oninr OF
the estate of
Professor William F. Meyer
TABLES OF LOGARITHMS
TO
FIVE PLACES OF DECIMALS,
WITH AUXILIARY TABLES.
EDITED BY
EDWIN S. CRAWLEY, PH.D.,
- THOMAS A. SCOTT PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF PENNSYLVANIA
E. S. CRAWLEY,
UNIVERSITY OF PENNSYLVANIA,
PHILADELPHIA,
1905.
COPYRIGHT, 1890,
BY
EDWIN S. CRAWLKY.
ELECTROTVPEO AND PHINTED BY J. B. LIPPINCOTT COMPANY, PHILADELPHIA. U.S. A
EDITOR'S NOTE.
THIS collection of logarithmic tables has been prepared to
accompany the editor's Elements of Trigonometry^ in response
to the demand of a number of teachers using the latter, who
prefer a text bound with tables. In commending the tables
to the use of educational institutions and the mathematical
public in general, the editor wishes to state that great care has
been taken to secure accuracy. The proof has been com-
pared twice, number by number, with different standard tables
(Yega's seven-place Tables, the 74th edition, edited by W. L.
F. Fischer; and Gauss's five-place Tables, the 20th edition),
and the method of differences was applied as a further check.
Besides these, other tests were applied to parts of the tables,
as in the case of Table III., where the log tan column was
checked by taking the difference of log sin and log cos, and
the log cot column was checked by taking the arithmetical
complement of log tan.
Should any errors be discovered, the editor will be glad to
be informed of them.
EDWIN S. CKAWLEY.
UNIVERSITY OF PENNSYLVANIA,
January, 1899.
C8
M577188
TABLE OF CONTENTS.
EXPLANATION or THE TABLES vii
TABLE I. Logarithms of Numbers 1
" II. Important Constants and their Logarithms 20
" III. Logarithms of the Sine, Cosine, Tangent, Cotangent
for every Minute of the Quadrant 21
" IV. Table for Computing the Log Sin and Log Tan of
Small Angles 67
" V. Natural Sines, Cosines, Tangents, and Cotangents . . 69
" VI. Circular Arcs, expressed in Kadians 74
* VII. Napierian Logarithms of Numbers 75
EXPLANATION OF THE TABLES.
1. Definitions and Rules. If three numbers n, a, x have
such values that the equation
n = a* (1)
is true, then x is called the logarithm of n to the base a. If,
without changing a, we give to n and x all possible values, con-
sistent with this equation, the values of x thus obtained form
a system of logarithms to the base a.
Hence : The logarithm of a number to a given base is the ex-
ponent of the power to which the base must be raised to produce the
number.
Suppose 9 is taken for the base, then
log 81 = 2, because 9 2 = 81
11 729= 3, " 9 3 =729
i = -l, -" 9~!= i
" 3= i, 9* = 3
" 9= 1, " 9 1 = 9
" 1= 0, " 9 = 1
In every system the logarithm of the base is 1, and the
logarithm of 1 is 0. This follows directly from the definition,
or from (1) ; for if n = a, x must be 1 ; and if n = 1, x must
be 0, without respect to the value of a.
It is plain, since any number will serve as the base of a
system of logarithms, that the number of such systems is
indefinite.
vii
Vlll EXPLANATION OF THE TABLES.
The systems of logarithms commonly used are :
(1.) The common or Briggian* system, with the base 10.
(2.) The natural or Napierianf system with the base
6 = 2.7182818285 ____
defined by the convergent infinite series
e = 1 + 1 + r2 + ii3+i^-4 + ---
Of these two systems, the first is used for all purposes of
numerical computation, and the second for purely analytical
purposes.
The logarithms of these tables (except in Table VII.) are
common or Briggian logarithms.
The corresponding logarithms of any two systems are in a
constant ratio to each other. Thus the relation between com-
mon and Napierian logarithms is
(This equation is read : " Logarithm of n to the base 10 equals
the reciprocal of the logarithm of 10 to the base e, multiplied
bv the logarithm of n to the base e.") The factor - is
log . 10
called the modulus of the common system. It is represented
by M, and its value to ten places is 0.4342944819.
The rules governing the use of logarithms in computation
are the following :
I. To multiply numbers, find the logarithm of each factor, and
add them ; the sum is the logarithm of the product.
II. To divide one number by another, subtract the logarithm
of the divisor from the logarithm of the dividend ; the difference is
the logarithm of the quotient.
III. To raise a number to any power multiply the logarithm of
the number by the exponent of the power ; the product is the log-
arithm of the required power of the number.
* Named for Henry Briggs (1666-1631), who first suggested the use of
the base 10.
f Named for John Napier, Baron of Merchiston, in Scotland (1560-
1617), the inventor of logarithms.
EXPLANATION OF THE TABLES. IX
IY. To extract any root of a number, divide the logarithm of
the number by the index of the root ; the quotient is the logarithm
of the required root of the number.
These statements and rules .are given without proof, as the
purpose here is simply to familiarize the student with the
mechanism and use of the tables. The theory of logarithms
is set forth in text-books on algebra, to which the student is
referred. In the same place will be found an explanation of
how logarithms are computed.
TABLE I. Common Logarithms of Numbers. (Pages 1-19)
2. Characteristic and Mantissa. A logarithm consists,
usually, of two parts : a whole number, called the characteristic,
and an incommensurable decimal fraction, called the mantissa.
The table gives only the mantissa ; the characteristic, which
may be positive, negative, or zero, must be supplied in every
case by the computer. The mantissa is always positive, ex-
cept in the logarithms of exact powers of 10, when it is zero.
Since 10 is the base we have :
log 100Q = 3, because 10 3 = 1000
log 100= 2, " 10 2 = 100
log 10= If " 10 l = 10
log 1= 0, " 10 = 1
log a = 1, " 10 ~ l = .1
log .01= 2, " 10- 2 = .01
log .001 = 3, " 10 ~ 3 = .001
(a)
This series of equations can be extended indefinitely in both
directions.
Let us now consider two numbers which contain the same
sequence of figures, with different positions of the decimal
point, say 72.936 and .72936. Now 72.936 = 100 X .72936.
Hence, by Eule I, 1 log 72.936 = log 100 -f log .72936,
or, by (a) = 2 + log .72936.
Hence, since any change in the position of the decimal
X EXPLANATION OF THE TABLES.
point in a number is equivalent to multiplication or division
by a power of 10, the effect produced upon the logarithm of
the number by a change of this kind is to increase it or
diminish it by a whole number ; that is, the characteristic is
affected by such a change, but not the mantissa. We have,
therefore, the following important fact :
I. The mantissa of the logarithm of a number depends only
upon the sequence of figures in the number.
Referring again to (a), we note that for all numbers greater
than 1 and less than 10 (all numbers with one significant
figure before the decimal point) the logarithm is greater than
and less than 1, that is, its characteristic is ; for all num-
bers greater than 10 and less than 100 (all numbers with two
significant figures before the decimal point) the logarithm is
greater than 1 and less than 2, that is, its characteristic is 1 ;
for all numbers greater than 100 and less than 1000 (all -num-
bers with three significant figures before the decimal point)
the logarithm is greater than 2 and less than 3, that is, its
characteristic is 2; and so on. Hence, we have the following
rule:
II. The characteristic of the logarithm of a number greater than
unity is one less than the number of significant figures preceding
the decimal point.
Again, from (#) it will be seen that if a number is greater
than .1 and less than 1, its logarithm is between and 1 ;
that is, using a positive mantissa, which we always do, it is
1 -f the mantissa, hence the characteristic is 1 ; if the
number is greater than .01 and less than .1, the logarithm is
between 1 and 2, which is written 2 -f- the mantissa,
that is, the characteristic is 2 ; if the number is greater than
.001 and less than .01, the logarithm is between 2 and 3,
which is written 3 -f- the mantissa, that is, the characteristic
is 3, and so on. Hence, we have the following rule :
III. The characteristic of the logarithm of a number less than
unity is negative, and is numerically one greater than the number
of ciphers between the decimal point and the first significant
figure.
EXPLANATION OF THE TABLES. XI
Verify the following statements :
characteristic of log 763.92 = 2
" " log 1.9841
*' "log .07296 = 2
" " log 26 = 1
" " log 400000= 5
" " log .9426 = !
" " log 3869= 3
" " log .00042 = 4
" " log .005 = 3 '
" " log 62893= 4
3. To Find the Logarithm of a Number of Four
Figures or Less.
If the number has less than four figures add ciphers on the
right until it has four figures, and then proceed in the manner
described below.
If the number has four figures, enter the table in the left
hand column of the page, the column marked N, with the
first three figures (the first three significant figures if the
number is a decimal fraction) and with the fourth figure in the
line running across the page at the extreme top or bottom.
Go across the page, in the line containing the first three
figures, until the column marked by the fourth figure is
reached. The three figures found at this point are the last
three figures of the mantissa. The first two figures of the
mantissa are printed only in the first column of the body of
the table, and if they are not found in the same line with the
last three figures they will be found a few lines above.
Suppose the number is 48.65. We find 486 in the N column
on page 9 ; and the column marked 5 at the top and bottom is
the one to the right of the heavy line down the middle of the
page. The three figures in this column and on the same line
with 486 are 708, which are the last three figures of the
mantissa; the first two figures are 68. Hence, mantissa of
log 48.65 is .68708. By II. 2 characteristic of log 48.65 is 1.
Hence, log 48.65 = 1.68708.
Find log 6.2. Annexing two ciphers, this becomes 6.200.
Xll EXPLANATION OF THE TABLES.
Proceeding then as above, we find that the mantissa is 79239.
Hence, log 6.2 = 0.79239.
Find log 431. Annexing one cipher this becomes 431.0.
Hence, the mantissa is 63448; and log 431. = 2.63448.
An important exception in one point of the usual procedure
is exemplified below. Find log .07416. Entering the table on
page 14, line 741, we find in the column marked 6, the figures
*017. The asterisk is inserted to indicate that the first two
figures of the mantissa are to be taken from the line below,
instead of from above. Hence, the mantissa of log .07416 is
.87017 ; and by III. 2 log .07416 = 2.87017. The negative
sign is written over the characteristic, instead of before it, as
it applies to the characteristic only, the mantissa being positive.
The reason for placing this asterisk in the table is easily
seen. The last logarithm that begins with 86 is 86999. The
next one in order is 87005, but as this comes in the middle of
the page there is not room to print 87 in the same column with
005, so the asterisk is inserted to call the computer's attention
to this fact and bid him take the first two figures from below.
Verify the following statements :
log 863.2 =2.93611 log 3 =0.47712
log 1.29 =0.11059 log 2758 =3.44059
log 18000 =4.25527 log 64.58 =1.81010
log .92 =1.96379 log .00006 = 5.77815
log .04312 = 2.63468 log .00183 = 3.26245
It is proper at this point to explain that in practical com-
putation negative characteristics are very rarely used. Their
use is avoided by adding 10 to the characteristic and writing
10 after the logarithm. In this way the true value of the
logarithm is not changed. With this modification the four
logarithms above with negative characteristics become
log .92 =9.96379 10 log .00006 = 5.77815 10
log .04312 = 8.63468 10 log .00183 = 7.26245 10
This method will be used exclusively in the examples which
follow. After a little practice the lO's written after the log-
arithm may be omitted without danger of error in the final
EXPLANATION OF THE TABLES.
result. Eule III. 2 can be changed, therefore, to the fol-
lowing:
The characteristic of the logarithm of a number less than unity
is formed by subtracting from 9 the number of ciphers between the
decimal point and the first significant figure, and writing 10
after the logarithm.
Verify the following statements :
log .3628 = 9.55967 10 log .0026 = 7.41497 10
log .0796 = 8.90091 10 log .007 =7.84510 10
4. To Find the Number to Four Figures which
Corresponds to a Given Logarithm.
The method is best explained by an example. Given log x
= 1.79683, to find x. Disregarding the characteristic for the
moment, we enter the table with the first two figures of the
mantissa, 79, looking for them in the column headed with 0.
We find them on page 12. We then look in that part of the
body of the table which contains the logarithms beginning
with 79, for the number nearest to 683 ; we find 685.
The logarithm in the table nearest to our given logarithm is
now located. The first three figures of the corresponding
number are taken from the column JV, on the same line with
685. They are 626. The fourth figure of the number is that
which stands at the top of the column containing 685. It is
4. Hence, the number is 6264. To insert the decimal point
we note that the characteristic of the given logarithm is 1 ;
hence, we must have two figures before the decimal point.
We have, therefore, x = 62.64.
Given log x = 7.14168 10 find x. The nearest logarithm
in the table is .14176, on page 2 (notice the asterisk). The
corresponding number is 1386. The real value of the charac-
teristic is 7 10 = 3. Hence by III. 2 there must be two
ciphers between the decimal point and the first significant
figure. We can also obtain the number of ciphers by sub-
tracting the augmented characteristic 7, from 9, according to
the rule above. The result is, therefore, x = .001386.
XIV . EXPLANATION OF THE TABLES.
Verify the following statements :
log x = 1.73682, x = 54.55 log x = 9.74464 10, x = .5554
log x = 5.41621, x = 260700 log x = 4.48493, x = 30540
log x = 8.91929 10, x = .08304 log x = 3.14139, x = 1385
log x = 2.43625, x = 273.1 log x = 7.79012 10, x = .006168
log x = .64443, x = 4.41 log x = 6.56822 10, x = .00037
5. Exercises and Examples.
1. Compute the value of (1.789) 5 .
By III. 1, we have log (1.789) 6 = 5 X log 1.789.
log 1.789 = .25261
5
log (1.789) 5 = 1.26305 .-. (1.789)* = 18.33
2. Compute the value of 728 X 63.86 X -4792
log 728= 2.86213
log 63.86= 1.80523
log .4792 = 9.68052 10
by I. 1, log (728 X 63.86 X -4792) = j 14 -34788 - 10
> or 4.34/88.
Hence 728 X 63.86 X -4792 = 22280.
3. Compute the value of V 73.
log 73 = 1.86332.
By IV. \ 1, log ^]T= * log 73 = .62111,
^'73= 4.179
In dividing log 73 by 3, the division is not exact. Such cases
arise with great frequency in logarithmic work ; and the student
must carefully observe the two following rules :
(1.) Never carry the work beyond the number of decimal places
given in the table, that is with this table, five places.
(2.) When the division is not exact, always take in the last place
the figure that is nearest to the true result.
Thus, in the case just above, where we divide 1.86332 by 3, the
last step of the division is 2 divided by 3. Now 3 goes into 2 more
nearly once than no times ; hence, we take 1 for the last figure.
Sometimes, when the divisor is an even number, the result falls
just half way between two integers in the last place. We then
take at pleasure either the larger or smaller of these two figures
for the last figure. The following example illustrates this :
4. Find 1/471L log 4711 = 3.67311,
.-. log V/4711 J log 4711 = 1.83655 or 1.83656.
Both of these logarithms give 68.64 as the result to four figures.
EXPLANATION OF THE TABLES. XV
5. Find y / .06398.
log .06398 = 8.80604 10.
We cannot divide this logarithm by 7 without getting an awk-
ward result. But if we add and subtract 60, we have
log .06398 = 68.80604 70,
where the number subtracted from the logarithm is now ten times
the number by which we must divide ; and hence, after the divi-
sion, it will be reduced to 10. This is the best practice for such
cases. Performing the division, we have
log >/.06398 = 9.82943 10, .'. 1/.06398 = .6752.
log l/"27" = log27 =* X 1.43136= .71568
log (9.261)? = f log 9.261 = f X 0.96666 = .41428
BylLgl logz = .30140
x = 2.002.
7. x= 68.96X^^228 ^
39 X (8.642)* X (.96) 2
log 68.96= 1.83860
log f .4228 = I log (.4228) = X 29.62613 30 = 9.87538 10
log of numerator = 11.71398 10
log 39 = 1.59106
log (8.642)* = f log 8.642 = f X 0.93661 = 1.56102
log (.96) 2 = 2 log (.96) = 2 X 9.98227 10 = 19.96454 20
f 23.11662-20
log of denominator = { QJ . Q U662
log x = log of numerator log of denominator = 8.59736 10.
Hence x = .03957.
In order to explain clearly each step in working this example,
the amount of written work set down is much greater than is
allowable in ordinary practice. The work for the same example
is arranged below in more concise form, and at the same time the
10's are omitted from the logarithms with negative character-
istics.
log 39 = 1.59106 log 68.96 = 1.83860
log (8.642) * =1.56102 log ^.4228 =9.87538
log (.96) 2 = 9.96454 log of num. = 1.71398
log of denom. = 3.11662 3.11662
x= .03957 log a; = 8.59736
XVI EXPLANATION OF THE TABLES.
EXAMPLES.
Find the values of the following numerical expressions, and
give the results to four significant figures :
1. 839.6 X /6129. Ans. 65730 5 21 . 38 X 6.296 X. 412. Ans . .2292
7 X ^ 41290
2. 19.63X^6^2. ^n, 173.4 4.19X6.2X^^67
(3.339) s X 142.9
7 298.7 X 563 X/TT Am . 7266
(2.96)*
. ft (9.8)3X^.4621X18.
F 12 X Vl7 1/41.63 X (2.649) 5
6. The Arithmetical Complement of the Logarithm
or Co-logarithm. To compute the value of - by log-
b
arithms, we may take either log a log b, or log a -f- log-.
b
Log - = log 1 log b = log b is called the co-logarithm of
b
b. We have, therefore, the following rule :
To form the co-logarithm of a given number subtract the log-
arithm of the number from 0.
It is customary in practice to subtract the logarithm from
10 instead of from 0, and then to write 10 after the result ;
that is, the logarithm is subtracted from 0, written in the form
10.00000 10. If the logarithm is one which has been itself
augmented by 10, the two 10's, that in the subtrahend and
that in the minuend, cancel each other.
Ex. Find colog 729.6. Log 728.6 = 2.86308. Subtracting this
from 10.00000 10, the result is colog 729.6 = 7.13692 10.
Ex. Find colog .0641. Log .0641 = 8.80686 10. Subtracting
this from 10.00000 10, the result is colog .0641 = 1.19314.
Verify the following statements :
colog 9986 = 6.00061, colog 3.9 = 9.40894,
colog 7.298 = 9.130SI ), colog 380.6 = 7.41953,
colog .4682 = .32957, colog .005 == 2.30103.
EXPLANATION OF THE TABLES. XV11
With a little practice the student can write down the colog
directly from the table, as readily as the log itself. The prac-
tical rule is to subtract each figure of the logarithm, beginning
at the left, from 9, except the last or right-hand figure, which
must be subtracted from 10. When the characteristic of the
logarithm is 0, care must be taken not to forget to subtract
this from 9, just as any other characteristic would be sub-
tracted.
The practical advantage of using cologs consists in the fact
that thereby the number of separate operations required to
obtain the log of the result is reduced. For example, suppose
we wish to calculate log ^ ^-^. Without using co = logs
three operations are required :
(1.) to find log a + log 6 -4- log c,
(2.) logd + loge + log/,
(3.) to subtract (2) from (1).
If, on the other hand, cologs are used, these three opera-
tions are reduced to one, viz. : to find log a -4- log b -\- log c -f-
colog d -f- colog e -f colog /.
Ex. By using cologs the work of Ex. 7, p. xv., may be arranged
in the following concise form :
log 68.96 = 1.83860
log 1^.4228 = 9.87538
colog 39 =8.40894
colog (8.642)1 =8.43898
colog (.96) 2 == 0.03546
logo; =8.59736
7. To Find the Logarithm of a Number which
Consists of Five Figures.
This is accomplished by the aid of the operation known as
interpolation. Let the given number be 31.687. The table
gives log 31.68 = 1.50079 and log 31.69 = 1.50092. To find
log 31,687 a small correction must either be added to log 31.68
or subtracted from log 31.69.
The whole difference between two consecutive logarithms in
XV111 EXPLANATION OF THE TABLES.
the table is called the tabular difference. In this case the
tabular difference is 13. That is, the logarithm increases by
13 for a change of unity in the fourth place in the number.
Hence, for 7 in the fifth place the proportional change in the
logarithm will be seven-tenths of 13, or 9.1, the nearest integer
to which is 9 ; hence, 9 is the correction to be added to log
31.68 to obtain 31.687. Therefore,
log 31.687 = 1.50079 + .00009 = 1.50088
This method of determining the correction for the fifth
figure is not theoretically correct, for it assumes that log-
arithms vary proportionally with the corresponding numbers;
but while this is not true, it is applied here for such a small
interval that no appreciable error arises from its use.
The work of computing corrections for the fifth figure is
performed in the little auxiliary tables in the column headed
Prop. Pts. (Proportional Parts). On the same page with log
31.68 we find one of these tables headed by the tabular differ-
ence 13. In this table we look in the column to the left of
the vertical line for the fifth figure, 7, of the given number.
The corresponding number to the right of the vertical line,
which is 9.1, is the required correction, the nearest integer to
which must be added to the logarithm corresponding to the
first four figures of the given number.
The student should accustom himself to apply the correc-
tion for the fifth figure mentally, and to write nothing on the
paper except the corrected logarithm.
Verify the following statements :
log 414.23 = 2.61724, log 69.426 = 1.84152,
log 3.8642 = 0.58706, log 1418.1 = 3.15171 ,
log .43007 = 9.63354, log 85672. = 4.93284.
8. To Find the Number to Five Figures Corres-
ponding to any Logarithm.
Let logo; = 2.38647. Look in the table for the nearest man-
tissa that is less than 38647, not for that which is absolutely
EXPLANATION OF THE TABLES. XIX
nearest, as when only four figures are required. This is found
to be 38632, which corresponds to the natural number 2434.
These are the first four figures of x. Next find the tabular
difference, which is 18. Then subtract the mantissa taken
from the table (38632) from the mantissa of the given log-
arithm (38647); the difference is 15. Hence, we have the
problem: If a difference of 18 in the mantissse makes a
change of a unit in the fourth figure of the number, what
change will be made by a difference of 15 in the mantissse ?
Evidently we have the proportion
18 : 1 = 15 : difference required
or difference = i-f = | = .8 ;
that is, the correction is .8 of a unit in the fourth place, or 8
units in the fifth place. Hence, the figures in the number x
are 24348, and inserting the point after the 3, because the
characteristic is 2, we have x = 243.48.
The work of determining the fifth figure is performed in
the marginal tables of Prop. Pts. Find the one correspond-
ing to the tabular difference IS^and look on the right of the
vertical column for the number nearest to 15, the difference
between the given log and the next smaller one in the table.
We find 14.4 and the corresponding number on the left of the
vertical line, which is 8, is the required fifth figure.
Verify the following statements :
log x = 3.28642, x = 1933.8 log x = 7.63419 10, x = .0043072
log x = 1.46010, x = 28.847 log x = 2.31419, x = 206.15
log x = 9.38642 10, x = .24346 log x = .76787, x = 5.8596
9. Exercises and Examples.
_ (36.842)* X. (1.6272)' X &_ -
~ 1/.062416 X 72.983 X ^189*
log (36.842)4 = 1.56634 X*= -52211
log (1.6272) 2 = .21144 X 2 = .42288
log 87 =1.93952
colog V .062416 = 1.20471 -s- 2 = .60235
colog 72.983 8.13678
colog ^189 = 7.72354 -=- 3 = 9.24118
x = 7.3252 log*= .86482
XX EXPLANATION OF THE TABLES.
EXAMPLES.
In working these examples use cologs wherever necessary, and
arrange the work as on preceding page.
67.284 X .10003
L f 74^99X6.7843' ^..10953
63.842 X
(42.32)* X(.02478) 3 - 1/2
(7.2S43) 8 X 1^.00067894
3 " (620.01)* X 489.62 ^m. 306.49
1986.1 X ff 92.836
3.403.75
5.
a
V 1000000
7. 1/IO~X ^100 X HOW! ^ m . 82.542
10. Numbers with Six Figures. As a general rule, we
cannot work to six figures in natural numbers with a table of
five-place logarithms, for when the correction for the sixth
figure is applied it will usually be too small to make any
difference in the logarithm. On the first page or two of the
table, however, where the logarithms vary rapidly, it can be
done with approximate accuracy.
The correction for the sixth figure is always one-tenth of the
correction for the same figure in the fifth place.
Ex. To find log 13.9647.
log 13.96 = 1.14489
correction for fifth figure = 12.4
" " sixth " = 2.17
total correction = 14.57, nearest integer = _ 15
log 13 9647 =1.14504
EXPLANATION OF THE TABLES. XXi
Ex. Find a, given log x 2.21647,
nearest log in table = .21643, corresponding to 1646
difference 4
nearest smaller prop.1
pt. under tab. diff. 26 / = ^ J corresponding to 1
difference remaining ~IA * for the fafth ^
1.4 X 10 (because sixth figure is required) = 14, corresponding to
5 for the sixth figure. Hence, x = 164.615.
Verify the following:
log 1219.35 = 3.08613. log x = 3.12964, x = 1347.84.
log 10.7642 = 1.03198. log x = 0.06432, x = 1.15963.
TABLE II. Constants and Their Logarithms. (Page #0.)
11. No description of this table is necessary. The loga-
rithms are given to seven places, instead of five, in case a
greater degree of accuracy should be required. If only the first
five places are used, the fifth figure must be increased by 1, if
the sixth figure is 5, or more.
TABLE III. Logarithmic Sines, Cosines, Tangents and
Cotangents. (Pages 21-66.)
12. The logarithms of the trigonometric functions are used
in computation much more frequently than the functions them-
selves, which are called natural functions. For this reason this
table is given more prominence than that of the natural func-
tions. The table gives the logarithms of the functions for each
minute from to 90. The functions of angles not expressed
evenly in minutes can be found by interpolation, as explained
below.
Since sec and esc are the reciprocals of cos and sin respec-
tively, their logs can always be found by taking the cologs of
the latter.
The sin and cos of all angles and the tan of angles less than
45 are less than unity ; hence, their logarithms have negative
characteristics. For this reason the characteristics of all these
logarithms are increased by 10 in the tables.
XXU EXPLANATION OF THE TABLES.
13. To Find the Logarithmic Function of an Angle
Less than 90.
Enter the table with the given number of degrees, which
will be found at the top of the page, if it is 44 or less, but at
the bottom of the page, if it is greater than 44. The func-
tion required is read at the top or bottom of the page, accord-
ing as the number of degrees is at the top or bottom, and the
required logarithm is taken from the corresponding column.
The minutes are read in the left hand column of the page, if
the degrees are read at the top, but in the extreme right hand
column of the body of the table if the degrees are read at the
bottom.
EXERCISES.
1. Find log sin 24 38'. 24 is at the top of page 46, and the log
sin column for 24 is the first column of logarithms on the page.
Running dow.n the page until we corne to 38', we find log-sin 24
38' = 9.61994.
2. Find log tan 57 16'. 57 is at the bottom of page 54. Run-
ning up the page in the column marked at the bottom log tan,
until we come to the line with 16' on the right, we find log tan
57 IV = 0.19192.
Verify the following statements:
log sin 39 16' = 9.80136, log cos 8 19' = 9.99541,
log tan 63 24' = 0.30037, log cot 54 9' = 9.85887,
log cos 41 31' = 9.87434, log tan 82 56' = 0.90670,
log cot 26 12' = 0.30798, log cot 7 = 0.91086,
log cos 31 = 9.93307, log sin 19 12' = 9.51702.
14. Interpolating for Seconds.
Find the logarithmic functions for the degrees and minutes as
before; then apply a correction for the seconds, as explained below.
This correction must be added if the function is sin or tan, and
subtracted if the function is cos or cot.
Find log sin 16 28' 35".
log sin 16 28' = 9.45249, and the tabular difference is 43 ; that
is, the log sin increases by 43, while the angle increases by 1'.
Hence, the proportional increase for 1" is f $ , and for 35" it is
U X 35 = J VV L = 25.08 . . . , the nearest integer to which is the
required correction. Hence,
log sin 16 28' 35" = 9.45249 + .00025 = 9.45274.
EXPLANATION OF THE TABLES. XX111
The auxiliary table of proportional parts for tabular differ-
ence 43 will give the same result. The column to the left of
the vertical line in these auxiliary tables gives the number of
seconds, arranged in the order 6, 7, 8, 9, 10, 20, 30, 40, 50.
If the correction for 1, 2> 3, 4, or 5 seconds is required it is
obtained by taking one-tenth of that for 10, 20, 30, 40, or 50
respectively. The work can be arranged concisely as follows,
but it is desirable in actual practice to compute the correction
mentally and to write only the complete logarithm :
log sin 16 28' = 9.45249
correction for 30" = 21.5
" " 5" = 3.58
log sin 16 28' 35" = 9.45274
Find log cot 61 13' 19". log cot 61 13' = 9.73987
correction for 10" (tab. diff. 30) = 5.0
u u g// __ 45
nearest integer to total correction = 10.0
Subtract correction because function is cot, 10
log cot 61 13' 19" = 9.73977
On pages 22 to 27 of the table, on account of the large
number of differences which occur, owing to the rapid varia-
tion of the logarithms, different arrangements of the tables
of Prop. Pts. are made. If the logarithm required falls on
pages 25 to 27, and it happens that the tabular difference is
one for which a table of proportional parts is given, the pro-
cedure is the same as above; otherwise as follows:
Find log tan 3 51' 26"
log tan 3 51' = 8.82799, tab. diff. = 188.
This tabular difference is not given, so we use the auxiliary
tables for 185 and 3 (because 185 -f 3 = 188) instead.
tab. diff. 185 ( Correction for 20" = 61.7
" 6" = 18.5
tab. diff. 3 { ' ?" 1-0
I " " 6"= 0.3
81.5
Hence, the total correction to be added is 82 and log tan 3
61' 26" = 8.82881.
XXIV EXPLANATION OF THE TABLES.
In a case of this kind it is, perhaps, just as easy to compute
the correction without using the auxiliary tables.
On pages 22 to 24 the Prop. Pt. is given for one second for
each tabular difference for log sin, log tan, and log cot. Log
cos varies so slowly in this part of the table that no auxiliary
tables are necessary.
Find log sin 1 48' 53".
log sin 1 48' = 8.49708, tab. diff. = 400
Prop. pt. for 1" (tab. diff. 400) = 6.67
" " 53" = 6.67X53 =353.51
. \ correction to be added = 354.
and log sin 1 48' 53" = 8.49708 + .00354 = 8.50062
On account of the very rapid variation in the log sin and
log tan at the beginning of the table, the theory that the varia-
tion of the log is proportional to that of the angle, leads to
results which are sometimes appreciably in error. For this
reason, when great precision is required, Table IV., pp. 67, 68,
should be used in finding the log sin and log tan of angles less
than 4. An explanation of this table is given below, 19.
Yerify the following statements :
log cos 17 38' 42" = 9. 97907, log tan 5 38' 5" = 8.99416,
log tan 84 9' 13" = 0.98972, ^log sin 1 12' 38" = 8.32482,
log sin 61 41' 31" = 9.94469, log cos 26 28' 37" = 9.95188,
log cos 87 6' 14" = 8.70351, log cot 9 1' 43" = 0.79889,
log cot 86 53' 34" = 8.73467, log sin 45 43' 28" = 9.85491.
15. To Find the Logarithmic Function of an Angle
>90.
According to the theorems demonstrated in Elements of
Trigonometry 28-31, and the rules on page 40, summariz-
ing the results, the functions of any angle can be found if
those of all angles less than 90 are known. These results
are given here in the form of the following rules:
I. To find the function of an angle between 90 and 180 sub-
tract the angle from 180 and look for the same function of the
difference, or subtract 90 from the angle and look for the co-function
of the difference.
II. To find a function of an angle between 180 and 270 sub-
tract the angle from 270 and look for the co-function of the differ-
EXPLANATION OF THE TABLES. XXV
ence, or subtract 180 from the angle and look for the same function
of the difference.
III. To find a function of an angle between 270 and 360 sub-
tract the angle from 360 and look for the same function of the differ-
ence, or subtract 270 from the angle and look for the co-function
of the difference.
The second alternative in each of these rules is better if the
angle has minutes and seconds, for there is less danger of
making a mistake in taking the difference.
EXERCISES.
1. Find log cos 117 19' 35".
By rule I. log cos 117 19' 35" = log (sin 27 19' 35").
NOTE. In taking the logarithm of a negative quantity we pro-
ceed as if the quantity were positive. To the logarithm when
found, we prefix the symbol ( ) or annex the symbol n. Neither
of these signs affect the operations to which the logarithm may be
subjected, but are used merely to remind the computer that the
corresponding numbers are negative.
log sin 27 19' 35" = 9.66187,
log cos 117 19' 35" = ( ) 9.66187.
2. Find log tan 242 20' 17".
By rule II. log tan 242 20' 17" = log tan 62 20' 17" = 0.28054.
Verify the following statements :
log sin 300 24' = ( ) 9.93577 log cot 200 30' 30" = 0.42707
log cos 216 14' 33" = ( ) 9.90662 log sin 138 48' 6" = 9.81867
log tan 101 6' 52" = ( ) 0.70674 log cos 342 38' 15" = 9.97975
16. To Find an Angle Given one of its Logarith-
mic Functions.
A further glance at the general constitution of the table is
first necessary. Upon each page of the table are four columns
of logarithms, the first and fourth are logarithmic sines and
cosines, the second and third are logarithmic tangents and
cotangents. The logarithms increase, going toward the back
of the table in the first and second columns, and then passing
into the fourth and third columns respectively, they increase,
going toward the front of the table. Eemembering this, the
place of any given logarithm in the table can be found readily.
XXVI EXPLANATION OF THE TABLES.
The rules for finding an angle from its logarithmic function
are as follows:
If the given function is log sin or log cos look for the nearest
smaller logarithm in the first or fourth column ; if it is log tan or
log cot, look in the second or third column.
Read the degrees at the top or bottom of the page, according as
the name of the given function is at the top or bottom of the column
in which the given logarithm is located.
Head the minutes on the left or right according as the degrees
are read at the top or bottom of the page, and in the same line
with the nearest logarithm smaller than the given one.
Determine the number of seconds by proportion and add them to
the degrees and minutes found, if the given function is log sin or
log tan, but subtract them if it is log cos or log cot.
EXERCISES.
1. Given log sin 6 = 9.86592, what is 6?
In the fourth column on p. 64 we find 9.86589, and log sin is
read at the bottom. Hence, the degrees and minutes are 47 15'.
The tabular difference is 11 and the difference between the given
log and log sin 47 15' is 3. Hence, exceeds 47 15' by T 3 T of one
minute. This fraction reduced to seconds is T 3 T X 60 16". Hence,
= 47 15' 16".
To use the auxiliary table to find the number of seconds, we
arrange the work as follows, using table for tabular difference 11.
whole difference = 3
nearest smaller prop. pt. = 1.8, corresponding to 10"
difference remaining = 1.2 " " 6"
whole number of seconds 16"
NOTE. The number of seconds corresponding to 1.2 under tabular
difference 11 is, according to the table, either 6" or 1" ; but 6" is
really a little nearer than 7", as we found above.
2. Given log cot = 0.72654, find 0.
On p. 32, in the third column, we find 0.72643, and log cot is
read at the top ; hence, the degrees and minutes are 10 38'. The
tabular difference is 70, and the difference between log cot 6 and
0.72643 is 11. Hence, using table of proportional parts, we have
whole difference =11
nearest smaller prop. pt. = 10.5, corresponding to 9"
difference remaining = .6,
EXPLANATION OF THE TABLES. XXV11
as this is less than half the prop. pt. for 1" (1.17), the entire correc-
tion is 9", which is subtracted from 10 38', giving = 10 37' 51".
3. Given log tan 6 = 8.61246, find 6.
On page 24, log tan 2 20' = 8.61009.
difference = 237, tab. diff. = 310, prop. pt. for 1" = 5.17,
no. of seconds = ^- = 46". .-.0 = 2 20' 46".
5.17
In these three exercises the results are incomplete, because we
know from Trigonometry that there are always two angles less
than 360 corresponding to any given trigonometric function.
The complete answers are as follows : 1. 6 = 47 15' 16" and 180
47 15' 16" = 132 44' 44", because sin 6 is positive in the first and
second quadrants. 2. 6 = 10 37' 51" and 180 -f 10 37' 51" = 190
37' 51". 3. 6 = 2 20' 46" and 180 + 2 20' 46" = 182 20' 46", be-
cause tan and cot 6 are positive in the first and third quadrants.
4. Given log cos 6 = ( ) 9.62983, find 0.
Assume that cos 6 is positive and find the angle corresponding
to it in the first quadrant. We find on p. 47 log cos 64 46' = 9.62972.
whole difference = 11
nearest smaller prop. pt. = 9.0, corresponding to 20"
difference remaining = 2.0 " " 4"
number of seconds to be subtracted, 24"
Hence, log cos 64 45' 36" = 9.62983.
Since the cos is negative in the second and third quadrants,
we have d - / 180 ~ 64 45/ 36// = 115 14/ 24//
\ 180 + 64 45' 36" = 244 45' 36".
When one or both values of the required angle are not in the
first quadrant, the following rules are to be followed :
To find an angle in the second quadrant, subtract the angle taken
from the table from 180.
To find an angle in the third quadrant, add the angle taken from,
the table to 180.
To find an angle in the fourth quadrant, subtract the angle taken
from the table from 360.
Verify the following statements :
log sin 0= 9.28642, 0= 11 9' 1" and 168 50' 59".
log cos 0= 8.46321, 0= 88 20' 6" " 271 39' 54".
log tan 0= 0.12983, 0= 53 26' 22" " 233 26' 22".
log cot 0= 9.62412, = 67 10' 36" " 247 10' 36".
log sin = ( ) 9.96419, = 247 3' 0" " 292 57' 0".
log cos = ( ) 9.78416, = 127 28' 15" " 232 31' 45".
log tan 0=( ) 9.42317, = 165 9' 36" " 345 9' 36".
log cot = ( ) 8.76432, = 93 19' 35" " 273 19' 35".
XXV111 EXPLANATION OF THE TABLES.
17. Functions of Negative Angles. To find the log-
arithmic functions of negative angles, follow the formulae
given in 31, Elements of Trigonometry.
18. General Remarks. In using a five-place table of
logarithmic functions the computer should remember that the
seconds in his results will be, in general, only approximately
correct. Nevertheless, angles can be determined in most parts
of the table more closely than to tenths of a minute ; so that
it seems preferable to give tables of proportional parts for
seconds, rather than for tenths of a minute.
Attention is here called to the fact that throughout all the
tables a final five is sometimes marked with a small dash over
it, thus 5, and sometimes it is not so marked. This mark is
used to indicate that if, for any reason, the computer wishes
to use a smaller number of decimal places than are gwen in
the table, the 5 is to be dropped without increasing the pre-
ceding figure by unity. If the 5 is not marked in this way
the preceding figure must be increased by unity if the 5 is
dropped.
The student may vary somewhat the procedure in the matter
of interpolation as he becomes accustomed to using the tables.
For example : in finding log 18769 he may take log 1877 from
the tables and subtract the correction for 1, instead of taking
log 1876 and adding the correction for 9. Again, in finding
log cos 78 38' 56" he may take log cos 78 39' and add the
correction for 4" instead of taking log cos 78 38' and sub-
tracting the correction for 56". Numerous points of this kind,
which in many cases will shorten the work, will 'suggest them-
selves, and need not be specified here.
EXAMPLES.
Find in each of the following examples :
1 n - 6 - 2984 8ip ' 63 18/ W 6 = / 127 l/ 7 "
" 7.5692 cot 116 36' 12" " 1 307 1' 1"
o 2.93 tan 48 6' 38" = / 121 34 ' 3 "
" 14.12 sin 26 13' 42" " i 238 25' 57"
EXPLANATION OF THE TABLES. XXIX
7 58' VI"
172 1' 43"
I sin 3 146 12' 19" X tan 78 12' 32" a _
8 - 8m := \ cot* 12 14' 6" X cos 64 4' 55" '
352 V 43"
. a _ .93862 cos 2 312 38' 40" e= f 32 55' 19"
4. cot a : _ -
TABLE IY. (Pages 67 and 68.)
19. Sine and Tangent of Small Angles. This table
derives its usefulness from the fact that when an angle (a) is
small the ratios ?!5L? and ^-^ vary but slowly. The quan-
a a
tities S and T in the table are the logarithms (increased by
10) of these ratios, where the angle is expressed in seconds.
Hence, to find log sin and log tan of a small angle we have
the formulae
log sin a = log a" + S
log tan a = log a" + T
and to find a small angle from its log sin or log tan we have
log a" = log sin a S
log a" =. log tan a T
Ex. Find log tan 26' 51".
26' 51" = 1611" log 1611 = 3.20710
T (for 270=4.68558
.-. log tan 26' 51" = 7.89268
(the same calculated from Table III. is 7.89264, which is thus
shewn to be in error four units in the fifth place).
Ex. Given log sin a = 8.36892, find a.
From Table III. we find that a = 1 20 7 approximately ;
hence, the proper value of S (from Table IY) is 4.68554. We
have, therefore, log sin a _ s= 3>68338 = log a //
a = 4824" -=1 20 / 24".
Yerify the following statements, by means of Table IY :
log sin 57' 36" = 8.22412.
log tan a = 8.19632, a = 54' 1".
To find the cosine or cotagent of an angle nearly 90 use
the same table, taking the sine or tangent, as the case may be,
of the complement of the given angle.
XXX EXPLANATION OF THE TABLES.
TABLE V. Natural Functions. (Pages 69-78.)
20. By the terms natural sine, cosine, etc., are meant the
actual values of these functions. The table is used compara-
tively seldom, and for that reason the functions are given for
every five minutes only. To find the functions for inter-
mediate minutes the process of interpolation by simple pro-
portion is used. Thus, to find sin 51 18', we have
sin 51 20' = .78079
sin 51 15' = .77988
difference for 5' = 91
hence, correction for 3' = f of 91 = 55,
and sin 51 18' = .77988 + .00055 = .78043.
The rules given above, for adding and subtracting correc-
tions and for finding functions of angles greater than 90,
apply here the same as in the case of Table III.
The results of interpolating minutes in that part of the
table which gives the cot of angles less than 15 and the tan-
gents of angles between 75 and 90 will, in general, not be
correct in the last place. Hence, when considerable .precision
is required in these cases the function should be found by
taking the natural number corresponding to the logarithm
found in Table III.
TABLE VI. Circular Arcs Expressed in Radians. (Page 74.)
This table gives to seven decimal places the number of
radians for every degree up to 180, with auxiliary tables for
minutes and seconds.
EXERCISES.
L How many radians in 126 38' 19" ? From the table we have
number of radians in 126 =2.1991149
38' = .0110538
" " 19"= .0000921
" " 126 38' 19" = 2.2102608
2. How many degrees, minutes and seconds in 4.6832964 radians ?
As this number of radians exceeds 180, we subtract the number of
EXPLANATION OF THE TABLES. XXXI
radians in 180 and find the degrees, minutes and seconds in the
remainder. This last added to 180 is the result :
Given number of radians = 4.6832964
Radians in 180 =3.1415927
Difference =1.5417037
Radians in 88 = 1.5358897
.0058140
Radians in 19' = .0055269
.0002871
Radians in 59" = 2860
Result = 268 19' 59" .0000011
The last difference, .0000011, corresponds to less than half a
second.
TABLE VII. Napierian Logarithms of Numbers. (Pages
75, 76.)
Although these logarithms are not used for purposes of
practical computation, their values are sometimes required in
calculating values of transcendental functions, and for other
purposes. -The table gives the logarithm of each number from
1 to 1000. As the value of the characteristic does not depend
upon the position of the decimal point, nor the value of the
mantissa solely upon the sequence of figures in the corres-
ponding number, we cannot use the table just as we do a
table of common logarithms. If log 363.8 is required we can
find it by interpolating between log 363 and log 364 ; but if
log 3638 is required we must find log 363.8 in the manner just
indicated, and then add log 10. The work is as follows :
log 363 =5.89440
log 364 = 5.89715
difference = 275
.8 of difference = 220
adding this to log 363 gives log 363.8 = 5.89660
log 10 = 2.30259
log 3638 = 8.19919
To find the number corresponding to a given Napierian
logarithm we first subtract as many times log 10 as may be
necessary to bring the logarithm within the limits of the
XXX11 EXPLANATION OF THE TABLES.
table. Then find the number corresponding to this difference
and multiply it by the power of 10, whose logarithm was
subtracted at the beginning. Thus, to find the number whose
Napierian logarithm is 9.62983 :
log 100 = 2 log 10 = 4.60517
9.62983 4.60517 = 5.02466
5.02466 is the logarithm of some number between 152 and 153.
Given log =5.02466
log 152 =5.02388
difference 78
tabular difference = 656
78 -4- 656 = .12.
5.02466 is the logarithm of 152.12.
Hence, 9.62983 is the logarithm of 152.12 X 100 = 15212.
TABLE I.
COMMON LOGARITHMS
OF NUMBERS.
If.
O
l
2
3
4
5
6 7
8
9
Prop. Pts.
100
00 000
043
087
130
173
217
260
303
346
389
01
432
475
518
561
604
647
689
732
775
817
44
43
42
02
03
860
oi 284
903
326
945
368
988
410
452
*0 7 2
494
*ii5
536
*I57
578
*i 99
620
*2 4 2
662
i
2
4.4
8.8
4-3
8.6
&
04
703
745
787
828
870
912
953
995
*o 3 6
*078
3
13.2
I2. 9
12.6
05
02 119
160
202
243
284
325
366
407
449
490
4
17.6
17.2
16.8
06
531
572
612
653
694
735
776
816
857
898
i
22.
26 A
21.5
21.0
25.2
07
08
09
938
03 342
743
979
383
782
423
822
*o6o
463
862
*IOO
503
902
*i 4 i
543
941
*i8i
583
981
*222
62 3
*O2I
*262
66 3
*o6o
*3<>2
703
*IOO
I
9
*v.|
30-8
35-2
39.6
30.1
38.7
29.4
33-6
37.8
110
04 139
179
218
258
297
336
376
415
454
493
II
532
57 i
610
650
689
727
766
80 5
* 844
* 883
41
40
39
12
922
961
999
*o 3 8
*o77
*II5
*i54
*I 9 2
i
4.1
4.0
7.0
13
05 308
346
385
423
461
500
538
576
614
652
2
8.2
8.0
O :?
7.8
14
15
690
06 070
729
108
767
145
805
183
843
221
88l
258
918
296
956
333
994
*0 3 2
408
3
4
12.3 12.0
16.4 16.0
ii. 7
15-6
16
446
483
558
595
633
670
707
744
781
5
20.5
20. o
19-5
6
24.6
24.0
23.4
17
819
856
893
930
967
*oo4
*o4 1
^078
*"5
^151
28.7
28.0
27.3
18
07 188
225
262
298
335
372
408
445
482
518
g
3.2.8
72.
31.2
19
555
628
664
700
737
773
809 | 846
882
9 36.9 36-0
35-1
120
918
954
990
*02 7
*o6 3
*099
*I35
*i 7 i
*207 *243
21
08 279
314
350
386
422
458
493
529
565 600
38
37
36
22
636
672
707
743
* 778
814
849
884
920
* 955
i
' 3-8
3-7
3-6
23
991
*026
*o6i
*o 9 6
*i6 7
202
*2 3 7
*2 7 2
2
7-6
7-4
7.2
24
09 342
377
412
447
482
517
552
587
621
* 656
3
11.4 ii. i
_ _ _ _ . o
10.8
25
26
691
10 037
726
072
760
106
795
140
830
175
864
209
8 99
243
934
278
9 68
312
346
4
1
15.2 14.8
19.0 18.5
22.8 22.2
14.4
18.0
21.6
27
380
415
449
483
517
55i
585
619
653
687
7
26.6 25.9 25.2
28
721
755
789
823
857
890
924
958
992
*02 5
8
30.4 2 9 .6 28.8
2 9
ii 059
093
126
160
193
227
26l
294
327
361
9
34-2 33-3! 32.4
180
394
428
461
494 528
561
594
628
661
694
31
727
760
793
826 860
893
926
959
992
*O24
35
34
33
32
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659
667
675
21
22
23
24
25
26
27
28
29
684
767
850
933
72 016
099
181
263
346
692
775
858
941
024
107
189
272
354
700
784
867
950
032
H5
362
709
792
875
958
041
123
206
288
370
717
800
883
966
049
132
214
296
378
725
809
892
975
057
140
222
304
387
734
817
900
983
066
148
230
313
395
742
825
908
991
074
156
239
321
403
750
834
917
999
082
165
247
329
411
759
842
925
*oo8
090
173
255
337
419
* i
2
3
4
I
8
9
8
0.8
1.6
2.4
3-2
4.0
4-8
5-6
6.4
7.2
580
428
436
444
452
460
469
477
485
493
501
3i
32
33
34
35
36
37
38
39
509
59i
673
754
835
916
997
73 078
159
5i8
599
681
762
843
925
*oo6
086
167
526
607
689
770
852
933
*OI 4
094
175
534
616
697
779
860
941
*022
102
183
542
624
705
787
868
949
*o3o
in
191
550
632
713
795
876
957
*o 3 8
119
199
558
640
722
803
884
965
*046
127
207
567
648
730
811
892
973
*54
135
215
575
656
738
819
900
981
*062
143
223
583
665
746
827
908
989
*070
151
231
i
2
3
7
0.7
1.4
2.1
540
239
24?
255
263
272
280
288
296
304
312
4
2.8
41
42
43
44
8
47
48
49
320
400
480
560
640
719
799
878
957
328
408
488
568
648
727
807
886
965
336
416
496
576
656
735
8i5
894
973
344
424
5>4
584
664
743
823
902
981
352
432
5"
592
672
751
830
910
989
360
440
520
600
679
759
838
918
997
368
448
528
608
687
767
846
926
00|
376
456
536
616
695
775
854
933
*oi 3
384
464
544
624
703
783
862
941
*020
392
472
552
632
711
791
870
949
*028
1
8
9
3-5
4-2
4-9
5-6
6.3
550
74 036
"44
052
060
068
076
084
092
099
107
N.
1 1
9
3
4
5
6
7
8
9
Pro]
. Pis.
II
N.
O
1
2
3
4
5
G
7 8
9
Prop. Pts.
550
51
52
53
54
P
57
58
59
560
61
62
63
64
65
66
3
69
570
7i
72
73
74
3
79
580
81
82
83
84
85
86
87
88
89
590
9i
92
93
94
95
96
97
98
99
600
74 036
044
052
060
068
076
084
092
099
107
I
2
3
4
5
6
8
9
i
2
'3
4
5
6
8
9
8
0.8
1.6
2.4
3-2
4.0
4.8
5-6
6.4
7.2
7
0.7
1.4
2.1
2.8
3-5
4.2
4-9
5-6
6.3
H5
194
273
35i
429
50?
586
663
74i
123
202
280
359
437
515
593
671
749
131
2IO
288
367
445
523
601
679
757
139
218
296
374
453
53i
609
687
764
147
225
304
382
461
539
617
695
772
155
233
312
39
468
547
624
702
780
162
241
320
398
476
554
632
710
788
170
249
327
406
484
562
640
718
796
178
257
335
414
492
570
648
726
803
881
186
265
343
421
500
578
656
733
811
819
827
834
842
850
858
865
873
889
896
974
75 051
128
205
282
358
435
5ii
904
981
059
136
213
289
366
442
5i9
912
i43
220
297
374
450
526
920
997
074
151
228
305
38i
458
534
927
*oo5
082
159
236
312
389
465
542
935
*OI2
089
166
243
320
397
473
549
943
*O2O
097
174
251
328
404
48l
557
950
*028
105
182
259
335
412
488
565
958
*035
"3
343
420
496
572
966
*43
120
197
274
351
427
504
580
587
595
603
610
618
626
633
641
648
656
664
740
815
8 9 r
967
76 042
118
193
671
747
823
899
974
050
^25
200
275
679
755
831
906
982
057
133
208
283
686
762
838
914
989
065
140
215
290
694
770
846
921
997
072
148
223
298
702
778
853
929
*oo5
080
155
230
305
709
785
861
937
*OI2
08 7
163
238
313
717
793
868
944
*O2O
095
170
245
320
724
800
876
952
*O27
103
178
253
328
732
808
884
959
*35
no
185
260
335
343
350
358
365
373
380
388
395
403
410
418
492
567
641
716
790
864
938
77 012
085
425
500
574
649
723
797
871
945
019
433
507
582
656
730
805
879
953
026
440
515
589
664
738
812
886
960
034
448
522
597
671
745
819
893
967
041
455
530
604
678
753
827
901
975
048
462
537
612
686
760
834
908
982
056
470
545
619
693
768
842
916
989
063
477
552
626
701
775
849
923
997
070
485
559
634
708
782
856
930
*oo4
078
093
100
107
H5
122
129
137
144
151
159
232
305
379
452
525
597
670
743
815
166
240
313
386
459
532
605
677
750
173
247
320
393
466
539
612
685
757
181
254
327
401
474
546
619
692
764
1 88
262
335
408
481
554
627
699
772
195
269
342
415
488
561
634
7 06
779
203
276
349
422
495
568
641
714
786
210
283
357
430
503
576
648
721
793
217
291
364
437
5io
583
656
728
801
225
298
37i
444
517
59
663
735
808
822
830
837
844
851
859
866
873
880
N.
| 1
2
3
4
5
6
7
8
9
Prop. Pts.
12
N.
1
2
3
4
5
6
7
8
9
Pro]
p. Pts.
600
77 8i5
822
830
837
844
8 5 I
859
866
873
880
01
02
03
04
05
06
07
08
09
887
960
78 032
104
176
247
319
390
462
895
967
039
in
183
254
326
398
469
902
974
046
118
190
262
333
405
476
909
981
053
125
197
269
340
412
483
916
988
061
132
204
276
347
419
490
924
996
068
140
211
283
355
426
497
931
*oo 3
075
147
219
290
362
433
504
938
*OIO
082
154
226
297
369
440
512
945
*oi7
089
161
233
305
376
447
519
952
*025
097
1 68
240
312
383
455
526
I
2
3
8
0.8
1.6
2.4
610
533
54o
547
554
56i
569
576
583
590
597
4
5
3-2
A O
ii
12
13
14
15
16
17
18
19
604
675
746
817
888
958
79 029
099
169
611
682
753
824
895
965
036
106
176
618
689
760
831
902
972
043
H3
183
625
696
767
838
909
979
050
120
190
633
704
774
845
916
986
057
127
197
640
711
78i
852
923
993
064
134
204
647
718
789
859
930
*000
071
141
211
654
725
796
866
937
*oo7
078
148
218
661
732
803
873
944
*oi4
085
155
225
668
739
810
880
* 951
*O2I
092
162
232
6
8
9
4.8
5-6
6.4
7.2
620
239
246
253
260
267
274
28l
288
295
302
21
22
23
24
25
26
27
28
29
309
379
449
5i8
657
727
796
865
3i6
386
456
525
595
664
734
803
872
323
393
463
532
602
671
741
810
879
330
4OO
470
539
609
678
748
817
886
337
407
477
546
616
685
754
824
893
344
414
484
553
623
692
761
831
900
351
421
491
560
630
699
768
837
906
358
428
498
567
637
706
775
844
913
365
435
505
574
644
713
782
851
920
372
442
5H
581
650
720
78 9
858
927
i
2
3
4
5
6
8
9
7
0.7
1.4
2.1
2.8
3-5
4.2
4-9
5-6
6.3
680
934
94i
948
955
962
969
975
982
989
99 6
3i
32
33
34
11
39
80 003
072
140
209
277
346
414
482
550
010
079
147
216
284
353
421
489
557
017
085
154
223
291
359
428
496
564
024
092
161
229
298
366
434
502
570
030
099
1 68
236
305
373
441
509
577
037
106
175
243
312
38o
448
5i6
584
044
"3
182
250
3i8
387
455
523
59i
051
120
188
257
325
393
462
53
598
058
127
195
264
332
400
468
536
604
065
134
2O2
271
339
407
475
543
611
i
2
3
6
0.6
1.2
1.8
640
618
625
632
638
645
652
659
665
672
679
4
2.4
4i
42
43
44
45
46
47
48
49
686
754
821
889
956
8 1 023
090
158
224
693
760
828
895
963
030
097
164
231
699
767
835
92
969
037
104
171
238
706
ffi
909
976
043
in
178
245
713
78i
848
916
983
050
117
184
251
720
787
855
922
990
057
124
I9 o
258
726
794
862
929
S 6
064
131
%
733
801
868
936
*oo3
070
137
20.}
271
740
808
875
943
*OIO
077
144
211
2 7 8
747
814
882
949
*oi7
084
151
218
285
8
9
3-o
3-6
4.2
4.8
5-4
650
291
298
305
3ii
3i8
325
331
33
345
35i
N.
1
2
3
1
5
G
7
8
9
Proj
. Pts.
N.
O
1
2
3
4
5
6
7 8
9
Prop. Pts.
650
51
52
53
54
55
56
57
58
59
660
61
62
63
64
65
66
67
68
69
670
72
73
74
75
76
77
78
79
680
81
82
83
84
85
86
87
88
89
690
92
93
94
9
9
99
700
81 291
298
305
311
318
325
331
338
345
351
418
485
551
617
684
750
816
882
948
i
2
3
4
6
I
9
i
2
3
4
5
6
8
9
7
0.7
1.4
2.1
2.8
3-5
4.2
4.9
5-6
6.3
6
0.6
1.2
1.8
2.4
3.o
3-6
4.2
4.8
5-4
358
425
491
558
624
690
757
823
889
365
431
498
564
631
697
763
829
895
37i
438
505
637
704
770
836
902
378
445
578
644
710
776
842
908
385
451
518
584
651
717
915
391
458
525
591
657
723
790
856
921
398
465
531
598
664
730
796
862
928
405
471
538
604
671
737
803
869
935
411
478
544
611
677
743
809
875
941
954
961
968
974
981
987
994
*ooo
*oo7
*oi4
82 020
086
151
217
282
347
413
478
543
027
092
158
223
289
354
419
484
549
033
099
164
230
295
360
426
491
556
040
105
171
236
302
367
432
497
562
627
046
112
I 7 8
243
308
373
439
504
569
053
119
184
249
315
380
445
575
060
125
191
256
321
387
452
517
582
066
132
197
263
328
393
458
523
588
073
138
204
269
334
400
465
530
595
079
210
2 7 6
341
406
471
536
601
607
614
620
633
640
646
653
659
666
672
737
802
866
930
995
83 059
123
187
679
743
808
872
937
*OOI
065
129
193
685
750
814
879
943
*oo8
072
136
200
692
756
821
885
078
142
206
698
763
827
892
956
*020
085
149
213
705
769
834
898
963
*027
091
155
219
711
776
840
905
969
*o 33
097
161
225
718
782
847
911
* 975
104
168
232
724
789
853
918
982
no
174
238
730
795
924
*f2
117
181
245
251
315
378
442
506
569
632
696
759
822
257
264
270
2 7 6
283
289
296
302
308
321
385
448
512
575
639
702
765
828
327
391
455
582
645
708
771
835
334
3f
461
525
651
715
778
841
340
404
467
531
594
658
721
784
847
347
410
474
537
601
664
727
790
853
353
417
480
544
607
670
734
797
860
359
423
487
550
613
677
740
i
366
429
493
556
620
683
746
809
872
372
436
499
563
626
689
753
816
879
885
891
897
904
910
916
923
929
935
942
948
84 on
073
136
198
261
323
386
448
954
017
080
142
205
267
330
392
454
960
023
086
148
211
273
336
398
460
967
029
092
155
217
280
342
404
466
973
036
098
161
223
286
348
410
473
979
042
105
167
230
292
354
417
479
985
048
in
173
236
298
423
485
992
055
117
180
242
305
367
429
491
998
061
123
186
248
311
373
435
497
067
130
192
255
317
379
442
504
5io
5i6
522
528
535
54i
547
553
559
566
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
N.
1
2
\ 3
4
5
6
7
8
9
Pro
). Pts.
700
84 510
5i6
522
528
535
541
547
553
559
566
01
02
03
04
05
06
07
08
09
572
634
696
757
819
880
942
85 003
065
578
640
702
763
825
887
948
009
071
584
646
708
770
831
893
954
016
077
590
652
7H
776
837
899
960
022
083
597
658
720
782
844
905
967
028
089
603
665
726
788
850
911
973
034
095
609
671
733
794
856
917
979
040
101
615
677
739
800
862
924
985
046
.107
621
683
745
807
868
930
991
052
114
628
689
751
813
874
936
997
058
120
I
2
3
7
0.7
1.4
2.1
710
126
132
138
144
150
156
163
169
175
181
4
2.8
7 C
ii
12
13
14
15
16
17
18
19
187
248
309
37o
43i
491
552
612
673
193
254
3i5
376
437
497
558
618
679
199
260
321
382
443
503
564
625
685
205
266
327
388
449
509
570
63i
691
211
272
333
394
455
5i6
576
637
697
217
278
339
400
461
522
582
643
703
224
285
345
406
467
528
588
649
709
230
291
352
412
473
534
594
655
715
236
297
358
418
479
540
600
661
721
242
303
364
425
485
546
606
667
727
6
9
O'O
4-2
4-9
5-6
6-3
720
733
739
745
75i
757
763
769
775
78i
788
21
22
23
24
25
26
27
28
29
794
854
9M
974
86 034
094
153
213
273
800
860
920
980
040
100
159
219
279
806
866
926
986
046
106
165
225
285
812
872
932
992
052
112
171
231
291
818
878
938
998
058
118
177
237
297
824
884
944
*oo4
064
124
183
243
303
830
890
950
*OIO
070
130
189
249
308
836
896
956
*oi6
076
136
195
255
3H
842
902
962
*O22
082
141
201
26l
320
848
908
968
*028
088
147
207
267
326
* i
2
3
4
1
8
9
6
0.6
1.2
1.8
2.4
3-0
3-6
4-2
4.8
5-4
730
332
338
344
350
356
362
368
374
380
386
3i
32
33
34
35
36
37
38
39
392
45i
5io
570
629
688
747
806
864
398
457
5i6
576
635
694
753
812
870
404
463
522
58i
641
700
759
817
876
410
469
528
587
646
705
764
823
882
4i5
475
534
593
652
711
770
421
481
540
717
776
835
894
427
487
546
605
664
723
782
841
900
433
493
552
611
670
729
738
847
906
439
499
558
617
676
735
794
853
911
445
504
564
623
682
74i
800
859
917
i
2
3
5
0-5
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740
923
929
935
941
947
953
958
964
970
976
4
2.0
4i
42
43
44
45
46
47
48
49
982
87 040
099
157
216
274
332
390
448
988
046
105
163
221
280
338
396
454
994
052
in
169
227
286
344
402
460
999
058
116
175
233
291
349
408
466
*oo 5
064
122
181
239
297
355
413
47i
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070
128
186
245
303
36i
419
477
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075
134
192
251
309
367
425
483
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081
140
198
256
315
373
$
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087
146
204
262
320
379
437
495
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093
151
210
268
326
384
442
500
9
2.5
3-0
3-5
4.0
4-5
750
506
5"
5i8
523
529
535
54i
547
552
558
N.
1
z
:*
t
5
6
7
N
9
I'roi
). I'tS.
N.
O 1 2 3
4
5
6
7
8
9
Prop. Pts.
750
5i
52
53
54
55
56
11
59
760
61
62
63
64
65
66
6?
68
69
770
7i
72
73
74
75
76
77
78
79
780
81
82
83
84
85
86
87
88
89
790
9i
92
93
94
95
96
97
98
99
800
87 506
512
5i8
523
529
535
541
547
552
558
I
2
3
4
6
8
9
i
2
3
4
5
6
8
9
6
0.6
1.2
1.8
2.4
3-0
3-6
4.2
4-8
54
5
o.5
I.O
1.5
2.0
2-5
3-0
3-5
4.0
4-5
564
622
679
737
795
852
910
967
88 024
570
628
685
743
800
858
915
973
030
576
633
691
749
806
864
921
978
036
58i
639
697
754
812
869
927
984
041
587
645
703
760
818
875
933
990
047
593
651
708
766
823
881
938
996
053
599
656
714
772
829
887
944
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058
604
662
720
777
835
892
950
*oo7
064
610
668
726
783
841
898
955
*OI 3
070
616
674
73i
789
846
904
961
*oi8
076
081
087
093
098
104
no
116
121
127
133
138
195
252
309
366
423
480
536
593
144
201
258
315
372
429
485
542
598
150
207
264
321
377
434
491
547
604
156
213
270
326
383
440
497
553
610
161
218
275
332
389
446
502
559
615
167
224
281
338
395
45i
508
564
621
173
230
287
343
400
457
513
570
627
I 7 8
235
2 9 2
349
406
463
519
576
632
184
241
298
355
412
468
525
58i
638
190
247
304
360
417
474
530
587
643
649
655
660
666
672
677
683
689
694
700
705
762
818
874
930
986
89 042
098
154
711
767
824
880
936
992
048
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159
717
773
829
885
941
997
053
109
165
722
779
835
891
947
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059
H5
170
728
784
840
897
953
*00 9
064
120
176
734
790
846
902
958
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070
126
182
739
795
852
908
964
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076
I 3 I
187
745
801
857
913
969
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081
137
193
750
807
863
919
975
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087
198
756
812
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9 2 5
981
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092
148
204
209
215
221
226
232
237
243
248
254
260
265
321
376
432
487
542
597
653
708
271
326
382
437
492
548
603
658
713
276
443
498
553
609
664
719
282
337
393
448
504
559
614
669
724
287
343
398
454
509
564
620
675
73
348
404
459
515
57o
625
680
735
298
354
409
465
520
575
631
686
741
304
360
415
470
526
58i
636
691
746
310
365
421
476
531
586
642
697
752
315
37i
426
481
537
592
647
702
757
763
768
774
779
785
79
796
801
807
812
818
873
927
982
90 037
091
146
200
255
823
878
933
988
042
097
151
206
260
829
883
938
993
048
102
157
211
266
834
889
944
998
053
108
162
217
271
840
894
949
*oo4
059
"3
168
222
2 7 6
845
900
955
*oo9
064
119
173
227
282
851
905
960
-015
069
124
179
233
287
856
911
966
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075
I2 9
184
238
293
862
916
971
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080
135
189
244
298
867
922
977
*0 3 I
086
140
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249
304
309
3i4
320
325
331
336
342
347
352
358
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
i6
N.
1
2
3
4
5
6
7
8
9
Pro]
[>. PtS.
800
90 309
314
320
325
331
336
342
347
352
358
OI
02
03
04
05
06
07
08
09
363
417
472
526
634
687
741
795
369
423
477
585
639
693
747
800
374
428
482
536
590
644
698
752
806
380
434
488
542
596
650
703
757
811
385
439
493
547
601
655
709
763
816
39
445
499
553
607
660
768
822
396
450
504
558
612
666
720
773
827
401
455
509
563
6i7
671
725
779
832
407
461
515
569
623
677
730
784
838
412
466
520
574
628
682
736
789
843
810
849
854
859
865
870
875
881
886
891
897
ii
12
14
II
17
18
19
902
956
91 009
062
116
169
222
275
328
907
961
014
068
121
174
228
28l
334
020
%
180
339
918
972
025
078
132
185
238
291
344
924
977
030
084
137
190
243
297
350
929
982
036
089
142
196
249
302
355
041
094
148
201
254
307
360
940
993
046
100
153
206
259
312
365
945
998
052
105
158
212
265
318
371
950
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057
1 10
164
217
270
323
376
I
2
3
4
6
8
6
0.6
1.2
1.8
2.4
3-o
3-6
4-2
4-8
820
381
387
392
397
403
408
413
418
424
429
9
5-4
21
22
23
24
25
26
27
28
29
434
487
593
803
855
440
492
545
598
651
703
756
808
861
445
498
603
656
709
761
814
866
450
609
66 1
714
766
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614
666
719
772
824
876
461
514
566
619
672
724
777
829
882
466
519
572
624
677
730
782
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47i
524
577
630
682
735
787
840
892
477
529
582
635
687
740
793
845
897
482
535
640
693
745
798
850
903
880
908
913
918
924
929
934
939
944
950
955
32
33
34
37
38
39
960
92 OI2
065
117
169
221
273
3 2 4
376
965
018
070
122
174
226
2 7 8
330
381
971
023
075
127
179
231
283
335
387
976
028
080
132
184
236
288
340
392
981
033
085
137
189
241
293
345
397
986
038
091
143
195
247
298
350
402
991
044
096
148
200
252
304
355
407,
997
049
101
153
205
257
309
412
*002
054
106
158
210
262
366
4l8
*oo7
059
in
163
215
267
319
37i
423
i
2
3
4
5
6
7
5
I.O
i-5
2.0
2.5
3-0
3-5
840
428
433
438
443
449
454
459
464
469
474
8
4.0
' 42
43
44
45
46
47
48
49
4 80
531
583
634
686
737
788
840
891
s
588
639
691
742
793
490
542
593
645
747
799
850
901
495
547
598
650
701
752
8-
500
655
706
758
809
860
911
505
557
609
660
711
763
814
865
916
5"
562
614
665
716
768
819
870
921
567
619
670
722
773
824
875
927
521
572
624
675
727
778
829
881
932
526
578
629
681
732
783
834
886
937
9
4-5
850
942
947
952
957
962
967
973
978
983
988
N.
1
2
3
I
I
;
7
8
Pro]
). PtS.
N.
1
2
3
4
5
6
7
8
9
Pro
p. Pts.
850
92 942
947
952
957
962
967
973
978
983
988
5i
52
53
54
55
56
57
58
59
993
93 044
095
146
197
247
298
349
399
998
049
IOO
151
202
252
303
354
404
*oo3
054
105
156
207
258
308
359
409
*oo8
059
no
161
212
263
313
364
414
*oi3
064
H5
166
217
268
3i8
369
420
"018
069
120
171
222
273
323
374
425
*024
075
125
176
227
278
328
379
430
*029
080
I3 1
181
232
283
334
384
435
*034
s
1 86
237
288
339
389
440
*039
090
141
192
242
293
344
394
445
i
2
3
6
0.6
1.2
1.8
860
450
455
460
465
470
475
480
485
490
495
4
5
2.4
7 O
61
62
63
64
65
66
67
68
69
500
55i
601
651
702
752
802
852
902
505
556
606
656
707
757
807
857
907
5io
56i
611
661
712
762
812
862
912
515
566
616
666
717
767
817
867
917
520
57i
621
671
722
772
822
872
922
526
576
626
676
727
777
827
877
927
53i
58i
631
682
732
782
832
882
932
536
586
636
687
737
787
837
887
937
54i
59i
641
692
742
792
842
892
942
546
596
646
697
747
797
847
897
947
6
1
9
1.6
4.2
4.8
5-4
870
952
957
962
967
972
977
982
987
992
997
7i
72
73
74
75
76
77
78
79
94 002
052
101
151
201
250
300
349
399
007
057
106
156
206
255
305
354
404
012
062
III
161
211
260
310
359
409
017
067
116
166
216
265
315
364
414
022
072
121
171
221
270
320
369
419
027
077
126
176
226
275
325
374
424
082
131
181
231
280
330
379
429
037
086
136
186
236
285
384
433
042
091
141
191
240
290
34o
047
096
146
196
245
295
345
394
443
i
2
3
4
5
6
8
9
5
0.5
I.O
i-5
2.O
2-5
3.0
3-5
4.0
4-5
880
448
453
458
463
468
473
478
483
488
493
81
82
83
84
85
86
87
88
89
498
547
596
645
694
743
792
841
890
503
552
601
650
797
846
895
507
557
606
655
704
753
802
851
900
512
562
611
660
709
758
807
856
905
517
567
616
665
714
763
812
861
910
522
57i
621
670
719
768
817
866
9*5
527
576
626
675
724
773
822
871
919
532
58i
630
680
729
778
827
876
924
537
586
635
685
734
783
832
880
929
542
59i
640
689
738
787
836
885
934
i
2
3
4
0.4
0.8
1.2
890
939
944
949
954
959
963
968
973
978
983
4
1.6
9 1
92
93
94
95
96
97
98
99
988
95 036
085
134
182
231
279
328
376
993
041
090
139
187
236
284
332
38i
998
046
095
143
192
240
289
337
386
*002
051
IOO
148
197
245
294
342
39
*00 7
056
105
153
202
250
299
347
395
*OI2
061
109
158
207
255
303
352
400
*oi7
066
114
163
211
260
308
357
405
*022
071
119
168
216
265
313
36i
410
*02 7
075
124
173
221
270
318
366
415
*032
080
129
177
226
274
323
371
419
6
8
9
2.O
2-4
2.8
11
900
424
429
434
439
444
448
453
458
463
468
N.
O
1
*
3
4
5
6
7
8
9
Proj
>. Pts.
i8
w.
0123
4
5
6
7
8
9
Prop. Pts.
900
01
02
03
04
05
06
07
08
09
910
ii
12
13
14
15
16
17
18
920
21
22
23
24
25
26
27
28
29
930
32
33
34
35
36
37
38
39
940
42
43
44
45
46
47
48
49
950
95 424
429
434
439
444
448
453
458
.463
468
I
2
3
4
6
8
9
i
2
3
4
6
8
9
5
0-5
I.O
1-5
2.0
2-5
3-o
3.5
4.0
4.5
4
0.4
0.8
1.2
1.6
2.O
2-4
2.8
3-2
3-6
472
521
569
617
665
713
76i
809
856
477
525
574
622
670
718
766
813
86 1
482
530
578
626
674
722
770
818
866
437
535
583
679
727
775
823
871
492
540
588
636
684
732
780
828
875
497
545
593
641
689
737
785
832
880
SGI
550
598
646
694
742
789
837
885
506
650
698
746
794
842
890
559
607
655
703
751
799
847
895
516
564
612
660
708
756
804
852
899
904
909
914
918
923
928
933
938
942
947
952
999
96 047
095
142
190
237
284
332
957
*OQ4
052
099
147
194
242
289
336
961
*oc>9
057
104
152
199
246
294
06 1
109
156
204
251
298
346
066
114
161
209
256
303
350
976
*02 3
071
118
166
213
261
308
355
980
*028
076
123
171
218
265
313
360
* 985
*o8o
128
175
223
270
317
365
990
*o 3 8
085
133
180
227
275
322
369
*o 9 !
090
137
185
232
280
327
374
_379_
426
473
520
' 614
661
708
755
802
384
388
393
398
402
407
412
417
421
43i
478
525
572
619
666
713
759
435
483
530
577
624
670
717
764
811
440
487
534
58i
628
675
722
769
816
445
492
539
586
633
680
727
774
820
450
497
544
638
685
778
825
454
501
548
595
642
689
736
783
830
459
506
553
600
647
694
741
788
834
464
558
605
652
699
745
792
839
468
515
562
609
656
703
750
797
844
848
853
858
862
867
872
876
881
886
890
895
942
988
97 035
08 1
1*8
220
267
900
946
993
039
086
132
179
225
271
904
95i
997
044
090
137
183
230
276
909
956
*002
049
095
142
188
234
280
914
960
*oo 7
053
IOO
146
192
239
285
918
965
*OII
058
104
151
197
243
290
923
970
*oi6
063
109
155
202
248
294
928
974
*02I
067
114
160
206
253
299
932
* 979
072
118
165
211
257
304
937
* 984
077
123
169
216
262
308
_3Jf3_
359
405
45i
497
543
589
727
317
322
327
331
336
340
345
350
354
364
410
456
502
548
594
640
685
368
414
460
506
552
598
644
690
736
373
419
465
557
603
649
695
740
377
424
470
607
9
745
791"
382
428
474
520
566
612
658
704
749
387
433
479
525
617
663
708
754
391
437
483
529
575
621
667
759
396
442
488
534
580
626
672
717
400
447
493
539
630
676
722
772
777
7*2
786
Soo
804
809
813
N.
1
2 :j i
5
7
8
Proi
.. Pis.
N.
o
1
2
3 4
5
6 | 7
8
9
Prop. Pts.
950
5i
52
53
- 54
55
56
57
58
59
OGO
61
62
63
64
65
66
67
68
69
970
7i
72
73
74
75
76
11
79
980
81
82
83
84
85
86
87
88
89
990
9i
92
93
94
95
96
.97
98
99
1000
97 772
777
782
786
791
795
800
804
809
8i3
i
2
3
4
5
6
9
i
2
3
4
5
6
8
9
5
0.5
I.O
i-5
2.0
2-5
3-o
3-5
4.0
4-5
4
0.4
0.8
1.2
1.6
2.0
2.4
2.8
%
818
864
909
955
98 ooo
046
091
137
182
823
868
914
959
005
050
096
141
186
827
873
918
964
009
055
100
146
191
832
877
923
968
014
059
105
150
195
836
882
928
973
019
064
109
155
200
841
886
932
978
023
068
114
159
204
845
891
937
982
028
073
118
164
209
850
896
941
987
032
078
123
1 68
214
855
900
946
991
037
082
127
173
218
859
95
950
996
041
087
132
177
223
227
232
236
241
245
250
254
259
263
268
272
3i8
363
408
453
498
543
588
632
277
322
367
412
457
502
547
592
637
281
327
372
417
462
507
552
597
641
286
331
376
421
466
5ii
556
601
646
290
336
381
426
471
516
561
60 5
650
295
340
385
430
475
520
565
610
655
299
345
390
435
480
525
57o
614
659
34
349
394
439
484
529
574
619
664
308
354
399
444
489
534
579
623
668
313
358
403
448
493
538
583
628
673
677
682
686
691
695
700
704
709
713
717
722
767
8x1
856
900
945
989
99 034
078
726
771
816
860
95
949
994
038
083
73i
776
820
865
909
954
998
043
087
825
869
914
958
-003
047
092
740
78 4
829
874
9l8
963
*00 7
052
096
744
789
834
878
923
967
*OI2
056
100
749
793
838
883
927
972
*oi6
061
105
753
798
843
887
932
976
*02I
065
I0 9
758
802
847
892
936
981
*025
069
114
762
807
851
896
941
985
*029
074
118
123
127 131
136
140
145
149
154
158
162
167
211
255
300
344
388
432
476
520
171
216
260
304
348
392
436
480
524
176
220
264
308
352
396
441
484
528
180
224
269
313
357
401
445
489
533
185
229
273
317
361
405
449
493
537
I8 9
233
277
322
366
410
454
498
542
193
238
282
326
370
414
458
502
546
I 9 8
242
286
330
374
419
463
506
550
202
247
291
335
379
423
467
5u
555
207
251
295
427
471
515
559
564
568
572
577
58i
585
590
594
599
603
607
651
695
739
782
826
870
913
957
612
656
699
743
787
830
874
917
961
616
660
704
747
791
835
878
922
965
621
664
708
752
795
839
883
926
970
625
669
712
756
800
843
887
930
974
629
673
717
760
804
848
891
935
978
634
677
721
765
808
852
896
939
983
638
682
726
769
813
856
900
944
987
642
686
730
774
817
861
904
948
991
647
691
734
778
822
865
909
952
996
00 000
004
009
013
017
022
026
030
035
039
N. ,
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
20
TABLE II.
CONSTANTS WITH THEIR LOGARITHMS.
Number.
Logarithm.
TT (ratio of circumference to diameter) . . .
7T 2 ...
3.14159265
9.86960440
1.77245385
0.31830989
O.IOI32II8
0.56418958
360
2i6oo /
I 296000"
57.2957795
3437^74677
206264". 806
.01745329
.00029089
.000004848
2.718281828
0.434294482
3.8197186
3963.296
3949.790
3956.
39-37
39.37079
39.37043
5280.
6080.290
1.466667
0.681818
0.49714 99
0.99429 97
0.24857 49
9.50285 01 10
9.00570 03 10
9.75142 5110
2.55630 25
4-33445 38
6.11260 50
1.75812 26
3-53627 39
5.31442 51
8.24187 74 10
6.46372 61 10
4.68557 4910
0.43429 45
9.63778 4310
0.58203 14
3-59805 65
3-59657 4o
3.59725 63
i.595i6 54
I.595I7 4i
I.595I7 oi
3.72263 39
3-78392 43
.16633 15
9.83366 8610
V*
I
I
I
v '
Number of degrees in circumference ....
minutes ....
seconds " ....
Degrees in arc equal to radius .
Minutes " " " ....
Seconds " " "
Length of arc of i degree
" " i minute
" " i second
Napierian base
Modulus of common logarithms . . .
Hours in which earth revolves through arc
equal to radius
Equat. radius of earth, miles (Clarke, 1878)
Polar " " " " "
Mean " " "
Inches in i metre (U. S. Standard)
" i " (British Standard) ....
" i " (Clarke 1866)
Feet in i mile . . ....
Feet in i nautical mile (U. S. Coast Survey)
Feet per second in i mile per hour
Miles per hour in i foot per second ....
21
TABLE III.
LOGARITHMS
OF THE
SINE, COSINE, TANGENT, AND COTANGENT
EACH MINUTE OF THE QUADRANT.
22
r
L. Sin.
d.
L.Tarig. c. d. L.Cotg.
L. Cos.
Prop. Pts.
O
i
2
3
4
6.46 373
6.76 476
6.94 085
7.06 579
30103
17609
12494
9691
7918
6694
5800
5"5
4576
4*39
3779
3476
3218
6.46 373
6.76 476
6.94 085
7.06 579
30103
17609
12494
9691
7918
6694
5800
5"5
4576
4139
3779
3476
3219
2996
2803
2633
2482
2348
2228
2119
2O2O
J 93i
1848
1773
1704
1639
1579
*524
1473
1424
1379
1336
1297
"59
1223
1190
"59
1128
IIOO
1072
1047
1022
99 8
976
955
934
9^5
895
878
860
843
828
812
797
782
769
756
742
73<>
3-53 627
3-23 524
3-05 915
2.93 421
o.oo ooo
0.00 000
o.oo ooo
0.00 000
o.oo ooo
6O
59
58
57
56
d.
I33 6
1297
1259
1223
1190
"59
1158
1128
IIOO
1072
1047
1046
IO22
999
998
976
955
954
934
d.
30103
17609
12494
9691
7918
6694
5800
5"5
4576
4139
3779
3476
3219
3218
2997
2996
2803
2802
2633
2483
2482
2348
2228
2227
2119
3021
2020
1931
1930
1848
1773
1704
1639
1579
1524
1473
1472
1424
1379
p.p.1"
22.27
21.62
20.98
20.38
19.83
19.32
19.30
18.80
18.33
17.87
17.45
17-43
17.03
16.65
16.63
16.27
15-92
15.90
15-57
p.p.l
501.7
293-4
208.^
l6i.e
I3I.9
111.5
96.6
85-2
7 6. 2
68.9
62.9
57-9
53.^
53-^
49-9
49-9
46.7
46.7
43-8
41.3
41-3
39- 1
37-1
37-1
35-3
33-^
33^
32.1
32.1
30.8
29.=
28.4
27-3
26.3
25.4
24-5
24-5
23-7
22.9
d.
9i5
914
896
895
878
877
860
843
828
827
812
797
782
769
756
755
743
742
73
2
8
3
2
7
7
7
5
7
8
8
3
5
3
5
3
2
O
8
8
7
3
3
2
2
8
7
8
7
5
o
2
2
5
3
3
8
P-P-1"
15-25
15-23
14-93
14.92
14-63
14.62
14-33
14-05
13.80
13-78
13-53
13.28
13-03
12.82
12.60
12.58
12.38
12.37
12.17
6
7
8
9
7.16 270
7.24 188
7.30 882
7.36 682
7.41 797
7.16 270
7.24 188
7.30 882
7.36 682
7.41 797
2.83 730
2.75 812
2.69 118
2.63 318
2.58 203
0.00 000
o.oo ooo
0.00 000
o.oo ooo
0.00 000
55
54
53
52
5i
10
ii
12
13
14
7-46 373
7-50 512
7-54 291
7-57 707
7.60 985
7-46 373
7-50 512
7-54 291
7-57 767
7.60 986
2.53 627
2.49 488
2.45 709
2.42 233
2.39 014
o.oo ooo
o.oo ooo
0.00 000
o.oo ooo
0.00 000
50
49
48
47
46
15
16
17
18
19
7.63 982
7.66 784
7.69 417
7.71 900
7-74 248
2802
2633
2483
2348
7.63 982
7.66 785
7.69 418
7.71 900
7-74 248
2.36 018
2-33 215
2.30 582
2.28 100
2.25 752
0.00 000
o.oo ooo
9-99 999
9-99 999
9-99 999
45
44
43
42
4i
20
21
22
23
24
7.76 475
7.78 594
7.80 615
7-82 545
7-84 393
2119
202 1
*93o
1848
1773
1704
1639
J579
1524
1472
1424
1379
1336
1297
.7.76 476
7-78 595
7.80 615
7-82 546
7-84 394
2.23 524
2.21 405
2.19 385
2.17 454
2.15 606
9-99 999
9-99 999
9-99 999
9-99 999
9-99 999
40
39
38
37
36
25
26
27
28
29
7.86 166
7.87 870
7.89 509
7.91 088
7.92 612
7.86 167
7.87 871
7-89 510
7.91 089
7.92 613
2.13 833
2.12 129
2.IO 490
2.08 911
2.07 387
9.99 999
9-99 999
9-99 999
9-99 999
9-99 998
35
34
33
32
3i
30
3i
32
33
34
7-94 084
7-95 508
7.96 887
7.98 223
7-99 520
7.94 086
7-95 5io
7.96 889
7.98 225
7-99 522
2.05 914
2.04 490
2.03 III
2.01 775
2.00 478
9-99 998
9-99 998
9-99 998
9-99 998
9-99 998
30
29
28
27
26
3 3 i
37
38
39
8.00 779
8. 02 002
8.03 192
8.04 350
8.05 478
12 59
1223
1190
1158
1128
8.00781
8.02 004
8.03 194
8.04 353
8.05 481
1.99 219
1.97 996
1.96 806
1.95 647
1-94 519
9-99 998
9-99 998
9-99 997
9-99 997
9.99 997
25
24
23
22
21
40
4i
42
43
44
8.06 578
8.07 650
8.08 696
8.09 718
8.10717
1072
1046
1022
999
976
954
934
914
896
877
860
843
827
812
797
782
769
755
743
73
8.06 581
8.07 653
8.08 700
8.09 722
8.10 720
1-93 419
1.92 347
1.91 300
1.90 278
1.89 280
9-99 997
9-99 997
9.99 997
9.99 997
9-99 996
20
19
18
17
16
45
46
47
48
49
8. 1 1 693
8.12 647
8.13 581
8.14 495
8.IS 39i
8. 1 1 696
8.12 651
8.13 585
8.14 500
8.15 305'
1.88 304
1.87 349
1.86 415
1.85 500
1.84 605
9.99 996
9-99 99 6
9-99 996
9-99 996
9.99 996
15
14
13
12
II
50
Si
52
53
54
8.16 268
8.17 128
8.17 971
8.18 798
8.19 610
8.16 273
8.17 133
8.17 976
8.18 804
8.19 616
1.83 727
1.82 867
1.82 024
1.81 196
x.8o 384
9-99 995
9-99 995
9-99 995
9-99 995
9-99 995
10
9
8
7
6
55
56
57
58
59
8.20 407
8.21 189
8.21 958
8.22713
8.23 456
8.20 413
8.21 195
8.21 964
8.22 720
8.23 462
i-79 587
1-78 805
1.78 036
1.77 280
1.76 538
9-99994
9-99 994
9-99 994
9-99 994
9-99 994
5
4
3
2
00
8.24 1 86
8.24 192
1.75 808
9-99 993
I,. ( os. d.
L. Cotg.
C. d.
L.Tniic.
JL. Sin.
f
Prop. Pts.
if
L. Sin.
d.
L.Tang.
c. d.
L. Cotg.
L. Cos.
Prop.
Pts
o
I
2
3
4
8.24 186
8.24 903
8.25 609
8.26 304
8.26 988
717
706
695
684
(,-~
8.24 192
8.24 910
8.25 616
8.26 312
8.26 996
*
706
696
68 4
fit?
1.75 808
1.75 090
1.74 384
1.73 688
1.73 004
9-99 993
9-99 993
9-99 993
9-99 993
9-99 992
GO
II
11
5
6
7
8
9
8.27 661
8.28 324
8.28 977
8.29 621
8.30 255
66 3
653
6 44
634
624.
8.27 669
8.28 332
8.28 986
8.29 629
8.30 263
73
663
654
643
634
6 2 e
1.72 331
1.71 668
1.71 014
1.70 371
1.69 737
9.99 992
9.99 992
9.99 992
9-99 992
9.99 991
55
54
53
52
5i
d.
P. D. 1"
d.
n TJ \"
10
ii
12
13
14
8.30 879
8-31 495
8.32 103
8.32 702
8-33 292
616
608
599
590
8.30 888
8-31 505
8.32 112
8.32711
8.33 302
6l 7
60 7
599
59i
-0 .
1.69 112
1.68 495
1.67 888
1.67 289
1.66 698
9-99 991
9-99 99 1
9-99 990
9.99 990
9-99 990
5O
49
48
47
46
718
717
706
696
605
11.97
n-95
11.77
1 1. 60
11.58
485
480
475
474
470
8.08
8.00
7.92
7.90
7 8q
15
16
17
18
19
8-33 875
8-34 450
8 35 018
8-35 578
8.36 131
53
575
568
560
553
8.33 886
8.34 461
8-35 029
8-35 590
8.36 143
54
575
568
56i
553
e/ifi
1.66 114
1-65 539
1.64 971
1.64 410
1.63 857
9-99 99
9.99 989
9-99 989
9-99 989
9.99 989
45
44
43
42
4i
684
673
663
654
653
11.40
11.22
11.05
10.90
10.88
t/ v
464
460
459
455
450
7.73
7.67
7.65
7.58
7.50
20
21
22
23
24
8.36 678
8.37 217
8-37 750
8.38 276
8.38 796
539
533
526
520
8.36 689
8.37 229
8.37 762
8.38 289
8.38 809
540
533
527
520
1.63 311
1.62 771
1.62 238
1.61 711
1.61 191
9.99 988
9-99 988
9.99 988
9-99 987
9-99 987
40
39
38
37
36
644
643
634
625
624
10.73
10.72
io.57
10.42
10.40
446
445
441
437
436
7.43
7.42
7.35
7.28
7.27
25
26
27
28
29
8.39 3io
8.39 818
8.40 320
8.40 816
8.41 307
508
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8.42 272
8.42 746
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56 768
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54 939
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8.83 075
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8.83 630
8.83 813
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8.84 177
8.83 547
8.83 732
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8.84 100
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55
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10
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8.87 156
8.87 325
8.87 494
8.87 661
171
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169
167
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8.87 277
8.87 447
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171
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1. 12 723
i.i2 553
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1. 12 215
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9-99 879
9-99 879
9.99 878
9.99 877
45
44
43
42
4i
9
10
20
30
40
25-7
28.5
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114.0
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56.3
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112.7
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25.1
27.8
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20
21
22
23
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8.87 829
8.87 995
8.88 161
8.88 326
8.88 490
166
166
165
164
8.87 953
8.88 120
8.88 287
8.88 453
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167
167
166
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1. 12 047
I. II 880
I. II 713
i. ii 547
i. n 382
9.99 876
9-99 875
9.99 874
9-99 873
9.99 872
40
39
38
37
36
5
6
7
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142.5
165
16.5
19-3
22.0
140.8
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1 60
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25
26
27
28
29
8 88 654
8.88 817
8.88 980
8.89 142
8.89 304
163
163
162
162
160
8.88 783
8.88 948
8.89 in
8.89 274
8.89 437
105
165
163
163
163
161
i. n 217
i. n 052
I. 10 889
i. 10 726
i.io 563
9-99 871
9-99 870
9 99 869
9.99 868
9-99 867
35
34
33
32
3i
9
10
20
3
40
24.8
27-5
55-0
82.5
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27.2
54-3
81.5
108.7
24.0
26.7
53-3
80.0
106.7
30
3i
32
33
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8.89 464
8.89 625
8.89 784
8.89 943
8.90 102
161
159
159
J 59
8.89 598
8.89 760
8.89 920
8.90 080
8.90 240
162
160
160
160
i.io 402
i.io 240
i.io 080
1.09 920
1.09 760
9.99 866
9-99 865
9-99 864
9-99 863
9.99 862
30
29
28
27
26
-5
6
7
8
J 37-5
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18.3
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155
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35
36
37
38
39
8.90 260
8.90 417
8.90 574
8.90 730
8.90 885
158
157
57
156
155
8.90 399
8.90 557
8.90 715
8.90 872
8.91 029
T 59
158
158
157
157
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1.09 60 1
1.09 443
1.09 285
1.09 128
i. 08 971
9 99 861
9.99 860
9-99 859
9-99 858
9-99 857
25
2 4
23
22
21
9
10
20
30
40
23.6
26.2
52.3
78.5
104.7
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25-8
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23.0
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40
4i
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43
44
8.91 040
8.91 195
8.91 349
8.91 502
8-91 655
!55
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1 53
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8.91 185
8.91 340
8.91 495
8.91 650
8.91 803
155
155
155
153
1.08 815
i. 08 660
i. 08 505
i. 08 350
1.08 197
9-99 856
9-99 855
9-99 854
9-99 853
9-99 852
20
19
18
17
16
b
6
7
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130.0
151
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20.1
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45
46
47
48
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8.91 807
8.91 959
8.92 no
8.92 261
8.92 411
152
152
151
151
150
8.91 957
8.92 no
8.92 262
8.92 414
8.92 565
X 54
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1.07 890
1.07 738
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9-99 851
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9-99 848
9-99 847
9.99 846
15
14
13
12
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9
10
20
30
40
22.7
25-2
50-3
75-5
100.7
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24.8
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22.1
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52
53
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8.92 561
8.92 710
8.92 859
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8-93 154
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8.92 716
8.92 866
8.93 016
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8.93 313
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150
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149
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1.07 134
i. 06 984
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9.99 842
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8.93 448
8-93 594
8.93 740
8.93 885
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8.93 609
8-93 756
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8.94 174
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8.94 340
8.94 485
8.94 630
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1.05 227
9-99 834
9-99 833
9-99 832
9.99 831
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8.95 170
8-95 3io
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8.94 917
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55
54
53
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9
10
20
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13
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8-95 450
8-95 589
8.95 728
8-95 867
8.96 005
139
139
139
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8.95 627
8-95 767
8.95 908
8.96 047
8.96 187
140
141
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1-04 373
1.04 233
1.04 092
1-03 953
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9.99 822
9.99 821
9.99 820
9-99 819
50
49
48
47
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6
7
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13.9
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15
16
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18
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8.96 143
8.96 280
8.96 417
8.96 553
8.96 689
137
137
136
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176
8.96 325
8 96 464
8.96 602
8.96 739
8.96 877
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139
138
137
138
lift
1.03 675
1.03 536
1.03 398
1.03 261
1.03 123
9-99 817
9.99 816
9-99 815
9-99 814
9 99 813
45
44
43
42
4i
9
10
20
30
40
20.9
23.2
46.3
69.5
92.7
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20.7
23.0
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69.0
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22.7
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68.0
90.7
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21
22
23
24
8.96 825
8.96 960
8-97 095
8.97 229
8-97 363
135
J 35
134
J 34
8.97 013
8.97 150
8.97 285
8.97 421
8-97 556
137
135
136
135
1.02 987
1.02 850
1.02 715
1.02 579
1.02444
9.99 812
9.99 810
9.99 809
9.99 808
9.99 807
40
39
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5
6
7
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25
26
27
28
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8.97 496
8.97 629
8.97 762
8.97 894
8.98 026
133
133
132
132
8.97 691
8.97 825
8-97 959
8.98 092
8.98 225
134
134
133
133
1.02 309
1.02 175
1.02 041
i.oi 908
i. oi 775
9.99 806
9-99 804
9.99 803
9.99 802
9.99 801
35
34
33
3 2
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9
10
20
30
40
20.3
22.5
45-0
67.5
90.0
20.0
22.2
44-3
66.5
88.7
19.7
21.8
43-7
65.5
87.3
30
3i
32
33
34
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8.98 288
8.98 419
8.98 549
8.98 679
131
131
130
130
8.98 358
8.98 490
8.98 622
8.98 753
8.98 884
J 33
132
132
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i.oi 642
i.oi 510
i.oi 378
i.oi 247
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9.99 800
9.99 798
9-99 797
9-99 79 6
9-99 795
30
29
28
27
26
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6
7
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112.5
129
12.9
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12.8
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8.98 937
8.99 066
8.99 194
8.99 322
129
129
128
128
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8.99 015
8-99 145
8.99 275
8.99 405
8-99 534
I 3 1
130
130
130
129
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i.oo 985
i.oo 855
i.oo 725
i.oo 595
I.OO 466
9-99 793
9.99 792
9.99 791
9-99 790
9-99 788
25
24
23
22
21
9
10
20
30
40
19.4
21.5
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64.5
86.0
19.2
21.3
42.7
64.0
85.3
18.9
21.0
42.0
63.0
84.0
40
41
42
43
44
8-99 450
8-99 577
8-99 704
8.99 830
8-99 956
127
127
126
126
126
8.99 662
8-99 79i
8.99 919
9.00 046
9.00 174
129
128
127
128
i.oo 338
i.oo 209
i.oo 081
0.99 954
0.99 826
9-99 787
9.99 786
9-99 785
9-99 783
9-99 782
20
19
18
17
16
50
6
7
8
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5
12.5
14.6
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123
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122
12.2
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45
46
47
48
49
9.00 082
9.00 207
9.00 332
9.00 456
9.00 581
125
125
124
125
9.00 301
9.00 427
9-00 553
9.00 679
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126
126
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0.99 699
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0.99 321
0.99 195
9-99 78i
9.99 780
9.99 778
9-99 777
9-99 776
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13
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9
10
20
3
40
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53
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9.00 704
9.00 828
9.00 951
9.01 074
9.01 196
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123
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9.01 055
9.01 179
9.01 303
9.01 427
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0.98 945
0.98 821
0.98 697
0.98 573
9-99 775
9-99 773
9-99 772
9-99 771
9-99 769
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9.01 550
9.01 673
9.01 796
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0.98 327
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0.98 082
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20
20
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118
117
117
116
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116
116
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114
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112
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110
110
110
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109
109
109
108
109
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107
107
106
107
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9.02 162
9.02 283
9.02 404
9.02 525
9.02 645
121
121
121
120
121
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118
119
118
118
117
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114
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112
112
112
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III
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109
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109
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107
108
107
106
107
106
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105
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0.97 838
0.97 717
0.97 596
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9-99 759
9-99 757
9-99 756
60
59
58
57
56
6
7
8
9
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6
8
9
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6
7
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9
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6
7
8
9
10
20
30
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6
7
8
9
10
20
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6
8
9
10
20
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40
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14.9
16.8
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10.6
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9.02 639
9.02 757
9.02 874
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9.04 181
9.04 297
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9-99 738
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9.04 376
9.04 490
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9.04 528
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0.95 127
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9-99 734
9-99 733
9-99 731
9-99 730
9-99 728
40
39
38
37
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26
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9.04 828
9.04 940
9.05 052
9.05 164
9-05 275
9.05 101
9-05 214
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9-99 727
9-99 726
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9.05 607
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9-o6 155
9.06 264
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9.07 103
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9.07 124
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9-07 337
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9.07 320
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0.92 572
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9-7 758
9.07 863
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9.08 071
9.08 177
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9-99 678
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30
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35
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0.61 743
9.98 783
9.98 780
9.98 777
9.98 774
9.98 771
3
3
3
3
30
29
28
27
26
6
7
8
> 45
53
5-3
6.2
7-i
5
5-2
6.1
6.9
35
36
37
38
39
9-37 081
9-37 133
9-37 185
9-37 237
9-37 289
52
52
52
52
9-38 313
9.38 368
9-38 423
9-38 479
9-38 534
55
55
56
55
0.61 687
0.61 632
0.61 577
0.61 521
0.61 466
9.98 768
9.98 765
9.98 762
9-98 759
9.98 756
3
3
3
3
25
24
23
22
21
9
10
20
3
40
8.0
8.8
17.7
26.5
35-3
7-8
8-7
17-3
26.0
34-7
40
4i
42
43
44
9-37 341
9-37 393
9-37 445
9-37 497
9-37 549
52
52
52
52
9-38 589
9.38 644
9.38 699
9-38 754
9.38 808
55
55
55
54
0.61 411
0.61 356
0.61 301
0.61 246
0.61 192
9-98 753
9.98 750
9.98 746
9-98 743
9.98 740
3
4
3
3
20
19
18
17
16
5
6
7
8
44.2
5i
5-i
6.0
6.8
43-3
4
0.4
0.5
-5
45
46
47
48
49
9-37 600
9-37 652
9-37 703
9-37 755
9-37 806
52
5i
52
5i
9-38 863
9.38 918
9.38 972
9-39 027
9.39 082
55
54
55
55
0.61 137
0.61 082
0.61 028
0.60 973
0.60 918
9-98 737
9-98 734
9.98 731
9.98 728
9.98 725
3
3
3
3
15
14
13
12
II
9
10
20
3
40
7-7
8-5
17.0
25.5
34-0
0.6
0.7
1-3
2.0
2. 7
50
Si
52
53
54
9-37 858
9-37 99
9-37 960
9.38 on
9.38 062
51
5i
Si
5i
9-39 I3 6
9-39 19
9-39 245
9-39 299
9-39 353
54
54
55
54
54
0.60 864
0.60 810
0.60 755
0.60 701
0.60 647
9.98 722
9.98 719
9-98 715
9.98 712
9.98 709
3
3
4
3
3
10
9
8
7
6
5
6
7
8
42.5
3
0.3
0.4
0.4
3-3
2
0.2
0.2
0-3
55
56
57
58
59
9-38 113
9.38 164
9-38 215
9.38 266
9-38 317
Si
5i
5i
51
9-39 407
9-39 461
9-39 515
9-39 569
9-39 623 j
54
54
54
54
0.60 593
0.60 539
0.60 485
0.60 431
0.60 377
9.98 706
9.98 703
9.98 700
9.98 697
9 98 694
3
3
3
3
3
5'
4
3
2
I
9
10
20
30
40
o-5
0-5
I.O
1-5
2.0
0-3
0-3
0.7
I.O
i-3
60
9.38 368
9-39 677
54
0.60 323
9.98 690
4
50
2-5
I -7
L. Cos.
d.
L.Cotg.
c. d.
L.Tang.
L. Sin.
d.
/
Pr
>|). I
ts.
76 C
r
L. Sin.
d.
L.Tang.
c. d.
L. Cotg.
L. Cos.
d.
Pi
op.
Pts.
i
2
3
4
9.38 368
9.38 418
9.38 469
9-38 519
9-38 57o
50
5 1
50
5i
9.39 677
9-39 73i
9-39 785
9-39 838
9.39 892
54
54
53
54
0.60 323
0.60 269
0.60 215
0.60 162
0.60 108
9.98 690
9.98 687
9.98 684
9.98 681
9.98 678
3
3
3
3
GO
59
58
57
56
6
7
54
5-4
6.3
53
5-3
6.2
5
6
8
9
9.38 620
9.38 670
9.38 721
9.38 771
9.38 821
50
5i
5o
50
9-39 945
9-39 999
9.40 052
9.40 106
9.40 159
54
53
54
53
0.60 055
0.60 ooi
0.59 948
0.59 894
0.59 841
9.98 675
9.98 671
9.98 668
9.98 665
9.98 662
4
3
3
3
55
54
53
52
5i
8
9
10
20
30
7-2
8.1
9.0
18.0
27.0
7-1
8.0
8.8
17.7
26.5
10
ii
12
13
14
9-38 871
9.38 921
9.38 971
9-39 02!
9-39 071
So
50
50
50
9.40 212
9.40 266
9.40 319
9.40 372
9.40 425
54
53
53
53
0.59 788
0-59 734
0.59 681
0.59 628
o-59 575
9.98 659
9.98 656
9.98 652
9.98 649
9 98 646
3
4
3
3
50
49
48
47
46
40
50
36.0
45-o
5
35-3
44.2
5i
15
16
17
18
19
9.39 121
9-39 170
9-39 220
9.39 270
9-39 319
49
50
50
49
9.40 478
9-40 531
9.40 584
9.40 636
9.40 689
53
53
52
53
0.59 522
0.59 469
0.59 416
0-59 364
0-59 3"
9.98 643
9.98 640
9.98 636
9-98 633
9.98 630
3
4
3
3
45
44
43
42
4i
6
7
8
9
10
5-2
6.1
6.9
7.8
8-7
&
6.8
7-7
8-5
20
21
22
23
24
9-39 369
9-39 4i8
9-39 467
9-39 517
9-39 566
5
49
49
So
49
9.40 742
9-40 795
9.40 847
9.40 900
9.40 952
53
52
53
S 2
0.59 258
0.59 205
o-59 153
0.59 100
0.59 048
9.98 627
9.98 623
9.98 620
9.98 617
9.98 614
4
3
3
3
40
39
38
37
36
20
30
40
50
17-3
26.0
34-7
43-3
17.0
25-5
34-o
42-5
25
26
27
28
29
9-39 615
9.39 664
9-39 713
9.39 762
9-39 8n
49
49
49
49
9.41 005
9.41 057
9.41 109
9.41 161
9.41 214
52
52
52
53
o-58 995
0-58 943
0.58 891
0.58 839
0.58 786
9.98 610
9.98 607
9.98 604
9.98 601
9 98 597
3
3
3
4
35
34
33
32
3i
6
7
8
5
5-o
5-8
6-7
49
4-9
5-7
6-5
30
3i
32
33
34
9.39 860
9.39 909
9-39 958
9.40 006
9-40 055
49
49
48
49
j.8
9.41 266
9.41 318
9.41 370
9.41 422
9.41 474
52
52
52
52
0-58 734
0.58 682
0.58 630
0.58 578
0.58 526
9.98 594
9.98 591
9.98 588
9-98 584
9.98 581
3
3
4
3
30
29
28
27
26
9
10
20
30
40
co
7-5
8-3
16.7
25.0
33-3
41.7
7-4
8.2
16.3
24-5
32-7
40 8
!i
37
38
39
9.40 103
9.40 152
9.40 200
9.40 249
9.40 297
49
48
49
48
9.41 526
9-41 578
9.41 629
9.41 681
9-41 733
52
5i
S 2
52
0.58 474
0.58 422
0.58 37i
0.58 319
0.58 267
9.98 578
9-98 574
9.98 571
9.98 568
9.98 565
4
3
3
3
25
24
23
22
21
6
48
4.8
C fi
47
4-7
c =
40
4i
42
43
44
9.40 346
9-40 394
9.40 442
9.40 490
9-40 538
48
48
48
48
A&
9.41 784
9.41 836
9.41 887
9-4i 939
9.41 990
52
Si
52
5i
0.58 216
0.58 164
0.58 113
0.58 061
0.58 oio
9.98 561
9-98 558
9-98 555
9-98 55i
9.98 548
3
3
4
3
3
20
19
18
17
16
7
8
9
10
20
3
5-
6-4
7.2
8.0
16.0
24.0
IS
7-1
7-8
15-7
23.5
45
46
47
48
49
9.40 586
9.40 634
9.40 682
9.40 730
9.40 778
4 8
4 8
4 8
48
9.42 041
9.42 093
9.42 144
9.42 195
9.42 246
52
5i
Si
Si
0-57 959
0.57 907
0.57 856
0.57 805
o.57 754
9-98 545
9.98 541
9-98 538
9-98 535
9-98 53i
4
3
3
4
15
14
13
12
.11
40
50
32.0
40.0
4
31-3
39-2
3
50
Si
52
53
54
9.40 825
9.40 873
9.40 921
9.40 968
9.41 016
47
48
48
47
48
9.42 297
9.42 348
9.42 399
9.42 450
9.42 501
5 1
5i
5i
5i
5i
0-57 703
0.57 652
0.57 601
0-57 550
o-57 499
9.98 528
9-98 525
9.98 521
9.98 518
9-98 515
3
4
3
3
10
9
8
7
6
6
8
9
10
0.4
o.5
o.S
0.6
0.7
0.3
0.4
0.4
o.5
o-S
P
57
58
59
9.41 063
9.4! in
9-41 158
9.41 205
9.41 252
47
48
47
47
47
9-42 552
9.42 603
9-42 653
9.42 704
9-42 75?
5 1
5*
So
5 1
5*
0.57 448
0.57 397
0-57 347
0.57 296
0-57 245
9.98 511
9.98 508
9-98 505
9-98 501
9.98 498
3
3
4
3
5
4
3
2
I
20
3
4
50
1-3
2.0
2.7
3-3
I.O
i-5
2.0
2-5
00
9-41 300
4
9-42 805
5
0-57 195
9.98 494
L. Cos.
d.
L. Cot.
c. d.
L.Taiii;.
1,. Sin.
d.
/
1'r
op. 1
Is.
75<
37
/
L. Sin.
d.
L.Tang. c. d.
L. Cotg.
JL. Cos.
d.
Prop. Pts.
I
2
3
4
9.41 300
9-4 1 347
9-41 394
9.41 441
9.41 488
47
47
47
47
47
47
46
47
47
46
47
46
47
46
47
46
46
47
46
46
46
46
46
46
45
46
46
46
45
46
45
46
45
46
45
45
46
45
45
45
45
45
45
45
44
45
45
45
44
45
44
45
44
45
44
44
44
45
44
44
9.42 805
9.42 856
9.42 906
9-42 957
9.43 007
51
50
51
50
5
51
5
So
So
50
So
50
So
5
So
49
50
50
49
50
49
50
49
So
49
49
49
So
49
49
49
49
49
49
49
48
49
49
48
49
49
48
49
48
48
49
48
48
48
49
48
48
48
48
48
48
47
48
48
48
0-57 195
0.57 144
0.57 094
0-57 043
0-56 993
9.98 494
9.98 491
9.98 488
9.98 484
9.98 481
3
3
4
3
4
3
3
4
3
4
3
4
3
3
4
3
4
3
4
3
4
3
4
3
3
4
3
4
3
4
3
4
3
4
4
3
4
3
4
3
4
3
4
3
4
4
3
4
3
4
3
4
4
3
4
3
4
4
3
4
60
59
58
57
56
6
8
9
10
20
30
40
50
6
7
8
9
10
20
3<>
40
5
6
7
8
9
10
20
30
40
50
6
7
8
9
10
20
3
40
50
6
7
8
9
10
20
3
40
50
51
5-1
6.0
6.8
7-7
8.5
17.0
25-5
34-
42.5
49
4.9
5-7
6-5
7-4
8.2
16.3
24-5
32-7
40.8
47
4-7
5-5
6-3
7-1
7.8
15-7
23-5
3i.3
39-2
45
4-5
5-3
6.0
6.8
7-5
15.0
22.5
30.0
37-5
4
0.4
o.S
0-5
0.6
0.7
i-3
2.0
2.7
3-3
50
S-o
5.8
6.7
7-5
8-3
16.7
25.0
33-3
41.7
48
4.8
5.6
6.4
7-2
8.0
16.0
24.0
32.0
40.0
46
4.6
5-4
6.1
6.9
7-7
15.3
23.0
30.7
38.3
44
4.4
5-i
5-9
6.6
7-3
14.7
22.O
29-3
36.7
3
0-3
0.4
0.4
o.S
o.S
I.O
i-5
2.0
2-5
t
7
8
9
1O
ii
12
13
14
9-4i 535
9.41 582
9.41 628
9-4i 675
9.41 722
9-43 057
9-43 i8
9-43 158
9-43 208
9-43 258
0.56 943
0.56 892
0.56 842
0.56 792
0.56 742
9.98 477
9.98 474
9.98 471
9.98 467
9.98 464
55
54
53
52
5i
9.41 768
9.41 815
9.41 861
9.41 908
9-4i 954
9-43 38
9 43 358
9-43 408
9.43 458
9 43 5o8
0.56 692
0.56 642
0.56 592
0.56 542
0.56 492
9.98 460
9-98 457
9-98 453
9.98 450
9.98 447
50
49
48
47
46
IS
16
17
18
19
9.42 ooi
9.42 047
9.42 093
9.42 140
9.42 186
9-43 558
9 43 607
9-43 657
9-43 707
9-43 756
0.56-442
0-56 393
0-56 343
0.56 293
0.56 244
9.98 443
9.98 440
9.98 436
9-98 433
9 98 429
45
44
43
42
41
20
21
22
23
24
9.42 232
9.42 278
9.42 324
9.42 370
9.42 416
9-43 806
9-43 855
9-43 95
9-43 954
9.44 004
0.56 194
0.56 145
0.56 095
0.56 046
0-55 996
9.98 426
9.98 422
9.98 419
9.98 415
9.98 412
40
39
38
37
36
25
26
27
28
29
9.42 461
9.42 507
9-42 553
9.42 599
9-42 644
9-44 53
9.44 102
9-44 I5i
9.44 201
9-44 250
o-55 947
0.55 898
0-55 849
o.55 799
0-55 750
9.98 409
9.98 405
9.98 402
9.98 398
9-98 395
35
34
33
32
3i
30
3i
32
33
34
9.42 690
9-42 735
9.42 781
9.42 826
9.42 872
9-44 299
9.44 348
9-44 397
9.44446
9-44 495
0-55 70i
0-55 652
0.55 603
o-SS 554
o-55 55
9.98 391
9.98 388
9.98 384
9.98 381
9-98 377
3O
29
28
27
26
35
36
37
38
39
9.42 917
9.42 962
9-43 8
9-43 53
9-43 098
9-44 544
9.44 592
9-44 641
9.44 690
9-44 738
0-55 456
0.55 408
0-55 359
0-55 3io
0.55 262
9-98 373
9.98 370
9.98 366
9-98 363
9-98 359
25
24
23
22
21
40
4i
42
43
44
9-43 143
9-43 188
9-43 233
9-43 278
9-43 323
9-44 787
9-44 836
9-44 884
9-44 933
9.44981
o-55 213
0.55 164
0.55 116
0.55 067
0.55 019
9-98 356
9 98 352
9-98 349
9-98 345
9.98 342
20
19
18
17
16
45
46
47
48
49
9-43 367
9-43 412
9-43 457
9-43 502
9-43 546
9.45 029
9-45 078
9.45 126
9-45 174
9-45 222
0.54 971
0.54 922
0.54 874
0.54 826
0-54 778.
9.98 338
9-98 334
9-98 33i
9.98 327
9.98 324
15
14
13
12
II
5O
Si
52
53
54
9-43 59i
9-43 635
9.43 680
9-43 724
9-43 769
9.45 271
9-45 319
9-45 367
9-45 415
9-45 463
0.54 729
0.54 681
0.54 633
0-54 585
-54 537
9.98 320
9.98 317
9-98 313
9.98 309
9.98 306
10
'9
8
7
6
55
56
57
58
59
9-43 813
9-43 857
9-43 901
9.43 946
9-43 990
9-45 5ii
9-45 559
9-45 606
9-45 654
9-45 72
0.54 489
0.54 441
-54 394
o-54 346
0.54 298
9.98 302
9.98 299
9.98 295
9.98 291
9.98 288
5
4
3
2
I
60
9.44 034
9-45 750
0.54 250
9.98 284
o
L. Cos.
d.
L.Cotg.
c, d.JL.Tang.
L. Sin.
d.
r
Prop. Pts.
74<
16'
/
L. Sin.
d.
L.Tang. c. d. L. Cotg.
L. Cos. d.
Prop. Pts.
o
I
2
3
4
9-44 034
9-44 078
9.44 122
9.44 166
9.44 210
44
44
44
44
43
44
44
44
43
44
44
43
43
44
43
44
43
43
43
43
43
44
43
42
43'
43
43
43
43
42
43
4 2
43
42
43
4 2
43
4 2
4 2
4 2
43
42
42
42
42
42
42
4 2
4i
4 2
42
42
4 1
42
4i
42
4i
42
4
42
9-45 750
9-45 797
9-45 845
9-45 892
9-45 940
47
48
47
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0-54 250
0.54 203
0-54 155
0.54 108
0.54 060
9.98 284
9.98 281
9.98 277
9.98 273
9.98 270
3
4
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3
4
4
3
4
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3
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9-44 559
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9.98 240
9.98 237
9-98 233
50
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9-44 733
9.44 776
9-44 819
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9.46 460
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9.98 229
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45
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9-44 948
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9-45 035
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9-46 835
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9-45 163
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30
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9-45 589
9-45 632
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25
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9-45 801
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20
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9.66 025
9.66 050
9.71 028
9.71 059
9.71 090
9.71 121
971 153
0.28 972
0.28 941
0.28 910
0.28 879
0.28 847
9-94 923
9-94 9 J 7
9-94 9H
9-94 94
9-94 898
50
49
48
47
46
15
16
17
18
19
9.66 075
9.66 099
9.66 124
9.66 148
9.66 173
24
25
24
25
9.71 184
9.71 215
9.71 246
9.71 277
9.71 308
0.28 816
0.28 785
0.28 754
0.28 723
0.28 692
9-94 891
9-94 885
9-94 878
9.94 871
9 94 865
45
44
43
42
4i
2O
21
22
23
24
9.66 197
9.66 221
9.66 246
9.66 270
9.66 295
24
25
24
25
9.71 339
9.71 370
9.71 401
9.71 431
9.71 462
0.28 661
0.28 630
0.28 599
0.28 569
0.28 538
9-94 858
9-94 852
9-94 845
9-94 839
9-94 832
40
39
38
37
36
25
26
27
28
29
9.66 319
9-66 343
9.66 368
9.66 392
9.66 416
24
25
24
24
9-7i 493
9-71 524
9-71 555
9.71 586
9.71 617
0.28 507
0.28 476
0.28 445
0.28 414
0.28 383
9.94 826
9-94 819
9-94 813
9.94 806
9-94 799
35
34
33
32
3i
30
3i
32
33
34
9.66 441
9.66 465
9.66 489
9.66 513
9-66 537
2 5
24
2 4
24
2 4
25
24
2 4
2 4
24
9.71 648
9.71 679
9.71 709
9.71 740
9.71 771
0.28 352
0.28 321
0.28 291
0.28 260
0.28 229
9.94 793
9-94 786
9.94 780
9-94 773
9.94 767
30
29
28
27
26
35
36
37
38
39
9.66 562
9.66 586
9.66 610
9.66 634
9.66 658
9.71 802
9-7i 833
9.71 863
9.71 894
9.71 925
0.28 198
0.28 167
0.28 137
0.28 106
0.28 075
9-94 76o
9-94 753
9-94 747
9-94 740
9-94 734
25
24
23
22
21
^O~
19
18
17
16
40
4i
42
43
44
9.66 682
9.66 706
9.66 731
9-66 755
9.66 779
2 4
24
25
24
24
9-71 955
9.71 986
9.72 017
9.72 048
9.72 078
0.28 045
0.28 014
0.27 983
0.27 952
0.27 922
9.94 727
9-94 720
9-94 714
9-94 707
9-94 700
45
46
47
48
49
9.66 803
9.66 827
9.66 851
9.66 875
9.66 899
2 4
2 4
24
24
24
9.72 109
9.72 140
9.72 170
9.72 201
9.72 231
0.27 891
0.27 860
0.27 830
0.27 799
0.27 769
9-94 694
9.94 687
9.94 680
9-94 674
9.94 667
15
14
13
12
II
5O
Si
52
53
54
9.66 922
9.66 946
9.66 970
9.66 994
9.67 018
2 3
24
24
24
2 4
9.72 262
9-72 293
9.72 323
9-72 354
9.72 384
0.27 738
0.27 707
0.27 677
0.27 646
0.27 616
9.94 660
9.94 654
9-94 647
9.94 640
9-94 634
10
9
8
\
11
57
58
59
9.67 042
9.67 066
9.67 090
9.67 113
9.67 137
24
. 24
23
24
9-72 415
9-72 445
9.72 476
9.72 506
9-72 537
0.27 585
0.27 555
0.27 524
0.27 494
0.27 463
9.94 627
9.94 620
9-94 614
9-94 607
9.94 600
5
4
3
2
I
60
9.67 161
9.72 567
0.27 433
9-94 593
7
L. Cos.
d.
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c. d.
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/
Prop. Pts.
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/
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d.
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c. d.
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I
2
3
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9.67 161
9.67 185
9.67 208
9.67 232
9.67 256
24
2 3
24
24
24
23
2 4
23
2 4
24
23
24
23
24
23
24
23
24
23
24
23
24
23
23
24
23
2 3
24
23
23
24
23
23
23
23
2 4
23
2 3
23
23
23
23
23
23
23
24
23
23
22
9.72 567
9.72 598
9.72 628
9.72 659
9.72 689
31
30
31
30
31
30
30
31
3
31
3
3
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30
3i
30
30
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30
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30
3
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30
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30
30
30
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29
30
0.27 433
0.27 402
0.27 372
0.27 341
0.27 311
9-94 593
9-94 587
9-94 58o
9-94 573
9-94 567
6
7
7
6
7
7
7
6
7
7
7
6
7
7
7
7
6
7
7
7
7
6
7
7
7
7
7
6
7
7
7
7
7
7
7
6
7
7
7
7
7
7
7
7
7
7
6
7
7
7
7
7
7
7
7
7
7
7
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60
59
58
57
56
6
7
8
9
10
20
3
40
50
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2
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c
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7
8
9
10
20
30
40
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2
3
4
5
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3
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0.6
0.7
0.8
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2.0
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5
6
7
8
9
9.67 280
9.67 303
9.67 327
9-67 350
9-67 374
9.72 720
9.72 750
9.72 780
9.72 8n
9.72 841
0.27 280
0.27 250
0.27 220
0.27 189
0.27 159
9.94 560
9-94 553
9-94 546
9-94 540
9-94 533
55
54
53
52
5i
1O
ii
12
13
14
9.67 398
9.67 421
9-67 445
9.67 468
9.67 492
9.72 872
9.72 902
9.72 932
9.72 963
9.72 993
0.27 128
0.27 098
0.27 068
0.27 037
O.27 OO7
9-94 526
9-94 519
9-94 513
9-94 506
9.94 499
5O
49
48
47
46
li
17
18
19
9-67 515
9-67 539
9.67 562
9.67 586
9.67 609
9.73 023
9-73 054
9-73 084
9-73 "4
9-73 144
0.26 977
0.26 946
0.26 916
0.26 886
0.26 856
9-94 492
9-94 485
9-94 479
9-94 472
9-94 465
45
44
43
42
4i
2O
21
29
23
24
9.67 633
9 67 656
9.67 680
9.67 703
9.67 726
9-73 175
9-73 205
9-73 235
9-73 265
9-73 295
0.26 825
0.26 795
0.26 765
0.26 735
0.26 705
9-94 458
9-94 451
9-94 445
9-94 438
9-94 431
40
39
38
37
36
25
26
27
28
29
9.67 750
9.67 773
9.67 796
9.67 820
9.67 843
9.73 326
9-73 356
9-73 386
9-73 4i6
9-73 446
0.26 674
0.26 644
0.26 614
0.26 584
0.26 554
9.94 424
9-94 417
9-94 4io
9.94 404
9-94 397
35
34
33
32
3i
30
3i
32
33
34
9.67 866
9.67 890
9-67 913
9.67 936
9-67 959
9 73 476
9-73 57
9-73 537
9-73 567
9-73 597
0.26 524
0.26 493
0.26 463
0.26 433
0.26 403
9-94 390
9-94 383
9-94 376
9.94 369
9-94 362
30
29
28
27
26
35
36
37
38
39
9.67 982
9.68 006
9.68 029
9.68 052
9-68 075
9.73 627
9-73 657
9-73 687
9-73 717
9-73747
0.26 373
0.26 343
0.26 313
0.26 283
0.26 253
9-94 355
9-94 349
9-94 342
9-94 335
9-94 328
25
24
23
22
21
40
4i
42
43
44
9.68 098
9.68 121
9.68 144
9.68 167
9.68 190
9-73 777
9 73 807
9-73 837
9-73 867
9-73 897
0.26 223
0.26 193
0.26 163
0.26 133
0.26 103
9-94 321
9-94 3H
9-94 307
9.94 300
9-94 293
20
19
18
17
16
45
46
47
48
49
9.68 213
9-68 237
9.68 260
9.68 283
9.68 305
9-73 927
9-73 957
9-73 987
9.74 017
9-74 047
0.26 073
0.26 043
0.26 013
0.25 983
0-25 953
9.94 286
9-94 279
9-94 273
9.94 266
9-94 259
15
14
13
12
II
5O
5i
52
53
54
9.68 328
9-68 351
9-68 374
9.68 397
9.68 420
2 3
23
23
23
23
23
23
23
23
22
23
9-74 077
9 74 107
9-74 137
9.74 166
9-74 196
0.25 923
0.25 893
0.25 863
0.25 834
0.25 804
9-94 252
9-94 245
9-94 238
9-94 231
9.94 224
10
9
8
7
6
6
7
8
9
10
20
30
40
5
55
56
57
58
59
9-68 443
9.68 466
9.68 489
9.68 512
9-68 534
9.74 226
9-74 256
9.74 286
9-74 3 l6
9-74 345
0.25 774
0.25 744
0.25 714
0.25 684
0-25 655
9.94 217
9.94 210
9-94 203
9-94 I9 6
9.94 189
5
4
3
2
I
60
9-68 557
9-74 375
0.25 625
9.94 182
7
L. Con.
d.
L.Cotg. c. d. L.Tnng.
L. Sin. d.
/
Prop. Pis.
61
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d.
L.Tang.c. d. L. Cotg.
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2
3
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9-68 557
9.68 580
9.68 603
9.68 625
9.68 648
23
23
22
23
23
23
22
23
23
22
23
22
23
23
9-74 375
9-74 405
9-74 435
9-74 465
9.74 494
30
3
30
2 9
3
30
29
30
30
30
29
30
30
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3
3
2 9
30
29
30
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30
3
29
30
29
30
29
30
29
30
29
30
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2 9
3
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30
29
29
30
29
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2 9
29
30
29
2 9
29
30
29
29
29
30
29
29
2 9
3
29
29
0.25 625
0-25 595
0-25 565
0.25 535
0.25 506
9.94 182
9-94 175
9.94 168
9.94 161
9-94 154
7
7
7
7
7
7
7
7
7
7
7
7
8
7
7
7
7
7
7
7
7
7
7
8
7
7
7
7
7
7
7
8
7
7
7
7
7
8
7
7
7
7
8
7
7
7
8
7
7
7
7
8
7
7
60
59
58
57
56
6
7
8
9
10
20
30
30
3-o
3-5
4.0
4-5
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10.0
15-0
20.0
25.0
29
2.9
3-4
3-9
4-4
4.8
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24.2
23
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2-7
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22
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2.6
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3-7
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3 L2
7 2.3
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6
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9.68 671
9.68 694
9.68 716
9.68 739
9.68 762
9-74 524
9-74 554
9-74 583
9.74 613
9-74 643
0.25 476
0.25 446
0.25 417
0.25 387
0.25 357
9-94 147
9-94 140
9-94 133
9.94 126
9-94 U9
55
54
53
52
Si
10
ii
12
13
14
9.68 784
9.68 807
9.68 829
9.68 852
9.68 875
9-74 6 73
9-74 702
9-74 732
9-74 762
9.74 791
0.25 327
0.25 298
0.25 268
0.25 238
0.25 209
9.94 112
9-94 i5
9.94 098
9-94 09
9.94 083
50
49
48
47
46
40
50
6
7
8
9
10
20
3
40
50
6
7
8
9
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20
30
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50
6
7
8
9
10
20
30
40
50
6 o
7 o
8 i
9 i
10 I
20 2
3 4
40 5
50 6
II
17
18
19
9.68 897
9.68 920
9.68 942
9.68 965
9.68 987
23
22
23
22
23
22
23
22
23
9.74 821
9-74 851
9.74 880
9.74 910
9-74 939
0.25 179
0.25 149
0.25 120
0.25 090
O.25 O6 I
9.94 076
9-94 069
9.94 062
9-94 055
9-94 048
45
44
43
42
41
20
21
22
23
24
9.69 oio
9.69 032
9-69 055
9.69 077
9.69 100
9-74 969
9.74 998
9-75 028
9-75 058
9-75 087
0.25 031
0.25 002
0.24 972
O.24 942
0.24 913
9.94 041
9.94 034
9-94 027
9.94 020
9-94 OI2
40
39
38
i
25
26
27
28
29
9.69 122
9.69 144
9.69 167
9.69 189
9.69 212
22
23
22
2 3
9-75 "7
9-75 146
9-75 176
9-75 205
9-75 235
0.24 883
0.24 854
0.24 824
0.24 795
0.24 765
9.94 005
9-93 998
9-93 99i
9-93 984
9-93 977
35
34
33
32
3i
30
3i
32
33
34
9- 6 9 234
9.69 256
9.69 279
9.69 301
9.69 323
22
23
22
22
22
23
22
22
22
22
2 3
22
22
22
9.75 264
9-75 294
9-75 323
9-75 353
9-75 382
0.24 736
0.24 706
0.24 677
0.24 647
0.24 618
9-93 970
9-93 963
9-93 955
9-93 948
9-93 94i
30
2 9
28
27
26
35
36
37
38
39
9- 6 9 345
9.69 368
9.69 390
9.69 412
9.69 434
9-75 4"
9-75 44i
9-75 470
9-75 500
9-75 529
0.24 589
0.24 559
0.24 530
0.24 500
0.24 471
9-93 934
9-93 927
9-93 920
9-93 912
9-93 905
25
24
23
22
21
4O
4i
42
43
44
9.69 456
9.69 479
9.69 501
9- 6 9 523
9- 6 9 545
9-75 558
9-75 588
9-75 617
9-75 647
9-75 676
0.24 442
0.24 412
0.24 383
0.24 353
0.24 324
9-93 898
9-93 891
9-93 884
9.93 876
9-93 869
*9
18
17
16
45
46
47
48
49
9.69 567
9.69 589
9.69 611
9.69 633
9-69 655
22
22
22
22
22
22
22
22
22
9-75 705
9-75 735
9-75 764
9-75 793
9-75 822
0.24 295
0.24 265
0.24 236
0.24 207
0.24 178
9.93 862
9-93 855
9-93 847
9-93 840
9-93 833
15
14
13
12
II
50
Si
52
53
54
9.69 677
9.69 699
9.69 721
9-69 743
9.69 765
9-75 852
9-75 881
9-75 9io
9-75 939
9-75 9 6 9
0.24 148
0.24 119
0.24 090
0.24 06 1
0.24 031
9.93 826
9-93 819
9-93 8n
9-93 804
9-93 797
10
9
8
7
6
3
57
58
59
9.69 787
9.69 809
9-69 831
9-69 853
9.69 875
22
22
22
22
22
9-75 998
9.76 027
9.76 056
9.76 086
9.76 115
0.24 002
0.23 973
0.23 944
0.23 914
0.23 885
9-93 789
9-93 782
9-93 775
9 93 768
9.93 760
7
7
7
8
5
4
3
2
I
60
9.69 897
9.76 144
0.23 856
9-93 753
7
L. Cos.
d.
L. Cotg.
c. d.
L.Tang.
L. Sin.
d.
r
Prop. Pts.
60 C
30 C
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L. Sin.
d.
L.Tang. c. d.' L. Cotg.
L. Cos. j d.
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Pts.
o
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2
3
4
9.69 897
9.69 919
9.69 941
9.69 963
9.69 984
22
22
22
21
9.76 144
9.76 173
9.76 202
9.76 231
9.76 261
29
29
29
3
29
29
29
29
29
29
29
29
29
29
29
29
30
29
29
28
29
29
29
29
29
29
29
29
29
29
29
29
28
29
29
29
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29
28
29
29
29
29
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29
29
29
28
29
29
28
29
29
29
28
29
28
29
29
28
0.23 856
0.23 827
0.23 798
0.23 769
0.23 739
9-93 753
9-93 746
9-93 738
9-93 73i
9-93 724
7
8
7
7
7
8
7
7
8
7
7
8
7
8
7
7
8
7
7
8
7
8
7
7
8
7
8
7
8
7
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
8
7
8
7
8
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59
58
57
56
3
6 3
7 3
8 4
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20 1C
30 is
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50 25
6
7
8
9
10
20
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6
7
8
9
10
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7
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10
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7 c
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9.70 006
9.70 028
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22
22
22
21
22
22
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21
22
22
21
22
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22
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9.76 290
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9.76 348
9.76 377
9.76 406
0.23 710
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0-23 594
9-93 717
9-93 709
9-93 702
9-93 695
9-93 687
55
54
53
52
51
1O
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12
13
14
9.70 115
9.70 137
9.70 159
9.70 180
9.70 202
9-76 435
9.76 464
9-76 493
9.76 522
9-76 551
0.23 565
0.23 536
0.23 507
0.23 478
0.23 449
9.93 680
9-93 673
9-93 665
9-93 658
9-93 650
50
49
48
47
46
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16
17
18
19
9.70 224
9.70 245
9.70 267
9.70 288
9.70 310
9.76 580
9.76 609
9.76 639
9.76 668
9.76 697
0.23 420
0.23 391
0.23 361
0.23 332
0.23 303
9-93 643
9-93 6 36
9-93 628
9-93 621
9-93 614
45
44
43
42
41
20
21
22
23
24
9.70 332
9-70 353
9-70 375
9.70 396
9.70 418
9.76 725
9-76 754
9.76 783
9.76 812
9.76 841
0.23 275
0.23 246
0.23 217
0.23 188
0.23 159
9-93 606
9-93 599
9-93 591
9-93 584
9-93 577
40
39
38
11
25
26
27
28
29
9-7 439
9.70 461
9.70 482
9.70 504
9-70 525
22
21
22
21
9.76 870
9.76 899
9.76 928
9-76 957
9.76 986
0.23 130
0.23 101
0.23 072
0.23 043
0.23 014
9-93 569
9-93 562
9-93 554
9-93 547
9-93 539
35
34
33
32
3i
30
3i
32
33
34
9-70 547
9.70 568
9.70 590
9.70 611
9.70 633
21
22
21
22
9.77 015
9-77 044
9-77 073
9.77 101
9-77 130
0.22 985
0.22 956
0.22 927
O.22 899
0.22 870
9-93 532
9-93 525
9-93 517
9-93 5io
9-93 502
30
29
28
27
26
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37
38
39
9.70 654
9.70 675
9.70 697
9.70 718
9.70 739
21
22
21
21
9-77 159
9.77 188
9-77 217
9.77 246
9.77 274
0.22 841
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0.22 783
0.22 754
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9-93 495
9-93 487
9-93 48o
9-93 472
9-93 465
25
24
23
22
21
4O
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42
43
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9.70 761
9.70 782
9.70 803
9.70 824
9.70 846
21
21
21
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9-77 303
9-77 332
9-77 361
9.77 390
9-77 4i8
O.22 697
0.22 668
O.22 639
0.22 6lO
O.22 582
9-93 457
9-93 450
9-93 442
9-93 435
9-93 427
20
19
18
17
16
45
46
47
48
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9.70 867
9.70 888
9.70 909
9.70 931
9.70 952
21
21
22
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21
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21
22
21
21
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21
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9-77 447
9-77 476
9-77 505
9-77 533
9-77 562
0.22 553
0.22 524
0.22 495
0.22 467
O.22 438
9-93 420
9-93 412
9-93 405
9-93 397
9-93 390
15
14
13
12
II
50
Si
52
53
54
9.70 973
9.70 994
9.71 015
9-71 036
9.71 058
9-77 59i
9-77 619
9-77 648
9-77 677
9.77 706
O.22 409
0.22 381
0.22 352
0.22 323
0.22 294
9-93 382
9-93 375
9-93 367
9-93 3
9-93 352
10
9
8
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55
56
57
58
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9.71 100
9.71 I2i
9.71 142
9-71 163
9-77 734
9-77 763
9-77 791
9 77 820
9-77 849
0.22 266
0.22 237
0.22 209
0.22 ISO
0.22 151
9-93 344
9-93 337
9-93 329
9.93 322
9-93 314
5
4
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9.71 184
9-77 877
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9.71 184
9-71 205
9.71 226
9.71 247
9.71 268
21
21
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9-77 906
9-77 935
9-77 963
9.77 992
29
29
28
29
28
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0.22 094
O.22 065
0.22 037
O.22 OO8
9-93 307
9-93 299
9-93 291
9-93 284
9.93 276
8
8
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60
59
58
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9.71 289
9.71 310
9-71 33i
9-7i 352
9-71 373
21
21
21
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9.78 049
9.78 077
9.78 106
9.78 135
29
28
29
29
28
0.21 980
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0.21 923
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0.21 865
9.93 269
9.93 26!
9-93 253
9-93 246
9-93 238
8
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9-71 393
9.71 414
9-7i 435
9-7i 456
9.71 477
21
21
21
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9.78 163
9.78 192
9.78 220
9.78 249
9.78 277
29
28
29
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0.21 808
O.2I 780
0.21 751
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9.93 230
9-93 223
9-93 215
9-93 207
9.93 200
7
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50
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16
17
18
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9.71 498
9-7i 519
9-7i 539
9.71 560
9.71 581
21
20
21
21
9.78 306
9-78 334
9.78 363
9.78 391
9.78 419
28
29
28
28
0.21 694
0.21 666
O.2I 637
0.21 609
O.2I 581
9-93 192
9-93 184
9-93 177
9.93 169
9-93 161
8
7
8
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45
44
43
42
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7
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9
10
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3-7
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20
21
22
23
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9.71 602
9.71 622
9.71 643
9.71 664
9.71 685
20
21
21
21
9.78 448
9.78 476
9-78 505
9-78 533
9.78 562
29
28
29
28
29
0.21 552
0.21 524
0.21 495
O.2I 467
0.21 438
9-93 154
9-93 X 4 6
9-93 138
9-93 131
9-93 123
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8
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40
39
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26
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9.71 705
9.71 726
9.71 747
9.71 767
9.71 788
21
21
20
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9.78 590
9.78 618
9.78 647
9.78 675
9.78 704
28
29
28
29
O.2I 4IO
0.21 382
0.21 353
0.21 325
O.2I 296
9-93 "5
9-93 108
9-93 ioo
9-93 092
9.93 084
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34
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20
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9.78 760
9.78 789
9.78 817
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28
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0.21 211
0.21 183
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9-93 077
9-93 069
9-93 061
9-93 053
9.93 046
8
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28
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9.71 932
9-7i 952
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28
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9-93 038
9-93 030
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0.20 957
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9.79 185
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9.72 238
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9-79 354
9.79 382
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9.92 921
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9.79 466
9-79 495
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0.20 253
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0.20 196
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9.80 279
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9.80 975
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9.79 176
9-79 192
9.79 208
9-79 224
9.79 240
16
16
16
16
16
9.89 671
9.89 697
9.89 723
9.89 749
9-89 775
26
26
26
26
26
o.io 329
o.io 303
o.io 277
o.io 251
o.io 225
9.89 504
9-89 495
9.89 485
9-89 475
9.89 465
9
10
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44
43
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7
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23
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9-79 256
9-79 272
9.79 288
9-79 304
9-79 319
16
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9.89 801
9.89 827
9-89 853
9-89 879
9.89 905
26
26
26
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o.io 095
9-89 455
9-89 445
9-89 435
9.89 425
9.89 415
10
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39
38
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25
26
27
28
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9-79 335
9-79 35i
9-79 367
9-79 383
9-79 399
16
16
16
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9.89 931
9-89 957
9.89 983
9.90009
9-90 035
26
26
26
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o.io 069
o.io 043
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0.09 991
0.09 965
9.89 405
9-89 395
9-89 385
9-89 375
9.89 364
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34
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9-79 415
9-79 431
9-79 447
9-79 463
9-79 478
16
16
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9.90 06 1
9.90 086
9.90 112
9.90 138
9.90 164
25
26
26
26
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0.09 939
0.09 914
0.09 888
0.09 862
0.09 836
9-89 354
9-89 344
9-89 334
9.89 324
9.89 314
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30
29
28
27
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9-79 526
9-79 542
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9.90 216
9.90 242
9.90 268
9.90 294
26
26
26
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0.09 784
0.09 758
0.09 732
0.09 706
9.89 304
9.89 294
9.89 284
9.89 274
9.89 264
10
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25
24
23
22
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43
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9-79 573
9-79 589
9.79 605
9.79 621
9-79 636
16
16
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9.90 320
9.90 346
9.90 371
9.90 397
9.90 423
26
25
26
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0.09 680
0.09 654
0.09 629
0.09 603
0.09 577
9.89 254
9.89 244
9-89 233
9.89 223
9.89 213
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10
10
20
19
18
17
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9-79 652
9.79 668
9-79 684
9.79699
9-79 715
16
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9.90 449
9-90 475
9.90 501
9.90 527
9-90 553
26
26
26
26
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0.09499
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0.09 447
9-89 203
9.89 193
9-89 183
9.89 173
9.89 162
10
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15
14
13
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4
50
7
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50
51
52
53
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9-79 731
9-79 746
9-79 762
9-79 778
9-79 793
5
16
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9.90 578
9.90604
9.90 630
9.90 656
9.90 682
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26
26
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0.09 422
0.09 396
0.09 370
0.09 344
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9-89 152
9.89 142
9.89 132
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10
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9-79 825
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9-79 856
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9-90 734
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26
25
26
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0.09 292
0.09 266
0.09 241
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0.09 189
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9.89 091
9.89 081
9.89 071
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9.87 568
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9.82 041
9.94 426
9-94 452
9-94 477
9-94 53
9.94 528
0.05 574
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9-87 535
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9.82 069
9.82 084
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9.82 112
9-94 554
9-94 579
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9-94 630
9-94 655
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9.82 212
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9.82 283
9.82 297
9.82 311
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9-82 354
9.82 368
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50
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45
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40
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30
29
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25
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9.83 120
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50
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45
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40
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35
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30
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25
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9.85 924
9.85 912
9-85 900
9.85 888
12
12
12
12
20
19
18
17
16
45
46
47
48
49
9.83 980
9-83 993
9.84 006
9.84 020
9-84 033
9.98 104
9.98 130
9 98 155
9.98 180
9.98 206
o.oi 896
o.oi 870
o.oi 845
o.oi 820
o.oi 794
9.85 876
9-85 864
9-85 851
9-85 839
9-85 827
12
13
12
12
IS
14
13
12
II
50
Si
52
53
54
9.84 046
9.84 059
9.84 072
9.84 085
9.84 098
9.98 231
9.98 256
9.98 281
9.98 307
9-98 332
o.oi 769
o.oi 744
o.oi 719
o.oi 693
o.oi 668
9-85 815
9-85 803
9.85 791
9.85 779
9.85 766
12
12
12
13
10
9
8
6
55
56
57
58
59
9.84 112
9.84 125
9.84 138
9.84 151
9.84 164
13
13
13
13
9-98 357
9-98 383
9.98 408
9-98 433
9.98 458
o.oi 643
o.oi 617
o.oi 592
o.oi 567
o.oi 542
9-85 754
9.85 742
9-85 730
9.85 718
9.85 706
12
12
12
12
5
4
3
2
I
60
9.84 177
9.98 484
o.oi 516
9-85 693
X 3
L. Cos.
d.
L. Cotg. |c. d.
L.Tang.
L. Sin.
d.
/
Prop. Pts.
46 C
66
44 e
/
I . Sin.
d.
L.Tang.
c. d.
L. Cotg.
L. Cos.
d.
Prop. Pts.
o
I
2
3
4
9.84 177
9.84 190
9.84 203
9.84 216
9.84 229
13
13
3
13
9.98 484
9.98 509
9-98 534
9 98 560
9-98 585
25
25
26
25
2 5
25
26
25
25
26
25
25
2 5
26
25
25
25
26
25
25
26
25
25
25
26
25
25
25
26
25
25
26
25
25
25
26
25
25
25
26
25
25
25
26
25
25
26
25
25
25
26
25
25
25
26
25
25
25
26
25
o.oi 516
o.oi 491
o.oi 466
o.oi 440
o.oi 415
9-85 693
9.85 681
9.85 669
9-85 657
9.85 645
12
12
12
12
GO
59
58
57
56
6
7
8
9
10
20
30
40
50
6
7
8
9
10
20
30
40
50
6
8
9
10
20
30
4
ero
26
2.6
3-9
4-3
8-7
13-0
17-3
21.7
35
2-5
2.9
3-3
3-8
4.2
8-3
w-S
16.7
20.8
14
1.4
1.6
1.9
2.1
2-3
4-7
7.0
9-3
11.7
13
i-3
i-S
i-7
2.0
2.2
$
8-7
10.8
13
1.2
1.4
1.6
1.8
2.0
4.0
6.0
8.0
10.0
5
6
7
8
9
9.84 242
9-84 255
9.84 269
9.84 282
9.84 295
J 3
13
14
13
13
13
13
13
13
13
13
12
13
13
13
9.98 610
9-98 635
9.98 661
9.98 686
9.98 711
o.oi 390
o.oi 365
o.oi 339
o.oi 314
o.oi 289
9.85 632
9.85 620
9.85 608
9-85 596
9-85 583
12
12
12
13
12
12
12
13
12
12
*3
12
12
J 3
12
12
13
12
12
13
12
13
12
12
J 3
12
13
12
13
12
12
13
12
*3
12
J 3
12
13
12
3
12
13
12
13
3
12
13
12
3
12
^3
13
12
'3
12
55
54
53
52
5i
10
II
12
13
14
9.84 308
9.84 321
9-84 334
9-84 347
9.84 360
9-98 737
9.98 762
9.98 787
9.98 812
9.98 838
o.oi 263
o.oi 238
o.oi 213
o.oi 188
o.oi 162
9-85 571
9-85 559
9-85 547
9-85 534
9.85 522
50
49
48
47
46
15
16
17
18
19
9-84 373
9-84 385
9 84 398
9.84411
9 84 424
9.98 863
9.98 888
9.98 913
9-98 939
9.98 964
o.oi 137
O.OI 112
o.oi 087
o.oi 061
o.oi 036
9-85 5io
9-85 497
9-85 485
9-85 473
9.85 460
45
44
43
42
41
20
21
22
23
24
9-84 437
9.84 450
9.84 463
9.84 476
9.84 489
J 3
13
13
13
3
9.98 989
9-99 oi5
9 99 040
9-99 065
9.99 090
O.OI Oil
o.oo 985
o.oo 960
o.oo 935
o.oo 910
9.85 448
9-85 436
9-85 423
9.85 411
9-85 399
4O
39
38
37
36
25
26
27
28
29
9.84 502
9-84 515
9.84 528
9.84 540
9-84 553
T 3
*3
13
12
13
13
13
J 3
13
13
12
13
13
13
13
12
13
13
13
12
13
13
*3
12
3
13
13
12
13
13
12
13
13
12
13
13
9.99 116
9-99 141
9.99 166
9 99 191
9-99 217
o.oo 884
o.oo 859
o.oo 834
o.oo 809
o.oo 783
9.85 386
9-85 374
9-85 361
9-85 349
9-85 337
35
34
33
32
3i
30
3i
32
33
34
9.84 566
9-84 579
9.84 592
9-84 605
9.84 618
9.99 242
9-99 267
9-99 293
9.99 318
9-99 343
o.oo 758
o.oo 733
o.oo 707
o.oo 682
o.oo 657
9-85 324
9-85 312
9.85 299
9-85 287
9.85 274
3O
29
28
27
26
35
36
37
38
39
9-84 630
9-84 643
9.84 656
9.84 669
9.84 682
9.99 368
9-99 394
9-99 419
9.99 444
9-99 469
o.oo 632
o.oo 606
o.oo 581
o.oo 556
o.oo 531
9.85 262
9.85 250
9-85 237
9.85 225
9.85 212
25
24
23
22
21
6
40
4i
42
43
44
9.84 694
9.84 707
9.84 720
9-84 733
9-84 745
9-99 495
9-99 520
9-99 545
9-99 570
9-99 596
o.oo 505
o.oo 480
o.oo 455
o.oo 430
o.oo 404
9.85 200
9.85 187
9-85 175
9.85 162
9.85 150
20
19
18
17
16
7
8
9
10
20
3
45
46
47
48
49
9.84 758
9.84 771
9.84 784
9.84 796
9.84 809
9.99 621
9-99 6 46
9-99 6 72
9-99 6 97
9-99 722
o.oo 379
o.oo 354
o.oo 328
o.oo 303
o.oo 278
9-85 137
9-85 125
9-85 "2
9.85 ioo
9.85 087
15
H
13
12
II
40
50
6
8
9
10
5O
5i
52
53
54
9.84 822
9-84 835
9.84 847
9.84 860
9.84 873
9-99 747
9-99 773
9-99 798
9-99 823
9-99 848
o.oo 253
o.oo 227
0.00 202
o.oo 177
o.oo 152
9-85 074
9.85 062
9.85 049
9-85 037
9-85 02 4
10
9
8
6
55
56
57
58
59
9 84 885
9.84 898
9.84911
9.84 923
9.84 936
9-99 874
9-99 899
9-99 924
9-99 949
9-99 975
o.oo 126
O.OO IOI
o.oo 076
o.oo 051
o.oo 025
9.85 012
9.84 999
9.84 986
9.84 974
9.84 9i
5
4
3
2
I
20
30
40
50
00
9.84 9 jr v
0.00 000
O.OO 000
9.84 949
L. Cos.
d.
L. Cot*.
c. d. L.Tang.
L. Sin.
d.
f
Prop. Pts.
45 C
o<
TABLE IV.
r r
r
s
T
60
120
1 80
240
o
i
2
3
4
4-68557
.68557
.68557
.68557
68557
4-68557
-68557
.68557
.68557
.68558
300
360
420
4 80
540
5
6
7
8
9
4.68557
.68557
68557
68557
.68557
4.68558
.68558
.68558
.68558
.68558
600
660
720
780
840
10
ii
12
13
14
4-68557
68557
.68557
.68557
.68557
4-68558
.68558
.68558
.68558
.68558
900
9 60
1020
I080
1140
II
17
18
19
4-68557
.68557
.68557
.68557
.68557
4-68558
.68558
.68558
.68558
.68558
1200
1260
1320
1380
I 44
20
21
22
23
24
4-68557
.68557
.68557
.68557
.68557
4-68558
.68558
.68558
.68558
.68558
1500
1560
1620
1680
1740
25
26
27
28
29
4-68557
.68557
68557
.68557
.68557
4-68558
.68558
.68558
.68558
68559
1800
1860
1920
1980
2040
30
31
32
33
34
4-68557.
.68557
.68557
.68557
-68557
4-68559
.68559
.68559
68559
68559
2100
2160
2220
228O
2340
35
36
37
38
39
4-68557
.68557
.68557
.68557
.68557
4-68559
.68559
.68559
.68559
68559
2400
2460
2520
2580
2640
40
41
42
43
44
4-68557
.68556
.68556
.68556
.68556
4-68559
.68560
.68560
.68560
.68560
2700
2760
2820
2880
2940
45
46
47
48
49
4-68556
.68556
.68556
.68556
.68556
4.68560
.68560
.68560
.68560
.68560
3OOO
3060
3120
3180
3240
50
51
52
53
54
4.68556
.68556
.68556
.68556
.68556
4.68561
.68561
.68561
.68561
.68561
3300
3360
3420
3480
3540
55
56
57
58
59
4.68556
.68556
.68555
.68555
-68555
4.68561
.68561
.6856!
.68562
.68562
3600
60
4-68555
4.68562
r r
r
S
T
3600
3660
3720
3780
3840
i
2
3
4
4.68555
68555
68555
.68555
68555
4.68562
.68562
.68562
.68562
.68563
3900
3960
4020
4080
4140
i
7
8
9
4.68555
.68555
.68555
.68555
.68555
4.68563
.68563
.68563
.68563
.68563
4200
4260
4320
4380
4440
10
ii
12
13
14
4-68554
.68554
.68554
68554
68554
4.68563
.68564
.68564
.68564
.68564
4500
456o
4620
4680
4740
15
16
17
18
19
4-68554
.68554
68554
.68554
68554
4-68564
.68565
.68565
.68565
.68565
4800
4860
4920
4980
5040
20
21
22
23
24
4.68554
.68553
.68553
.68553
68553
4-68565
.68566
.68566
.68566
.68566
5100
5160
5220
5280
5340
25
26
27
28
29
4-68553
68553
68553
68553
.68553
4.68566
.68567
.68567
.68567
.68567
5400
546o
5520
558o
5640
30
31
32
33
34
4-68553
.68552
.68552
.68552
.68552
4-68567
.68568
.68568
.68568
.68568
5700
576o
5820
5880
5940
CO CO CO CO CO
vO OOvj O\Cn
4.68552
.68552
.68552
.68552
.68551
4.68569
.68569
.68569
.68569
.68569
6000
6060
6120
6180
6240
40
42
43
44
4-68551
.68551
.68551
.68551
.68551
4.68570
.68570
.68570
.68570
.68571
6300
6360
6420
6480
6540
45
46
47
48
49
4-68551
.68551
.68550
.68550
.68550
4.68571
.68571
.68572
.68572
.68572
6600
6660
6720
6780
6840
So
51
52
53
54
4.68550
.68550
.68550
.68550
.68550
4-68572
.68573
.68573
.68573
68573
6900
6960
7020
7080
7140
55
56
57
58
59
4.68549
.68549
.68549
.68549
.68549
4.68574
.68574
.68574
.68575
.68575
7200
60
4.68549
4-68575
Log sin a = log a " -f S.
Log tan a log a " -j- T.
68
r r
A
S
T
7200
7260
7320
7380
7440
o
2
3
4
4.68549
.68549
.68548
.68548
.68548
4-68575
68575
.68576
.68576
.68576
7500
7560
7620
7680
7740
5
6
7
8
9
4.68548
.68548
.68548
.68547
.68547
4.68577
.68577
.68577
.68578
.68578
7800
7860
7920
7980
8040
10
ii
12
13
14
4.68547
68547
.68547
.68547
.68546
4.68578
.68579
68579
.68579
.68579
8100
8160
8220
8280
8340
15
16
17
18
19
4-68546
.68546
.68546
.68546
.68546
4.68580
.68580
.68580
.68581
.68581
8400
8460
8520
8580
8640
20
21
22
23
24
4-68545
68545
68545
68545
.68545
4.68582
.68582
.68582
.68583
.68583
8700
8760
8820
8880
8940
25
26
27
28
29
4-68545
68544
.68544
.68544
.68544
4-68583
.68584
.68584
.68584
.68585
9000
9060
9120
9180
9240
30
31
32
33
34
4.68544
.68544
68543
.68543
.68543
4.68585
.68585
.68586
.68586
.68587
9300
9360
9420
9480
9540
35
36
37
38
39
4-68543
68543
.68542
.68542
.68542
4.68587
.68587
.68588
.68588
.68588
9600
9660
9720
9780
9840
40
4i
42
43
44
4.68542
.68542
.68541
.68541
.68541
4.68589
.68589
.68590
68590
.68590
9900
9960
10020
I0080
IOI40
45
46
47
48
49
4-68541
.68541
.68540
.68540
.68540
4.68591
.68591
.68592
.68592
.68592
10200
10260
10320
10380
10440
50
Si
52
53
54
4.68540
.68540
.68539
68539
68539
4-68593
68593
.68594
.68594
68595
10500
10560
I062O
10680
10740
55
56
57
58
59
4.68539
.68539
.68538
.68538
.68538
4.68595
68595
.68596
.68596
.68597
10800
60
4.68538
4.68597
rr
r
S
T
10800
10860
10920
10980
11040
i
2
3
4
4-68538
.68537
68537
.68537
.68537
4.68597
.68598
.68598
.68599
.68599
IIIOO
11160
1 1 220
II280
II340
5
6
8
9
4-68537
.68536
.68536
.68536
.68536
4.68599
.68600
.68600
.68601
.68601
II400
II460
II520
II580
11640
10
ii
12
13
14
4-68535
68535
.68535
68535
68534
4.68602
.68602
.68603
.68603
.68604
II7OO
II760
II820
II880
II940
IS
16
17
18
19
4-68534
68534
.68534
.68533
-68533
4.68604
.68605
.68605
.68606
.68606
12000
I2O6O
I2I20
I2I80
12240
20
21
22
23
24
4-68533
68533
.68532
.68532
68532
4.68607
.68607
.68608
.68608
.68609
12300
12360
I242O
12480
12540
25
26
27
28
29
4-68532
.68531
.68531
.68531
.68531
4.68609
.68610
.68610
.68611
.68611
I26OO
12660
12720
12780
12840
3Q,
31
32
33
34
4.68530
.68530
.68530
68530
.68529
4.68612
.68612
.68613
.68613
.68614
12900
12960
13020
13080
I3I40
3
37
38
39
4.68529
.68529
.68529
.68528
.68528
4.68614
.68615
.68615
.68616
.68616
13200
13260
13320
13380
13440
40
4i
42
43
44
4.68528
.68528
.68527
.68527
.68527
4.68617
.68617
.68618
.68618
.68619
13500
13560
13620
13680
13740
45
46
47
48
49
4.68526
.68526
.68526
.68526
.68525
4.68620
.68620
.68621
.68621
.68622
13800
13860
13920
13980
14040
50
51
52
53
54
4-68525
.68^25
.68525
.68524
.68524
4.68622
.68623
.68623
.68624
.68625
I4IOO
I4I60
14220
14280
14340
55
56
57
58
59
4.68524
.68523
.68^23
.68523
.68522
4.68625
.68626
.68626
.68627
.68628
14400
60
4.68522
4.68628
Log sin a = log a " -f S.
Log tan a log a "
TABLE V.
NATURAL
SINES, COSINES, TANGENTS, AND COTANGENTS.
o /
N. Sin.
N. Tan.
N. Cot.
N. Cos.
/
N. Sin.
N. Tan.
N. Cot.
N. Cos.
o
5
10
15
20
25
.00 000
145
291
436
582
727
.00 OOO
145
291
436
582
727
Infinity.
687.55
343-77
229.18
171.89
I37.5I
Unity.
99 999
998
997
90 o
55
5o
45
40
35
230
35
40
45
50
55
.04 362
507
653
798
.04 943
.05 088
.04 366
512
658
803
.04 949
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5 o
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17 633
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15 o
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N. Cos.
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480
68 1
.61 882
.62 083
285
1.6319
.6265
.6212
.6160
.6107
6055
.85 264
188
112
85 035
84 959
882
30
25
20
15
10
5
27 o
5
10
15
20
25
45 399
529
658
787
45 917
.46 046
So 953
.51 136
319
503
688
.51 872
1.9626
9556
.9486
.9416
.9347
9278
.89 101
89 035
.88 968
902
835
768
68 o
55
So
45
40
35
32 o
5
10
15
20
25
52 992
53 "5
238
361
484
607
.62 487
689
.62 892
63095
299
503
1.6003
5952
.5900
.5849
5798
5747
.84 805
728
650
573
495
417
58 o
55
So
45
40
35
3
35
40
45
50
55
46 175
304
433
56i
690
819
52 057
242
427
613
798
52 985
1.9210
.9142
.9074
.9007
.8940
.8873
.88 701
634
566
499
431
363
30
25
20
15
10
5
30
35
4
45
50
55
53 73
853
53 975
54 097
220
342
.63 707
.63 912
.64 117
322
528
734
1.5697
5647
5597
5547
5497
.5448
84 339
261
182
104
.84 025
.83 946
30
25
20
15
10
5
28 o
5
10
15
20
25
.46 947
47 076
204
332
460
588
53 171
358
545
732
53 920
54 107
1.8807
.8741
.8676
.8611
.8546
.8482
.88 295
226
158
089
.88 020
87 951
62 o
55
So
45
40
35
33 o
5
10
15
20
25
54 464
586
708
829
54 95i
55 072
.64 941
65 148
355
563
771
.65 980
1-5399
5350
5301
5253
.5204
.5156
83 867
788
708
629
549
469
67 o
55
5o
45
40
35
30
35
40
45
So
55
47 7i6
844
47 97i
.48099
226
354
54 2 9 6
484
673
.54 862
55 051
241
1.8418
.8354
.8291
.8228
.8165
.8103
.87 882
812
743
673
603
532
30
25
20
15
10
5
3
35
40
45
50
55
55 194
3i5
436
557
678
799
.66 189
398
608
.66 818
.67 028
239
1.5108
.5061
5013
.4966
.4919
4872
83 389
308
228
147
.83066
.82 985
30
25
20
15
10
5
29 o
5
10
IS
20
25
.48 481
608
735
862
.48 989
.49 116
55 43i
621
55 812
.56003
194
385
1.8040
7979
.7917
.7856
.7796
7735
.87 462
391
321
250
178
107
61 o
55
So
45
40
35
34 o
5
10
15
20
25
55 9*9
.56 040
160
280
401
521
67 451
663
.67 875
.68 088
301
514
1.4826
4779
4733
4687
.4641
4596
.82 904
822
741
659
577
495
56 o
55
50
45
40
35
30
35
40
45
50
55
.49 242
369
495
622
748
49 874
56 577
769
.56962
57 155
348
54i
1.7675
7615
.7556
.7496
7437
7379
.87 036
.86 964
892
820
748
675
30
25
20
IS
10
5
30
35
40
45
So
55
641
760
.56 880
.57000
119
238
.68 728
.68 942
69 157
372
588
.69804
1-4550
.4505
.4460
.4415
4370
.4326
.82 413
330
248
165
.82 082
.81 999
30
25
20
15
10
5
30 o
.50 ooo
.57 735
1.7321
.86603
60 o
36 o
57 358
.70 021
1.4281
.81 915
55 o
N. Cos.
N. Cot.
N.Tan.
N. Sin.
o /
N. Cos.
N. Cot.
N. Tan.
N. Sin.
f
73
I
N. Sin.
N. Tan.
N. Cot.
N.Cos.
o t
N. Sin.
N. Tan.
N.Cot.
N.Cos.
35 o
5
10
15
2O
25
57 358
477
596
7iS
833
57 952
.70 021
238
455
673
.70 891
.71 no
1.4281
4237
4i93
.4150
.4106
.4063
.81 915
832
748
664
580
496
55 o
55
So
45
40
35
40 o
5
10
15
20
25
.64 279
390
501
612
723
834
.83 910
84 158
407
656
.84 906
85 i57
1.1918
.1882
.1847
.1812
.1778
1743
.76 604
5"
4i7
323
229
135
50 o
55
So
45
4
35
30
35
40
45
5o
55
.58 070
189
307
425
543
66 1
.71 329
549
769
71 9 9 p
.72 211
432
1.4019
3976
3934
.3891
3848
.3806
.81 412
327
242
157
.81 072
.80 987
30
25
20
15
10
5
30
35
40
45
50
55
.64 945
65 055
166
276
386
496
.85 408
660
85 912
.86 166
419
674
1.1708
.1674
.1640
.1606
1571
1538
.76 041
75 94 6
85 J
756
661
566
30
25
20
15
10
5
36 o
5
10
15
20
25
58 779
.58 896
.59014
131
248
365
.72 654
.72 877
73 I 00
323
547
771
1-3764
3722
.3680
3638
3597
3555
.80 902
816
730
644
558
472
54 o
55
So
45
40
35
41 o
5
10
15
20
25
.65 606
716
8 ^
65 935
.66 044
153
.86 929
.87 184
441
698
87 955
.88 214
1.1504
.1470
.1436
.1403
.1369
1336
75 47i
375
280
184
75 088
-74 992
49 o
55
So
45
40
35
30
35
40
45
50
55
.59 482
599
716
832
59 949
.60 065
73 996
.74 221
447
674
.74 900
75 128
I-35H
3473
3432
3392
3351
33"
.80 386
299
212
125
.80 038
79 95i
30
25
20
15
10
5
30
35
40
45
50
55
.66 262
37i
480
588
697
805
.88 473
732
.88 992
89 253
515
.89 777
1.1303
.1270
.1237
.1204
.1171
"39
.74 896
799
703
606
509
412
30
25
20
15
IO
5
37 o
5
10
15
20
25
.60 182
298
414
529
645
761
75 355
584
75 812
.76 042
272
502
1.3270
3230
.3190
3151
.3111
.3072
.79 864
776
688
600
512
424
53 o
55
So
45
40
35
42 o
5
10
15
20
25
.66 913
.67 021
129
237
344
452
.90 040
304
569
.90 834
.91 099
366
1.1106
.1074
.1041
.1009
.0977
0945
74 314
217
1 20
.74 022
73 924
826
48 o
55
50
45
40
35
30
35
40
45
50
55
876
.60 991
.61 107
222
337
451
76 733
.76964
77 I9 6
428
661
77 895
1.3032
2993
2954
2915
.2876
.2838
79 335
247
X f
.79 069
.78 980
891
30
25
20
15
10
5
3
35
40
45
50
55
67 559
666
773
880
.67 987
.68093
91 6 33
.91 901
.92 170
439
709
.92 980
1.0913
.0881
.0850
.0818
.0786
0755
.73 728
629
53i
432
333
234
30
?5
20
IS
10
5
38 o
5
10
IS
20
25
.61 566
68 1
795
.61 909
.62 024
138
.78 129
363
598
78 834
.79 070
306
1.2799
.2761
2723
2685
.2647
.2609
.78 801
711
622
532
442
35i
52 o
55
So
45
40
35
43 o
5
10
15
20
25
.68 200
306
412
5i8
624
730
93 252
524
93 797
94 071
345
620
1.0724
.0692
.0661
.0630
0599
0569
73 135
73 036
.72 937
837
737
637
47 o
55
So
45
40
35
30
35
40
45
50
55
.62 251
365
479
592
706
819
79 544
79 78i
.80 020
258
49 8
738
1.2572
2534
.2497
.2460
.2423
.2386
.78 261
170
.78 079
77 988
897
806
30
25
20
15
10
5
30
35
40
45
5o
55
.68 835
.68 941
.69 046
I5i
256
361
.94 896
95 173
451
95 729
.96008
288
1.0538
.0507
.0477
.0446
.0416
0385
72 537
437
337
236
136
72 035
30
25
20
15
10
5
39 o
5
10
15
20
25
.62 932
63 045
158
271
383
496
.80 978
.81 220
4 6l
703
.81 946
.82 igO
1.2349
.2312
.2276
2239
.2203
.2167
77 715
623
531
439
347
255
51 o
55
So
45
40
35
44 o
5
10
15
20
25
.69 466
570
675
779
883
.69 987
.96 569
.96 850
97 133
416
700
97 984
1-0355
0325
.0295
.0265
0235
.0206
7i 934
833
732
630
529
427
46 o
55
5o
45
40
35
30
35
40
45
50
55
.63 608
720
832
.63 944
.64 056
167
.82 434
678
.82 923
.83 169
415
662
1.2131
2095
.2059
.2024
.1988
1953
.77 162
.77 070
.76 977
884
791
698
30
25
20
15
10
5
30
35
40
45
So
55
.70 091
195
298
401
505
608
.98 270
556
98 843
99 131
420
99 7io
1.0176
.0147
.0117
.0088
.0058
.0029
71 325
223
121
.71 019
.70 916
813
30
25
20
IS
10
5
40 o
.64 279
.83 910
1.1918
.76 604
50 o
45 o
.70 711
1. 00 000
1. 0000
.707"
45 o
N.Cos.
N. Cot
N. Tan.
N. Sin.
o /
N. Cos.
N. Cot.
N. Tan.
N. Sin.
o /
74 TABLE VI. CIRCULAR ARCS EXPRESSED IN RADIANS.
DEGREES.
MINUTES.
SI < OM>S.
o
I
2
3
4
I
7
8
9
10
ii
12
13
14
15
16
17
18
19
20
21
22
33
24
25
26
2 7
28
29
30
3i
32
33
34
35
36
37
38
39
40
4i
42
43
44
45
46
47
48
49
50
Si
52
53
54
3
57
58
59
o.ooooo oo
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
9i
92
93
94
95
96
97
98
99
10O
101
IO2
103
104
IO I
106
107
108
109
110
in
112
"3
"4
"5
116
H7
118
119
120
1.04719 76
120
121
122
123
I2 4
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
I4 I
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
I7 6
177
178
179
1HO
2.09439 51
O'
I
2
3
4
5
6
7
8
9
10
ii
12
13
14
IS
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
3i
32
33
34
11
37
38
39
40
4i
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
00
0.00000 00
O"
I
2
3
4
6
7
8
9
10
ii
12
13
14
15
16
17
18
19
20
21
22
23
24
3
27
28
29
30
3i
32
33
34
35
36
37
38
39
40
4i
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
(10
o.ooooo oo
0.01745 33
0.03490 66
0.05235 99
0.06981 32
0.08726 65
0.10471 98
0.12217 3
0.13962 63
0.15707 96
1.06465 08
1.08210 41
i -9955 74
1.11701 07
1.13446 40
1.15191 73
1.16937 06
1.18682 39
1.20427 72
2.11184 84
2.12930 17
2.14675 50
2.16420 83
2.18166 16
2.19911 49
2.21656 82
2.23402 14
2.25147 47
0.00029 9
0.00058 I 8
0.00087 27
0.00116 36
0.00145 44
0.00174 53
0.00203 62
0.00232 71
0.00261 80
o.ooooo 48
o.ooooo 97
o.ooooi 45
o.ooooi 94
O.OOOO2 42
0.00002 91
0.00003 39
0.00003 88
0.00004 36
0.17453 29
1.22173 5
2.26892 80
0.00290 89
0.00004 85
0.19198 62
0.20943 95
0.22689 28
0.24434 61
0.26179 94
0.27925 27
0.29670 60
0.31415 93
0.33161 26
1.23918 38
1.25663 71
1.27409 04
1.29154 36
1.30899 69
1.32645 02
1-3439 35
1.36135 68
1.37881 01
2.28638 13
2.30383 46
2.32128 79
2.33874 12
2.35619 45
2.373 6 4 78
2.39110 ii
2.40855 44
2.42600 77
0.00319 98
0.00349 07
0.00378 15
0.00407 24
0.00436 33
0.00465 42
0.00494 51
0.00523 60
0.00552 69
0.00005 33
0.00005 82
0.00006 30
0.00006 79
0.00007 27
0.00007 76
0.00008 24
0.00008 73
0.00009 2I
0.34906 59
1.39626 34
2.44346 10
0.00581 78
0.00009 7
0.36651 91
0.38397 24
0.40142 57
0.41887 90
0.43633 23
0.45378 56
0.47123 89
0.48869 22
0.50614 55
1.41371 67
1.43117 oo
1.44862 33
1.46607 66
1.48352 99
1.50098 32
1.51843 64
1-53588 97
1-55334 30
2.46091 42
2.47836 75
2.49582 08
2.51327 41
2.53072 74
2.54818 07
2.56563 40
2-58308 73
2.60054 06
0.00610 87
0.00639 95
0.00669 4
0.00698 13
0.00727 22
0.00756 31
0.00785 40
0.00814 49
0.00843 S 8
o.oooio 18
o.oooio 67
o.oooii 15
o.ooon 64
0.00012 12
O.OOOI2 6l
0.00013 09
0.00013 57
0.00014 06
0.52359 88
1.57079 63
2.61799 39
0.00872 66
0.00014 54
0.54105 21
0.55850 54
0-57595 87
0.59341 19
0.61086 52
0.62831 85
0.64577 1 8
0.66322 51
0.68067 84
1.58824 96
1.60570 29
1.62315 62
1.64060 95
1.65806 28
1.67551 61
1.69296 94
1.71042 27
1.72787 60
2.63544 72
2.65290 05
2.67035 38
2.68780 70
2.70526 03
2.72271 36
2.74016 69
2.75762 02
2.77507 35
0.00901 75
0.00930 84
0.00959 93
0.00989 02
0.01018 ii
0.01047 20
0.01076 29
O.OII05 38
O.OII34 46
0.00015 3
0.00015 5i
0.00016 oo
0.00016 48
0.00016 97
0.00017 45
0.00017 94
0.00018 42
0.00018 91
0.69813 17
1.74532 93
2.79252 68
0.01163 55
0.00019 39
0.71558 50
0.73303 83
0.75049 16
0.76794 49
0.78539 82
0.80285 15
0.82030 47
0.83775 80
0.85521 13
1.76278 25
1.78023 58
1.79768 91
1.81514 24
1-83259 57
1.85004 90
1.86750 23
1.88495 56
1.90240 89
2.80998 01
2.82743 34
2.84488 67
2.86234 oo
2.87979 33
2.89724 66
2.91469 99
2.93215 31
2.94960 64
0.01192 64
0.01221 73
0.01250 82
0.01279 91
0.01309 oo
0.01338 09
0.01367 17
0.01396 26
0.01425 35
0.00019 88
0.00020 36
0.00020 85
0.00021 33
0.00021 82
0.00022 30
0.00022 79
0.00023 27
0.00023 76
0.87266 46
1.91986 22
2.96705 97
0.01454 44
0.00024 24
0.89011 79
0.90757 12
0.92502 45
0.94247 78
0-95993
0.97738 44
0.99483 77
1.01229 I0
1.02974 43
I-9373I 55
1.95476 88
1.97222 21
1.98967 53
2.00712 86
2.02458 19
2.04203 52
2.05948 85
2.07694 1 8
2.98451 30
3.00196 63
3.01941 96
3.03687 29
3.05432 62
3.07177 95
3.08923 28
3.10668 61
3.12413 94
0.01483 53
0.01512 62
0.01541 71
0.01570 80
0.01599 89
0.01628 97
0.01658 06
0.01687 15
0.01716 24
0.00024 73
0.00025 21
0.00025 70
0.00026 1 8
0.00026 66
0.00027 15
0.00027 63
0.00028 12
0.00028 60
1.04719 76
2.09431) M
3.14159 27
3.01745 ;
0.00029 09
TABLE VII.
NAPIERIAN LOGARITHMS OF NUMBERS.
75
N.
Log.
N.
Log.
N.
Log.
N.
Log.
N.
Log.
i
2
3
4
5
0.00000
0.69315
1.09861
1.38629
1.60944
21
22
23
24
25
3-04452
3.09104
3-13549
3-17805
3.21888
4i
42
43
44
45
3-7I357
3.73767
3.76120
3-78419
3.80666
61.
62
63-
64
65
4.11087
4.12713
4-14313
4.15888
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9
UNIVERSITY OF CALIFORNIA LIBRARY
BERKELEY
Return to desk from which borrowed.
This book is DUE on the last date stamped below.
ASTFIDNOMY LIB
111AJ .->
&& 1 5 1971
ARY
LD 21-100m-ll,'49(B7146sl6)476
YC 1 02260
MATH*.
STAT
UBftARY