Use of logarithms and logarithmic tables

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yiACHIKERTS REFERENCE 3E. -MO. 53
PUEMoHE. ", : T M/ CHINERY, NEW

MACHINERY'S REFERENCE SERIES

EACH NUMBER IS A UNIT IN A SERIES ON ELECTRICAL AND

STEAM ENGINEERING DRAWING AND MACHINE

DESIGN AND SHOP PRACTICE

NUMBER 53

USE OF LOGARITHMS AND
LOGARITHMIC TABLES

SECOND EDITION

CONTENTS

The Use of Logarithms, by ERIK OBERG - - 3

Tables of Logarithms - - 18

Copyright, 1912, The Industrial Press, Publishers of MACHINERY,
49-55 Lafayette Street, New York City

THE USB OP LOGARITHMS

It is not intended in the following pages to discuss the mathematical
principles on which logarithms and expressions containing logarithms
are based, but simply to impart a working knowledge of the use of
logarithms, so that practical men, unfamiliar with this means for
eliminating much of the work ordinarily required in long and cumber-
some calculations, may be able to make advantageous use of the tables
of logarithms given in the latter part of the book.

The object of logarithms is to facilitate and shorten calculations
involving multiplication, division, x the extraction of roots, and the
obtaining of powers of numbers, as will be explained later; but ordin-
ary logarithms cannot be applied to operations involving addition and
subtraction. Before entering directly upon the subject of the use of
logarithms in carrying out the various classes of calculations men-
tioned, it will be necessary to deal with the question of how to find
the logarithm of a given number from the tables; or, if the logarithm
is given, how to find the corresponding number.

A logarithm consists of two parts, a whole number and a decimal.
The whole number, which may be either a positive or a negative num-
ber,* or zero, according to the rules which will be given in the follow-
ing, is called the characteristic; the decimal is called the mantissa.
The decimal or mantissa only is given in the tables of logarithms on
pages 18 to 35, inclusive, and is always positive. The logarithm of 350,
for example is 2.54407. Here "2" is the characteristic, and "54407" is
the mantissa, this latter being found from the table on page 23.

Rules for Finding- the Characteristic

The characteristic is not given in the tables of logarithms, due
partly to the fact that it can so easily be determined without the aid
of tables, and partly because the tables, so to say, are universal when
the characteristic is left out.

For 1 and all numbers greater than 1 the characteristic is one less
than the number of places to the left of the decimal point in the given
number.

The characteristic of the logarithm of 237, therefore, is 2, because
2 is one less than the number of figures in 237. The characteristic
of the logarithm of 237.26 is also 2, because it is only the number of
figures to the left of the decimal point that is considered.

The characteristic of the logarithm of 7 is 0, because 0 is one less

* See MACHINERY'S Reference Series No. 54, Solution of Triangles, Chapter
III, Positive and Negative Quantities.

347608

4 No.. 53—UVE O'F LOGARITHMS

than 1, which is the number of places in the given number. Below
are given several numbers with the characteristics of their logarithms:

Characteristic of
Number Logarithm

31 1

3163 3

229.634 2

1,112,352.62 6

1000 3

100 2

1 0

For numbers smaller than 1, that is, for numbers wholly decimal,
the characteristic is negative, and its numerical value is one more
tJian the number of ciphers between the decimal point and the first
decimal which is not a cipher.

The numerical value of the characteristic of the logarithm of 0.036,
therefore, is 2; being negative it is — 2. Instead of writing the minus
sign ( — ) in front of or before the figure ( — 2), it is, however, written
over the figure ( 2 ) • This method is used because the minus sign
refers only to the characteristic, and not to the mantissa, this latter
always being positive. In the same way, the characteristic of the
logarithm of 0.36 is I ; and the characteristic of the logarithm of
0.0006 is i. Below are given several examples :

Characteristic of
Number Logarithm

0.00.00275 5

0.3 I

0.375 T

0.000812 4

0.01234 5

Finding1 the Logarithms of Numbers

After the characteristic has been found by the rules just given, the
mantissa must be found from the tables of logarithms. When finding
the mantissa the decimal point is entirely disregarded. The mantissa
of the logarithms of 2716, 271.6, 27.16, 2.716, or 0.02716, for example,
is the same; it is only necessary to find the given figures in the tables,
irrespective of the location of the decimal point.

Referring now to the tables on pages 18 to 35, it will be seen that
numbers from 100 to 1,000 are given in the left-hand column. In
addition, at the top of the tables, are figures from 0 to 9, each heading
a column of logarithms. These additional figures make it possible to
obtain directly from the tables the logarithms for all numbers with
four figures or less. The body of the tables gives the mantissa of the
logarithms.

While the tables do not give directly the mantissa of logarithms of
numbers with more than four figures, it is possible to approximate
the logarithm for numbers with a greater number of figures by methods
which will be explained later. At present, when the logarithm is
required for numbers with five or more figures, we will assume that

METHODS, RULES AND EXAMPLES 5

for practical results it is accurate enough to find the mantissa of the
logarithm of the first four figures of the number, remembering, of
course, that if the fifth figure is more than 5, then the fourth figure
should be increased by one unit.

To find the logarithm of a number from the tables, first locate the
three first figures of the number for which the logarithm is required
in the left-hand column, and then find the fourth figure at the top of
the columns of the page. Then follow the column down from this last
figure until opposite the three first figures in the left-hand column.
The figure thus found in the body of the table is the mantissa of the
logarithm, the characteristic having already been found by the rules
previously given.

If the number of which the logarithm is required does not contain
four figures, annex ciphers to the right so as to obtain four figures.
If the mantissa of the logarithm of 6 is required, for example, find
the mantissa for 6,000. The mantissa is the same for 6, 0.6, 0.06, 60,
600, 6,000, etc., as already explained. The difference in the logarithm
is taken care of by the change of characteristic for these various
values. In order to save space in the tables, it will be seen by
referring to them that the first two figures of the mantissa have been
given in the first columns of logarithms only, the 0-column. These
two figures should, however, always precede the three figures given
in each of the following columns.

A few examples will now make the use of the tables clearer.

Example 1. — Find the logarithm of 1852.

Following the rule already given, locate 185 in the left-hand column
of the tables (it will be found on page 19), and then following down-
ward the column headed "2" at the top of the page, find the required
mantissa opposite 185. It will be seen that the mantissa is .26764,*
the figures 26 being found in the column under "0" and prefixed to
the figures 764 found directly in the column under "2." The charac-
teristic of the logarithm, according to the rules previously given, is 3.
Hence the logarithm of 1852, or, as it is commonly written, log 1852
— 3.26764.

Example 2. — Find log 1.852.

As the figures in this number are the same as in that given in ex-
ample 1, the mantissa remains the same; but the characteristic is 0.
Therefore, the required logarithm, or log 1.852 = 0.26764.

Example 3. — Find log 93.14.

Locate 931 in the left-hand column of the tables (page 34), and
then following downward the column headed "4" at the top of the
page, find the required mantissa opposite 931. It will be found that
the mantissa is .96914. The characteristic is 1. Hence log 93.14
= 1.96914.

Example 4. — Find log 4.576.

Find as before the last three figures of the mantissa opposite 457

* All the mantissas, or the numbers in the tables, are decimals, and the
decimal point has, therefore, been omitted entirely, since no confusion could
arise from this; but it should always be put before the figures of the mantissa
as soon as taken from the table. The practice of eliminating the decimal point
from the tables is common to all logarithmic tables.

6 No. 53— USE OF LOGARITHMS

in the left-hand column, and in the column under "6" at the top of
the page. The figures are ^049. The sign # indicates that the two
figures to be prefixed are not 65, as would ordinarily be the case,
but 66, or the figures given in the next following line in the 0-column.
This rule should always be borne in mind. Hence, log 4.576 = 0.66049.

Example 5. — Find log 72.

To find the mantissa, proceed as if it were to be found for 7200.
This we find from the tables to be .85733. The characteristic of the
logarithm of 72 is 1. Hence log 72 = 1.85733.

Example 6. — Find log 0.007631.

To find the mantissa, proceed as if it were to be found for 7631.
This we find from the tables to be .88258. The characteristic is 3,
according to the rule given for characteristics of logarithms of num-
bers less than 1. Hence, log 0.007631 = 3-88258. i

Example 7. — Find log 37,262.

While we will later explain how to find more exactly the mantissa
for a number with five figures, at present we may consider it accurate
enough for our purpose to find the mantissa for four figures, or for
3726. This is .57124. The characteristic of the logarithm of 37,262 is
4. Hence log 37,262 = 4.57124. This, of course, is only an approxima-
tion, but is near enough for nearly all shop and general engineer-
ing calculations.

If the given number had been 37,267 instead of 37,262, the logarithm
should have been found for 3727, as the fourth figure then should
have been increased by 1, when dropping the fifth figure, which is
larger than o.

Below are given several examples of numbers with their logarithms.
A careful study of these examples, the student finding the logarithms
for himself from the tables, and checking them with the results given,
will tend to make the methods employed clearer and fix them in the

mincl.

Number Logarithm

16.95 1.22917

2 0.30103

966.2 2.98507

151 2.17898

3.5671 0.55230

12.91 1.11093

3803.8 3.58024 /^

0.007 3.84510

It should be understood that in logarithms of numbers less than 1,
the characteristic, only, is negative. The mantissa is always positive,
so that 3.84510 actually means (— 3) + 0.84510.

Finding the Number whose Logarithm is Given

When a logarithm is given, and it is^ required to find the corres-
ponding number, first find the first two figures of the mantissa in the
column headed "0" in the tables. Then find in the group of mantissas,
all having the same first two figures, the remaining three figures.

METHODS, RULES AND EXAMPLES 7

These may be in any of the columns headed "0" to "9." The number
heading the column in which the last three figures of the mantissa
were found, is the last figure in the number sought, and the number
in the left-hand column, headed "N," in line with the figures of the
mantissa, gives the three first figures in the number sought.

,When the actual figures in the number sought have thus been deter-
mined, locate the decimal point according to the rules given for the
characteristic of logarithms. If the characteristic is greater than 3,
ciphers are added. For example, if the figures corresponding to a
certain mantissa are 3765, and the characteristic is 5, then the num-
ber sought must have 6 figures to the left of the decimal point, and ^ \n
hence would be 376500. If the characteristic had been 3, then the •£* --
number sought, in this case, would have been 0.003765.

If the mantissa is not exactly obtainable in the tables, find the near- v(
est mantissa in the table to the one given, and determine the number
corresponding to this. In most cases this gives ample accuracy. A
method will be explained later whereby still greater accuracy may be
obtained, but for the present it will be assumed that the numbers
corresponding to the nearest mantissa in the tables are accurate
enough for practical purposes.

A few examples will now be given in which it is required to find
the number when the logarithm is given.

Example 1. — Find the number whose logarithm is 3.89382.

First find the firs.t two figures of the mantissa (89) in the column
headed "0" in the tables. Then find the remaining three figures (382)
in the mantissas which all have 89 for their first two figures. The
figures "382^ are found in the column headed "1," which thus is the
last figure in the number sought; the figures "382" are also opposite
the number 783 in the left-hand column, which gives the first three
figures in the number sought. The figures in this number, thus, are
7831, and as the characteristic is 3, it indicates that there are four
figures to the left of the decimal point, or, in other words, that 7831 is
a whole number.

Example 2. — Find the number whose logarithm is 2.75020.

First find the first two figures of the mantissa (75) in the column
head "0" in the tables. Then find the remaining three figures (020)
in the mantissas, which all have 75 for their first two figures. The
# in front of the figure ^020 in the line next above that in which 75
was found indicates that these figures belong to the group preceded
by 75. Therefore, as ^.020 is found in the column headed "6" and
opposite the number 562 in the left-hand column, the figures in the
number required to be found are 5626. As the characteristic is 2,
th«e decimal point is placed after the first three figures, and, hence,
the number whose logarithm is 2.75020 is 562.6.

Example 3. — Find the number whose logarithm is 2.45350.

After having located 45 iif the column headed "0," it will be found
that the last three figures (350) of the mantissa are not to be found
in the table in the group preceded by 45. The nearest value in the
table, which is 347, is, therefore, located, and the corresponding num-

8 No. 53— USE OF LOGARITHMS

ber is found to be 284.1, the decimal point being placed after the third
figure, because the characteristic of the logarithm is 2. Had the char-
acteristic of the logarithm been 5 instead of 2, the number to be feund
would have been 284,100.

Below are given a selection of examples of logarithms with tbybir
corresponding numbers. The student should find the numbers/for
himself from the tables, and check them with the results "given.
This will aid in fixing the rules and methods employed more firmly in
the mind.

Corresponding
Logarithm Number

1.43201 27.04

4.89170 77,930

2.76057 0.05762

0.12096 1.321

2.99099 979.5

T.60206 0.4

5.60206 400,000

It being now assumed that the student has mastered the methods
for finding the logarithms for given numbers, and the numbers for
given logarithms, from the tables, the use of logarithms in multipli
cation and division will next be explained.

Multiplication by Logarithms

// two or more numbers are to be multiplied' together, find the
logarithms of the numbers to be multiplied, and then add these
logarithms; the sum is the logarithm of the product, and the numbed
corresponding to this logarithm is the required product.

Example 1. — Find the product of 2831 X 2.692 X 29.69 X 19.4.
This calculation is carried out by means of logarithms as follows:
log 2831. =3.45194
log 2.692 = 0.43008
log 29.69 =1.47261
log 19.4 =1.28780

6.64243

The sum of the logarithms, 6.64243, is the logarithm of the product,
and from the tables we then find that the product equals 4.390,000.
This result is, of course, only approximately correct, at the last three
figures are added ciphers; but for most engineering calculations the
result would give all the accuracy required. In most engineering
calculations one or more factors are assumed from experimental
values, and as these assumed values evidently must often vary be-
tween wide limits, it would show lack of judgment to require calcula-
tions in which such assumed values enter, to be carried out with too
many "significant" figures. Such values are fully as well expressed
in round numbers, with ciphers annexed to give the required value
to the figures found from the tables.

If one or more of the characteristics of the logarithms are negative,
these are subtracted instead of added to the sum of the character-

METHODS, RULES AND EXAMPLES 9

istics. The mantissas, as already mentioned, are always positive,
so that they are always added in the usual manner. In order to
fully understand the adding of positive and negative numbers in the
following examples, the student should be* familiar with calculations
with positive and negative quantities, as explained in MACHINERY'S.
Reference Series No. 54, Solution of Triangles, Chapter III.
Example 2. — Find the product 371.2 X 0.0972 X 3.

log 371.2 =2.56961

log 0.0972=2.98767 7. *>

log 3. = 0.47712

2.03440

The number corresponding to the logarithm 2.03440 is 108.2. Note
that the first two figures of the mantissa of the logarithm are 03.
Example 3. — Find the product 12.76 X 0.012 X 0.6.
log 12.76 =1.10585
log 0.012=2.07918 7
log 0.6 =1.77815 "

2.96318
The product, hence, is 0.09187.

Division by Logarithms

When dividing one number by another, the logarithm of the divisor
is subtracted from the logarithm of the dividend. The remainder is
the logarithm of the quotient.

For example, if we are to find the quotient of 7568 -r- 935.3, we first
find log 7568 and then subtract from it log 935.3. The remainder
is then the logarithm of the quotient.

It is advisable, however, to make a modification, as explained in
the following, of the logarithm of the divisor so as to permit of its
addition to, instead of its subtraction from the logarithm of the divi-
dend. Assume, for instance, that an example, as below, were given:
375.2 X 97.2 X 0.0762 X 3

962.1 X 92 X 33.26

It would be perfectly correct to find the logarithms of all the fac-
tors in the numerator and add them together, and then the logarithms
of all the factors in the denominator and add them together; and
finally subtract the sum of the logarithms of the denominator from
the sum of the logarithms of the numerator. The remainder is the
logarithm of the result of the calculation. This method, however,
involves two separate additions and one subtraction. It is possible, by
a modification of the logarithms of the numbers in the denominator
to so arrange the calculation that a single addition will give the
logarithm of the final result.

In dealing with positive and negative numbers we learn that if
we add a negative number to a positive number, the sum will be
the same as if we subtract the numerical value of the negative

10 No. 53— USE OF LOGARITHMS

number from the positive number; that is 5 -f- ( — 2) = 5 — 2 •= 3.
If we reverse this proposition ve have 5 — 2 = 5 -f ( — 2). If we
now assume that 5 is the logarithm of a certain number a and 2 the
logarithm of another number Z>, and if we insert these values in the
last expression, instead of 5 and 2, we have:

log a — log 6 = log a 4- (— log &).

From this we see that instead of subtracting log & from log a we
can add the negative value of log 6 and obtain the same result. As
the mantissa always must remain positive, in order to permit direct
addition, the negative value of the logarithm cannot be obtained
by simply placing a minus sign before it. Instead, it is obtained in
the following manner:

If the characteristic is positive, add 1 to its numerical value and
place a minus sign over it. To obtain the mantissa, subtract the
given mantissa from 1.00000.

Example 1. — The logarithm of 950 = 2.97772. Find ( — log 950).

According to the rule given, the characteristic will be 3. The man-
tissa will be 1.00000 — .97772 = .02228. The last calculation can be
carried out mentally without writing it down at all, by simply finding
the figure which, added to the last figure in the given mantissa would
make the sum 10, and the figures which added to each of the other
figures in the mantissa, would make the sum 9. as shown below:

97772
02228

9 9 9 9 10

As this calculation is easily carried out mentally, the method
described, when fully mastered, greatly simplifies the vrork where
operations of both multiplication and division are to be performed in
the same example.

Example 2.— The logarithm of 2 is 0.30103. Find (— log 2).

According to the given rules the characteristic is T, and the
mantissa, .69897.

The following examples should be studied until thoroughly under-
stood:

log 270. =2.43136 - log 270. =3.56864

log 10. =1.00000 — log 10. = T.OOOOO

log 26.99 =1.43120 - log 26.99 " = 5.56880

In the example in the second line an exception from the rule for
obtaining the mantissa of the negative logarithm is made. It is
obvious, however, that if log 10 = 1.00000, then ( — log 10) =
T.OOOOO. In the example in the last line there is another deviation
from the literal understanding of the rule for the mantissa. As the
last figure in the positive logarithm is 0, the last figure in ( — log
26.99) is also 0, and the next last figure is treated as if it were the
last, making the next last figure in the negative logarithm 8. ,

If the characteristic of the logarithm is negative, subtract 1
its numerical value, and make it positive. The mantissa is obtained
by the same rule as before.

METHODS. RULES AND EXAMPLES

11

Example 1.— The logarithm of 0.003 = 3.47712. Find ( — log 0.003).
According to the rule just given the characteristic will be 2. The
mantissa will be .52288. Hence (— log 0.003) = 2.52288.

The following examples should be studied until fully understood:
log 0.3 — T.47712 —log 0.3 =0.52288

log 0.0006963=3.84280 -log 0.0006963 =3.15720

log 0.6607 = 1.82000 —log 0.6607 =0.18000

When sufficient practice has been obtained, the negative value of
a logarithm can be read off almost as quickly from the tables as the
positive value given, and the subsequent gain of time, and the ease of
the calculations following, more than justify this short-cut method.

Examples of the Use of Logarithms

We will now give a number of examples of the use of logarithms in
calculations involving multiplication and division. No comments
will be made, as it is assumed that the student has now grasped the
principles sufficiently to be able to follow the methods used without
further explanation.

Example 1.

0.0272 X 27.1 X 12.6.

2.371 X ,0.007
log 0.0272 = 2.43457
log 27.1 =1.43297
12.6 =1.10037
2.371 =T.€2507
0.007 =2.15490

log

— log
— log

2.74788
The result, then, is 559.6.

Example 2.

0.3752 X 0.063 X 0.012

0.092 X 1289

log 0.3752 = T.57426

log 0.063 =2.79934

log 0.012 =5.07918

-log 0.092 =1.03621

— log 1289.0 =4.88975

The result, then, is 0.000002392.
Example 3.

6.37874

3.463 X 1.056 X 14.7 X 144 X 10

log
log
log

log
log

The result, then, is 77,410.

3.463 = 0.53945

1.056 = 0.02366

14.7 =i:i6732

144.0
10.0

= 2'.15836
= 1.00000

4.88879

12 No. 53— USE OF LOGARITHMS

Example 4.

0.00005427 X 392 X 2.5 X 200 X 200
log 0.00005427 = 5.73456

log 392. —2.59329

log 2.5 =0.39794

log 200. =2.30103

log 200. =2.30103

3.32785
Hence, the result is 2127.

Obtaining: the Powers of Numbers

Expressions of the form 6.513 can easily be calculated by means of
logarithms. The small (3) is called exponent.* In this case the
"third power" of 6.51 is required.

A number may be raised to any power by simply multiplying the
logarithm of the number by the exponent of the number. The product
gives the logarithm of the value of the power.

Example 1. — Find the value of 6.51s.

log 6.51 = 0.81358
3 X 0.81358 = 2.44074

The logarithm 2.44074 is then the logarithm of 6.513. Hence 6.51*
equals the number corresponding to this logarithm, as found from the
tables, or 6.513 = 275.9. «

Example 2. — Find the value of 12 1-29.

log 12 = 1.07918
1.29 X 1.07918 = 1.39214

Hence, 12 1-29 = 24.67.

The multiplication 1.29 x 1.07918 is carried out in the usual arith-
metical way. The example above is one of a type which cannot be
solved by any means except by the use of logarithms. An expression
of the form 6.513 can be found by arithmetic by multiplying 6.51
X 6.51 X 6.51, but an expression of the form 121-29 does not permit
of being calculated by any arithmetical method. Logarithms are here
absolutely essential.

One difficulty is met with when raising a number less than 1 to a
given power. The logarithm is then composed of a negative term, the,
characteristic, and a positive term, the mantissa. For example: Find
the value 0.313. The logarithm of 0.31 = T.49136. In this case, multi-
ply, separately, the characteristic and the mantissa by the exponent,
as shown below. Then add the products.

log 0.31 =(1149136

V — JV^ — ^

Multiplying characteristic and mantissa separately by 5 we have:

5 Xl = 5
5 X .49136 = 2.45680

log 0.315 = 3.45680
Hence, 0.315 = 0.002863.

* See MACHINERY'S Reference Series No. 52,, Advanced Shop Arithmetic for
the Machinist, Chapter III.

METHODS, RULES AND EXAMPLES 13

If the exponent is not a whole number, the procedure will be some-
what more complicated. The principle of the method, however, re-
mains the same.

Example: Find the value of 0.062 -31

log 0.06 = 5.77815
Then

2.31 X" 2 = 2.31 X (—2) = — 4.62

2.31 X 0.77815 = 1.79753

In this case, tire first product, — 4.62, is negative both as regards the
whole number and the decimal. In order to make the decimal positive
so that we may be able to add it directly to the second product, 1.79753,
we must use the same rule as given for changing a logarithm with a
positive characteristic to a negative value. Hence — 4.62 = 5.38. We
can now add the products:

5.38
1.79753

log 0.062-31 =S.17753
Hence 0.062-81 = 0.001505.
As a further example, find 0.073-51.

log 0.07 = 5.84510
Then

3.51 X 5 = 3.51 X (—2)= — 7.02=3.98

3.51 X .84510 =2.96630

log 0.073-51= 5.94630
Hence 0.073-51 =0.00008837.

Extracting- Roots by Logarithms

Roots of numbers, as for example i/O$T, can easily be extracted by
means of logarithms. The small (5) in the radical (V) of the root-
sign is called the index of the root. In the case of the square root
the index is (2), but it is not usually indicated, the square root being
merely expressed by the sign V.

Any root of a number may be found by dividing its logarithm by
the index of the root; the quotient is the logarithm of the root.

Example 1. — Find -^ 276.

log 276 = 2.44091
2.44091 -=-3 = 0.81364

Hence log f 7~276~= 6.81364, and ^"276"= 6.511.

Example 2.— Find -^KiuJTT | j

log 0.67 = 1.82607

In this case we cannot divide directly, because we have a negative
characteristic and a positive mantissa. We then proceed as follows: Add
numerically as many negative units or parts of units to the character-
istic as is necessary to make it evenly contain the index of the root.
Then add the same number of positive units or parts of units to the
mantissa. Divide each separately by the index. The quotients give

14 No. 53— USE OF LOGARITHMS

the characteristic and mantissa, respectively, of the logarithm of the
root.

Proceeding with the example above according to this rule, we have:

1 + 2 = 3; 3 + 8 = I.

.82607 + 2 = 2.82607; 2.82607 -*- 3 = .94202.

Hence, log f/lK67 = T.94202, and ^ 0.67 = 0.875.

Example 3.— Find V °-2-

log 0.2 = 1.30103.

If we add ( — 0.7) to the characteristic of the logarithm found, it
will be evenly divisible by the index of the root.

Hence:

T + (—0.7) = —1.7; —1.7-^-1.7 = 1.
.30103 + 0.7 = 1.00103 ; 1.00103 -=- 1.7 = .58884.

Hence, log ™/~03 = T.58884, and l] ~OJJ = 0.388.
A number of examples of the use of logarithms in the solution of
everyday problems in mechanics, are given in MACHINERY'S Reference
Series No. 19, Use of Formulas in Mechanics, Chapter II, 2nd edition.

When exponents or indices are given in common fractions, it is
usually best to change them to decimal fractions before proceeding
further with the problem.

Interpolation

If the number for which the logarithm is required consists of five
figures, it is possible, by means of the small tables in the right-hand
column of the logarithm tables, headed "P. P." (proportional parts),
to obtain the logarithm more accurately than by taking the nearest
value for four figures, as has previously been done in the examples
given. The method by which the logarithm is then obtained is called
interpolation.

In the same way, if a logarithm is given, the exact value of which
cannot be found in the tables, the number corresponding to the logar-
ithm can be found to five figures by interpolation, although the main
tables contain only numbers of four figures.

The logarithm of 2853 is 3.45530, and the logarithm of 2854 is
3.45545, as found from the tables. Assume that the logarithm of 2853.6
were required. It is evident that the logarithm of this latter number
must have a value between the logarithms of 2853 and 2854. It must
be somewhat greater than the logarithm of the former number, and
somewhat smaller than that of the latter. While the logarithms, in
ceneral, are not proportional to the numbers to which they corres-
pond, the difference is very slight in cases where the increase in the
numbers is small; so that, in the case of an increase from 2853 to 2854,
the logarithms for the decimals 2853.1, 2853.2, etc., may be considered
proportional to the numbers. It is on this basis that the small tables in
the right-hand column headed "P.P." are calculated, and the logarithm
of 2853.6, for example, is found as follows:

Find first the difference between the nearest larger and the nearest
smaller logarithms. Log 2854 = 3.45545 and log 2853 = 3.45530. The

METHODS, RULES AND EXAMPLES 15

difference is 0.00015. Then in the small table headed "15" in the right-
hand column find the figure opposite 6 (6 being the last or fifth figure
in the given number). This figure is 9.0. Add this to the mantissa
of the smaller of the two logarithms already found, disregarding the
decimal point in the mantissa, and considering it, for the while being
as a whole number. Then 45530 -f 9.0 = 45539. This is the mantissa
of the logarithm of 2853.6, and the complete logarithm is 3.45539.

Example. — Find log 236.24.

Log 236.2 = 2.37328; log 236.3 = 2.37346; difference — 0.00018.
In table "18" the proportional part opposite 4 is 7.2. Then 37328 + 7.2
= 37335.2. The decimal 2 is not used, but is dropped. Hence log
236.24 = 2.37335.

If the proportional part to be added has a decimal larger than 5, it
should not be dropped before the figure preceding it has been raised
one unit. For example, if the logarithm of 236.26 had been required,
then the proportional part would have been 10.8 and the mantissa
sought 37328 + 10.8 = 37338.8. Now the decimal 8 cannot be dropped
before the figure 8 preceding it has been raised to 9. Then log
236.26 = 2.37339.

If the number for which the lorgarithm is to be found consists of
more than five figures, find the mantissa for the nearest number of
five figures, but choose the characteristic according to the total num-
ber of figures to the left of the decimal point. For example, if the
logarithm of 626,923 is required, find the mantissa, by interpolation,
for 62692. If the logarithm for 626,928 is required, find the mantissa
for 62693, always remembering to raise the value of the last figure,
if the figure dropped is more than 5. The characteristic in each of
these examples would, of course, be 5, as it is chosen according to the
total number of figures to the left of the decimal point in the given
numbers, which is 6.

To find a number whose logarithm is given more accurately than to
four figures, when the given mantissa cannot be found exactly in the
tables, find the mantissa which is nearest to, but less than the given
mantissa. Subtract this mantissa from the nearest larger mantissa
in the tables and find in the right-hand column the small table headed
by this difference. Then subtract the nearest smaller mantissa from
the given logarithm, and find the difference, exact or approximate, in
the "proportional part" table (in the right-hand column of this
table). The corresponding figure in the left-hand column of the "pro-
portional part" table is the fifth figure in the number sought, the other
four figures being those corresponding to the logarithm next mailer
to the given logarithm.

Example. — Find the number whose logarithm is 4.46262.

The mantissa can not be found exactly in the tables; therefore, fol-
lowing the rules just given, we see that the nearest smaller mantissa
in the tables equals 46255. The next larger is 46270. The difference
between them is 15. The difference between the mantissa of the given
logarithm, 46262 and the next smaller mantissa, 46255 is 7. Now, in
the proportional parts table opposite 7.5 in the right-hand column of

16 No. 53— USE OF LOGARITHMS

the table headed 15, we find that the fifth figure of the number sought
would be 5. The four first figures are 2901. Hence the number sought
is 29,015.

The following examples, if carefully studied, will give the student
a clear conception of the method of interpolation.

Number Logarithm

52,163 4.71736

26.913 1.42996

0.012635 5.10157

12.375 1.09254

6.9592 0.84256

The student should find for himself, first the logarithms correspond-
ing to the given numbers, and then the numbers corresponding to the
given logarithms. In this way a check on the accuracy of the work
can be obtained by comparing with the results given.

General Remarks

In the system of logarithms tabulated on pages 18 to 35, the base
of the logarithms is 10; that is, the logarithm is actually the exponent
which would be affixed to 10 in order to give the number correspond-
ing to the logarithm. For example log 20 = 1.30103, which is the
same as to say that 101-30103 = 20. Log 100 = 2, and, of course, we
know that 102 = 100. As 101 = 10, the logarithm of 10 = 1. The
logarithm of 1 = 0. The system of logarithms having 10 for its base
is called the Briggs or the common system of logarithms.

"While the accompanying logarithm tables are given to five decimals,
it should be understood that the logarithm of a number can be calcu-
lated with any degree of accuracy, so that large logarithm tables give
the logarithm with as many as seven decimal places, and some, used
for very accurate scientific investigations, give as many as ten deci-
mals. It will be noticed that in the accompanying tables the figure
5, when in the fifth decimal place, is either written 5 or 5. If the
sixth place is 5 or more, the next larger number is used in the fifth
place, and the logarithm is then written in the form 3.90855. The
dash over the 5 shows that the logarithm is less than given. If the
sixth figure is less than 5, the logarithm is written 3.91025, the dot
over the 5 showing that the logarithm is more than given. In calcu-
lations of the type previously explained, this, however, need not be
taken into consideration and these signs should be disregarded by the
student.

Hyperbolic Logarithms

In certain mechanical calculations, notably those involving the calcu-
lation of the mean effective pressure of steam in engine cylinders, use
is made of logarithms having for their base the number 2.7183, com-
monly designated e, and found by abstract mathematical analysis.
These logarithms are termed hyperbolic, Napierian or natural; the
preferable name, and that most commonly in use in the United States
is hyperbolic logarithms. The hyperbolic logarithms are usually desig-
nated "hyp. log." Thus, when log 12 is required, it always refers to

METHODS, RULES AND EXAMPLES 17

common logarithms, but when the hyp. log 12 is required, reference
is made to hyperbolic logarithms. Sometimes, the hyperbolic loga-
rithm is also designated "loge" and "nat. log."

To convert the common logarithms to hyperbolic logarithms, the
former should be multiplied by 2.30258. To convert hyperbolic loga-
rithms to common logarithms, multiply by 0.43429. These multipliers
will be found of value in cases where hyperbolic logarithms are re-
quired in formulas. Hyperbolic logarithms find extensive use in
higher mathematics.

SECTION II

TABLES OF
COMMON LOGARITHMS

1 TO 10,000

18

No. 53— USE OF LOGARITHMS

H.

L. 0 1 2 3 4

56789

P.P.

100

101
102
103
104

oo ooo 043 087 130 173

432 475 5l8 561 6<H
860 903 945 988 #030
01 284 326 368 410 452
703 745 787 828 870

217 260 303 346 389
647 689 732 775 817
#072 #115 *i57 #199 #242
494 536 578 620 662
912 953 225 #036 #078

i

2

3
4

I

9

i

2

3
4

7
8

9

i

2

3
4

i

9
i

2

3
4

y

S
9

i

2

3

4

i

9

44 43 42

4,4 4,3 4,2
8,8 8,6 8,4
13,2 12,9 I2,6
17,6 17,2 16,8
22,0 21,5 2I,°
26,4 25,8 25,2
30,8 30,1 29,4
35,2 34,4 33,6
39.6 38,7 37,8

41 40 39

•4,1 4,o 3,9
8,2 8,0 7,8
12,3 I2,° IJ,7
16,4 16,0 15,6
20,5 20,0 19,5
24,6 24,0 23,4
28,7 28,0 27,3
32,8 32,0 31,2
36,9 36,0 35,1

38 37 36

3,8 3,7 3,6
7,6 7,4 7,2
11,4 II,1 Io,8
15,2 14,8 14,4
19,0 18,5 18,0

22,8 22,2 21,6
26,6 25,9 25,2
30,4 29,6 28,8

34,2 33,3 32,4

35 34 33

3,5 3-4 3,3
7,0 6,8 6,6
10,5 10,2 9,9
14,0 13,6 13,2
17,5 17,0 16,5

2I,O 2O,4 19,8
24,5 23,8 23,I
28,0 27,2 26,4
31,5 30,6 29,7

32 31 30

3,2 3,1 3,0
6,4 6,2 6,0
9,6 9,3 9,o
12,8 12,4 I2,°
16,0 15,5 15,0
19,2 18,6 18,0
22,4 2I,7 21,0
25,6 24,8 24,0
28,8 27,9 27,0

107

108
109

02 119 l6o 202 243 284

531 572 612 653 694
938 979 *oi9 *o6o #ioo
03342 383 423 463 503
743 782 822 862 902

325 366 407 449 490
735 776 816 857 898
#141 #181 #222 ^262 ^302
543 583 623 663 703

941 981 *O2I #060 #IOO

110

in

112

"3

114

04 139 179 218 258 297
532 571 6 10 650 689
922 961 999 #038 *077
05 308 346 385 423 461
690 729 767 805 843

336 376 415 454 493
727 766 805 844 883
#115 #154 ^192 #231 ,269
500 538 576 614 6^2
88z 918 956 994 *Q32

"5
116
117
118
119

06 070T1J08 145 183 221

446 483 521 558 595
819 856 893 930 967

07.188 225 262 298 .335
555 59i 628 664 700

258 296 333 371 408
633 670 707 744 781
*oo4 #041 #078 .»ii5 #151
372 408 445 482 518
737 773 809 846 882

120

121
122
123
124

918 954 990 #027 #063
08 279 314 350 386 422
636 672 707 743 778
991 #026 *o6i ^096 #132
09 342 377 412 447 482

*099 *I35 #171 *207 *243
458 493 529 565 600
814 849 884 920 955
#167 *202 #237 #272 ^307
5J7 552 587 621 656

125
126
127
128
129

691 726 760 795 830
10 037 072 106 140 175
380 415 449 483 517
721 755 789 823 857
ii 059 093 126 160 193

864 899 934 968 *oo3
209 243 278 312 346
551 585 619 653 687
890 924 958 992 #025
227 261 294 327 361

130

I31

132

133
134

394 428 461 494 528
727 760 793 826 860
12057 090 123 156 189
385 418 450 483 516
710 743 775 808 840

561 594 628 661 ,694
893 926 959 902 #024
222 254 287 360 352
548 581 613 646 678
872 905 937 969 *ooi

135
136
137
138
139

13033 066 098 130 162
354 386 418 450 481
672 704 735 767 799
988 #019 ^051 #082 *H4
H30I 333 304 395 426

194 226 258 290 322

5J3 545 577 609 640
830 862 893 925 956
#145 #176 #208 #239 #270
457 489 520 551 582

140

141
142

143
144

613 644 675 706 737
922 953 983 #014 #045
15229 259 290 320 351-
534 564 594 625 655
836 866 897 927 957

768 799 829 860 891
#076 #106 #137 #108 #198
—381 412 442 473 503
685 7i5 746 776 806
987 #017 #047 #077 *I07

$

147
148
149

16 137 167 197 227 256

435 465 49? 524 554
732 761 791 820 8fo
17026 056 085 114 143
3J9 348 377 406 435

286 316 346 376 406
584 613 643 673 702

879 909 938 967 997
173 202 231 260 289
464 493 522 551 580

150

609 638 667 696 725

754 782 811 840 869

N.

L. 0 1 2 3 4

56789

p.p.

LOGARITHMIC TABLES

19

N.

L. 0 1 2 3 4

56789

P.P.

150

152

17 609 638 667 696 725
898 926 955 984 *oi3
18 184 213 241 270 298
469 498 526 554 583
752 780 808 837 865

754 782 811 840 869

327 355 384 412 441
611 639 667 696 724
893 921 949 977 *005

i

2

3

29 28

2,9 2,8

5,8 5,6
8/7 8,4

156

158
159

19033 061 089 117 14^
312 340 368 396 424
590 618 645 673 700
866 893 921 948 976

20 140 167 194 222 249

173 201 229 257 285
451 479 507 535 562
728 756 783 811 838

^003 ^030 $058 ^085 #112
276 303 330 358 385

4

1

9

11,6 11,2

14,5 14,0

17,4 16,8
20,3 19,6

23,2 22,4

26,1 25,2

160

161
162

163
164

412 439 466 493 520
683 710 737 763 790
952' 978 *oo5 ^032 ^059
21 219 245 272 299 325
484 511 537 564 590

54s 575 602 629 656
817 844 871 898 925

352 378 405 431 458
617 643 669 696 722

I

2

3

27 26

2,7 2,6

5,4 5,2
8,1 7,8

167
168
169

748 775 801 827 854

22 01 1 037 063 089 115
272 298 324 350 376

531 557 583 608 634
789 814 840 866 891

880 906 932 958 985
141 167 194 220 246
401 427 453 479 505
660 686 712 737 763
917 943 968 994 *oi9

4

I

9

10,8 10,4

J3,5 J3,o
16,2 15,6
18,9 18,2

21,6 2O,8

24,3 23,4

170

171
172
173
174.

23045 070 096 121 147
300 325 350 376 401
553 578 603 629 654
805 830 855 880 905
24055 080 105 130 155

172 198 223 249 274
426 452 477 502 528
679 704 729 754 779
930 953 98o *oo5 ^030
i 80 204 229 254 279

25

i 2,5

3 7.5

175
176
177
178
179

304 329 353 378 403
551 576 601 625 650
797 822 846 871 895
25 042 066 091 115 139
285 3io 334 358 382

428 452 477 502 527
674 699 724 748 773
920 944 969 993 *oi8
164 188 212 237 261
406 431 455 479 503

4 I0,°
5 12,5
6 15,0
7 17,5

8 20,0

9 22,5

180

181
182
183
184

527 55i 575 600 624
768 792 816 840 864
26 007 031 055 079 102
245 269 293 316 340
482 505 529 553 576

648 672 696 720 744
888 912 935 959 983

126 I^O 174 198 221

364 387 4ii 433 458
600 623 647 670 694

i

2

24 23

2,4 2,3
4,8 4,6
7,2 6,9

185
186
187
188
189

717 741 7631 788 811
951 973 998 *02i *045
27 184 207 231 254 277
416 439 462 485 508
646 609 692 715 738,

834 858 88 i 905 928

300 323 346 370 393
531 554 577 600 623
761 784 807 830 852

4

1

9

9/6 9,2
12,0 11,5

16*8 16^1
19,2 18,4
21,6 20,7

190

191
192
193
194

875 '898 921 944 967
28 103 126 149 171 194

.33° 353 375 398 421.
556 ^78 601 623 646
780 803 825 847 870

989 #012 $035 #058 #08 1
217 240 262 285 307
443 466 488 511 533
668 691 713 735 758
892 914 937 959 981

i

2

3

22 21

2/2 2,1
4/4 4/2

6/6 6,3

195
196
197
198
199

29 003 026 048 070 092
226 248 270 292 314

447 469 49i 513 533
667 688 710 732 754
885 907 929 951 973

115 137 159 181 203
336 358 380 403 423
557 579 601 623 645
776 798 820 842 863
994 *oi6 #038 *o6o *o8i

4

7
8

9

8/8 8,4
11,0 10,5
13,2 12,6

15,4 i4,7
17,6 16,8
19,8 18,9

200

3q 103 125 146 168 190

211 233 255 276 298

N.

L. 0 1 2 3 4

56789

P.P.

20

No. 53— USE OF LOGARITHMS

N.

L. 0 1 2 3 4

56789

P.P.

200

2OI
2O2
203
204

30 103. 125 146 168 190
320 341 363 384 406
535 557 578 600 621
750 771 792 814 835
963 984 *oo6 #027 ^048

211 233 255 276 298.

428 449 471 492 514
643 664 685 707 728
856 878 899 920 942
#069 #091 *H2 *I33 *I54

22

I 2,
2 4

3! 6,
4 8
5 ii,
6 13,
7 15,
8 17,

9li9,

I

2

3
4

9

I

2

3

4
5
6

7
8

9

I

2

3
4

7
8

9

I

2

3
4

7
8

9

21

2 2,1

4 4,2
6 6,3
8 8,4
0 I0,5

2 12,6

4 14,7
6 16,8
8 18,9

20

2,0

4,o
6,0
8,0

IO,O
I2,O
14,0

16,0
18,0

19

1,9
3,8

%

9,5
",4
13,3
15,2
17,1

18

1,8
3,6

5,4

7,2

9,o
10,8

12,6

14,4
16,2

17

i,7
3,4
5,1
6,8
8,5

10,2

n,9
13,6
15,3

205
206
207
208
209

31 175 197 218 239 260
387 4Q§ 429 450 471
597 618 639 660 681
806 827 848 869 890
32015 035 056 077 098

281 302 323 345 366
492 5*3 534 555 576
702 723 744 765 785
911 931 952 973 994
118 139 100 181 201

210

211
212
213
214

222 243 263 284 305

428 449 469 490 510

634 654 675 695 715
838 858 879 899 919

33 041 062 082 IO2 122

325 346 366 387 408
53i 552 572 593 613
736 756 777 797 818
940 960 980 #00 i #021
143 163 183 203 224

215

216

217

218

219

244 264 284 304 325

445 465 486 506 526
646 666 686 706 726
846 866 885 905 925
34044 064 084 104 124

345 365 385 405 425
546 566 586 606 626
746 766 786 806 826
945 96$ 985 *oo5 *025
143 163 183 203 223

220

221
222
223
224

242 262 282 301 321
439 459 479 498 518
63$ 655 674 694 713
830 850 869 889 908
35 025 044 064 083 102

341 361 380 400 420
537 557 577 596 616
733 753 772 792 811
928 947 967 986 *cx>5
122 141 160 180 199

22|
226
227
228
229

218 238 257 276 295
411 430 449 468 488
603 622 641 660 679
793 813 832 851 870
984 *oo3 *02i ^040 #059

315 334 353 372 392
507 526 545 564 583
698 717 736 755 774
889 908 927 946 965
^078 *097 #116 #135 *I54

230

231
232
233
234

36 173 192 211 229 248

361 380 399 418 436
549 568 586 605 624
736 754 773 791 810
922 940 959 977 996

267 286 305 324 342

455 474 493 5" 53O
642 661 680 698 717
829 847 866 884 903
*oi4 #033 ^051 ^070 #088

235

236

237
238
239

37 107 125 144 162 181
291 310 328 > 346 365

475 493 Sii 530 548
658 676 694 712 731
840 858 876 894 912

199 218 236 254 273
383 401 420 438 457
566 585 603 621 639
749 767 785 803 822
931 949 967 985 *003

240

241
242

243
244

38021 039 057 075 093
202 220 238 256 274

382 399 417 43$ 453
561 578 596 614 632
739 757 775 792 810

112 130 148 166 184
292 310 328 346 364
471 489 507 525 543
650 668 686 703 721
828 846 863 88 i 899

245
246
247
248
249

917 934 952 970 987
39094 in 129 146 164
270 287 305 322 340
445 463 480 498 515
620 637 655 672 690

*oo5 *023 #041 ^058 ^076
182 109 217 235 252
358 37$ 393 4io 428
533 55o 568 58$ 602
707 724 742 759 777

250

794 811 829 846 863

881 898 915 933 950

N.

L. 0 1 2 3 4

56789

P.P.

LOGARITHMIC TABLES

21

N.

L. 0 1 2 3 4

56789

I

.P.

250

251
252

253
254

39 794 811 829 846 863
967 985 *002 *oi9 ,,037
40 140 157 175 192 209
312 329 346 364 381
483 500 518 535 552

88 i 898 915 933 950
*054 ^071 #088 *io6 #123
226 243 261 278 295
398 415. 432 449 466
569 586 603 620 637

i

2

3

18

i 8
3>

5/4

in

3

259

654 671 688 705 722
824 841 858 875 892
993 *oio #027 *044 *o6i
41 162 179 196 212 229
330 347 363 380 397

739, 756 773 79° 807
909 926 943 960 976
#078 *095 *in *I28 *I45
246 263 280 296 313
414 430 447 464 481

4

8
9

7/2

9/o
10,8

12,6

14/4
16,2

260

261
262
263
264

497 514 53i 547 564
664 68 i 697 714 731
830 847 863 880 896
996 *oi2 #029 *04$ *o62
42 160 177 193 210 226

581 597 614 631 647
747 764 780 797 814
913 929 946 963 979
#078 *095 *m *I27 *I44
243 259 275 292 308

i

2

3

17

i/7
3/4

5/1

267
268
269

325 34i 357 374 39P
488 504 521 537 553
651 667 684 700 716
813 830 846 862 878
975 991 *oo8 *024 ^040

406 423 439 455 472
570 586 602 619 635
732 749 765 78i 797
894 911 927 943 959
#056 #072 #088 #104 #120

4

I

7
8

9

6/8
8,5

IO,2

«/9
13/6
15/3

270

271
272

273
274

43 136 152 169 185 201
•297 3J3 329 345 361
457 473 489 5°5" 52i
616 632 648 664 680
775 79i 807 823 838

217 233 249 265 281
377 393 409 425 44i
537 553 569 584 600
696 712 727 743 759
854 870 886 902 917

I

2

3

16

1/6
S/2
4/8

275
276

277
278
279

933 949 965 981 996

44091 107 122 138 154

248 264 . 279 295 311
404 420 436 .451 467

560 576 592 607 623

*oi2 #028 *044 #059 *075
170 i8£ 201 217 232
326 342 358 373 389
483 498 514 529 545
638 654 669 685 700

4

i

i

6/4
8/0
9/6

IIj2
12,8
14/4

280

281
282
283
284

716 731 747 762 778
871 886 902 917 932
45 025 040 056 071; 086
17^ 194 209 225 240
332 347 ,362 378 393

793 809 824 840 855
948 963 979 994 *oio
102 117 133 148 163
255 271 286 301 317
408 423 439 454 469

i

2

3

15

i/S
3/o
4/5

31

287
288
289

484 500 515 530 545
637 652 667 ^82 697
788 803 818 834 849
939 954 969 984 *ooo
46090 105 120 135 150

561 576 591 606 621
712 728 743 758 773
864 879 894 909 924
*oi5 #030 ^045: *o6o #075
165" 180 195 210 225

4
5
-6

7
8

9

6/0
7/5
9/o
10,5

12,0

13/5

290

291
292
293
294

240 255 270 285 300
389 404 419 434 449
538 553 568 583 598
687 702 716 731 746
835 850 864 879 894

315 330 345 359 374
464 479 494 509 523
613 627 642 657 672
761 776 790 805 820
909 923 938 953 967

I

3

14

i/4
2/8
4/2

297
298
299

982 997 #012 ^026 ^041
47 129 144 159 173 188
276 290 305 319 334
422 436 451 465 480
567 582 596 611 625

#056 #070 #085 #100 *ii4
202 217 232 246 261
349 363 378 391 4<V
494 509 524 538 553
640 654 669 683 698

4

1

9

5/6
7/o
8/4
9/8

11,2
12,6

300

712 727 741 756 770

784 799 813 828 842

N.

L. 0 1 2 3 4

56789

I

». P.

22

No. 53— USE OF LOGARITHMS

N.

L. 0 1 2 3 4

56789

I

.P.

300

301
302
303
304

47712 727 741 756 770
857 871 885 900 914
48 ooi 015 029 044 058

144 159 173 187 202

287 302 316 330 344

784 799 813 828 842
929 943 958 972 986
073 087 101 116 130
216 230 244 259 273
359 373 387 401 416

15

305
306
3°7
308
309

430 444 458 473 487
572 586 601 615 629
714 728 742, 756 770
855 869 883 897 911
996 *oio #024 ^038 #052

501 5!5 530 544 558
643 657 671 686 700
785 799 813 827 841
926 940 954 968 982
#066 *o8o ^094 *io8 *I22

2

3
4

7*5

9/o

310

312
3i4

49 136 150 164 178 192
276 290 304 318 332
415 429 443 457 471
554 568 582 596 610
693 707 721 734 748

206 220 234 248 262
346 360 374 388 402
485 499 513 527 541
624 638 651 66 c; 679
762 776 790 803 817

9

12,0
13/5

315

316

3J7
3i9

831 845 859 872 886
969 982 996 *oio #024
50 106 120 133 147 161
243 256 270 284 297
379 393 406 420 433

900 914 927 941 955
#037 #051 #065 #079 *O92
174 188 202 215 229
311 325 338 352 365
447 461 474 488 501

i

2

4

14

2^8
4/2

5/6

320

321

322
323
324

515 529 542 556 569
651 664 678 691 705
786 799 813 826 840
920 934 947 961 974
51055 068 081 095 108

583 596 610 623 637
718 732 745 759 772
853 866 880 893 907
987 *ooi #014 #028 #041

121 135 148 162 175

i

7

8

9

8,4
9/8

11,2
12,6

325

326

329

i 88 202 215 228 242
322 335 348 362 375
455 468 481 495 508
587 601 614 627 640
720 733 746 759 772

255 268 282 295 308
388 402 415 428 441

521 534 548 561 574
654 667 680 693 706
786 799 812 825 838

13

2/6

330

332
333
334

851 865 878 891 904
983 996 *oo9 *022 #035
52 114 127 140 153 166
244 257 270 284 297
375 388 401 414 427

917 930 943 957 970
#048 *o6i #075 #088 #101
179 192 205 218 231
310 323 336 349 362
440 453 466 479 492

3
4

1

3/9

5/2

§

9/1
10,4

337
338
339

504 517 530 543 556
634 647 660 673 686
763 776 789 802 815
892 905 917 930 943
53020 033 046 058 071

569 582 595 608 621
699 711 724 737 750
827 840 853 866 879
956 969 982 994 #007
084 097 no 122 135

9

«,7
1 2

340

342
343

344

148 161 173 186 199
27^ 288 301 314 326
403 415 428 441 453
529 542 555 567 580
656 668 68 i 694 706

212 224 237 250 263

339 352 364 377 390
466 479 491 504 517
593 605 618 631 643
719 732 744 757 769

i

2

3
4

1,2
2,4

3,6
4,8
6,0

itl
fg

349

782 794 807 820 832
908 920 933 945 958
54033 045 058 070 083
158 170 183 195 208
283 295 307 320 332

845 857 870 882 895
970 983 995 *oo8 *020
095 108 120 133 145

220 233 245 258 270

345 357 370 382 394

7
8

9

7,2
8/4
9/6
10,8

350

407 419 432 444 456

469 481 494 506 518

N.

L. 0 1 2 3 4

56789

I

'.P.

LOGARITHMIC TABLES

N.

L. 0 1 2 3 4

56789

I

.P.

350

352
353
354

54407 419 432 444 456
53i 543 555 568 580
654 667 679 691 704
777 790 802 814 827
900 913 923 937 949

469 481 494 506 518

593 603 617 630 642
716 728 741 753 765
839 851 864 876 888
962 974 986 998 *oi i

13

355
356
357
358
359

55023 035 047 060 072
143 157 169 182 194
267 279 291 303 315
388 400 413 425 437
509 522 534 546 558

084 096 108 121 133
206 218 230 242 253
328 340 352 364 376

449 461 473 485 497
570 582 594 606 618

2

3
*

7

2,6

3,9

5,2

7^8

9 *

360

361
362
363
364

630 642 654 666 678

75i 763 775 787 799
871 883 893 907 919
991 #003 #015 #027 $038
56 no 122 134 146 158

691 703 713 727 739
811 823 833 847 859

93i 943 955 967 979
#050 #062 #074 #086 #098
170 182 194 205 217

8
9

10,4
11,7

365
366

367
368

369

229 241 253 263 277
348 360 372 384 396
467 478 490 502 514
583 597 608 620 632
703 714 726 738 750

289 301 312 324 336
407 419 431 443 453
526 538 549 561 573
644 656 667 679 091
761 773 783 797 808

i

2

3
4

12

1,2

2,4

3,6

4,8

370

372
373
374

820 832 844 855 867

937 949 961 972 984
57 054 066 078 089 101
171 183 194 206 217
287 299 310 322 334

879 891 902 914 926
996 j|(Oo8 ^019 #031 #043
113 124 136 148 '159
229 241 252 264 276
345 357 368 380 392

I

9

6,0

7,2

8,4
9,6
10,8

375
376
377
378
379

403 415 426 438 449

519 530 542 553 563
634 646 657 669 680
749 761 772 784 795
864 8^5 887 898 910

461 473 484 496 507
576 588 600 6n 623
692 703 715 726 738
807 818 830 841 852
921 933 944 955 967

i

11

1,1

380

382
383
384

978 990 *OOI #013 *024

58 092 104 115 127 138
206 218 229 240 252
320 331 343 354 365
433 444 456 467 478

*O35 *O47 #058 4*070 jifOSi
149 161 172 184 193
263 274 286 297 309
377 388 399 4io 422
490 501 512 524 533

3
4

I

3,3
4,4

7,7
8,8

to to to to to

CO CO CO CO OO

vo oovi ONtn

546 557 569 58o 591
659 670 68 i 692 704
771 782 794 803 816
883 894 906 917 928
993 *oo6 *oi7 #028 #040

602 614^ 623 636 647
713 726" 737 749 760
827 838 850 861 872
939 95° 96i 973 984
^051 #062 #073 ^084 #095

9

9,9
1 ft

390

39i
392
393

394

59 106 118 129 140 151
218 229 240 251 262
329 340 35i 362 373
439 450 461 472 483
530 561 572 583 594

162 173 184 195 207
273 284 295 306 318
384 395 406 417 428
494 506 517 528 539
603 616 627 638 649

i

2

3
4

1,0

2,0
3,o
4,o
5,°

395
396
397
398
399

660 671 682 693 704
770 780 791 802 813
879 890 901 912 923
988 999 *oio *02i ^032
60097 108 119 130 141

713 726 737 748 759
824 835 846 857 868
934 945 956 966 977
^043 #054 #063 #076 *o86
152 163 173 184 195

7
8

9

6,0
7,o
8,0
9,o

400

206 217 228 239 249

260 271 282 293 304

N.

L. 0 1 2 3 4

56789

1

». P.

24

. 53— USE OF LOGARITHMS

N.

L. 0 1 2 3 4

56789

P.P.

400

401
402
403
404

60 206 217 228 239 249
314 325 336 347 358
423 433 444 455 4&6
531 54i 552 563 574
638 649 660 670 68 i

260 271 282 293 304
369 379 390 401 412
477 487 498 509 520
584 595 606 617 627
692 703 713 724 735

i

2

3
4

I

9
i

2

3
4

9

i

2

3

4

!

8
9

11

1,1

2,2

3,3
4,4

4;I
1%

9,9
10

!,0
2,O

3,o
4,o
5,°
6,0
7,o
8,0
9,o

9

o,9

1,8

2,7

3,6
4,5
5,4
6,3

7,2

8,1

405
406

407
408
409

746 756 767 778 788
853 863 874 885 895
959 970 981 991 *002
61 066 077 087 098 109

172 183 194 204 2lf

799 810 821 831 842
906 917 927 938 949
#013 #023 #034 *045 *055
119 130 140 151 162
225 236 247 257 268

410

411
412

4i3
414

278 289 300 310 321
384 395 405 416 426
490 500 511 521 532
595 606 616 627 637
700 711 721 731 742

331 342 352 363 374

437 448 458 469 479

542 553 563 574 584
648 658 669 679 690

752 763 773 784 794

4i|
416

417
418
419

805 815 826 836 847
909 920 930 941 951
62 014 024 034 045 055
118 128 138 149 159

221 232 242 252 263

857 868 878 888 899
962 972 982 993 *oo3
066 076 086 097 107

I7O l8o 190 2OI 211
273 284 294 304 315

420

421
422
423
424

325 335 346 356 366
428 439 449 459 469
53i 542 552 562 572
634 644 655 665 67^
737 747 757 767 778

377 387 397 408 418
480 490 5°° 511 521
583 593 603 613 624
685 696 706 716 726
788 798 808 818 829

425
426

427
428
429

839 849 859 870 880
941 951 961 972 982
63043 053 063 073 083
144 155 165 175 185
246 256 266 276 286

890 900 910 921 931

992 #002 *OI2 *022 #033
094 104 114 124 134
195 2O5 2l5 225 236

296 306 317 327 337

430

43i

432
433

434

347 357 367 377 387
448 458 468 478 488
548 558 568 579 589
649 659 669 679 689
749 759 769 779 789

397 407 417 428 438
498 508 518 528 538
599 609 619 629 639

699 709 719 729 739
799 809 819 829 839

435
436
437
438
439

849 859 869 879 889
949 959 969 979 988
64 048 058 068 078 088
147 157 167 177 187
246 256 266 276 286

899 909 919 929 939
998 #008 #018 #028 #038
098 108 118 128 137
197 207 217 227 237
296 306 316 326 335

440

441
442
443
444

345 355 365 375 385
444 454 464 473 483
542 552 562 572 582
640 650 660 670 680
738 748 758 768 777

395 404 4H 424 434
493 503 513 523 532
591 601 611 621 631
689 699 709 719 729
787 797 807 816 826

445
446
447
448
449

836 846 856 865 875

933 943 953 963 972
65 031 040 o^o 060 070
128 137 147 157 167
225 234 244 254 263

885 895 904 914 924

982 992 *002 *OII #021

079 089 099 108 118
176 186 196 205 215
273 283 292 302 312

450

321 331 341 350 360

369 379 389 398 408

N.

L. 0 1 2 3 4

56789

P.P.

LOGARITHMIC TABLES

25

N.

L. 0 1 2 3 4

56789

P.P.

450

45i

452
453
454

65321 331 341 350 360
418 427 437 447 456

514 523 533 543 552
610 619 629 639 648
706 715 725 734 744

369 379 389 398 408
466 475 485 495 504
562 571 581 591 600
658 667 677 686 696
753 763 772 782 792

i

2

3

4

i

9

i

2

3
4

7
8

9

i

2

3

4
5
6

I

9

10

1,0

2,0
3/°
4/o
5/°
6,0
7/o
8,0
9/o

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/o
4,«

5/6
6,4

7/2

455
456
457
458
459

801 811 820 830 839
896 906 916 92^ 935

992 *OOI *OII #020 ^030

66 087 096 106 115 124

l8l 191 200 210 219

849 858 868 877 887
944 954 963 973 982
*039 #049^058 *o68 *077
134 143 153 162 172
229 238 247 257 266

460

461
462
463
464

276 285 295 304 314
370 380 389 398 408

464 474 483 492 502
558 567 577 586 596
652 661 671 680 689

323 332 342 351 361
417 427 436 445 455
511 52* 530 539 549
605 614 624 633 642
699 708 717 727 736

465
466
467
468
469

74o 755 764 773 783
839 848 857 867 876
932 941 950 960 969
67025 034 043 052 062
117 127 136 145 154

792 801 811 820 829
88^ 894 904 913 922
978 987 997 *oo6 #015
071 080 089 099 108

164 173 182 191 201

470

47i

472
473
474

210 219 228 237 247

302 311 321 330 339
394 403 413 422 431
486 495 504 514 523
578 587 596 605 614

256 265 274 284 293

348 357 367 376 385
440 449 459 468 477
532 54i 550 560 569
624 633 642 651 660

475
476
477
478

479

669 679 688 697 706
761 770 779 788 797
852 861 870 879 888
943 952 961 970 979
68 034 043 052 061 070

7*£ 724 733 742 752
806 815 825 834 843
897 906 916 925 934
988 997 #006 *oi5 *024
079 088 097 106 115

480

481
482

483
484

124 133 142 151 160
215 224 233 242 251
305 314 323 332 341
395 404 413 422 431
485 494 502 511 520

169 178 187 196 205
260 269 278 287 296
35o 359 368 377 386
440 449 458 467 476
529 538 547 556 565

485
486

487
488
489

574 583 592 601 610
664 673 68 i 690 699
753 762 771 780 789
842 851 860 869 878
931 940 949 958 966

619 628 637 646 655
708 717 726 735 744
797 806 815 824 833
886 895 904 913 922
975 984 993 *oo2 *on

490

491

492
493
494

69 020 028 037 046 055

108 117 126 135 144
197 205 214 223 232
285 294 302 311 320
373 38i 390 399 408

064 073 082 090 099
152 161 170 179 188
241 249 258 267 276

329 338 346 35S 364
417 425 434 443 452

495
496
497
498
499

461 469 478 487 496
548 557 566 574 583
636 644 653 662 671
723 732 740 749 758
810 819 827 836 845

504 513 522 531 539

592 601 609 618 627
679 688 697 705 714
767 77$ 784 793 801
854 862 871 880 888

500

897 906 914 923 932

940 949 958 966 975

N.

L. 0 1 2 3 4

56789

P.P.

26

. 53— USE OF LOGARITHMS

N.

L. 0 1 2 3 4

56789

P.P.

500

501
502

5°3
504

69897 906 914 923 932
984 992 #ooi *oio *oi8-
70 070 079 088 096 105
157, 165 174 183 191
243 252 260 269 278

940 949 958 966 975
*027 ^036 *044 *053 *o62

114 122 131 140 148
200 209 217 226 234
286 295 303 312 321

9

i 0,9

2 1,8

3 2,7
4 3,6
5 4,5
6 5,4
7| 6,3
8 7,2
9! 8,1

8

i 0,8

2 1,6

3;2,4
4:3,2
5 4,o
6 4,8
7 5,6
8j6,4
9l7,2

f

7

i 0,7
2 1,4
3 2,1
4.2,8
5 3,5
6 4,2
7 4,9
8|5,6
9)6,3

507
508

509

329 338 346 355 364
415 424 432 441 449
501 509 518 526 535
586 595 603 612 621
672 680 689 697 706

372 381 389 398 406

458 467 475 484 492
544 552 561 569 578
629 638 64^ 65 s 663
714 723 731 740 749

510

5"
512
513
5i4

757 766 774 783 791
842 851 859 868 876
927 935 944 952 961

71 OI2 O2O 029 037 046
096 IO5 113 122 130

800 808 817 825 834
885 893 902 910 919
969 978 986 995 *oo3
054 063 071 079 088
139 147 155 164 172

5i5
5i6
5i7
5i8
5i9

181 189 198 206 214

265 273 282 29O 299

349 357 366 374 383
433 44i 45° 458 466
5J7 525 533 542 550

223 231 240 248 257
307 315 324 332 341
391 399 408 416 425
475 483 492 500 508
559 567 575 584 592

520

521
522
523

524

600 609 617 625 634
684 692 700 709 717
767 775 784 792 800
850 858 867 875 883
933 94i 950 958 966

642 650 659 667 675

725 734 742 750 759
809 817 825 834 842
892 900 908 917 925
975 983 99i 999 *oo8

525
526

527
528

529

72 016 024 032 041 049
099 107 115 123 132
181 189 198 206 214
263 272 280 288 296
346 354 362 370 378

057 066 074 082 090
140 148 156 165 173

222 230 239 247 255

304 313 321 329 337
387 395 4°3 4ii 419

530

53i

532
533
534

428 436 444 452 460
509 518 526 534 542
591 599 607 616 624
673 68 i 689 697 705
754 762 770 779, 787

469 477 485 493 501
550 558 567 575 583
632 640 648 656 665
713 722 730 738 746
795, 803 811 819 827

535
536
537
538
539

835 843 852 860 868
916 925 933 941 949
997 *oo6 #014 *O22 #030
73 078 086 094 102 in
159 167 175 183 191

876 884 892 900 908

957 96=; 973 981 989
^038 ^046 *054 *o62 ^070
119 127 135 143 151
199 207 215 223 231

540

54i
542
543
544

239 247 255 263 272
320 328 336 344 352
400 408 416 424 432
480 488 496 504 512
560 568 576 584 592

280 288 296 304 312
360 368 376 384 392
440 448 456 464 472
520 528 536 544 552
600 608 616 624 632

545
546
547
548
549

640 648 656 664 672
719 727 735 743 751
799 807 815 823 830
878 886 894 902 910
957 965 973 98i 989

679 687 695 703 711

759 767 775 783 791
838 846 854 862 870
918 926 933 941 949
997 »oo5 *oi3 *020 *028

550

74 036 044 052 060 068

076 084 092 099 107

N.

L. 0 1 2 3 4

56789

P.P.

LOGARITHMIC TABLES

27

N.

L. 0 1 2 3 4

56789

P.P.

550

55i
552
553
554

74 036 044 052 060 068
115 123 131 139 147
194 202 210 218 22=;
273 280 288 296 304
35i 359 367 374 382

076 084 092 099 107
155 162 170 178 186
233 241 249 257 265
312 320 327 335 343
390 398 406 414 421

8

i o,3

2 1,6

3 2,4
4 S,2
5 4,o
6 4,8
75,6
8 6,4

9 7,2

7

I 0,7
2 1,4
3 2,1
4 2,8
5 3,5
6 4,2
7 4,9
8 5,6
9 6,3

555
556
557
558
559

429 437 445 453 461
507 5J5 523 53i 539
586 593 601 609 617
663 671 679 687 695
741 749 757 764 772

468 476 484 492 500
547 554 5°2 57° 578
624 632 640 648 656
702 710 718 726 733
780 788. 796 803 8 ii

560

56i
562

563
564

819 827 834 842 850
896 904 912 920 927
974 981 989 997 *oo5
75 051 059 066 074 082
128 136 143 151 159

858 865 873 88 i 889
935 943 950 958 966
#012 *o^ *028 *035 *043
089 097 105 113 120
166 174 182 189 197

567

568
569

20^ 213 220 228 236
282 289 297 305 312

358 366 374 38i 389
435 442 450 458. 465
511 519 526 534 542

243 251 259 266 274
320 328 335 343 351
397 404 412 420 427
473 481 488 496 504
549 557 565 572 580

570

57i

572
573
574

587 595 603 610 6i£
664 671 679 686 094
740 747 755 762 770
815 823 831 838 846
891 899 906 914 921

626 633 641 648 656
702 709 717 724 732
778 785 793 800 808
853 861 868 876 884
929 937 944 952 959

§3

577
578
579

967 974 982 989 997
76 042 050 057 - 065 072
118 12=; 133 140 148

193 200 208 215 223

268 275 283 290 298

*oo5 *oi2 *020 #027 #035
080 087 095 103 no
J5S 163 17° J78 185
230 238 245 253 260
305 313 320 328 335

580

58i
582

583
584

343 35° 358 365 373
418 425 433 440 448
492 500 507 515 522
567 574 582 589 597
641 649 656 664 671

380 388 395 403 410

455 462 470 477 485

530 537 545 552 559
604 612 619 626 634
678 686 693 701 708

585
586
587
588
589

716 723 730 738 745
790 797 805 812 819
864 871 879 886 893
938 945 953 96o 967
77 012 019 026 034 041

753 76o 768 775 782
827 834 842 849 856
901 908 916 923 930
975 982 989 997 *oo4
048 056 063 070 078

590

59i
592
593
594

°85 093 loo 107 115
159 166 173 181 188
232 240 247 254 262
305 313 320 327 335
379 386 393 4°i 408

122 129 137 144 151
195 203 210 217 225
269 276 283 291 298

342 349 357 364 371
415 422 430 437 444

597
598
599

452 459 466 474 481
525 532 539 546 554
597 605 612 619- 627
670 677 685 692 699
743 75o 757 764 772

488 495 503 510 517
561 568 576 583 590
634 641 648 656 663
706 714 721 728 735
779 786 793 801 808

600

815 822 830 837 844

851 859 866 873 880

N.

L. 0 1 2 3 4

56789

P.P.

28

No. 53— USE OF LOGARITHMS

N.

L. 0 1 2 3 4

56789

P.P.

600

601
602

603
604

77815 822 830 837 844
887 895 902 909 916
960 967 974 981 988
78 032 039 046 053 061
104 in 118 125 132

851 859 866 873 880
924 931 938 945 952
996 *oo3 #010 *oi7 *025
068 075 082 089 097
140 147 154 161 168

i

2

3
4

9

i

2

3
4

9

i

2

3
4

i

7
8

9

8

0,8
1,6

2,4
3,2

4,0

$

6,4
7,2

7

o,7
i,4

2,1

2,8
3,5
4,2

$

6,3

6

0,6

1,2

1,8

2,4

i:!

%

5,4

607
608
609

176 183 190 197 204
247 254 262 269 276
319 326 333 340 347
390 398 405 412 419
462 469 476 483 490

211 219 226 233 240

283 290 297 305 312
355 362 369 376 383
426 433 440 447 455
497 5°4 5J2 5J9 526

610

611
612
613
614

533 540 547 554 561
604 611 618 625 633
675 682 689 696 704
746 753 76o 767 774
817 824 831 838 845

569 576 583 590 597
640 647 654 -66 I 668
711 718 725 732 739
#81 789 796 803 810
852 859 866 873 880

615
616
617
618
619

888 895 902 909 916
958 965 972 979 986
79029 036 043 050 057
099 106 113 120 127
169 176 183 190 197

923 930 937 944 951
993 *ooo #007 #014 *02i
064 071 078 085 092
134 141 148 155 162
204 211 218 22=; 232

620

621
622
623
624

239 246 253 260 267
309 316 323 330 337
379 386 393 400 407
449 456 463 470 477
518 525 532 539 546

274 281 288 295 302
344 35i 358 365 372
414 421 428 435 442
484 491 498 505 511
553 56o 567 574 58i

625
626
627
628
629

588 595 602 609 616
657 664 671 678 68$

727 734 74i 748 754
796 803 810 817 824
865 872 879 886 893

623 630 637 644 650
692 699 706 713 720
761 768 775 782 789
831 837 844 851 858
900 906 913 920 927

630

631
632
633
634

934 941 948 955 962
80003 OI° OI7 024 030
072 079 085 092 099
140 147 154 161 168
209 216 223 229 236

969 975 982 989 996
037 044 051 058 065
106 113 120 127 134
175 182 188 195 202

243 250 257 264 271

635

636

637
638
639

277 284 291 298 305
346 353 359 366 373
414 421 428 434 441
482 489 496 502 509
550 557 564 570 577

312 318 325 332 339
380 387 393 400 407
448 455 462 468 475
516 523 530 536 543
584 591 598 604 611

640

641
642

643
644

618 625 632 638 645
686 693 699 706 713
754 760 767 774 781
821 828 835 841 848
889 895 902 909 916

652 659 665 672 679
720 726 733 740 747
787 794 801 808 814
855 862 868 875 882
922 929 936 943 949

645
646

647
648

649

956 963 969 976 983
8 1 023 030 037 043 050
090 097 104 in 117
158 164 171 178 184
224 231 238 245 251

990 996 #003 #010 #017
057 064 070 077 084
124 131 137 . 144 151

191 198 2O4 211 2l8
258 265 271 278 285

650

291 298 305 311 318

325 331 338 345 35i

N.

L. 0 1 2 3 4

56789

P.P.

LOGARITHMIC TABLES

29

N.

L. 0 1 2 3 4

56789

P.P.

650

651

652

653
654

81 291 298 305 311 318
358 365 37i 378 385
425 431 438 445 451
491 498 505 511 518
558 564 571 578 584

325 33i 338 345 35i
391 398 405 411 418
458 465 471 478 485

525 53i 538 544 55i
591 598 604 6n 617

7
I 0,7
2 1,4
3 2,1
4 2,8
5 3/5
6 4,2

7 4/9
8 5/6
9 6,3

6

i 0,6

2 1,2

3 1/8
4 2,4
5 3/o
6 3/6
7 4/2
8 4/8
9 5/4

&
656

657
658

659

624 631 637 644 651
690 697 704 710 717
757 763 770 776 783
823 829 836 842 849
889 895 902 908 915

657 664 671 677 684

723 730 737 743 75°
790 796 803 809 816
856 862 869 875 882
921 928 935 941 948

660

661
662
663
664

954 961 968 974 981

82 020 027 033 040 046
086 092 099 105 112
151 158 164 171 178
217 223 230 236 243

987 994 *ooo 3*007 *oi4
053 060 066 073 079
119 125 132 138 145

184 igi 197 204 210
249 256 263 269 276

665
666
667
668
669

282 289 295 302 308

347 354 360 367 373
413 419 426 432 439
478 484 491 497 504
543 549 556 562 569

315 321 328 334 341
380 387 393 400 406
445 452 458 465 471
510 517 523 530 536
575 582 588 595 601

670

S1
£2

3S

607 614 620 627 633
672 679 685 692 698

737 743 75° 756 763
802 808 814 821 827
866 872 879 885 892

640 646 653 659 666
705 711 718 724 730
769 776 782 789 795
834 840 847 853 860
898 905 911 918 924

%i
677
678
679

930 937 943 95o 956
995 *ooi *oo8 ^014 *02o
83059 065 072 078 085
123 129 136 142 149

187 193 200 206 213

963 969 975 982 988
*027 *033 #040 #046 3,052
091 097 104 no 117
155 161 168 174 181
219 225 232 238 245

680

681
682
683
684

251 257 264 270 276
315 321 327 334 340
378 385 39i 398 404
442 448 455 461 467
506 512 518 525 531

283 289 296 302 308
347 353 359 366 372
410 417 423 429 436
474 480 487 493 499
537 544 550 556 563

685
686

688
689

569 575 582 588 594
632 639 645 651 658
696 702 708 715 721

759 765 77i 778 784
822 828 835 841 847

601 607 613 620 626
664 670 677 683 689

727 734 740 746 753
790 797 803 809 816
853 860 866 872 879

690

691
692

693
694

885 891 897 904 910
948 954 96o 967 973
84 on 017 023 029 036
073' 080 086 092 098
136 142 148 155 161

916 923 929 935 942
979 985 992 998 *oo4
042 048 055 061 067
105 in 117 123 130
167 173 180 186 192

695

§7
698
699

198 205 211 217 223

261 267 273 280 286

323 330 336 342 348

386 392 398 404 410
448 454 460 466 473

230 236 242 248 255
292 298 305 311 317
354 361 367 373 379
417 423 429 435 442
479 485 491 497 504

700

510 516 522 528 535

54i 547 553 559 566

N.

L. 0 1 2 3 4

56789

P.P.

. 53— USE OF LOGARITHMS

N.

L. 0 1 2 3 4

56789

P.P.

700

701
702

703
704

84510 516 522 528 535
572 578 584 590 597
634 640 646 652 658
696 702 708 714 720
757 763 770 776 782

54i 547 553 559 S66
603 609 615 621 628
665 671 677 683 689
726 733 739 745 75 i
788 794 800 807 813

i

8
3
4

I
I

9

i

2

3
4

7
8

9

I

2

3

4

7
8

9

7

o/7
1/4

2,1

2,8
3/5

4/2

P

6,3

6

0,6

1,2

1/8

2/4

3/o
3/6

4'o

4/8
5/4

5

o/S

1,0

i/5

2,0
2/5

3/o
3,5
4/o
4/5

&

707
708
709

819 825 831 837 844
880 887 893 899 905
942 948 954 960 967
85 003 009 016 022 028
065 071 077 083 089

850 856 862 868 874
911 917 924 930 936

973 979 985 991 997
034 040 046 052 058

095 101 107 114 120

710

711
712
713
7H

126 132 138 144 150

187 193 199 205 211

248 254 260 266 272
309 315 321 327 333
370 376 382 388 394

156 163 169 175 181
217 224 230 236 242
278 285 291 297 303

339 34$ 352 358 364
400 406 412 418 425

7i5
716
717

718
719

43i 43" 443 449 45?
491 497 503 509 516

552 558 564 570 576
612 618 625 631 637
673 6^ 685 691 697

461 467 473 479 485
522 528 534 540 546
582 588 594 600 606
643 649 655 661 667
703 709 715 721 727

720

721
722

723

724

733 739 745 75* 757
794 800. 806 812 818
854 860 866 872 878
914 920 926 932 938
974 980 986 992 998

763 769 775 781 788
824 830 836 842 848
884 890 896 902 908
944 950 956 962 968

*OO4 #OIO *Ol6 *O22 *O28

72|
726

728
729

86 034 040 046 052 058
094 loo 1 06 112 118
153 159 165 171 177
213 219 225 231 237
273 279 285 291 297

064 070 076 082 088
124 130 136 141 147
183 189 I9j 2OI 2O7
243 249 255 26l 267
303 308 314 320 326

730

73i
732
733
734

332 338 344 350 356
392 398 404^ 410 415
451 457 463 469 475
510 516 522 528 534
570 576 581 587 593

362 368 374 380 386
421 427 433 439 445
481 487 493 499 504
540 546 552 558 564
599 6oj 611 617 623

735
736
737
738
739

629 635 641 646 652
688 694 700 705 711

747 753 759 764 77°
806 812 817 823 829
864 870 876 882 888

658 664 670 676 682
717 723 729 735 741
776 782 788 794 800
835 841 847 853 859
894 900 906 911 917

740

74i
742

743
744

923 929 935 941 947

982 988 994 999 *oo$
87 040 046 052 058 064

099 105 III Il6 122

157 163 169 175 181

953 958 964 970 976
*on *oi7 #023 #029 ^035
070 07=; 08 1 087 093
128 134 140 146 151
186 192 198 204 210

745
746

747
748

749

' 2l6 221 227 233 239
274 280 286 291 297

332 338 344 349 355
390 396 402 408 413
448 454 460 466 471

245 251 256 262 268
3°3 309 3J3 320 326
361 367 373 379 384
419 425 431 437 442
477 483 489 493 500

750

506 512 518 523 529

535 54i 547 552 558

N.

L. 0 1 2 3 4

56789

P.P.

LOGARITHMIC TABLES

31

N.

L. 0 1 2 3 4

56789

P.P.

750

75i
752
753
754

87506 512 518 523 529
564 57° 576 581 587
622 628 633 639 645
679 685 691 697 703
737 743 749 754 760

535 54i 547 552 558
593 599 604 610 616
651 656 662 668 674
708 714 720 726 731
766 772 777 783 789

6

i 0,6

2 1,2

3 1,8
4 2,4

I 3'2
6 3,6

7 4,2
8 4,8

9 5/4

5

i o/5

2 1,0

3 i/S
4 2,0
5 2,5
6 3,o
7 3,5
8 4,0

9 4/5

757
758
759

793 800 806 812 818
852 858 864 869 875
910 915 921 927 933
967 973 978 984 990
88 024 030 036 041 047

823 829 833 841 846
881 887 892 898 904

938 944 950 955 961
996 *ooi *007 *oi3 *oi8
053 058 064 070 076

760

761
702

763
764

08 i 087 093 098 104
138 144 150 156 161
195 201 207 213 218
252 258 264 270 275
309 3*5 321 326 332

no 116 121 127 133
167 173 178 184 190
224 230 235 241 247
281 287 292 298 304
338 343 349 355 360

7S

766

769

366 372 377 383 389
423 429 434 440 446
480 485 491 497 502
536 542 547 553 559
593 598 604 610 615

393 400 406 412 417
451 457 463 468 474
508 513 519 523 530
564 570 576 581 587
621 627 632 638 643

770

771
772
773

774

649 655 660 666 672
705 711 717 722 728
762 767 773 779 784
818 824 829 833 840
874 880 885 891 897

677 683 689 694 700

734 739 745 75° 75$
790 795 801 807 812
846 852 857 863 868
902 908 913 919 923

775
776

777
778

779

930 936 941 947 953
986 992 997 *oo3 *oog
89 042 048 053 059 064

098 104 109 113 120
154 159 163 170 176

958 964 969 975 98i
*oi4 *02o *025 ^031 ^037
070 076 08 i 087 092

I2O 131 137 143 148
l82 187 193 198 2O4

780

781
782
783
784

2O9 215 221 226 232
265 271 276 282 287

321 326 332 337 343
376 382 387 393 398
432 437 443 448 454

237 243 248 254 26O
293 298 304 310 315

348 354 360 365 371
404 409 413 421 426
459 463 470 476 481

785
786

787
788
789

487 492 498 504 509.
542 548 553 559 564
597 603 609 614 620
653 658 664 669 673
708 713 719 724 730

513 520 526 531 537
57° 575 581 586 592
625 631 636 642 647
680 686 691 697 702
735 741 746 752 757

790

791
792
793
794

763 768 774 779 783
818 823 829 834 840
873 878 883 889 894

927 933 938 944 949
982 988 993 998 *004

790 796 801 807 812
845 851 856 862 867
900 905 911 916 922
953 960 966 971 977

*009 *OI3 *020 *026 *03 I

795
796

797
798

799

90037 042 048 053 059
091 097 102 108 113
146 151 157 162 168

200 206 211 217 222
253 260 266 271 276

064 069 073 080 086
119 124 129 233 140
173 179 184 189 193
227 233 238 244 249
282 287 293 298 304

800

309 314 320 325 331

336 342 347 352 358

N.

L. 0 1 2 3 4

56789

P.P.

32

No. 53— USE OF LOGARITHMS

N.

L. 0 1 2 3 4

56789

P.P.

800

801
802
803
804

90309 3*4 320 32$ 331
363 369 374 380 385
417 423 428 434 439
472 477 482 488 493
526 531 536 542 547

336 342 347 352 358
390 396 401 407 412
445 450 455 461 466
499 504 509 515 520
553 558 563 569 574

6

i 0,6

2 1,2

3 1,8
4 2,4

I 3'2
6 3,6

7 4/2
8 4,8

915,4

5

i 0,5

2 I,O

3 i,5
4 2,0
5 2,5
6 3,o
7 3,5
8 4,0

9 4,5

805
806
807
808
809

580 585 590 596 601
634 639 644 650 655
687 693 698 703 709
741 747 752 757 763
795 800 806 811 816

607 612 617 623 628
660 666 671 677 682
714 720 725 730 736
768 773 779 784 789
822 827 832 838 843

810

811
812
813
814

849 854 859 865 870
902 907 913 918 924
956 961 966 972 977

91 009 014 020 02§ 030
062 068 073 078 084

875 88 I 886 891 897
929 934 940 945 950
982 988 993 998 *004
036 041 046 052 057
089 094 100 105 no

8iS
816

817
818
819

116 121 126 132 137
169 174 180 185 190

222 228 233 238 243
275 28l 286 291 297

328 334 339 344 35o

142 148 153 158 164

196 201 206 212 217
249 254 259 265 270
302 307 3I2 3l8 323

355 36o 36$ 37i 376

820

821
822

g

381 387 392 397 403
434 440 445 450 455
487 492 498 503 508
540 54$ 55i 556 56i
593 598 603 609 614

408 413 418 424 429
461 466 471 477 482
514 519 524 529 535
566 572 577 582 587
619 624 630 635 640

£

827
828
829

830

831
832
833
834

64$ 651 656 661 666
698 703 709 714 719
751 756 761 766 772
803 808 814 819 824
855 861 866 871 876

672 677 682 687 693

724 730 735 740 745
777 782 787 793 798
829* 834 840 845 850
882 887 892 897 903

908 913 918 924 929
960 965 971 976 981
92 012 018 023 028 033
065 070 075 080 085

117 122 127 132 137

934 939 944 95o 955
986 991 997 *oo2 *007
038 044 049 054 059
091 096 101 106 in
143 148 153 158 163

III

837
838

839

169 174 179 184 189
221 226 231 236 241
273 278 283 288 293

324 330 335 340 345
376 381 387 392 397

195 200 205 210 215

247 252 257 262 267
298 304 309 314 319

35o 35$ 361 366 371
402 407 412 418 423

840

841
842

843
844

428 433 438 443 449
480 485 490 495 500
53i 53° 542 547 552
583 588 593 598 603
634 639 645 650 655

454 459 464 469 474
505 511 516 521 526
557 562 567 572 578
609 614 619 624 629
660 665 670 675 681

845
846
847
848
849

686 691 696 701 706

737 742 747 752 758
788 793 799 804 809
840 845 850 855 860
891 896 901 906 911

711 716 722 727 732
763 768 773 778 783
814 819 824 829 834
865 870 875 88 i 886
916 921 927 932 937

850

942 947 952 957 962

967 973 978 983 988

N.

L. 0 1 2 3 4

56789

P.P.

LOGARITHMIC TABLES

33

N.

L. 0 1 2 3 4

56789

P.P.

850

85i
852
853
854

92942 947 952 957 962
993 998 *oo3 *oo8 #013
93044 049 054 059 064
095 loo 105 no 115
146 151 156 161 166

967 973 978 983 988
*oi8 *024 *029 *034 *039
069 075 080 085 090

120 125 131 136 141

171 176 181 186 192

i

2

3
4

9

I

2

3

4

7
8

9

I

2

3
4

7
8

9

6

0,6

I/2

1,8

2,4

3,<?
3,6

4,2

4,8

5,4

5

o,5

1,0

i,5

2,0

2,5

3,0
3,5
4,o

4,5

4

4

o,4
0,8

1,2

1,6

2,0
2,4
2,8

11

SP

857
858

859

197 202 207 212 217
247 252 258 263 268
298 303 308 313 318
349 354 359 3^4 3^9
399 404 409 414 420

222 227 232 237 242
273 278 283 288 293

323 328 334 339 344
374 379 384 389 394
425 430 435 440 445

860

861
862
863
864

45o 455 460 465 470
50° 5°5 5io 5*5 520
551 556 561 566 571
601 606 611 616 621
651 656 661 666 671

475 480 485 490 495
526 53i 536 54i 546
576 581 586 591 596
626 631 636 641 646
676 682 687 692 697

865
866
867
868
869

702 707 712 717 722
752 757 762 767 772
802 807 812 817 822
852 857 862 867 872
902 907 912 917 922

727 732 737 742 747
777 782 787 792 797
827 832 837 842 847
877 882 887 892 897
927 932 937 942 947

870

871
872

873
874

952 957 962 967 972

94 002 007 012 017 022
052 057 062 OOJ 072

ioi 106 in 116 121
151 156 161 166 171

977 982 987 992 997
027 032 037 042 047
077 082 086 091 096
126 131 136 141 146
176 181 186 191 196

875
876

877
878
879

201 206 211 2l6 221
250 255 260 265 270
300 305 310 315 320

349 354 359 3^4 3^
399 404 409 414 419

226 231 236 240 245
275 280 285 290 295
325 330 335 340 345
374 379 384 389 394
424 429 433 438 443

880

881
882
883
884

448 453 458 463 468
498 5°3 507 512 517
547 \ 552 557 562 567
596 601 606 611 616
645 650 655 660 665

473 478 483 488 493
522 527 532 537 542
57i 576 581 586 59i
621 626 630 635 640
670 675 680 685 689

885
886
887
888
889

694 699 704 709 714
743 748 753 758 763
792 797 802 807 812
841 846 851 856 861
890 895 900 905 910

719 724 729 734 738
768 773 778 783 787
817 822 827 832 836
866 871 876 880 885
915 919 924 929 934

890

891
892

893
894

939 944 949 954 959
988 993 998 *oo2 *oo7
95 036 041 046 051 056
085 090 095 loo 105
134 139 143 148 153

963 968 973 978 983

*OI2 *OI7 *022 #027 ^032

061 066 071 075 080

109 114 119 124 129

158 163 168 173 177

895
896
897
898
899

182 187 192 197 202
231 236 240 245 250
279 284 289 294 299

328 332 337 342 347
376 381 386 390 395

207 211 2l6 221 226
255 260 265 270 274
303 308 313 318 323

SS2 357 361 366 37i

400 405 410 415 419

900

424 429 434 439 444

448 453 45s 463 468

N.

L. 0 1 2 3 4

56789

P.P.

34

No. 53— USE OF LOGARITHMS

N.

L. 0 1 2 3 4

56789

P.P.

900

901
902

903
904

95424 429 434 439 444
472 477 482 487 492
521 525 530 53$ 540
569 574 578 583 588
617 622 626 631 636

448 453 458 463 468
497 501 506 511 516

545 550 554 559 564
593 598 602 607 612
641 646 650 65$ 660

5

i 0,5

2 1,0

3 i/5
4 2,0

5:2,5

6; 3,0

7 3,5
8 4,0

9:4/5

4

ilo,4

2 0,8

3 1/2
4 1,6

5 2/o
6 2,4
7 2,8
83,2
9 3/6

907
908
909

665 670 674 679 684
713 718 722 727 732
761 766 770 775 780
809 813 818 823 828
856 861 866 871 875

689 694 698 703 708
737 742 746 75i 756
785 789 794 799 804
832 837 842 847 852
880 885 890 893 899

910

911
912

913
914

904 909 914 918 923
952 957 961 966 971
999 *oo4 ^009 #014 *oi9
96047 052 057 061 066
095 099 104 109 114

928 933 938 942 947
976 980 98$ 990 995
#023 #028 #033 $038 #042
071 076 080 085 090
118 123 128 133 137

9i5
916
917
918
919

142 147 152 156 161
190 194 199 204 209
237 242 246 251 256
284 289 294 298 303
332 336 34i 346 35°

166 171 17$ 180 185"
213 218 223 227 232
261 26$ 270 273 280
308 313 317 322 327
35S 360 363 369 374

920

921
922
923
924

379 384 388 393 398
426 431 43$ 440 445
473 478 483 487 492
520 525 530 534 539
567 572 577 581 586

402 407 412 417 421
430 454 459 464 468
497 501 506 511 515
544 548 553 558 562
591 59$ 600 603 609

92J

926

927
928
929

614 619 624 628 633
661 666 670 675 680
708 713 717 722 727

755 759 764 769 774
802 806 811 816 820

638 642 647 652 656
683 689 694 699 703
731 736 741 745 750
778 783 788 792 797
823 830 834 839 844

930

93i
932
933
934

848 853 858 862 867
895 900 904 909 914
942 946 951 956 960
988 993 997 *002 ^007
97035 039 044 049 053

872 876 881 886 890
918 923 928 932 937
963 970 974 979 984
#ou *oi6 *02i *02$ #030
058 063 067 072 077

935
936
937
938
939

08 1 086 090 095 too
128 132 137 142 146
174 179 183 i 88 192

220 225 230 234 239

267 271 276 280 285

104 109 114 118 123
151 155 160 163 169

197 202 206 211 2l6

243 248 253 257 262
290 294 299 304 308

940

941
942
943
944

313 317 322 327 331

359 364 368 373 377
4o§ 410 414 419 424
451 456 460 465 470
497 5°2 506 511 5l6

336 340 345 35o 354
382 387 391 396 400
428 433 437 442 447
474 479 483 488 493
520 523 529 534 539

$
S3

949

543 548 552 557 562
589 594 598 603 607
633 640 644 649 653
68 1 685 690 695 699
727 73i 736 740 745

566 571 575 58o 583
612 617 621 626 630
658 663 667 672 676
704 708 713 717 722
749 754 759 7^3 7^8

950

772 777 782 786 791

795 800 804 809 813

N.

L. 0 1 2 3 4

56789

P.P.

LOGARITHMIC TABLES

35

N.

L. 0 1 2 3 4

56789

P.P.

950

95i
952
953
954

97772 777 782 786 791
818 823 827 832 836
864 868 873 877 882
909 914 918 923 928
955 959 964 968 973

795 800 804 809 813

841 845 850 855 859
886 891 896 900 905

932 937 94i 946 950
978 982 987 991 996

955
956
957
958
959

98 ooo 005 009 014 019
046 050 055 059 064
091 096 100 105 109

137 141 146 150 155

182 186 191 195 200

023 028 032 037 041
068 073 078 082 087
114 118 123 127 132
159 164 168 173 177
204 209 214 218 223

960

961
962

963
964

227 232 236 241 24^
272 277 281 286 290
318 322 327 331 336
363 367 372 376 381
408 412 417 421 426

250 254 259 263 268
295 299 304 308 313
340 345 349 354 358
385 390 394 399 4°3
430 «5 439 444 448

5

ijo,5

2 1,0

3 i,S

965
966
967
968
969

453 457 462 466 471
498 502 507 511 516

543 547 552 556 561
588 592 597 601 605
632 637 641 646 650

475 480 484 489 493
520 525 529 534 538

565 57° 574 579 583
610 614 619 623 628
655 659 664 668 673

4 2,0
5 2,5
6 3/o
7 3,5
8 4,0

9 4,5

970

971
972
973
974

677 682 686 691 695
722 726 731 735 740
767 771 776 780 784
8il 816 820 825 829
856 860 865 869 874

700 704 709 713 717
744 749 753 758 762
789 793 798 802 807
834 838 843 847 851
878 883 887 892 896

975
976

977

978

979

980

981

982

983
984

900 905 909 914 918

945 949 954 958 963
989 994 998 #003 *oo7
99034 038 043 047 052
078 083 087 092 096

123 127 131 136 140
167 171 176 180 185

211 2l6 22O 224 229
255 260 264 269 273
300 304 308 313 317

923 927 932 936 941
967 972 976 981 985
#012 *oi6 #021 #02=; #029
056 061 065 069 074
100 105 109 114 118

145 149 154 158 162

189 193 198 202 207
233 238 242 247 251
277 282 286 291 295

322 326 330 335 339

4

I 0,4

2 0,8

3 1,2

985
986
987
988
989

344 348 352 357 361
388 392 396 401 4°5
432 436 44i 445 449
476 480 484 489 493
520 524 528 533 537

366 370 374 379 383
410 414 419 423 427
454 458 463 467 471
498 502 506 511 515
542 546 550 555 559

4 1,6
5 2,0
6 2,4
7 2,8
8 3,2
9 3,6

990

991
992
993
994

564 568 572 577 581
607 612 616 621 625
651 656 660 664 669
695 699 704 708 712
739 743 747 752 756

585 590 594 599 603
629 634 638 642 647
673 677 682 686 691
717 721 726 730 734

760 765 769 774 778

995
996

997
998

999

782 787 791 79$ 800
826 830 835 839 843
870 874 878 883 887
913 917 922 926 930
957 961 965 97° 974

804 808 813 817 822
848 852 856 861 86^
891 896 900 904 909
935 939 944 948 952
978 983 987 99i 996

1000

ooooo 004 009 013 017

022 026 030 035 039

N.

L. 0 1 2 3 4

56789

P.P.

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UNIVERSITY OF CALIFORNIA LIBRARY

CONTENTS OF DATA JHEI-T BOOKS

. i. Sc-ew Threads.— T nired

ail- '

>«i; C

Bri£- gs Pipe
'i • G; ^es: Fire Hose
T -read: AYorm
orms . Machine. "vVuod,
C^:-r:iare Brit

<M>IS; Tape] Tnrr.i g: Chang*
ing for <'.;e Latne, Borin:, B: -s and

No. 11. Milling Marine Indexin •,
damping Devices and. Plasar Jacks. —

r- Mes for Milling Ala-nine Indexing,
Ci. .use Gears foi 'nilhig- Spirals;
foi it t ting I.'.-Jexi.ig P"<v4d when to

Jolts antl NntJ. — Fil-

Clutches, Jig '-urni 1^0 vices; .-

-he;n

and Clamp?; I'lai^t ' ^ :KS.

x w, • St.°n<l-

.11

No. lv' fcT'j j.i'- * Pipcj Pittings.
ads a . lag " -iron Fii

.ze Fii v "s;

jJen<J|, Pi'>e .-imp

• I .Scr---\vh ; TJ- .) Drills;

:is of Pipe, foi Vc . .Services,

.

io. 13. Boilers *af 'aimncys.

»0

. ~ 0.- P. "i- 3. — B • i, iVlachi; ,

rfpac'ig iul Bracing i 'lers: St

-

of Boiler Joints; Rive'. ^ Boiler S>

L«i(

,.0-b- .- .v Mr ne

Chimneys

Tat <

r il jr "; s;

No. 14. I-ocon otive inf Railway "a.i.

Su.j

t-oU ' ns;

--Lr ITT i "ive B"il •!•<?; ^ aring Pr ,..•.

Pi

Hi -i " , 'itu •

>r L,^ ?• moth )t^ a

A<-

• • j«k '! '< ding

L.la« -ificei; ins. Rnij Sections:

:_i

^^^^H

Swit-, 'iep am; " iv. -. T

IT ;t .v Dr^il and

Force; i.iertin • ? ' ins. Brake 1

,1

T Faell

Br. ';. -. Rods, etc.

ij Pi-

.

o. 15. Steam a ^ das _ ng-ines
uraied Stej.ni; • :• m Pipe Si >s;

Ja • .. >1 i:

Engine .f Cyl -r

r-

StuiTiin.u .Setting Corliss . .

*'

vf

Vaj Air i

s

;: !.

l)cl 1 •El!.,,

C-arir^r. — Piameir-.

Automobile >"' ^in • •'|:.v'i,ats,

No. 16. MatV.m^ ,i.-. hies. S

of Mixe<: .\";

-1 Cast-iron C.-

Circles; 'fahV:
Solution of Tr. • ;?
. Regu, P ". j jiK-

; el. fc> *' ' -. d 'Vorm Q-*iai'-

gression, cio.

t?

and " j for Bavel
•! <.-ars; \.-

»o. li >*fichanics a^-I Strength J
terials.-fW- k: K re>; Cent

Force:

:

Spiral

.Materials: • ' i

lj.0 I'll1

i io of t /lit!-"'1 :> , 1 -0

lers

"!fC

Cey \ie Keyvrays. —

Jfo. 18. y?e; F <••'• v... ? r'ci .al

» t Til). Mti— "liis R,71u

Desi^ti.— • 1-j

tuiafrtit ' *e 1

uli 01 1

ni Runnn ,

Ni

'• v: «p!'' o ti.ii ! I

lilling K»-y-

•

1* . .9. Belt, B ne

^*< aring* Couplings, ClntclieK,

D'i., nifiioi.

.haiu and ^iooks. — Pil:

le\

i

(.".amp t ou: ing

ti'..;isin:
Dr;

Ci:.

"uni Scor«. .-.

an.

V.' '

.ngs, TMdes r.ud Machine

No. 20. Wiri i .grains. «eat3 r - I
Ventilation, n xscellaaeous Ti les 1

Typl<

11

-

I! OC( Ci. I

'•'a 1

on.-- !-^; Cent 'f;:g I

*

10 . ^.o.f * .ive. Speeds and Peeds,

lot -,'v <• I

v»hi

i a crearint a -in^ Uars.

at ling

j-si '•

Ta- les, Wei

• ftii, -i.>oil

mechanical

jf.'irnal, o;. gins 'or u< tbc R-ferenc< anl

.. published in foui

litioTi •- hp /S/it •< Edition. ^>L.OO a. .'a;B

oO a year;

fbe '\00. a year, u- trl

;

-

fl

^^•fcad'j , trial Press, Publishers o^ ' -"BINARY,

-g.

; ex York City, TJ S. 1