--- OHEOP
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.

STATISTICAL METHODS

WITH SPECIAL REFERENCE TO

BIOLOGICAL VARIATION.

BY

C. B. DAVENPORT, PH.D.,

Instructor in Zoology at Harvard University.

FIRST EDITION.

FIRST THOUSAND.

NEW YORK:

JOHN WILEY & SONS.
LONDON: CHAPMAN & HALL, LIMITED.

1899.

Copyright, 1899,

BY

C. B. DAVENPORT.

HOBERT DRUMMOND, PRINTER, NEW YORK.

PREFACE.

THIS book has been issued in answer to a repeated call for a
simple presentation of the newer statistical methods in their
application to biology. The immediate need which has called
it forth is that of a handbook containing the working formulae
for use at summer laboratories where material for variation-
study abounds. In order that the book should not be too
bulky the text has been condensed as much as is consistent
with clearness.

This book was already in rough draft when the work of
Duncker appeared in Roux's Archiv. I have made much use
of Duncker's paper, especially in Chapter IV. I am indebted
to Dr. Frederick H. Safford, Assistant Professor of Mathe-
matics at the University of Cincinnati and formerly Instructor
at Harvard University, for kindly reading the proofs and for
valuable advice. To Messrs. Keuffel and Esser, of New York,
I am indebted for the use of the electrotypes of Figures 1 and 2.
Finally, I cannot fail to acknowledge the cordial cooperation
which the publishers have given in making the book ser-
viceable.

C. B. DAVENPORT.

BIOLOGICAL LABORATORY OF THE BROOKLYN INSTITUTE,

COLD SPRING HARBOR, LONG ISLAND,

June 29, 1899.

Hi

CONTENTS.

ERRATA.

Page IN, bottom. Denominator of last term in equation for
// should rend

Pa^e 23, 9th line. iH-nominator of last term in equation 1W
f/o sliould read

/'(MI + I)l\m9 + 1).

?th line from bottom. Denominator of last term in
equation for //„ should read

\m -f 1).

Method of loaded ordinates 12

Method of rectangles 13

Certain constants of the Frequency Polygon 13

The mean 13

The mode 14

The median magnitude 14

The probable error of the mean 14

The index of variability 15

The probable error of the standard deviation .. 15

Average deviation and probable error 15

Coefficient of variability 15

31217

CONTENTS.

CHAPTER I.

ON METHODS OF MEASURING ORGANISMS.

PAGE

Preliminary Definitions 1

Methods of Collecting Individuals for Measurement 2

Processes Preliminary to Measuring Characters 2

The Dermination of Integral Variates— Methods of Counting 3

The Determination of Graduated Variates— Method of Measurement.. 4

Straight lines on a plane surface 4

Distances through solid bodies or cavities 4

Area of plane surfaces 4

Area of a curved surface 5

Form of a plane figure 6

Characters occupying three dimensions of space 9

Characters having weight , 9

Color characters 9

Marking- characters 10

CHAPTER II.

ON THE SERIATION AND PLOTTING OF DATA AND THE FREQUENCY

POLYGON.

Seriation 11

Plotting 12

Method of loaded ordinates 12

Method of rectangles 13

Certain constants of the Frequency Polygon 13

The mean 13

The mode 14

The median magnitude 14

The probable error of the mean 14

The index of variability 15

The probable error of the standard deviation . . 15

Average deviation and probable error 15

Coefficient of variability 15

31217

Vi CONTENTS.

CHAPTER III.

THH CLASSES OF FREQUENCY POLYGONS.

PAGE

Classification 16

To classify a simple frequency polygon 16

The normal curve 18

To compare any observed curve with the theoretical normal

curve 19

The index of abmodality 19

To determine the closeness of fit of a theoretical polygon to the

observed polygon 19

The normal curve as a binomial curve 20

Example of a nearly normal curve . 20

Other unimodal frequency polygons 21

Curves of limited range 22

Asymmetry or skewness 22

To compare any observed frequency polygon of Type I with its

corresponding theoretical curve — 22

To compare any observed frequency polygon of Type II with its

corresponding theoretical curve 23

To compare any observed frequency polj'gon of Type III with its

corresponding theoretical curve 23

To compare any observed frequency curve of Type IV with its

corresponding theoretical curve 24

Example of calculating the theoretical curve corresponding with ob-
served data 25

Multimodal curves 26

CHAPTER IV.

CORRELATED VARIABILITY.

General principles 30

Methods of determining coefficient of correlation 32

Gallon's graphic method 32

Pearson's method 32

Duncker's brief method 83

Spurious Correlation in Indices 35

Heredity 35

Uniparental inheritance 36

Biparental inheritance 36

Gallon's law of ancestral heredity 37

CHAPTER V.

SOME APPLICATIONS OF STATISTICAL BIOLOGICAL STUDY.

The laws of variation .... 38

The causes of variation 38

Selection 38

The origin of species 38

CONTENTS. Vll

PAGE

The definition of species 38

Distinction between species and varieties 39

Criterion for homology 39

Prepotency 39

SELECTED BIBLIOGRAPHY OP WORKS ON THE QUANTITATIVE STUDY OP

ORGANISM 40

EXPLANATION OP TABLES 43

LIST OP TABLES.

Table I. Formulas 53

" II. Certain constants and their logarithms 54

V

III. Table of ordinates of normal curve, or values of -=—

2/o

OS

corresponding to values of • — 55

CT

" IV. Table of values of the normal probability integral corre

oc
spending to values of -- ; or the fraction of the area of

(T

X OC

the curve between the limits 0 and -j or 0 and -56

cr a

" V. Table of Log r functions of p 57

" VI. Table of reduction of linear dimensions from common to

metric system 59

VII. First to sixth powers of integers from 1 to 30 60

" VIII. Squares, cubes, square-roots, cube-roots and reciprocals, 60

IX. Logarithms of numbers 77

" X. Logarithmic sines, cosines, tangents and cotangents 104

STATISTICAL METHODS

"WITH SPECIAL REFERENCE TO

BIOLOGICAL VAEIATION

CHAPTER I.
ON METHODS OF MEASURING ORGANISMS.

Preliminary I>efiuitions.

An individual is a segregated mass of living matter, capable
of independent existence. Individuals are either simple or
compound, i.e., stocks and conns. In the case of a compound
individual the morphological unit may be called a person.

A character is any quality common to a number of in-
dividuals.

The magnitude of a character is a quantitative expression of
the character.

A variate is a single magnitude-determination of a charac-
ter.

A class includes variates of the same or nearly the same
magnitude.

Integral variates are magnitude-determinations of charac-
ters "which from their nature are expressed in integers. Such
magnitudes are determined by counting; e.g. , the number of
teeth in a porpoise.

Graduated variates are magnitude-determinations of charac-
ters which do not exist as integers and which may couse-

2 STATISTICAL METHODS.

quently differ in different individuals by any degree of
magnitude however small; e g., the stature of man.

Methods of Collecting Individuals for Meas-
urement.

In collecting a lot of individuals for the study of the varia-
bility of any character undue selection must be avoided. The
rule is:

Having settled upon tlie general conditions, of race, sex,
locality, etc., which the individuals to be measured must fulfil,
take the individuals methodically at random and without possible
selection of individuals on the basis of the magnitude of the
character to be measured. If the individuals are simply not
consciously selected on the basis of magnitude of the character
they will often be taken sufficiently at random.

Processes Preliminary to Measuring1
Characters.

Some characters can best be measured directly; e.g., the
stature of a race of men. Often the character can be better
studied by reproducing it on paper. The two principal
methods of reproducing are by photography and by camera
drawings.

For photographic reproductions the organs to be measured
will be differently treated according as they are opaque or
transparent. Opaque organs should be arranged if possible
in large series on a suitable opaque or transparent back-
ground. The prints should be made on a rough paper so
that they can be written on ; blue-print paper is excellent.
This method is applicable to hard parts which may be studied
dry; e.g., mollusc shells, echiuoderms, various large arthro-
pods, epidermal markings of vertebrates and parts of the
vertebrate skeleton. Shadow photographs may be made of
the outlines of opaque objects, such as birds' bills, birds' eggs,
and butterfly wings, by usiug parallel rays of light and inter-
posing the object between the source of light* and the photo-

* A Welsbach burner or an electric light are especially good. Minute

MEASUREMENT OF ORGANISMS. 3

graphic paper. Move or less transparent organs, such as
leaves, petals, insect-wings, ami appendages of the smaller
Crustacea, may be reproduced either directly on blue-print
paper or by "solar prints," either of natural size or greatly
enlarged. For solar printing the objects should be mounted
in series on glass plates. They may be fixed on the plate by
means of balsam or albumen and mounted between plates either
dry or in Canada balsam or other permanent mounting media.
Wings of flies, orthoptera, ueuroptera, etc., may be prepared
for study in this way; twenty-five to one hundred sets of wings
being photographed on one sheet of paper, say 16 X 20 inches
in size. Microphotographs will sometimes be found service-
able in studying small organisms or organs, such as shells of
Protozoa or cytological details.

Camera drawings are a convenient although slow method of
reproducing on paper greatly enlarged outlines of microscopic
characters, such as the form and markings of worms and
lower Crustacea, sponge spicules, bristles, scales and scutes,
plant-hairs, cells and other microscopic objects. In making
such camera drawings a low-power objective, such as Zeiss A*,
will often be found very useful.

The Determination of Integral Variates. —
Methods of Counting.

While the counting of small numbers offers no special diffi-
culty, the counting becomes more difficult with an increase of
numbers. To count large numbers the general rule is to di-
vide the field occupied by the numerous organs into many
small fields each containing only a few organs. Counting
under the microscope, e.g. , the number of spines, scales or
plant-hairs per square millimetre, may be aided by cross-hair
rectangles in the eyepiece. The number of blood-corpuscles
in a drop of blood, or of organisms in a cubic centimetre of
water, have long been counted on glass slides ruled in small
squares.

electric lamps such as are fed by a single cell give sharp shadows of
small objects.

STATISTICAL METHODS.

The Determination of Graduated Variates.—
Methods of Measurement.

Straight lines on a plane surface are easily meas-
ured by means of a measuring-scale of some sort. The meas-
urement should always be metric because
this is the universal scientific system. Vari-
ous kinds of scales may be obtained of
optical companies and hardware dealers, —
such as steel measuring tapes, graduated to
millimetres (about $1.00), and steel rules
(6 cm. to 15 cm.) graduated to £ of a milli-
metre. Steel "spring-bow" dividers with
rnilled-head screw are useful for getting
distances which may be laid off on a scale.
Tortuous lines, e.g., the contour of the
serrated margin of a leaf or the outer
margin of the wing of a sphinx moth, may
be measured by a map-measurer ("Entfer-
nuugsrnesser," Fig. 1), supplied at artist's
and engineer's supply stores at about $3.50.
Distances through solid bodies
or cavities are measured by calipers of
some sort. Calipers for measuring diameters
of solid bodies are made in various styles.
Micrometer screw calipers ("speeded")
reading to one-hundredths of a millimetre
and sold by dealers in physical apparatus for
about $5.00 are excellent for determining diameters of bones,
birds' eggs, gastropod shells, etc. Leg calipers for rougher
work can be obtained for from 30 cents to $4.00. The
micrometer " caliper-square," available for inside or outside
measurements and measuring to huudredths of a millimetre,
is a useful instrument.*

The area of plane surfaces, as, e.g. , of a wing or leaf,
is easily determined by means of a sheet of colloidin scratched
in millimetre squares. By rubbing in a little carmine the

* Many of the instruments described in this section are made by the
Starrett Co., Athol, Mass., and by B rown and Sharpe, Providence, tool
cutters.

FIG. 1.

MEASUREMENT OF ORGANISMS. 5

•

scratches may be made clearer. The number of squares
covered by the surface is counted (fractional squares being
mentally summated) and the required area is at ouce obtained.
If the area has been traced on paper it may be measured by
the planimeter (Fig. 2). This instrument may be obtained at

FIG. 2.

engineer's supply shops. It consists of two steel arms hinged
together at one end; the other end of one arm is fixed by a
pin into the paper, the end of the second arm is provided with
a tracer. By merely tracing the periphery of the figure whose
area is to be determined the area may be read off from a drum
which moves with the second arm. This method is less
wearisome than the method of counting squares.

The area of a curved surface, like that of the elytra
of a beetle or the shell of a clam, is not always easy to find.
To get the area approximately, project the curved surface on
a plane by making a camera drawing or photograph of its
outline. By means of parallel lines divide the outline draw-
ing into strips such that the corresponding parts of the curved
surface are only slightly curved across the strips, but greatly
curved lengthwise of the strips. Measure the length of each
plane strip and divide the magnitude by the magnification of
the drawing. Measure also, with a flexible scale, the length
of the corresponding strip on the curved surface. Then, the
area of any strip of the object is to the area of the projection
as the length of the strip on the object is to the length of its
projection. The sum of the areas of the strips will give the
total area of the surface.

G

STATISTICAL METHODS.

The for in of a plane figure of irregular outline has
been expressed qualitatively by botanists, who have invented a
complicated nomenclature for the purpose; this is reproduced
in part here.

Linear, more than thrice longer than wide and of nearly the
same breadth throughout (Fig. 3).

Lanceolate, more than thrice longer than wide and tapering
towards one or both ends (Fig. 4).

Oblong, twice to thrice as long as broad (Fig. 5).

Elliptical, of the shape of an ellipse with an eccentricity
more than .5 (Fig. 6).

Oval, elliptical, with eccentricity from .5 to .1.

Orbicular, nearly circular, with eccentricity less than .1.

Ovate, with the outline of a hen's egg, one end broader than
the other (Fig. 7).

6

A

FIGS. 3-7.

Cuneate or cuneiform, wedge-shaped.

Spatulate, rounded at one end, long and narrow at the other,
like a spatula.

Acuminate, tapering to an angle of less than 15° (Fig. 8).

Acute, ending in an angle of from 15° to 90° (Fig. 9).

Obtuse, ending in an angle of over 90° (Fig. 10).

Truncate, terminating as though cut off (Fig. 11).

Retuse, with a re-entering obtuse end (Figs. 12-14).

Serrate, with small saw-like teeth (Fig. 15).

Dentate, with larger, more obtuse teeth (Fig. 16).

Crenate, rounded teeth (Fig. 17).

Repand, wavy margin, teeth broadly rounded, height less
than breadth (Fig. 18).

MEASUREMENT OF ORGANISMS.

8 9 10

Sinuate, still stronger waves, height equals or exceeds
breadth (Fig. 19).

Incised, with sharp, deep incisions (Fig. 20).

15

16

17 18 19

FIGS. 15-20.

The quantitative expression of variation in these forms can
usually be easily obtained by using an index, or ratio of two
dimensions.

greatest length

Index of Lmearuess, — — r-r .

greatest breadth

greatest length

" " Lauceolateness, — r- -^-T , also angle aoc.

greatest breadth

greatest length area

" " Oblongness, ^rrr, also — -r-r .

greatest breadth breadth

(greatest Igth.) — (greatest brdth.)
" " Ellipticity, - ->

(greatest length)

for values from 1 to .50.

"

8 STATISTICAL METHODS.

(greatest length) — (greatest breadth)

Index of Ovalness, ',

(greatest length)

for values from .50 to .1.

tt ~ , . , (greatest diam . ) — (greatest brdth. )

(greatest diameter)
for values from .1 to 0.

radius of curvature of

" " Ovateness or obovateness. ^ ^ ^ r .

radius of smaller end

diameter at £

Cuneateness, — — > °r angle abc (line a-c

diameter at £

passing through middle of major diameter),
length of radius of curve at broad

" " Spatulateness, ^ ^ . .

transverse diameter of narrow

part of organ

" " Acuminateness, angle abc at apex (Fig. 8).

" " Acuteness, angle abc at apex.

" " Obtuseness, angle abc at apex and radius of curva-
ture.

<« « Truncatedness, angle abc at apex and radius of curva-
ture.

cosine

" " Retuseness, — — — of | angle abc.

2 X sine

" " Serrateuess, number of teeth per linear unit of edge,

average angle of tooth.

" " Dentateness, number of teeth, average angle of tooth,
" " Crenateness, number of waves, average radius of

curvature of waves.

depth of waves

" Repandness, — -, average radius of cur-

length of waves

vature of waves.

depth of waves

" " Sinuateness, : — — -, average radius of cur-

length of waves

vature of waves.

depth of incision

" Incisedness, - — ^— —=-. — — — .
opening 01 incision

MEASUREMENT OF ORGANISMS. 9

Characters occupying three dimensions of
space m:iy be quantitatively expressed by volume. TLe
volume of water or sand displaced may be used to measure
volume in the case of solids. The volume of water or sand con-
tained will measure a cavity. Irregular form is best measured
by getting, either by means of photography or drawings, pro-
jections of the object on one or more of the three rectangular
fundamental planes of the organ, and then measuring these
plane figures as already described. Or two or more axes may
be measured and their ratio found.

Characters having weight are easily measured ; the
only precautions being those observed by physicists and
chemists.

Color Characters. Color may be qualitatively ex-
pressed by reference to named standard color samples. Such
standard color samples are given in Ridgeway's book,
" Nomenclature of Color," and also in a set of samples manu-
factured by the Milton Bradley Co., Springfield, Mass., costing
6 cents. The best way of designating a color character is by
means of the color wheel, a cheap form of which (costing 6
cents) is made by the Milton Bradley Co. The colors of this
"top" are standard and are of known wave-length as follows:

Red, 656 to 661 Green, 514 to 519

Orange, 606 to 611 Blue, 467 to 472

Yellow, 577 to 582 Violet, 419 to 424.

It is desirable to use Milton Bradley's color top as a standard.
Any color character can be matched by using the elementary
colors and white and black in certain proportions. The pro-
portions are given in perceuts. In practice the fewest possible
colors necessary to give the color character should be employed
and two or three independent determinations of each should
be made at different times and the results averaged. So far
as my experience goes any color character is given by only
one least combination of elementary colors. (See Science,
July 16, 1897.)

"When there is a complex color pattern the color of the
different patches must be determined separately. In case of
a close intermingling of colors, the colored area may be rapidly
rotated on a turntable so that the colors blend and the result-

10 STATISTICAL METHODS.

ant may then be compared with the color wheel. By this
means also the total melanism or albinism, viridesceuce, etc.,
may be measured.

Marking-characters. The quantitative expression of
markings or color patterns will often call for the greatest
ingenuity of the naturalist. Only the most general rules can
here be laid down. Study the markings comparatively in a
large number of the individuals, reduce the pattern to its
simplest elements, and find the law of the qualitative variation
of these elements. The variation of the elements can usually
be treated under one of the preceding categories. Find in how
far the variation of the color pattern is due to the variation of
some number or other magnitude, and express the variation in
terms of that magnitude. Remember that it is rarely a ques-
tion whether the variation of the character can be expressed
quantitatively but rather what is the best method of express-
ing it quantitatively.

SERIATIOK AND PLOTTING OF DATA. 11

CHAPTER II.

ON THE SERIATION AND PLOTTING OF DATA AND THE
FREQUENCY POLYGON.

The data obtained by measuring any character in a lot of
individuals consists either of amass of numbers for the charac-
ter in each individual ; or, perhaps, two numbers which are to
be united to form a ratio ; or, finally, a series of numbers such
as are obtained by the color wheel, of the order : TT 40%, tf
(Black) 38$, 7 12& G 101 The first operation is the simplifi-
cation of data. Each variate must be represented by one
number only. Consequently, quotients of ratios must be de-
termined and that single color of a series of colors which shows
most variability in the species must be selected, e.g.,N.

The process of seriation, which comes next, consists of the
grouping of similar magnitudes into the same magnitude
class. The classes being arranged in order of magnitude,
the number of variates occurring in each class is determined.
The number of variates in the class determines the frequency
of the class.

The method of seriation may be illustrated by two examples ; one of
integral variates. and the other of graduated variates.

Example 1. The magnitude of 21 integral variates are found to be as
follows : 12, 14, 11, 13, 12, 12, 14, 13, 12, 11, 12, 12, 11, 12, 10, 11, 12, 13, 12,
13, 12, 12. In seriation they are arranged as follows :

Classes : 10, 11, 12, 13, 14.
Frequency : 1, 4, 11, 4, 2.

Example 2. In the more frequent case of graduated variates our mag-
nitudes might be more as follows :

3.2 4.5 5.2 5.6 6.0
3.8 4.7 5.2 5.7 6.2
4.1 4.9 5.3 5.8 6.4

4.3 5.0 5.3 5.8 6.7
4.3 5.1 5.4 5.9 7.3

In this case it is clear that our magnitudes are not exact, but are merely
approximations of the real (forever unknowable) value. The question

]2 STATISTICAL METHODS.

arises concerning the inclusiveness of a class — the class range. An
approximate rule is : Make the classes only just large enough to have
no or very few vacant classes in the series. Following this rule we get

t 3.0-3.4;
Classes ... - 3.2

( 1
Frequency 1

I 5.5-5.9;
Classes.... - 5.7

(6
Frequency 5

The classes are named from their middle value, or better, for ease of
subsequent calculations, by a series of small integers (1 to 9).

In case the data show a tendency of the observer towards estimating
to the nearest round number, like 5 or 10, each class should include one
and only one of these round numbers.

As Fechner ('97) has pointed out, the frequency of the classes and all
the data to be calculated from the series will vary according to the
point at which we begin our seriation. Thus if, instead of beginning the
series with 3.0 as in our example, we begin with 3.1 we get the series :

3.5-3.9;

4.0-4.4;

4.5-4.9;

5.0-5.4;

3.7

4.2

4.7

5.2

o

3

4

5

1

3

3

7

6.0-6.4;

6.5-6.9;

V.0-7.4;

6.2

6.7

7.2

ft

i

8

9

3

1

1

Classes — -]

3.1-3.5;

3.6-4.0;

4.1-4.5;

4.6-5.0;

5.1-5.5;

3.3

3.8

4.3

4.8

3.5

Frequency

1

1

4

3

6

Classes ,;

5.6-6.0;

6.1-6.5;

6.6-7.0;

7.1-7.5;

i

5.8

6.3

6.8

7.3

Frequency

6

2

1

1

which is quite a different series. Fechner suggests the rule: Choose such
a position of the classes as will give a most normal distribution of fre-
quencies. According to this rule the first distribution proposed above
is to be preferred to the second.

In order to give a more vivid picture of the frequency of
the classes it is important to plot the frequency polygon.
This is done on coordinate paper.*

A different method should be adopted according as integral
or graduated variates are-unier consideration. In 1he case of
integral varia'cs proceed as follows : At equal intervals along
a horizontal line (axis of -X) draw a series of (vertical) ordinates
whose successive heights shall be proportional to the frequency
of the classes. Join the tops of the ordinates. Thus for the
example given, the curve will be as shown in Fig. 21. This
method of drawing the frequency polygon is known as the
method of loaded ordinates.

* This paper may be obtained at any artists' supply store.

SERIATION AND PLOTTING OF DATA.

13

111 the case of graduated varieties proceed as follows : Lay
off along a horizontal line equal contiguous spaces each of
which shall represent one class, number the spaces in order

. A

10

9

10

11

13

13

12
FIG. 21.

from left to right with the class magnitudes in succession,
and erect upon these bases rectangles proportionate in height
to the frequency of the respective clashes (Fig. 22).

i i

3.0

3.5 4.0 1.5 5.0 5.5 6.0 6.5 7.0 7.5

FIG. 2-2.

This method of drawing the frequency polygon is known as
the method of rectangles. If the tops of the middle
ordinates of successive contiguous rectangles be connected by
an oblique line a polygon made up of trapezia is obtained.
The outline of the polygon will be fairly close to that of a
curve passing through the tops of the central ordinates of the
rectangles.

CERTAIN CONSTANTS OF THE FREQUENCY POLYGON.

After the data have been gathered and arranged it is neces-
sary to determine the law of distribution of the variates. To
get at this law we must first determine certain constants.

The mean (M ) is the abscissa of the centre of gravity of
the variates or of the frequency polygon. It is found by
the formula

M=

V. f)

n

in which V is the magnitude of any class ; / its frequency ;

14 STATISTICAL METHODS.

2 indicates that the sum of the products for all classes into
frequency is to be got, and n is the number of variates.
Thus in the last example :

M = (3.2 X 1 + 3.7 X 1 + 4.2 X 3 + 4.7 X 3 + 5.2 X 7 + 5.7 X 5 + 6.2 X 3

+ 6.7 X 1 + 7.2 x 1) -s- 25 = 5.24,
or

J/i = (1X1+2X1+3X3+4X3+5X7+6X5+7X3+8X1+9X1) H- 25 = 5.08,
M = 5.2* + .08 (5.7 - 5.2) = 5.24

A still shorter method of finding Mis given on page 17.

The mode is the class with the greatest frequency.
In the example, the mode is 5.2.

The median magnitude is one above which and below
which 50$ of the variates occur. It is such a point on the axis
of X of the frequency polygon that an ordiuale drawn from it
bisects the polygon of rectangles or the continuous curve, but
not the polygon of loaded ordiuates.

To find its position: Divide the variates into three lots: those less than
the middle class, of which the total number is a; those of the middle
class, b; and those greater, c. Then a + b + c = n = the total number
of variates. Let I' = the lower limiting value of the middle class, and
I" = the upper limiting value, and let x = the abscissal distance of the
median ordinate above the lower limit or beloiv the upper limit of tJie
median class according as x is positive or negative. TJien %n - a : b =
x : I" — I' uhen x is positive, or Jn - c : b = x : I" — I' when x is negative.

Thus in the last example : 12.5 — 8 : 7 — x : 0.5; x = .32; the median
magnitude = 5.0 + .32 = 5.32. Or 12.5 - 10 : 7 = -x: 0.5; x = - .18; the
median magnitude = 5.5 - .18 = 5.32. (Cf. p. 11.)

Every determination of a constant of the frequency polygon
is an approximation only to the true value of the constant.
The closeness of the approximation to the truth is measured by
the so-called probable error of the determination. This is a
pair of values lying one above and one below the value deter-
mined. "We can say that there is an even chance that the true
value lies between these limits ; the chances are 4 to 1 that the
true value lies within twice these limits, and 19 to 1 that it lies
within thrice these limits.

The probable error of the mean is given by the for-
mula

standard deviation [see below] cr

± 0.6745 X - = ± 0.6745--=.

I/number of variates \n

It will be seen that the probable error is less, that is, that
the result is more accurate, the greater the number of variates

* 5.2 is the true class magnitude corresponding to the integer 5.

SERIATION AND PLOTTING OF DATA. 15

measured, but the accuracy does not increase in the same ratio
as the number of individuals measured, but as the square root
of the number. The probable error of the mean decreases as
the standard deviation decreases.

The index of the variability, or, of the variates when
they group themselves about one mode is found by adding
the products of the squared deviation-from-the-mean of each
class multiplied by its frequency, dividing by the total
number of variates, and extracting the square root of the
quotient, thus :

sum of [(deviation of class from mean)8

X frequency of class]

number of variates

This measure is known as the standard deviation.
The probable error of the standard deviation is

standard deviation . . cr

± 0.6745- =. = ± 0.6745 - -.

^2 X number of variates y 2n

Other Indices of Variation are the average deviation, or aver-
age departure, which is found thus:

sum of [deviations of class from mean X frequency]
~ number of variates

The probable error is the distance from the mode of that ordinate
which exactly bisects the half curve OMX or OMX1, Fig. 23; it is equal to
0.6745 X standard deviation = 0.6745o-. Neither of these last two indices
of variation is as good as the standard deviation when n is rather small.

The standard deviation, like the other indices of variation,
is a concrete number, being expressed in the same units as the
magnitudes of the classes. The standard deviation of one lot
of variates is consequently not comparable with the S. D. of
variates measured in other units. It has been proposed to re-
duce the index of variation to a concrete number, independent
of any particular unit, by dividing the index of variation of any
variates by the mean ; the quotient multiplied by 100 is called

the coefficient of variability. In a formula, CV = -^.
(Pearson, '96 ; Brewster, '97 )

16 STATISTICAL METHODS.

CHAPTER III.

THE CLASSES OP FREQUENCY POLYGONS.

The plotted curve may fall into one of the follow ing classes :

A. Unimodal.

I. Simple.

1. Range unlimited in both directions:

a. Symmetrical. The normal curve.

b. Unsymmetrical (Pearson's Type IV).

2. Range limited in one direction, together with

skewness (Type III).

3. Range limited in both directions :

a. Symmetrical, Type II.

b. Uusymrnetrical, Type I.
II. Complex.

B. Multimodal.

The classification of any given curve is not always an easy
task. "Whether the curve is unimodal or multimodal can be
told by inspection. Whether any unimodal curve is simple
or complex cannot be told by any existing methods without
great labor and uncertainty in the result.

Complex curves may be classified as follows :

1. Composed of two curves, whose modes are different but so near that
the component curves blend into one ; such curves are usually unsym-
metrical.

2. The sum of two curves having the same mode but differing varia-
bility.

3. The difference of two curves having the same mode but differing
variability.

If the material is believed to be homogeneous and the curve
is unimodal it is probably simple and its classification may be

carried further.

For classification the rule is as follows : Determine the mean
of the magnitudes. Take a class near the mean (call it Vm)

THE CLASSES OF FKEQUENCY POLYGONS. 17

as a zero point ; then the departure of all the other classes
will be - - 1, - - 2, - 3, etc., and + 1, + 2, -f- 3, etc.

Add the products of all these departures multiplied by the
frequency of tie corresponding class and divide by n; call
the quotient rt.

Add the products of the squares of all the departures multi-
plied by the frequency of the corresponding class and divide
by n; call the quotient r?.

Add the products of the cubes of all the departures multiplied
by the frequency of the corresponding class and divide by n;
call the quotient r3.

Add the products of the fourth powers of all the departures
multiplied by the frequency of the corresponding class and
divide by n; call the quotient v4. Or,

V V ~\

= departure of Vm from mean. Vm being

n

known, M may be found [J/ = Vm + vC\\ *

v - vmy

n

n

n

The values rlt r2, r3, r4, are called respectively the first,
second, third, and fourth moments of the curve about Vm.

To get the moments of the curve about the mean, either of
two methods (A or B) will be employed. Method A is used
when integral variates are under consideration ; method B
when we deal with graduated variates.

(A) To find moments in case of integral variates:

x/i = 0;

(B) To find moments in case of graduated variates :
* This is the'short method of finding M referred to on page 14.

18

STATISTICAL METHODS.

i — 0;

* = v* - rS -f- 1;
a = r3 — 3?'iK2 -f

Also,

= 0

and

F = 6 + 3/3, - 2/J2 = the "critical function."
Now the classification of any empirical curve depends upon
the value of its critical function, F.

,(/?i> 0, curve is of Type I.

When dispositive and \ ' '.* crn TT

] //! = 0, ^2 < 3, curve is of Type II.

ft i > 0, ySa > 3, curve is of Type III.
fjl = 0, /Jo = 3, curve is normal.
" F is negative and fii > 0, ^2 > 3, curve is of Type IV.
An important relation to be referred to later is

6(/?a- A - 1)

~T~ "'

in which s is an unknown, positive number.

Jf

5 i

THE NORMAL CURVE.

The normal curve is symmetrical about the mode ; con-
sequently the mode and the median and mean class coincide.

The mathematical formula of the normal curve, a formula
which one does not have to understand in order to make use

of it, is

<x

THE CLASSES OF FREQUENCY POLYGONS. 19

This formula gives the value of any ordiuate y (or auy
class) at any distance x (measured aiong the base, X, X', of
Fig, '23) from the mode, e is a constant number, 2.71828, the
base of the Naperian system of logarithms, a is the total area
of the curve or number of variutes, and cr is the Standard
Deviation, which is constant for uuy curve and measures the
variability of the curve, or the steepness of its slope.

To compare any observed curve with the theo-
retical normal curve we can make use of tables. For
the case of a polygon of integral variates the theoretical fre-

/>»

quency of any class at a deviation - - from the mean can be

taken directly from Table III. Here x is the actual deviation
from the mean expressed in the unit of the maximum, and cr
is the standard deviation.

For the case of a polygon of graduated variates built up of
rectangles representing the relative frequency of the variates,
Table IV gives the relation of the actual to the theoretical

2?

number of individuals occurring between the values -| and

x x

— . By looking up the given values of - the correspond-
ing theoretical percentage of variates between the limits

/ji sp SH

-\ and — will be found directly. The ratio — maybe

a a or

called the Index of Abmodality.

The normal curve may preferably be employed even when
/?i is not exactly equal to 0, nor /?2 exactly equal to 3, nor F
exactly equal to 0. Use the normal curve when

Qv t <),, 4

^X 7<23 < ± 1 and - -^- = 1 ± .2

?'4

To determine the closeness of fit of a theoreti-
cal polygon to the observed polygon. There are
two methods according as the variates are (A) integral or (B)
graduated.

(A.) Find for each class the percentage which the difference
between the theoretical value y and the observed frequency
/is of the frequency, and find the average of these percent-
ages, which is the index of closeness of fit sought.

20 STATISTICAL METHODS.

(B) Subtract in order each theoretical value of y from the
corresponding observed value, regarding signs. Call the dif-
ference di. "Whenever in the successive values of <5j there is
a change of sign, divide the product of these successive values
of di, in pairs, by their sum. Call this value £2; make its
sign always minus. Then the difference between the two
polygons in per cent of one of them is given by the equation

= . .

2n

where &i is summated without regard to sign, and n equals
the total number of variates. This is the method of Duucker,
'98. It may be considered a sufficient agreement between

observation and calculation when A < — —%.

yn

THE NORMAL CURVE OF FREQUENCY AS A BINOMIAL

CURVE.

The normal curve may also be expressed by the binomial
formula (p -f <?)*, where p = \,q — \, and I is the number
of terms, less 1, in the expansion of the binomial ; hence
approximately the number of classes into which the magni-
tudes of the variates should fall. If the standard deviation be
known, I may be found by the equation

I = 4 X (Standard Deviation)3 = 4crs.

Example of (nearly) normal curve. Number of spines in
dorsal fin of Acerina cernua, L. (Duncker, '99, p. 177).

F V - Vm f f(V-Vm)

11

- 3

1

- 3

9

- 27

81

12

- 2

2

- 4

8

- 16

32

13

- 1

189

- 189

189

- 189

189

14

0

1234

0

0

0

0

15

1

454

454

454

454

454

16

2

20

40

80

160

320

2

1900

298

740

382

1076

V, -

- 298 0

1568: vo= -^

-- 0.3895: v,-

382

-0.2041: v.—

1076 -o,

M = Vm + i>! = 14 + 0.1568 = 14.1568.
M3 = 0.3895 - 0.15682 = 0.3650.

M3 = 0.2011 - 3 X 0.1568 X 0.3895 -f 2 X 0.1568s = 0.0257.

THE CLASSES OF FREQUENCY POLYGONS. 21

= 0 5663 - 4X0.15G8 X 0.201 1 + 6 X 0.156S2 X 0.3895 - 3 X 0.;568< = 0.4929.

F = 6 4- .04074 - 7.3996 = - 1.3589. F . M23 = 1-3589 X 0.3653 _ .066.

W ~ ^ = 3 X 0.3895' - 2 X .15684 = ^ ^ =
P4 0.5663

n 1900

Maximum frequency = - — —.= = 1*255.

a- V2n -6041 X V%*

Although somewhat more closely of Type IV (see page 18) than of
the Normal Type, this example may be treated as Normal.

The difference between it and the normal is found below to be 1.39$.

To illustrate the method, and in accordance with Duncker's example,
A is here, exceptionally, calculated by rule page 20.

V— M

X

/

y

Si

Sa

<r

- 3.157

5.23

1

0.0

+ 1

-2.157

3.57

2

2.1

- 0.1

- 0.09

-- 1.157

1.92

189

200.4

- 10.6

-0.157

.26

1234

1213.0

+ 21.0

7.04

+ 0.843

1.40

454

474.0

-20.0

- 10.24

+ 1.843

3.50

20

11.9

+ 8.1

- 5.76

1901.5 60.8 23.1

100(60.8-88.1)
"1800

The values of y in the table above are calculated from the formula
y = y0 . e-^2/2^2. The sum of the theoretical y values should equal the
total number of variates.

OTHER UNIMODAL FREQUENCY POLYGONS.

The formulas of the remaining four types of unimodal simple fre-
quency polygons have a family resemblance with the formula

of the normal curve. They are as follows:
Curve of limited range on both sides:

Unsymmetrical, y - y0(l + — j7"1 (l - — )m'2, Type I.

V (I j' » Clj '

(x^\ in
1 ) , Type II.

(Jd *

Curve of range limited on one side:

Unsymmetrical, y = i/0(l +--)Pe ~ X' , Type III.

22 STATISTICAL METHODS.

Curves of unlimited range on both sides:

*))~fl — 1)Q {£

Unsyinmetrical, y = y0 cos 0" e , where tan 6 = -, Type IV.

CL

2(7-2

[Symmetrical, y = y0e , the normal curve.]

In these formulas :

y0 = modal ordinate, to be especially reckoned for each type.

y = the length of the ordinate (or area of rectangle) located at

the distance x from y0.
a = a part of the abscissa-axis XXT expressed in units of the

classes.
e = the base of the Naperian system of logarithms, 2.71828.

Curves of limited range are theoretically different from the
normal curve, which theoretically applies to cases where the classes
have an infinite range above and below the mean. Such an infinite
range is rare in biological statistics, although, as stated, the normal
curve often fits observational curves very closely. The range in
biological statistics may be limited at both extremes. Thus, the ratio
of carapace length to total length of the lobster is limited between 0
and 1.

The range may be limited on one side only. Thus the ratio

Antero-Post. Diam. .

— of a bivalve shell mar conceivably range from 0 to
Dorso-Veut. Diam.

co. The forms of the molluscan genera Pinna (or Malleus) and Solen
approach such extremes.

Asymmetry or skewness is found in Type I (of which Type II is
the symmetrical limit), 'l^-pe III and Type IV. In skew curves the mode
and the mean are separated from each other by a certain distance, d.
Asymmetry is measured by a factor

the result has the same sign as MS-
In Type I, A = ^ VPi~ •»-.
" " III, A = 14 V/F-

To compare any observed frequency polygon of Type
I with, its corresponding theoretical curve.

. x \?»i/ , a:\»i2

THE CLASSES OF FREQUENCY POLYGONS. 23

To find «j, «a, »»i, »"Q, ,'o

The total range, b, of the curve (along the abscissa axis) is found by
the equation

b = Vfrts + 2)»

Oj and a2 are the ranges to the one side and the other of t/0;

«! = y^b — da)\ d = a A = VVa • ^

a2 = i> — a i ;

nij = ~(s - 2); ?)*! -f m2 = s — 2;

To solve this equation it will be necessary to determine the value of
each parenthetical quantity following the r sign and find the corre-
sponding value of r from Table V. It is, however, sometimes easier to
calculate the value of yQ from the following approximate formula:

l/o = r •

(m, 4-

_L(_J_ _ _L _ JL)

"With these data the theoretical curve of Type I may be drawn. Fre-
quency polygons of Type I are found in biological measurements.

To compare any observed frequency polygon of Type
II \vith its corresponding theoretical curve.

This equation is only a special form of the equation of Type I in which
aj = «3 and ?»! = JH,. _

As from page 17, ^ = 0 in Type II, 6 = 2cr 4/s + 1 ; since the curve is
symmetrical, d = 0, and

b a rQn + 1.5)

a = — ; nt = J4(s-2); y0 = -

The r values will be found from Table V.

An approximate formula for y0 is given by Duncker as follows:

1

a s - 1 4(s- 2)

t/0 = -- =^ —
a VZ-ir V'(S

To compare any observed frequency curve of Type III
with its corresponding theoretical curve.

24 STATISTICAL METHODS.

The range at one side of the mode is infinite; at the other is found
by the formula

a = <r4 ~ fe. = cr1 ~ /*" (for Type III).
2 Yfr A

a a a. ]? + !

Also,

d «A ' a '

The value of r corresponding to p + 1 can be got from Table V,
Appendix.

To compare any observed frequency curve of Type
with its corresponding theoretical curve.

This is the commonest type of biological skew curves.

y =
0 is a variable, dependent upon x as shown in the equation

x = a tan 0.

The factor (cos 0)2m following y0 indicates that the curve is not calcu
lated from the mean ordinate (M), or the mode (M — d), but tliat the
zero ordinate is at M — md\ or at a distance m x d from the mean.

a = - |/ 16(8 -l)-^*-^; m = Y2(s + 2);

md = VW(S -

v= - - — - - — , with the opposite sign to MI ;

ir6°

0(arc of circle) =

180°'
(cos $Y*

r— e 3s 12.9

2/o = — 'V s-

v

= angle whose tangent is — .

s

* The foregoing value is approximate and is applicable when, as is
usually the case, s is greater than 2. The exact value is given by Pear-
son as

a e\w

Vo = —

a /»TT

/ (sin 0)'

t/0

Rev0d9

the formula for reducing which is to be gained from the integral calcu-
lus.

THE CLASSES OF FREQUENCY POLYGONS. 25

Example of calculating the theoretical curve corre-
sponding with observed data. (Fig. £4.)

Distribution of frequency of glands in the right fore leg of 2000 female
swine (integral variates):

Number of glands 012345678910
Frequency 15 209 365 482 414 277 134 72 22 8 2

Assume the axis yy' ( Vm) to pass through ordinate 4, then:

F

F - Ym f /(F — F»») f(V — F"02 /(?" — F»n)3

/(F- rm)«

0

— 4

15

— 60

240

— 960

3840

1

-3

209

— 627

1881

-5643

16929

2

— 2

365

-730

1460

— 2920

5840

3

— 1

482

-482

482

— 482

482

4

0

414

0

0

0

0

5

1

277

277

277

277

277

6

2

134

268

536

1072

2144

p*
i

3

72

216

648

1944

5832

8

4

22

88

352

1408

5632

9

5

8

40

200

1000

5000

10

6

2

12

72

432

2592

2 2000 —998 6148 —3872 48568

vj = — 998 -*- 2000 = - .499.
y2 = 6148 -f- 2000 = 3.074.
r, = — 3872 -*• 2000 = — 1.936.
v4 = 48568 -*- 2000 = 24.284.
Hl = M= 4 — .499 = 3.501.
M2 = 3.074 — (— .499)2 _ 0,304999.

M3 = - 1-036 - 8(- .499 X 3.074) + 2(- .499)' = 2.417278.
M4 = 24.284-4(-.499x - 1.936) + 6(.249001 X 3.074) - 3(- 499)* = 24.826297.
_ (2.417278)2 _ 5.843232929

PS — /.i oiijnnn.o — i- ••>.»->/•••».•»' — 3.110823.

(2.824999)3 22.545241683

24.826297 24.826297
(2.824999)2 ~~ ?.y«061935

F = 6 -f- 3 X 0.259178 - 2 X 3.110823 = -f 0.555888 (Type I).
6(3.11082 - 0.25918 - 1)

.55589

d = 1.680774 X .3111 = .5230.
d . s= .5230 X 19.9857 = 10.4519.

b = .840387 4/16 X 20.9857 + 0.25918 X (21.9857)2 = 18.0448.

18.0448 - 10.4519
cti = ~ = o.<96o.

26 STATISTICAL METHODS.

a2 = 18.0448 - 3.7965 = 14.2483.
3.7965 X 17.9857

18.0448

14.2483 X 17.9857
18.0448

=»• '8401-

2000 (18.9846) |/17\9846 0833(.0556 - .2643 - .0704)

X 2.1(1828

^n X 3.7840 X 14.2006
= 475.24, the number of cases in the modal class.

The equation of the theoretical curve is thus

3-784 / r \ 14-201

where x is the difference between the class magnitude and the
regarding signs.

Position of the mode, y0 = M — d = 3.501 - .523 - 2.978.

The mean percentage deviation of the theoretical ordinates from the

observed ordinates is 11. 4#* (Method A). This is calculated as follows:

V f y & %

observed theoretical

- 1 0 0.0 0.0

0 15 • 21.1 — 6.1 40.7

1 209 185.8 +23.2 11.1

2 365 395.1 —30.1 8.2

3 482 475.2 + 6.8 1.4

4 414 405.6 + 8.4 2.0

5 277 272.1 + 4.9 1.8

6 134 147.6 —13.6 10.2

7 72 65.9 -f 6.1 8.5

8 23 24.1 2.1 9.5

9 8 7.0 + 1.0 12.5
10 2 1.6 + 0.4 20.0

11 0 0.2 -

12 0 0.0 11.4*

MULTIMODAL, CURVES.

Multimodal curves are given when the frequency in the
different classes exhibits more than one mode. False multi-
modal curves result from too few observations, or when the
classes are made too numerous for the variates. By increas-
ing the number of variates or by making the classes more
inclusive some of the modes disappear.

* The mean percentage deviation by Duncker's determination with
method B using the same data is 1.73# of area.

THE CLASSES OJf

J X i-<J.b

. of
variateg

Classes..,

1

6

3

2345
FIG. 24.
Distribution of frequency in glands of swine.

, polygon of observed frequency.

— , polygon of theoretical frequency (Type I).

- - - -, normal frequency polygou.

10

STATISTICAL METHODS.

Multimodal curves differ iu degree. The modes may be so
close that only a single mode (usually in an asymmetrical
curve) appears in the result; or one of the modes may appear
as a hump on the other; or the two modes may even be far
apart and separated by a deep sinus (Figs. 25 to 28).

/

\

5.5 4.5 3.5 2.5 1.5 .5 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5
FIG. 25.

Pearson has offered a means of breaking up a compound
curve with apparently only one mode into two curves having
distinct modes; but this method is very tedious and rarely
applicable.

FIG. 26.

The index of divergence of two modes of a multi-
modal curve is the distance between the modes expressed in

THE CLASSES OF FREQUENCY POLYGONS.

terms of the standard deviation of the more variable of the
components.*

The index of isolation of two masses of variates
grouped about adjacent modes is the ratio of the depression
between the modes to the height of the shorter mode.

The meaning of rnultimodal curves is diverse. Sometimes

\

\

3210 12345678
FIG. 27.

they indicate a polymorphic condition of the species, the modes
representing the different type forms. This is the case with

32101234

01234

FIG. 28.

the number of ray flowers of the white daisy which has modes
at 8, 13, 21, 34, etc. Sometimes they indicate a splitting of a
species into two or more varieties.

* I have proposed (Science, VII, 685) to measure the divergence in a
unit = 3 X Standard Deviation, which has certain advantage in species
study.

30 STATISTICAL METHODS.

CHAPTER IV.

CORRELATED VARIABILITY.

tj

Correlated variation is such a relation between the magni-
tudes of two or more characters that any abmodality of the
one is accornpained by a corresponding abmodality of the
other or others.

The methods of measuring correlation depend upon the
assumption that the variates of the characters compared are
distributed normally about the mode. The method is approxi-
mately applicable to cases where the distribution of variates is
slightly skew.

TThe principles upon which the measure of correlated varia-
tion rest are these. "When we take individuals at random we
find that the mean magnitude of any character is equal to the
mean magnitude of this character in the whole population.
Deviation from the mean of the whole population in any lot of
individuals implies a selection. If we select individuals on the
basis of one character (A, called the subject) we select also any
closely correlated character (B, called the relative) (e.g. leg-
length and stature). If perfectly correlated, the index of
abmodality of B will be as great as that of A or

Index abmodality of relative 1
Index abmodality of subject

If there is no correlation, then whatever the value of the
index of modality of the subject, that of the relative will be
zero and the coefficient of correlation will be

Index of abmodality of relative 0
Index of abmodality of subject " m

The coefficient of correlation is represented in formulas by
the letter p. "\Ve cannot find the degree of correlation between
two organs by measuring a single pair only ; it is the correla-
tion '-'in the long run " which we must consider. Hence we
must deal with masses and with averages.

CORRELATED VARIABILITY.

31

1 I

w

t-O'WOrfCJOt-mrO
•^OSCOOJCOTTOlOi— i l-

02 ,M

Q 20

Cl

1

i-iOOOOi-iWCJCOCO

1 1 1

'tD "«
K >

g

tooooii'jTtiOi-'Csij;
c< i— c? c^ t^* T"* m t.-* T-H u-

^ 2Q

1

i-iOOOOi— ii-ii-iCiCO

1 1 1

4J —

c
n
o:

O^COGO'^O'— 'OOOO
OO CO "^ ^t* ^ l~ ^^ - ^ ^^ ^2
i-H??COCOC?Oi^3OSi-TT

0> o

c
1

C? T— ' O O T— < r-« CJ C? *~ CC 1O

1 1 1

cc

c

OOt'-COTtlO'-'OCOO
CT)OC5COOOi— f^OCOO

cocoi— i oo i- in •-< in oo o

l«

c

i— iCicoco^intocoi-o:

0
1—1

o
to

e* 1-1

OS

9

o in 03 c? i™<

00

*

r-l OS t- OS OJ W

fc-

-r

CO

CO i— O t- CO CO

1— » T— 1 1— 1

eo

0
TC

i-HOOococoinco

C^ lO "^ 1—1

o

O
1—1

I- CO I- !-• CO O CO — i

c« i.» o m d

-

d

COOOCOCOCJOOOi-i

i— i i— »

CO

1O

d
1

OS O CO O3 ^J* GO 1—1

OS I- 1-H Cl
1—1 1-H

-

Tf

in

TH

1

c

( oo ^r co t^ t-

m m oo c*

-

2

CJ

1

ir

3 1-1 m TJ< m i-H

m to 1-1

o

CO

1

a

D Tfi CJ

tub

0>

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"3

.2 n i

•<-> c T

cj C «

t, }~ I, CO CO CO CO CO CO V

J

1

i-i

"3

CH

4-1

O

'> d o

T 7 °

>

Classes <

Deviations

^ *™^ c

co 43
c3 fac

S'G

5 T^ d OO ^* *O ?O t^ QO Oi C

^

9

4

COCO

I- I-

a>
•a

•d

02

o in

coco

II i!
<s a>

"315

sa

43 ->-»

'C— •
co"

CO '

"Sb

"o-

32 STATISTICAL METHODS.

In studying correlation one (either one) of the characters is
regarded as subject and the other as relative. A correlation
table is then arranged as in the example on page 29, which
gives data for determining the correlation between the num-
ber of Miillerian glands on the right (subject) and left (rela-
tive) legs of male swine.

METHODS OF DETERMINING COEFFICIENT OF CORRELATION.

Galton's graphic method. On co-ordinate paper
draw perpendicular axes X and T '; locate a series of points
from the pairs of indices of abmodality of the relative and sub-
ject corresponding to each subject class. The indices of the
subjects are laid off as abscissae ; the indices of the relatives
as ordinates, regarding signs. Get another set of points by mak-
ing a second correlation table, regarding character B as subject
and character A as relative. Then draw a straight line through
these points so as to divide the region occupied by them into
halves. The tangent of the angle made by the last line with
the horizontal axis XX (any distance yp, divided by xp) is the
index of correlation.

A more precise method is given by Pearson as follows:
Sum of products (deviation subj. class X deviation each assoc.
rel. class X no. of cases in both)

total no. ofludivs. X Stand. Dev. of subject x Stand. Dev.

of relative ;

or, expressed in a formula :

2 (dev. x X dev. y X /)
p =

This method requires finding many products in the numera-
tor, as many sets of products as there are entries in the body of
the correlation table. A portion of the pioducts to be found
is indicated below ;

(- 3.540 X 8

- 3.547 X •!- 2.540 X 5

(- 1.540 X 2

f_ 3.540 X 4
| - 2.540 X 151

- 2.547 X -{ - 1-540 X 58

| - 0.540 X 9

L_ o.460 x 3

etc.

COKRELATED VARIABILITY. 33

A brief method of finding p is given by Duncker as follows:

i -, p _v.-/. ojXdev. .yx/) , 1

p is composed of two factors: - - and -

n cr,(

To find -v~~/.gxclev. y X /),

M

Separate the deviation from the mean of each class into its
integral and its fractional parts ; the fractional parts for all
classes below the mean will be equal to the fractional part of
the mean ; of all classes above the mean, to the complement
of that number. Designate the integral parts of the variants
of the subject by ± X^; of the relatives by ± X^, and the frac-
tional complement parts of the means of subject or relative by
£1, c2. Let /equal the frequency of any deviation in the com-
bination X.X2, as shown in the correlation table. Draw rect-
angular co-ordinates as shown on page 34 through the zero-
point of the correlation table. Number the N. W. quadrant,
which should include negative deviations of both subject and
relative variants, I ; the N. E. quadrant, II ; the S. "W.
quadrant containing solely positive deviations III ; and the
S. E. quadrant, IV. Then if 2i, 2U, etc., indicate a summa-
tion for the quadrant I, II, etc., and having regard to signs :

n n

The numerator of this fraction consists entirely of whole
numbers ; of them the following are on their own account

positive: 2I(fXlXJ, ^nifX.X^, 2i(f), 2n(fX,),
negative :

Rule : (1) Find products of integral parts of deviations of
both subject and relative and the combination frequency, for
all four quadrants, and take their sum.

(2) Subtract successively the sum. of the products of the sub-
ject deviations in the first quadrant multiplied by the fre-
quency, and the sum of the products of the relative deviations

34

STATISTICAL METHODS.

in the first quadrant multiplied by the frequency. Since these
are negative values they will be actually added.

(3) Add the sum of the numbers in the first quadrant ; sub.
tract the sum of the products of the integral parts of the rel-
ative deviations by the frequency in the second quadrant ;
subtract the sum of the products of the subject deviations of

he third quadrant multiplied by their frequency.

(4) Divide the algebraic sura of (1), (2), and (3) by the number
of variates, and from the quotient subtract the product of the
complement-fractional parts of the mean value of the subject
and relative.

To get p, divide

n

•by the product of <TI and o-2.

The probable error of the determination of /j is

0.6745(1 - p')

-

p)

Example. Correlation in number of Mullerian glands on
right and left legs of 2000 male swine.

Mean, right leg, = 3.5465 ;
<TJ = 1.7195;

Mean, left leg, = 3.5395
0-3 = 1.7304

Right leg, subject: Left leg, relative.

X,

- 3

2

- 1

0

0

1

o

tO

3

456

Rel.

class

0

1

2

o

O

4

5

6

7

8 9 10

Sub.

class

(I)

(ID

0 -

- 3

8

5

2

1 •

- 2

4

151

58

9

3

2 -

- 1

2

65

154

96

28

7

1

3

0

14

88

173

128

28

6

4

0

5

27

119

153

77

26

3

1

5

1

1

7

24

92

101

52

11

9

6

2

8

16

58

48

16

7 2

7

3

1

8

20

18

17

9 5

8

4

1

3

5

3

2 2

9

5

1

3

3

221

10

6 (III)

1 (IV)

CORRELATED VARIABILITY. 35

X2) = 1142-9— 9+1652 ")

= + 806* + 814 + 829 J- -*- n = ^ = 2.5625
- SnAfXi) 200°

= - 49 - 51 J

- 5l5, = - 4535 X .4605 = ^||f

o-lCr2 = 1.7195 X 1.7304 = 2.9754; p = -|4^ = -7919

£.0 1 04

4/2000 X 1.627

SPURIOUS CORRELATION IN INDICES.

When two characters A and 5 are measured in each individual
of a series of individuals, and each absolute magnitude is trans-
formed into an index by dividing it by the magnitude of a
third character (?as found in the same individual, a spurious
correlation will be found to exist between the indices of
A . B
C and C •

Let ?'i = the coefficient of variation of A ;
Da = " " " " « B;

V3 = " " " '< " C;

Po = " " " spurious correlation.

The precise method of using p0 in modifying any determina-
tion of p is uncertain. Pearson recommends using p — p0 as
the true measure of "organic correlation" in the case of
indices.

HEREDITY.

Heredity is a certain degree of correlation between the
abmodality of parent and offspring. The statistical laws of
heredity deal not with relations between one descendant and

36 STATISTICAL METHODS.

its parent or parents, but only with mean progeny of mean
parents. Any group of selected parents is called a parentage,
the progeny of a parentage is called a fraternity.

In uiiipareutal inheritance, as in budding or asexual
generation, heredity of any character is measured by the coef-
ficient of correlation between the abmodality in a parentage
arid the abmodality Jof the corresponding fraternity. More
strictly, since the variability of the character in the second
generation, cr2, may (as a result of selection or of environ-
mental change) be different from the variability of the char-
acter in the first generation, cr,, the index should be taken as

P —•

<T2

The probable error of this determination is

,67450-, /I - p,,2

-4/ — , m which p12 means the correlation coem-

cr2 n

cient between the filial character and that of the single parent
under consideration.

The variability of the fraternity is to variability of offspring

in general as |/1 — p- is to 1.

In bipareutal inheritance, if there is no evidence of
assortative mating, or correlation between the two parents in
the character in question, the mean abmodality of any frater-
nity will be

'7

11 + P-2 fi-3,

CT2 0-3

where 7ii .= average abmodality of fraternity ;

A2 = average abmodality of male parent ;

7i3 = average abmodality of female parent ;

p, — correlation coefficient between fraternity and

female parent ;
pa = correlation coefficient between fraternity and male

parent ;

o-j = standard deviation of fraternity ;
<r2 = standard deviation of male parent ;
<r3 = standard deviation of female parent.

CORRELATED VARIABILITY. 37

When assortative mating occurs, as is usually case, the
abmodality of a fraternity is given by

p3 — pip* cr, Pi — Pips

711 = ' A* +

where pi = correlation between male and female parents.
The other letters have the same signification as before.

The strength of heredity in assortative mating is measured
by the formula

1 - Pi9 ' oV

Galton ('97) has shown that an individual inherits not only
from his parents, but also from his grandparents, great-grand-
parents, and so on. The heritage from his 2 parents together
is, on the average, 50$ or £ of the whole ; from the 4 grand-
parents 25$ or ^ ; from the 8 great-grandparents 12.5$ or £ ;

from the ?ith ancestral generation — of the whole ; the total

/w

heritage adding up 100$. This law has been generalized by
Pearson ('98) as follows :

1 <TO 1 <TO 1 cr0 1 CTO

ill = n --- «i -f -#a +5 --- *a + rn — &4 + • •

2 <T! 4 o-2 8o-3 Ib cr4

where /i! = average abmodality of fraternity.
cr0 = standard deviation of fraternity.
(T! , (Ji . . . crs = standard deviation of mid-parent of

1st, 2d ... *th ancestral generation.
ki = abmodality of mid-parent of 1st ancestral genera-

tion.

•

&a, k3 . . . ks = abmodality of mid-parent of 2d, 3d

. . . sth ancestral generation.

The abmodality of the mid-parent of any degree of ancestry
may be taken as the average abmodality of all the contributory
ancestors of that generation.

38 STATISTICAL METHODS.

CHAPTER V.
SOME APPLICATIONS OF STATISTICAL BIOLOGICAL STUDY.

The Laws of Variation. Darwin and others have
formulated certain laws of variation, such as the law that
specific characters are more variable than generic ones ; that
highly aberrant characters are more variable than more usual
ones ; that males are more variable than females. These laws
can be established only by a determination of the Index or
Coefficient of Variation in critical cases.

The causes of variation can be determined only by t
quantitative study of the relation between specific change and
environmental change, or a knowledge of the degree and fre-
quency of sports.

The effect of selection in causing a greater death rate on
one side of the mean than on the other side — the production of
skewuess — requires the quantitative method for its complete
study. The change in the mode and in the index of skewness
measures the progress of the effect of selection.

The origin of species through geographical segrega-
tion can be studied by the determination of place-modes ; that
is, the modal condition of specific characters of oue and the
same species in various localities. The progress of specific
differentiation will be measured by the change in place-modes
from decade to decade, or by the formation of a binomial curve
in the place of a modal one ; and by the gradual separation of
the two modes of a binomial gurve.

The definition of species may be improved by being
rendered more quantitative. The relative importance of the
various criteria used in separating species may be determined
by finding that character in which there is least intergrading
between the modal condition characteristic of the two races.
Thus if for two species or varieties of birds both total length
and form of bill show two modes, the better criterion is that
in which the modes are farthest apart or in which the inter-
grades are fewest.

STATISTICAL BIOLOGICAL STUDY. 39

A basis for an arbitrary distinction between species

and varieties may be gained by determining a degree of
divergence and of isolation which shall be used to distinguish
the two. A degree of divergence of thrice the standard devia-
tion has been suggested as a convenient line between species
and varieties.

Quantitative studies in correlation will give us new cri-
teria for homology by telling us the relative morphoge-
netic kinship of the parts of the body.

Quantitative studies in heredity will give definitive informa-
tion on prepotency of sex or race. By examining hybrids
quantitatively and comparing them with their parents we shall
unravel the laws of inheritance in cross-breeding and the prin-
ciples of mixing characters in biparental inheritance.

In a word, by the use of the quantitative method biology
will pass from the field of the speculative sciences to that of
the exact sciences.

40 STATISTICAL METHODS.

SELECTED BIBLIOGRAPHY

OF WORKS ON THE QUANTITATIVE STUDY OF ORGANISMS.

AMANN, J., '96. Application du calcul des probabilites a
1'etude de la variation d'un type vegetal. Bull, de
1'Herb. Bossier. Geneve et Bale. IV, 578-590.

BREWSTER. E. T., '97. A Measure of Variability and the
Relation of Individual Variations to Specific Differences.
Proc. Amer. Acad. Arts and Sci., XXXII, 268-280.

BUMPUS, H. C., '97. The Variations and Mutations of the
Introduced Sparrow. Biol. Lect. Woods Holl, 1896, 1-15.

BUMPUS, H. C., '98, The Variations and Mutations of the
Introduced Littoriua. Zool. Bull., I, 247-259.

DAVENPORT, C. B., and J. W. BLANKINSHIP, '98. A Precise
Criterion of Species. Science, VII, 685-695.

DAVENPORT, C. B., and C. BULLARD, 96. Studies in Morpho-
genesis, VI. A Contribution to the Quantitative Study
of Correlated Variation and the Comparative Variability
of the Sexes. Proc. Amer. Acad. Arts and Sci. , XXXII,
85-97.

DUNCKER, G., '97. Correlation Studieu an den Strahlzahlen
eiuiger Flossen von Acerina cernua~L. Biol. Centralbl.,
XVII, 785-794 ; 815-831.

DUNCKER, G., '98. Bemerkung zu dem Aufsatz von H. C.
Bumpus " The Variations and Mutations of the Introduced
Littorina." Biol. Centralbl., XVIII, 569-573.

DUNCKER, G., '99. Die Methode der Variations-Statistik.
Arch. f. Entwickelungs-Mechan. d. Orgauismeu, VIII,
112-183. [The most important elementary presentation
of the subject ; extensive, nearly complete bibliography.]

EIGENMANN, C. H., '95. Leuciscus balteatus (Richardson),
a Study in Variation. Amer. Naturalist, XXIX, 10-25,
Pis. 1-5.

EIGENMANN, C. H., '96. The Study of Variation. Proc.
Indiana Acad. Sci., V, 265-278. [Extensive bibliography.]

FECHNER, G. T.,'97. Kollektivmasslehre. Im Auftrage der
Koniglich Sachsischen Gesellschaft der Wisseuschaften
herausgegebeu von Gottl. Friedr. Lipps. Leipzig : Engel
rnann. 483pp. [Important but too much neglected work.]

SELECTED BIBLIOGRAPHY. 41

FIELD, W. L. W., '98. A Contribution to the Study of Indi-
vidual Variation in the Wings of Lepidoptera. Proc.

Amer. Acad. Arts and Sci., XXXIII, 389-395.
GALTON, F.,'88. Correlations and their Measurement, chiefly

from Anthrdpornetric Data. Proc. Roy. Soc. London,

XLV, 136-145.

GALTON, F., '89. Natural Inheritance. London : Macmillan.
GALTON, F. '97. The Average Contribution of each several

Ancestor to the total Heritage of the Offspring. Proc.

Roy. Soc. London, LXI, 401-413.
LUCAS, F. C.,'98. Variation in the Number of Ray-flowers in

theWhiteDaisy. Amer. Naturalist, XXXII, 509-511. 2figs.
LUDWIG, F. , '95. Ueber Variatiouskurven uud Variations-

flachen der Pflanzen. Bot. Centralbl.., LXIV, 1-8 et folg.

2 Tafu.
LUDWIG, F., '96. Weiteres tiber Fibonacci- Kurven uud die

numerische Variation der gesammteu Bliithenstande der

Kompositeu. Bot. Centralbl. LXVIII, 1 et folg. 1 Taf.
LUDWIG, F., '96. Eine fiinfgipfplige Variations-Kurve. Ber.

deutsch. Bot. Ges., XIV, 204-207. 1 fig.
LUDWIG, F., '98. Die pflanzlichen Variation s-Kurven und die

Gauss 'sche "Wahrschemlichkeitskurve. Bot. Centralbl.,

LXXIII, 241-250 et folg. 1 Taf.
LUDWIG, F., '98. Ueber Variationskurven. Bot. Ceutralbl.,

LXXV, 97-107 ; 178-183. 1 Taf.
MOENKHAUS, W. J., '96. The Variation of Etheostoma cap-

rodes Rafinesque in Turkey lake and Tippecanoe lake.

Proc. Indiana Acad. Sci., V., 278-296.
PEARSON, K., '94. Contributions to the Mathematical Theory

of Evolution. [I. On the Dissection of Frequency

Curves.] Phil. Trans. Roy. Soc. London, CLXXXV,

A, 71-110. Pis. 1-5.
PEARSON, K., '95. Contributions, etc., II. Skew Variation

in Homogeneous Material. Phil. Trans. Roy. Soc.

London, CLXXXVI, A, 343-414. 10 Pis.
PEARSON, K., '96. Mathematical Contributions to the Theory

of Evolution, III. Regression, Heredity, and Panmixia.

Phil. Trans. Roy. Soc. London, CLXXXVII, A, 253-318.
PEARSON, K., '97. Mathematical Contributions, etc. On a

Form of Spurious Correlation, which may Arise when

42 STATISTICAL METHODS.

Indices are used in the measurement of Organs. Proc.
Roy. Soc. London, LX, 489-498.

PEARSON, K. '98. Mathematical Contributions, etc. On the
Law of Anrestral Heredity. Proc. Roy. Soc. London,
LXII, 386-412.

PEARSON, K., and L. N. G. FILON, '98. Mathematical Con-
tributions, etc., IV. On the Probable Errors of Frequency
Constants and on the Influence of Random Selection on
Variation and Correlation. Phil. Trans. Roy. Soc.
London, CXCI, A, 229-311.

THOMPSON, H., '94. On Correlations of Certain External
Parts of Palaemon serralus. Proc. Roy. Soc. London,
LV, 234-240.

VERSCHAFFELT, E., '95. Ueber Asymmetrische Variations-
kurven. Ber. deutsch. Bot. Ges., XIII, 348-356. 1 Taf.

DE VRIES, H., '94. Ueber halbe Galton-Kurven als Zeichnen
diskontinuirlichen Variation. Ber. deutsch. Bot. Ges.,
XII, 197-207. Taf. X.

DE VRIES, H., '95. Eine zweigipfelige Variatious-Kurve.
Arch. f. EntwickeluDgsraechauik, IL 52-65. 1 Taf.

WARREN, E., '96. Variation in Portunus depurator. Proc.
Roy. Soc. London, LX, 221-243.

WARREN, E., '97. An Investigation on the Variability of the
Human Skeleton with Especial Reference to the Naquada
Race. Phil. Trans. Roy. Soc. London, CLXXXIX, B,
135-227. PI. 22.

WELDON, W. F. R. , '90. The Variations occurring in Certain
Decapod Crustacea, I : Crangon vulgaris. Proc. Roy.
Soc. London, XLVII, 445-453.

WELDON, W. F. R., '92. Certain Correlated Variations in
Crangon vulgaris. Proc. Roy. Soc. London, LI, 2-21.

WELDON, W. F. R., '93. On Certain Correlated Variations in
Carcinus maenas. Proc. Roy. Soc. London, LIV, 318-
329.

WELDON, W. F. R., '95. Report of the Committee for Con-
ducting Statistical Inquiries into the measurable Char-
acteristics of Plants and Animals. Part I ; An Attempt
to Measure the Death-rate due to Selective Destruction of
Carcinus maenas with respect to a Particular Dimension.
Proc. Roy. Soc. London, LVII, 360-379.

EXPLANATION OF TABLES.

43

EXPLANATION OF TABLES.

I. Formulas. In this table the principal formulas used
in the calculation of curves are brought together for conven-
ient reference. The meanings of the letters are explained in
the text.

II. Certain constants and their logarithms.

This table includes the constants most frequently employed
in the calculations of this book.

III. Table of ordinates of normal curve. This
table is for comparison of a normal frequency polygon con-
sisting of weighted ordinates with the theoretical curve.

Example: M = 14.157 ; cr = 0.604 ; Iy0 = 1255.

(See page 19.)

Entries in Table
V - M corresponding to

2/o V f

X 1255 = 0.0 1
X 1255 = 1.8 2
X 1255 = 20L8 189

IV. Table of values of probability integral.

This table is for comparison of a normal frequency polygon
consisting of rectangles with the theoretical curve.

Example: M. 5.24 ; cr = 0.987. (See page 12).

X -4- cr

Y
11

12
13

- 3.157
- 2.157
- 1.157

<T

5.2
3.6
1.9

V — M

cr

.000004
.0015
.164

/

*

Class
limits

Deviation x
from
M = x

1

4

3.0

-2.24

-2.27

1

4

3.5

-1.74

-1.76

3

12

4.0

-1.24

-1.26

3

12

4.5

-0.74

-0.75

7

28

5.0

-0.24

-0.25

5

20

5.5

0.26

0.27

3

12

6.0

0.76

0.77

1

4

6.5

1.26

1.28

1

4

7.0

1.76

1.78

25

100

7.5

2.26

2.29

0

48 . 8

46.1

o9.o

27.3 '
The
9.91 20.5

lo.eUc?

Ob
27 9 .

oret
55.2

'do
serv
|

ical
79.6

•

ed

fre
08.4

}

Q2
fre

40.0

46 3

48.9..

quency
97.7

100
quency

44 STATISTICAL METHODS.

In the example, the curve of which is shown in Fig. 22, the
frequency between the limits is given in column /; the fre-

x
quency reduced to percents in column headed %. The - '- of

the limit is found and the entries in Table IV corresponding
to the quotient are taken. These are added in pairs as indi-
cated, one above and one below the mean, and the sum is
compared with the sum of the observed cases within those
limits (in italic figures). The closeness of agreement indicates
the closeness with which the observed frequency follows the
normal frequency.

V. Table of log T functions of q. This table
will enable one to solve the equations for y0 given on page 23.
The table gives the logarithms of the values of F functions
only within the range p = 1 to 2. As all values of the func-
tion within these limits are less than 1, the mantissa of the
logarithms is — 1 ; but it is given in the table as 10 — 1 = 9,
as is usually done in logarithmic tables.

Supposing the quantity of which we wish to find the value
reduced to the form r(4.273). The value cannot be found
directly because the value of p is larger than the numbers in
the table (1 to 2). The solution is made by aid of the equation

logr(1.273) = 9.955185
log 1.273 =0.104828

logT(2.273) = 0.060013
log 2.273 =0.356599

log r(3. 273) = 0.416612
log 3.273 =0.514946

log T(4. 273) = 0.931558

or, more briefly, log r(1.273) = 9.955185

log 1.273 = .104828
log 2.273 = .356599
log 3.273 = .514946

log T(4.273) = 0.931558 = log 8.542

EXPLANATION OF TABLES. 45

VI. Table of reduction from the common to
the metric system. This is given first for whole inches
from 1 to 99 excepting even tens, which may be got from the
first line of figures by shifting the decimal point one place
to the right. The table may be used for huudredths of an
inch by shifting the decimal point two places to the left.
Other fractious than decimals are given in the lower tables.

VII. First to sixth powers of integers from

1 to 3O. This table is useful in calculating moments.

VIII. Squares, cubes, square roots, and re-
ciprocals of numbers from 1 to 1O54. The use

of this table can be extended by using the principle that if auy
number be multiplied by n, its square is multiplied by ?is, its

cube by n3, and its reciprocal by — .

IX. Logarithms of numbers to six places.

The following explanation of the use of the logarithmic tables
is taken from Seaiies' Field Engineering, pp. 257-263 [ed.

1887].

APPENDIX IX. — The logarithm of a number consists of
two parts, a whole number called the characteristic, and a deci-
mal called the mantissa. All numbers which consist of the
same figures standing in the same order have the same man-
tissa, regardless of the position of the decimal point in the
number, or of the number of ciphers which precede or follow
the significant figures of the number. The value of the char-

o o

acteristic depends entirely on the position of the decimal point
in the number, It is always one less than the number of
figures in the number to the left of the decimal point. The
value is therefore diminished by one every time the decimal
point of the number is removed one place to the left, and vice
versa. Thus

Number. Logarithm.

13840. 4.141136

1384.0 3.141136

138.40 2.141136

13.84 1.141136

1.384 0.141136

.1384 —1.141136"

.01384 —2.141136

.001384 —3.141136

etc. etc.

46

STATISTICAL METHODS.

The mantissa is always positive even when the characteristic
is negative. We may avoid the use of a negative characteristic
by arbitrarily adding 10, which may be neglected at the closf
of the calculation. By this rule we have

Number. Logarithm.

1.384 0.141136

.1384 9.141136

.01384 8.141136

.001384 7.141136
etc, etc.

No confusion need arise from this method in finding a number
from its logarithm; for although the logarithm 6.141136 repre-
sents either the number 1,384,000, or the decimal .0001384, yet
these are so diverse in their values that we can never be uncer-
tain in a given problem which to adopt.

The table IX. contains the mantissas of logarithms, car-
ried to six places of decimals, for numbers between 1 and 9999,
inclusive. The first three figures of a number are given in the
first column, the fourth at the top of the other columns. The
first two figures of the mantissa are given only in the second
column, but these are understood to apply to the remaining
four figures in either column following, which are comprised
between the same horizontal lines with the two.

If a number (after cutting off the ciphers at either end) con-
sists of not more than four figures, the mantissa may be taken
direct from the table ; but by interpolation the logarithm of a
number having six figures may be obtained. The last column
contains the average difference of consecutive logarithms on
the same line, but for a given case the difference needs to be
verified by actual subtraction, at least so far as the last figure
is concerned. The lower part of the page contains a complete
list of differences, with their multiples divided by 10.

To find the logarithm of a number having six
figures :— Take out the mantissa for the four superior places
directly from the table, and find the difference between this
mantissa and the next greater in the table. Add to the man-
tissa taken out the quantity found in the table of proportional
parts, opposite the difference, and in the column headed by the
fifth figure of the number; also add ^ the quantity in the col-
umn headed by the sixth figure. The sum is the mantissa
required, to which must be prefixed a decimal point and the
proper characteristic.

EXPLANATION" OF TABLES. 47

Example.—Fiud the log of 23.4275.

For 2342 mantissa is 369587

" diff. 185 col. 7 129.5

" " " " 5 9.2

Ans. For 23.4275 log is 1.369726

The decimals of the corrections are added together to deter-
mine the nearest value of the sixth figure of the mantissa.

To find the number corresponding to a given
logarithm. — If the given mantissa is not in the table find the
one next less, and take out the four figures corresponding to it;
divide the difference between the two mantissas by the tabu-
lar difference in that part of the table, and annex the figures of
the quotient to the four figures already taken out. Finally,
place the decimal point according to the rule for characteristics,
prefixing or annexing ciphers if necessary. The division re-
quired is facilitated by the table of proportional parts, which
furnishes by inspection the figures of the quotient.

Example. — Find the number of which the logarithm is

8.263927 8.263927

First 4 figures 1836 from 263873

Diff.

Tabular diff. =236 .-. 5th fig. = 2

6th fig. = 3
Ans. No. = .0183623 or 183,623,000.

The number derived from a six-place logarithm is not
reliable beyond the sixth figure.

At the end of table XXIV. is a small table of logarithms of
numbers from 1 to 100, with the characteristic prefixed, for
easy reference when the given number does not exceed two
digits, But the same mantissas may be tound in the larger
table.

APPENDIX X.— The logarithmic sine, tangent,
etc. of an arc is the logarithm of the natural sine, tangent,
etc. of the same arc, but with 10 added to tne characteristic to
avoid negatives. This table gives log sines, tangents, cosines,
and cotangents for every minute of the quadrant. With the
number of degrees at the lef t side of the page are to be read
the minutes in the left-hand column ; with the degrees on

48 STATISTICAL METHODS.

the right-hand side are to be read the minutes in the right-hand
column. When the degrees appear at the top of the page the
top headings must be observed, when at the bottom those at
the bottom. Since the values found for arcs in the first quad-
rant are duplicated in the second, the degrees are given from
0° to 180°. The differences in the logarithms due to a change
of one second in the arc are given in adjoining columns.

To find the log. sin, cos, tan, or cot of a given
arc. : Take out from the proper column of the table the log-
arithm corresponding to the given number of degrees and
minutes. If there be any seconds multiply them by the ad-
joining tabular difference, and apply their product as a cor-
rection to the logarithm already taken out. The correction is
to be added if the logarithms of the table are increasing with
the angle, or subtracted if they are decreasing as the angle in-
creases. In the first quadrant the log sines and tangents in-
crease, and the log. cosines and cotangents decrease as the
angle increases.

Example.— Find the log sin of 9° 28' 20".

Log sin of 9° 28' is 9.216097

Add correction 20 X 12.62 252

Ans. 9.216349
Example.— Find the log cot of 9° 28' 20".

Log cotan of 9° 28' is 10.777948

Subtract correction 20 X 12.97 259

Ans, 10777689

To find the angle or arc corresponding to a
given logarithmic sine, tangent, cosine, or co-
tangent.— If the given logarithm is found in the proper
column take out the degrees and minutes directly; if not, find
the two consecutive logarithms between which the given
logarithm would fall, and adopt that one which corresponds to
the least number of minutes; which minutes take out with the
degrees, and divide the difference between this logarithm and
the given one by the adjoining tabular difference for a quo-
tient, winch will be the required number of seconds.

With logarithms to six places of decimals the quotient is
not reliable beyond the tenth of a second.

EXPLANATION" OF TABLKS. 49

Example. — 9.383731 is the log tan of what angle?
Next less 9.383682 gives 13' 36'

Diff. 49.00 -f- 9.20 = 05".3

Ans. 13° 36' 05".3

Example. — 9.249348 is the log cos of what angle?
Next greater 583 gives 79° 46'

Diff. 235 -r- 11.67 = 20M

Ans. 79° 46' 20 M

The above rules do not apply to the first two pages of this
table (except for the column headed cosine at top) because
here the differences vary so rapidly that interpolation made by
them in the usual way will not give exact results.

On tbe first two pages, the first column contains the number
of seconds for every minute from I'to2°; the minutes are
given in the second, the log. sin. in the third, and in the fourth
are the last three figures of a logarithm which is the difference
between the log sin and the logarithm of the number of sec-
onds in the first column. The first three figures and the char-
acteristic of this logarithm are placed, once for all, at the head
of the column.

To find the log sin of an arc less than 2° given
to seconds. — Reduce the given arc to seconds, and take the
logarithm of the number of seconds from the table of loga-
rithms, and add to this the logarithm from the fourth column
opposite the same number of seconds. The sum is the log sin
required.

The logarithm in the fourth column may need a slight inter-
polation of the last figure, to make it correspond closely to the
given number of seconds.

Example.— Find the log sin of 1° 39' 14". 4.

1° 39' 14".4 = 5954".4 log 3.774838

add (q — I) 4.685515

Ans. log sin 8.460353

Log tangents of small arcs are found in the same way, only
taking the last four figures of (q — I) from the fifth column.

50 STATISTICAL METHODS.

Example.— Find the log tan of 0° 52' 35".

52' 35" = (3120" -f 35") = 3155" log 5.498999

add (q — I) 4.685609

Am. log tan 8. 184608

To find the log- cotangent of an angle less than
2° given to seconds. — Take from the column headed ( q-\- 1)
the logarithm corresponding to the given angle, interpolating
for the last figure if necessary, and from this subtract the loga-
rithm of the number of seconds in the given angle.

Example.— Find the log cotan of 1° 44' 22". 5.

q -f I 15.314292
6240" + 22". 5 = 6262.5 log 3.796748

Am. 11.517544

These two pages may be used in the same way when the
given angle lies between 88° and 92° , or between 178° and 180°;
but if the number of degrees be found at the bottom of the page,
the title of each column will be found there also; and if the
number of degrees be found on the right hand side of the page,
the number of minutes must be found in the right hand col-
umn, and since here the minutes increase upward, the number
of seconds on the same line in the first column must be dimin-
ished by the odd seconds in the given angle to obtain the num-
ber whose logarithm Is to be used with ( q ± I) taken from the
table.

Example.— Find the log cos of 88° 41' 12". 5

(q-Z) 4.685537
4740" - 12'.5 = 4727.5 - log 3.674631

Ans. 8.360168

Example.— Find the log tan of 90° 30' 50".

q + I 15.314413
1800" -f 50" = 1850* log 3.267172

Ans. T27047241

To find the arc corresponding1 to a given log
sin, cos, tan, or cotan which falls within the
limits of the first two pages of Table X.

Find in the proper column two consecutive logarithms be-
tween which the given logarithm falls. If the title of the
given function is found at the top of that column read the

EXPLANATION OF TABLES. 51

degrees from the top of the page; if at the "bottom read from
the bottom.

Find the value of (q — 1} or (q -f- 1), as the case may require,
corresponding to the given log (interpolating for the last figure
if necessary). Then if q = given log and I = log of number of
seconds, n, in the required arc, we have at once I = q — (q — I)
or I = (q -\- 1) — q, whence n is easily found.

Find in the first column two consecutive quantities between
which the number n falls, and if the degrees are read from
the left hand side of the page, adopt the less, take out the
minutes from the second column, and take for the seconds
the difference between the quantity adopted and the number
n. But if the degrees are read from the right hand side of the
page, adopt the greater quantity, take out the minutes on the
same line from the right-hand column, and for the seconds
take the difference between the number adopted and the num-
ber n.

Example.— 11.734268 is the log cot of what arc?

q + I 15.314376

q 11.734268

.-. n = 3802.8 3.580108

For 1° adopt 3780. giving 03'

Difference 22". 8

Ans. 1° 03' 22".8 or 178° 56' 37".2.

Sample. — 8.201795 is the log cos of what arc?

q - I 4.685556

q 8.201795

/. TI = 3282". 8 3.516239

For 89° adopt 3300. giving 05'

Difference 17 ".2

Ans. 89° 05' 17". 2 or 90° 54' 42". 8.

52 STATISTICAL METHODS.

I. -FORMULAS.

M = - = Vm - Vl. P.E.M = ± 0.6745-4=. x = V - M.

.

_ . _ 0>7979(ri p£ _ o.6745<r.
n

"i2(+ I) = 2(aV/)(+ I).

« v

Ms2

% (for graduated variates) = i ~ a . 100*.

>v?l
S

A^ (for integral variates) = 2^,— 7. . 100$, where fc equals the number of classes.

/ . K

2(dev. x . dev. y . /)
p = -
?i(T1cr3

n n

0.6745(1 - p«)

Po (spurious correlation) =

(J-J

(index of heredity, uniparental inheritance) = p— .

ii = pj— 7i2 + PO— 7i3 [biparental inheritance; unassortative mating].

- -

_ Pi ~ PiPa ^ fj;, j^ Pa "" PiP« . ^i . /j. [biparental inheritance; assortative
1 - Pi ^a 1 ~ Pi °"a

mating],

CERTAIN" CONSTANTS AtfD THEIR LOGARITHMS. 53

II.— CERTAIN CONSTANTS AND THEIR LOGARITHMS.

Title.

Symbol

Number.

Log.

Ratio of circumference to diameter

n
1

7T

4/iT
l

3.1415927
0.3183099

1.7724538
0.5641896
2.506628

0.3989422

2.7182818
0.4342945

2.3025851

0.4971499
9.5028501

0.2485749
9.7514251
0.399090

9.6009100

0.4342945
9.6377843

0.3622157

Square root of same

Reciprocal of square root of same

V^T

4/2^

1

Square root of 2n-

Reciprocal of same

4/2T

e

m

1
m

Base of hyperbolic logarithms

Modulus of common system of logs = log e . . .
Reciprocal of same — hyp. log 10

Com. log x — m X hyp. log x, or
Com.log(com.logo;)=9.6377843+com.log(hyp.loga;)
Hyp. log x = com. log x x — , or

7/2' .

Com. log(hyp. log aO^com.logfcom.logaO-f 0.3622157
Circumference of circle —

2nr

JIT2

Yzlr

-2-irr"
360

najor axis
• axis of el

Area of sector (length of arc ~ 1)

Area of sector (angle of arc — a°)

/a,2 _ 52

minoi

; 6 — semi-
lipse.

54

STATISTICAL METHODS.

III.— TABLE OF ORDINATES OF NORMAL CURVE, OR VALUES_OF

— CORRESPONDING TO VALUES OF -.

2/o *

x = deviation from mean, y = frequency.

a- = standard deviation. y0 = — •_ = maximum frequency.

X/<r

y/Vo

x/<r

V/Vo

X/ff

V/Vo

X/tr

y/y0

0

i.

0.8

.7262

1.6

.2780

2.8

.0198

0.1

.9950

0.9

.6670

1.7

.2357

3.0

.0111

0.2

.9802

1.0

.6065

1.8

.1979

3.2

.0000

0.3

.9560

1.1

.5467

1.9

.1645

3.4

.0031

0.4

.9231

1.2

.4868

2.0

.1353

3.6

.0015

0.5

.8825

1.3

.4286

2.2

.0889

3.8

.0007

0.6

.8353

1.4

.3753

2.4

.0561

4.0

.0003

0.7

.7827

1.5

.3246

2.6

.0340

5.0

.000004

VALUES OF THE NORMAL PROBABILITY INTEGRAL. 5o

IV.— TABLE OF VALUES OF THE NORMAL PROBABILITY INTEGRAL

CORRESPONDING TO VALUES OF - OR THE FRACTION OF THE

AREA OF THE CURVE BETWEEN THE LIMITS 0 AND -f - OR 0

AND - - .

Total area of curve assumed to be 10000.

x = deviation from mean.
o- = standard deviation.

X

0

1

o

3

4

5

6

7

8

9

A

cr

0.0

0000

0040

0080

0120

0160

0200

0239

0279

0319

0359

40

0.1

0399

0438

0478

0517

0557

0597

0636

0676

0715

0754

40

0.2

0793

0832

0871

0910

0948

0987

1026

1064

1103

1141

39

0.3

1179

1217

1255

1293

1330

1368

1406

1443

1480

1517

38

0.4

1554

1591

1628

1664

1700

1737

1773

1808

1844

1879

36

0.5

1915

1950

1985

2020

2054

2089

2124

2157

2191

2225

34

0.6

2258

2291

2324

2357

2389

2422

2454

2486

2518

2549

32

0.7

2581

2612

2643

2672

2704

2734

2764

2794

2823

2853

30

0.8

2882

2910

2939

2967

2995

3023

3051

3078

3106

3133

28

0.9

3160

3186

3212

3238

3264

3290

3315

3340

3365

3389

*>6

1.0

3414

3438

3461

3485

3509

3532

3555

3577

3600

3622

23

1.1

3644

3665

3686

3708

3729

3750

3770

3791

3811

3830

21

1.8

3850

3869

3888

3906

3925

3944

3962

3980

3997

4015

19

1.3

4032

4049

4066

4083

4099

4115

4132

4147

4102

4178

17

1.4

4193

4208

4222

4237

4251

4265

4279

4292

4306

4319

14

1.5

4332

4345

4358

4370

4383

4395

4406

4418

4429

4441

12

1.6

4452

4463

4474

4485

4496

4506

4516

4526

4536

4545

10

1.7

4554

4564

4573

4582

4591

4600

4608

4617

4625

4633

9

1.8

4641

4648

4656

4664

4671

4678

4686

4693

4700

4706

7

1.9

4713

4720

4726

4732

4738

4744

4750

4756

4762

4767

6

2.0

4773

4778

4783

4788

4794

4799

4804

4808

4813

4817

5

2.1

4822

4826

4830

4834

4838

4842

4846

4850

4854

4858

4

2.2

4861

4865

4868

4872

4875

4878

4881

4884

4887

4890

3

2.3

4893

4896

4899

4901

4904

4906

4909

4911

4914

4916

3

2.4

4918

4921

4923

4925

4927

4929

4931

4933

4935

4936

2

2.5

4938

4940

4942

4943

4945

4946

4947

4949

4951

4952

2

2.6

4953

4955

4956

4958

4959

4960

4961

4962

4964

4965

1

2.7

4966

4967

4968

4969

4970

4970

4971

4972

4973

4974

1

2.8

4975

4975

4976

4977

4978

4978

4979

4980

4981

4981

0.5

2.9

4982

4982

4983

4983

4984

4984

4985

4985

4986

4986

0.5

3

4987

4991

4993

4995

4997

4998

4999

4999

4999

5000

QO

5000

56

STATISTICAL METHODS.

V.— TABLE OF LOG T FUNCTIONS OF p.

p

0

1

2

3

4

5

6

F"

8

9

1.00

9750

9500

90,<
0251

9003

8755

8509

8263

8017

r-r^-o
t i i O

1.01

9.997529

7285

7043

6801

6560

6320

6080

5841

5602

5365

1.02

5128

4892

4656

4421

4187

3953

3721

3489

3257

3026

1.03

2796

2567

2338

2110

1883

1656

1430

1205

0981

0775

1.04

0533

0311

0089

9868

§647

§427

§208

§989

§772

§554

1.05

9.988338

8122

7907

7692

7478

7265

7052

6841

6629

6419

1.06

6209

6000

5791

5583

5378

51G9

4963

4758

4553

4349

1.07

4145

3943

3741

3539

3338

3138

2939

2740

2541

2344

1.08

2147

1951

1755

1560

1365

1172

0978

0786

0594

0403

1.09

0212

0022

§833

6044

§456

§269

§082

§900

§710

§525

1.10

9.978341

8157

7974

7791

7G10

7428

7248

7068

6888

6709

1.11

6531

6354

6177

6000

5825

5650

5475

5301

5128

4955

1.12

4783

4612

4441

4271

4101

3932

3764

3596

3429

3262

1.13

3096

2931

2766

2602

2433

2275

2113

1951

1790

1629

1.14

1469

1309

1150

0992

0835

0677

0521

0365

0210

0055

1.15

9.969901

9747

9594

9442

9290

9139

8988

8838

8688

8539

1.16

8390

8243

8096

7949

7803

7658

7513

7369

7225

7082

1.17

6939

6797

6655

6514

6374

6234

6095

5957

5818

5681

1.18

5544

5408

5272

5137

5002

4868

4734

4601

4469

4337

1.19

4205

4075

3944

3815

3686

3557

3429

3302

3175

3048

1.20

2922

2797

2672

2548

2425

2302

2179

2057

1936

1815

1.21

1695

1575

1456

1337

1219

1101

0984

0867

0751

0636

1.22

0521

0407

0293

01SO

0067

9955

8843

9732

9621

9511

1.23

9.959401

9292

9184

9076

8968

8861

8755

8649

8544

8439

1.24

8335

8231

8128

8025

7923

7821

7720

7620

7520

7420

1.25

7321

7223

7125

7027

6930

6834

6738

6642

6547

6453

1.26

6359

6267

6173

6081

5989

5898

5807

5716

5627

5537

1.27

5449

5360

5273

5185

5099

5013

4927

4842

4757

4673

1.28

4589

4506

4423

4341

4-J59

4178

4097

4017

3938

3858

1.29

3780

3702

3624

3547

3470

3394

3318

3243

3168

3094

1.30

3020

2947

2874

2802

2730

2659

2588

2518

2448

2379

1.31

2310

2242

2174

2106

2040

1973

1907

1842

1777

1712

1.32

1648

1585

1522

1459

1397

1336

1275

1214

1154

1094

1.33

1035

0977

0918

0861

0803

0747

0690

0634

0579

0524

1.34

0470

0416

0362

0309

0257

0205

0153

0102

0051

0001

1.35

9.949951

9902

9853

9805

9757

9710

9663

9617

9571

9525

1.36

9480

9435

9391

9348

9304

9262

9219

9178

9136

9095

1.37

9054

9015

8975

8936

8898

8859

8822

8785

8T48

8711

1.3S

8676

8640

8605

8571

8537

8503

8470

8437

8405

8373

1.39

8342

8311

8280

8250

8221

8192

8163

8135

8107

8080

1.40

8053

8026

8000

7975

7950

7925

7901

7877

7854

7831

1.41

7808

77>G

7765

7744

7723

7703

7683

7664

7645

7626

1.42

7608

7590

7573

7556

7540

75-24

7509

7494

7479

7465

1.43

7451

7438

7425

7413

7401

7389

7378

7368

7358

7348

1.44

7338

7329

7321

7312

7305

7298

7291

7284

7278

7273

1.45

7268

7263

7259

7255

7251

7248

7246

7244

7242

7241

1.46

7240

7239

7239

7240

7241

7242

7243

7245

7248

7251

1.47

7254

7258

7262

7266

7271

r\~)r'f"1'
i6l t

7282

72>9

7295

7302

1.48

7310

7317

7326

7334

7343

7353

7363

7373

7384

7395

1.49

7407

7419

7431

7444

7457

7471

7485

7499

7515

7529

TABLE OF LOG T FUNCTIONS.

57

V.— TABLE OF LOG r FUNCTIONS OF p.

p

0

1

2

3

4

5

6

7

8

9

1.50

9.947545

7561

7577

7594

7612

7629

7647

7666

76^5

7704

1.51

7724

7744

7764

7785

7806

7828

7850

7873

7896

7919

1.53

7943

7967

7991

8016

8041

8067

8093

81-20

8146

8174

1.53

8201

8229

8-258

8287

8316

8346

8376

8406

8437

8468

1.54

8500

8532

8564

8597

8630

8664

8698

8732

8767

8802

1.55

8837

8873

8910

8946

8983

9021

9059

9097

9135

9174

1.56

9214

9254

9294

9334

9375

9417

9458

9500

9543

95S<;

1.57

95 .'9

9672

9716

9761

9806

9851

9896

9942

9989

5035

1.58

9.950082

0130

0177

0225

0274

0323

0372

04'22

047-2 052-2

1.59

0573

0624

0676

0728

07SO

0833

0886

0939

0993

1047

1.60

1102

1157

1212

1268

1324

1380

1437

1494

1552

1610

1.61

1668

1727

1786

1845

1905

1965

20-25

2086

2147

2209

1.62

2271

2333

2396

2459

2522

2586

2650

2715

2780

2845

1.63

2911

2977

3043

3110

3177

3244

3312

3380

3449

3517

1.64

3587

3656

3726

3797

3867

3938

4010

4081

4154

4226

1 65

4299

4372

4446

4519

4594

4668

4743

4819

4894

4970

1.66

5047

5124

5201

5278

5356

5434

5513

5592

5671

5740

1.87

5830

5911

5991

6072

6154

6235

6317

6400

6482

6566

1.6S

C649

6733

6817

6901

6986

7072

7157

7243

7322

7416

1.69

7503

7590

7678

7766

7854

7943

8032

812-2

8211

8301

1.70

8391

8482

8573

8664

8756

8848

8941

9034

9127

92-20

1.71

9314

9409

9502

9598

9t)93

9788

9884

9980

6077

5174

1.72

9.960271

0369

0467

0565

0(564

0763

0862

0961

1061

1162

1.73

1262

1363

1464

1566

1668

1770

1873

1976

2079

2183

1.74

2287

2391

2496

2601

2706

2812

2918

3024

3131

3-238

1.75

3345

3453

3561

3669

3778

3887

3996

4105

4215

43-26

1.76

4436

4547

4659

4770

4882

4994

5107

52-20

5333

5447

1.77

5561

5675

5789

5904

6019

6135

6251

6367

6484

6600

1.78

6718

6835

6953

7071

7189

7308

7427

7547

76P6

r"*-Qf-«
i iOi

1.79

7907

8023

8149

8270

8392

8514

8636

8759

8882

9005

1.80

9129

9253

9377

9501

9626

9751

9877

5008

5129

6255

1.81

9.970383

0509

0637

0765

0893

1021

1150

1-279

1408

153S

1.82

1668

1798

1929

2060

2191

2322

2454

2586

2719

285-2

1.83

2985

3118

3252

3386

3520

3655

3790

30-25

4061

4107

1.84

4333

4470

4606

4744

4881

5019

5157

5295

5434

5573

1.85

5712

5852

5992

6132

6-273

6414

6555

6607

6838

6980

1.86

71-23

7266

7408

7552

7696

7840

7984

8128

8273

8419

1.87

8564

8710

8856

9002

9149

9296

9443

9591

9739

9887

1.88

9.980036

0184

0333

0483

0633

0783

0933

1084

1234

1386

1.89

1537

1689

1841

1994

2147

2299

2453

2607

2761

2915

1.90

3069

3224

3379

3535

3690

3846

4003

4159

4316

4474

1.91

4631

4789

4947

5105

5264

5423

5582

5742

5902

6062

1.92

6223

6383

6544

6706

6867

70-29

7192

7354

7517

7680

1.93

7844

8007

8171

8336

8500

8665

8830

8996

9161

9327

1.94

9494

9660

9827

9995

5162

6330

6498

6666

6835

1004

1.95

9.991173

1343

1512

1683

1853

2024

2195

2366

2537

2709

1.96

2881

3054

32-27

3399

3573

3746

3920

4094

4269

4443

1.97

4618

4794

4969

5145

5321

5498

5674

5851

6029

6206

1.98

6384

6562

6740

6919

7078

7277

7457

7637

7817

7997

1.99

8178

8359

8540

8722

8903

9085

9268

9450

9633

9816

58

STATISTICAL METHODS.

VI.— TABLE OF REDUCTION FROM COMMON TO METRIC SYSTEM.

Inches to Millimeters.

1

o

3

*

5

6

7

8

9

25.40

50.80

76.20

101.60

127.00

152.40

177

.80

203.20

228.60

10

279.40

304.80

330.19

355.59

380.99

406.39

431

.79

457 19

482.59

20

533.39

55S.79

584.19

609.59

634.99

600.39

685

.79

711.19

736.59

30

787.39

812.

79

838 19

863.59

888.99

914.39

939

.78

965.18

990.58

40

1041.4

1066.

3

1092.2

11

17.6

1143.0

1168.4

1193

.8

1219.2^

1244.6

50

1295.4

1320.

3

1346.2

1371.6

1397.0

1422.4

1447

.8

14

1*3 2

1498.6

1549.4

1574.8

1600.2

1625.6

1651.0

1676.4

1701

.8

17:

17.2

1752.6

70

1803.4

1828.

s

1854.2

1879.6

1905.0

1930.4

1955

.8

1981.2

2006.6

80

2057.4

2082.8

2108.2

2133.6

2159.0

2184.4

2209

.8

2235.2

2260.6

90

2311.4

2336.8

2362.2

2387.6

2413.0

2438.4

2463

.8

2489.3

2514.6

Twelfths.

Sixteenths.

1/12
2/12
3/12

2.12
4.23
6.35

7/12
8/12
9/12

14.82
16.93
19.05

1/16

1/8
3/16

1.59 5/16
3.17 3/8
4.76 7/16

7.94 9/16
9.52 5/8
11.11 11/16

14.29
15.87
17.46

13/16 20.64
7/8 22.22
15/16 23.81

4/12

8.47

10/12

21.17

1/4

6.35 1/2

12.70 3/4

19.05

1 25.41

5/12

10.58

11/12

23.28

6/12

12.70

12/12

25.40

FIRST TO SIXTH POWERS OF INTEGERS.

59

TABLE VII.— FIRST TO SIXTH POWERS OF INTEGERS FROM 1 TO 30.

Powers.

First.

Second.

Third.

Fourth.

Fifth.

Sixth.

1

1

1

1

1

1

2

4

8

16

32

64

3

9

27

81

243

729

4

16

64

256

1024

4096

5

25

125

625

3125

15625

6

36

216

1296

ffrffyft
1 1 (D

46656

7

49

343

2401

16807

117649

8

64

512

4096

32768

2G2144

9

81

729

6561

59049

531441

10

100

1000

10000

100000

100000U

11

121

1331

14641

161051

1771561

12

144

1728

20736

248832

2985984

13

169

2197

28561

371293

4826809

14

196

2744

38416

537824

7529536

15

225

3375

50625

759375

11390625

16

256

4096

65536

1048576

16777216

17

289

4913

83521

1419857

24137569

18

324

5832

104976

1889568

34012224

19

361

6859

1303-21

2476099

47045881

20

400

8000

160000

3200000

64000000

21

441

9261

194481

4084101

85766121

22

484

10648

234256

5153632

113379904

23

529

12167

279841

6436343

148035889

24

576

13824

331776

7962624

191102976

25

625

156-J5

390625

9765625

244140625

26

676

17576

456976

11881376

308915776

27

7*9

19683

531441

14348907

387420489

28

784

21952

614656

17210368

481890304

29

841

24389

T07281

30511149

594823321

30

900

27000

810000

24300000

729000000

TABLE VIII. — SQUARES, CUBES, SQUARE ROOTS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

1

1

1

1.0000000

1.0000000

1.000000000

2

4

8

1.4142136

1.2599210

.500000000

3

9

27

1.7320508

1.4422496

.333*3:3333

4

16

64

2.0000000

1.5874011

.250000000

5

25

125

2.2360680

1.7099759

.200000000

6

36

216

2 4494897

1.8171206

.166666667

r*
4

49

343

2.6457513

1.9129312

.142857143

8

64

512

2.8284271

2.0000000

.125000000

9

81

729

3.0000000

2.0800837

.111111111

10

100

1000

3.1622777

2.1544347

.100000000

11

121

1331

3.3166248

2.2239801

.090909091

12

144

1728

3.4641016

2.2894286

.083333333

13

169

2197

3.6055513

2.3513347

.076923077

14

196

2744

3.7416574

2.4101422

.071428571

15

225

3375

3.87298:33

2.4662121

.066666667

16

256

4096

4.0000000

2.5198421

.062500000

17

289

4913

4.1231056

2.5712816

.058823529

18

324

5832

4.2426407

2.6207414

.055555556

19

361

6859

4.3588989

2.6684016

.052631579

20

400

8000

44721360

2.7144177

.050000000

21

441

9261

41S825757

2.7589243

.047619048

22

484

10648

4.6904158

2.8020393

.045454545

23

529

12167

4.7958315

2.8438670

.043478261

24

576

13824

4.8989795

2.8844991

.041666667

25

625

15625

5.0000000

2.9240177

.040000000

26

676

17576

5.0990195

2.9624960

.038461538

27

729

19683

5.1961524

3.0000000

.037037037

28

784

21952

5.2915026

3.0365889

.035714286

29

841

24389

5.3851648

3.0723168

.034482759

30

900

27000

5.4772256

3.1072325

.033333333

31

961

29791

5.5677644

3.1413806

.032258065

32

1024

32768

5.6568542

3.1748021

.031250000

33

1089

35937

5.7445626

3.2075343

030303030

34

1156

39304

5.8309519

3.2396118

.029411765

35

1225

42875

5.9160798

3.2710663

.028571429

36

1296

46656

6.0000000

3.3019272

.027777778

37

1369

50653

6.0827625

3.3322218

.027027027

38

1444

54872 .

6.1644140

3.3619754

.026315789

39

1521

59319

6.2449980

3.3912114

.025641026

40

1600

64000

6.3245553

3.4199519

.025000000

41

1681

68921

6.4031242

3.4482172

.024390244

42

1764

74088

6.4807407

3.4760266

.02:3809524

43

1849

79507

6.5574385

3.5033981

.023255814

44

1936 •

85184

6.633249£

3.5303483

.022727273

45

2025

91125

6.7082039

3.5568933

022222222

46

2116

97*36

6.7823300

3.5830479

.'021739130

47

2209

103823

6.8556546

3.6088261

.021276600

48

2304

110592

6.9282032

3.6342411

.0208:33333

49

2401

117649

7.0000000

3.6593057

.020408163

50

2500

125000

7.0710678

3.6840314

.020000000

51

2601

132651

7.1414284

3.7084298

.019607843

52

2704

140608

7.2111026

3.7325111

.019230769

53

2809

148877

7.2801099

3.7562858

.018867925

54

2916

157464

7.3484692

3.7797631

.018518519

55

3025

166375

7.4161985

3.8029525

.018181818

56

3136

175616

7.4833148

3.8258624

.017857143

57

3249

185193

7.5498344

3.8485011

.017543860

58

3364

195112

7.6157731

3.8708766

.017241379

59

3481

205379

7.6811457

3.8929965

.016949153

60

3600

216000

7.7459667

3.9148676

.016666667

61

3721

226981

7.8102497

3.9364972

.016393443

62

3844

2383-28

7.8740079

3.9578915

.016129032

60

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

63

3969

250047

7.9372539

3.9790571

.015873016

64

4096

262144

8.0000000

4.0000000

.015625000

65

4225

274625

8.0622577

4 . 0207256

.015384615

66

4356

287496

8.1240384

4.0412401

.015151515

67

4489

300763

8.1853528

4.0615480

.014925373

68

4624

314432

8.2462113

4.0816551

.014705882

69

4761

328509

8.3066239

4.1015661

.014492754

70

4900

343000

8.3666003

4.1212853

.014285714

71

5041

357911

8.4261498

4.1408178

.014084507

72

5184

373248

8.4852814

4.1601676

.013888889

73

5329

389017

8.5440037

4.1793390

013698630

74

5476

405224

8.6023253

4.1983364

.013513514

75

5625

421875

8.6602540

4.2171633

.013333333

76

5776

438976

8.7177979

4.2358236

.013157895

r"**
1 |

5929

456533

8.7749644

4.2543210

.012987013

78

6084

474552

8.8317609

4.2726586

.012820513

79

6241

493039

8.8881944

4.2908404

.012658228

80

6400

512003

8.9442719

4.3088695

.012500000

81

6561

5.31441

9.0000000

4.32G7487

.012345679

82

6724

551368

9.0553851

4.3444815

.012195122

83

6889

571787

9.1104336

4.3620707

.012048193

84

7056

592704

9.1651514

4.3795191

.011904762

85

7225

614125

9.2195445

4.3968296

.011764706

86

7396

636056

9.2736185

4.4140049

.011627907

87

7569

658503

9.3273791

4.4310476

.011494253

88

7744

681472

9.3808315

4.4479602

.011363636

89

7921

704969

9.4339811

4.4647451

.011235955

90

8100

729000

9.4868330

4.4814047

.011111111

91

8281

V53571

9.5393920

4.4979414

.010989011

92

8464

778688

9.5916630

4.5143574

.010869565

93

8649

804357

9.6436508

4.5306549

.010752688

94

8836

830584

9.6953597

4.5468359

.010638298

95

9025

857375

9.7467943

4.5629026

.010526316

96

9216

884736

9.7979590

4.5788570

.010416667

97

9409

912673

9.8488578

4.5947009

.010309278

98

9604

941192

9.8994949

4.6104363

.010204082

99

9801

970299

9.9498744

4.6260650

.010101010

100

10000

1000000

10.0000000

4 6415888

.010000000

101

10201

1030301

10.0498756

4.6570095

.009900990

102

10404

1061208

10.0995049

4.6723287

.609803922

103

10609

1092727

10.1488916

4.6875482

.009788738

104

10816

1124864

10.1980390

4.7026694

.0096153KK^.

105

1H)25

1157625

10.2469508

4.7176940

.009523810

106

11236

1191016

10.2956301

4.7326235

.009433962

107

11449

1225043

10.3440804

4.7474594

.009345794

108

11664

1259712

10.3923048

4.7622032

.00 259259

109

11881-

1295029

10.4403065

4.7768562

.009174312

110

12100

1331000

10.4880865

4.7914199

.009090909

111

12321

1367631

10.5356538

4.8058955

.009009009

112

12544

1404928

10.5830052

4.8202845

.0089285,1

113

12769

1442897

10.6301458

4.8345881

.008849558

114

12996

1481544

10.0770783

4.8488076

.008771930

115

13225

1520875

10.7238053

4.8629442

.008695652

116

13456

1560896

10.7703296

4.8769990

.008620690

117

13689

1601613

10.81665.-W

4.S909732

.008547009

118

13924

1643032

10.8627805

4.9048681

.008474576

119

14161

1685159

10.9087121

4.9186847

.008403361

120

14400

1728000

10.9544512

4.9324342

.008333333

121

14641

1771561

ll.OOJOOOO

4.9460874

.008264463

1*22

14884

1815848

11.0453610

4.9596757

.008196721

123

15129

1860867

11.0905365

4.9731898

.008130081

124

15376

1906624

11.1355287

4.9866310

.OOS064516

61

TABLE VIII. — SQUARES, CUBES, SQUARE ROOTS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

125

15625

1953125

11.1803399

5.0000000

.008000000

126

15876

2000376

11.2249722

5. 0132979 l

.007936508

137

16129

2048383

11.2694277

5.0265257

.007874016

128

16384

2097152

11.3137085

5.0396842

.007812500

129

16641

2146689

11.3578167

5.0527743

.007751938

130

16900

2197000

11.4017543

5.0657970

.007692308

131

17161

2248091

11.4455231

5.0787531

.007633588

132

17424

2299968

11.4891253

5.0916434

.007575758

133

17689

2352637

11.5325626

5.1044687

.007518797

134

17956

2406104

11.5758369

5.1172299

.007462687

135

18225

2460375

11.6189500

5.1299278

.007407407

136

18496

2515456

11.6619038

5.1425632

.007352941

137

18769

2571353

11.7046999

5.1551367

.007299270

138

19044

2628072

11.7473401

5.1676493

.007246377

139

19321

2685619

11.7898261

5.1801015

.007194245

140

19600

2744000

11.8321596

5.1924941

.007142857

141

19881

2803221

11.874:3421

5.2048279

.007092199-

142

20164

2863288

11.9163753

5.2171034

.007042254

143

20449

2924207

11.9582607

5.2293215

.006993007

144

20736

2985984

12.0000000

5.2414828

.006944444

145

21025

3048625

12.0415946

5.2535879

.006896552

146

21316

3112136

12.0830460

5.2656374

.006849315

147

21609

3176523

12.1243557

5.2776321

.006802721

148

21904

3241792

12.1655251

5.2895725

.006756757

149

22201

3307949

12.2065556

5.3014592

.006711409

150

22500

3375000

12.^474487

5.3132928

.006666667

151

22801

3442951

12.2882057

5.3250740

.006622517

152

23104

3511808

12.32S82SO

5.3368033

.006578947

153

23409

3581577

12.3693169

5.3484812

.006535948

154

23716

3652264

12.4096736

5.3601084

.006493506

155

24025

3723875

12.4498996

5.3716854

.006451613

156

24336

3796416

12.4899960

5.3832126

.006410256

157

24649

3869893

12.5299641

5 3946907

.006369427

158

24964

3944312

12.5698051

5.4061202

.006329114

159

23281

4019679

12.0095203

5.4175015

.006289308

160

25600

4096000

12.6491106

5.4288352

.006250000

161 25921

4173281

12.6885775

5.4401218

.006211180

162

. 26244

4251528

12.7279221

5.4513618

.006172840

163

26569

4330747

12.7671453

5 . 4625556

.006134969

164

26896

4410944

12.8062485

5.4737037

.006097561

165

27225

4492125

12.8452326

5.4848066

.006060606

166

27556

4574296

12.8840987

5.4958647

.006024096

167

27889

4657463

12 9228480

5.5068784

.005988024

168

28224

4741632

12.9614814

5.5178484

.005952381

169

28561

4826809

13.0000000

5.5287748

.005917160

170

28900

4913000

13.0384048

5.5396583

.005882353

171

29241

5000211

13.0766968

5.5504991

.005847953

172

29584

5088448

13.1148770

5.5612978

.005813953

173

29929

5177717

13.1529464

5.5720546

.005780347

174

30276

5268024

13.1909060

5.5827702

.005747126

175

30625

5359375

13 2287566

5.5934447

.005714286

176

30976

5451776

13.2664992

5.6040787

.005681818

177

31329

.0545233

13.3041347

5.6146724

.005649718

178

31684

5639752

13 3416641

5.6252263

.005617978

179

32041

5735339

13.3790882

5.6357408

.005586592

ISO

32400

5832000

13.4164079

5.6462162

.005555556

181

32761

5929741

13.4536240

5.6566528

.005524862

182

33124

6028568

13.4907376

5.6670511

.005494505

183

33489

6128487

13.5277493

5 6774114

.005464481

184

33856

6229504

13.5646600

5.6877340

.005434783

1S5

34225

6331625

13.6014705

5.6980192

.005405405

186

34596

6434856

13.6381817

5.7082675

.005370344

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

187

34969

6539203

13.6747943

5.7184791

.005347594

188

35344

6044672

13.7113092

5.7286543

.005319149

189

35721

6751269

13.7477271

5.7387936

.005291005

190

36100

6859000

13.7840488

5.7488971

.005263158

191

30481

6967871

13.8202750

5.7589652

.005235602

192

36864

7077888

13.8564065

5.7689982

.005208333

193

37249

7189057

13.8924440

5.7789966

.005181347

194

37636

7301384

13.9283883

5.7889604

.005154639

195

38025

7414875

13.9642400

5.7988900

.005128205

196

38416

7529536

14.0000000

5.8087857

.005102041

197

38809

7645373

14.0356688

5.8186479

.005076142

198

39204

7762392

14.0712473

5.8284767

.005050505

199

39601

7880599

14.1067360

5.8382725

.005025126

200

40000

8000000

14.1421356

5.8480355

.005000000

201

40401

8120601

14.1774469 5.8577660

.004975104

202

40804

8242408

14.2126704 5.8674643

.004950495

203

41209

8365427

14.2478068 5.8771307

.004926108

204

41616

8489664

14.2828569 5.8867653

.004901961

205

42025

8615125

14.3178211 5.8963685

.004878049

206

42436

8741816

14.3527001 5.9059406

.004854369

207

42849

8869743

14.3874946

5.9154817

.004830918

208

43264

8998912

14.4222051

5.9249921

.004807692

209

43681

9129329

14.4568323

5.9344721

.004784689

210

44100

9261000

14.4913767

5.9439220

.004761905

211

44521

9393931

14.5258390

5.9533418

.004739336

212

44944

9528128

14.5602198

5.9627320

.004716981

213

45369

9663597

14.5945195

5.9720926

.004694836

214

45796

9800344

14.6287388 . 5.9814240

.004672897

215

46225

9938375

14.6628783

5.9907264

.004651163

216

46656

10077696

14.6969385

6.0000000

.004629630

217

47089

10218313

14.7309199

6.G092450

.004608295

218

47524

10360232

14.7648231

6.0184617

.004587156

219

47961

10503459

14.7986486

6.0276502

.004566210

220

48400

10G48000

14.8323970 6.0368107

.004545455

221

48841

10793861

14.8660687 6.0459435

.004524887

222

49284

10941048

14.8996644 6.0550489

.004504505

223

49729

11089567

14.9331845

6.0641270

.004484305

224

50176

11239424

14.9666295

6.0731779

.004464286

225

50625

11390625

15.0000000

6.0822020

.004444444

226

51076

11543176

15.0332964

6.0911994

.004424779

227

51529

11697083

15.0665192

6.1001102

.004405286

228

51984

11852:352

15.0996689

6.1091147

.004385965

; 229

52441

12008989

15.1327460

6.1180332

.004366812

230

52900

12167000

15.1657509

6.1269257

.004347826

231

53361

12326391

15.1986842

6.1357924

.004329004

232

53824

12487168

15.2315462

6.1446337

.004310345

i 233

54289

12649337

15.2643375

6.1534495

.004291845

234

54756

12812904

15.2970585

6.1622401

.004273504

235

55225

12977875

15.3297097 6.1710058

.004255319

236

55696

13144256

15.3622915 6.1797466

.004237288

237

56169

13312053

15.3948043 6.1884628

.004219409

238

56644

13481272

15.4272486 6.1971544

.004201681

239

57121

13651919

15.4596248 6.2058218

.C04184100

240

57600

13824000

154919334 6.2144650

.004166667

i 241

58081

13997521

15.5241747 6.2230843

.004149378

242

58564

141724S8 1 5 . 5563492 6 . 231 6797

.004132231

243

59049

14348907 15.5884573 6.2402515

.004115226

244

59536

14526784

15.6204994

6.2487998

.004098361

245

60025

14706125

15.6524758

6.2573248

.004081633

246

60516

14886936

15.6843871

6.2658266

.004065041

247

61009

15069223

15.7162336

6.2743054

.004048583

1 248

61504

15252992

15.7480157

6.2827613

.004032258

63

TABLE YIII. — SQUARES, CUBES, SQUARE ROOTS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

249

62001

15438249

15.7797338

6.2911946

.004016064

250

62500

15625000

15.8113883

6.2996053

.004000000

251

03001

15813251

15.8429795

6.3079935

.003984064

252

63504

16003008

15.8745079

6.3163596

.003968254

253

64009

16194277

15.9059737

6.3247035

.003952569

254

64516

16387064

15.9373775

6.33:30256

.003937008

255

65025

16581375

15.9687194

6.3413257

.003921569

256

65536

16777216

16.0000000

6.3496042

.003906250

257

66049

16974593

16.0312195

6.3578611

.003891051

258

66564

17173512

16.0623784

6.3660968

.003875969

259

67081

17373979

16.0934769

6.3743111

.003861004

200

67600

17576000

16.1245155

6.3825043

.003846154

261

68121

17779581

16.1554944

6.3906765

.003831418

262

68644

17984728

16.1864141

6.3988279

.003816794

263

69169

18191447

16.2172747

6.4069585

.003802281

264

69696

18399744

16.2480768

6.4150687

.003787879

265

70225

18609625

16.2788206

6.4231583

.003773585

266

70756

18821096

16.3095064

6.4312276

.003759398

267

71289

19034163

16.3401346

6.4392767

.003745318

268

71824

19248832

16.3707055

6.4473057

.003731343

269

72361

19465109

16.4012195

6.4553148

.003717472

270

72900

19683000

16.4316767

6.4633041

.003703704

271

73441

19902511

16.4620776

6.4712736

.003690037

272

73981

20123648

16.4924225

6.4792236

.003676471

273

7452 J

20346417

16.5227116

6.4871541

.003663004

274

75076

20570824

16.5529454

6.4950653

.003649635

275

75625

20796875

16.5831240

6.5029572

.003636364

276

76176

21024576

16.6132477

6.5108300

.003623188

277

76729

21253933 ,

16.6433170

6.5186839

.003610108

278

77284

21484952

16.6733320

6.5265189

.003597122

279

77841

21717639

16.7032931

6.5343351

.003584229

280

78400

21952000

16.7332005

6.5421326

.003571429

281

78961

22188041

16.7630546

6.5499116

.003558719

282

79524

22425768

16.7928556

6.5576722

.003546099

283

80089

22665187

16.8226038

6.5654144

.003533569

284

80656

22906304

16.8522995

6.5731385

.00:3521127

285

81225

23149125

16.8819430

6.5808443

.003508772

286

81796

23393656

16.9115345

6.5885323

.003496503

287

82369

23639903

16.9410743

6.5962023

.003484321

288

82944

23887872

16.9705627

6.6038545

.003472222

289

83521

24137569

17.0000000

6.6114890

.003460208

290

84100

24389000

17.0293864

6.6191060

.003448276

291

84681

24642171

17.0587221

6.6267054

.003436426

292

85264

24897088

17.0880075

6.6342874

.003424658

293

85849

25153757

17.1172428

6.6418522

.003412969

294

86436

25412184

17.1464282

6.6493998

.003401361

295

87025

25672375

17.1755640

6.6569302

.003389831

296

87616

25934336

17.2046505

6.6644437

.003378378

297

88209

26198073

17.2.336879

6.6719403

.003367003

298

8SS04

26463592

17.2626765

6.6794200

.003355705

299

89401

26730899

17.2916165

C. 6868831

.003344482

300

90000

27000000

17.3205081

6. CO 13295

.003333333

301

90601

27270901

17.3493516

6.7017593

.003322259

302

01204

27543608

17.3781472

6.7091729

.003311258

303

91809

27818127

17.4068952

6.7165700

.003300330

304

92416

28094464

17.4:355958

6.7239508

.003289474

305

93025

28372625

17.4642492

6.7313155

.003278689

306

93636

28652616

17.4928557

6.7386641

.003267974

307

94249

28934443

17.5214155

6.7459967

.003257329

308

94864

29218112

17.5499288

6.7533134

.003246753

309

95481

29503629

17.5783958

6.7606143

.003236246

310

96100

29791000

17.6068169

6.7678995

.003225806

64

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

311

96721

30080231 17. 0351 921

6.7751090

.003215434

312

97344

30371328

17.0035217

6.7824229

.003205128

313

97969

30604297

17.6918060

6.7890013

.003194888

314

98596

30959144

17.7200451

6.7968S44

.003184713

315

99225

31255875

17.7482393

6.8040921

.003174603

316

99856

31554496

17.7763888

6.8112847

.003164557

317

100489

31855013

17.8044938

6.8184020

.003154574

318

101124

32157432

17.8325545

6.8256242

.003144654

319

101761

32401759

17.8005711

6.8327714

.003134796

320

102400

32708000

17.8885438

6.8399037

.003125000

321

103041

33070101

17.9104729

6.8470213

.003115265

322

103684

33380248

17.9443584

6.8541240

.003105590

323

104329

33698267

17.9722008

6.8612120

.003095975

324

104976

34012224

18.0000000

6.8682855

.003086420

325

105625

34328125

18.0277504

6.8753443

.003076923

320

106276

34645976

18.0554701

6.8823888

.003067485

327

106929

34965783

18.0831413 C. 8894188

.003058104

328

107584

35287552

18.1107703

6.8964345

.003048780

329

108241

35611289

18.1383571

6.9034359

.003039514

330

108900

35937000

18.1659021

6.9104232

.003030303

331

109561

36264691

18.1934054

6.9173964

.003021148

332

110224

30594308

18.2208072

6.9243556

.003012048

333

110889

36J26037

18.2482876

6.9313008

.003003003

334

111556

37259704

18.2750009

6.9382321

.002994012

335

112225

37595375

18.3030052

6.9451496

.002985075

330

112896

37933056

18.3303028

6.9520533

.002976190

337

113569

38272753

18.3575598

6.9589434

.002967359

338

114244

38614472

18.3847763

6.9658198

.002958580

339

114921

38958219

18.4119526

6.9726826

.002949853

340

115600

39304000

18.4390889

6.9795321

.002941176

341

116281

39651821

18.4661853

6.9803681

.002932551

342

116964 40001688

18.4932420

6.9931900

.002923977

343

117649

40353607

18.5202592

7.0000000

.002915452

344

118336

40707584

18.5472370

7.0067962

.002906977

345

119025 41003625

18.5741756

7 01:35791

.002898551

346

119716

41421736

18.6010752

7.0203490

.002890173

347

120409

41781923

18.6279360

7.0271058

.002881844

348

121104

42144192

18.6547581

7.0338497

.002873563

349

121801

42508549

18.6815417

7.0405806

.002865:330

350

122500

42875000

18.7082869

7.0T2987

.002857143

351

123201

43243551

18.7349940

7.0540041

.002849003

352

123904

43614208

18.7610630

7.0600907

.002840909

353

124609

43986977

18.7882942

7.0673767

.002832861

354

125316

44361804

18.8148877

7.0740440

.002824859

355

126025

44738875

18.8414437

7.0806988

.002816901

356

126736

45118016

18.8079623

7.0873411

.002808989

357

127449

45499293

18.8944436

7.0939709

.002801120

358

128164

45882712

18.9208879

7.1005885

.002793296

359

128881

46268279

18.9472953

7.1071937

.002785515

3GO

129600

46656000

18.9736660

7.1137866

.002777778

361

130321

47045881

19 0000000

7.1203674

.002770083

362

131044

47437928

19.0262976

7.1269360

.002762431

363

131769

47832147

19.0525589

7.1334925

.002754821

364

132496

48228544

19.0787840

7.1400370

.002747253

365

133225

48627125

19.1049732

7.1405695

.002739726

366

133956

49027896

19.1311265

7.1530901

.002732240

367

134689

49430863

19.1572441

7.1595988

.002724796

368

135424

49836032

19.1833261

7.1000957

.002717391

369

136161

50243409

19.2093727

7.1725809

.002710027

370

136900

50653000

19.2353841

7.1790544

.002702703

371

137641

51064811

19.2613603

7.1855162

.002695418

372

138384

51478848

19.2873015

7.1919603

.002688172

(JO

TABLE VIII. — SQUARES, CUBES, SQUARE ROOTS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

373

139129

51895117

19.3132079

7.1984050

.002680965

374

139876

52313624

19.3390796

7.2048322

.002673797

375

140625

52734375

19.3649167

7.2112479

.002666667

376

141376

53157376

19.3907194

7.2176522

.002659574

377

142129

53582633

19.4164878

7.2240450

.002652520

378

142884

54010152

19.4422221

7.2304268

.002645503

379

143641

54439939

19.4679223

7.2367972

.002638522

380

144400

54872000

19.4935887

7.2431565

.002631579

381

145161

55306341

19.5192213

7.2495045

.002624672

382

145924

55742968

19.5448203

7.2558415

.002617801

383

146689

56181887

19.5703858

7.2621675

.002610966

884

147456

56623104

19.5959179

7.2684824

.002604167

385

148225

57066625

19.6214169

7.2747864

.002597403

386

148996

57512456

19.6468827

7.2810794

.002590674

387

149769

57960603

19.6723156

7.2873617

.002583979

388

150544

58411072

19.6977156

7.2936330

.002577320

389

151321

58863869

19.7230829

7.2998936

.002570694

390

152100

59319000

19.7484177

7.3061436

.002564103

391

152881

59776471

19.7737199

7.3123828 .002557545

392

153664

60236288

19.7989899

7.3186114

.002551020

393

154449

60698457

19.8242276

7.3248295

.002544529

394

155236

61162984

19.8494332

7.3310369

.002538071

395

156025

61629875

19.8746069

7.3372339

.002531646

396

156816

62099136

19.8997487

7.3434205

.002525253

397

157609

62570773

19.9248588

7.3495966

.002518892

398

158404

63044792

19.9499373

7.3557624

.002512563

399

159201

63521199

19.9749844

7.3619178

.002506266

400

160000

64000000

20.0000000

7.3680630

.002500000

401

160801

64481201

20.0249844

7.3741979

.002493766

402

161604

64964808

20.0499377

7.3803227

.002487562

403

162409

65450827

20.0748599

7.3864373

.002481390

404

163216

65939264

20.0997512

7.3925418

.002475248

405

164025

66430125

20.1246118

7. 3986363

.002469136

406

164836

66923416

20.1494417

7.4047206

.002463054

407

165649

67419143

20.1742410

7.4107950

.002457002

408

166464

67917312

20.1990099

7.4168595

.002450980

409

167281

68417929

20.2237484

7.4229142

.002444988

410

168100

68921000

20.2484567

7.4289589

.002439024

411

168921

69426531

20.2731349

7.4349938

.002433090

412

169744

69934528

20.2977831

7.4410189

.002427184

413

170569

70444997

20.3224014

7.4470342

.002421308

414

171396

70957944

20.3469899

7.4530399

.002415459

415

172225

71473375

20.3715488

7.4590359

.002409639

416

173056

71991296

20.3960781

7.4650223

.002403846

417

173889

72511713

20.4205779

7.4709991

.002398082

418

174724

73034632

20.4450483

7.4769664

.002392344

419

175561

73560059

20.4694895

7.4829242

.002386635

420

176400

74088000

20.4939015

7.4888724

.002:380952

421

177241

74018J61

20.5182845

7.4948113

.002375297

422

178084

75151448

20.5426386

7.5007406

.002369668

423

178929

"5680967

20.5669638

7.5066607

.002364066

424

179776

,6225024

20 5912603

7.5125715

.002358491

425

180625

76765625

20.6155281

7.5184730

.002352941

426

181476

77308776

20.6397674

7.5243652

.002347418

427

182329

77854483

20.6639783

7.5302482

.002341920

428

183184

78402752

20.6881609

7.5361221

.002336449

429

184041

78953589

20.7123152

7.5419867

.002331002

430

184900

79507000

20.7364414

7.5478423

.002325581

431

185761

800G2991

20.7605395

7.5536888

.002320186

432

186624

80621568

20.7846097

7.5595263

.002314815

433

187489

81182737

20.8086520

7.5653548

.002309469

434

188356

81746504

20.&326667

7.5711743

.002304147

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Hoots.

Cube Roots.

Reciprocals.

4:35 189225

82312875

20.8566536

7.5769849

.002298851

436 190096

82881856

20.8806130

7.5827865

.002293578

437 190969

83453453

20.9045450

7.5885793

.002288330

438 191*14

84027672

20.9284495

7.5943633

.002283105

439 192721

84604519

20.9523268

7.6001385

.002277904

440 193600

85184000

20.9761770

7.6059049

.002272727

441 194481

85766121

21.0000000

7.6116626

.002267574

442

195364

86350888

21.0237960

7.6174116

.002262443

443

196249

86938307

21.0475652

7.6231519

.002257336

444 197136

87528:384

21.0713075

7.6288837

.002252252

445 198025 88121125

21.0950231

7.6346067

.002247191

446 198916 88716536

21.1187121

7.6403213

.002242152

447 199809

89314623

21.1423745

7.6460272

.002237136

448

200704

89915392

21.1660105

7.6517247

.002232143

449

201601 90518849

21 . 1896201

7.657413J

.002227171

450

202500 91125000

21.2132034

7.6630943

.002222222

451

203401 91733851

21.2367606

7.6687665

.002217295

452

204304 92345408

21.2602916

7.6744303

.002212389

453

205209 92959677

SI. 2837967

7.0800857

.002207506

454

206116 93576664

21.3072758

7.6857328

.002202643

455

207025 94196375

21.3307290

7.6913717

.002197802

456

207936 94818816

21.3541565

7.6970023

.002192982

457

208849 95443993

21.3775583

7.7026246

.002188184

458

209764

96071912

21.4009316

7.7082388

.00218:3406

459

210681

96702579

21.4242853

7.7138448

.002178649

460

211600

97336000

21.4476106

7.7194426

.002173913

461

212521

97972181

21.4709106

7.7250325

.002169197

462

213444

98611128

21.4941853

7.7306141

.002164502

463

214369

99252847

21.5174348

7.7361877

.002159827

464

215296

99897344

21.5406592

7.7417532

.002155172

465

216225

100544625

21.5638587

7.7473109

.002150538

466

217156

101194696

21.5870331

7.7528606

.002145923

467

218089

101847563

21.6101828

7.7584023

.002141328

468

219024

102503232

21.6333077

•7.7639361

.002136752

469

219961

103161709

21.6564078

7.7694620

.002132196

470

220900

103823000

21.6794834

7.7749801

.002127660

471

221841

104487111

21.7025344

7.7804904

.002123142

472

222784

105154048

21.7255610

7.7'859928

.002118644

473

223729

105823817

21.7485632

7.7914875

.002114165

474

224676

106496424

21.7715411

7.7969745

.002109705

475

225625

107171875

21.7944947

7.8024538

.002105263

476

226576

107850176

21.8174242

7.8079254

.002100840

477

227529

108531333

21.8403297

7.8133892

.002096436

478

228484

109215352

21 8632111

7.8188456

.002092050

479

229441

109902239

21.8860686

7.8242942

.002087683

480

230400

110592000

21.9089023

7.8297353

.002088333

481

231361

111284641

21.9317122

7.8351688

.002079002

482

232324

111980168

21.9544981

7.8405949

.002074689

483

233289

112678587

21.9772610

7.8460134

.002070393

484

234256

113379904

22.0000000

7.8514244

.002066116

485

235225

114084125

22.0227155

7.8568281

.002061856

486

236196

114791256

22.0454077

7.8622242

.002057613

487

237169

115501303

22.0680765

7.8676130

.002053388

i 488

238144

116214272

22.0907220

7.8729944

.002049180

489

239121

116930169

22.1133444

7 8783684

.002044990

490

240100

117649000

22.1.359436

7.8837352

.002040816

491

241081

1 18370771

22.1585198

7.S890946

.002036660

492

242064

119095488

22.1810730

7.8944468

.002032520

493

243049

11982:3157

22.2036033

7.8997917

.002028398

494

244036

120553784

22.2261108

7.9051294

.002024291

495

245025

121287375

22.2485955

7.9104599

.002020202

, 496

24(5016 122023936

22.2710575

7.9157832

.002016129

(37

TABLE VIII. — SQUARES, CUBES, SQUARE ROOTS.

Xo.

Squares.

Cubes.

Square
Koots.

Cube Roots.

Reciprocals.

407

247009

122703473 22.2034008

7.9210994

.002012072

498

248004

123505992

22.3159130

7.9204085

.002008032

499

249001

124251499

22.3383079

7.9317104

.002004008

500

250000

125000000

22.3606798

7.0370053

.002000000

501

251001

125751501

22.3830293

7.0422031

.001996008

502

252< :04

126506008

22.4053565

7.9475739

.001992032

503

253009

127263527

22.4276615

7.9528477

.001988072

504

254016

128(324064

22.4499443

7.9581144

.001984127

505

255025

128787625

22.4722051

7.9633743

.001980198

506

256036

129554216

22.4944438

7.9686271

.001976285

507

257049

13032:3843

22.5166605

7.9738731

.001972387

508

258064

131000512

22.5388553

7.9791122

.001968504

509

259081

131872229

22.5010283

7.9843444

.001964637

510

260100

132651000

22.5831796

7.0895697

.001960784

511

201 121

133432831

22.6053091

7.9947883

.001956947

512

262144

134217728

22.6274170

8.0000000

.001953125

513

263169

135005697

22.6495033

8.0052049

.001949318

514

204196

135796744

22.6715681

8.0104032

.001945525

515

265225

136590875

22.6936114

8.0155946

.001941748

516

260256

137:388096

22.7156334

8.0207794

.001937984

517

267289

138188413

22.7376340

8.0259574

.001934236

518

268324

138991832

22.7596134

8.0311287

.001930502

519

269361

139798359

22.7815715

8.0362935

.001926782

520

270400

140608000

22.8035085

8.0414515

.001923077

521

271441

141420761

22.8254244

8.0466030

.001910386

522

272484

142236648

22.8473193

8.0517479

.001915709

523

273529

143055667

22.8691933

8.0568862

.001912046

524

274576

143877824

22.8910403

8.0620180

.001908397

525

275625

144703125

22.9128785

8.0671432

.001904762

526

276076

145531576

22.9346899

8.0722620

.001901141

527

277729

146303183

22.9564806

8.0773743

.001897533

528

278784

147107052

22.9782506

8.0824800

.001893939

529

279841

148035889

23.0000000

8.0875794

.001890359

530

280900

148877000

23.0217289

8.0926723

.001886792

531

281961

149721291

23.0434372

8.0077589

.001883239

532

283024

150368768

23.0651252

8.1028390

.001879699

533

284089

151419437

23.0867928

8.1079128

.001876173

534

285156

152273304

23.1084400

8.1129803

.001872659

535

286225

153130375

23.1300670

8.1180414

.001869159

536

287296

153990656

23.1516738

8.1230962

.001865672

537

288369

154854153

23.1732605

8.1281447

.01)1862197

538

289444

155720872

23.1948270

8.1:331870

.001858736

539

290521

156590819

23.2163735

8.1382230

.001855288

540

291600

157464000

23.2379001

8.1432529

.001851852

541

292681

158340421

23.2594067

8.1482765

.001848429

542

293764

150220088

23.2808935

8.1532939

.001845018

543

294849

160103007

23.3023604

8.1583051

.001841621

544

295936

100989184

23.3238076

8.1633102

.0018:38235

545

297025

161878625

23.3452351

8.1083092

.001834862

546

29S116

162771336

23.3666429

8.1733020

.001831502

547

299209

103067323

23.3880311

8.1782888

.001828154

548

300304

104566502

23.4093998

8.1 $32695

.001824818

549

301401

165469149

23.4307490

8.1882441

.001821494

550

302500

106375000

23.4520788

8.1932127

.001818182

551

303001

167284151

23.47&3S92

8.1981753

.001814882

552

301704

1 OS 100008

23.4946802

8.2031319

.001811594

553

305809

100112377

23.5150520

8.2080825

.001808318

554

300916

170031464

23.5372046

8.2130271

.001805054

555

308025

170953875

23.5584380

8.2179657

.001801802

556

309136

171879616

23.5796522

8.2228985

.001798561

557

310249

172808093

23.6008474

8.2278254

.001795a32

558

311364

173741112

23.6220236

8.2327463

.001792115

68

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

559

312481

174676879

23.6431808

8.2376614

001788909

500

313600

175616000

23.6643191

8.2425706

.001785714

5G1

314721

176558481

23.6854386

8.2474740

.001782531

562

315844

177504328

23.7065392

8.2523715

001779359

563

316969

178453547

23.7276210

8.2572633

001776199

564

318096

179406144

23.7486842

8.2621492

.001773050

565

319225

180362125

23.7697286

8.2670294

.001769912

566

320356

I8132149o

23.7907545

8.2719039

.001766784

567

321489

182284263

23.8117618

8.2767726

.001763668

568

322624

183250432

23.8327506

8.2816355

.001760563

509

323761

184220000

2J.8537'209

8.2S64928

.001757469

570

324900

185193000

23.8746728

8.2913444

.001754386

571

326041

186169411

23.8956063

8.2961903

.001751313

572

327184

187149248

23.9165215

8.3010304

.001748252

573

328329

188132517

23.9374184

8.3058651

.001745201

574

329476

189119224

23.9582971

8.3106941

.001742160

575

330625

190109375

23.9791576

8.3155175

.001739130

576

331776

191102976

24.0000000

8.3203353

.001736111

577

332929

192100033

24.0208243

8.3251475

.001733102

578

334084

193100552

24.0416306

8.3299542

.001730104

579

335241

194104539

24.0624188

8.3347553

.001727116

580

336400

195112000

24.0831891

8.3395509

.001724138

581

337561

196122941

24.1039416

8.3443410

.001721170

582

338724

197137368

24.1246762

8.3491256

.001718213

583

339889

198155287

24.1453929

8.3539047

.001715266

584

341056

199176704

24.1660919

8.3586784

.001712329

585

342225

200201625

24.1867732

8.3634466

.001709402

586

343396

201230056

24.2074369

8.3682095

001706485

587

344569

202262003

24.2280829

8.3729668

.001703578

588

345744

203297472

24.2487113

8.3777188

.001700680

589

346921

204336469

24.2693222

8.3824653

.001697793

590

348100

205379000

24.2899156

8.3872065

.001694915

591

349281

206425071

24.3104916

8.3919423

.001692047

592

350464

207474688

24.3310501

8.3966729

.001689189

593

351649

208527857

24.3515913

8.4013981

.001686:341

594

352836

2G95S4584

24.3721152

8.4061180

.001683502

595

354025

210644875

24.3926218

8.4108326

001680672

596

355216

211708736

24.4131112

8.4155419

.001677852

597

356409

212776173

24.4335834

8.4202460

.001675042

598

357604

213847192

24.4540385

8.4249448

.001672241

599

358801

214921799

24.4744765

8.4296383

.001669449

600

360000

216000000

24.4948974

8.4343267

.001666667

601

361201

217081801

24.5153013

8.4390098

.001663894

602

362404

218167203

24.5356883

8.4436877

.001661130

603

363609

219256227

24.5560583

8.4483605

.001658375

604

364816

220348864

24.5764115

8.4530281

.001655629

605

366025

221445125

24.5967478

8.4576906

.001652893

606

367236

222545016

24.6170673

8.4623479

.OU1650165

607

368449

223648543

24.6373,00

8.4670001

.001647446

608

369664

224755712 I 24.6576560

8.4716471

.001644737

609

370881

225866529

24.6779254

8.4762892

.001642036

610

372100

226981000

24.6981781

8.4809261

.001639344

611

373321

228099131

24.7184142

8.4855579

.001636661

612

374544

229220928 , 24.7386338

8.4901848

001633987

613

375769

230346397

24.7588368

8.4948065

001631321

614

376996

231475544 24.7790234

8.4994233

.001628664

615

378225

232608375 24.7991935

8.5040350

.001626016

616

379456

233744896 24.8193473

8.5086417

.00162:3377

617

380689

234885113

24.8394847

8.5132435 .00162C746

618

381924

236029032

24.8596058

8.5178403 .001618123

619

383161

237176659 24.8797106

8.5224321

.001615509

630

384400

238328000 24.8997992 8.5270189

.001612903

69

TABLE VIII. — SQUARES, CUBES, SQUARE ROOTS.

No.

Squares

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

621

385641

239483061

24.9198710

8.5316009

.001610306

622

386884

240641848

24.9399278

8.5361780

.001607717

623 388129

241804367

24.9599679

8.5407501

.001605136

624 389376

242970624

24.9799920

8.5453173

.001602564

625 390625

244140625

25.0000000

8.5498797

.001600000

626 391876 245314376

25.0199920

8. J544372

.001597444

627 393129

246491883

25.0399681

8.5589899

.001594896

628 394384

247673152

25.0599282

8.5635377

.001592357

629

395641

248858189

25.0798724

8.5680807

.001589825

630

396900

250047000

25.0998008

8.5726189

.001587302

631 398161

251239591

25.1197134

8.5771523

.001584786

632 399424

252435968

25.1396102

8.5816809

.001582278

633 400689

253636137

25.1594913

8.5862047

.001579779

634 401956

254840104

25.1793566

8.5907238

.001577287

635 403225

256047875

25.1992063

8.5952380

.001574803

636 404496

257259456

25.2190404

8.5997476

.001572327

637

405769

258474853

25.2388589

8.6042525

.001569859

638

407044

259694072

25.2586619

8.6087526

.001567398

639

408321

260917119

25.2784493

8.6132480

.001564945

640

409600

262144000

25.2982213

8.6177388

.001562500

641 410881

263374721

25.3179778

8.6222248

.001560062

642 412164

264609288

25.3377189

8.6267063

.001557632

643 413449

265847707

25.3574447

8.6311830

.001555210

644 414736

267089984

25.3771551

8.6356551

.001552795

645 - 416025

268336125

25.3968502

8.6401226

.001550388

646 417316

269586136

25.4165301

8.6445855

.001547988

647 418609

270840023

25.4361947

8.6490437

.001545595

648 419904

272097792

25.4558441

8.6534974

.001543210

649 421201

273359449

25.4754784

8.6579465

.001540832

650

422500

274625000

25.4950976

8.6623911

.001538462

651

423801

275894451

25.5147016

8.C668310

.001536098

652 425104

277167808

25.5:342907

8.6712665

.001533742

653 426409

278445077

25.5538647

8.6756974

.001531394

654

427716.

279726264

25.5734237

8.6801237

.001529052

655

429025

281011375

25.5929678

8.6845456

.001526718

656

430336

282300416

25.6124969

8.6889630

.001524390

657

431649

283593393

25.6320112

8.6933759

.001522070

658

432964

284890312

25.6515107

8.6977843

.001519757

659

434281

286191179

25.6709953

8.7021882

.001517451

660

435600

287496000

25.6904652

8.7065877

.001515152

661

436921

288804781

25.7099203

8.7109827

.001512859

662

438244

290117528

25.7293607

8.7153734

.001510574

663

439569

291434247

25.7487864

8.7197596

.001508296

664

440896

292754944

25.7681975

8.7241414

.001506024

665

442225

294079625

25.7875939

8.7285187

.001503759

666

443556

295408296

25.8069758

8.7328918

.001501502

667

444889

296740963

25.8263431

8.7372604

.001499250

668

446224

298077632

25.8456960

8.7416246

.001497006

669

447561

299418309

25.8650343 8.7459846

.001494768

670

44890J

300763000

25.8843582

8.7503401

.001492537

671

450241

302111711

25.9036677

8.7546913

.001490313

672 4515S1

303464448

25.9229628 8.7590383

.001488095

673

45292;)

304821217

25.9422435 8.7633809

.001485884

674

454276

306182024

25.9615100 8.7677192

.001483680

675

455620

307546875

25.9S07621

8.7720532

.001481481

676

456976

308915776

26.0000000

8.7763830

.001479290

677

458329

310288733

26.0192237

8.7807084

.001477105

678

459684

311665752

26.0384331

8.785029(5

.001474926

679

461041

313046839

£6.0576284

8.7893466

.001472754

680

462400

314432000

26.0768096

8.7936593

.001470588

681

463761

315821241

26.0959767

8.7979679

.001468429

682

465124

317214568

26.1151297

8.8022721

.001466276

70

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Boots.

Cube Roots.

Reciprocals.

683

466489

318611987

26.1342687

8.8065722

.001464129

684

467856

320013504

26.15:53937

8.8108681

.001461988

685

469225

321419125

26.1725047

8.8151598

.001459854

686

470596

322828856

26.1916017

8.8194474

.001457720

687

471969

324242703

26.2106848

8.8237307

.001455604

688

473344

325660672

26.2297541

8.8280099

.001453488

689

474721

327082769

26.2488095

8.8322850

.001451379

690

476100

328509000

26.2678511

8.8365559

.001449275

691

477481

329939371

26.2868789

8.8408227

.001447178

692

478864

331373888

26.3058929

8.8450854

.001445087

693

480249

332812557

' 26.3248932

8.8493440

.001443001

694

481636

334255384

26.3438797

8.8535985

.001440922

695

483025

335702375

26.3628527

8.8578489

.001438849

696

48441(3

337153536

26.3818119

8.8620952

.001436782

697

485809

338608873

26.4007576

8.8663375

.001434720

698

487204

340068392

26.4196896

8.8705757

.001432065

699

488001

341532099

26.4386081

8.8748099

.001430615

700

490000

343000000

26.4575131

8.8790400

.001428571

701

491401

344472101

26.4764046

8.8832661

.001426534

702

492804

345948408

26.4952826

8.8874882

.001424501

703

494209

347428927

26.5141472

8.8917063

.001422475

704

495616

348913664

26.5329983

8.8959204

.001420455

705

497025

350402625

26.5518361

8.9001304

.001418440

706

498436

351895816

26.5706605

8.9043306

.001416431

707

499849

353393243

26.5894716

8.9085387

.001414427

708

501264

354894912

26.6082694

8.9127309

.001412429

709

502081

356400829

26 6270539

8.9109311

.001410437

710

504100

357911000

26.6458252

8.9211214

.001408451

711

505521

359425431

26.6645833

8.9253078

.001400470

712

506944

360944128

26.6833281

8.9294902

.001404494

713

508369

362467097

26.7020598

8.9336687

.001402525

714

509796

363994344

26.7207784

8.9378433

.001400560

715

511225

365525875

26.7394839

8.9420140

.001398601

716

512656

367061696

26.7581763

8.9461809

.001396648

717

514089

368601813

26.7768557

8.950:3438

.001394700

718

515524

370146232

26.7955220

8.9545029

.001392758

719

516961

371694959

26.8141754

8.9586581

.001390821

720

518400

373248000

26.8328157

8.9628095

.001388889

721

519841

374805361

26.8514432

8.9669570

001386963

722

521284

376367048

26.870057?

8.9711007

.001385042

723

522729

377933067

26.8886593

8.9752406

.001383126

724

524176

379503424

26.9072481

8.9793766

.001381215

725

525625

381078125

26.9258240

8.9835089

.001379310

726

527076

382657176

26.9443872

8.9876373

.001377410

727

528529

384240583

26.9629375

8.9917620

.001375516

728

529984

385828352

26.9814751

8.9958829

.001373026

729

531441

387420489

27.0000000

9.0000000

.001371742

730

532900

389017000

27.0185122

9.0041134

.001369863

731

534361

390617891

27.0370117

9.0082229

.001367989

732

535824

39222:3168

27.0554985

9.0123288

.001366120

733

537289

393832837

27.0739727

9.0164309

.001364256

734

538756

395446904

27.0924344

9.0205293

.001302398

735

540225

397065375

27.1108834

9.0246239

.001360544

736

541696

398688256

27.1293199

9.0287149

.001358696

737

543169

400315553

27.1477439

9.0328021

.001356852

738

544644

401947272

27.1661554

9.0368857

.001355014

739

546121

403583419

27.1845544

9.0409655

.001353180

740

547600

405224000

27.2029410

9.0450419

.001351:351

741

549081

406869021

27.2213152

9.0491142

.001349528

742

550564

408518488

27.2396769

9.0531831

.001347709

743

552049

410172407

27.2580263

9.0572482

.001345895

744 • 553536

411830784

27.2763034

9.0613098

.001344086

TABLE VIII. — SQUARES, CUBES, SQUARE ROOTS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

745

555025

413493625 •

27.2946881

9.0653677

.001342282

746

556516

415160936

27.3130006

9.0694220

.001340483

747

558009

416832723

27.3313007

9.0734726

.001338688

748

559504

418508992

27.3495887

9.0775197

.001&36898

749

561001

420189749

27.3678644

9.0815631

.001335113

750

562500

421875000

27.3861279

9.0856030

.001333333

751

564001

423564751

27.4043792

9.0896392

.001 331558

753

565504

425259008

27.4226184

9.0936719

.001329787

753

567009

426957777

27.4408455

9.0977010

.001328021

754

568516

428661064

27.4590604

9.1017265

.001326260

755

570025

430368875

27.4772633

9.1057485

.001324503

756

571536

432081216

27.4954542

9.1097669

.001322751

757

573049

433798093

27.5136330

9.1137818

.001321004

758

574564

435519512

27.5317998

9.1177931

.001319261

759

576081

437245479

27.5499546

9.1218010

.001317523

760

577600

438976000

27.5680975

9.1258053

.001315789

761

579121

440711081

27.5862284

9.1298061

.001314060

762

580644

442450728

27.6043475

9.1338034

.001312336

763

582169

444194947

27.62;.4546

9.1377971

.001310616

764

583696

445943744

27.6405499

9.1417874

.001308901

765

585225

447697125

27.6586334

9.1457742

.001:307190

766

586756

449455096

27.6767050

9.1497576

.001305483

767

588289

451217663

27.6947648

9.1537375

.001303781

768

589824

452984832

27.7128129

9.1577139

.001302083

769

591361

454756609

27.7308492

9.1616869

.001300390

770

592900

456533000

27.7488739

9.1656565

.001298701

771

594441

458314011

27.7668868

9.1696225

.001297017

C*TJi">

7*2

595984

460099648

27.7848880

9.1735852

.001295337

773

597529

461889917

27.8028775

9.1775445

.001293661

774

599076

463684824

27.8208555

9.1815003

.001291990

775

600625

465484375

27.8388218

9.1854527

.001290323

776

602176

467288576

27.8567766

9.1894018

.001288660

777

603729

469097433

27.8747197

9.1933474

.001287001

778

605284

470910952

27.8926514

9.1972897

.001285347

779

606841

472729139

27.9105715

9.2012286

.001283697

780

608400

474552000

27.9284801

9.2051641

.001282051

781

609961

476379541

27.9463772

9.2090962

.001280410

782

611524

478211768

27.9642629

9.2130250

.001278772

783

613089

480048687

27.9821372

9.2169505

.001277139

784

614656

481890304

28.0000000

9.2208726

.001275510

785

616225

483736625

28.0178515

9.2247914

.001273885

786

617796

485587656

28.0356915

9.2287068

.001272265

787

619369

487443403

28.0535203

9.2326189

.001270648

788

620944

489303872

28.0713377

9.2365277

.001269036

789

622521

491169069

28.0891438

9.2404333

.001267427

790

624100

493039000

28.1069386

9.2443355

.001265823

791

625681

494913671

28.1247222

9.2482344

.001264223

792

627264

496793088

28.1424946

9.2521300

.001262626

793

628849

498677257

28.1602557

9.2560224

.001261034

794

630436

500566184

28.1780056

9.2599114

.001259446

795

632025

502459875

28.1957444

9.2637973

.001257862

796

633616

504358336

28.2134720

9.2676798

.001256281

797

635209

506261573

28.2311884

9.2715592

.001254705

798

636804

508169592

28.2488938

9.2754352

.001253133

799

638401

510082399

28.2665881

9.2793081

.001251564

800

640000

513000000

28.2842712

9.2831777

.001250000

801

641601

513922401

28.3019434

9.2870440

.001248439

802

643204

515849608

28.3196045

9.2909072

.001246883

803

644809

517781627

28.3372546

9.2947671

.001245330

804

646416

519718464

28.3548938

9.2986239

.001243781

805

648025

521660125

28.3725219

9.3024775

.001242236

806

649636

523606616 28.3901391 9.3063278

.001240695

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

807

651249

525557943

28.4077454

9.3101750

.001239157

808

652864

527514112

28.4253408

9.3140190

.001237624

809

654481

529475129

28.4429253

9.3178599

.001236094

810

656100

531441000

28.4604989

9.3216975

.001234568

811

657721

533411731

28.4780617

9.3255320

.001233046

812

659344

535387328

28.4956137

9.32930.34

.001231527

813

660969

537367797

28.5131549

9.3331916

.001230012

814

662596

539353144

28.5306852

9.3370167

.001228501

815

664225

541343375

28.5482048

9.34081386

.001226994

816

665856

543338496

28.5657137

9.3446575

.001225490

817

667489

545338513

28.5832119

9.3484731

.001223990

818

669124

547343432

28.6006993

9.3522857

.001222494

819

670761

549353259

28.6181760

9. 3C 60952

.001221001

820

672400

551368000

28.6356421

9.3599016

.001219512

821

674041

553387661

28.6530976

9.3637049

.001218027

822

675684

555412248

28.6705424

9.3675051

.001216545

823

677329

557441767

28.6879766

9.3713022

.001215067

824

678976

559476224

28.7054002

9.3750963

.001213592

825

680625

561515625

28.7228132

9.3788873

.001212121

826

682276

563559976

28.7402157

9.3826752

.001210654

827

683929

565609283

28.7576077

9.3864600

.001209190

828

685584

567663552

28.7749891

9.3902419

.001207729

829

687241

569722789

28.7923601

9.3940206

.001206273

830

688900

571787000

28.8097206

9.3977964

.001204819

831

690561

573856191

28.8270706

9.4015691

.001203369

832

692224

575930368

28.8444102

9.4053387

.001201923

833

693889

578009537

28.8617394

9.4091054

.001200480

834

695556

580093704

28.8790582

9.4128690

.001199041

835

697225

582182875

28.8963666

9.4166297

.001197605

836

698896

584277056

28.9136646

9.4203873

.001196172

837

700569

586376253

28.9309523

9.4241420

.001194743

838

702244

588480472

28.9482297

9.4278936

.001193317

839

703921

590589719

28.9654967

9.4316423

.001191895

840

705600

592704000

28.9827535

9.4353880

.001190476

841

707281

594823321

29.0000000

9.4391307

.001189061

842

708964

596947688

29.0172363

9.4428704

.001187648

843

710649

599077107

29.0344623

9.4466072

.001186240

844

712336

601211584

29.0516781

9.4503410

.001184834

845

714025

603351125

29.0688837

9.4540719

.001183432

846

715716

605495736

29.0860791

9.4577999

.001182033

847

717409

607645423

29.1032644

9.4615249

.001180638

848

719104

609800192

29.1204396

9.4652470

.001179245

849

720801

611960049

29.1376046

9.4689661

.001177856

850

722500

614125000

29.1547595

9.4726824

.001176471

851

724201

616295051

29.1719043

9.4763957

.001175088

852

725904

618470208

29.1890390

9.4801061

.001173709

853

727609

620650477

29.2061637

9.4838136

.001172333

854

729316

622835864

29.2232784

9.4875182

.001170960

855

731025

625026375

29.2403830

9.4912200

.001169591

856

732736

627222016

29.2574777

9.4949188

.001168224

857

734449

629422793

29.2745623

9.4986147

.001166861

858

736164

631628712

29.2916370

9.5023078

.001165501

859

737881

633839779

29.3087018

9.5059980

.001164144

860

739600

636056000

29.3257566

9.5096854

.001162791

861

741321

638277381

29.3428015

9.51-33699

.001161440

862

743044

640503928

29.3598365

9.5170515

.001160093

863

744769

642735647

29.3768616

9.5207303

.001158749

864

746496

644972544

29.3938769

9.5244063

.001157407

865

748225

647214625

29.4108823

9.5280794

.001156069

866

749956

649461896

29.4278779

9.5317497

.001154734

867

751689

651714363

29.4448637

9.5354172

.001153403

868

753424

653972032

29.4618397

9.5390818

,001152074

TABLE VIII. — SQUARES, CUBES, SQUARE ROOTS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

869

755161 656234909

29.4788059

9.5427437 .001150748

870

756900

658503000

29.4957624

9.5464027 .001149425

871

758641

660776311

29.5127091

9.5500589

.001148106

872

760:384

663054848

29.5296461

9.5537123

.001146789

873

762129

665338617

29.5465734

9.5573630

.001145475

874

763876

667627624 29.5634910

9.5610108

.001144165

875

765625

669921875 29.5803989

9.5646559

.001142857

876

767376

672221376 29.5972972

9.5682982

.001141553

877

769129

674526133

29.6141858

9.5719377

.001140251

878

770884

676836152

29.6310648

9.5755745

.001138952

879

772641

679151439

29.6479342

9.5792085

.001137656

880

774400

681472000

29.6647939

9.5828397

.001136364

881

776161

683797841

29.6816442

9.5864682

.001L35074

882

777924

686128968

29.6984848

9.5900939

.001ia3787

883

779689

688465387

29.7153159

9.5937169

.001132503

884

781456

690807104

29.7321375

9.597a373

.001131222

885

783225

693154125

29.7489496

9.6009548

.001129944

886

784996

695506456

29.7657'521

9.6045696

.001128668

887

786769

697864103

29.7825452

9.6081817

.001127396

888

788544

700227072

29.7993289

9.6117911

.001126126

889

790321

702595369

29.8161030

9.6153977

.001124859

890

792100

704969000

29.8328678

9.6190017

.001123596

891

793881

707347971

29.8496231

9.6226030

.001122334

892

795664

709732288

29.8663690

9.6262016

.001121076

893

797449

712121957

29.8831056

9.6297975

.001119821

894

799236

714516984

29.8998328

9.6333907

.001118568

895

801025

716917375

29.9165506

9.6369812

.001117318

896

802816

719323136

29.9332591

9.6405690

,001116071

897

804609

721734273

29.9499583

9.6441542

.001114827

898

806404

724150792

29.9666481

9.6477367

.001113586

899

808201

726572699

29.9833287

9.6513166

.001112347

900

810000

729000000

30.0000000

9.6548938

.001111111

901

811801

731432701

30.0166620

9.6584684

.001109878

902

813604

733870808

30.0333148

9.6620403

.001108647

903

815409

736314327

30.0499584

9.6656096

.001107420

904

817216

738763264

30.0665928

9.6691762

.001106195

905

819025

741217625

30.0832179

9.6727403

.001104972

906

820836

743677416

30.0998339

9.6763017

.001103753

907

822649

746142643

30.1164407

9.6798604

.001102536

908

824464

748613312

30.1330383

9.6834166

.001101322

909

826281

751089429

30.1496269

9.6869701

.001100110

910

828100

753571000

30.1662063

9.6905211

.001098901

911

829921

756058031

30.1827765

9.6940694

.001097695

912

831744

758550528

30.1993377

9.6976151

.001096491

913

833569

761048497

30.2158899

9.7011583

.001095290

914

835396

763551944

30.2324329

9.7046989

.001094092

915

837225

766060875

30.2489669

9.7082369

.001092896

916

839056

768575296

30.2654919

9.7117723

.001091703

917

840889

771095213

30.2820079

9.7153051

.001090513

918

842724

773620632

30.2985148

9.7188354

.001089325

919

844561

776151559

30.3150128

9.7223631

.001088139

920

846400

778688000

so.asisois

9.7258883

.001086957

921

848241

781229961

30.3479818

9.7294109

.001085776

922

850084

783777448

30.3644529

9.7329309

.001084599

923

851929

786330467

30.3809151

9.7364484

.001083423

924

853776

788889024

30.3973683

9.7399634

.001082251

925

855625

791453125

30.4138127

9.7434758

.001081081

926

857476

794022776

30.4302481

9.7469857

.001079914

927

859329

796597983

30.4466747

9.7'504930

.001078749

928

861184

799178752

30.4630924

9.7539979

.001077586

9S9

863041

801765089

30.4795013

9.7575002

.001076426

930

864900

804357000

30.4959014

9.7610001 .001075269

74

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
lioots.

Cube Roots.

Reciprocals.

931

866761

806954491

30.5122926

9.7644974

.001074114

932

868624

809557568

30.5286750

9.7679922

.001072961

933

870489

812166237

30.5450487

9.7714845

.001071811

934

872356

814780504

30.5614136

9.77'49743

.001070664

935

874225

817400375

30.5777697

9.7784616

.001069519

936

876096

820025856

30.5941171

9.7819466

.001068376

937

877969

822656953

30.6104557

9.7854288

.001067236

938

879844

825293672

30.6267857

9.7889087

.001066098

S39

881721

827936019

30.6431069

9.7923861

.001064963

940

883600

830584000

30.6594194

9.7958611

.001063830

941

885481

833237621

30.6757233

9.7993336

.001062699

943

887364

835896888

30.6920185

9.8028036

.001061571

943

889249

838561807

30.7083051

9.8062711

.001060445

944

891136

841232384

:-,0. 7245830

9.8097362

.001059322

945

893025

843908625

30.7408523

9.8131989

.001058201

946

894916

846590536

30.7571130

9.8166591

.001057082

947

896809

849278123

30.7733651

9.8201169

.001055966

948

898704

851971392

30.7896086

9.8235723

.001054852

949

900601

854670349

30.8058436

9.8270252

.001053741

950

902500

857375000

30.8220700

9.8304757

.001052632

951

904401

860085351

30.8382879

9.8339238

.001051525

952

906304

862801408

30.8544972

9.8373695

.001050420

953

908209

865523177

30.8706981

9.8408127

.001049318

954

910116

868250664

30.8868904

9.8442536

.001048218

955

912025

870983875

80.9030743

9.8476920

.001047120

956

913936

873722816

30.9192497

9.8511280

.001046025

957

915849

876467493

30.9354166

9.8545617

.001044932

958

917764

879217912

30.9515751

9.8579929

.001043841

959

919681

881974079

30.9677251

9.8614218

.001042753

960

921600

884736000

30.9838668

9.8648483

.001041667

961

923521

887503681

31.0000000

9.8682724

.001040583

962

925444

890277128

31.0161248

9.8716941

.001039501

963

927369

893056347

31.0322413

9.8751135

.001038422

964

929296

895841344

31.0483494

9.8785305

.001037344

965

931225

898632125

31.0644491

9.8819451

.001036269

966

933156

901428696

31.0805405

9.8853574

.001035197

967

935089

904231063

31.0966236

9.8887673

.001034126

968

937024

907039232

31.1126984

9.8921749

.001033058

969

938961

909853209

31.1287648

9.8955801

.001031992

970

940900

912673000

31.1448230

9.8989830

.001030928

971

942841

915498611

31.1608729

9.9023885

.001029866

972

944784

918330048

31.1769145

9.9057817

.001028807

973

946729

921167317

31.1929479

9.9091776

.001027749

974

948676

924010424

31.2089731

9.9125712

.001026694

975

950625

926859375

31.2249900

9.9159624

.001025641

976

952576

929714176

31.2409987

9.9193513

.001024590

977

954529

932574833

31.2569992

9.9227379

.001023541

978

956484

935441352

31.2729915

9.9261222

.001022495

979

958441

938313739

31.2889757

9.9295042

.001021450

980

960400

941192000

31.3049517

9.9328839

.001020408

981

962361

944076141

31.3209195

9.9362613

.001019368

982

964324

94696G168

31.3368792

9.9396363

.001018330

983

966289

949862087

31.3528308

9.9430092

.001017294

984

968256

952763904

31.3687743

9.9463797

.001016260

985

970225

955671625

31. 3847097

9.9497479

.001015228

986

972196

958585256

31.4006369

9.9531138

.001014199

987

974169

961504803

31.4165561

9.9564775

.001013171

988

976144

964430272

31.4324673

9.9598389

.001012146

989

978121

967361669

31.4483704

9.9631981

.001011122

990

980100

970299000

31.4642654

9.9665549

.001010101

991

982081

973242271

31.4801525

9.9699095

.001009082

992

984064

976191488

31.4960315

9.9732619

.001008065

75

TABLE VIII. — SQUARES, CUBES, ETC.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

993

986049

979146657

31.5119025

9.9766120

.001007049

994

988036

982107784

31.5277655

9.9799599

.001006036

995

990025

985074875

31.5436206

9.9833055

.001005025

996

992016

988047936

31.5594677

9.9866488

.001004016

997

994009

991026973

31.5753068

9.9899900

.001003009

998

996004

994011992

31.5911380

9.9933289

.001002004

999

998001

997002999

31.6069613

9.9966656

.001001001

1000

1000000

1000000000

31.6227766

10.0000000

.001000000

1001

1002001

1003003001

31.6385840

10.0033322

.0009990010

1002

1004004

1006012008

31.6543836

10.0066622

.0009980040

1003

1006009

1009027027

31.6701752

10.0099899

.0009970090

1004

1008016

1012,148064

31.6859590

10.0133155

.0009960159

1005

1010025

1015075125

31.7017349

10.0166389

.0009950249

1006

1012036

1018108216

31.7175030

10.0199601

.0009940358

1007

1014049

1021147'343

31.7332633

10.0232791

.0009930487

1008

1016064

1024192512

31.7490157

10.0265958

.0009920635

1009

1018081

1027243729

31.7647603

10.0299104

.0009910803

1010

1020100

1030301COO

31.7804972

10.0332228

.0009900990

1011

1022121

103-3364331

31.7962262

10.0365330

.0009891197

T*

1012

1024144

1036433728

31.8119474

10.0398410

.0009881423

1013

1026169

1039509197

31.8276609

10.0431469

.0009871668

1014

1028196

1042590744

31.8433666

10.0464506

.0009861933

1015

1030225

1045678375

31.8590646

10.0497521

.0009852217

1016

1032256

1048772096

31.8747549

10.0530514

.0009842520

1017

1034289

1051871913

31.8904374

10.0563485

.0009&32842

1018

1036324

1054977832

31.9061123 .

10.0596435

.0009823183

1019

1038361

1058089859

31.9217794

10.0629364

.0009813543

1020

1040400

1061208000

31.9374388

10.0662271

.0009803922

1021

1042441

1064332261

31.9.530906

10.0695156

.0009794319

1022

1044484

1067462648

31.9687347

10.0728020

.0009784736

1023

1046529

1070599167

31.9843712

10.0760863

.0009775171

1024

1048576

1073741824

32.0000000

10.0793684

.0009765625

1025

1050625

10r6890625

32.0156212

10.0826484

.0009756098

1026

1052676

1080045576

32.0312348

10.0859262

.0009746589

1027

1054729

1083206683

32.0468407

10.0892019

.0009737098

1028

1056784

1086373952

32.0624391

10.0924755

.0009727626

1029

1058841

1089547389

32.0780298

10.0957469

.0009718173

1030

1060900

1092727000

32.0936131

10.0990163

.0009708738

1031

1062961

1095912791

32.1091887

10.1022835

.0009699321

1032

1065024

1099104768

32.1247568

10.1055487

.00 ,9689922

1033

1067089

1102302937

32.1403173

10.1088117

.0009680542

1034

1069156

1105507304

32.1558704

10.1120726

.0009671180

1035

1071225

1108717875

32.1714159

10.1153314

.0009661836

1036

1073296

1111934656

32.1869539

10.1185882

.0009652510

1037

1075369

1115157653

32.2024844

10.1218428

.0009643202

1038

1077444

1118386872

32.2180074

10.1250953

.0009633911

1039

1079521

1121622319

32.2335229

10 1283457

.0009624639

1040

1081600

1124864000

32.2490310

10.1315941

.0009615385

1041

1083681

1128111921

32.2645316

10.1348403

.0009606148

1042

1085764

1131366088

32.28002-18

10.1380845

.0009596929

1043

1087849

1134626507

32.2955105

10.1413266

.0009587738

1044

1089936

1137893184

32.3109888

10.1445667

.0009578544

1045

1092025

1141166125

32.3264598

10.1478047

.0009569378

1046

1094116

1144445336

32.3419233

10.1510406

.0009560229

1047

1096209

1147730823

32.3573794

10.1542744

.0009551098

1048

1098304

1151022592

32.3728281

10.1575062

.0009541985

1049

1100401

1154320649

32.3882695

10.1607359

.0009532888

1050

1102500

1157625000

32.4037035 10.1639636

.0009523810

1051

1104601

1160935651

32.4191301

10.1671893

.0009514748

1052

1106704

1164252608

32.4345495

10.1704129

.0009505703

1053

1108809

1167575877

32.4499615

10.1736344

.0009496676

1054

1110916

1170905464

32.4653662

10.1768539

.0009487666

76

TABLE IX. — LOGARITHMS OF NUMBERS.

No.

100 L. 000. J

.No. 109 L. 040.

N.

0

1284

5

6

7

8 9

Diff.

100

000000

0434 0868 1301 1734

2166

2598

3029

3461 3891

432

1

4321

4751 5181 5609 6038

6466

6894

7321

7748 8174

426

2

Rfiflfl

Q0°6 01^1 9K76

O\J\J\J

nsnn

0724

1147

1570

1993 2415

424

3

012837

3259 3680 4100 4521

4940

5360

5779

6197 6616

420

A

7033

7451 7868 8284 8700

9116

9^32

9947

*±

4 V/OO

4 ^*J J. 1 UU*J *J^*_ri (_ 4 \J\J

V*JtJ(W

*7i7T 1

0361 0775

416

5

021189

1603 2016 2428 2841

3252

3664

4075

4486 4896

412

6

5306

5715 6125 6533 6942

7350

7757

8164

8571 8978

408

r*

93K4

9789

'

*7tJLrx

019*5 0600 1001

1408

1812

001 fi

2619 3021

404

8

033424

3826 4227 4628 5029

xmsu

5430

J.*J A/V

5830

iSrfiSrf L\J

6230

*v*J 1 €7 (JV/iV 1

6629 7028

*±v^±

400

q

742fi

7825 8223 8620 9017

0414.

Q811

i7

1 1 >. U

04

I O<v*J \Jf*t*iJ \J\J&\J i3\J L 1

i7Tl^

«7<JA A

0207

0602 0998

397

PROPORTIONAL PARTS.

Vift.

1

2

3

4

5

6

7

8

9

434

43.4

86.8

130.2

173.6

217.0

260.

4

303.8

347.2

390.6

433

43.3

86.6

129.9

173.2

216.5

259.

8

303.1

346.4

389.7

432

43.2

86.4

129.6

172.8

216.0

259

2

302.4

345.6

388.8

431

43.1

86.2

129.3

172.4

215.5

258.

6

301.7

344.8

387.9

430

43.0

86.0

129.0

172.0 215.0

258

0

301.0

344.0

387.0

429

42.9

85.8

128.7

171.6

214.5

257

4

300.3

343.2

386.1

428

42.8

85.6

128.4

171.2

214.0

256

8

299.6

342.4 385.2

427

j 42.7

85.4

128.1

170.8

213.5

256

2

298.9

341.6

384.3

426

42.6

85.2

127.8

170.4

213.0

255

6

298.2

340.8

383.4

425

42.5

85.0

127.5

170.0

212.5

255

0

297.5

340.0

382.5

434

42.4

84.8

127.2

169.6

212.0

254

4

296.8

339.2

381.6

423

42.3

84.6

126.9

169.2

211.5

253

8

296.1

338.4

380.7

422

42.2

84.4

126.6

168.8

211.0

253

2

295.4

337.6

379.8

431

42.1

84.2

126.3

168.4

210.5

252

6

294.7

336.8

378.9

420

42.0

84.0

126.0

168.0

210.0

252

0

294.0

336.0

378.0

419

41.9

83.8

125.7

167.6

209.5

251

4

293.3

335.2

377.1

418

41.8

83.6

125.4

167.2

209.0

250.8

292.6

334.4

376.2

417

41.7

83.4

125.1

166.8

208.5

250

2

291.9

333.6

375.3

416

41.6

83.2

124.8

166.4

208.0

249

6

291.2

a32.8

374.4

415

41.5

83.0

124.5

166.0

207.5

249

0

290.5

332.0

373.5

414

41.4

82.8

124.2

165.6

207.0

248

4

289.8

331.2 372.6

413 41.3

82.6

123.9

165.2

206.5

247

8

289.1

330.4 ! 371.7

412

41.2

82.4

123.6

164.8

206.0

247

.2

288.4

329.6 370.8

411

41.1

82.2

123.3

164.4

205.5

246

.6

287.7

328.8 369.9

410

41.0

82.0

123.0

164.0

205.0 246

.0

287.0

328.0 369.0

409 40.9

81.8

122.7

163.6

204.5 245

.4

286.3

327.2 i 368.1

408 40.8

81.6

122.4

163.2

204.0 244

.8

285.6

326.4 367.2

407 40.7

81.4

122.1

162.8

203.5 244.2

284.9

325.6 366.3

406

40.6

81.2

121.8

162.4

203.0

243

6

284.2

324.8 365.4

405

40.5

81.0

121.5

162.0

202.5

243

.0

283.5

324.0 364.5

404

40.4

80 8

121.2

161.6

202.0

242

.4

282.8

323.2 363.6

403

40.3

80.6

120.9

161.2

201.5

241

.8

282.1

322.4 362.7

402

40.2

80.4

120.6

160.8

201.0

241

2

281.4

321.6 ; 361.8

401

40.1

80.2

120.3

160.4

200.5

240

.6

280.7

320.8 ! 360.9

400

40.0

80'0

120.0

160.0

200.0

240

.0

280.0

320.0 360.0

399

39.9

79.8

119.7

159.6

199.5

239

.4

279.3

319.2 359.1

398

39.8 79.6

119.4

159.2

199.0 238

.8

278.6

318.4 358.2

397 39.7 79.4

119.1

158.8

198.5 ) 238

.2 2

?7.9

317.6 ; 357.3

396 " 39.6 79.2

118.8

158.4

198.0 ' 237

.6 277.2

316.8 356.4

395 39.5 79.0

118.5

158.0 197.5 237

.0 276.5 316 0 355.5

TABLE IX. — LOGARITHMS OF NUMBERS.

No.

110 L. 041.]

•

[No. 119 L. 078.

N.

0

1

2

3 4

5

6

7

8

9

Diff.

110

041393

1787

2182

2576 2969

a362

3755

4148

4540

4932

393

1

5323

5714

6105

64'J5 6885

7275 ! 7664

8053

8442

8830

390

2

9218

9606

qqqo

iJ\J\J \J

0380 OTfi

1153

1538

-iqoj.

2SOQ

of-qi

38fi

3

053078

3463

3846

4230 4613

4996

5378

J. 3*r±

5760

ivOU.7

6142

6524

oou

383

4

f^Q05

r*oQiJ

7fififi

Oi"\ </» CJOI*

OOA~

qi85

Qt /?O

qqio

t \J\J\J

ij A.(J*J

i/«7^v

0390

37Q

5

060698

1075

1452

1829 2206

2582

2958

3333

3709

4083

Uf v

376

6

4458

4832

5206

5580 5953

6326

6699

7071

7443

7815

373

7

8186

8557

WOOQ

q*)qft q^fft

1

\J±\J\J

\J*J*J I

0038

0407

0776

114"

1514

370

8

071882

2250

2617

2985 3352

\J\jtJ(J

3718

\j~±\-> t
4085

VI 1 U

4451

4*16

1*-J J-Tt

5182

tJ 1 \J

366

9

5547

5912

6276

6640 7004

7368

7731

8094

8457

8819

363

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

7

8

9

395

39.5

79.0

118.5

158.0

197.5

237

.0

276.5

316.0

355.5

394

39.4

78.8

118.2

157.6

197.0

236

.4

275.8

315.2

354.6

393

39.3

78.6

11

7.9

157.2 196.5

235.8

275.1

314.4 353.7

392

39.2

78.4

117.6

156.8

196.0

235.2

274 .4

313.6 i 352.8

391

39.1

78.2

117.3

156.4

195.5

234

.6

273.7

312.8 351.9

390

39.0

78.0

117.0

156.0

195.0

234.0

273.0

312.0 351.0

389

38.9

77. £

!

116.7

155.6

194.5

233.4

272.3

311.2 350.1

388

38.8

77.6

116.4

155.2

194.0

232.8

271.6

310.4 349.2

387

38.7

77.4

116.1

154.8

193.5

232.2

270.9

309.6 348.3

386

38.6

77.2

115.8

154.4

193.0

231

.6

270.2

308.8 347.4

385

38.5

77.0

115.5

154.0

192.5

231.0

269.5

308.0 346.5

384

38.4

76.J

5

115.2

153.6

192.0

230.4

268.8

307.2 ! 345.6

383

38.3

76.6

114.9

153.2

191.5

229.8

268.1

306.4 344.7

382

38.2

76.4

114.6

152.8

191.0

229.2

267.4

305.6 343.8

381

38.1

76.2

114.3

152.4

190.5

228.6

266.7

304.8 342.9

380

38.0

76.0

114.0

152.0

190.0

228.0

266.0

304.0 342.0

379

37.9

75.8

113.7

151.6

189.5

221

'.4

265.3

303.2 341.1

378

37.8

75.6

113.4

151.2

189.0

226.8

264.6

302.4

340.2

37'

y
\

37.7

75.4

113.1

150.8

188.5

226.2

263.9

301.6

339.3

376

37.6

75.2

112.8

150.4

188.0

225.6

263.2

300.8

338.4

375

37.5

75.0

112.5

150.0

187.5

225.0

262.5

300.0

337.5

374

37.4

74.8

112.2

149.6

187.0

224.4

261.8

299.2

336.6

373

37.3

74.6

111.9

149.2

186.5

223.8

261.1

298.4

335.7

372

37.2

74.4

111.6

148.8

186.0

223.2

260.4

297.6

334.8

371

37.1

74.2

111.3

148.4

185.5

222.6

259.7

296.8

333.9

370

37.0

74.0

111.0

148.0

185.0

222.0

259.0

296.0

sas.o

369

36.9

73.8

110.7

147.6

184.5

221.4

258.3

295.2

332.1

368

36.8

73.6

110.4

147.2

184.0

220.8

257.6

294.4

331.2

867

36.7

73.4

110.1

146.8

183.5

220.2

256.9

293.6

330.3

366

36.6

73.2

109.8

146.4

183.0

219.6

256.2

292.8

329.4

365

36.5

73.0

109.5

146.0

182.5

219.0

255.7

292.0

328.5

364

36.4

72.8

109.2

145.6

182.0

218.4

254.8

291.2

327.6

363

36.3

72.6

108.9

145.2

181.5

21"

".8

254.1

290.4

326.7

362

36.2

72.4

108.6

144.8

181.0

217.2

253.4

289.6

325.8

361

36.1

72.2

108.3

144.4

180.5

216.6

252.7

288.8

324.9

360

36.0

72.0

108.0

144.0

180.0

216.0

252.0

288.0

324.0

359

35.9

71.8

107.7

143.6

179.5

215.4

251.3

287.2

323.1

358

35.8

71.

6

107.4

143.2

179.0

214.8

250.6

286.4

322.2

357

35.7

71.4

107.1

142 8

178.5

214.2

249.9

285.6

321.3

356

35.6

71.

2

106.8

142.4

178.0

213.6

249.2

284.8

320.4

78

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 120 L. 079.]

[No. 134 L. 130.

N.

0

1 2

3

4

5

6

7

8

9

Diff.

120

079181

9543 9904

!

1

0266

1 0626

0987

1347

1707

2067

2426

360

1

082785

3144 3.503

3861

4219

4576

4934

5291

5647

6004

357

2

6360

6716 7071

7426

7781

8136

8490

8845

9198

9552

355

QQAK

yyuo

0258 Of>1 1

0963

1315

1667

2018

2370

2721

3071

352

4

093422

\JAt*J\J \J\JA. J.

3772 4122

4471

4820

5169

5518

5866

A* 1 i-w A

6215

U\s 1 J.

6562

t^-/iV

349

5

6910

7257 7004

7951

8298

8644

8990

9335

9681

— - —

flO>fi

QJR

6

100371

0715 1059

1403

1747

2091

2434

2777

3119

\J\J&\J

3462

*^±\J

343

7

3804

4146 4487

4828

5169

5510

5851

6191

6531

6871

341

Q

7210

754Q 7S88

0007

8565

Rons

Q°41

Q57Q

QQIfi

O

t -V I ' '

1 *J^v I OiJMj

LM/M 1

\J*-f\J*J VWW^

t//^± A

• '• ' 4 «-'

t/«7 _1\J

flO^Q

QQU

9

110590

0926 1*63

1599

1934

2270

2605

2940

3275

V/^wOO

3609

ooo

335

130

3943

4277 4611

4944

5278

5611

5943

6276

6608

6940

333

1

7271

7603 7934

8265

8595

8926

9256

9586

9915

no 15

QQ(-)

2

120574

0903 1231

1560

1888

2216

2544

2871

3198

\//w t*J

3525

oov/

328

3

3852

4178 4504

4830

5156

5481

5806

6131

6456

6781

325

4

71(15

74.00 77^3

8076

8SQQ

R7'X>

Q045

Q3p,A

QtfClO

*±

i i < '• j

13

i -±^w J I I 'J'J

\J\J 1 \J

l_JOt7«7

\J i - •»

*J\J^±\J

S7OVO

iJ\J*J\J

0012

323

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

7

8

9

355

35.5

71.0

106.5

142.0

177.5

213.0

248.5

284.0

319.5

354

35.4

70.8

106.2

141.6

177.0

212.4

247.8

283.2

318.6

353

35.3

70.6

105.9

141.2

176.5

211.8

247.1

282.4

317.7

352

35.2

70.4

105.6

140.8

176.0

211.2

246.4

281.6

316.8

351

35.1

70.2

105.3

140.4

175.5

210.6

245.7

280.8

315.9

aso

35.0

70.0

105.0

140.0

175.0

210.0

245.0

280.0

315.0

349

34.9

69.8

104.7

139.6

174.5

209.4 244.3

279.2

314.1

348

34.8

69.6

104

.4

139.2

174.0

208.8 243.6

278.4

313.2

347

34.7

69.4

104.1

138.8

173.5

208.2 242.9

277.6

312.3

346

34.6

69.2

103.8

138.4

173.0

207.6

242.2

276.8

311.4

345

34.5

69.0

103.5

138.0

172.5

207.0

241.5

276.0

310.5

344

34.4

68.8

103.2

137.6

172.0

206.4

240.8

275.2

309.6

343

34.3

68.6

102.9

137.2

171.5

205.8

240.1

274.4

308.7

342

34.2

68.4

102

.6

136.8

171.0

205.2

239.4

273.6

307.8

341

34.1

68.2

102.3

136.4

170.5

204.6

238.7

272.8

306.9

340

34.0

68.0

102.0

136.0

170.0

204.0

238.0

272.0

306.0

339

33.9

67.8

101

.7

135.6

169.5

203 .4

237.3

271.2

305.1

338

33.8

67.6

101

.4

135.2

169.0

202.8

236.6

270.4

304.2

337

33.7

67.4

101

.1

134.8

168.5

202.2

235.9

269.6

303.3

336

33.6

67.2

100.8

134.4

168.0

201.6

235.2

268.8

302.4

335

33.5

67.0

100

.5

134.0

167.5 201.0

234.5

268.0

301.5

334

33.4

66.8

100

.2

ias.6

167.0

200.4 233.8

267.2

300.6

333

33.3

66.6

99

.9

ias.2

166.5

199.8

233.1

266.4

299.7

332

33.2

66.4

99

.6

132.8

166.0

199.2 232.4

265.6

298.8

asi

33.1

66.2

99

.3

132.4

165.5

198.6

231.7

264.8

297.9

aso

33.0

66.0

99

.0

132.0

165.0

198.0

231.0

264.0

297.0

329

32.9

65.8

98

.7

131.6

164.5

197.4

230.3

263.2

296.1

328

32.8

65.6

98

.4

131.2

164.0

196.8

229.6

262.4

295.2

327

32.7

65.4

98

.1

130.8

163.5

196.2

228.9

261.6

294.3

326

32.6

65.2

97

.8

130.4

163.0

195.6

228.2

260.8

293.4

325

32.5

65.0

97

.5

130.0

162.5

195.0

227.5

260.0

292.5

324

32.4

64.8

97

.2

129.6

162.0 194.4

226.8

259.2

291.6

323

32.3

64.6

96

.9

129.2

161.5 193.8

226.1

258.4

290.7

322

32.2 64.4

96.6

128.8 161.0 193.2

225.4

257.6

289.8

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 135 L. 130.]

[No. 149 L. 175.

N.

0

1 2

O

4

. .

7

8 9

Diff.

135 130334

0655 0977

1298

1619

1939 2260

2580

2900 3219

321

6

3539

3858 4177

4496

4814

5133 5451

5769

6086 6403

318

r«

i

6721

7037 7354

7071

7987

8303 8618

8934

9249 9564

316

Q

Q87Q

O

yoi a

01 04 0"i08

OS22

1136

1450 1763

2076

2389 2702

314

9

143015

V J. t/^C \J'J\J<^I

3327 3G39

V\J^fnf

3951

4263

4574 4885

5196

5507 5818

311

140

6128

6438 6748

7058

7367

7676 7985

8294

8603 8911

309

1

9219

9527 9835

•

0142

0-UQ

07*sfi 1 HAS

1370

1676 1982

307

2

152288

2594 2000

V 1 Tt>i*

3205

VT"1 t7

3510

\J I *J\J

3815

4120

.1 » J 1 \J

4424

4728 5032

305

3

5336

5640 5943

6246

6549

6852

7154

7457

7759 8061

303

A

»3fi>>

ftfifU ftOfi^

O-'fifi

Q=>fi7

QfifiS

f»

OOU.V

OUl^i Ot/VJtJ

y^\j\j

VtJ\J t

t7OVO

OlfiS

04fiO

07150 1 068

301

5

161368

1G67 1967

2266

2564

2863

V 1 UO

3161

\n\jtj
3460

\J t \J& -I \J\J\J

3758 4055

t/VA

299

6

4353

4050 4947

5244

5541

5838

6134

6430

6726 7022

297

r»
i

7317

7613 7908

8203

8497

8792

9086

9380

9674 9968

295

8

170262

0555 0848

1141

1434

1726

2019

2311

2603 2895

293

9

3186

3178 3769

4060

4351

4641

4932

5222

5512 5802

291

.PROPORTIONAL, PARTS.

Diff.

i

2

3

4

5

6

7

8

9

321

32.1 64.2

96

.3

128.4

160.5

192

6

224.7

256.8

288.9

320

32.0 64.0

96

.0

128.0

160.0

192

0

224.0

256.0

288.0

319

31.9 63.8

95

.7

127.6

159.5

191

4

223.3

255.2

287.1

318

31.8 63.6

95

.4

127.2

159.0

190

8

222.6

254. 4

286.2

317

31.7 63.4

95

.1

126.8

158.5

190

2

221.9

253.6

285.3

310

31.6 63.2

94

.8

126.4

158.0

189

6

221.2

252.8

284.4

315

31.5 63.0

94.5

126.0

157.5

189

0

220.5

252.0

283.5

314

31.4 62.8

94

.2

125.6

157.0

188

4

219.8

251.2

282.6

313

31.3 62.6

93

9

125.2

156.5

187

8

219.1

250.4

281.7

312

31.2 62.4

93

.6

124.8

156.0

187

2

218.4

249.6

280.8

311

31.1

62.2

93

.3

124.4

155.5

186

6

217.7

248.8

279.9

310

31.0 62.0

93

.0

124.0

155.0

186

0

217.0

248.0

279.0

309

30.9 61.8

92

r*
. t

123.6

154.5

185

4

216.3

247.2

278.1

308

30.8 61.6

92

.4

123.2

154.0

184

8

215.6

246.4

277.2

307

30.7 61.4

92

.1

1C2.8

153.5

184.2

214.9

245.6

276.3

306

30.6 ! 61.2

91

.8

122.4

153.0

183.6

214.2

244.8

275.4

305

30.5 61.0

91

.5

122.0

152.5

183

0

213.5

244.0

274.5

304

30.4 ; 60.8

91

.2

121.6

152.0

182

4

212.8

243.2

273.6

303

30.3 60.6

90

.9

121.2

151.5

181

8

212.1

242.4

272.7

302

30.2 60.4

90

.6

120.8

151.0

181

2

211.4

241.6

271.8

301

30.1

60.2

90

.3

120.4

150.5

180

6

210.7

240.8

270.9

300

30.0 60.0

90

.0

120.0

150.0

180

0

210.0

240.0

270.0

299

29.9 59.8

89

.7

119.6

149.5

179

4

209.3

239.2

269.1

298

29.8

59.6

89.4

119.2

149.0

178

8

208.6

238.4

268.2

297

29.7 59.4

89

.1

118.8

148.5

178

2

207.9

237.6

267.3

296

29.6 ' 59.2

88

.8

118.4

148.0

177

6

207.2

236.8

266.4

295

29.5 59.0

88

.5

118.0

147.5

177

0

206.5

236.0

265.5

294

29.4 58.8

88

.2

117.6

147.0

176

4

205.8

235.2

264.6

293

29.3

58.6

87

.9

117.2

146.5

175

8

205.1

234.4

263.7

292

29.2

58.4

87

.6

116.8

146.0

175

2

204.4

233.6

262.8

291

29.1

58.2

87

.3

116.4

145.5

174

6

203.7

232.8

261.9

290

29.0

58.0

87

.0

116.0

145.0

174

0

203.0

232.0

261.0

289

28.9 i 57.8

86

.7

115.6

144.5

173

4

202.3

231.2

260.1

288

28.8

57.6

86

.4

115.2

144.0

172

.8

201.6

230.4

259.2

287

28.7

57.4

86

.1

114.8

143.5

172

2

200.9

229.6

258.3

286

28.6

57.2 85

.8

114.4 143.0

171

6

200.2

228.8

257 .4

80

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 150 L. 176.]

[No. 169 L. 230.

N.

0

1

2

3

4

5

6

7

8 9

Diff.

150

176091

6381

6670

6959

7248

7536

7825

8113

8401 8689

289

1

8977

9264

9552

9839

0126 0413

0699

0986

1272 1558

287

2

181844

2129

2415

2700

2<J85 3270

3555

3839

4123 4407

w*tJ 1

285

3

4691

4975

5259

5542

5825

6108

6391

6674

6956 7239

283

4

7521

7803

8084

8366

8647

8928

9209

9490

9771

! 0051

281

5

190332 0612

0892

1171

1451 1730

2010

2289

2567 2846

279

6

3125 3403

3681

3959

4237 4514

4792

5069

5346 5623

378

7

5900 6176

6453

6729

7005 7281

7556

7832

8107 8382

276

8

8657

8932

9206

9481

9755

0029

0303

r)K'*j|*'

08^0 1 1 24

274

9

201397

1670

1943

2216

2488

2761

3033

3305

\,i\J*J\J i A i A/~f

3577 3848

fbf ^X

272

160

4120

4391

4663

4934

5204

5475

5746

6016

6286 6556

271

1

6826

7096

7365

7634

7904

8173 8441

8710

8979 9247

269

0 ./1UW

0051

0319

0586

0853

1121 1388

1654 1921

267

3

212188

2454

2720

2986

3252

3518

3783 4049

4314 4579

266

4

4844

5109

5373

5638

5902

6166

6430

6694

6957 7221

264

5

7484

7747

8010

8273

8536

8798

9060

9323

9585 9846

262

6

220108

0370

0631

0892

1153

1414

1675

1936

2196 2456

261

7

2716

2976

3236

3496

3755

i 4015

4274'

4533

4792 5051

259

8

5309

5568

5826

6084

6342

6600

6858

7115

7372 7630

258

9

7887

8144

8400

8657

8913

i 9170

9426

9682

99$

i.

23

0193

256

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

7

8

9

285

28.5

57.0

85.5

114.0

142.5

171.0

199.5

228.0

256.5

284

28.4

56.8

85.2

113.6

142.0

170.4

198.8

227.2

255.6

283

28.3

56.6

84.9

113.2

141.5

169.8

198.1

226.4

254.7

282

28.2

56.4

84.6

112.8

141.0

169.2

197.4

225.6

253.8

281

28.1

56.2

84.3

112 4

140.5

168.6

196.7

224.8

252.9

280

28.0

56.0

84.0

112.0

140.0

168.0

196.0

224.0

252.0

279

27.9

55.8

83.7

111.6

139.5

167.4

195.3

223.2

251.1

278

27.8

55.6

83.4

111.2

139.0

166.8

194.6

222.4

250.2

277

27.7

55.4

83.1

110.8

138.5

166.2

193.9

221.6

249.3

276

27.6

55.2

82.8

110.4

138.0-

165.6

193.2

220.8

248.4

275

27.5

55.0

82.5

110.0

137.5

165.0

192.5

220.0

247.5

274

27.4

54.8

82.2

109.6

137.0

164.4

191.8

219.2

246.6

273

27.3

54.6

81.9

109.2

136.5

163.8

191.1

218.4

245.7

272

27.2

54.4

81.6

108.8

136.0

163.2

190.4

217.6

244.8

271

27.1

54.2

81.3

108.4

135.5

162.6

189.7

216.8

243.9

270

27.0

54.0

81.0

108.0

135.0

162.0

189.0

216.0

243.0

269

26.9

53.8

80.7

107.6

134.5

161.4

188.3

215.2

242.1

268

26.8

53.6

80.4

107.2

134.0

160.8 187.6 214.4

241.2

267

26.7

53.4

80.1

106.8

133.5

160.2 186.9

213.6

240.3

266

26.6

53.2

79.8

106.4

133.0

159.6

186.2 212.8

239.4

265

26.5

53.0

79.5

106.0

132.5

159.0

185.5 212.0

238.5

264

26.4

52.8

79.2

105.6

132.0

158.4

184.8 211.2

237.6

263

26.3

52.6

78.9

105.2

131.5

157.8

184.1 210.4

236.7

262

26.2

52.4

78.6

104.8

131.0

157.2

183.4 209.6

235.8

261

26.1

52.2

78.3

104.4

130.5

156.6

182.7

208.8

234.9

260

26.0

52.0

78.0

104.0

130.0

156.0

182.0

208.0

234.0

259

25.9

51.8

77.7

103.6

129.5

155.4

181.3

207.2^

233.1

258

25.8

51.6

77.4

103.2

129.0

154.8

180.6

206.4

232.2

257

25.7

51.4

T7.1

102.8

128.5 i 154.2

179.9 205.6

231.3

256

25.6 51.2

76.8

103.4 128.0 153.6

179.2 204.8 230.4

255

25.5

51.0

76.5

102.0 1£7.5 153.0

178.5 204.0

229.5

81

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 170 L. 230.]

[No. 189 L. 278.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

170

230449

0704

0960

1215

1470

1724 1979

2234

2488

2742 255

1

2996

3250

3504

3757

4011

4264

4517

4770

5023

5276

253

2

5528

5781

6033

6285

6537

6789

7041

7292

7544

7795

252

a

8046

8297

8548

8799 9049

9299

9550

9800

U

\J\sX\J

0050

0300

250

4

240549

0799

1048

1297

1546

1795

2044

2293

2541

2790

249

5

3038

3286

3534

3782

4030"

4277

4525

4772

5019

5266

248

6

5513

5759

6006

6252

6499

6745

6991

7237

7482

7728

246

7

7973

U*)1 Q

8464

870')

HOVt

91 98

Q443

Qfi87

OQQO

8

| vi U

U 1 \JtJ

V JL. «7(J

*/Ti*J

*7vtJ 1

t/i7O/v

2610

243

250420

0664

0908

1151

1395

1638

1881

2125

2368

9

8853

3096

3338

3580

3822

4064

4306

4548

4790

5031

242

180

5273

5514

5755

5996

6237

6477

6718

6958

7198

7439

241

1

7679

7918

8158

8398

8637

8877

9116

9355

9594

9833

239

2

260071

0310

0548

0787

1025

1263

1501

1739

1976

2214

238

3

2451

2688

2925

3162

3399

3636

3873

4109

4346

4582

237

4

4818

5054

5290

5525

5761

5996

6232

6467

6702

6937

235

5

7172

7406

7641

7875

8110

8344

8578

8812

9046

9279

234

6

QC-| O

Q746

OQ80

'

t7 1 ^\J

t7*/tj\/

0213

0446

0679

0912

1144

1377

1609

233

W

271842

2074

2306

2538

2770

3001

3233

3464

3696

3927

232

8

4158

4389

4620

4850

5081

5311

5542

5772

6002

6232

230

9

6462

6692

6921

7151

7380

7609

7838

8067

8296

8525

229

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

7

8

9

255

25.5

51.0

76.5

102.0

127.5

153.

0

17S.5

204.0

229.5

254

25.4

50.8

76.2

101.6

127.0

152.

4

177.8

203.2

228.6

253

25.3

50.6

75.9

101.2

126.5

151.

8

177.1

202.4

227.7

252

25.2

50.4

75.6

100.8

126.0

151.

2

176.4

201.6

226.8

251

25.1

50.2

75.3

100.4

125.5

150.

6

175.7

200.8

225.9

250

25 0

50.0

75.0

100.0

125.0

150.

0

175.0

200.0

225.0

249

24.9

49.8

74.7

99.6

124.5

149.

4

174.3

199.2

224.1

248

24.8

49.6

74.4

99.2

124.0

148.

8

173.6

198.4

223.2

247

24.7

49.4

74.1

98.8

123.5

148.

2

172.9

197.6

222.3

246

24.6

49.2

73.8

98.4

123.0

147.6

172.2

196.8

221.4

245

24.5

49.0

73.5

98.0

122.5

147.

0

171.5

196.0

220.5

244

24.4

48.8

73.2

97.6

122.0

146.4

170.8

195.2

219.6

243

24.3

48.6

72.9

97.2

121.5

145.

8

170.1

194.4

218.7

242

24.2

48.4

72.6

96.8

121.0

145.

2

169.4

193.6

217.8

241

24.1

48.2

72.3

96.4

120.5

144.

(j

168.7

192.8

216.9

240

24.0

48.0

72.0

96.0

120.0

144.0

168.0

192.0

216.0

239

23.9

47.8

71.7

95.6

119.5

143.4

167.3

191.2

215.1

238

23.8

47.6

71.4

95.2

119.0

142.8

166.6

190.4

214.2

237

23.7

47.4

71.1

94.8

118.5

142.2

165.9

189.6

213.3

236

23.6

47.2

70.8

94.4

118.0

141.6

165.2

188.8

212.4

235

23.5

47.0

70.5

94.0

117.5

141.0

164.5

188.0

211.5

234

23.4

46.8

70.2

93.6

117.0

140.4

163.8

187.2

210.6

233

23.3

46.6

69.9

93.2

116.5

139.8

163.1

186.4

209.7

232

23.2

46.4

69.6

92.8

116.0

139.2

162.4

185.6

208.8

231

23.1

46.2

69.3

92.4

115.5

138.6

161.7

184.8

207.9

230

23.0

46.0

69.0

92.0

115.0

138.0

161.0

184.0

207.0

229

22.9

45.8

68.7

91.6

114.5

137.4

160.3

183.2

206.1

228

22.8

45.6

68.4

91.2

114.0

136.8

159.6

182.4

205.2

227

22.7

45.4

68.1

90.8

113.5

136.2

158.9

181.6

204.3

226

22.6

45.2

67.8 90.4

113.0 135.6

158 2

180.8

203.4

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 190 L. 278.] [No. 214 L. 332.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

190

278754

8982

9211

9439

9667

9895

0 1 °3

0351

0578

0806

228

1

281033

1261

1488

1715

1942

2169

V/.1 ••'J

2622

\J*j 4 1-7

2849

3075

227

0

/v

3301

3527

3753

3979

4205

4431

4u56

4882

5107

5332

226

3

5557

5782

6007

6232

6456

6681

CHU5

7130

7354

7578

225

4

7802

8026

8249

8473

8696

8920

9143

9366

9589

9812

223

5

290035

0257

0480

0702

0925

1147

1369

1591

1813

2034

222

6

2256

2478

2699

2920

3141

3383

3584

3804

4025

4246

221

r*
t

4466

4687

4907

5127

5:347

5567

5787

6007

6226

6446

220

8

6665

6884

7104

7323

7542

7761

7979

8198

8416

8635

219

Owe: O

9071

Qfc^XQ

QVI7

o^oPi

004-}

V *J<J I

i/i7-tO

(Vifil

0378

05Q5

0813

218

200

301030

1247

1464

1681

1898

2114

2331

VO t vJ

2547

\JiJ£J*J

2764

VfVJAU

2980

wJlU

217

1

3196

3412

3628

3844

4059

4275

4491

4706

4921

5136

216

2

5351

5566

5781

5996

6211

6425

6639

6854

7068

7282

215

3

7496

7710

7924

8137

8351

8564 8778

8991

9204

9417

213

00~>fi

0°f8

0481

0693

0006

1118

1330

1542

212

5

311754

1966

V/v'«J ' >

2177

2389

V~I'.-' A

2600

\S\Jt-l'J

2812

\J0\J\f

3023

J. 1 _l \J

3234

3445

3656

211

6

3867

4078

4289

4499

4710

4920

5130

5340

5551

5760

210

7

5970

6180

6390

6599

6809

7018

7227

7436

7(546

7854

209

8

8063

8272

8481

8689

8898

9106

9314

9522

9730

9938

208

9

320146

0354

0562

0769

0977

1184

1391

1598

1805

2012

207

210

2219

2426

2033

2839

3046

3252

3458

3665

3871

4077

206

1

4282

4488

4694

4899

5105

; 5310

5516

5721

5926

6131

205

2

6336

6541

6745

6950

7155

; 7359

7563

7767

7972

8176

204

0

ft^Sf)

CK no

8TST

cqni

01 Ql

Q3Q8

Q(>01

QOAK

O

tJOOVy

(_.

tot

Ot7«/l

t/J. t/T

£7(J«7O

i7Uv/J.

0008

0°11

203

4

330414

0617

0819

1022

1225

1427

1630

1832

\J\J\J\J

2034

2236

202

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

P*
I

8

9

225

22.5

45.0

67.5

90.0

112.5

135.0

157.5

180.0

202.5

224

22.4

44.8

67.2

89.6

112.0

134.4

156.8

179.2

201.6

223

22.3

44.6

66.9

89.2

111.5

133.8

156.1

178.4

200.7

222

22.2

44.4

66.6

88.8

111.0

133.2

155.4

177.6

199.8

221

22.1

44.2

66.3

88.4

110.5

132.6

154.7

176.8

198.9

220

22.0

44.0

66.0

88.0

110.0

132.0

154.0

176.0

198.0

219

21.9

43.8

65.7

87.6

109.5

131.4

153.3

175.2

197.1

218

21.8

43.6

65.4

87.2

109.0

130.8

152.6

174.4

196.2

217

21.7

43.4

65.1

86.8

108.5

130.2

151.9

173.6

195.3

216

21.6

43.2

64.8

86.4

108.0

129.6

151.2

172.8

194.4

215

21.5

43.0

64.5

86.0

107.5 129.0

150.5

172.0

193.5

214

21.4

42.8

64.2

85.6

107.0

128.4

149.8

171.2

192.6

213

21.3

42.6

63.9

85.2

106.5

127.8

149.1

170.4

191.7

212

21.2

42.4

63.6

84.8

106.0

127.2

148.4

169.6

190.8

211

21.1

42.2

63.3

84.4

105.5

126.6

147.7

168.8

189.9

210

21.0

42.0

63.0

84.0

105.0

126.0

147.0

168.0

189.0

209

20.9

41.8

62.7

83.6

104.5

125.4

146.3

167.2

188.1

208

20.8

41.6

62.4

83.2

104.0

124.8

145.6

166 4

187.2

207

20.7

41.4

62.1

82.8

103.5

124.2

144.9

165.6

186.3

206

20.6

41.2

61.8

82.4

103.0

123.6

144.2

164.8

185.4

205

20.5

4d.O

C1.5

82.0

102.5

103.0

143.5

164.0

184.5

204

20.4

40.8

61.2

81.6

102.0

122.4

142.8

163.2

183.6

203 20.3

40.6

60.9

81.2

101.5

12l!8

142.1 162.4

182.7

202 20.2

40.4

60.6

">0.8 101.0

121.2 141.4 161.6

181.8

83

TABLE IX. — LOGARITHMS OF NUMBERS.

— 1

No. 215 L. 332.] [No. 239 L. 380.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

215

332438

2640

2842

3044

3246

3447

3049

3850

4051

4253

202

6

4454

4055

4856

50.17

5257

5458

5058

5859

6059

6200

201

7

6460

6600

6860

7000

7260

7459

7059

7858

8058

8257

200

8

8456

8056

8855

9054

9253

9451

9050

984Q

\J\J^iJ

0047

0246

199

9

340444

0642

0841

1039

1237

1435

1032

1830

2028

2225

198

220

2423

2620

2817

3014

3212

3409

3606

3802

3999

4196

197

1

4392

4589

4785

4981

5178

5374

5570

5706

5962 6157

196

2

6353

6549

6744

6939

7135

7330

7525

7720

7915

8110

195

3

8305

8500

8094

8889

9083

9278

Q472

Q666

QKOO

4

Vf 1 nj

O\J\J\J

v{-J\J\J

1989

193

350248

0442

0636

0829

1023

1216

1410

1603

1796

5

2183

2375

2568

2761

2954

3147

3339

3532

3724

3916

193

6

4108

4301

4493

4085

4876

5008

5260

5452

5643

5834

192

7

6026

6217

6408 6599

6790

6981

7172

7363

7554

7744

191

8

7935

8125

8316 8506

8696

8886

9076

9266

9456

9646

190

9

9835

0025

0215

0404

0593

0783

0972

1161

1350

1539

189

230

361728

1917

2105

2294

2482

2671

2859

3048

3236

3424

188

1

3612

3800 3988

4176

4363

4551

4739

4926

5113

5301

188

2

5488

5675

5862

6049

6236

6423

6610

6796

6983

7169

187

3

7356

7542

7729

7915

8101

8287

8473

8659

8845

9030

186

4

9216

9401

9587

9772

9958

I

0143

0328

0513

0698

0883

185

5

371068

1253

1437

1622

1806

1991

2175

2360

2544

2728

184

' 6

2912

3096

3280

8464

3647

3831

4015

4198

4382

4565

184

7

4748

4932

5115

5298

5481

5664

5846

6029

6212

6394

183

8

6577

6759

6942

7124

7306

7488

7670

7852

8034

8216

182

9

8398

8580

8761

8943

9124

0306

Q487

Qfifift

38

%/ IftfTL

vWU

«TX<J 1

v\f\J\J

0030

181

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

7

8

9

202
201

20.2
"20.1

40.4
40.2

60.6
60.3

80.8
80.4

101.0
100.5

121.2
120.6

141.4
140.7

161.6
160.8

181.8
180.9

200

20.0

40.0

60.0

80.0

100.0

120.0

140.0

160.0

180.0

199

19.9

39.8

59.7

79.6

99 5

119.4

139.3

159.2

179.1

198

19.8

39.6 '

59.4

79.2

99.0

118.8

138.6

158.4

178.2

197

19.7

39.4

59.1

78.8

98.5

118.2

137.9

157.6

177.3

196

19 6

39.2

58.8

78.4

98 0

117.6

137.2

156.8

176.4

195

19.5

39.0

58.5

78.0

97.5

117.0

136.5

156.0

175.5

194

19.4

38 8

58.2

77.6

97.0

116.4

135.8

155.2

174.6

193

19 3

38.6

57.9

77.2

96.5

115.8

135.1

154.4

173.7

192

19 2

38.4

57.6

76.8

96.0

115.2

134.4

153.6

172.8

191

19 1

38.2

57.3

76.4

95.5

114.6

133.7

152.8

171.9

190

19.0

38.0

57.0

76.0

95.0

114 0

133.0

152.0

171.0

189"

18.9

37 8

56.7

75.6

94 5

113.4

132.3

151.2

170.1

188

18 8

37.6

56.4

75.2

94.0

112.8

131.6

150.4

169.2

187

18.7

37 4

56.1

74.8

93.5

112.2

130.9

149.6

168.3

186

18.6

37.2

55.8

74.4

93.0

111.6

130.2

148.8

167.4

185

18.5

37.0

55 5

74.0

92 5

111 0

129.5

148.0

166.5

184

18 4

36 8

55.2

73.6

92.0

110.4

128.8

147.2

165.6

183

18 3

36 6

54.9

73.2

91.5

109 8

128.1

146 4

164.7

182

18 2

36 4

54 6

72 8

91 0

109 2

127.4

145.6

163.8

181

18 1

36 2

54 3

72 4 90 5

108 6

126 7

144.8

162.9

180

18 0

36.0

54 0 72 0

90 0

108 0

126.0

144.0

102.0

179

17 9

35 8

53.7 71.6

89.5

107.4

125.3

143.2

101 1

TABLE IX. — LOGARITHMS OF NUMBERS.

Xo. 240 L. 380.]

L>o, 269 L. 431.

N.

0

1

2

3

4

5

G

7

8

9

Diffi.

240
1
2
3
4
5

6

r**
i

8
9

•250
1

2

3
4
5
6

ri

i

8
9

260
1
2
3

4
5

6

ty

i

8
9

380211
2017
3815
5606
7390
9166

0392

2197
3995
5785
7568
9343

0573
2377
4174
5964
7746
9520

07.54
2557
4353
6142
7924
9698

0934
2737
45:33
6321
8101
9875

1115
2917
4712
6499
8279

1296
3097
4891
6677
8456

1476
3277
5070
6856
86:34

1G56
3456
5249
70:34
8811

1837
3636
5428
7212
8989

181
180
179
178
178

177
176
176
175
174

173

173
172
171
171
170
169

169

103
167

167
166
165

165
164
164
163
162
162

161

0051
1817
3575
5326
7071

8808

0228
1993
3751
5501
7245

8981

0405
2169
3926
5676
7419

9154

0582
2345
4101
5850
7592

9328

0759
2521
4277
6025
7766

9501

390935
2697
4452
6199

7940
9674

1112
2873
4627
6374

8114
9847

1288
3048
4802
6548

8287

1464
322 1
4977
6722

8461

1641
3400
5152

6896

8634

0020
1745
3464
5176
6881
8579

0192
1917
3635
5:346
7051
8749

0365
2089
.3807
5517
7221
8918

0538
2261
3978
5688
7-391
9087

0711

2433
4149
5858
7561
9257

0883
2605
4320
6029
7731
9426

1056
2777
4492
6199
7'JOl
9595

1228

2U49
4663
6370
8070
9764

401401
3121
4834
6540
8240
99:33

1573
3292
5005
6710
8410

0102
1788
3467

5140

6807
8467

0271
1956
3635

5307
6973
86:33

0440
2124

3803

5474
7139
8798

0609
2293
3970

5641
7306
8964

0777
2461
; 4137

5808
7472
9129

0046
2621)
4305

5974
7638
9295

1114
2796
4472

61 11

7S04
9460

1283
2964

4639

6308
7970
9625

1451
3132
4806

6474
8135
9791

411620
3=300

4973
6641
8301
9956

0121
1768
3410
5045
6674
8297
9914

0286 0451
1933 2097
3574 3737
5208 5371
6836 6999
8459 8621

0616
2261
3901
5534
7161
8783

07S1
2426
4065
5697
7324
8944

0945
2590
4228
5860
7486
9106

1110
2754
4392
6023
7648
9268

1275
2918
4555
6186
7811
9429

1439
3082
4718
6349
7973
9591

421604
3246
4882
6511
8135
9752
43

0075 0236 0398

0559

0720 0881

1042

1203

PROPORTIONAL PARTS.

Diff. 1

2 3

4

5

6

7

8

9

178 17.8
177 17.7
176 17.0
175 17.5
174 17.4
173 17.3
172 17.2
171 17.1
170 17.0

109 16.9
168 16.8
167 16.7
166 16.6
165 16.5
164 16.4
163 16.3
162 16.2
161 16.1

35.6 53.4
35.4 53.1
35.2 52.8
35.0 52.5
34.8 52.2
34.6 51.9
34.4 51.6
34.2 51.3
34.0 51.0

33.8 50.7
33.6 50.4
33.4 50.1
33.2 49.8

m.o 49.5

32.8 49.2
32.6 48.9
32.4 48.5
32.2 48.3

71.2
70.8
70.4
70.0
69.6
69.2
68.8
68.4
68.0

67.6
67.2
66.8
66.4
66.0
65.6
65.2
64.8
61.4

89.0
88.5
88.0
87.5
87.0
86.5
86.0
85.5
85.0

84.5
84.0
83.5
83.0
82.5
82.0
81.5
81.0
80.5

100.8
106.2
105.6
105.0
104.4
103.8
103.2
102.6
102.0

101.4
100.8
100.3
99.6
99.0
98.4
97.8
97.2
96.6 |

124.6
123.9
123.2
122.5
121.8
121.1
120.4
119.7
119.0

118.3
117.6
116.9
116.2
115.5
114.8
114.1
113.4
112.7

142.4
141.6
140.8
140.0
139.2
138.4
137.6
136.8
136.0

135.2
134.4
1:33.6
132.8
132.0
131.2
130.4
129.6
128.8

160.2
159.3
158.4
157.5
156.6
155.7
154.8
153.9
153.. 0

152.1
151.2
150.3
149.4
148.5
147.6
146.7
145.8
144.9

85

TAIJLi: IX. — LOGARITHMS OF NUMBERS.

No. 270 L. 431.]

[No. 299 L. 476.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

270

431364

1525

1685

1846

2007

2167

2328

2488

2649

2809

161

1

2969

3130

3290

3450

3610

3770

3930

4090

4249

4409

160

2

4569

4729

4888

5048

5207

5367

5526

5685

5844

6004

159

3

6163 6322

6481

6640

6799

6957

7116

7275

74:33

7592

159

4

7751

7909

8067

8226

8384

8542 8701

8859

9017

9175

158

K

9333

9491

9648

9806

9964

*J

isij'J'J

£T7i t7J.

&\J^{J

*7<J\J\J

fJU\J ^t

0122 0279

0437

0594

0752

158

6

440909

1066

1224

1381

1538

1695 1852

2009

2166

2323

157

7

2480

2637

2793

2950

3106

3263 3419

3576

3732

3889

157

8

4045

4201

4357

4513

4669

4825 4981

5137

5293

5449

156

9

5604

5760

5915

6071

6226

6382 6537

6692

6848

7003

155

280

7158

7313

7468

7623

7778

7933 } 8088 8242

8397

8552

155

1

8706

8861

QD1 ^

91 '

'0

9324

9478

9033 9787

9941

J.

*J t \/\J

(JLJUA

_

" ' L

V

r*'Jiv^t

i/T 1 IJ

ij\J'-J'J t7 4 O 1

UtJ^Ll

0095

154

2

450249

0403

0557

0711

0865

1018

1172

1326

1479

1633

154

3

1786

1940

2093

2247

2400

2553

2706

2859

3012

3165

153

4

3318

3471

3624

3777

3930

4082

4235

4387

4540

4692

153

5

4845

4997

5150

5302

5454

5606

5758

5910

6062 , 6214

152

6

6366

6518

6670

6821

6973

7125

7276

7428

7579

7731

152

7

7882

8033

8184

8336

8487

8638

8789 8940

9091

9242

151

a

9392

9543

9fif)4

9845

9095

(_7

tJUviV

V^Jl^J

*J\JUX:

t7<_nt«-'

0146

0296 : 0447

0597

0748

151

9

460898

1048

1198

1348

1499

1649

1799

1948

2098

2248

150

290

2398

2548

2697

2847

2997

3146

3296

3445

3594

3744

150

1

3893

4042

4191

4:340

4490

4639

4788 4936

5085

5234

149

' 2

5383

5532

5680

5829

5977

6126

6274

6423

6571

6719

149

3

6868

7016

7164

7312

7460

7608

7756

7904

8052

8200

148

4

8347

8495

8643

8790

8938

9085 : 9233

9380

9527

9675

148

t

98'>2

9909

VW'W'V

&ts\Jts

fliifi

AOf

!Q

0410

0"-""

0704

ORM

none

114*1

147

6

471292

1438

\j 1 1 <j

1585

1732

\r±l\J

1878

2025

i \j^

2171

UOOJL

2318

V/.'t/O

2464

2610

146

7

2756

2903

3049

3195

3341

3487

3633

3779

3925

4071

146

8

4216

4362

4508

4653

4799

4944

5090

5235

5381

5526

146

9

5671

5816

5962

6107

6252

6397 6542

6687

6832

6976

145

I

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

<

8

9

161

16.1

32 2

48.3

64.4

80.5

96.6

112.7

128.8

144.9

160

16.0

32.0

48.0

64.0

80.0

96.0

112.0

128.0

144.0

159

15.9

31.8

47.7

63.6

79.5

95.4

111.3

127.2

143.1

158

15.8

31.6

47.4

63.2

79.0

94.8

110.6

126.4

142.2

157

15.7

31.4

47.1

62.8

78.5

94.2

109.9

125.6

141.3

156

15.6

31.2

46.8

62.4

78.0

93.6

109.2

124.8

140.4

155

15.5

31.0

46.5

62.0

77.5

93.0

108.5

124.0

139.5

154

15.4

30.8

46.2

61.6

77.0

92.4

107.8

123.2

1.38.6

153

15.3

30.6

45.9

61.2

76.5

91.8

107.1

122.4

137.7

152

15.2

30.4

45.6

60.8

76.0

91.2

106.4

121.6

136.8

151

15.1

30.2

45.3

60.4

75.5

90.6

105.7

120.8

135.9

150

15.0

30.0

45.0

60.0

75.0 -

90.0

105.0

120.0

135.0

149

14.9

29.8

44.7

59.6

74.5

89.4

104.3

119.2

134.1

148

14.8

29.6

44.4

59.2

74.0

88.8

103.6

118.4

133.2

147

14.7

29.4

44.1

58.8

73.5

88.2

102.9

117.6

132.3

146

14.6

29.2

43.8

58.4

73.0

87.6

102.2

116.8

131.4

145

14.5

29.0

43.5

58.0

72.5

87.0

101.5

116.0

130.5

144

14.4

28.8

43.2

57.6

72.0

86.4

100.8

115.2

129.6

143

14.3

28.6

42.9

57.2

71.5

85.8

100.1

114.4

128.7

142

14.2

28.4

42.6

56.8

71.0

85 2

99.4

113.6

127.8

141

14.1

28.2

42.3

56.4

70.5

84.6

98.7

112.8

126.9

140

14.0

28.0

42.0

56.0

70.0

81.0

98.0

112.0

126.0

86

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 300 L. 477.]

[No. 339 L. 531.

N.

0

1

2

o

tj

4

5

6

7

8

9

Diff.

300
1

2
3
4
5
6

r*
I

8
9

310
1
2
3
4
5
6

7
8
9

320
1
2
3

4
5
6
7
8
9

330
1

2

3
4
5
6

r*
t

8
9

477121
8566

7266
8711

7411

8855

7555
8999

7700
9143

7844
9287

7989
9431

8133
9575

8278
9719

8422
98G3

145
144

144
143
143
142
142
141
141

140

140
139
139
139
138
138

137

137
136
136

136
135
135

134
134
133
133
133
132
132

131

131

131
130
130
129
129
129

128
128

480007
1443
2874
4300
5721
7138
8551
9958

0151
1586
3016
4442

5863
7280
8692

0294
1729
3159
4585
6005
7421
8833

04-°,S
1872
3302
4727
6147
7563
8974

0582
20 1G
3445
4869
6289
7704
9114

0725
2159
3587
5011
6430
7845
9255

0869
2302
3730
5153
6572
7986
9396

1012
2445
3872
5295
6714
8127
9537

1156
2588
4015
5437
C855
8269
9677

1299
2731
4157
5579
6997
8410
9818

0099

1502
2900
4294
5683
7068
8448
9824

0239

1642
3040
4433

5822
7206
8586
9962

0380

1782
3179
4572
5960
7344
8724

0520

1922
3319
4711
6099
7483
8862

OG61

; 2062
3458
4850
6238
7G21
8999

0801

2201
3597
4989
6376
7759
9137

0941

2341
3737
5128
6515
7897
9275

1081

2481
3876
5267
6653
8035
9412

1222

2621
4015
5406
6791
8173
9550

491362
27GO
4155
5544
6930
8311
9GS7

0099
1470
2S37
41 99

5557
6911
8260
9G06

0236
1G07
2973
4335

5693
7046
8395
9740

0374
1744
3109
4471

5828
7181
8:>30
9.S74

0511

1880
3246
4607

5964
7316
8664

0648
2017
3382
4743

6099
7451
8799

0143
1482
2818
4149
5476
6800
8119

9434

0785
2154
3518
4818

6234

7586
8934

0922
2291
S655
5014

C370
1721

COGS

501059
2427
3791

5150
6505
7856
9203

1196
2564
3927

52S6
6640
7991
9337

1333

2700
40G3

5421
6776
8126
9471

0009
1349
2684
4016
5344
6668
7987

9303

0277
1616
2951
4282
£609
6932
8251

9566

0411
1750
£084
4415
5741
7064
8382

9697

510545

1883
3218
4548
5874
7196

8514

9828

0679
2017
3351
4681
6006
7328

8646
9959

OS13
2151
3484
4813
6139
7460

8r*r*p*
1 1 1

0947
2284
3617
4946
6271
7592

8909

1081
2418
3750
5079
6403
7724

9040

1215
2551
3883
5211
6535
7855

9171

0090-
1400
2705
4006
5304
6598
7888
9174

0221
1530
2835
4136
54:?4
6727
8016
9302

0353
1661
2966
4266
5563
6856
8145
9430

0484
1792
3096
4396
5G93
6985
8274
9559

0615
1922
3226
4526
6822
7114
8402
9G87

0745
2053
3356
4656
5951
7243
8531
9815

0876
2183
3486
4785
C081
7372
8GCO
</943

1C07
2314
3616
4915
C210
7"01
8788

521138
2444
3746
5045
6339
7630
8917

1269
2575
3876
5174
6469
7759
9045

0072
1351

530200

0328

0456

0584

0712 | OS40 0968 1 1(96 1223

PROPORTIONAL, PARTS.

Diff. 1

2

3

4

5

6

7

8

9

139 13.9
138 13.8
137 13.7
136 13.6
135 13.5
134 13.4
133 13.3
132 13.2
131 13.1
130 13.0
129 12.9
128 12.8
127 12 7

27.8
27.6
27.4
27.2
27.0
26.8
26.6
26.4
26.2
26.0
25.8
25.6
25.4

41.7
41.4
41.1
40.8
40.5
40.2
39.9
39.6
89.3
89.0
38.7
38.4
38.1

55.6
55.2
54.8
54.4
54.0
53.6
53.2
52.8
52.4
52.0
51.6
51.2
50.8

69.5
G9.0
68.5
68.0
67.5
67.0
66.5
66.0
65.5
65.0
64.5
64.0
63.5

83.4
82.8
82. 2
81.6
81.0
80.4
79.8
79.2
78.6
78.0
77.4
76.8
76.2

97.3
96.6
95.9
95.2
94.5
93.8
93.1
92.4
91.7
91.0
90.3
89.6
88.9

111.2
110.4
109.6
108.8
108.0
107.2
106.4
105.6
104.8
104.0
103.2
102.4
101.6

125.1
124.2
123.3
122.4
121.5
120.6
119.7
118.8
117.9
117.0
116.1
115.2
114.3

87

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 340 L. 531.]

LNo. 379 L. 579.

N.

0

1

o

3

4

5

6

7

8

9

Diff.

340

531479

1607

1731

1862

1990 2117 2245 2372

2500

2627

128

1

2754

2882

3009

3136

3264

3391

3518 3645

3772

3899

127

2

4026

4153

4280

4407

4534

4661

4787

4914

5041

5167

127

3

5294

5421

5547

56'

"4

5800 5927

6053

6180

6306

6432

126

4

6558

6685

6811

6937

7063 7189

7315

7441

750

i

7693

126

5

7819

7945

8071

8197

8322 ! 8448

8574

8699

8825

8951

126

n

Q07R

Q0()0

QO»)i*f

<U;

-,)

9578 • 9703

9829

u

jO1"*

i/Ul U

Or*\Jf*

£rz*

)<v

V

007

Q

n>2fu

•foe;

7

540329

0455

0580

0705

0830

0955

1080

1205

\j\j i »/
1330

1454

125

8

1579

1704

1829

1953

2078

2203

2327

2452

257

0

2701

125

9

2825

2950

3074

3199

3323

3447

3571

3090

3820

3944

124

350

4068

4192

4316

4440

4564

4688

4812

4936

5060

5183

124

1

5307

5431

5555

56'

-8

5802

5925

6049

6172

6296

6419

124

2

6543

6666

6789

6913

7036

7159

7282

7405

7529

7652

123

3

7775

7898

8021

8144

8207 8389

8512

8

i-JO^l

8758

8881

123

4

9003

9120

9249

93'

"1

9494 9016

9739

98B1

9984

T

0106

123

5

550228

0351

0473

0595

0717 0840

0962

1084

1206

1328

122

6

1450

1572

1694

1816

1938

2060

2181

2303

2425

2547

122

r*
1

2608

2790

2911

3033

3155

3276

3398

3519

3040

3762

121

8

3883

4004

4126

4247

4368

4489

4610

4731

4852

4973

121

9

5094

5215

5336

5457

5578

5699

5820

5940

6001

6182

121

360

6303

6423

6544

6664

6785

6905

7026

r-
t

146

726

fN
1

7387

120

1

7507

7627

7748

7808

7988

8108

8228

8349

8409

8589

120

2

8709

8829

8948

9068

9188

9308

9428

9548

960

1

9787

120

QQ07

0026

0146

0265

0385

0504

0024

0743

08(53

OQ89

11Q

4

501101

1221

1340

1459

1578

1098

1817

1936

X/U*JU

2055

2174

J.JL9

119

5

2293

2412

2531

2050

2709

! 2887

3006

3125

3244

3362

119

6

3481

3600

3718

3837

3955

4074

4192

4311

4429

4548

119

7

4006

4784

4903

5021

5139

5257

5376

5494

5612

5730

118

8

5848

5966

6084

6202

6320

6437

6555

6673

6791

6909

118

9

7026

7144

7262

73'

"9

7497

7614

7732

7849

7967

8084

118

370

8202

8319

8436

8554

8671

8788

8905

9023

9140

9257

117

i

9374

9491

9608

97^

9842

9959

JL

0076

0193

0309

0426

717

2

570543

0660

0776

0893

1010

1126

1243

1359

1476

1592

117

3

1709

1825

1942

2058

2174

2291

2407

2523

2639

2755

116

4

2872

2988

3104

3220

3336

3452

3568

3684

3800

3915

116

5

4031

4147

4263

43'

"9

4494

4610

4726

4841

4957

5072

116

6

5188

5303

5419

55:34

5650

5765

5880

5996

6111

6226

115

7

6341

6457

6572

6687

6802

6917

7032

7147

7202

7377

115

8

7492

7607

7722

7836

7951

8006

8181

8295

8410

8525

115

9

8639

8754

8868

8983

9097

9212

9326

9441

9555

9669

114

PROPORTIONAL PARTS.

Diff. 1

2 3

4

5

6

7

8

9

128 12.8

25.6 38.4

51.2

64.0

76.8

89.6

102.4

115.2

127 12 7

254 38.1

50.8

63.5

76.2

88.9

101.6

114.3

126 12 6

25.2 37.8

50.4

63.0

75.6

88.2

100.8

113.4

125 12.5

25.0 37.5

50.0

62.5

75.0

87.5

100.0

112.5

124 12.4

24.8 37.2

49.6

62.0

74.4

86 8

99.2

111.6

123 12.3

24.6 36.9

49.2

61.5

73.8

86.1

98.4

110.7

122 12.2

244 36.6

48.8

61.0

73.2

85.4

97.6

109.8

121 12.1

24.2 36.3

48.4 •

60.5

72.6

84.7

96.8

108.9

120 12.0

24 0 360

48.0

60.0

72.0

84.0

96.0

108.0

119 11 9

23.8 35.7

47.6

59.5

71.4

83.3

95.2

107.1

88

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 380. L. 579.]

r

[No. 414 L. 617.

N.

0

1

2

3

4

5

6

m

8

9

Diff.

380

579784

9898

0697

0012

0126

0241

0355

0469

0583

0811

114

1

580925

1039

1153

1267

*

1381

1495

1608

1722

1836

1950

2

20G3

2177

2291

2404

2518

2631

2745

2858

2972

3085

3

3199

3312

3426

3539

3652

3765

3879

3992

4105

4218

4

4331

4444

4557

4670

4783

4896

5009

5122

523:>

5348

113

5

54G1

5574

5086

5799

5912

6024

6137

6250

6362

64 i 5

6

6587

6700

6812

6925

7037

7149

7262

7374

7486

7599

7

7711

7823

7935

8047

8160

8272

8384

8496

8608

8720

112

8

8832
fiop^n

8944

9056

9167

9279

9391

9503

9G15

9726

9838

yysu

00(11

0173

0284

0396

0507 0619

0730

0842

0953

300

591065

VULI1

1176

\J 1. t ij

1287

1399

1510

1621

1732

1843

1955

2066

1

2177

22K8

2399

2510

2621

2732

2843

2954

30G4

3175

111

2

3286

3397

3508

3618

3729

3840

3950

4061

4171

4282

3

4393

4503

4614

4724

4834

4945

5055

5165

5276

5386

4

5496

5606

5717

582

y

5937

6047

6157

6267

6377

6487

1 in

5

6597

6707

6817

6927

7037

7146

7256

7366

7476

7586

ll(J

6

7695

7S05

7914

8024

8134

8243

8353

84G2

8572

8681

7

8791

f}OOO

8900

oono

9009

9119

9228

9337

9446

9556

9GG5

9774

•t no

ysoo

UV\)A

0101

0°10

C319

0428

0537

0646

0755

0864

lU'J

9

600973

1082

\J 1 ' ' 1

1191

Vis* 1\7

1299

1408

1517

1625

1734

1843

1951

400

20GO

2169

2277

2386

2494

2603

2711

2819

2928

3036

1

3144

3253

3361

3469

3577

3686

3794

3902

4010

4118

108

o

iV

4226

4334

4442

4550

4658

4766

4874

4982

5089

5197

3

5305

5413

5521

5628

5736

5844 5951

6059

61G6

6274

4

G3S1

6489

6596

6704

6811

6919

7026

7133

7241

7348

5

7455

7562

7(569

777

ft
1

7884

7991

8098

8205

8312

8419

107

6

8526

8633

8740

884

*»
i

8954

9061

9167

9274

9381

9488

7

9594

9701

9808

9914

00° 1

0128

0234

0341

0447

0554

8

G106GO

0767

0873

097

9

V/V'-w L

1086

1192

1298

1405

1511

1617

9

1723

1829

1936

2042

2148

2254

2360

24G6

2572

2678

106

410

2784

2890

2996

3102

3207

3313

3419

3525

3G30

3736

1

3842

3947

4053

4159

4264

4370

4475

4581

4686

4792

2

4897

5003

5108

5213

5319

5424

5529

5634

5740

5S45

3

5950

6055

61GO

6265

6370

6476

6581

6686

6790

6895

105

4

7000

7105

r-
i

210

7315

7420

7525

7629

7734

783.-)

7943

PROPORTIONAL PARTS.

Diff. 1

2

3

4

5

6

7

8

9

118 11.8

23.6

35.4

47.2

59.0

70.8

82.6

94.4

106.2

117 11.7

23.4

35.1

46.8

58.5

70.2

81.9

93. 6

105.3

11(5 11.6

23.2

34.8

46.4

58.0

69.6

81.2

92. S

101.4

115 11.5

23.0

34.5

46.0

57.5

69.0

80.5

92.0

103.5

114 11.4

22.8

34.2

45.6

57.0

68.4

79.8

91.2

102.6

113 11.3

22.6

33.9

45.2

56.5

67.8

79.1

90.4

101.7

112 11.2

22.4

33.6

44.8

56.0

67.2

78.4

89.6

100.8

111 11.1

22.2

33.3

44.4

55.5

66.6

77.7

88.8

99.9

110 11.0

22.0

33.0

44.0

55.0

66.0

77.0

88.0

99.0

109 10.9

21.8

32.7

43.6

54.5

65.4

76.3

87.2

98.1

108 10.8

21.6

32.4

43.2

54.0

64.8

75.6

86.4

97.2

107 10.7

21.4

32.1

42.8

53.5

64.2

71.9

K5.6

96.3

106 10.6

21.2

31.8

42.4

53.0

63.6

74.2

84.8

95.4

105 10.5

21.0

31.5

42.0

52.5

63.0

73.5

84.0

94.5

105 10.5

21.0

31.5

42.0

52.5

63.0

73.5

84.0

94.5

104 10.4

20.8

31.2

41.6

52.0

62.4

72.8

83.2

93.6

89

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 415 L. 618.] [No. 459 L. 662

N.

0

1

2

3

4

6

6

7

8

9

Diff.

415
6

r*
i

8
9

420
1
2
3
4
5
6

r*

I

8
9

430
1
2
3
4
5
6

7
8
9

440
1
2
3
4
5
6

r*

t

8
9

450
1
2
3
4
5
6
7

8
9

618048
9093

8153 8257
9198 9302

8362
9406

8466
9511

8571
9615

8676
9719

8780
9824

8884
9928

8989

105
104

103
102

101
100

99

98

97
96

95

0032
1072
2110
3146

4179
5210
6238
7263
8287
9308

620136
1176
2214

3249

4282
5312
6340
7366
8389
9410

0240

1280
2318

3353
4385
5415
6443
7468
8491
9512

0344

1384
2421

3456
4488
5518
6546
7571
8593
9613

0448
14S8
2525

3559
4591
5621
6648
7673
8695
9715

0552
1592

2628

3663
4695
5724
6751

7775
8797

9817

0656
1695
2732

3766
4798
5827
6853
7878
8900
9919

0760
1799
2835

3869
4901
5929
6956
7980
9002

0864
1903
2939

3973
5004
6032
7058
8082
9104

0968
2007
3042

4076
5107
6135
7161
8185
9206

0021
1038
2052
3064

4074

5081
6087
7089
8090

9088

0123
1139
2153
3165

4175
5182
6187
7189
8190
9188

0224
1241
2255
3266

4276

5283
6287
7290
8290

9287

0326
1342
2356
3367

4376
5383
6388
7390
8389
9387

630428
1444
2457

3468
4477
5484
6488
7490
8489
9486

0530
1545
2559

3569
4578
5584
6588
7590
8589
9586

0631
1647
2660

3670
4679
5685
6688
7690
8689
9686

0733
1748

2761

3771

4779
5785
6789
7790
8789
9785

0835
1849
2862

3872
4880
5886
6889
7890
8888
9885

0936
1951
2963

3973
4981
: 5986
6989
7990
8988
9984

0084
1077
2069
3058

4044
5029
6011
6992
7969
8945
9919

0183
1177
2168
3156

4143
5127
6110
7089
8067
9043

0283
1276
2267
3255

4242
5226
6208
7187
8165
9140

0382
1375
2366
3354

4340
5324
6306
7285
8262
9237

640481
1474
2465

3453

4439
5422
6404
7383
8360
9335

0581
1573
2563

3551
4537
5521

6502
7481
8458
9432

0680
1672
26G2

3650
4636
5619
6600
7579
8555
9530

0779
1771
2761

3749
4734
5717
6698
7676
8653
9627

0879
1871
2860

3847
4832
5815
6796
7774
8750
9724

0978
1970
2959

3946
4931
5913

6894
7872
8848
9821

0016
0987
1956
2923

3888
4850
5810
6769
7725
8679
9631

0113
1084
2053
3019

3984
4946
5906
6864
7820
8774
9726

0210
1181
2150
3116

4080
5042
6002
6960
7916
8870
9821

650308
1278
2246

3213

4177
5138
6098
7056
8011
8965
9916

0405
1375
2343

3309
4273
5235
6194
7152
8107
9060

0502
1472
2440

3405
4369
5331
6290
7247
8202
9155

0599
1569
2536

3502
4465
5427
6386
7343
8298
9250

0696
1666
2633

3598
4562
5523
6482
7438
8393
9346

0793
1762

2730

3695
4658
5619
6577
7534
8488
9441

0890
1859
2826

3791

4754
5715
6673
7629
8584
9536

0011
0960
1907

0106
1055
2002

0201
1150
2096

0296
1245
2191

G391
1339
2286

0486
1434
2380

0581
1529
2475

0676
1623
2569

0771
1718

2663

660865
1813

PROPORTIONAL PARTS.

Diff. 1

234

5

678

9

105 10 5
104 10 4
103 10 3
102 10 2
101 10 1
100 10.0
99 99

21.0 31 5 42.0
20 8 31 2 41 6
206 309 41.2
20 4 30 6 40.8
20 2 30 3 40.4
20 0 30.0 40 0
19 8 29 7 39 6

52 5
52.0
51 5
51 0
50 5
50 0
49 5

63 0 73.5 84 0
62 4 72 8 83 2
61 8 72 1 82.4
61 2 71 4 81 6
60 6 70 7 80 8
60.0 70 0 80 0
59 4 69 3 79 2

94.5
93.6

92 7
91 8
90.9
90 0
89.1

90

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 460 L. 662.]

LNo. 499 L. 698.

N

0

1

2

3

4

5

6

7

8

0

Diff.

460

662758

2852

2947

3041

3135

3230

3324 3418 3512

3607

1

3701

3795

3889

3983

4078

4172

4266

4360 4454

4548

o

4642

4736

4830

4924

5018

5112

5206

5299 5393

5487

94

3

5581

5675

5769

5862

5956

6050

6143 6237 6331

6424

4

6518

6612

6705

6799

6892

6986

7079 7173 7266

7360

5

7453

7546

7640

7733

7826

7920

8013 8106

8199

8293

6

8386

8479

8572

8665

8759

8852

8945

9038

9131

9224

r*
t

9317

9410

9503 9596 9689

9782

9875 j 9967

ATWt

n

0153

93

8

670246

0339

0431

0524

0617

0710

0802 0895 0988

\J L *J'J

1080

9

1173

1265

1358

1451

1543

1636

1728

1821 1913

2005

470

2098

2190

2283

2375

2467

2560

2652

2744

2836

2929

1

3021

3113

3205

3297

3390

3482

3574

3666 3758

3850

2

3942

4034

4126

4218

4310

4402

4494

4586 467

7

4769

92

3

4861

4953

5045

5137

5228

5320

5412

5503 5595

5687

4

5778

5870

5962

6053

6145

6236

6328

6419

6511

6602

5

6694

6785

6876

6968

7059

7151

7242

r<
t

333

7424

7516

6

7607

7698

7789

7881

7972

8063

8154

8245

8336

8427

7

8518

8609

8700

8791

8882

8973

9064

9155

9246

9337

91

a

Q4 Oji

Qr-| A

QfilO

Q70D

Q7Q1

OQOO

QQ1""-^

o

»i~0

iJUJ. V

17 i \J\J

o i & i

t'OO'w

flOftft

0154

0°45

9

680336

0426

0517

0607

0698

0789

0879

\J\J\J'J

0970

\J i iJ^.

1060

1151

480

1241

1332

1422

1513

1603

1693

1784

1874

1964

2055

1

2145

2235

2326

2416

2506

2596

2686

2777

2867

2957

2

3047

3137

3227

3317

3407

3497

3587

3677

3707

3857

90

3

3947

4037

4127

4217

4307

4396

4486

4576

4666

4756

4

4845

4935

5025

5114

5204

5294

5383

5473

5563

5652

5 5742

5831

5921

6010

6100

6189

6279

6368

6458

0547

6 6636

6726

6815

6904

6994

7083

7172

n
i

261

7351

7440

7 7529

7618

7707

7796

7886

7975

8064

8153

8242

8331

89

8 ' 8420

8509

8598

8687

8776

8865

8953

9042

9131

9220

9 9309

9398

9486

9575

9664

9753

9841

9930

0010

0107

490

690196

0285

0373

0462

0550

0639

0728

0816

W-I U

0905

V ±\J t

0993

1

1081

1170

1258

1347

1435

1524

1612

1

700

1789

1877

2

1965

2053

2142

2230

2318

2406

2494

2583

2671

2759

3

2847

2935

3023

3111

3199

3287

3375

3463

3551

3639

88

4

3727

3815

3903

3991

4078 ;

4166

4254

4342

4430

4517

5

4605

4693

4731

4868

4956

5044

5131 5219

5307

5394

6

5482

5569

5657

5744

5832

5919

6007 6094

6182

6269

r*

6356

6444

6531

6618

6706

6793

6880

6968

7055

7142

8

7229

7317

7404

7491

7578

7665

7752

7839

7926

8014

Oi™»

9

8100

8188

8275

8362

8449 i

8535

8622

8

709

8796

8883

87

PROPORTIONAL PARTS.

Diff. 1

2 3

4

5

6

7

8

9

98 9.8

19.6 29.4

39.2

49.0

58.8

68.6

78.4

88.2

97 9.7

19.4 29.1

38.8

48.5

58.2

67.9

77.6

87.3

96 9.6

19.2 28.8

38.4

48.0

57.6

67.2

76.8

86.4

95 9.5

19.0 28.5

38.0

47.5

57.0

66.5

76.0

85.5

94 9.4

18.8 28.2

37.6

47.0

56.4

65.8

75.2

84.6

93 9.3

18.6 27.9

37.2

46.5

55.8

65.1

74.4

83.7

92 9.2

18.4 27.6 1

36.8

46.0

55.2

64.4

73.6

82.8

91 9.1

18.2 27.3

36.4

45.5

54.6

63.7

72.8

81.9

90 9.0

18.0 27.0

36.0

45.0

54.0

63.0

72.0

81.0

89 8.9

17.8 26.7

35.6

44.5

53.4

62.3

71.2

80.1

88 8.8

17.6 26.4

35.2

44.0

52.8

61.6

70.4

79.2

87 8.7

17.4 26.1

34.8

43.5

52.2

60.9

69. B

78:3

86 8.6

17.2 25.8

34.4

43.0

51.6

60.2

68.8

77.4

91

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 500 L. 698.] [No. 544 L. 736.

N.

0

1

2

3

4

6

6

7

8

9

Diff.

500

698970

9057

9144

9231

9317

9404

9491

9578

9664

9751

1

9838

9Q24

*/t_MJIJ

VV*fT.

0011

0008

0184

0271

0^558

0444

05^1

Ofi17

2

700704

0790

Wl 1

0877

\J\J<J<J

0963

\J .ICr*

1050

ViV 1 1

1136

vuiJO

1222

\f L L 1

1309

\JlJIJl

1395

V/UJ. 1

1482

3

1568

1654

1741

1827

1913

1999

2086

2172

2258

2344

4

2431

2517

2603

2689

2775

2861

2947

3033

3119

3205

5

3291

3377

3463

3549

3635

3721

3807

3893

3979

4065

86

6

4151

4236

4322

4408

4494

4579

4665

4751

4837

4922

7

5008

5094

5179

5265

5350

5436

5522

5607

5693

5778

8

5864

5949

6035

6120

62C6

6291

6376

6462

6547

6632

9

6718

6803

6888

6974

7059

7144

7229

7315

7400

7485

510

7570

7655

7740

7826

7911

7996

8081

8166

8251

8336

Of.

1

8421

8506

8591

8676

8761

8846

8931

9015

9100

9185

oo

o

9^70

9355

Q440

95:24

ofioq

9fiQ4

Q77Q

Q8R3

QQ4S

i -» 1

i/Ov/iJ

iJ^ri.\J

isiJty^X

*J\J\Jis

U\JtJ^

V-t m v

*7OU(J

l/l/'XtJ

0033

3

710117

0202

0287

0371

0456

0540

0625

0710

0794

\J\JtJfJ

0879

4

0963

1048

1132

1217

1301

1385*

1470

1554

1639

1723

5

1807

1892

1976

2060

2144

2229

2313

2397

2481

2566

6

2650

2734

2818

2902

2986

3070

3154

3238

3323

3407

Rl

7

3491

3575

3659

3742

3826

3910

3994

4078

4162

4246

o-t

8

4330

4414

4497

4581

4665

4749

4S33

4916

5000

5084

9

5167

5251

5335

5418

5502

5586

5669

5753

5836

5920

520

6003

6087

6170

6254

6337

6421

6504

6588

6671

6754

1

6838

6921

-J004

7'088

7171

7254

7338

7421

7504

7587

2

7671

7754

7837

7920

8003

8086

8169

8253

8336

8419

oo

3

8502

8585

8668

8751

8834

8917

9000

9083

9165

9248

83

4

9331

9414

9497

9580

9663

9745

9828

9911

9994

0077

5

720159

0242

0325

0407

0490

0573

0655

0738

0821

\J\J 4 1

0903

6

0986

1068

1151

1233

1316

1398

1481

1563

1646

1728

7

1811

1893

1975

2058

2140

2222

2305

2387

2469

2552

8

2634

2716

2798

2881

2963

3045

3127

3209

3291

3374

9

3456

3538

3620

3702

3784

3866

3948

4030

4112

4194

82

530

4276

4358

4440

4522

4604

4685

4767

4849

49ol

5013

1

5095

5176

5258

5340

5422

5503

5585

5667

5748

5830

2

5912

5993

6075

6156

6238

6320

6401

6483

6564

6646

3

6727

6809

6890

6972

7053

7134

7216

7297

7379

7460

4

7541

7623

7704

7785

7866

7948

8029

8110

8191

8273

5

8354

8435

8516

8597

8678

8759

8841

8922

9003

9084

6

9165

9246

9327

9408

9489

9570

9651

9732

9813

9893

81

7.

QQ74

1

*7t/ 1 ^

0055

013fi

0*>17

0°Q8

0378

04 50

0540

Ofi°1

070^

8

730782

\J\J*J*J

0863

\J _l '-J\J

0944

V'.-« 1 |

1024

\/W«7(J

1105

v*J i O

1186

VAl«Jt7

1266

Vt-TTv/

1347

VU/srf A

1428

\J I \Jf*

1508

9

1589

1669

1750

1830

1911

1991

2072

2152

2233

2313

540

2394

2474

2555

2635

2715

2796

2876

2956

3037

3117

1

3197

3278

3358

3438

3518

3598

3679

3759

3839

3919

2

3999

4079

4160

4240

4320

4400

4480

4560

4640

4720

8fl

3

4800

4880

4960

5040

5120

5200

5279

5359

5439

5519

ou

4

5599

5679

5759

5838

5918

5998

6078

6157

6237

6317

PROPORTIONAL PARTS.

Diff. 1

234

5

678

9

87 8.7

17.4 26 1 34 8

43 5

52 2 60.9 69 6

78 3

86 8.6

17.2 258 34.4

43 0

51 6 60 2 68 8

77 4

85 8.5

17.0 25 5 34.0

42 5

51.0 595 680

76 5

84 8.4

16.8 252 33.6

42 0

50.4 58 8 67.2

75 6

92

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 545 L. 736.]

|No. 584 L. 767.

N.

0 1

2

8

4

6

V.

7

8

9

Diff.

545 736397 6476

6556

6635 6715 6795

6874

6954

7034 7113

6

7193 7272

7352

7431

7511 7590

7670

7749

7829 i 7908

7 7987 8067

8146

8225 8305 8384

8463

8543 8622 8701

8 8781

8860

8939

9018 i 9097 9177

9256

9335 9414 9493

9 Q<VM>

Qfir.1

9731

9810 9889 9968

V t ' J \.

0047 0126

0205

0284

79

550 740363

0442

0521

0600

0678 0757

case

0915

0994

1073

1 1152

1230

1309

1388

1467 1546

1624

1

ros

1782

1860

2 1939

2018

2096

2175

2254 2332

2411

2489

2568

2647

3 2725

2804 :

2882

2961

3039

3118

3196

3275

3353

3431

4 3510

3588

3667

3745

3823

3902

3980

4058

4136

4215

5 4293

4371

4449

4528

4606

4684

4762

4840

4919

4997

6

5075

5153

5231

5309

5387

5465

5543

5621

5699

5777

78

7

5855

5933

6011

6089

6167

6245

6323

6401

6479

6556

8

6634 6712

6790 6868

6945

7023 7101

r*
t

179

7256

7334

9

7412

7489

7567

7645

7722

7800 7878

7955

8033

8110

560

8188

8266

8343 8421 8498

8576 8653

8731

8808

8885

1

8963

1)040

9118 9195 9272

9350 9427

9504 ! 9582

9659

2

9736

9814

qoq-|

QQRQ

|

i '""

0045

0123 0200

0277 0354 0431

3

750508 0586

0663 0740 0817

C894 0971

1048 1125

1202

4

1279 1356

1433 1510 1587

16(J4 1741

1818 1895

1972

5

2048

2125

2202 2279 2356

2433 2509

2586 2663

2740

it

6

2816

2893

2970 3047 3123

3200 3277

3353 3430

3506

r*

^

3583

3660

3736 i 3813 3889

3966 4042

4119 ; 4195

4272

8

4348

4425

4501 ! 45'

"8 4654

4730 4807

4883 4960

5036

9

5112

5189

5265 5341

5417

5494 5570

5646 5722

5799

570

5875

5951

6027 6103

6180

6256 6332

6408 6484

6560

1

6636

6712

6788 : 6864

6940

7016 7092 7

168 7244

7320

76

2

7396

7472

7548 7624

7700

7775 7851

7927 8003

8079

3

8155

8230

8306 8382

8458

8533 8609

8685

8761

8836

4

8912

8988

9063 9139

9214

9290 9366

9441

951'

t
I

9592

5

9668

9743

9819

()8<U

Q^T)

iJ\J v^

1

004" 01 °1

1OA

no'-o

„„ .„

6

760422

0498

0573

0649

0724

0799 0875

0950 1025

1101

r*

i

1176

1251

1326

1402

1477

1552 1627

1702 1778

1853

8

1928

2003

2078

2153

2228

2303 2378

2453 2529

2604

9

2679

2754

2829

2904

2978

3053 3128 j 3203

3278

3353

(O

580

3428

3503

3578

3653

3727

3802 3877

3952

4027

4101

1

4176

4251

4326 4400

4475

4550 4624

4699

4774

4848

2

4923

4998

5072 5147

5221

5296 5370

5445

5520

5594

3

5669

5743

5818

5892

5966

6041

6115

6190

6264

6338

4

6413

6487

6562

6636

6710

6785 6859

6933

7007

7082

1

PROPORTIONAL PARTS.

Diff. 1

2

3

4

5

6

7

8

9
74.7

as 8.3

16.6

24.9

33.2

41.5

49.8

58.1

66.4

82 8.2

16.4

24.6

32.8

41.0

49.2

57.4

65.6

73.8

81 8.1

16.2

24.3

32.4

40.5

48.6

56.7

64.8

72.9

80 8.0

16.0

24.0

32.0

40.0

48.0

56.0

64.0

72.0

79 7.9

15.8

23.7

31.6

39.5

47.4

55.3

63.2

71.1

78 7.8

15.6

23.4

31.2

39.0

46.8

54.6

62.4

70.2

77 7.7

15.4

23.1

30.8

38.5

46.2

53.9

61.6

69.3

76 76

15.2

22.8

30.4

38.0

45.6

53.2

60.8

68.4

75 75

15.0

22.5

30.0

37.5

45.0

52.5

60.0

67.5

74 7.4

14.8

OO 9

***** • "•

29.6

37.0

44.4

51.8

59.2

66.6

93

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 585 L. 767.] [No. 629 L. 790.

N.

0

1

2

3

4

5 6

7

8

9

Diff.

585

767156

7230

7304

7379

7453

7527 7601

7675

7749 7823

6

7898

7972

8046

8120

8194

8268 8:342

8416

8490 8564

74

7

8638

8712

8786

8860

8934

9008 9082

9156

9230

9303

8

9377

9451

9525

9599

9673

9746 9820

9894

9968

0042

9

770115

0189

0283

0336

0410

0484 0557

0631

0705

\j\j^&
0778

590

0852

0926

0999

1073

1146

1220 1293

1367

1440

1514

1

1587

1661

1734

1808

1881

1955 2028

2102

2175

2248

2

2322

2395

2468

2542

2615

2688 2762

2835

2908

2981

3

3055

3128

3201

3274

3348

3421

3494

3567

3640

3713

4

3786

3860

3933

4006

4079

4152

4225

4298 i 4371

4444

73

5

4517

4590

4663

4736

4809

4882

4955

5028 i 5100

5173

6

5246

5319

5392

5465

5538

5610

5683

5756

5829

5902

7

5974

6047

6120

6193

6265

6338

6411

6483

6556

6629

8

6701

6774

6846

6919

6992

7064

7137

7209

7282

7354

9

7427

7499

7572

7644

7717

7789

7862

7934

8006

8079

600

8151

8224

8296

8368

8441

8513

8585

8658

8730

8802

1

8874

8947

9019

9091

9163

9236

9308

9380

9452

9524

QKOfi

QfifiO

974 1

Q813

988 "i

9957

*j\J<j\J

ijWtJ

\J t T J.

&\JA.tJ

«7O(Jt_J

OO'^Q

0101

M73

024*1

3

780317

0389

0461

0533

0605

0677

\J\JtviS

0749

VIVi

0821

t o

0893

\Jf*r±tj

0965

72

4

1037

1109

1181

1253

1324

1396

1468

1540

1612

1684

5

1755

1827

1899

1971

2042

2114

2186

2258

2329

2401

6

2473

2544

2616

2688

2759

2831

2902

2974

3046

3117

7

3189

3260

3332

3403

3475

3546

3618

3689

3761

3832

8

3904

3975

4046

4118

4189

4261

4332

4403

4475

4546

9

4617

4689

4760

4831

4902

4974

5045

5116

5187

5259

610

5330

5401

5472

5543

5615

5686

5757

5828

5899

5970

1

6041

6112

6183

6254

6325

6396

6467

6538

6609

6680

71

2

6751

6822

6893

6964

7035

7106

7177

7248

7319

7390

3

7460

7531

7602

7673

7744

7815

7885

7956

8027

8098

4

8168

8239

8310

8381

8451

8522

8593

8663

8734

8804

5

8875

8946

9016

9087

9157

9228

9299

9369

9440

9510

6

9581

9651

9722

9792

9863

9933

0004

0074

0144

0°15

7

790285

0356

0426

0496

0567

0637

\J\J\s^

0707

v"^' 1 rr

0778

V/A^^

0848

U'w.l tJ

0918

8

0988

1059

1129

1199

1269

1340

1410

1480

1550

1620

9

1691

1761

1831

1901

1971

2041

2111

2181

2252

2322

620

2392

2462

2532

2602

2672

2742

2812

2882

2952

3022

70

1

3092

3162

3231

3301

3371

3441

3511

3581

3651

3721

2

3790

3860

3930

4000

4070

4139

4209

4279

4349

4418

3

4488

4558

4627

4697

4767

4836 1 4906

4976

5045

5115

4

5185

5254

5324

5393

5463

5532 5602

5672

5741

5811

5

5880

5949

6019

6088

6158

6227 6297

6366

6436

6505

6

6574

6644

6713

6782

6852

6921 6990

7060

7129

7198

7

7268

7337

7406

7475

7545

7614 7683

7752

7821

7890

8

7960

8029

8098

8167

8236

8305 8374

8443

8513

8582

9

8651

8720

8789

8858

8927

8996

9065

9134

9203

9272

69

PROPORTIONAL PARTS.

Diff. 1

234

5078

9

i

75 7.5

15.0 22.5 30.0

37.5

45.0 52.5 60.0

67.5

74 7.4

14.8 22.2 29.6

37.0

44.4 51.8 59.2

66.6

73 7.3

14.6 21.9 29.2

36.5

43.8 51.1 58.4

65.7

72 7.2

14.4 21.6 28.8

36.0

43.2 50.4 57.6

64.8

71 7.1

14.2 21.3 28.4

35.5

42.6 49.7 56.8

63.9

70 7.0

14.0 21.0 28.0

35.0

42.0 49.0 56.0

63.0

69 6.9

13.8 20.7 27.6

34.5

41.4 48.3 55.2

62.1

94

TABLE IX. — LOGARITHMS OF LUMBERS.

Ko. 630 L. 799.] [No. 674 L. 829.

'N.

0

1

2

3

4 || 5

6

7

8

9

Diff.

'630

799341

9409

9478

9547

9616 : 9685

9754

9323

9892 9961

1

800029

0098

0167 0236

0305 0373

0442

0511

0580 0648

2

0717

0786

0854

0923

0992 1061

1129

1198

1266

1335

3

1404

1472

1541

1609

1678

1747

1815

1884

1952

2021

4

2089

2158

2226

2295

2363

2432

2500

2568

2637

2705

5

2774

2842

2910

2979

3047 3116

3184

3252

3321

3389

6

3457

3525

3594

3662

3730 3798

3867

39:35

4003

4071

7

4139

4208

4276

4344

4412 4480

4548

4616

4685

4753

8

4821

4889

4957

5025

5093 5161

5229

5297

5365

5433

68

9

5501

5569

5637

5705

5773

5841

5908

5976

6044

6112

640

806180

6248

6316

6384

6451

6519

6587

6655

6723

6790

1

6858

6926

6994

7061

7129

7197

7264

7332

7400

7467

0

IS/

7535

7603

7670

7738

7806

7873

7941

8008

8076

8143

3

8211

8279

8346

8414

8481

8549

8616

8684

8751

8818

4

8886

8953

9021

9088

9156

9223

9290

9358

9435

9492

5

9560

9627

9694

9762

9829

98Q6

9964

*7*Ji/Vl

tj *J\ji.

0031

0098

0165

6

810233

0300

0367

04:34

0501

0569

0636

0703

0770

0837

W

1

0904

0971

1039

1106

1173

1240

1307

1374

1441

1508

67

8

1575

1642

1709

1776

1843

1910

1977

2044

2111

2178

9

2245

2312

2379

2445

2512

2579

2646

2713

2780

2847

650

2913.

2980

3047

3114

3181

3247

3314

3:381

3448

a5i4

1

3581

3648

3714

3781

3848

3914

3981

4048

4114

4181

2

4248

4314

4381

4447

4514

4581

4647

4714

4780

4847

3

4913

4980

5046

5113

5179

5246

5312

5378

5445

5511

4

5578

5644

5711

5777

5843

5910

5976

6042

0109

6175

5

6241

6308

6374

6440

6506

6573

6639

6705

6771

6838

6

6904

6970

7036

7102

7169

7235

7301

7367

7433

7499

r*

i

7565

7631

7698

7764

7830

7896

7962

8028

8094

8160

8

8226

8292

8358

8424

8490

8556

8622

8688

8754

8820

£*£*

9

8885

8951

9017

9083

9149

9215

9281

9346

9412

9478

66

660

9544

9610

9676

9741

9807

9873

9939

fMVU

nrrft

n-i qc

1

820201

0267

0333

0399

0464

0530

0595

uu\>±
0661

UUt U

0727

U1OU

0792

2

0858

0924

0989

1055

1120

1186

1251

1317

1382 1448

3

1514

1579

1645

1710

1775

1841

1906

1972

2037 2103

4

2168

2233

2299

2364

2430

2495

2560

2626

2691 2756

5

2822

2887

2952

3018

3083

3148

3213

3279

3344 i 3409

6

3474

3539

3605

3670

3735

3800

3865

3930

3996

4061

7

4126

4191

4256

4321

4386 4451 4516

4581

4046

4711

£K

8

4776

4841

4906

4971

5036

5101

5166

5231

5296

5361

bo

9

5426

5491

5556

5621

5686

5751

5815

5880

5945

6010

670

6075

6140

6204

6269

6334

6399

6464

6528

6593

6658

1

6723

6787

6852

6917

6981

7046

7111

7175

7240

7305

2

7369

7434

7499

7503

7628

7692

7757

7821

7886

7951

3

8015

8080

8144

8209

8273

8338

8402

8467

8531

8595

4

8660

8724

8789

8853

8918

8982

9046

9111

9175

9239

PROPORTIONAL PARTS,

!Diff 1

234

5

678

9

68 68

13 6 20 4 27 2

34 0

40 8 47 6 54 4

61 2

67 67

13 4 20.1 26 8

S3 5

40 2 -Hi i) 53 6

60 3

66 66

13.2 19 8 26 4

33 0

39 6 -M 2 52 8

59 4

65 65

13 0 19 5 26 0

32.5

39 0 45 5 52 0

58 5

64 6.4

li 8 19.2 25 6

32 0

3S.1 44 8 51 2

57.6

95

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 675 L. 829.] [No. 719 L. 857.

N.

0

1

2

3 4

6

6

7

8

9

Diff,

I

675

829304

9368

9432

9497

9561

9625

9690

9754

9818

9882

6

9947

0011 007^ mao noru

0268

0332

0396

0460

0525

ry
4

830589

\J\JlJi \s\J t tJ

0653 0717

V^J-^/t/ VWW^

0781 0845

0909

0973

1037

1102

1166

8

1230

1294

1358

1422 1486

1550

1614

1678

1742

1806

64

9

1870

1934

1998

2062

2126

2189

2253

2317

2381

2445

680

2509

2573

2637

2700

2764

2828

2892

2956

3020

3083

1

3147

3211

3275

3338

3402

3466

3530

3593

3657

3721

2

3784

3848

3912

3975

4039

4103

4166

4230

4294

4357

3

4421

4484

4548

4611

4675

4739

4802

4866

4929

4993

4

5056

5120

5183

5247

5310

5373

5437

5500

5564

5627

5

5691

5754

5817

5881

5944

6007

6071

6134

6197

6261

6

6324

6387

6451

6514

6577

6641

6704

6767

6830

6894

7

6957

7020

7083

7146

7210

7273

7336

7399

7462

7525

8

7588

7652

7715

7778

7841

7904

7967

8030

8093

8156

/>ri

9

8219

8282

8345

8408

8471

85*4

8597

8660

8723

8786

63

690

8849

8912

8975

9038

9101

9164

9227

9289

9352

9415

1

9478

9541

9604

9667

9729

9792

9855

9918

9981

0043

2

840106

0169

0232

0294

0357

0420

0482

0545

0608

0671

3

0733

0796

0859

0921

0984

1046

1109

1172

1234

1297

4

1359

1422

1485

1547

1610

1672

1735

1797

1860

1922

5

1985

2047

2110

2172

2235

2297

2360

2422

2484

2547

6

2609

2672

2734

2796

2859

2921

2983

3046

3108

3170

7

3233

3295

3357

3420

3482

3544

3606

3669

3731

3793

8

3855

3918

3980

4042

4104

4166

4229

4291

4353

4415

9

4477

4539

4601

4664

4726

4788

4850

4912

4974

5036

700

5098

5160

5222

5284

5346

5408

5470

5532

5594

5656

62

1

5718

5780

5842

5904

5966

6028

6090

6151

6213

6275

2

6337

6399

6461

6523

6585

6646

6708

6770

6832

6894

3

6955

7017

7079

7141

7202

7264

7326

7388

7449

7511

4

7573

7634

7696

7758

7819

7881

7943

8004

8066

8128

5

8189

8251

8312

8374

8435

8497

8559

8620

8682

8743

6

8805

8866

8928

8989

9051

9112

9174

9235

9297

9358

7

9419

9481

9542

9604

9665

9726

9788

9849

9911

9972

8

850033

0095

0156

0217

0279

0340

0401

0462

0524

0585

9

0646

0707

0769

0830

0891

0952

1014

1075

1136

1197

710

1258

132.0

1381

1442

1503

1564

1625

1686

1747

1809

1

1870

1931

1992

2053

2114

2175

2236

2297

2358

2419

fi1

2

2480

2541

2602

2663

2724

2785

2846

2907

2968

3029

01

3

3090

3150

3211

3272

3333

3394

3455

3516

3577

3637

4

3698

3759

3820

3881

3941

4002

4063

4124

4185

4245

5

4306

4367

4428

4488

4549

4610

4670

4731

4792

4852

6

4913

4974

5034

5095

5156

5216

5277

5337

5398

5459

7

5519

5580

5640

5701

5761

5822

5882

5943

6003

6064

8

6124

6185

6245

6306

6366

6427

6487

6548

6608

6668

9

6729

6789

6850

6910

6970

7031

7091

7152

7212

7272

PROPORTIONAL PARTS.

Diff. 1

234

5

678

9

65 6.5

13.0 19.5 26.0

32.5

39.0 45.5 52.0

58.5

64 6.4

12.8 19.2 25.6

32.0

38.4 44.8 51.2

57.6

63 6.3

12.6 18.9 25.2

31.5

37.8 44.1 50.4

56.7

62 6.2

12.4 18.6 24.8

31.0

37.2 43.4 49.6

55.8

61 6.1

12.2 18.3 24.4

30.5

36.6 42.7 48.8

54.9

60 6.0

12.0 18.0 24.0

30.0

36.0 42.0 48.0

54.0

96

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 720 L. 857.] [No. 76* L. 883.

N.

0

1

2

3

4

6

6

•

8

9

Diff.

720

857332

7393

7453

7513

7574

7634

7694

7755

7815

7875

1

7935

7995

8056

8116

8176

8236

8297

8357

8417

8477

2

8537

8597

8057

8718

8778

8838

8898

8958

9018

9078

3

9138

9198

9258

9318

9379

9439

9499

9559

9619

9679

60

4

9739

9799

9859

9918

9978

0038

0098

0158

0218

0278

5

860338

0398

0458

0518

0578

0637

0697

0757

0817

0877

6

0937

0996

1056

1116

1176

1235

1295

1355

1415

1475

7

1534

1594

1654

1714

1773

1833

1893

1952

2012

2072

8

2131

2191

2251

2310

2370

2430

2489

2549

2608

2668

9

2728

2787

2847

2906

2966

3025

3085

3114

3204

3263

730

3323

3382

3442

3501

3561

3620

3680

3739

3799

3858

1

3917

3977

4036

4096

4155

4214

4274

4333

4392

4452

2

4511

4570

4630

4689

4748

: 4808

4867

4926

4985

5045

3

5104

5163

5222

5282

5341

5400

5459

5519

5578

5637

4

5696

5755

5814

5874

5933

5992

6051

6110

6169

6228

5

6287

6346

6405

6465

6524

6583

C642

6701

6760

6819

fen

6

6878

6937

6996

7055

7114

7173

7232

7291

7350

7409

59

7

7467

7526

7585

7644

7703

7762

7821

7880

7939

7998

8

8056

8115

8174

8233

8292

8350

8409

8468

8527

8586

9

8644

8703

8762

8821

8879

8938

8997

9056

9114

91.3

740

9232

9290

9349

9408

9466

9525

9584

9642

9701

9760

•t

O«1S

Qfi~7

OQ--IT

0(104

i

± «7U.LU

•7OI 1

iJ *J'J*J

«7t/i/T±

00^

0111

0170

0228

0°87

0345

2 870404

0462

0521

0579

\J\J*J'J

0638

\J 1 1 A

0696

\J 1 t \s

0755

Vivis/lJ

0813

V/'wO t

0872

UVXU

0930

3 0989

1047

1106

1164

1223

1281

1339

1398

1456

1515

4 1573

1631

1690

1748

1806

1865

1923

1981

2040

2008

5 i 2156

2215

2273

2331

2389

2448

2506

2564

2622

2681

6 2739

2797

2855

2913

2972

3030

3088

3146

3204

3262

7

3321

3379

3437

3495

3553

3611

3669

3727

3785

3844

8 3902

3960

4018

4076

4134

4192

4250

4308

4366

4424

58

9

4482

4540

4598

4656

4714

4772

4830

4888

4945

5003

750

5061

5119

5177

5235

5293

5351

5409

5466

5524

5582

1

5640

5698

5756

5813

5871

5929

5987

6045

6102

6160

2

6218

6276

6333

6391

6449

6507

6564

6622

6680

6737

3

6795

6853

6910

6968

7026

7083

7141

7199

7256

7314

4

7371

7429

7487

7544

7602

7659

7717

7774

7832

7889

5

7947

8004

8062

8119

8177

8234

8292

8349

8407

8464

6

8522

8579

8637

8694

8752

8809

8866

8924

8981

9039

7

9096

9153

9211

9268

9325

9383

9440

9497

9555

9612

D

QfifiQ

Q7-2fi

Q7S4

Q>U1

'IS'IS

QQ'"Sfi

o

iJ\J\J&

i7 1 <t\J

•7 1 CTT

i7O-*-l

*7O«7O

UiJ*J\J

001 3

0070

0127

01 85

9

880242

0299

0356

0413

0471

0528

V/w i'J

0585

\s\J t \J

0642

\J J./V I

0699

v/lOJ

0756

760

0814

0871

0928

0985

1042

1099

1156

1213

1271

1328

1

1385

1442

1499

1556

1613

1670

1727

1784

1841

1898

2

1955

2012

2069

2126

2183

2240

2297

2354

2411

2468

57

3

2525

2581

26:38

2695

2752

2809

2866

2923

2980

3037

4

3093

3150

3207

3264

3321

3377

3434

3491

3548

3605

1

PROPORTIONAL PARTS.

Diff. 1

234

5

678

9

59 5.9

11.8 17.7 23.6

29.5

35.4 41.3 47.2

53.1

58 5.8

11.6 17.4 23.2

29.0

34.8 40.6 46.4

52.2

57 5.7

11.4 17.1 22.8

28.5

34.2 39.9 45.6

51.3

56 5.6

11.2 16.8 22.4

28.0

33.6 39.2 44.8

50.4

97

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 765 L. 883.] [No. 809 L. 908.

N.

0

1

2

3

4

1
6

6

7

8

9

Dm.

765

883661

3718

3775

3832

3888

3945

4002

4059

4115

4172

6

4229

4285

4342

4399

4455

4512

4569

4625

4682

4739

7

4795

4852

4909

4965

5022

5078

5135

5192

5248

5305

8

5361

5418

5474

5531

5587

5644

5700

5757

5813

5870

9

5926

5983

6039

6096

6152

6209

6265

6321

6378

6434

770

6491

6547

6604

6660

6716

6773

6829

6885

6942

6998

1

7054

7111

7167

7223

7280

7336

7392

7449

7505

7561

2

7617

7674

7730

7786

7842

7898

7955

8011

8067

8123

3

8179

8236

8292

8348

8404

8460

8516

8573

8629

8685

4

8741

8797

8853

8909

8965

9021

9077

9134

9190

9246

5

9302

9358

9414

9470

9526

9582

96:38

9694

9750

9806

56

R

Q8(is>

99 IS

CW74

\J

i/(JU»v

v«7 J.O

t/«7 t ^

O0'¥>

OOSfi

0141

01 Q7

noivq

O^OQ

O^fi^i

r*
i

890421

0477

0533

\J\-I'J\S

0589

\J\JO\J

0645

\7lT±

0700

V/l «7 4

0756

vTvcJO

0812

\J«JV/«7

0868

\J'J\jKj

0924

8

0980

1035

1091

1147

1203

1259

1314

1370

1426

1482

9

1537

1593

1649

1705

1760

1816

1872

1928

1983

2039

780

2095

2150

2206

2262

2317

2373

2429

2484

2540

2595

1

2651

2707

2762

2818

28T3

2929

2985

3040

3096

3151

2

3207

3262

3318

3373

3429

3484

3540

3595

3651

3706

3

3762

3817

3873

3928

3984

4039

4094

4150

4205

4261

4

4316

4371

4427

4482

4538

4593

4648

4704

4759

4814

5

4870

4925

4980

5036

5091

5146

5201

5257

5312

5367

6

5423

5478

5533

5588

5644

5699

5754

5809

5864

5920

7

5975

6030

6085

6140

6195

6251

6306

6361

6416

6471

8

6526

6581

6636

6692

6747

6802

6857

6912

6967

7022

9

7077

7132

r*1 or*
llbl

7242

r-OA'*'
iZVl

7352

7407

7462

7517

7572

55

790

7627

7682

7737

7792

7847

7902

7957

8012

8067

8122

1

8176

8231

8286

8341

8396

8451

8506

8561

8615

8670

2

8725

8780

8835

8890

8944

8999

9054

9109

9164

9218

3

9273

9328

9383

9437

9492

9547

9602

9656

9711

9766

A

q«oi

Q875

WIO

qqcs

^

c/Oi-W J.

«7O t *J

iJiJ*J\J

i/«7O«J

003Q

0094

01 4Q

0°03

flOWJ

0312

5

900367

0422

0476

053!

W/*J«7

0586

\s\sis^

0640

V/l^i7

0695

V//VVL)

0749

Vi^tJO

0804

\J'J 1 *j

0859

6

0913

0968

1022

1077

1131

1186

1240

1295

1349

1404

7

1458

1513

1567

1622

1676

1731

1785

1840

1894

1948

8

2003

2057

2112

2166

2221

2275

2329

2384

2438

2492

9

2547

2601

2655

2710

2764

2818

2873

2927

2981

3036

800

3090

3144

3199

3253

3307

3361

3416

3470

3524

357'8

1

36:53

3687

3741

3795

3849

3904

3958

4012

4066

4120

2

4174

4229

4283

4337

4391

4445

4499

4553

4607

4661

3

4716

4770

4824

4878

4932

4986

5040

5094

5148

5202

tU

4

5256

5310

5364

5418

5472

5526

5580

56:34

5688

5742

tri

5

5796

5850

5904

5958

6012

6066

6119

6173

6227

6281

6

6335

6389

6443

6497

6551

6604

6658

6712

6766

6820

r*
*

6874

6927

6981

7035

7089

7143

7196

7250

7304

7:358

8

7411

7465

7519

7573

7626

7680

77:?4

7787

7841

7895

9

7949

8002

8056

8110

8163

8217

8270

8324

8378

8431

PROPORTIONAL PARTS.

Diflf. 1

234

5

678

9

57 5.7

11.4 17.1 22.8

28.5

34.2 39.9 45.6

51.3

56 5.6

11.2 16.8 22.4

28.0

33.6 39.2 44.8

50.4

55 5.5

11.0 16.5 22.0

27.5

33.0 38.5 44.0

49.5

54 5.4

10.8 16.2 21.6

27.0

32.4 37.8 43.2

48.6

TABLE IX. — LOGARITHMS OP NUMBERS.

No. 810 L. 908.] [No. 854 L. 931.

N.

0

1

2

3

4

5

6 7

8 9

Diff.

810

908485

8539

8592

8646

8699

8753

8807

8860

8914 8907

1

9021

9074

9128

9181

9235

9289

1)342

9396

9449 9503

2

9556

9610

9663

9716

9770

q«M'^

Q»77

QQSfl

QOS1

i7 1 J. \J

«7 1 1 \s

tJO'wO

i7O 1 I

i/«7Ov

HAST

3

910091

0144

0197

0251

0304 0358

0411

0464

0518

\J\J<J 1

0571

4

0624

0678

0731

0784

0838

0891

0944

0998

1051

1104

5

1158

1211

1264

1317

1371

1424

1477

1530

1584

1637

6

1690

1743

1797

1850

1903

1956

2009

2063

2116

2169

r*
t

2222

2275

2328

2381

2435

2488

2541

2594

2647

2700

8

27'53

2806

2859

2913

2966

3ul9

3072

3125

3178

3231

9

3284

3337

3390

3443

3496

3549

3602

3655

3708

3761

53

820

3814

3867

3920

3973

4026 4079

4132

4184

4237

4290

1

4343

4396

4449

4502

4555

4608

4660

4713

4766

4819

2

4872

4925

4977

5030

5083

5136 5189

5241 5294

5347

3

5400

5453

5505

5558

5611

5664

5716

5769 5822

5875

4

5927

5980

6033

6085

6138

6191

6243

6296

6349

6401

5

6454

6507

6559

6612

6664

6717

6770

(822

6875

M27

6

6980

7033

7085

7138

7190

7243

7295

7348

7400

7453

W
I

7506

7558

7611

7663

7716

7768

7820

7873

7925

7978

8

8030

8083

8135

8188

8240

8293

8345

8397

8450

8502

9

8555

8607

8659

8712

8764

8816

8869

8921

8973

9026

830

9078

9130

9183

9235

9287

9340

9392

9444

9406

9549

1

9601

Of.-.Q

9706

q/rnq

QS10 Qftfi0

no-) I ()Q(',7

2.

tJ\J\J JL

ts\JtJiJ

U I \J\J

t7 4 (JO

«7OJ.»-r t/OvJ/w

nm n

0071

2

920123

0176

0228

0280

0332

0384

0436

0489 0541

W 1 -I

0593

3

0645

0697

0749

0801

0853

0906

0958

1010

1062

1114

no

4

1166

1218

1270

1322

1374

1426

1478

1530

1582

1634

O--W

5

1686

1738

1790

1842

1894

1946

1998

2050

2102

2154

6

2206

2258

2310

2362

2414

2466

2518

2570

2622

2674

7

2725

2777

2829

2881

2933

2985

3037

3089

3140

3192

8

3244

3296

3:548

3399

3451

3503

3555 3607

3658

3710

9

3762

3814

3865

3917

3969

4021

4072

4124

4176

4228

840

4279

4331

4383

4434

4486

4538

4589

4641

4693

4744

1

4796

4848

4899

4951

5003

5054

5106

5157

5209

5261

2

5312

5364

5415

5467

5518

5570

5621

5673

5725

5776

3

5828

5879

5931

5982

6034

6085

6137

6188

6240

6291

4

6342

6394

6445

6497

6548

6600

6651

6702

6754

6805

5

6857

6908

6959

7011

7062

7114

7165

7216

7268

7319

6

7370

7422

7473

7524

7576

7627

7678

7730

7781

7832

7

7883

7935

7986

8037

8088

8140

8191

8242

8293

8345

8

8396

8447

8498

8549

8601

8652

8703

8754 8805

8857

9

8908

8959

9010

9061

9112

9163

9215

9266

9317

9368

850

9419

9470

9521

9572

9623

9674

9725

9776

9827

9879

fcfH

i

Q930

qqS1

51

J.

ijiJ^J\J

t7t7L'i

003-'

OOK^

0134

018s*

023fi

0287 i 0338 ns8Q

2

930440

0491

\J* >*)**

0542

WOtf

0592

\J JL'-r~f

0643

\Ij.\J\J

0694

\Jf**J\J

0745

0796 0847

v/^wt*

0898

3

0949

1000

1051

1102

1153

1204

1254

1305 1356

1407

4

1458

1509

1560

1610

1661

1712

1763

1814

1865

1915

PROPORTIONAL, PARTS.

Diff. 1

234

5

678

Q

53 5.3

10.6 15.9 21.2

26.5

31.8 37.1 42.4

47.7

52 5.2

10.4 15.6 20.8

26.0

31.2 36.4 41.6

46.8

51 5.1

10.2 15.3 20.4

25.5

30.6 35.7 40.8

45.9

50 5.0

10.0 15.0 20.0

25.0

30.0 35.0 40.0

45.0

1

99

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 855 L. 931.] [No. 899 L. 954.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

355

931966

2017

2068

2118

2169 2220 2271

2322

2372

2423

6

2474

2524

2575

2626

2677

2727

2778

2829

2879

2930

7

2981

3031

3082

3133

3183

3234

3285

3335

3386

3437

8

3487

3538

3589

3639

3690

3740 I 3791

3841

3892

3943

9

3993

4044

4094

4145

4195

4246

4296

434?

4397

4448

8GO

4498

4549

4599

4650

4700

4751

4801

4852

4902

4953

1

5003

5054

5104

5154

5205

5255

5306

5356

5406

5457

o

M

5507

5558

5608

5658

5709

5759

5809

5860

5910

5960

3

6011

6061

6111

6162

6212

6262

6313

6363

6413

6463

4

6514

6564

6614

6665

6715

6765

6815

6865

6916

6966

5

7016

7066

7116

7167

7217

7267

7317

7367

7418

7468

6

7518

7568

7618

7668

7718

7769

7819

7869

7919

7969

t A

7

8019

8069

8119

8169

8219 8209

8320 j 8370

8420

&470

50

8

8520

8570

8620

8670

8720

: 8770

8820 8870

8920

8970

9

9020

9070

9120

9170

9220

9270

9320

9309

9419

9469

870

9519

9569

9619

9669

9719

.9769

9819

9869

9918

9968

1

940018

0068

0118

0168

0218

0267

0317

0367

0417

0467

2

0516

0566

0616

0666

0716

0765

0815

0865

0915

0964

3 I 1014

1064

1114

1163

1213

1263

1313

1362

1412

1462

4

1511

1561

1611

1G60

1710

1760

1809

1859

1909

1958

5

2008

2058

2107

2157

2207

2256

2306

2355

2405

2455

6

2504

2£>54

2603

2653

2702

2752

2801

2851

2901

2950

7

3000

3049

3099

3148

3198

3247 3297

3346

asge

3445

8

3495

3544

3593

3643

3692

3742

3791

3841

3890

3939

9

3989

4038

4088

4137

4186

4236

4285

4335

4384

4433

880

4483

4532

4581

4631

4680

4729

4779

4828

4877

4927

1 1 4976

5025

5074

5124

5173

5222

5272

5321

5370

5419

2 5469

5518

5567

5616

5665

5715

5764

5813

5862

5912

3

5961

6010

6059

6108

6157

6207

6256

6305

6354

6403

4

6452

6501

6551

6600

6649

6698

6747

6796

6845

6894

5

6943

6992

7041

7090

7139

7189

7238

7287

7ase

7385

A f\

6

7434

7483

7532

7581

7630

7679

7728

7777

7826

7875

49

7

7924

7973

8022

8070

8119

8168

8217

8266

8315

8364

8

8413

8462

8511

8560

8608

8657

8706

8755

8804

8853

9

8902

8951

8999

9048

9097

9146

9195

9244

9292

9341

890

9390

9439

9488

9536

9585

9634

9683

9731

9780

9829

i

9878

9926

9975

J.

*J t \J

«7*7>v\J

t7t7 1 »J

0024

0073

0121

0170 0219

0267

0316

2

950365

0414

0462

0511

0560

0608

0657

0706

0754

0803

3

0851

0900

0949

0997

1046

1095

1143

1192

1240

1289

4

1.338

1386

1435

1483

1532

1580

1629

1677

1726

1775

5

1823

1872

1920

1969

2017

2066

2114

2163

2211

2260

6

2308

2356

2405

2453

2502

2550

2599

2647

2696

2744

7

2792

2841

2889

2938

2986

3034

3083

3131

3180

3228

8

3276

3325

3373

3421

3470

3518

3566

3615

3663

3711

9

3760

3808

3856

3905

3953

4001

4049

4098

4146

4194

PROPORTIONAL PARTS.

Dif£.

1

2

3 4

5

678

9

51

£• f\

5.1

10.2

15.3 20.4

25.5

30.6 35.7 40 8

45.9

50

j f\

5.0

10.0

15.0 20.0

25.0

30.0 &5.0 40.0

45.0

49

JO

4.9

9.8

14.7 19.6

24.5

29.4 34.3 39 2

44 1

48 | 4.8

9.6

14.4 19.2

24.0

28.8 33.6 38.4

43.2

100

TABLE IX. — LOGARITHMS OF NUMBERS.

No 900 L. 954.1 [No. 944 L. 975.

N.

0

1

2

3

4

5

6

1

8

9

Diff.

900

954243

4291

4339

4387

4435

4484

4532 1 4580

4628

4677

1

4725

4773

4821

4869

4918

4966

5014

5062

5110

5158

2

5207

5255

5303

5:351

5399

5447

5495

5543

5592

5640

3

5688

5736

5784

5832

5880

5928

5976

6024

6072

6120

4

6168

6216

6265

6313

6361

6409

6457

6505

6553

6601

AQ

5

6649

6697

6745

6793

6840

6888

6936

6984

7032

7080

4o

6

7128

7176

7224

7272

7320

7368

7416

7464

7512

7559

7

7607

7655

7703

7751

7799

7847

7894

7942

7990

8038

8

8086

8134

8181

8229

8277

8325

8373

8421

8468

8516

9

8564

8612

8659

8707

8755

8803

8850

8898

8946

8994

910

9041

9089

9137

9185

9232

9280

9328

9375

9423

9471

1

9518

9566

9614

9661

9709

9757

9804

9852

9900

9947

0

QQQK

4V

WJd

0042

0090

0138

0185

i 0233

0280

0328

0376

0423

3

960471

0518

0566

0613

0661

0709

0756

0804

0851

0899

4

0946

0994

1041

1089

1136

1184

1231

1279

1326

1374

5

1421

1469

1516

1563

1611

1658

1706

1753

1801

1848

6

1895

1943

1990

2038

2085

2132

2180

2227

2275

2322

7

2369

2417

2464

2511

2559

2606

2653

2701

2748

2795

8

2843

2890

2937

2985

3032

3079

3126

3174

3221

3268

9

3316

3363

3410

3457

3504

3552

3599

3646

3693

3741

920

3788

3835

3882

3929

3977

4024

4071

4118

4165

4212

1

4260

4307

4354

4401

4448

4495

4542

4590

4637

4684

2

4731

4778

4825

4872

4919

4966

5013

5061 5108

5155

3

5202

5249

5296

5:343

5390

5437

5484

5531

5578

5625

4

5672

5719

5766

5813

5860

5907

5954

6001

6048

6095

if

5

6142

6189

6236

6283

6329

6376

6423

6470

6517

6564

6

6611

6658

6705

6752

6799

6845

6892

6939

6986

7033

7

7080

7127

7173

7220

7267

7314

7361

7408

7454

7501

8

7548

7595

7642

7688

77:35

7782

7829

7875

7922

7969

9

8016

8062

8109

8156

8203

8249

8296

8:343

8390

8436

930

8483

8530

8576

8623

8670

8716

8763

8810

8856

8903

1

8950

8996

9043

9090

9136

9183

9229

9276

9323

9369

2

9416

9403

9509

9556

9602

9649

9695

9742

9789

9835

3

9882

9928

9975

00° 1

OOfiR

0114

0161

0°07 O9KJ.

0300

4

970347

0393

0440

\A/-w J.

0486

V/V/\JO

0533

V/A i^

0579

\J i\J A

0626

\Jm\J 1

0672

0719

0765

5

0812

0858

0904

0951

0997

1044

1090

1137

1183

1229

6

1276

1322

1369

1415

1461

1508

1554

1601

1647

1693

7

1740

1786

1832

1879

1925

1971

2018

2064

2110

2157

8

2203

2249

2295

2342

2388

2434

2481

2527

2573

2619

9

2666

2712

2758

2804

2851

2897

2943

2989

3035

3082

940

3128

3174

3220

3266

3313

3359

3405

3451

3497

3543

1

3590

3636

3682

3728

3774

3820

3866

3913

3959

4005

2

4051

4097

4143

4189

4235

4281

4327

4374

4420

4466

3

4512

4558 4604

4650

4696

4742

4788

4834

4880

4926

4

4972

5018

5064

5110

5156

5202

5248

5294

5340

5386

46

PROPORTIONAL PARTS.

Diff. 1

234

5

678

9

47 4.7

9.4 14.1 18.8

23.5

28.2 32.9 37.6

42.3

46 4.6

9.2 13.8 18.4

23.0

27.6 32.2 36.8

41.4

101

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 945 L. 975.] [No. 989 L. 995.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

945

975432

5478

5524

5570

5616

5662

5707

5753

5799

5845

6

5891

5937

5983

6029

6075

6121

6167

6212

6258

6304

7

6350

6396

6442

6488

6533

6579

6625

6671

6717

6763

8

6808

6854

6900

6946

6992

7037

7083

7129

7175

7220

9

7266

7312

7358

7403

7449

7495

7541

7586

7632

7678

950

7724

7769

7815

7861

7906

7952

7998

8043

8089

8135

1

8181

8226

8272

8317

8363

8409

8454

8500

8546

8591

o

M

8637

8683 8728

8774

8819

8865

8911

8956

9002

9047

3

9093

9138

9184

9230

9275

9321

9366

9412

9457

9503

4

9548

9594

9639

9685

9730

9776

9821

9867

9912

9958

5

980003

0049

0094

0140

0185

0231

0276

0322

0367 0412

6

C458

0503

0549

0594

0640

0685

0730

0776

0821 0867

7

0912

0957

1003

1048

1093

1139

1184

1229

1275 1320

8

1366

1411

1456

1501

1547

1592

1637

1683

1728 1773

9

1819

1864

1909

1954

2000

2045

2090

2135

2181

2226

960

2271

2316

2362

2407

2452

2497

2543

2588

2633

2678

1

2723

2769

2814

2859

2904

2949

2994

3040

3085

3130

2

3175

3220

3265

3310

3356

3401

3446

3491

3536

3581

3

3626

3671

3716

3762

3807

3852

3897

3942

3987

4032

4

4077

4122

4167

4212

4257

4302

4347

4392

4437

4482

5

4527

4572

4617

4662

4707

4752

4797

4842

4887

4932

45

6

4977

5022

5067

5112 5157

5202

5247

5292

5337

5382

7

5426

5471

5516

5561 5606

5651

5696

5741

5786

58:30

8

5875

5920

5965

6010

6055

6100

6144

6189

6234

6279

9

6324

6369

6413

6458

6503

6548

6593

6637

6682

6727

970

6772

6817

6861

6906

6951

6996

7040

7085

7130

7175

1

7219

7264

7;309

7353

7398

7443

7488

7532

7577

7622

2

7666

7711

7756

7800

7845

7890

7934

7979

8024

8068

3

8113

8157

8202

8247

8291

8336

8381

8425

8470

8514

4

8559

8604

8648

8693

8737

8782

8826

8871

8916

8960

5

9005

9049

9094

9138

9183

9227

9272

9316

9361

9405

6

9450

9494

9539

9583

9628

9672

9717

9761

9806

9850

7

nonp-

QQOfl

OflUQ

1

0028

0072

0117

0161

0206

0250

0294

8

990339

0383

0428

0472

0516

0561

0605

0650

0694

0738

9

0783

0827

0871

0916

0960

1004

1049

1093

1137

1182

980

1226

1270

1315

1359

1403

1448"

1492

1536

1580

1625

1

1669

1713

1758

1802

1846

1890

1935

1979

2023

2067

2

2111

2156

2200

2244

2288

2333

2377

2421

2465

2509

3

2554

2598

2642

2686

2730

2774

2819

2863

2907

2951

4

2995

3039

3083

3127

3172

3216

3260

3304

3:348

3392

5

3436

3480

3524

3568

3613

3657

3701

3745

3789

3833

6

3877

3921

3965

4009

4053

4097

4141

4185

4229

4273

7

4317

4361

4405

4449

4493

4537

4581

4625

4669

4713

44

8

4757

4801

4845

4889

41)33

4977

5021

5065

5108

5152

9

5196

5240

5284

5328

5372

5416

5460

55U4

5547

5591

PROPORTIONAL, FARTS.

Diff. 1

234

5

678

9

46 4.6

9.2 13.8 18.4

23.0

27.6 32.2 36.8

41.4

45 4.5

9.0 13.5 18.0

22.5

27.0 31.5 36.0

40.5

44 4.4

8.8 13.2 17.6

22.0

26.4 30.8 35.2

39.6

43 4.3

8.6 12.9 17.2

21.5

25.8 30.1 34.4

38.7

102

TABLE IX. — LOGARITHMS OF NUMBERS.

No. 990 L. 995.]

[No. 999 L. 999.

N.

0

1

2

3

4

5

6

7

8 9 Diff.

990

995635

5679

5723

5767

5811

5854

5898

5942

5986 6030

1

6074

6117

6161

6205

6249

6293

6337

b380

6424 6468

44

2

6512

6555

6599

6643

6687

6731

6774

6818

6862 6906

3

6949

6993

7037

7080

7124

r-
(

168

721 2

7255

7299 7343

4

7386

7430

7474

7517

7561

r
t

605

7648

7692

7736 7779

5

7823

7867

7910

7954

7998

8041

8085 8129

8172 8216

fi

8259

8303

8347

8390

8434

8477

8521

8564

8608 8652

7

8695

8739

8782

8826

8869

8913

8956

9000

9043 9087

8

9131

9174

9218

9261

9305

9348

9392

94X3

9479 9522

9

9565

9609

9652

9696

9739

9783

9826

98'

"0

9913 9957

43

LOGARITHMS OF NUMBERS

FROM 1 TO

100.

N.

Log.

N.

Log.

N.

Log.

N.

Log.

N.

Log.

1

0.000000

21

1.322219

41

1

.612784

61

1.785330

81

1.908485

2

0.301030

22

1.342423

42

1

.623249

62

1.

r92392

82

1.913814

3

0.477

121

23

1.361728

43

1

.633468

63

1.

r99341

83

1.919078

4

0.602060

24

1.380211

44

1

.643453

64

1.806180

84

1.924279

5

0.698970

25

1.397940

45

1

.653213

65

1.812913

85

1.929419

6

0.778151

26

1.414973

46

1

.662758

66

1.819544

86

1.934498

7

0.845098 ,

27

1.431364

47

1

.672098

67

1.826075

87

1.939519

8

0.903090

28

1.447158

48

1

.681241

68

1.832509

88

1.944483

9

0.954243

29

1.462398

49

1

.690196

69

1.838849

89

1.9-19390

10

1.000000

30

1.477121

50

1

.698970

70

1.845098

9J

1.954243

11

1.041393

31

1.491362

51

1

.707570

71

1.851258

91

1.959041

12

1.079181

32

1.505150

52

1

r«

. i

16003

72

1.857332

92

1.963788

13

1.113943

33

1.518514

53

1

.724276

73

1.86*323

93

1.968483

14

1.146128

34

1.531479

54

1

.732394

74

1.869232

94

1.973128

15

1.176091

35

1.544068

55

1

. i

40363

75

1.875061

95

1.977724

16

1.204120

36

1.556303

56

1

748188

76

1.880814

96

1.982271

17

1.230449

37

1.568202

57

1

.755875

77

1.886491

97

1.986772

18

1.255273

38

1.579784

58

1

.763428

78

1.892095

98

1.991226

19

1.278754

39

1.591065

59

1

.770852

79

1.897

627

99

1.995635

20

1.301030

40

1.602060

60

1

r»
. t

'"•O 1 *~ -t

• blol

80

1.903090

100 2.000000

(

Value
at 0°.

Sign
in 1st.
Quad.

Yalm
at 90°

, Sign
in2d
Quad.

Value
at

ISO0.

Sign
in 3d
Quad.

Value
at

270°

Sign
in 4th

Quad.

Value
at

360°.

Sin

R

_L

o

Tan

o

oo

o

i

00

o

Sec

R

QO

R

4

Versin

O

_i_

R

4-

2R

1

R

4

o

Cos

R

_ _

o

R

o

4.

Cot

00

o

oc

_L.

o

Cosec

00

R

4-

00

R signifies equal to rad ; oo signifies

infinite; O signifies evanescent.

103

TABLE X. — LOGARITHMIC SINES,

179<

II

/

Sine.

a-

-l

Tang.

Cotang.

q + l

Dl"

Cosine.

/

4.(

.85

15.314

0

0

Inf. neg.

575

1575

Inf. neg.

Inf. pos.

425

ten

60

60

1

6.463726

575

'575

6.463726

13.536274

425

ten

59

120

2

.764756

575

575

.764756

.235244

425

ten

58

180

3

6.940847

575

575

6.940847

13.059153

425

ten

57

240

4

7.065786

575

575

7.065786

12.934214

425

ten

56

300

5

.1U2696

575

575

.162696

.837304

425

ten

55

360

6

.241877

575

575

.241878

.758122

425

.02

f\f\

9.999999

54

420

7

.308824

575

575

.308825

.691175

425

.00

.999999

53

480

8

.366816

574

576

.366817

.633183

424

.00

.999999

52

540

9

.417968

574

576

.417970

.582030

424

.00

f\Ck

.999999

51

GOO

10

.463726

574

576

.463727

.536273

424

.02

.999998

50

660

11

7.505118

574

576

7.505120

12.494880

424

.00

i \ \

9.999998

49

720

12

.542906

574

577

.542909

.457091

423

.02

f\f\

.999997

48

780

13

.577668

574

577

.577672

.422328

423

.00

It.)

.999997

47

840

14

.609853

574

577

.609857

.390143

423

.02

f\f\

.999996

46

900

15

.639816

573

578

.639820

.360180

422

.00

.999996

45

960

16

.667845

573

578

.667849

.332151

422

.02

f\S\

.999995

44

1020

17

.694173

573

578

.694179

.305821

422

.00

f\C\

.999995

43

1080

18

.718997

573

579

.719003

.280997

421

.02

.999994

42

1140

19

.742478

573

579

.742484

.257516

421

.02

.999993

41

1200

20

.764754

572

580

.764761

.235239

420

.00

.999993

40

1260

21

7.785943

572

580

7.785951

12.214049

420

.C2

i 1. 1

9.999992

39

1320

22

.806146

572

581

.806155

.193845

419

.02

.999991

38

1380

23

.825451

572

581

.825460

.174540

419

.02

S\{\

.999990

37

1440

24

.843934

571

582

.843944

.156056

418

.02

f\S\

.999989

36

1500

25

.861662

571

583

.861674

.138326

417

.00

f\Cl

.999989

35

1560

26

.878695

571

583

.878708

.121292

417

.02

iki

.999988

34

1620

27

.895085

570

584

.895099

. 104901

416

.02

i~\ ~i

.999987

33

1680

28

.910879

570

584

.910894

.089106

416

.02

.999986

32

1740

29

.926119

570

585

.926134

.073866

415

.02

s\n

.999985

31

1800

30

.940842

569

586

.940858

.059142

414

.03

.999983

30

1860

31

7.955082

569

587

7.955100

12.044900

413

.02

iii

9.999982

29

1920

32

.968870

569

587

.968889

.031111

413

.02

.999981

28

1980

33

.982233

568

588

.982253

.017747

412

.02

.999980

27

2040

34

7.995198

568

589

7.995219

12.004781

411

.02

.999979

26

2100

35

8.007787

567

590

8.007809

11.992191

410

.03

.999977

25

2160

36

.020021

567

591

.020044

.979956

409

.02

y"vrt

.999976

24

2220

37

.031919

566

592

.031945

.968055

408

.02

f\Cl

.999975

23

2280

38

.043501

566

593

.043527

.956473

407

.03

.999973

22

2340

39

.054781

566

593

.054809

.945191

407

.02

i i »

.999972

21

2400

40

.065776

565

594

.065806

.934194

406

.02

.999971

20

2460

41

8.076500

565

595

8.076531

11.923469

405

.03

9.999969

19

2520

42

.086965

564

596

.086997

.913003

404

.02

/"YO

.999968

18

2580

43

.097183

564

598

.097217

.902783

402

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17

2640

44

.107167

563

599

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401

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16

2700

45

.116926

562

600

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400

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.999963

15

2760

46

.126471

562

601

.126510

.873490

399

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14

2820

47

.135810

561

602

.135851

.864149

398

.03

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13

2880

48

.144953

561

603

.144996

.855004

397

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i"\O

.999958

12

2940

49

.153907

560

604

.153952

.846048

396

.03

/"io

.999956

11

3000

50

.162681

560

605

.162727

.837273

395

.03

.999954

10

3060

51

8.171280

559

607

8.171328

11.828672

393

.03

/\o

9.999952

9

3120

52

.179713

558

608

.179763

.820237

392

.03

f\f\

.999950

8

3180

53

.187985

558

609

.188036

.811964

391

.03

/~>o

.999948

7

3240

54

.196102

557

611

.196156

.803844

389

.03

f\e\

.999946

6

3300

55

.204070

556

|612

.204126

.795874

388

.03

/\O

.999944

5

3360

56

.211895

556

613

.211953

.788047

387

.03

/\o

.999942

4

8120

57

.219581

555

615

.219641

.780359

385

.03

f\n

.999940

3

3480

58

.227134

554

616

.227195

.772805

384

.03

.999938

2

3.540

59

.234557

554

618

.234621

.765379

382

.03

.999936

1

3600

60

8.241855

553

619

8.241921

11.758079

381

.03

i

9.999934

0

4.<

185

15.314

//

i

Cosine.

9-

-I

Cotang.

Tang.

q + l

Dl"

Sine.

/

90°

104

89'

TABLE X. — LOGARITHMIC SIXES,

178°

/»

/

Sine.

q-l

Tang.

Cotang.

q + l

Dl'

Cosine.

/

4.685

15.314

3600

C

8.241855 i 553

619

8.241921

11.758079

381

r\t\

9.999934

60

3660

1

.249033

552

620

.249102

.750898

380

.Of>

f\f?

.999932

59

3720

2 .256094

551

622

.256165

. 743835

378

.999929

58

3780

3

.263042

551

623

.263115

.736885

377

•JS .999927

57

3840

4

.269881

550 -

625

.269956

.730044

375

•JS .999925

56

3900

5

.276614

549

627

.276691

.723309

373

.05

/\o

.999922

55

3960

6

.283243

548

628

.283323

.716677

372 -JS

.999920

54

4020

7

.289773

547

630

.289856

.710144

370 -J5

.999918

53

4080

8

.296207 546

632

.296292

.703708

368 $

.999915

52

4140

9

.302546 546

633

.302634

.697306

367 -X2

.999913

51

4200

10

.308794

545

635

.308884

.691116

365 -°°

.999910

50

4260

11

8.314954

544

637

8.315046

11.684954

363 -jj!>

9.999907

49

4320

12

.321027 543

638

.321122

.678878

o,>-> .0.3

oO/i ftp.

.999905

48

4380

13

.327016 ! 542

640

.327114

.672886

360 -JS

.999902

47

4440

14 | .332924 ! 541

642

.333025

.666975

358 -JS .999699

46

4500

15

.338753

540

644

.338856

.661144

356 '"2

.999897

45

4560

16

.344504

539

646

.344610

.655390

or i .05
o54 f.-

.999894

44

4620

17

.350181

539

648

.350289

.649711

352 'JS

.999891

43

4680

18

.355f83

538

649

.355895

.644105

or-. .OO

"51 ne;

.999888

42

4740

19

.361315

537

651

.361430

.638570

349 -JS

.999885

41

4800

20

.366777

536

653

.366895

.633105

347 -°°

.999882

40

4860

21

8.372171

535

655

8.372292

11.627708

•UK .05
o45 n-

9.999879

39

4920

22

.377499

534

657

.377622

.622378

343 • , -JS

.999876

38

4980

23

.382762

533

659

.382889

.617111

04-. .OO

o41 n~

.999873

37

5040

24

.387962

532

661

.388092

.611908

339 -JS

.999870

36

5100

25

.393101

531

663

.393234

.606766

337 ! -J2

.999867

35

5160

26

.398179

530

666

.398315

.601685

334 •)!-

.999864

34

5220

27

.403199

529

668

.403338

.596662

332

.uo

AK

.999861

33

5280

28

.408161

527

670

.408304

.591696

330

.05

f\t*r

.999858

32

5340

29

.413068

526'

672

.413213

.586787

328

.0*

f\K.

.999854

31

5400

30

.417919

525

674

.418068

.581932

326

.05

.999851

30

5460

31

8.422717

524

676

8.422869

11.577131

324

.05

rvy

9.999848

29

5520

32

.427462

523

679

.427618

.572382

321

.07

AE

.999844

28

5580

33

.432156

522

681

.432315

.567685

319

.05

f\K

.999841

27

5640

34

.436800

521

683

.436962

.563038

317

.05

/\ff

.999838

26

5700

35

.441394

520

685

.441560

.558440

315

.Ot

r\**

.999834

25

5760

36

.445941

518

688

.446110

.553890

312

.(Jo

r\ry

.999831 24

5820

37

.450440

517

690

.450613

.549387

310

.Oi
f\t'

.999827 ; 23

5880

38

.454893

516

693

.455070

.544930

307

.05

f\r~

.999824 22

5940

39

.459301

515

695

.459481

.540519

305

.Ol
f\r*

.999820 21

6000

40

.463665

514

697

.463849

.536151

303

.0<

.999816 20

6060

41

8.467985

512

700

8.468172

11.531828

300

.05

f\ff

9.999813

19

6120

42

.472263

511

702

.472454

.527546

298

,U7

/\r*

.999809 18

6180

43

.476498

510

705

.476693

.523307

295

.Oi

f\ty

.999805 17

6240

44

.480693

509

707

.480892

.519108

293

.Oi

f\r*

.999801 16

6300

45

.484848

50V

710

.485050

.514950

290

.Ot

f\K.

.94-9797 15

6360

46

.488963

506

713

.489170

.510830

287

.05

f\i*

.999794 14

6420

47

.493040

505

715

.493250

.506750

285

.Ot

Of

.999790 13

6480

48

.497078

503

718

.497293

.502707

282

t

r\r*

.999786 12

6540

49

.501080

502

720

.501298

.498702

280

.Oi

/-\r»

.999782 11

6600

50

.505045

501

723

.505267

.494733

277

.07

.999778 10

6660

51

8.508974

499

726

8.509200

11.490800

274

.07

no

9.999774

9

6720

52

.512867

498

729

.513098

.486902

271

.Oo

r\rt

.999769

8

6780

53

.516726

497

731

.516961

.483039

269

.Of
f\i*f

.999765

r*

1

6840

54

.520551

495

734

.520790

.479210

266

.o/

nr?

.999761

6

6900

55

.524343

494

737

524586

.475414

263

.07

f\ry

.999757

5

6960

56

.528102

492

740

.528349

.471651

260

.Oi

no

.999753

4

7020

57

.531828

491

743

.532080

.467920

257

.Oo

f\n

.999748

3

7080

58

.535523

490

745

.535779

.464221

255

.Oi

/\r*

.999744

2

7140

59

.539186

488

748

.539447

.460553

252

.Ol

f\Q

.999740

1

T200

60 8.542819

487 751

8.543084

11.456916

249

•Oo

9.999735

0

4.685

15.314

//

/

Cosine.

q-l

Cotang.

Tang.

q + l

Dl"

Sine.

'

105

COSINES, TANGENTS, AND COTANGENTS.

/

Sine.

D. r.

Cosine.

D. r.

Tang.

D. 1".

Cotang.

/

0

1

2
3
4
5
6

8 542319
.546422
.549995
.553539
.557054
.560540
.563999

60.05

59.55
59.07
58.58
58.10
57.65

K.*? 9fl

9.999735
.999731
.999726
.999722
.999717
.999713
.999708

.07

.08
.07
.08
.07
.08
07

8.5430S4
.546691
.550268
.553817
.557336
.560828
.564291

60.12
59.62
59.15

58.65
58.20
57.72
^7 *>7

11.456916
.453309
.449732
.446183
.442664
.439172
.435709

60
59
58
57
56
55
54

fV

i

8
9
10

.567431
.570836
.574214
.577566

Ol ./*l'

56.75
56.30
55.87
55.43

.999704
.999699
.999694
.999689

.08
.08
.08
.07

(r £*f*jr*c\**

. 5b i 4 2 t
.571137
.574520

C r*"**O^j»*'

.at <oi ,

56.83
56.38
55.95
55.52

.432273

.428863
.425480
.422123

53
52
51
50

11
12
13
14
15
16
17
18
19
20

8.580892
.584193
.587469
.590721
.593948
.597152
.600332
.603489
.606623
.609734

55.02
54.60
54.20
53.78
53.40
53.00
52.62
52.23
51.85
51.48

9.999685
.999680
.999675
.999670
.999665
.999660
.999655
.999650
.999645
.999640

.08
.08
.08
.08
.08
.08
.08
.08
.08
.08

8.581208
.584514
.587795
.591051
.594283
.597492
.600677
.603839
.606978
.610094

55.10
54.68
54.27
53.87
53.48
53.08
52.70
52.32
51.93
51.58

11 .418792
.415486
.412205
.408949
.405717
.402508
.399323
.396161
.393022
.389906

49

48
47
46
45
44
43
42
41
40

21
22

23
24
25
26
27
28
29
30

8.612823
.615891
.618937
.621962
.624965
.627948
.630911
.633854
.636776
.639680

51.13

50.77
50.42
50.05
49.72
49.38
49.05
48.70*
48.40
48.05

9.999635
.999629
.999624
.999619
.999614
.999608
.999603
.999597
.999592
.999586

.10

.08
.08
.08
.10
.08
.10
.08
.10
.08

8.613189
.616262
.619313
.622343
.625352
.628340
.631308
.6:34256
.637184
.640093

51.22
50.85
50.50
50.15
49.80
49.47
49.13
48.80
48.48
48.15

11.386811
.383738
.380687
.377657
.374648
.371660
.368692
.365744
.362816
.359907

39

38
37
36
35
34
33
32
31
30

31

8.642563

4r> r*»-

9.999581

•tn

8.642982

/C? OK

11.357018

29

32
33

.645428
.648274

* . < •)

47.43

.999575
.999570

. i(J
.08

.645853
.648704

4< .00

47.52

.354147
.351296

28
27

34
35
36
37

38
39
40

.651102
.653911
.656702
.659475
.662230
.664968
.667689

47.13

46.82
46.52
46.22
45.92
45.63
45.35
45.07

.999564
.999558
.999553
.999547
.999541
.999535
.999529

.10
.10
.08
.10
.10
.10
.10
.08

.651537
.654352
.657149
.659928
.662689
.665433
.668160

i .22
46.92
46.62
46.32
46.02
45.73
45.45
45.17

.348463
.345648
.342851
.340072
.337311
.334567
.331840

26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49

8.670393
.673080
. 675751
.678405
.681043
.6a3665
.686272
.688863
.6914^8

44.78
44.52
44.23
43.97
43.70
43.45
43.18
42.92

9.999524
.999518
.999512
.999506
.999500
.999493
.999487
.999481
.999475

.10
.10
.10
.10
.12
.10
.10
.10

8.670870
.673563
.676239
.678900
.681544
.684172
.686784
.689381
.691963

44.88
44.60
44.35
44.07
43.80
43.53
43.28
43.03

11.329130
.326437
.323761
.321100
.318456
.315828
.313216
.310619
.308037

19
18
17
16
15
14
13
12
11

50

.693998

42.67
42.42

.999469

.10
.10

.694529

42. ( i
42.53

.305471

10

51
52
53
54
55
56
57
58
59
60

8.696543
.699073
.701589
.704090
.706577
.709049
.711507
.713952
.716383
8.718800

42.17
41.93
41.68
41.45
41.20
40.97
40.75
40.52
40.28

9.999463
.999456
.999450
.fl!»9443
.999437
.999431
.999424
.999418
.999411
9.999404

.12
.10
.12
.10
.10
.12
.10
.12
.12

8.697081
.699617
.702139
.704646
.707140
.709618
.712083
.714534
.716972
8.719396

42.27
42.03
41.78
41.57
41.30
41.08
40.85
40.63
40.40

11.302919
.300383
.297861
.295354
.292860
.290382
.287917
.285466
.283028
11.280604

9
8
7
6
5
4
3
2
1
0

r

Cog;^e.

D. 1 .

Sine.

D. 1". 1

Cotang. D. 1". Tang.

r

92'

106

87'

TABLE X. — LOGARITHMIC SIXES,

176°

/

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. r.

Cotang.

/

0

1

2
3

4
5
6
7
8
9

8.718800
.721204
.723595
.725972
.728337
.730688
.733027
.735354
.737667
.739969

40.07

39.85
39.62
39.42
39.18
38.98
38.78
38.55
38.37

OQ -jr*

9.999404
.999398
.999391
.999384
.999378
.999371
.999364
.999357
.999350
.999343

.10
.12
.12
.10
.12
.12
.12
.12
.12
19

8.719396
.721806
.724204
.726588
.728959
.731317
.733663
.735996
.738317
.740626

40.17
39.97
39.73
39.52
39.30
39.10
38.88
38.68
38.48

oc 07

11.280604
.278194
.275796
.273412
.271041
.26WJS3
.266337
.264004
.261683
.259374

60
59
58
57
56
55
54
53
52
51

10

.742259

37.95

,999336

.12

.742922

38.08

.257078

50

11

8.744536

37 77

9.999329

to

8.745207

07 07

11.254793

49

12
13

14
15
16
17
18
19
20

.746802
.749055
.751297
.753528
.755747
.757955
.760151
.762337
.764511

1 .i 1

37.55
37.37
37.18
36.98
36.80
36.60
36.43
36.23
36.07

.999322
.999315
.999308
.999301
999294
.999287
.999279
.999272
.999265

.12
.12
.12
.12
.12
.13
.12
.12
.13

.747479
.749740
.751989
.754227
.756453
.758668
.760872
.763065
.765246

01 .o<

37.68
37.48
37.30
37.10
36.92
36.73
36.55
36.35
36.18

.252521
.250260
.248011
.245773
.243547
.241332
.239128
.2369:35
.234754

48
47
46
45
44
43
42
41
40

21
22
23
24
25

8.766675
.768828
.770970
.773101
.775223

35.88
35.70
35.52
35.37

QK: l-»

9.999257
.999250
.999242
.999235
.999227

.12
.13

.12

.13
10

8.767417
.769578
.771727
.773866
.775995

36.02
35.82
35.65

35.48
Q=; QO

11.232583
.230422
.228273
.226134
.224005

39

38
371
36'
35'

26
27
28
29
30

. i 77333
.779434
.781524
.783605
.785675

OO. 1 l

35.02
34.83
34.68
34.50
34.35

.999220
.999212
.999205
.999197
.999189

. JL«

.13
.12
.13
.13
.13

.778114
.780222
.782320
.784408
.786486

or) . o.v
35.13
34.97
34.80
34.63
34.47

.221886
.219778
.217680
.215592
.213514

34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

8.787736
.789787
.791828
.793859
.795881
.797894
.799897
.801892
.803876
.805852

34.18
34.02
33. 85
33.70
33.55
33.38
33.25
33.07
32.93
32.78

9.999181
.999174
.999166
.999158
.999150
.999142
.9991:34
.999126
.999118
.999110

.12
.13
.13
.13
.13
.13
.13
.13
.13
.13

8.788554
.790613
.792662
.794701
.796731
.798752
.800763
.802765
.804758
.806742

34.32
34.15
33.98

33. as

33.68
33.52
33.37"
33.22
33.07
32.92

11.211446
.209387
.207338
.205299
.203269
.201248
.199237
.197235
.195242
.193258

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47

8.807819
.809777
.811726
.813667
.815599
.817522
.819436

32.63

32.48
32.35
32.20
32.05
31.90

Q1 70

9.999102
.999094
.999086
.999077
.999069
.999061
.999053

.13
.13
.15
.13

.13
.13

1^

8.808717
.810683
.812641
.814589
.816529
.818401
.820384

32 7.7
32.03
32.47
32.33
32.20
32.05

01 Of)

11.191283

.189317
.187359
.185411
.183471
.181539
.179616

19
18
17
16
15
14
13

48
49
50

.821343
.823240
.825130

31.62
31.50
31.35

.9.99044
.999030
.999027

.13
.1.-)
.13

.822298
'.824205
.826103

31.78
31.63
31.48

.177702
.175795
.173897

12
11
10

51
52
53
54

8.827011
.828884
.830749
.832607

31.22
31.08
30.97

or* oo

9.999019
.999010
.999002
.998993

.15
.13
.15

8.827092
.829S74
.831748
.833613

31.37
31.23
31.08

11.172008
.170126
.168252
.166387

9
8

7
6

55
56
57
58
59
60

.834456
.836297
.838130
.839956
.841774
8.843585

O(J.O~

30.68
30.55
30.43
30.30
30.18

.998984
.11! 18976
.998967
.998958
.998950
9.998941

.10
.13
.15
.15
.13
.15

.835471
.837321
.839163
.840998
.842825
8.844644

oU . 9 1
30.83
30.70
30.58
30.45
30.32

.164529
.162679
.160837
.159002
.157175
11.155356

5
4
3
2

1
0

/

Cosine. D 1".

Sine.

D. 1'.

Cotang.

D. 1'. Tang.

/

107

86-

COSINES, TANGENTS, AND COTANGENTS.

'

Sine.

D. 1'.

Cosine.

D. r.

Tang.

D. 1".

Cotang.

'

0

1

2
3
4

8.843585
.845387
.847183
.848971
.850751

30.03
29.93
29.80
29.67

on ^fy

9.998941
.998932
.998923
.998914
.998905

.15
.15
.15

.15

1 t

8.R44644
.846455
.848260
.850057
.851H46

30.18
30.08
29.95
29.82

on '"'A

11.: fift356
. 153545
.151740
.149943
.148154

60

59
58
57
56

5
6
i

8
9
10

.852525
.854291
.856049
.857801
.859546
.861283

29 . o7
29.43
29.30
29.20
29.08
28.95
28.85

.998896
.998887
.998878
.998869
.998860
.998851

.15
.15
.15
.15
.15
.15
.17

.853628
.855403
.857171
.858932
.860686
.862433

29. (0
29.58
29.47
29.35
29.23
29.12
29.00

.146372
.144597
.142829
.141068
.139314
.137567

55
54
53
52
51
50

11

8.863014

4 i.i i**o

9.998841

1 K

8.864173

OQ

11.135827

49

12
13
14
15
16
17
18
19
20

.864738
.866455
.868165
.869868
.871565
.873255
.874938
.876615
.878285

28.73
28.62
28.50
28.38
28.28
28.17
28.05
27.95
27 .-83
27.73

.998832
.998823
.998813
.998804
.998795
.998785
.998776
.998766
.998757

.lo
.15
.17
.15
.15
.17
.15
.17
.15
.17

.865906
.867632
.869351
.871064
.872770
.874469
.876162
.877849
.879529

28^77
28.65
28.55
28.43
28.32
28.22
28.12
28.00
27.88

.134094
.132368
.130649
.128936
.127230
.125531
.123838
.122151
.120471

48
47
46
45
44
43
42
41
40

21

8.879949

^

9.998747

1 K

8.881202

O'*' '"'O

11.118798

39

22

.881607

2i .63

.998738

.lo

.882869

6t . *O
O7 fift

.117131 38

23

.883258

9~ iO

.998728

I*--

.884530

*• t . UO

.115470 37

24

.884903

0~ 90

.998718

.1 (
17

.886185

97 47

.113815 i 36

25

.886542

*V 1 . f J/W

Or* c\f\

.998708

. ( 1
•\ ».

.887833

*£< .rci

.112167 ; 35

26
27

.888174
.889801

2i .20
27.12

o^* r\f\

.998699
.998689

.la

.17

Irv

.889476
.891112

27^27

Of 17

.110524
.108888

34
33

28

.891421

2i .00

OC f\f\

.998679

t
1r*

.892742

tit . i 1

of A7

.107258

32

29
30

.893035
.894643

26.90
26.80
26.72

.998669
.998659

t

.17

.17

.894366
.895984

~i .Ul

26.97
26.87

.105634
.104016

31
30

31

32
33
34
35
36
37
38
39
40

8.896246
.897842
.899432
.901017
.902596
.904169
.905736
.907297
.908853
.910404

26.60
26.50
26.42
26.32
26.22
. 26.12
26.02
25.93
25.85
25.75

9.998649
.998639
.998629
.998619
.998609
.998599
.998589
.998578
.998568
.998558

.17
.17
.17
.17

.17
.17
.18
.17
.17
.17

8.897596
.899203
.900803
.902398
.903987
.905570
.907147
.908719
.910285
.911846

26.78
26.67
26.58
26.48
26.38
26.28
26.20
26.10
26.02
25.92

11.102404
.100797
.099197
.097602
.096013
.094430
.092853
.091281
.089715
.088154

29
28
27
26
25
24
23
22
21
20

41
42
43
44

8.911949
.913488
.915022
.916550

25.65
25.57
25.47

OK. OQ

9.998548
.998537
.998527
.998516

.18
.17
.18

]r*

8.913401
.914951
.916495
.918034

25.83
25.73
25.63

11.086599
.085049
.083505
.081966

19
18
17
16

45
46
47
48
49
50

.918073
.919591
.921103
.922610
.924112
.925609

25 . oo
25.30
25.20
25.12
25.03
24.95
24.85

.998506
.998495
.998485
.998474
.998464
.998453

i

.18
.17
.18
.17
.18
.18

.919568
.921096
.922619
.924136
.925649
.927156

25 '.47
25.38
25.28
25.22
25.12
25.03

.080432
.078904
.077381
.075864
.074351
.072844

15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

8.927100
.928587
.930068
.931544
.933015
.934481
.935942
.937398
.938850
8.940296

24.78
24.68
24.60
24.52
24.43
24.35
24.27
24^20
24.10

9.99R442
.998431
.998421
.998410
.998399
.998388
.998377
.998366
.998355
9.998344

.18
.17
.18
.18
.18
.18
.18
.18
.18

8.928658
.930155
.931647
.933134
.934616
.936093
.937565
.939032
.940494
8.941952

24.95
24.87
24.78
24.70
24.62
24.53
24.45
24.37
24.30

11.071342
.069845
.068353
.066866
.065384
.063907
.062435
.060968
.059506
11.058048

9

8
7
6
5

3"
2
1
0

'

Cosine.

D. r.

Sine.

D. 1".

Cotang. D. 1".

Tang. '

108

TABLE X. — LOGARITHMIC SINES,

174°

/

Sine.

D. r.

Cosine.

D. r.

Tang.

D. 1'.

Cotang.

/

0

1

2
3
4
5
6

rr

i

8
9
10

8.940296
.941738
.943174
.944606
.946034
.947456
.948874
.950287
.951696
.953100
.954499

24.03
23.93
23.87
23.80
23.70 |
23.63
23.55
23.48
23.40
23.32
23.25

9.998344
.998333
.998322
.998311
.998300
.998289
.998277
.998266
.998255
.998243
.998232

.18
.18
.18
.18
.18
.20
.18
.18
.20
.18
.20

8.941952
.943404
.944852
.946295
.947734
.949168
.950597
.952021
.953441
.954856
.956267

24. 2P
24.13
24.05
23.98
23.90
23.82
23.73
23.67
23.58
23.52
23.45

11.058048
.056596
.055148
.053705
.052266
.050832
.049403
.047979
.046559
.045144
.043733

60
59
58
57
56
55
54
53
52
51
50

11
12
13
14
15
16
17
18
19
20

8.955894
.957284
.958670
.960052
.961429
.962801
.964170
.965534
.966893
.968249

23.17
23.10
23.03
22.95
22.87
22.82
22.73
22.65
22.60
22.52

9.998220
.998209
.998197
.998186
.998174
.998163
.998151
.998139
.998128
.998116

.18
.20
.18
.20
.18
.20
.20
.18
.20
.20

8.957674
.959075
.960473
.961866
.963255
.964639
.966019
.967394
.968766
.970133

23.35
23.30
23.22
23.15
23.07
23.00
22.92
22.87
22.78
22.72

11.042326
.040925
.039527
.038134
.036745
.035361
.033981
.032606
.031234
.029867

49
48
47
46
45
44
43
42
41
40

21
22
23
24

8.969600
.970947
.972289
.973628

22.45

22.37
22.32

OO OQ

9.998104
.998092
.998080
.998068

.20

.20

.20
on

8.971496
.972855
.974209
.975560

22.65
22.57
22.52
22 4'3

11.028504
.027145
.025791
.024440

39
38
37
36

25
26
27
28
29
30

.974962
.976293
.977619
.978941
.980259
.981573

44. 4O

22.18
22.10
22.03
21.97
21.90
21.83

.998056
.998044
.998032
.998020
.998008
.997996

. **\j
.20
.20
.20
.20
.20
.20

.976906
.978248
.979586
.980921
.982251
.983577

/W^f . TO

22.37
22.30
22.25
22.17
22.10
22.03

.023094
.021752
.020414
.019079
.017749
.016423

35
34
33
32
31
30

31

8.982883

O1 I**1**

9.997984

on

8.984899

21 Q7

11.015101

29

32

.984189

4l.it
O1 *"*O

.997972

. 4U

90

.986217

«wl . t7 t

91 Q9

.013783

28

33
34

.985491
.986789

- J . i55

21.63

01 t^""*

.997959
.997947

. ^'V

.20
on

.987532

.988842

/v-L . &•*

21.83

91 7)3

.012468
.011158

27
26

35
36
37

38
39

.988083
.989374
.990660
.991943
.993222

41 .5<

21.52

21.43
21.38

21.32
01 o*c

.997935
.997922
.997910
.997897
.997885

. -wU

.22
.20
.22

.20
09

.990149
.991451
.992750
.994045
.995337

*vl . ( O

21.70
21.65
21.58
21.53

91 4R;

.009851
.008549
.007250
.005955
.004663

25
24
23
22
21

40

.994497

41 .43

21.18

.997872

!20

.996U24

s. 1 . ~t- )

21.40

.00:3376

20

41

8.995768

O1 1 <5

0.997860

99

8.997908

91 QQ

11.002092

19

42
43

.997036
.998299

-il . lO

21.05

O1 i I- 1

.997847
.997835

. &G

.20

OO

8.999188
9.000465

£1 . oo

21.28

91 99

11.000812
10.999535

18
17

44
45
46
47
48
49
50

8.999560
9.000816
.002069
.003318
.004563
.005805
.007044

y&l .\T&

20.93
20 88
20.82
20.75
20.70
20.05
20.57

.997822
.997809
.997797
.997784
.997771
.997758
.997745

. •<-<•£

.22
.20
.22
.22
.22
.22
.22

.001738
.003007
.004272
.005534
.006792
.008047
.009298

-- l . &&

21.15
21.08
21.03
20.97
20.92
20.85
20.80

.998262
.996993
.995728
.994466
.993208
.991953
.990702

16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

9.008278
.009510
.010737
.011962
.013182
.014400
.015613
.016824
.018031
9.019235

20 .53
20.45
20.42
20.33
20.30
20.22
20.18
20.12
20.07

9.997732
.997719
.997706
.997693
.997680
.097667
.997654
.997641
.997628
9.997614

.22
.22
.22
.22
.22
.22
.22
.22
.23

9.010546
.011790
.013031
.014268
.015502
.016732
.017959
.019183
.020403
9.021620

20.73
20.68
20.62
20 57
20.50
20.45
20.40
20.33
20.28

10.989454
.988210
.986969
.985732
.984498
.983268
.982041
.980817
.979597
10.978380

9

8
7
6
5
4
3
2
1
0

/

Cosine.

D. r.

: Sine.

D. 1'.

Cotang.

D. r.

Tang.

i

84*

109

COSINES, TANGENTS, AND COTANGENTS.

173'

1

'

Sine.

D. r.

Cosine.

D. r.

Tang.

D. r.

Cotang.

'

0

1

2
3
4
5
6
7
8
9

9.019235
.020435
.021632
.022825
.024016
.025203
.026386
.027567
.028744
.029918

20.00
19.95
19.88
19.85
19.78
19.72
19.68 ]
19.62
19.57

1Q V>

9.997614
.997601
. 997588
.997574
.997561
.997547
.997534
.997520
.997507
.997493

.22

00

!23
.22
.23
.22
.23
.22
.23
22

9.021620

! 024044
.025251
.026455
.027655
.028852
.030046
.031237
.032425

20.23
20.17
20.12
20.07
20.00
19.95
19.90
19.85

19.80
10 ^-i

10.978380
.977166
.975956
.974749
.973545
.972345
.971148
.969954
.968763
.967575

60
59

58
57
56
55
54
53
52
51

10

.031089

J «7 *J**

19.47

.997480

!23

.033609

1J . Id

19.70

.966391

50

11
12
13
14
15
16

9.032257
.033421
.034582
.035741
.036896
.038048

19.40
19.35
19.32
19.25
19.20

1Q 1^

9.997466
.997452
.997439
.997425
.997411
.997397

.23
.22
.23

.23
.23

9.034791
.035969
.037144
.038316
.039485
.040651

19.63
19.58
19.53
19.48
19.43

~l O Q1"*

10.965209
.964031
.962856
.961684
.960515
.959349

49

48
47
46
45
44

17

18
19
20

.039197
.040342
.041485
.042625

iy . 10
19.08
19.05
19.00
18.95

.997383
.997369
.997355
.997341

!23
.23
.23
.23

.041813
.042973
.044130
.045284

ly.oY
19.33
19.28
19.23

19.17

.958187
.957027
.955870
.954716

43

42
41
40

21

9.043762

18 88

9.997327

23

9.046434

10 1^

10.953566

39

22
23
24
25
26

.044895
.046026
.047154
.048279
.049400

J.fj . OO

18.85
18.80
18.75
18.68

-IQ flX

.997313
.997299
.997285
.997271
. 99725 r

!23
23
!23
.23

OK

.047582
.048727
.049869
.051008
.052144

J. \) . 1 0

19.08
19.03
18.98
18.93

1Q Qfl

.952418
.951273
.950131
.948992
.947856

38
37
36
35
34

27
28
29
30

.050519
.051635
.052749
.053859

JO . CM

18.60
18.57 .
18.50
18.45

.997242
.997228
.997214
.997199

. -JO

.23
23
!25
.23

.053277
.054407
.055535
.056659

lo.oo
18 83
18.80
18.73
18.70

.946723
.945593
.944465
.943341

33
32
31
30

31
32
33
34
35
36
37
38
39
40

9.054966
.056071
.057172
.058271
.059367
.060460
.061551
.062639
.063724
.064806

18.42
18.35
18.32
18.27
18.22
18.18
18.13
18.08
18.03
17.98

9.997185
.997170
.997156
.997141
.997127
.997112
.997098
.997083
.997068
.997053

.25
.23
.25
.23
.25
.23
.25
.25
.25
.23

9.057781
.058900
.060016
.061130
.062240
.063348
.064453
.065556
.066655
.067752

18.65
18.60
18.57
18.50
18.47
18.42
18.38
18.32
18.28
18.25

10.942219
.941100
.939984
.938870
.937760
.936652
.935547
.934444
.933345
.932248

29
28
27
26
25
24
23
22
21
20

41
42
43
44

9.065885
.066962
.068036
.069107

17.95

17.90 ,
17.85

17 82

9.997039
.997024
.997009
.996994

25

!25
.25

9.068846
.069938
.071027
.072113

18.20
18.15
18.10

1 D A1**

10.931154

.930062
.928973
.927887

19
18
17
16

45

.070176

i I . O-6

17 77

.996979

or

.073197

lo.Ui

1 O /"ifc~J

.926803

15

46
47

.071242
.072306

it.it

17.73

17 f\7

.996964
.996949

!25
tjfs

.074278
.075356

18.02
17.97

I1** no

.925722
.924644

14
13

48

.073366

1 f .Of

17 fi^

.996934

OK

.076432

< . 9o

1r- OQ

.923568

12

49

.074424

1 i . UO

.996919

2e

.077505

( .OO
1>~ O"

.922495

11

50

.075480

17 '. 55

.996904

D

.25

.078576

1 .8D

17.80

.921424

10

51

9.076533

17 P;O

9.996889

ott

9.079644

10.920356

9

52

.077583

1 t . OU
17 Af

.996874

.080710

17.77

1""* r^fl

.919290

8

53

.078631

I I .1|
17 /1O

.996858

Of

.081773

i .72

-, r. /»r»

.918227

7

54

.079676

i 1 . *±A
1r* OO

.996843

.x£j

.082833

17. oi

1r~ fn

.917167

6

55

.080719

< .OO
17 Q°.

.996828

Of

.083891

i .bo

1r* nf\

.916109

5

56

.081759

i / . oo

.996812

,6i

.084947

i .bO

If** C C

.915053

4

57
58

.082797
.083832

lf.25
1 T ^0

.996797
.996782

!25

O'**

.086000
.087050

i .55
17.50

1r* Afi

.914000
.912950

3
2

59
60

.084864
9.085894

IT! i?

.996766
9.996751

!25

.088098
9.089144

4 .47
17.43

.911902
10.910856

1
0

'

Cosine.

D. 1".

Sine.

D. 1'.

Cotaug.

D. 1".

Tang.

'

110

TABLE X. — LOGARITHMIC SINES,

172°

/

Sine.

D. 1'.

Cosine.

IX 1".

Tang.

D. r.

Cotang.

/

0

9 085894

17 -10

9.996751

O7

9.089144

-(7 QQ

10.910856

60

1

.086922

1 t . Id :
1— rio

.1)96735

-- 1
OK

.090187

1 * . OO
1 " QK

.909813

59

2

.087947

i .Uo

1*** f\K

.996720

,<6D

O'"*

.091228

1 1 . OO
1r* t\f\

.908772

58

3
4
5

.088970
.089990
.091008

i .Oo
17.00
16.97

•(« Q'J

.996704
.996688
.996073

.2i
.27
.25
27

.092266
.093302
.094336

i .6(1
17.27
17.23

17 18

.9077.34
.906698
.905664

57
56
55

6

.092024

10 . yo

-IK QQ

.996657

• <GJ

<>7

.095367

.1 i . JO

17 1S

.904633

54

7
8
9

.093037
.094047
.095056

1O . OO

16.83
16.82

IK 77

.996641
.996625
.996610

. **t

.27
.25

O7

.096395
.097422
.098446

11 .lo

17.12
17.07

17 flQ

.903605
.902578
.901554

53
52
51

10

.096062

10. t i

16.72

.996594

.lit

.27

.099468

1 1 .Uo

16.98

.900532

50

11
12
13
14
15
16

9.097065
.098066
.099065
.100062
.101056
.102048

16.68
16.65
16.62
16.57
16.53

-f s* A Q i

9.996578
.996562
.996546
.996530
.996514
. 996498

.27
.27
.27
.27
.27

O^*

9.100487
.101504
.102519
.103532
.104542
. 105550

16.95
16.92
16.88
16.83
16.80

1 i? r""*

10.899513
.898496
.897481
.896468
.895458
.894450

49

48
47
46
45
44

17
18
19
20

.103037
.104025
.105010
.105992

15.48 !
16.47
16.42 !
16.37
16.35

.996482
.996465
.996449
.996433

.2<
.28
.27
.27
.27

.106556
.107559
.108560
.109559

lt>. 7*
16.72
16.68
16.65
16.62

.893444
.892441
.891440
.890441

43
42
41
40

21

22
23
24
25
26
27
28
29
30

9.106973
.107951
. 108927
. 109901
.110873
.111842
.112809
.113774
.114737
.115698

16.30
16.27
16.23
16.20 •
16.15
16.12
16.08
16.05
16.02
15.97

9.996417
.996400
.996384
.996368
.996351
.996335
.996318
.996302
.996285
.996269

.28
.27

.27
.28
.27
.28
.27
.28
.27
.28

9.110556
.111551
.112543
.113533
.114521
.115507
.116491
.117472
.118452
.119429

16.58
16.53
16.50
16.47
16.43
16.40
16.35
16.33
16.28
16.25

10.889444
.888449
.887457
.886467
.885479
.884493
.883509
.882528
.881548
.880571

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9 116656
.117613
118567
.119519
.120469
.121417
.122362
. 123306
.124248
.125187

15.95
15.90
15.87
15.83
15.80
15.75
15.73
15.70
15.65
15.63

9.996252
.996235
. 996219
.996202
.996185
.996168
.996151
.996134
.996117
.996100

.28
.27

.28
.28
.28
.28
.28
.28
.28
.28

9.120404
.121377
. 122:348
.123317
.124284
.125249
.126211
.127172
.128130
.129087

16.22
16.18
16.15
16.12
16.08
16.03
16 02
15.97
15 95
15.90

10.879596
.878623
.877652
.876683
.875716
.874751
.873789
.872828
.871870
.870913

29

28
27
26
25
24
23
22
21
20

41
42

9 126125
.127060

15.58

1 ^ ftX

9.996083
.996066

.28

OQ

9.130041
.130994

15.88

-IK QO

10.869959
.869006

19
18

43
44
45
46
47
48
49
50

.127993
.128925
.129854
130781
.131706
132630
.133551
.134470

ID. DO

15 53
15.48
15 45
15.42
15.40
15 35
15.32
15.28

.996049
.996032
.996015
.995998
.995980
. 995963
.995946
.995928

.~O

.28
.28
.28
.30
.28
.28
.30
.28

.131944
.132893
. 133839
.134784
.135726
.136667
.137605
.138542

1O . oo

15.82
15.77
15.75
15.70
15.68
15 63
15 62
15.57

.868056
.867107
.866161
.865216
.864274
863333
.862395
.861458

17
16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

9 135387
.136303
.137216
.138128
139037
139944
140850
.141754
142655
3 143555

15 27
15 22
15.20
15 15
15 12
15 10
15 07
15.02
15.00

3 995911
.995894
.995876
.995859
.995841
995823
.995806
.995788
995771
9.995753

.28
.30
.28
30
.30
.28
.30
28
.30

9 139476
.140409
.141340
.142269
143196
.144121
.145044
.145966
.146885
9.147803

15.55
15 52
15.48
15 45
15.42
15.38
15 37
15.32
15.30

10 860524
.859591
.858660
.857731
.856804
.855879
.854956
.854034
.853115
10.852197

9
8
7
6
5
4
3
2
1
0

/

Cosine.

D. r.

Sine.

D.I'.

; Cotang.

D. 1".

Tang.

/

97'

111

82"

COSINES, TANGENTS, AND COTANGENTS.

'

Sine.

D. 1".

Cosine.

D. 1'.

Tang.

D. 1".

Cotang.

'

0

1

2
3
4
5
6
7
8
9
10

9.143555
.144453
. 145349
.146243
.147136
.148026
.148915
. 149802
. 150686
.151569
.152451

14.97
14.93
14.90
14.88
14.83
14.82
14.78
14.73
14.72
14.70
14.65

9.995753
.995735
.995717
.995699
.995681
.995664
.995646
.995628
.995610
.995591
.995573

.30
.30
.30
.30
.28
.30
.30
.30
.32
.30
.30

9.147803
.148718
.149632
.150544
.151454
.152363
.153269
.154174
.155077
.155978
.156877

15.25
15.23
15.20
15.17
15.15
15.10
15.08
15.05
15.02
14.98
14.97

10.852197
.851282
.850368
.849456
.848546
.847637
.846731
.845826
.844923
.844022
.843123

60
59
58
57
56
55
54
53
52
51
50

11
12
13
14
15
16
17
18
19
20

9.153330
.154208
.155083
.155957
.156830
.157700
.158569
.159435
.160301
.161164

14.63
14.58
14.57
14.55
14.50
14.48
14.43
14.43
14.38
14.35

9.995555
.995537
.995519
.995501
.995482
.995464
.995446
.995427
.995409
.995390

.30
.30
.30
.32
.30
.30
.32
.30
.32
.30

9.157775
.158671
.159565
.160457
.161:347
.162236
.163123
.164008
.164892
.165774

14.93
14.90
14.87
14.83
14.82
14.78
14.75
14.73
14.70
14.67

10.842225
.841329
.840435
.839543
.838653
.837764
.836877
.835992
.835108
.834226

49
48
47
46
45
44
43
42
41
40

21
22
23
24
25
26
27
28
29
30

9.162025

.162885
.163743
.164600
.165454
.166307
.167159
.168008
.168856
.169702

14.33
14.30
14.28
14.23
14.22
14.20
14.15
14.13
14.10
14.08

9.995372
.995353
.995334
.995316
.995297
.995278
.995260
.995241
.995222
.995203

.32
.32
.30
.32
.32
.30
.32
.32
.32
.32

9.166654
.167532
.168409
.169284
.170157
.171029
.171899
.172767
.173634
.174499

14.63
14.62
14.58
14.55
14.53
14.50
14.47
14.45
14.42
14.38

10.833346
.832468
.831591
.830716
.829843
.828971
.828101
.827233
.826366
.825501

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9.170547
.171389
.172230
.173070
.173908
.174744
.175578
.176411
.177242
. 178072

14.03
14.02
14.00
13.97
13.93
13.90
13.88
13 85
13.83
13.80

9.995184
.995165
. 995146
.995127
.995108
.995089
.995070
.995051
.995032
.995013

.32

.32
.32
.32
.32
.32
.32
.32
.32
.33

9.175362
.176224
.177084
.177942
.178799
. 179655
.180508
.181360
.182211
.183059

14.37
14.33
14.30
14.28
14 27
14.22
14.20
14.18
14.13
14.13

10.824638
.823776
.822916
.822058
.821201
.820345
.819492
.818640
.817789
.816941

29
28
27
26
25
24
23
22
21
20

41

9.178900

iq 77

9.994993

0.0

9.183907

1.1 OS

10.816093

19

42

.179726

.994974

QO

.184752

.815248

18

43
44
45
46
47
48
49
50

.180551
.181374
.182196
.183016
. 183834
.184651
.185466
.186280

16. <5
13.72
13.70
13.67
13.63
13.62
13.58
13.57
13.53

.994955
.994935
.994916
.994896
.994877
.994857
.994838
.994818

.&£

.33
.32
.33
.32
.33
.32
.33
.33

.185597
.186439
.187280
.188120
.188958
.189794
.190629
.191462

14. U8
14.03
14.02
14.00
13.97
13.93
13.92
13.88
13.87

.814403
.813561
.812720
.811880
.811042
.810206
.809371
.808538

17
16
15
14
13
12
11
10

51
52
53
54

9.187092
.187903
.188712
.189519

13.52 ;
13.48
13.45

9.994798
.994779
.994759
.994739

.32

.33
.33

9.192294
.193124
.193953
.194780

13.83
13.82
13.78

10.807706

.806876
.806047
.805220

9

8
7
6

55
56
57

58

.190325
.191130
.191933
192734

13. 43
13.42
13.38
13 .35
1 3 33

.994720
.994700
.994680
. 994660

.32
.33
.33
.33

QQ

.195606
.196430
.197253

.198074

Id. ii
13.73
13.72
13.68

iq «7

.804394
.803570
.802747
.801926

5
4
3
2

59
60

.1935:34
9.194332

13.30

.994640
9.994620

.33

.198894
9.199713

13.65

.801106
10.800287

1
0

'

Cosine.

D. 1'.

Sine.

D. 1". 1

Cotang.

D. 1'.

Tang.

'

112

81'

TABLE X. — LOGARITHMIC SINES,

170C

/

Sine.

D. r.

Cosine.

D. 1'.

Tang.

D. r.

Cotang.

/

0

1

2
3
4
5
6
7
8
9
10

9.194332
.195129
.195925
.19(5719
.197511
.198302
.199091
.199879
.200606
.201451
.202234

13.28
13.27
13.23
13.20
13.18
13.15
13.13
13.12
13.08
13.05
13.05

9.994620
.994000
.994580
.994500
.994540
.994519
.994499
.994479
.994459
.994438
.994418

.33
.33
.33
..33
.35
.33
.33
.33
.35
.33
.33

9.199713

.20052!)
.201345
.202159
.202971
.203782
.204592
.205400
.206207
.207013
.207817

13.60
13.60
13.57
13.53
13.52
13.50
13.47
13.45
13.43
13.40
13.37

10.800287
.799471
.798655
.797H41
.797029
.790218
.795408
.794600
.793793
.792987
.792183

60
59
58
57
56
55
54
53
52
51
50

11
12
18
14
15
16
17
18
19
20

9.203017
.2U3797
.204577
.205354
.200131
.200906
.207679
.208452
.209222
.209992

13.00
13.00
12.95
12.95
12.92
12.88
12.88
12.83
12.83
12.80

9.994398
.994377
.994357
.994336
.994316
.994295
.994274
.994254
.994233
.994212

.35
.33
.35
.33
.35
.35
.33
.35
.35
.35

9.208619
.209420
.210220
.211018
.211815
.212011
.213405
.214198
.214989
.215780

13.35
13.33
13.30
13.28
13.27
13.23
13.22
13.18
13.18
13.13

10.791381
.790580
.789780
.788982
.788185
.787389
.780595
.785802
.785011
.784220

49

48
47
46
45
44
•43
42
41
40

21
22
23
24
25
26
27
28
29
30

9.210760
.211526
.212291
.213055
.213818
.214579
.215338
.216097
.216854
.217609

12.77
12.75
12.73
12.72
12.68
12.65
12.65
12.62
12.58
12.57

9.994191
.994171
.994150
.994129
.994108
.994087
.994066
.994045
.994024
.994003

.33

.35
.35
.35
.35
.35
.35
.35
.35
.35

9.216568
.217356
.218142
.218926
.219710
.220492
.221272
.222052
.222830
.223607

13.13
13.10
13.07
13.07
13.03
13.00
13.00
12.97
12.95
12.92

10.7&3432

.782644
.781858
.781074
.780290
.779508
.778728
.777948
.777170
.776393

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9.218363
.219116
.219868
.220618
.221367
.222115
.222861
.223606
.224349
.225092

12.55
12.53
12.50
12.48 i
12.47
12.43
12.42
12.38
12.38
12.35

9.993982
.993960
.993939
.993918
.993857
.993875
.993854
.993832
.993811
.993789

.37
.35
.35
.35
.37
.35
.37
.35
.37
.35

9.224382
.225156
.225929
.226700
.227471
.228239
.229007
.229773
.230539
.231302

12.90

12.88
12.85
12.85
12.80
12.80
12.77
12.77
12.72
12.72

10.775618
.774844
.774071
.773300
.772529
.771761
.770993
.770227
.769461
.768698

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

9.225833
.226573
.227311
.228048
.228784
.229518
.230252
.230984
.231715
.232444

12.33
12.30
12.28
12.27
12.23
12.23
12.20
12.18
12.15
12.13

9.993768
.993740
.993725
.993703
.993681
.993660
.993638
.993616
.993594
.993572

.37
.35
.37
.37
.35
.37
.37
.37
.37
.37

9.232065
.232826
.233586
.234345
.235103
.235859
.236614
.237368
.238120
.238872

12.68
12.67
12.65
12.63
12.60
12.58
12.57
12.53
12.53
12.50

10.767935

.767174
.766414
.76565*
.764897
.764141
.763386
.762632
.761880
.761128

19
18
17
16
15
14
13
12
11
10

51

52
53
54
55
56
57
58

9.233172
.233899
.234625
.235349
.236073
.236795
.237515
.238235

12.12
12.10 ;
12.07 i
12.07
12.03
12.00
12.00

9.993550
.993528
.993506
.993484
.993462
.993440
.993418
.993396

.37
.37
.37
.37
.37
.37
.37

9.239622
.240371
.241118
.241865
.242610
.843354
.244097
.244839

12.48
12.45
12.45
12.42
12.40
12.38
12.37

10.760378
.759629
.758882
.758135
.757390
.756646
.755903
.755161

9
8
7
6
5
4
3
2

59
60

.238953
9.239670

.9'
11.95

.993374
9.993351

.67
.38

.245579
9.246319

2 . O-5

12.33

.754421
10.753681

1
0

/

Cosine. 1 D. 1'. i

Sine. D. 1".

Cotang.

P. r.

Tang.

/

89°

113

80s

10°

COSINES, TANGENTS, AND COTANGENTS.

169"

'

Sine.

D. 1".

Cosine.

D. 1'.

Tang. D. 1".

Cotang.

'

0

1

9.239670
.240386

11.93

nrhci

9.993351
.99:3329

.37

Of

9.246319
.247057

12.30

<t .-v t"JO

10.753681
.752943

60

59

2
3

4
5

.241101
.241814
.242526
.243237

.92
11.88
11.87
11.85

noo

.993307
.993284
.993262
.993240

.61
.38
.37
.37

OO

.247794
.248530
.249264
.249998

12.28
12.27
12.23
12.23

IL\ C%f\

.752206
.751470
.750736
.750002

58
57
56
55

6

r*
t

.243947
.244656

.83
11.82

nr*o

.993217
.993195

.60
.37

OO

.250730
.251461

2.20
12.18

i »"> i r*

.749270
.748539

54
53

8

.245363

. 10

.993172

.08

OO

.252191

12.17

1 - » i -

.747809

52

9

.246069

11.77

ni*Tf

.993149

.38

nff

.252920

12.15

-> . i -*O

.747080

51

10

.246775

. i7

nrv>

.993127

.67

OO

.253648

12.1-5

•« c\ -t f\

.746352

50

. i2

.38

12.10

11
12

9.247478
.248181

11.72

nr*f\

9.993104
.993081

.38

3r*

9.254374
.255100

12.10
10 A*"*

10.745626
.744900

49

48

13
14
15
16
17
18
19
20

.248883
.249583
.250282
.250980
.251677
.252373
.253067
.253761

. i()
11.67
11.65
11.63
11.62
11.60
11.57
11.57
11.53

.993059
.993036
.993013
.992990
.992967
.992944
.992921
.992898

i
.38
.38
.38
.38
.38
.38
.38
.38

.255824
.256547
.257269
.257990
.258710
.259429
.260146
.260863

2.0i
12.05
12.03
12.02
12.00
11.98
11.95
11.95
11.92

.744176
.743453
.742731
.742010
.741290
.740571
.739854
.739137

47
46
45
44
43
42
41
40

21

23
24
25
26
27
28

9.254453
.255144
.255834
.256523
.257211
.257898
.258583
.259268

11.52
11.50
11.48
11.47
11.45
11.42
11.42

noo

9.992875
.992852
.992829
.992806
.992783
.992759
.992736
.992713

.38

.38
.38
.38
.40
.38
.38

OO

9.261578
.262292
.263005
.263717
.264428
.265138
.265847
.266555

11.90
11.88
11.87
11.85
11.83
11.82
11.80

nf~rf

10.738422
.737708
.736995
.736283
.735572
.734862
.734153
.733445

39

38
37
36
35
34
33
32

29

.259951

.00
nnp*f

.992690

.08

Af\

.267261

. t7

nr*r**

.732739

31

30

.260633

.67

nOET

.992666

.40

OO

.267967

.7i

Hr*o

.732033

30

.65

.38

.73

31
32

9.261314
.261994

11.33

nOi"l

9.992643
.992619

.40

OO

9.268671
.269375

11.73

ni**/\

10.731329

.730625

29

28

33

.262673

.62

nt)f\

.992596

.38

Af\

.270077

. <0

ni*-f\

.729923

27

34
35
36
37
38
39
40

.263351
.264027
.264703
.265377
.266051
.266723
.267395

.60
11.27
11.27
11.23
11.23
11.20
11.20
11.17

.992572
.992549
.992525
.992501
.992478
.992454
.992430

.40
.38
.40
.40
.38
.40
.40
.40

.270779
.271479
.272178
.272876
.273573
.274269
.274964

. iO
11.67
11.65
11.63
11.62
11.60
11.58
11.57

.729221
.728521
.727822
.727124
.726427
.725731
.725036

26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

9.268065
.268734
.269402
.270069
.270735
.271400
.272064
.272726
.273388
.274049

11.15
11.13
11.12
11.10
11.08
11.07
11.03
11.03
11.02
10.98

9.992406
.992382
.992359
.992335
.992311
.992287
.992263
.992239
.992214
.992190

.40
.38
.40
.40
.40
.40
.40
.42
.40
.40

9.275658
.276351
.277043
.277734
.278424
.279113
.279801
.280488
.281174
.281858

11.55
11.53
11.52
11.50
11.48
11.47
11.45
11.43
11.40
11.40

10.724342
.723649
.722957
.722266
.721576
.720887
.720199
.719512
.718826
.718142

19
18
17
16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
.60

9.274708
.275367
.276025
.276681
.277337
.277991
.278645
.279297
.279948
9.280599

10.98
10.97
10.93
10.93
10.90
10.90
10.87
10.85
10.85

9.992166
.992142
.992118
.992093
.992069
.992044
.992020
.991996
.991971
9.991947

.40
,40
.42
.40
.42
.40
.40
.42
.40
1

9.282542
.283225

.283907
.284588
.285268
.285947
.286624
.287301
.287977
9.288652

11.38
11.37
11.35
11.33
11.32
11.28
11.28
11.27
11.25

10.717458
.716775
.716093
.715412
.714732
.714053
.713376
.712099
.712023
10.711348

9
8
7
6
5
4
3
2
1
0

'

Cosine.

D. 1". ll Sine. D. 1".

Cotang.

D. 1'. Tang.

'

100°

114

79C

11'

TABLE X. — LOGARITHMIC SINES,

168'

/

Sine.

D. 1'.

Cosine.

D. 1".

Tang.

D. r.

Cotang.

/

0

1

2
3
4
5

9.280599
.281248
.281897
.282544
.283190
.283836

:

10.82
10.82
10.78
10.77

10.77
in "5

9.991947
.991922
.991897
.991873
.991848
.991823

.42
.42
.40
.42
.42

A(\

9.288652
.289326

.289999
.290671
.291312
.292013

11.23

11.22
11.20
11.18
11.18

U1 f^

10.711348
.710674
.710001
.709329
. 708658
.707987

60
59
58
57
56
55

6
7
8
9
10

.284480
.285124
.285766
.286408
.287048

Ju. (a

10.73
10.70
10.70
10.67
10.67

.991799
.991774
.991749
.991724
.991699

.42
.42
.42
.42
.42

.292682
.293350
.294017
.294684
.295349

. 10

11.13
11.12
11.12
11.08
11.07

.707318
.706650
.705983
.705316
.704651

54
53
52
51
50

11
.12
13
14
15
16
17
18
19
20

9.287688
.288326
.288964
.289600
.290236
.290870
.291504
.292137
.292768
.293399

10.63
10.63
10.60
10.60
10.57
10.57
10.55
10.52
10.52
10.50

9.991674
.991649
.991624
.991599
.991574
.991549
.991524
.991498
.991473
.991448

.42
.42
.42
.42
.42
.42
.43
.42
.42
.43

9.296013
.296677
.297339
.298001
.298662
.299322
.299980
.300638
.301295
.301951

11.07
11.03
11.03
11.02
11.00
10.97
10.97
10.95
10.93
10.93

10.703987
.703323
.702661
.701999
.701338
.700678
.700020
.699362
.698705
.698049

49
48
47
46
45
44
43
42
41
40

21
22
23
24
25
26
27
28
29
30

9.294029
.294658
.295286
.295913
.296539
.297164
.297788
.298412
.259034
.299655

10.48
10.47
10.45
10.43
10.42
10.40
10.40
10.37
10.35
10.35

9.991422
.991397
.991372
.991346
.991321
.991295
.991270
.991244
.991218
.991193

.42
.42
.43
.42
.43
.42
.43
.43
.42
.43

9.302607
.303261
.303914
.304567
.305218
.305869
.306519
.307168
.307816
.308463

10.90
10.88
10.88
10.85
10.85
10.83
10.82
10.80
10.78
10.77

10.697393
.696739
.696086
.695433
.694782
.694131
.693481
.692832
.692184
.691537

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9.300276
.300895
.301514
.302132
.302748
.303364
.303979
.304593
.305207
.305819

10.32
10.32
10.30
10.27
10.27
10.25
10.23
10.23
10.20
10.18

9.991167
.991141
.991115
.991090
.991064
.991038
.991012
.990986
.990960
.990934

.43
.43
.42
.43
.43
.43
.43
.48
.43
.43

9.309109
.309754
.310399
.3110-42
.311685
.312327
.312968
.313608
.314247
.314885

10.75
10.75
10.72
10.72
10.70
10.68
10.67
10.65
10.63
10.63

10.690891
.690246
.689601
.688958
.688315
.687673
.687032
.686392
.685753
.685115

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

9.306430
.307041
.307650
.308259
.308867
.309474
.310080
.310685
.311289
.311893

10.18
10.15
10.15
10.13
10.12
10.10
10.08
10.07
10.07
10.03

9.990908
.990882
.990855
.990829
.990803
.990777
.990750
.990724
.990697
.990671

.43
.45
.43
.43
.43
.45
.43
.45
.43
.43

9.315523
.316159
.316795
.317430
.318064
.318697
3193.30
.319961
.320592
.321222

10.60
10.60
10.58
10.57
10.55
10.55
10.52
10.52
10.50
10.48

10.684477
.683841
.683205
.682570
.681936
.681303
.680670
.680039
.679408

ij^o^.^.0

.Dioi ID

19
18
17
16
15
14
13
12
11
10

51
52
53
54
65
56
57
58
59
60

9.312495
.313097
.313698
.314297
.314897
.315495
.316092
.316689
.317284
9.317879

10.03
10.02
9.98
10.00
9.97
9.95
9.95
9.92
9.92

9.990645
.990618
.990591
.990565
.990538
.990511
.990485
.990458
.990431
9.990404

.45
.45
.43
.45
.45
.43
.45
.45
.45

9.321851
.322479

.323106
.32373.3
.324358
.324983
.325607
.326231
.326853
9.327475

10.47
10.45
10.45
10.42
10.42
10.40
10.40
10.37
10.37

10.678149
.677521
.676894
.676267
.675642
.675017
.674393
.673769
.673147
10.672525 j

9
8
7
6
5
4
3
2
1
0

/

Cosine.

D. 1".

Sine.

D. 1".

Cotang. D. 1'. 1 Tang. '

101°

115

COSINES, TANGENTS, AND COTANGENTS.

167'

'

Sine.

D. 1".

Cosine.

D. 1'.

Tang.

D. 1'.

Cotang.

'

0

1

2
3
4
5
6
7
8

9.317879
.318473
.319066
.319658
.320249
.320840
.321430
.322019
.322607

9.90

9.88
9.87
9.85
9.85
9.83
9.82
9.80

9r*o

9.990404
.990378
.990351
.990324
.99029?
.990270
.990243
.990215
.990188

.43
.45
.45
.45
.45
.45
.47
.45

A K

9.327475
.328095
.328715
.329334
.329953
.330570
.331187
.331803
.332418

10.33
10.33
10.32
10.32
10.28
10.88
10.27
10.25

-4 f\ £)(•

10.672525
.671905
.671285
.670666
.670047
.669430
.668813
.668197
.667582

60
59
58
57
56
55
54
53
52

9

.323194

. <0

9r*ry

.990161

.4o

A K

.333033

10.J25

"I f\ OO

.666967

51

10

.323780

. <7
9.77

.990134

.45
.45

.333646

10. ««
10.22

.666354

50

11
12

9.324366
.324950

9.73

97Q

9.990107
.990079

.47

9.334259
.834871

10.20
10 18

10.665741
.665129

49

48

13

.325534

. ' O

.990052

•j?

.335482

J.V .IO

10 18

.664518

47

14

.326117

9ft£t

.990025

'AH

.336093

l\j . JO
"t f\ 1C

.663907

46

15
16

17
18
19
20

.326700
.327281
.327862
.328442
.329021
.329599

. IX

9.68
9.68
9.67
9.65
9.63
9.62

.989997
.989970
.989942
.989915
.989887
.989860

A5
.47
.45
.47
.45
.47

.336702
.337311
.337919
.338527
.339133
.339739

10.15
10.15
10.13
10.13
10.10
10.10
10.08

.663298
.662689
.662081
.661473
.660867
.660261

45
44
43
42
41
40

21
22
23
24
25
26
27
28

9.330176
.330753
.331329
.331903
.332478
.333051
.333624
.334195

9.62
9.60
9.57
9.58
9.55
9.55
9.52

9 to

9.989832
.989804
.989777
.989749
.989721
.989693
.989665
.989637

.47
.45
.47
.47
.47
.47
.47

AK

9.340344

.340948
.341552
.342155
.342757
.343358
.343958
.344558

10.07
10.07
10.05
10.03
10.02
10.00
10.00

9QQ

10.659656
.659052
.658448
.657845
.657243
.656642
.656042
.655442

39

38
37
36
35
34
33
32

29

.334767

. OO
9^0

.989610

.40

.345157

. t7O

q q7

.654843

31

30

.335337

. *JV

9.48

.989582

'AS

.345755

0. V I

9 97

.654245

30

31
32

9.3X5906
.336475

9.48

9A"7

9.989553
.989525

.47

9.346353
.346949

9.93

9QQ

10.653647
.653051

29

28

33

.337043

.4*

9 A K

.989497

.47

An

.347545

.yo
Q Q^

.652455

27

34

.337610

.45

94 O

.989469

At

A (**

.348141

o oft

.651859

26

35

.338176

.4o

9JQ

.989441

At

47

.348735

9 on

.651265

25

36

.338742

. **o

Q A.*")

.989413

At

Ary

.349329

. */ \j

9Q.Q

.650671

24

37
38
39

.339307
.339871
.340434

9i40
9.38

9orf

.989385
.989356
.989328

.47
.48
.47

iTf

.349922
.350514
.351106

.OO

9.87
9.87

9D«r

.650078
.649486
.648894

23
22

21

40

.340996

.67
9.37

.989300

At
.48

.351697

.OO
9QO
• OO

.648303

20

41
42
43
44

9.341558
.342119
.342679
.343239

9.35
9.33
9.33

9OA

9.989271
.989243
.989214
.989186

.47

.48
.47

AQ

9.352287
.352876
.353465
.354053

9.82
9.82
9.80

9r-o

10.647713
.647124
.646535
.645947

19
18
17
16

45

.343797

.oU

9OA

.989157

.48

AQ

.354640

. <0

9r*o

.645360

15

46

47

.344355
.344912

.oU

9.28

Q 9R

.989128
.989100

.48
.47

AQ

.355227
.355813

. IO

9.77

9r*»r

.644773
.644187

14
13

48
49

.345469
.346024

9^25

9 OK

.989071
.989042

.48
.48

•trt

.356398
.356982

. (O

9.73

9r«o

.643602
.643018

12
11

50

.346579

.£5

9.25

.989014

.47
.48

.357566

. to

9.72

.642434

10

51
52

9.347134

.347687

9.22
900

• 9.988985
.988956

.48

AQ

9.358149
.358731

9.70

97rt

10.641851
.641269

9

8

53
54

.348240
.348792

. . -

9.20

Q 18

.988927
.988898

.4o

.48

.359313
.359893

. t U

9.67
q KQ

.640687
.640107

7
6

55
56
57

.349343
.349893
.350443

y . 10
9.17

9.17

91 P*

.988869
.988840
.988811

!48
.48

AQ

.360474
.361053
.361632

y . DO
9.65
9.65

9 net

.639526
.638947
.638368

5
4
3

58
59

.350992
.351540

. IO

9.13

91 O

.988782
.988753

.48
.48

4 O

.362210
.362787

.00
9.62

• .637790
.637213

2

1

60

9.352088

.lo

9.988724

.48

9.303364

9'62 10.636636

0

'

Cosine.

D. r.

Sine,

D. r.

Cotang.

D. 1". Tang.

'

116

7T*

13°

TABLE X. — LOGARITHMIC SINES,

166'

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. IV

Cotang.

'

0

1

9.352088
.352635

9.12

91 A

9.988724
.988695

.48

4Q

9.363364
.363940

9.60

10.636636
.636060

60
59

2

.353181

.10
90S

.988666

.4o
fen

.364515

jj-jyj

.635485

58

3

.353726

.uo

9 AD

.988636

. ou

4 L>

.365090

y .00

.634910

57

4
5
6

7

.354271
.354815
.355358
.355901

.08
9.07
9.05
9.05

9e\n

.988607
.988578
.988548
.988519

.48
.48
.50

.48

1 1~\

.365664
.366237
.366810
.367382

9.5*
9.55
9.55
9.53

9 tO

.634336
.633763
.633190
.632618

56
55
54
53

8

.356443

.Oo

9AO

.988489

.50

4 LJ

.367953

.52

9 to

.632047

52

9

10

.356984
.357524

.02
9.00
9.00

.988460
.988430

.4o
.50
.48

.368524
.369094

.52
9.50
9.48

.631476
.630906

51
50

11
12

9.358064
.358603

8.98

8Q7

9.988401
.988371

.50

/is

9.369663
.370232

9.48

9AK

10.630337
.629768

49
48

13
14
15

.359141
.359678
.360215

. y t
8.95
8.95

.988342
.988312

.988282

.la

.50
.50

.370799
.371367
.371933

. ^o
9.47
9.43

9,1*}

.629201
.628633
.628067

47
46
45

16

.360752

o . yo

8 no

.988252

AQ

.372499

. HtO
Q 1 ' '

.627501

44

17

18
19
20

.361287
.361822
.362356
.362889

.92
8.92
8.90
8.88
8.88

.988223
.988193
.988163
.988133

.48
.50
.50
.50
.50

.373064
.373629
.374193
.374756

9.42
9.40
9.38
9.38

.626936
.626371
.625807
.625244

43
42
41
40

21

22
23

9.363422
.363954
.364485

8.87
8.85

8Ot

9.988103
.988073
.988043

.50
.50

t f\

9.375319
.375881
.376442

9.37
9.35

9Oc

10.624681
.624119
.623558

39
38
37

24

.365016

.85

8 no

.988013

.50

.377003

,o5

900

.622997

36

25

.365546

.00

8OO

.987983

t A

.377563

.00

9 'JO

.622437

35

26

.366075

,o3

8Qk>

.987953

.50

to

.378122

,0-i

900

.621878

34

27

.366604

.0%

O '**Q

.987922

.52

.378681

.0/w

Q Qi\

.621319

33

28
29

.367131
.367659

8^80

.987892
.987862

!50

t f\

.379239
.379797

9! 30

9tlO

.620761
.620203

32
31

30

.368185

8.77
8.77

.987832

.50
.52

.380354

.28
9.27

.619646

30

31

9.368711

8r*K

9.987801

p-A

9.380910

9ow

10.619090

29

32

.369236

. <5

8r*t

.987771

.50

.381466

.XI

Q 9^

.618534

28

33
34

.369761
.370285

. lO

8.72

8ryn

.987740
.987710

!50

.382020
.382575

9^25

.617980
.617425

27
26

35

.370808

. iX
8r-A

.987679

t A

.383129

O OO

.616871

25

36

.371330

.70

Q *"*Oj

.987649

.50

.383682

9OA

.616318

24

37

38
39

.371852
.372373
.372894

8^68
8.68

8/jrV

.987618
.987588
.987557

'.50
.52

5(1

.384234
.384786
.385337

.IvU

9.20

9.18
91 Q

.615766
.615214
.614663

23
22
21

40

.373414

.01

8.65

.987526

2

.50

.385888

.lo
9.17

.614112

20

41
42
43
44

9.373933
.374452
.374970

.375487

8.65
8.63
8.62

8nr\

9.987496
.987465
.987434
.987403

.52
.52
.52

fcO

9.386438
.386987
.387536

.388084

9.15
9.15
9.13

91 O

10.613562
.613013
.612464
.611916

19
18
17
16

45

.376003

.60

8/»A

.987372

.388631

. 1~
Q 1O

.611369 15

46
47

48

.376519
.377035
.377549

.oO
8.60
8.57

SK'V

.987341
.987310
.987279

,5xJ
.52
.52

.389178
.389724
.390270

g!io

9.10

9 no

.610822
.610276
.609730

14
13

12

49

.378063

.57

8t ^f

.987248

K O

.390815

.Uo

9AQ

.609185

11

50

.378577

.57
8.53

.987217

.52
.52

.391360

,0o

9.05

.608640

10

51

9-379089

8 to

9.987186

CO

9.391903

10.608097

9

52
53
54

.379601
.380113
.380624

.53

8.53
8.52

8Kf\

.987155
.987124
.987092

.52
.52
.53

fcO

.392447
.392989
.393531

9^03
9.03

9/\o

.607553
.607011
.606469

8
7
6

55
56

57
58
59

.381134
.381643
.382152
.382661
.383168

.50
8.48
8.48
8.48
8.45

8 A »

.987061
.987030
.986998
.986967
.986936

.52
.52
.53
52
'.52

to

.394073
.394614
.395154
.395694
.396233

Uo
9.02
9.00
9.00
8.98

SO1"*

.605927
.605386
.604846
.604306
.603767

5
4
3
2
1

60

9.383675

.4o

9.986904

.5o

9.396771

. y<

10.603229

0

'

Cosine.

D. r.

Sine.

D. 1".

Cotang.

D. r.

Tang.

'

103°

117

14°

COSINES, TANGENTS, AND COTANGENTS.

165=

'

Sine.

D. 1*.

Cosine.

D. 1".

Tang.

D.I".

Cotang,

'

0

1

2
3
4

9.383675
.384182
.384687
.385192
.385697

8.45
8.42
8.42

8.42

9.986904
.986873
.986841
.986809

.986778

.52
.53
.53

.52

KO

9.396771
.397309
.397846
.398383
.398919

8.97
8.95
8.95

8.93
8 no

10.603229
.602691
.602154
.601617
.601081

60
59
58
57
56

5

.386201

e qo

.986746

. OO

£0

.399455

. yo
8 92

.600545

55

6

7
8
9
10

.386704
.387207
.387709
.388210
.388711

O . OO

8.38
8.37
8.35
8.35
8.33

.986714
.986683
.986651
.986619
.986587

.00

.52
.53
.53
.53
.53

.399990
.400524
.401058
.401591
.402124

8.'90
8.90

8.88
8.88
8.87

.600010
.599476
.598942
.598409
.597876

54
53
52
51
50

11
12

9.389211
.389711

8.33

Q 00

9.986555
.986523

.53

CO

9.402656
.403187

8.85

8Ot

10.597344
.596813

49

48

13

.390210

O . O/w

C OA

.986491

. OO

PtO

.403718

. oO

8K^

.596282

47

14

.390708

o . OU
Q OA •

.986459

. OO

KO

.404249

.OO

8. ' . i

.595751

46

15

.391206

o . OU

.986427

. OO
r,o

.404778

. o&

8QO

.595222

45

16
17

18

.391703
.392199
.392695

8^27
8.27

897

.986395
.986363
.986331

. OO

.53
.53

KO

.405308
.405836
.406364

. oO

8.80
8.80

8QO

.594692
.594164
.593636

44
43
42

19
20

.393191
.393685

.^i

8.23
8.23

.986299
.986266

. OO

.55
.53

.406892
.407419

. ou

8.78
8.77

.593108
.592581

41
40

21
22

9.394179
.394673

8.23

Q OO

9.986234
.986202

.53

9.407945

.408471

8.77

8?t\

10.592055
.591529

39

38

23
24

.395166
.395658

8. '20
8 °0

.986169
.986137

'. 53

f;c.

.408996
.409521

. i O

8.75

8fO

.591004
.590479

37
36

25

26

.396150
.396641

8A8

Q 1Q

.986104
.986072

.OO

.53

.410045
.410569

. IO

8.73

8r~n

.589955
.589431

35
34

27

.397132

o . io
t; IK

.986039

KO

.411092

. i -.
8i~O

.588908

33

28

.397621

o.JO
817

.986007

. OO

.411615

. i 4
Q r-Q

.588385

32

29
30

.398111
.398600

. 1 l

8.15
8.13

.985974
.985942

.'53

.55

.412137
.412658

8. '68
8.68

.587863
.587342

31
30

31

9-399088

81 O

9.985909

t»

9.413179

8 nff

10.586821

29

32

.399575

. 14
81 O

.985876

.55

.413699

. Di
8 en

.586301

28

33

34

.400062
.400549

. j!v

8.12

.985843
.985811

,5o
.53

.414219
.414738

-Ol

8.65
8Kc.

.585781
.585262

27
26

35
36

.401035
.401520

8.' 08

8AQ

.985778
.985745

.'55

.415257
.415775

. DO

8.63

8i;q

.584743
.584225

25
24

37

.402005

.Uo
8r\*r

.985712

te

.416293

. Uo
8^0

.583707

23

38
39
40

.402489
.402972
.403455

.Oi

8.05
8.05
8.05

.985679
.985646
.985613

.OO

.55
.55
.55

.416810
.417326
.417842

.O-4

8.60
8.60
8.60

.583190
.582674
.582158

22
21
20

41
42

9.403938
.404420

8.03
t: AO

9.985580
.985547

.55

KK

9.418358
.418873

8.58

10.581642

.581127

19

18

43
44

45
46

.404901
.405382

.405S62
.406341

o.U-i
8.02
8.00
".98

*~ (IQ

.985514
.985480
.985447
.985414

.OO

.57
.55
.55

te

.419387
.419901
.420415
.420927

8.57
8.57
8.57
8.55

8KK

.580613
.580099
.579585
.579073

17
16
15
14

47
48
49

.406820
.407299
.407777

.yo
".98

^.97

.985381
.985347
.985314

.OO

.57

.55

.421440
.421952
.422463

.55

8.53
8.52
8 5°

.578560
.578048
.577537

13
12
11

50

.408254

-!95

.985280

'.55

.422974

8. ,50

.577026

10

51

9.408731

9.985247

9.423484

8<Q

10.576516

9

52
53

.409207
.409682

!93

no

.985213
.985180

.'55

p-rv

.423993
.424503

.io

8.50

8tfy

.576007
.575497

8

7

54

.410157

.92
90

.985146

.57

^K.

.425011

.47

8 At

.574989

6

55

.410632

(0

f\f\

.985113

.OO

.425519

.4i
8 it

.574481

5

56

.411106

.90

QQ

.985079

.57

.426027

.47

84t~

.573973

4

57

.411579

.88

GO

.985045

•**z

.426534

.45

8_<K

.573466

3

58

.412052

.00

o-v

.985011

KK

.427041

. -to

8 in

.572959

2

59
60

.412524
9.412996

.Oi

7.87

.984978
9.984044

.55
.57

.427547
9.428052

.43

8.42

. .572453
10.571948

1
0

/ a

y

Cosine.

D. 1". 1

Sine.

D. r.

Cotang.

D. 1'.

Tang.

'

118

75'

TABLE X. — LOGARITHMIC SINES,

164'

'

Sine.

D. r.

Cosine.

D. r.

Tang.

D. 1".

Cotang.

'

0

9.412996

7 oe

9.. 984944

r.7

9.428052

10.571948

60

1

.413467

.OO

N OK

.984910

. Ol

.428558

o . 4'j

8Af\

.571442

59

2

.413938

.85

go

.984876

'r~

.429062

. Q\J

Q Aft

.570938

58

3

.414408

.OO

OO

.984842

-E7

.429566

o . 4U

8A(\

.5704:34

57

4

.414878

.8-3

.984808

'<V7

.430070

.40

.569930

56

5

.415347

** OA

.984774

•57

.430573

Q Q1/*

.569427

55

6

r*

.415815
.416283

.80
7.80

7OA

.984740
.984706

.57

.431075
.431577

8 '.37

8Q-V

.568925
.568423

54

53

8

.416751

.80

.984672

57

c.7

.432079

.6l
89.=i

.567921

52

9

.417217

7.77

.984638

.0<
c.Q

.432580

.OO

Q 00

.567420

51

10

.417684

r* r<rt

( .77

.984603

.00
.57

.433080

O . OO

8.33

.566920

50

11

9.418150

r* r*t?

9.984569

9.433580

800

10.566420

49

12

.418615

i . (O

f -•• |

.984535

pro

.434080

. OO

Q QO

.565920

48

13

.419079

7. i3

.984500

.58

.434579

O . d&
800

.565421

47

14

.419544

7.75

l** ^*O

.984466

"eJL

.435078

. o*

8OA

.564922

46

15

.420007

7. 72

I** fO

.984432

.5*

CO

.435576

.oO

8OQ

.564424

45

16

.420470

f . f«

f *" 1

.984397

.00

jrrf

.436073

. -^O
8OQ

.563927

44

17

.420933

7. <2

r* F~n

.984363

.57

CO

.436570

.£<J

8Oi3

.563430

43

18

.421395

1 . lO
r* r*n

.984328

. Oo

.437067

. <vO

807

.562933

42

19
20

.421857
.422318

( . <U
7.68
7.67

.984294
.984259

'.58
.58

.437563
.438059

.*<

8.27

8 OK
.~O

.562437
.561941

41
40

81
22

9.422778
.423238

7.67

9.984224
.984190

.57

CO

9.438554
.439048

8.23

8 OX

10.561446
.560952

39

38

23

.423697

7.65

.984155

.58

rQ

.439543

.5iD
800

.560457

87

24

.424156

7.65

.984120

.58

.440036

.««

800

.559964

36

25

.424615

7.65

.984085

ro

.440529

.64

.559471

35

26
27

.425073
.425530

7.63
7.62

r* />o

.984050
.984015

!58

c.7

.441022
.441514

8^20

Son

.558978

.558486

34
33

28

425987

i .Q4

f £* A

.983981

.Ol

to

.442006

.~<J
81 Q

.557994

32

29

.426443

7.60

I"* £*f\

.983946

.58

CO

.442497

. lo

81 Q

.557503

31

30

.426899

i .60
7.58

.983911

.5o
.60

.442988

. lo

8.18

.557012

30

31
32

9.427354

.427809

7.58

9.983875
.983840

.58

CO

9.443479
.443968

8.15

81 *f

10.556521
.556032

29

28

33

.428263

7.57

.983805

.58

CQ

.444458

.17

8-f jr

.555542

27

34

.428717

7.57

*"* ce

.983770

.58

RQ

.444947

. lo

C 10

.555053

26

35

.429170

1 .OO

.983735

. OO
CO

.445435

0 . 1O

81 O

.554565

25

36

.429623

7.55

.983700

.58

.445923

.10

81 O

.554077

24

37

.430075

7.53

.983664

CO

.416411

.13

81 £>

.553589

23

38

.430527

7 53

.983629

.58

KQ

.446898

.12

81 A

.553102

22

39

.430978

1 . • * -v

r* p-o

.983594

.58
i\(\

.447384

.10
8in

.552616

21

40

.431429

i .5.2
7.50

.983558

.uu
.58

.447870

. 1U

8.10

.552130

20

41

9.431879

»" Kf\

9.983523

-_

9.448356

8AQ

10.551644

19

42
43

.432329
.432778

7^48

.983487
.983452

.'58
f»r\

.448841
.449326

. Uu

8.08

8n*v

.551159
.550674

18
17

44

.433226

7.47

.983416

.00

.449810

.07

.550190

16

45

.433675

7.48

rv .* f

.983381

p,f\

.450294

8.07

8AK

.549706

15

46

.434122

i .45

M J f

.983345

.u(J
f>f\

.450777

.UO
8f\K

.549223

14

47

.434569

7.45

f~- A f~

. 983309

.00

f*r\

.451260

.05
8 nee

.548740

13

48

.435016

7.45

f* A O

.983273

.00

CO

.451743

.05
8 no

.548257

12

49
50

.435462
.435908

7.4-3
7.43
7.42

.983238
.983202

.58
.60
.60

.452225
.452706

.Uo
8.02
8.02

..547775
.547294

11
10

51
52

9.436353
.436798

7.42

*** 4 A

9.983166
.983130

.60

9.453187
.453668

8.02

10.546813
.546332

9

8

53

.437242

< .40

.983094

f»f\

.454148

o . 00

8nn

.545852

7

54

.437686

7.40

pv OO

.983058

.00

£!A

.454628

.00

.545372

6

55

.438129

7.38

f«f OO

.983022

.bO

.455107

7 no

.544893

5

56

.438572

7.o8

r* o^>

.982986

S»f\

.455586

.98

.544414

4

57

.439014

< . .37

7 on

.982950

.bO

S*f\

.456064

wan

.543936

3

58

.439456

.37

'"• ocr

.982914

.60

Stf\

.456542

i -

.543458

2

59

.439897

7. .35

.982878

.bO
ftf\

.457019

f" A**

.542981

1

60

9.440338

7.35

9.982842

.60

9.457496

7.9o

10.542504

0

'

Cosine.

D. 1".

Sine.

D. r. 1

i Cotang.

D. r.

Tang.

'

105'

119

74°

16=

COSINES, TANGENTS, AND COTANGENTS.

163'

'

Sine.

D. 1". Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

9.440338

*"* QQ

9.982842

9.457496

10.542504

60

1

.440778

r* OO

.982805

An

.457973

A'QO

.542027

59

2

.441218

r* oo

.982769

.ou

en

.458449

>~ o'q

.541551

58

3
4

.441658
.442096

T! 30

1** OO

.982733
.982696

. ou

.62
An

.458925
.459400

7\92

.541075
.540600

57
56

5

.442535

r-'ot

.982660

.oil
An

.459875

r* f\f\

.540125

55

6

.442973

7 fc^ft

.982624

.oU

.460349

i .90

r- c\f\

.539651

54

7

.443410

'*' 9ft

.982587

no

.460823

t . yu

7 on

.539177

53

8

.443847

7 k^ft

.982551

. uu

.461297

i . yu

r* QQ

.538703

52

9

.444284

f* O^*

.982514

•jj?

.461770

4 .OO

7 87

.538230

51

10

.444720

t ,i£i

r* O""

.982477

'.do

.462242

4 . O4

7.88

.537758

50

11

9.445155

r* k~>"-

9.982441

ro

9.462715

*** O£

10.537285

49

12

.445590

it'oK

.982404

AO

.463186

t . OO

7 ft7

.536814

48

13
14

.446025
.446459

1 . ~-')

7.23

.982367
.982331

!eo

.463658
.464128

t . Ot

7.83

.536342
.535872

47
46

15

.446893

f OO

.982294

ro

.464599

r* DO

.535401

45

16

.447326

4 . t*4t

7 22

.982257

AO

.465069

4 . OO
7 ft^»

.534931

44

17

.447759

4 . *w<v
f* ^\{\

.982220

. \j'^
AO

.465539

^* GO

.534461

43

18

.448191

^ on

.982183

. \)<*

.466008

1 . O^

ft Oi,J

.533992

42

19

.448623

7 18

.982146

RO

.466477

r* Qf\

.533523

41

20

.449054

?!is

.982109

'.Q2

.466945

7^80

.533055

40

21

9.449485

7 17

9.982072

AO

9.467413

r- r-o

10.532587

39

22

.449915

t . A t

.982035

. \J-*

.467880

r-'r-D

.532120

38

23

.450345

7. 17
7 17

.981998

'AO

.468347

4 . 4 O

7 7R

.531653

37

24

.450775

i . j. i

.981961

AO

.468814

1 . 4 O

.531186

36

25

.451204

1 . 15

.981924

Aq

.469280

7.77

r* r*r*

.530720

35

26

.451632

i . J o
r* i o

.981886

ro

.469746

4.44

.530254

34

27

.452060

4 . 1O

7 10

.981849

ro

.470211

77C

.529789

33

28

.452488

4 . 1O
*** 10

.981812

cq

.470676

r" '*'""

.529324

32

29

.452915

r* i p

.981774

. Oo

.471141

i . • 0

7 7^

.528859

31

30

.453342

riio

.981737

!62

.471605

1 . 1 O

7.73

.528395

30

31

9.453768

r* -i A

9.981700

9.472069

7 79

10.527931

29

32

.454194

7 OS

.981662

K9

.472532

t • IJG

7 72

527468

28

33

.454619

« . v/o
7 Oft

.981625

cq

.472995

1 . l/b

7 70

.527005

27

34

.455044

1 . I/O

70ft

.981587

. OO
cq

.473457

< . * i/

.526543

26

35

.455469

. vO

7 07

.981549

.Oo

.473919"

7 70

.526081

25

36

.455893

4 . Ul

*** 0*5

.981512

cq

.474381

< . 1 V/

7 fift

.525619 24

37

.456316

4* OK

.981474

. OO
cq

.474842

1 . wO

7 fift

.525158 23

38

.456739

7 05

.981436

. Oo
AO

.475303

4 . wO

*** fi7

.524697 22

39

.457162

7 03

.981399

. U/v

p.q

.475763

r-'ci

.524237

21

40

.457584

7^03

.981361

.Oo

.63

.476223

1** /jr*
( .04

.523777

20

41

9.458006

7 OO

9.981323

r»q

9.476683

r- AC:

10.523317

19

42

.458427

1 .U*

7 OO

.981285

. DO
cq

.477142

4 . DO

'"* A^

.522858

18

43

.458848

4 . V^

7 00

.981247

. DO

«0

.477601

X'^q

.522399

17

44

.459268

t . UV
7 00

.981209

. DO

.478059

'"* 4-1Q

.521941

16

45

.459688

1 . vU

7 00

.981171

CO

.478517

4 . DO
i** AQ

.521483

15

46
47

.460108
.460527

4 . Uv

6.98

6OQ

.981133
.981095

. DO

.63

CO

.478975
.479432

7^62

7 fi2

.521025
.520568

14
13

48
49

.460946
.461364

yo
6.97

GQ7

.981057
.981019

.00
.63

.479889
.480345

7^60
7 fiO

.520111
.519655

12
11

50

.461782

. y 4
6.95

.980981

!65

.480801

4 . UV

7.60

.519199

10

51

9.462199

f* QX

9.980942

CO

9.481257

*** ^8

10.518743

9

52
53

.462616
.463032

6 !93

6QQ

.980904
.980866

.DO

.63

AKL

.481712
.482167

7. '58
7 57

.518288
.517833

8

ft
4

54

.463448

. yo

6QQ

.980827

.OO
60

.482621

.517379

6

55

.463864

. yo

6ru~>

.980789

O

.483075

7 ^\7

.516925

5

56
57

.464279
.464694

.J4

6.92

6nn

.980750
.980712

^63

.483529
.483982

7^55

.516471
.516018

4
3

58
59

.465108
.465522

.90

6.90

6ca

.980673
.980635

'.63

.484435
.484887

7^53

r* RO

.515565
.515113

2
1

60

9.465935

.00

9.980596

'

9.485339

10.514661

0

'

Cosine.

D. r.

Sine.

D. r.

Cotang.

D. 1".

Tang.

'

120

73=

17C

TABLE X. — LOGARITHMIC SINES,

162"

1

Sine.

D. r.

Cosine.

D. r.

Tang.

D. 1".

Cotang.

/

0

1

9.465935
.466348

6.88

6QQ

9.980596
.980558

.63

/••*

9.485339

.485791

7. no

^ ^\O

10.514661

.514209

60
59

e

3
4

.466761
.467173
.467585

.88
6.87
6.87

60~

.980519
.980480
.980442

.bo
.65
.63

.486242
.486693
.487143

7! 52

7.50

7 'SO

.513758
.513307

.512857

58
57
56

5
6

r*

.467996
.468407
.468817

. oO

6.85
6.83

.980403
.980364
.980325

'.Go

.65

.487593
.488043
.488492

4 . OU

7.50
7.48

.512407
.511957
.511508

55
54
53

8
9
10

.469227
.469637
.470046

6.83
6.83
6.82
6.82

.980286
.980247
.980208

'.65

.65
.65

.488941
.489390
.489838

7^48
7.47

.511059
.510610
.510162

52
51

50

11
12
13

9.470455
.470863
.471271

6.80
6.80

6 Of)

9.980169
.980130
.980091

.65
.65

9.490286
.490733
.491180

7.45
7.45

10.509714
.509267
.508820

49

48
47

14
15
16

.471679
.472086
.472492

. oU

6.78
6.77

6r/rp

.980052
.980012
.979973

167
.65

.491627
.492073
.492519

T! 43
7.43

*** AQ

.508373
.507927
.507481

46
45
44

17

.472898

.77

6ffH

.979934

ftK

.492965

r* 40

.507035

43

18

.473304

. <7

.979895

.6O
C^

.493410

t .4J

r* irt

.506590

42

19

.473710

6.77

6r*-

.979855

.01

.493854

i .40

.506146

41

20

.474115

. 10

6.73

.979816

!e?

.494299

7^40

.505701

40

21

9.474519

670

9.979776

9.494743

(** QQ

10.505257

39

22

.474923

. i •>

67Q

.979737

P7

.495186

7 in

.504814

38

23

.475327

. IO

6fO

.979697

f*~

.495630

t . *U

r- OU

.504370

37^

24

25

.475730
.476133

. i2
6.72

R 70

.979658
.979618

.DO

.67

.496073
.496515

t .08

7.37

.503927
.503485

36
35

26

.476536

U. IM

6fff\

.979579

£W

.496957

1 . *j(

.503043

34

27

.476938

. (0

6r*rv

.979539

.57

.497399

7.37

.502601

33

28

.477340

. ill

61 ' W

.979499

°fi7

.497841

7 ^^

.502159

32

29

.477741

. DO
6KA

.979459

R^

.498282

i . OO

7 SH

.501718

31

30

.478142

.Do

6.67

.979420

^67

.498722

7^35

.501278

30

31
32

9.478542

.478942

6.67

6r*iy

9.979380
.979340

.67

6**

9.499163
.499603

J-'^3

10.500837
.500397

29

28

33
34

.479342
.479741

.07

6.65

.979300
.979260

7
.67

.500042
.500481

7^32

.499958
.499519

27
26

35

.480140

D . t)O

.979220

/»«

.500920

i . O*v
t*1 OO

.499080

25

36

.480539

6.65
6/>o

.979180

.o<

.501359

7.o2

7 Qll

.498641

24

37

.480937

. Oo
(\ fi9

.979140

'(ft

.501797

t .OU

.498203

23

38
39

.481334
.481731

D . Qy<3

6.62

6fiO

.979100
.979059

'.G8

.502235

.502672

?! 28

^ OS

.497765
.497328

22
21

40

.482128

. ' » -

6.62

.979019

'.Q7

.503109

i . 40

7.28

.496891

20

41

; 9. 482525

9.978979

r"

9.503546

f: ~

10.496454

19

42

.482921

6 . b(J

6RQ

.978939

rs

.503982

i .2i
i- 07

.496018

18

43

.483316

. oo
6*»n

.978898

R7

.504418

^ 07

.495582

17

41

.483712

. uu
6rcQ

.978858

.D<

CO

.504854

i • AI

*"* O^

.495146

16

45

.484107

. Oo

.978817

.DO

.505289

I . lOu

7 o?;

.494711

15

46

.484501

b . Orf

.97'8777

R7

.505724

< . *O

*** O'%

.494276

14

47

.484895

b . ot
6^7

.978737

. D*

.506159

*"* OQ

.493841

13

48

.485289

. • * t
6.55

.9786D6

pQ

.506593

A o'l

.493407

12

49

.485682

.978655

"ft"

.507027

r- oo

.41)2973

11

50

.486075

D . OO

6.53

.{178615

!68

.507460

7^22

.492540

10

51
52

8 9. 486467
.486860

6.55

6r.o

9.978574
.978533

.68

9.507893
.508326

7.22

*" OO

10.492107
.491674

9

8

53
54
55
56

.487251
.487643
.488034
.488424

.•)£

6.53
6.52
6.50

'. 978452
.978411
.978370

.'68
.68
.68

.508759
.509191
.509622
.510054

1 • - -

7.20

7.18
7.20

r* -i o

.491241
.490809
.490378
.489946

7
6
5
4

57

.488814

6 . 50

6K.fl

978329

•jH

.510485

< .18

*7 1 U

.489515

3

58

.489204

. OU
6 to

.978288

no

.510916

< .lo

r' 1 r*

.489084

2

59

.489593

. ~rO

.978247

.On

/»o

.511346

( .1 1

.488654

1

60

9.489982

6.48

9.978206

.Oo

9.511776

7.17

10.488224

0

'

Cosine.

D 1".

Sine.

D. r.

Cotang.

D. r.

Tang.

'

107'

121

72'

18C

COSINES, TANGENTS, AND COTANGENTS.

'

Sine.

D. 1".

Cosine.

D. r.

Tang.

D. 1".

Cotang.

'

0

9.489982

61O

9.978206

C.Q

9.511776

17

10.488224

60

1

.490371

. 'iO

fi 47

.978165

. Do
AQ

.512206

. J 1

1 K.

.487794

59

2
3
4

.490759
.491147
.491535

u . -± i

6.47
6.47

.978124
.978083
.978042

. OO

.68
.68

f.Q

.512635
.513064
.513493

. JO

.15
.15

-10

.487365
.486936

.480507

58
57
56

5

.491922

0 . 4O
640

.978001

.CO

70

.513921

. lo

1 '>

.486079

55

C

7
8

.492308
.492095
.493081

. -iO

6.45
6.43

6dO

.977959
.977918
.977877

. t u

.68

.68

.514349
.514777
.515204

A3

12

.485051
.485223

.484796

54
53

52

9

.493406

. T'O

fi 49

.977835

. i i_l

CQ

.515631

10

.484369

51

10

.493851

\j . ~i**

6.42

.977794

. Do

.70

.516057

. 1U

19

. ! -

.483943

50

11

9.494230

649

9.977752

CO

9.516484

1O

10.483516

49

12

.494021

. ^r<v
a 40

.977711

.DO
r»{\

.516910

. 1U

nc

.483090

48

13

.495005

U . ~i\J
fi Qfi

.977669

. t \J
AQ

.517335

. uo
10

.482665

47

14

.495388

\j . OO
6A(\

.977628

. DO

.517761

. 1U

AO

.482239

46

15

.495772

, ^\J i

fi °,7

.977586

. i U
70

.518186

.Uo

07

.481814

45

16

.496154

U . O i
A QQ

.977544

. t U

f.Q

.518610

. Ul

u rv?

.481390

44

17

.490537

U . OO

607

.977503

. Oo

.519034

O.7

.480966

43

18

.496919

. Ol
6 or/

.977461

"r-n

.519458

.Ul

(Y7

.480542

42

19

.497301

. O t
60S

.977419

. i U

.519882

.\)i

.480118

41

20

.497682

. OO

6.35

.977377

!TO

.520305

^05

.479695

40

21

9.498064

6QQ

9.977335

70

9.520728

OT

10.479272

39

22

.498444

. OO
6 OK

.977293

. t\J

.521151

. UO
no

.478849

38

23

.498825

. OO

600

.977251

r'0

.521573

. Uo

.478427

37

24

.499204

. Oiw

6QO

.977209

. I U
r'A

.521995

no

.478005

36

25

.499584

.OO
GOO

.977167

. tO

r-A

.522417

.Oo

.477583

35

26

.499963

.66
6^,9

.977125

. (U

70

.522838

''09

.477162

34

27

.500342

. o&
60O

.977083

. * \J

.523259

09

.476741

33

28

.500721

. O/a

6 30

.977041

. i 0

T'A

.523680

. U**

no

.476320

32

29
30

.501(1!)'.)
.501476

6^28
6.30

.976999
.976957

!?o

.72

.524100
.524520

. uu
.00
.00

.475900
.475480

31
30

31
32
33

9.501854
.502231
.502607

6.28
6.27
6 28

9.976914
.976872
.976830

.70
.70

r*o

9.524940
.525359
.525778

6.98
6.98

6Qft

10.475060
.474641
.474222

29
28
27

34
35

.502984
.503360

6^27

fi QT

.976787
.976745

•70

.526197
.526615

. «7O

6.97

A Q7

.473803
.473385

26
25

36

.503735

U . f**j
6 OK

.976702

. i -

.527033

U . «7 1

607

.472967

24

37

.504110

. <vO

.976660

. < 0

.5274:,!

. y i

6 OK

.472549

23

38

.504485

6.25
6 ox

.976617

.72

ftf\

.527868

.95

6r\sr

.472132

22

39

.504860

.IvO
6OQ

.976574

.72

.528285

95

6QK

.471715

21

40

.505234

. - • >

6.23

.976532

^72

.528702

. yo
6.95

.471298

20

41
42

9.505608
.505981

6.22

699

9.976489
.976446

.72

9.529119
.529535

6.93

600

10.470881
.470465

19

18

43

.506354

. **&
600

.976404

r-fi

.529951

. yo

.470049

17

44
45
46

.506727
.507099
.507471

.»Ss

6.20
6.20

69O

.976361
.976318
.976275

.72

.72
.72
72

.530366
.530781
.531196

6^92
6.92

6Q9

.469634
.469219
.468804

16
15
14

47
48

.507843
.508214

. /v-U

6.18

6-1 O

.976232
.976189

!72

CVC1

.531611
.532025

. J7/V

6.90

.468389
.467975

13
12

49

.508585

.18

61Q

.976146

.72

.532439

6.90
6 on

.467561

11

50

.508956

. lo

6.17

.976103

!72

.532853

,bU

6.88

.467147

10

51
52

9.509326
.509696

6.17

6-1 K

9.976060
.976017

.72

9.533266
.533679

6.88

GOO

10.466734
.466321

9
8

53

.510065

.lo

6-i K

.975974

• '~

.534092

.OO

60^*

.465908

7

54

.510434

.15
61R

.975930

.73
^o

.534504

.O(
6QT"

.465496

6

55

.510803

. JO
fi IT,

.975887

. <*

79

.534916

.01

687

.465084

5

56
57

.511172
.511540

O.JO

6.13

6~t O

.975844
.975800

. * <£

.73

.535328
.535739

. 01

6.85

6O^

.464672
.464261

4
3

58

.511907

.12

619.

.975757

t i <V

.536150

.80
6 fie;

.463850

2

59
60

.512275

9.512042

. lo

6.12

.975714
9.975670

•/•q

.<3

.536561
9.536972

.oO

6.85'

.463439
10.463028

1

0

'

Cosine.

D. r.

Sine.

D. r.

Cotang.

D. r.

Tang.

'

103C

122

71'

19"

TABLE X. — LOGARITHMIC SINES,

ICO'

' Sine.

D. 1".

Cosine.

D. 1'.

Tang.

D. r.

Cotang.

'

0 9.512642

9.975670

9.536972

6 CO

10.463028

60

1 .513009

j! j0 .975627

• it*

r-o

.537382

.00
GttQ

.462618

59

2 .513375

61 A

.975583

. to

.537792

.OO
6R9.

.462208

58

3 .513741

.10

.975539

"p*n

.538202

.OO

.461798

57

4 .514107

6.10

G, LI

.975496

. i'~
r<o

.538611 ^

.461389

56

5

.514472

.*>»

6AO

.975452

. 10

r»o

.539020

.460980

55

6

.5148:17

.Oo

.975408

. 10

.5391-.".)

.460571

54

7

.515202

6.08 n~KOC'-

~ A~ .9i5obo

i ^

r*o

rOdL." •" D . nU

. DOcTOtJ J /» Qrt

.460163

53

8

.515506

6rvrf

.975321

r'O

.540245

.459755

52

9

10

.515930

.516294

.(Ji

6.07
6.05

.975277
.975233

. i O

.73
.73

.5406.:3
.5410t;i

D. «0

.459347
.458939

51
50

11

9.516657

6 A*'

9.975189

r»o

9.541468 A r.o

10.458532

49

12

.517020

.05

6 run

.975145

. to

r-»>

.541875 S'i«

.458125

48

13

.517382

.08

6 AC

.975101

. to

r-Q

.542281 °-ii

.457719

47

14
15

.517745
.518107

.05
6.03

6rwi

.975057
.975013

. IO

.73

r'O

.542688 • S'iS
.543094 *-iL

.457312
.456906

46
45

16

.5184(58

.02

GAO

.974969

. 10

r*o

.543499 °-X'2

.456501

44

17

.518829

.02

6AO

.974925

. 10

.543905 ^'Xi

.456095

43

18

.519190

.02

6/\.-v

.974880

r'O

.544310 ^'r-g

.455690

42

19

.519551

.02

6AA

.974836

. (0
r'O

.544715 r.'Xo

.455285

41

20

.519911

.00
6.00

.974792

. 10

.73

.545119

D. 10

6.75

.454881

40

21

22

9.520271
.520631

6.00

5fiQ

9.974748
.974703

.75

r*o

9.545524

.545928

6.73

6 r'O

10.454476
.454072

39

38

23

.520990

.9o

5AO

.974659

. 10

r*[r

.546331

. <~

6r-o

,453669

37

24

.521349

.98

5 A/"/

.974614

.75

r*o

.546735

. to

6 r*o

.453265

36

25

.521707

.97

.974570

. 10

r*tr

.547138

. 1^

6r*rv

.452862

35

26

.522066

5.98

5r\r>

.974525

.75

r'O

. 547540

. iO

6r/O

.452460

34

27

.522424

.97

SAC

.974481

. to

.547943

. 1^
6r*{\

.452057

33

28

.522781

.95

.974436

r*t

.548345

. t(J

6r<A

.451655

32

29

.523138

5.95

SAC

.974391

.75

r'O

.548747

.70

6r*A

.451253

31

30

.523495

.95
5.95

.974347

. ro

.549149

. <o
6.68

.450851

30

31
32
33

9.523852
.524208
.524564

5.93
5.93

SAO

9.974302
.974257
.974212

.75

.75

9.549550
.549951
.550352

6.68
6.68

/> or*

10.450450
.450049
.449648

29
28
27

34
35

.524920
.525275

. 93
5.92

5 A 1

.974167
.974122

.75
.75

.550752
.551153

6^68

6CK

.449248
.448847

26
25

36

.525630

.92

5t-\f\

.974077

.75

.551552

.or>

6r*ry

.448448

24

37

.525984

.90

.974032

.75

.551952

.o7

6LZK.

.448048

23

38

.526339

5.92

5f\f\

.973987

.75

fygf

.552351

.b5
6f*r

.447649

22

39

.526693

.90

5OO

.973942

.75

r*cr

.552750

.bo

6f»e

.447250

21

40

.527046

.88

5f\f\

.973897

.75

.553149

.bo

6/>er

.446851

20

.90

.75

.bo

41

9.527400

5QO

9.973852

ftft*

9.553548

6/>o

10.446452

19

42

43

.527753
.528105

.88
5.87

5OO

.973807
.973761

.75

.77

.553946
.554344

.bo
6.63

6i *•>

.446054
.445656

18
17

44

.528458

.88

5Qfy

.973716

r*c

.554741

.D/v

6£»O

.445259

16

45

.528810

.87

5 DC

.973671

.75

r*r/

.555139

.DO

6 CO

.444861

15

46

.529161

.85

5 or*

.973625

.77

.55'536

.b2

6^*O

.444464

14

47

48

.529513
.529864

.87
5.85

5 OK

.973580
.973535

.75
.75

rrff

.555933
.556329

.bis
6.60

6f*{\

.444067
.443671

13

12

49

.530215

>o
500

.973489

.77

r*K

.556725

.DO

A t\C\

.443275

11

50

.530565

.00
5.83

.973444

.75

.77

.557121

G'.GO

.442879

10

51
52

9.530915
.531265

5.83

f DiTk

9.973398
.973352

.77

9.557517
.557913

6.60

6 to

10.442483
.442087

9

8

53

.531614 2'o^

.973307

.75

.558308

.00
6 to

.441692

7

54
55
56

.531963
.532312
.532061

o.oa
5.82

5.82

5CA

.973261
.973215
.973169

.77
.77
.77

r-c

.558703
.559097
.559491

.00
6.57
6.57

6tr*

.441297
.440903
.440509

6
5
4

57
58

.533009
.533357

.oU

5.80

5 r'O

.973124
.973078

•i7

.559885
.560279

.5<
6.57

6tr*

.440115
.439721

3
o

59
60

.533704
9.534052

.78
5.80

.973032
9.972986

!77

.560673
9.561066

.5l

6.55

.439327
10.438934

1
0

'

Cosine.

D. r.

Sine. D. 1".

Cotang.

D. 1".

Tang. ! '

109'

123

70*

20°

COSINES, TAN-GENTS, AND COTANGENTS.

159-

'

Sine.

D. r.

Cosine.

D. r.

Tang.

D. r.

Cotang.

'

0

9.534052

5r*Q.

9.972986

77

9 561066

6tiK

10.438934

60

1

534399

. i O

5C"7

.972940

• 1 1

77

.561459

DO

6rq

.438541

59

2

.534745

. 4 t

5f**O

.972894

. 1 1

77

561851

OO

.438149

58

3

.535092

. <o

972848

1 1

77

562244

0 . OO

fi *S3

.437756

57

4

.535438

o.ii

5r*e

972802

i i

78

.562636

U . »JO

6 to

.437364

56

5

535783

. IO

5r*py

972755

IO

.563028

. oo

6trt

.436972

55

6

.536129

. (7

5r*t*

.972709

.77

tyry

563419

D<£

6K O

.436581

54

7 ! .536474

(5

5(~Q

.972663

. 1 1

77

563811

Oo

6 FLO

.436180

53

8

.536818

. IO

5rfe

972617

i i
78

.564202

1MB

6^9

.435798

52

9

.537163

. IO

5fQ

972570

. IO

77

.564593

. O<£
6^0

.435407

51

10

.537507

. ia

5.73

.972524

. 1 1

.77

.564983

OU

6.50

.435017

50

11

9.537851

'••O

9.972478

78

9.565373

6KA

10.434627

49

12
13
14

538194
538538
538880

5. '73
5.70

5r-t>

972431
972385
972338

. IO

.77
78

78

565763
.566153
566542

. c/U

6.50
6.48

6^0

.434237
.433847
.433458

48
47
46

15

.539223

• I JO

5r*f\

.972291

. 1 O

.566932

. *JL/

6-17

.433068

45

16

539565

. iO
5 '""A

.972245

i~C

.567320

. *i I
6JQ

.432680

44

17

.539907

. i(J

972198

. IO

567709

- T-O
6<lft

.432291

43

18

.540249

5.68

5/?O

.972151

• 1 O

.568098

. ^O

6Aiy

.431902

42

19

.540590

.68

r GO

.972105

.77

r-Q

.568486

.**t

.431514

41

20

.540931

5. bo
5.68

.972058

. IO

.78

.568873

QA7

.431127

40

21
22
23

9.541272
.541613
.541953

5.68
5.67

9.972011
.971964
971917

.78
.78

9.569261
.569648
.570035

6.45
6.45

fi 4^

10.430739
.430352
.429965

39

38
37

24

. 542293

5.67

f CK.

971870

r*o

.570422

\) . TcJ

.429578

36

25

.542632

5.05

.971823

. IO

r-o

.570809

K 4^

.429191

35

26

.542971

5 . 65

.971776

. IO

78

.571195

fi 4^

.428805

34

27

.543310

5 . 05

.971729

.10
r-o

.571581

64^

.428419

33

2«
29
30

543649
.543987
.544325

5 . 60
5.63
5.63
5.63

.971682
.971635
.971588

. 1 O

.78
.78
.80

.571967
.572352

.572738

.•iO

6.42
6.43
6.42

.428033
.427648
.427262

32
31
30

31

9.544663

9.971540

78

9.573123

fi 40

10.426877 29

32
33
34

35

.545000
.545338
.545674
.546011

5 '63
5.60
5.62

K CA

.971493
.971446
.971398
.971351

. 1 O

.78
.80
.78

80

.573507
.573892
.574276
.574660

U - Ttv

6.42
6.40
6.40
fi 40

.426493 28
.426108 27
.425724 26
.425340 25

36

.546347

O . DU

.971303

. ov/

78

.575044

u . ^v

600

.424956 24

37

.546683

r rrt

.971256

. 1 O

80

.575427

. OO

fi 3ft

.424573 23

38

.547019

5^Q

.971208

. OU
78

.575810

vl . »JO
fi 3ft

.424190 22

39

.547354

• "O O'"*'! "1 (*~\
pr ro .y ill 01

.10

QA

.576193

U . OO
fi 3ft

.423807

21

40

.547689

O. Oo n^-i 1 1 o

5.58 -9'1113

oO
.78

.576576

6^38

.423424

20

41
42
43

9.548024
. 548359
.548693

5.58
5 57

5*- fy

9.971066
.971018
.970970

.80

.80

QA

9.576959
.577341
.577723

6.37
6.37

10.423041
.422659
.422277

19
18
17

44

.549027

.57

.970922

.80
en

.578104

fi O*"*

.421896

16

45
46

.549360
.549693

o . o5
5.55

.970874
970827

. of

.78

OA

.578486

.578867

6!.35

6OC

.421514
.421133

15
14

47
48
49
50

. 550026
.550359
.550092
.551024

5 . 55
5.55
5.55
5.53
5.53

.970779
.<I7'0731
.970683
970635

.80
.80
80
,80

.82

.579248
579629
.580009
580389

. oO

6.35
6.33
6.33
6.33

.420752
.420371
.419991
.419611

13
12

11

10

R1

9.551356

5*0

9.970586

OA

9.580769

6QQ

10.419231

9

52

.55168?

. •>>*
5 5''

H70538

.OU

.581149

.00

639

.418851

8

53

.552C18

970490

CA

581528

. O^i

600

.418472

7

54
65

.552349

. 552USO

5 . 52
5.52

f- fc s\

.970442
.970394

.oO
.80

Ob~k

581907
.582286

.64

6.32
600

.418093
.417714

6
5

56

.553010

5 . oO

.970345

.82

OA

.582665

.32

600

.417335

4

57

.553311

^48

.970297

oO

OA

.583044

.oXJ
fi '•?(!

.416956

3

58

.553670

•J.4O

.970249

. oU

Oil

.583422

U . OU
6OA

.416578

2

59

.554000

t" id

.970200

.82

.583800

.dO

.416200

1

60

9.554329

G.4o

9.970152

.80

9.584177

6.28

10.415823

0

'

Cosine.

D. r.

Sine.

D. r.

Cotang. D. 1'.

Tang.

'

110a

124

69«

TABLE X. — LOGARITHMIC SINES,

158°

/

Sine.

D. r.

Cosine.

D. 1'.

Tang.

D. 1'.

Cotang.

/

0

9.554329

5 JO

9.970152

On

9.584177

BOA

10.415823

60

1

.554658

.48

.9701U3

.82

.584555

.60
600

.415445

59

2

.554987

5.48

5 Aft

. 970055

.80

On

.5S4932

.28

6OQ

.415068

58

3

.555315

.47

5 A *t

.970006

.82

Ob)

.585309

.28

6OQ

.414691

57

4

.555643

.47

5tri

.969957

.82

or\

.585686

.2o

GO'V

.414314

56

5
6

.555971
.556299

.47
5.47

54 K

.969909
.969860

.80

.82

on

.586062
.586439

.27

6.28

60^

.413938
.413561

55
54

7
8

.556626
.556953

.45
5.45

5 A "*

.969811
.969762

.82
.82

OA

.586815
.587190

.27
6.25

6 or*

.413185
.412810

53

52

9

.557280

.45

5 A O

.969714

.80

on

.587566

.27

6OC

.412434

51

10

.557606

.4.5

5.43

.969665

.82
.82

.5b;941

.25
6.25

.412059

50
r

11
12
13

9.557932
.558258
.558583

5.43
5.42

34 O

9.969616
.969567
.969518

.82

.82

on

9.588316
.588691
.589066

6.25

6.25
600

10.411684
.411309
.410934

49
48
47

14

.558909

.43

5m

.969469

.82

on

.589440

.23
600

.410560

46

15
16

.559234

.559558

.42
5.40

5IQ

.969420
.969370

.82
.83

OO

.589814
.590188

.2o
6.23

6 no

.410186
.409812

45
44

17

.559883

.4!w

54f\

.969321

.06
On

.590562

.43

6t"W)

.409438

43

18

.560207

.40

5A(\

.969272

.82

on

.590935

.22
6i~>o

.409065

42

19
20

.560531
.560855

.40
5.40
5.38

.969223
.969173

.82
.83
.82

.591308
.591681

.22
6.22
6.22

.408692
.408319

41
40

21
22

9.561178
.561501

5.38

5OO

9.9C9124
.969075

.82

OO

9.592054
.592426

6.20

6Ort

10.407946
.407574

89

38

23
24

.561824
.562146

.08
5.37

5f)rt

.969025
.968976

.00

.82

OO

.592799
.593171

.22
6.20

61 o

.407201
.406829

37
36

25

.562468

.67

5tjrf

.968926

.80

on

.593542

.lo

6OA

.406458

35

26

27

.562790
.563112

.67
5.37

5 OK

.968877
.968827

.82
.83

OO

.593914
.594285

.20
6.18

6-4 o

.406086
.405715

34
33

28
29
30

.563-133
.563755
.564075

.65
5.37
5.33
5.35

.968777
.968728
.968678

.83
.82
.83
.83

.594656
.595027
.595398

.18
6.18
6.18
6.17

.405344
.404973
.404602

32
31
30

31
32

9.564396
.564716

5.33
500

9.968628
.968578

.83

OO

9.595768
.596138

6.17

61 rf

10.404232
.403862

29

28

33
34
35
36
37
38

.565036
.565356
.565676
.565995
.566314
.566632

.63
5.33
5.33
5.32
5.32

5.30
500

.968528
.968479
.968429
.968379
.968329
.968278

.00

.82
.83
.83
.83
.85

OO

.596508
.596878
.557247
.597616
.597985
.598:354

.17
6.17
6.15
6.15
6.15
6.15

61 O

.403492
.403122
.402753
.402384
.402015
.401646

27
26
25
24
23
22

39
40

.566951
.567269

.62
5.30

5OA

.968228
.968178

.80
.83

OO

.598722
.599C91

.lo
6. 15

61 O

.401278
.400909

21
20

.oO

.83

.lo

41
42

43
44

9.567587
.567904
.568222
.568539

5.28
5.30
5.28

5OQ

9.968128
.968078
.968027
.967977

.83

.85
.83

OO

9.599459

.599827
.600194
.600562

6. 13
6.12
6.13

61 rt

10.400541
.400173
.399806
.399438

19

i 18
17
16

45

.568856

.2o

5£\iy

.967927

.00

OC

.600929

.12

61 O

.399071

15

46
47

.569172
.569488

.27

5.27

5c\ff

.967876

.967826

.00

.83

Of

.601296
.601663

.12
6.12

61 A

.398704
.398337

14
13

48

.569804

.27

50^

.967775

.85

OO

.602029

.10

61 A

.397971

12

49

.570120

.27

5 OK

. 967725

.83

OC

.602395

.10

61 A

.397605

11

50

.570435

.25
5.27

.967674

.80
.83

.602761

.10
6.10

.397239

10

51

9.570751

5nK

9.967624

oe

9.603127

61 A

10.396873

9

52

.571066

.535

5 no

. 967573

.85

OP

.603493

. 1(J

6AO

.396507

8

53

.571380

.26

5£~iE

.967522

.80

OST

.603858

.08

6r\o

.396142

ft
i

54

.571695

.25

5OO

.967471

.85

OO

.604223

.08

6AQ

.395777

6

55
56

.572009
.572323

.26
5.23

5 OO

.967421
.967370

.00

.85

O**

.604588
.604953

.08
6.08

6r\rr

.395412
.395047

5

4

57

.572636

.66
500

.967319

.80

f)ff

.605317

.Ol

6AO

.394683

3

58

.572950

.26
500

.967268

.85

OK

.605682

.08
6cft

.394318

2

59
60

.573263
9.573575

.22
5.20

.967217
[ 9.967166

.85

.85

.606046
9.606410

.07
6.07

.393954
10.393590

1
0

/

Cosine.

D. 1".

Sine.

D. r.

Cotang.

D. r.

Tang.

/

111'

125

68=

COSINES, TANGENTS, AND COTANGENTS.

157«

1

Sine.

D. 1".

' Cosine.

D. 1'.

Tang.

D. r.

Cotang.

'

0

1

2

3
4
5

6

7
8

9.573575
.573888
.574200
.574512
.574824
.575136
.575447
.575758
.576069

5.22
5.20
5.20
5.20
5.20
5.18
5.18
5.18

517

9.967166
.967115
.967064
.967013
.966961
.966910
.966859
.966808
.966756

.85
.85
.85
.87
.85
.85
.85
.87

CK

9.606410
.606773
.607137
.607500
.607863
.608225
.608588
.608950
.609312

6.05
6.07
6.05
6.05
6.03
6.05
6.03
6.03

6 flQ

10.393590
.393227
.392863
.392500
.392137
.391775
.391412
.391050
.390688

60
59
58
57
56
55
54
53
52

9

.576379

. 1 i

51 *7

,966705

.00

.609674

.UO

6AO

.390326

51

10

.576689

.ll

5.17

.966653

!85

.610036

.06
6.02

.389964

50

11
12
13
14
15
16
17
18

9.576999
.577309
.577618
577927
.578236
.578545
.578853
.579162

5.17
5.15
5.15
5 15
5.15
5.13
5.15

K 10

9.966602
.966550
.966499
.966447
.966395
.966344
966292
.966240

.87
.85
.87
.87
.85
.87
.87

Off

9.610397
.610759
.611120
.611480
611841
612201
612561
.612921

6.03
6.02
6.00
6.02
6.00
6.00
6.00

61 A

10 389603
.389241
.388880
.388520
.388159
.387799
.387439
.387079

49

48
47
46
45
44
43
42

19
20

.579470
.579777

O. 1O

5.12
5.13

.966188
.966136

Ol

.87
.85

.613281
.613641

.( 0

6.00
5.98

.386719
.386359

41
40

21
22

9.580085
580392

5.12

51 O

9.966085
966033

.87

Qri

9.614000
.614359

5.98

5OQ

10.386000 39
.385641 38

23
24

.580699
.581005

. 1/4

5.10

51 O

.965981
.965929

87

.87

QQ

.614718
.615077

.98
5 98

r nr*

.385282 37
384923 36

25

.581312

.1/5
K If)

.965876

.88
Q?

.615435 ?-£i

.384565 35

26

581618

O 1U

pr -j f)

.965824

-O<

orj"

.615793 ° XX

.384207

34

27
28
29

.581924

.582229
.582535

5." 08
5.10

SAG

.965772
.965720
.965668

87
.87
.87

QQ

.616151
.616509
616867

O.VI

5.97
5.97

5O.K

.383849 33
.383491 32
.383133 .11

30

.582840

.Uo
5.08

.965615

.88
.87

.617224

95

5 97

.382776

30

31
32

33
34
35
36

9.583145
.5a3449
583754
.584058
.584361
.584665

5.07
5.08
5.07
5.05
5.07

R AX

9 965563
965511
965458
965406
965353
965301

.87
.88
87
.88
.87

QQ

9 617582
617939

.618295
.618652
619008
.619364

5.95
5.93
5.95
5.93
5.93

5 no

10 382418
.382061
.381705
.381348
.380992
.380636

20

28
27
26
25
24

37

.584968

O . UO

5r\r*

965248

.OO

OQ

.619720

. */O
500

.380280

23

38
39
40

•.585272
.585574
.585877

.UY
5.03
5.05

5AQ

.965195
.965143
.965090

88
.87
.88

oo

.620076
.620432
.620787

y»i
5 93
5.92

5 no

.379924
.379568
.379213

22
21
20

.Uo

.88

.yis

41

9.586179

5AR

9.965037

QQ

9.621142

ft QQ

10.378858

19

42
43
44
45

.586482
.586783
.587085
.587386

. UO

5.02
5.03
5.02

5flQ

.964984
.964931
.964879
.964826

.80

.88

.87
88

QO

.621497

.621852
.622207
.622561

5^92
5.92
5.90

5f\f\

.378503
.378148
.377793
.377439

18
17
16
15

46

.587688

. Uo

5 no

964773

.88

QQ

.622915

.vO
5nn

.377085

14

47

48

.587989
.588289

.\)£

5.00

SAO

.964720
964666

.88
.90

OO

.623269
.623623

.yU
5.90

5OO

.376731
.376377

13

12

49

.588590

.02

fr r\ft

964613

.88

QQ

.623976

.88

5AA

.376024

11

50

.588890

O.UU

5.00

.964560

.88
.88

.624330

.90
5 88

.375670

10

51

9.589190

4O.Q

9.964507

OO

9.624683

5QQ

10.375317

9

52
53

.589489
.589789

.98
5.00

4OQ

.964454
.964400

.88
.90

QQ

.625036
.625388

.88
5.87

5 no

.374964
.374612

8

54

.590088

.98

4(\Q

.964347

.OO
OQ

.625741

.00
5 Of?

.374259

6

55
56

.590387
.590686

.98
4.98

A Q7

.964294
.964240

.88
.90

QQ

.626093
.626445

.87
5.87

t Q*7

.373907
.373555

5

4

57

58

590984
591282

*± . y i
4.97

A Qr*

.964187
.964133

.OO

.90

00

.626797
.627149

5^87

Sory

.373203
.372851

3

2

59
60

.591580
9.591878

4^97

964080
9.964026

.08
.90

.627501
9.627852

.87
5.85

.372499
10.372148

1
0

i

Cosine. D. 1".

Sine.

D. 1".

Cotang.

D. r.

Tang. | '

112°

126

67C

23*

TABLE X. — LOGARITHMIC SINES,

156'

i

Sine.

D. r.

Cosine. D. 1".

Tang.

D. r.

Cotang.

'

0

1

2
3
4

9.591878
.592176
.592473
.592770
.593067

4.97
4.95
4.95
4.95

4OQ

9.964026
.963972
.963919
.963865
.963811

.90
.88
.90
.90

9.627852
.628203
.628554
.628905
.629255

5.85
5.85
5.85
5.83

10.372148

.371797
.371446
.371095
.370745

60
59
58
57
56

1OCOI-OOO5O

.593363
.593659
.593955
.594251
.594547
.594842

.yo
4.93
4.93
4.93
4.93
4.92
4.92

.963757
.963704
.963650
.963596
.963542
.963488

'.m

.90
.90
.90
.90
.90

.629606
.629956
.630306
.630656
.631005
.631355

O . OO

5.83
5.83
5.83
5.82
5.83
5.82

.370394
.370044
.369694
.369344
.368995
.368645

55
54
53
52
51
50

11
12
13
14
15
16
17
18
19
20

9.595137
.595432
.595727
.596021
.596315
.596609
.596903
.597196
.597490 1
.597783

4.92
4.92
4.90
4.90
4.90
4.90
4.88
4.90
4.88
4.87

9.963434
.963379
.963325
.963271
.963217
.963163
.963108
.963054
.962999
.962945

.92
.90
.90
.90
.90
.92
.90
.92
.90
.92

9.631704
.632053
.632402
.632750
.6.33099
.633447
.633795
.634143
.634490
.634838

5.82
5.82
5.80
5.82
5.80
5.80
5.80
5.78
5.80
5.78

10.368296
.367947
.367598
.367250
.366901
.366553
.366205
.365857
.365510
.365163

49
48
47
46
45
44
43
42
41
40

21

9.598075 . A oo

9.962890

on

9.635185

K 7Q

10.364815

39

22

.598368

^t.oo

4Q"^

.962836

.yu
no

.635532

O . i O

.364468

38

23

.598660

.87

4Q*?

.962781

.92

.635879

9 'is

.364121

37

24
25

.598952
.599244

.8<
4.87

4OOf

.962727
.962672

!92

no

.636226
.636572

O . t o

5.77

5r/o

.363774
.363428

36
35

26

.599536

,oi

4O"

.962617

.92
no

.636919

. (8

.363081

34

27

.599827

.80

4Q"'

.962562

.92

.637265

5.77

577

.362735

33

28
29

.600118
.600409

.80

4.85

4O*r

.962508
.962453

!92

.637611
.637956

. i (

5.75

577

.362389
.362044

32
31

30

.600700

.OO

4.83

.962398

'.92

.638302

. l (

5.75

.361698

30

31

32

9.600990
.601280

4.83

4QO

9.962343

.962288

.92

OO

9.638647
.638992

5.75

S^K.

10.361353
.361008

29

28

33

.601570

.80
4QO

.962233

.9*
no

.639337

. (O

.360663

27

34
35
36

.601860
.602150
.602139

.83

4.83
4.82

4QO

.962178
.962123

.962067

.92
.92
.93

no

.639682
.640027
.640371

5.75
5.75
5.73

.360318
.359973
.359629

26
25
24

37

.602728

.08

4 82

.962012

.92

.640716

5.75

R 70.

.359284

23

38
39
40

.603017
.603305
.603594

4^80
4.82
4.80

.961957
.961902
.961846

!92
.93
.92

.641060
.641404
.641747

O . 1 0

5.73
5.72
5.73

.358940
.358596
.358253

22
21
20

41

9.603882

4 fin

9.961791

QQ

9.642091

R 70

10.357909

19

42

.604170

.DU
4r*o

.961735

. yo

no

.642434

\Jft&
5r*o

.357566

18

43

.604457

. (8

4Qf\

.961680

.92
no

.642777

. tii
5r*o

.357223

17

44

.604745

.oU

4f"*Q

.961624

.9o
9.-»

.643120

. r&

5r*o

.356880

16

45

.605032

. <8

4r-o

.961569

2

no

.643463

. <2

5r-o

.356537

15

46

.605319

. 18

4***Q

.961513

.9o
no

.643806

. 1 «

.356194

14

47

.605606

. rfO
4rtff

.961458

.92

.644148

5.70

.355852

13

48

.605892

. *7
J. '"ft

.961402

QQ

.644490

5.70
R 7A

.355510

12

49

.606179

4 . i O

.961346

'no

.644832

O . t \J
R 7f)

.355168

11

50

.606465

4^77

.961290

. yo
.92

.645174

O . 4 \J

5.70

.354826

10

51

9.606751

A 7K

9.961235

93

9.645516

R (\Q

10.354484

9

52

.607036

<± . t O

.961179

no

.645857

*J . UO

5r*rt

.354143

8

53

.607322

4.77

4W£

.961123

.9o
OQ

.646199

. <u

5(\Q

.353801

7

54

.607007

. <O

.961067

.yo

no

.646540

.Do
5^>o

.353460

6

55

.607892

4.75

4*~K

.961011

.93

QQ

.646881

.08

5 A3

.353119

5

56

.608177

. (O

.960955

.yo

no

.647222

.Do
5f*i*f

.a52778

4

57

.608161

4.73

4MB)

.960899

.93

.647562

. D t

5f»Q

.352438

3

58

.608745

. 73

4r*o

.960843

f\K

.647903

.68
5l*<*t

.352097

2

59

.609029

. <O

4^*»>

.960786

.95

f|O

.648243

.DY

5s*ry

.351757

1

60

9.609313

. 73

9.960730

.93

9.648583

.07

10.351417

0

' \ Cosine.

D. r.

Sine.

D. r.

Cotang.

D. r.

Tang.

'

113'

127

66a

24°

COSINES, TANGENTS, AND COTANGENTS.

155'

1

'

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. 1'.

Cotang.

'

0

9.609313

A ^Q

9.960730

no

9.648583

10.351417

60

1

.609597

*± . t O

4ruh}

.960674

. JO

.648923

0 . 0 1

5 e*ry

.351077

59

2

.609880

. -a

.960618

Q^

.649263

.o7

5ce

.350737

58

3

.610164

HO .960561

.yo

ClQ

.649602

.DO

t. Rff

.350398

57

4

.610447

r#; .960505

. U*J

.649942 ^

.350058 56

5

.610729

HX .960448

'no.

.650281

O.OO

.349719

55

6

.611012

*± . * <6

4r*r\

.960392

. yo

r\K

.650620

o . bo

f nr*

.349380

54

7

.611294

. (0

47(1

.960335

.95

qq

.650959 g'lg

.349041

53

8

.611576

. i U
A T»

.960279

. yo

OA

.651297

5CK

.348703

52

9

.611858

4. t\J

4f"f\

.960222

.yo

.651636

.DO
5nn

.348364

51

10

.612140

. tO
4.68

.960165

'.93

.651974

.bo
5.63

.348026

50

11
12

9.612421

.612702

4.68

4 AS

9.960109
.960052

.95

9.652312
.652650

5.63

10.347688
.347350

49

48

13
14
15

.612983
.613264
.613545

.Do

4.68
4.68

4 pry

.959995
.959938
.959882

^95
.93

.652988
.653326
.653663

5^63
5.62

5/jrx

.347012
.346674
.346337

47
46
45

16

.613825

.67

4A7

.959825

QK

.654000

.62

£ AO

.346000

44

17

18

.614105
.614385

.O<

4.67

4R7

.959768
.959711

. yo

.95

QK

.654337
.654674

O.GS

5.62

.345663
.345326

43
42

19

.614665

. \J t
4/jK

.959654

. yo
f\*t

.655011

O . Olv

K < '. i

.344989

41

20

.614944

.b5
4.65

.959596

.97
.95

.655348

5.b2
5.60

.344652

40

21
22
23
24
25

9.615223

.615502
.615781
.616060
.616338

4.65
4.65
4.65
4.63

4 CO

9.959539
.959482
.959425
.959368
.959310

.95
.95
.95
.97

9.655684
.656020
.656356
.656692
.657028

5.60
5.60
5.60
5.60

5CA

10.344316
.343980
.343644
.343308
.342972

39
38
37
36

26
27

28
29

.616616
.616894
.617172
.617450

.60
4.63
4.63
4.63

.959253
.959195
.959138
.959080

!97
.95
.97

.657364
.657699
.658034
.658369

.60
5.58
5.58
5.58

5 fro

.342636
.342301
.341966
.341631

34
33
32
31

30

.617727

4.62
4.62

.959023

!97

.658704

.58
5.58

.341296

30

31

32

' 9.618004
.618^81

4.62

4f>C\

9.958965

.958908

.95

9.659039
.659373

5.57

5 to

10.340961
.340627

29

28

33

.618558

.62

4CA

.958850

•7

.659708

.58

5frr»

.340292

27

34
35
36

.618834
.619110
.619386

.60
4.60
4.60

4f»f\

.958792
.958734
.958677

.'97

.95

fir'

.660042
.660376
.660710

.57
5.57
5.57

5tK

.339958
.339624
.339290

26
25
24

37

.619662

.bO

4f*f\

.958619

.97

.661043

.55

5K^f

.a38957

23

38

.619938

.bO

4 fro

.958561

'or?

.661377

.57

5fr cr

.338623

22

39
40

.620213
.620488

.58
4.58
4-58

.958503
.958445

!97
.97

.661710
.662043

.55
5.55
5.55

.338290
.337957

21
20

41
42
43
44
45
46
47
48

9.620763
.621038
.621313
.621587
.621861
.622135
.622409
.622682

4.58
4.58
4.57
4.57
4.57
4.57
4.55

9.958387
.958329
.958271
.958213
.958154
.958096
.958038
.957979

.97
.97
.97
.98
.97
.97
.98

07

9.662376
.662709
.663042
.663375
.663707
.664039
.664371
.664703

5.55
5.55
5.55
5.53
5.53
5.53
5.53

5KO

10.337624
.337291
.336958
.336625
.336293
.335961
.335629
.335297

19
18
17
16
15
14
13
12

49

.622956

4 . Of

.957921

. i7l

07

.665035

. OO

.334965

11

50

.623229

4^55

.957863

. y i
.98

.665366

5^53

.334634

10

51

9.623502

4 W

9.957804

07

9.665698

5KO

10.334302

9

52

.623774

4(r t;

.957746

. Jl

f\O

.666029

. \J&

5fr rt

.333971

8

53

54

.624047
.624319

.55
4.53
4^°.

.957687
.957628

.98
.98

.666360
.666691

.52
5.52

5CA

.333640
.333309

7
6

55

.624591

. OO
4 fro

.957570

f\O

.667021

. O\J

5t fc~k

.332979

5

56 1 .624863

.5-5 Qfr~ n1
4 fro . yO I'll

.98

no

.667352

.52

5t-f\

.332648

4

57

.625135

.53

.957452

. 98
08

.667682

.50

R KO

.332318

3

58

.625406

4 fro

.957393

. JO

.668013

O . O-w

5t f\

.331987

2

59

60

.625677
9.625948

.52
4.52

.957335
9.957276

!98

.668343
9.668673

.50
5.50

.a31657
10.331327

1
0

'

Cosine.

D. 1".

Sine.

D. 1".

Cotang.

D. 1". Tang.

'

114°

128

65'

25<

TABLE X. — LOGARITHMIC SIXES,

154°

/

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. r.

Cotang.

/

0

9.625948

A M

9.957276 nQ

9.668673 - ,

10.331327

60

1

.626219

A RO

.95721V

no

.669002 J'g

.330998

59

2

.626490

A *0

.957158

no

.669332 2'^

.330668

58

3
4
5
6

r*
1

8
9
10

.626760
.627030
.627300
.627570
.627840
.628109
.628378
.628647

4.50
4.50
4.50
4.50
4.48
4.48
4.48
4.48

.957099
.957040
.956981
.956921
.956862
.956803
.956744
.956684

.yo
.98
.98
1.00
.98
.98
.98
1.00
.98

.669661
.669991
.670320
.670649
.670977
.671306
.671635
.671963

D.'iO

5.50
5.48
5.48
5.47
5.48
5.48
5.47
5.47

.330339
.330009
.329680
.329351
.329023
.328694
.328365
.328037

57
56
55
54
53
52
51
50

11
13
13
14
15
16
17
18
19
20

9.628916
.629185
.629453
.629721
.629989
.630257
.630524
.630792
.631059
.631326

4.48
4.47
4.47
4.47
4.47
4.45
4.47
4.45
4.45
4.45

9.956625
.956566
.956506
.956447
.956387
.956327
.956268
.956208
.956148
.956089

.98
1.00
.98
1.00
1.00
.98
1.00
1.00
.98
1.00

9.672291
.672619
.672947
.673274
.673602
.673929
.674257
.674584
.674911
.675237

5.47
5.47
5.45
5.47
5.45
5.47
5.45
5.45
5.43
5.45

10.327709
.327381
.327053
.326726
.326398
.326071
.325743
.325416
.325089
.324763

49
48
47
46
45
44
43
42
41
40

21
22
23
24
25
26
27
28
29
30

9.631593
.631859
.632125
.632392
.632658
.632923
.633189
.633454
.633719
.633984

4.43
4.43
4.45
4.43
4.42
4.43
4.42
4.42
4.42
4.42

9.956029
955969
.955909
.955849
.955789
.955729
.955669
.955609
.955548
.955488

1.00
1.00
1.00
1.00
1.00
1.00
1.00
.98
1 00
1.00

9.675564
.675890
.676217
.676543
.676869
.677194
.677520
.677&46
.678171
.678496

5 43
5.45
5.43
5.43
5.42
5.43
5.43
5.42
5.42
5.42

10.324436
.324110
.323783
.323457
.323131
.322806
.322480
.322154
.321829
.321504

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9.634249
.634514
.634778
635042
.635306
. 635570
.635834
.636097
636360
.636623

4.42
4.40
4.40
4.40
4.40
4.40
4.38
4 38
4.38
4.38

9 955428
.955368
.955307
.955247
.955186
.955126
.955065
.955005
.954944
.954883

1.00
1.02
1.00
1.02
1.00
1.02
1.00
1.02
1.02
1.00

9.678821
679146
.679471
.679795
.680120
.680444
.680768
.681092
.681416
.681740

5 42
5 42
5.40
5.42
5.40
5.40
5.40
5.40
5.40
5.38

10.321179
.320854
.320529
.320205
.319880
.319556
.319232
.318908
.318584
.318260

29
28
27
26
25
24
23
22
21
20

41
42

43
44
45
46
47
48
49
50

9.636886
637148
.637411
.637673
.6379a5
.638197
.638458
.638720
.638981
.639242

4.37
4.38
4.37
4.37
4.37
4.35
4.37
4.35
4.35
4.35

9 954823
.954762
.954701
.954640
.954579
.954518
.954457
.954396
.954335
.954274

1 02
1.02
1.02
1.02
1.02
1.02
1 02
1 02
1.02
1.03

9.682063
.682387
.682710
.683033
. 683356
.683679
.684001
.684324
.684646
.684968

5.40
5.38
5.38
5 38
5.38
5.37
5.38
5.37
5.37
5.37

10 317937
.317613
.317290
.316967
.316644
.316321
.315999
.315676
.315354
.315032

19
18
17
16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

9.639503
.639764
.640024
.640284
.640544
.640804
.641064
.641324
.641583
9.641842

4.35
4.33
4.33
4.33
4.33
4.33
4.33
4.32
4.32

9 954213
.954152
.954090
.954029
.953968
.953906
.953845
.953783
.953722
9.953660

1.02
1.03
1.02
1.02
1 03
1.02
1.03
1.02
1.03

9.685290
.685612
.685934
686255
.686577
.686898
.687219
.687540
.687861
9.688182

5.37
5.37
5.35
5.37
5 35
5.35
5 35
5.35
5.35

10 314710
314388
314066
313745
.313423
.313102
.312781
.312460
.312139
10.311818

9
8
7
6
5
4
3
2
1
0

r

Cosine.

D. 1'.

Sine. I D. 1'. l| Cotang.

D. r. 1 Tang.

/

115'

129

64"

26°

COSINES, TANGENTS, AM) COTANGENTS.

153'

1

'

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. 1'.

Cotang.

'

0

1

2
3
4
5
6
7
8
9
10

9.641842
.642101
.642360
.642618
.642877
.643135
.643393
.643650
.643908
.644165
.644423

4.32
4.32
4.30
4.32
4.30
4.30
4.28
4.30
4.28
4.30
4.28

9.953660
.953599
.953537
.953475
.953413
.953352
.953290
.953228
.953166
.953104
.953042

1.02
1.03
1.03
1.03
1.02
1.03
1.03
1.03
1.03
1.03
1.03

9.688182
.688502
.688823
.689143
.689463
.689783
.690103
.690423
.690742
.691062
.691381

5.33
5.32
5.33
5.33
5.33
5.33

5. as

5.32
5.33
5.32
5.32

10.311818
.311498
.311177
.310857
.310537
.310217
.309897
.309577
.309258
.308938
.308619

60
59
58
57
56
55
54
53
52
51
50

11
12
13
14
15
16
17
18
19
20

9.644680
.644936
.645193
.645450
.645706
.645962
.646218
.646474
.646729
.646984

4.27
4.28
4.28
4.27
4.27
4.27
4.27
4.25
4.25
4.27

9.952980
.952918
,952855
.952793
.952731
.952669
.952606
.952544
.952481
.952419

1.03
1.05
1.03
1.03
1.03
1.05
1 03
1.05
1.03
1.05

9.691700
.692019
.6923-18
.692656
.692975
.693293
.693612
.693930
.694248
.694566

5.32
5.32
5.30
5.32
5.30
5.32
5 30
5.30
5.30
6.28

10.308300
.307981
.307662
.307344
.307025
.306707
.306388
.306070
.305752
.305434

49

48
47
46
45
44
43
42
41
40

21
22
23
24
25
26

9.647240
.647494
.647749
.648004
.648258
.648512

4.23
4.25
4.25
4.23
4.23

9.952356
.952294
.952231
.952168
.952106
.952043

1.03
1.05
1.05
1.03
1.05
1 0*1

9.694883
.695201
.695518
.695836
.696153
.696470

5 30

5.28
5.30
5.28
5.28

t OQ

10.305117
.304799
.304482
.304164

.303847
.303530

39
38
37
36
35
34

27

28
29
30

.648766
.649020
.649274
.649527

4! 23
4.23
4.22
4.23

.951980
.951917
.951854
.951791

i VAJ

1.05
1 05
1 05
1.05

.696787
.697103
.697420
.697736

O *o

5.27
5.28
5.27
5.28

.303213
.302897
.302580
.302264

33
32
31
30

31
32

as

34
35
36
37
38
39
40

9.649781
.650034
.650287
.650539
.650792
.651044
.651297
.651549
.651800
.652052

4.22
4.22
4.20
4.22
4.20
4.22
4.20
4.18
4.20
4.20

9.951728
.951665
.951602
.951539
.951476
.951412
.951349
.951286
.951222
.951159

1.05
1 05
1 05
1 05
1.07
1 05
1 05
1 07
l'05
1.05

9.698053
.698369
.698685
.699001
.699316
.699632
.699947
.700263
.700578
.700893

5.27
5.27
5.27
5.25
5.27
5.25
5.27
5.25
5.25
6.25

10.301947
.301631
.301315
.300999
.300684
.300368
.300053
.299737
.299422
.299107

29

28
27
26
25
24
23
22
21
20

41

9.652304

41Q

9.951096

1 07

9.701208

5 25

10.298792

19

42
43
44
45

.652555
.652806
.653057
653308

. lo

4.18
4.18

4.18
41"?

.951032
.950968
.950905
.950841

JL , vi

1.07
1 05
1 07
1 05

.701523
.701837
.702152
.702466

5^23
5.25

5.23
5 OK

.298477
.298163
.297848
.297534

18
17
16
15

46

.653558

. a

41 r*

.950778

-1 . "*•*

1 07

.702781

. -. • >

.297210 14

47
48

.653808
.654059

. if

4.18
4i*y

.950714
.950650

.1 . "•
1.07
1 07

.703095
.703409

5^23

.296005 13
.296591 , 12

49
50

.654309
.654558

. i i
4.15

4.17

.950586
.950522

1 ."'
1.07
1.07

.703722
.704036

5^23
5.23

.296278
.295964

11

10

51
52

9.654808
.655058

4.17

4-1 K

9.950458
.950394

1.07

9.704350
.704663

5.22

500

10.295650 9

.295337 8

53

.655307

.lo

41 *""

.950330

1f\i**

.704976

. «s»<W

5OO

.295024 ' 7

54
55

. 655556
.655805

.lo
4.15

A 1 X.

.950266
.950202

.VI

1.07

1 07

.705290
.705603

.553

5.22
5 22

.294710
.204307

6
5

56
57
58
59
60

. 65G054
656302
.656551
.656799
9.657047

4 . JO

4.13
4.15
4.13
4.13

.950138
.950074
.950010
.949945
9.949881

I . Ui

1.07
1.07
1.08
1.07

.705916
.706228
.706541
.706854
9.707166

5. '20
5.22
5.22
5.20

.294084
.293773

.293459
.293146
10.292834

4
3
2

1
0

'

Cosine.

D. 1".

Sine. D. r. Cotang.

D. r.

Tang.

'

116°

130

63"

27<

TABLE X.- -LOGARITHMIC SINES.

/

Sine.

D. r.

Cosine.

D. 1'.

Tang.

D. 1".

Cotang.

/

0

9.657047

41 O

9.949881

IAQ

9.707166

5OA

10.292KU

60

1

.657295

.la

41 o

.949816

.Uo

IA*™*

.707478

.20
5*~tA

.292522

59

2

.657542

.12

41 O

.949752

.0*

1A"**

.707790

.20

5OA

292210

58

3

4
5
6

r*
4

8
9
10

.657790
.658037
.658284
.658531
.658778
.659025
.659271
.659517

.lo

4.12
4.12
4.12
4.12
4.12
4.10
4.10
4.10

.949688
.949623
.949558
.949494
.949429
.949364
.949300
.949235

.0*
1.08
1.08
1.07
1.08
1.08
1.07
1.08
1.08

.708102
.708414
.708726
.709037
.709349
.709660
.709971
.710282

.20
5.20
5.20
5.18
5.20
5.18
5.18
5.18
5.18

.291898
.291586
.291274
.290963
.290651
.290340
.290029
.289718

57
56
55
54
53
52
51
50

11
12
13

9.659763
.660009
.660255

4.10
4.10

41 A

9.949170
. 949105
949040

1.08
1.08

1 AO

9 710593
.710904
711215

5.18

5.18

f 1 •»*

10.289407
.289096

.288785

49

48
47

14
15
16

17

660501
660746
.660991
. 661236

.10

4.08
4.08
4.08

4AQ

. 948975
.948910
.948845
948780

1.08
1.08
1.08
1.08

IAQ

.711525
.711836
.712140
.712456

5 1<
5.18
5.17

5.17
51 ff

.288475
.288164
.287854
.287544

46
45
44
43

18

.661481

.Uo

4AQ

.948715

.08

IAQ

.712766

.17

51 ^

.287234

42

19

.661726

.Uo
1 CV7

948650

.08

11 A

.713076

.17

51 >y

.286924

41

20

.661970

4.07

.948584

.10

1.08

.713386

.It

5.17

.286614

40

21
22
23
24
25

9.662214
.662459
.662703
662946
.663190

4.08
4.07
4 05

4.07
4/\*»

9.948519
948454
.948388
.948323
.948257

1.08
1.10
1.08
1.10

IAQ

9.713696
.714005
714:314
.714624
.714933

5.15
5.15
5.17
5.15

51 C

10.286304
.285995
.285686
285370
.285067

39

38
37
36
35

26

.663433

.Oo

4/\p*

.948192

.08

11 A

.715242

.15

51 f

.284758

34

27
28
29
30

.663677
.663920
.664163
.664406

.Oi
4.05
4.05
4.05
4.03

.948126
.948060
.947995
.947929

.10
1.10
1.08
1.10
1.10

.715551
715860
.716168
.716477

.lo
5. 15
5.13
5.15
5.13

284449
.284140
.283832
.283523

33
32
31
30

31
32

9.664648
.664891

4.05

4AO

9.947863
947797

1 10

11 f\

9.716785
.717093

5 13

51 O

10.283215

.282907

29

28

33

.665133

.Oo

4A9

.947731

.10

11 A

. 717401

.1-3

51 O

.282599

27

34

.665375

.0.3
4 no

947665

.10

1AO

.717709

.1-3

51 O

.282291

26

35
36
37

38
39

.665617
.665859
.666100
.666342
.666583

.Uo
4.03
4 02
4.03

4.02
4 no

.947600
. 947533
.947467
.947401
.947335

.08
1.12
1.10
1.10

1.10
11 n

.718017
. 718325
.718633
.718940
.719248

.13

5.13
5.13
5.12
5.13

51 hi

281983
.281675
.281367
281060
280752

25
24
23
22
21

40

.666824

.04
4.02

.947269

.10
1.10

.719555

.12
5.12

.280445

20

41
42

9.667065
.667305

4.00

4Ao

9 947203
.947136

1 12

11 A

9.719862
.720169

5.12

51 i

10.280138
279831

19
18

43
44

.667546
.667786

.02
4.00

A AO

.947070
947004

.10
1.10

11 Cl

.720476
.720783

.12
5.12

51 A

.279.V.24
.279217

17

16

45
46

.668027
668267

4^00

3f\Q

.946937
946871

.12
1.10

.721089
.721396

.10
5.12

51 f\

.278911
.278604

15
14

47
48
49
50

.668506
.668746
.668986
669225

.98
4 00
4.00
3.98
3.98

946804
.946738
.940071
.946604

1.12
1.10
1.12
1.12
1.10

.721702
.722009
.722315
.722021

.10
5.12
5.10
5.10
5.10

.278298
.277991
277085
.277379

13
12

11
10

51

9.669464

3{\Q

9.946538

11 t

9. 722! 127

5f\O

10.277073

9

52
53
54

55

669703
669942
670181
670419

.98
3.98
3.98
3.97

3 no

.946471
.940404
940337
.940270

.12
1.12
1.12
1.12

11 i^

723232
7'23538
.723844
724149

.08
5.10
5.10
5.08

51 t "'

.270768
276400
270156
275851

8
7
6
5

56

.670658

."o

Q O*'

940203

.12

11 fc">

.724454

.08

f -\ (\

.275546

4

58

670896
.671134

3^97

Sri'**

.946136
.946069

.12
1.12

11 hi

724760
725065

5.10
5.08

5AQ

275240
274935

3
2

59
60

.671372
9.671609

'it

3.95

946002
9.945935

.12
1.12

725370
9.725674

.08
5.07

274630
10.274326

1
0

'

Cosine.

D. 1".

Sine.

D. r.

Cotang.

D. r.

Tang.

'

117C

131

62°

28°

COSINES, TANGENTS, AND COTANGENTS.

151'

'

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

1

2
3
4
5
6

9.671609
.671847
.672084
.672321
.672558
.672795
.673032

3.97
3.95
3.95
3.95
3.95
3.95

9.945935
.945868
.945800
.945733
.945666
.945598
.945531

1.12
1.13

1.12
1.12
1.13
1.12

11 4~t

9.725674
.725979
.726284
.726588
.726892
.727197
.727501

5.08
5.08
5.07
5.07
5.05
5.07

10.274326
.274021

.273716
.273412
.273108
.272803
.272499

60

59
58
57
56
55
54

7

.673268

3.93

.945464

.12

11 O

.727805

5.07

.272195

53

8
9
10

.673505
.673741
.673977

3.95
3.93
3.93
3.93

.945396
.945328
.945261

.13

1.13
1.12
1.18

.728109
.728412
.728716

5.07
5.05
5.07
5.07

.271891
.271588
.271284

52
51
50

11

9.674213

3f\C\

9.945193

11 •>

9.729020

5ntf

10.270980

49

12
13
14

.674448
.674684
.674919

.92
3 93
3.92

.945125
.945058
.944990

.13
1.12
1.13

1-1 O

.729323
.729626
.729929

.05
5.05
5.05

.270677
.270374
.270071

48
47
46

15

. 675155

3.93

3f\f\

.944922

.13

1-t o

.730233

5.07

5/~\O

.269767

45

16
17
18
19

.675390
.675624
.675859
.676094

.92
3.90
3.92
3.92

3f\f\

.944854
.944786
.944718
.944650

.Id

1.13
1.13
1.13

11 O

.730535
.730838
.731141
.731444

.0.3
5.05
5.05
5.05

.269465
.269162
.268859
.268556

44
43
42
41

20

.676328

.90
3.90

.944582

.13

1.13

.731746

5.03
5.03

.268254

40

21
22

9.676562
.676796

3.90

9.944514
.944446

1.13

9.732048
.732351

5.05

5/"\O

10.267952
.267649

39

38

23
24
25

.677030
.677264
.677498

3.90
3.90
3.90

.944377
.944309
.944241

1 .15
1.13
1.13.

.732653
.732955
.733257

.03
5.03
5.03

.267347
.267045
.266743

37
36
35

26

.677731

3.88

3 no

.944172

1.15

11 O

.733558

5.02

5/-\n

.266442

34

27

.677964

.88

3OO

.944104

.13

11 O

.733860

.03

5f\Ct

.266140

33

28

.678197

.88

3OO

.944036

.13

.734162

.03

51 \, i

.265838

32

29

.678430

.88

3OO

.943967

1.15

11 O

.734463

.02

5* i. >

.265537

31

30

.678663

.88

3 ow

.943899

.13

11 K

.734764

.02

5f\t\

.265236

30

.87

.15

.Oo

31

9.678895

3OO

9.943830

11 "

9.735066

511,1

10.264934

29

32

.679128

88

.943761

.15

11 O

.735367

.02

.264633

28

33

.679360

3.87

.943693

.13

.735668

5.02

5f\Cl

.264332

27

34
35
36

.679592
.679824
.680056

3.87
3.87
3.87

3 Off

.943624
.943555
.943486

1.15
1.15
1.15

11 "

.735969
.736269
.736570

.02
5.00
5.02

5f\n

.264031
.263731
.263430

26
25
24

37

38

.680288
.680519

.87
3.85

3 OK

.943417
.943348

.lo
1. 15

.736870
.737171

.00
5.02

5f\f\

.263130
.262829

23
22

39

.680750

.85

3 Off

.943279

1.15

11 K

.737471

.00

5f\f\

.262529

21

40

.680982

.87

3 Of

.943210

.15

11 "

.737771

.00

5AA

.262229

20

.80

.15

.00

41

9.681213

3OO

9.943141

11 K

9.738071

5f\f\

10.261929

19

42

.681443

.83

3OCT

.94^072

.15

11 "

.738371

.00

5f\f\

.261629

18

43
44
45

.681674
.681905
.682135

.85
3.85
3.83

3OO

.943003
.942934
.942864

.15
1.15
1.17

11 K

.738671
.738971
.739271

.00
5.00
5.00

4f\Q

.261329
.261029
.260729

17
16
15

46
47
48
49

.682365
.682595
.682825
.683055

.83
3.83
3.83
3.83

3Ot~l

.942795
.942726
.942656

.942587

.15
1.15
1.17
1.15

11 •**

.739570
.739870
.740169
.740468

.98
5.00
4.98
4.98

4f\Q

.260430
.260130
.259831
.259532

14
13
12
11

50

.683284

.82
3.83

.942517

.17
1.15

.740767

.98
4.98

.259233

10

51

52

9.683514
.683743

3.82

3Ort

9.942448

.942378

1.17

11 **

9.741066
.741365

4.98

4(\Q

10.258934
.258635

9

8

53
54
55
56
57
58
59
60

.6a3972
.684201
.684430
.684658
. 684887
.685115
.685343
9.685571

.82
3.82
3.82
3.80
3.82
3.80
3 80
3.80

.942308
.942239
.942169
.942099
.942029
.941959
.941889
9.941819

.17
1.15
1.17
1.17
1.17
1.17
1.17
1.17

.741664
.741962
.742261
.742559
.742858
.743156
.743454
9.743752

.98
4.97
4.98
4.97
4.98
4.97
4.97
4.97

.258336
.258038
.257739
.257441
.257142
.256844
.256546
10.256248

7
6
5
4
3
2
1
0

'

Cosine.

D. r.

Sine.

D. r.

Cotang.

D. r.

Tang.

'

118'

132

TABLE X. — LOGARITHMIC SINES,

150°

'

Sine.

D. r.

Cosine.

D. r.

Tang.

D. 1'.

Cotang.

'

0

1

2

9.085571
. OS5T99
.686027

3.80

3.80

o r-Q

9.941819
.941749
.941079

1.17
1.17

1 17

9.743752
.744050
.744348

4.97
4.97

10.256248
.255950
.255652

60"
59

58

3
4
5
6

7

.686254
.680482
.686709
.686936
.687163

O. IO

3.80
3.78
3.78
3.78

3rffy

.941609
.941539
.941469
.941398
.941328

1 . J i

1.17
1.17

1.18
1.17

1 17

.744645
.744943
.745240
.745538
.745835

4. 97
4.95
4.97
4.95

.255355
.255057
.254760
.254462
.254165

57
56
55
54
53

8
9
10

.687389
.687616
.687843

. 1 t

3.78
3.78
3.77

.941258
.941187
.941117

I . It

1.18
1.17
1.18

.74ol32
.746429
.746726

4^95
4.95
4.95

.253868
.253571
.253274

52
51
50

11
12

9. 688069
.688295

3.77
3rfy

9.941046
.940975

1.18

11 ^

9.747023
.747319

4.93

4(\K

10.252977
.252681

49
48

13
14
15

.688521
.688747
.688972

.11
3.77
3.75

O f*"V

.940905
.940834
.940763

. I i

1.18

1.18
11 <**

.747016
.747913

.748209

. JD

4.95
4.93

4rio

.252384
.252087
.251791

47
46
45

16

.689198

3.77

3r-"*

.940693

.14
11 Q

.748505

.U3

.251495

44

17
18
19

.689423
.689648
.689873

. o
3.75
3.75

3/^n-

.940622
.940551
.940480

.18
1.18
1.18

11 Q

.748801
.749097
.749393

4^93
4.93

1 O*-l

.251199
.250903
.250607

43
42
41

20

.690098

. <O

3.75

.940409

.18
1.18

.749689

4^93

.250311

40

21
22

9.690323
.690548

3.75

37Q

9.940338
.940267

1.18

1 1*3

9.749985
.750281

4.93

4QO

10.250015
.249719

39
38

23
24

.690772
.690996

. 10

3.73

.940196
.940125

i . lo

1.18

1 18

.750576

.75087'2

. u&
4.93

4 Qh>

.249424

.249128

37
36

25
26
27

.691220
.691444
.691668

3^73
3.73

37*}

.940054

.939982
.939911

1 . lo

1.20

1.18

11 O

.751167
.751402
.751757

*± . t7/W

4.92

4.92

A QO

.248833
.248538
.248243

35
34
33

28

.691892

. i o
3*^O

.939840

. lo

1OA

.752052

^ . *J&

4QO

.247948

32

29

.692115

. <••*

3f***>

.939768

.£0

1-i O

.752347

. J£

4O»>

.247653

31

30

.692339

. 10

3.72

.939697

.18
1.20

.752642

.\)Z

4.92

.247358 30

31

9.692562

q ***<>

9.939625

1 18

9.752937

4 on

10.247063 29

32

.692785

q 70

.939554

l . Jo

.753231

*± . <7V

A no

.246709 28

33

.693008

6. i~

q 70

.939482

1 on

.753526

*i iff*

A on

.246474 27

34

.693231

O. i «

q 7d

.939410

i i&j

1 18

.753820

T1 . «7<J

4 Q9

.246180

26

35

.693453

o . 4 u
o 7.)

.939339

i . IO

i on

.754115

t . u&
A ()Q

.245885

25

36
37

.693676
.693898

3 '70

370

.939207
.939195

l'20

19f\

.754409
.754703

4^90
4 on

.245591
245297'

24
23

38

.694120

. t U
3r*t\

.939123

. £\J
11 &

754997

. •}* j
4 on

.245003

22

39
40

.694342
.694564

. tO
3.7'0
3.7'0

.939052
.938980

. ]f5

1.20
1.20

.755291
.755585

.JO

4.90

4.88

.'J447T)9
.244415

20

41
42

9.694786
.695007

3.68

9.938908
.938836

1.20

9.755878
.756172

4.90

4 ftR

10.244122

.243828

19
18

43
44

.695229
.695450

3 68

O ijii

.938763
.938691

1.20
I ''0

.756465
.756759

*T . OO

4.90

A 88

.243535
.243241

17

16

45
46
47

.695071
.695892
.696113

3.08
3.68

o *;£

.938619
938547
.938475

1.20
1.20

1 OO

.757052
.757345

.757638

4 .00

4.88
4.88

4QU

.242948
.242655
.242362

15
14
13

48
49
50

.696334
.696554
.096775

• * . oo
3.07
3.68
3.67

.938402
.938330

.938258

l'20
1.20
1.22

.757931
.758224

.758517

.OO

4.88
4.88
4.88

.242069
.241776
.241483

12

11

10

51
52
53

9.696995
.697215
.697435

3.67

3.67

9.938185
.938113
.938040

1.20
1.22

1 OO

9.758810
.759102
.75931)5

4.87

4.88

10.241190
.240898

. 240605

9

8

7

54
55

.697654
.697874

3.' 07

Q a*?

.937907
.937895

L20

14 >fc"l

.759087
.759979

4^87

4Ot>

240313
.240021

6
5

56
57
58
59
60

.698094
.698313
.698532
.698751
9.698970

3^65
3.65
3.65
3.05

.937'822
.937749
.937676
.937604
9.937531

.22
1 22
1.22
1.20
1.22

.760272
.760561
.760856
.761148
9.761439

.00

4.87
4.87
4.87

4.85

.239728
.239436
.239144
.238852
10.238561

4

3
o

1
0

'

Cosine.

D. r.

Sine. I). 1'. Cotang. 1 D. 1". Tang. '

60'

30°

COSINES, TANGENTS, AND COTANGENTS.

'

Sine.

D. 1".

Cosine.

D. r.

Tang.

D. 1".

Cotang.

'

0

1

2
3
4
5
6
7

9.698970
.699189
.699407
.699626
.699844
.700062
.700280
.700498

3.65
3.63
3.65
3.63
3.63
3.63
3.63

Q f.Q

9.937531
.937458
.937385
.937312
.937238
.937165
.93r092
.937019

1.22
1.22
1.22
1.23
1.22
1.22
1.22

1 OO

9.761439
.761731
.762023
.762314
.762606
.762897
.763188
.763479

4.87

4.87
4.85
4.87
4.85
4.85
4.85

10.238561
.238269
.237977
.237686
.237394
.237103
.236812
.236521

60
59
58
57
56
55
54
53

8

.700716

O . Llt>

q «o

.936946

.1 . *w/v

1 '73

.763770

4 . oo

.236230

52

9

10

.70<Jy33
.701151

O . O/v

3.63
3.62

.936872
.936799

1.'22
1.23

.764061
.764352

4'85
4.85

.235939
.235648

51
50

11

9.7013G8

3 62

9.936725

1 22

9.764643

A CQ

10.235357 49

12
13

.7 '1585
.701802

3^62

Q fiO

.936652
.936578

1:23

1 22

.764933
.7652-24

1 . OO

4.85
4 83

.2:35067 48
.2:34776 47

14
15
16
17
18
19

.702019
.702236
.702452
.702669

.702885
.703101

O . ' ' --

3.62
3.60
3.62
3.60
3.60

Q fiO

.936505
.936431
.936357
.936284
.936210
.936136

l'23
1.23
1.22
1.23
i.23

.765514
.765805
.766095
.766385"
.766675
.766965

4 '85
4.83
4 83
4.83
4.83

.234486
.234195
.233905
.233615
.233325
.233035

46
45
44
43
42
41

20

.703317

O . Uv

3.60

.936062

1^23

.767255

4^83

.232745 40

21
22
23
24
25
23

9.703533
.703749
.703964
.704179
.704395
.704610

3.60
3.58
3.58
3.60
3.58

9 K.Q

9.935988
.935914
.935840
.935766
.935692
.935618

1.23
1.23
1.23
1.23
1.23

9.767545

.767834
.768124
.768414
.768703
.768992

4.82
4.83
4.83
4.82

4.82

400

10.232455

.232166
.231876
.231586
.231297
.231008

39
38
37
36
35
34

27 .704825 O'RQ

. 935543

1 2'3

.769281

.c>4
A CO

.230719 33

28 .705040 q'=7

.935469

1 . <V'J

1 9°.

.769571

*± . OO

4 so

.230429 32

29 .705254 «•«

.935395

1 .xwO

1 OK

.769860

. Ox5

4 an

.230140 31

30 .705469 \ 3^

.935320

1 . -wO

1.23

.770148

.oU

4.82

.229852 30

31

32
33
34
35

9.705683
.705898
.706112
.706326
.706539

3.58
3.57
3.57
3.55

9.935246
.935171
.935097
.935022
.9:34948

1.25
1.23
1.25
1.23

1 ^)^

9.770437
.770726
.771015
.771303
.771592

4.82
4.82
4.80
4.82

4Qf\

10.229563
.229274

.228985
.228697
.228408

29

28
27
26
25

36 ' .706753
37 .706967
38 ; .707180
39 .707393
4J .707606

3.57
3.57
3.55
3-55
3.55
3.55

.934873
.9:34798
.934723
.934649
.934574

1 . /w*J

1.25
1.25
1.23
1.25
1.25

.771880
.772168
.772457
.772745
.773033

.oO
4.80
4.82
4.80
4.80
4.80

.228120
.227832
.227543
.227255
.226967

24
23
22
21
20

41 9-707819
42 : .708032
43 .708245
44 .708458
45 .708670
46 .708882
47 .709094

3.55
3.55
3.55
3.53
3.53
3.53

3 to

9.934499
.934424
.934349
.934274
.934199
.934123
.9:34048

1.25
1.25
1.25
1.25
1.27
1.25

9.773321

.773608
.773896
.774184
.774471
.774759
.775046

4.78
4.80
4.80
4.78
4.80
4.78

4f**Q

10.226679
.226392
.226104
.225816
.225529
.225241
.224954

19
18
17
16
15
14
13

48 .709306

.Oo
3^

.933973 i'~

.775333

. ro

A GO

.224667 12

49 .709518

. OO
3 co

.933898 ,o2

.775621

^t . OU

.224379 11

50 .709730

.OO

3.52

.933822

l".25

.775908 l'^

.224092

10

51 9.709941
52 .710153

3.53

9.933747 , 1>7

.933G71 :":,(

9.776195
.776482

4.78

4r-ff

10.223805 9
.223518 8

53 .710364 t?o

.933596 io7

.77'6768

. 1 i

A 70

.223232 7

54

.710575

0-0

.933520 i Z-

.777055

•i . (O

47Q

.222945 6

55

.710786

3 to

933445 iS

.777342

. <o

4r!'7

.222658 5

56 .710997

.5x4

3KO

.933369 ,i~

.777628

. i 1

4f**O

.222372 4

57 .711208

.52

o £O

.933293

i .at
10**

.777915

. (O

4 "77

.222085 3

58
59
60

.711419
.711629
9.711839

3^50
3.50

.933217
.933141
9.933066

.2<
1.27
1.25

.778201
.778488
9.778774

.11

4.78
4.77

.221799
.221512
10.221226

2
1
0

'

Cosine.

D. r.

Sine.

D. 1'.

Cotang.

D. r.

Tang. '

. >

120'

134

31'

TABLE X. — LOGARITHMIC SIXES,

148°

'

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

9.7im39

3 to

9.93:5066

1 ^7

9.778774

4r~w

10.221226

60

1

.712050

. o&

3KH

.932990

1 27

.779060

. 4 4

477

.220940

59

2
3

.712260
.712469

. ou
3.48

.WJ2914
.932838

l!27

1 97

.779346
.779632

. 4 1

4.77
4w

.220654
.220368

58
57

4
5

6

7
8

.712679
.712889
.713098
.713308
.713517

O . OU

3.50
3.48
3.50
3.48

q 40

.932762
.932685
.932609
.932533
.932457

1 ./vtf

1.28
1.27
1.27
1.27
1 9"^

.779918
.780203

.780489
.780775
.781060

. 4 4

4.75
4.77
4.77
4.75

415*5'

.220082
.219797
.219511
.219225
.218940

56
55
54
53
52

9

.713726

O . *±O

q 40

.932380

1 27

.781346

. 4 4

4l~f;

.218654

51

10

.713935

o . ^o

3.48

.932304

JL .Ml

1.27

.781631

. t o

4.75

.218369

D

50

11

9-714144

q 47

9.932228

1 98

9.781916

10.218084

49

12

.714352

0 . 14

q 10

.932151

1 . *vO

.782201

4.45

4r-p-

.217799

48

13

.714501

o . -fo

.932075

1 OR

.782486

. (O

4pj[-

.217514

47

14
15

.714769
.714978

3'48

q 17

.931998
.931921

l!28

.782771
.783056

. ID

4.75

.217229
.216944

46
45

16

.715186

o . ~± i

q 47

.931845

1 '*>8

.7a3341

4.75

47K

.216659

44

17
18

.715394
.715602

O . ~i 1

3.47

34^

.931763
.9316111

i'.as

.783626
.783910

. (O

4.73

4 we

.216374
.216090

43
42

19
20

.715809
.716017

. ~i<J

3.47
3.45

.931614
.931537

1^28
1.28

.784195
.784479

. <o
4.73
4.75

.215805
.215521

41
40

21
22
23
24
25
26

9-716224
.716432
.716639
.716846
.717053
.717259

3.47
3.45
3.45
3.45
3.43

q 4K

9.931460
.931383
.931306
.931229
.931152
.931075

1.28
1.28
1.28
1.28
1.28
1 28

9.784764
.785048
.785332
.785616
.785900
.786184

4.73
4.73
4.73
4.73
4.73

4r-q

10.215236

.214952
.214668
.214384
.214100
.213816

39
38
37
36
35
34

27
28
29
30

.717466
.717673
.717879

.718085

3.45
3.43
3.43
3.43

.930998
.930921
.930843
.930766

1 . ^O

1.28
1.30
1.28
1.30

.786468
.786752
.787036
.787319

. 40

4.73
4.73

4.72
4.73

.213532

.213248
.212964
.212681

33
32
31
30

31

9.718291

q 40

9.930688

1 28

9.787603

479

10.212397

29

32
33
34
35
36
37
38
39
40

.718497
.718703
.718909
.719114
.719320
.719525
.719730
.719935
.720140

O . TO

3.43
3.43
3.42
3.43
3.42
3.42
3.42
3.42
3.42

.930611
.930533
.930456
.930378
.930300
.930223
.930145
.930067
.929989

1 ,^>O

1 30
1.28
1.30
1 30
1.28
1.30
1 30
1.30
1.30

.787886
.788170
.788453
.788736
.789019
.789302
.789585
.789868
.790151

. 4/£

4.73

4.72
4.72
4.72
4.72
4.72
4.72
4.72
4.72

.212114
.211830
.211547
.211264
.210981
.210698
.210415
.210132
.209849

28
27
26
25
24
23
22
21
20

41

42

9.720345

.720549

3.40

3 '19

9.929911
.929833

1 30
1 30

9.790434
.790716

4.70
4 72

10.209566

.209284

19
18

43

.720754

. ~i&

3A(\

.929755

J O\/

1 SO

.790999

47(1

.209001

17

44

.720958

. "TV

34O

.929677

J ZJ\J

1 30

.791281

. t \/

A T»

.208719

16

45

.721162

. "~z\J

3 40

.929599

1 .'JU

i 30

.791563

4 . 1 U

4 72

.208437

15

46
47

.721366
.72157'0

3^40

.929521
.929442

J^ *J\J

l'32
1 30

.791846
.792128

4 '.70
4 70

.208154
.207872

14
13

48

.721774

O Af\

.929364

i SO

.792410

1 . t\j
47rt

.207590

12

49
50

.721978

.722181

3^38
3.40

.929286
.929207

_1 *.j\t

lisa

1.30

.792692
.792974

. t \J

4.70
4.70

.207308
.207026

11
10

51

9.722385

3OQ

9.929129

1 * ''"*

9.793256

4r*A

10.206744

9

52
53

.722588
.722791

.08

3.38

q qo

.929050
.928972

i!so

.7935:38
.793819

. <0

4.68

.206462
.206181

8
7

54
55
56
57
58
59
60

.722994
.723197
.723400
.723603
.723805
.724007
9.724210

O . OO

3.38
3.38
3.38
3.37
3.37
3.38

.928893
.928815
.928736
.928657
.928578
.928499
9.928420

L30
1.32
1 32
1.32
1.32
1.32

.794101
.794383
.794664
.794916
.795227
.795508
9. 79578 J

4.70
4.68
4.70
4.68
4.68
4.68

.205899
.205617
.205336
.205054
.204773
.204492
10.204211

6
5
4
3
2
1
0

'

Cosine. 1 D. 1'.

Sine.

D. r.

Cotang.

D. r.

Tang.

'

12V

135

58=

32°

COSINES, TANGENTS. AND COTANGENTS.

147=

/

Sine.

D. 1'.

Cosine.

D. 1'.

Tang.

D. r.

Cotang.

/

0

1

2
3
4
5
6
7
8
9

9.724210
.724412
.724614
.724816
.725017
.725219
.725420
.725622
.725823
.726024

3.37
3.37
3.37
3.35
3.37
3.35
3.37
3.35
3.35

0 OK

9.928420
.928342
.928263
.988183
.928104
.928025
.927946
.927867
.927787
.927708

1.30
1.32
1.33
1.32
1.32
1.32
1.32
1.33
1.32
1 ^o

9.795789
.796070
.796351
.796632
.796913
.797194
.797474
.797755
.798036
.798316

4.68
4.68
4.68
4.68
4.68
4.67
4.68
4.68
4.67

A A7

10.204211
.203930
.203649
.203368
.203087
.202806
.202526
.202245
.201964
.201684

CO
59
58
57
56
55
54
53
52
51

10

.726225

3.35

.927629

1.33

.798596

4.68

.201404

50

11
12
13
14
15
16
17
18
19
20

9.726426
.726626

.726827
.72J 027

.727228
.727428

.727628

.727828
.728027

.728227

3.33
3.35
3.33
3.35
3.33
3 33
3.33
3.32
3.33
3.33

9.927549
.927470
.927390
.927310
.927231
.927151
.927071
.926991
.926911
.926831

1.32
1.33

1.33
1.32
1.33
1.33
1.33
1.33
1.33
1.33

9.798877
.799157
.799437
.799717
.799997
.800277
.800557
.800836
.801116
.801396

4.67
4.67
4.67
4.67
4.67
4.67
4.65
4.67
4.67
4.65

10.201123
.200843
.200563
.200283
.200003
.199723
.199443
.199164
.198884
.198604

49

48
47
46
45
44
43
42
41
40

21
22
23

9.728427

.728626

.728825

3.32
3.32

39.9

9.926751
.926671
.926591

1.33
1.33

1 00

9.801675
.801955
.802234

4.67
4.65

4A£

10.198325
.198045
.197766

39
38
37

24

.729024

.OS

3qo

.926511

1 .00
1 °.°,

.802513

.00

4R!i

.197487

36

25
26

27
28
29
30

.729223
.729422

.729621
.729820
.730018
.730217

.««

3.32
3.32
3.32
3.30
3.32
3.30

.926431
.926351
.926270
.926190
.926110
.926029

1.33
1.35
1.33
1.33
1.35
1.33

.802792
.803072
.803351
.803630
.803909
.804187

.DO

4.67
4.65
4.65
4.65
4.63
4.65

.197208
.196928
.196649
.196370
.196091
.195813

35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9 ,730415
.730613
.730811
.731009
.731206
.73J404
.731602
.731799
.731996
.732193

3.30
3.30
3.30
3.28
3.30
3.30
3.28
3.28
3.28
3.28

9.925949
.925868
.925788
.925707
.925626
.925545
.925465
.925384
.925303
.925222

1.35
1.33
1.35
1.35
1.35
1.33
1.35
1.35
1.35
1.35

9.804466
.804745
.805023
.805302
.805580
.805859
.806137
.806415
.806693
.806971

4.65
4.63
4 155
4.63
4.65
4 63
4.63
4.63
4.63
4.63

10.195534
.195255
.194977
.194698
.194420
.194141
.193863
.193585
.19:3307
.193029

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49

9.732390
.732587
.732784
.732980
.733177
.733373
.733569
.733765
.733961

3.28
3.28
3.27
3.28
3.27
3.27
3.27

3.27
30?

9.925141
.925060
.924979
.924897
.924816
.924735
.924654
.924572
.924491

1.35
1.35
1-37
1.35
1.35
1.35
1.37
1.35

1 Q7

9.807249
.807527
.807805
.808083
.808361
.808638
.808916
.809193
.809471

4.63
4.63
4.63
4.63
4.62
4.63
4.62
4.63

4AO

10.192751
.192473
.192195
.191917
.191639
.191362
.191084
.190807
.190529

19
IS
17
16
15
14
13
12
11

50

.734157

.fit
3.27

.924409

1 -Ol

1.35

.809748

.0^

4.62

.190252

10

51
52
53

9.734353
.734549
.734744

3.27
3.25

9.924328
.924246
.924164

1.37
1.37

9.810025
.810302
.810580

4.62
4.63

10.189975
.189698
.189420

9

8

r*
t

54

.734939

.^5

.924083

.35

.810857

.o2

.189143

6

55
56

57

.735135
.735330
.735525

.2<
3.25
3.25

.924001
.923919
.923837

.87

1.37
1.37

.811134
.811410
.811687

.o2
4.60
4.62

.188866
.188590
.188313

5
4
3

58
59
60

.735719
.735914
9.736109

.2o
3.25

3.25

.923755
.923673
9.033591

of
1.37
1.37

.811964
.812241
9.812517

.o2
4.62
4.60

.188036
.187759
10.187483

2

1
0

i Cosine.

D. r.

Sine.

D. r.

Cotang. D. r. Tang.

/

122'

136

57*

83°

TABLE X. — LOGARITHMIC SINES,

1*6°

'

Sine.

D. r.

Cosine.

D. 1'.

Tang.

D. 1".

Cotang.

'

0

1

2
3

9.736109
.736303
.736498
.736692

3.23
3.25
3.23

300

9.923591
.923509
.923427
.923345

1.37
1.37
1.37

Inw

9.812517
.812794
.813070
.813347

4.62
4.60
4.62

4/>/-v

10.187483
.1872U6
.186930
.186653

60
59
58
57

4
5
6
7
8
9
10

.736886
.737080
.737274
.737467
.737661
.737855
.738048

.23
3.23
3.23
3.22
3.23
3.23
3.22
3.22

.923263
.923181
.923098
.923010
.922933
.922851
.922768

.at

1.37
1.38
1.37
1.38
1.37
1.38
1.37

.813623
.813899
.814176
.814452
.814728
.815004
.815280

.bO
4.60
4.62
4.60
4.60
4.60
4.60
4.58

.186377
.186101
.185824
.185548
.185272
.184996
.184720

56
55
54
53
52
51
50

11
12
13
14
15
16
17
18
19
20

9.738241
.738434
.738627
.738820
.739013
.739206
.739398
.739590
.739783
.739975

3.22
3 22
3.22
3.22
3.22
3.20
3.20
3.22
3.20
3.20

9.922686
.922603
.922520
.922438
.922355
.922272
.922189
.922106
.922023
.921940

1.38
1.38
1.37
1.38
1.38
1.38
1 38
1.38
1.38
1.38

9.815555
.815831
.816107
.816382
.816658
.816933
.817209
.817484
.817759
.818035

4.60
4.60
4.58
4.60
4.58
4.60
4.58
4.58
4.60
4.58

10.184445
.184169
.183893
.183618
.183342
.183067
.182791
.182516
.182241
.181965

49
48
47
46
45
44
43
42
41
40

21
22
23
24

25
26
27

28

9.740167
.740359
.740550
.740742
.740934
.741125
.741316
.741508

3.20
3.18
3.20
3.20
3.18
3.18
3.20

31 O

9.921857
.921774
.921691
.921607
.921524
.921441
.921357
.921274

1.38
1.38
1.40
1.38
1.38
1.40
1.38

Ilrt

9.818310
.818585
.818860
.819135
.819410
.819684
.819959
.820234

4.58
4.58
4.58
4.58
4.57
4.58
4.58

4t •**

10.181690
.181415
.181140
.180865
.180590
.180316
.180041
.179766

39
38
37
36
35
34
33
32

29

.741699

.18

.921190

.40
1 ^w

.820508

.5<

.179492

31

30

.741889

3.17
3.18

.921107

1^40

.820783

4.58
4.57

.179217

30

31
32
33

9.742080
.742271
.742462

3.18
3.18

31 T

9.921023
.920939
.920856

1.40
1.38

1A(\

9.821057
.821332
.821606

4.58
4.57

4 fry

10.178943
.178668
.178394

29
28
27

34

.742652

.17
o 17

.920772

.40
1 40

.821880

.57

.178120

26

35
36
37
38
39

.742842
.743033
.743223
.743413
.743602

O. it

3.18
3.17
3.17
3.15

31 '**

.920688
.920604
.920520
.920436
.920352

1 . riU

1.40
1.40
1.40

1.40

i ir\

.822154
.822429
.822703
.822977
.823251

4^58
4.57
4.57
4.57

4K ""

.177846

Ir^urw 1*4 4
i toil

.177^97
.177023
.176749

25
24
23
22
21

40

.743792

.It
3.17

.920268

1 .40
1.40

.823524

.55

4.57

.176476

20

41
42
43

9.743982
.744171
.744361

3.15
3.17

31 X

9.920184
.920099
.920015

1.42
1.40

14 (\

9.823798
.824072
.824345

4.57
4.55

4(r>v

10.176202
.175928
.175655

19
18
17

44
45

46

.744550
.744739

.744928

.ID
3.15
3.15

31 K

.919931
.919846
.919762

.40
1.42
1.40

IJrt

.824619
.824893
.825166

.5*
4.57
4.55

4CK

.175381
.175107
.174834

16
15
14

47

48

.745117
.745306

.15
3.15

31 O

.919677
.919593

.42
1.40

1A O

.825439
.825713

.55
4.57

4CC

.174561

.174287

13

12

49

.745494

.la

31 C

.919508

.42

1Af\

.825986

.55

4K.K

.174014

11

50

.745683

.15
3.13

.919424

.40
1.42

.826259

.55
4.55

.173741

10

51

52
53
54
55

9.745871
.746060
.746248
.746436
.746624

3.15
3.13
3.13
3.13

31 O

9.919339
.919254
.919169
.919085
.919000

1.42
1.42
1.40
1.42

1AC\

9.826532
.826805
.827078
.827351
.827624

4.55
4.55
4.55
4.55

4C "*

10.173468
.173195
.172922
.172649
.172376

9
8
7
6
5

56
57
58

.746812
.746999
.747187

.13

3.12
3.13

31 £1

.918915
.918830
.918745

.42
1.42
1.42

]4Q

.827897
.828170
.828442

.5o
4.55
4.53

4KK.

.172103
.171830
.171558

4
3
2

59
60

.747374
9.747562

.la
3.13

.918659
9.918574

.4-3
1.42

.828715
9.828987

.55
4.53

.171285
10.171013

1
0

'

Cosine.

D. 1".

Sine.

D. 1'.

Cotang. D. 1".

Tang.

'

123'

137

66°

34'

COSINES, TANGENTS, AND COTANGENTS.

145'

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

1

2
3
4
5
6

r*
i

8
9
10

9.747562
.747749
.747936
.748123
.748310
.748497
.748683
.748870
.749056
.749243
.749429

3.12
3.12
3.12
3.12
3.12
3.10
3.12
3.10
3.12
3.10
3.10

9.918574
.918489
.918404
.918318
.918233
.918147
.918062
.917976
.917891
.917805
.917719

1.42
1.42
1.43
1.42
1.43
1.42
1.43
1.42
1.43
1.43
1.42

9.828987
.829260
.829532
.829805
.830077
.830349
.830621
.830893
.831165
.831437
.831709

4.55
4.53
4.55
4.53
4.53
4.53
4.53
4.53
4.53
4.53
4.53

10.171013
.170740
.170468
.170195
.169923
.169651
.169379
.169107
.168835
.168563
.168291

60
59
58
57
56
55
54
53
52
51
50

11
12
13
14
15
16
17
18

9.749615
.749801
.749987
.750172
.750358
.750543
.750729
.750914

3.10
3.10
3.08
3.10
3.08
3.10
3.08

3f\Q

9.917634
.917548
.917462
.917376
.917290
.917204
.917118
.917032

1.43
1.43
1.43
1.43
1.43
1.43
1.43

9.831981
.832253
.832525
.832796
.833068
.833339
.833611
.833882

4.53
4.53
4.52
4.53
4.52
4.53
4.52

10.168019
.167747
.167475
.167204
.166932
.166661
.166389
.166118

49

48
47
46
45
44
43
42

19
20

.751099
.751284

.Uo
3.08
3.08

.916946
.916859

.4o
1.45
1.43

.834154
.834425

.0.3
4.52
4.52

.165846
.165575

41
40

21
22
23
24
25
26
27
28
29
30

9.751469
.751654
.751839
.752023
.752208
.752392
.752576
.752760
.752944
.753128

3.08
3.08
3.07
3.08
3.07
3.07
3.07
3.07
3.07
3.07

9.916773
.916687
.916600
.916514
.916427
.916341
.916254
.916167
.916081
.915994

1.43
1.45
1.43
1.45
1.43
1.45
1.45
1.43
1.45
1.45

9.834696
.834967
.835238
.835509
.835780
.836051
.836322
.836593
.836864
.837134

4.52
4.52
4.52
4.52
4.52
4.52
4.52
4.52
4.50
4.52

10.165304
.1650.33
.164762
.164491
.164220
.163949
.163678
.163407
.163136
.162866

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9.753312
.753495
.753679
.753862
.754046
.754229
.754412
.754595
.754778
.754960

3.05
3.07
3.07
3.07
3.05
3.05
3.05
3.05
3.03
3.05

9.915907
.915820
.915733
.915646
.915559
.915472
.915385
.915297
.915210
.915123

1.45
1.45
1.45
1.45
1.45
1.45
1.47
1.45
1.45
1.47

9.837405
.837675
.837946
.838216
.838487
.838757
.839027
.839297
.839568
.839838

4.50
4.52
4.50
4.52
4.50
4.50
4.50
4.52
4.50
4.50

10.162595
.162325
.162054
.161784
.161513
.161243
.160973
.160703
.160432
.160162

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

9.755143
.755326
.755508
.755690
.755872
.756054
.756236
.756418
.756600
.756782

3.05
3.03
3.03
3.03
3.03
3.03
3.03
3.03
3.03
3.02

9.915035
.914948
.914860
.914773
.914685
.914598
.914510
.914422
.914334
.914246

1.45
1.47
1.45
1.47
1.45
1.47
1.47
1.47
1.47
1.47

9.840108
.840378
.840648

1841187
.841457
.841727
.841996
.842266
.842535

4.50
4.50
4.48
4.50
4.50
4.50
4.48
4.50
4.48
4.50

10.159892
.159622
.159352
. 159083
.158813
.158543
.158273
.158004
.157734
.157465

19
18
17
16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

9.756963
.757144
.757326
.757507
.757688
.757869
.758050
.758230
.758411
9.758591

3.02
3 03
3.02
3.02
3.02
3.02
3.00
3.02
3.00

9.914158
.914070
.913982
.913894
.913806
.913718
.913630
913541
.913453
9.913365

1.47
1.47
1.47
1.47
1.47
1.47
1.48
1.47
1.47

9.842805
.843074
.843343
.843612
.843882
.844151
.844420
.844689
.844958
9.845227

4.48
4.48
4.48
4.50

4.48
4.48
4.48
4.48
4.48

io.ivri95

156926
.156657
.156388
.156118
.155849
.155580
.155311
.155042
10.154773

9
8
7
6
5
4
3
2
1
0

'

Cosine.

D. 1". j! Sine, i D. 1".

Cotang.

D. 1".

Tang.

1

134°

138

55'

35'

TABLE X. — LOGARITHMIC SIXES,

144'

/

Sine.

D. 1'.

Cosine.

D. r.

Tang.

D. r.

Cotang.

/

0

1

2
3
4

9.758591
.758772
.758952
.759132
.759312

3.02
3.00
3.00
3.00

3f\C\

9.913365
.913276
.913187
.913099
.913010

1.48
1.48
1.47
1.48

Ijir-f

9.845227
.845496
.845764
.846033
.846302

4.48
4.47
4.48
4.48

4jr*

10.154773
.154504
.154236
.153967
.153698

60
59
58
57
56

5
6

7

.759492
.759672

.759852

.00
3.00
3.00

9 OS

.912922
.912833
.912744

Ai

1.48
1.48

1 JS

.846570
.846839
.847108

.47

4.48
4.48

A J7

.153430
.153161
.152SJW

55
54
53

8
9
10

.760031
.760211
.760390

~.yo
3.00
2.98
2.98

.912655
.912566
.912477

1 . 4o

1.48
1.48
1.48

.847376
.847044
.847913

** . 4 i

4.47
4.48
4.47

.152624
.152356
.152087

52
51
50

11
12

9.760569
.760748

2.98

2OQ

9.912388
.912299

1.48

1IQ

9.848181
.848449

4.47

44ty

10.151819
.151551

49

48

13
14

.760927
.761106

.9o
2.98

2 no

.912210
.912121

.4o
1.48

It A

.848717
.848986

.4<

4.48

4A*y

.151283
.151014

47
46

15
16
17
18
19
20

.761285
.761464
.761642
.761821
.761999
.762177

.9o
2.98
2.97
2.98
2.97
2.97
2.98

.912031
.911942
.911853
.911763
.911674
.911584

.50
1.48
1.48
1.50
1.48
1.50
1.48

.849254
.849522
.849790
.850057
.850325
.850593

.4*

4.47
4.47
4.45
4.47
4.47
4.47

.150746
.150478
.150210
.149943
.149675
.149407

45
44
43
42
41
40

21
22
23
24
25

9.762356
.762534
.762712

.762889
.763067

2.97
2.97
2.95
2.97

2r\fy

9.911495
.911405
.911315
.911226
.911136

1.50
1.50

1.48
1.50

It A

9.850861
.851129
.851396
.851664
.851931

4.47
4.45
4.47
4.45

4|W

10.149139
.148871
.148604
.148336
.148069

39
38
37
36
35

26

.763245

.97

9 QA

.911046

.50
1 ^0

.852199

Ai

A A^

.147801

34

27
28
29

.763422
.763600
.763777

~.yo
2.97
2.95

2f\(-

.910956
.910866
.910776

Jl . OU

1.50
1.50

IP" A

.852466
.852733
.853001

rr . IJ

4.45
4.47

44 pr

.147534
.147267
.146999

33
32
31

30

.763954

.95
2.95

.910686

.50
1.50

.853268

.45
4.45

.146732

30

31
32

9.764131
.764308

2.95

2n;r

9.910596
.910506

1.50

1 no

9.853535

.853802

4.45

4Afi

10.146465
.146198

29

28

33
34
35

.764485
.764662

.764838

.yo
2.95
2.93

2f\K

.910415
.910325
.910235

1 . O-i

1.50
1.50

-I fO

.854069
.854336
.854603

.•1O

4.45
4.45

44 K

.145931
.145664
.145397

27
26
25

36
37
38
39

.765015
.765191
.765367
.765544

.9o
2.93
2.93
2.95

2 no

.910144
.910054
.909963
.909873

1 . 52
1.50
1.52

1.50
1~ ^

.854870
.855137
.855404
.855671

.4o
4.45
4.45
4.45

41 K.

.145130
.144863
.144596
.144329

24
23
22
21

40

.765720

.9o
2.93

.909782

.o2

1.52

.855938

.45
4.43

.144062

20

41
42

9.765896
.766072

2.93

2f\£\

9.909691
.909601

1.50

1Cr»"j

9.856204
.856471

4.45

4 JO

10.143796
.143529

19

18

43

.766247

.92

2QQ

.909510

.52

1 f,O

.856737

. 4o

4AK

.143263

17

44

45
46
47

48

.766423

.766598
.766774
.766949
.767124

.yo
2.92
2.93
2.92

2.92
2f\t\

.909419
.909328
.909237
.909146
.909055

1 .Ox!

1.52
1.52
1.52
1.52

It •""!

.857004
.857270
.857537
.857803
.858069

.4O

4.43
4.45
4.43
4.43

44 **

.142996
.142730
.142463
.142197
.141931

16
15
14
13
12

49

.767300

.9o
2 no

.908964

.52
i "o

.858336

.4o

44 n

.141664

11

50

.767475

.92
2.90

.908873

1 . 52
1.53

.858602

.4o
4.43

.141398

10

51
52
53
54
55
56
57
58
59
60

9.767649
.767824
.767999
.768173
.768348
.768522
.768697
.768871
.769045
9.769219

2.92
2.92
2.90
2.92
2.90
2.92
2.90
2.90
2.90

9.908781
.908690
.908599
.908507
.908416
.908324
.908233
.908141
.908049
9.907958

1.52
1.52
1.53
1.52
1.53
1.52
1.53
1.53
1.52

9.858868
.859134
.859400
.859666
.859932
.860198
.860464
.860730
.860995
9.861261

4.43
4.43
4.43
4.43
4.43
4.43
4.43
4.42
4.43

10.141132
.140866
.140600
.140334
.140068
.139802
.139536
.139270
.139005
10.138739

9
8
7
6
5
4
3
2
1
0

i

Cosine.

D. 1".

Sine.

D. r.

Cotang.

D. 1".

Tang.

/

125C

139

54°

36°

COSINES, TANGENTS, AND COTANGENTS.

143'

/

Sine.

D. 1'.

Cosine.

D. 1'. Tang.

D. 1'.

Cotang.

/

0

1

9.769219
.769393

2.90

2QQ

' 9.907S58
.907866

1.53
1 aq

9.861261
.861527

4.43

4yfO

10.138739
.138473

60
59

2
3
4

5
6

7
8

.769566
.769740
.769913
.770087
.770260
.770433
.770606

.50

2.90

2.88
2.90
2.88
2.88
2.88

2OO

.907774
.907682
.907590
.907498
.907406
.907314
.907222

i .00
1.53
1.53
1.53
1.53
1.53

1.53

It ••'

.861792
.862058
.862323
.862589
.862854
.863119
.863385

.\4
4.43
4.42
4.43
4.42
4.42
4.43

A J^k

. 138208
.137942
.137677
.137411
.137146
.136881
. 136615

58
57
56
55
54
53
52

9

.770779

.88

2OQ

.907129

.5o

1KQ

.863650 *•**

.136350

51

10

.770952

.08

2.88

.907037

.56
1.53

.863915

1.V4

4.42

.136085

50

11

12

9.771125

.771298

2.88

20^

9.906945
.906852

1.55

IJrO

9.864180
.864445

4.42

4A*\

10.135820
.135555

49

48

13
14

.771470
.771643

.01
2.88

2O^*

.906760
.906667

.5o
1.55

It O

.864710
.864975

.42
4.42

44C\

.135290
.135025

47
46

15

.771815

.87

2O^

.906575

.53

IK **

.865240

.42

4AC\

.134760

45

16
17
18
19

.771987
.772159
.772331
.772503

.87
2.87
2.87
2.87

2O^*

.906482
.906389
.906296
.906204

.5o
1.55
1.55
1.53

IK •*

.865505
.865770
.866035
.866300

.42
4.42
4.42

4.42

A \(\

.134495
.134230
.1:33965
.133700

44
43
42
41

20

.772675

.01
2.87

.906111

.55

1.55

.866564

4.40
4.42

.133436

40

21
22

9.772847
.773018

2.85

2Q*f

9.906018
.905925

1.55
1p-t

9.866829
.867094

4.42

4 A f\

lo.iasm

.132906

39

38

23
24
25
26
27
28
29
30

.773190
.773361
.773533
.773704
.773875
.774046
.774217
.774388

.01

2.85
2.87
2.85
2.85
2.85
2.85
2.85
2.83

.905832
.905739
.905645
.905552
.905459
.905366
.905272
.905179

.00
1.55
1.57
1.55
1.55
1.55
1.57
1.55
1.57

.867358
.867623
.867887
.868152
.868416
.868680
.868945
.869209

.40
4.42
4.40
4.42
4.40
4.40
4.42
4.40
4.40

.132642
.132377
.132113
.131848
.131584
.131320
.131055
.130791

37
36
35
34
33
32
31
30

31
32

9.774558
.774729

2.85

2OO

9.905085
.904992

1.55

Ierfv

9.869473

.869737

4.40

4AC\

10.130527
.130263

29

28

33
34
35
36
37
38
39
40

.774899
.775070
.775240
.775410
.775580
.775750
.775920
.776090

.00

2.85
2.83
2.83
2.83
2.83
2.83
2.83
2.82

.904898
.904804
.904711
.904617
.904523
.904429
.904335
.904241

.5<
1.57
1.55
1.57
1.57
1.57
1.57
1.57
1.57

.870001
.870265
.870529
.870793
.871057
.871321
.871585
.871849

.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.38

.129999
.129735
. 129471
.129207
.128943
.128679
.128415
.128151

27
26
25
24
23
22
21
20

41
42
43
44

9.776259
.776429
.776598
.776768

2.83
2.82
2.83

2QO

9.904147
.904053
.903959
.903864

1.57
1.57
1.58

IBrry

9.872112
.872376
.872640
.872903

4.40
4.40
4.38

4A(\

10.127888
.127624
.127360
.127097

19
18
17
16

45
46
47
48
49
50

.776937
.777106
.777275

.777444
.777613
.777781

.o2
2.82
2.82
2.82
2.82
2.80
2.82

.903770
.903676
.903581
.903487
.903392
.903298

.57
1.57
1.58
1.57
1.58
1.57
1.58

.873167
.873430
.873694

.873957
.874220
.874484

.40
4.38
4.40
4.38
4.38
4.40
4.38

.126833
.126570
.126306
.126043
.125780
.125516

15
14
13
12
11
10

51
52
53
54
55
56
57

9.777950

.778119
.778287
.778455
.778624
.778792
.778960

2.82
2.80
2.80
2.82
2.80
2.80

O QC\

9.903203
.903108
.903014
.902919
.902824
.902729
.902634

1.58
1.57
1.58
1.58
1.58
1.58

IerO

9.874747
.875010
.875273
.875537
.875800
.876063
.876326

4.38
4.38
4.40
4.38
4.38
4.38

4OQ

10.125253
.124990
.124727
.124463
.124200
.123937
.123674

9
8
7
6
5
4
3

58
59
60

.779128
.779295
9.779463

4. oO
2.78
2.80

.902539
.902444
9.902349

.OO

1.58
1.58

.876589
.876852
9.877114

.d8

4.38
4.37

.123411
. 123148
10.122886

2
1
0

/

Cosine.

D. 1'.

Sine.

D. I*.

Cotang. 1 D. 1*. 1 Tang.

/

126°

140

53'

S7<

TABLE X.— LOGARITHMIC SINES,

142'

'

Sine.

D. r.

Cosine.

D. r.

1

Tang.

D. 1".

Cotang.

'

0

1

o

A

3

4
5
6
7
S
9

9.779463
.779631
.779798
.779966
.780133
.780300
.780467
.780634
.780801
.780968

2.80
2.78
2.80
2.78
2.78
2.78
2.78
2.78
2.78

2i~7

9.902349
.902253
.902158
.902063
.901967
.901872
.901776
.901681
.901585
.901490

1.60
1.58
1.58
1.60
1.58
1.60
1.58
1.60
1.58
1 60

9.877114
.877377
.877640
.877903
.878165
.878428
.878691
.878953
.879216
.879478

4.38
4.38
4.38
4.37
4.38
4.38
4.37
4.38
4.37

A 90

10.122886
.122623
.122360
'. 122097
. 121835
. 121572
.121309
.121047
.120784
.120522

60
59
58
57
56
55
54
53
52
51

10

.781134

. I 1

2.78

.901394

i!oo

.879741

^r . OO

4.37

. 120259

50

11
12
13

9.781301
.781468
.781634

2.78

2.77

O 77

9.901298
.901202
.901106

1.60
1.60

1 All

9.880003
.880265
.880528

4.37
4.38

A Q7

10.119997
.119735
.119472

49

48
47

14

15

.781800
.781966

~. < <

2.77

O *"*7

.901010
.900914

1 . OU

1.60

1f*f\

.880790
.881052

4 . 01

4.37

A 9.7

.119210

.118948

46
45

16

.782132

~. 1 (

2f"ry

.900818

. ou

1(*r\

.881314

4.OI
49.13

.118686

44

17

18

.782298
.782464

. <7
2.77

'} 77

.900722
.900626

DU

1.60

.881577
.881839

.OO

4.37

A Q7

.118423
.118.61

43
42

19

.782630

li.il

.900529

1 . O^i

I/*/"*

.882101

*± . o»

A 07

.117899

41

20

.782796

2.77
2.75

.900433

. ou
1.00

.882363

4^37

.117637

40

21
92

9.782961
.783127

2.77

2W

9.900337
.900240

1.62

1fif\

9.882625

.882887

4.37

4 Or

10.117375
.117113

39

38

23

.783292

.7o

2r"?

.900144

.ul)

.883148

.OO
49.7

.116852

O***
01

24

.783458

. Tt

2fyt*

.900047

1 .02

1nr\

.88:3410

.01

.116590

36

25

.783623

. (0

275C

.899951

.01)

ICQ

.883672

4 . 37

A 97

.116328

35

26

.783788

. !•>

9 r--

.899854

. U.w

.883934

1 . o»
A 97

.116066

34

27

.783953

«. |O

2rytf

.899757

1 . D/i
1P»~>

.884196

ft . Ol
4 ox

.115804

33

28

.784118

.7o

2f*O

.899660

.02

1f»f\

.884457

.00

.115543

32

29

.784282

. <3

c\ p"*1*

.899564

.00

ICfl

.884719

4.37

4QX

.115281

31

30

.784447

2. To
2.75

.899407

.02
1.62

.884980

.OO

4.37

.115020

30

31

32

9.784612

.784776

2.73

2nW

9.899370
.899273

1.62

1/».~\

9.885242
.885504

4.37

10.114758
.114496

29

28

33
34

.784941
.785105

. <5

2.73

2r*o

.899176
.899078

.02
1.63

1rS~>

.885765
.886026

4.35
4.35

49"

.114235
.113974

27
26

35
36
37

.785269
.785433
.785597

. 10

2.73
2.73

2I**O

.898981
.898884
.898787

.63
1.62
1.62

1/JO

.886288
.886549
.886811

.01

4.35
4.37

4 OS

.113712
.113451
.113189

25
24
23

38

.785761

. (A

2r-O

.898689

.DO
Ion

.887072

.OO

49=1

.112928

22

39

40

.785925
.786089

. 10

2.73
2.72

.89,8592
.898494

.DA

1.63
1.62

.887333
.887594

.OO

4.35
4.35

.112667
.112406

21
20

41

42

9.786252
.786416

2.73

2r*,~\

9.898397
.898299

1.63

1/"*»~1

9.887855
.888116

4.35

10.112145

.111884

19
18

43

.786579

. <2

2r*o

.898202

.U-i
ICO

.888378

4.37

4 ox

.111622

17

44
45

46

.786742
.786906
.787069

. (2
2.73
2.72

O^%"4

.898104
.898006
.897908

.03
1.63
1.63

ICO

.888639
.888900
.889161

.00
4.35
4.35

4 CO

.111361

.iinoa

.110839

16
15
14

47

.787232

i •
2^**~)

.897810

.03

1{tn

.889421

.00

4 ox

.110579

13

48
49
50

.787395

.787557
.787720

. <2
2.70
2.72

.897712
.897614
.897516

.03
1.63
1.63

Inn

.889682
.889943
.890204

.00
4.35
4.35

4O"*

.110318
.110057
.109796

12
11
10

2.72

.03

.3o

51
52
53
54
55

9.787883
.788045
.788208
.788370
.788532

I

2.70
2.72
2.70
2.70

2r*f\

9.897418
.897320
.897222
.897123
.897025

1.63
1.63
1.65
1.63

1n~

9.890465
.890725
.890986
.891247
.891507

4.33
4.35
4.35
4.33

4Opr

10.109535
.109275
.109014
.108753
.108493

9

8
7
6
5

56
57

58

.788694
.788856
.789018

iO
2.70
2.70

.896926

.896828
.896729

.Oo
1.63
1.65

ICO

.891768
.892628
.892289

.05

4.33
4.35

499

.108232
.107972
.107711

4
3
2

59
GO

.789180
9.789342

2.70
2.70

.896631
9.896532

.03
1.65

.892549
9.892810

.33
4.35

. 107451
10.107190

1
0

1

Cosine.

D r. i

Sine.

D. r.

Cotang.

D. r.

Tang.

'

127°

141

52°

COSINES, TANGENTS, AND COTANGENTS.

141'

/

Sine.

D. 1'.

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

/

0

1

2

9.789342
.789504
.789665.

2.70
2.68

9 70

9.896532
.896433
.896335

1.65
1.63

1 fi^

9.892810
.893070
.893331

4.33
4.35

A QQ

10.107190
.106930
.106669

60

59
58

3

.789827

&. i\t

0 f\Q

.896236

1 . DO

1 r»T

.893591

4 . OO
A Q'^

. 106409

57

4
5
6

r*
1

8
9
10

.789988
.790149
.790310
.790471
.790632
.790793
.790954

& . oo
2.68
2.68
2.68
2.68
2.68
2.68
2.68

.896137
.896038
.895939
.895840
.895741
.895641
.895542

1 . UiJ

1.65
1.65
1.65
1.65
1.67
1.65
1.65

.893851
.894111
.894372
.894632
.894892
.895152
.895412

*± . 'JO

4.33
4.35
4.33
4.33
4.33
4.33
4.33

.106149
.105889
.105628
.105368
.105108
.104848
.104588

56
55
54
53
52
51
50

11
12
13
14
15
16
17
18
19
20

9.791115
.791275
.791436
.791596
.791757
.791917
.792077
.792237
.792397
.792557

2.67
2.68
2.67
2.68
2.67
2.67
2.67
2.67
2.67
2.65

9.895443
.895343
.895244
.895145
.895045
.894945
.894846
.894746
.894646
.894546

1.67
1.65
1.65
1.67
1.67
1.65
1.67
1.67
1.67
1.67

9.895672
.895932
.896192
.896452
.896712
.896971
.897231
.897491
.897751
.898010

4.33
4.33
4.33

4. as

4.32
4.33

4. as

4.33
4.32
4.33

10.104328
.104068
.103808
.103548
.103288
.103029
.102769
.102509
.102249
.101990

49
48
47
46
45
44
43
42
41
40

21
22
23
24
25
26
27
28

9.792716
.792876
.793035
.793195
.793354
.793514
.793673
.793832

2.67
2 65
2.67
2.65
2.67
2.65
2.65

O AK

9.894446
.894346
.894246
.894146
.894046
.893946
.893846
.893745

1.67
1.67
1.67
1.67
1.67
1.67
1.68

1 A7

9.898270
.898530
.898789
.899049
.899308
.899568
.899827
.900087

4.33
4.32
4.33
4.32
4.33
4.32
4.33

A QO

10.101730
. 101470
.101211
.100951
.100692
.100432
.100173
.099913

39
38
37
36
35
34
33
32

29
30

.793991
.794150

<i.OO

2.65
2.63

.893645
.893544

1 . D*

1.68
1.67

.900346
.900605

** . uZ

4.32
4.32

.099654
.099395

31
30

31
32
33
34
35
36
37
38
39
40

9.794308
.794467
.794628
.794784
.794942
.795101
.795259
.795417
.795575
.795733

2.65
2.65
2.63
2.63
2.65
2.63
2.63
2.63
2.63
2.63

9.893444
.893343
.893243
.893142
.893041
.892940
.892839
.892739
.892638
.892536

1.68
1.67
1.68
1.68
1.68
1.68
1.67
1.68
1.70
1.68

9.900864
.901124
.901383
.901642
.901901
.902160
.902420
.902679
.902938
.903197

4.33
4.32
4.32
4.32
4.32
4.33
4.32
4.32
4.32
4.32

10.099136
.098876
.098617
.098358
.098099
.097840
.097580
.097321
.097062
.096803

29
23
27
26
25
24
23
22
21
20

41
42
43
44

9.795891
.796049
.796206
.796364

2.63

2.62
2.63

2 CO

9.892435
.892,334
.892233
.892132

1.68
1.68
1.68

Iryrt

9.903456
.903714
.903973
.904232

4.30
4.32
4.32

4Ot~>

10.096544
.096286
.096027
.095768

19
18
17
16

45
46

.796521
.796679

.52
2.63

9 fi9

.892030
.891929

. (0

1.68

1 7f>

.904491
. 904750

.32
4.32

A QfJ

.095509
.095250

15
14

47
48

.796836
.796993

-. . D>£

2.62
2f*f\

.891827
.891726

1 . <U

1.68
1fft\

.905008
.905267

~t . OU

4.32
400

.094992
.0947*3

13
12

49
50

.797150
.797307

.62
2.62
2.62

.891624
.891523

. tO
1.68
1.70

.905526
.905785

.o2
4.32
4.30

.094474
.094215

11

10

51
52

9.797464
.797621

2.62

2f*S\

9.891421
.891319

1.70

1f^f\

9.906013
.906302

4.32
4 on

10.093957
.093698

9

8

53

.797777

.bO
2/».-»

.891217

. tO

1r^i\

.906500

.oO

4Ok">

.093440

7

54
55
56

.797934
.798091
.798247

.62
2.62
2.60

2S*f\

.891115
.891013
.890911

. iO
1.70
1.70

1r*rv

.906819
.907077
.907336

.o2
4.30
4.32

4O("l

.093181
.092923
.092664

6
5
4

57

.798403

.60

2s*c\

.890809

. <0
1WA

.907594

.ou

4Ort

.092406

3

58

.798560

.62

2ftf\

.890707

. iO
Irtn

.907853

.6%
4 on

.092147

o

iO

59
60

.798716
9.798872

.50
2.60

.890605
9.890503

. iO
1.70

.908111
9.908369

.o(J

4.30

.091889
10.091631

1
0

' Cosine.

D. r.

I Sine.

D. 1".

Cotang.

D. 1".

Tang.

/

128°

51"

39°

TABLE x. — LOGARITHMIC SINES,

140*

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

/

0

9.798872

2f*f\

9.890503

9.908369

4QO

10.091631

60

1

2
3

.799028
.799184
.799339

. ou
2.60

2.58
2ftf\

.890400
.890298
.890195

1 . i2
1.70
1.72

.908628
.90a886
.909144

tjyw

4^30

4.30
4 on

.091372
.091114
.090856

59

58
57

4
5
6

.799495
.799651
.799806

.60
2.60
2.58

2f»f\

.890093
.889990

.889888

l'.72
1.70

1**V>

.909402
.909660
.909918

.oO
4.30
4.30

4OO

.090598
.090340
.090082

56
55
54

r*
t

.799962

.00

2 to

.889785

i f£

.910177

.oa

4 on

.089823

53

8

.800117

.58

9 ^8

.889682

1.72

.910435

.oil

.089565

52

9

.800272

/i.OO
2 to

.889579

1 70

.910693

4 'in

.089307

51

10

.800427

.08

2.58

.889477

1.72

.910951

.ou
4.30

.089049

50

11
12
13
14

15

9.800582
.800737
.800892
.801047
.801201

2.58
2.58
2.58
2.57

O CO

9.889374
.889271
.889168
.889064
.888961

1.72
1.72
1.73
1.72

1j**O

9.911209
.911467
.911725
.911982
.912240

4.30
4.30
4.28
4.30

4 on

10.088791

.088533
.088275
.088018
.087760

49

48
47
46
45

16
17
18
19

.801356
.801511
.801665
.801819

2.58
2.58
2.57
2.57

.888858
.888755
.888651
.888548

. Tit
1.72
1.73

1.73

Iryn

.912498
.912756
.913014
.913271

.oO
4.30
4.30
4.28

4 on

.087502
.087244
.086986
.086729

44
43
42
41

20

.801973

M • O 1

2.58

.888444

. 16
1 .72

.913529

.60
4.30

.086471

40

21

9.802128

2 57

9.888341

1 ""I

9.913787

d 98

10.086213

39

22

.802282

O •" "*

.888237

1 . to

1 "^9

.914044

1 . <vO

4 on

.085956

38

23
24

.802436
.802589

2.oi
2.55

2C1"*

.888134

.888030

lira

1^*0

.914302
.914560

.60
4.30

4OQ

.085698
.085440

37'

36

25

.802743

.31

.887926

. 10
Iryn

.914817

.28
4 on

.085183

35

26

.802897

2.57

."> r ~

.887822

. id
Iryo

.915075

.60

4OLJ

.084925

34

27

.803050

2. a.)

.887718

. <0

1r*o

.915332

.2o

4 on

.084668

33

28

.803204

2.57

.887614

. (A

.915590

.30

.084410

32

29

.803357

o * •**

.887510

1 . 1 0

Ir-o

.915847

4 . ^o

4.">O

.084153

31

30

.803511

** . • t

2.55

.887406

. to
1.73

.916104

.28
4.30

.083896

30

31

9.803664

2 55

9.887302

1 7°.

9.916362

4 ^8

10.083638

29

32

.803817

.-) p- •?

.887198

1 . 1 O

.916619

4 on

.083381

28

33
34

.803970
.804123

2.OO
2.55

O K C

.887093
.886989

1 .75
1.73

IrVO

.916877
.9171:54

.oO

4.28

4,-JQ

.083123

.082866

27
26

35

.804276

2.55

r* f- *-

.886885

. (0

I^tK

.917391

.28
400

.082609

25

36
37

38

.804428
.804581
.804734

2.5o
2.55

2.55
9 BIQ

.886780
.886676
.886571

. i5
1.73
1.75

1 r^

.917648
.917906
.918163

.28
4.30
4.28

A OQ

.082352

.082094
.081837

24
23
22

39

.804886

/* . Oo

2KK

.886466

i . i O
1I**O

.918420

ft . -«o
400

.081580

21

40

.805039

.5o

2KO
. tJO

.886362

. (O

1.75

.918677

.28
4.28

.081323

20

41

9.805191

2 53

9.886257

"t ^^

9.918934

. QQ

10.081066 19

42

.805343

.886152

1 "^

.919191

4 9Q

.080809 , 18

43

.805495

O'KQ

.886047

1 . |O
Irye

.919448

^± . ->io
4 h^8

.080552 17

44

.805647

O to

.885942

. (O

.919705

400

.080295 16

45

46

.805799
.805951

2.5o
2.53

2cQ

.885837

.885732

l!75

1^*K

.919962
.920219

.28
4.28

4 no

.080038
.079781

15
14

47

.806103

. OO

o t^o

.885627

. (D

1 "~^

.920476

.28

49Q

.079524 13

48

.806254

2f* *>

.885522

1 . i O

.920733

. /CO
4rtO

.079267 12

49

.806406

.00

.885416

1 .77

IP***

.920990

.28

4r\rj

.079010 11

50

.806557

2^53

.885311

. <o
1.77

.921247

.28

4.27

.078753

10

51
52

9.806709
.806860

2.52

9.885205

.885100

1.75

-. p~rx

9.921503
.921760

4.28

4 no

10.078497

.078240

9

8

53

.807011

2 to

.884994 *•£

.922017

.28

.077983

7

54

.807163

.53

2 52

.884889 ; :}•;£

.922274

4.28

497

.077726

6

55
56

.807314
.807465

2i 52
o ^n

.884783
.884677

JL. 1 1

1.77

.922530

.922787

. At

4.28

49fl

.077470
.077213

5
4

57

.807615

O ",->

.884572 f-iU

.923044

. -iO

.076956

3

58

.807766

2KO

.884466 i'ii

.923300

4OQ

.076700

2

59
60

.807917
9.808067

.52
2.50

.884360 i Xi
9.884254

.923557
9.923814

.28

4.28

.076443
10.076186

1

0

'

Cosine.

D. 1".

Sine.

D. 1".

Cotang.

D. r.

Tang.

'

129C

143

50'

40°

COSINES, TANGENTS, AND COTANGENTS.

139°

'

Sine.

D. 1".

Cosine.

D. 1'.

Tang.

D. r.

Cotang.

'

0

9.808067

9 P^O

9.884254

9.923814

4O»f

10.076186

60

1

.808218

2t/~k

.884148

1 .77

Iryn

.924070

.27

4f)Q

.075930

59

2

.808368

.5U

cy f^O

.884042

. 1 I
1*7*9

.924327

.40

49'"'

.075673

58

3

4
5
6
7

.808519
.808669
.808819
.808960
.809119

2^50
2.50
2.50
2.50

2t A

.883936
.883829
.883723
.883617
.883510

. 11

1.78
1.77
1.77

1.78
Itw

.924583
.924840
.925096
.925352
.925609

.*<

4.28
4.27
4.27
4.28

4f\ri

.075417
.075160
.074904
.074648
.074391

57
56
55
54
53

8

.809269

.OU

9 ^0

.883404

. ii

1r~Q

.925865

.£1

A C)Q

.074135

52

9

10

.809419
.809569

2^50
2.48

.883297
.883191

. IO

1.77
1.78

.926122
.926378

4^27
4.27

.073878
.073622

51
50

11
12

9.809718
.809868

2.50

24 O

9.883084
.882977

1.78

1i**i"f

9.926634
.926890

4.27

4OQ

10.073366
.073110

49

48

13

.810017

.•-iO

o 50

.882871

. it

1r*o

.927147

.2o

4 Of

.072853

47

14
15

16
17

.810167
.810316
.810465
.810614

2^48
2.48
2.48

9 J8

.882764
.882657
.882550
.882443

. 10

1.78
1.78
1.78

1r-o

.927403
.927659
.927915
.928171

.tSi

4.27
4.27
4.27

A 97

.072597
.072341
.072085
.071829

46
45
44
43

18
19
20

.810763
.810912
.811061

/* . TO

2.48
2.48
2.48

.882336
.882229
.882121

. 1 O

1.78
1.80
1.78

.928427
.928684
.928940

*±.4&i

4.28
4.27
4.27

.071573
.071316
.071060

42
41
40

21
22
23
24
25
26
27
28
29

9.811210
.811358
.811507
.811655
.811804
.811952
.812100
.812248
.812396

2.47
2.48
2.47
2.48
2.47
2.47
2.47
2.47

2*ty

9.882014
.881907
.881799
.881692
.881584
.881477
.881369
.881261
.881153

1.78
1.80
1.78
1.80
1.78
1.80
1.80
1.80

Ir'O

9.929196
.929452
.929708
.929964
.930220
.930475
.930731
.930987
.931243

4.27
4.27
4.27
4.27
4.25
4.27
4.27
4.27

4O1"*

10.070804
.070548
.070292
.070036
.069780
.069525
.069269
.069013
.068757

39
38
37
36
35
34
33
32
31

30

.812544

.47

2.47

.881046

. IO

1.80

.931499

.XI

4.27

.068501

30

31
32
33
34
35
36
37

9.812692
.812840
.812988
.813135
.813283
.813430
.813578

2.47
2.47
2.45
2.47
2.45

2.47
2nz

9.880938
.880830
.880722
.880613
.880505
.880397
.880289

1.80
1.80
1.82
1.80
1.80
1.80

ICO

9.931755
.932010
.932266
.932522
.932778
.933033
.933289

4.25
4.27
4.27
4.27
4.25
4.27

4O7

10.068245
.067990
.067734
.067478
.067222
.066967
.066711

29
28
27
26
25
24
23

38
39

.813725

.813872

.•10
2.45

.880180
.880072

.O*

1.80

1 82

.933545
.933800

.lit

4.25

A 97

.066455
.066200

22
21

40

.814019

2^45

.879963

1 . o*

1.80

.934056

4 . - i

4.25

.065944

20

41
42
43
44
45
46
47
48
49
50

9.814166
.814313
.814460
.814607
.814753
.814900
.815046
.815193
.815339
.815485

2.45
2.45
2.45
2.43
2.45
2.43
2.45
2.43
2.43
2.45

9.879855
.879746
.879637
.879529
.879420
.879311
.879202
.879093
.878984
.878875

1.82
1.82
1.80
1.82
1.82
1.82
1.82
1.82
1.82
1.82

9.934311
.934567
.934822
.935078
.935333
.935589
.935844
.936100
.936355
.936611

4.27
4.25
4.27
4.25
4.27
4.25
4.27
4.25
4.27
4.25

10.065689
.065433
.065178
.064922
.064667
.064411
.064156
.063900
.063645
.063389

19

18
17
16
15
14
13
12
11
10

51
52
53
54
55
56
57

9.815632
.815778
.815924
.816069
.816215
.816361
.816507

2.43
2.43
2.42
2.43
2.43
2.43

o 10

9.878766
.878656
.878547
.878438
.87&S28
.878210
.878109

1.83
1.82
1.82
1.83

1.82

1.83
100

9.936866
.937121
.937377
.937632
.937887
.938142
.938398

4.25
4.27
4.25
4.25
4.25
4.27

4 OK

10.063134
.062879
.062623
.062368
.062113
.061858
.061602

9
8
7
6
5
4
3

58
59

.816652
.816798

2.42
2.43

O 4O

.877999
.877890

.5-3

1.82

"1 oo

.938653
.938908

.25
4.25

4OC

.061347
.061092

2

1

60

9.816943

2.45s

9.877780

9.939163

.25

10.060837

0

'

Cosine.

D. 1".

Sine.

D. r.

Cotang.

D. r.

Tang.

'

i30Q

144

40°

TABLE X. — LOGARITHMIC SINES,

138°

•

Sine.

D. 1'.

Cosine.

i>. r.

Tang.

D. 1'.

Cotang.

'

0

1

0

3
4
5
6

7
8
9
10

9.816943
.817088
.817233
.817379
.817524
.817663
.817813
.817958
.818103
.818247
.818392

2.42
2.42
2.43
2.42
2.40
2.42
2.42
2.42
2.40
2.42
2.40

9.877780
.877070
.877560
.877450
.877340
.877230
.877120
.877010
.876899
.876789
.876678

1.83
1.83
1.83
1.83
1.83
1.83
1.83
1.85
1.83
1.85
1.83

9.939163
.939418
.939073
.939928
.940183
.940439
.940094
.940949
.941204
.941459
.941713

4.25
4.25
4.25
4.25
4.27
4.25
4.25
4.25
4.25
4.23
4.25

10.060837
.060582
.060327
.060072
.059817
.059561
.059306
.059051
.058796
.058541
.058287

CO
59
58
57
56
55
54
53
52
51
LJ

11
13

13
It
15
16

17
18
19
20

9.818536
.818681
.818825
.818969
.819113
.819257
.819401
.819545
.819689
.819832

2.42
2.40
2.40
2.40
2.40
2.40
2.40
2.40
2.38
2.40

9.876568
.876457
.876347
.876236
.876125
.876014
.875904
.875793
.875682
.875571

1.85
1.83
1.85
1.85
1.85
1.83
1.85
1.85
1.85
1.87

9.941968
.942223
.942478
.942733
.942988
.943243
.943498
.943752
.944007
.944262

4.25
4.25
4.25
4.25
4.25
4.25
4.23
4.25
4.25
4.25

10.058032
.057777
.057522
.057267
.057012
.056757
.056502
.056248
.055993
.055738

49
48
47
46
45
44
4:5
42
41
40

21
22
23
24

25
26
27

28
29

9.819976
.820120
.820263
.820406
.820550
.820693
.820836
.820979
.821123

2.40

2.38
2.38
2.40
2.38
2.38
2.38
2.38

2QQ

9.875459
.875348
.875237
.875126
.875014
.874903
.874791
.874680
.874568

1.85
1.85
1.85
1.87
1.85
1.87
1.85
1.87

9.944517
.944771
.945026
.945281
.945535
.945790
.946045
.946299
.946554

4.23
4.25
4.25
4.23
4.25
4.25
4.23
4.25

A OQ

10.055483
.055229
.054974
.054719
.054465
.054210
.053955
.053701
.053446

CO
38
37
36
35
34
33
32
31

30

.821265

.GO

2.37

.874450

1 .81
1.87

.946808

4.25

.053192

30

31
32
33

9.821407
.821550
.821693

2.38
2.38

O Q"y

9.874344
.874232

.874121

1.87
1.85

1Q1"*

9.947063
.947318
.947572

4.25
4.23

4 OK

10.052937

.052682
.052428

29
28

27

34
35
36
37
38
39
40

.821835
.821977
.822120
.822262
.822404
.822546
.822688

2.37
2.38
2.37
2.37
2.37
2.37
2.37

.874009
.873890
.873784
.873672
.873560
.873448
.873335

.O*

1.88
1.87
1.87
1.87
1.87
1.88
1.87

.947827
.948081
.948335
.948590
.948844
.949099
.949353

.<,;>
4.23
4.23
4.25
4.23
4.25
4.23
4.25

.052173
.051919
.051665
.051410
.051156
.050901
.050647

26
25
24
23
22
21
20

41
42
43
44
45
46

9.822830
.822972
.823114
.823255
.823397
.823.539

2.37
2.37
2.35
2.37
2.37

Sort

9.873223
.873110
.872998
.872885
.872772
.872659

1.88
1.87
1.88
1.88

1.88

; 9.949608
.949862
.950116
.950371
.950625
.950879

4.23
4.23
4.25
4.23
4.23

A OQ

10.050392
.050138
.049884
.049629
.049375
.049121

19
18
17
16
15
14

47
48
49
50

.823680
.823821
.823963
.824104

.OO

2.35
2.37
2.35
2.35

.872547
.872434
.872321

.872208

1.88
1.88
1.88
1.88

.951133
.951388
.951642
.951896

4.25
4.23
4.23
4.23

.048867
.048612
.048358
.048104

13
12
11
10

51
52
53

9.824245
.824386
.824527

2.35
2.35

O OK

9.872095
.871981
.871868

1.90
1.88

1OQ

9.952150
.952405
.952659

4.25

4.23
400

10.047850
.047595
.047341

9

8

7

54
55

.824668
.824808

2 . oa
2.33

.871755
.871641

.OO

1.90

.952913
.953167

./fa

4.23

4OO

.047087
.046833

6
5

56
57
58
59
60

.824949
.825090
.825230
.825371
9.825511

2.35
2.33
2.35
2.33

.871528
.871414
.871301
.871187
9.871073

.00
1.90
1.88
1.90
1.90

. 953421
.953675
.953929
.954183
9.954437

.553

4.23
4.23
4.23
4.23

.046579
.046325
.046071
.045817
10.045563

4
3
2
1
0

-

Cosine.

D. r.

Sine.

D. 1".

Cotang.

D. 1'.

Tang.

'

131*

145

48'

42C

COSINES, TANGENTS, AND COTANGENTS.

137°

'

Sine.

D. r.

Cosine.

D. 1'.

Tang.

D. r.

Cotang.

'

0

9.825511

2QQ

9.871073

1 QQ

9.954137

4OQ

10.045563

60

1

2

3
4
5
6

7
8
9

.825651
.825791
.825931
.826071
.826211
.826351
.826491
.826631
.826770

.00

2.33
2.33
2.33
2.33
2.33
2.33
2.33
2.32

2OO

.870960
.870846
.870732
.870618
.870504
.870390
.870276
.870161
.870047

1^90
1.90
1.90
1.90
1.90
1.90
1.92
1.90

1f\f\

.954691
.954946
.955200
.955454
.955708
.955961
.956215
.956469
.956723

,/Sa

4.25
4.23
4.23
4.23
4.22
4.23
4.23
4.23

4f\n

.045309
.045054
.044800
.044546
.044292
.044039
.043785
.043531
.043277

59
58
57
56 1
55 '
54
53
52
51

10

.826910

.33
2.32

.869933

.90
1.92

.956977

.26
4.23

.043023

50

11
12
13
14
15
16
17
13
19
£0

9.827049
.827189
.827328
.827467
.827606
.827745
.827884
.828023
.828162
.828301

2.33
2.32
2.32
2.32
2.32
2.32
2.32
2.32
2.32
2.30

9.869818
.869704
.869589
.869474
.869360
.869245
.869130
.869015
.868900
.868785

1.90
1.92
1.92
1.90
1.92
1.92
1.92
1.92
1.92
1.92

9.957231
,957485
.957739
.957993
.958247
.958500
.958754
.959008
.959262
.959516

4.23
4.23
4.23
4.23
4.22
4.23
4.23
4.23
4.23
4.22

10.042769
.042515
.042261
.042007
.041753
.041500
.041246
.040992
.0407J8
.040484

49
48
47
46
45
44
43
42
41
40

21

9.828439

2OO

9.868670

9.959769

4OO

10.040231

39

22

.828578

. • ' -

2f\f\

.868555

1Ak"k

.960023

.26

4OO

.039977

38

23

.828716

.60

2Oi~>

.868440

.92

Ino

.960277

.26
400

.039723

37

24
25
26
27

28

.828855
.828993
.829131
.829269
.829407

.62
2.30
2.30
2.30
2.30

2O/\

.868324
.868209
.868093
.867978
.867862

.9.3
1.92
1.93
1.92
1.93

IfWl

.960530
.960784
.961038
.961292
.961545

.22
4.23
4.23
4.23

4.22
400

.039470
.039216
.038962
.038708
.038455

36
35
34
33
32

29

.829545

.60

2OA

.867747

.92

1AO

.961799

.26

4 Oft

.038201

31

30

.829683

.dO

2.30

.867631

.9o
1.93

.962052

.22
4.23

.037948

30

31
32
33
34
35
36
37
38
39
40

5.829821
.829959
.830097
.830234
.830372
.830509
.830646
.830784
.830921
.831058

2.30
2.30
2.28
2.30
2.28
2.28
2.30
2.28
2.28
2.28

9.867515
.867399
.867283
.867167
.867051
.866935
.866819
.866703
.866586
.866470

1.93
1.93
1.93
1.93
1.93
1.93
1.93
1.95
1.93
1.95

9.962306
.962560
.962813
.963067
.963320
.963574
.963828
.964081
.964335
.964588

4.23
4.22
4.23
4.22
4.23
4.23
4.22
4.23
4.22
4.23

10.037694
.037440
.037187
.036933
.036680
.036426
.036172
.035919
.035665
.035412

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

9.831195
.831332
.831469
.831606
.831742
.831879
.832015
.832152
.832288
.832425

2.28
2.28
2.28
2.27
2.28
2.27
2.28
2.27
2.28
2.27

9.866353
.866237
.866120
.866004
.865887
.865770
.865653
.865536
.865419
.865302

1.93
1.95
1.93
1.95
1.95
1.95
1.95
1.95
1.95
1.95

9.964842
.965095
.965349
.965602
.965855
.966109
.966362
.966616
.966869
.967123

4.22
4.23
4.22
4.22
4.23
4.22
4.23
4.22
4.23
4.22

10.035158
.034905
.034651
.034398
.034145
.033891
.033638
.033384
.033131
.032877

19
18
17
16
15
14
13
12
11
10

51

52
53
54
55
56
57

9-832561
.832697
.832833
.832969
.833105
.833241
.833377

2.27
2.27
2.27
2.27
2.27
2.27

Cl OC

9.865185
.865068
.864950
.864833
.864716
.864598
.864481

1.95
1.97
1.95
1.95
1.97
1.95

1f\fy

9.967376
.967629
.967883
.968136
.968389
.968643
.968896

4.22
4.23
4.22
4.22
4.23

4.22
400

10.032624
.032371
.032117
.031864
.031611
.031357
.031104

9

8
7
6
5
4
3

58

.833512

2.25

2cvy

.864363

.97

1f\i*f

.969149

.22

4,-)O

.030851

2

59
60

.833648
9.8-33783

.27
2.25

.864245
9.864127

.97
1.97

.969403
9.969656

.2o

4.22

.030597
10.030344

1

0

'

Cosine.

D. 1".

Sine. I D. 1°.

Cotang.

D. r.

Tang.

'

132«

146

47*

13°

TABLE X. — LOGARITHMIC SINES,

'

Sine.

D. 1'.

Cosine.

D. 1".

:

Tang. D. 1". Cotang.

'

0

9.833783

2 27

9.864127

9.969656

400

10.030344

60

1

.833919

2 OK

.864010

1O"*

.969909

. •*•*

400

.030091

59

2

.834054

. -^O
2 OK

.86:35 92

.9<

.970162

. -wXi
4OO

.029838

58

3

.834189

.25

o o'v

.863774

i . V» t

1/\*v

.970416

.23
40 >

.029584

57

I 4

.834325

6.41

2 OK

.863656

.9*

.970669

.21
400

.029331

56

5
6

7
8
9

.834460
.834595
.834730
.834865
.834999

.«O

2.25
2.25
2.25
2.23

.863538
.863419
.863301
.863183
.863064

l!98

1.97
1.97

1.98

Inr'

.970922
.971175
.971429
.971682
.971935

.22
4.22
4.23
4.22
4.22

4f\Cl

.029078
.028825
.028571
.028318
.028065

55
54
53
52
51

10

.835134

2^25

.862946

.y<
1.98

.972188

.22
4.22

.027812

50

11

9.835269

200

9.862827

9.972441

4OO

10.027559

49

12
13
14
15
16
17
18
19
20

.835403
.835538
.835672
.835807
.835941
.836075
.836209
.836343
.836477

,2o
2.25
2.23
2.25
2.23
2.23
2.23
2.23
2.23
2.23

.862709
.862590
.862471
.862353
.862234
.862115
.861996
.861877
.861758

1 .9<
1.98
1.98
1.97
1.98
1.98
1.98
1.98
1.98
2.00

.972695
.972948
.973201
.973454
.973707
.973960
.974213
.974466
.974720

.23
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.23
4.22

.027305
.027052
.026799
.026546
.026293
.026040
.025787
.025534
.025280

48
47
46
45
44
43
42
41
40

21
22

9.836611
.836745

2.23

2. )• »

9.861638
.861519

1.98

1f\O

9.974973
.975226

4.22

, ClCk

10.025027
.024774

39

38

23
24

25
26
27
28
29
30

.836878
.837012
.837146
.837279
.837412
.837546
.837679
.837812

.22
2.23
2.23
2.22
2.22
2.23
2.22
2.22
2.22

.861400
.861280
.861161
.861041
.860922
.860802
.860682
.860562

.98
2.00
1.98
2.00
1.98
2.00
2.00
2.00
2.00

.975479
.975732
.975985
.976238
.976491
.976744
.976997
.977250

4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22

.024521
.024268
.024015
.023762
.023509
.023256
.023003
.022750

37
36
35
34
33
32
31
30

31
32
33
34
35
36

9.837945
.838078
.838211
.838344
.838477
.838610

2.22
2.22
2.22
2.22
2.22
2 20

9.860442
.860322
.860202
.860082
.859962
.859842

2.00
2.00
2.00
2.00

2.00
2 no

9.977503
.977756

.978009
.978262
.978515
.978768

4.22
4.22
4.22
4.22
4.22

4OO

10.022497
.022244
.021991
.021738
.021485
.021232

29
28
27
26
25
24

37
38
39

.838742
.838875
.839007

2! 22

2.20

O Oi~)

.859721
.859601
.859480

. \j£

2.00
2.02

2r\f\

.979021
.979274
.979527

. £fy

4.22
4.22

.020979
.020726
.020473

23
22
21

40

.839140

£20

.859360

.00
2.02

.979780 J-g

.020220

20

41
42
43
44
45

9.839272
.839404
.839536
.839668
.839800

2.20

2.20
2.20
2.20

2f)n

9.859239
.859119
.858998
.858877
.858756

2.00
2.02
2.02
2.02

2/-w-»

9.980033
.980286
.980538
.980791
.981044

4.22
4.20
4.22
4.22

4.1.1

10.019967
.019714
.019462
.019209
.018956

19
18
17
16
15

46
47
48
49
50

.839932
.840064
.840196
.840328
.840459

.20
2.20
2.20
2.20
2.18
2.20

.858635
.85&514
.858393
.858272
.858151

.02
2.02
2.02
2.02
2.02
2.03

.981297
.981550
.981803
.982056
.982309

.22
4.22
4.22
4.22
4.22
4.22

.018703
.018450
.018197
.017944
.017691

14
13

12
11
10

51
52
53
54
55
56
57
58
59
60

9.840591
.840722
.84085-1
.840985
.841116
.841247
.841378
.841509
.841640
9.841771

2.18
2.20
2.18
2.18
2.18
2.18
2.18
2.18
2.18

9.858029
.857908
.857786
.857665
.857543
.857422
.857300
.857178
.857056
9.856934

2.02
2.03
2.02
2.03
2.02
2.03
2.03
2.03
2.03

9.982562
.982814
.983067
.983:320
.983573
.983826
.984079
.984332
.984584
9.984837

4.20
4.22
4.22
4.22
4.22
4.22
4.22
4.20
4.22

10.017438
.017186
.016933
.016680
.016427
.016174
.015921
.015668
.015416
10.015163

1

7
6
5
4
3
2
1
0

'

Cosine, i D. 1°. p Sine. D.I', li Cotang. D. 1". Tang. '

133'

147

44°

COSINES, TANGENTS, AND COTANGENTS.

135<

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. r.

Cotang.

'

0

1

2

9.841771
.841902
.842033

2.18
2.18

217

, 9.856934
.856812
.856690

2.03
2.03

o no

9.984837
.985090
.985343

4.22
4.22

499

10. 01 51 G3
.014910
.014657

GO
59

58

3
4
5

.842163
.842294
.842424

. 1 1

2.18
2.17

21ft

.856568
.856446
.856323

~ . \}>j

2.03
2.05

.985596
.985848
.986101

,tSt

4.20
4.22

499

.014404
.014152
.013899

57
56
55

6

.842555

. lo
o 17

.856201

f) Q5

986:354

./££
499

.013646

54

r*
i

8
9

.842685
.842815
.842946

A . 1 i

2.17
2.18

.856078
.855956
.855833

2! 03
2.05

2AQ

.986607
.986860
.987112

,rSi

4.22

4.20

4OO

.013393
.013140

.012888

53
52

51

10

.843076

2.' 17

.855711

.Uo

2.05

.987365

rw-V

4^22

.012635

50

11

9.843206

21 r*

9.855588

2 A*"

9.987618

4OO

10.012382

49

12

.843336

.1 /

917

.855465

.05

.987871

.22

.012129

48

13

.843466

rt.lt

O 1 Z.

.855342

2 At?

.988123

4 .20

.011877

47

14

.843595

A. lO
917

.855219

.05
2r\K

.988376

4.22

4OO

.011624

46

15
16
17

.843725
.843855
.843984

4.11

2.17
2.15

917

.855096
.854973
.854850

.05
2.05

2.05
o nx

.988629
.988882
.989134

.22
4.22
4.20

4OO

.011371
.011118
.010866

45
44
43

18

.844114

iG.H

91=;

.854727

Jc.Uo

.989387

. "w-V

4OO

.010613

42

19

.844243

£ . 1 • >
9 1K.

.854603 i ~ A".

.989640

. fv-m

A O')

. 010360

41

20

.844372

£. JLO

2.17

.854480

2! 07

.989893

4^0

.010107

40

21

9.844502

9 1«i

9.854356

9.990145

10.009855

39

22

.844631

o i-

.854233

o n~

.990398

^4 . /w

4fc~K~k

.009003

38

23
24

.844760
.844889

/*. 15
2.15

9 1 K

.854109
.853986

2^05
4) /•»»•*

.990651
.990903

. ^i^»

4 '20
/I oo

.009349
.009097

37
36

25

26

.845018
.845147

A, JO

2.15

21 ^

.853862
.853738

2'or

2 A.**

.991156
.991409

4 22
400

.008844
.008591

35
34

27

.845276

. 10

.853614

.()i

Q A**T

.991662

••"•

1OA

.008338

33

28
29
30

.845405
.845533
.845662

2J3
2.15
2.13

.853490
.853366
.853242

2^07
2.07
2.07

.991914
.992167
.992420

.20
4.22
4 22
4^20

.008086
.007833
.007580

32
31
30

31

32

9.845790
.845919

2.15

21 Q

9.853118
.852994

2.07

O AQ

9.992672
.992925

4.22

4OO

10.007328
.007075

29

28

33

.846047

. J.O
O 1Q

.852869

.993178

.ii4

4 OO

.006822

27

34
35
36
37
38

.846175
.846304
.846432
.846560
.846688

2.15
2.13
2.13
2.13

rt -1 O

.852745
.852620
.852496
.852371
.852247

2^08
2.07
2.08
2.07

.993431
.993683
.993936
.994189
.994441

4.20
4.22
4 22
4.20

A OO

.006569
.006317
.006064
.005811
.005559

26
25
24
23
22

39
40

.846816
.846944

2! 13

2.12

.852122
.851997

2^08
2.08

.994694
.994947

4 '22
4.20

.005306
.005053

21*
20

41

9.847071

O 1Q

9.851872

' ' OS

9.995199

4 oo

10.004801

19

42

.847199

w. 1O

9 1^

.851747

9 08

.995452

4. oo

.004548

18

43
44

.847327
.847454

iv. JLO

2.12

9 1Q

.851622
.851497

/£.Uo

2 08

.995705
.995957

4. '20

1 * )• )

.004295
.004043

17

16

45
46

.847582
.847709

it. JO

2.12

.851372
.851246

2JO

SAO

.996210
.996463

4 22

4. ii i

.003790
.003537

15
14

47
48
49
50

.847836
.847964
.848091
.848218

' ' 1 **

2J3
2.12
2.12
2.12

.851121
.850998
.850870
.850745

.08
2.08
2.10
2 08
2 10

.996715
.996968
.997221
.997473

.20
4 22
4.22
4.20
4 22

.003285
.003032
.002779
.002527

13
12
11
10

51
52
53
54
55
56
57
58
59
60

9.848345
.848472
.848599
.848726
.848852
.848979
.849106
.849232
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9.849485

2 12
2 12
2.12
2.10
2.12
2.12
2.10
2.12
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9.850619
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9 849485

2 10
2.08
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9.997726
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999495
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10 000000

4 22

4.20
4 22
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10.002274
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.000253
10.000000

9
8
7
6
5
4
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9

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