 

06124, 

06185 

06245 

06305 



., 9 03740 
^^ 03792; 



W 03792. 

|A .03845: ^^ 

S? .0389i; ?^ 

^\ 039501 ^^ 

J? 9 04002; 5§ 



6L 
6D 

6Q 
60 



"r 04473 " 
59 .04830 5? 



60 
6C 
60 



06366 
0642S 
06486 
0654i 
06606 

06667 
06727 
06787 
06847 
06907 

06967 
07027 
07087 
07146 
07206 

07266 
07326 
07386 
0744i 
07505 



/ 



0V565 
07325 
07684 
0774! 
07803 

07863 
07922 
07981 
08041 
08100 

08160 
08210 

08278 
08338 
08397 

0845^ 
0851^ 
08574 
08634 
08693 

08752 

lo^.Exs. 



60 
59 
60 

6j 
51 

51 
59 
60 

59 
59 
59 

5i 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 



Lg.Vere.; J> 



04630 VIL 
04682 \i 
04734 *^ 

0478r 



52 
52 

5r 

5! 




52 
52 

52 



9.0581S Ri 



I! 



en 9 0530^. \7y 

®° 05355 ?? 

05407 \ 

60 05458 ^' 
Rn 05510 

55 9.05561 % 

-- 05613 \ 

05664 ?V 

05715 2f 

05767 ^t 



Log.Exs. 



9 



9 



9 



9 



9 



9 



9 



9 



•UDOiS 

■05869 Ri 

.05921 2{ 

05972 W 

_06p2j °^ 

0607| \ 

06121 2 

06176 \ 

06227 % 

06279 °: 

9 06330 ^X 

0638?= ?V 

064?T ^} 

064P? \\ 

. .0653^ °^ 

9. 0658? \k 

•06635 ^V 

.06686 cA 

.0673P ?V 

-- .06787 ^). 

^^ 9 06838 ^" 

-» ILg. Vers.^ n 



9 



9 



9 



9 



9 



08752 
08811 
08870 
08929 
08988 

09047 

0910 

091 

0922 

0928_ 

09841 
09400 
09458 
09517 
09576 



09631 
09698 
09752 
09810 
00869 

09927 
09986 
10044 
10102 
10161 

10215 
10278 
10386 
10394 
10452 

10511 

10569 

1062 

1068! 

1074 



1080] 
10851 
1091* 
10971 
1103! 



11091 
11149 
11207 
11265 
11323 



11380 

1143 

1149 

11554 

11611 

11669 
1172Z 
11784 
11842 
11899 

11957 
12015 
12075 
1212? 
121.87 

1224? 



.59 
59 
59 
59 

59 
59 
58 
59 
59 

56 
59 
58 

59 
58 

t 

59 
58 
58 

5 

5 

ii 

11 
Ii 

58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 

5 

5 

58 

58 

57 
58 
58 

5? 
57 

58 

5 
5 
5 
5 

58 
57 

57 

52 
57 

57 



O 

1 
2 
8 

4 



10 

11 
12 

18 
14 

15 
16 
17 
18 
_19 

20 

21 
22 
23 
24 



25 
26 
27 
28 
29 



30 

81 
82 
83 
84 



? 



LogT 



xs. 



35 
86 
37 
38 
39 

40 

41 
42 
48 
44 

45 
46 
47 
48 
49 

50 
61 
52 
58 
54 

55 
56 
67 
58 
59 



\ 



D 



20 



P.P. 





61 


60 


6 


6.1 


6.0 


7 


7-1 


7.0 


8 


B- 


8.0 


9 


xS: 


9.0 


10 


10-0 


30 


20. 


20.0 


80 


80.5 


80.0 


40 


40.8 


40.0 


50 


60.8 


50.0 




6.9 

6-1 

7- 
8. 

9- 
19. 

20. 

89. 



57 



6. 



10 io 

28-6 

88.5 
47.0 




6 

7 

8 

9 

10 

20 

80 

40 

50 



6 

7 

8 

9 

10 

20 

80 

40 

60 





84.0 0.1 
42.6I0.4 



a 



tit: 



LOG. VERSED SINES AXD EXTEKXAL SEOAXTS 179 



■18631 t 
.16f87 ? 
18643 I 



■Sil • 



-18731 i 





ti 


! 


S G 

n 































54 J>4 

B I B.4 

B.| 6-3 

7 2 7.2 



BOI43 

4B, 4fl 



!!■? 


1 


s' 


4n f 




39- 



180 LOG. VERSED SINES AND EXTERISTAL SECANTS 



fl 



S3 S3 S3 

'3 5 2 - - 
3 61 

! l 

S 17 



if 47 la. 



■A-i^--/ 



LOG. VERSED SINES AND EXTERNAL SECANTS 181 



.87ii ; 

was ' 

.eeas J 

Mil I 

.sbbI ? 

■BOOJ * 



9.3B103 ? 
.2B16? ; 
'2B20g I 



■26518 ? 
,28570 ° 
9.26621 I 



30081 2 
_J0131 ' 
SSOIBI t 
■30231 ? 
802 S2 t 
30332 ^ 



fi3 

■1 


ft 


2B:i 


ll] 


43.3 


12. B 



50 fid 49 

6 9.D' a. 01 ' - 

7 5 9 5-1 I 



.0 33.6S3.3j33. 



•j 


t. 

j: 

















.23501 J 



.27749 I 
.a7B0C 2 
.27BS 2 " 



.23000 J 
9.230B3 j 



;|iiTE: 



.31333 ^ 
■313B3 ° 



182 LOG. VERSED SINES AND EXTERNAL SECANTS 



24273 
24316 
243 a [ 

24S20 
2*581 
a4B0l 

.3473S 

J481M 
.S4S4I 

■84881 

- aiBar' 



30 24-! 
40 32.3 
fi0i4D.4 



67 B. 
13 6 13- 
20-i 20 



)G. VERSED SINES AND EXTERNAL SECANTS 183 



a. 41810 
4ie6T 



[^^! 



24. 1 

ao-F 



184 LOG, VERSED SINES AND EXTERNAL SECANTS 



52 

atess 
■ swfls 

.3473n 
-347Bi^ 

aola 34803 



7«T5; 



i^rE 



36301 
36426 
3646! 
3648' 

3660(5 
56635 
36870 
31705 
B 36735 
3677? 
36809 
3eS4< 



f S-8 

I " 



|30-S m 
m7.t J 

4S f 



ni 



UW. VERSED SINES AND EXTERNAL SECANTS 185 



S 



i?m 









188 LOG. VERSED SINES AND EXTERNAL SECAXT3 



■ 48894 ' 
9-48923 I 
■18955 % 



^M473 ' 
■ MflOl J 



-«4r73 I 
-B4B1S J 
64556 \ 



-fl72U5 J 

S72« * 

8 87390 1 



.653^ ' 
9.a5S71 4 



9 49300 I 
4939! i 



51253 , 

51281 , 

"Sill % 

9 I 



■ flBJl! ' 

D-08SB3 ; 

68396 ; 

68488 i 

■98*81 * 



-49747 i 

- 49776 " 

9-49808 ^ 

40839 t 



9-08325 I 
■68288 1 

-saio : 

-06363 1 



LOG. VERSED SINES AND EXTERNAL SECANTS 189 



18S LOG. VERSED SINES AND EXTERNAL SECANTS 



(W^o/t^T^ 



6B905 

9 Bsgie 



WG. VERSED SINES AND EXTERNAL SECANTS 191 



3 


( 















190 LOG. \'ERSED SISTS AND EXTERNAL SECANTS 




LOG. VERSED SINES AND EXTERNAL SECANTS 193 



^8480* I 



.90136 J 

231 1 

.90384 * 

-90349 J 

; 90434 J 



.65083 n 

.65087 n 

9 asm a 

.651E7 ? 

.86181 i 

.65303 S 

■Bsaa a : 

9.66351 i 



aoB ? 

343 I 
9132S , 



.e54lS 2 

m t 

9.eS48S 3 

■flsaog i 

-66S3! 3 

-6BS56 i 

■ 66579 ' 

9. 66609 7 

.6SB36 2 



». 93011 4 
.9309fl i 



.87035 ? 
-B7055 n 
■ eTOBfi „ 



9 64151 1 
-8419^ 4 

! 94389 i 



,!i 



""^|Loe,E.='. 



192 LOG. VERSED SINES AND EXTERNAL SECANTS 



S8 -82448 









li 



1 2.7 
} 3.1 



U)G. VEKSED SINES AND EXTERNAL SECANTS 196 



.99919 ?T 
69940 ^^ 



15 3. 7032? ;» 

16 70246 ii 

17 70368, if 
IB .702891 i' 

Jk ToiiV^ 

20 } 70333' ;1 

21 70355 St 
-- .70371, 5i 

.703eBi if 

■ 70430, ' 

^ 70141 ik 

70463 ,T 
70485; ik 
70507 if 

■ 70521 '^ 
• 70550^ J 
■ ^0572 5f 



-71085 i 

■71111 ; 
.71135 ; 



?S 10. 01977 1' 



.03127 
.03171, 

03260 
10 . 03304 



J231ij 2 
7236! 2 

.72386! 2 
72408 2 



1! 

l6~ 03973 
^04055 
l04149 
i 04338 

i 04337 
riJ3?T 
.04416 
.04461 
-04505 

10.0463S 
04084 
04728 

04S1B 
10-04861 



05325. J 

05305; ' 
'""5310^ 



45 43 

8 4.JJ 4.4 
I B.3 B-a 



f i;! 

21.7 

32B.g 

6 36-3 



194 LOG. VERSED SINES AND EXTERNAL SECANTS 

68® 59° 



o 

1 

2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 



15 
16 
17 
18 

JSL 
»0 
21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 
36 
87 
38 

M 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



Lg. Vers. D Log. Exs 



9. 



9 



67558 
675811 
676051 
67626 
67649 1 

6767li 
676941 
67717, 
677391 
67762 



9 



9 



9 



9 



9 



9 



9 



9 



67217 
67240 
67263 
67281 
67308 



67331 
67354 
67378 
67399 
67422 



67445 
67467 
67490 
67513 
67535 



67785 
67807 
67830 
67852 
67875 



67897 
67920 
67942 
67965 
67987 



68010 
68032 
68055 
68077 



68122 

68145 

68167 

6819 

6821 



68235 

68257 

68280 

683071 

6832i 



68347 
6836? 
68392 
68414 
6843^ 



68459 
68481 
6850? 
68526 
6854S 



19. 94 791 
.9483 
.9488J 
• 94921 
.94961 



9-95011 

.95051 

.9509 

.9514 

.9518 



9. 95221 
.95269 
.95313 
.95356 
.95399 

9.95442 
.95485 
.9552 
• 9557 

mi 



2$ 



95657 
.95705 
.95744 
.95787 
.95830 



19.95873 
.95919 
.9595$ 
.96002 
.96046 



19.96089 
.96132 
.96171 
.96218 
.96261 



^.96305 
.96348 
.96391 
.96434 



9.96321 
.96564 
.96607 
.96650 
.96694 



^.96737 
.96780 
.96824 
.9686 
.9691 



i 



9-9695? 
.96997 
.9704 
-9708 
.97127 



S 



^.9717; 
.9721 
.97257 
.9730 
^9734: 

^•97387 



_6fH9. 68571 
^y^. Vers.l^D^Log. Exs.j 



43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
4? 
43 
43 

43 
43 

4? 
43 

43 

4? 
43 
43 

4i 
43 
43 
43 
45 

43 
43 
4f 
43 
43 

43 
43 
43 

43 

43 
43 
4? 

41 
43 

43 



Lg. Versi 



^-68571 
.6859 
.6861 
.68637 
-68660 



9.68793 
.68816 
.68838 
.68860 
.68882 



9.68905 
.68927 
.68949 
.68971 
.68995 

9.69016 
.69038 
.69060 
.69082 
.6910i 



9.6912? 
.69149 
.69171 
.69193 
.69215 



9.6923: 
.69251 
.6928] 
.6930] 
.6932! 



9.68682 
.6870i 
.68727 
.68749 
.68771 



^.6934 
.6936 
.69392 
.69414 



9.69458 
.69480 
.69502 
.69524 
-69546 



^.69568 
.69590 
.69612 
.69634 
.69656 



9.69673 

.6970 

.6972 

.6974 

.6976 



19.6978 
.6980 
.6983 
.69853 
.69875 



69897 



lg. Vers. 



22 
22 
22 
22 

22 
22 
22 
22 
2J 

22 
22 
22 
22 
22 

22 
22 
23 
22 
22 

2? 
22 
22 
22 
22 

22 
2? 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
25 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
2T 
22 
22 

22 
22 
22 
2T 
22 

22 



Log. Exs. D 



9-97387 
9743 
9747 
97517 
97565 



9 



9 



9 



9 



9 



9 



9 



9 



9 



9 



9 



9 



9760? 
97647 
97690 
97734 
97777 



97821$ 
97864 
97907 
97951 
9799i 



98038 
98081 
98125 
9816 

mi 



! 



98255 
9829? 
98342 
98885 
98429 



9847? 
98516 
9855| 
98603 
98647 



9869? 
98734 
98777 
98821 
98864 



98908 
98952 
98995 
99039 
99085 



99126 
99170 
99213 
99257 
9930? 



9934i 
99388 
99431 
9947? 
99519 



99562 
9960? 
99650 
99694 
99737 

9978T 
99825 
9986? 
99912 
99956 



^Q 00000 



Log. Exs. 



P.P. 



6 
7 
8 
9 

10 
20 
80 
40 
50 



5.1 

5.8 

§:i 

14.5 
21.7 
29. Q 
86.3 



6 
7 
8 
9 

10 
20 
80 
40 
50 




6 
7 
8 
9 

10 
20 
SO 
40 
50 



2.6 
3.0 

i1 

7. 
11. 
15. Q 
18.7 





22 


6 


2.2 


7 


2. 


8 


2.) 


9 


3.) 


10 


3. 


20 


7. 


80 


11.0 


40 


14-1 
18.9( 


50 



P.P. 



LOG. VERSED SINES AND EXTERNAL SECANTS 197 



15789 

-758flB 
-75SB8 



15-6 IS 3 

31-3 81. Q 
39.138-7 



I'JS LOG. VEItdJiii SINKS AXD' EXTEEN.VL SEtJASTS 



T^T? 



78 56 7 
3.78576 



9.79333 
-7fl35i 

iT339D 

-7B«8 
-7B485 

^79503 

9-79521 

^79555 

7Bfi9g 



i 



Lg. Ver;. 



IX)G. VERSED SINES AND EXTERNAL 8ECANTO 108 



000? ; 

■aoose I 

.80044 I 
.8QQB3 



9 8029? J 

!boso4 i 

""323 i 

341 • 



J0710 ' 
WriB-»073g H 



.22558 5 
-22008 i 
■22858 5 



W^' 



t" 


e 




T 




















10 


8 6 




27 


3C 


25 5 


it 


|p 


BO 


«.b!42: 



) 7 S 7-9 
] 8.6 g.f 
i 17.6 17-5 
) 28 5 2fl 3 
! 35-3.35.0 
)44-l!4a J 

I E.2 5.1 



10' B 3 

30 25.Q 
40 33 . 1 
60 4\.e 



200 I-Or.. VKHSEI> SINKS AND EXTERNAL SECIANTW 



1-8 

a. 4 


ll 




«:? 


n 




1B*S 







)G. VEBSED SINES AND EXTERNAL SECANTS 201 



3870? 

39881 
38933 
38996 
37054 
37111 
■871 89 

37284 
37342 



SO SO 
40I40. 
GOI6O. 



i\ '■ 
I 8. 



:!!: 



10| fl. 
S0 2B 
60,47^ 



2S 3 
,37.B 



'M2 TiOfl. VKTtWED KINES AND KXTliRNAL SEC;ANTS 




LOG. VERSED SINES AND EXTERNAL SECANTS 203 



-BSTSO i 

.Mm 



Si 8.88988 " 



■60308 „ 
D- 50375 S 
.6094 £ £ 
.G041D H 
.60475 a 
. 50545 ' 
O.SOfllS S 



.89044 i 

.89080 { 

- B9075 ' 

9.80DSI } 

-89107 i 

-B9I23 ! 

.89139 { 



.89265 1 
.B9SSI , 
.89291 1 



0.53325 
-5239J I 
.Q34B4 „ 

52603 " 



0.53021 J 

..31Bl J 

.E323T L 

.S330 I ' 

0. 63372 , 



o.saiso i 

.65232 i 
.55294 7 



.56311 , 
-56461 , 



^89627 ' 
9.89343 I 



.89768 ; 
. 89763 ' 
9.69791 I 



i 71 73 

B 7-4; 7-3 
1 8.3 B.B 
9.S 9-7 

5 11-1 10. S 

6 12.3 13.1 
34.6 34. S 
5 37.0 36-5 
49-3 48.8 

.5 61-el60-B 



8 10-6 11 
Oll.SlJ 



.". 


t 


6S 

e.B 


67 


!'.! 


I 


,8: 


10. 


II 


I 


H- 


33 i 























204 LOG. VERSED SINES AND EXTERNAL SECANTS 1 


78° 79° 


1 




Lg.Ven, 


/, 


bg.txs. 


» 


tg.Vefs. 


» 


Uk.E,=. 


„ 




PP 




9 38877 




I0-6S08 


75 

i 


9. 9080 




10 63745 


8 




86 as 84 1 




-Bfi393 










.82825 






S-l 


8.5; l.t 




.Beaoi 










■62906 








10- 




s 






5831 












1 


U^- 




i 


B.S999S 


e 


.5939 


j 90886 




Jeigef 


B 


_* 




in 


ill'i :' 


G 




6 


.89971 




■ 954 


7 














5?*S 


?Sil? i 




.g998G 




B81 


■ 9091 




-63310 






31 




42-5|42- 




.B0002 










-B3381 




B 


41 


67' 


roslV 


J_ 


.90015 




_j. ITB 


















9.90033 




10. B84 




9 90958 




io 635"5 




10 


61 f ^' . ". 1 




.90049 








.9097 










12 


-90064 




'■ Boa 
















B.i 




. BOOBt 


1 


9072 


76 


^9100. 




^8379 












.90095 




■ 9148 


^9101_9 




__ 63819 


li 


e:i2! 12^ 


u\ 




9.90111 




10. 9224 


9. 1034 










10 13. 13. 


13 -i 


16 


.901 ( 




. 9308 




. lOi 


I 


64043 














' liU 




■ 1 65 






17 


80I4I- i\. 






!901 '. 










I 




i 


18 


!8!l!: '& ii!:! 


il 


901 3 




9630 


7S 


'■ 1 9 


1 


:e42G9 


3U 






10. 9606 


9^^ 


10-64371 






. 9683 


11 






.B445S 


1 




SO 7» T8_ 


22 


:90a 3 II 


9760 


V, 


; 1 4 


1 


.64538 






sg I- 




23 




is 












23 




^.Ull 


"! 


25 


g": 90268 




10~99"9i 


! 


9-" i~87 


15 


10~6i7B4 


83 
83 


36 




Wi. 


11. 1 


2e 


.B02B1 




. 006B 








26 




26 D 


it 


'b031! 




.60145 
-60323 


: 1 3: 


Is 


'■ 84850 
-65033 


B3 






29 


.,90328 




.8030rt 




- 1247 




-65116 




1» 




IB is as 1 


66. B 




9.BD343 


1 


1D.6037B 


u 






10.65199 








3 


90359 




- 015 




'. 1278 




.65283 










.90374 






77 


. 129 




-6536^ 




32 


6 




7 




-9038? 








IB 




83 




7 


9:0 t-i 




S 












■Imi 




8 


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io~678 


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36 


9 






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! 65785 




20 


25.8125.S 




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it 

85 




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: 1393 






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_. 66377 


4A 






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is 




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: 65 


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10 0.1 




.80801 




. 701 


73 








1% 












■ 78 








\\ 




sd'o^S 


4 


! 90831 






79 






-6680 




49 


400-g 


80 


a-90fls: 




10- 194' 


B~r5B"5 


li 


10. 66 B 




50'0.4 






















S 










9 5B( 




-67 5 




52 


10 15,, IS. 


5 






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?? 








86 




S il 




i:i 


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5i 


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B 


8 90721 


li 


10.62346 


?§ 


9"~r64: 




10 67 1 


i 






a ( 


2a 


G 


90744 




-82424 




I 




56 


10 2^ 




25 


i 


8075! 




.6250? 


83 








i 


57 


ilr 


5 


il 




-9077: 




.62585 










7 ' 






_.90790 




.63865 






■57 61 






16: 


10 


f 


8-9080' 








8 8171i 




10. 6774 




ao 


50 i 13 J 


12.1 


13.(1 




CFv;^ 


— 


Log.Ex= 


TT 


LFViT 


■d 


TJirES- 


TT 






^7?. 



LOG. VERSED filNES AND EXTERNAL SECANTS 205 



m 

S 15 



206 LOG. VERSED SINES AND EXTEItNAL SECANTS 



I30 ISO 

la.o la-o 
IS. I u.o 



- 1 

100.0 



)G. VERSED SINES AND EXtEhNAL SECANTS 207 



0. 83281 ,, 

■ 93416}; 
■835S1 ); 

"III li 

0.93957!; 

MOSS j; 

i 94366 }i 

■ 94S Q5 '■ 
. 94641 ); 



0. 96035 [ 
■96176 i 
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■ 96603 " 
0.98746 !; 
9688! i: 
97033 i 
9717? [, 

.87332 ' 



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■ 02*87 }; 



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T.O443I )j 
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04773 1 
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■ 05106 ^' 



30 8S-D| BO.O 



93.3 

na S 



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1.07B65 [: 



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1. 00451 }; 
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■0I0 B5 j- 
1.01225 !i 



1. 08755 {] 

Hial " 

.9303 |! 

_^ 09486 '■' 

1.09689 !! 



200 LOG. VERSED SINES AND EXTERNAL SECANTS 





g.Vers 


n 


Log. El!. 


B 


Lg.Vers 


n 


L.S. E<,. 


n 

1 1 






p.p. 




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llt^l 




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r 


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mi; 




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79660 






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10li635 






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p:8 


8 


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997 
0083 


i 




9444; 

9443' 


J 
1 


^8971 
.BB63 


:l 


i 


8 


17. 


10 


.93636 






0189 


SI 




9449 






119 
31 




30 






11 


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0296 






3451 




■ 87070 






88-6 


80. 




-93865 






040 


OB 






.B7195 


ii 




60 OB. 3 


IM.O 




.93eBC 
















13 




H 


^9,3694 
















! 97431 


ii 






93709 




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94561 










bV^^O? 


1! 


! 93731 




B8 


% 








^8779 


■3 


17 


\\\ 


■ 


IB 


-93751 






04 




9461^ 
















IS 


93767 


I 




^ 


08 










3 


11 


\ 


:!: 


SO 


211 


-93781 








"9464( 


10.B816 


11 


:!|| 




37 


1 






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.1 


23 





'3- 


0^ 


2a 

21 


■93839 


w 




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09 




9468^ 




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23 






3. 










oi 




9471: 












3« 












94736 




.9988 


% 


;e 


3 2 


27 


-93882 






9202 






94741 










u-J 
























0.; 


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S2245 






9"li 


.8927 




39 




S-i 




3(1 


9392! 






If 




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121 




1 


C' 






- 93910 






le 




S479I 










1! 




32 


-S3953 




82577 






9481i 




.8964 


12 
121 


32 








.9396 






3268 
















[*» 


























li 


!l 


BE 


TasiT 






82910 






9436: 




10 90025 


?2i 






36 














9486' 
















11 




93 13. 






9488 










1 ii 


8B 


.9404 




8324 






9489 


.80406 


!2^ 


38 






on 


39. 


■9405 


i 




8335 






9490 




1^? 


ia 




8^ 


s 


s 






icT 


■bIItS 




9492 


10.90861 





4) 


;9WS 












.90789 


i2i 


41 














\ 












-S091? 










1 






\ 




8380 






B496< 




.91045 








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1 


M 


_9412 












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a 






1 




:941s 


fo" 


n 






lifi 


'";!)5S1 


i 




B 





J 


3 


47 


















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47 






^9418 
-9419 


_ 


i^m 


\ 




9506^ 




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_^B26 
10.91956 




6, ?^. ?*. 


SO 




? 




i^l 




!e4Z2 












9507? 




.9209^ 


3 


51 


8 




1 B 


B2 


-9434 






84837 




















G3 
















! 92351 










lij 


51 


^9437 






8 SO 68 






_i6m 




-9348! 






30 


4^ 


48 




■ 9438 




io" 


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3 


5 


30 


7- 


'8 


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8629 












3 


6 
































11 is 








8B53 






96177 


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10 93381 




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Vv«. 


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nr 



OG. VERSED SINES AND EXTERNAL SECANTS 2n7 



/«>, 


B 


Ug E«i. 


n 


Lg.VM 


7> 


Ug.Exs. 


D 




P.P. 


i20 




10. 338 










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9606; 






0316 












8808^ 






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sai 




: 388 










ibI 


3 


a 25 3i 24 '0 


m 




. 3821 








0364) 




9' 28.5' 27.0 








g-BBioi 








160 










. 409 










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30, 95.0 90. 








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544 




. 561 


96 7| 










17 


31 


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[)493 


165 
168 
16 
16 

165 




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150 140 










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563 








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-96 69 






OS95 






8 


30 


1 18. a 


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10. 574 


143 


9-9«3a; 


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17 




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i 




1 


558 
Ml 




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75 








S83 




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t 








6 13 


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36 


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984 


14 








0822 


m 






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4 


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1 






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S'5I= 




10.989 




rr 




7.5lfl- 










9660 










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188 


45 


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-96561 
















isaeo 








.9668 






1003 




47 




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5 


9. 96721 




u' 


106B5 








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10783 






1593! 






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5 


g H,, -", .'3. 






ioooi 




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B 




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3 


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9-9679S 


g 


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7 




1. 1 6 


55971 










■ 38 


.9660! 






1173( 








1 4 


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1 


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57 


20 


4 fi 


: ? ; 
















1211' 






I §' ^'^1 iX 


P 


13 


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16 


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a 96653 




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1_230; 
12601 




^ 


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BOia.liU 6:)1.5 


?7^ 


■» 


Loj.t... 




Ig.Vmi-IT 


La«.\i>V' 


'^ *rr 



ans i.oc\. v-RRSEn sinrr and external secants 




LOG. VERSED SINES AND EXTERNAL SECANTS 208 



46876 it 
47276 to 
47B77 '" 
I 11 4B0BS fS 
48193 !Y 
4BSDe ji 



52801 !i 
53255 l; 
5371S ! 



9951? { 

99543 } 

9 9955S I 

eB5Bg i 

995B1 , 

99594 j 

996 06 ' 

9 99615 1 

99632 t 

99945 t 



2.0D04e i 

.01858 ! 

■0270? j 

.04098 ' 

205535 I 

.07020 i 

.08567 { 



9975G } 

39785 } 
9979? ' 



7016B I; 

706'!4 
' 11 7150? Sj 
T3196 S 
7289i ,; 



99943 
9997^ 

io~ono"no 



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. 










o 




1« 


/ 




L. C. 


M. 


K. 


T. 


L. C. 


M. 


E. 


T. 





0.00 


0.000 


0.000 


0.00 


100.00 


0.218 


0.218 


60.00 





2 


8.33 


0.000 


0.000 


1.67 


103.33 


0.233 


0.283 


61.07 


2 


4 


6.67 


001 


0.001 


3.33 


106.66 


0.248 


248 


63.38 


4 


6 


10.00 


0.002 


0.002 


5.00 


110.00 


0.264 


0.264 


65.00 


6 


8 


13.33 


0.004 


0.004 


6.67 


113.83 


0.280 


0.280 


66.67 


8 


10 


16.67 


0.006 


0.006 


8.33 


116.66 


0.297 


0.297 


68.83 


10 


12 


20.00 


0.009 


0.009 


10.00 


120 00 


0.314 


0.814 


60.00 


12 


14 


23.33 


0.012 


0.012 


11.67 


123.83 


0.332 


0.832 


61.67 


14 


16 


26.67 


0.015 


0.015 


18.33 


126.66 


0.350 


0.850 


63.83 


16 


18 


80.00 


0.019 


0.019 


16.00 


130.00 


0.868 


0.368 


65.00 


18 


20 


33.33 


0.024 


0.024 


16.67 


138.33 


0.888 


0.388 


66.67 


SO 


22 


36 67 


0.029 


0.029 


18.33 


136.66 


0.407 


0.407 


68.33 


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 


73.83 


28 


80 


60.00 


0.054 


0.054 


25.00 


150.00 


0.491 


0.491 


76.00 


30 


32 


63 33 


002 


0.002 


26.67 


153.33 


0.518 


0.613 


76.67 


8S 


34 


66 67" 


•0 070 


070 


28.33 


156.66 


0.636 


0.636 


78.38 


84 


36 


60 00 


0.079 


o.o;9 


80.00 


160.00 


0.659 


0.669 


80.00 


86 


38 


63.33 


0.088 


0.088 


81.67 


163.33 


0.582 


0.6&3 


81.67 


88 


40 


66.67 


097 


0097 


83 .33 


166 66 


0.606 


0.606 


83.83 


40 


42 


70.00 


0.107 


0.107 


35.00 


170.00 


0.630 


0.680 


86.00 


42 


44 


73.33 


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 


88.38 


46 


48 


80.00 


0.140 


0.140 


40.00 


180.00 


0.706 


0.706 


90.00 


48 


50 


S3.:« 


0.151 


0.151 


41.67 


183 33 


738 


0.788 


91.67 


50 


52 


86 67 


0.164 


0.164 


43.33 


186.66 


0.760 


0.760 


93.83 


68 


54 


90 00 


176 


0.176 


45.00 


190.00 


0.788 


0.788 


05.00 


54 


60 


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 


98.88 


68 


60 


100.00 


0.218 


0.218 


50 00 


199.98 


0873 


878 


100.00 


60 





FUNCTIONS 


OF 


A ONE-DEGREE CURVE 


211 


. 




»° 


^ 


L.C. 


M. 


E. 


T. 


L. C. 


M. 


E. 









,873 0.673 


100.00 


S9B.M 


1,934 




w- 




a 


loaiai o.wn 




101. 6T 


303.29 


s.ooa 


a'.m 


151 :74 






£«I.M 


9W 


.833 


10q.34 


300. M 








4 




S».«T 


.008 


.m 






a^Doe 


aiooo 




B 


B 




.DOS 


.903 


10S:68 


aia.ao 










10 








10S.3S 


m.ti 


a.iss 


2.1«0 


158.42 






mvT 












a.aao 




IB 




M3,ai 




'.m 






a^Baa 
















iiaao 


Bw.sa 




a:33o 






IB 


sa'.K 








svo.us 


a.S76 


2.377 


185,00 






233.30 


.IBS 






B33.aB 








20 








'.sa 


iialsa 


S3S.fil 








22 










130.03 


sao.BS 


a.sOT 


a.5a8 






«8 










ais.a 


a. 673 




171,77 






aie!s3 








34B.S1 


a.aaa 








ao 




:39l 


'.m 


isslos 






sinTS 






39 


S63!S9 


.m 




1M.TO 


aM:a7 


s:7S4 


a.7J6 


178 72 


33 


SI 


au.ea 










a.77S 




178-39 




sa 


asa.x 








SrjOlM 








38 


3B 


m.» 


:6i3 


:si3 


131 ;7i 






i.m 




3S 


40 


sm.fig 






133 38 


3^6.60 


aoM 


a.B34 




40 




!M.M 








369.94 


g.9H7 


2.989 


185.07 








tw 












194,74 










'.67i 


isalss 


BTS^SD 


aioDB 


097 




48 


43 








140.06 


Sra.04 






190*07 


48 




avails 










3!™a 


:S0T 


101,74 


60 






!!04 


!794 




38e!eo 




.281 




52 




ast'.x 


.830 


.888 


i«:oB 


3B9B1 










M 


anas 






146.73 


S93.S7 


8.377 


■jTB 


1»8,TS 




S8 


»g.BS 




.K) 




snc.flo 






198. 4-J 


58 




»g.H 


low 






.^90.94 








00 




4= 


6- 




L. C. 


M. 


E. 


T. 




M. 


E, 


T, 


~^ 


amM 




.403 


iw.oo 


'mJe~ 


6 154 


Tlsfl" 


250 17 


~~^ 


S 




'.SSO 










sioM 


261 isi 






4oe!in 


.BOO 










5,807 




4 






.BOB 




soslio 










B 








:tso 


aOB.TT 


Bisiao 






analsB 




10 


4I«!W 


'.Tfn 


.TOO 






B:8a3 


5.830 






11 


4I>0? 


.848 












260.20 








.010 














4 




4!s!u 




.'m 


m'.a 


65«>a 








e 


IB 


4£BH 


!034 








fl.iao 


B.135 




s 


to 


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.OBB 












aaa.w 




ts 






:m 


sis:« 


S3flS1 








22 






!ae4 






639 HJ 


B;3Ba 


8.380 






«> 


443! £4 












8.4lt 


971 S8 




SS 


446.BS 


;363 


:3x 




64C..W 






KS..-H 






449.01 






sM.\s 


WO.tll 


0.599 






52 


M 


453. H 




Am 


^80 


BG3 17 








3a 


M 


4H.$7 


!b5D 








6;7BI 










450 90 


BIT 


>21 




GS9'b3 






aw 23 




n 


463. » 


.«81 


.088 


33181 


BBS IB 






»ioo 




40 


MO SB 






833 48 


SBfl.49 


7.00S 




283.57 


40 




400.80 


isfll 




aB5 IS 






7006 


1:86.24 


42 




iTSK 


m 


.riB3 


!tse.fa 


BTajs 






2WB1 


44 


40 


4TflH 


.098 




1.'3H.48 


678.48 






i88 60 


46 


ts 


4n.ss 










7339 






*S 




481 aa 


m 














50 




4».W 


.187 




a4S.4B 


BBflii? 




7;si7 


293.80 


62 






!3B 




ME IB 


B89.80 






2».a7 




H 


103. ai 






ai6 83 


5B3.I8 


7. 878 




290 06 


50 




4MM 


^381 








7.764 


7 77B 


i!sn 62 






40B.se 




.450 




7.B60 




300 30 





212 


FUNCTIONS OF A ONE-DEGREE CURVE 




/ 


6« 


7. 


/ 


L. C. M. E. T. 


L. C. M. E. T. 



2 

4 
6 
8 
10 
12 
14 
16 
13 

20 
22 
24 
26 
28 
SO 
32 
34 
86 
38 

40 
42 
44 
46 
48 
60 
52 
54 
66 
58 
60 


599.80 .7.850 7.861 300.30 
603.18 7.940 7.951 301.97 
606 46 8.030 8.041 303.64 
609.78 8.120 8.131 305.31 
613.11 8.210 8.221 306.98 
616.44 8.300 8.311 308.65 
619.76 8.390 8.401 310.32 
623.09 ' 8.480 8.491 311.99 
626.42 8.570 8.581 313.66 
629.74 8.660 8.671 315.33 

633.07 8.750 8.761 317.00 
636.40 8.844 8.856 818.67 
639.72 8.939 8.951 320.34 
643.05 9.033 9.046 322.01 
646.38 9.128 9.141 323.68 
649.70 9.222 9.236 325.35 
653.03 9.317 9.331 327.02 
656.36 9.411 9.426 828. 69 
659.69 9.506 9.521 330.37 
663.02 9.600 9.616 332 04 

666.34 9.695 9.712 383.71 
669.67 9 794 9.812 335.88 
673.00 9.894 9.912 337.05 
676.32 9.993 lO.Ol 338.78 
679.65 10.09 10.11 840.40 
682 98 10.19 10.21 842.07 
686.30 10.29 10.31 343.74 
689 63 10.39 10.41 845.41 
692.96 10.49 10.51 347.08 
696.28 10.59 10.61 848.76 
699.60 10.69 10.71 850.44 


699.60 10.69 10.71 850.44 
702.93 10.79 10.81 352.11 
706.26 10.90 10.92 353.79 
709.58 11.00 11.02 855.46 
712.91 11.11 11.13 367.18 
716.24 11.21 11.23 858.81 
719.56 11.31 11.33 360.48 
722.89 11.42 11.44 362.16 
726.21 11.52 11.54 363 88 
729.53 11.63 11.65 365.50 

732.86 11.73 11.75 367.17 
736.19 11.84 11.86 368.85 
739.51 11.95 11.97 870.52 
742.84 12.06 12 08 872 19 
746.17 12.17 12.19 373.86 
749.49 12.27 12.30 375.54 
752.82 12.38 12.41 377.22 
756.15 12.49 12 52 378.89 

759.47 12.60 12.63 380 57 
762.80 12.71 12.74 38J.24 

766.13 12.82 12.85 383.92 
769.45 12.98 12.96 385.60 
772.78 18.04 18.08 387.27 
776.11 18.15 13.19 388.95 

779.48 18.26 13.31 890.62 
782.76 18.37 13.42 392.80 
786.09 18.48 13.63 893.98 
789.41 13.59 13.66 396.66 
792.74 13.70 13.76 897.88 
796.07 18.81 18.88 899.01 
799.40 18.96 13.99 400.70 




2 

4 

6 

8 

10 

12 

14 

16 

18 

20 
22 
24 
26 
28 
30 
82 
34 
36 
38 

40 
42 
44 
46 
48 
50 
62 
54 
66 
68 
60 


/ 


8* 


9" 


/ 


L. C. M. E. T. 


L. C. M. E. T. 



2 
4 
6 

8 
10 
12 
14 
16 
18 

20 
22 
24 
26 
28 
80 
32 
34 
86 
88 

40 
42 
44 
46 
48 
50 
52 
54 
66 
58 
60 


799.40 13.96 13.99 400.70 
802.72 14.07 14.10 402.37 
806.04 14.19 14.22 404 05 

809.87 14.31 14.34 405.72 
812.69 14 43 14.46 407.89 
816.01 14.55 14.58 409.06 
819.34 14.66 14.70 410.74 

822.66 14.78 14.82 412.41 
825.98 14.90 14.94 414.03 
829.31 15.02 15.06 415.75 

882.63 15.14 15.18 417.48 
885.95 15.26 15.30 419.10 
839.28 15.38 15.48 420.77 
842.60 15.51 15.66 422.45 
845.92 15.63 15.68 424.12 
849.25 16.75 16.80 425.79 

862.67 15.88 15.93 427.47 
856.89 16.00 16.05 429.15 
859.22 16 12 16.18 430.82 
862.54 16.25 16.30 432.50 

865.86 16.38 16.43 434.18 
869.19 16.50 16.55 435.86 
872.51 16.63 16.08 437.54 

875.88 16.76 16.81 439.21 
879.16 16.89 16.94 440.89 
882.48 17.02 17.07 442.57 
886.80 17.14 17.19 444.25 
889.13 17.27 17.32 445.93 
892.46 17.40 17.46 447.60 
895.77 17.63 1758 449.28 
899.10 17.66 17.71 450.95 


899.10 l'-.66 17.71 460.95 
902.42 17.79 17.84 452.63 
905.74 17.92 17.98 464.81 
909.07 18.06 18.11 465.98 
912.39 18.19 18.25 467.66 
915.71 18.32 18.38 469.34 
919.04 18.46 18.52 461.02 
922.36 18.59 18.65 462.70 
925.68 18.72 18.79 464.87 
929.01 18.86 18.92 466.05 

932.83 18.99 19.06 467.73 
935.65 19.12 19.19 469.41 
938.98 19.26 19.33 471.06 
942.30 19.40 19.47 472.76 
945.62 19.54 19.61 474.43 
948.95 19.68 19.76 476.10 
952.27 19.82 19.89 477.78 
955.59 19.96 20.03 479.46 
958.92 20.10 20.17 481.14 
962.24 20.24 20.31 482.83 

965.56 20.38 20.45 484.51 
968.89 20.52 20.69 486.19 
972.21 20.66 20.74 487.87 
975.53 20.80 20.88 489.66 
978 86 20.94 21.03 491.34 
982.18 21.09 21.17 492. 9S 
985.50 21.23 21.81 494.60 
988.83 21.. 37 21.46 496.28 
992.15 21.51 21.60 497.96 
995.47 21.65 21.75 499.66 
998.80 21.80 21.89 601.81 



2 
4 
6 
8 

10 
12 
14 
16 
18 

90 
22 
24 
26 
28 
SO 
3» 
84 
36 
88 

40 
42 
44 
46 
48 
60 
68 
54 
56 
68 
GO 





■functions of a ONE-DEGllEE CURVE 


213 




10" 








L 0. 


M. 


B. 


T. 


L. C. M. 


E^ 


T. 










501.33 


1098 


4 SB.S8 










loaeii 


si!m 


isioa 






7 S6.M 




553:43 






t005.4 


sa.o* 


SB.18 


604:68 






SB:83 










S9.24 




306.36 


1108 3 SB.8B 


38,90 






8 






£3i48 






7 37.03 








ID 


ioi5;4 


bsIh 


Bs.ea 








37:33 




10 




lOlfl.T 


le.iM 


83.78 


511 40 




i ^7:35 


37.48 










le.BS 


ea.ys 


513,08 




5 87,31 


!!7,«B 






IB 


10S5.4 


















IB 


1038.7 


wlia 


la'aa 








B6fl:86 


18 


» 


loat.o 


sa.aa 




61H13 


1131 


a 38.00 


38.14 




30 


a 


1033.3 










) 3f.l7 




670 :3i 




14 


1038. B 












38147 






a 


104i.O 


in;73 


23,W 








38 64 


.173.58 


33 


18 






33.98 


334,85 




8 38,87 


38,81 




ia 














1 SflBl 






30 


aa 


imi.a 










i 39.1X1 








M 


10H.S 


iM:34 


W'4S 


32b:S9 






39:31 


.wo:*! 


34 


M 


im.t 


S1.4B 


34.30 




iH 


1 3fl;84 


39,48 


MS.00 


H 


S8 












4 30.50 


38.63 


683.(19 


18 


40 




















n 


loesls 


«:8s 


la'.ix 


N30;01 










43 




ion. 8 










4 30,01 




SB8.74 




' n 




S5!a7 








7 30.18 


30:34 


3D0.4ii 


46 


m 


10^! 
















4B 




1061. 


ssIqb 


li'.ro 








30 38 








106G. 


!B.r4 








(i 3070 




6B.l"47 
















) 30,87 


3i:04 


587.16 




M 




K.oe 


w'.ts 










598.8) 




U 


io»s; 


GO 2S 


3«.B4 








31.39 






10 


1088,4 


E6,88 














W 




1!" 


18' 




L.C, 


H, 


E. 


T. 


L. 


M, 


E 


T. 


~B~ 


IIST.II 


Sl.» 








3 36.83 


37,07 








1^1 .a 










J 37,0a 


37 in 






4 


I9M.S 








130: 






GM:a5 




(1 


1207. B 


sfoj 


at:o9 


Mrias 


131K 


i ai'.m 








B 


IMI 1 


Bi.OB 


33,« 


flOS.BT 




5 8; 59 


37,83 










SS.»7 


33.45 


310.33 






3S,0t 












S'J.BS 








38,3:1 






14 


iiHi;i 


kIk 






1r.3< 






384 69 




IB 


JiSi.i 


W,90 


k:o9 


015.73 




8 38.86 




366.*: 




IB 


ISBT.T 


si.or 


53.17 








38:83 










33.15 


33.35 








39(11 


069.75 


SO 


W 






S3,S3 


*-;;■'" 




7 38! OS 


a9,« 


671.44 








aaifii 








38.15 


30.40 


673.13 






1S4i:o 


33.sg 










SSI. 00 




se 
















39.80 




38 








34.3! 




im 


9 38.71 


40.(10 


678:30 


30 


U 




SliM 


34 43 






3 30,94 


40 18 


679. hO 






13M.! 


M.4i 








8 40,13 


40.80 






as 


12ST.S 


84. W 


31 U 






9 40.33 


4U 58 












3.-; 01 


034:30 




a 40,5i 


40.79 






« 


13M.3 


MBT 


SS.Ifl 


635.08 




5 40,71 


40 89 


6^.61 






1S6:.S 




33.37 


5.17.08 




8 40, ill 




088 33 






isro.8 


»!84 




















39.53 




64i:09 




4 41:3 


4i:co 


eOKTO 


43 






8S71 


S.i;04 


04i74 




' " 5 


41.B1 


683.30 


4B 


M 


1880.7 


35.00 




D44.48 






43,01 


OPS .08 




S3 


1£»,0 














0811.77 








X.S7 




(147 iai 






43, 4 J 


«U8.4G 








33 40 


3ii:ii9 


(119.48 


ISK 






700 14 




H 


IMf.O 


ss.u 


38.88 














« 


1«7.3 


M.ra 

















214 FUNCTIONS OF A ONE-DEGREE CURVE 



, 


14= 


16= 




L. C. 


M. 


E. 


T, 


L. C 


M. 


E. 


T. 




13%. fi 


42.71 


48. OS 










754. SS 








4S.W 


4.3^ 


705:23 


]4« 




24 


49:88 


58 


03 












lOfl.BS 








49.89 


57 


r4 












708,62 


















uw'.% 














50:34 












4S:?4 


44 07 


7ia:oi 








60-58 


'63 


s 










M.sa 


713.71 






34 




'M 
























'W 








M^a!] 


44!ll3 






iii 








-97 








1438.4 


44. £6 








fi fil 


00 


6i:4« 


69 


81 


IB 








45.19 


730.49 








si.es 






to . 










742.80 










76 


)1 


23 1 




43a. a 


« IB 












a!:i8 




TO 


U \ 






45 40 


45.76 


™:6B 


53t 


8 61 


89 


53.36 




40 


*6 \ 












727.38 












n 


38 \ 






















7S 


79 


SO 1 




44BS 


4< 




48 Uo 












81 


49 


S3 ^ 


34 


14K.B 




u 


48.61 


32'35 




K 


79 


Bs:i« 


89 


10 


S4 1 


86 


USB. a 






48.Ba 


34,05 






OS 




7S4 


w 


St 1 




UiD.a 




68 




35.74 














sa ^ 


40 


146^.8 


4( 
















786 


£9 




43 






D8 


47:48 


T3»:iB 




S 5! 


89 


54:30 


J8t 


99 


4« 










47, B8 


740.81 


56S 




93 






W 


44 






47 




47. SO 
















48 


4S 




i\ 


7a 




744:S0 












» 


48 


» 


HTflia 




SI 


48:m 


7<5.gg 


57; 


4 M 


81 


S5:n 


79« 


79 












48.58 


747.59 


581 




84 


55,37 


7M 


49 
















5S5 




07 








M 










40:60 




na 




30 






89 


» 




hwIb 




^ 


4fl.-ia 










Ba:o6 




59 




60 










751 :3S 


m, 


9 M 


76 


56.31 


805 


Xt 


60 




i«. 


IT 


~ 




M. 


E. 


T. 


L. C 


M. 


E. 


T. 






















3 


i'm.-i 


sslsB 










IS 




858 




t 






B6.iS 


:s6:78 






1 89 


IS 


84:i5 


869 


■8 










67.oa 


810.30 


i7oa 














e 


\m.\ 




67. »6 












883 






10 


19II.4 


MiflS 










IB 




8M 


(7 


10 




1614.7 


B7,17 


67:74 


815,49 


1713 


r 64 


43 


65:iS 


868 














817.10 














14 / 


















55:57 


869 


is 


18 / 


IB 


16a4:6 


57:b7 


5B:46 


820:59 






17 


flS.B3 


871 


6B 


18 / 




iBST.g 


B8 11 






i7ae 


8 65 




68 18 






SO / 


ss 
















68.43 






a 








5b'|B 














79 


« 


ss 


1B37;B 


ssiss 


60.43 


flSr'39 




7 6C 


18 




S7S 


W 


SS 


38 




M.06 


59. SS 










87:21 


880 




« 


30 
















87.47 




W 


30 


3* 


1647:7 




M,:8 


^yiA<i 


i74e 


fi 66 


M 


W.TS 


883 


SI 


St 


H 


1851 .0 


S9.78 


80 41 




174( 


9 87 




67,98 


8SG 




u 
















» 






s 


u 






















73 


SB 


40 


1860.8 


80 50 


81.14 






8 81 


96 


88,76 


890 




40 


& 




80.74 














893 




Vt 










m.-ii 








egiaa 


SOS 


88 




46 


Ifljo's 


81 .aa 


6i:S9 


844,43 


1781 


6 8( 




89.56 


sat 


U 








81. 4B 








9 at 


99 


89.ffi 








M 




61. rs 


e!:3g 






i 89 




70.09 


wt 










B1.96 






1771 


5 69 




n.M 


900 


g> 




H 


168i:(l 


B-!.ai 


83:89 


BSl 21 


178* 


a 89 


76 


T0.8S 


tos 


40 


St 










B5i.e4 


r78< 


1 T( 


OS 




904 


10 




i8M!e 


ti'.Vi 










SB 




901 


81 




60 


18B3.fi 


ta.M 


MM 


B58'a5 


I7« 


7 70.54 


Ti.4a 






w 



FUNCTIONS OF A ONE-DEGREE CURVE 215 







18* 


• 


19*» 1 




# 


L.C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 


/ 





17fl8.7 


70.64 


71.42 


907.52 


1891.5 


78.58 


79.65 


958.86 





2 


1796.0 


70.80 


71.69 


909.23 


1894.8 


78.86 


79.94 


960.57 


8 


4 


1799.8 


71.06 


71.96 


910.94 


1898.1 


79.13 


80.22 


963.30 


4 


6 


1803.6 


71.88 


73.23 


913.65 


1901.3 


79.41 


80.51 


964.00 


6 


8 


1806.9 


71.59 


72.50 


914.86 


1904.6 


79.68 


80.79 


965.72 


8 


10 


1809.8 


71.85 


72.77 


916.07 


1907.9 


79.96 


81.08 


967.43 


10 


13 


1818.6 


72.18 


73.04 


917.78 


1911.3 


80.24 


81.37 


969.15 


18 


14 


1815.7 


78.38 


73.31 


919.49 


1914.5 


80.51 


81.65 


970.86 


14 


16 


1819.0 


78.64 


78.58 


921.30 


1917.8 


80.79 


81.94 


978.58 


16 


18 


1838.8 


78 91 


73.85 


922.91 


1921.0 


81.07 


82.82 


974.29 


18 


80 


182S.6 


78.17 


74.13 


924.68 


1924.3 


81.35 


82.51 


976.01 


80 


88 


1888.9 


73.43 


74.89 


926.34 


1927.6 


81.63 


82.80 


977.72 


22 


84 


1832.3 


78.70 


74.67 


928.05 


1930.9 


81.91 


83.09 


979.44 


24 


86 


1886.5 


73.97 


74.94 


929.76 


1934.2 


82.20 


83.38 


981.15 


26 


88 


1888.8 


74.34 


75.28 


931.47 


1937.5 


82.48 


88.67 


982.86 


28 


80 


1848.1 


74.61 


75.49 


933.18 


1940.7 


82.76 


83.97 


984.58 


30 


88 


1845.4 


74.77 


75.77 


934.89 


1944.0 


83.05 


84.26 


986.30 


32 


84 


1848.7 


75.04 


76.04 


936.60 


1947.3 


83.33 


84.55 


988.02 


34 


86 


1858.0 


75.31 


76.33 


938.32 


1950.6 


83.61 


84.84 


989.74 


36 


88 


1855.3 


75.58 


76.59 


940.03 


1953.9 


83.90 


85.13 


991.46 


38 


40 


1858.6 


75.85 


76.87 


941.74 


1957.2 


84.18 


85.43 


993.18 


40 


48 


1861.9 


76.13 


77.14 


943.45 


1960.4 


84.47 


85.73 


994.90 


42 


44 


1865.1 


76.39 


77.43 


945.16 


1963.7 


84.75 


86.02 


996.68 


44 


46 


1868.4 


76.67 


77.70 


946.88 


1967.0 


85.04 


86.32 


998.34 


46 


48 


1871.7 


76.94 


77.98 


948.59 


1970.3 


85.32 


86.61 


1000.0 


48 


50 


1875.0 


77.31 


78.26 


950.30 


1973.6 


85.61 


86.91 


1001.8 


50 


68 


1878.8 


77.49 


78.53 


952.01 


1976.9 


85.90 


87.21 


1003.5 


53 


54 


1881.6 


77.76 


78.81 


953.72 


1980.1 


86.19 


87.50 


1005.2 


54 


60 


1884.9 


78.03 


79.09 


955.44 


1983.4 


86.47 


87.80 


1006.9 


56 


68 


1888.8 


78.31 


79.37 


957.15 


1986.7 


86.76 


88.09 


1008.6 


58 


60 


1891.5 


78.58 


79.65 


958.86 


1990.0 


87.05 


88.39 


1010.4 ' 


60 


/ 


20* 1 


210 


t 


L.C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. • 





1990.0 


87.06 


88.39 


1010.4 


2088.5 


95.95 


97.58 


1068.0 





8 


1998.3 


87.34 


88.69 


1012.1 


2091.8 


96.26 


97.90 


1063.7 


3 


4 


1996.6 


87.63 


88.99 


1013.8 


2095.0 


96.56 


98.21 


1065.4 


4 


6 


1999.8 


87.93 


89.29 


1015.5 


2098.8 


90.87 


98.53 


1067.2 


6 


8 


2003.1 


88.21 


89.59 


1017.2 


2101.6 


97.17 


98.84 


1068.9 


8 


10 


8006.4 


88.50 


89.89 


1019.0 


2104.9 


97.48 


99.16 


1070.6 


10 


18 


8000.7 


88.79 


90.19 


1020.7 


2108.1 


97.79 


99.48 


1072.4 


12 


14 


2018.0 


89.06 


90.49 


1022.4 


2111.4 


98.09 


99.79 


1074.1 


14 


16 


8016 8 


89.37 


90.79 


1024.1 


2114.7 


98.40 


100.1 


1075.8 


16 


18 


8019.5 


89.66 


91.09 


1025.8 


2118.0 


98.70 


100.4 


1077.5 


18 


80 


8028.8 


89.96 


91.40 


1087.6 


2121.2 


99.00 


J00.7 


1079.3 


20 


28 


8086.1 


90.25 


91.71 


1029.3 


2124.5 


99.30 


JOl.l 


1081.0 


88 


84 


8039.4 


90.55 


93.01 


1031.0 


2127.8 


99.60 


101.4 


1082.7 


84 


86 


8038.7 


90.85 


92.32 


1032.7 


2131.0 


99.90 


101.7 


1064.4 


26 


88 


8036.0 


91.15 


92.68 


1034.4 


2134.3 


100.2 


102.0 


1086.2 


28 


80 


8039.8 


91.45 


92.93 


1036.1 


2137.6 


100.5 


102.3 


1087.9 


80 


88 


8048.5 


91.74 


93.24 


1087.9 


2140.9 


100.8 


102.7 


1089.6 


32 


84 


8045.8 


92.04 


93.54 


1039.6 


8144.1 


101.1 


103.0 


1091.3 


34 


86 


8049.1 


92.34 


93.85 


1041.3 


2147.4 


101.4 


103.3 


1093.1 


86 


88 


8058.4 


92.64 


94.15 


1043.0 


2150.7 


101.7 


103.6 


1094.8 


88 


40 


2066.7 


93.94 


94.46 


1044.8 


2164.0 


102.1 


104.0 


1096.5 


40 


42 


8058.9 


98.24 


94.78 


1046.5 


2157.2 


102.4 


104.3. 


1098.3 


48 


44 


3068.8 


93.54 


95.09 


1048.2 


2160.5 


102.7 


104.6 


1100.0 


44 


46 


8066.5 


93.84 


95.40 


1049.9 


2163.8 


103.0 


104.9 


1101.7 


46 


48 


8068.8 


94.14 


95.71 


1051.7 


2167.1 


103.3 


105.3 


1103.4 


48 


50 


8078.1 


94.44 


96.03 


1053.4 


2170.3 


103.6 


105.6 


1105.2 


60 


58 


8075.4 


94.74 


96.34 


1055.1 


2173.6 


103.9 


105.9 


1106.9 


53 


64 


8078.6 


95.04 


96.65 


1056.8 


2176.9 


104.2 


106.3 


1108.6 


54 


66 


3081.9 


95.34 


96 96 


1058.6 


2180.1 


104.5 


106.6 


1110.3 


56 


68 


8066.8 


96.64 


97.87 


1060.3 


2183.4 


104.8 


106.9 


1112.1 


58 


ao 


8068.6 


96.96 


97.68 


1062.0 


2186.7 


105.8 


107.8 


1118.8 


60 



2ir, FUXCTIOXfi OF A ONE-DEGREE CURVE 



/ 



22» 1 


■ 


«8<» 






L._C._ 


M. 


E. 


T. 


L. 0. 


M. 


E. 


T. 


1 


2186.7 


ia->.3 


107.8 


1113.8 


2284.8 


115.0 


117.4 


1166.8 





a 


2190.0 


lOR.O 


107.6 


1115.5 


2288.1 


115.3 


117.7 


1167.6 


2 


4 


yi03.a 


105.9 


107.9 


1117.3 


2291.8 


115.7 


118.1 


1169.2 


4 


n 


21»0.5 


lOti.3 


108.2 


1119.0 


2294.6 


116.0 


118.4 


1171.0 


6 


H 


yi»9.8 


100.5 


108.6 


1120.7 


2297.8 


116.4 


118.8 


1172.7 


8 


10 


IWIW.O 


100.8 


10S.9 


1122.4 


2301.1 


116.7 


119.1 


1174.4 


10 


rj 


VJ2<HJ » 


107. 1 


109.3 


1124.2 


2304.4 


117.0 


119.5 


1178.2 


12 


It 


WOtt.6 


107.4 


109.6 


1125.9 


2307.6 


117.4 


119.8 


1177.9 


14 


la 


wr,» u 


107.7 


109.9 


1127.6 


2310.9 


117.7 


120.2 


1179.7 


16 


IH 


22ia.i 


llV.O 


110.3 


1129.4 


2814.1 


118.1 


120.6 


1181.4 


18 


Vl» 


\A}n).4 


iaM.4 


110.6 


1131.1 


2317.4 


118.4 


120.9 


1183.1 


20 


* • 


xMrw.7 


U>S.7 


110.9 


1132.8 


2320.7 


118.7 


121.2 


1184.9 


22 


V4 


•l-Wft.Q 


h>9 


lll.iS 


1134.6 


28*23.9 


119.1 


121 6 


1186.6 


24 


W 


'.nrju.'j 


n>a.4 


111.6 


1136.3 


2327.2 


119.4 


121.9 


1188.4 


26 


W 


•,M3',V5 


UV.7 


111.9 


113S.0 


2330.4 


119.8 


122.8 


1190.1 


28 


W 


'^W 7 


110.0 


112.3 


ll3y.7 


2833.7 


120.1 


122.6 


1191.8 


80 


:w 


•.•'JW.O 


no 4 


112 6 


1141.5 


2337.0 


120.4 


128.0 


1193.6 


^2 


M 


;■.*« » 


no. 7 


112,9 


1 14 ^ 3 


2340.2 


120.8 


128.8 


1195.3 


31 


;w 


x\m:».« 


1U.0 


ua » 


1144.9 


2343.5 


121.1 


128.7 


1197.1 


36 


1 ss 


^•IS.8 


111 4 


IIS 6 


114i» 7 


2*46.7 


121.5 


124.1 


1198.8 


88 


4V» 


v\\v.» I 


1M.7 


i:i 9 


lllS 4 


23.-10 


121.8 


124.4 


1200.5 


40 


l\J 


•:,\w 1 


i:: 


114 S 


ll.V.l 


2353.3 


122.1 


124.8 


1202.8 


42 


A\ 


;":>•* « 


IIC.S 


::4 6 


1!M 9 


2356.5 


122.6 


125.1 


1204.0 


44 


III 


v\Y.1 i^ 


nc: 


r.5 


1 W».6 


23.-9.8 


122.8 


125.5 


1205.8 


46 


I'i 


^XV^ 'J 


r.j.o 


115 S 


:::o.4 


231V3.0 


123.2 


125.8 


1207.5 


48 


\» 


'.\\«.4 


r.^ s 


V.^.T 


r.>: 1 


23J6 3 


123.5 


126.2 


1909.2 


50 


^\» 


»» » * 1 - 


r.;- : 


r.f.o 


i'*.< < 


2369.6 


123.8 


128.8 


1211.0 


52 


M 


•:::^ o 


i:* 


V.V.I* 


V.r-.- -5 


23^2 8 


124.2 


128.9 


1212.7 


54 


>* 


* • 1^ J* 


IM 5 




:.:: s 


eST-ii.l 


124.5 


127.8 


1214.5 


50 


N» 


CCS ^ 


1.4 : 


• • ■ * 


::■.* 


S379.S 


124.9 


127.8 


1216.2 


58 


tV 




•> 

V 

• • 


4' 

V 


: :« > 


?<^» 6 


125.8 


128.0 


1218.0 


M . 








t 


»• 




1 


1 


- 




M. 


E. 


T. 


f 





:i:> ; 


i-S.' i 


:i5 3 


189.1 


1270.3 







Cifcs^ ^ 




l> 4 


•.c..« - 


S;s< * 


.^2 


189.5 


1272.0 


2 


4 


■;^v ; 




v:s * 


^: 4 


ii?*-. .? 


:».5 


139.9 


1278.8 


4 


»» 


■:-^\* 4 






..■v-C .« 


i-i •: : 


:36 9 


140.3 


1275.5 


6 


^ 


CU^ 


•.Of ? 






i^ 4 


is: 2 


140.8 


1«7.8 


8 


^* 


;^Sci # 


•.;^ s: 


•.A < 




i-J«: < 


:r.4 


141.0 


1279.0 


10 


\*: 


■>*.>: c 


:r < 


:?» £ 


:^"^ 4 


iiX^ i 


13S J 


141.4 


1280.8 


IS 


\\ 


.^> 4 




•?», ? 


'C^' i 


•^S'V • 


:9?s 


141 9 


1293.5 


14 


* \ 


\>*> * 


'CS .* 


vv J 


:c- i 


i!S>. 4 


:s?.: 


1«.2 


12S«.8 


16 


^^ 


X'** '. # 


^> * 


'.? 5 


'.C-v* f 


■^.V* 


:Sr.O 


142.5 


1298.1 


18 


'V 


04 1 > i' 


•cs * 


•c • 


rs* 4 


«:s ? 


:».4 


142.9 


1287.8 


20 


^''O 


0* > > 


',cy »* 


\«ii : 




ji; - . ^ 


:k s 


:<s.8 


1289.8 


23 


\^« 


\^4C'. * 


-.cv * 


*?5C 4 


•c^ J 


?'-■;"« 


:«:.i 


14S.T 


1291.8 


24 


1 '^ 


\''».>^ ,'' 


*« * 


'SC < 


'*J. ? 




liTS 


141.1 


1298.1 


26 


' 1» 


1>«> C 


'!:V : 


Ji* *. 


*^^; « 


MM B 


:«o $ 


144.5 


12M.S 


88 


^'^ 


c^v y 


^^ « 


'X< «. 




SiS- : 


•-: s 


1«4 9 


1298.8 


80 


*^ 


<^*» > 


V s 


■S>- f 


•;.? > 


£SK « 


:*: < 


1«5 3 


1288.8 


St 


J 


♦rtN^ 




^* f 


*^" * 


as f 


:«£ P 


145.8 


laoo.i : 


81 




j S*'^ * 


^^ ^ 


■** * 




r*.^ t 


:«£« 


1«.« 


l8in.S 2 


86 


1 


1 tliM^ 


^3i* * 


V : 




iss : 


:<s r 


148.4 


1808.8 t 


8B 




1 « 


H** J 


".'i* « 


-£.r-^ < 


es.* 4 


:^F. 1 


14FS 


1808.8 i 


40 




1 


« 


^S5 - 


•3S i 


£S> f 


:■*: 5 


ir* 


180C.1 , 


41 




1 


^ 


^J»^ '. 


:»• ? 


£"",* f 


:«f. « 


MT.8 


1808.8 


44 




I 


£ 


^* 5 


:£^ . 


«... _ 


5*«.« 


3tCv.« 


1818.8 


46 




1 


i 


^:«» ? 


-;'jr * 


fS 4 


14* 5 


?4f 4 


1812.4 


4S 






r 


'^V i 




r^'' t 


14, » 


2«S 


1834.1 


SO 




J 


k 


>.V f 


•i^V .•■ 


tv- * 


:« s 


M(.S 


Xfe5.8 


» 




1 




^l^t 




i-^^ 1 


-^ T 


1«3 


m7.8 


54 








»^* 


-->,. ^ 


i'.~ * 


:j<- r 


3«i 


3818.4 


98 








'J^* 


*?»S i 


1" - 1 


:-4i.4 


1»J 


38B.1 


as 








>«r. • 


■$^- i 


iS" t 


?«f 


3n.r 


I8BL8 


•.. 





FUNCTIONS OF A ONE-DEGEEE CURVE 


217 








' 


L.C. H 


E 


T- 


t.C. M 


E. T 




1 

10 

1» 

IS 

Si 
U 

90 

i 

40 

ii 
1 
ii 

00 


Sill 

£587.0 117 
£990.0 148 
mt.l 148 
MB7.4 140 
HOOB 140 

ffi!,! IS 

0610,4 IM 
8016;o 5J 

saio.i (11 

ii 

SOSO.B 54 
8640.8 M 
0)40.1 U 

ii 

asTiii B7 
W75.S se 


s| 

1 IK 

S IIW 

9 M 

4 M 

7 M 

J M 

1 5! 

1 5( 

Ii 
ii 


ii'i 

1 I3sa:4 

7 1S40.4 

ij4s!a 

B IM6.T 

nil 

7 13S8.0 

) isoa'a 

J I8^o:4 


M75.8 168 
iiOTS.B 151 

i:1 i 

3091 5 Ct 
0707.7 B 

!«'i7:4 b; 

Si s 

2740-O M 
2753:0 67 

itsoIb oe 

1702.7 09 


U IS 

161 

a iflj 

) 161 

1 171 

ii 


7 1371 

;i 

5 13Bi 
1391 

S I3« 

fi 140! 

8 i4oe 
a 1410 

i 141! 

9 i4ie 

UiH 

H5C 






1 

is 

la 
» 

as 
as 

i 

38 
« 

i 

58 




SB- 


89" 




L. 0. M 


E 


T. 


L C W 


£ 




1 

ii 

18 
X 

1 

i 

I 


JTT5.0 m 

B778.0 ITI 

Si Bl 

ETee.o 172 
sni.8 iro 

1801. S IT! 

KOeiO 174 
HSILB 171 

Si 

»iT,4 177 
SSSO.S 177 
0B33.8 m 
SW7.1 17S 
8840 8 176 

isis 
II ii 

ISMii 18« 
1809 4 1B4 


I in 

4 171 

1 18< 

T IK 

4 184 
( 18! 

I la: 

J 18! 


I'M 

14488 
1457:0 

Meals 

fi 1402 3 
I4M.1 

i«7:t 

I4C0.5 

itn.B 

1478. fl 


SI ii 

a8»!:3 181 

83887 is; 
S8K.0 1)» 

!904:a im 

Siil 
ili ii 

»93T,4 IBO 
8930.6 IM 
aOBS.O 190 

511 iS 


! 1 

i » 

3 K 
7 9; 

s 1 

1 H 

! S 

i IK 

1 BOC 

11 


1483 
8 I46C 

a 140'. 

i i4n 

B 1407 

1401 

sjl 

1 1011 

ii 

3 152t 
1 IMS 




3 

U 
4 

8 
10 

ES 

1 

4a 

il 

i 



218 


FUNCTIONS OP A ONE-DEGREE CURVE 






^ 10° 


Sl- 




L.t 


M. 


E. 


T. 


ue. 


H. 


E. T. 





■■me 


1 ISB.a 


a«-J 


M7I 


aoM.f 


asit 


m 


3 1588.0 













ssalo 


80eg!i 






8 lS98:s 














sora.a 


so»:i 




7 JSW.4 




s 


H^TB 


) 197!o 






3076. < 




■M 




8 


10 




i M7 4 


^!b 












10 






1 197.E 




NB.O 


3081 It 




















3U85.( 


sii.a 


m 


fl lOOl.B 




IB 


B9BI 


I m'.e 








aia.o 


SBO 


I 1003.4 


10 


18 




1 IM.l 


8003 












18 




SOK 








30M.7 


iii.a 


S3I 




80 












3097. ( 




Sil 


B I0O8.S 






30M 


















Sfl 


3007 


) aw.t 


soe:a 


SSSiB 






n 




86 


■JB 






208.7 


5»0.4 


3107.; 
























wi 


S 1618.0 






















ta 


3t 


soa) 




sio:i 


saalT 


aiiT:i 










se 








BB7.B 


3IS0.! 


21B.S 




















aiB.9 


gas 


1083.8 




40 


soao 


4 a».s 














40 


41 


soss 


( SO).! 




STaio 


3iao:» 






B jm.a 


48 












aiaa.i 




ea7 


B88.B 




4a 

4B 


BOM 


1 wa'- 


l\fl 


5-p*8 




sie.i 


887 


S 050.5 


46 

46 


CO 


8046 


s iioe.i 


•m.i 


B80:i 


3i4-j:i 








50 






r 806. s 






8146.0 


SM.l 




8 635.9 














3H9.: 


a».! 


aa 


4 183J.7 






















Efl 


BS 


»59 


1 anis 




6er:8 


3ISN!f 








58 


BO 










BI.'.S.W 






B 1643.1 


00 




se° 


iS' 


-_!_ 


L. C 


M. 


E. 


v. 


L. C 


u. 


E 


T. 














HIIB.O 


HO 


1 8973 


i 












330.4 


t)4G 


G OH.l 






31BS 




gaiig 


Mfl.7 


sail. 8 








4 


s 


3108 






848.5 


KM.S 








6 


8 










3S6Ji 


B379 


S4B 


8 704.5 


8 




31 14 










>38.4 


Stf 


7 70B.4 






3178 


1 awe 


:i3»;B 




3274 [ 








iS 


u 


3181 








3377,8 








11 


It 










3^80 1 


!3B,6 




8 711.8 


16 














340.8 


250 


8 ria.1 


ia 




3180 


! SMB 




m 1 


assfl.f 








90 








i3e:4 




33901 




Wl 




88 














S4I5 


85* 


3 iriB.i 


94 






i Bi8!(] 




S66S 






851 


S 780.9 


96 






t W8.t 


S37B 




»m't 






4 ■98.7 


98 


» 




) sas 


a38.4 


ow.i 


33087 








30 


8S 










3305B 


«48.( 


aw 


4 -88.4 


3t 












3309 1 






1-«.B 






3^16 


1 S30.S 




D?»B 












IB 


tiW 


7 8307 


m.s 


B77.4 


3315! 


Mtt!] 






88 












3118.7 


1M&,( 


85) 


5 -33.8 


40 






1 !3I 7 














48 




30M 




m.o 


Ii8£8 






















3m,! 


247.0 




1 -39.1 


48 














S4TJ 


aw 


6 40.9 


48 
























iifi 


1 B34!o 




6B0O 


3337'.! 








5V 




8i4S 








3*11. ( 




seo 








S4IB 






BBS 7 




SWi 


800 


8 «.» 


58 
















Ml 


8 7B0.0 
















^.< 




S 51 .a 


60 





FUNCTIONS 


OF A ONE-DEGHEE CURVE 


219 




34- 






L.O. M. E. T. 


L.C. M. E, T, 


■i 

i 

19 

90 

22 

38 

1 

40 
41 

<I 
4B 

i 

H 

60 


3961 
3800 
336; 

1 
1 
1 

34* 

1 

MI7 

1 

34« 


a 250 

S 260 

!S1 

3 2B 

11 

3 35; 

1 361 

ii 


3 261 
7 26! 

3 26! 

?l 

7 38; 
7 361 

ii 
ii 
ii 


9 733;B 

i ?^:! 

8 761.0 

IE 

i Ml 

sill 

178B:5 
6 1788.4 

i 1793:9 
( IB0i:8 


84 3.1 305,2 27B,1 906.7 
31 9.3 365,7 378.6 8DB.5 

3452.5 366.8 379.8 810.3 

Si Si g;i tSi 

34 3,0 287.7 580.8 816.8 
8465.3 368.3 381.4 B17.7 
3468.3 268.7 2S1.B 819,5 
34 1.5 369.3 38i.B B3i:3 
3474.7 260.7 283.0 823.3 

348i:o 27017 284;3 lB3<>:a 
3481.3 371.3 284.7 l&S.? 

3493:7 378;? 386:4 im'l 
3496,9 373,2 387.0 1830.0 

3SO0:s 574:? m'} 1811.6 

3509.6 375.3 589.8 843.4 

mIJ:? 276.7 590:9 848:1 

35,31:9 378:? 3^:3 m'i 
33.38:2 370.8 391.3 8,39.9 


3D 

u 

! 

43 

43 

60 

SO 




SO- 


37° 




LO. M 


E. T. 


1..C. M. E. T. 


K 


8541 

1 

K7! 
337! 

1 

339( 
SBOl 
BOM 

3007 

HI 

i 

am 


4 il8C 
7 381 

si 

4 3« 

;l 

3SS 

2 3H 

4 38( 

7 w: 

3 381 
3 380 

5 8B0 
7 390 
a 291 

291 

II 

3 394 

1 895 
a 390 


4 391 

g 2Be 

6 m 

a wt 

391 

7 301 

8 3* 

9 30( 

11 

1 311 


9 1801.8 

I M 

a i8;°:4 

sill 

T 1908' 1 
S 1908.0 
9 1909. B 

1 19135 


iiiiiiiiii iiiiiiiiii iiiiiiiiigi 
iiiiiiiiii iiiiiiSiii iiiiiiiiiii 

iiiiiiiiii iiiiiiiiii iiiiiiiiiii 


I 

20 
33 

i 

38 
40 

1 

S2 
64 



220 FUNCTIONS OF A ONE-DEGREE CURVE 



3T3i.a syj.a S30.i 

3TU.1 3)a T 380. B 
a;3T.3 SIS. 3 391.4 



3TSI 31fl 334. G 
STSSl SIB. 6 389.1 
3TM a SlT.l 335 T 



s>8i.4 SS1.0 sio.o aws.a 

9764 5 K1.S SW.fl aoM.s 
SI8T.T 3I».I KI.3 SOM.S 



3SW.3 BM.3 »i;i T 
SSUa.t 334.0 344,3 



siiiii.s 32fiK 34t; i »>ii.s 

asioo 3«To sifl.8 swas.G 

3si».3 (liT.e 31? 4 xxa 4 

sttea.s ses.i aia i aoav.s 



349. a aais d 



tia.i «Ma 4 
« SKA Mxa 3 
! 358.7 »M4.1 



38:^6 S3? T SS8.B WAS. Id 

Bsrs.B ass 3 359 4 soei.i 
atiai.B S3SH 360.1 a)S3.a 



B8BI a 340.fi f 

ssei 4 34! 1 : 

SSOr 6 341.? } 

3903. S 3ti e "l 

39070 3434 ! 

3010. 1 34S.e e 

3SI1 a 344.9 E 

30l«.4 345 I E 



3919.5 Sl$.« 31!?.^ 

iva.e 3(3.1 3e!<.4 

soas.B 340 ? aoo ( 

.isae 34?.:: sua; 



UB.4 STl 9O0S c 



350 T 3?3.B 910£ U 



301.7 383.1 

3oa a 3Se a 

SW.9 867.4 



.5 303.4 838. 



.a sns.fl s»4. 



.8 3?5.j 401. 



ITJNCTIONS OF A ONE-DEGREE CURVH 221 



ll.B B8t 4 4J3. 



.8 SVll-O 

,s Kia.e 
,S siH.a 



14.2 B8T.B 410. U SlSi.B 
ITS 388.4 4ie.fl tM.4 



M.a an). 2 4is.t 



) 3IW.B 421 .» S33T.9 



) 433,5 1«7CI.B 

I 434.3 ssn.b 

i lU.a 1074. s 

i <SS.fl 8278.4 

I 43D.3 VS78.3 

r 4378 Z^'.i 

i 438. S 9fflB4.1 

) 439.9 awa.o 

i 439.9 xiea.o 

I 440. e atiBD.& 

I 441,4 3301.8 

1 442.1 £303.8 

1 443.8 SS9S.7 



n.Q 398 427, 



4293. 


41t.a 451 






430S, 




4J17. 


i-!i:4 4a; 


































*»l 










430. S 40. 


^ 


a:S J!. 



























.3 ssa.a 

a sMT.s 

.T £340 4 

.4 8251.3 

.1 «a8.3 

io n^7^ 



! 83^.0 
! a824!B 
I 9S-ja,7 



17.1 £307.3 
17.8 B309.3 
18,8 £311. £ 



t 478.1 £381 .B 



4410,1 4 
4410,3 4 



SO a 430 .9 £418, S 



4459,3 IRl 6 4M,3 



I £387,0 

i £BB9,fi 

I 23^1 5 

I 8873.4 



44«8.b 4S3 1 40!l.^ 



222 


FUNCTIONS OF A ONE-DISGREE CURVE 




/ 


46° 1 


470 





L. C. 


M. 


E. 


T. 
2432.2 


L. C. 


M. 


E. 


T. 





4477.8 


455.5 


494.8 


4569.7 


475.2 


518.3 


2491.5 


2 


4480.9 


456.1 


495.6 


2434.2 


4572.7 


475.9 


519.0 


2493.4 


2 


4 


4483.9 


456.8 


496.5 


2436.1 


4575.8 


476.5 


519.8 


2495.4 


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 


498.7 


2442.1 


4584.9 


478.5 


522.2 


2501.4 


10 


18 


4496.2 


459.4 


499.5 


2444.0 


4588.0 


479.2 


.528.0 


2508.4 


18 


14 


4499.2 


460.0 


500.3 


2446.0 


4591.0 


479.8 


528.8 


2505.4 


14 


16 


4502.3 


460.7 


501.0 


2448.0 


4594.1 


480.5 


524.6 


2507.8 


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 


2511.3 


80 


22 


4511.5 


462.7 


503.4 


2453.9 


4603.2 


482.3 


527.0 


2513.3 


22 


24 


4514.6 


463.3 


504.1 


2455.9 


4606.3 


483.0 


527.8 


2515.3 


24 


26 


4517.6 


464.0 


504.9 


2457.8 


4609.3 


483.7 


528.6 


2517.3 


26 


28 


4520.7 


464.6 


505.7 


2459.8 


4612.4 


484.3 


529.4 


2519.3 


28 


30 


4523.7 


465.8 


506.5 


2461.8 


4615.4 


485.0 


530.2 


2521.2 


80 


32 


4528.3 


466.0 


507.3 


2468.8 


4618.5 


485.7 


531.0 


2528.2 


32 


34 


4529.9 


466.6 


508.0 


2465.7 


4621.5 


486.3 


531.8 


2525.2 


84 


36 


4532.9 


467.3 


508.8 


2467.7 


4624.6 


487.0 


532.6 


8527.2 


36 


88 


4536.0 


467.9 


509.6 


2469.7 


4627.6 


487.7 


533.4 


2589.2 


38 


40 


4539.1 


468.6 


510.4 


2471.7 


4630.7 


488.4 


534.2 


2531.2 


40 


42 


4542.1 


469.3 


511.1 


2473.6 


4683.7 


489.1 


535.0 


2538.2 


42 


44 


4545.2 


469.9 


511.9 


2476.6 


4636.8 


489.8 


535.8 


2536.2 


44 


46 


4548.2 


470.6 


512.7 


2477.6 


4639.8 


490.5 


536.6 


2537.8 


46 


48 


4551.3 


471.2 


513.5 


2479.6 


4642.9 


491.2 


537.4 


8539.2 


48 


50 


4554.4 


471.9 


514.3 


2481.6 


4645.9 


491.9 


5as.2 


2541.2 


50 


52 


4557.4 


472.6 


515.1 


2483.5 


4649.0 


492.6 


539.0 


2548.1 


58 ; 


54 


4560.5 


473.2 


515.9 


2485.5 


4652.0 


493.3 


539.8 


2545.1 


54 


56 


4563.6 


473.9 


516.7 


2487.5 


4655.1 


494.0 


540.6 


2547.1 


56 


58 


4566.6 


474.6 


517.5 


2489.5 


4658.1 


494.7 


541.4 


8549.1 


58 


60 


4569.7 


476.2 


518.3 


2491.5 


4661.2 


495.4 


542.3 


8551.1 


60 


/ 


48° 1 


49° 1 


f 


L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





4661.2 


495.4 


542.8 


2551.1 


4752.3 


515.9 


567.0 


2611.3 





2 


4664.2 


496.0 


543.1 


2553.1 


4755.3 


516.5 


567.8 


2613.3 


2 


4 


4667.3 


496.7 


543.9 


2555.1 


4758.4 


517.2 


568.7 


2615.3 


4 


6 


4670.3 


497.4 


544.7 


2557.1 


4761.4 


517.9 


569.5 


2617.3 


6 


8 


4678.3 


498.1 


545.5 


2559.1 


4764.4 


518.6 


570.3 


8619.3 


8 


10 


4676.4 


498.8 


546.4 


2561.1 


4767.4 


519.3 


571.2 


2681.4 


10 


12 


4679.4 


499.4 


547.2 


2563.1 


4770.5 


520.0 


57T8.0 


26S3.4 


12 


14 


4682.5 


500.1 


548.0 


2565.1 


4773.5 


520.7 


572.8 


2685.4 


14 


16 


4685.5 


500.8 


548.8 


2567.1 


4776.5 


521.4 


573.7 


2687.4 


16 


18 


4688.5 


501.5 


549.6 


2569.1 


4779.6 


522.1 


574.5 


2629.4 


18 


20 


4691.6 


502.2 


550.6 


2571.1 


4782.6 


522.8 


575.3 


2631.4 


80 


22 


4694.6 


502.8 


551.3 


2573.1 


4785.6 


523.5 


676.2 


2833.5 


88 


24 


4697.6 


503.5 


552.1 


2575.1 


4788.7 


524.2 


577.0 


2685.5 


84 


26 


4700.7 


504.2 


552.9 


2577.1 


4791.7 


524.9 


677.9 


8637.5 


86 


28 


4703.7 


504.9 


553.7 


2579.1 


4794.7 


525.6 


578.7 


8689.5 


88 


80 


4706.7 


505.6 


554.6 


2.581.1 


4797.7 


526.3 


579.6 


8641.5 


30 


32 


4709.8 


506.2 


555.4 


2583.1 


4800.8 


527.0 


580.4 


8643.5 


88 


34 


4712.8 


506.9 


556.2 


2585.1 


4803.8 


527.7 


581.3 


8645.6 


34 


36 


4715.9 


507.6 


557.0 


2587.2 


4806.8 


528.4 


582.1 


8647.6 


86 


88 


4718.9 


508.3 


557.8 


2589.2 


4809.9 


529.1 


588.0 


8649.6 


88 


40 


4721.9 


509.0 


558.7 


2591.2 


4812.9 


529.8 


583.8 


8661.6 


40 


42 


47-25.0 


509.6 


559.5 


2593.2 


4815.9 


530.5 


584.7 


8658.7 


48 


44 


4728.0 


510.3 


560.3 


2595.2 


4819.0 


531.2 


586.5 


8655.7 


44 


46 


4731.0 


511.0 


561.2 


2597.2 


4822.0 


531.9 


586.4 


866T.7 


48 


48 


4734.1 


511.7 


562.0 


2599.2 


4825.0 


532.6 


587.8 


8669.7 


48 


50 


4737.1 


512.4 


562.8 


2601.2 


4828.0 


533.3 


588.1 


8861.8 


60 


52 


4740. -2 


513.1 


563.7 


2603.2 


4831.1 


534.0 


588.9 


8088.8 


68 


54 


4743.2 


513.8 


564.5 


2605.2 


4834.1 


534.7 


ooiT.o 


8686.8 


54 


56 


4746.2 


514.5 


565.3 


2607.2 


4837.1 


535.4 


590.6 


8667.8 


66 


58 


4749.3 


515.2 


566.2 


2609.3 


4840.2 


536.1 


591.5 


8600.9 


88 


60 


4752.3 


515.9 


567.0 


2611.3 


4843.2 


536.8 


592.4 


8671.0 


•0 



FUNCTIONS OF A ONE-DEGREE CURVE 223 



/ 


60' 


61° 


/ 


L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





4843.2 


536.8 


592.4 


2671.9 


4933 6 


558.2 


618.5 


2733.0 





2 


4846.2 


537.5 


593.2 


2673 9 


4936.6 


558.9 


619.3 


2735.1 


2 


4 


4849.2 


538.2 


594.1 


2676.0 


4939.6 


559.7 


620.2 


2737.1 


4 


6 


4852.2 


538.9 


594.9 


2678.0 


4942.6 


560.4 


621.1 


2739.2 


6 


8 


4856.2 


539.6 


596.8 


2680.0 


4945.6 


561.1 


622.0 


2741.2 


8 


10 


4858.3 


540.3 


596.7 


2682.1 


4948.6 


561.8 


622.9 


2743.3 


10 


Vi 


4861.3 


541.0 


597.5 


2684.1 


4951.6 


562.5 


623.7 


2745.3 


12 


H 


4864.3 


541.7 


598.4 


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 


4870.3 


543.1 


600.1 


2690.2 


4960.6 


564.7 


626.4 


2751 .5 


18 


SO 


4873.3 


543.9 


601.0 


2692.3 


4963.6 


565.4 


627.3 


2753.5 


20 


22 


4876.3 


544.6 


601.9 


2694.3 


4966.6 


666.2 


628.2 


2755.6 


22 


24 


4870.4 


545.3 


602.7 


2696.3 


4969.6 


566.9 


629.9 


2757.7 


24 


26 


4882.4 


546.0 


603.6 


2698.4 


4972.6 


567.6 


630.0 


2759.7 


26 


28 


4885.4 


546.7 


604.5 


2700.4 


4975.6 


568.3 


630.9 


2761.8 


28 


ao 


4888.4 


547.4 


605.3 


2702.4 


4978.6 


569.1 


631.8 


2763.8 


30 


9i 


4891.4 


548.1 


606.2 


2704.5 


4981.6 


569.8 


632.7 


2765.9 


32 


34 


4894.4 


548.8 


607.0 


2706.5 


4984.6 


570.5 


633.6 


2767.9 


34 


86 


4897.4 


549.5 


607.9 


2708.6 


4987.7 


571.2 


634.5 


2770.0 


36 


38 


4900.4 


550.a 


608.8 


2710.6 


4990.7 


572.0 


635.3 


2772.0 


38 


40 


4903.5 


551.0 


609.7 


2712.6 


4993.7 


572.7 


636.2 


2774.1 


40 


42 


4906.5 


551.7 


610.5 


2714.7 


4996.7 


573.4 


637.1 


2776.2 


42 


44 


4909.5 


56;2.4 


611.4 


2716.7 


4999.7 


574.1 


638.0 


2778.2 


44 


46 


4912.5 


553.1 


612.3 


2718.8 


5002.7 


574.9 


638.9 


2780.3 


46 


48 


4915.5 


553.8 


613.2 


2720.8 


5005.7 


575.6 


639.8 


2782.3 


48 


50 


4918.5 


564.5 


614.1 


2722.8 


5008.7 


576.3 


640.7 


2784.4 


50 


52 


4921.5 


555.2 


614.9 


2724.9 


5011.7 


577.0 


641.6 


2786.4 


52 


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 


5017.7 


578.5 


643.4 


2790.6 


56 


58 


4930.6 


557.4 


617.6 


2731 .0 


5020.7 


579.2 


644.3 


2792.6 


58 


,^ 


4933.6 


558.2 


618.5 


2733.0 


5023.7 


579.9 


645.2 


2794.7 


60 


/ 


62<» 1 


68» 1 


/ 


L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 





5023.7 


579.9 


645.2 


2794.7 


5113.5 


602.0 


672.7 


2a56.9 





2 


5026.7 


580.6 


646.1 


2796.8 


5116.5 


602.8 


673.7 


2858.9 


2 


4 


6029.7 


581.8 


647.0 


2798.8 


5119.4 


603.5 


674.6 


2861.0 


4 


6 


5032.7 


582.1 


647.9 


2800.9 


5122.4 


604.3 


675.5 


2863.1 


6 


8 


6035.7 


582.8 


648.9 


2803.0 


5125.4 


605.0 


676.4 


2865.2 


8 


10 


5038.7 


583.5 


649.8 


2805.0 


5128.4 


605.8 


677.4 


2867.3 


10 


12 


5041.7 


684.3 


650.7 


2807.1 


5131.3 


606.5 


678.3 


2869.4 


12 


14 


5044.7 


585.0 


651.6 


2809.2 


5134.3 


607.3 


679.2 


2871 .5 


14 


16 


5047.7 


585.7 


652.5 


2811.2 


5137.3 


608.0 


680.2 


2873.5 


16 


18 


5060.7 


58G.5 


65.3.4 


2813.3 


5140.3 


608.8 


681.1 


2875.6 


18 


20 


5053.6 


587.2 


654.3 


2815.4 


5143.2 


609.5 


682.0 


2877.7 


20 


22 


5056.6 


587.9 


655.2 


2817.4 


5146.2 


610.3 


683.0 


2879.8 


22 


24 


5059.6 


.^88.7 


656-. 2 


2819.5 


5149.2 


611.0 


683.9 


2881.9 


24 


26 


5062.6 


589.4 


657.1 


28-.M.6 


5152.1 


611.8 


684.9 


2884.0 


26 


28 


5065.6 


590.1 


658.0 


2823.6 


5155.1 


612.5 


685.8 


2886.1 


28 


90 


5068.6 


590.9 


658.9 


2825.7 


5158.1 


613.3 


686.7 


2888.1 


30 


^ 


5071.6 


591.6 


659.8 


2827.8 


5161.1 


614.0 


687.7 


2890.2 


32 


34 


5074.6 


592.3 


600.7 


2829.8 


.5164.0 


614.8 


688.6 


2892.3 


34 


36 


5077.6 


593.1 


661.6 


2a31.9 


5167.0 


615.5 


689.6 


2894.4 


36 


38 


5080.6 


593.8 


662.5 


2834.0 


5170.0 


616.3 


690.5 


2896.5 


38 


40 


5083.6 


594.5 


663.5 


2836.1 


5173.0 


617.0 


691.5 


2898.6 


40 


42 


5086.6 


595.8 


664.4 


2838.2 


.')175.9 


617.8 


692.4 


2900.7 


42 


44 


5069.6 


596.0 


665.3 


2840.2 


5178.9 


618. 5 


693.4 


2902.8 


4<i 


46 


5092.6 


596.7 


666.2 


2842.3 


51S1.9 


619.3 


694.3 


2904.9 


46 


48 


5095.6 


597.5 


667.2 


2844.4 


5184.'J 


620.1 


095.3 


2907.0 


48 


50 


5096.6 


598.2 


668.1 


2846..') 


51h7.8 


620.8 


696.2 


2909.1 


50 


52 


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 


5193.8 


62-J.:J 


098.1 


2013.3 


54 


56 


5107.6 


600.4 


670.9 


2852.7 


5196.7 


()23.() 


699. 


2915.4 


56 


68 


5110.6 


601.2 


671.8 


28.')4.8 


5199.7 


G-J3.8 


700.0 


2917.5 


58 


60 


5118.5 


602.0 


672.7 


m-)6.9 


.')-.>0J.7 


624.6 


700.9 


2919.5 


60 



224 FlINCTIONa OF A ONE-UEGREE CURVE 



' 


64- 






L.<: 


M. 


-?— 


T. 


I-C. 


Jl. 


E, 


T 














5 5201,7 




1 jB.e 


aaa 


























i-nx 


8 sae:! 


■m'.i 


xam 


a B!»7;( 








1 4 






5 BIB.fl 


Tos.e 


90«5 


B BSOO.e 


6)9.e 


73«.S 


MSB 


a 8 










Wif 






783.8 


SOS! 


3 8 
















"^.8 


1DB3 






Bsa: 


s e^ia 






2 5309i4 




-35,8 


90U5 








s Ba9.a 


jojIt 


5»a 


3 BSlS.a 




88.8 


»a97 
















flsals 


737.8 


aoo! 


i 18 


IB 




















SO 




4 BSa.! 


7)0-! 


!»40 


6 Mil -a 




73B.7 














aB«3 




BVila 


740,7 


suoo 




M 


OSSE 


3 Bssir 




SMI 






741,7 


soot 


a w 


SB 




3 834. G 
















SB 




I B3B.! 


714:4 


SB4E 


rusa'.o 


bmI'j 


743:? 


301:: 












efi5i 


1 533B.S 


688,0 


741.7 


IW14 








j eas!8 




au3 




05B.T 




3018 


B as 


H 


B2M 


1 euT.B 


iris 


zasB 








SD18 




98 




1 esa-t 


i8.e 


l»57 


11 684417 






BO-JI 


1 38 






1 em.Q 


10.3 


eBBB 


6 5347,7 


80S,0 


743,7 


aoffl 












SKI 




6«a 8 


740,7 


BOK 


a 40 


a 


i-xi 


Mo'.a 


731 !l 


l.-W) 




683 B 




802! 


5 4i 




i-Xi 


) flIl.S 


TSi.l 


1»UJ 


J B,5Kl'6 






















685:1 


753.8 


3031 


















753.8 




i 48 


eo 






7J5!o 






888,7 




303e 






firs 




r-'fl.o 




4 OaflB.a 






3038 










Tii.O 






68B:3 


750.8 


3040 
















BfiB.l 


757.8 


3043 


4 58 


















8044 


S 58 


00 


sai 


7 M?:* 












304^ 






6fl" 


67° 




I^I 


- ill. 


E. 


T. 


L, C, 


M. 


E. 


T. 












301C 




804.4 


760,8 




I 




















3 a 














<i8B:a 








e 


688e 


J BTSiB 


■.Hi.? 


3iB! 


1 W77:o 


«BB,» 


793:3 


3115 










783 .T 




a M7ii,fl 


807,8 


7BI.S 


811S 


7 8 


















8181 


9 10 








fiffllT 




5 5485!7 














7 BTIlio 




30BI 


8 Si'V.? 






8 tee 




IB 


'am 


S 078.8 


wIt 


8003 


8 5J01.8 


n»:B 


708:4 


I1S8 












8009 








3180 


a 18 


















SIS 






Wi 










On^!! 










Mlt 


3 Gm'J) 








w.o 




313: 




xa 


S4IF 


i m.- 










WM.e 


8I3E 


9 «e 


xs 


















t as 


so 












06:4 








3i 














808:8 


3140 








] BS4.U 








io8:o 


607.8 


S14i 


9 S4 
















808.3 




SB 


as 


6135 


1 Ba5:fl 






4 am'.r 








9 38 


40 


H3f 


3 680.4 


779.8 


3081 


8 BS«.7 


7 0,4 


810.9 




4 40 








TSO.S 






T 1.8 


HIJ.0 


3IH 


8 49 


















81BE 




W 


M41 
















B 40 


4S 


M» 




78410 


sm 


S 5538:4 


7 '.a 


815:i 


aioi: 


1 48 




,MM 




785.0 


sm 






818.9 


31U 


3 80 




















4 Et 












B 554::i 


7 lo 






8 84 




MW 




788:i 


uioi 




7 .a 


810:3 


8171 


8 58 


58 




B 093.8 












317J 


as 


M 


_SW 
















1 to 





FUNCTIONS OF A 


ONE-DEGREE CURVE 226 


rr 


GS* 


60= 






L. C. M._ 


e, 


T 


I.. C. 


M 


E 


T 








fl 718,4 


B-il 


4 SlTt 


1 5613.1 




8 B6! 


5 3S41 




1 




B 719. a 


Bm 














4 


USl 


r Tso.fl 












7 3341 







BSM 




m 


i 3182 


7 5B51.'8 


■45 


8 est 


8 3»8 




e 


6M7 


I ™:8 




r SIBJ 


9 66S4.T 


4f 


I 851 


Six 






«170 


' s?** 


^ 








7 H8( 


asx 


S 1 










1 3191 






8 861 


1 3*57 


S 1 




0S!» 








6 sm'.s 


74f 


1 86S 


a asKsE 






sssa 


1 745:6 






8 5669, a 








7 1 


sa 






esi 


1 8198 


5673.1 


751 




4 BSS! 






Ka? 




m 


1 aaoD 


8 tan.o 


761 




6 898C 






KW 


I 789.1 






i 6677.0 






1 am 


S 34 












G 6880,8 








i » 


a 






836 


3 3206 


T B8837 


764 




I seTs 




30 


6STO 


6 30B 




4 me 


t B8B6S 


755 




B 3£74 




Si 




I 781 .S 




i 3SI1 








3877 


1 BS 


«4 






8SB 










1 an9 


4 34 


X 






840 




S BB9S:3 


757 


7 873 




s ss 


38 


5flll 


7S3.7 


841 


a 3«i7 


r 68981 


768 


6 B74 


1 &183 




















4 39SII 


) 40 










1 Siti 














s«ie 






a mi 


I 6toc:b 




877 


1 32»Q 






B6« 


B 73T:o 


846 


1 awe 






) 878 


r SMS 








' 7SI8 














) 48 




SS^ 














) 3S07 




W 








i 3333 


I 67IB:4 


T« 




Sana 






EIS34 


1 ;4o:* 


m 


3 8235 


( 873), a 






1 3301 












4 3237 




m 






i 68 










B 31189 


r 6787:1 


766 








60 








6 3L>41 


6730,0 


767 








"^ 


80° 


01° 






L.C 


. M. 


E 


T. 


L,C, 


M 


E 


T. 






B780 




BM 










3 3875 






fi^ai 






















1 7«b!4 




T S31S 


7 K^'.l 










e 


6738 


B 770.8 


BBS 


I 3314 


9 BSM.O 


70! 


4 ^f. 


fl 33S1 




8 


B7J1 




ewi 










} 3SB1 


a 8 










) 3319 


3 B63a!? 








4 10 






1 77S!7 








TBf 










67S0 


a 773.B 


8M 


i 3.K3 


J BBsaia 


708 


a m 






IS 


BIM 


1 7T4.4 


SOB 


1 33!H 


] 6839.3 


790 


7 9« 


S 330! 


a 16 


IB 






890 






BOO 






4 18 












I 684S.1 






a 3397 


7 ao 


iU 


6781 


77B:fl 


ms 


) 333S 


r 6847.0 


80i 


a 9^ 


9 S30( 




M 


b;s4 


T77.8 








so; 






8 £4 


ffl 
















1 3404 


4 St 


^ 










4 S856!b 










80 


S77J 


78o!a 




' **'J 


8 6869.4 


BO.-. 


B Xi7 


4 34« 




M 


6778 


781.1 


m 




9 fisoa.a 


80il 


5 a* 


8 3411 


! 32 














80T 






5 34 


38 




















38 




TBs!? 




r 33SC 


8 5(t70:9 


8« 


B4S 






40 


BT87 


784.B 






i B87S,7 


B0( 




B 34W 




4S 


fl7» 




















Bna 


leelt 






i 687016 




e oie 






48 


57M 


7B7.0 


m 


1 3MB 


6 6SK.3 


eis 


4 Wf 


7 Star. 




48 


GT» 


T8T.9 


gia 




5 B«S5,S 






S 34i!0 


3 48 


W 


san 














3431 




u 








7 38ee 


I sseo^g 






a 3433 




M 


6807 


Tso!* 


oic 


8 3368 








3 348H 




M 


6818 


1 Tfli.a 


019 


) MTt 


J ^m^ 


8I( 


7 115> 


S MS 


4 sa 




















7 68 






1 79«!g 


930 












60 , 



226 FUNCTIONS OF A ONE-DEGREE CURVE 



# 


62« 1 


68- , 




L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. T. 





5902.4 


818.4 


954.8 


3442.9 


5987.8 


844.4 


990.8 8611.8 





2 


6905.2 


819.8 


956.0 


3445.2 


5990.6 


845.3 


991.5 8513.6 


2 


4 


5908.1 


820.1 


957.2 


3447.5 


5993.5 


846.2 


992.7 8516.9 


4 


6 


5910.9 


821.0 


958 8 


3449.7 


!S996.3 


847.1 


993.9 8518.2 


6 


8 


5913.8 


821.8 


959.5 


3452.0 


5999.1 


847.9 


995.1 8520.6 


8 


10 


5916.6 


822.7 


960.7 


34.54.3 


6002 


848.8 


996.3 8522.8 


10 


12 


5919.5 


823.6 


961.9 


3456.6 


6004.8 


849.7 


997.5 8525.1 


12 


14 


5922.3 


824.4 


963.0 


3458.8 


6007.7 


850.6 


998.7 3527.4 


14 


16 


5925.2 


825.3 


964.2 


3461.1 


6010.5 


851.4 


999.9 8529.7 


16 


18 


5928.0 


826.1 


965.4 


3463.4 


6013.3 


852.3 


1001.1 8532.0 


18 


20 


5930.9 


827.0 


966.6 


3465.7 


6016.2 


853.2 


1002.8 8634.8 


20 


22 


5933.7 


827.9 


967.8 


3167.9 


6019.0 


854.1 


1003.6 3536.6 


22 


24 


5936.6 


828.7 


968.9 


3470.2 


6021.8 


854.9 


1004.7 8538.9 


24 


26 


5939.4 


829.6 


970.1 


3472.5 


6024.7. 


855.8 


1005.9 8641.2 


26 


28 


5942.3 


830.4 


971.3 


3474.7 


6027.5 


856.7 


1007.1 8643.6 


28 


30 


5945.1 


831.3 


972.5 


3477.0 


6030.3 


857.6 


1008.4 8545.8 


30 


32 


5947.9 


832.2 


973.6 


3479.8 


6033.2 


858.4 


1009.6 8648.1 


32 


34 


5950.8 


8;i3.0 


974.8 


3481.6 


6036.0 


859.3 


1010.8 8650.4 


84 


36 


5953.6 


8a3.9 


976.0 


3483.9 


6038.9 


860.2 


1012.0 8552.7 


36 


:i8 


5956.5 


834.7 


977.2 


3486.2 


6041.7 


861.1 


1013.2 8655.0 


88 


40 


5959.3 


835.6 


978.4 


3488.5 


6044.5 


861.9 


1014.6 8657.3 


40 


42 


5962.2 


836.5 


979.6 


3490.7 


6047.4 


862.8 


1015.7 8559.6 


42 


44 


5965.0 


837.4 


980.8 


3493.0 


6050.2 


863.7 


1016.9 8562.0 


44 


46 


5967.9 


838.3 


982.0 


3495.3 


6053.0 


864.6 


1018.1 3564.3 


46 


48 


5970.7 


839.1 


983.2 


3497.6 


6055.9 


865.4 


1019.8 8566.6 


48 


50 


5973.6 


840.0 


984.4 


3499.9 


6058.7 


866.8 


1020.6 8668.9 


50 


52 


5976.4 


840.9 


985.5 


3502.2 


6061.6 


867.2 


1021.8 3571.2 


52 


54 


5979.3 


841.7 


986.7 


3504.5 


6064.4 


868.1 


1023.0 8578.5 


54 


56 


5982.1 


842.6 


987.9 


3506.8 


6067.2 


868.9 


1034.2 8675.8 


56 


58 


5985.0 


843.5 


989.1 


3509.0 


6070.1 


869.8 


1025.4 8578.1 


58 


60 


5987.8 


844.4 


990.3 


3511.3 

«_ 


6072.9 


870.7 


1026.7 8680.4 


60 




64o: 1 


66<» 




/ 


L. C. 


M. 


E. 


T. 


L. C. 


M. 


E. T. 


/ 





6072.9 


870.7 


1026.7 


8580.4 


6157.5 


897.8 


1064.0 8660.4 





2 


6075.7 


871.5 


1027.9 


3582.8 


6160.3 


898.2 


1066.2 8652.8 


8 


4 


6078.5 


872.4 


1029.2 


3585.1 


6163.1 


899.1 


1066.6 8655.1 


4 


6 


6081.4 


878.3 


1030.4 


3587.4 


6165.9 


900.0 


1067.7 8667.5 


6 


8 


6084.2 


874.2 


1031.7 


3589.7 


6168.7 


900.9 


1069.0 8659.8 


8 


10 


6087.0 


875.1 


1032.9 


3592.1 


6171 .5 


901.8 


1070.2' 8662.2 


10 


12 


6089.8 


8r5.9 


1034.1 


3594.4 


6174.8 


902.7 


1071.6 8664.6 


12 


14 


6092.6 


876.8 


1035.4 


8596.7 


6177.1 


903.6 


1072.7 8666.9 


14 


16 


6095.5 


877.7 


10.36.6 


3599.1 


6179.9 


904.5 


1074.0 8669.8 


16 


18 


6098.3 


878.6 


1037.9 


3601.4 


6182.7 


905.4 


1075.2 8671.6 


18 


20 


6101.1 


879.5 


1039.1 


3603.7 


6185.5 


906.8 


1076.6 8678.9 


80 


22 


6103.9 


880.3 


1040.3 


3606.0 


6188.8 


907.2 


1077.8 8676.S 


88 


24 


6106.7 


881.2 


1041.0 


3608.4 


6191.1 


908.1 


1079.1 8678.6 


94 


26 


6109.6 


882.1 


1042.8 


3610.7 


6193.9 


909.0 


1080.4 8680.9 


86 


28 


6112.4 


883.0 


1044.1 


3613.0 


6196.7 


909.9 


1061.7 8688.8 


88 


80 


6115.2 


883.9 


1045.3 


3615.3 


6199.5 


910.8 


1088.0 8686.6 


80 


32 


6118.0 


884.7 


1046.5 


3617.7 


6202.3 


911.7 


1084.2 8688.0 


88 


34 


6120.8 


885.6 


1047.8 


3620.0 


6'v»05.1 


912.6 


1086.5 8690.4 


84 


36 


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886.5 


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3622.3 


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913.5 


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86 


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6126.5 


887.4 


1050.3 


3624.7 


6210.8 


914.4 


1088.1 8606.1 


88 


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6129.8 


888.3 


1051.5 


3627.0 


6213.6 


915.3 


1089.4 8697.4 


40 


42 


6132.1 


889.2 


1052.7 


3629.4 


6216.4 


916.2 


1090.6 8699.8 


48 


44 


6134.9 


890.1 


1054.0 


3631.7 


6219.2 


917.1 


1091.9 3702.8 


44 


46 


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891.0 


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6222.0 


918.0 


1093.2 8704.5 


46 


48 


6140.6 


891.9 


1056.5 


3636.4 


6224.8 


918.9 


1094.6 8706.9 


48 


50 


6143.4 


892.8 


1057.7 


3638.7 


6227.6 


919.8 


1095.8 8709.8 


50 


52 


6146.2 


893.7 


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3641.1 


6230.4 


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1 eo 



KINCTIONS OF A ONE-DECUEE CURVE 227 



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FUNCTIONS OP A ONE-DEGREJ: QURVE 


22 








K. ' 




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117 


M. 


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4m. 


TIOO.O 


i^>5a.o 


160S.T 


4585.1 


30 


tt 




) 13SS.0 




45011. 


Tietl.S 


IJOT.O 


1610.4 


4087.9 


» 


M 














1613. 1 


4080.6 




CO 


TOM 


' !sB:i 


j'm.s 


401]! 


nm'.a 


i»u!i 




4084.8 




IS 


TOBt 


1 1«0.1 






T170.4 


imi.i 




459(1.0 




m 


TOM 


7 1«S0.« 






Ti:8.0 


Km a 


1617.8 


4M*'8 


30 


S-J 












]at!3 


1618.' 


4001.0 


ai 




not 


J laseia 


jm'.a 


4!ra. 




MK.Z 


1I13U. 


40013 




3« 


TIOS 


J im.3 




4545. 


TIHOM 








86 


M 








40«8. 


T1M.4 


1300.4 


IWI. 




ffl 






B 1236. a 




4530. T 


T18G.0 


i'jRa.4 


1035. 


461^.0 






TllO 


4 liiae. 


ioTe!4 


4033.4 


TiH<e 




1B3T. 


4ai5-8 




44 


TUB 


1 1*17. 




4030.1 


TlOl.S 


lawia 






44 


40 


TllS 


7 laas. 






Tioa.H 


l:!l».S 


1031 : 


4630!s 


40 






s itas 


168K0 


4511 !» 




liJTOfl 


1083. 


4Ba8..1 


48 


M 


TIM 


B 11140 


IMQ.S 


4.ML3 


Tiwio 






4036.3 


50 


13 


Tin 


i mi. 


IKHI.fl 




woia 


l^.i.r 


1686 






M 


TltB 


1 IMS. 


1080 A 


«Ib'7 


wois 


1378.7 


168: 


4U31 N 


5t 


BS 


Tin 




1588.3 






1371.8 


16:W. 


40i| 5 


-to 


[S 


TIJI 


1 i«4: 


IflOOO 


40m:i 


fjiw'4 










TIM 


1*10. 




j.y>7.8 








4i;hi!o 





230 PUNCTIONS OF A 0NE-DfX3REE CURVE 





78" 


7»' 


ao 

1 
i 

60 


L. C. M, E. T, 


L.C. SI, E. T. 


10 

IS 
90 
M 

1 
1 

i 

K 

56 

60 


7£li.O 1276.0 1043 

'^m.B isre.o OH 

ia7.8 1379.0 B4I 

7ii3a:i ia8a;4 losa 

TWr.B S87.4 1B60 
TM0.4 ^38.5 168: 

11 11 1 

SESla 393!7 D7] 
-aSB.O 1294. B B73 
■M8.5 liiBS.B 1874 
7iBl.l 1298. 9 1070 

72UB:-i 121)b!d 1U71 
7388.8 1300.0 1881 

iarels 1804:b losi 

7iffll.7 1305,3 16B0 
72fl4.8 1306.3 lOW 

72t«:s ISOsia IfiBf 


B 4B4i!B 

! S:! 

1 4851.1 
B 4BS8.B 

;s-: 

B 466e!o 
8 4667.7 

sii 

1 4Sm!4 

8 4889:9 
3 4B0i,7 

2 4095.5 
8 400B3 

; JS! 

3 4706.7 

8 471 ai a 

8 4715.0 
4788:4 


73a 

Si 

7291 

1 

Si 
1 

7331 

1 

-3JJ 

■TO 

■301 
-388 


G 1308 
I 308 
8 H 

a 1} 

8 IF 

i a!! 

8 SXI 

B ^ 

! S 

6 821 

! i 

! 1 

i 1 

8 839 
4 40 


8 1B98.0 47-^.4 
G 1687.7 47Se.« 
a 699.5 4729.0 
7 701.3 4781.1 

! ll 11 

1 718.8 4761.! 

! si iSi 

5 723.8 4765.1 

! is Ei 

7 7110. 4776.1 

i |s :|i 






81= 




L. C. M. E. T. 


L,C- M. B. T. 




8 

1 

« 
30 

3fl 

sa 

40 

a 
u 

80 


736U.4 1340.8 5C 

7371.3 1342!7 TS 

7374.0 1313.8 ■;X 
7878. « 1844.9 757 

7870.1 1346.0 TIE 

7388;? 134B.a ^ 
7888.8 18S0.3 "66 
TSBI.B I8.M 8 1 68 

7884.4 1%J.4 ITO 
^.B 1363. -72 
7808.B 1851. -78 
7402.0 18.18. -n 

11 if 1 

7417!8 tm. fH 
7418 8 1563 7S» 
742:1.8 1884. 790 
T424.B 1885. i-rvi 
7427.4 1306, 1701 
7130,0 1367, B 17a 

7435!l 13Be;0 17D1 


4BC8.0 

3 4810.9 

s S;: 

a 4822.8 

?»! 

a 4S33.7 

4 4888.5 

Si 
lis 

3 4K».S 

4^8.4 

4883.S 

4B69.1 

II 

4l<115'3 

S:i 


744f 

746C 
745; 

47i 

1 

491 
7483 
748! 

760! 

75i; 
75i; 


li 

ill 
I| 

J 1881 
1 I39t 

i 1884 
3 1881 

3 139( 

J 1401 

4 1401 


8 1805.5 4SB3.I 

1811.1 4902:i 
1818,0 480B.. 

ill i-i 

8 1824.2 48«.8 

! lS:i S:! 

8 1928.9 4831.1 

! II S:i 

3 1838:4 4048!i 

4 1841.3 4948.9 

5 1S48.2 4951 .8 

6 1845. 1 4964.7 

7 1847.0 48B1.8 

: ii i'l 


12 

20 
21 

at 

26 

1 

16 

SB 
40 

i 

i. 



INS OF A ONE-DRGREE CUllVE 231 



' 


... . ^° 








' 


L ft ii. 


E- 


T_ 


L, e. 


M. 


E, 


T. 


e 








JBrtl.O 


rSHS.B 


J43B5 






a 








40.S3.3 




1439 1 






8 




TSa 


4 1407!7 


issfl!i 


4iwn8 
















8 1408 8 


1868.1 


40BB,H 






v». 










4 ]4o».a 


187O.0 


4093 7 


7603, B 


1443.8 






B 


iO 










7606.0 




»30 




10 




T5ff 


4 1413.1 


1873 S 


48014! 6 






9SS, 


6087. 








9 1413.1 


im.9 


Mil .5 








SOBO. 




IB 


isai 




1B77.7 






1447 is 


rae! 




IS 


IB 


7M1 


1415:4 


187S.T 




7610.0 


1448,4 


1038. 




IB 


» 




S I41S.S 


1881 6 










60B0,3 


SO 


« 






1883 5 




76i;L.O 








8S 


M 


tMf 








76in.5 


1451. H 








18 


nsi 


Hldis 






7fi-:fi,U 


14.'. J i 


1W6. 


6108. 11 


86 


W 


raa 


i i4a).v 




MB^!] 










aa 


ao 






JSDI.B 


SOM-O 










30 


M 


mu 












10,13. 




31 


H 


7561 


It^lil 






7DS5.S 


I4,-.7.! 




5130.8 


31 


U 


7563 


5 11'».3 












S133.a 


SB 




THe 


14SG.4 


im'.o 


WX'.B 








513B,S 


38 


40 


TGAB 


5 1JK.5 


IDOI 


B033.8 


TW3.1 










M 




D HW.a 


1«». 




Tnw.o 


1481,1 






43 








IBOt. 








1064 


B13B. 


44 






1 143o!h 


I SOU. 


B0tM!7 












« 




B 1431.9 


1008. 


W51.7 


7B5a!4 


IWi.l 






48 


N 


TfiSI 








7655.B 


um.i 


197U 






U 








w57!b 




1407,3 


073. 


8 47 


68 




T96( 


1 435^2 




MMOO 




IIBS.i 




5 SO, 




U 


IB8f 


S 436.8 


1916; 




■;m.a 










M 




1 MT 4 




5000.5 


7665.8 


n:iili 


b;8. 




68 


BO 








MUll.4 






Bun. 




60 




84" 






_L. 


M, 


K. 


T. 


I., r. 


M, 


&_ 


1 




TWK 


S 471. 8 


1W».5 




7743.4 














B 17:1.0 




6iai:.i 




XB.i 








4 




1 474.0 






y'-i 


5U7,( 


3040. 






a 


TOTI 


7 475.1 












6«B8 


B 




?K( 


2 470.9 


lOBHic 




•rss'.'i 








B 


10 


TWO 




IMO.O 


BI7ji4 




5ii!i 








IS 


TB83 




10BSJ.7 


6177.0 


?"■' 


SlS.l 


axii. 


5aHI,0 




14 


TSBS 


6 llTBid 












^■', 


14 


le 


TBSe 


1 14W.7 






7TM.0 








10 




TWO 




iwsie 


BmO 


7TOJ4 




KOBi: 




la 


1 "* 


TWS 


11B1.0 


sooo.p 


r.l8»,D 


7760.3 


51B,i 


aws. 


sail 4 






TS<» 


A -1481 1 




BIW.fl 


7760.3 






6384.4 


ta 




TB97 


B 118.i. 










ssxe'. 


6«7.S 




M 




4 tIBfl.S 


WOflio 


BIW:7 


TTaia 


sjb!i 


9HM. 






% 




» 1IB7.4 


aooe.D 


1W0I.7 


7778,7 




8071 






! 30 


-70S 


3 488 






777B.I 




8073. 


(MBO-7 




1 SE 


-TOT 








77HI.J! 




a>T5 




3ii 


H 


-71 C 




soisio 


kiq'h 


7784.0 








31 


M 




g ISKB 






7790.4 


D-».a 








38 


ITI.' 


« 4B3-I 


wio!] 


wio!b 


77H8,U 


rm.i 


awi. 






40 






soBi.a 


B?M.O 






90P3, 


B-lli-a 




41 


TTa 






BaSB.i 


7703,8 


iiOti 




5!l1fi,4 


la 




w*s 






IKi«.] 




laaa.! 


»«7. 


.1318,6 








1 NO?: 


sojr!4 


ssjo.i 






BUM 


ssai.fl 


lU 






B HOS. 


aK9.4 


MH.i 


7HII1.1 


15»I.I 




M!!l,7 


18 


W 








BilM.3 




I!t.t9.7 




«97,B 


*2 




rw 


5 1900. 


amid 


BasH.a 


ww-n 


1M1 H 


:»SR. 


S3SUB 


^ 




77* 


lira. 


3%.^. 6 






IMS,! 




6384-0 


51 


' SB 




S 1503. 


OTa77 










n.'W7.i 




SB 


77M 




8030. s 


6il7.R 




jria'.! 


311 hi. 






SO 


?7« 






JS1»0,B 


TBI5-H 


HM.f. 


8104 




^». 



232 


FUNCTIONS OF A ONE-DEGREE CURVE 




90 

1 

34 
86 

i 

i 








1.. C. M. 1'.. ,T. 


L.C. M. E. T, 




7815.8 1539.3 2I04.S 5313,3 

B;! !tl!:i S\ S:! 

7833, l.M2,7 2111.3 6352.7 
7825. 1543.9 2113.4 6355,8 
7827. 1546.0 2116.5 5858.9 

TSSS! I547!3 2113!b 53a6!2 

S: ]^i S:! S:i 

7610.0 1550.7 2126.3 5.374.8 

Twiio i553:o am'.K 5"8o:b 

7817.8 1554.1 2132.8 5383.0 

7852^1 1.156!4 2ISe!0 539a!2 
7854. B 1557. S 2139.0 6393.4 
78,17,0 1558.7 2111.2 5396.6 
7859,4 1.159,8 2113,3 6099,7 

7804.3 1588,1 2147.7 5106.0 

787i;o ImS:? 8l5«:4 5ii8:8 

7873.4 1567.8 2158.8 5121.8 

7KSK2 ISToil 21fl2!9 5428!l 
7888.B 1.171.3 aioi.l 6131.2 
7Sid.l 1572.4 2107.8 6134.4 
TSS^.S 1578. 6 aiOO.S 64.37.5 


TBOoio 1674!8 2 71 

7893.3 1576.9 2 7B 

11 III 11 

Sli! Si S 

7917.4 1587.4 SIBJ 
7919.8 1688.8 2197 
7922.2 1589. T 8200 
7924.8 1690.9 aSOS 
79*7.1 1598.0 8304 
7B29.5 1503.2 OM 

7038.7 15ob!s mt 

1 li i 

S:! S:' S 

7BM.0 1M5.B IBBI 

iss..4 10OT.O aaai 


J 5443:9 

6447.1 

4 6453!4 

6 5456.6 

7 6169.8 
9 6486.0 

1 6466. 2 

il 

5 M88.S 
S 5401.7 

Sffl:! 

sil 

3 6614.1 

sS 

1 S628.B 

8 5610.1 
5 5538.3 


30 

1 

SO 

i 

i! 

s 

SO 






8S° 






L. !■. M, K. T. 


l.C. K. .. ,. 


19 
26 

1 

36 
,18 

i 


7960,8 16D8,V 2235,8 5533.3 

7963.2 1609,4 3237. 5.136.6 

7965.6 1610.6 2240. 5580.8 

7968.0 1D11.7 ii42. .W13.1 

7970.3 1613.8 3-344. 5516.8 

7972.7 1614.0 3248. 6549.5 

7975.1 1616.2 2249. 6552,8 

7984.7 161^o!h 9258' 6,165"7 

7989^4 15i!!l 2252! 5572!2 

7991. 8 ID23.3 -Jaoi, 6.175.4 
;W4.a 1024,1 2-J67. 5678.8 

7906.3 1!I!6,6 22110, .1581.9 

8oaK4 iii27!o i-m. nm.i 

8d03,8 1639.1 »-J7r., 5501.7 

Hooa.i I63U.2 ^ara. 65M,9 

Si SI i:s si! 

8015,7 634,0 2-:R7,6 6eOH.O 
)>ai8,l <l3fl.O 9:110 .11111.2 

S g;iS:iffiS 

MifJS,2 630,6 2396,7 6ii21,0 
N»27,6 640.7 22U1I.0 5524,8 
8I13O.0 041.8 2301.3 56'.'7.6 

8032.4 643.0 2303,6 5830.8 


iiiiiiiiii iiiiiiiiii iiiiiiiiii 

iiiiiiiiii iiiiiiiiii iiiiiiiiii 


a 6030.8 

iii 

1 6617 .8 
4 6660.6 

B 0080.4 

7 6063.7 

7 6673.0 

M 

7 oatB.s 

IE! 

btioIt 


i 

» 

IB 

« 

1 

00 



FUNCTIONS OF A ONE-DEGREE CURVE 233 





3 

4 

6 

8 

10 

13 

14 

16 

18 

20 

Hi 
26 
28 
80 
82 
84 
86 
38 

40 
4*^ 
44 
46 
48 
50 
52 
54 
56 
68 
60 



90« 



L. C. 



M. 



E. 



8108.5 
81(K}.8 
8106.2 
8110.5 
8112.0 
8115.2 
8117. 6 
8110.9 
8122.8 
8124.6 

8127.0 
81^.3 
8181.7 
8134.0 
81864 
8138.7 
8141.1 
8143.4 
8145.8 
8148.1 

8150.4 
8152.8 
8155.1 
8157.5 
8150.8 
8162.2 
8164.5 
8166.8 
8169.2 
8171.5 
8173.9 



1678.3 
1679.5 
1680.6 
1681.8 
1683.0 
1684.2 
1685.4 
1686.5 
1687.7 
1688.9 

1690.1 
1691.3 
1692.5 
1693.6 
1694.8 
1696.0 
1697.2 
1698.4 
1699.6 
1700.7 

1701.9 
1703.1 
1704.3 
1705.5 
1706.7 
1707.9 
1709.0 
1710.2 
1711.4 
1712.6 
1713.8 



2373.4 
2375.8 
23T8.2 
2380.5 
2382.9 
2385.3 
2387.6 
♦.'390.0 
2392.4 
2394.7 

2397.1 
2399.5 
2401.9 
2404.3 
2406.6 
2409.0 
2411.4 
2413.8 
2410.2 
2418.6 

2421.0 
2423.4 
2425.8 
2428.2 
2430.6 
2433.0 
2435.4 
2437.9 
2440.3 
2442.7 
2445.1 



5730.0 
5733.3 
5736.7 
5740.0 
5743.4 
5746.7 
5750.0 
5753.4 
5756.7 
5760.1 

5763.4 
5766.8 
5770.1 
5773.5 
5776.9 
5780.2 
5783.6 
5787.0 
5790.3 
5793.7 

5797.1 
5800.4 
5803.8 
5807.2 
.'iSlO.O 
5814.0 
5H17.3 
58-J0.7 
5824.1 
5827.5 
5830.9 



91« 



L. C. 



M. 



E. 



T. 



8173.9 
8176.2 
8178.5 
8180.9 
8183.2 
8185.5 
8187.9 
8190.2 
8192.5 
8194.8 

8197.2 
8199.5 
8201.8 
8204.2 
8206.5 
8208.8 
8211.1 
8213.5 
8215.8 
8218.1 

8220.4 
8222.8 
8225.1 
8227.4 
8229.7 
8232.0 
8234.3 
8236.7 
8239.0 
8241.3 
8243.6 



1713.8 
1715.0 
1716.2 
1717.4 
1718.6 
1719.7 
1720.9 
1722.1 
1723.3 
1724.5 

1725.7 
1726.9 
1728.1 
1729.3 
1730.5 
1731.7 
1732.9 
1734.1 
1735.3 
1736.4 

1737.6 
1738.8 
1740.0 
1741.2 
1742.4 
1743.6 
1744.8 
1746.0 
1747.2 
1748.4 
1749.6 



2445.1 
2447.5 
2450.0 
2452.4 
2454.8 
2457.2 
2459.7 
24G2.1 
2404.5 
2467.0 

24G9. 1 
2471.9 
2474.3 
2476.7 
2479.2 
2481 .6 
2484.1 
2486.5 
2489.0 
2491.5 

2493.9 
2496.4 
2498.9 
2501.3 
2508.8 
2506.3 
2508.7 
2511.2 
2513.7 
2516.2 
2518.7 



5830.9 
5834.3 
5837.7 
5841.1 
5844.5 
5847.9 
5851.3 
5854.7 
5858.1 
5861 .5 

5864.9 
5868.3 
5871.8 
5875.2 
5878.6 
5882.0 
5885.4 
5888.9 
5892.3 
5895.7 

5899.2 
5902.(5 
5906.0 
5909.4 
5912.9 
5916.3 
6919.8 
5923.2 
5926.7 
5930.1 
5933.6 





2 

4 

6 

8 

10 

12 

14 

16 

18 

20 
22 
24 
26 
28 
30 
32 
34 
36 
38 

40 
42 
44 
46 
48 
50 
52 
54 
56 
58 
60 





92«» 1 


93° 




/ 


L. C 


M. 


E. 


T. 


L. C. 


M. 


E. 


T. 







8243.6 


1749.6 


2518.7 


5933.6 


8312.8 


1785.7 


2594.2 


0088.2 





2 


8245.9 


1750.8 


2521.2 


5937.0 


8315.1 


1780.9 


2590.8 


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 


2(501.9 


6048.7 


6 


8 


8253.9 


1754.4 


2528.6 


5947.4 


8322.0 


1790.0 


2004.4 


00.52.2 


8 


10 


8255.3 


1755.6 


2531.1 


5950.9 


8324.3 


1791.8 


2007.0 


6055.8 


10 


13 


8257.5 


1756.8 


2533.6 


5954.4 


83v.'6.6 


1793.0 


20O9.C 


6059.3 


12 


14 


8259.8 


1758.0 


2536.1 


5957.8 


83-,.'8.8 


1794.2 


2612.1 


6062.8 


14 


16 


8362.2 


1759.2 


2538.6 


5901.3 


a331.1 


1795.4 


2614.7 


6066.4 


10 


18 


8264.5 


1760.4 


2541.1 


5904.8 


83:i3.3 


1796.6 


2617.3 


6069.9 


18 


20 


8266.8 


1761.6 


2543.6 


5968.2 


83.35.6 


1797.8 


2619.8 


6073.4 


20 


22 


8369.1 


1762.8 


2546.1 


5971.7 


8337.9 


1799.1 


2622.4 


6077.0 


22 


24 


8271 .4 


1764.0 


2548.6 


5975.2 


8340.2 


1800.3 


2625.0 


6080.5 


24 


26 


8278.7 


1765.2 


2551.2 


5978.7 


8342.5 


1801.5 


2(527.6 


6084.1 


26 


28 


8276.0 


1766.4 


2.553.7 


5982.2 


8344.8 


1802.7 


2630.2 


0087.6 


28 


80 


8278.8 


1767.6 


2556.2 


5985.6 


8347.1 


1803.9 


2632.7 


6091.2 


30 


83 


8380.6 


1768.8 


2558.7 


5989.1 


8319.4 


1805.1 


2635.3 


6094 7 


32 


84 


8282.9 


1770.0 


2501.2 


5992.0 


8351.7 


1806.3 


2037.9 


6098.3 


34 


36 


8285.2 


1771.2 


2563.8 


5996.1 


H354.0 


1807.0 


2610.5 


6101.8 


36 


88 


8287.5 


1772.5 


2506.3 


5999.6 


a-i56.3 


1808.8 


2043.1 


6105.4 


38 


40 


8289.8 


irr3.7 


2.568.8 


6003.1 


8358.5 


1810.0 


2645.7 


6109.0 


40 


42 


8292.1 


1774.9 


2571.3 


6000.6 


8300.8 


1811.2 


2648.3 


6112.5 


42 


44 


8294.4 


1776.1 


2573.9 


6010.1 


8303.1 


1812.4 


2050.9 


6110 1 


44 


46 


8296.7 


1777.3 


2576.4 


0013.0 


8305.4 


1813.0 


2053.5 


0119.7 


40 


48 


8299.0 


1778.5 


2578.9 


0017.1 


8367.7 


1814.9 


2050.1 


0123.2 


48 


50 


8301.3 


1779.7 


2581.5 


0020.0 


8309.9 


IMO.l 


2058.7 


01 --.'0. 8 


50 


52 


a303.6 


1780.9 


2584.0 


C021.1 


8372.2 


1817.3 


2(501.3 


0130.4 


5-» 


51 


8805.9 


1782.1 


2580.0 


(5027.0 


a374.5 


1818.5 


2663.9 


0133.9 


51 


56 


8308.3 


1783.3 


2589.1 


6031.1 


8370.8 


1819.7 


2(500.0 


6137.5 


5(5 


58 


8810.5 


1784.5 


2.591.7 


6034.6 


8379.1 


1820.9 


2(509.2 


6141.1 


58 


CO 


8S12.8 


17*5.7 


2594.2 


6038.2 


8381.3 


1822.2 


2(571.8 


6144.7 


60 



234 


FUNCTIONK OF A OXE-DEGREE CURVE 






9i' 






L.C, 


M. 


E. 


T. 


L.C. 


M. 


S. 


T. 






W2.S 




S1H.7 


8149.3 


858.B 


2751. E 












^a'' 




84.11.5 


860, 




8350:9 




4 


















4 


fl 
















B281:b 


8 




sm'A 




BBsa!; 


CiSslO 




883;i 


am'-i 


8267.B 






mi.7 








6460;i 














SS»'.i 


iSfSTA 










8876:2 






sssr'.a 














6378.8 






ftiBB.S 


831:: 




617314 




889:: 


3773:1 


8383.3 




18 


840L.7 


833.1 


2695.6 


6117.0 


846b:4 


B6B.S 


aj75.8 


8296.3 






















20 
















STBua 


«2ob:g 


28 


a* 


sm'.f, 


weie 


sroa: 


8187:8 


84TB:s 


87S:i 


3784.9 


8297.8 




» 


MIO.8 


SSB.O 


anw. 




B4TS.4 


874.1 


S7B8.7 




28 


















WM.a 


38 


3D 










8483:9 




2793:1 


•308.8 


ao 






Ml'.l 


aTMio 


c«o-i:a 


B4BS.I 


B78:i 


1(794.5 


flsn.9 


8d 




8419.9 


843.C 


ijTis.i 


6S05.9 


8487.4 


S79.I 






84 




















38 


38 


SiM.i 










B8s:i 




633S:o 


88 




8438.8 


848.0 


3724.5 


esiB.8 


8404. 1 


883.! 


SBffi.B 


flsoa.T 


40 












8496.3 




2808. g 




49 










&»'a 








6iu:i 


44 


iB 


84as:i 










B8r:a 


28n;i 


8387.8 


48 


46 


fttSS.7 


8S1.6 




K3]'.3 


BWS.O 


888.5 


2816.8 


8341,5 


48 












8505.3 


889.1 


2818.8 






ts 














2891.3 


B8J8:o 


SS 




««:4 










eBa;i 


2aB.1 


8958,7 




U 


8H4.T 


BW.I 


U746.T 


BB45.9 


8613:o 


S93.. 


3827.8 


8358,4 


58 


B8 






f-i^i 




8514.8 




2830.9 


8380.1 


58 


W 














3833.4 


8363.8 


80_ 




M- 


B7' 








M. 


E. 


T. 


L.C, 


M. 


E. 


T. 




8518.4 








BMa.o 


OSS.fl 


8917. 


8478,8 








rtor!! 


aSM',! 


03«7;5 


BMIS.S 


984.4 


3920. 


8480.4 




4 






1.-838.B 


6S7l.a 




w-t.: 


2828. 




4 




8^4 


899 ,'b 


3844!b 




8589:? 
%9 .B 






B\ 


s 








awT.i 




8594 1 


B39:< 


J93i: 


840S.8 


» 








S350.0 


6S88.3 


aioa.a 




2984, 


8489,4 


















2937. 


8503.1 


M 


















SUT.l 


S 




BsaaiB 




a8B8.4 




mi'-9 


944:1 


3948: 


0610.8 






08,a 




flIOl.S 


BCOSl 


915.7 


2M8. 


8S14.T 




^ 












H8.9 




081 S.E 




« 


8M8!2 




sbos't 






HH.2 


2981 : 


8888.3 




es 


8MG.4 


M.O 


8889.5 


04ia:4 


8811 :7 


949.4 


3964. 


8698.9 




£8 


SMT.e 






0416.3 


8(118.9 


950. 


2957, 


SSBO.O 
















952. 




ASn.B 




W 


BBsaio 










958. 


2088: 


«B7.7 






8564.8 




*i8(l8 


84iT6 


8B«),S 


9S4. 


3068, 


8Ml,a 




H 




















88 














sora: 


e6u:8 




U 


8580.8 




!»80.3 




8637.1 


SS8. 


3874. 


880.0 


t 












8899.3 




2B7T. 






Bssa'a 












20S0. 


taeo'.i 


,.fE 












8833'? 


Bau:. 


3983. 


8684.6 


m- 


JS 


BS89.9 


Bas.T 


S900..1 


6(ii».a 


863.1.8 


m. 


ma. 


8B88,« 


K* 


50 


B57S.0 




asos.a 


645r.fl 








8871.8 


n 






^!u 


2908.1 






tet,'. 


2001: 


esn-R 


8ff 






BM4 








997. 


3008 


8880,0 


W 


58 


WTSO 


B.W.7 


sill lis 




8644.8 


»g. 


2908 


BBB8.B 


m- 


sa 


WW.B 


M1.9 




UTi.e 


8618 8 






8587,7 




80 




»3aa 










SNM'i 


6591. B 


80 





FtJNCTIONS OF A ONE-tlEO.REE CURVE 


235 


J 

i 

88 
B> 

40 

£ 

BO 

i 

60 


88" 






1..0, M, ■, T. 


L. a M, E, T. 


S 

B69( 

1 
1 

889( 

i 

1 
i 


3 ism'.i sm'.a bsbI 

S 1974,8 3013.8 B60) 

9 im.i soisis 6011 

1 1918.8 8081.6 68H 

4 1980!9 a0-'7i6 CllSi 

8 1982.1 3030,4 B026 
B 1988,4 303;|.3 6B3( 

i III SI 1 
i is:? si! i 

a 1996.0 3063.0 6669 

5 B:! iw:." S 

1 1999.B 3071.9 06SI 
a 3001.1 3074,0 C685 
4 SDIS,B 3077,9 0683 

1 S:! S! S 

9 3006,1 3080.8 0701 
I 8007,4 3089.8 8705 
3 2008,7 3092.9 67O0 


6 

I 

S 

I 


8 le'.i sm'.a sm.a m%.a 

8 18.6 8011.3 3098.9 6716.9 
8-20.7 3013.5 3101,9 8780,8 

B-25;i SOIS.O 310719 0788!8 
8-87.8 8016,3 8111.0 67*2.7 
B-38,4 B017.5 3114,0 8738,7 

Bra3:7 S020!l 3120;o 6744:0 
8-31.8 8031,4 3133.1 8748,6 

B43;3 3085:8 313^:8 67G0:a 
8 44.5 2036,4 8135.8 (1764.0 

8-50:o S030:3 814i:4 6776:a 
8753.1 3031.6 3147.4 B7Wlfl 

B7.'.9:b 203S:4 3 96:0 6793:6 

8-^:1 lwo:b b 08:9 ^:o 

8776:7 2015:6 3 Si:8 6834.7 
8778.8 5046.8 3 84.3 0828.8 


a 

1 

as 

38 

1 


11 

90 
38 

i 

i 

41 

ii 
1 

M 
SB 

s 






■ 


LO. H. B, T. 


l..r. M, E. T, 


BT9! 

i 

1 


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I 

9 

a 


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i 

24 

36 
18 

i 

34 

! 

42 
40 

|, 

i 



23S 


FUNCTIONS OF A ONT>-DEGREE CURVE 




' 








1..C. M. K. T. 


l,.f. M. E. T. 


9 

10 

18 
SO 
22 

26 

30 
3-2 
34 
36 

40 

46 

48 
60 


SS08!2 2125J! 337b!3 7080^2 
88)0.3 2123 B 3881.6 7084.4 
B9I2 4 2197.9 3384.9 7083 D 
8911.5 2199 2 33.48.2 7082.8 

89I8> 9131:h 3394 "7 ilOlis 

em.a 2133.1 :iawo 7)0S.! 

8827.0 Eia-.O 84117.9 7118.9 

8929.1 2)SS3 BID. 2 7)22.4 

8931.2 2).)!l.6 8114.5 7126.7 

m]'4 2142i2 3131 is 71362 

Miii? 2i'47i4 siaiis flMi" 

8945.8 2148.7 3437.S 7156.5 

if ill 1 i ill 

S! Si Hi Piil 

8662,5 91SB,1 3404.5 7190,7 

-966.6 91flli7 S47l'2 7l9Bis 
BUOa.7 2)63 3171 6 7203.8 


Bi til i 

8879.1 2169.5 8491 

8983i2 ei72il 349i 
8895.3 9173.4 8501 

8887.3 2174.7 3506 
8889 4 9170.1 8508 
8891.5 9177.4 8511 
8893.5 2178.7 8515 
E095.6 9180.0 39IB 
8897.7 2)81.3 8693 
8999.7 9182.6 3525 
900) .8 2183.9 862t 

MOlio fm.l 358! 
9008.0 8187.8 3638 
9010.0 2189.1 3542 
9D12.1 2190.5 3540 

8014.2 3181.8 3549 
9910.2 9188.1 35K 

BO-Jois 9195 7 355t 

9022.4 £187.0 35K 

9030i6 9402i3 3577 


6 7208.6 

7 T916"b 

3 Ksais 

4 7946,3 

111 

s 7a99io 

Isil 


! 

M 
U 
40 
42 

S 

98 


19 
90 
30 


lat- 


106" 




I, G. M. E. T. 


I,. C. M. E. T. 


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0091.8 8941.8 3883 

0083.8 2243.1 8836 

9095.9 £914.4 36S0 

9097.8 S248.8 3893 

9089.9 994T.I 369e 

B]04!o £849i7 3704 
9106.0 9S51-1 8707 

9108.0 £253.4 8711 

0110.1 9963.7 8714 

9116.1 £257.7 872S 

9)£oie £26ai8 3733 

0129.2 9201.7 873(1 

9180 2 3e7i0 8751 

M Mi 1 

8 44i3 aaf'liis 8770 
I.J46 3 9977.6 3780 

0148.3 2278.0 ffTSS 

9150.4 2980.3 3787 


7 V476i 
1 7481. 

) 7494 i 
1 7480. 

i TSOS'. 

sir 

] TS9Bi 

IS: 

r 7A44. 
76BS! 
a TBBS. 
3 7669. 

II 

> tbm! 


8 
12 
16 
10 

1 

28 
80 

S 

8t 

IB 
40 

£1 

ij 



'■ 


FUNCTIONS OF A ONE-DEGREE CURVE 


237 










L,0. M. E. T. 


L. C. M. E. T. 


1 

10 

u 

la 

1 
a 

K 

to 

M 

S 

N 
U 

ii 

98 

SO 


9163.4 assi 

B1M.4 Smi 

B i 
i i 

nn. ssM 

sill 
111 i 

ii 
11 


si 
ii 

I 381! 

Ii 

ii 

9 38M 

t S8SI 
1 S8« 

ii 

3884 

Ii 


ill 

1 TiMa;6 

!S| 

S TS78:i 

B 7W1.7 

7 7fllK;i 
4 7B9a.T 

ii'i 

3 T730.0 


9913.3 3821,7 3903 

9311.3 3333.0 8908 

OJlsil 3335^7 3914 
9320.1 93S7.0 3918 
9333.1 3399.4 8933 
9iai.l 333S.7 393S 
9338.1 2331.1 3939 

aaffl.i mj.4 b93s 

9339.0 3335.1 3941 

S:S S;! S 

0237.9 9339.1 3953 
9339.9 3340.9 39.10 
0341.B 3SI1.8 3900 
9343 8 334S.1 39U 

934B:7 2317:2 3a7£ 
B3S1.7 334e.B 39T9 

9253.7 3349.9 3983 
9:^5.8 3331 a 8937 

g:! S3:! Si 

9031.5 E355.3 3999 
9263.5 3396.8 4009 

S-! S-S S 

0269.4 38G0.7 4014 

0271.8 3303.0 4018 


1 7743.7 
J 7748-4 

1 -767.3 
9 "773.0 

7 7776.7 
B "781.5 

4 7786.3 

3 '495!7 

; il 

9 820.1 

8 833.3 

•SI 

8 -853.0 

sl! 

8 Tsrs.a 

7 7877,0 

5 881.9 
B B8B.7 


10 

la 

IB 
30 

36 

84 

30 

40 
49 

4S 

n 

53 


■ 


J 08' 


1011" 




1..0. «. K. T. 


I..C. M. E, T. 


• 

a 

s 
s 

m 

18 

40 

S 

M 

1 

g 


on'i siTc 

1:1 

ii 

9804.S E38! 
93084 3887 
9310.4 2389 
BSia.S t3» 

E;! i 

8317 8 3401 

B8W.8 340e 


is 
?i 

4043 
4 40)S 

4057 
4061 

S 

407! 

i 

408C 

4001 

7 410! 

1 4109 

ti 

9 4l5 


I TOWiO 

8 S:? 

9 7990. fi 
8 79)11.4 
S 7930-8 

7 m-i.i 

ill 

4 706913 
4 7074.1 

lil 

3 7993.8 

3 7S98.7 

!B:! 

a Bitfl'a 

4 80-28 3 


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10 

i 
1 

33 
34 
86 

40 

i 

52 



238 FUNCTIONS OF A ONE-DEGREE CURVE 





no- 


in- 




L. C M. E. T. 


L.C. M. E. T. 


8 

32 

36 
38 
10 

46 

i 


M89:4 2444:8 *MA siw 
9391,3 2416.1 4268 3 8108 

9305,1 2418:9 4.46.6 8203 
9307.0 2)50.8 1280.8 890e 

S40S;7 245i:3 4333,3 8234 

9404.7 2455.7 4297.5 8329 
9)06.6 2)57.1 4301.7 B234 

9108.5 2158.4 4305.9 8339 
9)10.4 Oisae 4310.1 8314 

9)14!9 9469. 6 431B'5 8334 

941B:o 946S;9 432C:9 8261 
B4J0.9 3466.7 4331,1 8670 

9191.8 9468.0 4815.4 8275 
0433.7 3460.4 4380.6 B280 

9435.6 94T0.8 4343.8 OSBJ 

1 || 1 1 

048a!B 2480^4 4373!e 8331 

0440.7 2481,7 4377,9 8336 
0412.6 8483.1 4382 3 8332 
9441.fi 9484 5 43811.1 8337 




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■1 
ii 

n 

1 

98 

i 

SB 

IS 
10 
42 

18 
50 
61 

i 

60 




lie- 


Hi- 




L.O. M. E. T. 


L. C. M. E T. 


19 

90 

=8 
30 
33 

i 

40 

16 
IB 


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1 

19 

90 
32 

36 

98 

31 

1 
S 

44 

48 

S 

s 





FUNCTIONS OF A O 


NE-n 


RCRF 


E CURVE 


230 




lU' 








' 


L. C 


M 


E. 


T. 


ui:- 


u. 


E. 


T. 





Bsn 


a~iMM 




8823.4 


ms.s 


9631.3 


4931,4 


89B4.3 






















4 


9614 








9668!8 










e 


Hie 


S aSl! 


4 4«(n:a 


884d:3 


saro.e 










8 




asi4 


8 4SCW.6 


884(1.0 


XUA 


aess.s 


4954.0 


9017:4 


8 


10 


Mao 












4938.9 


D023.2 




11 


SB*t 


H» 


t 4Bir;i 


aasr'.-i 












14 


tsas 


«l( 


4S3S.B 


8868.9 


8677:8 


266i:i 


49S8:7 






1« 




xeo 


4 4Bea.e 


88fl8.B 


M7B.a 


3663.5 


4973.3 


9wo:t 


IB 




ttn 


G am 


a 4833.4 








4978.6 






*> 


SUI 


1 ssai! 


3 4838.1 


837»,9 


9683.1 


366S.4 


4988.4 






ft 


Mil 




e 4Ma.» 




9681.9 


36«a.B 


4988.3 


9068.1 


:£> 


M 


mi 




4tM7.r 








4993.3 


9064.0 




n 


WH 




4 iSSi.i 






3669:e 






20 


« 


MIS 




i 485r.S 


sm'.e 


flew: a 


2671.0 


50(B:a 


9075:7 




w 


M38 








9692.0 


2673.4 


B008.1 


B081.S 


30 






!631 


9 iSK.S 






^673.8 










9M1 


e seas 


4S7l.a 










9093:2 




X 


%iJ 


«84 


4 487fl.4 


smA 


OlJB7:4 


3676 :e 


50'.a:o 




36 


K 




8sas 


8 4S81.S 


8S3],1 






5028.0 


91050 


8S 


n 


3611 






8833.8 










40 


tt 


Ki9 


insE 


E 4Bw;s 


8912.3 












44 


gsu 


1 »40 


4S8B.7 


8fllB.a 


9704 :b 










M 


MH 


; SMI 












91!!8.5 


46 


4a 




















H 


WW 


1 aSH 














DO 


M 


MB8 


xu 


7 4ei5:o 


8971.3 


971i:6 


3687:6 




9146:« 




N 


WS9 


1 M', 






9713.3 


26893 


6067. B 




64 


W 




















K 


wn 


> se4( 
















W 


M« 


» wai 


3 4934:4 


889j;3 


9718:8 




5083:o 


9169:9 






llfl- 


IIJ" 


~ 


1.0 


M 


E. 


T. 


L. C. 


M. 


E. 


T. 








} EaeB.(i 




9711.3 


3780.1 


G33S.6 


9350.5 




3 


vao 


S6DS 










6241.8 








>7«e 


MM 


GoesJ 


9I8L8 




S73S:9 




9363:7 




S 


(ins 


m: 


Goge.i 


9188.8 


vm'.i 






9368.9 




B 


mt 




: 3103.3 




9778.8 


2741:8 




9375,0 






vm 


sma 














10 


II 


m» 


XlOi 


GllsiS 


twi'.i 












U 


tm 


me 


B[|8.4 


9311.3 


b:&<* 


27«0 












WW 






978.-i.3 


3747.5 


53?s:4 


9399:5 




IB 


n&i 






Vt-X.6 






5283,0 


9405.7 


18 


•> 


BTSS 


xim 


flIB3.8 


9K9fl 




2750,3 


6388.9 


0411-9 


30 


tt 


vm 


■7M 


6138.7 




B7S0.4 


2751,7 


6^91.2 


9418,0 




U 


vm 








9793.1 




B3B9.5 


91342 


34 


m 


V74I 


Jt7» 


3148 !b 


9*17:6 








9130.4 


36 


n 


irrtt 


STII 


SIM a 


9-J53.B 


979.^:6 






9430.6 


IS 










9M9.3 


9797.3 


2757,4 


5315.8 


0149,8 


30 




me 










2758,9 


B.^.6 


0449.0 




H 


ma 


»711 


tm.i 


fti71.S 


9800 :? 


2760.3 








H 








ftm.7 


9803.5 


3701,7 


esala 


Biori 


8<l 






nx 






9804.3 


2708.1 


B336.5 




38 


40 


na 


2731 


3184.8 


Ksn.e 




3704.0 








tt 


KK 


sma 




d^K.H 


0807:7 




5347:2 


948o:0 


42 


44 


Ktn 


STU 




mi. 9 




3767:3 


6352.5 


0486.8 




tt 


net 


amt 


am.a 


BSOT.e 








0492,5 




tt 


imo 


mr. 


B9CI3.4 


!M14,0 


mis 


8770 :s 








n 






taio.s 


03^.1 


B8i4.S 


2771.7 


5368.5 


9505:0 




n 


ffnt 


STSC 






9816.3 


2778.1 


5878.9 






14 


ms 


ST81 


3^1 !a 


93-')3:3 




2774.6 








M 


B?er 


273! 


Ba».3 


93883 


0819:7 


3776,0 








m 










ilKSI.) 


3777.4 


mn.n 


95SU,0 






m\ 


2!» 


iM3e!c 






3T7B,S 




9586.3 


00 



240 NATURAL SIXES, fOSIXES, TAXGENTS, COTAN. 





Sin. 


Cos. 


Tan. 


^^^ 


Sir.. 


Cos. 


Tan. 


Cot. 


. 




.CDOOO 














7.2900 


"55 


1 


.00028 






8437.76 


01774 


99984 


01775 


6 


S50S 


59 








.00068 


1718.37 


01803 


9S934 


01804 












One 


-00087 


1146.92 


01833 




01833 










ioOUB 


"ij^^ 


.001_iB_ 
.00145 


8i9..48B_ 


01891 


"999B2 


01862_ 
01891 


~i 


"ie2T 


_59 




.00115 












Qua 


.00175 


67 


.9 7 


01920 


99982 


01830 


2 


607 






! 00204 






49 


.1 6 


01949 


9998! 


01949 


1 




63 




.00333 


One 






.7 8 











485 


63 






One 


."00291 


"34. 


■ 7 i 


7o"203"6" 


"99979 


02007 
O"303e 


_4B 


157. 


_a 




.00381 




48 




SO 


ii 




98980 


003ZO 




.6 1 


.02065 


B0B79 


02068 


48 


131 






S8B89 












02095 


47 


395 


IB 


JL 




98999 


OOS78 


26 


.4 1 


: 03123 


B9B77 


02124 


47 


853 


47 




99989 


.00407 
00436 


Ji 


Ty-f- 


_021,52_ 


_9997_7_ 


02163 
02183 


46 


489 


_« 






46 


284 










004«5 


1 




:023U 


99978 


02211 


16 


261 










00496 





.2 9 


02240 


89B76 


02240 


44 


388 










00524 


9 










44 


661 




IS. 




MM8 


00553 


_£ 




02298 


99974 


02298 




_0B1. 




20 




msB 


00582 






02327 


99973 




—r 


611 




91 


ioOBIl 


998 9B 


00611 












4! 


336 






.00040 


88993 






3 9 


02335 


99972 


02336 




1158 








99988 


008fl9 




4 5 


024H 


99971 


02416 




4108 




24_ 


.00727 ■ 


98988 
89997 


pOlflB, 


— 


5 7" 


02443 
03473 


Jfl_B70_ 




_« 


174 






0247S 










.00766 . 


99887 


0O75B 




2 9 


03501 






9 


865 








89497 


00786 


2 






99B68 


02531 


a 


059 






.D0S14 


9Sa07 


00815 


2 


7 4 


02SBO 




02560 








an 


.00841 . 


99893 
98993 


_0CB44_ 
00373 




J40_ 

6 9 


03_5e9_ 


"99966" 


"026 9~ 








30 




835 




31 


' 008 02 








8 2 


02647 




026 fl 










.00931 


98996 








02676 






7 


579 




33 


.00980 , 


99986 




■ 


171 




9B963 


027 8 




660 




M 




989_95 


oiolf 




107 


02734. 




027 5 


6 


637 






_ 




"99962 


02764 


8 


776 






'O 047 


98995 


01O47 




395 


03793 




03793 


( 


008 






.0 076 


99994 


0107B 




■035 


03B31 








3 S 




38 






01105 








98959 


02861 


5 


6 6 




3£_ 


■0 134 ' 




01135 




'43fl 


02B79 


99959 


03831 


4 


1 1 








99983 


one 


, _ 


1393 


O2B0B 


99953 










41 








83. 






99957 


02939 




■273 




42 




99993 


0122 


81. 


!470 


02867 




03968 




9 S 


J 




.0 251 1 


99993 


0135 






0299B 












.0 2B0 ' 


99983. 


0123 


jsL 






_B9B54_ 


03026 


_3 


04 3 


is 






01309 














73 8 


li 




.0 338 


98981 




74 






99952 


03084 


32 


43 S 


14 




.0 3B7 


98991 


01367 


73. 


390 


03112 




03114 


S3 


11 1 


11 


M 




99990 


01398 












SI 


82 S 


J 


*». 


.0_425 


99990_ 
99939 


0142S. 
0H5S 


0- 


i3_3_ 


03199 


_9?B50_ 


03172 


31 


5234 








31 


34 


















98948 










62 


513 




01513 




055 


03257 


99947 




30 


BBS 








8988B 


01642 






032fie 






30 


41 


















99945 


_0S117_ 


SO 






86 




1b987~ 


oiooT 


~3T 


893" 


03346" 


99914 


03348 


2B 


BS 








999B7 


01839 




828 


03374 


9B943 


03376 


3B 


B3 






.01858 
















Id 








01887 


fliqae 
















1230 




5J_ 


S^ 


W 


01716 


8^ 


613 
2900 


■1?; 


99940 




38 


B77 




filL 


^^ 


^ 


^ 


■f 



NATURAL SINES, COSINES, TANOENTS, COTAN. 241 



_!. ■«■■ 


~c^ " 


"t^ 


C.,L ' 


fij.,. 


r,i^. 


T^ . 1 C,.:. 


- 


« .03 


.88938 


.03193 


28 5368 


05234 


98863 


052 1 






.OS 


.99938 


.03531 


38 


8994 


05383 






.B75B 






-81(937 


.03850 












.B711 




'03 




















Lm 


^B93S_ 


!0Sfi09_ 


_2_7_ 


7117 


0o350_ 


88657 




'6656 


























:o3Ba7 


27 


371S 


05408 


89854 










-99933 


-0368B 


2 


0566 


05437 


99852 


05415 


.3655 






.89931 


.03725 




8450 






054 4 


.2677 




. „ -03J a 


.9_99SO_ 


-03764 






OSiM. 


JL9.p4.a_ 


_055M_ 


- _-.1708_ 




IB -03 1 


.88929 


■03783 


2 














.03 D 






3 


22 S 










4 


.03 8 


.11111 


:03B43 


2 


03 7 


05583 


99941 


05581 


.8863 




.03 e 


.99925 


.03B71 


3 


83 8 






06620 


.7931 




-03 7 


.B9B24 


j)3aoo_ 

.03839 


_2_ 




05869" 


_9_984_ 


0564B 


JO 1 5 


_4 


-03 t 




06678 




.03 S 






3 














.03 1 


: 99931 


:o3ga7 




079 


05737 


9983 


06737 


.4314 




.04 3 


.89919 


.04016 




BB7 


0S756 


98831 


06786 


.3432 




-04 43 


.99918 


-04046 


J_ 


719 


057B5 


99833 


05785 


.2568 


_4 


r TZiTT 


.99917 


-04076 


2 


541 


05811 


98B3 


06824 


.1693 




.04 


.99916 


.04104 




367 


06844 




06864 


.0837 




.04 B 




-04133 






05873 








8 








3 












3 


!04 B 


199912 


-04191 


_2: 


8S9 


06S31 




06841 


.8318 


3 


- roT"^T 




.04220 




69 


059GD 




05870 


.749 


8 




.D9910 








05899 


9982 


05899 


.669 


3 


!04 S 




















-04 0* 






3: 








06058 




3 


■04_3. 


.99908 


;p4337L 


2 


05 


_06 76_ 


_L9Ji_ 


06087 




3 


' :8J f 




.04366 




90 8 


09 05 




ooiis 




3 


-99B04 


.04395 


2 


76 9 


06 34 


9981 




.272 




.044 
















.195 




.04449 


-lllol 


; 04454 


2; 


45 1 






062O4 






.044 J_ 
.04 07 




J148i 
.04613 


-23 


30 1 
1640 


_08 21_ 

oa 50 


9980 
9980 


062a3_ 
06282 




2 


.968 




.04 B 


: 98897 


.04B41 








9980 




.891 




-04 5 


-B9B9S 












06321 






.04 4 




:045fl9 




74 6 


06 37 


9979 


06350 










-04628 


J 


60 8 


QS366 


9979 


08378 


:676 




1 04 53 








09385 




06408 






.04 83 




-04687 


2 33B9 




99793 








-99889 




2 .3049 








.1638 




! 04740 




i 04745 


3 




08182 


88790 


06198 






_ -_047e9_ 
.047BB 


lam"" 


.04774 
-04603 


30 

31 


9160 
8IB8 


np.5iL 


_99.78B_ 
9978B 


06B3S 


"""2671 








.04837 




















-04SSB 


89B3? 


04862 












.1323 




.04885 


.89881 


-01891 


30 


4165 


08 37 


99780 


06643 


.0657 




-04914 


a9B7_fi_ 


M9_2a_ 




3263 




_887_7B. 


06671 


_1,98_88 
















88779 


08700 






^04 72 


.99878 


-04978 






08711 










.05 01 


.99875 


0S007 


10 


9703 


08743 


98779 


08769 






.05 30 








B54B 




H8770 


06798 


4" 7317 




_ J)50Be_ 


"99870" 


J5_0BB_ 


IS 




-^||^=- 


,99788. 


08817_ 
08847 






4.8058 




















4.6438 






: 99887 


"05 53 




4Q51 


0e8B9 






1.4839 








.05 92 




2959 


08918 


99790 


08934 






_ ]0S2J. 


lB9Bfl4. 


i.O_S 13_ 




1879 


.De9_47_ 


B876B 


_08963. 






s^H?- 


^^ 


^M. 


19 0811 


06976 


99756 
Sin. 


08983 


j^M. 


=4 




S 


,. 








1 


IB* 







242 NATURAL SINES, COSINES. TANGENTS. COTAN. 

4' S** 



9 


Sin. 


Cos. 


Tan. 


Cot. 1 


Sin. 


Cos. Tan. | 




Cot. 


1 




1 

2 
8 

4 


.06976 
.07005 
■07034 
.07063 
.07092 

.07121 
.07150 
.07179 
.07208 
.07287 


.99756 
.99754 
.99752 
.99750 
.99748 

.99746 
.99744 
.99742 
.99740 
.99788 


.06998 
.07022 
.07051 
.07080 
.07110 


14 

14 

14. 

14. 

14 


8007 
2411 
1821 
1285 
0655 


.08716 
.08745 
.08774 
■08808 
08881 


.99619 
.99617 
.99614 
.99612 
.99609 


.08749 
.08778 
.08807 
.08887 
.08866 


11.4801 
11.8810 
11.8640 
11.8168 
11.2780 


60 
N 
61 
ST 
Jl 


6 
6 
7 
8 

9 


.07139 
.07168 
.07197 
.07227 
.07256 


14 
13 
13 
13 
13 

13. 
13. 
13. 
13. 
18. 


0079 
9507 
8940 
8378 
7821_ 

7267 
6719 
6174 
5684 
.5098 


.08860 
.08889 
.08918 
.08947 
.08976 


.99607 
.99604 
.99602 
.99599 
.99596 


.08895 
.08926 
.08964 
.08988 
.09018 


11.2417 
11.2048 
11.1681 
11.1816 
11.0064 


H 
84 
61 
58 

51 


10 

11 

12 
18 
14 


.07266 
.07295 
.07824 
.07863 
.07382 

.07411 
.07440 
.07469 
.07498 
.07527 


.99786 
.99784 
.99781 
.99729 
.99727 


.07285 
.07314 
.07844 
.07873 
.07402 


.09005 
.09084 
.09068 
.09092 
.09121 


.99594 
.99591 
.99588 
.99586 
.99688 


.09042 
.09071 
.09101 
.09180 
.09159 


11.0694 
11.0287 
10.0883 
10.0680 
10.0178 


60 
41 
41 

47 

a 


15 
16 
17 
18 
19 


.99726 
.99728 
.99721 
.99719 
.99716 


.07481 
.07461 
.07490 
.07519 
.07548 


18. 
18. 
18. 
18. 
18 


4566 
4089 
8515 
2996 
2480 


.09150 
.09179 
.09208 
.09287 
■09266 


.99580 
.99578 
.99576 
.99672 
.99570 


.09189 
.09218 
.09247 
.09277 
.09806 


10.8830 
10.8488 
10.8188 
10.7787 
10.7467 


44 
41 


20 

21 
22 
28 
?4 


.07556 
.07585 
.07614 
.07643 
.07672 


.99714 
.99712 
.99710 
.99708 
.99705 


.07578 
.07607 
.07636 
.07665 
.07695 


13. 
13. 
18. 
18. 

12 


1969 
1461 
0958 
0458 
9962 


.09296 
.09824 
.09853 
.09882 
.09411 


.99567 
.99564 
.99562 
.99559 
■99556 


.09886 
.09866 
.09894 
.09428 
.09463 


10.7110 
10.6788 
10.6460 
10.6118 
10.6780 


40 
81 
88 

n 
Jl 


25 
26 
27 
28 
29 


.07701 
.07730 
.07759 
.07788 
.07817 


.99703 
.99701 
.99699 
.99696 
.99694 


.07724 
.07758 
.07782 
.07812 
.07841 


12 
12 
12 
12 
12 


9469 
8981 
8496 
8014 
7536 


.09440 
.09469 
.09498 
.09527 
■09556 


.99558 
.99551 
.99648 
.99645 
.99542 


.09482 
.09611 
.09641 
.09670 
.09600 

.09629 
.09668 
.09688 
.09717 
.09746 


10 6463 
10.6186 
10.4818 
10.4401 
10.4173 


81 
M 

It 

n 
n 


30 

81 
82 
88 
84 


.07846 
.07875 
.07904 
.07933 
.07962 


.99692 
.99689 
.99687 
.99686 
.99688 


.07870 
.07899 
.07929 
.07958 
.07987 


12 
12 
12 
12 
12 


.7062 
.6691 
.6124 
.5660 
.5199 


.09685 
.09614 
.09642 
.09671 
.09700 


.99640 
.99687 
.99584 
.99631 
.99628 


10.8864 
10.8688 
10.8224 
10.2018 
10.2802 


80 
SI 
91 

ar 


85 
86 
87 
88 
89 


.07991 
.08020 
.08049 
.08078 
.08107 


.99680 
.99678 
.99676 
.99678 
.99671 


.08017 
.08046 
.08075 
.08104 
■08134 


12 
12 
12 
12 
12 


.4742 
.4288 
.3838 
.3390 
.2946 


.09729 
.09758 
.09787 
.09816 
■09845 


.99626 
.99628 
.99520 
■99617 
. 99514 


.09776 
.00806 
.09884 
.09864 
.09898 


10.2204 
10.1088 
10.1688 
^0.1881 
Jip.1080 


a 
Jl 


40 

41 
42 
43 
44 


•08136 
.08165 
.08194 
.08223 
.08252_ 

.08281 
.08310 
.08339 
.08368 
.08397 


.99668 
.99666 
.99664 
.99661 
.99659 


.08163 
.08192 
.08221 
.08251 
.08280 


12 
12 
12 
12 
12 


.2505 
.2067 
.1632 
.1201 
.0772 


09874 
.09908 
.09932 
.09961 
.09990 


■99611 
■99508 
.99506 
.99608 
.99500 


.09928 
.09962 
.09981 
.10011 
.10040 


1( 
1( 
11 



).0780 

1.0488 

).0187 

08081 

0W9T 


80 
U 

u 

17 
Jl 


45 

46 
47 
48 
49 


.99657 
.99654 
.99652 
.99649 
.99647 

.99644 
.99642 
.99639 
.99637 
.99635 

.99632 
.99630 
.99627 
.99625 
.99622 

■99619 


.08309 
.08339 
.08368 
.08397 
.08427 

.08456 
.08485 
.08514 
.08544 
.08573 

.08602 
.08632 
.08661 
.08690 
.08720 

.08749 


12 
11 
11 
11 
11 


0346 

.9923 

.9504 

9087 

8673 


•10019 
.10048 
.10077 
.10106 
.10135 


.99497 
.99494 
.99491 
.99488 
.99485 

.99482 
.99479 
.99476 
.99478 
.99470 


.10069 
.10009 
.10128 
.10168 
.10187 









08101 
00211 
87888 

.84488 

81841 


U 
11 

u 


50 

51 
62 
53 
64 


.08426 
.08455 
.08484 
.08513 
.08542 

.08571 
.08600 
.08629 
.08658 
.08687 

.08716 


11 

11. 

11. 

11. 

11. 


8262 
7853 
7448 
7046 
6645 


.10164 
.10192 
.10221 
. 10250 
. 10279 


.10916. 
.10346 
.10376 
.10806 
.10884 




0. 


J. 


.78817 
70000 

.78817 
70441 


It 


65 
66 
57 
58 
69 


11. 
11. 
11. 
11. 

11. 


6248 
5853 
6461 
5072 
4686 


.10308 
.10337 
. 10366 
•10395 
. 10424 


.99467 
.99464 
.99461 
.99458 
.99456 

.99452 


.10868 
.10808 
.10422 
.10462 
.10481 



0. 

0. 


MSB 

.88208 
80400 
68781 
64108 




f^ 


11. 


4301 


.10453 


.10610 


± 


jittL 


-1 


'/ 


Cos. 


Sin. 


Cot. 


1 


'an. 


Cos. 


Sin. Cot. 1 




iSr\ 


^r 



85< 



84* 



NATURAL SINES, CX)SINES, TANGENTS. COTAN. 213 





Sm. 


c™. 


Tan. 


Cot. 


8iQ. 


Coo- 


Tan. 1 


Col. 


J_ 










9.51 38 


. 187 


6255 


2378 












. 0540 


B.48 81 


. 318 




3308 








- OBH 


9448 


. 0569 


fit 16 










68 




. 0640 


■B443 


. U599 










08800 


67 




■_066B_ 


-BiO. 


^ 0628. 


J JO ot 






2397 _ 


.06i74_ 


_6B 










9. 36 07 


" 3yf 


^ 6337' 












9434 


06 7 


9.36 4 


. 360 








54 






eisi 


- U7 S 


9.33 5 








00B48 


53 






9429 










12515 


69068 






- 0713 




■ 07 5_ 






__B_2!a 


3544 J 


_B717_B_ 






. 0743 


9421" 


. 0B06 


"97256 V 


r~47T 


8219 




96302 


fiO 




- 0771 


B418 


. 083 


a 230 8 


. S04 






93438 


49 




. OSOO 








. 533 


B211 


3B 3 








. 083B 


0412 


' 089 




. 682 


6208 










. 085B 
. 0Bfl7 


flB409 


. m\ . 


91.165 4_ 


_ AB.L 


~92bO 


2B 2 


8 80 84 


48 




27 2 


4S 




. OfllB 




'. 098 


o'lOfilfl 


'. 649 


6167 










. 0845 




- 101 


9.08 11 


. 878 












. 973 






9. OS B9 




9189 


28 


80632 


43 




- 003 


9393 




0-03 79 




8188 


28 X 


7B«?(i 


_il 




— HT 


9390 




3-00 63 


. 784 












. OBO 








. 793 




28 9 


7 254 


39 






B3S3 


: 115 






917B 


29 9 ■ 


7 480 


38 




; lie 


B3B0 


. 118 


e.B3 67 


: 861 


6171 


29 8 








^0*7. 


B377 




_8^Bi 20_ 


-_a80. 






















~91B3~ 


SO 7 


~6B308' 


35 




; zos 




: 127 


8^85 B 




6 IBO 


30 7 








334 


9367 


130 


8.84 5 


. 98B 












- 263 






.82 S 


. oes 




31 6 


63006 


33 




■ afl' 


" 9"357 


. 139 




■_ 024. 
. 063 


-gg^j*- 


__313_B . 


81287 
66 5 


31 




. 320 


30 




. a4S 


9354 


143 


; 5 2 


. 081 






57 a 












. 3 7 






3234 


6B 6 


28 










- 3 




96133 


3264 


54 7 


37 




- 4«G 


, 934_4_ 
9341 


-Ll51_ 


. S 


. 188 


96129 




51 3 






. B48 


- 167 




26 




. 491 










9B123 


3343 












. IBO 






96tl8 










: 692 


9331 


. 1B2 


. 989 


. 293 






48 4 


23 




- ^580 


9327 


■ IBS 


. 771 


■ 312 


99110 


3433 


44 9 


-ii 














89108 


3461 ; 








: ess 


9330 


: 171 


'■ 340 


: 370 






41 






867 


9317 




- 125 


. 369 




3531 


39 8 






690 


93U 








99094 


3660 


37 9 








~9S"07" 


. IBSfl" 






99091 


sm- 


38 9 
347 8 






." 7M 


-418B 


. 4BS 








. 78S 


9303 








99083 


3639 


33 






. 813 


9300 






- 643 


99079 


3669 








. 840 




. 1924 


.3882: 


- 573 














9398 


: IBM 


.sess 

. 449 


■_ BpO_ 




a768~ ~ 


2B4 3 
39873 






. 888 


9990 


96087 






. 837 


9386 




. 2 4 


: 658 


99083 


3737 








9sa 




. 3043 


. 


. flB7 












. 981 


9378 


. 3072 


. 8 7 




9B0S5 


3S4B 




7 




aoi* 

043 


_?_a7_fl_ 


^lOL 


- B G 
-2434 


■ '^^ 


.96051. 
99047 


_3B7B. . 


~19125 


B 










071 




: 2 60 


.23 4 








17 594 






. 100 


9285 


■ S ^° 








3BB3 


18071 






. 128 


9262 








99035 


3BB5 








■_m. 


_92S_a_ 


^?J.?- 






99031 


4024 






L 


■-"^ 


■s^ 


4^ 


^^ 


^ 


^ 


■^^ 


^ 


-^ 



244 NATURAL SINES, COSINES. TANGENTS, CX)TAN. 





in. 


Cos. 


Tan. 


C„t. 


in. 


C,>=. 


Tan 


Cot. 






91 




















94 












5B6 6 












411 7 






98780 


699 8 


29007 


Si 




00 


9901B 


411 7 


0705 


- 730 


98755 


592 6 


27829 


Si 




033 
406 


J9011, 


417 7 


0557 


- 758 


98731 


SB e 


26656 


SB 
















5) 




40fl 












60 8 


34321 







411 


98 98 


426 7 


0117 


. 945 


98737 


60 


:3ieD 


5! 




4148 


98 94 




9971 


. 873 


98732 




2003 


51 




417 


98 9Q 


T435* 6" 


98je_ 


__fl02_ 


98728 




0851 


ji 


10 


420 


98 BB 


9 623 




93723 


61 "" " 




SO 




423 


98 82 


113B 6 


9 38 


■ 959 


B8718 


ei 7 


8569 


11 


1 




98 7fi 




9 952 


. 98S 




61 6 


7419 


« 


1 


429 


















3_ 


4SS0 










98 704 






i 




4349 


"08 96" 




8968"" 




98700 


6 ~V 


4023 


11 


1 




98 Bl 












2899 




1 


4407 


SB 57 


455 6 








6 6 




« 


1 


4436 


9S 53 


458 B 


B547 


: 60 


BB6B6 


E 8 


0661 


« 


i_ 


4484. 


9B18 


461 6 


B40B 




9Beai 


6 05. 




A 


20 




98 14 








99676 




08144 


10 


31 




9B 40 


487 


8131 




98671 


B 65 


07340 


« 






98 36 


470 e 


7993 


. 275 


08887 


6 95 


06240 




33 




96 31 
















a* 




98 27 








98657 


6 6.5, 


04051 






4637" 


98 23 


479 ( 


758"3~ 


. "sbT 


98052 


6585 


'2962 




36 


4 ee 


BS IB 
















S7 










'■ 419 






0797 






23 




48B f 


717B 


. 447 


93638 


6674 


B7aO 


K 




.52, 


98 DJ_ 


_ 4fl_l_ J 


_70450_ 


■__416_ 


_9afl33. 


6704 




30 


30 


98629 




7576 




10 




497 € 






99924 


6764 


6510 


» 




39 


9889 


500 e 


66483 


- 663 


9B619 






a 


S3 




9B8B 






. 591 








n 


34_ 


4 90 

25 


"8B8B~ 




63831. 
62523 


~ 648" 




6854 


3335 


21 




609 6 


"986 dT 


6684 








BBS? 


612 « 


61219 


. 677 






123B 


11 


S7 














6 941 


OlBl 




38 






618 e 


5BG27 




98590 


6071 


9151 


i: 


39_ 


40 


9986 


531 i 


57339 


- 788 


.96565 


7004 


Bill 


Jl 


40 








58055 






7033 


7080 


« 


41 








64777 




98575 


7063 


6061 


1! 






8894 


30 C 


63503 


. 348 


98570 


7 B3 




11 




56 


9BB4 


33 ( 


62234 


. 678 


98565 




1001 


i: 


41 


_84_ 


9884 




60970 


^JQfl. 






2882 


.!! 


46 




9863 


39 


49710 


- 935 


98656 


7 83 


1966 




46 


41 


98S3 


42 fl 


48456 


. 964 


9B56I 


7 13 . 








70 


9882 


46 6 








7 43 








99 


9862 


48 8 


46961 


'■ 021 


BB541 


7 73 






49_ 


B3Tb" 


9881 


51 


44720_ 


■ _060_ 


_e853i 

96531 


7 03 




li 




9881 


540 6 


7 38 


8837 




5385 






42253 




98526 


7 63 


B9«l 


1 


62 




9880 


800 6 


41026 


: 7136 








1 


B3 


5142 


BBBO 


630 8 










8960 


' 


64^ 


-6471, 


_LB_7?_ 
9879 


e6Q_ 6 


38587 
37"371 




(18511 


7453 


72974 


J 






889 8 


. 7222 


68506 


7 63 


WM 


1 






9B787 


719 e 


36165 












67 




9BJB2 




34961 


'■ 7279 






0037 


1 


68 


5580 












J 73 


8081 


1 


B9_ 


_L815_ 


j.ms. 








_B84B_6_ 


_17 03^ _ 


8081, 


J 


£9- 


IBBM 
Cos. 


^^ 


:w^ 


31375" 


^ 


-^^ 


-"S-^ 


&- 


-T* 



NATDHAL SINES, COSINES, TANOENTS, OOTAN. 245 



■ 


Sn. 


c«. 


Tim. 


Co. 


Siu. 


Cos. 


Tan 




- 






; 1 


si 


;il 


gJi 48 
__6-63 96 

ill 

c; 5 77 


B1B5 


1.9,3 40_ 

■if 


ill 


I 
1 

J 


B 

jm 

08139 

05037 
04387 


00 

1 






8 


B iSb 

a us 


: S3 

. 843 

: B 


.■-931 - 
: 839 

'. Bse 


. 8589 

■Ji' 

J8B 
■ 9B2 




> 





Lii 

» 399 


-11 

ill 

B203 


08 07 

98080 
98034 

lii 


1 


i 


a 
a 

'7809 


6. 3 07 

iVi 


I'i 








ill 


: 18393 
48384 


i.lf. 




Si 

98037, 

lii 

797992" 

lii 


-2004 
^2016 




B8B40 






9 352 

9 339 
J_33L 

Hi 

B 304 


li i 
Hi 

5 37 05 


!l9IJ23 
-19851 


■.iiin 

20315 




E 






:!| 


19337 
-19B66 

.20051 


.20345 

■IS 


IVi 

4 8 013 
4 8 5B0 


^ 




r 


SIS 

: 8524 


B Z9B 

ii 

a" 272 

ji 

a 350 


-ISfld 

■ 877 

■ «80 

: 8B8 
-1895 


5 '33 87 
5 31 78 

ill 


: 20138 
.20186 

J03_36 

;Si 

20478 


11 
Ii 

'97910 

-87887 
.97881 


111 

. 20848 
.20670 

i 20770 
.20800 

20921 


" 






Hi 


. BIOS 




90769 




1 


;if 


IS 


IS 


!;ii 


-50535 


IS! 


\s 


J 


7 219 

ii 




s 

^ 


. BBSS 


ill 

W- 


. B287 

'■ B37S 
.1B«8 


6-18480 
5 ; 15258 


20649 


-97945 
.97839 


:iS! 




1 


^ 


«- 


JgL 


-'*^ 


-.a 



240 NATirRAL SINES, rORTXER. TAX0ENT8, COTAN. 





Bin, , C™. 


Tbq. 


Tot 


Bin. 


r-™. 


Ta«- 


Cot. 


L 





-2079 


-97818 


.2 268 


4- 04 3 


.22496 


7437 


TOBT- 




w 


1 


.3033 


.97809 


.2 336 


4. 7 1 


-32623 


74 




























3 








4' 84 2 








4 31430 


SI 


i 




"977^84" 


I "g- 


47 3 


326QB 




\2 ?09_ 


4-30BB0 


le 


S 


20B3 




-32637 


~^T 




6i 




.3063 


'97778 


3 38 


1. 4 8 


-33665 






4! 39734 








.97772 


-2 B9 


ill 






3 301 


4 ■29169 


U 


a 














-2 332 


4.28596 


51 


_B_ 


:3io4 

.3 07 


■97754 


i~i0" 


4. 8 5 


:237"78" 


_73_B_ 






Jl 






4.27471' 




U 




.97748 


^3 00 


4. 1 1 


.23307 




■ 2 424 


4. 26911 


1) 


13 






.3 21 






73 8 


-2 466 


4.38369 


a 


IS 


.J IB 










73 1 






a 


li_ 


A±f ^ 








;22B9J_ 




-2 647~ 


4^26239 


_u 






-97723 


r3~2" 


4. 5 2 




73 s" 


4. 24836 


a 


16 




.97717 


.3 43 


4- B 7 




73 1 


-2 678 


4. 24132 


u 












22977 








u 












.23006 






4- 23030 


41 


IS. 


:3i33 


,978Ba_ 


2 34 


4: 1 


33033 




-2 670 


4 22481 


-fl 


30 










-23063 






4. 21933 


to 




-3 38 






i. 7 6 










H 


S3 


-2 il 


.97680 


.2 35 


4. 1 


: 231 18 


73 Bl 


^2 782 


i. 2DS43 


11 


33 




.97373 










-3 793 


4.20293 




24 


-2 47 








:.33176 






4_LB7S6 


» 






















36 


.3 50 


.97861 


.3 17 


4. ISB 




7271 






as 


3B 








4. 563 




7264 


3 B85 




» 


37 


iz 65 








:2a 80 




2 BIB 


4^1 137 


N 




.2 SB 


.97643 


.3 108 


4^ 316 








4-1 800 


]] 


39 


.2 ai4 


.97BSO 


^ 13_B_ 


4 B93 




^7MA 
97337 


^3 BOB 




Ai 


30 


-23 45 


4-1 S80 


30 




-2 673 


.97628 


:33 00 


4: 461 


-23 73 




3 039 


*.! 997 


» 


82 




.97617 




4. 332 










a 


33 


i 729 


-97611 








97217 




4^1 934 


1 




-3 768_ 


-97604 






J3 SB 


_87210. 


134131- 


4-14406 






3 736 




.3 322"" 


"4; ~98e~ 








4 18877 


~3 


3« 


2 8H 
















21 


37 


.3 843 


:97BB 


'.2 383 




^23 43 


7 89 




4:12826 


1 






.B7B7 


.2 414 


4.4 155 


-33 71 


7 82 


-34 64 


4-12301 


1 


39 




.9757 


. 32444 


4 4 648 


23 99 


7 76 


24 85 


4.U77B 


J 




.3 933 


.B75fl 


.22475 
















.3 996 




-22606 


4 "44338 


-33666 


7 62 


.34 47 


4-10738 






2 939 






4.4 785 


23684 




24 77 


4-10316 


1 


13 


3 013 


:B754 


22667 










4-09898 




a. 


iO^L 


.9764 _ 


_2_35B7_ 


4 '42634. 


23740 


_ Till. 




4.09182 


J 




.2 070 












:24~70" 


4.DBeBB 


1 


u 


.3 038 


'.llll 


^22868 


4:41340 


-23 97 








1 


47 


.2 138 


-8763 


.22839 


4-40746 


23 26 


97120 


^34 32 


4:07688 










.32710 






97113 


-24 63 


4-07127 






ra~3i3" 


'b76D 
.9760 


.33731" 




■33 B3 


■ "7roo" 


..24_fl8. 
-24624 


4.08618 


J 


ST 


4-38983 










.2 240 


-B743 


.3:su 




^23 38 




.24665 


4-0SG99 














.23 66 






4-08082 




B3 




-9743 


! 33873 


4-37207 






■'24717 




I 


M_ 


iz 338 


.8747 


^3290S_ 


4-3flB23 






24747 


4:0*081 


_t 




.2 3S3 






■24061 




■ 24773 


4.03S7I 




GB 




!974B3 






-24 79 






4.08076 


1 


B7 




97457 


:22B9a 


4^34879 


.24 OB 




:24»40 


4-0aS74 


I 


G8 


.2 438 


.B74S0 


-23028 


4.34300 




7044 


-24371 


4.03074 




6L 


-2 487 


^7M4 


-330BB 


i_331.23.. 






.24902., 


4-01878 


_l 


60_ 


33496 




33087 




341B2' 


~ 7030 


34933 


-iiSiSl!- 


_J 




(Jos, 




T^r- 




"C^^ 


^iS^-c5tr 


^^T^ 





NATURAL SINES, COSINES, TANGENTS, COTAN. 247 





Sin. 


Cm. 


Tan 




■^in 


C<,^. 




i:>< 




f 

I 

1 


■ gMoa 

.24333 

awi8 

- 34SQ3 
.34531 

-3161B 
. Z4644 

'24726 


IE 

-97001 


ill 


'0053 
.0003 


ill 

^26022" 
:2ei35 


'S 

J8182_ 
96524 


2679S 

26920_ 

■ 2 044 
a 076 


.7 205 
. 771 

■ 1 

la 

7217_ 


si 


Ml 


5' 
5 


1 


^eBflS2 25273 
.8BB45 .2B304 
»eB37 .29335 
96930 '.253S6 

98884 1:2 521 


-98 b; 

^84 3 


■ii 


He 94 27201 


4 
4 


1 


9279 
9231 
9183 


is 

:26415 


^98471 
98483 

-98448 


11 

:2738a 


4 

4^ 


i 

s 


■ii 


99387 2 55S 
:9BS73 :2 B14 

968B3 '-25675 
-96SS1 1.25707 

96844 1.25738 

96322 ■ 25831 
98815 '.25882 

98800 ^25924 
96793 1 25955 
9678B .25981 


9136 
:B994 


. 26443 

26500 
2B52B 


9 6440 
.98433 

.98410 
96402 
98394 

B637B 
_9J37L 

96355 

98308 
98301 

96248 


27 51 

II 

: 27807 

: 27870 
37701_ 

:277e 

-3782 
-2785 _ 

2796 
2798 

jaoi_ 

:ii 

If" 

ii" 
JL 

jli 

28 6 75 


ii 

3 B304B 


3 




IE 

35010 


. 89004 

IB 

: 8 !3fl 


26584 

26688 
28896 

26752 
2678Q 

_2fl.B36_ 

26920 
28943 
2697F! 

IS 

II 

27 88 

27 2^ 

27 08 
27_3J_ 


: 62224 
80996 


3 
3 




ill 

2S161 


85745 


.5 57 
5 67 

:5 61 
-54 BB_ 

11 
II 

-50868 

^49509 

49125 

3. 48741 


2 
2 


r 


IE 

IS 


98771 
88784 

98749 
96742 

98 90 


23017 
2B04B 

28288 


■IE 
. liiL 

:81177 
.: -80376_ 

ill 

"77595" 
3640 


2 
-| 

1 




asiao 

III 
II 

if" 

25SS4 


26359 


i 


» 


98 80 ■ 285li 

96 45 ' 28577 
96 38 .28608 
98 30 28839 

ii -ii 


BB206 

96182 
98174 


i 

8 




98142 

_9B134, 

BB12^ 

Sin. 


5 

t 


¥■ 


^l#- 


-S^ 




^ 



:MS N,V|•na.^T, sinks, nnSlNRS. TANnRNTS. COTAN. 



-, 


Hi,>. 


'<:.■.. 


■r...i. 


,,„ " 


Si„, 




T.n ■ 


~~t~ 


T 


~ 




■ UUl^lH 


ait«TB 




-a9M7 -958 


-057r 


3. 27065 


"m 


1 




attllH 


JII70(( 




.38385 ,.flSe 3 


0606 


afl745 


a 


■J 




.aeiiu 


anTSB 






0637 






s 




.B«ll)J 








066B 


-26067 






gyaTR 
















'« 




■ HMri 


jetisj 




-asm" -85588 


" 0732" 


-29393 


"ss 


B 




.»Mt7B 


i»a«A 




JB404 1-35679 


07B4 


.35016 




J 




.»II071) 


JIRSS 












a 




.a«MJ 






: 1*9480 !»5583 












.VMM 






29487 ..BS6B4 


ISOBBO 


- 34049 


_ii- 




:>TIU 






s iSSfl 


.39516 |.B5545 


OBSl 


"2S7U 




1 


arin 






.39543 1-95536 














-iMSS 




.23871 -85528 














JWW 




396»9 I.BS518 


0887 


.22716 


17 






jtaii 


_.'»ii< 




:?eM.!^9S511_ 


^018- 


_^3S3B4 






■JTfS 






i.4MW 




.8BS0I 










vikut 




;»17* 


S «7IS 




.■MM 


83 


31733 






WW) 




JH.-IO 


J 4^'SU 




IMS. 


15 


.21392 


13 


■ 


j*>>«? 




.'V.'tJ 


341*71 


»T3T 




47 








^iHiU. 




-•<.'7* 


i AltiH 


-3tni 


-9^*87 


it « 




-11 


ti 


Jul .'it 


ii.Mt 


■753m" 


«.41JJ« 


.MTM 


.»M5B 


. 1 10 


.3D406 


40 




.'aix* 




.•Mi 7 


f.4M«# 


.m2i 






■ .<(K.79 


!) 


-' 


;«;.-i 


*i**t 


.-#$#1 


J.««OJ 


-nca 


95*4. 


1 74 


. B!«2 






.■».'.'* 


#M>40 


.■■94.V 


*.4i<li6 


.£M7S 


.9Sil3 


1 OB 


. B42B 




* 


.■*.■»* 




:«.« 


S MTTl. 


^m*- 


JMM. 


1 38 


liM, 


Jl 


« 


;*.■*.• 


*Mi.-i 


.;*iM 


5 5*4,''* 




.9S41S ! S1S70 


- B77B 


85 




.*.■*.' 


*^*•.J 






hi«o 


■ MOT ! 31403 


. B4fil 


H 




:tr.t 


*Mi': 




S Mf-S 


»MT 


-tSSM 3i*a* 


- 6127 


s 


it 


-■MM 


fMM 


:*J-'* 


J sm:: 


-HOlf 


.»s*t , gues 


7604 


32 




:m ■» 






J J-ii5J 


.WW- 


:^Ajjim^ 


_74«- 




**' 




*>**-■ 


.■*j.-. 


J J-ff4 


38071 


.M*T1 -31530 


. 71 5» 


30 




I-fi.-^ 




.-»?->* 




MOH 


.»sa iiitJ 


. eess 


"■ 






*i*J; 


.-;;** 


j JM-5 


WIN 


.|S*H SISM 


. 8617 


a 


JJ 


}«*-• 




•*":* 


' ';';• 




.•SHI -aioa 


. 8187 


1 








:i-it 




»LM.'J»»L JifSL 


. ■ 5B77 


J! 


*> 


.;*>*; 








s«» 


tSSSI 3100 


5S58 


« 


>» 








i >Mtj 


»»T 


fSUt 31723 


6240 






I'JM" 


f;«:i 






sraH 


■SSIO 31TS4 


- 923 


a 


f* 




*..*:* 




* ji-*.- 


MSU 


ISNl ' JITM 


805 


5? 


^ 




*.•*•■ 


^± 


i s««" 


a^B 


WW sifi» 


;m. 


Jl 










.( .«.;:! MM) 


•SSM S1B5Q 








>*'-.* 


A-^: 




J JJfV MOn 


»ITS IIWI 


6S8 


ij 






if -f: 




.<<<<:- MM 


MSW i-tlBl« 


. 341 








*■■■:■'* 


; .<.vs> SMH 


^^ SIMC 


037 


19 




;-»"': 


U-A 




f f-r* .jMRL 




»■» 


ans 


JJ 




.-*»■.• 




J.W 








. MOO 


If 




.•«».■ 




*■::•"* 








■ 3017 






«.5 












- 7K 


11 


*f 


if A': 


f': f: 










,1 


« 


•fits 














1^ 


J^- 


t^'t 














■ 






j;;;" 


s'-i'j 


< vjv jSS 








1 


y^ 


y.y^' 


«ju' 


:':>r: 


f ivt* JSSi 








1 


*, 


:■•-.;: 




*^; 


I "?.X- jSS 


jgji^S 


.'«S 




i;. 




.-■--aw 


*:■•? 


■' "■'" jSSS 




S.aMM 




K 












9 -own 




4^ 


-fr- 


"c"- 


^^;-- i 




'rr:^ ^ 


».|TW8 


J 



NATURAL SIOTS, COSINES, I'ANGENl'S, COTAN. 249 



, 


Sill. 


Cos. 


Tan. 


Cot. 


Si.l. 1 Co.. 


Tai,. 


Cot. 


, 




30903 


9S108 


32492 


3-07763 


-33557 








eo 




3D939 
















69 




S0flB7 










. 9153 3 






BB 




iOWb 




.33sas 




32639 


. 91523 


'■ 34530 3 




67 




31012_ 

aioio 


J 5070, 


aaeaL 


3: 06551. 


-3266_7_ 


.94614 


j;1663_ ^ 




_» 








IS 




31068 


05053 












8788 




310B5 


95013 


: 32717 


3:06649 


:3274B 


:9418S 


'■sieei 1 


3511 


Ga 




.31123 


95033 


.32749 


3.0S319 


-33777 




.34693 2 


3240 


53 


_ 


.311S1 


9 034 


327Ba. 


3-05049 


^2_80t, 


91466 


-34738 2 


7B_7i 


-61 




sina 


.9 015 


aaaTT 


3. 04749 




.914B7 


-347BB 3 


7700 


60 


L 


siaoe 




.32318 


3. 01150 


:32859 


.91447 


.31791 2 


7430 


19 


3 






-32873 


3-04162 


.33837 


91433 


31834 : 


7161 


48 


a 


312fil 


is OSB 


-32911 






. 91428 






17 


4 


S1389 


.a 870 


.33913 


3:035S6 


! 32912 


^91419 


318S9_ : 


6624. 


46 


I 




.94970 




3 03260 


-'32989 








45 


S 




.9496^ 


33007 


3.02963 


.33997 


:9i3Ba 


; 34954 i 








51372 


33040 


3. 02667 


33024 


.94390 


.34987 2 


6332 


43 




-31SB9 








33051 




-35020 3 




42 


ii. 


JHJL 


! 94933 


.33101 


3-02077 


■33079 


-94370 


,36052__2 




_^ 


Tm 


ll«t 




.33 a 




.33106 


.01361 


.35085 : 


6023 


40 


1 
















1768 


39 


i 






-33 1 


aOl 96 


33161 


: 91342 


.35150 i 








.94997 


-33 3 


3. 00 03 


33189 


.94332 


35183 2 


1329 


37 


« 


-31 es 


.94B8(I_ 
-94878 






-3324r 


:94313"" 


-35218 3 
.35346 2 


3703 


J^ 




-91S93 


-332 8 


3-00 19 








948B9 


.31330 


3.00023 


.33271 


-91303 


.35281 '. 


3139 


34 


^ 


51848 


94860 


33363 












33 


.31676 










94284 


^35346 '. 








S17D9 






_2J91i_ 


333_53_ 


.94371 


.35379 _2_ 






^ 




"sisaa" 


33490 












30 


■ i 


■ a 7M 






2. 985 


. 34DB 


: 91354 


.3541 i 






ii 
It 


.a 788 




33521 


2 982 2 


3439 


-94215 


.3547 : 


19 


28 


3 13 


B4S05 


.33557 


2. 980 1 




-91235 




18 


27 






33589 












.Tg 


i 

11 


.3 ea 




33621 


3-B"43"o" 


^a35ir 


-B4'2Tb" 


-3557 2 


10 I 


25 


9 OG 


9i777 


33654 


2.B 4 




.94208 


.3560 ! 


OB 3 


24 




91763 


33686 










06 4 




3 51 








: 33600 


^91 88 






23 


i_79. 

■ a OS 


9*749 


3375J. 


2 9 8 


-33627 


91 78 


38707 S 




_2I 


40 


94740 


ISTST 














s 


-3 34 


94730 




2-B 1 








95 5 




a 11 






SB 7 




^91 17 


^3580 2 


92 9 


la 


1 


-BMSS 






2-9 5 


: 33737 


.91 37 


.3583 2 


90 3 


17 


32 44 


J470_2, 
04893 


33913 


2 9 2_ 


.33^fl4_ 
33792 


9_1_27. 
91 18 


3537 S 


37 a 




s 




85 3 


)B 


■33 71 






2^94 09 


33919 


-94 08 


J593 S 


92 9 


11 




33" 


94S?4 


3401D 


2-94 28 










13 


-32 37 


94965 


34043 














_33_5L 


OJiaSB. 


3407S 


2-B3 BB 






^035_ 2 




H 


s 






34108 


2-93 99 




Tgi 69 


.36068 2 


"77354' 


to 




94937 


31140 








.36101 3 


77002 


9 




94637 














a 


H 










31011 








T 


iOi 




31238 


_2l9_2_7_6_ 


31038 


_?4p29_ 


:361B9 S 




-J 


i 


37 19 


94599" 


31270 








36232 2 


"75998 




32 47 


B43B0 














4 


37 74 
















a 


33 03 






2 90971 


-34]47 


.93989 




76246 


a 


i7 2B 


94581 


_31400, 


2.90B9S 


-34176 


93979 


36381 2 


71997 


_l 


!!«. 


■^ 


^e^ 


31433 




34303 


^e^ 


-T^- 


S^ 


-72 



250 NATUl'.AI. SIXES, COSINES, TANGENTS, COTAN. 





Sin. 


Ln^. 


Tm,. I 


Ol. 


Sin. 


C-OH. 


Tnn. 


Cot. 










36387 2 




583 


93353 


.38386 2 






1 


: 34229 


9 95 


36430 2 


4489 


586 


93348 








2 


.34267 


8 94 




4 51 








60057 


58 


3 
















58831 


67 


.4^ 


:34311_ 
.34339 


9 92 
9 91 




3 5fl 




_8_33ie. 


;38530_ ._2 


59608 


58 


6 


36583 2 


3 09 


597 






.34368 




38596 2 


3 03 








59156 




















58932 






: 34421 




38601 5 




606 












.3444a_ 
.34475 


AA7 _ 


^6fl84_ S. 








^3a6_67_ i_2_ 








3BJE7 ; 


a 81 






.3B731 3 


58361 






.34503 


8 as 




2 30 




9 43 


38764 ! 


58038 




n 


34530 


9 84 


387B3 




816 


9 32 


38787 2 






13 


.34557 


9 83 


30S23 


1 48 


. eiB 


9 23 




57593 






345B4 


9 i2 
B 61 


J6a5_9_ __ 


1 05 


■ 631 


.9 11 
.9 01 




67371 




iT 


34012 


36BBS 






-38886 i 








3463 B 


9 309 


30926 


19 


'. 027 






5B933 










36958 


77 




's 30 


-38855 i 


567 7 




la 


-34084 


9 78B 


389B1 






.9 69 


-38986 I 






19 






37031 


70 B4 


- 6352 






































BBBSa 




.8 46 


-39055 2 


560 8 




21 


:a4775 


8 75 


37090 






.9 37 


-39089 E 






S2 


-3400a 


9 74 


37123 


6B371 


6434 


-9 27 




5BB 8 








8 73 




69131 






:3B156 : 


653 B 




34. 


^B57_ 

.3 384 


J J.2 _ 


37180 i 


86653 


"e"5l5" 


'S 06_ 


.39_19i J 


661 








548 3 








9 70 


37356 






■ a 084 


39357 


647 4 






.3 63B 




37289 






.9 74 


.39290 








.3 9Bfl 






67917 


. 6596 


.9 63 












8 87 




6770^ 


. 6623_ 






540 3 




3U 


T"02l' 




37383 


67402 


. 8B50 




39391 


638 5 


: 


31 


.a 04a 








. 6677 














9 64 


37465 






6 20 








33 


:3 102 




37488 


6B7B2 




93 10 


-39493 






3L- 


■ 030. 


9 e_2_ 




6851S 


0756 


.B3_9_S_ 
.93 88 


-39628 










9 61 


37554 


68261 




8B669 








i 84 


8 80 


37B88 


66040 




.92 78 


.395B8 
















683 




.3B«3a 








-3 as 


9 38 


3 664 




8887 












■3 26t 


9 67 


3 flR7 


6S343 


38884 


: 83945 


38694 


6 8 B 


" 






B 58 






8E3 




39727 


S 7 5 








9 59 


3 7M 














42 


■ S 347 


S 544 


a 787 : 


8 843 


eB7 


B2 13 




6 3 8 




43 


-3 375 


9 53 


a 820 : 


8 410 


700 


93 02 


-38629 


B 6 




44_ 


.3 102_ 
3 429 




3 663 


6 177 


^ '259, 


BS 92 


-39_892, _ 


5 864 






9 51 


3 887 


6 945 








a 


3 458 


9 50 


3 920 


6 714 


708 


92 70 




60440 














711 


92 59 


39863 2 


B022B 










3 980 








39997 2 


SOOIB 




9_ 


-35505" 




38053" 




718 


"fl"2"37" 


40031 2 








B 48 


62781 


71B 


40085 2 


"48697 








9 45 


36060 




791 




-40088 2 


49386 


















.40132 2 










a 43 


38153 




727 














_BJ2_ 


aaias 


61874 


_'2J_ 


B2_84_ 
93773 


:4020o a 


«87£8 






86701 








.40384 a 


4aUfl 


















.40967 3 


48840 










38388 




': 738 




40301 9 






a 


^35783 


a 37 


38320 




740 


83740 


40SSB 3 


fi&i 




J_ 


35BK)- 


J- ae . 


_3I353 _ 




743 




-40.888 ■ ± 


imi 




02. 


^ 


-•■a^ 


jgLi 


1^ 


-¥S^ 


^ 


4040a 3 


«* 


■^ 



NATURAL SINES, COSINES, TANGENTS, COTAN. 251 
Z2^ 23' 



f 


Sin. 


Cob. 


Tan. 


Cot. ( 


Sin. 


Cos. 


Tan. 


Cot. 


$ 


o 

1 

2 
8 

i 


87461 
.87488 
.87616 
.87642 
.87669 


.92718 
.92707 
.92697 
.92686 
.93675 


.40408 
.40486 
.40470 
.40504 
.40588 


2. 
2. 
2. 
2. 
,2 


47509 
47302 
47095 
46888 
46682 


39073 
.39100 
.39127 
.39153 

39180 


•92050 
.92039 
.92028 
.92016 
•92005 


•42447 
•42482 
.42516 
•42551 
42585 


2. 

2. 

2. 

2 

2 


35585 
35395 
35205 
35015 
34825 


60 

59 
58 
57 
56 


6 
8 

T 
8 
9 


.87695 
.87622 
.87649 
.87676 
.87708 


.92664 
.92658 
.92643 
.92681 
93620 


.40572 
.40606 
.40640 
.40674 
.40707 


2 
2 
2 
2 
2 


46476 
46270 
46065 
45860 
■45655 


•39207 
•39234 
•39260 
•39287 
•39314 


.91994 
.91982 
•91971 
•91959 
.91948 


.42619 
•42654 
42688 
.42722 
.42757 


2 
2 
2 
2 
2 


84636 
34447 
34258 
34069 
33881 


55 
54 
53 
52 
51 


10 

11 
12 
18 
14 


.87780. 
87767 
.87784 
.87811 
.87888 


.92609 
.93598 
.93587 
.93576 
.93565 


.40741 
.40775 
.40809 
.40843 
.40877 


2 
2 
2 
2 
2 


.45451 

45246 

45043 

.44839 

.44636 


•89341 
•39367 
•39394 
•39421 
.39448 


•91936 
.91925 
.91914 
.91902 
.91891 


.42791 
•42826 
.42860 
.42894 
.42929 


2 
2 
2 
2 
2 


33693 
33505 
33317 
33130 
32943 


50 

49 
48 
47 
46 


16 
16 
17 
U 


.87866 
.87892 
.87919 
.87946 

JZ2ZL. 

.87099 

.88036 

.88068 

.88080 

.88107 


.93554 
.92548 
.93583 
.92621 
.93610 


.40911 
.40945 
.40979 
.41013 
.41047 


2 
2 
2 
2 
2 


•44433 

44230 

.44027 

.43825 

43623 


.39474 
.39501 
.39528 
.39655 : 
.39581 ! 


•91879 
•91866 
.91856 
•91845 
91833 


.42963 
•42998 
.43032 
•43067 
.43101 


2 
2 
2 
2 
2 


32756 
32570 
32383 
32197 
32012 


45 
44 
43 
42 


90 
21 
22 
28 

94 


.93499 
.93488 
.93477 
.92466 
.93455 


.41081 
.41115 
.41149 
.41183 
.41217 


2 
2 
2 
2 
2 


.43422 
.43220 

43019 
.42819 

42618 


•39608 

■39635 

.39661 

39688 

39715 


•91822 
•91810 
•91799 
•91787 
•91775 


•43136 
•43170 
.43205 
•43239 
•43274 


2 
2 
2 
2 
2 


31826 
31641 
31456 
31271 
31086 


40 

39 
38 
37 
36 


26 
26 
27 
98 


.88184 
.88161 
.88188 
.88216 
.88341 


.93444 
.93483 
.92421 
.92410 
.93899 


.41351 
.41285 
.41319 
.41353 
.41387 


2 
2 
2 
2 
2 


•42418 
42218 
•42019 
.41819 
.41620 


89741 
39768 
.39795 
39822 
39848 


•91764 
•91752 
•91741 
.91729 
•91718 


43308 
•43343 

43378 
•43412 
.43447L 

•43481 
•43516 
.43550 
•43585 
•43620 


2 
2 
2 
2 
2 


30902 
■30718 
30534 
30351 
30167 


35 

34 
33 
32 
31 


80 

81 

82 

88 

84 


.88268 
.88396 
.88833 
.88849 
.88876 


.93388 
.92377 
.92866 
.93856 
.93348 


.41421 
.41455 
.41490 
.41524 
.41553 


2 
2 
2 
2 
2 


41421 
•41223 
•41025 
.40827 
.40629 


•39875 

39902 

.39928 

.39955 

39982 


•91706 
•91694 
•91683 
.91671 
•91660 


2 
2 
2 
2 
2 


29984 
29801 
29619 
29437 
■29254 


30 

29 
28 
27 
26 


86 
86 
87 
88 

Si. 


.88408 
.88480 
.88466 
.88488 

jmsL 

.88687 
.88564 
.88591 
.88617 
.88644 


.93883 
.93831 
.93310 
.93399 
.93387 


.41592 
.41626 
.41680 
.41694 
.41728 


2 
2 
2 
2 
_2, 


.40432 
.40235 
.40038 
.39841 
39645 


.40008 
•40035 
■40062 
•40088 
•40115 


.91648 
.91636 
.91625 
•91613 
.91601 


•43654 
•43689 
•43724 
•43758 
.43793 


2 
2 
2 
2 


■29073 

■28891 

■28710 

28528 

28348 


25 
24 
23 
22 
21 


40 
41 
42 
48 


.93376 
.93365 
.93354 
.93348 
.93381 


.41763 
.41797 
.41881 
.41865 
.41899 


2 
2 
2 
2 
2 

2 
2 
2 
2 
2 


.39449 
39253 
39058 
38863 
38668 

■38473 
38279 
38084 
37891 
37697 


40141 

•40168 

40195 

40221 

.40248 


.91590 
91578 
91566 
91555 

.91543 


.43828 

43862 

43897 

•43932 

•43966 

.44001 
.44036 
.44071 
•44105 
.44140 


2 
2 
2 
2 
2 


28167 
27987 

■27806 
27626 

. 27447 


SO 

19 
18 
17 
16 


46 
46 

47 
48 
49 


.88671 
.88898 
.88735 
.88762 
.88778 


.93330 
.92209 
.92198 
.93186 
.93175 


.41983 
.41968 
.42002 
.42036 
.42070 


•40275 
40301 

.40328 ; 
40355 

•40381 


.91531 
.91519 
.91508 
.91496 
.91484 


2 
2 
2 
2 
2 


■27267 
27088 
26909 
26730 

■26552 

26374 
26196 
26018 
25840 
25663 


15 
14 
13 
12 
11 


50 

61 

62 

68 

64 


.88805 
.88832 
.88889 
.88886 

.88912 


.93164 
.93163 
.92141 
.93180 
.93119 


.42105 
.42139 
.42173 
.42207 
.42242 


2 
2 
2 
2 
2 


37504 

37311 

■37118 

■36925 

36733 


40408 

40434 

•40461 

•40488 

.40514 


•91472 
.91461 
.91449 
.91437 
.91425 

.91414 

91402 

91390 

.91378 

.91366 ; 

.91355 : 


•44175 
.44210 
.44244 
.44279 
.44314 

•44349 
44384 
•44418 
.44453 
■44488 


2 
2 
2 
2 
2 


10 

9 
8 
7 
6 


66 
68 

67 
68 

69 


.88989 
.88966 
.88998 
.89020 
.89046 


.93107 
.93096 
.93086 
.93073 
.93062 


.42276 
•42310 

.42345 
.42379 
.42413 


2 
2 
2 
2 
2 


■36541 
■36349 
■36158 
■35967 
35776 


.40541 
40567 
40594 
40621 
40847 


2 
2 
2 
2 
2 


■25486 

25309 

■25132 

■ 24956 

24780 

24604 


5 

4 
3 

2 

1 


fiO 


.89078 


.93050 


•42447 


^ 


■35585 


•40674 ' 


44523 


2 


O 


^"i— 


6o8. 


Sin. 


Cot. 


Tan. 1 


Cos. 1 Sin. Cot. Tan. 


t 



er 



M" 



252 NATITRAI. SINES, COSINES, TANGENTS, COTAN. 



. 


Sin. 


Co3. 


Tan. 


Cot. 


Sin. 


Ccs. 


Tan. _ 


Ot. 






10671 










.90B31 








I 


40700 


- 1S43 








-90E1B 










10727 


- 1331 


-44693 


3:24252 










68 


3 






.11627 


3-21077 


12311 


: 90591 


1873 3 


963 


BT 




_4078Q_ 

iDSoe 


J 307 


■44883 


3.23903 


42397 


_B0pa_3_ 
90569 


4877 2 


801 


_5e 


"6~ 


















40B33 










.9QB67 


46843 a 


477 








. 1372 


.11797 




42448 


.90615 


1887 3 


318 


53 


S 




. 1280 


44803 




13173 


.90633 


IBBl 3 


151 




A. 


"imsb" 


7"l23B 


11B37_ 
.44873 




_42499_ 


J0_63D_ 


4696 3 


993 








.489 3 


B32 


"so 








.11907 




12563 


■ 90196 


170 2 


671 




12 










43573 


90183 


470 a 






18 


11019 


- 1200 




2.23337 




90170 








1* 


_11045_ 


. il8«- 




2.23181_ 






471 S 




_M 








.16047 


3.21993 


~m57~ 




471 2 


2030 




IS 


llOSB 


-31 61 






136BS 






1871 




17 


41126 


.91 53 


^15 17 


3^21817 


.12708 


9012 


472 2 


1711 


18 




msi 


-91 40 


.15 53 


3.3117 


43738 


BMO 


473 2 


1652 


12 


18- 


tll7B 


J1_1L 






^??3 


oom 


jn — a. 


;?93 


-ii 


ao 


4J201 


.91 16 


-45 22 


2. 1 3 


4 788 


9038 


1734 2 


1233 










.45 57 




4 BIS 


9037 


4737 2 


1075 


3S 










i: 9 








0916 


38 


23 








2. I 












21_ 


liaio 


'9ioe8_ 


:i6 63_ 


3. 0M_ 


_42B91_ 


80331 


Al^ - _2 


0800 






41337 


91056 


.46397 










0142 


'Js 


36 










43948 


90309 








3J 


41390 






a. 993 


42972 


90290 


.4769 a 


0138 


33 


SB 


11416 


.91020 


-45502 


3. 978 




00281 


.47828 3 


8968 


33 


2S 


41413 


.91008 


-4553_a_ 
.15573 


3. 960 


43035_ 
43051 


902 71 
9025B 


,1718a_ J. 
178B8 2 


^18II_ 
09361 


_31 


30 


11169 


.90599 


3. 943 






41499 




.45 OB 








7733 a 


09498 
























S3 








: 9B2 


43130 


: 90321 


17805 a 


09184 




S4_ 


11675 


^90948, 


15 13 


. ^875 ^ 


_*_ai5e_ 


B020B 


_4781_n_ _2_ 


0902B 








.90938 














~35 










■ 841 


43200 


: 90133 














16 19 


- 33S 


43235 


90171 


47948 2 


08560 






1 631 


90899 


45 51 


. BOB 






17981 3 


03405 


2 


39_ 


4 707 


30987 


45 89 


791B 


43287 


90148 


48019 3 


08350 


_1 


40 






45 34 


- 771 


43313 


aoi33 


18055 3 


08094 




il 


4 760 




.45 60 


. 768 






48091 2 


07939 


9 


13 


1 7B7 


90851 


.46 95 












B 














' 90095 






7 


« 


"l 86b" 


TaoBir 






13_418_ 


_900B2. 


18198 2 


07478 


_B 


4S 


.48 01 






07331 


6 






.90803 














1 












43 97 


90015 




07011 




48 


4 945 




:ie 06 


3' 642 


43 23 


-M0S2 


48343 3 


06860 


2 


49_ 


11972 


;90_7J_B_ 


^fi._42_ 


2. B,255_ 


43 49 


-B0QI3 


48378 _2_ 


08706 














1^ 








■lfl313 


I'. 6926 


43802 


: 89994 


18450 2 






63 


43051 


.90729 


.43348 


2. 6760 


48838 




48488 3 






53 


12077 


.90717 
















M_ 


43101 


90701 


jeii3_ 


_!: 6432_ 
2. 5268 


T370r 






M7fll 


_-8 




43180 




■16454 


-89943 


48593 3 


S 


SB 


12156 


: 90880 


-46489 


a- SiOl 


43733 


89930 


48B3B 3 


35687 




67 


43133 


.90883 






4376B 


.89918 




)5485 




SS 


















2 








.486?6_ 


W 


13811 


89893 






^ 


^ 


Cat. J 


^IP- 


-■"^ 


43B37. 


.89879 
Sin. 


-^c^ — 


S^ 


-^ 



NATDRAL SINES, COSINES, TANGENTS, COTAN. 253 



. 


6iD. 


c™. 


Tun. 


Cot. 


Sin^ 


Cqa. 


Tan^ 


Cot^ 


. 


o 


48887 




-48773 






■89101 


60953 


-98261 


fto 


i 


'S% 




.18808 


a' 01879 


■45125 


-89087 










.48845 


2.01729 


■45151 










» 


■ 48UB 
















67 


A. 


-jam. 














'95898 


_S6 




.13968 




.18953 


2.04278 


T5529" 


Tb9035" 




.85567 




t 


.43884 




-13989 


2 125 


.45551 


.89021 








7 


.44020 






2.0 975 






-61 08 




63 


e 






! 19063 








-61 46 


: 85137 


52 




i*.i072 
.MOBB 


^ stbT 


^9098 




15632 


;8Bm_ 


6! B3 


__e4Be7_ 


51 


10 


2-0 536 


50 






. B739 


■49170 


2^0 37B 










19 


u 


;t4ifli 




.19206 




'.iiv 


; 68912 


■51393 


:8157a 


IB 


IS 


.14177 


'. S713 


.19242 


2.0 076 


■ 457 6 


-6892B 




. 94140 




1* 


.44203 


. S700 


.18315 


_2_-.0 929 


-157 2 
■ 157 7 


■88902 


61 03' 


. 8430 J 
.811B2 




16 




. BB87 


~15 


IB 


;J^ 


■ 9674 


.49351 


2:oa63i 


.158 3 


-88888 


51 40 


.94023 




17 


. SBSS 


■ 49S8 


a ■ 02183 












!L 


:|^ 


A 


iss. 


2 ■02335 
3 02187 


lis 1 


'8BB48 


1 i 


! 93603 


i 


M 




■ 9838 


■4919 


3.02039 


■1B9I7 


■88835 




-83170 


40 




.413BS 




.4868 


2^01S81 














.41111 


: 9597 


.1856 


■01718 


!459SS 




761 


: 831 95 


SB 




.Mia7 


. 8584 




2-01S9B 


45994 


-B8785 


798 


-83057 


37 




■44461 


. 9571 
.8955 


-19B r 


2.01119 


^1J020_ 
■4604fl 


-88782 


1872 


. ,9^920_ 


3_6 






88 






.3961 


.487 a 


ioiiss 


-18072 


:687 5 


1809 


: 92615 


81 






.8963 




.01008 


.46097 


■ fl87 1 




■82508 








■ 1961 


!497 B 


: .00882 
















.»BD 


.188 2 




46149 












u/ao 


.8919 


.49869 


"'. TOOSBO" 


46175 


T8701 


"2067 


'~92098" 


30 




■**& 


.8948 


















.^n 


.8948 


















.UBBS 


.8946 


:i9987 


■-00131 


: 16352 


:8e 61 


2168 


.» 890 


27 




:M731_ 


■IHi . 


. SOOOl 


._999Ba_ 


^627B_ 


,98 171 


2205 


9 651 


28 


8i 






se 


: 14778 


:B911 




:9aB95 


: 46330 




2279 


ifl 282 




87 


.11802 


.8940 


50113 


99550 


-463 5 


-86 07 


2316 


■ 9 117 


23 


ss 


-11B3S 


.8838 




.99408 






2353 




23 


«_ 


■44BM 


■M"B 


!501SS 


_e926L 


'lai 7 




_2390. 


^9 976 


_31 


to 


.44880 


.BB3B 


-50222 


.89116 


-iSi 3 


■ 86 66 


2427 


-90711 






.14806 


.8936 


.50258 


.BB972 


■ 484 B 


88 53 






19 


a 




.88387 
















a 


.88391 


: 50331 




:48S 






-90837 




44 




■89811 


.50388 


-88 540. 


-4B6 6 


-88 12 
■68498 


2578 
2913 


_B02PB_ 


18 


is 








T5 


u 


■fflJJ 
















11 


17 




^50177 


:68UO 


'.t«e 3 








13 


48 


■tsota 




.50511 


■B78BB 


-486 9 


,88458 


734 


-68867 






.4B1M. 


■■B3« 
.8B23Z 


-50_5Bg_ 
.60687 


■87823 
■87681 


■ 16B 


■ 8BH5 


__L81 


88533 




Bii~ 








10 


61 




.8B219 


■60823 


■97538 


.187 B 




83 


■89266 


8 






.8BZD6 




■97396 


■ 487 2 


88404 


287 


.89133 


8 


9S 










.187 7 


.8B380 


91 




7 


M 


J^u 


: 89180 


: 50733 






.38377 






_a 


H 




.69187 


-60788 


■99968 


"i'is 9 


88383 




^8731 


s 


H 




.19163 








-83519 


3022 


88602 


1 


BT 




-89140 














3 


tl 




. 81 127 
















N 




.88114 


5091B 


^96402 












SB. 


4MM, 


89101 


; 50958 


I ■96261 


«6B^7" 


-BBiBS' 


5317r 


1 88073 


Za 




^r 


-si^ 






Cos. 




Ci,^ 







2r>4 NATXIRAI, SINES, COSINES, TANOENTS, CX)TAN. 



, 


Sin. 


c™ 


T^n. 


Ci.(.. 


Sin. 


r,«. 


Tun. 


Cot. 




-o- 


46847 












431 


804 5 


"bT. 


1 


49973 


! 88281 










469 


-902 1 


69 


3 


4eBSB 


.88287 


53248 






^87434 


607 






8 




.88254 


-53283 


. 7377 


.48567 


87420 


545 






JL 


A70_B0_ 
.47070 


_B8240. 


^63^20 


7546 


.48683 


-S7408 


mT 


"~~7 7"S8~ 




B 






"bs 


e 


.47101 


8SS13 








: 87377 


860 


7 669 




1 


.47127 


.83199 


.63433 




.43659 


.37363 












83136 


634S0 


- 021 


49894 












■4J17B 


89172 


^350X 




jano^ 


■8733X 




_l7 296_ 


Jl 





.47204 


8B1B8 








-87821 












.89144 




. 6B30 


4876 


















. 6 98 


.4878 








48 


3 


: 47231 


'89117 










'. 926 


■ 7 807 


47 


!L 


.47306 


88103. 
.B8089 








:872fl4_ 


B64_ 


^7_e85_ 


4B 






.53732 


- e oo"" 
























44 


7 










■48913 


: 87221 


: 079 


■ 7 319 


4S 


8 






.63344 




.48988 


87207 




.7 198 




L 






.63382 


■ 5 01 


48964 


-e?193 


—ifi 


_j.g?,T 


-U 


8« 










48989 


.97179 


. 194 


7 965 


40 


ai 






; 63967 




.49014 


-97164 


. 282 


-7 8S4 






.471)11 




.GS995 


. :304 


.49040 


.97150 


























37 


34 






■B4070 












_S.B 


as 


"47588 


T7fl5r 


.64107 


~^ 4918 


.49 "iB 


^I'or 


"T35"" 


~ r773sT 






































- 462 




33 


38 




37909 


6422 




Ua 92 


:B7064 


. 601 


.76890 


32 


a8_. 


otib" 


87fl.Ge_ 


^425B_ 


1 4177 


^Ji. 


J7050. 


589 


J_7e86?_ 








~30 


31 


.47741 


: 87363 


643S3 




^49263 


:b7021 


'. BIB 




29 


32 


47737 


.87364 


64371 


. 3932 


.49293 


.37007 


. 964 


.76610 


38 








54400 


. 3794 


.49813 


.88993 




78390 




84 


4.78 B 


"87312 


"54"4a"4" 


. 3667 
. 3540 


"49369 


jeB7i_ 
89664 




7637] 


36 


3S 


rTe?" 


76151 


33 






.37793 


54622 




.49394 


.86949 


303 


.76032 




















.7581S 


38 


38 


479 D 
















32 


|2_ 


.479 9 


87766. 


64635 




49470 


B990B 


829 


-TBB76 


M 








54B73 


. 2909 






962 


-7S868 




41 




















42 


^480 3 


■8771B 


64743 












B 




.48043 


-H7701 


64789 


2528 


-49 71 






. 76300 


7 


44 


.48073_ 

.48 es 


.B7687 


64324 


. 2402_ 


49 9_e_ 


: 88834 


19 


.75083. 


_B 


is 






4e 


.48 24 




549O0 




'49647 










47 


48 GO 


: 97945 


54938 


2026 


.49672 


■b67»1 




74738 








97831 






-49697 


.88777 


71 


-74810 




49 


492"26" 


9791 7_ 
37 OS 


6505r 


. 649' 






_ 00. 


..744B3. 




fill 


.49748 




~io 


51 




37 99 


S603B 


524 


.49773 


-86733 


36 


^«3B7 


B 






37 75 


65127 








25 


-711*0 


i 
















64 


. 74033 




B4 


.48_aai_ 














-78906 




BS 




: 87532" 


65241 


. 1025 


4B374" 


'smW 


41 


.78788 


~6 








65279 








30 


.73971 
























OB 


4B430 


















SB 


4B4S6 




56^93 


. 0529 




'■nen 




:78S31 


1 


"^ 


4H4al 1.37482 


■«^ 


j^. 


^ 


B6B03 '67735 


c^ff^ 


-^ 



NATURAL SINES, COSINES, TANGENTS, COTAN. 255 





Sin. 


c™. 


Tsn. 


Co,. 


.in. 


C..», 


Tnn. 


(v... 






















6D 




60026 














.86318 


59 


a 


6005 


86573 


67813 


.73973 




■66687 


: 60185 


.66209 


58 


8 




86959 


67851 


.72857 


.6 579 


86672 


-60205 






±. 




86544_ 


57890 


_^A74!_ 


■ 6 804 


. 8B667 


■60245 




ae 




501 2~~ 


86 30 












66881 


85 


e 




86 15 




: 2509 


:5 B53 


: 35627 


60324 


.65772 




7 




86 01 




. 2S91 


-6 678 


85612 


-6036* 






8 


'g020 








.6 703 


.85697 










-5023 _ 


86 71 


S3D8S 






-8566" " 


."60183" 




-51 


»" 


86457 




. 2047 


■5 753 


-65337 














.6 778 




.80522 




49 






86427 


5820 












43 






fiflllS 


S824 


: 1702 


:& 828 






.65011 


17 


4_ 




86998, 




. 15BB 


5 852 


8550 


_60e42_ 


__6*.9fl3_ 


-M 








5831 ~ 








8 






5B35 


'■ 1358 


: 51902 


8547B 


60721 




44 


n 








. 12*4 


.51927 




60 61 


64579 


43 


U 
















8*471 




a_ 


iBC^ 






71015 


.51977 






.64363 


41 




















M 


?mc* 


aasio 




. 70901 






.60 Bi 


.6*256 


40 


11 


■waai 












-60 21 


.6*1*8 




33 


-BUU 


aeiiai 








!S 36 








31 


.mn 


iB3as 


6883 


.70560 


52078 


.8 70 






37 


tt. 




iaaei 


6887 


.70446 


.52101 


.8 40 


jioiq. 


63826 


36 


U 












35 


se 




86312 


S874 


"70319 




.8 25 




.63612 


31 






86207 


6878 


.70106 


S2175 




-61 60 


.83505 


38 


la 




86193 






. 52200 


8 94 


.61 00 






8B_ 


,fi0739_ 










■ a. 2_79_ 




^3?92_ 




30 




86163 


"689" " 


69766 


"522"50 


8 184 


:61280"" 


63185 


"30 


11 




86148 




69653 






.61320 


.63079 




g 




861S3 








8 234 


.61360 






M 




laiiB 






.62324 


8 218 








B. 




.mo*- 


'590 


_ 16931 8_ 


523*9 


.862g3_ 


■eiii _ 


. .62760_ 


^8 


M 




seoBB 








36 88 








H 




86074 








.85 73 




leasts 




g 




aaOGS 


.691 


:6B9T9 


: 52423 


.85 7 


:ei66 


-62*1 


23 


H 




8804t 


592 


.68866 


52448 




-6160 




22 


1- 




jsm. 


-592 8 


6B764 


_2473_ 


;85 7 


.MM. 


_;6223 . 


-SI 


40 












.85! 2 




.6212 


2n. 


«1 


'Sffi 


•8000 


"693 


: 68531 


2523 




'■a 72 




9 


u 




.693 


.88410 


. 25*7 




.6 78 




8 


a 




elsTO 




.68308 




^850 6 








u 




85866 








^86.0 1_ 




-Aim _ 


B 


4E 


~iiiaf 


85941 


"594 


".08085 










5 




■SllSl 


85926 


5BS 




.5 6*6 


'850 










.S1178 








■ 5 671 


-850 5 


G1962 


^6138 


3 


U 




85898 




:fi7753 






-62003 


.6128 




iB_ 


Jigw. 


S6«ai 


.596 


.97841 


.5 7*5 


J4g_4 
-849 9 


_62043_ 


■ 6117 


_n 


M 


SG888 




-67630 






n 








-67*19 






-62124 


'6097 




« 






597 


-67309 








floee 








598 9 




^5 '19 


849 3 






1 


IL 






69849 




^844_ 


B4B 7 


-622*5 




J 










.86978" 




B4B 2 


.82285 


.8055 




se 


81*04 


B5777 


59928 










.804* 




>7 


■61129 


85762 


5B967 










8034 




ta 


.51464 
















3 


M_ 


J14je 




# 




-6 967_ 


846 


li- 


6013 
1 60033 


^ 


a^ 


W 


I^ 


■ 62992 


8*805 


-^ 



25fi NATURAL SINES, COSINES, TANGENTS, COTAN. 



, 


Bin. 


Cr,. 


Tail. 


Cr>t 


Sin. 


C.^. 


Tun. 


C..t. 




~ 




84805 


.82487 


.80083 


-54 B 




-64B41 


- 3986 























i 




















6 


» 


30BB 




:B3B0B 


.59723 


■54 3 


93819 




. 3683 


5 


.« 


309 


84743_ 


62848 


-5S620 


■JiA^ 


93304 


-63106 


3695 


J_ 






84728 




.69517 










6 




314 








^S461 








5 


7 


sie 


81397 


: 83770 


^5931! 


5463 


9375 


-66231 


: 330 


63 


8 


318 


84981 


.82811 


-59209 


5465 


8374 


.65272 


. 320 


B- 


B 




J41flL 


62852 








-66314 


_ -- -3-ia_ 


J. 


10 


823 




.82B92 


"759002 


"5470 


"63701" 


-66S65 


. 301 


S 


11 


328 


84635 


-62933 


1.58900 


5173 


8369 


S6397 


- 291 




13 


32B 


84 19 


.62878 


.58797 






-66438 


. zei 






















i 


14 


_333_ 








:64BD 




65621 






IB 




84 "73 


.""83 96~ 


~58 90" 




"8382 " 




^263"" 


"i 


16 




B4G57 














4 


17 


341 






isS 86 


^6487 




ieSBlfl 


'- 333 


4 


IB 


343 


845ZB 


-63 17 


-58 84 


.5490 


8358 


G66BB 


- 223 


t 


U_ 


34B 


84611 


-63 58 


.58 83 


6492 


8356 


.65739 


■ 213 


J 


20 


348 


B419B 


.63 98 


.67981 






.66771 






ai 


360 


84480 


.83 40 


.57979 


-6497 


8353 


.65813 


: 1 4 




22 








.67778 




8351 


.66834 


-61 5 




















■ 1 5 




84 


3583 


84433 












- 1 6 


! 


ae 


3807 










"8346 " 


TOBBSO" 


- 1 6 


8 




















! 




3fiE8 


84383 


-63684 














a« 


3681 


94370 


B3625 


:B7176 


:B511 




'66105 






Bfl 


_3705^ 


9135^ 


■63838 


■ 57089 


J.BIL 


_B3_40 _ 


.88147 


. 1 7 


3 








.63707 












3( 




375S 


84324 


-63748 


: 66889 


: 65218 








29 




377S 


84308 


.83789 


.S6767 


.65 12 


8335 


.663 2 


. 9: 










.63830 


.68Q87 




8334 


.683 4 


. 078 


2 




3838 
3853 


84261" 


Te3913 




■5S.91_ 

.5 3 6 


8332 


.663 B_ 




_2 




33Q 








3877 


84345 


.63963 


:B6366 


.B 3 B 




;66440 


.SO 13 






S902 


84230 




-68285 


-6 a 3 


3 7 


-6 4 2 


-60417 














.6 3 8 


3 6 




-80323 




sa_ 


39fll 




jms. 




"4 2 


_U_ 


■6 5 6 


^- 50328 




40 


397S 




.6^117 


.56988 


.5 1 6 


3 2 


.6 6 3 


.50 3S 














.540 


a 1 


.8 e 


-50 3B 




















-49 4 




43 


4048 




'■ 64240 














«_ 


4073 


_B4.iao_ 


_e4_28i_ 


_l5_B5a7 






-6 7 8 


^49 5 


















.6 8 8 


























47 


148 




: 64404 


: 65269 


5 605 




!b 9 2 






48 


171 


84057 


84446 


.65170 


5 830 


3 9 


.6 9 1 


-19378 




*L 


IflS 


81041. 


.64487 






3(18: 


ii_i 8 


,4928. 






230 


84025 




_ 




2H 


bIodb 




'b4B73 


56702 










sa 


zes 


83994 


.84810 


-B4774 


66726 


3034 


- 11 


.4900 




«s 


293 


83978 


.84862 




55750 


3017 


. 16 


.4BS0 1 




6^ 


317 


83982 








3001 


. 19 


_^48B1 _ 


















. 23 






GB 








64379 


56823 




. 38 


.'4862 






4391 






.64281 


66847 


29B3 


. 32 


.4853 




G8 


4415 




-64958 








. 38 


.4844 




B9_ 


4440 


jjg. 


J4899_ 
-64941 


: 51065 


# 


_29_30_ 


. 40 


- J834 _ 




8^ 




-cfh 


■ Tanf ■ 


^ 



NATURAL SINKS, C0PINK8, TAN'GENTS, COTAN. 257 





Sin. 1 r..5. 1 Tan. 1 


rti„. 


C..J.. 


T^n- 


Col. 




■ r 






.48259 


57358 


8 916 






8 




-S5BM 


.83887 


.8749 


.48183 


.57381 


S 899 






5 




.6S89B 


.82871 


-6753 


.43070 




























42650 


5 














■ 8 848 
81332 


Loi»_ 


-42*62 


Jl 




-66040 aaaaa 


.8786 


.47792 


"57477"" 








.82806 


.8770 


.47899 










64 


















'42IBB 


63 




-BBiia 


: 82773 






67548 


.81782 


! 70368 


.43110 






^Bi3e_ 


A2_?5L 


-8783 


■47422 


J7672_ 


-617.65_ 






_6i 


o" 




83 41 


.8787 


.47330 








.41934 


GO 




■sei6t 










-8 7 1 


'70499 


-41847 


4S 




.SB20B 


'-S2 08 


'8796 


1*47146 


; 57843 


-6 7 4 














8600 


1.47053 












- 


;6ezsB_ 


^82 75_ 




" :4687"0 


757715 " 


6 6 1 
81684 


-70673" 


'~iim~ 


46 




.82 SB 


.6608 " 


4S 






.83 43 




-48778 




81647 










.M 38 














48 




.83 10 


:883l 


:46595 


57786 




70804 


^41235 


43 






-P^8 


JB25_ 


46503 


57810 


31597 


70843 


41146 


_ii 


to 




■at 77 








81 BO 








11 




.83 81 


:6634 


.48320 




31 63 






38 


32 


'5B449 


.82 44 


.6836 


.46 29 


57381 




70979 


40387 


38 


33 












81 30 






37 


t±. 






















. 58521 


"82495 




" :46""65" 


57"9"53" 


8:496"* 


-"7 llT 


40827 


~U 


ie 


.SB54S 


.82478 






.5'976 


81479 


.7 154 






37 






-68 














38 




^82448 


-68 42 








7 242 




33 


3S 


:56B17 


_82«?. 


6B985_ 


_ L*S_5?2_ 


■5B047_ 


_BH?a^. 


.7 285 


„40381. 


Ji 


10 


-56811 


























.A5 10 












(3 


■ 56888 


: 23SO 


^66 14 


-46 20 


:5s 8 


8 378 


.7 4 7 


.40022 




IS 


.5B71S 


. 2363 


88 S7 


.45 29 


.58 1 


8 361 


.7 4 1 






w 


5673_e_ 
5e7BB 


: 23"o"" 


788942 








-7 5_5_ 
.7 649 


.39784' 






.45 49 


"26 




.5B7Bf. 


. 23 1 


6BB85 


.44 53 


58 12 


81 10 


.7 693 


-39679 












.44 88 




81 93 












'. 23 1 


'89071 


















■aaaw 


69114 


44 sa 




:81 69 


'■7 726 


: 38421 


-a. 












58307 


-81242 










' 33 1 




















. 33 4 


. 69243 


.44 18 


58364 




-7 857 


.39166 








. 21 B 






68379 


.81191 


.7 901 


-39079 


17 


«_ 


- r7000" 


- 21 5 




.44 49 


-68401_ 


Jim. 


J_e48_ 


^3899.4 


Jfl 




69372 








IB 


.57034 


. 21 8 




144 80 


■ 8449 




72034 


: 38824 




47 










. 8472 


81 23 


.72076 


-38738 


3 




; 57071 


: 21 S 


89502 


iil a? 










3 




. 570B5 


. 2g_fl_ 


:3sm- 


_ 43 92 


8543 


"81072"" 


7722 1" 


. J 8588. 


-- 1 


so" 








II 






.89831 


43 14 








:38399 


9 




:57 87 


: 2048 


69675 


.43 25 










8 


SB 


.57 91 








6614 


81021 








8*. 










^B637_ 


61004 


.723 8 






G5 


.57:13s 












.72432 


~SS060" 


~B 


06 


. 57362 


982 


.69847 


■43IBB 








37978 


4 


G7 


57266 








870B 


80963 








SS 










8731 


60936 


: 72565 




3 


SB 


ii- 


032 


■70021 


"i:«8i6 




= 


-72810 


37722 


^ 


S»- 


, ,iai^ 


- 8779 


^ 



268 NATURAL SINES, COSINES, TANGENTS, COT AN. 





Sin. 


c™. 


Tan. 


Cot. 


Sin. 1 C™. 1 


Tan. 


Cot. 







58779 


8090 


72654 


.37638 








. 2701 


60 






9098 


72699 


.37554 






75401 


. 2624 




a 






72748 


.37470 


60228 '. 


6929 


75117 


. 3544 




3 










60251 . 


79BI1 


7S493 






_4_ 


5B873 






^37303_ 


60374 ^ 


379^ 






_5fl 






5081~ 


72877' 










. 3301 










72B21 


^37134 




'9 sa 


75629 


. 3321 






60e4S 


9018 




.37050 




79 41 






53 




sen 67 


6076 


73010 


.36987 




9 23 






63 




_esflio_ 


BQ74 


73100 




60390 


9 06 


75787 


. 19B1 


_^1 






■36800 


60414 


9688 


75813 








59037 




73141 


.39716 


60437 


9671 






49 




69061 




731B9 












43 










: 36540 




79836 


75960 


: 1698 


47 






aOB6 


73378 


JMiS. 


_6050a_ 


79618 


75996 


^ ^J,5ia_ 




B~ 


59131 




73333 




~45 


e 


59151 




73868 




flOSBS 






:ai427 




7 


SBm 




73418 




80576 ' 


7 665 


7613 


-31348 






B9301 




7S1B7 






7 547 


.7616 


-81369 




L 


mV 


SSSSS. 


73803 


^MBJ- 


60933 


7 530 


7622 




-ih 


::o 


69218 


BDBSB 


7S547 


.35968 


606 5 1 


7 512 


.7627 


81110 




1 


50273 






.85889 






.7881 


■81081 




3 










606 1 






■30952 


38 


!3 




806O7 




* 35719 


107 1 








37 


4_ 


59343_ 


80489 




.35637 




.7 441 


^16 _ 


30785 


_II 




3flS 


B0479 




.35654 








3071B 














BG7 4 




!7654 




3 


27 




8043B 


73861 








.7B59 


:30668 




!8 


436 


80420 


73906 






7 37 


.7861 


.30480 






. 459_ 


BO 40 3 


73B51 








.7668 


■ 30401 




3o" 






- 736143 


"elsT 


7 33s" 


■ TB73 




"3 


SI 


606 


80388 


74011 


.35060 




7 318 


.7877 


■^ 30241 


2 




9529 


80351 


74 86 


-34978 




7 300 


.7982 


.30168 


2 




. flS62 








60 9 IS 








2 


34 


9575 








60983 


79284 


17691 ^ 




_2 


^S 


8539 


bo'sm' 


"71 21 


^ 7s"4"7"sr 




79247 




-29931 






. 9622 


80282 


74 67 


.34850 








.29893 






. 9646 








6 033 










SB 




80217 


^74 67 


^34487 


6 OBI 


7919S 


:7710 


;2flase 




SL. 


JmL 


80230 


.mo2_ 


.84405 


-B 84 


79176 


,7714. 


_iBfil8. 






. 9716 




.71447 




-6 07 












. 973B 






: 34242 




;6140 


:7721 


: 29493 




13 




.80178 


;7453B 


.34160 




79123 


.7728 


.39395 






, 97S6 


.90160 


.74593 


■34079 


.8 76 






,28307 




M_ 


■ 8609 


■80143 


.71628 
.74674 




.61 99 
.B 322 


"79099" 








15 


.33916 


.7742 " 


.29153 




46 


. 9866 


Iboiob 


.74719 


.33835 




790S1 


■ 7747 


29071 




17 


. 9879 


.60091 




■33754 














. 9902 








.6 391 










49- 


. 9926 


laooee 


.79855 


.335 2 


Alii 


78998 


i77Bl , 


.28B12 




SO 


. 9949 


.B003B 


.74900 


.3" 5 1 




.78980 




.28781 






. 9973 








.i 3BD 










na 


. 9995 


! 80003 






.6 333 




■ 7775.. 


^38810 




ES 


. 0019 


.7B98B 


.75037 


.3 3 8 




.78926 


.7780 


.28533 




§4_ 


- 0013 


.T99BB 


.7508a. 


.3 1 7 


-8 439 


.78908 


^7.84 


_^45J. 














.6 151 






se 






■751 8 


;33028 


.8 474 


:7B873 


*7791 






67 


. 0112 


.79916 


.752 9 


.32946 




.79855 


.7798 






SB 


. 0135 


.79699 


.752 4 


.S2B 6 




.79837 


.7803 








. 015J_ 


•79891 


.753 0^ 






7asi_9_ 


^808 _ 


.3B071_ 




M. 


. 0182 


^79864. 


.75335 


.33704 


~e"66T 


■78801_ 


■7813 


37991 


_, ! 




Cm. j Sin. 


(^ 1 W 




Sin." 


1 Cot, 


tLh, 





NATURAL STNES, CORINKR, TANaRNTS, COTAN. 2r>!) 





Sia. 


c™. 


Tnn. 


fnl. 


Hm. 


CCH. 


Tuu 


Col. 


' 


■l 




- 880 


.78129 


.27964 


.6 932 


77715 






Btt 




.SlSSfl 


. 878 


.7817B 


.27917 


-B 955 


77888 


102 


33418 


sa 




a flis 










77878 


107 


23343 


58 




-fi BS6 


'■ 874 












.33370 






-6 858 


■ 873 


: 78316 










23196 




_ 


-e 881 




.78363 


"r27Bll 


6 046" 


"77823" 


" 132~ 








■ 6 704 










77805 


128 


:230B0 


54 




e 726 




-7B4S7 




B 090 






.22077 








- 865 


78504 


.27883 


.6 113 




I3t 










. 664 


78551 


__2_7308_ 


6 IS5_ 










i r 


.e 796 










77631 




.22768 






6 BIB 


'7860 


' 78645 
















fi 841 


.7858 


78893 


: 27077 


8 203 


77494 






46 










37001 


6 235 


77476 


180 




47 












-6 248 


7746B_ 
77439 


. J65_ 
1703 




-^ 




81908 


.7853 


78334 


.26849 




.2 394" 












.267 4 


-0 293 


77421 


175 


.2 321 


44 




















43 




. G1B7B 


















'. - 


. 83001 


-7848 


9022 


265 6 


6 361 


77368 


189 


■ 2 104 


1 









































1 


63046 


















3 


82088 




:791 1 


^26319 


.6 438 


77310 


204 


: 886 


8 


S 


smi 


-7838 


-792 2 


-26244 


-6 451 


77292 


209 




7 




8211B 


. 76389 




■26169 


6 473 


77273 


214 


■ 742 


6 


r 


.S21S8 




.79sT 






773 




. 670 


9 


a 


-62160 


i 78333 


.79354 


i: 28018 


6 518 


773 


2 3 


. 598 


4 


7 




78316 














3 


t 






















:6a22_e 




.79496 




.6 585 


771 


? !-_ 


. 362 




_ 




. 78261" 


.79544 


1-2S7T 


.8 608 










1 




















1 


.82297 




: 79619 


I'aSB 7 


:b 653 


7 


3 3 


;21166 




1 


. 62330 


78206 


.79688 




-8 675 


7 


2 8 


.21094 




1 


. 82342 


78XB8_ 


._797_34_ 


.1 J54 _7_ 




-J 

77 


2 2 


21023 




1 


.82365 


3 7 


.20951 




! 


82368 


: 76152 


: 79829 


■ 2 268 


637 2 


770 


2 2 


-20879 








781 4 


.79877 


.2 1B3 








.20806 




1 














282 






i. 


jm5B_ 


'?80 8 


iS. 


1 ' 2504.^ 




769 


367 


1.2068B 




I > 


-63479 


-780 9 


80020 


















-780 1 
















L 




-780 3 










302 


1:20451 








.760 B 


.80163 


-3 746 


-63899 


769 


307 


1-20379 




! _ 




iTBOJ, 


.B021L 


^ ^7_2_ 


63922. 


_7e9 _ 


312 


1.203_0^ 




1 














31 


1. 20237 




1 


:B261B 




! 60306 




63966 


7B8 6 


32 


1-201 6 






63638 


-779 2 


.80854 


.24449 


63980 




32 










-779 4 
















1 


-82708" 


77916 
. 77897 






-64056 




33 . 


_i:i9g 3_ 

I-IBB 2 




50 


80498" 


-24 27 


7fl7"T 


34 








77879 






-64078 










il 




















S3 


. 6377* 












35 4 






S4_| 


-62796 


'. 77824 


JP690. 


■2 ?3l_ 


6414B 


767 7 


36_3_ 


1 '19699 


_B 






-77806 


















i 89842 




















■62884 












37 1 


; 9387 


8 




.62887 


.777B1 


.80883 


.2 637 


64234 


75642 


38 1 


9316 


3 


iS 


.easDB 


-7773S 


.80930 


-2 S63 


-84256 


76_B23 




9246 




5 


jgL 


-«^ 


80978 


igii 


J1279. 


78604 

am. 


83910" 


-i-^ 


^ 



NATURAT, RI\E8, COSINES, TANGENTS, COTAN. 





Sin. 


Co,. 


Tn.. 


Cot- 


Sin. 


Cos. 


Ted. 


Col. 






64279 


76601 


3910 


1.19175 








■ 6037 




1 






3960 














2 


64323 










7S433 


87031 


- 4902 




3 


61346 


78B46 


4059 


MS9B4 


86672 


■76111 


87082 


- 4831 




4 


J43eB_ 


'8530 


4108 


1. 18894 


86694 




_871S3. 
.67184 


- - *'87- 
■ 4699 
















e 


84412 


6492 








: 75366 


-87338 






7 


61435 


6173 


4268 


8864 


■ 6 759 


.76337 












. 8455 


1307 


■ BfiU 


.8 781 




87338 


■ H4B6 






81479 




_43B7, 


_ . 8S44 






B7389 


„_«3-lL 






-61601 








TiisT 














6398 


4157 


■ 8404 


-658 7 




^87493 






12 




■ 8380 


4507 


. B334 




■ 75241 


-87613 


■ 4239 


a 


J3 


:645B8 








^868 1 


.7S222 




- 4162 


41 


14 


- 84590 


^ 6342_ 


-im. 


■ 81B4 


_659 3. 


-75203 




■ 4095 




ii 


.648 3 






.TB1B4 


. B789B~ 


4028 


~a 


16 


.848 5 




B4708 


! 18055 


: 86966 


■76166 


-87749 












81768 


■ 7886 


.66976 






■ ^'fli 






















(3 


iL. 


■817 1 




.SiSM, 


.' 7816 




J£i22. 


;,87B04 


M37B1 


Ji 










. 7777 




■76086 










■84748 


.78 














bs 


33 


-61763 


76 82 


: 86006 


■ 7638 


86088 


: 76050 


.8B0B9 


. 3681 


H 


Z3 




78 3 




. 7669 


■86108 


■76030 


■BBllO 


■ S491 


t1 


a*. 




778" 5" 




7600 


l6_613J_ 


_7Bp_n. 


88182 


■ 3428 


J! 


25 


.64834 


■85167 




B8304 


3361 




28 




■ 7 6 


86207 


:i 


^66 5 


^74978 


■88265 


. 8396 


39 


2B 


■64901 








■ 66 8 








s: 


29 


■84923 


■-J-5-- 




'■ 715 


.68 40 


■74916 


86421 


13096 


ai 


30 






. 708 


.862 2 


■74696 


SB 73 




30 


















- 2B63 


li 


3 


81989 








^863 8 








31 


3 




■7 98 


:B56 9 


. 687 


663 7 


:74B3B 


:86 28 


■ 2831 


31 


3 




■ 7 98 


■956 9 


!■ 680 _ 


.863 g 


lT4818_ 


68 80 


2786 


21 




■65055" 
















-» 


36 


■85077 


■ 7 27 


^857 




; 86393 


: 71780 


:8B 84 


: 2633 


31 


37 




7 08 


■ 867 1 


- 660 


.664 4 


74760 


■ 88 38 


■ 3887 


as 




















33 


39_ 


: 66144 


7 670 


'S5B 2 


- Me 


-864 8 




J8£ig. 


l- IJ35 


Ji 






■ 7 51 


■ 869 2 


639 


■ 664 


71703 


.88 92 


. ssee 


















-89 IB 


.13308 


i( 




■86210 














■ 9383 


11 




86233 


■ 7 94 






■ 865 B 


7464 


^BB « 


. aiM 


17 


M_ 


■86354 


_7_ Z5_ 


86115 


_^ B12 _ 


■ 666 B 


7462 . 


■ BBgBl 


■ 3011 








■ 6BS S 






1) 




'65298 


■ ■> 38 






■ 886 


:7ieB 


lis OB 


■ 975 


1( 


47 


.65320 


■ 7 19 


■86267 


■ 591 


■ 666 2 


.7168 


.SB SB 


. 909 












. 686 


686 9 


.71B4B 


■ BB 10 


■ 814 






















J! 


sT 


766388" 


7 61" 


.86419 


^5"7i " 


■ 666 7 


■"71609" 


■"bbbib" 


. 713 






■65406 


■ 7 42 


.88170 


. 5847 


687 8 




■89S67 


■ 848 




5 




















S 


; 65453 


.76 04 








■ 74461 


'bB872 


: B17 


7 


5 _ 


J517I_ 


■75 86 


86823 




^7_3. 


74431 


_8_9126, 


. 4611 


J 








.88674 






























■ SSI 


4 










. 53 40 


.68818 




■89883 


. 2S6 


1 


53 


: 86562 


^76 OB 


; 86827 


■ 5172 


86970 


71363 


■89936 




! 


59_ 


L.65_584. 


.76490 


,8887a_ 


_^l(M_ 


,e8_BJL 


^1334 


_8998i 




_l 


sa. 


■65608 


-75471 


86929 




■66913. 


■71314 


BOOIO 


■ 081 


J 






Si.. C.t. 




(-■(1^. 


-siT- 


^mT 


^T^. 





NATUHAT. SINES, COSINES, TANGENTS, COTAN. 261 





Sm. 


ra. 


T-^„. 




,si„. 


<■.,<- 


T^n. 


Cot^ 




S" 


. 6913 


.743 4 


.90040 




66200 


.73136 


-93352 


■07237 


DO 


1 




■ 743 S 




1 10996 












I 








1.10931 










68 


: BB78 


:743 6 










■9341B 


:0704 


57 


6 


- 6899 


.742 7 






JP8JI_ 


.7306L 


.-93_46L 


. .0698, 


J6 






"90304" 


""10737" 












« 


7043 






.10672 










6 


V 


. 7064 


74 8 














S 


a 




74 8 


.90463 








93688 


1^0873 


a 


£_ 




.74 6 


J05ie_ 




08391 


2957 


..937J3_ 


1 0967 








.74130 


1104 i 






"fi 


1 




.74100 
















3 


717a 




.90674 






7389 


93903 


.0648 




a 


7194 




90727 


I.IOS 


-68476 


7287 




06127 








^04 1_ 


OOTBL 


1-101 9 


.66497 




' 94016 


0636 




a" 






80334 








94071 


.0630 




e 


7358 


'■7 0D2 


90387 




: 68639 


72B1 ; 


-B4125 


0624 


i 


7 


7280 


.7 9BS 




.09963 


.68561 








i 








-9 993 














2_ 


7123 






-09634 


■68603, 


72757 


: 94290 


0605 


_3 , 






.7 B24' 














~ 








.1 53 






72717 


! 94400 


:069S 








7 685 




1-09642 


-83666 


7269 


94455 


0687 


3 




740e 


! 865 


9 59 


1 09578 










3 


i_ 


7«30 


.7 846_ 
.7 636 


'IHr 


1 0a5,14_ 
1-09460 


■66709 


J365_ 


i94565 


-0574 


3 


e 










s 


e 


^7473 


.7 806 


9 41B 






7361 


.94676 


:o5e2 




7 






9 473 




68773 


726B 


91731 




s 


S 








09258 










3 


s_ 




"_7_74Z- 




.09196_ 


-63814 


73557 


J4.84!. 


-0643 




to 






'9 633 


.09131 


-08935 








"3 


1 








-09067 








0531 


i 


a 


■ 7603 


.7 88 


9 740 






72497 


9 007 


■ 0B25 




a 


7623 


.7 69 




: 09910 


68899 


72477 






i 


i^ 


7945 


-J 4S_ 


9_B47_ 


_08_87a_ 


68920 


J24_57_ 




_ 0513_ 


a 


« 




.7 39 


9 901 




63941 


72437 


-6 17a" 


■ 0BD7 




4 


768 


-7 10 


9 956 


I 08749 


68962 








i 




7708 


■ 7 90 










■ 9 294 


■ 0494 




» 








r 03623 


:S9004 


7237' 


.9 310 


0188 




a~ 


Jllj. 


JmL 


±2iS- 


1 08S59 


69025, 


_7235_ 


J_395_ 


_om_ 




10 


. 777 


.7 31 








7333 


.9 451 


0476 




J 






B 234 


1-03432 


: 8B067 


7231'; 


.9 608 






3 


. 781 


17 91 


B 277 














S 


. 7ia 


.7 472 




I 06306 


69100 


7227' 


■ a 618 


1^045S 




4_ 


- 785 


■7 453 


9 395 


1-03243^ 


. 69130 . 


73257 


■ 9 673 


_1^04B_a2_ 




E 


. 786 


.73 33 




1.0817B 


68151 


7233 








« 


. 790 


.73 13 




1-08118 


-69172 


7221 




^0440 




7 


. 793 


.73 93 


: 93547 








'.9 841 


04340 








.73 73 












01271 




s_ 


■ 796 
.6798 


-73 53 
.73333 


'93656 


_1 ■.07927. 


le9235_ 


;7.3_ 6 . 


79 008 






» 


" .04158 








.73314 












■040B7 




3 


6802 


73394 


92817 


1 07738 


69 98 


^73 8 








X 


6806 


-73374 


-92873 


1-07676 


-es 19 


73 7 


-S 176 


: 03976 




4- 


JSP7?. 


^S2M_ 


■9392|_ 






■73 51 


:99233_ 


^03916. 


- 


s 




















4 




-73216 


-93034 


i 07487 


69382 










7 


'68136 


73195 


.93068 




69403 


^71 9i 




'03734 




^ 












71974 


98457 


■03974 














69449 


■71951 


-96513 


■03613 




1^ 


^a. 


W- 


"93353" 


X^§L 


■^s^ 


"^ 


- 96569 


-^f^ 


-Jl 



2«2 XATT'RAT, SIXES, COSIXES. TAXfiENTS. COTAX. 





m- 


r... 1 T^, 


rm. 






Sm. 1 C-oj. 




C. 






69496 






60 




70091 -TiImI 


.93271 


-01761 




1 












31 


13.71305 








3 












33 


321-71284 




-0184; 


3i 


8 


S9538 


7 873 


8738 


:03372 


57 


33 




-98441 


-01S8 


27 


_i_ 


_B9_M 


7 853 


_8794 


- 03312 


_5fl_ 


3^ 


74': 71343 


_9B4Bt 


.0162 






















.0146 






6958 


■ 7 81 


6983 






38 




:fl861! 


-01401 






B»B1: 


.7 79 


.03132 


53 


37 


703381 -7 16 


.B8B7I 


.0134 








7 77 




-03072 


52 


33 


702571.7 16 












-7 5 


_7_g76 


■030i; 


_1L 


J9_ 


70277 1 7 H 


-98781 


:012a 




io" 


'89675 


7 733 










702981-7 12 


9884! 


-01I7| 


'20 


11 


69691 


.7 U 


7 89 


02892 


19 


41 


70319!. 7110 


98901 










.7 9 




02832 






70339' -n08C 








13 


















!00994 


11 


li_ 


E97SS 
















-99073 


.00835 


J8 


IG 


89779 


.7 30 














-99131 






IS 


B98O0 








44 








-88189 


-0081 1 


14 


17 


6B8ai 


■ 7 a 


7525 


:02 33 




47 


■7044; 




.88347 


-007Si 


13 


U 


■ema 


.7 6 




-02 i 








■ 708 7 


-SB304 




13 


12- 


•88883 


7 49 


. 7643 


.02 




Ji-L 


■704a! 


JOfl 7 


■98363 


!00642 


_n 


lo- 


ISSTa 


1^5 


7700 








-70soBi"-roBie 


-9B42( 


.0058! 


10 


st 




■ 7 06 










-70525I.70B96 






9 


32 


"sflS?: 


7 68 


781! 


■ 02? 


3 


53 


-70548 -70B75 






a 


33 


8994 


-7 68 




.021 6 


3 




-70687 


70855 


-98594 


1.0040B 


7 


2*. 




J_''7 




^03117 




^i- 


70687 


ziom 


-99852 


1J03_5P 










7984 








70608 








~i 


38 


-7000 


'■■! 07 


804] 


;0199i 


3 


66 


70621 


; 70795 






i 


27 


-70O2 




8098 


-01931 


3 


57 


7084 


.7077; 


-99836 


1.00175 


» 








6155 




3 








.99884 


1.O0116 




3B_ 






8213 






59 






■99B43 


100068 




sa. 


J009 


i^iaas 


8370 










^7071 1 


1.00000 




Zo 










-TT^n- 




■ 




-kt- 


tw. 


T^ 





NATURAL SECANTS AND COSECANTS 268 



, 


SECANTS, 


• 




0" 


1- 


a- 


8° 


4° 


6" 


e° 




l.DOQDO 








I. 00344 


1.00:383 




W 










'ooisa 


00MB 






SO 




OOMO 


00018 


OOOM 


. 00140 






00557 






00000 


00017 








00390 


00560 


67 




awDo 




00066 


001 ta 


Do:n3 


00303 








00000 




oooea 


oous 


00-»4 










00000 


oooie 


00067 








OD560 






00000 


oooia 








ootoo 




53 




ooooo 






ODTGO 


DOJfil 


00403 










oooao 




00161 


00383 










00000 


OOOIil 


00073 




00S65 




0O58S 


50 




1.00001 


i.ooosi 






1.00387 


1.00411 


i.ocesE 






00001 




'00074 


'00156 


00369 


















00371 




00503 


47 




00001 










OD410 


00606 






00001 








00376 


OOiSl 








OODOl 


OO0S4 


00073 


00168 


ooars 












oooas 


00079 


O0164 






00004 






00001 


oooao 




ooieo 




00439 


00608 














003&I 


ooos 










00027 


OOOBS 


OOlBB 


003S7 












lOOOK 


I.000B4 


i.oom 










1 


oooos 


mm 






00301 


00440 












DOOST 


00175 
















00OS8 


00176 






O0B37 








00031 


00089 


00178 






00630 






oooca 


OOOSl 








00161 








oooos 














Si 










00183 


0030S 












OOOM 


OOOM 


00185 






00044 




» 


ooo« 


000S4 








'00463 


00047 






1.00004 




1.00097 




.mjs 


1.00)65 




sa 








OOOBB 


00100 






' 006^4 








00037 


00099 








00657 






ocoos 










004T4 








00005 








aim 






^ 


* 






00103 


00198 






00607 




1 




DOOM) 








0M83 


00671 






OOMl 








00485 






9S 


00008 








00330 


00483 






10 


00007 




00108 


ooara 






OOOBI 




I 


1.00O07 


1.00W3 


1.00110 








1 0n684 




19 












00407 








00008 








00340 








H 


oooos 




001 14 


oftjia 


00843 




00095 




B 




00017 


00116 






00506 


00698 


J 


ts 




ODMS 


00117 






00600 






00009 






ooais 


00860 








(8 


00010 




ooiao 


oosso 










u 




OO06O 










00713 





N 


00011 


00061 




D0«4 




OOSil 




51 


i.Doini 




1.001 a* 




,00350 


1.005» 


1.00719 




U 




00063 


ooiaB 


ooas 






00733 




U 




OOOM 








00530 










00066 


00138 




003C7 












ooosfl 


00130 


00S34 


00369 




00733 




H 


ocois 


00057 


00181 


OOaSB 










57 


00014 








00374 


00548 








oooii 




00:34 


00:140 


00377 






3 








00136 


IXfiK 






00747 




M_ 


0001 s 


oofloi 


00137 








00751 






BO- 


88° 


87" 


MB" 


86° 


84- 


SB* 






C09KCANTB, 



264 


NATURAL 


SECANTS AND COSECANTS 




"^ 


SECANTS. 




J. 


8° 


9= 


10- 


II" 


!«■ 


13° 














1.03034 


1,03630 


60 




00755 


















iXI?S8 




013G6 
















00895 


oiaei 


ol^5S 


01889 


oasss 






1 


00765 








01895 


0M50 


02658 




B 














03665 


6S 


6 


00773 










OtSTS 








OOTTfl 


oioia 


oip 


79 


oiBie 


















0IB18 


0«85 


03686 




g 












02291 


02693 


SI 




00787 


010^4 


01S94 






09396 




10 




1.007B1 


I.OIOSB 


1.01398 


1.0 60 


1.01B36 


1.03304 




49 










060 


OIWI 


03811 


0S7H 


48 


13 
















47 




00802 










033S3 




46 






0.040 


01318 


o«s 


aiD.'« 


033W 










OlOM 






01W5 


03336 


0*742 




IT 


ooeia 










03343 


01749 


43 




0081 T 


01059 






019TT 


02349 




42 




oosei 


O106.1 


01337 


043 


01BWS 


0S356 








00833 








oim 


03309 


OBTTO 


40 




1.0083? 


l.oiori 










1.0K77 


TO 




oofisa 


01078 


'oi.lBl 


'0B5B 




'oasra 








D0S3S 


0.090 


01356 




imn 


03382 








OOBIO 








0=013 


D338a 




» 




00844 


DI0S9 








023S5 




as 




00818 


oio'Ja 




681 


l»U5 






u 




00851 




01376 


687 


0S031 


01408 




33 




00895 


Olio: 


01381 




03037 


03416 


0282T 


33 


^B 


0OS3S 














n 


30 


0086S 


mill 












30 


HI 


1.00867 




1.0I39S 


1.01T00 


1,05055 


1.034.15 


1.03819 




S3 












0*4(1 


(txaa 


28 




110875 










o»4a 


friscs 


S7 




00879 


1«8 












26 




0088'^ 




0141S 


omi 




03461 


03879 




3S 


a)3S6 








(,ao«5 


0346S 


03983 


24 




00890 










0«74 


KSH 


S3 


38 


0D894 


146 














39 


OOSBS 




OHM 


01763 




owes 


0(907 




« 


009D3 




01440 


017B8 


oiiio 


OUM 


0«914 


20 




1.0O90A 


1.0 ICO 






1.05116 


1.03501 


1 .03981 


10 






lU 


01450 








02938 








lOB 




0I77S 


OBias 


03515 






44 








OliSI 


osist 


03521 


02S4S 


10 














02538 


02050 


IS 




000» 


182 


014T1 


017IK 




03535 














01T9S 




03543 


03M5 




48 








01S03 


0SI5B 


03548 


03973 


It 










oieng 






moo 










01491 


0.815 








10 


M 








1.0.880 


1.05178 


1.025BS 


.OKM 


















mat 


8 




009M 




01,'iOG 


0183-; 






030DS 






0DPS8 






01837 


03196 


0^89 


OMIT 


e 














03B96 


00034 


5 




00056 


012'J8 
















00970 


01;JM 


oisn 


018M 






03031 












OIHOO 


0»i31 


09017 


08W6 


3 










01860 


oassa 


nEB34 


03054 




M 


ooiwa 


01347 


01 .'.13 















8!^~ 


HI" 


^~80'"' 


"je^"" 


V«" 


j"j^ 


70° 


~ 


J- 


COJJECAN-TB. 





NATURAL SECANTS AND COSECANTS 


265 


' 


SECANTS, 


.' 


14° 


16" 


10° 


1 7" 


18" 


10° 


SO" 




LOaoti 


i.oMas 


1,01030 


1.0456B 


1,05146 




.08418 


60 




oaoee 


oasse 










054^ 






03078 








05106 


D578S 










ossEfi 




01587 


05178 


05704 


061.52 




4 


OBOBl 


03560 


OWK 


04008 


05160 




06463 






09000 


















OS10« 






D4625 




05836 














W635 


05210 


05830 




CB 


e 


oBiai 








06iW8 




06508 




» 


03190 




OIIW 












JO 


OS! 37 


03600 




01863 




05860 






11 




1.0SB17 


i,Mia6 




1,05288 


1, 05979 


l.OOMB 




19 


OBIM 


038S5 


04135 




05a6ii 








IS 


mm 


0363.1 










06565 






08167 






04700 




05911 


06577 














OflSB7 










03182 


086S8 


WHO 












17 


01100 


03606 


oino 


UT-JO 




C0941 


00611 






03188 






04738 






088>S 


43 




atxs 


03683 




OI74S 


05337 










08-Jia 


OSflBI 


waio 












11 


1.08M1 


1.03600 


l.owil 






1.05987 


,06657 


















05908 






3S 




o; 






0*780 


05378 








« 


(SUM 


Of 


a 


04M1 


04705 


053SS 




06891 






flSffil 




-m 


D43HI 








0670.1 






oseso 












06041 






la 










04824 


054IB 


06052 






S8 


OOKS 


0! 


B8 


012T7 


01B34 


05420 










0S3^ 




M 










08749 


















06085 


00781 




SI 


I.OSS08 


1,0! 


•83 


1.MS04 


1.04803 


l.DMBO 


i.oeoM 






K 


CSSM 




■Bl 


W313 


WSW 


05470 
























06796 














05490 


06130 








03SM 


03818 


013« 


04901 


05501 




06810 




36 








04911 


05511 




06S31 




S7 














06843 








0S812 
















08380 


03SSO 








061B4 






« 


01368 




0)385 


04050 


05653 












1.03867 




1.04B50 






1.06880 




4( 








O4D60 




"08317 






a 


03393 








oum 














OH*! 




05504 


OB'^S 


O00.'5 




M 




oaoi 


M43I 






08=50 














06008 




O0361 
























(S 


OMBi 




OHiS 




05556 




D60;3 


















06984 












06047 




08306 






Bl 


l.OSiM 




1.04186 


1.05IK7 


1.05067 










0S4Sa 


03901 










"o70ia 
















06339 








OSITS 








OSfiSB 


06350 








0S«7 


03BB7 


015'J3 


0)097 


05700 








H 




03005 


01S3a 








07067 




m 




otooi 






0.^730 


06.3M 








OBII 






OSI'JO 


05741 








B» 


DSUU 


OMWI 


Oisqo 


05138 










K 


D39SS 








057BS 


oaiiH 


07115 


"T 


78- 


74-^~ 


78* 


SECANT 


-Jiii 


JO" 






CO 









266 


NATURAL 


SECANTS AND COSECANTS 




~ 


eECANI*. 




ei- 


IE* 


_98°__ 


04. 


aa° 


sa- 


a?" 










.0046) 1. 








eo 




'oiise 


0786* 


oswfl 


00478 














07870 


OBMi! 


0910-J 


0388 


11293 


2200 












0950U 


0383 




K83 


57 












0398 






EO 




071T4 


07017 


0STO3 




0413 




3310 










08717 


0»49 


)428 


11355 


2333 


















234B 


















2366 


52 




07328 


07M8 


08757 






11403 


2383 










08771 


09S06 


0488 


11419 


2400 










1.08784 






.11435 1. 




49 




075Sfl 














4H 




07-J71 


0801S 


08811 


09619 




1407 




4T 








08825 


OB063 


MJB 


1483 










O80t» 


08839 










4S 




07307 




08863 








2501 






OTSaO 


08071 


08800 


DB707 


C691 


irai 


2518 


48 






0B084 


OBsao 


09731 


0O08 




25M 








08097 


08803 














DTSse 














40 


£1 


1.0JS88 


i.oeiM 


1.089S1 


1 ,09764 1, 


066S 


. 15S5 I 


25ffi 


89 


23 




06135 




09770 


KTO 




3002 


S8 








08948 






16-;7 








07405 


oa:Gi 












90 


as 


07417 


08174 


08076 






IS5B 












ososa 


00S37 




1675 


so™ 


84 












0747 




2087 


33 




07154 
















39 


074SO 


082^6 














30 




08239 


00044 


09895 




1740 


2738 






1 07-191 


1,0^5S 






080B 








St 


orws 


08205 








17T2 






M 




(BST8 


0908B 










SI 


» 








09963 


9854 


1805 


2807 


28 












0870 










0^53 


0331B 














S7 




U8331 










3868 




M 








10012 




1870 


2B75 


22 


3a 


OTSSM 








00.^2 


1880 


3892 




40 


07008 


08370 


09183 










» 


n 








l.IOffiO 1. 




I.IIBIO 1, 


3927 


>s 


a 






09211 




0978 


1936 


2944 


16 




O7&I0 








0991 












08423 










2979 




45 






00252 








BOBO 




a 








10130 


041 


2001 


30IS 


14 












050 


■J018 










08476 








2034 






4,9 






09308 










31 


BO 


07737 




09333 


10189 


103 


2097 


8083 


10 




1.07740 


1 08616 


1.00.137 






. 3084 1. 




e 






08529 


09351 












53 






09305 








3135 


I 








09379 




100 


S133 




e 






0S569 






161 












08582 


09)07 


(K7S 










57 




0B9g« 




020.1 


213 


«83 


3205 


I 


68 


ama 








320 


2100 




a 




07S41 


08623 








3310 










08ft» 


094S4 




ifio 


2233 


3257 







08" 


07° 


uo- 


65- 


64° 


as° 


flS" 




C03E(?ANTa. 





NATORAL SECANTS AND COSECANTS 267 



la's? I. 






.ie6«3 l.I 


i«M 


4»M 


HBfl 


16884 1 


1S»3 


4ara 




1B704 1 


IS31D 












S54S 




18345 


44^8 


5587 


IflTW 1 


18862 








isaso 






13808 1 




4488 




IBHS7 1 




iSK 




16ftJ8 1 


13133 












58S4 


.18888 1.1 


tuns 


txa 


s™ 


I0«« 1 


IHBS 


4578 






I3BM 




li 


l^* 1 




4sas 


ST82 


imi I 


lasj? 


IMl 




iTOia 1 


mra 


4870 


SSffi 














4707 


5801 


i:075 1 


1M2S I. 


4746 1. 


5881 


. 7095 1.1 


I864B 








I3M4 












^B40 


7158 1 


ISTOO 


4801 






18718 




















80IB 


TiMl 1 


ISWl 


4877 




7262 1 


ISTBK 


4SM 










i!a70 


.7301 1.1 


■see 




OM 




13M8 


49K 






lassi 














7388 1 


13807 


WOB 


17D 


7408 1 


18918 








lasM 














17178 ^f 


I»70 


5085 


0MB 


17198 1 


11868 1. 








t«ns 








IMM 


5143 


aaiB 


7558 1 


IWU 


5183 


aaw 




lUIl 


5181 
















5818 


6400 


Ml 1 


14118 


H-M 


84 iM 
















8480 






sm I. 


8481 


■ ;?S '■' 


141B8 


5316 












77SS 1 


ma 






7790 1 


HS4a 


5373 


85(W 


7811 1 


uaes 


6393 


B6S2 


78SS 1 


14IH0 








14SN 






1787S 1 


i4«ir 


8451 


8&13 


17SB0 1 


1433S 


5470 


8883 





loaai aowia 



1B53I sooai 



ie8a« 21IK7 



1W91 eiuos 



IB783 E1XI5 



19874 ivias 



1.21385 
31389 

xan 81438 

afm SI -ISC 

aM5S 21487 

*B83 B15II 

20108 (1535 

18787 20129 31580 



»a 20292 SI 731 

18945 ao3ie ei756 



sojgtt Siioss 

20822 lfiiU77 



268 



NATURAL SECANTS AND COSECANTS 



1 






SECANTS 


. 






/ 






86« 


86« 


87" 


88"» 


89° 


40° 


41* 




1 


1.22077 


1.23607 


1.25214 


1.26902 


1. 28076 


1.30541 


1.88501 


60 


1 


1 


22102 


23633 


25241 


26931 


28706 


80573 


82585 


59 


2 


22127 


23659 


25269 


26960 


28737 


80605 


82568 


58 


3 


22152 


28685 


25296 


26988 


28767 


80686 


82802 


57 




4 


22177 


23711 


25324 


27017 


28797 


80668 


82686 


56 




5 


22202 


23738 


25351 


27046 


28828 


80700 


82669 


55 




6 


22227 


28764 


25379 


27075 


28858 


80732 


82703 


54 




7 


22252 


23790 


25406 


27101 


28889 


30764 


82787 


58 i 


8 


22277 


28816 


25434 


27133 


28919 


80796 


82770 


52 




9 


22802 


28843 


25462 


27162 


28950 


30829 


32804 


51 




10 


22327 


23869 


25489 


27191 


28980 


80861 


82S88 


50 




11 


1.22352 


1.23S95 


1.25517 


1 .27221 


1.29011 


1.30893 


1.82872 


49 




12 


22377 


23922 


25545 


27250 


29042 


30925 


82905 


48 




13 


22402 


23948 


25572 


27279 


29072 


80957 


82989 


47 




14 


22428 


23975 


25600 


27808 


29108 


80989 


82973 


46 




15 


22453 


24001 


25628 


27337 


29138 


81022 


88007 


45 




16 


22478 


24028 


25656 


27366 


29164 


31054 


88041 


44 




17 


22503 


24054 


25683 


27396 


29195 


81066 


88075 


48 




18 


22528 


24081 


25711 


27425 


29226 


31119 


88109 


42 




19 


22554 


24107 


25739 


27454 


29256 


81151 


88148 


41 




20 


22579 


24134 


25767 


27483 


29287 


81188 


88177 


40 


21 


1.22604 


1.24160 


1.25795 


1.27513 


1.29318 


1.31216 


1.88211 


39 


22 


22629 


24187 


25823 


27542 


29849 


31248 


88245 


38 




23 


22G55 


24213 


25851 


27572 


29380 


81281 


88279 


37 




24 


22680 


24240 


25879 


27601 


29411 


81818 


83314 


86 




25 


22706 


24267 


25907 


27630 


29442 


81846 


88848 


85 




26 


22731 


24293 


25935 


27660 


29478 


81878 


88882 


84 




27 


22756 


24320 


25963 


27689 


29504 


81411 


88416 


S3 




28 


22782 


24347 


25991 


27719 


29585 


81443 


88451 


82 




29 


22807 


24373 


26019 


27748 


29566 


81476 


88485 


81 




30 


22833 


24400 


26047 


27778 


29597 


81509 


88519 


80 




31 


1.22858 


1.24427 


1.26075 


1.27807 


1.29628 


1.81541 


1.88A54 


29 




32 


22884 


21454 


2C104 


27837 


29659 


81574 


83588 


28 




33 


22909 


24481 


26132 


27867 


29690 


81607 


88822 


27 




34 


2^935 


24508 


26160 


27896 


29721 


81640 


88657 


26 




35 


22960 


24534 


. 26188 


27926 


2975.2 


81672 


88691 


25 




36 


22986 


24561 


26216 


27956 


29784 


81705 


88726 


24 1 


37 


23012 


24588 


26245 


279^5 


29815 


81788 


88780 


28 


38 


28037 


24615 


26273 


28015 


29846 


81771 


88795 


22 


39 


23063 


24642 


26301 


28045 


29877 


81804 


88880 


21 


40 


23089 


24669 


26330 


28075 


S9909 


81887 


83864 


20 ! 


41 


1.23114 


1.24696 


1.26358 


1.28105 


1.29940 


1.81870 


1.88809 


19 




42 


23140 


24723 


26387 


28134 


29971 


81908 


88984 


18 




43 


23166 


21750 


26415 


2SI64 


80008 


81936 


88968 


17 




44 


23192 


24777 


26443 


28194 


80034 


81969 


84008 


16 : 


45 


23217 


24804 


26472 


28224 


80066 


82002 


84068 


15 1 


46 


23243 


24832 


26500 


28254 


80097 


82085 


8407S 


14 


47 


23269 


24859 


26529 


28284 


80129 


82068 


84108 


18 




48 


23295 


24886 


26557 


28314 


80160 


82101 


84142 


12 




49 


23321 


24913 


26586 


28844 


80192 


82184 


84177 


" - 




60 


23347 


24940 


26615 


28374 


80228 


82168 


84212 


10 • 


51 


1.23373 


1.24967 


1.26643 


1.28404 


1.80255 


1.82201 


1.84247 


9 , 


52 


23399 


24995 


26672 


28434 


80287 


82284 


84282 


8 


53 


23424 


25022 


26701 


28464 


80318 


82287 


84817 


7 


54 


23450 


25049 


26729 


28495 


80350 


82801 


84858 


6 ' 


55 


23476 


25077 


26758 


28525 


30382 


82884 


84887 


6 


56 


S«502 


25104 


26787 


28555 


80413 


82868 


84428 


4 


57 


23529 


25181 


26815 


28585 


80445 


82401 


84458 


8 . 


58 


23555 


25159 


26844 


28615 


80477 


82484 


84498 


2 


59 


23581 


25186 


26873 


28646 


80509 


82468 


84523 


1 


60 


23607 


25214 


26902 


28676 


80541 


82501 


84568 


1 


/ 


64» 


68» 


62» 


610 


60» 


40« 


480 


1 








COSECANTS. 



















NATURAL 


SECANTS AND COSECANTS 


269 




8ECAKT8. 






*»' 


it' 


w 


45° 


48° 


47" 


48" 












1.4'ffi56 


.466^ 








■«6W 


' 38710 


■ 39065 


■«483 


43989 


46674 


■48IOO 






«SS4 


MB07 


S90B5 










98 




meas 




























48611 


49841 






M7» 


36819 


3921B 


41837 


44178 


46897 


48680 


55 




urn 


36858 


30S61 


41888 


44817 


48903 


497SS 


H 








39381 




44360 




487B7 






S4g4« 










489B5 


49839 






um 


3706S 




4179.1 


44847 


470J1 


49884 


51 




B«17 


87105 


SOIOO 


41835 




47087 


49938 


60 
































'47190 


'sooao 






3S0M 
35000 


B7J18 
37255 


3056C 


43001 


S 


47336 
47278 


s 


! 










45^4 






50328 






SS1S7 


37308 






44fl9H 


47411 


90W5 


43 




83303 


37408 


m-a 


4«68 


44743 


4745S 


90S34 


4» 




KS!B 
































40 




1,85310 


I.J7S1B 


1.398+1 




H4e7S 


1,47608 


.90471 


89 




B5S4fl 




asBSj 


42835 








38 
























3763a 








4773S 


50619 


36 




3MM 


37870 


■10003 


4'im 


450(13 


■;7T84 


90889 


S9 




3M90 


37703 


«>04S 




45068 














■tooes 








50767 


as 












451 B5 


47935 


60817 


93 




SSSBB 




40163 




4S2ffl 


47978 


50M8 


31 




3S38J 


S7M0 


40S03 


42678 






SOBta 


SO 




I.SseTO 


1,37S98 








1.48068 












'403S3 




'4E88a 




'51OIB 


38 








40S!M 


42:88 


45408 


48160 




27 






smi 


*JS51 


4S941 


4945a 


48307 




38 






38061 




4-S83 




48294 
















49542 




61 31 6 


34 




ISBSB 




404B5 






46349 




23 




aS9M 


SSIW 


406SB 


43011 




48888 








35M1 


B8204 




























51415 


30 




1.3WB4 


i.ssasQ 


l.fOfllB 


1.43139 


1.45788 


.48938 


1.91466 


IB 




BOOTO 


88310 


40B87 


431 SI 


4681 1 


4B5B8 


































49901 


48681 


61816 


16 




SBIBO 


S8I31 


4(»08 


43S10 


45941! 


4BT38 


91065 






M817 






433W 


45Ma 










saasa 








46037 




























a<i8i7 


3SSSB 


40B7I 




48137 


48819 


51887 


11 




Mses 


8tlfl2S 




43aS4 


46178 


48987 


91918 


10 






t.secw 




1 .43riB7 












'»I437 














8 




W*74 


387M 


41134 


438B3 


48309 


49111 


S3069 


7 






3,1783 


4117S 


43896 


463M 


49199 


B2ia) 


e 












48400 










3E&SS 








48445 




G2-»2 






se<i?i 


S 


WISH 
413S8 


4arao 


48481 
48937 


493D3 
40351 


63373 
52383 






MflM 












B3874 


















53435 






47> 


te- 


46" 


" i4'~ 


"'43"'' 


4S" 


41" 






COSECANTS. 



270 



NATURAL SECANTS AND COSECANTS 



/ 


SECANTS. 


/ 


49« 


60«» 


61° 


62° 


58° 


64° 


65° 





1.52426 


1.56672 


1.68902 


1.62427 


1.66164 


1.70130 


1.74345 


60 


1 


52476 


66626 


58959 


62487 


66228 


70198 


74417 


59 


2 


52627 


65680 


69016 


62548 


66292 


70267 


74490 


68 


3 


52579 


65734 


69073 


62609 


66a')7 


70885 


74662 


57 


4 


52630 


66789 


69180 


62669 


66421 


70403 


74685 


56 


5 


62681 


56843 


69188 


62730 


66486 


70472 


74708 


55 


6 


52732 


55897 


69245 


62791 


66550 


70540 


74761 


54 


7 


52784 


65951 


69302 


62852 


66616 


70609 


74651 


58 


8 


52835 


56005 


59360 


62913 


66679 


70677 


74927 


62 


9 


52886 


66060 


69418 


62974 


66744 


70746 


75000 


61 


10 


52988 


66114 


59475 


63035 


66809 


70616 


7S0:3 


60 


11 


1.62989 


1.56169 


1.69633 


1.63096 


1.66873 


1.70684 


1.75146 


49 


12 


63041 


56223 


69590 


63157 


66938 


70953 


75219 


48 


13 


63092 


56278 


69648 


63218 


67003 


71022 


75293 


47 


14 


63144 


66332 


59706 


63279 


67008 


71091 


75366 


46 


15 


58196 


66387 


69764 


63341 


67133 


71160 


75440 


45 


16 


63247 


56442 


69822 


63402 


67199 


71229 


75518 


44 


17 


68299 


66497 


69880 


63464 


67264 


71298 


75587 


43 


18 


53361 


66551 


69988 


63526 


67329 


71868 


75661 


42 


19 


53403 


66606 


69996 


63687 


67394 


71437 


76734 


41 


20 


63465 


56661 


60054 


63648 


67460 


71506 


75808 


40 


21 


1.63507 


1.66716 


1.60112 


1.63710 


1.67525 


1.71576 


1.75882 


39 


22 


63669 


66771 


60171 


63772 


67591 


71646 


75966 


86 


23 


63611 


66826 


60229 


63834 


67656 


71716 


76031 


87 


24 


63663 


66881 


60287 


63896 


67722 


71785 


76105 


86 


25 


53716 


66937 


60346 


63957 


67788 


71865 


76170 


35 


26 


63768 


66992 


60404 


64019 


67853 


71925 


76258 


84 


27 


63820 


67047 


60463 


64081 


67919 


71996 


76828 


88 


28 


63872 


57103 


60621 


64144 


67985 


78066 


76402 


32 


29 


63924 


67158 


60580 


64200 


68051 


72136 


76477 


81 


80 


63977 


67218 


60639 


64268 


68117 


72206 


76652 


80 


81 


1.640-29 


1.57269 


1.00698 


1.61330 


1.68183 


1.72275 


1.76626 


29 


32 


64082 


57324 


60750 


64393 


68250 


72346 


78701 


88 


33 


54134 


67380 


60815 


64455 


68316 


72416 


76776 


87 


34 


64187 


67436 


60874 


64618 


68382 


72487 


76851 


86 


36 


61240 


57491 


60933 


64680 


68449 


72667 


76086 


85 


36 


54292 


67547 


6099;! 


64643 


68515 


72628 


77001 


84 


37 


54345 


57603 


61051 


64705 


68682 


72696 


77077 


88 


38 


54398 


67669 


61111 


64768 


68648 


72769 


77158 


88 


39 


54451 


67715 


61170 


648:11 


68715 


72640 


77887 


81 


40 


64504 


67771 


61229 


64894 


6878;^ 


72911 


77808 


80 


41 


1.54657 


1.57827 


1.61288 


1. 04957 


1.68846 


1.72962 


1.77878 


19 


42 


54610 


67883 


61348 


65020 


68915 


73058 


77454 


18 


43 


54663 


67939 


61407 


65088 


68982 


78124 


77580 


17 


44 


64716 


67996 


61467 


66146 


69049 


78195 


77800 


16 


45 


64769 


68061 


61526 


66209 


69116 


78267 


77881 


15 


46 


64822 


58108 


61586 


66272 


69183 


73838 


77757 


14 


47 


54876 


68164 


61646 


65336 


69250 


7IM09 


77888 


13 


48 


64929 


68221 


61705 


66399 


69818 


78481 


7TB10 


13 


49 


64982 


58277 


61765 


65462 


69385 


78562 


77986 


11 


60 


55036 


58333 


61825 


65526 


69452 


73624 


78062 


10 


61 


1.56089 


1.68390 


1.61886 


1.65689 


1.69520 


1.78696 


1.78188 


9 


52 


65143 


68447 


61946 


66658 


69687 


78768 


78815 


8 


63 


55196 


68608 


62005 


65717 


69655 


78640 


7889] 


7 


64 


66260 


68660 


62065 


65780 


69723 


73911 


78868 


6 


66 


65803 


68617 


62125 


65644 


69790 


78983 


7844& 


5 


56 


55867 


68674 


02185 


65908 


69658 


74066 


78681 


4 


67 


65411 


68781 


62246 


66972 


69926 


74128 


7%08 


8 


68 


66465 


68788 


62306 


66036 


69994 


74200 


78676 


8 


69 


65618 


58846 


62366 


66100 


70062 


74272 


78769 


1 


60 


65672 


68902 


62427 


66164 


70130 


74845 


76889 





/ 


40* 


89» 


88° 


87° 


86° 


86° 


84* 


r • 








COSECANTa 









NAT 


URAL 


SECA 


NTS A 


lND C 


OSECi' 


USTTS 


27] 





SECANTS. 


f 


66» 


67« 


58° 


69*» 


60° 


61° 


62° 




1.78639 


1.83606 


1.88708 


1.94160 


8.00000 


8.06267 


8.13005 


60 


1 


78906 


63690 


88796 


94854 


00101 


06375 


13188 


59 


2 


78984 


83773 


88684 


94349 


00808 


06483 


13239 


58 


3 


79061 


88855 


88978 


94443 


00303 


06592 


13356 


57 


4 


79186 


68938 


69060 


94637 


00404 


06701 


13473 


56 


5 


79816 


64080 


69148 


94638 


00505 


06609 


13590 


55 


6 


79898 


64103 


69237 


94726 


00607 


06918 


13707 


54 


7 


79871 


84166 


69385 


94821 


00708 


07087 


13885 


53 


8 


79449 


64869 


89414 


94916 


00810 


07137 


13948 


58 


9 


79587 


64358 


89503 


95011 


00918 


07846 


14060 


51 


10 


79604 


84486 


89591 


95106 


01014 


07356 


14178 


60 


11 


1.79682 


1.64516 


1.89680 


1.95201 


2.01116 


2.07465 


8.14896 


49 


18 


79761 


64601 


89769 


95296 


01818 


07575 


14414 


48 


IS 


7S639 


64685 


89658 


95392 


01380 


07685 


14533 


47 


14 


79917 


64768 


89948 


95487 


01422 


07795 


14651 


46 


15 


79996 


64862 


90037 


95583 


01525 


07905 


14770 


45 


16 


80074 


64935 


90186 


95678 


01628 


08015 


14889 


44 


17 


80158 


85019 


90816 


95774 


01730 


08126 


15008 


43 


18 


80881 


85103 


90im 


95870 


01833 


08286 


15187 


48 


19 


80809 


65187 


90395 


95966 


01936 


08347 


15246 


41 


HO 


M0888 


85871 


90485 


96062 


02039 


08458 


15366 


40 


21 


1.80467 


1.85355 


1.90575 


1.96158 


8.02143 


2.08569 


2.15485 


39 


28 


80646 


85439 


90665 


96255 


08246 


08680 


15605 


38 


88 : 


80625 


85583 


90755 


96351 


08349 


08791 


15725 


37 


84 • 


80704 


65608 


90845 


96448 


08458 


08903 


15845 


36 


25 


80783 


85698 


90935 


06544 


02557 


09014 


15965 


35 


26 


80668 


85777 


91086 


96641 


02661 


09186 


16085 


34 


27 


80948 


85861 


91116 


96738 


08765 


09838 


16206 


33 


88 


81021 


65946 


91207 


96835 


08869 


00350 


16326 


38 


29 


81101 


86031 


91297 


96932 


02973 


09468 


16447 


31 


80 


61180 


86116 


91388 


97089 


03077 


09574 


16568 


30 


81 


1.8i860 


1.66801 


1.91479 


1.97127 


8.03188 


8.09686 


2.16689 


89 


88 


. 81340 


66886 


91570 


97224 


03886 


09799 


16810 


88 


83 


81419 


86371 


91661 


97328 


03391 


09911 


16932 


27 


84 


81499 


66457 


91752 


97420 


03496 


10024 


17053 


26 


85 


61579 


66548 


91844 


97517 


03601 


101B7 


17175 


85 


86 


61659 


66687 


91935 


97615 


08706 


10250 


17297 


84 


87 


81740 


66713 


92027 


97713 


03811 


10363 


17419 


83 


88 


61880 


66799 


92118 


97811 


03916 


10477 


17541 


28 


89 


81900 


66885 


98210 


97910 


04022 


10590 


17663 


81 


40 


81981 


66990 


92302 


98008 


04128 


10704 


17786 


20 


41 


1.88061 


1.87056 


1.92394 


1.98107 


2.04233 


2.10817 


2.17909 


19 


42 


88142 


87148 


92486 


98205 


04339 


10931 


18031 


18 


43 


68882 


87229 


98578 


98304 


04445 


11045 


18154 


17 


44 


68303 


67315 


98670 


98403 


04551 


11159 


18277 


16 


45 


88384 


87401 


92762 


9a502 


04658 


11274 


18401 


15 


46 


88465 


87488 


92855 


98601 


04764 


11388 


18524 


14 


47 


88546 


87574 


92947 


98700 


04870 


11503 


18648 


13 


48 


68627 


87661 


93040 


98799 


04977 


11617 


18772 


12 


49 


68709 


87748 


93133 


98899 


05084 


11782 


18895 


11 


50 


68790 


87834 


93286 


98998 


05191 


11847 


19019 


10 


51 


1.88671 


1.87921 


1.93319 


1.99098 


8.05298 


2.11963 


2.19144 


9 


58 


68953 


88008 


93412 


99198 


05405 


12078 


19263 


8 


58 


63034 


88095 


93505 


99298 


05512 


18193 


19393 


7 


54 


83116 


88183 


93598 


99398 


05619 


18309 


19517 


6 


55 


63198 


68870 


93692 


99498 


05727 


12425 


19642 


5 


56 


63880 


88357 


93785 


99598 


06835 


12540 


19767 


4 


57 


63368 


88445 


938T9 


99698 


05942 


12657 


19898 


3 


56 


68444 


88538 


93973 


99799 


06050 


12773 


80018 


8 


59 


63686 


68680 


9406G 


99.S99 


06158 


12889 


80143 


1 


60 


63606 


88708 


94160 


8.00000 


06267 


13005 


80269 





/ 


88» 


82<» 


81** 


SO- 


29° 


28° 


27° 


/ 


COSECANTS. 





272 


NATURAL SECANTS AND COSECANTS 






SRC ANTS. 






8S' 


M" 


lls-' ««' 


61' 


68- 


BO- 
















2.79041 


60 




'20385 


"ss-iia 






'87139 




69 




awai 


28360 


seeiu «i8i 


tatsi 


0783! 




68 




MB47 




370ai J03K 


561S8 


6T5JS 


79679 


s; 










56684 


6m8 


79991 








£B80D 










M . 




sinau 


aee»T 


S760IJ 4flfflf7 




66106 










sBo;i 


37658 48ii89 


*:'":'! 


68299 


80531 
















80748 






















81635 


■J»i87 


38107 47477 








60 




a.aisfs 




s.aaaM s.j7Bto 


s.Msrfl 


2.09079 


a.8189U 


49 








S84MI 47»«I4 




M275 


81605 




la 


aiuis 




asBBs 4nK7 












K045 


30W0 


88707 4BI31 




09667 


S2aa7 


48 








888,^7 4B«5 


6859! 


09864 


82254 


46 


a 








68771 


70MI 










3MST 


39159 48Sa4 








48 




Nnr>9 


30aiH 


9931] 48789 






aiSOQ 


48 




a^HSS 


30r85 


3ms 489.-4 


69311 


TOflfia 


83124 




w 


^IT 




3W14 49119 


69491 


TW«l 


ss34a 


49 




s.iem 




£.39760 a. 49-^84 


3.6Do:a 




8.83601 


39 








see 18 49J.W 






B87B0 


88 




2S:M,') 




40070 49010 


COOSB 


1448 


89999 


»7 


a* 








eo-ii7 




8iil9 


86 




ItUSI 






003119 










B3ASI 


aiTir 


405-J8 60115 


60581 


11047 


84^ 










40081 6oasa 


00763 


m? 


84880 


S 










60940 




B5102 


a 


fa 


!»eea 








n649 


853-^3 








3«8J 


4I1I3 50784 


6131B 




85545 


30 








SliaflS 8.50952 


3.01)90 


a.73oas 


8.83707 


S 


M 


'iMsrs 








"8254 


89ei» 


19 




!4»» 


33708 


4IU1S 61:.'l9 


01864 


8460 








UIMO 


sa850 


417« 61J57 


G<W9 


8659 










S»93 


41914 Moae 


oa2»4 


-3808 




85 


H 


ai903 








4006 






a: 




33«:8 




OMM 


4209 










S3«ti 


44390 MI34 




4478 




U 


■it 






12586 B3WJ 


ftfSTfl 


4077 




s. 


40 


iisiSJ 


83708 










80 






3.3SS9a 


!. 42848 a. 63045 




S 76086 


9.88011 








33B06 


430l>5 BS815 


03*3.1 


JB29» 


88998 




a 






jsifl'j Bao86 


6S72i 


nHB7 


»405 


17 




■»Wi 




43318 581.^7 


08909 












3H-J9 


43l71i S33l!9 










le 




WKS 


43833 63.W0 


»4a»5 


^81 10 


89148 




« 


ami 




43790 638TJ 




T61Ha 


80370 


3 


48 


M4X 


3I8D3 


43948 53815 




TSMO 


89006 




4S 




:am 


4410U 54017 


64851 


:<St37 






U 






44204 5)190 


65040 


T094S 


90CMS 





fia 


s.xm 


S.ar-MO 


a.4<4t!3 3 54303 
4468'.' 51630 


S^ 


8,77164 


8.90203 


I 


na 






4)741 64709 


66609 








M 








6*799 


77780 


909HS 




M 


Km 


3r>8ai 


45059 660S7 


66989 


77990 


om7 










46S1B 55if3l 


66180 


78200 












4.5378 55405 


CI13TI 


78410 


91081 








aes!5 


45539 It»80 


66608 


78621 


aisu 




s» 




IKt73 


4.'i09g 56755 




78832 


B2147 


I 








4,1859 56930 


669*7 


79048 


«»90 





"T 


8<" 


Hi- 


St' SB- 


ea- 


ei- 


10° 










COSECANTS. 











NATUHAL SECANTS AND COSECANTS , 


273 


'~~^ 


PKCiN'TS. 


' 




•SO' 


II" 


7e° 


13' 


W 


Jo" 


76° 





s.Biaso 


B.inss 




a. 4:^30 


3.62796 


B.86S70 


4.13357 












44356 


63164 


88:911 


13S39 


69 




99810 






4*683 




87211 




6S 


■ 


noes 






430J0 












ouie 


08197 


B47?D 


43337 




BSDG6 


16865 






SKM 




mei 


43066 


6464S 


88178 


167S8 


66 










48995 


95018 


88904 






1 


«OES 


D8S8S 












68 


B 


MHI3 


09a4S 












S2 








Mi'sr 


44e8n 




9UTB1 


17744 


61 










45317 


■ 66515 






60 


II 


a.WBTa 


*'ic^ 


*'a??aB 


'45983 


8-eW9J 


^'illi-'i 


4.18733 
19988 


48 


13 






87480 


«Slfi 




9iB0'4 


1BT85 


47 


11 


Dsafli 




ST7I7 


46651 


6BCW5 




ao844 




IS 


B»S1 


]I^T 




46980 


68785 


98770 
93204 


91284 


44 


11 






mji 




69)117 


93640 


81786 




18 


leess 






47906 


60549 




2*i80 




IS 


IH.>W 










94■^i4 














48671 






83239 




SI 




B. '^«9 


8.39814 


S.49010 


8.70700 


8.95398 


4.837W 


39 


a 


'«7ei8 








71085 


96832 


24S5S 






fiTBW 


S4B 




49691 




99374 








«8ioe 


S^ 




5003-; 




96716 






» 


Bsua 




31084 


60374 








36 


u 


tMBH 








7S635 


976M 


89300 




ST 


seras 






filMO 






MB14 






eeoss 












87380 






«B«B 


mi 


3M)4 




73803 


»eiH4 


27847 


31 


30 








6S094 


74198 


99393 




39 




s.ewEi 


S. 5i3» 














« 


s.oooer 


701 








4!doeri 




88 


S3 




»T9 


33471 


53134 






8993B 


87 








B37K3 


6348a 


75775 


01198 


3015* 


86 




oaeio 








76171 












' eoa 














ffr 


OIWS 


1033 


84713 


MMl 


76966 






S3 


SB 








64Sfi3 


77305 


03080 


88MO 


29 












77765 


08479 


SSOOO 




« 


ososr 

S.OiSOH 


a.iHiM 


B.SW 


3.5r,Bio 


78166 




33983 
4.34154 


SO 
1-9 










50894 


'78970 


'''oisoo 


Mm 






o«ia 








79374 












19040 












16 


4S 




18BSS 


37ffil 






06851 


xm 






03HB 




STS37 


6T718 


8QBSfl 


06717 


3683S 






OSSSI 


19480 






SM90 








4S 
49 


0«JB 

o«aw 


II 


S»4W> 


M7»1 




fmi 


3WM 










3SSU9 








.39018 






3.<VIS39 


i.sioii 






a.8s63a 


4.0006.3 


4.39558 








s\m 


'89448 












U 






89798 


60838 


PS457 
















60601 


83871 


10481 








OSBU 


£fira 




90965 
















407SI 


















4.057 


6169.'. 


85116 








AS 


OSUI 








SBB33 


18394 


48184 






OWM 










1287 


439H;.' 




1 " 




aSM7 


48030 


68796 


86370 








|~ 


!•• 


18' 


17" 


OSECAh 


16' 


14° 


II' 










"c 


TS 







274 



NATURAL SECANTS AND COSECANTS 



/ 






SECANTS 


• 






1 

/ 
60 


77* 


78» 


79° 


80° 


SV 


82° 


88° 





4.44541 


4.80973 


5.24084 


5.75877 


6.39245 


7.18530 


8.20551 


1 


4510^ 


81633 


24870 


76829 


40422 


20020 


22500 


59 


2 


45664 


82294 


25658 


77784 


41602 


21517 


24457 


58 


8 


46228 


82956 


26448 


78742 


42787 


23019 


26425 


57 


4 


46798 


83621 


27241 


79703 


43977 


24529 


28402 


56 


5 


47360 


84288 


28036 


80667 


45171 


26044 


80388 


55 


6 


479-28 


84956 


28833 


81635 


46369 


27566 


82384 


54 


7 


48498 


85627 


29634 


82606 


47572 


29095 


84390 


53 


8 


49069 


86299 


30436 


83581 


48779 


30630 


86405 


52 


9 


49642 


86973 


31241 


84558 


49991 


32171 


88431 


51 


10 


50216 


87649 


32049 


85539 


51208 


33719 


40466 


50 


11 


4.50791 


4.88327 


5.32859 


5.86524 


6.52429 


7.35274 


8.42511 


49 


12 


51368 


89007 


33071 


87511 


5.3655 


30836 


44566 


48 


13 


51947 


89689 


34486 


88502 


54886 


38403 


46632 


47 


14 


52527 


90373 


35304 


89497 


56121 


39978 


48707 


46 


15 


53109 


91058 


36121 


90495 


57361 


41560 


60793 


45 


16 


53692 


91746 


36947 


91496 


58606 


43148 


•62889 


44 


17 


54277 


92436 


37772 


92501 


598.55 


44743 


64996 


43 


18 


54863 


93128 


38000 


93509 


61110 


46846 


67118 


4i 


19 


55451 


93821 


39130 


94521 


62369 


47955 


69241 


41 


20 


56041 


94517 


4021)3 


95530 


63033 


49571 


61879 


40 


21 


4.56632 


4.95215 


5.41099 


5.91:555 


6.64902 


7.51194 


8.63528 


S9 


22 


57224 


95914 


41937 


97577 


66176 


52825 


65688 


88 


23 


57819 


96616 


42778 


98603 


67454 


54462 


67859 


87 


24 


58414 


97320 


43622 


99633 


68738 


56107 


70041 


36 


25 


59012 


980:.'5 


44468 


6.00606 


70027 


57759 


72234 


85 


26 


59611 


98733 


45317 


01703 


71321 


59418 


•74488 


84 


27 


60211 


99443 


46169 


02743 


7i620 


61085 


7B658 


83 


28 


60813 


5.00155 


47023 


03787 


739S4 


62759 


78880 


82 


29 


61417 


00869 


47881 


04a34 


75233 


64441 


81118 


81 


30 


0.i023 


01585 


48740 


05886 


76547 


66180 


88867 


80 


31 


4 C2030 


6.02303 


5.49603 


6.06941 


6.77866 


7.67886 


8.&'>628 


29 


32 


032:^8 


03021 


50468 


08000 


79191 


69580 


87901 


28 


33 


63849 


03746 


51337 


09062 


80521 


71242 


90186 


27 


34 


64461 


04471 


52208 


10129 


818.^6 


72962 


92482 


26 


35 


65074 


05197 


53081 


11199 


83196 


74689 


94791 


25 


36 


65690 


05926 


63958 


12273 


84542 


76424 


97111 


24 


37 


66307 


06657 


54e»37 


13350 


85893 


78167 


99444 


28 


38 


66925 


07390 


55720 


14432 


87250 


79918 


9.01788 


22 


. 39 


67545 


08125 


56605 


15.517 


88612 


81677 


01146 


21 


40 


68167 


08863 


57493 


16607 


89979 


83448 


08515 


SO 


41 


4.68791 


5.09602 


5.58:383 


6.17700 


6.91352 


7.85218 


9.08897 


19 


i'i 


69417 


10344 


59277 


18797 


92731 


87001 


11292 


18 


43 


70044 


11088 


60174 


19898 


94115 


88792 


18099 


17 


44 


70673 


11835 


61073 


21004 


95505 


90602 


10120 


16 


45 


71303 


125K» 


61976 


22113 


96900 


92400 


1R66R 


15 


46 


71935 


13334 


62881 


23226 


98301 


94216 


20999 


14 


47 


72569 


14087 


63790 


24343 


99708 


96040 


88460 


18 


48 


73205 


14842 


64701 


25464 


7.01120 


97873 


25081 


12 


49 


73843 


15599 


65616 


2()590 


02538 


99714 


28417 


11 


50 


74482 


16359 


66533 


27719 


03962 


8.01565 


80917 


10 


51 


4.75123 


5.17121 


5.67454 


6.28853 


7.06392 


8.0S428 


9.88480 


9 


52 


75766 


17886 


68377 


29991 


06828 


05291 


8B057 


8 


53 


76411 


18652 


69304 


31133 


08-^9 


07167 


88497 


7 


54 


77057 


19421 


702134 


32279 


09717 


09052 


41052 





55 


77705 


20193 


71166 


33429 


11171 


10946 


48020 


5 


56 


78355 


209(i6 


72102 


34584 


12630 


12849 


40208 


4 


57 


79007 


21742 


73041 


35743 


14096 


14760 


48600 


8 


58 


79661 


22521 


73983 


36906 


15508 


16681 


61411 


2 


59 


80316 


23301 


74929 


38073 


17046 


18612 


640S7 


1 


60 


80973 


24084 


75877 


89245 


18530 


20651 


60077 





/ 


12° 


11° 


10** 


9° 


8° 


70 


f 


/ 






COSECANTS. 









NATUHAL SECANTS AND 


COSBOANTS 


275 




SECANTS. 




84° 


SS° 


86° 


8;° 


88° 


89^ _ 




B.. 16077 


11.47371 








K .£98119 






omj 


61199 


'3W47 


■^1397 


■p9440 


r-i.W9;B 










.155Sa 


ajiB-j 


39.13017 


50.27461 












43088 


8881^ 


00.81411 


57 














61.39106 




6 


T01M 


66769 


61011 






8I!.8071S 




« 


7iB33 




70!U8 


7C560 


Sfl.lOlSO 


63 66460 


64 


































> 


81119 


Ba76S 


893^8 


H). 11208 




07:40997 




10 


smia 


B6bS7 


9&788 


awas 


31.Si7S8 


68.75730 


BO 








15,02810 






















7i!62a)r 




18 


BSS89 


B9-il4 




»S4l 




73.14683 






»5M8 


18.08307 


asm 


71737 


43071 


74.78586 










28979 






76.39655 
















™. 13*74 






'osm 






-ji.09t«a 


■38IIB 


79.949U« 




IS 


(ie84> 


S04U'7 


49011 


W85-J 


70H35 


81.85815 




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34.01109 






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10. 16790 










88.14!l£4 


80 


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187» 


s?bTb 


780<B 


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90.40886 




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BITM 


4U409 


85268 








ar 


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40000 








«i4g47] 










99906 




30.10141 


9H.t>«303 




H 


30887 


S51«i 




-3.3050 




101.11185 




K 


33973 


8057* 


'149119 




00953 




3.1 


ts 
















M 




69838 


3(r397 






iioisoai.ii 




U 


43SM 


74660 


38011 


9£M 


3B,aoi5s 






91 


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16.45899 




























6S9Ba 






39310 




iirrisa^ 


27 


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66108 


93617 


69808 


653-.^ 


97797 


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ai 


SO 


0»34« 




77914 




40.4ia!O 


















148!2406i 






66885 








41.4'.;206 


149 46837 




M 


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i7.wa35 


'a 1037 


9S77a 


156.26!I1!B 


h 


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rao? 


18411 




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42.445W5 


163 708iiS 


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76B49 










171.S8831 


so 




lO.TKia 






24.73873 




180.98496 


10 


*3 


mx 


■3371-J 


37196 


91790 


44107746 






43 


86001 


3BS91 


46005 


25.09960 


64980 








WHS 




51903 






214!8»95 










63898 






2». 18385 






atiw 


ai07S 


7-.1fi5 


601.3S 


40. 4500:1 






«7 


99841 


Booai 


ftilW 










4B 


11.01860 










280! 47948 






06894 






VNWiO 




31U.5^U97 






10468 


7IBU 


'lOBOi 


45051 


i9.nm 






Gl 


tl.IMS9 


la.BifciB 
























4-J9 171873 




51 




a»99 






51! 31290 






M 


Stan 


OMIM 




■»981 


5:!.D90*7 




8 


60 


98610 




59180 


50S04 




687164060 














58.7™ 


8.-lD.43eS9 






S6010 




70488 






114.V9157 


3 


08 


3B79;i 


21730 




28,18417 


5r,:4.W53 


171H.S78S 


a 


69 


48669 


IITMO 


lo.ooiss 


41700 








SO 








65371 


57.29869 


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COSEOASTfl. 



276 



NATURAL VERSED SIXES AXD EXTERNAL SEC. 
O* 1° 2° 3* 



/ 


Vers. Ex. sec. 


Vers. 


Ex. sec. 


Vera. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
8 

4 


. 00000 
.00000 
. 00000 
• 00000 
'. 00000 


.00000 
.00000 
.00000 
.00000 
.00000 

.00000 
.00000 
.00000 
.00000 
.00000 

.00000 
.00001 
.00001 
.00001 
.00001 

.00001 
.00001 
.00001 
.00001 
.00002 


•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 


00187 
■00139 
00140 
00142 
00143 


.00187 
.00139 
■00140 
.00142 
.00143 



1 
2 
3 
4 


6 
6 
7 
6 

e 


• 00000 
•00000 
.00000 
. 00000 
. 00000 


•00018 
00018 
.00019 
.00020 
.00020 


.00018 
.00018 
.00019 
.00020 
.00020, 

. 00021 
.00021 
.00022 
.00023 
00023 

.00024 
00024 
.00025 
.00026 
.00026 


. 00066 
.00067 
.00068 
.00069 
.00070 


00145 

00146 

00148 

■00150 

■00151 


.00145 
.00147 
.00148 
.00160 
.00161 


5 
6 
7 
8 
9 


10 

11 

12 
13 
14 


. 00000 
.00001 
.00001 
.00001 
•00001 


•00021 
.00021 
.00022 
.00023 
•00023 


00071 

00073 

. 00074 

•00075 

00076 


00072 
.00073 
•00074 
.00076 

00076 


00153 
.00164 
.00166 
.00158 
.00169 


.00163 
.00166 
.00166 
.00168 
.00169 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


•00001 
•00001 
.00001 
00001 
•00002 


• 00024 
.00024 
.00025 
.00026 
•00026 


•00077 
00078 
00079 
00081 
00082 


.00077 

00078 

00079 

.00081 

.00082 


.00161 
.00163 
.00164 
.00166 
00168 


.00161 
.00163 
.00164 
.00166 
■00168 


IS 
18 
17 
18 

-is 


20 

21 
22 
23 
24 


00002 
•00002 

00002 
•00002 

00002 


.00002 
■00002 
.00002 
.00002 
00002 

00003 
.00003 
.00003 
.00003 
.00004 


•00027 

00028 

00028 

.00029 

•00030 


.00027 
.00028 
.00028 
.00029 
.00030 

.00031 
00031 
.00032 
.00033 
.00034 


.00083 

00084 

00085 

■ 00087 

•00088 


00083 
00084 
00085 
00087 
.00088 


.00169 

00171 

00178 

■00174 

■00176 


.00169 
.00171 
.00178 
00176 
.00176 


20 
21 

22 
3S 


25 
26 
27 
28 
29 


• 00003 
00003 

• 00008 

• 00003 
00004 


•00031 
•00031 

00032 
• 00033 

00034 


•00089 
00090 
•00091 
•00093 
•00094 


.00089 

.00090 

.00091 

00093 

00094 


.0017d 

00179 

00181 

.00188 

.00185 


-00178 
.00180 
.00183 
00188 
.00186 


2S 
2t 
21 
28 
28 


30 

31 
32 
33 
34 


■ 00004 
•00004 
■00004 
■00005 
■00005 

00005 
•00005 
•00006 
• 00006 
■00006 


.00004 
.00004 
.00004 
.00005 
.00005 


00034 
• 00035 
■ 00036 
•00037 
•00037 


.00034 
.00035 
■ 00036 
.00037 
.00037 


00095 
00096 
00098 
00099 
•00100 


.00095 
00097 
00098 

.00099 
00100 


00187 
00188 
00190 
00192 
00194 


.00187 
.00189 
.00190 
.00193 
.00194 


80 
81 
83 
88 
84 


35 
36 
37 
38 
89 


.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 
.00300 
.00301 

.00305 

00307 

.00309 

.00311 

_J)0318. 

00315 

■^0316 

..00318 

.00330 

.00323 


88 
88 

87 
88 

Ji 


40 

41 
42 
43 
44 


. 00007 
00007 

■ 00007 
00008 

■00008 


.00007 

00007 

• 00007 

.00008 

00008 

00009 
.00009 
.00009 
.00010 
.00010 

.00011 
00011 
.00011 
.00012 
.00012 

.00013 
.00013 
■00014 
.000] 4 
•00015 


•00042 

•00043 

00044 

• 00045 

• 00046 


.00042 

00043 

.00044 

.00045 

00046 


•00108 
•00110 
00111 
•00112 
•00114 


00108 
.00110 
00111 
00113 
00114 

.00116 

•00117 

00118 

00120 

00121 

.00122 
00124 
.00125 
.00127 
•00128 

.00130 
.00131 

00133 
.00134 

00136 


00206 
00207 
00208 
00210 
00212 


40 
41 
42 
48 
44 


45 
46 
47 
47 
49 


00009 
00009 
00009 
•00010 
00010 


•00047 
00048 
■00048 
•00049 
■00050 

■00051 
•00052 
•00053 
•00054 
•00055 

00056 
•00057 
•00058 
•00059 

00060 


.00047 
00048 
■00048 
.00049 
. 00050 

00051 
.00052 
.00053 
.00054 

00055 

.00056 

.00057 

.00058 

00059 

00060 


•00115 
•00'17 
•00118 
•00119 
■00121 


00214 
■00216 
■00218 
■00220 
■00222 


45 
48 

47 

a 

4S 


50 

51 
52 
53 
54 


00011 
■00011 
00011 
00012 
00012 

• 00013 
■00013 
00014 
■00014 
■00015 


■00122 
•00124 
•00125 
■00127 
00128 

ooiso 

•00131 
■00133 
•00134 
.00138 


■00234 

■00226 

■00328 

00380 

00383 


.00224 
.00336 
.00338 
.00330 
00332 


50 
81 
88 
68 
64 


55 
56 
57 
58 
59 


■00384 
■00386 
00338 
•00240 
■00343 


.00334 
.00336 
.00338 
.00340 


88 
81 

67 
68 


60 


■00015 


•00015 


•00061 


.00061 


00137 


00137 


•00344 


.00344 


eo 



NATURAL VERSED SINES AND l^]XTl!]RNAL SEC. 
4* «• 6* 7* 



277 



f 


Vera. 


Ex. sec. 


Vers. 


f]x. sec. 


Vera. 


Ex. t*ec. 


Vera. 


Ex. sec. 


t 


o 

1 

2 
8 

4 


00244 

00246 

00248 

.00250 

•00252 


00244 
.00346 
.00348 
.00250 

00262 

.00254 

.00257 

.00259 

00261 

00363 

00366 
.00267 
.00269 
.00371 
.00274 

.00276 
.00278 
00280 
.00262 
■00284 


00381 

00383 

■00380 

■00388 

■00391 

.00393 
.00396 
■00398 
■00401 
■00404 

00406 
00409 

■00412 
00114 

.00417 




00382 
00385 
00387 
00890 
00392 


•00548 
•00551 
•00554 
00557 
•00560 




00551 
00554 
00557 
00560 
00563 

00566 
■00569 
■00573 
■00576 

00579 

■00582 
■00585 
■00588 
■00592 
00595 

00598 
00601 
00604 
00608 
00611 


• 00745 

• 00749 
00752 
00756 
00760 

00763 
00767 
•00770 
00774 
00778 




■00751 
■00755 

00758 
■00762 

00765 




] 
2 
3 
4 


5 
6 
7 
8 
9 


.00254 
00266 
00258 
00360 
00263 




00395 
00397 
00400 
00403 
00405 


•00563 
.00566 
•00569 
•00572 
■00576 

00579 

•00582 

00585 

00588 

•00591 




■00769 
00773 
00776 
00780 
00784 


5 
8 
7 

8 

9 


10 

11 
12 
18 
14 


00264 
.00266 
00269 
00271 
00278 




00408 
00411 
■00413 
00416 
00419 


•00781 
00785 

■ 00789 
00792 

•00796 




.00787 
00791 

■00795 
00799 

■00802 


10 

11 
12 
13 
14 


15 
16 
17 
18 

iL 


00276 

00277 

00279 

.00281 

ppm 

00386 
00288 
■00290 
00298 
00295 


.00420 

•00422 

■00425 

00428 

00430 




00421 
•00424 
00427 
00429 
00432 


00594 
•00598 
•00601 
•00604 
•00807 


•00800 
•00803 
•00807 
•00811 
• 00814 




00806 
■00810 
■00813 
■00817 

00821 


"'l5 
16 
17 
18 
19 


SO 

81 
22 
28 
24 


.00287 

.00289 

.00291 

00298 

00296 


■00433 
■00436 

00433 
•00441 

00444 

•00447 

00449 

00452 

•00455 

•00458 

.00460 
•00463 
00466 
•00469 
iP0472_ 

.00474 
00477 
■00480 
■00483 
■00486 




■00435 
00438 
■00440 
■00443 
■00446 


■00810 
■00614 
00617 
•00620 
•00623 




00614 
00617 
00621 
00624 
00627 


•00818 
•00822 
•00825 
•00829 
•00833 




■00825 
00828 
00832 
00886 

■00840 


20 

21 
22 
28 
24 


26 
26 
27 
28 
29 


00397 
■00299 
.00801 

00804 
■00806 


.00298 
.00800 
.00802 
.00805 
00307 




■00449 
00451 

■00454 
00457 
00460 


•00626 
■ 00630 
00633 
•00636 
• 00640 




00630 

■00634 

■00637 

00640 

00644 

00647 
■00650 
00654 
00657 
00660 


■ 00837 
•00840 
. 00844 

■ 00848 
•00852 




■00844 
00848 
00851 
00855 
00859 


25 
26 
27 
28 
29 


80 
81 
82 
88 

M 


00808 
00811 
00818 
00816 
■00817 


.00809 
.00312 
.00814 
.00316 
00818 




■004b3 
■00465 
00468 
■00471 
■00474 


00643 
• 00646 
•00649 
■00653 
•00656 

00659 
■00663 
■00666 
■00669 
•00673 


•00856 
00859 
•00863 
•00867 
•00871 




00863 
00867 
00871 
00875 
00878 

00882 
00886 
00890 
00894 
00898 


30 

31 
32 
33 
34 


16 
86 

17 
66 


■00820 

00823 

■00324 

■00827 

■00332 
■00384 
00886 
■00889 
■00841 


.00821 
.00828 
.00326 
.00828 
■00830 




00477 
■00480 
■00482 
■00485 
■00488 




00664 
00667 
00671 
00674 
00677 


•00875 
• 00878 
•00882 
■ 00886 
•00890 




35 
36 
37 
38 

39 


40 
41 
42 
O 
i4 


00333 
00335 

.00887 
00340 

■00342 


.00489 
■00492 
00494 
■00497 
•00500 

•00503 
■00506 
■00509 
■00512 
■00515 




■00491 
.00494 
.00497 
■00500 
00503 

■00506 
■00509 
.00512 
00515 
J)0518_ 

.00521 

00524 

00527 

00530 

_P0533 

00536 
00539 
00542 
00545 
00548 


00676 
•00680 
•00683 
•00686 
•00690 




00681 
00684 
00688 
00691 
00695 


•00894 
■00898 
• 00902 

■ 00906 
■00909 

•00913 
00917 
00921 
00925 

•00929. 

00933 

00937 

•00941 

•00945 

■ 00949 

•00953 
00957 

•00961 
00965 

•00969 




00902 
00906 
00910 
00914 
00918 

00922 
00926 
00930 
00934 
_00938 

'00942 
00946 
00950 
00954 
00958 

00962 
00966 
00970 
00975 
00979 


40 

41 
42 
43 
44 


46 
46 

47 
tt 


■00843 
■00346 
■00848 
C0851 
_PP368 

■00356 
00858 
00361 

■00864 
00865 


00346 

00347 

00850 

.00352 

.00364 


•00693 
00607 

■ 00700 
00703 
00707 

•00710 
•00714 
•00717 
00721 
• 00724 


■00698 

■00701 

00705 

00708 

00712 


45 
46 
47 
48 
49 


60 
61 
82 
BS 
64_ 


.00857 

.00359 

.00862 

00864 

00867 


■00518 
.00521 
■00524 
.00527 
■00530 




00715 
00719 
00722 
00726 
00730 


50 

51 
52 
53 
54 


66 
66 

67 
BS 


■00868 

■00870 

00878 

00875 

00878 


.00869 

.00373 

00874 

00377 

00879 


00533 
■00536 
.00539 
■00542 

00545 


•00728 
■00731 
•00735 
•00738 
•00742 




00733 
00737 
00740 
00744 
00747 


55 
56 
57 
58 
59 


60 


■00881 


.00382 


00548 


« 


00551 


•00745 




00751 


•00973 




00983 


60 



278 NATURAL VERSED SINES AND 'EXTERNAL SEC. 





Vera. 


.... 


Vers. 


.... 


Vera. 


Et,„. 


Vera. 


Ei-aec. 






Si! 

lit 

iQ 

II 

043 




Hi 

S99 


II 

_OL2.7J_ 

iiii 




si! 
ill 

II 


II 

;ii 

585 

:i| 


- 


si 

sli 

BOfl 


II 
III 

_01_868_ 

;i| 




I'i 


i 




- 


so 

iii 

92 
S3 

iii 

95: 








a 031 


11 




ii 


01300 

Hi 




01318 


-0 5BB 

:iii! 




S ?, 
8 1 


Iii 




95 
98 

is! 


15 

i 




80 
77 




lis 

D o_a4_ 

Hi 


si 

0134 

II 

OlSB 
0137 

iii 


- 


sii 

pi_3_ai_ 

II 


-ii 

Hi 

.0 875 
:o 890 




iii 


:i i!! 

973 
.01984 

OMOT 

Si 




98 

99 
00 
00 

"0 01 ~ 

03 

IE 








iii 

89B 
703 

sis 






- 


01391 
01410 


33 


. 


.9137 

.0 15B 




8ii 


39 

91 

ai4i5 

1 


III 
0! 3,5. 

11 


■si 
HI 

.0 750 


- 


Hi 

758 
7B4 

]| 

ii 
ii 

83L 

iii 

B72 


:8|i 

02080 
-0208B 

: 02084 


- 


p 10 

D3_fl_ 


_39 




11 




8!g 


44.4 

. !!! 


' 


01466 

l i 

43 


■ 804 

11 

.0 m 


II 


n 




189 

:sis 




ii! 


11 

■8? ii 


i 


k-l 


li 
"""/ 




si 

01211 


si 

SIB 


.02 55 

!0B17^ 
02 BS 


- 


1 


1 

_69 



NATURAL VERSED STNRf! AND EXTERNAL SEC. 2?9 





v=«. 


Ex.£ef. 


Vers. 


...e.. 


\^rs. 


Ei. mn- 


-V-^r.- 


Ei-«c, 







.021 5 


03231 


"0258 


02630 


02970 


-03061 




■03538 


"3 






.032 


.0257 


D2837 


02B77 






.03530 


1 
















.03423 




a 














.03084 


.03430 














02668^ 


_0i_9a9 


_03091_ 


.-Q-3-438_ 


Jo3B60_ 


_4 




.023 S 


.03288 


0358 












B 








.0280 








■C3463 




e 




!023 8 










^03114 










■ 032 * 


.03285 


ioaoi 


03636 


■03027 


-03121 




■03592 






pawn 


um9i_ 


lOlfiS- _ 


036B3_ 




03 37 


.05483 


-03601 


_9 


M 


.D2348 


-03308 


















'03 48 


.03 44 










'■9V» 


■uau 


-02642 




03 55 






■03625 


13 




■Sm 


■03817 


-0234 


037 1 






03 03 


-03633 


3 




■JSK. 


,.>j838 


•°gB5 _ 




-OS 70 


:03 67_ 


03 14 


■03642 


_* 




oaaTT 




-ozafl 


027 5 


-08 77 














.03888 


.0288 


03743 






.03 29 


i 03653 






-03848 


.0287 


037 9 


!03 91 


^03 90 


03 37 




7 




■oaass 


■03340 




037 8 


-03 98 


.OS 98 






a 


^ 


.03308 


■02303 


:03B3 


0" 3 


■08 06 






03683 


_9 


w 


0269 




^rrr 


.03 13 


.03 60 


03691 


30 


31 


.03314 


.02360 




027 7 


03 20 


-03221 


03 37 




1 








0370 


027 4 


03 27 










ss 


-03327 


: 02332 


037! 












a 


M- 


J»3333 


-03383 


J3I3 _ 






T^sasi 


A3_90 


_l03724_ 


34 


26 




.03739 


028 6 


-03 4 


35 






.03402 


02738 


















-03403 










03 14 


03749 








.03415 










03 21 


■03758 


28 


M- 


^02370" 


Tci342r 




_03e_L 


,03_7 


:03282 


03 29 


03766 


—39 


SO 


02783 










81 


-02377 


.03435 














31 


» 












.03303 


;036 3 


03791 


32 


M 


:023BS 


: 031 43 


03733 


028 3 


.0320 


08313 








U 


03399 


■03454 


02790 


02B_0_ 


i.0A3-14_ 


■03321 


03676 


~03816~ 


34 


8b 






35 


l« 




-03453 




028 5 


:032 9 


:03337 


036 4 


03825 


30 


1? 


.02418 


.03474 


02811 


028 3 


.032 8 


03345 




.03833 




n 


02411 


.03481 


02818 














JL 


"Ma? 


fa«» 




020 7 












«o 


02434 


.03494 


.02851 


029 


■ 032 a 


.03368 


037 5 


03868 




u 


.02440 


-03501 


02838 




.03286 


.03378 










02447 






















: 03515 


03853 


039 


: 03281 


.03392 


037 9 


.03834 


43 


Ji^ 


:p_2*59_ 


_:0352L 


:03B59 


029 _ 


^032.88 


_-O340O. 


_oa_7 7 


_03892_ 


_« 


iS 




















4B 








03B58 


■03303 










4T 




: 03543 


: 03380 




-03310 


-0342 








«S 


03185 


■03548 


-02887 


o^sf 




-0343 


-037 8 


.03927 


48 


i?- 


03492 


036S5 


^Be*_ _ 








J)37 8_ 


_03935. 


_4B 


M 


.03493 










^0344 










03504 


: 03589 


03907 


03994 


03340 


■0345 


038 2 






83 


03311 


■03573 


-02914 


08002 


08347 




-038 


.0 981 


53 


G3 


02S17 












-038 a 




63 


B- 






l029_2S_ _ 








J38_6, 


-JJ78_ 


_S4 


K 


03S30 


~03B9r 


-0?935 








033 4 






w 


02537 


■03003 


02943 




03377 


0349 


038 2 


-03995 


68 


87 


-02543 








08385 


-0350 






57 


M 




















^ 


°a"8 


03 324 


02963 


D3054_ 


03400 




038 8 


04021 


_6£ 


W 


.03833 


03330 


-03970 




03407 




033 4 


-04030 


BO 



280 NATURAL VERSED SIXES AND EXTERNAL SEC. 





V..™. 


E^..cc, 


Vers. 


>;-x.s<.i^ 


Vern. I. 


x.s«. 


Vers. 


Es. SK 







0SB74 


.04030 


04370 


.04589 








.05762 




1 
















06773 






OMBB 


04056 


04395 


'04597 


-SI 


ST. 


^05 B7 
.05 77 


.0S7B3 


3 




"MOB 




.04404 


_.04fipfl_ 


0493 _ 


05 a8 


.05436 


-05805 


_1 














-05819 


















!05 05 








: 03830 


"Ol 91 






; 04957 


05318 


.06 16 


05838 






.03333 


-U4 00 


.04438 


.01B41 


-04967 






05847 






OSML 


.04 08 




^04e53_ 


.04978 ■ 






05868 










04455 




.04985 


"0524T 


r05 43 


06889 


lo 


11 


bosses 


04 28 




'04873 


.04934 


062S8 




■05879 


11 


12 




04135 




.04882 


























06 73 








-03905 


04 bT 


0448 9_ 
.04498 


".04710" 


031 


05388 


_Q5 -e?_ 












-05922 






.04003 


.D4 70 


























048 


05S17 


0681 




17 








: 04524 


: 04738 


.0 057 


0S337 


0562 


-05055 


1! 






'04 97 


.04533 


.04748 


087 


05337 


■ 06 3 


.05985 


Jl 


2o" 












05347 






SO 












085 


06357 


'05 4 










.04 33 


.04559 


.04778 








-05998 




33 


04080 


.04 33 




04788 






05 6 


.08009 




24_ 


QjQflB 


-04 41 




.04705 


~^ - 


."5388 


Tfe~ 


_0jp30_ 








585 


.04305 


"25 


28 


04085 


:04 59 


.0 5»3 


.04815 








oeoii 




37 


04093 


.04 SB 








05418 




.08052 




3B 










^ il9 


05439 






2B 










.01B43 


■ 15B 


05439 


^«5 3 _ 




.2_9 




.04113 


.04 95 


■ 338 


.01853 








.06085 




31 


04130 










05 3 60 




.08098 














-0 138 


05470 


-05 5 




33 










01882 


.0 195 


054BO 






38 


34 






04B83 


.04B91 






L05 .7 _ 


06 39 


_M 


3G 


.04159 


.04 40 








05501 




.oe 40 


35 










■ 01911 


323 


05511 


-05 9 




38 






ioi 58 


04890 


01930 






05 






38 




04 87 












-06 73 




39. 


04133 


■ Jj-LL 


-Q4707 




:0 351 






■ 06 84 


-Si 










04950 






05 












04735 














*3 


04313 


.04 03 


04734 










08 17 




la 


04228 








;o 283 


05581 




■ 08 28 


43 


-*t 


Oiaai 


^«1L 




^1989_ 


.053BB 


055B4 


.06 _^. 


06 39 


_44 






.04 31 


.O47B0 














IB 


04351 


.04 40 






!05318 


058 5 




06 81 


46 


17 


04280 






: 0501 a 


.05328 


056 5 


9 


■ 08 73 


47 








.04787 


.05028 








.06 83 








: 04483 


04798 






066 6 




■08 BB 




so" 


04283 








To 53 51" ~ 


058 7 


^D 


.06 06 


~S0 


61 








: 05057 


-05363 






.06 17 


51 








.04833 


.06067 








.06 33 






04310 


.04504 


-04B33 














fl4_ 


.0431?_ 


_0461i_ 






: 05391 


05fl 9 


:o 








~ofoW 


-05401 







T08"6"9" 


""55 








04858 


05107 








■ 06 73 






.04344 


04541 


048B7 










■ OB 84 




S8 


04353 















08 D6 


68 


^ 


■Q«3B^ 


' 04580 


04835 


.0B138 


■05139 


05 51 


fif- 


■08107 


_5i 


Mf 


04370 j 


■OtSSB 


.01894 


.05148 


.D914B 






.08418 


60 



NATURAL VERSED SINES AND EXTERNAL SEC. 281 



Vera. 


Ei.s8g. 


Ver.^^ 


..=^. 


Veri. 


Ex.™c. 


^'era. 


Ei.Beo- 










071 5 


















071 B 




.078 a 








OtOfil 


06440 




071 8 










2 


OBoai 


.08462 








:07B 2 


:079B4 


08676 


3 


oeo7i_ 


_0.ei63 


^eeB4_ _ 

.06684 


07174" 


■07325_ 


073 4 


P7995_ 


_-.P«flflp_ 








08091 














i 08717 


8 


08 01 


06497 




07199 




07B43 


: 08039 


.08730 


7 




.08508 


06726 


07311 


.07889 


-079S5 


06041 


.08744 




08 21 
08 31 




.0B7_3B_ _ 


oim. 


07SBD 


_J)796a 




0B757 








.08771 


~ib 


OS 41 


: 08543 


0B7b7 




: 07403 


07994 


08075 


0878 


11 






-0B76B 


07359 


.07413 


.06008 


OB0B6 


.0679 










07271 












-OBIII. 












.""08"l"2"l" 


.0883 




-OaS3B 


- 0T79"9" " 


07 95 


.07446 


.08045 


"a 








07 07 


.07457 




























08211 


; 08622 


0883 








08 S-j 


.0888 


18 


OBzai 


■06634 


■ 0684 


07 44 


■07490 


.08 97 


08 87 


0869 . 


19 




















06341 












*08 90 






08252 


:0866B 


:0687 


07 80 


:07633 


■ 08 35 


-08 01 


0898 


33 




.08630 




07 83 


-075S4 






.08948 


23 










i07545_ 


■08 74"" 




08975" 




06282 


.06703 


■ 0890 


"07 17 


.08238 


"35 




.06715 


■ 06S1 


07 39 


; 07568 


■03037 




.08889 


38 




















06312 




^0693 














_0_6333_ 


.087J9 


-M^i _ - 


07466 


JVBQl^ 


_Lo_B_23G_ 


'_ge2t2_ 


..■O803O 


_38 








07479 


.07612 




.06294 




30 




















06353 


: 06784 


:o69a 




.0763 


■0326B 


: 08317 


.09073 




08363 


.06796 


.06980 


07518 


.0764 


-03278 


-08329 


.09088 


S3 


~083B4 


_.p6B07_ 




_07.528_ 


■ 0768 


_^pa3Bi_ 


.£-8340, 


~ 09 m 


^4 












083M 


ioBBSl 


:0702 






:083 B 


.08364 


09127 


38 








075B5 


.07890 


■ 083 1 


08375 


09141 


37 








07578 


0770 










08*25 


■08866 




07590 




'083 7 


-08399 




_39 


06436 


.08878 


"OTOe 


O7.i02 


.07724 


.083 


.08410 


09183 


4U 


08*45 


■08889 




07815 


.0773 


.083 3 


.06422 






oats 6 


















08488 


:08B18 




07O40 








.09224 


48 


08476 


.08926 


-0710 


07362 


.0776 


„■ 08421, 


;o8ML 


_.0923a_ 


44 


08488 












08*97 


















08507 


;o69eo 


^0714 


07890 


■ 0780 


: 08463 


.08192 


.09280 


47 


08617 


■06973 


■ 0716 


07702 


07B1 


08476 


.03504 


.09294 


48 


08529 








r0783 " 


^0848b. 
-08603 


:oF538"" 


09308 
.09323 




0863B 


"08995" 


■ 0717 


"07727" 




0854B 


■07007 




07740 


07B4 




.OB5S9 


.09337 


61 
















.09551 




ofiaee 


: 07031 








.08542 


ioBBes 






08680 


■07048 


^0721 


07779. 


.0788 


_,0Ba_6B_ 


^06575, 


09379 




























56 


0861 


.07079 


:0724 


078 8 






08610 




67 


0662 


■07001 


■ 0728 


078 8 


927 




08623 


09435 




OBIjS 


■0710S 


■ 0737 




.0 936 




08634 


09449 


_69 




^m 


.0738 |- 


078 3 




mw 


\ 





282 NATURAL VERSED SINES AND EXTERNAL SEC. 





Vera. 


Etboo. 


Vers. 


El. 6E.C- 


Ve,^. 


EX-HPC, 


Vera. 


El. SRC. 












.1()S3B- 








. 2233 






.08657 












0913 


. 334u 


I 




.08659 


■09492 


.09394 


: 10388 


: 0148 


'. 1 93 


. 0928 


. 2288 


2 




.08681 


.09506 


-09406 


.10383 


. 0159 


- 1 08 


0939 




a 




roB76B~ 


■03520 


■PM1I_ 
-09431 


.10398 


^2- 
0181 


_J- =-3- 


■ 0953 














. 23 IS 






.08717 


^09549 


094 3 


: 10428 


- 0197 


■ ] 65 


'■ 097B 


. 3333 


6 






.08663 


.094 5 


-10443 




71 


0993 








^08790 














'■ 2386 






.08753 








. 0238 






. 2383 


-J 






.OBBOa 


r09tS3~ 


^048"b" 


. 0248 


16 


.1 oaT 






















'■ 211 S 






:087GB 


!09S3S 






'■ 0374 






. Z433 


3 




.08800 


-09849 


.06S30 


.10533 


. 0287 


487 


1 073 


. S450 


3 






■09663 


.09543 


.10549 


^0300_ 


■ 483 


■.1_085_ 


. 2487 




















2481 


~6 




!0B83e 


:09oB2 


.06567 


; 0579 


'■ 0328 


615 


1 112 


. 2501 


fl 






.09707 


-08579 


- 0594 


. 0338 


. 31 


. 125 










-09721 








47 




! 125^4 


B 






-09735 


'03604 


' 0625 






153 


.12551 


_9 
























.0B8B4 


.097S0 


.09617 




-10377 







































. 0870 






:i 163 


. 3603 


3 




.08920 


.09793 


09654 


. 0688 


- 0416 


. 627 


. 206 


- 2819 


3 




. 0B132_ 


09808 


■09666 


. D70] 


_0429_ 


. 643 




2638 


4 














.i 659 




- 2683 






:0SB68 




' 09891 


: 0731 


; 0465 


.1 675 




. 2670 


6 




-0B9B8 


.09851 


.09704 


. 0747 


. 0488 








7 




-OBsao 


.09888 












'■ 3704 


B 




QBBflZ 
.09004 


-06880 
.09885 


ro9TT 






:i 724 
.1 740 




■ 3731 


__S 




. 0793 


. 0507 


. 299 








.09018 


.06909 
































.1377 


33 




lOBOlO 






. 0839 


: Q546 


'l 789 


■ l 339 


-1278 


33 




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097 2 


. 0354 


0556 


_1J05_ 


■ a63 


■ 1380 


34 


g 




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-1383 


35 










'■ 0885 






■ 380 


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38 




-09 88 


: 9997 


09829 


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- 393 




37 




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. 0013 


















■09 13 


■ ooaB 


-09854 


'■ 0932 


0624 




: 420 


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_39 




.09 3B 




06867 


. 947 


■ 0637 


19 S 


434 


1291 






.09 37 




-09B80 




0G5O 




447 








.09 49 


. 0071 












^2944 








- 0085 






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-129 


43 


M 






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■To 70 a 












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30 




it 


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598 


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2 


sa 


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M_ 


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81 


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31 


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56 


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851 


- 31 






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SB 












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S 


ss. 






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60 



NATURAL VERSED SINES AND EXTERNAL SEC. 283 





Yen-. 


EI.WC. 


Vere. 




V.,.. 


Ex.^, 


V.r.. 


K.^..<:. 










. 3388 


4335 


97 


. 6470 


142 3 








- 719 


. 375 


. 3562 


4354 


- 12 


- 5489 


.142 a 




1 


2 


. S3 


. 292 




4372 


427 








3 


3 
























■ 327 




_440B 






-143 3 








74 




360B " 


4438 


70 


. 5567 


.143 B 


~1 66 


~i 


S 


37 


. 333 


- 3823 














7 




, 380 


























14 


6636 




37 


e 




: 42" 


J— 4I5_ 




_4_603 


- 29 


6B45 


,U4I8_ 
14433 


_: 48 


_a 


To 


. 433 


. Z67a 


643 


5663 








. 56 


. 451 




4639 


- 653 


- 6684 


. 444! 


89 










'■ 2708 


14 38 








. 09 


13 


3 


33 


: 48B 


. 2733 


14 76 


- 587 








13 


i_ 


97 


. B04 


. 2750 


_AAS. 


- 602 


.■ 5743 


4494 




^ 




11 


. 621 








71 




e 


26 


, 539 




4 33 


■ 36 1 






92 












4 51 


. 36 fl 


'. 803 


: 453! 




7 




' 52 


: 67B 


. 279S 








455 


: 033 




2- 




13593 


■ SS07 _ 


_4_89. 




- 841 


. 466 


"54 


_3 










4 07 


. 36 


. 861 


. 468 


73 


80 




'- 94 


■ 628 


; 3836 


4 ae 




. 881 


- 469 




21 


2 


07 


. 64B 


2350 






- BOl 


. 461 


16 




3 






. 3884 




57 i 






- 87 




?4^ 










. 37 9 






^_5B. 


Ji_4 


35 


- 049 


700 


. 2893 


4 01 


- 3763 


. 960 


"486 




S 


SB 


. 063 




. 2907 








. 467 


99 




















. 30 




29 


: OBI 


: 753 


: 298B 


4 58 


■ 3808 


■ 019 




41 


3 


2i_ 


104 


. 771 


. 2960 


4 77 




_. _03Jl 


"i473 
.1473 


83" 


_fl 








. 2964 












- 807 




14 








04 




33 


: 146 


. 335 


: 39B3 


33 


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99 






2 


33 


IflO 


. 43 


. 3007 


62 


. 831 


19 


.14781 






3J_ 


174 




■ 3022 




^ 898, 
. 911 


. 6 39 


0479 . 


^ 7_ 














38 


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936 


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023 




. 6 99 


484 




7 














. 8 19 


. 436 


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3 94 


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6 39 


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40 


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70 




85 


985 


~TW 


. 486 


-17 3 




41 


2271 


■ 88 


: 3 23 1 


?-■ 




. 6 79 


- 49C 










. 4006 


. 3 37 






. 6 89 




- 7 5 








■ 4024 










'. 493 




43 






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4_044. 




4949 


- V 7 


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. 4061 


. 3 80 ■ 


131 




~ b" 69 




. 7 8 










3 96 










. 7 




47 


'. 3355 














- 7 1 


47 


48 


. 2S69 


: 4115 


i 3 33 


239 


4104 


'■ 8420 


6011 


- 7 2 


48 


*»„ 


2383 


■ 4134 


■ 3 38 1 


S25a_ 


4119 


_B440_ 


5036 


7 83 


-4? 














SI 


'■ 3411 




'. 3 67 


5298 


4 49 








61 


53 


. 42B 


■ 4188 


. 3 Bl 


5315 


i 64 


- 6601 




. 7 7 


62 


63 


- 439 


4S07 


.13 96 


5335 


4 79 


. 6521 




7 8 




54_ 


__454_ 
. 4B3 




.13 35 r 




4 OS 






7 














~56 




. 432 




. 13 SB 


5393 


4223 


: e582 


15134 


'. 78 3 


6fl 


57 


■ 436 




-13 =4 




4 38 


. 6602 


16149 


. 78 4 


57 


SB 


















BB 


52_ 






:i3 83 ! 


6461 




: 6643 




-Vl**^ 


^. 


BO 


- 538 


■ 4S35 


.13 97 1 


£470 


.14m 


\\^\% 




_? 



286 NaTURXL versed SINES AND EXTERNAL SEC. 





Ver,. 


x.«c. 


Vara. 1 Ei. Iff, 


Vers. 


I-U. MC. 


Vera. 


:-.«o. 






23saa 




.34529 


82501 






TeiT" 


36 33 












32535 






384 


36 70 








30609 




33588 


35724 


.34631 


604 


SS 07 








30639 


.24538 
















^MIQ. - 


30efl|_ 
070 


.246M__ 




"26733" 


_i470_l. 
.34710 


B44 










33669 


984 


36 19 








073 


'24644 






.34775 


984 


36 56 


























2I51 






3i!770 


25811 






37 30 






33 5M 


0B2 


.24720" 


33804_ 


_25afli_ 


.34882 
.3"4817 


013 


.37068. 




■ 






; 063 






2360 






32873 


36800 














82 




32805 


35820 


: 31988 


103 


371B0 








65 


.24778 






.35034 


123 


37ai8 






23es 

23877 


S8_ 


^247ii: _ 


32073 
33007 


26_8M 
35876 


^3_50_60_ 
■35095 


143 


_37_25S 
37283 






163 








05 


;24835 






.35131 


183 


37330 








03 


-24854 




25017 












2373a 






33109 


36037 


: 36203 


323 


37406 






23752 


15 


■24893 


33143 


36056 


35238 


7343 












-24813 


33177 






3B3 


37481 


ai 




23 780 






33211 






3 383 


37518 






238 08 






33245 




.35346 


3 303 


37656 






23827 


2S 


"21970 










37594 


3 






313 


-34889 


33314 


38154 






37832 


_2 










33348 


23174 


.35454 


"2"r6r ~ 


17670 






238 SI 


378 














2 




23(103 


411 


.35047 




28213 






(7748 


a 






443 


.35066 


33451 


3B333 


.36582 


2 123 


37781 






23B58~ " 


jm.. 


.J60i5_ _ 
-35104 


33485 


36353 


.35586 


2 443 


37.a.22_ 


3 




33519 


39372 






3> 










33551 


38393 


: 35670 


2 483 


1788B 












33588 




.35707 


2 603 


57936 


3 




2401 B 


1607 














3 


31 


2_4035._ 


e40_ 


^231 a2_ 
.25201 


33691" 


16371" 


". 35815"" 


27"56r~ 




_3 


8 






S 


24078 


705 




33726 


36390 












24 93 


738 






S6410 




27803 


J8137 


3 




24 11 


771 


-25259 


3379B 


£6430 


■35924 


27823 


86165 


3 


a 


24 30 _ 


604 


.25278 


33830 


36449 


■35061 


27B43 


36301 


_3 


4U 


24 4fl 


837 


















24 68 


870 


:25317 


33899 


26489 


■36034 


27683 


J82S0 






24 87 


803 


.25336 


33834 


26509 


■36070 


37703 






43 


24 OB 


















4_ 


_34 2B_ _ 


868 














_4 






2003 


.25394 


34036" 


"26668" 


TaeiBo" 


277B4 "" 






4 




















4 


34 81 












27804 






4S 


34 00 


2101 


35453 


54142 


28627 


i 36290 


27B21 


38550 






24330 _ 


3134 


-_a_5473 . _ 


_S41_7X 


28647 


■36337 


_2_7B44_ _ 


38688. 
























61 


34 68 


2201 


^35611 


34247 


26686 


'. 36100 


27B81 


36868 




12 


24 77 


3234 


.25530 


34383 


26706 


-36437 


37B05 


JB7D5 








33367 




34317 














34 '5 


















65" 


34434 


^"233T 


ll" 


it 


WefT 


".3654"r 


li" 


36823 


_ 






















58 


244B1 








36825 










59. 


I4S10 


3468 


;3B6e6._ 




26815. 


■m^^ 


28046 


38877 




60 


S4S9S 


3601 


■asBse 


84883 


36868 


.88734 


38088 


3M1B 


2 



NATURAL VERSED SINES AND EXTERNAL SEC. 285 





Vera. 


Ex. ser. 


Vers. 


Ex-seo. 


Vers, i 


..^. 


Ven,- 


Ki. sec. 




o 














.23285 


















23931 


-23304 






3 










1336 


28960 








S 




23889 


201 B 


26288 


1253 






: 28787 




_4_ 


9167 


.23711 


202 7 
202 * 


_.25_32t 


_ Wl_ 


37040 


;32359_ 
23377 


_28797 








.23738 
















27075 






6 










.26406 


21324 


27ID1 




^28389 


? 


B 


1M3S 


.23816 


202 7 


.25434 






33132 






_s_ 


_9_2_S3_ 


_J384S_ 


_Z02 4 


_.2S46a_ 


31360 __ 


271Br 


■22450 


"28980 










To 


11 










21396 


27221 


2348 


28011 


11 


13 


8304 


-23922 


203 7 


26545 






3260 


28043 




18 




.23818 




.26672 






.2252 






ii. 


_|3II_ 










37ioa_ 


_2354 




14 


IB 






'20400' 


~2B638 


""r468" " 






~2913S" 


15 


IS 


8373 


.24028 


204 7 


.2Q6S6 








-29184 


18 


17 


8880 


-2*054 


204 9 








22598 


















27425 








21. 




-24107 


304 


; 36738 


1540 


27454 


■22634 


39356 


19 


20 


8442 


.24134 


204 8 


39787 








29387 


^m 






-241B0 










22971 
















1595 


S7542 






23 


38 


1949S 






26851 


BIS 


37572 




-333B0 


23 


M 


_mL 


.^34240 . 


^0559_ 


_ 2587B_ 


Ml 


3760L 


^22715" 


^M!l_ 












ae 










667 


37660 


33784 




28 


37 


91)83 






-26963 


885 


37889 






37 


28 


9630 


24347 


208 9 


-25891 








29535 


2B 


29_ 


H597 


24373 


BOa 7 


3B01B 






:328ie 


_3_95_86 


28 


3U 






; 7S9~ ~ 


277 8 




31 






30882 


36076 


757 


378 7 




-39638 


31 


32 


9348 


.24454 


20700 


26104 


77B 






-28859 


33 


SS 




-24481 


20718 








.33893 






3* 




■ 24608_ 










32913 






35 


9~70r 


24634 


"20763" 


26188 


880 


T7TT 






~3B 


80 


971B 


24361 


20771 


26318 








2978 


38 
















22867 






88 




















sa- 


9770 




2DB34 


26301 


.. 902 


28045 


23001 


3887 


_3S 




978B 
















40 
















33041 
















2 967 


28134 








ls 


19840 




2089fi 


28415 


2 975 


28184 






43 


**_ 


_98_57_ 


:a4777_ 


30B18 


28443 


3 983 , 


28^94. 




_;3p034 


« 
















23 16 






46 








: 26500 


2203 


38264 








47 


9908 


2435B 


2 967 


28528 




38284 






47 


48 


9837 


24886 












30160 


48 










"38615 


^3208 






^0182 


49 


so" 


9933 


-2 940 


2 020 


2210 


28374 


-23209 




Bl 


3878 


2 967 


3 088 


-26643 




38404 








ea 
















!ao287 




















-30318 




84 


0032 


"a 077" 




38728 


^31.7 . ^ 


28496 


ja B3_ 


_aO350. 




BB 


0048 


2 109 


26758 








~5S 


















^30413 


58 
















.23339 






ts 












28615 


-23358 




Be 


M. 


0118 


2 186 


Mm. 


-^M73 


3236 
.2..«\ 


=8646 


.3^377 


gQMt 


LS. 


80 




.2 214 


2 lee 


. zflBOa 




\ -wsw.-^ 


;^ 



286 NaTURXL versed SINES AND EXTERNAL i 



. 30829 
.30BS1 
. 30BS3 



.24682 
.J170I 
34720 



-32669 
.33703 
.32737 
■32770 
.32SD4_ 
.32B3B 
.82872 
.3290S 
. 32939 
_^S2D74^ 
.33007 
.33011 

^33109 



2S8B0 
3B»0a 
SSB20 

3Si^J9 
"25978 
269 9 B 



.35187 
. 36203 
-35239 



^2*989. 
. 2SobB 
.35027 



.33270 
T3J34r 



.SaBU . 

. 83899 . 

.33831 . 

.33988 . 

31003 . 



.28233 I - 

i28363-l ^ 

.36372 

.28292 

.26312 

.28331 

Tawi" 

.23390 

.aeuo 

.36430 

jam.. 



■85997 . 

.88034 . 

.88070 . 

.36107 . 

-36143 . 



_l3_4177 
-34212 
.54247 



?6|47_ 
26667 
.26686 



.36337 
36363 
36400 



27964 
27905 



' -34472 
I -34491 

/ -3*539 



33434 .25647 

25666 . 

SaSOll 2BBBe 



NATURAL VERSED SINES AXD EXTKllXAL SEC. 287 





Vera. 


JLSCI!. 


Vers. 


-(.«... 


Vers. Ex. sec. 


Vei=. 


Ei.aec. 





380«e 






41421 


. 0534 


43968 


-31800 


46633 


I 


3I08B 


S9065 






. 0556 


43999 




.46674 




asiOB 


39096 


39330 




- 0678 








8 






28351 








.31864 




J._ 


3aH7 


39173 


38372 _ 


416J6_ 

41827 


. 0539 




31885 


46811 




28187 


39213 


■-44173 










39251 


2B4 3 




. 0660 




:31B3B 


! 46803 








2B433 


41710 












28328 






41762 


. 0702 


.44304 


^31671 






MMJL _ 


39SSB- 




J17B3_ 


■ 0733 


44347 


.31892 


47041 


o~ 


89409 


2bTT" 


41B3S 




.443B1 


33013 






38389 






41876 


. 0766 


■ 44435 


-32035 






28808 


3B187 












'47180 






39627 


2B5 7 


41BSB 


■ 0807 


^44523 




-47228 


i_ 


38390 


39656 
3B606 


3957B 


42001 


OB28 


.44587 


32099 


47373 




38370 






- 0849 


44610 




IS 






2B619 


42084 


. O870 






-4 365 


17 


28110 












32163 






38431 


39726 


2B661 




: 912 


.14742 






12_ 


38461 - 


msi. 


3BB81 l_ 


42210 


. 933 


44787 




_4_5D4. 




3847 


3B804 


28702 ■ 








33227 


-4 651 


2t 








42283 


975 


: 44875 


-S2248 


-4 688 


33 


28S1 
















33 




39924 


29764 












34 


3855 


3B9G3 


_39T86__ 


42419 


03B 




:32312 


:4 738 


IS 


3857 


lOOOS 


29805 






.46063 










40043 


298 B 














38814 






4 545 


. 101 


.45141 


.33977 




31 


3883 


40123 






. 122 


.45185 


.32398 


.4 935 


M_ 




40163 


298 a _ 


4 630 


^Ji3_ 


■45239 


■ 33430 


-4 972 




"3887 






4 673 










91 


2889 


40243 






86 


;453!B 


.32462 




S2 




40333 


299 I 






.46368 


32484 


.4 113 










4 798 












2575 


409 B4 




4 841 


49 _ 








ir 


3877 




300 3 " 


4 836 




. 45497 


.32548 
33670 


1 254 


a? 


2881 


404B6 




4 968 


313 








S8 


38B3 




300 5 






.45681 


32913 


.4 396 


22_ 


2885 


405P,5 




4 063 


Mm, 


-45676 


32B34 


4 443 


to 


2897 ~ 


loeoe 




4 096 


- 376 








11 


2890 




301 a 




. 397 


:457Bfl 


82877 








40687 


301 8 


4 iBl 






.828B9 


■! tSS 


u 


39941 


40727 


301 9 


4 224 


; 43B 






.4 633 


M_ 


gBBBl 


40768 


302 




■- -J-" 


.46B01 




4 881 






40809 




4 310 


483 




F^S! 


i lil 


<B 


29002 


4 849 


30242 




. 603 


-45992 


BS795 


.4 77B 


47 






ao 63 






46037 


32808 


4 824 




29 043 


40930 




4 438 




46082 








29053 


4 971 


3D 04 


4 481 


587 








so" 






30 25 






46173 


.33B71 


4 967 


ei 


39104 






4 587 




48218 




4 015 




29125 


4 083 




4 610 






.32914 




S3 










. 851 




.32836 


* HI 


M_ 




""4'TF 


30 0?_ _ 


4 flfl6_ 


.~694 


4B354 


.32"979" 


-4 169 








29197 


B« 


S9S07 


4 57 


30 5 




- 715 


46445 




-* o^SS 


57 






30 7 


4 838 




48491 




* l°\ 


U 


3BS48 






4 aaa 






ll^ii 




«!- 


28289 


SSI. 


J!U__ 


.912 




4B6B2 


-SJ""? 


_4_fl9S. 


80 


282«| 


4 21 


30 3 1 




.31800 


4862B 


33087 



288 NATURAL VERSED SINES AND EXTERNAL SEC. 





V,.r.. 


C...... 


Vers. 


Ri. BOO. 


VOTK. 


Ex.-.«. 






1 


ill 


49493 


B 


iii 


57B 


isi 


!i 


s 


I 


I 


IB 


49390 
49884 


4504 

4570 
4693 


52531 


'i 


1 


!!i 


59418 


8 
8 


? 


III 


ii 


1B3S 

SiS 

4703 
4734 

is 

4813 
4834 

4300 
_ 4_92S_ 

if? 

_^033_ 
5059 
5077 


52938 

-11 

ii 

53403 

II 

Jli 
ill 


B0S1_ 

ill 

oiaa 

B190 
fl3B8 


.53 

■ 56 
.58 

^^ - 

:i 

.5Ba 

:t?, 

.587 


7 94 


ill! 


1 


s 

L 

I 


ii 
11 

33716 


60 3B 
60 75 

50423 


49a 

521 
544 

813 


ill 

' 69998 

:B!S 

'80387 
. BO 680 


J 

3 


1 


III 


si 

fl370_ 

II 

3B4S2 


.BBS 
-569 

.J71B_ 


703 
7736 


25 

i 


Is 


S3738 


il 


:607lfl 


33 


i 


33 47 


;iii 


■i 




30501 
:SB67 


Ii 

:6771 


SI 

i7664 


.60933 


J! 

38 

9i 


11 


ii 

31044 


lii 


ii 


i54B10 
.54863 
-5471B 


.3583 


:57B3 


■ ii 
ill 

111 


Ii 


s 


1? 


i 

■'si 


lis 
11 

^5 oes 


■ii 

Ii 


-51709 

■ b 196 
-5 350 


.3673 
.38752 

Sfl797 
_3e82g. 

"38365 
-3Bfl87 

1 36933 
.38855 
-38976 

3704S 


-6805 

■6832 
_5fl37_ 

'.5SU 


■(11735 


S 


r 


:ii 


;ii 


so 

1 


SB 


■lii 

34372 


; IP. 


s 


; II 


.6S12E 
-82185 


i 


60 


.85731 


.5657 |.3708B 


■56903 


8131 


.63137 


no 



NATURAL VEHSED SINES AND EXTERNAL SEC. 289 





Vbb. 


^^ 


Vers. 


...... 


Verr.. 


I'jl.SBC. 


v„. 


Etpso. 


. 




■38480 
l_52e_ 

:i 




i2_8_9_ 


II 

40028 




Eli 

86815 
8687 

r 

si 


! l 

: ! 
"4~y~ 


Ill 

_10_403. 

;S 

! 748 
7 BIS 


43718 
.42782 

111 




74346 


' 






is 






i 

755 




r-i 


Si 


lis 




B07S 

Ji 

8031 
6109 


1 




JSi 

847 
1 




! S! 

fl 84 

jJL 

! 1 
111 


4016S 

Si 

40261 

IS 

'403 78 
40401 

40471 


ill 

i 1 

1 i 

Pi 

, 38 


41- 


.43000 
: 43088 

ill 


_1 






si 

J.805L 
68117 
68183 


^135. 

2ja7. 

jJi 
pi 




" 


6338 

J.8I 

8652 

78778 
J6B51 

ii 

77727 

E 






: p 




84398 

sal 


II 






is 

3.S 
1 


II 

in 


40B35 

Sf 

40 B 

40775 
407B9 


R87 5 

ill 


It 
M 

"43730" 

Is 






i 


0869 

sis 

4 9 8 

sl 


89! 8 

M 

i 


Hi 

i-li 
1 

42810 


_^355a 

i 


" 


1 

787S2 


i 

4 




111 

891 


11 

li 

■ 44 57 


S 








i:i 


■s 
s 




3 81S 


8B1 * 


331 


70130 


43843 


74346 


■440B1 


.7B839 


80 



290 NATURAL VERSED SINES AND EXTERNAL SEC. 





VwiEx.,.o. 


Vpr,. 


K*.-.., 


Ver^. 


Ex.^ec. 


Vera. 


Z^^. 






-44081 


.73829 


45336 


















.78903 


45580 


: 83690 


17033 














.7898 




.83773 


-17057 


.88834 


48545 






3 












-83972 


18571 


4443 




* 


^41 77 






: 83938 




■89060 




4537 








.7921 


45858 


. 84020 


r4713r 














.7929 


45BB3 


.84103 


-17158 




■18645 


4728 












.84136 




9325 


18671 


4821 




8 


-44274 


















_?- 


44Z9B 


■ T3K 


-45756 




"-i725s" 






_501L. 










-34435 




18746" " 






11 


-4434B 










9880 


18771 


B201 




12 


-44370 


■79731 












95396 




13 


.44395 


.79839 


















44443' 


^_79917_ 


4SB73 


■647fli 


:i735t 


~. 0037 


■MJ.lt, 










5583 






44437 


. 80074 
















17 


44491 




45951 


: 85018 


: 47428 














■801|f 




-35103 


-47453 




48948 


5870 




IS- 


'. 44540 


80309 




-35187 




:903n5 


48971 


59 6 


-H 


20 


-445E 




4B025 














ai 


4453 


80487 


48049 


:B5355 


: 527 
















48074 


-B5439 


552 


.80683 


49046 


82 5 


22 












. 577 


.90755 


49071 








■im , 


.30704 




-BSBQB 


eou 


_B0B45_ 


-lOOBB. _ 


6448 


M 


W 




-80783 


48147" 


.85692 


828 
















.46179 


-35777 


651 


:b 026 


"4B148 


68 1 


26 














-B 1]8 


49171 




27 




447B8 
















28 


3fl 


44732 
















3fl 


30 






.46270 




- 'iW 


.9 383 


49248 


70 9 


.■10 














-fl 179 


49271 














-382 e 














4487: 


















31_ 


4^90: 


:b 19!) 


: 48363 


'384 7 


■J B4J_ 




LiiS-iL -. 


J4 0_ 


_31 




















a 






















37 


4 978 




48443 


.88713 






■ 49432 


7713 




!_ 


J_035_ 


3 320 


46486 
.46491 


88835 


.47949 
.47974 


.B 1 8 


19447 
_4917I__ 


7811 
97910 


3 

-5 






-3 BHI 


















4 073 


.82081 










: 19333 


8107 




42 






46665 


-B7142 


- 8048 


.921 6 


■49547 


8205 
















-325 B 








il_ 






Ami. 


873' 3. 
.8740 




.93670 




8403 


_4 


45 


4~"7r 


.82384 






.49633 


8502 




te 


4 33 


.32155 


46363 


-8718 


-43148 




-4SS18 


8301 
















.939 7 












-83827 


















4 53 


.82709 












3899 




so" 


.4 332 


.33790 




.B7B31 


-48217 


.91228 




3998 














































4 385 


: 83031 


48836 


: 88095 


: 48322 




■49821 


9298 




51- 


4_39_0_ 


83118 


.4636p_ 


. 881 i3_ 


_48347_ 


-93598 


49846 _ 


9398 




















9498 


























9 483 


'83 62 


■411934 


^88445 












58 








.83532 


.48448 




■49950 


9799 




.5B_ 


4 512 


83 35 


46983 


.88620 


48471 


: 91088 


49B7S , _ 


9899 




^ 




.83 OB 




.88708 










60 



NATURAL VERSED SINES AND EXTERNAL SEC. 29! 





Vern, 


Ex. MC. 


Vera. 


Ex. sec. 


\'«». 


Es. BBC. 


Vers. 


E»,aec 






-5UO0O 






































3 














■54653 


■30521 




3 


■Jmo* 






.06593 


-53 30 


; 13356 


64678 


20647 




_4. 




■00401 


31 


06701 


■ 53 58 


■13473 


4JC6 


■2D773 




I 


■ uiae 


1-DDGOa 












-3D90O 




i 


.Mill 


1 -00807 








^ 3707 


'■ 4767 


.2102E 




t 


-BMW 


1.0070B 






:53 33 


, SB2B 


- 4782 






i 


"i^ 


.00110 


33 


.07137 


-53 58 










^ 




.00913 


: «_ 


_072_46_ 


.63 B4 
-63 10 


-i-4oeo. 




"21536 






- 48ed~ 




■ 1 






9» 




53 38 


: 14296 








2 




01218 


35 


.07576 


53 61 














.01320 




-076BS 










1; 


i_ 




■01423 




■07795 


53 13 




4964 
. 9»0 


^23046 


_J 








01 




-53 39 


■ 4770" 


-23174 




e 


.50404 


i.oieas 


27 


.08015 


























■ 042 


-33430 


1 












:53 16 


■ 5127 


- 088 


-22659 


1 


L 


,50479, 




03 


:08347 


.63_4i. 




"" 


2Vm 






.tOKOS 


















1 








losses 


:53 93 


.' 5485 


146 


: 23946 




:2 


■satat 


1.02346 


. 030 


-03680 




- 5605 


172 


.23075 


2 


:i 


.a»Bi 


1.02S49 














2 


L 


www 


1.02453 


!„J31, 


J8903_ 






B 324 


llssaai 








.02657 


168 


-09014 


"sSBflT 








~2 






.02861 






























■ 6 303 


















' 6336 


6 S28 


1 23856 


2 








: 25S. 


09462 


53799 


—644 7 


l1_|54 


LzaBBS 


_3 


; IT 


-50758 


077 


































'3 








-b 335 




: 53877 


: 8810 


-5 432 


1 ■24378 




S3 


-508S4 


-0 391 


.6 361 


1. 09911 


63B03 






1-24509 


S 


i_ 


S0B59 


Q 438 


■6 338 


.100S4 


Mm. 


1-'- T063 
7176 


■56484 
■55510 


.34840 


_a 






-0 601 


-5 4 




63954 






'bob 10 


-0 706 




: 10350 


■ 58980 




-66638 


.■24803 




7 


-S0flS5 














.35036 


3 








624 














.9. 


■soasg 


04023 


fH 


"195,90 


54058 


LIi£63, 




55300 


M 


40 


.61011 


04128 










-55841 


-26132 










535 












41 






; 04339 


525 


i 10931 


54136 


-18031 


:B66B3 






43 


.51087 












55719 


.25830 




**_ 


,B1_113^ 




"5268" 






18277 


56745 


-36983 


_44 






"0465S" 


1.11274 


54213 


- 8401 






46 




^siies 












55797 






47 
















.36364 




i» 




-04977 


5274 


ill817 


54290 




66849 


-264B8 


48 




51239 




J.S77_ 


ajraa. 


54316 


: 889B 


S687B 
55903 1 


^2ee3?_ 


J? 


50" 




■19010 




51 




05298 


5 823 










: 36800 






^61318 


-05406 




i 12078 


54394 






.27036 


62 




-61941 








54420 


: 19393 






63 


M 


.61388 




-l f^ 


^lllll- 


_64146_ 


-1BS17 


_560P6_ 


^27804 
.27436 


_6* 


56 








Bi 


ee 


^61417 


i-05835 


5 960 


.12540 


54487 








68 






I-05B42 


6 97 


-:2e57 


54523 


; 9893 






ST 


98 












0018 


66111 1 




1 


iS. 








12389 


54575 


0143 


l>6'3? 


'27881 


SO 


-5161B 


1.08307 


5 05 


113005 


54601 








CO 



292 NATURAL VERSED SINES AND EXTERNAL f 





Vers. 


Et sec. 


V«rs. 


E:s. sec. 


Vet-. 


j:i. .oe. 


\>er3 


Ex.H^. 




o 


- eies 



















1 


. eiBQ 


128253 


















. eaii 


.283B0 


: 57791 


l'3691'' 


.68379 


:« 81 




1:06283 










.57817 


1.37064 


-B940B 


.16 42 


.81007 


1 56458 




4 


LjafiL 


-28863 


^57e44_ 


1.37212 


- 66433 


.46 01 


_aip34_ 


1 . 56631 






. 28800 






46 65 












1.36937 


578Sa 


1:37509 


:B948e 






.96888 










57923 


1 37658 


.59512 


:i6 89 








S 


. B372 






1.37floa 




47 62 


6 HI 






t 


_e3aB_ 


29349 


57978 


! 37957 


'59566 


^i7 1-*- 


6 laa 










29"4a7 


58003 


" 88 107 










1 










-38256 


: 59619 


:47 40 




:678 9 


1 


3 


. B4T7 










. 7 04 


6 348 


.580 1 


1 


3 


. 6503 


'- 290O1 


58081 






. 7 67 


.6 275 


-882 3 


1 


4. 


e528_ 


__ap9*a_ 


58108. 


: 38707 




_S_31_ 


6 303 


_6MJ. 


_1 












59725 


. a 95 




1-685 1 


1 


B 












. a 69 


6 858 


1 587 1 


1 


7 


6608 


- 167 


5ai87 


39159 






.0 aas 


1588 






. 6634 


. 596 


53213 


39311 


59805 


: a a9 




11*3 \ 


1 


, £. 


°w 


m 


58241). 


38463 


59BS3 


■ S 54 






J 





. aB87 


■ 875 


58 SB 8 














1 


S71S 


015 


58293 


1:39768 


59885 








21 


2 


. 8739 










49150 


.8 517 


598 3 


33 


: a 














8 514 




23 


4 




'. 43e_ 




l_aP3Z2_ 


J9965_ 




9 570 


602 7 


21 


E 


6B18 
. 6B44 


5!! 


5B3bT 


-.40375 


69992 


" 9 48 

■ I 


a 597 


1|! 


a 




















a 










:: 40835 












. S- 


Bfl23_ 


" 2140 


58504 


1-40988 


JPP-iL 


LoS-L 


8 7 


^11 9. 


J 


; 














8 7 S 


.aia 3 




















leii 8 


3 




7001 






>41460 


60178 


: 1120 


8 7 5 


aiB 


:3 


7028 


2708 


58810 


1-41605 


60305 


- 128 


8 8 a 


1. 81881 




. 4_ 


7054_ 


_za5CL 


SBJ37_ 


^41760, 


60333 




8 8 9 


JI20 9 


I. 








3 










;: 12070 


60385 


: 179 


8 8 3 


: 8^418 


3 


7 


7 33 


- 278 


58718 


1.4322B 


60313 


. 198 


8 9 


.63004 




8 


7 SB 




58743 


1.42380 












-L 


-Lis. 






uam. 


s. 


ijliy. 


B 9 4 


:a 076 


J 


40 




708 




1-42892 


.80883 


. S47 








1 


- 7238 


852 








. ZB4 








3 










:8044S 


■ 381 










: 7291 






1:48183 


.60473 


. 398 


:6 1 


:e 722 




.*. 


■ ',31!L 


■ aaj. 


5BB03_ 


l_43i.l8_ 


,60499_ 


^31S_ 


^!_IL 


i_J_90B. 


J 


4S 


7343 


























1:43033 


: 60552 


3500 


62 83 


1:8 285 


48 


47 






58981 


1.43790 


.80679 




-8 9 




17 


48 




!- 4883 


. 5900a 














.i5_ 


57443 


1. 500_9_ 


- 590S4 
59081 


L -44106 


: 60B33 


U 401 T 


J 43_ 


.68040 






1.44261 


-60659 


■50 


fii 




: 5300 


59087 












81 


62 


57527 












82 24 






Si 








1. 44741 


: 60740 


: 4709 


62 51 


: 65609 


58 




;57Gag_ 


1- 5air 


..5Sie7_ 


\_Mm- 


J076a . 


_18BS_ 


l6_ .7.8 


1.65799 


8* 


"66" 






56 


: 57633 














i:6Eiao 














.60847 




:a3468 


1 66371 












1:45539 


-60873 


: 580 


62486 


1.66563 


58 


IS. 




1 «473 


69300 


!_4SS91. 


60B0Q 


m. 


»2 1'^ 


1 86756 


_S£ 


aol S773B 


i.smo 




I.458BS 


.BOSS! 


r"**l. 


^"™ 


\^ ***" 


\Wt 



NATURAL VERSED SINES AND EXTERNAL SEC. 





Ven. 


llx..... 


Ver.. 


En. s«- 


V.^, 


E.- .... 


\«r,. Ex- «.. 




o 








.79043 


.657 9 


1-92380 


87 43 






1 


8 5Bfl 


-6713 


.84190 


-79254 




.93614 




2. 0741 5 




3 


-e 5S3 


-8733 












3 07378 




















2. 07936 






B 847 






r 80104" 


-659 5 




:87 53 


2_0ai9J_ 




~b" 




8791 " 


.84399 


F9"3"564" 






-e 70 


6110 


64328 


1-80313 








2. 08731 






-8 73 












.87838 


3 08933 
















1. 94383 


-87883 








-B 78: 




84435 




_L80 4 


1 94500 


.87891 


2 09510 
3.09774 




io" 


6 809 


-8388 


1.81175 




87718 






e S3B 












67746 


3 1003B 
















952 3 








13 


!fl BSD 


18947 




1 81831 


56154 










11_ 


■ a 917 


LBflefl- 


U*544_ 


1 82037 


_&6111_ 
66J0B 


r9"59 




2 10B34 
2 llBl 








9 86B 












1 82471 


68236 










17 




7035 


-64625 


1 B2B88 












IS 


.(3025 


7045 










8 939 






iL 


-eaosa 


-1265- 


'84680 


1 83124 


68318 


96893 


8- 8 






ao 






84707 














n 


: 83108 


^7105 


-84734 
















.esiaa 


7134 




83780 


66400 


1 97819 


8 9 












: 64789 


83999 




1 97882 


80 7 






?t 




:71B47 


-84816 


B4319 


66455 


1 98108 


8 04 


3 3520 






-flS216 


.71847 
















S8 






:8487D 


8485B 


66510 


98594 


6 9 


! 406 




37 


flSiea 


: 73347 


-84898 


84880 


66537 










aa 


63398 


.73M8 
















39 


83333 


73849 












3 488 




3« 




.738 


"6 97T 


"85543" 


'66619 


99574 


6 3 






SI 


;B3S77 


.730 2 




B6767 


868 7 


96821 


6 2 7 


i 5429 






. 83404 






































sir 




.lZII?- 


J 0_B8_ 


88437 


667 B 


00583 




2 8255 
2 8531 




. 834S5 








3 810 






























^74369 


B5 70 


1 87J09 












ss 


-83588 


74473 




1 87334 


66 39 










i2_ 


8^594 


_7i877. 




87680 


08 88 


LoaL 


858 


? 7641 






. 83831 




65252 


87786 












<i 


-83948 


!7S08B 




BBOn 


68 21 






















2 559 


6 6 1 


3 B47B 










85334 


88485 






8 833 


2 8759 




-M- 


83739 


-75703 


85381 


.88692 


-67 ■ " 


2-03315 


J_e56 


2 9040 
























8 711 






^7 


83810 


-78323 


^65443 


89 76 


67 










ts 


83838 


-7B530 




89 05 


87 1; 


2 04075 




3 20169 




«- 


.8J_8_8B 








_87_ 


2_0432_9_ 




2_ 30^53 












~900B3 


87 




8 922 




5 


SI 


83919 




85553 


90293 


87 










t3 




:77 83 








2 06094 


8 B77 


2 3 6 


I 
















B 905 




S 












87 




_oa=_ 






"55" 


-8M37 


-77990- 


^5fler 


"9"l3!7 


67 


"osSBr 




3 22185 


'6 


H 


B405S 


-78200 








2 06131 


6 9BB 




5 


57 








91881 




08379 


8 015 


2 23740 


















6 043 


2 33038 




». 






6 771 


flzi" 




08896 




3 23317 


_S. 


60 


841 6S 


-78043 


-B 798 


93380 


6741 


3 07159 


8 098 




_1 



291 XATrRAL ^^!;Rs^:D sixes and external sec. 







«, 


v..U_. 


1 ^ 


r.. 


Es.s«. 








! 


-7G7B3 '. 
7 071 2 


42030 

Jsak 

Si 


" 


13 

jiL 

i 


2.8SS70 

2J80J6_ 
2.8B17B 

■is'l 


1 




- 


-7 802 

■HI 

il 


I 


04 


40 


5 

! 




.' 




111 

Isli 
Ii. 

SOO 2 


44 




-El 


! 


IS. 




1 


;j8! 
Ui 


1 

3052 
SOSO 


s. 


71085 
71853 


r 

80 


■lllll 

ill! 


i 

J 

20 

J 




— 


71571 




SOS 4 


313B 


i 


72216 

il 


71321 

74B73 
71906 


a 97160 
i:931S7 


Ii 

1 


- 


— ^ 


71598 

II 

71794 
71850 
71877 

7iBB» 




III 
1 ii 

Li- 
sa 94 

57 flf 

sa 94 

BO 01 

ii 


ii 

78383 

ii 

735^9 
73557 
73535 

max 

II 
ii 

78950 

74062 
74090 


2 

! 
t 

i 
I 

i 


74198 
71SS1 
76775 
76568 
7776B 

ii 


US 

II 

7BalB 




11 

J 

1 




, 117 ,^ 


ii 




i 

I 
i 

I 

I 


S 

B1B13 

ill 
II 

86951 


75413 
75111 

75497 
7552B 

If 
J578D. 


-062S1 

31 


48 

ii 


~ 


ii 
.11 

-7Z408 
-7Z4SB 




51 
55 

1 



NATURAL VERSED SINES AND EXTERNAL SEC. 
76* 77* 78' 79* 



295 



t 


Vera. 


Ex. sec. 


Vers. 


Ex. sec. 


Vera. 


Ex. sec. 


Vera. 


Ex. sec. 


1 


o 

1 

2 
8 

4 


75808 

■76886 

•75864 

76882 

75921 


S 
S 
8 
8 
8 


■13357 

. 13839 

■14828 

14809 

15295 


■77505 
■77533 
■77562 
•77590 
77618 

77647 

■ 77675 

■ 77703 
• 77782 

■ 77760 


3.44541 
3 ■ 45102 
3 45664 
8 •46228 
3-46793 


.79209 
•79237 
•79266 
• 79294 
- 79323 


3 80973 
3 •81633 
3 82294 
3 82956 
3-83621 


•80919 
80948 
•80976 
•81005 
-81033 


4-24084 
4-24870 
4-25658 
4 •26448 
4^27241 




1 
2 
3 
4 


6 

6 

y 

8 

8 


■75949 

75977 

76005 

•76084 

•76062 


8 
3 
8 
8 
3 


15782 
16271 
16761 
17252 
17744 


3 47360 
3-47928 
3 •48498 
3 49069 
3 49642 


•79351 

- 79380 

- 79408 
•79437 

- 79465 


3 . 84288 
3.84956 
3.85627 
8. 86299 
3-86973 


.81062 
•81090 
•81119 
•81148 
81176 


4.28036 
4 •28833 
4 29634 
4 •30436 
4- SI 241 


5 
6 
7 
8 
9 


10 

11 
12 
18 
14 


•76090 
•76118 
76147 
.76175 
•76208 


8 

8. 

8 

8 

8 


18238 
18733 
19228 
19725 
20224 


■ 77788 
•77817 

77845 

■ 77874 
77902 


3-50216 
3 50791 
8 •51868 
3. 51947 
3-52527 


• 79493 

• 79522 
. 79550 

• 79579 
- 79607 


3 - 87649 
3. 88327 
3.89007 
3-89689 
3-90373 


•81205 
•81233 
•81262 
•81290 
-81319 


4 32049 
4^ 32859 
433671 
4. 34486 
4 •35304 


10 

11 
12 
13 
14 


15 
16 
17 
18 

IS. 


■76281 
■76280 
■76288 
■ 7681-6 
76844 

76878 

76401 

•76429 

•76458 

■76486 


3 
8 
8 
3 
3 


20723 
21224 
21726 
22229 
22734 


■77930 

■ 77959 

■ 77987 

• 78015 

• 78044 


3- 53109 
3 53692 
3 . 54277 
3 • 54863 
3 55451 


• 79636 

• 79664 
. 79693 

79721 

• 79750 


3-91058 
3-91746 
3-92436 
3-93128 
3 93821 


•81348 
-81376 
-81405 
•81433 
.81462 


4.36124 
4 36947 
4 37772 
4 •38600 
4-39430 


15 
16 
17 
18 

-12 


20 
21 
22 
28 
24 


8 
8 
3 
3 
3 


23239 

23746 

■24255 

•24764 

25275 


• 78072 

■ 78101 

■ 78129 
■78157 
•78186 


3. 56041 
3 . 56632 
3.57224 
3. 57819 
3. 58414 


•79778 

- 79807 

- 79835 
■79864 
-79892 


3-94517 
3-95215 
3- 95914 
3-96616 
3-97320 


•81491 
•81519 
81548 
-81576 
-81605 


4.40263 
4. 41099 
4.41937 
4-42778 
4-43622 


20 

2.. 
2l 
23 
24 


26 
26 
27 
28 


•76514 
•76542 
• 76571 
■76599 
•76627 


3 
8 
3 
3 
.8 


■25787 
•26300 
•26814 
•27330 
27847 


■ 78214 
78242 

■ 78271 

■ 78299 
■78328 


3 59012 
3-59611 
3-60211 
3-60813 
3. 61417 


• 79921 

• 79949 
79978 

• 80006 
•80035 


3.98025 
3-98733 
3-99443 
:4-00155 
4 00869 


81633 
-81662 
-81691 
-81719 
-81748 


4-44468 
4-45317 
4-46169 
4-47023 
4-47881 


25 
26 
27 
28 
29 


80 
81 
82 
88 
84 


■ 76655 
•76684 
•76712 
•76740 
• 76769 


3 
8 
3 
3 
8 


28366 

28885 

•29406 

.29929 

■30452 


■78356 

■ 78384 

■ 78413 
•78441 
• 78470 


3 • 62023 
3 • 62630 
3. 63238 
3-63849 
3-64461 

3-65074 
3. 65690 
3-66307 
3-66925 
3-67545 


-80063 
•80092 
•80120 
•80149 
-80177 


401585 
4-02303 
4-03024 
4 03746 
4-04471 


-81776 
-81805 
-81834 
•81862 
•81891 


4-48740 
4-49603 
4-50468 
451337 
4 •52208 


30 

31 
32 
33 
34 


86 

86 
87 
88 

is. 


76797 

76825 

■76854 

.76882 

.76910 


3 
3 
8 
3 

L 


30977 
31503 
32031 
32560 
33090 


78498 
•78526 

■ 78555 

■ 78588 

■ 78612 


80206 
•80234 
.80263 
•80291 
■ 80320 


4-05197 
4 05926 
4-06657 
4.07390 
4.08125 


■81919 
•81948 
•81977 
•82005 
-82034 


4-53081 
4.53958 
4 •54837 
4. 55720 
4.56605 


35 
36 
37 
38 
39 


40 
41 
42 
48 
U 


•76988 
76967 
•76995 
.77028 
.77052 


3. 
3. 
3. 
8- 
3. 


33622 
34154 
34689 
35224 
35761 


•78640 
78669 

■ 78697 
- 78725 

■ 78754 


3 -68167 
3.68791 
3.69417 
3.70044 
3 . 70673 


-80348 

80377 

• 80405 

- 80434 

- 80462 


4-08863 
4-09602 
4-10344 
4 11088 
4-11835 

412583 
4.13334 
4-14087 
4-14842 
4- 15599 


-82063 
•82091 
•82120 
•82148 
•82177 

• 82206 

• 82234 

• 82263 
•82292 
■ 82320 


4^57493 
4 •58383 
4 •59277 
4-60174 
4^61073 

4-61976" 
4-62881 
4 63790 
4. 64701 
4-65616 


40 

41 
42 
43 
44 


45 
46 

47 
48 

48 


.77080 
•77108 
.77187 
■ 77165 
-77193 


8. 
%. 
3. 
3- 
3^ 

3. 
3. 
3- 
3. 
3. 


36299 
36839 
37380 
37923 
38466 

39012 
39558 
40106 
40656 
41206 


•78782 
•78811 
•78839 
■78868 
• 78896 

■ 78924 
■78953 
■78981 
■79010 

■ 79038 


3.71303 
3.71935 
3.72569 
3.73205 
3 • 73843 


8049: 
• 8052U 

- 80548 
-80577 

- 80605 


45 
46 
47 
48 
-49 


50 
51 
52 
58 
54 


.77222 i 

77260 

77278 

■77807 

■77885 


3 • 74482 
3 • 75123 
3.75766 
3 •76411 
3.77057 

3. 77705 
3.78355 
3.79007 
3.79661 
3. 80316 


-80634 
■80662 
•80691 
•80719 
- 80748 


4-16359 
4-17121 
4-17886 
4-18652 
4 19421 


-82349 
•82377 

- 82406 
■ 82435 
i824e3_ 

- 82492 
-82521 
. 82549 
-82578 

82607 


4-66533 
4-67454 
468377 
4-69304 
4-70234 

4-71166 
472102 
4-73041 
4-73983 
4.749?9 


50 

51 
52 
53 
54 


55 
56 

67 
58 


■77868 
■77892 
•77420 

■ 77448 

■ 77477 


3. 
3^ 
3. 
3^ 

L 


41759 
42312 
42867 
43424 
43982 


■79067 
■ 79095 
■79123 
■79152 
•79180 


80776 
80805 
80833 
80862 
80891 


4-20193 
4-20966 
4-21742 
4-22521 
4-23301 


55 
56 
57 
58 
59 


eo 


77505 


8- 


44541 


.79209 


3. 80973 


80919 


4 24084 


82635 


4-75877 


60 



296 NATURAL VERSED SINES AND EXTERNAL SEC. 





Vers. 


EX..,.. 


Vers. 


» 


S.BB.. 


V.„. 


..™. 


Vers. 


^.^. 




a 

! 


: 82661 


liii 


Ss 


i 
5 


39245 

42787 
43977 


[bbisb, 

sn 

86342 


S- 


ii? 

30630 


"6"7"957" 


T 


ii 


1 

-J 

T 

1 

1 

i 


I 




J:ffi 

i-szeoe 


HI 


1 

5, 

i 

L 
1 
r 

L 

1 


7572 
48779 
49991 
1208 
2429 

6131 

9855 
1110 

Bi7a 

72830 
73924 


? 
I 

7 
7 


3038 
3640 
4251 




-83039 

ii 

aaiBQ 

833Bfl 
83284 
83323_ 

83409 
33487 
83485 

ill 


I.S5S3B 


liiis 
II 

84815 
84903 
84931 

iii 

85190 

ii 

85334 


■ 86400 

^83436, 

-86573 
J6R30, 

- 6669 

'■ B717 

- 6746 
^_6774_ 

:li 


l 

fl 

6 

8 
6 
6 
6 


35274 
39 78 


88 31 


IS 
IB 

1 


m 

:|i 

TOSSBB 
:1012a 


44 43 
47955 
49 71 


lis 

88363 
88391 

88606 


7 


DOTS 

1? 
592 
813 

is 


11 
1 

11 


1 


6 
6 
8 
6 

H 


ii! 


88693 
_8865!_ 

IISSS 

iii 


1 

? 
7 
7 
7 


?T 


i 


1 


5 

L 

i 

L 

i 

I 

J 
fl_ 

1 

6 

! 

s 


76547 
77869 

3iB5_6_ 
II 

II 
98301 

03983 

ii; 


;87149 
.87288 

11 

87351 

III 


6 
6 
6 

! 


66130 


8 S 

isl 


1 


S 
t 


if 

838B7 

Si 
:EI 


: !S 
:_loo4_ 

: *343 
5I..85J.O. 


ii 

85478 

ii 

85622 
85708 

Ism" 


6 

! 

8 
i 

a 

8 

e 

f 

I 
3 


83443 

88792 

B0592 
92400 
94216 

071 7 
J90 2 

ii 

0551 


H 

89056 
89084 


I 
1 
I 

i 


9844 

1! 


3S 

31 

Jl 
40 


1 
1 

ES 


M 

89258 
89345 


1 

I 
i 

\ 


i 

in 


!! 

48 

-il 

1 


I 


■ U918 

M 

■84IBT 


■KEH 

B-8B2*B 


-BflOBS 


11 

_7049_ 
IBBSO 


^877 6 
877 4 
.878 3 


89480 
89547 


1 

L 

B 


436 

54S 7 
686 7 


si 



NATURAL VERSED SINES AND EXTERNAL SBC. 297 



Vsts. 


Eiceec. 


V„. 


Ei.«c. 


1'or.. E(..ec. 




:i!| 




S 11 

e 03 




ii 

91400 




! 


4 371 
G 337 


- 


II 
IT 

J3431 _ 

B34 

SI! 


\ 


11 


S 


:|| 




7 03 
7 33 

7 II 




Kfi 


C 


ii 




s 


8 


:!!!! 




B 517 




II 




c 


90934 




is 

23223 




■son 


- 


11 




9171B 

St 
lii 




- 


07310 

30427 
24761 

335Z1 




28970 

11 

85369 


5 
32 


■SI 


- 


11 




S| 






til 


- 


in 

938" S~ 
940 1 


\ 


s 

30287 


1 


.0047 
:B0S3 


13 43 
SB 08 


- 


III 

93270 






ii 

93B 7 


J 


III 

779m" 

lii 


1 


s 

■ma 


- 


69 e 

u 

79 2 

B2 fl 


ill 

83415 

II 


- 


2 


H 
fil 




940 

iii 


i! 

38 
_39 
iO 

43 






93 7 

a1 




93S89 

is! 




a 


493 3 




i 




ill 

00794 


!! 




11 
\ 



lis 

JS6_2. 
173 1 


- 


II 

IB" 

^3995 
93024 


- 


3 


ii 

II 

33BS9 


' 






111 

49153 


1 

94 


1 

.S13B1 


ii! 

947 7 


- 


It 


57 

Js 



298 NATURAL VERSED SINES AND EXTERNAL SEC. 
87* 88** 89* 



r 


Vers. 


£x.seo. 


Vera. 


Ex. sec. 


Vera. 


Ex. Reo. 


< 




i 

2 
3 
4 


.94766 
.94795 
.94825 
.94854 
.94883 


18 
18 
18 
18 
18 


10732 
21397 
.32182 
43088 
54119 


.96510 
.96539 
.96568 
.96597 
.96626 


27 
27 
28 
28 
28 

28 
29 
29 
29 
29 


65371 
89440 
13917 
■38812 
64137 

89903 

■16120 

42802 

69960 

.97607 


.98255 
•98284 
.98313 
98342 
•98371 


63.29869 
67.26976 
68.27431 
69.81411 
60.89105 




1 
2 
S 

4 


5 
6 
7 
8 
9 


.94912 
.94941 
.94970 
.94999 
.95028 


18 

18. 

18. 

18. 

19. 

19. 

19. 

19. 

19 

19. 


65275 
76560 
87976 
99524 
11208_ 

23028 
34989 
47093 
59341 
71737 


.96655 
.96684 
.96714 
.96743 ' 
.96772 


•98400 
.98429 
.98458 
.98487 
•98517 


61.60715 
82.66460 
83.86572 
66.11304 
66. 40927 


5 
6 
7 
8 

9 


10 

11 
12 
13 
14 


.95057 
.95086 
.95115 
.95144 
.95173 


.96801 
.96830 
.96859 
.96888 
.96917 


30 

30 

30 

31. 

31 


.25758 

■54425 

83623 

13366 

43671 


•98546 
.98575 
.98604 
.98683 
.98662 


67.75736 
69.16047 
70.62207 
73.14583 
73. 73586 


10 

11 
12 
IS 
14 


15 
16 
17 
18 


.95202 
.95231 
.95260 
.95289 
.95318 


19. 
19. 
20. 
20. 
20. 


84283 
96982 
09838 
22852 
36027 


.96946 
.96975 
.97004 
.97033 
.97062 


31 

32 

32 

32. 

33 


74554 
06030 
38118 
70835 
04199 


.98691 
.98720 
.98749 
.98778 
•98807 


75.89655 
77.18374 
78. 94968 
80.85315 
82-84947 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


.95347 
.95377 
.95406 
.95435 
.95464 


20. 
20. 
20. 
20. 

21. 


49368 
62876 
76555 
90409 
04440 


.97092 
.97121 
.97150 
.97179 
.97208 


33 

33. 

34 

34 

34 


38232 
72952 
08380 
44539 
81452 


.98836 
.98866 
.98895 
.98924 
.98953 


84. 94661 
87.14924 
89. 48886 
91. 91387 
94-49471 


20 

21 
22 
23 
24 


25 
86 
27 
28 
29 


.95493 
.95522 
.95551 
.95580 
.95609 


21. 
21. 
21. 
21. 
21. 


18653 
33050 
47635 
62413 
77386 


.97237 
.97266 
.97295 
.97324 
.97353 


35 

35 

35 

36. 

36. 


19141 
57633 
96953 
37127 
78185 


.98982 
.99011 
.99040 
.99069 
.99098 


97 •23803 
100.1119 
108.1767 
106.4811 
109. 8966 


25 
26 
27 
21 
29 


30 

31 
32 
33 
34 


95638 
.95667 
.95696 
.95725 
.95754 


21. 
22. 
22. 
22. 
22. 


92559 
07935 
23520 
39316 
55328 


.97382 
.97411 
.97440 
.97470 
.97499 


37. 

37 

38 

38 

38 


20155 
63068 
06957 
51855 
97797 


.99127 
.99156 
.99186 
.99215 
.99244 


118.5980 
117.6444 
131.7780 
136.8358 
181.3323 


30 

81 
82 
88 
84 


35 
36 
37 
38 

2L. 


.95783 
.95812 
.95842 
.95871 
-95900 


22. 
22. 
23- 
23- 
23. 


71563 
88022 
04712 
21637 
38802 


.97528 
.97557 
.97586 
.97615 
•97644 


39 
39 
40 
40 
41 


44820 
92963 
42266 
92772 
44525 


.99278 
.99302 
.99831 
.99360 
•99389 


186.5111 
143. 3406 
148.4684 
155.2638 
163.70AS 


85 
Sfl 

87 
81 

JS 


40 

41 
42 
43 
44 


.95929 
.95958 
.95987 
.96016 
.96045 


23 

23. 

23. 

24 

24 


56212 
73873 
91790 
09969 
28414 


.97673 
.97702 
.97731 
.97760 
.97789 

.97819 
.97848 
.97877 
•97906 
.97935 

•97964 
•97993 
•98022 
•98051 
.98080 

.98109 
•98138 
•98168 
.98197 
•98226 


41 
42 
43 
43 
44 

44 
45 
46 
46 
47 


97571 
51961 
07748 
•64980 
23720 

■ 84026 
45963 

■09596 
74997 

■42241 


.99418 
.99447 
.99476 
.99505 
•99535 


170.8883 
179.0850 
189.9868 
301.3213 
318.8800 


40 
41 
42 
48 
44 


45 
46 
47 
48 
49 


.96074 
.96103 
.96132 
.96161 
.96190 


24- 

24 

24 

25 

25 


47134 
66132 
85417 
04994 
24869 


.99564 
.99593 
.99622 
.99651 
.99680 


338.1889 
344.6540 
368.4437 
386.4795 
811.5330 


48 
46 
47 
46 
49 


50 

51 
52 
53 
54 


.96219 
.96248 
•96277 
.96307 
.96336 


25 
25 
25 
26 
26 


45051 

65546 

86360 

.07503 

■28981 


48 

48 

49 

50. 

51 

51 
52 
53 
54 
55 


■11406 

■82576 

55840 

^1290 

09027 

89156 
■71790 
57046 
45053 
35946 


.99709 
.99738 
.99787 
.99796 
•99825 

.99855 
.99884 
.99913 
.99942 
.99971 


843.7753 
880.9723 
438.7187 
490.1070 
671.0581 


50 
61 
62 
68 
64 


55 
56 
57 
58 

59 


.96365 
.96394 
.96423 
.96452 
.96481 


26 
26 
26 
27 
27 


. 50804 

72978 

.95513 

.18417 

41700 


688.5406 
858.4869 
1144.016 
1717.874 
8436.747 


55 
96 

57 
61 

ja 


GO 


.96510 


27 


.65371 


.98255 


56 


29869 


1.00000 


Infinite 





TABLES 



299 



RADII — METRIC 
R = lOcsc^ 



Dee. 


Radius. 


Log. rad. 


Deg. 


Radius. 


IvOg. rad. 


Deg. 


Radius. 


Log. rad. 


0-00' 


00 


• 
00 


roo' 


1145.930 


3.05910 


2*00' 


572.987 


2.75814 


01 


68754.931 


4.83730 


01 


1127.145 


3.05198 


01 


508.252 


2.75454 


02 


34377.468 


4.53627 


02 


1108.900 


3.04492 


02 


563.595 


2.75097 


03 


22918.310 


4.36018 


03 


1091.304 


3.03797 


03 


559.013 


2.74742 


04 


17188.735 


4.23524 


04 


1074.311 


3.03113 


04 


554.505 


2.74391 


05 


13750.987" 


4.13833 


05 


1057.785 


3.02440 


05 


550.070 


2.74042 


06 


11459.157 


4.05912 


00 


1041.757 


3.01777 


06 


545.705 


2.73090 


07 


9822.134 


3.99221 


07 


1020.209 


3.01124 


07 


541.408 


2.73352 


06 


8594.369 


3.93421 


08 


1011.118 


3.00480 


08 


537.179 


2.73012 


09 


7639.438 


3.88306 


09 


990.465 


2.99840 


09 


533.015 


2.72074 


10 


6875.496 


3.83730 


10 


982.230 


2.99221 


10 


528.916 


2.72339 


11 


6250.451 


3.79591 


11 


908.396 


2.98605 


11 


524.879 


2.72000 


12 


5729.582 


3.75812 


12 


954.947 


2.97998 


12 


520.903 


2.71070 


13 


5288.845 


3.72336 


13 


941.800 


2.97399 


13 


510.985 


2.71348 


14 


4911.070 


3.69118 


14 


929.139 


2.96808 1 


14 


513.129 


2.71023 


15 


-4583.666 


3.66121 


15 


910.751 


2.96225 


15 


509.328 


2.70700 


18 


4297.187 


3.63318 


10 


904.089 


2.95650 


10 


505.584 


2.70379 


17 


4044.412 


3.60680 


17 


892.940 


2.95082 


17 


501.894 


2.70001 


18 


3819.723 


3.58203 


18 


881.492 


2.94522 


18 


498.258 


2.09745 


19 


3618.685 


3.55855 


19 


870.335 


2.93969 


19 


494.074 


2.09432 


20 


3437.752 


3.53627 


•20 


859.456 


2.93422 


20 


491.141 


2.69121 


21 


3274.050 


8.51509 


21 


848.840 


2.92883 


21 


487.058 


2.08812 


22 


3125.230 


3.49488 


22 


838.495 


2.92350 


22 


484.224 


2.08505 


23 


2989.351 


3.47558 


23 


828.393 


2.91824 


23 


480.838 


2.08200 


24 


2864.795 


3.46709 


24 


818.532 


2.91304 


24 


477.500 


2.07897 


26 


2760.203 


3.43936 


25 


808.902 


2.90790 


25 


474.207 


2.07597 


26 


2644.427 


3.42233 


20 


799.497 


2.90282 


20 


470.900 


2.07298 


27 


2546.485 


3.40594 


27 


790.308 


2.89780 


27 


407.750 


2.07002 


28 


2455.540 


3.39015 


28 


781.327 


2.89283 


28 


404.590 


2.00708 


29 


2370.867 


3.37491 


29 


772.549 


2.88793 


29 


401.479 


2.00415 


30 


2291.838 


3.30018 


30 


703.900 


2.88307 


30 


458.403 


2.60125 


31 


2217.909 


3.34594 


31 


755.571 


2.87828 


31 


455.307 


2.05830 


32 


2148.600 


3.33210 


32 


747.359 


2.87353 


32 


452.372 


2.05550 


33 


2083.491 


3.31879 


33 


739.323 


2.86883 


33 


449.410 


2.05205 


84 


2022.212 


3.30583 


34 


731.458 


2.80419 


34 


440.498 


2.04982 


85 


1964.435 


3.29324 


35 


723.759 


2.85959 


35 


443.018 


2.04701 


86 


1909.868 


3.28100 


30 


710.221 


2.85505 


30 


440.775 


2.04422 


87 


1858.251 


3.20910 


37 


708.837 


2.85055 


37 


437.908 


2.04144 


88 


1809.350 


3.25752 


38 


701.005 


2.84009 


38 


435.190 


2.03808 


80 


1762.957 


3.24024 


39 


094.518 


2.84108 


39 


432.400 


2.03595 


40 


1718.883 


3.23525 


40 


087.574 


2.83732 


40 


429.757 


2.03322 


41 


1676.059 


3.22452 


41 


080.700 


2.83300 


41 


427.088 


2.63052 


42 


1637.032 


3.21400 


42 


074.093 


2.82872 


42 


424.452 


2.62783 


43 


1598.962 


3.20384 


43 


007.549 


2.82448 


43 


421.849 


2.02510 


44 


1562.623 


3.19385 


44 


001.130 


2.82029 


44 


419.277 


2.02250 


45 


1527.899 


3.18409 


45 


054.834 


2.81013 


45 


410.737 


2.01980 


46 


1494.084 


3.17455 


40 


048.057 


2.81202 


; 40 


414.227 


2.01724 


47 


j 1462.883 


3.10521 


47 


042.595 


2.80794 


47 


411.747 


2.01463 


48 


! 1432.406 


3.15007 


48 


036.616 


2.80390 


48 


409.290 


2.01204 


49 


- 1403.174 


3.14711 


49 


630.800 


2.79990 


49 


400.875 


2.00940 


50 


■ 1375.111 


3.13834 


50 


625.072 


2.79593 


50 


404.482 


2.00090 


51 


1348.148 


3.12974 


51 


019.441 


2.79200 


51 


402.117 


2.00435 


.52 


1322.223 


3.12130 


52 


013.910 


2.78810 


52 


399.780 


2.60182 


53 


1297.276 


3.11303 


53 


008.478 


2.78424 


53 


397.409 


2.59930 


54 


1273.253 


3.10491 


54 


603.141 


2.78042 


54 


395.180 


2.59680 


55 


1250.103 


3.09095 


55 


597. S97 


2.77663 


55 


392.928 


2.59431 


56 


1227.780 


3.08912 


50 


592.743 


2.77287 


50 


3W.090 


2. 591 84 


57 


1206.241 


3.08143 


57 


587.677 


2.76914 


57 


388.489 


2 58938 


58 


1185.444 


3.07388 


58 


582.608 


2.76.544 


58 


380.307 j 


2.58693 


69 


1165.352 


3.00040 


59 


577.802 


2.76178 


59 


384.149 


2.58450 



300 



THE RAILROAD TAPER 



RADII — METRIC — (Continued) 



Deg. 


Radius. 


1 
Log. rad. 

i 


Deg. 


Radius. 


1 
Log. rad. 

1 


Dog. 


Radius, i 


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 1 


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 


2.45179 


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 


05 


225.500 


2.35315 


06 


369.695 


2.56784 


06 


279.551 


2.44646 


06 


224.764 


2.35173 


07 


367.719 


2.56552 


07 


278.420 


3.44470 


07 


224.032 


2.35031 


08 


365.763 


2.56320 


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 


2.346U9 


11 


360.020 


2.55633 


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 1 


15 


218.346 


2.33915 


16 


350.838 


2.64511 


16 


268.636 


2.42916 


16 


217.656 


2.33777 


17 


349.058 


2.54290 


17 


267.691 


2.42747 


17 


216.969 


2.33640 


18 


347.295 


2.54070 


18 


266.555 


2.42579 


18 


216.288 


2.33503 


19 


345.550 


2.538.51 


19 


265.526 


2.42411 


19 


215.610 


2.33367 


20 


343.823 


2.53633 


20 


264.505 


2.42243 , 


20 


214.937 


2.33231 


21 


342.113 


2.53417 


21 


263.492 


2.42077 1 


21 


214.268 


2.33096 


22 


340.420 


2.53202 


22 


262.487 


2.41911 1 


22 


213.603 


2.32961 


23 


338.743 


2.52987 


23 


261.489 


2.41745 1 


23 


212.943 


2.32826 


24 


337.084 


2.52774 


24 


260.499 


2.41581 


24 


212.285 


2.32092 


25 


335.440 


2.52561 


25 


259.517 


2.41417 


25 


211.632 


2.32558 


26 


333.812 


2.52350 


26 


258.642 


2.41253 


26 


210.984 


2.32425 


27 


332.200 


2.52140 


27 


257.574 


2.41090 


27 


210.339 


2.32292 


28 


330.603 


2.51931 


28 


256.613 


2.40928 1 


28 


209.698 


2.32159 


29 


329.022 


2.51722 


29 


255.660 


2.40766 


29 


200.061 


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 


2.40445 


31 


207.799 


2.31764 


32 


324.367 


2.51104 


32 


252.841 


2.40285 


32 


207.174 


2.31634 


33 


322.845 


2.60899 


33 


251.916 


2.40126 


33 


206.552 


2.31503 


34 


321.337 


2.60696 


34 


250.997 


2.39967 


34 


205.934 


2.81373 


35 


319.842 


2.50494 


35 


250.085 


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 


2.60092 


37 


248.280 


2.39494 


37 


204.108 


2.30085 


38 


315.442 


2.49892 ' 


38 


247.387 


2.39338 


38 


203.499 


2.30866 


39 


314.003 


2.49693 1 


39 


246.501 


2.39182 


39 


202.899 


2.30728 


40 


312.576 


2.49496 


40 


245.621 


2.39027 


40 


202.303 


2.30600 


41 


311.162 


2.49299 


41 


244.748 


2.38872 


41 


201.710 


2.30178 


42 


309.761 


2.49103 


42 


243.880 


2.38718 


43 


201.121 


2.30346 


43 


308.372 


2.48907 


43 


' 243.019 


2.38564 


43 


200.635 


2.30219 


44 


306.996 


2.48713 


44 


1 242.164 


2.38411 


44 


199.952 


2.30093 


45 


305.632 


2.48520 


45 


, 241.314 


2.38258 


45 


199.873 


2.29067 


46 


304.280 


2.48327 


46 


■ 240.471 


2.38100 


46 


198.798 


2.29841 


47 


302.940 


2.48136 


47 


' 239.634 


2.37955 


47 


198.225 


2.20716 


48 


301.612 


2.47945 


48 


1 238.802 


2.37804 


48 


197.AM 


2.20501 


49 


3(K).295 


2.47755 


49 


1 237.976 


2.37653 


49 


197.090 


2.20486 


50 


1 298.990 


2.47566 ' 


50 


1 237.156 


2.37503 


50 


196.528 


2.29842 


51 


297.696 


2.47377 i 


51 


236.342 


2.373-)4 


51 


195.968 


2.29219 


52 


296.414 


2.47190 1 


52 


■ 235.533 


2.37205 


52 


195.412 


2.29005 


53 


295.142 


2.47(K)3 ' 


1 53 


234. 73U 


2.37057 


53 


194.859 


2.28972 


54 


293.881 


2.4GS17 


54 


233.932 


2.36909 


54 


194.S0Q 


2.28849 


55 


292.631 


1 2.4(Hi:i2 


1 55 


233.139 


2.36761 


56 


198.782 


2.28727 


56 


291.392 


2.46448 


1 56 


. 232.352 


2.36615 


56 


198.218 


2.28005 


57 


290.163 


2.462W 


1 ^^ 


, 231.570 


' 2.36468 


57 


192.877 


2.28483 


58 


288.944 


2Aimi 


, 58 


■ 230.794 


. 2.36.322 


58 


192.140 


2.28862 


59 


287.735 


2.45S9U 


! 5!) 

1 


' 230.022 


2.36177 

1 


59 


191.005 


2.28241 



TABLES 



301 







RADII — METRIC 


— (Continued) 




Dec. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


••00' 


191.073 


2.28120 


7*00' 


163.804 


2.21432 


8'* 00' 


143.356 


2.15642 


01 


190.544 


2.28000 


01 


163.415 


2.21329 


01 


143.058 


2.15551 


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 


05 


161.880 


2.20919 


05 


141.880 


2.15192 


00 


187.944 


2.27403 


06 


161.500 


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 


140.435 


2.14748 


11 


185.413 


2.26814 


11 


159.629 


2.20311 


11 


140.150 


2.14659 


12 


184.915 


2.26697 


12 


159.260 


2.20211 


12 


139.865 


2.14571 


13 


184.420 


2.26581- 


13 


158.892 


2.20110 


13 


139.582 


2.14483 


U 


183.927 


2.26465 


14 


158.527 


2.20010 


14 


139.300 


2.14395 


15 


183.437 


2.26349 


15 


158.163 


2.19910 


15 


139.019 


2.14307 


16 


182.950 


2.26233 


16 


157.800 


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 


10 


181.503 


2.25888 


19 


156.724 


2.19514 


19 


137.906 


2.13958 


ao 


181.026 


2.25774 


20 


156.368 


2.19415 


20 


137.631 


2.13872 


21 


180.552 


2.25660 


21 


156.014 


2.19316 


21 


137.367 


2.13785 


22 


180.079 


2.25546 


22 


155.661 


2.19218 


22 


137.084 


2.13699 


28 


179.610 


2.25433 


23 


155.310 


2.19120 


23 


136.812 


2.13612 


24 


179.142 


2.25320 


24 


154.961 


2.19022 


24 


136.541 


2.13526 


» 


178.678 


2.25207 


25 


154.613 


2.18925 


25 


136.271 


2.13440 


26 


178.215 


2.25094 


26 


154.267 


2.18827 


26 


136.002 


2.13355 


27 


177.755 


2.24982 


27 


153.923 


2.18730 


27 


135.734 


2.13269 


28 


177.298 


2.24870 


28 


153.580 


2.18633 


28 


135.468 


2.13184 


29 


176.842 


2.24759 


29 


153.238 


2.18537 


29 


135.202 


2.13098 


80 


176.389 


2.24647 


30 


152.898 


2.18440 


30 


134.937 


2.13013 


31 


175.939 


2.24536 


31 


152.559 


2.18344 


31 


134.674 


2.12928. 


32 


175.490 


2.24425 


32 


152.222 


2.18248 


32 


134.411 


2.12843 


83 


175.044 


2.24315 


33 


151.887 


2.18152 


33 


134.150 


2.12759 


84 


174.600 


2.24204 


34 


151.553 


2.18056 


34 


133.889 


2 12674 


35 


174.159 


2.24095 


35 


151.220 


2.17961 


35 


133 630 


2.12590 


36 


173.720 


2.23985 


36 


150.889 


2.17866 


36 


133.371 


2.12506 


37 


173.283 


. 2.23876 


37 


150.559 


2.17771 


37 


133.114 


2.12422 


38 


172.848 


2.23766 


38 


150.231 


2.17676 


38 


132.857 


2.12338 


39 


172.415 


2.23658 


39 


149.904 


2.17581 


39 


132.602 


2.12255 


40 


171.984 


2-. 23549 


40 


149.579 


2.17487 


40 


132.347 


2.12171 


41 


171.556 


2.23441 


41 


149.255 


2.17393 


41 


132.094 


2.12088 


42 


171.130 


2.23333 


42 


148.932 


2.17299 


42 


131.841 


2.12005 


43 


170.705 


2.23225 


43 


148.611 


2.17205 


43 


131.589 


2.11922 


44 


170.284 


2.23117 


44 


148.291 


2.17111 


44 


131.339 


2.11839 


45 


109.863 


2.23010 


45 


147.973 


2.17018 


45 


131.089 


2 11757 


46 


169.446 


2.22903 


46 


147.656 


2.16925 


46 


130.840 


2 11674 


47 


169.030 


2.22796 


47 


147.340 


2.16832 


47 


130.593 


2.11592 


48 


168.616 


2.22690 


48 


147.026 


2.16739 


48 


130.346 


2.11510 


49 


168.204 


2.22584 


49 


146.713 


2.16647 


49 


130.100 


2.11428 


50 


167.794 


2.22478 


50 


146.401 


2.16554 


50 


129.855 


2.11346 


51 


167.387 


2.22372 


51 


146.091 


2.16462 


51 


129.611 


2.11264 


52 


166.981 


2.22267 


52 


145.782 


2.16370 


52 


129.368 


2 11183 


58 


166.677 


2.22162 


53 


145.474 


2.16278 ; 


53 


129.125 


2 11101 


54 


166. 175 


2.22057 


54 


145.168 


2.16187 . 


5t 


12S.884 


2 11020 


55 


165.775 


2.21952 


55 


144.862 


2.16095 ' 


1 55 


128.644 


2 10939 


66 


165.377 


2.21848 


56 


144.559 


2. mm 


56 


12S.4()4 


2.10858 


57 


164.981 


2.21743 


67 


144.256 


2.1.5913 


57 


128.166 


2.10777 


58 


164.587 


2.21640 


58 


143.955 


2.15823 


58 


127.928 


2.10697 


60 


164.195 


2.21536 


69 


143.655 


2.15732 


59 


127.691 


2.10616 



302 



THE RAILROAD TAPER 



RADII — METRIC — (Continued) 



Deg. 


Radius. 


Log. rad. 


Deg. 
10*00' 


Radius. 


Tx)g. rad. 


Deg. 
11*00' 


Radius. 


Log. rad. 


9° 00' 


127.455 


2.10536 


114.737 


2.05970 


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 


2.06826 


02 


104.020 


2.01712 


03 


126.752 


2.10295 


03 


114.168 


2.05754 


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 


2.05611 


05 


103.552 


2.01516 


06 


126.057 


2.10057 


06 


113.604 


2.05639 


06 


103.397 


2.01451 


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 


126.142 


2.09740 


10 


112.861 


2.05264 


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 


2.05113 


12 


102.477 


2.01063 


13 


124.465 


2.09506 


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 


15 


102.023 


2.00670 


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 


2.09115 


18 


111.404 


2.04690 


18 


101.673 


2.00678 


19 


123.132 


2.09037 


19 


111.224 


2.04620 


19 


101.424 


2.00614 


20 


122.912 


2.08959 


20 


111.046 


2.04550 


20 


101.276 


2.00550 


21 


122.694 


2.08882 


21 


110.867 


2.04480 


21 


101.127 


2.00487 


22 


122.476 


2.08805 


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 


2.08574 


26 


110.160 


2.04202 


25 


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 


2.08345 


28 


109.635 


2.03996 


28 


100.102 


2.00044 


29 


120.973 


2.08269 


29 


109.461 


2.03926 


29 


99.957 


1.99981 


30 


120.761 


2.08193 


30 


109.288 


2.03867 


30 


99.812 


1.99918 


31 


120.550 


2.08117 


31 


109.116 


2.03788 


31 


99.608 


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 


2.03683 


34 


99.239 


1.99668 


35 


119.713 


2.07814 


35 


108.430 


2.03615 


36 


99.097 


1.99606 


36 


119.506 


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 


2.07589 


38 


107.921 


2.03311 


38 


98.072 


1.99419 


39 


118.888 


2.07514 


39 


107.753 


2.03243 


39 


98.532 


1.99358 


40 


118.684 


2.07439 


40 


107.585 


2.03175 


40 


98.391 


1.99296 


41 


118.480 


2.07364 


41 


107.418 


2.03108 


41 


98.251 


1.99284 


42 


118.277 


2.07290 


42 


107.251 


2.03040 


42 


98.112 


1.99172 


43 


'; 118.074 


2.07215 i 


43 


107.084 


2.02972 


43 


97.978 


1.09111 


44 


117.873 


2.07141 


44 


: 106.919 


2.02905 


44 


97.834 


1.99049 


45 


117.672 


2.07067 


45 


106.753 


2.02838 


45 


97.690 


1.98088 


46 


117.471 


2.06993 


46 


106.588 


2.02771 


46 


97.S58 


1.98926 


47 


117.272 


2.06919 


47 


106.424 


2.02704 


47 


97.420 


1.08865 


48 


1 117.073 


2.00846 


48 


106.261 


2.02637 


48 


97.283 


1.96801 


40 


116.875 


2.06772 


49 


106.097 


2.02670 


49 


97.147 


1.08743 


50 


116.677 


2.06699 


50 


105.935 


2.02604 


50 


97.010 


1.98682 


51 


116.480 


2.06625 


51 


105.772 


2.02437 


51 


96.874 


1.98621 


52 


. 116.284 


2.06552 


52 


105.611 


2.02371 


52 


96.739 


1.08560 


5:} 


110.088 


2.06479 


53 


105.449 


2.02304 


53 


96.004 


1.98600 


54 


1 115.893 


2.06406 


64 


105.289 


2.02238 


64 


96.409 


1.98439 


55 


115.099 


! 2.06.333 


1 55 


105.128 


2.02172 


55 


96.334 


1.98S78 


56 


' 115.505 


2.06260 


' 56 


104.968 


2.02106 


56 


96.200 


1.98318 


57 


1 115.312 


2.06187 


57 


104.809 


2.02040 


67 


96.067 


1.98257 


58 


115.120 


2.06115 


58 


104.650 


2.01974 


58 


95.083 


1.96197 


5f) 


114.928 

1 


2.06043 

1 


59 


104.492 


2.01908 


59 


95.800 


1.96137 



TABLES 



303 



RADII — METRIC — (Continued) 



Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 

1 


Deg. 


Radius. 


Log. rad. 


12-00' 


95.668 


1.98077 


w-oo' 


88.337 


1.94614 


14*00' 


82.055 


1.91410 


01 


95.536 


1.98017 


01 


88.224 


1.94559 


01 


81.958 


1.91359 


08 


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 


06 


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 


94.750 


1.97658 


07 


87.554 


1.94228 


07 


81.380 


1.91052 


Ofi 


94.620 


1.97598 


08 


87.444 


1.94173 


08 


81.285 


1.91001 


09 


94.491 


1.97539 


09 


87.333 


1.94118 


09 


81.190 


1.90950 


10 


94.362 


1.97480 


10 


87.223 


1.94063 1 


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 


1.90090 


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 


90 


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 


82 


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 


92.593 


1.96658 


24 


85.711 


1.93304 . 


24 


79.787 


1.90193 


25 


92.469 


1.96600 


25 


85.605 


1.93250 1 


i 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 


80 


91.855 


1.96310 


30 


85.079 


1.92982 


30 


79.240 


1.89894 


81 


91.733 


1.96253 


31 


84.975 


1.92929 


31 


79.149 


1.89845 


82 


91.612 


1.96195 


32 


84.871 


1.92876 
1.92823 ' 


32 


79.0.59 


1.8979.5 


- 83 


91.491 


1.96138 


33 


84.767 


33 


78.909 


1.89746 


84 


91.370 


1.96080 


34 


84.663 


1.92769 i 


34 


78.879 


1.89090 


85 


91.249 


1.96023 


35 


84.560 


1.92716 1 


1 35 


78.790 


1.89047 


36 


91.129 


1.95966 


36 


84.457 


1.92664 1 


36 


78.700 


1.89.597 


37 


91.009 


1.95908 


37 


84.354 


1.92611 1 


37 


78.611 


1.89548 


88 


90.890 


1.95852 


38 


84.251 


1.92558 1 


38 


78.522 


1.89499 


39 


90.770 


1.95794 


39 


84.149 


1.92505 


39 


78.433 


1.89450 


40 


90.652 


1.95738 


40 


84.047 


1.92452 

1 


40 


78.344 


1.89401 


41 


90.533 


1.95681 


41 


83.945 


1.92399 1 


41 


78.2.'>0 


1.89352 


42 


90.415 


1.95624 


42 


83.843 


1.92347 


42 


78.108 


1.89303 


43 


90.297 


1.95567 


43 


83.742 


1.92294 . 


43 


78.080 


1.892.54 


44 


90.179 


1.95511 


44 


8.3.640 


1.92241 


44 


77.992 


1.S9205 


45 


90.061 


1.95454 


45 


83., 540 


1.92189 ' 


45 


77.9(H 


1.891.50 


46 


89.944 


1.95397 


46 


83.439 


1.92137 1 


46 


77.817 


1.89107 


47 


89.828 


1.95341 


47 


83.339 


1.92085 ' 


47 


77.729 


1.890.5S 


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.. WO 


1.88902 


50 


89.479 


1.95172 


50 


83.039 


1.91928 , 


50 


77.469 


1.8S913 


51 


89.363 


1.95116 


51 


82.939 


1.91876 


51 


77.382 


1.8.S8(H 


52 


89.248 


1.95060 


52 


82.840 


1.91824 


52 


77.290 


1 88810 


53 


89.133 


1.950O4 . 


53 


82.741 


1.91772 


.W 


77.210 


1.88707 


54 


89.019 


1.94948 


54 


82.642 


1.91720 , 


54 


77.121 


1.88719 


55 


88.904 


1.94892 


55 


82.544 


1.91669 


55 


77.0.39 


1.88071 


56 


88.790 


1.94836 


56 


82.440 


1.91617 ' 


56 


70.9.53 


1.88023 


57 


88.676 


1.94781 


57 


82.348 


1 91565 ' 


57 


76.808 


1 88575 


58 


88.563 


1.94725 


oS 


S2.2r)() 


; 1 91514 


58 


76.78.3 


1.88,527 


59 


88.450 


1.94670 


1 59 


82.152 


1.91462 


59 


76.698 


1.88478 



304 



THE RAILROAD TAPER 







RADII METRIC {Con 


iinued) 






Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


Log. rad. 


Deg. 


Radius. 


L<%.Tad. 


15° 00' 


76.613 


1.88430 


19° 00' 


60.589 


1.78239 


23° 00' 


50.159 


1.70035 


05 


76.192 


1.88191 


05 


60.326 


1.78050 


15 


49.626 


1.69571 


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 


45 


48.596 


1.68660 


20 


74.957 


1.87481 


20 


59.554 


1.77491 


24° 00' 


48.097 


1.68212 


25 


74.654 


1.87247 


25 


59.300 


1.77306 


15 


47.609 


1.67769 


30 


74.156 


1.87015 


30 


69.050 


1.77122 


30 


47.130 


1.67330 


35 


73.762 


1.86783 


35 


58.801 


1.76938 


45 


46.662 


1.66896 


40 


73.372 


1.86553 


40 


58.564 


1.76756 


26° 00' 


46.202 


1.66466 


45 


72.986 


1.86324 


45 


58.309 


1.76574 


16 


45.752 


1.66041 


50 


72.604 


1.86096 


50 


58.067 


1.76393 


30 


45.311 


1.65620 


55 


72.227 


1.85870 


55 


57.826 


1.76212 


46 


44.878 


•1.65203 


16" 00' 


71.853 


1.85644 


20*00' 


57.588 


1.76033 


26° 00' 


44.454 


1.64791 


05 


71.483 


1.85420 


05 


57.351 


1.75854 


16 


44.038 


1.64383 


10 


71.117 


1.85197 


10 


57.117 


1.75677 


30 


43.630 


1.63979 


15 


70.755 


1.84976 


15 


56.884 


1.75499 


46 


43.230 


1.63579 


20 


70.396 


1.84755 


20 


56.6.53 


1.75322 


27° 00' 


42.837 


1.63182 


25 


70.041 


1.84535 


25 


56.425 


1.75147 


15 


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 


65.973 


1.74798 


45 


41.701 


1.62015 


40 


68.998 


1.83884 


40 


55.749 


1.74624 


28° 00' 


41.336 


1.61633 


45 


68.657 


1.83668 


45 


55.528 


1.74451 


15 


40.977 


1.61254 


50 


68.320 


1.83455 


50 


55.308 


1.74279 


30 


40.625 


1.60879 


55 


67.986 


1.83242 


55 


65.090 


1.74107 


46 


40.279 


1.60506 


17*00' 


67.655 


1.83030 


21*00' 


54.874 


1.73937 


29° 00' 


39.939 


1.60140 


05 


67.327 


1.82819 


05 


54.660 


1.73767 


15 


39.605 


1.69775 


10 


67.003 


1.82609 


10 


54.447 


1.73597 


30 


39.277 


1.59414 


15 


66.682 


1.82401 


15 


64.236 


1.73429 


45 


38.954 


1.59055 


20 


66.363 


1.82193 


20 


64.026 


1.73260 


30° 00' 


88.687 


1.58700 


25 


66.058 


1.81986 


25 


53.819 


1.73094 




• 




30 


65.736 


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 


1.72696 


■ 






45 


64.818 


1.81170 


45 


53.003 


1.72430 








50 


64.517 


1.80967 


50 


62.804 


1.72267 








55 


64.219 


1.80766 


55 


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 


63.056 


1.79973 


15 


51.827 


1.71456 








20 


62.772 


1.79777 


20 


51.636 


1.71295 








25 


62.490 


1.79581 


25 


61.446 


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 








50 


61 . 120 


1 78618 


50 


50.520 


1.70346 








55 


60.853 


1.78428 


55 


50.338 


1.70190 









TABLES 305 

NOTATION 

A = Curvature increment of taper per 10 meters of taper. 
M = Degree of main circular curve. 
{ = Length of taper from the B.C. of the taper. 
/ = Deflection from tangent at the B.C. of the taper. 
ft fu fif 6^' = 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 -\- df 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 Une 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 = 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 -\' D tan ^ /. (7) Both ends the same 

taper. 

t = t -{■ D tan i I — d esc /. (8) Tapered end — other 

end not tapered. 

n = iJtan i / -f (Zcsc /. (9) Untapered end — 

other end tapered. 

1 = <i -f A tan i I — {di — di) esc /. (10) Both ends tapered, 

with different ta- 
pers. 

J = (2 -f A tan J- 1 — {d'l — di) esc /. (11) Both ends tapered, 

with different ta- 
pers. 

? = D sec i / — -R. (12) Both ends the same 

taper. 



306 THE RAILROAD TAPER 

EMPIRICAL EQUATIONS DERIVED FROM THE 

METRIC TABLES 

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

{I + 5)(l + 10) .-.. 

/= 20 . ^^^^' 

A 

., _ a + 2g + i5)a-s) + 5o ._ . 

A 

o = ill^ . . (108, 

a" 

The equations under paragraph 18 as derived from the met 
taper tables become : 

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

h = ^{Mi-A). (46i 

h = —^ . (47i 

Zc = ^ {M2 - Ml - A). (48i 

15 (M2 + Ml) (M2 - Ml - A) ,_^ 

A 

15 Ml (M2 - M, - A) ,_, 

a= - — (51a 1 

A 

, (3Z.4-?c4- 15)^0 + 50 ,.. 

•^'^^ 20 ^^^' 

A 

, 5 r(4 M, + 2 M. + A) ( M2 - Ml - A) . . "1 ._. 

/c = 2 L A "^ ^J • (^^' 



TABLES 



307 



I 



o 

>^ 

> 

V 



O 

O 

)^ 
>-\ 
m 

< 

O 

"A 
>-\ 

H 
< 
> 

O 

d 

H 



< 





qS 




» o "Tt^ CO 00 1^ c^ »-H to »o 05 Op CO c^ r>* CO 

i-H C^ CO CO CO CO C^ O '* T-H "^ ,-1 CO Tt< lO . 
o lO »0 lO »0 »0 »0 to to TjH Tt< CO CO C^ i-H o 


• 


•is 

0!S 

dSo 


O 


^ C^ lO CO CO '«f l> lO OC l> O 00 »-H 05 C^ .CO 

CO "^ lo o iO -^ CO i-H lo CO »o c^ CO lo . u:> 

O Tt< Tt< Tt< Tt< ""^ ""^ Tj< TjH CO CO C^ (N T-l O "O 




OS 

OSo 




^COOO»Or>*CO>OC^'^i-iCOO(NOO .c^o 

iOO»-HrHrHO>OCO'— ITfirHCO'^ .»OlO 

CO '•^ '* "^ ''t ''f CO CO CO C^ (N T-H O "O^ 




ds 

dio 




-^CQCOOSQJOCOC^C^OOOOTtHiO ,OOCOC<I 
<N CO CO ^ CO (N i-H lO (N lO C^ 'sf .TjiTjirt? 

o CO CO CO CO CO CO CO C^ (N T-H .-H O * O »-H C^ 




dS 
dSo 


o 


^ CN|(N O lO Q 05 CO CO t^ t^ T-H .lOiOi-HC^ 

lO O O O O Tt< CO i-H rj< T-i Tj< .Tj<C0COC0 

C^ CO CO CO CO C<l C<I(N — 1 T-H O •O'-H(NC0 




d*c 


8 


* Tt< (N lo CO r>» lo 00 CO 05 1^ .i-ioQor>»oo 
CM CO CO CO <N T-H lo CO o CO . TjH c<i c^ i-H T-H 

C^ (M (N C<I(N (N i-H i-H »-H O •O»-hC^C0'* 




d§ 
d*c 


s 


*00iOt^Tt<COC0>Q(NC0 .t^OOOCMQCO 

O T-H CM CM CM rH T-H T-H T-l O ■ O »-H CM CO "^ lO 


s 

i 


ds 
d*c 


s 


* lO T-H CM t^ 00 CO Tt< O .COCO t^t^— 1 OiO 

CO Tt< TJH CO CM T-l »0 CO . CO i-H to TtH Tt< 'Tf TtH 
OT-H,-lT-HrHT-HrHOO "Oi-HrHCMCOTfiO 


1 

5 


d8 
d*c 


g 


H««|^ iH|« "hN . iH|M>-il-* HN «|-*Hc« 

* lOOSOOCOCMt^CO .O^COCMC^OOOO'* 

^^^^©"^CM . CO O -* CO C^ 1-H -H CM 

^^,-,^^00 • O i-H »-i CM CO Tt< lO CO 




o<^ 

dtn 


s 


* t^ Q 00 i-H 05 CM . CO 00 lO t^ CO LO CM Tf 1-H 
iOO»OiOCOCM .CM lOeO 1-H OiO^OiOO 

o Oi-HOOOO • O O 1-1 CM CO CO Tt< to t^ 




dk 
d^ 


§ 


-^i-HCOQCMOO .C<IOCOCM>OCOCOrhl^itC 
TjHTt*^CO.-i . C^ lO CM O "* CO CM C<l CM CO 

OOOOO •OO^CMCMCO'^iOCOt^ 




dS 
d*^ 




- OOOQrh":! .OOCO CMt^COOO'^COOOt^ 
CMC^Cv|»-< .T-H'rf»-iTj<CMr^OiOiOiOO 

OOOOO 'OO^'^CMCO'^'^iOCOQC 




dS 
dt. 

d^5 
d*o 


o 

CO 

8 


•> l> t^ .-H . lO ^ --I CM t^ 00 CO -^ O O COI> 
^^^ .r^COOeOOTj^cOCMCM C^C^ICO 

o OOO "OO 1-H rHCMCM CO"* iOCOt^OO 




^ Oil^ .i-iOOOI>'00>CI^'^COCOiOCMCO 
OO .T-iCM'Oi-iTpC^lOiOTtHTfrtoocO 

o OO ■ O O O -I 1-H CM CO CO ''f iO CO t^ 05 




dg 
d*o 


o 


*C0 .I^OXCMQ?0i-H'«tCM>0C0t^»O00 
O .OCMCOOCOO'^C<1^000^CM 

« O • O O O '^ i-H CM CM CO Tt< lO CO t^ 00 O 




d2 


o 


.COCOI-^I>''-hQiO'^OOOOCMC<ICO»0 
.O— •CM-t^'-H^r-^iOrOCMCM^lCMCOir: 

« •OOOOi-H-HC^lCMCOtiOCOt^aCCi 


•s 

I 


L 

its 


•«* 


OOpOQQQOOQQOOOQO 
i-l?5c0^u5cOl^00050'-'CMCO^iO 

T-H T-H r-H 1— 1 1— 1 1— 1 



308 



THE RAILROAD TAPER 



II 
< 



<3 



a> 00 »o Oi 05 c^ t^ 

Oi Oi Oi CO t^ ^ CO 
Od Od O) O) O Oi Oi 



OQUSQO^ 
OOt^CO: 



O O O O) O O Oi Oi Oi Oi OP O) O O^ O) 
t-hC^ CO CO Tt<U3«Ot>.00 0>0»-HO^CO'^ 



> 

T-l 

o 

O 

O 

< 

w 

H 
-si 
> 

O 






H 



O 00 rt4 CO C<l 00 
~ ~ ~ OOt^'^ 



05 
05 



o> o> d 



^ CO CO t^ ^ ^ 

00 t>. CO Tt< C<l OS 



T-ic^?ico'^»ocot>.ooa)Ot-HCSico^ 



CO 



^ 



b*cOC<lt*T-i 

O 05 1— I In. T— I 



T-iioS5c<i©ofi^c>iioo5^i^T-ic6cp 

OO»-HC0C00i»0C^»— iT-liOOOSO^ 

oooooo»-<c<coTti»dt>o6i-^co 



o^oot*cO'«fNo6^oo 

OiOiOiOiOiOiOiOOOOt^ 



osoococ^cot^ 

0S05 



100*005^05^05^05^05^0S^ 
T-lT-HrHC<IWC000"^Tt<»O»OCOCOt^ 




illi 



OOOOOOOOOi-Ht-ht-hC^cOCO 



g* 



OOOOOOOOCO^UJOi-* 

T— 11— ll— IC0»-HC0'«t<W5T-HT— I 



COoOt^C^f-i 

^uoococsi 



tH«0 



_ 00»OCO 
CO '-H 005 




^-^ss 



00l^t*COCOlOW5»O 

cococo<N<Nc<ic^c<icsic^c^csic^c^c5i 



CO »-H Tt< Q CO 05 »0 C^ 
Tt<C^00CO»OC0^0p 
00O5O500»-HT-HrJ<O5 



r>»coNioi 

CO l^ C^ CO ' 

OS lO CO OS' 



T-H b* »0 CO '^ lO CO 
05 C^ TjH 1-1 CO IQ l>» 

M lo »-H OS ^* CO w3 



OSl>. 
^1 ^1 



CO 



p^ 

pLi 



HC4'H|C4 H|C«HC4 HMiHlrn HmHn 

OC^'^'-iiOCOCOCOCOiO»-HTfC^OO 
o O O O »-i tH C<1 CO -^ lO CO 00 OS »-H 00 »o 



oo 



co^ioSt^SSoi-iSw^S 



CO'^O'-HCOTfO'-iCO'^Oi-HCO^O 

ooi-i»-Hi-4T-HCic^c<c^cococoeO'^ 



TABLES 309 

TAPER J METRIC — INTERMEDIATE VALUES d=i 



» 


i 


/ 


b 


' • 


D 


logD 




. 


' 


. 


tc 


D« 


30.000 


OOOJ 


13 


221 


1527 021 


J 18410 




0.032 


10 000 


0.066 


20.000 


20 000 




13.3J3 


out 


017} 


0201 


137S.141 


3,13835 


0,03(1 


11.667 


0.080 


33.333 


33,333 




29.MT 


Dlt) 




03&I 


1350.144 










26.888 


38,887 


100 


aO-000 


171 


211 




1I4E.0B4 


3.03B18 


IZ 


15.000 


0.153 


20. OOO 


30.000 


OB 


33.S33 


30} 


331 


641 


10.17-855 


1.02443 




18.867 


0,301 


33,333 


33,333 


10 




2*1 




lOti 




2.99226 










36.806 




M.OOO 






in 


016.860 


2.99330 


0.109 


10.999 


0,327 


30.908 


39.099 


30 




32i 


OHl 


136} 














43,332 


IS 




381 


1021 


1301 


800.082 










46,883 


48.8SS 




50.000 


0*U 


mi 


1621 


764.156 


2.8S318 


0^100 


24 008 


0600 


40.094 


49. 9M 


3fi 


IU.333 


tm 


1201 


30CI 


723.084 


3.SS973 


.,- 


28,664 


0.718 


53.325 


53,331 


10 




osU 


1301 


2 21} 




2.83740 










66.663 


Ab 




57 


MOl 


2 371 




2.81633 




39.096 






59.9*8 




M.333 


im 


Ifili 


3S4i 


63S.423 


3.70617 


0.3SI 


31.662 


1,154 


63.318 


63,336 






1081 




311} 


598.399 


3.77802 






1.332 




68.868 


200 


70.000 


US 


2 IS 




673.445 


2.75840 


0,458 


34,992 


1.527 


"•" 


69.989 


05 


73 333 


121! 


2 371 


3 401 


650.683 


2.74083 


518 


36,857 


1.740 




73.310 




76 H7 


1281 


2 401 


4 091 


529.499 


3.72387 


0.583 


38,331 












13S! 


2 641 








0864 


39,088 




70.048 


79,979 








3 081 


4 511 














83,307 


26 


M.M7 


IJOl 


3 231 


6141 


475.020 


2.67671 


0.813 


43,314 


3,791 




88 836 


30 


TC.OOO 


1S81 


3 381 


S37i 


459.303 


2.66310 


0,900 


44,977 


3,107 


S9.009 


80.962 




93.333 


207 


3 Ml 


6 011 


444.812 


2,64798 


0.9M 


48,630 


3,446 


03.324 


03.388 


40 




2151 


4 Hi 


6 281 






1,094 


48 303 






08.814 


IS 


100. OOO 


2 Ml 


428 


5S21 


417. B37 


2,63111 




49.064 








50 




3 33J 


4 45) 


7 191 


405,705 






61026 




103.157 


103.259 




105.667 


1421 




7 48] 






1.433 


63.286 








sw) 


110 000 


3 521 


a 221 


8 IS 


383.576 


2.SS385 


1.560 


54,046 


6,512 


109,783 


100.900 


M 


113 333 


3 021 


S411 


S441 


373 38T 


3,57216 


1.894 


56 805 


6,005 




113 217 




US 88? 


3 121 


BOH 


141 


363 750 


2.560S0 


1.836 


68,363 


8,826 


116.351 














3S4.B32 




1.985 


60,930 






110.848 


20 


113.333 


334 


5 431 


1018) 


34S.9SS 


3.539fH 




81578 






123.160 




128.897 


34fi 


7 041 


10 491 


337.747 


2.53859 


3l307 


68.331 


8,369 


126.199 




30 


130 000 


3 5«1 


7 28J 


11221 


329.935 


2.S1843 


3.480 


64,885 


8.911 


120,470 


120.778 


j5 


133 333 


4 071 


7 40 


1158} 


332.503 


1 


■ IM 734 


133.078 


M 


1)B6S7 


4 101 




12 31} 


315.427 




■•■ n-,.1>93 


136.380 




IW.OOO 


132 


S351 


13 07J 


siB.ees 




■ ■■ 1 ■'! J«! 






1(3.000 


1441 


900 


13 441 








142.973 




i« m 


*67 


9 241 


14 31} 


S96.107 






148.386 


4 m 


flooon 


tio 


BM 


16 00 


300 24(1 


2.46278 3. 7il3 74. ;S1 


13,4I|7|I4S.M2 


140.550 



THE RAILBOAD TAPEB 



s 




1 


- SSSgi'i3 = R5ssf2S : 




s 


■SSSlssSSSSSSS :S 


OS 


a 


;SS2S = ||SSKSS :R| 


0;= 


s 


■ «83 = S|s|||a :S58 


us 

"1= 


S 


■S33s|gs223 ;ass| 


qS 


s 


; sSslllgge issfis 


t 


3 


-SSSSSSSS :6S1SS8SS 


u3 


£ 




d?. 


S 


■Sggfss iSssHsssS 


dfe 
d^ 


s 


■ KSS3R isJsSSSiSiS 


03 


s 


■ssis :S;si'assa8l'«4' 


qs 


3 


■ SSI :S2S32SfeIS3S 
.000 :o-««»a^«33. 


ds 

u°- 


g 


: Is If !gf l;Hf ?5! 


qS 


3 




OS 


_, 


■ iSsssassSsSsfsa 


1 


.. 




=2SgSSSgSSS2|gg 



TABLES 



311 



II 
<J 






o 

T-H 

> 



O 

o 
O 

-Si 

w 

O 

H 
< 

O 



O 

I— I 



w 

PL, 
-«1 



^ 



§QOr>*C^OO<OOOQT-H^iOQOOO 

000)0^0030^00)0)0)0^0300 
»-HC5?lCO"**<»OCOt^OOOOi-HC<W 



oooot^«coo»ooo 



8 



o 

o 



OOOOOOOOOOOQOh-O 
»-iC^0lC0'^»OC0t^000)O»-iC^C0 



2300*0 

O»-HC0CD 




OOOOrH 



rHCO'*COOOO'<1*t*lH 

1—1 T-H T-H C^ 



t* c<i :* o «p »-i 
ooobO"^T-H 
o o o o o o 






»00'rt<0'^0)"^0"^0'«t'0"^0 



T-HCOQOt^QOiOl^QOl^»OrHt:2T-J 
T-H"^O»-i00O'-iO0)O'-HOTt<Q0 
OOi-iC^WOOCOt^COT-HOOO 

ooodooc>i-HT-ic^cocoTjJ<d 



OOCQiOT-HCOt^t^ 

i-HT-HC^Tf<»00«5C^ 

Q)C5C)0i-iCSi»O00Q5 
fO to CO 00 1— I *o 
O b* «0 «0 lO '* 



-^^t^gggjasjs 



00rHC^»O00g0COCO 

iocooo»-<»oooc^ 



coco C^ C^ CSI ^ 

coc^c^wwc^c^c^c^c^c^c^c^cci 






^ ^ loocp 



o 



- -O 

oco 



CNp'* iOOQQ»-iO 

ioc5ioCNioo»o»oi> 

-^OOOOi-HCOl^O 



T-H r>- lo 



iO«OQQ»OiOQQiO«OQQiO»0 
»-i'^COCO"n<i-HOO'-''^COCO'<t<T-H 

00»-<C^CO«Ot^O»-<COCOOC9SP 

T-H T-H T-H T-H C^ W 



T-i8w^SSt^SooT-i8w 



o 



O^O WO wo^o^o wo^ 
o i-iT-HC^c<icocO'^'«j<»o»oocoi>r>* 



312 THE RAILROAD TAPER 

TAPER 1 METRIC. — TNTERJMEDIATS VALUES 4=i 



M 


i / 


6 


\-D log 




( 


I , 


fc 


130 


" ■ 


26 


T^ 


1 


10 003 


109 20 001 


20000 


36 


!iM7]on 




t) 






1867 




23.838 Om 


31 




Mi 




50 


25.000iO2fli 


39 


I'l 




I81{ 4 999 26 000 
225' 26 065 26 6a« 




28.333 32 












30.000 




065 






03 "9998 


29.888 


o; 


31.Mfi 
38.333 
35.000 


lf,\ 


if 






HM 


is 








19 








30 U5 


1! 


33^333 












3S33I 


a 


40.uai: 


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lO ^5^ ^ toScoco^co 'S? 



O^coi* 



•«* 

T 

w 

> 
w 

< 
Q 






o 



CO 



05 



W5O5C0 



a> oi i>- CO 00 -^ th th ^ 1-H c^ 
*o ^ ^ »— • (55 ^ 1— • " 



SOOOt-h 



I CD 
I CO 



CO CO -^ -^ "^ »0 »0 lO »0 CO t* 00 00 Oa O tH d CO -^ 



COt^OOOJiH 
1-^ »— I tH iHN 



wc3C^'-i05coco®'«*i 
SOQt^co:a!S5C*5QO 
oooooocoobcooocoiN. 



tHtJ^COCO 
CO C^ 00 CO 
t^l^COCO 






CO CO ' 






8SS8Sc5c5SiSi nSS5c5SSco?ocSS ScoeoSocS 



00'^" 
?1C0 



8 00 05 CO 
ops 

cot* In. 00 



SOCS|CO»HOQCDS5t^« 
COiOlN.OC^wSoOi-i'^ 



iioSi 



C^ C^ C^ C^ CO CO CO CO '^ "^ ^lOiOiOCD 



1-tCOt^CO 

N^coob 

lO^CO^ 
O 05 Oi 05 



tH 1-H C3 ^ O 
C^ lOQd »-H -^ 



05 



00 




^ CO M i~< 

oboooooo 




fcf: 



>c« 



t^t* 




r^ Tfi io o »-• 

•^ iO »-H 1-t 40 < 

C^l fh C^l "^ t^ 



i§5^ 



^SP" 



ot^ CO:* CO 53 
00 00 00 ob 00 00 



gS JSSffgSfeSSSS 



CoSioSaO 



f-)|e« 



f-)|e« 



OiTt^i-HOOCPTJH'^'^lO 
C0'*'«!flOCOCOt*0000 



cocposic^co^®© 

C0OC0i^»0C001OO 

ST-(C0^COt*O5i-lC0»O 
cqc^c^c^wficococo 



SStoSS 

CO W) ^ ^ ^ 



He«Hc«H<«H|-« 



eoNieoNiHn 



C0C0O"^05'^O»0C^ 

co^oc^-^oco^ocsio 

OOOOOOOOO^T-H 



Hi-* WHiHtNH-* 

c^co'«j<wDcot^oooi-He3 






^ iOOOOCOt^O"^XC^ 
* O ^ CO TP iO r-i C^l CO »0 

g ioio»oio»ococococo 



T-(CSC0i60QC0Cib"^^O 

csiioc^u5c<iocorHioeo 

t*t*00000500rHi-HC^ 



COCO'^^tOiO 



^SS^^Tji'^^rjiS iOio»oSScoSSSt>. f»r*SSS 



5» 



'•j^ »o cor>.oooiO rSM 



e CO 



TABLES 



isai 



TAPER 3} METRIC. CURVATURE INCREASING 3* 00' 

EVERY 10 M. A =3 



TraQut 
at 




10 
20 
30 
40 
50 
60 
70 



B.C. 

3'00' 



045 
2 37J 
530 
9 22} 
14 15 
20 07 J 
27 00 



Deflections to 



C.C. 

6' 00' 



10 



045 

130' 

4 07i 

745 

12 22} 

18 00 

24 37J 



C.C. 


C.C. 


C.C. 


roo' 


12" 00' 


16' 00' 


20 


30 


40 


e t 


o / 


o / 


152} 


330 


5 37} 


130 


3 22} 


5 45 




2 15 


4 52} 


2 15 


• • • • • 


300 


5 37} 


300 


• « • • • 


10 00 


7 07} 


345 


15 22} 


12 15 


8 37} 


2145 


18 22} 


14 30 



C.C. 

18-00' 



50 



8 15 
8 37} 
800 
6 22} 

3 45 I 

4 30 I 
10 07} 



C.C. 

21° 00' 



60 



1122} 
12 00 
1137} 
10 15 
7 52} 
430 

5 15 



C.C. 

24° 00' 



70 



15 00 
15 52} 
15 45 
14 37} 
12 30 
9 22} 
5 15 



TAPER 3i METRIC. CURVATURE INCREASING 3*^00' 



EVERY 10 M. — {Continued) 



A=3 



M 


/ 





D 


m r 


• / 




600 


10 


130 


191 . 138 


900 


20 


4 30 


127.716 


12 00 


30 


900 


96.320 


15 00 


40 


15 00 


77.912 


18 00 


50 


22 30 


66.186 


2100 


60 


3130 


58.471 


24 00 


70 


42 00 


53.451 



log D 



2 

2. 

1. 

1. 

1. 

1. 

1. 



28135 
10625 
98372,0, 
891601. 
82077 2. 
76694 3 
72796 5 



065 
261 
652 
299 
262 
597 



4 
9 
14 
19 
24 
29 



,998 
,986 
951 
868 
703 
405 



9 
19 



0.131 
0.654 
1.830|29 
3.910 39 
7.129149 
11.684'58 



354 33.906 17.709,66 



999 
986 
916 
697 
166 
076 
090 



Ic 



10.000 
19.997 
29.972 
39.889 
49.680 
59.240 
68.421 



338 



THE RAILROAD TAPER 



CO 

II 
< 



t>. 05 1— t CO »0 t>. 00 
00 O Tt< CO 1-H 05 



1-H tH 05 1^ »o 03 00 
O CO »0 00 1^ Tl? 50 
00 CO tP W i-H o> t^ 



sScis^sss; 



iO«Ot«00OiO5 



fH C^ CO 3 Tt< »0 
CO CO CO CO CO CO 



coco 

00 ^ "^ l>- O - - 
05 00 CO "«^ CO 1-t 



COCOiQ 



t^CO 



WCO 

So 



CO CO 00 

t^»oco 



00 '^O 

oS8S 



O O I-* Cq CO T}^ tH 

fh cq cs c^ csi w c^ 



w5coi>.oog50 



o »-• c^ CO :* -^ WD 

CO CO CO CO CO CO CO 



OQ 



> 

Q 






o 



CO 
W 



Tt< Tf 00 l>. 1-t Q "^ 

' cq OS r^ CO »o -^ 

t^t^OOOSOi-i 






OOOOOi-tTH tH fH tH 1— I 1-t 1-H tH C^ C^ W C^ C^ C^ 



CO ^ »0 Q "^ 00 C<l COQCOCOOfH CO '^ CO tN. t^ t^ t» 

00 Q 1-1 CO "^ up t^ OQOi-tWCOiO CO t^ 00 o> Q i-< 53 

O^OOCSJCOO-^ 00 CO t^ 1-1 »0 05 CO t^ i-< »o O Tt< 00 

oJoo^^c^cq c<icoco'«j<"^'^ »o»ccdcdt>^t>^t>^ 



fH UD 1-H 
COOQfH 

wesico 



C0 05 

CO CO 



COOSCOt^FHlO Oi^OOCOOO 
^rJiuSiOCOCO COt^lN.0000 



ooooooo oooooo ooooooo 






05 O 




(N Cq M C^ C^ C^ Cq 



lO O CO CO 00 C<l 
§§fhQ^S 

w cvi c^ c^ 1-H 1-J 




cocgp^o^w^ 



FHOt^ _ - 

l>- CO O O ^ CO l>- 



s 



1— t CO »ft 00 
C^00»O^C0C0 



00 lO 00 CO 

l-IOS^lN. 

-^tOObi-H 



^odcSoooooo w 



o Tt< "f UDio »0 CO CO CO l>- l>- 00 00 Oi Oi OS O O i-< »-< w 



rO 



o (M C^ CO CO CO CO CO '«1< "^ rj< T}^ lO iC »0 CO CO CO «o r* t^ 



C^ OS O Tfi 1-H Oi l>- iO-^OJ^OQ 



CO ^ >oo^H 85 ^ 



o ^ fH C^ Cq C^ C^ C^l (NC^COCOCOCO CO CO CO Tj« o^i* -^ Tj« 



CO t^ O CO t^ 
COCOOCO CO 
00 CO»OCO »-i 



cot^c 
cococ 
O0CO»c 



cot* 
coco 
CO fH 



COIiQCOti* 



s 



O i-H C<1 CO T}^ lO 

cq w cq csi c<i cs 



»OCOt*OQOSO 
C<1 W C<1 C<l fl CO 



Q;H^cO;3J»OiO 
CO CO COCO CO OO CO 



^ 



- QU5Q"5Q"50 OQ^QupQ ^Q^Q*00»0 
O T-( CO "f O 1-1 CO "^O'-HCO'n'O 1-1 CO ^ O »-• CO ^ 



o CS 



c^ 



eo 



TABLES 



339 



^ Oi cj* t^ o> 



88g 



T-( OOrf O 



CO Qi t>» t^ 05 



1-H tH Oi '^ i-H 
CO 



^8859 



CO CO CO CO 



tH CO ^ CO0QO5 
^^i ^J^ ^^ ^T^ ^^^ ^'7^ 



•-H C<l "^ O t* 05 

lO o »o »c »o »o 



ssss 



MCO 



Gi Gi 
00 CO 



CO 05 

CO 00 



OOCdCO 
lO 1— I CO 
OOfH 



00C0Ot*'«t*O 

05 '-H r-t X -^ t* 

coc^t^^ coo 



COOCO 
CO(M'<t' 



CO t^ 00 00 05 
CO CO CO CO CO 



T-(^'^cot*a> 

^Ji ^^^ ^T^ ^'7^ ^J^ ^^ 



oc^co »ocooo o 

lO «0 »0 ^ ^ W5 ilO 



C3 92^ 

CO CO CO 



O C5 C» 05 vH 
rHflrJ^COO 



C^ CO CO CO 



CO coco 05 



SCO 



005CSI 



COTf CO-^QOCO 

53 (» 00 C3 Q CO ^ 

OOCO iO»OiQiOt>- 



COCOCOCOCO "^ "^ lO »0 CO t>- l>-OOOSOO'-i C^CO-fiCt* 



tO»fiC0i-HG0 

eQ"**w5cpco 



rHC^ 



$^ 



S4O.C0 
OQ 



t>.l>.'^?OCOO 1— ii— I 



Tj^C^THOSt* -^(NOX 



QS OOt;? ^ COQ 



CO 
t>» CO 00 CO 
1-H 05 COrf 



0000O5O5O5 Q^CSCOCOt*< 

^,-1^^,-, ^4 04 04 csi csi c<i 



lOCOt^t-OOOi 



OOi-hCSCO 
co co co co co 



»0 1-H t>* CO 9^ 
C5 1-H tH CSJ ?! 



CO 04 00 



1^04 
l^CO 

O04 



»OC0 000^05 
"^ CO 00 »— I CO lO 



^s 



05CO O5Q0 "^ 

lO CO »- • T-( »o 

00 '-H ^ l>» CO 



0404 O4O4O4C0COCO CO Tt< tJI Tt< lO 



I? 



eoQ04co ^ 
O^COOCD 
004^00 fH 

1-H »-H O piCft 

0> O) ^ 



^OSOOi 

oo»o 



cot* 



0000 0000 



»^«^sss? 



coo Oi »-i op QJ 

o I— • I— • CO "^ 

»-HQO00l>- _ 

00 00 1^ t* i>. i>. 



iC CO 00 C5 
•^ CO coco . 

OiO^ioast:^ 

10»0"^ C0 04 

t^l>.t*t*t^ 



SC0 1-HOQ04 
CO tH CO *—< 
1-1 In. "^04 05 



04»O00t^C0CO 
OiCO C^ ^ ^ 00 
»0 rf rf lO 00 i-H 



s 



04OO 

_ lOCO 05 

CO 04 05 CO 



T^ tH 1— I l>. t>. »-H 

b- t^"«t t*t*»0 

T}^ Tt^iCCOOO'^ 



COi-HQOit^ 

ao 00 00 i 



lt>.l>- 



lo CO '-H 05 1>- CO 

l>t^t^COCOCO 



SCO i-hQOSOO 
CO CO CO US ^ 



t>. CO lO-f CO 

lO »o »o »o »o 



^ Qi-lCOCO 
* S504U304 

o 04C0C0'^*O 



ml-* eol-* 



ooo 
oco 

CO t* 00 05 i-H 04 
rH rH »— I 1—t 04 04 



«!■* «!■• eoW 

copoopcoo 

CO lO CO 00 05 r-i 
04 04 04 04 04 CO 



«H 



nl-* 



oSc6o4 

CO^COOOOI 
CO CO CO CO ^ 



ihNi«N>H««N"h|n 

• tN.ooooa»05 



t^CO^ 004 »0 

oo-^coo^^ 

OOi-H 04C0t}^ 



iHfe* esl'* esl'* H'* "-ilw Hw 

S»0 CO CO Th t* 
ooooo 

»0 CO t* 00 05 o 

T-( ^ 1-1 ,-1 ,-i(M 



04 00 coo 

i-H i-H C4 CO 



i-H 04C0 Tf t^ 

O^ 01 O^ 04 04 



eoNieoNiHHinHiHe* 

* loS^Sco 



c9Hi«MH|e«H'*«W 
rH CO 04 05C0 40 
O 04 O 1— I '^ 1— I 



i-i|'*i-il'* cj|'*i-i|'*Hn 
Tt^ tJ<IlOCOC5 04 
Tti t-H rtl T-( Tt< 04 



COi-H>. "^Q 
iCCO O '^O 



COCOCOt^t^OO 0005000^ i-i(MC0C0"5 



SQCOCO 
u5 cO T-( 



g 



t^COQt^COQ t>.cOQt>.co 
COCOQCOCOQ CO CO © CO CO 

cocoococoo COCOOCOCO 



t^co 
coco 
coco 



CO CO CO CO ^ 



1-HCO ic CO ooo 

T}^ Tf rj< Tf "^ lO 



,-1 CO O CO 00 
lO »f3 »0 O »0 



»-i CO »C CO o 
CO CO CO CO t> 



«83g^8 

W3 



g8* 



CO 



o 

CO 



o 

CO 



o 

CO 



o 

CO 



o 

CO 



no CO t>- 00 05 p *-H 

vH tH rH 1-1 1-1 ^ 04 



§88 



340 



THE RAILROAD TAPER 



TAPER 3J METRIC. CURVATURE INCREASING 

E\ ERY 10 M. 



TraiMit 
at 



Deflections to 



B.C. 

4*00' 



c.c. 



c.c. 

12* 00' 







10 



20 



C.C. 

woo* 



90 



C.C. 
20*00' 



40 



C.C. 

24' CO* 



50 





• / 


• / 


• ' 


• . 


o t 


• r 





■ 


100 


230 


440 


730 


1100 


10 


100 




200 


430 


740 


1130 


20 


330 ! 


200 




300 


630 


;io4o 


30 


720 ' 


530 


300 




400 


830 


40 


12 30 


10 20 


730 


400 




: 500 


50 


19 00 


16 30 


13 20 


930 


500 




60 


2650 : 


24 00 


20 30 


16 20 


1130 


600 

1 



TAPER 3J METRIC. CURVATURE INCREASING i 

EVERY 10 M. — (Continued) 



M 


/ 





D . 


log D 


i 


t 


z 


1 
f 


e / 

800 


10 


1 

200 


143.443 


2.15668 


0.087 


4.996 


0.175 


9. «7t7«7 


1200120 600 96.0161 1.98234 


0.348 9.974 0.872 


19.974 


16 00 1 30 12 00 


72.719 


1.86165 


0.86614.912 


2.437 


29.851 


2000 40 2000 


59.309 


1.77312 


1.72119.767 


5.195 


39.463 


24 00 50130 00 51.083 


1.70828 


2.98624.477 


9.432 


48.526, 


28 00 


60 


42 00 


46.067 


1.66339 


4.73128.957 


15.347 


56.616 



TABUiS 



341 



TAPER 3} METRIC. INTERMEDIATE VALUES 


A-«4 


M 


I 


/ 


b 


o 


D 


log D 


i 


t 


z 


y 


u 


• / 




/ 


/ 


f 
















12 00 


20.000 


230 


330 


600 


96.016 


1.98234 


0.348 


9.974 


0.872 


19.974 


19.993 


15 


20.625 


2 37 


342 


619 


94.095 


1.97357 


0.372 


10.284 


0.941 


20.596 


20.617 


30 


21.250 


244 


3 53} 


638} 


92.252 


1.96498 


0.397 


10.594 


1.013 


21.217 


21.241 


45 


21.875 


2 5U 


4 07 


6 58} 


90.484 


1.95657 


0.423 


10.904 


1.089 


21.838 


21.865 


13 00 


22.500 


2 58} 


420 


7 18} 


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 


11.523 


1.252 


23.077 


23.111 


30 


23.750 


314 


4 47 


8 01 


85.587 


1.93241 


0.506 


11.833 


1.339 


23.696 


23.734 


45 


24.375 


322 


500} 


8 22} 


84.078 


1.92468 


0.538 


12.142 


1.430 


24.315 


24.357 


14 00 


25.000 


330 


5 15 


8 45 


82.625 


1.91711 


0.570 


12.450 


1.525 


24.933 


24.979 


15 


25.625 


3 38} 


529} 


9 07} 


81.225 


1.90969 


0.602 


12.759 


1.624 


25.550 


25.602 


80 


26.250 


346} 


5 44} 


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 


13.375 


1.834 


26.783 


26.846 


15 00 


27.500 


4 031 


615 


1018} 


77.321 


1.88830 


0.708 


13.683 


1.945 


27.398 


27.467 


30 


28.750 


4 21} 


6 47 


1108} 


74.940 


1.87471 


0.784 


14 298 


2.182 


28.626 


28.710 


16 00 


30.000 


440 


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 


0.953 


15.524 


2.710 


31.072 


31.190 


17 00 


32.500 


518} 


830 


13 48} 


68.700 


1.83696 


1.045 


16.135 


3.002 


32.288 


32.427 


30 


33.750 


5 39 


• 07 


14 46 


66.879 


1.82529 


1.143 


16.745 


3.315 


33.499 


33.662 


18 00 


35.000 


600 


9 45 


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} 


62.060 


1.79281 


1.471 


18.563 


4.377 


37.097 


37.355 


30 


38.750 


706} 


1147 


18 53} 


60.643 


1.78278 


1.593 


19.166 


4.775 


38.284 


38.581 


20 00 


40.000 


730 


12 30 


^00 


59.309 


1.77312 


1.721 


19.767 


5.195 


39.463 


39.804 


30 


41.250 


754 


13 14} 


2108} 


58.053 


1.76382 


1.855 


20.365 


5.639 


40.633 


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 


43.750 


844 


14 47 


23 31 


5S.755 


1.74628 


2.143 


21.554 


6.598 


42.945 


43.449 


22 00 


45.000 


9 10 


15 35 


24 45 


54.706 


1.73804 


2.298 


22.145 


7.114 


44.086 


44.656 


30 


46.250 


9 36} 


16 24} 


26 01 


53.717 


1.73011 


2.459 


22.733 


7.655 


45.215 


45.858 


23 00 


47.500 


10 03} 


17 15 


27 18} 


52.787 


1.72253 


2.628 


23.317 


8.222 


46.332 


47.056 


24 00 


50.000 


1100 


19 00 


30 00 


51.083 


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 


51.787 


26 00 


55.000 


13 00 


22 45 


35 45 


48.248 


1.68348 


3.794 


26.753 


12.173 


52.725 


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 


2S00 


60.000 


15 10 


26 50 


42 00 


46.067 


1.66339 


4.731 


28.957 


15 347 


56 616 


58.660 



342 



THE RAILROAD TAPER 



TAPER 3f METRIC. CURVATURE INCREASING 5° 00' 

EVERY 10 M. A=5 



Transit 
at 


Deflections to 


B.C. 

S'^OO' 


C.C. 

10° 00' 


C.C. 

15° 00' 


C.C. 

20° 00' 


C.C. 

26° 00' 


C.C. 

30° 00' 


/ 





10 


20 


30 


40 


50 



10 
20 
30 
40 
50 


o / 

i is " 

4 22i 

9 10 
15 371 
23 45 


e / 

1 15 

2 30 " 
6 521 

12 55 
20 371 


e / 

3 071 
230 

"345" 
9 221 
16 40 


/ 

550 
5 37} 
345 

'566" 
11521 


/ 

-9 221 
9 35 
8 071 
500 

615 ' 


/ 

13 45 

14 22§ 
13 20 
10 371 

6 15 



TAPER 3J METRIC. CURVATURE INCREASING 5*^ 00' 

EVERY 10 M. — (Continued) A=5 



M 


I 
10 





D 


log D 


i 


t 


z 


y 


ie 


1 

10 00 


/ 
2 30 


114.847 


2.00012 


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 



TABLES 



343 



to 

n 

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OONVERSION OF BNQLISH INCHES IHTO CB 


NTIMBrnSRi 


i. 


Ins 







1 


5 




s 


' 




30 
SO 

w 

TO 
100 


»6 * 
jl 


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83BB sa b; 


il 

1 fi« 



OF CENTI^ETilKb I\l ) EN( 1 



4.U1^ 4.T42| 5. lis 



ao.on 20.4TS aowr 



M.Oia M.llO, 24.8 



SS.59I l».96&,EB.3 



CONVElRaiON OF KNOLISH FEET INTO METEBS. 



Wtet. 





1 


a 


» 


4 


B 


6 


T 


B 


, 
































































































































































































































31.898 K. «MiB2. 80833. fliaaa.BlSlaS 22* 



CONVERSION OF METERS INTO EN0LI8H FEET. 



9 H.OBO a 

a 08.693 T 

7 101 TI 1 

1 134. sa 1 



)3 88!b84 01 



352 



THE RAILROAD TAPER 



TRIGONOMETRIC AND MISCELLANEOUS 

FORMULAS. 

SOLUTION OF RIGHT TRIANGLES. 



Required. 


Giren. 


A, C, c 


a, h 


A, 0,6 


a, c 


C, b, c 


A^ a 


C, a, c 


A,b 


C, a, 6 


Ay c 



Formuliis. 



sin^ = COS O = ^ J c = >^(6 4- a) (6 — a), 

tan^=cotB = -; 6 = N/aa4-c«. 

c 

C=^90°-— -4; c = acot^ ; &=:«coaec.4. 

C = 90° — -4; a = &8in^; c = 6co8inil. 

C = 90° — ^; a = ctan-4; 5 = csec.4.. 



SOLUTION OF OBLIQUE TRIANGLES. 



Required. 
b 

B 

i(A - ^) 

A 
B 



Area 
Area 

Area 



Given. 

A, B, a 
A, a, b 



r 



a, 6, C 



^ 



a, 6, c 



-! 



A, by C 

A, B, c 



Formulas. 



6 = 



sinB 



a sing 
sin .4 

b&inA 



a 



i(^ + B)=i(180-C) 

tani(^ - B) = ^Lz:^tani(i4 + B> 

A=i(A-^B)-\'^{A-B) 
B^H^-{-B)-i(A-B) 



If »=|(a+64-c), sini.4 
\ 6c 

_ / (3-6)(8-C) 

\ 8{8-a) 



-J<^ 



c) 



tan 4-^ 



sin 



in.i — ^^^(^ -«)(«- b)(8-e} 

be 

Area = '^s {s — a) (« — 6) (« — cj 
Area = i 6c sin A 

Area = .^i^iBA«!ILB 



TABLES 353 

TRIGONOMETRIC AND MISCELLANEOUS 

FORMULAS. 

GENERAL FORMULAS. 
BbiA = ^1 — coe^A = taja.A cob A. 

sin^ = — 1— . == v^Jd - cos2^). 
cosec^ 

C06A == = n/i — 8in2^ = cot^ sin^, 

sec^ 

C06^ = l — 28in2|^ = l — vers^. 

C08-4 == -/^ + icos2^ = co82^^ — sin2^^ 

tBOiA = ^^ = ^aec^A - L 
cos J. 



^j^^__2^1 — cos^^ 8in2^ 



cos^ 1 + CO&2A 

t»s,A L_=l-co82^^ 

cot A sin 2^ 

cot-4 = -J— = 5^ = >/co8ec«T^L 
tan J. sin^ 

cot^ = _«?Ilid_ = l±cos2^. 
1 — co82^ sin 2^ 

secil = = the reciprocal of any expression for ^osA 

cos A 

cosec A = = the reciprocal of any expression for sin J.. 

sm^ 

vera-4 = l — cos-4 =2sin2|^. 

ex8ec^ = sec^ - 1 =Xem^. 

cos^ 

vers -4 



'^*^=/T^=>n 



i