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Full text of "Surveying and tables"

UC-NRLF 




75D 









SURVEYING AND TABLES 



BY 

G. A. WENTWORTH 

AUTHOR OF A SERIES 6r TEXT-BOOKS IK MATHEMATICS 



SECOND REVISED EDITION 



GINN & COMPANY 

BOSTON NEW YORK . CHICAGO LONDON 






MATHEMATICAL TEXT-BOOKS 

BY 

GEORGE A. WENTWORTH 



Elementary Arithmetic 

Practical Arithmetic 

Mental Arithmetic 

Primary Arithmetic (Wentworth and Heed) 

Grammar School Arithmetic 

Advanced Arithmetic 

Exercises in Arithmetic (Wentworth and Hill) 

First Steps in Algebra 

School Algebra 

New School Algebra 

Higher Algebra 

Elements of Algebra 

Complete Algebra 

Shorter Course in Algebra 

College Algebra (Revised Edition) 

Exercises in Algebra (Wentworth and Hill) 

First Steps in Geometry (Wentworth and Hill) 

Plane and Solid Geometry (Revised) 

Plane Geometry (Revised) 

Solid Geometry (Revised) 

Plane and Solid Geometry and Plane Trigonometry 

(Second Revised Edition) 
Analytic Geometry 
Logarithms and Metric Measures 
Geometrical Exercises 
Syllabus of Geometry 

Examination Manual in Geometry (Wentworth and Hill) 
Exercise Manual in Geometry (Wentworth and Hill) 
Plane Trigonometry (Second Revised Edition) 
Plane Trigonometry and Tables (Second Revised 

Edition) 
Plane and Spherical Trigonometry (Second Revised 

Edition) 
Plane and Spherical Trigonometry, and Tables 

(Second Revised Edition) 
Plane Trigonometry and Surveying, and Tables 

(Second Revised Edition) 

Surveying and Tables (Second Revised Edition) 
Plane and Spherical Trigonometry and Surveying, 

and Tables (Second Revised Edition) 
Plane and Spherical Trigonometry, Surveying, and 

Navigation (Secpnd Revised Edition) 
Logarithmic and Trigonometric Tables 

Seven Tables (Wentworth and Hill) 

Complete 



COPYRIGHT, 1882, 1895, 1896, 1903, BY 
G. A. WENTWORTH 



ALL BIGHTS RESERVED 



PBEFACE 

THE object of this work on Surveying is to present the subject in 
a clear and intelligible way, according to the best methods in actual 
use, and in so small a compass that students in general will find 
time to acquire a competent knowledge of this important study. 

The author is under obligation to G. A. Hill, A.M., of Cambridge, 
Mass. ; to Professor James L. Patterson, of Chestnut Hill, Pa. ; to 
Dr. F. N. Cole, of New York, N.Y. ; to Professor S. F. Norris, of 
Baltimore, Md. ; and to Professor B. F. Yanney, of Alliance, Ohio. 
Professor Yanney has done most of the work on the second revision, 
and Miss M. Gertrude Cross, of Boston, has furnished the drawings. 

G. A. WEOTWORTH. 
EXETER, N.H., 1903. 



iii 



CONTENTS 

SURVEYING 

[The numbers refer to the pages.] 

CHAPTER I. FIELD INSTRUMENTS: 

Definitions, 1 ; classification, 1 ; operations comprised, 2 ; the sur- 
veyor's chain, 3; the engineer's chain, 3; accompanying pieces, 4; 
how to chain, 4 ; special constructions by means of the chain, 5 ; 
obstacles to chaining, 7 ; the tape, 9 ; the compass, 10 ; kinds of 
compasses, 11 ; bearing of a line, 12 ; checking bearings, 13; obsta- 
cles, 14 ; measurement of horizontal angles, 14 ; measurement of 
vertical angles, 15; verniers, 15; uses of the compass vernier, 17; 
magnetic declination, 19; surveyor's transit, 23; uses, 24; measure- 
ment of horizontal angles, 26 ; measurement of vertical angles, 26 ; 
stadia measurements, 26 ; the solar compass, 28 ; to establish a true 
meridian, 32 ; the Y level, 36 ; the leveling rod, 36 ; substitutes for 
the Y level, 39 ; the plane table, 40 ; to orient the table, 42 ; to plot 
any point, 43 ; to plot a field, 43 ; the three-point problem, 44. 

CHAPTER II. OFFICE INSTRUMENTS : 

Definitions, 46 ; the diagonal scale, 46 ; the circular protractor, 47 ; 
constructions, 48 ; the planimeter, 49 ; the slide rule, 49. 

CHAPTER III. LAND SURVEYING: 

Definitions, 50 ; special methods of surveying, and of computing 
areas, 51 ; rectangular system of co-ordinates, 52 ; general method 
for farm surveys, 57 ; field notes, 58 ; computation of the area, 58 ; 
balancing the work, 60 ; supplying omissions, 61 ; to make a plot, 
63 ; modification of the latitude and departure method, 66 ; location 
surveys, 67 ; illustrative problems, 67 ; laying out the public lands, 
71 ; reference lines, 71 ; townships, 71 ; subdivision of townships, 
73 ; meander lines, 73. 



vi CONTENTS 

CHAPTER IV. TRIANGULATION: 

Definitions, 74 ; classification, 75 ; measurement of base lines, 75 ; 
measurement of angles, 76. 

CHAPTER V. LEVELING: 

Definitions, 77 ; corrections for curvature and refraction, 77 ; dif- 
ferential leveling, 78 ; single setting of the level, 78 ; several settings 
of the level, 79 ; profile leveling, 80 ; field work, 81 ; making the 
profile, 84 ; topographic leveling, 85 ; drainage surveying, 86 ; field 
work, 86 ; plot and profile, 86. 

CHAPTER VI. RAILROAD SURVEYING : 

Laying out the route, 89 ; establishing the roadbed, 89 ; excava- 
tions, 89 ; embankments, 90 ; curves, 91 ; methods of laying out the 
curve, 92. 

CHAPTER VII. CITY SURVEYING: 

Field-work instruments, 94 ; streets, 94 ; blocks and lots, 96 ; plots, 
96 ; records, 96. 



SURVEYING 

CHAPTER I 

FIELD INSTRUMENTS 

SECTION I 

DEFINITIONS 

Definition. Surveying is the art of determining and repre- 
senting distances, areas, and the relative position of points on 
the surface of the earth. 

Classification. Of surveying there are various kinds, depend- 
ing upon the extent, the purpose, or the method of the survey. 
The following are the principal divisions : 

1. Plane Surveying, in which the part of the earth's surface 
surveyed is regarded as a plane ; Geodetic Surveying, in which 
the true figure of the earth is regarded. 

2. Land Surveying, in which boundary lines, contents, and 
outline maps are the chief things aimed at ; Topographic Sur- 
veying, in which differences in elevation and contour maps are 
chiefly sought ; Hydrographic Surveying, in which the purpose 
is to determine the configuration and topography of the bed or 
basin of a body of water ; Mine Surveying, in which the posi- 
tion and extent of underground excavations are determined and 
graphically represented. 

3. Rectangular Surveying, in which a system of perpen- 
dicular lines is used as reference lines ; Triangular Surveying, 
which proceeds by means of a system of triangles referred to 
a well established base line. 

1 



2 SURVEYING 

Operations Comprised. Surveying commonly comprises the 
following three distinct operations : 

1. The Field Measurements, or the determining certain lines 
and angles by direct measurement. 

2. The Computation of the required parts from the meas- 
ured lines and angles. 

3. The Plotting, or representing on paper the measured and 
the computed parts in relative extent and position. 

Historic Note. Surveying is undoubtedly one of the oldest of the arts 
of civilized man. The Bible contains several admonitions not to 
remove " the ancient landmark," as in Proverbs xxii. 28. To the Baby- 
lonians is credited the division of the circle into 360 degrees. The Egyp- 
tians were known to survey frequently the valley of the Nile, a necessity 
owing to the periodic overflow of that river. Thence came Geometry. The 
Egyptians also possessed rules for finding the area of land of various shapes. 
Moreover, on Egyptian soil the Greek mathematician Eratosthenes made 
the first attempt at determining the circumference of the earth by meas- 
uring an arc of the circumference. This was in 276 B.C. Among the 
Romans Surveying was considered one of the liberal arts, and received 
impetus in the time of Julius Caesar from his sweeping order that the 
entire empire should be surveyed for the purpose of equitable adjustment 
of taxes, and also from the introduction of the more practical parts of 
Greek Geometry. The works of Roman surveyors served as models for 
centuries, and much that we have to-day is only improvements on what has 
been handed down from them. For a brief account of surveying in the 
United States, see Cajori's "The Teaching and History of Mathematics 
in the United States," pp. 92, 286. 



FIELD INSTRUMENTS 3 

SECTION II 
THE CHAIN 

Surveyor's Chain. The Surveyor's Chain, or Gunter's Chain 
as it is often called, is made of iron or steel wire and is 4 
rods or 66 feet long, composed of 100 links connected by small 
rings, and provided with a tally mark at the end of every 10 
links. A link as a unit of measure includes a link of the chain 
and half the rings that connect it with adjoining links. Each 
link is 7.92 inches long. Since a chain is 4 rods long, a square 
chain contains 16 square rods, and since an acre contains 160 
square rods, a square chain is one-tenth of an acre. A square 
chain contains also 10,000 square links and, therefore, an acre 
contains 100,000 square links. Hence, if a given area is 
expressed in square chains, it is reduced to acres by pointing 
off the last figure, and, if expressed in square links, it is reduced 
to acres by pointing off the last five figures. The tally marks 
are appropriately notched to facilitate counting links from 
either end, the one at the middle of the chain being rounded 
so as to be distinguished readily from the others. Handles 
form part of the end links, to which they are so attached as 
to prevent twisting and to allow lengthening or shortening 
of the chain. The Surveyor's Chain is used in measuring 
land. 

Engineer's Chain. The Engineer's Chain differs from the 
ordinary Surveyor's Chain chiefly in that it is 100 feet in 
length, the length of each link being 1 foot. It is used in 
surveying railroads and canals, and often in other surveys 
where extensive lines are to be run. 

Both the Surveyor's Chain and the 'Engineer's Chain are 
generally provided with attachments, so that from the full 
chains half-chains can be made up, to be used in case of rough 
or hilly country. 



4 SURVEYING 

Accompanying Pieces. Usually eleven, sometimes ten, Mark- 
ing pins go with the chain. These are of iron or steel, about 
14 inches long, pointed at one end and formed into a ring at 
the other end. In case eleven pins are used, the first one is 
placed at the beginning of the line to be measured, and there- 
after one at the end of each chain. The last pin in the ground 
is, therefore, not to be counted. In case ten pins are used, 
the first one is placed at the end of the first chain, and so on, 
the last pin in the ground being counted. Strips of red cloth 
should be fastened to the ring ends of the pins so as to make 
them easily visible. Ranging poles, which are of various 
lengths, are necessary for alignment. These are commonly 
made of wood, and are steel shod, graduated to feet, and 
painted in alternate red and white stripes. 

How to chain. Ranging poles should be placed, one at each 
end of the line to be measured, and at such intermediate 
points as the necessities of the case require. A head chain- 
man or leader, and a rear chainman or follower are required. 
The follower takes one end of the chain, and one pin, which 
he thrusts into the ground at the beginning of the line. The 
leader takes the other end of the chain and the remaining ten 
pins, and moves forward until the word " Halt " from the fol- 
lower warns him that he has advanced nearly the length of 
the chain. At this signal he stops, and the follower, mean- 
while having placed his end of the chain against the pin at 
the beginning of the line, directs the leader by the words 
"Right" and "Left" until he is exactly in the line. This 
being accomplished, and the chain tightly stretched in a hori- 
zontal position, the follower says, " Down." The leader then 
puts in a pin at the end of the chain and answers, " Down " ; 
after which the follower withdraws the pin at his end of the 
chain, and the chainmen move forward, repeating the process 
just described until the end of the line is reached. 

If the marking pins in the hands of the leader are all placed 



FIELD INSTRUMENTS 5 

before the end of the line is reached, after putting the last pin 
in the ground he waits until the follower comes up to him, 
gives him the ten pins in his hands and records the fact that 
ten chains have been measured. The measuring then proceeds 
as before. If the distance from the last pin to the end of the 
line is less than a chain, the leader places his end of the chain 
at the end of the line, and the follower stretches tightly such 
part of the chain as is necessary to reach the last pin, and 
the number of links is counted. If the ground slopes, one end 
of the. chain must be raised until the horizontal position is 
attained. By means of a plumb line or a slender staff or, less 
accurately, in case of the leader by dropping a pin (heavy end 
downwards), the point vertically under the raised end of the 
chain may be determined!. If the slope is considerable, half 
a chain or less may be used ; in which case care must be taken 
that the correct number of full chains and links is found. For 
instance, if a tally shows 15 half chains and 35 links, the 
appropriate measure is 7 chains and 85 links, or, as it is 
usually expressed, 7.85 chains. 

Special Constructions by Means of the Chain. 1. At a given 
point in a given line to construct a perpendicular to that 
line. 

Let LE (Fig. 1) be the given line, and P the given point. 
On LE measure off PB = PA = 20 links. Then place one end of 
the chain at B and the other end at A. 



Stretch the chain from the middle point, / \ 



and mark that point, as C. PC is the 
perpendicular required. (Why ?) / 



Or, make PB = 30 links. Place one 



E 



A P B 
end of the chain at P, and the end of 

the 90th link at B. Then, taking the 

chain at the end of the 40th link from P and stretching 
both portions tightly, mark that point, as C. Then PC is the 
perpendicular required. (Why ?) 



6 SURVEYING 

2. Through a given point without a given line to construct 
a perpendicular to that line. 

Let LE (Fig. 1) be the given line, and C the given point. 
Take any point as B in the line and stretch the chain between 
C and B ; then swing the chain about C until the point at B is 
again in the line, as at A. Measure the distance between A 
and B. Then P, the mid-point of AB, is a second point in the 
required perpendicular. (Why ?) 

Or, let the middle of the part of the chain between C and B 
be held in place, and swing the end at C until it meets the 
line as at P. PC is the required perpendicular. (Why ?) 

3. At a given point in a given line to construct an angle 
equal to a given angle. 





D E 

FIG. 2 



Let P (Fig. 2) be the given point in the given line LE, 
and angle A the given angle. Make PD = AB. At D and B, 
respectively, construct perpendiculars DF and EC. Make 
DO = BC. Then angle OPD is the angle required. (Why ?) 

4. To construct any given angle, as 25 40'. 

Find from the tables the tangent of 25 40', which is 0.4806. 
Lay off PD (Fig. .2) = 100 links. Construct the perpendicular 
DF and lay off DO = 48.06 links. Then angle OPD is the 
angle required. (Why ?) 

5. Through a given point to construct a line parallel to a 
given line. 

Let P (Fig. 3) represent the given point, and LE the 
given line. Through P lay out any convenient line as BA 



FIELD INSTRUMENTS 7 

intersecting LE. Construct angle BPD = angle PAE. Then 
the line CD is the required line. (Why ?) 

/B 

C P/ D 






E 



FIG. 3 

Obstacles to chaining. In general practice various obstacles 
are encountered in chaining. The circumstances in each case 
must decide the best method to be used. Only a few sugges- 
tive cases can be considered in this work. 

1. To measure a line when a building, or other object, 
stands in the way. 

In Fig. 4 construct the 

perpendicular AB, the per- -^ 

pendicular BC, the perpen- 
dicular CD = AB, then the 



D 



E 



perpendicular DE, which FIG. 4 

will be in line LA prolonged. 

Then, LA +BC + DE = LE. (Why ?) As a check, another 
series of perpendiculars may be constructed. 
2. To measure across a body of water. 

At A (Fig. 5) lay 
out AP, making angle 
P,4=:60 . This can 
be done by laying out 
the equilateral tri- 
angle ABD. At P 
: range out PC, mak- 
ing angle APC = 60. 
FIG. 5 Then measure A P. 




B 



8 SURVEYING 

The line AC is equal to AP. (Why ?) If C is some fixed 
point in LE, across the stream, accessible or inaccessible, we 
may proceed as follows : After laying out AP, as already 
described, with 90 links of the chain stretched in the form 
of an equilateral triangle, and with one side of this triangle 
in AP, move the triangle until the point C is in line with 
the forward side of the triangle. Then proceed as before. 

3. To measure a line the end of which is invisible from 
the beginning, and the intermediate points are unknown. 

C 

L D! K 



FIG. 6 

Let LE (Fig. 6) represent the line. Lay out the line LR so 
that R shall be beyond E and visible from L. Construct 
from E the perpendicular EA to LR. Measure LA and AE. 
LE can then be computed. (How ?) If intermediate points 
on LE are to be sought, take any point in LA, as B ; construct 
EC perpendicular to LA ; then measure off BD of such length 
that BD:AE = LB:LA. The line LR is called a random 

L ^__ E line. 

4. To measure the dis- 
tance between two inaccessi- 
/ ble points. 

N v' Let L and E (Fig. 7) be 

/ \ two inaccessible points. 

/ \ Select some point asP from 

/ XN \ N which both L and E are 

visible. Measure PL and PE 




E/ ' by the method in 2. Kange 



FIELD INSTRUMENTS 9 

out PL' in line with LP and equal to LP ; similarly, RE' = ER. 
Then measure L'E', which is equal to LE. (Why ?) 

EXERCISE I 

1. Range out a line which, by estimation, is more than 
10 chains long. Then measure it with the chain out and 
back. 

2. Prolong a line beyond a building, or other obstacle which 
prevents continuous alignment. 

3. Find the distance from a point to a line when the dis- 
tance is more than a chain. 

4. Lay out a square field each side of which shall be 5.76 
chains long. 

5. Find the length of a line by means of a random line. 
Then, as a check, find its length by direct measurement. 

SECTION III 
THE TAPE 

Kinds of Tape. The tape measure used by the surveyor 
or engineer consists of a thin ribbon of steel, or of linen 
with interwoven wires of brass, wound upon a reel, often in 
a leather or metal case. Tapes vary in length from 25 feet 
to 500 feet or more. They are variously graduated to links, 
to feet and inches, to feet and tenths of a foot, to metric units, 
or to a combination of these. A common combination is feet 
and tenths of a foot on one side, and links on the reverse side. 

Uses. The kind of tape determines to a great extent the 
use to which it is to be put. If 33 feet or 66 feet long and 
graduated to links, the evident purpose is for land survey- 
ing. If 50 feet or 100 feet long and graduated to feet and 



10 



SURVEYING 



tenths of a foot, the tape is especially designed for city work 
Other kinds are employed in bridge, road, or mining work, in 
very accurate measurements of base lines, or as standards of 
comparison for other instruments of measurement. 



SECTION IV 
THE COMPASS 

Parts and their Uses. The essentials of the compass, one 
style of which is shown in Fig. 8, are : the compass circle, 
graduated to half degrees and figured from to 90 each way 




FIG. 8. THE SURVEYOR'S COMPASS 

NOTE. The letters E and W on the face of the compass are reversed 
from their true positions. The reason for this is that if the sights are 
turned towards the west, the north end of the needle is turned towards 
the letter W, and if the north end of the needle is turned towards E, the 
sights are turned towards the east. 

If the north end of the needle points exactly towards E or W, the 
sights range east or west. 



FIELD INSTRUMENTS 11 

from the north and south points, for indicating the directions 
of lines ; the magnetic needle, pivoted on a pin at the centre 
of the compass circle, for showing the direction of the mag- 
netic meridian ; and the sight standards, attached to the ends 
of the main plate, for alignment. To the main plate are 
attached two spirit levels at right angles to each other for 
leveling the instrument; underneath is a needle-lifting screw 
which, by actuating a concealed spring, lifts the needle from 
the pivot and presses it against the glass covering of the 
compass circle when the instrument is not in use ; a tangent 
screw, and almost directly under it a clamp screw, which 
operates the vernier ; and a small dial plate for keeping tally 
in chaining. The north end of the needle usually has some 
ornamentation to distinguish it from the south end, and a 
coil of tine wire is wound on the south end to prevent the 
needle from dipping. The sight standards have fine slits 
nearly their whole length, with circular openings at intervals 
to facilitate sighting upon an object ; on the edges of the north 
standard are tangent scales for reading vertical angles, and 
on the outside of the south standard are two eyepieces at the 
same distance from the main plate as the zeros of the tangent 
scales, respectively. The telescopic sight is an attachment to 
the south standard, now often used. The instrument entire 
turns horizontally upon the upper end of a ball spindle, the 
lower end of which rests in a spherical socket in the top of 
a Jacob's staff, or a tripod, which supports the instrument. 
The socket of the compass which fits to the ball spindle is 
provided with a clamp screw and a spring catch. From the 
centre of the plate at the top of the tripod a plummet is 
suspended by which the centre of the compass can be placed 
directly over a definite point on the ground. 

Kinds of Compasses. The compass described is the vernier 
compass, or surveyor's compass, and is the one in general use. 
If there is no vernier attachment, the compass is called a plain 



12 



SURVEYING 



compass and is used in running new lines and the preparation 
of maps. A railroad compass has all the features of the vernier 
compass, and has also a vernier plate and graduated limb for 
measuring horizontal angles. 

Hints on the Use and Care of Instruments. The instruments 
described in this work are adjusted by the maker. If they 
should require readjustment, full directions will be found 
in the manual furnished with the instruments. Before begin- 
ning to use any instrument, make a thorough study of its 
various parts and their uses. In moving or adjusting any 
part always know what you are doing and why you are doing 
it. When an instrument is not in use keep it in a place that 
is free from moisture and dust. 

Bearing of a Line. The magnetic 'meridian of a place is the 
direction which a bar magnet assumes when freely supported 

in a horizontal position. 
The magnetic bearing of a 
line is the angle it makes 
with the magnetic merid- 
ian. To take the bearing 
of a line proceed as fol- 
lows : Place the compass 
so that the Jacob's staff, 
or plummet of the tripod, 
is directly over one end of 
the line, and level by press- 
ing with the hands on the 
main plate until the bub- 
bles are brought to the 
centres of the spirit levels. 
Turn the south end of the instrument toward you, and sight 
at the ranging pole at the other end of the line. Eead the 
bearing from the north end of the needle. First, write N. or 
S. according as the north end of the needle is nearer N. or S. 




FIG. 9 



FIELD INSTRUMENTS 13 

of the compass circle ; secondly, write the number of degrees 
between the north end of the needle and the nearest zero mark ; 
thirdly, write E. or W. according as the north end of the needle 
is nearer E. or W. of the compass circle. Thus, in Fig. 9 (&), the 
bearing is K 45 W. ; (&), K 60 E. ; (c), S. 60 W. ; (d), S. 45 E. 

If the needle coincides with the KS. or E.W. line, the 
bearing is 1ST., S., E., or W. according as the north end of the 
needle is over K, S., E., or W. As the compass circle is 
divided into half degrees, the bearing may be determined 
pretty accurately to quarter degrees. 

It will be noticed that the letters E and W on the face of 
the compass are reversed from their true positions. These are 
so placed in order that when the sights are turned towards 
the west the north end of the needle will point towards the 
letter W, or if the sights are turned towards the east, the 
north end of the needle will point towards the letter E. It 
turns out that if the south sight standard is always turned 
towards the observer, the reading at the north end of the 
needle will indicate the true bearing of the line. Should the 
north sight standard be turned towards the observer, the read- 
ing at the south end of the needle would then be taken. 

Checking Bearings. When the bearing of a line has been 
taken, the instrument should be removed to the other end of 
the line and the reverse bearing taken. The number of 
degrees should be the same, but the letters should be reversed. 
For instance, if the direct bearing is K 35f W., the reverse 
bearing should be S. 35f E. In case the reverse bearing is 
not what it ought to be, there is some mistake, or some local 
disturbance, or both. To detect errors a second trial at the 
direct bearing should be taken. To detect local disturbances 
take the direct and reverse bearings of other lines ranged out 
from the beginning of the line whose bearing is sought. If 
they all show the same difference between their two respective 
bearings, the evidence of some local disturbance, as iron, 



14 SURVEYING 

iron ore, etc., is pretty conclusive. In this case the true bear- 
ing of the line can be obtained by making the necessary cor- 
rection. In all cases, precautions should be taken to have the 
chain, pins, and other instruments that would affect the direc- 
tion of the needle sufficiently removed from the compass. 

Obstacles. When a fence or other obstruction interferes 
with placing the instrument over the line the instrument may 
be placed at one side, the ranging pole being correspondingly 
placed at the other end. If one end of the line cannot be seen 
from the other end, run a random line. Then (Fig. 6, p. 8) 
tan EL A = AE -=- LA, whence the angle EL A can be found. 
This angle combined with the bearing of the random line will 
give the bearing required. Or some point can be selected 
from which the ends of the line are visible. The distances to 
the ends may be measured, and the angle between the two 
auxiliary lines may also be measured. Of the triangle thus 
formed, the angle at the beginning of the given line may be 
computed, and, when properly combined with the bearing of the 
first auxiliary line, will give the required bearing. If a single 
triangle is not sufficient, a series of triangles may be employed 
until the end of the line is reached. 

Measurement of Horizontal Angles. To measure a horizontal 
angle by means of the needle, place the compass over the 
vertex of the angle, take the bearing of each line separately, 
and combine these bearings according to the following rules, 
as suggested by Eig. 10 : 

1. If the first letters of the bearings are alike, and also the 
last letters, find the difference of the bearings. 

2. If the first letters are alike, and the last letters unlike, 
add the bearings. 

3. If the -first letters are unlike, and the last also unlike, 
subtract the difference of the bearings from 180. 

4. If the first letters are unlike, and the last alike, subtract 
the sum of the bearings from 180. 



FIELD INSTRUMENTS 



15 




FIG. 10 



W 



Measurement of Vertical Angles. A vertical angle is an 
angle the sides of which are in a vertical plane. If one side 
of a vertical angle is horizontal and the other ascends, the 
angle is called an angle of elevation; if one side is horizontal 
and the other descends, the angle is called an angle of depres- 
sion. To measure an angle of elevation by means of the com- 
pass, sight through the lower eyepiece to a point that is as 
far above the point whose elevation is sought as the instru- 
ment is above the point from which the elevation is to be 
taken. Kead off the degrees of the right-hand tangent scale, 
marked by a card placed squarely across the face of the south 
standard, the top of the card being in the line of sight. To 
measure an angle of depression, proceed in the same manner, 
using the upper eyepiece and the left-hand tangent scale. If 
the compass is provided with a telescopic sight that has a 
vertical circle attachment, these should be used instead of 
the eyepieces and tangent scales. 

Verniers. A vernier is a contrivance for measuring portions 
smaller than those into which a line is divided. We shall 
describe two kinds. 

Let AB (Fig. 11) be a portion of a line graduated to tenths 
and hundredths of a foot. VR is the vernier. 

In (a), nine parts of the line are divided into ten equal parts 
on the vernier. Hence, a division on the vernier is less than 
a division on the line by the difference between T ^ of a foot 
and T L of T ^ of a foot, or T ^Vtf of a foot. Now, if the vernier 



16 



SURVEYING 



is moved so that 1 of the vernier coincides with 1 of the scale, it 
has moved over a space equal to yo 1 ^ of a foot. If the vernier 
is moved so that 2 of the vernier coincides with 2 of the scale, 
it has moved over a space equal to T ^ 2 ff F of a foot ; and so on. 
In (7>), 6 of the vernier coincides with 9 of the scale, which 
indicates that the zero of the vernier has moved past 3 of the 
scale a space equal to T Q G O cr f a foot. The reading, then, is 



uuurumu 



L 



I, I.I ,1 



(w/iaaaaiaaaaan? 



o/raaaarr rm u 



~* <to op 



(aaaaaiaaaaj.n( 



FIG. 11 



0.536 foot. This form of the vernier is known as the direct 
form, since the figuring on the vernier proceeds in the same 
direction as that on the scale. 

In (c), eleven parts of the line are divided into ten equal parts 
on the vernier. Hence, a division on the vernier is greater 
than a division on the line by the difference between T ^ of 
jfa of a foot and yi^ of a foot, or TTJ ^^ of a foot. Now, if the 
vernier is moved so that 1 of the vernier coincides with 10 



FIELD INSTRUMENTS 



17 



of the scale, i.e. t the end of the 6th tenth, the vernier has 
moved over a space equal to TIJ ^ of a foot. If the vernier is 
so moved that 2 of the vernier coincides with 9 of the scale, 
the vernier has moved over a space equal to T Q S ^O f a ^^ 5 
and so on. 

In (d), 6 of the vernier coincides with 7 of the scale, which in- 
dicates that the zero of the vernier has moved past 3 of the scale 
a space equal to T o 6 o o ^ a ft. The reading here is 0.636 foot. 

This form of the vernier is known as the retrograde form, 
since the figuring on the vernier proceeds in the opposite direc- 
tion from that on the scale. In either form the following 
rule for using and reading the vernier may be adopted : 

Move the vernier until its zero line, or index, is at the point 
to which the required measurement is to be taken; read the 
main scale to the nearest division below the index, and that 
number of the division line of the vernier which stands opposite 
a line of the main scale. 




FIG. 12 



Compass Vernier and its Uses. Let LL 1 (Fig. 12) repre- 
sent the limb of the compass graduated to half degrees, and 
W the vernier divided into thirty equal spaces, equal to 
twenty-nine spaces of the limb. Then, one space of the vernier 
is less than one space of the limb by l'(=30' ^ of 29x30 f ), 
and the reading may be obtained to single minutes. 



18 



SURVEYING 



In Fig. 12 the index, or zero, of the vernier stands between 
32 and 32 30', and the line of the vernier marked 9 coincides 
with a line of the limb. Hence, the reading is 32 9'. 

When the index moves from the zero line of the limb in a 
direction opposite to that in which run the numbers of the 
limb, the number of minutes obtained as above must be sub- 
tracted from 30' to obtain the minutes required. (Why ?) If, 
however, the vernier is made double, that is, if it has thirty 
spaces on each side of the zero line, it is always read directly. 
The usual form of the double vernier, shown in Fig. 13, has 




only fifteen spaces on each side of the zero line. When the 
vernier is turned to the right less than 15' past a division line 
of the limb, read the lower figures on the left of the zero line 
at any coincidence ; if moved more than 15' past a division 
line of the limb, read the upper figures on the right of the 
zero line at any coincidence ; and vice versa. In this form of 
the double vernier it will be observed that the spaces on the 
vernier are larger than those on the limb, since the 30 equal 
spaces of the former are equal to 31 half-degree spaces of the 
latter. 



FIELD INSTRUMENTS 19 

Tho most important use of the vernier compass is in setting 
off the variation of the needle explained just below. If the 
variation of the needle at any place is known, by means of the 
vernier screw the compass circle may be turned through an arc 
equal to the variation. If the observer stands at the south 
end of the instrument, the vernier is turned to the right or left 
according as the variation is west or east. The compass now 
gives the bearings of the lines with the true meridian. 

In order to retrace the lines of an old survey, turn the 
sights in the direction of a known line and move the vernier 
until the needle indicates the old bearing. If no line is defi- 
nitely known, the change of variation from the time of the 
old survey will give the arc to be set off. 

Magnetic Declination. The magnetic declination, or varia- 
tion of the needle, at any place is the angle which the magnetic 
meridian makes with the true meridian, or north and south 
line. The variation is east or west, according as the north end 
of the needle lies east or west of the true meridian. Western 
variation is indicated by the sign +, and eastern by the sign . 
The kinds of magnetic decimation are put under three heads : 

1. Irregular variations, which are sudden deflections of 
the needle due to magnetic storms or other causes not well 
understood. 

2. Solar-diurnal variations, which in northern latitudes reach 
their farthest point east about 8 o'clock A.M., and their farthest 
point west about 2 o'clock P.M., varying from 5' in the winter 
in some localities to 20' in the summer in other localities. 

3. Secular variation, which is a change in the same direction 
for a period of years, then in the opposite direction for about 
the same time. 

It is not accurately known how long it takes a complete 
secular variation to run its course, but from data already 
obtained it seems probable that the period of time covered is 
not less than two and a half or three centuries. 



20 



SURVEYING 



O 
Si a 



HOO ' i- O rH o O 



I + 



5 t O r-i Ci -tf _ CO <* O O <* OS iC ^ TJH rH rH OS G<1 t- O O -HH Ci O 

oftM^rH^r-l ' ^ ,! r-l r-i rH Jin ,_ ,_, ^ * ^ ^ ^^^CO 



. 

T i Ol ^f O -^ 



t O? O r- 1 



rH <M O rH OS -^ OS O O t^rHOOOl^J OCO Ci O O (N 

L)Ci'H/II>-CDQOOO i 5<lOOC<lOCJC;'rJHTt<OOOO-*OOi lt^CO5<lrHl^-CCO7 

c5dSCOod>"3COt^Oijt^O^^c6ocOrHC5(Nr-5OrHrH 



O 00 l^ rHCO 00 C^rHCO 7* O GO (N O rH . 

p Th GO rH ?N T^H CC t^ OS CO l~ O O t~- CO OS O O "^_ rH '-^i C<| t- CO O| GO CO O TJH 

M^^'^COO5odl^t^(NOOOt^OI>^^^OOrHC<ioC<ioOOOrHO 

^ rH ^ rH I I .' 



OOOOO ^ Tt*-HHt^ T<1 

OSC<IOSQOOOSrH! lr^rHt>-(MI>-OOCOOQOr-ia<ICOOrHOT Ir- l(7<J 



5oCil^rHt^c<|^QO^TtHOOC<ICOOOOl^->OOOOL't)COCCOCO^CC' 
^COOTtlrHrHTtlOOT-!Sq^i-OrHrHCOTjlOC<lOrHO(>ieO^OOOOC<l 



t-?O?OCOJt*OOS 

(Ni i T-H I-H iO O O CO 



X 



'S >!2 1 

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^d 

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H ..rZ & - ^ 2 fcc^ 2 "K-o* 3 '-<H- : *CCo'- l ha'a;'"i-i-iJ - 

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tJ 

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B tf.S 



ci 

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FIELD INSTRUMENTS 



21 



^gioi- 




ii 



1111 777^77 



O O: -i OC rH CO CO -* O CO 'NLCOOd 

CiOCCOC<JCiCOCiOOCCCOOCCC<lC<IOCC 
" to O O CJ 




00?OCDt~OSt-~Ot-OOO''tlr~rHrH'<$t(NaO 




O 

rH TtH 
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Sj CS TJH rt^ (>| O T-H CD 
O O *O t^" CO " t>~ 00 GO 

I I I I I 'III 



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O t to T i i^ c: 

o ' t-^ cd ' cc' t^ 

I 'II 'II 



co t- 

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i i 



00 

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CO O CS CC rH O 

O O O -* <M O 




22 SURVEYING 

The agonic line, or line of no variation, is a line joining 
those places at which the magnetic meridian coincides with 
the true meridian. At the beginning of the present century 
this line crossed the United States in an irregular way from 
Michigan to South Carolina. It is gradually moving west- 
ward, so that the variation is increasing at places east of this 
line, and decreasing at places west of the line. East of this 
line the variation is westerly, and west of this line the varia- 
tion is easterly. Lines that join places of equal magnetic 
declination are called isogonic lines. 

Table of Magnetic Declination. On pp. 20, 21 will be found 
a table showing the variation in magnetic declination at 
different places in the United States and contiguous territory 
during the nineteenth century ; also the annual change for the 
epoch of 1900. 

EXERCISE H 

1. Lay out a field of five sides and take the bearings and 
measures of the sides in order, beginning at the most westerly 
point and going about the field clockwise. 

2. From the bearings obtained in Example 1 find the value 
of each of the interior angles. What is their sum ? 

3. Lay out the field the bearings and distances of whose 
sides are given in Example 1 of Exercise YI, p. 64. 

4. Eange out a line whose bearing is X. 38 30' W., and at 
some point in this line range out another line making a right 
angle with it. What is the bearing of the second line ? 

5. Set up the compass at a spot near which there is known 
to be some local disturbance, as iron in a building, or an 
iron fence, and find the variation of the needle due to such 
disturbance. 



FIELD INSTRUMENTS 23 

SECTION Y 
THE TRANSIT 

Surveyor's Transit. The transit is the most important 
instrument used in surveying. There are many modifications 
of it, each adapted to its own particular use. All, however, 
have about the same essential features. The one described 
here, and shown in Fig. 14, is the surveyor's transit, the one 
of most general use. The essential parts are the telescope 
with its axis and two standards, the circular plates with their 
attachments, the sockets upon which the plates revolve, the 
leveling head, and the tripod. Within the telescope are two 
fine cross wires, at right angles to each other, whose intersection 
determines the optical axis, or line of collimation, of the tele- 
scope. Under the telescope, and attached to it, is a spirit 
level by which horizontal lines may be run, or the difference 
of level between two stations be found. The axis of the tele- 
scope carries a vertical circle which measures vertical angles 
to single minutes by means of a vernier. The vernier plate, 
which carries the telescope and also the compass circle, has 
two verniers diametrically opposite to each other, and it moves 
entirely around the graduated limb of the main plate. The 
sockets are compound ; the interior spindle attached to the 
vernier plate turning in the exterior socket, when an angle is 
taken on the limb, but when the plates are clamped together 
the exterior socket itself, and with it the whole instrument, 
revolves in the socket of the leveling head. The transit is 
leveled by four leveling screws which pass through a plate 
firmly fastened to the ball spindle and rest in small sockets, 
these resting in turn on the upper side of the tripod plate. 
On the underside of this lower or tripod plate is an arrange- 
ment called a shifting centre, which enables the surveyor to 
change the position of the vertical axis horizontally without 



24 SURVEYING 

moving the tripod ; besides this there is, if specially ordered, 
a device called a quick-leveling attachment to bring the transit 
quickly to an approximately level position by the pressure of 
the hands after which the leveling screws are used. 

Uses. The transit may be used for all the purposes for 
which the compass may be used, but with much greater 
precision. The principal use, however, is in measuring hori- 
zontal angles by means of the graduated limb and verniers. 
It may be used, furthermore, in obtaining differences of level ; 
also, provided there is the attachment to the telescope known 
as the stadia, in measuring distances, especially over broken 
ground. A still further use, when the transit is supplied 
with what is known as a gradienter attachment, is in fixing 
grades as well as measuring distances. 

Getting the Transit Ready. The instrument should be set 
up so as to be firm, the tripod legs being pressed into the 
ground until the plates are as nearly level as can conveniently 
be done by this means. For the subsequent leveling turn the 
instrument until the spirit levels on the vernier plate are 
parallel to the vertical planes passing through opposite pairs 
of the leveling screws. Take hold of opposite screw heads 
with the thumb and forefinger of each hand, and turn both 
thumbs in or out as is necessary to bring the bubble to its 
proper place, the left thumb always moving in the direction 
that the bubble is to move. For precise work, in addition to 
leveling by the leveling screws, it is advisable to level the 
plates by the telescope level, as this is much more sensitive 
than the levels on the plate. In this operation the position 
of the level on the telescope must be observed over both sets 
of leveling screws, one half the correction being made by the 
axis tangent screw, the other half by the leveling screws. 
Before an observation is made with the telescope, the eye- 
piece should be focused by its pinion until the cross wires 
appear distinct ; the object glass is then focused by its pinion 



FIELD INSTBUMENTS 



25 




FIG. 14. THE SUUVEYOR'C TRANSIT 



26 SURVEYING 

until the object to be observed appears well defined. This 
latter process must be repeated when the distance to the 
object is changed. 

Measurement of Horizontal Angles. Place the instrument 
directly over the vertex of the angle, and level. Set the 
limb at zero by the tangent screw of the plates, and turn the 
telescope in the direction of one of the sides of the angle, 
directing it to the object by the tangent screw of the leveling 
head. Then unclamp the main plate and turn the telescope 
until it is in the direction of the other side of the angle, 
and read the angle by the verniers. The object of the two 
verniers on the vernier plate is to correct any mistakes that 
might arise from the want either of exact coincidence in the 
centres of the verniers and the limb or of exact graduations 
on the limb. The correct reading may be obtained by adding 
to the reading of one vernier the supplement of the reading of 
the other, and taking half their sum. 

Measurement of Vertical Angles. Direct the telescope to the 
object ; clamp, and read the angle indicated on the vertical 
circle by the vernier. The angle read will be an angle of 
elevation or depression as the case may be, the horizontal line 
being the line of collimation of the telescope when in a hori- 
zontal position. 

Stadia Measurements. As already stated on page 24, the 
stadia is an attachment to the telescope used in measuring 
distances, especially over rough ground. It consists essen- 
tially of two horizontal wires fastened to small movable 
slides, and so adjusted as to include a given space, say 
one foot on a rod 100 feet distant. These wires will then 
include two feet on a rod 200 feet away, or a half-foot at a 
distance of 50 feet, and so on. Usually the instrument is so 
adjusted that the zero of the indicated distance is in front 
of the centre of the instrument; hence, the true distance 
is the indicated distance plus the distance of this zero from 



FIELD INSTRUMENTS 27 

the centre of the instrument. This latter distance is deter- 
mined for each instrument by the maker, and noted on a card 
placed on the inside of the instrument box. It is known as 
the constant of the instrument. The readings are taken on a 
rod, specially designed for the purpose, known as the stadia 
rod. This is graduated to feet, and tenths and hundredths of 
a foot. Any ordinary leveling rod, if similarly graduated, will 
answer the same purpose. Obviously in taking stadia meas- 
urements the rod must always be held at right angles to 
the line of sight. This statement has special reference to 
measurements taken up or down hill-slopes. In this case, if 
horizontal distance is required, the measured distance must 
be multiplied by the cosine of the angle of elevation or 
depression. (Why ?) 



EXERCISE HI 

1. By means of the transit, measure the interior angles of 
the field of Example 1, Exercise II, p. 22, and compare with 
the results obtained in Example 2 of the same exercise. 

2. Lay out the entire angular magnitude about some point 
into four or more angles, and measure each one of them. What 
should the sum of them equal ? 

3. If the constant of a transit adjusted to one foot 100 feet 
away is 3.8 inches, what is the true length of a line when the 
indication on the rod is 2.35 feet ? 

4. Measure a line by the stadia, and compare with measure- 
ments taken by the chain and also by the tape. 

5. Compute the height of a tall object, as a tree or steeple, 
by first measuring its distance from some convenient point and 
measuring the angle of elevation at that point. 

6. Lay out a square field containing just one acre. 



28 SURVEYING 

SECTION VI 
THE SOLAR COMPASS 

Description and Uses. A full description of the solar com- 
pass, or Burtfs solar compass, as it is often called from its 
inventor, with its principles, adjustments and uses, forms the 
subject of a considerable volume, which should be in the hands 
of the surveyor who uses this instrument. The limits of our 
space will allow only a brief reference to its principal features. 
Fig. 15 exhibits the instrument by itself; Fig. 16, p. 31, is a 
graphical illustration of the solar apparatus as an attachment to 
the transit, the circles shown being intended to represent those 
supposed to be drawn upon the concave surface of the heavens. 
The form of the solar compass shown in Fig. 15 has the 
arrangement of its sockets and plates similar to that of the 
transit, the standards similar to those of the compass, the solar 
apparatus being placed on the upper vernier plate and taking 
the place of the needle, for which it operates as a substitute 
in the field. 

The solar compass consists mainly of three arcs of circles, a 
the latitude arc, by which is set off the latitude of the place, b 
the declination arc, by which is set off the declination of the 
sun, and c the hour arc, by which is set off the hour of the day. 
The arm h is movable about a point at the extremity of the 
piece containing the declination arc, there being at each end a 
solar lens having its focus on a silvered plate on the other end. 
The arc of the declination limb turns on an axis, and one or 
the other solar lens is used, according as the sun is north or 
south of the equator. Fig. 15 shows the position of the decli- 
nation arc when the sun is south; Fig. 16, when it is north. 
The needle box is moved about its centre by a slow-motion 
screw. It contains a magnetic needle, and is furnished with a 
graduated arc about 36 in extent. 



FIELD INSTRUMENTS 



29 




FIG. 15. BUBT'S SOLAR COMPASS 



FIELD INSTRUMENTS 



31 




FIG. 16. TRANSIT WITH SOLAR ATTACHMENT 

The circles shown in the cut are intended to represent in miniature circles supposed 
to be drawn upon the concave surface of the heavens. 



32 SURVEYING 

The solar compass may be used for most of the purposes of a 
compass or transit. Its most important use, however, is to run 
north and south lines, especially in laying out the public lands. 
It may be used also in determining the latitude of a place. 

To establish a True Meridian. Set off on the latitude arc 
the latitude of the place. Set off on the declination arc the 
declination of the sun, corrected for refraction. Set the 
instrument over the station ; level, and turn the sights in a 
north and south direction by the needle. The surveyor then 
turns the solar lens to the sun, and with one hand on the 
instrument and the other on the revolving arm, moves both 
from side to side until the sun's image is made to appear on 
the silvered plate, precisely between the equatorial lines. The 
line of sights then indicates the true meridian. 

The bearing of any line from the meridian may be read by 
the verniers of the horizontal limb. When a due east and 
west line is to be run, these verniers are set at 90, and the 
sun's image is kept between the lines as before. 

Other Methods. By North Star at Culmination. The North 
Star, or Polaris, at present revolves about the north pole of 
the heavens at the distance of about 1^ ; hence, it is on the 
meridian twice in 23 h. 56 m. 4 s. (a sidereal day), once above 
the pole, called the upper culmination, and 11 h. 58 m. 2 s. 
later below the pole, called the lower culmination. 

The time of the upper culmination of Polaris may be found 
by means of the star Mizar, the middle one of the three stars 
in the handle of the Dipper (in the constellation of the Great 
Bear). It crosses the meridian at nearly the same time as 
Polaris. Suspend a plumb line, placing the bob in a pail of 
water to lessen its vibrations. South of the plumb line, upon 
a horizontal board firmly supported, place a compass sight, or 
any upright with a small opening or slit, fastened to a board 
a few inches square. At night, when Mizar by estimation 
approaches the meridian, place the compass sight in line with 



FIELD INSTRUMENTS 



33 



Cassiopeiae 



Polaris 



Polaris and the plumb line, and move it so as to keep it in 
this line until the plumb line falls also on Mizar (Fig. 17). 
Note the time ; then (1903) 3 in. 39 s. later Polaris will be 
on the meridian. If then Polaris, the plumb line and the com- 
pass sight are brought into line, the plumb line and compass 
sight will give two points in the meridian ; or if the telescope 
of the transit is brought to bear on Polaris, 
and a light is held near to make the wires 
visible if necessary, the telescope will 
then lie in the plane of the meridian, 
which may be marked by bringing the 
telescope to a horizontal position. 

For each year subsequent to 1903 add 
21 s. to 3 m. 39 s. If the lower culmina- 
tion takes place at night, the time may be 
found in a similar manner. When Mizar 
cannot conveniently be used, 8 Cassiopeiae 
(Fig. 17) may be employed, the method 
being the same as in the case of Mizar. 
The interval, however (1903), is 4 m. 24 s. 
and the annual increase of the interval Mizar * 
about 20 s. 

By North Star at Greatest Elongation. 
When Polaris is at its greatest apparent 
angular distance east or west of the pole, 
it is said to be at greatest elongation. It 
attains its greatest eastern elongation and 
western elongation, respectively, 5 h. 59 m. 1 s. after lower and 
upper culmination. The azimuth of a star is the angle which 
the meridian plane makes with the vertical circle passing 
through the star and the zenith of the observer. 

If now we know the time of either extreme elongation and 
also the azimuth of Polaris at an extreme elongation, we can 
from these data establish a true meridian. The latter of these 



Pole 



Great 



Bear 



FIG. 17 



34 SURVEYING 

data is given for various latitudes and for years to come in 
tables, to which the surveyor is supposed to have access. To 
obtain a line in the direction of Polaris at greatest elongation, 
we may proceed as follows : A few minutes before the time 
of greatest elongation, place the compass sight in line with 
the plumb line and Polaris, keeping it in line with these until 
the star begins to 'recede. At this moment the sight and 
plumb line are in the required line. Or bring the telescope 
of the transit to bear on the star, and follow it keeping the 
vertical wire over the star until it begins to recede. The 
telescope will then be in the required line. In either case, 
after having the transit sighted in the direction of the line 
just found, turn it in the proper direction through an angle 
equal to the azimuth as found from the tables. 

The accompanying table * gives the Washington mean time 
of each tenth transit of Polaris (upper culmination) at the 
meridian of Washington, D.C. The last column contains the 
variation per day, to facilitate the interpolation of the time 
for any intermediate transit. 

The transit which occurs October 17 is the tenth transit 
following that which occurs on October 8. This is because 
two transits occur on October 13 ; the interval separating them 
being 23 h. 56 m. 4 s. of mean time. These two transits are 
introduced in the table for greater convenience, and as a safe- 
guard against error respecting the particular day of transits in 
that vicinity. The double lines merely call attention to the 
break thus caused in the series. 

By interpolation we may, by taking account of the longitude 
of any given station, find the local mean time of transit of 
Polaris at that station for any particular day. Thus, to find 
the Cincinnati mean time of the upper culmination of Polaris 
at Cincinnati, on May 15, 1902, we have (p. 36) : 

* Furnished by the Director of the Nautical Almanac Office, Wash- 
ington, D.C. 



FIELD INSTRUMENTS 



35 



DAY OF 


LOCAL MEAN TIME 


CHANGE IN 


THE YEAR 


OF EVERY lOrn TRANSIT 


1 DAY 


1902 


(Civil Time.) 




Jan. 1 


6 h 41 m 


19 s P.M. 


- 3 m 56 s . 8 


11 


6 1 


51 " 


56.9 


21 


5 22 


22 ' ' 


56.9 


31 


4 42 


53 ' 


56.9 


Feb. 10 


4 3 


24 ' 


56.8 


20 


3 23 


56 ' 


56.7 


Mar. 2 


2 44 


29 ' 


56.6 


12 


2 5 


4 " 


56.4 


22 


1 25 


41 " 


56.2 


Apr. 1 


12 46 


20 " 


56.0 


11 


12 7 


1 " 


55.8 


21 


11 27 


44 A.M. 


55.7 


May 1 


10 48 


28 " 


55.7 


11 


10. 9 


14 " 


55.3 


21 


9 30 


2 " 


55.1 


31 


8 50 


51 ' 


55.0 


June 10 


8 11 


41 ' 


55.0 


20 


7 32 


31 ' 


54.9 


30 


6 53 


22 ' 


54.9 


July 10 


6 14 


13 ' 


54.8 


20 


5 35 


5 ' 


54.9 


30 


4 55 


56 ' 


54.9 


Aug. 9 


4 16 


47 ' 


55.0 


10 


3 37 


37 ' 


55.1 


29 


2 58 


26 ' 


55.2 


Sept. 8 


2 19 


14 ' 


55.3 


18 


1 40 


1 ' 


55.4 


28 


1 


46 ' 


55.5 


Oct. 8 


12 21 


30 ' 


55.7 


Oct. 13 


12 1 


51 A.M. 




Oct. 13 


11 57 


55 P.M. 












Oct. 17 


11 42 


12 P.M. 


55.8 


27 


11 2 


53 " 


56.0 


Nov. 6 


10 23 


32 " 


56.1 


16 


9 44 


10 " 


56.3 


26 


9 4 


46 " 


56.4 


Dec. 6 


8 25 


21 " 


56.5 


16 


7 45 


55 " 


56.7 


26 


7 6 


27 " 


56.8 


36 


6 26 


58 " 


-3 56.9 



36 SURVEYING 

Local mean time of transit at Washington, May 11, 1902 

= 10 h 9 m 14 s A.M. 
Longitude of Cincinnati west of Washington 

= + O h 29 m 40 s = + O d .021. 
May 15 d + O d .021 = May 15 d .021. 
Preceding tabular date = May 11. 
Therefore, interval = 4 d .021. 

Daily variation = - 3 m 55 8 .3 = - 235 8 .3. 

Total change = 4.021 x (- 235 8 .3) = - 15 m 46 s . 

10 h 9 m 14 s A.M. 

-15 46 
9 h 53 m 28 s A.M. 

Therefore, the required time is 9 h .53 m 28 s A.M., May 15, 1902. 



SECTION YII 
THE Y LEVEL 

Description. The essential parts of the Y level (Fig. 18) 
are, the telescope, which is of various lengths, usually about 
20 inches, and rests on supports called Y's, from their shape ; 
the spirit level, which is under the telescope and attached to 
it ; and the leveling head and tripod, which are similar to the 
same parts of the transit. 

Leveling Rod. There are several kinds of leveling rods, 
each possessing some merit peculiar to its purpose. The one 
shown in Fig. 19 is known as the Philadelphia leveling rod, 
and is the one in most common use. It is made of two pieces 
of wood, sliding upon each other, and held in position by a 
clamp. The front surface of each piece is graduated to hun- 
dredths of a foot up to 7 feet ; the back surface of the real- 
piece is figured from 7 to 13 feet, reading from the top down, 



FIELD INSTRUMENTS 



37 




FIG. 18. THE Y LEVEL 



38 SURVEYING 

and it also has a scale by which the rod is read to half hun- 
dredths of a foot as it is extended. The target 
slides along the front of the rod and is held in 
place by two springs which press upon the sides 
of the rod. It has a square opening at the centre, 
through which the division line of the rod oppo- 
site to the horizontal line of the target may be 
seen. It also carries a scale by which heights 
may be read to half hundredths of a foot. For 
heights not greater than 7 feet, the target is 
moved up or down the front surface, the rod 
being closed and clamped; but when a greater 
height i-s required the target is fixed at 7 feet 
and the rear half of the rod extended to the 
required height. The rod thus becomes a self- 
reading rod 13 feet long. 

How to use Level and Rod. When the leveling 
instrument is used, the tripod should be set firm ; 
the spirit level should then be brought succes- 
sively over each opposite pair of leveling screws 
and leveled in each position, the operation being 
repeated until the bubble remains in the middle 
of the tube through an entire rotation of the tele- 
scope. Each time before taking an observation 
the instrument should be examined to see if it 
is still level. Care should be taken to bring the 
cross wires of the telescope precisely in focus and 
the object into such perfect view that the wires 
will appear to be fastened to the surface, how- 
ever the eye is moved. For very accurate work 
the instrument should be shielded from the direct 
rays of the sun. 
FIG. 19 The leveling rod should be held in a truly 

vertical position, the rodman standing squarely behind it. 



FIELD INSTRUMENTS 



39 



The target is then raised or lowered at the signal of the 
leveler until its horizontal line is cut by the intersection of 
the cross wires of the telescope. The reading is done by the 
leveler or the rodinan according to the kind of rod used. 

Substitutes for the Y Level. For ordinary work, the Sur- 
veyor's or Engineer's Transit is often used. 

The plumb level (Fig. 20) consists of two pieces of wood 
joined at right angles. A straight line is drawn on the upright 
perpendicular to the upper edge of the crosshead. The instru- 
ment is fastened to a support by a screw through the centre of 
the crosshead. The upper edge of the crosshead is brought 
to $> level by making the line on the upright coincide with a 
plumb line. 



n 



9 

FIG. 20 



FIG. 21 



FIG. 



A carpenter's square can be made into a level by being 
supported by a post (Fig. 21), the top of which is split or sawed 
so as to receive the longer arm. The shorter arm is made 
vertical by a plumb line, which brings the longer arm to a 
level. 

Tlie water level, as shown in Fig. 22, consists of two 
upright glass tubes cemented into a connecting tube of any 
material. The whole is nearly filled with water and sup- 
ported at a convenient height. The surface of the water 
in the uprights determines the level. The water should be 
colored. 



40 SURVEYING 

A level line may be obtained by sighting along the upper 
surface of the block in which an ordinary spirit level is 
mounted. 

For many purposes not requiring great accuracy, any of the 
foregoing simple instruments in connection with any graduated 
rod will be sufficient. 



EXERCISE IV 

1. Set up the level and take the readings on the leveling 
rod at two stations equally distant from the instrument. What 
does the difference of these readings indicate ? 

2. Set up the level successively at the two stations in 
Example 1, taking the readings on the leveling rod placed 
where the instrument was first. What does the difference of 
these readings indicate? Ought this difference to be the 
same as that in Example 1 ? Explain. 

3. In the field of Example 1, Exercise II, p. 22, set up the 
level successively at the middle of each of the five sides, taking 
the readings on the rod each time at both adjacent stations of 
the field. Find the difference between the sum of the hind- 
sights and the sum of the foresights. What should this 
difference equal? 



SECTION VIII 
THE PLANE TABLE 

Description and Uses. The plane table, an approved form of 
which is shown in Fig. 23, consists mainly of a drawing 
board made of well-seasoned wood, arranged in sections to pre- 
vent warping, and supported at a convenient height by a tripod 
and leveling head, with attachments for horizontal movement. 



FIELD INSTRUMENTS 



41 




FIG. 23. THE PLANE TABLE 



42 SURVEYING 

The board is provided with rollers or clamps or both, for 
keeping the paper secure and even. The plumbing arm has 
its end brought to a point which, however placed on the 
paper, is directly above the corresponding point on the ground 
determined by the plummet. The alidade is a ruler of brass 
or steel supporting a telescope with stadia or sight standards, 
whose line of sight is in or parallel to the same vertical plane 
with the beveled edge of the ruler. A compass with two 
spirit levels serves both to level the table and, when applied 
by the edges parallel to the zero, line of the compass circle, 
to determine the magnetic bearing of the lines drawn on the 
paper, or the direction of the table itself. 

After the principal lines of a survey have been determined 
and plotted, the details of the plot may be filled in by means 
of the plane table ; or, when a plot only of a tract of land is 
desired and extreme accuracy is not required, this instrument 
affords the most expeditious means of obtaining it. There is 
little use for it outside of the United States Coast and Geodetic 
Survey and the United States Geological Survey. 

To orient the Table. This operation consists in placing the 
table so that the lines of the plot shall be parallel to the 
corresponding lines on the ground. 

This may be accomplished approximately by turning the 
table until the needle of the compass indicates the same bear- 
ing as at a previous station, the edge of the compass coinciding 
with the same line on the paper at both stations. 

If, however, the line connecting the station at which the 
instrument is placed with another station is already plotted, 
the table may be placed in position accurately by placing it 
over the station so that the plotted line is by estimation over 
and in the direction of the line on the ground ; then making 
the edge of the ruler coincide with the plotted line, and turn- 
ing the board until the line of sight bisects the signal at the 
other end of the line on the ground. 



FIELD INSTRUMENTS 



43 




To plot any Point. Let ab on the paper represent the line 
AB on the ground; it is required to plot c, representing C on 
the ground. 

1. By intersection. 

Place the table in position at A (Fig. 24), plumbing a over A, and 
making the fiducial edge of the 
ruler pass through a; turn the ]$ 

alidade about a until the line of / V N 

sight bisects the signal at C, and j X 

draw a line along the fiducial edge 
of the ruler. Place the table in 
position at 5, plumbing b over B, 
and repeat the operation just 
described. Then c is the intersec- 
tion of the two lines thus drawn. 

2. By resection. 

Place the table in position at A (Fig. 25), and draw a line in the direc- 
tion of (7, as in the former case ; then remove the instrument to C, place 

it in position by the line drawn 
from a, make the edge of the 
ruler pass through &, and turn 
the alidade about b until B is in 
the line of sight. A line drawn 
along the edge of the ruler will 
intersect the line from a in c. 

3. By radiation. 

Place the table in position at A 
(Fig. 26), and draw a line from a 
toward C, as in the former cases. 
Measure AC, and lay off ac to 
the same scale as ab. 

To plot a Field AB CD 

By radiation. 

Set up the table at any point 
P, and mark p on the paper over 
FIG. 26 P. Draw indefinite lines from p 




44 



SURVEYING 



toward J., B, C, Measure PA, PJB, , and lay off pa, 
suitable scale, and join a and 6, 6 and c, c and d, 



to a 



By progression. 

Set up the table at A, and draw 
a line from a toward B. Measure 
AB, and plot ab to a suitable 
scale. Set up the table in position 
at B, and in like manner deter- 
mine and plot be ; and so on. 



FIG. 27 



By intersection. 

Plot one side as a base line. 

Plot the other corners by the method of intersection, and join these points 
in proper order by straight lines. 

By resection. 

Plot one side as a base line. Plot the other corners by the method of 
resection, and join these points in proper order by straight lines. 

The Three-Point Problem. Let A, B, C represent three field 
stations plotted as a, 5, c, respectively (Fig. 28) ; it is required 




FIG. 28 



to plot d representing a fourth field station D, from which A, 
B, and C are visible. 

Place the table over D, level and orient approximately by 
the compass. Determine d by resection as follows : Make the 



FIELD INSTRUMENTS 45 

edge of the ruler pass through a and lie in the direction a A, 
and draw a line along the edge of the ruler. In like manner, 
draw lines through b toward B and through c toward C. If 
the table is oriented perfectly, these lines meet at the required 
point d, but ordinarily they will form the triangle of error, ab, 
ac, be. In this case, through a, b, and ab ; a, c, and ac ; and 
b, c, and be, respectively, draw circles ; these circles will inter- 
sect in the required point d. For at the required point the 
sides ab, ac, be must subtend the same angle as at the points 
ab, ac, be, respectively. Hence, the required point d lies at 
the intersection of the three circles mentioned. The plane 
table may now be oriented accurately. 

The three-point problem may also be solved by fastening on the board 
a piece of tracing paper and marking the point d representing D, after 
which lines are drawn from d toward A, B, and C. The tracing paper 
is then moved until the lines thus drawn pass through a, 6, c. respectively, 
when by pricking through d the point is determined on the plot below. 
This method, however, is impracticable in case the wind blows. 



CHAPTER II 
OFFICE INSTRUMENTS 

SECTION IX 
PLOTTING INSTRUMENTS 

Definitions. A map is a representation by means of points, 
lines, and conventional signs on a plane surface, as on paper 
of a surveyed portion of the earth's surface, including objects 
upon it. If only the boundary lines are drawn, the repre- 
sentation is called an outline map, or plot. The plot is a 
figure similar to the original, and the ratio of a line of the 
field to the corresponding line of the plot is called the scale. 
In surveying it is customary to designate the scale as so many 
chains to the inch. 

Principal Minor Instruments. The principal minor instru- 
ments used in plotting are a ruler, pencil, straight-line pen, 
hair-spring dividers, compasses, a right triangle of wood or 
hard rubber, a T-square, and a parallel ruler. 

The Diagonal Scale. A portion of this scale is shown in 
Fig. 29. AB is the unit. AB and A'B' are divided into ten 
equal parts, and B is joined with h, the first division point to 
the left of B'; the first division point to the left of B is joined 
with the second to the left of B', and so on. The part of the 
horizontal line numbered 1 intercepted between BB' and Bh is 
evidently y 1 ^ of y 1 ^ = T ^ of the unit ; the part of the hori- 
zontal line numbered 2 intercepted between BB' and Bh is 

of the unit, and so on. 

46 



OFFICE INSTRUMENTS 



47 



The method of using this scale is as follows : 
Let it be required to lay off the distance 1.43. 



A' 



10 | 



h B' 



C' 




10 9 

A 



o 
D 

FIG. 29 



C 



Place one foot of the dividers at the intersection of the horizontal line 
numbered 3 and the diagonal numbered 4, and place the other foot at 
the intersection of the vertical line numbered 1 (CC") and the horizontal 
line numbered 3 ; the distance between the feet of the dividers will be 
the distance required. For, measuring along the horizontal line num- 
bered 3, from CC" to BB' is 1 ; from BB' to Bh is 0.03 ; and from Bh ta 
the diagonal numbered 4 is 0.4 ; and 1 + 0.03 + 0.4 = 1.43. 

The Circular Protractor. This instrument (Fig. 30) usually 
consists of a semicircular piece of brass or german silver, with 
its arc divided into degrees and its centre marked. 

Some protractors have an arm which carries a vernier, loy 
which angles may be constructed to single minutes. Still 
others embrace an entire circle and have several arms with 
verniers. 

A rectangular protractor, having the degrees marked off on 
three sides of a plane scale, is sometimes used. Often this 
form of the protractor is found on the reverse side of the 
diagonal scale. 



48 SURVEYING 

Constructions. 1. To lay off an angle with the circular 
protractor. Place the centre over the vertex of the angle, 
and make the diameter coincide with the given side of the 
angle. Mark off the number of degrees in the given angle, 
and draw a line through this point and the vertex. 




FIG. so 

2. To draw through a given point a line parallel to a given 
line with a right triangle and ruler. 

Make one of the sides of the triangle coincide with the 
given line, and, placing the ruler against one of the other sides, 
move the triangle along the ruler until the first side passes 
through the given point ; then draw a line along this side. 

3. To draw through a given point a line perpendicular to 
a given line with a right triangle and ruler. 

Make the hypotenuse of the right triangle coincide with 
the given line, and, placing a ruler against one of the other 
sides of the triangle, revolve the triangle about the vertex of 
the right angle as a centre until its other perpendicular side 
is against the ruler ; then move the triangle along the ruler 
until the hypotenuse passes through the given point, and 
draw a line along the hypotenuse. 



OFFICE INSTRUMENTS 49 

SECTION X 
COMPUTING INSTRUMENTS 

The Planimeter. This is an instrument for measuring the 
area of any irregular field, by applying it to a plot of the 
field drawn accurately to scale. The form in most common 
use is that known as the polar planimeter. The essential 
parts are two arms, one fixed in length, the other adjustable, 
and a rolling wheel mounted on an axis parallel to the adjust- 
able arm. The outer end of the arm of fixed length is made 
fast to the plot by means of a needle point, and the free end 
of the other arm is made to trace the perimeter of the figure 
to be measured. A disk records the area in the unit for 
which the instrument is set. 

The Slide Rule. This is an instrument for effecting the 
processes of multiplication, division, involution, and evolution 
by means of logarithms. It consists of a series of scales so 
arranged that by sliding one upon the other the addition or 
subtraction of logarithms is mechanically performed. For a 
full description of this labor-saving device in its various forms, 
the student is referred to some treatise on the subject. 



CHAPTER III 
LAND SURVEYING 

SECTION XI 
DEFINITIONS 

Land Surveying is the art of measuring, laying out, and 
dividing land, computing parts and areas from measured parts, 
and preparing a plot. An original survey includes laying out 
the boundary lines and establishing the corners. A resurvey 
is the retracing of old boundary lines and the finding of corner 
monuments, or the relocating of them when lost. 
Rules for Areas. The unit of land measure is the 

acre = 10 square chains = 4 roods 

= 160 square rods, perches, or poles. 

Areas are referred to the horizontal plane, no allowance 
being made for inequalities of surface. 

Let .4, B, and C be the angles of a triangle, and a, , and 
c the opposite bides, respectively, and let s = %(a + b -j- c). 
Area of triangle ABC base x altitude 
= |- be sin A 

a 2 sin B sin C 
~ 2 sin (B + C) 
= Vs (s a) (s b) (s c). 
Area of rectangle = base x altitude. 
Area of trapezoid = sum of parallel sides x altitude. 

NOTE. Spanish American units are in use in Texas, California, and 
Mexico. In this system the vara is the unit of length, which in Texas is 

50 



LAND SURVEYING 51 

reckoned 33 inches, in California 33 inches, in Mexico 32.9927 inches. 
The area of a square 1000 varas on a side is called a labor, and of a 
square 5000 varas on a side is called a league. 



SECTION XII 
SPECIAL METHODS OF SURVEYING, AND COMPUTING AREAS 

Triangular Fields. Measure, as may be most convenient, 
the three sides, two sides and the included angle, two angles 
and the included side, or a side and the altitude upon that 
side, and compute the area by the appropriate formula. 

Fields having More than Three Straight Sides. Divide the 
field into triangles and take the sum of the areas of the 
triangles. Or, run a diagonal and perpendiculars to it from 
the opposite vertices ; take the sum of the areas of the right 
triangles, rectangles, and trapezoids thus formed. 

A third method is as follows: Let ABCD (Fig. 31) repre- 
sent a field, and P and P' two stations within it. (They may 
be without the field.) Measure D 
PP' with great exactness. Meas- 
ure the angles between PP' and 
the lines from P and P' to the 
corners of the field. 

In the triangle P'PD, PP' and 
the angles PP'D and P'PD are 
known ; hence, PD may be found. 
In like manner, PC may be found. 
Then, in the triangle PDC, PD, 
PC, and the angle DPC are known ; hence, the area of PDC 
may be computed. In like manner, the areas of all the trian- 
gles about P or P' may be determined. 

Area ABCD = PAD + PDC + PCB + PEA', 
also, area ABCD = P'AD + P'DC + P'CB + P'BA. 




52 



SURVEYING 



Fields having Irregular Boundary Lines. Let A GBCD (Fig. 32) 
represent a field having a stream AEFGHKB as a boundary 
line. Run the line AB. From E, F, G, H, K, prominent 
points on the bank of the stream, let fall perpendiculars EE', 
FF', GG' 9 etc., upon AB. Regarding AE, EF, etc., as straight 





FIG. 32 



FIG. 33 



lines, the portion of the field cut off by AB is divided into 
right triangles, rectangles, and trapezoids, the necessary ele- 
ments of which can be measured and the areas computed. 
The sum of these areas added to the area of A BCD gives 
the area required. If the offsets are at regular intervals, then 
the area of the part cut off by AB may be found by adding the 
offsets and multiplying by the common distance between them. 

When the irregular boundary line crosses the straight line 
that joins its extremities, as in Fig. 33, the areas of EFH and 
HGB may be found separately, as in the preceding case. Then, 
the area required = ABCD + HGB AEFH. 

Rectangular System of Co-ordinates. Let XX' and YY' (Fig. 
34) be two fixed perpendicular lines intersecting at the point 0. 
Let the four parts into which these lines divide the plane be 
called Quadrants, as in Trigonometry, and be distinguished by 
naming them, respectively, first, second, third, 'and fourth 
quadrants. 

Suppose the position of a point is described by saying that 
its distance from YY', expressed in terms of some chosen unit 



LAND SURVEYING 53 

of length, is 3, and its distance from XX' is 4. Then there 
is in each quadrant one point and only one which will 
satisfy these conditions. The position of the point in each 
quadrant may be found by drawing parallels to YY' at the 
distance 3 from YY', and parallels -^ D 
to XX' at the distance 4 from XX' ; E \p 
then the intersections P lt P 2 , P 3 , 
and P 4 satisfy the given condi- 
tions. X- rr 



O 

In order to determine which ] 



IV 



-H 



one of the four points, P 1? P 2 , P 3 , 

P 4 , is meant, we adopt the rule O- 

that distances measured from YY 1 & Y/ 

to the right are positive; to the 

left, negative. Distances measured from XX' upward are 

positive ; downward, negative. Then, the position of P x will 

be denoted by + 3, + 4 ; of P 2 , by 3, + 4 ; of P 8 , by 3, 4; 

of P 4 , by -f 3, - 4. 

The fixed lines XX' and YY' are called the Axes of Co-ordi- 
nates ; XX' is called the Axis of Abscissas, or Axis of x ; YY', 
the Axis of Ordinates, or Axis of y. The intersection is 
called the Origin. 

The two distances (with signs prefixed) which determine 
the position of a point are called the Co-ordinates of the 
point ; the distance of the point from YY 1 is called its 
Abscissa ; and the distance from XX', its Ordinate. 

Abscissas are usually denoted by x, and. ordinates by y, 
and a point is represented algebraically by simply writing 
the values of its co-ordinates within parentheses, that of the 
abscissa being always written first. 

Thus, P! (Fig. 34) is the point (3, 4), P 2 the point 
(- 3, 4), P 3 the point (- 3, - 4), and P 4 the point (3, - 4). 
In general the point whose co-ordinates are x and y is the 
point (x, y). 



54 



SURVEYING 



This system of co-ordinates may be applied to the determi- 
nation of areas in the following manner : 

Suppose the field to be ABODE (Fig. 35). Lay out the two 
axes so that the field shall lie wholly within the first quad- 
rant. Then measure the co-ordinates of each of the vertices 




M 



N P 
FIG. 35 



R 



and designate them as follows : for A, (x^ 2/1) ; for B, (x 2 , 2/2) ; 
for Cj (x s , 2/3) ; for D, (aj 4 , y^ ; for E, (x 6 , y 5 ). Evidently each 
of these co-ordinates is positive. Then, 

area ABODE = area LABM -f area MB OP + area PCDR 

- area NEDR area LAEN-, 

or, in terms of the co-ordinates, 

area ABODE = (2/1 + 2/2) (x 9 - x,} + (y a + 7/ 3 ) (x s - x 2 ) 



+ #4 (2/3 - 2/5) + #5(2/4 - 2/l) ? 

This method can be put in the form of a general rule : 
Take one-half the algebraic sum of the products obtained 
by multiplying the abscissa of each vertex by the difference 
between the ordinates of the two adjacent vertices, taken in 
the clockwise order. 



LAND SURVEYING 



55 



EXERCISE V 

1. Kequired the area of a triangular field whose sides are 
13 chains, 14 chains, and 15 chains. 

2. Required the area of a triangular field if it has two 
angles 48 30' and 71 45', and the included side 20 chains. 

3. Required the area of a triangular field whose base is 
12.60 chains, and altitude 6.40 chains. 

4. Required the area of a triangular field which has two sides 
4.50 chains and 3.70 chains, and the included angle 60. 

5. Required the area of a field in the form of a trapezium, 
one of whose diagonals is 9 chains, and the two perpendicu- 
lars upon this diagonal from the oppo- 
site vertices 4. 50 chains and 3. 25 chains. 

6. Required the area of the field 
ABCDEF (Fig. 36), if 

AE= 9.25 chains, FF* = 6.40 chains, B 
BE = 13.75 chains, DD' = 1 chains, 
DB = 10 chains, CC' = 4 chains, 

and A A' 4.75 chains. FIG. 36 




P'PD = 165 40', 
P'PC = 303 15'. 



7. Determine the area of the field ABCD from two interior 
stations P and P', if PP' = 1.50 chains, 

PP'C= 89 35', PP'D = 349 45', 
PP'B = 185 30', P'PB = 3 35', 
PP'A = 309 15', P'PA = 113 45', 

8. Required the area of the field 
ABCDEF (Fig. 37), if 

AF' = 4 chains, FF' = 6 chains, A> 
EE' = 6.50 chains, AE' = 9 chains, 
AD = 14 chains, AC' = 10 chains, 
AB' = 6.50 chains, BB' = 7 chains, 
CC' = 6.75 chains. 




56 SURVEYING 

9. Required the area of the field AGBCD (Fig. 32, p. 52), 
if the diagonal A C = 5, BB' (the perpendicular from B to A C) 
= 1, DD' (the perpendicular from D to A C) = 1.60, EE' = 
0.25, FF' = 0.25, GG' = 0.60, HH' = 0.52, M'= 0.54, AE' = 
0.2, F'F' = 0.50, F'G 1 = 0.45, G7T = 0.45, #'#' = 0.60, and 
K'B = 0.40. 

10. Required the area of the field AGBCD (Fig. 33, p. 52), 
if AD = 3, AC = 5, AB = 6, angle DAC = 45, angle BAG = 
30, ylF' = 0.75, AF' = 2.25, AH = 2.53, 4' = 3.15, EE' = 
0.60, FF' = 0.40, and GG 1 = 0.75. 

11. Determine the area of the field A BCD from two exterior 
stations P and P', if PP' = 1.50 chains, 

P'P - 41 10', P'P> = 104 45', PP'B = 132 15', 
P'PA = 55 45', PP'D = 66 45', PP'A = 103 0'. 
p'PC = 77 20', PP'C= 95 40', 

12. Find the area of the field ABODE (Fig. 35, p. 54), if 
the co-ordinates, in chains, of the vertices taken in order are 
(1.40, 6.75), (4.60, 8.32), (9.00, 9.05), (12.15, 5.58), and 
(5.27, 1.16). 

13. Find the area of the field ABCDE (Fig. 35, p. 54), by 
measuring distances as follows : 

AL = 400 feet ; EM = 700 feet ; CP = 680 feet ; 
DR = 380 feet ; EN = 200 feet; LM = 150 feet ; 
MN = 250 feet; NP = 200 feet ; PR = 220 feet. 

14. Lay out a field of four sides, and find its area by the 
method of triangles and also by the method of rectangular 
co-ordinates. 

15. Lay out a field of six sides, and find its area by the 
method of triangles and also by the method of rectangular 
co-ordinates. 



LAND SURVEYING 57 

SECTION XIII 
GENERAL METHOD FOR FARM SURVEYS 

Definitions. A course is the bearing and length of a 
line. The latitude of a course is the distance between the 
parallels through its extremities, and is called a northing or a 
southing, as the course is northward or southward. The 
departure of a course is the distance between the meridians 
through its extremities, and is called an easting or a westing, 
as the course is eastward or westward. The meridian dis- 
tance of a point is its distance from a meridian. The double 
meridian distance of a course is double the meridian distance of 
its mid-point, and therefore equal to the sum of the meridian 
distances of the extremities of the 
course. 

Let AB (Fig. 38) represent a line, 
whose bearing and length are known. 
Let MN be a reference meridian ; 
and let p and p' be parallels through 
A and B, and m and m' meridians 
through* the same points. Then, 
angle mAB represents the bearing 

of line AB. The latitude of the course AB is AE, and its 
departure EB. The meridian distance of the point B is BC 
and of A, AD. Evidently, the double meridian distance of 
the course AB is (BC + AD). 

Again, in the triangle AEB, 

AE = AB x cos EAB, and EB = AB x sin EAB. 
Hence, latitude = distance x cos of bearing, and departure = 
distance x sin of bearing. From these formulas, the latitude 
and departure of any course may be found by means of a 
table of natural sines and cosines. They may be found also 




58 



SURVEYING 



from the Traverse Table, which is merely the tabulated results 
of the foregoing method for given courses. 

Field Notes. The field notes are kept in a book provided 
for the purpose. The page is commonly ruled in three col- 
umns, in the first of which is written the number of the 
station ; in the second, the bearing of the side ; and in the 
third, the length of the side. 

FIELD NOTES 

N 



E 




I 


N. 20 E. 


8.66 


2 


S. 70 E. 


5.00 


3 


S. 10 E. 


10.00 


4 


N. 70 W. 


10.00 



FIG. 39 



To obtain the field notes, say of field 
ABCD (Fig. 39), place the compass 
at A, the first station, and take the 
bearing of AB (p. 12) ; suppose it to 
be N. 20 E. Write the result in 
the second column of the field notes 
opposite the number of the station. 
Measure AB = 8.66 chains, and write 
the result in the third column of the 
field notes. Place the compass at J5, 
and, after testing the bearing of AB (p. 13), take the bear- 
ing of BC, measure BC, and write the results in the field 
notes ; and so continue until the bearing and length of each 
side have been recorded. 

Computation of the Area. The survey may begin at any 
corner of the field ; but, for computing the area, the field notes 
should be arranged so that the most eastern or the most western 
station shall stand first. For the sake of uniformity, we shall 
always begin with the most western station and keep the field 
on the right in passing around it. 



LAND SURVEYING 



59 



The field notes occupy the first three of the eleven columns in the tablet 
below. Columns IV, V, VI, and VII contain the latitudes and departures 
corresponding to the sides, taken from the Traverse Table. The line 
represented by each number is indicated immediately above that number. 
Column VIII contains the meridian distances of the points B, C, D, and 
A, taken in order. Column IX contains the double meridian distances 



I 


II 


Ill 


IV 


V 


VI 


VII 


VIII 


IX 


X 


XI 


SIDE 


BEARING 


DlST. 


N. 


S. 


E. 


W. 


M.D. 


D.M.D. 


N.A. 


S.A. 


AB 


N.20E. 


8.66 


AB' 
8.14 




BB' 

2.96 




BB' 

2.96 


BB' 

2.96 


2 ABB 1 
24.0944 




BC 


S. 70 E. 


'5.00 




B'C' 
1.71 


C"C 
4.70 




CC' 
7.66 


BB' + CC' 
10.62 




2 C'CBB' 
18.1602 


CD 


S. 10 E. 


10.00 




CD' 
9.85 


D"D 

1.74 




DD' 
9.40 


CC' + DD' 
17.06 




1 D'DCC 
168.0410 


DA 


N.70W. 


10.00 


D'A 
3.42 






DD' 

9.40 





DD' 
9.40 


2 ADD 

32.1480 


.... 






33.66 


11.56 


11.56 


9.40 


9.40 






56.2424 


186.2012 



(186.2012 sq. ch. - 56.2424 sq. ch.) -=- 2 = 64.98 sq. ch. = 6.50 acres. 

of the courses. Their composition is indicated by the letters immedi- 
ately above the numbers. Column X contains the products of the double 
meridian distances by the northings in the same line. The first number, 
24.0944 = 2.96 x 8.14 = BB' x AB' = twice area of triangle ABB' 
32.1480 = 9.40 x 3.42 = DD' x AD' = twice area of triangle ADD'. 
Column XI contains the products of the double meridian distances by 
the southings in the same line. The first number, 

18.1602 = 10.62 x 1.71 = (BB' + CC') x B'C' 

twice area of trapezoid C'CBB' 
168.0410 = 17.06 x 9.85 = (CC' + DD') x D'C' 

= twice area of trapezoid D'DCC'. 
The sum of the north areas in column X 

= 56.2424 = 2 (ABB' + ADD'). 
The sum of the south areas in column XI 

= 186.2012 = 2 (C'CBB' + D'DCC'). 

But (C'CBB' + D'DCC') - (ABB' + ADD') = ABCD. 

Hence, 2 (C'CBB' + D'DCC') - 2 (ABB' + ADD') = 2 ABCD ; 
that is, 186.2012 - 56.2424 = 129.9588 = 2 ABCD. 

Hence, area ABCD = of 129.9588 sq. ch. = 64.98 sq. ch. = 6.50 A. 



60 SURVEYING 

Balancing the Work. In the survey, we pass entirely around 
the field ; hence, we move just as far north as south. There- 
fore, the sum of the northings should equal the sum of the 
southings. In like manner, the sum of the eastings should 
equal the sum of the westings. In this way the accuracy of 
the field work may be tested. 

In the example on page 59 the sum of the northings is 
equal to the sum of the southings, being 11.56 in each case ; and 
the sum of the eastings is equal to the sum of the westings, 
being 9.40 in each case. Hence, the work balances. 

In actual practice the work seldom balances. When it 
does not balance, corrections are generally applied to the 
latitudes and departures by the following rules : 

1. The perimeter of a field is to any one side as the total 
error in latitude is to the correction required. 

2. The perimeter of a field is to any one side as the total 
error in departure is to the correction required. 

EXAMPLE. The perimeter of a field measured 306.62 chains 
and one side 72.47 chains, with a total error of 22 links in 
latitude and of 18 links in departure. 

Then 306.62 : 72.47 = 22 links : x = 18 links : y. 

Whence x 5 links and y = 4 links. 

Hence the correction in latitude applied to the given side is 
0.05 chains, and the correction in departure is 0.04 chains. 

If special difficulty was found in taking a particular bear- 
ing, or in measuring a particular line, the corrections should 
be applied to the corresponding latitudes and departures. 

The amount of error allowable varies in the practice of dif- 
ferent surveyors, and according to the nature of the ground. 
An error of 1 link in 8 chains would not be considered too great 
on smooth, level ground ; while on rough ground an error of 
1 link in 3 chains might be allowed. If the error is consider- 
able, the field measurements should be repeated. 



LAND SURVEYING 



61 



As another example let it be required to find the area of 
field ABCDEF from the following 



FIELD NOTES 



1 


N. 73 30' W. 


5.00 


2 


S. 10 30' W. 


5.00 


3 


N. 28 30' W. 


7.07 


4 


N. 20 00' E. 


11.18 


5 


S. 43 30' E. 


5.00 


6 


S. 13 30' E. 


10.00 



SIDE 


BEARING 


DlST. 


N. 


S. 


E. 


W. 


M.D. 


D.M.D. 


N.A. 


S.A. 


AE 


N. 2000'E. 


11 18 


10.51 




3.82 




B'B 
382 


B'B 

3 82 


1ABB' 

40.1482 




BC 
CD 


S.4330'E. 
S 13 30' E 


5.00 
10 00 




3.63 
9 72 


3.44 
2 33 


... 


crc 

7.26 
D'D 
9 59 


B'B+ C'C 
11.08 
C'C + D'D 

16 85 




2 C'CBB' 
40.2204 
ID'DCC 
163 780 


DE 


N. 7330'W 


5.00 


1 42 






4.79 


E'E 
4.79 


D'D + E'E 
14.38 


2 D'DEE' 
20.4196 




EF 
FA 


S.1G30'W. 
N 2830'W 


5.00 
707 


6 21 


4.79 


... 


4.80 
1.42 

3 37 


F'F 
3.37 

00 


E'E + F'F 
8.16 
F'F 
3 37 


2AFF' 
20 9277 


2 F'FEE' 
39.0864 




























43.25 


18.14 


18.14 


9.59 


9.58 
9.59 






81.4955 


243.0888 



43.25 : 5 = 0.01 : x. Area = 8.08 acres. 

The first station in the field notes is Z), but we rearrange the numbers 
in the tablet so that A stands first. The northings and southings balance, 
but the eastings exceed the westings by 1 link. We apply the correction 
to the westing 4.79 (the distance DE being in doubt), making it 4.80, and 
write the correction. In practice, the corrected numbers are written in 
red ink, and often all the latitudes and departures are rewritten in four 
additional columns, headed, respectively, N', S', E', W'. 

Supplying Omissions. If for any reason the bearing and the 
length of any side do not appear in the field notes, the lati- 
tude and departure of this side may be found in the following 
manner : 



62 



SURVEYING 



Find the latitudes and departures of the other sides as 
usual. The difference between the northings and southings 
gives the northing or southing of the unknown side, and the 
difference between the eastings and westings gives the easting 
or westing of the unknown side. 

If the length and the bearing of the unknown side are de- 
sired, they may be found by solving the right triangle, whose 
sides are the latitude and departure found by the method just 
explained, and whose hypotenuse is the length required. 

Obstructions. If the end of a line is not visible from 
its beginning, or if the line is inaccessible, its length and 
bearing may be found as follows : 

By means of a random line (p. 8). 

When it is impossible to run a random line, which is fre- 
quently the case on account of the extent of the obstruction, 
the following method may be used : 



N 




Let AB (Fig. 40) represent an inaccessible line 
whose extremities A and B only are known, and B 
invisible from A. 

Set flagstaffs at convenient points, C and D. 
Find the bearings and lengths of AC, CD, and DB, 
and then proceed to find the latitude and departure 
of AB. 



FIG. 40 



EXAMPLE. 
lowing notes (see Fig. 40) : 



Suppose that we have the f ol- 



SIDE 


BEARING 


DlST. 


N. 


S. 


E. 


w. 


AC 


S. 45 E. 


3.00 




2.12 


2.12 




CD 


E. 


3.50 






3.50 




DB 


N. 30 E. 


4.83 


4.18 




2.42 






4.18 


2.12 


8.04 






LAND SURVEYING 



63 



The northing of AB is AE = 2.06, and the easting, EB = 8.04. These 
numbers may be entered in the tablet in the columns N. and E., opposite 
the side AB. 

If the bearing and length of AB are required, 

tan BAE = ?^_ = ^ = 3.903. 
AE 2.06 

Hence, the angle BAE = 75 38'. 

Also, 



AB = ^AE + ~BE* = V8.04 2 + 2.06* = 8.30. 
Therefore, the bearing and length of AB are N. 75 38' E. and 8.30. 

To make a Plot. A plot or map may be drawn to any 
desired scale. If a line 1 inch in length in the plot represents 
a line 1 chain in length, the plot is 
said to be drawn to a scale of 1 chain 
to an inch. In this case (Fig. 41) the 
plot is drawn to a scale of 8 chains 
to an inch. 

Draw the line NAS to represent 
the meridian, and lay off the first 
northing ^' = 8.14. Through B 
draw an indefinite line perpendicular 
to NS and lay off B'B, the first easting, 
= 2.96. Draw AB then the line AB 
represents the first side of the field. 
Through B draw BC" perpendicular 
to BB', and make BC" = 1.71, the first 
southing. Through C" draw C"C 
perpendicular to BC", and equal to 
4.70, the second easting. Draw BC. 
The line BC represents the- second 
side of the field. Proceed in like 
manner until the field is completely 
represented. The extremity of the 
last line F'A, measured from F', should fall at A. 
test of the accuracy of the plot. 




F F 



FIG. 41 



This is a 



64 



SURVEYING 



By drawing AC, AE, and EC, the hexagonal figure 
ABCDEFA is divided into triangles, the bases and altitudes 
of which may be measured and the area computed approxi- 
mately. 

Another method is as follows : 
Draw MN (Fig. 42) to represent a 
meridian. Let the point A in this 
line be taken as the first station in 
the rearranged field notes of page 
61. With the circular protractor 
mark off each of the bearings as I, c, 
dj e, fj and a. Draw AE to scale 
through b. With triangle and ruler 
(p. 48) or with parallel ruler draw to 
scale EC parallel to Ac ; and so on. 

After some practice, still other 
methods will be suggested, but the 
methods given are among the best. 




EXERCISE VI 

Find the areas of the following and make a plot of each. 
In 3 and 7, detours were made on account of obstructions 
(p. 62). The notes of the detours are written in braces. 

123 



STA. 


BEARINGS 


DlST. 


1 


S. 75 E. 


6.00 


2 


S. 15 E. 


4.00 


3 


S. 75 W. 


6.93 


4 


N. 45 E. 


5.00 


5 


N. 45 W. 


5.19* 



STA. 


BEARINGS 


DlST. 


1 


N. 45 E. 


10.00 


2 


S. 75 E. 


11.55 


3 


S. 15 W. 


18.21 


4 


N. 45 W. 


19.11 



STA. 


BEARINGS 


DlST. 


1 


S. 215 / E. 


9.68 


J 


N. 5145'W. 
S. 8500'W. 


2.39 
6.47 


[ 


S. 5510 / W. 


1.62 


3 


N. 345'E. 


6.39 


4 


S. 6645'E. 


1.70 


5 


N. 1500'E. 


4.98 


6 


S. 8245'E. 


6.03 



LAND SURVEYING 



STA. 


BEARINGS 


DlST. 


1 


N. 530'W. 


6.08 


2 


S. 8230 / W. 


6.51 


3 


S. 300'E. 


5.33 


4 


E. 


6.72 



STA. 


BEARINGS 


DlST. 


1 


N. 615'W. 


6.31 


2 


S. 8150'W. 


4.06 


3 


S. 500'E. 


5.86 


4 


N. 8830'E. 


4.12 



STA. 


BEARINGS 


DlST. 


1 


N. 2000'E. 


4.62i 


2 


N. 7300'E. 


4.16| 


3 


S. 4515'E. 


6.18i 


4 


S. 3830'W. 


8.00 


5 


Wanting 


Wanting 



STA. 


BEARINGS 


DlST. 


, r 


S. 8120'W. 


4.28 


M 


N. 7630'W. 


2.67 


2 


N. 500'E. 


8.68 


3 


S. 8730 / E. 


5.54 


r 


S. 700'E. 


1.79 


4 


S. 2700'E. 


1.94 


| 


S. 1030'E. 


5.35 


I 


N. 7645 / W. 


1.70 



STA. 


BEARINGS 


DlST. 


1 


N. 8945'E. 


4.94 


2 


S. 700 / W. 


2.30 


3 


S. 2800 / E. 


1.52 


4 


S. 045'E. 


2.57 


5 


N. 8445 / W. 


5.11 


6 


N. 230'W. 


5.79 



9. An Ohio farm is bounded and described as follows : 
Beginning at the southwest corner of lot No. 13, thence N. 1^ 
E. 132 rods and 23 links to a stake in the west boundary 
line of said lot ; thence S. 89 E. 32 rods and 15 T 4 o links to 
a stake 5 thence N. 1 J E. 29 rods and 15 links to a stake 
in the north boundary line of said lot ; thence S. 89 E. 61 
rods and 18/o links to a stake; thence S. 32 W. 54 rods 
to a stake; thence S. 35 E. 22 rods and 4 links to a 
stake ; thence S. 48 E. 33 rods and 2 links to a stake ; 
thence S. 7 W. 76 rods and 20 links to a stake in the south 
boundary line of said lot ; thence N. 89 W. 96 rods and 10 
links to the place of beginning. Containing 85.87 acres, more 
or less. 

Verify the area given and plot the farm. 



66 SURVEYING 

Modification of the Latitude and Departure Method. The area 
of a field may be found by a modification of the latitude and 
departure method, if its sides and interior angles are known. 

Let A, B, Cj D represent the interior angles of the field 
ABCD (Fig. 43). Let the side AB determine the direction of 
reference. The bearing of AB, with reference to AB, is 0. 
The bearing of BC, with reference to AB, is the angle 
b == 180 - B. The bearing of CD, with reference to AB, is 
the angle c = C b. The bearing of DA, with reference 

to AB, is the angle d = A. 

The area may now be com- 
puted by the latitude and depar- 
ture method, regarding AB as the 
meridian. 

In practice, the exterior angles, 
when acute, are usually measured. 
As the interior angles may be 
measured with considerable accu- 
racy by the transit, the latitudes 

and departures should be obtained by using a table of natural 
sines and cosines. 

EXERCISE VII 

1. Find the area of the field ABCD, in which the angle 
A = 120, B = 60, C = 150, and D = 30 ; and the side 
AB = 4 chains, BC = 4 chains, CD = 6.928 chains, and DA = 
8 chains. 

Keep three decimal places, and use the Traverse Table. 

2. Find the area of the farm ABCDE, in which the angle 
A = 106 19', B = 99 40', C = 120 20', D = 86 8', and E = 
127 33'; and the side AB = 79.86 rods, BC = 121.13 rods, 
CD = 90 rods, DE = 100.65 rods, and EA = 100 rods. 

Use the table of natural sines and cosines, keeping two decimal places 
in the results. 




LAND SURVEYING 67 

General Remarks on determining Areas. Operations depend- 
ing upon the reading of the magnetic needle must lack 
accuracy. Hence, when great accuracy is required (which is 
seldom the case in land surveying) the method of pp. 58-61 
cannot be employed. 

The best results are obtained by the methods explained on 
pp. 51-54 and 66, the horizontal angles being measured with 
the transit, and great care exercised in measuring the lines. 



SECTION XIV 
LOCATION SURVEYS 

Definition. In surveying proper we measure lines and 
angles as we find them, while in location surveys we mark them 
out on -the ground where they are required to be in order to 
inclose a given area, or conform to a specified shape, or meet 
some other given condition. Laying out, parting off, and 
dividing up land are included in this class of surveys. The 
surveyor must, for the most part, depend on his general knowl- 
edge of Geometry and Trigonometry, and his own ingenuity, 
for the solutions of problems that arise in location surveys. 

Illustrative Problems. PROBLEM 1. To divide a trian- 
gular field into two parts having a given 
ratio, by a line through a given vertex. 

Let ABC (Fig. 44) be the triangle, and A the 
given vertex. 

T> T\ 

Divide BC at Z>, so that equals the given 

X/(_/ 

ratio, and draw AD. ABD and ADC are the parts fc ________. _______ 

required; for D 

ABD : ADC = BD : DC. - ** 

PROBLEM 2. To cut off from a triangular field a given 
area, by a line parallel to the base. 




68 



SURVEYING 




Let ABC (Fig. 45) be the triangle, and 
let DE be the division line required. 

Then ABC : A DE = AS? : ZZ> 2 . 

" VABC : VADE = AB . AD. . 



.-. AD = 



ADE 

ABC' 



PROBLEM 3. To cut off from a 
FIG. 45 triangular field a given fraction of 

the field, by a line from a given point in a side. 

Let ABC (Fig. 46) be the triangle, and P the point from which the 
line PD is to be located so as to cut off, say, one-third the area of the 
triangle. 

AD = AB x AC + SAP. 

For ABC:APD = AB x AC : AP x AD = 3:1. 





FIG. 4G 

PROBLEM 4. To divide any field into two parts having a 
given ratio, by a line through a given point in the perimeter. 

Let ABCDE (Fig. 47) represent the field, P the given point, and PQ 
the required division line. 

The areas of the whole field and of the required parts having been 
determined, run the line PD from P to a corner D, dividing the field, 
approximately, as required. Determine the area PBCD. 

The triangle PDQ represents the part which must be added to PBCD 
to make the required division. 



LAND SURVEYING 



69 



Hence, 



NOTE. 



Area PDQ = i x PD x DQ x sin PDQ. 

2 x 



PD x sin PDQ 



2 x area PDQ 



This perpendicular from 



perpendicular from P on DE 
P on DE may be run and measured directly. 

PROBLEM 5. To divide a field into a given number of parts, 
so that access to a pond of water is 
given to each. 

Let ABODE (Fig. 48) represent the field, 
and P the pond. Let it be required to divide 
the field into four parts. Find the area of 
the field and of each part. 

Let AP be one division line. Run PE, 
and find the area APE. Take the differ- 
ence between APE and the area of one 
of the required parts ; this gives the area of 
the triangle PQ#, from which QE may be 
found, as in Problem 4. Draw PQ; PAQ is 
one of the required parts. In like manner, 
PQR and PAS are determined; whence, 
PSR must be the fourth part required. 




E 



EXERCISE Vm 

1. From the square A BCD, containing 6 acres 1 rood 24 
perches, part off 3 acres by a line EF parallel to AB. 

2. From the rectangle ABCD, containing 8 acres 1 rood 
24 perches, part off 2 acres 1 rood 32 perches by a line EF 
parallel to AD which is equal to 7 chains. Then, from the 
remainder of the rectangle, part off 2 acres 3 roods 25 perches, 
by a line GH parallel to EB. 

3. Part off 6 acres 3 roods 12 perches from a rectangle ABCD, 
containing 15 acres, by a line EF parallel to AB ; AD being 
10 chains. 



70 SURVEYING 

4. From a square ABCD, whose side is 9 chains, part off a 
triangle which shall contain 2 acres 1 rood 36 perches, by a 
line BE drawn from B to the side AD. 

5. From ABCD, representing the rectangle, whose length is 
12.65 chains, and breadth 7.58 chains, part off a trapezoid 
which shall contain 7 acres 3 roods 24 perches, by a line BE 
drawn from B to the side DC. 

6. In the triangle ABC, AB = 12 chains, AC = 10 chains, 
and BC = 8 chains ; part off a trapezoid of 1 acre 2 roods 16 
perches, by the line DE parallel to AB. 

1. In the triangle ABC, AB = 26 chains, AC = 20 chains, 
and BC = 16 chains ; part off a trapezoid of 6 acres 1 rood 24 
perches, by the line DE parallel to AB. 

8. It is required to divide the triangular field ABC among 
three persons whose claims are as the numbers 2, 3, and 5, so 
that they may all have the use of a watering place at C; AB 
= 10 chains, AC = 6.85 chains, and CB = 6.10 chains. 

9. Divide the five-sided field ABCHE among three persons, 
X, Y, and Z, in proportion to their claims, X paying $500, Y 
paying $750, and Z paying $1000, so that each may have the 
use of an interior pond at P, the quality of the land being 
equal throughout. Given AB = 8.64 chains, BC = 8.27 chains, 
CH = 8.06 chains, HE = 6.82 chains, and EA = 9.90 chains. 
The perpendicular PD upon AB = 5.60 chains, PD 1 - upon BC 
= 6.08 chains, PD" upon CH = 4.80 chains, PD'" upon HE 
= 5.44 chains, and PD"" upon EA = 5.40 chains. Assume 
PH as the divisional fence between the shares of X and Z, it 
is required to determine the position of the fences PM and PN 
between the shares of X and Y and between the shares of Y 
and Z. 

10. Divide the triangular field ABC, whose sides AB, AC, 
and BC are 15, 12, and 10 chains, respectively, into three equal 



LAND SURVEYING 71 

parts, by fences EG and DF parallel to BC, without finding 
the area of the field. 

11. Divide the triangular field ABC, whose sides AB, BC, 
and AC are 22, 17, and 15 chains, respectively, among three 
persons, A, B, and C, by fences parallel to the base AB, so that 
A may have 3 acres above the line AB, B 4 acres above A's 
share, and C the remainder. 



SECTION XV 
LAYING OUT THE PUBLIC LANDS 

Reference Lines. The public lands north of the Ohio Eiver 
and west of the Mississippi are generally laid out in accord- 
ance with what is known as the rectangular system of 
surveying. First, an initial point is selected with great care, 
and then astronomically established. Through this point a 
principal meridian, or true north and south line, is run by 
means of the solar compass, or the transit with observations 
on Polaris ; and also an east and west line, called a base line. 
Crossing the principal meridian at intervals of 24 miles, both 
north and south of the initial point, are run other east 
and west lines, called standard parallels, or correction lines. 
Northward from the base line and from each of the standard 
parallels, at intervals of 24 miles, both ways from the princi- 
pal meridian, are run true north and south lines, called guide 
meridians. Thus, the land is divided into blocks approxi- 
mately 24 miles square. Six principal meridians have been 
established, in addition to which and connected with wfyich 
there are twenty or more independent meridians in the 
western states and territories. 

Division from Reference Lines; Townships. Within each 
block parallels to the base line, or to a standard parallel, are 
run at intervals of 6 miles. These are called township lines. 



72 



SURVEYING 



At the same intervals are also run north and south lines, 
called range lines. Thus, the tract would be divided into 
townships exactly 6 miles square if it were not for the con- 
vergence of the meridians on account of the curvature of the 
earth. An east and west series of townships is called a tier, 
and a north and south series is called a range. A township 
is designated by giving the number of the tier north or south 
of the base line and the number of the range east or west of 



TF- 



D- 


M' 


1 


O 


i\k A i\ M 


T 












t" 








*o" 






B 






t 1 








p' 




. 













A 




p 














r 


r' 


r'' 


\ 


G 








u 








G 




































D 






C 

















-E 



FIG. 49 

the principal meridian. Thus, T. 3 N., R. 2 W., read town- 
ship three north, range two west, means that the township is 
in the third tier north of the base line, and in the second tier 
west of the principal meridian. 

Let NS (Fig. 49) represent a principal meridian ; WE a base 
line; DL and D'L' standard parallels; GM and G'M 1 guide 



LAND SURVEYING 



73 



meridians ; rl, r'l', . . . , range lines ; tp t t'p', . . . , township 
lines. If Or is taken as 6 miles, then O'l will be less than 
6 miles. O'k being equal to 6 miles and O'l being less, 
it will be observed that there will be offsets on the base line 
and on standard parallels at intervals of 6 miles. 

Township A would be designated thus : T. 2 X., R. 3 E. 
How would townships B and C be designated? 

Subdivision of Townships. The townships are divided into 
sections approximately 1 mile square, and the sections are 
divided into quarter sections. The 
township, section, and quarter-section 
corners are permanently marked. The 
sections are numbered, beginning at 
the northeast corner, as in Fig. 50, 
which represents a township divided 
into sections. The quarter sections are 
designated, according to their position, 
as N.E., N.W., S.E., and S.W. Section 
lines are surveyed in such an order as 

to throw the errors on the northwest quarter sections, which 
are carefully measured and their areas calculated. 

Meander Lines. If in running a line a navigable stream 
or a lake more than 1 mile in length is encountered, it is 
meandered by marking the intersection of the line with the 
bank and running lines from this point along the bank to 
prominent points which are marked, and the lengths and 
bearings of the connecting lines recorded. 

Manual. For detail of methods, see the "Manual of Sur- 
veying Instructions," issued by .the Commissioner of the 
General Land Office, at Washington, D.C., for the use of 
Surveyors-General. 



G 


5 


4 


3 


2 


1 


7 


8 


9 


10 


It 


12 


18 


17 


16 


15 


14 


13 


19 


20 


21 


22 


23 


24 


30 


29 


28 


27 


26 


25 


31 


32 


33 


34 


35 


36 



FIG. so 



CHAPTER IV 



TRIANGULATION 

SECTION XVI 

DEFINITIONS 

The third method of surveying explained on pa^; 51 is 
an example of triangulation on a small scale. The simple 
principle there involved is elaborately worked out in hydro- 
graphic or topographic- surveys, or in the measurement of 
terrestrial arcs, as in the " Transcontinental Triangulation 
and American Arc of the Parallel," recently completed by 
the United States Coast and Geodetic Survey. 

Let F (Fig. 51) represent a point whose position with 
reference to the base line AB is required. Connect AB with 
F by the series of triangles Q 

ABC, A CD, ADE, and DEF, 
so that a signal at C is visible 
from A and B, a signal at D 
visible from A and C, a signal 
at E visible from A and Z>, 
and a signal at F visible from 
D and E. In the triangle AB C, 
the side AB is known, and the angles at A and B may be 
measured; hence, AC may be computed. In the triangle 
ACD, AC is known, and the angles at A and C may be 
measured ; hence, AD may be computed. In like manner, 
DE and EF or DF may be determined. DF, or some suitable 
line connected with DF, may be measured, and this result 

74 




FIG. 51 



TRIANGULATION 75 

compared with the computed value to test the accuracy of 
the field measurement. This net or chain of triangles enables 
us to determine the relative position of all the points with 
respect to each other. If the point A is, furthermore, 
astronomically located, and the azimuth of line AB is known, 
then we have sufficient data also to determine the absolute 
geographical position of each of the points. 

Classification. Three orders of triangulation are recognized, 
viz. : primary, in which the sides are from 20 to 190 miles in 
length; secondary, in which the sides are from 5 to 40 miles 
in length, and which connect the primary with the tertiary ; 
tertiary, in which the sides are seldom over 5 miles in length, 
and which bring the survey down to such dimensions as to 
admit of the minor details being filled in by the compass and 
plane table. 

Measurement of Base Lines. Base lines should be measured 
with a degree of accuracy corresponding to their importance. 
Suitable ground must be selected and cleared of all obstruc- 
tions. Each extremity of the line may be marked by cross 
lines on the head of a copper tack driven into a stub which 
is sunk to the surface of the ground. Poles are set up in 
line about half a mile apart, the alignment being controlled 
by a transit or theodolite placed over one end of the line. 
The preliminary measurement may be made with an iron 
wire about one-eighth of an inch in diameter and 60 meters 
in length, or with a steel chain of the same length. 

The final measurement is made with the tape line, or with 
bars 6 meters long, which are supported upon trestles when in 
use. These bars are placed end to end, and brought to a 
horizontal position, if this can be quickly accomplished ; if 
not, the angle of inclination is taken by a sector, or a vertical 
offset is measured with the aid of a transit, so that the exact 
horizontal distance can be computed. A thermometer is 
attached to each bar, so that the temperature of the bar may 



76 SURVEYING 

be noted and a correction for temperature applied. Some- 
times the bars are laid in melting ice, in which case accuracy 
to at least one five-millionth part of the length measured is 
attainable. 

Measurement of Angles. Angles are measured by means of 
the transit with much greater accuracy than with the com- 
pass, since the reading of the plates of the transit is taken to 
minutes, and by means of microscopes to seconds, while the 
reading of the needle of the compass is to quarter or half- 
quarter degrees. 

In order to eliminate errors of observation and of adjust- 
ment, and errors arising from imperfect graduation of the 
circles, a large number of readings is made and their mean 
taken. Two methods are in use, viz., repetition and series. 

The method of repetition consists essentially in taking as 
many readings of an angle as is desired, the reading in each 
case after the first being from the index of the next preceding 
reading, and then taking the mean. 

The method of series is the one generally used when several 
angles about the same point are to be measured. It con- 
sists essentially in taking the readings successively on each 
station, then reversing the telescope and repeating the obser- 
vations in the reverse order, which completes a series. This 
process is repeated a number of times, each series beginning 
with a different index. Then the mean of the different series 
is found. 

On account of the curvature of the earth, the sum of the 
three angles of a triangle upon its surface exceeds 180. 
This spherical excess, as it is called, becomes appreciable only 
when the sides of the triangle are about 5 miles in length. 
To determine the angles of the rectilinear triangle having the 
same vertices, one-third of the spherical excess is generally 
deducted from each spherical angle. 



CHAPTER V 

LEVELING 
SECTION XVII 

DEFINITIONS 

A level surface is a surface parallel with the surface of 
still water, and is, therefore, slightly curved owing to the 
spheroidal shape of the earth. A level line is a line in a 
level surface. The line of apparent level of a place is a 
tangent to the level line at that place. Hence, the line of 
apparent level is perpendicular to the plumb line. 

Leveling is the process of finding the difference of level of 
two places, or the distance of one place above or below a level 
line through another place. 

Corrections for Curvature and Refraction. In ordinary leveling 
no distinction is made between true aad apparent levels. In 
precise leveling the difference between the two is measured, 
i.e., correction is made for curvature of the earth. There is 
sometimes also a correction made for refraction of light. 

Let t (Fig. 52) represent the line of apparent level of the 
place P, a the level line, d the diameter t p 

of the earth ; then c represents the cor- 
rection for curvature. To compute the 
correction for curvature : 

t 2 = c(c + d). (Geometry, 381.) 



2 



Therefore, c = - ; = ? approxi- 
c + d d 

mately, since c is very small compared 

with d, and t = a, very nearly. FJ G . 52 

77 




78 SURVEYING 

Since d is constant (= 7920 miles, nearly), the correction 
for curvature varies as the square of the distance. 

EXAMPLE. What is the correction for curvature for 1 mile ? 
By substituting in the formula deduced above, 

miles = 8 inches, nearly. 



7920 

Hence, the correction for curvature for any distance may 
be found in inches, approximately, by multiplying 8 by the 
square of the distance expressed in miles. 

If correction for refraction is also made, it is customary to 
diminish the above by about one-sixth of itself; or, c = f of 8 a 2 . 

SECTION XVIII 
DIFFERENTIAL LEVELING 

Single Setting of Instrument. To find the difference of 
level between two places when both are visible from some 
intermediate point, and the difference of level does not 
exceed 13 feet, only one setting of the level will usually be 
necessary. 

Let A and B (Fig. 53) represent the two places. Set the 
Y level at a station equally distant, or nearly so, from A and 



FIG. 53 



B, but not necessarily on the line AB. After leveling the 
instrument, bring the telescope to bear upon the rod (p. 38), 
and by signal direct the rodman to move the target until its 
horizontal line is in the line of apparent level of the telescope. 



LEVELING 



79 



Let the rodman now record the height A A' of the target. In 
like manner find BB'. The difference between A A' and BB' 
is the difference of level required. If the instrument is 
equally distant from A and B, or nearly so, the curvature and 
the refraction on the two sides of the instrument balance, and 
no correction for curvature or refraction is necessary. 

Several Settings of Instrument. When both places are not 
visible from the same place, or when the difference of level 
between them is considerable, two or more settings of the 
level may be necessary. 

Let A and D (Fig. 54) represent the two places. Place the 
level midway between A and some intermediate station B. 




FIG. 54 

Find A A' and BB', as in the preceding case, and record the 
former as a backsight and the latter as a foresight. Select 
another intermediate station C, and in like manner find the 
backsight BB" and the foresight CC"; and so continue until 
the place D is reached. 

The difference between the sum of the foresights and the sum 
of the backsights ivill be the difference of level required. 

Since, BB' + CC' + DD' - (A A' + BB" + CC") 
= BB' - BB" + CC' - CC" + DD' - A A' 
= B'B" + C'C" + D'D - AA' = A'A" - AA' = AA". 



80 SURVEYING 

SECTION XIX 
PROFILE LEVELING 

Definitions. The intersection of a vertical plane with the 
surface of the earth is called a section, or profile. The term 
" profile/' however, usually designates the plot, or representa- 
tion of the section on paper. 

Profile leveling is leveling to obtain the data necessary for 
making a profile or plot of any required section. 

A profile is made for the purpose of exhibiting in a single 
view the inequalities of the surface of the ground for great 
distances along the line of some proposed improvement, such 
as a railroad, canal, or ditch, thus facilitating the establishment 
of the proper grades. 

The data necessary for making a profile of any required 
section are the heights of its different points above some 
assumed horizontal plane, called the datum plane, together 
with their horizontal distances apart or their distances from 
the beginning of the section. 

The position of the datum plane is fixed with reference to 
some permanent object near the beginning of the section, 
called a bench mark, and in order to avoid negative heights 
is assumed at such a distance below this mark that all the 
points' of the section shall be above it. 

The heights of the different points of the section above 
the datum plane are determined by means of the level and 
leveling rod ; and the horizontal length of the section is 
measured with an engineer's chain or tape, and divided into 
equal parts, usually 100 feet in length, called stations, marked 
by stakes numbered 0, 1, 2, 3, and so on. 

Where the ground is very irregular, it may be necessary, 
besides taking sights at the regular stakes, to take occasional 
sights at points between them. If, for instance, at a point 



LEVELING 



81 



40 feet in advance of stake 3 (Fig. 55) there is a sudden 
rise or fall in the surface, the height of this point would be 
determined and recorded as at stake 3.40. 

The readings of the rod are ordinarily taken to the nearest 
tenth of a foot, except on bench marks and points called 
turning points, where they are taken to thousandths of a foot. 

A turning point is a point on which the last sight is taken 
just before changing the position of the level, and the first 
sight from the new position of the instrument. A turning 
point may be coincident with one of the stakes, but must 
always be a hard point, so that the foot of- the rod may stand 
at the same level for both readings. 






1 




1 




t 




J^ 


^1 







J.U 


\ 

11 


1 
1 




1 






5 














1 




1 




















1 
1 




1 

, 1 





















FIG. 55 

Field Work. To explain the method of obtaining the field 
notes necessary for making a profile, let 0, 1, 2, 3, -, 11 
(Fig. 55) represent a portion of a section to be leveled and 
plotted. Establish a bench mark at or near the beginning of 
the line, measure the horizontal length of the section, and set 
stakes 100 feet apart, numbering them 0, 1, 2, 3, and so on. 
Place the level at some point, as between 2 and 3, and take 
the reading of the rod on the bench 4.832. Let PP' repre- 
sent the datum plane, say 15 feet below the bench mark; then 

15 + 4.832 = 19.832 

is the height of the line of sight AB, called the height of the 
instrument, above the datum plane. 



82 



SURVEYING 



Now take the reading at = 5.2 = A, and subtract the 
same from 19.832, which leaves 14.6 = P, the height of the 
point above the datum plane. Next take sights at 1, 2, 3, 
3.40, and 4, equal, respectively, to 3.7, 3.0, 5.1, 4.8, and 8.3, and 
subtract the same from 19.832 ; the remainders 16.1, 16.8, 




J. 




sT 
i 

I 

i 


4 


t 


5 


^6 


^^1 








11 



FIG. 56 

14.7, 15.0, and 11.5 are respective heights of the points 1, 2, 
3, 3.40, and 4. 

Then, as it is necessary to change the position of the 
instrument, select a point in the neighborhood of 4 suitable 
as a turning point (t.p. in the figure), and take a careful 
reading on it 8.480 ; subtract this from 19.832, and the 
remainder, 11.352, is the height of the turning point. 

Now carry the instrument forward to a new position, as 
between 5 and 6, shown in the figure, while the rodman 
remains at t.p,\ take a second reading on t.p. = 4.102, and add 
it to 11.352, the height of t.p. above PP';- the sum 15.454 is 
the height of the instrument CD in its new position. 

Take sight upon 5, 6, and 7, equal, respectively, to 4.9, 2.8, 
and 0.904 ; subtract these sights from 15.454, and the results 
10.6, 12.7, and 14.550 are the heights of the points 5, 6, and 
7, respectively. 

The point 7, being suitable, is made a turning point, and 
the instrument is moved forward to a point between 9 and 10. 
The sight at 7 = 6.870, added to the height of 7 gives 21.420 
as the height of the instrument EF in its new position. The 



LEVELING 



83 



readings at 8, 9, 10, and 11, which are, respectively, 5.4, 3.6, 
5.8, and 9.0, subtracted from 21.420 give the heights of these 
points, namely, 16.0, 17.8, 15.6, and 12.4. 

Proceed in like manner until the entire section is leveled, 
establishing bench marks at intervals along the line to serve 
as reference points for future operations. The bench marks 
should be described with sufficient minuteness to enable any 
one not connected with the survey to locate them easily and 
unmistakably. A record of the work is given in the following 
table : 



STATION 


+ s. 


H.I. 


-s. 


H.S. 


REMARKS 


B 


4.832 






15.0 


Bench on rock 20 ft. 







19.832 


5.2 


14.6 


south of 


1 






3 7 


16.1 




2 






3.0 


16.8 




3 






5.1 


14.7 


3 to 3.40 turnpike road 


3.40 






4.8 


15.0 




4 






8.3 


11.5 




t.p. 


4.102 




8.480 


11.352 




5 




15.454 


4.9 


10.6 




6 






2.8 


12.7 




7 


6.870 




0.904 


14.550 




8 




21.420 


5.4 


16.0 




9 






3.6 


17.8 




10 






5.8 


15.6 




11 






9.0 


12.4 




B 










Bench on oak stump 


12 










27 ft. KE. of 12, 


etc. 










etc. 



The first column contains the numbers or names of all the 
points on which sights are taken. The second column con- 
tains the sight taken on the first bench mark, and the sight 
on each turning point taken immediately after the instrument 



84 SURVEYING 

has been moved to a new position. These are called plus 
sights (+ 5.) because they are added to the heights of the 
points on which they are taken to obtain the height of the 
instrument given in the third column (#./.). The fourth 
column contains all the readings except those recorded in the 
second column. These are called minus sights ( S.) because 
they are subtracted from the numbers in the third column to 
obtain all the numbers in the fifth column except the first, 
which is the assumed depth of the datum plane below the 
bench. The fifth column (H.S., height of surface) contains 
the required heights of all the points of the section named 
in the first column together with the heights of all benches 
and turning points. 

Making the Profile. Draw a line PP 1 (Fig. 56), to repre- 
sent the datum plane, and beginning at some point as P, lay 
off the distances 100, 200, 300, 340, 400 feet, and so on, to the 
right, using some convenient scale, say 200 feet to the inch. 
At these points of division erect perpendiculars equal in length 
to the height of the points 0, 1, 2, 3.40, 4, , given in the 
fifth column of the above field notes, using in this case a larger 
scale, say 20 feet to the inch. Through the extremities of 
these perpendiculars draw the irregular line 0, 1, 2, 3, , 11, 
and the result, with some explanatory figures, is the required 
plot or profile. 

The making of a profile is much simplified by the use of 
profile paper, which may be had by the yard or roll. 

If a horizontal plot is required, the bearings of the differ- 
ent portions of the section must be taken. Such a plot should 
be made, if it will assist in properly understanding the field 
work, or if it is desirable for future reference in connection 
with the field notes. Sometimes both the profile and the plot 
are drawn side by side on the same sheet ; in this case, if 
the line leveled over is not straight, the profile will be longer 
than the plot. 



LEVELING 



85 



SECTION XX 
TOPOGRAPHIC LEVELING 

The principal object of topographic surveying is to show 
the contour of the ground. This operation, called topographic 
leveling, is performed by representing on paper the curved 
lines in which parallel horizontal planes at uniform distances 
from each other would meet the surface. It is evident that 
all points in the intersection of a horizontal plane with the 
surface of the ground are at the same level. Hence, it is 
necessary only to find points at the same level and join these 
to determine a line of intersection. 

The method commonly employed will be understood by 
reference to Fig. 57. The ground ABCD is divided into 
equal squares, and a numbered 
stake driven at each intersection. 
By means of a level and leveling 
rod the heights of the other sta- 
tions above m and Z>, the lowest 
stations, are determined. A plot 
of the ground with the intersect- 
ing lines is then drawn, and the 
height of each station written as 
in the figure. 

Suppose that the horizontal 
planes are 2 feet apart ; if the 
first passes through m and D, the 

second will pass through p, which is 2 feet above m ; and 
since n is 3 feet above m, the second plane will cut the line 
mn in a point s determined by the proportion mn : ms = 3:2. 
In like manner, the points t, q, and r are determined. 

The irregular line tsp - qr represents the intersection of 
the second horizontal plane with the surface of the ground. 




86 SURVEYING 

In like manner, the intersections of the planes, respectively, 
4, 6, and 8 feet above m are traced. The more rapid the 
change in level the nearer these lines approach each other. 

SECTION XXI 
DRAINAGE SURVEYING 

Preliminaries. The locality to be drained should first be 
carefully reconnoitered, with the view of ascertaining the 
general feature of the land so as to enable the surveyor 
properly to locate the drains ; the beginning, route, and 
terminus of which should all be definitely planned. By the 
beginning of a drain is meant its highest point. 

Field Work. The field work is essentially the same for 
under drains and for open drains. The first thing is to 
establish the line of a drain. This includes the setting of 
stakes at intervals of from 50 feet to 100 feet, and also 
wherever there is an angle in the line ; the bearings and 
lengths of the successive straight-line sections, beginning 
with the instrument set over the beginning of the drain ; 
and the designation by distances of the points of meeting 
of roads and land lines. Levels of the lines are then taken 
in accordance with the method described on pp. 81-83. If 
circumstances will permit, it is sometimes of advantage to 
have the leveling process go hand in hand with the estab- 
lishing of the line. 

Plot and Profile. If a considerable region is to be drained, 
a plot should be made of the entire tract, and on this plot 
should be drawn, in proper position, the lines of the drain 
and its branches. In a suitable place on the sheet should be 
noted the courses of the various sections of the drain and 
the number of linear feet belonging to each owner of land 
within the tract drained. A profile should also be made, 



LEVELING 



87 



as shown on page 84. From this profile inspection will 
determine whether a single grade will suffice, or whether a 
succession of different grades will be better. 



EXERCISE IX 

1. Find the difference of level of two places from the fol- 
lowing field notes: backsights, 5.2, 6.8, and 4.0; foresights, 
8.1, 9.5, and 7.9. 

2. Stake of the following notes stands at the lowest point 
of a pond to be drained into a creek ; stake 10 stands at the 
edge of the bank, and 10.25 at the bottom of the creek. Make 
a profile, draw the grade line through and 10.25, and fill out 
the columns H.G. and C., the former to show the height of 
grade line above the datum, and the latter, the depth of cut 
at the several stakes necessary to construct the drain. 



STATION 


+ S. 


H.I. 


-s. 


H.S. 


H.G. 


C. 


REMARKS 


B 


6.000 






25 






Bench on rock 









10.2 




20.8 


0.0 


30 ft. west of 


1 






5.3 






5.3 


stake 1 


2 






4.6 










3 






4.0 










4 






6.8 










5 


4.572 




7.090 










6 






3.9 










7 






2.0 










8 






4.9 










9 






4.3 










10 






4.5 










10.25 






11.8 











Horizontal scale, 2 ch. = 1 in. 
Vertical scale, 20 ft. = 1 in. 



88 



SURVEYING 



3. Find the difference in altitude between the highest 
point and the lowest point of the campus or of a field. 

4. Obtain the data necessary for a profile of a half mile of 
highway, and make the profile. 

5. Write the proper numbers in the third and fifth columns 
of the following table of field notes, and make a profile of the 
section. 



STATION 


+ S. 


H.I. 


-s. 


H.S. 


REMARKS 


B 




6.944 




7.4 


20 


Bench on post 22 ft. 
north of 


1 






5.6 






2 






3.9 






3 






4.6 






*.*. 

4 


3.855 




5.513 
4.9 






5 






3.5 






6 






1.2 







CHAPTER VI 
RAILROAD SURVEYING 

SECTION XXII 
LAYING OUT THE ROUTE 

Preliminary Survey. After it has been decided which of 
several feasible lines is the best, a preliminary survey for 
final location should be made. This should include, among 
other things, data referring to elevations, depressions, streams 
to be crossed, highways, buildings obstructing, character of 
soil, and natural resources affording materials for construc- 
tion ; also data referring to proximity to towns, titles of 
land, rights of way, and so on. 

Establishing the Roadbed. When the general route of a 
railroad has been determined, a middle surface line is run 
with the transit. A profile of this line is determined, as 
on page 84. The leveling stations are commonly 1 chain 
(100 feet) apart. Places of different level are connected by 
a gradient line, which intersects the perpendiculars to the 
datum line at the leveling stations in points determined by 
simple proportion. Hence, the distance of each leveling 
station above or below the level or gradient line which 
represents the position of the roadbed is known. 

SECTION XXIII 
CROSS-SECTION WORK 

Excavations. If the roadbed lies below the surface, an 
excavation is made. Let ACBD (Fig. 58) represent a cross 

89 



90 



SURVEYING 



section of an excavation,/ a point in the middle surface line, 
/' the corresponding point in the roadbed, and CD the width 
of the excavation at the bottom. The slopes at the sides are 
commonly made so that A A' = f A'C, and BB' = f DB'. When 
ff and CD are known, the points A, B, C 1 , and D' are readily 
determined by a level and tape measure. 



A' 



f 
FIG. 58 




If from the area of the trapezoid ABB'A' the areas of 
the triangles A A'C and BB'D are deducted, the remainder 
is the area of the cross section. In like manner the cross 
section at the next station may be determined. These two 
cross sections are the bases of a solid whose volume will 
be the amount of the excavation. Since the cross sections 
are not similar, the computations, to be accurate, should 
be made by means of the Prismoidal Formula (Geometry, 
733). 

Embankments. If the roadbed lies above the surface, an 
embankment is made, the cross section of which is like that 
of the excavation, but inverted. 




Fig. 59 represents the cross section of an embankment which 
is lettered so as to show its relation to the excavation of Fig. 58. 



RAILROAD SURVEYING 



91 



SECTION XXIV 
CURVES 

Principles. When it is necessary to change the direction 
of a railroad it is done gradually by a curve, usually the 
arc of a circle. Let AF and 
AO (Fig. 60) represent two 
lines to be thus connected. 
Take any convenient length 
AB = AE = t. The inter- 
section of the perpendicu- F 
lars EC and EC determines 
the centre C, and the radius 
of curvature EC = r. The 
length of the radius de- 
pends on the angle A and the tangent AB. 
triangle ABC, 




FIG. 



For, in the right 



tan BA C 



AB 



or tan A=- 



Hence, 



The degree of a railroad curve is the angle subtended at the 
centre of the curve by a chord of 100 feet. If D is the degree 
of a curve and r its radius, 



. . 50 

sin 4- D = 3 
r 



and r = 50 esc -J- D. 



For example, a 6 curve has a radius of 955.37 feet. 

Sometimes the topography of the route is such as to neces- 
sitate a successive series of curves of different radii, in which 
case the whole series of curves is called a compound curve, the 
principles involved being the same for each component as for 
a simple curve. 



92 



SURVEYING 



Methods of laying out the Curve. 1. Let Em (Fig. 61) 
represent a portion of the tangent. It 
is required to find mP, the perpendicu- 
lar to the tangent meeting the curve 
at P. 

mP = En = CB Cn. 




FIG. 61 



Cn = "V CP 2 - Pn 





Hence, 

2. It is required to find mP (Fig. 62) 
in the direction of the centre. 

raP = mC PC. 
But mC = 
Hence, 



FIG. 62 



3. Place transits at E and E (Fig. 63). Direct the tele- 
scope of the former to E, 

and of the latter to A. Turn A 

each toward the curve the 

same number of degrees, 

and mark P, the point of 

intersection of the lines of 

sight. P is a point in the 

circle to which AB and AE 

are tangents at B and E, 

respectively. 

4. If the degree D of the curve is given and the tangent 
BA at B (Fig. 64), place the transit at B and direct the tele- 
scope toward A. Turn off successively the angles ABP, PBP', 




FIG. 63 



RAILROAD SURVEYING 



93 



P'BP", , each equal to 
%D, and take BP, PP', P'P", 

., each 100 feet, the length 
of the tape. Then, P, P', 
P", lie on the required 
curve. 

If the angle A and the 
tangent distance BA = t 
are given, D can be found 
from the formulas 

50 




FIG. 61 



sin -J- D = j and r = t tan % A. 
50 



Whence, sin D 



EXERCISE X 

1. The cross-section areas at five stations, 100 feet apart, 
of a railroad cut are, respectively, 576.8 square feet, 695.1 
square feet, 809.5 square feet, 652.0 square feet, and 511.7 
square feet. Compute the volume of material in this portion 
of the cut : (i) on the hypothesis that the cross sections are 
similar ; (ii) on the hypothesis that they are dissimilar, the 
alternate cross sections being regarded as mid-sections. 

2. Find the radius of a curve of 1, of 2, of 3, of 4, of 5. 

3. Two adjacent straight sections of a railroad form an 
angle of 148 16'. They are joined by a curve touching each 
of them at the distance of 388 feet from the vertical point. 
Find the radius and the degree of the curve. 

4. Lay out a curve by the first or second method, and 
check the work by means of one of the transit methods. 



CHAPTER VII 

CITY SURVEYING 

SECTION XXV 

FIELD WORK 

Instruments. Since the principles in city surveying are 
essentially the same as those in land surveying, instruments 
of the same general character as the instruments already 
described may be used, except that in this class of work the 
ordinary compass and the chain are set aside. For tho smaller 
cities, an instrument such as the surveyor's transit is suffi- 
cient in accuracy for the purposes of angle measurement and 
for leveling. However, when extreme accuracy is demanded, 
as in the case of large cities, specially made instruments 
should be used : a transit reading to 30 seconds, or even to 
10 seconds ; a high-grade Y level of at least 20-inch length ; 
and a standard tape, tested for sag and temperature. 

Streets. In most cases the city engineer must take the 
streets as he finds them. When a city has outgrown its 
original plan, if indeed it had any, sheer necessity may 
demand the location of additional streets or changes in 
existing streets. If a proposed town or city is to be laid 
out, the general contour of the ground and location of the 
site determine to a great extent the system of streets to be 
adopted. Experience has shown that wherever possible a 
rectangular system of street lines, with a few well-located 
diagonal streets, is the most satisfactory. Streets ordinarily 
vary in width from 50 to 100 feet, and each sidewalk from 7 

04 



CITY SURVEYING 




n 



to 15 feet. The principal improvements of streets are grad- 
ing, paving, setting curbs, laying sidewalks, constructing 
sewers, and laying water pipes. 



96 SURVEYING 

The field work necessary for all these may be included 
under the heads of leveling, locating lines, and locating 
points, which have already been described. 

Blocks and Lots. There is no established rule for the 
size of either blocks or lots. Fig. 65 gives some idea of their 
dimensions. The location of a block is described by refer- 
ence to the streets which bound it. A lot is described by 
number and block, or by number alone, or by giving the 
location and length of its bounding lines. The co-ordinate 
system of location of points, described on page 53, has 
much in its favor for use in city surveying. Monuments 
at points of reference and at intersections of streets and 
corners of lots should be of permanent character, and set 
with extreme care. 

SECTION XXVI 
OFFICE WORK 

Plots. Among the more important plots that should be 
prepared by the city engineer are a complete city map, drawn 
to scale, showing the streets and alleys, blocks and lots, with 
dimensions, and the location of railroads, street-car lines, sew- 
age system, water-pipe lines, and so on ; a topographical map 
of the entire city, including as may be found desirable por- 
tions of the surrounding region ; a profile map of the streets. 
These are made from the field notes, which should be amply 
and carefully prepared. 

Records. No work of importance, whether done in the field 
or in the office, should fail to be recorded in some perma- 
nent form. Field notes, computations, plots, and copies of 
work specially prepared should be properly indexed and filed 
away. 



ANSWERS 97 



SURVEYING 



Exercise II. Page 22 

2. 540. 4. N. 51 30' E. 

Exercise III. Page 27 

2. 360. 3. 235 ft. 3.8 in. 

Exercise V. Page 55 



1. 8 A. 64 P. 


6. 13 A. 6 T V P. 


11. 4 A. 35 P. 


2. 16 A. 74|f P. 


7. 2 A. 581 p. 


12. 4 A. 110 P. 


3. 4 A. 5sV P. 


8. 11 A. 157 P. 


13. 6 A. 23i| P. 


4. 115oV p - 


9. 7.51925. 




5. 3 A. 78 P. 


10. 13.0735. 





Exercise VI. Page 64 

1. 2 A. 26 P. 3. 8 A. 54 P. 5. 2 A. 78 P. 7. 5 A. 42 p. 

2. 20 A. 12 P. 4. 3 A. 122 p. 6. 6 A. 2 p. 8. 2 A. 151 p. 

Exercise VII. Page 66 
1. 2 A. 121 p. 2. 98 A. 92 p. 

Exercise VIII. Page 69 

1. J.#=3.75ch. 4. AE=5.5Qch. 

2. ^# = 3.50ch.; 5. CE = 4.456 ch. 

EG = 3.42 ch. 6 AD _ 2 .275 ch. ; BE = 1.82 ch. 

3. ^ = 4.55ch. 7. AD = 4. 51 ch.; # = 3.61 ch. 

8. The distances on AB are 2 ch., 3 ch., and 5 ch. 

9. EM (on EA) = 2.5087 ch. ; AN (on AB) = 6.4390 ch. 



98 



SURVEYING 



10. LetEG>DF; then AE = 12.247 ch., AG = 9.798 ch., AD = 8.660 

ch., AF= 6.928 ch. 

11. LetDG>EF- then CG= 14.862 ch., CD = 13.113 ch., CF= 11.404 

ch., CE = 10.062 ch. 

Exercise IX. Page 87 

1. 9.5. 

2. Column H.G. 20.8, 20.4, 20.0, 19.6, 19.2, 18.8, 18.4, 18.0, 17.6, 

17.2, 16.8, 16.7. 
Column C. 0.0, 5.3, 6.4, 7.4, 5.0, 5.1, 6.2, 8.5, 6.0, 7.0, 7.2, 0.0. 




9 10 10.25 



5. Third column : 26.944 opposite 0; 25.286 opposite 4. 

Fifth column : 20, 19.5, 21.3, 23, 22.3, 21.431, 20.4, 21.8, 24.1. 




Exercise X. Page 93 

1. 9986. 5 cu. yd.; 9994.9 cu. yd. 

2. 5730ft.; 2865ft.; 1910ft.; 1433ft.; 1146ft. 

3. 1365ft.; 4 11' 53". 



FIVE -PLACE 



LOGARITHMIC AND TRIGONOMETRIC 



TABLES 



ARRANGED BY 



G. A. WENTWORTH, A.M. 



AND 



G. A. HILL, A.M. 



GINN & COMPANY 

BOSTON NEW YORK . CHICAGO . LONDON 



Entered according to Act of Congress, in the year 1882, by 

G. A. WENTWORTH AND G. A. HILL 
in the office of the Librarian of Congress at Washington 



Copyright. 1895, by G. A. WENTWORTH and G. A. HILL. 

210.2 



Scltfjenaum 



GIXN & COMPANY. PRO- 
PRIETORS BOSTON U.S.A. 



INTRODUCTION. 



1. If the natural numbers are regarded as powers of ten, the 
exponents of the powers are the Common or Briggs Logarithms of 
the numbers. If A and B denote natural numbers, a and b their 
logarithms, then 10 a A, 10 & B ; or, written in logarithmic form, 

log A = a, log B = b. 

2. The logarithm of a product is found by adding the logarithms 
of its factors. 

For, AX B = 10 X 10 & = 10 + 6 . 

Therefore, log ( A X B) = a + b = log A + log B. 

3. The logarithm of a quotient is found by subtracting the 
logarithm of the divisor from that of the dividend. 

l=i!=-->- 

Therefore, log = a b = log A log B. 

_D 

4. The logarithm of a power of a number is found by multiply- 
ing the logarithm of the number by the exponent of the power. 

For, A n = (10) = 10 an . 

Therefore, log-4 n = an = n log A. 

5. The logarithm of the root of a number is found by dividing 
the logarithm of the number by the index of the root. 

n, n, ? 

For, VZ = VlO" 10". 

Therefore, log tfZ = S = !4. 
n n 

6. The logarithms of 1, 10, 100, etc., and of 0.1, 0.01, 0.001, 
etc., are integral numbers. The logarithms of all other numbers 
are fractions. 



IV LOGARITHMS. 

For, 10 = 1, hence log 1 = ; 10- 1 = 0.1, hence log 0.1 = 1 

10 1 = 10, hence log 10 = 1 ; 10~ 2 = 0.01, hence log 0.01 = 2 ; 

102 = 100, hence log 100 = 2 ; 10- 3 = 0.001, hence log 0.001 = - 3 ; 

10 3 = 1000, hence log 1000 = 3 ; and so on. 

If the number is between 1 and 10, the logarithm is between and 1. 

If the number is between 10 and 100, the logarithm is between 1 and 2. 

If the number is between 100 and 1000, the logarithm is between 2 and 3. 

If the number is between land 0.1, the logarithm is between and 1. 

If the number is between 0.1 and 0.01, the logarithm is between 1 and 2. 
If the number is between 0.01 and 0.001, the logarithm is between 2 and 3. 
And so on. 

7. If the number is less than 1, the logarithm is negative ( 6), 
but is written in such a form that the fractional part is always positive. 

For the number may be regarded as the product of two factors, one of 
which lies between 1 and 10, and the other is a negative power of 10 ; the 
logarithm will then take the form of a difference whose minuend is a positive 
proper fraction, and whose subtrahend is a positive integral number. 
Thus, 0.48 = 4.8X0.1. 

Therefore ( 2), log 0.48 = log 4.8 + log 0.1 = 0.68124 - 1. (Page 1.) 
Again, 0.0007 = 7 X 0.0001. 

Therefore, log 0.0007 = log 7 + log 0.0001 = 0.84510 4. 

The logarithm 0.84510 4 is often written 4.84510. 

8. Every logarithm, therefore, consists of two parts : a positive 
or negative integral number, which is called the Characteristic, and 
a positive proper fraction, which is called the Mantissa. 

Thus, in the logarithm 3.52179, the integral number 3 is the characteristic, 
and the fraction .52179 the mantissa. In the logarithm 0.78254 2, the inte- 
gral number 2 is the characteristic, and the fraction 0.78254 is the mantissa. 

9. If the logarithm is negative, it is customary to change the 
form of the difference so that the subtrahend shall be 10 or a multiple 
of 10. This is done by adding to both minuend and subtrahend a 
number which will increase the subtrahend to 10 or a multiple of 10. 

Thus, the logarithm 0.78254 2 is changed to 8.78254 10 by adding 8 to 
both minuend and subtrahend. The logarithm 0.92737 13 is changed to 
7.92737 20 by adding 7 to both minuend and subtrahend. 

10. The following rules are derived from 6 : 

If the number is greater than 1, make the characteristic of the 
logarithm one unit less than the number of figures on the left of 
the decimal point. 

If the number is less than 1, make the characteristic of the loga- 
rithm negative, and one unit more than the number of zeros between 
the decimal point and the first significant figure of the given number. 



INTRODUCTION. V 

If the characteristic of a given logarithm is positive, make the 
number of figures in the integral part of the corresponding number 
one more than the number of units in the characteristic. 

If the characteristic is negative, make the number of zeros between 
the decimal point and the first significant figure of the correspond- 
ing number one less than the number of units in the characteristic. 

Thus, the characteristic of log 7849.27 = 3 ; 

the characteristic of log 0.037 = 2 = 8.00000 10. 

If the characteristic is 4, the corresponding number has five figures in its inte- 
gral part. If the characteristic is 3, that is, 7.00000 10, the corresponding 
fraction has two zeros between the decimal point and the first significant figure. 

11. The logarithms of numbers that can be derived one from 
another by multiplication or division by an integral power of 10 
have the same mantissa. 

For, multiplying or dividing a number by an integral power of 10 will 
increase or diminish its logarithm by the exponent of that power of 10 ; and 
since this exponent is an integer, the mantissa of the logarithm will be 
unaffected. 

Thus, log 4.6021 =0.66296. (Page 9.) 

log 460.21 = log (4.6021 X 10 2 ) = log 4.6021 + log 10 2 

= 0.66296 + 2 = 2.66296. 
log 460210 = log (4.6021 X 10 5 ) = log 4.6021 + log 10 6 

= 0.66296 + 5 = 5.66296. 
log 0.046021 = log (4.6021 -=- 10*) = log 4.6021 - log 10 2 

= 0.66296 2 = 8.66296 - 10. 



TABLE I. 

12. In this table (pp. 1-19) the vertical columns headed N con- 
tain the numbers, and the other columns the logarithms. On page 1 
both the characteristic and the mantissa are printed. On pages 
2-19 the mantissa only is printed. 

The fractional part of a logarithm can be expressed only approx- 
imately, and in a five-place table all figures that follow the fifth are 
rejected. Whenever the sixth figure is 5, or more, the fifth figure 
is increased by 1. The figure 5 is written when the value of the 
figure in the place in which it stands, together with the succeeding 
figures, is more than 4-J-, but less than 5. 

Thus, if the mantissa of a logarithm written to seven places is 5328732, it is 
written in this table (a five-place table) 53287. If it is 5328751, it is written 
53288. If it is 5328461 or 5328499, it is written in this table 53285.. 

Again, if the mantissa is 5324981, it is written 53250 ; and if it is 4999967, it 
is written 50000. 



Vi LOGARITHMS. 

This distinction between 5 and 5, in case it is desired to curtail 
still further the mantissas of logarithms, removes all doubt whether 
a 5 in the last given place, or in the last but one followed by a 
zero, should be simply rejected, or whether the rejection should 
lead us to increase the preceding figure by one unit. 

Thus, the mantissa 13925. when reduced to four places should be 1392 ; but 
13925 should be 1393. 

To FIND THE LOGARITHM OF A GIVEN NUMBER. 

13. If the given number consists of one or two significant 
figures, the logarithm is given on page 1. If zeros follow the 
significant figures, or if the number is a proper decimal fraction, 
the characteristic must be determined by 10. 

14. If the given number has three significant figures, it will be 
found in the column headed 1ST (pp. 2-19), and the mantissa of its 
logarithm in the next column to the right, and on the same line. 
Thus, 

Page 2. log 145 = 2. 16137, log 14500 = 4.16137. 

Page 14. log 716 = 2.85491, log 0.716 = 9.85491 10. 

15. If the given number has four significant figures, the first 
three will be found in the column headed N, and the fourth at the 
top of the page in the line containing the figures 1, 2, 3, etc. The 
mantissa will be found in the column headed by the fourth figure, 
and on the same line with the first three figures. Thus, 

Page 15. log 7682 = 3.88547, log 76.85 = 1.88564. 
Page 18. log 93280 = 4.96979, log 0.9468 = 9.97626 10. 

16. If the given number has five or more significant figures, a 
process called interpolation is required. 

Interpolation is based on the assumption that between two con- 
secutive mantissas of the table the change in the mantissa is directly 
proportional to the change in the number. 

Required the logarithm of 34237. 

The required mantissa is ( 11) the same as the mantissa for 3423.7 ; mere- 
fore it will be found by adding to the mantissa of 3423 seven-tenths of the 
difference between the mantissas for 3423 and 3424. 

The mantissa for 3423 is 53441. 

The difference between the mantissas for 3423 and 3424 is 12. 

Hence, the mantissa for 3423.7 is 53441 + (0.7 X 12) = 5344 

Therefore, the required logarithm of 34237 is 4.53449. 



INTRODUCTION. Vll 

Required the logarithm of 0.0015764. 

The required mantissa is the same as the mantissa for 1576.4 ; therefore 
it will be found by adding to the mantissa for 1576 four-tenths of the difference 
between the mantissas for 1576 and 1577. 

The mantissa for 1576 is 19756. 

The difference between the mantissas for 1576 and 1577 is 27. 

Hence, the mantissa for 1576.4 is 19756 + (0.4 X 27) 19767. 

Therefore, the required logarithm of 0.0015764 is 7.19767 10. 

Required the logarithm of 32.6708. 

The required mantissa is the same as the mantissa for 3267.08; therefore 
it will be found by adding to the mantissa for 3267 eight-hundredths of the 
difference between the mantissas for 3267 and 3268. 

The mantissa for 3267 is 51415. 

The difference between the mantissas for 3267 and 3268 is 13. 

Hence, the mantissa for 3267.08 is 51415 + (0.08 X 13) = 51416. 

Therefore, the required logarithm of 32.6708 is 1.51416. 

17. When the fraction of a unit in the part to be added to the 
mantissa for four figures is less than 0.5 it is to be neglected ; when 
it is 0.5 or more than 0.5 it is to be taken as one unit. 

Thus, in the first example, the part to be added to the mantissa for 3423 is 
8.4, and the .4 is rejected. In the second example, the part to be added to the 
mantissa for 1576 is 10.8, and 11 is added. 



To FIND THE ANTILOGARITHM ; THAT is, THE NUMBER CORRE- 
SPONDING TO A GIVEN LOGARITHM. 

18. If the given mantissa can be found in the table, the first 
three figures of the required number will be found in the same line 
with the mantissa in the column headed N, and the fourth figure at 
the top of the column containing the mantissa. 

The position of the decimal point is determined by the charac- 
teristic ( 10). 

Find the number corresponding to the logarithm 0.92002. 

Page 16. The number for the mantissa 92002 is 8318. 

The characteristic is ; therefore, the required number is 8.318. 

Find the number corresponding to the logarithm 6.09167. 

Page 2. The number for the mantissa 09167 is 1235. 

The characteristic is 6 ; therefore, the required number is 1235000 

Find the number corresponding to the logarithm 7.50325 10. 

Page 6. The number for the mantissa 50325 is 3186. 

The characteristic is 3 ; therefore, the required number is 0.003186. 



Vlll LOGARITHMS. 

19. If the given mantissa cannot be found in the table, find 
in the table the two adjacent mantissas between which the given 
mantissa lies, and the four figures corresponding to the smaller of 
these two mantissas will be the first four significant figures of the 
required number. If more than four figures are desired, they may 
be found by interpolation, as in the following examples : 

Find the number corresponding to the logarithm 1.48762. 

Here the two adjacent mantissas of the table, between which the given man- 
tissa 48762 lies, are found to be (page 6) 48756 and 48770. The corresponding 
numbers are 3073 and 3074. The smaller of these, 3073, contains the first four 
significant figures of the required number. 

The difference between the two adjacent mantissas is 14, and the difference 
between the corresponding numbers is 1. 

The difference between the smaller of the two adjacent mantissas, 48756, 
and the given mantissa, 48762, is 6. Therefore, the number to be annexed to 
3073 is T 6 of 1 = 0.428, and the fifth significant figure of the required number 
is 4. 

Hence, the required number is 30.734. 

Find the number corresponding to the logarithm 7.82326 10. 

The two adjacent mantissas between which 82326 lies are (page 13) 82321 
and 82328. The number corresponding to the mantissa 82321 is 6656. 

The difference between the two adjacent mantissas is 7, and the difference 
between the corresponding numbers is 1. 

The difference between the smaller mantissa, 82321, and the given mantissa, 
82326, is 5. Therefore, the number to be annexed to 6656 is f of 1 = 0.7, and 
the fifth significant figure of the required number is 7. 

Hence, the required number is 0.0066567. 

In using a five-place table the numbers corresponding to man- 
tissas may be carried to five significant figures, and in the first 
part of the table to six figures.* 

20. The logarithm of the reciprocal of a number is called the 
Cologarithm of the number. 

If A denotes any number, then 

colog A = log = log 1 log A ( 3) = log A. 
A 

Hence, the cologarithm of a number is equal to the logarithm of 
the number with the minus sign prefixed, which sign affects the 
entire logarithm, both characteristic and mantissa. 

*In most tables of logarithms proportional parts are given as an aid to 
interpolation ; but, after a little practice, the operation can be performed nearly 
as rapidly without them. Their omission allows a page with larger-faced type 
and more open spacing, and consequently less trying to the eyes. 



INTRODUCTION. IX 

In order to avoid a negative mantissa in the cologarithm, it is 
customary to substitute for log A its equivalent 

(10 - log A) 10. 

Hence, the cologarithm of a number is found by subtracting the 
logarithm of the number from 10, and then annexing 10 to the 
remainder. 

The best way to perform the subtraction is to begin on the left 
and subtract each figure of log A from 9 until we reach the last 
significant figure, which must be subtracted from 10. 

If log A is greater in absolute value than 10 and less than 20, 
then in order to avoid a negative mantissa, it is necessary to write 
log A in the form 

(20 log A) 20. 

So that, in this case, colog A is found by subtracting log A from 
20, and then annexing 20 to the remainder. 

Find the cologarithm of 4007. 

10 10 

Page 8. log 4007 = 3.60282 

colog 4007= 6.39718-10 

Find the cologarithm of 103992000000. 

20 -20 

Page 2. log 103992000000 = 11.01700 

colog 103992000000 = 8.98300 20 

If the characteristic of log A is negative, then the subtrahend, 
10 or 20, will vanish in finding the value of colog A. 

Find the cologarithm of 0.004007. 

10 -10 

log 0.004007 = 7.60282 - 10 
colog 0.004007 = 2.39718 

With practice, the cologarithm of a number can be taken from 
the table as rapidly as the logarithm itself. 

By using cologarithms the inconvenience of subtracting the log- 
arithm of a divisor is avoided. For dividing by a number is 
equivalent to multiplying by its reciprocal. Hence, instead of 
subtracting the logarithm of a divisor its cologarithm may be 
added. 



LOGARITHMS. 



EXERCISES. 



Find the logarithms of : 



1. 6170. 

2. 0.617. 

3. 2867. 



4. 85.76. 

5. 296.8. 

6. 7004. 



7. 0.8694. 

8. 0.5908. 

9. 73243. 



10. 67.3208. 

11. 18.5283. 

12. 0.0042003. 



Find the cologarithms of : 



13. 72433. 

14. 802.376. 

15. 15.7643. 



16. 869.278. 

17. 154000. 

18. 70.0426. 



19. 0.002403. 

20. 0.000777. 

21. 0.051828. 



Find the antilogarithms of : 



22. 2.47246. 

23. 7.89081. 

24. 2.91221. 



25. 1.26784. 

26. 3.79029. 

27. 5.18752. 



28. 9.7902910. 

29. 7.62328-10. 

30. 6.15465 10. 



COMPUTATION BY LOGARITHMS. 

21. (1) Find the value of x, if x = 72214 X 0.08203. 

Page 14. log 72214 = 4.85862 

Page 16. log 0.08203 = 8.91397 - 10 

By 2. logx =3.77259 

Page 11. x = 5923.63 



(2) Find the value of x, if x = 5250 -j- 23487. 

Page 10. log 5250 = 3.72016 

Page 4. colog 23487 = 5.62917 - 10 

Page 4. logx = 9.34933 - 10 = log 0.2235S 

.-. x = 0.22353 



/ox v j 4.1, ^ f - f 7.56X4667X567 

(3) Find the value of x, if = 399^ X0 .oo337 X 23435* 

Page 15. log 7.56 = 0.87852 

Page 9. log 4667 = 3.66904 

Page 11. log 567 = 2.75358 

Page 17. colog 899.1 = 7.04619 10 

Page 6. . colog 0.00337 >= 2.47237 

Page 4. colog 23435 = 5.63013 10 

Page 6. log x 2.44983 = log 281.73 

,:x =281.73. 



INTRODUCTION. Xl 

C4) Find the cube of 376. 

Page 7. log 376 = 2.57519 

Multiply by 3 ( 4), 3 

Page 10. log 376 3 = 7.72557 = log 53158600 

.-. 376 3 = 53158600. 

(5) Find the square of 0.003278. 

Page 6. log 0.003278 = 7.51561-10 

2 

Page 2. log 0.003278 2 = 15.03122 - 20 = log 0.000010745 

.-. 0.003278 2 = 0.000010745. 

(6) Find the square root of 8322. 

Page 16. log 8322 = 3.92023 

Divide by 2 ( 5), 2)3.92023 

log V8322 = 1.96012 = log 91.226 

.-. V8322 = 91.226. 

If the given number is a proper fraction, its logarithm will have 
as a subtrahend 10 or a multiple of 10. In this case, before divid- 
ing the logarithm by the index of the root, both the subtrahend and 
the number preceding the mantissa should be increased by such a 
number as will make the subtrahend, when divided by the index of 
the root, 10 or a multiple of 10. 

(7) Find the square root of 0.000043641. 

Page 8. log 0.000043641 = 5.63989-10 

10 10 

Divide by 2 ( 5), 2)15.63989-20 

Page 13. log VO. 000043641 = 7.81995 10 = Log 0.0066062 

.-. V0.000043641 = 0.0066062. 

(8) Find the sixth root of 0.076553. 

'Page 15. log 0.076553 = 8.88397-10 

50 -50 

Divide by 6 ( 5), 6)58.88397-60 

Page 13. log VO. 076553 = 9.81400 - 10 = log 0-65163 

.-. \/0. 076553 = 0.65163. 

EXERCISES. 

Find by logarithms the value of : 

1 45607 5.6123 2.567 

' 31045' A 0.01987' 0.05786* 



LOGARITHMS. 
0.06547 



' 74.938 X 0.05938 
4.657 X 0.03467 



3.908 X 0.07189 

0.0075389 X 0.0079 
0.00907 X 0009784* 

312 X 7.18 X 31.82 
519 X 8.27 X 5.132* 

0.007 X 57.83 X 28.13 
9.317 X 00.28 X 476.5* 

5.55 X 0.0007632 X 0.87654 
2.79 X 0.0009524 X 1.46785 



/0. 003457 X 43.387 X 99. S 
\ 0.005824 X 15.724 X 1.3* 



2 X 0.00025 
38 X 0.00089 



8/23.815 X 29.36 X 0.007 X 0.62487 
' \ 0.00072 X 9.236 X 5.924 X 3.0007 



/3.1416 X 0.0314 
' \ 1.7285 X 0.01 75 



031416 X 0.0031416 



017285 X 0.0017285 



TABLE II. 

22. This table (page 20) contains the value of the number TT, 
its most useful combinations, and their logarithms. 

Find the length of an arc of 47 32' 57" in a unit circle. 

47 32' 57" =171177" 
log 171177 = 5.23344 

log 4> = 4.68557 - 10 



log arc 47 32' 57" = 9.91901 - 10 = log 0.82994 
.-. length of arc = 0.82994. 

Find the angle if the length of its arc in a unit circle = 0.54936. 
log 0. 54936 = 9. 73986 - 10 

colog = log a" = 5.31443 

log angle = 5.05429 = log 113316 

.-. angle = 113316" = 31 28" 36". 



INTRODUCTION. 



23. The relations between arcs and angles given in Table II. 
are readily deduced from the circular measure of an angle. 

In Circular Measure an angle is defined by the equation 

arc 

angle = r= > 
radius 

in which the word arc denotes the length of the arc corresponding 
to the angle, when both arc and radius are expressed in terms of 
the same linear unit. 

Since the arc and radius for a given angle in different circles 
vary in the same ratio, the value of the angle given by this equa- 
tion is independent of the value of the radius. 

The angle which is measured by a radius-arc is called a Radian, 
and is the angular unit in circular measure. 

C 4- C 

Since C = 2 irR, it follows that = 2 TT, and ^ = TT. Therefore, 

H H 

If the arc = circumference, the angle = 2 TT. 

If the arc = semicircumference, the angle = TT. 

If the arc = quadrant, the angle = % TT. 

If the arc = radius, the angle = 1. 

Therefore, TT = 180, TT = 90, TT = 60, TT = 45, J TT = 30", 
7T = 22i, and so on. 

Since 180 in common measure equals TT units in circular measure, 

1 in common measure = T-^- units in circular measure ; 

loO 

180 . 
1 unit in circular measure = - in common measure. 

7T 

By means of these two equations, the value of an angle expressed 
in one measure may be changed to its value in the other measure. 
Thus, the angle whose arc is equal to the radius is an angle of 

180 
1 unit in circular measure, and is equal to - , or 57 17' 45", 

very nearly. 

TABLE III. 

24. This table (pp. 21-49) contains the logarithms of the trigo- 
nometric functions of angles. In order to avoid negative character- 
istics, the characteristic of every logarithm is printed 10 too large. 
Therefore, 10 is to be annexed to each logarithm. 

On pages 28-49 the characteristic remains the same throughout 
each column, and is printed at the top and the bottom of the column, 



XIV LOGARITHMS. 

But on pp. '60, 49, the characteristic changes one unit in value at the 
places marked with bars. Above these bars the proper characteristic 
is printed at the top, and below them at the bottom, of the column. 

25. On pages 28-49 the log sin, log tan, log cot, and log cos, of 
1 to 89, are given to every minute. Conversely, this part of the 
table gives the value of the angle to the nearest minute when 
log sin, log tan, log cot, or log cos is known, provided log sin or 
log cos lies between 8.24186 and 9.99993, and log tan or log cot 
lies between 8.24192 and 11.75808. 

If the exact value of the given logarithm of a function is not 
found in the table, the value nearest to it is to be taken, unless 
interpolation is employed as explained in 26. 

If the angle is less than 45 the number of degrees is printed at 
the top of the page, and the number of minutes in the column to 
the left of the columns containing the logarithm. If the angle 
is greater than 45, the number of degrees is printed at the bottom 
of the page, and the number of minutes in the column to the right 
of the columns containing the logarithms. 

If the angle is less than 45, the names of its functions are 
printed at the top of the page ; if greater than 45, at the bottom 
of the page. Thus, 

Page 38. log sin 21 37' = 9.56631 10. 

Page 45. log cot 36 53' - 10. 12473 10 = 0.12473. 

Page 37. log cos 69 14' = 9.5496910. 

Page 49. log tan 45 59' = 10.01491 10 = 0.01491. 

Page 48. If log cos = 9.87468 - 10, angle = 41 28'. 

Page 34. If log cot = 9.39353 - 10, angle = 76 6'. 

If log sin = 9.47760 10, the nearest log sin in the table is 5.47774 10 
(page 36), and the angle corresponding to this value is 17 29'. 

If log tan = 0.76520 = 10.76520 10, the nearest log tan in the table is 
10.76490 10 (page 32), and the angle corresponding to this value is 80 15'. 

26. If it is desired to obtain the logarithms of the functions of 
angles that contain seconds, or to obtain the value of the angle in 
degrees, minutes, and seconds, from the logarithms of its functions, 
interpolation must be employed. Here it must be remembered 
that, 

The difference between two consecutive angles in the table 
is 60". 

Log sin and log tan increase as the angle increases ; log cos and 
log cot diminish as the angle increases- 



INTRODUCTION. XV 

Find log tan 70 46' 8". 

Page 37. log tan 70 46' 0.45731. 

The difference between the mantissas of log tan 70 46' and log tan 70 47' 
is 41, and ^ of 41 = 5. 

As the function is increasing, the 5 must be added to the figure in the fifth 
place of the mantissa 45731 ; and 

Therefore log tan 70 46' 8" = 0.45736. 

Find log cos 47 35' 4". 

Page 48. log cos 47 35' = 9.82899 - 10. 

The difference between this mantissa and the mantissas of the next log cos 
is 14, and -fo of 14 = 1. 

As the function is decreasing, the 1 must be subtracted from the figure in the 
fifth place of the mantissa 82899 ; and 

Therefore log cos 47 35' 4" = 9.82898 - 10. 

Find the angle for which log sin = 9.45359 10. 

Page 35. The mantissa of the nearest smaller log sin in the table is 45334. 

The angle corresponding to this value is 16 30'. 

The difference between 45334 and the given mantissa, 45359, is 25. 

The difference between 45334 and the next following mantissa, 45377, is 43, 
and ff of 60" = 35". 

As the function is increasing, the 35" must be added to 16 30'; and the 
required angle is 16 30' 35". 

Find the angle for which log cot = 0.73478. 

Page 32. The mantissa of the nearest smaller log cot in the table is 73415. 

The angle corresponding to this value is 10 27'. 

The difference between 73415 and the given mantissa is 63. 

The difference between 73415 and the next following mantissa is 71, and ff 
of 60" = 53". 

As the function is decreasing, the 53" must be subtracted from 10 27'; and 
the required angle is 10 26' 7". 



EXERCISES. 
Find 

1. log sin 30 8' 9". 9. log tan 25 27' 47". 

2. log sin 54 54' 40". 10. log cos 56 11' 57". 

3. log cos 43 32' 31". 11. log cot 62 0' 4" 

4. log cos 69 25' 11". 12. log cos 75 20' 58* 

5. log tan 32 9' 17". 13. log tan 33 27' 13". 

6. log tan 50 2' 2". 14. log cot 81 55' 24". 

7. log cot 44 33' 17". 15. log tan 89 46' 35". 

8. log cot 55 9' 32". 16. log tan 1 25' 56". 



XVI LOGARITHMS. 

Find the angle A if 

17. log sin A- 9.70075. 25. log cos A = 940008: 

18. logsinJ.= 9.91289. 26. log cot A - 9.78815. 

19. log cos A 9.86026. 27. log cos A = 9.34301. 

20. log cos A- 9.54595. 28. log tan A = 10. 52288. 

21. logtan^l = 9.79840. 29. log cot A = 965349. 

22. log t&nA = 10.07671. 30. log sin A = 8.39316. 

23. log cot A = 10.00675. 31. log sin A = 8.06678. 

24. log cot A- 9.84266. 32. logtan^l = 8.11148. 

27. If log sec or log esc of an angle is desired, it may be found 
from the table by the formulas, 

sec A = ; hence, log sec A = colog cos A. 

COS ^L 

esc A = 7 ; hence, log esc A = colog sin A. 
sin A 

Page, 31. log sec 8 28' = colog cos 8 28' =0.00476. 
Page 42. log esc 59 36' 44" = colog sin 59 36' 44" = 0.06418. 

28. If a given angle is between and 1, or between 89 and 90; 
or, conversely, if a given log sin or log cos does not lie between the 
limits 8.24186 and 9.99993 in the table; or, if a given log tan or 
log cot does not lie between the limits 8.24192 and 11.75808 in the 
table ; then pages 21-24 of Table III. must be used. 

On page 21, log sin of angles between and 3', or log cos of 
the complementary angles between 89 57' and 90, are given to 
every second; for the angles between and 3', log taii = log sin, 
and log cos = 0.00000 ; for the angles between 89 57' and 90, 
log cot = log cos, and log sin = 0.00000. 

On pages 22-24, log sin, log tan, and log cos of angles between 
and 1, or log cos, log cot, and log sin of the complementary 
angles between 89 and 90, are given to every 10". 

Whenever log tan or log cot is not given, they may be found by 
the formulas, 

log tan = colog cot. log cot = colog tan. 

Conversely, if a given log tan or log cot is not contained in the 
table, then the colog must be found ; this will be the log cot or 
log tan, as the case may be, and will be contained in the table. 

On pages 25-27 the logarithms of the functions of angles 
between 1 and 2, or between 88 and 90, are given in the manner 
employed on pages 22-24. These pages should be used if the angle 
lies between these limits, and if not only degrees and minutes, but 
degrees, minutes, and multiples of 10" are given or required. 



INTRODUCTION. XV11 

When the angle is between '0 and 2, or 88 and 90, and a 
greater degree of accuracy is desired than that given by the table, 
interpolation may be employed ; but for these angles interpolation 
does not always give true results, and it is better to use Table IV. 

Find log tan 2' 47", and log cos 89 37' 20". 

Page 21. log tan 2' 47" ~ log sin 2' 47" = 6.90829 10. 
Page 23. log cos 89 37' 20" = 7.81911 - 10. 

Find log cot 2' 15". 

10 -10 

Page 21. log tan 2' 15" - 6.81591 10 
Therefore, log cot 2' 15" = 3.18409 

Find log tan 89 38' 30". 

10 10 

Page 23. log cot 89 38' 30" = 7.79617-10 
Therefore, log tan 89 38' 30" - 2.20383 

Find the angle for which log tan = 6.92090 10. 

Page 21. The nearest log tan is 6.92110 10. 
The corresponding angle for which is 2' 52". 

Find the angle for which log cos = 7.70240 10. 

Page 22. The nearest log cos is 7.70261 10. 
The corresponding angle for which is 89 42' 40". 

Find the angle for which log cot = 2.37368. 

This log cot is not contained in the table. 
The colog eot = 7.62632 10 = log tan. 

The log tan in the table nearest to this is (page 22) 7.62510 10, and the 
angle corresponding to this value of log tan is 14' 30". 

29. If an angle x is between 90 and 360, it follows, from 
formulas established in Trigonometry, that, 

between 90 and 180, between 180 and 270, 

log sin x = log sin (180 x), log sin x = log sin (x 180) n , 

log cos x = log cos (180 x) n , log cos x = log cos (x 180) n > 

log tan x = log tan (180 ), log tan x = log tan (x 180), 

log cot x = log cot (180 x) n ; log cot x = log cot (aj 180) j 

between 270 and 360, 
log sin x = log sin (360 aj) B , 
log cos x = log cos (360 x), 
log tail x = log tan (360 x) n , 
log cot a: = log cot (360 ). 



XV111 LOGARITHMS. 

The letter n is placed (according to custom) after the logarithms 
of those functions which are negative in value. 

The above k formulas show, without further explanation, how to 
find by means of Table III. the logarithms of the functions of any 
angle between 90 and 360. 

Thus, log sin 137 45' 22" = log sin 42 14' 38" = 9.82756 10. 
log cos 137 45' 22" - log n cos 42 14' 38" = 9.86940 n - 10. 
log tan 137 45' 22" = log,, tan 42 14' 38" = 9.95815 n - 10. 
log cot 137 45' 22" = log n cot 42 14' 38" = 0.04185 n . 
log sin 209 32' 50" = log n sin 29 32' 50" = 9.69297 n 10. 
log cos 330 27' 10" = log cos 29 32' 50" - 9.93949 - 10. 

Conversely, to a given logarithm of a trigonometric function 
there correspond between and 360 four angles, one angle in 
each quadrant, and so related that if x denote the acute angle, the 
other three angles are 180 x, 180 + x, and 360 x. 

If besides the given logarithm it is known whether the function 
is positive or negative, the ambiguity is confined to two quadrants, 
therefore to two angles. 

Thus, if the log tan = 9.47451 10, the angles are 16 36' 17" in Quadrant I. 
and 196 36' 17" in Quadrant III. ; but if the log tan = 9.47451 n 10, the angles 
are 163 ^3' 43" in Quadrant II. and 343 23' 43" in Quadrant IV. 

To remove all ambiguity, further conditions are required, or a 
knowledge of the special circumstances connected with the problem 
in question. 

TABLE IV. 

30. This table (page 50) must be used when great accuracy is 
desired in working with angles between and 2, or between 88 
and 90. 

The values of S and T are such that when the angle a is 
expressed in seconds, 

S = log sin a log a", 
T = log tan a log a". 

Hence follow the formulas given on page 50. 

The values of S and T are printed with the characteristic 10 too 
large, and in using them 10 must always be annexed. 



Find log sin 58' 17". 

58' 17" = 3497" 
log 3497 = 3.54370 

8 = 4.68555 10 
log sin 68' 17" = 8.22925 - 10 



Find log cos 88 26' 41.2". 

90 - 88 26' 41.2" = 1 33' 18.8" 

= 5598.8" 
log 5598.8 = 3.74809 

S = 4.68552 10 
log cos 88 26' 41.2" = 8.43361 - 10 



INTRODUCTION. XIX 



Find log tan 52' 47.5", 

52' 47. 5" = 3167. 5" 

log 3167. 5 = 3. 50072 

T = 4.68561 - 10 
log tan 52' 47.5" = 8.18633 - 10 



Find log tan 89 54' 37.362". 

90 89 54' 37.362" = 5' 22.638" 

= 322.638" 
log 322.638 = 2.50871 

T= 4.68558 -10 

log cot 89 54' 37.362" = 7.19429 - 1O 
log tan 89 54' 37.362" = 2.80571 



Find the angle, if log sin = 6.72306 10. 

6.72306-10 
S = 4.68557 - 10 
Subtract, 2.03749 = log 109.015 

109.015" = I' 49.015". 

i 

Find the angle for which log cot = 1.67604. 

colog cot = 8.32396 10 
T = 4.68564 - 10 

Subtract, 3.63832 = log 4348.3 

4348.3" = 1 12' 28.3". 

Find the angle for which log tan = 1.55407. 

colog tan = 8.44593 10 
T = 4.68569-10 
Subtract, 3.76024 = log 5757.6 

5757.6" = 1 35' 57.6", 
and 90 - 1 35' 57.6" = 88 24' 2.4". 
Therefore, the angle required is 88 24' 2. 4". 

TABLE V. 

31. This table (p. 51), containing the circumferences and areas 
of circles, does not require explanation. 

TABLE VI. 

32. Table VI. (pp. 52-69) contains the natural sines, cosines, 
tangents, and cotangents of angles from to 90, at inter- 
vals of 1'. If greater accuracy is desired it may be obtained 
by interpolation. 

NOTE. In preparing the preceding explanations, we have made free use 
of the Logarithmic Tables by F. G. Gauss. For Table VI. we are indebted 
to D. Carhart. 

TABLE VII. 

33. This table (pp. 70-75) gives the latitude and departure to 
three places of decimals for distances from 1 to 10, corresponding 
to bearings from to 90 at intervals of 15'. 



XX 



LOGARITHMS. 



If the bearing does not exceed 45 it is found in the Ze/Miand 
column, and the designations of the columns under " Distance " 
are taken from the top of the page ; but if the bearing exceeds 
45, it is found in the right-h&nd. column, and the designations 
of the columns under " Distance " are taken from the bottom of 
the page. 

The method of using the table will be made plain by the follow- 
ing examples : 

(1) Let it be required to find the latitude and departure of the 
course N. 35 15' E. 6 chains. 

On p. 75, left-hand column, look for 35 15' ; opposite this bearing, in the 
vertical column headed "Distance 6," are found 4.900 and 3.463 under the 
headings "Latitude" and "Departure" respectively. Hence, latitude or 
northing = 4.900 chains, and departure or easting = 3.463 chains. 

(2) Let it be required to find the latitude and departure of the 
course S. 87 W. 2 chains. 

As the bearing exceeds 45, we look in the right-hand column of p. 70, and 
opposite 87 in the column marked ' ' Distance 2 ' ' we find (taking the designa- 
tions of the columns from the bottom of the page) latitude = 0.105 chains, and 
departure = 1.997 chains. Hence, latitude or southing = 0. 105 chains, and 
departure or westing = 1.997 chains. 

(3) Let it be required to find the latitude and departure of the 
course 1ST. 15 45' W. 27.36 chains. 

In this case we find the required numbers for each figure of the distance 
separately, arranging the work as in the following table. In practice, only the 
last columns under "Latitude " and " Departure " are written. 



DISTANCE. 


LATITUDE. 


DEPARTURE. 


20 = 2 X 10 
7 
0.3 =3 -=-10 
0.06 = 6 + 100 


1.925 X 10 = 19.25 
6.737 
2.887 -r 10 =0.289 
5. 775 -r 100 = 0.058 


0.543 X 10 = 5.43 
1.90 
0.814 -r 10 =0.081 
1.628 -r 100 = 0.016 


27.36 


26.334 


7.427 



Hence, latitude = 26.334 chains, and departure = 7.427 chains. 



TABLE L 


THE 


COMMON OE BEIGGS LOGAEITHMS 


OF THE 


NATURAL NUMBERS 


From 1 to 10000. 


1-100 


N log 


N log 


N log 


N log 


N log 


1 0. 00 000 
2 0. 30 103 
3 0.47712 
4 0. 60 206 
5 0.69897 


21 1.32222 
22 1.34242 
23 1.36173 
24 1.38021 
25 1.39794 


41 1. 61 278 

42 1. 62 325 
43 1. 63 347 
44 1.64345 
45 1.65321 


61 1. 78 533 

62 1. 79 239 
63 1. 79 934 
64 1.80618 
65 1. 81 291 


81 1.90849 
82 1.91381 
83 1. 91 908 
84 1. 92 428 
85 1. 92 942 


6 0.77815 

7 0.84510 
8 0.90309 
9 0. 95 424 
10 1.00000 


26 1. 41 497 

27 1.43136 
28 1.44716 
29 1.46240 
30 1.47712 


46 1. 66 276 

47 1. 67 210 
48 1.68124 
49 1.69020 
50 1.69897 


66 1. 81 954 

67 1. 82 607 
68 1.83251 
69 1.83885 
70 1. 84 510 


86 1.93450 
87 1. 93 952 
88 1.94448 
89 1.94939 
90 1. 95 424 


11 1.04139 
12 1.07918 
13 1.11394 
14 1. 14 613 
15 1. 17 609 


31 1.49136 
32 1.50515 
33 1. 51 851 
34 1.53148 
35 1.54407 


51 1.70757 
52 1. 71 600 
53 L 72 428 
54 1. 73 239 
55 1.74036 


71 1. 85 126 

72 1.85733 
73 1. 86 332 
74 1.86923 
75 1.87506 


91 1. 95 904 

92 1.96379 
93 1. 96 848 
94 1.97313 
95 1. 97 772 


16 1.20412 
17 1.23045 
18 1. 25 527 
19 1.27875 
20 1.30103 


36 1.55630 
37 1.56820 
38 1.57978 
39 1.59106 
40 1.60206 


56 1. 74 819 

57 1.75587 
58 1.76343 
59 1.77085 
60 1. 77 815 


76 1.88081 
77 1.88649 
78 1. 89 209 
79 1.89763 
80 1.90309 


96 1.98227 
97 1.98677 
98 1. 99 123 
99 1.99564 
100 2. 00 000 


N log 


N log 


N log 


N log 


N log 



1-100 



100-150 



3T 


01234 


56789 


100 

101 
102 
103 
104 


00000 00043 00087 00130 00173 
00432 00475 00518 00561 00604 
00860 00903 00945 00988 01030 
01284 01326 01368 01410 01452 
01703 01745 01787 01828 01870 


00217 00260 00303 00346 00389 
00647 00689 00732 00775, 00817 
01 072 01 115 01 157 01 199 01 242 
01494 01536 01578 01620 01662 
01912 01953 01995 02036 02078 


105 

106 
107 
108 
109 


02119 02160 02202 02243 02284 
02531 02572 02612 02653 02694 
02938 02979 03019 03060 03100 
03342 03383 03423 03463 03503 
03743 03782 03822 03862 03902 


02325 02366 02407 02449 02490 
02735 02776 02816 02857 02898 
03141 03181 03222 03262 03302 
03543 03583 03623 03663 03703 
03941 03981 04021 04060 04100 


no 

111 

112 
113 
114 


04139 04179 04218 04258 04297 
04532 04571 04610 04650 04689 
04922 04961 04999 05038 05077 
05308 05346 05385 05423 05461 
05690 05729 05767 05 805. 05843 


04336 04376 04415 04454 04493 
04727 04766 04805 04844 04883 
05 115 05 154 05 192 05 231 05 269 
05500 05538 05576 05614 05652 
05881 05918 05956 05994 06032 


115 

116 
117 
118 
119 


06070 06108 06145 06183 06221 
06446 06483 06521 06558 06595' 
06819 06856 06893 06930 06967 
07188 07225 07262 07298 07335 
07555 07591 07628 07664 07700 


06258 06296 06333 06371 06408 
06633 06670 06707 06744 06781 
07004 07041 07078 07115 07151 
07372 07408 07445 07482 07518' 
07737 07773 07809 07846 07882 


120 

121 
122 
123 
124 


07918 07954 07990 08027 08063 
08279 08314 08350 08386 08422 
08636 08672 08707 08743 08778 
08991 09026 09061 09096 09132 
09342 09377 09412 09447 09482 


08099 0813^ 08171 08207 08243 
08458 08493 08529 08565 08600 
08814 08849 08884 08920 08955 
09167 09202 09237 09272 09307 
09517 09552 09587 09621 09656 


125 

126 
127 
128 
129 


09691 09726 09760 09795 09830 
10037 10072 10106 10140 10175 
10380 10415 10449 10483 10517 
10721 10755 10789 10823 10857 
11059 11093 11126 11160 11193 


09864 09899 09934 09968 10003 
10209 10243 10278 10312 10346 
10551 10585 10619 10653 10687 
10890 10924 10958 10992 11025 
11227 11261 11294 11327 11361 


130 

131 
132 
133 
134 


11394 11428 11461 11494 11528 
11727 11760 11793 11826 11860 
12057 12090 12123 12156 12189 
12385 12418 12450 12483 12516 
12710 12743 12775 12808 12840 


11561 11594 11628 11661 11694 
11893 11926 11959 11992 12024 
12222 12254 12287 12320 12352 
12548 12581 12613 12646 12678 
12 372 12905 12937 12969 13001 


135 

136 
137 
138 
139 


13 033 13 066 13 098 13 130 13 162 
13354 13386 13418 13450 13481 
13672 13704 13735 13767 13799 
13988 14019 14051 14082 14114 
14301 14333 14364 14395 14426 


13 194 13 226 13 258 13 290 13 322 
13513 13545 13577 13609 13640 
13830 13862 13893 13925 13956 
14145 14176 14208 14239 14270 
14457 14489 14520 14551 14582 


140 

141 
142 
143 
144 


14613 14644 14675 14706 14737 
14922 14953 14983 15014 15045 
15229 15259 15290 15320 15351 
15534 15564 15594 15625 15655 
15836 15866 15897 15927 15957 


14768 14799 14829 14860 14891 
15 076 15 106 15 137 15 168 15 198 
15 381 15 412 15 442 15 473 15 503 
15685 15715 15746 15776 15806 
15987 16017 16047 16077 16107 


145 

146 
147 
148 
149 


16137 16167 16197 16227 16256 
16435 16465 16495 16524 16554 
16732 16761 16791 16820 16850 
17026 17056 17085 17114 17143 
17319 17348 17377 17406 17435 


16286 16316 16346 16376 16406 
16584 16613 16643 16673 16702 
16879 16909 16938 16967 16997 
17173 17202 17231 17260 17289 
17464 17493 17522 17551 17580 


150 


17609 17638 17667 176% 17725 


17754 17782 17811 17840 17869 


N 


O 1 2 3 4 


56789 



100-150 



150-200 



N 


01234 


56789 


150 

151 
152 
153 
154 


17609 17638 17667 17696 17725 
17898 17926 17955 17984 18013 
18184 18213 18241 18270 18298 
18469 18498 18526 18554 18583 
18752 18780 18808 18837 18 865. 


17754 17782 17811 17840 17869 
18041 18070 18099 18127 18156 
18327 18355 18384 18412 18441 
18611 18639 18667 18696 18724 
18893 18921 18949 18977 19005 


155 

156 
157 

158 
159 


19033 19061 19089 19117 19145 
19312 19340 19368 19396 19424 
19590 19618 19645 19673 19700 
19866 19893 19921 19948 19976 
20140 20167 20194 20222 20249 


19173 19201 19229 19257 19285 
19451 19479 19507 19535 19562 
19728 19756 19783 19811 19838 
20003 20030 20058 20085 20112 
20276 20303 20330 20358 20385. 


160 

161 
162 
163 
164 


20412 20439 20466 20493 20520 
20683 20710 20737 20763 20790 
20952 20978 21005 21032 21059 
21219 21245 21272 21299 21325 
21484 21511 21537 21564 21590 


20548 20575 20602 20629 20656 
20817 20844 20871 20898 20925 
21085 21112 21139 21165 21192 
21352 21378 21405 21431 21458 
21617 21643 21669 21696 21722 


165 

166 
167 
168 
169 


21748 21775 21801 21827 21854 
22011 22037 22063 22089 22115 
22272 22298 22324 22350 22376 
22531 22557 22583 22608 22634 
22789 22814 22840 22866 22891 


21880 21906 21932 21958 21 985. 
22141 22167 22194 22220 22246 
22401 22427 22453 22479 22505 
22660 22686 22712 22737 22763 
22917 22943 22968 22994 23019 


170 

171 
172 

173 

174 


23 045 23 070 23 096 23 121 23 147 
23300 23325 23350 23376 23401 
23553 23578 23603 23629 23654 
23805 23830 23855 23880 23905 
24055 24080 24105 24130 24155 


23172 23198 23223 23249 23274 
23426 23452 23477 23502 23528 
23679 23704 23729 23754 23779 
23930 23955 23980 24005 24030 
24180 24204 24229 24254 24279 


175 

176 
177 
178 
179 


24304 24329 24353 24378 24403 
24551 24576 24601 24625 24650 
24797 24822 24846 24871 24895 
25042 25066 25091 25115 25139 
25285 25310 25334 25358 25382 


24428 24452 24477 24502 24527 
24674 24699 24724 24748 24773 
24920 24944 24969 24993 25018 
25 164 25 188 25 212 25 237 25 261 
25406 25431 25455 25479 25503 


180 

181 
182 
183 

184 


25527 25551 25575 25600 25624 
25768 25792 25816 25840 25864 
26007 26031 26055 26079 26102 
26245 26269 26293 26316 26340 
26482 26505 26529 26553 26576 


25648 25672 25696 25720 25744 
25888 25912 25935 25959 25983 
26126 26150 26174 26198 26221 
26364 26387 26411 26435 26458 
26600 26623 26647 26670 26694 


185 

186 
187 
188 
189 


26717 26741 26764 26788 26811 
26951 26975 26998 27021 27045 
27184 27207 27231 27254 27277 
27416 27439 27462 27485 27508 
27646 27669 27692 27715 27738 


26834 26858 26881 26905. 26928 
27068 27091 27114 27138 27161 
27300 27323 27346 27370 27393 
27531 27554 27577 27600 27623 
27761 27784 27807 27830 27852 


19O 

191 
192 
193 
194 


27875 27898 27921 27944 27-967 
28103 28126 28149 28171 28194 
28330 28353 28375 28398 28421 
28556 28578 28601 28623 28646 
28780 28803 28825 28847 28870 


27989 28012 28035 28058 28081 
28217 28240 28262 28285 28307 
28443 28466 28488 28511 28533 
28668 28691 28713 28735 28758 
28892 28914 28937 28959 28981 


195 

196 
197 
198 
199 


29003 29026 29048 29070 29092 
29226 29248 29270 29292 29314 
29447 29469 29491 29513 29*535 
29667 29688 29710 29732 29754 
29885 29907 29929 29951 29973 


29115 29137 29159 29181 29203 
29336 29358 29380 29403 29425 
29557 29579 29601 29623 2964*5 
29776 29798 29820 29842 29863 
29994 30016 30038 30060 30081 


2OO 


30103 30125, 30146 30168 30190 


30211 30233 30255 30276 30298 


N 


01234 


56789 



150-200 



200-250 



K 


O 1 2 3 4 


56789 


2OO 


30103 30123 30146 30168 30190 


30211 30233 30255 30276 30298 


201 


30320 30341 30363 30384 30406 


30428 30449 30471 30492 30514 


202 


30535 30557 30578 30600 30621 


30643 30664 30685 30707 30728 


203 


30750 30771 30792 30814 30835 


30856 30878 30899 30920 30942 


204 


30963 30984 31006 31027 31048 


31 069 31 091 31 112 31 133 31 154 


205 


31 175 31 197 31 218 31 239 31 260 


31281 31302 31323 31345 31366 


206 


31387 31408 31429 31450 31471 


31492 31513 31534 31555 31576 


207 


31597 31618 31639 31660 31681 


31 702 31 723 31 744 31 765 31 785 


208 


31806 31827 31848 31869 31890 


31911 31931 31952 31973 31994 


209 


32015 32035 32056 32077 32098 


32118 32139 32160 32181 32201 


21O 


32222 32243 32263 32284 32305 


32325 32346 32366 32387 32408 


211 


32428 32449 32469 32490 32510 


32531 32552 32572 32593 32613 


212 


32634 32654 32675 32695 32715 


32736 32756 32777 32797 32818 


213 


32838 32858 32879 32899 32919 


32940 32960 32980 33001 33021 


214 


33041 33062 33082 33102 33122 


33143 33163 33183 33203 33224 


215 


33244 33264 33284 33304 33325 


33345 33365 33385 33405 33425 


216 


33445 33465 33486 33506 33526 


33546 33566 33586 33606 33626 


217 


33646 33666 33686 33706 33726 


33746 33766 33786 33806 33826 


218 


33846 33866 33885 33905 33925 


33945 33965 33985 34005 34025 


219 


34044 34064 34084 34104 34124 


34143 34163 34183 34203 34223 


22O 


34242 34262 34282 34301 34321 


34341 34361 34380 34400 34420 


221 


34439 34459 34479 34498 34518 


34537 34557 34577 34596 34616 


222 


34635 34655 34674 34694 34713 


34733 34753 34772 34792 34811 


223 


34830 34850 34869 34889 34908 


34928 34947 34967 34986 35005 


224 


35025 35044 35064 35083 35102 


35 122 35 141 35 160 35 180 35 199 


225 


35218 35238 35257 35276 35295 


35315 35334 35353 35372 35392 


226 


35411 35430 35449 35468 35488 


35507 35526 35545 35564 35583 


227 


35603 35622 35641 35660 35679 


35698 35717 35736 35755 35774 


228 


35793 35813 35832 35851 35870 


35889 35908 35927 35946 35965 


229 


35984 36003 36021 36040 36059 


36078 36' 097 36116 36135 36154 


230 


36173 36192 36211 36229 36248 


36267 36286 36305 36324 36342 


231 


36361 36380 36399 36418 36436 


36455 36474 36493 36511 36530 


232 


36549 36568 36586 36605 36624 


36642 36661 36680 36698 36717 


233 


36736 36754 36773 36791 36810 


36829 36847 36866 36884 36903 


234 


36922 36940 36959 36977 36996 


37014 37033 37051 37070 37088 


235 


37107 37125 37144 37162 37181 


37199 37218 37236 37254 37273 


236 


37291 37310 37328 37346 37365 


37383 37401 37420 37438 37457 


237 


37475 37493 37511 37530 37548 


37566 37585 37603 37621 37639 


238 


37 658 37 676 3* 594 37 712 37 731 


37749 37767 37785 37803 37822 


239 


37840 37858 3', '76 37894 37912 


.37931 37949 37967 37985 38003 


240 


38021 38039 38057 38075 38093 


38112 38130 38148 38166 38184 


241 


38 202 38 220 38 238 38 256 38 274 


38292 38310 38328 38346 38364 


242 


38382 38399 38417 38435 38453 


38471 38489 38507 38525 38543 


243 


38561 38578 38596 38614 38632 


38650 38668 38686 38703 38721 


244 


38739 38757 38775 38792 38810 


38828 38846 38863 38881 38899 


245 


38917 38934 38952 38970 38987 


39005 39023 39041 39058 39076 


246 


39094 39111 39129 39146 39164 


39182 39199 39217 39235 39252 


247 


39270 39287 39 305. 39322 39340 


39358 39375 39393 39410 39428 


248 


39445 39463 39480 39498 39515 


39533 39550 39568 39585 39602 


249 


39620 39637 39655 39672 39690 


39707 39724 39742 39759 39777 


250 


39794 39811 39829 39846 39863 


39881 39898 39915 39933 39950 


N 


O 1 2 3 4 


56789 



200-250 



250-300 



N 


O 1 2 3 4 


56789 


25O 

251 
252 

253 

254 


39794 39811 39829 39846 39863 
39967 39985 40002 40019 40037 
40140 40157 40175 40192 40209 
40312 40329 40346 40364 40381 
40483 40500 40518 40535 40552 


39881 39898 39915 39933 39950 
40054 40071 40088 40106 40123 
40226 40243 40261 40278 40295 
40398 40415 40432 40449 4O466 
40569 40586 40603 40620 40637 


255 

256 

257 
258 
259 


40654 40671 40688 40705 40722 
40824 40841 40858 40875 40892 
40993 41010 41027 41044 41061 
41 162 41 179 41 196 41 212 41 229 
41330 41347 41363 41380 41397 


40739 40756 40773 40790 40807 
40909 40926 40943 40960 40976 
41 078 41 095 41 111 41 128 41 145 
41 246 41 263 41 280 41 296 41 313 
41414 41430 41447 41464 41481 


260 

261 
262 
263 
264 


41497 41514 41531 41547 41564 
41 664 41 681 41 697 41 714 41 731 
41830 41847 41863 41880 41896 
41996 42012 42029 42045 42062 
42 160 42 177 42 193 42 210 42 226 


41581 41597 41614 41631 41647 
41 747 41 764 41 780 41 797 41 814 
41913 41929 41946 41963 41979 
42078 42095 42111 42127 42 H4 
42243 42259 42275 42292 42306 


265 

266 
267 
268 
269 


42325 42341 42357 42374 42390 
42488 42504 42521 42537 42553 
42651 42667 42684 42700 42716 
42813 42830 42846 42862 42878 
42975 42991 43008 43024 43040 


42406 42423 42439 42455 42472 
42570 42586 42602 42619 42635 
42732 42749 42765 42781 42797 
42894 42911 42927 42943 42959 
43 056 43 072 43 088 43 104 43 120 


270 

271 
272 
273 
274 


43 136 43 152 43 169 43 185 43 201 
43297 43313 43329 43345 43361 
43457 43473 43489 43505 43521 
43616 43632 43648 43664 43680 
43 775 43 791 43 807 43 823 43 838 


43217 43233 43249 43265 43281 
43377 43393 43409 43425 43441 
43537 43553 43569 43584 43600 
43696 43712 43727 43743 43759 
43854 43870 43886 43902 43917 


275 

276 

277 
278 
279 


43933 43949 43965 43981 43996 
44091 44107 44122 44138 44154 
44248 44264 44279 44295 44311 
44404 44420 44436 44451 44467 
44560 44576 44592 44607 44623 


44012 44028 44044 44059 44075 
44170 44185 44201 44217 44232 
44326 44342 44358 44373 44389 
44483 44498 44514 44529 44545 
44638 44654 44669 44685 44700 


280 

281 

282 
283 
284 


44716 44731 44747 44762 44778 
44871 44886 44902 44917 44932 
45025 45040 45056 45071 45086 
45 179 45 194 45 209 45 225 45 240 
45332 45347 45362 45378 45393 


44793 44809 44824 44840 44855 
44948 44963 44979 44994 45010 
45 102 45 117 45 133 45 148 45 163 
45255 45271 45286 45301 45317 
45408 45423 45439 45454 45469 


285 

286 

287 
288 
289 


45484 45500 45515 45530 45545 
45637 45652 45667 45682 45697 
45788 45803 45818 45834 45849 
45939 45954 45969 45984 46000 
46090 46105 46120 46135 46150 


45561 45576 45591 45606 45621 
45 712 45 728 45 743 45 758 45 773 
45864 45879 45894 45909 45924 
46015 46030 46045 46060 46075 
46165 46180 46195 46210 46225 


29O 

291 
292 
293 

294 


46240 46255 46270 46285 46300 
46389 46404 46419 46434 46449 
46538 46553 46568 46583 46598 
46687 46702 46716 46731 46746 
46835 46850 46864 46879 46894 


46315 46330 46345 46359 46374 
46464 46479 46494 46509 46523 
46613 46627 46642 46657 46672 
46761 46776 46790 46805 46820 
46909 46923 46938 46953 46967 


295 

296 
297 
298 
299 


46982 46997 47012 47026 47041 
47129 47144 47159 47173 47188 
47276 47290 47305 47319 47334 
47422 47436 47451 47465 47480 
47567 47582 47596 47 61J. 47625 


47056 47070 47085 47100 47114 
47202 47217 47232 47246 47261 
47349 47363 47378 47392 47407 
47494 47509 47524 47538 47553 
47640 47654 47669 47683 47698 


3OO 


47712 47727 47741 47756 47770 


47784 47799 47813 47828 47842 


N 


O 1 2 3 4 


56789 



250 - 300 



300-350 



N 


01234 


56789 


3OO 

301 
302 
303 
304 


47712 47727 47741 47756 47770 
47857 47871 47885 47900 47914 
48001 48015 48029 48044 48058 
48144 48159 48173 48187 48202 
48287 48302 48316 48330 48344 


47784 47799 47813 47828 47842 
47929 47943 47958 47972 47986 
48073 48087 48101 48116 48130 
48216 48230 48244 48259 48273 
48359 48373 48387 48401 48416 


3O5 

306 
307 
308 
309 


48430 48444 48458 48473 48487 
48572 48586 48601 48615 48629 
48714 48728 48742 48756 48770 
48855 48869 48883 48897 48911 
48996 49010 49024 49038 49052. 


48501 48515 48530 48544 48558 
48643 48657 48671 48686 48700 
48785 48799 48813 48827 48841 
48926 48940 48954 48968 48982 
49066 49080 49094 49108 49122 


310 

311 
312 
313 
314 


49136 49150 49164 49178 49192 
49276 49290 49304 49318 49332 
49415 49429 49443 49457 49471 
49554 49568 49582 49596 49610 
49693 49707 49721 49734 49748 


49206 49220 49234 49248 49262 
49346 49360 49374 49388 49402 
49485 49499 49513 49527 49541 
49624 49638 49651 49665 49679 
49762 497T& 49790 49803 49817 


315 

316 
317 
318 
319 


49831 49845 49859 49872 49886 
49969 49982 49996 50010 50024 
50106 50120 50133 50147 50161 
50243 50256 50270 50284 50297 
50379 50393 50406 50420 50433 


49900 49914 49927 49941 49955 
50037 50051 50065 50079 50092 
50174 50188 50202 50215 50229 
50311 50325 50338 50352 50365 
50447 50461 50474 50488 50501 


320 

321 

322 
323 
324 


50515 50529 50542 50556 50569 
50651 50664 50678 50691 50705 
50786 50799 50813 50826 50840 
50920 50934 50947 50961 50974 
51 055 51 068 51 081 51 095 51 108 


SO 583 50596 50610 50623 50637 
50718 50732 50745 50759 50772 
50853 50866 50880 50893 50907 
50987 51001 51014 51028 51041 
51121 51135 51148 51162 51175 


325 

326 
327 

328 
329 


51 188 51 202 51 215 51 228 51 242 
51322 51335 51348 51362 51375 
51455 51468 51481 51495 51508 
51587 51601 51614 51627 51640 
51720 51733 51746 51759 51772 


51255 51268 51282 51 295 51308 
51388 51402 51415 51428 51441 
51521 51534 51548 51561 51574 
51654 51667 51680 51693 51706 
51786 51799 51812 51825 51838 


330 

331 
332 
333 
334 


51851 51865 51878 51891 51904 
51983 51996 52009 52022 52035 
52 114 52 127 52 140 52 153 52 166 
52244 52257 52270 52284 52297 
52375 52388 52401 52414 52427 


51917 51930 51943 51957 51970 
52048 52061 52075 52088 52101 
52179 52192 52205 52218 52231 
52310 52323 52336 52349 52362 
52440 52453 52466 52479 52492 


335 

336 
337 
338 
339 


52504 52517 52530 52543 52556 
52634 52647 52660 52673 52686 
52763 52776 52789 52802 52815 
52892 52905 52917 52930 52943 
53020 53033 53046 53058 53071 


52569 52582 52595 52608 52621 
52699 52711 52724 52737 52750 
52827 52840 52853 52866 52879 
52956 52969 52982 52994 53007 
53084 53097 53110 53122 53135 


340 

341 
342 
343 
344 


53 148 53 161 53 173 53 186 53 199 
53275 53288 53301 53314 53326 
53403 53415 53428 53441 53453 
53 529 53 542 53 555 53 567 53 580 
53656 53668 53681 53694 53706 


53212 53224 53237 53250 53263 
53339 53352 53364 53377 53390 
53466 53479 53491 53504 53517 
53593 53605 53618 53631 53643 
53719 53732 53744 53757 53769 


345 

346 
347 
348 
349 


53782 53794 53807 53820 53832 
53908 53920 53933 53945 53958 
54033 54045 54058 54070-54083 
54158 54170 54183 54195 54208 
54283 54295 54307 54320 54332 


53845 53857 53870 53882 53895 
53970 53983 53995 54008 54020 
54095 54108 54120 54133 54145 
54220 54233 54245 54258 54270 
54345 54357 54370 54382 54394 


350 


54407 54419 54432 54444 54456 


54469 54481 54494 54506 54518 


N. 


01234 


56789 



300-350 



350-400 



N 


01234 


56789 


350 

351 
352 
353 
354 


54407 54419 54432 54444 54456 
54531 54543 54555 54568 54580 
54654 54667 54679 54691 54704 
54777 54790 54802 54814 54827 
54900 54913 54 923 54937 54949 


54469 54481 54494 54506 54518 
54593 54605 54617 54630 54642 
54716 54728 54741 54753 54765 
54839 54851 54864 54876 54888 
54962 54974 54986 54998 55011 


355 

356 
357 

358 
359 


55023 55035 55047 55060 55072 
55 145 55 157 55 169 55 182 55 194 
55267 55279 55291 55303 55315 
55388 55400 55413 55425 55437 
55509 55522 55534 55546 55558 


55 084 55 096 55 108 55 121 55 133 
55206 55218 55230 55242 55255 
55328 55340 55352 55364 55376 
55449 55461 55473 55485 55497 
55570 55582 55594 55606 55618 


360 

361 
362 
363 
364 


55630 55642 55654 55666 55678 
55751 55763 55773 55787 55799 
55871 55883 55895 55907 55919 
55991 56003 56015 56027 56038 
56110 56122 56134 56146 56158 


55691 55703 55715 55727 55739 
55811 55823 55835 55847 55859 
55931 55943 55955 55967 55979 
56050 56062 56074 56086 56098 
56170 56182 56194 56205 56217 


365 

366 

367 
368 
369 


56229 56241 56253 56265 56277 
56348 56360 56372 56384 56396 
56467 56478 56490 56502 56514 
56585 56597 56608 56620 56632 
56703 56714 56726 56738 56750 


56289-56301 56312 56324 56336 
56407 56419 56431 56443 56455 
56526 56538 56549 56561 56573 
56644 56656 56667 56679 56691 
56761 56773 56785 56797 56808 


370 

371 
372 
373 
374 


56820 56832 56844 56855 56867 
56937 56949 56961 56972 56984 
57054 57066 57078 57089 57101 
57171 57183 57194 57206 57217 
57287 57299 57310 57322 57334 


56879 56891 56902 56914 56926 
56996 57008 57019 57031 57043 
57113 57124 57136 5714*8 57159 
57229 57241 57252 57264 57276 
57345 57357 57368 57380 57392 


375 

376 
377 
378 
379 


57403 57415 57426 57438 57449 
57519 57530 57542 57553 57565 
57634 57646 57657 57669 57680 
57749 57761 57772 57784 57795 
57864 57875 57887 57898 57910 


57461 57473 57484 57496 57507 
57576 57588 57600 57611*57623 
57692 57703 57715 57726 57738 
57807 57818 57830 57841 57852 
57921 57933 57944 57955 57967 


38O 

381 
382 
383 
384 


57978 57990 58001 58013 58024 
58092 58104 58115 58127 58138 
58206 58218 58229 58240 58252 
58320 58331 58343 58354 58365 
58433 58444 58456 58467 58478 


58035 58047 58058 58070 58081 
58149 58161 58172 58184 58195 
58263 58274 58286 58297 58309 
58377 58388 58399 58410 58422 
58490 58501 58512 58524 58535 


385 

386 
387 
388 
389 


58546 58557 58569 58580 58591 
58659 58670 58681 58692 58704 
58771 58782 58794 58805 58816 
58883 58894 58906 58917 58928 
58995 59006 59017 59028 59040 


58602 58614 58625 58636 58647 
58715 58726 58737 58749 58760 
58 827 58 838 58 850 58 861 58 872 
58939 58950 58961 58973 58984 
59051 59062 59073 59084 59095 


390 

391 
392 
393 

394 


59106 59118 59129 59140 59151 
59218 59229 59240 59251 59262 
59329 59340 59351 59362 59373 
59439 59450 59461 59472 59483 
59550 59561 59572 59583 59594 


59162 59173 59184 59195 59207 
59273 59284 59295 59306 59318 
59384 59395 59406 59417 59428 
59494 59506 59517 59528 59539 
59605 59616 59627 59638 59649 


395 

396 
397 
398 
399 


59660 59671 59682 59693 59704 
59770 59780 59791 59802 59813 
59879 59890 59901 59912 59923 
59988 59999 60010 60021 60032 
60097 60108 60119 60130 60141 


59715 59726 59737 59748 59759 
59824 59835 59846 59857 59868 
59934 59945 59956 59966 59977 
60043 60054 60065 60076 60086 
60152 60163 60173 60184 60195 


4OO 


60206 60217 60228 60239 60249 


60260 60271 60282 60293 60304 


N 


O 1 2 3 4 


5 67 8 9 



350-400 



400-450 



N 


01234 


56789 


400 

401 
402 
403 
404 


60206 60217 60228 60239 60249 
60314 60325 60336 60347 60358 
60423 60433 60444 60455 60466 
60531 60541 60552 60563 60574 
60638 60649 60660 60670 60681 


60260 60271 60 282 60293 60304 
60369 60379 60390 60401 60412 
60477 60487 60498 60509 60520 
60584 60595 60606 60617 60627 
60692 60703 60713 60724 60 735. 


4O5 

406 
407 
408 
409 


60746 60756 60767 60778 60788 
60853 60863 60874 60885 60895 
60 959 60 970 60 981 60 991 61 002 
61066 61077 61087 61098 61109 
61 172 61 183 61 194 61 204 61 215 


60799 60810 60821 60831 60842 
60906 60917 60927 60938 60949 
61013 61023 61034 61045 61055 
61 119 61 130 61 140 61 151 61 162 
61225 61236 61247 61257 61268 


41O 

411 
412 
413 
414 


61278 61289 61300 61310 61321 
61384 61395 61405 61416 61426 
61490 61500 61511 61521 61532 
61595 61606 61616 61627 61637 
61700 61711 61721 61731 61742 


61331 61342 61352 61363 61374 
61437 61448 61458 61469 61479 
61542 61553 61563 61574 61584 
61648 61658 61669 61679 61690 
61752 61763 61773 61784 61794 


415 

416 
417 
418 
419 


61805 61815 61826 61836 61847 
61909 61920 61930 61941 61951 
62014 62024 62034 62045 62055 
62118 62128 62138 62149 62159 
62221 62232 62242 62252 62263 


61857 61868 61878 61888 61899 
61962 61972 61982 61993 62003 
62066 62076 62086 62097 62107 
62170 62180 62190 62201 62211 
62273 62284 62294 62304 62315 


420 

421 
422 

423 
424 


62325 62335 62346 62356 62366 
62428 62439 62449 62459 62469 
62531 62542 62552 62562 62572 
62634 62644 62655 62665 62675 
62737 62747 62757 62767 62778 


62377 62387 62397 62408 62418 
62480 62490 62500 62511 62521 
62583 62593 62603 62613 62624 
62685 62696 62706 62716 62726 
62788 62798 62808 62818 62829 


425 

426 
427 

428 
429 


62839 62849 62859 62870 62880 
' 62941 62951 62961 62972 62982 
63043 63053 63063 63073 63083 
63 144 63 155 63 165 63 175 63 185 
63246 63256 63266 63276 63286 


62890 62900 62910 62921 62931 
62992 63002 63012 63022 63033 
63 094 63 104 63 114 63 124 63 134 
63 195 63 205 63 215 63 225 63 236 
63296 63306 63317 63327 63337 


430 

431 
432 
433 
434 


6334Z 63357 63367 63377 63387 
63448 63458 63468 63478 63488 
63 548 63 558 63 568 63 579 63 589 
63649 63659 63669 63679 63689 
63749 63759 63769 63779 63789 


63397 63407 63417 63428 63438 
63498 63508 63518 63528 63538 
63599 63609 63619 63629 63639 
63 699 63 709 63 719 63 729 63 739 
63799 63809 63819 63829 63839 


435 

436 
437 
438 
439 


63849 63859 63869 63879 63889 
63949 63959 63969 63979 63988 
64048 64058 64068 64078 64088 
64147 64157 64167 64177 64187 
64246 64256 64266 64276 64286 


63899 63909 63919 63929 63939 
63998 64008 64018 64028 64038 
64098 64108 64118 64128 64137 
64197 64207 64217 64227 64237 
64296 64306 64316 64326 64335 


440 

441 
442 

443 
444 


64345 64355 64365 64375 64385 
64444 64454 64464 64473 64483 
64542 64552 64562 64572 64582 
64640 64650 64660 64670 64680 
64738 64748 64758 64768 64777 


64395 64404 64414 64424 64434 
64493 64503 64513 64523 64532 
64591 64601 64611 64621 64631 
64689 64699 64709 64719 64729 
64787 64797 64807 64816 64826 


445 

446 
447 
448 
449 


64836 64846 64856 64865 64875 
64933 64943 64953 64963 64972 
65031 65040 65050 65060 65070 
65128 65137 65147 65157 65167 
65225 65234 65244 65254 65263 


64885 64895 64904 64914 64924 
64982 64992 65002 65011 65021 
65 079 65 089 65 099 65 108 65 118 
65 176 65 186 65 196 65 205 65 215 
65273 65283 65292 65302 65312 


450 


65321 65331 65341 65350 65360 


65369 65379 65389 65398 65408 


N 


01234 


56789 



400-450 



450-500 



N 


O 1 2 3 4 


56789 


450 


65321 65331 65341 65350 65360 


65369 65379 65389 65398 65 406 


451 


65418 65427 65437 65447 65456 


65466 65475 65485 65495 65504 


452 


65514 65523 65533 65543 65552 


65562 65571 65581 65591 65600 


453 


65610 65619 65629 65639 65648 


65658 65667 65677 65686 65696 


454 


65706 65715 65725 65734 65744 


65 753 65 763 65 772 65 782 65 792 


455 


65801 65811 65820 65830 65839 


65849 65858 o5 868 65877 65887 


456 


65896 65906 65916 65925 65935 


65^44 65954 65963 65973 65982 


457 


65992 66001 66011 66020 66030 


66039 66049 66058 66068 66077 


458 


66087 66096 66106 66115 66124 


66134 66143 66153 66162 66172 


459 


66181 66191 66200 66210 66219 


66229 66238 66247 66257 66266 


460 


66276 66285 66295 66304 66314 


66323 66332 66342 66351 66361 


461 


66370 66380 66389 66398 66408 


66417 66427 66436 66445 66455 


462 


66464 66474 66483 66492 66502 


66511 66521 66530 66539 66549 


463 


66558'66567 66577 66586 66596 


66605 66614 66624 66633 66642 


464 


66652 66661 66671 66680 66689 


66699 66708 66717 66727 66736 


465 


66745*66755 66764 66773 66783 


66792 66801 66811 66820 66829 


466 


66839 66848 66857 66867 66876 


66 885 66894 66904 66913 66922 


467 


66932 66941 66950 66960 66969 


66978 66987 66997 67006 67015 


468 


67025 67034 67043 67052 67062 


67071 67080 67089 67099 67108 


469 


67117 67127 67136 67145 67154 


67164 67173 67182 67191 67201 


470 


67210 67219 67228 67237 67247 


67256 67265 67274 67284 67293 


471 


67302 67311 67321 67330 67339 


67348 67357 67367 67376 67385 


472 


67394 67403 67413 67422 67431 


67440 67449 67459 67468 67477 


473 


67486 67495 67504 67514 67523 


67532 67541 67550 67560 67569 


474 


67578 67587 67596 67605 67614 


67624 67633 67642 67651 67660 

^ 


475 


67669 67679 67688 67697 67706 


67715 67724 67733 67742 67752 


476 


67761 '67770 67779 67788 67797 


67806 67815 67825 67834 67843 


477 


67852 67861 67870 67879 67888 


67897 67906 67916 67925 67934 


478 


67943 67952 67961 67970 67979 


67988 67997 68006 68015 68024 


479 


68034 68043 68052 68061 68070 


68079 68088 68097 68106 68115 


48O 


68124 68133 68142 68151 68160 


68169 68178 68187 68196 68205 


481 


68215 68224 68233 68242 68251 


68260 68269 68278 68287 68296 


482 


68305 68314 68323 68332 68341 


68350 68359 68368 68377 68386 


483 


68395 68404 68413 68422 68431 


68440 68449 68458 68467 68476 


484 


68485 68494 68502 68511 68520 


68529 68538 68547 68556 68565 


485 


68574 68583 68592 68601 68610 


68619 68628 68637 68646 68655 


486 


68664 68673 68681 68690 68699 


68708 68717 68726 68735 68744 


487 


68753 68762 68771 68780 68789 


68797 68806 68815 68824 68833 


488 


68842 68851 68860 68869 68878 


68886 68895 68904 68913 68922 


489 


68931 68940 68949 68958 68966 


68975 68984 68993 69002 69011 


490 


69020 69028 69037 69046 69055 


69064 69073 69082 69090 69099 


491 


69108 69117 69126 69135 69144 


69152 69161 69170 69179 69188 


492 


69197 69205 69214 69223 69232 


69241 69249 69258 69267 69276 


493 


69285 69294 69302 69311 69320 


69329 69338 69346 69355 69364 


494 


69373 69381 69390 69399 69408 


69417 69425 69434 69443 69452 


495 


69461 69469 69478 69487 69496 


69504 69513 69522 69531 69539 


496 


69548 69557 69566 69574 69583 


69592 69601 69609 69618 69627 


497 


69636 69644 69653 69662 69671 


69679 69688 69697 69705 69714 


498 


69723 69732 69740 69749 69758 


69767 69775 69784 69793 69801 


499 


69810 69819 69827 69836 69845 


69854 69862 69871 69880 69888 


5OO 


69897 69906 69914 69923 69932 


69 $40 69949 69958 69966 69975 


N 


O 1 2 3 4 


56789 



450-500 



10 



500-550 



N 


01234 


56789 


500 


69897 69906 69914 69923 69932 


69940 69949 69958 69966 69975 


501 


69984 69992 70001 70010 70018 


70027 70036 70044 70053 70062 


502 


70070 70079 70088 70096 70105 


70114 70122 70131 70140 70148 


503 


70157 70165 70174 70183 70191 


70200 70209 70217 70226 70234 


504 


70243 70252 70260 70269 70278 


70286 70295 70303 70312 70321 


5O5 


70329 70338 70346 70355 70364 


70372 70381 70389 70398 70406 


506 


70415 70424 70432 70441 70449 


70458 70467 70475 70484 70492 


507 


70501 70509 70518 70526 70535 


70544 70552- 70561 70569 70578 


508 


70586 70595 70603 70612 70621 


70629 70638 70646 70655 70663 


509 


70672 70680 70689 70697 70706 


70714 70723 70731 70740 70749 


51O 


70757 70766 70774 70783 70791 


70800 70808 70817 70825 70834 


511 


70842 70851 70859 70868 70876 


70885 70893 70902 70910 70919 


512 


70927 Z0935 70944 70952 70961 


70969 70978 70986 70995 71003 


513 


71012 71020 71029 71037 71046 


71054 71063 71071 71079 71088 


514 


71096 71105 71113 71122 71130 


71139 71147 71155 71164 71172 


515 


71181 71',189 71198 71206 71214 


71223 71231 71240 71248 71257 


516 


71265 71273 71282 71290 71299 


71307 71315 71324 71332 71341 


517 


71349 71357 71366 71374 71383 


71391 71399 71408 71416 71425 


518 


71433 71441 71450 71458 71466 


71475 71483 71492 71500 71508 


519 


71517 71.525 71533 71542 71550 


71559 71567 71575 71584 71592 


52O 


71600 71609 71617 71625 71634 


71642 71650 71659 71667 71675 


521 


71 684 71 692 71 700 71 709 71 717 


71 725 71 734 71 742 71 750 71 759 


522 


71 767 71 775 71 784 71 792 71 800 


71809 71817 71825 71834 71842 


523 


71850 71858 71867 71875 71883 


71892 71900 71908 71917 71925 


524 


71933 71941 71950 7J 958 71966 


71 975 71 983 71 991 71 999 72 008 


525 


72016 72024 72032 72041 72049 


72057 72066 72074 72082 72090 


526 


72099 72107 72115 72123 72132 


72140 72148 72156 72165 72173 


527 


72181 72189 72198 72206 72214 


72222 72230 72239 72247 72255 


528 


72263 72272 72280 72288 72296 


72304 72313 72321 72329 72337 


529 


72346 72354 72362 72370 72378 


72387 72395 72403 72411 72419 


530 


72428 72436 72444 72452 72460 


72 469 72 477 72 485 72 493 72 501 


531 


72509 72518 72526 72534 72542 


72550 72558 72567 72575 72583 


532 


72591 72599 72607 72616 72624 


72632 72640 72648 72656 72665 


533 


72673 72681 72689 72697 72705 


72713 72722 72730 72738 72746 


534 


72754 72762 72770 72779 72787 


72795 72803 72811 72819 72827 


535 


72835 72843 72852 72860 72868 


72876 72884 72892 72900 72908 


536 


72916 72925 72933 72941 72949 


72957 72965 72973 72981 72989 


537 


72997 73006 73014 73022 73030 


73038 73046 73054 73062 73070 


538 


73078 73086 73094 73102 73111 


73119 73127 73135 7314.3 73151 


539 


73 159 73 167 73 175 73183 73 191 


73199 73207 73215 73223 73231 


54O 


73 239 73 247 73 255 73 263 73 272 


73280 73288 73296 73304 73312 


541 


73320 73328 73336 73344 73352 


73360 73368 73376 73384 73392 


542 


73400 73408 73416 73424 73432 


73440 73448 73456 73464 73472 


543 


73480 73488 73496 73504 73512 


73520 73528 73536 73544 73552 


544 


73560 73568 73576 73584 73592 


73600 73608 73616 73624 73632 


545 


73640 73648 73656 73664 73672 


73 679 73 687 73 695 73 703 73 711 


546 


73719 73727 73735 73743 73751 


73759 73767 73775 73783 73791 


547 


73799 73807 73815 73823 73830 


73838 73846 73854 73862 73870 


548 


73878 73886 73894 73902 73910 


73918 73926 73933 73941 73949 


549 


73957 73965 73973 73981 73989 


73997 74005 74013 74020 74028 


550 


74036 74044 74052 74060 74068 


74076 74084 74092 74099 74107 


N 


01234 


56789 



500-550 



550-600 



ii 



N 


O 1 2 3 4 


56789 


55O 

551 

552 
553 
554 


74036 74044 74052 74060 74068 
74115 74123 74131 74139 74147 
74194 74202 74210 74218 74225 
74273 74280 74288 74296 74304 
74351 74359 74367 74374 74382 


74076 74084 74092 74099 74107 
74155 74162 74170 74178 74186 
74233 74241 74249 74257 74265 
74312 74320 74327 74335 74343 
74390 74398 74406 74414 74421 


555 

556 

557 
558 
559 


74429 74437 74445 74453 74461 
74507 74515 74523 74531 74539 
74586 74593 74601 74609 74617 
74663 74671 74679 74687 74695 
74741 74749 74757 74764 74772 


74468 74476 74484 74492 74500 
74547 74554 74562 74570 74578 
74624 74632 74640 74648 74656 
74 702 74 710 74 718* 74 726 74 733 
74780 74788 74796 74803 74811 


56O 

561 
562 
563 

564 


74819 74827 74834 74842 74850 
74896 74904 74912 74920 74927 
74974 74981 74989 74997 75005 
75051 75059 75066 75074 75082 
75 128 75 136 75 143 75 151 75 159 


74858 74865 74873 74881 74889 
74935 74943 74950 74958 74966 
75012 75020 75028 75035 75043 
75 089 75 097 75 105 75 113 75 120 
75166 75174 75182 75189 75197 


565 

566 
567 
568 
569 


75205 75213 75220 75228 75236 
75282 75289 75297 75305 75312 
75358 75366 75374 75381 75389 
75435 75442 75450 75458 75465 
75511 75519 75526 75534 75542 


75243 75251 75259 75266 75274 
75320 75328 75335 75343 75351 
75397 75404 75412 75420 75427 
75473 75481 75488 75496 75504 
75549 75557 75565 75572 75580 


57O 

571 
572 
573 
574 


75587 75595 75603 75610 75618 
75664 75671 75679 75686 75694 
75 740 75 747 75 755 75 762 75 770 
75815 75823 75831 75838 75846 
75891 75899 75906 75914 75921 


75626 75633 75641 75648 75656 
75702 75709 75717 75724 75732 
75778 75785 75793 75800 75808 
75853 75861 75868 75876 75884 
75929 75937 75944 75952 75959 


575 

576 

577 
578 
579 


75967 75974 75982 75989 75997 
76042 76050 76057 76065 76072 
76118 76125 76133 76140 76148 
76193 76200 76208 76215 76223 
76268 76275 76283 76290 76298 


76005 76012 76020 76027 76035 
76080 76087 76095 76103 76110 
76155 76163 76170 76178 76185 
76230 76238 76245 76253 76260 
76305 76313 76320 76328 76335 


580 

581 

582 
583 
584 


76343 76350 76358 76365 76373 
76418 76425 76433 76440 76448 
76492 76500 76507 76515 76522 
76567 76574 76582 76589 76597 
76641 76649 76656 76664 76*671 


76380 76388 76395 76403 76410 
76455 76462 76470 76477 76485 
76530 76537 76545 76552 76559 
76604 76612 76619 76626 76634 
76678 76686 76693 76701 76708 


585 

586 
587 
588 
589 


76716 76723 76730 76738 76745 
76790 76797 76805 76 812 76819 
76864 76871 76879 76886 76893 
76938 76945 76953 76960 76967 
77012 77019 77026 77034 77041 


76753 76760 76768 76775 76782 
76827 76834 76842 76849 76856 
76901 76908 76916 76923 76930 
76975 76982 76989 76997 77004 
77048 77056 77063 77070 77078 


59O 

591 
592 
593 
594 


77085 77093 77100 77107 77115 
77159 77166 77173 77181 77188 
77232 77240 77247 77254 77262 
77305 77313 77320 77327 77335 
77379 77386 77393 77401 77408 


.77122 77129 77137 77144 77151 
77195 77203 77210 77217 77225 
77269 77276 77283 77291 77298 
77342 77349 77357 77364 77371 
77415 77422 77430 77437 77444 


595 

596 
597 
598 
599 


77452 77459 77466 77474 77481 
77525 77532 77539 77546 77554 
77597 77605 77612 77619 77627 
77670 77677 77685 77692 77699 
77743 77750 77757 77764 77772 


77488 77495 77503 77510 77517 
77561 77568 77576 77583 77590 
77634 77641 77648 77656 77663 
77706 77714 77721 77728 77735 
77779 77786 77793 77801 77808 


600 


77815 77822 77830 77837 77844 


77851 77859 77866 77873 77880 


N 


1 2 3 4 


56789 



550-600 



12 



600-650 



N 


01234 


56789 


6OO 

601 
602 
603 
604 


77815 77822 77830 77837 77844 
77887 77895 77902 77909 77916 
77960 77967 77974 77981 77988 
78032 78039 78046 78053 78061 
78104 78111 78118 78125 78132 


77851 77859 77866 77873 77880 
77924 77931 77938 77945 77952 
77996 78003 78010 78017 78025 
78068 78075 78082 78089 78097 
78140 78147 78154 78161 78168 


605 

606 
607 
608 
609 


78176 78183 78190 78197 78204 
78247 78254 78262 78269 78276 
78 319 78 326 78 333 78 340 78 347 
7839G 78398 78405 78412 78419 
78462 78469 78476 78483 78490 


78211 78219 78226 78233 78240 
78283 78290 78297 78305 78312 
78355 78362 78369 78376 78383 
78426 78433 78440 78447 78455 
78497 78504 78512 78519 78526 


61O 

611 
612 
613 
614 


78533 78540 78547 78554 78561 
78604 78611 78618 78625 78633 
78675 78682 78689 78696 78704 
78746 78753 78760 78767 78774 
78817 78824 78831 78838 78845 


78569 78576 78583 78590 78597 
78640 78647 78654 78661 78668 
78711 78718 78725 78732 78739 
78781 78789 78796 78803 78810 
78852 78859 78866 78873 78880 


615 

616 
617 
618 
619 


78888 78895 78902 78909 78916 
78958 78965 78972 78979 78986 
79029 79036 79043 79050 79057 
79099 79106 79113 79120 79127 
79169 79176 79183 79190 79197 


78923 78930 78937 78944 78951 
78993 79000 79007 79014 79021 
79064 79071 79078 79085 79092 
79134 79141 79148 79155 79162 
79204 79211 79218 79225 79232 


62O 

621 
622 
623 
624 


79239 79246 79253 79260 79267 
79309 79316 79323 79330 79337 
79379 79386 79393 79400 79407 
79449 79456 79463 79470 79477 
79518 79525 79532 79539 79546 


79274 79281 79288 79295 79302 
79344 79351 79358 79365 79372 
79414 79421 79428 79435 79442 
79484 79491 79498 79505 79511 
79553 79560 79567 79574 79581 


625 

626 
627 
628 
629 


79588 79595 79602 79609 79616 
79657 79664 79671 79678 79685 
79727 79734 79741 79748 79754 
79796 79803 79810 79817 79824 
79865 79872 79879 79886 79893 


79 623 79 630 79 637 79 644 79 650 
79692 79699 79706 79713 79720 
79761 79768 79775 79782 79789 
79831 79837 79844 79851 79858 
79900 79906 79913 79920 79927 


630 

631 
632 
633 
634 


79934 79941 79948 79955 79962 
80003 80010 80017 80024 80030 
80072 80079 80085 80092 80099 
80140 80147 80154 80161 80168 
80209 80216 80223 80229 80236 


79969 79975 79982 79989 79996 
80037 80044 80051 80058 80065 
80106 80113 80120 80127 80134 
80175 80182 80188 80195 80202 
80243 80250 80257 80264 80271 


635 

636 
637 
638 
639 


80277 80284 80291 80298 80305 
80346 80353 80359 80366 80373 
80414 80421 80428 80434 80441 
80482 80489 80496 80502 80509 
80550 80557 80564 80570 80577 


80312 80318 80325 80332 80339 
80380 80387 80393 80400 80407 
80448 80455 80462 80468 80475 
80516 80523 80530 80536 80543 
80584 80591 80598 80604 80611 


640 

641 
642 
643 
644 


80618 80625 80632 80638 80645 
80686 80693 80699 80706 80713 
80754 80760 80767 80774 80781 
80821 80828 80835 80841 80848 
80889 80895 80902 80909 80916 


80652 80659 80665 80672 80679 
80720 80726 80733 80740 80747 
80787 80794 80801 80808 80814 
80855 80862 80868 80875 80882 
80922 80929 80936 80943 80949 


645 

646 
647 
648 
649 


80956 80963 80969 80976 80983 
81023 81030 81037 81043 81050 
81 090 81 097 81 104 81 111 81 117 
81158 81164 81171 81178 81184 
81224 81231 81238 81245 81251 


80990 80996 81003 81010 81017 
81057 81064 81070 81077 81084 
81 124 81 131 81 137 81 144 81 151 
81191 81198 81204 81211 81218 
81258 81265 81271 81278 81285 


65O 


81291 81298 81305 81311 81318 


81325 81331 81338 81345 81351 


N 


01234 


56789 



600-650 



650-700 



13 



N 


O 12 3 4 


5 6 7 8 9 


65O 

651 
652 
653 
654 


81291 81298 81305 81311 81318 
81358 81365 81371 81378 81385 
81425 81431 81438 81445 81451 
81491 81498 81505 81511 81518 
81558 81564 81571 81578 81584 


81325 81331 81338 81345 81351 
81391 81398 81405 81411 81418 
81458 81465 81471 81478 81485 
81525 81531 81538 81544 81551 
81591 81598 81604 81611 81617 


655 

656 

657 
658 
659 


81624 81631 81637 81644 81651 
81 690 81 697 81 704 81 710 81 717 
81 757 81 763 81 770 81 776 81 783 
81823 81829 81836 81842 81849 
81889 81895 81902 81908 81915 


81657 81664 81671 81677 81684 
81723 81730 81737 81743 81750 
81790 81796 81803 81809 81816 
81856 81862 81869 81875 81882 
81921 81928 81935 81941 81948 


660 

661 
662 
663 
664 


81954 81961 81968 81974 81981 
82020 82027 82033 82040 82046 
82086 82092 82099 82105 82112 
82151 82158 82164 82171 82178 
82217 82223 82230 82236 82243 


81987 81994 82000 82007 82014 
82053 82060 82066 82073 82079 
82119 82125 82132 82138 82145 
82184 82191 82197 82204 82210 
82249 82256 82263 82269 82276 


665 

666 
667 
668 
669 


82282 82289 82295 82302 82308 
82347 82354 82360 82367 82373 
82413 82419 82426 82432 82439 
82478 82484 82491 82497 82504 
82543 82549 82556 82562 82569 


82315 82321 82328 82334 82341 
82380 82387 82393 82400 82406 
82445 82452 82458 82465 82471 
82510 82517 82523 82530 82536 
82575 82582 82588 82595 82601 


670 

671 
672 
673 
674 


82607 82614 82620 82627 82633 
82672 82679 82685 82692 82698 
82737 82743 82750 82756 82763 
82802 82808 82814 82821 82827 
82866 82872 82879 82885 82892 


82640 82646 82653 82659 82666 
82705 82711 82718 82724 82730 
82769 82776 82782 82789 82795 
82834 82840 82847 82853 82860 
. 82898 82905 8291] 82918 82924 


675 

676 

677 
678 
679 


82930 82937 82943 82950 82956 
82995 83001 83008 83014 83020 
83059 83065 83072 83078 83085 
83 123 83 129 83 136 83 142 83 149 
83187 83193 83200 83206 83213 


82963 82969 82975 82982 82988 
83027 83033 83040 83046 83052 
83091 83097 83104 83110 83117 
83 155 83 161 83 168 83 174 83 181 
83219 83225 83232 83238 83245 


68O 

681 
682 
683 
684 


83251 83257 83264 83270 83276 
83315 83321 83327 83334 83340 
83378 83385 83391 83398 83404 
83442 83448 83455 83461 83467 
83506 83512 83518 83525 83531 


83283 83289 83296 83302 83308 
83347 83353 83359 83366 83372 
83410 83417 83423 83429 83436 
83474 83480 83487 83493 83499 
83537 83544 83550 83556 83563 


685 

686 
687 
688 
689 


83569 83575 83582 83588 83594 
83632 83639 83645 83651 83658 
836% 83702 83708 83715 83721 
83 759 83 765 83 771 83 778 83 784 
83822 83828,83835 83841 83847 


83601 83607 83613 83620 83626 
83664 83670 83677 83683 83689 
83727 83734 83740 83746 83753 
83790 83797 83803 83809 83816 
83853 83860 83866 83872 83879 


690 

691 
692 
693 
694 


83885 83891 83897 83904 83910 
83948 83954 83960 83967 83973 
84011 84017 84023 84029 84036 
84073 84080 84086 84092 84098 
84136 84142 84148 84155 84161 


83 916 83 923 83 929 83 935 83 942 
83979 83985 83992 83998 84004 
84042 84048 84055 84061 84067 
84105 84111 84117 84123 84130 
84167 84173 84180 84186 84192 


695 

696 
697 
698 
699 


84198 84205 84211 84217 84223 
84261 84267 84273 84280 84286 
84323 84330 84336 84342 84348 
84386 84392 84398 84404 84410 
84448 84454 84460 84466 84473 


84230 84236 84242 84248 84255 
84292 84298 84305 84311 84317 
84354 84361 84367 84373 84379 
84417 84423 84429 84435 84442 
84479 84485 84491 84497 84504 


70O 


84510 84516 84522 84528 84535 


84541 84547 84553 84559 84566 


N 


O 1 2 3 4 


56789 



650-700 



14 



700-750 



N 


01234 


56789 


TOO 

701 
702 
703 
704 


84510 84516 84522 84528 84535 
84572 84578 84584 84590 84597 
84634 84640 84646 84652 84 658 
84696 84702 84708 84714 84720 
84757 84763 84770 84776 84782 


84541 84547 84553 84559 84566 
84603 84609 84615 84621 84628 
84665 84671 84677 84683 84689 
84726 84733 84739 84745 84751 
84788 84794 84800 84807 84813 


7O5 

706 
707 
708 
709 


84819 84825 84831 84837 84844 
84880 84887 84893 84899 84905 
84942 84948 84954 84960 84967 
85003 85009 85016 85022 85028 
8506^ 85071 85077 85083 85089 


84850 84856 84862 84868 84874 
84911 84917 84924 84930 84936 
84973 84979 84985 84991 84997 
85034 85040 85046 85052 85058 
85 095 85 101 85 107 85 114 85 120 


710 

711 
712 
713 
714 


85126 85132 85138 85144 85150 
85 187 85 193 85 199 85 205 85 211 
85248 85254 85260 85266 85272 
85309 85315 85321 85327 85333 
85370 85376 85382 85388 85394 


85156 85163 85169 85175 85181 
85217 85224 85230 85236 85242 
85278 85285 85291 85 297 85 303 
85339 85345 85352 85358 85364 
85400 85406 85412 85418 85425 


715 

716 
717 
718 
719 


85431 85437 85443 85449 85455 
85491 85497 85503 85509 85516 
85552 85558 85564 85570 85576 
85612 85618 85625 85631 85637 
85673 85679 85685 85691 85697 


85461 85467 85473 85479 85485 
85522 85528 85534 85540 85546 
85582 85588 85594 85600 85606 
85643 85649 85655 85661 85667 
85703 85709 85715 85721 85727 


720 

721 
722 
723 

724 


85733 85739 85745 85751 85757 
85794 85800 85806 85812 85818 
85854 85860 85866 85872 85878 
85914 85920 85926 85932 85938 
85974 85980 85986 85992 85998 


85763 85769 85775 85781 85788 
85824 85830 85836 85842 85848 
85884 85890 85896 85902 85908 
85944 85950 85956 85962 85968 
86004 86010 86016 86022 86028 


725 

726 

727 
728 
729 


86034 86040 86046 86052 86058 
86094 86100 86106 86112 86118 
86153 86159 86165 86171 86177 
86213 86219 86225 86231 86237 
86273 86279 86285 86291 86297 


86064 86 070 '86 076 86082 86088 
86124 86130 86136 86141 86147 
86183 86189 86195 86201 86207 
86243 86249 86255 86261 86267 
86303 86308 86314 86320 86326 


730 

731 
732 
733 
734 


86332 86338 86344 86350 86356 
86392 86398 86404 86410 86415 
86451 86457 86463 86469 86475 
86510 86516 86522 86528 86534 
86570 86576 86581 86587 86593 


86362 86368 86374 86380 86386 
86421 86427 86433 86439 86445 
86481 86487 86493 86499 86504 
86540 86546 86552 86558 86564 
86599 86605 86611 86617 86623 


735 

736 

737 
738 
739 


86629 86635 86641 86646 86652 
86688 86694 86700 86705 86711 
86747 86753 86759 86764 86770 
86806 86812 86817 86823 86829 
86864 86870 86876 86882 86888 


86658 86664 86670 86676 86682 
86 717 .86 723 86 729 86 735 86 741 
86776 86782 86788 86794 86800 
86835 86841 86847 86853 86859 
86894 86900 86906 86911 86917 


74O 

741 
742 
743 
744 


86923 86929 86935 86941 8694,7 
86982 86988 86994 86999 87005 
87040 87046 87052 87058 87064 
87099 87105 87111 87116 87122 
87157 87163 87169 87175 87181 


86953 86958 86964 86970 86976 
87011 7017 87023 87029 87035 
87070 87075 87081 87087 87093 
87128 87134 87140 87146 87151 
87186 87192 87'198 87204 87210 


745 

746 

747 
748 
749 


87216 87221 87227 87233 87239 
87274 87280 87286 87291 87297 
87332 87338 87344 87349 87355 
87390 87396 87402 87408 87413 
87448 87454 87460 87466 87471 


87245 87251 87256 87262 87268 
87303 87309 87315 87320 87326 
87361 87367 87373 87379 87384 
87419 87425 87431 87437 87442 
87477 87483 87489 87495 87500 


75O 


87506 87512 87518 87523 87529 


87535 87541 87547 87552 87558 


N 


O 1 2 3 4 


56789 



700-750 



750-800 



15 



N 


O 1 2 3 4 


56789 


75O 

751 
752 
753 
754 


87506 87512 87518 87523 87529 
87564 87570 87576 87581 87587 
87622 87628 87633 87639 87645 
87679 87685 87691 87697 87703 
87737 87743 87749 87754 87760 


87535 87541 87547 87552 87558 
87593 87599 87604 87610 87616 
87651 87656 87662 87668 87674 
87708 87714 87720 87726 87731 
87766 87772 87777 87783 87789 


755 

756 

757 
758 
759 


87795 87800 87806 87812 87818 
87852 87858 87864 87869 87875 
87910 87915 87921 87927 87933 
87967 87973 87978 87984 87990 
88024 88030 88036 88041 88047 


87823 87829 87835 87841 87846 
87881 87887 87892 87898 87904 
87938 87944 87950 87955 87961 
87996 88001 88007 88013 88018 
'88053 88058 88064 88070 88076 


76O 

761 
762 
763 
764 


88081 88087 88093 88098 88104 
88138 88144 88150 88156 88161 
88195 88201 88207 88213 88218 
88252 88258 88264 88270 88275 
88309 88315 88321 88326 88332 


88110 88116 88121 88127 88133 
88167 88173 88178 88184 88190 
88224 88230 88235 88241 88247 
88281 88287 88292 88298 88304 
88338 88343 88349 88355 88360 


765 

766 
767 
76S 
769 


88366 88372 88377 88383 88389 
88423 88429 88434 88440 88446 
88 480 88 485 88 491 88 497 88 502 
88536 88542 88547 88553 88559 
88593 88598 88604 88610 88615 


88395 88400 88406 88412 88417 
88451 88457 88463 88468 88474 
88508 88513 88519 88525 88530 
88564 88570 88576 88581 88587 
88621 88627 88632 88638 88643 


77O 

771 
772 
773 
774 


88649 88655 88660 88666 88672 
88705 88711 88717 88722 88728 
88762 88767 88773 88779 88784 
88818 88824 88829 88 835 '88840 
88874 88880 88885 88891 88897 


88677 88683 88689 88694 88700 
88734 88739 88 745_ 88750 88756 
88790 88795 88801 88807 88812 
88846 88852 88857 88863 88868 
88902 88908 88913 88919 88925 


775 

776 

777 
778 
779 


88930 88936 88941 88947 88953 
88986 88992 88997 89003 89009 
89042 89048 89053 89059 89064 
89098 89104 89109 89115 89120 
89154 89159 89165 89170 89176 


88958 88964 88969 88975 88981 
89014 89020 89025 89031 89037 
89070 89076 89081 89087 89092 
89126 89131 89137 89143 89148 
89182 89187 89193 89198 89204 


78O 

781 
782 
783 
784' 


89209 89215 89221 89226 89 232 
89265 89271 89276 89282 89287 
89321 89326 89332 89337 89343 
89376 89382 89387 89393 89398 
89432 89437 89443 89448 89454 


89237 89243 89248 89254 89260 
89293 89298 89304 89310 89315 
89348 89354 89360 89365 89371 
89404 89409 89415 89421 89426 
89459 89465 89470 89476 89481 


785 

786 

787 
788 
789 


89487 89492 89498 89504 89509 
89542 89548 89553 89559 89564 
89597 89603 89609 89614 89620 
89653 89658 89664 89669 89675 
89708 89713 89719 89724 89730 


89515 89520 89526 89531 89537 
89570 89575 89581 89586 89592 
S962;> 89631 89636 89642 89647 
89680 89686 89691 89697 89702 
89735 89741 89746 89752 89757 


790 

791 
792 
793 
794 


89763 89768 89774 89779 89785 
89818 89823 89829 89834 89840 
89873 89878 89883 89889 89894 
89927 89933 89938 89944 89949 
89982 89988 89993 89998 90004 


89790 89796 89801 89807 89812 
89845 89851 89856 89862 89867 
89900 89905 89911 89916 89922 
89955 89960 89966 89971 89977 
90009 90015 90020 90026 90031 


795 

796 

797 
798 
799 


90037 90042 90048 90053 90059 
90091 90097 90102 90108 90113 
90146 90151 90157 90162 90168 
90200 90206 90211 90217 90222 
90255 90260 90266 90271 90276 


90064 90069 90075 90080 90086 
90119 90124 90129 90135 90140 
90173 90179 90184 90189 90195 
90227 90233 90238 90244 90249 
90282 90287 90293 90298 90304 


8OO 


90309 90314 90320 90325 90331 


90336 90342 90347 90352 90358 


N 


01234 


56789 



750-800 



16 



800-850 



N 


01234 


56789 


8OO 

801 

802 
803 
804 


90309 90314 90320 90325 90331 
90363 90369 90374 90380 90385 
90417 90423 90428 90434 90439 
90472 90477 90482 90488 90493 
90526 90531 90536 90542 90547 


90336 90342 90347 90352 90358 
90390 90396 90401 90407 90412 
90445 90450 90455 90461 90466 
90499 90504 90509 90515 90520 
90553 90558 90563 90569 90574 


8O5 

806 
807 
808 
809 


90580 90585 90590 90596 90601 
90634 90639 90644 90650 90655 
90687 90693 90698 90703 90709 
90 741 90 747 90 752 90 757- 90 763 
90795 90800 90806 90811 90816 


90607 90612 90617 90623 90628 
90660 90666 90671 90677 90.682 
90714 90720 90725 90730 90736 
90768 90773 90779 90784 90789 
90822 90827 90832 90838 90843 


81O 

811 
812 
813 
814 


90849 90854 90859 90865 90870 
90902 90907 90913 90918 90924 
90956 90961 90966 90972 90977 
91009 91014 91020 91025 91030 
91062 91068 91073 91078 91084 


90875 90881 90886 90891 90897 
90929 90934 90940 90945 90950 
90 982 90 988 90 993 90 998 91 004 
91036 91041 91046 91052 91057 
91089 91094 91100 91105 91110 


815 

816 

817 
818 
819 


91 116 91 121 91 126 91 132 91 137 
91 169 91 174 91 180 91 185 91 190 
91 222 91 228 91 233 91 238 91 243 
91 275 91 281 91 286 91 291 91 297 
91328 91334 91339 91344 91350 


91 142 91 148 91 153 91 158 91 164 
91 196 91 201 91 206 91 212 91 217 
91249 91254 91259 91265 91270 
91302 91307 91312 91318 91323 
91355 91360 91365 91371 91376 


82O 

821 
822 
823 
824 


91381 91387 91392 91397 91403 
91434 91440 91445 91450 91455 
91487 91492 91498 91503 91508 
91540 91545 91551 91556 91561 
91593 91598 91603 91609 91614 


91408 91413 91418 91424 91429 
91461 91466 91471 91477 91482 
91514 91519 91524 91529 91535 
91 566 91 572 91 577 91 582 91 587 
91619 91624 91630 91635 91640 


825 

826 
827 
828 
829 


91645 91651 91656 91661 91666 
91 698 91 703 91 709 91 714 91 719 
91751 91756 91761 91766 91772 
91803 91808 91814 91819 91824 
91855 91861 91866 91871 91876 


91672 91677 91682 91687 91693 
91 724 91 730 91 735 91 740 91 745 
91777 91782 91787 91793 91798 
91 829 91 834 91 840 91 845 91 850 
91882 91887 91892 91897 91903 


830 

831 
832 
833 
834 


91908 91913 91918 91924 91929 
91960 91965 91971 91976 91981 
92012 92018 92023 92028 92033 
92065 92070 92075 92080 92085 
92117 92122 92127 92132 92137 


91934 91939 91944 91950 91955 
91986 91991 91997 92002 92007 
92038 92044 92049 92054 92059 
92091 92096 92101 92106 92111 
92143 92148 92153 92158 92163 


835 

836 
837 
838 
839 


92169 92174 92179 92184 92189 
92221 92226 92231 92236 92241 
92273 92278 92283 92288 92293 
92324 92330 92335 92340 92345 
92376 92381 92387 92392 92397 


92 195 92 200 92 205 92 210 92 215 
92247 92252 92257 92262 92267 
92298 92304 92309 92314 92319 
92350 92355 92361 92366 92371 
92402 92407 92412 92418 92423 


840 

841 
842 
843 
844 


92428 92433 92438 92443 92449 
92480 92485 92490 92495 92500 
92531 92536 92542 92547 92552 
92583 92588 92593 92598 92603 
92634 92639 92645 92650 92655 


92454 92459 92464 92469 92474 
92505 92511 92516 92521 92526 
92557 92562 92567 92572 92578 
92609 92614 92619 92624 92629 
92660 92665 92670 92675 92681 


845 

846 
847 
848 
849 


92686 92691 92696 92701 92706 
92737 92742 92747 92752 92758 
92788 92793 92799 92804 92809 
92840 92845 92850 92855 92860 
92891 92896 92901 92906 92911 


92 711 92 716 92 722* 92 727 92 732 
92763 92768 92773 92778 92783 
92814 92819 92824 92829 92834 
92865 92870 92875 92881 92886 
92916 92921 92927 92932 92937 


850 


92942 92947 92952 92957 92962 


92967 92973 92978 92983 92988 


N 


01234 


56789 



800-850 



850-900 



17 



N 


O 1 2 3 4 


56789 


85O 

851 
852 
853 
854 


92942 92947 92952 92957 92962 
92993 92998 93003 93008 93013 
93044 93049 93054 93059 93064 
93 095 93 100 93 105 93 110 93 115 
93146 93151 93156 93161 93166 


92967 92973 92978 92983 92988 
93018 93024 93029 93034 93039 
93069 93075 93080 93085 93090 
93 120 93 125 93 131 93 136 93 141 . 
93 171 93 176 93 181 93 186 93 192 


855 

856 
857 
858 
859 


93197 93202 93207 93212 93217 
93247 93252 93258 93263 93268 
93298 93303 93308 93313 93318 
93349 93354 93359 93364 93369 
93399 93404 93409 93414 93420 


93222 93227 93232 93237 93242 
93273 93278 93283 93288 93293 
93323 93328 93334 93339 93344 
93374 93379 93384 93389 93394 
93425 93430 93435 93440 93445 


86O 

861 
862 
863 
864 


93450 93455 93460 93465 93'470 
93500 93505 93510 93515 93520 
93551 93556 93561 93566 93571 
93601 93606 93611 93616 93621 
93651 93656 93661 93666 93671 


93475 93480 93485 93490 93495 
93526 93531 93536 93541 93546 
93576 93581 93586 93591 93596 
93626 93631 93636 93641 93646 
93676 93682 93687 93692 93697 


865 

866 
867 
868 
869 


93702 93707 93712 93717 93722 
93752 93757 93762 93767 93772 
93802 93807 93812 93817 93822 
93852 93857 93862 93867 93872 
93902 93907 93912 93917 93922 


93727 93732 93737 93742 93747 
93777 93782 93787 93792 93797 
93827 93832 93837 93842 93847 
93877 93882 93887 93892 93897 
93927 93932 93937 93942 93947 


870 

871 
872 
873 
874 


93952 93957 93962 93967 93972 
94002 94007 94012 94017 94022 
94052 94057 94062 94067 94072 
94101 94106 94111 94116 94121 
94151 94156 94161 94166 94171 


93977 93982 93987 93992 93997 
94027 94032 94037 94042 94047 
94077 94082 94086 94091 94096 
94126 94131 94136 94141 94146 
94176 94181 94186 94191 94196 


875 

876 
877 
878 
879 


94201 94206 94211 94216 94221 
94250 94255 94260 94265 94270 
94300 94305 94310 94315 94320 
94349 94354 94359 94364 94369 
94399 94404 94409 94414 94419 


94226 94231 94236 94240 94245 
94275 94280 94285 94290 94295 
94325 94330 94335 94340 94345 
94374 94379 94384 94389 94394 
94424 94429 94433 94438 94443 


880 

881 
882 
883 
884 


94448 94453 94458 94463 94468 
94498 94503 94507 94512 94517 
94547 94552 94557 94562 94567 
94596 94601 94606 94611 94616 
94645 94650 94655 94660 94 6&5 


94473 94478 94483 94488 94493 
94522 94527 94532 94537 94542 
94571 94576 94581 94586 94591 
94621 94626 94630 94635 94640 
94670 94675 94680 94685 94689 


885 

886 
887 
888 
889 


94694 94699 94704 94709 94714 
94743 94748 94753 94758 94763 
94792 94797 94802 94807 94812 
94841 94846 94851 94856 94861 
94890 94895 94900 94905 94910 


94719 94724 94729 94734 94738 
94768 94773 94778 94783 94787 
94817 94822 94827 94832 94836 
94866 94871 94876 94880 94885 
94915 94919 94924 94929 94934 


89O 

891 
892 
893 
894 


94939 94944 94949 94954 94959 
94988 94993 94998 95002 95007 
95036 95041 95046 95051 95056 
95 085 95 090 95 095 95 100 95 105 
95 134 95 139 95 143 95 148 95 153 


94963 94968 94973 94978 94983 
95012 95017 95022 95027 95032 
95061 95066 95071 95075 95080 
95 109 95 114 95 119 95 124 95 129 
95158 95163 95168 95173 95177 


895 

896 
897 
898 
899 


95 182 95 187 95 192 95 197 95 202 
95231 95236 95240 95245 95250 
95279 95284 95289 95294 95299 
95328 95332 95337 95342 95347 
95376 95381 95386 95390 95395 


95207 95211 95216 95221 95226 
95255 95260 95265 95270 95274 
95303 95308 9S 313 95318 95323 
95352 95357 95361 95366 95371 
95400 95405 95410 95415 95419 


900 


95424 95429 95434 95439 95444 


95448 95453 95458 95463 95468 


N 


O ' 1 2 3 4 


56789 



850-900 



18 



900-950 



N 


01234 


56789 


9OO 


95424 95429 95434 95439 95444 


95448 95453 95458 95463 95468 


901 


95472 95477 95482 95487 95492 


95497 95501 95506 95511 95516 


902 


95521 95525 95530 95535 95540 


95545 95550 95554 95559 95564 


903 


95569 95574 95578 95583 95588 


95593 95598 95602 95607 95612 


904 


95617 95622 95626 95631 95636 


95641 95646 95650 95655 95660 


905 


95665 95670 95674 95679 95684 


95 689 95 694 95 698 95 703 95 708 


906 


95713 95718 95722 95727 95732 


95 737 95 742 95 746 95 751 95 756 


907 


95 761 95 766 95 770 95 775 95 780 


95 785 95 789 95 794 95 799 95 804 


908 


95809 95813 95818 95823 95828 


95832 95837 95842 95847 95852 


909 


95856 95861 95866 95871 95875 


95880 95885 95890 95895 95899 


910 


95904 95909 95914 95918 95923 


95 928 95 933 ^95 938 95 942 95 947 


911 


95952 95957 95961 95 966 95 971 


95976 95980 95985 95990 95995 


912 


95999 96004 96009 96014 96019 


96023 96028 96033 96038 96042 


913 


96047 96052 96057 96061 96066 


96071 96076 96080 96085 96090 


914 


96095 96099 96104 96109 96114 


96118 96123 96128 96133 96137 


915 


96142 96147 96152 96156 96161 


96166 96171 96175 96180 96185 


916 


96190 96194 96199 96204 96209 


96213 96218 96223 96227 96232 


917 


96237 96242 96246 96251 96256 


96261 96265 96270 96275 96280 


918 


96284 96289 96294 96298 96303 


96308 96313 96317 96322 96327 


919 


96332 96336 96341 96346 96350 


96355 96360 96365 96369 96374 


92O 


96379 96384 96388 96393 96398 


96402 96407 96412 96417 96421 


921 


96426 96431 96435 96440 96445 


96450 96454 96459 96464 96468 


922 


96473 96478 96483 96487 96492 


96497 96501 96506 96511 96515 


923 


96520 96525 96530 96534 96539 


96544 96548 96553 96558 96562 


924 


96567 96572 96577 96581 96586 


96591 96595 96600 96605 96609 


925 


96614 96619 96624 96628 96633 


96638 96642 96647 96652 96656 


926 


96661 96666 96670 96675 96680 


96685 .96689 96694 96699 96703 


927 


96708 96713 96717 96722 96727 


96731 96736 96741 96745 96750 


928 


96755 96759 96764 96769 96774 


96778 96783 96788 96792 96797 


929 


96802 96806 96811 96816 96820 


96825 96830 96834 96839 96844 


930 


96848 96853 96858 96862 96867 


96872 96876 96881 96886 96890 


931 


96895 96900 96904 96909 96914 


96918 96923 96928 96932 96937 


932 


96942 96946 96951 96956 96960 


96965 96970 96974 96979 96984 


933 


96988 96993 96997 97002 97007 


97011 97016 97021 97025 97030 


934 


97035 97039 97044 97049 97053 


97058 97063 97067 97072 97077 


935 


97081 97086 97090 97095 97100 


97104 97109 97114 97118 97123 


936 


97128 97132 97137 97142 97146 


97151 97155 97160 97165 97169 


937 


97174 97179 97183 97188 97192 


97197 97202 97206 97211 97216 


938 


97220 97225 97230 97234 97239 


97243 97248 97253 97257 97262 


939 


97267 97271 97276 97280 97285 


97290 97294 97299 97304 97308 


940 


97313 97317 97322 97327 97331 


97336 97340 97345 97350 97354 


941 


97359 97364 97368 97373 97377 


97382 97387 97391 97396 97400 


942 


97405 97410 97414 97419 97424 


97428 97433 97437 97442 97447 


943 


97451 97456 97460 97465 97470 


97474 97479 97483 97488 97493 


944 


97497 97502 97506 97511 97516 


97520 97525 97529 97534 97539 


945 


97543 97548 97552 97557 97562 


97566 97571 97575 97580 97585 


946 


97589 97594 97598 97603 97607 


97612 97617 97621 97626 97630 


947 


97635 97640 97644 97649 97653 


97658 97663 97667 97672 97676 


948 


97681 97685 97690 97695 97699 


97704 97708 97713 97717 97722 


949 


97727 97731 97736 97740 97745 


97749 97754 97759 97763 97768 


95O 


97772 97777 97782 97786 97791 


97795 97800 97804 97809 97813 


N 


O 1 2 3 4 


56789 



900-950 



950-1000 



19 



N 


O 1 2 3 4 


56789 


95O 


97772 97777 97782 97786 97791 


97795 97800 97804 97809 97813 


951 


97818 97823 97827 97832 97836 


97841 97845 97850 97855 97859 


952 


97864 97868 97873 97877 97882 


97886 97891 97896 97900 97903 


953 


97909 97914 97918 97923 97928 


97932 97937 97941 97946 97950 


954 


97955 97959 97964 97968 97973 


97978 97982 97987 97991 97996 


955 


98000 98005 98009 98014 98019 


98023 98028 98032 98037 98041 


956 


98046 98050 98055 98059 98064 


98068 98073 98078 98082 98087 


957 


98091 98096 98100 98105 98109 


98114 98118 98123 98127 98132 


958 


98137 98141 98146 98150 98155 


98159 98164 98168 98173 98177 


959 


98182 98186 98191 98195 98200 


98204 98209 98214 98218 98223 


96O 


98227 98232 98236 98241 98245 


982^0 98254 98259 98263 98268 


961 


98272 98277 98281 98286 98290 


98295 98299 98304 98308 98313 


962 


98318 98322 98327 98331 98336 


98340 98345 98349 98354 98358 


963 


98363 98367 98372 98376 98381 


98385 98390 98394 98399 98403 


964 


98408 98412 98417 98421 98426 


98430 98435 98439 98444 98448 


965 


98453 98457 98462 98466 98471 


98475 98480 98484 98489 98493 


966 


98498 98502 98507 98511 98516 


98520 98525 98529 98534 98538 


967 


98543 98547 98552 98556 98561 


98565 98570 98574 98579 98583 


968 


98588 98592 98597 98601 98605 


98610 98614 98619 98623 98628 


969 


98632 98637 98641 98646 98.650 


98655 98659 98664 986,68 98673 


970 


98677 98682 98686 98691 98695 


98700 98704 98709 98713 98717 


971 


98722 98726 98731 98735 98740 


98744 98749 98753 98758 98762 


972 


98767 98771 98776 98780 98784 


98789 98793 98798 98802 98807 


973 


98811 98816 98820 98825 98829 


98834 98838 98843 98847 98851 


974 


98856 98860 98865 98869 98874 


98878 98883 98887 98892 98896 


975 


98900 98905 98909 98914 98918 


98923 98927 98932 98936 98941 


976 


98945 98949 98954 98958 98963 


98967 98972 98976 98981 98985 


977 


98989 98994 98998 99003 99007 


99012 99016 99021 99025 99029 


978 


99034 99038 99043 99047 99052 


99056 99061 99065 99069 99074 


979 


99078 99083 99087 99092 99096 


99100 99105 99109 99114 99118 


980 


99123 99127 99131 99136 99140 


99145 99149 99154 99158 99162 


981 


99167 99171 99176 99180 99185 


99189 99193 99198 99202 99207 


982 


99211 99216 99220 99224 99229 


99233 99238 99242 99247 99251 


983 


99255 99260 99264 99269 99273 


99277 99282 99286 99291 99295 


984 


99300 99304 99308 99313 99317 


99 322 99 326 99 330 99 335 99 339 


985 


99344 99348 99352 99357 99361 


99366 99370 99374 99379 99383 


986 


99388 99392 99396 99401 99405 


99410 99414 99419 99423 99427 


987 


99432 99436 99441 99445 99449 


99454 99458 99463 99467 99471 


988 


99476 99480 99484 99489 99493 


99498 99502 99506 99511 99515 


989 


99520 99524 99528 99533 99537 


99542 99546 99550 99555 99559 


99O 


99564 99568 99572 99577 99581 


99585 99590 99594 99599 99603 


991 


99607 99612 99616 99621 99625 


99629 99634 99638 99642 99647 


992 


99651 99656 99660 99664 99669 


99673 99677 99682 99686 99691 


993 


99695 99699 99704 99708 99712 


99717 99721 99726 99730 99734 


994 


99739 99743 99747 99752 99756 


99760 99765 99769 99774 99778 


995 


99782 99787 99791 99795 99800 


99804 99808 99813 99817 99822 


996 


99826 99830 99835 99839 99843 


99848 99852 99856 99861 99865 


997 


99870 99874 99878 99883 99887 


99891 99896 99900 99904 99909 


998 


99913 99917 99922 99926 99930 


99935 99939 99944 99948 99952 


999 


99957 99961 99965 99970 99974 


99978 99983 99987 99991 99996 


1OOO 


00000 00004 00009 00013 00017 


00022 00026 00030 00 033 00039 


N 


O 1 2 3 4 


56789 



950-1000 



20 TABLE II. -LOGARITHMS OF CONSTANTS. 



Circumference of th 
Circumference of th 
Circumference of th 
If the radius r = 1, 1 
TT = 3. 14 159 265 3 




log 
2.55630250 
4. 33 445 375 
6.11260500 

0. 49 714 987 


e Circle in minutes .... 21 600 


e Circle in seconds 1 296 000 


lalf the Circumferenc 
58 979 323 846 264 338 


e of the Circle is 
328 




Also: 
27r = 6.28318531 
47r = 12.56637061 
^= 1.57079633 

= 1.04719755 
3 

i^= 4.18879020 
3 

= 0.78539816 
4 

= 0.52359878 
i= 0.31830989 
~ = 0.15915494 
|= 0.95492966 
-= 1.27323954 

7T 

= 0.23873241 

4?r 


log 
0. 79 817 987 

1. 09 920 986 
0. 19 611 988 

0. 02 002 862 
0. 62 208 861 
9. 89 508 988 - 10 
9. 71 899 862 .-10 
9. 50 285 013 - 10 
9. 20 182 013 - 10 
9. 97 997 138 - 10 
0.10491012 
9. 37 791 139 - 10 


-r 2 = 9. 86 960 440 
1-0. 10 132 US 
VTT = 1. 77 245 385 

= 0. 56 418 958 
VT 

J^ = 0. 97 720 502 

J 4 =1.12837917 
*/TT = 1.46 459 189 

= 0. 68 278 406 
& 
^/7r 2 = 2. 14 502 940 

| /A = 0.62 035 049 

\47T 

^ = 0. 80 599 598 


log 
0.99429975 

9.00570025-10 
0.24857494 
9. 75 142 506 - 10 

9.98998569-10 

0. 05 245 506 
0.16571662 
9. 83 428 338 - 10 

0. 33 143 325 
9. 79 263 713 - 10 

9. 90 633 287 - 10 


Arc a, whose length is equal to the radius r, is : 
in degrees ... a . 18 5729577951. 


log 
1. 75 812 263 

3. 53 627 388 
5. 31 442 513 

2. 05 915 263 
3.83730388 
5. 61 545 513 

8 24 187 737 10 


7T 

in minutes a' 10 ^ 3 437 74 677' 


7T 

in seconds " - 648 - ?H6 9.64 R06" 


Arc 2 a, whose lengt 
in degrees 

in minutes 
in seconds ..... 
If the radius r = 1, 


7T 

h is equal to twice the radius, 2 r, is : 
. 2 a . . . . = = 114. 59 155 903 

2 a' ^ 6 875 49 354 f 


7T 

a" 1 2 ^ 6 ^ 41 59 61" 


7T 

ihe length of the arc is : 
1 . - "" n m 74"; -*9Q 


for 1 minute "" . 00 09 089 


6.46372612 10 


. a' 10 800 
for 1 second T 00 000 48 "5 


4 68557487 10 


"a"'" 648000" 
for degree ...- * - "" 000879. 66S 


7. 94 084 737 - 10 
616 969 61 2 10 


t for minute. . . 
for $ second . . . 
Sin 1" in the unit ci 


2a 360 
"* Q 00014 544 


'2 a' 21600 
"" 00 000 94^ 


4.38454487-10 
4.68557487-10 


"2a 1296000 a00(X 
rcle - 0. 00 000 485. . 





21 



TABLE III. 


THE LOGAEITHMS 


OF THE 


TRIGONOMETRIC FUNCTIONS : 


Prom to 3', or 89 57' to 90, for every second ; 


From to 2, or 88 to 90, for every ten seconds ; 


From 1 to 89, for every minute, 


NOTE. To all the logarithms 10 is to be appended. 


i /\O log tan = log sin. 
lOg SlU U log cos = 10. 00 000 


ft 


O' 1' 2' 


r f 


r t 


O' 1' 2' 


f r 


o 


6. 46 373 6. 76 476 


6O 


30 


6.16270 6.63982 6.86167 


30 


1 


4. 68 557 6. 47 090 6. 76 836 


59 


31 


6.17694 6.64462 6.86455 


29 


2 


4. 98 660 6. 47 797 6. 77 193 


58 


32 


6.19072 6.64936 6.86742 


28 


3 


5.16270 6.48492 6.77548 


57 


33 


6.20409 6.65406 6.87027 


27 


4 


5. 28 763 6. 49 175 6. 77 900 


56 


34 


6.21705 6.65870 6.87310 


26 


5 


5. 38 454 6. 49 849 6. 78 248 


55 


35 


6.22964 6.66330 6.87591 


25 


6 


5.46373 6.50512 6.78595 


54 


36 


6.24188 6.66785 6.87870 


24 


7 


5.53067 6.51165 6.78938 


53 


37 


6.25378 6.67235 6.88147 


23 


8 


5.58866 6.51808 6.79278 


52 


38 


6.26536 6.67680 6.88423 


22 


9 


5. 63 982 6. 52 442 6. 79 616 


51 


39 


6.27664 6.68121 6.88697 


21 


10 


5.68557 6.53067 6.79952 


5O 


40 


6.28763 6.68557 6.88969 


2O 


11 


5.72697 6.53683 6.80285 


49 


41 


6.29836 6.68990 6.89240 


19 


12 


5.76476 6.54291 6.80615 


48 


42 


6.30882 6.69418 6.89509 


18 


13 


5. 79 952 6. 54 890 6. 80 943 


47 


43 


6.31904 6.69841 6.89776 


17 


14 


5. 83 170 6. 55 481 6. 81 268 


46 


44 


6.32903 6.70261 6.90042 


16 


15 


5. 86 167 6. 56 064 6. 81 591 


45 


45 


6.33879 6.70676 6.90306 


15 


16 


5.88969 6.56639 6.81911 


44 


46 


6.34833 6.71088 6.90568 


14 


17 


5. 91 602 6. 57 207 6. 82 230 


43 


47 


6.35767 6.71496 6.90829 


13 


18 


5.94085 6.57767 6.82545 


42 


48 


6.36682 6.71900 6.91088 


12 


19 


5.96433 6.58320 6.82859 


41 


49 


6.37577 6.72300 6.91346 


11 


20 


5.98660 6.58866 6.83170 


4O 


5O 


6.38454 6.72697 6.91602 


10 


21 


6.00779 6.59406 6.83479 


39 


51 


6.39315 6.73090 6.91857 


9 


22 


6.02800 6.59939 6.83786 


' 38 


52 


6.40158 6.73479 6.92110 


8 


23 


6. 04 730 6. 60 465 6. 84 091 


37 


53 


6.40985 6.73865 6.92362 


7 


24 


6. 06 579 6. 60 985 6. 84 394 


36 


54 


6.41797 6.74248 6.92612 


6 


25 


6.08351 6.61499 6.84694 


35 


55 


6.42594 6.74627 6.92861 


5 


26 


6. 10 055 6. 62 007 6. 84 993 


34 


56 


6.43376 6.75003 6.93109 


4 


27 


6. 11 694 6. 62 509 6. 85 289 


33 


57 


6.44145 6.75376 6.93355 


3 


28 


6.13273 6.63006 6.85584 


32 


58 


6.44900 6.75746 6.93599 


2 


29 


6.14797 6.63496 6.85876 


31 


59 


6.45643 6.76112 6.93843 


1 


30 


6.16270 6.63982 6.86167 


30 


60 


6.46373 6.76476 6.94085. 





ff 


59' 58' 57' 


tf 


rr 


59' 58' 57' 


tt 



log COt = log 008 

log sin - 10, 00 000 



89 



log cos 



22 







t tt 


log sin log tan log cos 


// t 


t tt 


log sin log tan log cos 


tt r 


O 


10.00000 


06O 


1OO 


7. 46 373 7. 46 373 10.00000 


05O 


10 


5.68557 5.68557 10.00000 


50 


10 


7. 47 090 7. 47 091 10.00000 


50 


20 


5.98660 5.98660 10.00000 


40 


20 


7. 47 797 7. 47 797 10.00000 


40 


30 


6. 16 270 6. 16 270 10.00000 


30 


30 


7. 48 491 7. 48 492 10.00000 


30 


40 


6. 28 763 6/28 763 10.00000 


20 


40 


7. 49 175 7. 49 176 10.00000 


20 


50 


6.38454 6.38454 10.00000 


10 


50 


7. 49 849 7. 49 849 10.00000 


10 


1 


6.46373 6.46373 10.00000 


059 


110 


7.50512 7.50512 10.00000 


049 


10 


6. 53 067 6. 53 067 10.00000 


50 


10 


7.51165 7.51165 10.00000 


50 


20 


6.58866 6.58866 10.00000 


40 


20 


7. 51 808 7. 51 809 10.00000 


40 


30 


6.63982 6.63982 10.00000 


30 


30 


7. 52 442 7. 52 443 10.00000 


30 


40 


6. 68 557 6. 68 557 10.00000 


20 


40 


7.53067 7.53067 10.00000 


20 


50 


6. 72 697 6. 72 697 10.00000 


10 


50 


7. 53 683 7. 53 683 10.00000 


10 


2 


6.76476 6.76476 10.00000 


058 


120 


7. 54 291 7. 54 291 10.00000 


048 


10 


6. 79 952 6. 79 952 10.00000 


50 


10 


7. 54 890 7. 54 890 10.00000 


50 


20 


6.83170 6.83170 10.00000 


40 


20 


7. 55 481 7. 55 481 10.00000 


40 


30 


6. 86 167 6. 86 167 10.00000 


30 


30 


7. 56 064 7. 56 064 10.00000 


30 


40 


6. 88 969 6. 88 969 10.00000 


20 


40 


7. 56 639 7. 56 639 10.00000 


20 


50 


6.91602 6.91602 10.00000 


10 


50 


7. 57 206 7. 57 207 10.00000 


10 


3 


6. 94 085 6. 94 085 10.00000 


057 


130 


7. 57 767 7. 57 767 10.00000 


047 


10 


6.96433 6.96433 10.00000 


50 


10 


7.58320 7.58320 10.00000 


50 


20 


6.98660 6.98661 10.00000 


40 


20 


7. 58 866 7. 58 867 10.00000 


40 


30 


7. 00 779 7. 00 779 10.00000 


30 


30 


7. 59 406 7. 59 406 10.00000 


30 


40 


7.02800 7.02800 10.00000 


20 


40 


7. 59 939 7. 59 939 10.00000 


20 


50 


7. 04 730 7. 04 730 10.00000 


10 


50 


7. 60 465 7. 60 466 10.00000 


10 


4 


7. 06 5 79 7. 06 5 79 10.00000 


056 


140 


7.60985 7.60986 10.00000 


046 


10 


7.08351 7.08352 10.00000 


50 


10 


7. 61 499 7. 61 500 10.00000 


50 


20 


7.10055 7.10055 10.00000 


40 


20 


7. 62 007 7. 62 008 10.00000 


40 


30 


7. 11 694 7. 11 694 10.00000 


30 


30 


7.62509 7.62510 10.00000 


30 


40 


7.13273 7.13273 10.00000 


20 


40 


7. 63 006 7. 63 006 10.00000 


20 


50 


7. 14 797 7. 14 797 10.00000 


10 


50 


7. 63 496 7. 63 497 10.00000 


10 


5 


7.16270 7.16270 10.00000 


055 


150 


7.63982 7.63982 10.00000 


045 


10 


7. 17 694 7. 17 694 10.00000 


50 


10 


7. 64 461 7. 64 462 10.00000 


50 


20 


7. 19 072 7. 19 073 10.00000 


40 


20 


7. 64 936 7. 64 937 10.00000 


40 


30 


7.20409 7.20409 10.00000 


30 


30 


7.65406 7.65406 10.00000 


30 


40 


7. 21 705 7. 21 705 10.00000 


20 


40 


7.65870 7.65871 10.00000 


20 


50 


7. 22 964 7. 22 964 10.00000 


10 


50 


7. 66 330 7. 66 330 10.00000 


10 


6 


7. 24 188 7. 24 188 10.00000 


054 


160 


7. 66 784 7. 66 785 10.00000 


044 


10 


7. 25 378 7. 25 378 10.00000 


50 


10 


7. 67 235 7. 67 235 10.00000 


50 


20 


7.26536 7.26536 10.00000 


40 


20 


7.67680 7.67680 10.00000 


40 


30 


7.27664 7.27664 10.00000 


30 


30 


7. 68 121 7. 68 121 10.00000 


30 


40 


7. 28 763 7. 28 764 10.00000 


20 


40 


7. 68 557 7. 68 558 9.99999 


20 


50 


7. 29 836 7. 29 836 10.00000 


10 


50 


7.68989 7.68990 9.99999 


10 


7 


7. 30 882 7. 30 882 10.00000 


053 


170 


7.69417 7.69418 9.99999 


043 


10 


7. 31 904 7. 31 904 10.00000 


50 


10 


7.69841 7.69842 9.99999 


50 


20 


7.32903 7.32903 10.00000 


40 


20 


7. 70 261 7. 70 261 9. 99 999 


40 


30 


7. 33 879 7. 33 879 10.00000 


30 


30 


7. 70 676 7. 70 677 9. 99 999 


30 


40 


7. 34 833 7. 34 833 10.00000 


20 


40 


7.71088 7.71088 9.99999 


20 


50 


7.35767 7.35767 10.00000 


10 


50 


7.71496 7.71496 9.99999 


10 


8 


7. 36 682 7. 36 682 10.00000 


052 


180 


7.71900 7.71900 9.99999 


042 


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7.37577 7.37577 10.00000 


50 


10 


7. 72 300 7. 72 301 9. 99 999 


50 


20 


7. 38 454 7. 38 455 10.00000 


40 


20 


7. 72 697 7. 72 697 9. 99 999 


40 


30 


7.39314 7.39315 10.00000 


30 


30 


7.73090 7.73090 9.99999 


30 


40 


7.40158 7.40158 10.00000 


20 


40 


7.73479 7.73480 9.99999 


20 


50 


7. 40 985 7. 40 985 10.00000 


10 


50 


7.73865 7.73866 9.99999 


10 


9 


7. 41 797 7. 41 797 10.00000 


051 


190 


7.74248 7.74248 9.99999 


041 


10 


7. 42 594 7. 42 594 10.00000 


50 


10 


7.74627 7.74628 9.99999 


50 


20 


7.43376 7.43376 10.00000 


40 


20 


7.75003 7.75004 9.99999 


40 


30 


7. 44 145 7. 44 145 10.00000 


30 


30 


7.75376 7.75377 9.99999 


30 


40 


7. 44 900 7. 44 900 10.00000 


20 


40 


7. 75 745 7. 75 746 9. 99 999 


20 


50 


7.45643 7.45643 10.00000 


10 


50 


7.76112 7.76113 9.99999 


10 


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7.46373 7.46373 10.00000 


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7.76475 7.76476 9.99999 


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log cos log cot log sin 


ft t 


t ft 


log cos log cot log sin 


ft f 



89 C 



23 



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log sin log tan log cos 


/; t 


r ft 


log sin log tan log cos 


rt t 


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7.76475 7.76476 9.99999 


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7.94084 7.94086 9.99998 


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10 


7. 76 836 7. 76 837 9. 99 999 


50 


10 


7.94325 7.94326 9.99998 


50 


20 


7.77193 7.77194 9.99999 


40 


20 


7.94564 7.94566 9.99998 


40 


30 


7.77548 7.77549 9.99999 


30 


30 


7.94802 7.94804 9.99998 


30 


40 


7.77899 7.77900 9.99999 


20 


40 


7.95039 7.95040 9.99998 


20 


50 


7.78248 7.78249 9.99999 


10 


50 


7.95274 7.95276 9.99998 


10 


210 


7.78594 7.78595 9.99999 


039 


310 


7.95508 7.95510 9.99998 


029 


10 


7.78938 7.78938 9.99999 


50 


10 


7.95741 7.95743 9.99998 


50 


20 


7.79278 7.79279 9.99999 


40 


20 


7.95973 7.95974 9.99998 


40 


30 


7.79616 7.79617 9.99999 


30 


30 


7. 96 203 7. 96 205 9. 99 998 


30 


40 


7.79952 7.79952 9.99999 


20 


40 


7.96432 7.96434 9.99998 


20 


50 


7. 80 284 7. 80 285 9. 99 999 


10 


50 


7. 96 660 7. 96 662 9. 99 998 


10 


220 


7.80615 7.80615 9.99999 


038 


320 


7.96887 7.96889 9.99998 


028 


10 


7. 80 942 7. 80 943 9. 99 999 


50 


10 


7.97113 7.97114 9.99998 


50 


20 


7.81268 7.81269 9.99999 


40 


20 


7.97337 7.97339 9.99998 


40 


30 


7. 81 591 7. 81 591 9. 99 999 


30 


30 


7.97560 7.97562 9.99998 


30 


40 


7.81911 7.81912 9.99999 


20 


40 


7.97782 7.97784 9.99998 


20 


50 


7.82229 7.82230 9.99999 


10 


50 


7. 98 003 7. 98 005 9. 99 998 


10 


230 


7.82545 7.82546 9.99999 


037 


330 


7. 98 223 7. 98 225 9. 99 998 


027 


10 


7.82859 7.82860 9.99999 


50 


10 


7.98442 7.98444 9.99998 


50 


20 


7.83170 7.83171 9.99999 


40 


20 


7. 98 660 7. 98 662 9. 99 998 


40 


30 


7.83479 7.83480 9.99999 


30 


30 


7.98876 7.98878 9.99998 


30 


40 


7.83786 7.83787 9.99999 


20 


40 


7.99092 7.99094 9.99998 


20 


50 


7. 84 091 7. 84 092 9. 99 999 


10 


50 


7. 99 306 7. 99 308 9. 99 998 


10 


240 


7.84393 7.84394 9.99999 


036 


340 


7. 99 520 7. 99 522 9. 99 998 


026 


10 


7. 84 694 7. 84 695 9. 99 999 


50 


10 


7.99732 7.99734 9.99998 


50 


20 


7. 84 992 7. 84 994 9. 99 999 j 40 


20 


7.99943 7.99946 9.99998 


40 


30 


7. 85 289 7. 85 290 9. 99 999 ! 30 


30 


8.00154 8.00156 9.99998 


30 


40 


7.85583 7.85584 9.99999 


20 


40 


8. 00 363 8. 00 365 9. 99 998 


20 


50 


7. 85 876 7. 85 877 9. 99 999 


10 


50 


8.00571 8.00574 9.99998 


10 


250 


7. 86 166 7. 86 167 9. 99 999 


035 


350 


8.00779 8.00781 9.99998 


025 


10 


7.86455 7.86456 9.99999 


50 


10 


8. 00 985 8. 00 987 9. 99 998 


50 


20 


7. 86 741 7. 86 743 9. 99 999 


40 


20 


8. 01 190 8. 01 193 9. 99 998 


40 


30 


7.87026 7.87027 9.99999 


30 


30 


8. 01 395 8. 01 397 9. 99 998 


30 


40 


7.87309 7.87310 9.99999 


20 


40 


8.01598 8.01600 9.99998 


20 


50 


7. 87 590 7. 87 591 9. 99 999 


10 


50 


8.01801 8.01803 9.99998 


10 


260 


7. 87 870 7. 87 871 9. 99 999 


034 


360 


8. 02 002 8. 02 004 9. 99 998 


024 


10 


7.88147 7.88148 9.99999 


50 


10 


8.02203 8.02205 9.99998 


50 


20 


7.88423 7.88424 9.99999 


40 


20 


8. 02 402 8. 02 405 9. 99 998 


40 


30 


7. 88 697 7. 88 698 9. 99 999 


30 


30 


8. 02 601 8. 02 604 9. 99 998 


30 


40 


7.88969 7.88970 9.99999 


20 


40 


8. 02 799 8. 02 801 9. 99 998 


20 


50 


7. 89 240 7. 89 241 9. 99 999 


10 


50 


8.02996 8.02998 9.99998 


10 


270 


7.89509 7.89510 9.99999 


033 


370 


8.03192 8.03194 9.99997 


023 


10 


7.89776 7.89777 9.99999 


50 


10 


8.03387 8.03390 9.99997 


50 


20 


7. 90 041 7. 90 043 9. 99 999 


40 


20 


8.03581 8.03584 9.99997 


40 


30 


7. 90 305 7. 90 307 9. 99 999 


30 


30 


8. 03 775 8. 03 777 9. 99 997 


30 


40 


7. 90 568 7. 90 569 9. 99 999 


20 


40 


8.03967 8.03970 9.99997 


20 


50 


7.90829 7.90830 9.99999 


10 


50 


8. 04 159 8. 04 162 9. 99 997 


10 


280 


7.91088 7.91089 9.99999 


032 


380 


8.04350 8.04353 9.99997 


022 


10 


7. 91 346 7. 91 347 9. 99 999 


50 


10 


8. 04 540 8. 04 543 9. 99 997 


50 


20 


7. 91 602 7. 91 603 9. 99 999 


40 


20 


8.04729 8.04732 9.99997 


40 


30 


7.91857 7.91858 9.99999 


30 


30 


8.04918 8.04921 9.99997 


30 


40 


7.92110 7.92111 9.99998 


20 


40 


8. 05 105 8. 05 108 9. 99 997 


20 


50 


7.92362 7.92363 9.99998 


10 


50 


8. 05 292 8. 05 295 9. 99 997 


10 


290 


7. 92 612 7. 92 613 9. 99 998 


031 


390 


8. 05 478 8. 05 481 9. 99 997 


021 


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7.92861 7.92862 9.99998 


50 


10 


8.05663 8.05666 9.99997 


50 


20 


7.93108 7.93110 9.99998 


40 


20 


8.05848 8.05851 9.99997 


40 


30 


7.93354 7.93356 9.99998 


30 


30 


8.06031 8.06034 9.99997 


30 


40 


7. 93 599 7. 93 601 9. 99 998 


20 


40 


8.06214 8.06217 9.99997 


20 


50 


7.93842 7.93844 9.99998 


10 


50 


8.06396 8.06399 9.99997 


10 


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7.94084 7.94.086 9.99998 


030 


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8.06578 8.06581 9.99997 


020 


/ tt 


log cos log cot log sin 


t? f 


t ff 


log cos log cot log sin 


tr r 



24 



f ft 


log sin log tan log cos 


ff t 


t tf 


log sin log tan log cos 


ff f 


400 


8.06578 8.06581 9.99997 


02O 


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8.16268 8.16273 9.99995 


01O 


10 


8. 06 758 8. 06 761 9. 99 997 


50 


10 


8.16413 8.16417 9.99995 


50 


20 


8.06938 8.06941 9.99997 


40 


20 


8.16557 8.16561 9.99995 


40 


30 


8.07117 8.07120 9.99997 


30 


30 


8. 16 700 8. 16 705 9. 99 995 


30 


40 


8.07295 8.07299 9.99997 


20 


40 


8.16843 8.16848 9.99995 


20 


50 


8.07473 8.07476 9.99997 


10 


50 


8.16986 8.16991 9.99995 


10 


410 


8.07650 8.07653 9.99997 


019 


510 


8.17128 8.17133 9.99995 


9 


10 


8.07826 8.07829 9.99997 


50 


10 


8. 17 270 8. 17 275 9. 99 995 


50 


20 


8.08002 8.08005 9.99997 


40 


20 


8.17411 8.17416 9.99995 


40 


30 


8.08176 8.08180 9.99997 


30 


30 


8.17552 8.17557 9.99995 


30 


40 


8.08350 8.08354 9.99997 


20 


40 


8. 17 692 8. 17 697 9. 99 995 


20 


50 


8.08524 8.08527 9.99997 


10 


50 


8.17832 8.17837 9.99995 


10 


420 


8.08696 8.08700 9.99997 


018 


520 


8.17971 8.17976 9.99995 


8 


10 


8.08868 8.08872 9.99997 


50 


10 


8.18110 8.18115 9.99995 


50 


20 


8.09040 8.09043 9.99997 


40 


20 


8.18249 8.18254 9.99995 


40 


30 


8.09210 8.09214 9.99997 


30 


30 


8. 18 387 8. 18 392 9. 99 995 


30 


40 


8.09380 8.09384 9.99997 


20 


40 


8. 18 524 8. 18 530 9. 99 995 


20 


50 


8.09550 8.09553 9.99997 


10 


50 


8. 18 662 8. 18 667 9. 99 995 


10 


430 


8.09718 8.09722 9.99997 


017 


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8. 18 798 8. 18 804 9. 99 995 


7 


10 


8.09886 8.09890 9.99997 


50 


10 


8.18935 8.18940 9.99995 


50 


20 


8.10054 8.10057 9.99997 


40 


20 


8. 19 071 8. 19 076 9. 99 995 


40 


30 


8.10220 8.10224 9.99997 


30 


30 


8. 19 206 8. 19 212 9. 99 995 


30 


40 


8.10386 8.10390 9.99997 


20 


40 


8.19341 8.19347 9.99995 


20 


50 


8. 10 552 8. 10 555 9. 99 996 


10 


50 


8.19476 8.19481 9.99995 


10 


440 


8.10717 8.10720 9.99996 


016 


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8. 19 610 8. 19 616 9. 99 995 


6 


10 


8. 10 881 8. 10 884 9. 99 996 


50 


10 


8. 19 744 8. 19 749 9. 99 995 


50 


20 


8.11044 8.11048 9.99996 


40 


20 


8. 19 877 8. 19 883 9. 99 995 


40 


30 


8.11207 8.11211 9.99996 


30 


30 


8.20010 8.20016 9.99995 


30 


40 


8.11370 8.11373 9.99996 


20 


40 


8.20143 8.20149 9.99995 


20 


50 


8.11531 8.11535 9.99996 


10 


50 


8. 20 275 8. 20 281 9. 99 994 


10 


450 


8.11693 8.11696 9.99996 


015 


550 


8.20407 8.20413 9.99994 


5 


10 


8.11853 8.11857 9.99996 


50 


10 


8.20538 8.20544 9.99994 


50 


20 


8.12013 8.12017 9.99996 


40 


20 


8. 20 669 8. 20 675 9. 99 994 


40 


30 


8.12172 8.12176 9.99996 


30 


30 


8.20800 8.20806 9.99994 


30 


40 


8. 12 331 8. 12 335 9. 99 996 


20 


40 


8.20930 8.20936 9.99994 


20 


50 


8. 12 489 8. 12 493 9. 99 996 


10 


50 


8.21060 8.21066 9.99994 


10 


460 


8.12647 8.12651 9.99996 


014 


560 


8. 21 189 8. 21 195 9. 99 994 


4 


10 


8.12804 8.12808 9.99996 


50 


10 


8.21319 8.21324 9.99994 


50 


20 


8. 12 961 8. 12 965 9. 99 996 


40 


20 


8.21447 8.21453 9.99994 


40 


30 


8.13117 8.13121 9.99996 


30 


30 


8.21576 8.21581 9.99994 


30 


40 


8.13272 8.13276 9.99996 


20 


40 


8.21703 8.21709 9.99994 


20 


50 


8.13427 8.13431 9.99996 


10 


50 


8. 21 831 8. 21 837 9. 99 994 


10 


470 


8.13581 8.13585 9.99996 


013 


570 


8.21958 8.21964 9.99994 


3 


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8. 13 735 8. 13 739 9. 99 996 


50 


10 


8.22085 8.22091 9.99994 


50 


20 


8. 13 888 8. 13 892 9. 99 996 


40 


20 


8.22211 8.22217 9.99994 


40 


30 


8. 14 041 8. 14 045 9. 99 996 


30 


30 


8. 22 337 8. 22 343 9. 99 994 


30 


40 


8. 14 193 8. 14 197 9. 99 996 


20 


40 


8. 22 463 8. 22 469 9. 99 994 


20 


50 


8. 14 344 8. 14 348 9. 99 996 


10 


50 


8.22588 8.22595 9.99994 


10 


480 


8.14495 8.14500 9.99996 


012 


580 


8.22713 8.22720 9.99994 


2 


10 


8.14646 8.14650 9.99996 


50 


10 


8.22838 8.22844 9.99994 


50 


20 


8.14796 8.14800 9.99996 


40 


20 


8.22962 8.22968 9.99994 


40 


30 


8.14945 8.14950 9.99996 


30 


30 


8.23086 8.23092 9.99994 


30 


40 


8.15094 8.15099 9.99996 


20 


40 


8.23210 8.23216 9.99994 


20 


50 


8.15243 8.15247 9.99996 


10 


50 


8.23333 8.23339 9.99994 


10 


490 


8.15391 8.15395 9.99996 


Oil 


590 


8.23456 8.23462 9.99994 


1 


10 


8.15538 8.15543 9.99996 


50 


10 


8.23578 8.23585 9.99994 


50 


20 


8.15685 8.15690 9.99996 


40 


20 


8.23700 8.23707 9.99994 


40 


30 


8.15832 8.15836 9.99996 


30 


30 


8.23822 8.23829 9.99993 


30 


40 


8.15978 8.15982 9.99995 


20 


40 


8.23944 8.23950 9.99993 


20 


50 


8. 16 123 8. 16 128 9. 99 995 


10 


50 


8. 24 065 8. 24 071 9. 99 993 


10 


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8. 16 268 8. 16 273 9. 99 995 


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8.24186 8.24192 9.99993 


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/ tt 


log cos log cot log sin 


ft r 


f n 


log cos log cot log sin 


ff f 



89< 



25 



f ft 


log sin log tan log cos 


// t 


iff 


log sin log tan log cos 


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8.24186 8.24192 9.99993 


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8. 30 879 8. 30 888 9. 99 991 


05O 


10 


8.24306 8.24313 9.99993 


50 


10 


8. 30 983 8. 30 992 9. 99 991 


50 


20 


8.24426 8.24433 9.99993 


40 


20 


8.31086 8.31095 9.99991 


40 


30 


8.24546 8.24553 9.99993 


30 


30 


8.31188 8.31198 9.99991 


30 


40 


8. 24 665 8. 24 672 9. 99 993 


20 


40 


8. 31 291 8. 31 300 9. 99 991 


20 


50 


8.24785 8.24791 9.99993 


10 


50 


8.31393 8.31403 9.99991 


10 


1 


8.24903 8.24910 9.99993 


059 


110 


8.31495 8.31505 9.99991 


049 


10 


8. 25 022 8. 25 029 9. 99 993 


50 


10 


8.31597 8.31606 9.99991 


50 


20 


8. 25 140 8. 25 147 9. 99 993 


40 


20 


8.31699 8.31708 9.99991 


40 


30 


8. 25 258 8. 25 265 9. 99 993 


30 


30 


8.31800 8.31809 9.99991 


30 


40 


8. 25 375 8. 25 382 9. 99 993 


20 


40 


8.31901 8.31911 9.99991 


20 


50 


8. 25 493 8. 25 500 9. 99 993 


10 


50 


8. 32 002 8. 32 012 9. 99 991 


10 * 


2 


8.25609 8.25616 9.99993 


058 


120 


8.32103 8.32112 9.99990 


048 


10 


8.25726 8.25733 9.99993 


50 


10 


8. 32 203 8. 32 213 9. 99 990 


50 


20 


8.25842 8.25849 9.99993 


40 


20 


8 32 303 8. 32 313 9. 99 990 


40 


30 


8.25958 8.25965 9.99993 


30 


30 


8. 32 403 8. 32 413 9. 99 990 


30 


40 


8. 26 074 8. 26 081 9. 99 993 


20 


40 


8.32503 8.32513 9.99990 


20 


50 


8. 26 189 8. 26 196 9. 99 993 


10 


50 


8. 32 602 8. 32 612 9. 99 990 


10 


3 


8.26304 8.26312 9.99993 


057 


130 


8.32702 8.32711 9.99990 


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10 


8.26419 8.26426 9.99993 


50 


10 


8. 32 801 8. 32 811 9. 99 990 


50 


20 


8. 26 533 8. 26 541 9. 99 993 


40 


20 


8. 32 899 8. 32 909 9. 99 990 


40 


30 


8. 26 648 8. 26 655 9. 99 993 


30 


30 


8. 32 998 8. 33 008 9. 99 990 


30 


40 


8.26761 8.26769 9.99993 


20 


40 


8.33096 8.33106 9.99990 


20 


50 


8.26875 8.26882 9.99993 


10 


50 


8. 33 195 8. 33 205 9. 99 990 


10 


4 


8. 26 988 8. 26 996 9. 99 992 


056 


140 


8. 33 292 8. 33 302 9. 99 990 


046 


10 


8. 27 101 8. 27 109 9. 99 992 


50 


10 


8.33390 8.33400 9.99990 


50 


20 


8. 27 214 8. 27 221 9. 99 992 


40 


20 


8. 33 488 8. 33 498 9. 99 990 


40 


30 


8. 27 326 8. 27 334 9. 99 992 


30 


30 


8.33585 8.33595 9.99990 


30 


40 


8. 27 438 8. 27 446 9. 99 992 


20 


40* 


8. 33 682 8. 33 692 9. 99 990 


20 


50 


8.27550 8.27558 9.99992 


10 


50 


8. 33 779 8. 33 789 9. 99 990 


10 


5 


8.27661 8.27669 9.99992 


055 


150 


8.33875 8.33886 9.99990 


045 


10 


8. 27 773 8. 27 780 9. 99 992 


50 


10 


8. 33 972 8. 33 982 9. 99 990 


50 


20 


8. 27 883 8. 27 891 9. 99 992 


40 


20 


8. 34 068 8. 34 078 9. 99 990 


40 


30 


8. 27 994 8. 28 002 9. 99 992 


30 


30 


8. 34 164 8. 34 174 9. 99 990 


30 


40 


8.28104 8.28112 9.99992 


20 


40 


8.34260 8.34270 9.99989 


20 


50 


8. 28 215 8. 28 223 9. 99 992 


10 


50 


8.34355 8.34366 9.99989 


10 


6 


8. 28 324 8. 28 332 9. 99 992 


054 


160 


8. 34 450 8. 34 461 9. 99 989 


044 


10 


8. 28 434 8. 28 442 9. 99 992 


50 


10 


8.34546 8.34556 9.99989 


50 


20 


8.28543 8.28551 9.99992 


40 


20 


8.34640 8.34651 9.99989 


40 


30 


8. 28 652 8. 28 660 9. 99 992 


30 


30 


8.34735 8.34746 9.99989 


30 


40 


8. 28 761 8. 28 769 9. 99 992 


20 


40 


8.34830 8.34840 9.99989 


20 


50 


8. 28 869 8. 28 877 9. 99 992 


10 


50 


8. 34 924 8. 34 935 9. 99 989 


10 


7 


8. 28 977 8. 28 986 9. 99 992 


053 


170 


8.35018 8.35029 9.99989 


043 


10 


8. 29 085 8. 29 094 9. 99 992 


50 


10 


8.35112 8.35123 9.99989 


50 


20 


8. 29 193 8. 29 201 9. 99 992 


40 


20 


8. 35 206 8. 35 217 9. 99 989 


40 


30 


8. 29 300 8. 29 309 9. 99 992 


30 


30 


8.35299 8.35310 9.99989 


30 


40 


8.29407 8.29416 9.99992 


20 


40 


8. 35 392 8. 35 403 9. 99 989 


20 


50 


8.29514 8.29523 9.99992 


10 


50 


8. 35 485 8-. 35 497 9. 99 989 


10 


8 


8. 29 621 8. 29 629 9. 99 992 


052 


180 


8.35578 8.35590 9.99989 


042 


10 


8.29727 8.29736 9.99991 


50 


10 


8. 35 671 8. 35 682 9. 99 989 


50 


20 


8. 29 833 8. 29 842 9. 99 991 


40 


20 


8. 35 764 8. 35 775 9. 99 989 


40 


30 


8. 29 939 8. 29 947 9. 99 991 


30 


30 


8. 35 856 8. 35 867 9. 99 989 


30 


40 


8. 30 044 8. 30 053 9. 99 991 


20 


40 


8.35948 8.35959 9.99989 


20 


50 


8.30150 8.30158 9.99991 


10 


50 


8.36040 8.36051 9.99989 


10 


9 


8. 30 255 8. 30 263 9. 99 991 


051 


190 


8.36131 8.36143 9.99989 


041 


10 


8. 30 359 8. 30 368 9. 99 991 


50 


10 


8.36223 8.36235 9.99988 


50 


20 


8. 30 464 8. 30 473 9. 99 991 


40 


20 


8.36314 8.36326 9.99988 


40 


30 


8. 30 568 8. 30 577 9. 99 991 


30 


30 


8. 36 405 8. 36 417 9. 99 988 


30 


40 


8. 30 672 8. 30 681 9. 99 991 


20 


40 


8.36496 8.36508 9.99988 


20 


50 


8. 30 776 8. 30 785 9. 99 991 


10 


50 


8. 36 587 8. 36 599 9. 99 988 


10 


1OO 


8.30879 8.30888 9.99991 


05O 


2O 


8.36678 8.36689 9.99988 


040 


f tr 


log cos log cot log sin 


tf f 


f t f 


log cos log cot log sin 


rf f 



88 C 



26 



f ft 


log sin log tan log cos 


n t 


t tr 


log sin log tan log cos 


ff r 


20 o 


8.36678 8.36689 9.99988 


04O 


3OO 


8.41792 8.41807 9.99985 


03O 


10 


8.36768 8.36780 9.99988 


50 


10 


8. 41 872 8. 41 887 9. 99 985 


50 


20 


8.36858 8.36870 9.99988 


40 


20 


8. 41 952 8. 41 967 9. 99 985 


40 


30 


8.36948 8.36960 9.99988 


30 


30 


8. 42 032 8. 42 048 9. 99 985 


30 


40 


8.37038 8.37050 9.99988 


20 


40 


8.42112 8.42127 9.99985 


20 


50 


8.37128 8.37140 9.99988 


10 


50 


8. 42 192 8. 42 207 9. 99 985 


10 


210 


8.37217 8.37229 9.99988 


039 


310 


8. 42 272 8. 42 287 9. 99 985 


029 


10 


8,37306 8.37318 9.99988 


50 


10 


8.42351 8.42366 9.99985 


50 


20 


8.37395 8.37408 9.99988 


40 


20 


8.42430 8.42446 9.99985 


40 


30 


8.37484 8.37497 9.99988 


30 


30 


8.42510 8.42525 9.99985 


30 


40 


8.37573 8.37585 9.99988 


20 


40 


8. 42 589 8. 42 604 9. 99 985 


20 


50 


8.37662 8.37674 9.99988 


10 


50 


8. 42 667 8. 42 683 9. 99 985 


10 


220 


8.37750 8.37762 9.99988 


038 


320 


8.42746 8.42762 9.99984 


028 


10 


8.37838 8.37850 9k 99 988 


50 


10 


8.42825 8.42840 9.99984 


50 


20 


8.37926 8.37938 9.99988 


40 


20 


8. 42 903 8. 42 919 9. 99 984 


40 


30 


8.38014 8.38026 9.99987 


30 


30 


8. 42 982 8. 42 997 9. 99 984 


30 


40 


8.38101 8.38114 9.99987 


20 


40 


8.43060 8.43075 9.99984 


20 


50 


8.38189 8.38202 9.99987 


10 


50 


8.43138 8.43154 9.99984 


10 


230 


8.38276 8.38289 9.99987 


037 


330 


8. 43 216 8. 43 232 9. 99 984 


027 


10 


8.38363 8.38376 9.99987 


50 


10 


8.43293 8.43309 9.99984 


50 


20 


8.38450 8.38463 9.99987 


40 


20 


8. 43 371 8. 43 387 9. 99 984 


40 


30 


8.38537 8.38550 9.99987 


30 


30 


8. 43 448 8. 43 464 9. 99 984 


30 


40 


8.38624 8.38636 9.99987 


20 


40 


8.43526 8.43542 9-99984 


20 


50 


8.38710 8.38723 9.99987 


10 


50 


8.43603 8.43619 9.99984 


10 


240 


8.38796 8.38809 9.99987 


036 


340 


8.43680 8.43696 9.99984 


026 


10 


8. 38 882 8. 38 895 9. 99 987 


50 


10 


8.43757 . 8.43773 9.99984 


50 


20 


8.38968 8.38981 9.99987 


40 


20 


8.43834 8.43850 9.99984 


40 


30 


8.39054 8.39067 9.99987 


30 


30 


8. 43 910 8. 43 927 9. 99 984 


30 


40 


8.39139 8.39153 9.99987' 


20 


40 


8. 43 987 8. 44 003 9. 99 984 


20 


50 


8.39225 8.39238 9.99987 


10 


50 


8.44063 8.44080 9.99983 


10 


250 


8.39310 8.39323 9.99987 


035 


350 


8.44139 8.44156 9.99983 


025 


10 


8. 39 395 8. 39 408 9. 99 987 


50 


10 


8. 44 216 8. 44 232 9. 99 983 


50 


20 


8.39480 8.39493 9.99987 


40 


20 


8. 44 292 8. 44 308 9. 99 983 


40 


30 


8.39565 8.39578 9.99987 


30 


30 


8.44367 8.44384 9.99983 


30 


40 


8. 39 649 8. 39 663 9. 99 987 


20 


40 


8.44443 8.44460 9.99983 


20 


50 


8.39734 8.39747 9.99986 


10 


50 


8.44519 8.44536 9.99983 


10 


260 


8. 39 818 8. 39 832 9. 99 986 


034 


360 


8.44594 8.44611 9.99983 


024 


10 


8. 39 902 8. 39 916 9. 99 986 


50 


10 


8.44669 8.44686 9.99983 


50 


20 


8.39986 8.40000 9.99986 


40 


20 


8.44745 8.44762 9.99983 


40 


30 


8. 40 070 8. 40 083 9. 99 986 


30 


30 


8. 44 820 8. 44 837 9. 99 983 


30 


40 


8. 40 153 8. 40 167 9. 99 986 


20 


40 


8. 44 895 8. 44 912 9. 99 983 


20 


50 


8. 40 237 8. 40 251 9. 99 986 


10 


50 


8.44969 8.44987 9.99983 


10 


270 


8.40320 8.40334 9.99986 


033 


370 


8.45044 8.45061 9.99983 


023 


10 


8.40403 8.40417 9.99986 


50 


10 


8.45119 8.45136 9.99983 


50 


20 


8.40486 8.40500 9.99986 


40 


20 


8. 45 193 8. 45 210 9. 99 983 


40 


30 


8.40569 8.40583 9.99986 


30 


30 


8. 45 267 8. 45 285 9. 99 983 


30 


40 


8.40651 8.40665 9.99986 


20 


40 


8. 45 341 8. 45 359 9. 99 982 


20 


50 


8.40734 8.40748 9.99986 


10 


50 


8. 45 415 8. 45 433 9. 99 982 


10 


280 


8.40816 8.40830 9.99986 


032 


380 


8. 45 489 8. 45 507 9. 99 982 


022 


10 


8. 40 898 8. 40 913 9. 99 986 


50 


10 


8. 45 563 8. 45 581 9. 99 982 


50 


20 


8. 40 980 8. 40 995 9. 99 986 


40 


20 


8.45637 8.45655 9.99982 


40 


30 


8. 41 062 8. 41 077 9. 99 986 


30 


30 


8.45710 8.45728 9.99982 


30 


40 


8.41144 8.41158 9.99986 


20 


40 


8.45784 8.45802 9.99982 


20 


50 


8.41225 8.41240 9.99986 


10 


50 


8.45857 8.45875 9.99982 


10 


290 


8. 41 307 8. 41 321 9. 99 985 


031 


390 


8.45930 8.45948 9.99982 


021 


10 


8. 41 388 8. 41 403 9. 99 985 


50 


10 


8. 46 003 8. 46 021 9. 99 982 


50 


20 


8.41469 8.41484 9.99985 


40 


20 


8.46076 8.46094 9.99982 


40 


30 


8.41550 8.41565 9.99985 


30 


30 


8. 46 149 8. 46 167 9. 99 982 


30 


40 


8.41631 8.41646 9.99985 


20 


40 


8.46222 8.46240 9.99982 


20 


50 


8.41711 8.41726 9.99985 


10 


50 


8.46294 8.46312 9.99982 


10 


300 


8.41792 8.41807 9.99985 


03O 


4OO 


8.46366 8.46385 9.99982 


02O 


9 rf 


log cos log cot log sin 


tt r 


f ft 


log cos log cot log sin 


tr t 



88 



27 



t ft 


log sin log tan log cos 


tt f 


t tt 


log sin log tan log cos 


ft t 


4OO 


8.46366 8.46385 9.99982 


02O 


500 


8. 50 504 8. 50 527 9. 99 978 


01O 


10 


8.46439 8.46457 9.99982 


50 


10 


8.50570 8.50593 9.99978 


50 


20 


8.46511 8.46529 9.99982 


40 


20 


8.50636 8.50658 9.99978 


40 


30 


8.46583 8.46602 9.99981 


30 


30 


8. 50 701 8. 50 724 9. 99 978 


30 


40 


8.46655 8.46674 9.99981 


20 


40 


8. 50 767 8. 50 789 9. 99 977 


20 


50 


"8. 46 727 8.46745 9.99981 


10 


50 


8.50832 8.50855 9.99977 


10 


410 


8.46799 -8.46817 9.99981 


019 


510 


8.50897 -8.50920 9.99977 


9 


10 


8.46870 8.46889 9.99981 


50 


10 


8. 50 963 8. 50 985 9. 99 977 


50 


20 


8.46942 8.46960 9.99981 


40 


20 


8.51028 8.51050 9.99977 


40 


30 


8. 47 013 8. 47 032 9. 99 981 


30 


30 


8.51092 8.51115 9.99977 


30 


40 


8. 47 084 8. 47 103 9. 99 981 


20 


40 


8.51157 8.51180 9.99977 


20 


50 


8.47155 8.47174 9.99981 


10 


50 


8.51222 8.51245 9.99977 


10 


420 


8. 47 226 8. 47 245 9. 99 981 


018 


520 


8.51287 8.51310 9.99977 


8 


10 


8.47297 8.47316 9.99981 


50 


10 


8.51351 8.51374 9.99977 


50 


20 


8. 47 368 8. 47 387 9. 99 981 


40 


20 


8.51416 8.51439 9.99977 


40 


30 


8.47439 8.47458 9.99981 


30 


30 


8.51480 8.51503 9.99977 


30 


40 


8.47509 8.47528 9.99981 


20 


. 40 


8.51544 8.51568 9.99977 


20 


50 


8.47580 8.47599 9.99981 


10 


50 


8.51609 8.51632 9.99977 


10 


430 


8. 47 650 8. 47 669 9. 99 981 017 


530 


8.51673 8.51696 9.99977 


7 


10 


8.47720 8.47740 9.99980 


50 


10 


8.51737 8.51760 9.99976 


50 


20 


8.47790 8.47810 9.99980 


40 


20 


8.51801 8.51824 9.99976 


40 


30 


8.47860 8.47880 9.99980 


30 


30 


8.51864 8.51888 9.99976 


30 


40 


8.47930 8.47950 9.99980 


20 


40 


8.51928 8.51952 9.99976 


20 


50 


8.48000 8.48020 9.99980 


10 


50 


8.51992 8.52015 9.99976 


10 


440 


8.48069 8.48090 Q. 99 980 


016 


540 


8.52055 8.52079 9.99976 


6 


10 


8.48139 8.48159 9.99980 


50 


10 


8.52119 8.52143 9.99976 


50 


20 


8. 48 208 8. 48 228 9. 99 980 


40 


20 


8.52182 8.52206 9.99976 


40 


30 


8. 48 278 8. 48 298 9. 99 980 


30 


30 


8.52245 8.52269 9.99976 


30 


40 


8. 48 347 8. 48 367 9. 99 980 


20 


40 


8.52308 8.52332 9.99976 


20 


50 


8.48416 8.48436 9.99980 


10 


50 


8.52371 8.52396 9.99976 


10 


450 


8. 48 485 8. 48 505 9. 99 980 


015 


550 


8.52434 8.52459 9.99976 


5 


10 


8. 48 554 8. 48 574 9. 99 980 


50 


10- 


8. 52 497 8. 52 522 9. 99 976 


50 


20 


8. 48 622 8. 48 643 9. 99 980 


40 


20 


8.52560 8.52584 9.99976 


40 


30 


8.48691 8.48711 9.99980 


30 


30 


8. 52 623 8. 52 647 9. 99 975 


30 


40 


8.48760 8.48780 9.99979 


20 


40 


8.52685 8.52710 9.99975 


20 


50 


8.48828 8.48849 9.99979 


10 


50 


8. 52 748 8. 52 772 9. 99 975 


10 


460 


8.48896 8.48917 9.99979 


014 


560 


8. 52 810 8. 52 835 9. 99 975 


4 


10 8.48965 8.48985 9.99979 


50 


10 


8. 52 872 8. 52 897 9. 99 975 


50 


20 


8. 49 033 8. 49 053 9. 99 979 


40 


20 


8.52935 8.52960 9.99975 


40 


30 


8. 49 101 8. 49 121 9. 99 979 


30 


30 


8. 52 997 8. 53 022 9. 99 975 


30 


40 


8.49169 8.49189 9.99979 


20 


40 


8. 53 059 8. 53 084 9. 99 975 


20 


50 


8.49236 8.49257 9.99979 


10 


50 


8.53121 8.53146 9.99975 


10 


470 


8. 49 304 8. 49 325 9. 99 979 


013 


570 


8. 53 183 8. 53 208 9. 99 975 


3 


10 


8. 49 372 8. 49 393 9. 99 979 


50 


10 


8.53245 8.53270 9.99975 


50 


20 


8.49439 8.49460 9.99979 


40 


20 


8.53306 8.53332 9.99975 


40 


30 


8.49506 8.49528 9.99979 


30 


30 


8. 53 368 8. 53 393 9. 99 975 


30 


40 


8.49574 8.49595 9.99979 


20 


40 


8. 53 429 8. 53 455 9. 99 975 


20 


50 


8. 49 641 8. 49 662 9. 99 979 


10 


50 


8.53491 8.53516 9.99974 


10 


480 


8.49708 8.49729 9.99979 


012 


580 


8.53552 8.53578 9.99974 


2 


10 


8. 49 775 8. 49 796 9. 99 979 


50 


10 


8.53614 8.53639 9.99974 


50 


20 


8. 49 842 8. 49 863 9. 99 978 


40 


20 


8.53675 8.53700 9.99974 


40 


30 


8.49908 8.49930 -9.99978 


30 


30 


8. 53 736 8. 53 762 9. 99 974 


30 


40 


8. 49 975 8. 49 997 9. 99 978 


20 


40 


8.53797 8.53823 9.99974 


20 


50 


8. 50 042 8. 50 063 9. 99 978 


10 


50 


8.53858 8.53884 9.99974 


10 


490 


8.50108 8.50130 9.99978 


Oil 


590 


8.53919 8.53945 9.99974 


1 


10 


8.50174 8.50196 9.99978 


50 


10 


8. 53 979 8. 54 005 9. 99 974 


50 


20 


8. 50 241 8. 50 263 9. 99 978 


40 


20 


8.54040 8.54066 9.99974 


40 


30 


8. 50 307 8. 50 329 9. 99 978 


30 


30 


8. 54 101 8. 54 127 9. 99 974 


30 


40 


8. 50 373 8. 50 395 9. 99 978 


20 


40 


8. 54 161 8. 54 187 9. 99 974 


20 


50 


8.50439 8.50461 9.99978 


10 


50 


8. 54 222 8. 54 248 9. 99 974 


10 


5OO 


8.50504 8.50527 9.99978 


01O 


600 


8. 54 282 8. 54 308 9. 99 974 


O 


/ f t 


log cos log cot log sin 


tt t 


9 /' 


log cos log cot log sin 


tt t 



88 



28 



/ 


log sin log tan log cot log cos 


f 




8 S 11 ') 




o 


24186 24192 75808 99993 


6O 


1 


24903 24910 75090 99993 


59 


2 


25609 25616 74384 99993 


58 


3 


26304 26312 73688 99993 


57 


4 


26988 26996 73004 99992 


56 


5 


27661 27669 72331 99992 


55 


6 


28324 28332 71668 99992 


54 


7 


28977 28986 71014 99992 


53 


8 


29621 29629 70371 99992 


52 


9 


30255 30263 69737 99991 


51 


1C 


30879 30888 69112 99991 


50 


11 


31495 31505 68495 99991 


49 


12 


32103 32112 67888 99990 


48 


13 


32702 32711 67289 99990 


47 


14 


33292 33302 66698 99990 


46 


15 


33875 33886 66114 99990 


45 


16 


34450 34461 65539 99989 


44 


17 


35018 35029 64971 99989 


43 


18 


35578 35590 64410 99989 


42 


19 


36131 36143 63857 99989 


41 


2O 


36678 3*6689 63311 99988 


40 


21 


37217 37229 62771 99988 


39 


22 


37750 37762 62238 99988 


38 


23 


38276 38289 61711 99987 


37 


24 


38796 38809 61191 99987 


36 


25 


39310 39323 60677 99987 


35 


26 


39818 39832 60168 99986 


34 


27 


40320 40334 59666 99986 


33 


28 


40816 40830 59170 99986 


32 


29 


41307 41321 58679 99985 


31 


30 


41792 41807 58193 99985 


3O 


31 


42272 42287 57713 99985 


29 


32 


42746 42762 57238 99984 


28 


33 


43216 43232 56768 99984 


27 


34 


43680 43696 56304 99984 


26 


35 


44139 44156 55844 99983 


25 


36 


44594 44611 55389 99983 


24 


37 


45044 45061 54939 99983 


23 


38 


45489 45507 54493 99982 


22 


39 


45930 45948 54052 99982 


21 


40 


46366 46385 53615 99982 


2O 


41 


46799 46817 53183 99981 


19 


42 


47226 47245 52755 99981 


18 


43 


47650 47669 52331 99981 


17 


44 


48069 48089 51911 99980 


16 


45 


48485 48505 51495 99980 


15 


46 


48896 48917 51083 99979 


14 


47 


49304 49325 50675 99979 


13 


48 


49708 49729 50271 99979 


12 


49 


50108 50130 49870 99978 


11 


50 


50504 50527 49473 99978 


10 


51 


50897 50920 49080 99977 


9 


52 


51287 51310 48690 99977 


8 


53 


51673 51696 48304 99977 


7 


54 


52055 52079 47921 99976 


6 


55 


52434 52459 47541 99976 


5 


56 


52810 52835 47165 99975 


4 


57 


53183 53208 46792 99975 


3 


58 


53552 53578 46422 99974 


2 


59 


53919 53945 46055 99974 


1 


60 


54282 54308 45692 99974 


O 




8 ft 11 '* 




f 


<^ JL JL *7 

log COB log cot log tan log sin 


t 



f 


log sin 


log tan 


log cot 


log cos 


/ 


o 


54282 


54308 


45692 


99974 


6O 


1 


54642 


54669 


45331 


99973 


59 


2 


54999 


55027 


44973 


99973 


58 


3 


55354 


55382 


44618 


99972 


57 


4 


55705 


55734 


44266 


99972 


56 


5 


56054 


56083 


43917 


99971 


55 


6 


56400 


56429 


43571 


99971 


54 


7 


56743 


56773 


43227 


99970 


53 


8 


57084 


57114 


42886 


99970 


52 


9 


57421 


57452 


42548 


99969 


51 


1O 


57757 


57788 


42212 


99969 


5O 


11 


58089 


58121 


41879 


99968 


49 


12 


58419 


58451 


41549 


99968 


48 


13 


58747 


58779 


41221 


99967 


47 


14 


59072 


59105 


40895 


99967 


46 


15 


59395 


59428 


40572 


99967 


45 


16 


59715 


59749 


40251 


99966 


44 


17 


60033 


60068 


39932 


99966 


43 


18 


60349 


60384 


39616 


99965 


42 


19 


60662 


60698 


39302 


99964 


41 


2O 


60973 


61009 


38991 


99964 


40 


21 


61282 


61319 


38681 


99963 


39 


22 


61589 


61626 


38374 


99963 


38 


23 


61894 


61931 


38069 


99962 


37 


24 


62196 


62234 


37766 


99962 


36 


25 


62497 


62535 


37465 


99961 


35 


26 


62795 


62834 


37166 


99961 


34 


27 


63091 


63131 


36869 


99960 


33 


28 


63385 


63426 


36574 


99960 


32 


29 


63678 


63718 


36282 


99959 


31 


3O 


63968 


64009 


35991 


99959 


30 


31 


64256 


64298 


35702 


99958 


29 


32 


64543 


64585 


35415 


99958 


28 


33 


64827 


64870 


35130 


99957 


27 


34 


65110 


65154 


34846 


99956 


26 


35 


65 391 


65435 


34565 


99956 


25 


36 


65670 


65715 


34285 


99955 


24 


37 


65947 


65993 


34007 


99955 


23 


38 


66223 


66269 


33731 


99954 


22 


39 


66497 


66543 


33457 


99954 


21 


4O 


66769 


66816 


33184 


99953 


2O 


41 


67039 


67087 


32913 


99952 


19 


42 


67308 


67356 


32644 


99952 


18 


43 


67575 


67624 


32376 


99951 


17 


44 


67841 


67890 


32110 


99951 


16 


45 


68104 


68154 


31846 


99950 


15 


46 


68367 


68417 


31583 


99949 


14 


47 


68627 


68678 


31322 


99949 


13 


48 


68886 


68938 


31062 


99 948 


12 


49 


69144 


69196 


30804 


99948 


11 


50 


69400 


69453 


30547 


99947 


1O 


51 


69654 


69708 


30292 


99946 


9 


52 


69907 


69962 


30038 


99946 


8 


53 


70159 


70214 


29786 


99945 


7 


54 


70409 


70465 


29535 


99944 


6 


55 


70658 


70714 


29286 


99944 


5 


56 


70905 


70962 


29038 


99943 


4 


57 


71 151 


71208 


28792 


99942 


3 


58 


71395 


71453 


28547 


99942 


2 


59 


71638 


71697 


28303 


99941 


1 


6O 


71880 


71940 


28060 


99940 











1 -i 






' 


log cos 


log cot 


JL JL 

log tan 


log sin 


t 



88 



87 



3 



29 



f 


log sin 


log tan 


log cot 


log cos 


r 


o 


71880 


71940 


28060 


99940 


6O 


1 


72120 


72181 


27819 


99940 


59 


2 


72359 


72420 


27580 


99939 


58 


3 


72597 


72659 


27341 


99938 


57 


4 


72834 


72896 


27104 


99938 


56 


5 


73069 


73132 


26868 


99937 


55 


6 


73303 


73366 


26634 


99936 


54 


7 


73 535 


73600 


26400 


99936 


53 


8 


73767 


73832 


26168 


99935 


52 


9 


73997 


74063 


25 937 


99934 


51 


1O 


74226 


74292 


25708 


99934 


50 


11 


74454 


74521 


25479 


99933 


49 


12 


74680 


74748 


25 252 


99932 


48 


13 


74906 


74974 


25026 


99932 


47 


14 


75130 


75199 


24801 


99931 


46 


15 


75353 


75423 


24577 


99930 


45 


16 


75575 


75645 


24355 


99929 


44 


17 


75795 


75867 


24133 


99929 


43 


18 


76015 


76087 


23913 


9^928 


42 


19 


76234 


76306 


23694 


99927 


41 


2O 


76451 


76525 


23475 


99926 


40 


21 


76667 


76742 


23 258 


99926 


39 


22 


76883 


76958 


23042 


99925 


38 


23 


77 097 


77173 


22 827 


99924 


37 


24 


77310 


77387 


22613 


99923 


36 


25 


77522 


77600 


22400 


99923 


35 


26 


77733 


77811 


22189 


99922 


34 


27 


77943 


78022 


21978 


99921 


33 


28 


78152 


78232 


21768 


99920 


32 


29 


78360 


78441 


21559 


99920 


31 


30 


78 568 


78649 


21351 


99919 


3O 


31 


78774 


78 855 


21145 


99918 


29 


32 


78979 


79061 


20939 


99917 


28 


33 


79183 


79266 


20734 


99917 


27 


34 


79386 


79470 


20530 


99916 


26 


35 


79588 


79673 


20327 


99915 


25 


36 


79789 


79875 


20125 


99914 


24 


37 


79990 


80076 


19924 


99913 


23 


38 


80189 


80277 


19723 


99913 


22 


39 


80388 


80476 


19 524 


99912 


21 


40 


80 585 


80674 


19326 


99911 


2O 


41 


80782 


80872 


19128 


99910 


19 


42 


80978 


81 068 


18932 


99909 


18 


43 


81173 


81264 


18736 


99909 


17 


44 


81367 


81459 


18541 


99908 


16 


45 


81560 


81653 


18347 


99907 


15 


46 


81752 


81846 


18154 


99906 


14 


47 


81944 


82038 


17962 


99905 


13 


48 


82134 


82230 


17770 


99904 


12 


49 


82324 


82420 


17580 


99904 


11 


50 


82513 


82610 


17390 


99903 


10 


51 


82701 


82799 


17201 


99902 


9 


52 


82888 


82987 


17013 


99901 


8 


53 


83075 


83175 


16825 


99900 


7 


54 


83261 


83361 


16639 


99899 


6 


55 


83446 


83547 


16453 


99898 


5 


56 


83630 


83732 


16268 


99898 


4 


57 


83813 


83916 


16084 


99897 


3 


58 


83996 


84100 


15900 


99896 


2 


59 


84177 


84282 


15718 


99895 


1 


60 


84358 


84464 


15536 


99894 


O 








-i -i 






r 


log cos 


log cot 


JL _1_ 

log tan 


log sin 


t 



r 


log sin 


log tan 


log cot 


log cos 


f 




g 


g . 


-\ -i 


Q 







84358 


84464 


J- -lr 

15536 


99894 


6O 


1 


84539 


84646 


15354 


99893 


59 


2 


84718 


84826 


15174 


99892 


58 


3 


84897 


85006 


14994 


99891 


57 


4 


85075 


85185 


14815 


99891 


56 


5 


85252 


85363 


14637 


99890 


55 


6 


85429 


85540 


14460 


99889 


54 


7 


85605 


85717 


14283 


99888 


53 


8 


85780 


85893 


14107 


99887 


52 


9 


85955 


.86 069 


13931 


99886 


51 


1O 


86128 


86243 


13757 


99885 


50 


11 


86301 


86417 


13583 


99884 


49 


12 


86474 


86591 


13409 


99883 


48 


13 


86645 


86763 


13237 


99882 


47 


14 


86816 


86935 


13065 


99881 


46 


15 


86987 


87106 


12894 


99880 


45 


16 


87156 


87277 


12723 


99879 


44 


17 


87325 


87447 


12553 


99879 


43 


18 


87494 


87 616 


12384 


99878 


42 


19 


87661 


87785 


12215 


99877 


41 


20 


87829 


87953 


12047 


99876 


4O 


21 


87995 


88120 


11880 


99875 


39 


22 


88161 


88287 


11713 


99874 


38 


23 


88 326 


88453 


11547 


99873 


37 


24 


88490 


88618 


11382 


99872 


36 


25 


88654 


88783 


11217 


99871 


35 


26 


88817 


88948 


11052 


99870 


34 


27 


88980 


89111 


10889 


99869 


33 


28 


89142 


89274 


10726 


99868 


32 


29 


89304 


89437 


10563 


99^67 


31 


3O 


89464 


89598 


10402 


99866 


30 


31 


89625 


89760 


10240 


99865 


29 


32 


89784 


89920 


10080 


99864 


28 


33 


89943 


90080 


09920 


99863 


27 


34 


90102 


90240 


09760 


99862 


26 


35 


90260 


90399 


09601 


99861 


25 


36 


90417 


90557 


09443 


99860 


24 


37 


90574 


90715 


09 285 


99859 


23 


38 


90730 


90872 


09128 


99 858 22 


39 


90885 


91029 


08971 


99857 


21 


40 


91040 


91185 


08815 


99856 


2O 


41 


91195 


91340 


08660 


99855 


19 


42 


91349 


91495 


08505 


99854 


18 


43 


91502 


91650 


08350 


99853 


17 


44 


91655 


91803 


08197 


99852 


16 


45 


91807 


91957 


08043 


99851 


15 


46 


91959 


92110 


07890 


99 850 


14 


47 


92110 


92262 


07738 


99848 


13 


48 


92261 


92414 


07586 


99847 


12 


49 


92411 


92565 


07435 


99846 


11 


50 


92561 


92716 


07284 


99845 


1O 


51 


92710 


92866 


07134 


99844 


9 


52 


92859 


93016 


06984 


99843 


8 


53 


93007 


93165 


06835 


99842 


7 


54 


93154 


93313 


06687 


99841 


6 


55 


93301 


93462 


06538 


99840 


5 


56 


93448 


93609 


06391 


99839 


4 


57 


93594 


93756 


06244 


99838 


3 


58 


93740 


93 903 


06097 


99837 


2 


59 


93885 


94049 


05951 


99836 


1 


6O 


94030 


94195 


05805. 


99834 


O 




fi 


s 


1 1 






' log cos 


log cot 


JL JL 

log tan 


log sin 


r 



86 C 



85' 



30 



r 


log sin 


log tan 


log cot 


log oos 


r 




g 


. g 


JJ 


9 







94030 


94195 


05805 


99834 


6O 


1 


94174 


94340 


05660 


99833 


59 


2 


94317 


94485 


05515 


99832 


58 


3 


94461 


94630 


05370 


99831 


57 


4 


94603 


94773 


05227 


99830 


56 


5 


94746 


94917 


05083 


99829 


55 


6 


94887 


95060 


04940 


99828 


54 


7 


95029 


95202 


04798 


99827 


53 


8 


95 170 


95344 


04656 


99825 


52 


9 


95310 


95486 


04514* 


99824 


51 


10 


95450 


95627 


04373 


99823 


5O 


11 


95589 


95767 


04233 


99822 


49 


12 


95728 


95908 


04092 


99821 


48 


13 


95867 


96047 


03953 


99820 


47 


14 


96005 


96187 


03813 


99819 


46 


15 


96143 


96325 


03675 


99817 


45 


16 


96280 


96464 


03 536 


99816 


44 


17 


96417 


96602 


03398 


99815 


43 


18 


96553 


96739 


03261 


99814 


42 


19 


96689 


96877 


03123 


99813 


41 


2O 


96825 


97013 


02987 


99812 


4O 


21 


96960 


97150 


02850 


99810 


39 


22 


97095 


97285 


02715 


99809 


38 


23 


97229 


97421 


02579 


99808 


37 


24 


97363 


97556 


02444 


99807 


36 


25 


97496 


97691 


02309 


99806 


35 


26 


97629 


97825 


02175 


99804 


34 


27 


97762 


97959 


02041 


99803 


33 


28 


97894 


98092 


01908 


99802 


32 


29 


98026 


98225 


01775 


99801 


31 


30 


98157 


98 358 


01642 


99800 


30 


31 


98288 


98490 


01510 


99798 


29 


32 


98419 


98622 


01378 


99797 


28 


33 


98549 


98753 


01.247 


99796 


27 


34 


98679 


98884 


01116 


99795 


26 


35 


98808 


99015 


00985 


99793 


25 


36 


98937 


99145 


00855 


99792 


24 


37 


99066 


99275 


00725 


99791 


23 


38 


99194 


99405 


00595 


99790 


22 


39 


99322 


99534 


00466 


99788 


21 


40 


99450 


99662 


00338 


99787 


2O 


41 


99577 


99791 


00209 


99786 


19 


42 


99704 


99919 


00081 


99785 


18 


43 


99830 


00 046 


99 954 


99783 


17 


44 


99956 


00174 


99826 


99782 


16 


45 


00082 


00301 


99699 


99781 


15 


46 


00207 


00427 


99573 


99780 


14 


47 


00332 


00553 


99447 


99778 


13 


48 


00456 


00679 


99321 


99777 


12 


49 


00581 


00802 


99195 


99776 


11 


50 


00704 


00930 


99070 


99775 


1O 


51 


00828 


01055 


98945 


99773 


9 


52 


00951 


01179 


98821 


99772 


8 


53 


01074 


01303 


98697 


99771 


7 


54 


01196 


01427 


98573 


99769 


6 


55 


01318 


01550 


98450 


99768 


5 


56 


01440 


01673 


98327 


99767 


4 


57 


01561 


01796 


98204 


99765 


3 


58 


01682 


01918 


98082 


99764 


2 


59 


01803 


02040 


97960 


99763 


1 


60 


01923 


02162 


97838 


99761 









9 


1O 


9 




r 


- 

log cos 


log cot 


log tan 


log sin 


r 



t 


log sin 


log tan 


log cot 


log cos 


/ 




9 


9 


10 


9 




o 


01923 


02162 


97838 


99*761 


6O 


1 


02043 


02283 


97717 


99760 


59 


2 


02163 


02404 


97596 


99759 


58 


3 


02283 


02525 


97475 


99757 


57 


4 


02402 


02645 


97355 


99756 


56 


5 


02520 


02766 


97234 


99755 


55 


6 


02639 


02885 


97115 


99 753 


54 


7 


02757 


03005 


96995 


99752 


53 


8 


02874 


03124 


96876 


99751 


52 


9 


02992 


03242 


96758 


99749 


51 


1O 


03109 


03361 


96639 


99 748 


5O 


11 


03226 


03479 


96521 


99747 


49 


12 


03342 


03597 


96403 


99 745 


48 


13 


03458 


03714 


96286 


99744 


47 


14 


03574 


03832 


96168 


99742 


46 


15 


03690 


03948 


96052 


99741 


45 


16 


03805 


04065 


95935 


99740 


44 


17 


03920 


04181 


95819 


99738 


43 


18 


04034 


04297 


95703 


99737 


42 


19 


04149 


04413 


95587 


99736 


41 


2O 


04262 


04528 


95472 


99734 


40 


21 


04376 


04643 


95357 


99733 


39 


22 


04490 


04 758 


95242 


99731 


38 


23 


04603 


04873 


95127 


99730 


- 37 


24 


04715 


04987 


95013 


99728 


36 


25 


04828 


05101 


94899 


99727 


35 


26 


04940 


05214 


94786 


99726 


34 


27 


05052 


05328 


94672 


99724 


33 


28 


05164 


05441 


94559 


99723 


32 


29 


05275 


05553 


94447 


99721 


31 


30 


05386 


05666 


94334 


99720 


3O 


31 


05497 


05778 


94222 


99718 


29 


32 


05607 


05890 


94110 


99717 


28 


33 


05717 


06002 


93998 


99716 


27 


34 


05827 


06113 


93887 


99714 


26 


35 


05937 


06224 


93776 


99713 


25 


36 


06046 


06335 


93665 


99711 


24 


37 


06155 


06445 


93555 


99710 


23 


38 


06264 


06556 


93444 


99708 


22 


39 


06372 


06666 


93334 


99707 


21 


40 


06481 


06775 


93225 


99705 


2O 


41 


06589 


06885 


93115 


99 704 


19 


42 


06696 


06994 


93006 


99 702 


18 


43 


06804 


07103 


92897 


99701 


17 


44 


06911 


07211 


92789 


99699 


16 


45 


07018 


07320 


92680 


99698 


15 


46 


07124 


07428 


92572 


99696 


14 


47 


07231 


07536 


92464 


99695 


13 


48 


07337 


07643 


92357 


,99 693 


12 


49 


07442 


07751 


92249 


99692 


11 


50 


07548 


07 858 


92142 


99690 


10 


51 


07653 


07964 


92036 


99689 


9 


52 


07758 


08071 


91929 


99687 


8 


53 


07863 


08177 


91823 


99686 


7 


54 


07968 


08283 


91717 


99684 


6 


55 


08072 


08389 


91611 


99683 


5 


56 


08176 


08495 


91505 


99681 


4 


57 


08280 


08600 


91400 


99680 


3 


58 


08383 


08705 


91 295 


99678 


2 


59 


08486 


08810 


91190 


99677 


1 


60 


08589 


08914 


91086 


99675 











1 ft 






f 


log 008 


log cot 


JL vF 

log tan 


log sin 


t 



84 C 



83 C 



7 



8 C 



31 



r 


log sin 


log tan 


log cot 


log cos 


f 




< I 


c) 


10 










08589 


08914 


91086 


99675 


6O 


1 


08692 


09019 


90981 


99674 


59 


2 


08795 


09123 


90877 


99672 


58 


3 


08897 


09227 


90773 


99670 


57 


4 


08999 


09330 


90670 


99669 


56 


5 


09 101 


09434 


90566 


99667 


55 


6 


09 202 


09 537 


90463 


99666 


54 


7 


09 304 


09640 


90360 


99664 


53 


8 


09405 


09742 


90258 


99663 


52 


9 


09506 


09845 


90155 


99661 


51 


1O 


09606 


09947 


90053 


99659 


50 


11 


09707 


10049 


89951 


99658 


49 


12 


09807 


10 150 


89850 


99656 


48 


13 


09 907 


10252 


89748 


99655 


47 


14 


10006 


10353 


89647 


99653 


46 


15 


10106 


10454 


89546 


99651 


45 


16 


10205 


10555 


89445 


99650 


44 


17 


10 304 


10 656 


89344 


99648 


43 


18 


10402 


10756 


89244 


99647 


42 


19 


10 501 


10856 


89144 


99645 


41 


2O 


10599 


10956 


89044 


99643 


4O 


21 


10697 


11056 


88944 


99642 


39 


22 


10795 


11 155 


88845 


99640 


38 


23 


10 893 


11254 


88746 


99638 


37 


24 


10990 


11353 


88647 


99637 


36 


25 


11087 


11452 


88 548 


99635 


35 


26 


11 184 


11551 


88449 


99633 


34 


27 


11281 


11649 


88351 


99632 


33 


28 


11377 


11747 


88253 


99630 


32 


29 


11474 


11 845' 


88155 


99629 


31 


30 


11570 


11943 


88057 


99627 


30 


31 


11666 


12040 


87960 


99625 


29 


32 


11761 


12138 


87862 


99624 


28 


33 


11857 


12235 


87765 


99622 


27 


34 


11952 


12332 


87668 


99620 


26 


35 


12047 


12428 


87572 


99618 


25 


36 


12142 


12525 


87475 


99617 


24 


37 


12 236 


12621 


87379 


99 615 


23 


38 


12331 


12717 


87283 


99613 


22 


39 


12425 


12813 


87187 


99612 


21 


40 


12519 


12909 


87091 


99610 


2O 


41 


12612 


13004 


86996 


99608 


19 


42 


12706 


13099 


86901 


99607 


18 


43 


12799 


13 194 


86806 


99605 


17 


44 


12 892 


13289 


86711 


99603 


16 


45 


12985 


13384 


86616 


99601 


15 


46 


13078 


13478 


86522 


99600 


14 


47 


13171 


13573 


86427 


99598 


13 


48 


13263 


13667 


86333 


99596 


12 


49 


13355 


13761 


86239 


99595 


11 


50 


13447 


13854 


86146 


99593 


1O 


51 


13539 


13948 


86052 


99591 


9 


52 


13630 


14041 


85959 


99589 


8 


53 


13722 


14134 


85866 


99588 


7 


54 


13813 


14227 


85773 


99586 


6 


55 


13904 


14320 


85680 


99584 


5 


56 


13994 


14412 


85588 


99582 


4 


57 


14085 


14504 


85496 


99581 


3 


58 


14175 


14597 


85403 


99579 


2 


59 


14266 


14688 


85312 


99577 


1 


6O 


14356 


14780 


85220 


99.575 


O 








1 O 






t 


leg cos 


log cot 


JL VF 

log tan 


log sin 


r 



r 


log sin 


log tan 


log cot 


log cos 


t 




<) 





10 


C) 







14356 


14780 


85220 


99575 


6O 


1 


14445 


14872 


85128 


99574 


59 


2 


14535 


14963 


85037 


99572 


58 


3 


14624 


15 054 


84946 


99570 


57 


4 


14714 


15145 


84855 


99568 


56 


5 


14803 


15236 


84764 


99566 


55 


6 


14891 


15327 


84673 


99565 


54 


7 


14980 


15417 


84583 


99563 


53 


8 


15069 


15508 


84492 


99 561 


52 


9 


15157 


15598 


84402 


99559 


51 


1O 


15245 


15688 


84312 


99557 


5O 


11 


15333 


15777 


84223 


99556 


49 


12 


15421 


15867 


84133 


99554 


48 


13 


15508 


15956 


84044 


99552 


47 


14 


15596 


16046 


83954 


99 550 


46 


15 


15683 


16135 


83865 


99 548 


45 


16 


15770 


16224 


83776 


99546 


44 


17 


15857 


16312 


83688 


99545 


43 


18 


15944 


16401 


83599 


99543 


42 


19 


16030 


16489 


83511 


99541 


41 


2O 


16116 


16577 


83423 


99539 


4O 


21 


16203 


16665 


83335 


99537 


39 


22 


16289 


16753 


83247 


99535 


38 


23 


16374 


16841 


83159 


99533 


37 


24 


16460 


16928 


83072 


99532 


36 


25 


16545 


17016 


82984 


99530 


35 


26 


16631 


17103 


82897 


99528 


34 


27 


16716 


17190 


82810 


99526 


33 


28 


16801 


17277 


82723 


99524 


32 


29 


16886 


17363 


82637 


99522 


31 


3O 


16970 


17450 


82 550 


99520 


3O 


31 


17055 


17536 


82464 


99518 


29 


32 


17139 


17622 


82378 


99517 


28 


33 


17223 


17708 


82292 


99515 


27 


34 


17307 


17794 


82206 


99513 


26 


35 


17391 


17880 


82120 


99511 


25 


36 


17474 


17965 


82035 


99509 


24 


37 


17558 


18051 


81949 


99507 


23 


38 


17641 


18136 


81864 


99505 


22 


39 


17724 


18221 


81779 


99503 


21 


40 


17807 


18306 


81694 


99501 


20 


41 


17890 


18391 


81609 


99499 


19 


42 


17973 


18475 


81525 


99497 


18 


43 


18055 


18560 


81440 


99495 


17 


44 


18137 


18644 


81356 


99494 


16 


45 


18220 


18728 


81272 


99492 


15 


46 


18302 


18812 


81188 


99490 


14 


47 


18383 


18896 


81 104 


99488 


13 


48 


18465 


18979 


81021 


99486 


12 


49 


18547 


19063 


80937 


99484 


11 


50 


18628 


19146 


80854 


99482 


1O 


51 


18709 


19229 


80771 


99480 


9 


52 


18790 


19312 


80688 


99478 


8 


53 


18871 


19395 


80605 


99476 


7 


54 


18952 


19478 


80522 


99474 


6 


55 


19033 


19561 


80439 


99 472 


5 


56 


19113 


19643 


80357 


99470 


4 


57 


19193 


19725 


80275 


99468 


3 


58 


19273 


19807 


80 193 


99466 


2 


59 


19353 


19889 


80111 


99464 


1 


6O 


19433 


19971 


80029 


99462 











1 n 






/ 


log cos 


log cot 


J. vf 

log tan 


log sin 


f 



82 



sr 



32 



9 



10 



t 


log sin 


log tan 


log cot 


log cos 


f 




g 


j) 


10 


<) 




o 


19433 


19971 


80029 


99462 


6O 


1 


19513 


20053 


79947 


99460 


59 


2 


19592 


20134 


79866 


99458 


58 


3 


19672 


20216 


79784 


99456 


57 


4 


19751 


20297 


79703 


99454 


56 


5 


19830 


20378 


79622 


99452 


55 


6 


19909 


20459 


79541 


99450 


54 


7 


19988 


20540 


79460 


99448 


53 


8 


20067 


20621 


79379 


99446 


52 


9 


20145 


20701 


79299 


99444 


51 


1C 


20223 


20782 


79218 


99442 


50 


11 


20302 


20862 


79138 


99440 


49 


12 


20380 


20942 


79058 


99438 


48 


13 


20458 


21022 


78978 


99436 


47 


14 


20535 


21102 


78898 


99434 


46 


15 


20613 


21182 


78818 


99432 


45 


16 


20691 


21261 


78739 


99429 


44 


17 


20768 


21341 


78659 


99427 


43 


18 


20845 


21420 


78 580 


99425 


42 


19 


20922 


21499 


78501 


99423 


41 


2O 


20999 


21578 


78422 


99421 


40 


21 


21076 


21657 


78343 


99419 


39 


22 


21153 


21736 


78264 


99417 


38 


23 


21229 


21814 


78186 


99415 


37 


24 


21306 


21893 


78107 


99413 


36 


25 


21382 


21971 


78029 


99411 


35 


26 


21458 


22049 


77951 


99409 


34 


27 


21534 


22127 


77873 


99407 


33 


28 


21610 


22205 


77795 


99404 


32 


29 


21685 


22283 


77717 


99402 


31 


3D 


21761 


22361 


77639 


99400 


30 


31 


21836 


22438 


77562 


99398 


29 


32 


21912 


22516 


77484 


99396 


28 


33 


21987 


22593 


77407 


99394 


27 


34 


22062 


22670 


77330 


99392 


26 


35 


2213-7 


22747 


77253 


99390 


25 


36 


22211 


22824 


77176 


99388 


24 


37 


22286 


22901 


77099 


99385 


23 


38 


22361 


22977 


77023 


99383 


22 


39 


22435 


23054 


76 946 


99381 


21 


4O 


22 509 


23130 


76870 


99379 


2O 


41 


22583 


23206 


76794 


99377 


19 


42 


22657 


23283 


76717 


99375 


18 


43 


22731 


23359 


76641 


99372 


17 


44 


22805 


23435 


76565 


99370 


16 


45 


22878 


23510 


76490 


99368 


15 


46 


22952 


23586 


76414 


99366 


14 


47 


23 025 


23661 


76339 


99364 


13 


48 


23098 


23737 


76263 


99362 


12 


49 


23171 


23812 


76188 


99359 


11 


5O 


23244 


23887 


76113 


99357 


1O 


51 


23317 


23962 


76038 


99355 


9 


52 


23390 


24037 


75963 


99353 


8 


53 


23462 


24112 


75888 


99351 


7 


54 


23535 


24186 


75814 


99348 


6 


55 


23607 


24261 


75739 


99346 


5 


56 


23679 


24335 


75665 


99344 


4 


57 


23752 


24410 


75590 


99342 


3 


58 


23823 


24484 


75516 


99340 


2 


59 


23895 


24558 


75442 


99337 


1 


60 


23967 


24632 


75368 


99335 


O 








1 O 






t 


log cos 


log cot 


A Vr 

log tan 


log sin. 


f 



f 


log sin 


log tan 


log cot 


log cos 


t 







<) 


1 O 


<) 




O 


23967 


24632 


J. Vr 

75368 


99335 


6O 


1 


24039 


24706 


75294 


99333 


59 


2 


24110 


24779 


75221 


99331 


58 


3 


24181 


24853 


75147 


99328 


57 


4 


24253 


24926 


75 074 


99326 


56 


5 


24324 


25000 


75000 


99324 


55 


6 


24395 


25073 


74927 


99322 


54 


7 


24466 


25146 


74854 


99319 


53 


8 


24536 


25219 


74781 


99317 


52 


9 


24607 


25292 


74708 


99315 


51 


10 


24677 


25365 


74635 


99313 


50 


11 


24748 


25437 


74563 


99310 


49 


12 


24818 


25510 


74490 


99308 


48 


13 


24888 


25582 


74418 


99306 


47 


14 


24958 


25655 


74345 


99304 


46 


15 


25028 


25727 


74273 


99301 


45 


16 


25098 


25799 


74201 


99299 


44 


17 


25 168 


25871 


74129 


99 297 


43 


18 


25237 


25943 


74057 


99294 


42 


19 


25307 


26015 


73985 


99292 


41 


2O 


25376 


26086 


73914 


99290 


4O 


21 


25445 


26158 


73842 


99 288 - 


39 


22 


25514 


26229 


73771 


99285 


38 


23 


25583 


26301 


73699 


99283 


37 


24 


25 652 


26372 


73628 


99281 


36 


25 


25721 


26443 


73557 


99278 


35 


26 


25790 


26514 


73486 


99276 


34 


27 


25858 


26585 


73415 


99274 


33 


28 


25927 


26655 


73345 


99271 


32 


29 


25995 


26726 


73 274 


99269 


31 


3O 


26063 


26797 


73203 


99267 


3O 


31 


26131 


26867 


73133 


99264 


29 


32 


26199 


26937 


73063 


99262 


28 


33 


26267 


27008 


72992 


99260 


27 


34 


26335 


27078 


72922 


99257 


26 


35 


26 403 


27148 


72852 


99255 


25 


36 


26470 


27218 


72782 


99252 


24 


37 


26538 


27288 


72712 


99250 


23 


38 


26605 


27357 


72643 


99248 


22 


39 


26672 


27427 


72573 


99 245 


21 


4O 


26739 


27496 


72504 


99243 


2O 


41 


26806 


27566 


72434 


99241 


19 


42 


26873 


27635 


72365 


99238 


18 


43 


26940 


27704 


72296 


99236 


17 


44 


27007 


27773 


72227 


99233 


16 


45 


27073 


27842 


72158 


99231 


15 


46 


27 140 


27911 


72089 


99229 


14 


47 


27206 


27980 


72020 


99226 


13 


48 


27273 


28049 


71951 


99224 


12 


49 


27339 


28117 


71883 


99221 


11 


5O 


27405 


28186 


71814 


99219 


10 


51 


27471 


28254 


71746 


99217 


9 


52 


27537 


28323 


71677 


99214 


8 


53 


27602 


28391 


71609 


99212 


7 


54 


27668 


28459 


71541 


99209 


6 


55 


27734 


28527 


71473 


99207 


5 


56 


27799 


28595 


71405 


99204 


4 


57 


27864 


28662 


71338 


99202 


3 


58 


27930 


28730 


71270 


99200 


2 


59 


27995 


28798 


71202 


99197 


1 


60 


28060 


28865 


71135 


99195 


O 








1O 






t 


log cos 


log cot 


A Vr 

log tan 


log sin 


f 



80< 



79 ( 



ir 



12 



33 



f 


log sin 


log tan 


log cot 


log cos 


f 




9 


9 


1 O 


9 




o 


28 06'0 


28865 


_L \J 

71 135 


99195 


60 


1 


28 125 


28933 


71067 


99 192 


59 


2 


28 190 


29000 


71000 


99 190 


58 


3 


28 254 


29067 


70933 


99187 


57 


4 


28319 


29134 


70866 


99185 


56 


5 


28384 


29201 


70799 


99182 


55 


6 


28448 


29268 


70732 


99 180 


54 


7 


28512 


29335 


70665 


99177 


53 


8 


28577 


29402 


70598 


99175 


52 


9 


28641 


29468 


70532 


99172 


51 


1O 


28705 


29535 


70465 


99170 


5O 


11 


28769 


29 601 


70399 


99167 


49 


12 


28833 


29668 


70332 


99165 


48 


13 


28896 


29734 


70266 


99162 


47 


14 


28960 


29800 


70200 


99160 


46 


15 


29024 


29866 


70134 


99157 


45 


16 


29087 


29932 


70068 


99 155 


44 


17 


29150 


29998 


70002 


99152 


43 


18 


29214 


30064 


69936 


99150 


42 


19 


29277 


30130 


69870 


99147 


41 


2O 


29340 


30195 


69805 


99145 


4O 


21 


29403 


30261 


69739 


99142 


39 


22 


29466 


30326 


69674 


99140 


38 


23 


29529 


30391 


69609 


99137 


37 


24 


29591 


30457 


69543 


99135 


36 


25 


29654 


30522 


69478 


99132 


35 


26 


29716 


30587 


69413 


99130 


34 


27 


29779 


30652 


69348 


99127 


33 


28 


29841 


30717 


69283 


99124 


32 


29 


29903 


30782 


69218 


99122 


31 


3O 


29966 


30846 


69154 


99119 


30 


31 


30028 


30911 


69089 


99117 


29 


32 


30090 


30975 


69025 


99114 


28 


33 


30151 


31040 


68960 


99112 


27 


34 


30213 


31104 


68896 


99109 


26 


35 


30275 


31 168 


68832 


99106 


25 


36 


30 336 


31233 


68767 


99104 


24 


37 


30398 


31297 


68703 


99101 


23 


38 


30459 


31361 


68639 


99099 


22 


39 


30521 


31425 


68575 


99096 


21 


4O 


30582 


31489 


68511 


99093 


2O 


41 


30643 


31552 


68448 


99091 


19 


42 


30704 


31616 


68384 


99088 


18 


43 


30765 


31679 


68321 


99086 


17 


44 


30826 


31743 


68257 


99083 


16 


45 


30887 


31806 


68194 


99080 


15 


46 


30947 


31870 


68130 


99078 


14 


47 


31008 


31933 


68067 


99075 


13 


48 


31068 


31996 


68004 


99072 


12 


49 


31 129 


32059 


67941 


99070 


11 


5O 


31 189 


32122 


67878 


99067 


1O 


51 


31250 


32185 


67815 


99064 


9 


52 


31310 


32248 


67752 


99062 


8 


53 


31370 


32311 


67689 


99059 


7 


54 


31430 


32373 


67627 


99056 


6 


55 


31490 


32436 


67564 


99 054 


5 


56 


31549 


32 498 


67502 


99051 


4 


57 


31609 


32 561 


67439 


99048 


3 


58 


31669 


32623 


67377 


99046 


2 


59 


31728 


32685 


67315 


99043 


1 


6O 


31788 


32747 


67253 


99040 


O 




9 


9 


1 O 






r 


log cos 


log cot 


A VF 

log tan 


log sin 


f 



t 


log sin 


log tan 


log cot 


log cos 


r 




9 


9 


1 O 


9 




O 


31788 


32747 


A VF 

67253 


99040 


6O 


1 


31847 


32810 


67190 


99038 


59 


2 


31907 


32872 


67128 


99035 


58 


3 


31966 


32933 


67067 


99032 


57 


4 


32025 


32995 


67005 


99030 


56 


5 


32084 


33 057 


66943 


99027 


55 


6 


32143 


33119 


66881 


99024 


54 


7 


32202 


33 180 


66820 


99022 


53 


8 


32261 


33242 


66758 


99019 


52 


9 


32319 


33303 


66697 


99016 


51 


1O 


32378 


33365 


66635 


99013 


50 


11 


32437 


33426 


66574 


99011 


49 


12 


32495 


33487 


66513 


99008 


48 


13 


32553 


33 548 


66 452 


99 005 


47 


14 


32612 


33609 


66391 


99002 


46 


15 


32670 


33670 


66330 


99000 


45 


16 


32728 


33731 


66269 


98997 


44 


17 


32786 


33792 


66208 


98994 


43 


18 


32844 


33853 


66147 


98991 


42 


19 


32902 


33913 


66087 


98989 


41 


2O 


32960 


33974 


66026 


98986 


40 


21 


33018 


34034 


65966 


98983 


39 


22 


33075 


34095 


65 905 


98980 


38 


23 


33 133 


34155 


65845 


98978 


37 


24 


33 190 


34 215 


65785 


98975 


36 


25 


33248 


34276 


65724 


98972 


35 


26 


33305 


34336 


65664 


*98 969 


34 


27 


33362 


34396 


65604 


98967 


33 


28 


33420 


34456 


65544 


98964 


32 


29 


33477 


34516 


65484 


98961 


31 


30 


33534 


34576 


65424 


98958 


30 


31 


33591 


34635 


65365 


98955 


29 


32 


33647 


34695 


65305 


98953 


28 


33 


33704 


34755 


65245 


98950 


27 


34 


33761 


34814 


65 186 


98947 


26 


35 


33818 


34874 


65 126 


98944 


25 


36 


33874 


34933 


65067 


98941 


24 


37 


33931 


34992 


65008 


98938 


23 


38 


33987 


35051 


64949 


98936 


22 


39 


34043 


35111 


64889 


98933 


21 


4O 


34100 


35 170 


64830 


98930 


2O 


41 


34156 


35 229 


64771 


98927 


19 


42 


34212 


35288 


64712 


98924 


18 


43 


34268 


35347 


64653 


98921 


17 


44 


34324 


35405 


64595 


98919 


16 


45 


34380 


35464 


64536 


98916 


15 


46 


34436 


35523 


64477 


98913 


14 


47 


34491 


35581 


64419 


98910 


13 


48 


34547 


35640 


64360 


98907 


12 


49 


34602 


35698 


64302 


98904 


11 


50 


34658 


35757 


64243 


98901 


10 


51 


34713 


35815 


64185 


98898 


9 


52 


34769 


35873 


64127 


98 896 


8 


53 


34824 


35931 


64 069- 


98893 


7 


54 


34879 


35989 


64011 


98890 


6 


55 


34934 


36047 


63953 


98887 


5 


56 


34 989 


36105 


63895 


98884 


4 


57 


35044 


36163 


63837 


98881 


3 


58 


35099 


36221 


63779 


98878 


2 


59 


35154 


36279 


63721 


98875 


1 


60 


35 209 


36336 


63664 


98872 


O 








1 O 






f 


log cos 


log cot 


A VF 

log tan 


log sin 


f 



78 C 



13 



F 


log sin 


log tan 


log cot 


log cos 


t 




9 


<) 


10 


9 




O 


35209 


36336 


63664 


98872 


6O 


1 


35263 


36 394 


63 606 


98869 


59 


2 


35318 


36 452 


63548 


98867 


58 


3 


35373 


36509 


63491 


98864 


57 


4 


35427 


36566 


63434 


98861 


56 


5 


35481 


36624 


63376 


98858 


55 


6 


35536 


36681 


63319 


98855 


54 


7 


35590 


36738 


63262 


98852 


53 


8 


35644 


36795 


63205 


98849 


52 


9 


35698 


3.6 852 


63148 


98846 


51 


1O 


35752 


36909 


63091 


98843 


50 


11 


35806 


36966 


63034 


98840 


49 


12 


35 860 


37023 


62977 


98837 


48 


13 


35914 


37080 


62920 


98834 


47 


14 


35968 


37137 


62863 


98831 


46 


15 


36022 


37193 


62807 


98828 


45 


16 


36075 


37250 


62750 


98825 


44 


17 


36129 


37306 


62694 


98822 


43 


18 


36182 


37363 


62637 


98819 


42 


19 


36236 


37419 


62581 


98816 


41 


20 


36289 


37476 


62524 


98813 


4O 


21 


36342 


37532 


62468 


98810 


39 


22 


36395 


37588 


62412 


98807 


38 


23 


36449 


37644 


62356 


98804 


37 


24 


36502 


37700 


62300 


98801 


36 


25 


36555 


37756 


62244 


98798 


35 


26 


36608 


37812 


62188 


98795 


34 


27 


36660 


37868 


62132 


98792 


33 


28 


36713 


37924 


62076 


98789 


32 


29 


36766 


37980 


62020 


98786 


31 


3O 


36819 


38035 


61965 


98783 


30 


31 


36871 


38091 


61909 


98780 


29 


32 


36924 


38147 


61853 


98777 


28 


33 


36976 


38202 


61798 


98774 


27 


34 


37028 


38257 


61743 


98771 


26 


35 


37081 


38313 


61687 


98768 


25 


36 


37133 


38368 


61632 


98765 


24 


37 


37185 


38423 


61 577 


98762 


23 


38 


37237 


38479 


61521 


98759 


22 


39 


37289 


38534 


61466 


98 756 


21 


4O 


37341 


38589 


61411 


98753 


2O 


41 


37393 


38644 


61356 


98750 


19 


42 


37445 


38699 


61301 


98746 


18 


43 


37497 


38754 


61246 


98743 


17 


44 


37549 


38808 


61192 


98740 


16 


45 


37600 


38863 


61137 


98737 


15 


46 


37652 


38918 


61082 


98734 


14 


47 


37703 


38972 


61028 


98731 


13 


48 


37755 


39027 


60973 


98728 


12 


49 


37806 


39082 


60918 


98725 


11 


50 


37858 


39136 


60 864 


98722 


10 


51 


37909 


39190 


60810 


98719 


9 


52 


37960 


39 245 


60755 


98715 


8 


53 


38011 


39299 


60701 


98712 


7 


54 


38062 


39353 


60647 


98709 


6 


55 


38113 


39407 


60593 


98706 


5 


56 


38164 


39461 


60539 


98703 


4 


57 


38215 


39515 


60485 


98700 


3 


58 


38266 


39569 


60431 


98697 


2 


59 


38317 


39623 


60377 


98694 


1 


6O 


38368 


39677 


60323 


98690 


O 








1 O 






f 


. 
log cos 


log cot 


A Vr 

log tan 


log sin 


f 



f 


log sin 


log tan 


log cot 


log cos 


r 




g 


9 


10 


9 







38368 


39677 


60323 


98690 


6O 


1 


38418 


39731 


60269 


98687 


59 


2 


38469 


39785 


60215 


98684 


58 


3 


38519 


39838 


60162 


98681 


57 


4 


38570 


39892 


60108 


98678 


56 


5 


38620 


39945 


60055 


98675 


55 


6 


38670 


39999 


60001 


98671 


54 


7 


38721 


40052 


59948 


98668 


53 


8 


38771 


40106 


59894 


98665 


52 


9 


38821 


40159 


59841 


98662 


51 


1O 


38871 


40212 


59788 


98659 


5O 


11 


38921 


40266 


59734 


98656 


49 


12 


38971 


40319 


59681 


98652 


48 


13 


39021 


40372 


59628 


98649 


47 


14 


39071 


40425 


59575 


98646 


46 


15 


39121 


40478 


59522 


98643 


45 


16 


39170 


40531 


59469 


98640 


44 


17 


39220 


40 584 


59416 


98636 


43 


18 


39270 


40636 


59364 


98633 


42 


19 


39319 


40689 


59311 


98630 


41 


20 


39369 


40742 


59258 


98627 


40 


21 


39418 


40795 


59205 


98623 


39 


22 


39467 


40847 


59153 


98620 


38 


23 


39517 


40 900 


59100 


98617 


37 


24 


39566 


40952 


59048 


98614 


36 


25 


39615 


41005 


58995 


98 610 


35 


26 


39664 


41057 


58943 


98607 


34 


27 


39713 


41109 


58891 


98604 


33 


28 


39762 


41161 


58839 


98601 


32 


29 


39811 


41214 


58786 


98597 


31 


3O 


39860 


41266 


58734 


98594 


30 


31 


39909 


41318 


58682 


98591 


29 


32 


39958 


41370 


58630 


98588 


28 


33 


40006 


41422 


58 578 


98584 


27 


34 


40055 


41474 


58526 


98581 


26 


35 


40103 


41526 


58474 


98578 


25 


36 


40152 


41578 


58422 


98574 


24 


37 


40200 


41629 


58371 


98571 


23 


38 


40249 


41681 


58319 


98568 


22 


39 


40297 


41733 


58267 


98565 


21 


4O 


40346 


41784 


58216 


98561 


2O 


41 


40394 


41836 


58164 


98 558 


19 


42 


40442 


41887 


58113 


98555 


18 


43 


40490 


41939 


58061 


98551 


17 


44 


40538 


41990 


58010 


98548 


16 


45 


40586 


42041 


57959 


98545 


15 


46 


40634 


42093 


57907 


98541 


14 


47 


40682 


42144 


57856 


98538 


13 


48 


40730 


42195 


57805 


98535 


12 


49 


40778 


42246 


57754 


98531 


11 


50 


40825 


42297 


57703 


98528 


1O 


51 


40873 


42348 


57652 


98 525 


9 


52 


40921 


42399 


57601 


98521 


8 


53 


40968 


42450 


57550 


98518 


7 


54 


41016 


42501 


57499 


98515 


6 


55 


41063 


42 552 


57448 


98511 


5 


56 


41111 


42603 


57397 


98508 


4 


57 


41 158 


42653 


57347 


98 505 


3 


58 


41205 


42704 


57296 


98-501 


2 


59 


41 252 


42755 


57245 


98498 


1 


6O 


41 300 


42805 


57195 


98494 


O 








1 O 






t 


log cos 


log cot 


JL VF 

log tan 


log sin 


t 



76 



75' 



15 C 



16 



35 



f 


log sin 


log tan 


log cot 


log cos 


f 




9 


9 


1 O 


9 




o 


41300 


42805 


JL VF 

57195 


98494 


6O 


1 


41347 


42856 


57144 


98491 


59 


2 


41394 


42906 


57094 


98488 


58 


3 


41441 


42957 


57043 


98484 


57 


4 


41488 


43007 


56993 


98481 


56 


5 


41535 


43057 


56943 


98477 


55 


6 


41 582 


43108 


56892 


98474 


54 


7 


41628 


43158 


56842 


98471 


53 


8 


41 675 


43208 


56792 


98467 


52 


9 


41722 


43258 


56742 


98464 


51 


1O 


41768 


43308 


56692 


98460 


50 


11 


41815 


43358 


56642 


98457 


49 


12 


41861 


43408 


56592 


98453 


48 


13 


41908 


43458 


56542 


98450 


47 


14 


41954 


43508 


56492 


98447 


46 


15 


42001 


43558 


56442 


98443 


45 


16 


42047 


43607 


56393 


98440 


44 


17 


42093 


43657 


56343 


98436 


43 


18 


42140 


43707 


56293 


98433 


42 


19 


42186 


43756 


56244 


98429 


41 


20 


42232 


43806 


56194 


98426 


40 


21 


42278 


43855 


56145 


98422 


39 


22 


42324 


43905 


56095 


98419 


38 


23 


42370 


43954 


56046 


98415 


37 


24 


42416 


44004 


55996 


98412 


36 


25 


42461 


44053 


55947 


98409 


35 


26 


42507 


44102 


55898 


98405 


34 


27 


42553 


44151 


55849 


98402 


33 


28 


42599 


44201 


55799 


98398 


32 


29 


42644 


44250 


55750 


98395 


31 


30 


42690 


44299 


55701 


98391 


30 


31 


42735 


44348 


55652 


98388 


29 


32 


42781 


44397 


55603 


98384 


28 


33 


42826 


44446 


55554 


98381 


27 


34 


42872 


44495 


55505 


98377 


26 


35 


42917 


44544 


55456 


98373 


25 


36 


42962 


44592 


55408 


98370 


24 


37 


43008 


44641 


55359 


98366 


23 


38 


43053 


44690 


55310 


98363 


22 


39 


43098 


44738 


55262 


98359 


21 


4O 


43143 


44787 


55213 


98356 


2O 


41 


43188 


44836 


55164 


98352 


19 


42 


43233 


44884 


55 116 


98349 


18 


43 


43278 


44933 


55067 


98345 


17 


44 


43323 


44981 


55019 


98342 


16 


45 


43367 


45029 


54971 


98338 


15 


46 


43412 


45078 


54922 


98334 


14 


47 


43457 


45126 


54874 


98331 


13 


48 


43502 


45 174 


54826 


98327 


12 


49 


43546 


45222 


54778 


98324 


11 


50 


43 591 


45271 


54729 


98320 


1O 


51 


43635 


45319 


54681 


98317 


9 


52 


43680 


45367 


54633 


98313 


8 


53 


43724 


45415 


54585 


98309 


7 


54 


43769 


45463 


54 537 


98306 


6 


55 


43813 


45511 


54489 


98302 


5 


56 


43857 


45559 


54441 


98299 


4 


57 


43901 


45606 


54394 


98295 


3 


58 


43946 


45654 


54346 


98291 


2 


59 


43990 


45702 


54298 


98 288 


1 


60 


44034 


45750 


54250 


98284 







9 


9 


1 O 






t 


log cos 


log cot 


JL Vr 

log tan 


log sin 


r 



t 


log sin 


log tan 


log cot 


log cos 


f 




9 


9 


1 O 


9 




O 


44034 


45750 


-LvF 

54250 


98284 


60 


1 


44078 


45797 


54203 


98281 


59 


2 


44122 


45845 


54155 


98277 


58 


3 


44166 


45892 


54108 


98273 


57 


4 


44210 


45940 


54060 


98 270 56 


5 


44253 


45987 


54013 


98266 


55 


6 


44297 


46035 


53965 


98262 


54 


7 


44341 


46082 


53918 


98 259 


53 


8 


44385 


46130 


53870 


98 255 


52 


9 


44428 


46177 


53823 


98251 


51 


1O 


44472 


46224 


53776 


98248 


5O 


11 


44516 


46271 


53729 


98244 


49 


12 


44559 


46319 


53681 


98240 


48 


13 


44602 


46366 


53634 


98237 


47 


14 


44646 


46413 


53 587 


98233 


46 


15 


44689 


46460 


53540 


98229 


45 


16 


44733 


46507 


53493 


98226 


44 


17 


44776 


46554 


53446 


98222 


43 


18 


44819 


46601 


53399 


98218 


42 


19 


44862 


46648 


53352 


98215 


41 


2O 


44905 


46694 


53306 


98211 


4O 


21 


44948 


46741 


53259 


98207 


39 


22 


44992 


46788 


53212 


98204 


38 


23 


45035 


46835 


53165 


98200 


37 


24 


45077 


46881 


53119 


98196 


36 


25 


45 120 


46928 


53072 


98192 


35 


26 


45163 


46975 


53025 


98189 


34 


27 


45206 


47021 


52979 


98185 


33 


28 


45249 


47068 


52932 


98181 


32 


29 


45292 


47114 


52886 


98177 


31 


3O 


45334 


47160 


52840 


98174 


30 


31 


45377 


47207 


52793 


98170 


29 


32 


45419 


47253 


52747 


98166 


28 


33 


45462 


47299 


52701 


98162 


27 


34 


45504 


47346 


52654 


98159 


26 


35 


45547 


47392 


52608 


98155 


25 


36 


45589 


47438 


52562 


98151 


24 


37 


45632 


47484 


52516 


98147 


23 


38 


45674 


47530 


52470 


98144 


22 


39 


45716 


47576 


52424 


98140 


21 


40 


45758 


47622 


52378 


98136 


2O 


41 


45801 


47668 


52332 


98132 


19 


42 


45843 


47714 


52286 


98129 


18 


43 


45885 


47760 


52240 


98125 


17 


44 


45927 


47806 


52194 


98121 


16 


45 


45969 


47852 


52 148 


98117 


15 


46 


46011 


47897 


52103 


98113 


14 


47 


46 053 


47943 


52057 


98110 


13 


48 


46095 


47989 


52011 


98106 


12 


49 


46136 


48035 


51965 


98102 


11 


50 


46178 


48080 


51920 


98098 


1O 


51 


46220 


48126 


51874 


98094 


9 


52 


46262 


48171 


51829 


98090 


8 


53 


46303 


48217 


51783 


98087 


7 


54 


46345 


48262 


51738 


98083 


6 


55 


46386 


48307 


51693 


98079 


5 


56 


46428 


48353 


51647 


98075 


4 


57 


46469 


48398 


51602 


98071 


3 


58 


46511 


48 443 


51557 


98067 


2 


59 


46552 


48489 


51511 


98063 


1 


6O 


46594 


48 534 


51466 


98060 


O 




9 


9 


1O 






r 


log cos 


log cot 


JL VF 

log tan ' 


log sin 


f 



74 C 



36 



17 



18 C 



r 


log sin 


log tan 


log cot 


log cos 


r 







<) 


1 O 


<) 




O 


46594 


48534 


A vF 

51466 


98060 


6O 


1 


46635 


48579 


51421 


98056 


59 


2 


46676 


48624 


51376 


98052 


58 


3 


46717 


48669 


51331 


98048 


57 


4 


46758 


48714 


51286 


98044 


56 


5 


46800 


48759 


51241 


98040 


55 


6 


46841 


48804 


51 196 


98036 


54 


7 


46882 


48849 


51151 


98032 


53 


8 


46923 


48894 


51106 


98029 


52 


9 


46964 


48939 


51061 


98025 


51 


10 


47005 


48984 


51016 


98021 


5O 


11 


47045 


49029 


50971 


98017 


49 


12 


47086 


49073 


50927 


98013 


48 


13 


47127 


49118 


50882 


98009 


47 


14 


47168 


49163 


50837 


98005 


46 


15 


47209 


49207 


50793 


98001 


45 


16 


47249 


49252 


50748 


97 997 


44 


17 


47290 


49296 


50704 


97993 


43 


18 


47330 


49341 


50659 


97989 


42 


19 


47371 


49385 


50615 


97986 


41 


2O 


47411 


49430 


50570 


97982 


4O 


21 


47452 


49474 


50526 


97978 


39 


22 


47492 


49519 


50481 


97974 


38 


23 


47533 


49563 


50437 


97970 


37 


24 


47573 


49607 


50393 


97966 


36 


25 


47613 


49652 


50348 


97962 


35 


26 


47654 


49696 


50304 


97958 


34 


27 


47694 


49740 


50260 


97954 


33 


28 


47734 


49784 


50216 


97950 


32 


29 


47774 


49828 


50172 


97946 


31 


30 


47814 


49872 


50128 


97942 


3O 


31 


47854 


49916 


50084 


97938 


29 


32 


47894 


49960 


50040 


97934 


28 


33 


47934 


50004 


49996 


97930 


27 


34 


47974 


50048 


49952 


97926 


26 


35 


48014 


50092 


49908 


97922 


25 


36 


48054 


50136 


49 864 


97918 


24 


37 


48094 


50180 


49820 


97914 


23 


38 


48133 


50223 


49777 


97910 


22 


39 


48173 


50 267 


49733 


97906 


21 


4O 


48213 


50311 


49689 


97902 


2O 


41 


48252 


50355 


49645 


97898 


19 


42 


48292 


50398 


49602 


97894 


18 


43 


48332 


50 442 


49558 


97890 


17 


44 


48371 


50485 


49515 


97886 


16 


45 


48411 


50529 


49471 


97882 


15 


46 


48450 


50572 


49428 


97878 


14 


47 


48490 


50616 


49384 


97874 


13 


48 


48529 


50659 


49341 


97870 


12 


49 


48568 


50703 


49297 


97866 


11 


5O 


48607 


50746 


49254 


97861 


1O 


51 


48647 


50 789 


49211 


97857 


9 


52 


48686 


50833 


49167 


97853 


8 


53 


48725 


50876 


49124 


97849 


7 


54 


48764 


50919 


49081 


97845 


6 


55 


48 803 


50962 


49038 


97 841 


5 


56 


48842 


51005 


48995 


97837 


4 


57 


48881 


51048 


48952 


97833 


3 


58 


48920 


51092 


48908 


97829 


2 


59 


48959 


51135 


48865 


97825 


1 


60 


48998 


51178 


48822 


97821 


O 






<) 


1O 


9 




t 


log cos 


log cot 


log tan 


log sin 


t 



t 


log sin 


log tan 


log cot 


log cos 


/ 




9 


9 


1 O 


cj 




O 


48998 


51178 


J. \9 

48822 


97821 


6O 


1 


49037 


51221 


48779 


97817 


59 


2 


49 076 


51264 


48736 


97 812 


53 


3 


49115 


51306 


48694 


97808 


57 


4 


49153 


51349 


48651 


97804 


56 


5 


49192 


51392 


48608 


97800 


55 


6 


49231 


51435 


48565 


97796 


54 


7 


49269 


51478 


48522 


97792 


53 


8 


49308 


51520 


48480 


97788 


52 


9 


49347 


51563 


48437 


97784 


51 


1O 


49385 


51606 


48394 


97779 


5O 


11 


49424 


51648 


48352 


97775 


49 


12 


49462 


51691 


48309 


97771 


48 


13 


49500 


51734 


48266 


97767 


47 


14 


49539 


51776 


48224 


97763 


46 


15 


49577 


51819 


48181 


97759 


45 


16 


49615 


51861 


48139 


97754 


44 


17 


49654 


51903 


48097 


97750 


43 


18 


49692 


51946 


48054 


97746 


42 


19 


49730 


51988 


48012 


97742 


41 


2O 


49768 


52031 


47969 


97738 


'40 


21 


49806 


52073 


47927 


97734 


39 


22 


49844 


52115 


47885 


97729 


38 


23 


49882 


52 157 


47843 


97725 


37 


24 


49920 


52200 


47800 


97721 


36 


25 


49958 


52242 


47 758 


97717 


35 


26 


49996 


52284 


47716 


97713 


34 


27 


50034 


52 326 


47674 


97708 


33 


28 


50072 


52368 


47632 


97704 


32 


29 


50110 


52410 


47590 


97700 


31 


30 


50148 


52452 


47548 


97696 


30 


31 


50185 


52494 


47506 


97691 


29 


32 


50223 


52536 


47464 


97687 


28 


33 


50261 


52578 


47422 


97683 


27 


34 


50298 


52620 


47380 


97679 


26 


35 


50336 


52 661 


47339 


97674 


25 


36 


50374 


52703 


47297 


97670 


24 


37 


50411 


52745 


47255 


97666 


23 


38 


50449 


52787 


47213 


97662 


22 


39 


50486 


52829 


47171 


97657 


21 


40 


50 523 


52870 


47 130 


97653 


2O 


41 


50 561 


52912 


47088 


97649 


19 


42 


50 598 


52953 


47047 


97645 


18 


43 


50635 


52995 


47005 


97640 


17 


44 


50673 


53037 


46963 


97636 


16 


45 


50710 


53078 


46922 


97632 


15 


46 


50747 


53120 


46880 


97628 


14 


47 


50784 


53161 


46839 


97623 


13 


48 


50821 


53202 


46798 


97619 


12 


49 


50858 


53244 


46756 


97615 


11 


50 


50 896 


53285 


46715 


97610 


10 


51 


50 933 


53327 


46673 


97 606 


9 


52 


50970 


53368 


46632 


97602 


8 


53 


51007 


53409 


46591 


97 597 


7 


54 


51043 


53450 


46550 


97593 


6 


55 


51080 


53492 


46508 


97589 


5 


56 


51117 


53533 


46467 


97 584 


4 


57 


51154 


53574 


46426 


97580 


3 


58 


51191 


53 615 


46385 


97576 


2 


59 


51227 


53656 


46344 


97571 


1 


GO 


51264 


53697 


46303 


97567 


O 




9 


9 


1O 


9 





r 


log cos 


log cot 


log tan 


log sin 


t 



72 



19 C 



20 



37 



f 


log sin 


log tan 


log cot 


Iqg cos 


r 




9 


j> 


10 


9 




o 


51264 


53697 


46303 


97567 


60 


1 


51301 


53 738. 


46262 


97563 


59 


2 


51338 


53779 


46221 


97558 


58 


3 


51374 


53820 


46180 


97554 


57 


4 


51411 


53861 


46139 


97550 


56 


5 


51447 


53902 


46098 


97545 


55 


6 


51484 


53943 


46057 


97541 


54 


7 


51520 


53984 


46016 


97536 


53 


8 


51557 


54025 


45975 


97532 


52 


9 


51593 


54065 


45935 


97528 


51 


10 


51629 


54106 


45894 


97523 


5O 


11 


51666 


54147 


45853 


97519 


49 


12 


51702 


54187 


45813 


97515 


48 


13 


51738 


54228 


45772 


97510 


47 


14 


' 51 774 


54269 


45731 


97506 


46 


15 


51811 


54309 


45691 


97501 


45 


16 


51847 


54350 


45 650 


97497 


44 


17 


51883 


54390 


45610 


97492 


43 


18 


51919 


54431 


45569 


97488 


42 


19 


51955 


54471 


45529 


97484 


41 


2O 


51991 


54512 


45488 


97479 


4O 


21 


52027 


54552 


45448 


97475 


39 


22 


52 063 


54593 


45407 


97470 


38 


23 


52099 


54633 


45367 


97466 


37 


24 


52135 


54 673 


45-327 


97461 


36 


25 


52171 


54714 


45286 


97457 


35 


26 


52207 


54754 


45 246 


97453 


34 


27 


52242 


54794 


45206 


97448 


33 


28 


52278 


54835 


45165 


97444 


32 


29 


52314 


54875 


45125 


97439 


31 


30 


52350 


54915 


45085 


97435 


3O 


31 


52385 


54955 


45045 


97 430 


29 


32 


52421 


54995 


45005 


97426 


28 


33 


52456 


55035 


44965 


97421 


27 


34 


52492 


55075 


44925 


97417 


26 


35 


52527 


55115 


44 885 


97412 


25 


36 


52563 


55 155 


44845 


97408 


24 


37 


52598 


55 195 


44805 


97403 


23 


38 


52634 


55235 


44765 


97399 


22 


39 


52669 


55 275 


44725 


97394 


21 


40 


52705 


55315 


44685 


97390 


2O 


41 


52740 


55355 


44645 


97385 


19 


42 


52775 


55395 


44605 


97381 


18 


43 


52811 


55434 


44566 


97376 


17 


44 


52846 


55 474 


44526 


97372 


16 


45 


52881 


55514 


44486 


97367 


15 


46 


52916 


55554 


44446 


97363 


14 


47 


52951 


55593 


44407 


97358 


13 


48 


52986 


55633 


44367 


97353 


12 


49 


53021 


55673 


44327 


97349 


11 


50 


53056 


55712 


44288 


97344 


10 


51 


53092 


55 752 


44248 


97340 


9 


52 


53126 


55 791 


44209 


97 335 


8 


53 


53161 


55831 


44169 


97331 


7 


54 


53196 


55870 


44130 


97326 


6 


55 


53 231 


55 910 


44090 


97322 


5 


56 


53 266 


55949 


44051 


97317 


4 


57 


53301 


55989 


44011 


97312 


3 


58 


53336 


56028 


43972 


97308 


2 


59 


53370 


56067 


43933 


97303 


1 


60 


53405 


56107 


43893 


97299 


O 




9 




1 O 






f 


log cos 


log cot 


A VF 

log tan 


log sin 


r 



t 


log sin 


log tan 


log cot 


log cos 


r 




9 


9 


1 O 


9 




O 


53405 


56 107 


JL Vf 

43893 


97299 


60 


1 


53440 


56146 


43854 


97294 


59 


2 


53475 


56185 


43 815 


97289 


58 


3 


53509 


56224 


43776 


97285 


57 


4 


53544 


56264 


43736 


97280 


56 


5 


53578 


56 303 


43697 


97276 


55 


6 


53613 


56342 


43658 


97271 


54 


7 


53647 


56381 


43619 


97266 


53 


8 


53682 


56420 


43580 


97262 


52 


9 


53716 


56 459 


43541 


97257 


51 


10 


53751 


56498 


43502 


97252 


5O 


11 


53785 


56 537 


43463 


97248 


49 


12 


53819 


56576 


43424 


97243 


48 


13 


53854 


56615 


43 385 


97238 


47 


14 


53888 


56654 


43346 


97234 


46 


15 


53922 


56693 


43307 


97229 


45 


16 


53957 


56732 


43268 


97224 


44 


17 


53991 


56771 


43229 


97220 


43 


18 


54025 


56810 


43 190 


97215 


42 


19 


54059 


56849 


43151 


97210 


41 


20 


54093 


56887 


43113 


97206 


4O 


21 


54127 


56926 


43074 


97201 


39 


22 


54161 


56965 


43035 


97196 


38 


23 


54195 


57004 


42996 


97192 


37 


24 


54229 


57042 


42958 


97187 


36 


25 


54263 


57081 


42919 


97182 


35 


26 


54297 


57120 


42880 


97178 


34 


27 


54331 


57158 


42842 


97173 


33 


28 


54365 


57197 


42803 


97168 


32 


29 


54399 


57235 


42765 


97163 


31 


3O 


54433 


57 274 


42726 


97159 


30 


31 


54466 


57312 


42688 


97154 


29 


32 


54500 


57351 


42649 


97149 


28 


33 


54534 


57389 


42611 


97145 


27 


34 


54567 


57428 


42572 


97140 


26 


35 


54 601 


57466 


42534 


97135 


25 


36 


54635 


57504 


42496 


97130 


24 


37 


54668 


57543 


42457 


97126 


23 


38 


54702 


57581 


42419 


97121 


22 


39 


54735 


57619 


42381 


97116 


21 


40 


54769 


57658 


42342 


97111 


20 


41 


54802 


57 696 


42304 


97107 


19 


42 


54836 


57 734 


42266 


97102 


18 


43 


54869 


57772 


42228 


97097 


17 


44 


54903 


57810 


42190 


97092 


16 


45 


54936 


57849 


42151 


97087 


15 


46 


54969 


57887 


42113 


97083 


14 


47 


55003 


57925 


42075 


97078 


13 


48 


55036 


57963 


42037 


97073" 


12 


49 


55069 


58001 


41999 


97068 


11 


5O 


55102 


58039 


41961 


97063 


1O 


51 


55 136 


58077 


41923 


97059 


9 


52 


55169 


58 115. 


41885 


97054 


8 


53 


55202 


58153 


41847 


97049 


7 


54 


55235 


58191 


41809 


97044 


6 


55 


55268 


58229 


41771 


97039 


5 


56 


55301 


58267 


41733 


97035 


4 


57 


55334 


58304 


41696 


97030 


3 


58 


55367 


58342 


41658 


97025 


2 


59 


55400 


58380 


41620 


97020 


1 


6O 


55433 


58418 


41582 


97015 


O 








1 O 






f 


log cos 


log cot 


_I_ VF 

log tan 


log sin 


f 



70 



69 



38 



22 



t 


log sin log tan log cot log cos 


t 




i> 9 1O 9 







55433 58418 41582 97015 


60 


1 


55466 58455 41545 97010 


59 


2 


55499 58493 41507 97005 


58 


3 


55532 58531 41469 97001 


57 


4 


55564 58569 41431 96996 


56 


5 


55597 58606 41394 96991 


55 


6 


55630 58644 41356 96986 


54 


7 


55663 58681 41319 96981 


53 


8 


55695 58719 41281 96976 


52 


9 


55728 58757 41243 96971 


51 


1O 


55761 58794 41206 96966 


50 


11 


55 793 58 832 41 168 96 962 


49 


12 


55826 58869 41131 96957 


48 


13 


55858 58907 41093 96952 


47 


14 


55891 58944 41056 96947 


46 


15 


55923 58981 41019 96942 


45 


16 


55956 59019 40981 96937 


44 


17 


55988 59056 40944 96932 


43 


18 


56021 59094 40906 96927 


42 


19 


56053 59131 40869 96922 


41 


2O 


56085 59168 40832 96917 


40 


21 


56118 59205 40795 96912 


39 


22 


56150 59243 40757 96907 


38 


23 


56182 59280 40720 96903 


37 


24 


56215 59317 40683 96898 


36 


25 


56247 59354 40646 96893 


35 


26 


56279 59391 40609 96888 


34 


27 


56311 59429 40571 96883 


33 


28 


56343 59466 40534 96878 


32' 


29 


56375 59503 40497 96873 


31 


30 


56408 59540 40460 96868 


30 


31 


56440 59577 40423 96863 


29 


32 


56472 59614 40386 96858 


28 


33 


56504 59651 40349 96853 


27 


34 


56536 59688 40312 96848 


26 


35 


56568 59725 40275 96843 


25 


36 


56599 59762 40238 96838 


24 


37 


56631 59799 40201 96833 


23 


38 


56663 59835 40165 96828 


22 


39 


56695 59872 40128 96823 


21 


4O 


56727 59909 40091 96818 


2O 


41 


56759 59946 40054 96813 


19 


42 


56790 59983 40017 96808 


18 


43 


56822 60019 39981 96803 


17 


44 


56854 60056 39944 96798 


16 


45 


56886 60093 39907 96793 


15 


46 


56917 60130 39870 96788 


14 


47 


56949 60166 39834- 96783 


13 


48 


56980 60203 39'797 96778 


12 


49 


57012 60240 39760 96772 


11 


50 


57044 60276 39724 96767 


1O 


51 


57075 60313 39687 96762 


9 


52 


57107 60349 39651 96757 


8 


53 


57138 60386 39614 96752 


7 


54 


57169 60422 39578 96747 


6 


55 


57201 60459 39541 96742 


5 


56 


57232 60495 39505 96737 


4 


57 


57264 60532 39468 96732 


3 


58 


57295 60568 39432 96727 


2 


59 


57326 60605 39395 96722 


1 


6O 


57358 60641 39359 96717 







9 9 1O 9 




f 


log cos log cot log tan log sin 


r 



t 


log sin log tan log cot log cos 


r 




9 9 1O 9 







57358 60641 39359 96717 


6O 


1 


57389 60677 .39323 96711 


59 


2 


57420 60714 39286 96706 


58 


3 


57451 60750 39250 96701 


57 


4 


57482 60786 39214 96696 


56 


5 


57514 60823 39177 96691 


55 


6 


57545 60859 39141 96686 


54 


7 


57576 60895 39105 96681 


53 


8 


57607 60931 39069 96676 


52 


9 


57638 60967 39033 96670 


51 


1O 


57669 61004 38996 96665 


5O 


11 


57700 61040 38960 96660 


49 


12 


57731 61076 38924 96655 


48 


13 


57762 61112 38888 96650 


47 


14 


57793 61148 38852 96645' 


46 


15 


57824 61184 38816 96640 


45 


16 


57855 61220 38780 96634 


44 


17 


57885 61256 38744 96629 


43 


18 


57916 61292 38708 96624 


42 


19 


57947 61328 38672 96619 


41 


20 


57978 61364 38636 96614 


4O 


21 


58008 61400 38600 96608 


39 


22 


58039 61436 38564 96603 


38 


23 


58070 61472 38528 96598 


37 


24 


58101 61508 38492 96593 


36 


25 


58131 61544 38456 96588 


35 


26 


58162 61579 38421 96582 


34 


27 


58192 61615 38385 96577 


33 


28 


58223 61651 38349 96572 


32 


29 


58253 61687 38313 96567 


31 


3O 


58284 61722 38278 96562 


3O 


31 


58314 61758 38242 96556 


29 


32 


58345 61794 38206 96551 


28 


33 


58375 61830 38170 96546 


27 


34 


58406 61865 38135 96541 


26 


35 


58436 61901 38099 96535 


25 


36 


58467 61936 38064 96530 


24 


37 


58497 61972 38028 96525 


23 


38 


58527 62008 37992 96520 


22 


39 


58557 62043 37957 96514 


21 


4O 


58588 62079 37921 96509 


20 


41 


58618 62114 37886 96504 


19 


42 


58648 62150 37850 96498 


18 


43 


58678 62185 37815 96493 


17 


44 


58709 62221 37779 96488 


16 


45 


58739 62256 37744 96483 


15 


46 


58769 62292 37708 96477 


14 


47 


58799 62327 37673 96472 


13 


48 


58829 62362 37638 96467 


12 


49 


58859 62398 37602 96461 


11 


50 


58889 62433 37567 96456 


10 


51 


58919 62468 37532- 96451 


9 


52 


58949 62504 37496 96445 


8 


53 


58979 62539 37461 96440 


7 


54 


59009 62574 37426 96435 


6 


55 


59039 62609 37391 96429 


5 


56 


59069 62645 37355 96424 


4 


57 


59098 62680 37320 96419 


3 


58 


59128 62715 37285 96413 


2 


59 


59158 62750 37250 96408 


1 


60 


59188 62785 37215 96403 


O 




90 I (k 




f 


*J A Vr *J 

log cos log cot log tan log sin 


f 



68 ( 



67 



23 ( 



24 C 



39 



f 


log sin 


log tan 


log cot 


log cos 


t 




<j 


9 


1 O 


<) 







59*188 


62*785 


JL vr 

37215 


96403 


6O 


1 


59218 


62820 


37180 


96397 


59 


2 


59247 


62855 


37145 


96392 


58 


3 


59277 


62890 


37110 


96387 


57 


4 


59307 


62926 


37074 


96381 


56 


5 


59336 


62961 


37039 


96376 


55 


6 


59366 


62996 


37004 


96370 


54 


7 


59396 


63031 


36969 


96365 


53 


8 


59425 


63066 


36934 


96360 


52 


9 


59455 


63101 


36899 


96354 


51 


1O 


59484 


63135 


36865 


96349 


50 


11 


59514 


63170 


36830 


96343 


49 


12 


59543 


63205 


36795 


96338 


48 


13 


59573 


63240 


36760 


96333 


47 


14 


59602 


63275 


36725 


96327 


46 


15 


59632 


63310 


36690 


96322 


45 


16 


59661 


63345 


36655 


96316 


44 


17 


59690 


63379 


36621 


96311 


43 


18 


59720 


63414 


36586 


96305 


42 


19 


59749 


63449 


36551 


96300 


41 


20 


59778 


63484 


36516 


96294 


40 


21 


59808 


63519 


36481 


96289 


39 


22 


59837 


63553 


36447 


96284 


38 


23 


59866 


63588 


36412 


96278 


37 


24 


59895 


63623 


36377 


96273 


36 


25 


59924 


63657 


36343 


96267 


35 


26 


59954 


63692 


36308 


96262 


34 


27 


59983 


63726 


36274 


96256 


33 


28 


60012 


63761 


36239 


96251 


32 


29 


60041 


63796 


36204 


96245 


31 


30 


60070 


63830 


36170 


96240 


3O 


31 


60099 


63865 


36135 


96234 


29 


32 


60128 


63899 


36101 


96229 


28 


33 


60157 


63934 


36066 


96223 


27 


34 


60186 


63968 


36032 


96218 


26 


35 


60215 


64003 


35997 


96212 


25 


36 


60244 


64037 


35963 


96207 


24 


37 


60273 


64072 


35928 


96201 


23 


38 


60302 


64106 


35894 


96196 


22 


39 


60331 


64140 


35 860 


96190 


21 


4O 


60359 


64175 


35825 


96 185 


2O 


41 


60388 


64209 


35791 


96179 


19 


42 


60417 


64243 


35757 


96174 


18 


43 


60446 


64278 


35722 


96168 


17 


44 


60474 


64312 


35 688 


96162 


16 


45 


60 503 


64346 


35654 


96157 


15 


46 


60532 


64381 


35619 


96151 


14 


47 


60561 


64415 


35585 


96146 


13 


48 


60589 


64449 


35551 


96140 


12 


49 


60618 


64483 


35517 


96135 


11 


50 


60646 


64517 


35483 


96129 


1O 


51 


60675 


64552 


35448 


96123 


9 


52 


60704 


64586 


35414 


96118 


8 


53 


60732 


64620 


35 380 


96112 


7 


54 


60761 


64654 


35346 


96107 


6 


55 


60789 


64688 


35312 


96101 


5 


56 


60818 


64722 


35278 


96095 


4 


57 


60846 


64756 


35244 


96090 


3 


58 


60875 


64790 


35210 


96084 


2 


59 


60903 


64824 


35176 


96079 


1 


60 


60931 


64858 


35142 


96073 











1 O 






f 


log cos 


log cot 


A \J 

log tan 


log sin 


t 



r 


log sin 


log tan 


log cot 


log cos 


f 




9 


9 


10 


9 







60931 


64858 


35 142 


96073 


6O 


1 


60960 


64892 


35108 


96067 


59 


2 


60988 


64926 


35074 


96062 


58 


3 


61016 


64960 


35040 


96056 


57 


4 


61045 


64994 


35006 


96050 


56 


5 


61073 


65028 


34972 


96045 


55 


6 


61101 


65 062 


34938 


96039 


54 


7 


61129 


65096 


34904 


96034 


53 


8 


61158 


65 130 


34870 


96028 


52 


9 


61186 


65164 


34836 


96022 


51 


1O 


61214 


65 197 


34803 


96017 


5O 


11 


61242 


65231 


34769 


96011 


49 


12 


61270 


65265 


34735 


96005 


48 


13 


61298 


65299 


34 701 


96000 


47 


14 


61326 


65333 


34667 


95994 


46 


15 


61354 


65366 


34634 


95988 


45 


16 


61382 


65400 


34600 


95982 


44 


17 


61411 


65 434' 


34566 


95977 


43 


18 


61438 


65467 


34533 


95971 


42 


19 


61466 


65501 


34499 


95965 


41 


20 


61494 


65535 


34465 


95960 


4O 


21 


61522 


65568 


34432 


95954 


39 


22 


61550 


65602 


34398 


95948 


38 


23 


61578 


65636 


34364 


95942 


37 


24 


61606 


65669 


34331 


95937 


36 


25 


61634 


65703 


34297 


95931 


35 


26 


61662 


65736 


34264 


95925 


34 


27 


61689 


65770 


34230 


95920 


33 


28 


61717 


65803 


34197 


95914 


32 


29 


61745 


65837 


34163 


95908 


31 


3O 


61 773 


65870 


34130 


95902 


30 


31 


61800 


65904 


34096 


95897 


29 


32 


61828 


65937 


34063 


95891 


28 


33 


61856 


65971 


34029 


95885 


27 


34 


61 883 


66004 


33996 


95879 


26 


35 


61911 


66038 


33962 


95873 


25 


36 


61939 


66071 


33929 


95868 


24 


37 


61966 


66104 


33896 


95862 


23 


38 


61994 


66138 


33862 


95856 


22 


39 


62021 


66171 


33829 


95850 


21 


40 


62049 


66204 


33796 


95844 


2O 


41 


62076 


66238 


33 762 


95839 


19 


42 


62104 


66271 


33729 


95833 


18 


43 


62131 


66304 


33696 


95827 


17 


44 


62159 


66337 


33663 


95 821 


16 


45 


62186 


66371 


33629 


95815 


15 


46 


62214 


66404 


33596 


95810 


14 


47 


62241 


66437 


33663 


95804 


13 


48 


62268 


66470 


33530 


95798 


12 


49 


62296 


66503 


33497 


95792 


11 


50 


62323 


66537 


33463 


95786 


1O 


51 


62350 


66570 


33430 


95780 


9 


52 


62377 


66603 


33397 


95 775 


8 


53 


62405 


66636 


33364 


95769 


7 


54 


62432 


66669 


33331 


95763 


6 


55 


62459 


66702 


33298 


95757 


5 


56 


62 486 


66735 


33265 


95751 


4 


57 


62513 


66768 


33232 


95745 


3 


58 


62541 


66801 


33199 


95739 


2 


59 


62568 


66834 


33166 


95 733 


1 


6O 


62595 


66867 


33133 


95728 


O 








1 O 






r 


log cos 


log cot 


J.U 

log tan 


log sin 


r 



66 C 



65< 



40 



25 C 



26< 



t 


log sin 


log tan 


log cot 


log cos 


r 







<) 


1 1 > 


9 




o 


62595 


66867 


JL Vr 

33133 


95728 


6O 


1 


62622 


66900 


33100 


95722 


59 


2 


62649 


66933 


33067 


95716 


58 


3 


62676 


66966 


33034 


95710 


57 


4 


62703 


66999 


33001 


95704 


56 


5 


62730 


67032 


32968 


95 698 


55 


6 


62 757 


67065 


32935 


95692 


54 


7 


62784 


67098 


32902 


95686 


53 


8 


62811 


67131 


32869 


95680 


52 


9 


62838 


67163 


32837 


95674 


51 


1O 


62865 


67196 


32804 


95668 


50 


11 


62892 


67229 


32771 


95663 


49 


12 


62918 


67262 


32738 


95657 


48 


13 


62945 


67295 


32705 


95651 


47 


14 


62972 


67327 


32673 


95645 


46 


15 


62999 


67360 


32 640 


95639 


45 


16 


63026 


67393 


32607 


95633 


44 


17 


63052 


67426 


32574 


95627 


43 


18 


63079 


67458 


32542 


95621 


42 


19 


63106 


67491 


32509 


95 615 


41 


20 


63133 


67524 


32476 


95609 


4O 


21 


63159 


67556 


32444 


95 603 


39 


22 


63186 


67589 


32411 


95597 


38 


23 


63213 


67622 


32378 


95591 


37 


24 


63239 


67654 


32346 


95585 


36 


25 


63266 


67687 


32313 


95579 


35 


26 


63292 


67719 


32281 


95 573 


34 


27 


63319 


67752 


32248 


95 567 


33 


28 


63345 


67785 


32215 


95561 


32 


29 


63372 


67817 


32183 


95555 


31 


3O 


63398 


67850 


32 150 


95549 


30 


31 


63425 


67882 


32118 


95543 


29 


32 


63451 


67915 


32085 


95537 


28 


33 


63478 


67947 


32053 


95531 


27 


34 


63504 


67980 


32020 


95525. 


26 


35 


63531 


68012 


31988 


95 519 


25 


36 


63557 


68044 


31 956 


95513 


24 


37 


63583 


68077 


31923 


95 507 


23 


38 


63610 


68109 


31891 


95500 


22 


39 


63636 


68142 


31858 


95494 


21 


4O 


63662 


68 174 


31826 


95488 


20 


41 


63689 


68206 


31794 


95482 


19 


42 


63715 


68239 


31761 


95476 


18 


43 


63741 


68271 


31729 


95470 


17 


44 


63767 


68303 


31697 


95464 


16 


45 


63794 


68336 


31664 


95458 


15 


46 


63820 


68368 


31632 


95 452 


14 


47 


63846 


68400 


31600 


95446 


13 


48 


63872 


68432 


31568 


95440 


12 


49 


63898 


68465 


31535 


95434 


11 


50 


63924 


68497 


31503 


95 427 


1O 


51 


63950 


68529 


31471 


95421 


9 


52 


63976 


68561 


31439 


95 415 


8 


53 


64002 


68593 


31407 


95 409 


7 


54 


64028 


68626 


31374 


95403 


6 


55 


64054 


68658 


31342 


95397 


5 


56 


64 080 


68690 


31310 


.95391 


4 


57 


64106 


68722 


31278 


95384 


3 


58 


64132 


68754 


31246 


95378 


2 


59 


64158 


68786 


31214 


95372 


1 


60 


64184 


68818 


31 182 


95366 











1 O 






r 


log cos 


log cot 


.!_ Vr 

log tan 


log sin 


r 



? 


log sin 


log tan 


log cot 


log cos 


t 




<) 


9 


10 


4) 




O 


64184 


68818 


31182 


95*366 


6O 


1 


64210 


68850 


31150 


95360 


59 


2 


64236 


68882 


31118 


95354 


58 


3 


64262 


68914 


31086 


95348 


57 


4 


64288 


68946 


31 054 


95341 


56 


5 


64313 


68978 


31022 


95335 


55 


6 


64339 


69010 


30990 


95329 


54 


7 


64365 


69042 


30958 


95323 


53 


8 


64391 


69074 


30926 


95 317 


52 


9 


64417 


69106 


30894 


95310 


51 


1O 


64442 


69138 


30862 


95304 


5O 


11 


64468 


69170 


30830 


95298 


49 


12 


64 494 


69202 


30798 


95292 


48 


13 


64519 


69234 


30 766 


95286 


47 


14 


64545 


69266 


30734 


95279 


46 


15 


64 571. 


69298 


30702 


95273 


45 


16 


64596 


69329 


30671 


95267 


44 


17 


64622 


69361 


30639 


95261 


43 


18 


64647 


69393 


30607 


95254 


42 


19 


64673 


69425 


30575 


95248 


41 


20 


64698 


69457 


30543 


95242 


40 


21 


64724 


69488 


30512 


95236 


39 


22 


64749 


69520 


30480 


95229 


38 


23 


64775 


69 552 


30448 


95223 


37 


24 


6480G 


69584 


30416 


95217 


36 


25 


64826 


69615 


30385 


95211 


35 


26 


64851 


69647 


30353 


95204 


34 


27 


64877 


69679 


30321 


95 198 


33 


28 


64902 


69 710 


30290 


95192 


32 


29 


64927 


69742 


30258 


95185 


31 


30 


64953 


69774 


30226 


95179 


30 


31 


64978 


69805 


30195 


95 173 


29 


32 


65 003 


69837 


30163 


95 167 


28 


33 


65029 


69868 


30132 


95 160 


27 


34 


65054 


69900 


30100 


95 154 


26 


35 


65079 


69932 


30068 


95148 


25 


36 


65 104 


69963 


30037 


95 141 


24 


37 


65 130 


69995 


30005 


95135 


23 


38 


65155 


70026 


29974 


95129 


22 


39 


65 180 


70058 


29942 


95122 


21 


4O 


65205 


70 089 


29911 


95116 


2O 


41 


65230 


70121 


29879 


95 110 


19 


42 


65255 


70 152 


29848 


95103 


18 


43 


65281 


70184 


29816 


95097 


17 


44 


65306 


70215 


29785 


95090 


16 


45 


65331 


70247 


29753 


95084 


15 


46 


65356 


70278 


29722 


95078 


14 


47 


65381 


70309 


29691 


95071 


13 


48 


65406 


70341 


29 659 


95065 


12 


49 


65431 


70372 


29628 


95 059 


11 


5O 


65 456 


70404 


29596 


95052 


1O 


51 


65481 


70435 


29565 


95046 


9 


52 


65506 


70466 


29534 


95039 


8 


53 


65531 


70498 


29502 


95033 


7 


54 


65556 


70529 


29471 


95027 


6 


55 


65580 


70 560 


29440 


95 020 


5 


56 


65605 


70592 


29408 


95014 


4 


57 


65630 


70623 


29377 


95007 


3 


58 


65 655 


70654 


29346 


95001 


2 


59 


65680 


70685 


29315 


94995 


1 


60 


65705 


70717 


29283 


94988 


O 








1 *\ 






t 


log cos 


log cot 


JL\f 

log tan 


log sin 


f 



64 C 



63 C 



28 C 



41 



f 


log sin 


log tan 


log cot 


log cos 


f 




9 


9 


1 O 


9 




o 


65705 


70717 


_L vF 

29283 


94988 


60 


1 


65729 


70748 


29252 


94982 


59 


2 


65754 


70779 


29221 


94975 


58 


3 


65779 


70 810 


29190 


94969 


57 


4 


65 804 


70841 


29159 


94962 


56 


5 


65828 


70873 


29127 


94956 


55 


6 


65 853 


70904 


29096 


94949 


54 


7 


65878 


70935 


29065 


94>943 


53 


8 


65902 


70966 


29034 


94936 


52 


9 


65927 


70997 


29003 


94930 


51 


1O 


65952 


71028 


28972 


94923 


5O 


11 


65 976 


71059 


28941 


94917 


49 


12 


66001 


71090 


28910 


94911 


48 


13 


66 025 


71121 


28879 


94904 


47 


14 


66050 


71153 


28847 


94898 


46 


15 


66075 


71] 84 


28816 


94891 


45 


16 


66099 


71215 


28785 


94 885 


44 


17 


66124 


71246 


28754 


94878 


43 


18 


66148 


71277 


28723 


94871 


42 


19 


66173 


71308 


28692 


94865 


41 


2O 


66197 


71339 


28661 


94858 


4O 


21 


66221 


71370 


28630 


94852 


39 


22 


66246 


71401 


28599 


94845 


38 


23 


66270 


71431 


28569 


94839 


37 


24 


66295 


71462 


28538 


94832 


36 


25 


66319 


71493 


28507 


94826 


35 


26 


66343 


71524 


28476 


94819 


34 


27 


66368 


71555 


28445 


94813 


33 


28 


66392 


71586 


28414 


94806 


32 


29 


66416 


71617 


28383 


94799 


31 


30 


66441 


71648 


28352 


94793 


30 


31 


66465 


71679 


28321 


94786 


29 


32 


66489 


71709 


28291 


94780 


28 


33 


66513 


71740 


28260 


94773 


27 


34 


66537 


71771 


28229 


94767 


26 


35 


66562 


71802 


28198 


94760 


25 


36 


66586 


71833 


28167 


94753 


24 


37 


66610 


71863 


28137 


94747 


23 


38 


66634 


71894 


28106 


94740 


22 


39 


66658 


71925 


28075 


94734 


21 


4O 


66682 


71955 


28045 


94727 


2O 


41 


66706 


71986 


28014 


94 720 


19 


42 


66731 


72017 


27983 


94714 


18 


43 


66755 


72048 


27 952 


94707 


17 


44 


66779 


72078 


27922 


94700 


16 


45 


66803 


72109 


27891 


94694 


15 


46 


66827 


72140 


27860 


94 687 


14 


47 


66851 


72170 


27830 


94680 


13 


48 


66875 


72201 


27799 


94674 


12 


49 


66899 


72231 


27 769 


94667 


11 


50 


66922 


72262 


27738 


94660 


10 


51 


66946 


72293 


27707 


94654 


9 


52 


66970 


72323 


27677 


94647 


8 


53 


66994 


72354 


27646 


94640 


7 


54 


67018 


72384 


27616 


94634 


6 


55 


67042 


72415 


27585 


94627 


5 


56 


67066 


72445 


27 555 


94620 


4 


57 


67090 


72476 


27524 


94614 


3 


58 


67113 


72506 


27494 


94607 


2 


59 


67137 


72537 


27463 


94600 


1 


6O 


67161 


72567 


27433 


94593 


O 








1 O 






t 


log cos 


log cot 


JL\J 

log tan 


log sin 


f 



r 


log sin 


log tan 


log cot 


log cos 


f 




9 


9 


1 O 


9 




O 


67161 


72567 


J_ Vr 

27433 


94593 


60 


1 


67185 


72598 


27402 


94587 


59 


2 


67208 


72628 


27372 


94580 


58 


3 


67232 


72659 


27341 


94573 


57 


4 


67256 


72689 


27311 


94567 


56 


5 


67280 


72720 


27280 


94560 


55 


6 


67303 


72750 


27250 


94553 


54 


7 


67327 


72780 


27220 


94546 


53 


8 


67350 


72811 


27 189 


94540 


52 


9 


67374 


72841 


27159 


94533 


51 


1O 


67398 


72872 


27128 


94526 


5O 


11 


67421 


72902 


27098 


94519 


49 


12 


67445 


72932 


27068 


94513 


48 


13 


67468 


72963 


27037 


94506 


47 


14 


67492 


72993 


27007 


94499 


46 


15 


67515 


73023 


26977 


94492 


45 


16 


67539 


73054 


26946 


94485 


44 


17 


67562 


73084 


26916 


94479 


43 


18 


67586 


73114 


26886 


94472 


42 


19 


67609 


73144 


26856 


94465 


41 


20 


67633 


73175 


26825 


94458 


4O 


21 


67656 


73205 


26795 


94451 


39 


.22 


67680 


73235 


26765 


94445 


38 


23 


67 703 


73265 


26735 


94 438 


37 


24 


67726 


73295 


26705 


94431 


36 


25 


67750 


73326 


26674 


94424 


35 


26 


67773 


73356 


26644 


94417 


34 


27 


67796 


73386 


26614 


94410 


33 


28 


67820 


73416 


26584 


94404 


32 


29 


67843 


73446 


26554 


94397 


31 


30 


67866 


73476 


26524 


94390 


30 


31 


67890 


73507 


26493 


94383 


29 


32 


67913 


73537 


26463 


94376 


28 


33 


67936 


73567 


26433 


94369 


27 


34 


67959 


73597 


26403 


94362 


26 


35 


67982 


73627 


26373 


94355 


25 


36 


68006 


73657 


26343 


94349 


24 


37 


68029 


73687 


26313 


94342 


23 


38 


68052 


73717 


26283 


94335 


22 


39 


68075 


73747 


26253 


94328 


21 


4O 


68098 


73777 


26223 


94321 


2O 


41 


68121 


73807 


26193 


94314 


19 


42 


68144 


73837 


26163 


94307 


18 


43 


68167 


73867 


26133 


94300 


17 


44 


68190 


73897 


26103 


94293 


16 


45 


68213 


73927 


26073 


94286 


15 


46 


68237 


73957 


26043 


94279 


14 


47 


68260 


73987 


26013 


94273 


13 


48 


68283 


74017 


25983 


94266 


12 


49 


68 305 


74047 


25953 


94259 


11 


50 


68328 


74077 


25923 


94252 


10 


51 


68351 


74107 


25893 


94245 


9 


52 


68374 


74137 


25863 


94238 


8 


53 


68397 


74166 


25834 


94231 


7 


54 


68420 


74196 


25804 


94224 


6 


55 


68443 


74226 


25774 


94217 


5 


56 


68 466 . 


74 256 


25744 


94210 


4 


57 


68489 


74286 


25714 


94203 


3 


58 


68512 


74316 


25684 


94196 


2 


59 


68534 


74345 


25655 


94189 


1 


6O 


68557 


74375 


25625 


94182 


O 




9 




1 O 






f 


log cos 


log cot 


J_ Vr 

log tan 


log sin 


f 



er 



42 



r 


log sin 


log tan 


log cot 


log cos 


f 




4) 


g 


10 


4) 




O 


68557 


74375 


25625 


94182 


6O 


1 


68580 


74405 


25595 


94175 


59 


2 


68603 


74435 


25565 


94168 


58 


3 


68625 


74465 


25535 


94161 


57 


4 


68648 


74494 


25506 


94154 


56 


5 


68671 


74524 


25476 


94147 


55 


6 


68694 


74554 


25446 


94140 


54 


7 


68716 


74583 


25417 


94133 


53 


8 


68739 


74613 


25387 


94126 


52 


9 


68762 


74643- 


25357 


94119 


51 


1C 


68784 


74673 


25327 


94112 


5O 


11 


68807 


74702 


25298 


94105 


49 


12 


68829 


74732 


25268 


94098 


48 


13 


68852 


74762 


25238 


94090 


47 


14 


68875 


74791 


25209 


94083 


46 


15 


68897 


74821 


2S179 


94076 


45 


16 


68920 


74851 


25 149 


94069 


44 


17 


68942 


74880 


25120 


94062 


43 


18 


68965 


74910 


25090 


94055 


42 


19 


68987 


74939 


25 061 


94 048 


41 


20 


69010 


74969 


25031 


94041 


4O 


21 


69032 


74998 


25002 


94034 


39 


22 


69055 


75028 


24972 


94027 


38 


23 


69077 


75058 


24942 


94020 


37 


24 


69100 


75087 


24913 


94012 


36 


25 


69 122 


75 117 


24 883 


94005 


35 


26 


69144 


75146 


24854 


93998 


34 


27 


69167 


75176 


24824 


93991 


33 


28 


69189 


75205 


24795 


93984 


32 


29 


69212 


75235 


24765 


93977 


31 


30 


69234 


75264 


24736 


93970 


3O 


31 


69256 


75294 


24706 


93963 


29 


32 


69279 


75323 


24677 


93955 


28 


33 


69301 


75353 


24647 


93948 


27 


34 


69323 


75382 


24618 


93941 


26 


35 


69345 


75411 


24589 


93934 


25 


36 


69368 


75 441 


24559 


93927 


24 


37 


69390 


75470 


24530 


93920 


23 


38 


69412 


75500 


24500 


93912 


22 


39 


69434 


75529 


24471 


93905 


21 


4O 


69456 


75558 


24442 


93898 


2O 


41 


69479 


75588 


24412 


93891 


19 


42 


69501 


75617 


24383 


93884 


18 


43 


69523 


75647 


24353 


93876 


17 


44 


69545 


75676 


24324 


93869 


16 


45 


69567 


75705 


24295 


93862 


15 


46 


69589 


75735 


24265 


93855 


14 


47 


69611 


75 764 


24236 


93847 


13 


48 


69633 


75793 


24207 


93840 


12 


49 


69655 


75822 


24178 


93833 


11 


50 


69 377 


75852 


24148 


93826 


10 


51 


69699 


75881 


24119 


93819 


9 


52 


69721 


75910 


24090 


93811 


8 


53 


69743 


75 939 


24061 


93804 


7 


54 


69765 


75969 


24031 


93797 


6 


55 


69787 


75998 


24002 


93789 


5 


56 


69809 


76027 


23973 


93782 


4 


57 


69831 


76056 


23944 


93775 


3 


58 


69853 


76086 


23914 


93768 


2 


59 


69875 


76115 


23885 


93760 


1 


60 


69897 


76144 


23856 


93753 


O 








1 O 






f 


log cos 


log cot 


_I_ VF 

log tan 


log sin 


t 



f log sin 


log tan 


log cot 


log cos 


/ 


<) 


<> 


10 


<) 




O 69 897 


76144 


23 856 


93753 


6O 


1 


69919 


76173 


23827 


93746 


59 


2 


69941 


76202 


23798 


93738 


58 


3 


69963 


76231 


23769 


93731 


57 


4 


69984 


76261 


23739 


93724 


56 


5 


70006 


76290 


23710 


93717 


55 


6 


70 028 


76319 


23681 


93709 


54 


7 


70050 


76348 


23652 


93702 


53 


8 


70072 


76377 


23623 


93695 


52 


9 


70093 


76406 


23594 


93687 


51 


10 


70115 


76435 


23565 


93680 


5O 


11 


70137 


76464 


23536 


93673 


49 


12 


70159 


76493 


23507 


93665 


48 


13 


70180 


76522 


23478 


93658 


47 


14 


70202 


76551 


23449 


93650 


46 


15 


70224 


76580 


23420 


93643 


45 


16 


70245 


76609 


23391 


93636 


44 


17 


70267 


76639 


23361 


93628 


43 


18 


70288 


76668 


23332 


93621 


42 


19 


70310 


76697 


23303 


93614 


41 


2O 


70332 


76725 


23275 


93606 


4O 


21 


70353 


76754 


23246 


93599 


39 


22 


70375 


76783 


23217 


93591 


38 


23 


70396 


76812 


23188 


93584 


37 


24 


70418 


76841 


23159 


93577 


36 


25 


70439 


76870 


23130 


93569 


35 


26 


70461 


76899 


23101 


93562 


34 


27 


70482 


76928 


23072 


93554 


33 


28 


70504 


76957 


23043 


93547 


32 


29 


70 525 


76986 


23014 


93539 


31 


30 


70547 


77015 


22985 


93532 


30 


31 


70 568 


77044 


22956 


93525 


29 


32 


70590 


77073 


22927 


93517 


28 


33 


70611 


77101 


22899 


93510 


27 


34 


70633 


77 130 


22870 


93502 


26 


35 


70654 


77159 


22841 


93495 


25 


36 


70675 


77188 


22812 


93487 


24 


37 


70697 


77217 


22783 


93480 


23 


38 


70718 


77246 


22754 


93472 


22 


39 


70739 


77274 


22726 


93465 


21 


4O 


70761 


77303 


22697 


93457 


2O 


41 


70782 


77332 


22668 


93450 


19 


42 


70803 


77361 


22639 


93442 


18 


43 


70824 


77390 


22610 


93435 


17 


44 


70846 


77418 


22582 


93427 


16 


45 


70867 


77447 


22553 


93420 


15 


46 


70888 


77476 


22524 


93412 


14 


47 


70909 


77505 


22495 


93405 


13 


48 


70931 


77533 


22467 


93397 


12 


49 


70952 


77 562. 


22438 


93390 


11 


5O 


70973 


77591 


22409 


93382 


10 


51 


70994 


77619 


22381 


93375 


9 


52 


71015 


77648 


22352 


93367 


8 


53 


71036 


77677 


22323 


93360 


7 


54 


71 058 


77706 


22294 


93352 


6 


55 


71079 


77734 


22266 


93344 


5 


56 


71100 


77763 


22237 


93337 


4 


57 


71121 


77791 


22209 


93329 


3 


58 


71142 


77820 


22180 


93322 


2 


59 


71163 


77849 


22151 


93314 


1 


60 


71 184 


77877 


22123 


93307 











1 O 






f 


log cos 


log cot 


A Vr 

log tan 


log sin 


t 



60 C 



59 C 



31 C 



32' 



43 



t 


log sin 


log tan 


log cot 


log cos 


r 




<) 


9 


10 


<j 




o 


71 184' 


77877 


22123 


93307 


60 


1 


71205 


77906 


22094 


93299 


59 


2 


71226 


77935 


22065 


93291 


58 


3 


71247 


77963 


22037 


93284 


57 


4 


71268 


77992 


22008 


93276 


56 


5 


71289 


78020 


21980 


93269 


55 


6 


71310 


78049 


21951 


93261 


54 


7 


71331 


78077 


21923 


93253 


53 


8 


71352 


78106 


21894 


93246 


52 


9 


71373 


78135 


21865 


93238 


51 


1O 


71393 


78163 


21837 


93230 


5O 


11 


71414 


78192 


21808 


93223 


49 


12 


71435 


78220 


21780 


93215 


48 


13 


71456 


78249 


21751 


93207 


47 


14 


71477 


78277 


21723 


93200 


46 


15 


71498 


78306 


21694 


93192 


45 


16 


71519 


78334 


21666 


93184 


44 


17 


71539 


78363 


21637 


93177 


43 


18 


71560 


78391 


21609 


93169 


42 


19 


71581 


78419 


21581 


93161 


41 


2O 


71602 


78448 


21552 


93154 


4O 


21 


71622 


78476 


21524 


93 146 


39 


22 


71643 


78505 


21495 


93138 


38 


23 


71664 


78533 


21 467 


93131 


37 


24 


71685 


78562 


21438 


93123 


36 


25 


71705 


78590 


21410 


93115 


35 


26 


71726 


78618 


21382 


93108 


34 


27 


71747 


78647 


21353 


93100 


33 


28 


71767 


78675 


21325 


93092 


32 


29 


71788 


78704 


21296 


93084 


31 


3O 


71809 


78732 


21268 


93077 


3O 


31 


71829 


78760 


21240 


93069 


29 


32 


71850 


78789 


21211 


93061 


28 


33 


71870 


78817 


21183 


93053 


27 


34 


71891 


78845 


21155 


93046 


26 


35 


71911 


78874 


21126 


93038 


25 


36 


71932 


78902 


21098 


93030 


24 


37 


71952 


78930 


21070 


93022 


23 


38 


71973 


78959 


21041 


93014 


22 


39 


71994 


78987 


21013 


93007 


21 


4O 


72014 


79015 


20985 


92999 


2O 


41 


72034 


79043 


20957 


92991 


19 


42 


72055 


79072 


20928 


92983 


18 


43 


72075 


79100 


20900 


92976 


17 


44 


72096 


79128 


20872 


92968 


16 


45 


72116 


79156 


20844 


92960 


15 


46 


72137 


79185 


20815 


92952 


14 


47 


72157 


79213 


20787 


92944 


13 


48 


72177 


79241 


20759 


92936 


12 


49 


72198 


79269 


20731 


92929 


11 


50 


72218 


79297 


20703 


92 921. 


10 


51 


72238 


79326 


20674 


92913 


9 


52 


72259 


79354 


20646 


92905 


8 


53 


72279 


79382 


20618 


92897 


7 


54 


72299 


79410 


20590 


92889 


6 


55 


72320 


79438 


20562 


92881 


5 


56 


72340 


79466 


20534 


92874 


4 


57 


72360 


79495 


20505 


92866 


3 


58 


72381 


79523 


20477 


92858 


2 


59 


72401 


79551 


20449 


92850 


1 


6O 


72421 


79579 


20421 


92842 


O 








1 O 






t 


log cos 


log cot 


JL vF 

log tan 


log sin 


f 



f 


log sin 


log tan 


log cot 


log cos 


f 




9 


9 


10 


<) 







72421 


79579 


20421 


92842 


6O 


1 


72441 


79607 


20393 


92834 


59 


2 


72461 


79635 


20365 


92826 


58 


3 


72482 


79663 


20337 


92818 


57 


4 


72502 


79691 


20309 


92810 


56 


5 


72522 


79719 


20281 


92803 


55 


6 


72542 


79747 


20253 


92795 


54 


7 


72562 


79776 


20224 


92787 


53 


8 


72582 


79804 


20196 


92779 


52 


9 


72602 


79832 


20168 


92771 


51 


1O 


72622 


79860 


20140 


92763 


5O 


11 


72643 


79888 


20112 


92755 


49 


12 


72663 


79916 


20084 


92747 


48 


13 


72683 


79944 


20056 


92739 


47 


14 


72703 


79972 


20028 


92731 


46 


15 


72723 


80000 


20000 


92723 


45 


16 


72743 


80028 


19972 


92715 


44 


17 


72763 


80056 


19944 


92707 


43 


18 


72783 


80084 


19916 


92699 


42 


19 


72803 


80112 


19888 


92691 


41 


2O 


72823 


80140 


19860 


92683 


4O 


21 


72843 


80168 


19832 


92675 


39 


22 


72863 


80195 


19805 


92667 


38 


23 


72883 


80223 


19777 


92659 


37 


24 


72902 


80251 


19749 


92651 


36 


25 


72922 


80279 


19721 


92643 


35 


26 


72942 


80307 


19693 


92635 


34 


27 


72962 


80335 


19665 


92627 


33 


28 


72982 


80363 


19637 


92619 


32 


29 


73002 


80391 


19609 


92611 


31 


30 


73022 


80419 


19581 


92603 


30 


31 


73041 


80447 


19553 


92595 


29 


32 


73061 


80474 


19526 


92587 


28 


33 


73081 


80502 


19498 


92579 


27 


34 


73101 


80530 


19470 


92571 


26 


35 


73121 


80 558 


19442 


92563 


25 


36 


73140 


80586 


19414 


92555 


24 


37 


73160 


80614 


19386 


92546 


23 


38 


73180 


80642 


19358 


92538 


22 


39 


73200 


80669 


19331 


92530 


21 


40 


73219 


80697 


19303 


92522 


20 


41 


73239 


80725 


19275 


92514 


19 


42 


73259 


80 753 


19247 


92506 


18 


43 


73278 


80781 


19219 


92498 


17 


44 


73 298 


80808 


19192 


92490 


16 


45 


73318 


80836 


19164 


92482 


15 


46 


73337 


80864 


19136 


92473 


14 


47 


73357 


80892 


19108 


92465 


13 


48 


73377 


80919 


19081 


92457 


12 


49 


73396 


80947 


19053 


92449 


11 


5O 


73416 


80975 


19025 


92441 


to 


51 


73435 


81003 


18997 


92433 


9 


52 


73455 


81030 


18970 


92425 


8 


53 


73474 


81058 


18942 


92416 


7 


54 


73494 


81086 


18914 


92408 


6 


55 


73513 


81113 


18887 


92400 


5 


56 


73533 


81141 


18859 


92392 


4 


57 


73552 


81169 


18831 


92384 


3 


58 


73572 


81196 


18804 


92376 


* 2 


59 


73591 


81224 


18776 


92367 


1 


6O 


73611 


81252 


18748 


92359 


O 








1 O 


9 




r 


log cos 


log oot 


_1. vF 

log tan 


log sin 


t 



58 



57 C 



44 



33 



34 



f 


log sin 


log tan 


log cot 


log cos 


t 




) 


<) 


10 


C) 




o 


73 6!1 


81252 


18748 


92359 


6O 


1 


73630 


81279 


18721 


92351 


59 


2 


73650 


81307 


18693 


92343 


58 


3 


73669 


81335 


18665 


92335 


57 


4 


73689 


81362 


18638 


92326 


56 


5 


73708 


81390 


18610 


92318 


55 


6 


73727 


81418 


18582 


92310 


54 


7 


73747 


81445 


18555 


92302 


53 


8 


73766 


81473 


18527 


92293 


52 


9 


73785 


81500 


18500 


92285 


51 


1C 


73805 


81528 


18472 


92277 


5O 


11 


73 824 


81556 


18444 


92269 


49 


12 


73843 


81583 


18417 


92260 


48 


13 


73863 


81611 


18389 


92252 


47 


14 


73882 


81638 


18362 


92244 


46 


15 


73901 


81666 


18334 


92235 


45 


16 


73921 


81693 


18307 


92227 


44 


17 


73940 


81721 


18279 


92219 


43 


18 


73959 


81748 


18 252 


92211 


42 


19 


73978 


81776 


18224 


92202 


41 


20 


73997 


81803 


18197 


92194 


4O 


21 


74017 


81831 


18169 


92186 


39 


22 


74036 


81 858 


18142 


92177 


38 


23 


74055 


81886 


18114 


92169 


37 


24 


74074 


81913 


18087 


92161 


36 


25 


74093 


81941 


18059 


92152 


35 


26 


74113 


81968 


18032 


92144 


34 


27 


74132 


81996 


18004 


92136 


33 


28 


74151 


82023 


17977 


92127 


32 


29 


74170 


82051 


17949 


92119 


31 


30 


74189 


82078 


17922 


92111 


30 


31 


74208 


82106 


17894 


92102 


29 


32 


74227 


82133 


17867 


92094 


28 


33 


74246 


82161 


17839 


92086 


27 


34 


74265 


82188 


17812 


92077 


26 


35 


74284 


82215 


17785 


92069 


25 


36 


74303 


82243 


17757 


92060 


24 


37 


74322 


82270 


17730 


92052 


23 


38 


74341 


82298 


17702 


92044 


22 


39 


74 360 


82325 


17675 


92035 


21 


4O 


74379 


82352 


17648 


92027 


2O 


41 


74398 


82380 


17620 


92018 


19 


42 


74417 


82407 


17593 


92010 


18 


43 


74436 


82435 


17565 


92002 


17 


44 


74455 


82462 


17538 


91993 


16 


45 


74474 


82489 


17511 


91985 


15 


46 


74493 


82517 


17483 


91976 


14 


47 


74512 


82544 


17456 


91968 


13 


48 


74531 


82 571 


17429 


91959 


12 


49 


74549 


82599 


17401 


91951 


11 


50 


74568 


82626 


17374 


91942 


10 


51 


74587 


82653 


17347 


91934 


9 


52 


74606 


82681 


17319 


91925 


8 


53 


74625 


82708 


17292 


91917 


7 


54 


74644 


82735 


17265 


91908 


6 


55 


74662 


82762 


17238 


91900 


5 


56 


74681 


82790 


17210 


91891 


4 


57. 


74700 


82817 


17183 


91883 


3 


58 


74719 


82844 


17156 


91874 


2 


59 


74737 


82871 


17129 


91866 


1 


6O 


74756 


82899 


17101 


91857 


O 








1 O 






r 


log cos 


log cot 


JL vf 

log tan 


log sin 


f 



t I log sin 


log tan 


log cot 


log cos 


r 




9 


9 


1 O 


- , 




O 


74756 


82899 


J. vF 

17101 


91857 


6O 


1 


74 775 


82926 


17074 


91849 


59 


2 


74794 


82953 


17047 


91840 


58 


3 


74812 


82980 


17020 


91832 


57 


4 


74831 


83008 


16992 


91823 


56 


5 


74850 


83035 


16965 


91815 


55 


6 


74868 


83062 


16938 


91806 


54 


7 


74887 


83089 


16911 


91798 


53 


8 


74906 


83117 


16883 


91789 


52 


9 


74924 


83144 


16856 


91781 


51 


1O 


74943 


83171 


16829 


91772 


5O 


11 


74961 


83198 


16802 


91763 


49 


12 


74980 


83225 


16775 


91755 


48 


13 


74999 


83252 


16748 


91746 


47 


14 


75017 


83280 


16720 


91^38 


46 


15 


75036 


83307 


16693 


91729 


45 


16 


75054 


83334 


16666 


91720 


44 


17 


75073 


83361 


16639 


91712 


43 


18 


75091 


83388 


16612 


91703 


42 


19 


75110 


83415 


16585 


91 695 


41 


2O 


75128 


83442 


16558 


91686 


40 


21 


75147 


83470 


16530 


91677 


39 


22 


75165 


83497 


16503 


91669 


38 


23 


75 184 


83524 


16476 


91660 


37 


24 


75202 


83551 


16449 


91651 


36 


25 


75221 


83578 


16422 


91643 


35 


26 


75239 


83605 


16395 


91634 


34 


27 


75258 


83632 


16368 


91625 


33 


28 


75276 


83659 


16 341 


91617 


32 


29 


75294 


83686 


16314 


91608 


31 


3O 


75313 


83713 


16287 


91599 


3O 


31 


75331 


83740 


16260 


91 591 


29 


32 


75 350 


83768 


16232 


91582 


28 


33 


75368 


83795 


16205 


91573 


27 


34 


75386 


83 822 


16178 


91565 


26 


35 


75405 


83849 


16151 


91556 


25 


36 


75423 


83876 


16124 


91547 


24 


37 


75441 


83903 


16097 


91 538 


23 


38 


75459 


83930 


16070 


91530 


22 


39 


75478 


83957 


16043 


91521 


21 


40 


75496 


83984 


16016 


91512 


2O 


41 


75514 


84011 


15989 


91504 


19 


42 


75533 


84038 


15962 


91495 


18 


43 


75551 


84065 


15935 


91486 


17 


44 


75569 


84092 


15908 


91477 


16 


45 


75587 


84119 


15881 


91469 


15 


46 


75605 


84146 


15854 


91460 


14 


47 


75624 


84173 


15827 


91451 


13 


48 


75642 


84200 


15800 


91442 


12 


49 


75660 


84227 


15773 


91433 


11 


50 


75678 


84254 


15746 


91425 


10 


51 


75 696 


84280 


15 720 


91416 


9 


52 


75 714 


84307 


15693 


91407 


8 


53 


75733 


84334 


15666 


91398 


7 


54 


75751 


84361 


15639 


91389 


6 


55 


75769 


84388 


15612 


91 381 


5 


56 


75787 


84415 


15585 


91372 


4 


57 


75805 


84 442 


15558 


91363 


3 


58 


75823 


84469 


15531 


91354 


2 


59 


75841 


84496 


15504 


91345 


1 


6O 


75859 


84523 


15477 


91336 











1 O 






f 


log cos 


log cot 


A vF 

log tan 


log sin 


t 



56 C 



55 C 



35 C 



36 



45 



f 


log sin 


log tan 


log cot 


log cos 


f 




9 


c) 


1 O 


9 




O 


75859 


84523 


__ Vr 

15477 


91336 


6O 


1 


75877 


84550 


15450 


91328 


59 


2 


75895 


84576 


15424 


91319 


58 


3 


75913 


84603 


15397 


91310 


57 


4 


75931 


84630 


15370 


91301 


56 


5 


75949 


84657 


15343 


91292 


55 


6 


75967 


84684 


15316 


91283 


54 


7 


75985 


84711 


15289 


91274 


53 


8 


76003 


84738 


15262 


91266 


52 


9 


76021 


84764 


15236 


91257 


51 


1O 


76039 


84791 


15209 


91248 


50 


11 


76057 


84818 


15 182 


91239 


49 


12 


76075 


84845 


15155 


91230 


48 


13 


76093 


84872 


15128 


91 221 


47 


14 


76111 


84899 


15 101 


91212 


46 


15 


76129 


84925 


15075 


91203 


45 


16 


76146 


84952 


15048 


91194 


44 


17 


76164 


84979 


15021 


91185 


43 


18 


76182 


85006 


14994 


91176 


42 


19 


76200 


85033 


14967 


91167 


41 


20 


76218 


85059 


14941 


91158 


40 


21 


76236 


85086 


14914 


91149 


39 


22 


76253 


85113 


14887 


91141 


38 


23 


76271 


85 140 


14860 


91132 


37 


24 


76289 


85 166 


14834 


91123 


36 


25 


76307 


85193 


14807 


91114 


35 


26 


76324 


85220 


14780 


91105 


34 


27 


76342 


85247 


14753 


91096 


33 


28 


76360 


85273 


14727 


91087 


32 


29 


76378 


85300 


14700 


91078 


31 


30 


76395 


85327 


14673 


91069 


30 


31 


76413 


85354 


14646 


91060 


29 


32 


76431 


85380 


14620 


91051 


28 


33 


76448 


85407 


14593 


91042 


27 


34 


76466 


85434 


14566 


91033 


26 


35 


76484 


85460 


14540 


91023 


25 


36 


76501 


85487 


14513 


91014 


24 


37 


76519 


85514 


14486 


91005 


23 


38 


76537 


85540 


14460 


90996 


22 


39 


76554 


85567 


14433 


90987 


21 


40 


76572 


85594 


14406 


90978 


20 


41 


76590 


85620 


14380 


90969 


19 


42 


76607 


85647 


14353 


90960 


18 


43 


76625 


85674 


14326 


90951 


17 


44 


76642 


85700 


14300 


90942 


16 


45 


76660 


85727 


14273 


90933 


15 


46 


76677 


85754 


14246 


90924 


14 


47 


76695 


85780 


14220 


90915 


13 


48 


76712 


85807 


14193 


90906 


12 


49 


76730 


85834 


14166 


90896 


11 


50 


76747 


85860 


14140 


90887 


1O 


51 


76765 


85887 


14113 


90878 


9 


52 


76782 


85913 


14087 


90869 


8 


53 


76800 


85940 


14060 


90860 


7 


54 


76817 


85967 


14033 


90851 


6 


55 


76835 


85993 


14007 


90842 


5 


56 


76852 


86020 


13980 


90832 


4 


57 


76870 


86046 


13954 


90823 


3 


58 


76887 


86073 


13927 


90814 


2 


59 


76904 


86100 


13900 


90805 


1 


6O 


76922 


86126 


13874 


90796 


O 








1 O 






f 


log cos 


log cot 


A VF 

log tan 


log sin 


r 



r 


log sin 


log tan 


log cot 


log cos 


f 










10 


9 




O 


76922 


86126 


_1_ VF 

13874 


90796 


6O 


1 


76939 


86153 


13847 


90787 


59 


2 


76957 


86179 


13821 


90777 


58 


3 


76974 


86206 


13794 


90 7.68 


57 


4 


76991 


86232 


13768 


90759 


56 


5 


77009 


86 259 


13741 


90750 


55 


6 


77026 


86285 


13715 


90741 


54 


7 


77043 


86312 


13688 


90731 


53 


8 


77061 


86338 


13662 


90722 


52 


9 


77078 


86365 


13635 


90713 


51 


1C 


77095 


86392 


13608 


90704 


50 


11 


77112 


86418 


13582 


90694 


49 


12 


77130 


86445 


13555 


90685 


48 


13 


77147 


86471 


13529 


90676 


47 


14 


77164 


86498 


13502 


90667 


46 


15 


77181 


86524 


13476 


90657 


45 


16 


77199 


86551 


13449 


90648 


44 


17 


77216 


86 577 


13423 


90639 


43 


18 


77233 


86603 


13397 


90630 


42 


19 


77250 


86630 


13370 


90620 


41 


2O 


77268 


86656 


13344 


90611 


40 


21 


77285 


86683 


13317 


90602 


39 


22 


77302 


86709 


13291 


90592 


38 


23 


77319 


86736 


13264 


90583 


37 


24 


77336 


86762 


13238 


90574 


36 


25 


77353 


86789 


13211 


90565 


35 


26 


77370 


86815 


13185 


90555 


34 


27 


77387 


86842 


13158 


90546 


33 


28 


77405 


86868 


13132 


90537 


32 


29 


77422 


86894 


13106 


90527 


31 


30 


77439 


86921 


13079 


90518 


3O 


31 


77456 


86947 


13053 


90509 


29 


32 


77473 


86974 


13026 


90499 


28 


33 


77490 


87000 


13000 


90490 


27 


34 


77507 


87027 


12973 


90480 


26 


35 


77524 


87053 


12947 


90471 


25 


36 


77541 


87079 


12921 


90462 


24 


37 


77558 


87106 


12894 


90452 


23 


38 


77575 


87132 


12868 


90443 


22 


39 


77 592" 


87158 


12842 


90434 


21 


4O 


77609 


87185 


12815 


90424 


2O 


41 


77626 


87211 


12789 


90415 


19 


42 


77643 


87238 


12762 


90405 


18 


43 


77660 


87264 


12736 


90396 


17 


44 


77677 


87290 


12710 


90386 


16 


45 


77694 


87317 


12683 


90377 


15 


46 


77711 


87343 


12657 


90368 


14 


47 


77728 


87369 


12631 


90358 


13 


48 


77744 


87396 


12604 


90349 


12 


49 


77761 


87422 


12578 


90339 


11 


50 


77778 


87448 


12552 


90330 


1O 


51 


77795 


87475 


12525 


90320 


9 


52 


77812 


87501 


12499 


90311 


8 


53 


77829 


87527 


12473 


90301 


7 


54 


77846 


87554 


12446 


90292 


6 


55 


77862 


87580 


12420 


90282 


5 


56 


77879 


87606 


12394 


90273 


4 


57 


77896 


87633 


12367 


90263 


3 


58 


77913 


87659 


12341 


90254 


2 


59 


77 930 


87685 


12315 


90244 


1 


6O 


77946 


87711 


12289 


90235 


O 








1 O 






f 


log cos 


log cot 


JL VF 

log tan 


log sin 


f 



53< 



46 



37 



38' 



f 


log sin 


log tan 


log cot 


log cos 


r 




4) 


g 


1 ( \ 


Q 







77946 


87711 


A vF 

12289 


90235 


60 


1 


77963 


87738 


12262 


90225 


59 


2 


77980 


87764 


12236 


90216 


58 


3 


77997 


87790 


12210 


90206 


57 


4 


78013 


87817 


12183 


90197 


56 


5 


78030 


87843 


12157 


90187 


55 


6 


78047 


87869 


12131 


90178 


54 


7 


78063 


87895 


12105 


90168 


53 


8 


78080 


87922 


12078 


90159 


52 


9 


78097 


87948 


12052 


90149 


51 


10 


78113 


87974 


12026 


90139 


5O 


11 


78130 


88000 


12000 


90130 


49 


12 


78147 


88027 


11973 


90120 


48 


13 


78163 


88053 


11947 


90111 


47 


14 


78180 


88079 


11921 


90101 


46 


15 


78197 


88 105 


11895 


90091 


45 


16 


78213 


88131 


11869 


90082 


44 


17 


78230 


88158 


11842 


90072 


43 


18 


78246 


88184 


11816 


90063 


42 


19 


78263 


88210 


11790 


90053 


41 


2O 


78280 


88236 


11764 


90043 


40 


21 


78296 


88262 


11738 


90034 


39 


22 


78313 


88289 


11711 


90024 


38 


23 


78329 


88315 


l.i. 685 


90014 


37 


24 


78346 


88341 


11659 


90005 


36 


25 


78362 


88367 


11633 


89995 


35 


26 


78379 


88393 


11607 


89985 


34 


27 


78395 


88420 


11580 


89976 


33 


28 


78412 


88446 


11554 


89966 


32 


29 


78428 


88472 


11528 


89956 


31 


30 


78445 


88498 


11502 


89947 


30 


31 


78461 


88524 


11476 


89937 


29 


32 


78478 


88550 


11450 


89927 


28 


33 


78494 


88577 


11423 


89918 


27 


34 


78510 


88603 


11397 


89908 


26 


35 


78527 


88629 


11371 


89898 


25 


36 


78 543 


88655 


11345 


89888 


24 


37 


78560 


88681 


11319 


89879 


23 


38 


78576 


88707 


11293 


89869 


22 


39 


78592 


88733 


11267 


89859 


21 


4O 


78609 


88759 


11241 


89849 


20 


41 


78625 


88786 


11214 


89840 


19 


42 


78642 


88812 


11188 


89830 


18 


43 


78658 


88838 


11 162 


89820 


17 


44 


78674 


88864 


11136 


89810 


16 


45 


78691 


88890 


11110 


89801 


15 


46 


78707 


88916 


11084 


89791 


14 


47 


78723 


88942 


11058 


89781 


13 


48 


78739 


88968 


11032 


89771 


12 


49 


78756 


88994 


11006 


89761 


11 


50 


78772 


89020 


10980 


89752 


10 


51 


78788 


89046 


10 954 


89742 


9 


52 


78805 


89073 


10927 


89732 


8 


53 


78821 


89099 


10901 


89722 


7 


54 


78837 


89125 


10875 


89712 


6 


55 


78853 


89151 


10849 


89702 


5 


56 


78869 


89177 


10823 


89693 


4 


57 


78886 


89203 


10797 


89683 


3 


58 


78902 


89229 


10771 


89673 


2 


59 


78918 


89255 


10745 


89663 


1 


60 


78934 


89281 


10719 


89653 


O 








1O 






t 


log oos 


log oot 


__ vr 

log tan 


log sin 


r 



r 


log sin 


log tan 


log cot 


log cos 


/ 




<) 


9 


1 A 


o 







78934 


89281 


JL vf 

10719 


89653 


6O 


1 


78950 


89307 


10693 


89643 


59 


2 


78967 


89333 


10667 


89633 


58 


3 


78983 


89359 


10641 


89624 


57 


4 


78999 


89385 


10615 


89614 


56 


5 


79015 


89411 


10589 


89604 


55 


6 


79031 


89437 


10563 


89594 


54 


7 


79047 


89463 


10537 


89584 


53 


8 


79063 


89489 


10511 


89574 


52 


9 


79079 


89515 


10485 


89564 


51 


1O 


79095 


89541 


10459 


89554 


5O 


11 


79 111 


89567 


10433 


89544 


49 


12 


79128 


89593 


10407 


89534 


48 


13 


79144 


89619 


10381 


89524 


47 


14 


79160 


89645 


10355 


89514 


46 


15 


79176 


89671 


10329 


89504 


45 


16 


79192 


89697 


10303 


89495 


44 


17 


79208 


89723 


10277 


89485 


43 


18 


79224 


89749 


10251 


89475 


42 


19 


79240 


89775 


10225 


89465 


41 


20 


79256 


89801 


10199 


89455 


4O 


21 


79272 


89827 


10173 


89445 


39 


22 


79288 


89853 


10147 


89435 


38 


23 


79304 


89879 


10121 


89425 


37 


24 


79319 


89905 


10095 


89415 


36 


25 


79335 


89931 


10069 


89405 


35 


26 


79351 


89957 


10043 


89395 


34 


27 


79367 


89983 


10017 


89385 


33 


28 


79383 


90009 


09991 


89375 


32 


29 


79399 


90035 


09965 


89364 


31 


30 


79415 


90061 


09939 


89354 


3O 


31 


79431 


90086 


09914 


89344 


29 


32 


79447 


90112 


09888 


89334 


28 


33 


79463 


90138 


09862 


89324 


27 


34 


79478 


90164 


09836 


89314 


26 


35 


79494 


90190 


09810 


89304 


25 


36 


79510 


90216 


09784 


89294 


24 


37 


79526 


90242 


09 758 


89284 


23 


38 


79542 


90268 


09732 


89274 


22 


39 


79558 


90294 


09706 


89264 


21 


40 


79573 


90320 


09680 


89254 


20 


41 


79589 


90346 


09654 


89244 


19 


42 


79605 


90371 


09629 


89233 


18 


43 


79621 


90397 


09603 


89223 


17 


44 


79636 


90423 


09577 


89213 


16 


45 


79652 


90449 


09551 


89203 


15 


46 


79668 


90475 


09525 


89193 


14 


47 


79684 


90501 


09499 


89183 


13 


48 


79699 


90527 


09473 


89173 


12 


49 


79715 


90553 


09447 


89162 


11 


5O 


79731 


90578 


09422 


89152 


1O 


51 


79746 


90604 


09396 


89142 


9 


52 


79762 


90630 


09370 


89132 


8 


53 


79778 


90656 


09344 


89122 


7 


54 


79793 


90682 


09318 


89112 


6 


55 


79809 


90708 


09292 


89101 


5 


56 


79825 


90734 


09266 


89091 


4 


57 


79840 


90759 


09241 


89081 


3 


58 


79856 


90785 


09215 


89071 


2 


59 


79872 


90811 


09189 


89060 


1 


60 


79887 


90837 


09163 


89050 


O 








1O 






f 


log oos 


log cot 


A VF 

log tan 


log sin 


t 



52 



51 C 



39 



40 



47 



t 


log sin 


log tan 


lag cot 


log cos 


t 







9 


10 


9 




O 


79887 


90837 


09163 


89050 


60 


1 


79903 


90863 


09 137 


89040 


59 


2 


79918 


90889 


09111 


89030 


58 


3 


79934 


90914 


09086 


89020 


57 


4 


79950 


90940 


09060 


89009 


56 


5 


79965 


90966 


09034 


88999 


55 


6 


79981 


90992 


09008 


88989 


54 


7 


79996 


91018 


08982 


88978 


53 


8 


80012 


91043 


08957 


88968 


52 


9 


80027 


91069 


08931 


88958 


51 


1O 


80043 


91095 


08905 


88948 


50 


11 


80058 


91121 


08879 


88937 


49 


12 


80074 


91147 


08853 


88927 


48 


13 


80089 


91172 


08828 


88917 


47 


14 


80105 


91198 


08 802 


88906 


46 


15 


80120 


91224 


08776 


88896 


45 


16 


80136 


91250 


08750 


88886 


44 


17 


80151 


91276 


08724 


88875 


43 


18 


80166 


91301 


08699 


88865 


42 


19 


80182 


91327 


08673 


88 855. 


41 


2O 


80197 


91353 


08647 


88844 


40 


21 


80213 


91379 


08621 


88834 


39 


22 


80228 


91404 


08596 


88824 


38 


23 


80244 


91430 


08570 


88813 


37 


24 80 259 


91456 


08544 


88803 


36 


25 SO 274 


91482 


08518 


88793 


35 


26 


80290 


91507 


08493 


88782 


34 


27 


80305 


91533 


08467 


88772 


33 


28 


80320 


91559 


08441 


88761 


32 


29 


80336 


91585 


08415 


88751 


31 


30 


80351 


91610 


08390 


88741 


3O 


31 


80366 


91636 


08364 


88730 


29 


32 


80382 


91662 


08338 


88720 


' 28 


33 


80397 


91688 


08312 


88709 


27 


34 


80412 


91713 


08287 


88699 


26 


35 


80428 


91739 


08261 


88688 


25 


36 


80443 


91765 


08235 


88678 


24 


37 


80458 


91791 


08209 


88668 


23 


. 38 


80473 


91816 


08184 


88657 


22 


39 


80489 


91842 


08158 


88647 


21 


40 


80504 


91868 


08132 


88636 


2O 


41 


80519 


91893 


08107 


88626 


19 


42 


80534 


91919 


08081 


88615 


18 


43 


80550 


91945 


08 055. 


88605 


17 


44 


80565 


91971 


08029 


88594 


16 


45 


80580 


91996 


08004 


88584 


15 


46 


80595 


92022 


07978 


88 573 


14 


47 


80610 


92048 


07952 


88563 


13 


48 


80 625 


92073 


07927 


88 552 


12 


49 


80641 


92099 


07901 


88542 


11 


5O 


80656 


92125 


07875 


88 531 


1O 


51 


80671 


92150 


07850 


88521 


9 


52 


80686 


92176 


07824 


88510 


8 


53 


80701 


92202 


07798 


88499 


7 


54 


80716 


92227 


07773 


88489 


6 


55 


80731 


92253 


07747 


88478 


5 


56 


80746 


92279 


07721 


88468 


4 


57 


80762 


92304 


07696 


88457 


3 


58 


80777 


92330 


07670 


88447 


2 


59 


80792 


92356 


07644 


88436 


1 


6O 


80807 


92381 


07619 


88425 


O 








1 O 






r 


log cos 


log cot 


J-VF 

log tan 


log sin 


f 



f 


log sin 


log tan 


log cot 


log cos 


t 







<) 


10 







o 


80807 


92381 


07619 


88425 


6O 


1 


80822 


92407 


07593 


88415 


59 


2 


80837 


92433 


07567 


88404 


58 


3 


80852 


92458 


07542 


88394 


57 


4 


80867 


92484 


07516 


88383 


56 


5 


80882 


92510 


07490 


88372 


55 


6 


80897 


92535 


07465 


88362 


54 


7 


80912 


92561 


07439 


88351 


53 


8 


80927 


92587 


07413 


88340 


52 


9 


80942 


92612 


07388 


88330 


51 


10 


80957 


92638 


07362 


88319 


50 


11 


80972 


92663 


07337 


88308 


49 


12 


80987 


92689 


07311 


88298 


48 


13 


81002 


92715 


07 285 


88287 


47 


14 


81017 


92740 


07260 


88276 


46 


15 


81032 


92766 


07234 


88266 


45 


16 


81047 


92792 


07208 


88255 


44 


17 


81061 


92817 


07183 


88244 


43 


18 


81076 


92843 


07157 


88234 


42 


19 


81091 


92868 


07132 


88223 


41 


20 


81106 


92894 


07106 


88212 


40 


21 


81121 


92920 


07080 


88201 


39 


22 


81136 


92945 


07055 


88191 


38 


23 


81151 


92971 


07029 


88180 


37 


24 


81166 


92996 


07004 


88169 


36 


25 


81180 


93022 


06978 


88158 


35 


26 


81195 


93048 


06952 


88148 


34 


27 


81210 


93073 


06927 


88137 


33 


28 


81225 


93099 


06901 


88126 


32 


29 


81240 


93124 


06876 


88115 


31 


3O 


81254 


93150 


06850 


88105 


3O 


31 


81269 


93175 


06825 


88094 


29 


32 


81284 


93201 


06799 


88083 


28 


33 


81299 


93227 


06773 


88072 


27 


34 


81314 


93252 


06748 


88061 


26 


35 


81328 


93278 


06722 


88051 


25 


36 


81343 


93303 


06697 


88040 


24 


37 


81358 


93329 


06671 


88029 


23 


38 


81372 


93354 


06 646 


88018 


22 


39 


81387 


93380 


06620 


88007 


21 


40 


81402 


93406 


06594 


87996 


2O 


41 


81417 


93431 


06569 


87 985 


19 


42 


81431 


93457 


06543 


87975 


18 


43 


81446 


93482 


06518 


87964 


17 


44 


81461 


93508 


06492 


87953 


16 


45 


81 475 


93533 


06467 


87942 


15 


46 


81490 


93559 


06441 


87931 


14 


47 


81505 


93584 


06416 


87920 


13 


48 


81519 


93610 


06390 


87909 


12 


49 


81534 


93636 


06364 


87898 


11 


50 


81549 


93661 


06339 


87 887 


10 


51 


81563 


93687 


06313 


87877 


9 


52 


81578 


93712 


06288 


87866 


8 


53 


81592 


93738 


06262 


87855 


7 


54 


81 607 


93763 


06237 


87844 


6 


55 


81 622 


93789 


06211 


87833 


5 


56 


81636 


93 814 


06186 


87822 


4 


57 


81651 


93840 


06160 


87811 


3 


58 


81665 


93865 


06135 


87800 


2 


59 


81680 


93891 


06109 


87789 


1 


6O 


81694 


93916 


06084 


87778 











1 O 






r 


log cos 


log cot 


JL \J 

log tan 


log sin 


r 



50' 



49' 



48 



42 



r 


log sin 


log tan 


log cot 


log cos 


t 










10 










81694 


93916 


06084 


87778 


60 


1 


81709 


93942 


06058 


87767 


59 


2 


81723 


93967 


06033 


87756 


58 


3 


81738 


93993 


06007 


87745 


57 


4 


81752 


94018 


05982 


87734 


56 


5 


81767 


94044 


05956 


87723 


55 


6 


81781 


94069 


05931 


87712 


54 


7 


81796 


94095 


05905 


87701 


53 


8 


81810 


94120 


05880 


87690 


52 


9 


81825 


94146 


05854 


87679 


51 


10 


81839 


94171 


05829 


87668 


5O 


11 


81854 


94197 


05803 


87657 


49 


12 


81868 


94222 


05778 


87646 


48 


13 


81882 


94248 


05752 


87635 


47 


14 


81897 


94273 


05727 


87624 


46 


15 


81911 


94299 


05701 


87613 


45 


16 


81926 


94324 


05676 


87601 


44 


17 


81940 


94350 


05650 


87590 


43 


18 


81 955 


94375 


05625 


87579 


42 


19 


81969 


94401 


05599 


87568 


41 


2O 


81983 


94426 


05574 


87557 


4O 


21 


81998 


94452 


05548 


87546 


39 


22 


82012 


94477 


05523 


87535 


38 


23 


82026 


94503 


05497 


87524 


37 


24 


82041 


94528 


05472 


87513 


36 


25 


82055 


94554 


05446 


87501 


35 


26 


82069 


94579 


05421 


87490 


34 


27 


82084 


94604 


05396 


87479 


33 


28 


82098 


94630 


05370 


87468 


32 


29 


82112 


94655 


05345 


87457 


31 


3O 


82126 


94681 


05319 


87446 


3O 


31 


82141 


94706 


05294 


87434 


29 


32 


82155 


94732 


05268 


87423 


28 


33 


82169 


94757 


05243 


87412 


27 


34 


82184 


94783 


05217 


87401 


26 


35 


82198 


94808 


05192 


87390 


25 


36 


82212 


94834 


05 166 


87378 


24 


37 


82226 


94 859 


05141 


87367 


23 


38 


82240 


94884 


05116 


87356 


22 


39 


82255 


94910 


05090 


87345 


21 


4O 


82269 


94935 


05065 


87334 


2O 


41 


82283 


94961 


05039 


87322 


19 


42 


82297 


94986 


05014 


87311 


18 


43 


82311 


95012 


04988 


87300 


17 


44 


82326 


95037 


04963 


87288 


16 


45 


82340 


95062 


04938 


87277 


15 


46 


82354 


95088 


04912 


87266 


14 


47 


82368 


95113 


04887 


87255 


13 


48 


82382 


95139 


04861 


87243 


12 


49 


82396 


95164 


04836 


87232 


11 


5O 


82410 


95 190 


04810 


87221 


1O 


51 


82424 


95215 


04785 


87209 


9 


52 


82439 


95240 


04760 


87198 


8 


53 


82453 


95266 


04734 


87187 


7 


54 


82467 


95291 


04709 


87175 


6 


55 


82481 


95317 


04683 


87164 


5 


56 


82495 


95342 


04 658 


87153 


4 


57 


82509 


95368 


04632 


87141 


3 


58 


82523 


95393 


04607 


87130 


2 


59 


82537 


95418 


04582 


87119 


1 


6O 


82551 


95444 


04556 


87107 


O 








1O 






r 


log cos 


log cot 


J. V7 

log tan 


log sin 


f 



t 


log sin 


log tan 


log cot 


log cos 


/ 




9 


<) 


1 O 


<) 







82551 


95444 


JL vf 

04556 


87107 


6O 


1 


82565 


95469 


04531 


87096 


59 


2 


82579 


95495 


04505 


87085 


58 


3 


82593 


95520 


04480 


87073 


57 


4 


82607 


95545 


04455 


87062 


56 


5 


82621 


95571 


04429 


87050 


55 


6 


82635 


95596 


04404 


87039 


54 


7 


82649 


95622 


04378 


87028 


53 


8 


82663 


95647 


04353 


87016 


52 


9 


82677 


95672 


04328 


87005 


51 


1O 


82691 


95698 


04302 


86993 


50 


11 


82705 


95723 


04277 


86982 


49 


12 


82719 


95748 


04252 


86970 


48 


13 


82733 


95 774 


04226 


86959 


47 


14 


82747 


95799 


04201 


86947 


46 


15 


82761 


95825 


04175 


86936 


45 


16 


82775 


95850 


04150 


86924 


44 


17 


82788 


95875 


04125 


86913 


43 


18 


82802 


95901 


04099 


86902 


42 


19 


82816 


95926 


04074 


86890 


41 


2O 


82830 


95952 


04048 


86879 


4O 


21 


82844 


95977 


04023 


86867 


39 


22 


82858 


96002 


03998 


86855 


38 


23 


82872 


96028 


03972 


86844 


37 


24 


82885 


96053 


03947 


86832 


36 


25 


82899 


96078 


03922 


86821 


35 


26 


82913 


96104 


03896 


86809 


34 


27 


82927 


96129 


03871 


86798 


33 


28 


82941 


96155 


03 845 


86786 


32 


29 


82955 


96180 


03820 


86775 


31 


3O 


82968 


96205 


03795 


86763 


30 


31 


82982 


96231 


03769 


86752 


29 


32 


82996 


96256 


03744 


86740 


28 


33 


S3 010 


. 96 281 


03719 


86728 


27 


34 


83023 


96307 


03693 


86717 


26 


35 


83037 


96332 


03668 


86705 


25 


36 


83051 


96357 


03643 


86694 


24 


37 


83065 


96383 


03617 


86682 


23 


38 


83078 


96408 


03592 


86670 


22 


39 


83092 


96433 


03567 


86659 


21 


4O 


83106 


96459 


03541 


86647 


2O 


41 


83120 


96484 


03516 


86635 


19 


42 


83133 


96510 


03490 


86624 


18 


43 


83147 


96535 


03465 


86612 


17 


44 


83161 


96 560 


03440 


86600 


16 


45 


83174 


96586 


03414 


86589 


15 


46 


83188 


96611 


03389 


86577 


14 


47 


83202 


96636 


03364 


86565 


13 


48 


83215 


96662 


03338 


86554 


12 


49 


83229 


96687 


03313 


86542 


11 


5O 


83242 


96712 


03288 


86530 


1O 


51 


83256 


96738 


03262 


86518 


9 


52 


83270 


96763 


03237 


86507 


8 


53 


83283 


96788 


03212 


86495 


7 


54 


83297 


96814 


03186 


86483 


6 


55 


83310 


96839 


03161 


86472 


5 


56 


83324 


96864 


03136 


86460 


4 


57 


83338 


96890 


03110 


86448 


3 


58 


83351 


96915 


03 085 


86436 


2 


59 


83365 


96940 


03060 


86425 


1 


6O 


83378 


96966 


03034 


86413 


O 








1 f\ 






f 


log cos 


log cot 


JL vF 

log tan 


log sin 


f 



48 C 



47 



43 C 



44 C 



49 



t 


log sin 


log tan 


log cot 


log cos 


f 




<) 


C) 


10 


g 




o 


83378 


96966 


03034 


86413 


60 


1 


83392 


96991 


03009 


86401 


59 


2 


83405 


97016 


02984 


86389 


58 


3 


83419 


97042 


02958 


86377 


57 


4 


83432 


97067 


02933 


86366 


56 


5 


83446 


97092 


02908 


86354 


55 


6 


83459 


97118 


02882 


86342 


54 


7 


83473 


97143 


02857 


86330 


53 


8 


83486 


97168 


02832 


86318 


52 


9 


83500 


97193 


02807 


86306 


51 


1O 


83513 


97219 


02781 


86295 


50 


11 


83527 


97244 


02756 


86283 


49 


12 


83540 


97269 


02731 


86271 


48 


13 


83554 


97295 


02705 


86259 


47 


14 


83567 


97320 


02680 


86247 


46 


15 


83581 


97345 


02655 


86235 


45 


16 


83594 


97 371 


02629 


86223 


44 


17 


83608 


97396 


02604 


86211 


43 


18 


83621 


97421 


02579 


86200 


42 


19 


83634 


97447 


02553 


86188 


41 


20 


83648 


97 472 


02528 


86176 


40 


21 


83661 


97497 


02503 


86164 


39 


22 


83674 


97523 


02477 


86152 


38 


23 


83688 


97548 


02452 


86140 


37 


24 


83701 


97573 


02427 


86128 


36 


25 


83715 


97598 


02402 


86116 


35 


26 


83728 


97624 


02 376 


86104 


34 


27 


83741 


97649 


02351 


86092 


33 


28 


83755 


97674 


02326 


86080 


32 


29 


83768 


97700 


02300 


86068 


31 


30 


83781 


97725 


02275 


86056" 


30 


31 


83795 


97750 


02250 


86044 


29 


32 


83808 


97776 


02224 


86032 


28 


33 


83821 


97801 


02199 


86020 


27 


34 


83834 


97826 


02174 


86008 


26 


35 


83848 


97851 


02149 


85996 


25 


36 


83861 


97877 


02123 


85984 


24 


37 


83874 


97902 


02098 


85972 


23 


38 


83887 


97927 


02073 


85960 


22 


39 


83901 


97953 


02 047 


85948 


21 


40 


83914 


97978 


02022 


85936 


2O 


41 


83927 


98003 


01997 


85924 


19 


42 


83940 


98029 


01971 


85912 


18 


43 


83954 


98054 


01946 


85900 


17 


44 


83967 


98079 


01921 


85888 


16 


45 


83980 


98104 


01896 


85876 


15 


46 


83993 


98130 


01870 


85864 


14* 


47 


84006 


98 155 


01845 


85851 


13 


48 


84020 


98180 


01820 


85839 


12 


49 


84033 


98206 


01794 


85827 


11 


50 


84046 


98231 


01769 


85815 


1O 


51 


84059 


98256 


01744 


85803 


- 9 


52 


84072 


98281 


01719 


85791 


8 


53 


84085 


98307 


01693 


85779 


7 


54 


84098 


98332 


01668 


85766 


6 


55 


84112 


98357 


01643 


85754 


5 


56 


84125 


98383 


01617 


85742 


4 


57 


84138 


98408 


01592 


85730 


3 


58 


84151 


98433 


01567 


85718 


2 


59 


84164 


98458 


01542 


85706 


1 


60 


84177 


98484 


01516 


85693 











1 O 






f 


log cos 


log cot 


A VF 

log tan 


log sin 


'1 



t 


log sin 


log tan 


log cot 


log cos 


f 




9 


) 


10 


9 







84177 


98484 


01516 


85693 


60 


1 


84190 


98509 


01491 


85681 


59 


2 


84203 


98534 


01466 


85669 


58 


3 


84216 


98560 


01440 


85657 


57 


4 


84229 


98585 


01415 


85645 


56 


5 


84242 


98610 


01390 


85632 


55 


6 


84255 


98635 


01365 


85620 


54 


7 


84269 


98661 


01339 


85608 


53 


8 


84282 


98686 


01314 


85596 


52 


9 


84295 


98711 


01289 


85583 


51 


1O 


84308 


98737 


01263 


85571 


50 


11 


84321 


98762 


01238 


85559 


49 


12 


84334 


98787 


01213 


85547 


48 


13 


84347 


98812 


01 188 


85534 


47 


14 


84360 


98838 


01162 


85522 


46 


15 


84373 


98863 


01 137 


85 510 


45 


16 


84385 


98888 


01 112 


85497 


44 


17 


84398 


98913 


01087 


85485 


43 


18 


84411 


98939 


01061 


85473 


42 


19 


84424 


98964 


01036 


85460 


41 


20 


84437 


98989 


01011 


85448 


4O 


21 


84450 


99015 


00985 


85436 


39 


22 


84463 


99040 


00960 


85 423 


38 


23 


84476 


99065 


00935 


85411 


37 


24 


84489 


99090 


00910 


85399 


36 


25 


84502 


99116 


00884 


85386 


35 


26 


84515 


99141 


00859 


85374 


34 


27 


84528 


99166 


00834 


85361 


33 


28 


84540 


99191 


00809 


85349 


32 


,29 


84553 


99217 


00783 


85337 


31 


3O 


84 566 


99242 


00758 


85324 


30 


31 


84 579 


99267 


00733 


85312 


29 


32 


84592 


99293 


00707 


85299 


28 


33 


84605 


99318 


00682 


85287 


27 


34 


84618 


99343 


00657 


85274 


26 


35 


84630 


99368 


00632 


85262 


25 


36 


84643 


99394 


00606 


85250 


24 


37 


84656 


99419 


00581 


85237 


23 


38 


84669 


99444 


00556 


85225 


22 


39 


84682 


99469 


00531 


85212 


21 


4O 


84694 


99495 


00505 


85200 


2O 


41 


84707 


99520 


00480 


85 187 


19 


42 


84720 


99545 


00455 


85 175 


18 


43 


84733 


99570 


00430 


85162 


17 


44 


84745 


99596 


00404 


85150 


16 


45 


84758 


99621 


00379 


85137 


15 


46 


84771 


99646 


00354 


85 125 


14 


47 


84784 


99672 


00328 


85 112 


13 


48 


84796 


99697 


00303 


85100 


12 


49 


84809 


99722 


00278 


85087 


11 


50 


84822 


99747 


00253 


85074 


10 


51 


84835 


99773 


00227 


85062 


9 


52 


84847 


99798 


00202 


85049 


8 


53 


84860 


99823 


00177 


85037 


7 


54 


84873 


99848 


00152 


85024 


6 


55 


84885 


99874 


00126 


85012 


5 


56 


84898 


99899 


00101 


84999 


4 


57 


84911 


99924 


00076 


84986 


3 


58 


84923 


99949 


00051 


84974 


2 


59 


84936 


99 975 


00025 


84961 


1 


60 


84949 


00000 


00000 


84949 


O 






i o 


1 O 






t 


log cos 


J_ VF 

log cot 


A vF 

log tan 


log sin 


t 



46 C 



45 ( 



50 



TABLE IV. 


1 f" 

FOB DETERMINING WITH GREATER 


r 

ACCURACY THAN CAN BE DONE BY 


MEANS OF TABLE III. : 


1. 


log sin, log tan, and log cot, when 


the angle is between and 2 ; 


2. 


log cos, log tan, and log cot, when 


the angle is between 88 and 90 ; 


3. 


The value of the angle when the logarithm of the function does not 




lie between the limits 8. 


54 684 and 11. 45 316. 



FORMULAS FOE THE USE OF THE NUMBERS S AND T. 




I. When the angle a 


is between and 2 : 




log sin a = log a' f + S. 


log o" = log sin a S, 




log tan a = log a" -f T. 


= log tan a T, 




log cot a = colog tan a. 


= COlog COt a T. 


II. When the angle a is between 88 and 90 : 




log cos a = log (90 - a)" + S. 


log (90 - a)" = log COS a - S, 




log cot a = log (90 - a)" + T. 


= log COt a T, 




log tan a = colog cot a. 


= colog tan a T, 




<*>X 


and o = 90- (90 -a). 


VALUES OF S AND T, 


a" 


S log sin a 




a" T log tan a 


a T log tan a 





__ 







_ 


5 146 8. 39 713 




4. 68 557 




4. 68 557 


4. 68 567 


2409 


'8.06740 




200 


6. 98 660 


5 424 8. 41 999 




4. 68 556 




4. 68 558 


4. 68 568 


3417 


8. 21 920 




1726 


7. 92 263 


5 689 8. 44 072 




4. 68 555 




4. 68 559 


4. 68 569 


3823 


8. 26 795 




2432 


8. 07 156 


5 941 8. 45 955 




4. 68 555 




4. 68 56.0 


4. 68 570 


4190 


8. 30 776 




2976 


8. 15 924 


6 184 8. 47 697 




4. 68 554 




4. 68 561 


4. 68 571 


4840 


8.37038 




3434 


8. 22 142 


6417 8.49305 




4. 68 553 




4. 68 562 


4. 68 572 


5414 


8. 41 904 




3838 


8. 26 973 


6 642 8. 50 802 




4. 68 552 




4. 68 563 


4. 68 573 


5932 


8. 45 872 




4204 


8.30930 


6859 8.52200 




4.68551 




4. 68 564 


4. 68 574 


6408 


8. 49 223 




4540 


8. 34 270 


7 070 8. 53 516 




4. 68 550 




4. 68 565 


4. 68 575 


6633 


8. 50 721 




4699 


8. 35 766 


7 173 8. 54 145 




4. 68 550 




4. 68 565 


4. 68 575 


6851 


8. 52 125 




4853 


8. 37 167 


7 274 8. 54 753 




4. 68 549 




4. 68 566 




7267 


8.54684 




5146 


8. 39 713 




a" 


8 log sin a 




a" T log tan a 


a T log tan a 



TABLE V. CIRCUMFERENCES AND AREAS OF CIRCLES. 



If N = the radius of the circle, the circumference = 2 Tr-ZV. 


If N = the radius of the circle, the area = irN 2 . 


If N = the circumference of the circle, the radius = N. 


IfN= the circumference of the circle, the area = N' 2 . 




47T 


N 


27TJV IT}?* JLjVT J_^2 
27T 47T 


N 


* ** &* * 


O 


0.00 0.0 0.000 0.00 


50 


314.16 7854 7.96 198.94 


1 


6. 28 3. 1 0. 159 0. 08 


51 


320.44 8171 8.12 206.98 


2 


12.57 12.6 0.318 0.32 


52 


326.73 8495 8.28 215.18 


3 


18.85 28.3 0.477 0.72 


53 


333.01 8825 8.44 223.53 


4 


25.13 50.3 0.637 1.27 


54 


339.29 9161 8.59 232.05 


5 


31.42 78.5 0.796 ].99 


55 


345.58 9503 8.75 240.72 


6 


37.70 113.1 0.955 2.86 


56 


351.86 9852 8.91 249.55 


7 


43.98 153.9 1.114 3.90 


57 


358.14 10207 9.07' 258.55 


8 


50.27 201.1 1.273 5.09 


58 


364.42 10568 9.23 267.70 


9 


56.55 254.5 1.432 6.45 


59 


370.71 10936 9.39 277.01 


1O 


62.83 314.2 1.592 7.96 


60 


376.99 11310 9.55 286.48 


11 


69.12 380.1 1.751 9.63 


61 


383.27 11690 9.71 296.11 


12 


75.40 452.4 1.910 11.46 


62 


389.56 12076 9.87 305.90 


13 


81.68 530.9 2.069 13.45 


63 


395.84 12469 10.03 315.84 


14 


87.96 615.8 2.228 15.60 


64 


402.12 12868 10.19 325.95 


15 


94.25 706.9 2.387 17.90 


65 


408.41 13273 10.35 336.21 


16 


100.53 804.2 2.546 20.37 


66 


414.69 13685 10.50 346.64 


17 


106.81 907.9 2.706 23.00 


67 


420.97 14103 10.66 357.22 


18 


113.10 1017.9 2.865 25.78 


68 


427.26 14527 10.82 367.97 


19 


119.38 1134.1 3.024 28.73 


69 


433.54 14957 10.98 378.87 


20 


125.66 1256.6 3.183 31.83 


70 


439.82 15394 11.14 389.93 


21 


131.95 1385.4 3.342 35.09 


71 


446.11 15837 11.30 401.15 


22 


138.23 1520.5 3.501 38.52 


72 


452.39 16286 11.46 412.53 


23 


14*4.51 1661.9 3.661 42.10 


73 


458.67 16742 11.62 424.07 


24 


150.80 1809.6 3.820 45.84 


74 


464.96 17203 11.78 435.77 


25 


157.08 1963.5 3.979 49.74 


75 


471.24 17671 11.94 447.62 


26 


163. 36 2 123. 7 4. 138 53. 79 


76 


477.52 18146 12.10 459.64 


27 


169.65 2290.2 4.297 58.01 


77 


483.81 18627 12.25 471.81 


28 


175.93 2463.0 4.456 62.39 


78 


490.09 19113 12.41 484.15 


29 


182.21 2642.1 4.615 66.92 


79 


496.37 19607 12.57 496.64 


30 


188.50 2827.4 4.775 71.62 


8O 


502.65 20106 12.' 73 509.30 


31 


194. 78 3 019. 1 4. 934 76. 47 


81 


508.94 20612 12.89 522.11 


32 


201.06 3217.0 5.093 81.49 


82 


515.22 21124 13.05 535.08 


33 


207.35 3421.2 5.252 86.66 


83 


521.50 21642 13.21 548.21 


34 


213.63 3631.7 5.411 91.99 


84 


527.79 22167 13.37 561.50 


35 


219.91 3848.5 5.570 97.48 


85 


534.07 22698 13.53 574.95 


36 


226.19 4071.5 5.730 103.13 


86 


540.35 23235 13.69 588.55 


37 


232.48 4300.8 5.889 108.94 


87 


546.64 23779 13.85 602.32 


38 


238. 76 4 536. 5 6. 048 114. 91 


88 


552.92 24328 14.01 616.25 


39 


245.04 4778.4 6.207 121.04 


89 


559.20 24885 14.16 630.33 


40 


251.33 5026.5 6.366 127.32 


9O 


565.49 25447 14.32 644.58 


41 


257.61 5281.0 6.525 133.77 


91 


571.77 26016 14.48 658.98 


42 


263.89 5541.8 6.685 140.37 


92 


578.05 26590 14.64 673.54 


43 


270. 18 5 808. 8 6. 844 147. 14 


93 


584.34 27172 14.80 688.27 


44 


276. 46 6 082. 1 7. 003 154. 06 


94 


590.62 27759 14.96 703.15 


45 


282.74 6361.7 7.162 161.14 


95 


596.90 28353 1512 718.19 


46 


289.03 6647.6 7.321 168.39 


96 


603.19 28953 15,28 733.39 


47 


295.31 6939.8 7.480 175.79 


97 


609.47 29559 15.44 748.74 


48 


301.59 7238.2 7.639 183.35 


98 


615.75 30172 15.60 764.26 


49 


307.88 7543.0 7.799 191.07 


99 


622.04 30791 15.76 779.94 


50 


314.46 7854.0 7.958 198.94 


1OO 


628.32 31416 15.92 795.77 


* 


2ir 4ir 


N 


27T 4K 



52 



TABLE VI. -NATFEAL FUNCTIONS. 



/ 


O 


1 


2 o 


3 


4 


r 




sin cos 


sin cos 


sin cos 


sin cos 


sin cos 




o 


0000 1.000 


0175 9998 


0349 9994 


0523 9986 


0698 9976 


6O 


1 


0003 1.000 


0177 9998 


0352 9994 


0526 9986 


0700 9975 


59 


2 


0006 1.000 


0180 9998 


0355 9994 


0529 9986 


0703 9975 


58 


3 


0009 1.000 


0183 9998 


0358 9994 


0532 9986 


0706 9975 


57 


4 


0012 1.000 


0186 9998 


0361 9993 


0535 9986 


0709 9975 


56 


5 


0015 1.000 


0189 9998 


0364 9993 


0538 9986 


0712 9975 


55 


6 


0017 1.000 


0192 9998 


0366 9993 


0541 9985 


0715 9974 


54 


7 


0020 1.000 


0195 9998 


0369 9993 


0544 9985 


0718 9974 


53 


8 


0023 1.000 


0198 9998 


0372 9993 


0547 9985 


0721 9974 


52 


9 


, 0026 1.000 


0201 9998 


0375 9993 


0550 9985 


0724 9974 


51 


1O 


0029 1.000 


0204 9998 


0378 9993 


0552 9985 


0727 9974 


50 


11 


0032 1.000 


0207 9998 


0381 9993 


0555 9985 


0729 9973 


49 


12 


0035 1.000 


0209 9998 


0384 9993 


0558 9984 


0732 9973 


48 


13 


0038 1.000 


0212 9998 


0387 9993 


0561 9984 


0735 9973 


47 


14 


0041 1.000 


0215 9998 


0390 9992 


0564 9984 


0738 9973 


46 


15 


0044 1.000 


0218 9998 


0393 9992 


0567 9984 


0741 9973 


45 


16 


0047 1.000 


0221 9998 


0396 9992 


0570 9984 


0744 9972 


44 


17 


0049 1.000 


0224 9997 


0398 9992 


0573 9984 


0747 9972 


43 


18 


0052 1.000 


0227 9997 


0401 9992 


0576 9983 


0750 9972 


42 


19 


0055 1.000 


0230 9997 


0404 9992 


0579 9983 


0753 9972 


41 


2O 


0058 1.000 


0233 9997 


0407 9992 


0581 9983 


0756 9971 


4O 


21 


0061 1.000 


0236 9997 


0410 9992 


0584 9983 


0758 9971 


39 


22 


0064 1.000 


0239 9997 


0413 9991 


0587 9983 


0761 9971 


38 


23 


0067 1.000 


0241 9997 


0416 9991 


0590 9983 


0764 9971 


37 


24 


0070 1.000 


0244 9997 


0419 9991 


0593 9982 


0767 9971 


36 


25 


0073 1.000 


0247 9997 


0422 9991 


0596 9982 


0770 9970 


35 


26 


0076 1.000 


0250 9997 


0425 9991 


0599 9982 


0773 9970 


34 


27 


0079 1.000 


0253 9997 


0427 9991 


0602 9982 


0776 9970 


33 


28 


0081 1.000 


0256 9997 


0430 9991 


0605 9982 


0779 9970 


32 


29 


0084 1.000 


0259 9997 


0433 9991 


0608 9982 


0782 9969 


31 


3O 


0087 1.000 


0262 9997 


0436 9990 


0610 9981 


0785 9969 


3O 


31 


0090 1.000 


0265 9996 


0439 9990 


0613 9981 


0787 9969 


29 


32 


0093 1.000 


0268 9996 


0442 9990 


0616 9981 


0790 9969* 


28 


33 


0096 1.000 


0270 9996 


0445 9990 


0619 9981 


0793 9968 


27 


34 


0099 1.000 


0273 9996 


0448 9990 


0622 9981 


0796 9968 


26 


35 


0102 9999 


0276 9996 


0451 9990 


0625 9980 


0799 9968 


25 


36 


0105 9999 


0279 9996 


0454 9990 


0628 9980 


0802 9968 


24 


37 


0108 9999 


0282 9996 


0457 9990 


0631 9980 


0805 9968 


23 


38 


0111 9999 


0285 9996 


0459 9989 


0634 9980 


0808 9967 


22 


39 


0113 9999 


0288 9996 


0462 9989 


0637 9980 


0811 9967 


21 


40 


0116 9999 


0291 9996 


0465 9989 


0640 9980 


0814 9967 


20 


41 


0119 9999 


0294 9996 


0468 9989 


0642 9979 


0816 9967 


19 


42 


0122 9999 


0297 9996 


0471 9989 


0645 9979 


0819 9966 


18 


43 


0125 9999 


0300 9996 


0474 9989 


0648 9979 


0822 9966 


17 


44 


0128 9999 


0302 9995 


0477 9989 


0651 9979 


0825 9966 


16 


45 


0131 9999 


0305 9995 


0480 9988 


0654 9979 


0828 9966 


15 


46 


0134 9999 


0308 9995 


0483 9988 


0657 9978 


0831 9965 


14 


47 


0137 9999 


0311 9995 


0486 9988 


0660 9978 


0834 9965 


13 


48 


0140 9999 


0314 9995 


0488 9988 


0663 9978 


0837 9965 


.12 


49 


0143 9999 


0317 9995 


0491 9988 


0666 9978 


0840 9965 


11 


5O 


0145 9999 


0320 9995 


0494 9988 


0669 9978 


0843 9964 


1O 


51 


0148 9999 


0323 9995 


0497 9988 


0671 9977 


0845 9964 


9 


52 


0151 9999 


0326 9995 


0500 9987 


0674 9977 


0848 9964 


8 


53 


0154 9999 


0329 9995 


0503 9987 


0677 9977 


0851 9964 


7 


54 


0157 9999 


0332 9995 


0506 9987 


0680 9977 


0854 9963 


6 


55 


0160 9999 


0334 9994 


0509 9987 


0683 9977 


0857 9963 


5 


56 


0163 9999 


0337 9994 


0512 9987 


0686 9976 


0860 9963 


4 


57 


0166 9999 


0340 9994 


0515 9987 


0689 9976 


0863 9963 


3 


58 


0169 9999 


0343 9994 


0518 9987 


0692 9976 


0866 9962 


2 


59 


0172 9999 


0346 9994 


0520 9986 


0695 9976 


0869 9962 


1 


6O 


0175 9999 


0349 9994 


0523 9986 


0698 9976 


0872 9962 







cos sin 


cos sin 


'cos sin 


cos sin 


cos sin 




f 


89 


88 


87 


86 


85 


r 



NATURAL SINES AND COSINES. 



58 



f 


5 


6 


7 


8 


9 


f 




sin cos 


sin cos 


sin cos 


sin cos 


sin cos 




o 


0872 9962 


1045 9945 


1219 9925 


1392 9903 


1564 9877 


60 


1 


0874 9962 


1048 9945 


1222 9925 


1395 9902 


1567 9876 


59 


2 


0877 9961 


1051 9945 


1224 9925 


1397 9902 


1570 9876 


58 


3 


0880 9961 


1054 9944 


1227 9924 


1400 9901 


1573 9876 


57 


4 


0883 9961 


1057 9944 


1230 9924 


1403 9901 


1576 9875 


56 


5 


0886 9961 


1060 9944 


1233 9924 


1406 9901 


1579 9875 


55 


6 


0889 9960 


1063 9943 


1236 9923 


1409 9900 


1582 9874 


54 


7 


0892 9960 


1066 9943 


1239 9923 


1412 9900 


1584 9874 


53 


8 


0895 9960 


1068 9943 


1241 9923 


1415 9899 


1587 9873 


52 


9 


0898 9960 


1071 9942 


1245 9922 


1418 9899 


1590 9873 


51 


1C 


0901 9959 


1074 9942 


1248 9922 


1421 9899 


1593 9872 


50 


11 


0903 9959 


1077 9942 


1250 9922 


1423 9898 


1596 9872 


49 


12 


0906 9959 


1080 9942 


1253 9921 


1426 9898 


1599 9871 


48 


13 


0909 9959 


1083 9941 


1256 9921 


1429 9897 


1602 9871 


47 


14 


0912 9958 


1086 9941 


1259 9920 


1432 9897 


1605 9870 


46 


15 


0915 9958 


1089 9941 


1262 9920 


1435 9897 


1607 9870 


45 


16 


0918 9958 


1092 9940 


1265 9920 


1438 9896 


1610 9869 


44 


17 


0921 9958 


1094 9940 


1268 9919 


1441 9896 


1613 9869 


43 


18 


0924 9957 


1097 9940 


1271 9919 


1444 9895 


1616 9869 


42 


19 


0927 9957 


1100 9939 


1274 9919 


1446 9895 


1619 9868 


41 


2O 


0929 9957 


1103 9939 


1276 9918 


1449 9894 


1622 9868 


40 


21 


0932 9956 


1106 9939 


1279 9918 


1452 9894 


1625 9867 


39 


22 


0935 9956 


1109 9938 


1282 9917 


1455 9894 


1628 9867 


38 


23 


0938 9956 


1112 9938 


1285 9917 


1458 9893 


1630 9866 


37 


24 


0941 9956 


1115 9938 


1288 9917 


1461 9893 


1633 9866 


36 


25 


0944 9955 


1118 9937 


1291 9916 


1464 9892 


1636 9865 


35 


26 


0947 9955 


1120 9937 


1294 9916 


1467 9892 


1639 9865 


34 


27 


0950 9955 


1123 9937 


1297 9916 


1469 9891 


1642 9864 


33 


28 


0953 9955 


1126 9936 


1299 9915 


1472 9891 


1645 9864 


32 


29 


0956 9954 


1129 9936 


1302 9915 


1475 9891 


1648 9863 


31 


30 


0958 9954 


1132 9936 


1305 9914 


1478 9890 


1650 9863 


30 


31 


0961 9954 


1135 9935 


1308 9914 


1481 9890 


1653 9862 


29 


32 


0964 9953 


1138 9935 


1311 9914 


1484 9889 


1656 9862 


28 


33 


0967 9953 


1141 9935 


1314 9913 


1487 9889 


1659 9861 


27 


34 


0970 9953 


1144 9934 


1317 9913 


1490 9888 


1662 9861 


26 


35 


0973 9953 


1146 9934 


1320 9913 


1492 9888 


1665 9860 


25 


36 


0976 9952 


1149 9934 


1323 9912 


1495 9888 


1668 9860 


24 


37 


0979 9952 


1152 9933 


1325 9912 


1498 9887 


1671 9859 


23 


38 


0982 9952 


1155 9933 


1328 9911 


1501 9887 


1673 9859 


22 


39 


0985 9951 


1158 9933 


1331 9911 


1504 9886 


1676 9859 


21 


40 


0987 9951 


1161 9932 


1334 9911 


1507 9886 


1679 9858 


20 


41 


0990 9951 


1164 9932 


1337 9910 


1510 9885 


1682 9858 


19 


42 


0993 9951 


1167 9932 


1340 9910 


1513 9885 


1685 9857 


18 


43 


0996 9950 - 


1170 931 


1343 9909 


1515 9884 


1688 9857 


17 


44 


0999 9950 


1172 9931 


1346 9909 


1518 9884 


1691 9856 


16 


45 


1002 9950 


1175 9931 


1349 9909 


1521 9884 


1693 9856 


15 


46 


1005 9949 


1178 9930 


1351 9908 


1524 9883 


1696 9855 


14 


47 


1008 9949 


1181 9930 


1354 9908 


1527 9883 


1699 9855 


13 


48 


1011 9949 


1184 9930 


1357 9907 


1530 9882 


1702 9854 


12 


49 


1013 9949 


1187 9929 


1360 9907 


1533 9882 


1705 9854 


11 


50 


1016 9948 


1190 9929 


1363 9907 


1536 9881 


1708 9853 


1O 


51 


1019 9948 


1193 9929 


1366 9906 


1538 9881 


1711 9853 


9 


52 


1022 9948 


1196 9928 


1369 9906 


1541 9880 


1714 9852 


8 


53 


1025 9947 


1198 9928 


1372 9905 


1544 9880 


1716 9852 


7 


54 


1028 9947 


1201 9928 


1374 9905 


1547 9880 


1719 9851 


6 


55 


1031 9947 


1204 9927 


1377 9905 


1550 9879 


1722 9851 


5 


56 


1034 9946 


1207 9927 


1380 9904 


1553 9879 


1725 9850 


4 


57 


1037 9946 


1210 9927 


1383 9904 


1556 9878 


1728 9850 


3 


58 


1039 9946 


1213 9926 


1386 9903 


1559 9878 


1731 9849 


2 


59 


1042 9946 


1216 9926 


1389 9903 


1561 9877 


1734 9849 


1 


60 


1045 9945 


1219 9925 


1392 9903 


1564 9877 


1736 9848 







cos sin 


cos sin 


cos sin 


cos sin 


cos sin 




' 84 


83 


82 


81 


80 


t 



54 



NATURAL SINES AND COSINES. 



/ 


10 


11 


12 


13 


14 


r 




sin cos 


sin cos 


sin cos 


sin cos 


sin cos 







1736 9848 


1908 9816 


2079 9781 


2250 9744 


2419 9703 


6O 


1 


1739 9848 


1911 9816 


2082 9781 


2252 9743 


2422 9702 


59 


2 


1742 9847 


1914 9815 


2085 9780 


2255 9742 


2425 9702 


58 


3 


1745 9847 


1917 9815 


2088 9780 


2258 9742 


2428 9701 


57 


4 


1748 9846 


1920 9814 


2090 9779 


2261 9741 


2431 9700 


56 


5 


1751 9846 


1922 9813 


2093 9778 


2264 9740 


2433 9699 


55 


6 


1754 9845 


1925 9813 


2096 9778 


2267 9740 


2436 9699 


54 


7 


1757 9845 


1928 9812 


2099 9777 


2269 9739 


2439 9698 


53 


8 


1759 9844 


1931 9812 


2102 9777 


2272 9738 


2442 9697 


52 


9 


1762 9843 


1934 9811 


2105 9776 


2275 9738 


2445 9697 


51 


10 


1765 9843 


1937 9811 


2108 9775 


2278 9737 


2447 9696 


50 


11 


1768 9842 


1939 9810 


2110 9775 


2281 9736 


2450 9695 


49 


12 


1771 9842 


1942 9810 


2113 9774 


2284 9736 


2453 9694 


48 


13 


1774 9841 


1945 9809 


2116 9774 


2286 9735 


2456 9694 


47 


14 


1777 9841 


1948 9808 


2119 9773 


2289 9734 


2459 9693 


46 


15 


1779 9840 


1951 9808 


2122 9772 


2292 9734 


2462 9692 


45 


16 


1782 9840 


1954 9807 


2125 9772 


2295 9733 


2464 9692 


44 


17 


1785 9839 


1957 9807 


2127 9771 


2298 9732 


2467 9691 


43 


18 


1788 9839 


1959 9806 


2130 9770 


2300 9732 


2470 9690 


42 


19 


1791 9838 


1962 9806 


2133 9770 


2303 9731 


2473 9689 


41 


20 


1794 9838 


1965 9805 


2136 9769 


2306 9730 


2476 9689 


4O 


21 


1797 9837 


1968 9804 


2139 9769 


2309 9730 


2478 9688 


39 


22 


1799 9837 


1971 9804 


2142 9768 


2312 9729 


2481 9687 


38 


23 


1802 9836 


1974 9803 


2145 9767 


2315 9728 


2484 9687 


37 


24 


1805 9836 


1977 9803 


2147 9767 


2317 9728 


2487 9686 


36 


25 


1808 9835 


1979 9802 


2150 9766 


2320 9727 


2490 9685 


35 


26 


1811 9835 


1982 9802 


2153 9765 


2323 9726 


2493 9684 


34 


27 


1814 9834 


1985 9801 


2156 9765 


2326 9726 


2495 9684 


33 


28 


1817 9834 


1988 9800 


2159 9764 


2329 9725 


2498 9683 


32 


29 


1819 9833 


1991 9800 


2162 9764 


2332 9724 


2501 9682 


31 


30 


1822 9833 


1994 9799 


2164 9763 


2334 9724 


2504 9681 


30 


31 


1825 9832 


1997 9799 


2167 9762 


2337 9723 


2507 9681 


29 


32 


1828 9831 


1999 9798 


2170 9762 


2340 9722 


2509 9680 


28 


33 


1831 9831 


2002 9798 


2173 9761 


2343 9722 


2512 9679 


27 


34 


1834 9830 


2005 9797 


2176 9760 


2346 9721 


2515 9679 


26 


35 


1837 9830 


2008 9796 


2179 9760 


2349 9720 


2518 9678 


25 


36 


1840 9829 


2011 9796 


2181 9759 


2351 9720 


2521 9677 


24 


37 


1842 9829 


2014 9795 


2184 9759 


2354 9719 


2524 9676 


23 


38 


1845 9828 


2016 9795 


2187 9758 


2357 9718 


2526 9676 


22 


39 


1848 9828 


2019 9794 


2190 9757 


2360 9718 


2529 9675 


21 


40 


1851 9827 


2022 9793 


2193 9757 


2363 9717 


2532 9674 


2O 


41 


1854 9827 


2025 9793 


2196 9756 


2366 9716 


2535 9673 


19 


42 


1857 9826 


2028 9792 


2198 9755 


2368 9715 


2538 9673 


18 


43 


1860 9826 


2031 9792 


2201 9755 


2371 9715 


. 2540 9672 


17 


44 


1862 9825 


2034 9791 


2204 9754 


2374 9714 


2543 9671 


16 


45 


1865 9825 


2036 9790 


2207 9753 


2377 9713 


2546 9670 


15 


46 


1868 9824 


2039 9790 


2210 9753 


2380 9713 


2549 9670 


14 


47 


1871 9823 


2042 9789 


2213 9752 


2383 9712 


2552 9669 


13 


48 


1874 9823 


2045 9789 


2215 9751 


2385 9711 


2554 9668 


12 


49 


1877 9822 


2048 9788 


2218 9751 


2388 9711 


2557 9667 


11 


50 


1880 9822 


2051 9787 


2221 9750 


2391 9710 


2560 9667 


10 


51 


1882 9821 


2054 9787 


2224 9750 


2394 9709 


2563 9666 


9 


52 


1885 9821 


2056 9786 


2227 9749 


2397 9709 


2566 9665 


8 


53 


1888 9820 


2059 9786 


2230 9748 


2399 9708 


2569 9665 


7 


54 


]891 9820 


2062 9785 


2233 9748 


2402 9707 


2571 9664 


6 


55 


1894 9819 


2065 9784 


2235 9747 


2405 9706 


2574 9663 


5 


56 


1897 9818 


2068 9784 


2238 9746 


2408 9706 


2577 9662 


4 


57 


1900 9818 


2071 9783 


2241 9746 


2411 9705 


2580 9662 


3 


58 


1902 9817 


2073 9783 


2244 9745 


2414 9704 


2583 9661 


2 


59 


1905 9817 


2076 9782 


2247 9744 


2416 9704 


2585 9660 


1 


6O 


1908 9816 


2079 9781 


2250 9744 


2419 9703 


2588 9659 







cos sin 


cos sin 


cos sin 


cos sin 


cos sin 




f 


79 


78 


77 


76 


75 


f 



NATURAL SINES AND COSINES. 



55 



/ 


15 


16 


17 


18 


19 


f 




sin cos 


siii cos 


sin cos 


sin cos 


sin cos 




o 


2588 9659 


2756 9613 


2924 9563 


3090 9511 


3256 9455 


60 


1 


2591 9659 


2759 9612 


2926 9562 


3093 9510 


3258 9454 


59 


2 


2594 9658 


2762 9611 


2929 9561 


3096 9509 


3261 9453 


58 


3 


2597 9657 


2765 9610 


2932 9560 


3098 9508 


3264 9452 


57 


4 


2599 9656 


2768 9609 


2935 9560 


3101 9507 


3267 9451 


56 


5 


2602 9655 


2770 9609 


2938 9559 


3104 9506 


3269 9450 


55 


6 


2605 9655 


2773 9608 


2940 9558 


3107 9505 


3272 9449 


54 


7 


2608 9654 


2776 9607 


2943 9557 


3110 9504 


3275 9449 


53 


8 


2611 9653 


2779 9606 


2<H6 9556 


3112 9503 


3278 9448 


52 


9 


2613 9652 


2782 9605 


2949 9555 


3115 9502 


3280 9447 


51 


10 


2616 9652 


2784 9605 


2952 9555 


3118 9502 


3283 9446 


50 


11 


2619 9651 


2787 9604 


2954 9554 


3121 9501 


3286 9445 


49 


12 


2622 9650 


2790 9603 


2957 9553 


3123 9500 


3289 9444 


48 


13 


2625 9649 


2793 9602 


2960 9552 


3126 9499 


3291 9443 


47 


14 


2628 9649 


2795 9601 


2963 9551 


3129 9498 


3294 9442 


46 


15 


2630 9648 


2798 9600 


2965 9550 


3132 9497 


3297 9441 


45 


16 


2633 9647 


2801 9600 


2968 9549 


3134 9496 


3300 9440 


44 


17 


2636 9646 


2804 9599 


2971 9548 


3137 9495 


3302 9439 


43 


18 


2639 9646 


2807 9598 


2974 9548 


3140 9494 


3305 9438 


42 


19 


2642 9645 


2809 9597 


2977 9547 


3143 9493 


3308 9437 


41 


20 


2644 9644 


2812 9596 


2979 9546 


3145 9492 


3311 9436 


4O 


21 


2647 9643 


2815 9596 


2982 9545 


3148 9492 


3313 9435 


39 


22 


2650 9642 


2818 9595 


2985 9544 


3151 9491 


3316 9434 


38 


23 


2653 9642 


2821 9594 


2988 9543 


3154 9490 


3319 9433 


37 


24 


2656 9641 


2823 9593 


2990 9542 


3156 9489 


3322 9432 


36 


25 


2658 9640 


2826 9592 


2993 9542 


3159 9488 


3324 9431 


35 


26 


2661 9639 


2829 9591 


2996 9541 


3162 9487 


3327 9430 


34 


27 


2664 9639 


2832 9591 


2999 9540 


3165 9486 


3330 9429 


33 


28 


2667 9638 


2835 9590 


3002 9539 


3168 9485 


3333 9428 


32 


29 


2670 9637 


2837 9589 


3004 9538 


3170 9484 


3335 9427 


31 


30 


2672 9636 


2840 9588 


3007 9537 


3173 9483 


3338 9426 


30 


31 


2675 9636 


2843 9587 


3010 9536 


3176 9482 


3341 9425 


29 


32 


2678 9635 


2846 9587 


3013 9535 


3179 9481 


3344 9424 


28 


33 


2681 9634 


2849 9586 


3015 9535 


3181 9480 


3346 9423 


27 


34 


2684 9633 


2851 9585 


3018 9534 


3184 9480 


3349 9423 


26 


35 


2686 9632 


2854 9584 


3021 9533 


3187 9479 


3352 9422 


25 


36 


2689 9632 


2857 9583 


3024 9532 


3190 9478 


3355 9421 


24 


37 


2692 9631 


2860 9582 


3026 9531 


3192 9477 


3357 9420 


23 


38 


2695 9630 


2862 9582 


3029 9530 


3195 9476 


3360 9419 


22 


39 


2698 9629 


2865 9581 


3032 9529 


3198 9475 


3363 9418 


21 


4O 


2700 9628 


2868 9580 


3035 9528 


3201 9474 


3365 9417 


2O 


41 


2703 9628 


2871 9579 


3038 9527 


3203 9473 


3368 9416 


19 


42 


2706 9627 


2874 9578 


3040 9527 


3206 9472 


3371 9415 


18 


43 


2709 9626 


2876 9577 


3043 9526 


3209 9471 


3371 9414 


17 


44 


2712 9625 


2879 9577 


3046 9525 


3212 9470 


3376 9413 


16 


45 


2714 9625 


2882 9576 


3049 9524 


3214 9469 


3379 9412 


15 


46 


2717 9624 


2885 9575 


3051 9523 


3217 9468 


3382 9411 


14 


47 


2720 9623 


2888 9574 


3054 9522 


3220 9467 


3385 9410 


13 


48 


2723 9622 


2890 9573 


3057 9521 


3223 9466 


3387 9409 


12 


49 


2726 9621 


2893 9572 


3060 9520 


3225 9466 


3390 9408 


11 


5O 


2728 9621 


2896 9572 


3062 9520 


3228 9465 


3393 9407 


1O 


51 


2731 9620 


2899 .9571 


3065 9519 


3231 9464 


3396 9406 


9 


52 


2734 9619 


2901 9570 


3068 9518 


3234 9463 


3398 9405 


8 


53 


2737 9618 


2904 9569 


3071 9517 


3236 9462 


3401 9404 


7 


54 


2740 9617 


2907 9568 


3074 9516 


3239 9461 


3404 9403 


6 


55 


2742. 9617 


2910 9567 


3076 9515 


3242 9460 


3407 9402 


5 


56 


2745 9616 


2913 9566 


3079 9514 


3245 9459 


3409 9401 


4 


57 


2748 9615 


2915 9566 


3082 9513 


3247 9458 


3412 9400 


3 


58 


2751 9614 


2918 9565 


3085 9512 


3250 9457 


3415 9399 


2 


59 


2754 9613 


2921 9564 


3087 9511 


3253 9456 


3417 9398 


1 


6O 


2756 9613 


2924 9563 


3090 9511 


3256 9455 


3420 9397 







cos sin 


cos sin 


cos sin 


cos sin 


cos sin 




t 


74 


73 


72 


71 


70 


f 



56 



NATURAL SINES AND COSINES. 



/ 


2O 


21 


22 


23 


24 


f 




sin cos 


sin cos 


sin cos 


sin cos 


sin cos 







3420 9397 


3584 9336 


3746 9272 


3907 9205 


4067 9135 


60 


1 


3423 9396 


3586 9335 


3749 9271 


3910 9204 


4070 9134 


59 


2 


3426 9395 


3589 9334 


3751 9270 


3913 9203 


4073 9133 


58 


3 


3428 9394 


3592 9333 


3754 9269 


3915 9202 


4075 9132 


57 


4 


3431 9393 


3595 9332 


3757 9267 


3918 9200 


4078 9131 


56 


5 


3434 9392 


3597 9331 


3760 9266 


3921 9199 


4081 9130 


55 


6 


3437 9391 


3600 9330 


3762 9265 


3923 9198 


4083 9128 


54 


7 


3439 9390 


3603 9328 


3765 9264 


3926 9197 


4086 9127 


53 


8 


3442 9389 


3605 9327 


3768 9263 


3929 9196 


4089 9126 


52 


9 


3445 9388 


3608 9326 


3770 9262 


3931 9195 


4091 9125 


51 


1O 


3448 9387 


3611 9325 


3773 9261 


3934 9194 


4094 9124 


5O 


11 


3450 9386 


3614 9324 


3776 9260 


3937 9192 


4097 9122 


49 


12 


3453 9385 


3616 9323 


3778 9259 


3939 9191 


4099 9121 


48 


13 


3456 9384 


3619 9322 


3781 9258 


3942 9190 


4102 9120 


47 


14 


3458 9383 


3622 9321 


3784 9257 


3945 9189 


4105 9119 


46 


15 


3461 9382 


3624 9320 


3786 9255 


3947 9188 


4107 9118 


45 


16 


3464 9381 


3627 9319 


3789 9254 


3950 9187 


4110 9116 


44 


17 


3467 9380 


3630 9318 


3792 9253 


3953 9186 


4112 9115 


43 


18 


3469 9379 


3633 9317 


3795 9252 


3955 9184 


4115 9114 


42 


19 


3472 9378 


3635 9316 


3797 9251 


3958 9183 


4118 9113 


41 


2O 


3475 9377 


3638 9315 


3800 9250 


3961 9182 


4120 9112 


40 


21 


3478 9376 


3641 9314 


3803 9249 


3963 9181 


4123 9110 


39 


22 


3480 9375 


3643 9313 


3805 9248 


3966 9180 


4126 9109 


38 


23 


3483 9374 


3646 9312 


3808 9247 


3969 9179 


4128 9108 


37 


24 


3486 9373 


3649 9311 


3811 9245 


3971 9178 


4131 9107 


36 


25 


3488 9372 


3651 9309 


3813 9244 


3974 9176 


4134 9106 


35 


26 


3491 9371 


3654 9308 


3816 9243 


3977 9175 


4136 9104 


34 


27 


3494 9370 


3657 9307 


3819 9242 


3979 9174 


4139 9103 


33 


28 


3497 9369 


3660 9306 


3821 9241 


3982 9173 


4142 9102 


32 


29 


3499 9368 


3662 9305 


3824 9240 


3985 9172 


4144 9101 


31 


30 


3502 9367 


3665 9304 


3827 9239 


3987 9171 


4147 9100 


30 


31 


3505 9366 


3668 9303 


3830 9238 


3990 9169 


4150 9098 


29 


32 


3508 9365 


3670 9302 


3832 9237 


3993 9168 


4152 9097 


28 


33 


3510 9364 


3673 9301 


3835 9235 


3995 9167 


4155 9096 


27 


34 


3513 9363 


3676 9300 


3838 9234 


3998 9166 


4158 9095 


26 


35 


3516 9362 


3679 9299 


3840 9233 


4001 9165 


4160 9094 


25 


36 


3518 9361 


3681 9298 


3843 9232 


4003 9164 


4163 9092 


24 


37 


3521 9360 


3684 9297 


3846 9231 


4006 9162 


4165 9091 


23 


38 


3524 9359 


3687 9296 


3848 9230 


4009 9161 


4168 9090 


22 


39 


3527 9358 


3689 9295 


3851 9229 


4011 9160 


4171 9088 


21 


4O 


3529 9356 


3692 9293 


3854 9228 


4014 9159 


4173 9088 


2O 


41 


3532 9355 


3695 9292 


3856 9227 


4017 9158 


4176 9086 


19 


42 


3535 9354 


3697 9291 


3859 9225 


4019 9157 


4179 9085 


18 


43 


3537 9353 


3700 9290 


3862 9224 


4022 9155 


4181 9084 


17 


44 


3540 9352 


3703 9289 


3864 9223 


4025 9154 


4184 9083 


16 


45 


3543 9351 


3706 9288 


3867 9222 


4027 9153 


4187 9081 


15 


46 


3546 9350 


3708 9287 


3870 9221 


4030 9152 


4189 9080 


14 


47 


3548 9349 


3711 9286 


3872 9220 


4033 9151 


4192 9079 


13 


48 


3551 9348 


3714 9285 


3875 9219 


4035 9150 


4195 9078 


12 


49 


3554 9347 


3716 9284 


3878 9218 


4038 9148 


4197 9077 


11 


50 


3557 9346 


3719 9283 


3881 9216 


4041 9147 


4200 9075 


1O 


51 


3559 9345 


3722 9282 


3883 9215 


4043. 9146 


4202 9074 


9 


52 


3562 9344 


3724 9281 


3886 9214 


4046 9145 


4205 9073 


8 


53 


3565 9343 


3727 9279 


3889 9213 


4049 9144 


4208 9072 


7 


54 


3567 9342 


3730 9278 


3891 9212 


4051 9143 


4210 9070 


6 


55 


3570 9341 


3733 9277 


3894 9211 


4054 9141 


4213 .9069 


5 


56 


3573 9340 


3735 9276 


3897 9210 


4057 9140 


4216 9068 


4 


57 


3576 9339 


3738 9275 


3899 9208 


4059 9139 


4218 9067 


3 


58 


3578 9338 


3741 9274 


3902 9207 


4062 9138 


4221 9066 


2 


59 


3581 9337 


3743 9273 


3905 9206 


4065 9137 


4224 9064 


1 


6O 


3584 9336 


3746 9272 


3907 9205 


4067 9135 


4226 9063 







cos sin 


cos sin 


cos sin 


cos sin 


cos sin 




f 


69 


68 


67 


66 


65 


f 



NATURAL SINES AND COSINES. 



57 



/ 


25 


26 


27 


28 


29 


t 




sin cos 


sin cos 


sin cos 


sin cos 


sin cos 




o 


4226 9063 


4384 8988 


4540 8910 


4695 8829 


4848 8746 


6O 


1 


4229 9062 


4386 8987 


4542 8909 


4697 8828 


4851 8745 


59 


2 


4231 9061 


4389 8985 


4545 8907 


4700 8827 


4853 8743 


58 


3 


4234 9059 


4392 8984 


4548 8906 


4702 8825 


4856 8742 


57 


4 


. .4237 9058 


4394 8983 


4550 8905 


4705 8824 


4858 8741 


56 


5 


4239 9057 


4397 8982 


4553 8903 


4708 8823 


4861 8739 


55 


6 


4242 9056 


4399 8980 


4555 8902 


4710 8821 


4863 8738 


54 


7 


4245 9054 


4402 8979 


4558 8901 


4713 8820 


4866 8736 


53 


8 


4247 9053 


4405 8978 


4561 8899 


4715 8819 


4868 8735 


52 


9 


4250 9052 


4407 8976 


4563 8898 


4718 8817 


4871 8733 


51 


1O 


4253 9051 


4410 8975 


4566 8897 


4720 8816 


4874 8732 


50 


11 


4255 9050 


4412 8974 


4568 8895 


4723 8814 


4876 8731 


49 


12 


4258 9048 


4415 8973 


4571 8894 


4726 8813 


4879 8729 


48 


13 


4260 9047 


4418 8971 


4574 8893 


4728 8812 


4881 8728 


47 


14 


4263 9046 


4420 8970 


4576 8892 


4731 8810 


4884 8726 


46 


15 


4266 9045 


4423 8969 


4579 8890 


4733 8809 


4886 8725 


45 


16 


4268 9043 


4425 8967 


4581 8889 


4736 8808 


4889 8724 


44 


17 


4271 9042 


4428 8966 


4584 8888 


4738 8806 


4891 8722 


43 


18 


4274 9041 


4431 8965 


4586 8886 


4741 8805 


4894 8721 


42 


19 


4276 9040 


4433 ' 8964 


4589 8885 


4743 8803 


4896 8719 


41 


2O 


4279 9038 


4436 8962 


4592 8884 


4746 8802 


4899 8718 


40 


21 


4281 9037 


4439 8961 


4594 8882 


4749 8801 


4901 8716 


39 


22 


4284 9036 


4441 8960 


4597 8881 


4751 8799 


4904 8715 


38 


23 


4287 9035 


4444 8958 


4599 8879 


4754 8798 


4907 8714 


37 


24 


4289 9033 


4446 8957 


4602 8878 


4756 8796 


4909 8712 


36 


25 


4292 9032 


4449 8956 


4605 8877 


4759 8795 


4912 8711 


35 


26 


4295 9031 


4452 8955 


4607 8875 


4761 8794 


4914 8709 


34 


27 


4297 9030 


4454 8953 


4610 8874 


4764 8792 


4917 8708 


33 


28 


4300 9028 


4457 8952 


4612 8873 


4766 8791 


4919 8706 


32 


29 


4302 9027 


4459 8951 


4615 8871 


4759 8790 


4922 8705 


31 


30 


4305 9026 


4462 8949 


4617 8870 


4772 8788 


4924 8704 


3O 


31 


4308 9025 


4465 8948 


4620 8869 


4774 8787 


4927 8702 


29 


32 


4310 9023 


4467 8947 


4623 8867 


4777 8785 


4929 8701 


28 


33 


4313 9022 


4470 8945 


4625 8866 


4779 8784 


4932 8699 


27 


34 


4316 9021 


4472 8944 


4628 8865 


4782 8783 


4934 8698 


26 


35 


4318 9020 


4475 8943 


4630 8863 


4784 8781 


4937 8696 


25 


36 


4321 9018 


4478 8942 


4633 8862 


4787 8780 


4939 8695 


24 


37 


4323 9017 


4480 8940 


4636 8861 


4789 8778 


4942 8694 


23 


38 


4326 9016 


4483 8939 


4638 8859 


4792 8777 


4944 8692 


22 


39 


4329 9015 


4485 8938 


4641 8858 


4795 8776 


4947 8691 


21 


4O 


4331 9013 


4488 8936 


4643 8857 


4797 8774 


4950 8689 


20 


41 


4334 9012 


4491 8935 


4646 8855 


4800 8773 


4952 8688 


19 


42 


4337 9011 


4493 8934 


4648 8854 


4802 8771 


4955 8686 


18 


43 


4339 9010 


4496 8932 


4651 8853 


4805 8770 


4957 8685 


17 


44 


4342 9008 


4498 8931 


4654 8851 


4807 8769 


4960 8683 


16 


45 


4344 9007 


4501 8930 


4656 8850 


4810 8767 


4962 8682 


15 


46 


4347 9006 


4504 8928 


4659 8849 


4812 8766 


4965 8681 


14 


47 


4350 9004 


4506 8927 


4661 8847 


4815 8764 


4967 8679 


13 


48 


4352 9003 


4509 8926 


4664 8846 


4818 8763 


4970 8678 


12 


49 


4355 9002 


4511 8925 


4666 8844 


4820 8762 


4972 8676 


11 


50 


4358 9001 


4514 8923 


4669 8843 


4823 8760 


4975 8675 


1O 


51 


4360 8999 


4517 8922 


4672 8842 


4825 8759 


4977 8673 


9 


52 


4363 8998 


4519 8921 


4674 8840 


4828 8757 


4980 8672 


8 


53 


4365 8997 


4522 8919 


4677 8839 


4830 8756 


4982 8670 


7 


54 


4368 8996 


4524 8918 


4679 8838 


4833 8755 


4985 8669 


6 


55 


4371 8994 


4527 8917 


4682 8836 


4835 8753 


4987 8668 


5 


56 


4373 8993 


4530 8915 


4684 8835 


4838 8752 


4990 8666 


4 


57 


4376 8992 


4532 8914 


4687 8834 


4840 8750 


4992 8665 


3 


58 


4378 8990 


4535 8913 


4690 8832 


4843 8749 


4995 8663 


2 


59 


4381 8989 


4537 8911 


4692 8831 


4846 8748 


4997 8662 


1 


60 


4384 8988 


4540 8910 


4695 8829 


4848 8746 


5000 8660 


O 




cos sin 


cos sin 


cos sin 


cos sin 


cos sin 




f 


64 


63 


62 


61 


60 


f 



58 



NATURAL SINES AND COSINES. 



/ 


3O 


31 


32 


33 


34 


f 




sin cos 


sin cos 


sin cos 


sin cos 


sin cos 







5000 8660 


5150 8572 


5299 8480 


5446 8387 


5592 8290 


60 


1 


5003 8659 


5153 8570 


5302 8479 


5449 8385 


5594 8289 


59 


2 


5005 8657 


5155 8569 


5304 8477 


5451 8384 


5597 8287 


58 


3 


5008 8656 


5158 8567 


5307 8476 


5454 8382 


5599 8285 


57 


4 


5010 8654 


5160 8566 


5309 8474 


5456 8380 


5602 8284 


56 


5 


5013 8653 


5163 8564 


5312 8473 


5459 8379 


5604 8282 " 


55 


6 


5015 8652 


5165 8563 


5314 8471 


5461 8377 


5606 8281 


54 


7 


5018 8650 


5168 8561 


5316 8470 


5463 8376 


5609 8279 


53 


8 


5020 8649 


5170 8560 


5319 8468 


5466 8374 


5611 8277 


52 


9 


5023 8647 


5173 8558 


5321 8467 


5468 8372 


5614 8276 


51 


1O 


5025 8646 


5175 8557 


5324 8465 


5471 8371 


5616 8274 


5O 


11 


5028 8644 


5178 8555 


5326 8463 


5473 8369 


5618 8272 


49 


12 


5030 8643 


5180 8554 


5329 8462 


5476 8368 


5621 8271 


48 


13 


5033 8641 


5183 8552 


5331 8460 


5478 8366 


5623 8269 


47 


14 


5035 8640 


5185 8551 


5334 8459 


5480 8364 


5626 8268 


46 


15 


5038 8638 


5188 8549 


5336 8457 


5483 8363 


5628 8266 


45 


16 


5040 8637 


5190 8548 


5339 8456 


5485 8361 


5630 8264 


44 


17 


5043 8635 


5193 8546 


5341 8454 


5488 8360 


5633 8263 


43 


18 


5045 8634 


5195 8545 


5344 8453 


5490 8358 


5635 8261 


42 


19 


5048 8632 


5198 8543 


5346 8451 


5493 8356 


5638 8259 


41 


20 


5050 8631 


5200 8542 


5348 8450 


5495 8355 


5640 8258 


4O 


21 


5053 8630 


5203 8540 


5351 8448 


5498 8353 


5642 8256 


39 


22 


5055 8628 


5205 8539 


5353 8446 


5500 8352 


5645 8254 


38 


23 


5058 8627 


5208 8537 


5356 8445 


5502 8350 


5647 8253 


37 


24 


5060 8625 


5210 8536 


5358 8443 


5505 8348 


5650 8251 


36 


25 


5063 8624 


5213 8534 


5361 8442 


5507 8347 


5652 8249 


35 


26 


5065 8622 


5215 8532 


5363 8440 


5510 8345 


5654 8248 


34 


27 


5068 8621 


5218 8531 


5366 8439 


5512 8344 


5657 8246 


33 


28 


5070 8619 


5220 8529 


5368 8437 


5515 8342 


5659 8245 


32 


29 


5073 8618 


5223 8528 


5371 8435 


5517 8340 


5662 8243 


31 


30 


5075 8616 


5225 8526 


5373 8434 


5519 8339 


5664 8241 


3O 


31 


5078 8615 


5227 8525 


5375 8432 


5522 8337 


5666 8240 


29 


32 


5080 8613 


5230 8523 


5378 8431 


5524 8336 


5669 8238 


28 


33 


5083 8612 


5232 8522 


5380 8429 


5527 8334 


5671 8236 


27 


34 


5085 8610 


5235 8520 


5383 8428 


5529 8332 


5674 8235 


26 


35 


5088 8609 


5237 8519 


5385 8426 


5531 8331 


5676 8233 


25 


36 


5090 8607 


5240 8517 


5388 8425 


5534 8329 


5678" 8231 


24 


37 


5093 8606 


5242 8516 


5390 8423 


5536 8328 


5681 8230 


23 


38 


5095 8604 


5245 8514 


5393 8421 


5539 8326 


5683 8228 


22 


39 


5098 8603 


5247 8513 


5395 8420 


5541 8324 


5686 8226 


21 


40 


5100 8601 


5250 8511 


5398 8418 


5544 8323 


5688 8225 


2O 


41 


5103 8600 


5252 8510 


5400 8417 


5546 8321 


5690 8223 


19 


42 


5105 8599 


5255 8508 


5402 8415 


5548 8320 


5693 8221 


18 


43 


5108 8597 


5257 8507 


5405 8414 


5551 8318 


5695 8220 


17 


44 


5110 8596 


5260 8505 


5407 8412 


5553 8316 


5698 8218 


16 


45 


5113 8594 


5262 8504 


5410 8410 


5556 8315 


5700 8216 


15 


46 


5115 8593 


5265 8502 


5412 8409 


5558 8313 


5702 8215 


14 


47 


5118 8591 


5267 8500 


5415 8407 


5561 8311 


5705 8213 


13 


48 


5120 8590 


5270 8499 


5417 8406 


5563 8310 


5707 8211 


12 


49 


5123 8588 


5272 8497 


5420 8404 


5565 8308 


5710 8210 


11 


50 


5125 8587 


5275 8496 


5422 8403 


5568 8307 


5712 8208 


10 


51 


5128 8585 


5277 8494 


5424 8401 


5570 8305 


5714 8207 


9 


52 
53 


5130 8584 
5133 8582 


5279 8493 
5282 8491 


5427 8399 
5429 8398 


5573 8303 
5575 8302 


5717 8205 
5719 8203 


8 

7 


54 


5135 8581 


5284 8490 


5432 8396 


5577 8300 


5721 8202 


6 


55 


5138 8579 


5287 8488 


5434 8395 


5580 8299 


5724 8200 


5 


56 


5140 8578 


5289 8487 


5437 8393 


5582 8297 


5726 8198 


4 


57 


5143 8576 


5292 8485 


5439 8391 


5585 8295 


5729 8197 


3 


58 


5145 8575 


5294 8484 


5442 8390 


5587 8294 


5731 8195 


2 


59 


S148 8573 


5297 8482 


5444 8388 


5590 8292 


5733 8193 


1 


60 


5150 8572 


5299 8480 


5446 8387 


5592 8290 


5736 8192 


O 




cos sin 


cos sin 


cos sin 


cos sin 


cos sin 




f 


59 


58 


57 


56 


55 


t 



NATURAL SINES AND COSINES. 



59 



t 


35 


30 


37 


38 


39 


t 




sin cos 


sin cos 


sin cos 


sin cos 


sin cos 




o 


5736 8192 


5878 8090 


6018 7986 


6157 7880 


6293 7771 


60 


1 


5738 8190 


5880 8088 


6020 7985 


6159 7878 


6295 7770 


59 


2 


5741 8188 


5883 8087 


'6023 7983 


6161 7877 


6298 7768 


58 


3 


5743 8187 


5885 8085 


6025 7981 


6163 7875 


6300 7766 


57 


4 


5745 8185 


5887 8083 


6027 7979 


6166 7873 


6302 7764 


56 


5 


5748 8183 


5890 8082 


6030 7978 


6168 7871 


6305 7762 


55 


6 


5750 8181 


5892 8080 


6032 7976 


6170 7869 


6307 7760 


54 


7 


5752 8180 


5894 8078 


6034 7974 


6173 7868 


6309 7759 


53 


8 


5755 8178 


5897 8076 


6037 7972 


6175 7866 


6311 7757 


52 


9 


5757 8176 


5899 8075 


6039 7971 


6177 7864 


6314 7755 


51 


10 


5760 8175 


5901 8073 


6041 7969 


6180 7862 


6316 7753 


50 


11 


5762 8173 


5904 8071 


6044 7967 


6182 7860 


6318 7751 


49 


12 


5764 8171 


5906 8070 


6046 7965 


6184 7859 


6320 7749 


48 


13 


5767 8170 


5908 8068 


6048 7964 


6186 7857 


6323 7748 


47 


14 


5769 8168 


5911 8066 


6051 7962 


6189 7855 


6325 7746 


46 


15 


5771 8166 


5913 8064 


6053 7960 


6191 7853 


6327 7744 


45 


16 


5774 8165 


5915 8063 


6055 7958 


6193 7851 


6329 7742 


44 


17 


5776 8163 


5918 8061 


6058 7956 


6196 7850 


6332 7740 


43 


18 


5779 8161 


5920 8059 


6060 7955 


6198 7848 


6334 7738 


42 


19 


5781 8160 


5922 8058 


6062 7953 


6200 7346 


6336 7737 


41 


2O 


5783 8158 


5925 8056 


6065 7951 


6202 7844 


6338 7735 


40 


21 


5786 8156 


5927 8054 


6067 7950 


6205 7842 


6341 7733 


39 


22 


5788 8155 


5930 8052 


6069 7948 


6207 7841 


6343 7731 


38 


23 


5790 8153 


5932 8051 


6071 7946 


6209 7839 


6345 7729 


37 


24 


5793 8151 


5934 8049 


6074 7944 


6211 7837 


6347 7727 


36 


25 


5795 8150 


5937 8047 


6076 7942 


6214 7835 


6350 7725 


35 


26 


5798 8148 


5939 8045 


6078 7941 


6216 7833 


6352 7724 


34 


27 


5800 8146 


5941 8044 


6081 7939 


6218 7832 


6354 7722 


33 


28 


5802 8145 


5944 8042 


6083 7937 


6221 7830 


6356 7720 


32 


29 


5805 8143 


5946 8040 


6085 7935 


6223 7828 


6359 7718 


31 


30 


5807 8141 


5948 8039 


6088 7934 


6225 7826 


6361 7716 


30 


31 


5809 8139 


5951 8037 


6090 7932 


6227 7824 


6363 7714 


29 


.32 


5812 8138 


5953 8035 


6092 7930 


6230 7822 


6365 7713 


28 


33 


5814 8136 


5955 8033 


6095 7928 


6232 7821 


6368 7711 


27 


34 


5816 8134 


5958 8032 


6097 7926 


6234 7819 


6370 7709 


26 


35 


5819 8133 


5960 8030 


6099 7925 


6237 7817 


6372 7707 


25 


36 


5821 8131 


5962 8028 


6101 7923 


6239 7815 


6374 7705 


24 


37 


5824 8129 


5965 8026 


6104 7921 


6241 7813 


6376 7703 


23 


38 


5826 8128 


5967 8025 


6106 7919 


6243 7812 


6379 7701 


22 


39 


5828 8126 


5969 8023 


6108 7918 


6246 7810 


6381 7700 


21 


4O 


5831 8124 


5972 8021 


61] 1 7916 


6248 7808 


6383 7698 


2O 


41 


5833 8123 


5974 8020 


6113 7914 


6250 7806 


6385 7696 


19 


42 


5835 8121 


5976 8018 


6115 7912 


6252 7804 


6388 7694 


18 


43 


5838 8119 


5979 8016 


6118 7910 


6255 7802 


6390 7692 


17 


44 


5840 8117 


5981 8014 


6120 7909 


6257 7801 


6392 7690 


16 


45 


5842 8116 


5983 8013 


6122 7907 


6259 7799 


6394 7688 


15 


46 


5845 8114 


5986 8011 


6124 7905 


6262 7797 


6397 7687 


14 


47 


5847 8112 


5988 8009 


6127 7903 


6264 7795 


6399 7685 


13 


48 


5850 8111 


5990 8007 


6129 7902 


6266 7793 


6401 7683 


12 


49 


5852 8109 


5993 8006 


6131 7900 


6268 7792 


6403 7681 


11 


50 


5854 8107 


5995 8004 


6134 7898 


6271 7790 


6406 7679 


10 


51 


5857 8106 


5997 8002 


6136 7896 


6273 7788 


6408 7677 


9 


52 


5859 8104 


6000 8000 


6138 7894 


6275 7786 


6410 7675 


8 


53 


5861 8102 


6002 7999 


6141 7893 


6277 7784 


6412 7674 


7 


54 


5864 8100 


6004 7997 


6143 7891 


6280 7782 


6414 7672 


6 


55 


5866 8099 


6007 7995 


6145 7889 


6282 7781 


6417 7670 


5 


56 


5868 8097 


6009 7993 


6147 7887 


6284 7779 


6419 7668 


4 


57 


5871 8095 


6011 7992 


6150 7885 


6286 7777 


6421 7666 


3 


58 


5873 8094 


6014 7990 


6152 7884 


6289 7775 


6423 7664 


2 


59 


5875 8092 


6016 7988 


6154 7882 


6291 7773 


6426 7662 


1 


6O 


5878 8090 


6018 7986 


6157 7880 


6293 7771 


6428 7660 


O 




cos sin 


cos sin 


cos sin 


cos sin 


cos sin 




t 


54 


53 


52 


51 


50 


t 



60 



NATURAL SIXES AND COSINES. 



/ 


4O 


41 


42 


43 


44 


f 




sin cos 


sin cos 


sin cos 


sin cos 


sin cos 




o 


6428 7660 


6561 7547 


6691 7431 


6820 7314 


6947 7193 


60 


1 


6430 7659 


6563 7545 


6693 7430 


6822 7312 


6949 7191 


59 


2 


6432 7657 


6565 7543 


6696 7428 


6824 7310 


6951 7189 


58 


3 


6435 7655 


6567 7541 


6698 7426 


6826 7308 


6953 7187 


57 


4 


6437 7653 


6569 7539 


6700 7424 


6828 7306 


6955 7185 


56 


5 


6439 7651 


6572 7538 


6702 7422 


6831 7304 


6957 7183 


55 


6 


6441 7649 


6574 7536 


6704 7420 


6833 7302 


6959 7181 


54 


7 


6443 7647 


6576 7534 


6706 7418 


6835 7300 


6961 7179 


53 


8 


6446 7645 


6578 7532 


6709 7416 


6837 7298 


6963 7177 


52 


9 


6448 7644 


6580 7530 


6711 7414 


6839 7296 


6965 7175 


51 


1O 


6450 7642 


6583 7528 


6713 7412 


6841 7294 


6967 7173 


5O 


11 


6452 7640 


6585 7526 


6715 7410 


6843 7292 


6970 7171 


49 


12 


6455 7638 


6587 7524 


6717 7408 


6845 7290 


6972 7169 


48 


13 


6457 7636 


6589 7522 


6719 7406 


6848 7288 


6974 7167 


47 


14 


6459 7634 


6591 7520 


6722 7404 


6850 7286 


6976 7165 


46 


15 


6461 7632 


6593 7518 


6724 7402 


6852 7284 


6978 7163 


45 


16 


6463 7630 


6596 7516 


6726 7400 


6854 7282 


6980 7161 


44 


17 


6466 7629 


6598 7515 


6728 7398 


6856 7280 


6982 7159 


43 


18 


6468 7627 


6600 7513 


6730 7396 


6858 7278 


6984 7157 


42 


19 


6470 7625 


6602 7511 


6732 7394 


6860 7276 


6986 7155 


41 


20 


6472 7623 


6604 7509 


6734 7392 


6862 7274 


6988 7153 


40 


21 


6475 7621 


6607 7507 


6737 7390 


6865 7272 


6990 7151 


39 


22 


6477 7619 


6609 7505 


6739 7388 


6867 7270 


6992 7149 


38 


23 


6479 7617 


6611 7503 


6741 7387 


6869 7268 


6995 7147 


37 


24 


6481 7615 


6613 7501 


6743 7385 


6871 7266 


6997 7145 


36 


25 


6483 7613 


6615 7499 


6745 7383 


6873 7264 


6999 7143 


35 


26 


6486 7612 


6617 7497 


6747 7381 


6875 7262 


7001 7141 


34 


27 


6488 7610 


6620 7495 


6749 7379 


6877 7260 


7003 7139 


33 


28 


6490 7608 


6622 7493 


6752 7377 


6879 7258 


7005 7137 


32 


29 


6492 7606 


6624 7491 


6754 7375 


6881 7256 


7007 7135 


31 


30 


6494 7604 


6626 7490 


6756 7373 


6884 7254 


7009 7133 


30 


31 


6497 7602 


6628 7488 


6758 7371 


6886 7252 


7011 7130 


29 


32 


6499 7600 


6631 7486 


6760 7369 


6888 7250 


7013 7128 


28 


33 


6501 7598 


6633 7484 


6762 7367 


6890 7248 


7015 7126 


27 


34 


6503 7596 


6635 7482 


6764 7365 


6892 7246 


7017 7124 


26 


35 


6506 7595 


6637 7480 


6767 7363 


6894 7244' 


7019 7122 


25 


36 


6508 7593 


6639 7478 


6769 7361 


6896 7242 


7022 7120 


24 


37 


6510 7591 


6641 7476 


6771 7359 


6898 7240 


7024 7118 


23 


38 


6512 7589 


6644 7474 


6773 7357 


6900 7238 


7026 7116 


22 


39 


6514 7587 


6646 7472 


6775 7355 


6903 7236 


7028 7114 


21 


4O 


6517 758S 


6648 7470 


6777 7353 


6905 7234 


7030 7112 


2O 


41 


6519 7583 


6650 7468 


6779 7351 


6907 7232 


7032 7110 


19 


42 


6521 7581 


6652 7466 


6782 7349 


6909 7230 


7034 7108 


18 


43 


6523 7579 


6654 7464 


6784 7347 


6911 7228 


7036 7106 


17 


44 


6525 7578 


6657 7463 


6786 7345 


6913 7226 


7038 7104 


16 


45 


6528 7576 


6659 7461 


6788 7343 


6915 7224 


7040 7102 


15 


46 


6530 7574 


6661 7459 


6790 7341 


6917 7222 


7042 7100 


14 


47 


6532 7572 


6663 7457 


6792 7339 


6919 7220 


7044 7098 


13 


48 


6534 7570 


6665 7455 


6794 7337 


6921 7218 


7046 7096 


12 


49 


6536 7568 


6667 7453 


6797 7335 


6924 7216 


7048 7094 


11 


5O 


6539 7566 


6670 7451 


6799 7333 


6926 7214 


7050 7092 


10 


51 


6541 7564 


6672 7449 


6801 7331 


6928 7212 


7053 7090 


9 


52 


6543 7562 


6674 7447 


6803 7329 


6930 7210 


7055 7088 


8 


53 


6545 7560 


6676 7445 


6805 7327 


6932 7208 


7057 7085 


7 


54 


6547 7559 


6678 7443 


6807 7325 


6934 7206 


7059 7083 


6 


55 


6550 7557 


6680 7441 


6809 7323 


6936 7203 


7061 7081 


5 


56 


6552 7555 


6683 7439 


6811 7321 


6938 7201 


7063 7079 


4 


57 


6554 7553 


6685 7437 


6814 7319 


6940 7199 


7065 7077 


3 


58 


6556 7551 


6687 7435 


6816 7318 


6942 7197 


7067 7075 


2 


59 


6558 7549 


6689 7433 


6818 7316 


6944 7195 


7069 7073 


1 


60 


6561 7547 


6691 7431 


6820 7314 


6947 7193 


7071 7071 


O 




cos sin 


cos sin 


cos sin 


cos sin 


cos sin 




t 


49 


48 


47 


46 


45 


f 



NATURAL TANGENTS AND COTANGENTS. 



61 



t 









1 




2 




3 




4 


t 




tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 







0000 


Infinite 


0175 


57.2900 


0349 


28.6363 


0524 


19.0811 


0699 


14.3007 


6O 


1 


0003 


3437.75 


0177 


56.3506 


0352 


28.3994 


0527 


18.9755 


0702 


14.2411 


59 


2 


0006 


1718.87 


0180 


55.4415 


0355 


28.1664 


0530 


188711 


0705 


14.1821 


58 


3 


0009 


1145.92 


0183 


54.5613 


0358 


27.9372 


0533 


18.7678 


0708 


14.1235 


57 


4 


0012 


859.436 


0186 53.7086 


0361 


27.7117 


0536 


18.6656 


0711 


14.0655 


56 


5 


0015 


687.549 


0189 


52.8821 


0364 


27.4899 


0539 


18.5645 


0714 


14.0079 


55 


6 


0017 


572.957 


0192 


52.0807 


0367 


27.2715 


0542 


18.4645 


0717 


13.9507 


54 


7 


0020 


491.106 


0195 


51.3032 


0370 


27.0566 


0544 


18.3655 


0720 


13.8940 


53 


8 


0023 


429.718 


0198 


50.5485 


0373 


26.8450 


0547 


18.2677 


0723 


13.8378 


52 


9 


0026 


381.971 


0201 


49.8157 


0375 


26.6367 


0550 


18.1708 


0726 


13.7821 


51 


1O 


0029 


343.774 


0204 


49.1039 


0378 


26.4316 


0553 


18.0750 


0729 


13.7267 


50 


11 


0032 


312.521 


0207 


48.4121 


0381 


26.2296 


0556 


17.9802 


0731 


13.6719 


49 


12 


0035 


286.478 


0209 


47.7395 


0384 


26.0307 


0559 


17.8863 


0734 


13.6174 


48 


13 


0038 


264.441 


0212 


47.0853 


0387 


25.8348 


0562 


17.7934 


0737 


13.5634 


47 


14 


0041 


245.552 


0215 


46.4489 


0390 


25.6418 


0565 


17.7015 


0740 


13.5098 


46 


15 


0044 


229.182 


0218 


45.8294 


0393 


25.4517 


0568 


17.6106 


0743 


13.4566 


45 


16 


0047 


214.858 


0221 


45.2261 


0396 


25.2644 


0571 


17.5205 


0746 


13.4039 


44 


17 


0049 


202.219 


0224 


44.6386 


0399 


25.0798 


0574 


17.4314 


0749 


13.3515 


43 


18 


0052 


190.984 


0227 


44.0661 


0402 


24.8978 


0577 


17.3432 


0752 


13.2996 


42 


19 


0055 


180.932 


0230 


43.5081 


0405 


24.7185 


0580 


17.2558 


0755 


13.2480 


41 


20 


0058 


171.885 


0233 


42.9641 


0407 


24.5418 


0582 


17.1693 


0758 


13.1969 


40 


21 


0061 


163.700 


0236 


42.4335 


0410 


24.3675 


0585 


17.0837 


0761 


13.1461 


39 


22 


0064 


156.259 


0239 


41.9158 


0413 


24.1957 


0588 


16.9990 


0764 


13.0958 


38 


23 


0067 


149.465 


0241 


41.4106 


0416 


24.0263 


0591 


16.9150 


0767 


13.0458 


37 


24 


0070 


143.237 


0244 


40.9174 


0419 


23.8593 


0594 


16.8319 


0769 


12.9962 


36 


25 


0073 


137.507 


0247 


40.4358 


0422 


23.6945 


0597 


16.7496 


0772 


12.9469 


35 


26 


0076 


132.219 


0250 


39.9655 


0425 


23.5321 


0600 


16.6681 


0775 


12.8981 


34 


27 


0079 


127.321 


0253 


39.5059 


0428 


23.3718 


0603 


16.5874 


0778 


12.8496 


33 


28 


0081 


122.774 


0256 


39.0568 


0431 


23.2137 


0606 


16.5075 


0781 


12.8014 


32 


29 


0084 


118.540 


0259 


38.6177 


0434 


23.0577 


0609 


16.4283 


0784 


12.7536 


31 


30 


0087 


114.589 


0262 


38.1885 


0437 


22.9038 


0612 


16.3499 


0787 


12.7062 


3O 


31 


0090 


110.892 


0265 


37.7686 


0440 


22.7519 


0615 


16.2722 


0790 


12.6591 


29 


32 


0093 


107.426 


0268 


37.3579 


0442 


22.6020 


0617 


16.1952 


0793 


12.6124 


28 


33 


0096 


104.171 


0271 


36.9560 


0445 


22.4541 


0620 


16.1190 


0796 


12.5660 


27 


34 


0099 


101.107 


0274 


36.5627 


0448 


22.3081 


0623 


16.0435 


0799 


12.5199 


26 


35 


0102 


98.2179 


0276 


36.1776 


0451 


22.1640 


0626 


15.9687 


0802 


12.4742 


25 


36 


0105 


95.4895 


0279 


358006 


0454 


22.0217 


0629 


15.8945 


0805 


12.4288 


24 


37 


0108 


92.9085 


0282 


35.4313 


0457 


21.8813 


0632 


15.8211 


0808 


12.3838 


23 


38 


0111 


90.4633 


0285 


35.0695 


0460 


21.7426 


0635 


15.7483 


0810 


12.3390 


22 


39 


0113 


88.1436 


0288 


34.7151 


0463 


21.6056 


0638 


15.6762 


0813 


12.2946 


21 


4O 


0116 


85.9398 


0291 


34.3678 


0466 


21.4704 


0641 


15.6048 


0816 


12.2505 


20 


41 


0119 


83.8435 


0294 


34.0273 


0469 


21.3369 


0644 


15.5340 


0819 


12.2067 


19 


42 


0122 


81.8470 


0297 


33.6935 


0472 


21.2049 


0647 


15.4638 


0822 


12.1632 


18 


43 


0125 


79.9434 


0300 


33.3662 


0475 


21.0747 


0650 


15.3943 


0825 


12.1201 


17 


44 


0128 


78.1263 


0303 


33.0452 


0477 


20.9460 


0653 


15.3254 


0828 


12.0772 


16 


45 


0131 


76.3900 


0306 


32.7303 


0480 


20.8188 


0655 


15.2571 


0831 


12.0346 


15 


46 


0134 


74.7292 


0308 


32.4213 


0483 


20.6932 


0658 


15.1893 


0834 


11.9923 


14 


47 


0137 


73.1390 


0311 


32.1181 


0486 


20.5691 


0661 


15.1222 


0837 


11.9504 


13 


48 


0140 


71.6151 


0314 


31.8205 


0489 


20.4465 


0664 


15.0557 


0840 


11.9087 


12 


49 


0143 


70.1533 


0317 


31.5284 


0492 


20.3253 


0667 


14.9898 


0843 


11.8673 


11 


5O 


0146 


68.7501 


0320 


31.2416 


0495 


20.2056 


0670 


14.9244 


0846 


11.8262 


1O 


51 


0148 


67.4019 


0323 


30.9599 


0498 


20.0872 


0673 


14.8596 


0849 


11.7853 


9 


52 


0151 


66.1055 


0326 


30.6833 


0501 


19.9702 


0676 


14.7954 


0851 


11.7448 


8 


53 


0154 


64.8580 


0329 


30.4116 


0504 


19.8546 


0679 


14.7317 


0854 


11.7045 


7 


54 


0157 


63.6567 


0332 


30.1446 


0507 


19.7403 


0682 


14.6685 


0857 


11.6645 


6 


55 


0160 


62.4992 


0335 


29.8823 


0509 


19.6273 


0685 


14.6059 


0860 


11.6248 


5 


56 


0163 


61.3829 


0338 


29.6245 


0512 


19.5156 


0688 


14.5438 


0863 


11.5853 


4 


57 


0166 


60.3058 


0340 


29.3711 


0515 


19.4051 


0690 


14.4823 


0866 


11.5461 


3 


58 


0169 


59.2659 


0343 


29.1220 


0518 


19.2959 


0693 


14.4212 


0869 


11.5072 


2 


59 


0172 


58.2612 


0346 


28.8771 


0521 


19.1879 


0696 


14.3607 


0872 


11.4685 


1 


60 


0175 


57.2900 


0349 


28.6363 


0524 


19.0811 


0699 


14.3007 


0875 


11.4301 







cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 




f 


89 


88 


87 


86 


85 


/ 



62 



NATURAL TANGENTS AND COTANGENTS. 



t 


5 


6 


7 


8 


9 


t 




tan cot 


tan cot 


tan cot 


tan cot 


tan cot 




o 


0875 11.4301 


1051 9.5144 


1228 8.1443 


1405 7.1154 


1584 6.3138 


6O 


1 


0878 11.3919 


1054 9.4878 


1231 8.1248 


1408 7.1004 


1587 6.3019 


59 


2 


0881 11.3540 


1057 9.4614 


1234 8.1054 


1411 7.0855 


1590 6.2901 


58 


3 


0884 11.3163 


1060 9.4352 


1237 8.0860 


1414 7.0706 


1593 6.2783 


57 


4 


0887 11.2789 


1063 9.4090 


1240 8.0667 


1417 7.0558 


1596 6.2666 


56 


5 


0890 11.2417 


1066 9.3831 


1243 8.0476 


1420 7.0410 


1599 6.2549 


55 


6 


0892 11.2048 


1069 9.3572 


1246 8.0285 


1423 7.0264 


1602 6.2432 


54 


7 


0895 11.1681 


1072 9.3315 


1249 8.0095 


1426 7.0117 


1605 6.2316 


53 


8 


0898 11,1316 


1075 9.3060 


1251 7.9906 


1429 6.9972 


1608 6.2200 


52 


9 


0901 11.0954 


1078 9.2806 


1254 7.9718 


1432 6.9827 


1611 6.2085 


51 


10 


0904 11.0594 


1080 9.2553 


1257 7.9530 


1435 6.9682 


1614 6.1970 


50 


11 


0907 11.0237 


1083 9.2302 


1260 7.9344 


1438 6.9538 


1617 6.1856 


49 


12 


0910 10.9882 


1086 9.2052 


1263 7.9158 


1441 6.9395 


1620 6.1742 


48 


13 


0913 10.9529 


1089 9.1803 


1266 7.8973 


1444 6.9252 


1623 6.1628 


47 


14 


0916 10.9178 


1092 9.1555 


1269 7.8789 


1447 6.9110 


1626 6.1515 


46 


15 


0919 10.8829 


1095 9.1309 


1272 7.8606 


1450 6.8969 


1629 6.1402 


45 


16 


0922 10.8483 


1098 9.1065 


1275 7.8424 


1453 6.8828 


1632 6.1290 


44 


17 


0925 10.8139 


1101 9.0821 


1278 7.8243 


1456 6.8687 


1635 6.1178 


43 


18 


0928 10.7797 


1104 9.0579 


1281 7.8062 


1459 6.8548 


1638 6.1066 


42 


19 


0931 10.7457 


1107 9.0338 


1284 7.7883 


1462 6.8408 


1641 6.0955 


41 


2O 


0934 10.7119 


1110 9.0098 


1287 7.7704 


1465 6.8269 


1644 6.0844 


4O 


21 


0936 10.6783 


1113 8.9860 


1290 7.7525 


1468 6.8131 


1647 6.0734 


39 


22 


0939 10.6450 


1116 8.9623 


1293 7.7348 


1471 6.7994 


1650 6.0624 


38 


23 


0942 10.6118 


1119 8.9387 


1296 7.7171 


1474 6.7856 


1653 6.0514 


37 


24 


0945 10.5789 


1122 8.9152 


1299 7.6996 


1477 6.7720 


1655 6.0405 


36 


25 


0948 10.5462 


1125 88919 


1302 7.6821 


1480 6.7584 


1658 6.0296 


35 


26 


0951 10.5136 


1128 8.8686 


1305 7.6647 


1483 6.7448 


1661 6.0188 


34 


27 


0954 10.4813 


1131 88455 


1308 7.6473 


1486 6.7313 


1664 6.0080 


33 


28 


0957 10.4491 


1134 8.8225 


1311 7.6301 


1489 6.7179 


1667 5.9972 


32 


29 


0960 10.4172 


1136 8.7996 


1314 7.6129 


1492 6.7045 


1670 5.9865 


31 


30 


0963 10.3854 


1139 8.7769 


1317 7.5958 


1495 6.6912 


1673 5.9758 


30 


31 


0966 10.3538 


1142 8.7542 


1319 7.5787 


1497 6.6779 


1676 5.9651 


29 


32 


0969 10.3224 


1145 8.7317 


1322 7.5618 


1500 6.6646 


1679 5.9545 


28 


33 


0972 10.2913 


1148 8.7093 


1325 7.5449 


1503 6.6514 


1682 5.9439 


27 


34 


0975 10.2602 


1151 8.6870 


1328 7.5281 


1506 6.6383 


1685 5.9333 


26 


35 


0978 10.2294 


1154 8.6648 


1331 7.5113 


1509 6.6252 


1688 5.9228 


25 


36 


0981 10.1988 


1157 8.6427 


1334. 7.4947 


1512 6.6122 


1691 5.9124 


24 


37 


0983 10.1683 


1160 8.6208 


1337 7.4781 


1515 6.5992 


1694 5.9019 


23 


38 


0986 10.1381 


1163 8.5989 


1340 7.4615 


1518 6.5863 


1697 5.8915 


22 


39 


0989 10.1080 


1166 8.5772 


1343 7.4451 


1521 6.5734 


1700 5.8811 


21 


40 


0992 10.0780 


1169 8.5555 


1346 7.4287 


1524 6.5606 


1703 5.8708 


2O 


41 


0995 10.0483 


1172 8.5340 


1349 7.4124 


1527 6.5478 


1706, 5.8605 


19 


42 


0998 10.0187 


1175 8.5126 


1352 7.3962 


1530 6.5350 


1709 5.8502 


18 


43 


1001 9.9893 


1178 8.4913 


1355 7.3800 


1533 6.5223 


1712 5.8400 


17 


44 


1004 9.9601 


1181 8.4701 


1358 7.3639 


1536 6.5097 


1715 5.8298 


16 


45 


1007 9.9310 


1184 8.4490 


1361 7.3479 


1539 6.4971 


1718 5.8197 


15 


46 


1010 9.9021 


1187 8.4280 


1364 7.3319 


1542 6.4846 


1721 5.8095 


14 


47 


1013 9.8734 


1189 8.4071 


1367 7.3160 


1545 6.4721 


1724 5.7994 


13 


48 


1016 9.8448 


1192 8.3863 


1370 7.3002 


1548 6.4596 


1727 5.7894 


12 


49 


1019 9.8164 


1195 8.3656 


1373 7.2844 


1551 6.4472 


1730 5.7794 


11 


5O 


1022 9.7882 


1198 8.3450 


1376 7.2687 


1554 6.4348 


1733 5.7694 


1O 


51 


1025 9.7601 


1201 8.3245 


1379 7.2531 


1557 6.4225 


1736 5.7594 


9 


52 


1028 9.7322 


1204 8.3041 


1382 7.2375 


1560 6.4103 


1739 5.7495 


8 


53 


1030 9.7044 


1207 8.2838 


1385 7.2220 


1563 6.3980 


1742 5.7396 


7 


54 


1033 9.6768 


1210 8.2636 


1388 7.2066 


1566 6.3859 


1745 5.7297 


6 


55 


1036 9.6499 


1213 8.2434 


1391 7.1912 


1569 6.3737 


1748 5.7199 


5 


56 


1039 9.6220 


1216 8.2234 


1394 7.1759 


1572 6.3617 


1751 5.7101 


4 


57 


1042 9.5949 


1219 8.2035 


1397 7.1607 


1575 6.3496 


1754 5.7004 


3 


58 


1045 9.5679 


1222 8.1837 


1399 7.1455 


1578 6.3376 


1757 5.6906 


2 


59 


1048 9.5411 


1225 8.1640 


1402 7.1304 


1581 6.3257 


1760 5.6809 


1 


60 


1051 9.5144 


1228 8.1443 


1405 7.1154 


1584 6.3138 


1763 5.6713 







cot tan 


cot tan 


cot tan 


cot tan 


cot tan 




f 


84 


83 


82 


81 


8O 


t 



NATURAL TANGENTS AND COTANGENTS. 



63 



1 


1O 


11 


12 


13 


14 


t 




tan cot 


tan cot 


tan cot 


tan cot 


tan cot 




o 


1763 5.6713 


1944 5.1446 


2126 4.7046 


2309 4.3315 


2493 4.0108 


60 


1 


1766 5.6617 


1947 5.1366 


2129 4.6979 


2312 4.3257 


2496 4.0058 


59 


2 


1769 5.6521 


1950 5.1286 


2132 4.6912 


2315 4.3200 


2499 4.0009 


58 


3 


1772 5.6425 


1953 5.1207 


2135 4.6845 


2318 4.3143 


2503 3.9959 


57 


4 


1775 5.6330 


1956 5.1128 


2138 4.6779 


2321 4.3086 


2506 3.9910 


56 


5 


1778 5.6234 


1959 5.1049 


2141 4.6712 


2324 4.3029 


2509 3.9861 


55 


6 


1781 5.6140 


1962 5.0970 


2144 4.6646 


2327 4.2972 


2512 3.9812 


54 


7 


1784 5.6045 


1965 5.0892 


2147 4.6580 


2330 4.2916 


2515 3.9763 


53 


8 


1787 5.5951 


1968 5.0814 


2150 4.6514 


2333 4.2859 


2518 3.9714 


52 


9 


1790 5.5857 


1971 5.0736 


2153 4.6448 


2336 4.2803 


2521 3.9665 


51 


10 


1793 5.5764 


1974 5.0658 


2156 4.6382 


2339 4.2747 


2524 3.9617 


5O 


11 


1796 5.5671 


1977 5.0581 


2159 4.6317 


2342 4.2691 


2527 3.9568 


49 


12 


1799 5.5578 


1980 5.0504 


2162 4.6252 


2345 4.2635 


2530 3.9520 


48 


13 


1802 5.5485 


1983 5.0427 


2165 4.6187 


2349 4.2580 


2533 3.9471 


47 


14 


1805 5.5393 


1986 5.0350 


2168 4.6122 


2352 4.2524 


2537 3.9423 


46 


15 


1808 5.5301 


1989 5.0273 


2171 4.6057 


2355 4.2468 


2540 3.9375 


45 


16 


1811 5.5209 


1992 5.0197 


2174 4.5993 


2358 4.2413 


2543 3.9327 


44 


17 


1814 5.5118 


1995 5.0121 


2177 4.5928 


2361 4.2358 


2546 3.9279 


43 


18 


1817 5.5026 


1998 5.0045 


2180 4.5864 


2364 4.2303 


2549 3.9232 


42 


19 


1820 5.4936 


2001 4.9969 


2183 4.5800 


2367 4.2248 


2552 3.9184 


41 


2O 


1823 5.4845 


2004 4.9894 


2186 4.5736 


2370 4.2193 


2555 3.9136 


4O 


21 


1826 5.4755 


2007 4.9819 


2189 4.5673 


2373 4.2139 


2558 3.9089 


39 


22 


1829 5.4665 


2010 4.9744 


2193 4.5609 


2376 4.2084 


2561 3.9042 


38 


23 


1832 5.4575 


2013 4.9669 


2196 4.5546 


2379 4.2030 


2564 3.8995 


37 


24 


1835 5.4486 


2016 4.9594 


2199 4.5483 


2382 4.1976 


2568 3.8947 


36 


25 


1838 5.4397 


2019 4.9520 


2202 4.5420 


2385 4.1922 


2571 3.8900 


35 


26 


1841 5.4308 


2022 4.9446 


2205 4.5357 


2388 4.1868 


2574 3.8854 


34 


27 


1844 5.4219 


2025 4.9372 


2208 4.5294 


2392 4.1814 


2577 3.8807 


33 


28 


1847 5.4131 


2028 4.9298 


2211 4.5232 


2395 4.1760 


2580 3.8760 


32 


29 


1850 5.4043 


2031 4.9225 


2214 4.5169 


2398 4.1706 


2583 3.8714 


31 


30 


1853 5.3955 


2035 4.9152 


2217 4.5107 


2401 4.1653 


2586 3.8667 


30 


31 


1856 5.3868 


2038 4.9078 


2220 4.5045 


2404 4.1600 


2589 3.8621 


29 


32 


1859 5.3781 


2941 4.9006 


2223 4.4983 


2407 4.1547 


2592 3.8575 


28 


33 


1862 5.3694 


2044 4.8933 


2226 4.4922 


2410 4.1493 


2595 3.8528 


27 


34 


1865 5.3607 


2047 4.8860 


2229 4.4860 


2413 4.1441 


2599 3.8482 


26 


35 


1868 5.3521 


2050 4.8788 


2232 4.4799 


2416 4.1388 


2602 3.8436 


25 


36 


1871 5.3435 


2053 4.8716 


2235 4.4737 


2419 4.1335 


2605 3.8391 


24 


37 


1874 5.3349 


2056 4.8644 


2238 4.4676 


2422 4.1282 


2608 3.8345 


23 


38 


1877 5.3263 


2059 4.8573 


2241 4.4615 


2425 4.1230 


2611 3.8299 


22 


39 


1880 5.3178 


2062 4.8501 


2244 4.4555 


2428 4.1178 


2614 3.8254 


21 


40 


1883 5.3093 


2065 4.8430 


2247 4.4494 


2432 4.1126 


2617 3.8208 


20 


41 


1887 5.3008 


2068 4.8359 


2251 4.4434 


2435 4.1074 


2620 3.8163 


19 


42 


1890 5.2924 


2071 4.8288 


2254 4.4374 


2438 4.1022 


2623 3.8118 


18 


43 


1893 5.2839 


2074 4.8218 


2257 4.4313 


2441 4.0970 


2627 3.8073 


17 


44 


1896 5.2755 


2077 4.8147 


2260 4.4253 


2444 4.0918 


2630 3.8028 


16 


45 


1899 5.2672 


2080 4.8077 


2263 4.4194 


2447 4.0867 


2633 3.7983 


15 


46 


1902 5.2588 


2083 4.8007 


2266 4.4134 


2450 4.0815 


2636 3.7938 


14 


47 


1905 5.2505 


2086 4.7937 


2269 4.4075 


2453 4.0764 


2639 3.7893 


13 


48 


1908 5.2422 


2089 4.7867 


2272 4.4015 


2456 4.0713 


2642 3.7848 


12 


49 


1911 5.2339 


2092 4.7798 


2275 4.3956 


2459 4.0662 


2645 3.7804 


11 


50 


1914 5.2257 


2095 4.7729 


2278 4.3897 


2462 4.0611 


2648 3.7760 


10 


51 


1917 5.2174 


2098 4.7659 


2281 4.3838 


2465 4.0560 


2651 3.7715 


9 


52 


1920 5.2092 


2101 4.7591 


2284 4.3779 


2469 4.0509 


2655 3.7671 


8 


53 


1923 5.2011 


2104 4.7522 


2287 4.3721 


2472 4.0459 


2658 3.7627 


7 


54 


1926 5.1929 


2107 4.7453 


2290 4.3662 


2475 4.0408 


2661 3.7583 


6 


55 


1929 5.1848 


2110 4.7385 


2293 4.3604 


2478 4.0358 


2664 3.7539 


5 


56 


1932 5.1767 


2113 4.7317 


2296 4.3546 


2481 4.0308 


2667 3.7495 


4 


57 


1935 5.1686 


2116 4.7249 


2299 4.3488 


2484 4.0257 


2670 3.7451 


3 


58 


1938 5.1606 


2119 4.7181 


2303 4.3430 


2487 4.0207 


2673 3.7408 


2 


59 


1941 5.1526 


2123 4.7114 


2306 4.3372 


2490 4.0158 


2676 3.7364 


1 


60 


1944 5.1446 


2126 4.7046 


2309 4.3315 


2493 4.0108 


2679 3.7321 


O 




cot tan 


cot tan 


cot tan 


cot tan 


cot tan 




f 


79 


78 


77 


76 


75 


i 



NATURAL TANGENTS AND COTANGENTS. 



f 


15 


16 


17 


18 


19 


r 




tan cot 


tan cot 


tan cot 


tan cot 


tan cot 




o 


2679 3.7321 


2867 3.4874 


3057 3.2709 


3249 3.0777 


3443 2.9042 


6O 


1 


2683 3.7277 


2871 3.4836 


3060 3.2675 


3252 3.0746 


3447 2.9015 


59 


2 


2686 3.7234 


2874 3.4798 


3064 3.2641 


3256 3.0716 


3450 2.8987 


58 


3 


2689 3.7191 


2877 3.4760 


3067 3.2607 


3259 3.0686 


3453 2.8960 


57 


4 


2692 3.7148 


2880 3.4722 


3070 3.2573 


3262 3.0655 


3456 2.8933 


56 


5 


2695 3.7105 


2883 3.4684 


3073 3.2539 


3265 3.0625 


3460 2.8905 


55 


6 


2698 3.7062 


2886 3.4646 


3076 3.2506 


3269 3.0595 


3463 2.8878 


54 


7 


2701 3.7019 


2890 3.4608 


3080 3.2472 


3272 3.0565 


3466 2.8851 


53 


8 


2704 3.6975 


2893 3.4570 


3083 3.2438 


3275 3.0535 


3469 2.8824 


52 


9 


2708 3.6933 


2896 3.4533 


3086 3.2405 


3278 3.0505 


3473 2.8797 


51 


10 


2711 3.6891 


2899 3.4495 


3089 3.2371 


3281 3.0475 


3476 2.8770 


50 


11 


2714 3.6848 


2902 3.4458 


3092 3.2338 


3285 3.0445 


3479 2.8743 


49 


12 


2717 3.6806 


2905 3.4420 


3096 3.2305 


3288 3.0415 


3482 2.8716 


48 


13 


2720 3.6764 


2908 3.4383 


3099 3.2272 


3291 3.0385 


3486 2.8689 


47 


14 


2723 3.6722 


2912 3.4346 


3102 3.2238 


3294 3.0356 


3489 2.8662 . 


46 


15 


2726 3.6680 


2915 3.4308 


3105 3.2205 


3298 3.0326 


3492 2.8636 


45 


16 


2729 3.6638 


2918 3.4271 


3108 3.2172 


3301 3.0296 


3495 2.8609 


44 


17 


2733 3.6596 


2921 3.4234 


3111 3.2139 


3304 3.0267 


3499 2.8582 


43 


18 


2736 3.6554 


2924 3.4197 


3115 3.2106 


3307 3.0237 


3502 2.8556 


42 


19 


2739 3.6512 


2927 3.4160 


3118 3.2073 


3310 3.0208 


3505 2.8529 


41 


2O 


2742 3.6470 


2931 3.4124 


3121 3.2041 


3314 3.0178 


3508 2.8502 


4O 


21 


2745 3.6429 


2934 3.4087 


3124 3.2008 


3317 3.0149 


3512 2.8476 


39 


22 


2748 3.6387 


2937 3.4050 


3127 3.1975 


3320 3.0120 


3515 2.8449 


38 


23 


2751 3.6346 


2940 3.4014 


3131 3.1943 


3323 3.0090 


3518 2.8423 


37 


24 


2754 3.6305 


2943 3.3977 


3134 3.1910 


3327 3.0061 


3522 2.8397 


36 


25 


2758 3.6264 


2946 3.3941 


3137 3.1878 


3330 3.0032 


3525 2.8370 


35 


26 


2761 3.6222 


2949 3.3904 


3140 3.1845 


3333 3.0003 


3528 2.8344 


34 


27 


2764 3.6181 


2953 3.3868 


3143 3.1813 


3336 2.9974 


3531 2.8318 


33 


28 


2767 3.6140 


2956 3.3832 


3147 3.1780 


3339 2.9945 


3535 2.8291 


32 


29 


2770 3.6100 


2959 3.3796 


3150 3.1748 


3343 29916 


3538 2.8265 


31 


30 


2773 3.6059 


2962 3.3759 


3153 3.1716 


3346 2.9887 


3541 2.8239 


3O 


31 


2776 3.6018 


2965 3.3723 


3156 3.1684 


3349 2.9858 


3544 2.8213 


29 


32 


2780 3.5978 


2968 3.3687 


3159 3.1652 


3352 2.9829 


3548 2.8187 


28 


33 


2783 3.5937 


2972 3.3652 


3163 3.1620 


3356 2,9800 


3551 2.8161 


27 


34 


2786 3.5897 


2975 3.3616 


3166 3.1588 


3359 2.9772 


3554 2.8135 


26 


35 


2789 3.5856 


2978 3.3580 


3169 3.1556 


3362 2.9743 


3558 2.8109 


25 


36 


2792 3.5816 


2981 3.3544 


3172 3.1524 


3365 2.9714 


3561 2.8083 


24 


37 


2795 3.5776 


2984 3.3509 


3175 3.1492 


3369 2.9686 


3564 2.8057 


23 


38 


2798 3.5736 


2987 3.3473 


3179 3.1460 


3372 2.9657 


3567 2.8032 


22 


39 


2801 3.5696 


2991 3.3438 


3182 3.1429 


3375 2.9629 


3571 2.8006 


21 


40 


2805 3.5656 


2994 3.3402 


3185 3.1397 


3378 2.9600 


3574 2.7980 


2O 


41 


2808 3.5616 


2997 3.3367 


3188 3.1366 


3382 2.9572 


3577 2.7955 


19 


42 


2811 3.5576 


3000 3.3332 


3191 3.1334 


3385 2.9544 


3581 2.7929 


18 


43 


2814 3.5536 


3003 3.3297 


3195 3.1303 


3388 2.9515 


3584 2.7903 


17 


44 


2817 3.5497 


3006 3.3261 


3198 3.1271 


3391 2.9487 


3587 2.7878 


16 


45 


2820 3.5457 


3010 3.3226 


3201 3.1240 


3395 2.9459 


3590 2.7852 


15 


46 


2823 3.5418 


3013 3.3191 


3204 3.1209 


3398 2.9431 


3594 2.7827 


14 


47 


2827 3.5379 


3016 3.3156 


3207 3.1178 


3401 2.9403 


3597 2.7801 


13 


48 


2830 3.5339 


3019 3.3122 


3211 3.1146 


3404 2.9375 


3600 2.7776 


12 


49 


2833 3.5300 


3022 3.3087 


3214 3.1115 


3408 2.9347 


3604 2.7751 


11 


50 


2836 3.5261 


3026 3.3052 


3217 3.1084 


3411 2.9319 


3607 2.7725 


10 


51 


2839 3.5222 


3029 3.3017 


3220 3.1053 


3414 2.9291 


3610 2.7700 


9 


52 


2842 3.5183 


3032 3.2983 


3223 3.1022 


3417 2.9263 


3613 2.7675 


8 


53 


2845 3.5144 


3035 3.2948 


3227 3.0991 


3421 2.9235 


3617 2.7650 


7 


54 


2849 3.5105 


3038 3.2914 


3230 3.0961 


3424 2.9208 


3620 2.7625 


6 


55 


2852 3.5067 


3041 3.2880 


3233 3.0930 


3427 2.9180 


3623 2.7600 


5 


56 


2855 3.5028 


3045 3.2845 


3236 3.0899 


3430 2.9152 


3627 2.7575 


4 


57 


2858 3.4989 


3048 3.2811 


3240 3.0868 


3434 2.9125 


3630 2.7550 


3 


58 


2861 3.4951 


3051 3.2777 


3243 3.0838 


3437 2.9097 


3633 2.7525 


2 


59 


2864 3.4912 


3054 3.2743 


3246 3.0807 


3440 2.9070 


3636 2.7500 


1 


60 


2867 3.4874 


3057 3.2709 


3249 3.0777 


3443 2.9042 


3640 2.7475 


O 




cot tan 


cot tan 


cot tan 


cot tan 


cot tan 




t 


74 


73 


72 


71 


70 


9 



NATURAL TANGENTS AND COTANGENTS. 



65 



t 


2O 


21 


22 


23 


24 


t 




tan cot 


tan cot 


tan cot 


tan cot 


tan cot 




o 


3640 2.7475 


3839 2.6051 


4040 2.4751 


4245 2.3559 


4452 2.2460 


6O 


1 


3643 2.7450 


3842 2.6028 


4044 2.4730 


4248 2.3539 


4456 2.2443 


59 


2 


3646 2.7425 


3845 2.6006 


4047 2.4709 


4252 2.3520 


4459 2.2425 


58 


3 


3650 2.7400 


3849 2.5983 


4050 2.4689 


4255 2.3501 


4463 2.2408 


57 


4 


3653 2.7376 


3852* 2.5961 


4054 2.4668 


4258 2.3483 


4466 2.2390 


56 


5 


3656 2.7351 


3855 2.5938 


4057 2.4648 


4262 2.3464 


4470 2.2373 


55 


6 


3659 2.7326 


3859 2.5916 


4061 2.4627 


4265 2.3445 


4473 2.2355 


54 


7 


3663 2.7302 


3862 2.5893 


4064 2.4606 


4269 2.3426 


4477 2.2338 


53 


8 


3666 2.7277 


3865 2.5871 


4067 2.4586 


4272 2.3407 


4480 2.2320 


52 


9 


3669 2.7253 


3869 2.5848 


4071 2.4566 


4276 2.3388 


4484 2.2303 


51 


1C 


3673 2.7228 


3872 2.5826 


4074 2.4545 


4279 2.3369 


4487 2.2286 


50 


11 


3676 2.7204 


3875 2.5864 


4078 2.4525 


4283 2.3351 


4491 2.2268 


49 


12 


3679 2.7179 


3879 2.5782 


4081 2.4504 


4286 2.3332 


4494 2.2251 


48 


13 


3683 2.7155 


3882 2.5759 


4084 2.4484 


4289 2.3313 


4498 2.2234 


47 


14 


3686 2.7130 


3885 2.5737 


4088 2.4464 


4293 2.3294 


4501 2.2216 


46 


15 


3689 2.7106 


3889 2.5715 


4091 2.4443 


4296 2.3276 


4505 2.2199 


45 


16 


3693 2.7082 


3892 2.5693 


4095 2.4423 


4300 2.3257 


4508 2.2182 


44 


17 


3696 2.7058 


3895 2.5671 


4098 2.4403 


4303 2.3238 


4512 2.2165 


43 


18 


3699 2.7034 


3899 2.5649 


4101 2.4383 


4307 2.3220 


4515 2.2148 


42 


19 


3702 2.7009 


3902 2.5627 


4105 2.4362 


4310 2.3201 


4519 2.2130 


41 


2O 


3706 2.6985 


3906 2.5605 


4108 2.4342 


4314 2.3183 


4522 2.2113 


40 


21 


3709 2.6961 


3909 2.5533 


4111 2.4322 


4317 2.3164 


4526 2.2096 


39 


22 


3712 2.6937 


3912 2.5561 


4115 2.4302 


4320 2.3146 


4529 2.2079 


38 


23 


3716 2.6913 


3916 2.5539 


4118 2.4282 


4324 2.3127 


4533 2.2062 


37 


24 


3719 2.6889 


3919 2.5517 


4122 2.4262 


4327 2.3109 


4536 2.2045 


36 


25 


3722 2.6865 


3922 2.5495 


4125 2.4242 


4331 2.3090 


4540 2.2028 


35 


26 


3726 2.6841 


3926 2.5473 


4129 2.4222 


4334 2.3072 


4543 2.2011 


34 


27 


3729 2.6818 


3929 2.5452 


4132 2.4202 


4338 2.3053 


4547 2.1994 


33 


28 


3732 2.6794 


3932 2.5430 


4135 2.4182 


4341 2.3035 


4550 2.1977 


32 


29 


3736 2.6770 


3936 2.5408 


4139 2.4162 


4345 2.3017 


4554 2.1960 


31 


3O 


3739 2.6746 


3939 2.5386 


4142 2.4142 


4348 2.2998 


4557 2.1943 


3O 


31 


3742 2.6723 


3942 2.5365 


4146 2.4122 


4352 2.2980 


4561 2.1926 


29 


32 


3745 2.6699 


3946 2.5343 


4149 2.4102 


4355 2.2962 


4564 2.1909 


28 


33 


3749 2.6675 


3949 2.5322 


4152 2.4083 


4359 2.2944 


4568 2.1892 


27 


34 


3752 . 2.6652 


3953 2.5300 


4156 2.4063 


4362 2.2925 


4571 2.1876 


26 


35 


3755 2.6628 


3956 2.5279 


4159 2.4043 


4365 2.2907 


4575 2.1859 


25 


36 


3759 2.6605 


3959 2.5257 


4163 2.4023 


4369 2.2889 


4578 2.1842 


24 


37 


3762 2.6581 


3963 2.5236 


4166 2.4004 


4372 2.2871 


4582 2.1825 


23 


38 


3765 2.6558 


3966 2.5214 


4169 2.3984 


4376 2.2853 


4585 2.1808 


22 


39 


3769 2.6534 


3969 2.5193 


4173 2.3964 


4379 2.2835 


4589 2.1792 


21 


40 


3772 2.6511 


3973 2.5172 


4176 2.3945 


4383 2.2817 


4592 2.1775 


2O 


41 


3775 2.6488 


3976 2.5150 


4180 2.3925 


4386 2.2799 


4596 2.1758 


19 


42 


3779 2.6464 


3979 2.5129 


4183 2.3906 


4390 2.2781 


4599 2.1742 


18 


43 


3782 2.6441 


3983 2.5108 


4187 2.3886 


4393 2.2763 


4603 2.1725 


17 


44 


3785 2.6418 


3986 2.5086 


4190 2.3867 


4397 2.2745 


4607 2.1708 


16 


45 


3789 2.6395 


3990 2.5065 


4193 2.3847 


4400 2.2727 


4610 2.1692 


15 


46 


3792 2.6371 


3993 2.5044 


4197 2.3828 


4404 2.2709 


4614 2.1675 


14 


47 


3795 2.6348 


3996 2.5023 


4200 2.3808 


4407 2.2691 


4617 2.1659 


13 


48 


3799 2.6325 


4000 2.5002 


4204 2.3789 


4411 2.2673 


4621 2.1642 


12 


49 


3802 2.6302 


4003 2.4981 


4207 2.3770 


4414 2.2655 


4624 2.1625 


11 


5O 


3805 2.6279 


4006 2.4960 


4210 2.3750 


4417 2.2637 


4628 2.1609 


1O 


51 


3809 2.6256 


4010 2.4939 


4214 2.3731 


4421 2.2620 


4631 2.1592 


9 


52 


3812 2.6233 


4013 2.4918 


4217 2.3712 


4424 2.2602 


4635 2.1576 


8 


53 


3815 2.6210 


4017 2.4897 


4221 2.3693 


4428 2.2584 


4638 2.1560 


7 


54 


3819 2.6187 


4020 2.4876 


4224 2.3673 


4431 2.2566 


4642 2.1543 


6 


55 


3822 2.6165 


4023 2.4855 


4228 2.3654 


4435 2.2549 


4645 2.1527 


5 


56 


3825 26142 


4027 2.4834 


4231 2.3635 


4438 2.2531 


4649 2.1510 


4 


57 


3829 2.6119 


4030 2.4813 


4234 2.3616 


4442 2.2513 


4652 2.1494 


3 


58 


3832 2.6096 


4033 2.4792 


4238 2.3597 


4445 2.2496 


4656 2.1478 


2 


59 


3835 2.6074 


4037 2.4772 


4241 2.3578 


4449 2.2478 


4660 2.1461 


1 


6O 


3839 2.6051 


4040 2.4751 


4245 2.3559 


4452 2.2460 


4663 2.1445 







cot tan 


cot tan 


cot tan 


cot tan 


cot tan 




t 


69 


68 


67 


66 


65 


t 



NATURAL TANGENTS AND COTANGENTS. 



f 


25 


26 


27 


28 


29 


T 




tan cot 


tan cot 


tan cot 


tan cot 


tan cot 




o 


46t>3 2.1445 


4877 2.0503 


5095 1.9626 


5317 1.8807 


5543 1.8040 


6O 


1 


4667 2.1429 


4881 2.0488 


5099 1.9612 


5321 1.8794 


5547 1.8028 


59 


2 


4670 2.1413 


4885 2.0473 


5103 1.9598 


5325 1.8781 


5551 1.8016 


58 


3 


4674 2.1396 


4888 2.0458 


5106 1.9584 


5328 1,8768 


5555 1.8003 


57 


4 


4677 2.1380 


4892 2.0443 


5110 1.9570 


5332 1.8755 


5558 1.7991 


56 


5 


4681 2.1364 


4895 2.0428 


5114 1.9556 


5336 1.8741 


5562 1.7979 


55 


6 


4684 2.1348 


4899 2.0413 


5117 1.9542 


5340 1.8728 


5566 1.7966 


54 


7 


4688 2.1332 


4903 2.0398 


5121 1.9528 


5343 1.8715 


5570 1.7954 


53 


8 


4691 2.1315 


4906 2.0383 


5125 1.9514 


5347 1.8702 


5574 1.7942 


52 


9 


4695 2.1299 


4910 2.0368 


5128 1.9500 


5351 1.8689 


5577 1.7930 


51 


1C 


4699 2.1283 


4913 2.0353 


5132 1.9486 


5354 1.8676 


5581 1.7917 


50 


11 


4702 2.1267 


4917 2.0338 


5136 1.9472 


5358 1.8663 


5585 1.7905 


49 


12 


4706 2.1251 


4921 2.0323 


5139 1.9458 


5362 1.8650 


5589 1.7893 


48 


13 


4709 2.1235 


4924 2.0308 


5143 1.9444 


5366 1.8637 


5593 1.7881 


47 


14 


4713 2.1219 


4928 2.0293 


5147 1.9430 


5369 1.8624 


5596 1.7868 


46 


15 


4716 2.1203 


4931 2.0278 


5150 1.9416 


5373 1.8611 


5600 1.7856 


45 


16 


4720 2.1187 


4935 2.0263 


5154 1.9402 


5377 1.8598 


5604 1.7844 


44 


17 


4723 2.1171 


4939 2.0248 


5158 1.9388 


5381 1.8585 


5608 1.7832 


43 


18 


4727 2.1155 


4942 2.0233 


5161 1.9375 


5384 1.8572 


5612 1.7820 


42 


19 


4731 2.1139 


4946 2.0219 


5165 1.9361 


5388 1.8559 


5616 1.7808 


41 


20 


4734 2.1123 


4950 2.0204 


5169 1.9347 


5392 1.8546 


5619 1.7796 


4O 


21 


4738 2.1107 


4953 2.0189 


5172 1.9333 


5396 1.8533 


5623 1.7783 


39 


22 


4741 2.1092 


4957 2.0174 


5176 1.9319 


5399 1.8520 


5627 1.7771 


38 


23 


4745 2.1076 


4960 2.0160 


5180 1.9306 


5403 1.8507 


5631 1.7759 


37 


24 


4748 2.1060 


4964 2.0145 


5184 1.9292 


5407 1.8495 


5635 1.7747 


36 


25 


4752 2.1044 


496S 2.0130 


5187 1.9278 


5411 1.8482 


5639 1.7735 


35 


26 


4755 2.1028 


4971 2.0115 


5191 1.9265 


5415 1.8469 


5642 1.7723 


34 


27 


4759 2.1013 


4975 2.0101 


5195 1.9251 


5418 1.8456 


5646 1.7711 


33 


28 


4763 2.0997 


4979 2.0086 


5198 1.9237 


5422 1.8443 


5650 1.7699 


32 


29 


4766 2.0981 


4982 2.0072 


5202 1.9223 


5426 1.8430 


5654 1.7687 


31 


30 


4770 2.0965 


4986 2.0057 


5206 1.9210 


5430 1.8418 


5658 1.7675 


3O 


31 


4773 2.0950 


4989 2.0042 


5209 1.9196 


5433 1.8405 


5662 1.7663 


29 


32 


4777 2.0934 


4993 2.0028 


5213 1.9183 


5437 1.8392 


5665 1.7651 


28 


33 


4780 2.0918 


4997 2.0013 


5217 1.9169 


5441 1.8379 


5669 1.7639 


27 


34 


4784 2.0903 


5000 1.9999 


5220 1.9155 


5445 1.8367 


5673 1.7627 


26 


35 


4788 2.0887 


5004 1.9984 


5224 1.9142 


5448 1.8354 


5677 1.7615 


25 


36 


4791 2.0872 


5008 1.9970 


5228 1.9128 


5452 1.8341 


5681 1.7603 


24 


37 


4795 2.0856 


5011 1.9955 


5232 1.9115 


5456 1.8329 


5685 1.7591 


23 


38 


4798 2.0840 


5015 1.9941 


5235 1.9101 


5460 1.8316 


5688 1.7579 


22 


39 


4802 2.0825 


5019 1.9926 


5239 1.9088 


5464 1.8303 


5692 1.7567 


21 


4O 


4806 2.0809 


5022 1.9912 


5243 1.9074 


5467 1.8291 


5696 1.7556 


20 


41 


4809 2.0794 


5026 1.9897 


5246 1.9061 


5471 1.8278 


5700 1.7544 


19 


42 


4813 2.0778 


5029 1.9883 


5250 1.9047 


5475 1.8265 


5704 1.7532 


18 


43 


4816 2.0763 


5033 1.9868 


5254 1.9034 


5479 1.8253 


5708 1.7520 


17 


44 


4820 2.0748 


5037 1.9854 


5258 1.9020 


5482 1.8240 


5712 1.7508 


16 


45 


4823 2.0732 


5040 1.9840 


5261 1.9007 


5486 1.8228 


5715 1.7496 


15 


46 


4827 2.0717 


5044 1.9825 


5265 1.8993 


5490 1.8215 


5719 1.7485 


14 


47 


4831 2.0701 


5048 1.9811 


5269 1.8980 


5494 1.8202 


5723 1.7473 


13 


48 


4834 2.0686 


5051 1.9797 


5272 1.8967 


5498 1.8190 


5727 1.7461 


12 


49 


4838 2.0671 


5055 1.9782 


5276 1.8953 


5501 1.8177 


5731 1.7449 


11 


5O 


4841 2.0655 


5059 1.9768 


5280 1.8940 


5505 1.8165 


5735 1.7437 


10 


51 


4845 2.0640 


5062 1.9754 


5284 1.8927 


5509 1.8152 


5739 1.7426 


9 


52 


4849 2.0625 


5066 1.9740 


5287 1.8913 


5513 1.8140 


5743 1.7414 


8 


53 


4852 2.0609 


5070 1.9725 


5291 1.8900 


5517 1.8127 


5746 1.7402 


7 


54 


4856 2.0594 


5073 1.9711 


5295 1.8887 


5520 1.8115 


5750 1.7391 


6 


55 


4859 2.0579 


5077 1.9697 


5298 1.8873 


5524 1.8103 


5754 1.7379 


5 


56 


4863 20564 


5081 1.9683 


5302 1.8860 


5528 1.8090 


5758 1.7367 


4 


57 


4867 2.0549 


5084 1.9669 


5306 1.8847 


5532 1.8078 


5762 1.7355 


3 


58 


4870 2.0533 


5088 1.9654 


5310 1.8834 


5535 1.8065 


5766 1.7344 


2 


59 


4874 2.0518 


5092 1.9640 


5313 1.8820 


5539 1.8053 


5770 1.7332 


1 


60 


,4877 2.0503 


5095 1.9626 


5317 1.8807 


5543 1.8040 


5774 1.7321 


O 




cot tan 


cot tan 


cot tan 


cot tan 


cot tan 




f 


64 


63 


62 


61 


6O 


t 



NATURAL TANGENTS AND COTANGENTS. 



67 



t 


3O 


31 


32 


33 


34 


t 




tan cot 


tan cot 


tan cot 


tan cot 


tan cot 




o 


5774 1.7321 


6009 1.6643 


6249 1.6003 


6494 1.5399 


6745 1.4826 


6O 


1 


5777 1.7309 


6013 1.6632 


6253 1.5993 


6498 1.5389 


6749 1.4816 


59 


2 


5781 1.7297 


6017 1.6621 


6257 1.5983 


6502 1.5379 


6754 1.4807 


58 


3 


5785 1.7286 


6020 1.6610 


6261 1.5972 


6506 1.5369 


6758 1.4798 


57 


4 


5789 1.7274 


6024 1.6599 


6265 1.5962 


6511 1.5359 


6762 1.4788 


56 


5 


5793 1.7262 


6028 1.6588 


6269 1.5952 


6515 1.5350 


6766 1.4779 


55 


6 


5797 1.7251 


6032 1.6577 


6273 1.5941 


6519 1.5340 


6771 1.4770 


54 


7 


5801 1.7239 


6036 1.6566 


6277 1.5931 


6523 1.5330 


6775 1.4761 


53 


8 


5805 1.7228 


6040 1.6555 


6281 1.5921 


6527 1.5320 


6779 1.4751 


52 


9 


5808 1.7216 


6044 1.6545 


6285 1.5911 


6531 1.5311 


6783 1.4742 


51 


1O 


5812 1.7205 


604S 1.6534 


6289 1.5900 


6536 1.5301 


6787 1.4733 


5O 


11 


5816 1.7193 


6052 1.6523 


6293 1.5890 


6540 1.5291 


6792 1.4724 


49 


12 


5820 1.7182 


6056 1.6512 


6297 1.5880 


6544 1.5282 


6796 1.4715 


48 


13 


5824 1.7170 


6060 1.6501 


6301 1.5869 


6548 1.5272 


6800 1.4705 


47 


14 


5828 1.7159 


6064 1.6490 


6305 1.5859 


6552 1.5262 


6805 1.4696 


46 


15 


5832 1.7147 


6068 1.6479 


6310 1.5849 


6556 1.5253 


6809 1.4687 


45 


16 


5836 1.7136 


6072 1.6469 


6314 1.5839 


6560 1.5243 


6813 1.4678 


44 


17 


5840 1.7124 


6076 1.6458 


6318 1.5829 


6565 1.5233 


6817 1.4669 


43 


18 


5844 1.7113 


6080 1.6447 


6322 1.5818 


6569 1.5224 


6822 1.4659 


42 


19 


5847 1.7102 


6084 1.6436 


6326 1.5808 


6573 1.5214 


6826 1.4650 


41 


20 


5851 1.7090 


6088 1.6426 


6330 1.5798 


6577 1.5204 


6830 1.4641 


4O 


21 


5855 1.7079 


6092 1.6415 


6334 1.5788 


6581 1.5195 


6834 1.4632 


39 


22 


5859 1.7067 


6096 1.6404 


6338 1.5778 


6585 1.5185 


6839 1.4623 


38 


23 


5863 1.7056 


6100 1.6393 


6342 1.5768 


6590 1.5175 


6843 1.4614 


37 


24 


5867 1.7045 


6104 1.6383 


6346 1.5757 


6594 1.5166 


6847 1.4605 


36 


25 


5871 1.7033 


6108 1.6372 


6350 1.5747 


6598 .1.5156 


6851 1.4596 


35 


26 


5875 1.7022 


6112 1.6361 


6354 1.5737 


6602 1.5147 


6856 1.4586 


34 


27 


5879 1.7011 


6116 1.6351 


6358 1.5727 


6606 1.5137 


6860 1.4577 


33 


28 


5883 1.6999 


6120 1.6340 


6363 1.5717 


6610 1.5127 


6864 1.4568 


32 


29 


5887 1.6988 


6124 1.6329 


6367 1.5707 


6615 1.5118 


6869 1.4559 


31 


30 


5890 1.6977 


6128 1.6319 


6371 1.5697 


6619 1.5108 


6873 1.4550 


30 


31 


5894 1.6965 


6132 1.6308 


6375 1.5687 


6623 1.5099 


6877 1.4541 


29 


32 


5898 1.6954 


6136 1.6297 


6379 1.5677 


6627 1.5089 


6881 1.4532 


28 


33 


5902 1.6943 


6140 1.6287 


6383 1.5667 


6631 1.5080 


6886 1.4523 


27 


34 


5906 1.6932 


6144 1.6276 


6387 1.5657 


6636 1.5070 


6890 1.4514 


26 


35 


5910 1.6920 


6148 1.6265 


6391 1.5647 


6640 1.5061 


6894 1.4505 


25 


36 


5914 1.6909 


6152 1.6255 


6395 1.5637 


6644 1.5051 


6899 1.4496 


24 


37 


5918 1.6898 


6156 1.6244 


6399 1.5627 


6648 1.5042 


6903 1.4487 


23 


38 


5922 1.6887 


6160 1.6234 


6403 1.5617 


6652 1.5032 


6907 1.4478 


22 


39 


5926 1.6875 


6164 1.6223 


6408 1.5607 


6657 1.5023 


6911 1.4469 


21 


4O 


5930 1.6864 


6168 1.6212 


6412 1.5597 


6661 1.5013 


6916 1.4460 


20 


41 


5934 1.6853 


6172 1.6202 


6416 1.5587 


6665 1.5004 


6920 1.4451 


19 


42 


5938 1.6842 


6176 1.6191 


6420 1.5577 


6669 1.4994 


6924 1.4442 


18 


43 


5942 1.6831 


6180 1.6181 


6424 1.5567 


6673 1.4985 


6929 1.4433 


17 


44 


5945 1.6820 


6184 1.6170 


6428 1.5557 


6678 1.4975 


6933 1.4424 


16 


45 


5949 1.6808 


6188 1.6160 


6432 1.5547 


6682 1.4966 


6937 1.4415 


15 


46 


5953 1.6797 


6192 1.6149 


6436 1.5537 


6686 1.4957 


6942 1.4406 


14 


47 


5957 1.6786 


6196 1.6139 


6440 1.5527 


6690 1.4947 


6946 1.4397 


13 


48 


5961 1.6775 


6200 1.6128 


6445 1.5517 


6694 1.4938 


6950 1.4388 


12 


49 


5965 1.6764 


6204 1.6118 


6449 1.5507 


6699 1.4928 


6954 1.4379 


11 


50 


5969 1.6753 


6208 1.6107 


6453 1.5497 


6703 1.4919 


6959 1.4370 


10 


51 


5973 1.6742 


6212 1.6097 


6457 1.5487 


6707 1.4910 


6963 1.4361 


9 


52 


5977 1.6731 


6216 1.6087 


6461 1.5477 


6711 1.4900 


6967 1.4352 


8 


53 


5981 1.6720 


6220 1.6076 


6465 1.5468 


6716 1.4891 


6972 1.4344 


7 


54 


5985 1.6709 


6224 1.6066 


6469 1.5458 


6720 1.4882 


6976 1.4335 


6 


55 


5989 1.6698 


6228 1.6055 


6473 1.5448 


6724 1.4872 


6980 1.4326 


5 


56 


5993 1.6687 


6233 1.6045 


6478 1.5438 


6728 1.4863 


6985 1.4317 


4 


57 


5997 1.6676 


6237 1.6034 


6482 1.5428 


6732 1.4854 


6989 1.4308 


3 


58 


6001 1.6665 


6241 1.6024 


6486 1.5418 


6737 1.4844 


6993 1.4299 


2 


59 


6005 1.6654 


6245 1.6014 


6490 1.5408 


6741 1.4835 


6998 1.4290 


1 


60 


6009 1.6643 


6249 1.6003 


6494 1.5399 


6745 1.4826 


7002 1.4281 







cot tan 


cot tan 


cot tan 


cot tan 


cot tan 




t 


59 


68 


57 


56 


55 


t 



68 



NATURAL TANGENTS AND COTANGENTS. 



f 


35 


36 


37 


38 


39 


t 




tan cot 


tan cot 


tan cot 


tan cot 


tan cot 







7002 1.4281 


7265 1.3764 


7536 1.3270 


7813 1.2799 


8098 1.2349 


6O 


1 


7006 1.4273 


7270 1.3755 


7540 1.3262 


7818 1.2792 


8103 1.2342 


59 


2 


7011 1.4264 


7274 1.3747 


7545 1.3254 


7822 1.2784 


8107 1.2334 


58 


3 


7015 1.4255 


7279 1.3739 


7549 1.3246 


7827 1.2776 


8112 1.2327 


57 


4 


7019 1.4246 


7283 1.3730 


7554 1.3238 


7832 1.2769 


8117 1.2320 


56 


5 


7024 1.4237 


7288 1.3722 


7558 1.3230 


7836 1.2761 


8122 1.2312 


55 


6 


7028 1.4229 


7292 1.3713 


7563 1.3222 


7841 1.2753 


8127 1.2305 


54 


7 


7032 1.4220 


7297 1.3705 


7568 1.3214 


7846 1.2746 


8132 1.2298 


53 


8 


7037 1.4211 


7301 1.3697 


7572 1.3206 


7850 1.2738 


8136 1.2290 


52 


9 


7041 1.4202 


7306 1.3688 


7577 1.3198 


7855 1.2731 


8141 1.2283 


51 


1O 


7046 1.4193 


7310 1.3680 


7581 1.3190 


7860 1.2723 


8146 1.2276 


50 


11 


7050 1.4185 


7314 1.3672 


7586 1.3182 


7865 1.2715 


8151 1.2268 


49 


12 


7054 1.4176 


7319 1.3663 


7590 1.3175 


7869 1.2708 


8156 1.2261 . 


48 


13 


7059 1.4167 


7323 1.3655 


7595 1.3167 


7874 1.2700 


8161 1.2254 


47 


14 


7063 1.4158 


7328 1.3647 


7600 1.3159 


7879 1.2693 


8165 1.2247 


46 


15 


7067 1.4150 


7332 1.3638 


7604 1.3151 


7883 1.2685 


8170 1.2239 


45 


16 


7072 1.4141 


7337 1.3630 


7609 1.3143 


7888 1.2677 


8175 1.2232 


44 


17 


7076 1.4132 


7341 1.3622 


7613 1.3135 


7893 1.2670 


8180 1.2225 


43 


18 


7080 1.4124 


7346 1.3613 


7618 1.3127 


7898 1.2662 


8185 1.2218 


42 


19 


7085 1.4115 


7350 1.3605 


7623 1.3119 


7902 1.2655 


8190 1.2210 


41 


2O 


7089 1.4106 


7355 1.3597 


7627 1.3111 


7907 1.2647 


8195 1.2203 


4O 


21 


7094 1.4097 


7359 1.3588 


7632 1.3103 


7912 1.2640 


8199 1.2196 


39 


22 


7098 1.4089 


7364 1.3580 


7636 1.3095 


7916 1.2632 


8204 1.2189 


38 


23 


7102 1.4080 


7368 1.3572 


7641 1.3087 


7921 1.2624 


8209 1.2181 


37 


24 


7107 1.4071 


7373 1.3564 


7646 1.3079 


7926 1.2617 


8214 1.2174 


36 


25 


7111 1.4063 


7377 1.3555 


7650 1.3072 


7931 1.2609 


8219 1.2167 


35 


26 


'7115 1.4054 


7382 1.3547 


7655 1.3064 


7935 1.2602 


8224 1.2160 


34 


27 


7120 1.4045 


7386 1.3539 


7659 1.3056 


7940 1.2594 


8229 1.2153 


33 


28 


7124 1.4037 


7391 1.3531 


7664 1.3048 


7945 1.2587 


8234 1.2145 


32 


29 


7129 1.4028 


7395 1.3522 


7669 1.3040 


7950 1.2579 


8238 1.2138 


31 


30 


7133 1.4019 


7400 1.3514 


7673 1.3032 


7954 1.2572 


8243 1.2131 


30 


31 


7137 1.4011 


7404 1.3506 


7678 1.3024 


7959 1.2564 


8248 1.2124 


29 


32 


7142 1.4002 


7409 1.3498 


7683 1.3017 


7964 1.2557 


8253 1.2117 


28 


33 


7146 1.3994 


7413 1.3490 


7687 1.3009 


7969 1.2549 


8258 1.2109 


27 


34 


7151 1.3985 


7418 1.3481 


7692 1.3001 


7973 1.2542 


8263 1.2102 


26 


35 


7155 1.3976 


7422 1.3473 


7696 1.2993 


7978 1.2534 


8268 1.2095 


25 


36 


7159 1.3968 


7427 1.3465 


7701 1.2985 


7983 1.2527 


8273 1.2088 


24 


37 


7164 1.3959 


7431 1.3457 


7706 1.2977 


7988 1.2519 


8278 1.2081 


23 


38 


7168 1.3951 


7436 1.3449 


7710 1.2970 


7992 1.2512 


8283 1.2074 


22 


39 


7173 1.3942 


7440 1.3440 


7715 1.2962 


7997 1.2504 


8287 1.2066 


21 


4O 


7177 1.3934 


7445 1.3432 


7720 1.2954 


8002 1.2497 


8292 1.2059 


2O 


41 


7181 1.3925 


7449 1.3424 


7724 1.2946 


8007 1.2489 


8297 1.2052 


19 


42 


7186 1.3916 


7454 1.3416 


7729 1.2938 


8012 1.2482 


8302 1.2045 


18 


43 


7190 1.3908 


7458 1.3408 


7734 1.2931 


8016 1.2475 


8307 1.2038 


17 


44 


7195 1.3899 


7463 1.3400 


7738 1.2923 


8021 1.2467 


8312 1.2031 


16 


45 


7199 1.3891 


7467 1.3392 


7743 1.2915 


8026 1.2460 


8317 1.2024 


15 


46 


7203 1.3882 


7472 1.3384 


7747 1.2907 


8031 1.2452 


8322 1.2017 


14 


47 


7208 1.3874 


7476 1.3375 


7752 1.2900 


8035 1.2445 


8327 1.2009 


13 


48 


7212 1.3865 


7481 1.3367 


7757 1.2892 


8040 1.2437 


8332 1.2002 


12 


49 


7217 1.3857 


7485 1.3359 


7761 1.2884 


8045 1.2430 


8337 1.1995 


11 


5O 


7221 1.3848 


7490 1.3351 


7766 1.2876 


8050 1.2423 


8342 1.1988 


10 


51 


7226 1.3840 


7495 1.3343 


7771 1.2869 


8055 1.2415 


8346 1.1981 


9 


52 


7230 1.3831 


7499 1.3335 


7775 1.2861 


8059 1.2408 


8351 1.1974 


8 


53 


7234 1.3823 


7504 1.3327 


7780 1.2853 


8064 1.2401 


8356 1.1967 


7 


54 


7239 1.3814 


7508 1.3319 


7785 1.2846 


8069 1.2393 


8361 1.1960 


6 


55 


7243 1.3806 


7513 1.3311 


7789 1.2838 


8074 1.2386 


8366 1.1953 


5 


56 


7248 1.3798 


7517 1.3303 


7794 1.2830 


8079 1.2378 


8371 .1.1946 


4 


57 


7252 1.3789 


7522 1.3295 


7799 1.2822 


8083 1.2371 


8376 1.1939 


3 


58 


7257 1.3781 


7526 1.3287 


7803 1.2815 


8088 1.2364 


8381 1.1932 


2 


59 


7261 1.3772 


7531 1.3278 


7808 1.2807 


8093 1.2356 


8386 1.1925 


1 


6O 


7265 1.3764 


7536 1.3270 


7813 1.2799 


8098 1.2349 


8391 1.1918 


O 




cot tan 


cot tan 


cot tan 


cot tan 


cot tan 




t 


54 


53 


52 


51 


5O 


r 



NATURAL TANGENTS AND COTANGENTS. 



69 



t 


40 


41 


42 


43 


44 


t 




tan cot 


tan cot 


tan cot 


tan cot 


tan cot 




o 


8391 1.1918 


8693 1.1504 


9004 1.1106 


9325 1.0724 


9657 1.0355 


60 


1 


8396 1.1910 


8698 1.1497 


9009 1.1100 


9331 1.0717 


9663 1.0349 


59 


2 


8401 1.1903 


8703 1.1490 


9015 1.1093 


9336 1.0711 


9668 1.0343 


58 


3 


8406 1.1896 


8708 1.1483 


9020 1.1087 


9341 1.0705 


9674 1.0337 


57 


4 


8411 1.1889 


8713 1.1477 


9025 1.1080 


9347 1.0699 


9679 1.0331 


56 


5 


8416 1.1882 


8718 1.1470 


9030 1.1074 


9352 1.0692 


9685 1.0325 


55 


6 


8421 1.1875 


8724 1.1463 


9036 1.1067 


9358 1.0686 


9691 1.0319 


54 


7 


8426 1.1868 


8729 1.1456 


9041 1.1061 


9363 1.0680 


9696 1.0313 


53 


8 


8431 1.1861 


8734 1.1450 


9046 1.1054 


9369 1.0674 


9702 1.0307 


52 


9 


8436 1.1854 


8739 1.1443 


9052 1.1048 


9374 1.0668 


9708 1.0301 


51 


10 


8441 1.1847 


8744 1.1436 


9057 1.1041 


9380 1.0661 


9713 1.0295 


50 


11 


8446 1.1840 


8749 1.1430 


9062 1.1035 


9385 1.0655 


9719 1.0289 


49 


12 


8451 1.1833 


8754 1.1423 


9067 1.1028 


9391 1.0649 


9725 1.0283 


48 


13 


8456 1.1826 


8759 1.1416 


9073 1.1022 


9396 1.0643 


9730 1.0277 


47 


14 


8461 1.1819 


8765 1.1410 


9078 1.1016 


9402 1.0637 


9736 1.0271 


46 


15 


8466 1.1812 


8770 1.1403 


9083 1.1009 


9407 1.0630 


9742 1.0265 


45 


16 


8471 1.1806 


8775 1.1396 


9089 1.1003 


9413 1.0624 


9747 1.0259 


44 


17 


8476 1.1799 


8780 1.1389 


9094 1.0996 


9418 1.0618 


9753 1.0253 


43 


18 


8481 1.1792 


8785 1.1383. 


9099 1.0990 


9424 1.0612 


9759 1.0247 


42 


19 


8486 1.1785 


8790 1.1376 


9105 1.0983 


9429 1.0606 


9764 1.0241 


41 


2O 


8491 1.1778 


8796 1.1369 


9110 1.0977 


9435 1.0599 


9770 1.0235 


40 


21 


8496 1.1771 


8801 1.1363 


9115 1.0971 


9440 1.0593 


9776 1.0230 


39 


22 


8501 1.1764 


8806 1.1356 


9121 1.0964 


9446 1.0587 


9781 1.0224 


38 


23 


8506 1.1757 


8811 1.1349 


9126 1.0958 


9451 1.0581 


9787 1.0218 


37 


24 


8511 1.1750 


8816 1.1343 


9131 1.0951 


9457 1.0575 


9793 1.0212 


36 


25 


8516 1.1743 


8821 1.1336 


9137 1.0945 


9462 1.0569 


9798 1.0206 


35 


26 


8521 1.1736 


8827 1.1329 


9142 1.0939 


9468 1.0562 


9804 1.0200 


34 


27 


8526 1.1729 


8832 1.1323 


9147 1.0932 


9473 1.0556 


9810 1.0194 


33 


28 


8531 1.1722 


8837 M316 


9153 1.0926 


9479 1.0550 


9816 1.0188- 


32 


29 


8536 1.1715 


8842 1.1310 


9158 1.0919 


9484 1.0544 


9821 1.0182 


31 


3D 


8541 1.1708 


8847 1.1303 


9163 1.0913 


9490 1.0538 


9827 1.0176 


30 


31 


8546 1.1702 


8852 1.1296 


9169 1.0907 


9495 1.0532 


9833 1.0170 


29 


32 


8551 1.1695 


8858 1.1290 


9174 1.0900 


9501 1.0526 


9838 1.0164 


28 


33 


8556 1.1688 


8863 1.1283 


9179 1.0894 


9506 1.0519 


9844 1.0158 


27 


34 


8561 1.1681 


8868 1.1276 


9185 1.0888 


9512 1.0513 


9850 1.0152 


26 


35 


8566 1.1674 


8873 1.1270 


9190 1.0881 


9517 1.0507 


9856 1.0147 


25 


36 


8571 1.1667 


8878 1.1263 


9195 1.0875 


9523 1.0501 


9861 1.0141 


24 


37 


8576 1.1660 


8884 1.1257 


9201 1.0869 


9528 1.0495 


9867 1.0135 


23 


38 


8581 1.1653 


8889 1.1250 


9206 1.0862 


9534 1.0489 


9873 1.0129 


22 


39 


8586 1.1647 


8894 1.1243 


9212 1.0856 


9540 1.0483 


9879 1.0123 


21 


4O 


8591 1.1640 


8899 1.1237 


9217 1.0850 


9545 1.0477 


9884 1.0117 


2O 


41 


8596 1.1633 


8904 1.1230 


9222 1.0843 


9551 1.0470 


9890 1.0111 


19 


42 


8601 1.1626 


8910 1.1224 


9228 1.0837 


9556 1.0464 


9896 1.0105 


18 


43 


8606 1.1619 


8915 1.1217 


9233 1.0831 


9562 1.0458 


9902 1.0099 


17 


44 


8611 1.1612 


8920 1.1211 


9239 1.0824 


9567 1.0452 


9907 1.0094 


16 


45 


8617 1.1606 


8925 1.1204 


9244 1.0818 


9573 1.0446 


9913 1.0088 


15 


46 


8622 1.1599 


8931 1.1197 


9249 1.0812 


9578 1.0440 


9919 1.0082 


14 


47 


8627 1.1592 


8936 1.1191 


9255 1.0805 


9584 1.0434 


9925 1.0076 


13 


48 


8632 1.1585 


8941 1.1184 


9260 .1.0799 


9590 1.0428 


9930 1.0070 


12 


49 


8637 1.1578 


8946 1.1178 


9266 1.0793 


9595 1.0422 


9936 1.0064 


11 


50 


8642 1.1571 


8952 1.1171 


9271 1.0786 


9601 1.0416 


9942 1.0058 


1O 


51 


8647 1.1565 


8957 1.1165 


9276 1.0780 


9606 1.0410 


9948 1.0052 


9 


52 


8652 1.1558 


8962 1.1158 


9282 1.0774 


9612 1.0404 


9954 1.0047 


8 


53 


8657 1.1551 


8967 1.1152 


9287 1.0768 


9618 1.0398 


9959 1.0041 


7 


54 


8662 1.1544 


8972 1.1145 


9293 1.0761 


9623 1.0392 


9965 1.0035 


6 


55 


8667 1.1538 


8978 1.1139 


9298 1.0755 


9629 1.0385 


9971 1.0029 


5 


56 


8672 1.1531 


8983 1.1132 


9303 1.0749 


9634 1.0379 


9977 1.0023 


4 


57 


, 8678 1.1524 


8988 1.1126 


9309 1.0742 


9640 1.0373 


9983 1.0017 


3 


58 


8683 1.1517 


8994 1.1119 


9314 1.0736 


9646 1.0367 


9988 1.0012 


2 


59 


8688 1.1510 


8999 1.1113 


9320 1.0730 


9651 1.0361 


9994 1.0006 


1 


60 


8693 1.1504 


9004 1.1106 


9325 1.0724 


9657 1.0355 


1.000 1.0000 


O 




cot tan 


cot tan 


cot tan 


cot tan 


cot tan 




t 


49 


48 


47 


46 


45 


/ 



70 



TABLE VII.-TRAVEKSE TABLE. 



Bearing, 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing. 


f 


Lat, Dep, 


Lat. Dep, 


Lat, Dep, 


Lat, Dep, 


Lat, Dep, 


o / 


CIS 


1.000 0.004 


2.000 0.009 


3.000 0.013 


4.000 0.017 


5.000 0.022 


8945 


30 


1.000 0.009 


2.000 0.017 


3.000 0.026 


4.000 0.035 


5.000 0.044 


30 


45 


1.000 0.013 


2.000 0.026 


3.000 0.039 


4.000 0.052 


5.000 0.065 


15 


1 


1.000 0.017 


2.000 0.035 


3.000 0.052 


3.999 0.070 


4.999 0.087 


89 


15 


1.000 0.022 


2.000 0.044 


2.999 0.065 


3.999 0.087 


4.999 0.109 


45 


30 


1.000 0.026 


1.999 0.052 


2.999 0.079 


3.999 0.105 


4.998 0.131 


30 


45 


1.000 0.031 


1.999 0.061 


2.999 0.092 


3.998 0.122 


4.998 0.153 


15 


2 


0.999 0.035 


1.999 0.070 


2.998 0.105 


3.998 0.140 


4.997 0.174 


88 


15 


0.999 0.039 


1.998 0.079 


2.998 0.118 


3.997 0.157 


4.996 0.196 


45 


30 


0.999 0.044 


1.998 0.087 


2.997 0.131 


3.996 0.174 


4.995 0.218 


30 


45 


0.999 0.048 


1.998 0.096 


2.997 0.144 


3.995 0.192 


4.994 0.240 


15 


3 


0.999 0.052 


1.997 0.105 


2.996 0.157 


3.995 0.209 


4.993 0.262 


87 


15 


0.998 0.057 


1.997 0.113 


2.995 0.170 


3.994 0.227 


4.992 0283 


45 


30 


0.998 0.061 


1.996 0.122 


2.994 0.183 


3.993 0.244 


4.991 0.305 


30 


45 


0.998 0.065 


1.996 0.131 


2.994 0.196 


3.991 0.262 


4.989 0.327 


15 


40 


0.998 0070 


1.995 0.140 


2.993 0.209 


3.990 0.279 


4.988 0349 


86 


15 


0.997 0.074 


1.995 0.148 


2.992 0.222 


3.989 0.296 


4.986 0.371 


45 


30 


0.997 0.078 


1.994 0.157 


2.991 0.235 


3.988 0.314 


4.985 0.392 


30 


45 


0.997 0.083 


1.993 0.166 


2.990 0.248 


3.986 0.331 


4.983 0.414 


15 


5 


0.996 0.087 


1.992 0.174 


2.989 0.261 


3.985 0.349 


4.981 0.436 


85 


15 


0.996 0.092 


1.992 0.183 


2.987 0.275 


3.983 0.366 


4.979 0.458 


45 


30 


0.995 0096 


1.991 0192 


2.986 0.288 


3.982 0.383 


4.977 0.479 


30 


45 


0.995 0.100 


1.990 0.200 


2.985 0.301 


3.980 0.401 


4.975 0.501 


15 


6 


0.995 0.105 


1.989 0209 


2.984 0.314 


3.978 0.418 


4.973 0.523 


84 


15 


0.994 0.109 


1.988 0.218 


2.982 0.327 


3.976 0.435 


4.970 0.544 


45 


30 


0.994 0.113 


1.987 0.226 


2.981 0.340 


3.974 0.453 


4.968 0.566 


30 


45 


0.993 0.118 


1.986 0.235 


2.979 0.353 


3.972 0.470 


4.965 0.588 


15 


7 0. 


0.993 0.122 


1.985 0.244 


2.978 0.366 


3.970 0.487 


4.963 0.609 


83 


15 


0.992 0.126 


1.984 0.252 


2.976 0.379 


3.968 0.505 


4.960 0.631 


45 


30 


0.991 0.131 


1.983 0.261 


2.974 0.392 


3.966 0.522 


4.957 0.653 


30 


45 


0.991 0.135 


1.982 0.270 


2.973 0.405 


3.963 0.539 


4.954 0.674 


15 


8 


0.990 0.139 


1.981 0.278 


2.971 0.418 


3.961 0.557 


4.951 0.696 


82 


15 


0.990 0.143 


1.979 0.287 


2.969 0.430 


3.959 0.574 


4.948 0.717 


45 


30 


0.989 0.148 


1.978 0.296 


2.967 0.443 


3.956 0.591 


4.945 0.739 


30 


45 


0.988 0.152 


1.977 0.304 


2.965 0.456 


3.953 0.608 


4.942 0.761 


15 


9 


0.988 0.156 


1.975 0.313 


2.963 0.469 


3.951 0.626 


4.938 0.782 


81 


15 


0.987 0.161 


1.974 0.321 


2.961 0.482 


3.948 0.643 


4.935 0.804 


45 


30 


0.986 0.165 


1.973 0.330 


2.959 0.495 


3.945 0.660 


4.931 0.825 


30 


45 


0.986 0.169 


1.971 0.339 


2.957 0.508 


3.942 0.677 


4.928 0.847 


15 


1O 


0.985 0.174 


1.970 0.347 


2.954 0.521 


3.939 0.695 


4.924 0.868 


8O 


15 


0984 0.178 


1.968 0.356 


2.952 0.534 


3.936 0.712 


4.920 0.890 


45 


30 


0.983 0.182 


1.967 0.364 


2.950 0.547 


3.933 0.729 


4.916 0.911 


30 


45 


0.982 0.187 


1.965 0.373 


2.947 0.560 


3.930 0.746 


4.912 0.933 


15 


11 


0.98?, 0.191 


1.963 0.382 


2.945 0.572 


3.927 0.763 


4.908 0.954 


79 


15 


0.981 0.195 


1.962 0.390 


2.942 0.585 


3.923 0.780 


4.904 0.975 


45 


30 


0.980 0.199 


1.960 0.399 


2.940 0.598 


3.920 0.797 


4.900 0.997 


30 


45 


0.979 0.204 


1.958 0.407 


2.937 0611 


3.916 0.815 


4.895 1.018 


15 


12 


0.978 0.208 


1.956 0.416 


2.934 "0624 


3.913 0.832 


4.891 1.040 


78 


15 


0.977 0.212 


1.954 0.424 


2.932 0.637 


3.909 0.849 


4886 1.061 


45 


30 


0.976 0.216 


1.953 0.433 


2.929 0.649 


3.905 0.866 


4.881 1.082 


30 


45 


0.975 0.221 


1.951 0.441 


2.926 0662 


3.901 0883 


4.877 1.103 


15 


13 


0.974 0.225 


1.949 0.450 


2.923 0.675 


3.897 0.900 


4.872 1.125 


77 


15 


0.973 0.229 


1.947 0.458 


2.920 0.688 


3.894 0.917 


4.867 1.146 


45 


30 


0.972 0.233 


1.945 0.467 


2.917 0.700 


3.889 0.934 


4.862 1.167 


30 


45 


0971 0.238 


1.943 0.475 


2.914 0.713 


3.885 0.951 


4.857 1.188 


15 


14 


0.970 0.242 


1.941 0.484 


2.911 0.726 


3.881 0.968 


4.851 1.210 


76 


15 


0.969 0.246 


1.938 0.492 


2.908 0.738 


3.877 0.985 


4.846 1.231 


45 


30 


0.968 0.250 


1.936 0.501 


2.904 0.751 


3.873 1.002 


4.841 1.252 


30 


45 


0.967 0.255 


1.934 0.509 


2.901 0.764 


3.868 1.018 


4.835 1.273 


15 


15 


0.966 0.259 


1.932 0.518 


2.898 0.776 


3.864 1.035 


4.830 1.294 


75 C 


f 


Dep, Lat, 


Dep, Lat, 


Dep, Lat, 


Dep, Lat, 


Dep. Lat, 


o t 


Bearing, 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing, 



75-90 < 



71 



Bearing, 


Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 1O. 


Bearing, 


o r 


Lat, 


Dep, 


Lat, 


Dep, 


Lat, 


Dep, 


Lat. 


Dep, 


Lat, 


Dep, 


o r 


O15 


6.000 


0.026 


7.000 


0.031 


8.000 


0.035 


9.000 


0.039 


10.000 


0.044 


8945 


30 


6.000 


0.052 


7.000 


0.061 


8.000 


0.070 


9.000 


0.079 


10.000 


0.087 


30 


45 


5.999 


0.079 


6.999 


0.092 


7.999 


0.105 


8.999 


0.118 


9.999 


0.131 


15 


1 


5.999 


0.105 


6.999 


0.122 


7.999 


0.140 


8.999 


0.157 


9.999 


0.175 


89 


15 


5.999 


0.131 


6.998 


0.153 


7.998 


0.175 


8.998 


0.196 


9.998 


0.218 


45 


30 


5.998 


0.157 


6.998 


0.183 


7.997 


0.209 


8.997 


0.236 


9.997 


0.262 


30 


45 


5.997 


0.183 


6.997 


0.214 


7.996 


0.244 


8.996 


0.275 


9.995 


0.305 


15 


2 


5.996 


0.209 


6.996 


0.244 


7.995 


0.279 


8.995 


0.314 


9.994 


0.349 


88 


15 


5.995 


0.236 


6.995 


0.275 


7.994 


0.314 


8.993 


0.353 


9.992 


0.393 


45 


30 


5.994 


0.262 


6.993 


0.305 


7.992 


0.349 


8.991 


0.393 


9.991 


0.436 


30 


45 


5.993 


0.288 


6.992 


0.336 


7.991 


0.384 


8.990 


0.432 


9.989 


0.480 


15 


3 


5.992 


0.314 


6.990 


0.366 


7.989 


0.419 


8.988 


0.471 


" 9.986 


0.523 


87 


15 


5.990 


0.340 


6.989 


0.397 


7.987 


0.454 


8.986 


0.510 


9.984 


0.567 


45 


30 


5.989 


0.366 


6.987 


0.427 


7.985 


0.488 


8.983 


0.549 


9.981 


0.611 


30 


45 


5.987 


0.392 


6.985 


0.458 


7.983 


0.523 


8.981 


0.589 


9.979 


0.654 


15 


4 


5.985 


0.419 


6.983 


0.488 


7.981 


0.558 


8.978 


0.628 


9.976 


0.698 


86 


15 


5.984 


0.445 


6.981 


0.519 


7.978 


0.593 


8.975 


0.667 


9.973 


0.741 


45 


30 


5.982 


0.471 


6.978 


0.549 


7.975 


0.628 


8.972 


0.706 


9.969 


0.785 


30 


45 


5.979 


0.497 


6.976 


0.580 


7.973 


0.662 


8.969 


0.745 


9.966 


0.828 


15 


5 


5.977 


0.523 


6.973 


0.610 


7.970 


0.697 


8.966 


0.784 


9.962 


0.872 


85 


15 


5.975 


0.549 


6.971 


0.641 


7.966 


0.732 


8.962 


0.824 


9.958 


0.915 


45 


30 


5.972 


0.575 


6.968 


0.671 


7.963 


0.767 


8.959 


0.863 


9.954 


0.959 


30 


45 


5.970 


0.601 


6.965 


0.701 


7.960 


0.802 


8.955 


0.902 


9.950 


1.002 


15 


6 


5.967 


0.627 


6.962 


0.732 


7.956 


0.836 


8.951 


0.941 


9.945 


1.045 


84 


15 


5.964 


0.653 


6.958 


0.762 


7.952 


0.871 


8.947 


0.980 


9.941 


1.089 


45 


30 


5.961 


0.679 


6.955 


0.792 


7.949 


0.906 


8.942 


1.019 


9.936 


1.132 


30 


45 


5.958 


0.705 


6.951 


0.823 


7.945 


0.940 


8.938 


1.058 


9.931 


1.175 


15 


7 


5.955 


0.731 


6.948 


0.853 


7.940 


0.975 


8.933 


1.097 


9.926 


1.219 


83 


15 


5.952 


0.757 


6.944 


0.883 


7.936 


1.010 


8.928 


1.136 


9.920 


1.262 


45 


30 


5.949 


0.783 


6.940 


0.914 


7.932 


1.044 


8.923 


1.175 


9.914 


1.305 


30 


45 


5.945 


0.809 


6.936 


0.944 


7.927 


1.079 


8.918 


1.214 


9.909 


1.349 


15 


8 


5.942 


0.835 


6.932 


0.974 


7.922 


1.113 


8.912 


1.253 


9.903 


1.392 


82 


15 


5.938 


0.861 


6.928 


1.004 


7.917 


1.148 


8.907 


1.291 


9.897 


1.435 


45 


30 


5.934 


0.887 


6.923 


1.035 


7.912 


1.182 


8.901 


1.330 


9.890 


1.478 


30 


45 


5.930 


0913 


6.919 


1.065 


7.907 


1.217 


8.895 


1.369 


9.884 


1.521 


15 


9 


5.926 


0.939 


6.914 


1.095 


7.902 


1.251 


8.889 


1.408 


9.877 


1.564 


81 


15 


5.922 


0.964 


6.909 


1.125 


7.896 


1.286 


8.883 


1.447 


9.870 


1.607 


45 


30 


5.918 


0.990 


6.904 


1.155 


7.890 


1.320 


8.877 


1.485 


9.863 


1.651 


30 


45 


5.913 


1.016 


6.899 


1.185 


7.884 


1.355 


8.870 


1.524 


9.856 


1.694 


15 


1O 


5.909 


1.042 


6.894 


1.216 


7.878 


1.389 


8.863 


1.563 


9.848 


1.737 


80 


15 


5.904 


1.068 


6.888 


1.246 


7.872 


1.424 


8.856 


1.601 


9.840 


1.779 


45 


30 


5.900 


1.093 


6.883 


1.276 


7.866 


1.458 


8.849 


1.640 


9.833 


1.822 


30 


45 


5.895 


1.119 


6.877 


1.306 


7.860 


1.492 


8.842 


1.679 


9.825 


1.865 


15 


11 


5.890 


1.145 


6.871 


1.336 


7.853 


1.526 


8.835 


1.717 


9.816 


1.908 


79 


15 


5.885 


1.171 


6.866 


1.366 


7.846 


1.561 


8.827 


1.756 


9.808 


1.951 


45 


30 


5.880 


1.196 


6.859 


1.396 


7.839 


1.595 


8.819 


1.794 


9.799 


1.994 


30 


45 


5.874 


1.222 


6.853 


1.425 


7.832 


1 .629 


8.811 


1.833 


9.791 


2.036 


15 


12 


5.869 


1.247 


6.847 


1.455 


7.825 


1.663 


8.803 


1.871 


9.782 


2.079 


78 


15 


5.863 


1.273 


6.841 


1.485 


7.818 


1.697 


8.795 


1.910 


9.772 


2.122 


45 


30 


5.858 


1.299 


6.834 


1.515 


7.810 


1.732 


8.787 


1.948 


9.763 


2.164 


30 


* 45 


5.852 


1.324 


6.827 


1.545 


7.803 


1.766 


8.778 


1.986 


9.753 


2.207 


15 


13 


5.846 


1.350 


6.821 


1.575 


7.795 


1.800 


8.769 


2.025 


9.744 


2.250 


77 


15 


5.840 


1.375 


6.814 


1.604 


7.787 


1.834 


8.760 


2.063 


9.734 


2.292 


45 


30 


5.834 


1.401 


6.807 


1.634 


7.779 


1.868 


8.751 


2.101 


9.724 


2.335 


30 


45 


5.828 


1.426 


6.799 


1.664 


7.771 


1.902 


8.742 


2.139 


9.713 


2.377 


15 


14 


5.822 


1.452 


6.792 


1.693 


7.762 


1.935 


8.733 


2.177 


9.703 


2.419 


76 


15 


5.815 


1.477 


6.785 


1.723 


7.754 


1.969 


8.723 


2.215 


9.692 


2.462 


45 


30 


5.809 


1.502 


6.777 


1.753 


7.745 


2.003 


8.713 


2.253 


9.682 


2.504 


30 


45 


5.802 


1.528 


6.769 


1.782 


7.736 


2.037 


8.703 


2.291 


9.671 


2.546 


15 


15 


5.796 


1.553 


6.761 


1.812 


7.727 


2.071 


8.693 


2.329 


9.659 


2.588 


75 


f 


Dep, 


Lat, 


Dep, 


Lat, 


Dep. 


Lat, 


Dep, 


Lat, 


Dep, 


Lat, 


f 


Bearing. 


Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 10. 


Bearing. 



75-90 c 



72 



15-30 ( 



Bearing. 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing. 


o r 


Lat. Dep, 


Lat. Dep. 


Lat. Dep, 


Lat, Dep, 


Lat. Dep. 


f 


1515 


0.965 0.263 


1.930 0.526 


2.894 0.789 


3.859 1.052 


4.824 1.315 


7445 


30 


0.964 0.267 


1.927 0.534 


2.891 0.802 


3.855 1.069 


4.818 1.336 


30 


45 


0.962 0.271 


1.925 0.543 


2.887 0.814 


3.850 1.086 


4.812 1.357 


15 


16 


0.961 0.276 


1.923 0.551 


2.884 0.827 


3.845 1.103 


4.806 1.378 


74 


15 


0.960 0.280 


1.920 0.560 


2.880 0.839 


3.840 1.119 


4.800 1.399 


45 


30 


0.959 0.284 


1.918 0.568 


2.876 0.852 


3.835 1.136 


4.794 1.420 


30 


45 


0.958 0.288 


1.915 0.576 


2.873 0.865 


3.830 1.153 


4.788 1.441 


15 


17 


0.956 0.292 


1.913 0.585 


2.869 0.877 


3.825 1.169 


4.782 1.462 


73 


15 


0.955 0.297 


.910 0.593 


2.865 0.890 


3.820 1.186 


4.775 1.483 


45 


30 


0.954 0.301 


.907 0.601 


2.861 0.902 


3.815 1.203 


4.769 1.504 


30 


45 


0.952 0.305 


.905 0.610 


2.857 0.915 


3.810 1.220 


4.762 1.524 


15 


18 


0.951 0.309 


1.902 0.618 


2.853 0.927 


3.804 1.236 


4.755 1.545 


72 


15 


0.950 0.313 


1.899 0.626 


2.849 0.939 


3.799 1.253 


4.748 1.566 


45 


30 


0.948 0.317 


1.897 0.635 


2.845 0.952 


3.793 1.269 


4.742 1.587 


30 


45 


0.947 0.321 


1.894 0.643 


2.841 0.964 


3.788 1.286 


4.735 1.607 


15 


19 


0.946 0.326 


1.891 0.651 


2.837 0.977 


3.782 1.302 


4.728 1.628 


71 


15 


0.944 0.330 


1.888 0.659 


2.832 0.989 


3.776 1.319 


4.720 1.648 


45 


30 


0.943 0.334 


1.885 0.668 


2.828 1.001 


3.771 1.335 


4.713 1.669 


30 


45 


0.941 0.338 


1.882 0.676 


2.824 1.014 


3.765 1.352 


4.706 1.690 


15 


2O 


0.940 0.342 


1.879 0.684 


2.819 1.026 


3.759 1.368 


4.698 1.710 


7O 


15 


0.938 0.346 


1.876 0.692 


2.815 1.038 


3.753 1.384 


4.691 1.731 


45 


30 


0.937 0.350 


1.873 0.700 


2.810 1.051 


3.747 1.401 


4.683 1.751 


30 


45 


0.935 0.354 


1.870 0.709 


2.805 1.063 


3.741 1.417 


4.676 1.771 


15 


21 


0.934 0.358 


1.867 0.717 


2.801 1.075 


3.734 1.433 


4.668 1.792 


69 


15 


0.932 0.362 


1.864 0.725 


2.796 1.087 


3.728 1.450 


4.660 1.812 


45 


30 


0.930 0.367 


1.861 0.733 


2.791 1.100 


3.722 1.466 


4.652 1.833 


30 


45 


0.929 0.371 


1.858 0.741 


2.786 1.112 


3.715 1.482 


4.644 1.853 


15 


22 


0.927 0.375 


1.854 0.749 


2.782 1.124 


3.709 1.498 


4.636 1.873 


68 


15 


0.926 0.379 


1.851 0.757 


2.777 1.136 


3.702 1.515 


4.628 1.893 


45 


30 


0.924 0.383 


1.848 0.765 


2.772 1.148 


3.696 1.531 


4.619 1.913 


30 


45 


0.922 0.387 


1.844 0.773 


2.767 1.160 


3.689 1.547 


4.611 1.934 


15 


23 


0.921 0.391 


1.841 0.781 


2.762 1.172 


3.682 1.563 


4.603 1.954 


67 


15 


0.919 0.395 


1.838 0.789 


2.756 1.184 


3.675 1.579 


4.594 1.974 


45 


30 


0.917 0.399 


1.834 0.797 


2.751 1.196 


3.668 1.595 


4.585 1.994 


30 


45 


0.915 0.403 


1.831 0.805 


2.746 1.208 


3.661 1.611 


4.577 2.014 


15 


24 


0.914 0.407 


1.827 0.813 


2.741 1.220 


3.654 1.627 


4.568 2.034 


66 


15 


0.912 0.411 


1.824 0.821 


2.735 1.232 


3.647 1.643 


4.559 2.054 


45 


30 


0.910 0.415 


1.820 0.829 


2.730 1.244 


3.640 1.659 


4.550 2.073 


30 


45 


0.908 0.419 


1.816 0.837 


2.724 1.256 


3.633 1.675 


4.541 2.093 


15 


25 


0.906 0.423 


1.813 0.845 


2.719 1.268 


3.625 1.690 


4.532 2.113 


65 


15 


0.904 0.427 


1.809 0.853 


2.713 1.280 


3.618 1.706 


4.522 2.133 


45 


30 


0.903 0.431 


1.805 0.861 


2.708 1.292 


3.610 1.722 


4.513 2.153 


30 


45 


0.901 0.434 


1.801 0.869 


2.702 1.303 


3.603 1.738 


4.503 2.172 


15 


26 


0.899 0.438 


1.798 0.877 


2.696 1.315 


3.595 1.753 


4.494 2.192 


64 


15 


0.897 0.442 


1.794 0.885 


2.691 1.327 


3.587 1.769 


4.484 2.211 


45 


30 


0.895 0.446 


1.790 0892 


2.685 1.339 


3.580 1.785 


4.475 2.231 


30 


45 


0.893 0.450 


1.786 0.900 


2.679 1.350 


3.572 1.800 


4.465 2.250 


15 


27 


0.891 0.454 


1.782 0.908 


2.673 1.362 


3.564 1.816 


4.455 2.270 


63 


15 


0.889 0.458 


1.778 0.916 


2.667 1.374 


3.556 1.831 


4.445 2.289 


45 


30 


0.887 0.462 


1.774 0.923 


2.661 1.385 


3.548 1.847 


4.435 2.309 


30 


45 


0.885 0.466 


1.770 0.931 


2.655 1.397 


3.540 1.862 


4.425 2.328 


15* 


28 


0.883 0.469 


1.766 0.939 


2.649 1.408 


3.532 1.878 


4.415 2.347 


62 


15 


0.881 0.473 


1.762 0.947 


2.643 1.420 


3.524 1.893 


4.404 2.367 


45 


30 


0.879 0.477 


1.758 0.954 


2.636 1.431 


3.515 1.909 


4.394 2.386 


30 


45 


0.877 0.481 


1.753 0.962 


2.630 1.443 


3.507 1.924 


4.384 2.405 


15 


29 


0.875 0.485 


1.749 0.970 


2.624 1.454 


3.498 1.939 


4.373 2.424 


61 


15 


0.872 0.489 


1.745 0.977 


2.617 1.466 


3.490 1.954 


4.362 2.443 


45 


30 


0.870 0.492 


1.741 0.985 


2.611 1.477 


3.481 1.970 


4.352 2.462 


30 


45 


0.868 0.496 


1.736 0.992 


2.605 1.489 


3.473 1.985 


4.341 2.481 


15 


3O 


0.866 0.500 


1.732 1.000 


2.598 1.500 


3.464 2.000 


4.330 2.500 


6O 


o r 


Dep. Lat, 


Dep, Lat. 


Dep, Lat, 


Dep, Lat, 


Dep. Lat. 


f 


Bearing, 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing. 



60-75< 



15 -30' 



73 



Bearing. Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 1O. 


Bearing, 


| 
o r 


Lat, Dep. 


Lat, Dep, 


Lat, Dep, 


Lat, Dep, 


Lat, Dep, 


t 


1515 


5.789 1.578 


6.754 1.841 


7.718 2.104 


8.683 2.367 


9.648 2.630 


7445 


30 


5.782 1.603 


6.745 1.871 


7.709 2.138 


8.673 2.405 


9.636 2.672 


30 


45 


5.775 1.629 


6.737 1.900 


7.700 2.172 


8.662 2.443 


9.625 2.714 


15 


16 


5.768 1.654 


6.729 1.929 


7.690 2.205 


8.651 2.481 


9.613 2.756 


74 


15 


5.760 1.679 


6.720 1.959 


7.580 2.239 


8.640 2.518 


9.601 2.798 


45 


30 


5.753 1.704 


6.712 1.988 


7.671 2.272 


8.629 2.556 


9.588 2.840 


30 


45 


5.745 1.729 


6.703 2.017 


7.661 2.306 


8.618 2.594 


9.576 2.882 


15 


17 


5.738 1.754 


6.694 2.047 


7.650 2.339 


8.607 2.631 


9.563 2.924 


73 


15 


5.730 1.779 


6.685 2.076 


7.640 2.372 


8.595 2.669 


9.550 2.965 


45 


30 


5.722 1.804 


6.676 2.105 


7.630 2.406 


8.583 2.706 


9.537 3.007 


30 


45 


5.714 1.829 


6.667 2.134 


7.619 2.439 


8.572 2.744 


9.524 3.049 


15 


18 


5.706 1.854 


6.657 2.163 


7.608 2.472 


8.560 2.781 


9.511 3.090 


72 


15 


5.698 1.879 


6.648 2.192 


7.598 2.505 


8.547 2.818 


9.497 3.132 


45 


30 


5.690 1.904 


6.638 2.221 


7.587 2.538 


8.535 2.856 


9.483 3.173 


30 


45 


5.682 1.929 


6.629 2.250 


7.575 2.572 


8.522 2.893 


9.469 3.214 


15 


19 


5.673 1.953 


6.619 2.279 


7.564 2.605 


8.510 2.930 


9.455 3.256 


71 


15 


5.665 1.978 


6.609 2.308 


7.553 2.638 


8.497 2.967 


9.441 3.297 


45 


30 


5.656 2.003 


6.598 2.337 


7.541 2.670 


8.484 3.004 


9.426 3.338 


30 


45 


5.647 2.028 


6.588 2.365 


7.529 2.703 


8.471 3.041 


9.412 3.379 


15 


2O 


5.638 2.052 


6.578 2.394 


7.518 2.736 


8.457 3.078 


9.397 3.420 


7O 


15 


5.629 2.077 


6.567 2.423 


7.506 2.769 


8.444 3.115 


9.382 3.461 


45 


30 


5.620 2.101 


6.557 2.451 


7.493 2.802 


8.430 3.152 


9.367 3.502 


30 


45 


5.611 2.126 


6.546 2.480 


7.481 2.834 


8.416 3.189 


9.351 3.543 


15 


21 


5.601 2.150 


6.535 2.509 


7.469 2.867 


8.402 3.225 


9.336 3.584 


69 


15 


5.592 2.175 


6.524 2.537 


7.456 2.900 


8.388 3.262 


9.320 3.624 


45 


30 


5.582 2.199 


6.513 2.566 


7.443 2.932 


8.374 3.299 


9.304 3.665 


30 


45 


5.573 2.223 


6.502 2.594 


7.430 2.964 


8.359 3.335 


9.288 3.706 


15 


22 


5.563 2.248 


6.490 2.622 


7.417 2.997 


8.345 3.371 


9.272 3.746 


68 


15 


5.553 2.272 


6.479 2.651 


7.404 3.029 


8.330 3.408 


9.255 3.787 


45 


30 


5.543 2.296 


6.467 2.679 


7.391 3.061 


8.315 3.444 


9.239 3.827 


30 


45 


5.533 2.320 


6.455 2.707 


7.378 3.094 


8.300 3.480 


9.222 3.867 


15 


23 


5.523 2.344 


6.444 2.735 


7.364 3.126 


8.285 3.517 


9.205 3.907 


67 


15 


5.513 2.368 


6.432 2.763 


7.350 3.158 


8.269 3.553 


9.188 3.947 


45 


30 


5.502 2.392 


6.419 2.791 


7,336 3.190 


8.254 3.589 


9.171 3.988 


30 


45 


5.492 2.416 


6.407 2.819 


7.322 3.222 


8.238 3.625 


9.153 4.028 


15 


24 


5.481 2.440 


6.395 2.847 


7.308 3.254 


8.222 3.661 


9.136 4.067 


66 


15 


5.471 2.464 


6.382 2.875 


7.294 3.286 


8.206 3.696 


9.118 4.107 


45 


30 


5.460 2.488 


6.370 2.903 


7.280 3.318 


8.190 3.732 


9.100 4.147 


30 


45 


5.449 2.512 


6.357 2.931 


7.265 3.349 


8.173 3.768 


9.081 4.187 


15 


25 


5.438 2.536 


6.344 2.958 


7.250 3.381 


8.157 3.804 


9.063 4.226 


65 


15 


5.427 2.559 


6.331 2.986 


7.236 3.413 


8.140 3.839 


9.045 4.266 


45 


30 


5.416 2.583 


6.318 3.014 


7.221 3.444 


8.123 3.875 


9.026 4.305 


30 


45 


5.404 2.607 


6.305 3.041 


7.206 3.476 


8.106 3.910 


9.007 4.345 


15 


26 


5.393 2.630 


6.292 3.069 


7.190 3.507 


8.089 3.945 


8.988 4.384 


64 


15 


5.381 2.654 


6.278 3.096 


7.175 3.538 


8.072 3.981 


8.969 4.423 


45 


30 


5.370 2.677 


6.265 3.123 


7.160 3.570 


8.054 4.016 


8.949 4.462 


30 


45 


5.358 2.701 


6.251 3.151 


7.144 3.601 


8.037 4.051 


8.930 4.501 


15 


27 


5.346 2.724 


6.237 3.17S 


7.128 3.632 


8.019 4.086 


8.910 4.540 


63 


15 


5.334 2.747 


6.223 3.205 


7.112 3.663 


8.001 4.121 


8.890 4.579 


45 


30 


5.322 2.770 


6.209 3.232 


/.096 3.694 


7.983 4.156 


8.870 4.618 


30 


45 


5.310 2.794 


6.195 3.259 


7.080 3.725 


7.965 4.190 


8.850 4.656 


15 


fc8 


5.298 2.817 


6.181 3.286 


7.064 3.756 


7.947 4.225 


8.829 4.695 


62 


15 


5.285 2.840 


6.166 3.313 


7.047 3.787 


7.928 4.260 


S.809 4.733 


45 


30 


5.273 2.863 


6.152 3.340 


7.031 3.817 


7.909 4.294 


8.788 4.772 


30 


45 


5.260 2.886 


6.137 3.367 


7.014 3.848 


7.891 4.329 


8.767 4.810 


15 


29 


5.248 2.909 


6.122 3.394 


6.997 3.878 


7.872 4.363 


8.746 4.848 


61 


15 


5.235 2.932 


6.107 3.420 


6.980 3.909 


7.852 4.398 


8.725 4.886 


45 


30 


5.222 2.955 


6.093 3.447 


6.963 3.939 


7.833 4.432 


8.704 4.924 


30 


45 


5.209 2.977 


6.077 3.474 


6.946 3.970 


7.814 4.466 


8.682 4.962 


15 


3D 


5.196 3.000 


6.062 3.500 


6.928 4.000 


7.794 4.500 


8.66/) 5.000 


60 


f 


Dep, Lat, 


Dep, Lat. 


Dep, Lat,. 


Dep, Lat, 


Dep, Lat, 


f 


Bearing, 


Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 1O. 


Bearing, 



60-75 C 



74 



30-45 ( 



Bearing, 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing, 


o t 


Lat, Dep, 


Lat. Dep, 


Lat, Dep, 


Lat, Dep, 


Lat, Dep, 


f 


3O15 


0.864 0.504 


1.728 1.008 


2.592 1.511 


3.455 2.015 


4.319 2.519 


5945 


30 


0.862 0.508 


1.723 1.015 


2.585 1.523 


3.447 2.030 


4.308 2.538 


30 


45 


0.859 0.511 


1.719 1.023 


2.578 1.534 


3.438 2.045 


4.297 2.556 


15 


31 


0.857 0.515 


1.714 1.030 


2.572 1.545 


3.429 2.060 


4.286 2.575 


59 


15 


0.855 0.519 


1.710 1.038 


2.565 1.556 


3.420 2.075 


4.275 2.594 


45 


30 


0.853 0.522 


1.705 1.045 


2.558 1.567 


3.411 2.090 


4.263 2.612 


30 


45 


0.850 0.526 


1.701 1.052 


2.551 1.579 


3.401 2.105 


4.252 2.631 


15 


32 


0.848 0.530 


1.696 1.060 


2.544 1.590 


3.392 2.120 


4.240 2.650 


58 


15 


0.846 0.534 


1.691 1.067 


2.537 1.601 


3.383 2.134 


4.229 2.668 


45 


30 


0.843 0.537 


1.687 1.075 


2.530 1.612 


3.374 2.149 


4.217 2.686 


30 


45 


0.841 0.541 


1.682 1.082 


2.523 1.623 


3.364 2.164 


4.205 2.705 


15 


33 


0.839 0.545 


1.677 1.089 


2.516 1.634 


3.355 2.179 


4.193 2.723 


57 


15 


0.836 0.548 


1.673 1.097 


2.509 1.645 


3.345 2.193 


4.181 2.741 


45 


30 


0.834 0.552 


1.668 1.104 


2.502 1.656 


3.336 2.208 


4.169 2.760 


30 


45 


0.831 0-.556 


1.663 1.111 


2.494 1.667 


3.326 2.222 


4.157 2.778 


15 


34 


0.829 0.559 


1.658 1.118 


2.487 1.678 


3.316 2.237 


4.145 2.796 


56 


15 


0.827 0.563 


1.653 1.126 


2.480 1.688 


3.306 2.251 


4.133 2.814 


45 


30 


0.824 0.566 


1.648 1.133 


2.472 1.699 


3.297 2.266 


4.121 2.832 


30 


45 


0.822 0.570 


1.643 1.140 


2.465 1.710 


3.287 2.280 


4.108 2.850 


15 


35 


0.819 0.574 


1.638 1.147 


2.457 1.721 


3.277 2.294 


4.096 2.868 


55 


15 


0.817 0.577 


1.633 1.154 


2.450 1.731 


3.267 2.309 


4.083 2.886 


45 


30 


0.814 0.581 


1.628 1.161 


2.442 1.742 


3.257 2.323 


4.071 2.904 


30 


45 


0.812 0.584 


1.623 1.168 


2.435 1.753 


3.246 2.337 


4.058 2.921 


15 


36 


0.809 0.588 


1.618 1.176 


2.427 1.763 


3.236 2.351 


4.045 2.939 


54 


15 


0.806 0.591 


1.613 1.183 


2.419 1.774 


3.226 2.365 


4.032 2.957 


45 


30 


0.804 0.595 


1.608 1.190 


2.412 1.784 


3.215 2.379 


4.019 2.974 


30 


45 


0.801 0.598 


1.603 1.197 


2.404 1.795 


3.205 2.393 


4.006 2.992 


15 


37 


0.799 0.602 


1.597 1.204 


2.396 1.805 


3.195 2.407 


3.993 3.009 


53 


15 


0.796 0.605 


1.592 1.211 


2.388 1.816 


3.184 2.421 


3.980 3.026 


45 


30 


0.793 0.609 


1.587 1.218 


2.380 1.826 


3.173 2.435 


3.967 3.044 


30 


45 


0.791 0.612 


1.581 1.224 


2.372 1.837 


3.163 2.449 


3.953 3.061 


15 


38 


0.788 0.616 


1.576 1.231 


2.364 1.847 


3.152 2.463 


3.940 3.078 


52 


15 


0.785 0.619 


1.571 1.238 


2.356 1.857 


3.141 2.476 


3.927 3.095 


45 


30 


0.783 0.623 


1.565 1.245 


2.348 1.868 


3.130 2.490 


3.913 3.113 


30 


45 


0.780 0.626 


1.560 1.252 


2.340 1.878 


3.120 2.504 


3.899 3.130 


. 15 


39 


0.777 0.629 


1.554 1.259 


2.331 1.888 


3.109 2.517 


3.886 3.147 


51 


15 


0.774 0.633 


1.549 1.265 


2.323 1.898 


3.098 2.531 


3.872 3.164 


45 


30 


0.772 0.636 


1.543 1.272 


2.315 1.908 


3.086 2.544 


3.858 3.180 


30 


45 


0.769 0.639 


1.538 1.279 


2.307 1.918 


3.075 2.558 


3.844 3.197 


15 


4O 


0.766 0.643 


1.532 1.286 


2.298 1.928 


3.064 2.571 


3.830 3.214 


50 


15 


0.763 0.646 


1.526 1.292 


2.290 1.938 


3.053 2.584 


3.816 3.231 


45 


30 


0.760 0.649 


1.521 1.299 


2.281 1.948 


3.042 2.598 


3.802 3.247 


30 


45 


0.758 0.653 


1.515 1.306 


2.273 1.958 


3.030 2.611 


3.788 3.264 


15 


41 


0.755 0.656 


1.509 1.312 


2.264 1.968 


3.019 2.624 


3.774 3.280 


49 


15 


0.752 0.659 


1.504 1.319 


2.256 1.978 


3.007 2.637 


3.759 3.297 


45 


30 


0.749 0.663 


1.498 1.325 


2.247 1.988 


2.996 2.650 


3.745 3.313 


30 


45 


0.746 0.666 


1.492 1.332 


2.238 1.998 


2.984 2.664 


3.730 3.329 


15 


42 


0.743 0.669 


1.486 1.338 


2.229 2.007 


2.973 2.677 


3.716 3.346 


48 


15 


0.740 0.672 


1.480 1.345 


2.221 2.017 


2.961 2.689 


3.701 3.362 


45 


30 


0.737 0.676 


1.475 1.351 


2.212 2.027 


2.949 2.702 


3.686 3.378 


30- 


45 


0.734 0.679 


1.469 1.358 


2.203 2.036 


2.937 2.715 


3.672 3.394 


15 


43 


0.731 0.682 


1.463 1.364 


2.194 2.046 


2.925 2.728 


3.657 3.410 


47 


15 


0.728 0.685 


1.457 1.370 


2.185 2.056 


2.913 2.741 


3.642 3.426 


45 


30 


0.725 0.688 


1.451 1.377 


2.176 2.065 


2.901 2.753 


3.627 3.442 


30 


45 


0.722 0.692 


1.445 1.383 


2.167 2.075 


2.889 2.766 


3.612 3.458 


15 


44 


0.719 0.695 


1.439 1.389 


2.158 2.084 


2.877 2.779 


3.597 3.473 


46 


15 


0.716 0.698 


1.433 1.396 


2.149. 2.093 


2.865 2.791 


3.582 3.489 


45 


30 


0.713 0.701 


1.427 1.402 


2.140 2.103 


2.853 2.804 


3.566 3.505 


30 


45 


0.710 0.704 


1.420 1.408 


2.131 2.112 


2.841 2.816 


3.551 3.520 


15 


45 


0.707 0707 


1.414 1.414 


2.121 2.121 


2.828 2.828 


3.536 3.536 


45 


f 


Dep, Lat, 


Dep, Lat, 


Dep, Lat, 


Dep, Lat, 


Dep, Lat, 


f 


Bearing, 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing, 



45 -60 



30 -45 - 



75 



Bearing. 


Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 1O. 


Bearing, 


o r 


Lat, Dep, 


Lat, Dep, 


Lat, Dep, 


Lat. Dep. 


Lat, Dep, 


f 


3O15 


5.183 3.023 


6.047 3.526 


6.911 4.030 


7.775 4.534 


8.638 5.038 


5945 


30 


5.170 3.045 


6.031 3.553 


6.893 4.060 


7.755 4.568 


8.616 5.075 


30 


45 


5.156 3.068 


6.016 3.579 


6.875 4.090 


7.735 4.602 


8.594 5.113 


15 


31 


5.143 3.090 


6.000 3.605 


6.857 4.120 


7.715 4.635 


8.572 5.150 


59 


15 


5.129 3.113 


5.984 3.631 


6.839 4.150 


7.694 4.669 


8.549 5.188 


45 


30 


5.116 3.135 


5.968 3.657 


6.821 4.180 


7.674 4.702 


8.526 5.225 


30 


45 


5.102 3.157 


5.952 3.683 


6.803 4.210 


7.653 4.736 


8.504 5.262 


15 


32 


5.088 3.180 


5.936 3.709 


6.784 4.239 


7.632 4.769 


8.481 5.299 


58 


15 


5.074 3.202 


5.920 3.735 


6.766 4.269 


7.612 4.802 


8.457 5.336 


45 


30 


5.060 3.224 


5.904 3.761 


6.747 4.298 


7.591 4.836 


8.434 5.373 


30 


45 


5.046 3.246 


5.887 3.787 


6.728 4.328 


7.569 4.869 


8.410 5.410 


15 


33 


5.032 3.268 


5.871 3.812 


6.709 4.357 


7.548 4.902 


8.387 5.446 


57 


15 


5.018 3.290 


5.854 3.838 


6.690 4.386 


7.527 4.935 


8.363 5.483 


45 


30 


5.003 3.312 


5.837 3.864 


6.671 4.416 


7.505 4.967 


8.339 5.519 


30 


45 


4.989 3.333 


5.820 3.889 


6.652 4.445 


7.483 5.000 


8.315 5.556 


15 


34 


4.974 3.355 


5.803 3.914 


6.632 4.474 


7.461 5.033 


8.290 5.592 


56 


15 


4.960 3.377 


5.786 3.940 


6.613 4.502 


7.439 5.065 


8.266 5.628 


45 


, 30 


4.945 3.398 


5.769 3.965 


6.593 4.531 


7.417 5.098 


8.241 5.664 


30 


45 


4.930 3.420 


5.752 3.990 


6.573 4.560 


7.395 5.130 


8.217 5.700 


15 


35 


4.915 3.441 


5.734 4.015 


6.553 4.589 


7.372 5.162 


8.192 5.736 


55 


15 


4.900 3.463 


5.716 4.040 


6.533 4.617 


7.350 5.194 


8.166 5.772 


45 


30 


4.885 3.484 


5.699 4.065 


6.513 4.646 


7.327 5.226 


8.141 5.807 


30 


45 


4.869 3.505 


5.681 4.090 


6.493 4.674 


7.304 5.258 


8.116 5.843 


15 


36 


4.854 3.527 


5.663 4.115 


6.472 4.702 


7.281 5.290 


8.090 5.878 


54 


15 


4.839 3.548 


5.645 4.139 


6.452 4.730 


7.258 5.322 


8.064 5.913 


45 


30 


4.823 3.569 


5.627 4.164 


6.431 4.759 


7.235 5.353 


8.039 5.948 


30 


45 


4.808 3.590 


5.609 4.188 


6.410 4.787 


7.211 5.385 


8.013 5.983 


15 


37 


4.792 3.611 


5.590 4.213 


6.389 4.815 


7.188 5.416 


7.986 6.018 


53 


15 


4.776 3.632 


5.572 4.237 


6.368 4.842 


7.164 5.448 


7.960 6.053 


45 


30 


4.760 3.653 


5.554 4.261 


6.347 4.870 


7.140 5.479 


7.934 6.088 


30 


45 


4.744 3.673 


5.535 4.286 


6.326 4.898 


7.116 5.510 


7.907 6.122 


15 


38 


4.728 3.694 


5.516 4.310 


6.304 4.925 


7.092 5.541 


7.880 6.157 


52 


15 


4.712 3.715 


5.497 4.334 


6.283 4.953 


7.068 5.572 


7.853 6.191 


45 


30 


4.696 3.735 


5.478 4.358 


6.261 4.980 


7.043 5.603 


7.826 6.225 


30 


45 


4.679 3.756 


5.459 4.381 


6.239 5.007 


7.019 5.633 


7.799 6.259 


15 


39 


4.663 3.776 


5.440 4.405 


6.217 5.035 


6.994 5.664 


7.772 6.293 


51 


15 


4.646 3.796 


5.421 4.429 


6.195 5.062 


6.970 5.694 


7.744 6.327 


45 


30 


4.630 3.816 


5.401 4.453 


6.173 5.089 


6.945 5.725 


7.716 6.361 


30 


45 


4.613 3.837 


5.382 4.476 


6.151 5.116 


6.920 5.755 


7.688 6.394 


15 


4O 


4.596 3.857 


5.362 4.500 


6.128 5.142 


6.894 5.785 


7.660 6.428 


5O 


15 


4.579 3.877 


5.343 4.523 


6.106 5.169 


6.869 5.815 


7.632 6.461 


45 


30 


4.562 3.897 


5.323 4.546 


6.083 5.196 


6.844 5.845 


7.604 6.495 


30 


45 


4.545 3.917 


5.303 4.569 


6.061 5.222 


6.818 5.875 


7.576 6.528 


15 


41 


4.528 3.936 


5.283 4.592 


6.038 5.248 


6.792 5.905 


7.547 6.561 


49 


15 


f.Sll 3.956 


5.263 4.615 


6.015 5.275 


6.767 5.934 


7.518 6.594 


45 


30 


4.494 3.976 


5.243 4638 


5.992 5.301 


6.741 5.964 


7.490 6.626 


30 


45 


4.476 3.995 


5.222 4.661 


5.968 5.327 


6.715 5.993 


7.461 6:659 


15 


42 


4.459 4.015 


5.202 4.684 


5.945 5.353 


6.688 6.022 


7.431 6.691 


48 


15 


4.441 4.034 


5.182 4.707 


5.922 5.379 


6.662 6.051 


7.402 6.724 


45 


30 


4.424 4.054 


5.161 4.729 


5.898 5.405 


6.635 6.080 


7.373 6.756 


30 


45 


4.406 4.073 


5.140 4.752' 


5.875 5.430 


6.609 6.109 


7.343 6.788 


15 


43 


4.388 4.092 


5.119 4.774 


5.851 5.456 


6.582 6.138 


7.314*6.820 


47 


15 


4.370 4.111 


5.099 4.796 


5.827 5.481 


6.555 6.167 


7.284 6.852 


45 


30 


4.352 4.130 


5.078 4.818 


5.803 5.507 


6.528 6.195 


7.254 6.884 


30 


45 


4.334 4.149 


5.057 4.841 


5.779 5.532 


6.501 6.224 


7.224 6.915 


15 


44 


4.316 4.168 


5.035 4.863 


5.755 5.557 


6.474 6.252 


7.193 6.947 


46 


15 


4.298 4.187 


5.014 4.885 


5.730 5.582 


6.447 6.280 


7.163 6.978 


45 


30 


4.280 4.206 


4.993 4.906 


5.706 5.607 


6.419 6.308 


7.133 7.009 


30 


45 


4.261 4.224 


4.971 4.928 


5,681 5.632 


6.392 6.336 


7.102 7.040 


15 


45 


4.243 4.243 


4.950 4.950 


5.657 5.657 


6.364 6.364 


7.071 7.071 


45 


f 


Dep. Lat. 


Dep. Lat, 


Dep, Lat, 


Dep. Lat. 


Dep, Lat, 


f 


Bearing, 


Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 1O. 


Bearing, 



45-60 c 




A TABLE OF THE ANGLES 

Which every Point and Quarter Point of the Compass makes with the Meridian. 



North. 


Points. 

0-1/4 

o-% 

Q-% 
1 


f If 

2 48 45 
5 37 30 
8 26 15 
11 15 


Points. 

S3 

-4 


South.. 


N. by E. 


N. by W. 


S. by E. 


S. by W. 


N.N.E. 


N.N.W. 


l Vk 

2 


14 3 45 
16 52 30 
19 41 15 
22 30 


2 


S.S.E. 


S.S.W. 


N.E. by 'N. 


N.W. by N. 


't\ 
r l 


25 18 45 
28 7 30 
30 56 15 
33 45 


l-l 

H 


S.E. by S. 


S.W. by S. 


N.E. 


N.W. 


1:8 
r l 


36 33 45 
39 22 30 
42 11 15 
45 


!:'$ 

J-4 


S.E. 


S.W. 


N.E. by E 


N.W. by W. 


\--\ 

* 


47 48 46 
50 37 30 
53 26 15 
56 15 


5 * 


S.E. by E. 


S.W. by W. 


E.N.E. 


W.N.W. 


!:& 
f% 


59 3 45 
61 52 80 
6441 15 
67 30 


6 


E.S.E. 


W.S.W. 


E. by N. 


W. by N. 


6-J4 

-4J 

7 


70 18 45 
73 7 30 
75 56 15 
78 45 


6-4 

ft 3/ 

7 


E. by S. 


W. by S. 


East. 


West. 


7-V 4 
8 


81 33 46 
84 22 30 
87 11 15 
90 


8~ * 


East. 


West. 



THIS BOOK IS DUE ON THE LAST DATE 
STAMPED BELOW 



AN INITIAL FINE OF 25 CENTS 

WILL BE ASSESSED FOR FAILURE TO RETURN 
THIS BOOK ON THE DATE DUE. THE PENALTY 
WILL INCREASE TO SO CENTS ON THE FOURTH 
DAY AND TO Sl.OO ON THE SEVENTH DAY 
OVERDUE. 



OCT 22 1932 



21 1938 




2 1947 
Y 19 J348 



' <*: 

" 3l >5&ru v 
B