/4ZA*r7\
ASTRONOMY DEPT.
NEWCOMB'S
MATHEMATICAL COURSE.
/. SCHOOL COURSE.
Algebra for Schools.
Key to Algebra for Schools.
Plane Geometry and Trigonometry, with Tables.
The Essentials of Trigonometry.
II. COLLEGE COURSE.
Algebra for Colleges.
Key to Algebra for Colleges.
Elements of Geometry.
Plane and Spherical Trigonometry, with Tables.
Trigonometry (separate).
Tables (separate).
Elements of Analytic Geometry.
Elements of the Differential and Integral Calculus.
Astronomy, Advanced Course, by Newcomb and Holden.
Astronomy, Briefer Course, by Newcomb and Holden.
HENRY HOLT & CO., Publishers, New York.
NE WC OMB' 8 MATHEMATICAL COURSE
LOGARITHMIC AND OTHER
MATHEMATICAL TABLES
WITH EXAMPLES OF THEIR USE AND HINTS ON THE ART OK
COMPUTATION
BY
SIMON NEWCOMB
f Mathematics, in the Johns Hopkins University.
NEW YORK
HENRY HOLT AND COMPANY
LC I ? I
ASTRONOMY DEPT
COPYRIGHT, 1883,
BY
HENRY HOLT & COl
PBEFACE.
IK the present work an attempt is made to present to computers
and students a set of logarithmic and trigonometric tables which
shall have all the conveniences familiar to those who use German
tables. The five-figure tables of F. G. GAUSS, of which fifteen edi-
tions have been issued, have, after long experience with them, been
taken as the basis of the present ones, but modifications have been
introduced wherever any improvement could be made.
Five places of decimals have been adopted as an advantageous
mean. The results obtained by them, being nearly always reliable to
the 10,000th part, are amply accurate for most computations, while
the time of the student who uses them is not wasted in unnecessary
calculation.
The Introduction is intended to serve not only as an explanation
of the tables, but as a little treatise on the art of computation, and
the methods by which the labor of computation may be abridged.
To avoid fostering the growing evil of nearsightedness among
students, the author and publishers have spared neither pains nor
expense in securing clearness of typography.
M192073
CONTENTS.
INTRODUCTION TO TABLES.
TABLE I.
LOGARITHMS OP NUMBERS.
KAMI
Introductory Definitions 3
The Use of Logarithms 4
Arrangement of the Table of Logarithms 6
Characteristics of Logarithms 8
Interpolation of Logarithms 10
Labor-Saving Devices ' 11
Number Corresponding to Given Logarithm 13
Adjustment of Last Decimal 14
The Arithmetical Complement 16
Practical Hints on the Art of Computation 18
Imperfections of Logarithmic Calculation 20
Applications to Compound Interest and Annuities 25
Accumulation of an Annuity 28
TABLE II.
MATHEMATICAL CONSTANTS.
Explanation 31
TABLES III. AND IV.
LOGARITHMS OP TRIGONOMETRIC FUNCTIONS.
Angles less than 45 32
Angles between 45 and 90 33
Angles greater than 90 35
Methods of Writing the Algebraic Signs 36
Angle Corresponding to a Given Function 37
Cases when the Function is very Small or Great 38
TABLE V.
NATURAL SINES AND COSINES.
Explanation 42
CONTENTS.
TABLE VI.
ADDITION AND SUBTRACTION LOGARITHMS.
PAGE
Use in Addition ........................................................ 43
Use in Subtraction .................. t .................................. 44
Special Cases ........................................................... 45
TABLE
SQUARES OP NUMBERS.
Explanation. .............. .................. .......................... 49
TABLE VIII.
HOURS, MINUTES, AND SECONDS INTO DECIMALS OP A DAY.
Explanation ........................................... . ................ 51
TABLE IX.
To CONVERT TIME INTO ARC.
Explanation ............................................................ 53
TABLE X.
MEAN AND SIDEREAL TIME.
Explanation ............................................................ 55
OF DIFFERENCES AND INTERPOLATION.
General Principles ........ . ...... .' ...................................... 56
Fundamental Formulae .................................................. 61
Transformations of the Formulae ......................................... 62
Formulas of Stirling and Bessel ..................................... ..... 68
Special Cases of Interpolation Interpolation to Halves .................... 64
Interpolation to Thirds ................................................ 66
Interpolation to Fifths ........................... , ..................... 70
FORMULAE FOR THE SOLUTION OF PLANE AND SPHERICIAL
TRIANGLES.
Remarks ............................................................... 74
Formulae ............................................................... 75
TABLES I. TO X.
TABLE I.
LOGARITHMS OF NUMBERS.
1. Introductory Definitions.
Natural numbers are numbers used to represent quantities.
The numbers used in arithmetic and in the daily transactions of
life are natural numbers.
To every natural number may be assigned a certain other number,
called its logarithm.
The logarithm of a natural number is the exponent of the
power to which some assumed number must be raised to produce the
first number. The assumed number is called the base. E.g. f the
logarithm of 100 with the base 10 is 2, because 10 2 = 100; with the
base 2, the logarithm of 64 would be 6, because 2 fl = 64.
A system of logarithms means the logarithms of all POSN
tive numbers to a given base.
Although there may be any number of systems of logarithms,
only two are used in practice, namely:
1. The natural or Napierian system, base = e = 2. 718 282.
2. The common system, base = 10.
The natural system is used for purely algebraic purposes.
The common system is used to facilitate numerical calculations
and is the only one employed in this book.
If the natural number is represented by n, its logarithm is called
log n.
A logarithm usually consists of an integer number and a decimal
part.
The integer is called the characteristic of the logarithm.
The decimal part is called the mantissa of the logarithm.
A table of logarithms is a table by which the logarithm of
any given number, or the number corresponding to any given loga-
rithm, may be found.
4 LOGARITHMIC TABLES.
The moot simple form of table is that on the first page of Table I., which
gives the logarithms of all entire numbers from 1 to 150; each logarithm being
found alongside its number. The student may begin his exercises with this
table.
Mathematical tables in general enable us, when one of two related
quantities is given, to find the other.
In such tables the quantity supposed to be given is called the
argument.
The argument is usually printed on the top, bottom, or side of
the table.
The quantities to be found are called functions of the argu-
ment, and are found in the same columns or lines as the argument,
feut in the body of the table.
In a table of logarithms the natural number is the argument,
and the logarithm is the function.
2. The Use of Logarithms.
The following properties of logarithms are demonstrated in
treatises on algebra.
I. The logarithm of a product is equal to the sum of the loga-
rithms of its factors.
II. The logarithm of a quotient is found by subtracting the loga-
rithm of the divisor from that of the dividend.
III. The logarithm of any power of a number is equal to the loga-
rithm of the number multiplied by the exponent of the power.
IV. The logarithm of the root of a number is equal to the loga-
rithm of the number divided by the index of the root.
"We thus derive the following rules:
To find the product of several factors by logarithms.
EULE. Add the logarithms of the several factors. Enter the
table with the sum as a new logarithm, and find the number corres-
ponding to it.
This number is the product required.
Example 1. To multiply 7x8.
We find from the first page of Table I.
log? = 0.84510
" 8 = 0.90309
Sum of logs = 1.748 19 = log of product.
Having added the logarithms, we look in column log for a num-
THE USE OF LOGARITHMS. 5
ber corresponding to 1.788 19 and find it to be 56, which is the pro-
duct required.
Ex. 2 To find the continued product 2x6x8.
log 2, 0.30103
" 6, 0.77815
" 8, 0.90309
Sum of logs, 1.982 27 = log product.
The number corresponding to this logarithm is found to be 96,
which is the product required.
Ex. 3. To find the quotient of 147 -f- 21.
log 147, 2.16732
" 21,1.32222
Difference, 0.845 10
We find this difference to be the logarithm of 7, which is tha
required quotient.
Ex. 4. To find the quotient arising from dividing the continued
/roduct 98 X 102 X 148 by the continued product 21 X 37 X 68.
log 21, 1.322 22 log 98, 1.991 23
" 37, 1.56820 " 102, 2.00860
" 68, 1.83251 " 148, 2.17026
Sum = log divisor, 4.722 93 Sum = log dividend, 6.170 09
log divisor, 4.72293
Difference = log quotient, 1.447 16
Looking into the table, we find the number corresponding to this
logarithm to be 28, which is the required quotient.
NOTE. The student will notice that we have found this quotient without
actually determining either the divisor or dividend, having used only their loga-
rithms. If he will solve the problem arithmetically, he will see how much
shorter is the logarithmic process.
Ex. 5. To find the seventh power of 2.
We have log 2 = 0.301 03
7
2.10721 = log 128
Hence 128 is the required power.
Ex. 6. To find the cube root of 125.
3 | 2.09691
0.69897
6 LOGARITHMIC TABLES.
The index of the root being 3, we divide the logarithm of 125 by
it. Looking in the tables, we find the number to be 5, which is the
root required.
EXEKCISES.
Compute the following products, quotients, powers, and roots by
logarithms.
OO C
1. 11 . 13. Ans. 143. 5. ~~. Ans. 128.
2. 12'. Ans. 144. 6. 51 ' 98 ^ 81 . Ans. 31.
O4: . DO
8. -^-. Ans. 48. 7. ' . Ans. 144.
D D
2. 9'. 91. 78 , 54.48
4 13'. 21. 3 * AnS ' 108 - 8 --8T9-' Ans ' 36 '
3. Arrangement of the Table of Logarithms.
A table giving every logarithm alongside its number, as on the
first page of Table I., would be of inconvenient bulk. For numbers
larger than 150 the succeeding parts of Table I. are therefore used.
Here the first three figures of the natural number are given in the
left-hand column of the table. The first figure must be understood
where it is not printed. The fourth figure is to be sought in the
horizontal line at the top or bottom. The mantissa of the logarithm
is then found in the same line with the first three digits, and in the
column having the fourth digit at the top.
To save space the logarithm is not given in the column,
but only its last three figures. The first two figures are found in
the first column, and are commonly the same for all the logarithms
in any one line.
Example 1. To find the logarithm of 2090.
. We find the number 209, the figure 2 being omitted in printing,
in the left-hand column of the table, and look in the column having
the fourth figure, 0, at its top or bottom. In this column we find
320 15, which is the mantissa of the logarithm required.
Ex. 2. To find the logarithm of 2092.
Entering the table with 209 in the left-hand column, and choos-
ing the column with 2 at the top, we find the figures 056. Te
these we prefix the figures 32 in column 0, making the total logarithm
320 56. Therefore
Mantissa of log 2092 = .32056.
ARRANGEMENT OF THE TABLE. 7
EXERCISES.
Find in the same way the mantissas of the logarithms of the fol-
lowing numbers:
2240; 5133;
2242; 5256;
2249; 5504;
2895; 8925;
3644; 9557;
4688; 9780.
When the first two figures of the mantissa are not found in the
same line in which the number is sought, they are to be found in the
first line above which contains them.
Example. The first two figures of log 6250 are 79, which be-
longs to all the logarithms below as far as 6309. Therefore mantissa
of log 6250 = .795 88.
EXEECISES.
Find the mantissae of the logarithms of
6300; answer, .79934.
6309; " .79996.
6434;
6653;
6755;
6918;
7868.
Exception. There are some cases in which the first two figures
change in the course of the line. In this case the first two figures
are to be sought in the line above before the change and in the line
next below after the change.
Example. The mantissa of log 6760 is .82995. But the man-
tissa of log 6761 is .83001. In this case the figures 83 are to be
found in the next line below. To apprise the computer of these
cases, each of the logarithms in which the two first figures are found
in the line below is indicated by an asterisk.
EXERCISES,
Find the mantissa of
log 1022; answer, .009 45.
log 1024; " .01030.
8 LOGARITHMIC TABLES.
1231; 1999;
1387; 3988;
1419; 4675;
1621; 4798;
1622; 5377;
1862; 8512;
1863; 1009.
4. Characteristics of Logarithms.
The part of the table here described gives only the mantissa oj
<each logarithm. The characteristic must be found by the general
theory of logarithms.
The following propositions are explained in treatises on algebra:
The logarithm of 1 is 0.
" " " 10 " 1.
" " " 100 " 2.
" " " 1000 " 3.
10* " n.
Since any number of one digit is between and 10, its logarithm
is between and 1; that is, it is plus some fraction. In the same
way, the logarithm of a number of two digits is 1 + a fraction. And
in general,
The characteristic of the logarithm of any number greater than 1 is
less by unity than the number of its digits preceding the decimal point.
Example. The characteristic of the logarithm of any number
between 1 and 10 is 0; between 10 and 100 it is 1; between 100 and
1000 it is 2, etc.
Characteristic of log 1646 is 3.
" " " 164.6 " 2.
" " " 16.46 " 1.
" " " 1.646 " 0.
It is also shown in algebra that if a number be divided by 10 we
diminish its logarithm by unity.
Logarithms of numbers less than unity are most conveniently ex-
pressed by making the characteristic alone negative.
For example:
log 0.2 = log 2 - 1 = - 1 + .301 03;
" 0.02 = log2-2 = -2-f .301 03.
Hence: The mantissa of the logarithms of all numbers which
differ only in the position of the decimal point are the same.
CHARACTERISTICS OF LOGARITHMS. 9
Hence, also, in seeking a logarithm from the table we find the
mantissa without any reference to the decimal point. Afterward we
affix the characteristic according to the position of the decimal point.
For convenience, when a negative characteristic is written the
minus sign is put above it to indicate that it extends only to the
characteristic below it and not to the mantissa. Thus we write
log .02 = 2. 301 03.
In practice, however, it is more common to avoid the use of
negative characteristics by increasing them by 10. We then write
log. 02 = 8. 301 03 -10.
If we omitted to write 10 after the logarithm, the latter would,
in strictness, be the log of 2 X 10 8 . But numbers so great as this
product occur so rarely in practice that it is not generally neces-
sary to write 10 after the logarithm. This may be understood.
A convenient rule for remembering what characteristic belongs to
the logarithm of a decimal fraction is:
The characteristic is equal to 9, minus the number of zeros after
the decimal point and before the first significant figure.
Examples, log 34060 =4.53224
" 340.60 =2.53224
" 3.4060 =0.53224
" .03406 =8.53224-10
" .000 340 6 = 6.532 24 - 10
It will be seen that we can find the logarithms of numbers from
1 to 150 without using the first page of the table at all, since all the
mantissas on this page are found on the following pages as loga-
rithms of larger numbers.
EXERCISES.
Find the logarithms of the following numbers:
1.515 .003 899
.01 702 0.4276
18.62 464700
.03 735 98.030
Find the numbers corresponding to the following logarithms:
3.241 80;
1.19145;
A65321;
.74827;
7.560 03;
ans.
ans.
ans.
450 000
5 601 000
36 310-000
8.75035
7.411 28
.6.88997
9.116 94
7.250 18
-10;
-10;
-.I?;
-10;
9.99991 -
5.99996;
2.96028;
"0788627;"
0.00087.
10;
10 LOGARITHMIC TABLES.
5. Interpolation of Logarithms.
In all that precedes we have used only logarithms of numbers
containing not more than 4 significant digits. But in practice
numbers of more than four figures have to be used. To find the
logarithms of such numbers the process of interpolation is necessary.
This process is one of simple proportion, which can be seen from the
following example.
To find log. 1167.23.
The table gives the logarithms of 1167 and of 1168, which we find
to be as follows:
log 1167 = 3.06707
" 1168 = 3.06744
Difference of logarithms = .000 37
Now the number of which we wish to find the logarithm being
between these numbers, its logarithm is between these logarithms;
that is, it is equal to 3.067 07 plus a fraction less than .000 37.
Since the difference 37 corresponds to the difference of unity in
the two numbers, we assume that the quantity to be added to the
logarithm bears the same proportion to .23 that 37 does to unity.
We therefore state the proportion
1 : .23 :: 37 : increase required.
The solution of this proportion gives .23 X 37 = 8.51, which is
the quantity to be added to log 1167 to produce the logarithm
required.* The result is 3.067 155 1.
But our logarithms extend only to five places of decimals, while
the result we have written has seven. "We therefore take only five
places of decimals. If we write the mantissa 3.06715, the result will
be too small by .51. If we write 3.067 16, it will be too great by .49.
Since the last result is nearer than the first, we give it the prefer,
ence, and write for the required logarithm
log 1167. 23 = 3.06716.
We thus have the following rule for interpolating:
Take from the table the logarithm corresponding to the first four
significant digits of the number.
Considering the following digits as a decimal fraction, multiply
the difference between the logarithm and the next one following by
such decimal fraction.
* In this multiplication we have used a decimal point to mark oil the
fifth order of decimals. This is a convenient process in all such computations.
LABOR-SAVING DEVICES. H
This product "being added to the logarithm of the table will give
the logarithm required.
The whole operation by which we have found log 1167.23 would
then be as follows:
log 1167 = 3.06707
37 X 0.2 7.4
X 0.03 1.11
log 1167.23 = 3.06716
The products for interpolation, 7.4 and 1.11, may be found by
multiplying by the fifth and sixth figures of the number separately.
To facilitate this multiplication, tables of proportional parts are
given in the margin. Each difference between two logarithms will
be readily found in heavy type not far from that part of the table
which is entered, and under it is given its product by .1, .2, etc., . . .9.
We therefore enter this little table with the fifth figure, and take out
the corresponding number to be added to the logarithm. Then if
there is a sixth figure, we enter with that also and move the decimal
one place to the left. We then add the two sums to the logarithm.
6. Labor-saving Devices.
In using a table of logarithms, the student should accustom
himself to certain devices by which the work may be greatly facili-
tated.
In the first place it is not necessary to take the whole difference
between two consecutive logarithms. He has only to subtract the
last figure of the preceding logarithm from the last one of the fol-
lowing, increased by 10 if necessary, and thus find the last figure of
the difference.
The nearest difference in the margin of the table having this
same last figure will always be the difference required.
Example. If the first four figures of the number are 1494, in-
stead of subtracting 435 from 464 we say 5 from 14 leaves 9, and
look for the nearest difference which has 9 for its last figure. This
we readily find to be 29, at the top of the next page.
NOTE. In nearly all cases the difference will be found on the same page
with the logarithm. The only exception is at the bottom of the first page, where t
owing to the number of differences, they cannot all be printed.
In the preceding examples we have written down the numbers in
full, which it is well that the beginner should do for himself. But
after a little practice it will be unnecessary to write down anything
12 LOGAEITHMIG TABLES.
but the logarithm finally taken out. The student should accustom
himself to take the proportional parts mentally, adding them to the
logarithm of the table and writing down the sum at sight The
habit of doing this easily and correctly can be readily acquired by
practice.
Exercises. Find the logarithms of
792 638; 0.99997;
1000.77; 949.916;
1000.07; 20.8962;
100 007; 660 652;
181 982; 77.642;
281.936; 8.8953.
As a precaution in taking out logarithms, the computer should
always, after he has got his result, look into the table and see that
it does really fall between two consecutive logarithms in the table.
If the fraction to be interpolated is nearly unity, especially if it
is equal to or greater than 9, it will generally be more convenient to
multiply the difference of the logarithms by the complement* of the
fraction and subtract the product from the logarithm next succeed-
ing. The following are examples of the two methods, which may
always be applied whether the fraction be large or small:
Example 1. log 1004.28 = log (1005 - .72).
log 1004, .00173 log 1005, .00217
pr. pt. for .2, 8.8 pr. pt. for .7, - 30.8
" " " .08, 3.5 " " " .02, .9
log, 3.001 85 log, 3.001 85
Ex. 2. log 154 993 = 155 000 - 7.
log 1549, .19005 1550, .19>033
pr. pt. for .9, 25.2 pr. pt. for .07, 1.9b
" " " .03, 0.8
log, 5.19031
log, 5.19031
* By the complement or arithmetical complement of a decimal fraction is here
meant the remainder found by subtracting it from unity or from a unit of the
next order higher than itself. Thus :
co. .723 = .277
co. .1796 = .8204
co. .9932 = .0068.
NUMBER CORRESPONDING TO A GIVEN LOGARITHM. 13
7. To find the Number corresponding to a given
Logarithm.
The reverse process of finding the number corresponding to a
given logarithm will be seen by the following example:
To find the number of which the logarithm is 2.027 90.
Entering the table, we find that this logarithm does not exactly
occur in the table. We therefore take the next smaller logarithm,
which we find to be as follows:
log 1066 = 2.02776.
Subtracting this from the given logarithm we find the latter to be
greater by 14, while the difference between the two logarithms of
the table is 40. We therefore state the proportion
40 : 14 : : 1 to the required fraction.
The result is obtained by dividing 14 by 40, giving a quotient .35.
The required number is therefore 106.635. It will be remarked that
we take no account of the characteristic and position of the decimal
until we write down the final result, when we place the decimal in
the proper position.
The table of proportional parts is used to find the fifth and sixth
figures of the number by the following rule:
If the given logarithm is not found in the table, note the ex-
cess of the given logarithm above the next smaller one in the table,
which call A.
Take the difference of the two tabular logarithms, and fjrui it
among the large figures which head the proportional parts.
That proportional part next smaller than A will be the fifth
figure of the required number.
Take the excess of A above this proportional part; imagine its
decimal point removed one place to the right, and find the nearest
number of the table.
This number will be the sixth figure of the required number.
Example. To find the number of which the logarithm is 2.193 59.
Entering the table, we find the next smaller logarithm to be
.193 40. Therefore A = 19.
Also its tabular difference = 28.
Entering the table of proportional parts under 28, we find 16.8
opposite 6 to be the number next smaller than 19 the value of A.
Therefore the fifth figure of the number is 6.
The excess of 19 above 16.8 is 2.2. Looking in the same tablt
for the number 22, we find the nearest to be opposite 8.
14 LOGARITHMIC TABLES.
Therefore the fifth and sixth figures of the required number are
68. Now looking at the log .193 40 and taking the corresponding
number, we find the whole required number to be
156 168.
The characteristic being 2, the number should have three figures
before the decimal point. Therefore we insert the decimal point at
the proper place, giving as the final result 156.168.
8. Number of Decimals necessary.
In the preceding examples we have shown how with these tables
the numbers may be taken out to six figures. In reality, however,
it will seldom be worth while to write down more than five figures.
That is, we may be satisfied by adding only one figure to the four
found from the table. In this case, when we enter the table of
proportional parts, we take only the number corresponding to the
nearest proportional part.
To return to the last preceding example, where we find the num-
be/ corresponding to 2. 193 59. We find under the difference 28 that
th^ number nearest 19 is 19.6, which is opposite 7.
Therefore the number to be written down would be 156.17.
In the following exercises it would be well for the student to
4n ite six figures when the number is found on one of the first two
pa^es of the table and only five when on one of the following page*
Tl -e reason of this will be shown subsequently.
EXAMPLES AND EXERCISES.
\. To find the square root of f .
We have log 3, 0.477 12
" 2, 0.30103
log |, 0.17609
-f- 2, log V$, 0.08804
Here we have a case in which the half of an odd number is
required. We might have written the last logarithm 0.088045, but
we should then have had six decimals, whereas, as our tables only
give five decimals, we drop the sixth. If we write 4 for the fifth
figure it will be too small by half a unit, and if we write 5 it will
be too large by half a unit. It is therefore indifferent which figure
we write, so far as mere accuracy is concerned.
NUMBER OF DECIMALS NECESSARY. 15
A good rule to adopt in such a case is to write the nearest EVEN"
number. For example,
for the half of .261 81 we write .130 90;
" " .26183 " .13092;
" " .26185 " .13092;
" " .26187 " .13094;
" " .26189 " .13094;
" " .26197 " .13098;
" " .26199 " .13100.
Returning to our example, we find, by taking the number corre-
sponding to 0.088 04,
Vt = 1.224 72.
2. To find the square root of f .
log 2, 0.301 03
" 3, 0.47712
logf, 9.82391 - 10
| log |, 4.911 96 - 5 = log |/f.
The last logarithm is the same as
9.911 96 - 10,
which is the form in which it is to be written in order to apply the
rule of characteristics. The corresponding number is 0.816 50.
We have here a case in which, had we neglected considering the
surplus 10 as we habitually do, the characteristic of the answer
would have been 4 instead of 9 or 1. The easiest way to treat
such cases is this:
When we have to divide a logarithm in order to extract a root,
instead of increasing the characteristic by 10, increase it by 10 X
index of root.
Thus we write log ^= 19.823 91 - 20.
Dividing by 2, log i/f = 9.911 96 - 10,
which is in the usual form. , >
3. To find the cube root of |.
logl, 0.00000
" 2, 0.30103
log|, 9.69897 -10,
which we write in the form
log | = 29.69897-30.
Dividing this by 3,
J log 1 = log VT = 9-899 66 - 10.
16 LOGARITHMIC TABLES.
This logarithm is in the usual form, and gives
V?= 0.793 70.
The affix 30, or 10 x divisor, can be left to be understood
in these cases as in others. All that is necessary to attend to is that
instead of supposing the characteristic to be one or more units less
than 10, as in the usual run of cases, we suppose it to be one or more
nnits less than 10 X divisor.
Find: 4. The square root of -J;
5. The cube root of 2;
6. The fourth root of f ;
7. The fifth root of 20;
8. The tenth root of 10;
9. The tenth root of .
9. The Arithmetical Complement.
When a logarithm is subtracted from zero, the remainder is
called its arithmetical complement.
If L be any logarithm, its arithmetical complement will be L.
Hence if
L = log n,
then
arith. comp. = L = log -;
that is,
The arithmetical complement of a given logarithm is the logarithm
tfthe reciprocal of the number corresponding to the given logarithm.
Notation. The arithmetical complement of a logarithm is writ-
ten co-log. It is therefore defined by the form
co-log n = log .
Finding the arithmetical complement. To find the arithmetical
Complement of log 2 = 0.301 03, we may proceed thus:
0.00000
log 2, 0,30103
co-log 2, 9.69897-10.
We subtract from zero in the usual way; but when we come to
the characteristic, we subtract it from 10. This makes the re*
mainder too large by 10, so we write 10 after it, thus getting a
quantity which we see to be log 0.5.
We may leave the 10 to be understood, as already explained.
THE ARITHMETICAL COMPLEMENT. 17
The arithmetical complement may be formed by the following
rule:
Subtract, each figure of the logarithm from 9, except the last sig-
nificant one, which subtract from 10. The remainders will form the
arithmetical complement.
For example, having, as above, the logarithm 0.301 03, we form,
mentally, 9-0 = 9; 9-3 = 6; 9-0 = 9; 9-1 = 8; 9-0 = 9;
10 3 = 7; and so write
9.698 97
as the arithmetical complement.
To form the arithmetical complement of 3.284 00 we have 9 3
= 6; 9 2 = 7; 9 8 = 1; 10 4 = 6. The complement is
therefore
6.71600.
The computer should be able to form and write down the arith-
metical complement without first writing the tabular logarithm, the
subtraction of each figure being performed mentally.
Use of the arithmetical complement. The co-log is used to substi-
tute addition for subtraction in certain cases, on the principle: To
add the co-logarithm is the same as to subtract the logarithm.
Example. We may form the logarithm of J in this way by ad-
dition:
log 3, 0.47712
co-log 2, 9.69897
logj, 0.17609
Here there is really no advantage in using the co-log. But there
is an advantage in the following example:
97fi3 N/ J.1Q 9J.
To find the value of P = ;L We ad <* to the loga-
yy
rithms of the numerator the co-log of the denominator, thus:
log 2763, 3.44138
log 419.24, 2.62246
co-log 99, 8.00436
logP, 4.06820
'.P = 11700.
The use of the arithmetical complement is most convenient when
^o divisor is a little less than some power of 10.
J8 LOQAEITHMIC TABLES.
EXERCISES.
Form by arithmetical complements the values of:
109 X 216.26
1.
2.
3.
0.99316
8263 X 9162.7
92 X 99.618
4 X 6 X 8219
9X992
1C. Practical Hints on the Art of Computation.
The student who desires to be really expert in computation
should learn to reduce his written work to the lowest limit, and to
perform as many of the operations as possible mentally. We have
already described the process of taking a logarithm from the table
without written computation, and now present some exercises which
will facilitate this process.
1. Adding and subtracting from left to right. If one has but
two numbers to add it will be found, after practice, more easj and
natural to write the sum from the left than from the right. The
method is as follows:
In adding each figure, notice, before writing the sum, whether
the sum of the figures following is less or greater than 9, or equal
to it.
If the sum is less than 9, write down the sum found, or its last
figure without change.
If greater than 9, increase the figure by 1 before writing it down.
If equal to 9, the increase should be made or not made accord-
ing as the first sum following which differs from 9 is greater or less
than 9.
If the first sum which differs from 9 exceeds it, not only must we
increase the number by 1, but must write zeros under all the places
where the 9's occur. If the first sun different from 9 is less than 9,
write down the 9's without change.
The following example illustrates the process:
7502768357858892837
8239171645041102598
15741940002899995435
Here 7 and 8 are 15. 5 + 2 being less than 9, we write 15 without
change. 3 + being less than 9, we write 7 without change. 9 + 2 being
greater than 9, we increase the sum 3 + by 1 and write down 4. 7 + 1 being
PRACTICAL HINTS ON THE ART OF COMPUTATION. 19
less than 9, we write the last figure of 9 -|- 2, or 1, without change. 6 + 7 being
greater than 9, we increase 7 + 1 by 1 and write down 9. Under 6 + 7 we
write down 3 or 4. To find which, 8+ 1 = 9; 3 + 6 = 9; 5+4= 9; 7 + 5 =
12. This first sum which is different from 9 being greater than 9, we write 4
under 6 + 7, and O's in the three following places where the sums are 9. 7+5
= 12. Since 8 + < 9, we write down 2. Before deciding whether to put 8 or
9 under 8 + 0, we add 5 + 4 = 9; 8 + 1 = 9; 8 + 1 =9; 9 + = 9; 2 + 2 = 4
This being less than 9, we write 8 under 8 + 0, and 9's in the four following
places. Since 5 + 8 = 13 > 9, we write 5 under 2 + 2. Since 9+ 3 = 12 > 9, we
write 4 under 5 + 8. Since 8 + 7 = 15 > 9, we write 3 under 9 + 3. Finally,
under 8 + 7 we write 5.
This process cannot be advantageously applied when more than
two numbers are to be added.
EXERCISES.
Let the student practise adding each consecutive pair of the fol-
lowing lines, which are spaced so that he can place the upper margin
of a sheet of paper under the lines he is adding and write the sum
upon it.
250917285316981208
251235964692184368
791615832316646891
208532164379102909
868588964342944825
987654321012345674
Subtracting. We subtract each figure of the subtrahend from
the corresponding one of the minuend (the latter increased by 10 if
necessary), as in arithmetic.
Before writing down the difference, we note whether the follow-
ing figure of the subtrahend is greater, less, or equal to the corre-
'gponding figure of the minuend.
If greater, we diminish the remainder by 1 and write it down.*
If less, we write the remainder without change.
If equal, we note whether the subtrahend is greater or less than
the minuend in the first following figure in which they differ.
If greater, we diminish the remainder by 1, as before, and write
9's under the equal figures.
* If the student is accustomed to carrying 1 to the figures of the minuend
when he has increased the figure of his subtrahend by 10, he may find it easier
to defer each subtraction until he sees whether the remainder is or is not to be
diminished by 1, and, in the latter case, to increase the minuend by 1 before
subtracting.
20 LOGARITHMIC TABLES.
If less, write the remainder unchanged, putting O's under the
equal figures.
Example.
72293516214394
24268518014198
48024998200196
Here 7 2 =5; because 4 > 2, we write 4 12 4 = 8; because 2 = 2
and 6 < 9, we write 8; and write in the following place. 96 = 3; be-
cause 8 > 3, we write 2. 13 8 = 5; 5 = 5; 1 = 1; 8>6; so under 13 8 we
write 4, with 9's in the two next places. 16 8 = 8; because < 2, we write
8. 2 = 2; 1 = 1; 4 = 4; 1 < 3; so under 2 we write 2, followed by O's.
3 1 = 2; because 9 = 9, 8 > 4, we write 1, with 9 in the next place. 14 8 =
6, which we write as the last figure.
EXERCISES.
The preceding exercises in addition will serve as exercises in sub-
traction by subtracting each line from that above or below it. The
student should be able to subtract with equal facility whether the
minuend is written above or below the subtrahend.
Mental addition and subtraction. When an expert computer has
to add or subtract two logarithms, as in forming a product or quo-
tient of two quantities, he does not necessarily write both of them,
but prefers to write the first and, taking the other mentally, add (or
subtract) each figure in order from left to right, and write down the
sum (or difference). He thus saves the time spent in writing one
number, and, sometimes, the inconvenience of writing it where
there is not sufficient room for it.
This process of inverted addition is most useful in adding the
proportional part in taking a logarithm from the table. It is then
absolutely necessary to save the computer the trouble of copying
both logarithm and proportional part.
Expert computers can add seven-figure logarithms in this way
without trouble. But with those who do not desire to become ex-
perts it will be sufficient to learn to add two or three figures, so as
to be able to take a five-figure or seven -figure logarithm from the
table without writing anything but the result.
11. Imperfections of Logarithmic Calculations.
Nearly all practical computations with logarithms are affected
by certain sources of error, arising from the omission of deci-
mals. It is important that these errors should be understood in
IMPERFECTIONS OF LOGARITHMIC CALCULATIONS. 21
order not only to avoid them so far as possible, but to avoid spend-
ing labor in aiming at a degree of accuracy beyond that of which the
numbers admit.
Mathematical results may in general be divided into two classes:
(1) those which are absolutely exact, and (2) those which are only
to a greater or less degree approximate.
As an example of the former case, we have all operations upon
entire numbers which involve only multiplication and division. For
example, the equations
16 s = 256
j^_16
6 a ~ 9
are absolutely exact.
But if we express the fraction | as a decimal fraction, we have
^ = .142857. ., etc., ad infinitum.
Hence the representation of \ as a decimal fraction can never be
absolutely exact. The amount of the error will depend upon how
many decimals we include. If we use only two decimals we shall
certainly be within one hundredth; if three, within one thou-
sandth, etc. Hence the degree of accuracy to which we attain de-
pends upon the number of decimals employed. By increasing the
number of decimals we can attain to any degree of accuracy. As an
example, it is shown in geometry that if the ratio of the circumfer-
ence of a circle to its diameter be written to 35 places of decimals,
the result will give the whole circumference of the visible universe
without an error as great as the minutest length visible in the most
powerful microscope.
There are no numbers, except the entire powers of 10, of which
the logarithms can be exactly expressed in decimals. We must
therefore omit all figures of the decimal beyond a certain limit. The
number of decimals to be used in any case depends upon the degree
of accuracy which is required. The large tables of logarithms con-
tain seven decimal places, and therefore give results correct to the
ten-millionth part of the unit. This is sufficiently near the truth
in nearly all the applications of logarithms.
With five places of decimals our numbers will be correct to the
hundred-thousandth part of a unit. This is sufficiently near for
most practical applications.
Accumulation of errors. When a long computation is to be
made, the small errors are liable to accumulate so that we cannot
rely upon this degree of accuracy in the final result. The manner
22 LOGARITHMIC TABLES.
in which the tables are arranged so as to reduce the error to a mini-
mum may be shown as follows:
We have to seven places of decimals
log 17 = 1.230 448 9
" 18 = 1.2552725
When the tables give only five places of decimals the two last
figures must be omitted. If the tables gave log 17=. 230 44, the
logarithm would be too small by 89 units in the seventh place. It is
therefore increased by a unit in the fifth place, and given .23045.
This quantity is then too large by 11, and is therefore nearer the
truth than the other. The nearest number being always given, we
have the result:
Every logarithm in the table differs from the truth ~by not more
than one half a unit of the last place of decimals.
Since the error may range anywhere from zero to half a unit, and
is as likely to have one value as another between those limits, we
-conclude:
The average error of the logarithms in the tables is one fourth of
<a unit of the last place of decimals.
Errors in interpolation. When we interpolate the logarithm we
add to the tabular logarithm another quantity, the proportional part,
which may also be in error by half a unit, but of which the average
error will only be one fourth of a unit.
As most logarithms have to be interpolated, the general result
will be:
An interpolated logarithm may possibly be in error by a unit in
the last place of decimals.
The sum of the average errors will, however, be only half a unit.
But these errors may cancel each other, one being too large and the
other too small. The theory of probabilities shows that, in conse-
quence of this probable cancellation of errors, the average error only
increases as the square root of the number of erroneous units added.
The square root of 2 is 1.41.
If, therefore, we add two quantities each affected with a probable
error, .25, the result will be, for the probable error of the sum,
1.41 X .25 = 0.35.
We therefore conclude:
The average error of a logarithm derived from the table by inter-
polation is 0.35 of a unit of the last place.
Applying the above rule of the square root to the case in which
IMPERFECTIONS OF LOGARITHMIC CALCULATIONS. 23
several logarithms are added or subtracted to form a quotient, we?
find the results of the following table:
No. of logs added or subtracted.
Average error.
1
0.35
2
0.50
3
0.63
4
0.72
5
0.81
6
0.88
7
0.95
8
1.02
9
1.08
10..
, 1.14
From this table we see that if we form the continued product of
eight factors, by adding their logarithms the average error of th&
sum of the logarithms will be more than a unit in the last place.
As an example of the accumulation of errors, let us form the*
product 11 . 13.
We have from the table
log 11 = 1.041 39
" 13 = 1.11394
log product, 2.155 33
We see that this is less than the given logarithm of the product
143 by a unit of the fifth order. But if we use seven decimals we,
have log 11, 1.0413927
" 13, 1.1139434
2.1553361
Comparing this with the computation to five places, we see the
source of the error.
If the numbers with which we enter the tables are affected by
errors, these errors will of course increase the possible errors of the
logarithms.
In determining to what degree of accuracy to carry our results,
we have the following practical rule :
It is never worth while to carry our decimals beyond the limit of
precision given ~by the tables, which limit may be a considerable frac-
tion of the unit in the last figure of the tables.
Let us have a logarithm to five places of decimals, 1.92949, of
which we require the corresponding number. Entering the table, wa
24 LOGARITHMIC TABLES.
perceive that the corresponding number is between 85. 01 an4 85.0&
If this logarithm is the result of adding a number of logarithms,
each of which may be in error in the way pointed out, we may sup-
pose it probably affected by an error of half a unit in the last figure
and possibly by an error of a whole unit or more. That is, its true
value may be anywhere between 92 948 and 92 950.
The number corresponding to the former value is 85. C13, and
that corresponding to the latter 85.016. Since the numbtrs may
fall anywhere between these limits, we assign to it a mean \alue of
85. 014, which value, however, may be in error by two units in the
last place. It is not, therefore, worth while to carry the interpolation
further and to write more than five digits.
Next suppose the logarithm to be 2.021 70. Entering the table,
we find in the same way that the number probably lies between the
limits 105.121 and 105.126. There is therefore an uncertainty of
five units in the sixth place, or half a unit in the fifth place. If the
greatest precision is desired, we should write 105. 124. But our last
figure being doubtful by two or three units, the question might arise
whether it were worth while to write it at all. As a general rule, if
the sixth figure is required to be exact, we must use a six- or seven-
place table of logarithms.
Still, near the beginning of the table, the probable error will be
diminished by writing the sixth figure.
Now knowing that at the beginning of the table a difference of
one unit in the number makes a change ten times as great in the
logarithm as at the end of the table, we reach the conclusions :
In talcing out a number in the first part of the table, it can never
be worth while to write more than six significant figures^ and very
little is added to the precision by writing more than five.
In the latter part of the table it is never worth while to write more
than five significant figures.
Sometimes no greater accuracy is required than can be gained by
irj/ng four-figure logarithms. There is then no need of writing the
last figure. If, however the printed logarithm is used without
change, the fourth figure must be increased by unity whenever the
fifth figure exceeds 5. When the fifth figure is exactly 5, the increase
should or should not be made according as the 5 is too small or too
great. To show how the case should be decided, a stroke is printed
above the 5 when it is too great. In these cases the fourth figure
should be used as it stands, but, when there is no stroke, it should
be increased by unity.
THE COMPUTATION OF ANNUITIES, ETC. 25
12. Applications of Logarithms to the Computation of
Annuities and Accumulations of Funds at Com-
pound Interest.
One of the most useful applications of logarithms is to fiscal
calculations, in which the value of moneys accumulating for long
periods at compound interest is required.
Compound interest is gained by any fund on which the interest
is collected at stated intervals and put out at interest.
As an example, suppose that $10 000 is put out at 6 per cent
interest, and the interest collected semi-annually and put out at the
same rate. The principal will then grow as follows:
Principal at starting 810 000.00
Six months' interest = 3 per cent 300.00
Amount at end of 6 months $10 300.00
Interest on this amount = 3 per cent. . 309.00
Amount at end of 1 year $10 609.00
Interest on this amount = 3 per cent. . 318.27
Amount at end of H years $10 927.27
Interest on this amount for 6 months. . 327.82
Amount at end of 2 years $11 255.09
Although in business practice interest is commonly payable semi-
annually, it is in calculations of this kind commonly supposed to be
collected and re-invested only at the end of each year. This makes
the computation more simple, and gives results nearer to those ob-
tained in practice, because a company cannot generally invest its
income immediately. If it had to wait three months to invest each
semi-annual instalment of interest collected, the general result would
be about the same as if it collected interest only once a year and in-
vested it immediately.
If r be the rate per cent per annum, the annual rate of increase
will be . Let us put
1UU
p, the annual rate of increase = -r^;
1UU
p, the amount at interest at the beginning of the time, or the
principal;
a, the amount at the end of one or more years.
26 LOGARITHMIC TABLES.
Then, at the beginning of first year, principal p
Interest during the year pp
Amount at end of year p (1 -f- p)
Interest on this amount during second year pp (I -f- p)
Amount at end of second year, (1 + p)p (1 + P) P (1 + PY
Continuing the process, we see that at the end of n years the
amount will be
a=p(l + py. (1)
To compute by logarithms, let us take the logarithms of both
members. We then have
log a = logp + n log (1 + p). (2}
Example. Find the amount of $1250 for 30 years at 6 per cent
per annum.
Here p = .06
1 + p = 1.06
log (1 -f p) = 0.025 306 (end of Table I.)
30
nlog(l+p), 0.75918
Iogj9, 3.09691
log a, 3.85609
a, $7179.50 = required amount.
EXERCISES.
1. Find the amount of $100 for 100 years at 5 per cent compound
interest.
2. A man bequeathed the sum of $500 to accumulate at 4 per
cent interest for 80 years after his death. After that time the annual
interest was to be applied to the support of a student in Harvard
College. What would be the income from the scholarship?
3. If the sum of one cent had been put out at 3 per cent per
annum at the Christian era, and accumulated until the year 1800,
what would then have been the amount, and the annual interest on
this amount?
It is only requisite to give three significant figures, followed by the necessary
number of zeros.
4. Solve by logarithms the problem of the horseshoeing, in which
a man agrees to pay 1 cent for the first nail, 2 for the second, and so
on, doubling the amount for every nail for 32 nails in all.
NOTE. It is only necessary to compute the amount for the 32d nail, be-
cause it is easy to see that the amount paid for each nail is 1 cent more than fnr
all the preceding ones.
COMPUTATION OF ANNUITIES, ETC. . 27
5. A man lays aside $1000 as a marriage-portion for his new-born
daughter, and invests it so as to accumulate at 8 per cent compound
interest. The daughter is married at the age of 25. What does the
portion amount to?
6. A man of 30 pays $2000 in full for a $5000 policy of insurance
on his life. Dying at the age of 80, his heirs receive $7000, policy
and dividends. If the money was worth 4 per cent to him, how
much have the heirs gained or lost by the investment?
7. What would have been the answer to the previous question,
had the man died at the age of 40, and the amount paid been
$6000?
Other applications of the formula. By means of the equations
(1) and (2) we may obtain any one of the four quantities a, p, p, and
n when the other three are given.
CASE I. Given the principal, rate of interest, and time, to
find the amount.
This case is that just solved.
CASE II. Given the amount, time, and rate per cent, to find
the principal.
Solution. Equation (1) gives
Taking the logarithms,
log p = log a n log (1 + p),
by which the computation may be made.
CASE III. Given the principal, amount, and time, to find the rate.
Solution. Equation (2) gives
n n p
Example. A man wants a principal of $600 to amount to $1000
in 10 years. At what rate of interest must he invest it?
Solution. log a = 3.000 00
logp = 2.77815
log - = 0.221 85
~ log|- = 0.022 185 = log (1 + P).
Hence, from last page of logarithms,
1+ p = 1.05241;
and rate = 5.241,
or 5^ per cent, nearly.
28 LOGARITHMIC TABLES.
EXEKCISES.
1. At what rate of interest will money double itself every ten
years? Ans. 7.177.
2. At what rate will it treble itself every 15 years? Ans. 7.599.
3. A man having invested $1000, with all the interest it yielded
him, for 25 years, finds that it amounts to $3386. What was the
rate of interest? Ans. 5 per cent.
4. A life company issued to a man of 20 a paid-up policy for
$10,000, the single premium charged being $3150. If he dies at the
age of 60, at what rate must the company invest its money to make
itself good? Ans. 2.93 per cent.
5. A man who can gain 4 per cent interest wants to invest such a
sum that it shall amount to $5000 when his daughter, now 5 years old,
attains the age of 20. How much must he invest? Ans. $2776. 62.
6. How much must a man leave in order that it may amount to
$1,000,000 in 500 years at 2J per cent interest? Ans. $4.36
7. How much if the time is 1000 years, the rate being still 2-J
per cent, and the amount $1,000,000? Ans. 0.0019 of a cent.
8. A man finds that his investment has increased fivefold in 25
years. What is the average rate of interest he has gained ?
Ans. 6.65.
9. An endowment of $7500 is payable to a man when he attains
the age of 65. What is its value when he is 45, supposing the rate
of interest to be 4 per cent? Ans. $3423.
13. Accumulation of an Annuity.
It is often necessary to ascertain the present or future value of a
series of equal annual payments. Thus it is very common to pay a
constant annual premium for a policy of life insurance. The value
of such a series of payments at any epoch is found by reducing the
value of each one to the epoch, allowing for interest, and taking the
sum. Supposing the epoch to be the present time, the problem may
be stated as follows:
A man agrees to pay p dollars a year for n years, the first pay*
ment being due in one year, and the total number of payments n.
What is the present value of all n payments 9
Putting, as before, p = rate the present value of p
dollars payable after y years will, by 12, Case II., be
P
ACCUMULATION OF AN ANNUITY. 29
Putting in succession, y = 1, y = 2, . . . y = w, the sum of the
present values is
p p , p , , m p
1 4- p "" (i -}- p) a ' (1 -f p) 3 "* *" (1 -f p)*
This is a geometrical progression in which
First term = ^- ;
1 +/>
Common ratio = ^ ;
Number of terms = n.
By College Algebra, 212, the sum of this progression will be
/ 1 \n
, '-(iT-J
(1 + <)-!
If the first payment is to be made immediately, instead of at the
end of a year, the last or th payment will be due in n 1 years,
and the progression will be
p + r+p + (i + />)* + ' ' * + (i+P)' 1 - 1 *
We find the sum of the geometric progression to be
EXERCISES.
1. What is the present value of 15 annual payments of $85 each,
of which the first is due in one year, the rate being 5 per cent?
We find by substitution
Present value = 85 __!05^-1__ J5_ 1.05- 1
1.05 16 - 1.05 16 1.05" ' .05
_1700(1.05 16 -1)
(1.05) 16 '
log 1.05, 0.021189 1.05", 2.07895
_ 15 1.05" -1, 1.07895
log 1.05", 0.31784 log, 0.03300
co-log 1.05 15 , 9.68216
log 1700, 3.23045
Value, $882.28 log value, 2L945 6?
30 LOQAR1THM10 TABLES.
2. The same thing being supposed, what would be the present
value if the rate of interest were 4 per cent ? Ans. $945.80
3. What is the present value of 25 annual payments of $1000
each, the first due immediately, if the rate of interest is 3 per cent ?
Ans. $17,935
4. A debtor owing $10,000 wishes to pay it in 10 equal annual
instalments, the first being payable immediately. If the rate of
interest is 6 per cent, how much should each payment be?
Ans. $1281.76.
NOTE. This problem is the reverse of the given one, since, in the equation
(2), we have given 2 a = 10000, p = 0.06, and n = 10, to find p.
5. The same thing being supposed, what should be the annual
payment in case the payments should begin in a year?
Ans. $1358.69.
Perpetual annuities. If the rate of interest were zero, the
present value of an infinity of future payments would be infinite.
But with any rate of interest, however small, it will be finite. For if,
in the first equation (1), we suppose n infinite, f J will converge
toward zero, and we shall have
This result admits of being put into a concise form, thus:
Since 2 is the present value of the perpetual annuity p, the
annual interest on this value will be p*S. But the equation (3) gives
p2=p.
Hence:
The present value of a perpetual annuity is the sum of which the
annuity is the annual interest.
Example. If the rate of interest were 3| per cent, the present
value of a perpetual annuity of $70 would be $2000.
EXERCISES.
1. A government owing a perpetual annuity of $1000 wishes to
pay it off by 10 equal annual payments. If the rate of interest is 4
per cent, what should be the amount of each payment?
Ans. $3082.30.
2. A government bond of $100 is due in 15 years with interest at
6 per cent. The market rate of interest having meanwhile fallen to
3| per cent, what should be the value of the bond?
NOTE. We find, separately, the present value of the 15 annual instalments
of interest, and of the principal.
TABLE II.
MATHEMATICAL CONSTANTS.
14. In this table is given a collection of constant quantities
which frequently occur in computation, with their logarithms.
The logarithms are given to more than five decimals, in order to
be useful when greater accuracy is required. When used in five-
place computations, the figures following the fifth decimal are to be
dropped, and the fifth decimal is to be increased by unity in case
the figure next following is 5 or any greater one.
TABLES III. AND IT.
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS.
15. By means of these tables the logarithms of the six trigono-
metric functions of any angle may be found.
The logarithm of the function instead of the function itself is
given, because the latter is nearly always used as a factor.
We begin by explaining Table IV., because Table III. is used only
in some special cases where Table IV. is not convenient.
I. Angles less than 45. If the angle of which a function is
soiight is less than 45, we seek the number of degrees at the top of
the table and the minutes in the left-hand column. Then in the
line opposite these minutes we find successively the sine, the tan-
gent, the cotangent, and the cosine of the angle, as given at the
heading of the page.
Example. log sin 31 27' = 9.717 47;
log tan 31 27' = 9.78647;
log cotan 31 27' = 0.213 53;
cos 31 27' = 9.93100.
The sine, tangent, and cosine of this angle being all less than
unity, the true mantissas of the logarithm are negative; they are
therefore increased by 10, on the system already explained.
If the secant or cosecant of an angle is required, it can be found
by taking the arithmetical complement of the cosine or sine. It is
shown in trigonometry that
secant = : ,
cosine
and
cosecant = -; .
sine
Therefore log secant = log cosine = co-log cosine;
log cosec = log sine = co-log sine.
We thus find log sec 31 27' = 0.069 00;
log cosec 31 27' = 0.282 53.
After each column, upon intermediate lines, is given the differ-
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 33
ence between every two consecutive logarithms, in order to facilitate
interpolation.
In the case of tangent and cotangent, only one column of differ-
ences is necessary for both functions.
If we use no fractional parts of minutes, no interpolation is
necessary; but if decimals of a minute are employed, we can inter-
polate precisely as in taking out the logarithms of numbers.
"Where the differences are very small they are sometimes omitted.
Tables of proportional parts are given in the margin, the use of
which is similar to those given with the logarithms of numbers.
Example 1. To find the log sin of 31 27'.7.
We have from the tables, log sin 31 27' = 9.717 47
Under diff. 20, P.P. for 7, 14
log sin 31 27'. 7 = 9.71761
Ex. 2. To find log cot 15 44'. 34.
The tables give log cot 15 44' = 0.550 19
Under diff. 48, opposite 0.3, P.P., - 14.4
" " 0.4 -7- 10, - 1.9
log cot 15 44'. 34, 0.55003
Since the tabular quantity diminishes as the angle increases, the
proportional parts are subtractive.
EXEECISES.
Find from the tables :
1. log cot 43 29'. 3;
2. log tan 43 29'. 3;
3. log cos 27 10'. 6;
4. log sin 27 10'. 6;
5. log tan 12 9'. 43;
6. log cot 12 9'. 43.
In the case of sines and tangents of small angles the differences
vary so rapidly that in most cases the exact difference will not be
found in the table of proportional parts. In this case, if the pro-
portional parts are made use of, a double interpolation will generally
be necessary to find the fraction of a minute corresponding to a given
sine or tangent. If only tenths of minutes are used, an expert com-
puter will find it as easy to multiply or divide mentally as to refer to
ihe table.
II. Angles between 45 and 90. It is shown in trigonometry
that if we compute the values of the trigonometric functions for the
34 LOGARITHMIC TABLES.
first 45, we have those for the whole circle by properly exchanging
them in the different parts of the circle. First, if we have
a + ft = 90,
then a and ft are complementary functions, and
|. sin ft = cos a\
tan ft = cotan a.
Therefore if our angle is between 45 and 90, we may find its
complement. Entering the table with this complement, the com-
plementary function will then be the required function of the angle.
Example. To find the sine of 67 23', we may enter the table
with 22 37' (= 90- 67 23') and take out the cosine of 22 37',
whichfis the required sine of 67 23.
To save the trouble of doing this, the complementary angles and
the complementary denominations of the functions are given at the
bottom of the page.
The minutes corresponding to the degrees at the bottom are given
on the right hand. Therefore:
To find the trigonometric functions corresponding to an angle
between 45 an d 90, we take the degrees at the bottom, of the page and
the minutes in the right-hand column. The values of the four func-
tions log sine, log tangent, log cotangent, and log cosine, as read at
the bottom of the page, are then found in the same line as the
minutes.
Example 1. For 52 59' we find
log sin = 9.90225;
log tan = 0.12262;
log cot = 9.877 38;
log cos = 9.77963.
Ex. 2. To find the trigonometric functions of 77 17'. 28.
77 17' 9.
sin. tan. cot. cos.
98921 0.64653 9.35347 9.34268
P.P. for 0.2
+ 0.6
+ 11.8
- 11.8
- 11.2
" 0.08
+ 0.2
+ 4.7
- 4.7
- 4.5
9. 989 22 0. 646 70 9. 353 30 9. 342 52
Then log sec = co-log cos = 0.657 48;
log cosec = co-log sin = 0.010 78.
EXERCISES.
Find the logarithms of the six functions of the following angles:
1. 45 50'. 74; 3. 74 0'.68;
2. 4849'.37; 4. 83 59'.62.
LOGARITHMS OF TRIGONOMETRIC
III. When the angle exceeds 90.
EULE. Subtract from the angle the greatest multiple of 90
which it contains.
If this multiple is 180, enter the table with the excess of the angle
over 180 and take out the functions required, as if this excess were
itself the angle.
If the multiple is 90 or 270, take out the complementary func-
tion to that required.
By then assigning the proper algebraic sign, as shown in trigo-
nometry, the complete values of the function will be obtained.
The computer should be able to assign the proper algebraic
sign according to the quadrant, without burdening his memory with
the special rules necessary in each +
case. This he can do by carrying Sine positive
in his mind's eye the following
scheme. He should have at com-
mand the arrangement of the four
quadrants as usually represented in
trigonometry, so as to know, when
an angle is stated, where it will fall
relatively to the horizontal and ver-
tical lines through the centre of the
circle. Then, in the case of
Sine or cosecant. If the angle Sine negative
is above the horizontal line (which
it is between and ISO ), the sine is positive; if below, negative.
Cosine or secant. If the angle is to the right of the vertical
central line (as it is in the first and fourth quadrants), the cosine and
secant are positive; if to the left (as in the second and third quad-
rants), negative.
Tangent or cotangent. Through the opposite first and third quad-
rants, positive; through the opposite second and fourth quadrants,
negative.
Example 1. To find the tangent and cosine of 122 44'. Sub-
tracting 90, we enter the table with 32 44' and find
log cot 32 44' = 0.19192;
log sin 32 44' = 9.73298.
T nerefore, writing the algebraic sign before the logarithm, we hare
log tan 122 44' = - 0.191 92;
log cos 122 44' = 9.732 98.
86 LOGARITHMIC TABLES.
Ex. 2. To find the sine and cotangent of 322 58'.
Entering the table with 52 58' = 322 58' 270, and taking
out the complementary functions, we find
log sin 322 58' = - 9.779 80;
log cot 322 58' = 0.122 36.
Ex. 3. To find the sine and tangent of 253 5'.
Entering with 73 5', we take out the sine and tangent, finding
log sin 253 5' = 9.890 79;
log tan 253 5' = -f 0.516 93.
Ex. 4. To find the six trigonometric functions of 152 38'. We
have
log sin 152 38' = log cos 62 38' pos. = + 9.662 46;
log cos 152 38' = log sin 62 38' neg. = 9.948 45;
log tan 152 38' = log cot 62 38' neg. = 9.71401;
log cot 152 38' = log tan 62 38' neg. = 0.285 99;
log sec = co-log cos = 0.05155;
log cosec = co-log sin = + 0.337 54.
EXERCISES.
Find the six trigonometric functions of the following angles:
276 29'.3;
66 0'.5;
96 59'.8;
252 20'.3;
318 10'. 7;
- 25 22'.2;
-155 30'. 7.
16. Method of Writing the Algebraic Signs.
As logarithms are used in computation, they may always be con-
sidered positive. It is true that the logarithms of numbers less than
unity are in reality negative, but, for convenience in calculation, we
increase them by 10, so as to make them positive.
The number corresponding to a given logarithm may, in compu-
tation, be positive or negative. There are two ways of distinguishing
the algebraic sign of the number, between which the computer may
choose for himself.
I. Write the algebraic sign of the number before the logarithm.
As usually interpreted, the algebraic sign written thus would apply
to the logarithm, which it does not. It is therefore necessary for the
ANGLE CORRESPONDING TO A GIVEN FUNCTION. 37
computer to bear in mind that the sign belongs, not to the loga-
rithm, as written, but to the number.
II. Write the letter n after the logarithm when the number i&
negative. This plan is theoretically the best, but, should the com-
puter accidentally omit the letter, the number will be treated as
positive, and a mistake will be made. It therefore requires vigilance
on his part. An improvement would be to write a letter not likely
to be mistaken for n, s for instance, after all positive logarithms.
17. To Find the Angle Corresponding to a Given
Trigonometric Function.
Disregarding algebraic signs, there will always be four angles
corresponding to each function, one in each quadrant. These angles,
will be:
The smallest angle, as found in the table;
This angle increased by 180;
The complementary angle increased by 90;
The complementary angle increased by 270.
For instance, for the angle of which log tan is 0.611 92, we find
76 16'. But we should get this same tangent for 103 44', 256 16',
and 283 44'.
Of the four functions corresponding to the four angles, two will
always be positive and two negative; so that, in reality, there will
only be two angles corresponding to a function of which both the-
sign and the absolute value are given. These values are found by
selecting from the four possible ones the two for which the functions
have the given algebraic sign. After selecting them, they may be
checked by the following theorems, which are easily deduced from:
the relations between the values of each function as given in trigo-
nometry:
The sum of the two angles corresponding to the same sine is ISO '
vr 540.
The sum of the two angles corresponding to the same cosine is
360.
The difference of the two angles corresponding to the same tangenk
is 180.
Which of the two possible angles is to be chosen depends upon
the conditions of the problem or the nature of the figure to which
the angle belongs. If neither the conditions nor the figure decida
the question, the problem is essentially ambiguous, and either ^~
Doth angles are to be taken.
38 LOGARITHMIC TABLES.
EXEECISES.
Find tne pairs of values of the angle a from the following values
of the trigonometric functions:
1. log sin a = + 9.902 43; 12. log sec a = -{- 0.221 06;
2. log sin a = - 9.902 43; 13. log sec a = - 0.221 06;
3. log cos a = + 9.902 43; 14. log sec a - - 0.099 20;
4. log cos a = 9.902 43; 15. log sec a + 0.123 46;
5. log tan a = + 0.143 16; 16. log sin a = -f 8.990 30;
6. log tan a = 0.143 16; 17. log sin a ~ 8.990 30;
7. log cot a = -f 0.143 16; 18. log cos a = + 9.218 67;
8. log cot a 0.143 16; 19. log cos a = 9.218 67;
9. log tan a = - 9.024 81; 20. log tan a = - 9.136 90;
10. log tan a = - 0.975 19; 21. log tan a = -f- 9.136 90;
11. log tan a = + 0.975 19; 22. log cot a = + 9.136 90.
18. Cases when the Function is very Small or Great.
When the angle of which we are to find the functions approaches
to zero, the logarithms of the sine, tangent, and cotangent vary so
.rapidly that their values to five figures cannot be readily interpolated.
The same remark applies to the cosine, cotangent, and tangent of
angles near 90 or 270. The mode of proceeding in these cases will
depend upon circumstances.
In the use of five-place logarithms, there is little advantage in
carrying the computations beyond tenths of minutes, though the
hundredths may be found when the tangent or cotangent is given.
Where greater accuracy than this is required, six- or seven-place
tables must be used.
If the angles are only carried to tenths of minutes, there is no
necessity for taking out the sine, tangent, or cotangent to more than
four decimals when the angle is less than 3, and three decimal?
suffice for angles less than 30'. The reason is that this number of
decimals then suffice to distinguish each tenth of minute.
When the decimals are thus curtailed, an expert computer will be
able to perform the multiplication and division for the tenths o?
minutes mentally. If, however, this is inconvenient, the following
rule may be applied.
To find the log sine or log tangent of an angle less than 2 to
four places of decimals:
EULE. Enter the table of logarithms of numbers with the valu*
WHEN THE FUNCTION IS VERY SMALL OR GREAT. 39
of the angle expressed in minutes and tenths, and take out the loga-
rithm.
To this logarithm add the quantity 6.4637.
The sum will be the log sine, and the log tangent may be assumed
to have the same value.
Example 1. To find log sin 1 2$'. 6.
1 22'. 6 = 82'.6
log 82'.6 = 1.9170
constant, 6.4637
log sin 1 22'. 6, 8.3807
This rule is founded on the theorem that the sines and tangents
of very small arcs may be regarded as equal to the arcs themselves.
Since, in using the trigonometric functions, the radius of the circle
is taken as unity, an arc must be expressed in terms of the unit
radius when it is to be used in place of its sine or tangent. Now, it
is shown in trigonometry that the unit radius is equal to 57. 2958 or
3437'. 747 or 206 264". 8. Hence we must divide the number of
angular units in the angle by the corresponding one of these coef-
ficients to obtain the length of the corresponding arcs in unit
radius. Now,
log 3437. 747 = 3.5363
co-log 6.4637
which may be added instead of subtracting the logarithm.
To find the cosine of an angle very near 90, we find the sine of
its complement, which will then be a very small angle, positive or
negative.
EXERCISES.
Find to four places of decimals:
1. log sin 22'. 73;
2. log sin 1 1M2;
3. log cos 90 0'.78;
4. log tan 88 59'. 35;
5. log cot 90 28'. 76;
6. log cos 89 22'.23;
7. log sin 0'.25.
If an angle corresponding to a given sine or tangent is required,
the rule is:
From the given log sine or tangent subtract 6.4637 or add 3.5363.
The result is the logarithm of the number of minutes.
Of course this rule applies only to angles less than 2, in the
value of which only tenths of minutes are required.
40 LOGARITHMIC TABLES.
EXERCISES.
Find a from:
1. log sin a = 7.2243; 3. log tan a = - 3.8816;
2. log cot a = 2.8816; 4. log cos a = 6.9218.
When the small angle is given in seconds. Although the com,
puter may take out his angles to tenths of minutes, cases often arise
in which a small angle is given in seconds, or degrees, minutes, and
.seconds, and in which the trigonometric function is required to five
decimals. In this case the preceding method may not always give
= accurate results, because the arc and its sine or tangent may differ by
.a greater amount than the error we can admit in the computation.
Table III. is framed to meet this case. The following are the
quantities given:
In the second column : The argument, in degrees and minutes, as
already explained for Table IV.
In the first column : This argument reduced to seconds. From
this column the number of seconds in an arc of less than 2, given in
degrees, minutes, and seconds, may be found at sight.
Example. How many seconds in 1 28' 39' ? In the table, before
1 28', we find 5280*, which being increased by 39" gives 5319", the
number required.
Col. 3. The logarithm of the sine of the angle. This is the same
as in Table IV.
Col. 4. The value of log sine minus log arc; that is, the difference
between the logarithm of the sine and the logarithm of the number
of seconds in the angle.
Col. 5. The same quantity for the tangent.
Cols. 6 and 7. The complements of the preceding logarithms, dis-
tinguished by accents.
The use of the tables is as follows.
To find the sine or tangent of an angle less than 2:
Express the angle in seconds ~by the first two columns of the table.
Write down the logarithm in column 8 or column T, according as
the sine or a tangent is required.
Find from Table I. the logarithm of the number of seconds.
Adding this logarithm to S or T, the sum will be the log sine or
log tangent.
Example. Find log sin 1 2' 47'.9.
8, 4.68555
1 2' 47*. 9 = 3767*.9; log, 3.576 10
log sin 1 2' 47*.9, 8.261 65
WHEN THE FUNCTION IS VERY SMALL OR GREAT. 4J
To find the arc corresponding to a given sine or tangent:
Find in the column L. sin. the quantity next greater or next
smaller than the given logarithm.
Take the corresponding value of S' or T' according as the given
function is a sine or tangent, and add it to the given function.
The sum is the logarithm of the number of seconds in the required
angle.
Example. Given log tan x = 8.401 25, to find x.
log tan x, 8.401 25
T', 5.31433
logz, 3.71558
x = 5194'. 9 = 1 26' 34'. 9, from col. 2.
EXERCISES.
Find: 1. log sin 20' 20'.25;
2. log tan 0' 1'.2273;
3. log sin 1 59' 22'. 7;
4. log tan 1 0'59'.7.
Find x from:
1. log tans = 8.42796;
2. log tan x = 7.42796;
3. log tan x = 6.427 96;
4. log sin x = 5.35435;
5. log sin x = 4.226 19;
6. log sin x = 8.540 78.
When the cosine or cotangent of an angle near 90 or 270 is re-
quired, we take its difference from 90 or 270, and find the comple-
mentary function by the above rules.
Remark. The use of the logarithms of the trigonometric func-
tions is so much more extensive than that of the functions themselves
that the prefix "log" is generally omitted before the designation of
the logarithmic function, where no ambiguity will result from the
omission.
TABLE V.
NATURAL SINES AND COSINES.
19. This table gives the actual numerical values of the sine and
cosine for each minute of the quadrant.
To find the sine or cosine corresponding to a given angle less than
45, we find the degrees at the top of a pair of columns and the
minutes on the left.
In the two columns under the degrees and in the line of minutes
we find first the sine and then the cosine, as shown at the head of
the column.
A decimal point precedes the first printed figure in all cases, ex-
cept where the printed value of the function is unity.
If the given angle is between 45 and 90, find the degrees at the
bottom and the minutes at the right.
Of the two numbers above the degrees, the right-hand one is the
sine and the left-hand one the cosine.
For angles greater than 90 the functions are to be found ac-
cording to the precepts given in the case of the logarithms of the
sines and tangents.
TABLE VL
ADDITION AND SUBTRACTION LOGARITHMS.
20. Addition and subtraction logarithms are used to solve the
problem: Having given the logarithms of two numbers, to find the
logarithm of the sum or difference of the numbers.
The problem can of course be solved by finding the numbers
corresponding to the logarithms, adding or subtracting them, and
taking out the logarithm of their sum or difference. The table
under consideration enables the result to be obtained by an abbrevi-
ated process.
I. Use in addition. The principle on which the table is con-
structed may be seen by the following reasonings. Let us put
8=*a + b,
a and b being two numbers of which the logarithms are given. "We
shall have
putting, for clearness, x = -.
We then have
log S = log a + log (1 + x).
Since log a and log b are both given, we can find log x from the
equation
log x = log b log a y
which is therefore a known quantity.
Now, for every value of log x there will be one definite value of
each of the quantities x, 1 + #> and log (I-{- x). Therefore a table
may be constructed showing, for every value of log x, the correspond-
ing value of log (1 -f- x).
Such a table is Table VI.
The argument, in column A, being log x, the quantity B in the
table is log (1 -j- x).
Example, log 0.25 = 9. 397 94.
Entering the table with A = 9.397 94, we find
J3 = 0.096 91,
which is the logarithm of 1.25.
44 LOGARITHMIC TABLES.
Therefore, entering the table with log x as the argument, we
take out log (1 -}- x), which added to log a will give log S.
We have therefore the following precept for using the table in
addition:
Take the difference of the two given logarithms.
Enter the table with this difference as the argument A, and take
out the quantity B.
Adding B to the subtracted logarithm, the sum will be the required
logarithm of the sum.
It is indifferent which logarithm is subtracted, but convenience
in interpolating will be gained by subtracting the greater logarithm
from the lesser increased by 10. The number B will then be added
to the greater logarithm.
Example. Given log m = 1.62974, log n = 2.203 86 ; find,
log (m + n).
The required logarithm is found in either of the following two
ways:
Jog m, 1.629 74 (1) log , 0.676 76 (4)
log 'n, 2.203 86 (2) log m, 1.629 74 (1)
B, 0.102 64 (4) log , 2.203 86 (2)
A = log w -r- n, 9.425 88 (3) log n -=- ra, 0.574 12 (3)
log ( m + n ), 2.306 50 (5) log (m + ), 2.306 50 (5)
The figures in parentheses show the order in which the numbers
are written.
EXERCISES.
Log a and log b having the following values, find log (a -f- b).
1. log a = 1.700 37; log b = 0.921 69.
2. log a = 0.624 60; log b = 9.881 26.
3. log a = 9.791 86; log b = 9.322 09.
4. log a = 1.601 62; log I = 1.306 06.
5. log a = 0.792 90; log b = 9.221 27.
6. log a = 0.601 32; log b = 9.001 68.
7. log a = 4.796 43; log b = 3.981 86.
II. Use in subtraction. The problem is, having given log a and
log I, to find the logarithm of
D - a - b.
Wehave
ADDITION AND SUBTRACTION LOGARITHMS. 45
Since log -r is found by subtracting log b from log a, if we can
find log IT- l) from log j-, the problem will be solved.
From the construction of the table already explained, if we have
we must have
We now have the following precept for subtraction:
Subtract the lesser of the given logarithms from the greater.
Enter the table so as to find the difference of the logarithms in the
numbers B of the table.
Add the corresponding value of A to the lesser of the given loga-
rithms. The sum will be the logarithm of the difference.
. Example. Find log (n m) in the example of the preceding
section.
log n, 2.203 86 (1)
log m, 1.62974 (2)
^,0.43945 (4)
log = B, 0.57412 (3)
m
log (n - m), 2.069 19 (5)
EXERCISES.
Find the logarithms of the differences of the quantities a and b
in the preceding section.
Remark. In the use of addition and subtraction logarithms,
the precepts apply to numerical sums and differences, without
respect to the algebraic signs of the quantities. For example, the
algebraic difference between -f- 1473 and 29 462 is to be found by
addition, and the algebraic sum of a positive and negative quantity
by subtraction.
Case where the quotient is large. Near the end of the table, A
and B become nearly equal; the structure of the table is therefore
changed so as to simplify its use. It is evident that if b is very
small compared with a, the logarithms of a -f- b and a b will
not differ much from the logarithm of a itself. Hence, in this case,
we shall have smaller numbers to use if we can find the quantity
which must be added to log a to give log (a -f- b), or subtracted from
46 LOGARITHMIC TABLES.
log a to give log (a 5). Now, the equations already written give,
when a > b, log a = log ft -{- A,
log (a + 1) = log b + ;
whence, by subtraction,
log (a + b) log a = B A,
or log (a + b) = log a + - A. (with Arg. ^)
We find in the same way,
log (a - b) = log a - (B - A), (with Arg. B)
Now, whenever log a log ~b is greater than 1.65, we shall find
it more convenient to take out B A from the table than either
A or B. We notice that the last two figures of B in this part of
the table vary slowly, and we need only attend to them in interpolate
ing. For instance, in the horizontal line corresponding to A = 1.66
we find:
for A = 1.660 00; B = 1.669 40; B - A = .009 40;
.66100; .67038; .00938;
.66200; .67136; .00936;
.66300; .67233; ,00933;
.66400; .67331; .00931;
.66500; .67429; .00929;
etc. etc. etc.
The interpolation of B A is now very easy whether the quan-
tity given is A or B. We note that B A has but three significant
figures, of which the first is found in column zero, and the other two
are the last two figures of B as printed.
As an example, let us find log (a + b) from
logfl = 2.79163
log = 1.12819
A = 1.66344
Entering the table with this value of A, we find by column
that B A falls between .009 40 and .009 19. Following the hori-
zontal line A = 1.66 to column 3 and interpolating the last two
figures between 33 and 31 for .44, with the difference 2, we find
B - A = .00932
Then log a = 2. 791 63
log(0 + b) = 2.80095
Next, if log (a b) is required, we have to find the difference
1.663 44 in the part B of the table. We find in the table:
for B = 1.662 55; B - A = .009 55;
for B = 1.663 53; B - A = .009 53.
ADDITION AND SUBTRACTION LOGARITHMS. 47
Therefore
for B = 1.663 44; B - A = .009 53.
Subtracting this from log a, we have
log (a - b) = 2.78210.
EXEECISES.
^'nd log (a + ) and log (a ) from:
8. log a = 0.367 02; log b = 8.462 83.
9. log a = 0.001 26; log b = 8.329 07.
10. log a = 2.069 23; log b = 0.110 85.
11. log a = 5.807 35; log b = 3.83809.
For values of A and B greater than 2.00, the table is so arranged
that no interpolation at all is necessary. The computer has only to
find what value of A or B given in the table comes nearest his value
of log a log b and take the corresponding value of B A. He
must remember that column A is to be entered for addition, and B
for subtraction.
In this part of the table A and B are given to fewer than five
decimals; because five decimals are not necessary to give B A with
accuracy. The nearer the end of the table is approached, the fewer
the decimals necessary in taking the difference.
Example. Find log (a -f b) and log (a b) from
log a = 1.265 32
log b = 9.22230
log a log, 2.04302
Entering column A with this difference, we find the nearest tabu-
lar value of A to be 2.0425, to which corresponds B A = .003 92.
Hence
log (a + b) = 1.265 32 -f .003 92 = 1.269 24.
Entering column B with the same difference, we find B A SB '
.00395; whence
log (a - b) = 1.265 32 - .003 95 = 1.261 37.
EXERCISES.
Find log (a -f- b) and log (a b) from:
1. log a = 4.069 05; log b = 2.001 32.
2. log a = 3.926 93; log b = 1.201 59.
3. log a = 3.061 64; log b = 0.126 15.
4. log a = 1.22C 68; log b = 7.321 56.
5. log a = 0.693 17; log 5 = 6.010 23.
6. log a = 2.306 20; log b = 7.023 01.
48 LOGARITHMIC TABLES.
Case of nearly equal numbers. Near the beginning of the table
the reverse is true: it is not possible to find A with accuracy to five
places of decimals. But here the value of A taken from the tables,
though it be found to only two, three, or four places of decimals,
will give as accurate a result as the computation of a and b to five
places will admit of. Let us suppose, for example, that we have to
find log (a b) from
log a = 9.883 15
log b = 9.88296
B = 0.00019
We find ^4 = 6.64-10;
whence log (a b) = 6.52 10.
We note that the value of A may be 6.63 or 6.65 as well as 6.64,
so that the result cannot be carried beyond two decimals. To show
that these two are as accurate as the work admits of* we find the
natural numbers a and b from Table I.
a= 0.76410
b = 0.76377
a-b = 0.00033
Since a b has but two significant figures, and the first of these
is less than 5, two figures in the logarithm are all that can be
accurate.
TABLE YIL
SQUARES OF NUMBERS.
21. By means of this table the square of any number less than
1000 may be found at sight, and that of any number less than 10 000
by a simple and easy interpolation.
The first page gives the squares of the first 100 numbers, which
it is often convenient to have by themselves.
On the second and third pages (98 and 99) the hundreds of the
number to be squared are found at the tops of the several columns,
and the tens and units in the left-hand column. The first three or
four figures of the square are in the column under the hundreds,
and opposite the tens and units, and the last two figures on the right
of the page after the column 9 + +
Examples. The square of 634 is 401 956;
" " 329 " 108241;
" " 265 " 70225;
" " 153 " 23409;
999 " 998001.
The same table may be used for any number of three significant
figures by attention to the position of the decimal-point. Thus:
51100 9 = 2611210000;
511 s = 261121;
51.1' = 2611.21;
5.11' = 26.1121;
0.511 9 = 0.261121.
When there are four significant figures, an interpolation may be
executed in several ways. If n be the nearest number the square of
^hich is found in the table, and h the excess of the given number
over this, so that n -{ his the number whose square is required, we
shall have
where N = n + h 9 the given number.
50 LOGARITHMIC TABLES.
We may therefore find the square of 257.4 in the following way:
257 s = 66 049
514.4 X .4 = 205.76
(257. 4) 2 = 66254.76
To find the square of 9037 we proceed thus:
9037
9030 a = 81 540 900
18067 X 7 = 126469
9037 a = 81667369
In many cases only one more figure will be required in the square
than in the given number. The square can then be interpolated with
all required accuracy by the differences, the last two figures of which
are found in the last column of the table, while the remaining figures
are found by taking the difference between two consecutive numbers
in the principal column.
To return to the last example, we find the difference between
257 a and 258 3 to be 515, the first figure being the difference between
660 and 665, and the last two, 15, in the last column. Then
257 3 = 66 049
515 X 0.4 = 206
(257. 4) a = 66255
which is correct to the nearest unit.
It will be remarked that the two methods are substantially the
same when only five figures are sought in the result. The substantial
identity rests upon the general theorem that
The difference of the squares of two consecutive numbers is equal
to the sum of the numbers.
We prove this theorem thus:
(n + l) a - n* = 2n + 1 = n + (n + 1).
When the tabular difference is taken in the way already described,
it will often happen that the difference between the numbers in the
columns of hundreds is to be diminished by unity. Thus, although
4173 4160 = 13, the difference between 645 s and 646 2 is not 1391,
but 1291. These cases are noted by the asterisk after the number in
the last column.
The squares of numbers of more than four figures may be found
in the same way, but in such cases it will generally be easier to use
logarithms than the table of squares.
TABLE VIII.
TO CONVERT HOURS, MINUTES, AND SECONDS
INTO DECIMALS OF A DAY, AND VICE VEKSA.
. The familiar method of solving this problem is to convert
the seconds into decimals of a minute, and the minutes into decimals
of an hour, by dividing by 60, and then the hours into decimals of a
day by dividing by 24. The reverse problem is solved by multiply-
ing by 24, 60,- and 60.
Table VIII. enables us to perform these operations without divi-
sion. Column D gives each hundredth of a day, but its numbers may
also be regarded as ten thousandths or millionths of a day, according
to which of the following three columns is used. In column H.M.S.
are found the hours, minutes, and seconds corresponding to these
hundredths. In the next column is one hundredth of column H. M. S. 9
or the minutes and seconds in the number of ten thousandths of a
day in column D. Finally, column ' ' shows the number of
iuu
seconds in the number of millionths of a day found in column D.
Example. To convert O d .532 946 into hours, minutes, and seconds.
O d .53 = 12 h 43 m 12 s
.002 9 = 4 m 10 8 .56
.000046= 3 8 .97
O d .532 946 = 12 h 47 m 26 8 .53
It will be seen that we divide the figures of the given decimal of
a day into pairs, and enter the three columns of time with these
three pairs in succession.
If seven decimals are given, we may interpolate the last number,
as in taking out a logarithm.
Example. Convert O d .050 762 7.
O d .05 = l h 12 m O 8
.000 7 = l m s . 48
.000062 = 5 s . 36
.0000007 = .7X.08= 0-.06
l h 13 m 5 8 .90
52 LOGARITHMIC TABLED
In practice the computer will perform the interpolation mentally,
adding .7 X .08 = .06 to the number 5.36 of the table in his head,
and writing down 5 s . 42 as the last quantity to be added.
EXERCISES.
Convert into hours, minutes, and seconds:
1. O d .2030792;
2. O d . 783 605 8;
3. O d .0102034;
4. O d . 990 990 9.
To use the table for the reverse operation, we proceed as in the
following example:
It is required to convert 17 h 29 m 30 s . 93 into decimals of a day.
Looking in the table, we find that the required decimal is between
0.72 and 0.73. Hence the first two figures are 0.72, the equivalent
of 17 h 16 m 48 s . Subtracting the lat- 1711 ggm 30*. 93
ter from the given number, we 0.72 = 17 h 16 m 48 s
have a remainder 12 m 42 8 .93, to be 12 m 42 s . 93
,,. . . H.M.S. _. -0088 = 12 m 40 s . 32
sought for in column m ~. This >000 03 2 = " ~2^61
gives 88 as the next two figures. Subtracting the equivalent of
.0088 or 12 m 40 8 .32, we have left 2 s . 61, which we are to seek in
TT -r nr
column ' ' '. We find the corresponding number of column D to
100
be 302. Hence
17 h 29 m 30 8 .93 = O d . 728 830 2.
In solving this problem the computer should be able, after a little
practice, to perform the subtractions and carry the remainders men-
tally, thus saving himself the trouble of writing down the numbers.
EXERCISES.
Take the answers obtained from the four preceding exercises,
subtract each result from 24 h O m B , change the remainder to deci-
mals of a day, and see if when added to the decimals of the preceding
exercises the sum is l d . 000 000 0, as it should be.
TABLE IX.
TO CONVERT TIME INTO ARC, AND VICE VERSA.
23. In astronomy the right ascensions of the heavenly bodies
are commonly given in hours, minutes, and seconds, the circumfer-
ence being divided into 24 hours, each hour into 60 minutes, and
each minute into 60 seconds.
Since 360 r = one circumference,
ve have l h = 15;
l m = 15';
1 s = 15';
the signs h , m , and 8 indicating hours, minutes, and seconds of time.
Hence we may change time into arc by multiplying by 15, and
arc into time by dividing by 15, the denominations being changed in
each case. Table IX. enables us to do this by simple addition and
subtraction by a process similar to that employed in changing hours,
minutes, and seconds into decimals of a day.
To turn time into arc, we find in the table the whole number of
degrees contained in the time denomination next smaller than the
given one, and subtract the former time denomination from the
latter.
Next we find the minutes of arc corresponding to the given time
next smaller than the remainder, and again subtract.
Lastly we interpolate the seconds corresponding to the second
remainder.
Example. Change 15 h 29 m 46 8 .24 to arc.
Given time, 15 h 29 m 46 8 .24
The table gives 232 = 15 h 28 m
Remainder, l m 46 8 .24
The table gives 26' = l m 44 8
Remainder, 2 s . 24 = 33'. 6
Hence
15 h 29 m 46 8 .24 = 232 26' 33'. 6.
54 LOGARITHMIC TABLES.
The computer should be able to go through this operation with-
out writing down anything but the result.
The operation of changing arc into time is too simple to require
description, but it is more necessary to write down the work.
EXERCISES.
Change the following times to arc, and then check the results by
changing the arcs into time and seeing whether the original times
are reproduced:
1. 7 h 29 m 17 8 .86;
2. O h 4 m 8 .25;
3. 12 h 4 m s . 25;
4. 13 h 48 m 16 9 .40;
5. 19 h 7 m 59 8 .92.
TABLE X.
TO CONVERT MEAN TIME INTO SIDEREAL TIME,
AND SIDEREAL INTO MEAN TIME.
24. Since 365 solar days = 366^- sidereal days (very nearly),,
any period expressed in mean time may be changed to sidereal time-
by increasing it by its - part, and an interval of sidereal time-
uDO./oO
may be changed to mean time by diminishing it by its - part..
ODD. -CO
The first part of the table gives, for each 10 minutes of the argu-
ment, its Q part, by which it is to be increased. The second:
part of the table gives the O^FITH P ar ^ f the argument.
The small table in the margin shows the change for periods of
less than 10 minutes.
Example 1. To change 17 h 48 m 36 s . 7 of mean time to sidereai
time.
Given mean time, 17 h 48 m 36 s . 70
Corr. for 17 h 40 m , 2 m 54M3
Corr. for 8 m 37', 1 8 .41
Sidereal time, 17 h 51 m 32 s . 24
Ex. 2. To change this interval of sidereal time back to mean
time.
Corr. for 17 h 50 m , - 2 m 55". 29
Corr. for l m 32% - 8 .25
2 m 55 8 .54
Sidereal time, 17 h 51 m 32 8 .24
Mean time, 17 h 48 m 36 8 .70
EXERCISES.
Change to sidereal time:
1. 3 h 42 m 36". 5 m. t.; 3. 22 h 3 m 5 .61 m. t*
2. 18 h 46 m 29 8 .82 " 4. O h l m 12 B .55 "
Change to mean time:
5. O h 7 m 16 8 .3 sidereal time;
6. 22 h 17 m 29 8 .65 "
56 OF INTERPOLATION.
OF DIFFERENCES AND INTERPOLATION.*
25. General Principles.
"We call to mind that the object of a mathematical table is to
enable one to find the value of a function corresponding to any value
whatever of the variable argument. Since it is impossible to tabulate
the function for all values of the argument, we have to construct the
table for certain special values only, which values are generally equi-
distant. For example, in the tables of sines and cosines in the
present work the values of the functions are given for values of the
argument differing from each other by one minute.
The process of finding the values of functions corresponding to
values of the argument intermediate between those given is called
interpolation.
We have already had numerous examples of interpolation in its'
most simple form; we have now to consider the subject in a more
general and extended way.
In the first place, we remark that, in strictness, no process of
interpolation can be applicable to all cases whatever. From the
mere facts that
To the number 2 corresponds the logarithm 0.301 03,
" " " 3 " " " 0.477 12,
we are not justified in drawing any conclusion whatever respecting
"the logarithms of numbers between 2 and 3. Hence some one or
more hypotheses are always necessary as the base of any system of
interpolation. The hypotheses always adopted are these two:
1. That, supposing the argument to vary uniformly, the function
varies according to some regular law.
2. That this law may be learned from the values of the function
given in the table.
These hypotheses are applied in the process of differencing, the
* The study of this subject will be facilitated by first mastering so much of
it as is contained in the author's College Algebra, 299-302.
It is also recommended to the beginner in the subject that, before going
over the algebraic developments, he practise the methods of computation
according to the rules and formulae, so as to have a clear practical understand
ing of the notation. He can then more readily work out the developments.
GENERAL PRINCIPLES. 57
nature of which will be seen by the following example, where it is
applied to the logarithms of the numbers from 30 to 37:
Function. A' 4" A"< A"
log 30. 1.47712
"
31. 1.491 36 ~ - 45 ,
" 32. 1.505 15 T JoS - 43 J * + 2
" 33. 1.518 51 J i9Q7 - 39 J * - 8
34. 1.531 48 J J*JJ _ 38 + + 1
35. 1.544 07 + }f* _ 36 + * + 1
" 36. 1.556 30 "tifS - 33 + '
"37. 1.568 20 ~*
The column A' gives each difference between two consecutive
values of the function, formed by subtracting each number from that
next following. These differences are called first differences.
The column A" gives the difference between each two consecu-
tive first differences. These are called second differences.
In like manner the numbers in the succeeding columns, when
written, are called third differences, fourth differences, etc.
Now if, in continuing the successive orders of differences, we find
them to continually become smaller and smaller, or to converge to-
ward zero, this fact shows that the values of the functions follow a
regular law, and the first hypothesis is therefore applicable.
In order to apply interpolation we must then assume that the
intermediate values of the function follow the same law. The truth
of this assumption must be established in some way before we can
interpolate with mathematical rigor, but in practice we may suppose
it true in the absence of any reason to the contrary.
26. Effect of errors in the values of the functions. In the pre-
ceding example it will be noticed that if we continue the orders of
differences beyond the fourth, they will begin to increase and become
irregular. This arises from the imperfections of the logarithms,
owing to the omission of decimals beyond the fifth, already described
in 11.
When we find the differences to become thus irregular, we must
be able to judge whether this irregularity arises from actual errors in
the original numbers, which ought to be corrected, or from the small
errors necessarily arising from the omission of decimals*
The great advantage of differencing is that any error, however
small, in the quantities differenced, unless it follows a regular law,
will be detected by the differences. To show the reason of this, we
investigate what effect errors in the given functions will have upon
the successive orders of differences.
68 OF INTERPOLATION.
THEOREM. The differences of the sum of two quantities are equal
to the sums of their differences.
General proof. Let
/ /, /, etc., be one set of functions;
//, /,',/,', etc., another set.
fi +//> /a + //> /s + /'> etc., wil1 then be their sums.
In the first of the following columns we place the first differences
of/, in the second those of/', and in the third those of / + /', each
formed according to the rule :
etc. etc. etc.
It will be seen that the quantities in the third column are the
sums of those in the first two.
NUMERICAL EXAMPLE.
/ A f A' f+f A
lt+n l + i ll + n
*n ~r H c + 3 t-f. + 14
_ 5 ?- 51 10 + 4 9~ 47
We see that the third set of values of A r follow the theorem.
Because the second differences are the differences of the first, the
third the differences of the second, etc., it follows that the theorem
is true for differences of any order.
Now when we write a series of functions in which the decimals ex-
ceeding a certain order are omitted, we may conceive each written num-
ber to be composed of the algebraic sum of two quantities, namely:
1. The true mathematical value of the function.
2. The negative of the omitted decimals.
Example. In the preceding collection of logarithms, since the
true value of log 30 is 1.477 121 3 . . . , we may conceive the quantity
written to be
1.477 12 = log 30 - .000 001 3 ____
Hence the differences actually written are the differences of the
true logarithms minus the differences of the errors. Now suppose
the errors to be alternately + 0.5 and 0.5 = the point marking
off the last decimal. Their differences will then be as follows:
/' J' A" 4'"
- 0.5 , x + 2 __ 4
+ 0.5 i-2
- 0.5 ~ \ + 2 '
+ 0.5 " L - 2 "
etc. etc. etc. etc.
GENERAL PRINCIPLES. 59
It is evident that the wth order of differences of the errors are
equal to 2"- 1 . Hence, in this case, if the nth order of differences
of the true values of the function were zero, still, in consequence of
the omission of decimals, the actual differences of the nth order would
be2- 1 .
This, however, is a very extreme case, since it is beyond all proba-
bility that the errors should alternate in this way. A more probable
average example will be obtained by supposing a single number to have
an error of 0.5, while the others are correct. We shall then have:
f A 1 4" d'" ^ 1T ^ T
A + 0-5 o
__
o ' + 0.5
In this case the maximum value of the difference of the nth order
is 1.5 in the differences of the third order, 3 in those of the fourth,
5 in those of the fifth, etc. Its general expression is
1 n (n - 1) (n - 2) ____ (n - s -f 1)
2 1.2. 3.... s
where n is the order of differences, and
n
n
-1
* = 2
or
2
according as n is even or odd. Thus:
A' =1
.
2
>
' - * L
2
' 1
=
i;
1
3
I/
^'" = g-
" 1
=
.<
& - 1
4.
3
3.
2
* 1.
2
y
1
5.
4
5.
2
' 1.
2 "~
,
etc. etc.
This being about the average case, in actual practice the differ-
ences may be two or three times as great without necessarily imply-
ing an error greater than 0.5 in the numbers written.
We have now the following general rule for judging whether a
series of numbers do really follow a uniform law:
Difference the series until we reach an order of differences in which
the 4* and signs either alternate or follow each other irregularly.
60 OF INTERPOLATION.
If none of the differences of this order expressed in units of the
last place of decimals exceed the limit
n (n 1) . . . _. (n s + 1)
1. 2. 3 .... s
that is, the value of the largest binomial coefficient of the nth order
the given numbers may be assumed to follow a regular law, and
therefore to be correct to a unit in the last figure.
If some differences exceed this limit, their quotient by the above
binomial coefficient may be considered to show the maximum error
with which the number opposite it is probably affected.
We can thus detect an isolated error in a series of numbers with
great certainty. Suppose, for example, an error of 2 in some number
of the series. Differencing the series 0, 0, 0, 2, 0, 0, 0, we shall
find the four largest differences of the fifth order to be 10, -j- 20,
20, -[- 10, which would enable us to hit at once upon the erro-
neous number and judge of the magnitude of its error.
An error near the beginning and end of the series of numbers of
which the differences are taken cannot be detected by the differences
unless it is considerable. If, for instance, the first or last number
is in error by 1, the error of each order of differences will only be 1,
as we may easily see by the following example:
/' A' A" A'"
~ I + I - 1 etc.
It is only in those differences which are on or near the same line
as the numbers which are magnified in the way we have shown. But
at the beginning and end of the series we cannot determine theso
differences.
Examining the various tables of differences, we see that n numbers
have n 1 first differences, n 2 second differences, and so on, the
number diminishing by 1 with each succeeding order. Hence, unless
the number of given functions exceeds the index expressing the order
of differences which we have to form, no certain conclusion can be
drawn.
What is here said of the correctness of the numbers when the
differences run properly must be understood as applicable to isolated
errors only. If all the numbers were subject to an error following a
regular law, this error would not be detected by the differences be-
cause, from the nature of the case, the latter only show deviations
from some regular law.
FUNDAMENTAL FORMULA. 61
27. Fundamental Formulae of Interpolation.
We suppose a series of numbers to be differenced in the way already
shown, and the various differences to be designated as in the follow-
ing scheme, which is supposed to be a selection from a series preceding
and- folio wing it.
Function. 1st Diff. 2d Diff. 3d Diff. 4th Diff. 5tn Diff.
a /f' 2 /f"' 3 yfv
A -I A ,, A _, J-.j
' "
,
3
A'",
etc. etc. etc. etc. etc. etc.
It will be seen that the lower indices are chosen so as to
on which line a difference of any order falls. Thus all quantities
with index 2 are on one horizontal line, those with index |- = 2 are
half a line below, etc. This notation is a little different from that
used in algebra, but the change need not cause any confusion. ,
It is shown in algebra that if n be any index, we have
the notation being changed as in the preceding scheme.
Now the fundamental hypothesis of interpolation is that this
iormula, which can be demonstrated only for integral values of ^>k
true also for fractional values; that is, for values of the function u
between those given in the table or in the above scheme. We there-
fore suppose this formula to express the value of the function u for
any value of n between and 1. r . , .
For values between -f 1 and -f 2 we have only to increase the
indices, of u and its differences by unity, thus:
. + etc,,
and by supposing n to increase from to 1 in this formula we shall
have values of u from u l to w a .
62 OF INTERPOLATION.
Increasing the indices again that is, applying our general foi
mulae to a row of quantities one line lower we shall have
etc.
,
The equation (a) is known as Newton's formula of interpolation.
28. Transformations of the Formula of Interpolation.
In the equation (a) and those following it, the formula of inter-
polation is not in its most convenient form. We shall therefore
transform it so that the differences employed shall be symmetrical
with respect to the functions between which the interpolation is to
be made.
In working these transformations we shall suppose the sixth and
following orders of differences to be so small as not to affect the
result. These differences being supposed zero, any two consecutive
fifth differences may be supposed equal.
First transformation. Let us first find what the original formula
(a) will become when, instead of using the series of differences
J'*, J" lf ^'"j, A^\, etc.,
we use
J'l, J" , J'"i, J* f etc.
To effect the transformation we must find the values of the first
series of differences in terms of the second, and substitute them in
the formula (a).
We find, by the mode of forming the differences,
for which, because we suppose the values of J T to be equal, we may put
#\ = *\ + 8J;
/}', = A\.
Making these substitutions in (a), we have
=. + J' t + * (* ~ l) (A>\ + J'",)
(-!) (-*) ,,
1.2.3.4.5 **
TRANSFORMATIONS OF FORMULAE. 63
Reducing by collecting the coefficients of equal differences, we find
- = //'* + n ( n ~ l l A" + ( + l) (-*) A ,n.
**n **o !l> t i 12 oi 123
( + l)n(*-l)(-2)
1.2.3. 4
. v .
1.2.3.4.5 **
Second transformation. Next, instead of the series of this last
formula, (J),
J't, J"., J'" b ^ , etc.,
let us use
/J'_ t , J"., J"'_ 4 , J\, etc.
To effect this transformation we substitute in (S) for d\, 4"i, etc.,
The series (b) then changes into
*-^^*
- . i
1.2.3.4
1.2.3.4.5 '-*'
tV<f transformation. Stirling's formula. We effect a third
transformation by taking the half sum of the equations (b) and (c),
Which gives us a formula perfectly symmetrical with respect to the
lines of differences, namely,
*-^4^+^
*>'-!) n(it--l)(n'-4)k.H-^ . ^ ^
1.2.3.4 ^ 1.2.3.4.5 2 T 0.| W
which is known as Stirling 9 s formula of interpolation.
It will be seen that we have put
n* - I for (n + 1) (n - 1),
w 9 - 4 for (n + 2) (w - 2),
etc. etc.
Fourth transformation. In the equation (5), instead of the series
of differences
J'h J"., J'",, J"., etc.,
let us use
A' k , A', 4"', & etc.
64 OF INTERPOLATION.
-. To effect this we put
J" = A'\ - J'" i5
JiV o = Jl Vi _ JV^
Making these substitutions in (), it becomes
* - *J' 11 .f" ^
-- , iv
1.2.3.4" ' '
- . v
1.2.3.4.5
transformation. BesseVs formula. Let us take half the
sum of the equations (e) and (b). We then have
1.2.3.4.5
which is commonly known as BesseVs formula of interpolation, and
which is the one most convenient to use in practice.
, In applying this formula to find a value of the function inter-
mediate between two given values, we must always suppose ,^ the
index to apply to the given value next preceding that to be found,
and the index 1 to apply to that next following. The quantity n
will then be a positive proper fraction.
29. Example of interpolation to halves. If we increase the loga-
rithms of 30, 31, etc., already given, by unity, we shall have the
logarithms of 300, 310, 320, etc. It is required to find, by interpola-
tion, the logarithms of the numbers half way between the given ones
(omitting the first and last), namely, the logarithms of 315, 325, 335,
etc.
Here, the required quantities depending upon arguments half way
between the given ones, we have n = -J, and the values of the Bessel-
ian coefficient, so far as wanted, are
n (n - 1) _ !_
2 " 8 J
log (a, - 5) = log a, - -
TRANSFORMATIONS OF FORMULA. 65
The subsequent terms are neglected, being insensible. So, if we
put a and a l for any consecutive two of the numbers 300, 310, etc.,
we have
(*)
where we put A for that first difference between a and a lt
These two formulae are two expressions for the same quantitj
because a -f- 5 = a l 5. They are both used in such a way as to
provide a check upon the accuracy of the work. For this purpose we
compute the two quantities
log (a. + 5) - log a = -^A\ - - 1, 1
1 A" A" f W
log a, - log (..+ 5) = -J'i + - 1 1. J
The most convenient and expeditious way of doing the work is
shown in the accompanying table, where we give every figure which
it is necessary to write, besides those found on p. 57. The following
is the plan of computation:
Ko. Log. Difl. ^', *'" + '"'. ^+X
310
2.49136
315
.498 31
RS4.
+ 689.5
- 5.5
-44
320
.50515
UOTt
325
.51188
aao
668.0
- 5.1
- 41
330
.51851
DOO
r* Ef o
335
.52504
bOo
/? A 4
648.5
-4.8
- 38
340
.53148
b44
AQ/f
345
.53782
OO4:
629.5
- 4.6
- 37
350
.54407
/1-f />
355
360
.55023
2.55630
616
607
+ 611.5
- 4.3
-34
We compute the right-hand column by the formula
using the values of A given in the scheme, p. 57.
This mode of computing the half sum of two numbers which are
nearly equal is easier than adding and dividing by 2.
In the next two columns to the left, the sixth place of decimals
66 OF INTERPOLATION.
is added in order that the errors may not accumulate by the addition
of several quantities. This precaution should always be taken when
the interpolated quantities are required to be as accurate as the given
ones.
The fourth column from the right is formed by adding and sub-
tracting the numbers of the second and third columns according to
the formula (k). The additional figure is now dropped, because no
longer necessary for accuracy. The numbers thus formed are the
first differences of the series of logarithms found by inserting the
interpolated logarithms between the given ones, as will be seen by
equation (&).
We write the first logarithm of the series, namely,
log 310 = 2.49136,
and then form the subsequent ones by continual addition of the dif-
ferences, thus:
log 315 = log 310 + 695;
log 320 = log 315 + 684;
log 325 = log 320 + 673;
etc. etc. etc.
If the work is correct, the alternate logarithms will agree with the
given ones in the former table.
The continuance of the above process for a few more numbers,
say up to 450, is recommended to the student as an exercise.
3O. Interpolation to thirds. Let us suppose the value of a
quantity to be given for every third day, and the value for every
day to be required. By putting n = -j- and applying formula (/) to
each successive given quantity, we shall have the value for each day
following one of those given, and by putting n = } we shall have
values for the second day following, which will complete the series .
But the interpolation can be executed by a much more expeditious
process, which consists in computing the middle difference of the
interpolated quantities and finding the intermediate differences by a
secondary interpolation.
Let us put
/ / / f , etc., the given series of quantities;
/ /u f*> /> /4> etc -> tne required interpolated series;
A' t A", etc., the first differences, second differences, etc., of the
given series;
$', ", etc., the first differences, second differences, etc., of the
interpolated series.
TRANSFORMATIONS OF FORMULAS. 67
We may then put
/,/, = ^'* (in the given series);
/,-/.-*')
/,/! = ^ 'f > (in the interpolated series).
/.-/.= *Y)
We shall then have
<*'* + <*', + f| = 4'*.
The value of /i / = b\ is given by putting n = $ in the Bes-
selian formula (/). Thus we find
,, 1 1 J".+ J". , 1 .,
** = 3 J *~9 2 + i62 J
^^ + ^_J_
r 243 2 1458
Putting w = |, we have the value of/, ~/ , that is, of
Thus we find
5 ^ t +^. 1
r 2 r !458
Subtracting these expressions, we have
^ = 3 L ^-^'"*
which is most easily computed in the form
We see that the computation of tf'f, the middle difference of the
interpolated quantities, is much simpler than that of <?V It will
therefore facilitate the work to compute only these middle differ-
ences, and to find the others by interpolation.
This process is again facilitated, in case the second differences are
considerable, by first computing the second differences of the inter-
polated series on the same plan. The formulae for this purpose are
derived as follows:
Let us put
<*'! =/, -/,-
The second difference of which we desire the value is then
<*,= ?'{ - ff
The value of S\ is given by the equation
tf'j^'j-OJ'j + <?',),
OS LOGARITHMIC TABLES.
and the value of d'$ is found from that of d' by simply increasing, the
indices of the differences by unity, because it belongs to the next
lower line.
We thus find
I
^
243 a 1458
5 A\ + 4\ 1
243 2 1458
Then by subtraction,
4- (^-^
__ . I ._ _1_ _
r 243 2 1458 ^ f W *
Eeducing the first of these terms, we have
^Y-^V;M",
For the second term,
whence
^". + ^" a = ^^ + A'"\ - A '"\ = 2^", + A\,
and
^". + aj " + ^"> = 2 J + L j..
2 > ^ 2 '*
For the third term,
j'" f _ A'" k = J 1 ^.
For the fourth term, dropping the terms in d* as too small iu
practice, we may put
j". + aj". + ^'% =
g
The difference of the fifth terms may also be dropped, because
they contain only sixth differences.
Making these substitutions in the value of #" 3 , we find
OF INTERPOLATION. 69>
By this formula we may compute every third value of #", and
then interpolate the intermediate values. By means of these values,
we find by addition the intermediate values of d', of which every
third value has been computed by formula (m). Then, by continu-
ally adding the values of 6', we find those of the function/.
As an example of the work, we give the following values of the.
sun's declination for every third day of part of July, 1886, for Green*
wich mean noon:
Date. Q'sDec. A' A" A'"
1886 o / // in n it
6.. .
...22
41
9
>
16
28.
3
212.
4
9.. .
...22
8.
5
20
0.
7
207.
9
+ 4.5
12.. .
...21
57
39
9
23
28.
6
203.
4
-f 4.5
15.. .
...21
30
47,
9
26
52.
197.
7
+ 5.7
18..
21
38.
2
30
9.
7
The values of J iv are too small to hav-e any influence.
The whole work of interpolation is shown in the following table>
fhere, as before, the right-hand column is that first computed, and
gives the value of A' -faA'" according to formula (m) :
Date. o'sDec. <?' d" A' -
1886. ' " ' " "
July 6 ...... 22 41 9.2 R , , RR - 23.60
7 ...... 22 34 52.4 ' ' * * - 23.43 20 Q ,
8 ...... 22 28 12.1 ; ^'?* - 23.27 " ' 87
9 ...... 22 21 8.5 "I ft 22 - 23 - 10
10 ...... 22 13 41.9 " ] * - 22.93 ^ 2g 77
11 ...... 22 5 52.3 " *5J - 22.78 ^ ^ <77
12 ...... 21 57 39.9 " g- - 22.61
13 ...... 2149 4.9 " I *** - 22.42 2 -
14 ...... 21 40 7.5 ' I - 22. 19 " 26 52 ' 21
15 ...... 21 30 47.9 " 19M - 21.97
To make the process in the example clear, the computed differ-
ences, etc., are printed in heavier type than the interpolated ones.
It is also to be remarked that the sum of the three consecutive
values of d", formed of one computed value and the interpolated
values next above and below it, should be equal to the difference
between the corresponding computed first differences. For instance,
23".27 + 23".10 + 22".93 = 7' 49".59 - 6' 40".29.
But in the first computation this condition will seldom be exactly
fulfilled, owing to the errors arising from omitted decimals and other
sources. If the given quantities are accurate, the errors should never
70 LOGARITHMIC TABLES.
exceed half a unit of the last decimal in the given quantities, or five
units in the additional decimal added on in dividing.
To correct these little imperfections after the interpolation of the
second differences, but before that of the first differences, the sum of
the last two figures in each triplet of second differences should be
formed, and if it does not agree with the difference of the first differ-
ences, the last figures of the second difference should each be slightly
altered, to make the sum exact.
The first differences can then be formed by addition.
In the same way, the sum of three consecutive first differences
should be equal to the difference between the given quantities. If,
as is generally the case, this condition is not exactly fulfilled, the
differences should be altered accordingly. This alteration may, how-
ever, be made mentally while adding to form the required inter-
polated functions.
As an exercise for the student we give the continuance of the
sun's declination for the remainder of the month, to be interpolated
for the intermediate dates from July 15th onward:
o i n
July 21 20 27 16.5
24 19 50 49.1
27 19 11 22.7
30 18 29 4.8
Aug. 2 17 44 3.1
As another exercise the logarithms of the intermediate numbers
from 998 to 1014 may be interpolated by the following table:
Number. Logarithm.
994 2. 997 386 4
997 2.9986952
1000 3.0000000
1003 3.001 3009
1006 3.002 598
1009 3. 003 891 2
1012 3.005 180 5
1015 3. 006 466
1018 3.007 747 8
32. Interpolation to fifths. Let us next investigate the formulas
when every fifth quantity is given and the intermediate ones are to
be found by interpolation. By putting n = in the Besselian for-
mula, we shall have the value of the interpolation function second
OF INTERPOLATION. 71
following one of the given ones, and by putting n = f that third
following. The difference will be the middle interpolated first dif-
ference of the interpolated series. Putting n = in (/ ), we have
2 ., 2.3 J" + J", 2.3.1 .,
m = * c + r J'* - ^ ^ - + gr^'"*
2.3.7.8 A\ + ^ _ 2.3.7.8.1
^2.3.4.5* 2 2 a .3.4.5.5 6 **
Putting n = $, we have
, 3 . 2.3 J" + A'\ 2.3.1 .,
' "'
5
2.3.7.8 ^ 1Y + ^ lv , 8.3.2.7.1 .
h 2.3.4.5 4 2 i "2'.3.4.5.5'
The difference of these expressions, being reduced, gives
w - u\ = J'* - 125^'"* + 15625 JV *
The term in ^ y will not produce any effect unless the fifth differ-
ences are considerable, and then we may nearly always, in practice,
put $ instead of -Jfa.
The interpolated second differences opposite the given functions
are most readily obtained by Stirling's formula, (d). Putting n = \>
we have the following value of the interpolated first differences im-
mediately following a given value of the function:
2 50 6.5.25 2
24
Again, putting n = -fr, and changing the signs, we find for the
first difference next preceding a given function
50 6.5.25
24
6.5.20 -
The difference of these quantities gives the required second dif-
ference, which we find to be
72 LOGARITHMIC TABLES.
As an example and exercise we show the interpolation of loga*
rithms when every fifth logarithm is given:
Number.
Logarithm.
6'
3"
A p A"
1000
3.0000000
+ 21 661
1005
1006
3.002 166 1
.0025980
4319.2
A 0-1 A Q
-4.32
- 4.31
- 108
1007
1008
1009
1010
.003 029 5
.0034606
.0038912
3.0043214
'ioi^b. y
4310.6
4306.3
4302.0
A CIA W tV
- 4.30
- 4.30
- 4.29
- 4.28
+ 21 553
- 107
1011
.0047512
4297.7
A Cir\f*i f
- 4.27
1012
.1013
1014
1015
.0051805
.0056094
.006 037 9
3.0064660
4293.5
4289.2
4285.0
4280.8
- 4.26
- 4.23
- 4.20
-4.16
+ 21 446
+ 21 342 ~" 104
1020
3.0086002
1025
3.0107239
1030
3.0128372
1035
3.0149403
1040
3.0170333
FORMULAE
OE THE SOLUTION OF
PLANE AND SPHERICAL TRIANGLES.
REMARKS.
1. It is better to determine an angle by its tangent than by its
sine or cosine, because a small angle or an angle near 180 cannot be
accurately determined by its cosine, nor one near either 90 or 270
by its sine,
Sometimes, however, the data of the problem are such that the
angle can. be determined only through its sine or cosine. Any un-
certainty which may then arise from the source pointed out is then
inherent in the problem; e.g., if the hypothenuse and one side of a
right triangle are 0.39808 and 0.39806 respectively (sixth and follow-
ing decimals being omitted), the value of the included angle may be
anywhere between 25' and 42', no matter what method of com-
putation be adopted.
2. If the sine and cosine can be independently computed, their
agreement as to the angle will generally serve as a check on the
accuracy of the computation. If they agree, their quotient will give
the tangent.
3. It is desirable, when possible, to have a check upon the accu-
racy of the computation; that is, to make a computation which must
give a certain result if the work is right. But no check can give a
positive assurance of accuracy: all it can do is to make it more or
less improbable that a mistake exceeding a certain limit exists.
4. In the following list several formulae are sometimes given as
applicable to the same problem. In such cases, the most convenient
for the special purpose must be chosen.
PLANE TRIAXGLE8.
Notation, a, b, and c are the three sides.
A, B, and C are the opposite angles.
PLANE TRIANGLES.
Given.
Required.
s a+
a, b, c,
^,
the three
one angle.
tin A- -4 ~ \/
sides.
o
s(s a)
A, B, C,
TT A/( S ~ a ) ( S 0)(S C) m
all the
s '
angles.
tan^ = -*-
tan 45- H
s V
tan -i G
tCUl T ^
s <?
Checks: A + B + C = 180;
a b c
^ ____
sin A sin .5 sin (7
b, c,A,
B and <7,
7) /
rwo sides
the other
tan ItCR r\ ont 4- A
tu/JJ ^ ^ jj - - ^ 1 ^u i, -j ^n -
~T~ C
and the
included
angles.
!(+ C) = 90 -|^;
angle.
t7=|(^+(7)I|(^Z (7). 5
Check, as before.
a, B, C,
a sin -J (B C ) = (b c) cos -J A ;
the
rt cos - (B C) = (b -\- c) sin \A.
remaining
Having found a and (J? (7), proceed
parts.
as in the last case.
o, 5, A,
two sides
c, B, C,
the re-
sin B = sin A; (two values of B.)
and the
maining
/T _ -i QA / A 1 Z?\
O = loU {A. -f- x>j;
angle oppo-
parts.
__ sin C' sin C
site one of
sin 5 " sin ^4 "
them.
76
RIGHT SPHERICAL TRIANGLES.
Given.
Required.
a, A, B,
a, c, o,
C = 180 - (A + 5);
one side
the re-
, _ a sin ^
and any
maining
sin J. '
two angles.
parts.
a sin (7 a sin (^4 -f- J9)
sin A sin ^4
RIGHT SPHERICAL TRIANGLES.
c is the hypothenuse.
, b,
A, B, or c.
cot A = cot a sin #;
the sides
cot B = cot J sin a\
containing
cos c = cos a cos 5;
the right
sin #
CJl f^i /
angle.
bill t/ . -
Bin -4
A and c.
sin c sin A = sin a;
sin c cos ^4 = cos a sin 5;
cos c = cos a cos 5*
sin c sin ^ = sin J;
B and c
sin c cos B = sin a cos 5.
a, c,
A, B } or b.
. sin G^
sin A . -,
one side
sin c
and the hy-
cos .Z? = tan a cot C|
pothenuse.
^ cos c
C/wb C/ ~
cos a
a, A,
b, c, or B.
sin = tan a cot ^4;
one side
sin a
and the
sin c = rj
sin -4
opposite
. cos A '
r*i >i A.
angle.
bill X>
*,&,
b, c, or A.
tan b = sin tan B;
one side
tan a
and the
tan c = ^;
cos 2r
adjacent
cos ^4 = cos a sin .#.
angle.
c and A.
sin A sin c = sin a;
sin ^4 cos c = cos a cos 5;
cos A = cos a sin B.
QUADRANTAL SPHERICAL TRIANGLES.
77
i*iven.
Required.
0, B.
b and A.
sin .4 sin b = sin a sin B;
sin -4 cos # = cos B.
c,A,
a, b, or B.
sin a = sin c sin ^4;
the hypo-
tan b = tan c cos A;
thenuse
cot j5 = cos c tan A
and one
angle.
a and B.
cos a sin j5 = cos A;
cos a cos J? = sin A cos c;
sin a = sin -4 sin c.
a and b.
cos a sin b = cos -4 sin c\
cos a cos # = cos c.
A,B,
a, b, or c.
cos .4
the two
sin ^ J
angles.
, cos jB
cos b = -. -. ;
sm ^1
cos c = cot ^4 cot B.
QUADEANTAL SPHERICAL TEIANGLES.
a, 5,
the two
sides.
a, (7,
one side
and the
angle oppo-
site the
right side.
A, B, or C,
either
angle.
A, B, or b.
A and b.
A and B.
cos A =
c is the omitted side equal to 90.
C is the angle opposite this side.
cos a t
sin b'
cos
cos B =
sin a
cos C = cot
cot
sin -4 = sin a sin (7;
tan B = cos a tan (7;
cot # = tan a cos (7.
cos ^4 sin Z = cos a;
cos A cos b = sin a cos C.
sin -4. = sin a sin &
cos A sin ^ = cos a sin (7;
cos ^4 cos B = cos C.
78
QUADRANTAL SPHERICAL TRIANGLES.
Given.
one angle
and the
adjacent
side.
0, A,
one side
and the
opposite
angle.
one angle
and the
angle oppo-
site the
right side.
A,B,
two angles.
Required.
a, B, or (7.
a and B.
a and C.
b, B, or C.
a, b, or B.
a, b 9 or C.
a and C.
I and a
cos a = cos A sin 5;
tan B = sin A tan b;
cot C = cot A cos .
sin a sin B = sin J. sin J;
sin a cos .Z? = cos &;
cos a = cos A sin .
sin a sin (7 = sin A \
sin a cos C = cos J. cos
5 =
cos a
cos A'
sin .# = cot a tan .4;
. ~ sin A
sm # = .
sin a
sin A
sm a%= - ^
sm C"
cos b = tan ^4 cot C";
D cos (7
cosjS = 3.
cot a = cot A sin B;
cot # = sin A cot .5;
cos (7 = cos A cos #.
sin 7 sin a sin ^4;
sin (7 cos a = cos ^4 sin 5;
cos (7 = cos A cos 5.
sin C sin 5 = sin B\
sin (7 cos b = sin ^4 cos B.
SPHERICAL TRIANGLES IN GENERAL.
SPHERICAL TRIANGLES IN GENERAL.
Given.
Required.
a, b, c,
A, B, G,
- \ ),
the three
sides.
the three
angles.
rr A/ Sm ( S ~~ a ) SI* 1 ( S ~#) Sm ( 5 C )
sins
tan 4 A
sin (s a)
fin 47?
tail TT -O : 7T>
sm (5 b)'
tnr 4 ^
sm (s )
p, , sin a sin 5 sin c
sin J[ sin B sin (7*
a, 5, C 9
A and <?,
sin c sin A = sin a sin C;
two sides
one angle
sin c cos A = cos a sin b sin cos b cos (7;
and the
and the
cos c = cos a cos 5 + sin a sin 5 cos C.
included
remaining
angle.
side.
B and c.
sin c sin B = sin 5 sin C;
sin c cos .Z? = sin a cos 5 cos a sin 5 cos C.
If addition and subtraction logarithms
are not available for this computation, we
may compute Jc and K from
Tc sin K = sin a cos (7;
& cos K = cos a.
Then
sin c cos -4 =; Jc sin (# JT);
cos c = k cos (b JT).
Also,
A sin H = sin # cos C\
h cos If = cos J.
Then
sin c cos ^ = h sin (# If);
cos c = h cos (a 5").
A, B, c,
sin -J c sin -J ( J. .Z?) = cos % Csin % (a b) ;
all the
sin I c cos | (^4 ^) = sin-J- C7sinJ( + );
remaining
cos|csin- (A-\-B)= cos-J (7cos| (a b);
pasts.
cosij ccos-J (-44-2?) = sin J C cos J (a -\-b).
80
SPHERICAL TRIANGLES.
Given.
Required.
a, I, A,
two sides
B, C, c,
all the
. D sin A sin b , ,
sm B = : (two values of );
sin CL
and an
remaining
4-0^, i n cos i( a ^) c t %(A -\- B)
opposite
parts.
cos % (a + b)
angle.
cos 4 (A 4- i?) tan !( + &)
tan * c <s \ i ^
A, B, c,
a and C,
sin C sin a = sin A sin c;
two angles
one side
sin (7 cos a = cos ^4 sin B -\- sin J. cos B cos 0;
and the
and the
cos (7 = cos A cos .5-f-sin A sin^B cos c.
included
third angle.
side.
1 and C.
sin Csmb = sin .# sin c\
sin (7 cos 5 = sin A cos ^ -f- cos A sin J5 cos c.
If we compute & and K from
& sin K = cos ^4,
& cos -5T = sin ^4. cos c,
then sin (7 cos a = k cos (^ K)\
cos (7 = k sin (# -ZT).
If we compute h and H from
A sin ^T = cos B y
h cos If = sin j9 cos c,
;hen sin (7 cos = h cos (^4 #);
-
cos C h sin (A H).
a, 1, C,
sin (7sin(a-|- Z>) = sinc cosj(^4 B);
all the
sin tfcos I (a -f Z>) = cos | c cos J (^4 + B);
remaining
cos^(7sinj( b) = sinc sin %(A B);
parts.
cos -J- C'cos J ( b) = cos ^ c sin (^4 + ).
A, B, a,
two angles
&, c, C,
all the
. .. sin a sin B
en T| /) /-f-TTT/^ iTrtli-i/%ci r\-r A\*
Dill U : 1 LWU YdtlUUo \JL U}*
sin A
and an
opposite
remaining
parts.
^*- _ cos 1 (^ + B) tan i (a + b) m
wall -g- (/ ' " 1 / /f " 7?\ '
side.
cos -|- (^ ^) cot -|- (^4 -j- ^)
COS % ( a + ^)
A, B, (7,
a, I, c,
#=(^ + +(7);
the three
the three
P-4/ -cos-S'
angles.
sides.
K cos(^-^) cos (#-.#) cos (#-(7)'
tan Ja = P cos ($ ^4);
tan %b = P cos ($ ^)^
tan^c = Pcos (S C).
TABLES.
TABLE I.
COMMON LOGARITHMS
OF NUMBERS.
X.
Log.
N.
Log.
N.
Log.
N.
Log.
N.
Log.
1
2
3
Infinity.
30
31
32
33
I.477I2
60
61
62
63
1.77815
90
91
92
93
1.95424
120
121
122
123
2.07 918
O.OO OOO
0.30 103
0.47 712
1.49 136
I.505I5
1.51 851
1.78533
1.79239
1.79934
.95904
.96 379
.96 848
2.08 2>9
2.08636
2.08 991
4
5
6
0.60 206
0.69 897
0.77815
34
35
36
I.53I43
1.54407
1.55630
64
65
66
i. 80618
1.81 291
1.81 954
94
95
96
97 313
.97 772
.98 227
124
125
126
2.09342
2.09691
2.10037
7
8
9
10
11
12
13
0.84 510
.0.90 309
0.95 424
37
38
39
40
41
42
43
1 . 56 820
1.57978
1.59 106
67
68
69
70
71
72
73
1.82607
1.83251
1.83 885
97
98
99
100
101
102
103
.98 677
.99 123
99 564
127
128
129
130
131
132
133
2.10 380
2.IO 721
2. 1 1 059
I.OOOOO
1. 60 206
1.84 510
2.00000
2. ii 394
.04 139
.07 918
.11 394
1.61 278
1.62323
1.63347
1.85 126
1.85733
1.86332
2.00432
2.00 860
2.01 284
2. 1 1 727
2.12057
2.12385
14
15
16
.14613
.17609
.20412
44
45
46
1.64345
1.65321
1.66 276
74
75
76
1.86923
1.87 506
i. 88081
104
105
106
2.01 703
2.02 119
2.02 531
134
135
136
2.12 710
2,13033
2.13354
17
18
19
20
21
22
23
.23043
.25 527
.27 875
47
48
49
50
51
52
53
1.67 210
1.68 124
1.69 020
77
78
79
80
81
82
83
1.88649
1.89 209
1^89763
107
108
109
110
111
112
113
2.02 938
2.03 342
2.03743
137
138
139
140
141
142
143
2.13672
2.13988
2.I430I
.30 103
1.69897
1.90309
2.04139
2.I46I3
.32 222
.34242
.36 173
1.70757
1.71 600
1.72428
i .90 849
1.91 381
1.91 908
2.04532
2.04922
2.05 308
2.14 922
2.15229
2.15534
24
25
26
.38021
39794
41 497
54
55
66
1.73239
1.74036
1.74819
84
85
86
i .92 428
1.92942
1.93450
114
115
116
2.O5 600
2.06070
2.06446
144
145
146
2.15836
2.16 137
2.16435
27
28
29
30
.43 136
.44716
.46 240
57
58
59
60
1.75587
I 76 343
1.77085
87
88
89
90
1.93952
1.94448
1-94939
117
118
119
120
2.06 819
2.07 188
2.07 555
147
148
149
150
2.16732
2.17 026
2.17 319
1.477"
1.77815
1.95424
2.07918
2.17 609
TABLE I.
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
100
01
02
03
04
05
06
07
08
09
110
11
12
13
14
15
16
17
18
19
120
21
22
23
24
25
26
27
28
29
130
31
32
33
34
35
36
37
38
39
140
41
42
43
44
45
46
47
48
49
150
oo ooo
043
087
130
173
217
260
303
346
389
i
2
3
4
5
6
7
8
9
i
2
3
4
5
6
7
8
9
i
2
3
4
5
6
7
8
9
i
2
3
4
5
6
7
8
9
i
2
3
4
5
6
7
8
9
44
4-4
8.8
13 a
17-6
22.0
26-4
30.8
35-2
39-6
4
4'
8.2
12.3
16.4
20.5
24.6
28.7
32.8
36-9
38
3-8
7.6
ix. 4
15-2
19.0
22.8
26.6
3 4
34 .3
35
3-5
7-o
10.5
14.0
17 5
21.
4 5
28.0
31-5
32
3-2
6-4
9-6
12.8
16.0
19.2
22-4
25-6
28.8
43
43
8.6
12.9
17.2
21. S
2 5 .8
30.1
34 4
38.7
40
4-o
8.0
12.0
16.0
2O-
24.0
28.0
32.0
36.0
37
3-7
7-4
IX. I
14-8
18-5
22.2
25.9
2 9 .6
33-3
.34
3-4
6-8
10 2
I 3 -6
17.0
20.4
23.8
27.2
30.6
31
31
6.2
93
12.4
IS 5
18.6
21.7
24-8
27.9
49
4
8.
12.
16.
21.
25
29.
33-6
37-8
39
39
7-8
11.7
156
'9 5
23.4
27 3
31-2
35
36
3.6
7-a
10.8
14 4
18 o
21 6
25.3
28.8
32 4
33
33
6.6
99
13 3
I6 '5
19-8
23. x-
26.4
29 7
30
3-o
6.0
9-o
12.
15-0
18.0
21.
24.0
37-0
432
860
oi 284
703
02 119
531
938
03 342
743
475
903
326
7 2 5
1 60
572
979
383
782
518
945
368
787
202
612
*oi9
423
822
561
988
410
828
243
653
*o6o
463
862
604
*030
452
870
284
694
*IOO
503
902
647
*072
494
912
325
735
*I 4 I
543
941
689
*ii5
536
953
366
776
*i8i
583
981
732
*I57
578
995
407
816
*222
623
*02I
775
*i99
620
*O36
449
857
*262
663
*o6o
817
*242
662
*078
490
898
* 3 Q2
703
*IOO
04 139
179
218
258
297
336
376
415
454
493
532
922
05 308
690
06 070
446
819
07 1 88
555
5 2'
9 6i
346
729
1 08
483
856
225
591
610
999
385
767
H5
521
893
262
628
650
*038
423
805
183
558
930
298
664
689
*077
461
843
221
595
967
335
700
727
*ii S
500
881
258
633
*oo4
372
737
766
*I54
538
918
296
670
*04i
408
773
805
="192
576
956
333
707
*078
445
809
844
*23I
614
994
371
744
*ii5
482
846
883
*269
652
*032
408
781
*I 5 I
518
882
918
954
990
*027
*o6 3
*099
*I35
*i 7 i
*2O7
*243
08 279
636
991
09 342
691
10 037
380
721
n 059
3H
672
*026
377
726
072
415
755
093
350
707
*o6i
412
760
1 06
449
789
126
386
743
*096
447
795
140
483
823
1 60
422
778
*I 3 2
482
830
175
517
857
193
458
814
*i67
517
864
209
551
890
227
493
849
*202
552
8 99
243
585
924
261
529
884
*2 3 7
587
934
278
619
958
294
565
920
* 27 2
621
968
312
653
992
327
600
955
*3<>7
656
*oo3
346
687
*025
36 1-
394
428
461
494
528
561
594
628
661
694
727
12 057
385
710
13033
354
672
988
14 301
760
090
418
743
066
386
704
*oi9
333
793
123
450
775
098
418
735
*o5i
364
826
483
808
130
450
767
*082
395
860
189
516
840
162
481
799
*ii4
426
893
222
548
8 7 2
194
513
830
*I45
457
926
254
581
3
545
862
*i76
489
959
287
613
937
258
577
893
*208
520
992
320
646
969
290
609
925
*239
55i
*O24
$
*OOI
322
640
956
*27O
582
613
644
675
706
737
768
799
829
860
891
922
15 229
534
836
16 137
435
732
17 026
319
953
259
564
866
167
465
761
056
348
983
290
594
897
197
495
791
085
377
*oi4
320
625
927
227
524
820
114
406
*045
351
655
957
256
554
850
H3
435
*076
38i
685
987
286
584
879
f ?
464
*io6
412
715
*oi7
316
613
909
202
493
*i37
442
746
*047
346
643
938
231
522
*i68
473
776
*077
376
673
967
260
55i
*i98
503
806
*io7
406
702
289
580
609
O
638
667
2
696
3
725
4
754
MIBMMMM
5
782
-
6
811
7
840
8
869
9
N.
1
Prop. Pts.
LOGARITHMS OF NUMBERS.
N.
O
1
2
3
4
5
6
r
8
9
Prop. Pts.
150
17 609
638
667
696
725
754
782
811
840
869
51
898
926
955
984
*oi3
"041
*o7o
*O99
*I2 7
*i$6
29
28
52
18 184
213
241
270
298
327
355
384
412
441
53
469
498
526
554
583
611
639
667
696
724
1
2
2.9
5.8
2.8
5.6
54
752
780
808
837
865
893
921
949
977
*oos
3
8.7
8.4
55
19033
061
089
117
145
173
201
229
257
285
4
11.6
11.2
56
312
340
368
396
424
45 l
479
507
535
562
5
14.5
14.0
57
590
618
645
673
700
^728
* 756
783
811
838
6
17.4
16.8
58
866
893
921
948
976
*o 5 8
*o85
*II2
7
20.3
19.6
59
20 140
167
194
222
249
276
303
330
358
385
8
23.2
22.4
OK
160
412
439
466
493
520
548
575
602
629
656
<J
*
<iJ , t
61
68 3
710
* 737
763
* 79
817
844
871
898
925
27
20
62
952
978
*O32
*o85
*II2
*I39
*i65
+192
1
o 7
9 ft
63
21 219
245
272
299
325
352
378
405
43i
458
J.
2
& . f
5.4
4. V
5.2
64
484
5"
537
564
590
617
643
669
696
722
3
8.1
7.8
65.
748
775
80 1
827
854
880
9 06
932
958
985
4
10.8
10.4
66'
22 Oil
037
063
089
115
141
167
194
220
246
5
13.5
13.0
Ifi 9
1C
67
68
69
272
531
, 789
298
557
814
324
583
840
350
608
866
376
634
891
401
660
917
427
686
943
453
712
968
479
737
994
7*3
7
8
9
ID .
18.9
21.6
24.3
10 . o
18.2
20.8
23.4
170
23 045
070
096
121
147
172
198
223
249
274
71
300
325
350
376
401
426
452
477
502
528
25
72
553
603
629
654
679
704
729
754
779
1 2.5
73
805
830
855
880
905
930
955
980
*<x>5
*030
2 5.0
74
24 055
080
105
130
155
180
204
229
254
279
3 7.5
75
304
329
353
378
403
428
452
477
502
527
4 10.0
76
576
601
625
650
674
699
724
748
773
5 12.5
77
797
822
846
871
895
920
944
969
993
*oi8
6 15.0
7 17.5
78
25 042
066
091
115
139
164
1 88
212
237
261
8 20.0
79
285
310
334
358
382
406
431
455
479
503
180
527
551
573
600
624
648
672
696
720
744
81
768
792
816
840
864
888
912
935
959
983
34
H
82
26 007
031
055
079
102
126
150
174
198
221
1
2.4
2.3
83
245
269
293
3 I6
340
364
387
411
435
458
2
4.8
4.6
84
482
55
529
553
576
600
623
647
670
694
3
7.2
6.9
85
717
764
788
8n
834
858
881
905
928
4
9.G
9.2
86
951
975
998
*02I
*045
*o68
"091
"114
*i 3 8
*i6i
5
6
12.0
14.4
11.5
13.8
87
27 184
207
231
254
277
300
323
346
370
393
7
16.8
16.1
88
89
416
646
439
669
g
485
715
508
738
531
761
554
784
577
807
600
830
623
852
8
9
19.2
21.6
18.4
20.7
190
875
898
921
944
967
989
*OI2
*Q35
*os8
*o8i
99
91
91
28 103
126
149
171
194
217
240
262
285
307
mm
21
92
330
353
398
421
443
466
488
533
1
2.2
2.1
93
556
578
601
623
646
668
691
713
735
758
2
4.4
4.2 i
94
780
803
825
847
870
892
914
937
959
981
3
4
6.6
80
6.3
Q 4 ,
95
96
29 003
226
026
248
048
270
070
292
092
336
137
358
58
ill
403
203
425
rr
5
6
* O
11.0
13.2
O . rr
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
16.3
99
885
907
929
951
973
994
*oi6
*o 3 8
*o6o *o8i
9
19.818.9
200
30 103
125
146
168
190
211
233
255
276 298
N.
O
1
2
3
4
5
6
7
8 1 9
Prop. Pts.
TABLE I.
N.
O
1
9
3
4
5
6
7
8
O
Prop.
Pts. 1
200
01
02
03
04
05
06
07
08
09
210
11
12
13
14
15
16
17
>18
19
220
21
22
23
24
25
26
27
28
29
280
31
32
33
34
35
36
37
38
39
240
41
42
43
44
45
46
47
48
49
250
MM
N.
30 103
123
146
168
190
211
233
255
276
298
2
1 2
2 4
3 6
4 8
511
613
7 15
817
9 19
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
~Pr
2 21
.2 2.1
.4 4.2
.6 6.3
.8 8.4
.0 10.5
.2 12.6
.4 14.7
..6 16.8
.818.9
20
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
19
1.9
3.8
5.7
7.6
9.5
11.4
13.3
15.2
17.1
18
1.8
3.6
5.4
7.2
9.0
10.8
12.6
14.4
16.2
17
1.7
3.4
5.1
6.8
8.5
10.2
11.9
13.6
15.3
mmm*mwiBmmmmv
op. Pts.
320
535
750
963
3i 175
387
597
32 015
34i
557
771
984
197
408
618
827
035
363
578
792
*oo6
218
429
sis
040
056
&
814
*027
239
450
660
869
077
406
621
835
*048
260
471
681
890
098
428
643
8 5 6
*o6 9
281
492
702
9ii
118
449
664
878
*09i
302
513
723
93i
139
471
685
899
*II2
323
534
744
952
1 60
492
707
920
*I33
345
555
765
973
181
5H
728
942
*I54
366
576
785
994
20 1
222
243
263
284
305
325
346
366
387
408
428
634
838
33 041
244
445
646
846
34 044
449
654
858
062
264
465
666
866
064
469
6/5
879
082
284
486
686
885
084
490
695
899
102
304
5 06
7 06
90S
104
510
715
919
122
325
526
726
925
124
53i
736
940
H3
345
546
746
945
U3
552
756
960
163
3 S
566
766
965
163
572
777
980
183
385
586
786
985
183
593
797
*OOI
203
405
606
806
*oo5
203
613
818
*02I
224
425
626
826
*O25
223
242
262
282
301
321
34i
361
380
400
420
439
635
830
35 025
218
411
603
793
984
459
655
850
044
238
430
622
813
*oc>3
479
674
869
064
257
449
641
832
*02I
498
083
2 7 6
468
660
* 85 '
*O4O
5 l8
713
908
I O2
679
870
*059
537
733
928
122
315
507
698
889
*078
557
753
947
141
334
526
717
908
*097
577
772
967
160
353
545
736
927
*u6
596
792
986
1 80
372
564
755
946
*i35
616
Su
*oo5
199
392
583
774
965
*I 5 4
36 173
192
211
229
248
267
286
305
324
342
361
549
736
922
37 107
291
S
840
380
568
754
940
125
310
493
676
858
773
959
144
328
694
876
418
605
791
977
162
346
530
712
894
436
624
810
996
181
365
548
73i
912
i 55
642
829
*oi4
199
383
566
749
93i
474
661
847
*0 33
218
401
%*
767
949
493
680
866
*o5i
236
420
603
785
967
511
*070
254
438
621
803
985
530
717
903
*o88
273
457
639
822
*oo3
38 021
039
057
075
093
112
130
148
166
184
2O2
382
561
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
I 35
614
792
970
146
322
498
672
274
453
632
Sio
987
164
340
I 15
690
292
471
650
828
*oo5
182
358
533
707
310
846
*023
199
375
550
724
328
507
686
863
*04i
217
393
568
742
346
525
703
881
*os8
235
410
585
759
364
543
721
899
*076
252
428
602
777
794
MMMMBBMHI
Sn
|T
829
2
846
^ :
3
863
4
881
M
5
898
6
915
mammmmm
7
933
8
950
O
LOGARITHMS OF NUMBERS.
N.
O
1
2
3
4
5
6
7
8
9
Pr<
>p. PK
250
39 794
811
829
846
863
88 1
898
915
933
950
MMMMHM
MBMMMMMM
51
967
985
*002
*io6
*I2 3
18
52
53
40 140
312
157
329
I7 I
346
192
364
209
226
398
243
415
261
432
278
449
295
466
1
2
1.8
3 6
54
483
500
5 I8
535
552
569
586
603
620
637
3
5.4
55
654
671
688
705
722
739
756
773
790
8^7
4
7.2
56
824
841
858
875
892
909
926
943
960
976
5
9.0
57
993
*OIO
*027
*044
*o6i
* 07 g
*095
*m
*I28
*i45
6
10.8
58
m
41 162
179
196
212
229
246
263
280
296
3^3
7
12.6
59
330
347
363
380
397
414
430
447
464
481
8
14.4
260
497
5H
53 1
547
564
581
597
614
631
647
'
61
664
68 1
697
7H
731
747
764
780
797
814
17
62
830
847
863
880
896
913
* 92 2
946
963
979
i
1 7
63
996
*OI2
*029
*045
*062
*o 7 8
*m
"127
*I44
J.
2
-l.i
3.4
64
65
42 160
325
177
341
193
357
210
374
226
390
406
259
423
275
439
292
455
308
472
3
4
5.1
6.8
66
488
504
521
537
553
570
586
602
619
635
5
8.5
67
651
66 7
684
700
716
732
749
765
781
797
6
7
10.2
UQ
68
8i3
830
846
862
878
* 894
* 9 "
927
* 943
959
t
. y
1 Q ft
69
975
991
*oo8
*034
*040
*o88
*I20
9
lo . O
15.3
270
43 136
152
169
185
201
217
233
249
265
281
71
297
313
329
345
361
377
393
409
425
441
16
72
457
473
489
505
521
537
553
569
584
600
1
1.8
73
616
632
648
664
680
696
712
727
743
759
2
74
775
791
807
823
838
854
870
886
902
* 9 ' 7
3
4^8
75
933
949
965
981
996
*OI2
*028
*44
*o 59
4
6.4
76
44091
107
122
138
154
170
185
20 1
217
232
5
Q
8.0
(i
77
248
264
279
295
3"
326
342
358
373
389
7
/ vl
11.2
78
79
404
560
420
576
436
592
45i
607
467
623
483
638
498
654
669
S2Q
685
545
700
8
9
12.8
14.4
280
716
73 i
747
762
778
793
809
824
840
855
81
871
886
902
917
932
948
963
979
994
*OIO
15
82
45 025
040
056
071
086
1 02
117
133
148
163
1
1.5
83
179
194
209
225
240
255
271
286
301
317
2
3.0
84
332
347
362
378
393
408
423
439
454
469
3
4"
4.5
6f\
85
484
500
515
530
545
561
576
606
621
.0
7e
86
637
652
667
682
697
712
728
743
758
773
6
.5
9.0
87
788
803
818
834
849
864
879
894
909
924
7
10.5
88
939
954
969
984
*000
*oi5
*O3O
*45
*o6o
"075
8
12.0
89
46 090
105
1 20
135
150
165
1 80
195
210
225
9
13.5
200
240
255
270
285
300
315
330
345
359
374
91
389
404
419
434
449
464
479
494
509
5 2 3
14
92
93
g
553
702
568
716
583
746
613
761
627
776
642
790
657
805
820
1
2
1.4
2.8
94
835
850
864
879
^894
* 9 9
* 923
* 938
953
967
3
A
4.2
6 a
95
982
997
*OI2
*026
*IOO
*n 4
TE
ft
U
7 O
96
47 129
144
159
*73
1 88
202
217
232
246
261
*J
6
1 Vf
8.4
97
276
290
305
319
334
349
363
378
392
407
7
9.8
98
422
436
451
465
480
494
509
'524
538
553
8
11.2
99
567
582
59 6
6n
625
640
654
669
683
698
9
12.6
800
712
727
741
756
770
784
799
813
828
842
N.
O
1
2
3
4
5
6
7
8
9
Pr
p. Pts.
TABLE I. "
N.
1
2
3
4
5
6
7
8
Prop. Pts.
800
47 712
727
741
756
770
784
799
813
828
842
01
8 5 7
871
885
900
914
929
943
958
972
986
02
48 ooi
015
029
044
058
073
087
101
116
130
03
144
159
173
187
202
216
230
244
259
273
15
04
287
302
316
330
344
359
373
387
401
416
1
1.5
05
06
430
572
AAA
M r*r
586
458
601
6if
487
629
643
515
657
530
671
in
558
700
2
3
3.0
4.5
07
714
728
742
756
770
78J
799
813
827
841
4
6.0
08
855
869
883
J*&
911
926
940
954
968
982
5
7.5
09
996
*OIO
*O24
*o66
*o8o
*io8
*I22
6
7"
9.0
1ft K
810
49 136
150
164
178
192
206
220
234
248
262
8
1U.O
12.0
11
276
290
304
318
332
346
360
374
388
402
9
13.5
12
13
554
429
568
443
582
457
596
471
610
485
624
513
651
679
14
693
707
721
734
748
762
776
790
803
8l 7
' 15
831
845
859
872
886
900
* 9 * 4
927
941
* 9 "
14
16
969
982
996
*OIO
*024
"037
"065
1
1.4
17
50 106
120
133
H7
161
174
1 88
202
215
229
2
2.8
18
243
2 5 6
270
284
297
325
338
352
365
3
4.2
19
379
393
406
420
433
447
461
474
488
SOI
4
5.6
7/v
820
515
529
542
556
569
583
596
610
623
637
6
.0
8 4
21
22
23
X
920
664
799
934
678
813
947
691
826
961
70S
840
974
718
987
732
866
*OOI
745
880
759
893
*028
772
% 907
7
8
9
9.8
11.2
12.6
24
51 055
068
08 1
095
1 08
121
135
148
162
175
25
26
188
322
202
335
348
228
362
242
375
388
402
282
415
2 95
308
441
27
455
468
481
495
508
521
534
548
561
574
18
28
587
601
614
627
640
654
667
680
693
706
1
1.3
29
720
733
746
759
772
7 86
799
812
825
838
2
2.6
880
851
865
878
891
904
917
930
943
957
970
3
3.9
31
983
996
"009
*022
*Q35
*048
*o6i
*075
*o88
*IOI
4
5.2
6K
32
52 114
127
140
153
1 66
179
192
205
218
231
O
7Q
33
244
257
270
284
297
310
323
336
349
362
7
.0
9.1
34
375
388
401
414
427
440
453
466
479
49*
8
10.4
35
36
504
634
I 17
647
543
673
556
686
569
699
582
711
595
724
608
737
621
750
9
11.7
37
763
776
789
802
815
827
840
853
866
* 879
38
892
905
917
930
943
956
969
982
994
39
53 020
033
046
058
071
084
097
no
122
135
12
840
148
161
173
1 86
199
212
224
237
250
263
1
1.2
41
275
288
301
3H
326
339
352
364
377
390
2
2.4
3/>
42
403
415
428
441
466
479
491
504
.6
43
529
542
555
567
580
593
605
618
631
643
4
5
4.8
6
44
45
656
782
668
794
681
807
694
820
706
832
719
845
732
857
744
870
882
769
895
6
7
7.2
8.4
46
908
920
933
945
958
970
983
995
*oo8
*020
8
9.6
47
54033
045
058
070
083
095
108
120
133
145
9
10.8
48
158
170
183
195
208
220
233
245
258
270
49
283
295
307
320
332
345
357
370
382
394
850
407
419
432
444
456
469
481
4941
506
518
N.
1
2
3
4
5
6
7 1 8
9
Prop. Pts.
LOGARITHMS OF NUMBERS.
N.
BMMMMM
350
51
52
53
54
, 55
56
57
58
59
360
61
62
63
64
65
66
67
68
69
370
71
72
73
74
75
76
77
78
79
880
81
82
83
84
85
86
87
88
89
390
91
92
93
94
95
96
97
98
99
400
N.
O
MHMMMMMMM
54 407
lj_
419
2
432
3
444
4
456
5
469
6
481
7
494
8
~
9
~
Prc
<MMMMKH
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
1
6
7
S
1
1
2
3
4
5
6
7
8
9
~Pro
>p. Pts.
13
1.3
2.6
3.9
5.2
6.5
7.8
9.1
10.4
11.7
12
1.2
2.4
3.6
4.8
6.0
7.2
8.4
9.6
10.8
11
1.1
2.2
3.3
4.4
5.5
6.6
7.7
8.8
9.9
10
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
p. Pts.
53i
654
777
900
55 023
145
267
388
509
630
75i
871
991
56 1 10
467
585
703
543
667
790
913
035
157
279
400
522
555
679
802
925
047
169
291
4i3
534
568
691
814
937
060
182
303
425
546
580
704
827
949
072
194
315
437
558
593
716
839
962
084
206
328
449
570
605
728
851
974
096
218
340
461
582
617
741
864
986
1 08
230
352
473
594
630
753
876
998
121
242
364
485
606
642
765
888
*OII
133
255
376
497
618
642
654
666
678
691
703
715
727
739
763
883
*OO3
122
241
360
478
597
7H
775
895
*oi5
134
253
372
490
608
726
787
907
*027
146
265
384
502
620
738
799
919
"038
158
277
396
5H
632
750
811
93i
*o5o
170
289
407
526
644
761
823
* 9 J 3
*062
182
301
419
1%
656
773
835
955
"074
194
312
43i
549
667
785
847
967
*o86
205
324
443
561
679
797
859
979
*098
217
336
453
573
691
808
820
832
844
855
867
879
891
902
914
926
937
57 054
171
287
403
519
634
749
864
978
9 i?
066
183
299
415
530
646
7 6i
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
"3
229
345
461
576
692
807
921
*oo8
124
241
357
473
588
88
933
*oi9
136
252
368
484
600
715
830
944
"031
148
264
380
496
61-1
726
841
955
*043
159
276
392
507
623
738
8f
967
990
*OOI
*OI 3
*O24
*Q35
*047
"058
*O7O
*o8i
58 092
206
320
433
546
659
771
883
995
104
218
331
444
557
670
782
894
*oo6
"5
229
343
456
569
681
794
906
*OI 7
127
240
354
467
e
805
917
*028
138
252
365,
478
591
704
816
928
*O4o
149
263
377
490
602
715
827
939
*o5i
161
274
388
501
614
726
838
172
286
399
512
625
737
850
961
*073
184
297
410
524
636
749
861
973
*o84
195
309
422
I 3 *
647
760
872
984
*o 9 s
59 106
118
129
140
151
162
173
184
195
207
218
329
439
770
879
60 097
229
340
450
I 61
671
780
890
240
461
III
791
901
*OIO
119
2 I'
362
472
a
802
912
*02I
130
262
373
483
594
704
813
923
"032
141
273
384
494
605
715
824
934
"043
152
284
395
506
616
726
835
945
*Q54
163
295
406
517
627
737
846
956
*o6ij
173
306
417
528
638
748
857
966
"076
184
3i8
428
539
649
759
868
977
*o86
195
206
O
217
mfmm^mmmm
1
228
2
239
3
249
4
260
5
271
6
282
7
293
8
304
9
TABLE I.
1 N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
400
60 206
217
228
239
249
260
271
282
293
304
01
3H
325
336
347
358
369
379
390
401
412
02
423
433
444
455
466
477
487
498
509
520
03
531
541
552
563
574
584
595
606
617
627
04
638
649
660
670
68 1
692
703
713
724
735
05
746
756
767
778
788
799
810
821
83' 842
06
853
863
874
885
895
906
917
927
938
949
11
07
959
970
981
991
*O02
"013
*023
*034
*045
*055
1
1.1
08
61 066
077
087
098
I0 9
119
130
140
151
162
2
2.2
09
172
183
194
204
215
225
236
247
257
268
3
A
3.3
A A
410
278
289
300
310
321
33i
342
352
363
374
I
5
tb.4
5.5
11
384
395
405
416
426
437
448
458
469
479
6
6.6
12
13
490
595
500
606
5"
616
SJ
532
637
542
648
553
658
g
I 74
679
584
690
7
8
7.7
8.8
14
15
700
805
711
815
721
826
731
836
742
847
752
857
763
868
III
784
888
794
899
9
9.9
16
909
920
930
941
951
962
972
982
993
*c3
17
62 014
024
034
045
055
066
076
086
097
107
18
118
128
138
149
159
170
1 80
190
201
211
19
221
232
242
252
263
273
284
294
304
315
420
325
335
346
356
366
377
387
397
408
4 l8
21
428
439
449
459
469
480
490
500
511
521
IV
22
23
531
634
542
644
IS
562
665
572
675
583
685
603
706
716
624
726
1
2
1.0
2.0
24
737
747
757
767
778
788
798
808
818
829
3
3.0
4 A
25
26
839
941
849
95i
s?
870
972
880
982
890
992
900
*002
910
*OI2
921
*022
931
*033
4
5
6
.0
5.0
6.0
27
63 043
053
063
073
083
094
IO4
114
124
134
7
7.0
28
144
155
165
175
185
195
205
215
225
236
8
8.0
29
246
256
266
276
286
296
306
317
327
337
9
9.0
430
347
357
367
377
387
397
407
417
428
438
31
448
458
468
478
488
498
5 08
5 l8
528
538
32
548
558
568
579
589
599
60 9
619
629
639
33
649
659
669
679
689
699
709
719
729
739
34
35
849
759
859
769
869
779
879
789
889
899
80 9
909
819
919
829
929
839
939
36
949
959
969
979
988
998
*oo8
*oi8
*028
*o 3 8
9
37
64 048
058
068
078
088
098
108
118
128
137
1
0.9
38
147
157
167
177
187
197
207
217
227
237
2
1.8
39
246
256
266
276
286
296
306
316
326
335
3
2.7
440
345
355
365
375
385
395
404
414
424
434
4
3.6
4C
41
444
454
464
473
483
493
503
513
523
532
6
.0
5.4
42
542
552
562
572
582
59i
601
611
621
631
7
ft a
43
640
650
660
670
680
689
699
709
719
729
1
8
u %}
7.2
44
738
748
758
768
777
787
797
807
816
826
9
8.1
45
836
846
856
865
875
885
904
914
924
46
933
943
953
963
972
982
992
*O02
*OII
*O2I
47
65 031
040
050
060
070
079
089
099
108
118
48
128
137
H7
157
167
176
1 86
I 9 6
205
215
^ \
49
225
234
244
254
263
273
283
292
302
312
450
321
33i
341
350
360
369
379
389
398
408
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
LOGARITHMS OF NUMBERS.
N.
450
51
52
53
54
55
56
57
58
59
460
61
62
63
64
65
66
67
68
69
470
71
72
73
74
75
76
77
78
79
480
81
82
83
84
85
86
87
88
89
490
91
92
93
94
95
90
97
93
99
500
O
65 321
1
2
MM^M
341
3
350
4
5
~
6
-
379
7
m*mmm*mmm
389
8
"398"
9
408
Pro]
i
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
IMMHHMB
Pro
). PtS.
MMMi^M
10
1.0
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
8
0.8
1.6
2.4
3.2
4.0
4.8
5.6
6.4
7.2
p. Pts.
418
5H
610
706
801
896
992
66 087
181
276
370
464
558
652
745
839
932
67 025
117
427
523
619
715
811
906
*OOI
096
191
437
533
629
725
820
916
*OII
106
200
447
543
639
734
830
925
*O2O
2IO
456
III
744
839
935
124
219
466
562
658
753
849
944
134
229
475
57i
667
763
858
954
238
485
|8l
677
772
868
963
153
247
495
686
782
877
973
*o68
162
257
504
600
696
792
887
982
172
266
285
295
304
3 T 4
323
332
342
35 1
361
380
474
567
661
755
848
941
034
127
389
483
577
671
764
857
950
043
136
398
492
5 86
680
773
867
960
052
MS
408
502
596
689
783
876
969
062
154
417
33
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
20 1
210
219
228
237
247
256
265
274
284
293
302
394
486
H 8
669
761
852
*o 943
68 034
3"
403
495
587
679
770
861
952
043
321
504
596
688
779
870
961
052
330
422
605
788
879
970
06 1
339
523
614
706
797
888
979
070
348
440
532
624
897
988
079
357
449
541
633
724
906
997
088
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
124
133
142
I 5 I
160
169
178
187
196
205
215
305
395
485
574
664
753
842
224
3H
404
494
583
673
762
851
940
233
323
413
502
592
68 1
771
860
949
242
332
422
601
690
780
869
958
251
431
520
610
699
789
878
966
260
350
440
529
619
708
975
269
359
449
538
628
717
806
895
984
278
368
458
547
637
726
815
904
993
287
377
467
S A
646
735
824
913
*OO2
$
st
744
833
922
*OII
69 020
028
037
046
055
064
073
082
090
099
108
197
285
3 P
461
548
636
810
897
^^ MHHHMHBBBMC
O
117
205
294
469
557
644
732
819
126
214
302
390
478
566
653
740
827
135
223
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
338
425
513
601
688
&
170
2 58
346
434
522
609
6 97
784
871
179
267
355
443
705
793
880
1 88
276
364
452
539
627
7H
801
888
906
914
923
3
932
4
940
5
949
6
958
7
966
975
N.
1
2
8
9
10
TABLE I.
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
500
01
02
03
04
05
06
07
08
09
510
11
12
13
14
15
16
17
18
19
520
21
22
23
24
25
26
27
28
29
580
31
32
33
34
35
36
37
38
39
540
41
42
43
44
45
46
47
48
49
550
N.
69897
984
70 070
157
243
329
415
501
586
672
906
914
923
932
940
949
958
966
975
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
^M^BM^
Pro
9
0.9
1.8
2.7
3.6
4.5
5.4
6.3
7.2
8.1
8
0.8
1.6
2.4
3.2
4.0
4.8
5.6
6.4
7.2
7
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5.6
6 3
P. Pts.
992
079
165
252
338
424
509
595
680
*OOI
088
174
260
346
432
li 8
603
689
*OIO
096
183
269
355
441
526
612
697
*oi8
105
191
278
364
449
I 35
621
706
*02 7
114
2OO
286
372
458
544
629
7H
*036
122
.209
295
381
467
III
723
*044
131
217
303
389
475
561
646
73i
*Q53
140
226
312
398
484
569
655
740
*062
148
234
32 1
406
492
578
663
749
757
842
927
71 012
096
181
265
( 349
433
517
766
851
935
020
105
189
273
357
441
525
774
783 ; 791
800
808
817
825
834
859
944
029
113
366
450
533
868
952
037
122
206
290
374
458
542
876
961
046
130
214
299
383
466
550
885
969
054
139
223
307
391
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
500
584
919
*oo3
088
172
257
341
s
592
600
609
617
625
634
642
650
659
667
675
684
767
850
933
72 016
099
181
"^28"
692
III
941
024
107
189
272
354
700
784
867
950
032
"5
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
148
230
313
395
742
8
991
074
156
239
321
403
750
834
917
163
247
329
411
759
842
925
*oo8
090
173
255
337
419
436
444
452
460
469
477
485
493
501
509
59i
673
754
835
916
997
73 078
159
518
599
681
762
843
925
*oo6
086
167
526
607
689
770
852
933
*oi4
094
175
I 3 !
616
697
779
860
941
*022
102
I8 3
542
624
705
787
868
949
*03o
in
191
1 S
632
713
795
876
957
*038
119
199
558
640
722
803
884
965
*046
127
207
567
648
730
811
892
973
*054
135
215
575
656
738
819
900
981
*062
143
223
583
665
746
827
908
989
*O7O
151
231
239
247
255
263
272
280
288
296
304
312
320
400
480
560
640
719
1%
957
74 036
-
O
328
408
488
568
648
727
807
886
965
336
416
496
i^
656
735
815
894
973
344
424
504
584
664
743
823
902
981
352
432
512
592
672
751
830
910
989
360
440
520
600
679
759
838
918
997
368
448
528
608
687
767
846
926
*oo3
376
456
536
616
695
775
854
933
*oi3
384
464
544
624
703
783
862
* 941
*O2O
392
472
552
632
711
791
870
949
*028
044
MMMM
1
052
2
060
3
068
4
076
^MiMB
5
084
6
092
7
099
8
107
9
LOGARITHMS OF NUMBERS.
1 1
fmmmmv^
N.
550
5.1
52
53
54
55
56
57
58
59
660
61
62
63
64
65
66
67
68
69
570
71
72
73
74
75
76
77
78
79
580
81
82
83
84
85
86
87
88
89
590
91
92
93
94
95
96
97
98
99
600
N.
mmm^mttmrmm
74 036
1
044
2
052
3
060
4
^M^^M
068
5
^^MHMK
O76
6
084
r
092
8
099
9
107
Pro]
^M^H
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
"Pr7
>.Pte.
8
0.8
1.6
2.4
3.2
4.0
4.8
5.6
6.4
7.2
7
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5.6
6.3
p.Pts.
US
194
273
351
429
507
586
663
74i
123
202
280
359
437
515
593
671
749
131
210
288
367
445
523
601
679
757
139
218
296
374
453
531
609
687
764
147
225
304
382
46l
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
1 86
265
343
421
500
578
656
733
SH
819
827
834
842
850
858
865
873
881
889
896
974
75051
128
205
282
358
435
5"
904
981
059
136
213
289
366
442
519
912
989
066
143
220
297
374
450
526
920
997
074
I5i
228
305
38i
458
534
927
*ool
082
'59
236
312
389
465
542
935
*OI2
089
166
243
320
397
473
549
943
*020
097
174
251
328
404
481
557
950
*028
105
182
259
335
412
488
565
958
*035
"3
189
266
343
420
496
572
966
*043
120
197
274
351
427
504
580
587
595
603
610
618
626
633
641
648
656
664
740
815
891
967
76 042
118
193
268
343
671
747
823
899
974
050
125
200
275
679
755
83'
906
982
057
133
208
283
686
762
-838
914
989
o6J
140
215
290
694
770
846
921
997
072
148
223
298
702
778
853
929
*oos
080
155
230
305
709
785
861
937
*OI2
087
163
238
313
717
793
868
944
*020
095
170
245
320
724
800
876
952
*O27
103
178
253
328
732
808
884
959
*<>35
no
'
260
335
350
358
365
373
380
388
395
403
410
486"
I 59
634
708
782
856
930
*oo4
078
418
492
567
641
716
790
77 012
425
500
574
649
723
797
871
945
019
433
507
582
656
730
805
879
953
026
440
515
589
664
738
812
886
960
034
448
522
597
671
745
819
^
967
041
455
530
604
678
753
827
901
975
462
537
612
686
760
834
908
982
056
470
545
619
693
768
842
916
989
063
477
g
%
849
923
997
070
085
093
IOO
107
H5
122
129
137
144
151
& 159-
232
305
379
452
525
597
670
743
166
240
313
386
459
532
605
677
750
173
247
320
3 ?I
466
539
612
685
757
181
254
327
401
474
546
619
692
764
188
262
335
408
481
554
627
699
772
195
269
342
415
488
5 6l
634
706
779
203
276
349
422
495
568
641
7H
786
210
28 3
357
430
5>3
576
648
721
793
217
291
364
437
5io
583
656
728
801
225
371
444
517
590
663
7 8ol
815
^I^HH^^B^B
822
1
830
3
837
3
844
4
851
5
859
6
866
7
873
8
880
9
12
TABLE I.
N.
O
1
2
3
4
5
6
7
8
9
Pro]
E>.Pts.
600
77 815
822
830
837
844
851
859
866
873
880
01
02
03
04
05
06
07
08
09
887
960
78 032
104
176
247
319
390
462
8 9 5
967
039
in
183
254
326
398
469
902
974
046
118
190
262
333
405
476
909
981
053
125
197
269
340
412
483
916
988
061
132
204
276
347
419
490
924
996
068
140
211
283
355
426
497
* 931
*oo3
075
147
219
290
362
433
504
* 938
*OIO
082
154
226
297
369
440
512
945
*oi7
089
161
233
305
376
447
519
952.
*025
097
1 68
240
312
383
455
526
1
2
3
8
0.8
1.6
2.4
39
610
533
540
547
554
561
569
576
583
590
597
5
4.0
11
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
1 20
190
633
704
774
845
916
986
057
127
197
640
711
78i
852
923
993
064
134
204
647
718
789
859
930
*000
071
141
211
654
725
796
866
937
*oo7
078
148
218
661
732
803
873
* 944
*oi4
085
155
225
668
739
810
880
95i
*02I
092
162
232
6
7
8
9
4.8
5.6
6.4
7.2
620
239
246
253
260
267
274
28l
288
295
302
21
22
23
24
25
26
27
28
29
309
379
449
518
588
657
727
796
865
316
386
456
525
595
664
734
803
872
323
393
463
532
602
671
741
810
879
330
400
470
539
678
748
817
886
337
407
477
546
616
685
754
824
893
344
414
484
553
623
692
761
831
900
351
421
491
S 60^
630
699
768
837
906
358
428
498
567
637
706
775
844
913
365
435
505
I 74
644
713
782
851
920
372
442
5ii
58i
650
720
789
858
927
1
2
3
4
5
C
7
8
9
7
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5.6
6.3
630
934
941
948
955
962
969
975
982
989
996
-
31
32
33
34
1 35
36
37
38
39
80 003
072
140
209
277
346
414
482
550
OIO
079
H7
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
516
584
044
H3
182
250
3i8
387
455
523
59i
051
120
188
257
325
393
462
530
598
058
127
195
264
332
400
468
536
604
065
134
202
271
339
407
475
543
611
1
2
3
6
0.6
1.2
1.8
640
618
625
632
638
645
652
659
665
672
679
4
2.4
41
42
43
44
45
46
47
48
49
686
754
821
889.
956
8 1 023
090
158
224
693
760
828
895
963
030
097
164
231
699
767
835
902
969
037
104
171
238
706
774
841
909
976
043
in
178
245
713
781
848
916
983
050
117
184
251
720
787
855
922
990
057
124
191
258
726
794
862
929
996
064
I3J
198
265
733
801
868
93<3
*oo"j
070
137
204
271
740
808
875
* 943
*OIO
077
144
211
2 7 8
747
814
882
949
*oi7
084
151
218
285
5
6
7
8
9
3.0
3.6
4.2
4.8
5.4
650
N.
291
O
298
1
305
2
3ii
3
318
4
325
5
33i
6
338
KMWMM
7
345
8
351
9
Pro
p Pts.
LOGARITHMS OF NUMBERS.
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
650
81 291
298
305
311
3i8
325
33i
338
345
35i
51
358
365
371
378
385
391
398
405
411
418
52
425
438
445
458
465
47i
478
485
53
491
498
505
5i8
525
538
544
551
54
55
558
624
564
631
637
578
644
584
651
657
Li
604
671
611
677
617
684
56
690
697
704
710
717
723
730
737
743
750
57
757
763
770
776
783
790
796
803 809
816
58
823
829
836
842
849
856
862
869
875
882
59
889
895
902
908
915
921
928
935
941
948
660
954
961
968
974
981
987
994
*ooo
*oo7
*oi4
61
82 020
027
033
040
046
053
060
066
073
079
62
086
092
099
105
112
"9
125
132
138
143
1
0.7
63
151
158
164
171
I 7 8
184
191
197
204
210
2
1.4
64
65
217
282
223
289
230
295
236
302
308
249
315
256
321
263
328
269
334
2 7 6
341
3
4
2.1
2.8
3e
66
347
354
36o
367
373
380
387
393
400
406
6
O
4.2
67
413
419
426
432
439
445
452
458
465
471
7
4.9
68
478
484
491
497
504
510
523
530
53 6
8
5.6
69
543
549
556
562
569
575
582
588
595
601
9
6.3
670
607
614
620
627
633
640
646
653
659
666
71
672
679
685
692
698
705
711
718
724
730
72
73
737
802
808
750
814
756
821
763
827
769
834
776
840
782
847
789
853
795
860
74
866
872
879
885
892
898
905
911
918
924
75
930
937
943
* 95
956
963
* 969
975
982
988
76
995
*OOI
*oo8
*O2O
*027
*046
"052
77
83 059
065
072
078
085
091
097
104
no
117
78
W 123
129
136
142
149
155
161
168
174
181
79
187
193
200
206
213
219
225
232
238
241
680
251
257
264
270
2 7 6
283
289
296
302
308
81
315
321
327
334
340
347
353
359
366
372
6
82
83
378
442
448
391
455
398
461
404
467
410
474
480
423
487
429
493
436
499
1
2
0.6
1.2
84
506
512
518
525
531
537
544
55
556
563
3
1.8
2 A
85
86
569
632
575
639
645
588
651
594
658
664
607
670
613
677
620
683
626.
689
5
6
.4
3.0
3.6
87
696
702
708
715
721
727
734
740
746
753
7
4.2
88
759
765
771
778
784
790
797
809
816
8
4.8
89
822
828
835
841
847
853
860
866
872
879
5.4
690
885
891
897
904
910
916
923
929
935
942
91
948
954
960
967
973
979
985
992
998
*oo4
92
84 on
017
023
029
036
042
048
055
061
067
93
073
080
086
092
098
105
in
117
123
130
94
95
136
142
205
148
211
155
217
161
223
167
230
III
180
242
186
248
192
251
96
261
267
273
280
286
292
2 9 8
305
3"
317
97
98
$
330
392
336
398
342
404
348
410
354
417
3 6i
423
367
429
373
435
379
442
99
448
454
460
466
473
479
485
491
497
504
700
510
516
522
528
535
54i
547
553
559
566
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
TABLE I.
N.
i
2
3
4
5
6
7
8
Prop. Pts.
700
01
1 02
03
1 04
05
OG
i 07
1 08
09
710
11
12
13
14
15
16
17
18
19
720
21
22
23
24
25
26
27
28
29
780
31
32
33
34
35
36
37
38
39
740
41
42
43
44
45
46
47
48
49
750
N.
84 510
516
522
528
535
541
547
553
559
566
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
MI^M
Pro]
7
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5.6
6.3
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
).PtS.
572
634
696
757
819
880
942
85 003
065
126
I 78
640
702
763
825
887
948
009
071
584
646
708
770
831
893
954
016
077
590
652
7H
776
837
899
960
022
083
597
658
720
782
844
905
967
028
089
603
665
726
788
850
911
973
034
095
609
671
733
794
856
917
979
040
101
6i5
677
739
800
862
924
107
621
683
743
807
868
93<>
991
052
114
628
689
75i
8i3
874
936
997
058
1 20
132
138
144
150
156
163
169
175
181
187
248
309
370
43i
491
552
612
673
193
254
315
376
437
497
558
618
679
199
260
321
382
443
503
564
625
685
205
266
327
388
449
509
570
631
691
211
272
333
394
455
516
576
6 37
697
217
278
339
400
461
522
582
643
703
224
285
345
406
467
528
588
649
709
230
291
352
412
473
534
594
655
715
236
297
358
418
479
540
600
661
721
242
303
364
425
485
546
606
667
727
733
739
745
751
757
763
769
775
78i
788
794
854
914
^ 974
86 034
094
153
213
273
800
860
920
980
040
IOO
159
219
279
338"
806
866
926
986
046
106
165
225
285
812
872
932
992
052
112
171
231
291
818
878
938
998
058
118
177
237
297
824
884
944
*(X>4
064
124
183
243
303
830
890
950
*OIO
070
130
189
249
308
836
896
956
*oi6
076
136
195
255
3H
842
902
962
*O22
082
141
201
26l
320
848
908
968
*028
088
147
207
267
326
332
344
404
463
522
I 81
641
700
759
817
876
350
356
362
368
374
380
386
392
45'
510
570
629
688
747
806
864
39o
457
516
576
635
694
753
812
870
410
469
528
587
646
705
764
823
882
415
475
534
593
652
711
770
829
888
421
481
540
599
658
717
776
835
894
427
487
546
605
664
723
782
841
goo
433
493
552
611
670
729
788
847
906
439
499
558
617
676
735
794
853
911
445
504
564
623
682
741
800
859
917
923
929
935
941
947
953
958
964
970
976
982
87 040
099
I5 2
216
274
332
390
448
988
046
105
163
221
280
338
396
454
994
052
in
169
227
286
344
402
460
999
058
116
'75
233
291
349
408
466
*00 5
064
122
181
239
297
355
4i3
47i
*on
070
128
186
245
303
361
419
477
*oi7
075
134
192
251
309
367
425
483
*023
08 1
140
198
256
315
373
489
*O29
087
146
204
262
320
379
437
495
*35
093
151
210
268
326
384
442
500
558
9
5 o6
o
*
518
2
523
3
529
4
535
5
54i
6
547
7
552
8
LOGARITHMS OF NUMBERS.
N.
>*m~~
750
51
52
53
54
55
56
57
58
59
7 CO
61
62
63
64
65
66
67
68
69
770
71
72
73
74
75
76
77
78
79
780
81
82
83
84
85
86
87
88
89
790
91
92
93
94
95
96
97
98
99
800
O
87 506
1
512
2
~
3
523
4
529
5
^HMMMM
535
6
MHMMM*
541
7
MMMM^
547
8
i^mmmtmm
552
9
Iss"
Pro
MMMM
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
p. Pts.
-^
O.G
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
622
679
737
795
852
910
967
88 024
570
628
685
743
800
858
915
973
030
576
633
691
749
806
864
921
978
036
|8i
639
697
754
812
869
927
984
041
587
645
703
760
818
875
933
990
047
593
65 i
708
766
823
881
938
996
053
599
656
714
772
829
887
944
*OOI
058
604
662
720
777
835
892
950
*oo7
064
610
668
726
783
841
898
* 955
*oi3
070
616
674
73i
789
846
904
961
*oi8
076
081
087
093
098
104
1 10
116
121
127
133
138
195
252
309
366
423
480
536
593
144
20 1
258
315
372
429
485
542
598
150
207
264
321
377
434
491
%
156
213
270
326
383
440
497
553
610
161
218
275
332
389
446
502
559
615
167
224
281
338
395
45i
508
564
621
173
230
287
343
400
457
513
570
627
173
235
292
349
406
463
519
576
632
184
241
298
355
412
468
525
581
638
190
247
3>4
360
417
474
530
I 87
643
649
655
660
666
672
677
683
689
694
700
70S
III
874
89042
098
154
711
767
824
880
936
992
048
104
159
717
773
829
885
941
997
053
109
165
722
779
835
891
947
*oo3
059
H5
170
728
784-
840
897
953
*oc9
064
120
176
734
79
846
902
958
*oi4
070
126
182
739
795
852
8
964
*020
076
I 3 I
745
801
857
9 I 3
969
*025
08 1
137
193
750
807
863
919
* 975
*03i
087
$
756
812
868
925
981
*037
092
148
204
209
215
221
226
232
237
243
248
254
260
265
%
542
597
$
271
326
382
437
492
548
603
658
713
2 7 6
$
$
553
609
664
719
282
337
393
448
504
559
614
669
724
287
$
454
509
564
620
675
730
$
404
459
5i5
570
625
680
735
298
354
409
465
520
575
686
741
304
360
415
470
526
581
636
691
746
310
365
421
476
53i
5 b<3
642
697
752
807
315
Z7 \
426
481
537
592
647
702
757
763
768
774
779
785
790
796
801
812
818
873
927
982
90037
091
146
200
253
823
878
933
988
042
097
'*!
206
260
829
883
938
993
048
1 02
'.57
211
266
$
944
998
33
162
217
271
840
894
949
*004
059
"3
168
222
2 7 6
845
900
955
*oo9
064
119
173
227
282
851
905
960
*oi"5
069
124
179
233
287
856
911
966
*O2O
075
129
I8 4
2 3 8
293
862
916
971
*026
080
135
189
244
298
867
922
977
*03i
086
140
195
249
304
309
314
320
325
331
336
342
347
352
358
N.
i
2
3
4
5
<* 7 8
Prop. Pts.
1 6
TABLE I.
N.
1
2
3
4
5
7
8
9
~
Prop. Pts.
800
90 309
3H
320
325
331
~
342
347
352
01
363
369
374
380
385
390
396
401
407
412
02
417
423
428
434
439
445
450
455
461
466
03
472
477
482
488
493
499
504
509
515
520
04
526
53I
536
542
547
553
558
563
569
574
05
580
58?
590
596
601
607
612
617
623
628
00
634
639
644
650
655
660
666
671
677
682
07
687
693
698
703
709
714
720
725
730
736
08
74 1
747
752
757
763
768
773
779
784
789
09
795
800
806
Sii
816
822
827
832
838
843
810
849.
854
~8~59~
865
870
875
88 1
886
891
897
11
12
902
956
907
961
966
918
972
924
977
929
982
934
988
940
993
998
950
*oo4
1
0.6
13
91 009
014
020
025
030
036
041
046
052
057
2
1.2
14
062
068
073
078
084
089
094
IOO
105
no
3
1.8
2 A
15
116
121
126
132
137
142
148
153
158
164
,*t
3 A
16
169
174
180
185
190
196
20 1
206
212
217
6
u
3.6
17
222
228
233
238
243
249
254
259
265
270
7
4.2
18
275
281
286
291
297
302
307
312
318
323
8
4.8
19
328
334
339
344
350
355
360
365
371
376
9
5.4
820
381
387
392
397
403
408
413
418
424
429
21
434
440
445
450
455
461
466
47 i
477
482
22
487
492
498
503
508
5H
519
524
529
535
23
540
545
556
561
566
572
577
582
587
24
593
598
603
609
614
619
624
630
635
640
25
645
651
656
661
666
672
677
682
687
693
26
f 698
703
709
7H
719
724
730
735
740
745
27
75 1
756
761
766
772
777
782
787
793
798
28
803
808
814
819
824
829
834
840
845
850
29
855
861
866
871
876
882
887
892
897
93
830
908
913
918
924
929
934
939
944
950
955
31
960
965
971
976
981
986
991
997
*OO2
*oo7
32
02 OI2
018
023
028
033
038
044
049
054
059
1
0.5
33
065
070
075
080
085
091
096
101
106
in
2
1.0
34
117
122
127
132
137
14^
148
153
158
163
3
1.5
2i\
35
36
I6 9
221
174
226
179
231
184
236
189
241
19?
247
200
252
205
257
2IO
262
215
267
5
6
.V
2.5
3.0
37
273 : 278
283
288
293
298
304
309
314
319
7
3.5 '
38
324 330
335
340
345
350
355
361
3 66
371
8
4.0 1
39
376
381
3^7
392
397
402
407
412
4l8
423
4.5
840
428
433
438
443
449
454
459
464
469
474
41
480
485
490
495
500
505
5 11
516
521
526
42
531
536
542
547
557
562
567
572
578
43
588
593
598
603
609
614
619
624
629
44
634
639
645
650
655
660
665
670
675
68 1
45
46
686
737
691
742
747
701
752
706
758
711
763
716
768
722
773
727
778
732
783
47
48
788
840
793
845
799
850
804
85?
809
860
814
865
819
870
824
875
829
881
834
8S6
49
891
896
901
906
911
916
921
927
932
937
850
942
947
952
957
962
967
973
978
983
988
N.
O
1
MBMMIM
3
MMM
4
5
6
7
8
9
Prop.Pts.
LOGARITHMS OF NUMBERS.
N.
1
2
3
4
5
6
7
8
9
Pro]
). PtS.
850
92 942
947
952
957
962
967
973
978
983
988
51
52
53
54
i 55
50
57
58
59
993
93 044
095
146
197
247
298
349
399
998
049
IOO
151
202
252
303
354
404
*oo3
054
105
156
207
258
308
359
409
*oo8
059
no
161
212
263
3 < 3
364
414
*OI 3
064
"5
166
217
268
3 i 8
369
420
*oi8
069
120
171
222
273
323
374
425
*024
075
125
176
227
278
328
379
430
*029
080
131
181
232
283
334
384
435
*Q34
085
136
1 86
237
288
339
389
440
*Q39
090
141
192
242
293
544
394
445
1
2
3
0.6
1.2
1.8
24.
860
450
455
460
465
470
475
480
485
490
495
5
3.0
61
62
63
64
65
66
67
68
60
500
551
601
651
702
752
802
852
902
505
656
707
757
807
857
907
S x
561
611
661
712
762
812
862
912
&
616
666
717
767
817
867
917
520
57i
621
671
722
772
822
872
922
52 ^
576
626
676
727
777
827
877
927
531
|8i
631
682
732
782
832
882
932
536
586
636
687
737
787
837
887
937
541
59i
641
692
742
792
842
892
942
546
596
646
697
747
797
847
897
947
6
7
8
9
3.6
4.2
4.8
5.4
870
952
957
962
967
972
977
982
987
992
997
6'
71
72
73
74
75
76
77
78
79
94 002
052
101
151
201
250
300
349
399
007
5 2
106
156
206
255
305
354
404
012
062
III
161
211
260
310
359
409
017
067
116
166
216
265
315
364
4H
022
072
121
171
221
270
3 20
369
419
027
077
126
176
226
275
325
374
424
032
082
I3i
181
280
330
379
429
037
086
136
1 86
236
285
335
384
433
042
091
141
191
240
290
340
389
438
047
096
146
196
245
295
345
394
443
1
2
3
4
5
6
7
8
9
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
880
448
453
458
463
468
473
478
483
488
493
81
82
83
84
85
86
87
88
89
498
547
596
645
694
743
792
841
890
503
I 52
601
650
699
748
797
846
895
507
557
606
655
704
753
802
851
900
512
562
660
709
758
807
856
905
517
567
616
665
7H
763
812
86 1
910
522
571
621
670
719
768
817
866
915
52 2
576
626
675
724
773
822
871
919
532
630
680
729
778
827
876
924
537
586
635
685
734
783
832
880
929
542
I 91
640
689
738
787
836
885
934
1
2
3
4
0.4
0.8
1.2
890
939
944
949
954
959
963
968
973
978
983
4
5
1.6
2
91
92
93
94
95
96
97
98
99
988
95 036
085
134
182
231
279
328
376
993
041
090
139
187
236
284
332
38i
998
046
095
143
192
240
289
&
*002
051
IOO
148
197
245
294
342
390
*<x>7
056
105
153
202
250
209
347
395
*OI2
061
109
158
207
255
303
352
400
*oi7
066
114
163
211
260
308
357
405
*022
071
119
1 68
216
265
3 < 3
361
410
*027
075
124
173
221
270
318
366
415
*O32
080
129
177
226
274
323
371
419
6
7
8
9
2.4
2.8
3.2
3.6
900
424
429
434
439
444
448
453
458
463
468
N.
1
2
3
4
5
6
7
8
9
Pro
p. Pts.
i8
TABLE I.
N.
O
1 2
3
4
5
6
7
8
9
Prop. Ptg.
900
95 424
429 434
439
444
448
453
"458"
"463"
"468"
01
472
477
482
487
492
497
501
506
5"
516
02
03
521
569
525
574
530
578
535
540
588
545
593
550
598
602
559
607
564
612
04
617
622
626
631
636
641
646
650
655
660
05
665
670
674
679
684
689
694
698
703
708
06
713
718
722
727
732
737
742
746
756
07
761
766
770
775
780
785
789
794
799
804
08
809
813
818
823
828
o
837
842
847
852
09
856
861
866
871
875
880
885
890
895
899
910
904
909
914
918
923
928
933
938
942
947
11
952
957
* 961
966
* 971
976
980
985
990
995
12
999
*oo 4
*oi4
*023
*028
*O33
*o 3 8
*042
1
0.5
13
96047
052
057
061
066
071
076
080
085
090
2
1.0
14
15
095
142
099
147
104
152
109
156
114
161
118
166
123
128
175
III
137
185
3
4
p.
1.5
2.0
9 f\
16
190
194
199
204
209
213
218
223
227
232
o
6
.O
3.0
17
18
as
242
289
246
294
251
298
256
303
261
308
265
313
270
317
275
322
280
327
7
8
3.5
4.0
19
332
336
341
346
350
355
360
365
369
374
9
4.5
920
379
384
388
393
398
402
407
412
417
421
21
426
431
435
440
445
450
454
459
464
468
22
473
478
483
487
492
497
501
506
5"
515
23
520
525
530
534
539
544
548
553
558
562
24
25
4
572
619
577
624
581
628
586
633
591
638
I 95
642
600
647
605
652
609
656
26
661
666
670
675
680
685
689
694
699
703
27
28
708
713
759
717
764
722
769
727
774
778
736
783
788
745
792
750
29
802
806
811
816
820
825
830
834
839
844
930
848
853
858
862
867
872
876
881
886
890
31
32
895
942
900
946
904
909
956
914
#9 6o
918
965
923
970
928
974
932
979
984
1
0.4
33
988
993
997
*OO2
*OII
*oi6
*02I
*O2S
*O3O
2
0.8
34
97 035
039
044
049
053
058
063
067
072
077
3
1.2
10
35
08 1
086 1 090
095
100
104
109
114
118
123
.0
2(1
36
128
132 1 137
142
146
155
1 60
165
169
6
.V
2.4
37
174
179
183
188
192
197
202
206
211
216
7
2.8
38
220
225
230
234
239
243
248
253
257
262
8
3.2
39
267
271
276
280
285
290
294
299
304
308
9
3.6
940
313
317
322
327
331
336
340
345
350
354
41
359
364
368
373
377
382
387
391
396
400
42
405
410
414
419
424
428
433
437
442
447
43
456
460
465
470
474
479
483
488
493
44
497
502
! 506
5"
516
520
525
529
534
539
45
543
548
552
557
562
566
575
580
585
46
589
594
598
603
607
612
617
621
626
630
47
633
640
644
649
653
658
663
667
672
676
48
68 1
685
690
695
699
704
708
713
717
722
49
727
736
740
745
749
754
759
763
768
950
772
777
782
786
791
795
800
804
809
813
N.
O
1 | 2
3
4
5
6
r
8
9
Prop. Pte.
LOGARITHMS OF NUMBERS.
N.
O
1
2
3
4
5
6
7
8
9
Prop.Pte.||
950
97 772
777
782
786
791
795
800
804
809
813
51
818
823
827
832
836
841
845
850
855
859
52
864
868
873
877
882
886
891
896
900
90S
53
909
914
918
923
928
932
937
941
946
950
54
955
959
964
968
973
978
982
987
991
996
55
98 ooo
00$
009
014
019
023
028
032
037
041
56
046
050
055
059
064
068
073
078
082
087
'
57
58
091
137
096
141
100
146
105
150
!SS
114
159
118
164
III
127
173
132
177
59
182
186
191
195
200
204
209
214
218
223
960
227
232
236
241
245
250
254
259
263
268
61
272
277
281
286
290
295
2 99
304
308
313
6
62
318
322
327
331
336
340
345
349
354
358
1
0.5
63
363
367
372
376
381
385
390
394
399
403
2
1.0
64
65
408
412
457
4J7
462
421
466
426
47i
430
475
480
439
484
444
489
448
493
3
4
1.5
2.0
2 ft
66
498
502
507
5"
516
520
525
529
534
538
6
3.0
67
68
69
F
632
547
592
637
552
597
641
I* 6
601
646
I?
605
650
565
610
655
570
614
659
574
619
664
579
623
668
673
7
8
9
3.&
4.0
4.6
970
677
682
686
691
695
700
704
709
713
717
71
722
726
731
735
740
744
749
753
758
762
1
72
767
771
776
780
784
789
793
798
802
807
1
73
fell
816
820
825
829
834
838
843
847
851
1
74
856
860
865
869
874
878
883
887
892
896
|
75
900
9^5
909
914
918
923
927
932
936
941
I
76
945
949
954
958
963
967
972
976
981
985
1
77
989
994
098
"003
*oo7
*OI2
*oi6
*O2I
*025
*029
73
79
99034
078
038
083
043
087
047
092
052
096
056
100
061
105
06 5
109
069
114
074
118
I
980
123
127
131
136
140
145
149
154
158
162
81
167
171
176
1 80
185
189
193
I 9 8
202
207
II
82
211
216
220
224
229
233
238
242
247
251
1
0.4
83
255
260
264
269
273
277
282
286
291
295
2
0.8
84
300
304
308
313
317
322
326
330
335
339
3
1.2
1 ct
85
344
348
352
357
361
366
370
374
379
383
9 n
86
388
392
396
401
405
410
414
419
423
427
6
24
87
432
436
441
445
449
454
458
463
467
471
7
2.8
88
476
480
484
489
493
498
502
506
5 11
515
8
3.2
89
520
524
528
533
537
542
546
550
555
559
9
3.6
990
564
568
572
577
581
585
590
594
599
603
91
607
612
616
621
62$
629
634
638
642
647
1
92
651
656
660
664
669
673
677
-682
686
691
93
695
699
704
708
712
717
721
726
730
734
94
739
743
747
752
756
760
765
769
774
778
95
782
787
791
795
800
804
808
813
8i7
822
1
96
826
830
835
839
843
848
852
856
861
865
1
97
870
874
878
883
887
801 896
000
904
909
1 98
99
913
957
^61
922
965
926
970
930
974
935
978
939
983
944
987
94**
991
952
996
1000
00 000
004
009
013
017
022
026
030
035
039
9
N.
O
1
2
3
4
5
6
7
8
9
Prop.Pta.ll
|l
2O
TABLE I.
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
1000
ooo ooo
043
087
130
174
217
260
304
347
391
1001
434
477
521
564
608
651
694
738
78i
824
1002
868
911
954
998
*O4i
*o84
*I28
*I 7 I
*2I 4
*2 5 8
1003
ooi 301
344
388
43i
474
517
561
604
647
690
44
1004
734
777
820
863
907
950
993
*036
*o8o
*I2 3
1
4.4
1005
002 1 66
209
252
296
339
425
468
512
555
2
8.8
1006
598
641
684
727
771
814
857
900
943
986
3
13.2
1007
003 029
073
116
159
202
245
288
331
374
417
4
17.6
1008
461
504
547
590
633
676
719
762
805
848
5
22.0
1009
891
934
977
*020
*o6 3
*io6
*i 49
*I92
*235
*278
6
IT
26.4
OA Q
1010
004 321
364
407
450
493
536
579
622
665
708
f
8
oU. o
35.2
1011
75i
794
837
880
923
966
*oo9
"052
*095
*I38
9
39.6
1012
005 i 80
223
266
309
352
395
438
481
524
567
1013
609
652
695
738
781
824
867
909
952
995
1014
006 038
08 1
124
166
209
252
295
338
380
423
1015
466
509
552
594
637
680
723
765
808
851
43
1016
894
936
979
*022
*o6s
*io7
*i5o
*I93
'236
*278
1
4.3
1017
007 321
364
406
449
492
534
577
620
-662
705
2
8.6
1018
748
790
833
876
918
961
*oo4
*046
*o89
*I 3 2
3
12.9
1019
008 174
217
259
302
345
387
430
472
515
558
4
17.2
O1 K
1020
600
643
_6_8s_
728
770
813
856
898
941
983
6
21. o
25.8
1021
009 026
068
in
153
196
238
281
323
366
408
7
30.1
1022
45i
493
536
578
621
f\f\^
706
748
791
833
8
34.4
1023
876
918
961
*oo3
*45
*o88
*I30
*I73
*2I 5
*2 5 8
9
38.7
1024
oio 300
342
385
427
470
512
554
597
639
681
1025
724
766
809
851
893
936
978
*020
*o6 3
*ios
1026
on 147
190
232
274
317
359
401
444
486
528
1027
570
613
655
697
740
782
824
866
909
95i
42
1028
993
*035
*o 7 8
*I20
*I62
*2O4
*247
*289
*33i
*373
1
4/2
1029
012 415
458
500
542
584
626
669
711
753
795
2
8.4
1030
837
879
922
964
*oo6
*048
*O9O
*I 3 2
*I74
*2I 7
8
12.6
1031
013 259
301
343
385
427
469
5ii
553
596
638
4
16.8
91 A
1032
680
722
764
806
848
890
932
974
*oi6
="058
XI . v
OK O
1033
014 100
142
184
226
268
310
352
395
437
479
7
4t). i
29.4
1034
521
563
605
647
689
730
772
814
856
898
8
33.6
1035
940
982
*024
*o66
*io8
*i5o
*I 9 2
*234
*276
* 3 i8
9
37.8
1036
015 360
402
444
485
527
569
611
653
695
737
1037
779
821
863
904
946
988
*O3O
*O72
*ii4
*I56
1038
016 197
239
281
323
365
407
448
490
532
574
1039
616
657
699
74i
783
824
866
908
950
992
41
1040
017 033
075
117
159
200
242
284
326
367
409
1
4.1
1041
45i
492
534
576
618
659
701
743
784
826
2
8.2
10i2
868
909
95i
993
*034
*076
*n8
*I 59
*20I
*243
3
12.3
1043
018 284
326
368
409
451
492
534
576
6I 7
659
4
5
16.4
20.5
1044
700
742
784
825
867
908
950
992
*033
*075
6
24.6
1045
019 116
158
199
241
282
324
366
407
449
490
7
28.7
1046
532
573
615
656
698
739
781
822
864
905
8
32.8
1047
947
988
*030
*07I
*ii3
*I54
*I 95
*237
*2 7 8
*320
9
36.9
1048
1049
020 361
775
403
817
444
858
486
900
527
94i
568
982
610
*O24
651
"065
693
*io7
734
*I48
1050
021 189
231
272
313
355
396
437
479
520
561
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
LOGARITHMS OF NUMBERS.
21
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
1050
1051
1052
1053
1054
1055
]056
1057
1058
1059
10CO
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
021 189
231
272
3i3
355
396
437
479
520
561
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
i
2
3
4
5
6
7
8
9
"FT
42
4.2
8.4
12.6
16.8
21.0
25.2
29.4
33.6
37.8
41
4.1
8.2
12.3
16.4
20.5
24.6
28.7
32.8
36.9
40
4.0
8.0
12.0
16.0
20.0
24.0
28.0
32.0
36.0
89
3.9
7.8
11.7
15.6
19.5
23.4
27.3
31.2
35.1
M^M^MMMV
op. Pts.
603
022 Ol6
4 2S
841
023 252
664
024 075
486
896
644
057
470
882
294
705
116
527
937
685
5"
923
335
746
157
568
978
727
140
552
964
376
787
198
609
*oi9
768
181
593
*oos
417
828
239
650
*o6o
809
222
635
*047
458
870
280
691
*IOI
851
263
676
*oS8
499
911
321
732
*I42
892
3^5
717
*I2 9
541
952
363
773
*i8 3
933
346
758
*I70
582
993
404
814
*22 4
974
387
799
*2II
623
*034
445
855
*26 5
025 306
347
388
429
470
SH
552
593
634
674
, 715
026 125
533
942
027 350
757
028 164
57i
978
029 384
756
165
574
982
390
798
205
612
*oi8
797
20<$
615
*023
431
839
246
653
*059
838
247
656
"064
472
879
287
693
*IOO
879
288
697
*io5
5i3
920
327
734
*I4O
920
329
737
"146
II?
368
775
*i8i
961
370
778
*i86
594
*002
409
* 8 ' 5
S^T) j
*002
411
819
*227
635
*O42
449
856
*262
*043
452
860
*268
676
*o83
490
896
*3Q3
708
*o84
492
901
* 309
716
*I2 4
53i
937
*343
424
465
506
546
587
627
668
749
789
030 195
600
031 004
408
812
032 216
619
033 021
830
* 35
640
045
449
853
256
659
062
871
276
681
085
489
893
296
699
102
9l \
316
721
126
530
933
337
740
142
952
357
762
166
570
974
377
780
182
77
992
397
802
206
610
*oi4
417
820
223
*Q33
438
843
247
651
*054
458
860
263
665
*073
478
883
287
691
*95
498
901
303
705"
*H4
519
923
328
732
*i 35
538
941
343
*I54
559
964
368
772
*i75
' 5 Z 8
981
384
424
464
504
544
625
745
785
826
034 227
628
035 029
43
830
036 230
629
037 028
866
267
669
069
470
870
269
906
308
709
109
510
9IO
309
709
108
946
348
749
149
550
950
349
749
148
986
388
789
190
590
990
389
789
187
*O27
428
829
230
630
*O3O
227
*o67
468
869
270
670
*070
469
868
267
*io7
508
909
310
710
*IIO
509
908
307
*I 47
548
949
350
750
*i5o
549
948
347
"187
588
989
390
790
*i9o
9
387
426
466
506
546
586
984
382
779
176
573
969
365
761
*i$6
626
665
705
745
*H3
54i
938
335
73i
*I2 7
523
919
*3U
708
8
785
825
038 223
620
039 017
414
811
040 207
602
998
041 393
B^-M^ ^B
O
865
262
660
057
454
850
246
642
*Q37
904
302
700
097
493
890
286
68 1
*077
914
342
739
136
533
929
325
721
*ii6
*O24
421
819
216
612
*oo9
405
800
*'95
"064
461
859
? 55
652
*o 4 8
444
840
I 2 3i
630
6
*io3
501
295
484
879
*2 7 4
*i83
580
978
374
771
*i67
563
958
*353
432
MMMMW
1
472
2
5H
3
55i
4
590
5
669
7
748
9
TABLE II.
TABLE II.
CONSTANTS WITH THEIR LOGARITHMS.
Number.
Logarithm.
Ratio of circumference to diameter, TV,
3.14159265
0.49714 99
.. n\
9.86960440
0.99429 97
. 27T,
6.28318531
0.79817 99
.. ^,
I-77245385
0.2485749
Number of degrees in circumference,
360
2-5563025
minutes
21600'
4-33445 38
seconds
1296000"
6.11260 50
Degrees in arc equal to radius,
57- 2957795
1.75812 26
Minutes .. ..
3437'. 74677
3-53627 39
Seconds
2o6264 // .8o6
5.31442 51
Length of arc of i degree,
.01745329
8.24187 7410
..i minute,
.00029089
6.46372 61 10
.. ..i second,
.000004848
4.68557 4910
Number of hours in i day,
24
1.38021 12
minutes
1440
3.15836 25
. seconds
86400
4-9365 37
Number of days in Julian year,
365.25
2.5625902
Naperian base,
2.718281828
0.43429 45
Modulus of common logarithms,
0.434294482
9.637784310
Hours in which earth revolves through
arc equal to radius,
3.8197186
0.58203 14
Minutes of time .. .. ..
229.18312
2.36018 26
Seconds of time
'3750-987
4-1383339
III SINES AND TANGENTS OF SMALL ANGLES. 25
TABLE III.
FOR
SINES AND TANGENTS OF SMALL ANGLES.
TO FIND THE SINE OB TANGENT I
Log sin a = log a (in seconds) + &
Log tan a = log a (in seconds) + T.
TO FIND A SHALL ANGLE FROM ITS SINE OB TANGENTs
Log a (in seconds) = log sin a + ?'
Log a (in seconds) as log tan a + 2*.
26
TABLE III.
99
9
L. Sin.
8
T
S'
T'
60
1 2O
1 80
240
I
2
3
4
6.4^73
76476
7.06579
4-68557
.68557
.68557
.68557
.68557
4.68557
.68557
.68557
.68557
.68558
S-3I443
.31443
.31443
.31443
.31443
5-31443
.31443
.31443
.31443
.31442
300
360
420
4 80
540
1
I
9
7.16270
.24188
.30882
.36682
.41797
4.68557
.68557
.68557
.68557
.68557
4.68558
.68558
.68558
.68558
.68558
5-31443
31443
.31443
31443
.31443
5-31442
3 I 442
31442
.31442
.31442
600
660
720
780
840
10
ii
12
13
H
7.46373
.50512"
54291
n
-4:68557
.68557
68557
.68557
68557
4-68558
.68558
.68558
.68558
.68558
S.3I443
.31443
.31443
31443
.31443
5-3I442
.31442
.31442
3 I 442
. 3*442
s
960
I02O
1080
1140
!!
[I
19
7 :S&
.69417
.71900
.74248
4 .6557
.68557
.68557
.68557
.68557
4.68558
.68558
.68558
.68558
.68558
5-31443
.31443
.31443
.31443
.31443
5-31442
.31442
.3H42
.3*442
.3*442
1200
1260
1320
1380
1440
20
21
22
23
24
7.76475
.78594
.80615
.82545
.84393
4.68557
.68557
.68557
.68557
.68557
4-68558
.68558
.68558
.68558
.68558
5-3I443
.31443
.31443
.31443
.31443
5-31442
3 I 442
.31442
..31442
.31442
1500
1560
1620
1680
1740
3
%
29
7.86166
.87870
.89509
.91088
.92612
4.68557
.68557
.68557
.68557
.68557
4-68558
.68558
.68558
.68558
.68559
5-31443
.31443
.31443
.31443
31443
5.31442
.31442
.31442
3 I 442
.3*44*
1800
1860
1920
1980
2040
30
3i
32
33
34
7.94084
.95508
.96887
.98223
99520
4-68557
.68557
.68557
.68557
.68557
4-68559
.68559
.68559
.68559
.68559
5-31443
.31443
.31443
3 I 443
3 I 443
5.3I44I
3 I 44i
3*441
.3*441
3*44*
SHOO
2l6o
2220
2280
2340
8
8
39
8.00779
.02002
.03192
04350
05478
4.68557
68557
.68557
.68557
.68557
4-68559
.68559
.68559
.68559
.68559
5.3I443
.31443
.31443
.31443
.31443
5.3*44*
3*441
.3*44*
.31441
.31441
2400
2460
2520
2580
2640
40
4i
42
43
44
8.06578
.07650
.09718
.10717
4 'S 55 1
.68556
.68556
.68556
.68556
4-68559
.68560
68560
.68560
.68560
5-31443
3 I 444
3 I 444
.31444
.31444
5-3*44*
.3*440
.31440
.3*440
3*440
2700
2760
2820
2880
2940
9
47
48
49
8.11693
.12647
.13581
.14495
.15391
4.68556
.68556
.68556
.68556
.68556
4.68560
.68560
.68560
.68560
.68560
5-3 I 444
3 I 444
.31444
.3!444
. 3*444
5.3I440
.3*440
.31440
3 I 440
.3*440
3000
3060
3120
3180
3240
50
51
52
S3
54
8.16268
.17128
-I797I
.18798
. 19610
4-68556
.68556
.68556
.68556
.68556
4.68561
.68561
.68561
.68561
.68561
5-31444
3 I 444
3M44
.51444
3 I 444
5.31439
.31439
.3H39
.31439
.3H39
3300
3360
3420
3480
3540
P
P
59
8.20407
.21189
.21958
.22713
23456
4-68556
.68556
.68555
.68555
.68555
4.68561
.68561
.68561
.68562
.68562
5-31444
3 I 444
3 I 445
.31445
31445
5-3I439
.3H39
.31439
.3H38
.31438
3600
60
8.24186
4-68555
4.68562
5-3I445
5 3H38
SINES AND TANGENTS OF SMALL ANGLES.
1
M
t
L.Sin.
8
T
8'
r
3600
3660
3720
3780
3840
o
I
2
3
4
8.24186
.24907
.25609
.26304
.26988
4-68555
.68555
.68555
.68555
.68555
4.68562
.68562
.68562
.68562
.68563
5-3I445
.31445
31445
.31445
.31445
5.31438
31438
.31435
.31438
3H37 i
3900
3960
4020
4080
4140
i
I
9
8.27661
.28324
.28977
.29621
.30255
4-68555
.68555
.68555
.68555
.68555
4.68563
.68563
.68563
.68563
.68563
5-31445
.31445
.31445
31445
.31445
5-3H37 1
.3H37
.3H37
.3H37
31437
4200
4260
4320
4440
10
ii
12
13
H
8.30879
.3H95
.32103
.32702
.33292
4-68554
.68554
-68554
.68554
.68554
4-68563
.68564
.68564
.68564
.68564
5-31446
.31446
.31446
.31446
.31446
5.3H37
.31436
.31436
4620
4680
4740
%
19
8.33875
34450
.35018
.35578
.36131
4-68554
.68554
.68554
.68554
.68554
4.68564
.68565
.68565
.68565
.68565
5-31446
.31446
.31446
.31446
.31446
5.3I436
.3H3S
.31435
.31435
.31435
4800
4860
4920
498o
5040
20
21
22
23
24
8.36678
.37217
'38276
.38796
4.68554
.6S553
.68553
.68553
.68553
4.68565
.68566
.68566
.68566
.68566
5- 3M46
.31447
.31447
.31447
3 H47
5.31435
.31434
.31434
.3H34
.31434
5100
5160
5220
5280
5340
11
11
29
'40320
.40816
4-68553
.68553
.68553
.68553
.68553
4.68566
.68567
.68567
.68567
.68567
5-3I447
.31447
.31447
.31447
31447
5-3I434
^433
.31433
.31433
5400
5520
558o
5640
30
31
32
33
34
8 41792
.42272
42746
.43216
.43680
4-68553
.68552
.68552
.68552
.68552
4-68567
.6856$
.68568
.68568
.68568
5-31447
.31448
.31448
.31448
31448
5-3H33
.3H32
'3H32
5700
5760
5820
5880
CO CO CO CO CO
8.44139
.44594
.45044
.45489
.45930
4-68552
.68552
.68552
.68552
.68551
4-68569
.68569
.68569
.68569
.68569
5-3I448
.31448
.31448
.31448
.31449
^31431
3H3I
.3H3I
3I43I
6060
6120
6180
6240
40
41
42
43
44
8.46366
.46799
.47226
.4p5o
.48069
4.68551
.68551
.68551
.68551
.68551
4.68570
.68570
.68570
.68570
.68571
5.31449
.31449
3 '449
.31449
.31449
5-3I430
.31430
.3M30
'31429
6300
6360
6420
6480
6540
49
'49304
.49708
.50108
4.68551
.68551
.68550
.68550
.68550
4.68571
.68571
.68572
.68572
.68572
5-3I449
31449
'3H50
5-3I429
.31429
.31428
.31428
.31428
6600
6660
6720
6780
6840
50
51
52
53
54
8.50504
'51673
.52055
4.68550
.68550
.68550
.68550
.68550
4.68572
.68573
.68573
.68573
.68573
5.3I450
.3H50
.3H50
.3H50
5.31428
.31427
31427
.31427
.31427
6900
6960
7020
7080
7I4P
11
11
59
8.52434
.52810
.53183
53552
.53919
4.68549
.68549
.68549
.68549
.68549
4.68574
.68574
.68574
.68575
68575
5.3I45I
.3H5I
.31451
.3H5I
.31451
5.3I426
.31426
.31426
.31425
31425
7200
60
8.54282
4.68549
4-68575
53H5I
5 31425
TABLE IV. LOGARITHMS, ETC. 29
TABLE IV.
LOGARITHMS
OF THE
SINE, COSINE, TANGENT AND COTANGENT
FOR
EACH MINUTE OF THE QUADRANT.
TABLE IV.
9
L. Sin.
d.
L. Tang.
c. d.
L. Cotg.
L. Cos.
Proj
). Pt
S.
I
2
3
4
6.46373
6 . 76 476
6.94085
7.06 579
30x03
17609
12494
9601
6.46373
o . 76 476
6.94085
7.06579
30103
17609
12494
0601
3.53627
3.23524
3 05915
2.93421
o.ooooo
o.ooooo
o.ooooo
o.ooooo
0.00000
00
3
3
.1
.2
3
3476
348
695
1043
3218
322
644
965
2997
300
599
899
i
I
9
7.16 270
7 24188
7.30882
7.36682
7.41 797
7918
6694
5800
55
4576
7.16270
7.24 188
7.30882
7.36682
7.41 797
7918
6694
5800
S5
2.83730
2.75812
2.69 118
2.63318
2.58203
o.ooooo
o.ooooo
o.ooooo
0.00000
o.ooooo
55
54
53
52
5i
4
5
.x
1390
1738
2&M
280
X28 7
1609
2633
263
1199
1498
2483
248
10
ii
12
13
14
7.46373
7.50512
7-54291
7-57 767
7.60985
4139
3779
3476
32x8
7-46373
7.50512
7.5429I
7.57767
7.60986
4139
3779
3476
3219
2.53627
2.49488
2.45 709
2.42233
2.39014
o.oo ooo
o.ooooo
o.ooooo
o.ooooo
o.ooooo
50
4 2
4 8
47
46
.2
3
4
5
500
84X
ZI2X
S401
2227
527
790
1053
1316
497
745
993
1242
1848
ii
Is 7
19
7.63 982
7.66 784
7.69417
7.71900
7.74248
2802
2633
2483
2348
7.63982
7.66 ?8|
7.69418
7.71900
7.74248
2803
2633
2482
2348
2228
2.36018
2.33215
2.30582
2.28 100
2.25 752
0.00000
o.ooooo
9-99999
9-99999
9 99999
45
44
43
42
41
.x
.2
3
4
.5
22 3
445
668
Sgx
11x3
202
404
606
808
XOIO
370
554
739
924
20
21
22
23
24
7.76475
7- 78*>4
7.80615
7-82545
7-84393
2119
2O2I
X930
x8 4 8
7.76476
7.78595
7.80615
7-82546
7 84394
2119
2O2O
1931
1848
2.23524
2.21 405
2.19385
2.17454
2.15 606
9 99999
9-99999
9-99999
9-99999
9-99999
40
37
36
.1
.2
3
1704
170
5"
1579
158
3x6
474
I47
47
294
442
11
11
29
7.86 166
7.87870
7.89 509
7 91088
7.92 612
1704
x6 39
579
524
7.86 167
7-87871
7.89510
7.91 089
7.92613
1704
1639
579
1524
2.13833
2.12 129
2 . 10 490
2.08911
2.07387
9.99999
9-99999
9 99999
9-99999
9.99998
35
34
33
31
4
5
.1
682
852
1379
138
632
789
1297
130
589
736
1223
122
30
32
33
34
7.94084
7-95 58
7.96 887
7.98223
7-99 520
1472
1424
379
X336
1297
7.94086
7-95 5io
7.96889
7-98225
7.99522
*473
1424
'379
1336
"97
2.05914
2.04490
2.03 III
2.01 77<j
2.00478
9.99998
9.99998
9.99998
9.99998
9.99998
80
S
.2
3
4
5
276
552
690
11*8
259
389
5*9
649
45
367
489
6x2
39
8.00 779
8.02002
8.03 192
8.04350
8.05478
1259
X223
II9O
XI58
XI28
8.00 781
8.02004
8.03 194
8.04353
8.05481
1223
1190
"59
1x28
99 219
.97996
.96806
.95 647
945*9
9.99998
9.99998
9-99997
9-99 997
9 99997
25
24
23
22
21
.x
.2
3
.4
.5
"5
116
232
347
463
570
xxo
220
33
440
55
105
209
418
523
40
42
43
44
8.06578
8.07650
8.08696
8.09 718
8.10 717
IO72
1046
X022
999
n-rfi
8.06581
8.07653
8.08 700
8.09 722
8 . 10 720
1072
1047
XO22
O?6
93 419
.92347
.91300
9-99997
9-99997
9-99997
9-99997
9-99996
20
19
18
\l
.1
.2
^
999
100
200
300
954
95
191
286
914
9 1
183
274
46
47
48
49
8.11693
8.12647
8.13581
8.14495
8.I539I
97
954
934
914
896
8. ii 696
8.12651
8.13585
8.14500
8.15395
97
955
934
9i5
895
0-0
88304
87349
.86415
.85500
.84605
9.99996
9.99996
9.99996
9-99996
15
13
12
II
4
5
.1
400
500
877
88
382
477
843
84
366
457
812
8x
50
5'
52
53
54
8.16268
8.17 128
8.17971
8.18798
8 19610
77
860
843
827
812
8.16273
8.I7I33
8.17976
8.18804
8.19616
860
843
828
812
83 727
.82867
.82 024
.81 196
.80384
9-99995
9 99995
9-99995
9 99995
9 99995
10
I
.2
3
4
5
175
263
438
169
253
337
422
162
244
325
406
3
57
58
59
8.20407
8.21 189
8.21 958
8.22713
8.23456
797
. 782
769
755
743
8.20413
8.21 195
8.21 964
8.22 720
8.23462
797
782
769
756
742
79587
.78805
.78036
.77280
.76538
9-99994
9.99994
9.99994
9.99994
9.99994
5
4
2
I
.2
3
4
78
56
755
75
226
730
73
x 4 6
219
292
i 00
8.24 186
.73
8.24 192
73
i . 75 808
9 99993
3 5
L. Cos.
d.
L. Cotg.
c. d.
L. Tang.
L. Sin.
9
Pro]
>.Pt
S.
89
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 31
1
9
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
Prop. Pts.
I
2
3
4
8.24 186
8.24903
8.25609
8.26304
8.26988
717
7 o6
695
68 4
6 73
66 3
653
6 44
634
624
616
608
599
590
583
575
568
560
553
547
539
533
526
520
5H
508
502
496
49 1
485
480
474
470
464
459
455
450
445
441
436
433
427
424
419
416
411
408
404
400
396
393
39
386
382,
379
376
373
369
367
363
8.24 192
8.24910
8.25616
8.26312
8.26996
7 o6
696
68 4
673
66 3
654
643
634
625
617
607
599
584
575
568
553
546
54<>
533
527
520
509
502
496
49 1
486
480
475
470
464
460
455
450
446
437
432
428
424
420
416
413
408
4<H
401
397
393
390
386
383
380
376
373
370
367
363
.75808
75090
'73688
73004
9-99993
9-99993
9-99993
9 99 993
9-99992
00
59
58
P
.X
.2
3
4
5
.1
.3
3
4
5
.x
.9
3
4
5
.x
.3
3
4
5
.x
.3
3
4
5
.1
.9
3
4
.5
.1
.3
3
4
5
.1
.3
3
4
5
.2
3
4
5
717
71.7
143-4
215.1
286.8
358-5
653
65-3
130.6
195.9
261.9
396.5
599
59-9
119.8
179.7
939.6
299.5
553
55-3
110.6
165.9
221.2
276.5
5-4
102.8
154.2
205.6
957.0
480
48
96
44
199
940
9
135
1 80
225
420
4*
84
126
x68
310
390
39
78
117
'95
695
69-5
139.0
208.5
278.3
347 5
634
63-4
126.8
190.2
253.6
317.0
583
58.3
116.6
174-9
933.2
291.5
539
53-9
107.8
161.7
215.6
269.5
Soa
50.2
100.4
150.6
200.8
251.0
470
47
94
188
*35
^
i88
132
176
220
4 IO
41
82
123
164
205
3 80
38
76
1X4
190
673
67.3
134.6
201.9
269.3
336.5
6x6
61.6
123. a
184.8
246.4
308.0
568
56.8
113.6
170.4
227.3
284.0
5a6
S a.6
105.3
X57-8
310.4
263.0
490
49
98
147
196
45
460
46
93
38
184
330
430
43
86
X29
179
915
400
40
80
120
1 60
20O
370
37
74
XXX
148
185
9
10
ii
12
13
14
8.27661
8.28324
8.28977
8.29621
8-3025?
8.27669
8.28332
8.28986
8.29 629
8 . 30 263
7233 1
.71668
.71014
.70371
69 737
9.99992
9.99992
9.99992
9-99992
9.99991
55
54
53
52
8.30879
8.31 495
8.32 103
8.32 702
8.33292
8.30888
8-31 So?
8.32 112
8.327II
8.33302
.69 112
9.99991
9.99991
9-99990
9.99990
9.99990
50
It
19
8-33 875
8.34450
8.35018
8-35 578
8.36131
8.33886
8.34461
8.35029
8-35 590
8.36 143
.66 114
65 539
.64971
.64419
63 857
9.99990
9-99989
9.99989
9.99989
45
44
43
42
41
20
21
22
23
24
8.36678
8.37217
8.37750
8.38276
8.38796
8.36689
8.37229
8.37 762
8.38289
8.38809
633"
.62 771
.62 238
.61 711
.61 191
9.99988
9-99988
9.99988
9.99987
9.99987
40
39
38
i
29
"30"
32
33
34
8.39310
8.39818
8.40320
8.40816
8.41 307
8.39323
8.39832
8.40334
8.40 830
8.41 321
.60677
.60168
.59666
.59170
58679
9.99987
9.99986
9.99986
9.99986
9.99984
9.99984
35
34
33
32
8.41 792
8.42272
8.42 746
8.43216
8.43680
8.41 807
8.42287
8.42 762
8.43232
8.43696
58 193
57713
57238
56 768
56304
30
29
28
11
35
36
39
8.44 139
8-44594
8.45044
8.45489
8-45 93
8.44 156
8.44611
8.45061
8-45 507
8.45948
-55844
55389
54939
54493
54052
9-99983
9.99982
9.99982
25
24
23
22
21
40
42
43
44
8.46366
8.46 799
8.47 226
8.47650
8.48069
8.46385
8.46817
8.47245
8.47669
8.48089
.53615
53 183
52 755
52331
Si 9ii
9.99982
9-9998I
9.99981
9.99981
9.99980
20
18
\l
1
49
8.48485
8.48896
8.49304
8.49 708
8.50 108
8.48505
8.48917
8.49325
8.49 729
8.50 130
5 1 495
5i 083
50675
.50271
49 870
9.99980
9-99979
9-99979
9-99979
9.99978
15
14
13
12
II
50
1 5I
52
53
59
8.50 504
8.50 897
8.51287
8.51 673
8.52055
8-50527
8.50920
8.51 310
8.51 696
8.52079
49473
.49 080
.48 690
.48 304
47 921
9.99978
9-99977
9-99977
9-99977
9.99976
10
i
5
4
3
2
I
8.52434
8.52810
8-53 183
8.53552
8.53919
8.52459
8.52835
8.53 208
8-53578
8.53945
47 541
47 165
.46 792
.46422
46055
9.99976
9-99975
9-99975
9-99974
9-99974
00
8.54282
8.54308
1.45692
9-99974
L. Cos.
(1.
L. Cotg.
c.d.
L. Tang.
L. Sin.
/
Prop. Pts.
88
TABLE IV.
2
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
Pro]
). PtS
>.
I
2
3
4
8.54282
8 . 54 642
8-54999
8-55354
8-55 705
360
357
355
35i
349
8.54308
8.54669
8.55027
8.55382
8-55734
361
358
355
352
349
1.45692
I.4533I
44973
.44618
.44266
9-99974
9-99973
9-99973
9.99972
9.99972
GO
59
58
11
.1
.2
3
360
36
72
108
350
35
70
105
340
3*
a
102
I
9
8.56054
8 . 56 400
8.56743
8.57084
8.57421
346
343
341
337
336
8.56083
8.56429
8.56773
8.57114
8-57452
346
344
34i
338
006
.43917
43 571
43 227
.42886
42 548
9.99971
9.99971
9.99970
9.99970
9.99969
55
54
53
52
5i
4
5
.6
7
.8
144
180
216
252
288
140
75
210
245
280
I 3 6
170
204
2 3 8
2 7 2
10
it
12
13
14
8-57757
8 . 58 089
8.58419
8.58747
8.59072
333
330
328
325
323
8.57 788
8.58121
8 58451
8.58779
8.59105
333
330
3?8
y
32*
.42 212
.41 879
41 549
.41 221
.40895
9.99969
9.99968
9-99968
9.99967
9-99967
50
8
%
9
.x
.3
3
324
330
33
66
99
315
320
32
6 4
9 6
306
310
3*
6a
93
15
10
17
18
19
8-59395
8.59715
8.60033
8.60349
8.60662
320
3i3
316
313
311
8.59428
8 59749
8.60068
8 . 60 384
8.60698
321
319
316
314
.40572
.40251
-39932
.396l6
.39302
9-99967
9.99966
9.99966
9-99965
9.99964
45
44
43
42
41
*4
5
.6
7
.8
132
165
108
231
264
128
160
192
224
256
124 1
55
186 .
217
248 .
20
21
22
2 3
24
8.60973
8.61 282
8.61 589
8.61 894
8.62 196
309
307
305
302
8.61 009
8.61 319
8.61 626
8.61 931
8.62234
310
307
305
303
.38991
.38681
.3f374
.38069
37 766
9.99964
9.99963
9.99963
9.99962
9.99962
40
39
38
y
y
.X
.a
3
297
300
30
60
90
288
290
29
58
87
27$
285
28.-
57-<
85-5
s
3
29
8.62497
8.62 79!
8.63091
8.63385
8.63678
298
296
294
293
8-62 535
8.62834
8.63 131
8.63426
8.63 718
299
297
295
292
.37465
.37 166
36869
.36574
.36282
9.99961
9.99961
9.99960
9.99960
9-99959
35
34
33
32
3*
4
.5
.6
7
.8
120
ISO
180
210
240
116
145
174
203
232
114.0
142.5
171.0
199.5
228.0
BO
3i
32
33
34
8.63968
8.64256
8.64543
8.64827
8.65 no
288
287
284
283
281
8.64009
8.64298
8.64585
8.64870
8.65 154
289
,.87
"85
284
281
35 99i
35 702
35415
35 130
.34846
9-99959
9.99958
9-99958
9-99957
9.99956
80
3
11
9
.x
.2
3
270
280
28.0
S 6.0
84.0
275
27.5
55-0
82.5
256.5
270
27.0
54-0
81.0
35
36
*
1 39
8-65391
8.65670
8.65947
8.66223
8.66497
279
277
276
274
8.65435
8.65 715
8.65993
8.66269
8.66543
280
278
276
274
.34565
34285
.34007
33 73i
33457
9.99956
9-99955
9-99955
9-99954
9-99954
25
24
23
22
21
4
5
.6
7
.8
112.
I40.O
168.0
196.0
224.0
IIO.O
137.5
165.0
192.5
22O.O
108.0
135-0
162.0
189 o
2ld
40
4i
42
43
44
8.66769
8.67039
8.67308
8.67575
8.67841
270
269
267
266
267
8.66816
8.67087
8.67356
8.67624
8.67890
271
269
268
266
_g.
33 184
32913
.32644
.32376
.32 no
9-99953
9.99952
9-99952
9-99951
9-99951
20
JQ
!2
.1
.2
3
265
.26.5
53-0
79-5
260
.26.O
.52.0
. 7 8.0
255
25.5
.51.0
.76.5
9
%
49
8.68 104
8.68367
8.68627
8.68886
8.69 144
263
260
259
258
2-5
8.68 154
8.68417
8.68678
8.68938
8.69 196
263
261
260
258
.31 846
31 583
.31 322
.31 062
30804
9-99950
9-99949
9.99949
9.99948
9.99948
15
H
13
12
II
-4
5
.6
7
.8
132-5
159.0
185-5
212.0
2^8 <;
I04.O
130.0
156.0
182.0
208.0
127.5
153.0
178.5
204.0
50
5i
52
S3
54
8.69400
8.69654
8.69907
8.70159
8.70409
254
253
253
250
8.69453
8.69708
8.69962
8.70214
8.70465
25S
254
252
251
.30547
.30292
.30038
.29786
29535
9 99947
9.99946
9.99946
9-99945
9-99944
10
1
I
.1
.a
3
250
.25.0
.50.0
.75-0
345
.24.5
.49.0
73-5
240
.24.0
.48.0
.72.0
56
Ji
59
8.70658
8.70905
8.71 151
8.71395
8.71 638
949
247
246
344
243
8.70714
8.70962
8.71 208
71453
8.71697
249
248
246
245
244
.29286
.29038
.28 792
28 547
.28 303
9-99944
9-99943
9.99942
9.99942
9.9994:
5
4
3
i
.4
5
.6
7
125.0
150.0
175.0
200.0
295.O
122.5
X47.0
X7I-5
Z96.O
22O.5
I2O.O
144.0
168.0
193.0
2x6.0
GO
8.71 880
8.71940
1.28060
9.99940
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
t
Pro
P. Pfc
i.
1
87
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 33
3
t
L. Sin.
d.
L. Tang.
e.d.
L. Cotg.
L. Cos.
P0p. PtS.
I
8.71880
8.72 120
340
8.71940
8.72 181
241
1.28060
1.27819
9.99940
9.99940
GO
338
334
339
2
8.72359
339
2-0
8 . 72 420
239
1.27580
9-99939
58
.1
33.8
83-4
33.9
3
4
8.72$97
8.72834
330
837
8.72659
8.72896
239
237
2T.6
1.27341
1.27 104
9-99938
9-99938
y
.3
3
47.6
46.8
70.2
45-8
68.7
I
8.73069
8.73303
234
8.73132
8.73366
1.26868
1.26634
9 99937
9.99936
55
54
-4
5
95-2
119.0
93-6
117.0
91.6
"4-5
I
8-73535
8.73767
232
232
8.73600
8.738J2
234
232
i . 26 400
1.26 1 68
9-999J6
9-99935
53
52
.6
7
142.8
166.6
140.4
163.8
137-4
160.3
if
8-73997
229
228
8.74063
231
229
1-25937
9 99934
51
.8
9
190.4
314.2
187.2
2X0.6
183.3
206. x
8.74226
8.74292
1.25 708
9 99934
50
ii
12
8-74454
8.74680
226
226
8.74521
8.74748
229
227
__
1-25479
1.25 252
9 99933
9-99932
4-Q
.,
22.5
22O
32.0
310
3X.6
13
8.74906
8.74974
1.25 026
9.99932
47
.3
45-0
44.0
43-a
H
8.75130
224
8-75 199
225
i . 24 801
9-99931
46
3
67-5
66.0
64.8
16
8-75353
8-75575
222
8.75423
8.75645
222
1-24577
1-24355
9.99930
9 99 9-9
45
44
4
5
90.0
112.5
88.0
XIO.O
86.4
108.0
ii
19
8-75795
8.76015
8.76234
220
8X9
217
8.75867
8.76087
8.76306
220
2X9
1.24133
1.23913
1.23694
9 99929
9.99928
9.99927
43
42
41
.6
7
.8
135-0
157.5
180.0
133.0
154.0
176.0
_-0 n
139.6
151.2
173.8
20
21
22
8.76451
8. 76667
8.76883
216
3l6
8.76525
8.76742
8.76958
3J7
2x6
1-23475
1.23258
1.23 042
9-99926
9.99926
9.99925
40
39
38
9
.1
313
21.2
198.0
308
20.8
304
20.4
23
8.77097
8X4
8.77173
8X5
1.22 827
9.99924
37
.3
42.4
41.6
40.8
24
8.77310
213
8.77387
8X4
I.226I3
9.99923
36
3
6 3 .6
63.4
61.3
25
26
8.77522
8-77 733
311
8.77600
8.77811
813
211
I . 22 400
1.22 189
9.99923
9.99922
35
S4
4
5
84.8
1 06.0
83.2
104.0
81.6
103.0
%
29
8-77943
8.78152
8.78360
309
208
208
8 . 78 022
8.78232
8.78441
211
2IO
209
208
I. 21 978
1. 21 768
I-2I559
9.99921
9 99 920
9.99920
33
32
w
.6
7
.8
9
127.2
148.4
169.6
190.8
124.8
145.6
166.4
187.2
122.4
142.8
163.2
183.6
30
8.78508
8.78649
I- 21 351
9.99919
S 2
8.78774
8.78979
205
8.78855
8.79061
206
206
I 21 145
1.20939
9.99918
9.99917
28
.1
301
2O. I
197
19.7
193
19.3
33
34
8.79183
8.79386
204
203
2O2
8 . 79 266
8.79470
205
204
1.20734
1 . 20 530
9 99917
9.99916
z
.3
3
40.2
60.3
39-4
59- 1
38.6
57-9
35
36
9
39
10"
8.79588
8.79789
8.79990
8.80 189
8.80388
201
201
199
199
197
197
8.79673
8.79875
8 80076
8.80277
8.80476
203
202
201
201
199
I 9 8
x 9 8
1.20327
1.20 125
1 . 19 924
1 . 19 723
1 . 19 524
9-999I5
9.99914
9 999'3
9-999I3
9.99912
25
24
23
22
21
20"
19
4
5
.6
7
.8
9
100.5
120.6
140.7
160.8
180.9
189
98.5
1x8.2
137-9
157.6
185
772
96-5
1x5.8
I35.I
154-4
173-7
181
8.80585
8.80782
8 80674
8.80872
1 . 19 326
I.I9 128
9.99911
9.99910
42
8.80978
196
8.81068
196
1.18932
9.99909
18
i
18.9
18.5
18.1
43
8.81 173
X 95
8.81 264
196
I.I8736
9.99909
17
.2
37.
37.0
36.3
44
8.81 367
194
8.81459
195
1.18 541
9.99 908
16
3
56.7
__ f
55-5
54-3
45
46
49
8.81 560
8.81 752
8.81 944
8.82 134
8.82324
192
192
190
190
1 80
8.81 653
8 81 846
8.82038
8.82 230
8.82420
194
193
192
192
190
1.18347
1.18154
1.17962
1.17 770
1.17 580
9.99907
9.99906
9-99905
9.99904
9.99904
15
13
12
II
4
5
.6
7
.8
75.0
94-5
132.3
151.9
170.1
74.0
92.5
XXX.
129.5
148.0
166.5
73.4
90.5
108.6
126.7
144-8
162.0
50
8.82513
8.82 701
188
8.82610
8.82 799
190
x89
1.17390
I.I7 201
9.99903
9.99 902
10
4
3 a x
52
53
8.82888
8.83075
8.83261
187
x8 7
186
8.82 987
8.83 175
8.83361
1 88
188
1 86
-QJC
I.I70I3
i . 16 825
i . 16 639
9.99901
9.99 900
9.99899
I
.1
.3
3
0.4
0.8
.2
0.3 O.2 O.X
0.6 0.4 o.a
0.9 0.6 0.3
59
8.83446
8.83630
8.83813
8.83996
8.84177
184
183
183
181
x8x
8-83547
8-83732
8.83916
8 . 84 100
8.84282
185
184
x8 4
182
182
i 16 453
1 . 16 268
i . 16 084
1.15900
1.15 718
9-99898
9.99898
9.99897
9.99896
9-99895
5
4
3
2
I
4
5
.6
7
.8
9
.O
4
.8
.2
3.6
1.5 .0 0.5
x.8 .2 0.6
a. i .4 0.7
3.4 .608
9.7 .809
60
8.84358
8 . 84 464
I-I5536
9.99894
o
L. Cos.
d.
L. Cotg.
c.d
L. Tang.
L. Sin.
f
Prop. Pts.
86
TABLE IV.
4
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
Prop. Pts.
I
2
3
4
8.84358
8.84539
8.84 718
8.84897
8.85075
181
179
179
178
177
177
176
US
175
173
173
173
171
171
171
169
169
169
167
168
166
1 66
165
164
164
163
163
162
162
160
161
59
J 59
159
158
157
157
156
155
155
55
154
153
*52
152
152
151
15*
150
150
149
149
148
147
M7
147
146
146
4S
145
8.84464
8.84646
8.84826
8.85006
8.85 185
182
180
180
179
178
15536
15354
15 *74
.14994
.14815
9.99894
9-99893
9.09892
9.99891
9.99891
60
11
H
.x
.2
3
.4
-5
.6
7
.8
9
.1
.2
3
4
5
.6
7
.8
9
.t
.2
3
4
5
.6
7
.8
9
")
.i
.2
3
4
5
.6
7
.8
9
.1
.2
3
4
5
.6
7
.8
9
.2
3
4
5
.6
7
.8
9
z8z
18.1
36.2
54-3
72.4
90.5
108.6
126.7
144-8
162.9
175
i7-5
35-0
53-5
70.0
87-5
105.0
122.5
140.0
157-5
168
.16.8
33-6
50.4
779
17.9
35-8
53-7
71.6
89-5
107.4
125-3
143.2
161.1
173
17-3
34-6
51-9
69.2
86.5
103.8
121. 1
138.4
155-7
166
16.6
/33-2
49.8
66.4
83.0
99.0
116.2
132.8
149.4
159
15.9
31-8
47-7
63.6
79-5
95-4
111.3
127.2
i43-
153
15-3
30.6
45-9
61.2
76.5
91.8
107.1
122.4
"37-7
M7
14.7
29.4
44.1
58.8
73-5
88.2
102.9
117.6
132.3
177
17.7
35-4
53-1
70.8
88.5
106.9
123.9
141.6
*59-3
171
17.1
34-
Si-3
68.4
85.5
102.6
119.7
136.8
153-9
164
16.4
32.8
49.8
65.6
820
98-4
114.8
131-3
147.6
157
15-7
31-4
47-
62.8
78-5
94-3
109.9
125.6
MI- 3 '
IS*
15.1
30.2
45-3
60.4
75-5
90.6
105.7
120.8
135.9
z
O.I
O.S
0.3
0.4
0.5
0.6
0.7
0.8
0.9
I
I
9
8.85 252
8.85429
8.85 605
8.85 780
8.8595!
8-85 363
8.85 540
8.85 717
8.85893
8.86069
177
177
176
176
174
174
174
172
172
171
171
170
169
169
168
167
167
1 66
165
165
165
163
163
163
161
162
160
1 60
1 60
!59
158
158
157
157
156
155
155
155
153
154
152
152
152
151
151
150
'So
149
148
149
147
H7
147
146
146
H 637
.14460
. 14 283
.14107
I393 1
9.99890
9.99 889
9.99888
9-99887
9.99886
55
54
53
S 2
5i
10
ii
12
13
14
8.86 128
8.86301
8.86474
8.86645
8-86816
8.86243
8.86417
8.86591
8.86 763
8.86935
13757
!3 5 8 3
13 409
.13237
.13065
9.99885
9.99884
9.99883
9.99882
9.99881
50
49
48
47
46
15
1 6
\l
I I9
8.86987
8.87 156
8.87325
8.87494
8.87661
8.87 106
8.87277
8.87447
8.87616
8.87785
.12894
. 12 723
.12553
. 12 384
.12 215
9.99880
9.99879
9.99879
9.99878
9.09877
45
4.
43
42
4i
20
21
22
23
2 4
8.87829
8.87995
8.88 ?6i
8.88326
8.88490
8-87953
8.88 126
8.88287
8.88453
8.88618
. 12 047
.11 880
.11 713
II547
.11 382
9-99876
9 99875
9.99874
9.99873
9.99872
40
i
H
2
II
31
32
33
34
8.88654
8.88817
8.88980
8.89 142
8.89304
8.88 783
8.88948
8.89 in
8.89274
8.89437
.11 217
. 1 1 052
. 10 889
. 10 726
. 10 563
9.99871
9.99870
9 . 99 869
9.99868
9.99867
35
34
33
32
3i
67.5
84.0
100.8
117.6
134-4
8.89464
8.89625
8.89 784
8.89943
8.90 102
8.89598
8.89 760
8.89920
8.90080
8.90240
. 10 402
. 10 240
. 10 080
.09920
.09760
9.99866
9.99865
9.99864
9.99863
9 . 99 862
30
2 9
28
%
151.2
; 162
16.2
33-4
4 8.6
6 4 .8
81.0
97.2
"3-4
129.6
145-8
155
iS-5
31-0
46.5
62.0
77-5
93-o
108.5
124.0
139-5
149
14.9
29.8
44-7
59-6
74-5
89.4
104.3
119.2
I34-J
it
3
39
8.9O260
8.90417
8.90 574
8.90 730
8.90885
8.90399
8.90557
8.90 715
8.90872
8.91 029
.09601
.09443
.09285
.09 128
.08 971
9.99861
9.99860
9-99859
9.99858
.9 99857
25
24
23
22
21
"20"
ii
'to
i 4i
i 4*
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
.08815
.08 660
.08 505
.08350
.08 197
9.99856
9.99855
9-99854
9.99853
9.99852
9
9
49
8.91 807
8.91 959
8.92 no
8.92261
8.92411
8-91 957
8.92 no
8 . 92 262
8 92 414
8.92565
8.92 716
.8.92866
8.93016
8.9.3 165
8-93 313
.08043
.07890
.07 738
.07 586
07435
9.99851
9.99850
9-99848
9.99847
9.99846
15
14
13
12
II
50
Si
52
53
54
8.92 561
8.92 710
8.92859
8.93007
8-93 154
.07 284
07 134
.06 984
.06 835
.06 687
9-99845
9.99844
9-99843
9.99842
9.99841
10
1
I
55
56
R
59
8.93301
8.93448
8-93594
8.93 740
8.93 885
8.93462
8 9^609
8-9.3756
8.93903
8.94049
.06 538
.06 391
.06244
.06 097
05951
9.99840
9-99839
9.99838
9.99837
9.99836
5
4
3
2
I
60
8.94030
8.94195
1.05805
9-99834
L. Cos.
(1.
L. Cotg.
1C. d.
L. Tang.
L. Sin.
f
Prop. Pts.
85 . |
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 35
5
9
L. Sin.
d.
L. Tang-.
c. d.
L. Cots.
L. Cos.
Pro
p. Pfe
1.
I
2
3
4
8.94030
8-94 174
8.94317
8.94461
8.94603
144
MS
144
142
8-94 195
8.94340
8.94485
8.94630
8-94 773
145
145
145
43
144
05805
.05 660
05 5i5
05 370
.05 227
9-99834
9-99833
9.99832
9.99831
9.99830
60
3
3
.X
.3
3
145
14.5
ay o
43-5
143
4-3
38.6
43.9
14X
X4.x
28.9
42-3
\
I
9
8.94746
8.94887
8.95 029
8.95 170
8-95 3io
4
142
4
140
8.94917
8.95060
8.95 202
8.95 344
8.95 486
143
142
142
143
.05 083
.04 940
.04 798
.04 656
04 5H
9.99829
9.99828
9-99827
9.99825
9-99824
55
54
53
52
5i
4
5
.6
7
.8
58.0
73.5
87.0
101.5
116.0
57-2
71.5
85.8
1 00. 1
1x4.4
56.4
70.5
84.6
98.7
XI3.8
10
ii
12
13
14
8-95 450
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0.88253
o 88155
9 99635
9 99633
9.99632
9.99630
9 99629
35
34
33
32
3i
w
3
11
BO
3i
32
33
34
9 "570
9.11 666
9.11 761
9 ii857
9.11 952
9 ii 943
9.12040
9 .12 138
9.1223?
9.12332
0.88057
0.87960
0.87862
o 87765
o 87 668
9 99627
9.99625
9.99624
9 99622
9 . 99 620
35
36
H
39
9-12047
9.12 142
9.12 2^6
9.12331
9.12425
9.12428
9 12 525
9.12621
9.12717
9.12813
o 87 572
o 87475
o 87379
o 87283
0.87 187
9.99618
9.99617
9.99615
9.99613
9.99612
25
24
23
22
21
20"
!1
\l
40
4i
42
' 43
1 44
9.12519
9.12 612
9 12 706
9. 12 799
9 12 892
9 12909
9.13004
9 13099
9 13 194
9.13289
o 87091
0.86996
0.86901
o 86 806
o 86 711
9.99 610
9 99608
9 99607
9 99605
9 99603
8
?
49
50
51
52
53
54
9.12.985
9 13078
9 U i/i
9 13263
9 U355
9 i33 8 4
9 13478
9 13573
9.13667
9.13 761
o 86616
o 86 522
o 86 427
086333
o 86 239
9 99 601
9 99 600
9 99 598
9 99 596
9 99595
15
14
13
12
II
9 13447
9 13539
9.13630
9.13722
9 13813
9 13854
9.13948
9.14041
9-I4I34
9.14227
0.86 146
0.86052
0.85 959
0.85 866
0-85 773
9 99593
9-99591
9-99589
9-99 588
9.99586
10
1
I
P
y
59
9.13904
9-13994
9.14085
9-I4I75
9.14266
9.14320
9.14412
9.14504
9-14597
9.14688
0.85 680
0.85 588
0.85 496
o 85403
0.85 312
9.99584
9.99582
9.99581
9-99 579
9-99577
5
4
3
2
I
"0"
loo
9.I4356
9.14780
0.85 220
9-99575
L. Cos.
d.
L. Cotg.
c. d.
L. Tang.
L. Sin.
f
Prop. Pts.
82
8
TABLE I\
r
8
9
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
F
ro
P.;
Pts
b
2
3
4
9-I4356
9-14445
9-14535
9.14624
9.14714
89
90
8 9
90
80
9.14780
9.14872
9.14963
9-15054
9-i5 i45
99
9*
9*
9
0.85 220
0.85 128
0.85 037
o . 84 946
0.84855
9 99575
9-99574
9-99572
9-99570
9.99568
(JO
5 si
1
9
i 9
.2 18
.3 27
a
.2
1
i
.1
27
x
.1
.2
^
go
9.0
27.0
i
1
9
9.14803
9.14891
9.14980
9.15069
9-15 157
88
89
89
88
88
9.15236
9-15327
9-15417
9.15508
9 15 598
9 1
90
9
90
o . 84 764
0.84673
0.84583
o . 84 492
0.84402
9.99566
9.99565
9 99563
9.99561
9-99559
55
54
53
52
5 1
.436
5 46
6 55
.7 64
18 I 3
.8
.0
.2
4
.6
36
45
8
P
4
i
i
36.0
45-Q
54.0
63.0
P'
10
II
12
'3
H
9.15245
9-15333
9.15421
9-I5508
9-I5596
88
88
87
83
87
9.15688
9-15777
9.15867
9.I5956
9.16046
89
90
89
90
80
0.84312
0.84223
0.84133
0.84044
0.83954
9-99557
9-99556
9-99554
9-99 SS 2
9 99550
50
8
S
9 82
.2
3
.8
!
i
2(
81
*9
5-9
7-8
37
9
i
i
81.0
M
$.8
a
hi
jS'
19
9-15683
.9.15 770
9-I5857
9-15944
9.16030
87
87
87
86
86
9-16135
9.16224
9.16312
9.16401
9.16489
89
88
89
88
88
0.83865
0.83 776
0.83688
0.83 599
0.83 511
9-99548
9.99546
9-99545
9 99543
9-99541
45
44
43
42
41
-4
1
.:i
3.
*
i
5-0
1-5
5-4
z-3
1.2
3.
4^
r
7<
) 2
:i
1.6
>-4
20
21
22
23
2 4
9.16 116
9 16203
9.16289
9-16374
9 16 460
87
86
85
86
8e
9-I6577
9.16665
9-16753
9.16841
9.16928
88
88
88
87
88
0.83423
0-83335
0.83247
0.83159
0.83072
9-99539
9-99537
9 )9 535
9i 99 533
9 99532
40
3.
11
9
.1
.2
3
a
ck
1
}
I
J. 1
17
5-7
7-4
^ (
J
i
2
^.2
36
?.6
tl
'2
2
29
9-16545
9.16631
9.16 716
9.56801
9.16886
86
85
85
85
BA.
9.17016
9.17 103
9.17 190
9.17277
9 17363
87
87
87
86
8?
0.82984
o 82897
0.82 810
0.82 723
o 82 637
9 99530
9.99528
9-99526
9 99524
9-99 522
35
34
33
32
3i
3
is
3<
4.
I
6(
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^&
J-5
Z.2
3.9
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3<
4.
1
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J-o
[.6
5.2
$.8
80
31
S
34
9.16970
9-I7055
9.17139
9-17223
9.17307
85
84
84
84
4
9.I7450
9.17536
9.17 622
9.17708
9.17794
86
86
86
86
86
0.82 550
0.82464
0.82378
0.82 292
0.82 206
9.99520
9.99518
9-995I7
9 99515
9 99513
30
1
9
.1
.2
3
7 J
i
i
2.
>-o
!5
5-5
7-0
>-5
7
J
i<
2
7-4
H
^:S
J -?
i
39
9.17391
9-17474
9.17558
9.17641
9.17724
83
84
8 3
8 3
8l
9.17880
9.17965
9.18051
9.18 136
9.l8 221
85
86
85
85
a-
O.82 I2O
0.82035
0.81 949
0.81 864
0.81 779
9.99511
9 99509
9 99507
9 99505
9 99503
25
24
23
22
21
:!
i
#
*
5
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2-5
[.0
1:1
3r
3.
4^
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7
J-6
s.o
^
7-2
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40
4?
42
43
44
9.17807
9.17890
9-17973
9.18055
9.18137
8 3
8 3
82
82
8?
9.18306
9.I839I
9-I8475
9.18560
9.18644
85
84
85
84
g.
0.81 694
0.81 609
0.81 525
0.81 440
0.81 356
9.99501
9.99499
9 99497
9 99495
9 99494
20
18
\l
.1
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3
/ l
1
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K
2*.
J
3
1:2
t 9
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1
i
K
2i
w
to
$.2
il
45
46
49
9.18220
9.18302
9-18383
9-18465
9-18547
82
81
82
82
81
9.18728
9.I88I2
9.18896
9.18979
9.19063
84
84
8 3
84
Q,
0.81 272
0.81 188
0.81 104
0.81 021
0.80937
9.99492
9.99490
9.99488
9.99486
9.99484
15
14
13
12
II
'4
3.
4
I
7
i- 2
[ -5
).8
J.I
> 4
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4
4<
i
7'
S.o
[.O
)-2
^
}g
SO
5'
52
53
54
9:18628
9.18709
9.18790
9.18871
9-18952
8x
Si
81
81
81
9.I9I46
9.19229
9.I93I2
9-19395
9.19478
8 3
83
8 3
83
83
0-
0.80854
0.80771
0.80688
0.80605
0.80522
9.99482
9.99480
9.99478
9.99476
9 99474
10
I
I
1
.1 8
.2 16
3 24
I
.1
.2
3
/
a
S
16
24
/.
.0
.O
.0
0.2
04
0.6
A X
11
H
r59
9 1933
9.19113
9.19 193
9 19273
9 19353
80
80
80
80
flrt
9.I956I
9.19643
9.19725
9 19807
9.19889
*3
82
82
82
82
0.80439
0.80357
0.80275
0.80 193
0.80 in
9.99472
9.99470
9.99468
9.99466
9.99464
5
4
3
2
I
4 3 2
5 4Q
.6 48
7 56
.8 64
.0 72
4
i
. y
3 2
?
7?
.0
.0
:.o
.0
o
1.0
1.2
M
1.8 1
j60
9 19433
9.19971
0.80029
9.99462
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
f
p
ro
p.]
Pfe
.
81
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 39
9
t
L. Sin.
d.
L. Tangr.
c.d.
L. Cotg.
L. Cos.
I
to
P.:
Pfe
b
I
2
3
4
9-19433
9-i95i3
9.19592
9 19672
9 I975I
80
79
80
79
70
9.19971
9.20053
9 20 134
9.20216
9.20297
8a
81
82
81
8x
0.80029
0.79 947
0.79 866
0.79784
0.79703
9.99462
9.99460
9-99458
9-99456
9-99454
00
59
58
11
8
.1 8
2 16
3 24
2
2
4
.6
!
8
16
24
I
.1
.2
3
80
8.0
16.0
24.0
i
t
9
9.19830
9.19909
9.19988
9.20067
9.20145
79
79
79
78
.0
9.20378
9.20459
9.20540
9.20621
9.20 701
8x
81
81:
80
RT
0.79622
0.79541
0.79460
0.79379
0.79299
9-99452
9-99450
9-99448
9-99446
9-99444
55
54
53
52
5i
4 32
5 41
-6 49
ig
.8
.0
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.4
.6
32
4 c
4*
4
. ^
o
i
32.0
40.0
48.0
56.0
64.0
10
ii
12
'3
14
9.20223
9.20302
9.20380
9 . 20 458
9 20535
79
78
78
77
.0
9.20 782
9.20862
9.20942
9.21 022
9.21 102
80
80
80
80
0-
0.79218
0.79138
0.79058
0.78978
0.78898
9.99442
9.99440
9-99438
9 99436
9-99434
50
3
3
9 73
.1
.2
3
.8
,
t
I
2;
yu
79
-1
J-7
y
i
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2:
72.0
rt
3
J-4
II!
!1
19
9.20613
9.20691
9 . 20 768
9.20845
9 . 20 922
78
77
77
77
9-21 l82
9.21 26l
9 21 341
9.21 420
9 21 499
79
80
79
79
0.78818
0.78739
0.78 659
0.78580
0.78501
9-99432
9.99429
9.99427
9 99425
9 99423
45
44
43
42
41
4
i
1
3
3 (
4
I:
[.to
?-5
7-4
5-3
5-2
3
3<
1
t.2
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J.6
20
21
22
23
24
9.20999
9 21 076
9 21 153
9 21 229
9 21 306
77
77
76
77
_g
9 21 578
9 21 657
9 21 736
9 21 814
9 21 893
79
79
79
78
79
_0
0.78422
0.78343
0.78 264
0.78 186
0.78 107
9.99421
9.99419
9;994i7
9 99415
9.994I3
40
I
9
.1
.2
3
7
-
i
2.
77
r-7
5-4
J -J
7 (
|
i
i.
2:
?
T-6
Ii
2 5
26
3
29
9 21 382
9 21 45 8
9 21 534
9 21 610
9.21 685
76
76
76
75
76
9 21 971
9.22049
9.22 127
9 22 2O5
9 22 283
78
78
78
78
_Q
0.78029
0.77951
0.77873
0-77795
o 77 717
9 99411
9.99409
9.99407
9.99404
9 99402
35
34
33
32
3i
4
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6
3.8
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4.
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d
;.6
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so
31
32
33
34
9 21 761
9 21 836
9 21 912
9.21 987
9 . 22 O62
7
75
76
75
75
9.22361
9 22438
9.22 516
9-22593
9.22 670
77
78
77
77
o 77 639
o 77562
0.77484
o 77407
0.77330
9 99400
9 99398
9.99396
9 99394
9.99392
ao
29
28
n
.2
3
I.
2:
3
75
7-S
>.o
2-5
I
I
I;
2:
J -4
74
[1
1.2
P
ii
39
9 22137
9 22 21 1
9.22286
9 22361
9 22435
75
74
75
75
74
9 22 747
9.22824
9.22901
9.22977
9 23054
77
77
77
76
77
o 77253
o 77176
0.77099
o 77023
o 76946
9 99390
9.99388
9 99385
9 99383
9 99381
25
24
23
22
21
4
ii
:I
3<
3
4.
g
6
J.O
7-S
5-0
2-5
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2C
3'
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5
1!
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7.0
f:|
i 6
40
4i
42
1 43
i 44
9 22509
9 22583
9 22657
9 22731
9.22 805
74
74
74
74
74
9 23 130
9.23 206
9 23283
9 23359
9-23435
76
76
77
7
76
0.76870
0.76794
0.76717
o 76 641
0.76565
9 99379
9 99377
9 99375
9 99372
9 99370
20
19
18
17
16
V
.1
.2
3
D
1.
2
5
73
I'l
'9
i
i4
2
.<
^a
7.2
50
11
;i
49
9.22878
9.22952
9 23025
9.23098
9.23 171
73
74
73
73
73
9.23510
9 23586
9.23661
9 23737
9.23812
75
76
75
76
75
o 76 490
o 76414
o 76339
0.76263
o 76188
9 99368
9 99366
9 99 364
9 99 3 6 2
9 99 359
5
14
13
12
11
4
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II
2<
3<
4.
1
H
j'8
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5-4
7
2c
3<
4:
5<
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5-2
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, 8
50
5i
52
53
54
9.23244
9 23317
9.23390
9.23462
9 23535
73
73
73
72
73
9-23887
9-23962
9.24037
9.24 112
9.24 186
75
75
75
75
74
0.76 113
o . 76 038
0-75 963
0.75888
0.75814
9 99357
9 99355
9-99353
9 99351
9 99348
10
I
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i 7
2 14
3 21
. -0
".
i
.1
2
-3
C
c
C
3
'3
6
9
O.2
4
0.6
r 8
it
12
59
9.23607
9.23679
9.23752
9.23823
923895
72
72
73
7i
7*
9.24261
9-24335
9.24410
9-24484
9.24558
75
74
75
74
74
0-75 739
0.75665
0.75590
o.755i6
0.75442
9 99346
9 99344
9 99342
9-9934C
9 99337
5
4
3
2
I
.4 2%
:i2
:I3
96?
4
I
I
n
I
I
2
2
1
I
i
4
7
I.O
1.2
Ji
1.8
60
9.23967
72
9.24632
74
0.75368
9-99335
L. Cos.
d.
L. Cot?.
c.d.
L. Tang.
L. Sin.
f
r
ro
P
Pfc
u
80
4
TABLE IV.
10 I
/
L. Sin.
d.
L. Tang.
c.d.
L. Cot?.
L.Cos.
Prop. Pts. |
i
2
6
4
9-23967
9-24039
9.24 no
9.24 181
9 24253
72
7*
7*
72
7
7
7
70
7*
70
7
70
70
70
70
70
70
69 '
7<>
69
69
69
69
69
69
69
63
69
68
68
68
68
68
68
68
67
68
67
67
67
67
67
67
67
66
67
66
67
66
66
66
66
65
66
66
65
65
66
65
65
"dT
9-24632
9.24706
9-24779
9-24853
9 . 24 926
74
73
74
73
74
73
73
73
73
73
72
73
72
73
7*
7
72
72
72
7
7*
7*
73
7*
7
7*
7
70
7*
7 1
70
7<>
7*
70
70
70
70
69
70
69
70
69
69
69
69
69
69
69
63
69
68
69
68
68
68
68
67
68
68
67
0.75368
0.75294
0.75 221
0-75 H7
0.75074
9-99335
9-99333
9-99331
9.99328
9.99326
>0
P
11
.1
.2
3
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9
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A
e
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3
4
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74
il'i
22.2
29.6
37-o
Si
11:1
7
7-2
14.4
21.6
28.8
36.0
43 2
50.4
III
70
7-0
14.0
21.0
28.0
35-0
42.0
490
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9-2439?
9.24466
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9.24607
9.25 ooo
9-25073
9.25 146
9-25219
9-25292
0.75 ooo
0.74927
0.74854
0.74781
0.74708
9 99324
9-99322
9-993I9
9-993I7
9-9931?
55
54
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9.24677
9.24748
9.24818
9.24888
9.24958
9-25365
9-25437
9.25 5io
9-25 582
9-25655
0.74635
0.74563
0.74490
0.74418
0.74345
9 993*3
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9-99308
9.99306
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50
49
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47
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21
22
23
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9 . 25 028
9-25098
9.25 1 68
9 25237
9 25307
9.25 727
9-25 799
9.25871
9-25 943
9.26 015
0.74273
0.74201
0.74129
0.74057
0.73985
9.99301
9.99299
9-99297
9.99294
9-99292
45
44
43
42
41
16"
P
35
34
33
32
3i
9-25 376
9-25445
9.25514
9-25583
9 25652
9.26086
9.26 158
9.26229
9.26301
9.26372
0.73914
0.73842
0.73771
o 73699
0.73628
9-99290
9.99 288
9.99285
9.99283
9.99281
3
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9.25 721
9-25 790
9-25858
9-25927
9-25995
9.26443
9.26514
9-26585
9.26655
9.26 726
0.73557
0.73486
0.73415
0.73345
o 73 274
9.99278
9 99276
9-99274
9-99271
9.99269
80
3i
32
33
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9.26063
9.26 131
9.26199
9.26 267
9 26335
9-26 797
9.26867
9.26937
9.27008
9.27078
0.73203
0.73133
0.73063
0.72992
0.72 922
9.99267
9-99264
9.99 262
9.99260
9 99257
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41
42
43
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9 . 26 403
9.26470
9 26538
9.26605
9.26672
9.27 148
9.27218
9.27288
9-27357
9.27427
0.72 852
0.72 782
0.72 712
0.72643
0.72573
9.9925?
9.99252
9.99250
9.99248
9 99245
25
24
23
22
21
9 . 26 806
9.26873
9 . 26 940
9.27007
9-27496
9-27566
9-27635
9.27704
9-27 773
0.72504
0.72434
0.72365
0.72 296
0.72 227
9 99243
9.99241
9.99238
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9.27 140
9.27 206
9.27273
9 27339
9.27842
9.27911
9.27980
9.28049
9.28 117
0.72 158
0.72 089
0.72 020
0.71 951
0.71 883
9.99231
9-99229
9.99226
9 99 224
9 99 221
15
14
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9
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9.27405
9.27471
9-27537
9.27 602
9.27668
9.28 186
9-28254
9.28323
9-28391
9.28459
0.71 814
0.71 746
0.71677
0.71 609
0.7I54I
9.99219
9.992I7
9-99214
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10
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9-27799
9.27864
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9-27995
9 -*So6o
9-28527
9.2859!
9.28662
9.28730
9.28 798
0.7M73
0.71405
0.71338
0.71 270
0.71 202
9.99207
9.99204
9.99202
9.99200
9-99 197
5
4
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9.28865
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L* Cos.
L. Cotg.
c. d.
L. Tang.
L. Sin.
t
Prop. Pts.
79
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 41
i 11
f i iu MIL d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
Prop. Pts.
1 9.28060
i 1 9.28 125
2 1 9.28 190
3 9-28254
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65
6 4
65
65
64
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64
64
64
63
64
6 3
63
64
63
63
63
63
63
63
62
63
62
62
63
6a
62
61
6a
62
61
62
61
6a
61
61
61
61
61
61
60
61
60
61
60
61
60
60
60
60
59
60
60
59
60
9.28865
9-28933
9.29000
9.29067
9.29 134
68
67
67
67
67
67
67
67
66
67
66
67
66
66
66
66
66
66
66
65
65
65
65
65
65
64
65
64
65
64
65
64
64
64
64
63
64
63
64
6 3
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63
63
63
63
63
63
63
62
62
63
62
62
62
0.71 i35
0.71 067
0.71 ooo
0.70933
0.70 866
9-99 195
9-99 192
9-99 190
9-99 187
9-99 185
GO
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9-99 172
55
54
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47
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9-29535
9.29 601
9.29668
9-29 734
9.29 800
0.70465
0.70399
0.70332
0.70 266
0.70 200
9-99 170
9.99 167
9-99 165
9.99 162
9.99 160
15 1 9.29024
16 I 9.29087
17 9.29 150
18 1 9.29214
19 1 9.29277
9.29866
9.29932
9-29998
9.30064
9.30 130
0.70 134
0.70068
0.70002
0.69936
0.69 870
9-99 157
9-99 155
9-99 152
9.99150
9 99 H7
45
44
43
42
41
1T
P
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20
21
22
2 3
24
9.29340
9.29403
9.29466
9 29 529
9.29591
9-30I95
9.30261
9.30326
9-30391
9-30457
0.69 805
0-69 739
0.69674
0.69609
0.69 543
9 99H5
9-99 '42
9-99 HO
9-99 137
9-99 135
2 7
29
30
32
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9-29654
9.29716
9.29 779
9.29841?
9-29903
9 . 29 966
9 . 30 028
9.30090
9 30151
9 30213
9.30275
9-30336
9-30398
9.30521
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9-30587
9.30652
9.30717
9.30782
0.69478
0.69413
0.69348
0.69 283
0.69 218
9-99 132
9 99 13
9 99 127
9-99 124
9.99122
35
34
33
32
9.30846
9.30911
9-30975
9.31 040
9.31 104
0.69 154
0.69 089
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9.99119
9.99117
9 99 "4
9.99 112
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30
11
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9 -31 233
9.31 297
9.31 361
0.68832
0.68 767
0.68 703
0.68639
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9.99 106
9-99 104
9.99 101
9.99099
9.99096
25
24
23
22
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9-30643
9.30 704
9-30765
9 . 30 826
9.31489
9-31 55|
9.31 616
9.31 679
9-3i 743
0.68511
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0.68321
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9.99093
9.99091
9.99088
9.99086
9.99083
20
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46
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9-30947
9.31 008
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9-32059
0.68 194
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9-99075
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52
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9-32248
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0.67878
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0.67 752
0.67 689
0.67 627
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9.99064
9 . 99 062
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9-99056
10
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9-31 549
9.31 609
9.31 669
9.31 728
9-32436
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9.32561
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0.67 564
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0.67439
0.67377
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9-99054
9.99051
9.99048
9-99046
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5
4
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9.99040
L. Cos.
d.
L. Cotg. c. d.
L. Tans.
L. Sin.
t
Prop. Pts.
78
TABLE IV.
12
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
p
rop. ]
Pts.
I
2
3
4
9.31 788
9-3 1 847
9.31 907
9.31 966
9.32025
59
60
59
59
rg
9-32747
9.32810
9.32872
9.32933
9-32995
63
62
61
62
62
0.67253
0.67 190
0.67 128
0.67067
0.67 005
9.99040
9.99038
9-99035
9.99032
9.99030
(JO
3
11
.1
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3
63
6.3
12.6
18.9
62
6.2
12 4
18.6
I
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9
9.32084
9-32 H3
9.32202
9.32261
9-32 3*9
59
59
59
58
9-33057
9-33 "9
9-33 * 80
9-33242
9.33303
62
61
62
61
62
o . 66 943
0.66881
0.66 820
0.66758
0.66697
9.99027
9.99024
9 . 99 022
9.99019
9.99 016
55
54
53
S 2
5i
4
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I
25.2
3J:i
44.1
50.4
24.8
31.0
37-2
43-4
49-6
10
ii
12
13
14
9-32378
9.32437
9-32495
9-32553
9.32 612
59
58
58
59
eg
9.33365
9.33426
9.33487
9-33548
9-33609
61
61
61
61
61
0.66635
0.66574
0.66 513
0.66452
0.66391
9.99013
9.99011
9.99008
9.99005
9.99002
50
3
s
9
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3
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61
6.1
12.2
18-3
55- 8
60
6.0
12.0
18.0
II
ii
19
9-32670
9.32 728
9-32 786
9-32844
9.32902
58
58
58
5
t-8
9.33670
9-33 73 1
9-33 792
9.33853
9.33913
61
61
61
60
61
0.66 330
0.66 269
0.66 208
0.66 147
0.66087
9.99 ooo
9.98997
9.98994
9.98991
9 . 98 989
45
44
43
42
41
4
ii
24.4
30-5
3 6.6
42-7
48.8
24.0
30.0
36.0
42.0
48.0
20
21
22
\ 23
24
9.32960
9.33018
9-33075
9-33 133
9-33 190
58
57
58
57
eg
9-33974
9-34034
9-34095
9.34155
9.34215
60
61
60
60
6z
0.66 026
0.65 966
0.65 905
0.65 845
0.65 785
9.98 986
9.98983
9.98980
9.98978
9-98975
40
3
11
9
54-9
5
.i 5
.2 II
3 IJ
54-
9
'I
2 5
26
3
29
9-33248
9-33305
9-33362
9-33420
9-33477
57
57
58
57
9.34276
9.34336
9.34396
9.34456
9-345I6
60
60
60
60
60
0.65 724
0.65 664
0.65 604
0.65 544
0.65 484
9-98972
9.98969
9.98967
9-98964
9.98961
35
34
33
32
31
4 *J
.5 29
6 35
.7 4i
.8 45
Q H^
.6
5
4
-3
.2
IT
30
3i
32
33
34
9-33 534
9-33591
9-33647
9-33 704
9-3376I
57
56
57
57
9.34576
9.34635
9.34695
9-34755
9.34814
59
60
60
59
60
0.65 424
0.65 365
0.65 305
0.65 245
0.65 186
9.98958
9.98955
9.98953
9.98950
9.98947
30
3
Z
.1
.2
3
y jj
58
5-8
n. 6
17-4
57
5-7
11.4
17.1
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35
36
%
39
9.338i8
9-33 874
9-33931
9-33987
9-34043
57
56
57
56
56
9-34874
9-34933
9-34992
9-3505I
9-35IH
59
59
59
60
0.65 126
0.65 067
0.65 008
0.64 949
0.64889
9.98944
9.98941
9.98938
9.98936
9-98933
25
24
23
22
21
ii
ii
.0
23.2
29.0
34-8
40.6
46.4
tJ2.2
28.5
34-2
39-9
45-6
51.3
40
4i
42
43
44
9-34 loo
9.34I5 6
9.34212
9.34268
9-34324
56
5
56
56
efi
9-35 170
9-35 229
9-35 288
9-35347
9-35 405
59
59
59
58
o . 64 830
0.64 771
0.64 712
0.64653
0.64595
9.98930
9.98927
9.98924
9.98 921
9.98919
20
ii
11
.1
.2
3
56
5-6
II. 2
16.8
122 4
55
5-5
II.
16.5
22 O
$
9
49
9-3438o
9-34436
9-34491
9-34547
9.34602
5
56
55
56
55
9.35464
9.35 5J3
9-35 58i
9-35 640
9-35698
59
58
59
58
0.64536
0.64477
0.64419
0.64 360
0.64 302
9.98 916
9.98913
9.98910
9.98907
9.98904
15
14
13
12
II
ii
.9
28.0
33-6
39-2
44-8
50.4
27-5
33-0
38-5
44-0
49-5
50
5i
52
53
54
9-34658
9-347I3
9-34769
9-34824
9-34879
5
55
56
55
55
9-35 757
9-35815
9-35 873
9-35931
9-35 989
58
58
58
58
eg
0.64243
0.64 185
0.64 127
0.64069
0.64 on
9.98 901
9.98898
9.98896
9.98893
9.98890
10
I
I
.1
.2
4
3
-3
0.6
0.9
1.2
a
10.2
0.4
0.6
0.8
9
9
1 59
9-34934
9.34989
9-35044
9.35099
9-35 154
55
55
55
55
55
9.36047
9.36 105
9-36 163
9.36221
9.36279
58
58
58
58
0.63953
0.63 895
0.63 837
0.63 779
0.63 721
9.98887
9.98884
9.98881
9.98 878
9.98875
5
4
3
2
I
I
I
i
4
ii
2.1
2.4
2.7
I.O
1.2
H
1.8
60
9-35209
55
9.36336
0.63 664
9.98 872
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
t
I
>rop.
Pts.
77
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 43
1 3 I
I ,
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
Prop. Pts.
9-35209
9-36336
eg
0.63 664
9.98872
60
i
9-35 263
9.36394
eg
o 63 606
9.98869
59
58
57
2
9-35 3 l8
55
9-36452
0.63 548
9.98867
S8
.1
S-8
5-7
3
9-35373
55
9-36509
0.63491
9.98864
57
.2
11.6
II.4
4
9-35427
54
9.36566
58
0.63434
9.98861
56
3
17.4
I7.I
I
9-3548i
9-35 S3 6
55
9.36624
9.36681
57
0.63 376
0.63319
9.98858
9-98855
55
S4
4
23.2
29.0
22.8
28.5
7
9-35590
54
9-36738
57
0.63 262
9.98852
53
.6
34-8
34-2
8
9-35 644
54
9.36 795
57
0.63 205
9.98849
S2
7
40.6
39-9
9
9-35698
54
54
9.36852
57
57
0.63 148
9.98846
51
.8
46.4
45-6
10
9-35 752
9-36909
0.63091
9-98843
50
9
52.2
5^3
ii
9.358o6
54
9.36966
57
0.63 034
9.98 840
49
56
55
12
13
H
9-35 860
9-359H
54
54
9.37023
9.37080
9-37 137
57
57
eg
0.62977
0.62 920
0.62863
9-98837
9.98834
9-98831
48
47
46
.2
-3
5-6
II. 2
16.8
5.5
II.
16.5
11
17
9.36 022
9-36075
9 3 6 129
53
54
9-37 193
9-37250
9.37306
57
56
0.62 807
0.62 750
0.62694
9.98828
9.98825
9.98822
45
44
43
4
22.4
28.0
33-6
22.0
27-5
33-o
18
9.36 182
53
9-373 6 3
57
0.62637
9.98 ^19
42
I
39-2
38-5
19
9-36236
54
53
9.37419
5
0.62 581
9.98816
.8
44.8
44-0
20
9.36289
9-37476
0.62524
9-988i 3
40
9
5'4
49-5
21
9-36342
53
9-37532
e
0.62468
9.98 810
39
54
22
23
24
9-36395
9.36449
9-36502
53
54
53
CO
9.37588
9-37644
9-37 700
5
56
56
0.62412
0.62 356
0.62 300
9.93 807
9.98804
9.98801
38
11
.1 5-4
.2 10.8
3 16.2
%
9.36555
9.36608
53
9.37756
9.37812
56
0.62244
0.62 188
9.98 798
9 98795
35
34
.4 21.6
.5 27.0
27
9.36660
5~
9.37868
50
0.62 132
9.98 792
33
.6 32.4
29
9.36713
9.36766
53
53
51
9-37924
9.37980
5
56
0.62076
o . 62 020
9.98 789
9.98786
32
'I 37-8
.8 43-2
9..Q 6
30
9.36 819
9-38035
0.61 965
9-98783
80
40.0
9.36871
52
9.38091
56
0.61 909
9.98780
29
53
S*
32
33
34
9.36924
9.36976
9.37028
53
52
52
9.38 147
9.38202
9-38257
50
55
55
0.61 853
0.61 798
0.61 743
9.98 777
9.98 774
9.98771
28
11
.1
.2
3
5-3
10.6
15-9
5-2
10.4
15.6
1
39
9-37o8i
9-37I33
9.371S5
9 37237
9-37289
52
52
52
52
9-383I3
9.38368
9-38423
9-38479
9.38534
55
55
56
55
0.61 687
0.61 632
0.61 577
o.Ci 521
0.61 466
9.98768
9.98 765
9.98 762
9-98 759
9.98 756
25
24
23
22
21
4
7
21 .2
26.5
3? 8
37-i
42.4
20.
26.0
3 1.7
36.4
41.6
46 8
40
9-37341
9-37393
52
9-38589
9.38644
55
55
0.61 411
0.61 356
9.98 753
9.98 750
20
19
.
51
4
42
9-37445
5 2
9.38699
55
o.Ci 301
9.98 746
18
.1
b- 1
0.4
43
44
9-37497
9-37549
S 2
52
9! 38 808
55
54
0.61 246
0.61 192
9-98 743
9.98740
\l
.2
3
10.2
15-3
1.2
45
46
9.37600
52
9.38863
9.38918
55
55
0.61 137
0.61 082
9-98 737
9-98 734
15
14
4
25-5
2.0
11
9-37703
9-37755
Si
52
9-38972
9.39027
54
55
0.61 028
0.60973
9.98 73i
9.98 728
13
12
:J
Hi
49
9.37806
5 1
9.39082
55
0.60918
9-98 725
II
O
AC Q
\ 6
50
9.37858
9-39 136
54
0.60864
9.98 722
10
51
9-37909
5 X
9.39 190
54
0.60810
9.98 719
9
3
52
53
54
9.37960
9.38011
9.38062
5*
5
9.39245
9.39299
9-39353
55
54
54
0.60 755
0.60 701
0.60647
9.98 715
9.98 712
9.98 709
I
.1
.2
-3
0.3
0.6
0.9
I*
O.2
O.O
o 8
P
9.38 "3
9.38 164
S
9.39407
9.39461
54
54
0.60593
0.60539
9.98 706
9.98 703
5
4
1:1
1.0
1.2
9-38215
9.38266
5 X
5
9.39515
54
54
0.6048?
0.60431
9.98700
9.98697
3
i
2.1
2 A
59
9-38317
5
9-39623
54
0.60377
9.98694
i
.9
if .4
2.7
1.8
60
9-38368
9-39677
54
0.60323
9 . 98 690
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
t
Prop. Pts.
76
44
TABLE IV.
14
t
L. Sin.
d.
L. Tang.
.d.
L. Coter.
L. Cos.
d.
T]
rop. ]
>ts.
1
I
2
3
4
9-38368
9.38418
9.38469
9-38519
9-38570
so
51
50
51
5
9.39677
9-39 73i
9.39785
9-39838
9-39892
54
54
53
54
53
0.60323
o . 60 269
0.60 215
0.60 162
0.60 108
9.98690
9.98687
9.98684
9.98681
9.98678
3
3
3
3
00
59
58
y
.1
2
54
5-4
10 8
53
IO O
1
I
9
9.38620
9.38670
9-38721
9.38 771
9.38821
So
Si
So
5
SO
9-39945
9-39999
9.40052
9.40 106
9-40 159
54
53
54
53
53
0.60055
0.60001
0.59948
0.59894
0.59841
9-98675
9.98671
9.98668
9.98665
9.98662
4
3
3
3
55
54
53
52
5i
3
-4
.7
16.2
21.6
27.0
SI
15-9
21.2
26.q
318
37-1
10
ii
12
13
H
9.38871
9.38921
9.38971
9.39021
9.39071
So
So
So
So
9.40212
9.40266
9.40319
9.40372
9.40425
54
53
53
53
0.59788
0-59734
0.59681
0.59 628
0-59575
9.98659
9.98 656
9-98652
9.98649
9 . 98 646
3
4
3
.3
60
3
%
.8
9
&6
5*
42-4
47-7
5
\l
11
19
9-39 121
9 39 170
9.39220
9.39270
9.393I9
49
50
50
49
9.40478
9-40531
9.40584
9.40 636
9.40689
53
53
52
53
53
0.59522
0.59469
0.59416
0.59364
0-593"
9-98643
9.98640
9.98636
9-98633
9.98630
3
4
3
3
45
44
43
42
41
.1
.2
3
4
5-2
10.4
15.6
20.8
26.0
5-x
IO.2
15.3
20.4
25-5
20
21
22
23
24
9-39369
9.39418
9-39467
9.395I7
9.39566
49
49
So
49
9.40 742
9-40 795
9.40847
9.40900
9.40952
53
53
53
53
0.59258
0.59205
0.59153
0.59 100
0.59048
9.98627
9.98 623
9 98 620
9.98617
9.98 614
4
3
3
3
40
39
38
11
i
9
31.2
36.4
41.6
46.8
30.0
35-7
40.6
45-9
>
8
3
29
9-396i5
9.39664
9-397I3
9.39762
9.39811
49
49
49
49
9-41 005
9-4i 057
9.41 109
9.41 161
9.41 214
53
53
53
53
52
0.58995
0.58943
0.58891
0.58839
0.58786
9.98 610
9.98607
9.98604
9.98601
9-98597
3
3
3
4
35
34
33
32
3i
.1
2
3
A
50
5-0
10.
15.0
20 o
49
4-9
9.1
14.7
IQ.6
80
3i
32
33
34
9.39860
9.39909
9-3995?
9.40006
9-40055
49
49
48
49
48
9.41 266
9.41 318
9.41 370
9.41 422
9.41 474
53
53
53
53
5 a
0.58 734
0.58682
0.58630
0.58578
0.58 526
9.98594
9.98591
9-98588
9.98584
9.98581
3
3
4
3
80
3
2
:!
.9
25.0
30.0
35.0
40.0
45.0
24.5
29.4
34-3
39.2
44.1
P
H
39
9-40 103
9-40 152
9.40200
9.40249
9.40297
49
48
49
48
9-41 526
9.41 578
9.41 629
9.41 68 i
9-41 733
5
5
5
53
0.58474
0.58422
0.58371
0.58319
0.58267
9.98578
9.98574
9-9857I
9-98568
9-98565
4
3
3
3
25
24
23
22
21
.1
.2
48
4-8
9.6
47
4-7
9.4
40
4i
42
43
44
9.40346
9 40394
9-40442
9.40490
9-40538
48
48
48
48
A a
9.41 784
9.41 836
9.41 887
9.4I939
9.41 990
5*
Si
53
S
0.58 216
0.58 164
0.58 113
0.58061
0.58010
9.98 561
9.98558
9.98555
9.98551
9-98548
3
3
4
3
20
18
II
3
4
14.4
19.2
24.0
28.8
3,V6
ls',8
S:ll
32.9 1
3
s
49
9.40586
9.40634
9.40682
9.4073
9.40 778
48
48
48
48
9.42041
9.42093
9.42 144
9.42 195
9.42 246
53
51
Si
5*
0-57959
0.57907
0.57856
0.57805
0-57754
9-98545
9.98541
9.98538
9.98535
9-98531
4
3
3
4
15
14
13
12
II
9
38.4
43 2
4
37.6
:
sr
! 5I
1 C2
1 53
54
9.40825
9.40873
9.40921
9 40 968
9.41 016
47
48
48
47
48
9.42297
9-42348
9.42399
9-42450
9.42501
5i
5
Si
5
0-57 703
0.57652
0.57601
0.57550
0.57499
9.98528
998525
9 98521
9-98518
9 985*5
3
3
4
3
3
10
1
.1
.2
3
4
0.4
o.S
1.2
1.6
2.C
0.3
0.6
0.9
1.2
;-j
5 5^
?
59
9.41063
9.41 in
9.41 158
9.41 205
9 41 252
48
47
47
47
.0
9.42552
9-42603
9-42653
9-42 704
9-42 755
5*
50
S
Si
0.57448
0-57397
0-57347
0.57296
0.57245
9.98511
9-98508
9-98505
9.98501
9.98498
4
3
3
4
3
5
4
3
2
I
:l
9
2.4
2.8
3:!
1.8
2.1
2-4
2-7
it jo
9.41 300
9.42805
0-57 195
9.98494
L. Cos.
d.
L. Cotg.
c.d
L. Tang
L. Sin.
d.
t
1
*rop.
Pts.
75
LOGARITHMS OF SINE, CO.Bl^, TANGENT AND COTANGENT, ETC.
45
15
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pts.
9.41 300
9.42805
O-57 J 95
9.98494
00
I
9-4i 347
9-42856
0.57 144
9.98491
S9
2
9-41 394
9.42906
0.57094
9.98488
3
S8
SI
5
3
4
9.41441
9.41 488
47
47
9-42957
9.43007
50
50
0.57043
0-56993
9.98484
9.98481
3
4
H
.1
.2
IO.2
5-o
IO.O
i
I
9
9-4i 535
9.41 582
9.41 628
9-4i 675
9.41 722
47
46
47
47
46
9-43057
9-43 108
9-43 158
9.43208
9-43258
50
50
50
5
0-56943
0.56 892
0.56 842
0.56 792
0.56 742
9.98477
9.98474
9.98471
9.98467
9.98464
3
3
4
3
55
54
53
52
.3
4
15-3
20.4
25-5
30.6
3S-7
15-0
20. o
25-0 ;
30.0
35-o
10
9.41 768
9-43308
0.56692
9 . 98 460
50
.8
40.8
40.0
12
9.41815
9.41 861
47
46
9-43358
9.43408
5
50
0.56642
0.56592
9-98457
9-98453
3
4
49
48
9
45-9
45-0
13
14
9.41 908
9-41 954
47
46
9-43458
9-43508
5
50
5
0.56542
0.56492
9.98450
9.98447
3-
3
47
46
49
48
ii
9.42001
9.42047
9.42093
9.42 140
46
46
47
9.43558
9.43607
9.43657
9-43 707
49
50
50
0.56442
0.56393
0.56343
0.56293
9-98443
9-98440
9.98436
9-98433
3
4
3
45
44
43
42
.1
.2
3
4
49
9-8
14.7
19.6
4-8
9-6
14.4
19.2
19
9.42 186
40
46
9-43756
49
5
0.56244
9.98429
4
41
5
24-5
21
22
23
9.42232
9.42278
9.42324
9.42370
46
46
46
.f.
9-43855
9-43954
49
50
49
0.56 194
0.56 145
0.56095
0.56046
9.98 426
9.98422
9.98419
9.98415
4
3
4
40
39
38
37
9
29.4
34.3
39-2
44.1
pU
43-2
1 24
9.42416
AC
9.44004
40
0.55996
9.98412
3
36
f
?
29
9.42461
9.42507
9-42553
9.42599
9.42644
4 6
4 6
46
45
46
9-44053
9-44 102
9.44I5I
9.44201
9.44250
49
49
So
49
0-55947
0.55898
0.55849
0-55 799
0.55750
9.98409
9.98405
9.98402
9.98398
9-98395
4
3
4
3
35
34
33
32
31
.1
.2
3
A
47
4-7
9-4
14.1
18 8
4.6
9-2
n.8
18.4
30
9.42690
9.42735
45
.e.
9.44299
49
0.55 701
0-55652
9.98391
9.98388
3
29
23.5
28.2
23.0
27.6
32
33
34
9.42 781
9.42 826
9.42872
40
45
46
45
9-44397
9.44446
9.44495
49
49
49
0.55603
0-55554
0.55505
9-98384
9.98381
9.98377
4
3
4
28
11
.9
32.9
37-6
32.2
36.8
41.4
P
9.42917
9.42962
45
A f.
9-44544
9-44592
48
0.55456
0.55408
9-98373
9.98370
3
25
24
ii
39
9.43008
9.43053
9.43098
4
45
45
9.44641
9-44690
9.44738
49
49
48
0-55359
0.55310
0.55 262
9-98366
9-98363
9.98359
4
3
4
23
22
21
2
45
4-5
9O
44
it
40
9-43 143
9.44787
49
0-55213
9.98356
3
20
.3
41
9-43 l8 8
45
9.44836
49
0-55 164
9.98 352
4
19
4
18.0
17 6
42
43
44
9.43233
9.43278
9.43323
45
45
45
9-44884
9-44933
9.44981
48
49
48
A 9
0.55 116
0.55067
0.55019
9-98349
9-98345
9.98342
3
4
3
\l
22.5
27.0
v-s
22.
26.4
30.8
?
9.43367
9.43412
9-43457
45
45
9-45029
9-45078
9-45 126
49
48
0-54971
0.54922
0.54874
9-98338
9-98334
9-9833I
4
4
3
15
H
13
9
36.0
40.5
35-2
39-6
48
9-43502
45
9-45 174
4 8
0.54826
9-98327
4
12
49
9-43546
44
9-45222
48
0.54778
9.98324
3
II
4
3
50
9-43591
9-45 271
49
0.54729
9.98320
4
10
.1
04
03
5i
52
9-43635
9.43680
44
45
9-45 319
9-45 367
4 8
48
0.54681
0.54633
9.98317
9-983I3
3
4
1
.2
3
0.8
1.2
0.6
0.9
53
9-43 724
44
9.45 4i5
48
0-54585
9.98309
4
7
4
1.6
1.2
9-43 769
45
9-45463
48
.0
0-54537
9.98306
3
6
2.O
1-j
55
9-43813
9-455"
4
0.54489
9.98302
4
5
.b
2.4
1.8
5<>
9-43857
44
9-45559
48
0.54441
9.98299
3
4
7
2.8
2.1
57
9.43901
44
9-45 606
47
0-54394
9.98295
4
3
.8
3.2
2.4
58
9-43 946
45
9-45 654
48
0.54346
9.98291
4
2
9
3-6
2.7
59
9-43990
44
9.4^702
48
0.54298
9.98288
3
I
GO
9-44034
9-45 750
4
o 54 250
9.98284
4
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
,
Prop. P^.
74
TABLE IV.
16
/
L. Sin.
d.
L. Tan?.
c.d.
L. Cot?.
L, Cos.
d.
p
rop. 1
Pis.
I
2
3
4
9.44034
9.44078
9.44 122
9.44 166
9.44210
44
44
44
44
43
9-45 750
9-45 797
9.4584?
9-45892
9-45 940
47
48
47
48
47
0.54250
0.54203
0.54155
0.54 1 08
0.54060
9 . 98 284
9.98281
9.98277
9.98273
9.98270
3
4
4
3
00
9
9
.1
48
4-8
9f.
47
4-7
I
9
9-44253
9.44297
9-44341
9.44385
9.44428
44
44
44
43
44
9-45 987
9-46035
9.46082
9.46 130
9.46 177
48
47
48
47
0.54013
0.53965
o.539i8
0.53870
0.53823
9.98266
9.98 262
9.98259
9.98255
9.98251
4
3
4
4
55
54
53
52
|l
3
4
7
14.4
19.2
24.0
28.8
33 6
9-4
14.1
18.8
III
72 Q
110
ii
12
13
14
9-44472
9 44 5 l6
9-44559
9.44602
9.44646
44
43
43
44
43
9.46224
9.46271
9-463I9
9-46366
9.46413
47
48
47
47
47
0.53776
0.53729
0.53681
0.53634
0.53587
9.98248
9.98244
9.98240
9.98237
9-98233
3
4
4
3
4
60"
%
%
8
9
P-4
43- 2
46
37-6
42.3
45
15
16
\l
19
9.44689
9-44 733
9-44 776
9-448I9
9.44862
44
43
43
43
9.46460
9.46 507
9.46554
9.46601
9.46648
47
47
47
47
<i6
0-53540
0-53493
0.53446
0-53399
0.53352
9.98 229
9 . 98 226
9.98 222
9.98218
9.98215
3
4
4
3
45
44
43
42
41
.1
.2
3
4
4.6
9-2
13.8
18.4
23.0
4-5
9.0
'3-5
18.0
22.5
20
21
22
23
24
9-44905
9.44948
9.44992
9-45035
9-45 077
43
44
43
43
9-46694
9.46 741
9.46 788
9-46835
9.46881
47
47
47
46
0.53306
0.53259
0.53212
0-53 165
0-53 "9
9.98 211
9.98 207
9.98204
9.98 200
9.98 196
4
4
3
4
4
40
9
9
*
9
2y. 6
32.2
36.8
41.4
27.0
3J-5
36.0
40.5
%
%
29
9-45 "O
9-45 163
9-45 206
9-45 249
9-45 292
43
43
43
43
4 2
9.46 928
9-46 975
9.47021
9.47068
9-47 "4
47
46
47
46
4 6
0.53072
0.53025
0.52979
0.52932
0.52886
9.98 192
9.98 I9
9.98 IS?
9.98 181
9-98 177
3
4
4
4
35
34
33
32
31
.1
.2
3
44
31
I?1
43
3:1
12.9
172
130
31
62
33
34
9-45 334
9-45 377
9.45419
9.45462
9-45504
43
42
43
42
9.47 160
9.47207
9.47253
9.47299
9-47346
47
46
46
47
d.6
0.52840
0.52 793
0.52 747
0.52 701
0.52654
9-98 174
9.98 170
9.98 166
9.98 162
9-98 159
4
4
4
3
30
27
26
1
.0
I/.U
22.0
26.4
30.8
35-2
39-6
l/.^
21. 5
25.8
30.1
9
9
39
9-45 547
9-45 589
9-45632
9-45674
9-45 7i6
42
43
42
42
9-47392
9.47438
9.47484
9-47530
9-47576
46
46
46
46
46
0.52608
0.52562
0.52 516
0.52470
0.52424
9-98 155
9.98 151
9.98 147
9.98 144
9.98 140
4
4
4
3
4
25
24
23
22
21
.1
2
4
4-2
8 4
41
40
4i
42
43
44
9-45 758
9.45801
9-45 843
9-45885
9-45927
43
42
42
42
9.47622
9.47668
9-47 7H
9.47 760
9.47806
46
46
46
46
A.6
0.52378
0.52332
0.52 286
0.52 240
0.52 194
9.98 136
9.98 132
9.98 129
9-98 12?
9.98 121
4
4
3
4
4
20
JQ
il
3
4
i
.7
12.6
16.8
21.0
25.2
29.4
"3
16.4
20.5
24.6
28.7
45
46
47
48
49
9 45969
9.46011
9-46053
9.46095
9.46 136
42
42
42
4i
9.47852
9.47897
9-47943
9.47989
9-48035
45
46
46
46
0.52 148
0.52 103
0.52057
0.52011
0.51965
9.98 117
9.98113
9.98 1 10
9.98 106
9.98 IO2
4
4
3
4
4
'5
H
13
12
II
.8
9
33-6
37-8
4
32.8 1
36.9
3
50
Si
52
53
1 54
9 46 178
9.46 220
9.46 262
9-46303
9-46345
42
42
4i
43
9.48080
9.48 126
9.48 171
9.48217
9.48262
46
45
46
45
0.51 920
0.51874
0.51 829
0.51 783
0.51 738
9,98098
9.98094
9.98090
9.98087
9.98083
4
4
4
3
4
10
I
I
.1
.2
3
4
5
0.4
0.8
1.2
1.6
2.0
0.3
0.0
0.9
1.2
-J
9
9
59
9.46386
9 . 46 428
9.46469
9.46511
9-46552
42
4i
42
4
9.48307
9.48353
9.48398
9-48443
9.48489
46
45
45
46
0.51 693
0.51 647
0.51 602
0.51557
0.51 5ii
9.98079
9.98075
9.98071
9.98067
9.98063
4
' 4
4
4
4
5
4
3
2
I
.6
9
li
H
1.8
2.1
2.4
2.7
60
9.46594
9.48534
0.51 466
9.98 060
3
L. Cos.
d.
L. Cot?.
c.d.
L. Tan?.
L. Sin.
d.
f
r
top.
Pts.
73
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 47
17 1
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pts. \
2
3
4
9-46594
9-46635
9.46 676
9.46717
9-46758
4 1
4*
41
41
42
4
4*
41
4<
4
4
41
41
4<
41
40
4>
40
4
40
4>
40
4'
40
40
41
40
40
4<>
40
4
4<>
40
4<>
40
4<>
40
39
40
40
39
4<>
40
39
40
39
40
39
39
39
40
39
39
39
39
39
39
39
39
39
9-48534
9.48579
9.48 624
9.48669
9.48 714
45
45
45
45
45
45
45
45
45
45
45
44
45
45
44
45
44
45
44
45
44
45
44
44
45
44
44
44
44
44
44
44
44
44
44
44
44
43
44
44
44
43
44
43
44
43
44
43
44
43
43
44
43
43
43
43
43
44
43
43
0.51 466
0.51 421
0.51 376
0.51331
0.51 286
9 . 98 060
9.98056
9.98052
9 . 98 048
9.98044
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
4
4
4
4
4
4
4
4
4
4
00
f*
y
.1
.2
3
4
:i
.7
.8
9
.1
.2
3
4
j
9
I
.2
3
4
;2
9
.1
.2
3
4
.b
:l
9
.2
3
4
j
-9
45
4-5
9.0
!3i
22.5
27.0
3J-S
36.0
40-5
43
tt
12.9
17.2
21-5
258
30.1
34-<
38-7
V
41
8^2
12.3
16.4
20.5
24.6
28.7
32.8
36-9
39
?:l
11.7
15.6
iQ-5
23-4
27-3
31.2
35-i
4
0.4
0.8
-1.2
1.6
2.0
2-4
2.8
1-6
44
si
o.o
'3-2 ,
17 6
22
26.4
30.8
35-2
39-6
42
4-2
8.4
12.6
16.8
21.
25-2
29.4
336
37-8
t
40
4.0
8.0
12.0
16.0
20. o
24.0
28.0
32.0
36.0
5
0.5
1.0
15
2.0
2-5
30
35
4-0
4-5
3
-3 |
0.6
0.9
1.2
!:i
2.1
2.4
2.7
I
I
9
10
ii
12
13
14
9 . 46 800
9.46841
9.46882
9.46923
9.46964
9-48 759
9.48804
9.48849
9-48894
9-48939
0.51 241
0.51 196
0.51 151
0.51 106
0.51 061
9 . 98 040
9 . 98 036
9.98032
9.98029
9.98025
55
54
53
52
5i
9.47005
9-47045
9.47086
9-47 127
9.47168
9.48984
9.49029
9-49073
9.49 118
9-49 103
0.51 016
0.50971
0.50927
O.5O 8\>2
0.50837
9.98021
9.98017
9.98 013
9 98 oc i
9 . 98 005
50
3
8
15
16
17
18
19
9.47209
9.47249
9.47290
9-47330
9-47371
9-49 207
9-49252
9.49296
9-49341
9.49385
0.50793
0.50 748
o . 50 704
0.50659
0.50 615
9.98001
9-97997
9-97993
9.97989
9.97986
45
44
43
42
41
20
21
22
23
24
9.47411
9.47452
9-47492
9-47533
9-47573
9-49430
9-49474
9.49519
9-49 563
9.49607
0.50570
0.50526
0.50481
0.50437
0.50393
9.97982
9.97978
9-97974
9.97970
9.97966
40
3
11
II
%
29
9-47613
9.47654
9.47694
9 47 734
9-47 774
9.49652
9.49696
9-49 740
9.49784
9.49828
0.50 348
0.50304
0.50 260
0.50216
o . 50 1 72
9.97962
9-97958
9-97954
9-97950
9.97946
35
34
33
32
31
mT
?8
11
30
3i
32
33
34
9.47814
9-47854
9.47894
9-47934
9 47974
9.49872
9.49916
9.49960
9 . 50 004
9 . 50 048
0.50 128
o . 50 084
o . 50 040
0.49996
0.49952
9.97942
9-97938
9-97934
9.97930
9.97926
9
%
39
9.48014
9-48054
9.48094
9 48 133
9-48 173
9.50092
9.50 136
9.50 180
9.50223
9.50267
0.49 908
0.49 864
0.49820
0.49777
0-49 733
9.97922
9.97918
9.97914
9.97910
9.97906
25
24
23
22
21
40
4i
42
43
44
9.48213
9-48252
9.48292
9-48332
9-4837I
9-503"
9.50355
9-50398
9.50442
9-50485
0.49 689
0.49645
0.49 602
0.49558
OA951S
9.97902
9-97898
9.97894
9.97890
9.97886
20
:
\i
45
46
47
48
49
9.48411
9.48450
9.48490
9-48529
9.48568
9-50529
9.50572
9.50616
9-50659
9-50703
0.49471
0.49428
0.49 384
0.49341
0.49 297
9.97882
9.97878
9.97874
9.97870
9.97866
15
14
13
12
II
~w
I
5
4
3
2
I
~0"
50
5i
52
53
54
9.48 607
9.48647
9.48686
9-48725
9.48 764
9.50746
9-50789
9-50833
9.50876
9.50919
0.49 254
0.49 211
0.49 167
0.49 124
0.49081
9.97861
9-97857
9-97853
9-97849
9-97845
P
57
58
59
w
9-48803
9.48842
9.48881
9.48920
9-48959
9.50962
9-5IOOS
9-51048
9-51092
9-51 135
0.49038
0.48995
o 48 952
0.48908
0.48865
9.97841
9-97837
9-97833
9.97829
9-97825
9-48998
9-51 178
0.48822
9.97821
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
1
Prop. Pts.
72
TABLE IV.
18
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pts.
Y
I
2
3
4
9.48998
9.49037
9.49076
9-49 H5
9 49 153
39
39
39
38
39
39
38
39
39
33
39
38
28
39
38
38
39
38
38
38
38
38
38
38
38
38
38
38
38
38
37
38
38
37
38
38
37
38
37
37
38
37
37
38
9-5i 178
9.51 221
9.51 264
9.51 306
9-51 349
43
43
42
43
43
43
43
43
43
42
43
43
42
43
42
42
43
42
43
42
42
42
43
42
42
42
42
42
42
42
42
42
42
42
42
42
42
42
4>I
42
42
41
4*
42
41
42
41
42
4*
41
4 1
0.48822
0.48 779
0.48 736
0.48 694
0.48 651
9.97821
9.97817
9.97812
9.97808
9.97804
4
5
4
4
4
4
4
4
4
5
4
4
4
4
4
5
4
4
4
00
59
58
JL
55
54
53
52
.1
.2
3
4
^9
.1
.2
3
4
9
.1
.2
3
4
'9
.1
.2
3
4
ii
9
4
8
12
17
21
25
30
3
4
: i
9
l
2
/
ii
i<
ic
2<
23
3'
3f
:
^
\
i
ii
\\
22
2
2(
3;
(
t
t
i
3
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9
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i
4
7
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\
12
ie
2C
2^
2*
I
9
9
3
>-5
4
'3
.2
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17
1 7
r-4
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1:1
1.2
>-9
)-6
J-3
5
).f
I.G
z.o
2-5
J-o
5-5
l-
1-5
43
4-2
8-4
12.6
16.8
21.0
25.2
29.4
33 6
37-8
x
I
f 2
3
'4
>.5
5
>_9
38
3-8
7-6
ii. 4
15-2
19.0
22.8
26.6
30.4
34-2
36
3 .6
10.8
18 o
21 6 ;
25.2
28.8
32-4
4
; 3-4 i
0.8
1.2
1.6
2.O
2.4
2.8
I
I
9
12
14
9.49 192
9.49231
9.49269
9.49308
9-49 347
9.51392
9.5I435
9-5 1 478
9-51 520
9 5 1 563
0.48608
0.48 565
0.48 522
z 48480
? 48 437
9.97800
9.97796
9.97792
9-97 788
9.97784
9.49424
9.49462
9.49500
9 49539
9.51 606
9.51 648
9.51691
9-5 1 734
9.51 776
0.48394
0.48352
0.48309
0.48 266
0.48 224
9-97779
9-97775
9.97771
9-97767
9-97 763
50
49
48
15
16
\l
19
"20"
21
22
23
24
9-49577
9.49615
9 49654
9-49692
9-49730
9.51819
9.51 861
9-5 1 903
9.51946
9-51988
0.48 181
0.48 139
0.48097
0.48 054
0.48012
9-97 759
9-97754
9-97750
9-97746
9-97 742
45
44
43
42
41
9.49768
9.49 806
9.49844
9.49882
9.49920
9.52031
9-52073
9-52 157
9.52 200
0.47969
0.47927
0.47 885
0-47843
0.47 800
9.97738
9-97734
9.97725
9-97 72i
4
5
4
4
4
4
5
4
4
4
5
4
4
4
5
4
4
4
5
4
4
4
5
4
4
4
5
4
4
5
4
4
5
4
4
5
4
4
5
4
40
9
9
?
29
31
32
33
f
9
39
9 49958
9.49996
9.50034
9.50072
9.50 no
9.52242
9.52284
9.52326
9-52368
9.52410
0-47 758
0.47716
0.47674
0.47632
0.47590
9.97717
9-97713
9-97 7oS
9.97704
9.97700
35
34
33
32
W
2Q
28
9.50 148
9.50185
9-50223
9.50261
9.50298
9.52494
9.52536
9.52 620
0.47548
0.47 56
0.47464
0.47422
0.47380
9.97696
9.97691
9-97687
9.97683
9.97679
9-50336
9.50374
9.50411
9 50449
9.50486
9.52 661
9-52 703
9.52745
9-52 787
9.52829
0-47 339
0.47297
0-47255
0.47213
0.47 171
9.97670
9.97666
9.97662
9-97657
25
24
23
22
21
40
42
43
1 44
9-50523
9 50 561
9.50598
9 50 635
9-50673
9.52870
9 52912
9 52953
9 52 995
9-53037
0.47 130
o 47 088
o 47 047
o 47005
0.46963
9-97653
9.97649
9-97645
9-97640
9-97636
20
19
17
16
45
46
9
49
9.50710
9.50747
9.50784
9.50 821
9.50858
37
37
37-
37
38
37
37
37
36
9-53078
9-53 120
9 53 161
9.53202
0.46 922
0.46880
0.46 839
0.46 798
0.46 756
9-97632
9.97628
9-97623
9.97619
9-976I5
IS
14
13
12
II
50
52
53
54
9 . 50 896
9.50933
9.50970
9.51 007
9.51 043
9-53285
9.53327
9.53368
9-53409
9.53450
0.46 715
0.46673
0.46 632
0.46591
0.46 550
9.97 610
9.97606
9.97602
9 97597
9-97593
10
I
59
9.51 080
9 5i "7
9-51 154
9.51 191
9.51 227
37
37
37
36
9-53492
9-53533
9-53574
9-53615
9-53656
0.46 508
0.46 467
0.46 426
0.46 385
0.46344
9-97589
9-97584
9.97580
9-97576
9-97571
5
4
3
2
I
GO
9.51 264
37
9-53697
0.46303
9-97567
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
t
Prop. Pts.
! 71
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 49
19 !
f
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pts. |
I
2
3
4
9.51 264
9-5i3oi
9-5I338
9-5 1 374
9-5 1 4"
37
37
36
37
36
37
36
37
36
36
37
36
36
36
37
36
36
36
36
36
36
36
36
36
36
36
35
36
36
36
35
36
35
36
35
36
35
36
35
36
35
35
36
35
35
35
35
35
35
35
36
34
35
35
35
35
35
35
34
35
9-53697
9-53 738
9-53779
9-53820
9-5386I
41
41
41
41
4
4<
4<
4*
40
4i
4
40
41
41
4<>
4
4<>
4^
40
41
0.46303
0.46 262
0.46 221
0.46 180
0.46 139
9-97567
9-97563
9.97558
9-97554
9-97550
4
5
4
4
5
4
5
4
4
5
4
4
5
4
5
4
5
4
4
5
4
5
4
5
4
4
5
4
5
4
5
4
5
4
5
4
5
4
5
4
5
4
5
4
5
4
5
5
4
5
4
5
4
5
4
5
5
4
5
4
(JO
59
58
H
.1
.2
.3
.4
i
9
.1
.2
3
4
:S
:i
9
.1
.2
3
4
:i
:l
9
.1
.2
3
4
:1
9
41
4-1
8.2
"3
16.4
20.5
24.6
28.7
32.8
36.9
3
.' 3
.2 7
3 "
4 15
5 19
.6 23
.7 27
8 3i
9 35
37
3-7
7-4
ii. i
14.8
18.5
22.2
25.9
29.6
33-3
35
3-5
7-o
10.5
14.0
17-5
21.0
24-5
28.0
31.5
5
o-5
I.O
15
2.0
2-5
3-o
3-5
4.0
4-5
40
4.0
8.0
12.
16.0
20.0
24.0
28.0
32.0
36.0
9
1
iiii
-5
4
3
.2
.1
36
3-6
10.8
14.4
18.0
21.6
25.2
28.8
32.4
34
i: 4 8
10.2
136
17.0
20.4
23-8
2 7 .2
30.6
4
0.4
0.8
1.2
1.6
2.0
:i
i:l
I
I
9
'To
ii
:2
13
14
9-5 1 447
9.51 484
9-5 1 5 2 o
9.51557
9-5i 593
9-51629
9.51 666
9.51 702
9 5' 738
9 5i 774
9-53902
9-53943
9-53984
9-54025
9.54065
0.46 098
0.46057
o 46016
0-4597
0.45 93
9-97545
9-97541
9-97536
9-97532
9.97528
55
54
53
52
5i
9.54106
9-54147
9-54 187
9.54228
9.54269
0.45894
0-45 853
0.45813
0.45 772
0-45 73 1
9-97523
9 975*9
9-975I5
9.97510
9.97506
50
49
48
47
46
IS
\l
ft
21
22
23
24
9.51 811
9-5I847
9.51883
9-5i 9i9
9-5I955
9-54309
9-54350
9 54390
9 5443 1
9-54471
0.45 691
0.45 650
0.45 610
0-45 569
0-45 5 2 9
9.97501
9-97497
9.97492
9.97488
9.97484
45
44
43
42
41
40~
39
38
P
9 5 1 99i
9.52027
9.52063
9.52099
9-5 2 135
9 54512
9 54552
9 54593
9 54633
9 54673
40
41
40
40
4
40
40
4 1
40
4>
40
4<>
40
4
4<>
40
4<>
40
4
4
4<>
4<>
39
4
4>
4
39
40
4<>
39
40
39
40
39
o . 45 488
0.45 448
0.45407
0-45 367
0.45327
9-97479
9-97475
9.97470
9.97466
9.97461
111
3
29
9.52 171
9.52207
9.52242
9.52278
9 5 2 3H
9 547H
9-54754
9-54794
9.54835
9 54875
0.45 286
0.45 246
0.45 206
0.45 165
0.45 125
9 97457
9 97453
9.97448
9-97444
9-97439
35
34
33
32
3i
80
3i
32
33
_34_
9
3
39
9-52350
9-52385
9.52421
9-52456
9-52492
9-54915
9-54955
9-54995
9-55035
9.55075
0.45 08
0.4504
0.4500;
0.4496;
0.4492
9 97435
9 9743
9.97426
9-97421
9.97417
30
2 9
28
3
9-52527
9-52563
9-52598
9-52634
9.52669
9-55 "5
9-55 155
9-55 195
9.55235
9 55275
0.44885
0.44845
0.44805
0.44 765
0-44 725
9 97412
9.97408
9-97403
9 97399
9 97394
25
24
23
22
21
20"
19
18
!|
40
4i
42
43
44
9 52 705
9.52740
9.52775
9.52 811
9.52 846
9 553J5
9-55355
9-55395
9-55434
9-55474
0.44685
0.44645
0.44605
0.44566
0.44526
9.97390
9 97385
9.97381
9.97376
9.97372
11
%
49
Iso'
1 5 1
52
53
54
9.52881
,9.52916
9.52951
9-52986
9 53021
9-555H
9-55554
9-55593
9.55633
o . 44 486
0.44446
0.44407
0.44367
0.44327
9 97367
9 97363
9-97358
9 97353
9 97349
15
H
13
12
II
9-53056
9-53092
9-53 126
9-53i6i
9-53I96
9-55712
9 55752
9-55 79i
9-55831
9 55870
0.44288
0.44248
0.44209
0.44 169
0.44 130
9-97344
9-97340
9-97335
9-97331
9.97326
10
6
P
P
59
9-53231
9-53266
9-53301
9.53336
9-53370
9-55910
9-55949
9.55989
9.56028
9.56067
39
4<>
39
39
4<>
0.44090
0.44051
0.44011
0.43972
0-43933
9-97322
9-973I7
9.97312
9.97308
9 97303
5
4
3
2
I
60
9 534^5
9-56107
0.43893
9.97299
L. Cos.
(1.
L. Cots.
c.d.
L. Tang.
L. Sin.
d.
' f
Prop. Pts.
70
TABLE IV.
20
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
p
rop.
Pts.
I
2
3
4
9.53405
9-53440
9-53475
9.53509
9-53544
35
35
34
35
34
9.56 107
9.56 146
9-56185
9.56224
9 . 56 264
39
39
39
40
39
0.43893
0.43 854
0.43 815
0.43 776
0.43 736
9.97299
9-97294
9.97 289
9.97285
9.97280
5
5
4
5
00
59
58
i
.1
2
40
\l
39
3 -i
7
I
I
9
9.53578
9.53$i3
9.53647
9-53682
9-53 7i6
35
34
35
34
35
9-56303
9.56342
9-56381
9.56420
9.56459
39
39
39
39
39
0.43 697
0.43658
0.43619
0.43 580
0.43 54i
9.97276
9.97271
9.97266
9.97262
9-97257
5
5
4
5
55
54
53
52
Si
3
:!
.7
12.0
16.0
20.0
7.0
ii. 7
15.6
19-5
23 4
27.7
10
ii
12
13
14
9-53 751
9-53 75
9-53819
9.53854
9.53888
34
34
35
34
34
9.56498
9.56537
9.56576
9.56615
9-56654
39
39
39
39
39
0.43 502
0.43463
0.43 424
0-43 385
0-43 346
9-97252
9.97248
9-97243
9-97238
9-97234
4
5
5
4
w
8
%
.8
9
32.0
36.0
38
31-2
35-1
37
15
16
!i
19
9-53922
9-53957
9-53991
9-54025
9 54059
35
34
34
34
34
9.56693
9.56732
9.56771
9.56810
9.56849
39
39
39
39
38
0-43 307
0.43 268
0.43 229
0.43 190
0.43 I5i
9.97229
9-97224
9.97220
9-97215
9.97210
5
4
5
5
45
44
43
42
41
.2
3
4
3 ^
7-6
11.4
15-2
19.0
3-7
7-4
ii. i
14.8
18.5
20
21
22
23
24
9.54093
9-54I27
9.54i6i
9-54I95
9-54229
34
34
34
34
34
9-56887
9.56926
9-56965
9.57004
9.57042
39
39
39
38
39
0-43 "3
0.43074
0.43035
0.42 996
0.42958
9.97206
9.97201
9-97 196
9.97192
9-97 187
5
5
4
5
40
3
11
i
9
22.8
26.6
30.4
34-2
22.2
25-9
29.6
33-3
>
3
2
29
9-54263
9-54297
9-54331
9-5436?
9-54399
34
34
34
34
34
9.57081
9.57 120
9.57158
9-57197
9.57235
39
38
39
38
39
0.42919
0.42880
0.42 842
0.42 803
(^42 765
9.97 182
9.97178
9-97 173
9.97 168
9-97 163
4
5
5
5
35
34
33
32
3i
.1
.2
3 i<
A \.
35
J-S
7-0
>-5
1 O
80
3i
32
33
34
9-54433
9.54466
9-54500
9-54534
9.54567
33
34
34
33
OJ
9.57274
9.57312
9-57351
9.57389
9.57428
38
39
38
39
08
0.42 726
0.42688
0.42 649
0.42 611
0.42 572
9-97 159
9-97 154
9-97 149
9-97 H5
9-97 HO
5
5
4
5
30
1
5 *
.6 2
.7 2,
.8 2!
9 3
7-3
I.O
ti
[.5
9
!?
39
9-54601
9-54635
9.54668
9.54702
9-54735
34
33
34
33
34
9.57466
9.57504
9-57543
9.57581
9-57619
38
39
38
38
30
0.42534
0.42496
0.42457
0.42419
0.42 381
9 97135
9-97 130
9.97 126
9.97121
9.97 116
5
5
4
5
5
25
24
23
22
21
.1
2
34
n
33
3 6' 3 6
40
4i
42
43
44
9-54769
9.54802
9-54836
9-54869
9-54903
33
34
33
34
33
9.57658
9.57696
9-57734
9-57772
9.57810
38
38
38
38
30
0.42342
0.42304
0.42 266
0.42 228
0.42 190
9.97 in
9-97 107
9.97 102
9.97097
9.97092
5
4
5
5
5
20
J 9
II
3
4
.7
10.2
13-6
17.0
20.4
23.8
9-9
13.2
i6.g
19.8
23-1
s
47
48
49
9-54936
9.54969
9-55003
9-55036
9-55069
33
34
33
33
9.57849
9 57887
9.57925
9-57963
9.58 ooi
38
38
38
38
^s
0.42 151
0.42 113
0.42075
0.42037
0.41 999
9.97087
9-97083
9.97078
9.97073
9.97068
5
4
5
5
5
15
14
13
12
II
.8
9
2 7 .2
3O.6
5
26.4
29.7
4
50
5i
5 2
53
54
9-55 102
9-55 136
9-55 169
9-55202
9.55235
34
33
33
33
q-a
9-58039
9.58077
9-58 n5
9.58153
9-58 191
38
38
38
38
38
0.41 961
0.41 923
0.41 885
0.41 847
0.41 809
9.97063
9-97059
9-97054
9.97049
9-97044
5
4
5
5
5
10
I
.1
.2
3
4
5
o-5
I.O
i-5
2.0
2-5
0.4
0.8
1.2
1.6
2.0
55
56
ii
59
9-55268
9-55301
9-55334
9.55367
9-55400
33
33
33
33
9-58229
9-58267
9-58304
9-58342
9-58380
38
37
38
38
,Q
0.41 771
0.41 733
0.41 696
0.41 658
0.41 620
9-97039
9.97035
9.97030
9.97025
9.97020
5
4
5
5
5
5
4
3
2
I
.b
:l
9
3-o
3-5
4-0
4-5
111
5.8
60
9-55433
9.58418
0.41 582
9.97015
5
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
t
F
rop. ]
Pte.
69
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 5 ,
21
r
L. Sill.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop.Pte.
I
2
3
4
9-55433
9.55466
9-55499
9-55 532
9.55564
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
33
3*
3
33
33
3
3
33
33
33
33
33
3'
33
33
33
3'
33
3*
33
33
3*
33
3'
3i
33
3
33
3*
3
33
9-58418
9-58455
9.58493
9-58531
9-58569
37
38
38
38
37
38
37
38
38
37
38
37
38
37
37
38
37
38
37
37
37
38
37
37
37
37
38
37
37
37
37
37
37
37
37
37
37
36
37
37
37
37
36
37
37
37
36
37
37
36
37
36
37
36
37
36
37
36
37
36
0.41 582
0.41 545
0.41 507
0.41 469
0.41 431
9.97015
9.97010
9.97005
9.97001
9.96996
5
5
4
5
5
5
5
5
5
5
4
5
5
5
5
5
5
5
5
5
5
5
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
5
5
5
5
5
5
5
5
5
5
5
60
P
g
.1
.2
3
4
:i
i
9
.1
.2
3
:!
is
9
.1
.2
3
4
:i
9
.1
.2
3
4
i
9
38
3-8
7-6
11.4
15-2
19.0
22.8
26.6
304
34.2
3 6
3-6
10.8
14.4
18.0
21.6
25.2
28.8
32.4
J"
.2 <
3 <
.4 i:
5 I(
.6 K
7 *
.8 l
.9 2\
3>
6.2
93
12.4
21.7
24.8
27-9
5
-5
I.O
1-5
2.0
2.5
3-o
3-5
4.0
4-5
37
3-7
74
II. 1
14.8
18.5
22.2
25.9
29.6
33-3
33
i:i
99
15.2
i&.J
19.8
23.1
26.4
29.7
J
J- a
>-4
)-6
2.8
3.0
?-2
|1
3.8
|
o'.6
1.2
I 8
2.4
3-o
36
4-2
48
54
4
0.4 j
0.8
1.2
1.6
2.O
;i
3:8
\
9
\w
II
12
13
H
9-55 597
9-55630
9-55663
9-55695
9.55728
9.58606
9.58644
9.58681
9-58719
9-58757
0.41 394
0.41 356
0.41 319
0.41 281
0.41 243
9.96991
9.96986
9.96981
9-96976
9.96971
55
54
53
52
5i
9-55 76i
9-55 793
9.55 826
9.55858
9-5589I
9.58794
9-58832
9-58869
9-58907
9-58944
0.41 206
0.41 168
0.41 131
0.41 093
0.41 056
9.96966
9.96962
9-96957
9-96952
9.96947
60
8
8
3
:;
19
9.55923
9^5956
9.56021
9-56053
9.58981
9.59019
9-59056
9-59094
9-59131
0.41 019
0.40981
0.40 944
0.40 906
0.40 869
9.96942
9.96937
9.96932
9.96927
9.96922
45
44
43
42
4i
IT
1
20
21
22
23
24
9.56085
9.56 na
9-56 150
9.56 182
9-56215
9-59 1 68
9-59205
9-59243
9.59280
9.59317
0.40832
0.40 795
0.40 757
0.40 720
0.40 683
9.96917
9.96912
9.96907
9 . 96 903
9.96898
2
29
9.56247
9.56279
9-56311
9-56343
9.56375
9-59354
9.59391
9.59429
9.59466
9-59503
0.40 646
0.40609
0.40571
0.40 534
0.40497
9 . 96 893
9.96888
9-96883
9.96878
9.96873
35
34
33
32
3i
30
3i
32
33
34
9.56408
9.56440
9.56472
9-56504
9-56536
9-59540
9-59577
9-59614
9-5965I
9.59688
9.59725
9.59762
9 59799
9.59835
9.59872
0.40 460
0.40423
0.40386
0.40349
0.40312
9.96868
9-96863
9-96858
9-96853
9.96848
30
11
11
25
24
23
22
21
20-
19
18
g
35
36
9
39
9.56568
9-56599
9-56631
9.56663
9.56695
0.40275
0.40 238
0.40 2OI
O.40 165
O.40 125
9 96 843
9-96838
9-96833
9.96828
9-96823
j40
4i
42
43
44
9-56727
9-56759
9-56790
9 56 822
9.56854
9-59909
9^9946
9.59983
9.60019
9.60 056
0.40091
0.40054
0.40017
0.39981
0-39944
9.96818
9-96813
9.96808
9.96803
9.96 798
45
46
47
48
49
50
5i
52
53
54
9.56886
9.56917
9 56949
9 . 56 980
9.57012
9.60093
9.60 130
9.60 1 66
9.60203
9 . 60 240
0.39907
0.39870
o 39834
0-39 797
0.39760
9-96 793
9.96 788
9-96 783
9.96 778
9.96772
15
H
13
12
II
"10"
6
9 57044
9-57075
9.57107
9.57138
9.57169
9.60276
9.60313
9.60349
9.60386
9.60422
0.39 724
0.39687
0.39651
0.39614
0-39 578
9.96767
9.96 762
9.96757
9.96 752
9.96 747
55
56
?
ft
9.57201
9-57232
9.57264
9.57295
9-57326
9.60459
9.60495
9.60532
9.60 568
9.60605
o 39 54i
0.39505
0.39468
0.3943 2
0-39395
9.96742
9.96 737
9.96 732
9.96727
9.96 722
5
4
3
2
I
9-57358
9.60641
0-39359
9.96717
L. Cos.
d.
L. Cots.
o. d.
L. Tang.
L. Sin.
d.
/
Prop. Pts.
68
TABLE IV.
22
I
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pts.
I
9.57358
9-57389
31
9.60641
9.60677
36
0-39359
0.39323
9.96717
9.96711
6
60
>9
2
9.57420
9.60714
,6
0.39 286
9.96 706
5
S8
37 1
36
3
4
9 57451
9.57482
3 2
9.60750
9.60 786
36
37
0.39250
0.39214
9.96 701
9.96696
5
5
.1
2
3-7
7 4
3-6
7.2
i
9.57514
9-5754?
9.60823
9 . 60 859
36
,6
0.39177
o-39 HI
9.96691
9.96686
5
55
S4
3
.4
II. I
14.8
14.4
I
9
9-57576
9-57607
9-57638
3
3*
9.60895
9.60931
9-60967
36
36
37
0.39 105
0.39069
0.39033
9.96681
9.96676
9.96670
5
5
6
53
52
1
18-5
22.2
25.9
18.0
21.6
2 5- 2
10
9.57669
9.61 004
,6
0.38996
9.96665
50
.8
29.6
28.8
ii
9.57700
9.61 040
og
0.38960
9.96 660
5
49
9
33-3
32-4
12
13
14
9-57731
9.57762
9-57793
$
9.61 076
9.61 112
9.6l 148
36
36
36
0.38924
0.38888
0.38852
9-96655
9 . 96 650
9.96645
5
5
48
8
35
\l
9.57824
9.57855
9.57885
30
9.6l 184
9.61 22O
9.61 256
36
36
,A
0.38816
0.38 780
0.38 744
9 . 96 640
9.96634
9 . 96 629
6
5
45
44
43
.2 7-0
3 lo-S
18
9.57916
*
9.61 292
,jt
0.38 708
9.96624
5
42
.4 I4.O
19
9-57947
31
9.61 328
3
36
0.38672
9.96619
5
41
20
21
9.57978
9.58008
30
9-6l 364
9.6l 400
36
0.38636
0.38600
9.96614
9.96608
6
*?
.7 24.5
X 28 O
22
23
24
9-58039
9.58070
9.58 101
9.6l 436
9.6l 472
9.6l 508
3
36
36
36
0.38 564
0.38528
0.38492
9.96603
9.96598
9-96593
5
5
5
11
9 3'-5
11
9-58131
9.58 162
9.61 544
9.61 579
35
-e.
0.38456
0.38421
9.96588
9.96582
6
35
34
3
31
3 |
%
29
9.58192
9.58223
9-58253
30
30
31
9.61 615
9.61 651
9.61 687
3
36
36
35
0-38385
0.38349
0.38313
9-96577
9.96572
9.96567
5
5
5
33
32
31
.2
3
A
T? 8
6.2
9.3
12.4
30
9 S 8 284
9.61 722
ofi
0.38278
9 . 96 562
30
16 o
31
9-583I4
3
9.61 758
30
,6
0.38242
9 96556
29
.6
19.2
18.6
34,
9-58345
9.58406
30
9.61 794
9.61 830
9,61 865
36
35
16
0.38 206
0.38 170
0.38135
9 96551
9 96546
9.96541
5
5
6
11
9
22.4
25.6
28.8
21.7
24.8
27.9
9-58436
9.58467
3'
9.61 901
9.61 936
35
0.38099
0.38064
9 96535
9 96530
5
25
24
39
9-58497
9-58527
9 58557
30
30
30
9.61972
9.62 008
9.62043
36
36
35
,6
0.38028
0.37992
0-37957
9-9652?
9.96520
9.96514
5
5
6
23
22
21
.1
2
30
ft
29
2.9
(0
9-58588
3
9 . 62 079
0.37921
9.96509
20
.3
9
8 7
41
9.58618
3
9.62 114
35
0.37886
9-96504
5
19
4
12.0
ii. 6
42
9.58648
3
9.62 150
30
0.37850
9.96498
18
15
14-5
43
9.58678
3
9.62 185
35
-f.
0-37815
9.96493
5
17
.6
18.0
17-4
44
9.58 709
3 1
9.62 221
3
0.37 779
9.9648$
5
ib
.7
21.0
20.3
|
9-58739
9.58769
30
9.62 256
9 . 62 292
36
0-37744
0.37708
9.96483
9.96477
6
15
14
9
24.0
27.0
23.2
26.1
9.58799
9 - 58 829
30
30
9.62327
9.62 362
35
35
0.37673
0.37638
9.96472
9.96467
5
5
13
12
49
9-58859
3
9.62 398
0.37602
9.96461
II
6
5
50
9.58889
9.58919
30
9.62433
9.62468
35
0.37567
0.37532
9.96451
5
10
.1
.2
0.6
1.2
0-5
I.O
52
9-58949
3
9.62 504
36
0.37496
9-96445
6
8
3
1.8
15
53
9-58979
30
9.62 539
35
0.37461
9.96440
5
7
4
2.4
2
54
9.59009
30
9.62574
35
0.37426
5
5
6
3-o
2-5
55
9.59039
9.62609
0.37391
9.96429
5
b
3-6
3-0
56
9.59069
30
9 . 62 645
36
0-37355
9.96424
5
4
'I
4-2
3-5
57
9.59098
29
9.62680
35
0.37320
9.96419
5
3
.8
4-8
4.0
58
59
9-59 128
9-59 IS 8
3
30
9-62 715
9.62 750
35
35
0.37285
0.37250
9-96413
9.96408
6
5
2
I
9
5-4
4-5
60
9.59188
9.62 785
0.37215
9.96403
5
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
t
Prop. Pts.
67 |
LOGARITHMS OF SINE, COSINE, TAff -tfT AND COTANGENT, ETC.
i 23
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pts.
I
2
3
4
9.59 188
9.59218
9-59247
9-59277
9- 5937
30
29
30
30
29
30
30
29
30
29
30
29
30
29
30
29
39
30
29
9
30
29
29
29
*9
30
9
29
29
29
29
29
29
29
9
29
29
29
29
28
29
29
29
28
29
29
29
28
29
28
29
38
29
28
39
28
29
28
28
9-62785
9 . 62 820
9.62 855
9 . 62 890
9.62 926
35
35
35
36
35
35
35
35
35
34
35
35
35
35
35
35
34
35
35
35
35
34
35
35
34
35
34
35
35
34
35
34
35
34
35
34
35
34
34
35
34
34
35
34
34
35
34
34
34
34
35
34
34
34
34
34
34
34
34
34
0.37215
0.37 180
0-37 145
0.37 no
0.37074
9.96403
9.96397
9-96392
9.96387
9.96381
6
5
5
6
5
6
5
5
6
5
6
5
5
6
5
6
5
6
5
6
5
5
6
5
6
5
6
5
6
5
6
5
6
5
6
5
6
5
6
5
6
5
6
6
5
6
5
6
5
6
6
5
6
5
6
6
5
6
5
6
00
CQ
i
.1
4
3
4
i
9
.1
.2
3
:l
i
9
.1
.2
3
-4
i
i
9
3
3
7
ID
11
21
25
28
32
.1
.2
3
4
:l
.1
9
2
1
s
12
IE
ll
21
2<:
2;
.1
.2
3
.4
i
:l
.9
(
:
i
i
i
6
.6
2
.8
4-
.6
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4
I
\
1C
':
i;
2C
2;
2;
3<
.0
.0
>.o
.0
:S
.0
^o
r.o
i
i
i
2
2
6
>.6
[.2
[.8
i-4
M
1:5
;-4
35
3-5
7.0 1
10.5
14.0
17-5
21.0
24-5
28.0
31-5
14
S;i
>.2
t-6
r.O
;i
r.2
>.6
39
2.9
l l
II. 6
14-5
17-4
20.3
2:?
8
2.8
I' 6
1.4
1.2
4..0
5.8
9.6
2.4
52
5
0.5
1.0
'5
20
2-5
30
35
40
4-5
i
I
9
lo-
ii
12
13
H
9.59336
9-59366
9-59396
9-59425
9-59455
9.62 961
9.62 996
9-63031
9 . 63 066
9.63 101
0.37039
0.37004
0.36969
0.36934
o . 36 899
9-96376
9.96370
9-96365
9-96360
9-96354
55
54
53
52
5i
9-59484
9-595H
9-59543
9 59573
9.59602
9-63 135
9-63 170
9.63205
9.63240
9 63275
0.36865
0.36830
0.36 795
0.36 760
0.36 725
9-96349
996343
9 96338
9 96333
9.96327
60
*
%
15
10
11
19
9-59632
9 59661
9.59690
9.59720
9-59749
9.63310
9.63345
9.63379
9.63414
9-63449
0.36690
0.36655
0.36621
0.36586
0.36551
9-96322
9.96316
9.96311
9-96305
9.96300
45
44
43
42
4i
40"
39
38
8
20
21
22
23
24
9-59778
9-59808
9 59837
9.59866
9 59895
9.63484
9.63519
9.63 553
9-63588
9-63623
0.36516
0.36481
0.36447
0.36412
0.36377
9.96294
9.96 289
9.96284
9.96278
9.96273
3
2
29
9-59924
9-59954
9-59983
9.60012
9.60041
9-63657
9.63692
9.63726
9-63 76i
9-63796
0.36343
0.36308
0.36274
0.36239
0.36204
9.96267
9.96262
9 96256
9 96251
9.96245
35
34
33
32
_1L
80
3
27
26
30
3i
32
33
34
9.60070
9.60099
9.60 128
9-60157
9.60186
9 63 830
9-63865
9.63899
9.63934
9.63968
0.36 170
0.36 135
0.36 101
0.36066
0.36032
9 . 96 240
9-96234
9 96 229
9-96223
9.96 218
9
12
i-
41
42
43
44
9.60215
9.60244
9.60273
9.60302
9 60331
9 64003
9.64037
9.64072
9.64 106
9.64 140
0-35 997
0-35 963
0.35 928
0.35894
0.35 860
9.96 212
9.96207
9.96 201
9.96 196
9.96 190
25
24
23
22
21
~w
19
i!
9-60359
9.60388
9.60417
9.60446
9-60474
9.64I75
9.64209
9-64243
9.64278
9.64312
0-35825
0-35 79i
o.35 757
0.35 722
0.35688
9.96 185
9.96 179
9.96174
9.96 168
9.96 162
i*
iti
49
9.60503
9.60532
9.60 561
9.60589
9.60618
9 64 346
9-64381
9.64415
9.64449
9.64483
0-35654
0.35619
0-35 585
o.3555i
0-35 5i7
9.96 157
9.96151
9.96146
9.96 140
9-96I35
15
14
13
12
II
lo"
I
60
5i
52
53
54
9.60646
9-60675
9.60 704
9.60 733
9.60 76.
9-64517
9 6^552
9-64586
9.64620
9.64654
0-35 483,
0.35448
0.354J4
0.35380
0-35 346
9.96129
9.96 123
9.96 1 18
9.96 112
9.96 107
11
%
59
9.60 789
9.60818
9.60846
9.60875
9.60903
9.64688
9.64722
9.64756
9.64790
9.64824
0.35312
0.35 278
0.35 244
0.35210
0.35 176
9.96 ioi
9.96095
9.96090
9 . 96 084
9.96079
5
4
3
2
I
~TT
00
9-60931
9.64858
0-35 142
9.96073
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
t
Prop. Pts.
66 j
TABLE IV.
24
t
L. Sin.
d.
L. Tang.
c.d.
L. Cots.
L. Cos.
d.
Proi
>.
Pts.
2
3
4
9.60931
9.60960
9.60988
9.61 016
9.61045
29
28
28
29
28
9.64858
9 . 64 892
9 . 64 926
9 . 64 960
9.64994
34
34
34
34
34
0-35 J 42
0.35 108
0-35 074
0.35 040
0.35006
9.96073
9.96067
9 . 96 062
9 . 96 056
9 . 96 050
6
5
6
6
60
CQ
I
3
.1 3
2 6
4
*
S3
i:I
I
I
9
9.61073
9.61 101
9.61 129
9.61 158
9.61 186
28
28
29
28
28
9 . 65 028
9 . 65 062
9 . 65 096
9.65 130
9.65 164
34
34
34
34
33
0.34972
0.34938
0.34904
0.34870
0.34836
9-96045
9-96039
9.96034
9 . 96 028
9 . 96 022
6
5
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LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 57
27
9
L. Sin.
d.
L. Tang.
.d.
L. Cotgr.
L. Cos.
d.
p
rop
,1
>ts.
I
2
3
4
9.65 705
9.65 729
9-65754
9-65 779
9.65804
24
25
25
25
24
9.70717
9.70748
9.70779
9.70 810
9.70841
3
3
3
3
o . 29 283
0.29252
0.29 221
O.29 190
0.29 159
9.94988
9.94982
9-94975
9.94969
9.94962
6
7
6
7
6
60
5?
58
Ii
.1
.2
32
I
2
/)
31
ii
1
I
9
9.65828
9-65853
9-65878
9.65902
9.65927
25
25
24
25
25
9.70873
9 . 70 904
9.70935
9.70966
9.70997
3i
3
31
3
31
0.29 127
0.29096
0.29065
0.29034
0.29003
9.94956
9.94949
9-94943
9.94936
9-94930
7
6
7
6
55
54
53
52
5i
3
4
:!
7
9
12
16
19
22
6
8
2
4
9-3
12.4
21.7
10
ii
12
13
14
9-65952
9.65976
9 . 66 ooi
9.66025
9.66050
24
23
24
25
25
9.71 028
9.71059
9.71090
9.71 121
9-71 153
3
31
31
32
31
0.28972
0.28941
0.28910
0.28879
0.28 847
9.94923
9.94917
9.94911
9.94904
9.94898
6
6
7
6
50
3
t?
.8
9
3
6
8
3
24.8
27.9
15
10
\l
19
9-66075
9.66099
9.66 124
9.66 148
9-66173
24
25
24
5
2 4
9.71 184
9.71 2l|
9.71 246
9.71277
9.71308
3*
3i
3*
3*
0.288l6
0.28 785
0.28 754
0.28 723
0.28 692
9.94891
9.9488;
9.94878
9.94871
9.94865
6
7
7
6
45
44
43
42
41
i
.2
3
4
i
I
$
12
;s
.0
.O
.0
.0
.0
! o
20
21
22
23
24
9.66 197
9.66 221
9.66246
9.66270
9.66293
24
25
24
25
9.71339
9.71370
9.7I40I
9.7I43I
9.71 462
3
3
30
3
0.28 661
o . 28 630
0.28599
o 28 569
0.28 538
9.94858
9.94852
9.94845
9-94839
9.94832
6
7
6
7
6
40
fs
11
i
9
21
24
2;
.O '
[0
r.o
25
20
3
29
9-66319
ris
IM
24
25
24
4
9$
9.7I493
9-71 5 2 4
9.7I555
9.71586
9.7I6I7
3i
3
3
3*
0.28 507
0.28476
0.28445
0.28414
0.28383
9 . 94 826
9.94819
9.94813
9.94806
9-94799
7
6
7
7
6
35
34
33
32
3i
.1
.2
3
A
a
2
5
7
10
5
5
.0
5
o
4
2 4
4.8
&
30
3i
32
33
34
9.66441
9.6646?
9.66489
9-66513
9-66537
24
24
24
24
25
9 71 648
9.71679
9.71 709
9.71 740
9.71 771
3*
30
3i
3i
0.28352
0.28 321
0.28 291
0.28 260
0.28 229
9-94793
9.94786
9.94780
9-94773
9.94767
7
6
7
6
so
29
28
27
26
''I
9
12
15
17
20
22
5
.0
-5
.0
-S
12.
14 4
16.8
192
21.6
II
9
39
9.66 562
9.66586
9.66610
9.66634
9.66658
24
24
24
24
9.71 802
9-71833
9.71863
9.71894
9.71 925
3i
30
3*
3
0.28 198
0.28 167
0.28 137
0.28 1 06
0.28075
9.94760
9 94753
9-94 747
9.94740
9 94 734
7
6
7
6
25
24
23
22
21
.1
?
23
a
40
41
42
43
44
9.66 682
9 . 66 706
9.66731
9-66755
9.66779
24
25
2 4
2 4
9.7I955
9.71 986
9.72017
9 . 72 048
9.72078
3i
3
3i
30
o . 28 045
0.28 014
0.27983
0.27952
0.27922
9.94727
9.94720
9.94714
9-94 707
9.94700
7
6
7
7
|
20
19
il
3
4
:i
7
,
i
5, 9
9.2
il
3
4 J
49
9.66803
9.66827
9.66851
9 66875
9.66 899
24
24
24
24
9.72 109
9.72 140
9.72170
9 . 72 201
9.72231
3i
30
3i
30
o 27891
0.27 860
0.27 830
0.27 799
0.27 769
9.94694
9.94687
9 . 94 680
9.94674
9.94667
7
7
6
7
15
14
13
12
II
.8
9
i
2
7
8.4
0.7
6
50
5i
52
53
54
9.66 922
9 . 66 946
9.66970
9.66994
9.67018
24
24
24
24
9 . 72 262
9.72293
9-72323
9.72354
9.72384
3i
30
3
30
0.27738
0.27 707
0.27677
0.27646
0.27616
9 . 94 660
9-94654
9.94647
9.94640
9-94634
7
6
7
7
6
10
I
.1
.2
3
4
c
i
2
2
3
-7
4
. i
.S
5
0.6
1.2
1.8
2.4
3-0
I
s
59
9.67042
9.67066
9.67090
9.67113
,9.67137
24
24
23
24
9-72415
9.72445
9.72476
9.72506
9.72537
30
3i
30
3i
0-27585
0-27555
0.27524
.0.27494
0.27463
9.94627
9 . 94 620
9.94614
9.94607
9.94600
7
6
7
7
5
4
3
2
I
:1
9
4
4
1
.2
1
3
3-6
4.2
4-8
5-4
60
9.67 161
9.72 567
0.27433
9-94593
L. Cos.
d.
L. Cotg.
c.d
L. Tang
L. Sin.
d.
t
]
VO]
>
Pts.
62
g TABLE IV.
28
1
I. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pts.
9.67 161
9.72567
0.27433
9-94593
00
I
2
3
4
9.67 185
9.67208
9.67232
9.67256
23
24
24
24
9.72598
9.72 628
9.72659
9.72 689
3 1
3
31
0.27402
0.27372
0.27341
0.27311
9-94587
9-9458o
9-94573
9-94567
7
7
6
.1
2
31
3- 1
6 2
30
I
9.67280
9.67303
23
24
9.72 720
9.72750
30
0.27 280
0.27 250
9-9456o
9-94553
7
55
54
3
-4
9-3
12.4
9.0
12.0
I
9.67327
9.67350
23
9.72 780
9.72811
3i
0.27 220
0.27 189
9.94546
9.94540
6
53
S2
i
5
15.0
18.0
9
9-67374
24
9.72841
31
0.27159
9-94533
7
7
21
.7
21 .0
10
9.67398
9.72872
0.27 128
9.94 526
50
.8
24
.8
24.0
ii
12
9.67421
9-67445
9.67468
24
23
9.72902
9-72932
9-72963
3
30
3*
0.27098
0.27068
0.27037
9-94 5 J 9
9-94513
9.94506
7
6
7
49
48
9
27
9
27.0
*4
9.67492
23
9-72993
30
0.27007
9-94499
7
46
39
!<>
9.675I5
9.73023
0.26977
9.94492
45
.1
2.9
16
9-67539
9-73054
3 1
0.26 946
9-94485
7
44
.2
5-8
17
18
9.67562
9.67 586
24
9.73084
9-73ii4
30
0.26916
0.26886
9-94479
9. 94 472 ,
7
43
42
3
4
&
19
9.67609
23
24
9-73 144
3
0.26856
9-94465
7
4i
5
14-5
20
9.67633
9 73 175
0.26 825
9-94458
40
7
17-4
20 7
21
9.67656
9-73205
0.26 795
9-94451
39
'I
22
9.67680
9.73235
30
0.26 765
9-94445
38
Q
23.2
26 i
23
24
9.67703
9.67726
23
24
9-73265
9.73295
30
3
0-26735
0.26 705
9-94438
9-9443 1
7
7
y
25
9.67750
9.73326
0.26 674
9-94424
35
34
26
29
9.67773
9-67796
9.67820
9.67843
23
24
23
23
9.73356
9.73386
9.73416
9-73446
3
30
30
30
0.26 644
0.26 614
0.26 584
0.26554
9.94417
9.94410
9.94404
9-94397
7
7
6
7
34
33
32
.1
.2
3
A
7-2
o 6
V 6
6.9
92
30
9.67866
9-73476
0.26 524
9-94390
30
12.0
II. j
32
33
34
9.67890
9.67913
9.67936
9 67959
23
23
23
23
9-73507
9-73537
9-73567
9-73597
3 1
30
30
30
0.26 493
o . 26 463
0.26433
0.26403
9-94383
9-94376
9-94369
9-94362
7
7
7
7
1
i
.9
14.4
16.8
19.2
21.6
20.7
%
9.67982
9.68006
24
9.73627
30
0.26373
0.26343
9-94355
9 94 349
6
25
24
B
9.68029
9.68052
23
23
9-73687
9-73 717
30
30
0.26313
0.26 283
9-94342
9-94335
7
7
23
22
33
2 2
39
9.68075
23
9-73 747
30
0.26253
9-94328
7
21
.2
4-4
40
9.68098
9-73777
o . 26 223
9-94321
20
-3
6.6
41
9.68 121
9.73807
3
0.26 193
9-943H
7
19
.4
8.8
42
43
9.68 144
9.68 167
23
23
9.73867
3
30
0.26 163
0.26 133
9-94307
9.94300
7
7
18
17
i
II.
13.2
44
9.68 190
23
9.73897
3
0.26 103
9-94293
7
16
.7
9
9.68 213
24
9-73927
9-73957
30
0.26073
0.26043
9.94286
9.94279
7
15
9
17.6
19.8
47
9.68260
9- 73 987
3
0.26013
9-94273
13
48
9.68283
9.74017
30
0.25983
9 . 94 266
7
12
49
9.68305
9.74047
30
0.25 953
9-94259
7
II
7
6
50
9.68328
9.74077
0.25923
9-94252
10
.1
0.7
0.6
5'
9-6835I
23
9-74 107
3
0.25893
9-94245
7
9
.2
]
[-4
1.2
52
53
9-68374
9.68397
23
23
9.74137
9.74166
30
29
0.25863
0.25 834
9.94238
9.94231
7
7
7
-3
4
2.1
2.8
1.8
2.4
54
9.68420
23
23
9.74196
30
0.25 804
9.94224
7
6
5
3-5
8-8
B
ii
9.68443
9.68466
9.68489
9-68 512
23
23
23
9.74226
9.74256
9 . 74 286
9.74316
30
30
30
0.25 774
0.25 744
0.25 714
0.-25 684
9.94217
9.94 2IO
9.94203
9.94196
7
7
7
5
4
3
2
.b
i
9
4.2
4-9
I' 6
6-3
3- 6
5-4
59
9-68534
23
9-74345
29
0-25655
9-94 189
7
I
00
9.68557
9-74375
0.25 625
9.94 182
L. Cos.
d.
L. Cots.
c.d.
L. Tancr.
L. Sin.
d.
t
Prop. Pts.
I 61
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. r
29 i
9
I
2
3
4
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pts.
9-68557
9.68580
9 68603
9.68625
9.68648
23
23
22
23
3
23
22
23
23
22
23
22
23
23
23
23
29
23
22
23
32
23
22
23
22
22
23
22
23
22
22
23
22
22
22
23
33
93
33
22
3
32
33
33
33
33
32
22
22
23
32
22
22
22
22
22
23
22
22
23
9-74375
9-74405
9-74435
9.74465
9-74494
3
3
3
29
30
30
29
30
30
30
29
30
30
29
30
30
29
3<>
29
30
29
30
30
29
30
29
30
29
30
29
30
29
30
29
29
30
29
30
29
29
30
29
30
29
29
30
29
29
29
30
29
29
29
30
29
29
29
30
29
29
o . 25 625
25 595
0-25565
0-25 535
0.25 506
9.94 182
9.94175
9.94 168
9.94161
9 94154
7
7
7
7
7
7
7
7
7
7
7
7
8
7
7
7
7
7
7
7
7
7
7
8
7
7
7
7
7
7
7
8
7
7
7
7
7
8
7
7
7
7
8
7
7
7
8
7
7
7
7
8
7
7
8
7
7
7
8
7
00
3
11
.1
.2
3
4
i
9
.1
.2
3
:I
:|
9
.1
.2
3
4
:!
i
9
.1
.2
3
4
:i
:l
9
.1 C
.2 1
.3 '
.4 :
i :
:l \
.9 ;
30
3
6.0
9.0
12.0
I 5-
18.0
21.0
24.0
27.0
9
2.9
H
n. 6
14.5
17-4
20.3
2J.2
20.1
93
i 1
6. 9
9-2
"5
'3-8
IO.I
18.4
20.7
99
2.2
to
8.8
II. O
13-2
15-4
17.6
|I 9 .8
8 7
).8 0.7
.6 1.4
(.4 2.1
5.2 2.8
[.o 3.5
t.8 4.2
;.6 4-9
-4 5-6
r.2 6.3
1
9
9.68 671
9 . 68 694
9.68 716
9.68739
9.68 762
9.74524
9^74554
9-74583
9.74613
9.74643
0.25 476
0.25 446
0.25417
0.25387
0.25357
9.94147
9.94 140
9-94I33
9.94 126
9.94 119
55
54
53
52
Si
10
ii
12
3
14
9.68 784
9.68807
9.68829
9.68852
9.68875
9.74673
9.74702
9-74732
9.74762
9-74791
0.25327
0.25 298
0.25 268
0.25 238
0.25 209
9.94 112
9.94105
9.94098
9.94090
9.94083
50
4 2
4 8
47
46
45
44
43
42
41
!2
!2
19
9.68897
9.68920
9.68942
9.68965
9.68987
9.74821
9-74851
9.74880
9.74910
9-74939
0.25 179
0.25 149
O.25 120
O.25 09O
0.25 06 1
9.94076
9.94069
9.94062
9-94055
9.94048
20
21
22
23
24
9.69010
9.69032
9 69055
9.69077
9.69 100
9.74969
9.74998
9.75028
9.75058
9.75087
0.25 031
O.25 002
0.24972
0.24942
0.24913
9.94041
9-94034
9.94027
9.94020
9.94012
40
39
38
1
35
34
33
32
3i
%
11
29
9.69 122
9.69144
9.69167
9.69 189
9.69212
9-75 "7
9-75 H6
9-75 176
9-75205
9-75 235
0.24883
0.24854
0.24824
0.24795
0.24765
9.94005
9-93998
9-93991
9-93984
9-93977
n
31
32
33
34
9.69234
9.69256
9.69279
9.69301
9.69323
9-75 264
9-75294
9.75323
9-75353
9.75382
0-24736
o . 24 706
0.24677
0.24647
0.24618
9.93970
9.93963
9-93955
9-93948
9-93941
30
27
26
i
39
IT
41
42
43
44
9-69345
9.69368
9.69390
9.69412
9-69434
9-754"
9-75441
9-75470
9-75500
9.75529
0.24589
0.24559
0.24530
0.24500
0.24471
9-93934
9.93927
9.93920
9.93912
9-93905
25
24
23
22
21
"20"
! 9 8
\l
9-69456
9-69479
9.69501
9-69523
9.69545
9-75558
9-75588
9-756I7
9-75647
9.75676
0.24442
0.24412
0.24383
0.24353
0.24324
9-93898
9.93891
9.93 884
9.93 876
9.93869
9
8
49
9.69567
9.69589
9.69 611
9-69633
9-69655
9-75705
9-75735
9-75 764
9-75793
9-75822
0.24295
0.24 265
0.24236
0.24207
0.24 178
9.93862
9.93855
9.93847
9.93840
9-93833
15
H
13
12
II
50
5i
52
53
54
9.69677
9.69699
9.69721
9 69743
9 69765
9-75852
9.75881
9.75910
9-75939
9.75969
0.24 148
0.24 119
0.24090
0.24061
0.24031
9.93826
9.93819
9.93811
9.93804
9-93 797
10
1
I
55
56
9
59
1ST
9 69787
9.69809
9.69831
9 69853
9 69875
9-75998
9.76027
9.76056
9.76086
9.76 115
0.24002
c 23973
0.23944
0.23 914
0.23885
9-93 789
9-93 782
9-93 775
9.93 768
9 93 76o
5
4
3
2
I
"0"
9 69 897
9.76 144
o 23 856
9-93753
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
t
Prop. Pts.
60
TABLE IV*
i 80 1
t
L. Sin.
d.
L. Tang.
.d.
L. Cotg.
L. Cos.
d.
Prop. Pis.
i '
\
9.69897
9.69919
9.69941
9 69963
9.69984
23
33
33
21
23
9.76 144
9.76 173
9 . 76 202
9.76231
9.76 26l
39
29
29
30
29
39
29
29
9
29
39
29
29
39
29
29
30
29
39
28
39
39
29
29
39
29
29
29
29
39
29
29
28
29
29
29
29
29
28
29
29
29
29
28
29
29
29
28
29
29
28
29
29
29
28
29
28
29
29
28
0.23 856
0.23 827
o 23 798
0.23 769
o 23 739
9 93 753
9 93 746
9 93 738
9-9373"
9 93 724
7
8
7
7
7
8
7
7
8
7
7
8
7
8
7
7
8
7
7
8
7
8
7
7
8
7
8
7
8
7
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
8
7
8
7
8
7
00
3
11
3<
:i I
3 9
.4 12
:i !I
its
.9 27
.1
.2
3
:I
i
9
.1
.2
3
.4
i
:J
9
.1
.2
3
4
:i
i
9
.1 (
.2
3 '
4 ,
5 *
.6 -
:2 .
9
>
o 2.9
o j.8
.0 8.7 1
.0 ii. 6
.0 14.5
.0 17.4
.0 20.3
.0 23.2
.0 20.1
38
2.8
s- 6
1-4
II. 2
14 o
16.8
19.6
22.4
25.2
sa
2.2
ft
8.8
II.
13 2
"5-4
17.6
I 9 .8
at
2.1
4 2
63
8.4
10 5
12.6
2
18.9
8 7
5.8 0.7
[.6 1.4
2.4 2.1
J.2 2.8
*o 3.5
^.8 4-2
C.6 49
3-4 5-6
7.2 6.3
i
2
9
10
ii
12
13
H
\l
11
19
21
22
23
24
9 . 70 006
9 . 70 028
9 . 70 050
9.70072
9.70093
33
32 ,
22
21
23
33
33
91
32
33
31
22
81
23
32
21
23
21
22
21
22
31
33
21
33
21
33
81
33
31
31
23
21
31
33
31
31
21
22
21
21
21
22
21
21
21
21
21
23
21
XX
21
21
21
21
9.76290
9.76319
9.76348
9.76377
9.76406
0.23 710
0.23 68 1
0.23652
0.23623
o 23 594
9 93717
9 93 709
9.93702
9-93695
9-93687
9.93680
9 93673
9 93 66 J
9-93658
9 93 650
55
54
53
52
5"
60"
3
2
9.70115
9.70137
9.70159
9.70 180
9 . 70 202
9.76435
9.76464
9.76493
9.76522
9-7655I
0.23565
0.23536
0.23 507
0.23478
0.23449
9.70224
9.70245
9.70 267
9.70288
9.70310
9.76580
9.76609
9.76639
9.76668
9.76697
o . 23 420
0.23391
0.23361
0.23332
0.23303
9-93643
9 93636
9 93 628
9-9362I
9.93614
45
44
43
42
4"
9-70332
9.70352
9.70375
9.70396
9.70418
9.76725
9-76 754
9-76783
9.76 812
9.76841
0.23275
0.23 246
0.23217
0.23 188
0.23 159
9.93606
9-93599
9-9359"
9 93 584
9 93577
40
3
1
35
34
33
32
i
I
s
2
29
31
32
33
Jl_
$
12
39
4i
42
43
1 44
9.70439
9.70461
9.70482
9.70504
9-70525
9.76870
9.76899
9.76928
9.76957
9.76986
0.23 130
0.23 101
0.23072
0.23043
0.23 014
9-93569
9 93562
9-93554
9-93547
9 93539
9-70547
9-70568
9.70590
9.70611
9-70633
9.77015
9.77044
9.77073
9.77 101
9.77130
0.22985
0.22 956
0.22 927
. 22 899
0.22 870
9-93532
9-935 2 5
9-93 5"7
9-93510
9-93 502
9.70654
9.70675
9.70697
9.70718
9-70739
9.77159
9.77188
9.77217
9.77246
9.77274
0.22 841
0.22 8l2
0.22 783
0.22 754
0.22 726
9-93495
9 93487
9-9348o
9-93472
9 93465
25
24
23
22
21
w
19
ii
9.70761
9.70782
9.70803
9.70824
9 . 70 846
9.77303
9-77332
9.77361
9-77390
9.77418
0.22 697
. 22 668
O.22 639
O.22 6lO
0.22 582
9 93457
9 93450
9 93442
9 93435
9 93427
4
46
%
49
10"
5i
52
53
J
9
5 5 I
I
9.70 867
9.70888
9.70909
9.70931
9.70952
9-77447
9.77476
9- 7755
9-77533
9.77562
0.22 553
0.22 524
0.22495
0.22 467
0.22438
9.93420
9 93412
9 93405
9 93397
9 93390
15
14
"3
12
II
lo~
1
5
4
3
2
I
~0"
9-70973
9.70994
9.71 015
9.71036
9.71058
9-77591
9.77619
9-77648
9.77677
9.77706
. 22 409
22 381
0.22352
0.22 323
22 294
9 93382
9 93 375
9 93 367
9-93360
9 93 352
9.71 079
9.71 loo
9.71 121
9.71 142
9 7i I 6 3
9-77734
9-77 763
9.77791
9.77820
9.77849
0.22 266
0.22237
O.22 2O9
0.22 l80
0.22 151
9-93344
9 93337
9 933 2 9
9 93322
9 93314
9 71 184
9.77877
0.22 123
9 93 37
L. Cos.
d.
L. Cots
c.d,
L. Tang
L. Sin.
d.
/
Prop. Pts.
59 1
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 6Y
31
/
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
W
CQ
57
56
Prop. Pts.
I
2
3
4
9.71 184
9.71 205
9.71 226
9.71247
9.71 268
21
21
21
21
21
21
21
21
21
20
21
21
21
21
21
21
20
21
21
21
2O
21
si
91
2O
81
21
20
21
21
20
21
20
21
20
21
20
21
21
20
20
21
20
21
20
21
2O
2O
21
20
20
21
20
20
21
30
2O
M
90
30
9.77877
9.77906
9-7793?
9.77963
9-77992
29
29
28
29
28
0.22 123
0.22094
0.22065
0.22037
0.22008
9-93307
9.93299
9-93 291
9.93284
9.93276
8
8
7
8
7
8
8
7
8
8
7
8
8
7
8
8
7
8
8
7
8
8
7
8
8
7
8
8
8
7
8
8
8
7
8
g
8 '
.1
.2
3
4
ii
9
.1
.2
3
:!
:J
9
.1
.2
3
4
ii
9
.1
.2
.3
4
ii
ii
9
.1 <
.2 ]
3 -
.4 :
:i ;
:i i
9 ;
39
S*
,!:i
H-5
17 4
20.3
li.l
38
2.8
5.6
8.4
II. 2
14-0
16 8
19.6
22.4
25.2
ax
2.1
X
8-4
S:i
Z
18.9
20
2.O
4-0
6.0
8.0
IO.O
12.
14-0
16.0
180
8 7
).8 0.7
1.6 1.4
J.4 2.1
J.2 2.8
^o 3-5
k8 4-2
;.6 4.0
)-4 5- 6
r 2 63
1
I
9
9.71289
9.71310
9-7i33i
9-71352
9 7i 373
9 . 78 020
9.78049
9.78077
9.78 106
9.78135
29
28
29
29
28
0.21 980
0.21 951
0.21 923
0.21 894
0.21 865
9.93269
9.93261
9-93253
9-93246
9-93238
55
54
53
5 2
51
"50"
49
48
8
10
ii
12
'3
14
9.71393
9.71414
9-7i 435
9-71 45 6
9.71 477
9-78163
9.78192
9 . 78 220
9.78249
9.78277
29
28
29
28
29
28
29
28
28
29
38
29
98
29
38
28
29
28
29
28
28
29
28
28
29
28
28
29
28
28
28
29
28
28
28
29
28
28
28
28
29
28
28
28
28
28
39
28
28
28
0.21 837
0.21 808
0.21 780
0.21 751
0.21 723
9.93230
9.93223
9-932I5
9-93207
9-93200
11
II
19
9.71 498
9.7i 5*9
9-71 539
9.71 560
9.71 581
9.78306
9-78334
9-78363
9.78391
9.78419
0.21 694
0.21 666
0.21 637
O.2I 609
0.21 581
9-93 192
9-93 l8 4
9-93 177
9-93 169
9.93 161
45
44
43
42
*
9
11
20
21
22
23
24
9.71 602
9.71 622
9-71 643
9.71664
9.71 685
9.78448
9-78476
9-78505
9.78533
9-78562
0.21 552
O.2I 524
0.21 495
0.21 467
0.21 438
9 93 154
9-93 H6
9-93 138
9 93 131
9 93 123
3
27
28
3
3i
32
33
34
9.71 705
9.71 726
9.71 747
9.71 767
9.71 788
9.78590
9.78618
9.78647
9.78675
9.78 704
0.21 410
O.2I 382
0-21353
0.21325
0.21 296
9-93 "5
9.93 1 08
9.93 loo
9.93092
9.93084
35
34
33
32
*
%
2
25
24
23
22
21
~w
19
18
\l
9.71 809
9.71 829
9-71850
9.71 870
9.71 891
9-78732
9.78760
9-78789
9-73817
9-78845
0.21 268
0.21 240
O.2I 211
O.2I 183
0.21 155
9.93077
9.93069
9.93061
9.93053
9.93046
9
II
39
9.71 911
9.71932
9.71952
9-7i 973
9.71994
9-78874
9.78902
9.78930
9-78959
9-78987
0.21 126
0.21 098
0.21 07O
O.2I 041
0.21 013
9-93038
9.93030
9.93022
9.93014
9.93007
40
4i
42
43
44
9.72014
9.72034
9.72055
9.72075
9.72096
9.79015
9-79043
9.79072
9.79 loo
9.79 128
0.20985
0.20957
0.20928
O.2O 9OO
0.2072
9.92999
9.92991
9.92983
9-92976
9 . 92 968
S
S
49
9.72 116
9.72 137
9.72 157
9.72177
9.72 198
9-79I56
9-79 185
9-792I3
9.79241
9.79269
o . 20 844
0.20815
0.20 787
0.20759
o 20 731
9 . 92 960
9.92952
9.92944
9.92936
9.92929
15
14
J 3
12
II
ICT
z
50
5i
52
53
54
9.72 218
9.72238
9.72259
9.72279
9-72299
9-79297
9.79326
9-79354
9.79382
9.79410
0.20 703
O.2O 674
. 2O 646
O.206l8
0.20 590
9.92921
9.92913
9.92905
9.92 897
9.92 889
55
56
9
59
9.72320
9.72340
9.72360
9.72381
9.72401
9-79438
9-79466
9-79495
9.795 2 3
9-79551
0.20 562
0.20534
0.20 505
0.20477
0.20449
9.92881
9.92874
9.92866
9-92858
9.92 850
5
4
3
2
I
~0"
60
9.72421
9-79579
O.2O 421
9.92842
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
/
Prop. Pts.
58
TABLE IV.
32
9
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
p
roi
). ]
Pts.
I
2
3
4
9.72421
9.72441
9.72461
9.72482
9.72502
20
20
21
2O
20
9-79579
9.79607
9.79635
9.79663
9.79691
28
28
28
28
98
0.20421
0.20393
0.20 365
0.20337
0.20309
9.92842
9.92834
9.92 826
9.92818
9.92 810
8
8
8
8
GO
59
58
%
.1
2
z
2
9
38
2.8
<: 6
i
I
9
9.72522
9-72542
9-72562
9.72582
9 . 72 602
20
20
2O
2O
2O
9-79 719
9-79 747
9-79 776
9.79804
9-79832
28
29
28
28
28
O.2O 28l
0.20253
0.20224
0.20 196
0.20 168
9.92 803
9-92 795
9.92 787
9-92 779
9-92 77i
8
8
8
8
g
55
54
53
52
3
4
7
14
17
2C
5
-4
.T,
:> - u
8.4
II. 2
14-0
16.8
19.6
12
13
H
9.72622
9.72643
9-72663
9.72683
9-72703
21
2O
20
20
2O
9 . 79 860
9.79888
9.79916
9-79944
9.79972
28
28
28
28
28
0.20 140
0.20 112
0.20084
0.20056
0.20028
9-92 763
9.92755
9.92747
9-92 739
9.92 731
8
8
8
8
3
1
%
.8
9
%
.2
a
22.4
25.2
7
19
9.72723
9-7? 743
9.72763
9.72783
9.72803
20
20
20
20
2O
9.80000
9.80028
9 . 80 056
9.80084
9.80 112
28
28
28
28
28
0.20000
0.19972
0.19944
O.I9 916
0.19888
9.92 723
9.92 715
9 92 707
9.92699
9.92691
8
8
8
8
3
45
44
43
42
.1
.2
3
4
1
2
i
1C
7
.4
iis
20
21
22
23
24
9-72823
9-72843
9.72863
9-72883
9.72902
2O
20
20
19
9.80 140
9.80168
9-80 195
9.80223
9.80251
28
27
28
28
28
0.19 860
0.19832
0.19805
0.19777
o. 19 749
9.92 683
9.92675
9.92667
9.92659
9.92651
8
8
8
8
3
40
1
1
9
I*
2]
2;
*-9
1.6
t-3
27
28
29
9.72922
9-72942
9.72962
o . 72 982
9.73002
20
20
2O
2O
9.80279
9.80307
9.80335
9.80363
9.80391
28
28
28
28
28
0.19 721
0.19 693
0.19 665
0.19637
0.19 609
9-92643
9-92635
9.92627
9.92 619
9.92 611
8
8
8
8
3
35
34
33
32
.1
.2
3
4
-.
2
4
(
J
II
!.I
I 2
H
30
2.0
4.0
6.0
8.0
30
32
33
34
9.73022
9.73041
9.73061
9.73081
9.73101
20
20
20
9.80419
9.80447
9.80474
9 . 80 502
9.80530
28
27
28
28
28
0.19581
0.19553
0.19 526
0.19498
0.19470
9.92603
9-92 595
9.92587
9-92579
9.92571
8
8
8
8
3
30
29
28
%
9
1C
i:
i,
K
I.
11
1-7
3.8
5-9
IO.O
12.
14.0
16.0
18.0
9
9
39
9.73121
9.73140
9-73 160
9.73180
9.73200
J 9
20
20
20
10
9.80558
9.80586
9.80614
9 . 80 642
9.80669
28
28
28
27
28
0.19 442
0.19414
0.19 386
0.19358
0.19331
9-92563
9-92555
9.92 546
9-92 538
9-92530
8
9
8
8
3
25
24
23
22
21
.2
[ . C
9
40
41
42
43
44
9.73219
9.73239
9-73259
9.73278
9.73298
20
20
20
9.80697
9-80725
,9.80753
9.80 781
9.80808
28
28
28
27
28
0.19303
0.19275
0.19247
0.19 219
0.19 192
9.92522
9.92514
9-92506
9.92498
9-92490
8
8
8
8
3
20
19
18
17
16
3
7
i
i
\l
?-5
i.4
3-3
1:1
4-5
54
6-3
49
9-733I8
9-73337
9-73357
9-73377
9 73396
19
20
20
'9
9.80836
9.80864
9.80892
9.80919
9.80947
28
28
27
28
28
o. 19 164
0.19 136
0.19 108
0.19081
0.19053
9.92482
9-92473
9.92465
9-92457
9.92449
9
8
8
8
3
15
14
13
12
II
.8
9
i
i
5-2
7-1
8
21
7
W
53
54
9-734i6
9-73435
9-73455
9-73474
9-73494
20
20
9.80975
9.81 003
9.81 030
9.81 058
9.81 086
28
27
28
28
0.19025
0.18997
o. 18 970
0.18 942
0.18 914
9.92441
9-92433
9.92425
9.92416
9 . 92 408
8
8
9
8
3
10
I
.1
.2
3
4
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t
D.8
1.6
2.4
3-2
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o-7
2.1
2.8
3-5
59
9 73513
9-73533
9 73552
9 73572
9 73591
2O
20
19
9.81 113
9.81 141
9.81 169
9.81 196
9!8i 224
28
28
27
28
28
0.18887
0.18859
0.18831
0.18804
o. 18 776
9.92 400
9.92 392
9.92384
9-92367
8
8
8
9
3
s
4
3
2
I
9
t
(
\.t>
\\
7-2
4.2
4-9
1.1
!()()
9 73 6n
9.81 252
0.18 748
9-92359
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
f
1
To
1>.
Pte.
57
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC.
33
/
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pte.
I
2
3
4
9.73611
9.73630
9-73 6 50
9.73 669
9.73689
19
20
*9
30
19
'9
20
19
J 9
20
19
19
2O
19
*9
20
9
J 9
19
19
20
19
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J 9
J 9
20
9
9
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9
19
19
J 9
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X 9
19
19
9
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19
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19
i9
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19
9
19
18
19
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19
19
19
18
19
19
19
18
*9
9.81 252
9.81 279
9.81 307
9-8i 335
9.81 362
7
28
28
27
28
28
27
28
27
28
0.18 748
0.18 721
0.18 693
0.18665
0.18638
9 92359
9-9235 1
9-92343
9-92335
9.92326
8
8
8
9
8
8
8
9
8
8
8
9
8
8
9
. 8
8
8
9
8
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9
8
8
9
8
8
9
8
8
9
8
8
9
8
9
8
8
9
8
9
8
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
60
3
11
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5 H
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7 19
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9 25
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90
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14-0
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18.0
9
5:1
M
95
H.4
J3.3
15.2
17.1
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1.8
36
5-4
7.2
9.o
10.8
12 6
14.4
16.2
9
>-9 0.8
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.6 3.2
5 4.o
4 4.8
3 5-6
.2 6.4
.1 72
I
I
9
9.73708
9.73727
9-73747
9.73766
9-73785
9-8i 390
9.81418
9.81445
9 8i473
9.81 500
0.18 610
0.18 582
0.18555
0.18 527
0.18 500
9.92318
9.92310
9.92302
9.92293
9-92285
55
54
53
52
5i
10
ii
12
13
14
9.73805
9.73824
9.73843
9.73863
9.73882
9.81 528
9 81 556
9.81 583
9.81 611
9.81 638
28
27
28
27
28
0.18472
0.18444
0.18417
0.18389
0.18 362
9.92277
9.92269
9.92 260
9 92252
9.92244
50
3
3
15
10
\l
19
,20
21
22
23
24
9.73901
9.73921
9-73940
9 73959
9.73978
9.81 666
9.81 693
9.81 721
9.81 748
9.81 776
27
28
27
28
27
28
27
28
27
28
27
28
27
28
27
28
27
28
27
27
28
27
28
27
27
28
27
28
27
27
28
27
27
28
27
27
28
27
27
27
28
27
27
27
28
0.18334
0.18307
0.18 279
0.18252
0.18 224
9-92235
9.92227
9.92 219
9.92 211
9 . 92 202
45
44
43
42
41
40"
i
9-73997
9.74017
9.74036
9-74055
9.74074
9.81 803
9.81 831
9.81858
9.81 886
9.81913
0.18 197
0.18 169
0.18 142
0.18 114
0.18087
9.92 I 9 4
9.92 186
9-92 177
9.92 169
9.92 161
3
i 27
28
29
9-74093
9.74"3
9-74132
9-74 IS 1
9.74170
9.81941
9.81 968
9.81 996
9.82023
9.82051
0.18 059
0.18032
0.18004
0.17977
0.17949
9-92 152
9.92 144
9.92 136
9.92 127
9.92119
35
34
33
32
3 1
30
29
28
27
26
30
3 1
32
33
34
9.74189
9.74208
9.74227
9.74246
9.74265
9.82078
9.82 106
9.82 133
9.82 161
9.82 1 88
0.17922
0.17894
0.17867
0.17839
0.17 812
9.92 in
9.92 102
9 . 92 094
9 . 92 086
9.92077
9
i?
39
9.74284
9 -7433
9.74322
9-74341
9.74360
9.82 215
9.82243
9.82 270
9.82 298
9.82325
0.17785
0.17757
0.17730
0.17 702
0.17675
9.92069
9.92060
9-92052
9.92044
9-92035
25
24
23
22
21
W
il
40
4i
42
43
44
9-74379
9-74398
9.744I7
9.74436
9-74455
9.82352
9.82380
9.82407
9-82435
9 . 82 462
0.17 648
0.17 620
0.17593
0.17565
0.17538
9.92027
9.92 018
9.92010
9 . 92 002
9-91 993
3
s
49
9-74474
9-74493
9-74512
9-74531
9-74549
9.82489
9.82517
9.82544
9.82571
9.82599
0.17511
0.17483
0.17456
0.17429
0.17401
9.91 985
9.91 976
9.91 968
9-9i 959
9-9i 95i
15
H
13
12
II
50
5i
52
53
54
9.74568
9-74587
9.74606
9.74625
9.74644
9.82 626
9.82653
9.82681
9.82 708
9-82 735
0.17374
0.17347
0.17319
0.17 292
0.17265
9.91 942
9-9i 934
9.91 925
9.91 917
9.91 908
10
I
I
5
4
3
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55
56
H
59
9.74662
9.74681
9.74700
9.74719
9-74737
9.82 762
9.82 790
9.82 817
9.82 844
9.82871
0.17238
0.17 210
0.17 183
O.I7I56
O.I7 129
9.91 900
9.91 891
9-91 f3
9.91 874
9.91 866
00
9-74756
9.82899
0.17 ioi
9-91 857
L. Cos.
d.
L. Cotg.
c. d.
L. Tang.
L. Sin.
d.
/
Prop. Pts.
56
TABLE IV.
34
/
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Pro]
0.
Pte.
I
2
3
4
9-74756
9-74775
9-74794
9.74812
9-74831
X 9
X 9
18
X 9
9.82899
9.82 926
9-82953
9.82980
9.83008
27
27
27
28
27
0.17 101
0.17074
0.17047
0.17 020
0.16992
9.91 857
9.91 849
9.91 840
9.91 832
9.91 823
8
9
8
9
8
58
11
g
.1 2
2 e
8
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6
27
2.7
r A
i
I
9
9.74850 J
9.74868
9-74887
9.74906
9.74924
18
X 9
X 9
18
to
9-83035
9.83062
9-83089
9-83117
9-83 144
27
27
28
27
27
0.16 965
0.16938
0.16911
0.16883
0.16856
9.91815
9.91 806
9.91 798
9.91 789
9.91 781
9
8
9
8
55
54
53
5i
3 8
.4 ii
5 14
.6 16
.7 19
4
.2
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11
10.8
13-5
16.2
18.9
10
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12
13
9-74943
9.74961
9.74980
9-74999
9.75017
18
X 9
x8
X 9
9-83 171
9.83 198
9-83225
9-83252
9.83280
27
27
27
28
27
0.16 829
0.16 802
0.16775
0.16 748
0.16 720
9.91 772
9-91 763
9-9i 755
9.91 746
9.91 738
9
9
8
9
8
50
8
3
.8 22
9 25
4
.2
i
21.6
24.3
(6
19
9-75036
9-75054
9.75073
9.75091
9 75 "
18
X 9
18
X 9
18
9-83307
9-83334
9-83361
9.83388
9-83415
27
27
27
0.16693
0.16666
0.16639
0.16 612
0.16 585
9.91 729
9.91 720
9.91 712
9.91 703
9.91 695
9
8
9
8
45
44
43
42
41
.1
.2
3
4
1
2
c
i
i
1C
i;
>..6
>-4
'6
20
21
22
23
24
9-75 128
9-75 147
9-75 165
9-75 184
9.75202
X 9
18
X 9
18
9.83442
9.83470
9-83497
9-83524
9-83551
28
2?
27
27
0.16 558
o. 16 530
0.16503
0.16476
o 16 449
9.91 686
9.91 677
9.91 669
9.91 660
9.91 651
9
9
8
9
9
40"
P
H
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9
ij
2<
2;
>- 6
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M
29
9-75221
9.75239
9-75258
9-75276
9.75294
18
X 9
18
18
9.83578
9.83605
9-83659
9.83686
27
27
27
27
27
0.16422
0.16395
0.16368
0.16341
0.16 314
9.91 643
9.91 634
9.91 625
9.91617
9.91 608
9
9
8
9
35
34
33
32
.1
.2
-3
A
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[9
7 6
80
32
33
34
9-753I3
9-75.331
9-75350
9-75368
9 75386
18
X 9
18
18
ig
9-83 7i3
9.83 740
9-83 768
9-83 795
9.83822
27
28
2 7
27
27
0.16287
0.16260
0.16 232
0.16 205
0.16 178
9-9i 599
9-91 59i
9.91582
9.91573
9 -9i S 6 ?
9
8
9
9
8
30
27
26
9
(
I
i;
i
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>-5
1-4
5-3
5-2
7-1
P
3^
39
9 75405
9-75423
9 75441
9-75459
9 75478
18
18
18
X 9
18
9-83849
9-83876
9-83903
9-83930
9.83957
27
27
2 7
27
0.16 151
0.16 124
0.16097
0.16 070
o. 16 043
9-91 556
91 547
9.91 530
9.91 521
9
9
8
9
25
24
23
22
21
.1
.2
IS
[.8
z.6
40
42
43
44
9-75496
9-755I4
9-75533
9-75551
9-75 569
18
18
18
18
9-83984
9.84 on
9.84038
9.84065
9 . 84 092
27
27
27
27
o. 16 016
0.15989
0.15 962
0.15935
0.15 908
9.91512
9.91 504
9-91 495
9.91 486
9.91 477
9
8
9
9
9
3
20
19
18
17
16
-3
4
i
K
I
5-4
7-2
?-o
D.8
2.6
47
48
49
9-75587
9-75605
9 75 624
9.75642
9.75660
It
18
18
18
9.84 119
9 . 84 146
9.84173
9 . 84 200
9.84227
7
27
27
27
0.15881
0.15854
0.15 827
0.15 800
0.15 773
9.91 469
9.91 460
9-91 451
9.91 442
9-91433
9
9
9
9
8
15
14
13
12
II
9
9
8
50
5*
52
53
54
9.75696
9-757H
9-75733
9-75751
18
18
X 9
18
18
9.84254
9.84280
9.84307
9.84334
9.84361
27
27
27
0.15 746
o. 15 720
0.15 693
o.'is 666
0.15639
9.91 425
9.91 416
9.91407
9.91 398
9.91 389
9
9
9
9
8
10
I
.1 C
.2 1
3 '<
.4 :
5 4
> 9
.8
5-7
\-6
[5
0.8 i
i 6
2 4
3-3
11
56
59
9.75769
9.75787
9.75805
9-75823
9-75841
z8
18
x8
18
18
9.84388
9.84415
9.84442
9.84469
9.84496
27
27
27
0.15 612
0-15585
0.15558
O.I553I
0.15 504
9.91 381
9.91 372
9-91 363
9-91 354
9-91 345
9
9
9
9
5
4
3
2
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:5 1
-4
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r.2
;.i
4 .S
I 6
6.4
7.2
60
9-75 859
9.84523
0.15477
9-9i S3 6
9
L. Cos.
d.
L. Cotg.
c.d.
L. Tancr.
L. Sin.
d.
t
Pro
>
Pis.
55
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC.
85
t
L. Sin.
<1.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Proj
.1
?ts.
I
2
3
4
9-75859
9.75877
9.75895
9.75913
9-75931
18
It
18
18
18
9-84523
9-8455o
9.84576
9 . 84 603
9.84630
27
26
27
27
27
o.i5477
0.15450
0.15424
0.15397
0.15370
9-91 S3 6
9.91 328
9.91319
9.91 310
9.91 301
8
9
9
9
9
60
5 2
58
H
2
.1 2
.2 5
7
-7
4
26
2.6
C.2
I
I
9
9-75949
9.75967
9.75985
9/76003
9.76021
18
18
18
18
18
9-84657
9.84684
9.84711
9.84738
9.84764
27
27
27
26
27
0-15343
0.15 316
0.15 289
0.15 262
0.15 236
9.91 292
9.91 283
9.91 274
9.91 266
9.91 257
9
9
8
9
9
55
54
53
52
5i
3 8
.4 10
: 7 Is
.1
.8
5
.2
9
7.8
10.4
I 3 .0
i.6
18.2
10
ii
12
13
14
9.76039
9.76057
9.76075
9.76093
9 . 76 1 1 1
18
18
18
18
18
9.84791
9.84818
9.84845
9 . 84 872
9.84899
27
27
27
27
26
0.15 209
0.15 182
0-15 155
0.15 128
0.15 101
9.91 248
9.91 239
9.91 230
9.91 221
9.91 212
9
9
9
9
50
49
48
47
46
.8 21
9 24
.6
3
i
20.8
23 4
8
11
\l
19
9.76129
9.76146
9.76164
9.76 182
9 . 76 200
7
18
18
18
18
9.84925
9.84952
9.84979
9.85 006
9-85033
27
27
27
27
26
0-15075
0.15 048
0.15 021
0.14994
0.14967
9.91 203
9.91 194
9.91 185
9.91 176
9.91 167
9
9
9
9
45
44
43
42
4i
.1
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3
4
\
1
t
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1-6
5-4
r-2
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20
21
22
23
24
9.76 218
9.76236
9-76253
9.76271
9.76289
18
7
18
18
18
9-85059
9.85086
9-85 "3
9.85 140
9.85 166
27
27
27
26
0.14941
O.I49I4
0.14887
0.14860
0.14834
9.91 158
9.91 149
9.91 141
9.91 I 3 2
9.91 123
9
8
9
9
40
li
II
i
9
i:
i.
K
.5
1.6
11
11
27
28
29
9.76307
9.76324
9.76342
9.76360
9.76378
7
18
18
18
9-85 193
9.85 220
9-85 247
9.85273
9.85300
27
27
26
27
0.14807
O.I4 780
0-14753
0.14727
0.14 700
9.91 114
9.91 IO5
9 91 096
9.91087
9.91 078
9
9
9
9
35
34
33
32
3i
.1
.2
3
A
t7
t-7
J-4
Ij
30
3i
32
33
34
9-76395
9.76413
9.76431
9.76448
9 . 76 466
18
18
17
18
18
9-85327
9-85354
9-85380
9.85407
9-85434
27
26
27
27
26
O.I4 673
0.14 646
0.14620
0.14593
0.14566
9.91069
9.91 060
9.91051
9.91042
9 91 033
9
9
9
9
30
2 2 |
11
1
i
9
i
i
i
i
*-5
D.2
1 1
36
5-3
9
9
39
9.76484
9.76501
9 76519
9.76537
9.76554
17
18
18
17
18
9.85460
9-85487
9-855I4
9.85540
9-85 567
27
27
26
27
0.14540
O.I45I3
0.14486
0.14460
0-14433
9.91023
9.91014
9.91 005
9.90996
9.90987
9
9
9
9
25
24
23
22
21
.1
.2
zo
I.O
2.O
40
41
42
43
44
9.76572
9.76590
9.76607
9-76625
9 . 76 642
18
7
18
*7
x8
9 85594
9.85 620
9.85647
9.85674
9.85 700
26
27
27
26
0.14406
0.14380
0-14353
o. 14 326
0.14300
9.90978
9.90969
9.90960
9.90951
9.90942
9
9
9
9
20
19
ii
3
-4
:i
.7
30
4.0
|-o
5.0
7.0
9
9
49
9.76 660
9.76677
9-76695
9.76712
9.76730
17
18
7
18
9.85 727
9-85 754
9.85 780
9-85807
9-85834
27
27
26
27
27
ofi
0.14273
0.14246
0.14220
0.14193
0.14 166
9 90933
9.90924
9.90915
9.90906
9 . 90 896
9
9
9
9
xo
15
14
13
12
II
.8
9
9
8.0
30
8
50
5i
52
53
_5!_
9.76747
9.76765
9.76782
9.76800
9.76817
18
17
18
7
18
9.85860
9.85887
9-859I3
9.85940
9.85967
27
26
2 7
2 7
0.14 140
0.14113
0.14 087
0.14060
0.14033
9.90887
9.90878
9 . 90 869
9 . 90 860
9 . 90 85 1
9
9
9
9
9
10
I
.1 C
.2 1
3 2
4 2
'} A
>-9
'I
r-5
0.8
1.6
2.4
32
4 'S
55
56
12
59
9-76835
9-76852
9.76870
9.76887
9.76904
17
18
7
17
18
9 85993
9 . 86 020
9.86046
9.86073
9.86 loo
27
26
27
27
0.14007
0.13980
0-13954
0.13927
o 13 900
9.90842
9.90832
9.90823
9.90814
9 90805
9
xo
9
9
9
5
4
3
2
I
:? i
.8 I
. 9 i
4
3
.2
.1
4-8
56
6.4
7 2
GO
9.76922
9.86 126
o 13874
9 90796
9
L. Cos.
d.
L. Cotgr.
c.d.
L. Tang.
L. Sin.
d.
t
Pro]
>
Pfs.
54
66
TABLE IV.
36
1
L. Sin.
(1.
L. Tang.
C.<1.
L. Cot!?.
L. Cos.
<1.
Pro]
>.
Pts.
I
2
3
4
9.76922
9 76939
9-76957
9.76974
9.76991
17
18
J7
7
18
9.86 126
9-86 153
9.86179
9.86206
9.86232
27
26
27
26
27
0.13874
0.13847
0.13 821
0.13794
0.13 768
9.90 796
9.90787
9.90777
9.90768
9.90 759
9
10
9
9
GO
3
11
3
I 2
2 C
7
7
|
26
2.6
IT 2
I
I
9
9.77009
9.77026
9-77043
9.77061
9.77078
17
17
18
7
17
9.86 259
9.86285
9.86312
9.86338
9.86365
26
2 7
96
27
27
o. 13 741
0.13 7i5
0.13688
0.13 662
0.13635
9.90750
9.90 741
9.90731
9.90 722
9.90 713
9
9
x>
9
9
55
54
53
52
5i
3 *
.4 ic
* \l
7 iS
.1
.8
5
.2
9
fl
10.4
ill
10
ii
12
13
'4
9-77095
9.772
9.77130
9-77 H7
9.77164
7
18
7
*7
*7
9 86392
9.86418
9.86445
9.86471
9.86498
26
27
26
27
26
0.13608
0.13582
0.13555
0.13529
0.13 502
9.90704
9-90694
9.90685
9.90 676
9.90667
9
10
9
9
9
50
3
%
.8 21
.9 24
.6
3
20.8
23.4
18
III
17
18
19
9.77181
9.77199
9.77216
9.77233
9-77250
18
17
7
17
18
9-86524
9.86551
9.86577
9.86603
9.86630
27
26
26
27
2<>
0.13476
0.13449
0.13423
0.13397
0.13370
9.90657
9.90648
9-90639
9.90630
9 . 90 620
9
9
9
10
45
44
43
42
41
.1
.2
3
4
I
I
>,
i
c
.8
1-6
!-4
r.2
).o
20
21
22
23
24
9.77268
9.77285
9.77302
9-773I9
9-77336
17
7
17
7
17
9.86656
9.86683
9.86709
9-86 736
9.86 762
27
26
27
26
0.13344
0.13317
0.13291
0.13264
0.13 238
9.90611
9.90602
9.90592
9-90583
9-90574
9
9
xo
9
9
~w
fs
11
.0
:l
9
K
Ii
1^
I(
).o
j.6
M
) 2
2
3
29
9-77353
9-77370
9-77387
9.77405
9.77422
7
17
18
17
17
9.86789
9.86815
9.86842
9.86868
9.86894
26
27
26
26
0.13 211
0.13 185
0.13 158
O.I3I32
0.13 106
9-90565
9.90555
9.90546
9-90537
9.90527
9
10
9
9
xo
35
34
33
32
3i
.1
.2
3
]
17
[-7
5-4
ij
30
3i
32
! 33
1 34
9-77439
9-77456
9-77473
9.77490
9-77507
7
7
7
'7
17
9.86921
9.86947
9.86974
9.87000
9.87027
26
27
26
2 7
26
0.13079
0.13053
0.13026
0.13000
0.12973
9.90518
9.90509
9-90499
9.90490
9.90480
9
9
10
9
xo
80
1
:!
.9
!
1C
i
;
i
J-5
5.2
'i
j.6
53
!$
12
39
9-77524
9-77541
9-77558
9-77575
9-77592
'7
7
7
17
17
9-87053
9.87079
9.87 106
9-87 132
9.87 158
26
27
26
26
0.12947
0.12 921
0.12894
0.12868
O.I2 842
9.90471
9.90462
9.90452
9-90443
9-90434
9
9
xo
9
9
25
24
23
22
21
.1
.2
16
[.6
i. 2
40
4i
42
43
44
9 . 77 609
9.77626
9-77643
9 77660
9.77677
17
17
i?
17
17
9-87 185
9.87211
9.87238
9.87264
9.87290
26
2 7
26
26
0.12 815
O.I2 789
O.I2 762
0.12 736
0.12 710
9.90424
9.90415
9.90405
9.90396
9.90386
9
xo
9
xo
20
19
il
3
4
:I
.7
*
i
(
f
3
;:o-
>.6
1.2
45
46
4 4 I
49
9.77694
9.77711
9.77728
9-77 744
9.77761
17
17
16
17
9.87317
9-87343
9.87369
9.87396
9.87422
26
26
2 7
26
26
0.12 683
0.12 657
0.12 631
0.12 604
0.12 578
9.90377
9.90368
9-90358
9-90349
9-90339
9
9
xo
9
xo
15
14
13
12
II
.8
9
i
u
i<
2.8
J-4
9
50
5i
5 2
53
54
9.77778
9-77795
9.77812
9-77829
9.77846
J7
7
17
'7
16
9.87448
9.87475
9-87501
9.87527
9.87554
27
26
26
27
26
0.12552
O.J2525
0.12499
0.12473
0.12446
9-90330
9.90320
9.90311
9.90301
9.90292
9
xo
9
xo
9
10
I
.1 i
.2 2
3 3
4 A
'? f
.0
.0
.0
.0
.0
?:2
ai
4-5
11
11
59
9.77862
9.77879
9.77896
9.779I3
9.77930
'7
17
'7
'7
16
9.87580
9.87606
9-87633
9-87659
9.87685
26
2?
26
26
26
O. 12 42O
0.12394
0.12367
O.I234I
O.I23I5
9.90 282
9.90273
9 . 90 263
9.90254
9.90244
9
xo
9
xo
5
4
3
2
I
.6 c
:l I
9 9
.0
.0
.0
.0
1.1
C
00
9.77946
9.87711
0.12 289
9-90235
9
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
9
Pro]
). .
Pts.
53
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 67
1 37
t
1 Sin.
d.
L. Tang 1 .
c.d.
L. Cotg-.
L. Cos.
d.
6JT
Prop. Pts.
I
2
3
4
9.77946
9.77963
9.77980
9-77997
9.78013
17
17
16
'7
16
17
7
16
7
17
7
16
16
7
17
16
7
16
17
16
17
16
16
17
16
17
16
16
7
16
7
16
i7
16
16
16
17
16
16
16
16
16
7
16
16
16
16
17
16
16
16
9.87711
9-87738
9-87764
9.87 790
9-87817
27
26
26
27
26
26
26
27
26
26
0.12 289
0.12 262
0.12 236
0.12 210
0.12 183
9-90235
9.90225
9.90 216
9.90206
9.90197
xo
9
xo
9
xo
9
xo
9
xo
xo
9
xo
9
xo
xo
9
xo
9
xo
xo
9
xo
xo
9
10
xo
9
xo
xo
9
xo
xo
9
xo
10
xo
.9
10
xo
xo
9
xo
10
xo
9
xo
IO
xo
xo
9
xo
10
xo
xo
xo
9
xo
xo
10
xo
.1
.2
3
4
i
:l
-9
.1
.2
3
4
4
.1
.2
3
4
:I
9
.2
3
4
'.S
9
.1 l
.2 i
3 :
4 4
V
91 *
7
2.7
H
10.8
'3-5
16.2
18.9
21.6
24.3
2.6
5-2
7.8
10.4
13.0
15.6
18.2
20.8
23 4
17
1-7
3-4
5.1
6.8
8-5
10.2
"I
13-6
15-3
16
1.6
|!
8.0
9-6
II. 2
12.8
14.4
to g
.0 0.9
5.0 1.8
5-0 2.7
[.o 3.6
>.o 4-5
>-o 5.41
r.o 6.3
5.0. 7.2
i.o[ 8.1
I
I
9
10
12
13
9.78030
9.78047
9-78063
9.78080
9.78097
9.87843
9.87869
9.87895
9.87922
9.87948
O.I2I57
0.12 131
0.12 ID?
0.12078
0.12052
9.90187
9.90 178
9.90 168
9-90 159
9-90 149
55
54
53
52
9.78113
9-78130
9.78147
9-78163
9.78 180
9.87974
9.88000
9.88027
9-88053
9.88079
26
27
26
26
0.12026
0.12000
O.II973
O.II947
o.ii 921
9.90139
9-90 13
9.90 120
9.90 III
9.90 ioi
50
4 2
4 8
~iT
44
43
42
41
!i
\l
19
"20"
21
22
23
24
9.78197
9-78213
9.78230
9.78246
9.78263
9.88 105
9 .88 131
9.88 158
9.88 184
9.88210
26
8 7
26
26
26
26
27
26
26
26
26
27
26
26
26
26
26
27
26
26
26
26
26
26
26
27
26
26
26
26
26
26
26
26
26
26
27
26
26
26
26
26
26
26
o.ii 895
O.II 869
o.ii 842
o.ii 816
o.ii 790
9.90091
9.90 082
9.90072
9-90063
9-90053
9.78280
9.78296
9-78313
9.78329
9.78346
9.88236
9.88262
9.88289
9.88315
9.88341
o.ii 764
o.ii 738
o.ii 711
o.ii 685
o.ii 659
9.90043
9.90034
9 . 90 024
9.90014
9.90005
1
27
28
29
9.78362
9.78379
9-78395
9.78412
9.78428
9.88367
9.88393
9.88420
9.88446
9.88472
o.ii 633
o.ii 607
o.ii 580
o.ii 554
o.ii 528
9-89995
9.89 985
9.89976
9.89966
9.89956
35
34
33
32
3'
30
32
33
34
9.78445
9.78461
9.78478
9.78494
9.78510
9.88498
9.88524
9-88550
9.88577
9.88603
o.ii 502
o.ii 476
o.ii 450
o.ii 423
0.11397
9.89947
9-89937
9 '.899i8
9.89908
30
29
28
%
I
39
41
42
43
44
9.78527
9.78543
9-78560
9-78592
9.88629
9-88655
9.88681
9.88707
9-88 733
o.ii 371
o.ii 345
o.ii 319
o.ii 293
o.ii 267
9.89898
9.89888
9.89879
9.89869
9-89859
25
24
23
22
21
"20"
|2
9.78609
9-78625
9 . 78 642
9.78658
9-78674
9-88 759
9.88 786
9.88812
9.88838
9.88864
o.ii 241
o.ii 214
o.ii 188
o.ii 162
o.ii 136
9.89849
9.89840
9.89830
9 . 89 820
9.89810
47
48
49
9.78691
9.78 707
9.78723
9.88.890
9.88916
9.88942
9.88968
9.88994
O.II IIO
o.ii 084
o.ii 058
o.ii 032
O.II 006
9.89801
9.89 791
9.89 781
9.89 771
9.89 761
15
14
13
12
II
lo~
I
I
50
52
53
54
9.78772
9.78788
9-78805
9.78821
9.78837
9.89020
9.89046
9.89073
9.89099
9-89 125
0.10980
0.10954
o.io 927
0.10901
0.10875
9.89 752
9.89 742
9.89732
9.89 722
9.89 712
1
58
59
9-78853
9.78869
9.78886
9.78902
9.78918
9.89 151
9.89 177
9.89203
9 . 89 229
9-8925?
0.10849
0.10823
0.10797
o.io 771
0.10745
9.89 702
9.89693
9.89683
9.89673
9-89663
5
4
3
2
I
nr
GO
9-78934
9.89281
o.io 719
9 89653
L. Cos.
d.
L. Cotfir.
c.d.
L. Tang-.
L. Sin.
d.
t
Prop. Pts.
52
68
TABLE IV.
38
1
L. Sin.
d.
L. Tanar.
c.d.
L. Cotg.
L. Cos.
d.
r
ro]
).
Pts.
2
3
4
9-7*934
9.78950
9.78967
9.78983
9.78999
16
17
16
16
16
9.89281
9-89307
9.89333
9-89359
9-89385
26
26
26
26
26
o.io 719
0.10693
0.10667
0.10641
0.10615
9-89653
9.89643
9-89633
9 . 89 624
9.89 614
xo
xo
xo
xc
xo
00
9
11
.1
2
2
2
e
6
.6
2
25
2-5
e o 1
I
i
9
9.79015
9.79031
9.79047
9.79063
9.79079
16
*6
16
16
16
9.89411
9.89437
9.89463
9.89489
9-89515
76
26
26
26
26
0.10589
o.io 563
0.10537
o.io 511
0.10485
9.89604
9.89594
9-89584
9-89574
9.89564
xo
xo
xo
xo
55
54
53
52
51
.3
:S
.7
7
10
13
;i
.8
-4
.0
.6
.2
75
10.
12.5
15.0
^S i
10
ii
12
13
H
9-7995
9.79111
9.79128
9.79 144
9.79160
16
17
16
16
16
9.89541
9.89567
9.89593
9.89619
9.89645
26
26
26
26
26
0.10459
0.10433
0.10407
0.10381
0.10355
9-89554
9.89544
9.89534
9.89524
9.89514
xo
xo
xo
xo
50
3
8
.8
9
20
23
.8
4
i
20. o
22.5
7
it
\l
19
9.79176
9.79192
9 . 79 208
9.79224
9.79240
16
16
16
16
16
9.89671
9.89697
9.89723
9.89749
9.89775
26
26
26
26
26
0.10329
0.10303
0.10277
0.10251
0.10225
9.89504
9-89495
9.89485
9-89475
9.89465
9
xo
xo
xo
45
44
43
42
41
.1
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3
4
5
i
i
!
-7
1-4
si
.s
"20
21
22
23
24
9.79256
9.79272
9.79288
9-79304
9.79319
16
x6
x6
IS
16
9.89801
9.89827
9.89853
9-89879
9.89905
26
26
26
26
26
o.io 199
o.io 173
o.io 147
O.IO 121
0.10095
9-89455
9-89445
9-89435
9.89425
9.89415
xo
xo
xo
xo
40
It
9
:l
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I
i;
i.
[ -9
*-6
5-3
3
3
29
9-79335
9-79351
9.79367
9-79383
9-79399
16
16
16
16
x6
9.89931
9.89957
9,89983
9.90009
9-90035
26
26
26
26
26
0.10069
0.10043
0.10017
0.09991
0.09965
9.89405
9.89395
9-89385
9.89375
9.89364
IO
xo
xo
XI
35
34
33
32
31
.1
.2
3
.4
i
i
i
^
f
6
.6
3
> 4
15
'5
3-o
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30
3i
32
33
34
9-794I5
9-79431
9-79447
9-79463
9.79478
16
16
16
15
16
9.90061
9 . 90 086
9.90 112
9.90138
9.90 164
25
26
26
26
26
0.09 939
0.09 914
0.09 888
0.09 862
0.09 836
9.89354
9.89344
9.89334
9.89324
9.89314
xo
10
xo
IO
30
2!
11
1
i
.9
i
c
ii
12
It
1.0
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5.8
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7-5
9.0
10.5
12.
'3 5
9
9
39
9-79494
9.79510
9.79526
9-79542
9-79558
16
16
16
16
9.90190
9.90216
9.90242
9.90268
9.90294
26
26
26
26
26
0.09810
0.09 784
0.09 758
0.09 732
0.09 706
9.89304
9.89294
9.89284
9.89274
9.89 264
xo
xo
10
xo
25
24
23
22
21
.1
.2
ii
[.i
2.2
40
4i
42
43
44
9-79573
9-79589
9-79605
9.79 621
9.79636
x6
16
16
IS
16
9.90320
9.90346
9.90371
9.90397
9.90423
26
25
26
26
26
0.09 680
0.09 654
0.09 629
o . 09 603
0.09577
9.89254
9.89244
9-89233
9.89223
9.89213
xo
XX
xo
xo
20
ii
3
4
7
i
(
3-3
[>4
7-7
2
;* 7
49
9.79652
9 79668
9.79684
9.79699
9-797I5
16
16
IS
16
16
9.90449
9-90475
9-90501
9.90527
9-90553
26
26
26
26
0.09551
0.09525
0.09499
0.09473
0.09447
9.89 203
9-89193
9.89 183
9-89 173
9.89 W2
zo
xo
xo
XX
15
14
13
12
II
.8
9
3
<
i
3.8
)-9
9
50
5i
52
53
54
9-79 73 1
9.79746
9-79 762
9.79778
9 79 793
IS
16
16
IS
16
9.90578
9.90604
9.90630
9.90656
9 . 90 682
26
26
26
26
26
0.09 422
0.09 396
0.09370
0.09344
0.09318
9 .8 9 I52
9.89142
9.89 132
9.89 122
9.89 112
xo
xo
xo
xo
10
I
I
.1
.2
3
4
]
2
3
A
I
.0
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0.9
ii
4-5
55
56
fi
59
9.79809
9-79825
9.79840
9.79856
9.79872
x6
15
x6
16
9 . 90 708
9-90734
9-90759
9.90 785
9.90 811
26
25
26
26
26
0.09 292
0.09 266
0.09 241
0.09 215
0.09 189
9.89 101
9.89 091
9.89081
9.89071
9 . 89 060
xo
10
xo
IX
5
4
3
2
I
:1
.9
I
9
.0
.0
.0
n
K
60
9.79887
9.90837
0.09 163
9 . 89 050
L. os.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
/
p
roj
).
PtSr
51
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 69
39
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Proj
.1
ts.
I
2
3
4
9.79887
9-79903
9.79918
9-79934
9-79950
16
15
16
16
15
9.90837
9.90863
9.90889
9.90914
9-90940
26
26
as
26
26
0.09 163
0.09 137
0.09 in
0.09 086
0.09060
9 . 89 050
9 . 89 040
9.89030
9.89020
9.89009
xo
IO
10
II
IO
00
ii
11
.1
.2
21
2
5
5
.6
.2
\
I
9
9.79981
9.79996
9.80012
9.80027
16
IS
16
15
16
9.90966
9.90992
9.91 018
9.91 043
9.91069
26
26
as
26
26
0.09034
0.09 008
0.08982
0.08957
0.08931
9.88999
9.88989
9.88978
9.88968
9.88958
10
XI
xo
xo
xo
55
54
53
52
51
3
4
'7
7
10
13
!i
.8
4
.0
.6
.2
10"
ii
12
'3
14
9.80043
9.80058
9 . 80 074
9.80089
9.80 105
IS
16
IS
16
15
9.91095
9.91 121
9.91 147
9.91 172
9.91 198
26
26
26
26
0.08905
0.08879
0.08853
0.08828
0.08802
9.88948
9-88937
9.88927
9.88917
9.88906
XX
xo
IO
XX
xo
50
4 i
48
.8
9
20
23
a
.8
4
5
19
9.80 120
9.80 136
9.80151
9.80 166
9.80 182
16
IS
IS
16
15
9.91 22 4
9 9i 250
9.91 276
9-91 301
9.91 327
26
26
as
26
26
0.08 776
0.08 750
0.08 724
0.08 699
0.08673
9 . 88 896
9.88886
9.88875
9.88865
9.88855
IO
XX
xo
xo
jj
45
44
43
42
41
.1
.2
3
4
2
5
7
10
12
5
.0
5
.0
5
20
21
22
23
24
9.80 197
9.80213
9.80228
9.80244
9-80259
16
IS
16
9 9i 353
9-91 379
9.91404
9.91430
9.91 456
26
85
26
26
26
0.08 647
0.08 621
0.08 596
0.08 570
0.08544
9.88844
9-88834
9.88824
9.88813
9.88803
xo
IO
XX
IO
40
3
11
9
J 5
I?
2C
22
5
.0
-5
29
9.80274
9 . 80 290
9-80305
9.80320
9-86336
x6
15
IS
x6
9.91 482
9.91 507
9 9i 533
9-91 559
9 9i 585
as
26
26
26
0.08518
0.08493
0.08467
0.08441
0.08 415
9.88793
9.88782
9.88772
9.88 761
9.88751
XX
IO
IX
xo
35
34
33
32
31
.1
.2
3
A
1
1
3
i
6
.6
1:5
30
32
33
34
980351
9.80366
9.80382
9.80397
9.80412
IS
16
IS
IS
16
9.91 610
9.91 636
9.91 662
9.91 688
9.91 713
26
26
26
as
26
0.08 390
0.08 364
0.08 338
0.08 312
0.08287
9.88741
9 88730
9.88 720
9.88709
9.88699
IX
IO
IX
IO
30
i
I
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J
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i:
K
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1.8
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39
9.80428
9.80443
9-80458
9.80473
9.80489
IS
15
IS
16
9-9i 739
9.91 765
9 9i 791
9.91 816
9 91 842
26
26
as
26
26
0.08 261
0.08235
0.08 209
0.08 184
0.08 158
9.88688
9.88678
9.88668
9.88657
9.88647
xo
xo
II
IO
25
24
23
22
21
.1
.2
]
<5
5
4i
42
43
44
9 . 80 504
9 80519
980534
9 80550
9 80565
IS
16
IS
9 91 868
9 91 893
9.91 919
9-9 945
9.91971
as
26
26
26
0.08 132
0.08 107
0.08081
0.08055
0.08 029
9.88636
9 . 88 626
9 88615
9.88605
9.88594
IO
II
10
II
20
il
3
4
7
i
(
(
1C
kS
) O
r.5
) o
>-5
s
49
9 . 80 580
9 80 595
9 80 610
9 80 625
9 80 641
15
'5
16
9.91 996
9 92 022
9 92 048
9 92073
9.92099
26
26
25
26
26
0.08004
o 07 978
o 07952
0.07927
0.07 901
9 88 584
988573
9.88563
9 88552
9-88542
II
IO
II
IO
15
14
13
12
II
.8
9
i:
i;
d
s.o
J 5
xo
50
52
53
54
9 80 656
9.80 671
9.80686
9.80 701
9.80 716
IS
15
IS
15
9.92 125
9-92150
9.92176
9 . 92 202
9.92227
85
26
26
as
06
0.07875
0.07 850
0.07 824
0.07 798
0.07 773
9.88531
9.88521
9.88510
9-88499
9.88489
xo
II
II
10
10
I
I
.1 1
.2 :
.3 :
.4 /
5 l
t.i
1.2
5-3
t-4
i-j
I.O
2.O
3-o
4.0
i-
55
56
59
9 80731
9 80 746
9 80 762
9.80777
9 . 80 792
15
16
IS
9.92279
9.92304
9.92330
9.92356
26
85
26
26
0.07 747
0.07 721
0.07696
0.07 670
0.07 644
Q 88 478
9 . 88 468
9.88457
9.88447
9.88436
10
XI
10
XX
5
4
3
2
I
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9 S
>.t>
7 -7
5.8
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6.0
7.0
8.0
9.0
(iO
9.80807
9.92381
2 S
0.07 619
9.88425
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
t
Pro]
P.:
Pts.
50
TABLE IV.
40
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pte.
9.80807
9.92381
26
0.07619
9.88425
60
i
9.80822
9.92407
0.07 593
9.88415
59
2
9.80837
is
9-92433
0.07567
9.88404
58
36
3
9.80852
15
9.92458
25
26
0.07 542
9.88394
10
57
j
2 6
4
9.80867
9.92484
26
0.07 516
9.88383
M
56
.2
e.2
1
9.80882
9.80897
9.80912
9.80927
15
15
15
9.92 510
9-92535
9-92561
9.92587
25
26
26
0.07490
0.07465
0.07439
0.07413
9.88372
9.88362
9-88351
9.88340
10
II
II
55
54
53
S2
3
4
7-8
10.4
130
15.6
9
9.80942
15
9.92 612
25
26
0.07388
9.88330
II
7
lS.2
ii
12
9.80957
9.80972
9.80987
15
IS
9.92 638
9.92663
9.92 689
25
26
0.07362
0.07337
0.07311
9.88319
9.88308
9.88298
II
IO
50
49
48
.8
9
20.8
23-4
13
9.8l 002
15
9.92 715
20
0.07285
9.88287
47
14
9.8l 017
15
15
9-92 740
25
26
0.07260
9.88276
IO
46
5
!i
9.8l 032
9.8l 047
IS
9.92 766
9.92 792
26
0.07 234
0.07 208
9.88266
9.88255
XX
45
44
.1
.2
2.5
5-0
IS
9.81 061
9.81 076
14
IS
9.92817
9.92 843
25
26
0.07 183
0.07 157
9.88244
9-88234
to
43
42
3
4
7-5
IO.O
19
9.81 091
IS
15
9.92868
25
26
0.07 132
9.88223
I]
41
5
12.5
20
21
9.81 106
9.81 121
IS
9.92894
9.92 920
26
0.07 106
0.07080
9.88212
9.88201
XI
*9
i
17-5
20 o
22
9.81 136
*5
9 92945
25
?f>
0.07055
9.88 191
38
o
22. C
23
24
9.81 151
9.81 166
IS
9.92971
9.02 996
25
26
0.07029
o . 07 004
9.88 180
9.88 169
II
11
11
9.81 180
9.81 195
9.93022
9.93048
26
0.06 978
0.06952
9.88 158
9 . 88 148
IO
35
34
* f
11
9.8l 210
9.81 225
15
IS
9-93073
9.93099
25
26
0.06927
0.06901
9.88137
9.88126
xr
33
32
.2
J
3-0
29
9.8l 240
15
9 93 124
25
26
0.06876
9.88 115
'.
.4
60
30
3'
32
9.8l 254
9.8l 269
9.8l 284
15
IS
9-93 ISO
9 93 175
9.93 2OI
25
26
0.06 850
0.06 825
0.06 799
9.88105
9.88094
9.88083
ii
IX
30
3
i
7-5
9.0
10.5
33
34
9.8l 299
9.8l 314
15 .
15
9-93227
9.93252
26
25
26
0.06 773
0.06 748
9.88072
9.88061
IX
10
2
9
12.0
135
P
8
9.81328
9-8i 343
9-8i 358
9.81 372
15
15
14
9.93278
9-93303
9-93329
9-93354
25
26
25
0.06 722
0.06 697
0.06 671
0.06 646
9.88051
9.88040
9 88 029
9.88018
XI
II
II
25
24
23
22
N
I A.
39
9.81387
15
9.9338o
26
0.06 620
9.88007
21
.2
I'i
40
9.81 402
9.93406
0.06 594
9.87996
20
3
4-2
41
9.81417
15
9-93431
25
0.06 569
9.87985
19
.4
5-6
42
9.81 431
14
9 93457
26
0.06 543
9 87975
18
7.0
43
44
9.81446
9.81 461
15
15
9.93482
9-93 5o8
25
26
0.06 518
0.06492
9.87964
9.87953
II
11
7
8.4
9.8
45
9 81475
9-93533
25
0.06467
9.87942
IS
.8
II. 2
46
9.81 490
9-8i 505
IS
15
9-93 559
9-93 584
26
25
0.06441
0.06416
9.87931
9.87920
II
14
13
9
12.6
48
9.81 519
14
9.93610
26
0.06390
9.87909
12
49
9 81 534
IS
9.93636
26
0.06 364
9.87898
II
IX 10
50
5 2
9.81 549
9-81563
9.81 578
14
15
9.93661
9-93687
9 93 7 12
25
26
25
0.06339
0.06313
0.06 288
9.87887
9.87877
9.87866
xo
II
10
.1 I.I I.O
.2 2.2 2.0
3 33 3-0
53
9.81 592
14
9-93 73 s
26
0.06 262
9-87855
IX
7
.4 4.4 4.0
54
9.81 607
15
9-93 763
25
0.06237
9.87844
6
-I n I'i
1
9.81 622
9.81 636
9.81 651
9.81 665
14
IS
14
9 93 789
9-93 814
9.93840
25
26
25
0.06211
0.06 186
0.06 160
0.06 135
9-87833
9.87822
9.87811
9.87800
XX
XX
XX
5
4
3
2
.6 6.6 6.0
1 11 I:S
9 9.9 9.0
59
9.81 680
15
9.93891
26
0.06 109
9.87789
XX
I
GO
9 81 694
9.93916
25
0.06084
9.87778
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
t
Prop. Pts.
49 |
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 7,
1
41
,
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos. '
d.
Proj
). Pts.
ho"
2
*
9.81 694
9.81 709
9.81 723
9.81 738
9.81 752
s
14
15
14
15
9.93916
9-93942
9-93967
9-93 993
9.94018
26
25
26
25
26
0.06084
0.06058
0.06 033
0.06007
0.05 982
9.87778
9.87 767
9.87 756
9-87 745
9-87 734
IX
zz
ZI
IZ
ZI
60
9
9
.1
.2
a6
2.6
C.2
*
9
9 81 767
9.81 781
9 81 796
9.81 810
9.81825
14
15
14
15
14
9.94044
9.94069
9-94095
9.94 120
9-94 H6
25
26
25
26
25
0.05 956
0.05 931
0.05 905
0.05 880
0.05 854
9.87723
9 87 712
9.87 701
9.87690
9.87679
zz
zz
zz
zz
JI
55
54
53
52
51
3
:!
.7
7-8
10.4
13.0
15.6
18.2
10"
ii
1 I2
*3
14
9.81 839
9-81854
9.81868
9.81 882
9.81897
IS
14
14
15
14
9.94171
9-94 197
9.94222
9.94248
9-94273
26
25
26
25
26
0.05 829
0.05 803
0.05 778
0.05 752
0.05 727
9.87 668
9.87657
9.87646
9.f7635
9.87624
zz
zz
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9.
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9
20.8
23.4
m
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9.81 926
9.81 940
9 81955
9.81 969
S
H
15
14
14
9.94299
9-94324
9-94350
9 94375
9,94401
S
26
25
26
25
0.05 701
0.05 676
0.05 650
0.05 625
0.05 599
9-87613
9.87601
9.87590
9 87579
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12
zz
zz
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45
44
43
42
41
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3
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5-o
7-5
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20
21
22
1 23
24
9 81 983
9.81 998
9.82 OI2
9 . 82 026
9.82 041
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M
14
S
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9 94 426
9 94452
9-94477
9 94503
9.94528
26
25
26
25
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0.05 574
0.05 548
0.05 523
0.05497
0.05472
9.87557
9.87546
9 87535
9.87524
9 87513
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9.82055
9 82 069
9.82084
9.82098
9.82 112
14
IS
14
14
9-94 554
9-94579
9.94604
9 94630
9 94655
25
25
26
25
26
0.05 446
0.05421
0.05 396
0.05 370
0-05 345
9.87501
9.87490
9.87479
9.87468
9 87 457
tz
II
II
ZI
ZI
35
34
33
32
31
i
.2
3
.4,
5
"5
30
n
30
31
I 32
33
34
9.82 126
9.82 141
9-82 155
9.82 169
9.82 184
IS
14
4
S
9.94681
9.94706
9-94732
9-94757
9-94 783
25
26
25
26
25
0.05319
0.05 294
0.05 268
0.05 243
0.05 217
9 87446
9 87434
9.87423
9 87412
9.87401
IS
II
ZI
II
zz
30
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11
ii
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9
7.5
9 .o
lo-S
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9
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9.82 198
9 82 212
9 82 226
9 . 82 240
9 82255
14
M
4
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9.94808
9-94 834
9.948$9
9.94884
9.94910
26
25
S
26
o 05 192
0.05 166
o 05 141
o 05 116
o 05 090
9 87390
9.87378
9 87 367
9 87356
9 87345
13
zz
zz
ZI
ZI
25
24
23
22
21
.1
.2
14
;i
40
41
42
43
44
9 82 269
9 82283
9 82297
9.82 311
9.82326
14
14
14
IS
9 94935
9.94961
9 94986
9.95012
9 95037
26
25
26
25
0.05 065
005039
0.05 014
o 04988
0.04963
9 87334
9.87322
9 87311
9.87300
9.87288
12
II
II
12
IZ
20
19
il
3
4
i
7
g
i;:
9.8
9
9-
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9 82 340
9 82354
9.82 368
9.82382
9 82396
4
14
14
14
9.95062
995088
9 95 3
9 95 '39
9-95 164
26
25
26
25
26
o 04938
0.04912
0.04 887
0.04 861
o 04 836
9 87277
9 87266
9 87255
9 87 243
9.87232
zz
IZ
Z2
ZI
15
14
13
12
II
.8
9
i
II. 2
12.6
a II
150
5 1
52
53
54
9.82 410
9 82424
9.82439
9 82453
9.82467
14
s
14
4
9-95 !90
9-952I5
9-95 240
9-95 266
9-95 291
25
as
26
5
26
0.04 810
0.04 785
0.04 760
0.04 734
0.04 709
9.87 221
9.87209
9.87 I 9 8
9.87 187
9 87175
la
zz
zz
za
ff
10
6
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.2 s
3 c
4 4
.2 I.I
(.4 2.2
[6 3.3
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56
19
59
9.82481
9.82495
9-82509
9-82523
9 82537
4
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14
4
14
9-953I7
9-95342
9-95 368
9-95393
9.95418
as
26
as
5
96
0.04683
0.04 658
0.04 632
o . 04 607
0.04 582
9.87 164
9-87153
9.87 141
9-87 130
9.87 119
XI
12
xz
11
5
4
3
2
I
Ii
9
r.2 6.6
U 7-7
> 6 8.8
>.8 9.9
9-82551
9-95444
0.04 556
9.87 107
1
L. Cos.
d.
L. Cotgr.
c.d.
L. Tang.
L. Sin.
d.
/
Pro]
p. Pts.
4:8
2
LAKLZ, IV
42
/
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Pro]
). ]
Pts.
I
2
3
4
9.82551
9.82565
9.82579
982593
9.82 607
14
14
14
14
9-95444
9 95469
9-95495
9-95 520
9-95 545
25
26
25
25
26
0.04556
0.04531
0.04505
0.04480
0.04455
9.87 107
9.87096
9-87085
9.87073
9.87062
ii
IX
12
II
60
5 2
58
8
2
a
2
i
.6
2
1
1
9
9.82 621
9-82635
9 . 82 649
9.82663
9.82677
4
14
14
4
14
9-95571
9-95 596
9-95 622
9-95 647
9-95 672
25
26
25
25
26
o 04 429
o 04 404
0.04 378
0.04353
o 04328
9.87 050
9.87039
9.87028
9.87 016
9 - 87 005
II
IX
12
II
12
55
54
53
52
5i
3
.4
:I
.7
7
1C
13
!!
.8
4
.0
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10
ii
12
13
H
9.82 691
9.82 705
9.82 719
9-82 733
9-82747
H
14
J4
4
14
9.95698
9-95 723
9-95 748
9-95 774
9-95 799
25
25
26
25
36
o . 04 302
0.04277
0.04252
0.04 226
0.04 201
9.86993
9 . 86 982
9.86 970
9.86959
9.86947
II
l
IX
12
50
3
11
.8
9
2C
23
3
.8
4
5
II
\l
19
9.82 761
9-82775
9.82 788
9.82802
9.82816
4
13
14
14
1^
9.95 825
9-95850
9-95875
9-95 901
9 95 926
25
25
26
25
26
0.04 175
0.04 150
0.04 125
0.04099
0.04074
9 . 86 936
9 . 86 924
9 86 913
9.86902
9.86890
12
XI
II
is
ii
45
44
43
42
41
.1
.2
3
4
J
2
5
5
1C
12
5
.0
'5
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5
20
21
22
23
24
9.82830
9.82844
9.82858
9 . 82 872
9.82885
>4
4
14
3
14
9-95 952
9-95 977
9.96002
9 . 96 028
9-96053
25
95
6
*5
o 04 048
0.04023
0.03998
0.03 972
0.03947
9.86879
9.86867
9.86855
9.86844
9 . 86 832
13
12
XX
19
40
P
H
:i
9
i;
2C
22
O
r-5
5 O
'5
2 5
26
3
29
9.82899
9.82913
9.82927
9.82941
9 82955
4
14
14
H
13
9.96078
9.96 104
9.96 129
9.96 155
9.96 180
26
25
26
25
0.03 922
0.03 896
0.03871
0.03 845
o 03 820
9.86821
9.86 809
9.86 798
9.86786
9-86775
12
IX
12
XX
35
34
33
32
3i
.1
.2
3
A
1
1
t
14
1:1
H
80
3i
32
33
34
9.82968
9.82982
9.82996
9.83 oio
9.83023
14
14
14
3
14
9.96205
9.96231
9.96256
9.96 281
9.96 307
26
25
25
26
0.03 795
0.03 769
0.03 744
0.03 719
0.03693
9.86763
9.86752
9.86 740
9.86 728
9.86717
XX
12
12
XX
12
30
3
27
26
:i
:I
.9
I
(
I
i:
ii:
>.i
[.2
2.6
9
9
39
9-83037
9.83051
9.8300?
9.83078
9.83092
4
14
13
14
14
9.96332
9 96357
9-96383
9 . 96 408
9-96433
25
26
25
25
26
0.03 668
0.03 643
0.03 617
0.03592
c 03 567
9.86705
9 . 86 694
9.86682
9.86670
9 86 659
XX
12
13
II
25
24
23
22
21
.1
.2
>3
1-3
2.6
40
41
42
43
44
9.83 106
9.83 I2O
9 83 133
9 83 147
9.83 161
14
13
14
4
*3
9-96459
9.96484
9-96510
9.96535
9.96560
25
26
25
25
26
0.03541
0.03 516
0.03490
0.03 465
0.03440
9 86647
9 86635
9 86 624
9.86612
9.86600
12
XI
12
12
20
19
ii
3
4
:!
7
3 9
1
M
4 S
<6
47
48
49
9-83 174
9.83 188
9 . 83 202
9 83215
9.83229
4
14
13
M
9.96586
9.96 61 1
9 . 96 636
9 . 96 662
9.96687
25
25
26
25
0.03 414
0.03389
0.03 364
0.03338
0.03313
9 86 589
9 86577
9 .86 565
9-86554
9.86542
12
12
II
12
15
14
13
12
II
.8
9
3
I
i
2
3-4
1.7
XI
60
5i
52
53
54
9.83 242
9 83256
9-83270
9.83283
9.83297
14
14
3
4
9.96 712
9.96738
9-96763
9.96 788
9.96 814
2 5
26
25
25
26
0.03 288
0.03 262
0.03 237
0.03 212
0.03 186
9.86530
9.86 518
9.86 507
9.86495
9.86483
12
II
12
12
10
I
.1 1
.2 2
-3 I
\ \
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4
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I.I
2.2
3-3
4-4
5-5
fi
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59
9.83310
9 83324
983338
9-8335I
983365
14
14
13
14
9.96839
9.96864
9.96890
9.96915
9 9 6 940
25
25
26
25
25
0.03 161
o 03 136
0.03 no
0.03 085
o . 03 060
9.86472
9 . 86 460
9.86448
9.86436
9.86425
12
12
12
II
5
4
3
2
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.8 c,
9 ic
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Ii
9-9
60
9.83378
9.96 966
0.03034
9.86413
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
/
Pro]
>
Pts.
47
LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 73
1 43
,
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pts.
i
2
3
4
9 83378
9-83392
9-83405
9.83419
9-83432
14
13
13
14
13
14
13
13
13
14
13
13
14
13
13
14
3
13
13
13
3
13
4
3
U
14
13
J3
13
13
13
13
'3
13
14
13
13
'3
13
13
9.96966
9.96991
9.97016
9.97042
9.97067
25
25
26
25
25
0.03034
0.03 009
O.02 984
O.O2 958
0.02 933
9-86413
9.86 401
9-86389
9-86377
9.86366
12
12
12
II
12
S3
S3
13
12
II
S3
12
13
12
13
12
S3
IS
S3
GO
ii
i
55
54
53
52
.1
.2
3
4
.1
:i
-9
.1
.2
3
4
:l
9
.1
.2
3
4
:I
9
.1
.2
3
4
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9
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-3 :
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;! i
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aff
2.6
w
10.4
111
18.2
20.8
23 4
25
5-0
7.5
10.
12.5
17^5
20. o
22.5
14
1:2
U
9-8
II. 2
12.6
13
39
7$
9-1
10.4
11.7
13 II
[.2 I.I
Z.4 2.2
J-6 3-3
1-8 4-4
>- 5-5
7.2 6.6
!:* Ii
).8 9.9
I
8
9
9.83446
9 83459
9.83473
9.83486
9.83500
9.97092
9-97 "8
9-97 143
9.97 168
9-97 193
26
25
25
25
26
25
25
26
25
25
26
25
25
26
25
25
26
25
25
25
26
25
25
26
25
25
26
25
25
25
26
25
25
26
25
25
26
25
25
25
26
25
25
26
25
25
25
26
25
25
26
25
5
25
26
o.o2"9o8
0.02 882
0.02 857
O.02 832
O.O2 807
9-86354
9-86342
9.86330
9.86318
9.86306
10
ii
12
13
H
9.83513
9.83540
9.83554
9-83567
9.97219
9-97244
9-97269
9.97320
0.02 781
O.02 756
0.02 731
0.02 705
0.02 680
9.86295
9.86283
9.86271
9-86259
9.86247
50
4 2
4 8
ii
1 L.
21
22
23
24
9.83581
9-83594
9.83608
9.83 621
9-83634
9-97345
9-97371
9.97396
9-97421
9-97447
0.02 655
O.02 629
O.O2 604
0.02 579
0.02 553
9.86235
9.86223
9.86211
9.86 200
9.86 188
45
44
43
42
9-83648
9.83661
9.83674
9.83688
9.83 701
9-97472
9-97497
9-97548
9-97573
O.O2 528
0.02 503
0.02477
0.02452
O.O2 427
9.86176
9.86 164
9.86 152
9.86 146
9.86 128
S3
S3
S3
S3
S3
13
13
S3
S3
S3
S3
S3
S3
12
S3
S3
13
S3
12
S3
S3
S3
12
12
12
13
13
12
S3
S3
S3
S3
S3
13
S3
S3
S3
S3
S3
40
39
38
37
Jl
35
34
33
32
?
2 9
9.83715
9-83728
9.83741
9.83755
9-83 768
9-97 598
9-97624
9.97649
9.97674
9.97 700
O.O2 4O2
0.02376
0.02351
0.02 326
0.02 300
9.86 116
9.86 104
9.86092
9.86080
9.86068
3
32
i 33
34
9.83781
9 83 795
9.83808
9.83821
9-83834
9-97 725
9=97750
9-97 776
9.97801
9.97826
O.O2 275
0.02 2JJO
0.02 224
O.O2 199
0.02 174
9.86056
9.86044
9.86032
9.86020
9.86008
30
3
11
35
3 6
39
9.83848
983861
9.83874
9-83887
9.83901
9-97851
9.97877
9.97902
9.97927
9-97953
O.O2 149
0.02 123
0.02098
0.02073
0.02 047
9.85996
9.85984
9.85972
9.85960
9.85948
25
24
23
22
21
w
19
i!
40
42
43
44
45
46
i 48
1 49
9.83914
9.83927
9.83940
9 83954
9.83967
9-97978
9.98003
9.98029
9.98054
9-98079
O.O2 O22
o.oi 997
o.oi 971
o.oi 946
o.oi 921
9-85936
9.85924
9.85912
9.85 900
9.85888
9.83980
9.f3993
9 . 84 006
9 . 84 020
9 84 033
9.98 104
9-98 130
9-98155
9.98 180
9.98 206
o.oi 896
o.oi 870
o.oi 845
o.oi 820
o.oi 794
9.85876
9.85 864
9-85851
9-85839
9.85827
15
13
12
II
50
i 5 '
1 53
1 54
9 84 046
9 84059
9 84072
9-8408$
9 . 84 098
9-98231
9-98256
9.98281
9.98307
9-98332
o.oi 769
o.oi 744
o.oi 719
o.oi 693
o.oi 668
9.85815
9-85803
9.85 791
9-85 779
9-85 766
10
S
5
4
3
i
55
5 6
58
59
9.84 112
9.84 125
9.84138
9.84151
9.84 164
9 '.98 p|
9.98408
9.98458
o.oi 643
o.oi 617
o.oi 592
o.oi 567
o.oi 542
9-85 754
9-85742
9-85 730
9.85 718
9-85 7o6
9.84 177
9.98484
o.oi 516
9-85693
L. Cos.
d.
L. Cotsr.
c.d.
L. Tang.
L. Sin.
d.
f
Prop. Pts.
46
TABLE IV.
44
t
L. Sin.
(!.
L. Tang.
e.d.
L. Cot.
L. Cos.
d.
Proi
). Pts.
I
2
3
4
9.84 177
9.84 190
9-84203
9.84 210
9.84229
3
13
*3
3
9.98484
9.98509
9-98534
9.98 560
9-98585
25
25
26
25
o.oi 516
o.oi 491
O.OI 466
o.oi 440
o.oi 415
9-85693
9.85 68 1
9-85669
9-85657
9-85645
12
12
xa
12
(>0
9
11
,i
26
2.6
1 2
I
I
9
9 . 84 242
9 84255
9 . 84 269
9 84282
9.84295
*3
4
>3
*3
9.98 610
9-98635
9.98661
9.98686
9.98711
25
26
25
25
26
o.oi 390
o.oi 365
o.oi 339
o.oi 314
o.oi 289
9.85632
9.85 620
9.85608
9.85596
9.85 583
12
12
12
3
55
54
53
52
5i
3
4
:I
.7
f:
10.4
13.0
15.6
18.2
10
ii
12
*3
H
9.84308
9.84321
9- ^4334
9.84347
9.84360
3
3
13
3
9.98737
9.98 762
9-98 787
9.98812
9.96838
25
25
s
26
o.oi 263
o.oi 238
o.ci 213
o.oi 188
o.oi 162
9.85571
9.85559
9-85 547
9.85534
9-85 522
12
12
13
12
50
49
48
47
46
.8
9
20.8
23.4
25
3-
!1
19
9.84373
9.84385
9.84398
9.84411
9.84424
12
3
3
13
9.98863
9.98808
9.98913
9 9*939
9.98964
25
25
26
25
o.oi 137
O.OI 112
o.oi 087
o.oi obi
o.oi 036
9.85 510
9.85497
9.85485
9-85473
9.85460
3
12
12
3
45
44
43
42
41
.1
.2
3
4
I
2.5
5-0
7-5
10.
12.5
20
21
22
23
24
9-84437
9.84450
9.84463
9.84476
9.84489
3
3
J3
13
9.98989
9.99015
9.99040
9.99065
9.99090
26
25
25
25
26
O.OI Oil
0.00985
o.oo 960
0.00935
0.00910
9-85448
9-85436
9-85423
9.85411
9.85399
12
'3
12
12
40
I
.0
:1
9
I S
17-5
20.0
22.5
S
27
28
29
9-8450^
9.84515
9-84528
9.84540
9.84553
13
*3
12
13
9.99 116
9-99 HI
9.99 166
9-99 191
9.99217
25
25
85
26
0.00884
0.00859
o.oo 834
0.00809
o.oo 783
9-85386
9-85374
9-85361
9 85349
9-85337
19
<3
12
12
35
34
33
32
3i
.1
.2
3
4
M
5:1
4.2
5-6
80
3
32
33
34
9-84566
9 84579
9 84592
9.84605
9.84618
3
3
3
*3
9-99242
9.99267
9-99293
9.99318
9-99343
25
26
25
25
o.oo 758
o.oo 733
o.oo 707
0.00682
0.00657
9-85 324
9-85312
9-85 299
9-85287
9.85274
12
3
12
3
12
80
1
:i
i
9
7-0
8-4
9.8
II. 2
12.6
9
9
39
9 84 630
9.84 643
9 . 64 656
9.84669
9.84082
*3
3
3
3
9.99368
9-99394
9.99419
9.99444
9.99469
26
25
25
25
26
0.00632
0.00606
o.oo 581
0.00556
o.oo 531
9 . 85 262
9-85250
9-85237
9-85 225
9.85 212
12
3
12
>3
12
25
24
23
22
21
.1
.2
3
11
40
41
42
43
44
9-84694
9-84707
9.84720
9 84733
9-4 745
3
3
13
' 12
9-99495
9-99 520
9-99545
9-99570
9-99 59&
25
25
25
26
o.oo 505
o.oo 480
o.oo 455
0.00430
0.00404
9.85 200
9.85 187
9-85 175
9.85 162
9-85 15
3
12
3
12
i>0
;i
3
3
!i
7
39
l: i
7-8
9-i
3
J2
49
9-84758
9.84 771
9.84 784
9.84 796
9 . 84 809
X 3
13
13
12
3
9.99621
9.99646
9.99672
9.99697
9.99722
8 S
25
26
25
25
0.00379
0.00354
0.00328
0.00303
0.00278
9-85 137
9-85 125
9.85 112
9.85 ioo
9.85087
12
*3
12
13
15
14
13
12
II
.8
9
10.4
11.7
xa
50
5i
52
53
54
9.84822
9.8483$
9-84847
9.84860
9-84873
X 3
3
12
3
3
9-99747
9-99 773
9.99798
9-9982;?
9.99848
25
26
25
25
25
0.00253
0.00227
O.OO 2O2
o.oo 177
o.oo 152
9.85074
9.85062
9.85049
9.85037
9.85024
18
3
12
>3
10
1
I
.1
.2
3
4
1
1.2
'I
tt
P
12
59
9-84885
9.8489$
9.849"
9.84923
9.84936
3
3
19
3
9.99874
9-99899
9-99924
9.99949
9-99975
25
25
25
96
o.oo 126
0.00 101
0.00076
o.oo 051
0.00025
9.85012
9.84999
9.84986
9.84974
9.84961
3
3
19
3
ft
5
4
3
i
.0
i
9
*1
,2:1
00
9 84949
*3
o.ooooo
25
o.oo ooo
9.84949
L. Cos.
d.
L. Cotg.
c.d.
L. Tang:.
L. Sin.
d.
i
Pro]
[>. PtS.
45
TABLE V NATURAL SINES AND COSINES. 75
TABLE V.
NATURAL
SINES AND COSINES
) 1 Alil^r* V
1
2
3
40
40
60
59
58
H
55
54
53
52
5i
50
2
t
tf. sine
N. cos.
N. sine
S T. cos.
N. sine
N. cos.
N. sine
^. cos.
N. sine|N. cos.
O
I
2
3
4
I
'I
9
1C
ii
12
~^3~
14
II
1
19
20
21
22
23
24
.00000
.00029
.00058
.00087
.00116
.00145
.00175
.00000
.00000
.00000
.00000
.00000
.00000
.00000
01745
.01774
.01803
.01832
.01862
.01891
.01920
.99985
.99984
.99984
.99983
.99983
.99982
.99982
.03490
03519
03548
-03577
.03606
03635
.03664
99939
.99938
99937
9993 6
99935
99934
99933
35234
.05263
.05292
05321
05350
05379
.05408
.99863
99861
.99860
.99858
99857
99855
99854
.06976
.07005
.07034
.07063
.07092
.07121
.07150
99756
99754
99752
99750
99748
99746
99744
.00204
.00233
.00262
.00291
.00320
.00349
.00000
.00000
.00000
.00000
99999
99999
.01949
.01978
.02007
.02036
.02065
.02094
.99981
.99980
.99980
99979
99979
99978
.03693
.03723
03752
.03781
.03810
03839
99932
99931
.99930
.99929
99927
.99926
5437
.05466
05495
05524
05553
.05582
.99852
.99851
99849
.99847
.99846
.99844
.07179
.07208
.07237
.07266
.07295
07324
.99742
.99740
99738
99736
99734
99731
.00378
.00407
.00436
.00465
.00495
.00524
99999
99999
99999
99999
99999
99999
.02123
.02152
.02181
.02211
.O224O
.02269
99977
99977
.99976
99976
99975
99974
.03868
.03897
.03926
03955
.03984
.04013
99925
.99924
.99923
.99922
.99921
.99919
.05611
.05640
.05669
.05698
05727
05756
.99842
.99841
99839
.99838
99836
99834
07353
07382
.07411
.07440
.07469
.07498
.99729
.99727
99725
99723
99721
.99719
9
45
44
43
42
005*3
.00582
.00611
.00640
.00669
.00698
99998
.99998
.99998
99998
.99998
.99998
.02298
.02327
02356
02385
.02414
.02443
99974
99973
99972
.99972
.99971
.99970
.04042
.04071
.04100
.04129
.04159
.04188
.99918
.99917
.99916
99915
99913
.99912
05785
05814
.05844
05873
.05902
05931
99833
.99831
.99829
.99827
.99826
.99824
.07527
07556
07585
.07614
07643
.07672
.99716
.99714
.99712
.99710
.99708
99705
41
40
39
38
37
36
2
3
29
jp_
3i
32
33
34
35
_36_
11
39
40
4i
42
43
44
45
46
47
48
.00727
.00756
.00785
.00814
.00844
.00873
99997
99997
99997
99997
.99996
.99996
.02472
.02501
.02530
.02560
.02589
.02618
99969
.99967
.99966
.99966
.04217
.04246
.04275
.04304
04333
.04362
.99911
.99910
.99909
.99907
.99906
.99905
.05960
05989
.06018
.06047
.06076
.06105
.99822
.99821
.99819
99817
.99815
.99813
.07701
.07730
07759
.07788
.07817
.07846
99703
99701
.99699
.99696
99694
.99692
35
34
33
32
3i
30
.00902
.00931
.00960
.00989
.01018
.01047
.99996
.99996
99995
99995
99995
99995
.02647
.02676
.02705
.02734
.02763
.02792
99965
.99964
99963
.99963
.99962
.99961
.04391
.04420
04449
.04478
04507
04536
99904
.99902
.99901
99897
.06134
.06163
.06192
.06221
.06250
.06279
1.99812
.99810
.99808
.99806
.99804
99803
07875
.07904
07933
.07962
.07991
.08020
.99689
.99687
.99685
9968 3
.99680
.99678
%
11
25
24
.01076
.01105
.01134
.01164
.01193
.01222
.99994
99994
99994
99993
99993
99993
.02821
.02850
.02879
.02902
.02938
.02967
.99960
99959
99959
.99958
99957
99956
04565
.04594
.04623
04653
.04682
.04711
.99896
.99894
99893
.99892
.06308
.06337
.06366
06395
.06424
06453
.99801
99799
99797
99795
99793
.99792
.08049
.08078
.08107
.08136
.08165
.0819^
.99676
99673
.99671
.99668
.99666
.99664
23
22
21
2O
*9
.01251
.01280
.01309
01338
.01367
.01396
99992
99992
.99991
.99991
.99991
.99990
.02996
.03025
03054
.03083
.O3II2
.03141
99955
99954
99953
.99952
99952
99951
.04740
.04769
.04798
.04827
.04856
.04885
.99888
.99886
.99885
.99883
.99882
.99881
.06482
.06511
.06540
.06569
.06598
.06627
.99790
.99788
99786
.99784
99782
.99780
.08223
.08252
.08281
.08310
.0836*
.99661
.99659
99657
.99654
.99652
99649
II
15
H
13
12
II
10
I
49
50
5i
! 52
, 53
5 *
.01425
.01454
.01483
.01513
.01542
.01571
99990
99989
99989
.99989
.99988
.99988
.03170
.03199
.03228
03257
.03286
.03316
.99950
99949
99948
99947
99946
99945
.04914
.04943
.04972
.05001
05030
05059
.99879
.99878
.99876
99875
99873
99872
.06656
.06685
.0671^
06743
.06773
.06802
.99778
99776
99774
99772
.99770
.99768
.08397
.08426
.08455
.08484
S&3
99647
.99644
99642
99639
.99637
99635
1 55
56
57
5
8
.OI6OO
.01629
.01658
.01687
.01716
01745
N. cos.
.99987
.99987
.99986
.99986
.99985
.99985
N. sine
03345
03374
03403
03432
.03461
.03490
N. cos.
99944
99943
.99942
.99941
.99940
99939
N. sine
.05088
.05117
.05146
05175
.05205
05234
N. cos.
.99870
.99869
.99867
.99866
.99864
99863
N. sine
.06831
06860
.00889
.06918
.06947
.06976
N. cos.
.99766
.99764
.99762
.99760
99758
99756
N. sine
S
.08629
.08658
.08687
.08716
N. cos.
1.99632
.99630
.90627
99625
.99622
.99619
N. sine
5
4
3
2
I
f
89'
88
87
86
85
NATURAL SINES AND COSINES.
77
5
6
7
8
9
/
o
I
2
3
4
1
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
.08716
.08745
.08774
.08803
.08831
.08860
.08889
.99619
.99617
.99614
.99612
.99609
.99607
.99604
10453
. 10482
.10511
10540
.10569
.10597
. 10626
.99452
99449
99446
99443
.99440
99437
99434
.12187
.12216
.12245
.12274
.12302
12331
.12360
99255
.99251
.99248
99244
.99240
.99237
99233
I39I7
.13946
13975
.14004
14033
.14061
. 14090
.99027
.99023
.99019
.99015
.99011
.99006
.99002
15643
.15672
.15701
15730
15758
15787
.15816
.98769
.98764
.98760
98/55
9875'
.98746
.98741
60
9
9
55 '
54 ,
I
9
10
ii
12
.08918
.08947
.08976
.09005
.09034
.09063
.99602
99599
.99596
99594
99591
.99588
! 10684
.10713
.10742
.10771
.10800
9943 1
.99428
.99424
.99421
.99418
99415
.12389
.12418
.12447
.12476
.12504
12533
99230
.99226
.99222
.99219
99215
.99211
.14119
.14148
.14177
.14205
.14234
.14263
.98998
98994
.98990
.98986
.98982
.98978
15845
15873
.15902
I593I
15959
.15988
98737
98732
.98728
.98723
.98718
.98714
53 i
52
Si
50
49
48
13
14
15
11
.09092
.09121
.09150
.09179
.09208
.09237
.99586
99583
.99580
99578
99575
99572
. 10829
. 10858
. 10887
.10916
.10945
.10973
.99412
.99409
.99406
.99402
99399
99396
.12562
.12591
.12620
.12649
.12678
.12706
.99208
.99204
.99200
.99197
99193
.99189
.14292
.14320
14349
14378
.I44CJ
.14436
98973
.98969
.98965
.98961
98957
98953
.16017
.16046
.16074
.16103
.16132
.16160
.98709
.98704
.98700
.98695
.98690
.98686
47
46
45
44
43
42
19
2O
21
22
23
24
.09266
.09295
.09324
09353
.09382
.09411
9957
99567
.99564
.99562
99559
99556
.IIOO2
.11031
.11060
.11089
.IIIlS
.11147
99393
.99390
.99386
99383
99380
99377
12735
.12764
.12793
.12822
.12851
.12880
.99186
.99182
.99178
99175
.99171
.99167
.14464
.14493
.14522
I455I
.14580
.14608
.98948
.98944
.98940
.98936
.98931
.98927
.16189
.16218
.16246
.16275
.16304
16333
.98681
.98676
.98671
.98667
.98662
.98657
41
40
39
38
37
36
2
3
29
30
.09440
.09469
.09498
.09527
09556
09585
99553
99551
.99548
99545
99542
.99540
. 1 1 1 76
.11205
.11234
.11263
.11291
.II32O
99374
99370
99367
99364
9936o
99357
.12908
12937
.12966
.12995
.13024
13053
.99163
.99160
.99156
.99152
.99148
.99144
14637
.14666
.14695
H723
14752
.14781
.98923
.98919
.98914
.98910
.98906
.98902
.16361
.16390
.16419
.16447
.16476
16505
.98652
.98648
98643
.9863$
98633
.98629
35
34
33
32
31
30
3 1
32
33
34
I
.09614
.09642
.09671
.09700
.09729
.09758
99537
99534
99531
.99528
99526
99523
II349
.11378
.11407
.11436
.11465
.11494
99354
99351
99347
99344
99341
99337
.13081
.13110
I3 J 39
.13168
I3I97
.13226
.99141
99137
99133
.99129
.99125
.99122
.14810
14838
.14867
.14896
.14925
14954
.98897
! 98884
.98880
.98876
16533
.16562
'SB!
.16620
.16648
.16677
.98624
.98619
.98614
.98609
.98604
.98600
%
11
25
24
9
39
40
4i
42
.09787
.09816
.09845
.09874
.09903
.09932
.99520
99517
99514
995"
.99508
.99506
H523
"552
.11580
.11609
.11638
.11667
99334
9933 1
99327
99324
.99320
99317
13254
13283
I33I2
I334I
13370
13399
.99118
.99114
.99110
.99106
.99192
.99098
.14982
.15011
.15040
.15069
.15097
.15126
.98871
.98867
.98863
.98858
.98854
.98849
.16706
16734
.16763
.16792
.16820
.16849
98595
98585
.98580
98575
.98570
23
22
21
20
43
44
9
3
.09961
.09990
.10019
.10048
.10077
.10106
99503
.99500
99497
99494
.99491
.99488
.11696
.11725
.11754
.11783
.11812
.11840
993M
.99310
99307
99303
.99300
99297
13427
13456
13485
I35I4
13543
13572
99094
.99083
99079
99075
I5I55
.15184
.15212
.15241
.15270
.15299
98845
.98841
.98836
.98832
.98827
.98823
.16878
.16906
'6935
.1696^
.16992
.17021
98565
.98561
98556
98551
.98546
.98541
\l
15
H
13
12
49
50
51
52
53
54
IOI 35
.10164
.10192
.10221
.10250
.10279
.99485
.99482
99479
99476
99473
.99470
.11869
.11898
.11927
.11956
.11985
.12014
99293
.99290
.99286
99283
99279
-99276
.13600
.13629
13658
13687
.13716
13744
.99071
.99067
.99063
.99059
99055
.99051
15327
15356
15385
I54H
.15442
I547I
.98818
.98814
.98809
.98805
.98800
.98796
.17050
.17078
.17107
.17136
.17164
17193
98536
98531
98526
.98521
98516
.98511
11
10
I
55
56
9
S
. 10308
10337
.10366
10395
.10424
10453
.99467
.99464
.99461
.99458
99455
9945 2
.12013
.12071
.12100
.12129
.12158
.12187
.99272
.99269
99265
.99262
.99258
99255
13773
.13802
:&
13889
13917
.99047
99043
.99039
99035
.99031
.99027
.15500
15529
15557
.15586
.15615
15643
.98791
.98787
.98782
98778
98773
.98769
. i 7222
17250
.17279
.17308
17336
17365
.98506
.98501
.98496
.98491
.98486
.98481
5
4
3
2
I
O
(
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
9
84
83
82
81
80
7 8 TABLE V.
10
11 %
12
13
14
/
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
o
i
2
3
1
17365
17393
.17422
I745I
17479
.17508
17537
.98481
.98476
.98471
.98466
1.98461
.9f455
9845
.19081
.19109
.19138
.19167
.19195
.19224
.19252
98163
98i57
.98152
.98146
.98140
98135
.98129
.20791
.20820
.20848
.20877
.20905
.20933
.20962
978i5
.97809
.97803
97797
.97791
97784
97778
.22495
.22523
22552
.22580
.22637
.22665
97437
9743
97424
97417
.97411
97404
97398
.24192
.24220
.24249
.24277
24305
24333
.24362
.97030
.97023
97015
.97008
.97001
60
59
58
!?
55
54
I
9
10
ii
12
17565
17594
.17623
:$
.17708
98445
.98440
98435
.98430
.98425
.98420
.19281
.19309
I 933 8
.19366
19395
.19423
.98124
.98118
.98112
.98107
.98101
.98096
.20990
.21019
.21047
.21076
.21104
.21132
.97772
.97766
.97760
97754
97748
.97742
.22693
.22722
.22750
.22778
.22807
22835
97391
97384
97378
97371
97365
97358
.24390
.24418
.24446
24474
24503
24531
.96980
.96973
.96966
.96959
.96952
96945
53
52
5i
50
1
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H
\l
II
:!$6 7
17794
'17852
! i 7880
.98414
.98409
.98404
.98399
.98394
.98389
.19452
.19481
.19509
.I953f
.19566
19595
.98079
98073
.98067
.98061
.21161
.21189
.21218
.21246
.21275
.21303
97735
97729
97723
.97717
.97711
97705
.22863
.22892
.22920
.22948
.22977
.23005
97351
97345
9733 s
97331
97325
97318
24559
.24587
.24615
.24644
.24672
.24700
.96937
.96930
.96923
.96916
.96909
.96902
9
45
44
43
42
19
20
21
22
23
24
3
2
29
30
.17909
17937
.17966
17995
.18023
.18052
3
'&
.98362
98357
.19623
.19652
.19680
.19709
.19737
.19766
.98056
.98050
.98044
.98039
98033
.98027
21331
.21360
.21388
.21417
21445
.21474
.97698
.97692
.97680
97673
.97667
23033
.23062
.23090
.23118
.23146
23175
973H
97304
.97298
97291
.97284
97278
.24728
24756
24784
.24813
.24841
.24869
.96894
.96887
.96880
96873
.96866
.96858
41
40
P
11
.18081
.18109
.18138
.18166
18195
. 18224
98352
9f347
.98341
98336
98331
98325
.19794
.19823
.19851
.19880
.19908
19937
.98021
.98016
.98010
.98004
97998
97992
.21502
21530
21559
.21587
.21616
.21644
.97661
2$
3&
.97630
.23203
.23231
.23260
.23288
.23316
23345
.97271
.97264
.97257
97251
97244
97237
.24897
24925
24954
.24982
.25010
.25038
.96851
.96844
.96837
.96829
.96822
96815
35
34
33
32
31
30
31
32
33
34
$
.18252
.18281
.18309
18338
18367
.18395
.98320
98315
.98310
.98304
.98299
.98294
.19965
.19994
.20022
.20051
.20079
.20108
.97987
.97981
97975
97969
.97963
97958
.21672
.21701
.21729
.21758
.21786
.21814
.97623
.97617
.97611
.97604
97598
97592
.23373
.23401
.23429
23458
.23486
23514
.97230
97223
.97217
.97210
.97203
.97196
.25066
.25094
.25122
.25151
25179
.25207
.96807
.96800
.96793
.96786
.96778
.96771
1
25
24
S
39
40
4i
42
.18424
.18452
.18481
.18509
18538
18567
.98288
.98283
.98277
.98272
.98267
.98261
.20136
.20165
.20193
-2O222
.20250
.20279
97952
.97946
97940
97934
.97928
.97922
.21843
.21871
.21899
.21928
.21956
.21985
97585
97579
97573
97566
97560
97553
.23542
.23571
23599
.23627
.23656
.23684
.97189
.97182
.97176
.97169
.97162
.97155
25235
.25263
25291
25320
25348
25376
.96764
.967.56
.96749
.96742
96734
96727
23
22
21
2O
43
44
i 45
46
2
18595
.18624
.18652
.18681
.18710
.18.738
.98256
.98250
98245
.98240
.98234
.98229
.20307
.20336
20364
20393
.20421
.20450
.97916
.97910
97905
97899
97893
97887
.22013
.22041
.22070
.22098
.22126
22155
97547
97541
97534
97528
97521
97515
.23712
.23740
.23769
.23797
.23825
23853
.97148
.97141
97134
.97127
.97120
97"3
25404
25432
.25460
.25488
.25516
25545
.96719
.96712
.96705
.96697
\l
15
H
13
12
49
50
5i
52
53
54
.18767
i8795
.18824
.18852
.18881
.18910
.98223
.98218
.98212
.98207
.98201
.98196
.20478
.20507
20535
.20563
.20592
.2O62O
.97881
97875
.97869
.97863
97857
97851
.22183
.22212
.22240
.22268
.22297
22325
.97508
.97502
.97496
97489
97483
97476
.23882
.23910
20
23995
.24023
.97106
.97100
97093
.97086
.97079
.97072
25573
.25601
.25629
25657
.25685
25713
96075
.96667
.96660
.96653
.96645
.96638
II
10
I
I
55
56
H
8
.18938
.18967
.18995
.19024
.19052
.19081
N. cos.
.98190
.98185
.98179
.98174
.98168
98163
N. sine
.20649
.20677
.20706
20734
20763
.20791
N. cos.
97845
97839
97833
97827
.97821
978i5
N. sine
22353
.22382
.22410
.22438
.22467
22495
N. cos.
97470
97463
97457
.97450
97444
97437
N. sine
.24051
.24079
.24108
.24136
.24164
.24192
N. cos.
.97065
.97058
.97051
.97044
.97037
97030
N. sine
25741
.25769
.25826
-25854
.25882
N. cos.
.96630
'12
[96600
96593
M. sine
5
4
3
2
O
/
79
7
77
76
75
NATURAL SINES AND COSINES.
79
15
16
17
18
19
t
N". sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
o
I
2
3
4
.25882
.25910
25938
.25966
25994
.26022
.26050
.96593
.96585
.96578
.96570
.96562
96555
96547
27564
.27592
.27620
.27648
.27676
.27704
.27731
.96126
.96118
.96110
.96102
^96078
29237
.29265
29293
29321
.29348
.29376
.29404
95630
.95622
95613
95605
955? 6
.95588
95579
.30902
.30929
3957
30985
.31012
.31040
.31068
.95106
95097
.95088
95079
95070
95061
95052
32557
32584
.32612
.32639
.32667
32694
.32722
94552
94542
94533
94523
94514
94504
94495
60
59
58
11
55
54
9
10
ii
12
.26079
.26107
26135
.26163
.26191
.26219
.96540
96532
.96524
96517
.96509
96502
27759
.27787
.27815
.27843
.27871
.27899
.96070
.96062
.96054
.96046
96037
.96029
.29432
.29460
.29487
29515
29543
29571
95571
95562
95554
95545
95536
95528
31095
3 II2 3
.31151
.31178
.31206
3^233
9543
95033
.95024
95015
.95006
94997
32749
.32777
.32804
.32832
32859
32887
.94485
.94476
.94466
94457
94447
.94438
53
52
5i
50
4 J
13
14
ii
||
.26247
.26275
.26303
-26331
26359
.26387
.96494
.96486
.96479
.96471
.96463
.96456
.27927
27955
.27983
.28011
.28039
.28067
.96021
.96013
.96005
95997
.95989
.95981
2Q5C9
.29626
.29654
.29682
.29710
29737
95519
955"
95502
95493
95485
95476
.31261
.31289
31316
31344
31372
31399
.94988
94979
94970
.94961
94952
94943
.32914
32942
.32969
32997
33024
33051
.94428
.94418
94409
94399
.94390
.94380
9
45
44
43
42
19
20
21
22
3
24
26415
.26443
.26471
.26500
.26528
.26556
.96448
.96440
96433
96425
.96417
.96410
.28095
.28123
.28150
.28178
.28206
.28234
95972
95964
95956
.95948
.95940
95931
29765
29793
.29821
.29849
.29876
.29904
95467
95459
9545
95441
95433
.95424
31427
3H54
.31482
3i5io
31537
31565
94933
94924
94915
.94906
.94897
.94888
33079
.33106
33!34
33161
33189
.33216
94370
.94361
94351
94342
94332
94322
41
40
9
H
%
27
28
29
3
.26584
.26012
'.2.6696
.26724
.96402
.96394
.96386
96379
.96371
96363
.28262
.28290
.28318
28346
28374
.28402
95923
95915
95907
.95898
.95890
.95882
.29932
.29960
.29987
30015
30043
.30071
95415
95407
95398
95589
95380
95372
31593
.31620
.31648
3 l6 75
31703
31730
.94878
.94869
.94860
.94851
.94842
.94832
33JM4
332/1
3329?
33326
33353
33381
94313
94303
94293
.94284
.94274
.94264
35
34
33
32
3i
30
3i
32
33
34
i
.26752
.26780
.26808
.26836
.26864
.26892
96355
96347
.96340
.96332
.96324
.96316
.20429
28457
.28485
2C 5 I 3
.28541
.28569
95*74
.95865
95857
.95849
.95841
95832
.30098
.30126
3 OI 54
.30182
.30209
.30237
95363
95354
95345
95337
95328
95319
31758
31786
31813
.31841
.31868
.31896
.94823
.94814
.94805
94795
.94786
91-777
33408
33436
33463
33490
33518
33545
94254
94245
94235
94225
94215
.94206
11
3
25
24
9
39
40
41
1 42
.26920
.26948
.26976
.27004
.27032
.27060
.96308
.96301
96293
96285
.96277
.96269
.28597
.28625
.28652
.28680
.28708
.28736
.95824
.95816
95807
95799
95791
95782
30265
.30292
.30320
30348
.30376
.30403
95310
95301
95293
.95284
95275
.95266
31923
3J95 1
31979
.32006
32034
.32061
.94768
94758
94749
94740
94730
94721
33573
33600
.33627
33655
33682
33710
.94196
.94186
.94176
.94167
94157
.94147
23
22
21
20
11
43
44
45
i 46
i:i
.27068
.27116
.27144
.27172
.27200
.27228
.96261
96253
.96246
96238
.96230
.96222
.287^4
.28792
28820
.2^847
28875
.20905
95774
95766
-95757
95749
95740
95732
30431
:$
30514
30542
30570
95257
.95248
95240
95231
.95222
95213
.32089
.32116
.32144
.32171
32199
.32227
.94712
.94702
.94603
.9461)4
.94674
.CJ46CS
33737
33764
33792
33819
33846
33874
94137
.94127
.94118
.94108
.94098
.94088
!!
15
14
13
12
49
50
51
52
53
54
.27256
.27284
.27312
.27340
27368
27396
.96214
.96206
.96198
.96190
.96182
96174
28931
.28959
.28987
29015
.29042
.29070
95724
95715
.95698
.95690
95681
30597
.30625
30653
.30680
.30708
30736
.95204
95*95
.95186
95177
.95168
95159
32254
.32282
32309
32337
32364
32392
94656
.94646
94637
.94627
.94618
.94609
33901
33929
33956
33983
.34011
34038
.94078
.94068
.94058
.94049
94039
.94029
II
10
I
i
g
.27424
.27452
.27480
.27508
27536
27564
.96166
.96158
.96150
.96142
.96134
.96126
.29098
.29126
.29154
.29182
.29209
.29237
.95673
.95664
95656
95647
95639
95630
N. sine
30763
.30791
30819
.30846
.30874
.30902
N. cos.
95150
95142
-95*33
95124
95^5
.95106
.32419
32447
32474
32502
.32529
32557
94599
.94590
.94580
94571
9456i
94552
34065
34093
34120
34H7
34175
.34202
.94019
.94009
93999
.93989
93979
.93969
5
4
3
2
I
O
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
9
74
73
72
71
70
8o
TABLE V.
2O
21
22
23
24
/
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
60
59
58 y
1
55
54
o
I
2
3
i
.34202
.34229
.34257
.34284
543"
34339
34366
93969
93959
93949
93939
93929
939 J 9
93909
35837
.35864
35891
35918
35945
35973
.36000
93358
93348
93337
93327
93316
93306
93295
.37461
37488
37515
37542
.37569
37595
.37622
37649
.37676
37703
37730
37757
37784
.92718
.92707
.92697
.92686
.92675
.92664
92653
39073
.39100
.39127
39153
.39180
.39207
39234
.92050
.92039
.92028
.92016
.92005
.91994
.91982
.40674
.40700
.40727
40753
.40780
.40806
40833
.40860
.40886
.40913
.40939
.40992
91355
91343
91331
913*9
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92587
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39314
39341
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39394
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52
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49
48
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46
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41
40
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11
2
2
29
30
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34939
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34993
35021
93718
93708
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93688
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93667
36515
36542
36569
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36623
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93052
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38134
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91752
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41337
41363
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41443
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35
34
33
32
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30
3i
32
33
34
8
35048
35075
35102
35130
35157
35184
93657
93647
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27
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10
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50
51
52
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90753
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ft
5 58 7
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35701
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35755
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35810
35837
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93389
93379
93368
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37353
37380
37407
37434
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92773
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92751
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38939
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39046
39073
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42235
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90655
90643
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5
4
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N. cos.
N. sine
N. cos.
N. sine
N. cos.
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N. sine
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69
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67
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NATURAL SINES AND COSINES.
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48532
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87434
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60
59
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57
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55
54
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9
10
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42473
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42525
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44151
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89777
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47255
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48659
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87363
87349
87335
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53
52
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50
49
48
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.42604
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42683
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42736
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90433
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44177
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44255
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44307
89713
.89700
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45736
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45813
45839
45865
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88915
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88875
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47306
47332
47358
4733
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.88103
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.88075
.88062
.88048
.48811
48837
.48862
.48888
.48913
48938
.87278
.87264
.87250
87235
.87221
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47
46
45
44
43
42
19
20
21
22
23
24
.42762
.42788
42815
.42841
.42867
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.90396
90383
.90371
90358
.90346
90334
44333
44359
44385
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44437
44464
.89636
.89623
.89610
89597
.89584
89571
.45891
45917
.45942
.45968
45994
.46020
.88848
.88835
.88822
.88808
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.88782
47434
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475
47537
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87993
87979
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.48989
.49014
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.49065
.49090
87193
.87178
.87164
87150
.87136
.87121
41
40
P
M
2
3
29
30
.42920
.42946
42972
.42999
43025
43051
.90321
.90309
.90296
.90284
.90271
.90259
.44490
.44516
.44542
.44568
44594
.44620
.89558
89545
89532
89519
.89506
.89493
.46046
.46072
.46097
.46123
.46149
46i75
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88755
.88741
.88728
88715
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47588
476i4
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47665
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.47716
8795!
87937
.87923
.87909
.87896
.87882
.49116
.49141
.49166
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.49217
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.87107
.87093
.87079
.87064
.87050
.87036
35 ,
34
33
32
3i
30
31
32
33
34
35
36
4377
.43104
43130
43156
.43182
.43209
.90246
.90233
.90221
.90208
.90196
.90183
.44646
.44672
.44698
44724
44750
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.89454
.89441
.89428
.89415
.46201
.46226
.46252
.46278
.46304
46330
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.88674
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.88647
.88634
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47741
47767
47793
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47844
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.87854
.87840
.87826
.87812
.87798
.49268
.49293
.49318
49344
49369
49394
.87021
.87007
.86993
.86978
.86964
.86949
i
27
26
25
24
11
39
40
4i
42
43235
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.43287
43313
43340
43366
.90171
.90158
.90146
.90133
.90120
.90108
.44802
.44828
.44854
.44880
.44906
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.89402
.89389
89376
89363
89350
89337
46355
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.46407
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46458
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88553
88539
47895
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47971
47997
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87743
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87715
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49445
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49495
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86935
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23
22
21
20
19
43
44
45
46
47
1 48
43392
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43445
43471
43497
43523
90095
.90082
.90070
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45010
45036
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89324
.89311
.89298
.89285
.89272
.89259
.46510
46536
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.46587
.46613
46639
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.88512
.88499
.88485
.88472
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.48048
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.48124
.48150
48i75
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87673
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49571
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49697
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.86805
.86791
86777
||
*5
14
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12
49
50
51
52
53
54
43549
43575
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43628
43654
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.89981
.89968
.89956
45 "4
.45140
.45166
.45192
.45218
45243
89245
.89232
.89219
.89206
.89193
.89180
.46664
.46690
.46716
.46742
.46767
46793
.88445
.88431
.88417
.88404
.88390
88377
.48201
.48226
.48252
.48277
48303
.48328
.87617
.87603
.87589
87575
.87561
.87546
49723
49748
49773
.49798
.49824
.49849
.86762
.86748
86733
.86719
.86704
.86690
II
10
7
58
60
.43706
43733
43759
43785
438"
.43837
.89943
.89930
.89918
89879
.45269
.45295
45321
45347
45373
45399
.89167
.89153
.89140
.89127
.89114
.89101
.46819
.46844
.46870
.46896
.46921
46947
.88363
.88349
88336
.88322
.88308
.88295
.48354
.48379
.48405
.48430
.48456
.48481
87532
.87518
87504
.87490
.87476
.87462
.49874
.49899
.49924
.49950
49975
.50000
.86675
.86661
.86646
.86632
.86617
.86603
5
4
3
2
I
O
N. cos
N. sine
N. cos
N. sine
N. cos.
N. sine
N. cos
N. sine
N. cos.
N.sine
/
1
4
6
3
6
2
e
1
TABLE V.
30
31
32
33
34
f
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
I
2
3
4
|
.50000
50025
.50050
.50076
.50101
.50126
.50151
.86603
.86588
86573
86559
.86544
.86530
.86515
51504
51529
51554
51579
.51604
.51628
5 l6 53
.85717
.85702
.85687
.85672
85657
.85642
-85627
.52992
53017
53041
.53066
53091
53H5
53 HO
.84805
.84789
.84774
84759
84743
.84728
.84712
54464
.54488
54513
54537
.54561
54586
.54610
.83867
83851
83835
83819
.83804
83788
.83772
.55919
55943
55968
55992
.56016
.56040
.56064
^82871
.82855
.82839
.82822
.82806
60
II
H
55
54
2
9
10
ii
12
50176
.50201
.50227
.50252
.50277
50302
.86501
.86486
.86471
.86457
.86442
.86427
.51678
STO
.51728
51753
.85612
85597
85582
85567
85551
85536
53164
53189
53214
53238
53263
5328S
.84697
.84681
.84666
.84650
84635
.84619
54635
54659
54683
.54708
54732
54756
.83750
.83740
.83724
.83708
.83692
.83676
.56088
.56112
56136
.56160
.56184
.56208
.82790
82773
.82757
.82741
.82724
.82708
53
52
5i
50
49
48
13
14
is
[I
50327
50352
50377
50403
.50428
5453
.86413
.86398
.86384
.86369
86354
.86340
.51828
.51852
51877
.51902
51927
5!952
85521
.85506
.85491
.85476
.85461
.85446
533 12
53337
5336i
53386
-534"
53435
.84604
.84588
84573
84557
84542
.84526
5478i
54805
54829
54854
.54878
.54902
.83660
83645
.83629
83613
83597
.83581
.56232
.56256
.56280
56305
56329
.56353
.82692
.82675
.82659
.82643
.82626
.82610
47
46
45
44
43
42
19
20
21
22
23
24
.50478
50503
50528
50553
.50578
.50603
.86325
.86310
! 86266
.86251
51977
.52002
.52026
52051
.52076
.52101
85431
85416
.85401
85385
85370
85355
53460
.53484
53509
53534
53558
53583
.84511
84495
.84480
.84464
.84448
84433
54927
54951
54975
54999
55024
55048
83565
83549
83533
83517
83501
83485
56377
.56401
56425
56449
56473
56497
82593
82577
.82561
82544
.82528
.82511
41
40
1
II
3
29
30
.50628
50654
.50679
.50704
.50729
.50754
86237
.86222
.86207
.86192
.86178
.86163
.52126
52151
52175
.52200
.52225
52250
85340
85325
.85310
.85294
.85279
.85264
53607
53632
53656
.53681
53705
5373
.84417
.84402
.84386
.84370
.84355
84339
55072
55097
55I2I
55145
55169
55194
.83469
83453
83437
.83421
83405
83389
56521
56545
56569
56593
56617
.56641
.82495
.82478
.82462
.82446
.82429
.82413
35
34
33
32
31
30
'11
11
25
24
3i
32
33
34
ii
.50779
.50804
.50829
50854
.50879
.50904
.86148
86133
.86119
.86104
.86089
.86074
52275
52299
52324
52349
52374
52399
.85249
85234
.85218
.85203
.85188
85173
53754
53779
53804
.53828
53853
53877
.84324
.84308
.84292
.84277
.84261
84245
.55218
55242
55266
55291
55315
55339
83373
83356
.83340
83324
.83308
.83292
.56665
.56689
56736
.56760
.56784
.82396
.82380
82363
82347
82330
82314
11
39
40
4i
42
.50929
50954
50979
.51004
.51029
51054
.86059
.86045
.86030
.86015
.86000
85985
5 2 423
.52448
52473
.52498
.52522
52547
.85157
85142
.85127
.85112
.85096
.85081
53902
53926
53951
53975
.54000
.54024
84230
.84214
.84198
.84182
.84167
.84151
:P
55412
55436
55460
55484
83276
.83260
.83244
.83228
.83212
83195
.56808
.56832
'56904
.56928
.82297
.82281
.82264
.82248
.82231
.82214
23
22
21
2O
43
44
1
2
51079
.51104
.51129
5H54
5H79
.51204
.85970
85956
.85941
.85926
.85911
.85896
5 2 572
52597
52621
.52646
.52671
.52696
.85066
85051
85035
.85020
.85005
.84989
.54049
54073
54097
.54122
.54146
.54171
84135
.84120
.84104
.84088
.84072
.84057
55509
55533
55557
5558i
55605
55630
83179
83163
83147
83131
.83115
.83098
.56952
.56976
.57000
57024
57047
57071
.82198
.82181
.82165
.82148
.82132
.82115
\l
15
H
'3
12
49
50
51
52
53
54
.51229
51254
51279
5 I 34
5 I 3 2 9
:5i354
.85881
.85866
.85851
.85836
.85821
.85806
.52720
52745
52770
.52794
.52819
52844
84974
.84959
.84943
.84928
.84913
.84897
54195
.54220
54244
.54269
54293
54317
.84041
.84025
.84009
83994
.83978
.83962
55654
55678
55702
55726
55750
55775
.83082
.83066
.83050
83034
83017
.83001
57095
57"9
57143
.57167
.57191
57215
.82098
.82082
.82065
.82048
.82032
.82015
II
10
|
P
R
e
5 I 379
.51404
.51429
51454
S479
51504
.85792
85777
.85762
85747
85732
85717
.52869
52893
.52918
52943
52967
.52992
.84882
.84866
.84851
.84836
.84820
.84805
54342
.54366
54391
.54415
.54440
.54464
.83946
83930
83915
.83899
.83883
83867
55799
55823
.55847
55871
55895
55919
.82985
.82969
82953
.82936
.82920
.82904
5723*
.57262
.57286
57310
.57334
57358
.81999
.81982
81965
.81949
.81932
81915
5
4
3
2
I
O
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
f
59
. 68
57
56
55
NATURAL SINES AND COSINES. 8
35
36
37
38
39
1
f
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
i
2
3
4
I
57358
573^1
57405
57429
57453
57477
57501
81915
.81899
.81882
.81865
.81848
.81832
.81815
.58779
.58802
.58826
.58849
58873
.58896
.58920
.80902
.80885
.80867
.80850
.80833
.80816
.80799
.60182
.60205
.60228
.60251
.60274
.60298
.60321
.79864
.79846
.79829
.79811
79793
79776
79758
.61566
61589
.61612
61635
.61658
.61681
.61704
.78801
78783
78765
.78747
.78729
.78711
.78694
.62932
62955
.62977
.63000
.63022
63045
.63068
.77715
.77696
77678
.77660
.77641
.77623
.77605
60
*
H
55 '
54 I
I
9
1C
ii
12
13
14
3
17
18
57524
57548
57572
57596
57619
57643
.81798
.81782
.81765
.81748
.81731
.81714
58943
58967
.58990
.59014
59037
.59061
.80782
.80765
.80748
.80730
.80713
.80696
.60344
.60367
.60390
.60414
.60437
.60460
79741
79723
.79706
.79688
.79671
79653
.61726
6i749
.61772
.61795
!6i84i
.78676
.78658
.78640
.78622
.78604
.78586
.63090
63113
I 3 ! 3 !
.63180
63203
77586
77568
77550
77531
77513
77494
53
52
5i
So
49
48
.57667
57691
57715
57738
57762
57786
.81698
.81681
.81664
.81647
.81631
.81614
.59084
.59108
59I3I
59154
.59178
.59201
.80679
.80662
.80644
.80627
.80610
80593
.60483
.60506
60529
60553
.60576
.60599
.79635
.79618
.79600
.79583
79565
79547
.61864
.61887
.61909
.61932
.61955
.61978
. 78568
78550
78532
.785H
.78496
.78478
63225
.63248
.63271
63293
63316
63338
.77476
77458
77439
.77421
.77402
77384
47
46
45
44
43
42
19
20
21
22
23
2 4
57810
57833
57857
.57881
57904
.57928
igg
.81563
.81546
.81530
81513
59225
.59248
.59272
59295
59318
59342
.80576
.80558
.80541
.80524
.80507
.80489
.60622
.60645
.60668
.60691
.60714
60738
79530
79512
79494
79477
79459
.79441
.62001
.62024
.62046
.62069
.62092
.62115
.78460
.78442
.78424
.78405
78387
.78369
.63361
* 338 2
.63406
.63428
63451
63473
.77366
77347
77329
.77.310
.77292
.77273
41
40
I
3
2
29
30
57952
.57976
57999
-58023
.58047
.58070
.81496
.81479
.81462
.81445
.81428
.81412
59365
59389
.59412
.59436
-59459
.59482
.80472
80*455
80438
.80420
.80403
.80386
.60761
.60784
.60807
.60830
.60853
.60876
.79424
.79406
.79388
79371
79353
79335
.62138
.62160
.62183
.62206
.62229
.62251
78351
78333
78315
.78297
.78279
.78261
.63496
.63518
63540
63563
63585
.63608
77255
77236
.77218
.77199
.77181
.77162
35
34
33
32
3i
30
31
32
33
34
g
.58094
.58118
.58141
.58165
.58189
.58212
.81395
81378
.81361
.81344
81327
.81310
59506
-59529
59552
59576
59599
.59622
.80368
.80351
80334
.80316
.80299
.80282
.60899
.60922
.60945
.60968
.60991
.61015
793 l8
.79300
.79282
.79264
.79247
.79229
.62274
.62297
.62320
.62342
62365
.62388
.78243
.78225
.78206
.78188
.78170
78152
.63630
.63653
63675
.63698
.63720
63742
77144
77125
.77107
.77088
.77070
77051
2
25
24
S
39
40
4i
42
58236
.58260
.58283
58307
58330
58354
81293
.81276
.81259
.81242
.81225
.81208
59646
.59669
59693
59716
59739
59/63
.80264
.80247
.80230
.80212
.80195
.80178
.61038
.61061
.61084
.61107
.61130
6"53
.79211
.79193
.79176
79158
.79140
.79122
.62411
62433
.62456
.62479
.62502
.62524
78i34
.78116
.78098
.78079
.78061
.78043
63765
.63787
.63810
.63832
.63854
.63877
77033
.77014
.76996
.76977
76959
.76940
23
22
21
2O
"il
15
H
13
12
43
44
45
46
47
48
58378
.58401
58425
.58449
.58472
.58496
.81191
.81174
8n57
.81140
.81123
.81106
59786
.59809
59832
59856
598/9
.59902
.80160
.80143
.80125
.80108
.80091
.80073
.61176
.61199
.61222
61245
.61268
.61291
79105
79087
.79069
79051
79033
.79016
.62547
.62570
.62592
.62615
.62638
.62660
.78025
. 78007
77988
779/0
77952
77934
.63899
.63922
.63944
.63966
.63989
.64011
-.76921
.76903
.76884
.76866
.76847
.76828
49
50
51
52
53
54
.58519
58543
58567
58590
.58614
58637
.81089
.81072
81055
.81038
.81021
.81004
.59926
59949
59972
59995
.60019
.60042
.80056
.80038
.80021
.80003
.79986
.79968
.61314
:$
^
.61406
.61429
.78998
.78980
.78962
.78944
.78926
.78908
.62683
.62706
.62728
.62751
62774
.62796
.77916
77897
.77879
.77861
77843
.77824
64033
.64056
.64078
.64100
.64123
.64145
.76810
.76791
.76772
76754
76735
.76717
II
IO
I
55
56
8
.58661
.58684
58708
58731
.58755
.58779
.80987
.80970
80953
.80936
.80919
.80902
.60065
.60089
.60112
60135
.60158
.60182
79951
79934
.79916
.79899
.79881
.79864
.61451
61474
.61497
.61520
61543
.61566
.78891
78873
78855
78837
.78819
.78801
.62819
.62842
.62864
.62887
.62909
.62932
.77806
.77788
.77769
77751
77733
77715
.64167
.64190
.64212
.64234
.64256
.64279
.76698
76679
.76661
.76642
.76623
.76604
N. sine
S
4
3
2
I
O
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
i
54
53
52
51
50
TABLE V.
, 40
41
42
43
44
9
N. sine
N. cos.
N. sine N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
60
%
H l
55 .
54
O
I
2
3
4
I
.64279
.64301
64323
.64346
.64368
.64390
.64412
. 76604
76586
7 6 567
.76548
76530
.76511
.76492
.65606
.65628
.65650
.65672
.65694
65716
65738
75471
75452
75433
75414
75395
75375
75356
66913
66935
.66956
.66978
.66999
.67021
.67043
743H
74295
74276
.74256
74237
.74217
.74198
.68200
.68221
.68242
.68264
.68285
.68306
.68327
73135
.73116
73096
.73076
73056
73036
.73016
.69466
.69487
.69508
.69529
.69549
.69570
.69591
71934
.71914
.71894
71873
71853
71833
.71813
i
9
10
ii
12
64435
64457
64479
.64501
.64524
.64546
76473
76455
76436
. 7641 7
. 76398
76380
65759
.65781
65803
.65825
-65847
.65869
75337
753i8
75299
.75280
75261
75241
.67064
.67086
.67107
.67129
.67151
.67172
.74178
.74159
74139
.74120
.74100
.74080
.68349
68370
68391
.68412
.68434
68455
.72996
.72976
72957
72937
.72917
72897
.69612
69633
.69654
'%&
.69696
.69717
.71792
.71772
.71752
.71732
.71711
.71691
53
52
5i
50
49
48
13
H
:i
17
18
.64568
.64590
.64612
64635
64657
.64679
.76361
.76342
.76323
76304
.76286
.76267
.65891
65913
65935
.65956
.65978
.66000
.75222
75203
75184
75165
75H6
.75126
.67194
67215
.67237
.67258
.67280
.67301
.74061
.74041
.74022
.74002
73983
73963
.68476
.68497
.68518
.68539
.68561
.68582
72877
72857
72837
.72817
.72797
72777
.69737
.69758
.69779
.69800
.69821
.69842
.71671
.71650
.71630
.71610
71590
71569
47
46
45
44
43
42
19
20
21
22
23
24
.64701
.64723
.64746
.64768
.64790
.64812
.76248
.76229
.76210
.76192
76i73
76i54
.66022
.66044
.66066
.66088
.66109
.66131
75107
.75088
.75069
75050
75030
.75011
67323
%m
67387
.67409
6743
73944
73924
73904
73885
73865
.73846
.68603
.68624
.68645
.68666
.68688
.68709
72757
.72737
.72717
.72697
.72677
.72657
.69862
.69883
.69904
.69925
.69946
.69966
71549
71529
.71508
.71488
.71468
71447
41
40
11
H
II
2
29
30
.64834
.64856
.64878
.64901
.64923
.64945
76135
.76116
.76097
76078
76059
.76041
66153
I7S
.66197
.66218
.66240
.66262
.74992
74973
74953
74934
74915
.74896
.67452
67473
67495
.67516
67538
67559
73826
.73806
73787
.73767
73747
73728
.68730
.68751
.68772
68793
.68814
68835
.72637
.72617
72597
72577
72557
72537
.69987
.70008
.70029
.70049
.70070
.70091
.71427
.71407
71386
.71366
71345
71325
35
34
33
32
31
30
3*
32
33
34
P
.64967
.64989
.65011
65033
65055
.65077
. 76022
.76003
75984
75965
75946
75927
.66284
.66306
66327
.66349
66371
.66393
.74876
74857
.74838
.74818
74799
.74780
95
.67623
.67645
.67666
.67688
73708
.73688
.73669
73649
73629
.73610
.68857
.68878
.68899
.68920
.68941
.68962
72517
72497
72477
72457
72437
.72417
.70112
.70132
70153
.70174
70195
.70215
71305
.71284
.71264
71243
.71223
.71203
3
11
25
24
P
39
40
4i
42
.65100
.65122
3%
.65188
.65210
75870
.75851
75832
758i3
.66414
.66436
.66458
.66480
.66501
66523
.74760
74741
.74722
74703
.74683
.74664
.67709
.67730
.67752
67773
67795
.67816
73590
73570
73551
73531
735"
73491
.68983
.69004
.69025
.69046
.69067
.69088
72397
.72377
72357
72337
.72317
.72297
.70236
70257
.70277
.70298
.70319
70339
.71182
.71162
.71141
.71121
.71100
.71080
23
22
21
2O
lo
43
44
$
i
65232
8
.65298
.65320
65342
75794
75775
75756
75738
75719
.75700
66545
.66566
66588
.66610
.66632
.66653
74644
.74625
.74606
74586
74567
74548
.67837
67859
.67880
.67901
67923
.67944
73472
73452
-73432
73413
73393
73373
.69109
.69130
69151
.69172
.69193
.69214
.72277
72257
.72236
.72216
.72196
.72176
.70360
.70381
.70401
.70422
70443
70463
71059
.71039
.71019
.70998
.70978
70957
11
15
H
13
12
49
50
51
52
53
54
65364
65386
.65408
65430
.65452
65474
.75680
.75661
.75642
75623
.75604
75585
.66675
.66697
.66718
.66740
.66762
.66783
.74528
74509
.74489
.74470
74451
74431
.67965
.68029
.68051
.68072
73353
73333
733!4
73294
73274
73254
69235
.69256
.69277
.69298
.69319
69340
72156
.72136
.72116
.72095
72075
72055
.70484
70505
70525
70546
70567
70587
.70937
.70916
.70896
70875
70855
.70834
II
10
1
6
P
II
g
.65496
.65518
65540
.65562
l&
.75566
75547
75528
75509
75490
75471
.66805
.66827
.66848
.66870
.66891
66913
.74412
74392
74373
74353
74334
743H
.68093
.68115
.68136
.68157
.68179
.68200
73234
.73215
73*95
73*75
73155
73135
69361
.69382
69403
.69424
69445
.69466
72035
72015
.71995
.71974
.71954
.71934
.70608
.70628
.70649
.70670
.70690
.70711
70813
70793
.70772
.70752
-7073 1
.70711
5
4
3
2
I
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
N. cos.
N. sine
9
49
48
47
46
45
TABLE VI. ADDITION AND SUBTRACTION LOGARITHMS.
8
TABLE VI.
; ADDITION AND SUBTRACTION LOGARITHMS.
PRECEPTS.
I. Wfan difference of given logarithms is less than 2.OO.
ADDITION. Enter table with difference between
logarithms
as Arg. A, and take out B.
Add B to subtracted logarithm.
SUBTRACTION. Subtract lesser from greater logarithm;
enter
with the difference as B, and take out A.
Add A to the subtracted logarithm.
II. When difference of given logarithms exceeds 2.OO.
Subtract lesser from greater.
ADDITION. Enter table with difference as Arg. A t take out
BA and add it to the greater logarithm.
SUBTRACTION. Enter column B with difference of
logarithms ;
take out BA, and subtract it from greater logarithm.
A.
B.
1
2
3
4
5
6
7
8
9
Prop.
Pts.
5-
o.oo ooo
001
OOI
OOI
OOI
OOI
002
002
"003
"ooj
m^mmm
G.O
004
004
005
005
005
005
005
005
005
005
6.1
005
006
006
006
006
006
006
006
007
007
3
4
5
6
6.2
007
007
007
007
008
008
008
008
008
008
i
0.3
o 4
0.3
0.6
6.3
009
009
009
009
OIO
OIO
OIO
OIO
OIO
Oil
2
0.6
08
I.O
1.3
3
0.9
Z 2
1.5
1.8
6.4
on
on
Oil
OI2
OI2
012
013
013
013
013
4
1.3
i 6
3.O
a 4
6.5
014
014
014
015
015
015
016
016
017
017
5
i. 5
3
9 *5
3.0
6.6
017
018
018
019
019
019
020
020
02 1
02 1
6
1.8
a 4
3-0
3-6
6.7
022
022
023
023
024
024
025
026
026
027
7
8
3.1
8.4
3 8
3 2
3-5
4.0
4-a
4.8
68
027
028
029
O2Q
O3O
O3I
Oil
O32
o^*
014
6.9
034
035
036
X
037
038
039
J
040
*J
O4I
ijj
041
JT^
042
7.0
043
044
045
047
048
049
050
051
052
053
7-1
055
056
057
059
060
061
063
064
066
06 7
j
7
O 7
0.8
9
TO
7-2
069
070
072
074
075
077
079
08 1
083
085
2
w.y
1.4
1.6
0.9
1.8
2.O
7-3
087
089
091
093
095
097
099
102
104
106
3
2.1
3-4
2.7
3-o
7-4
109
in
ii/
117
119
122
125
128
131
134
4
2.8
3-2
3-6
4.0
7-5
137
140
144
147
150
154
157
161
165
169
5
6
3-5
4- 2
4.0
4.8
4-5
5 A
5-o
6.0
7.6
173
177
181
185
I8 9
194
198
203
207
212
7
4-9
5-6
'^
6-3
7.0
7-7
217
222
227
233
2 3 8
244
249
255
261
267
8
5-6
6-4
7.2
8.0
7-8
273
280
286
293
299
3 06
313
321
328
336
9
3
7.3
8.1
9.0
79
344
352
360
3 68
377
385
394
403
413
422
1 8.0
432
442
452
463
474
485
496
507
519
531
1 A>
B.
1
2
a
. 4
5
6
7
8
9
Prop.
Pts.
86
TABLE VL
ADD i lo S<* ~ lo S* = A > Qrm / lo S* ~ lo S<* = &
Am Mlo g (0 + ) = log0 + ^. SU3 'Uog(0~) = log + ^.
A.
rsloo
8.01
8.02
8.03
8.04 ;
8.05
8.06
8.07
8.08
8.09
8.10
8. ii
8.12
8.13
8.14
8.15
8.16
8.17
8.18
8.19
8.20
8.21
8.22
1 8-23'
8.24
8.25
8.26
8.27
8.28
8.29
8.30
.3i
8.32
I-
Sip
l :5
8.39
8.40
Itt
8.43
8.44
I' 4 !
8.46
8.47
8.48
8.49
8.50
11.
o.oo 432
1
433
i
434
a
435
4
~
5
~
M
438
7
m^m^mm
439
8
440
9
44i
"451
462
473
483
495
506
518
530
542
Proi
^^^w
I
2
3
i
i
9
i
2
3
4
i
I
9
i
2
3
4
1
I
9
).Pt8. 1
'
*
0.8
J:5
\i
:\
3
0.6
?:I
1:1
*
'.'
*1
0.8 \
1.2
1.6
2.0
:i
!:J
442
452
463
474
485
496
507
519
53i
443
453
464
475
486
497
508
520
532
444
454
465
476
487
498
510
521
533
445
456
466
477
488
499
5"
523
535
446
457
467
478
489
500
512
524
536
447
458
468
479
490
502
513
525
537
448
459
469
480
49 i
503
5H
526
538
449
460
470
481
492
504
515
527
540
450
461
47i
482
494
505
517
529
541
543
5|6
569
582
595
609
623
638
652
667
683
545
546
547
548
550
55i
552
553
555
557
570
583
597
611
625
639
654
669
558
57i
585
598
612
626
641
655
671
560
$
599
613
628
642
657
672
561
574
587
601
6i5
629
644
658
674
562
575
589
602
616
630
645
660
675
564
577
590
604
618
632
646
661
677
565
578
59i
605
619
633
648
663
678
566
579
593
606
620
635
649
664
680
567
58i
594
608
622
636
&
681
684
686
688
689
691
692
694
696
697
699
715
731
748
766
783
801
820
839
858
"8^8
898
919
940
962
984
o.oi 006
030
Q53
077
700
716
733
750
767
785
803
822
841
702
718
735
752
769
787
805
823
842
703
720
736
753
771
789
807
825
844
705
721
738
755
773
790
809
827
846
707
723
740
757
774
792
810
829
848
708
725
741
759
776
794
812
831
850
710
726
743
760
778
796
814
833
852
712
728
745
762
780
798
816
835
854
713
730
747
764
78i
799
818
837
856
860
862
864
866
868
870
872
874
876
880
900
921
942
964
986
009
032
056
882
902
923
944
966
988
on
034
058
884
904
925
946
968
99 o
013
037
060
886
906
927
948
970
993
016
039
063
888
908
929
95i
973
995
018
041
065
890
910
93i
953
975
997
020
044
068
892
912
933
955
977
999
022
046
070
894
915
936
957
979
*002
O23
048
073
896
917
938
P?
*<x>4
027
51
075
080
082
085
087
090
092
095
097
IOO
1 02
128
153
1 80
207
23
263
292
322
352
105
130
156
183
2IC
23*
266
295
32]
107
133
159
185
21;
240
269
298
328
HO
135
161
1 88
215
243
272
30
33
112
138
164
I 9 I
218
246
275
304
334
H5
140
167
193
221
249
2 7 *
307
337
117
143
169
196
22;
252
280
310
340
120
146
172
199
226
25"
283
313
343
122
148
175
2O2
229
257
286
316
346
125
151
177
204
232
260
289
319
349
35!
358
36
364
36*
37
374
377
380
II A -
B.
1
2
8
4
5
6
7
8
9
Prop. Pts.
o
ADDITION AND SUBTRACTION LOGARITHMS. g
Ann J lQ % b "" lo % a ^' STTB J lo S<* lo ^ = A
' I log ( + *) = log* -f A bIJ i log (a - J) = log* + A.
A.
~&60
8.5i
8.52
8-53
8.54
8-55
8.56
8.57
8.58
8-59
8.60
8.61
8.62
8.63
8.64
8.65
8.66
8.67
8.68
8.69
8.70
8.7i
8.72
8.73
8.74
8.75
8.76
*.77
8.78
8.79
8.80
8.81
8.82
8.83
8.84
8.85
8.86
8.87
8.88
8.89
8.90
8.91
8.92
8-93
8.94
8.95
8.96
8.97
8.98
8.99
9.00
B.
o.oi 352
383
415
447
480
5H
549
584
621
658
695
1
"355
2
IP
a
~i
4
364
5
368
6
37i
7
374
8
377
9
380
P
I
2
3
4
i
I
9
I
2
3
i
i
9
i
2
3
I
I
9
i
2
3
i
I
9
rop.
MHM^
s
0.3
0.6
0.9
1.2
*:I
2.1
2.4
2.7
5
0-5
I.O
-5
2.O
2-5
3-o
3-5
4-0
4-5
7
0.7
1-4
2.1
2.8
3-5
4.2
4-9
I' 6
0-3
9
!:!
11
4-5
*1
*;
Pts.
mmm^^m^mmmm
4
0.4
0.8
1.2
1.6
2.O
2:3
&
6
0.6
1.2
1.8
2.4
3-o
3-6
4.2
4.8
54
8
0.8
1.6
2.4
3-2
4.0
4.8
I' 6
6.4
7.2
so
I.O
2.O
30
4-0
5-0
6.0
7.0
8.0
9.0
386
418
450
484
518
552
588
624
661
389
421
454
487
521
556
I 9 l
628
665
"703
393
424
457
490
5^5
559
595
632
669
39 6
428
460
494
5 2 8
563
599
635
673
399
43i
464
497
531
566
602
639
676
402
434
467
501
535
570
606
643
680
405
437
470
504
538
574
610
646
684
408
441
474
507
542
577
613
650
688
412
444
477
5"
545
581
617
654
692
699
707
711
715
719
722
726
730
734
774
814
856
898
941
985
O.O2 030
077
124
738
778
8.'8
860
902
945
990
035
08 1
742
782
822
864
906
950
994
040
086
746
786
827
868
911
954
999
044
091
750
790
831
872
9i5
959
*oo3
049
095
754
794
835
877
?i 9
03
*oo8
053
100
758
798
839
881
924
967
*OI2
058
105
762
802
843
885
928
972
*oi7
063
no
766
806
847
889
932
976
*O2I
06 7
114
770
8ro
851
894
937
981
*026
072
119
129
133
138
143
148
153
158
162
167
172
221
272
323
376
430
485
54i
599
177
226
277
329
38i
435
490
547
604
182
231
282
334
387
441
496
I 52
610
187
236
287
339
392
446
502
558
616
192
241
292
344
397
452
507
564
622
197
246
297
350
403
457
513
570
628
2O2
252
303
355
408
463
518
I 75
634
207
257
308
360
414
468
524
581
639
211
262
3'3
365
419
474
%
645
216
267
3i8
37i
424
479
535
593
651
657
717
779
841
905
971
0.03 037
106
175
247
663
669
675
68 1
687
693
699
705
711
723
785
848
912
977
044
"3
183
254
729
791
854
918
984
051
120
100
26l
735
797
860
925
991
058
126
197
268
742
803
867
931
22
065
133
204
276
748
810
873
938
*oo4
071
140
211
283
754
816
879
944
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078
147
218
290
760
822
886
95i
"017
085
'54
3
766
829
892
957
*024
092
161
232
305
772
835
899
964
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099
168
240
312
320
327
334
342
349
357
364
37i
379
386
394
470
548
627
708
790
5*
961
0.04 049
401
478
555
635
7i6
799
883
970
058
409
485
563
643
724
807
892
979
067
417
493
57i
651
73 2
816
901
987
076
424
501
579
659
74i
824
909
996
085
432
509
587
667
749
832
918
*00 5
094
439
516
595
675
757
841
926
*oi4
103
447
524
603
683
765
849
935
*023
112
455
532
611
691
774
858
944
*0 3 2
121
462
540
619
700
782
866
953
*o4o
130
139
148
'57
167
176
i*5
194
203
213
222
A.
B.
2
a
4
6
tf
7
8
9
Prop. Pts. !
88
TABLE VI.
ADD f Io 8* lo % a = A * SUB i lo % a "* lo S^ = A
x i log( + J) = log* + ^. bU i log( - *) = log + ^.
A.
B.
1
2
a
4
5
6
7
8
9
Prop. Pts.
0.00
9.01
9.02
9.03
9.04
9.05
9.06
9.07
9.08
9.09
9.10
9.11
9.12
9-13
9-H
9.15
9.16
9-17
9.18
9.19
9.20
9.21
9.22
9-23
9-24
9.25
9.26
9-27
9.28
9.29
9.30
9.31
9-32
9-33
9-34
9-35
9.36
9-37
9-38
9-39
9.40
9.41
9.42
9-43
9-44
9-45
9.46
9-47
9.48
9-49
9.50
0.04 139
148
157
167
176
185
194
203
213
306
401
499
598
700
803
909
017
127
222
z
3
3
4
5
6
7
8
9
z
3
3
4
5
6
7
8
9
i
2
3
4
5
6
7
8
9
X
2
3
4
5
6
7
8
9
z
3
3
4
5
6
7
8
9
z
2
3
4
5
6
7
8
9
9
0.9
1.8
2.7
3-6
4-5
5-4
6-3
7.2
8.z
xa
Z.2
2.4
3.6
4.8
6.0
7.2
8.4
9.6
10.8
'5
-5
3-0
4-5
6.0
7-5
9.0
10.5
13.0
3-5
18
z.8
3-6
S-4
7.2
9.0
10.8
13.6
H.4
16.2
at
zo
z.o
2.0
3.c
4.0
5.0
6.0
7.0
8.0
9.0
'3
1-3
2.6
3-9
5-2
6-5
7.8
9.1
10.4
zz.7
16
z.6
3.2
4.8
6.4
8.0
9.6
IX. 2
13.8
x 4-4
9
1.9
3.8
5-7
7.6
9-5
II. 4
T 3-3
15.2
17. x
23
XI
z.z
3.il
3-3
44
5-5
6.6
7-7
8.8
9-9
M
*-4
3.8
4.
5-6
7.0
8.4
9.8
ii. a
12.6
17 ;
f
3-4
5-1
6.8
8-5
10. a
11.9
13.6
'5-3
2O
a.o
4-0
6.0
8.0
IO.O
12.0
14.0
16.0-
18.0
as
231
325
421
5 ! 9
618
720
824
93i
0.05 039
240
334
43
528
628
73i
835
941
050
250
344
440
I 38
639
74i
845
952
061
259
353
450
548
649
75i
856
963
072
268
363
460
I 58
659
762
867
974
083
278
373
469
568
669
772
877
985
094
287
382
479
I 78
679
782
888
995
105
297
392
489
588
689
793
898
006
116
315
411
509
608
710
814
92O
028
139
150
161
172
183
195
206
217
229
240
251
263
378
496
616
738
863
991
0.06 121
254
274
390
508
628
$
004
134
267
286
401
519
640
763
889
*oi7
147
281
297
4i3
53i
652
775
901
*030
161
294
308
425
543
664
788
914
1*043
174
308
320
436
555
677
800
927
*o 5 6
187
321
332
448
567
689
813
939
*o69
200
335
343
460
579
701
825
952
*082
214
348
355
472
59i
7H
838
965
*095
227
362
366
484
604
726
851
978
*io8
240
376
389
403
417
430
444
458
472
486
500
513
13
812
959
0.07 108
261
416
575
736
54i
683
827
973
123
276
432
59i
753
I 55
697
841
988
138
291
448
607
769
569
711
856
*oc>3
154
307
463
623
785
583
725
870
*oi8
169
322
479
639
802
597
740
885
*033
184
338
495
655
818
612
754
900
*048
199
354
I 11
671
835
626
769
914
*o63
215
369
527
687
851
640
783
929
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230
385
543
704
868
654
798
944
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245
400
559
720
884
901
918
934
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968
985
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0.08 069
240
415
592
774
958
0.09 146
338
533
086
257
432
610
792
977
165
357
553
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275
450
628
810
996
184
377
573
120
292
468
646
829
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204
396
593
137
309
485
664
847
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223
416
612
154
327
503
683
865
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242
I 35
632
171
344
521
701
884
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261
455
652
188
362
539
719
206
379
557
737
921
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299
494
692
223
397
574
755
940
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319
5H
712
902
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280
474
672
4-2
6.3
8.4
10.5
12.6
14.7
16.8
18.9
24
3.4
4.8
7.2
9.6
12.0
14.4
z6.8
19.2
21.6
4-4
6.6
8.8
II.
13-2
iS-4
17.6
19.8
5
2-5
5.0
7-5
10.
12.5
15.0
17-5
20.0
22. 5
4-6
6.9
9.3
XZ.J
13.8
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18.4
20.7
26
2.6
5-2
7.8
10.4
13-0
15.6
lS.2
20.8
23-4
732
935
o.io 141
35i
565
783
o.n 005
23
46
69-
933
752
773
793
813
833
853
874
894
914
955
162
373
587
805
028
%
715
976
183
394
609
827
050
277
507
742
996
204
415
630
849
073
3 oc
53i
766
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225
437
652
872
095
323
554
790
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246
458
674
894
118
345
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267
479
696
916
140
368
60 1
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288
501
718
938
163
392
62;
86
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309
522
739
960
1 86
415
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885
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330
544
761
983
208
438
671
909
957
98
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*054
*o 7 8
*I02
*I27
*i 5 i
A.
B.
1
2
8
|
6
6
7
8
9
Prop. Pts.
ADDITION AND SUBTRACTION LOGARITHMS.
89
A ( log loga = A. 9 i loga log = B. \
x \ log (a + t) = loga + B. bUB * i log(a - b] = log^ + A
A.
B.
1
2
a
4
5
6
7
*I02
8
9
Prop. Pts.
9.50
9.51
9.52
9-53
9-54
9-55
9.56
9-57
9.58
9-59
9.60
9.61
9.62
9-63
9.64
9.65
9.66
9.67
9.68
9.69
9.70
9.71
9-72
9-73
9-74
9-75
9.76
9-77
9.78
9-79
9.80
9.81
9.82
9-83
9.84
9.85
9.86
9-8'/
9.88
9.89
9.90
9.91
9.92
9-93
9-94
9-95
9.96
9-97
9.98
9-99
0.00
o." 933
0.12 175
422
673
928
0.13 188
452
721
994
0.14 272
~554
841
0.15 133
43
, 73i
o.io 037
349
665
986
o.i7_3i2
643
980
0.18 322
668
0.19 020
378
740
0.20 108
481
860
957
981
*oo5
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*54
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"151
i
2
3
4
5
6
7
8
9
i
2
3
4
5
6
7
8
9
i
2
3
4
5
6
7
8
9
i
2
3
4
5
6
7
8
9
i
5
3
4
5
e
7
8
9
i
2
3
4
6
7
8
9
37
2.7
5-4
8.1
10.8
13-5
16.2
18.9
21.6
24-3
31
3-i
6.2
9-3
12.4
15-5
18.6
21.7
24.8
27.9
35
3-5
7.0
10.5
14.0
17-5
21.0
24-5
28.0
31-5
39
3-9
7.8
11.7
15-6
19-5
23-4
27-3
31-2
35- 1
43
4-3
8.6
12.9
17.2
21.5
25.8
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34-4
33.7
47
4-7
9-4
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18.8
23-5
28.2
32.9
37-6
43.3
38
2.8
5-6
8.4
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I 4 .0
16.8
19.6
22.4
25.2
32
3-2
6.4
9-6
12.8
16.0
19.2
22.4
25.6
28.8
36
3-6
7-2
10.8
14.4
18.0
21.6
25.2
28.8
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40
29
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5-8
8.7
ix. 6
14-5
17.4
20.3
23.2
26.1
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9-9
13.2
16.5
19.8
23.1
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29.7
37
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18.5
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29.6
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41
30
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2 3 .8
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30.6
38
3-8
7.6
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22.3
26.6
30.4
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43
200
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698
954
214
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300
224
472
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240
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328
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267
532
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356
274
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293
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384
298
547
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3 J 9
586
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*I32
412
323
572
826
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346
613
884
*i6o
441
348
597
851
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372
640
911
*i88
469
372
622
877
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399
667
939
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497
397
648
903
*l62
425
694
966
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526
583
611
640
668
697
726
755
783
812
870
162
460
7 6l
068
380
697
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345
899
192
489
792
099
411
729
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378
928
221
520
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443
761
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411
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251
550
8 I 3
161
474
793
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444
986
281
580
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192
506
825
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310
610
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224
538
857
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510
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340
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255
569
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370
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976
286
601
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577
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400
701
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317
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610
677
710
744
777
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494
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198
558
923
294
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845
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529
879
234
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331
708
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878
912
946
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356
703
056
414
777
145
519
898
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390
738
091
450
813
182
557
937
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425
773
127
486
850
220
594
975
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460
808
163
522
887
257
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564
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270
631
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369
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306
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406
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633
985
342
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444
822
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8.0
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16.0
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24.0
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32.0
36.0
44
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8.8
13-2
17.6
22.0
26. 4
30.8
35-2
39.6
48
4.8
9.6
14.4
19.2
24.0
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38.4
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8.2
12.3
16.4
20.5
24-6
28.7
32.8
36-9
45
4-5
9.0
13-5
18.0
22.5
27.0
3i-5
36.0
40-5
49
4.9
9.8
14.7
19.6
24-5
29.4
34-3
39-2
44.1
8.4
12.6
16.8
21.0
25.2
29.4
33-6
37-8
46
4.6
9.2
13-8
18.4
23.0
27.6
32.2
36.8
41.4
50
5-0
IO.O
iS-o.
20. o
25.0
30.0
35-0
40.0
45-0
0.21 244
634
O.22 O29
430
8 3 6
0.23 247
665
0.24 088
5 I6
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283
322
361
399
438
477
516
556
595
989
3S9
795
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623
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473
907
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673
069
470
877
289
707
130
559
994
712
109
510
918
330
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173
603
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752
149
55i
959
372
791
216
646
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791
189
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414
833
258
689
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831
229
632
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455
875
301
733
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870
269
673
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497
918
344
776
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910
309
713
*I2 3
539
960
387
819
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949
349
754
*i65
581
*oo3
430
863
*302
0.25 390
434
479
523
568
612
657
701
746
791
836
0.26 287
744
0.27 207
675
0.28 149
629
0.29 115
606
881
332
790
253
722
197
677
163
655
926
378
836
300
769
245
726
212
705
970
423
882
346
817
292
7 2 4
261
754
*oi6
469
928
393
864
340
822
310
804
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515
974
440
911
388
871
359
854
*io6
560
*02I
487
959
436
920
409
903
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606
"067
534
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484
968
458
953
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652
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581
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532
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507
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698
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628
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581
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556
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0.30 103
153
2O3
253
303
354
404
454
505
555
1 A.
B.
1
2
a
4
5
6
7
8
9
Prop. Pts.
ADD { ^ga-logd = A. g ( loga - log = B.
' \ log (a + &) = log + -#. | log (a ) = log + A.
A.
B.
1
2
3
4
6
7
8
9
Prop. Pts.
0.00
O.OI
O.O2
0.03
0.04
O.O5
0.06
0.07
0.08
O.O9
0.10
O.II
O.I2
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
O.2J
O.26
0.27
0.29
0.30
0.31
0.32
0.33
0-34
0.35
0.36
0.37
0.38
0.39
0.40
0.4
0.42
0.43
0.4
0.4
0.46
0.4
0.4
0.4
0.5
0.30 103
606
0.31 115
629
0.32 149
675
0.33 207
744
0.34 287
836
153
203
253
303
354
404
454
505
555
5
5-0
10.0
15.0
20.0
25.0
30.0
35-o
40.0
> 5-o
54
5-4
! 10.8
JI6.2
J2I.6
527.0
532.4
737-8
3 43-2
?4 8.6
58
' 5-8
2 II. 6
3 17-4
4 23.2
5 29.0
634.8
740.6
846.4
952.2
62
I 6.2
2 12.4
3 18.6
424.8
53i-o
637.2
743-4
849-6
955.8
66
i 6.6
213.2
3 19-8
426.4
533-c
639-6
746.2
852.8
959-4
70
2 14. c
32I.C
4 28. c
535-c
6 42. c
749 <
8 5 6.c
963.*
51
5-i
tO. 2
'5-3
20.4
25-5
30.6
J5-7
to. 8
15-9
55
5-5
II.
'6.5
22.0
27.5
33-o
38.5
44-o
49-5
59
5-9
ii. 8
17.7
23-6
29-5
35-4
4i-3
47-2
S3-i
63
6-3
12.6
18.9
25.2
37-8
44.1
5-4
56-7
67
6-7
13-4
20.1
26.8
33-5
40.2
46-9
53-6
60.3
71
14.2
21.3
28.4
35.5
42. e
49-3
56.8
6 3 -<3
5-2
to. 4
20.8
26.0
;i.2
56.4
56
5-6
[1.2
[6.8
22.4
28.0
33-6
39-2
44-8
50.4
60
6.0
12.0
18.0
24.0
30.0
36.0
42.0
48.0
54-0
6.4
12.8
19.2
25.6
32.0
38.4
44.8
51.2
57-6
68
6.8
13.6
20.4
27.2
34-c
40.8
47-C
54-4
61.2
73
7.2
14.4
21. e
28.
43-s
50.4
57-<
64.*
53
5-3
0.6
5-9
1.2
6-5
1.8
7-J
2-4
7-7
57
5-7
M
7-i
22.8
28.5
34-a
39-9
45-6
Si-3
6x
6.x
12.2
I8. 3
24.4
30-5
36.6
42.7
4 8.8
54-9
65
6-5
13-0
19.5
26.0
32-5
39-0
45.5
52.0
58.5
69
6.9
13-8
20.7
27.6
34-5
41.4
48.3
55-2
62.1
73
7-3
14.6
21.9
29.9
36.5
43-8
51.1
58.4
65-7
656
1 66
68 1
20 1
728
260
798
342
891
707
217
732
254
852
396
946
758
268
784
306
834
367
906
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809
320
836
359
887
421
960
* 5 6
859
37i
888
411
940
474
'5
561
112
910
422
940
464
993
528
069
616
168
961
474
992
5I 2
046
582
123
670
223
012
526
045
569
100
636
178
726
279
063
577
097
622
153
690
232
78i
334
0.35 390
446
502
558
614
6 7
726
782
838
894
* 95 ?
0.36 516
0.37 088
665
0.38 247
836
0.39 430
0.40 029
634
0.41 244
007
573
723
306
895
489
089
695
*o6 3
630
203
363
954
549
149
756
119
687
260
839
423
*oi3
609
2IO
816
"428
669
297
570
214
864
?J!
744
897
482
*073
669
270
877
*2 33
80 1
375
955
*I 3 2
729
33 o
938
+289
858
433
*oi4
600
*I9I
789
999
* 34 6
916
491
*072
/ 59
849
452
*o6i
403
973
549
718
*3io
909
512
*I22
*459
*O3O
607
777
*370
969
573
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306
367
490
552
613
1
487
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763
4 08
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716
"377
675
737
798
860
0.42 481
0.43 108
740
o.44 378
0.45 020
668
0.46 322
980
922
544
171
804
442
085
387
O/1 (
984
606
234
867
506
149
799
453
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360
995
634
279
929
584
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794
423
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698
344
994
650
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920
550
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827
473
782
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982
613
89?
538
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8 4 8
* 5 IO
*045
677
*3H
956
603
*2 5 6
914
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Q.47 643
0.48 312
986
0.49 665
0.50 349
0.51 037
0.52 430
o.53 133
84
o.54 554
710
777
844
910
977
O4^
*iii
*I 7 8
*245
918
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968
661
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770
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* 379
73:
417
107
801
500
204
912
447
*I2I
Soi
486
176
870
570
274
983
869
555
245
940
640
345
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938
624
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710
416
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*oo6
692
384
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78
486
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716
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761
453
"150
85
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783
* 4 6i
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830
522
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92
628
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851
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899
592
*289
992
699
*4i
626
697
769
841
912
984
*os6
*I28
*2OO
0.55 272
, 994
0.56 72
o.57 45
0.58 18
92
0.59 67
0.60 42
0.61 17
344
*o66
794
262
748
497
251
416
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86
59
3
82
57
32
488
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940
672
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897
648
402
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746
484
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972
723
478
632
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819
558
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79
554
704
*?59
893
63
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874
6 3 c
777
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*2 3 2
967
706
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94
70
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849
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78 4
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*379
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854
* S9 8
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857
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*i6
*237
*3M
* 39 c
*54
*6i9
B.
1
2
3
4
5
1
8
9
Prop. Pts.
ADDITION AND SUBTRACTION LOGARITHMS.
A f loga log b= A. q ( loga log = B.
ADD ' i log(a + b) = log* + B. 3 ' i log(0 - b) = log + ^.
A.
B.
1
2
a
4
5
G
7
8
9
Prop. Pts.
0.50
0.51
0.52
o.53
0.54
o.55
0.56
0.57
0.58
0.59
0.60
0.61
0.62
0.63
0.64
0.65
0.66
0.67
0.68
0.69
0.70
0.71
0.72
o.73
0.74
0.75
0.76
0.77
0.78
o.79
0.80
0.81
10.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1.00
0.61 933
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+085
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2
3
4
5
6
7
8
9
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3
3
4
5
6
7
8
9
t
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3
4
5
6
7
8
9
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2
3
4
5
6
7
8
9
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3
3
4
5
6
7
8
9
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a
3
4
5
6
7
8
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37.0
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51.8
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66.6
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7-7
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30.8
38.5
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61.6
69.3
80
8.0
16.0
24.0
32.0
40.0
48.0
56.0
64.0
72.0
83
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24.9
33-2
41.5
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58.1
66.4
74-7
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51.6
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89
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65.6
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63.7
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0.62 695
0.63 461
0.64 231
0.65 005
783
0.66 565
0.67 351
0.68 141
935
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538
308
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861
644
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160
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0.70 533
0.71 338
0.72 146
958
0.73 774
o.74 592
o.75 415
0.76 240
0.77 069
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0.78 736
o.79 575
0.80 416
0.81 261
0.82 108
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0.83 812
0.84 668
0.85 527
614
419
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855
674
497
323
152
694
499
308
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937
757
579
406
235
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580
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911
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703
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476
562
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0.87 254
0.88 121
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0.89 863
0.90 738
0.91 616
0.92 49 6
0.93 378
0.94 263
340
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0.98 720
0.99 618
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1.02 325
1.03 231
239
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914
810
708
609
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322
327
416
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594
683
772
861
217
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798
699
601
506
413
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692
597
503
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879
782
687
594
485
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778
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230
321
412
503
594
685
776
867
958
1 A.
B.
1
2
8
4
6
G
7
8
9
Prop. Pts.
2 TABLE VI.
ADD ) l Z a ~~ 10g b = A ' QTTR f lo g * ~ lo * = #
* t log (a + b) == log ^+ A i log( - J) = log^ 4- A.
; A.
B.
1
2
a
4
5
6
7
8
9
Prop. Pts.
1.00
01
02
03
04
l
06
:>7
08
09
10
ii
12
13
14
15
16
17
18
19
.20
21
22
.23
.24
.25
.26
!28
.29
.30
31
3 2
< -33
: -34
i -35
36
37
38
39
1.04 139
230
321
412
503
594
685
776
867
958
9
i 9
2 18
3 27
4 36
5 45
6 54
9 Si
I
2
3
4
i
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9
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2
3
4
1
9
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8 7
9 8
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2
3
4
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3
4
9
93
27
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1
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4
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96
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1.07 790
i. 08 708
1.09 627
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1. 12 394
140
053
966
882
800
719
640
562
486
232
144
058
974
891
811
732
655
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323
235
149
065
983
903
824
747
671
414
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418
332
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167
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857
596
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259
179
101
024
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607
524
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285
209
134
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616
535
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301
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1.13 320
412
503
598
690
783
876
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i 14 247
I.I5 175
1.16 106
1.17 J7
97i
1.18 905
1.19 841
1.20 779
1. 21 717
1.22 657
1.23 599
1.24 54i
1.25 485
1.26 430
1.27 376
1.28 323
1.29 272
1.30 221
1.31 J 72
340
268
199
131
064
999
935
872
811
432
361
292
224
157
092
029
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525
454
385
317
251
1 86
122
060
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618
547
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563
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316
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462
411
362
88 1
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714
660
608
557
507
458
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755
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652
602
553
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1.32 124
219
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505
600
695
791
886
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1.33 77
1.34 3
985
1.35 94i
1.36 898
1.37 856
1.38 814
1.39 774
1.40 734
172
126
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910
870
830
267
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458
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508
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744
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840
794
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622
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4
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792
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1.42 658
1.43 6 2 J
1.44 584
1.45 549
I.465H
1.47 480
1.48 447
1.49 4i
1.50 383
754
717
681
645
61
577
544
%
850
813
777
742
707
674
64
608
577
946
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874
8 3 8
804
770
737
705
674
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970
935
901
867
834
802
771
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997
964
93
895
868
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964
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1.51 35
449
546
64:
740
837
934
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A.
B.
1
2
3
4
5
1
8
t)
Prop.
Pts.
ADDITION AND SUBTRACTION LOGARITHMS.
93
A ( log a - log b = A. Q ( log a log b = B.
x \ log (a + J) = log<* + ^. bu B ' i log( - ) = log* + A.
A.
1.50
51
52
53
54
:ii
?
59
.GO
.61
.62
63
.64
65
.66
.67
.68
.69
.70
71
.72
73
74
:3
77
.78
79
.80
.81
.82
.8 3
.84
.85
.86
.87
.88
.89
1.90
91
.92
93
94
95
.96
97
.98
99
2.00
B.
1
2
546
3
643
4
5
7
8
9
Prop. Pts.
1-51 352
1.52 322
1.53 292
i . 54 263
1-55 235
1.56 207
1.57 180
1-58 153
1.59 128
I. 60 102
1.61 077
449
740
837
934
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i
2
3
1
I
9
i
2
3
4
i
I
9
i
2
3
4
1
9
97
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19.4
38.8
485
58.2
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77.6
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98
9-8
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49.5
1
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389
360
332
304
277
251
225
200
516
486
457
429
402
375
348
322
297
613
583
555
526
499
472
446
420
395
710
680
652
624
596
569
543
517
492
807
778
749
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693
667
640
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875
846
818
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273
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1.63 o 3 o
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1.05 962
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1.67 919
1.68 898
1.69 878
151
127
104
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202
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639
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737
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1.71 859
1.72 820
1.73 801
1-74 783
1.75 766
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1.78 715
1-79 699
937
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881
864
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i. 80 68 3
781
880
978
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1.81 667
1.82 652
i.8 3 6 3 8
1.84 625
1.85 609
1.86 595
1.87 582
1.88 569
1.89 556
766
75 i
736
722
708
694
681
667
655
864
849
835
820
806
793
779
766
753
963
948
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i 9Q 543
1.91 53i
1.92 519
1-93 507
1.94 496
1.95 485
1.96 474
1-97 463
1.98 452
1.99442
2.00 432
642
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840
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630
618
606
PI
573
562
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717
705
69^:
682
671
66 1
650
640
827
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804
770
760
749
739
926
914
903
891
880
869
859
848
838
*025
*OI 3
*002
990
979
968
958
947
937
*I2 4
*II2
*IOO
*o8 9
*078
*o6 7
*o 57
*o 4 6
*o 3 6
*22 3
*2II
*I99
*i88
*i77
*i66
*I 5 6
*I 4 5
*i35
* 3 2I
* 3 IO
*2 9 8
*28 7
*2 7 6
*26s
*254
*244
*2 34
*420
*4o8
*397
* 3 86
! 3 ^
* 3 64
*353
*343
*333
53i
630
729
828
927
*026
*I25
*22 4
*323
B.
1
2
8
4
5
7
8
9
Prop. Pts.
94
TABLE VI.
log a log b =
log( + *) =
= A. log a log b = B.
log* + (B - A). log (a - b) = log a - (B - A)
A.
B.
B-A.
A.
B.
B-A.
A.
B.
B-A.
1.9823
.9833
.9842
.9852
.9862
1.9868
.9878
.9887
.9897
.9907
.00450
449
448
447
446
2.0337
.0348
.0359
.0370
.0381
2.0377
.0388
0399
.0410
.0421
.00400
399
398
397
396
2.0920
.0932
.0945
.0957
.0970
2.0955
.0967
.0980
.0992
.1005
.00350 |
349
348
347
346
1.9872
.9882
.9891
.9901
.9911
1.9917
.9926
9935
9945
9955
.00445
444
443
442
441
2.0392
.0403
.0414
.0425
0437
2.0432
.0443
.0454
.0465
.0476
.00395
394
393
392
39i
2.0982
.0995
.1008
.1020
1033
2.1017
.1029
.1042
.1054
.1067
.00345
344
343
342
34i
1.9921
9931
.9941
.9951
.9961
1.9965
9975
.9985
9995
2.0005
.00440
439
438
437
436
2.0448
.0459
.0470
.0481
.0493
2.0487
.0498
.0509
.0520
.0532
.00390
389
388
387
386
2.1046
.1059
.1072
.1085
.1098
2.1080
.1093
.1106
.1119
.1132
.00340
339
338
337
336
1.9971
.9981
.9991
2.0001
.OOI I
2.0015
.0024
.0034
.0044
.0054
.00435
434
433
432
431
2.0504
.0515
.0527
.0538
.0550
2.0543
0553
.0565
.0576
.0588
.00385
384
383
382
38i
2. mi
.1124
1137
.1150
.1163
2.1144
.1157
.1170
.1183
.1196
00335
334
333
332
33i
2.OO2I
.OO32
.OO42
.0052
.0062
2.0065
.0075
.0085
.0095
.0105
.00430
429
428
427
426
2.0561
.0573
.0584
.0596
.0607
2.0600
.0611
.0622
.0634
.0645
.00380
379
378
377
376
2.1176
.1190
.1203
.1216
.1229
2.1209
.1223
.1236
.1249
.1262
00330
329
328
327
326
2.0073
.0083
.0093
.OIO4
.0114
2.0115
.0125
.0135
.0146
.0156
.00425
424
423
422
421
2.0619
.0630
.0642
.0654
.0666
2.0656
.0667
.0679
.0691
.0703
.00375
374
373
372
37i
2.1243
.1256
.1270
.1283
.1297
2.1275
.1288
.1302
.1315
.1329
.00325
324
323
322
321
2.0124
.0135
.0145
.0156
.0166
2.0166
.oi7/
.0187
.0198
.0208
.00420
419
418
417
416
2.0677
.0689
.0701
.0713
.0725
2.0714
.0726
.0738
.0750
.0762
.00370
369
368
367
366
2.1310
.1324
.1338
.1351
.1365
2.1342
.1356
.1370
.1383
1397
.00320
319
3i8
317
316 1
2.0177
.0187
.0198
.0208
.0219
2.0218
.0228
.0239
.0249
.0260
.00415
414
413
412
411
2.0737
.0749
.0761
.0773
.0785
2.0773
.0785
.0797
.0809
.0821
.00365
3 6 4
363
362
361
2.1379
.1393
.1407
.1421
H35
2.1410
.1424
.1438
.1452
.1466
.00315
3'4
313
312
3"
2.0229
.0240
.0251
.O26l
.O272
2.0270
.0281
.0292
.0302
.0313
.00410
409
408
407
406
2.0797
.0809
.0821
0833
.0845
2.0833
.0845
.0857
.0869
.0881
.00360
359
358
35 2
356
2.1449
.1463
.1477
.1491
.1505
2.1480
.1494
.1508
.1522
1536
.00310
309
308
307
306
2.O283
.0294
.0305
OS'S
.0326
2.0324
.0334
.0345
3 II
.0366
.00405
404
403
402
401
2.0858
.0870
.0882
.0895
.0907
2.0893
.0905
.0917
.0930
.0942
.00355
354
353
352
35i
2.1520
.1534
.1548
.1563
.'577
2.1550
.1564
.1578
.1593
. 1607
.00305
304
303
302
30i
2.0337
2.0337
.00400
2.0920
2.0955
.00350
2.1592
2 1622
.00300
A.
B.
B-A.
A.
B.
B-A.
A.
B.
B-A.
ADDITION AJND 5UJJTKAUT1U.N -LUUAK.1T.HM5. g
Iog0 log as A. loga log = B.
log(0 + V] = loga + (B A). log(a V) as loga (B A).
A.
B.
B-A.
A.
B.
B-A.
A.
B.
B A.
2.1592
.1606
.1621
.1635
.1650
2.1622
.1636
.1651
.1665
.1680
.00300
299
298
297
296
2.2386
.2403
.2421
.2439
.2456
2.2411
.2428
.2446
.2464
.2481
.00250
249
248
247
246
2.3358
3379
.3401
.3423
.3446
2.3378
3399
.3421
3443
.3466
.00200
199
198
197
196
2.1665
.1680
.1694
.1710
.1724
2.1694
.1709
.1723
.1739
.1753
.00295
294
293
292
291
2.2474
.2492
.2510
.2528
.2546
2.2498
.2516
.2534
.2552
.2570
.00245
244
243
242
241
2.3468
.3490
.3513
3535
.3558
2.3487
.3509
3532
3554
3577
.00195
194
193
192
191
2.1739
.1754
.1770
.1785
.1800
2.1768
.1783
.1799
.1814
.1829
.00290
289
288
287
286
2.2564
.2582
.2600
.2618
.2637
2.2588
.2606
.2624
.2642
.2661
.00240
239
238
237
236
2.3581
.3604
.3627
.3650
.3673
2.3600
.3646
.3669
.3692
.00190
187
1 86
2.1815
.1830
.1846
.1861
.1877
2.1844
.1858
.1874
.1889
.1905
to to to to to
00 00 0000 OO
1-4 to CO 4^> Cn
2.2656
.2674
.2693
.2711
.2730
2.2679
.2697
.2716
.2734
2753
.00235
234
233
232
231
2.3697
.3720
3744
.3768
.3792
(0
CO CO CO CO OJ
<-> OO ONCO i-i
O ON tO OOCn
.00185
184
183
182
181
2.1892
.1908
.1923
.1939
.1955
2.1920
.1936
.1951
.1967
.1983
.00280
279
278
277
276
2.2749
.2768
.2787
.2806
.2825
2.2772
.2791
.2810
.2829
.2848
.00230
229
228
227
226
2.3816
.3840
.3865
.3889
.39H
.3883
.3907
.3932
.00180
179
178
177
176
2.1971
.1987
.2002
.2019
.2035
2.1998
.2014
.2029
.2046
.2062
.00275
274
273
272
271
2.2845
.2864
.2884
.2903
.2923
2.2867
.2886
.2906
.2925
.2945
.00225
224
223
222
221
2-3939
.3964
.3989
.4014
.4039
2.3956
.3981
.4006
.4031
.4056
.00175
174
173
172
171
2 . 205 1
.2067
.2083
.2099
.2116
2 . 2078
.2094
.2110
.2126
.2143
.00270
269
268
267
266
2.2943
.2962
.2982
.3002
.3022
2.2965
.2984
.3004
.3024
.3044
.OO22O
219
218
217
216
2.4065
.4090
.4116
.4142
.4168
2.4082
.4107
.4133
.4159
.4185
.00170
\6 7
1 66
2.2132
.2149
.2165
.2182
.2198
2.2159
.2175
.2191
.2208
.2224
00265
264
263
262
261
2.3043
.3063
.3083
.3104
.3124
2.3064
.3084
.3104
.3125
.3H5
.00215
214
213
212
211
2.4195
.4221
.4248
.4275
.4302
2.4211
.4237
.4264
.4291
.4318
.00165
164
'g
162
161
2.2215
.2232
.2249
.2266
.2283
2.2241
.2258
.2275
.2292
.2309
.00260
259
258
256
.3166
.3187
.3208
.3229
2.3166
.3187
.3208
.3229
.3250
20g
208
207
206
2.4329
.4356
.4383
.4411
4439
2.4345
.4372
4399
.4427
4455
.00160
159
158
157
156
2.2300
.2317
.2334
.2369
2.2325
.2342
.2359
.2376
.2394
.00255
254
253
252
251
to
CO CO CO CO CO
CO CO tO tO tO
CO >-i\O Vjen
O\4^co - O
2.3271
.3291
.3313
3334
.3356
.OO2O5
204
20 3
202
201
2.4467
4495
.4523
4552
.4581
2.4482
.4510
.4538
.4567
.4596
.00155
154
153
152
2.2386
2.2411
.00250
2.3358
2.3378
.O0200
2.4609
2 4624
.00150
A.
B.
B-A.
A.
B.
B-A.
A.
B.
B-A.
TABLE VI.
1
Iog0 log b = A. log a log b = B.
log (a + ) = log* + (^ ^). log (a ) = log a (^ ^4).
A.
B.
B-A.
A.
B.
B-A.
A.
B.
B-A.
2.4609
.4638
.4668
.4697
.4727
2.4624
.4653
.4683
.4712
.4742
.00150
149
148
H7
146
2.6373
.6416
.6461
.6505
.6550
2.6383
.6426
.6471
5 S i 5
.6560
.OOIOO
.00099
98
9
96
2.9385
9474
.9563
.9655
.9748
2.9390
9479
.9568
.9660
9753
.00050
J3
%
2-4757
.4787
.4817
4848
4878
2.4772
.4801
.4831
.4862
.4892
.00145
144
143
142
141
2.6596
.6642
.6688
.6735
.6783
2.6606
.6651
.6697
.6744
.6792
.00095
94
93
92
9i
2.9844
2.9941
3.0041
.0143
.0248
2.9848
2.9945
3-0045
.0147
.0252
.00045
44
43
42
41
4910
4941
4972
5004
5036
2.4924
4955
.4986
.5018
.5050
.00140
139
138
137
136
2.6831
.6880
.6928
.6978
.7028
2 . 6840
.6889
.6937
.6987
.7037
.00090
89
88
87
86
3 ' 3 ^
.0466
.0578
.0694
.0813
3-0360
.0470
.0582
.0698
.0817
.00040
P
!
5068
5100
5 J 33
5165
5199
2.5081
.5H|
.5146
.5178
.5212
.00135
134
133
132
131
2.7079
7131
.7183
.7236
.7289
2.7088
.7139
.7191
.7244
.7297
.00085
84
83
82
81
3-0935
.1061
.1191
.1324
.1463
3-0939
.1064
.1194
.1327
.1466
.00035
34
33
32
3i
5232
5266
5299
5333
5368
2.5245
.5279
.5312
.5346
.538i
.00130
129
128
127
126
2.7343
.7398
7453
.7509
.7566
2.7351
.7406
.7461
.7517
7574
.00080
79
78
8
3.1606
.1753
.1905
.2063
.2226
3.1609
.1756
.1908
.2066
.2229
.00030
29
28
27
26
5402
5437
5472
5508
5544
2.5415
5449
.5484
.5520
.5556
.00125
124
123
122
121
2.7623
.7682
.7741
.7801
.7862
2.7631
.7689
.7748
.7808
.7869
.00075
74
73
72
7i
3-2396
.2575
.2760
.2952
.3154
3-2399
.2577
.2762
.2954
.3156
.00025
24
23
22
21
5580
5616
5653
5690
5727
2.5592
.5628
.5665
.5702
5739
.OOI20
119
118
117
116
2.7923
.7985
.8050
.8114
.8180
2.7930
.7992
.8057
.8121
.8187
.00070
69
68
67
66
3.3366
3590
.3825
.4072
4335
3.3368
3592
.3827
.4074
4337
.OO020
19
I/
16
2.5765
5803
5841
5880
.5919
2.5776
.5814
5852
.5891
.5930
.00115
114
H3
112
III
2.8245
.8313
.8381
.8451
.8521
2.8252
.8319
.8387
.8457
.8527
.00065
64
3
62
61
3-4617
49 J 7
.5237
.5587
.5964
3.46i9
.4918
.5238
.5588
.5965
.00015
14
13
12
II
2.5958
5998
.6038
.6079
.6120
2.5969
.6009
.6049
.6090
.6131
.OOIIO
107
106
2.8593
.8666
.8741
.8816
.8893
2.8599
.8672
.8747
.8822
.8899
.00060
59
58
i
3.6377
.6835
7345
.7925
.8595
3.6378
.6836
.7346
.7926
.8596
.OOOIO
09
08
07
06
2.6161
.6202
.6244
.6287
.6329
2.6172
.6212
.6254
6297
.0339
.00105
104
103
102
101
2.8971
.9051
.9132
.9215
.9300
2.8977
.9056
9'37
.9220
.9305
.00055
54
53
52
5i
3.9390
4.0355
4.1600
4.3375
4.6367
3.9391
4.0355
4.1600
4.3375
4.6367
.00005
04
03
02
01
2.6373
2.6383
.OOIOO
2.9385
2.9390
.00050
oo
00
.00000
A.
B.
B-A.
A.
B.
B-A.
A.
B.
B-A.
TABLE VII. SQUARES OF NUMBERS.
97
TABLE VTL>
SQUARES OF NUMBERS.
No.
Square.
No.
Square.
No.
Square.
No.
Square.
No.
Square.
I
2
3
20
21
22
23
400
40
4i
42
43
1600
60
61
62
63
3600
80
Si
82
83
6400
I
4
9
441
484
529
1681
1764
1849
3721
3844
3969
6561
6724
6889
4
5
6
16
25
36
24
25
26
576
625
676
44
45
46
1936
2025
2116
64
65
66
4096
4225
4356
84
85
86
7056
7225
7396
7
8
9
10
ii
12
13
49
64
81
27
28
29
80
3i
32
33
729
784
841
47
48
49
50
5i
52
53
2209
2304
2401
67
68
69
70
7i
72
73
4489
4624
4761
87
88
89
90
9i
92
93
7569
7744
7921
100
900
2500
4900
8100
121
144
I6 9
961
1024
1089
2601
2704
2809
5041
5184
5329
8281
8464
8649
14
15
16
J96
225
2 5 6
34
35
36
1156
1225
1296
54
55
56
2916
3025
3136
74
75
76
5476
5625
5776
94
95
96
8836
9025
9216
17
18
19
20
289
324
3 6l
37
38
39
40
1369
1444
7521
57
58
59
60
3249
3364
348i
77
78
79
80
5929
6084
6241
97
98
99
100
9409
9604
9801
400
1600
3600
6400
10000
o8
SQUARES OF NUMBERS FROM 100 TO 1000.
1<*
9~
3*+
4**
5~
6~
1~
$
94*
Diff.
00
100
40O
900
1600
2500
3600
4900
6400
8100
00
X
01
102
404
906
1608
2510
3612
4914
6416
8118
01
O2
03
104
106
408
412
912
918
1616
1624
252O
2530
&
4928
4942
6432
6448
8136
8154
04
09
3
5
7
04
108
416
924
1632
2540
3648
495 6
6464
8172
16
05
no
420
930
1640
2550
3660
4970
6480
8190
2 5
j j
06
112
424
936
1648
2560
3672
4984
6496
8208
36
07
114
428
942
1656
2570
3684
4998
6512
8226
49
08
116
432
948
1664
2580
3696
5012
6528
8244
64
*
09
118
436
954
1672
2590
3708
5026
6544
8262
81
19*
10
121
441
961
1681
2601
3721
5041
6561
8281
oo
21
II
123
445
967
1689
26ll
3733
5055
6577
8299
21
12
125
449
973
1697
2621
3745
5069
6593
8317
44
25
13
127
453
979
1705
2631
3757
5083
8335
69
27
14
129
457
985
1713
2641
3769
5097
6625
8353
96
20*
"
15
I 3 2
462
992
1722
2652
3782
5112
6642
8372
2 5
OT
16
134
466
998
1730
2662
3794
5126
6658
8390
5 6
J*
33
17
136
470
1004
1738
2672
3806
5140
6674
8408
89
#
18
139
475
IOII
1747
2683
3819
5*55
6691
8427
24
19
141
479
1017
1755
2693
3831
5i 6 9
6707
8445
61
39*
20
144
484
1024
1764
2704
3844
5184
6724
8464
00
4
21
22
I 4 6
145
488
492
1030
1036
1780
2714
2724
3856
3868
5198
5212
6740
6756
8482
8500
84
43
AS*
23
IS'
497
1043
1789
2735
3881
5227
6773
8519
29
45
47
24
3
$
I 5 8
501
506
1049
1056
IO02
1797
1806
1814
2745
2756
2766
3893
3906
5241
5256
5270
6789
6806
6822
P
8574
76
49*
5*
53*
27
28
161
515
519
1069
1075
1823
1831
2777
2787
3931
3943
5285
5299
6839
6855
??
29
84
55
29
166
524
1082
1840
2798
3956
53H
6872
8630
4i
59*
30
169
529
1089
1849
2809
3969
5329
6889
8649
00
61
3 i
171
533
1095
1857
2819
3981
5343
6905
8667
61
6,*
32
33
:n
542
1102
1108
1866
1874
2830
2840
3994
4006
5358
5372
6922
6938
8686
8704
n
"j
65 |
67*
34
9
ig
547
55J
556
1115
1 122
1128
1883
1892
1900
2851
2862
2872
4019
4032
4044
5387
5402
6972
6988
8723
8742
8760
56
96
69*
7 /
73*
8
1*0
566
"35
1142
1909
1918
2883
2894
4057
4070
5431
5446
7005
7022
8779
69
44
75 *
39
J93
57i
"49
1927
2905
4083
7039
88? 7
21
79*
40
196
576
1156
1936
29l6
4096
5476
7056
8836
00
81
41
198
580
1162
1944
2926
4108
5490
7072
8854
81
8 *
42
2OI
585
1169
'953
2937
4121
5505
7o8 9
8873
64
8 *
43
204
590
1176
1962
2948
4134
5520
7106
8892
49
87*
44
207
595
1183
1971
2959
4M7
5535
7123
8911
36
89"
$
210
213
600
605
1190
1197
1980
1989
2970
298l
4160
4173
5550
5565
7140
7i57
8930
8949
3
9t"
93*
47
216
610
1204
1998
2992
4186
558o
7174
8968
09
OS*
48
219
615
I2II
2007
4199
5595
7191
8987
04
95
07*
49
222
620
1218
2016
3 OI 4
4212
5610
7208
9006
01
97
99*
50
225
625
1225
2025
3025
4225
5625
7225
9025
oo
SQUARES OF NUMBERS FROM 100 TO 1000 (Continued).
99
1~
2~
3~
4**
5~
6~
r+
8~
0*^
Diff.
60
225
625
1225
2025
3025
4225
5625
7225
9025
00
i
5 i
228
630
1232
2034
3036
4238
5640
7242
9044
01
52
53
231
234
635
640
1239
1246
2043
2052
3047
3058
4251
4264
5655
5670
7259
7276
9063
9082
04
09
3
5
7
54
237
645
1253
2061
3069
4277
5685
7293
9101
16
55
240
650
1260
2070
3080
4290
5700
7310
9120
25
56
243
655
1267
2079
3091
4303
5715
7327
9U9
36
131 '
57
246
660
1274
2088
3102
43l6
5730
7344
9158
49
58
249
665
1281
2097
31*3
4329
5745
9177
64
59
252
670
1288
2106
3124
4342
5760
7378
9196
Si
19*
60
256
676
1296
2116
3136
4356
5776
7396
9216
00
I
61
259
681
1303
2125
3U7
4369
5791
7413
9235
21
62
63
265
686
691
1310
1317
2134
2143
3158
3169
4382
4395
5806
5821
7430
7447
9254
9273
%
35
27
64
268
696
1324
2152
3l80
4408
5836
7464
9292
96
29*
272
702
1332
2162
3192
4422
5852
7482
9312
25
66
275
707
'339
2171
3203
4435
5867
7499
9331
56
33
%
278
712
718
1346
1354
2180
2190
32H
3226
4448
4462
5882
5898
7534
9350
9370
8 9
24
*
69
285
723
1361
2199
3237
4475
59^3
755'
9389
61
39*
70
289
729
1369
2209
3249
4489
5929
7569
9409
00
41
72
292
295
734
739
1376
1383
2218
2227
3260
3271
4502
4515
5944
5959
7586
7603
9428
9447
41
84
43
73
299
745
I39i
2237
3283
45 2 9
5975
7621
9467
29
47
74
9
302
306
309
750
1398
1406
1413
2246
2256
.2265
3294
3306
4542
4556
4569
5990
6006
6021
7638
7656
7673
9486
9506
9525
76
49*
s-
78
313
316
767
772
1421
1428
2275
2284
3329
3340
4583
4596
6037
6052
7691
7708
9545
95 6 4
29
84
55
79
320
778
1436
2294
3352
4610
6068
7726
9584
41
57*
59*
80
324
784
1444
2304
33 6 4
4624
6084
7744
9604
00
61
81
327
789
I45I
2313
3375
4637
6099
7761
9623
61
fia*
82
331
795
U59
2323
3387
4651
6115
7779
9643
24
03
6.
83
334
800
1466
2332
3398
4664
6130
7796
9662
89
J
67*
f 4
ii
338
342
345
806
812
817
1474
1482
1489
2342
2352
2361
3422
3433
4678
4692
4705
6146
6162
6177
78H
7832
7849
9682
9702
9721
56
69*
71
73*
87
349
823
1497
2371
3445
4719
6193
7867
9741
69
88
353
829
1505
2381
3457
4733
6209
7885
9761
44
75
89
357
835
'513
2391
3469
4747
6225
793
9781
21
77*
79*
90
361
841
1521
2401
348i
4761
6241
7921
9801
00
81
9 1
364
846
1528
2410
3492
4774
6256
793
9820
81
92
93
368
372
852
858
1536
1544
2420
2430
3504
3516
4788
4802
6272
6288
7956
^Q74
9840
0860
64
49
8 3*
85'
87*
94
384
864
870
876
1560
1568
2440
2450
2460
3528
3540
3552
4816
4830
4844
6304
6320
6336
79^*
8010
8028
9880
9900
9920
36
s
89*
9**
93*
11
388
392
882
888
1576
2470
2480
35 6 4
3576
4858
4872
6352
636*
8046
8064
9940
9960
09
04
95*
99
396
894
1592
2490
4886
6384
$082
9980
01
97*
99*
100
400
900
1600
2500
3600
4900
6400
8100
1 0000
oo
TABLE VIII. DECIMALS OF DAY INTO HOURS, ETC.
101
H. M. S.
H.M.S.
H. M. S.
H.M.S.
D.
H. M. S.
~"~ o
D.
H. M. S.
IOO
IOO
IOO
IOO*
d.
h. m. s.
M. S.
s.
d.
h. m. s.
m, s.
*.
O.OI
o 14 24
o 8.64
0.09
0.51
12 14 24
7 20.64
4.41
0.02
o 28 48
o 17.28
0.17
0.52
12 28 48
7 29.28
4.49
0.03
o 43 12
o 25.92
0.26
0.53
12 43 12
7 37.92
4.58
O.O4
o 57 36
o 34.56
0.35
0-54
12 57 36
7 46.56
4.67
O.O5
I 12
o 43.20
o.43
0.55
13 12
7 55-20
4.75
O.O6
I 26 24
o 51.84
0.52
0.56
13 26 24
8 3.84
4.84
0.07
I 40 48
0.48
0.60
0.57
13 40 48
8 12.48
4.92
0.08
I 55 12
9.12
0.69
0.53
13 55 12
8 21.12
5.01
O.O9
0.10
2 9 36
2 24
17.76
26.40
0.78
0.86
0-59
0.60
14 9 36
14 24 o
8 29.76
8 38.40
5.10
5.18
O.II
2 38 24
35.04
0.95
0.61
14 38 24
8 47.04
5.27
0.12
2 52 48
43-68
.04
0.62
14 52 48
8 55.68
5.36
0.13
3 7 12
52.32
.12
0.63
15 7 12
9 4.32
5.44
0.14
0.15
3 21 36
3 36 o
2 0.96
2 9.60
.21
.30
0.64
0.65
15 21 36
15 36 o
9 12.96
9 21.60
III
0.16
0.17
3 50 24
4 4 48
2 18.24
2 26.88
.38
47
0.66
0.67
15 50 24
16 448
9 30-24
9 38.88
5.70
5.79
0.18
4 19 12
2 35.52
.56
0.68
16 19 12
9 47.52
5.88
0.19
4 33 36
2 44.16
.64
0.69
16 33 36
9 56.16
5.96
O.2O
448 o
2 52.80
73
0.70
16 48 o
10 4.80
6.05
0.21
5 2 24
3 L44
.81
0.71
17 2 24
10 13.44
6.13
0.22
5 16 48
3 10.08
.90
0.72
17 16 48
10 22.08
6.22
0.23
5 3 1 I2
3 18.72
99
0.73
17 31 12
10 30.72
6-31
0.2l
0.2$
5 45 36
600
3 27.36
3 36.00
2.07
2.16
0.74
o.75
17 45 36
1800
10 39.36
10 48.00
6.39
6.48
O.26
6 14 24
3 44.64
2.25
0.76
18 14 24
10 56.64
6-57
0.27
6 28 48
3 53.28
2-33
o.77
18 28 48
II 5.28
6.6 5
0.28
6 43 12
4 1.92
2.42
0.78
18 43 12
II 13.92
- 6.74
0.29
6 57 36
4 10.56
2.51
0.79
18 57 36
II 22.56
6.83
0.30
7 12
4 19.20
2.59
0.80
19 12 O
II 31.20
6.91
0.31
7 26 24
4 27.84
2.68
0.81
19 26 24
II 39.84
7.00
0.32
7 40 48
4 36.48
2.76
0.82
19 4O 48
II 48.48
7.08
0-33
7 55 12
4 45.12
2.85
0.83
19 55 12
II 57.12
7-17
8 9 36
2.94
0.84
20 9 36
12 5.76
7.26
0.35
8 24 o
5 2.40
3.02
0.85
20 24 o
12 14.40
7.34
0.36
8 38 24
5 ".04
3 TI
0.86
20 38 24
12 23.04
7-43
0.37
8 52 48
5 19.68
3-20
0.87
20 52 48
12 31.68
7.52
0.38
9 7 12
5 28.32
3.28
0.88
21 7 12
12 40.32
7.6o
o-39
9 21 36
5 36.96
3-37
0.89
21 21 36
12 48.96
7.6 9
0.40
9 36 o
5 45-60
3-46
0.90
21 36
12 57.6O
7.78
0.41
9 50 24
5 54-24
3-54
0.91
21 50 24
13 6.?A
7.86
0.42
10 4 48
6 2.88
3.63
0.92
22 4 48
13 14.88
7.95
o.43
IO 19 12
6 11.52
3-72
0.93
22 19 12
I? 13.52
8.04
0.44
10 33 36
6 20.16
0.94
22 33 36
15, jz.ib
8.12
0.45
10 48 o
6 28.80
3-89
0.95
22 48
15 40.80
8.21
0.46
0.47
II 2 24
ii 16 48
6 37-44
6 46.08
3-97
4.06
0.96
o.97
23 2 24
23 16 48
13 49-44
13 58.08
8.29
8.38
0.48
II 31 12
6 54.72
4.15
0.98
23 31 12
14 6.72
8.47
0.49
ii 45 36
7 3.36
4.23
0-99
23 45 36
14 15.36
8.55
0.50
12
7 12.00
4-32
1. 00
24 o o
14 24.00
8.64
i
102
TABLE IX. ARC INTO TIME AND VICE VERSA.
o
h, m.
o
/i. m.
h. m.
h. m.
o
h. m.
h. m.
i
m. s.
j.
o
O O
60
4 o
1 20
8
180
12 O
240
16 o
300
20
o
0.000
I
o 4
61
4 4
121
8 4
181
12 4
241
16 4
301
20 4
i
o 4
i
0.066
2
o 8
62
4 8
122
8 8
182
12 8
242
16 8
302
20 8
2
o 8
2
*33
3
12
63
4 12
123
8 12
183
12 12
243
16 12
33
2O 12
3
12
3
0.200
4
o 16
64
4 16
I2 4
8 16
184
12 16
244
16 16
304
20 16
4
o 16
4
0.266
I
o 20
o 24
66
420
I2 5
126
8 20
824
185
186
12 20
12 24
III
16 20
16 24
305
306
20 20
20 24
I
O 20
o 24
\
o 333
0.400 i
7
028
67
4 28
127
8 28
187
12 28
247
16 28
20 28
7
28
7
0.466
8
032
68
4 32
128
832
188
12 3 2
248
16 32
308
2o 32
8
o 32
8
o-533
9
o 36
69
436
129
836
189
12 36
249
1636
309
20 36
9
o 36
9
0.600
10
o 40
70
4 40
130
8 40
190
40
250
16 40
310
2O 40
IO
o 40
10
0.666
n
o 44
4 44
844
191
12 44
251
16 44
3"
20 44
n
o 44
n
0-733
12
o 48
72
448
132
8 48
192
12 48
252
1648
312
2048
12
o 48
12
0.800
13
052
73
452
133
852
193
12 52
253
16 52
313
20 52
3
o 52
13
0.866 i
'4
o 56
74
456
134
8 56
194
12 56
254
16 56
20 56
M
o 56
M
0-933
11
o
4
76
c o
5 4
135
136
Q O
9 4
'95
196
13 o
13 4
255
256
17 o
17 4
3'5
316
21 O
21 4
15
16
o
4
15
1 6
1. 000
1.066
11
8
12
78
5 8
5 '2
137
138
9 8
9 12
197
198
13 8
13 12
257
258
17 8
17 12
317
21 8
21 12
\l
8
12
M
I 133
1.200
*9
16
79
516
139
9 16
199
13 16
259
17 16
319
21 16
19
16
19
1.266
20
20
80
5 20
140
9 20
200
13 20
260
17 20
320
21 20
20
20
20
'333
21
22
24
28
Si
82
528
141
142
9 24
9 28
201
202
13 24
n 28
261
262
17 24
17 28
321
322
21 24
21 28
21
22
3
21
22
i 400
1.466
23
32
83
5 3?
143
9 32
203
'3 32
263
17 32
323
21 32
23
32
23
' 533
2 4
36
84
536
144
9 36
204
13 36
264
17 36
324
21 36
24
36
24
i. 600
11
40
44
II
5 40
5 44
MS
146
9 40
9 44
205
206
3 40
3 44
266
1740
'7 44
1
21 40
21 44
3
40
44
11
1.666
1-733
11
48
52
11
548
5 52
H8
948
9 5 2
207
208
13 48
13 52
267
268
1748
17 52
21 48
21 52
11
48
52
11
i. 806
1.866
29
56
89
5 56
149
9 S^
20 9
*3 5 6
269
17 56
329
21 56
29
56
29
* 933
30
2 O
90
6 o
ISO
1C O
210
14 o
270
18 o
330
22 O
30
2 O
30
2 OOO
31
2 4
6 3
10 4
211
14 4
271
18 4
331
22 4
2 4
31
2.066
32
2 8
92
6 8
152
ic 8
212
14 8
272
18 8
332
22 8
32
2 8
32
2-133
33
2 12
93
6 12
153
10 12
213
14 12
273
18 12
333
22 12
33
2 12
33
2.200
34
2 16
94
6 16
154
ic 16
2I 4
14 16
274
18 16
334
22 16
34
2 16
34
2.266
35
36
2 20
2 24
95
96
6 20
624
155
IO 20
ic 24
215
216
14 20
14 24
III
18 20
18 24
335
336
122 20
22 24
35
3 6
2 20
2 24
1
2-333
2.4OO
37
2 28
6 28
*57
10 28
217
14 28
277
18 28
337
22 28
2 28
2.466
38
2 3 2
98
632
158
10 32
218
M 3 2
278
1832
338
22 32
38
2 32
38
2-533
39
2 3 6
99
6 36
159
10 36
219
M 36
279
18 36
339
22 36
39
2 36
39
2.600
40
240
IOO
640
160
10 40
22O
M 40
280
18 40
340
22 40
40
2 40
40
2.666
4'
2 44
101
644
161
10 44
221
M 44
281
18 44
22 44
2 44
2 733
4?
248
102
6 48
162
10 48
222
1448
282
18 48
342
22 48
42
2 4 8
42
2.800
43
2 52
103
652
163
10 52
223
14 52
283
18 52
343
22 S2
43
2 52
43
2.866
44
2 56
104
r ;6
164
10 56
224
14 56
284
18 56
344
22 5 6
44
2 5 6
44
2 933
45
46
3 o
3 4
I0 5
1 06
7 4
II
ii 4
III
15 o
15 4
III
19 o
19 4
345
346
23 o
23 4
46
3 o
3 4
46
3.000
3.066
47
3 8
107
7 8
167
n 8
227
15 8
287
19 8
347
23 8
47
3 8
47
3 133
48
3 I2
108
7 12
1 68
II 12
228
15 12
288
19 12
348
23 12
48
3 I2
4^
3.200
49
316
109
716
169
ii 16
229
15 16
289
19 16
349
23 16
49
316
49
3.266
50
320
no
7 20
170
II 20
230
15 20
290
19 20
350
2 3 20
50
3 20
50
3-333
51
3 24
in
724
171
II 24
2 3 I
15 24
291
19 24
351
23 24
51
51
3.400
52
3 28
112
728
172
II 28
232
15 28
292
19 28
352
23 28
52
3 28
52
3.466
53
3 32
"3
732
173
II 32
233
15 32
293
19 32
353
23 32
53
3 32
53
3 533
54
336
114
736
174
II 36
234
15 36
294
19 36
354
2336
54
336
54
3-600
55
3 40
115
740
175
II 40
235
'5 40
295
19 40
355
23 40
55
3 40
55
3.666
56
3 44
no
7 44
176
ii 44
236
15 44
296
19 44
356
23 44
56
3 44
56
3 733
348
117
748
II 48
237
15 48
297
1948
357
2348
348
57
3.800
58
3 52
118
7 52
178
II 52
238
15 52
298
19 52
358
23 52
58
3 52
58
3-866
.59
356
119
756
179
II 56
239
15 56
299
19 5 6
359
2356
59
356
59
3 933
TABLE Xa. TO CONVERT MEAN INTO SIDEREAL TIME.
103
Mean T.
h. m.
Correction.
m. s.
Mean T.
h. m.
Correction.
/. s.
Mean T
h. m.
Correction.
m. s.
Corr. for min.
and sec.
m. s. s.
o o
0.00
8 o
i 18.85
16 o
2 37.70
10
0.03
10
1.64
10
20.50
10
39.35
20
O.Oj
20
3.29
20
22.14
20
40.99
30
0.08
30
4-93
30
23.78
30
42.63
40
O.II
40
6.57
40
25-42
40
44.28
5
0.14
50
8.21
50
27.07
50
45.92
I
0.16
IO
0.19
I
o 9.86
9 o
i 28.71
17 o
2 47.56
20
3
0.22
C.25
IO
11.50
IO
30.35
IO
49.20
40
0.27
20
13.14
20
31-99
20
50.85
0.30
30
14.78
30
33.64
30
52.49
2 O
0.33
40
50
16.43
16.07
40
50
35.28
36.92
40
50
54.13
55-77
XO
20
0.36
0. 3 3
30
0.41
2
10
o 19.71
21.36
10
10
i 38.56
40.21
18 o
10
2 57.42
59.06
40
50
0.44
0.47
20
23.00
20
41.85
20
3 0.70
3 o
o 49
30
c 40
So
24.64
26.28
27.93
30
40
50
43-49
45.14
46.78
30
40
50
2.34
3.99
5.63
IO
20
30
40
0.52
o 55
o 60
50
0.63
3 o
o 29.57
II
i 48.42
19 o
3 7.27
4
0.66
10
31.21
10
50.06
10
8.92
10
0.68
20
32.86
20
51.71
20
10.56
20
0.71
30
34.50
3
53-35
30
12. 2O
30
0.74
40
36.14
40
54-99
40
13.84
40
0.77
50
37.78
56.64
50
15.49
5
0.79
c o
0.82
4 o
o 39-43
12
I 58.28
20 o
3 I7.I3
IO
o 85
n RS
10
20
30
41.07
42.71
44.35
10
20
30
59-92
2 1.56
10
20
30
18.77
20.42
22.06
20
30
40
5
o.oo
0.90
093
0.96
40
46.00
40
4*85
40
23.70
50
47.64
50
6.49
50
25.34
6 o
10
o 99
.01
20
04
5 o
o 49.28
13 o
2 8.13
21
3 26.99
30
.07
10
50.92
10
9.78
10
28.63
40
IO
20
52 57
20
11.42
20
30.27
50
.12
30
54.21
30
13.06
30
31.91
7 O
j-
40
55.85
40
14.70
40
33.56
/
10
'18
50
57.50
50
16.35
50
35-20
20
.21
30
23
6 o
10
o 59.H
i 0.78
14 o
10
2 17.99
19.63
22
10
3 36.84
38.48
40
50
.29
20
2.42
20
21.28
20
40.13
8 o
.31
30
4.07
30
22.92
30
4L77
IO
34
40
5.71
40
24.56
40
43-41
20
37
50
7-35
50
26.20
50
45.o6
3>
40
.40
.42
So
45
7 o
10
I 9.00
10.64
15 o
10
2 27.85
29.49
23 o
10
3 46.70
48.34
9 o
48
rf\
20
12.28
20
20
49.98
20
5
30
13.92
30
32.77
30
51-63
3
?o
40
15.57
40
34.42
40
53.27
40
Pg
50
17.21
50
36.06
50
50
.6,
TABLE X. TO CONVERT SIDEREAL INTO MEAN TIME.
Sid. T.
k. m.
Correction.
m. s.
Sid. T.
A. m.
Correction.
m. s.
Sid. T.
k. m.
Correction,
w. s.
Corr. for min.
and sec.
m. s. s.
o o.oo
8 o
I 18.64
16 o
2 37-27
O IO
0.03
10
20
1.64
3.28
10
20
20.28
21 .91
IO
20
38.91
40.55
20
30
0.05
0.08
30
4.92
30
23.55
30
42.19
40
O.1I
40
6-55
40
25.19
40
43.83
5
0.14
50
8 19
50
26.83
50
45.46
I
o. 16
10
o. 19
t
o 9.83
9 o
I 28.47
17 o
2 47.10
20
3
0.22
0.25
IO
11.47
IO
30.10
10
48.74
40
O.27
20
13."
20
3L74
20
50.38
50
0.30
30
H.74
30
33.38
30
52.02
2 O
o ? 3
40
16.38
40
35-02
40
53-66
10
0.35
50
18.02
50
36.66
50
55-29
20
0.38
30
0.41
t
IO
o 19.66
21.30
10
10
i 38.30
39-93
18 o
IO
2 56.93
58.57
40
50
0.44
0.47
20
22.94
20
4L57
20
3 0.21
3 O
0.49
30
40
24.57
26.21
30
40
43-21
44-85
30
40
1.85
3.48
IO
90
0.52
o-55
50
27.85
50
46.49
50
5.12
3
40
0.57
0.60
50
0.63
^ o
o 29.49
II
i 48.12
19 o
3 6.76
4- o
0.66
10
31.13
10
49.76
10
8.40
10
0.68
20
32.76
20
51.40
20
10.04
20
0.71
30
34.40
30
53.04
30
11.68
30
0.74
40
36.04
40
54.68
40
13.32
40
0.76
50
37.68
50
56.32
50
J4-95
5
0.79
5
0.82
4 o
10
20
o 39.32
40.96
42.60
12
IO
2O
i 57.96
59-59
2 1.23
20 o
10
20
3 16.59
18.23
19.87
10
20
30
4
0.85
0.87
0.90
0.93
30
44.23
30
2.87
30
21.51
5
0.96
40
45.87
40
4-5'
40
23.14
30
47.51
50
6.15
50
24.78
6 o
10
0.98
i .01
20
i .04
C o
o 49.15
13 o
2 7.78
21
3 26.42
30
i. 06
10
50.79
IO
9.42
IO
28.06
40
1.09
20
52 42
20
II. 06
20
29.70
5
I . 12
30
54.06
30
12.70
3
31.34
7
.15
40
55-70
40
14-34
40
32.97
IO
1 7
50
57-34
50
15.98
50
34.6i
20
.20
30
2 3
6 o
o 58.98
14 o
2 I7.6l
22
3 36.25
40
fr\
.26
10
I 0.62
10
19.25
10
37.89
5
l2
20
2.25
20
20.89
20
39-53
8 o
31
- 3
3.89
30
22.53
30
41.16
10
34
40
5-53
40
24.17
40
42.80
20
37
50
7.17
50
25.80
50
44-44
3
40
39
.42
5
45
7 o
i 8.81
15 o
2 27.44
23 o
3 46.08
4*7
IO
10.44
lo
29.08
10
47.72
IO
*t/
20
12.08
20
30.72
20
49.36
20
C?
30
13.72
30
32.36
30
51.00
3
56
40
15.36
40
34-00
40
52.63
40
.58
5
17.00
50
35.64
50
54.27
50
.61
i
UNIVERSITY OF CALIFORNIA LIBRARY
BERKELEY
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THE UNIVERSITY OF CALIFORNIA LIBRARY