\-
6
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. CRAWLEY.
ELECTROTYPED AND PRINTED 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
(Vega 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. CEAWLEY.
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
u V. Natural Sines, Cosines, Tangents, and Cotangents . . 69
" VI. Circular Arcs, expressed in Radians 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
729 =
3,
((
9 3
= 729
3 =
~v
((
gi
= 3
9 =
i;
II
9 1
= 9
1 =
o,
II
9
-<
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 Napierian-)- system with the base
6 = 2.7182818285....
defined by the convergent infinite series
6= 1 + l + IV + 1^3 + l34 +
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 (exce pt 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
by the logarithm of n to the base e.") The factor - is
log e 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 (1656-1631), who first suggested the use of
the base 10.
f Named for John Napier, Baron of Merchiston, in Scotland (1550-
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 1 = 10
(a)
log 1= 0, " 10 = 1
log a = -i, " 10 - = .1
log .01= 2, " 10- 2 = .01
log .001 = 3, " 10 - 3 = .001
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 + 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.
Eeferring again to (<2), 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 -j- 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=
" u log .07296 = 2
" " log 26 = 1
" " log 400000= 5
" " log .9426 = !
" " log 3869= 3
" " log .00042 = 4
" " log .005 = 3
" u 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.
Yerify 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 10 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 reul 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. g 1, we have log (1.789) 5 = 5 X log 1.789.
log 1.789 = .25261
_ 5
log (1.789) 5 = 1.26305 /. (1. 789) 5 = 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) = ( 14-34788 10
I or 4.34788.
Hence 728 X 63.86 X -4792 = 22280.
3. Compute the value of $ 73.
log 73 = 1.86332.
By IV. \ 1, log f W= \ 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 m. 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 V/.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, . . y. 06398= .6752.
log 1/27 = | log 27 = J X 1.43136 = .71568
log (9.261)* = f log 9.261 = f X 0.96666 = .41428
By II. g 1 loga; = .30140
* = 2.002.
7. x= 68.96 X >flndg .
39 X (8.642)^ X (-96) 2
log 68.96= 1.83860
lg ^.4228 = 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
. A / 23.11662 20
log of denominator = \ or 3>11662
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) s =1.56102 log f. 4228 = 9.87538
log (.96) 2 = 9.96454 log of num. = 1.71398
log of denom. = 3.11662 3.11662
x= .03957 log x = 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 1/6129. Am. 65730 5 21 . 38 X 6.296 X. 412. Ans. .2292
7 X ^ 41290
2. 19.63 X ^689.2. Ans. 173.4 4 19 y 6 2 x f "^7
6 (3.S39) 3 X 142.9
3. 2 X " ;~* -^ins. o. <yo _ _ _ NX i/ n
^O QA9\ 7 ^0-< X 00<> X V li .
(2.96)*
4. I__. An*
. 4021X18 ^ ns . 197 .
1/41.63 X (2.649) 5
6. The Arithmetical Complement of the Logarithm
or Co-logarithm. To compute the value of ^ by log
arithms, we may take either log a log b, or log a -f- log-.
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 729.6 = 2.86308. Subtracting this
from 10.00000 10, the result is colog 729.6 == 7.13692 10.
Ex. Find colog .0641. Log .0641 a 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 = 9.13680, 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
d X e X /
three operations are required :
(1.) to find log a -f log b -\- 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 -f- log b -f log c -j-
colog d -4- colog e + 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 f .4228 = 9.87538
colog 39 =8.40894
colog (8.642)1 =8.43898
colog (.96) 2 = 0.03546
log x = 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 -f- .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 log x = 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 mantissa 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 mantissas?
Evidently we have the proportion
18 : 1 = 15 : difference required
or difference = if = = .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) 2 X 87
~ X 72.983 X
log (36.842)* = 1.56634 Xi = -52211
log(1.6272) 2 = .21144X2= .42288
log 87 _ =1.93952
colog V .062416 = 1.20471 -4- 2 = .60235
colog 72.983 = 8.13678
colog 1^189 = 7.72354 + 3 = 9.24118
x = 7.3252 log 2;= .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
1 f7tiX 6.7843 ^..10953
63.842 X
(42.32)* X (.02478) 3 -*- 1/2 An8 * 93 38
(7.2S43) 8 X 1^.00067894
3 " (620.01)* X 489.62 ^m. 306.49
1986.1 X ^92.836
~~F" Ana. 403.75
V 11 X V 22 X
5. .064219 X - ^*w- -17641
6. V 124.37 . Am. 25.243
I 7 1000000
7. l/ 10 X ^100 X flOGO. ^ns. 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.
Ex. Find a;, given log x 2.21647,
nearest log in table = .21643, corresponding to 1646
difference 4
nearest smaller prop.)
pt. under tab. dlff. 26 / = 3,6 1 corresponding to 1
difference remaining ~Ti l for the fifth fig.
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
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.
XX11 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 down the page until we come 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 16 = 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
4J x 35 _3jn_ 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
it it g// tt it 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 -j- 3 = 188) instead.
correction for 20" = 61.7
tab. diff. 3
tab. diff. 185 {
20"= 1.0
u n 6// = 0.3
81.5
Hence, the total correction to be added is 82 and log tan 3 C
51 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 V (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.
Verify the following statements :
log cos 17 38 42" = 9. 97907, log tan 5 38 5" = 8.99416,
log tan 84 V 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 ( siu 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. Remembering 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.
Read 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 8 == 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, 6 exceeds 47 15 by T 3 T of one
minute. This fraction reduced to seconds is T 3 T X 60 = 16 /x . Hence,
6 = 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 /x
difference remaining = 1.2 " " 6 X/
whole number of seconds 16 X/
NOTE. The number of seconds corresponding to 1.2 under tabular
difference 11 is, according to the table, either 6" or 7 /x ; but 6" is
really a little nearer than 7 X/ , as we found above.
2. Given log cot 6 = 0.72654, find 6.
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 X . 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 /x
difference remaining = .5,
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 V = 5.17,
no. of seconds = ^L = 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. 0*=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 + 10 37 51" = 190
37 51". 3. = 2 20 46" and 180 + 2 20 46" = 182 20 46", be
cause tan 6 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,
n _ f 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, subtfact the angle taken
from the table from 360.
Verify the following statements :
log sin 6= 9.28642, 0= 11 9 1" and 168 50 59".
log cos 6= 8.46321, 6= 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, 0= 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 given 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 tan 6 = 6>2984 sin2 63 18/ 20// 0=1 127 V 7//
7.5692 cot 116 36 12" " 1 307 V 7"
2 POS 6 = - 2 93 tan 48 6 38" = f 121 34 3"
14.12 sin 26 13 42" 1 238 25 57"
EXPLANATION OF THE TABLES. XXIX
7 58 17"
//
Q a I sin3 146 12/ 19// X tan 78 12 32" A _ J 172 1 43
3. am - 187 o 53,
352 V 43
"
4 t g^ .93862 cos 2 312 38 / 40" fl = f 32 55 19"
.86471 tan s 214 26 31" " \ 212 55 19"
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 ^5LJ? 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" -f- S
log tan a = log a" -f 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 tail a T
Ex. Find log tan 26 51".
26 51" = 1611" log 1611 = 3.20710
T (for 27 ) = 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 r 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 69-73.)
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.
1. 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 =1.6358897
.0058~140
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. I
N.
1
2 3
4
5
6
7
8
9
Prop. Pts.
100
oo ooo
043
087
130 173
217
260
303
346
389
01
432
475
518
561 604
647
689
732
775
817
44
43
42
02
860
903
945
988
^030
*0 7 2
*H5
*i57
*I99
*242
^
4-4
4.3
4.2
03
oi 284
326
368
410
452
494
536
578
620
662
2
8.8
8.6
8.4
04
703
745
787
828
870
912
953
995
036
*o78
3
13-2
12.9
12.6
05
02 119
1 60
202
243
284
325
366
407
449
49
4
17.6
17.2
16.8
06
531
572
612
653
694
735
776
816
857
898
b
22.0
21-5
21.0
6
26.4
2S.8
25.2
07
938
979
*oi9
*o6o
*IOO
*i 4 i
*i8i
*222
*262
*302
7
30 8
.O.I
2Q.4
08
03 342
383
423
463
503
543
583
623
663
703
8
35- 2
.4.4
7
33-6
09
743
782
822
862
902
941
981
-021
*o6o
*IOO
9
39.6 38.7
V7-8
110
04 139
179
218
258
297
336
376
415
454
493
ii
532
571
610
650
689
727
766
80 5
844
883
4i
40
39
12
922
961
999
038
*o77
*H5
*I54
*IQ2
^231
^269
T
4.1
4.0
..Q
13
05 38
346
385
423
461
500
538
576
614
652
2
8.2 8.0
7.8
14
690
729
767
805
843
881
918
956
994
*032
3
12.3 12.0
11.7
15
06 070
108
145
183
221
258
296
333
408
4
16.4 16.0
15.6
16
446
483
521
558
595
633
670
707
744
781
5 20.5 20.0
19-5
6 24.6
24.C
23.4
17
819
856
893
930
967
*oo4
*O4 1
^078
*H5
^151
7
28.7
28.0
27.3
18
07 1 88
225
262
298
335
372
408
445
482
518
8
.2.8
.2-0
31.2
19
555
628
664
700
737
773
809
846
882
9 36.9 36.0
35-1
120
918
954
990
*027
*o6 3
^99
*i35
*i7i
*207 *243
21
08 279
314
35o
386
422
458
493
529
565
600
3*
37
3&
22
636
672
707
743
778
814
849
884
920
955
T
3-8 3-7
3-6
23
991
*026
*o6i
*i67
*202
*2 3 7
*2 7 2
*37
2
7.6 7.4
7.2
24
09 342
377
412
447
482
517
552
587
621
656
3
11.4 ii. i
10.8
25
26
691
10 037
726
072
760
106
795
140
830
175
864
209
899
243
934
278
9 68
312
*oo 3
346
4
5
6
15.2 14.81 14.4
19.0 18.5 18.0
22.8, 22.2 21.6
27
380
415
449
483
517
55i
585
619
653
687
7
26.6 25.Q 25.2
28
721
755
78 9
823
857
890
924
958
992
*025
8
30.4
29.6; 28.8
29
ii 059
093
126
160
193
227
261
294
327
361
9
34-2
33-3 32.4
130
394
428
461
494! 528
561
594
628
66 1
694
31
727
760
793
826 860
893
926
959
992
*024
35
34
33
32
12 057
090
123
156 189
222
254
287
320
352
i
3-5
3-4
3-3
33
385
418
450
483 ! 516
548
581
613
646
678
2
7.0
6.8
6.6
34
710
743
775
808
840
8 7 2
905
937
969
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3
10.5
IO.2
9-9
35
36
13 033
354
066
386
098
418
130
450
162
481
194
513
226
545
258
577
290
609
322
640
4
5
6
14.0
17-5
21.0
13.6
17.0
20.4
13.2
16.5
19.8
11
672
988
704
^019
* 735
767
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799
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830
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862
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893
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925
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956
-270
8
24-5
28.0
23.8
27.2
23.1
26.4
39
14 301
333
364
395
426
457
489
520
55 1
582
9
31.5
30.6
29.7
140
613
644
675
706
737
768
799
829
860
891
41
922
953
983
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32
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30
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15 229
259
290
320
351
381
412
442
473
503
I
3-2
3.1
3-
43
534
564
594
625
655
685
715
746
776
806
2
6.4
6.2
6.0
44
45
46
836
16 137
435
866
167
465
897
197
495
927
227
524
957
256
554
987
286
584
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613
346
643
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376
673
*io 7
406
702
3
4
1
9.6
12.8
16.0
19.2
9.3
12.41
15.5
T8.6
9.0
12.0
15.0
18.0
47
732
761
791
820
850
879
909
938
967
997
7
22.4
21.7
21.
48
17 026
056
085
114
143
173
202
231
260
289
8
25.6
24.8
24.0
49
319
348
377
406
435
464
493
522
55i
58o
9
28.8
27.9
27.0
150
609
638
667
696
725
754
782
811
840
869
N.
1
2
3
4
5 6
7 I 8
a
Prop. Pts.
N.
O
1 2
3 4
5
6 7 | 8
9
Prop. Pts.
150
17 609
638
667
696
725
754
782
811
840
869
5i
898
926
955
984
*oi3
*o4i
^070
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*i56
29
28
52
18 184
213
241
270
298
327
355
384
412
441
I
2.9
2.8
53
469
498
526
554
583
611
639
667
696
724
2
5-8
5.6
54
752
780
808
837
865
893
921
949
977
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3
8.7
8.4
55
19 33
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089
117
145
173
201
229
257
285
4
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56
312
340
368
396
424
45i
479
507
535
562
5
6
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17.4
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16.8
57
58
590
866
618
893
645
921
948
700
976
728
*oo3
756
*030
783
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811
*o85
838
*II2
8
20.3
23.2
19.6
22.4
59
20 140
167
194
222
249
276
303
330
358
385
9
26.1
25-2
1GO
412
439
466
493
520
548
575
602
629
656
61
683
710
737
763
790
817
844
871
898
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27
20
62
952
978
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*032
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*I39
*i65
^192
i
2.7
2.6
63
21 219
245
272
299
325
352
378
405
431
458
2
5-4
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64
484-
5ii
537
564
590
617
643
669
696
722
3
8.1
7.8
65
748
775
801
827
854
880
906
932
958
985
4
10.8
10.4
66
22 Oil
037
063
089
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141
167
194
220
246
5
6
13-5
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13.0 -
15.6
67
272
298
324
350
376
401
427
453
479
505
7
18.9
18.2
68
531
557
583
608
634
660
686
712
737
763
8
?T.6
20.8
69
789
814
840
866
891
917
943
968
994
*OI 9
9
24-3
23.4
170
23 45
070
096
121
147
172
198
223
249
274
7i
300
325
350
376
401
426
452
477
502
528
25
72
553
578
603
629
654
679
704
729
754
779
i 2.5
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830
855 ! 880
95
930
955
980
*oo5
*030
2 5-0
74
24 055
080
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155
1 80
204
229
254
279
3 7-5
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329
353
378
403
428
452
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502
527
4 10.0
76
55i
576
601
625
650
674
699
724
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773
5 12.5
6 15.0
77
797
822
846
8 7 I
895
920
944
969 993
*oi8
7 17-5
78
25 042
066
091
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139
164
188
212
237
261
8 20. o
79
285
310
334
358
382
406
431
455
479
503
9 22.5
180
527
551
575
600
624
648
672
696
720
744
81
768
792
816
840
864
888
912
935
959
983
24
23
82
26 007
031
055
079
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126
150
174
198
221
i
2.4
2.3
3
245
269
293
316
340
3 6 4
387
411
435
458
2
4-8
4.6
84
482
505
529
553
576
600
623
647
670
694
3
7-2
6.9
85
717
741
764 788
811
834
8,S8
88 1
95
928
4
9.6
9.2
86
95i
975
998
*02I
*045
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*i38
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6
12.
14.4
ii. 5
13-8
87
27 184
207
231
254
277
300
323
346
37o
393
7
16.8
16.1
88
416
439
462
485
508
53i
554
577
600
623
8
19.2
18.4
89
646
669
692
715
738
761
784
807
830
852
9
21.6
20.7
190
875
898
921
944
967
989
*OI2
*o 3 5
*o 5 8
*o8i
9i
28 103
126
149: 171 194
217
240
262
285
307
22
21
92
330
353
375 398 421
443
466
488
5ii
533
i
2.2
2.1
93
556
578
601
623 I 646
668
6 9 I
713
735
758
2
4.4
4.2
94
780
803
825
847
870
892
914
937
959
.981
3
6.6
6.3
95
29 003
026
048
070 092
H5
137
159
181
203
4
8.4
96
226
248
270
292 ; 314
336
358
38o
403
425
II.O
13.2
10.5
12.6
97
447
469
491
513 ! 535
557
579
601
623
645
7
15.4
14.7
98
667
688
710
732
754
776
798
820
842
863
8
17.6
16.8
99
885
907
929
95i
973
994
*oi6
*o 3 8
*o6o
*o8i
9
19.8
18.9
200
30 103
125
146
1 68
190
211
233
255
276
298
N.
O
1
2
3
4
5
6
7 8
9
Prop. Pts.
jr.
1
2
3
4
5
6
7
8
9
Pro
p. Vis.
200
30 103
125
146
1 68
190
211
233
255
276
298
OI
02
03
04
05
06
07
08
09
320
535
750
963
31 175
387
597
806
32 015
341
557
771
984
197
408
618
827
035
363
578
792
*oo6
218
429
639
848
056
384
600
814
*02 7
239
450
660
86 9
077
406
621
835
260
681
890
098
428
643
856
*o6 9
281
492
702
911
118
449
664
878
302
513
723
139
471
685
899
*II2
323
534
744
952
160
49 2
707
920
*I33
345
555
765
973
181
514
728
942
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366
576
785
994
2OI
2
I 2
2 4
3 6
4 *
5 ii
6 13
7 if
8 17
9 IS
2 21
.2 2.1
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.6 6.3
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4 14-7
.6 16.8
.8 18.9
210
222
243
263
284
305
325
346
366
387
408
ii
12
13
1
17
18
19
428
634
838
33 041
244
445
646
846
34 044
449
654
858
062
264
465
666
866
064
469
675
879
082
284
486
686
885
084
49
695
899
102
304
506
7 06
905
104
715
919
122
325
526
726
925
124
736
940
143
345
546
746
945
143
552
756
960
^63
365
566
766
965
163
572
777
980
183
385
586
786
985
183
593
797
*OOI
203
405
606
806
203
613
818
*O2I
224
425
626
826
223
i
2
3
4
5
6
8
9
20
2.0
4.0
6.0
O.O !
10.0
12.0
I4.O
16.0
18.0
220
242
262
282
301
321
34 i
361
380
400
42O
21
22
23
24
25
26
27
28
29
439
635
830
35 025
218
411
603
793
984
459
655
850
044
238
430
622
* 813
*00 3
479
674
869
064
257
449
641
* 832
*02I
498
694
88 9
08 3
2 7 6
468
660
*o 5 o
518
713
908
102
295
488
679
870
537
733
928
122
315
507
6 9 8
889
557
753
947
141
334
526
717
908
577
772
967
1 60
353
545
736
927
*n6
596
792
986
180
372
564
755
946
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616
811
*oo 5
199
392
583
774
965
*i54
i
2
3
4
5
6
8
9
19
1.9
3-8
5-7
7-6
9-5
11.4
13-3
15-2
17.1
280
36 173
192
211
229
248
267
286
305
324
342
32
33
34
35
36
37
38
39
361
549
736
922
37 107
291
475
658
840
380
568
754
940
125
310
493
676
858
399
586
773
959
144
328
694
876
418
605
791
977
162
346
530
712
894
436
624
810
996
181
365
548
912
455
642
829
*oi4
566
749
474
661
847
*o 33
218
401
585
767
949
493
680
866
*0 5 I
236
420
603
785
967
254
438
621
803
985
717
903
*o88
273
457
639
822
*00 3
i
2
3
4
6
7
18
1.8
3-6
5-4
7.2
9.0
10.8
12.6
14.4
16.2
240
38 02 1
039
057
075
093
112
130
148
1 66
184
42
43
44
45
46
47
48
49
202
382
739
917
39 94
270
445
620
220
399
578
757
934
in
287
463
637
238
417
596
775
952
129
305
480
655
256
435
614
792
970
146
322
498
672
274
453
632
810
987
164
515
690
2 9 2
471
650
828
182
358
533
707
310
846
*02 3
I 99
375
550
724
328
507
686
* 863
217
393
568
742
346
525
703
881
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235
410
585
759
364
543
721
* 8 "
252
428
602
777
i
2
3
4
6
8
9
3-4
6.8
8.5
10.2
$
15-3
250
794
811
829
846
863
881
898
9i5
933
950
N.
o
1
2
3
4
5
6
7
8
9
Pro
p. Pts.
N.
1
2
3
4
5
6
7
8
9
Pro
p. Pts.
250
39 794
811
829
846
863
881
898
915
933
950
5i
52
53
54
55
56
57
58
59
967
40 140
312
483
654
824
993
41 162
330
985
157
329
500
67 1
841
*OIO
179
347
*OO2
175
346
518
688
858
*02 7
196
363
*oi9
192
364
535
705
875
*044
212
380
*o 3 7
209
381
552
722
892
*o6i
229
397
*Q54
226
398
569
739
909
*o78
246
414
*o 7 i
243
415
586
756
926
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263
430
*o88
261
432
603
773
943
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280
447
*io6
278
449
620
790
960
*I28
296
464
*I2 3
295
466
637
807
976
*I45
313
481
I
2
3
4
6
9
18
1.8
3.6
5-4
7.2
9.0
10.8
12.6
14.4
16.2
260
497
5i4
53i
547
564
58i
597
614
631
647
61
62
63
64
65
66
67
68
69
664
830
996
42 160
325
651
813
975
68 1
847
*OI2
177
341
504
66 7
830
99 I
697
863
*029
193
357
521
684
846
*oo8
714
880
*Q45
210
374
537
700
862
*024
731
896
*062
226
390
553
716
878
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747
913
*o 7 8
243
406
57o
732
894
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764
929
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259
423
586
749
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780
946
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275
439
602
765
927
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797
963
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292
455
619
78i
943
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814
979
**44
308
472
635
797
959
*I2O
i
2
3
4
6
I
9
17
1.7
3.4
5-i
6.8
8.5
10.2
11.9
13.6
15.3
270
43 136
152
169
185
201
217
233
249
265
28l
7i
72
73
74
75
76
77
78
79
297
457
616
775
933
44 091
248
404
56o
313
473
632
791
949
107
264
420
576
329
807
965
122
2 7 9
436
592
345
505
664
823
981
138
295
45i
607
361
521
680
838
996
154
311
467
623
377
537
696
854
*OI2
170
326
483
6 3 8
393
553
712
870
*028
185
342
498
654
409
569
727
886
*044
20 1
358
5H
669
425
584
743
902
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217
373
529
685
441
600
759
917
*Q75
232
389
545
700
i
2
3
4
5
6
8
9
16
1.6
is
6. 4
8.0
9.6
II. 2
12.8
14.4
280
7i6
73i
747
762
778
793
809
824
840
855
81
82
83
84
85
86
87
88
89
871
45 025
179
332
484
637.
788
939
46 090
886
040
194
347
500
652
803
954
105
902
056
209
362
5i5
667
818
969
120
917
071
225
378
530
682
834
984
135
932
086
240
393
545
697
849
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150
948
102
255
408
561
712
86 4
*oi5
165
963
117
271
423
576
728
879
*030
1 80
979
133
286
439
59i
743
894
*<H5
195
994
148
301
454
606
758
909
*o6o
210
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163
317
469
621
773
924
*Q75
225
i
2
3
4
5
6
8
9
15
i.5
3-0
4-5
6.0
7-5
9.0
10.5
12.0
13-5
290
240
255
270
285
300
315
330
345
359
374
9i
9 2
93
94
9
96
97
98
99
835
982
47 129
276
422
567
404
553
702
850
997
144
290
436
582
419
568
7 l6
864
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159
305
451
596
434
583
73i
879
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173
3J9
465
611
449
598
746
894
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188
334
480
625
464
613
761
909
*o 5 6
202
349
494
640
479
627
776
923
*o7o
217
363
509
654
494
642
790
938
*o8s
232
378
524
669
509
657
805
953
*IOO
246
392
538
683
f3
672
820
967
*ii 4
261
407
553
698
i
2
3
4
8
9
14
1.4
2.8
4.2
5.6
7.0
8.4
9.8
II. 2
12.6
BOO
712
727
741
756
770
784
799
813
828
842
N.
1
a
3
4
5
G
7
8
9
Pro
p. Pts.
N.
1
2
3
4
5
6
7
8
9
Pro
p. Pts.
300
47 712
727
741
756
770
784
799
813
828
842
01
02
03
04
05
06
07
08
09
857
48 ooi
144
287
430
572
714
855
996
871
015
159
302
444
586
728
869
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885
029
173
3i6
458
601
742
883
*O24
900
044
187
330
473
615
756
897
*o 3 8
914
058
202
344
487
629
770
9i i
*052
929
073
216
359
SGI
643
785
926
*o66
943
087
230
373
515
657
799
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244
387
530
671
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116
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544
686
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130
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416
558
700
841
982
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2
3
4
6
15
1-5
3-o
4-5
6.0
7-5
9.0
310
49 136
150
164
178
192
206
220
234
248
262
10.5
ii
12
13
14
15
16
17
18
19
276
415
554
693
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969
50 106
243
379
290
429
568
707
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393
304
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133
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406
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457
596
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147
284
420
332
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161
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346
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624
762
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174
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360
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215
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488
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229
365
501
9
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3
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14
1.4
2.8
4.2
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320
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529
542
556
569
583
596
610
623
637
5
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21
22
23
24
25
&
27
28
29
651
786
920
5i 055
1 88
322
455
587
720
664
799
934
068
202
335
468
601
733
678
813
947
08 1
215
348
481
614
746
691
826
961
095
228
362
495
627
759
705
840
974
108
242
375
508
640
772
718
853
987
121
255
388
521
654
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732
866
*OOI
135
268
402
534
667
799
745
880
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148
282
415
548
680
812
759
893
*028
162
295
428
561
693
825
772
907
*o4i
175
308
441
574
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8
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9.8
II. 2
12.6
13
i-3
2.6
330
851
865
878
891
904
917
930
943
957
970
3
3-9
3i
32
33
34
35
36
37
38
39
983
52 114
244
375
504
634
763
892
53 020
996
127
257
388
517
647
776
905
033
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140
270
401
530
660
789
917
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153
284
414
543
673
802
930
058
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166
297
427
556
686
815
943
071
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179
310
440
569
699
827
956
084
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192
323
453
582
711
840
969
097
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205
336
466
595
724
853
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218
349
479
608
737
866
994
122
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231
362
492
621
750
879
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135
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6
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9
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7.8
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12
340
148
161
173
1 86
199
212
224
237
250
263
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2
1.2
2 A
4i
42
43
44
45
46
47
48
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275
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529
656
782
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54 033
158
283
288
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542
668
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301
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441
567
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195
320
326
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580
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958
083
208
332
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466
593
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220
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108
233
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364
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618
744
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120
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377
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133
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390
517
643
769
895
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145
270
394
3
4
5
6
8
9
4
3-6
4.8
6.0
7-2
8.4
9.6
10.8
350
407
419
432
444
456
469
481
494
506
5i8
N.
1
2
3
4
5
6
7
8
9
Pro
p. Pts.
N.
o
1
2
3
4
*
6
7
8
9
Pro
p. Pts.
350
54 407
419
432
444
456
469
481
494
506
5i8
51
52
53
54
55
56
57
58
59
53i
654
111
900
55 023
145
267
388
509
543
667
790
913
035
157
279
400
522
555
679
802
925
047
169
291
413
534
568
691
814
9
060
182
303
425
546
58o
704
827
949
072
194
315
437
558
593
716
839
962
084
206
328
449
57o
605
728
851
974
096
218
340
461
582
617
741
864
986
108
230
352
473
594
630
753
876
998
121
242
364
485
606
642
765
888
*OII
133
255
376
497
618
i
2
3
4
5
6
13
3
3-9
5-2
6.5
7-8
360
630
642
654
666
678
691
703
715
727
739
9.1
61
62
63
64
65
66
67
68
69
75i
871
99 1
56 no
229
348
467
585
703
763
883
*oo3
122
241
360
478
597
714
775
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134
253
372
490
608
726
787
907
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146
265
384
502
620
738
799
919
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158
277
39 6
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632
750
811
93i
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170
289
407
526
644
761
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182
301
419
538
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773
835
955
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194
312
43i
549
667
785
847
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205
324
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679
797
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979
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217
336
455
573
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9
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2
3
4
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12
1.2
2.4
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4.8
370
820
832
844
855
867
879
891
902
914
926
6.0
7i
72
73
74
75
76
77
78
79
937
57 054
171
287
403
519
634
749
864
949
066
183
299
415
530
646
76!
875
961
078
194
310
426
542
657
772
887
972
089
206
322
438
553
669
784
898
984
101
217
334
449
565
680
795
910
996
H3
229
345
461
576
692
807
921
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124
241
357
473
588
703
818
933
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136
252
368
484
600
715
830
944
*c>3i
148
264
380
496
611
726
841
955
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J 59
276
392
507
623
738
852
967
8
9
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2
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8.4
9.6
10.8
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2.2
380
978
990
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*oi3
*024
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*o 5 8
*o7o
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3
3-3
81
82
83
84
85
86
87
88
89
58 092
206
320
433
546
659
771
883
995
104
218
33i
444
557
670
782
894
*oo6
115
229
343
456
569
68 1
794
906
*oi 7
127
240
354
467
580
692
805
917
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138
252
365
478
591
704
816
928
^040
149
263
377
49
602
715
827
939
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161
274
388
5oi
614
726
838
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172
286
399
512
625
737
850
961
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184
297
410
524
636
749
86 1
973
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195
309
422
535
647
760
872
984
*0 95
4
5
6
8
9
4
5-5
6.6
7-7
8.3
9-9
10
390
59 106
118
129
140
151
162
173
184
195
207
i
I.O
9i
92
93
94
95
96
97
98
99
218
329
439
550
660
770
879
988
60 097
229
34o
450
56i
671
780
890
240
35i
461
572
682
791
901
*OIO
119
251
362
472
583
693
802
912
*02I
130
262
373
483
594
704
813
923
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141
273
384
494
605
715
824
934
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152
284
395
506
616
726
835
945
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163
295
406
517
627
737
846
956
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173
306
417
528
638
748
857
966
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184
3i8
428
539
649
759
868
977
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195
3
4
5
6
8
9
3-0
4.0
5.o
6.0
7.0
8.0
9.0
400
206
217
228
239
249
260
271
282
293
304
N.
O
1
I
3
4
5
6
7
8
9
Pro
p. Pts.
N.
1
ft
3 4
5
67 8
9
Prop. Pts.
400
01
02
03
04
05
06
07
08
9
410
ii
12
13
14
15
16
17
18
19
420
21
22
23
24
25
26
27
28
29
430
3i
32
33
34
35
36
37
38
39
440
4i
42
43
44
45
46
47
48
49
450
60 206
217
228
239
249
260
271 282
293
304
i
2
3
4
I
8
9
i
2
3
4
6
9
i
2
3
4
6
8
9
ii
i.i
2.2
3-3
4.4
5-5
6.6
8.S
9-9
10
I.O
2.0
3-0
4.0
5-0
6.0
7.0
8.0
9.0
9
0.9
1.8
2.7
3-6
4-5
5-4
6.3
7.2
8.1
3H
423
531
638
746
853
959
61 066
172
325
433
541
649
756
863
970
077
183
336
444
552
660
767
874
981
087
194
347
455
563
670
778
885
991
098
204
358
466
574
681
788
895
*002
109
215
369
477
584
692
799
906
*oi 3
119
225
379
487
595
703
810
917
*02 3
130
236
390
498
606
713
821
927
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140
247
401
509
617
724
831
938
*45
151
257
412
520
627
735
842
949
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162
268
278
289
300
310
321
33i
342
352
363
374
384
490
595
700
805
909
62 014
118
221
395
500
606
711
815
920
024
128
232
.405
511
6i
721
826
930
034
138
242
416
52i
627
731
836
941
045
149
252
426
532
637
742
847
951
055
159
263
437
542
648
752
857
962
066
170
273
448
553
658
763
868
972
076
1 80
284
458
563
669
773
878
982
086
190
294
469
574
679
784
888
993
097
201
304
479
584
690
794
899
*oo3
107
211
315
325
335
346
356
366
377
387
397
408
418
428
53"
634
737
839
941
63 043
144
246
439
542
644
747
849
95i
053
155
256
449
552
655
757
859
961
063
165
266
459
562
665
767
870
972
073
175
276
469
572
675
11*
982
08 3
185
286
480
583
685
788
890
992
094
195
296
49
593
696
798
900
*002
104
205
306
500
603
706
808
910
*OI2
114
215
317
511
613
7 l6
818
921
*022
124
225
327
521
624
726
829
931
*033
134
236
337
347
357
367
377
387
397
407
417
428
438
448
548
649
749
849
949
64 048
147
246
~345
458
558
659
759
859
959
058
157
256
468
568
669
769
869
969
068
167
266
478
579
679
779
879
979
078
177
276
488
589
689
789
88 9
9 88
088
I8 7
286
498
599
699
799
899
998
098
197
296
508
609
709
809
108
207
306
518
619
719
819
919
*oi8
118
217
3i6
528
629
729
829
929
*028
128
227
326
538
639
739
839
939
*o 3 8
137
237
335
355
365
375
385
395
493
59i
689
787
885
982
079
176
273
404
414
424
434
444
542
640
738
836
933
65 031
128
225
454
552
650
748
846
943
040
137
234
33i
464
562
660
758
856
953
050
H7
244
473
572
670
768
865
963
060
157
254
35o
483
582
680
777
875
972
070
167
263
503
601
699
797
895
992
089
186
_?3
379
513
611
709
807
904
*002
099
I 9 6
292
523
621
719
816
914
*OII
108
205
302
532
631
729
826
924
*02I
118
215
312
321
341
360
369
389
398
408
N.
O
1
2
3
4
5
6 7
8
9
Prop. Pts.
N.
1
Z
3
4
5
6
7
8
9
Fro]
p. Pts.
450
65 321
331
341
350
360
369
379
389
398
408
51
52
53
54
55
56
57
58
59
418
5H
610
706
801
896
992
66 087
181
427
523
619
715
8n
906
*OOI
096
191
437
533
629
725
820
916
*OII
106
200
447
543
639
734
830
925
*020
"3
210
456
552
648
744
839
935
*030
124
219
466
562
658
753
849
944
*o 39
134
229
475
57i
667
763
858
954
*049
143
238
485
58l
677
772
868
963
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153
247
495
59 1
686
782
877
973
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162
257
504
600
696
792
887
982
*o 7 7
172
266
I
2
3
10
I.O
2.0
3-0
460
276
285
295
304
3H
323
332
342
35i
36i
4
c
4.0
c o
61
62
63
64
65
66
67
68
69
370
464
558
652
745
839
. 932
67 025
117
380
474
567
661
755
848
941
034
127
389
483
577
671
764
857
95o
043
136
398
492
586
680
773
867
960
052
145
408
502
596
689
783
876
969
062
154
417
5ii
605
699
792
885
978
071
164
427
521
614
708
801
894
987
080
173
43 6
530
624
717
811
904
997
089
182
445
539
633
727
820
913
*oo6
099
191
455
549
642
736
829
922
*oi5
108
201
6
8
9
6.0
7.o
8.0
9.0
470
210
219
228
237
247
256
265
274
284
293
7i
72
73
74
75
76
77
78
79
302
394
486
578
669
76i
852
943
68 034
3ii
403
495
f7
679
770
861
952
043
321
413
504
779
870
961
052
330
422
514
605
697
788
879
970
06 1
339
43i
523
614
706
797
888
979
070
348
440
532
624
7i5
806
079
357
449
54i
633
724
815
906
367
459
550
642
733
825
916
*oo6
097
376
468
560
651
742
834
925
*oi5
106
385
477
569
660
752
843
934
*024
115
i
2
3
4
6
9
9
0.9
1.8
2-7
3-6
4-5
5.4
6.3
7-2
8.1
480
124
133
142
151
160
169
178
187
196
205
81
82
83
84
85
86
87
88
89
215
305
395
485
574
664
753
842
93i
224
3H
404
494
583
673
762
851
940
233
323
4*3
502
592
681
771
860
949
242
332
422
5ii
601
690
780
869
958
251
34i
431
520
610
699
789
878
966
260
350
440
529
619
708
797
886
975
269
359
449
538
628
717
806
895
984
278
368
458
547
637
726
815
904
993
287
377
467
556
646
735
824
913
*002
476
565
655
744
833
922
*OII
i
2
3
8
0.8
1.6
2.4
490
69 020
028
037
046
055
064
073
082
090
099
4
3-2
9 1
92
93
94
95
96
99
108
197
285
373
461
548
636
723
810
117
205
294
38i
469
557
644
732
819
126
214
302
390
478
566
653
740
827
135
223
3U
399
487
574
662
749
836
144
232
320
408
496
583
671
758
845
152
241
329
417
504
592
679
767
854
161
249
425
513
601
688
775
862
170
258
346
434
522
609
697
784
871
179
267
355
443
&
705
793
880
188
276
364
452
539
627
7M
801
888
5
6
8
9
4.0
4-8
5-6
6.4
7.2
500
897
906
914
923
932
940
949
958
966
975
N.
-|
2
3
4
5
6
7
8
9
Proj
. Pts.
IO
N.
1
2
3
4
5
6
7
8
9
Proj
). PtS.
500
69 897
906
914
923
932
940
949
958
966
975
01
02
03
04
05
06
07
08
09
984
70 070
157
243
329
415
501
586
672
992
079
165
252
338
424
509
595
680
*OOI
088
174
260
346
432
5i8
603
689
*OIO
096
183
269
355
441
526
612
697
*oi8
105
191
278
364
449
535
621
706
*02 7
114
2OO
286
372
458
544
629
714
*o 3 6
122
209
295
381
467
552
6 3 8
723
*044
131
217
303
389
475
56i
646
73i
*o 53
140
226
312
398
484
569
655
740
*062
148
234
321
406
492
578
663
749
I
2
3
9
0.9
1.8
2.7
i f\
510
757
766
774
783
791
800
808
817
825
834
4
c
3- b
A Z
ii
12
13
14
II
17
18
19
842
927
71 012
9 6
181
265
349
433
517
851
935
020
105
I8 9
273
357
441
525
859
944
029
H3
198
282
366
45
533
868
952
037
122
206
2 9
374
458
542
876
961
046
130
214
299
383
466
550
885
969
054
139
223
37
39 1
475
559
893
978
06 3
147
231
315
399
483
567
902
986
071
155
240
324
408
492
575
910
995
079
164
248
332
416
^oo
584
919
*oo 3
088
172
257
34i
425
508
592
6
8
9
5-4
6-3
7-2
8.1
520
600
609
617
625
634
642
650
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
$
950
032
H5
198
280
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
J73
255
337
419
i
2
3
4
6
8
9
8
0.8
1.6
2.4
3-2
4.0
4.8
5-6
6-4
7-2
530
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
68 1
762
843
925
*oo6
086
167
526
607
689
770
852
933
*oi4
094
175
534
616
697
779
860
941
*022
IO2
183
542
624
705
787
868
949
*03o
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
*Q54
135
215
575
656
738
819
900
981
*062
143
223
583
665
746
827
908
989
*o7o
151
231
i
2
3
7
0.7
1.4
2.1
540
239
247
255
263
272
280
288
296
304
312
4
2.8
4i
42
43
44
45
46
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
815
894
973
344
424
504
584
664
743
823
902
981
352
432
512
592
672
75i
830
910
989
360
440
520
600
679
759
838
918
997
368
448
528
608
687
767
846
926
*oos
376
456
536
616
6 95
775
854
933
*oi3
384
464
544
624
703
783
862
941
*O2O
392
472
552
632
711
791
870
949
*028
5
6
I
9
3-5
4.2
4-9
5-6
6.3
550
74 036
044
052
060
068
076
084
092
099
107
N.
O
1
2
3
4
5
6
7
8
9
Proi
. Pts.
II
N.
O
1
2
3
4
5
7
8
9
Pro
p. Pts.
550
74 036
044
052
060
068
076
084
092
099
107
52
53
54
55
56
57
58
59
194
273
35i
429
50?
586
663
741
123
202
280
359
437
515
593
671
749
2IO
288
367
445
523
601
679
757
139
218
296
374
453
609
687
764
147
225
304
382
461
539
617
695
772
155
233
312
390
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
186
265
343
421
500
578
656
733
811
560
819
827
834
842
850
858
865
873
881
889
61
62
63
64
65
66
67
68
69
896
974
75 051
128
205
282
358
435
904
981
059
136
213
289
366
442
519
912
989
066
220
297
374
450
526
920
997
074
151
228
305
458
534
^927
082
159
236
312
389
465
542
935
*OI2
089
1 66
243
320
397
473
549
943
*020
097
174
251
328
404
48l
557
950
*028
105
182
259
335
412
488
565
* 958
189
266
343
420
496
572
966
120
I 9 7
274
351
427
504
580
i
2
3
4
5
6
8
8
0.8
1.6
2.4
3-2
4.0
4.8
5.6
6.4
570
587
595
603
610
618
626
633
641
648
656
9
7-2
72
73
74
75
76
11
79
664
740
815
891
967
76 042
118
193
268
671
747
823
899
974
050
125
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
"005
080
155
230
305
709
785
861
* 937
087
163
238
313
717
111
944
*02O
095
170
245
320
724
800
876
* 952
103
178
253
328
732
808
884
* 959 -
110
185
260
335
580
343
35
358
365
373
380
388
395
403
410
81
82
83
84
85
86
87
88
89
418
492
567
641
716
790
864
938
77 012
425
500
574
649
723
797
871
945
019
433
507
582
656
730
805
879
953
026
440
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
* 93
078
i
2
3
4
i
7
7
0.7
1.4
2.1
2.8
3.5
4.2
4-9
590
085
093
100
107
115
122
129
137
144
151
8
5.6
9 2
93
94
95
96
97
98
99
159
232
305
379
452
525
597
670
743
1 66
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
188
262
335
408
481
554
627
699
772
195
26 9
342
4T5
488
561
634
7 06
779
203
276
349
422
495
568
641
714
786
2IO
283
357
430
503
576
648
721
793
217
291
364
437
583
656
728
80 1
225
298
444
517
590
663
735
808
9
6-3
600
815
822
830
837
844
851
859
866
873
880
N.
O
1
2
3
4
5
6
7
8
9
Prop
. Pts.
12
1 N.
O
1
2
3
4
5
6
7
8
9
Pro]
[). PtS.
600
77 815
822
830
837
844
8 5 I
859
866
873
880
01
02
03
04
05
06
3
09
887
960
78 032
104
176
247
319
390
462
895
967
039
III
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
*oo3
075
147
219
290
362
433
504
938
*OIO
082
154
226
297
369
440
512
945
*OI 7
089
161
233
305
376
447
5i9
* 95 -
*025
097
168
240
312
383
455
526
I
2
3
8
0.8
1.6
2.4
610
533
540
547
554
56i
56 9
576
583
590
597
4
3-2
40
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
1 06
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
781
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
95i
*02I
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
s
27
28
29
309
379
449
5i8
588
657
727
796
865
3i6
386
456
525
595
664
734
803
872
323
393
463
532
602
671
74i
810
879
330
400
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
4 9 I
560
630
699
768
837
906
358
428
498
567
6 37
706
775
844
913
365
435
505
574
644
713
782
851
920
372
442
511
581
650
720
789
858
927
i
2
3
4
6
8
9
7
0.7
1.4
2.1
2.8
3-5
4.2
4-9
5-6
6.3
680
934
941
948
955
962
969
975
982
989
996
3i
32
33
34
35
36
37
38
39
80 003
072
140
209
277
346
414
482
550
OIO
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
380
448
5i6
584
044
"3
182
250
3i8
387
455
523
59i
051
120
1 88
257
325
393
462
530
598
058
127
195
264
332
400
468
536
604
065
134
202
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
81 023
090
158
224
693
760
828
895
963
030
097
164
231
699
767
835
902
969
037
104
171
238
706
774
841
909
976
043
in
178
245
713
78i
848
916
983
050
117
184
251
720
787
855
922
990
057
124
191
258
726
794
862
929
996
064
I3i
198
265
733
80 1
868
936
*oo3
070
137
204
271
740
808
875
943
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077
144
211
2 7 8
747
814
882
949
*oi7
084
151
218
285
5
6
8
9
3-o
3-6
4-2
4-8
5-4
650
291
298
305
3ii
3i8
325
331
338
345
35i
N.
O
1
2
3
4
5
6
7
8
9
Prop
.Pts.
N.
1
2
3
4
5
6
1 7
8
9
Pro]
l. Pts.
650
81 291
298
305
311
318
325
33i
338
345
351
5i
52
53
54
55
56
57
58
59
358
425
49 1
558
624
690
757
823
889
365
431
498
564
631
697
763
829
895
371
438
505
571
637
704
770
836
902
378
445
5ii
578
644
710
776
842
908
385
451
5i8
584
651
717
783
849
915
391
458
525
591
657
723
79
856
921
398
465
53i
598
664
730
796
862
928
405
471
538
604
671
737
803
869
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411
478
544
611
677
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809
875
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418
485
551
617
684
750
816
882
948
660
954
961
968
974
981
987
994
*000
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61
62
63
64
65
66
67
68
69
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
046
112
I 7 8
243
308
373
439
504
569
053
119
184
249
3*5
380
445
5io
575
060
125
191
256
321
387
452
517
582
066
132
197
263
328
393
458
523
588
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138
204
269
334
400
465
530
595
079
145
210
2 7 6
341
406
471
536
601
I
2
3
4
6
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7
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1.4
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5- 6
670
607
614
620
627
633
640
646
653
659
666
9
6-3
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72
73
74
75
76
77
78
79
672
737
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866
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995
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123
187
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743
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065
129
193
685
750
814
879
943
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072
136
200
692
756
821
885
950
*oi4
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
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104
168
232
724
789
853
918
982
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no
174
238
730
795
860
924
988
*0 5 2
117
181
245
680
251
257
264
270
2 7 6
283
289
296
302
308
81
82
83
84
85
86
87
88
89
315
378
442
506
569
632
696
759
822
321
385
448
512
575
639
702
765
828
327
39i
455
5i8
582
645
708
771
835
334
398
461
525
588
651
715
778
841
340
404
467
531
594
658
721
784
847
347
410
474
537
60 1
664
727
790
853
353
417
480
544
607
670
734
797
860
359
423
487
550
613
677
740
803
866
366
429
493
556
620
683
746
809
872
372
436
499
563
626
689
753
816
879
i
2
3
4
6
7
6
0.6
1.2
1.8
2.4
3.0
3.6
4.2
690
885
891
897
904
910
916
923
929
935
942
8
4-8
9i
92
93
94
95
96
97
98
99
948
84 on
073
136
198
261
|
448
954
017
080
142
205
267
330
392
454
960
31
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
36i
423
485
992
055
117
180
242
305
367
429
491
998
061
123
186
248
3ii
373
435
497
*oo4
067
130
192
255
317
379
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N.
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3
4
5
6
7
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Prop
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1
2
3
4
5
6
7
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Pro
p. Pts.
800
90 309
314
320
325
331
336
342
347
352
358
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02
03
04
05
06
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363
417
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2
3
4
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8
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Proj
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N.
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1
2
3
4
5
6
7
8
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850
92 942
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56
57
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1
2
3
4
5
6
7
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Proj
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18
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1
2
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4
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02
03
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1
2
3
4
5 I
6
7
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Prop
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N.
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1
2
3
4
5
6
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950
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009
013
017
022
026
030
035
039
N. .
1
2
3
4
5
6
7
8
9
Proj
). PtS.
2O
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
36o
21600
i296ooo //
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 0310
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 43 10
0.58203 14
3-59805 65
3-59657 4o
3.59725 63
i.595i6 54
L595I7 4i
1.59517 01
3.72263 39
3.78392 43
.16633 15
9.83366 86io
v
I
I
I
Vir :
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
FOR
EACH MINUTE OF THE QUADRANT.
22
/
Li. Sin.
d.
L.. Tang. 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
6.46 373
6.76 476
6.94 085
7.06 579
30103
17609
12494
9691
79 i8
6694
5800
5"5
457 6
4139
3779
3476
3219
2996
2803
2633
2482
2348
2228
2119
2020
1931
I8 4 8
1773
1704
l6 39
1579
1524
1473
1424
1379
1336
1297
1259
1223
1190
"59
1128
IIOO
1072
1047
IO22
99 8
976
955
934
9i5
895
878
860
843
828
812
797
782
769
756
742
730
3-53 627
3-23 524
3.05 915
2.93 421
o.oo ooo
o.oo ooo
o.oo ooo
0.00 000
o.oo ooo
60
59
58
57
56
d.
1336
1297
1259
1223
1190
H59
1158
1128
IIOO
1072
1047
1046
1022
999
998
976
955
954
934
d.
30103
17609
12494
9691
7918
6694!
5800 :
5"5
4576
4 X 39
3779
3476
3219
3218
2997
2996
2803
2802
2633
2483
2482
2348
2228
2227
2119
2O2I
2020
J 93i
1930
1848
1773
1704
1639
1579
1524
1473
1472
1424
1379
p.p.l"
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
SOL?
293-4
208.2
161.5
131-9
111.5
96.6
85.2
76.2
68.9
62.9
57-9
53-6
53-6
49-9
49-9
46.7
46.7
43-8
4L3
41-3
39-1
37-1
37-1
35-3
33-6
S3- 6
32.1
32.1
30.8
29-5
28.4
27-3
26.3
25-4
24-5
24-5
23-7
22.9
d.
915
914
896
895
878
877
860
843
828
827
812
797
782
769
756
755
743
742
73
a
2
8
3
2
7
7
7
5
7
8
8
3
5
3
5
3
2
8
8
7
3
3
2
2
8
7
8
7
o
5
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
5
6
7
8
9
7.16 270
7.24 188
7.30 882
7.36 682
7.41 797
79 i8
6694
5800
5"5
4576
4139
3779
3476
3218
2997
2802
2633
2483
2348
2227
2119
2021
1930
1848
1773
1704
1639
1579
!5 2 4
1472
1424
1379
1336
1297
1259
1223
1190
1158
1128
IIOO
1072
1046
IO22
999
976
954
934
914
896
877
860
843
827
812
797
782
769
755
743
730
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
o.oo ooo
0.00 000
o.oo ooo
o.oo ooo
0.00 000
55
54
53
52
5i
10
ii
12
13
14
7-4 6 373
7-5 512
7-54 291
7-57 767
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
0.00 000
o.oo ooo
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
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
o.oo ooo
0.00 000
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
,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.OO 478
9-99 998
9.99 998
9-99 998
9.99 998
9-99 998
30
29
28
27
26
35
36
37
38
39
8.00 779
8.02 002
8.03 192
8.04 350
8.05 478
8.00 781
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
8.10 717
8.06 581
8.07 653
8.08 700
8.09 722
8.10 720
i-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
8.15 39i
8. ii 696
8.12 651
8.13 585
8.14 500
8.15 395
1.88 304
1.87 349
1.86 415
1.85 500
1.84 605
9-99 99 6
9-99 99 6
9-99 99 6
9-99 99 6
9-99 996
15
14
13
12
II
50
5i
52
53
54
55
56
57
58
59
8.16 268
8.17 128
8.17 971
8.18 798
8.19 610
8.16 273
8-17 133
8.17976
8.18 804
8.19 616
1.83 727
1.82 867
1.82 024
1.81 196
1.80 384
9-99 995
9-99 995
9-99 995
9-99 995
9-99 995
10
9
8
7
6
8.20 407
8.21 189
8.21 958
8.22 713
8.23 456
8.20 413
8.21 195
8.21 964
8.22 720
8.23 462
1-79 587
1.78 805
i.78 036
1.77 280
1-76 538
9-99 994
9.99 994
9.99 994
9-99 994
9.99 994
5
4
3
2
60
8.24 186
8.24 192
1.75 808
9-99 993
O
li. Cos.
d.
L. Cotg. c. d. L.Tang.
I,. Sin.
r
Prop. Pts.
89
r
L. Sin.
d.
L.Tang.
c. d.
L.Cotg.
JL. Cos.
Prop.
Pts
O
i
2
3
4
8.24 186
8.24 903
8.25 609
8.26 304
8.26 988
717
7 o6
695
684
6-7Q
8.24 192
8.24 910
8.25 616
8.26 312
8.26 996
,-
706
696
68 4
dii
1.75 808
1.75 090
i-74 384
1.73 688
1.73 004
9-99 993
9-99 993
9-99 993
9-99 993
9.99 992
60
59
58
57
56
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
(..
8.27 669
8.28 332
8.28 986
8.29 629
8.30 263
73
663
654
643
634
621;
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 99i
55
54
53
52
5i
d.
P. P. 1"
d.
p, n, 1"
10
ii
12
13
14
8.30 879
8-31 495
8.32 103
8.32 702
8.33 292
616
608
599
59
CO-
8.30 888
8-31 505
8.32 112
8.327II
8-33 32
617
607
599
59i
,0 .
1.69 112
1.68 495
1.67 888
1.67 289
1.66 698
9-99 99 1
9-99 99i
9-99 99
9.99 990
9-99 99
50
49
48
47
46
718
717
706
696
605
11.97
11-95
11.77
1 1. 60
11.58
485
480
475
474
47
8.08
8.00
7.92
7-90
7.83
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
575
568
56i
553
f.f.
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
H.05
10.90
10.88
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
4O
39
38
37
36
644
643
634
625
624
10.73
10.72
10.57
10.42
10.40
446
445
441
437
436
7-43
7.42
7-35
7.28
7.27
25
26
27
28
29
8.39 310
8.39 818
8.40 320
8.40 816
8.41 307
5*4
508
502
496
491
8-39 323
8.39 832
8-4 334
8.40 830
8.41 321
509
502
496
491
.Of.
i. 60 677
i. 60 168
1.59 666
1.59 170
1.58 679
9-99 987
9-99 986
9.99 986
9.99 986
9-99 985
35
34
33
32
3i
617
616
608
607
599
10.28
10.27
10.13
IO.I2
9.98
o 8^
433
432
428
427
424
7.22
7.20
7.13
7.12
7.07
30
3i
32
33
34
8.41 792
8.42 272
8.42 746
8.43 216
8.43 680
480
474
47
464
8.41 807
8.42 287
8.42 762
8.43 232
8.43 696
480
475
470
464
i-58 193
1-57 713
1-57 238
1.56 768
1.56 304
9-99 985
9-99 985
9-99 984
9-99 984
9.99 984
30
29
28
27
26
590
584
583
575
568
v-oi)
9.83
9-73
9-72
9.58
9-47
419
416
412
411
408
6.98
6-93
6.87
6.85
6.80
35
36
37
38
39
8.44 139
8.44 594
8.45 044
8.45 489
8-45 930
459
455
45
445
441
8-44 156
8.44 611
8.45 061
8-45 507
8.45 948
455
45
446
441
i-55 844
i-55 389
i-54 939
i-54 493
1.54 052
9-99 983
9.99 983
9-99 983
9-99 982
9-99 982
25
24
23
22
21
56i
560
553
547
546
9-35
9-33
9.22
9.12
9.10
404
401
400
397
396
6-73
6.68
6.67
6.62
6.60
40
4i
42
43
44
8.46 366
8.46 799
8.47 226
8.47 650
8.48 069
43
433
427
424
419
Alft
8.46 385
8.46 817
8-47 245
8.47 669
8.48 089
437
432
428
424
420
1-53 615
1-53 183
1-52 755
1-52 33i
1.5191!
9.99 982
9-99 98i
9-99 98i
9.99 981
9-99 98o
20
19
18
17
16
540.
539
533
527
526
9.00
8.98
8.88
8.78
8-77
393
39
386
383
382
6.55
6.50
6-43
6.38
6-37
45
46
47
48
49
8.48 485
8.48 896
8.49 304
8.49 708
8.50 108
410
411
4 08
404
400
Ql-lfi
8.48 505
8.48 917
8-49 325
8.49 729
8.50 130
412
408
404
401
i-5i 495
1.51 083
1-5 675
1.50 271
1.49 870
9-99 98o
9.99 979
9-99 979
9.99 979
9-99 978
15
J 4
13
12
II
520
514
59
508
502
4Q6
O.O7
8-57
8.48
8.47
8-37
8 27
380
379
376
373
370
q6o
6-33
6.32
6.27
6.22
6.17
6 i<
50
5i
52
53
54
8.50 504
8.50 897
8.51 287
8.51 673
8.52 055
39
393
39
386
382
8.50 527
8.50 920
8.51 310
8.51 696
8.52 079
397
393
39
386
383
oRn
1-49 473
1.49 080
1.48 690
1.48 304
1.47 921
9.99 978
9-99 977
9-99 977
9.99 977
9-99 976
10
9
8
7
6
491
486
8.18
8.10
367
363
6.12
6.05
55
56
57
58
59
8.52 434
8.52 810
8-53 183
8-53 552
8-53 919
376
373
369
367
8.52 459
8.52 835
8-53 208
8-53 578
8-53 945
300
376
373
370
367
1-47 54i
i-47 165
1.46 792
1.46 422
1.46 055
9-99 976
9-99 975
9-99 975
9-99 974
9-99 974
5
4
3
2
I
60
8.54 282
33
8.54 308
33
1.45 692
9-99 974
O
L. Cos.
d.
L.Cotg.
c. d.
L.Tang.
L. Sin.
/
Prop.
Pts
1
88 C
r
L. Sin. d.
L.Tang.
c. d.
L. Cotg.
L. Cos.
Prop. Pts.
O
i
2
3
4
8.54 282
8.54 642
8.54 999
8-55 354
8.55 70S
360
357
355
35i
349
346
343
34i
337
336
332
330
328
325
323
320
3i8
316
3i3
3 11
39
37
35
302
301
298
296
294
2 93
290
288
287
284
283
8.54 308
8.54 669
8-55 027
8-55 382
8-55 734
361
358
355
352
349
346
344
34i
338
336
333
33
328
326
323
321
319
316
3M
3"
310
307
305
33
301
299
297
295
292
291
289
287
285
284
281
280
278
276
274
273
271
269
268
266
.264-
263
261
260
258
257
255
254
252
251
249
248
246
245
244
*43
1.45 692
1-45 33i
1-44 973
1.44 618
1.44 266
9-99 974
9-99 973
9-99 973
9.99 972
9.99 972
GO
59
58
57
56
p. p. i"
6.02
6.00
5-97
5-95
5-92
5.87
5.85
5-82
5-77
5-73
5-72
5-68
5-63
<C.6 2
5-6o
5-55
5-53
5-50
5-47
5-43
5.42
5.38
5-35
5-33
5-32
5-30
5-27
5.23
5.22
5-i8
5-17
5.15
5-12
5-o8
5-05
5-03
5.02
4.98
4-97
4-95
4-93
4.92
4.90
4.88
4-87
d.
291
290
289
288
287
285
284
283
281
280
279
278
277
276
274
273
272
271
270
269
268
267
266
264
263
261
260
259
258
257
256
255
254
253
252
251
250
249
248
247
246
245
244
243
242
p. p .1"
4-85
4.83
4.82
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8.89 304
163
3
162
162
160
161
X 59
159
59
158
157
157
156
!55
155
155
J 54
153
*53
152
152
J5 1
15*
150
150
149
149
148
M7
H7
147
146
146
145
145
8.88 783
8.88 948
8.89 in
8.89 274
8.89 437
i. n 217
i. n 052
1. 10 889
i.io 726
1. 10 563
9-99 871
9.99 870
9 99 869
9.99 868
9-99 867
3O
3i
32
33
34
8.89 464
8.89 625
8.89 784
8.89 943
8.90 102
8.89 598
8.89 760
8.89 920
8.90 080
8.90 240
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
35
36
37
38
39
8.90 200
8.90 417
8.90 574
8.90 730
8.90 885
8.90 399
8-9 557
8.90 715
8.90 872
8.91 029
1.09 601
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
40
4i
42
43
44
8.91 040
8.91 195
8.91 349
8.91 502
8-91 655
8.91 185
8.91 340
8.91 495
8.91 650
8.91 803
1.08 815
i. 08 660
i. 08 505
i. 08 350
i. 08 197
9-99 856
9-99 855
9-99 854
9-99 853
9-99 852
45
46
47
48
49
8.91 807
8.91 959
8.92 no
8.92 261
8.92 411
8.91 957
8.92 no
8.92 262
8.92 414
8.92 565
i. 08 043
1.07 890
1.07 738
1.07 586
1.07 435
9-99 851
9 99 850
9-99 848
9.99 847
9.99 846
50
5i
52
53
54
8.92 561
8.92 710
8.92 859
893007
8-93 154
8.92 716
8.92 866
8.93 016
8-93 165
8-93 313
1.07 284
1.07 134
1.06 984
1.06 835
i. 06 687
9-99 845
9.99 844
9.99 843
9.99 842
9-99 841
55
56
57
58
59
8-93 301
8.93 448
8-93 594
8.93 740
8.93 885
8-93 462
8.93 609
8-93 756
8-93 93
8-94 049
i .06 538
1.06 391
i. 06 244
i. 06 097
1.05 95i
9.99 840
9-99 839
9.99 838
9-99 837
9-99 836
60
8.94 030
8.94 195
1.05 805
9-99 834 1
L. Cos.
d.
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c. d.
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Prop.
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2
3
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8.94 030
8.94 174
8-94 317
8.94 461
8.94 603
144
143
i 44
142
143
141
142
141
140
8.94 195
8.94 340
8.94 485
8.94 630
8-94 773
145
i45
US
M3
144
M3
142
142
142
141
140
141
139
140
138
139
138
137
138
136
137
135
136
135
i35
134
134
133
133
133
132
132
131
131
131
130
130
130
129
128
129
128
127
128
127
126
126
126
126
125
125
124
124
124
123
123
123
122
122
122
1.05 805
1.05 660
1-05 515
1.05 370
1.05 227
9-99 834
9-99 833
9-99 832
9.99 831
9.99 830
GO
59
58
57
56
6
7
8
9
10
20
3
4
5
6
7
8
9
10
20
3
40
50
6
7
8
9
10
20
3
40
50
6
7
8
9
10
20
30
4
50
6
7
8
9
10
20
3
40
50
I
2
3
4
5
i45
14-5
16.9
19-3
21.8
24.2
48.3
72.5
96.7
120.8
139
13-9
16.2
18.5
20.9
23.2
46.3
69-5
92.7
115.8
135
13-5
15.8
18.0
20.3
22.5
45-0
67.5
90.0
112.5
129
12.9
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17.2
19.4
21.5
43-0
64.5
86.0
107.5
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12.5
14.6
16.7
18.8
20.8
41.7
62.5
83-3
104.2
121
6 12.
7 14-
8 16.
9 18.
o 20.
o 40.
o 60.
o 80.
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143
14.3
16.7
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21.5
23.8
47.7
71.5
95.3
119.2
138
13.8
16.1
18.4
20.7
23.0
46.0
69.0
92.0
115.0
133
13-3
15-5
17.7
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22.2
44-3
66.5
88.7
110.8
128
12.8
14.9
17.1
19.2
21.3
42.7
64.0
85-3
106.7
123
12.3
14.4
16.4
141
14.1
16.5
18.8
21.2
23-5
47.0
70-5
94.0
II7-5
136
13-6
15-9
18.1
20.4
22.7
45-3
68.0
90.7
"33
131
I3-I
15-3
17-5
19.7
21.8
43-7
65.5
87.3
109.2
126
12.6
14.7
16.8
18.9
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42.0
63.0
84.0
105.0
122
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16.3
18.3
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40.7
61.0
81.3
101.7
I
) O.I
> O.I
) O.I
> 0.2
> 0.2
> 0.3
> 0.5
5 0.7
3 0.8
6
7
8
9
8.94 746
8.94 887
8-95 029
8.95 170
8-95 3io
8.94 917
8.95 060
8.95 202
8-95 344
8.95 486
1.05 083
1.04940
1.04 798
1.04 656
1.04 514
9-99 829
9.99 828
9-99 827
9-99 825
9-99 824
55
54
53
52
5i
10
ii
12
13
14
8-95 450
8-95 589
8.95 728
8.95 867
8.96 005
140
J 39
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138
3(8
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J 37
136
136
136
J 35
J 35
134
134
8.95 627
8.95 767
8.95 908
8.96 047
8.96 187
1-04 373
1.04 233
1.04 092
1-03 953
1.03 813
9.99 823
9.99 822
9.99 821
9.99 820
9-99 819
5O
49
48
47
46
IS
16
17
18
19
8.96 143
8.96 280
8.96 417
8-96 553
8.96 689
8.96 325
8.96 464
8.96 602
8.96 739
8.96 877
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
999813
45
44
43
42
4i
20
21
22
23
24
8.96 825
8.96 960
8-97 095
8.97 229
8-97 363
8.97 013
8.97 150
8-97 285
8.97 421
8-97 556
i. 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
38
i
25
26
27
28
29
8.97 496
8.97 629
8.97 762
8-97 894
8.98 026
J 33
133
133
132
132
8.97 691
8-97 825
8-97 959
8.98 092
8.98 225
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
32
3i
30
3i
32
33
34
8.98 157
8.98 288
8.98 419
8.98 549
8.98 679
J3 1
J3 1
131
130
130
129
129
129
128
128
128
127
127
126
126
126
125
125
124
125
123
124
123
123
122
22
22
21
21
21
2O
8.98 358
8.98 490
8.98 622
8.98 753
8.98 884
i.oi 642
i.oi 510
i.oi 378
i.oi 247
i.oi 116
9.99 800
9-99 798
9-99 797
9-99 796
9-99 795
30
29
28
27
26
35
36
37
38
39
8.98 808
8.98 937
8.99 066
8.99 194
8.99 322
8.99 015
8.99 145
8-99 275
8.99 405
8-99 534
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
40
4i
42
43
44
8.99 450
8-99 577
8.99 704
8.99 830
8.99 956
8.99 662
8.99 791
8.99 919
9.00 046
9.00 174
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
45
46
47
48
49
9.00 082
9.00 207
9.00 332
9.00 456
9.00 581
9.00 301
9.00 427
9-00 553
9.00 679
9.00 805
0.99 699
0.99 573
0.99 447
0.99 321
0.99 195
9-99 78i
9-99 78o
9-99 778
9.99 777
9-99 776
15
14
13
12
II
18.5
20.5
41.0
61.5
82.0
; 102.5
120
I I2.C
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i i6.c
2 l8.C
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53
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9.00 704
9.00 828
9.00 951
9.01 074
9.01 196
9.00 930
9.01 055
9.01 179
9.01 303
9.01 427
0.99 070
0.98 945
0.98 821
0.98 697
0.98 573
9-99 775
9-99 773
9-99 772
9-99 77i
9-99 769
10
9
8
7
6
55
56
57
58
59
9.01 318
9.01 440
9.01 561
9.01 682
9.01 803
9- 01 550
9.01 673
9.01 796
9.01 918
9.02 040
0.98 450
0.98 327
0.98 204
0.98 082
o 97 960
9.99 768
9.99 767
9-99 765
9.99 764
9-99 763
5
4
3
2
I
GO
Q.OT Q2 3
9.02 162
0.97 838
9-99 761
O
L. Cos. d.
L. Cotg.
c. d
L.Tang
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Prop. Pts.
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28
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/
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L.Tang.
c. d.
L. Cotg.
L. Cos.
Prop. Pts.
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2
3
4
9.01 923
9.02 043
9.02 163
9.02 283
9.02 402
120
120
120
II 9
Il8
9
18
*7
18
17
J 7
16
116
116
116
"5
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114
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9.02 162
9.02 283
9.02 404
9.02 525
9.02 645
121
121
121
120
121
II 9
1 2O
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118
119
n8
118
117
118
116
117
116
116
116
"5
"5
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114
114
"3
114
"3
112
II 3
112
112
112
III
III
III
no
in
no
109
1 10
109
109
108
109
108
108
107
108
107
106
107
106
106
106
106
105
105
105
104
0.97 838
0.97 717
0.97 596
0.97 475
0-97 355
9-99 76i
9-99 76o
9-99 759
9-99 757
9-99 756
6O
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
3
40
So
6
7
8
9
10
20
30
4
50
6
7
8
9
10
20
30
40
SO
6
8
9
10
20
30
4
50
121
I2.I
I4.I
16.1
18.2
20. 2
40-3
60.5
80. 7
100.8
118
11.8
13-8
15-7
17.7
19.7
39-3
59-0
78.7
98.3
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13-4
15.3
17-3
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38.3
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76.7
95-8
112
II. 2
I3-I
14.9
16.8
18.7
37-3
56.0
74-7
93-3
109
10.9
12.7
14-5
16.4
18.2
36.3
54-5
72.7
90.8
106
10.6
12.4
14.1
15-9
17.7
35-3
53-o
70.7
88.3
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I2.C
I4.c
i6.c
i8.c
20.C
40.0
6o.c
8o.c
IOO.C
117
11.7
13-7
15-6
17.6
19-5
39-0
58-5
78.0
97-5
114
11.4
13-3
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17.1
19.0
38.0
57-0
76.0
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14.8
16.7
18.5
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55-5
74.0
92.5
108
10.8
12.6
14.4
16.2
18.0
36.0
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72.0
90.0
105
10.5
12.3
14.0
15-8
17.5
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52.5
70.0
87.5
119
> 11.9
> 13-9
> 15-9
> 17-9
> 19.8
> 39-7
> 59-5
> 79-3
> 99- 2
xx6
n.6
13-5
15-5
17.4
19-3
38.7
58.0
77-3
96.7
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17.0
18.8
37-7
56.5
75-3
94.2
no
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12.8
14-7
16.5
2* 3
36-7
55-o
73-3
91.7
107
10.7
12.5
14-3
16.1
17.8
35-7
53-5
71-3
89.2
104
10.4
12. 1
13-9
15-6
17-3
34-7
52.0
69-3
86.7
5
6
7
8
9
9.02 520
9.02 639
9 02 757
9.02 874
9.02 992
9.02 766
9.02 885
9.03 005
9.03 124
9.03 242
0.97 234
0.97 115
0.96 995
0.96 876
0.96 758
9-99 755
9-99 753
9-99 752
9-99 751
9-99 749
55
54
53
52
5i
1O
ii
12
13
14
9.03 109
9.03 226
9-3 342
9-03 458
9-03 574
9.03 361
9-03 479
9-03 597
9.03 714
9.03 832
0.96 639
0.96 521
0.96 403
0.96 286
0.96 168
9-99 748
9-99 747
9-99 745
9-99 744
9-99 742
50
49
48
47
46
15
16
17
18
19
9.03 690
9.03 805
9.03 920
9.04 034
9.04 149
9.03 948
9.04 065
9.04 181
9.04 297
9.04 413
0.96 052
0-95 935
0.95 819
0-95 73
0-95 587
9-99 74i
9.99 740
9-99 738
9.99 737
9-99 73 6
45
44
43
42
4i
20
21
22
23
24
9.04 262
9.04 376
9.04 490
9.04 603
9-04 715
114
114
"3
112
"3
112
112
112
III
III
III
1 10
no
no
no
109
109
109
108
109
108
107
1 08
107
107
106
107
106
105
106
105
105
105
105
9.04 528
9.04 643
9.04 758
9.04 873
9.04 987
0.95 472
0-95 357
0.95 242
0.95 127
0.95 013
9-99 734
9-99 733
9-99 73i
9-99 730
9-99 728
40
39
38
37
36
25
26
27
28
29
9.04 828
9.04 940
9.05 052
9.05 164
9-05 275
9.05 101
9.05 214
9.05 328
9.05 441
9-05 553
0.94 899
0.94 786
0.94 672
o-94 559
0.94 447
9-99 727
9-99 726
9.99 724
9-99 723
9-99 72i
35
34
33
32
3i
30
3i
32
33
34
9.05 386
9-05 497
9.05 607
9.05 717
9.05 827
9.05 666
9.05 778
9.05 890
9.06 002
9.06 113
0-94 334
0.94 222
0.94 no
0.93 998
0.93 887
9.99 720
9-99 7i8
9-99 717
9.99 716
9 99 714
30
29
28
27
26
35
36
37
38
39
9-05 937
9.06 046
9-o6 155
9.06 264
9.06 372
9.06 224
9-o6 335
9.06 445
9.06 556
9.06 666
0.93 776
0.93 665
0-93 555
Q-93444
0-93 334
9-99 713
9.99711
9-99 7io
9-99 708
9-99 707
25
24
23
22
21
40
4i
42
43
44
9.06 481
9.06 589
9.06 696
9.06 804
9.06 911
9.06 775
9.06 885
9.06 994
9.07 103
9.07 211
0.93 225
0-93 US
0.93 006
0.92 897
0.92 789
9-99 705
9.99 704
9-99 702
9.99 701
9-99 6 99
20
19
18
17
16
45
46
47
48
49
9.07 018
9.07 124
9.07 231
9-7 337
9.07 442
9.07 320
9.07 428
9-07 536
9.07 643
9.07 751
0.92 680
0.92 572
0.92 464
0.92 357
0.92 249
9.99 698
9-99 6 9 6
9-99 6 95
9-99 6 93
9-99 6 92
IS
14
13
12
II
5O
Si
52
53
54
9.07 548
9-07 653
9.07 758
9.07 863
9.07 968
9.07 858
9.07 964
9.08 071
9.08 177
9.08 283
0.92 142
0.92 036
0.91 929
0.91 823
0.91 717
9.99 690
9.99 689
9-99 6 87
9.99 686
9-99 6 84
10
9
8
7
6
P
57
58
59
9.08 072
9.08 176
9.08 280
9.08 383
9.08 486
104
104
103
103
103
9.08 389
9.08 495
9.08 600
9.08 705
9.08 810
0.91 611
0.91 505
0.91 400
0.91 295
0.91 190
9-99 6 83
9.99 681
9.99 680
9-99 6 78
9.99 677
5
4
3
2
I
60
9.08 589
9.08 914
0.91 086
9-99 6 75
L. Cos.
d.
L. Cotg. c. d.
L.Tang.
L. Sin.
r
Prop. Pts.
83<
2 9
/
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9-3i 370
9-3i 43
61
60
60
60
9.32 122
9.32 185
9.32 248
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9-32 373
3
63
63
63
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0.67 878
0.67 815
0.67 752
0.67 689
0.67 627
9.99 067
9.99 064
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3
3
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8
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56
57
58
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9.31 490
9-3i 549
9.31 609
9.31 669
9.31 728
59
60
60
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9-32 43 6
9.32 498
9.32 561
9.32 623
9.32 685
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63
62
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0.67 564
0.67 502
0.67 439
0.67 377
0.67 315
9.99 054
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59
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59
59
59
59
59
59
58
59
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58
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9.32 810
9.32 872
9-32 933
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63
62
61
62
62
62
61
62
61
62
61
61
61
61
61
61
61
61
60
61
60
61
60
60
61
60
60
60
60
60
59
60
60
59
60
59
59
59
60
59
59
59
59
58
59
59
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59
58
59
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58
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0.67 253
0.67 190
0.67 128
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9-99 038
9-99 035
9-99 032
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2
3
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3
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9.32 202
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9-32 319
9-33 057
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9-33 180
9-33 242
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0.66 943
0.66 881
0.66 820
0.66 758
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9-32 437
9-32 495
9-32 553
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9-33 426
9-33 487
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9-33 609
0.66 635
0.66 574
0.66 513
0.66 452
0.66 391
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15
16
17
18
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9.32 728
9.32 786
9-32 844
9.32 902
9.33 670
9-33 73i
9-33 792
9-33 853
9-33 913
0.66 330
0.66 269
0.66 208
0.66 147
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9.98 997
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44
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21
22
23
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9-33 133
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9-33 974
9-34 034
9-34 95
9-34 155
9-34 215
0.66 026
0.65 966
0.65 905
0.65 845
0.65 785
9.98 986
9.98 983
9.98 980
9.98 978
9-98 975
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38
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26
27
28
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9-33 305
9-33 362
9-33 420
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9-34 276
9-34 336
9-34 396
9-34 456
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0.65 724
0.65 664
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35
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0.65 245
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30
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27
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36
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25
24
23
22
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9
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9.34 268
9-34 324
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0.63 376
0.63 319
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0.63 205
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56
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0.62 244
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0.62 076
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9.38 147
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56
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3
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22
23
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9.39 418
9-39 4 6 7
9-39 517
9-39 566
49
49
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9.40 742
9-40 795
9.40 847
9.40 900
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53
52
53
52
0.59 258
0.59 205
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0.59 100
0.59 048
9.98 627
9.98 623
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39
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26
27
28
29
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9-39 664
9-39 713
9-39 762
9.39 811
49
49
49
49
9.41 005
9.41 057
9.41 109
9.41 161
9.41 214
52
52
52
53
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0.58 943
0.58 891
0.58 839
0.58 786
9.98 610
9.98 607
9.98 604
9.98 601
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3
3
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34
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9-39 909
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9.40 006
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49
49
48
49
4.8
9.41 266
9.41 318
9.41 370
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52
52
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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
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30
29
28
27
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10
20
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36
37
38
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9.40 103
9.40 152
9.40 200
9.40 249
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49
48
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9.41 526
9-41 578
9.41 629
9.41 681
9-41 733
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Si
52
52
0.58 474
0.58 422
0.58 371
0.58 319
0.58 267
9.98 578
9-98 574
9.98 571
9.98 568
9-98 565
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0.58 164
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9.98 548
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9.40 634
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9.40 730
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9.42 144
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0.57 907
0.57 856
0.57 805
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9-98 545
9.98 541
9-98 538
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9-98 53i
4
3
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9.40 825
9.40 873
9.40 921
9.40 968
9.41 016
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0.57 652
0.57 601
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3
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9.41 205
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9.41 628
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9.43 108
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3
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21
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9.42 232
9.42 278
9.42 324
9.42 370
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9-43 95
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0.56 194
0.56 145
0.56 095
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9.98 426
9.98 422
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4
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9.42 507
9-42 553
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9-44 397
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49
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9.98 388
9-98 384
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3
4
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29
28
27
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9
10
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0.55 408
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0.55 262
9-98 373
9.98 370
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4
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24
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9-43 188
9-43 233
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9.44 836
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0.55 164
0.55 116
0-55 067
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9-98 349
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4
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19
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9-43 412
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45
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9-45 029
9-45 078
9.45 126
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9-98 338
9-98 334
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9.98 327
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4
3
4
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14
13
12
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40
50
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52
53
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9-43 591
9-43 635
9.43 680
9-43 724
9-43 769
45
44
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9-45 271
9-45 319
9-45 3 6 7
9-45 415
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0.54 681
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0-54 585
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9.98 320
9.98 317
9-98 313
9.98 309
9.98 306
4
3
4
4
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10
9
8
7
6
6
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8
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56
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9-43 813
9-43 857
9-43 9i
9-43 946
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44
44
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9-45 559
9.45 606
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3
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25
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9.97 448
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9-52 563
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9-55 195
9-55 235
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9-55 355
9-55 395
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9-97 376
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9-97 363
9-97 358
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9-53 126
9-53 161
9-53 196
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9-55 752
9-55 791
9-55 831
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0.44 248
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9-97 340
9-97 335
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9.53 266
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9-53 336
9-53 370
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9-55 949
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9-97 271
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37
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37
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37
38
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32
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37
38
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9-59 577
9-59 614
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37
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9.61 662
9.61 689
9.61 717
9-6i 745
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9-65 736
9.65 770
9.65 803
9-65 837
0.34 297
0.34 264
0.34 230
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35
34
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9.61 800
9.61 828
9.61 856
9.61 883
9.65 870
9-65 94
9-65 937
9.65 971
9.66 004
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0.34 096
0.34 063
0.34 029
0.33 996
9-95 902
9-95 897
9-95 891
9-95 885
9-95 879
30
29
28
27
26
35
36
37
38
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9.61 911
9.61 939
9.61 966
9.61 994
9.62 O2I
9.66 038
9.66 071
9.66 104
9.66 138
9.66 171
0.33 962
0.33 929
0.33 896
0.33 862
0.33 829
9-95 873
9-95 868
9.95 862
9-95 856
9-95 850
25
24
23
22
21
40
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42
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9.62 049
9.62 076
9.62 104
9.62 131
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9.66 204
9.66 238
9.66 271
9.66 304
9.66 337
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9-95 839
9-95 833
9-95 827
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19
18
17
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45
46
47
48
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9.62 214
9.62 241
9.62 268
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28
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9.66 404
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15
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9.62 350
9.62 377
9.62 405
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27
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9-66 735
9.66 768
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26
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9.66 900
9.66 933
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33
33
33
33
33
33
33
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32
33
33
33
33
32
33
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32
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40
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35
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30
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27
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35
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37
38
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45
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40
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30
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40
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9-75 146
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1.4
1.6
1.9
2.1
2-3
4-7
7.0
9-3
11.7
13
1-3
i-5
i-7
2.0
2.2
*!
8-7
10.8
12
1.2
1.4
1.6
1.8
2.0
4.0
6.0
8.0
IO.O
5
6
7
8
9
9.84 242
9-84 255
9.84 269
9.84 282
9.84 295
13
14
J 3
13
13
13
13
I 3
13
*3
12
13
13
13
J 3
13
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
13
12
12
13
12
12
13
12
12
13
12
13
12
12
13
12
13
12
13
12
12
13
12
13
12
13
12
13
12
13
12
13
12
13
*3
12
13
12
13
12
13
13
12
13
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
5O
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
2O
21
22
23
24
9-84 437
9.84 450
9.84 463
9.84 476
9.84 489
9.98 989
9-99 oi5
9.99 040
9-99 6 5
9 99 f>9
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
9-845I5
9.84 528
9.84 540
9-84 553
13
13
12
13
13
13
13
13
13
12
13
13
13
J 3
12
13
13
13
12
13
13
13
12
J 3
13
J 3
12
13
13
12
13
13
12
T 3
13
9.99 116
9.99 141
9.99 166
9 99 I9 1
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 3 6 i
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 3i8
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
9
10
20
3
40
5
11
37
38
39
9.84 630
9.84 643
9.84656
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 ISO
20
19
18
17
16
7
8
9
10
20
3
40
50
6
7
8
9
IO
45
46
47
48
49
9.84 758
9.84 771
9.84 784
9.84 796
9.84 809
9.99 621
9.99 646
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
9.85 112
9.85 ioo
9.85 087
15
14
13
12
II
50
Si
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
O.OO 2O2
o.oo 177
o.oo 152
9.85 074
9.85 062
9.85 049
9-85 037
9.85 024
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 OI2
9.84 999
9.84 986
9.84 974
9.84 961
5
4
3
2
I
20
30
40
50
60
9.84 949
0.00 000
0.00 000
9.84 949
O
L. Cos.
d.
JL. Cotg.
c. d. L.Tang.
L. Sin.
d.
/
Prop. Pts.
45
o<
TABLE IV.
67
//
r
S
T
60
120
1 80
240
i
2
3
4
4-68557
.68557
.68557
-68557
.68557
4-68557
-68557
.68557
.68557
.68558
3 00
360
420
480
540
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
H
4-68557
.68557
68557
.68557
.68557
4-68558
.68558
.68558
68558
68558
900
9 60
1020
1080
1140
15
16
17
18
19
4-68557
.68557
.68557
.68557
.68557
4.68558
.68558
.68558
.68558
.68558
1200
1260
1320
1380
1440
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
2280
2340
35
36
37
38
39
4-68557
.68557
.68557
.68557
.68557
4-68559
.68559
68559
.68559
.68559
2400
2460
2520
2580
2640
40
4i
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
3000
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
.68561
.68562
.68562
3600
60
4-68555
4.68562
r r
/
S
T
3600
3660
3720
378o
3840
i
2
3
4
4.68555
.68555
.68555
.68555
68555
4.68562
.68562
.68562
.68562
.68563
3900
3960
4020
4080
4140
6
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
45oo
4560
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
5460
5520
558o
5640
30
31
32
33
34
4-68553
.68552
.68552
.68552
.68552
4-68567
.68568
.68568
.68568
.68568
5700
5760
5820
5880
5940
35
36
37
38
39
4.68552
68552
.68552
.68552
.68551
4.68569
.68569
.68569
.68569
.68569
6000
6060
6120
6180
6240
40
4i
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
50
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
.6857!
.68575
7200
60
4-68549
4-68575
Log sin a = log a " -f S.
Log tan a log a " + T.
68
3 C
r r
r
S
T
7200
7260
7320
7380
7440
o
i
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
P
37
38
39
4-68543
68543
.68542
.68542
.68542
4.681587
.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
IOO2O
10080
IOI40
45
46
47
48
49
4-68541
.68541
.68540
.68540
.68540
4.68591
68591
.68592
.68592
.68592
10200
IO26O
IO32O
10380
10440
50
51
52
53
54
4.68540
.68540
68539
68539
68539
4-68593
68593
.68594
.68594
-68595
IO5OO
10560
10620
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
r r
r
S
T
10800
10860
10920
10980
11040
o
i
2
3
4
4-68538
-68537
68537
68537
.68537
4-68597
.68598
.68598
.68599
.68599
IIIOO
11160
1 1 220
11280
11340
6
7
8
9
4-68537
.68536
.68536
-68536
.68536
4.68599
.68600
.68600
.68601
.68601
11400
11460
11520
11580
11640
10
ii
12
13
14
4-68535
68535
68535
68535
68534
4.68602
.68602
.68603
.68603
.68604
11700
11760
11820
11880
11940
15
16
17
18
19
4-68534
68534
.68534
68533
-68533
4.68604
.68605
.68605
.68606
.68606
12000
I2O6O
I2I20
I2I80
I224O
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
12600
I266O
12720
12780
12840
39,
31
32
33
34
4.68530
.68530
.68530
.68530
.68529
4.68612
.68612
.68613
.68613
.68614
12900
12960
I3O2O
13080
I3I40
lOvO Cxoo O
CO CO CO CO CO
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
.68525
.68525
.68524
.68524
4.68622
.68623
.68623
.68624
.68625
14100
; 14160
14220
14280
14340
55
56
57
58
59
4.68524
.68523
.68523
.68523
.68522
4.68625
.68626
.68626
.68627
.68628
14400
60
4.68522
4.68628
Log sin a = log a " -f
Log tan a log a " + T.
TABLE V.
NATURAL
SINES, COSINES, TANGENTS, AND COTANGENTS.
o /
N. Sin.
N. Tan.
N. Cot.
N. Cos.
o f
N, Sin.
N. Tan.
N. Cot.
N. Cos.
o
5
10
15
20
25
.00 000
145
291
436
582
727
.00 000
145
291
436
582
727
Infinity.
687.55
343-77
229.18
171.89
I37.5I
Unity.
99 999
998
997
90 o
55
So
45
40
35
230
35
40
45
5o
55
.04 362
507
653
798
04 943
.05 088
.04 366
512
658
803
.04 949
05 095
22.904
22.164
21.470
20.819
20.206
19.627
99 90S
898
892
885
878
870
87 30
25
20
15
10
5
3
35
40
45
50
55
.00 873
.01 018
164
39
454
600
.00873
.01 018
164
309
455
600
114-59
98.218
85.940
76.390
68.750
62.499
99 996
995
993
991
989
987
30
25
20
IS
10
5
3 o
5
10
15
20
25
05 234
379
524
669
814
.05 960
.05 241
387
533
678
824
.05 970
19.081
18.564
18.075
17.611
17.169
16.750
.99 863
855
847
839
831
822
87 o
55
50
45
40
35
1
5
10
IS
20
25
oi 745
.01 891
.02 036
181
327
472
.01 746
.01 891
.02 036
182
328
473
57.290
52.882
49.104
45.829
42.964
40.436
99 985
982
979
976
973
969
89 o
55
50
45
40
35
30
35
40
45
50
55
.06 105
250
395
540
685
831
.06 116
262
408
554
700
847
16.350
15.969
605
15-257
14.924
.606
99 813
804
795
786
776
766
30
25
20
15
10
5
30
35
40
45
So
55
.02 618
763
.02 908
03 054
199
345
.02 619
764
.02 910
03 055
201
346
38.188
36.178
34-368
32.730
31.242
29.882
.99 966
962
958
953
949
944
30
25
20
15
10
5
4 o
5
10
15
20
25
.06 976
.07 121
266
411
556
701
.06 993
07 139
285
43i
578
724
14.301
14.008
13.727
457
13.197
12.947
99 756
746
736
. 725
7H
703
86 o
55
50
45
40
35
2 o
5
10
15
20
25
03 49
635
781
.03 926
.04 071
217
.03 492
638
783
.03 929
.04 075
2 2O
28.636
27.490
26.432
25-452
24.542
23-695
99 939
934
929
923
917
911
88 o
55
50
45
40
35
30
35
40
45
50
55
.07 846
.07 99 I
.08 136
28l
426
571
.07 870
.08 017
163
309
456
602
12.706
474
.251
12.035
11.826
.625
.99 692
680
668
657
644
632
30
25
20
15
10
5
2 30
.04 362
.04 366
22.904
-99 905
87 30
5 o
.08 716
.08 749
11.430
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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
39i
321
250
178
107
61 o
55
So
45
40
35
34 o
5
10
IS
20
25
55 919
.56 040
160
280
401
52i
67 45i
663
67 875
.68 088
301
514
1.4826
4779
4733
.4687
.4641
4596
.82 904
822
74i
659
577
495
56 o
55
50
45
40
35
3
35
40
45
50
55
.49 242
369
495
622
748
49 874
56 577
769
.56 962
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
.69 804
1-4550
4505
.4460
4415
4370
4326
.82413
33
248
165
.82 082
8 1 999
30
25
20
15
10
5
30 o
.50000
57 735
1.7321
.86 603
60 o
35 o
57 358
.70 021
1.4281
.81 915
55 o
N. Cos.
N. Cot.
N. Tan.
N. Sin.
o /
N. COB.
N. Cot.
N. Tan.
N. Sin.
o /
73
o t
N. Sin.
N. Tan.
N. Cot.
N.Cos.
/
N. Sin.
N. Tan.
N.Cot.
N.Cos.
35 o
5
10
15
20
25
57 358
477
596
715
833
57 952
.70 021
238
455
673
.70 891
.71 no
1.4281
4 2 37
4i93
.4150
.4106
.4063
.81 915
832
748
664
580
496
55 o
55
50
45
40
35
40 o
5
IO
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
5ii
4i7
323
229
135
50 o
55
So
45
40
35
3
35
40
45
So
55
.58 070
189
37
425
543
66 1
.71 329
549
769
71 990
.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
3
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
75 946
851
756
661
566
30
25
20
15
10
5
36 o
5
10
15
20
25
58 779
.58 896
59 OI 4
131
248
365
72 654
.72 877
73 I0
323
547
771
1-3764
.3722
.3680
3638
3597
3555
.80 902
8x6
73
644
558
472
54 o
55
5o
45
40
35
41 o
5
10
15
20
25
.65 606
716
825
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
5o
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-35I4
3473
3432
3392
3351
33II
.80 386
299
212
125
.80 038
79 95i
3
25
20
15
10
5
3
35
4
45
5o
55
.66 262
37i
480
588
697
805
88 473
732
.88 992
89 253
515
.89 777
1.1303
.1270
.1237
.1204
.1171
ii39
.74 896
799
703
606
509
412
30
25
20
15
10
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
5o
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
120
.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
77 196
428
661
77 895
1.3032
2993
2954
2915
.2876
.2838
79 335
247
158
.79 069
.78 980
891
30
25
20
15
10
5
30
35
40
45
50
55
67 559
666
773
880
.67 987
.68 093
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
3
25
20
15
IO
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
5o
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
50
45
40
35
30
35
40
45
50
55
.62 251
365
479
592
706
819
79 544
79 78i
.80 020
258
498
738
1.2572
2534
.2497
.2460
.2423
2386
.78 261
170
.78 079
77 988
897
806
30
25
20
15
IO
5
30
35
40
45
5o
55
.68 835
.68 941
.69 046
I5i
256
361
.94 896
95 173
45i
95 729
.96 008
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
IS
20
25
.62 932
63 045
158
271
383
496
.80 978
.8l 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
5o
45
40
35
44 o
5
10
IS
20
25
.69 466
570
675
779
883
.69 987
.96 569
.96 850
97 133
416
700
97 984
i-355
0325
.0295
.0265
0235
.0206
7i 934
833
732
630
529
427
46 o
55
50
45
40
35
30
35
40
45
5o
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
15
10
5
40 o
.64 279
.83 910
1.1918
.76 604
50 o
45 o
.70 711
1. 00 000
1. 0000
.70 711
45 o
N.Cos.
N. Cot
N. Tan.
N. Sin.
o /
N. Cos.
N. Cot.
N. Tan.
N. Sin.
o f
74 TABLE VI. CIRCULAR ARCS EXPRESSED IN RADIANS.
DEGREES.
MINUTES.
SECONDS.
O
I
2
3
4
6
8
9
10
ii
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
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
55
56
57
58
59
60
0.00000 00
6O
61
62
63
64
65
66
67
68
69
70
7i
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
100
101
102
103
104
105
106
107
108
109
110
in
112
"3
114
"5
116
117
118
119
12O
1.04719 76
120
121
122
I2 3
I2 4
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
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
176
177
178
179
180
2.09439 5i
O
I
2
3
4
8
9
10
ii
12
13
14
IS
16
17
18
19
2O
21
22
23
24
25
26
27
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
55
56
57
58
59
60
o.ooooo oo
0"
0.00000 00
o.oi745 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
1-09955 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 09
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
I
2
3
4
5
6
8
9
10
ii
12
13
14
15
16
17
18
19
20
21
22
23
2 4
25
26
27
28
2 9
30
3i
32
33
34
35
36
37
38
39
40
4i
42
43
44
45
46
47
48
49
50
5i
52
53
54
55
56
57
58
59
60
o.ooooo 48
o.ooooo 97
o.ooooi 45
o.ooooi 94
0.00002 42
0.00002 91
0.00003 39
0.00003 88
0.00004 36
0.17453 29
1.22173 05
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
1.34390 35
1.36135 68
1.37881 01
2.28638 13
2.30383 46
2.32128 79
2.33874 12
2.35619 45
2.37364 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 Si
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
1.4311700
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
2.53 72 74
2.54818 07
2-56563 40
2-58308 73
2.60054 6
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.ooon 15
o.ooon 64
0.00012 12
0.00012 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 18
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
O.OIOI8 II
O.OIO47 2O
O.OIO76 29
O.OII05 38
O.OII34 46
0.00015 03
0.00015 5 1
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 9 1
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
O.OOO22 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 ii
0.97738 44
0.99483 77
1.01229 10
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 18
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 7
0.00026 1 8
0.00026 66
0.00027 *5
0.00027 63
O.OOO28 12
0.00028 60
1.04719 76
2.09439 51
3.14159 27
0.01745 33
0.00029 09
75
TABLE VII.
NAPIERIAN LOGARITHMS OF NUMBERS.
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. 4452
3.09104
3-13549
3-17805
3.21888
41
42
43
44
45
3-7I357
3-73767
3.76120
3-78419
3.80666
6i
62
63-
64
65
4.11087
4.12713
4-I43I3
4.15888
4-17439
81
82
83
84
85
4-39445
4.40672
4.41884
4.43082
4.44265
6
7
8
9
10
1.79176
1.94591
2.07944
2.19722
2.30259
26
27
28
2 9
30
3.25810
3-29584
3.33220
3-36730
3.40120
46
47
48
49
50
3.82864
3-85015
3.87120
3.89182
3.91202
66
67
68
69
70
4.18965
4.20469
4.21951
4.23411
4.24850
86
87
88
89
90
4-45435
4.46591
4-47734
4.48864
4.49981
ii
12
13
14
15
2.39790
2.48491
2.56495
2.63906
2.70805
31
32
33
34
35
3-43399
3-46574
3-49 6 5i
3-52636
3-55535
51
52
53
54
55
3-93183
3-95124
3.97029
3.98898
4- 00 733
7i
72
73
74
75
4.26268
4.27667
4.29046
4.30407
4-3I749
91
92
93
94
95
4.51086
4-52I79
4-5326o
4-54329
4.55388
16
17
18
19
20
2.77259
2.83321
2.89037
2-94444
2-99573
36
37
38
39
40
3-58352
3.61092
3.63759
3-66356
3.68888
56
57
58
59
60
4-02535
4.04305
4.06044
4-07754
4.09434
76
77
78
79
80
4-33073
4.3438i
4-3567I
4-36945
4.38203
96
97
98
99
100
4.56435
4-57471
4-58497
4-59512
4-60517
N.
O
1
2
3
4
5
6
7
8
9
10
ii
12
13
14
4-6 0517
4.7 0048
8749
4-8 6753
4-9 4164
1512
0953
9579
7520
4876
2497
1850
#0402
8280
5583
3473
2739
#1218
9035
6284
4439
3620
#2028
9784
6981
5396
4493
#2831
*0 5 2 7
7673
6344
5359
#-3628
#1265
8361
7283
6217
*44I9
*I 99 8
9043
8213
7068
*52Q3
#2725
9721
9135
7912
*S98i
*3447
*Q395
15
16
17
18
19
5.0 1064
7517
5-i 358o
9296
5-2 4702
1728
8140
4166
9850
5227
2388
8760
4749
#0401
5750
3044
9375
5329
#0949
6269
3695
9987
5906
#1494
6786
4343
*059S
6479
#2036
7300
4986
#1199
7048
*2575
7811
5625
#1799
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UNIVERSITY OF CALIFORNIA LIBRARY
BERKELEY
Return to desk from which borrowed.
This book is DUE on the last date stamped below.
ASTFIDNOMY LIB
HIM
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LD 21-100m-ll, 49(B7146sl6)476
YC 1 02260
MATH*.
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