QH105

STATISTICAL METHODS

WITH SPECIAL REFERENCE TO

BIOLOGICAL VARIATION.

BY

C. B. DAVENPORT,

Director of Department of Experimental Evolution,
Carnegie Institution of Washington.

THIRD, REVISED EDITION.
FIRST THOUSaJSTI'. •

NEW YORK

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

Copyright, 1899, 1904,

BY

C. B. DAVENPORT.

PRESS OF

BRAUNWORTH & CO.

BOOK MANUFACTURERS

BROOKLYN, N. Y.

PKEFACE.

THIS book has been issued iu 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.

iii

383952

PREFACE TO THE SECOND EDITION.

THE first edition of this book having been favorably re-
ceived, the 'publishers have authorized a revised edition
embodying many of the new statistical methods elaborated
chiefly by Professor Karl Pearson and his students and
associates, and presenting a summary of the results gained
by these methods. These, while increasing somewhat the
bulk of the book, have, it is hoped, rendered it more service-
able to investigators. Too much emphasis can hardly be
laid on the debt that Biometriciajis owe to Professor Pear-
son's indefatigable researches in the new science of Biome-
try— especially in the development of Statistical Theory.

The publishers, also, of this book are deserving of credit
for the courage they have shown in reproducing expensive
tables for the use of a still very limited body of statistical
workers. Especial attention is called to Table IV, which
is an extension of Table IV of the first edition that was cal-
culated by Professor Frederick H. Safford, and appears to
have been the first published table of the normal probability
integrals based on the standard deviation. More recently
Mr. W. F. Sheppard has published in Biometrika a similar
table in which, however, the tabular entries are given to
seven places of decimals, while the arguments are given
to two decimal places only. In the present table the argu-
ments are subdivided to three places of decimals and with
the aid of the table of proportional parts interpolation is
easily effected.

Especial acknowledgment must be made of assistance
received from my friend Mr. F. E. Lutz, who read over the
entire manuscript and contributed certain of the numerical
examples.

STATION FOR EXPERIMENTAL EVOLUTION

CARNEGIE INSTITUTION OF WASHINGTON.

COLD SPRING HARBOR,

March 27, 1904.

iv

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 determination 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

Characters occupying three dimensions of space 6

Characters having weight 6

Color characters 6

Marking-characters 7

Aids in calculating 7

Precautions in arithmetical work 8

CHAPTER II.

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

Seriation 10

Plotting 11

Method of rectangles 11

Method of loaded ordinates 12

The rejection of extreme variates 12

Certain constants of the frequency polygon 13

The average or mean 13

The mode 13

The median magnitude 14

The probable error of the determination 14

The probable difference between two averages 15

The probable error of the mean 15

The probable error of the median 15

The geometric mean 15

The index of variability 15

The probable error of the standard deviation 16

Average deviation and probable departure 16

Y

VI CONTENTS.

PAGE

Coefficient of variability 16

The probable error of the coefficient of variability 16

Quick methods of roughly determining average and variability. ... 17

CHAPTER III.

THE CLASSES OF FREQUENCY POLYGONS.

Classificatfon 19

To classify a simple frequency polygon 19

The normal curve 22

To compare any observed curve with the theoretical normal

curve 23

The index of abmodality 23

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

observed polygon 24

To determine the probability of a given distribution being

normal 24

The probable range of abscissae 25

The normal curve as a binomial curve 25

Example of a normal curve 26

To find the average difference between the pth and the (p+l)th

individual in any seriation 27

To find the best fitting normal frequency distribution when only a

portion of an empirical distribution is given 28

Other unimodal frequency polygons 30

The range of the curve 30

Asymmetry or skewness 30

To compare any observed frequency polyg-on of Type I with its

corresponding theoretical curve 31

To compare any observed frequency polygon of Type II with

its corresponding theoretical curve 32

To compare any observed frequency polygon of Type III with

its corresponding theoretical curve 33

To compare any observed frequency curve of Type IV with its

corresponding theoretical curve 33

To compare any observed frequency polygon of Type V with

its corresponding theoretical curve 34

To compare any observed frequency polygon of Type VI with

its corresponding theoretical curve 34

Example of calculating the theoretical curve corresponding with

observed data 35

The use of logarithms in curve fitting 36

General 38

Type IV. .,

Multimodal curves 39

CHAPTER IV.

CORRELATED VARIABILITY.

General principles 42

Methods of determining coefficient of correlation 44

CONTENTS. Til

PAGE

Galton's graphic method 44

Pearson's method 44

Brief method 45

Probable error of r 45

Example 45

Coefficient of regression 47

The quantitative treatment of characters not quantitatively meas-
urable 47

The correlation of non-quantitative qualities 49

Example. 51

Quick methods of roughly determining the coefficient of correlation. 54

Spurious correlation in indices 54

Heredity 55

Uniparental inheritance 55

Biparental inheritance 55

To find the coefficient of correlation between brethren from the

means of the arrays 56

Galton's law of ancestral heredity 57

Mendel's law of inheritance in hybrids 57

A dissymmetry index 60

CHAPTER V.
SOME RESULTS OF STATISTICAL BIOLOGICAL STUDY.

General 62

Variability 62

General 62

Man , 63

Mammalia 65

Aves 65

Amphibia 66

Pisces 66

Tracheata 66

Crustacea 66

Annelida 67

Brachiopoda 67

Bryozoa 67

Mollusca 67

Echinodermata 68

Ccelenterata 68

Protista 69

Plants 69

Some types of biological distributions. . 71

Type 1 71

Type IV 72

Type V 72

Normal 72

Skewness 72

Complex distributions 73

Vlll COKTEOTS.

PAGE

Correlation 73

General 73

Man 73

Lower animals 76

Plants 78

Heredity 78

General 78

Parental 79

Grandparental -80

Fraternal 80

Theoretical coefficient of heredity between, relatives 81

Homotyposis 81

Mendelism 82

Telegony 82

Fertility 82

Selection 82

Dissymmetry 82

Direct effect of environment 83

Local races 83

Useful tables 84

BIBLIOGRAPHY 85

EXPLANATION OF TABLES 105

LIST OF TABLES.

The Greek alphabet 114

Index to the principal letters used in the formulse of this book. . . 115

Table I. Formulas 116

II. Certain constants and their logarithms 117

*' III. Table of ordinates of normal curve, or values of —

2/o

corresponding to values of — 118

a

•' IV. Table of half-class index values (£a) or the values
of the normal probability integral corresponding to

values of — ; or the fraction of the area of the curve
a

between the limits 0 and -\ — - or 0 and — - 119

a -a

" V. Table of Log r functions of p 126

" VI. Tat>le of reduction of linear dimensions, from common

to metric system 128

" VII. Minutes and seconds in decimals of a degree 128

•' VIII. First to sixth powers of integers from 1 to 50 129

*' IX. Probable errors of the coefficient of correlation 130

•' X. Squares, cubes, square-roots, cube-roots, and recip-
rocals 131

** XI. Logarithms of numbers 149

•• XII. Logarithmic sines, cosines, tangents, and cotangents. 176

STATISTICAL METHODS

WITH SPECIAL REFERENCE TO

BIOLOGICAL VAKIATION.

CHAPTER I.

ON METHODS OF MEASURING ORGANISMS. .
Preliminary Definitions.

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

A multiple organ is one that is repeated many times on the
same individual. Example, the leaves on a tree, the scales
on a fish.

A character is any quality common to a number of indi-
viduals or to a number of multiple organs of one individual.

A variate is a single magnitude-determination of a character.

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

Graduated variates are magnitude-determinations of char-
acters which do not exist as integers and which may conse-
quently differ in different variates by any degree of magni-
tude however small; e.g., the stature of man.

A variant, among integral variates, is a single number-con-
dition, e.g., 5 (flowers), 13 (ray-flowers), etc.

A class, among graduated variates, includes variates of
the same or nearly the same magnitude. The class range
gives the limits between which the variates of any class fall.

Individual variation deals with diversity in the characters
of individuals.

Organ variation, or partial variation, deals with diversity in
multiple organs in single individuals.

1

2 , STATISTICAL METHODS.

Met5:orto 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 the general conditions, of race, sex,
locality, age, 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 tJie
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.

The number ot variates to be obtained should be large; if
possible from 200 to 2000, depending on abundance and
variability of the material.

Processes Preliminary to Measuring
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 photograpliic 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, echinoderms, various large arthnx
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 using 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. More or less transparent organs, such as
leaves, petals, insect-wings, and 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, neuroptera, 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 ZeissA*,
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 measurin-g 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
milled-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 ("Eutfer-
nungsmesser," Fig. 1), supplied at artist's
and engineer's supply stores at about $8.50.
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-huudredths 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 hundredths 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 Brown and Sharpe, Providence, tool
cutters.

FIG. 1.

MEASUREMENT OF ORGAKISMS.

scratches may be made clearer. The number of squares
covered by the surface is counted (fractional squares being
mentally sum mated) and the required area is at once obtained.
If the area has been traced on paper it may be measured by
the plauimeter (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.

6 STATISTICAL METHODS.

Characters occupying three dimensions of
space may be quantitatively expressed by volume. The
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:
Bed, 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 percents. 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-

MEASUREMENT OF ORGANISMS. 7

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

Aids in Calculating. An indispensable aid in multi-
plying and dividing is a book of reckoning tables of which
Crelle's Rechnungstafeln (Berlin: Geo. Reimer) is the best.
This work enables us to get directly any product to 999 X 999
and indirectly, but with great rapidity, any higher product or
any quotient.

The tables of Barlow (" Tables of Squares, Cubes, Square
Roots, Cube Roots, and Reciprocals of all Integer Numbers
up to 10,000") are like our Table X, but more extended.

The tedious work of adding columns of numbers is greatly
simplified by the use of some one of the better adding ma-
chines. There are many forms; of which the best are made
in the United States. The author has used the " Comp-
tometer" made by the Felt and Tarrent Manufacturing Co.,
Chicago ($225), and found it perfectly satisfactory. This
machine is manipulated by touching keys, as in a typewriter,
but it does not print the numbers touched off. In this respect
it is inferior to the Burroughs Adding Machine of the Ameri-
can Arithometer Co., St. Louis, Mo., which costs $250 to $350,
or to the Standard Adding Machine, St. Louis ($185).

For the multiplication and division of large numbers the
Baldwin Calculator is well spoken of (Science, xvn, 706). It
is sold by the Spectator Company, 95 William Street, New
York, price $250. The same firm is agent for Tate's Im-

8 STATISTICAL METHODS.

proved Arithometer ($300 to $400). The "Brunsviga" cal-
culating machine (Herrn Grimme, Natalia & Co., Brunswick,
Germany, Manufacturers; price $- 140 to v 75) is highly recom-
mended by Pearson.

To draw logarithmic curves and for the mechanical solu-
tion of arithmetical problems the instrument of Brooks
(Science, xvn, 690, not yet marketed) should be found useful.

Precautions in Arithmetical Work. Even the
most careful computers make mistakes in arithmetical wrork.
It is absolutely necessary to take such precautions that errors
may be detected. The best method is for statistical workers
to compute in pairs, but absolutely independently, comparing
results as the work progresses, so that time shall not be
wasted by elaborate work done with erroneous values. In
case of disagreement botli workers should recompute, start-
ing from that point of the work where their results check. In
cases where it is not feasible for the work to be done by two
people, it should be calculated on distinct pages of the note-
book— proceeding through several steps on the one page and
then independently through the same steps on another page;
checking the work as it progresses. It will be found useful
as the work progresses to make rough checks by comparing
the results with the original data to see that the results are
probable.

Neatness in arrangement of work and in the making of
figures is essential. It is best to make all calculations in a
book with pages about 20 cm. by 30 cm., quadruple ruled,
with about three squares to the centimetre, so that each
figure may occupy a distinct square. I like to work with a
pencil, of 2H grade, so that slight errors may be erased and
rectified. In case of larger errors running through several
steps of the work, the erroneous calculations should not be
erased but cancelled.

In using logarithms with the six-place table given in this
book, it is ordinarily necessary to write the entire mantissa
to six places, and to determine the number corresponding to
any logarithm to at least six places by use of the table of
proportional parts given at the bottom of the page. Upon
the completion of the calculation the number of decimal
places to be recorded will depend upon the probable* error of

MEASUREMENT OF ORGANISMS. 0

each constant. It will ordinarily suffice if the probable error
contain two significant figures, e.g., ±0.17 or ±O.OOS9; then
the constant will be carried out to the same number of places
and not farther.

iQ STATISTICAL METHODS.

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 : W 40$, N
(Black) 38$, 7 12$, O 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. Each class has a central value, an inner and an
outer limiting value, and a certain range of values.

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, 13,
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

SERIATION AND PLOTTING OF DATA.

11

arises concerning the inclusiveiiess 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

3.0-3.4; 3.5-3.9; 4.0-4.4; 4.5-4.9; 5.0-5.4;
Classes....

Frequency

Classes....

Frequency

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 :

[ 3.1-3.5; 3.6-4.0; 4.1-4.5; 4.6-5.0; 5.1-5.5;

3.2

3.7

4.2

4.7

5.2

1

2

3

4

5

1

1

3

3

7

5.5-5.9;

6.0-6.4;

6.5-6.9;

Y.0-7.4;

5.7

6.2

6.7

7.2

6

7

8

9

5

3

1

1

Classes....
Frequency
Classes ....

3.3

1

( 5.6-6.0;
( 5.8

3.8
1

6.1-6.5;
6.3

4.3

4

6.6-7.0;
6.8
1

4.8

3.5
6

7.1-7.5;

7.3

Frequency 6211

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.*

The best method, especially when the number of classes
is less than 20, is to represent the frequencies by rectangles
of equal base and of altitude proportional to the frequencies.
Lay off along a horizontal line equal contiguous spaces each
of which shall represent one class, number the spaces in order
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 classes (Fig. 3).

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

12 STATISTICAL METHODS.

This method of drawing the frequency polygon is known as
the method of rectangles.

When the number of classes is large the frequencies may be
represented by ordinates as follows : At equal intervals along

~n , ,

, , 1

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5

FIG. 3.

a horizontal line (axis of X) draw a series of (vertical) ordi-
nates whose successive heights shall be proportional to the
frequency of the classes. Join the tops of the ordinates as
shown in Fig. 4. This method of drawing the frequency
polygon is known as the method of loaded ordinates.

2600 LEAVES

-NORMAL CURVE

600
6CO
620
480

440

i

>•?

s

2

\,

'/

\,

/

^

7

i

\

,

Q400

I

j

s

Z

\

/

i

^

2

1;

i

\

^"00

\x>

H
'

V

120

-

\\

/

\

40

/

i

\

^-

' -

^5.

9 10 11 12 13 1-i
NUMBER OF VEINS

15

10 u

MEAN

18

19

20

21 22"'

23 J

FIG. 4. — VEINS IN BEECH LEAVES, AFTER PEARSON, '02'.

The rejection of extreme variates in calculating
the constants of a distribution polygon is to be done only
rarely and with caution. In many physical measurements
Chauvenet's criterion is used to test the suspicion that a
single extreme variant should be rejected. A limiting devia-
tion (KG) is calculated. K is the argument in Table IV cor-
responding to a tabular entry equal to - — •

471

SERIATIOK AND PLOTTING OF DATA. 13

EXAMPLE. — In 1000 minnows from one lake there are found the
following frequencies of anal fin-rays:

789 10 11 12 13

1 2 15 279 554 144 5

A = 10.835 ; a = .728 fin-rays.
1999 .__„

K=:*ooo==-49975-

Looking in Table IV we find 3.48 corresponding to the entry 49975.
Then the limiting deviation = 3. 48 X. 728 = 2.5334 and the limiting class
is 10.835 — 2.533=8.302; hence the observation at 7 might be excluded
in calculating the constants of the seriation ; but it should not be sup-
pressed in publishing the data.

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 average or mean (A) is the abscissa of the centre of
gravity of the frequency polygon. It is found by the formula

in which V is the magnitude of any class; / its frequency;
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 example on p. 10:
A =(3.2X1+3.7X1+4.2X3+4.7X3 + 5.2X7+5.7X5+6.2X3

+ 6.7 X 1 +7.2 XI)-*- 25 = 5.24,
or
AI=* (IX 1+2 XI +3X3 + 4X3 +5X7 +6X5 +7X3+8X1+ 9X1)

-*- 25=5.08,
A = 5.2* + .08(5.7-5.2) = 5.24.

A still shorter method of finding A is given on page 20.

The mode (M) is the class with the greatest frequency.
It is necessary to distinguish sharply between the empirical
and the theoretical mode. The empirical mode is that mode
which is found on inspection of the seriated data. In the
example, the empirical mode is 5.2. The theoretical mode is
the mode of the theoretical curve most closely agreeing with
the observed distribution. Pearson 1902b, p. 261) gives this

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

14 STATISTICAL METHODS.

rule for roughly determining the theoretical mode. The
mode lies on the opposite side of the median from the mean ;
and the abscissal distance from the median to the mode is
double the distance from the median to the mean; or,
mode=mean — 3 X (mean — median). More precise directions
for finding the mode in the different types of frequency poly-
gons are given in the discussion of the types.

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 ordinate drawn
from it bisects the polygon of rectangles or the continuous
curve, but not the polygon of loaded ordinates.

To find its position: Divide the variates into three lots: those less than
the middle class, i.e., the one that contains the median magnitude, 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 below the upper limit of the median class according as x
is positive or negative. Then \n — a : b = x : I" — V when x is positive,
or %n — c : b = x : I" — V when x is negative.

Thus in the last example: (12.5-8): 7 =x : 0.5; #=.32;. the median
magnitude = 5.0 + .32 = 5.32. Or (12.5-10): 7 = -x : 0.5; z=-.18;
the median magnitude =5. 5 -.18 = 5. 32. (Cf. p. 10.)

The probable error (E) of the determination of

any value gives the measure of unreliability of the determina-
tion; and it should always be found. For, any determination
of a constant of a frequency polygon is only an approximation
to the truth. The probable error (E) is a pair of values lying
one above and the other below the value determined. We
can say that there is an even chance that the true value lies
between these limits. The chances that the true value lies
within :*

±2£are 4,5:1 ±5E are 1,310:1

±3Eare21 :1 ±6E are 19,200:1

±4E are 142 :1 ±7E are 420,000:1

±8£are 17,000,000:1

±9E are about a billion to 1.

The probable error should be found to two significant
* These values are easily deduced from Table IV.

SEKIATIOST AND PLOTTING OF DATA. 15

figures. The determination of which it is the error should
be carried out to the same number of places as the probable
error and no more.

The probable difference between two averages (Al and
•A 2) of which the probable errors (El and E2) are known is
the square root of the sum of the squared probable errors, or
(Pearson, '02):

Probable Difference of Al-A2 is \/E* + E*.

The probable error of the mean is given by the
formula

tion [gee bglow]

Vnumber 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
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_ probable error of the median is ±.84535<r
+\/n (Sheppard, '98).

The geometric mean of a series of values (v) is the
number corresponding to the average of the logarithms of
the values. Thus,

J_(log^)

\JT — IV •

n

The index of the variability, a, 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:

V

sum of [(deviation of class from mean)2

X frequency of class]

number of variates

where X is the number of units in the class range, frequently
unity.

16 STATISTICAL METHODS.

This measure is known as the standard deviation. It
is a concrete number expressed in the units of the classes.
This, the best measure of variability, is expressed geomet-
rically as the half parameter, or the abscissa of the point on
the frequency curve where the change of curvature (from
concave to convex toward the centre) occurs.

The probable error of the standard deviation is

+ n.B74K ^andard deviation _ ±0-6745 *
V2 X number of variates \/2n

Other Indices of Variation. The average deviation,
or average departure, is found thus:

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

The average deviation is equal to .7 97 9 X standard deviation, or

The probable (or mid) departure is the distance from the mode
of that ordinate which exactly bisects the half curve QMX or OMX1,
Fig. 5, 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 devia-
tion 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 pro-
posed to reduce the index of variation to an abstract number,
independent of any particular unit, by dividing the index of
variation of any variates by the mean; the quotient multi-
plied by 100 is called the coefficient of variability. In

a formula, C=-^XlOO% (Pearson, '96; Brewster, '97).

The probable error of the coefficient of vari-
ability is given by Pearson as:

SERIATIOX AND PLOTTING OF DATA. 17

When C is small, say less than 10%, the factor in brackets
may be omitted, especially as only two significant figures
of the probable error need be recorded.

The average, standard deviation, coefficient of correlation,
and their probable errors may be conveniently calculated al-
together by logarithms, as shown in the paradigm on page 38.

QUICK METHODS OF ROUGHLY DETERMINING AVERAGE AND
VARIABILITY.*

1. Arrange the specimens in a series according to the mag-
nitude of the character, simply judging the order by the eye.
Then pick out those two that will divide the series into thirds
and measure them. Their average will be the average of the
whole series. Then,

Mean — the smaller of the two measures _
.43

(.43 is the value of ± — , at which the area of the curve

included between these limits of x equals one-third of the
whole) .

Or, 2. Select roughly two specimens that seem to be about
one-third of the distance from the two extremes £nd group
all others as larger than the larger one, smaller than the
smaller one, or between the. two. Measure the two speci-
mens. Count the number in each group and determine o
by aid of Table IV (p. 120) as follows: Taking as origin the
middle of the whole series, call the number of leaves from
the middle to the smaller n2. and the number from the
middle to the larger w2". Also, the x distance to the lower
division point h^ and to the upper division point h2. Then
(/i1+/i2) = the range covered by the middle division or the
difference between the upper and lower value. As we know
the areas of the curve between the origin and h^ on the one
hand and h2 on the other (percentage of individuals between

the middle and h, and h2), we can find — and — from Table IV,

a a

x
since they are the values — corresponding to the percentage

* See Macdonell, 1902.

18 STATISTICAL METHODS.

areas determined. But — +— =• : thus a is deter-

G a a

mined. Knowing a we can get h^ or h2, and hence the mean.
Or the value of the character of the middle specimen may
be taken as the mean value.

EXAMPLE. — Seventy-six beech-leaves which had fallen from one
tree were picked up. They were sorted out as in the second method.
It was found that 22 were smaller than the smaller type leaf, which
was 1.78 inches in length; and 23 were larger than the larger type leaf
(2.22 inches in length). The 38th leaf is the middle of the series, and
so the smaller type leaf was distant 16 leaves from the middle, and
the larger 15.

From Table IV:

.21223

.20884

Therefore ^ = .555.

Similarly = .5

h2

A! = .2278, A2=-2122.

Mean is at 1.78 + .2278 =2.01.

THE CLASSES OF FREQUENCY POLYGONS, 19

CHAPTER III.
THE CLASSES OF FREQUENCY POLYGONS.

The plotted curve may fall into one of the following'classes :

A. Unimodal.

I. Simple.

1. Range unlimited in both directions:

a. Symmetrical. The normal curve.

b. Unsym metrical (Pearson's Type IV).

2. Range limited in one direction, together with

skewness (Types III, V, and VI).

3. Range limited in both directions :

a. Symmetrical, Type II.

b. Unsymmetrical, 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 Fo)

20 STATISTICAL METHODS.

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 the corresponding class and divide by n\ call
the quotient VL

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 v*.

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 vz.

Add the products of ike fourth powers of all the departures
multiplied by the frequency of the corresponding class and
divide by n\ call the quotient v±. Or,

— y }

— — = departure of FQ from mean. F0 being

known, A may be found [A

The values rlt r3) r$, r4, are called respectively the first,
second, third, and fourth moments of the curve about Fo.

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:

* This is the short method of finding A referred to on page 13.

THE CLASSES OF FREQUENCY POLYGONS. 21

71

^5 ~ 5"i»4 + lOVva - lOVv., + 4y15 ;

(B) To find moments in case of graduated variates:

y^ - 10V*2 + 4y15 - f /£JA5;
in which A is the class range expressed in the same unit as
the average.

The probable error of the preceding constants in the special
case of the normal curve is as follows:

Efi2= .67449<724/- ; E^= .67449

71 '

= .6T4494 -;

T 71

= •67449V//i<r (p- 31) ;

# of Skewness= .67449y 7^. (See page 30.)
(From Pearson, 1903C).

The classification of any empirical frequency polygon
depends upon the value of its " critical function," F* (Pear-
son, 1901d).

4(4A-3/?1)(2/?2-3A-6)<

* This value of F is general. For the special case of Types I-IV
the following critical function was given by Pearson and has been

STATISTICAL METHODS.

Value of F.

Corresponding Frequency Curve.

F>1 and <oo
F=l

F>0and <1
F-O/A-OtA-S

F=Q, ft = 0, 02 not = 3
.F<0

Type III. Transitional between Type

I and Type VI.
Type VI.
Type V. Transitional between Type

IV and Type II.
Type IV.
Normal curve.
Type II.
Type I.

An important relation to be referred to later is-

3A-2& + 6'
M

f

\

1

\

I

\

1

\

1

\

1

\

1

\

/

\

•

/

\

/

i i

5.

\

& 4 3 2 1 v

FIG.

\. 2 .

5 1 2

THE NORMAL CURVE.

The normal curve is symmetrical about the mode; con-
sequently the mode and the median and mean coincide.
The mathematical formula of the normal curve, a formula

much used. Fi = 2p2— 3/?i — 6. The classification was given as follows:

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

When F u negative and ^ ^ ^ ^ ^ .g rf Typ<j „

-| *>»• *>»• CUrVe Is °f Typf IIL

< /?i=0, ^2 = 3, curve is normal.

When F is positive and ft> 0, 52> 3, curve is of Type IV,

When .

and

THE CLASSES OF FREQUENCY POLYGONS. 23

of which one does not have to understand the development
in order to make use of it, is

This formula gives the value of any ordinate y (or any class)
at any distance x (measured along the base, X, X', of Fig. 5)
from the mode, e is a constant number, 2.71828, the base
of the Naperian system of logarithms, n is the total area
of the curve or number of variates, and o is the Standard
Deviation, which is constant for any 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 loaded ordinates the theoretical fre-

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

taken directly from Table III. Here — is the actual devia-

o

tion from the mean expressed in units of the standard devia-

tion, and 3L the corresponding ordinate, yQ being taken as

2/o
equal to 1, and <r is the standard deviation.

For the case of a polygon built up of rectangles represent-
ing the relative frequency of the variates, Table IV gives
immediately the theoretical number of individuals occurring

between the values z=0 and x= ±— . By looking up the

x
given values of — the corresponding theoretical percentage

G

of variates between the limits rc=0 and x= ±— will be found

a

directly. The ratio — may be called the Index of Abmodality.

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

and

24 STATISTICAL METHODS.

also the skewness (p. 30) should be less than twice the value
.67449 /-

To determine the closeness of fit of a theoreti-
cal polygon to the observed polygon. Find for
each class the difference (<^) between the theoretical value (y)
and the observed frequency (/). Divide the square of this
difference in each case by y. The square root of the sum of the

/~d~2
quotients is the index of closeness of fit (/). Or, A= 4/ I— •

a

The probability (P:l) that the observed distribution is truly
represented by the theoretical polygon may be calculated from
the following formula, to use which the number of classes
(A) must be odd or must be made odd by the addition of a
class with 0 frequency.

__
2 2.42.4.6

4£ZL_\
.4.6...,l-3/'

This is the method of Pearson, 1900&.

To determine the probability of a given dis-
tribution being normal. Having found, in units of the
standard deviation, the deviation (7) of the inner limiting
value (L) of each class from the average, look up the
corresponding class-index a from Table IV. Or, better, find a
directly for each class by dividing the half of the total num-
ber of variates minus all those lying beyond the inner limit-
ing value of the class in question by the half of the total

I */

number of variates; or, in a formula, -y— -; where JT0Y/ means

?n

add all the frequencies from the median value to ^, and n
is the number of variates. Next find for each class the sum
of A-}- a%. This should equal L. The difference is the
actual discrepancy. The probable discrepancy should next be
calculated for all but the extreme values. It is calculated
by use of the formula

0.6745,

THE CLASSES OF FREQUENCY POLYGONS.

where the value of z corresponding to % is got from Table III,
or from the formula

1

The ratio of actual to probable discrepancy is next to be
calculated for each class. The probable limit (P.L.) of the
ratios varies with the number (A) of ratios found, according
to the following table :

Ai

P.L.

A\

P.L.

Ai

P.L

Ai

P.L.

I
2
3
4
5

1.000
1.559
1.874
2.088
2.248

6
7
8
9
10

2.375

2.481
2.570
2.648
2.716

11
12
13
14
15

2.777
2.832
2.882
2 928
2 970

16
17
18
19
20

3.009
3.046
3.080
3.112
3.142

The foregoing method is from Sheppard (1898).

The probable range of abscissae (2xt) of a normal dis-
tribution, or that beyond which the theoretical frequency (y)
is less than 1, varies with the number of variates (ri) as well
as with a, in accordance with the following formula derived

by the transposition of y= — -p=-e~x2/2<r* by putting y=l:

a\/2x

V l~6

2xt= 2(7

Example. For the ventricosity of 1000 shells of Lit-
tornea littorea from Tenby, Wales, A = 90.964% and o=
2.3775%. What is the probable range of ventricosity
expressed in per cent.?

/
.

1 000

The observed range was 15 (Duncker, '98). See also the
criterion of Chauvenet ('88) for the rejection of extreme
variates (page 12).

THE NORMAL CURVE OF FREQUENCY AS A BINOMIAL
CURVE.

The normal curve may also be expressed by the binomial
formula (pXq)A, where p=J, g=i, and A is the number of

26 STATISTICAL METHODS.

terms, less 1, in the expansion of the binomial; hence approx-
imately the number of classes into which the magnitudes of
the variates should fall. If the standard deviation be known,
A may be found by the equation

J=4X (Standard Deviation) 2=4<r2.

Example of Normal Curve. — Number of rays in lower valve
of Pecten opercularis from Firth of Forth:

V

14
15
16
17
18
19
20
21

/

F-FO

f(V—Vo)

/(F-T

1

-3

-3

9

8

-2

-16

32

63

-1

-63

63

154

0

0

0

164

1

164

164

96

2

192

384

20

3

60

180

2

4

8

32

= 508

342

864

-27

-64

63

0

164
768
540
128

1446

81

128

63

0

164

1S36

1620

512

4104

A = F0+ vi = 17 +.6732 = 17.6732. „

/m2= 1.7008 -0.67322 =1.2475; <r = V^ =1.1169.

^ =2.8465 -3 X0.6732 Xl.7008 +2 X0.67323 =0.0217.

/x4 =8.0787 -4 X0.6732 X2.8465 +6X0.67322 Xl.7008 -3 X0.67324

0 02172 4 4223 =4.4223.

U.u^J.* r\ (\f\f\n. a -z.-m&'J QA~\ A

2 ~ i 1 i £»7To =^.o414;.

-0.00047 ; F/jL23 = 0.0009.

0.0002 X5.84142

4XH.3650X(-0.3178)
3v>2-2v14_3(1.7008)2-2X.70594_
v4 "L8.0787

508

Theoretical maximum frequency, 2/0 = —

cV2« 1.1169V2^

= 181.5.

The probable discrepancy, based on the five larger values
of y, is found as follows, the Xi values being taken from a
tablelike Table IV:

L

a

Xi

A +<r\i

Actual
Dis-
crepancy.

Probable
Dis-
crepancy.

Ratio of
Actual to
Probable
Dis-

14.5

-0.99606

crepancy.

15.5

-0.96457

-2.11

15.34

+ 0.17

.083

2.05

16.5

-0.71654

-1.07

16.51

-0.01

.032

0.31

17.5

-0.11023

— 0.138

17.55

-0.05

.025

2.00

18.5

+ 0.53543

0.73

18.51

—0.01

.027

0.37

19.5

+ 0.91439

1.72

19.62

-0.12

.054

2.22

20.5

+ 0.99213

THE CLASSES OF FREQUENCY POLYGONS. 2?

The extreme values are not calculated for the relations
indicated by the formula do not hold well there where the
frequencies are small and the proportionate values of y are
changing rapidly for small changes of x For the five values
considered the actual discrepancy is less than the probable
discrepancy in three cases and less than the probable limit
in all.

To find the average difference between the
pt\\ and the (pi l)th individual in any seriation
(Gait oil's difference problem). Let xp be the aver-
age interval between the pth and (p-fl)th individual; n the
total number of variates; and a their standard deviation.

Then, (1) when n is large and p small:

. _ V2np pVe'V

P_ nym*

where ym— / — e~^m .

m can be found from Table IV by the use of the formula

where the value of m sought is the argument correspond-
ing to the tabular entry ( -- — j .

-=?- + 1.875^2. (^V;

ym n3 \ym]

.-
-p)p n1 y

57_±

ym

28 STATISTICAL METHODS.

_ocfa-p)5 + p5 ~r(n-pY-p* w_
-^ -*nn- '

31m*-101raM-28
2/m

The solution of the equations for cuc2, and c3 will be facili-
tated by finding, once for all, the logarithms of n, (n — p),

tn
(n-2p))(n-p)p, and— .

i/m

(2). When n and /> are both large and not nearly equal:

(3). When n is small the unsimplified form of the equa"-
tion must be used.

X-- (1 + ^ + ^+03+ ...)•

ym

\n means the products of all integers from 1 to n. The
series clt c2, c3 is not complete, but the values of c with higher
subscripts are so small that they may be neglected.

Let Ip'p" be the difference measured in units of o between
the p'th and the p"th individual, then

The foregoing method is that of Pearson (1902k) based
upon some considerations of Galton (1902).

To find the best fitting normal frequency dis-
tribution when only a portion of an empirical
distribution is given.

First apply the following parabola of the second order:

THE CLASSES OF FREQUENCY POLYGONS. 29

2 1

+£ij+s*\T) \>

where Z is the half range and

2/0 07 1 A) **'-2 i r£2? A 7 J* *1**;Z 72'

61 1O 7/1^ 77? Ot

To find m0 arrange the frequencies in the usual manner
(p. 26) and find the logarithm of each; their sum is equal
to ra0. Making the class situated at the middle of the
range 0, find the deviation of each of the other classes from
this class. The algebraic sum of the product of the loga-
rithms by the deviations gives mr The second moment
about the same zero point gives m2. Or,

.

Substituting in (1) we get a numerical quadratic equation
which can be put in the form

If the normal curve be y=z0e
(3) y

whence, by comparison of right-hand expressions in equa-
tions (2) and (3),

(£ 2\
'o-jjjjj

Then the required normal curve is

i-vr***

(Pearson, 1902m.)

30 STATISTICAL METHODS.

OTHER UNIMODAL FREQUENCY POLYGONS.
The formulas of Pearson's Types I to VI are as follows:

Type I. y=4+Q

/ X2\

Type II. y=y0(l—fi) .

Typelll. y=y0(l + jYe~x/d'

Type IV. y=y0cos62me~1:0, where tan 6=-r.

Type V. y=y0x~pe~r/x.

Type VI. y=y0(x-l)«*/3«*.

In these formulas:
rr, abscissae;
y0, the ordinate at the origin, to be especially reckoned for

each type;
y, the height of the ordinate (or rectangle) located at the

distance x from 2/0;
I, a part of the abscissa-axis XX' expressed in units of the

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

The other letters stand for relations that are explained in
the sections below treating of each type separately.

The range of the curve is limited in both directions in
Types I and II, is limited in one direction only in Types III,
V, and VI, and is unlimited in both directions in Type IV
and the normal curve. The normal curve may give the best
fit, however, notwithstanding the fact that in biological
statistics the range is ordinarily limited at both extremes.
Thus the range of carapace length to total length of the
lobster is limited between 0 and 1. The ratio of carapace
length to abdominal length in various crustaceans may, how-
ever, conceivably take any value from + oo to 0. In the ratio
of dorso ventral to antero-posterior diameter the forms of the
molluscan genera Pinna or Mafleus on the one hand and
Solen on the other approach such extremes.

Asymmetry or Skewiiess (a) is found in Types I, III,
IV, V, and VI. In skew curves the mode and the mean are

THE CLASSES OF FKEQUEKCY POLYGONS. 31

ated from each other by a certain distance D; or D =

an — mode. Asymmetry is measured by the ratio a= — .

a

If the mean is greater than the mode, skewness is positive;
if the mean is less than the mode, skewness is negative. D,
and hence skewness, may be calculated when the theoretical
mode is known (see pages 13, 14, and below).

In Types I and III skewness is measured also by the

raf.o -jVSjff, where '-%£& When

5/92 — 6/?t — 9 is positive, a has the sign of /*3; if negative,
a has the opposite sign to /(3 (Duncker, '00b).

InType i.

" " III, a=%\/Ji= — ±fi/3—, where the sign is the
+ 2V 1*2* same as that of /*3.

P
since p — 4 is the positive root of the quadratic:

p is readily found.
InType VI, a=

(ql

where (1—qJ and (q2+l) are the two roots of the equation

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

2/=

32 STATISTICAL METHODS.

To find 119 12, ml9 m2, yQ. *

The total range, Z, of the curve (along the abscissa axis]
is found by the equation

1=

Zj and 12 are the ranges to the one side and the other of t/0;

2. a;

™1 . m™2

=s — 2;

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

_n (m1 + m2+l)\m1 + m2 12
I \/27tmlm2

_J1___L\
2 m\ m2/

With these data the theoretical curve of Type I maybe
drawn. Frequency polygons of Type I are often found in
biological measurements.

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

This equation is only a special form of the equation of Type
I in which ZX=Z2 and m1 = m2.

As from page 22, ^ = 0 in Type II, l=2a\/s+l; since the
curve is symmetrical, D— 0, and

t/ ON n F(m+1.5)

m=i(«-2); 7/0=-— A— -f-.

The F values will be found from Table V.

THE CLASSES OF FREQUENCY POLYGONS. 33

An approximate formula for yQ is given by Duncker as fol-
lows:

s-1

. _ -

\/(* +!)(*- 2)

4(8-2).

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

X\ P -x/d

"// e

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

l,= a ^^= <2—^- (for Type III).
2\/ft

I, I, n "p+l

Also, p=j-= — ; yQ=-

j\ j
D aa'

7-- — — •
I, epr(p+l)

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

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

This is the commonest type of biological skew curves.

6 is & variable, dependent upon x as shown in the equation

The factor (cos 6)2m following y0 indicates that the curve
is not calculated from the mean ordinate (A), or the mode
(A — D), but that the zero ordinate is at A —mD; or at a dis-
tance mXD from the mean.

the opposite sign to

34 STATISTICAL METHODS.

0 (arc of circle) = 5, •

nA/s e
2/o=y4/ oT-

___

<j>= angle whose tangent is — .
s

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

To find p solve the quadratic equation
and take the positive root.

r=a(p -2)v/Pz:3; ?/o=rrLiv D==S^^Y

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

y=y0(x-W2/*qi-

1 — ql and q2-{- 1 are the two roots of the equation

li=s i^-' <, where (1-qJ and s are negative;

-

D=;

* The foregoing value is approximate and is applicable when, as is
usually the case, a is greater than 2. The exact value is given by
Pearson as

/„ >

(sin 6feT0dd
.

the formula for reducing which is to be gained from the integral cal-
culus.

THE CLASSES OF FREQUENCY POLY<i()\S. 35

Example of calculating tlie theoretical curve corre-
sponding vvitli observed data. (Fig. 6.)

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

Number of glands 0123456789 10
Frequency 15 209 365 482 414 277 134 72 22 8 2

Assume the axis yy' ( Vni) to pass through ordinate 4, then:
V V-Vm f /(F-Fm) f(V— F«>)2 /(F- F»*)3 /(F — Fm)«

0

4

15

— 60

240

— 960

3840

1

3

209

— 627

1881

— 5643

16929

2

o

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

7

3

72

216

648

1944

5832

8

4

22

88

352

1408

5632

9

5

8

40

200

1000

5000

10

6

o

12

72

432

2592

2 2000 —998 6148 —3872 48568

Vl = — • 998 -*- 2000 = — .499.

ra = 6148 -*- 2000 = 3.074.

v, = — 3872 -*• 2000 = — 1.936.

v4 = 48568 -*- 2000 = 24.284.

/n=0; A = 4-. 499 =3.501.

Ma = 3.074 — (— .499)2 = 2.824999.

M3 = - 1.936 - 3(- .499 X 3.074) + 2(- .499)' = 2.417278.

M4 = 24.284 -4(-. 499 X- 1.936) -f 6(.249001 X 3.074) - 3(- 499)* = 24.826297,

(2.417278)* _ 5.843232929 _
Pl (2.824999)3 " 22.545241683 ~
_ 24.826297 _ 24.826297 _

^^

_
(2.824999^ - 7.98061935

.259 X (6.111)2

4(12.443 - .778)(6.222 - 6.778)
6(3.11082- 0^25918 -
.555o9

,21.9857

a- K V.259178 ~^- = .31115.

D- 1.680774 X .3111 = .5230.
Z).a- .5230 X 19.9857 = 10.4519.

Z= .840387 4/16 X 20.9857 -f 0.25918 X (21.9857)2 = 18.0448.
18.0448- 10.4519 = ^^

36 STATISTICAL METHODS.

Ja= 18.0448 -3.7965 = 14.2483;
3.7965X17.9857

- 18.0448
14.2483X17.9857
1^0448 -

.
378401'

17.9846 vo1_1QO_.0833(.0556-.2643-J0704)

A 4.L

18.0448 \/2 K x 3.7840 X 14.2006

= 475.24, the frequency of the modal class.

Position of the mode, ' y0=*A -D=3.50l -.523 = 2.978. The close-
ness of fit to the theoretical curve is calculated below by Pearson's
method (page 24).

V f Theoretical (y} d d*

-1 0 0.0 0.0

0 15 21.1 - 6.1 37.21 1.76

1 209 185.8 +23.2 538.24 2.90

2 365 395.1 -30.1 906.01 2.30

3 482 475.2 + 6.8 46.24 .10

4 414 405.6 + 8.4 70.56 .17

5 277 272.1 + 4.9 24.01 .09

6 134 147.6 -13.6 184.96 1.25

7 72 65.9 + 6.1 37.21 .57

8 22 24.1 - 2.1 4.41 .18

9 8 7.0 + 1.0 1.00 .14

10 2 1.6 + 0.4 .16 .10

11 0 0.2 - 0.2 .04

12 0.0

^—=9.56

y

That is, the probability is that in one out of every two random series
belonging to Type I we should expect a fit not essentially closer
than that given by our series, which, of course, assures us that this
distribution is properly classified under Type I.

THE USE OF LOGARITHMS IN CURVE-FITTING.

Most of the statistical operations can be greatly facilitated
by the use of logarithms. In curve-fitting their use becomes

.01234.56T

FIG. 6.
Distribution of frequency in glands of swine.

» polygon of observed frequency.

— - — -, polygon of theoretical frequency (Type I).
- • - -, normal frequency polygon.

38 STATISTICAL METHODS.

necessary. The following paradigm will be found of assist-
ance:

GENERAL.

log vj = log 2(V- F0) - log n. A = Vm + vr

log v2= log 2(V— V0) 2 — log n. log a= J log fiy.

log i/3= log 2(V- F0)3 -log n. log C= J log /*2-log A.

Iogv4=log ^'(F-F0)4 -logn.

log E. A = 9.828982 + log a - J log n.

log E.a = log E.A - 0. 150515.

logE.c=logE.ff-log A.

log 2= 0.301030 TV= .08333 Find 2 log ^

log 3= 0.477121 ^=.02916 3 log vl

log 4=0.602060 Tjf 0 = .0125 4 log vx

log 6=0.778151 log i= 9. 98970

V2=N(\og v2) -AT (2 log Vl) -[.0833]. Find: log /£2; 2 log /z2;

3 log M2.
j«3= AT(log v3) - JV(log 3 + log v, + log v2) + AT (log 2 + 3 log vx)

Find: log /£,; 2 log py
fi4= N(\og y4) -TV (log 4 + log yt + log v3)

+ AT (log 6 + 2 log vj + log v2) - A^dog 3 + 4 log vj
-7V[9.698970 + log ^2]-^. Find log /i4.
log /?!= 2 log /£3 - 3 log /£2.
log/?2=log//4-21og/M2.
T(;==5/32-6/3i-9 (Types I, IV).

Skewness:

Type I: log «=1 .og ft + log w- log (ft + 3) -0.301030.
Type III: log a=J log ft-0.301030.

THE CLASSES OF FREQUENCY POLYGONS. 39

Type IV: log a= J log ft + log (/?2-f 3) - log w - 0.301030.

Type V: log a=log 2 + i log (p-3)-log p.

Type VI: log a=log (& + &) + J log (ft-&-3)-log (&-&)

TYPE IV.

This is the most difficult of all the types to be fitted. The
work of fitting is carried out by the use of logarithms, as
follows :

log /=i log ft + log («-2). log fc=l

log a=log /-log (s + 2) -0.301030.

log 1= J log /£2 + i log { N[log (*-!) + 1 .204120]

-JV[log A + 2 log(s-2)]| -0.602060.

s + 2

log ml) = log A: -0.602060.

log r=log /b + log s- 0.602060 -log

log tan 6— log T — log s.

— log 3s)

—N (8.920819 -log s)-N(log r+log ^)] + 9.637784)
-0.399090-log Z-(s + l) log cos 0.
log y=logy0+N [log (s + 2) + log log cos 6]

T].

MULTIMODAL CURVES.

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

* Tn degrees and fractions of a degree; see Table VII.

40 STATISTICAL METHODS.

Multimodal curves differ in 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. 7 to 10).

6.5 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. 7.

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.

2
FIG. 8.

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

THE CLASSICS OF FREQUENCY POLYGONS. 41

terms of the standard deviation of the Linore 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 multimodal curves is diverse. Sometimes

FIG. 9.

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

3 i

321012

FIG. 10.

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 advantages >n species
itudy.

42 STATISTICAL METHODS.

CHAPTER IV.
CORRELATED VARIABILITY.

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

The methods of measuring correlation given below are
applicable to cases where the distribution of variates is
either symmetrical or skew.

The principles upon which the measure of correlated varia-
tion rests 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 (p. 23) of any class of B will be as great as that of
the corresponding class of A , or

Index abmodality of relative class _
Index abmodality of subject class ~~

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

Index of abmodality of relative class _ 0 _
Index of abmodality of subject class ~~ m ~ '

The coefficient of correlation is represented in formulas by
the letter r. We cannot find the degree of correlation be-
tween two organs by measuring a single pair only; it is the
correlation "in the long run" which we must consider.
Hence we must deal with masses and with averages.

CORRELATED VARIABILITY.

CI»-iOOOOi-<C»<?*

I I I I

lO SO
CO CO

• 8 8 8 '•£• 3

s z s

*-* o o o o

I I I

as s

I- CO Tt» O ~

OS GO GO i— I ^«

O »-i CM CO CO

I

Cl C* »-i

CO »i «O I- CO CO

oo QO »o co

28 a

C* QO rj« QO
O »O QO

> d

I I I I

14 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 43, which
gives data for determining the correlation between the num-
ber of Mulleriaii glands on the right (subject) and left (rela-
tive) legs of male swine. The selected subject class is called
the type; the corresponding distribution of the relative mag-
nitudes is called the array.

METHODS OF DETERMINING COEFFICIENT OF CORRELATION.

Gallon's graphic method. On co-ordinate paper
draw perpendicular axes JTand 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.

r_el. class X PP. of cases in both)

total no. of indivs. X Stand. Dev. of subject x Stand. Dev.
of relative ;

or, expressed in a formula :

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

T =

n<Ti<Ji

This method requires finding many products in the numera-
lor, as many sets of products as there are entries in the body of
the correlation table. A portion of the products to be found
in correlation table, p. 43, is indicated below:

(- 3.540 X 8

- 3.547 X <- 2.540 X 5

(- 1.540 X 2

f- 3.540 X 4
"1.540 X 161
,540 X 58
etc*

- 2.547 X «( ~ 2.540 X 161

|-i.r-

CORRELATED VARIABILITY. 45

The handling of long decimal fractions may be avoided by
the use of a method similar to that used at page 26 for find-
ing the average and standard deviation. The formula for r
may be written

Assuming the class including or nearest to the true mean
of the subject values as the mean of the subjects, and the
class including or nearest to the true mean of the relative
values as the mean of the relatives, find for each variate the
product of its deviations xf and yf from the respective assumed
means, and (having regard for signs) find the algebraic sum
of these products. Divide this sum by the number of vari-
ates; the quotient is the average of the deviation products about
the assumed axes. To refer to the true axes, passing through
the true means, find the average moments, v1 (as on page 26),
both for the subject and the relative distributions about their
respective assumed means, and subtract the product of the
two values of vl from the average of the approximate devia-
tion products already found. Divide the difference by the
product of the standard deviations of the two frequency dis-
tributions. (Compare Yule, '97b, pp. 12-17.)

The probable error of the determination of r is

E =0.6745(l-r2)

\/n

(Pearson and Filon, '98, p. 242.)

Example. Correlation in number of Miillerian glands
on right and left legs of 2000 male swine. (See table on next
page.)

For + quadrants Z(x'y')= 5243
" - " J Cry) =-118

5125

— 0= 2.5625=

STATISTICAL METHODS.

5s rfi (N r-t CO
N (N O •* •*

b. O <M cO O CO
Oi <N O »O O CO
C<l CO |> (N CO

(N O

CO O
CO (M

"^ N, o >o co i>

CO H ® £ g §
b»

00 ^ CO (N

"H 00 t>. O ^ »O CC
r>. 1> O O CO (M U2

ca GO 05 O «o cop

^ | C-) •<* i-KN

C5iC

i-l 0

1 1 1 * M '

1 II I CO

C^JT-i

, TJ4 CO <N rH

if J I 1 1 1

rH (N CO Tf 1C CO *T II v <N
S'S'

INI

- <0

5c bio

•c ^

COl> ^_,

rH ^-O ool> ^05 £<N SN

w co^-t oO ,-!> SCO ^^0

CO (NN TOO 000

"

096
^ cog —

IN g«N g^-i c g OS
w t OST
o 8 6SI

6 818

^ -HCO

CO O5 rnrf MOO WH

-l^ rH | (N | j

, r

TOOT

E2 1881 —

o ^6

g i-oet- g

' Z-ZIQ- ^

CORRELATED VARIABILITY. 47

-W -i- =(2.5625-. 4535 X. 4605)

*L7T95X 1.730 a=s°-7911-

A/2000

The average variability of an array is =<r\/l— r2.
The coefficient of regression marks the proportional
change of the relative organ for a unit's change of the sub-

ject organ. It is given by the equation p=r — , where a^ is

<72

the standard deviation of the subject, o2 that of the relative.

THE QUANTITATIVE TREATMENT OF CHARACTERS NOT QUAN-
TITATIVELY MEASURABLE.

Even qualities that do not lend themselves to a quanti-
tative expression may be expressed in a roughly quantitative
fashion. The fundamental assumption is made that the
frequencies would obey the normal law of frequency more
or less closely, provided a quantitative scale could be found.
This assumption will not, in most biological data, lead us far
astray.

Divide the data into three classes (e.g., in eye-color we may
have black, brown and gray, and blue), and let the frequency
of these classes be n}, n2, n.3, in which nt and n.3 are each less
than \n, so that n2 contains the median. Let Llf L.f be the
(unknown) distances of the mean from the two boundaries
of n2. Call Li/a=hl and L3/a=h3, then

7T./0

and

nr+n2-n3
n

/2 f
V n JQ

48

STATISTICAL METHODS.

Now the left-hand side in these equations is known ; it is \a
of Table IV. From this table the right-hand value of the

FIG. 11.

equations is found; it is the entry corresponding to the argu-
ment Ja. Thus hj_ and 7i3 I = — j are found, and hence LJo

— —

and L3/(j and the entire range

of the middle class,

in terms of o, is known. Call the range in absolute units Z.
Then l=L3 + ^ and I/ a is known and for a second series I /a'
can be similarly determined. Hence a /a' ', the ratio of the
variabilities of the two series, is determined.

Again, since LJa and — - * are known, Ll/(L3 + L1) is

known, and this gives us the ratio in which the mean divides
the true range of the central class. (Pearson and Lee, 1900.)
The foregoing method may sometimes be advantageously
employed where the data are quantitative. In this case
the numerical value of I is known. (Macdonell, 1902.)

Consequently /it + h2 = —
= * 3

is known and hence

"i

, the standard deviation, is found. Since Lv

the distance of the mean from the left-hand boundary of n2f
the position of the mean is known.
The probable error of a is

3

where

and

COKRELATED VARIABILITY.

49

The values of the last two equations may be obtained
directly from Table III.

The probable error of Llt or rf the mean, is

= .67449 -

_ / Ea V v 2 _
- ' A"

_n3(n-n3)

THE CORRELATION OF NON-QUANTITATIVE QUALITIES.

Pearson (1900C) has ingeniously discovered a method of ex-
pressing correlation quantitatively when the variables cannot
be so expressed, as, for example, in the case of effectiveness
of vaccination. Strictly, this method assumes normal vari-
ation in variables, but it can be employed generally, in
default of a better method, with fairly accurate results.

The prime requisite is that the qualities to be compared
shall be separable into two grades, an upper and a lower.
For example, in the case of the result of vaccination: on
the one hand, either presence or absence of a scar; on
the other, either recovery or death. As either of the
second pair may occur with either of the first pair, four
classes, a, 6, c, d, will be formed altogether and a correlation
surface like the following may be made:

a

b

a + b

c

d

c + d

a + c

b + d

n

The axes y,—y and x,— x probably do not coincide with the
axes y and x passing through the "origin" of the correlation

50 STATISTICAL METHODS.

surface, but may be regarded as situated from those axes at
the respective distances h and k. These values may be
found from the formulae

(a+b)_(c + rf) /2M
n * TT •/ o

a, b, c, and d being known, h and fc are found from Table IV.
Then

of which the values may be looked up in Table III, or, better,
their product may be calculated by logarithms as follows:

Find also log hk, h2, and k2. To find r solve the following
equation to as many terms as may be necessary:

4 - 6/i2 + 3) (k* - G/c2 + 3)r5

This gives us a numerical equation of the nth degree which
can be solved by ordinary algebraic methods, using Sturm's
functions and Horner's method. Or it can be solved by
successive approximations as follows: The first approxima-
tion is made by neglecting all powers of r above the second
and solving the quadratic (remembering, that if ax2 + bx + c == 0,

CORRELATED VARIABILITY. 51

— 2^ — " ) » anc* taking the positive root. Substi-
tute this value in the whole equation to the 4th power for
/(r), and in the first derivative of the same equation for /'(r)
(remembering that the first derivative of /(#) is obtained by
multiplying each term in f(x) by the exponent of x in that
term and diminishing the exponent of x by 1). The correc-
tion ^7-7^7 should be added to the value of r used in substi-
tuting. Repeat this process as often as the correction affects
the fourth place of decimals, and go to r5 if necessary.
The probable error of r as thus determined is

found as follows: First calculate the relations &= _=

k — rh
and /?2 = ". — . Also find

and

from Table TV. Moreover,

1

Then,

Prob. error of r=-. i[j(o

(a + b)(d + c) &* + 2(ad - be) fa

which can be easily solved by substitution. In using the
foregoing formula, it must be noted that "a is the quadrant
in which the mean falls, so that h and k are both positive."
In other words, a + c > b + d and a + 6 > c + d. (Pearson, '00°.)

Example. The eye-colors of a certain set of people (see Bio-
metrika, II, 2 pp. 237-240) and of their great -grandparents were
found to be distributed as follows

STATISTICAL METHODS.
Offspring.

1

2

3

4

5

6

7

8

6

C
3

|

J^

o

Q.

«

pq

Q

<B

ofj

pq

j

s

pq

m

"S

1

§«

^l

|w

!>

g

1

1

1

1

3

«

O

(5

3

pq

P

S

H

•a

1 . Light blue

4

3

8

5

1

21

§

2 Blue— dark blue . .

8

177

95

76

' 's

39

'si'

"17"

448

J?

3. Gray — blue-green.
4. Dark gray — hazel.

1
6

69
30

85
21

52
27

2
2

20

7

26
15

1
1

256
109

5 Light brown

4

4

»H

6 Brown . . .

2

37

27

17

3

30

20

4

140

O

7. Dark brown

15

20

24

3

4

9

9

84

8 Black

10

13

12

2

2

7

5

51

Totals

21

345

269

213

17

103

108

37

1113

It was desired to determine the correlation between the eye-color
of the offspring and that of their great-grandparents. Clearly the
ranges of the classes given above are not quantitatively equal nor
determinable Consequently a fourfold table was formed by dividing
the population into those having eyes whose color was gray blue-green,
or lighter, and those having dark gray, hazel, or darker eyes. This
gives a good basis for calculation If the dark gray and hazel eyes
had been grouped with the lighter eyes it would have made quadrant
a entirely too large; and there is nothing in the nature of the data
that strongly favors one division more than another.

7 25 -388
1113

.302785

From the tables:

Offspring

1-3

4-8

Totals.

1-3

450

275

725

4-8

185

203

388

Totals.

635

478

1113

Ol

h

.31

.30

. 39886
.38532

.01

.01354

«2

k

.15
.14

.18912
. 17637

.01

.01275

h = .38532 + ( 1 .354 X .002785) = .389091
k = .17637 + (1.275 X .001060) = .177722

CORRELATED VARIABILITY.

53

Log £=9.5900512
Log k = 9.2497412

Log A2 = 9.1801024
Log k2 = 8. 499 4824

Jthlf— 0-34 *7* h

A2 = . 151392
A;2 = .031585

* + k2 .182t77

4-A-2
HQI^CQ

Log ta -8.8397924

h1f.= ORQI.^n

Log (450X203 -27 5X185) =4.607 1869

Log HK = - log 2 rr - .091489 log e

= 9.2018201 -M8.9613689 + 9.63778428]
= 9.2018201-0.0397332 = 9.1620869

= 4.6071869-(9.1620869 + 2 log 1113) =9.3521096

.224962 =r

Solving .034575r2+r-. 224962 = 0,

1 ± X/f+4T034575 X .224962)
-2(7634575)—

2_ ! = _ .848608

Coeff. r4 =

A:2- 1 = - .968415 CoeflF.

+ -069150 X 2.848608 X 2.968415

.136967

.024363r4 + . 136967r3 + .03457 5r2 + r - .224962 = 0.
Applying Newton's approximation, we reach the result

r = .2217.
(7 5095 + 303530&>2 + 281 300^t

Log w0=log*ff-*log(l-r2)

-Iog(l-r2)-log2]

ht + k*-2rhk = 0.152315, l-r2 = 0.950850.

Logw0=9.20182-9.989056-M9.637784 + 9.18274-9.978112-0.30103]
= 9.1779797

Log

9.828975-4.569743-9.177980 = 4.081253.

/?!= 0.358614
Table IV:

.358 .13983
22.2
.4

fa = .14006

Log E .r = 4 . 08 1 2530 + \ log 7 4426 . 858
E.r = 0.03289

#2=0.093794

.093

.03705
27.3
3.5

4>2 = .03736

54 STATISTICAL METHODS.

QUICK METHODS OF ROUGHLY DETERMINING THE COEFFI-
CIENT OF CORRELATION.
•
The method just described may be used in lieu of the rela-

2x 11
tion r= — — whenever the distributions of frequencies of

na1a2

the two correlated organs are normal. An exceedingly sim-
ple relation that is independent of the assumption of a normal
distribution has been given by 'Yule ('00b) as

_^ ad — be
T*~~ ad + bc'

and this may be used as a rough approximation to the coeffi-
cient of correlation.

But Pearson ('00C) has shown that this simple relation is
not nearly as close to the true r as the following:

where

4abcd . \

The superiority of the value rs as an approximation to r^
justifies the additional work its determination demands.

SPURIOUS CORRELATION IN INDICES.

When two characters a and b are measured in each indi-
vidual of a series of individuals, and each absolute magnitude
is transformed into an index by dividing it by the magnitude
of a third character c as found in the same individuals, a
spurious correlation will be found to exist between the indices

of - and - (Pearson, '97).
c c

Let Cj=the coefficient of variability of a;

C_ f f < ( ( ( (I I ( -L.

2= b;

C__<< t f t ( 1 1 ( ( .
3~~ C,

rQ= " " spurious correlation.

C 2

COERELATED VARIABILITY. 55

The precise method of using r0 in modifying any determi-
nation of r is uncertain. Pearson recommends using r — r0
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
its parent or parents, but only with mean progeny of
parents. Any group of selected parents is called a parentage,
the progeny of a parentage is called a fraternity.

Three categories of inheritance have long been recognized
(Galton, 1888, p. 12). These are: (1) blending heritage illus-
trated by stature in man; (2) alternative heritage, illustrated
by human eye-color; and (3) mixed heritage, illustrated by
the piebald condition of the progeny of mice of different
colors. The immediately following statistical laws of inherit-
ance hold especially for blending heritage.

In u 11 i parental 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
and the abmodality of the corresponding fraternity. More
strictly, since the variability of the character in the second
generation, a2, may (as a result of selection or of environ-
mental change) be different from the variability of the char-
acter in the first generation, <71? the index should be taken as

r— , called the coefficient of regression.

The probable error of this determination is

.6745^. /I -r122 .

— I/ — , in which r12 means the correlation coem-

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\/l—r2 is to 1.

In bipareiital 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 f rater-

56 STATISTICAL METHODS.

nity will be

where 7^= average abmodality of fraternity;

h2= average abmodality of male parent;

h3= average abmodality of female parent;

r2= correlation coefficient between fraternity and

female parent;
r3= correlation coefficient between fraternity and male

parent;

o1 = standard deviation of fraternity;
o2= standard deviation of male parent;
<73= standard deviation of female parent.
When assortative mating occurs, as is usually the case, the
abmodality of a fraternity is given by

where rx= 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

To find the coefficient of correlation between
brethren from the means of the arrays.

This is given by the formula

where rij is the number of the brethren in an array [and there-
fore JrijCnj — 1) is the number of possible pairs of brothers in
that array]; Al is the mean value of the array; a is the
standard deviation of the character in the brethren taken
all together, n is the total number of variates, and A2 is the
average of the brethren. This method will be found useful
where to take all possible pairs of brethren would be found
a work of too great magnitude (Pearson, Lee, etc.,'99, p. 271).

CORRELATED VARIABILITY. 57

Galtou ('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 J ; from the 8 great-grandparents 12. 5# or £ ;

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

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

where /i! = average abmodality of fraternity.
CTO = standard deviation of fraternity*
cTj , o-a . . . crs = standard deviation of mid-parent of
1st, 2d ... sth ancestral generation.
ki = abmodality of mid-parent of 1st ancestral genera-

tion.
&2, #3 . . . Jcs = abmodality of mid-parent of 2d, 3d

. . . sth ancestral generation.

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

MENDEL'S LAW OF ALTERNATIVE INHERITANCE.

In 1865 Gregor Merdel published an account of his experi-
ments in Plant Hybridization and reached the following laws,
which have been abundantly confirmed in certain experi-
ments.

First Case. The two parents differ in one character (the
antagonistic peculiarity) — case of monohybrids.

Of the two antagonistic peculiarities the cross exhibits
only one; and it exhibits it completely, so as not to be dis-
tinguishable in this regard from one of the parents. Inter-
mediate conditions do not occur [in alternative heritage].

2. In the formation of the pollen and the egg-cell the two
antagonistic peculiarities are segregated; so that each ripe
germ-cell carries only one of these peculiarities.

58 STATISTICAL METHODS.

Of the two antagonistic peculiarities united in the cross,
that which becomes visible in the soma is called by Mendel the
dominating, that which lies latent is called the recessive char-
acter. What determines which character shall be dominating
is still unknown, and the determination of this point offers an
enticing field of inquiry. In some cases the dominating form
is the systematically higher, in others it is the older or ances-
tral form.

The law of dichotomy may now be developed. When a
mongrel (monohybrid) fertilization takes place the zygote con-
tains both the dominant quality (abbreviated d) and the re-
cessive quality (r). In the early cleavages d and r are both
passed over into both the daughter-cells; but apparently, at
the time of segregation of the germ-cells, the somatic cells
are provided with d only, while the germ-cells retain both
qualities. In the ripening of these germ-cells, probably in
the' reduction division, d and r come to reside in distinct cells,
so that we have

of the female cells 50%d + 50%r, and
of the male cells 50%d + 50%r.

If now mongrels are crossed haphazard, a male d cell may
unite with either a female d cell or with a female r cell; like-
wise a male r cell may unite with a female d or a female r cell.
Consequently in the long run we shall have of all the zygotes

25%d, d + 50%d, r + 25%r, r,

or 50% of the zygotes hybrid and 50% of pure blood, and of
the latter half exclusively maternal and half paternal. But
since the soma developed from the hybrid germ-cell has the
dominant character, we shall have

75% of the cases with the dominant character;
25% " " " " " recessive

and this agrees with various empirical results, of which the
following from Correns is instructive. A cross was obtained
between a variety of pea with a green (</) germ and one having
a yellow (y) germ. Yellow is dominating.

CORRELATED VARIABILITY.

59

Gen. 1.

31 y (hybrid) peas produced 12 plants*
these bore:

Gen. 2.

775 y (hybrid + y) peas ( =75.8%) 247 g (pure-blooded)
21 plants were produced: peas ( =24.2%).

7 (33%) pure- 14 (66%) hybrids,
blooded y, because they 20 plan
because they bore :
bore: |

ts bore :

Gen. 3.

292 y peas' 462 y 149 g 670 green peas,
(hybrid + y) (pure-blooded)
peas (=76.4%) peas (=23.6%)

It is clear that if this process of crossing of the hybrids
continues, the proportion of hybrids to the whole population
will diminish; for the share of pure-blooded forms breeds
true; while the originally equal share of hybrids is repeatedly
halved.

If the hybrid is crossed with one of the parents instead of
with another hybrid, we will get

(1) (d + r)d=d, d+d, r, and

(2) (d + r)r=d, r+r,r.

In (1) all of the progeny will appear of the dominant type.
In (2) one-half will appear of that type. This again agrees
with experiment.

Second Case. The two parents differ in respect to two
characters — case of dihybrids. Imagine a lot of ripe germ-
cells with the antagonistic qualities of any pair separated
according to the second principle stated at the outset. A
indicates the one pair of qualities and B the other; then we
shall have nine classes of zygotes, the proportion of each of
which is as follows:

A.

B. 6.25%d, d; 12.5%d, r; 6.25%r, r.
A. 50%d, r

B. 12.5%d,d; 25%d, r; 12.5%r, r.

A. 25%r, r

B. 6.25%d, d; 12.5%d, r; 6.25%r, r.

60 STATISTICAL METHODS.

Thus the first class has 6.25% purely dominant in both
characters; the second class, 12.5% purely dominant in one
character and hybrid in the other, and so on. Recalling that
hybrid zygotes produce somas with the dominant character,
it follows that the progeny appear as follows:

Ratios
A. dom. + B. rec ........... 18.75% 3

'A. rec. +B. dom .......... 18.75% 3

A. dom. + 5. dom .......... 56.25% 9

A. rec. +B. rec ........... 6.25% 1

This result again agrees with experiment. The resulting
mixture of characters in tri- to poly hybrids may be likewise
predicted, by extending the principles already laid down.

MEASURE OF DISSYMMETRY IN ORGANISMS.

A Dissymmetry-Index, 3, measuring the average de-
gree of asymmetry in the right and left organs of bilateral
organisms, has been proposed by Duncker (1903).

First a series of integral differences —3, —2, —1, 0, 1, 2,
3, 4, etc., between the right- and left-side measurements of
the organ in question is made, and the frequencies of each
integral difference (reckoning to the nearest integer) is found.
The average of the difference series is the difference of the
averages of the right- and left-side measurements, and the
standard deviation of the difference is given by

in which the subscripts, refer to the bilateral series of which
the asymmetry is to be found, and r is the coefficient of cor-
relation between the two sides.

Let d' represent any positive differences in the series, and
d" any negative differences; and let /,/, //, etc., represent
the frequencies of the negative-difference, classes, and //',
//', etc., the frequencies of the positive-difference classes.
Then the asymmetry-index

_

CORRELATED VARIABILITY. 61

Example. Absolute difference between dextral (d) and
sinistral (s) lateral edges (L) of carapace of right-handed
fiddler-crabs — Gelasimus pugilator (Yerkes, 1901; Duncker,
1903)'

d=Ld-Ls: -10 1 23
/: 1 63 310 23 3

310X 1 + 23X2 + 3x3 = 365, ^(/0 = 336.
l, 2<j") = \, n=400.

336X365-1X1_ 122639
400X366 " ~ 146400

62 STATISTICAL METHODS.

CHAPTER V.

SOME RESULTS OF STATISTICAL BIOLOGICAL STUDY.

It is hoped that the following analysis of the literature,
although not complete, will prove suggestive and otherwise
useful. Numerical results are occasionally given. These are
intended to be used in making comparisons with numerical
results obtained in the same field and thus to assist in the
interpretation of such results. The literature references are
to the Bibliography which follows this chapter, in which the
titles are arranged by author and date.

GENERAL.

EXPOSITIONS, ADDRESSES, ETC.: Amann, '96; Ammon, '99;
Camerano, '00b, '01, '02; Davenport, '00, '00d, Olb;
Duncker, '99b; Eigenmann, '96; Galton, '01; Gallardo,
'00, '01, 'Olb; Ludwig, '00, '03; Redeke, '00; Volterre,
'01.

TEXT-BOOKS: Galton, '89; Bateson, '94; Duncker, '00;
Pearson, '00; Vernon, '03.

METHOD: Camerano, '00; Engberg, '03; Fechner, '97;
Galton, '89, '02; Heincke, '97; Johannsen, '03; Pear-
son, '94, '95, '96, '97, '97b, '98, '00°, 'Old, '02C, '02f, '02e,
'02m, '02n, '03e; Pearson and Lee, '00; Sheppard, '98,
'98b, '03; Verschaffelt, '95; Wasteels, '99, '00; Yule,
'97, '97b, '00, '00b, '03.

VARIABILITY.
General.

Frequency polygon, its significance; its dependence on
time, place, and conditions: Burkill, '95; Kellerman, '01;
Tower, '02; Shull, '02; Yule, '02; Johannsen, '03.

Proper value of ratio of first to second prizes: Galton, '02;
Pearson, '02k.

STATISTICAL BIOLOGICAL STUDY. 63

Coefficient of variability; significance: Pearson, '96; Brew-
ster, '97; Duncker, '00b; Davenport, '00e.

Mutations: Bateson, '94; Howe, '98; deVries, '01-' 03;
Weldon, '02°.

Individual vs. specific variation: Brewster, '97, '99; Field,
'98; Mayer, '02; Davenport '03*>.

Variability independent of sexual reproduction: Warren,
;99, '02; Pearson and others, '01 c, pp. 359-362.

Relative variability of the sexes: — in man, Pearson, '97C;
Brewster, '99; Pearl, '03; in crabs, Schuster, '03.

Relative variability of primitive and modern races: — in man,
primitive races less variable: Pearson, '96, p. 281; Pearson
(and others), '01 c, p. 362.
Man.

Stature. — Seriation for adults of different races: Bavari-
ans, Ammon, '99; United States, recruits, Baxter, '75, Pear-
son, '95, p. 385; various, Macdonell, '02; English middle
upper classes, Galton, '89, Pearson, '96, p. 270; Germans,
Pearson, '96, p. 278; French, Pearson, '96, p. 281; Cam-
bridge University students, Pearson, '99.

Lot.

n

A

0

C

Engl. upper middle class 1

683

69.215" ±. 066

2.592" ±.047

do. husbands .

200

69. 135" ±.126

2.628" ±. 089

3.66

Cambridge Univ. students

68. 863" ±.054

2.522" ±.048

cm.

cm.

English fathers

1078

171.95

6.81

3.99

English sons. . .

1078

174.40

6.94

3.98

U. S. recruits

25878

170.94

6.56

3.84

N. S. Wales, criminals.. . .

2862

169.88

6.58

3.80

Frenchmen . . .

284

166.80

6.47

3.88

English criminals

3000

166 . 46

6.45

3.88

French, Lyons

166. 26 ±.53

5.50± .37

Germans

390

156.93

6.68

4.02

in.

in.

Engl . upper middle class ?

652

64.043 ±.061

2.325 ±.043

do. wives. . . .

200

63.869±.110

2. 303 ±.078

Cambridge Un. students ?

63. 883 ±.130

2.361 ±.092

3.69

French, Lyons ? . .

154.02 cm. ±.52

5 5.45 ±.37

Seriation at different ages: British infant at birth, Pearson,
'99; school children, Bowditch, '91; St. Louis schoolgirls,
Porter, '94, Pearson, '95, p. 386; Australian adult whites,
Powys, '01.

64

STATISTICAL METHODS.

Lot. Average. a

New-born infant, British $ . 20 . 503 ± . 028 in. 1 . 332 ± . 020

?. 20.124± .025 " 1.117 ±.018

St. Louis schoolgirls 118.271 cm. 2.776

Australian whites:

Age,

Years

20-25
25-30
30-40
40-50
50-60

Average.

66.95
67.30
67.15
66.91
66.74

60 & over 66.26

62.50
62.76
62.44
62.96
62.22
61.31

2.475

2.562
2.587
2.618
2.633

2.365
2.432
2.303
2.555

2.591
2.300

$

3.70
3.81
3.86
3.91
3.95
4.04

C

6.500
5.849

3.79
3.87
3.69
4.06
4.16
3.75

Weight. — Seriations at different ages, British: Infants,
Pearson, '99; University students, Pearson, '99; 5552 Eng-
lishmen, Sheppard, '98.

Lot. Average. o C

New-born infants, $ 7 .301 ± .0241b. 1.144±.017 15.66%

? 7.073±.021 1.006±.015 14.23

Cambridge Univ. students, $ 152. 783 ± .35 16. 547 ±.25 10.83

?125.605±.77 14.030±.57 11.17

Skull. — Cephalic index: Bavarians, Ranke, '83; 6800 20-
year old Badeners, working class, Ammon, '99, p. 85; various
races, Pearson, '96, p. 230, Macdonell, '02.

Lot. n

Bavarian peasants 100

Baden recruits 6748

Modern Parisians

French peasants 53

Cambridge students 1000

Criminals (British) 100

Brahmans of Bengal 100

Whitechapel English. . , 107

Maquada race

Skull capacity: coefficients of variability. Fawcett and
Lee, '02.

Lot. $

Andamanese 5 . 04

Ainos 6.89

Negroes 7.07

Low-caste Pun jabs . . 7.24

Parisian French 7 . 36

Kanakas 7 . 37

17th Century English. 7 . 68

A

a

C

83.41

3.58

4.29

81.15

3.63

4.48

79.82

3.79

4.74

79.79

3.84

4.81

78.33

2.90

3.70

76.86

3.65

4.75

75.77

3.37

4.44

74.73

3.31

4.43

72.94

2.98

3.95

?

Lot.

$

9

5.59

Naquadas ... .

7 72

6.92

6.82

Germans

. 7.74

8.19

6.90

Egyptian mummies. ,

. 8.13

8.29

8.99

Polynesians

. 8.20

5.55

7.10

Italians

8 34

8.99

6.68

Modern Egyptians. .

. 8.59

7.17

8.15

Etruscans . .

9.58

8.54

STATISTICAL BIOLOGICAL STUDY. 65

Various cranial dimensions, Lee and Pearson, '01.

Other Organs. — Coefficient of variability of bones of skele-
ton of French and Naquada (C. of limb-bones, 4.53-5.57),
Warren, '97; appendicular skeleton, Pearson, '96; finger-
bones, Lewenz and Whiteley, '02; seriation of position of
spinal nerves, Bardeen and Elting, '01; various organs in
diverse races, Brewster, '97, '99.
Mammalia.

Relative variability of specific and generic characters in
various mammals the former being greater, Brewster, '97;
seriation of number of Miillerian glands in Sus scrofa, n, 2000;
A, 3.501 ±.025; a, 1.6SO±.018; C, 48.0, Davenport and Bui-
lard, '96.
Aves.

Seriations of various proportions of N. A. birds, Allen, '71;
characters of Lanius ("shrike") and its races, Strong, '01;

Lot. n A a C

Shrike, length L. wing $ 168 99.06mm. 2.74mm. 281

' ? 112 97.98 2.64 2*69

tail length 4 141 101.57 3.48 343

? 95 99 55 3.63 3.65

bill length, $ 164 12.01 0.71 589

" " ? 112 11.71 0.63 5^35

" depth, a 126 9.27 0.42 457

? 85 8.95 0.41 4.61

melanism of crown, $ 144 83.57% 3.0% 358

? 99 83.66 3.19 3.81

'" upper tail-coverts i 142 53.13 15.42 2902

1 ? 104 47.98 18.99 39^58

Curvature of cuimen 29.94° 2.74° 9. 15

Eggs, proportions: Passer domesticus, Bumpus, '97, Pear-
son, '02e; various species, Latter, '02.

Av.

Length, Length, mm. Breadth, mm.

Species. Bird, n A a C A a C

Cuckoo

in.
14

243

2°

40

1.

059

4.

72

16.54

.650

3 93

Blackbird

10

114

29.

44

1 .

357

4

01

21 73

!787

3 62

Song-thrush ....
Starling .

9

8-8.5

151
27

27
29

,44

78

0
1.

999
097

3,
3

.04
08

20.69
21 76

.516
.423

2. '50
1 94

Yellowhammer. .
Tree-pipit
Meadow-pipet . .
House-sparrow
(English)
House-sparrow
(American) . . .
Hedge-sparrow. .
Robin .

7
6.5
6

6

6
6
6

32
27

74

687

868
26
57

21 .
20
19.

21

21
20
20

55

01

72

.82

.32
.12
.22

0.
0.

1.
1.

1
0

()

682
698
250

195

05

.810
,857

3.

3
6.

5.

4.
4
4

17
49
37

47

92
.02
24

16.04
15.09
14.56

15.51

15.34
14.73
15 43

.405
.449
.561

.525

.415
.477

2'53
2.97
3.84

3.38

2.81
3 09

Linnet . ..

5.5-6

65

17

.14

0,

598

3.

.49

13.33

.358

2^69

66 STATISTICAL METHODS.

Amphibia.

Seriations of variations in position of pelvic girdle in
Necturus, Bumpus, '97.
Pisces.

Geographical races: in Leuciscus, Eigenmann, '95; in
adjacent lakes, Moenkhatis, '96; in schools of herring, Heincke,
'97; in flounders, Bumpus, '98; in mackerel, Williamson, '00.
See under Local Races.

Various species: Pimephales fin-rays and scales of lateral
line, Voris, '99; Zeus faber, an ancestral Pleuronectid, has
its plates symmetrical in only 23.6% of the individuals,
Byrne, '02; dimensions of 141 PetromyzorT, Lonnberg, '93.
Tracheata.

Lepidoptera. — Seriations of wing dimensions of Thyreus
abbotti. Field, '98; number of "eye-spots" on wing of Epi-
nephele, Bachmetjew, '03; number of spots on different
species of the genus Papilio, Mayer, '02; breadth of wing,
985 Strenia clathrata C=4.57, Warren, '02.

Aphidce. — Asexually produced offspring show an average
variability of 60% that of the race, Warren, '02, p. 144;
seriation of fertility, empirical mode =7 young, Warren, '02,
p. 133; reduced variability of the earlier generations, because
they include only such as can produce fertile offspring,War-
ren, '02.

Dimension. Grandmothers. Children.

a C a C

Frontal breadth 2.28mm. 6.07% 2.96mm. 8.26

Length R. antenna 7.36 8.77 10.94 12.97

Lengthantenna
Frontal breadth

Myriapoda, — Lithobius: seriations of length of adults,
C, for 4 's= 10.97; ?'s= 11.25; number of prosternal teeth;
of antennal joints; of coxal pores in which C varies from 9.9
to 15.4, Williams, '03.
Crustacea.

Podophthalmata. — Seriations of 12 dimensions of right-
handed and left-handed lt fiddler-crabs," Gelasimus pugilator,
C varies from 7.0 to 11.1, Yerkes, '01; relative variability of
male and female Eupagurus prideauxi from deep and from
shallow water. Schuster, '03; forehead breadths of Carcinus

STATISTICAL BIOLOGICAL STUDY. 67

moenas, Weldon, '93, Pearson, '94; various dimensions, Cran-
gon, Weldon, '90; length of rostrum, Palaemon serratus,
Thompson, '94, Pearson, '94; number of rostral teeth of
PalaBinonetes, Weldon, '92b, Pearson, '95, Duncker, '00.

Lot. A, mm. n, mm. C, %

Eupagurus, short edge of R. chela:

$ deep water 9. 708 ±.085 2.76 28.5

$ shallow water 10. 272 ± .075 2.59 25.2

? deep water 7. 400 ±.033 1.06 14.3

$ shallow water 7.485±.029 1.02 13.6

Eupagurus, long edge of R. chela:

$ deep water 17. 97 ±.14 4.73 27.8

$ shallow water 18. 68+. 13 4.38 23.5

? deep water 14.14±.06 1.67 11.9

9 shallow water 13. 97 ±.05 1.82 13.0

Eupagurus, carapace length:

$ deep water 8. 59 ±.05 1.67 19.4

$ shallow water 7. 54 ±.03 0.94 12.5

? deep water 7.12±.03 0.86 12.1

Palaemonetes vulgar is, dorsal spines . 8.28 0.81 9.83

ventral spines. 2.98 0.45 15.03

Palaemonetes, varians, dorsal spines . 4.31 0.86 20.00

ventral spines. 1.70 0.48 28.26

Amphipoda. — Seriations of lengths of body, of second
antennse, and of ratio of second antenna? to body-length,
Smallwood, '03.
Annelida.

Chcetopoda. — Teeth on jaws of Nereis virens. Right: A =
10.055 ±.045, o= 1.339 ±.032, (7=13.3%; Left: A = 10.00 ±
.044, <7=1.306±.031, (7=13.1%, Hefferan, '00.
Brachiopocla.

Seriation of widths breadth, width of sinus -f- depth, num-
ber of plications on ventral and dorsal valves in sinus and on
fold, Cummings and Mauck, '02.
Bryozoa.

Number of spines on statoblasts of Pectinatella magnifica.
A = 13.782±.031, o= 1.318 ±.022, C=9.57±.16, Davenport,
'00«.
Moll u sea.

Gastropoda. — Frequency polygons of ventricosity, weight,
and index of Littorina littorea for 3 British and 10 American
localities — greater variability in America. Index: <7£=2.3%,

68 STATISTICAL METHODS.

B=2.6%, Cj = 3.0%, Bumpus, '98, Duncker, '98;
critical, Bigelow and Rathbun, '03; seriations of length,
ratio of diameter to length, ratio of aperture to length,
apical angle, number of whorls, color of aperture lip, and
depth of suture between whorls in Nassa, Dimon, '02; seria-
tions of shell-index and spinosity of lo in different parts of a
river system, Adams, '00; variability of adult Clausilia
laminata less than that of young, 15:13, ascribed to periodic
selection, although average size not altered, Weldon, ''01;
variability of bands of Helix nemoralis in one spot of America,
Howe, '98; in different localities near Strasburg, Hensgen, '02.
Lamellibranchiata. — Seriation of number of ribs of Car-
dium, Baker, '03; Pecten; ray-frequency, Lutz, '00, Daven-
port, '00, '03 a '03b; change in proportions with age, acquisi-
tion of new symmetry about transverse axis; definition of
form units from different localities, Davenport, '03, '03b.

Lot. Number of Rays.

Pecten irradians' A a C

Cold Spring Har.,L. I., R. valve 17. 353 ±.018 0.876 ±.013 5.05 ±.07

Cutchogue, L. I., R. valve .... 16.534 ± .034 0.852 ± .024 5.32 ± .36

Cold Spring Har., L. valve ... 16.790±.022 0.916±.015 5.46±.09

Cutchogue, L. valve ......... 15.954±.105 0.881±.075 5.52±.49

Pecten opercularis:

Eddystone, R valve ........ 17.478±.029 1.000±.020 5.72 ±.12

Irish Sea, R. valve .......... 18.101 ±.029 1.074±.021 5.93±.ll

Firth of Forth, R. valve ..... 17.673±.027 1.117±.019 6.32±.ll

Pecten gibbus-

Tampa, Fla~,R valve ........ 20.512±.030 0.991±.021 4.83±.10

Pecten ventricosus:

San Diego, Cal., R. valve .... 19.495 ±.087 0.885 ±.019 4.55 ±.10

Ecliiiiodermata.

Seriation of ray-frequency in starfish, Crossaster papposus:
A = 12.391, C=0.788, 0=6.36%, Ludwig, '93b.
Coeleiiterata.

Scyphomedusce. — Seriation of number of tentaculocysts of
Aurelia aurita: n=3000, empirical range 4-15; empirical
mode=8, genital sacs, M'=4, range, 2-10, Browne, '95, '01.

Hydromedusce. — Seriation of number of radial canals,
gonads, gastric lobes, and tentacles of Gonionemus, Hargitt,
'01; radial canals and lips of Pseudoclytia pentata, Mayer, '01,
Davenport, '02; radial canals, etc., of Eucope, Agassiz and
Woodworth, '96.

STATISTICAL BIOLOGICAL STUDY. 69

Lot. A a C

[ocyltia, num. radial canals 5.004± .094 0.441 8.81

lips 4.868±.012 0.556 11.4

Protista.

Paramecium recently divided, Simpson, '02; seriation of
diameter of Actinospherium and number of cysts and nuclei
in body, Smith, '03; outer and inner diameters of shell of
502 Arcella vulgaris, Pearl and Dunbar, '03; various diatoms,
Schroter and Vogler, '01.

Lot. A a C

Paramecium, length n 229.05 19. 15 8.36%

breadth 68.13 9.16 13.44

index 29.91 4.03

Arcella, outer diameter 55.79±.17 5.73±.12 10.27±.22

inner diameter 15.91±.07 2.17±.05 13.66±.30

Plants.

GENERAL. — Multimodal polygons especially frequent in
plants, Ludwig, '97; critical, Lee, '02; Pearson, '02h.

RAY-FLOWERS IN COMPOSITE. — Seriation of ray-frequency
of Coreopsis, de Vries, '94; of Senecio nemorensis, S. fuchsii,
Cent urea cyanus, C. jacea, Solidago virga aurea, Achilla mille-
folium, Ludwig, '96; ray-frequency in Chrysanthemum,
Ludwig, '97C, Lucas, '98, Tower, '02, Pearson and Yule, '02;
Helianthus, Wilcox, '02; Bellis perennis, Ludwig, '9Sb; Soli-
dago serotina, Ludwig, '00b; Arnica montana, Ludwig, '01;
Aster, Shull, '02.

Num. Ray-flowers. A o C

Aster shortii 14.000±.068 1.526±.048 10.90

A. novaj-anglise 42.874±.302 6.308±.213 14.71

A.punicens 36.672±.107 4.480±.076 12.22

A. prenanthoides 28.080± .107 4.070± .077 14.52

OTHER SERIATIONS OF FLORAL ORGANS: Ranunculacece. —
Petals, Ranunculus bulbosus, de Vries, '94, Pearson, '95;
calyx, coralla, stamens, and pistils of Ficaria verna, Ludwig,
'01; number of Ficaria pistils, early flowers, A = 17.448, a=
3.89; late flowers, A = 12.147, <r=3.88; number of stamens,
early, A = 26.731, (7=3.761 and late, A = 17,863, <y=3.298,

70 STATISTICAL METHODS.

MacLeod, '99, Weldon, '01; number of petals of Caltha
palustris, de Vries, '94; number of calyx parts and petals
of Trollius europseus and number of fruits per head of Ranun-
culus acris, Ludwig, '9Sb, '00b; number of seeds per capsule-
compartment of Helleborus foetidus, Ludwig, '97.

Crucifercc. — Number of flowers, Cardamine pratensis, em-
pirical modes at 2, 5, 8, 11, 13, 16, 19, 22, not in Fibonacci
series, Vogler, '03.

Papaveracece. — Number of floral organs in Papaver, Mac-
Leod, '00; number of sepals and petals in the lesser Celan-
dine, various species, Pearson and others, '03.

Caryophyllacece. — Number of stamens in Stellaria media,
varies with season and position on plant, Burkill, '95; num-
ber of anthers in 44,542 flowers of Stellaria media — a com-
plex polygon due to effect of age and environment, Reinohl,
'03.

Sapidacece. — Number of compartments in fruit of Acer
pseudoplatanus, de Vries, '94.

Leguminosce. — Number of blossoms in clover plants, Type I:
(7=2.788, de Vries, '94, Pearson, '95, p. 402; number of ele-
vated flowers in blossoms of Trifolium repens perumbellatum,
de Vries, '94; floral organs of Lotus uliginosus, L. cornicu-
latus, Medicago saliva, M. falcata, Ludwig, '97; flowers per
head of Lathyrus, Ludwig, '00b.

Rosacew. — Number of stamens of Prunus spino^ a and Cra-
taBgus, Ludwig, '01; sepals of 1000 Potentilla lormentilla
and petals of 4097 Potentilla anserina, de Vriep, '94.

Cornacece. — Number of flowers in head of Cornus mas and
C. sanguinea, not in Fibonacci series, Vogler, '03.

Capri foliacece. — Number of petals of 1167 Weigelea ama-
bilis, de Vries, '94: number of flowers in inflorescence and
number of petals on flower of Adoxa moschatellina, White-
head, '02.

DipsaccB. — Number of flowers per head in Knautia arven-
sis, maximum at 64, Vogler, '03.

Composite. — Number of male and female flowers in umbel
of Homogyne, Ludwig, '01.

Primulacece. — Number of flowers per umbel, Primula,
multimodal, Ludwig, '97, '98b, '00; rays in Primula farinosa,
Vogler, '01.

STATISTICAL BIOLOGICAL STUDY. 71

Scrophulariacece. — Number of parts in peloria of Lenaria
spuria, Yost, '99; number of stamens, Digitalis, Gallardo, '00.

Orchidacece. — Extremes in variability of number of spots
on flower, Chodat, '01.

LEAVES. — Seriation of numbers of paired leaflets of Pirus
aucuparia, Fraxinus excelsior, Senecio nemorencis, and Pole-
monium, Ludwig, '97, '98b. Length and breadth of leaves
of Fagus silvatica and Carpinus betulus, Ludwig, '99. Leaf-
dimensions, Sanguinaria, Liriodendron, Ampelopsis, and
Ailanthis (n, small), Harshberger, '01. Number of side ribs
on leaves of Fagus silvatica, Carpinus betulus, and Quercus
monticola, Ludwig, '99; on leaves of beech, Pearson, '00;
leaves of mulberry, Fry, '02; dimensions of Typha leaves,
Davenport and Blankinship,-'9S; pine needles, Ludwig, '01;
from various branches of Pinus silvestris, Lee, -'02.

Lot> length of pine needles A mm. a mm. C

Pinus silv. , lower branches . . 22 . 163 db . 048 4 . 474 ± . 034 20 . 19

" middle branches . 26. 524 ±.055 5. 167 ±.039 19.48

" upper branches . 25. 949 ±.062 5. 858 ±.044 22.59

Fruit. — Number of ears in head of Agropyrum repens and
Brachypodium, Ludwig, '01 ; of the grass Lolium, Ludwig,
'00b; fruits per head of Ranunculus acris Ludwig, '00b; num-
ber of seeds per capsule-compartment, Helleborus, Ludwig,
;97; fruit length, Oenothera Lamarckiana, and Helianthus,
de Vries, '94; dimensions of beans in masses and in succes-
sive generations of same family, Johannsen, '03.

BRYOPHYTA. — Seriations of length of capsule-stalk, Bryum
cirratum, Amann, '96; parts in sexual organs of Marchantea
and Lonicera, Ludwig, '00b.

SOME TYPES OF BIOLOGICAL DISTRIBUTIONS.

General. — Pearson, '95 'Old. a modified by selection,
Reinohl, '03.
Type I.

Petals of 222 flowers of Ranunculus bulbosus, de Vries, '94,
Pearson, '95, p. 401.

Number of glands of fore legs of swine, Davenport and
Bullard, '96, Pearson, '96, p. 291: a=. 311 ±.016.

72 STATISTICAL METHODS.

Fertility (percentage of births with one year of marriage)
of wives at different ages, Powys, '01.

Rays in dorsal fin of Pleuronectes 5 , Duncker, '00.
" " anal " " " ?, " "

Type IV.

Stature of St. Louis schoolgirls, Pearson, '95, p. 386.
a=- 0.489.

Number of teeth, Palsemonetes varians Plymouth, Pear-
son, '95, p. 404. a= 0.134.

Stature of Australian whites, Powys, '01.
Rays in dorsal fin of Pleuronectes, ? , Duncker, '00.
" " anal " " i( & " ll

" " pectoral " " " $ " "

Type V.

Number of lips of medusa, P. pentata, Mayer, '01, Pearson,
'Old. a=.549.

Normal.

Stature, U. S. recruits, Baxter, 75, Pearson, 95, p. 385.
Ray frequency, Pectens, Davenport, '00, '03b.

Skewiiess.

GENERAL. — Mathematical Analysis. — Pearson, '95, 'Old, '02f,
'02^, '02ra. Biological Interpretation.— Davenport, ;01b, 'Olc.

Quantitative Results.

Numerous cranial characters, Naquada race, Fawcett, '02.

Nearly always -K

Num. lips of medusa, P. pentata (Mayer, '01; Pearson, '01(i) + .549

Num. Miillerian glands, legs of swine (Pearson arid Filon, '98). . . + .311

Num. dorsal teeth, Palsemonetes varians (Pearson, '95) + . 130

Num. rays, Pecten opercularis, Irish Sea (Davenport, '031')- ..... + .087
Eddystone (Davenport, '03h) . . . . + .080

" hooks on statoblasts, Pectinatella (Davenport, '00e) + .077

Weldon's crab measurements, "No. 4 " (Pearson, '95) + .077

Num. rays lower valve, Pecten irradians, L. I (Davenport, '00C)+ .023

" " " " P. opercularis, F. of Forth + .007

' upper valve, P. irradians (Davenport, '00C) db .000

Height, British criminals (Macdonell, '02) — .023

Baxter's height of U. S. recruits (Pearson, '95) — .038

Porter's height of 2192 St. Louis schoolgirls (Pearson, '95) — .049

Head breadth, British criminals (Macdonell, '02) - .051

STATISTICAL BIOLOGICAL STUDY.

Index of Littorina, Casco Bay (Bumpus. '98) + . 13

Index of Littorina Newport (Bumpus. '98) -f .25

Humber " ' + .048

So Kincardineshire (Bumpus '98) + .068

21-rayed Chrysanthemum (de Vries, '99) — . 13

13- " " " " " + .12

Selected 12- (and 13-) rayed Chrysanthemum (de Vries, '99) .... +1 .9

Rays of Pectenirradians, fossil, Va oldest (Davenport, 'Olb) . . . . — .22

'* youngest — . 16

" " " " recent, NC -.09

•* " " " recent, L I. . + .023

Length of wings of long-winged chinch- bug (Davenport, 'Olb). .. — .43
" " " " short-winged chinch- bug " ...+ .44

Length horns rhinoceros-beetle , long-horned (Davenport, '011}). . — .03
" " " " short-horned " .. + .48

Complex Distributions.

Bimodal Polygons. — Discontinuity in hairiness of Biscu-
tella, Saunders, '97; of Lychnis, Bateson and Saunders, '02,
Weldon, '02C.

Length of cephalic horns of rhinoceros-beetle, and forceps
length of male earwigs, Bateson, '94; explanation of di-
morphism, Giard, '94.

Multimodal Polygons. — Modes fall in Fibonacci series, Lud-
wig, '96, '96b, '96C, '97, '97b, '97C

Modes of Chrysanthemum segetum at 13, 21, de Vries, '95.

Opposed to Fibonacci series, complex polygon due to lack
of homogeneity, Lucas, '93, Shull, '02, Pearson, '02* , Lee, '02,
Reinohl, '03, Vogler, '03.

CORRELATION.

General and Method. — Galton, '88, '89, Pearson,
'96, Yule, '97, '97b; spurious correlation, Pearson,'97; non-
quantitative characters, Pearson, '00C, Pearson and Lee, '00,
Yule, '00, '00b, '02; index not constant in related races,
Weldon, '92, Pearson, '96, '98b p. 175, '02n p. 2, Daven-
port, '03b.
Man.

General — Galton, '88; British criminals, various dimen-
sions, r=.13 to .84, Macdonell, '02.

Skull — Correlated with cranial capacity in living persons,
Lee and Pearson, '01 ; breadth and length, Naquada, Bavari-
ans, French, Pearson, '96, p. 280; N. A. Indians, Boas, '99;

74 STATISTICAL METHODS.

various dimensions, Aino and German, Lee and Pearson, '01;

Naquadas, Fawcett and Lee, '02. With civilization woman's

correlation tends to gain on man's, Lee and Pearson, '01,

Pearson, '02n.

Lot. r

Breadth and Length:

German, * 49± .05

Smith Sound Eskimo 47

Aino, a 43 ± .06

Aino, ? 37 ± .07

German, $ 29±.06

Modern Bavarian peasants 28 ± . 06

Naquada race 27

Sioux Indians. 24

Modern French peasants 13 ± . 09

British Columbian Indians $ 08

Modern French (Parisians) 05± .06

Shuswap Indians 04

Lot. r$ r?
Aino:

Capacity and length 89 ± . 01 . 66 ± . 05

11 breadth 56±.05 .50±.07

" height 54±.05 .52±.07

Length and height 50± .05 .35± .07

Breadth and height 35 ± . 06 . 18 ± . 08

Cap. and ceph. index -.31 ±.07 -.25±.09

German :

Capacity and breadth 67 d= .04 .70± .03

11 length 51±.05 .69±.04

" height 24±.06 .45±.05

Cap. and ceph. index 20 ± . 06 - . 03 ± . 07

Breadth and height 07 ± . 06 . 28 db . 06

Length and height - . 10± .07 .31 ± .06

Skeletal. — Rollet, '89; stature correlated with length of
long bones, reconstruction of stature of extinct races, Pear-
son, '98b; various coefficients of correlation, Pearson, '99, '00,
p. 402; in hand-bones, Whiteley and Pearson, '99, Lewenz
and Whiteley, '02.

STATISTICAL BIOLOGICAL STUDY. 75

Lot. r

Right and left femur 95

Metacarpals, ii and iii digits right 94

First joints, iv digit, R. and L. hands 93

First joints, ii and iii, right 90

Metacarpals, ii and v digits, right 89

Femur and humerus 84 to . 87

Femur and tibia 81 to . 89

First joints, ii and v, right 82

Stature and femur 80( ? ) to . 81( $ )

Stature and humerus 77(?) to .81(4)

Stature and tibia 78 ( a ) to . 80( ? )

Humerus and ulna. . . . 75 to .86

Humerus and radius 74 to .84

Radius and stature 67 ( « ) to 70( a )

Clavicle and humerust 44 to . 63

Forearm and stature 37

Clavicle and scapula 12 to . 16

Stature and cephalic index — .08

Various: Pearson, '99; intelligence not correlated with
size or shape of head, Pearson, '02.

Weight and length of new-born infant $ 644 ± .012

!t " " " ? 622±.013

Weight and stature of Cambridge (Engl.) students, $ . . . .486± .016

" " ? 721 ±.026

Breadth of head (reduced to 12th yr.) and intelligence,

youth 084± .024

Length of head (reduced to 12th yr.) and intelligence,

youth 044 ± . 024

Cephalic index and intelligence, youth 005 ± .024

Breadth of head and ability, adults 045 ± .032

Cephalic index and ability, University men 031 ± .035

" length of head, University men - .086± .033

Vaccination and Recovery. — Pearson, '00C; Macdonell, '02,
'03. r= .23 to .91.

Assortative Mating. — Pearson, '96, '99b, '00, Pearson and
Lee, '00.

Stature of husbands and wives r— .093± .047

ditto, another determination r= .28 ± .02

Eye-color, husbands and wives r= . 100 ± .038

Age at death of consorts r= . 22

76 STATISTICAL METHODS.

Lower Animals.

ANTIMERTCALLY SYMMETRICAL ORGANS:

Paired organs. — Number of Miillerian glands on R. and L.
fore legs of swine, Davenport and Bullard, '96; R. and L.
fins of fishes, Duncker, '97, '00; number of coxal pores on R.
and L. legs of the centipede Lithobius, Williams, '03; R. and
L. dimensions of Gelasimus, Yerkes, '01, Duncker, '03; num-
ber of teeth on R. and L. jaws of Nereis, Hefferan, '00;
breadth of R. and L. valves of Pecten, Davenport, '03b;
skeletal spicules on R. and L. half of Echinus larva.

Subject and Relative. r

Length R, and L. sides of carapace, Gelasimus 947 ± .003

" " " M meropodite, first walking leg 918 ± .005

Breadth R. and L. valve of Pecten opercularis, Irish Sea. . . .858± .006

Num. of teeth R. and L jaws of Nereis 820 ± .008

" " fin-rays R. and L pectoral, Acerina 710

" coxal pores R. and L. 14th pair legs, Lithobius 69 ± . 02

" " " 13th pair legs, Lithobius 686±.029

" " " 12th pair legs, Lithobius 58 ±.04

" " " " " " anal pair legs, Lithobius 575±.039

Other antimeric organs:

r
Num of dorsal and ventral spines, Palsemonetes vulgaris

(Duncker, 'OQb) 380± .019

Num. of lips and canals of the medusa, Pseudoclytia

(Mayer, '01 ; Davenport, '02) 325 ± .019

SECONDARILY ANTIMERIC ORGANS. — (Median organs in
animals that lie on one side.)

r
Num, of dorsal.and anal fin-rays in flounder, $ 651

" " " " " " " " ? 690

Length antero-posterior and dorso-ventral diameters, Pecten . 970 ± .001

Unsymmetrical paired organs. — Pleuronectes, Duncker,
'00; Gelasimus, the fiddler-crab, Yerkes, '01, Duncker, '03.

Length of meropodite, R. and L. chelae of Galasimus 754 ± .014

" " carpopodite, R. and L. chelae of Gelasimus 698 ± .017

" " propodite, R. and L. chelse of Gelasimus 473 i .026

Num. rays R. and L. pectoral fin, flounder, Pleuronectes, $ . .594
" " " " " " " " ?. .582

" of dorsal fin-rays at which lateral line ends, R. and L.

Pleuronectes, $ 467

Num. rays R, and L. ventral fin, Pleuronectes, $ 243

STATISTICAL BIOLOGICAL STUDY. 77

METAMERICALLY REPEATED ORGANS. — Fin-rays of fishes,
Duncker, '97; coxal pores centipede, Williams, '03; seg-
ments of shrimp Crangon, Weldon, '92.

Num. dorsal spines and soft fin-rays, Acerina — .379

" " " Cot t us 11C

" coxal pores R. anal and 14th segment, Lithobius 440

R. 13th and 14th segments, Lithobius 722

R. 13th and 12th segments, Lithobius 464

Length carapace and post-spinous portion rostrum, Crangon 81

" tergum VI abd. seg., Crangon 09

" tergum VI and telson, Crangon — .11

MIXED AND CROSS CORRELATION. — Length of wing and tail of
Lanius " shrike," Strong, '01; in fishes, Duncker, '97, '99; pro-
portions of aphids, "plant-lice," Warren, '02; coxal pores
of centipede, Williams, '03; length of carapace and of chela?
in Eupagurus, "hermit-crab," Schuster, '02; diameter of
cell and body length, Daphnia, Warren, ;03; cross correla-
tion in teeth on jaws of Nereis, Hefferan, '00; various char-
acters of the mud-snail, Nassa, Dimon, '02; circumference to
number of spines, statoblast of Bryozoa, Davenport, '00e;
diameter of body of the Heliozoan Actinosphaerium Echorni
and the number of cysts and of nuclei, Smith, '03; inner and
outer diameters and color of the shell of Arcella, Pearl and
Dunbar, '03.

Organs . r

Carapace length and chela length, Eupagurus, $ 9389 ± 0036

" " ? 8626±.()080

Diameter of body of Actinospherian and num. of nuclei .854 ± . 017

Inner and outer diameter shell of Arcella 836 ± .007

Diam. of body of Actinosphasrium and num. of cysts. . .769 ± .026

Wing length and tail length, Lanius 569

Diam. of cell and body length, Daphnia, hatching to

3d molt 551

Diam.. of cell ai\d body length, Daphnia, 3:1 to 4th

molt 393

Diam. of cell and body length, Daphnia, after 4th molt.. . 248

Num. coxal pores, R. anal and L. 12th seg. , Lithobius. . . . 427 ± . 046

Frontal breadth and antennal length (Warren, '02) 320- ± .032

Ccxal pores, R. 14th leg and body length, Lithobius.. . .308 ± .059
Num. rays dorsal fin and end-point of L. lateral line,

Pleuronectes, $ '. -208

Outer diameter and color Arcella 012

Num. dorsal spines and L. pectoral rays, Pleuronectes. .004

78 STATISTICAL METHODS.

Organs. r

Body length and number antennal joints — .013± .067

Circumference of statoblast and number spines,

Pectinatella . — .092 ± .006

Num. R. definite teeth and L. indefinite, Nereis — .524 ± .023

Carapp-ce length and chela index, Eupagurus — .522± .022

Num. of cysts and their diam., Actinosphserium — .669± .040

Plants.

Between various parts of flowers, Ludwig, '01.

Floral parts. — Stamens and pistils of Ficaria, MacLeod,
'98, '99, Ludwig, '01, Weldon, '01, Lee, '02; rays and bracts
and rays and disc florets of Aster, Shull, '02; various organs
on Lesser Celandine, Pearson and others, '03.

Organs. r

Num. rays and bracts. Aster 856 to .799

stamens and pistils Ficaria ranunculoides, early. .. .507± .031

late 749 ± .015

rays and disc florets, Aster 574 to .353

" petals and sepals Ficaria verna + . 34 to — . 18

" stamens and pistils, Celandine 43 to .75

" petals. Celandine 38 to .22

pistils and petals, Celandine 35 to .19

sepals, Celandine 25 to .03

" stamens and sepals, Celandine 06 to .02

Other parts. — Size of leaves of same rosette of Bellis peren-
nis, Verschaffelt, '99; various pairs of dimensions of fruits
and leaves, Harshberger, '01; parts of Syndesmon, Keller-
man, '01.

HEREDITY.
General.

Treatises.— Galton, '89, Pearson, '00.

Classification. — Galton, '89, pp. 7, 12, Pearson and Lee, '00,
pp. 89, 91, 98.

Law of ancestral heredity. — Galton, '97, Pearson, '98; esti-
mate of heredity from a single ancestral generation, Pearson,
'96, p. 306.

Inequality in parental transmission. — Father prepotent in
sons; mother in daughters, Pearson and Lee, '00, p. 115;
heredity weakened by change of sex, Pearson and Lee, '00,
p. 115, Lutz, '03.

STATISTICAL BIOLOGICAL STUDY.

79

Inheritance of Eye-color, Homo,
s.son; d, daughter; /.father; w, mother.

( Average of rg/ and rdn
Parental •< ,, „ rJ an(jr

Grand-
pare:

( " Vandr<*

intal 1 "t rsfm' Tdffi

' " rsm/» rd/m

m/^ rsmm •
Great-grand-parental inheritance, average . .

No . of Changes of Sex.

0
.530

.370

.347

.459
.300
.222

.296
.145

.038

Parental.

Exceptional fathers produce exceptional sons at a rate
three to six times that of non-exceptional fathers and ex-
ceptional pairs at ten times the rate of non-exceptional pairs,
Pearson, '00C, pp. 38, 47.

x y Cor. Reg.

Longevity: r PXy

Father and son (Beeton and Pearson, '99) .12

*' adult son (Beeton and Pearson, '01) .135 .10

" adult dau. " " ' 130 .08

Mother and adult son " " " " . 131 . 12

•• " dau. " " " " 149 .12

Eye-color (Pearson and Lee, '00) 55 to .44

Stature, English middle class:

Father and son (Pearson, '96, p. 270) 396 .352

•• dau. " " " -360 .419

Mother anjd son " " " 302 .269

" dau. " " " 284 .275

Head index, N. Amer. Indian:

Mother and son (Pearson, '00, p. 458) .370

" dau. " 300

Coat-color, thoroughbred horses:

Sire, foal (Pearson. '00, p. 458)

Dam, foal " 527

Fertility :

Mother and daughter, British upper class

Father and son, -051 ± -009

Mother and daughter, British peerage .210

Father and son,

Mother and daughter, landed gentry

Father and son -116

r P

Frontal breadth, Hyalopterus (Warren, '02)

Length R. antenna, Hyalopterus '

Ratio: R. antenna + frontal breadth (Warren, '02) . . .
Ratio- Length protopodite-J- length body, Daphnia

(Warren, '02) 466 -619

80

STATISTICAL METHODS.

Graiidpareiital.

Coat color, thoroughbred race-horses „

" Basset hounds

Frontal breadth, Hyalopt3rus, Aohidse (Warren, '02)

Length, R. antenna, Aphidae (Warren, '02)

Ratio R. antenna -4- frontai breadth, Aphidse (Warren, '02
Ratio Length protopodite -r- length body, Daphnia (War-
ren. '02)

Gr'dson and gr'df.. homo male line (Pearson, '96)

" " " female l>ne (Pearson, '96).

Ortgr'dson and grtgr'df. homo $ line

.339
.113
.321
.177
.231

.269
.192
.295

[.27 .5]

.199
.089
.105
.031

Eye-color, homo, f., grandfather, and son (Blanchard, '03) .421

.324
.380
.360
.372
.359
.297
.311
.272
.309
.221
.204
.262
.261
.318
.239

Coat
Eye
Coat
Eye
Coat
Eye
Coat
Eye
Coat
Eye
Coat
Eye
Coat
Eye
Coat

horse,

homo, "

horse, "

homo, m.,

horse, "

homo, "

horse, "

homo, f., grandmother, and son

horse, "

homo, " " dau.

horse, "

homo, m.,

horse, "

homo, "

horse, "

dau.

son
dau.

son

dau.

Fraternal. r

Daphnia, length of spine (Warren, '99; Pearson, 'Olc) 693

Aphis, antennal length (Warren, '02) 679

" frontal breadth (Warren, '02) 666

Paramecium, index of just separated fission pairs (Simpson, '02). .664
Horse, coat-color (Pearson, Lee, and Moore), average of 3 sets. . .633

Man, forearm, English (Pearson, 'Olc) 542

Hound, coat-color, Bassett (Pearson and Lee, '00) 526

Man, eye-color, English (Pearson, 'Olc). Average of 2 sets 475

Pectinatella, statoblast hooks (Pearson, '01 c) 430

Man, stature Average of 3 sets. . .403

cephalic index, N. A. Ind. Average of 3 sets. . .403

" longevity, Quakers (Beeton and Pearson, '01) 332

*' temper, British (Pearson, 'Olc) 317

longevity, British peerage (Pearson, '01) 260

•* " Quakers '* " " 197

Average of 23 sets 476

Mean of 42 fraternal correlations (Pearson, '02k) 495

Some mental characteristics, inherited exactly like physical
characters (Pearson, 'Ole):

Conscientiousness . . 593 Popularity 504

Self-consciousness. . 592 Vivacity 470

Shyness 528 Intelligence 456

Average of 6 507

STATISTICAL BIOLOGICAL STUDY. 81

Theoretical coefficient of correlation be-
tween relatives. — Pearson, '00, Pearson and Lee, '00.

Blended Alternative

Inherit- Inherit-

ance, ance.

Offspring and Parent 3000 . 5000

" grandparent 1500 .250

" " great-grandparent 0750 .123

" gt.-gt. -grandparent. . . .0375
" nth order grandparent .6X(i)n

Brothers 4000 .4 to 1 .0

Half-brothers 2000 .2 to 0.5

Uncle and nephew 1500 .250

First cousins 0750

First cousins once removed 0344

Second cousins . 0172

Third cousins 0041

Homotyposis.

Correlation in non-sexual reproduction, as in production of
homologous uridifferentiated physiologically independent
parts, Pearson, 'Olc; criticism, Bateson, '01; reply, Pearson,
'021; rejoinder, Bateson, '03; correlation between differen-
tiated homologous organs, Pearson, '02e.

Lot.

Ceteract , Somersetshire
Hartstongue, Somersetshire
Shirley poppy Chelsea

(y •

Character. \

. Lobes on fronds ...
. Sori on fronds
Stigmatic bands

Var. to
rar. of
Race.
. 78
. 78
. 79
79

Corre-
lation.

.631
.630
.615
.611
.599
.591
.570
.562
.549
.533
.524
.466
.416
.405
.400
.396
.374
.355
.273
.190
.183
.173

.457

English onion. Hampden

.Veins in tunics . .

Holly Dorsetshire . '
Soanish chestnut, mixed
Beech, Buckinghamshire
Papaver rhooas, Hampden

. Prickles on leaves . .
. Veins in leaves
.Veins in leaves ....
Stigmatic bands

. 80
. 81
. 82
. 83
84
. 85
. 85
. 89
, 91
, 91
92
92
93
93
96
98
98
98

87.4

Mushroom Hampden

Grill indices

Papaver rh<ras, Quantocks
Shirley poppy. Hampden
Spanish chestnut, Buckinghamshire
Broom, Yorkshire
Ash. Monmouthshire
Papaver rhooas. Lower Chilterns. . ,.
Ash, Dorsetshire

.Stigmatic bands. . . .
.Stigmatic bands. . . .
. Veins in leaves , ....
.Seeds in pods ,
. Leaflets on leaves, - . ,
.Stigmatic bands
Leaflets on leaves. - . .

Ash Buckinghamshire
Holly. Somersetshire
\Vild ivy, mixed localities

. Leaflets on leaves
. Prickles on leaves. . . .
Leaf indices

Nigella hispanica Slough

.Seg of seed -capsules.
.Seg. of seed-vessels....
. Members of whorls .

Malva rotundi folia, Hampden
Woodruff, Buckinghamshire

Mean of 22 cases . . ,

.

Bands of capsules of Shirley poppies, msan of 8 crops (Pear-
son, and others, '02) 498

Mean of 39 cases of homotyposis (Pearson, '02' ) 499

82 STATISTICAL METHODS.

Mendelism.

General Statement.— Mendel, '66, de Vries, '00, '00b, '00C,
'03, Correns, '00, Davenport, '01, Bateson, '02, Castle, '03;
critical, Weldon, '02, '03, Pearson, W.

Plants.— Correns, '00, '00b, '01, '02-'02C, '03-'03C, de Vries,
'02, '01-'03, Bateson and Saunders, '02

Animals. — Echinoids, Doncaster, '03; poultry, Bateson
and Saunders, '02; mice, Darbishire, '02, '03, '03b, Castle, '03b,
Bateson, '03b; rabbits, Woods, '03.
Telegoiiy.

No evidence of, in human statures, Pearson and Lee, '96.
Fertility.

Inherited in man and race-horses, Pearson, Lee, and Bram-
ley-Moore, '99; greater fertility in poppy of seeds from cap-
sules with a high number of stigmatic bands, Pearson, '02;
fertility of medusae with symmetrical bands exceeds that of
the unsymmetrical as 3 to 4, Mayer, '01.
SELECTION.

General. — Intensity of selection connotes a lessening tf
correlation, Pearson, '02d, p. 23; mediocre individuals not
the fittest to survive, Pearson, '02n, p. 50.

Man. — 50% to 80% of human death-rate selective, Beeton
and Pearson, '01.

Other Animals. — Annihilation of the extremes in the. spar-
row, Bumpus, '99; percentage death-rate of families of
Aphids has inverse correlation with length of antenna of
mother (r= -.201 ±.084), with frontal breadth of mother
(r= —.184 ±.084), and with number in newly born brood
(r= -.188 ±.084); in Carcinus moenas, Weldon, '95, '99;
in Clausilia, Weldon, '01.

Plants. — Transformation of skew frequency curve to a sym-
metrical one by selection, de Vries, '94, '98; shifting of the
mode by selection, de Vries, '99.

Sexual. — Pearson, '96: A a

Stature of husbands, inches 69.136±.126 2.628±.089

" males in general 69 . 215 ± . 066 2 . 592 ± . 047

" wives 63.869± .110 2.303±.078

" adult females in general . . 64. 043 ±.061 2. 325 ±.043

See also Correlation: Assortative mating (p. 75).

DISSYMMETRY.

The following values for A have been determined by
Duncker, '00 and '03:

STATISTICAL BIOLOGICAL STUDY. 83

Pleuronectes flesus L., 1060 R.-eyed and 60 L.-eyed: Right- Left-
eyed, eyed.

Num. of pectoral divided rays 997 — .983

Total num. pectoral rays 604 — .583

Num. of ventral divided rays 326 — .374

Total num. of ventral fin-rays 019 — .083

Gelasimus pugilator Latr. (fiddler-cnJt)). Right- Left-
handed, handed.

Lateral edge of carapace 838 .793

Length of meropodite, first ambulacral appendage. .813 .872
Length of meropodite, of carpopodite, and of pro-

podite of chelse, all 1 .00 1 .00

Num. of rays on R. and L. pectoral fins, Acer ina —0.111

" " glands 011 wrists of swine .0053

DIRECT EFFECT OF ENVIRONMENT.

Animals. — Aphids reared in successive generations in in-
creasingly unfavorable conditions have reduced dimensions,
Warren, '02:

Grandmother. Grandchildren.

Frontal breadth, Aphid. . A = 37 . 56 33 . 93

Length of R. antenna. ... A = 83 . 91 76 . 59

Ratio ^-5-*. . A = 22.46 22.57

K. A.

Depauperization of mud-snail, Nassa, in diluted sea-water,
Dimon, ;02.

Plants.— Conditions of life affect number of floral parts in
poppy, de Vrics, '99, MacLeod, '00, Pearson and others, '03;
number of ray-flowers of Primula farinosa increases with
moisture, Vogler, '01 ; empirical mode in number of anthers
in Stellaria in poor environment is 3; in good environment
5, Reinohl, '03; leaf -blade smaller in light than in shade,
MacLeod, '98.

LOCAL RACES.

General. — Davenport and Blankenship, '98, Davenport, '99.
Pisces. — Leuciscus from different altitudes, Eigenmann,
'95; herring from different sea-areas distinguishable, Heincke,
'97, 98; mackerel from three Scotch localities differ, Wil-
liamson, '00; fin-rays of Pleuronectes from New England
shore, Bumpus, '98:

Wood H oil. Waquoit. Bristol, R.I.
Dorsal fin-i ays. . . A = 66 . 1 65 . 2 64 . 9
Anal " . A = 49.7 48.6 48.7

84 STATISTICAL METHODS.

Number of fin-rays of Pleuronectes flesus from Western
Baltic, Af'=39, southern North Sea 41^, Plymouth 44,
Duncker, '99.

Fish in similar and adjacent lakes belonging to different
drainage-basins have marked difference in scales on nape,
number of fin-rays and of dorsal spines, Moenkhaus, '96.

Invertebrata. — Mean and variability of deep- and shallow-
water Eupagurus differ, Schuster, '03; proportions, variability,
and correlation coefficients of Pecten opercularis differ at
Eddystone, Irish Sea, and Firth of Forth, Davenport, '03b.
Plants. — Lesser celandine, Pearson and others, '03.

USEFUL TABLES.

Probability Integral. — Area and ordinate of normal curve
in terms of abscissa, Sheppard, '93, '03; abscissa of normal
curve in terms of ordinate, Sheppard, '93; abscissa and ordi-
nate in terms of difference of area, Sheppard, '03; abscissa
of normal curve in terms of class index, Sheppard, '98.

Probability of fitted curve being the true one:

__ . . . 24fi ^,_

Elderton, '02.

Values of log ] /I/ — e~~^2 c for various values of ^2.

Elderton, '02.

Table of log —. - — - - -- . Elderton, '02.

— —

Table of 4/ — I e~*x2d%, for different values of /, Elder-

r * J\

ton, '02.

Table of Iog10 (!+#)— x Iog10 e for various values of a*, for
use with curves of Type III.

Tables for calculating probable error, Sheppard, ;98.

Table of values of 1— r2 and \/l—r2 for all values of r
from 0 to 1 proceeding by hundredths, Yule, '97.

Probable errors of r for all values of n, Yule, '97.

BIBLIOGKAPHY. 85

BIBLIOGRAPHY.

Note. — An effort has been made to include all recent
works containing usable quantitative data in botany and
zoology; but the literature on the mathematical treatment
of statistics and that affording data in anthropology are
by no means completely listed.

ABBREVIATIONS.

The following names of journals often referred to have

been much abbreviated:

Amer. Nat. = American Naturalist.

Ber. d. deutsch. bot. Ges. = Berichte der deutschen botanischen
Gesellschaft.

Biom.= Biometrika.

Bot. Centralbl. = Botanisches Centralblatt.

Phil. Trans. = Philosophical Transactions of the Royal
Society of London.

Proc. Roy. Soc. = Proceedings of the Royal Society of Lon-
don.

The references are scattered through fifty-seven periodi-
cals.

ADAMS, C. C. '00. Variation in lo. Proc. Amer. Assoc. for

the Adv. of ScL, XLIX, 18 pp., 27 plates.
AGASSIZ, A., and W. McM. WOODWORTH, '96. Some varia-
tions in the Genus Eucope. Bull. Mus. Comp. Zool.,

XXX, 123-150. Plates I-IX. Nov.
ALLEN, J. A., '71. On the Mammals and Winter Birds of

East Florida, etc. Bull. Mus. Comp. Zool., II, 161-450.

Plates IV-VIII.
AMANN, J., '96. Application du calcul des probability a

1' etude de la variation d'un type vegetal. Bull, de

1'Herb. Bossier. IV, 578-590.
AMMON, OTTO, '99. Zur Anthropologie der Badener. Jena:

G. Fischer, 707 pp, 15 Tab.
BACHMETJEW, P., '03. Ueber die Anzahl der Augen auf der

Unterseite der Hinterfliigel von Epinephele jurtina L.

Allgemeine Zeitschr. f. Entomologie, VIII, 253-256.
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XXXVII, 481-488, July.
BALLOWITZ, E., '99. Ueber Hypomerie und Hypermerie

bei Aurelia aurita. Arch. f. Entw. Mech. d. Organis-

men, VIII, 239-252.

86 STATISTICAL METHODS.

BARDEEN, C. R., and A. W. ELTING, '01. A Statistical Study
of the Variations in the Formation and Position of the
Lumbo-sacral Plexus in Man. Anatom. Anz., XIX, 124-
135, 209-238, Mar., Apr.

BATESON, W., '89. On some Variations of Cardium edule
apparently Correlated to the Conditions of Life. Phil.
Trans., 1889 B., 297-330, PI. 26.

BATESON, W., '94. Materials for the Study of Variation.
London and New York, xvi-f-598 pp.

BATESON, W., '01. Heredity, Differentiation, and Other
Conceptions of Biology: a consideration of Professor
Karl Pearson's paper "On the Principle of Homoty-
posis." Proc. Roy. Soc., LXIX, 193-205.

BATESON, W., and E. R. SAUNDERS, '02. Reports to the
Evolution Committee — Report I. Royal Society. Lon-
don: Harrison & Sons. 160 pp.

BATESON, W., '02. Mendel's Principles of Heredity. Cam-
bridge [Engl.]: Univ. Press. 212 pp.

BATESON, W., '03. Variation and Differentiation in Parts
and Brethren. Cambridge [Engl.]: J. and C. F. Clay.

BATESON, W., '03b. The Present State of our Knowledge of
Colour-heredity in Mice and Rats. Proc. Zool. Soc.,
London, 1903, II, 71-99, Oct. 1.

BAXTER, J. H., '75. Statistics, Medical and Anthropological,
of the Provost-Marshal-General's Bureau. 2 vols. Wash-
ington: Gov't Printing Office. 568 + 767 pp.

BEETON, MARY, and K. PEARSON, '99. Data for the Problem,
etc. II. A First Study of the Inheritance of Longevity,
and the Selective Death-rate in Man. Proc. Roy. Soc.,
LXV, 290-305.

BEETON, MARY, and KARL PEARSON, '01. On the Inheritance
of the Duration of Life, and on the Intensity of Natural
Selection in Man. Biom., I, 50-89, Oct.

BEETON, M., G. U. YULE, and K. PEARSON, '00. Data for the
Problem of Evolution in Man. V. On the Correlation
between Duration of Life and the Number of Offspring.
Proc. Roy. Soc., LXVII, 159-179, Oct. 31.

BIGELOW, R. P., and ELEANOR P. RATHBUN, '03. On the
Shell of Littorina littorea as Material for the Study of
Variation. Amer. Nat., XXXVII, 171-183, Mar.

BIBLIOGRAPHY. 87

BLANCHARD, N., '02. On the Inheritance of Coat-colour of
Thoroughbred Horses (Grandsire and Grandchildren).
Biom., I, 361-364, April.

BLANCHARD, NORMAN, '03. On Inheritance (Grandparent
and Offspring) in Thoroughbred Horses. Biom., II,
229-234, Feb.

BOAZ, FRANZ, '99. The Cephalic Index. American An-
thropologist, N. S., I, 448-461, July.

BOWDITCH, H. P., '01. The Growth of Children, Studied by
Galton's Method of Percentile Grades. Twenty-second
Annual Report of the State Board of Health of Massachu-
setts (for 1890).

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

BREWSTER, E. T., '99. Variation and Sexual Selection in
Man. Proc. Boston Soc. Nat. Hist., XXIX, 45-61,
July, '99.

BROWNE, E. T., '95. On the Variation of the Tentaculocysts
of Aurelia aurita. Quart. Jour. Micros. Soc., XXXVII,
245-251.

BROWNE, E. T., '01. Variation in Aurelia aurita. Biom., I,
90-108, Oct.

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

BUMPUS, H. C., '97b. A Contribution to the Study of Varia-
tion. Jour, of Morphol., XII, 455-480. Pis. A-C.

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

BUMPUS, H. C., '98b. On the Identification of Fish Arti-
ficially Hatched. Amer. Nat., XXXII, 407-412.

BUMPUS, H. C., '99. The Elimination of the Unfit, as Illus-
trated by the Introduced Sparrow, Passer domesticus.
Biol. Lects. Woods Holl for '97 and '98, pp. 209-226.

BURKTLL, I. H., '95. On some Variations in the Number of
Stamens and Carpels. Proc. Linn. Soc., Botany, XXXI,
216-245.

BYRNE, L. W., '02. On the Number and Arrangement of
the Bony Plates of the Young John Dory. Biom., II,
113-120. Nov.

88 STATISTICAL METHODS.

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100 STATISTICAL METHODS.

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EXPLANATION OF TABLES. 105

EXPLANATION OF TABLES.

I. Formulas. In this table the principal formulas used
in the calculation of curves are brought together for con-
venient reference. The meanings of the letters are explained
in the text. This table is preceded by an index to the prin-
cipal letters used in the formulae of this book.

II. Certain constants and their logarithms.

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

III. Table of orclinates of normal curve. Th's
table is for comparison of a normal frequency polygon con-
sisting of weighted ordiriates with the theoretical curve.

Example: 4 = 17.673; a= 1.117; y0=181.4.
(See page 26.)

Entries in Table
V — M corresponding to

V

14
15
16

V —M

-3.673
-2.673
-1.673

a

3.29
2.39
1.50

V —M

2/0

X181.4 =
X181.4 =
X181.4 =

y

0.8
10.4
58.9

;

1
8
63

a

.00449
.05750
.32465

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: A = 17.673; <r = 1.1169. (See page 26.)

Deviation

(£_£a)Xll

Class.

X

Per

Class

from

Xi

r»oo

a

cent.

Limits.

A=XI

a

less 2, x _._

14

-3.29

.2

.225

14.5

-3.173

-2.841

15

-2.39

1.6

2.364

15.5

-2.173

-1.945

16

-1.50

12.4

12.097

16.5

-1.173

-1.050

17

- .60

30.3

29.155

17.5

-0.173

-0.155

18

.29

32.3

33.194

18.5

0.827

0.740

19

1.19

18.9

17.873

19.5

1.827

1.636

20

2.08

3.9

4.524

20.5

2.827

2.531

21

2.98

0.4

.568

100.0

100.000

106 STATISTICAL METHOPS.

In the example, the data of which are given on p. 26, the

frequency between the limits is given in % column. The — of

a

each limit (as an inner class limit) is found and the entries
in Table IV corresponding to the limits are taken. Each
such entry is subtracted from 0,50000, 'is multiplied by
100, and from the product is subtracted the total theoretical
percentage of variates lying between the outer limit of the
class and the corresponding extremity of the half curve.
This gives the theoretical frequency of the class in question.
The closeness of agreement of the last column with the
" Per cent." column indicates the closeness of the observed
frequency to the theoretical.

V. Table of log T functions of p. This table
will enable one to solve the equations for yQ given on page 32.
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 JT(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

log T(l. 273) = 9. 955185
log 1.273 =0.104828

log 1X2.273) = 0.060013
log 2.273 =0.356599

log T(3.273) = 0.416612
log 3.273 =0.514946

log T(4.273) = 0.931558

or, more briefly, log T(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. 107

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 hundredths of an
inch by shifting the decimal point two places to the left.
Other fractious than decimals are given in the lower tables.

VII. Table of minutes and seconds of arc in
decimals of a degree. This table will be found of use
in the fitting of curves of Type IV (p. 33).

VIII. First to sixth powers of integers from 1
to 3O. This table is useful in calculating moments.

IX. Table of the probable errors of the coeffi-
cient of correlation for various numbers of ob-
servations or variates (ri) and for various values
of r. The probable error of the coefficient of correlation

, . 0.6745(1 -r2) f

being — — := -, a table for the varying values of n and r

Vn

is easily constructed, and for large values of n is accurate
with interpolation by inspection to two significant figures,
which are all that are required.

X. Squares, cubes, square roots, and recip-
rocals of numbers from 1 to 1O54. The use of
this table can be extended by using the principle that if any
number be multiplied by n, its square is multiplied by n2, its

cube by n3, and its reciprocal by — .

XI. Logarithms of numbers to six places.

The following explanation of the use of the logarithmic tables
-is taken from Searles'* Field Engineering, pp. 257-263 [ed.
1887].

The logarithm of a number consists of two parts,
a whole number, called the characteristic, and a decimal,
called the mantissa. All numbers which consist of the
same figures standing in the same order have the same man-
tiss"a, 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-
&cteristic depends entirely on the position of the decimal point

108 STATISTICAL METHODS.

iii 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 1.141138

1.384 0.141136

.1384 —1.141136

.01384 —2.141136

.001384 —3.141136

etc. etc.

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 close
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.

TABLE XI, 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

EXPLANATION OF TABLES. 109

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 T*o 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.

Example.— Find 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 L369726

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. 54J)
Tabular diff. =236 , • . 5th fig. =2 47. 2

6.80

6th fig. = 3 7.08

Ans. No. = .0183623 or 183,623,000.

110 STATISTICAL METHODS.

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

At the end of Table XI 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 found in the larger
table.

TABLE XII. — 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 the 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 left side of the page are to be read
the minutes in the left-hand column; with the degrees on
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.

Exampk.— 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, which 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.

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

Diff. 49.00 -*- 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 -J- 11.67 = 20M

Ans. 79° 46 20M

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 1' to 2° ; the minutes are
given in the second, the log. sin. in the tJiird, 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.

STATISTICAL METHODS.

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 - 1) 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.

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

52' 35" = (3120" 4- 35") = 3155" log 5.498999

add (q - I) 4.685609

Ans. log tan 8. 184608

To find the log cotangent of an angle less than
2° given to seconds.— Take from the column headed ( q-{- Z)
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 + I 15.314292
6240" + 22". 5 = 6262.5 log 3.796748

Ans. 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±l) taken from the
table.

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

fe - J) 4.685537
4740" - 12".5 = 4727.5 log 3.674631

Ans. 8.360168

EXPLANATION OF TABLES. 113

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

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

ATM. 12.047241

To find the arc corresponding- 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
degrees from the top of the page; if at the "bottom read from
the bottom.

Find the value of (q — 1} or (q -f- 0, 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 — 1}
or I — (q -[- 0 — 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 11.734668

.-. n = 3802.8 "3.580108

For 1° adopt 3780. giving 03'

Difference 22". 8

Ans. 1° 03' 22".8 or 178° 56' 37".2.

J&cample. — 8.201795 is the log cos of what arc?
q - I 4.685556

q 8.201795

/. n =t 3282". 8 3.51623d

For 89° adopt 3300. giving 05r

Difference 17 ".2

Ans, 89° 05' 17".2 or 90° 54' 42". 8.

114

STATISTICAL METHODS.

THE GREEK ALPHABET.

A a

Alpha

/ i

Iota

Pp

Rho

B/3

Beta

K K

Kappa

3 cr<

> Sigma

r y

Gamma

A A

Lamba

T r

Tau

A d

Delta

M/t

Mu

TV

Upsilon

E €

Epsilon

N v

Nu

<P (f)

Phi

z c

Zeta

S 1

Xi

X x

Chi

Hr;

Eta

Oo

Omicron

W 0

Psi

GQ$

•• Theta

nit

Pi

£L GO

Omega

EXPLANATION OF TABLES.

115

INDEX TO THE PRINCIPAL LETTERS USED IN THE
FORMULAE OF THIS BOOK.

A, average, mean.

a, class index (p. 24); also upper

left-hand quadrant (p. 49).
a, skewness index.

6, the frequency of the upper
right quadrant (p. 49).

/?, ratio of moments.

C, coefficient of variability.

c, the frequency of the lower left
quadrant (p. 49).

D, distance from mean to mode.

d, a difference; differential; the
frequency of lower right quad-
rant (p. 49).

4, index of closeness of fit.

d, difference between y and /.

E, probable error.

e, base of Naperian logarithms,
1 =2.718282.

F, critical function.
/, class frequency.

G, geometric mean.
H, a function of h.

h, a fixed value of x\ also, index of
heredity.

7, interval between the p'th and
j>"th individual.

t, interval between the pth and

(p + l)th individual (p. 27).
K, a function of k.
k, a fixed value of x.
L, limiting value of class.
Z, range of curve along x.
l\, ^2, portions of the curve range.
A , number of classes.
A, class range.
M , abscissal value of the mode

(theoretical).
M', abscissal value of the mode

(empirical).
H, moment about A.
N, the number corresponding to

a log.

n, number of variates; area of

polygon; any, not specified,

number.
|rc_, product of all integers from

1 to n.

v, average moment about VQ.
H, index of dissymmetry.
P, probability
p, ordinal rank of a particular

individual or case (p. 27); a

root or power.
ic, circumference in units of diame-

ter, 3.14159.
q, a root or power.
r, coefficient of correlation.
p, coefficient of regression.
s, a relation of @'s (p. 22).
2", summation sign.
<?, standard deviation; index of

variability.
T, transmuting factor, o into E,

.67449.
r, in Type IV.

' f angles.

9» )

V, magnitude of any class.

FO, magnitude of central class.

v, any variate or value.

™ = 5/?2-6/?i-9 (p. 31).

X, the horizontal axis or base of

polygon.

x, a varying abscissal value.
x\, X2, etc., definite values of x.

Y, the vertical axis of polygons ;

also the log of / (p. 29).
y, a varying ordinate value.
2/0, value of the ordinate at the

origin.
z, ordinate value.

116 STATISTICAL METHODS.

I. FORMULAS.

= ±0.6745—^=. x = V-J.
\/n

= 0.6745-^,

V2n

C=j-XlOO%. E,

A. D.=^^ = 0.7979(7. ^ =0.6745(7.

1 I

(*4./) , j ^(rr2./) , 7 j.
71^^2/1 ^240 J *

£|f (Types I, IV). a = 2Vp 3 (Type y)>

Probable discrepancy, ™^' \ ± . (-^} - A+^ I *
Vn / 2 2/ \ J7 )

J(dev. rcXdev. yX/) = S(x&2f) _ 0.6745(1 -r2)

(72
ro (spurious correlation) = — — —

.

71 o2 n

To solve any equation of the second degree,

-b±\/b2-4ac
Q; x= =~ .

CERTAIN CONSTANTS AND THEIR LOGARITHMS. 117
II.— CERTAIN CONSTANTS AND THEIR LOGARITHMS.

Title.

Symbol.

7T
1

7T

VT

1

VT

\/2ir
1

\/2~x
1

27T

VT

1

V2~

V?

e
1
VT
m

1
m

T

Number.

Log.

Ratio of circumference to diameter

3.1415927
0.3183099

1.7724538
0.5641896
2.506628
0.3S89422

0.159155
1.4142136
0.707105
0.797816

2.7182818
0.606530
0.4342945
2.3025851
0.67449

0.4971499
9.502CSC1

0.248574i
9.75K-2fl
0 . 3C909G
9.6009101
9.201820
0.150515
9.8494849
9.9019401

0.4342945
9.78285281
9.6377843
0.3622157
9.828976

Reciprocal of same . .

Reciprocal of sQiiare root of same .

Reciprocal of same ... . ...

Reciprocal of 2?r . .

Square root of 2

Reciprocal of same . ...

2
Square root of —

7T

Base of hyperbolic logarithms

Reciprocal of square root of same

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

Factor to reduce a to probable error. . . .

Com. log:r = raXhyp. log x, or

Com. log (com. log x)
= 9.6377843 + com. log (hyp. log*)

Hyp. log x = com. logo: X— , or
m

Com. log(hyp. logs)
= com. log (com. log) x + 0. 3622157

Circumference of circle

2xr
«r«

lAlr
a 9
360^

where a
semi-n

= semi-majoi
linor axis of

• axis: 6 =
ellipse.

Area of circle

Area of sector (length of arc =Z)

Area of sector (angle of arc =a°)

F ' *t f I!' A/a2 — &2

a2

118

STATISTICAL METHODS.

TABLE III.— TABLE OF ORDINATES (2) OF NORMAL CURVE,

OR VALUES OF — CORRESPONDING TO VALUES OF — . •

yb a

y= frequency.
2/o = — 7= = maximum frequency.

x = deviation from mean.
a = standard deviation.

X/a

0

1

2

3

4

99920
99025
97161
94387
90774

5

99875
98881
96923
94055
90371

6

99820
98728
96676
93723
89961

7

8

9

0.0
0.1
0.2
0.3
0.4

100000
99501
98020
95600
92312

99995
99396
97819
95309
91939

99980
99283
97609
95010
91558

99955
99158
97390
94702
91169

99755
98565
96420
93382
89543

99685
98393
96156
93034
89119

99596
98211
95882
92677

88688

0.5
0.6
0.7
0.8
0.9

88250
83527
78270
72615
66698

87805
83023
77721
72033
66097

87353
82514
77167
71448
65494

86896
82010
76610
70861
64891

86432
81481
76048
70272
64287

85962
80957
75484
69681
63683

85488 85006
80429 79896
74916 74342
69087 68493
63077 62472

84519
79359
73769
67896
61865

84060
78817
73193
67298
61259

.0
.1
.2
.3

.4

60653
54607
48675
42956
37531

60047
54007
48092
42399
37007

59440
53409
47511
41845
36487

58834
52812
46933
41294
35971

58228
52214
46357
40747
35459

57623
51620
45783
40202
34950

57017
51027
45212
39661
34445

56414
50437
44644
39123
33944

55810
49848
44078
38589
33447

55209
49260
43516
38058
32954

.5
.6

.7
.8
.9

32465
27804
23575
19790
16448

31980
27361
23176
19436
16137

31500
26923
22782
19086
15831

31023
26489
22392
18741
15530

30550
26059
22008
18400
15232

30082
25634
21627
18064
14939

29618
25213
21251
17732
14650

29158
24797
20879
17404
14364

28702
24385
20511
17081
14083

28251
23978
20148
16762
13806

2.0
2.1
2.2
2.3
2.4

13534
11025

08892
07100
05614

13265
10795
08698
06939
05481

13000
10570
08507
06780
05350

12740
10347
08320
06624
05222

12483
10129
08136
06471
05096

12230
09914
07956
06321
04973

11981
09702
07778
06174
04852

11737
09495
07604
06029
04734

11496
09290
07433

05888
04618

11259
09090
07265
05750
04505

2.5
2.6
2.7
2.8
2.9

04394
03405
02612
01984
01492

04285
03317
02542
01929
01449

04179
03232
02474
01876
01408

04074
03148
02408
01823
01367

03972
03066
02343
01772
01328

03873
02986
02280
01723
01289

03775
02908
02218
01674
01252

03680
02831
02157
01627
01215

03586
02757
02098
01581
01179

03494
02684
02040
01536
01145

3
4
5

01111
00034
00000

00819
00022

00598
00015

00432
00010

00309
00006

00219
00004

00153
00003

00106
00002

00073
00001

00050
00001

VALUES OF NORMAL PROBABILITY INTEGRAL. 119

TABLE IV.-TABLE OF THE HALF CLASS INDEX (*a) VALUES
OF THE NORMAL PROBABILITY INTEGRAL CORRESPOND-
ING TO VALUES OF -; OR THE FRACTION OF THE AREA

a

OF THE CURVE BETWEEN THE LIMITS 0 AND +-, OR 0

a

AND -— .

a

Total area of curve assumed to be 100,000.
x = deviation from mean.
a — standard deviation.

x/a

0

1

2

3

4

5

6

7

8

9

J

0.00

00000

40

80

120

159

199

239

279

319

359

40

0.01

0399

439

479

519

559

598

638

678

718

758

0.02

0798

838

878

917

957

997

1037

1077

1117

1157

0.03

1197

1237

1276

1316

1356

1396

1436

1476

1516

1555

0.04

1595

1635

1675

1715

1755

1795

1834

1874

1914

1954

0.05

1994

2034

2074

2113

2153

2193

2233

2273

2313

2352

0.06

2392

2432

2472

2512

2551

2591

2631

2671

2711

2751

0.07

2790

2830

2870

2910

2949

2989

3029

3069

3109

3148

0.08

3188

3228

3268

3307

3347

3387

3427

3466

3506

3546

0.09

3586

3625

3665

3705

3744

3784

3824

3864

3903

3943

0.10

3983

4022

4062

4102

4141

4181

4221

4261

4300

4340

0.11

4380

4419

4459

4498

4538

4578

4617

4657

4697

4736

0.12

4776

4815

4855

4895

4934

4974

5013

5053

5093

5132

0.13

5172

5211

5251

5290

5330

5369

5409

5448

5488

5527

0.14

5567

5606

5646

5685

5725

5764

5804

5843

5883

5922

0.15

5962

6001

6041

6080

6119

6159

6198

6238

6277

6317

0.16

6356

6395

6435

6474

6513

6553

6592

6631

6671

6710

0.17

6750

6789

6828

6867

6907

6946

6985

7025

7064

7103

0.18

7142

7182

7221

7260

7299

7338

7378

7417

7456

7495

0.19

7535

7574

7613

7652

7691

7730

7769

7809

7848

7887

0.20

7928

7965

8004

8043

8082

8121

8160

8199

8238

8278

0.21

8317

8356

8395

8434

8473

8512

8551

8590

8628

8667 39

0.22

8706

8745

8784

8823

8862

8901

8940

8979

9018

9057

0.23

9095

9134

9173

9212

9250

9289

9328

9367

9406

9445

0.24

9483

9522

9561

9600

9638

9677

9716

9754

9793

9832,

0.25

9871

9909

9948

9986

10025

10064

10102

10141

10180

10218

0.26

10257

10295

10334

10372

10411

10449

10488

10526 10565

10603'

0.27

10642

10680

10719

10757

10796

10834

10872

10911 10949

10988

0.28

11026

11064

11103

11141

11179

11217

11256

11294

11333

11371

0.29

11409

11447

11485

11524

11562

11600

11638

11676

11715

11753

0.30

11791

11829

11867

11905

11943

11981

12019

12058

12096

12134

0.31

12172

12210

12248

12286

12324

12362

12400

12438

12476

12514

38

0.32
0.33

12552
12930

12589 12627
12968 13005

12665
13043

12703
13081

12741
13118

12778
13156

12816
13194

12854
13232

12892
13269

0.34

13307

13344 13382

13420

13457

13495

13533

13570

13608

13645

0.35

13683

13720

13758

13795

13833

13870

13908

13945

13983

14020

PROPORTIONAL PARTS.

A

1

2345

6

789

40

4.0

8.0 12.0 16.0 20.0

24.0

28 . 0 32 . 0 36 . 0

39

3.9

7.8 11.7 15.6 19.5

23.4

27.3 31.2 35.1

38

3.8

7.6 11.4 15.2 19.0

22.8

26.6 30.4 34.2

37

3.7

7.4 11.1 14.8 18.5

22.2

25.9 29.6 33.3

120

STATISTICAL METHODS.

TABLE IV.— Continue*

x/a

0

1

2

3

4

5

6

7

8

9

A

0.36

14058

14095

14132 14169

14207

14244

14281

14319

14356

14393

0.37

14431

14468

14,

305 145<

12

14579

14617

14654

14691

14

728

147

65

0.38

14803

14840

14*

*77 149

14

14951

14988

15025

15062

15

099

151

36

37

0^39

15173

15210

15247

15284

15321

15357

15394

15431

15468

15505

0.40

15542

15579

15(

516

156

v_>

15689

15726

15763

15799

15

836

158

73

0.41

15910

15946

15983

16019

16056

16093

16129

16166

16202

16239

0 42

16276

16312

16:

548

163!

35

16421

16458

16494

16531

16

567

166

04

0.43

16640

16676

16713

16749

16785

16821

16858

16894

16930

16967

0 44

17003

17039

17(

)75

171

11

17147

17184

17220

17256

17

292

173

28

0.45

17364

17400

17436

17472

17508

17544

17580

17616

17652

17688

36

0.46

17724

17760

17'

r96

178

31

17867

17903

17939

17975

IS

Oil

18C

46

0.47

18082

18118

18153

18189

18225

18260

18296

18332

18367

18403

0.48

18439

18474

18,

509

185^

15

18580

18616

18651

18687

18

722

187

58

0 49

18793

18829

18*

364

188<

39

18934

18969

19005

19040

IS

075

191

11

0.50

19146

19181

1C216

19251

19287

19322

19357

19392

19427

19462

0.51
0.52

19497
19847

19532
19881

19567
19916

19602 19637
19951 119986

19672
20020

19707
20055

19742
20090

19777
20125

19812
20160

35

0.53

20194

20229

20263

20298

20332

20367

20402

20436

20471

20505

0.54

2054020574

20(

509

206

13

20678

2C712

20746

20781

2C

815

20*

$50

0.55

20884 20918

20952

20986

21021

21055

21089

21123

21158

21192

0.56

2122621260

2L

294

213

28

21362

21396

21430

21464

21

498

211

532

34

0.57

21566 21600

21634

21667

21701

21735

21769

21803

21836

21870

0.58

21904 21938

21<

371

220

35

22039

22072 22106

22139

^

>173

22S

>()7

0.59
0.60
0.61
0.62

22240122274
225751 22608
22907 j 22940
23237 23270

22307 22341
22641 22674
22973 23006
23303 '23335

22374
22707
23039
23368

22407 22441
22741 22774
23072 23105
23401 23434

22474
22807
23138
23467

22508
22840
23171
23499

22541
22874
23204
23532

33

0.63

23565 23598

23630 23663

23695

23728 23761

23793

23826

23859

0.64

23891 23924

23

356

239

S8

24021

24053 24085

24118

2^

U50

24

L83

0.65

24215 24247

24

280

243

12

24344

24376 24408

24441

2^

t473

24,

SOS

0.66

24537

24569

24601

24633

24665

24697 24729

24761

24793

24825

32

0.67

24857

24889

24

920

249

->2

24984

25016 25048

25079

2,

5111

25

143

0.68

25175

25206

25238 25269

25301

25332 25364

25395

25427

25459

0.69

25490

25521

25

,;>:;

255

84

25615

25647

25678

25709

2,

5741

25'

111

0.70

25804

25835

25

S66

258

97

25928

25959

25990

26021

2(

3052

26(

384

0.71

26115

26146

26176

26207

26238

26269

26300

26331

26362

26393

31

0.72

26424

26454

26

485

265

16

26546

26577

26608

26638

2(

1661

26'

roo

0.73

26730

26761

26791

26822

26852

26883

26913

26943

26974

27004

0.74

27035

27065

27

095

271

25

27156

27186

27216

27246

2'

^277

27

307

0.75

27337

27367

27

397

274

27

27457

27437

27517

27547

2'

r577

27(

307

30

0.76

27637

27667 : 27697

27726

27756

27786

27816

27845

27875

27905

0.77

27935

27964

27

994

280

23

28053

28082

28112

28142

2i

3171

28

201

0.78

28230

28260

28289

28318

28347

28377

28406

28435

28465

28494

0.79

28524

28553

28

582

286

11

2864C

28669

28698

28727

2*

3756

28

rss

0.80

28814

28843

28872

28901

2893C

28958

28987

29016

29045

29074

29

PROPORTIONAL, PARTS.

J

1

2

3

4

5

6

7

8

9

37

3.7

7.4

11.1

14.8

18.5

22.2

25.9

29.6

33.3

36

3.6

7.2

10.8

14.4'

18.0

21.6

25.2

28.8

32.4

35

3.5

7.0

10.5

14.0

17.5

21.0

24.5

28.0

31.5

34

3.4

6.8

10.2

13.6

17.0

20.4

23.8

27.2

30.6

33

3.3

6.6

9.9

13.2

16.5

19.8

23.1

26.4

29.7

32

3.2

6.4

9.6

12.8

16.0

19.2

22.4

25.6

28.8

31

3.1

6.2

9.3

12.4

15.5

18.6

21.7

24.8

27.9

30

3.0

6.0

9.0

12.0

15.0

18.0

21.0

24.0

27.0

29

2.9

5.8

8.7

11.6

14.5

17.4

20.3

23.2

26.1

YALUES OF NORMAL PROBABILITY INTEGRAL. 121

TABLE IV.— Continued.

x/a

Q

1

2

3

4

5

6

7

8

9

J

0.81
0.82

29103 \
29389 \

29132
29417

29160
29446

29189
29474

29217
29502

29246
29531!

29274 \
29559M

29303

29588

29332

29616

29360
29645

0.83

29673 \

29701

297

29

>7

29785

29814J29842

29870

29898

299

.>()

0.84

29954 \

29982

30010

30038

30066

30094

30122

30150

30178

30206

28

0.85

30234 30261

302

89

3031

30344

30372

30400

30427

30455

304

83

0.86

30510 ,

30538

30£

)(>5

305?

)3

J0620

30648

30675

30702

30730

307

57

0.87

30785

30812

30839

30866

30894

30921

30948

30975

31002

31030

0.88

31057

31084

311

11

311C

,s

31165

31192

31219

31246

31273

313

)()

27

0.89

31327

31353

31380

31407

31433

31460

31487

31514

31540

31567

0.90

31594

31620

316

>47

316'

ra

31700

31726

31753

31780

31806

318

;_'

0.91

31859

31885

31911

31937

31964

31990

32016

32042

32069

32095

0:92

32121

32147

321

73

321'

)9

32225

32251

32277

32303

32329

323

5.5

26

0.93

32381

32407

32^

L33

324,

59

32484

32510

32536

32562

32587

326

13

0.94

32639

32665

32690

32715

32741

32766

32792

32818

32843

32869

0.95

32894 32919

32<

)45

329'

T()

32995

33021

33046

33071

33096

331

22

0.96

33147:33172

331

L97

332S

l-l

33247

33272

33297

33322

33347

333

73

25

0.97

33398 33422

33447

33472

33497

33521 33546

33571

33596 33621

0.98

33646 '33670

33(

595

337

19

33744 33768 33793

33817

33842 338

(17

0.99

33891

33915 33940

33964

33988 34013

34037

34061

34086

34110

1.00

34134

34158(34

L82

342(

)0

34230

34255

34279

34303

34327

343

51

24

1.01

34375

34399 34423

34446

34470

34494

34518

34542

34566

34590

1.02

34613

34637

34(

li.i

346?

$4

34708

34731

34755

34778

34802

348

26

1.03

34849

34873

34896

34919

34943

34966

34989

35013

35036

35059

1.04

35083

35106

35

129

351

>2

35175

35198

35221

35245

35268

352

91

23

1.05

35314

35337

35360

35382

35405

35428

35451

35474

35497

35520

1.06

35543

35565

35

588

356

10

35633

35656

35678

35701

35724

357

40

1.07

Irjo

35769

OCTQQO

35791

35814

35836

35858

35881

35903

35926

35948

35970

. Uo

ooyyo

01 ^

037

059

081

103

125

1J.8

170

192

22

Y.09

38214

U1O

236

258

280

302

324

345

367

389

411

1.10

433

455

477

4

is

520

541

563

585

607

6

>2S

1.11

650

671

693

714

735

757

778

OQ1

800

821

843

1 . 12

864

885

tJUU

928

94i.

970

yyi

012

034

055

1.13

37176 097

118

139

160

181

202

223

244

265

21

1.14

286 306

327

348

368

389

410

430

451

472

1.15

493 513

534

5

-.1

574

595

615

636

656, (

>77

1.16

697

718

738

758

778

798

819

839

859

880

1 . 17

900

92C

94(

96(

yol

ooc

02C

04C

06C

080

2C

1.18

38100

120

139

159

17S

199

218

238

258

278

1.19

298

317

337

3

r,(

37e

395

415

434

454

t

i":

1.20

493

512

531

551

57C

589

609

628

647

667

PROPORTIONAL PARTS.

A

1

2

3

4

5

6

7 8

9

29

2.9

5.8

8.7

11.6

14.5

17.4

20.3 23.2

26.1

28

2.8

5.6

8.4

11.2

14.0

16.8

19.6 22.4

25.2

27

2.7

5.4

8.1

10.8

13.5

16.2

18.9 21.6

24.3

26

2.6

5.2

7.8

10.4

13.0

15.6

18.2 20-8

23.4

25

2.5

5.0

7.5

10.0

12.5

15.0

17.5 20.0

22.5

24

2.4

4.8 •

7.2

9.6

12.0

14.4

16.8 19.2

21.6

23

2.3

4.6

6.9

9.2

11.5

13.8

16.1 18.4

20.7

22

2.2

4.4

6.6

8.8

11.0

13.2

15.4 17.6

19.8

21

2.1

4.2

6.3

8.4

10.5

12.6

14.7 16.8

18.9

20

2.0

4.0

6.0

8.0

10.0

12.0

14.0 16.0

18.0

19

1.9

3.8

5.7

7.6

9.5

11.4

13.3 15.2

17.1

122

STATISTICAL METHODS.

TABLE IV. — Continued.

x/a

0

1

2

3

4

5

6

800
990

7
819

8

9

J

1.21
1.22

1.23

1.24
1.25
1.26
1.27
1.28

1.29
1.30
1.31
1.32
1.33
1.34

.35
.36
.37
.38
.39
.40

.41
.42
.43
.44
.45
.46
.47

.48
.49
.50
.51
.52
.53
.54
.55

.56
.57

.58
.59

38686
876

705
895

724
914

743
933

762
952

139
324
507

688
866

042
216
388
557
725
889

052
213
371
527
683
834
984

781
971

158
343
525

706
884

060
233
405
574
742
906

838

857

19

18
17

16
15

14

13
12

008
195
380
562
742
920

095
268
439
608
775
938

027
214
398
580
760
937

112
285
456
625
792
955

117
277
434
590
744
894

046
232
417
598
778
955

39065
251
435
617
796
973

084
270
453
634
813
990

102
288
471
652
831

121

306
489
670
849

177
361
544
724
902

077
251
422
591
758
922

008
182
354
524
692
857

025
199
371
540
709
873

130
303
473
641
808
971

40147
320
490
658
825 j
987

41149
308
466
621
774
924

42073
220
364
507
647
785
922

165
337
507
675
841

004
165
324
481
637
789
939

088
234
378
521
661
799
935

020
181
340
497
652
804
954

102
248
393
535
675
813
949

083
215
345
473
599
724
846
967

036
197
355
512
667
819
969

117
263
407
549

688
826
962

068
229
387
543
698
849
998

084
245
403
558
713
864

101
261
418
574

728
879

133
292
450
605
759
909

013
161
306
449
591
730
867

002
136
267
396
524
649
773
894

028
175
321
464

605
744
881

016
149
.280
409
536
662
785
906

043
190
335

478
619
758
895

029
162
293
422
549
674
797
919

038
156
271
385
498

058
205
350
492
633
772
908

043
175
306
435
562
687
810
931

050
167
283
397
509

131
277
421
563
702
840
975

109
241
371
498
624
748
870
990

109
225
340
453

146
292
435
577
• 716
854
989

122
254
383
511
637
760
882

43056
189
319
448
574
699
822
943

069
202
332
460
587
711
834
955

096
228
358
486
612
736
858
978

002
120
237
351
464

014
132
248
363
475

026
144
260
374
486

44062
179
295
408

074
191
306
419

085
202
317
430

097
214
329
. 442

PROPORTIONAL PARTS.

J

1

2 3

4

5

6

7 8 9

19
18
17
16
15
14
13
12
11

1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1

3.8 5.7
3.6 5.4
3.4 5.1
3.2 4.8
3.0 4.5
2.8 4.2
2.6 3.9
2.4 3.6
2.2 3.3

7.6
7.2
6.8
6.4
6.0
5.6
5.2
4.8
4.4

9.5
9.0
8.5
8.0
7.5
7.0
6.5
6.0
5.5

11.4
10.8
10.2
9.6
9.0
8.4
7.8
7.2
6.6

13.3 15.2 17.1
12.6 14.4 16.2
11.9 13.6 15.3
11.2 12.8 14.4
10.5 12.0 13.5
9.8 11.2 12.6
9.1 10.4 11.7
8.4 9.6 10.8
7.7 8.8 9.9

VALUES OF KORMAL PROBABILITY INTEGRAL. 123

TABLE IV —Continued.

x/a

0

1

2

3

4

5

6

7

8

9

J

1.60

44520

531

542

553

564

575

586

597

608

619

11

1.61

630

641

652

662

673

684

695

706

717

727

1.62

738

749

760

770

781

791

802

813

823

834

1.63

845

855

866

876

887

897

908

918

929

939

1 .64

950

960

97 C

980

991

001

on

022

032

042

1.65

45053

063

073

083

093

103

114

124

144

1.66

154

164

174

184

194

204

214

224

234

244

10

1.67

254

264

274

283

293

303

313

323

332

342

1.68

352

362

371

381

391

400

410

419

429

439

1.69

449

458

467

477

486

496

505

515

524

534

1.70

543

553

562

571

581

590

599

609

618

627

1.71

637

646

655

664

673

682

692

701

710

719

1.72

728

737

746

755

764

773

782

791

800

809

9

1.73

818

827

836

845

854

863

871

880

889

898

1.74

907

916

924

933

942

950

959

968

977

985

1.75

994

003

on

020

028

037

045

054

062

071

1.76

46080

088

096

105

113

121

130

138

147

155

1.77

164

172

180

188

196

205

213

221

230

238

1.78

246

254

262

270

279

287

295

303

311

319

1.79

327

335

343

351

359

367

375

383

391

399

8

1.80

407

415

423

430

438

446

454

462

% 469

477

1.81

485

493

500

508

516

523

531

539

547

554

1.82

562

570

577

585

592

600

607

615

622

630

1.83

638

645

652

660

667

674

682

689

697

704

1.84

712

719

726

733

741

748

755

762

770

777

1.85

784

791

798

806

813

820

827

834

841

849

1.86

856

863

870

877

884

891

898

905

912

919

7

1.87

926

933

939

946

953

960

967

974

981

988

1 .88

QQK

yyo

nm

nns

nl ^

n9i

O9S

035

040

fl4Q

055

1.89

47062

UUl

069

UUo

075

uio

082

LWl

088

U^o

095

102

\jt&
108

U^*7

115

122

1.90

128

135

141

148

154

161

167

174

180

187

1.91

193

200

206

212

219

225

231

238

244

251

1.92

257

263

270

276

282

288

294

301

307

313

1.93

320

326

332

338

344

350

356

362

369

375

1.94

381

387

393

399

405

411

417

423

429

435

6

1.95

441

447

453

459

465

471

476

482

488

494

1.96

500

506

512

517

523

529

535

541

546

552

1,97

558

564

569

575

581

586

592

598

603

609

1.98

615

620

626

631

637

643

648

654

659

665

1.99

670

676

681

687

692

698

703

709

714

719

2.00

725

730

735

741

746

752

757

762

768

772

2.01

778

784

789

794

799

804

810

815

820

826

2.02

831

836

841

846

851

856

862

867

872

877

2.03

882

887

892

897

902

907

912

917

922

927

5

2.04

932

937

942

947

952

957

962

967

972

977

PROPORTIONAL PARTS.

A

1

2345

6

7

8 9

11

1.1

2.2 3.3 4.4 5.5

6.6

7.7

8.8 9.9

10

1.0

2.0 3.0 4.0 5.0

6.0

7.0

8.0 9.0

9

0.9

1.8 2.7 3.6 4.5

5.4

6.3

7.2 8.1

8

0.8

1.6 2.4 3.2 4.0

4.8

5.6

6.4 7.2

7

0.7

1.4 2.1 2.8 3.5

4.2

4.9

$.6 6.3

6

0.6

1.2 1.8 2.4 3.0

3.6

4.2

4.8 5.4

124

STATISTICAL METHODS.

TABLE IV.— Continued.

X/a

0

1

2

3

4

5

6

7

8

9

a

2rvt

47QR9

007

QQ1

QQft

. UO

1 (Ma£

ys /

yy l

yyo

noi

onfi

m i

mr

H9n

(\f)K

2.06

48030

035

039

044

UU-l-

049

uuo
054

ui i

058

o

063

u^u
068

\JAO

073

2.07

077

082

087

091

096

100

105

110

114

119

2.08

124

128

133

137

142

146

151

155

160

165

2.09

169

173

178

182

187

191

196

200

205

209

2.10

g!4

218

222

227

231

235

240

244

248

253

2.11

257

261

266

270

274

27.8

283

287

291

295

2.12

300

304

308

312

316

320

325

329

333

337

2.13

341

345

350

354

358

362

366

370

374

378

2.14

382

386

390

394

398

402

406

410

414

418

4

2.15

422

426

430

434

438

442

446

450

453

457

2.16

461

465

469

473

477

480

484

488

492

496

2.17

500

503

507

511

515

518

522

526

530

533

2.18

537

541

544

548

552

555

559

563

566

570

2.19

574

577

581

584

588

592

595

599

602

606

2.20

610

613

617

620

624

627

631

634

638

641

2.21

645

648

652

655

658

662

665

669

672

676

2.22

679

682

686

689

692

696

699

702

7C6

709

2.23

713

716

719

722

726

' 729

732

736

739

742

2.24

745

749

752

755

758

761

765

768

771

774

2.25

778

781

784

787

790

793

796

799

803

806

2.26

809

812

815

818

821

824

827

83C

833

837

2.27

840

843

846

849

852

855

858

861

864

867

3

2.28

870

872

875

878

881

884

887

890

893

896

2.29

899

902

905

907

• 910

913

916

919

922

925

2.30

928

930

933

936

939

942

944

917

950

953

2.31

956

958

961

964

966

969

972

975

977

980

2qo

983

OCA

988

QQ1

QQ4

QQA

999

. • > —

yoO

yy i

yy^

yyo

nr>9

_

2.33

49010

012

015

017

020

023

025

\J(jz

028

004
031

007
033

2.34

036

038

041

043

046

048

051

054

056

059

2.35

061

064

066

069

071

074

076

079

081

084

2.36

086

089

092

094

096

098

101 103

106

108

2.37

111

113

115

118

120

122

125 127

130

132

2.38

134

137

139

141

144

146

148

151

153

155

2.39

158

160

162

164

167

169

171

173

176

178

2.40

180

182

185

187

189

191

193

196

198

200

2.41

202

205

207

209

211

213

215

217

220

222

2.42

224

226

228

230

232

234

237

239

241

243

2.43

245

247

249

251

253

255

257

259

261

264

2.44

266

268

270

272

274

276

278

280

282

284

2

2.45

286

288

290

292

294

295

297

299

301

303

2.46

305

307

309

311

313

315

317

319

321

323

2.47

324

326

328

330

332

334

336

337

339

341

2.48

343

345

347

349

350

352

354

356

358

359

2.49

361

363

365

367

368

370

372

374

375

377

2.5

379

396

413

430

446

461

477

492! 506

520

16

2.6

534

547

560

573

585

59S

609

621

632

643

12

2.7

653

664

674

683

693

702

711

720

728

736

9

2.8 1

744

752

760

767

774

781

788

795

801

807

7

PROPORTIONAL PARTS.

A

1

2345

6

7

9 9

16

1.6

3.2 4.8 6.4 8.0

9.6

11.2

12.8 14.4

12

1.2

2.4 3.6 4.8 6.0

7.2

8.4

9.6 10.8

9

0.9

1.8 2.7 3.6 4.5

5.4

6.3

7.2 8.1

7

0.7

1.4 2.1 2.8 3.5

4.2

4.9

5.6 6.3

•

VALUES OF NORMAL PROBABILITY INTEGRAL. 125

TABLE IV.— Continued.

x/a

0

1

2

3

4

5

6

7

8

9

,

2.9

49813

819

825! 831

836

841

846

851

856

861

5

3.0

865

869

873 878

882

886

889

893

897

900

4

3.1

903

906

910

913

916

918

921

924

926

929

3

3.2

931

934

936

938

940

942

944

946

948

950

2

3.3

952

953

955

957

958

960

961

962

964

965

1

3.4

966

968

969

970

971

972

973

974

975

976

1

3.5

977

978

978

979

980

981

981

982

982

983

1

3.6

984

985

985

986

986

987

987

988

988

989

1

3.7

989

990

990

990

991

991

992

992

992

992

0

3.8

993

993

993

994

994

994

994 995

995

995

0

3.9

995

995

996

996

996

996

996 996

997

997

0

4

997

998

999

999

999

000

000

000

00.

000

0

PROPORTIONAL, PARTS.

A

1

234

5

6

789

5

0.5

1.0 1.5 2.0

2.5

3.0

3.5 4.0 4

.5

4

0.4

0.8 1.2 1.6

2.0

2.4

2.8 3.2 3.6

3

0.3

0.6 0.9 1.2

1.5

1.8

2.1 2.4 2

.7

2

0.2

0.4 0.6 0.8

1.0

1.2

1.4 1.6 1

.8

1

0.1

0.2 0.3 0.4

0.5

0.6

0.7 0.8 0.9

126 STATISTICAL METHODS.

V.— TABLE OF LOG r FUNCTIONS OF p (see pages 32-34).

P

0

1

2

3

4

5

6

7

8

9

1.00

97'50

9500

9251

9003

8755

8509

8263

8017

7773

1.01

"9! 997529

7285

7043

6801

6560

0320

6080

5H41

5602

5365

1.02

5128

4892

4656

4421

4187

3953

3721

3489

3257

3026

1.03

2796

2567

2338

2110

1883

1056

1430

1205

0981

0757

1.04

0533

0311

0089

9868

§647

§427

§208

§989

grw

§554

1.05

9.988338

8122

7907

7692

7478

7265

7052

6841

6629

6419

1.06

6209

6000

5:91

5583

5376

5169

4903

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

§333

§644

§456

§269

§082

8896

8? 10

8^25

1.10

9.978341

8157

7974

7791

7610

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

3704

3596

3429

3262

1.13

3096

2931

2766

2602

2438

2275

2113

1951

1790

1029

1.14

1469

1309

1150

0992

0835

0077

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

5081

1.18

5544

5408

5272

5137

5002

4808

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

0180

0067

9955

8843

9732

9621

9511

1.23

9.959401

9292

9184

9076

8968

8801

8755

8019 .

8544

8439

1.24

8335

8231

8128

8025

7923

7821

7720

7620

7520

7420

1.25

7321

7223

7125

7027

6930

6S34

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

4259

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

0090

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

9430

9435

9391

9348

9304

9202

9219

9178

9136

9095

1.37

9054

9015

8975

8936

8898

8859

8822

8785

8748

8711

1.33

8676

8640

8005

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

7786

7765

7744

7723

7703

7683

7064

7645

7626

1.42

7608

7590

7573

7556

7540

7524

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

7277

7282

72>9

7295

7302

1.48

7310

7317

7326

7334

7343

7353

7303

7373

7384

7395

1.49

7407

7419

7431

7444

7457

7471

7485

7499

7515

7529

TABLE OF LOG r FUNCTIONS. 127

V.— TABLE OF LOG r FUNCTIONS OP p (see pages 32-34).

p

0

1

2

3

4

5

6

7

8

9

.50

9.947545

7561

7577

7594

7612

7629

7647

7666

76S5

7704

.51

7724

7744

7764

7785

7806

7828

7850

7873

7896

7919

.52

7943

7967

7991

8016

8041

8067

8093

8120

8146

8174

.53

8201

8229

8258

8287

8316

8346

8376

8406

8437

8468

.54

8500

8532

8564

8597

8630

8664

8698

8732

8767

8802

.55

8837

8873

8910

8946

8983

9021

9059

9097

9135

9174

.56

9214

9254

9294

9334

9375

9417

9458

9500

9543

9586

.57

93 .'9

9672

9716

9761

9806

9851

9896

9942

9989

6035

.58

9.950082

0130

0177

0225

0274

0323

0372

0422

0472

0522

.59

0573

0624

0676

0728

0780

0833

0886

0939

0993

1047

.60

1102

1157

1212

1268

1324

1380

1437

1494

1552

1610

.61

1668

1727

1786

1845

1905

1965

2025

2086

2147

2209

.62

2271

2333

2396

2459

2522

2586

2650

2715

2780

2845

.63

2911

2977

3043

3110

3177

3244

3312

3380

3449

3517

.64

3587

3656

3726

3797

3867

3938

4010

4081

4154

4226

.65

4299

4372

4446

4519

4594

4668

4743

4819

4894

4970

.66

5047

5124

5201

5278

5356

5434

5513

5592

5671

5750

.67

5830

5911

5991

6072

6154

6235

6317

6400

6482

6566

.68

6649

6733

6817

6901

6986

7072

7157

7243

7329

7416

.69

7503

7590

7678

7766

7854

7943

8032

8122

8211

8301

.70

8391

8482

8573

8664

8756

8848

8941

9034

9127

9220

.71

9314

9409

9502

9598

9ii93

9788

9884

9980

6077

6174

.72

9.960271

0369

0467

0565

0664

0763

0862

0961

1061

1162

.73

1262

1363

1464

1566

1668

1770

1873

1976

2079

2183

.74

2287

2391

2496

2601

2706

2812

2918

3024

3131

3238

.75

3345

3453

3561

3669

3778

3887

3996

4105

4215

4326

.76

4436

4547

4659

4770

4882

4994

5107

5220

5333

5447

.77

5561

5675

5789

5904

6019

6135

6251

6367

6484

6600

.78

6718

6835

6953

7071

7189

7308

7427

7547

7666

7787

.79

7907

8028

8149

8270

8392

8514

8636

8759

8882

9005

.80

9129

9253

9377

9501

9626

9751

9877

5003

5129

6255

.81

9.970383

0509

0637

0765

0893

1021

1150

1579

1408

1538

.82

1668

1798

1929

2060

2191

2322

2454

2586

2719

2852

.83

2985

3118

3252

3386

3520

3655

3790

3925

4061

4197

.84

4333

4470

4606

4744

4881

5019

5157

5295

5434

5573

.85

5712

5852

5992

6132

6273

6414

6555

6697

6838

6980

.86

7123

7266

7408

7552

7696

7840

7984

8128

8273

8419

.87

8564

8710

8856

9002

9149

9296

9443

9591

9739

9887

.88

9.980036

0184

0333

0483

0633

0783

0933

1084

1234

1386

.89

1537

1689

1841

1994

2147

2299

2453

2607

2761

2915

.90

3069

3224

3379

3535

3690

3846

4003

4159

4316

4474

.91

4631

4789

4947

5105

5264

5423

5582

5742

5902

6062

.92

6223

6383

6544

6706

6867

7029

7192

7354

7517

7680

.93

7844

8007

8171

8336

8500

8665

8830

8996

9161

9327

.94

9494

9660

9827

9995

6162

5330

5498

6666

5835

1004

1.95

9.991173

1343

1512

1683

1853

2024

2195

2306

2537

2709

1.96

2881

3054

3227

3399

3573

3746

3920

4094

4269

4443

1.97

4618

4794

4969

5145

5321

5498

5674

5851

6029

6206

1.98
1.99

6384
8178

6562
8359

6740
8540

6919

8722

7098
8903

7277
9085

7457
9268

7637
9450

7817
9633

7997
9816

128

STATISTICAL METHODS.

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

Inches to Millimeters.

1

2

3

4

5

6

7

8

9

25.40

50.

RO

76.20

101.60

127.00

152.40

177

.80

203.20

228.60

10

279.40

304.

SO

330.19

355.59

380.99

406.39

431

.79

457.19

482.59

?0

533.39

558.

79

584.19

6

09.59

634.99

660. 3^

685

.79

7

1.19

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.

8

1092.2

1117.6

1143.0

1168.4

1193

.8

1219.2

1244.6

50

1295.4

1320.

R

1346.2

13

71.6

1397.0

1422.4

1447

.8

14

"3.2

1498.6

60

1549.4

1574.

R

1600.2

1625.6

1651.0

1076.4

1701

.8

1727.2

1

752.6

70

1803.4

1828.

R

1854.2

1879.6

1905.0

1930.4

1955

.8

1981.2

2006.6

80

2057.4

2082.

8

2108.2

21

33.6

2159.0

2184.4

2209

.8

22

35.2

2

260.6

90

2311.4

2336.8

2362.2

2387.6

2413.0

2438.4

2463

.8

2489.2

2514.6

Twelfths.

Sixteenths.

1/12

2/12
3/12
4/12

2.12
4.23
6.35
8.47

7/12
8/12
9/12
10/12

14.82
16.93
19.05
21.17

1/16

1/8
3/16

1/4

1.59
3.17
4.76
6.35

5/16

3/8
7/16
1/2

7.94
9.52
11.11

12.70

9/16

5/8
11/16

3/4

14.29

15.87
17.46
19.05

13/16

7/8
15/16
1

20.64
22.22
23.81

25.40

5/12

10.58

11/12

23.28

6/12

12.70

12/12

25.40

TABLE VII— MINUTES AND SECONDS IN DECIMALS OF A DEGREE.

'

o

'

0

'

0

-

o

"

o

"

o

1

.016666

21

. 350000

41

.683333

1

.000278*

21

.005833

41

.011389

2

.033333

22

. 366666

42

.700000

2

.000556

22

.006111

42

.011667

3

.050000

23

.383333

43

.716666

3

.000833

23

.006389

43

.011944

4

.066666

24

.400000

44

.733333

4

.001111

24

.006667

44

.012222

5

.083333

25

.416666

45

.750000

5

.001389

25

.006944

45

.012500

6

. 100000

26

.433333

46

.766666

6

.001667

26

.007222

46

.012778

7

.116666

27

. 450000

47

.783333

7

.001944

27

.007500

47

.013056

8

.133333

28

.466666

48

.800000

8

.002222

28

.007778

48

.013333

9

. 150000

29

.483333 49

.816666

9

. 002500

29

.008056

49

.013611

10

. 166666

30

. 500000

50

.833333

10

.002778

30

.008333

50

.013889

11

. 183333

31

.516666

51

.850000

11

. 003056

31

.008611

si

.014167

12

. 200000

32

. 533333

52

.866666

12

. 003333

32

.008889

52

.014444

13

.216666

33

. 550000

53

.883333

13

.003611

33

.009167

53

.014722

14

. 233333

34

. 566666

54

. 900000

14

. 003889

34

. 009444

54

.015000

15

. 250000

35

.583333

55

.916666

15

.004167

35

.009722

55

.015278

16

. 266666

36

.600000

56

.933333

16

.004444

36

.010000

56

.015556

17

. 283333

37

.616666

57

.950000

17

.004722

37

.010278

57

.015833

18 .300000

38

. 633333

58

.966666

18

. 005000

38

.010556

58

.016111

19

.316666

39

.650000

59

.983333

19

.005278

39

.010833

59

.016389

20

.333333

40

.666666

60

1.000000

20

.005556

40

.011111

60

.016667

* .0002777778.

FIRST TO SIXTH POWERS OF INTEGERS.

129

TABLE VIII.— FIRST TO SIXTH POWERS OF INTEGERS FROM 1 TO 50.

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

7776

46656

7

49

343

2401

16807

117649

8

64

512

4096

32768

202 144

9

81

729

6561

59049

531441

10

100

1000

10000

100000

1000000

11

121

1331

14641

161051

1771561

12

144

1728

20736

248832

2985984

13

169

2197

28561

371293

4826809

14

196

2744

38416

5378-24

7529536

15

225

3375

50625

759375

11390625

16

256

4096

65536

1048576

16777216

17

289

4913

83521

1419857

24137569

18

324

5832

104976

1889568

34012-J24

19

361

6859

180821

2476099

47045881

20

400

8000

160000

3200000

64000000

21

441

9261

194481

4084101

85766121

22

484

10648

234256

5153632

113379904

23

529

12167

279841

643U343

148035889

24

576

13824

331776

7962624

191102976

25

625

156','5

390625

9765625

244140025

26

676

17576

456976

11881376

308915776

27

7*9

19683

531441

14348907

387420489

28

784

21952

614656

17210368

481890304

29

841

24389

707281

30511149

594823321

30

900

27000

810000

24300000

729900000

31

961

29791

923521

28629151

887503681

32

1024

32768

1048576

33554432

1078741824

33

1089

35937

1185921

39I35W3

1:*) 14(57%'.)

34

1156

39304

1336336

45435424

1544SW4N5

35

1225

42875

1500625

52521875

1S3SXJ65625

36

1296

46656

1679616

60464176

2176782336

37

1369

50653

1874161

88648967

2565726409

38

1444

54872

2085136

79886168

30KMM53S4

39

1521

59319

2313441

90224199

351S7437J5I

40

1600

64000

2560000

102400000

4IM5000000

41

1681

68921

2825761

ItJWOBOJ

4750104241

42

1764

74088

3111696

13(H5912U2

54*9081744

43

1849

79507

3418801

147008448

15321 3»KW«.»

44

1936

85184

3748096

164916884

725631

45

202.")

91125

4100625

1H4528125

KHH765625

46

2116

97336

4477456

:>o:><x>-»i:<;

9471:.'

47

2209

103823

4S796H1

8292)49007

1077»ttl5888

4S
49
50

2304
2401
2500

110592
117649
125000

5WS416
57154801
6250000

864808968

282475249

312500000

12230590464

13S4I2S72W
15T,:.'50l)0000

130

STATISTICAL METHODS.

TABLE IX.— PROBABLE ERRORS OF THE COEFFICIENT OF COR-
RELATION FOR VARIOUS NUMBERS OF OBSERVATIONS OR
VARIATES (n) AND FOR VARIOUS VALUES OF r.

Decimal point, properly preceding each entry, is omitted. (Specially Cal-
„._, culated.)

Number

Correlation Coefficient r.

of Obser-

vations-

0.0

0.1

0.2

0.3

0.4

0.5

0.6

20

1508

1493

1448

1373

1267

1131

0965

30

1231

1219

1182

1121

1035

0924

0788

40

1067

1056

1024

0971

0896

0800

0683 .

50

0954

0944

0915

0868

0801

0715

0610

60

0871

0862

0836

0793

0731

0653

0557

70

0806

0798

0774

0734

0677

0605

0516

80

0754

0747

0724

0686

0633

0566

0483

90

0711

0704

0683

0647

0597

0533

0455

100

0674

0668

0648

0614

0567

0506

0432

150

0551

0546

0529

0501

0463

0413

0352

200

0477

0472

0458

0434

0401

0358

0305

250

0426

0421

0409

0387

0358

0319

0272

300

0389

0386

0374

0354

0327

0292

0249

400

0337

0334

0324

0307

0283

0253

0216

500

0302

0299

0290

0274

0253

0226

0193

600

0275

0272

0264

0251

0232

0207

0176

700

0255

0252

0245

0232

0214

0191

0163

800

0239

0236

0229

0217

0200

0179

0153

900

0225

0222

0216

0205

0189

0169

0144

1000

0213

0211

0205

0194

0179

0160

0137

2000

0151

0149

0145

0137

0127

0113

0097

5000

0095

0094

0092

0087

0080

0072

0061

0.65

0.7

0.75

0.8

0.85

0.9

0.95

20

0871

0769

0660

0543

0419

0287

0147

30

0711

0628

0539

0444

0342

0234

0120

40

0616

0544

0467

0384

0296

0203

0104

50

0551

0486

0417

0343

0265

0181

0093

60

0503

0444

0381

0313

0241

0165

0085

70

0466

0411

0353

0290

0224

0153

0079

80

0436

0385

0330

0271

0209

0143

0074

90

0411

0363

0311

0256

0197

0135

0069

100

0391

0345

0294

0242

0187

0128

0066

150

0318

0281

0241

0198

0153

0105

0054

200

0275

0243

0209

0172

0133

•0091

0047

250

0246

0218

0187

0154.

0118

0081

0042

300

0225

0199

0170

0140

0108

0074

0038

400

0195

0172

0148

0122

0094

0064

0033

500

0174

0154

0132

0109

0084

0057

0029

600

0159

0140

0121

0099

0076

0052

0027

700

0147

0130

0112

0092

0071

0049

0025

800

0138

0122

0105

0086

0066

0045

0023

900

0130

0114

0098

0081

0062

0043

0022

1000

0123

0109

0093

0077

0059

0041

0021

2000

0087

0077

0066

0054

0042

0029

0014

5000

0055

0049

0042

0034

0026

0018

0009

TABLE X.— SQUARES, CUBES, ETC.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

" " —"I
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

.333333333

4

16

64

2.0000000

1.5874011

.250000000

5

25

125

2.2360680

1.7099759

.200000000

6

36

216

2 4494897

1.8171206

.166600667

7

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

3.8729833

2.4662121

.000000007

16

256

4096

4.0000000

8.5196401

.002500000

17

289

4913

4.1231056

2.5712816

.05882:1529

18

324

5832

4.3426407

2.6207414

.055555556

19

361

0859

4.3588989

2.6684016

.0520:31579

20

400

8000

4.4721360

2.7144177

.050000000

21

441

1)261

4.5825?T>7

2.7589243

.047019048

22

484

10648

4.6904158

2.8020393

.045454545

23

529

12167

4.7958315

2.8438670

.043478261

24

576

13824

4.8989795

2.8844991

.041000007

25

025

15625

5.0000000

2.1)240177

.040000000

26

676

17576

5.0990195

2.9624960

.038401538

27

729

19683

5.1961524

3.0000000

.037037037

28

784

21952

5.2915026

8.0865889

.035714286

29

841

24389

5.3851648

3.0723168

.034482759

30

900

27000

5.4772256

3.1072325

.033333333

31

961

29791

5.5677644

3.1413806

.032258005

32

1024

32768

5.6568542

3.1748021

.031250000

33

1089

35937

5.7445626

3.2075343

030303030

34

1156

39304

5.8309519

3.2396118

.020411765

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

0.1644140

3.3619754

.026315789

39

1521

59319

6.2449980

3.3912114

.025041020

40

1600

64000

6.3245553

3.4199519

.025000000

41

1681

68921

6.4031242

3.4482172

.(121:590344

42

17G4

74088

0.4807407

3 4760266

.023809524

43

184!)

79507

6.5574385

3.5033981

.023255814

44

1936

85184

6.0332496

3.5303483

.022727273

45

2025

91125

6.7082039

3.5568933

.022222222

46

2116

07336

6.7823300

3.5830479

.021739130

47

2209

103823

6.8556546

3.6088261

.(121276600

48

2304

110592

6. 92820: W

3.6342411

.(M:s:«333

49

2401

I 117649

7.0000000

3.0593057

.020408163

50

2500

125000

7.0710678

3.6840314

.020000000

51

2G01

132651

7.1414284

3.7084298

.010(507843

52

2704

140608

7.2111026

3.7325111

.019330769

68

2809

148877

7.2801099

8.7662858

,018607935

54

2916

157464

7.3484(592

3.7797631

.018518519

55

3025

166375

7.4161985

8.8009535

.018181818

56

3136

1?'5616

7.4833148

8.8856634

.017857148

57

3249

185193

7.549S::u

8.8485011

58

3304

105112

7.6157781

3 8708766

.017241879

59

3481

206379

7.6811457

8.8999965

.016940153

60

3600

21BOOO

7.7459667

3.0148676

.01 6666( :» .7

61

3721

226981

7.8102497

3.0.304072

.0163931 »:J

63

3844

238838

7.8740079

3.0578915

.010120032

131

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

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

77

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

512000

8.9442719

4.3088695

.012500000

81

6561

531441

9.0000000

4.3267487

.012345679

83

6724

55136S

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

753571

9.5393920

4.4979414

.010989011

92

8464

778688

9.5916G30

4.5143574

.010869565

93

8649

804357

9.643G508

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.C570095

.009900990

102

10404

1061208

10.0995049

4.6723287

.009803922

103

10609

1092727

10.1488916

4.G875482

.009708738

104

10816

1124864

10.1980390

4.7026694

.009615385

105

11025

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

10.4880885

4.7914199

.009090909

111

12321

1307631

10.535C538

4.8058955

.009009009

112

12544

1404928

10.5830052

4.8202845

.0089285:1

113

12769

1442897

10.6301458

4.8345881

.008849558

114

12996

1481544

10.(.7707&3

4.8488076

.008771930

115

13225

152087'5

10.7238053

4.8629442

.008095652

116

13456

1560896

10.7703296

4.8769990

.008020690

117

13689

1601613

10.8166538

4.8909732

.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

11.00 0000

4.9460874

.0082G4463

1S2

14884

1815848

11.0453610

4.9596757

.008196721

123

15129

1860867

11.0905365

4.9731898

.008130081

124

15376

1906624

11.1355287

4.98G6310

.008064516

132

CUBE ROOTS, AtfD RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

125

15625

1953125

11.1803399

5.0000000

.008000000

126

15876

2000376

11.2249722

5.0132979

.007936508

127

16129

2048383

11.2694277

5.0265257

.007874016

128

16384

2097152

11.3137085

5.0396842

.007812500

128

16641

2146689

11.3578167

5.0527743

.007751938

130

16900

2197000

11.4017543

5.0657970

.007692308

131

i;iei

22480U1

11.4455231

6.0787531

.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

11.8743421

5.2048279

.007092199

142

2Q104

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

.006890552

146

21316

3112136

12.0830460

5.2656374

.006849315

147

21009

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

12.3288280

5.3368033

.006578947

153

23409

3581577

12.3693169

5.3484812

.006535948

154

23716

3652264

12.4096736

5.3601084

.006493506

155

24025

37238?o

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

25281

4019679

12.6095202

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

.006134909

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

17'0

28900

4913000

13.03S4048

5.5396583

.00588235:5

171

29241

5000211

13.0766968

5.5504991

. 005847 •'.):>:*

172

29584

5088448

13.1148770

5.5612978

.005813953

173

29929

5177717

13.1529464

5.5720546

.005780347

174

30276

5268024

13 1909060

5.5827702

.00574712(5

175

30625

5359375

13 2287566

5.5934447

.0057142SU

176

30976

5451776

13.2664992

5.60407S7

.005681818

177

31329

5545233

13.3041347

5.6146724

.005649718

178

31684

5639752

13 3410041

6.6252263

.005617978

179

32041

5735339

13.3790882

5.6357408

.005586592

180

32400

5&32000

13.4164079

5.6462162

.005.V

181

82761

59297'41

13.4536240

5.6566528

.005524H* tt

182

33124

6028568

13.4907376

5.6670511

.0054!)}:."5

183

33489

6128487

13.5277493

5 6774114

.0054011S1

184

33856

6229504

13.5646600

5.6877340

.006481788

185

34225

6331625

13.6014705

5.69801!»2

.0054054it:>

186

34596

6434856

13.6381817

6.7082675

.005376344

133

TABLE X. -SQUARES, CUBES, SQUARE ROOTS,

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

187

34969

6539203

13.6747943

5.7184791

.005347594

188

35344

6644672

13.7113092

5.7286543

.005319149

189

35721

6751269

13.7477271

5.7387936

.005291005

190

36100

6859000

13.7840488

5.7488971

.005263158

191

36481

6967871

13.8202750

5.7589652

.005235602

192

36864

7077888

13.8564065

5.76891)82

.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

397

38809

7645373

14.0356688

5.8186479

.005076142

198

39204

7762392

14.0712473

5.8284767

.005030505

199

39601

7880599

14.1067360

5.8382725

.005025126

200

40000

8000000

14.1421356

5.8480355

.005000000

201

40401

8120601

14.1774469

5.8577660

.004975124

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

.0048,°,0918

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

9938875

14.6628783

5.9907264

.004651163

216

46656

10077696

14.6969385

6.0000000

.004629630

217

47089

10218313

14.7309199

6.0092450

.004608295

218

47524

10360232

14.7648231

6.0184617

.004587156

219

47961

10503459

14.7986486

6.0276502

.004566210

220

48400

10648000

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

.004484306

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

6.1001702

.004405286

228

51984

11852352

15.0996689

6.1091147

.004385965

229

52441

12008989

15.1327460

6.1180332

.004366812

230

52900

12167000

15.1657509

6.1269257

.004347826

251

53361

12326391

15.1986842

6.1357924

.004329004

232

53824

12487168

15.2315462

6.1446337

.004310345

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

.004184100

240

57600

13824000

15 4919334

6.2144650

.004166667

241

58081

13997521

15.5241747

6.2230843

.004149378

242

58564

14172488

15.5563492

6.2316797

.004132231

243

59049

14348907

15.5884573

6.2402515

.004115226

244

59536

14526784

15.6204994

6.2487998

.004098561

245

60025

14706125

15.6524758

6.2573248

.004081633

246

60516

14886936

15.6843871

6.2658266

.004065041

247

61009

15069223

15.7162336

6.2743054

.004048583

248

61504

15252992

15.7480157

6.2827613

.004032258

134

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

249

62001

15438249

15.7797338

6.2911946

.004010064

250

62500

15625000

15.8113883

6.29960.53

.004000000

251

0300 1

15813251

15.8429795

6.307!

.003lfc$4U04

252

63504

16003008

15.8745079

6.31035%

.008968254

253

64009

16194277

15.9059737

6.3247035

.00.7.):

254

64516

16387064

15.9373775

6.3330256

.003937008

255

65025

16581375

15.9087194

6.3413257

.003921509

256

65536

16777216

16.0000000

6.3490042

.003900250

257

66049

16974593

16.0312195

6.3578011

. 003891 U51

258

66564

17173512

16.0023784

6.3600908

.003875909

259

67081

17373979

16.0934769

6. 3743111

.003801004

260

67600

17576000

16.1245155

6.3825043

.003846154

261

68121

17779581

16.1554944

0.31)00705

.003831418

262

68644

17984723

16.1864141

6.3988279

.003810794

263

69169

18191447

16.2172747

6.4009585

.003802281

264

69696

18399744

16.2480768

6.4150687

.003787879

265

70225 '

18609625

16.2788206

6.4231583

.0037;

266

70756

18821096

16.3095064

6.4312276

.003759398

267

71289

19034163

16.3401346

6.43927'07

.003745318

268

71824

19248332

16.3707055

6.4473057

.003731343

269

72361

19465109

16.4012195

6.4553148

.003717472

270

72900

19683000

16.4316767

6.4033041

.003703704

271

73441

19902511

16.4620776

6.4712736

.003090037

272

7'398t

20123648

16.4924225

6.4792236

.00307-0471

273

74529

20346417

16.5227116

6.4871541

.00366^3004

274

75076

20570824

16.5529454

6.4950653

.003049035

275

75625

20796875

16.5831240

6.5029572

.003031)304

276

76176

21024576

16.6132477

6.5108300

.003623188

277

76729

21253933

16.6433170

6.5186839

.0013010108

278

77284

21484952

16.0733320

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

.003540099

283

80089

22665187

16.8220038

6.5654144

.003533509

284

80656

22906304

16.8522995

6.5731385

.003521127

285

81225

23149125

16.8819430

6.5808443

.003508772

286

81796

23393656

16.9115345

6.5885323

.003496503

287

82369

23639903

16.9410743

6.5902023

.00348 4:{\!1

288

82944

23887872

16.9705027

6,6088545

.003t;

289

83521

24137509

17.0000000

G. 01 14690

.003 100208

290

84100

24389000

17.0293864

6.6191060

.003418270

291

84681

24042171

1 .0587221

6. 02071).-, l

.008486426

292

&5204

24897088

1 .0880075

6.63I2S1

.006424658

293

85849

25153757

1 .1172428

6.6418522

.003412%'.)

294

86433

25412184

1 .1464282

6.6493998

.00340i:;i;i

295

87025

256?'237'5

1 .1755640

6.6509302

.0033S!)>:;i

296

87616

25931336

1 .2046505

6.6644437

1 00337*

297

88209

26198073

1 .2336879

6.6719403

.«>n:::507003

298

8880 1

26463:.!* 3

1 .2620705

6.6794200

.0038!

299

89401

26730899

1 .2910105

C.G808831

.003.-J1

300

90000

27000000

1 .3205081

6 crm-m")

.00888

301

90601

27270901

1 .3493516

6.7017588

.008888859

302

91204

27'543008

1 .3781472

6.7001789

; i -j:,s

303

91809

27818127

1 .4008952

6.7165700

304

92416

2809 41(51

1 .4:355958

6.72395US

s'i; i

305

93025

28372625

1 .4642492

6.7818159

.00327*

306

93636

28652616

1 .4928557

6.7:386641

. 003207 !«;t

307

94249

28934443

1 .5214155

6.7459967

308

94864

29218112

1 .54992SS

0.7533134

46758

309

95481

295031 i--".)

1 .57K5'.r,S

G. 70001 i:5

8246

310

96100

29791000

i .0008109

0.7078995

.003225806

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

No.

Squares.

Cubes.

Square
Kuots.

Cube Roots.

Reciprocals.

311

96721

30080231

17.6351921

6.7751690

.003215434

312

97344

30371328

17.6035217

6.7824229

.003205128

313

97969

30664297

17.6918060

6.7896613

.003194888

314

98596

30959144

17.7200451

6.7968844

.003184713

315

99225

31255875

17.7482393

6.8040921

.003174603

316

99856

31554496

17.7763888

6.8112847

.003164557

317

100489

31855013

17.8044938

6.8184620

.003154574

318

101124

32157432

17.8325545

6.8256242

.003144654

319

101761

32461759

17.8605711

6.8327714

.003134796

320

102400

32768000

17.8885438

6.8399037

.003125000

321

103041

33076161

17.9164729

6.8470213

.003115205

322

103684

33386248

17.9443584

6.8541240

.003105590

323

104329

33698267

17.9722008

6.8612120

.003095975

324

104976

34012224

18.0000000

6.8682855

.003086420

325

105625

34328125

18.0277564

6.8753443

.003076923

320

106276

34645976

18.0554701

6.8823888

.003067485

327

106929

34965783

18.0831413

6.8894188

.003058104

328

107584

35287552

18.1107703

6.8964345

.003048780

o29

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

36594368

18.2208672

6.9243556

.003012048

333

110889

36926037

18.2482876

6.9313008

.003003003

334

111556

37259704

18.2756669

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

6.9863681

.002932551

342

116964 40001688

18.4932420

6.9931906

.002923977

343

117649 | 40353607

18.5202592

7.0000000

.002915452

344

118336

40707584

18.5472370

7.0067962

.002906977

345

119025

41063625

18.57417'56

7 0135791

.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

.002865330

350

122500

42875000

18.7082869

7.04^2987

.002857143

351

123201

43243551

18.7349940

7.0540041

.002849003

352

123904

43614208

18.7616630

7.0600907

.002840909

353

124609

4398G977

18.7882942

7.0073707

.002832861

354

125316

44361864

18.8148877

7.0740440

.002824859

355

126025

44738875

18.8414437

7.0800988

.002816901

356

126736

45118016

18.8679623

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

360

129GOO

46656000

18.9736660

7.1137866

.002777778

361

130321

47045881

19,0000000

7.1203074

.002770083

362

131044

47437928

19.0262976

7.1269360

.002762431

363

131769

47832147

19.0525589

7.1334925

.002754821

364

132496

48228544

19.0787840

7.140037'0

.002747253

365

133225

48627125

19.1049732

7.1465695

.002739726

366

133956

49027896

19.1311265

7.1530901

.0027'32240

367

134689

49430863

19.1572441

7.1595988

.002724796

368

135424

49836032

19.1833261

7.1660957

.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

7.1919663

.002688172

136

CUBE ROOTS, AND RECIPROCALS.

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 1 7.2240450

.002652520

378

142884

54010152

19.4422221 1 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

147'456

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

7.33?2339

.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

17'3056

71991296

20.3960781

7.4650223

.002403846

417

173889

72511713

20.4205779

7.4709991

.(Kt'398082

418

174724

73034632

20.4450483

7.4769664

.002392344

419

175561

73560059

20.4694895

7.4829242

.002386635

420

176400

74088000

20.4939015

7.4888724

.002380052

421

177241

74618461

80.5188845

7.4948113

.002375'J'ir

422

178084

75151448

80.6486888

7.5007406

iMi-j:«59668

423

178929

75686967

80.6669688

7.5066607

.002364066

424

179776

AJ225024

20 591 OT5

7.5125715

.002358491

425

180625

76765625

20.6155281

7.5184730

.002352! HI

426

181476

7730877'6

20.6397674

7.5243652

.002347418

427

182329

77854483

20.6639783

7.5808488

.002341! 120

428

183184

78402752

80.6881609

7.5861881

86448

429

184041

78953589

80.7188169

7.5410667

.002331002

430

184900

79507000

20.7364414

7.5478423

85581

431

185761

800(52991

W). 7605395

7.5586888

,0088$

432

186624

80021568

80.7846097

7.5595863

.01)2:11 IS 15

433

187489

81188787

• 80. 80W r,;>()

7.5658548

09469

434

188356

81746504

80.8886667

7.5711743

.0'fJ304147

137

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

No.

Squares.

Cubes.

Square
lioots.

Cube Roots.

Reciprocals.

435

189225

82312875

20.8566536

7.5769849

.002298851

436

i 190096

82881856

20.8806130

7.5827865

.002293578

437

190969

83453453

20.9045450

7.5885793

.002288330

438

191844

8402<67'2

20.9284495

7.5943633

.002283105

439

192721

84604519

.20.9523268

7.6001385

.002277904

440

193600

85184000

20.9761770

7.6059049

.002272727

441

194481

8576U121

21.0000000

7.6116626

.002267574

442

195364

86350888

21.0237960

7.6174116

.002262443

443

196249

86938307

21.0475652

7.6231519

.002257'336

444

197136

87528384

21.0713075

7.6288837

.002252252

445

198025

88121125

21.0050231

7.6346067

.002247191

446

198916

88716536

21.1187121

7.6403213

.002242152

447

199809

89314623

21.1423745

7.6460272

.002237136

448

200704

89915392

21.1660105

7.6517'247

.002232143

449

201601

90518849

21.1896201

7.6574133

.002227171

450

202500

91125000

21.2132034

7.6630943

.002222222

451

203401

91733851

21.2367606

7.6687665

.002217295

452

204304

923454J8

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

21.4009346

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

7.747'3109

.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

7.7859928

.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 8032111

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

21.9544984

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

488

238144

116214272

22.0907220

7.8729944

.002049180

489

239121

116930169

22.1133444

7 8783684

.002044990

490

240100

117649000

22.1&59436

7.8837352

.002040816

491

241081

118370771

22.1585198

7.8890946

.002036660

492

242064

119095488

22.1810730

7.8944463

.002032520

493

243049

119823157

22.2036033

7.8997917

.002028398

494

244036

120553784

22.2261108

7.9051294

.002024291

495

245025

121287375

22.2485955

7.9104599

.002020202

496

246016

122023936

22.2710575

7.9157832

.002016129

138

CUBE ROOTS, AtfD RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

497

247009

122763473

22.2934968

7.9210994

.002012072

498

248004

123505992

22.3159136

7.9264085

.002008032

499

249001

124251499

22.3383079

7.9317104

.002004008

500

250000

125000000

22.3606798

7.9370053

.002000000

501

251001

125751501

22.3830293

7.9422931

.001996008

502

252004

126506008

22.4053565

7.9475739

.001992032

503

253009

127263527

22.4276615

7.9528477

.001988072

504

254016

128024064

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

131096512

22.5388553

7.9791122

.001968504

509

259081

131872229

22.5610283

7.9843444

.001964637

510

260100

132651000

22.5831796

7.9895697

.001960784

511

261121

133432831

22.6053091

7.9947883

.001956947

512

262144

134217728

22.6274170

8.0000000

.001953125

513

263169

135005697

22.6495033

8.0052049

.001949318

514

264196

135796744

22.6715681

8.0104082

.001945525

515

265225

136590875

22.6936114

8.0155946

.001941748

516

266256

137388096

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

.0019193S6

522

272484

142236648

22.8473193

8.0517479

.001915709

523

273529

143055667

22.8691933

8.0568862

.001912046

524

274576

143877824

22.8910463

8.0620180

.001908397

525

275625

144703125

22.9128785

8.0671432

.001904762

526

276676

145531576

22.9346899

8.0722620

.001901141

527

277729

146363183

22.9564806

8.0773743

.001897533

528

278784

147197952

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

8.0977589

.001883239

532

283024

150568768

23.0651252

8.1028390

.001879699

533

284089

151419437

23.0867928

8.1079128

.001876173

534

285156

152273304

23.1084400

8.1129803

.001872(559

535

286225

153130375

23.1300670

8.1180414

.001869159

536

287296

153990656

23.1516738

8.1230962

.001805(572

537

288369

154854153

23.1732605

8.1281447

.001802197

538

289444

155720872

23.1948270

8.1331870

.001K58736

539

290521

156590819

23.2163735

8.1382230

.001855288

540

291600

157464000

23.2379001

8.1432529

.00185ia52

541

292681

158340421

23 . 2594067

8.1488705

.001848429

542

293764

159220088

23.2808935

8.1532939

.001845018

543

294849

160103007

23.3023604

8.1583051

.001841021

544

295936

160989184

23.3238076

8.1033102

.00183*

545

297025

161878625

23. 3452351

S.10S3092

.001834^2

546

298116

162771336

23.3666429

8.1733020

.00183151),!

547

299209

163667323

23.3880311

8.173

.001898151

548

300304

1045005<>2

23.40'.)399S

8 1S32695

.001924818

549

301401

1U5469149

23.4307490

8.1882441

.001821494

550

302500

166375000

23.4520788

8.1932127

.001S1R182

551

303601

167284151

23.4733892

8.1981758

.80181

552

304704

16819000S

23.49WSO:>

8.2031319

.IM1SH594

553

805809

169112377

23.5159520

B.203

.001808818

554

30001 6

1700314(54

83.5372046

8.2130271

.001805054

555

308025

170953875

23.5584380

8.2179657

.001801802

556

309136

171879616

23.579C.522

8.2228985

.00179*

557

310249

172808(593

23. (5008474

8.2-j;

.00179

558

311364

173741112

23.6220236

8.2327463

.001792115

139

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

559

312481

174676879

23.6431808

8.2376614

001788909

560

313600

175616000

23.6643191

8.2425706

.001785714

561

314721

176558481

23.6854386

8.2474740

.001782531

562

315844

177504328

23.7065392

8.2523715

001779359

563

316969

178453547

23.7'276210

8.2572633

001776199

564

318096

179406144

23.7486842

8.2621492

.001773050

565

319225

180362125

23.7697286

S.267'0294

.001769912

566

320356

181321496

23.7907545

8.2719039

.001766784

567

321489

182284263

23.8117618

8.2767726

001763668

568

322624

183250432

23.8327506

8.2816355

,001760563

569

323761

184220000

&i.8537'209

8.2864928

.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

.001686341

594

352836

20S5S4584

24.3721152

8.4061180

.001683502

595

354025

210644875

24.3926218

8.4108326

001680672

596

355216

211708736

24.4131112

8.4155419

.001677852

597

356409

2127761 7'3

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

8.4348267

.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

.001055629

605

366025

221445125

24.5967478

8.4576906

.001652893

606

367236

222545016

24.6170673

8.4623479

.001650165

607

368449

223648543

24.6373,00

8.4670001

.001047446

608

369664

224755712

24.0576560

8.4716471

.001644737

609

3?'0881

225866529

24.6779254

8.47'62b92

.001642036

610

372100

226981000

24.6981781

8.4809261

.001639344

611

373321

228099131

24.7184142

8.4855579

.001030661

612

374544

229220928

24.7386338

8.4901848

001633987

613

375769

230346397

24.7588368

8.4948065

001631321

614

376996

231475544

24.7790234

8.4994233

.001028664

615

378225

232608375

24.7991935

8.5040350

.001626016

616

379456

233744896

24.8193473

8.5086417

.001623377

617

380689

234885113

24.8394847

8.5132435

.001620746

618

381924

236029032

24.8596058

8.5178403

.001618123

619

383161

237176659

24.8797106

8.5224321

.001615509

620

384400

238328000

24.8997992

8.5270189

.001612903

140

CUBE BOOTS, AKD RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

621
622
623
624
625
626
627
628
629

385641
386884
388129
389376
390625
391876
393129
394384
395641

239483061
240041848
241804307
242970024
244140625
245314376
246491883
247673152
248858189

24.9198710
24.9399278
24.9599079
24.9799920
25.0000000
25.0199920
25.0399081
25.0599282
25.0798724

8.5316009
8.5361780
8.5407501
8.5453173
8.5498797
8.5544372
8.5589899
8.5635377
8! 5680807

.001610300
.001007717
.001005130
.001002564
.001600000
.001597444
.001594896
.001592357
.001589825

630
631
632
633
634
635
636
637
638
639

396900
398101
399424
400089
401956
403225
404496
405709
407044
408321

250047000
251239591
252435968
253636137
254840104
256047875
257259456
258474853
259694072
260917119

25.0998008
25.1197134
25.1390102
25.1594913
25.1793500
25.1992003
25.2190404
25.2388589
25.2580019
25.2784493

8.5726189
8.5771523
8.5816809
8.5862047
8.5907238
8.5952380
8.5997470
8.6042525
8.6087526
8.6132480

.001587302
.001584786
.001582278
.001579779
.001577287
.001574803
.001572327
.001509859
.001567398
.001564945

640
641
642
643
644
645
646
647
648
649

409000
410881
412164
413449
414736
416025
417316
418609
419904
421201

262144000
263374721
264609288
265847707
267089984
268330125
269586136
270840023
272097792
273359449

25.2982213
25.3179778
25.3377189
25.3574447
25.3771551
25.3968502
25.4165301
25.4361947
25.4558441
25.4754784

8.6177388
8.6222248
8.6267063
8.6311830
8.6356551
8.6401220
8.6445855
8.6490437
8.6534974
8.6579465

.001562500
.001560062
.001557632
.001555210
.001552795
.001550388
.001547988
.001545595
.001543210
.001540832

650
651
652
653
654
655
656
657
658
659

422500
423801
425104
426409
427716
429025
430336
431649
432964
434281

274625000
. 275894451
277167808
278445077
279726264
281011375
282300416
283593393
284890312
286191179

25.4950976
25.5147016
25.5342907
25.5538647
25.5734237
25.5929678
25.6124969
25.6320112
25.6515107
25.6709953

8.6623911
8.6668310
8.6712665
8.6756974
8.6801237
8.6845456
8.6889630
8.6933759
8.6977843
8.7021882

.001538462
.001536098
.001533742
.001531394
.001529052
.001526718
.001524390
.001522070
.001519757
.001517451

660
661
662
663
664
665
666

435600
436921
438244
439569
440896
442225
443556

287496000
288804781
290117528
291434247
292754944
294079625
295408296

25.6904652
25.7099203
25.7293607
25.7487864
25.7681975
25.7875939
25.8069758

8.7065877
8.7109827
8.7153734
8.7197596
8.7241414
8.7285187
8.7328918

.001515152
.001512a59
.001510574
.001508296
.001506024
.001503759
.001501502

667
668

444889
446224

296740963
298077632

25.8263431
25.8456960

8.7372604
8.7416246

.001499250
.001497006

669

447561

299418309

25.8650343

8.7459846

.001494768

670
671
672
673
674
675
670

448900
450241
451584
452929
454276
455625
450976

300763000
302111711
303464448
304821217
300182024
307546875
308915776

25.8843582
25.9036677
25.9229028
25.9422435
25.9615100
25.9807621
26.0000000

8.7503401
8.7546913
8.7590383
8.7633809
8.7677192
8.7720532
8.7763830

.001492537
.001490313
.001488095
.001485884
.00148:3080
.001481481
.001479290

677

678

458329
459084

310288733
311665752

26.0192237
26.0384331

8.7807084
8.7850296

.001477105
.001474926

679

401041

313046839

26.0576284

8.7893466

.001472754

680

402400

314432000

26.0708096

8.7936593

.001470588

681

403701

315821241

26.0959767

8.7979679

.001408429

682

405124

317214568

26.1151297

8.8022721

.001400276

141

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

6&3

466489

318611987

26.1342687

8.8065722

.001464129

684

467856

320013504

26.153-3937

8.8108081

.001401988

685

469225

321419125

26.1725047

8.8151598

.001459854

686

470596

322828856

26.1916017

3.8194474

.001457720

687

471969

324242703

26.2106848

8.8237307

.001455004

688

473344

3251360672

26.2297541

8.8280099

.001453488

689

474721

3270827(59

26.2488095

8.8322850

.001451379

690

476100

328509000

26.2678511

8.8305559

.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

484416

337153536

26.3818119

8.8020952

.001430782

697

485809

338608873

26.4007576

8.8003375

.001434720

698

487204

340068392

26.4196896

8.8705757

.001432065

699

488601

341532099

26.4386081

8.8748099

.001430015

700

490000

343000000

26.4575131

8.8790400

.001428571

701

491401

344472101

26.4764046

8.8832001

.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

8.9043366

.001416431

707

499849

353393243

26.5894716

8.9085387

.001414427

708

501264

354894912

20.6082094

8.9127309

.001412429

709

502081

356400829

20 6270539

8.9109311

.001410437

710

504100

a57911000

26.6458252

8.9211214

.001408451

711

505521

359425431

20.0045833

8.9253078

.001400470

712

506944

360944128

26.6833281

8.9294902

.001404494

713

508369

362467097

26.7020598

8.9330087

.001402525

714

509796

363994344

26.7207784

8.9378433

.001400500

715

511225

365525875

20.7394839

8.9420140

.001398001

716

512656

367061696

20.7581703

8.9401809

.001390648

717

514089

368601813

20.7768557

8.9503438

001394700

718

515524

370146232

26.7955220

8.9545029

.001392758

719

516961

371694959

26.8141754

8.9580581

.001390821

720

518400

373248000

26.8328157

8.9628095

.001388889

721

519841

374805361

20.8514432

8.9009570

001386963

722

521284

376367048

20.8700577

8.9711007

.001385042

723

522729

377933067

20.8886593

8.9752400

.001383126

724

524176

379503424

20.9072481

8.9793706

.001381215

725

525625

381078125

20.9258240

8.9835089

.001379310

726

527076

382657176

26.9443872

8.9876373

.001377410

727

528529

384240583

26.9029375

8.9917020

.001375516

728

529984

385828352

20.9814751

8.9958829

.001373026

729

531441

387420489

27.0000000

9.0000000

.001371742

730

532900

389017000

27.0185122

9.0041134

.001369803

731

534361

390617'891

27.0370117

9.0082229

.001307989

732

535824

392223168

27.0554985

9.0123288

.001300120

733

537289

393832837

27.0739727

9.0104309

.001304256

734

538756

395446904

27.0924344

9.0205293

.001302398

735

540225

397065375

27.1108834

9.0240239

.001300544

736

541696

398688256

27.1293199

9.0287149

.001358090

737

543169

400315553

27.1477439

9.0328021

.001350852

738

544644

401947272

27.1001554

9.0308857

.001355014

739

546121

403583419

27.1845544

9.0409055

.001353180

740

547COO

405224000

27.2029410

9.0450419

.001351351

741

549081

406869021

27.2213152

9.0491142

.001349528

742

550564

408518488

27.2390709

9.0531831

.001347709

. 743

552049

410172407

27.2580203

9.0572482

.001345895

744

553536

411830784

27.2703034

9.0013098

.001344086

\42

CUBE ROOTS, AND RECIPROCALS.

fNo.

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

.001336898

749

561001

420189749

27.3678644

9.0815631

.001335113

750

562500

421875000

27.3861279

9.0856030

.001333333

751

564001

423564751

27.4043792

9.0890392

.Oi' 1331558

752

565504

425259008

27.4226184

9.0936719

.001329787

753

567009

426957777

27.4408455

9.0977010

.001328021

754

568516

428661064

27.4590604

9.1017205

.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

.001307190

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

27.76688C8

9.1696225

.001297017

772

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

9.2090902

.001280410

782

611524

478211768

27.9642629

9.2130250

.001278772

783

613089

480048687

27.982137'2

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

.00127'2265

787

619369

4874434CS

28.0535203

9.2326189

.00127'0648

788

620944

480906879

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

9.25213DO

.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

512000000

28.2842712

9.2831777

.001250000

801

641601

513922401

28.3019434

9.2870440

.001248439

802

643204

515849(508

28.3196045

9.2909072

.001246883

803

644809

517781627

28.3372546

9.2947671

.001245330

804

046416

519718464 1 28.3548938

9.2986239

.001243781

805

648025

521660125 i 28.3725219

9.3024775

.001242236

806

649636

523606616 28.3901391

9.3063278

.001240695

143

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

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

9.3293634

.001231527

813

660969

537367797

28.5131549

9.3331916

.001230012

814

662596

539353144

28.5306852

9.3370167

.001228501

815

664225

541343375

28.5482048

9.3408386

.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

9.3560952

.001221001

820

672400

551368000

28.6356421

9.3599016

.001219512

821

674041

553387661

28.6530976

9.3637049

.001218027

822

675684

555412248

28.6705424

9.3675051

.001*216545

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

9.5133699

.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
r

753424

653972032

29.4618397

9.5390818

,001152074

144

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

8G9

755161

656234909

29.4788059

9.5427437 .001150748

870

756900

658503000

29.4957624

9.5464027

.001149425

871

758641

660776311

29.5127091

9.5500589

.001148106

872

760384

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

.001135074

882

777924

686128968

29.6984848

9.5900939

.001133787

883

779689

688465387

29.7153159

9.5937169

.001132503

884

781456

690807104

29.7321375

9.5973373

.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

7097'32288

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

7'33870808

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

9.6&34166

.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

30.3315018

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

9.7899684

.001082251

925

855625

791453125

30.4138127

9.7434758

.001081081

926

857476

794022776

30.4302481

9.7469857

.001079914

927

859329

796597983

30.4466747

9.7504930

.001078749

928

861184

79! >1 78752

30.4630924

9.7539979

.001077586

929

863041

801765089

30.4795013 i 9.7575002

.001076426

930

864900

804357000

30.4959014

9.7610001 .001075269

145

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

931

866761

806954491

30.5122926

9.7644974

.001074114

932

868624

809557568

30.5286750

9.7679922

.00107'2961

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

.00100tt)98

839

881721

827936019

30.6431069

9.7923861

.001064963

940

883600

830584000

30.6594194

9.7958611

.001063830

941

885481

833237621

30.6757233

9.7993336

.001062699

942

887364

835896888

30.6920185

9.8028036

.001061571

943

889249

838561807

80.7083051

9.8062711

.001060445

944

891136

841232384

cO. 7245830

9.8097'362

.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

80.8382879

9.8339238

.001051525

952

906304

862801408

80.8544972

9.8373695

.001050420

953

908209

865523177

30.8706981

9.8408127

.001049318

954

910116

868250664

30.8868904

9.8442536

.001048218

955

912025

87098387'5

30.9030743

9.8476920

.001047120

956

913936

873722816

80.9192497

9.8511280

.001040025

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

81.0322413

9.8751135

.001038422

9&4

929296

895841344

31.0483494

9.87'85305

.001037344

965

931225

896682125

31.0644491

9.8819451

.001036269

966

933156

901428000

31.0805405

9.8853574

.001035197

967

935089

904231063

31.0966236

9.8887673

.001034126

968

937024

907039232

81.1126984

9.8921749

.001033058

969

938961

909853209

31.1287648

9.8955801

.001031992

970

940900

912673000

31.1448230

6. 8989830

.001030928

971

942841

915498611

31.1608729

9.9028835

.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

946966168

31.3368792

9.9396363

.001018330

983

966289

949862087

31.3528308

9.9430092

.001017294

984

968256

952763904

31.3687743

9.9463797

.001016260

985

970225

956671625

31.3847097

9.9497479

.001015228

986

972196

958585256

31.4006369

9.9531138

.001014199

987

974169

901504803 31.4165561

9.9564775

.001013171

988

976144

964430272 : 31.4324673

9.9598389

.001012146

989

978121

967361669 31.4483704

9.9631981

.001011122

990

980100

97'OS99000 31.4642654

9.9665549

.001010101

991

982081

073242271 31.4801525

9.9699095

.001009082

992

984064

9?'6191488 31.4960315

9.9732619

.001008065

146

CUBE ROOTS, AtfD RECIPROCALS.

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

991036973.

31.5753068

9.9899900

.001003009

998

996004

994011992

31.5911380

9.9933289

.001002004

999

998001

9970029SJ9 ! 31 .6069613

9.9966656

.001001001

1000

1000000

1000000000

31.6227766

10.0000000

.001000000

1001

1002001

1003003001

31.6385840

10.00&3322

.0009990010

1002

1004004

1006012008

31.6543836

10.0066622

.0009980040

1003

1006009

' 1009027027

31.6701752

10.0099899

.0009970090

loot

1008016

1012J48064

31.6859590

10.0133155

.0009960159

1003

1010025

1015075125 31.7017349

10.0166389

.0009950249

1006

1012036

1018108216 31.7175030

10.0199601

.0009940358

ioor

1014049

1021147343 31.7332633

10.0232791

.0009930487

1003

1016064

1024192512 j 31.7490157

10.0265958

.0009920635

1009

1018081

1027243729 31.7647603

10.0299104

.0009910803

1010

1020100

1030301COJ i 31.7804972

10.0332228

.0009900990

1011

1022121

1033364331 31.7962262

10.0365330

.0009891197

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

.0009832842

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

31.9530906

10.0695156

.0009794319

1022

1044484

1067462648

31.9637317

10.0728020

.0009784736

1023

1046529

1070599167

31.9843712

10.0760863

.0009775171

1021

1048576

1073741824

32.1000000

10.0793684

.0009765625

1025

1050625

1076890625

32.0156212

10.0826484

.0009756098

10215

1052676

1080045576

32.0312348

10.0859262

.0009746589

1027

1054729

1083206683

32.0468407

10.0892019

.0009737098

1028

1056784

1036373952

32.0624391

10.0924755

.0009727626

1029

1058841

1039547339

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

1103717875

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

.000963:3911

1039

1079521

1121622319

32.2335229

10.1283457

.0009624639

1040

1081600

1124864000

32.2490310

10.1315941

.0009615385

1041

10a3681

1128111921

32.2645316

10.1348403

.0009606148

1042

1085764

1131366088

32.2800248

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

32.3728981

10.1575062

.0009541985

1049

1100401

1154320649

32.3882<>95

10.1607359

.0009532888

1050

1102500

1157625000

32.4037035

10.1639636

.0009523810

1051

1104601

1160966651

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

147

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 100 L. 000.]

_Xo. 109 L. 040.

N.

0

1

284

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

428

2

8600

9026

9151 9876

0300

0724

1147

1570

1993

2415

AOA

3

012a37

3259

3680 ! 4100 4521

4940

5360

5779

6197

6616

420

4

4

7033

7451

7868 8284 8700

9116

9532

9947

0361

0775

416

i 5

021189

1603

2016 2428 2841

3252

3664

4075

4486

4896

412

6

5306

5715

6125 6533 6942

7350

7757

8164

8571

8978

408

7

9384

9789

1

0195 0600 1004

1408

1812

2216

261S

3021

404

8 '

033424

3826

4227 4628 5029

5430

5830

6230

6629

7028

400

0

7426

7825

8223 8620 9017

9414

9811

04

0207

0602

0998

397

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

7

8

9

434

43.4

86.8

130.2

173.6

217.0

260.4

3C

)3.8

347.2

390.6

433

43.3

86.6

129.9

173.2

216.5

259.8

31

18.1

346.4

389.7

432

43.2

86.4

129.6

172.8

216.0

259

2

ac

>2.4

345.6

388.8

431

43.1

86.2

129.3

172.4

215.5

258

6

301.7

344.8

387.9

430

i 43.0

86.0

129.0

172.0

215.0

258

0

3(

)1.0

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

42.7

85.4

128.1

170.8

213.5

256

2

2J

)8.9

341.6

384.3

426

i 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

424

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

21

\Q 1

338.4

380.7

422

i 42.2

84.4

126.6

168.8

211.0

253

2

2<

»*.4

337.6

379.8

421

! 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

i 41.9

83.8

125.7

167.6

209.5

251

4

2(

J3.3

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

as. 4

125.1

166.8

208.5

250

2

2

)1.9

333.6

375.3

416

41.6

83.2

124.8

166.4

208.0

249

6

291.2

332.8

374.4

415

41.5

as.o

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

2

39.1

330.4

371.7

412

41.2

82.4

123.6

164.8

206.0

247

2

2

38.4

329.6

370.8

411

41.1

82.2

123.3

164.4

205.5

246

6

21

37.7

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

a

36.3

327.2

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

21

34.2

324.8 i 365.4

405

40.5 81.0

121.5

162.0

202.5

243.0

283.5

324.0 , 364.5

404

! 4Q.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

21

31.4

321.6 361.8

401

40.1

80. g

120.3

160.4

200.5

240

.6

21

30.7.

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. £

1

119.7

159.6

199.5

239

.4

21

j*9 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

o

r7.9

317.6 357.3

396

39.6

79. i

i

118.8

158.4

198.0

237

.6

2

rr.2

316.8 1 356.4

395

39.5

79.0 118.5

158.0

197.5 237

.0

276.5 316.0 1 355.5

149

TABLE XI. — 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

3362

3755

4148

4540

4932

393

1

5323

5714

6105

6495 6885

7275

7664

8053

8442

8830

390

2

9218

9606

9993

0380 0766

1153

1538

1924

2309

2694

386

3

053078

3463

3846

4230 4613

4996

5378

5760

6142

6524

383

4

6905

7286

7666

8046 8426

8805

9185

9563

9942

0320

379

5

060698

1075

1452

1829 2206

2582

2958

3333

37'09

4083

376

6

4458

4832

5206

5580 5953

6326

6699

7071

7443

7815

373

7

8186

8557

8928

9298 9668

0038

0407

f)7T<

1145

1514

370

8

071882

2250

2617

2985 3352

3718

4085

\Jt i O

4451

4816

5182

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
394

39.5
39.4

79.0

78.8

118.5
118.2

158.0
157.6

197.5
197.0

237
236

.0
.4

276.5
275.8

316.0
315.2

355.5
354.6

393

39.3

78.6

11

7.9

157.2

196.5

235

.8

2

75.1

314.4

353.7

392

39.2

78.4

11

7.6

156.8

196.0

235

.2

" 274.4

313.6

352.8

391

39.1

78.2

11

7.3

156.4

195.5

234

.6

2

73.7

312.8

351.9

390

39.0

78.0

11

7.0

156.0

195.0

234

.0

2

73.0

312.0

351.0

389

38.9

77.8

116.7

155.6

194.5

233.4

272.3

311.2

350.1

388

38.8

77.6

11

6.4

155.2

194.0

232

.8

2

71.6

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

11

5.8

154.4

193.0

231

.6

2

70.2

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

76.8

115.2

153.6

192.0

230.4

268.8

307.2

345.6

383

38.3

76.6

11

4.9

153.2

191.5

229

.8

2

68.1

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

11

4.3

152.4

190.5

228

.6

2

66.7

304.8

342.9

38C

38.0

76. C

11

4.0

152.0

190.0

228

.0

2

66.0

304.0

342.0

379

37.9

75.8

113.7

151.6

189.5

227.4

265.3

303.2

341.1

378

37.8

75.6

11

3.4

151.2

189.0

226

.8

2

64.6

302.4

340.2

377

37.7

75.4

11

3.1

150.8

188.5

226

.2

2

63.9

301. e

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. e

111.9

149.2

186.5

223.8

261.1

298.4

335.7

37$

1

37.2

74.4

1

11

1.6

148.8

186.0

223

.2

2

60.4

297.6

334.8

371

37.1

74.2

111.3

148.4

185.5

222.6

259.7

296.8

333 9

37C

)

37.0

74. (

)

11

1.0

148.0

185.0

222

.0

2

59.0

296.0

333^0

36<

)

36.9

73. *

i

11

0.7

147.6

184.5

221

.4

2

58.3

295.2

332.1

36£

;

36.8

73.6

110.4

147.2

184.0

220.8

257.6

294.4

331.2

S6r

36.7

73.4

[

11

0.1

146.8

183.5

22C

.2

2

56.9

293.6

330.3

36(

5

36.6

73.$

\

1(

)9.8

146.4

183.0

21S

.6

2

56.2

292.8

329.4

S65

36.5

73.0

109.5

146.0

182.5

219.0

255.7

292.0

328.5

364

36.4

72.?

\

109.2

145.6

182.0

218.4

254.8

291.2

327.6

1

36.3

72. e

5

1(

)8.9

145.2

181.5

217

.8

2

54.1

290.4

326.7

36$

!

36.2

72.4

(

1(

)8.6

• 144.8

181.0

217

.2

2

53.4

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

35<

)

35.9

71.1

]

1(

)7.7

143.6

179.5

215

.4

2

51.3

287.2

323.1

3&

J

35.8

71. (

>

1(

)7.4

143.2

179.0

214

.8

2

50.6

286.4

322.2

357

35.7

71.4

107.1

142.8

178.5

214

.2

249.9

285.6

321.3

85

>

35.6

71.2

106.8

142.4

178.0

213.6

249.2

284.8

320.4

150

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 120 L. 079.]

[No. 134 L. 130.

N.

0

1

2

3

4 | 5

6

7

S

9

Diff.

I

120

079181 y&43

9904

0266

0626

0987

1347

1707

2007

2426 j 300

1

082785

3144

3503

3861

4219

4576

4934

5291

5647

6004

357

2
g

6360
9905

6716

7071

7426

7781

8136

8490

8845

9198

9552

355

0258

0611

0963

1315

1667

2018

2370

2721

3071

352

4

093422

3772

4122

4471

4820

5169

5518

5866

6215

6562

349

5

6910

7257

7004

7951

8298 8644

8990

9335

9681

j

0026

346

6

100371

0715 1059

1403

1747

2091

2434

2777

3119

3402

343

7

3804

4146

4487

4828

5169

5510

5851

6191

6531

6871

341

8 7210

7549

7888

8227

8565 8903 9241

9579

9916

0253

338

9 110590

0926

1363

1599

1934 2270

2005

2940

3275

3609

335

130 3943

4277

4611

4944

5278

5611

5943

6276

6608

6940

333

I 79171

7603

7934

8265

8595

8926

9.250

9586

9915

0245

330

2 120574

0903

1231

1560

1888

2216

2544

2871

3198

3525

328

3

3852

4178

4504

4830

5156

5481

5806

6131

6456

6781

325

4

7105

7429

7753

8076

8399

8722

9045

9368

9090

13

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

1

41.2

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

1

40.4

175.5

210.

0

245.7

280.8

315.9

350

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

1

39.6

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

1

38.8

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

1

37.6

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

1

36.8

171.0

205

239.4

273.6

307.8

341

34.1

68.2

102

.3

1

36.4

170.5

204.

5

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

1

35.6

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

1

34.8

168.5

202.

235.9

269.6

303.3

336

33.6

67.2

100.8

134.4

168.0

201.

5

235.2

268.8

302.4

335

83.5

67.0

100

.5

134.0

167.5

201.0

234.5

268.0

301.5

334

as. 4

66.8

100

.2

1

33.6

167.0

200.

4

233.8

267.2

300.6

333

as.3

66.6

99

.9

ias.2

166.5

199.

3

233.1

266.4

299.7

332

33.2

66.4

99

.6

i

32.8

166.0

199.

2

232.4

265.6

298.8

331

as.i

66.2

99.3

132.4

165.5

198.

3

231.7

264.8

297.9

aso

33.0

66.0

99

.0

]

32.0

165.0

198.

}

231.0

264.0

297.0

329

32.9

65.8

98

.7

31.6

164.5

197. <

t

230.3

263.2

296.1

328

32.8

65.6

98

.4

131.2

164.0

196.

3

229.6

262.4

295.2

327

32.7

65.4

98

.1

1

30.8

103.5 196.

2

228.9

201.6

294.3

326

32.6

05.2

97

.8

130.4

103.0 195.6

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

1

29.2

161.5 193. J

J

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

151

TABLE XI. — LOGARITHMS OF KtlMBERS.

No. 135 L. 130.]

[No. 149 L. 175.

N.

0

1

2

3

4

5

6

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

7
g

6721

9879

7037

7354

7671

7987

8303

8618

8934

9249 9564

316

0194

0508

0822

1136

1450

1763

2076

2389 2702

314

9

143015

3327

3639

3951

4263

4574

4885

5196

5507 5818

311

140

6128

6438

6748

7058

7367

7676

7985

8294

8603 8911

309

9219

9527

9835

0142

0449

0756

1063

1370

1676 1982

307

2

152288

2594

2900

3205

3510

3815

4120

4424

4728 5032

305

3

4

5336

8362

5640
8664

5943
8965

6246
9266

6549
9567

6852
9868

7154

7457

7759 8001

303

0168

04G9

0769 1068

301

5

161368

1667

1967

2266

2564

2863

3161

3460

3758 4055

299

6

4353

4650

4947

5244

5541

5838

6134

6430

6726 7022

297

7

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

3478

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

2£

3.3

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

2S

1.9

253.6

285.3

316

31.6

63.2

94

.8

126.4

158.0

189

6

2$

1.2

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

21

9.8

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

21

7.0

248.0

279.0

309

30.9

61.8

92.7

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

21

5.6

246.4

277.2

307

30.7

61.4

92

.1

L£2 8

153.5

184

0

214.9

245.6

276.3

306

30.6

61.2

91

.8

122 .'4

153.0

183

6

21

4.2

244.8

275.4

305

30.5

61.0

91

.5

122.0

152.5

183

0

21

3.5

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

21

2.1

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

2C

)9.3

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

2(

)7.9

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

2(

)6.5

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

2C

)5.1

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

2(

)3.7

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

57.8

86

.7

115.6

144.5

173

4

2C

2.3

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

2C

>0.9

229.6

258.3

286

28.6

57.2

85

.8

114.4

143.0

171

6

200.2

228.8

257.4

152

TABLE XI. — 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

8977

9264

9552

9839

0126

0413

0699

0986

19^9

tf&R

007

2

181844

2129

2415

2700

2985

3270

3555

3839

4123

4407

mot

285

3

4691

4975

5259

5542

5825

6108

6391

6674

6956 7239

283

7521

7803

8084

8366

8647

8928

9209

9490

9771

AAK-J

931

5

190332

0812

0892

1171

1451

1730

2010

2289

— — — UUO1

2567 2846

SBJL

279

6

3125

3403

3681

3959

4237

4514

4792

5069

5346 5623

378

7

5900

6176

6453 6729

7005

7281

7'556

7832

8107 8382

276

g

8657

8932

9206

9481

9755

0029

0303

0577

0850 1124

97 A.

9

201397

1670

1943

2216

2488

2761

3033

3305

3577 3848

iCn

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

2

9515

9783

0051

0319

0586

0853

1121 | 1388

1654 ' 1QO1

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

4015

4274

4533

4792

5051

259

8

5309

5568

5826

6084

6342

6600

6858

7115

7372

7630

258

9

7887

8144

8400

8657

8913

9170

9426

9682

9938

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. g

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. g

79.2

105.6

132.0

158.4

184.8

211.2

237.6

263

26.3

52. €

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

61.1

77.7

103.6

129.5

155.4

181.3

207.2

233.1

258

25.8

51. e

>

77.4

103.2

129.0

154.8

180.6

206.4

232.2

257

25.7

51.4

102.8

128.5

154.2

179.9

205.6

231.3

256

25.6

51. S

5

76 '.8

102.4

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

153

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 170 L. 230.] [No. 189 L. 278.

N.

0

1

2

8

4

6

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

3

8046

8297

8548

8799

9049

9299

9550

9800

0050

0300

OKA

4

240549

0799

1048

1297

1546

1795

2044

2293

2541

2790

/<iOU

249

5

3038

3286

35:34

3782

4030

4277

4525

4772

5019

5266

248

6

5513

5759

6006

6252

6499

6745

6991

7237

7482

7728

246

7

7973

8219

8464

8709

8954

9198

9443

9687

9932

0176

245

8

250420

0664

0908

1151

1395

1638

1881

2125

2368

2610

243

9

2853

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

g

9513

9746

9980

0213

0446

0679

0912

1144

1377

1609

oqq

7

271842

2074

2306

2538

2770

3001

3233

3464

3696

3927

600

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
254

25.5

25.4

51.0

50.8

76.5
76.2

102.0
101.6

127.5
127.0

153.0
152.4

178.5

177.8

204.0
203.2

229.5

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

144.6

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

154

TABLE XI. — LOGAEITHMS 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

I Ki li'

ftlOQ

rvoe-j

nK»"Q

OOQ

1

281033

1261

1488

1715

1942

2169

&m

a ,-j-j

2849

3075

227

2

3301

3527

3753

3979

4205

4431

4056

4882

5107

5332

226

3

5557

5782

6007

6232

6456

6681

6905

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

20:34

222

6

2256

2478

2699

2920

3141

3363

3584

3804

4025

4246

821

7

44G6

4687

4907

5127

5347

5567

5787

6007

6226

6446

220

8

6665

68&4

7104

7323

7542

7761

7979

8198

8416

8635

219

g

8853

9071

9289

9507

9725

9943

0161

0378

0595

0813

218

200

301030

1247

1464

1681

1898

2114

2331

2547

2764

2980

217

1

3196

*3412

36.<28

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

877'8

8991

9204

9417

213

4

9630

9843

0056

0268

0481

0693

0906

1118

1330

1542

212

5

311754

1966

2177

2389

2600

2812

3023

3234

3415

3656

211

6

3867

4078

4289

4499

4710

! 4920

5130

5340

5551

5760

210

7

5970

6180

6390

6599

6809

7018

72:37

7436

7646

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

2633

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

3

8380

8583

8787

8991

9194

9398

9601

9805

0008

0211

203

4

330414

0617

0819

1022 I 1225

1427

1630

1&32

2034

2236

202

PROPORTIONAL PARTS.

Biff.

1

2

3

4

5

6

7

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

1,33.2

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

198.fi

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

180.9

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

&3.2

104.0

124.8

145.6

166 4

187 J

207

20.7

41.4

62.1

82.8

103.5

124.2

144.9

165.6

186.3

206
205
204
203
202

20.6
20.5
20.4
20.3
20.2

41.2
44.0
40.8
40.6
40.4

61.8
C1.5
61.2
60.9
60.6

82.4
82.0
81.6
81.2
'X).8

103.0
102.5
102.0
101.5
101.0

123.6
123.0
122.4
121.8
121.2

144.2
143.5
142.8
142.1
141.4

164.8
164.0
163.2
162.4
161.6

185.4
184.5

183.6
182.7
181.8

155

TABLE XI. — LOGARITHMS OF NUMBERS.

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

3649

3850

4051

4253

202

6

4454

4655

4856

5057

5257

5458

5658

5859

6059

6260

201

7

6460

6660

6860

7060

7260

7459

7659

7858

8058

8257

200

g

8456

8656

8855

9054

9253

9451

9650

9849

0047

0246

199

9

340444

0642

0841

1039

1237

1435

1632

1830

2028

2225

198

220

2423

2620

2817

3014

3212

3409

3606

3802

3999

4196

197

1

4392

4589

4785

4981

5178

5374

5570

5766

5962

6157

196

2

6353

6549

6?'44

6939

7135

7330

7525

7720

7915

8110

195

g

8305

8500

8694

8889

9083

9278

9472

9666

9860

- 0054

194

4

350248

0442

0636

0829

1023

1216

1410

1603

1796

1989

193

5

2183

2375

2568

2761

2954

3147

3339

&532

3724

3916

193

6

4108

4301

4493

4685

4876

5068

5260

5452

5643

5834

192

7

6026

6217

6408

6599

6790

6981

7172

7363

7554

7744

- 191

8

7935
9835

8125

8316

8506

8696

8886

9076

9266

9456

9646

190

0025

0215

0404

0593

0783

097°

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

977°

9958

0143

0328

0513

0698

0883

18*4

5

371068

1253

1437

1622

1806

1991

2175

2360

2544

2728

184

6

2912

3096

3280

3464

3647

3831

4015

4198

43H2

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

g

8398

8580

8761

8943

9124

9306

9487

9668

9849

38

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

724 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

162.0

179

17.9

35.8

53.7

71.6 89.5

107.4

125.3

143.2

161.1

156

TABLE XI. — LOOAKITHMS OF Xt'MBERS.

No. 240 L. 380.]

INo. 269 L. 431.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

240

380211

0392

0573

0754

0934

1115

1296

1476

1656

1837

181

1

2017

2197

2377

25.

)7

2737

2917

3097

3

277

345

6

3636

180

2

3815

3995

4174

43,

33

4533

4712

4891

5

070

524

>

5428

179

3

5606

5785

5964

6142

6321

6499

6677

6856

7034

7212

178

4

7390

7568

7746

7$

U

8101

8279

8456

8

634

881

1

8989

178

5

9166

9343

9520

90(

)8

9875

0051

ft9OQ

f\

4(\K

-...

6

390935

1112

1288

1464

1641

1817

1993

(J-HJO

2169

058«
2345

() 1 90

XJ.VJ1

17 i

176

7

2697

2873

3048

32S

U

3400

3575

3751

3

926

410

1

4277

176

8

4452

4027

4802

49

"7

5152

5326

5501

5

676

585

0

0025

175

9

6199

6374

6548

6722

6896

7071

7245

7

419

7592

7766

174

250

7940

8114

8287

8461

8634

8808

8981

9154

9328

9501

173

1

9674

9847

0020

01

V}

0365

_„_„

ft

ooo

oi>»i

•*T>

2

401401

1573

1745

1917

2089

2261

2433

Uooo
2605

105o
2777

2949

178

172

3

3121

3292

3464

3635

.3807

3978

4149

4320

4492

4663

171

4

4834

5005

5176

5346

5517

5688

5858

6029

6199

6370

171

5

6540

6710

6881

70,

U

7221

7391

7561

731

790

1

8070

170

6

8240

8410

8579

8749

8918

9087

9257

9426

9595

9764

109

7

9933

0102

0271

0440

0609

0777

0946

1114

1283

1451

169

8

411620

1788

1956

2124

2293

2461

2629

2796

2964

3132

168

9

3300

3467

3635

3803

3970

4137

4305

4472

4639

4806

167

260

4973

5140

5307

5474

5641

5808

5974

0141

6308

6474

167

1

6641

6807

6973

71'

*9

7306

7472

7638

804

797

0

8135

166

2

8301

8467

8633

8798

8964

9129

9295

9460

9625

9791

165

3

9956

0121

0286

04.

^1

0616

0781

0945

•i

i1A

197

K

14VJ

165

4

421604

1768

1933

2097

2261

2426

2590

iLL\J

27.54

l*lu

2918

3082

164

5

3246

3410

3574

3737

3901

4065

4228

4392

4555

4718

164

6

4882

5045

5208

53

"1

5534

5697

5860

6

023

618

0

6349

163

7

6511

6674

6836

69<

)9

7161

! 7324

7486

7

648

781

1

7973

162

8
g

8135
9752

8297
9914

8459

8621

8783

| 8944

9106

9268

9429

9591

162

43

0075 0236 0398

! 0559

0720

0881

1042

1203

161

PROPORTIONAL PARTS.

Diff

1

2

3

4

5

6

7

8

9

178

17.8

35.6

53.4

71.2

89.0

106.8

124.6

142.4

160.2

177

17.7

35.4

53.1

70.8

88.5

106.2

123.9

141.6

159.3

176

17.6

35.2

52.8

70.4

88.0

105.6

123.2

140.8

158 I

175

17.5

35.0

52.5

70.0

87.5

105.0

122.5

140.0

174

17.4

34.8

52.2

69.6

87.0

104.4

121.8

139.2

156.6

173

17.3

34.6

51.9

69.2

86.5

103.8

121.1

138.4

155.7

172

17.2

34.4

51.6

68.8

86.0

103.2

1.20.4

137.6

154.8

171

17.1

34.2

51.3

68.4

85.5

102.6

119.7

136.8

153.9

170

17.0

34.0

51.0

68.0

85.0

102.0

119.0

136.0

153.0

169

16.9

33.8

50.7

67.6

84.5

101.4

118.3

13.-). 2

152.1

168

16.8

33.6

50.4

67.2

84.0

100.8

117.6

134.4

151.2

167

16.7

as. 4

50.1

66.8

83.5

100.2

116.9

133.6

150.3

166

16.6

33.2

49.8

66.4

83.0

99.6

116.2

132.8

149.4

165

16.5

33.0

49.5

66.0

82.5

99.0

115.5

132.0

148.5

164

16.4

32.8

49.2

65.6

82.0

98.4

114.8

131.2

147.6

163

16.3

32.6

48.9

65.2

81.5

97.8

114.1

130.4

no. 7

162

16.2

32.4

48.5

64.8

81.0

97.2

113.4

129.6

161

16.1

32.2

48.3

64.4

80.5

96.6

112.7

128.8

n » . (j

157

TABLE XI. — 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

&

m

2649

2809

161

1

2969

3130

8290

3450

3610

3770

3930

4(

)90

4249

4409

160

2

4569

4729

4888

5048

5207

5367

5526

5685

5844

6004

159

3

6163

6322

6481

6640

6799

6957

7116

7'

275

7433

7592

159

4

7751

7909

8067

8226

8384

8542 8701

8

359

9017

9175

158

5

9333

9491

9648

9806

9964

0122

0279

o^

137

0594

0752

158

6

440909

1066

1224

1381

1538

1695

1852

2009

2166

2323

157

7

2480

2637

2793

2950

3106

3263

3419

3,

376

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

9015

9170

9324

9478

9633

9787

9941

0095

154

2

450249

0403

0557

0711

0865

1018

1172

1326

1479

1633

154

3

1786

1940

2093

2247

2400

2553

2706

2i

359

3012

3165

153

4

3318

3471

3624

3777

3930

4082

4235

4387

4540

4692

153

5

4845

4997

5150

5302

5454

5606

5758

5

310

6062 , 6214

152

6

6366

6518

6670

6821

6973

7125

727'6

7428

7579

7731

152

7
g

7882
9392

8033
9543

8184
9694

8336
9845

8487
9995

8638

8789

8940

9091

9242

151

0146

0296

0%

i/17

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

4340

4490

4639

4788

4

J36

5085

5234

149

2

5383

5532

5680

5829

5977

6126

6274

6423

6571

6719

149

3

6868

7016

7164

7312

7460

7608

7756

7

J04

8052

8200

148

4

8347

8495

8643

8790

8938

9085 9233

9380

9527

9675

148

5

9822

9969

0116

0263

0410

0557

0704

0

asi

0998

1145

147

6

471292

1438

1585

1732

1878

2025

2171

2318

2464

2610

146

7

2756

2903

3049

3195

3341

3487

3633

3779

3925

4071

146

8

4216

4362

4508

4653

4799

4944

5090

5

235

5381

5526

146

9

5671

5816

5962

6107

6252

6397

6542

6687

6832

6976

145

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

7

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

138.6

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

84.0

98.0

112.0

126.0

158

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 300 L. 477.]

[No. 339 L. 531.

N.

0

1

2

3

4

5

6

7

8

9

Diflf.

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

300

1

2
3
4
5
6

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
7
8

9

477121
8566

7266
8711

7411

8855

7555
8999

7700
9143

7844
9287

7989
9431

8133
9575

8278
9719

8422
9863

480007
1443
2874
4300
5721
7138
8551
9958

0151
1586
3016
4442

5863

7280
8692

0294
1729
3159
4585
6005
7421
8833

0438
1872
3302
4727
6147
7563
8974

0582
2016
3445
4869
6289
7704
9114

0725
2159
3587
5011
6430
7845
9255

0869
2:302
3730
5153
6572
7986
9396

1012
2445
3872
5295
6714
8127
9537

1156
2588
4015
5437
C855
8269
9077

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

0661

2062
3458
4850
6238
7621
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
0791
8173
9550

491362
2760
4155
5544
6930
8311
9687

0099
1470
2837
4199

5557
6911
8260
9606

0236
1607
2973
4:335

5693
7046
8395
9740

0374
1744
3109
4471

5828
7181
8530
9874

0511
1880
3246
4607

5964
7316
8664

0648
2017-
3382
4743

6099
7451
8799

0785
2154
3518
4878

6234
7586
8934

0922
2291
3655
5014

6370
7721
9068

501059
2427
3791

5150
6505
7&56
9203

1196
2564
3927

5286
6640
7991
9337

ias3

2700
4063

5421
6776
8126
9471

0009
1349
2684
4016
5344
6668
7987

9303

0615
1922
3226
4526
5822
7114
8402
9687

0143
1482
2818
4149
5476
6800
8119

9434

0745
2053
3356
4656
5951
7243
8531
9815

0277
1616
2951
4282
5609
6932
8251

9566

0876
2183
3486
4785
C081
7372
80CO
9943

0411
1750
£084
4415
5741
7064
8382

9697

1C07
2314
3616
4915
6210
7501
8788

510545
1883
3218
4548
5874
7196

8514
9828

0679
2017
3351
4681
6006
7328

8646
9959

0813
2151
3484
4813
6139
7460

8777

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
5434
6727
8016
9-302

0353
1661
2966
4266
5563
6856
8145
9430

0484
1792
3096
4396
5693
6985
8274
9559

521138
2444
3746
5045
6339
7630
8917

1269
2575
3876
5174
6469
7759
9045

0072
1351

530200

0328

0456

0584

0712

0840

0968

1096

1223

PROPORTIONAL PARTS.

Diff. 1

2

3

4

5

6

7

8

9
125.1

123.3
122.4
121.5
120.6

11'.' 7
118.8
117.9

117.0
116.1

11. VJ
111 .:;

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
69.0
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
%..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
in: .-J
106.4

ior..6

104.8
104.0
103.2
102.4
101.6

"" 159

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 340 L. 531.]

[No. 379 L. 579.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

340
1
2
3
4
5
6

7
8
9

350
1
2
3

4

5
6

r»

8
9

360
1
2
3

4
5

6
7
8
9

370

1

2
3
4
5
6
7
8
9

531479
2754
4026
5294
6558
7819
9076

1607
2882
4153
5421
6685
7945
9202

1734

3009
4280
5547
6811
8071
9327

1862
3136
4407
5674
6937
8197
9452

1990 |
3264
4534
5800
7063
8322
9578

2117
j 3391
I 4661
5927
7189
8448
9703

2245
3518
4787
6053
7315
8574
9829

2372
3045
4914
6180
7441
8699
9954

1205
2452
309G

4936
61?'2
7405
8035
9861

2500
3772
5041
6306
7567
8825

2627
3899
5167
6432
7693
8951

128
127
127
126
126
126

125
125
125
124

124
124

123
123

123
123

122
121
121
121

120
120
120

119
119
119
119
118
118
118

117

J17
117
116
116
116
115
115
115
114

0079
1330
2576
3820

5060
6290
7529
8758
9984

0204
1454
2701
3944

5183
6419
7052
8881

540329
1579
2825

4068
5307
6543

7775
9003

0455
1704
2950

4192
5431
6666
7898
9126

0580
1829
3074

4316
6555
6789
8021
9249

0705
1953
3199

4440
5678
6913
8144
9371

0830
2078
3323

4564

5802
7036
8267
9494

0955
2203
3447

4688
5925
7159
8389
9616

1080
2327
3571

4812
6049
7282
8512
9739

0106
1328
2547
3702
4973
6182

7387
8589
9787

550228
1450
2668
3883
5094

6303
7507
8709
9907

0351
1572
2790
4004
5215

6423
7627

8829

0473
1694
2911
4126
5336

6544

7748
8948

0595
1816
3033
4247
5457

6664

7868
90(58

0717
1938
3155
4368
5578

6785

7988
9188

0840
2000
3276
4489
5699

6905
8108
9308

0962
2181
3398
4610
5820

7026
8228
9428

1084
2303
3519
4731
5940

7146
8349
9548

1206
2425
3040
4852
6001

7267
8469
9007

0026
1221

2412
3600
4784
5966
7144

8319
9491

0146
1340
2531
3718
4903
6084
7262

8436
9608

0265
1459
2650
3837
5021
6202
7379

8554
9725

0385
1578
2769
3055
5139
6320
7497

8671

9842

! 0504
1098
2887
4074
5257
6437
7614

8788
9959

0624
1817
3006
4192
5376
6555
7732

8905

0743
1936
3125
4311
5494
6673
7849

9023

0863
2055
3244
4429
5612
6791
7967

9140

0982
2174
3362
4548
5730
6909
8084

9257

561101
2293
3481
4666
5848
7026

8202
9374

0076
1243

2407
3568
4726
5880
7032
8181
9326

0193
1359
2523
3684
4841
5996
7147
8295
9441

0309
1476
2639
3800
4957
6111
7262
8410
9555

0426
1592
2755
3915
5072
6226
7377
8525
9069

570543
1709
2872
4031
5188
6341
7492
8639

0660
1825
2988
4147
5303
6457
7607
8754

0776
1942
3104
4263
5419
6572
7722
8868

0893
2058
3220
4379
5534
6687
7836
8983

1010
2174
3336
4494
5650
6802
7951
9097

1126
2291
3452
4610
5765
6917
8066
9212

PROPORTIONAL PARTS.

Diff. 1

2 3

4

5

6

7

8

9

128 12.8
127 12 7
126 12 6
125 12.5
124 12.4
123 12.3
122 12.2
121 12.1
120 12.0
119 11.9

25.6 38.4
25 4 38.1
25.2 37.8
25.0 37.5
24.8 37.2
24.6 36.9
24.4 36.6
24.2 36.3
24,0 36,0
23.8 35.7

51.2
50.8
50.4
50.0
49.6
49.2
48.8
48.4
48.0
47.6

64.0
63.5
63.0
62.5
62.0
61.5
61.0
60.5
60.0
59.5

76.8
76.2
75.6
75.0
74.4
73.8
73.2
72.6
72.0
71.4

89.6

88.9
88.2
87.5
86 8
86.1
85.4
84.7
84.0
83.3

102.4
101.6
100.8
100.0
99.2
98.4
97.6
96.8
96.0
95.2

115.2
114.3
113.4
112.5
111.6
110.7
109.8
108.9
108.0
107.1

160

TABLE XI.— LOGARITHMS OF NUMBERS.

No. 380. L. 579.]

[No. 414 L. 617.

N.

0

1

2

3

*

5

6

7

8

9

Diff.

380

579784

9898

0012

01

26

0241

0355

0469

0583

069

7

0811

1
2

580925
2063

1039
2177

1153
2291

1267
2404

1381
2518

1495
2631

1608
2745

1722
2858

1836
2972

1950
3085

3

3199

3312

3426

3539

3652

3765

3879

3992

4105

4218

4

4331

4444

4557

46

70

4783

4896

5009

5122

523

5

5348

113

5

5461

5574

5686

57

99

5912

6024

6137

6250

636

I

6475

6

6587

6700

6812

69

25

7037

7149

7262

7374

7486

7599

7

7711

7823

7935

80

47

8160

8272

8384

8496

860

s

8720

112

8
9

8832
9950

8944

9056

9167

9279

9391

9503

9615

9726

9838

0061

0173

0284

0396

0507

0619

0730

0842

0953

390

591065

1176

1287

1399

1510

1621

1732

1843

1955

2066

1

2177

2288

2399

25

10

2621

2732

2843

2954

306

4

3175

111

2

3286

3397

3508

36

18

3729

3840

3950

4061

41?

1

4282

3

4393

4503

4614

47

24

4834

4945

5055

5165

52?

li

5386

4

5496

5606

5717

5827

5937

6047

6157

6267

6377

6487

5

6597

6707

6817

69

27

7037

7146

7256

7366

74?

ti

7586

110

6

7695

7805

7914

8024

8134

8243

8353

8462

8572

8681

7

8791

8900

9009

9119

9228

9337

9446

9556

9665

9774

8

9883

9992

0101

0210

0319

0428 ! 0537

0646

0755

0864

109

9

600973

1082

1191

1299

1408

1517

1625

1734

1843

1951

400

2060

2169

2277

23

36

2494

2603

2711

2819

2928

3036

1

3144

3253

3361

34

EJ9

3577

3686

3794

3902

4010

4118

108

2

422(5

4334

4442

45

~>o

4658

4766

4874

4982

508

9

5197

3

5305

5413

5521

56

28

5736

5844

5951

6059

610

6

6274

4

6381

6489

6596

6704

6811

6919

7026

7133

7241

7348

5

7455

7562

7669

77

7884

7991

8098

8205

831

0

8419

107

6

8526

8633

8740

8847

8954

9061

9167

9274

9381

9488

9594

9701

9808

99

14

0021

0128

0234

0341

044

7

0554

8

610660

0767

0873

0979

1086

1192

1298

1405

1511

1617

9

1723

1829

1936

2042

2148

2254

2360

2466

2572

2678

105

410

2784

2890

2996

3102

3207

3313

3419

3525

3630

3736

1

3842

3947

4053

4159

4264

4370

4475

4581

4686

4792

2

4897

5003

5108

52

13

5319

5424

5529

5634

574

0

5845

3

5950

6055

6160

62

i5

6370

6476

6581

6686

679

0

6895

105

4

7000

7105

7210

7315

7420

7525

7629

7734

7839

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

116

11.6

23.2

34.8

46.4

58.0

69.6 81.2

92.8

ldl.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

1<>'2 (5

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

08.1

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 74.9

85.6

96.3

106

10.6

21 2

31.8

42.4

53.0

63.6 74.2

84.8

06.4

105
105

10.5
10.5

2l!5
21.0

31.5
31.5

42.0
42.0

52.5
52.5

63.0 73.5
63.0 73.5

84.0
84.0

94.5
94.5

104

10.4

20.8

31.2

41.6

52.0

62.4 72.8

83.2

93.6

161

TABLE XI. — LOGARITHMS OF LUMBERS.

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

N.

0

1

2

3

4

6

6

7

8

9

Diff.

415

618048

8153

8257

8362

8466

8571

8676 8780 8884

8989

105

6

9093

9198

9302

9406

9511

9615

9719 9824

9928

rjAOO

-

620136

0240

0344

0448

0552 | 0656

0760

0864 0968

UlMM

1072

104

8

1176

1280

1384

1488

1592 1695

1799

1903 2007

2110

9

2214

2318

2421

2525

2628 2732

2835

2939

3042

3146

420

3249

3353

3456

3559

3663

3766

3869

3973

4076

4179

1

4282

4385

4488

4591

4695

4798

4901

5004 I 5107

5210

103

2

5312

5415

5518

5621

5724

5827

5929

6032

6135

6238

3

6340

6443

6546

6648

6751

6853

6956

7058

7161

7203

4

7366

7468

7571

7673

7775

787'8

7980

8082

8185

8287

5

8389

8491

8593

8695

8797

8900

9002

9104

9206

9308

102

9410

9512

9613

9715

9817

9919

0021

0123

0224

0326

y

630428

0530

0631

0733

0835

0936

• 1038

1139

1241

1342

8

1444

1545

1647

1748

1849

1951

2052

2153

2255

2356

9

2457

2559

2660

2761

2862

2963

3064

3165

3266

3367

430

3468

3569

3670

3771

3872

3973

4074

4175

4276

4376

101

1

4477

4578

4679

4779

4880

4981

5081

5182

5283

5383

2

5484

5584

5685

5785

5886

5986

6087

6187

6287 6388

3

6488

6588

6688

6789

6889

6989

7089

7189

7290 7390

4

7490

7590

7690

7790

7890

7990

8090

8190

8290 8389

100

5

8489

8589

8689

8789

8888

8988

9088

9188

9287 9387

6

9486

9586

9686

9785

9885

9984

'

0084

0183

0283

0382

7

640481

0581

0680

0779

0879

0978

1077

1177

1276

1375

8

1474

1573

1672

1771

1871

1970

2069

2168

2267

2366

9

2465

2563

2662

2761

2860

2959

3058

3156

3255

3354

99

440

3453

3551

3650

3749

3847

3946

4044

4143

4242

4340

1

4439

4537

4636

4734

4832

4931

5029

5127

5226

5324

2

5422

5521

5619

5717

5815

5913

6011

6110

6208

6306

3

6404

6502

6600

6698

6796

6894

6992

7089

7187

7285

98

4

7383

7481

7579

7676

7774

7872

7969

8067

8165

8262

5

8360

8458

8555

8653

8750

8848

8945

9043

9140

9237

6

9335

9432

9530

9627

9724

9821

9919

0016

0113

0210

7

650308

0405

0502

0599

0696

0793

0890

0987

1084

1181

8

1278

1375

14V2

1569

1666

1762

1859

1956

2053

2150

97

9

2246

2343

2440

2536

2633

2730

2826

2923

3019

3116

450

3213

3309

3405

3502

3598

3695

3791

3888

3984

4080

1

4177

4273

4369

4465

4562

4658

4754

4850

.4946

5042

2

5138

5235

5331

5427

5523

5019

5715

5810

5906

6002

96

3

6098

6194

6290

6386

6482

6577

6673

6769

6864

6960

4

7056

7152

7247

7343

7438

7534

7629

7725

7820

7916

5

8011

8107

8202

8298

8393

8488

8584

8679

8774

8870

6

8965

9060

9155

9250

9346

9441

9536

9631

9726 | 9821

7

9916

I

0011

0106

0201

0296

0391

0486

0581

0676 0771

95

8

660865

0960

1055

1150

1245

1339

1434

1529

1623 1718

9

1813

1907

2002

2096

2191

2286

2380

2475

2569 2603

PROPORTIONAL PARTS.

Diff 1

234

5

678

9

105 10 5

21.0 31 5 42.0

52 5

63 0 73.5 84 0

94,5

104 lo 4

20 8 31 2 41 6

52.0

62 4 72 8 83 2

93.6

103 10 3

206 309 41.2

51 5

61 8 72 1 82.4

92 7

102 10 2

20 4 30 6 ! 40 8

51 0

61 2 71.4 81 6

91 8

101 10 1

20 2 30 3 40.4

50 5

60 6 70 7 80 8

90.9

100 10.0

20 0 30.0 40 0

50 0

60.0 70 0 80 0

90 0

99 99

19 8 29 7 39 6

49 5

59 4 69 3 79 2

89.1

162

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 460 L. 662.]

[No. 499 L. 698.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

460
1
o

3
4
5
6

8
9

470
1
2
3
4
5
6
7
8

9

480
1
2
3
4
5
6
7
8
9

490
1
2
3
4
5
6
7
8
9

662758
3701
4642
5581
6518
7453
8386
9317

2852
3795
4736
5675
6612
7546
8479
9410

2947
3889
4830
5769
6705
7640
8572
9503

3041
3983
4924

7733

8665
9596

3135

4078
5018
5956
6892
7826
8759
9689

3230
4172
5112
6050
6986
7920
8852
9782

3324

5206
6143
7079
8013
8945
9875

3418
4300
5299

7173
8106
9038
9967

3512

1 15 1

5898

6331
7260
8199
9131

3607
4548
5487
6424
7360
8293
9224

94

93

92
91

90
89

88
87

0060
0988
1913

2836
3758
4677
5595
6511
7424
8336
9246

0153
1080
2005

2929
3850
4769
5687
6602
7516
8427
9337

670246
1173

2098
3021
3942

4861
5778
6094
7607
8518

0339
1265

2190
3113
4034
4953
5870
6785
7698
8609
9519

0431
1358

2283
3205
4126
5045
5962
6876
7789
8700
9610

0524
1451

2375
3297
4218
5137
6053
6968

8791
9700

0617
1543

2467
3390
4310
5228
6145
7059
7972
8882
9791

0710
1636

2560
3482
4402
5320
6236
7151
8063
8973
9882

0802
1728

2652
3574
4494
5412
6328
7242
8154
9064
9973

0895
1821

2744
3666
4586
5503
6419
7333
8245
9155

0063"
0970

1874
2777
3677
4576
5473
6368

8153
9042

0154
1060

1964
3867
3767
4666
5563
6458
7351
8242
9131

0245
1151

2055
2957
3857
4756
5652
6547
7440
8331
9220

680336

1241
2145
3047
3947
4845
5742
6636
7529
8420
9309

690196
1081
1965
2847
3727
4605
5482
6356
7229
8100

0426

1332
2235
3137
4037
4935
5831
6726
7618
8509
9398

0517

1422
2326
3227
4127
5025
5921
6815
7707
8598

0607

1513
2416

asi7

4217
5114
6010
6904
7796
8687
9575

0698

1603
2506
3407
4307
5204
6100
6994
7886
8776
9664

0550
1435
2318
3199
4078
4956
5832
6706
7578
8449

0789

1693
2596
3497
4396
5294
6189
7083
7975
8865

0879

1784
2686
3587
4486
5383
6279
7172
8064
8953

0019

0905
1789
2671
3551
4430
5307
6182
7055
7926
8796

0107

0993
1877
2759
3639
4517
5394
6269
7142
8014
8883

0285
1170
2053
2935
3815
4693
5569
6444
7317
8188

0373
1258
2142
3023
3903
4781
5657
6531
7404
8275

0462
1347
2230
3111
3991
4868
5744
6618
7491
8362

0639
1524
2406
3287
4166
5044
5919
6793
7665
8535

0728
1612
2494
3375
4254
5131
6007
6880
7752
8622

0816
1700
2583
3463
4342

6094
6968
7839
8709

PROPORTIONAL PARTS.

Diff

1

2 3

4

5

6

7

8

9

98
97
96
95
94
93
92
91
90
89
88
87
86

9.8
9.7
9.6
9.5
9.4
9.3
9.2
9.1
9.0
8.9
8.8
8.7
8.6

19.6 29.4
19.4 29.1
19.2 28.8
19.0 28.5
18.8 28.2
18.6 27.9
18.4 27.6
18.2 27.3
18.0 27.0
17.8 26.7
17.6 26.4
17.4 26.1
17.2 25.8

39.2
38.8
38.4
38.0
37.6
37.2
36.8
36.4
36.0
35.6
35.2
34.8
34.4

49.0
48.5
48.0
47.5
47.0
46.5
46.0
45.5
45.0
44.5

ftg

43.0

58.8
58.2
57.6
57.0
56.4
55.8
55.2
54.6
54.0
58.4
52.8
'52.2
51.6

68.6
67.9
67;2

65.'8
65.1
64.4
63.7
63.0
62.3
61.6
60.9
60.2

78.4
77.6
76.8
76.0
75.2
74.4
73.6
72.8
72.0
71.2
70.4
69.6
68.8

88.2
87.3
86.4
85.5
84.6
83.7
82.8
81.9
81.0
80.1
79.2

n

163

TABLE XI. — 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

9924

0011

0098

0184

0271

0358

0444

0531

0617

2

700704

0790

0877

0963

1050

113G

1222

1309

1395

1482

3

1568

1654

1741

1827

1913

1990

2085

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

62G6

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

1

8421

8506

8591

8676

8761

8846

8931

9015

9100

9185

85

2

9270

9355

9440

9524

9609

9694.

9779

9863

9948

0033

3

710117

0202

0287

0371

0456

0540

0625

0710

0794

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

7

3491

3575

3659

3742

3826

3910

3994

4078

4162

4246

84

8

4330

4414

4497

4581

4665

4749

4833

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

7004

7088

7171

7254

7338

7421

7504

7587

2

7671

7754

7837

7920

8003

8086

8169

8253

8336

8419

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

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

49bl

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

9974

0055

0136

0217

0298

0378

0459

0540

0621

0702

8

730782

0863

0944

1024

1105

1186

1266

1347

1428

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

8598

3679

3759

3839

3919

2

3999

4079

4160

4240

4320

4400

4480

4560

4640

4720

Qf\

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

2

3 4

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

25.8 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 59.5 68.0

76 5

84

8.4

16.8

25.2 33.6

420

50.4 58 8 67.2

75.6

164

TABLE XI.— LOGARITHMS OP NUMBERS.

No. 545 L. 736.]

|No. 584 L. 707.

N.

0

1

2

8

4 || 6

C

7

8

9 Diff.

545

7
8
9

736397
7193
7987
8781
9572

6476
7272

8067
8860
9651

6556
7352
8146
8939
9731

6635
7431
8225
9018
9810

6715
7511
8806

9097
9889

, 6795
1 7590
i 8384
91 77
9968

6874
7670
8463
9256

6954
7749
8543
9335

7034 7113

7829 7908
8622 8701
9414 9493

550
1
2
3

4
5
6
7
8
9

740363
1152
1939
2725
3510
4293
5075
5855
6634
7412

0442
1230
2018
2804
3588
4371
5153
5933
6712
7489

0521
1309
2096
2882
3667
4449
5231
6011
6790
7567

0600
1388
2175
2961
3745
4528
5309
6089
6868
7645

0678
1467
2254
3039
3823
4606
5387
6167
6945
7722

0757
! 1546
, 2332
3118
3902
4684
5465
6245
7023
7800

0047

0836
1624
2411
3196
3980
4762
5543
6323
7101
7878

0126

0915
1703
2489
3275
4058
4840
5621
6401
7179
7955

0205

0994
1782
2568
3353
4136
4919
5699
6479
7256
8033

0284

1073
1860
2647
3431
4215
4997
5777
6556
7334
8110

79

78

560
1

8188
8963

8266
9040

8343
91J8

8421
9195

8498
9272

8576 8653
9350 9427

8731
9504

8808
QfiflB

8885
9659

2

9736

9814

9891

qc

96

3

4
5
6

7
8

0045
0817
1587
2356
3123
3889
4654

0123 0200
0894 0971
! 16U4 1741
| 2433 2509
3200 3277
3966 4042
4730 4807

0277
1048
1818
2586
3353
4119
4883

0354
1125
1895
2663
3430
4195
4960

0431
1202
1972
2740
3506
4272
5036

77

750508
1279
2048
2816
3583
4348

0586
1356
2125
2893
3660
4425

0663
1433
2202
2970
3736
4501

0740
1510
2279
3047
3813
4578

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

68

>4

6940

7016 7093

7168

734

4

7320

76

2

7396

7472

7548

7(5

24

7700

7775 7851

7927

800

R

8079

3

8155

8230

8306

83

*2

8458

8533 8609

8685

8761

8836

4

8912

8988

9063

91

J,9

9214

i 9290 9366

9441

951

1»

9592

5

9668

9743

9819

06

9-1

9970

n/v<pr , nioi

1! 1 III'

6

760422

0498

0573

0649

0724

0799 0875

0950

1025

1101

7

1176

1251

1326

14

)2

1477

i 1552 1627

1702

177

8

1853

8

1928

2003

2078

21

53

2228

2303 2378

2453

252

D

2604

9

2679

2754

2829

2904

2978

i 3053 j 3128

3203

3278

3353

73

580

3428

3503

&578

36

>3

3727

! 3802

3877

3952

4027

4101

1

4176

4251

4326

4400

4475

4550 i 4624

4699

477

4

4848

4923

4998

5072

51

47

5221

5296 j 5370

5445

552

0

5594

3

5669

5743

5818

58

12

5966

6041

6115

6190

626

4

6338

4

6413

6487

6562

66

36

6710

6785

6859

6933

7007

7082

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

7

8

9

83

8.3

16.6

24.9

33.2

41.5

49.8

58.1

66.4

74.7

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

7 6

15.2

22.8

30.4

38.0

45.6

53.2

60.8

68.4

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

165

TABLE XI. — LOGAKITHMS OF NUMBERS.

No. 585 L. 767.1 [No. 629 L. 799.

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

i 8268

8342

8416

8490

8564

74

7

8638

8712

8786

8860

8934

9008

9082

9156

9230

9303

g

9377

9451

9525

9599

9673

9746

9820

9894

9968

0042

9

770115

0189

02C3

0336

0410

0484

0557

0631

0705

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

4371

4444

73

5

4517

4590

4663

4736

4809

4882

4955

5028

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

6S46

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

2

9596

9669

9741

9813

9885

9957

0029

0101

0173

0245

3

780317

0389

0461

0533

0605

0677

0749

0821

0893

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

2T59

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

9581

9651

9722

9792

9863

9933

0004

0074

0144

QOIPJ

7

790285

0356

0426

0496

0567

0637

0707

0778

0848

0918

8

0988

1059

1129

1199

1269

1340

1410

1480

1*50

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

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

7i29

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

5

678

9

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

166

TABLE XL— LOGARITHMS OF NUMUEUS.

No. 630 L. 799.]

[No. 674 L. 829.

.-N.

0

1

2

3

4

5

6

7

8

9

Diflf.

'630

799341

9409

9478

9547

9616

9685

9754

9323

9892

9961

1
2

800029
0717

0098

0786

0167
0854

0236
0923

0305
0992

0373
1061

0442
1129

0511
1198

0580

0648
1335

3

1404

1472

1541

16

J9

1678

1747

1815

1884

1062

2021

4

2089

2158

2226

22

J5

2363

2432

2500

2

508

281

7

2705

5

2774

2842

2910

29

re

3047

3116

3184

3

>:>i>

332

i

3389

6

3457

3525

3594

3662

3730

3798

3867

3935

4003

4071

7

4139

4208

4276

4&

14

4412

4480

4548

4

616

468-

r>

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

70

31

7129

7197

7264

332

740

0

7467

2

7535

7603

7670

7738

7806

7873

7941

8008

8076

8143

3

8211

8279

8346

84

14

8481

8549

8616

8

684

875

i

8818

4

8886

8953

9021

9088

9156

9223

9290

9358

9425

9492

5

9560

9627

9694

97

W

9829

9896

9964

(\

no-i

A/yi

o

6

810233

0300

0367

0434

0501

0569

0636

UVOA

0703

UU»O

0770

0165
0837

7

0904

0971

1039

1106

1173

1240

1307

1374

1441

1508

67

8

1575

1642

1709

17

~6

1843

1910

1977

2

044

211

1

2178

9

2245

2312

2379

2445

2512

2579

2646

2713-

2780

2847

650

2913

2980

3047

3114

3181

3247

3314

3381

3448

3514

1

3581

3648

3714

37

SI

3848

3914

3981

4

048

411

4

4181

2

4248

4314

4381

4447

4514

4581

4647

4714

4780

4847

3

4913

4980

5046

51

13

5179

5246

5312

5

378

544

5

5511

4

5578

5644

5711

5777

5843

5910

5976

6042

6109

6175

5

6241

6308

6374

64

40

6506

6573

6639

6

705

677

1

6838

6

6904

6970

7036

7102

7169

7235

7301

7367

7433

7499

7

7565

7631

7698

77

54

7830

7896

7962

8

02S

SOS

4

8160

8

8226

8292

8358

8424

8490

8556

8622

8688

8754

8820

aa

9

8885

8951

9017

9083

9149

9215

9281

9346

9412

9478

DO

660

9544

9610

9676

97

11

9807

9873

9939

0

nai

007

n

0136

1

820201

0267

0333

0399

0464

0530

0595

0661

0727

0792

2

0858

0924

0989

10

>5

1120

1186

1251

1

317

138

2

1448

3

1514

1579

1645

17

10

1775

1841

1906

1

972

203

M

2103

4

2168

2233

2299

2364

2430

2495

2560

2626

2691

2756

5

2822

2887

2952

3018

3083

3148

3213

3279

3344

3409

6

3474

3539

3605

36

70

3735

3800

3865

3

930

39S

a

4061

7

4126

4191

4256

43,

21

4386

4451

4516

4

581

484

(5

4711

65

8

4776

4841

4906

4971

5036

5101

5166

5231

528

0

5361

9

5426

5491

5556

5621

5686

5751

5815

5880

5945

6010

670

6075

6140

6204

6269

6334

6399

6464

6528

659

8

6658

1

6723

6787

6852

69

7

6981

7046

7111

7

175

724

0

7:305

2

7369

7434

7499

7563

7628

7692

7757

7821

7886

7951

3

8015

8080

8144

82(

)',)

8273

8338

8402

8

467

853

i

8595

4

8660

8724

8789

8853

8918

8982

9046

9111

9175

9239

PROPORTIONAL PARTS.

.'Biff 1

2

3

4

5

6

7

8

9

68 68

13 6

20 4

27 2

34 0

40 8

47 6

54 4

61 3

67 67

13 4

20.1

26 8

33 5

40 2

4li !>

IK) -i

66 66

13.2

19 8

26:4

33 0

39 0

41 i 2

52 8

r>:> 4

65 65

13 0

19 5

26 0

32.5

39 0

i:> :>

58 5

64 6.4

12 8

19.2

25 6

32.0

;;s. i

il N

51 2 ov.o

167

TABLE XI. — LOGARITHMS OF NUMBERS.

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

N.

0

1

2

I

4

6

6

7

8

9

Diff.

675
g

829304
9947

9368

9432

9497

9561

9625

9690

9754

9818

9882

0011

0075

0139

0204

0268

0332

0396

0460

0525

7

830589

0653

0717

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

6310

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

9

8219

8282

8345

8408

8471

8534

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

G090

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 i 9297

9358

7

9419

9481

9542

9604

9665

9726

9788

9849 | 9911

9972

8

850033

0095

0156

0217

0279

0340

0401

0462 i 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

2

2480

2541

2602

2663

2724

2785

2846

2907

2968 3029

61

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 I 4852

6

4913

4974

5034

5095

5156

5216

5277

5337

5398

5459

7

5519

5580

5640

5701

5761 i

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

168

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 720 L. 857.] [No. 764 L. 883.

N.

0

1

2

8

4

6

e

7

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

8657

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

( NI'IS!

m f^ft

5

860338

0398

0458

0518

0578

0637

wyo
0697

UltX)

0757

0218
0817

0278
0877

6

0937

0996

1056

1116

1176

1236

1295

1355

1415

1475

7

1534

1594

1654

1714

1773

1833

1893

1952

2012

2072

8

2131

2191

2251

2310

2370

2430

2439

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

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

1

9818

9877

9935

9994

0053

0111

0170

0228

0287

0345

2

870404

0462

0521

0579

0638

0696

0755

0813

0872

0930

3

0989

1047

1106

1164

1223

1281

1339

1398

1456

1515

4

1573

1631

1690

1748

1806

1865

1923

1981

2040

2008

5

2156

2215

2273

2331

2389

2448

2506

2564

2622

2681

6

2739

2797

2855

2913

2972

3030

3088

8146

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
g

9096
9669

9153
9726

9211

9784

9268
9841

9325

9898

9383
9956

9440

9497

9555

9612

0013

0070

0127

0185

9

880242

0299

0356

0413

0-171

0528

0585

0642

0699

0756

760

0814

0871

0928

0985

1042

1099

1156

1213

1271

ms

1

1385

1442

1499

1556

1613

1670

1727

1784

1841

1SJJS

K7

2

1955

2012

2069

2126

2183

2240

2297

2354

2411

2468

Oi

3

2525

2581

2638

2695

2752

2809

2866

2923

2980

3037

4

3093

3150

3207

3264

3321

3377

3434

3491

3548

3605

PROPORTIONAL PARTS.

Diff

1

2

3 4

5

6 7. 8

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

169

TABLE XI. — LOGARITHMS OF NUMBERS.

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

N.

0

1

2

3

4

5

6

7

8

9

Diff.

765

883661

3718

3775

3832

3888

3945

4002

4059

4115

4172

6

4229

4285

4342

4399

4455

4512

4569

4625

4682

4739

7

4?95

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
6

9302
9862

9358
9918

9414
9974

9470

9526

9582

9638

9694

9750

9806

56

0030

0086

0141

0197

0253

0309

0365

7

890421

0477

0533

0589

0645

0700

0756

0812

0868

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

2873

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

7187

7242

7297

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

8&S5

8890

8944

8999

9054

9109

9164

9218

3
4

92Y3

9821

9328
9875

9383
9930

9437
9985

9492

9547

9602

9656

9711

9766

0039

0094

0149

0203

0258

0312

5

900367

0422

0476

0531

0586

0640

0695

0749

0804

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

3578

1

3633

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

54

4

5256

5310

5364

5418

5472

5526

5580

5634

5688

5742

5

5796

5850

5904

5958

6012

6066

6119

6173

6227

6281

6

6335

6389

6443

6497

6551

6604

6658

6712

6766

6820

7

6874

6927

6981

7035

7089

7143

7196

7250

7304

7358

8

7411

7465

7519

7573

7626

7680

7734

77'87

7841

7895

9

7949

8002

8056

8110

8163

8217

8270

8324

8378

8431

PROPORTIONAL PARTS.

Diff. 1

234

5

6 7

8

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

170

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 810 L. 908.] [No. a>i~L~9^T

N.

0

1

2

3

4

6

6

7

8

9

Diff.

810
1
2

908485
9021
9556

8539
9074
9610

8592
9128
9663

8646
9181
9716

8699
9235
9770

8753
9289
9823

8807
9342
9877

8860
9396
9930

8914
9449
9984

8967
9503

3
4
5
6

8
9

910091
0624
1158
1690
2222
2753
3284

0144
0678
1211
1743
2275
2806
3337

0197
0731
1264
1797
2328
2859
3390

0251
0784
1317
1850
2381
2913
3443

0304
0838
1371
1903
2435
2966
3496

1 0358
0891
1424
1956
2488
3U19
3549

0411
0944
1477
2009
2541
3072
3602

0464
0998
1530
2063
2594
3125'
3655

0518
1051
1584
2116
2647
3178
3708

0037
0571
1104
1G37
2169
2700
3231
3761

53

820
1
2

3814
4343

4872

3867
4396
4925

3920
4449
4977

3973
4502
5030

4026
4555
5083

4079
4608
5136

4132
4660
5189

4184
4713
5241

4237
4766
5294

4290
4819
5347

3
4
5
6

5400
5927
6454

6980

5453

5980
6507
7033

5505
6033
6559

7085

5558
6085
6612
7138

5611
6138
6664
7190

5664
6191
6717
7243

5716
6243
6770

7295

5769
6296
1822
7348

5822
6349
6875
7400

5875
6401
6927
7453

7

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

87&4

8816

8869

8921-

8973

9026

830

9078

9130

9183

9235

9287

9340

9392

9444

9496

9549

1

9601

9653

9706

9758

9810

9802

9914

9907

2

920123

0176

0228

0280

0332

0384

0436

0489

UU1\»

0541

OOil
0593

3

0645

0697

0749

0801

0853

0906

0958

1010

1062

1114

4

1166

1218

127'0

1322

1374

1426

1478

1530

1582

1634

52

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

3348

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

SS57

9

8908

8959

9010

9061

9112

9163

9215

9266

9317

9888

850

9419

9470

9521

9572

9623

9674

9725

9776

9827

9879

1 1 9930

9981

51

0032

0083

0134

0185

0236

0287

0338

0389

2

930440

0491

0542

0592

0643

0694

0745

0796

0847

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

2

3 4

5

678

9

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

171

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 855 L. 931.1 [No. 899 L. 954.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

855

931966

2017

2068

2118

2169

2220

»r.

2322

2372

2423

6

2474

2524

2575

2626

2677

2727

2778

2829

2879

2930

7

2981

3031

3082

31:33

3183

3234

3285

3335

3386

3437

8

3487

3538

3589

3639

3690

! 3740

3791

3841

3892

3943

9

3993

4044

4094

4145

4195

| 4246

4296

4347

4397

4448

860

4498

4549

4599

4650

4700

4751

4801

4852

4902

4953

1

5003

5054

5104

5154

5205

5255

5306

5356

5406

5457

5507

5558

5608

5658

5709

5759

5809

5800

5'JIO

5960

3

6011

6061

6111

6162

6212

6262

6313

6863

6413

6463

4

6514

6564

6614

6665

6715

67'65

6815

6865

6910

6966

5

7016

7066

7116

7167

7217

7267

7317

7367

7418

7468

6

7518

7568

7618

7668

7718

7769

7819 7869

7919

7969

en

8019

8069

8119

8169

8219

8269

8320 ! 8370

8420

8470

DU

8

8520

8570

8620

8670

8720

8770

8820 8870

8920

8970

9

9020

9070

9120

9170

92fO

9270

9320 9369

9419

9469

87'0

9519

9569

9^1S

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

1014

1064

1114

1163

1213

1263

1313

1362

1412

1462

4

1511

1561

1611

1660

1710

1760

1809

1859

1909

1958

5

2008

2058

2107

2157

2207

2256

2306

2355

2405

2455

6

2504

2t>54

2603

2653

2702

2752

2301

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

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

7336

7385

40

6

7434

7483

7532

7581

7630

7679

7728

777*7

7826

7875

t±u

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
9878

9439
9926

9488
9975

9536

9585

9634

9683

9731

9780

9829

0024

0073

0121

0170

0219

0267

0316

2

950365

0414

0462

0511

0560

0608

0657

0706

0754

0803

3

0&51

0900

0949

0997

1046

1095

1143

1192

1240

1289

4

1338

1386

1435

1483

1532

1580

1629

1677

1726

1775

5

1823

1872

1920

1969

2017

2066

2114

21 63

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.

Diff

1

2

3 4

5

678

9

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 a5.0 40.0

45.0

49

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

172

TABLU XI. — LOGARITHMS OF XUMBERS.

No 900 L. 954.1

[No. 944 L. 975.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

900

954243

4291

4339

4387

4435

4484

4532 4580 4628

4677

1

4725

4773

4821

4869

4918

1866

5014 5062 5110 5158

2

5207

5255

5.'

503

5351

5399

5447

5495

13 5592

5640

3

5688

5736

5784

5832

5880

5928

5976

6024 6072

6120

4

6168

6216

65

>65

631J

5

6361

6409

6457

65(

>:> r,.v,:{

6601

5

6649

6697

6745

6793

6840

6888

6936

69

34 7038

7080

48

6

7128

7176

7;

224

7275

>

7320

7368

7416

74

34

7512

7559

7

7607

7655

ro3

7751

7799

7847

7894

7942

7990

8038

8

8086

8134

8

181

822

1

8277

8325

8373

84

n

84(58

8516

9

8564

8G12

8659

8707

8755

8803

8850

88

e

8946

8994

910

9041

9089

9137

9185

9232

9280

9328

9375

9423

9471

1
2

9518
9995

9566

9614

9661

9709

9757

9804

9852

9900

9947

0042

0090

0138

0185

0233

0280

0328

0376

0423

3

960471

0518

0566

0613

0(561

0709

0756

0804

0851

0899

4

0946

0994

1

041

108

)

1136

1184

1231

12

71)

1326

1374

5

1421

1469

1516

1563

1611

1658

1706

1753

1801

1848

6

1895

1943

1

990

203

3

2085

2132

2180

22

'27

2275

2322

7

2369

2417

2

464

251

1

2559

2606

2653

27

01

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

4

K-J5

487

2

4919

4966

5013

5C

161 5108

5155

3

5202

5249

5296

5343

5390

5437

5484

5531

5578

5625

4

5672

5719

5

766

581

i

5860

5907

5954

6C

01

6048

6095

4T

5

6142

6189

6236

6283

6329

6376

6423

6470

6517

6564

6

6611

6658

6

705

675

2

6799

6845

6892

6£

BO

6986

7033

7

7080

7127

7

173

7220

7267

7314

7361

7408

7454

7501

8

7548

7595

i~

642

768

8

7735

7782

7829

7?

75

7922

7969

9

8016

8062

8109

8156

8203

8249

8296

8343

8390

8436

930
1

8483
8950

8530
8996

8576
9043

8623
9090

8670
9136

8716
9183

8763
9229

8810
9276

8856
9323

8903
9369

2

9416

94(53

9509

9556

9602

9649

9695

B

•42

9789 9835

3

9882

9928

9975

0021

0068

0114

0161

0207

0234

0300

4
5

6
7
8
9

970347
0812
1276
1740
2203
2666

0393
0858
1322
1786
2249
2712

0440
0904
1369
1832
2295
2758

0486
0951
1415
1879
2342
2804

0533
0997
1461
1925

2388
2851

0579
1044
1508
1971
2434
2897

0626
1090
1554
2018
2481
2943

0672
1137
1601
2064
2527
2989

0719
1183
1647
2110
2573
3035

0765
1229
1693
2157
2619
3082

940
1
2
3
4

3128
3590
4051
4512
4972

3174
3636
4097
4558
5018

3220

3682
4143
4604
5064

3266
3728
4189
4650
5110

3313
3774
4235
4696
5156

3359
3820
4281
4742
5202

3405
3866
4327
4788
5248

3451
3913
4374

4834
521)4

3497
3959
4420
4880
5340

3543
4005
4466
4926
5386

46

f»ROPORTIONAL PARTS.

Diff. 1

2

3

4

5

6

7

8

9

47 4.7
46 4.6

9.4
9.2

14.1
13.8

18.8
18.4

23.5
23.0

28.2
27.6

32.9
32.2

37.6
36.8

42.3
41.4

173

TABLE XT. — LOGARITHMS OF LUMBERS.

No. 945 L. 975.] [No. 989 L. 895.

N.

0

1

i

a

4

6

6

7

S

9

Diff.

945

975432

5478

5524

5570

5616

5662

5707

5753

5799

5845

6

5891

£937

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

2

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 I 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

5i>92

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

7309

7353

7398

7443

7488

7532

7577

7622

2

7666

7711

7756

7800

7845

7890

7934

7979

8024

8068

3

8113

8157

8202

8247

8291

&336

8331

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

9895

9939

9983

0028

0072

0117

0161

0206

0250

0294

8

990a39

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

1835

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

3173

3216

3260

3304

3348

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

4403

4537

4581

4625

4669

4713

44

8

4757

4801

4845

4889

4933

4977

5021

5065

5108

5152

9

5196

5240

5284

5328

5372

5416

5460

55U4

5547

5591

PROPORTIONAL, FARTS.

Diff

1

23 4

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

9.0 135 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

174

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 990 L. 995.]

[No. 999 L. 999.

N.

0

1

2

3

4 5

6

7

8

9

Diflf.

990

995635

5679

5723

5767

5811 5854 5

898

5942

5986

6030

1

6074

6117

6161

6205

6249 6

293 6

337

0380

6424

6468

44

2

6512

6555

6599

6643

6687 6

"31 C

0818

6862

6906

3

6949

6993

7037

7080

7124 7

168 7

212

7255

7299

7343

4

7386

7430

7474

7517

7561 7

305 7

648

7692

7736

7779

5

7823

7867

7910

7954

7998 8041 £

085

8129

8172

8216

6

8259

8303

8347

8390

8434 a

177 e

1521

8564

8608

8652

7

8695

8739

8782

8826

8869 8913 £

956

9000

9043

9087

8

9131

9174

9218

9261

9305 9

348 £

392

9435

9479

9522

9

9565

9609

9652

9696

9739 9783 {

1826

9870

9913

9957

43

LOGARITHMS OP 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

i.noaisr.

2

0.301030

22

1.342423

42

1.68

>3249

62

1.7£

12392

82

1.913814

3

0.477121

23

1.361728

43

1.633468

63

1.799341

as

.919078

4

0.602060

24

1.380211

44

1.6-

13453

64

1.8

)6180

84

.924279

5

0.698970

25

1.397940

45

1.653213

65

1.812913

85

.029419

6

0.778151

26

1.414973

46

1.662758

66

.819544

86

.934498

7

0.845098

27

1.431364

47

1.672098

67

.826075

87

.939519

8

0.903090

28

1.447158

48

1.6

31241

68

.Si

12509

88

.!) His:}

9

0.954243

29

1.462398

49

1.690196

69

.a38849

89

.949390

10

1.000000

30

1.477121

50

1.698970

70

.845098

90

.954243

11

1.041393

31

1.491362

51

1.707570

71

.851258

91

1.959041

12

1.079181

32

1.505150

52

1.716003

72

.857332

92

1.1W37S8

13

1.113943

33

1.518514

53

1.7

24276

73

.8(

33323

93

1.%84S3

14

1.146128

34

1.531479

54

1.7

32394

74

.8(

59232

94

1.973128

15

1 176091

35

1.544068

55

1.740363

75

.875061

95

1.977724

16

1.204120

36

1.556303

56

1.748188

76

.880814

96

1.982271

17

1.230449

37

1.568202

57

1.755875

77

.886491

97

1.9K6772

18

1.255273

38

1.579784 !

58

1.7

53428

78

.8<

8095

98

1.991226

19

1.278754

39

1.591065 !

59

1.770852

79

.897027

!)!>

1 . 995635

20

1.301030

40

1.602060 !

60

1.778151

80

.903090

100

2.000000

Value .

at 0°. ;

Sign
n 1st
3uad.

Vain
at 90°

, .Sign
9 in2d
• Quad.

Valu
at

180°.

3 Sign
in 3d
Quad.

Value
at
270°

Sign
in 4th
Quad.

Value
at
360°.

Sin

0

R

4-

O

R

__

O

Tan

o

00

o

4-

00.

_

O

Sec

R

oo

R

00

4

R

Versin....

0

R

4-

2R

4.

R

4

0

Cos

R

o

R

__

O

4

R

Cot

00

__

o

00

-f

O

_

00

Cosec

00

--

R

4-

00

R

~~

00

R signifies equal to rad; oo signifies infinite ; O signifies evanescent.

175

TABLE XII. — LOGARITHMIC SINES,

179°

"

'

Sine.

q-l

Tang.

Cotang.

*+i

Dl"

Cosine.

/

4.685

15.314

0

0

Inf. neg.

575

575

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

.162696

575

575

.162696

.837304

425

ten

55

360

6

.241877

575

!575

.241878

.758122

425

.02

9.999999

54

420

7

.308824

575

1575

.308825

.691175

425

.00

.999999

53

480

8

.366816

574

1576

.366817

.633183

424

.00

.999999

52

540

g

.417968

574

576

.417970

.582030

424

.00

.999999

51

600

10

.463726

574

|576

.463727

.536273

424

.02

.999998

50

660

11

7.505118

574

1576

7.505120

12.494880

424

.00

9.999998

49

720

12

.542906

574

! 577

.542909

.457091

423

.02

.999997

48

780

13

.577668

574

577

.577672

.422328

423

.00

.999997

47

840

14

.609853

574

i 577

.609857

.390143

423

.02

.999996

46

900

15

.639816

573

578

.639820

.360180

422

.00

.999996

45

960

16

.667845

573

J578

.667849

.332151

422

.02

.999995

44

1020

17

.694173

573

!578

.694179

.305821

422

.00

.999995

43

1080

18

.718997

573 | ! 579

.719003

.280997

421

.02

.999994

42

1140

19

.742478

573

i 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

9.999992

39

1320

22

.806146

572

581

.806155

. 193845

419

.02

.999991

38

1380

23

.825451

572

581

.825460

.174540

419

.02

.999990

37

1440

24

.843934

571

582

.843944

.156056

418

.02

.999989

36

1500

25

.861662

571

583

.861674

.138326

417

.00

.999989

35

1560

26

.878695

571

583

.878708

.121292

417

.02

.999988

34

1620

27

.895085

570 584

.895099

.104901

416

.02

.999987

33

1680

28

.910879

570 1 i 584

.910894

.089106

416

.02

.999986

32

1740

29

.926119

570

585

.926134

.073866

415

.02

.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

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 I 589

7.995219

12.004781

411

.02

.999979 1 26

2100

35

8.007787

567 j 590

8.007'809

11.992191

410

.03

.9i.;9977

25

2160

36

.020021

567 I 591

.020044

.979956

409

.02

.999976

24

2220

37

.031919

566

592

.031945

.968055

408

.02

.999975

23

2280

38

.043501

566

593

.043527

.956473

407

.03

.999973

22

2340

39

.054781

566

593

.054809

.945191

407

.02

.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

i596

.086997

.913003

404

.02

.999968

18

2580

43

.097183

564

J598

.097217

.902783

402

.03

.999966

17

2640

44

.107167

563

1599

.107203

.892797

401

.03

.999964

16

2700

45

.116926

562

1600

.116963

.883037

400

.02

.999963

15

2760

46

.126471

562

|601

.126510

.873490

399

,03

.999961

14

2820

47

.135810

561

602

.135851

.864149

398

.03

.999959

13

2880

48

.144953

561

J603

.144996

.855004

397

.02

.999958

12

2940

49

.153907

560

604

.153952

.846048

396 | «5g

.999956

11

3000

5C

.162681

560

1605

.162727

.837273

395 1 -03

.999954

10

3060

51

8.171280

559

J607

8.171328

11.828672

393

• .03

9.999952

9

3120

52

.179713

558 608

.179763

.820237

392

1 .03

.999950

8

3180

53

.187985

558 | i 609

.188036

.811964

391

.03

.999948

7

3240

54

.196102

557

611

.196156

.803844

389

.('3

.999946

6

3300
3360

55
56

.204070
.211895

556
556

612
1613

.204126
.211953

.795874

.788047

388
387

.03
.03

.899944
.999942

5
4

8120

57

.219581

555 i 615

.219641

.780359

385

.03

.999940

3

3480

58

.227134

554 616

.227195

.772805

384

.03

.999938

2

3540

59

.234557

554

618

.234621

.765379

382

.03

.999936

1

3600

60

8.241855

553

619

8.241921

11.758079

381

.03

9.999934

0

4.685

15.314

"

t

Cosine.

q-l

Cotang.

Tang.

ff-M

Dl"

Sine.

'

90°

176

89-

OSINES, TANGENTS, AND COTANGENTS.

178°

"

'

Sine.

i-i

Tang.

Cotang.

q + l

Dr

Cosine.

,

I 4.685

15.314

1

3600
3660
3720
3780
3840
3900
3960
4020
4080
4140

1

2
3
4
5

6

7
8
9

8.241855
.249033
. 256094
.263042

! .269881
.276614
.283243
i .289773
.296207
. 302546

553
552
551
551
550
549
548
547
546
546

019
020
022
623
625
627
628
630
032
033

8.241921
.24910-2
.250105
.203115
.269956
.276691
.283323
.289856
.296292
.302634

11. 7-58070
.750898

.786885

.730044
.723309
.710077
.710144
.7037'08
.697866

381
380
378
377
375

372
370
368
367

.oa

.06
.08
.08
.06

.03

.05

.03

9.999931
.999988

.99998!

.999985

.999915
999918

00
59
58
57
50
55
54
58
52
51

4200 10 .308794 545

635

,308884

.691116

365

.05

! 999910

5C

4260
4320
4380
4440
4500
4560
4620
4680

11
12
13
14
15
16
17
18

8.314954
.321027
.32701(5
.332924
.338753
.344504
.350181
.355783

544
543

542
541
540

539
539
538

637
638
640
642
644
646
648
649

8.315046
.321122
.327114
.333025
.338856
.344610
.350289
.355895

11.684954
.67'8878
.672886
.666975
.661144
.655390
.649711
.644105

363
302
360
358
356
354
352
351

.05
.03
.05
.05
.03
.05
.05
.05

9.999907
.999906
.999908
.999899

48
47
46

43
42

4740

19

.361315

537

651

.361430

.638570

349 •"•?

QQQRBK

41

4800

20

.366777

536

653

.366895

.633105

347 'Uo .9WW

40

4860
4920
4980

21
22
23

8 372171
.377499

.382762

535
534
533

655
657
659

8.372292
.377622
.382889

11.627708
.622378
.617111

345 a!

.'543 •;;?

341 'SS

9.999873
.999876

.999873

38

5040

24

.387962

532

661

.388092

.611908

339 •<*?

30

5100

25

.393101

531

663

.393234

.606766

337 ; 'SB

.999^07

35

5160

26

.398179

530

666

.398315

.601685

334

.uo

.999664

34

5220

27

.403199

529

668

.403338

.596662

332

.05

.999861

33

5280

28

.408161

527

670

.408304

.591696

330

t .05

.998858

32

5340

29

.413068

526

672

.413213

.586787

328

.07

.998854

31

5400

30

.417919

525

674

.418068

.581932

326

.05

.999851

30

5460

31

8.422717

524

676

8.422869

11.577131

324

.05

9.90HK

5520

32

.427462

523

679

.427618

.572382

321

.07

5580

33

.432156

522

681

.432315

.567685

319

.05

5640

34

.436800

521

683

.436962

.563038

317

.05

18 !j>

5700

35

.441394

520

685

.441560

.558440

315

.07

.99989

5760

36

.445941

518

688

.446110

.553890

312

.05

.999881 24

5820

37

.450440

517

690

.450613

.549387

310

.07

.998W7 23

5880

38

.454893

516

693

.455070

.544930

307

.05

5940

39

.459301

515

695

.459481

.540519

305

.07

.99982d 21

6000

40

.463665

514

697

.463849

.536151

303

.07

.999810 20

6060

41

8.467985

512

700

8.468172

11.531828

300

.05

9.998813 11)

6120

42

.472263

511

702

.472454

.527546

298

.

6180

43

.476498

510

705

.476693

.523307

295

.07

.998N

6240

44

.480693

509

707

.480892

.519108

293

.07

91 i 10

6300

45

.484848

50v'

710

.485050

.514950

290

.07

.tt9?tf

6360

46

.488963

506

713

.489170

.510830

287

.05

/\r-

.999794 14

6420

47

.493040

505

715

.493250

.506750

285

•JE .99MJ.O 13

6480

48

.497078

503

718

.497293

.502707

888

•}{i .9897HJ 12

6540

49

.501080

502

720

.501298

.498702

280

.UY

-2 11

6600

50

.505045

501

723

.505267

.494733

277

• '

.999778 10

6660

51

8.508974

499 '

726

8.509200

11 .490800

274

.07 8.c,n<)774 '.»
.08

6720

52

.512867

498

729

.513098

.486902

271

.99974

6780

53

.516726

497

731

.510901

.483039

269

• '

6840

54

.520551

495

734

.520790

.479210

266

*07

.999761

6

6000

55

.524343

494

737

524586

.475414

263

.999767

5

6960

56

.528102

492

740

.528349

.471651

260

*08

.999768

4

7020

57

.531828

491

743

.532080

.467920

257

*07

.999748

3

7080

58

.535523

490

745

.5:35779

.464221

255

*07

.999744

2

7140

59

.539186

488

748

.539447

.460553

252

•08

.999740

1

7200

60

8.542819

487

751

8.543084 1

11.456916

249

9.990786

0

4.685

15.314

~"~~

~

Cosine.

q-l

Cotang.

Tang.

q + l

Dr

Sine.

'j

91°

177

88

TABLE XII. — LOGARITHMIC SINES

177°

'

Sine.

D. r.

Cosine.

D. r.

Tang.

D. r.

Cotang.

'

0

1

2
3
4
5
6
7

8 542319
.546422
.549995
.553539
.557054
.560540
.563999
.567431

60.05
59.55
59.07
58.58
58.10
57.65
57.20

KR i"K

9.999735
.999731
.999726
.999722
.999717
.999713
.999708
.999704

.07
.08
.07
.08
.07
.08
.07

AQ

8.5430S4
.546691
.550268
.553817
.557336
.560828
.564291
.567727

60.12
59.62
59.15
58.65
58.20
57.72
57.27

t£» OQ

11.456916
.453309
.449732
.446183
.442664
.439172
.435709
.432273

60
59
58
57
56
55
54
53

8
9

.570836
.574214

OO. <O

56.30

.999699
.999694

.Uo

.08

.571137
.574520

OO.oO

56.38

.428863
.425480

52
51

10

.577566

55.87
55.43

.999689

.08
.07

.577877

55 . 95
55.52

.422123

50

11
12
13
14
15
16

8.580892
.584193
.587469
.590721
.593948
.597152

55.02
54.60
54.20
53.78
53.40

tO AT)

9.999685
.999680
.999675
.999670
.999665
.999660

.08
.08
.08
.08
.08

AQ

8.581208
.584514
.587795
.591051
.594283
,597492

55.10
54.68
54.27
53.87
53.48

tO AQ

11.418792
.415486
.412205
.408949
.405717
.402508

49

48
47
46
45
44

17
18
19

.600332
.603489
.606623

Do.UU

52.62
52.23

.999655
.999650
.999645

.Uo

.08
.08

.600677
.603839
.606978

oo.Uo
52.70
52.32

.399323
.396161
.393022

43
42
41

20

.609734

51.85
51.48

.999640

.08
.08

.610094

51 .93
51.58

.389906

40

21
22
23
24

8.612823
.615891
.618937
.621962

51.13
50.77
50.42

9.999635
.999629
.999624
.999619

.10
.08
.08

8.613189
.616262
.619313
.622343

51.22
50.85
50.50

11.386811
.383738
.380687
.377657

39

38
37
36

25

26
27
28
29
30

.624965
.627948
.630911
.633854
.636776
.639680

50.05
49.72
49.38
49.05

48.70
48.40
48.05

.999614
.999608
.999603
.999597
.999592
.999586

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

.625352
.628340
.631308
.634256
.637184
.640093

50.15
49.80
49.47
49.13
48.80
48.48
48.15

' .374648
.371660
.368692
.365744
.362816
.359907

35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

8.642563
.645428
.648274
.651102
.653911
.656702
.659475
.662230
.664968
.667689

47.75
47.43
47.13
46.82
46.52
46.22
45.92
45.63
45.35
45.07

9.999581
.999575
.999570
.999564
.999558
.999553
.999547
.999541
.999535
.999529

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

8.642982
.645853
.648704
.651537
.654352
.657149
.659928
.662689
.6654.33
.668160

47.85
47.52
47.22
46.92
46.62
46.32
46.02
45.73
45.45
45.17

11.357018
.354147
.351296
.348463
.345648
.342851
.340072
.337311
.334567
.331840

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49

8.670393
.673080
.675?'51

.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

1ft

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

. 1U

.10

.694529

42.77
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
.999443
.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

Ooe^e.

D. r. i

Sine. ! D. 1". 1

Cotang.

D. 1'.

Tang.

r

92'

178

COSINES, TANGENTS, AND COTANGENTS.

176°

'

Sine.

D.I".

Cosine.

D. r.

Tang.

D. I'.

Cotang.

-

0

1

2
3
4
5

6

8
9

8.718800
.721204
.723595
.725972
.728337
.730688
.733027
.7&53S4
.737667
.739969

40.07
39.85
39.62
39.42
39.18
38.98
38.78
38.55
38.37

9.999404
.999398
.999391
.91)9384
.999378
.999371
.999364
.999357
.999350
.999343

.10
.12
.12
.10
.12
.12 i
.12 i
.12
.12

8.719306
.721806
.724204
.7205S8
.728959
.731317
.733663
.735996
.738317
.740626

40.17
89.97
89.78
89.52
89.80
39.10
38.88
38.68
88.48

11.280604
,278194
.275796
.278412
.271041

.266887
364004
.261668
.259874

60

59
58

r.7

56
59
54

53

u

51

10

.742259

37.95

.999336

!l2 i -7429;^

38.08

.257078

50

11
12

8.744536
.746802

37.77

9.999329

.999322

.12

8.745207
.747479

37.87

! 252521

49

48

13
14
15
16
17
18
19
20

.749055
.751297
.753528
.755747
.757955
.760151
.7623137
.764511

37.37
37.18
36.98
36.80
36.60
36.43
36.23
36.07

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

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

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

37.48
37.30
37.10
36.92
36.73
36.55
86.85
36.18

.250260
.248011
.245773
.248547

!2391:iS
.286985
.284754

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

9.999257
.999250
.999242
.999235
.999227

.12
.13
.12
.13

8.767417
.769578
.771727
.773866
.775995

86 JOB

as. 82

a5.65
85.48

oe oo

11.232583
.230422
.228273
.226134
.224005

39
38
371
36
35

26
27
28
29
30

.777333
.779434
.781524
.783605
.785675

35.17
35.02
34.83
34.68
34.50
34 35

.999220
.999212
.999205
.999197
.999189

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

.778114
.780222
i .782320
.784408
.786486

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
3° 78

9.999181
.999174
.999166
.999158
.999150
.999142
.999134
.999126
.999118
.999110

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

8.788554
.790613
.792662
.794701
.796731
.796752
.800763
.802765
.804758
.8067'42

34.32
34.15
33.98

as. as
as. 68

33.52

as. 37

33.22

as. 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
48
49
50

8.807819
.809777
.811726
.813667
.815599
.817522
.819436
.821343
.823240
.825130

32.63
32.48
32. a5
32.20
32.05
31.90
31.78
31.62
31.50
31 35

9.999102
.999094
.999086
.999077
.999069
.999061
.999053
.999044
.999036
.999027

.13
.13
.15
.13
.13
.13
.15
.13
.15
13

8.808717
.810683
.812641
.814589
.816529
.818461
.820384
.822298
.824205
.826103

88.W

32.63
32.47
32.33
32.20
32.05
31.90
31.78
31.63
31.48

11.191283
.189317
.187359
.185411
.183471
.181539
.179616
.177702
.175795
.173897

19
18
17
16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

8.827011
.828884
.a30749
.832607
.834456
.836297
.838130
.839956
.841774
8.843585

31.22
31.08
30.97
30.82
30.68
30.55
30.43
30.30
30.18

9.999019
.999010
.999002
.998993
.998984
.998976
.998967
.998958
.998950
9.998941

.15
.13
.15
.15
.13
.15
.15
.13
.15

8.827992
.829874
.831748
.833613
.835471
.837321
.839163
.840998
.842825
8.844644

31.37
31.23
31.08
30.97
30.83
30.70
30.58
30.45
30.32

11.172008
.170126
.166852
.166867

.164529
.162679
.160687
.159002
.157175
11.155356

9
8
7
6
5
4
3
2
1
0

/

Cosine.

D 1*.

Sine.

D. I'.

Cotang.

D. r. Tang.

L_ 86-

179

TABLE XII. — LOGARITHMIC SINES,

174°

'

Sine.

D. 1'.

Cosine.

D. 1\

Tang.

D. 1".

Cotang.

t

0

1

2
3
4
5
6

8
9
10

8.8435a5
.845387
.847183
.848971
.850751
.852525
.854291
.856049
.857801
.859546
.861283

30.03
29.93

29.80
29.67
29.57
29.43
29.30
29.20
29.08
28.95
28.85

9.998941
.998932
.998923
.998914
.998905
.998896
.998887
.998878
.998869
.998860
.998851

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

8.844644
.846455
.848260
.850057
.851846
.853628
.855403
.857171
.858932
.860686
.862433

30.18
30.08
29.95
29.82
29.70
29.58
29.47
29.35
29.23
29.12
29.00

'. 153545
.151740
.149943
.148154
.146372
.144597
.142829
.141068
.139314
.137567

60

59
58
57
56
55
54
53
52
51
50

11
12
13
14
15
16

8.863014
.864738
.866455
.868165
.869868
.871565

28.73
28.62
28.50
28.38

28.28

90 -\rt

9.998841
.998832
.998823
.998813
.998804
.998795

.15
.15
.17
.15
.15

8.864173
.865906
.867632
.869351
.871064
.872770

28.88
28.77
28.65
28.55
28.43

no oo

11.135827
.134094
. 132368
.130649
.128936
.127230

49
48
47
46
45
44

17

18
19
20

.873255
.874938
.876615

.878285

tCO. i i

28.05
27.95
27,83
27.73

.998785
.998776
.998766
.998757

. 5

* k

.874469
.876162
.877849
.879529

/CO . Ofi

28.22
28.12
28.00
27.88

. 125531
.123838
.122151
.120471

43
42
41
40

21
22

8.879949
.881607

27.63

9.998747
.998738

. 5

• n

8.881202
.882869

27.78

11.118798
.117131

39

38

23
24
25

.883258
.884903

.886542

27.52
27.42
27.32

.998728
.998718
.998708

K

.884530

.886185
.887833

27! 58
27.47
27 38

.115470
.113815
.112167

37
36
35

26

.888174

/«< • ""

.998699

. •)

.889476

.110524

34

27

.889801

27.12

.998689

• ^

.891112

27.27

.108888

33

28

.891421

27.00

.998679

' i

.892742

27.17

.107258

32

29

.893035

26.90

.998669

- ^

.894366

27.07

.105634

31

30

.894643

26.80
26.72

.998659

; f

.895984

26.97
26.87

.104016

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

8.911949
.913488
.915022

25.65
25.57

9.998548
.998537
.998527

.18
.17

8.913401
.914951
.916495

25.83
25.73
25 63

11.086599
.085049
.083505

19
18
17

44
45
46
47
48
49
50

.916550
.918073
.919591
.921103
.922610
.924112
.925609

25 .'38
25.30
25.20
25.12
25.03
24.95
24.85

.998516
.998506
.998495
.998485
.998474
.998464
.998453

'i?

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

.918034
.919568
.921096
.922619
.924136
.925649
.927156

25! 57
25.47
25.38
25.28
25.22
25.12
25.03

.081966
.080432
.078904
.077381
.075864
.074351
.072844

16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

8.927100
.928587
.930068
.931544
.933015
.9:34481
.935942
.937398
.938850
8.940296

24.78
24.68
24.60
24.52
24.43
24.35
24.27
24.20
24.10

9.998442
.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.I".

Tang.

'

94'

180

85*

COSINES, TANGENTS, AND COTANGENTS. 174°

'

Sin«.

D. 1". ! Cosine.

D. 1".

Tang.

D. r.

Cotang.

•

0

1

2

3
4
5
6
7
8
9
10

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

24.03
88.93

23.87
23.80
23.70
23.63
23.55
23.48
23.40
23.32
23.25

1 9.998344

! 998322
.998311
.998300
.998289
.998277
.998266
.998255
.998243
| .998232

.18
.18
.18

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

! 948404

.944852
.946295
.947734
.949168
.950697
.952021
.953441
.954856

24.20
84.18
84.05

23.98
23.90
23.82
23.73
23.67
23.58
23.52
23 45

11.058048
.066698

.055148
.053705

.060888

.049-403
.047979
.046569
.045144
.043733

60
59
68
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
.993116

.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
.040985
.039527
.088184

.036745
.035361
.033981
.032606
.081884
.029867

49
48
47
46
45
44
43
42
41
40

21
22
23
24
25
26
27
28
29
30

8.969600
.970947
.972289
.973628
.974962
.976293
.977619
.978941
.980259
.981573

22.45
22.37
22.32
22.23
22.18
22.10
22.03
21.97
21.90
21.83

9.998104
.998092
.998080
.998068
.998056
.998044
.998032
.998020
.998008
.997996

.20
.20
.20
.20
.20
.20
.20
.20
.20
.20

8.971496
.972855
.974209
.975560
.976906
.978248
.979586
.980921
.982251
.983577

22.65
22.57
22.52
22.43
22.37
22.30
22.25
22.17
22.10
22.03

11.028504
.027145
.025791
.024440
.023094
.021752
.020414
.019079
.017749
.016423

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

8.982883
.984189
.985491
.986789
.988083
.989374
.990660
.991943
.993222
.994497

21.77
21.72
21.63
21.57
21.52
21.43
21.38
21.32
21.25
21.18

9.997984
.997972
.997959
.997947
.997935
.997922
.997910
.997897
.997885
.997872

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

8.984899
.986217
.987532
.988842
.990149
.991451
.992750
.994045
.995337
.996624

21.97
21.92
21.83
21.78
21.70
21.65
21.58
21.53
21.45
21.40

11.015101
.013783
.012468
.011158
.009851
.008549
.007250
.005955
.004663
.003376

29
28
27
26
25
24
23
22
21
20

41

8.995768

01 i <*

9.997860

oo

8.997908

91 3H

11.002092

19

42

.997036

.997847

on

8.999188

01 oo

11.000812

18

43
44
45

46
47
48
49
50

.998299
8.999560
9.000816
.002069
.003318
.004563
.005805
.007044

21.02
20.93
20 88
20.82
20.75
20.70
20.65
20.57

.997835
.997822
.997809
.997797
.997784
.997771
.997758
.997745

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

9.000465
.001738
.003007
.004272
.005534
.006792
.008047
.009298

21.22
21.15
21.08
21.03
20.97
20.92
20.85
20.80

10.999535
.998262
.996993
.995728
.994406
.993208
.991953
.990702

17
16
15
14
13
12
11
10

51
52

9.008278
.009510

20.53 !

9.997732
.997719

.22

9.010546
.011790

20.73

on «a

10.989454
.988840

9

8

53
64
55
56
57
58
59
60

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

20.42
20.33
20.30
20.22
20.18
20.12
20.07

.997706
.997693
.997680
.997667
.997654
.997641
.997628
9.997614

.22
.22
.22
.22
.22 i
.22 1
.23 i

.013031
.014268
.015502
.016732
.017959
.0191&3
.020403
9.021620

20.62
20 57
20.50
20.45
20.40
20.33
20.28

.986969
.986788
.984498

.988868
.988041
.980617
.979697

10.978380

7
6
5
4
3
2
1
0

•

Cosine.

D. r.

Sine.

D. r.

Cotang.

D. r.

Tang.

1

05°

181

81*

TABLE XII. — LOGARITHMIC SINES,

173°

'

Sine.

D. 1'.

Cosine.

D.I-.

Tang.

D. r.

Cotang.

•

0

1

2
3
4
5
6

8
9
10

9.019235
.020435
.021632
.022825
.024016
.025203
.026386
.027567
.028744
.029918
.031089

20.00
19.95
19.88
19.85
19.78
19.72
19.68
19.62
19.57
19 52
19.47

9.997614
.997601
.997588
.997574
.997561
.997547
.997534
.997520
.997507
.997493
.997480

.22
.22

.23

!23
.22
.23
.22
.23
22

9.021620
.022834
.024044
.025251
.026455
.027655
.028852
.030046
.031237
.032425
.033609

20.23
20.17
20.12
20.07
20.00
19.95
19.90
19.85
19.80
19.73
19.70

10.978380
.977166
.975956
.974749
.973545
.972345
.971148
.969954
.968763
.967575
.966391

60
59
58
57
56
55
54
53
52
51
50

11
12
13
14
15
16
17
18
19
20

9.032257
.033421
.034582
.035741
.036896
.038048
.039197
.040342
.041485
.042625

19.40
19.35
19.32
19.25
19.20
19.15
19.08
19.05
19.00
18.95

9.997466
.997452
.997439
.997425
.997411
.997397
.997383
.997369
.997355
.997341

.23
.22
.23
.23
.23

!23
.23
.23

.23

9.034791
.035969
.037144
.038316
.039485
.040651
.041813
.042973
.044130
.045284

19.63
19.58
19.53
19.48
19.43
19.37
19.33
19.28
19.23
19.17

10.965209
.964031
.962856
.961684
.960515
.959349
.958187
.957027
.955870
.954716

49
48
47
46
45
44
43
42
41
40

21
22
23
24

9.043762
.044895
.046026
.047154

18.88
18.85
18.80

9.997327
.997313
.997299
.997285

.23
.23
.23

9.046434
.047582

.048727
.049869

19.13
19.08
19.03

18 Oft

10.953566
.952418
.951273
.950131

39
38
37
36

25

26
27

28

.048279
.049400
.050519
.051635

18.75
18.68
18.65
18.60

.997271
.997257
.997242
.997228

.23
.23
.25
.23

.051008
.052144
.053277
.054407

Jo.yo
18.93
18.88
18 83

18 8ft

.948992
.947856
.946723
.945593

35
34
33
32

29
30

.052749
.053859

18.57
18.50
18.45

.997214
.997199

.23
.25
.23

.055535
.056659

Jo.oU

18.73
18.70

,944465
.943341

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
45
46

9.065885
.066962
.068036
.069107
.070176
.071242

17.95
17.90
17.85
17.82

1H7

9.997039
.997024
.997009
.996994
.996979
.996964

.25
.25
.25
.25
.25

9.068846
.069938
.071027
.072113
.073197
.074278

18.20
18.15
18.10
18.07
18.02

10.931154
.930062
.928973
.927887
.926803
.925722

19
18
17
16
15
14

47

.072306

<•» A7

.996949

.25

| .075356

17.97

.924644

13

48

.073366

17* AQ

.996934

.25

.076432

17.93

.923568

12

49
50

.074424
.075480

li .OO

17.60
17.55

.996919
.996904

.25
.25
.25

.077505
.078576

17.88
17.85
17.80

.922495
.921424

11
10

51
52
53
54

9.076533
.077583
.078631
.079676

17.50

17.47
17.42

1r» QQ

9.996889
.996874
.996858
.996843

.25
.27
.25

9.079644

.080710
.081773

.082833

17.77
17.72
17.67

10.920356
.919290
.918227
.917167

9

8
7
6

55
56

.080719
.081759

< . oo

17.33

.996828
.996812

.27
27

.083891
.084947

17 '.60

.916109
.915053

5
4

57

58

.082797
.083832

17.30
17.25

.996797
.996782

'.25

.086000
.087050

17.55
17.50

17 /17

.914000
.912950

3
2

59

.084864

17.20

.996766

•~*

.OS8098

( .4<

.911902

1

60

9.085894

17.17

9.996751

.2o

9.089144

17.43

10.910856

0

'

Cosine.

D. r.

Sine.

D. r.

Cotang. | D. 1". | Tang.

'

98°

182

83*

COSINES, TANGENTS, AND COTANGENTS. 172°

•

Sine.

D. 1'.

Cosine.

B

Tan,.

D. r.

Cotang.

/

0

1

2

9 085894
.086922

.087947

17.13

17.08

9.996751
. 990735
990?'20

27

9.089144
.090187

17.38

17 :>5

10.910856
.909818

60
59

3
4
5
6
7
8
9
10

11
12
13
14
15
16
17
18
19
20

. 088970
.089990
.091008
.092024
.093037
.094047
.095056
.096062

9 097065
.098066
. 099065
. 100002
.101056
. 102048
. 103037
. 104025
.105010
. 105992

17.00
10.97
10.93
10.88
16.83
16.82
16.77
16.72

16.68
16.65
16.62
16.57
16.53
16.48
16.47
16.42
16.37
16.35

.990704
. 990088
.9900?:}
.990057
.990041
.990025
.990010
.990594

9.996578
.996562
.996546
.996530
.996514
996498
.996482
.990405
.990449
.996433

' o~

/25
.27

jr

.27

i

27
.'27
.28
27
.'27
27

.092266
.098302

.094386
.095307
.090395
.097422
.098446
.099468

| 9.100487
I .101504
.102519
. 103532
.104542
. 105550
.106556
.107559
.108500
.109559

7.30
7.27
7.23
7.18
7.13
7.12
7.07
7.03
16.98

16.95
10.92
16.88
16.83
16.80
16.77
16.72
16.68
16.65
if? f\9

.908772

.'.HKifiiW

.905664

.904688

.903005
.902578
.901554

.900532

10.899513
.898496

.897481
.896468
.895458
.894450
.893444
.892441
.891440
.890441

58
57
56
55
54
53
52
51
50

49
48
47
46
45
44
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 i
15.97

9.996417
.996400
.996384
.996368
.990351
.996335
.996318
.996302
.996285
.996269

.28
.27
.27

.28

3

.27
.28
.27
28

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

16.58
16.53
16.50
10.47
16.43
16.40
16.35
16.33
16.28
16 25

10.889444
.888449
.887457
.886467
.885479
.884493
.888509
.882528
.881548
.880571

39
38
37
36
35
84
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9 116656
.117613
118567
.119519
. 120469
.121417
. 122302
.123306
.124248
.125187

15 95 i
15.90 i
15 87
15.83
15.80
15.75
15.73
15.70
15.05
15 63

9.996252
.996235
996219
.996202
.996185
.996168
.996151
.990134
.990117
.996100

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

9.120404
.121377
.122348
.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
43
44

9 126125
.127000
.127993
.128925

15.58 >
15.55
15 53
15 48

9.996083
.990066
.996049
.996032

.28
.28
.28

oo

9.130041
.130994
.131944
.132893

15 88
15.83
15.82

10.869959
.869006
.808056
.867107

19
18
17
16

45
46
47
48
49
50

129854
130781
131706
132030
133551
134470

15 45
15.42 !
15.40
15 35
15.32
15.28

.996015
. 995998
.995980
. 995963
995946
.995928

.28
' .30
.28
28
30
.28

133839
.134784
. 135726
.136667
.137605
.138542

15.75
15.70
15 68
15 63
15 62
15 57

.866161
.865216
.864274
803333
862395
.861458

15
14
13
12
11
10

51
52
53

9 135387
.136303
137210

15 27

15 22

3 995911
995894
.995876

28
.30

9 139476
. 140409

.141340

15.55

15 52

10 860524
.859591
866660

9

8

7

54

138128 i

. 995859

28

.142269

15.48

.857731

6

55

139037

995841

143196

S50N >4

5

56
57
58
59
60

139944
140850
.141754
142055
3 143555

15 10 j
15 07
15 02
15.00

995823
995806
995788
995771
9 995753

.28
.30
28
.30

.144121
145044
.145966
146885
9.147S03

15 38
15 37
15.32
15.30

. s:,.>79
854956

854084
858115

4
3
2

0

i

Cosine.

D. r. i

Sine. 1

D. r. i

Cotang.

D. r.

Tang.

'

97"

183

82"

TABLE XII. — LOGARITHMIC SINES,

171°

'

Sine.

D. r.

Cosine.

D. 1'.

Tang.

D. 1".

Cotang.

'

0

9.143555

9.995753

qA

9.147803

1 fi 9^

10.852197

60

1

2
3
4
5
6

.144453
.145349
.146243
.147136
.148026
.148915

14.97
14.93
14.90

14.88
14.83
14.82

.995735
.995717
.995699
.995681
.995664
.995646

.OU

.30
.30
.30
.28
.30

OA

.148718
. 149632
.150544
.151454
.152363
.153269

lO.^D

15.23
15.20
15.17
15.15
15.10

.851282
.850368
.849456
.848546
.847637
.846731

59
58
57
56
55
54

7
8
9

.149802
.150686
.151569

14.78
14.73

14 -I2

.995628
.995610
.995591

.OU

.30
.32
30

.154174
.155077
.155978

15.08
15.05
15.02

.845826
.844923
.844022

53
52
51

10

.152451

14.70
14.65

.995573

.156877

14^97

.843123

50

11
12

9.153330
.154208

14.63

9.995555
.995537

.30

9.157775

.158671

14.93

10.842225
.841329

49

48

13
14
15
16
17
18
19

.155083
.155957
.156830
.157700
.158569
.159435
.160301

14.58
14.57
14.55
14.50
14.48
14.43
14.43

.995519
.995501
.995482
.995464
.995446
.995427
.995409

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

qo

.159565
.160457
.161347
.162236
.163123
.164008
.164892

14.90
14.87
14.83
14.82
14.78
14.75
14.73
14 70

.840435
.839543
.838653
.837764
.836877
.835992
.835108

47
46
45
44
43
42
41

20

.161164

14.38
14.35

.995390

.O.4

.30

.165774

14^67

.834226

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
14.24
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

9.170547

.171389

14.03

9.995184
.995165

.32

9.175362
.176224

14.37

10.824638
.823776

29

28

33
34
35
36
37
38
39
40

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

14.02
14.00
13.97
13.93
13.90
13.88
13 85
13.83
13.80

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

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

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

14.33
14.30
14.28
14 27
14.22
14.20
14.18
14.13
14.13

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

27
26
25
24
23
22
21
20

41
42

9 178900
.179726

13.77

9.994993
.994974

.32
32

9.183907

.184752

14.08
14 08

10.816093
.815248

19
18

43
44
45
46

47

.180551
.181374
.182196
.183016
.183834

13! 72
13.70
13.67
13.63

1 ^ P>9

.994955
.994935
.994916
.994896
.994877

!33
.32
.33
.32

qq

.185597
.186439
.187280
.188120
.188958

14.' 03
14.02
14.00
13.97

•jq oq

.814403
.813561
.812720
.811880
.811042

17
16
15
14
13

48
49

.184651
.185466

Jo. \>6

13.58
13 57

.994857
.994838

.00

.32
33

.189794
.190629

Jo.yo

13.92

-«q OQ

.810206
.809371

12
11

50

.186280

13! 53

.994818

.191462

JO.oo

13.87

.808538

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
.1947'80

13.83

13.82
13.78

IQ p"v

10.807706

.806876
.806047
.805220

9

8
7
6

55
56
57
58
59
60

.190325
.191130
.191933
.192734
.193534
9.194332

13.43
13.42
13.38
13.35
13 33
13 30

.994720
.994700
.994680
.994660
. 994640
9.994620

.'33
.33
.33
.33
.33

.195606
.196430
.197253
.198074
.198894
9.199713

Jo. i i
13.73
13.72
13.68
13.67
13.65

.804394
.803570
.802747
.801926
.801106
10.800287

5
4
3
2
1
0

'

Cosine.

D. r.

Sine.

D. 1".

Cotang.

D. r.

Tang.

'

98«

184

81*

COSINES, TANGENTS, AND COTANGENTS.

170°

'

Sine.

D.r.

Cosine.

D.r.

Tang.

D.r.

Cotang.

'

0

1

2
3
4
5
6
7
8
9
10

9.194332
.195129
.195925
.196719
.197511
.198302
.199091
.199879
.200666
.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
.88
.88
.88

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

! 9.199713
.200529
.201345
.202159
.202971
.2037K2
.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

.797841

.796818
.795408
.794600
.798798
.798987
.79X188

60
BO

58
57
56
55

58
51

50

11
12
18
14
15
16
17
18
19
20

9.203017
.203797
.204577
.205354
.206131
.206906
.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
.212611
.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

.786595
.785802
.785011

.784220

40
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

10.783432
.782644
.781858
.781074
.780290
.779508
.778?28
.777948
.777170
.770393

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38

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

12.55
12.53
12.50
12.48
12.47
12.43
12.42

19 QQ

9.993982
.993960
.993939
.993918
.9938&T
.993875
.993854
.993832

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

9.224382
.225156
.225929
.2267'00
.227471
.228239
.229007
.229773

12.90

12.88
12.85
12.85
12.80
12.80
12.77

10.775618
.774844
.774071
.773300
.778588
.771701
.770993
770227

29
28
27
26
85
84
88
88

39
40

.224349
.225092

12.38
12.35

.993811
.993789

.37
.35

.230539
.231302

12.77
12.72
12.72

[769461

.768698

81

20

41
42
43
44

9.225833
.226573
.227311

.228048

12.33
12.30
12.28

19 97

9.993768
.993746
.993725
.993703

.37
.35
.37

07

9.232065
.232826
.233586
.234345

12.68
12.67
12.65

19 A5t

10.767935

.707174
.766414

.70511:,*

19
18
17
16

45
46
47
48
49
50

.228784
.229518
.230252
.230984
.231715
.232444

12.23
12.23
12.20
12.18
12.15
12.13

.993681
.993660
.993638
.993616
.993594
.993572

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

.235103
.235859
.236614
.237368
.238120
.238872

12.60
12.58
12.57
12.53
12.53
12.50

.764897
.704141
.788886
.788688

.761880
.761128

15

14

18

12
11
10

51
52
53
54
55
56
57
58
69

9.233172
.233899
.234625
.235349
.236073
.236795
.237515
.238235
.238953

12.12
12.10
12.07
12.07
12.03
12.00
12.00
11.97

9.993550
.993528
.993506
.993484
.993462
.993440
.993418
.993396
.993374

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

9.239622
.240371
.241118
.241865
.242610
.243354
.244097
.244839
.24557!)

32.48
12.45
12.45
12.42
12.40
12.38
12.37
12.33

10.760378
.759688
.758888
.758185

9
8
7
6
5
4
8
8
1

60

9.239670

11.95

9.993351

.38

9.246319

0

'

Cosine.

D.r. 1

Sine.

D.r. i!

Cotang.

D.r.

Tang.

'

89°

185

80"

10°

TABLE XII. — LOGARITHMIC SINES,

169°

1

Sine.

D. 1".

Cosine.

D.I".

Tang.

D.I".

Cotang.

'

0

1

9.239670
.240386

11.93

9.993351
.993329

.37

9.246319
.247057

12.30

10.753681
.752943

60

59

2

.241101

11.92

.993307

.37

.247794

12.28

.752206

58

3
4
5

.241814
.242526
.24323?

11.88
11.87
11.85

noq

.993284
.993262
.993240

.38
.37
.37

OQ

.248530
.249264
.249998

12.27
12.23
12.23

.751470
.750736
.750002

57
56
55

6

7

.243947
.244656

.00

11.82

niro

.993217
.993195

.OO

.37

OQ

.250730
.251461

12.20
12.18

.749270
.748539

54
53

8
9
10

.245363
.246069
.246775

. 10

11.77
11.77
11.72

.993172
.993149
.993127

.OO

.38
.37

.38

.252191
.252920
.253648

12.17
12.15
12.13
12.10

.747809
.747080
.746352

52
51
50

11
12
13
14

9.247478
.248181

.248883
.249583

11.72
11.70
11.67

9.993104
.993081
.993059
.993036

.38
.37
.38

9.254374
.255100
.255824
.256547

12.10
12.07
12.05

10.745626
.744900
.744176
.743453

49

'48
47
46

15
16

.250282
.250980

11.65
11.63

.993013
.992990

.38
.38

.257269
.257990

12.03
12.02

.742731
.742010

45

44

17

.251677

11.62

.992967

.38

.258710

12.00

.741290

43

18

.252373

11 .60

.992944

.38

.259429

11 .98

.740571

42

19
20

.253067
.253761

11.57
11.57
11.53

.992921
.992898

.38
.38
.38

.260146
.260863

11 .95
11.95
11.92

.739854
.739137

41
40

21

9.254453

nKO

9.992875

9.261578

10.738422

39

22

.255144

,ue

.992852

.38

.262292

11 .90

.737708

38

23

.255834

11.50

.992829

.38

.263005

11.88

.736995

37

24

.256523

11 .48

.992806

.38

.263717

11.87

.736283

36

25
26

.257211

.257898

11.47
11.45

.992783
.992759

.38
.40

.264428
.265138

11.85
11.83

.735572

.734862

35
34

27

.258583

11.42

.992736

.38

.265847

11.82

.734153

33

28

.259268

11 .42

.992713

.38

.266555

11.80

.733445

32

29

.259951

11.38

.992690

.38

.267261

11.77

.732739

31

30

.260633

11.37
11.35

.992666

.40
.38

.267967

11.77
11.73

.732033

30

31

32

9.261314
.261994

11.33

9.992643
.992619

.40

9.268671
.269375

11.73

10.731329

.730625

29

28

33
34

.262673
.263351

11.32
11.30

HO7

.992596
.992572

.38
.40

.270077
.270779

11.70
11.70

.729923
.729221

27
26

35
36
37
38
39
40

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

>iff

11.27
11.23
11.23
11.20
11.20
11.17

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

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

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

11.67
11.65
11.63
11.62
11.60
11.58
11.57

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

25
24
23
22
21
20

41
42

9.268065
.2687:34

11.15

9.992406

.992382

.40

9.275658
.276351

11.55

10.724342
.723649

19

18

43
44

45

.269402
.270069
.270735

11.13
11.12
11.10

nftS

.992359
.992335
.992311

.38
.40
.40

.277043
.277734
.278424

11.53
11.52
11.50

.722957
.722266
.-721576

17
16
15

46
47
48

.271400
.272064
.272726

,\Jo

11.07
11.03

.992287
.992263
.992239

.40
.40
.40

.279113
.279801
.280488

11.48
11.47
11.45

.720887
.720199
.719512

14

13
12

49
50

.273388
.274049

11.03
11.02
10.98

.992214
.992190

.42
.40

.40

.281174
.281858

11.43
11.40
11.40

.718826
.718142

11
10

51

9.274708

1A Qft

9.992166

9.282542

10.717458

9

52
53

.275367
.276025

lU.yo
10.97

.992142
.992118

.40
.40

.283225
.283907

11.38
11.37

.716775
.716093

8

7

54

.276681

10.93

.992093

.42

.284588

11.35

.715412

6

55
56

.277337
.277991

10.93
10.90

.992069
.992044

.40
.42

.285268
.285947

11.33
11.32

.714732
.714053

5

4

57
58

.278645
.279297

10.90
10.87

.992020
.991996

.40
.40

.286624
.287301

11.28
11.28

.713376
.712699

3
2

59
.60

.279948
9.280599

10.85
10.85

.991971
9.991947

.42 '
.40

.287977
9.288652

11 .27
11.25

.712023
10.711348

1
0

'

Cosine.

D.I".

Sine.

D. r.

Cotang.

D. 1". Tang.

'

100°

186

79*

11°

COSINES, TANGENTS; AND COTANGENTS. lee"

/

Sine.

D. 1".

Cosine.

D.I".

Tang.

D. r.

Cotang.

/

0

1

2
3
4
5
6
7
8
9
10

9.280599
.281248
.281897
.282544
.283190
.283836
.284480
.285124
.285766
.286408
.287048

10.82
10.82
10.78
10.77
10.77
10.73
10.73
10.70
10.70
10.67
10.67

9.991947
.991922
.991897
.991873
.991848
.991823
.991799
.991774
.991749
.991724
.991699

.42
.42
.40
.42
.42 !
.40
.42
.42
.42
.42
.42

9.288652
.880606

.289999
.290671
.291342
.292013
.292682
.293350
.294017
.294684
.295349

11.23
1 1 . -,'-j
11.20
11.18
11.18
11.15
11.13
11.12
11.12
11.08
11.07

10.711348
.710074
.710001
.709329
.708658
.707987
.707318
.706650
.706868
.705316
.704651

60
68

58
67

56
55
51
53
52
51
50

11
12

9.287688
.288326

10.63
in fi^

9.991674
.991649

.42

9.296013
.296677

11.07

10.703987
.703323

49

48

13

.288964

1ft Aft

.991624

.297339

.702661

47

14
15
16
17
18
19

.289600
.290236
.290870
.291504
.292137
.292768

10.60
10.57
10.57
10:55
10.52

1ft ^9

.991599
.991574
.991549
.991524
.991498
.991473

.42
.42
.42
.43
.42

A9

.298001
.298662
.299322
.299980
.300638
.301295

11.03
11.02
11.00
10.97
10.97
10.95

.701999
.701338
.700678
.700020
.699362
.698705

46
45
44
43
42
41

20

.293399

10.50

.991448

.43

.301951

10.93

.698049

40

21

9.294029

1ft A&

9.991422

A9

9.302607

10.697393

39

22
23
24
25
26
27
28
29
30

.294658
.295286
.295913
.296539
.297164
.297788
.298412
.299034
.299655

10.47
10.45
10.43
10.42
10.40
10.40
10.37
10.35
10.35

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

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

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

10.88
10.88
10.85
10.85
10.83
10.82
10.80
10.78
10.77

.696739
.696086
.695433
.694782
.694131
.098481
.692832
.892184
.691537

38
37
36
35
34
33
32
M
30

31
32
33
34
35
36
37
38

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

10.32
10.32
10.30
10.27
10.27
10.25
10.23

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

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

9.309109
.309754
.310399
.311042
.311685
.312327
.312968
.313608

10.75
10.75
10.72
10.72
10.70
10.68
10.67

10.690891
.690246
.689601
.868968

.688315
.687673
.687032
.686392

29
28
27
26
25
24
23
22

39
40

.305207
.305819

10.23
10.20
10.18

.990960
.990934

.43
.43

.314247
.314885

10.63
10.63

.886758

.685115

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
319330
.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
.882670
.681936
.681303

.oswro

.I5SM39
JJT'.MOS
.678778

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 i
.45

9.321851
.322479
.323106
.323733
.324358
.3249a3
.325607
.326231
.826858
9.827175

10.47
10.45
10.45
10.42
10.42
10.40
10.40
10.37
10.37

10.678149
.677521
.678894
.878887

.CHOI; 10

.675017
.674393

.878147

9
8
7
6
5
4
3
2

0

/

Cosine.

D.I-.

Sine.

D. I', i

Cotang. D. 1'. 1

Tang.

/

101'

187

78*

12°

TABLE XII. — LOGARITHMIC SINES,

167°

'

Sine/

D. r.

Cosine.

D.r.

Tang.

D.l".

Cotang.

'

0

1

9.317879
.318473

9.90

9QO

9.990404
.990378

.43

9.327475

.328095

10.33
in v%

10.672525
.671905

60
59

2
3

.-319066
.319658

.GO

9.87

9 OK

.990351
.990324

!45
45

.328715
.329334

1U.OO

10.32
10 32

.671285
.670666

58
57

4
5

.320249

.320840

.OO

9.85

.99029?
.990270

'.45

.329953
.330570

10'.28

.670047
.669430

56
55

6

7
8
9
10

.321430
.322019
.322607
.323194
.323780

9.83
9.82
9.80
9.78
9.77
9.77

.990243
.990215
.990188
.990161
.990134

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

.331187
.331803
.332418
.333033
.333646

10.28
10.27
10.25
10.25
10.22
10.22

.668813
.668197
.667582
.666967
.666354

54
53
52
51
50

11
12
13
14
15
16

9.324366
.324950
.325534
.326117
.326700
.327281

9.73
9.73
9.72
9.72

9.68
9fift

9.990107
.990079
.990052
.990025
.989997
.989970

.47
.45
.45
.47
.45

9.334259
.334871
.335482
.336093
.336702
.337311

10.20
10.18
10.18
10.15
10.15
in 1 Q

10.665741
.665129
.664518
.663907
.663298
.662689

49
48
47
46
45
44

17

18
19
20

.327862
.328442
.329021
.329599

.Do

9.67
9.65
9.63
9.62

.989942
.989915
.989887
.989860

'.45

.47
.45
.47

.337919
.338527
.339133
.339739

lU. lo
10.13
10.10
10.10
10.08

.662081
.661473
.660867
.660261

43
42
41

40

21
22
23
24

9.330176
.330753
.331329
.331903

9.62
9.60
9.57

9.989832
.989804
.989777
.989749

.47
.45
.47

9.340344
.340948
.341552
.342155

10.07
10.07
10.05

10.659656
.659052
.658448
.657845

39
38
37
36

25
26

27
28
29
30

.332478
.333051
.333624
.334195
.334767
.335337

9.58
9.55
9.55
9.52
9.53
9.50
9.48

.989721
.989693
.989665
.989637
.989610
.989582

.47
.47
.47
.47
.45
.47
.48

.342757
.343358
I .343958
.344558
.345157
.345755

10.03
10.02
10.00
10.00
9.98
9.97
9.97

.657243
.656642
.656042
.655442
.654843
.654245

35
34

32
31
30

31

9.335906

9.989553

9.346353

10.653647

29

32

.336475

•*• ^

.989525

.47

.346949

9QQ

.653051

28

33
34
35
36

.337043
.337610
.338176
.338742

9.47
9.45
9.43
9.43

Q 49

.989497
.989469
.989441
.989413

.47
.47
.47
.47

.347545
.348141
.348735
.349329

. yo
9.93
9.90
9.90
9 88

.652455
.651859
.651265
.650671

27
26
25
24

37

.339307

y .e±&

Q 4fi

.989385

*4ft

.349922

.650078

23

38

.339871

y .4u

Q ^8

.989356

,4o

.350514

9 87

.649486

22

39
40

.340434
.340996

y .00
9.37
9.37

.989328
.989300

!47
.48

.351106
.351697

9^85
9.83

.648894
.648303

21
20

41

42

9.341558
.342119

9.35

Q VI

9.989271
.989243

.47

9.352287
.352876

9.82

10.647713
.647124

19
18

43

.342679

y.oo
900

.989214

Art

.353465

9 DA

.646535

17

44
45
46

.343239
.343797
.344355

.00
9.30
9.30

Q 9ft

.989186
.989157
.989128

.47

.48
.48

.354053
.354640
.355227

.oU

9.78
9.78

9r»r>

.645947
.645360
.644773

16
15
14

47

.344912

y .Aio

.989100

.47

.355813

. < <

9r<(r

.644187

13

48

.345469

0 OK

.989071

.48

.356398

. < O

.643602

12

49

.346024

9.XO

n OK

.989042

"^

.356982

9»~o

.643018

11

50

.346579

9^25

.989014

.47

.48

.357566

. <o

9.72

.642434

10

51

52

9.347134

.347687

9.22

999

9.988985
.988956

.48

9.358149
.358731

9.70

q I~A

10.641851
.641269

9

S

53 .348240

.MB

9 20

.988927

4ft

.359313

9'fff

.640687

7

54 .348792

91ft

.988898

.4o
4ft

.359893

.D<
9fift

.640107

6

55
56
57
58
59

.349343
.349893
.350443
.350992
.351540

. 15

9.17
9.17
9,15
9.13

n 1 o

.988869
.988840
.988811
.988782
.988753

.4o

.48
.48
.48
.48

/4ft

.360474
.361053
.361632
.362210

.362787

.uo
9.65
9.65
9.63
9.62

.639526
.638947
.638368
.637790
.637213

5
4
3
2
1

60 9.352088 ">10

9.988724

.4o

9.363364

9.62

10.636636

0

' I Cosine. D. 1". |l Sine, | D. 1".

Cotang.

D. 1*. Tang.

'

188

77*

COSINES, TANGENTS, AND COTANGENTS. 166°

'

Sine.

D.I'.

Cosine.

D. r. | Tang.

D. r.

Cotang.

•

0 9.352088
1 .352635

9.12
9 10

9.988724
.988695

,o 9.363364
•Jjj .363040

9.60

10.636036
.030000

60
59

2
3

.853181
.353726

9[ 08

.988666

.988036

! 365090

9.56

9.58

.035485

57

4

354271 ' v'-

.988607

4s -;^5C6i

9.57

.034336

56

5

'.354815 o'ar

.988578

9 . 55

55

6

.355358 V "V!

.988548

! 866810

9.66

.688190

54

7

.355901

j . \-><)

.988519

•?;: .367382

9.53

.032018

58

8

.356443

o no

.988489

.867968

9.52

.032047

52

9

.356984

0 00

.988460

•g : .368524

!' . 52

.031476

51

10

.357524

9[oo

.988430

•Jg ; .309094

9.48

.630906

50

11

9.358064

8 no

9.988401

',n 9.309663

10.630337

49

12

.358603

.yo

.988371

.370232

9.48

.629768

48

13
14
15

.359141
.359678
.360215

8.97
8.95
8.95

8Qt

.988:342
.988312

.988282

•Jg i] .370799
•JX -371367

.371 1(33

9.45
9.47
9.43

.029201
.628688
.688067

47
46
45

16

.360752

.yo

8 GO

; 988252

•JH ; .372499

9.43

.627501

44

17
18
19
20

.361287
.361822
.362356
.362889

.We

8.92
8.90

8.88
8.88

.988223
.988193
.988163
.988133

•J2 i -373064
•2J .373629
i .374193
•;*; i .374756

9 '.42
9.40
9.38
9.38

.626936
.620371
.625807
.625244

43
42
41

40

21
22
"23
24

9.363422
.363954
.364485
365016

8.87
8.85
8.85

9.988103
.988073
.988043
.988013

'" 9.375319
.375881
•rj , -376442
'I., .377003

9.37
9. '35
9.35

900

10.624681
.024119
.623558
.622997

39
38
87
36

26
27
28
29
30

.365546
.366075
.366604
.367131
.367659
.368185

8.83
8.82
8.82
8.78
8.80
8.77
8.77

.9879813
.987953
1 .987922
.987892
.987862
.987832

[50
.52
.50
.50
.50
.52

.377563
.378122
.378681
.379239
.379797
.380354

.Oo

9.32
9.32
9.30
9.30
9.28
9 27

.622437
.621878
.621319
.620761
.620203
.619646

35
84

33
32
31
30

31
32
33
34
35
36
37
38
39
40

9.368711
.369236
.369761
.370285
.370808
.371,330
.371852
.372373
.372894
.373414

8.75
8.75
8.72
8.72
8.70
8.70
8.68
8.68
8.67

8C.fi

9.987801
1 .987771
.987740
.987710
.987679
.987649
.987618
.987588
.987557
.987526

.50
.52
.50
.52

.50
.52
.50
.52
.52
.50

9.380910
.381466
.382020
.382575
.383129
.383682
.384234
.384786
.385337
.385888

9.27
9.23
9.25
9.23
9.22
9.20
9.20
9.18
9.18
9.17

10.619090
.618534
.617980
.617425
.616871
.616318
.015700
.615214
.614663
.614112

29
28
27

25
24

a

22

n

20

41
42
43
44
45
46
47
48
49
50

51
52
53
54
55
56
57
58
59
60

9.373933
.374452
.374970
.375487
.376003
.376519
.3770,35
.377549
.378063
.378577

9-379089
.379601
.380113
.380624
.381134
.381643
.382152
.382661
.383168
9.383675

.DO

8.65
8.63
8.62
8.60
8.60
8.60
8.57
8.57
8.57
8.53

8.53
8.53
8.52
8.50
8.48
8.48
8.48
8.45
8.45

9.987496
.987465
.987434
.987403
.987372
.987341
.987310
.987279
.987248
.987217

9.987186
.987155
.987124
.987092
.987061
.987030
.986998
.986967
.986936
9.986904

.52
.52
.52
.52
.52
.52
.52
.52
.52
.52

.52
.52
.53
.52
.52
.53
.52
.52
.53

9.386438
.386987
.387536
.388084
I .388631
.389178
! .389724
I .390270
.390815
.391360

9.391903
.392447
.302989
.31*3531
.394073
.394614
.395154
.895694
.896388
9.396771

9.15
9.15
9.13
9.12
9.12
9.10
9.10
9.0f
9.08
9.05

9.07
9.03
9.03
9.03
9.01
9.00
9.00
8.0
8.97

10.613562
.613013
.612464
.611916
.611869
.610822
.610270

.609186

.608640

10.608087
.607558
.607011
.606469
.605927
.606186
[604840
[604806
.608707
10.603229

19
18
17
16
15
14
13

11
10

9

8
7
6

I

8

1

0

.

/

Cosine.

D. r.

Sine.

D. 1". Cotang.

D. r.

Tang. I '

1 7.6'

103'

189

14°

TABLE XII. — LOGARITHMIC SINES,

165°

'

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. r.

Cotang.

'

0

9.383675

8AK.

9.986904

52

9.396771

8Q7

10.603229

60

1

.384182

.10
8 An

.986873

.397309

.y<

8 QK

.602691

59

2
3

4

.384687
.385192
.385697

,*Bi

8.42
8.42

.986841
.986809
.986778

!53
.52

KO

.397846
.398383
.398919

.yo

8.95
8.93

8 QO

.602154
.601617
.601081

58
57
56

5
6

.386201
.386704

8^38

8OQ

.986746
.986714

.Do

.53

.399455
.399990

. yo
Son

.600545
.600010

55

54

7
8

.387207
.387709

. OO

8.37

8 OK

.986683
.986651

!53

KO

.400524
.401C58

.yu
8.90

8QO

.599476
.598942

53

52

9

.388210

.oD

.986619

.Do

KO

.401591

.OO
800

.598409

51

10

.388711

8.35
8.33

.986587

.Do

.53

.402124

.OO

8.87

.597876

50

11

9.389211

800

9.986555

KO

9.402656

8QK

10.597344

49

12
13

.389711
.390210

. OO

8.32

8 on

.986523
.986491

.Do

.53

KO

.403187
.403718

.OD

8.85
8 85

.596813
.596282

48
47

14
15

.390708
.391206

.ou

8.30

.986459
.986427

.Oo

.53

.404249
.404778

8. '82

8QO

.595751
.595222

46
45

16

.391703

8.28

897

.986395

.53

to

.405308

.OO
8 Oft

.594692

44

17
18
19
20

.392199
.392695
.393191
.393685

.4(

8.27
8.27
8.23
8.23

.986363
.986331
.986299
.986266

.Do

.53
.53
.55
.53

.405836
.406364
.406892
.407419

. oil

8.80
8.80
8.78
8.77

.594164
.593636
.593108
.592581

43
42
41
40

21

9.394179

9.986234

KO

9.407945

877

10.592055

39

22
23

.394673
.395166

8^22

.986202
.986169

.Do

.55

pro

.408471
.408996

. < <

8.75
8 75

.591529
.591004

38
37

24
25

26

.395658
.396150
.396641

8^20
8.18

.986137
.986104
.986072

.Do

.55
.53

.409521
.410045
.410569

8.73
8.73

ft "~O

.590479
.589955
.589431

36
35
34

27
28

.397132
.397621

8.18
8.15
81"

.986039
.986007

.55
.53

.411092
.411615

8 '.72
8 70

.588908
.588385

33
32

29

.398111

.14

.985974

.55

.412137

8 Aft

.587863

31

30

.398600

8.15
8.13

.985942

.53
.55

.412658

. Do

8.68

.587342

30

31
32

9.399088
.399575

8.12

9.985909
.985876

.55

9.413179
.413699

8.67

10.586821
.586301

29

28

33

.400062

8.12

.985843

.55

KO

.414219

8f>K

.585781

27

34

.400549

8. 12
8m

.985811

.Do

.414738

. OO
8 OK

.585262

26

35

.401035

.10
8 Aft

.985778

.55

.415257

. DD

8(\r>

.584743

25

36
37

38

.401520
.402005
.402489

.Uo
8.08
8.07

.985745
.985712
.985679

.55
.55
.55

.415775
.416293
.416810

.Do

8.63
8.62

80ft

.584225
.583707
.583190

24
23
22

39
40

.402972
.403455

8.' 05
8.05

.985646
.985613

.55
.55
.55

.417326
.417842

. DU

8.60
8.60

.582674
.582158

21
20

41

42
43

9.403938
.404420
.404901

8.03
8.02

9.985580
.985547
.985514

.55
.55

9.418358

.418873
.419387

8.58
8.57

8K7

10.581642
.581127
.580613

19

18
17

44
45

.405382
.405862

8.02
8.00

rj QQ

.985480
.985447

!55

KK

.419901
.420415

.04

8.57

Q KK

.580099
.579585

16
15

46
47
48
49
50

.406341
.406820
.407299
.407777
.408254

< .yo
7.98
7.98
7.97
7.95
7.95

.985414
.985381
.985347
.985314
.985280

.DO

.55
.57
.55
.57
.55

.420927
.421440
.421952
.422463
.422974

o.DD

8.55
8.53
8.52
8.52
8.50

.579073
.578560
.578048
.577537
.577026

14
13
12
11
10

51

9.408731

7QQ

9.985247

9.423484

8AQ

10.576516

9

52

.409207

.yo

.985213

***

.423993

.4O

8Kfl

.576007

8

53
54

.409682
.410157

7.93
7.92

.985180
.985146

.55
.57

.424503
.425011

. DU

8.47

8 Art

.575497
.574989

7
6

55

.410632

7.92

.985113

.55

.425519

.44

847

.574481

5

56

.411106

7.90

.985079

*£»

.426027

.4<

8AK

.573973

4

57

.411579

7.88

7QQ

.985045

.5<

K7

.426534

.4D

.573466

3

58

.412052

.00

.985011

.D<

.427041

j>" j2

.572959

2

59
60

.412524
9.412996

7.87
7.87

.984978
9.984944

.55
.57

.427547
9.428052

8.'43

.572453
10.571948

1
0

'fl

Cosine.

D. r.

Sine.

D. 1'.

Cotang.

D. r.

Tang, j

'

104°

190

75°

CQSINES, TANGENTS, AND COTANGENTS. 164°

'

Sine.

D. 1'.

Cosine.

D. 1'.

Tang.

D. r.

Cotang.

'

0

1

2

3
4
5
6

7
8
9
10

9.412996
.413467
.413938
.414408
.414878
.415347
.415815
.416283
.416751
.417217
.417684

7.85

7.85
7.83
7.83
7.82
7.80
7.80
7.80
7.77

7^77

9.984944
.984910
.984876
.984842
.984808
.984774
.984740
.984706
.984672
.984638
.984603

.57
.57
.57
.57
.57
.57
.57
.57
.57
.58
.57

9.428052
.428558
.429062
.429566
.430070
.430573
.431075
.431577
.432079
.432580
.433080

8.43
8.40
8.40
8.40
8.38
8.37
8.37
8.37
8.35
8.33
8.33

10.571948
.571442
.570938
.570434
.569930
.569427
.668085
.568423
.507981
.567420
.566920

60
59
58
57
56
55
54
53
B8
51
50

11

9.418150

7 75

9.984569

57

9.433580

Q OQ

10.566420

49

12

.418615

.984535

.434080

O.oO

.565920

48

13
14
15
16
17
18
19
20

.419079
.419544
.420007
.420470
.420933
.421395
.421857
.422318

7! 75

7.72
7.72
7.72
7.70
7.70
7.68
7.67

.984500
.984466
.984432
.984397
.984363
.984328
.984294
.984259

!57
.57
.58
.57
.58
.57
.58
.58

.434579
.435078
.435576
.436073
.436570
.437067
.437563
.438059

8.32
8.32
8.30
8.28
8.28
8.28
8.27
8.27
8.25

.565421
.564922
.564424
.563927
.563430
.562933
.562437
.561941

47
46
45
44
43
42
41
40

21
22
23
24

9.422778
.423238
.423697
.424156

7.67
7.65
7.65

9.984224
.984190
.984155
.984120

.57
.58
.58

to

9.438554
.439048
.439543
.440036

8.2a
8.25
8.22

899

10.561446
.560952
.560457
.559964

39
38
37
36

25
26

.424615
.425073

7.65
7.63

.984085
.984050

.DO

.58

.440529
.441022

.£&

8.22

.559471
.558978

35
34

27

.425530

7.62

.984015

.58

.441514

8.20

.558486

33

28

425987

7.62

.983981

.57

.442006

8.20

.557994

32

29

.426443

7.60

.983946

.58

fro

.442497

8.18

81B

.557503

31

30

.426899

7.60
7.58

.983911

.Do

.60

.442988

. io
8.18

.557012

30

31

9.427354

9.983875

to

9.443479

81K

10.556521

29

32

.427809

7.58

.983840

.Do
to

.443968

. ID
817

.556032

28

33

.428263

7.57

.983805

.DO

tQ

.444458

.if

81^

.555542

27

34

.428717

7.57

.983770

.Do
KB

.444947

. ID

.555053

26

35

.429170

7.55

.983735

.DO

.445435

8.13

.354565

25

36

.429623

7.55

.983700

.58
fift

.445923

8.13

8-jq

.554077

24

37
38
39
40

.430075
.430527
.430978
.431429

7.53
7 53
7.52
7.52
7.50

.983664
.983629
.983594
.983558

.bU

.58
.58
.60
.58

.446411

.446898
.447384
.447870

. lo

8.12
8.10
8.10
8.10

.553589
.553102
.552616
.552130

23
22
21
20

41

9.431879

9.983523

ftn

9.448356

81 K

10.551644

19

42
43
44
45
46
47
48
49
50

.432329
.432778
.4a3226
.433675
.434122
.434569
.435016
.435462
.435908

7.50
7.48
7.47
7.48
7.45
7.45
7.45
7.43
7.43
7.42

.983487
.983452
.983416
.983381
.983345
.983309
.983273
.983238
.983202

.bU
.58
.60
.58
.60
.60
.60
.58
.60
.60

.448841
.449326
.449810
.450294
.450777
.451260
.451743
.452225
.452706

.Uo

8.08
8.07
8.07
8.05
8.05
8.05
8.03
8.02
8.02

.551159
.550674
.550190
.549706
.540223
.548740
.548257
.547775
.547294

18
17
16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

9.436353
.436798
.437242
.437686
.438129
.438572
.439014
.439456
.439897
9.440338

7.42
7.40
7.40
7.38
7.38
7.37
7.37
7.35
7.35

9.983166
.983130
.983094
.983058
.983022
.982986
.982950
.982914
.982878
9.982842

.60
.60
.60
.60
.60
. .60
.60
.60
.60

9.453187
.453668
.454148
.454628
.455107
.455586
.456064
.456542
.457019
9.457496

8.02
8.00
8.00
7.98
7.98
7.97
7.97
7.95
7.95

10.546813
.546332
.M.J.SM

!544898

.544414
.543936

.5-i.ms
.548981
10.542504

9
8
7
6
5
4

2
1
0

* \ Cosine.

D. I'.

Sine.

D. r.

Cotang.

D. r.

Tang.

105-

191

16°

TABLE XII. — LOGARITHMIC SINES,

163°

'

Sine.

D.I".

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

1

9.440338
.440778

7.33

7 33

9.982842
.982805

.62

9.457496
.457973

7.95

10.542504
.542027

60
59

2
3
4

.441218
.441658
.442096

7^33
7.30
7 32

.982769
.982733
.982696

'.60
.62

.458449
.458925
.459400

7.93
7.93
7.92

.541551 58
.541075 57
.540600 56

5
6

7
8

.442535
.442973
.443410
.443847

7^30
7.28
7.28
7 28

>! .982660
.982624
.982587
.982551

!eo

.62
.60

.459875
.460349
.460823
.461297

7.92
7.90
7.90
7.90

r- on

.540125 55
.539651 54
.539177 i 53
.538703 52

9

.444284

.982514

*RO

.461770

< .OO

.538230 51

10

.444720

7^25

.982477

'.60

.462242

7.87
7.88

.537758 50

11

9.445155

7 25

9.982441

9.462715

10.537285 49

12

.445590

7*25

.982404

Rf>

.463186

Z'52

.536814 48

13

.446025

7 23

.982367

.uSB

60

.463658

1 .O<
r- oq

.536342 47

14

.446459

7*2,3

.982331

62

.464128

t .OO

.535872 46

15

.446893

7 22

.982294

62

.464599

ii*S|

.535401 45

16
17

.447326
.447759

l'-22

.982257
.982220

.*62

..465069
.465539

7^83

.534931 44
.534461 43

18

.448191

7*90

.982183

RO

.466008

7.82

7 BO

.533992

42

19
20

.448623
.449054

riia

7.18

.982146
.982109

!62
.62

.466477
.466945

t .0/4

7.80
7.80

.533523
.533055

41
40

21

9.449485

7 17

9.982072

9.467413

r» r>Q

10.532587

39

22

.449915

.982035

ro

.467880

t . t 0

f r-o

.532120

38

23

.450345

717

.981998

RO

.468347

I . <O
r- r-o

.531653

37

24

.450775

7 15

.981961

CO

.468814

i . tO

.531186

36

25

.451204

7 13

.981924

. D/4

CO

.469280

7.77

.530720

35

26

.451632

i . 13

.981886

.DO

.469746

7.77

.530254

34

27

.452060

7 13

.981849

. 62

RO

.470211

7.75

.529789

33

28

.452488

7 12

.981812

.O/6
co

.470676

7. <5

.529324

32

29

.452915

.981774

.Do

RO

.471141

7.75

r> i~o

.528859

31

30

.453342

7.' 10

.981737

.D^

.62

.471605

i .to

7.73

.528395

30

31

9.453768

7 10

9.981700

RP.

9.472069

10.527931

29

32

.454194

r-'no

.981662

.Do

.472532

Z*i?

527468

28

33

.454619

i .Uo

7 08

.981625

.62

! .472995

» . fit

.527005 ! 27

34

.455044

.981587

CO

.473457

(t'r-r!

.526543 26

35

36

.455469
.455893

7\ 07

.981549
.981512

.DO

.62

.473919
.474381

7. iO

7.70

7 Rft

.526081 25
.525619 24

37

.456316

7 05

.981474

.63

RP

.474842

i .Do

.525158 ' 23

38
39

'.456739
.457162

7^05
7 03

.981436
.981399

.Do

.62

RP.

.475303
.4T57G3

7.67

.524697 , 22
.524237 i 21

40

.457584

.981361

.DO

.63

.476223

7. '67

.523777

20

41

9.458006

7 02

9.981323

9.476683

r< R^

10.523317

19

42

.458427

.981285

CO

.477142

t .DO

*" R^

.522858

18

43

.458848

7 00

.981247

.DO

CO

.477601

i .DO

.522399

17

44

.459268

.981209

.DO

.478059

£*S

.521941

16

45

.459688

7nn

.981171

no

.47'8517

Z*2

.521483

15

46

.460108

.uu
6 98

.981133

.DO
RP.

.478975

< .DO

.521025

14

47

.460527

6 no

.981095

.Do

.479432

'Z*S

. 520568*

13

48

.460946

.yo

6Q7

.981057

.63

.479889

< .o2

.520111

12

49

50

.461364
.461782

. y<
6.97
6.95

.981019
.980981

!63
.65

.480345
.480801

7\60
7.60

.519655
.519199

11
10

51

9.462199

6 95

9.980942

cq

9.481257

* KQ

10.518743

9

52

.462616

6oq

.980904

. Do

.481712

7 ^ft

.518288

8

53

54

.463032
.463448

. yo
6.93

.980866
.980827

!65

.482167
.482621

7^57

.517833
.517379

6

55

.463864

6.93

.980789

.63

.483075

7.57

.516925

5

56

.464279

6. 92

.980750

.65

.483529

7.57

.516471

4

57

.464694

6.92
6 90

.980712

.63
65

.4&S982

7.55

.516018

3

58

.465108

6 on

.980673

.484435

7 ^P

.515565

2

59

.465522

. yu

600

.980635

'<£

.484887

r* KO

.515113

1

60

9.465935

.00

9.980596

.DO

9.485339

( .OO

10.514661

0

'

Cosine.

D. 1".

Sine.

D. r.

Cotang.

D. r.

Tang.

'

100*

192

73*

17°

COSINES, TANGENTS, AND COTANGENTS.

162°

'

Sine.

D. 1".

Cosine.

D. r.

Tang.

D. r.

Cotang.

'

0

1

3
4
5
6
7
8
9
10

9.485935
.466348
.466761
.467173
.467585
.467996
.468407
.468817
.469227
.469637
.470046

6.88
6.88
6.87
6.87
6.85
6.85
6.83
6.83
6.83
6.82
6.82

i 9.980596
.980558
.980519
.980480
.980442
.980403
.980364
.980325
.980286
.980247
.980208

.63

.05

!<J3
.65

.05
.05
.05
.65
.65
.05

9.485339
.485791

.486242
.486693
.487143
.487593
.488043
.488492
.488941
.489390
.489838

7.53

7.58

7.50
7.50
7.50
7.48
7.48
7.48
7.47
7.47

10.514661
.514209

.513307
.518857

.511957
.511508
.511059
.510010
.510162

60
59
58
57
56
55
54
56
52
51
50

11

12
13
14
15
16
17
18
19
20

9.470455
.470863
.471271
.471679
.472086
.472492
.472898
.473304
.473710
.474115

6.80
6.80
6.80
6.78
6.77
6.77
6.77
6.77
6.75
6.73

9.980169
.980130
.980091
.980052
.980012
.979973
.979934
.979895
.979855
.979816

.65
.65
.65
.67
.65
.05
.05
.67
.05
.67

9.490286
.490733
.491180
.491627
.492073
.492519
.492965
.493410
.493854
.494299

7.45
7.45
7.45
7.43
7.43
7.43
7.42
7.40
7.42
7.40

10.509714
.509867
.508880

.508373
.507927
.507481
.507035
.506590
.506146
.505701

49
48
47
46
45
44
43
48
41
40

21
22
23
24

9.474519
.474923
.475327
.475730

6.73
6.73
6.72

6r»»>

9.979776
.979737
.979697
.979658

.65
.67
.65

9.494743
.495186
.495630
.496073

7.38
7.40
7.38

10.505257 39
.50481 l
.504370 37 '
.50890S

25
26

27
28
29

.476133
.476536
.476938
.477340
.477741

. i£

6.72
6.70
6.70
6.68
6 68

.979618
.979579
.979539
.979499
.979459

.67
.65
.67
.67
.67

6K

.496515
.496957
.497399
.497841
.498282

7.37
7.37
7.37
7.37
7.35

7 '45*

.506486

.50301
.502601 :«
.502159 32
.501718 31

30

.478142

6^67

.979420

0

.67

.498722

i .00

7.35

.501278

30

31

9.478542

6A7

9.979380

R7

9.499163

ft qq

10.500837 29

32

.478942

.O<

6tV7

.979340

.Of

.499603

I .33

.500397 28

33

.479342

. Oi

.979300

.67

.500042

i .91

.499958 27

34

.479741

6.65

6 AX

.979260

.67

.500481

7.32

f qo

.499519 26

35

36
37

.480140
.480539
.480937

.DO

6.65
6.63

.979220
.979180
.979140

167
.67

.500920
.501359
.501797

1 •'.

7.32
7.30

.499080 25
.498641 24
.498203 23

38

.481334

6.62

61-.)

.979100

.67

no

.502235

7.30

7 9ft

.497765 22

39

40

.481731
.482128

.0/0
6.62
6.62

.979059
.979019

.Do

.67
.67

.502672
.503109

< .*o

7.28
7.28

.49732S -l
.496891 -JO

41

9.482525

6 Art

9.978979

fty

9.503546

7 27

10.496454 1!)

42
43

.482921
.483316

.OU

6.58

.978939
.978898

.O«

.68

.503982
.504418

7^27

.496018 18
.49558-3 17

41

.483712

6.60

.978858

.67

AP.

.504854

7.27

r» OK

.495146 16

45
46
47

48

.484107
.484501
.484895
.485289

6.58
6.57
6.57
6.57

.978817
.978777
.978737
.978696

.DO

.67
.67
.68

.505289
.505724
.506159
.506593

i . -•>

7.25
7.25

7.23

7.).)

.494711 15

.4!ir.'Ti; it
.493841 13
.493407 12

49

.485682

6.55

.978655

.68

.507027

.*B

; 11

50

.486075

6.55
6.53

.978615

.67
.68

.507460

7.'22

;•) 10

51
52
53
54
55
56
57
58
59
60

9.486467
.486860
.487251
.487643
.488034
.488424
.488814
.489204
.489593
9.489982

6.55
6.52
6.53
6.52
6.50
6.50
6.50
6.48
6.48

9.978574
.978533
.978493
.978452
.978411
.978370
978329
.978288
.978247
9.978206

.68
.67
.68
.68
.68
.68
.68
.68
.68

9.507893
.508326
.508759
.509191
.509622
.510054
.510485
.510916
.511346
9.511776

7.22
7.23
7.20
7.18
7.20
7.18-
7.18
7.17
7.17

10.492107
.491674
. J'.M-JH
.490809

. I1MKJ7S

.489946
.489515
.489084
.488654
10.4688M

9
8
7
6
5
4
3
2

0

'

Cosine.

D r.

Sine.

D. 1".

Cotang.

D. 1'.

Tang.

'

107-

193

72-

18°

TABLE XII. — LOGARITHMIC SINES,

161°

'

Sine.

D. 1".

Cosine.

D.I".

Tang.

D. 1".

Cotang.

'

0 9.489982

9.978206

pQ

9.511776

r» -j 7

10.488224

60

1 .490371

6.48

.978165

.Do
fift

.512206

I . 1 1

.487794

59

2 j .490759

6.47

.978124

.Do
Aft

.512635

71^

.487365

58

3 .491147

6.47

.978083

.DO

.513064

t . 1O
71 K.

.486936

57

4 .491535

6.47

.978042

fift

.513493

. lo

71°.

.486507

56

5 .491922

6.45

.978001

.Do
r-A

.513921

. lo

7 -jq

.486079

55

6

7
8
9

.492308
.492695
.493081
.493466

6.43
6.45
6.43
6.42

.977959
.977918

.977877
.977835

. t\J

.68
.68
.70

.514349
.514777
.515201
.515631

« . lo

7.13

7.12
7.12

.485651
.485223
.484796
.484369

54
53
52
51

10

.493851

6.42
6.42

.977794

!70

.516057

7.12

.483943

50

11

9.494236

649

9.977752

68

9.516484

r. 1rt i 10.483516

49

12

.494621

.4/4
A 4ft

.977711

7ft

.516910

,1'Ao .483090

48

13

.495005

D.4U

6qO

.977669

. <U

.517335

£• JQ .482665

47

14

.495388

.OO

.977628

•JJJ-

.517761

r> AQ

.482239

46

15
16

.495772
.496154

6.40
6.37

6qo

.977586
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. <0
.70

Aft

.518186
.518610

< .UO

7.07

7 07

.481814
.481390

45
44

17

.496537

.00
6°.7

.977503

.Do

.519034

r- ryy

.480966

43

18

.496919

.of

.977461

'r-A

.519458

t .U<

.480542

42

19

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6.37

6qK

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. <o

I~A

.519882

7 ft^.

.480118

41

20

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6.35

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. <u
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t .UO

7.05

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21

9.498064

6qq

9.977335

i 9.520728

10.479272

39

22

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6qK

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70

.521151

7 03

.478849

38

23

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.977251

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.478427

37

24

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6.32

6qq

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'7ft

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7.03
7 03

.478005

36

25

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A °.9

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7 ft9

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35

26

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u.o/*

.977125

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< .U^

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34

27

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6.32

6qo

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7ft

.523259

7 02

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33

28

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6qft

.977041

. <U

.523680

7 ftft

.476320

32

29

.501099

.oil

690

.976999

'•~n

.524100

i .UU
7 ftft

.475900

31

30

.501476

,<£O

6.30

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i .UU

7.00

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30

31

9.501854

69ft

9.976914

9.524940

no

10.475060

29

32

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6 27

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7ft

.525359

6 98

.474641

28

33

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69ft

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.525778

6 no

.474222

27

34

.502984

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697

.976787

'r-A

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.473803

26

35
36
37
38
39

.503360
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.iff

6.25
6.25
6.25
6.25

.976745
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. <o

.72
.70

.72
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.527868
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6^97
6.97
6.95
6.95

.473385
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25
24
23
22
21

40

.505234

6^23

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.70

.72

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6.95
6.95

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20

41

9.505608

OO

9.976489

r-O

9.529119

10.470881

19

42
43
44
45
46

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e!22

6.22
6.20
6.20

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.976404
.976361
.976318
.976275

'.70

.72
.72
.72

.529535
.529951
.530366
.530781
.531196

b.9o
6.93
6.92
6.92
6.92

.470465
.470049
.469634
.469219
.468804

18
17
16
15
14

47
48
49

.507843
.508214
.508585

6^18
6.18
61ft

.976232
.976189
.976146

.72
.72
.72

.531611
.532025
.532439

6.92
6.90
6.90

.468389
.467975
.467561

13

12
11

50

.508956

. lo

6.17

.976103

.72
.72

.532853

6.90
6.88

.467147

10

51

9.509326

617

9.976060

9.533266

6QQ

10.466734

9

52

.509696

. 1 •

6 15

.976017

.72

79

.533679

.OO
600

.466321

8

53

.510065

.975974

. i A

79.

.534092

. GO

.465908

7

54
55
56

.510434
.510803
.511172

e!is

6.15
6-«q

.975930

.975887
.97'5844

. (6

.72
.72

.534504
.534916
.535328

6.87
6.87
6.87

.465496
.465084
.464672

6
5
4

57

.511540

. lo
6 12

.975800

.73

.535739

6.85

6QK

.464261

3

58

.511907

61 °.

.975757

"An

.536150

.00

.463850

2

59
60

.512275
9.512642

. lo

6.12

.975714 -Ao
9.975670 j -'d

.536561
9.536972

6.85
6.85

.463439
10.463028

1

0

' 1 Cosine.

D. 1'.

Sine. D. 1*.

Cotang.

D. r.

Tang.

'

108*

194

71°

COSINES, TANGENTS, AND COTANGENTS. 160°

'

Sine.

D. 1".

Cosine.

D. 1'.

Tang.

D. 1'.

Cotang.

'

0

9.512642

6 12

9.975670

72

9.536972

6QQ

10.463028

60

1

.513009

.975627

r-o

.537382

.00

.462618

59

2

.513375

A ift

.975583

. <•>
r o

.537792

6.83

.462208

58

3

.513741

6 10

.975539

. 1 O

.538202

6.83

.461798

57

4

.514107

.975496

'r-o

.538611

6.82

.461389

56

5

6

.514472
.514837

6 . 08
6.08

6flK

.975452
.975408

.73
.73

.539020
.539429

6.82
6.82

.460980
.460571

55

54

7
8
9

.515202
.515566
.515930

.Uo

6.07
6.07

6/17

.975365
.975321
.975277

.73
.73

r-o

.539837
.540245
.540653

6.80
6.80
6.80

.460163
.459755
.459347

53
52
51

10

.516294

.Ui

6.05

.975233

. to

.73

.541061

6.80
6.78

.458939

50

11

9.516657
.517020

6.05

9.975189
.975145

.73

9.541468
.541875

6.78

10.458532
.458125

49

48

13

.517382

6.03

.975101

.73

.542281

6.77

.457719

47

14
15

.517745
.518107

6.05
6.03

.975057
.975013

.73
.73

.542688
.543094

6.78
6.77

.457312
.456906

46
45

16

.518468

6.02

.974969

.73

.543499

6.75

.456501

44

17

.518829

6.02

.974925

.73

.543905

6.77

.456095

43

18

.519190

6.02

.974880

.75

.544310

6.75

.455690

42

19

.519551

6.02

.974836

.73

.544715

6.75

41

20

.519911

6.00

.974792

.73

.545119

6.73

'.454881

40

6.00

.73

6.75

21
22

9.520271
.520631

6.00

9.974748
.974703

.75

9.545524
.545928

6.73

10.454476

.454072

39

38

23

.520990

5.98

.974659

.73

.546331

6.72

.453669

37

24

.521349

5.98

.974614

.75

.546735

6.73

.453265

36

25

.521707

5.97

.974570

.73

.547138

6.72

.452862

35

26

27
28

.522066
.522424
.522781

5.98
5.97
5.95

.974525
.974481
.974436

.75
.73
.75

.547540
.547943
.548345

6.70

6.72
6.70

.452460
.452057
.451655

34
33
32

29

.523138

5.95

.974391

.75

.548747

6.70

.451253

31

30

.523495

5.95

.974347

.73

.549149

6.70

.450851

30

5.95

.75

6.68

31

9.523852

9.974302

9.549550

10.450450

29

32

.524208

5.93

.974257

.75

.549951

6.68

.450049

28

33

34

.524564
.524920

5.93
5.93

.974212
.974167

.75
.75

.550352
.550752

6.68
6.67

.449648
.449248

27
26

35

36

.525275
.525630

5.92
5.92

.974122
.974077

.75
.75

.551153
.551552

6.68
6.65

.448847
.448448

25
24

37

38

.525984
.526339

5.90
5.92

.974032
.973987

.75

.75

.551952
.552351

6.67
6.65

.448048
.447649

23
22

39
40

.526693
.527046

5.90
5.88
5.90

.973942
.973897

.75
.75
.75

.552750
.553149

6.65
6.65
6.65

.447250
.446851

21

20

41
42
43
44
45
46
47
48
49
50

9.527400
.527753
.528105
.528458
.528810
.529161
.529513
.529864
.530215
.530565

5.88
5.87
5.88
5.87
5.85
5.87
5.85
5.85

5. as

5.83

9.973852
.973807
.973761
.973716
1 .973671
.973625
.973580
.973535
.973489
.973444

.75
.77
.75
.75
.77
.75
.75
.77
.75
.77

9.553548
.553946
.554344
.554741
.555139
.55f.536
.555933
.556329
.556725
.557121

6.63
6.63
6.62
6.63
6.62
6.62
6.60
6.60
6.60
6.60

10.446452
.446054
.445656
.445259
.444861
.444464
.444067
.443671
.443275
.442879

19
18
17
16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

9.530915
.531265
.531614
.531963
.532312
.532661
.533009
.533357
.533704
9.534052

5.83

5.82
5.82
5.82
5.82
5.80
5.80
5.78
5.80

9.973398
.973352
.973307
.973261
.973215
.973169
.973124
.973078
.973032
9.972986

.77
.75

.77
.77
.77
.75
.77
.77
.77

9.557517
.557913
.558308
.558703
.559097
.559491
.559885
.560279
.560673
9.561066

6.60
6.58
6.58
6.57
6.57
6.57
6.57
6.57
6.55

10.442483
.442087
.441692
.441297
.440903
.440509
.440115
.439721
.439327
10.488984

9
8
7
6
5
4
3
2
1
0

'

Cosine.

D. r.

Sine.

D. r.

Cotang.

D. 1'.

Tang, i '

109'

195

70'

20°

TABLE XII. — LOGARITHMIC SINES,

159°

'

Sine.

D. 1".

Cosine.

D. 1*.

Tang.

D. r.

Cotang.

•

0

1

9.534052
534399

5.78

9.972986
.972940

.77

9 561066
.561459

6 55

6 tq

10.438934
.438541

60
59

2
3

.534745
.535092

5 77

5.78

.972894
.972848

'77

77

561851
562244

Do

6.55

6 CO

438149
.437756

58
57

4

.535438

5.77

972802

. I (

78

.562636

.Do

6 53

.437364

56

5

.535783

5.75

972755

<o

.563028

.436972

55

6

.536129

5.77

.972709

77

563419

6 tq

.436581

54

7

.536474

5 75

.972663

. i 1

563811

Do

436180

53

8

.536818

5.73

972617

78

564202

6 52

o to

.435798

52

9
10

.537163
.537507

5! 73
5.73

9r< 2570
.972524

. <o

,77

564593
564983

D. D/*

6 50
6.50

,435407
.435017

51
50

11
12
13
14

9.537851
538194
538538
536880

5.72
5 73
5.70

9.972478
972431
972385
972338

.78

.77
.78

r-o

9-565373
565763
.566153
566542

6.50
6 50
6.48

6KA

10.434627
.434237
.433847
,433458

49

48
47
46

15

.539223

5.72

.972291

. 1 O

77

.566932

.OU

6 47

.433068

45

16

539565

5.70

.972245

< t
78

.567320

648

.432680

44

17
18

.539907
.540249

5.70
5.68

972198
.9721M

. <o

.78

567709

.568098

.<*O

6.48

647

.432291
.431902

43

42

19
20

.540590
.540931

5.68
5.68
5.68

.972105
.972058

i78

568486
568873

.4<

6.45
6.47

.431514
.431127

41
40

21

9.541272

9.972011

r-o

-9.569261

10.430739

39

22

23

.541613
.541953

5.68
5.67

.971964
971917

. <o

.78

.569648
.570035

6^45

.430352
.429965

38
37

24

.542293

5.67

.971870

.78

.570422

6.45
6 45

,429578

36

25

.542632

5.65

.971823

r-o

.570809

64P.

.429191

35

~6

.542971

5.65

.971776

. iO
78

.571195

.'lo
64'}

.428805

34

27
28
29
30

.543310
543649
.543987
.544325

5 '.65
5.63
5.63
5.63

.971729
.971682
.971635
.971588

. <o

.78
.78
.78
.80

.571581
.571967
.572352
572738

•*o
6.4S
6.42
6.43
6.42

,428419
.428033
.427648
.427262

33
32
31
30

31
32
33
34
35

9.544663
.545000
.545338
.545674
.546011

5.62
5.63
5.60
5.62

9.971540
.971493
.971446
.971398
.971351

.78
.78
.80
.78

9.573123

.573507
.573892
.574276
.574660

6 40
6.42
6.40
6.40

10.426877
.426493
.426108
.425724
.425340

29

28
27
26
25

36
37
38

546347
.546683
.547019

5.60
5.60
5.60

.971303
.971256

.971208

!78
.80

.575044

.575427
.575810

6^38
6 38

6 op

.424956
.424573
.424190

24
23

22

39
40

.547354
.547669

5.58
5.58
5.58

.971161
.971113

.78
80
.78

.576193
.576576

.00

6.38
6.38

.423807
.423424

21
20

41

9.548024

Sfcrp

9.971066

PA

9.576959

6 37

10.423041

19

42

. 548359

.OO

.971018

.oU
PA

.577341

6 37

.422659

18

43

.548693

5 57

.970970

.oU

OA

.577723

.422277 17

44

.549027

5.57

.970922

.oU

.578104

607

.421896

16

45

46

.549360
.549693

5.55
5.55

5fK

.970874
.970827

.80
.78

PA

.578486
.578867

.o<
6.35
6 35

.421514
.421133

15
14

47

48

.550026
.550359

.OO

5.55

pr te

.970779
.970731

. oU

.80

.579248
.579629

6^35
6 33

.420752
.420371

13
12

49
50

.550692
.551024

O .DD

5.53
5.53

.970683
.970635

.'80
.82

.580009
580389

eias

6.33

.419991
.419611

11
10

61

9.551356

9.970586

OA

9.580769

6qq

10.419231

9

52

.551687

5.52

.970538

.oU

.581149

.00

.418851

8

53
54

.552018
.552349

5.52
5.52

.970490
.970442

.80
.80

,581528
581907

6.32
6.32

.418472
.418093

7
6

55

.552680

5.52

.970394

.80

.582286

6.32

.417714

5

56

.553010

5.50

.970345

.82

PA

.582665

6.32
69.9

.417335

4

57
58

.553341
.553670

5. '48

5tA

.970297
.970249

.OU

.80

.583044
.583422

. O/i

6.30
6 30

.416956
.416578

3
2

59

.554000

.OU

.970200

•5?

.583800

.416200

1

60

9.554329

5.48

9.970152

.80

9.584177

6.28

10.415823

0

'

Cosine.

D. 1".

Sine.

D. 1". 1

Cotang.

D. i".

Tang.

'

110°

196

21°

COSINES, TANGENTS, AND COTANGENTS. 168°

'

Sine.

D. 1".

Cosine.

D. 1'.

Tang.

D. 1-.

Cotang.

'

0

1

2
3
4
5
6
7
8
9
10

9.554329
.554658
.554987
.555315
.555643
.555971
.556299
.556626
.556953
.557280
.557606

5.48
5.48
5.47
5.47
5.47
5.47
5.45
5.45
5.45
5.43
5.43

9.970152

.970103
. 97 0055
.970006
.909957
.969909
.909800
.969811
.969762
.969714
.969665

.82
.SO
.82
.82 j
.80 i
.82
.82
.82
.80
.82
.82

9.584177
.584555
.584982
.585809
.585686
.580002
.586439
.586815
.587190
.587566
.5S7941

6.30
6.28
6.28
6.28
6.27
6.28
6.27
6.25
6.27
6.25
6.25

10.415823
.415446

.414601

.414314
.413938
.413561
.413185
.412810
.412484
.412069

60

66

57
56
55
54
53
52
51
60

11
12
13
14
15
16

9.557932
.558258
.558583
.558909
.559234
.559558

5.43
5.42
5.43
5.42
5.40
5.42

9.969616
.969567
.969518
.969469
.969420
.969370

.82
.82
.82
.82
.83

9.588316
.588691
.589066
.589440
.589814
.590188

6.25
6.25
6.23
6.23
6.23

10.411684
.411809
.410984
.410660

.410186

49

48
47
40
45
44

17

18
19
20

.559883
.560207
.560531
.560855

5. 40
5.40
5.40
5.38

.969321
.969272
.969223
.969173

!82
.82

.as

.82

.590562
.590935
.591308
.591681

6^22
6.22
6.22
6.22

.409438
.409005
.408692
.408319

43
42

41
40

21
22
23
24

9.561178
.561501
.561824
.562146

5.38
5.38
5.37
5 37

9.969124
.969075
.969025
.968976

.82
.83
.82

oq

9.592054
.592426
.592799
.593171

6.20
6.22
6.20

10.407946
.407574
.407201
.406829

39
88
37
36

25
26

27

.562468
.562790
.563112

5^37 '
5.37

5 OK

.968926
.968877
.968827

.OO

.82
.83

.593542
.593914
.594285

6.18
6.20
6.18

.406458
.40608(5
.405715

35
84

M

28

.563433

.OO

5q7

.968777

.83

.594656

6.18

.405344

29

.563755

.O<

.968728

.82

.595027

6.18

.404973

31

30

.564075

5.33
5.35

.968678

.83
.83

.595398

6.18
6.17

.404602

30

31

9.564396

K qq

9.968628

9.595768

10.404532

29

32

.564716

O. oo

.968578

.83

.596138

6.17

.403802 28

33

.565036

5.33

5qq

.968528

.83

.596508

6.17

617

.403492 57

34
35
36

.565356
.565676
.565995

.00

5.33
5.32

.968479
.968429
.968379

.82
.83
.83

.596878
.597247
.597616

.It

6.15
6.15

.403122 5ft
.402753 2.1
.4023K1 51

37

.566314

5.32

.968329

.83

.597985

6.15

.4020 K

38

.566632

5.30

.968278

.85

.598354

6.15

.40101'

39

.566951

5.32

.968228

.83

.598722

6.13

.401278

21

40

.567269

5.30
5.30

.968178

.83
.83

.599C91

6.15
6.13

.400909

20

41
42

43
44
45
46

9.567587
.567904
.568222
.568539
.568856
.569172

5.28
5.30
5.28
5.28
5.27

597

9.968128
.968078
.968027
.967977
.967927
.967876

.83

.85
.83
.83 :

.85 i

QQ

9.599459
.599827
.600194
.600562
.600929
.601296

6.13
6.12
6.13
6.12
6.12
6 12

10.400541
.400173
.399806
.399438
.399071
.398704

19

1H
17
10
15
14

47
48
49

.569488
.569804
.570120

.lilt

5.27
5.27

.967826
.967775
.967725

.OO

.85

.83

.601663
.602029
.602395

6.' 10
6.10
6 10

.896897

.397971
.397605

13
12
11

50

.570435

5^27

.967674

!83

.602761

e!io

.397239

10

61

52
53
54
55
56
57

9.570751
.571066
.571380
.571695
.572009
.572323
.572636

5.25
5.23
5.25
5.23
5.23
5.22

9.967624
.967573
.967522
.967471
.967421
.967370
.967319

.85
.85
.85
.83
.85
.85

9.603127
.603493
.603858
.604223
.604588
.604953
.605317

6.10
6.08
6.08
6.08
6.08
6.07
6 08

10.396873
.896607

.396142
.895777
.896412

.894688

9
8
7
6
5
4
3

58

.572950

5.23

.967268

•§3

.605682

2

59
60

.573263
9.573575

5.22
5.20

.967217
9.907100

.85

.85 !

.606046
9.606410

6.07

10.393590

1
0

'

Cosine.

D. 1".

Sine. D. r. Cotang. D. 1'.

Tang.

L

111°

197

68*

22°

TABLE XII. — LOGARITHMIC SINES,

157°

'

Sine.

D. i*.

Cosine.

D. 1'.

Tang.

D. r.

Cotang.

'

0

1

2
3

4

9.573575

.573888
.574200
.574512

.574824

5.22
5.20
5.20
5.20

59O

9.967166
.967115
.967064
.967013
.966961

.85

.85
.85

.87

QC

9.606410
.606773
.607137
.607500
.607863

6.05
6.07
6.05
6.05

A no.

10.393590
.393227
.392863
.392500
.392137

60

59
58
57
56

5

.575136

.<*U

.966910

.OO

.608225

O.Uo

.391775

55

6

8
9

.575447
.575758
.576069
.576379

5.18
5.18
5.18
5.17

517

.966859
.966808
.966756
.966705

.85
.85
.87
.85

87

.608588
.608950
.609312
.609674

6.05
6.03
6.03
6.03

6 no

.391412
.391050
.390688
.390326

54
53
52
51

10

.576689

. 1 1

5.17

.966653

.of

.85

.610036

.Uo

6.02

.389964

50

11

9.576999

517

9.966602

87

9.610397

6AO

10 389603

49

12
13
14
15

.577309
.577618

577927
.578236

. 1 1

5.15
5.15
5 15

.966550
.966499
.966447
.966395

• of

.85
.87
.87

.610759
.611120
.611480
.611841

.Uo

6.02
6.00
6.02

.389241
.388880
.388520
.388159

48
47
46
45

16
17

.578545

.578853

5.15
5.13

5 -IK

.966344
.966292

.85
.87

87

.612201
. 612561

6.00
6.00

6AA

.387799
.387439

44
43

18

.579162

. 10

.966240

.01

.612921

.UU

.387079

42

19
20

.579470

.579777

5.13
5.12
5.13

.966188
.966136

.87
.87
.85

.613281
.613641

6.(;0
6.00
5.98

.386719
.386359

41
40

21

9.580085

9.966085

87

9.614000

10.386000

39

22
23

.580392
.580699

5 '.12
5in

.966033
.965981

.OY

.87

87

.614359
.614718

5.'98

5QQ

.385641

.385282

38
37

24

.581005

. 1U

.965929

.01

88

.615077

yo

,384923

36

25
26
27

28
29

.581312
581618
.581924
.582229
.582535

5.12
5.10
5.10
5.08
5.10

5AQ

.965876
.965824
.965772
.965720
.965668

.00

.87
.87
.87
.87

88

.615435
.615793
.616151
.616509
.616867

5^97
5.97
5.97
5.97

K QK

.384565
.384207
.383849
.383491
.383133

85
34
33
32
31

30

.582840

.Uo

5.08

.965615

.00

.87

.617224

o . yo
5 97

.382776

30

31
32

9.583145
.583449

5.07
5na

9 965563
.965511

.87

88

9.617582
617939

5.95

fr QO.

10 382418
.382061

29

28

33
34
35
36

583754
.584058
.584361
.584665

.Uo
5.07
5.05
5.07

5AK

. 965458
.965406
965353
.965301

.OO

.87
.88
.87

QQ

.618295
.618652
.619008
.619364

o. yo
5.95
5.93
5.93

5QO

.381705
.381348
.380992
.380636

27
26
25
24

37

.584968

.UO
5O7

.965248

.OO
QQ

.619720

.yo

5 QO

.380280

23

38
39

.585272
.585574

.U<

5.03

5AK

.965195
.965143

.OO

.87

QQ

.620076
.620432

.yo
5 93
5 92

.379924
.379568

22
21

40

.585877

.UO

5.03

.965090

.OO

.88

.620787

5192

.379213

20

41

9.586179

5A.K

9.965037

88

9.621142

10.378858

19

42

.586482

.UO

.964984

.00
88

.621497

5 no

.378503

18

43
44

586783
.587085

5.03

.964931
.964879

.OO

.87

88

.621852
.622207

.»/*
5.92
5 on

.378148
.377793

17
16

45

.587386

5.02

5AO

.964826

.OO
88

.622561

.yu

5 Of)

.377439

15

46

.587688

.Uo

.964773

.00

.622915

.yu

.377085

14

47
48
49

.587989
.588289
.588590

5.02
5.00
5.02
5 00

.964720
.964668
.964613

.88
.90

.88

QQ

.623269
.623623
.623976

5.90
5.90

5.88

5QA

.376731
.376377
.37'6024

13
12
11

50

.588890

5!oo

.964560

.OO

.88

.624330

.yu
5.88

.375670

10

51

9.589190

4 no

9.964507

QQ

9.624683

5 CO

10.375317

9

52
53

.589489
.589789

.yo
5.00

4Q8

.964454
.964400

.OO

.90

.625036
.625388

.00
5.87

.374964
.374612

8

7

54

.590088

.yo

.964347

.88

.625741

5.88

.374259

6

55

56

.590387
.590686

4.98
4.98

.964294
.964240

.88
.90

.626093
.626445

5.87
5.87

.373907
.373555

5

4

57
58

.590984
591282

4.97
4.97

4Q7

.964187
.964133

.88
.90

88

.626797
.627149

5.87
5.87

587

.373203
.372851

3

2

59
60

.591580
9.591878

.y«
4.97

.964080
9.964026

.00

.90

.627501

9.627852

.of

5.85

.372499
10.372148

1
0

'

Cosine.

D. 1".

I Sine.

D. 1".

Cotang.

D. r.

Tang.

~

198

67°

23°

COSINES, TANGENTS, AND COTANGENTS. 166°

'

Sine.

D. 1*.

Cosine.

D. r.

Tang.

D. r.

Cotang.

'

0

1

2

3

4
5
6
7
8
9
10

9.591878
.592176
.592473
.592770
.593067
.593363
.593659
.593955
.594251
.594547
.594842

4.97
4.95
4.95
4.95
4.93
4.93
4.93
4.93
4.93
4.92
4.92

9.964026
.963972
.963919
.963865
.963811
.963757
.963704
.963650
.963596
.963542
.963488

.00
.88
.90
.90
.90
.88
.90
.90
.90
.90
.90

9.627863
.688808
.628554
.628905
.629856
.629606
.629956
.630306
.630656
.631005
.631355

5.85
5.85
5.85
5.83
5.85
5.83
5.83
5.83
5.89
5.83
5.82

10. 372 118
,871707
.871440
.871095
.870745
.370394
.370044
.869604
.369344
.868995
.368645

80

59

58
57
56
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
.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
.633099
.633447
.633795
.634143
.634490
.634838

5.82
5.82
5.80
5.82
5. SO
5.80
5.80
5.78
5.80
5.78

10.368296
.887947
.867598

.367250
.366901
.366553
.366205
.865857
.365510
.365162

49

48
47
46
45
44
43
42
41
40

21
22

23
24
25

9.598075
.598368
.598660
.598952
.599244

4.88
4.87
4.87

4.87
407

9.962890
.962836
.962781
.962727
.962672

.90
.92
.90
.92

9.635185
.635532
.635879
.636226
.636572

5.78
5.78
5.78
5.77

10.364815
.864468

.364121
.363774
.363428

39
38
37
36
35

26
27

28
29

.599536
.599827
.600118
.600409

:.OI

4.85
4.85
4.85
4 85

.962617
.962562
.962508 .
.962453

.92
.92
.90
.92

.636919
.637265
.637611
.637956

5.78
5.77
5.77
5.75

5r>ry

.363081
.362735
.362389
.362044

34
33

32
31

30

.600700

4.83

.962398

!92

.638302

. It

5.75

.361698

30

31

9.600990

4 83

9.962343

9.638647

57K

10.361353

29

32
33

.601280
.601570

4^83

400

.962288
.962233

!92

.638992
.639337

. 1 •)

5.75

.361008-
.360663

28
27

34

.601860

.OO
4QQ

.962178

.92

.639682

5.75

5<~K

.360318

26

35

36
37

38

.602150
.602439
.602728
.603017

.oO

4.82
4.82
4.82

480

.962123
.962067
.962012
.961957

!93
.92
.92

.640027
.640371
.640716
.641060

. 1.)

5.73
5.75
5.73

5r»O

.359973
.359629
.359284
.358940

25
24
23
22

39

.603305

. of

.961902

•~|

.641404

. 1 • )

.358596

21

40

.603594

4^80

.961846

!92

.641747

5173

.358253

20

41

9.603882

4 on

9.961791

9.642091

10.357909

19

42

.604170

.oU

.961735

•*2

.642434

5.72

.357566

18

43

.604457

4.78

.961680

.92

.642777

5.72

5r»O

.35?223

17

44
45
46
47
48

.604745
.605032
.605319
.605606
.605892

4.78
4.78
4.78
4.77

.961624
.961569
.961513
.961458
.961402

!92
.93
.92
.93

.643120
.643463
.643806
.644148
.644490

. i ~

5.72
5.72
5.70
5.70

.356880
.356537

! 855862
.355510

16
15
14
13
12

49
50

.606179
.606465

4.78
4.77

4.77

.961346
.961290

.93
.93
.92

.644832
.645174

5.70
5.70
5.70

.355168

.354S-G

11
10

51

9.606751

9.961235

9.645516

5(*Q

10.354484

9

52

.607036

4.75

.961179

.93

.645857

.OO

.854149

8

53
54
55
56

.607322
.607607
.607892
.608177

4.77
4.75
4.75
4.75

.961123
.961067
.961011
.960955

.93
.93
.93
.93

.646199
.646540
.646881
.647222

5.70
5.68
5.68
5.68

5K7

.858801
.858440

.353119
.858778

7
6
5
4

57

.608461

4.73

.960899

.93

.647562

.Ol
5 Aft

3

58
59
60

.608745
.609029
9.609313

4.73
4.73
4.73

.960843
.960786
9.960730

.93
.95
.93

.647903
.648243
9.648583

.08

5.67
5.67

.858097

.351757
10.351417

2

1
0

9

Cosine.

D. 1'.

Sine.

D. 1". |i Cotang.

D. r.

Tang.

'

113'

199

66-

24°

TABLE XII. LOGARITHMIC SINES,

155°

/

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

i

0

1

9.609313
.609597

4.73

9.960730
.960674

.93

9.648583

.648923

5.67

10.351417
.351077

60
59

2

.609880

4. 72

4*~q

.960618

.93
05

.649263

5.67

.350737 ! 58

3

.610164

. IO

.960561

.649602

5 , 65

.350398 57

4
5

.610447
.610729

4.72
4.70

.960505
.960448

!95

.649942
.650281

5.67
5.65

.350058
.349719

56
55

6

.611012

4.72

47rt

.960392

.93

QK.

.650620

5.65

.349380

54

7

.611294

. t(J

.960335

.yo

.650959

5.65

.349041

53

8
9
10

.611576
.611858
.612140

4.70
4.70
4.68

.960279
.960222
.960165

.93
.95
.95
.93

.651297
.651636
.651974

5^65
5.63
5.63

.348703
.348364
.348026

52
51
50

11

9.612421

4 CQ

9.960109

QK

9.652312

5f>O

10.347688

49

12
13
14

.612702
.612983
.613264

.Do

4.68
4.68

4 CO

.960052
.959995
.959938

.yo
.95
.95

QO

.652650
.652988
.65a326

.Do

5.63
5.63

.347350
.347012
.346674

48
47
46

15
16
17

.613545
.613825
.614105

. Do

4.67
4.67

4K7

.959882
.959825
.959768

.yo
.95
.95

QK

.653663
.654000
.654337

5^62
5.62

.346337
.346000
.345663

45
44
43

18
19
20

.614385
.614665
.614944

.Ol

4.67
4.65
4.65

.959711
.959654
.959596

.yo

.95
.97
.95

.654674
. .655011
.655348

5.62
5.62
5.62
5.60

.345326
.344989
.344652

42
41
40

21

22
23
24
25
26
27
28
29
30

9.615223
.615502
.615781
.616060
.616338
.616616
.616894
.617172
.617450
.617727

4.65
4.65
4.65
4.63
4.63
4.63
4.63
4.63
4.62
4.62

9.959539
.959482
.959425
.959368
.959310
.959253
.959195
.959138
.959080
.959023

.95
.95
.95
.97
.95
.97
.95
.97
.95
.97

9.655684
.656020
.656358
.656692
.657028
.657364
.657699
.658034
.658369
.658704

5.60
5.60
5.60
5.60
5.60
5.58
5.58
5.58
5.58
5.58

10.344316
.343980
.343644
.343308
.342972
.342636
.342301
.341966
.341631
.341296

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39

9.618004
.618S81
.618558
.618834
.619110
.619386
.619662
.619938
.620213

4.62
4.62
4.60
4.60
4.60
4.60
4.60
4.58

9.958965
.958908
.958850
.958792
.958734
.958677
.958619
.958561
.958503

.95
.97
.97
.97
.95
.97
.97
.97

9.659039
.659373
.659708
.660042
.660376
.660710
.661043
.661377
.661710

5.57
5.58
5.57
5.57
5.57
5.55
5.57
5.55

10.340961
.340627
.340292
.339958
.339624
.339290
.338957
.338623
.338290

29
28
27
26
25
24
23
22
21

40

.620488

4.58
4.58

.958445

.97
.97

.662043

5.55
5.55

.337957

20

41
42
43
44

9.620763
.621038
.621313
.621587

4.58
4.58
4.57

9.958387
.958329
.958271
.958213

.97
.97
.97

9.662376
.662709
.663042
.663375

5.55
5.55
5.55

10.337624
.337291
.336958
.336625

19

18
17
16

45

.621861

4.57

.958154

*]5»

.663707

5.53

.336293

15

46
47

.622135
.622409

4 '.87

.958096
.958038

.97
.97

.664039
.664371

5.53
5.53

.335961
.335629

14
13

48
49
50

.622682
.622956
.623229

4.55
4.57
4.55
4.55

.957979
.957921
.957863

.98
.97
.97
.98

.664703
.665035
.665366

5.53
5.53
5.52
5.53

.335297
.334965
.334634

12
11
10

51

52

9.623502
.623774

4.53

9.957804
.957746

.97

QQ

9.665698
.666029

5.52

10.334302
.333971

9

8

53
54
55
56
57

.624047
.624319
.624591
.624863
.625135

4.55
4.53
4.53
4.53
4.53

.957687
.957628
.957570
.957M1
.957452

.yo
.98
.97
.98

.98

.666360
.666691
.667021
.667352
.667682

5.52
5.52
5.50
5.52
5.50

.333640
.333309
.332979
.332648
.332318

7
6
5
4
3

58
59
60

.625406
.625677
9.625948

4.52
4.52
4.52

.957393
.957335
9.957276

.98
.97
.98

.668013
.668343
9.668673

5.52
5.50
5.50

.331987
.331657
10.331327

2
1
0

'

Cosine.

D. 1*.

Sine.

D. 1".

Cotang.

D. 1".

Tang.

'

200

65*

COSINES, TANGENTS, AND COTANGENTS.

164*

I

Sine.

D. 1".

Cosine.

D. r.

Tang.

D. 1'.

Cotang.

'

0

1

2
3

4
5
6

7

9.625948
.626219
.626490
.626760
.627030
.627300
.627570
.627840

4.52
4.52
4.50
4.50
4.50
4.50
4.50
4 48

9.957276
.957217
.957158
.957099
.957040
.956981
.956921
.956862

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

QO

1MJI5S073
.669002
.669332
.669661
.669991
.670320
.670649
.670977

5.48
5.50
5.48
5.50
5.48
5.48
5.47

.880996
.880668
.880689
.880009

.829680
.829851
.829028

60
59
58
57
56
55
54
53

8

.628109

440

.956803

QQ

.671306

5.48

.826694

B2

9

10

.628378
.628647

4.48
4.48

.956744
.956684

1.00

.98

.671635
.671963

5.48
5.47
5.47

.828865

.826097

51
50

11
12
13
14
15
16

9.628916
.629185
.629453
.629721
.629989
.630257

4.48
4.47
4.47
4.47
4.47

4AK.

9.956625
.956566
.956506
.956447
.956387
.956327

.98
1.00
.98
1.00
1.00

no

9.672291
.672619
.672947
.673274
.673602
.673929

5.47
5.47
5.45
5.47
5.45

5Aff

10.327709
.327381
.327053
.826726
.826896
.326071

49
48
47
46
45
44

17
18
19

.630524
.630792
.631059

4.47
4.45

44K

.956268
.956208
.956148

1.00
1.00

QQ

.674257
.674584
.674911

5.45
5.45

t An

.325743
.325416
.825060

48

42
41

20

.631326

4.45

.956089

1.00

.675237

5.45

.324763

40

21
22
23
24
25
26

9.631593
631859
.632125
.632392
.632658
.632923

4.43
4.43
4.45
4.43
4.42

9.956029
.955969
.955909
.955849
.955789
.955729

1.00
1.00
1.00
1.00
1.00

9.675564
.675890
.676217
.676543
.676869
.677194

5 43
5.45
5.43
5.43
5.42

10.324436
.324110
.828788

.323457
.323131
4 .322806

39
38
37
36
35
34

27

28

.633189
.633454

4.43

4.42

.955669
.955609

1.00

.677520
.677846

5.43

.822480

,322154

33
32

29
30

.633719
.633984

4.42
4.42
4.42

.955548
.955488

1 00
1.00

.678171
.678496

5.42
5.42

.321829
.321504

31
30

31
32
33
34
35
36
37

9.634249
.634514
.634778
635042
.635306
.635570
.635834

4.42
4.40
4.40
4.40
4.40
4.40

4OQ

9 955428
.955368
.955307
.955247
.955186
.955126
.955065

1.00
1.02
1.00
1.02
1.00
1.02
1 00

9.678821
. 679146
.679471
.679795
.680120
.680444
.680768

5.42
5 42
5.40
5.42
5.40
5.40
5 40

10.321179
.820654

.820529

.320205
.319880
.319556
.319232

29
28
27
26
25
24
23

38
39
40

.636097
636360
.636623

438
4.38
4 38

.955005
.954944
.954883

1.02
1.02
1 00

.681092
.681416
.681740

5.40
5.40
5.38

.318908
.318584
.318260

22
21
20

41
42

43
44
45

46
47
48
49
50

9.636886
637148
.637411
.637673
.637935
.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 02

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
.816967
.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
1 .685934
686255
.686577
686898
.687219
.687540
.687861
9.688182

5.37
5.37
5.35
5.37
5 35
5.35
5 35
5.88
5.35

10 314710
.314388
814066
313745
.313423
.313102
.812781
.312460
.312139
10.311818

9
8
7
6
5
4
3
2
1
0

'

Cosine.

D. r.

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

D. 1'. I Tang.

115*

201

26°

TABLE XII. — LOGARITHMIC SINES,

153°

'

Sine.

D. I*.

Cosine.

D.l'.

Tang.

D. 1'.

Cotang.

>

0

1

2
3
4
5
6

9.641842
.642101
.642360
.642618
.642877
.643135
.643393

4.32
4.32
4.30
4.32
4.30
4.30

4 Oft

9.953660
.953599
.953537
.953475
.953413
.953352
.953290

1.02
1.03
1.03
1.03
1.02
1.03
1 03

9.688182
.688502
.688823
.689143
.689463
.689783
.690103

5.33
5.32
5.33
5.33
5.33
5.33

5qq

10.311818
.311498
.311177
.310857
.310537
.310217
.309897

60
59
58
57
56
55
54

8
9
10

.643650
.643908
.644165
.644423

.60

4.30
4.28
4.30
4.28

.953228
.953166
.953104
.953042

l.'OS
1.03
1.03
1.03

.690423
.690742
.691062
.691381

.00

5.32
5.33
5.32
5.32

.309577
.309258
.308938
,308619

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';8
.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
5.28

10.308300
.307981
.307662
.307344
.307025
.306707
.306388
.306070
.305752
.305434

49
48
47
46
45
44
43
42
41
40

21

9.647240

4oq

9.952356

1 03

9.694883

10.305117

39

22

.647494

.6-J

.952294

| .695201

5 'oft

.304799

38

23

.647749

4.25

.952231

1 05

1 .695518

.60

K qn

.304482

37

24
25

.648004
.648258

4 .25
4.23

.952168
.952106

1^03
1 05

! .695836
| .696153

O.oU

5.28

K Oft.

.304164
.303847

36
35

26

.648512

A' c*n

.952043

1 05

! .696470

O .60

50ft

.303530

34

27
28
29
30

.648766
.649020
.649274
.649527

4.6O

4.23
4.23
4.22
4.23

.951980
.951917
.951854
.951791

lios

1.05
1.05
1.05

; .696787
i .697103
.697420
.697736

.60

5.27
5.28
5.27
5.28

.303213
.302897
.302580
.302264

33
32
31
30

31
32
33
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
105
1.07
1 05
l!05

1 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
5.25

10.301947
.301631
.301315
.300999
.300684
.300368
.300053
.299737
.299422
.299107

29

28
27
26
25
24
23
22
21
20

41
42
43
44
45
46

9.652304
.652555
.652806
.653057
653308
.653558

4.18
4.18
4.18
4.18
4.17

9.951096
.951032
.950968
.950905
.950841
.950778

1.07
1.07
1.05
1.07
1.05
1 07

9.701208
.701523
.701837
.702152
.702466
.702781

5.25
5.23
5.25
5.23
5.25

10.298792

.298477
.298163
.297848
.297534
.297219

19
18
17
16
15
14

47
48
49
50

.653808
.654059
.654309
.654558

4.17
4.18
4.17
4.15
4.17

.950714
.950650
.950586
.950522

1 ."<

1.07
1.07
1.07
1.07

.703095
.703409
.7037'22
.704036

5 .23
5.23
5.22
5.23
5.23

.296905
.296591
.296278
.295964

13
12
11
10

51

52
53
54
55
56
57
58

9.634808
.655058
.655307
.655556
.655805
.656054
656302
.656551

4.17
4.15
4.15
4.15
4.15
4.13
4.15

9.950458
.950394
.950330
.950266
.950202
.950138
.950074
.950010

1.07
1.07
1.07
1.07
1.07
1.07
1.07
Ifift

9.704350
.704663
.704976
.705290
.705603
.705916
.706228
.706541

5.22
5.22
5.23
5.22
5.22
5.20
5.22

10.295650
.295337
.295024 i
.294710
.294397
.294084
.293772
.293459

9
8
7
6
5
4
3
2

59
60

.656799
9.657047

4.13
4.13

.949945
9.949881

. Uo

1.07

.706854
9.707166

5. '20

.293146

10.292834

1
0

' 1

Cosine.

D.l\

Sine. D. 1". Cotang.

D.r. I

Tang.

it

116*

202

63*

27°

COSINES, TANGENTS, AND COTANGENTS. 162°

•

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. r.

Cotang.

'

0

1

2
3

9.657047
.657295
.657542
.657790

4.13 i
4.12
4.13

9.949881
.949816
.949762
.949688

1.08
1.07
1.07 !

9.707166
.707478
.707790
.708102

5.20
5.20
5.20

.898210
.891898

60
59
58
57

4
5
6

.658037
.658284
.658531

4.12
4.12
410

.9496213
.949558
.949494

1.08
1.07

1 Oft

.708414
.708788

.709037

5.80

5.18
Son

.891586
.890968

56
65

54

.658778

.949429

.709349

518

.200861

58

8
9
10

.659025
.659271
659517

4.10
4.10
4.10

.949364
.949300
.949235

1.07
1.08
1.08

.709660
.709971
.710282

5.18
5.18
6.18

.890640
.890089

.289718

51

51
50

11
12
13
14
15
16
17
18
19
20

9.659763
.660009
.660255
.660501
660746
.660991
.661236
.661481
.661726
.66197'0

4.10
4.10
4.10
4.08
4.08
4.08
4.08
4.08
4.07
4 07

9.949170
.949105
.949040
.948975
.948910
.948845
,948780
.948715
.948650
.948584

1.08
1.08
1.08
1.08
1.08
1.08
1.08
1.08
1.10
1 08

9.710593
.710904
.711215
.711525
.711836
.712146
.712456
.712766
.713076
.713386

5.18
5.18
5.17
5.18
5.17
5.17
5.17
5.17
5.17
5.17

10.289407
.889096
.888785
.888475
.888164
.887854
.887544

.286614

40
H
47
46
45
44
43
42
41
40

21

9.662214

4AQ

9.948519

1 Aft

9.713696

5 15

10.286304

30

22
23

24
25

26
27
28
29
30

.662459
.662703
. 662946
.663190
.663433
.663677
.663920
.664163
.664406

4.07
4.05
4.07
4.05
4.07
4.05
4.05
4.05
4 03

.948454
.948388
.948323
.948257
.948192
.948126
.948060
.947995
.947929

1.10
1.08
1.10
1.08
1.10
1.10
1.08
1.10
1 10

.714005
.714314
.714624
.7149313
.715242
.715551
.715860
.716168
.716477

5.15
5.17
5.15
5.15
5.15
5.15
5.13
5.15
5.13

.886996
.885686

.285376
.885067

.284758
.284449
.2K41 !0
.283832
.88800

as

37
36
,35
34
33
32
31
30

31

32
33
34
35
36
37
38
39
40

41
42
43
44
45
46
47
48
40
50

51
52
53
54
55
56
57
58
59
60

9.664648
.664891
.665133
.665375
.665617
.665859
.666100
.666342
.666583
.666824

9.667065
.667305
.667546
.667786
.668027
.668267
.668506
.668746
.668986
.669225
' 9.660464
669703
(509912
670181
670419
.670(558
670896
.671134
.671372
9.671609

4.05
4.03
4.03
4.03
4.03
4.02
4.03
4.02
4.02
4.02

4.00
4.02
4.00
4.02
4.00
3.98
4.00
4.00
3.98
3.98

3.98
3.98
3.98
3.97
3.98
3.97
3.97
3.97
3.95

9.947863
.947797
.947731
.947665
.947600
.947533
.947467
.947401
.947335
.947269

9.947203
.947136
.947070
.947004
.946937
.946871
946804
.946738
.946671
.946604

9.946538
.946471
.946404

946337
.946270
946203
.946136
.946069
.946002
9.945935

1 10
1.10

1.10
1.08
1.12
1.10
1.10
1.10
1.10
1.10

1.12
1.10
1.10
1.12
1.10
1.12
1.10
1.12
1.12
1.10

1.12
1.12
1.12
1.12
1.12
1.18
1.18
1.12
1.12

9.716785
.717093
.717401
.717709
.718017
.718325
.718633
.718940
.719248
.719555

9.719862
.720169
.720476
.720783
.721089
.721396
.721702
.722009
.722315
.788681

9.722027
723232

.788844
7*4141
784464
78476!
785060

5 13
5.13
5.13
5.13
5.13
5.13
5.12
5.13
5.12
5.12

5.12
5.12
5.12
5.10
5.12
5.10
5.12
5.10
5.10
5.10

5.08
5.10
5.10

5.10

5. OS
5. OS
5.07

10.283215
.888907
.888599
.888891
881988
.281675
.2811367
.881060
880758
.280446

10.280138
879881

.279217
.278911
[878601
.878898

•

29
28
27
26
25
24
23
22
21
20

19
18
17
16
15
14
13
18
11
10

9
H
7
6
5
4

i
9

'

Cosine.

D. 1".

Sine, i D. 1". Cotang

i). r.

Tang. | '
~^fi&

117°

203

28°

TABLE XII. — LOGAKITHMIC SINES,

161°

'

Sine.

D. 1".

Cosine.

D. r.

Tang.

D. r.

Cotang.

'

0

1

9.671609
.671847

3.97

9.945935
.945868

1.12

9.725674

.725979

5.08

10.274326
.274021

60
59

2
3

.672084
.672321

3.95
3.95

.945800
.945733

1.12

.726284
.726588

5.08
5.07

.273716
.273412

58
57

4

.672558

3.95

.945666

1 .12

.726892

5.07

.273108

56

5
6

7

.672795
.673032
.673268

3.95
3.95
3.93

.945598
.945531
.945464

1.13
1.12

1.12

.727197
.727501

.727805

5.05
5.07
5.07

.272803
.272499
.272195

55
54
53

8

.673505

3.95

.945396

1 .13

.728109

5.07

.271891

52

9
10

.673741
.673977

3.93
3.93
3.93

.945328
.945261

1 .13
1.12
1.13

.728412
.728716

5.05
5.07
5.07

.271588
.271284

51
50

11
12

9.674213
.674448

3.92

9.945193
.945125

1.13

9.729020
.729323

5.05

10.270980
.270677

49
48

13

.674684

3.93

.945058

1.12

.729626

5.05

.270374

47

14

.674919

3.92

.944990

1.13

.729929

5.05

.270071

46

15

.675155

3.93

.944922

1.13

.730233

5.07

.269767

45

16

.675390

3.92

.944854

1.13

.730535

5.03

.269465

44

17

.675624

3.90

.944786

1.13

.730838

5.05

.269162

43

18

.675859

3.92

.944718

1.13

.731141

5.05

.268859

42

19
20

.676094
.676328

3.92
3.90

.944650
.944582

1.13
1.13

.731444
.731746

5.05
5.03

.268556
.268254

41
40

3.90

1.13

5.03

21

9.676562

9.944514

9.732048

10.267952

39

22

.676796

3.90

.944446

1 .13

.732351

5.05

.267649

38

23

.677030

3.90

.944377

1 .15

.732653

5.03

.267347

37

24
25

26
27

28
29
30

.677264
.677498
.677731
.677964
.678197
.678430
.678663

3.90
3.90
3.88
3.88
3.88
3.88
3.88
3.87

.944309
.944241
.944172
.944104
.944036
.943967
.943899

1.13
1.13
1.15
1.13
1.13
1.15
1.13
1.15

.732955
.733257
.733558
.733860
.734162
.734463
.734764

5.03
5.03
5.02
5.03
5.03
5.02
5.02
5.03

.267045
.266743
.266442
.266140
.265838
.265537
.265236

36
35
34
33
32
31
30

31
32

9.678895
.679128

3.88

9.943830
.943761

1.15

9.735066
.735367

5.02

10.264934
.264633

29

28

33
34
35
36

.679360
.679592
.679824
.680056

3.87
3.87
3.87
3.87

.943693
.943624
.943555
.943486

1.13
1.15
1.15
1.15

.735668
.735969
.736269
.736570

5.02
5.02
5.00
5.02

.264332
.264031
.263731
.263430

27
26
25
24

37

38

.680288
.680519

3.87
3.85

.943417
.943348

1.15
1.15

.736870
.737171

5.00
5.02

.263130

.262829

23

22

39
40

.680750
.680982

3.85
3.87
3.85

.943279
.943210

1.15
1.15
1.15

.737471
.737771

5.00
5.00
5.00

.262529
.262229

21
20

41
42
43
44

9.681213
.681443
.681674
.681905

3.83

3.85
3.85

9.943141
.943072
.943003
.942934

1.15
1.15
1.15

9.738671
.738371
.738671
.738971

5.00
5.00
5.00

10.261929
.261629
.261329
.261029

19

18
17
16

45
46

47
48
49
50

.682135
.682365
.682595
.682825
.683055
.683284

3.83
3.83
3.83
3.83
3.83
3.82
3.83

.942864
.942795
.942726
.942656
.942587
.942517

1.17
1.15
1.15
1.17
1.15
1.17
1.15

.739271
.739570
.739870
.740169
.740468
.740767

5.00
4.98
5.00

4.98
4.98
4.98
4.98

.260729
.260430
.260130
.259831
.259532
.259233

15

14
13
12
11
10

51

52

9.683514

.683743

3.82

9.942448
.942378

1.17

9.741066
.741365

4.98

10.258934
.258635

9

8

53
54
55

.683972
.684201
.684430

3.82
3.82
3.82

.942308
.942239
.942169

1.17
1.15
1.17

.741664
.741962
.742261

4.98
4.97
4.98

.258336
.258038
.257739

7
6
5

56
57
58
59
60

.684658
.684887
.685115
.685343
9.685571

3.80
3.82
3.80
3.80
3.80

.942099
.942029
.941959
.941889
9.941819

1.17
1.17
1.17
1.17
1.17

.742559
.742858
I .743156
.743454
9.743752

4.97
4.98
4.97
4.97
4.97

.257441
.257142
.256844
.256546
10.256248

4
3
2
1
0

'

Cosine.

D. r.

Sine.

D. 1".

Cotang.

D. 1".

Tang.

'

118*

204

61-

COSINES, TANGENTS, AND COTANGENTS. 150°

/

Sine.

D. 1".

Cosine.

D. 1*.

Tang.

D. r.

Cotang.

•

0

9.<KV>71

3Qf\

9.941R19

11*"*

40"

GO'

1

2

! 685799
.686027

3.80

0 r-Q

.941749
.941679

1.17
1 17

.744050
744348

4 Mr

5H

3
4
5

.686254
.686482

.680709

3. SO
3.78

0 r~Q

.941609

.941539
.941469

1.17 i
1.17
1 18

.744645

.744943
.745240

4.87

4.95
4 97

.255057
.254760

57

56
55

6

7

.686936
.687163

3.78

.941398
.941328

1.17

1 17

. 74*5885

4.95

.254462

54
58

8
9

10

.687389
.687616
.687843

3.78
3.78
3.77 1

.941258
.941187

.941117

1.18
1.17

1.18

.74ol32
.746489
.746726

4.95
4.95
4.95

.

92

51
50

11
12
13
14
15
16
17
18
19
20

9. 688069
.688295
.688521
.688747
.688972
.689198
.689423
.689648
.689873
.690098

3.77
3.77
3.77
3.75
3.77
3 . 75
3.75
3.75 ,
3.75 |
3.75

9.941040
.940975
.940905
.940834
.940763
.940693
.940022
.940551
.940480
.940409

1.18
1.17
1.18
1.18
1.17
1.18
1.18
1.18
1.18
1.18

9.747023
.747319
.747010
.747913
.748209
.74&505
.748801
.749097
.749393
.749689

4.93
4.95
4.95
4.93
4.93
4.93
4.93
4.93
4.03
4.93

259087
.251791
.251495

.251199
.250903
.250807

.250311

49
48
47
46
45
44
43
42
41
40

21
22
23
24
25
26
27
28

9.690323
.690548
.690772
.690996
.691220
.691444
.691668
.691892

3.75
3.73

3.73 I
3.73 |
3.73 1
3.73 j
3.73 I

9.940338
.940207
.940196
.940125
.940054
.939982
.939911
.939840

1.18
1.18
1.18
1.18
1.20
1.18
1.18

9.749985
.750281
.750576
.750872
.751167
.751462
.751757
.752052

4.93
4.92
4.93
4.92
4.92
4.92
4.92

4Q9

10 250015
.249719
.248424
.249128
.248833
.248588
.248243
.247948

39
38

86

35
34
33
32

29
30

.692115
.692339

3.72 1
3.73
3 72

.939768
.939697

1.18
1.20

.752347
.752642

4.92
4.92

.247653
.247358

31
30

31
32
33
34
35
36
37
38
39
40

9.692562
.692785
.693008
.693231
.693453
.693676
.693898
.694120
.694342
.694564

3.72
3.72
3.72
3.70
3.72
3.70
3.70
3.70
3.70
3 70

9.939625
.939554
.939482
.939410
.939339
.939207
.939195
.939123
.939052
.938980

1.18
1.20
1 20
1.18
1.20
1.20
1.20
1.18
1.20
1.20

9.752937
.753231
.753526
.753820
.754115
.754409
.754703
.754997
.755291
.755585

4.90
4 92
4.90
4.92
4.90
4.90
4.90
4.90
4.90
4.88

10.247063
.246709
.240474
.246180
.245885
.245591
245297
.245003
.244709
.244415

29
28
27
26
29
xl4
23
22
21
20

41
42

43
44
45
46
47
48
49
50

9.694786
.695007
.695229
.695450
.695671
.695892
.696113
.696334
.696554
.090775

3.68
3.70
3.68
3 68
3.68
3.68
3.68
3. (57
3.68
3 07

9.938908
.938836
.938763
.938691
.938619
.938547
.938475
.938402
.938330
.938258

1.20
1.22
1.20
1.20
1.20
1.20
1.22
1.20
1.20
1 °2

9.755878
.756172
.756465
.756759
.757052
.757345
.757638
.757931
.758224
.758517

4.90
4.88
4.90
4.88
4.88
4.88
4.88
4.88
4.88
4.88

10.244122
.243535
.242948
.242862

'. 241 776
.241483

19
18
17
16
15
14
18
12
11
10

51
52
53
54
55
56
57
58
59
60

9.696995
.697215
.697435
.697654
.697874
.698094
.698313
.698532
.698751
9.698970

3.07
3.67
3.65
3.67
3.67
3.65
3.65
3.65
3.65

9.938185
.938113
.938040
.937967
.937895
.937822
.937749
.937676
.937601
9.937531

1.20
1.22
1.22
1.20
1.22
1.22
1.22
1.20
1.22

9.758810
.759102
.759395
.759687
759979
.760272
.760564
.760856
.761148
8.761489

4.87
4.KS
4.87
4.87
4.88
4.87
4 ^7

4.85

10.241190
.240898
840605
240818

.240021

.888808

10.238561

9
8
7
6
5
4
3
2
1
0

'

Cosine.

D.r.

Sine.

D. 1'.

Cotang. D. 1'. Tang.

119°

205

30°

TABLE XII. — LOGARITHMIC SINES,

149°

'

Sine.

D. 1".

Cosine.

D. r.

Tang.

D. 1".

Cotang.

'

0

1

9.698970
.699189

3.65

3 no

9.937531
.937458

1.22

9.761439
.761731

4.87

10.238561

.238269

60
59

2
3

.699407
.699626

. Do

3.65

3 CO

.937385
.937312

.22 |
.22

00

.762023
.762314

4.87
4.85

.237977
.237686

58
57

4

.699844

.DO
3K°.

.937238

.9*9

.762606

4.87

.237394

56

5

.700062

.Do

3 63

.937165

.22

.762897

4.85

.237103

55

6
7
8
9

.700280
.700498
.700716

3^63
3.63
3.62
3 63

.937092
.937019
.936946
.936872

!22
.22
.88
oo

.763188
.763479
.763770
.764061

4.85
4.85
4.85
4.85

.236812
.236521
.236230
.235939

54
53

52
51

10

! 701 151

3^62

.936799

'.23

.764352

4.85
4.85

.235648

50

11

9.701368

3 62

9.936725

9.764643

10.235357

49

12

.7J1585

.936652

go

.764933

4.83

.235067

48

13

.701802

3 62

.936578

. 4iO

.765224

4.85

.234776

47

14
15
16

.702019
.702236
.702452

3^62
3.60
3 62

.936505
.936431
.936357

^23
.23
22

.765514

.765805
.766095

4.83

4.85
4.83

4 DO

.234486
.234195
.233905

46
45
44

17

18

.702669

.702885

3^60
3 60

.936284
.936210

^23
23

.766385
.766675

OO

4.83

400

.233615
.233325

43

42

19

.703101

3 60

.936136

oq

.766965

.00

.233035

41

20

.703317

3^60

.936062

.6<J

.767255

4.83
4.83

.232745

40

21
22

9.703533
.703749

3.60
3 58

9.935988
.935914

1.23

9.767545
.767834

4.82

4 DO

10.232455
.232166

39

38

23
24
25
26

.703964
.704179
.704395
.704610

3^58
3.60
3.58
3 58

.935840
.935766
.935692
.935618

1.23
1.23
1.23

.768124
.768414
.768703
.768992

.OO

4.83
4.82
4.82

.231876
.231586
.231297
.231008

37

36
35
34

27

28

.704825
.705040

3^58
3 57

.935543
.935469

1^23

1 9°,

.769281
.769571

4.82
4.83

.230719
.230429

33
32

29
30

.705254
.705469

3^58
3.57

.935395
.935320

1 ,/4O

1.25
1.23

.769860
.770148

4.82
4.80
4.82

.230140
.229852

31
30

31

9-705683

3 58

9.935246

9.770437

4 DO

10.229563

29

32
33
34
35
36

.705898
.706112
.706326
.706539
.706753

3^57
3.57
3.55
3.57

3KTf

.935171
.935097
.935022
.934948
.934873

l'.23
1.25
1.23
1.25

.770726
.771015
.771303
.771592

.771880

.O-*

4.82
4.80
4.82
4.80

.229274

.228985
.228697
.228408
.228120

28
27
26
25
24

37

.706967

.Of

3tpr

.934798

i '25

.772168

4.80

.227832

23

38
39

.707180
.707393

.OO

3-55
3 55

.934723
.934649

1^23

.772457

.772745

4.82
4.80

4 on

227543
! 227255

22
21

40

.707606

3^55

.934574

1.'25

.773033

.OU

4.80

.226967

20

41
42
43

9.707819
.708032
.708245

3.55
3.55
3 55

9.934499
.934424
.934349

1.25
1.25

1OK

9.773321

.773608

.773896

4.78
4.80

10.226679
.226392
.226104

19
18
17

44
45

.708458
.708670

3^53
3 53

.934274
.934199

.^D

1.25

197

.774184
.774471

4>8

4 DA

.225816
.225529

16
15

46
47
48
49
50

.708882
.709094
.709306
.709518
.709730

3^53
3.53
3.53
3.53
3.52

.934123
.934048
.933973
.933898
.933822

.*!

1.25
1.25
1.25
1.27
1.25

.774759
.775046
.775333
.775621
.775908

.OU

4.78
4.78
4.80
4.78
4.78

.225241
.224954

.224667
.224379
.224092

14
13
12
11
10

51
52
53
54
55

9.709941
.710153
.710364
.710575
.710786

3.53
3.52
3.52
3.52

9.933747
.933671
.933596
.933520
933445

1.27
1.25
1.27
1.25

9.776195

.776482
.776768
.777055
.777342

4.78
4.77
4.78
4.78

477

10.223805
.223518
.223232
.222945

.222658

9
8
7
6
5

56
57
58
59
60

.710997
.711208
.711419
.711629
9.711839

3.52
3.52
3.52
3.50
3.50

.933369
.933293
.933217
.933141
9.933066

1.27
1.27
1.27
1.27
1.25

.777628
.777915

.778201
.778488
9.778774

. I (

4.78
4.77
4.78
4.77

.222372

.222085
.221799
.221512
10.221226

4
3
2
1
0

'

Cosine.

D. 1". |

I Sine.

D. r.

Cotang.

D. 1*.

Tang.

'

120°

206

59'

COSINES, TANGENTS, AND COTANGENTS. 148°

/

$ine.

D. r.

Cosine.

D. r.

Tang.

D. r.

Cotang.

/

0

9.711839

3 to

9.033066

197

9.778774

4r»7

60

1

2
3

4
5
6

.712050
.712260
.712469
.712679
712889
713098
713308

3.50
3.48
3.50
3.50
3.48
3.50

3AO

.988990
.682914

.932838
.932762
.U*J»>«5
.932609
.932533

1.27
1.27
1.27
1.28 :
1.27
1.27 :

1O7 i

.779080

.779346
.779883
.779918

.780203
.780489
.780775

4.77
4.77
4.77
4.75
4.77
4.77

.220940

.220088

.219797
.219611

.219225

59
58
57
56
55
54
53

8
9
10

713517
713726
.713935

3.48
3.48
3.48

.932457
.932380
.932304

1.28 !
1.27 I
1.27 !

.781060
.781346
.781631

4.77
4.75
4.75

.218940
.218864

.218369

•

52
51
50

11

12
13
14
15
16
17
18
19
20

9.714144
.714352
.714561
.714769
.714978
.715186
.715394
.715602
.715809
.716017

3.47
3.48
3.47
3.48
3.47
3.47
3.47
3.45
3.47
3.45

9.932228
.932151
.932075
.931998
.931921
.931845
.931768
.931691
.931614
.931537

1.28 !
1.27
1.28
1.28
1.27
1.28
1.28
1.28
1.28
1.28

9.781916
.782201
.782486
.782771
.783056
.783341
.783626
.783910
.784195
.784479

4.75
4.75
4.75
4.75
4.75
4.75
4.73
4.75
4.73
4.75

10.218084
.217799
.217514
.217229
.218944
.216659
.216374
.216000
.215805
.215521

49

48
47
46
45
44
43
42
41
40

21
22
23
24
25
26
27
28
29
30

9.716224
.716432
.7J6639
.716846
.717053
.717259
.717466
.717673
.717879
.718085

3.47
3.45
3.45
3.45
3.43
3.45
3.45
3.43
3.43
3 43

9.931460
• .931383
.93130(5
.931229
.931152
.931075
.930998
.930921
.930843
.930766

1.28
1.28
1.28
1.28
1.28 [
1.28
1.28
1.30 !
1.28
1.30

9.784764
.785048
.785332
.785616
.785900
.786184
.786468
.786752
.787036
.787319

4.73
4.73
4.73
4.73
4.73
4.73
4.73
4.73
4.72
4.73

10.215236
.214952
.214668
.214384
.214100
.213816
.213532
.213248
.212984
.212881

39
38
37
36
35
34
33
82
31
30

31
32
33
34
35
36
37
38
39
40

9.718291
.718497
.718703
.718909
.719114
.719320
.719525
.719730
.719935
.720140

3.43
3.43
3.43
3.42
3.43
3.42
3.42
3.42
3.42
3 42

9.930688
.930611
.930533
.930456
.930378
.930300
.930223
.930145
.930067
.929989

1.28
1.30
1.28
1.30
1.30
1.28
1.30
1.30
1.30
1 30

9.787603
.787886
.788170
.788453
.788736
.789019
.789302
.789585
.789868
.790151

4.72
4.73
4.72
4.72
4.72
4.72
4.72
4.72
4.72
4.72

10.212397
.212114

.211830
.211547
.211264
.210981
.210698
.210415
.210132
.209849

29
28
27
26
25
24
23
22
81
80

41

42
43
44
45
46
47
48
49
50

9.720345
.720549
.720754
.720958
.721162
.721366
.721570
.721774
.721978
.722181

3.40
3.42
3.40
3.40
3.40
3.40
3.40
3.40
3.38

3A(\

9.929911
.929833
.929755
.929677
.929599
.929521
.929442
.929364
.929286
.929207

1.30
1.30
1.30
1.30
1.30
1.32
1.30
1 30
1.32
1 30

9.790434
.790716
.790999
.791281
.791563
.791846
.792128
.792410
.792692
.792974

4.70
4.72
4.70
4.70

4'.70
4.70
4.70
4.70
4.70

10.209566
.20928*

.209001

1208487

.208154
.207872

.207308
.207088

19

18

18

15
14
13
18
11
10

51
52
53
54
55
56
57
58
59
60

9.722385
.722588
.722791
,722994
.723197
.723400
.723603
.723805
.724007
9.724210

3.38
8.88

3.38
3.38
3.38
3.38
3.37
3.37
3.38

9.929129
.929050
.928972
,928898
.988815
.928736
.928657
.928578
.02S4W
9.928420

1.32
1.30
1.32
1.30
1.32
1.32
1.32

lisa

1.32

9.79325f
.798538
.79881!
.794101
.7943*3
79488
.794914
. 79582-3
.796608
9.7957K.

4.70
4.68
4.70
4.70
4.68
4.70
4.68
4.68
4.68

10.206744

.208181
.206890
.205817
; 206888
.206064

.804498

9

8

6
5

4
3

1
0

/

Cosine.

D. r.

Sine.

D. r.

Cotang

D. 1'.

Tang.

68*

121

207

32°

TABLE XII. — LOGARITHMIC SINES,

147°

1

'

Sine.

D. 1".

Cosine.

D. 1'.

Tang.

0.1*.

Cotang.

'

0

1

9.724210

.724412

3.37

39.7

9.928420

.928342

1.30

9.795789
.796070

4.68
4 AS

10.204211
.203930

CO
50

2

.724614

• Ol

.928263

?*£»

.796351

.Oo

4 Aft

.203649

58

3

.724816

3.37

39.K

.938183

1 99

.796632

.Do
4 Aft

.203368

57

4

.725017

.OO
39.7

.928104

Iqo

.796913

.Do
4 Aft

.203087

56

5

.725219

.of

.928025

,«£•

.797194

.Do
4A7

.202806

55

6

.725420

3.35

39.7

.927946

1 99

.797474

.Of

4 Aft

.202526

54

7

.725622

.ot

q qK

.927867

1 33

.797755

. Do
4 gg

.202245

53

8

.725823

O.OO
39.K

.927787

.798036

.201064

52

9

.726024

.OO
3 OK

.927708

1 99

.798316

4A1"*

.201684

51

10

.726225

.OO

3.35

.927629

lias

.798596

.Oi

4.68

.201404

50

11

9.726426

9.927549

1 9.9

9.798877

4A1**

10.201123

40

12

.726626

o.oo

3qK

.927470

Iqq

.799157

. Ol
4A7

.200843 ; 48

13

.726827

. OO

.927390

.00

Iqq

.799437

.VI

4A7

.200563

47

14
15

.727027
.727228

3.33
3.35

.927310
.927231

.00

1.32

.799717
.799997

-0<

4.67

.200283
.200003

46
45

16
17
18
19
20

.727428
.727628
.727828
.728027

.728227

3.33
3 33
3.33
3.32
3.33
3.33

.927151
.927071
.926991
.926911
.926831

li33
1.33
1.33
1.33
1.33

.800277
.800557
.800836
.801116
.801396

4i67
4.65
4.67
4.67
4.65

.199723
.199443
.190164

.108884
.108604

4-1

43
42
41
40

21

fl. 728427

9.926751

1 9.9.

9.801675

10.198325

30

22
23
24

,728626
.728825
.729024

3^32
3.32

.926671
.926591
.926511

1 . OO

1.33
1.33

1 99

.801955
.802234
.802513

4^65
4.65

.198045
.107766
.107487

38
37
36

25
26
27

.729223
.729422
.729621

3.32
3.32
3.32

3qo

.926431
.926351
, 926270

1 .OO

1.33
1.35

Iqq

.802792
.803072
.803351

4.65
4.67
4.65

40-r

.107208
.106028
.106640

35
34
33

28
29
30

.729820
.730018
.730217

.CM

3.30
3.32
3.30

.926190
.926110
.926029

. OO

1.33
1.35
1.33

.803630
.803909
.804187

. OO

4.65
4.63
4.65

.106370
.106091
.195813

32
31
30

31

9 730415

39.fl

9.925949

Q-

9.804466

10.195534

29

32

.730613

. OU

.925868

1 ,OO

.804745

4. 60

.105255

28

33
34
35
36
37
38
39
40

.730811
.731009
.731206
.73J404
.731602
.731799
.731996
.732193

3.30
3.30
3.28
3.30
3.30
3.28
3.28
3.28
3.28

.925788
.925707
.925626
.925545
.925465
.925384
.925303
.925222

1.33
1.35
1.35
1.35
1.33
1.35
1.35
1.35
1.35

.805023
.805302
.805580
.805859
.806137
.806415
.806693
.806971

4.63
4 65
4.63
4.65
4 63
4.63
4.63
4.63
4.63

.194977
.194698
.194420
.194141
.193863
.103585
.103307
.193029

27
26
25
24
23
22
21
20

41
42

9.732390
.7'32587

3.28

39ft

9.925141
.925060

1.35

9.807249
.807527

4.63

4A9.

10.102751
.102473

19

IS

43
44
45

46
47
48

.732784
.732980
.733177
.733373
.733569
.733765

. *o

3.27
3.28
3.27
3.27
3.27

.924979
.924897
.924816
.924735
.924654
.924572

1:37
1.35
1.35
1.35
1.37

IqK

.807805
.808083
.808361
.808638
.808916
.809193

.OO

4.63
4.63
4.62
4.63
4.62

4A9.

.102195
.191017
.101639
.191362
.191084
.190807

17
16
15
14
13
12

49
50

.733961
.734157

3^27
3.27

.924491
.924409

.OD

1.37
1.35

.809471
.809748

.Oo

4.62
4.62

.100520
.190252

11
10

51
52
53
54
55
56
57
58
59
60

9.734353
.734549
.734744
.734939
.735135
.735330
.735525
.735719
.735914
9.736109

3.27
3.25
3.25
3.27
3.25
3.25
3.23
3.25
3.25

9.924328
.924246
.924164
.924083
.924001
.923919
.923837
.023755
.923673
9.023501

1.37
1.37
1.35
1.37
1.37
1.37
1 37
1.37
1.37

9. SI 0025
.810302
.810580
.810857
.811134
.811410
.811687
.811964
.812241
9.812517

4.62
4.63
4.62
4.62
4.60
4.62
4.62
4.62
4.60

10.1 ROOTS
. 180608
.189420
.180143
.188866
.188500
.188313
.188036
.187759
10.187483

9
8
7
6
5
4
3
2
1
0

' Cosine.

D. 1". i Sine. D. 1'. i Cotang. D. 1".

Tang.

'

1230

208

57-

COSINES,, TANGENTS, AND COTANGENTS.

146°

'

Sine.

D. 1'.

Cosine.

D. r.

Tang.

D. r.

Cotang.

'

0

1

2
3

9.736109
.736303
.736498
.736692

3.23
3.25 |
3.23

9.923591
.923509
.923427
.923345

1.37
.37
.37

a 7

9.812517
.812794

.813070
.813347

4.02
4.60
4.62

10.1S7J83
.186668

60
59
58
67

4

.736886

Q 9^

.923203

1

37

.813623

4.60
4 fin

.186377

56

5
6

.737080
.737274

3^23

399

.9231! 1
.9230U8

.818899

.814176

.ou
4.62

.186101

55
54

7

.737467

.*<*

.92301(5

1 oft

.814452

4.60

1 185548

58

8
9

.737661
.737855

3.23
3.23

.922933
.922851

J .00
1.37

1OQ

.814728
.815004

4.60
4.60

.184996

52
51

10

.738048

3.22
3.22

.922768

.OO

1.37

.815280

4.60
4.58

. 1847','U

50

11

9.738241

399

9.922686

9.815555

4 on

10.184445

49

12

.738434

.44

.922603

1 .38

.815831

.OU

.184169

48

13
14

.7:38627
.738820

3.22
3.22

.922520
.922438

1 .38
1.37

.816107
.816382

4.60
4.58
4 fin

.188808

.183618

47
46

15

.739013

3.22

.922355

1 .38

.816658

.w

4RQ

.183342

45

16

.739206

3.22

.922272

1 .38

.816933

.OO

4 fin

.188067

44

17

.7'39398

3.20

.922189

i .00

.817209

.DU

.182791

4:}

18
19

.739590
.739783

3.20
3.22

.922106
.922023

1 .38
1.38

1OQ

.817484
.817759

4.58
4.58
4 fin

.188516
.188241

41

20

.739975

3.20
3.20

.921940

.00

1.38

.818035

.ou
4.58

.181965

40

21
22
23
24
25
26
27
28
29

9.740167
.740359
.740550
.740742
.740934
.741125
.741316
.741508
.741699

3.20
3 18
3.20
3.20
3.18
3.18
3.20
3.18

9.921857
.921774
.921691
.921607
.921524
.921441
.921357
.921274
.921190

1.38
1.38
1.40
1.38
1.38
1.40
1.38
1.40

9.818310
.818585
.818860
.819135
.819410
.819684
.819959
.820234
.820508

4.58
4.58
4.58
4.58
4.57
. 4.58
4.58
4.57

10.181690
.181415
.181140
.180865
.180580
.180316
.180041
.179766
.179492

39
38

36
86
M
88

32
31

30

.741889

3.17
3.18

.921107

1AO

.820783

4^57

.179217

30

31
32
33
34
35
36
37
38
39
40

9.742080
.742271
.742462
.742652
.742842
.743033
.743223
.743413
.743602
.743792

3.18
3.18
3.17
3.17
3.18
3.17
3.17
3.15
3.17
3.17

9.921023
.920939
.920856
.920772
.920688
.920604
.920520
.920436
.920352
.920268

1.40
1.38
1.40
1.40
1.40
1.40
1.40
1.40
1.40
1.40

9.821057
.821332
.821606
.821880
.822154
.822429
.822703
.822977
.823251
.823524

4.58
4.57
4 57
4.57
4.58
4.57
4.57
4.57
4.55
4.57

10.178943
.178668
.178:394
.178120
.177846
.177571
.177^97
.177023
.176749
.176476

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

51
52
53
54
55
56
57
58
59
60

9.743982
.744171
.744361
.744550
.744739
.744928
.745117
.745306
.745494
.745683

9.745871
.746060
.746248
.746436
.746624
.746812
.746999
.747187
.747374
9.747562

3.15
3.17
3.15
3 15
3.15
3.15
3.15
3.13
3.15
3.13

3.15
3.13
3 13
3.13
3.13
3.12
3.13
3.12
3,13

9.920184
.920099
.920015
.919931
.919846
.919762
.919677
.919593
.919508
.919424

9.919339
.919254
.919169
.919085
.919000
.918915
.918830
.918745

9 '.91 8574

1.42
1.40
1.40
1.42
1.40
1.42
1.40
1.42
1.40
1.42

1.42
1.42
1.40
1.42
1.42
1.42
1.42
1.48
1.42

9.823798
.824072
.824345
.824619
.824893
.825166
.825439
.825713
.825986

'. 826803
.827078
.827351
827684

.827897
.828170

4.57
4.55
4.57
4.57
4.55
4.55
4.57
4.55
4.55
4.55

4.55
4.55
4.55
4.55
4.55
4.55
4.53

4.53

10.176202
.175928
.175655
.175381
.175107
.174834
.174561
.174287
.171011
.173741

10.173468
.173195

.172649
.172103

10.171013

19
18
17
16
15
14
13

n

n

10

9
8
7
6
5
4
3
2
1
0

/

Cosine.

D. 1".

Sine.

D. r.

('otang.

D. r.

Tang.

r

66'

209

34°

TABLE XII. LOGARITHMIC SINES,

145°

'

Sine.

D. 1".

Cosine.

D. r.

Tang.

D. 1".

Cotang.

'

0

1

9.747562
.7477'49

3.12

9.918574
.918489

1.42

9.828987
.829260

4.55

10.171013
. 170740

60

59

2
3

.747936
.748123

3.12
3.12
3 12

.918404
.918318

1 .42
1.43

1 49

.829532

.829805

4.53
4.55

4 to

.170468
.170195

58
57

4
5

6

8
9
10

.748310
.748497
.748683
.748870
.749056
.749243
.749429

3.12
3.10
3.12
3.10
3.12
3.10
3.10

.918233
.918147
.918062
.917976
.'917891
.917805
.917719

1 .4/6

1.43
1.42
1.43
1.42
1.43
1.43
1.42

.830077
.830349
.830621
.830893
.831165
.831437
.831709

.00

4.53
4.53
4.53
4.53
4.53
4.53
4.53

.169923
.169651
.169379
.169107

.168835
.168563
.168291

56
55
54
53

52
51
50

11
12
13

9.749615

.749801
.749987

3.10
3.10

3AQ

9.917634
.917548
.917462

1.43
1.43

9.831981
.832253
.832525

4.53
4.53

10.168019
.167747
.167475

49

48
47

14
15

750172
! 750358

.Uo
3.10
3ns

.917376
.917290

1 .43
1.43

.832796
.833068

4.52
4.53

.167204
.166932

46
45

16
17

.750543
. 750729

.Uo
3.10
3ns

.917204
.917118

1 .43
1.43

.833339
.833611

4.52
4.53

.166661
.166389

44
43

18
19

.750914
.751099

.Uo
3.08

3 no

.917032
.916946

1 .43
1.43

.833882
.834154

4.52
4.53

.166118
.165846

42

41

20

.751284

.Uo

3.08

.916859

1 .45
1.43

.834425

4.52
4.52

.165575

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
.165033
.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

9.753312
.753495
.753679
.753862
.754046
.754229
.754412

3.05
3.07
3.07
3.07
3.05
3.05

9.915907
.915820
.915733
.915646
.915559
.915472
.915385

1.45
1.45
1.45
1.45
1.45
1.45

9.837405
.837675
.837946
.838216

.838487
.838757
.839027

4.50
4.52
4.50
4.52
4.50
4.50

10.162595
.162325
.162054
.161784
.161513
.161243
.160973

29
28
27
26
25
24
23

38
39
40

.754595

.754778
.754960

3.05
3.05
3.03
3.05

.915297
.915210
.915123

1.47
1.45
1.45

1.47

.839297
.839568
.839838

4.50
4.52
4.50
4.50

.160703
.160432
.160162

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
.1,40917
.841187
.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

9.756963
.757144
.757326
.757507

3.02
3 03
3.02
3 no

9.914158
.914070
.913982
.913894

1.47
1.47
1.47

9.842805
.843074
.843343
.843612

4.48
4.48
4.48

10.157195
156926
.156657
.156388

9

8
7
6

55
56
57

.757688
.757869
.758050

.U/v

3.02
3.02

.913806
.913718
.913630

1.47
1.47
1.47

.843882
.844151
.844420

4.50
4.48
4.48

.156118
.155849
.155580

5
4
3

58
59
60

.758230
.758411
9.758591

3.00
3.02
3.00

913541
.913453
9.913365

1 .48
1.47
1.47

.844689
.844958
9.845227

4.48
4.48
4.48

.155311
.155042
10.154773

2
1
0

'

Cosine.

D. 1". |

Sine. 1 D. 1". i

Cotang. D. 1*.

Tang.

'

210

$5"

35°

COSINES, TANGENTS, AND COTANGENTS.

144°

'

Sine.

D. 1'.

Cosine.

D. r.

' Tang.

D. 1'.

Ootaag,

/

0

9.758591

3 0°

9.0133(55

1 48

9.845287

44ft

10.154778

60

1

2
3

4
5
6

7

.758772
.758952
.759132
.759312
.759492
.759672
.759852

S^OO
3.00
3.00
3.00
3.00
3.00

.913276
.913187
.913099
.913010
.912922
.912833
.912744

1.48
1.47
1.48
1.47
1.48
1.48

1 4ft

'.845764
.846033
.846302
.846570
.846839
.847108

4.47
4.48
4.48
4.47
4.48
4.48

.154504
.154286

.158696
.153430
.158161

59

57
56
55
54

53

8
9

.760031
.760211

3.00

2Qft

.912655
.912566

1.48

14ft

.847376
.847644

4.47

44ft

.152624
.152856

52
51

10

.760390

2.98

.912477

1.48

.847913

4.47

.150067

50

11

9.760569
.760748

2.98

9.912388
.912299

1.48

9.848181
.848449

4.47

10.151819
.151651

49

48

13
14
15

.760927
.761106
.761285

2.98
2.98
2.98

.912210
.912121
.912031

1.48
1.48
1.50

14ft

.848717
.848986
.849254

4.48
4.47

447

.151283
.151014
.150746

47
40
45

16
17
18
19
20

.761464
.761642
.761821
.761999
.762177

2.97
2.98
2.97
2.97
2 98

.911942
.911853
.911763
.911674
.911584

1.48
1.50
1.48
1.50
1.48

.849522
.849790
.850057
.850325
.850593

4.47
4.45
4.47
4.47
4.47

.150478
.150210
.149943
.149675
.149407

44
43
42
41
40

21
22
23
24
25
26
27
28
29
30

9.762356
.762534
.762712
.762889
.763067
.763245
.763422
.763600
.763777
.763954

2.97
2.97
2.95
2.97
2.97
2.95
2.97
2.95
2.95
2 95

9.911495
.911405
.911315
.911226
.911136
.911046
.910956
.910866
.910776
.910686

1.50
1.50
1.48
1.50
1.50
1.50
1.50
1.50
1.50
1.50

9.850861
.851129
.851396
.851664
.851931
.852199
.852466
.852733
.853001
.853268

4.47
4.45
4.47
4.45
4.47
4.45
4.45
4.47
4.45
4.45

10.149139
.148871
.148604
.148336
.148069
.147801
.147534
.147267
.146999
.146732

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9.764131
.764308
.764485
.764662
.764838
.765015
.765191
.765367
.765544
.765720

2.95
2.95
2.95
2.93
2.95
2.93
2.93
2.95
2.93
2 93

9.910596
.910506
.910415
.910325
.910235
.910144
.910054
.909963
.909873
.909782

1.50
1.52
1.50
1.50
1.52
1.50
1.52
1.50
1.52
1 52

9.853535
.853802
.854069
.854336
.854603
.854870
.855137
.855404
.855671
.855938

4.45
4.45
4.45
4.45
4.45
4.45
4.45
4.45
4.45
4.43

10.146465
.146198
.145981
.145664
.145397
.145130
.144863
.144596
.144329
.144062

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

9.765896
.766072
.766247
.766423
.766598
.766774
.766949
.767J24
.767300
.767475

2.93
2.92
2.93
2.92
2.93
2.92
2.92
2.93
2.92
2 90

9.909691
.909601
.909510
.909419
.909328
.909237
.909146
.909055
.908964
.908873

1.50
1.52
1.52
1.52
1.52
1.52
1.52
1.52
1.52
1 53

9.856204
.856471
.856737
.857004
.857270
.857537
.857803
.858069
.858336
.858602

4.45
4.43
4.45
4.43
4.45
4.43
4.43
4.45
4.43
4.43

10.143796

J48263

.142996
.142780
.142463
.142197
.141981
!l41664
.1413'JS

19
18
17
16
15
14
13
12
11
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

! 908141
.908049

1.52
1.52
1.53
1.52
1.53
1.52
1.53
1.53
1.52

9.858868
.859134
.859400
.859666
.859932
960198
.860464
.860730
.860995
9.861261

4.43
4.48
4 43
4.43
4.48
443
4.43
4.42
4.43

10.141132
.140866

.140600

.140068
! 189609
J80686

.139270
.139005

9
8
7
6
5
4
3
2
1
0

'

Cosine.

D. r.

Sine.

D. r.

Cotang.

D. r.

Tang.

125'

211

36°

TABLE XII. — LOGARITHMIC SINES,

143°

'

Sine.

D.I".

Cosine.

D. r.

Tang.

D. 1".

Cotang.

'

0

1

2
3

4
5
6
7
8

9.769219
.769393
.769566
.769740
.769913
.770087
.770260
.770433
.770606

2.90

2.88
2.90
2.88
2.90
2.88
2.88
2.88

9.907S58
.907866
.907774
.907682
.907550
.907498
.907406
.907314
.907222

1.53
1.53
1.53
1.53
1.53
1.53
1.53
1.53

9.861261
.861527
.861792
.862058
.862323
.862589
.862854
.863119
.863385

4.43
4.42
4.43
4.42
4.43
4.42
4.42
4.43

10.138739
.138473
.138208
.137942
.137677
.137411
.137146
.136881
.136615

60
59
58
57
56
55
54
53
52

9
10

.770779
.770952

2.88
2.88
2.88

.907129
.907037

1.55
1.53
1.53

.863650
.863915

4.42
4.42
4.43

.136350
.136085

51
50

11
12
13
14
15
16
17
18
19

9.771125
.771298
.771470
.771643
.771815
.771987
.772159
.772331
.772503

2.88
2.87
2.88
2.87
2.87
2.87
2.87
2.87

9.906945
.906852
.906760
.906667
.906575
.906482
.906389
.906296
.906204

1.55
1.53
1.55
1.53
1.55
1.55
1.55
1.53

9.864180
.864445
.864710
.864975
.865240
.865505
.865770
.866035
.866300

4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42

10.135820
.135555
.135290
.135025
.134760
.134495
.134230
.133965
.133700

49
48
47
46
45
44
43
42
41

20

.772675

2.87
2.87

.906111

1.55
1.55

.866564

4.40
4.42

.133436

40

21
22
23
24
25
26
27
28
29
30

9.772847
.773018
.773190
.773361
.773533
.773704
.773875
.774046
.774217
.774388

2.85

2.87
2.85
2.87
2.85
2.85
2.85
2.85
2.85
2.83

9.906018
.905925
.905832
.905739
.905645
.905552
.905459
.905366
.905272
.905179

1.55
1.55
1.55
1.57
1.55
1.55
1.55
1.57
1.55
1.57

9.866829
.867094
.867358
.867623
.867887
.868152
.868416
.868680
.868945
.869209

4.42
4.40
4.42
4.40
4.42
4.40
4.40
4 42
4.40
4.40

10.133171
.132906
.132642
.132377
.132113
.131848
.131584
.131320
.131055
.130791

39
38
37
36
35
34
33

31
30

31
32
33
34
35
36
37
38
39
40

9.774558
.774729
.774899
.775070
.775240
.775410
.775580
.775750
.775920
.776090

2.85
2.83
2.85
2.83
2.83
2.83
2.83
2.83
2.83
2.82

9.905085
.904992
.904898
.904804
.904711
.904617
.904523
.904429
.904335
.904241

1.55
1.57
1.57
1.55
1.57
1.57
1.57
1.57
1.57
1.57

9.869473
.861)737
.870001
.870265
.870529
.870793
.871057
.871321
.871585
.871849

4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.38

10.130527
.130263
.129999
.129735
.129471
.129207
.128943
.128679
.128415
.128151

29

28
27
26
25
24
23
22
21
20

41

9.776259

200

9.904147

9.872112

4 /in

10.127888

19

42
43

.776429
.776598

.00
2.82

2 DO

.904053
.903959

l!57

1KQ

.872376
.872640

.4U

4.40

4QQ

.127624
J 27360

18
17

44
45

46

.776768
.776937
.777106

.OO

2.82
2.82

.903864
.903770
.903676

.Do

1.57
1.57

.872903
.873167
.873430

.OO

4.40
4.38

.127097
.126833
.126570

16
15
14

47
48

.777275
.777444

2^82

2Qp

.903581
.903487

1 .58
1.57
1 58

.873694
.873957

4.40
4.38

4QQ

.126306
.126043

13

12

49

50

.777613

.777781

. O&

2.80

2.82

.903392
.903298

1.'57
1.58

.874220

.874484

.OO

4.40
4.38

.125780
.125516

11
10

51

9.777950

CO

9.903203

IRQ

9.874747

4OQ

10.125253

9

52

.778119

OA

.903108

.Do

.875010

.OO

.124990

8

53
54

.778287
.778455

2.80

2.80
209

.9030*4
.902919

1.57
1.58
1 58

.875273

.875537

4.38
4.40

4OQ

.124727
.124463

7
6

55
56

.778624

.778792

.0/5
2.80

.902824
.902729

l.&B

.875800
..876063

.OO

4.38

.124200
.123937

5
4

57
58

.778960
.779128

2.80
2.80

.902634
.902539

l!58
1 58

.876326
.876589

4.38
4.38

4OQ

.123674
.123411

3
2

59

.779295

Ort

.902444

.876852

. OO

.123148

1

60

9.779463

2.oO

9.902349

1.58

9.877114

4.37

10.122886

0

'

Cosine.

D. r.

Sine.

D. r.

Cotang. 1 D. 1'.

Tang.

'

126°

212

COSINES, TANGENTS, AND COTANGENTS.

142°

'

Sine.

D. 1'.

Cosine.

D.I'.

Tang.

D. r.

Cotang.

'

0

1

2

3
4
5
6

9.779463
.779631
.779798
.779966
.780133
.780300
.780467

2.80
2.78
2.80
2.78
2.78
2.78

9.902349
.902253
.902158
.902063
.901967
.901872
.901776

1.60
1.58
1.58
1.60
1.58
1.60

9.877111

.877640
.877903
.878165
.878428
.878691

4.38
4.38
4.38
4.37

4.38

4. as

10.122886

.128087

.121835
.121578

.121309

60
59

58

56
66

54

7
8
9

10

.780634
.780801
.780968
.781134

2.78
2.78
2.77
2.78

.901681
.901585
.901490
.901394

1.60
1.58
1.60
1.60

.878953
.879216
.879478
.879741

4.38
4.37
4.38
4.37

.121047
.120784
.120522

.120259

68

68
61

50

11
12
13

9.781301
.781468
.781634

2.78

2.77

277

9.901298
.901202
.901106

1.60
1.60

9.880003
.880265
.880528

4.37
4.38

10.119997
.110786

.119472

49
48
47

14
15
16
17
18
19

.781800
.781966
.782132
.782298
.782464
.782630

2.77 !
2.77 !
2.77 i
9 77

2>7

2C-7

.901010
.900914
.900818
.900722
.900626
.900529

1.60
1.60
1.60
1.60
1.62

1 Art

.880790
.881052
.881314
.881577
.881839
.882101

4.37
4.37
4.38
4.37

4.37
407

.119210
.118948
,118686
.118423
.118161
.117899

46
45
44
43
42
41

20

.782796

2.75

.900433

1.60

.882363

4.37

.117637

40

21
22
23
24
25
26
27
28
29

9.782961
.783127

.783292
.783458
.783623
.783788
.783953
.784118
.784282

2.77
2.75
2.77
2.75
2.75
2.75
2.75
2.73

9.900337
.900240
.900144
.900047
.899951
.899854
.899757
.899660
.899564

1.62
1.60
1.62
1.60
1.62
1.62
1.62
.60

9.882625

.882887
.883148
.883410
.883672
.883934
.884196
.884457
.884719

4.37
4.35
4.37
4.37
4.37
4.37
4.35
4.37

4 OK

10.117375
.117113
.116852
.116590
.116328
.116066
.115804
.115543
.115281

39
38
37
36
35
34

as

32
31

30

.784447

2.75
2.75

.899467

.62

.884980

4.37

.115020

30

31
32
33
34
35

9.784612
.784776
.784941
.785105
.785269

2.73
2.75
2.73
2.73

9.899370
.899273
.899176
.899078
.898981

.62
.62
.63
.62

9.885242
.885504
.885765
.886026
.886288

4.37
4.35
4.35
4.37

4 OK

10.114758
.114496
.114235
.113974
.113712

29
28
27
26
25

36
37
38
39
40

.785433

.785597
.785761
.785925
.786089

2.73
2.73
2.73
2.73
2.73
2.72

.898884
.898787
.898689
.89&592
.898494

.62
.63
.62
.63
.62

.886549
.886811
.887072
.887333
.887594

4.37
4.35
4.35
4.35
4.35

.113451
.113189
.112928
.112667
.112406

24
23
22
21
20

41
42
43

44
45
46

47
48
49
50

9.786252
.786416
.786579
.786742
.786906
.787069
.787232
.787395
.787557
.787720

2.73
2.72
2.72
2.73
2.72
2.72
2.72
2.70
2.72
* 72

9.898397
.898299
.898202
.898104
.898006
.897908
.897810
.897712
.897614
.897516

.63
.62
.63
.63
.63
.63
.63
.63
.63
63

9.887855
.888116
.888378
.888639
.888900
.889161
.889421
.889682
.889943
.890204

4.35
4.37
4.35
4.35
4.35

4. as

4.35
4.35
4.35
4.35

10.112145
.111884
.111622
.111361
.111100
.110839
.110579
.110318
.110057
.109796

19
18
17
16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

9.787883
.788045
.788208
.788370
.788532
.788694
.788856
.789018
.789180
9.789342

2.70
2.72
2.70
2.70
2.70
2 70
2.70
2.70
2.70

9.897418
.897320
.897222
.897123
.897025
.896926
.896828
.896729
.896631
9.896532

.63
.63
.65
.63
.65
.63
1.65
1.63
1.65

9.890465
.890725

.89098(5
.891247
.891507
.891768

! 892289
.892549
9.892810

1

4.33
4.35
4.35

4. as

4.35

4. as
4. as

10.109535
.109275
.109014
1108758
.106188
.108888
! 107878
.107711
.107451
10.107190

9

8

7
6
5
4
3
2
1
0

"/"

Cosine.

D 1".

Sine.

D. r.

i Cotang.

D. r.

Tang.

'

127°

213

52'

38°

TABLE XII. — LOGARITHMIC SINES,

141°

'

Sine.

D. 1*.

Cosine.

D. 1".

Tang.

D.I".

Cotang.

•

0

1

2
3
4
5

9.789342
.789504
.789665
.789827
.789988
.790149

2.70

2.68
2.70
2.68
2.68

9.896532
.896433
.896335
.896236
.896137
.896038

1.65
1.63
1.65
1.65
1.65

CK

9.892810
.893070
.893331
.893591
.893851
.894111

4.33
4.35
4.33
4.33
4.33

4 OR

10.107190
.106930
.106669
.106409
.106149
.105889

GO
59
58
57
56
55

6

.790310
.790471

2^68

2 CO

.895939
.895840

. .DO ,

.894372
.894632

.00

4.33

4qo

.105628
.105368

54
53

8

.790632

. Do

2 AS

.895741

67

.894892

. o<*
4qq

.105108

52

9
10

.790793
.790954

.Do

2.68
2.68

.895641
.895542

f

.65
.65

.895152
.895412

.00

4.33
4.33

.104848
.104588

51
50

11

9.791115

9.895443

R7

9.895672

400

10.104328

49

12
13
14
15
16
17
18
19

.791275
.791436
.791596
.791757
.791917
.792077
.792237
.792397

2^68
2.67
2.68
2.67
2.67
2.67
2.67

2A7

.895343
.895244
.895145
.895045
.894945
.894846
.894746
.894646

.Of

.65
.65
.67
.67 !
.65
.67

: .67

.895932
.896192
.896452
.896712
.896971
.897231
.897491
.897751

.00

4.33
4.33
4.33
4.32
4.33
4.33
4.33

.104068
.103808
.103548
.103288
.103029
.102769
.102509
.102249

48
47
46
45
44
43
42
41

20

.792557

. Of

2.65

.894546

:!e?

.898010

4^33

.101990

40

21

22
23
24
25
26
27

9.792716
.792876
.793035
.793195
.793354
.793514
.793673

2.67
2.65
2.67
2.65
2.67
2.65

9.894446
.894346
.894246
.894146
.894046
.893946
.893846

1.67
, 1.67
1.67
1.67
1.67
1.67

9.898270
.898530
.898789
.899049
.899308
.399568
.899827

4.33
4.32
4.33
4.32
4.33
4.32

10.101730
.10147C
.101211
.100951
.100692
.100432
.100173

39
38
37
36
35
34
33

28

.793832

2.65

.893745

1 .68

.900087

4.33

.099913

32

29

.793991

2.65

2 Of

.893645

.67

.900346

4.32
409

.099654

31

30

.794150

.DO
2.63

.893544

!67

.900605

,(S6

4.32

.099395

30

31

9.794308

9.893444

9.900864

400

10.099136

29

32
33

.794467
.794626

2.65
2.65

.893343
.893243

'.67

AQ

.901124
.901383

.00

4.32

.098876
.098617

23
27

34

.794784

2.63

.893142

.Do

.901642

^'e\n

.098358

26

35
36

.794942
.795101

2.63
2.65

.893041
.892940

.68

.68

.901901
.902160

4.O4

4.32

.098099
.097840

25
24

37
38

.795259
.795417

2.63
2.63

.892839
.892739

.68
.67

.902420
.902679

4.33

4.32

.097580
.097321

23
22

39

40

.795575
.795733

2.63
2.63
2.63

.892638
.892536

.68

.70
.68

.902938
.903197

4.32
4.32

4.32

.097062
.096803

21
20

41
42
43
44
45
46
47
48
49

9.795891
.796049
.796206
.796364
.796521
.796679
.796836
.796993
.797150

2.63

2.62
2.63
2.62
2.63
2.62
2.62
2.62

9.8924a5
.892334
.892233
.892132
.892030
.891929
.891827
.891726
.891624

.68
.68
.68
.70
.68
.70
.68
.70

Aft

9.903456
.903714
.903973
.904232
.904491
.904750
.905008
.905267
.905526

4.30
4.32
4.32
4.32
4.32
4.30
4.32
4.32

4OO

10.096544
.096286
.096027
.095768
.095509
.095250
.094992
.094733
.094474

19
18
17
16
15
14
13
12
11

50

.797307

2! 62

.891523

.Do

.70

.905785

.Die

4.30

.094215

10

51
52
53
54
55
56
57
58
59
60

9.797464
.797621

.797777
.797934
.798091
.798247
.798403
.798560
.798716
9.79887'2

2.62

2.60
2.62
2.62
2.60
2.60
2.62
2.60
2.60

9.891421
.891319
.891217
.891115
.891013
.890911
.890809
.890707
.890605
9.890503

.70
.70
.70
1.70
1.70
1.70
1.70
1.70
1.70 ,

9.906043
.906302
.906500
.906819
| .907077
.907336
.907594
.907853
.908111
9.908369

4.32
4.30
4.32
4.30
4.32
4.30
4.32
4.30
4.30

10.093957
.093698
.093440
.093181
.092923
.092664
.092406
.092147
.091889
10.091631

9

8
7
6
5
4
3
2
1
0

'

Cosine.

D. r.

Sine.

D. i". i

Cotang.

D. r.

Tang.

'

128°

214

COSINES, TANGENTS, AND COTANGENTS.

140°

'

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. r.

Cotang.

/

0

9.798872

2 60

9.890503

1 72

9.908369

4 o0 10.MH1,:',!

60

1

.799028

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.890400

1 7fl

.906628

rS i .«)'.» i:;:-'

59

2

.799184

2 58

.890298

1 . I\J

!X>KHSI) 1'™ ! O'jnit

58

3

4
5

.799339
.799495
.799651

2^60
2.60
2 58

.890195
.890093
.889990

l'.70
1.72
1 70

'.909144

.909402
.909660

4.30
4.30

4OA

.1 '.MS.M1

.090696
.090840

57
56
56

6

.799806

.889888

I'M

.909918

.oU

.090068

54

r»

.799962

2.60

.889785

l!72

.910177

4.32

4OA

58

8

.800117

jj'jjS

.889682

.910435

.ou

58

9
10

.800272
.800427

2 '.58
2.58

.889579
.889477

l'.70
1.72

.910693
.910951

4.30
4.30
4.30

.069807

51

50

11
12
13
14

9.800582
.800737
.800892
.801047

2.58
2.58
2.58

9.889374

.889271
.889168
.889064

1.72
1.72
1.73

1 72

9.911209
.911467
.911725
.911982

4.30
4.30
4.28
4 30

10.088791
066583

49
48
47
46

15
16
17
18
19
20

.801201
.801356
.801511
.801665
.801819
.801973

2 '.58
2.58
2.57
2.57
2.57
2.58

.888961
.888858
.888755
.888651
.888548
.888444

1>2
1.72
1.73
1.72
1.73
1.72

.912240
.912498
.912756
.913014
.913271
.913529

4.' 30
4.30
4.30
4.28
4.30
4.30

.067760
.067508

! 086988

.086471

4B
44
48

41

40

21
22
23
24
25
26
27
28
29
30

9.802128
.802282
.802436
.802589
.802743
.802897
.803050
.803204
.803357
.803511

2.57
2.57
2.55
2.57
2.57
2.55
2.57
2.55
2.T.7
2.55

9.888341
.888237
.888134
.888030
.887926
.887822
.887718
.887614
.887510
.887406

1.73
1.72
1.73
1.73
1.73
1.73
1.73
1.73
1.73
1.73

9.913787
.914044
.914302
.914560
.914817
.915075
.915332
.915590
.915847
.916104

4.28
4.30
4.30
4.28
4.30
4.28
4.30
4.28
4.28
4.30

10.086213
065956
.085698
.065440

.OK51S3
.084925
.084668
.084410
.084153
.083896

39
38
37
36
35
84
33
32
31
30

31
32
33
34

9.803664
.803817
.803970
.804123

2.55
2.55
2.55
2 55

9.887302
.887198
.887093
.886989

1.73
1.75
1.73
1.73

9.916362
.916619
.916877
.917134

4.28
4.30
4.28
4 28

10.083638
.083:581
.083123
.068866

29

28
27
86

35
36

.804276
804428

2^55

.886885
886780

.917391
.917648

4.28

.IK.WH)

I068858

M

37
38
39
40

'.804581
.804734
.804886
.805039

2.55
2.55
2.53
2.55
2.53

! 886676
.886571
.886466
.886362

1.73
1.75
1.75
1.73
1.75

.917906
.918163
.918420
- .918677

4.30
4.28
4.28
4.28
4.28

.068094

.081837
.081580
.081323

23

a

20

41
42
43
44
45
46
47
48
49
50

9.805191
.805343
.805495
.805647
.805799
.805951
.806103
.806254
.806406
.806557

2.53
2.53
2.53
2.53
2.53
2.53
2.52
2.53
2.52
2 53

9.886257
.886152
.886047
.885942
.885837
.885732
.885627
.885522
.885416
.885311

1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.77
1.75
1.77

9.918934
.919191
.919448
.919705
.919962
.920219
.920476
.920733
.920990
.921247

4.28
4.28
4.28
4.28
4.28
4.28
4.28
4.28
4.28
4.27

10.081066
080809

.060895

.nsou-is
.079781

.079524

.079010
.078753

19

18
17
16
15
14
18
18
11
10

51
52
53
54
55
56
57
58
59
60

9.806709
.806860
.807011
.807163
.807314
.807465
.807615
.807766
.807917
9.808067

2.52
2.52
2.53
2.52
2.52
2.50
2.52
2.52
2.50

9.885205
.885100
.884994
.884889
.884783
.884677
.884572
.884466
.884360
9.884254

1.75
1.77
1.75
1 1.77
1.77
1.75
1.77
j 1.77
1.77

9.921503
.921760
.922017
.988874
.988580
.^8787
.923044
.923300

4.28
4.28
4.28
4.«?

4.28

4.88
4.28

10.07R497

.077813

.070700
.079448

10.076186

9

s
7
6
5
4
3
2

0

'

Cosine.

D. 1".

Sine.

D. r.

Cotang.

D. r.

Tang.

129°

215

40°

TABLE XII. — LOGARITHMIC SINES,

139°

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

1

2

9.808067
.808218
.808368

2.52
2.50
2 52

9.884254
.884148
.884042

1.77
1.77

1 77

9.923814
.924070
.924327

4.27
4.28
4 27

10.076186
.075930)
.075673

60

59
58

3

.808519

.883936

1.14

78

.924583

.075417

57

4

5

.808669
.808819

2^50

.883829
.883723

. <O

'I7

.924840
.925096

4.27

.075160
.074904

56

55

6
7
8
9
10

.808960
.809119
.809269
.809419
.809569

2.50
2.50
2.50
2.50
2.50
2.48

.883617
.883510
.883404
.883297
.883191

.77
.78
.77
.78
.77
.78

.925352
.925609
.925865
.926122
.926378

4.27

4.28
4.27
4.28

4.27
4.27

.074648
.074391
.074135
.073878
.073622

54
53
52
51
50

11
12

9.809718
.809868

2.50

9.883084

.882977

1.78

9.926634

.926890

4.27

10.073366
.073110

49

48

13

.810017

2.48

.882871

1 .77

.927147

S

.072853

47

14
15
16
17
18
19
20

.810167
.810316
.810465
.810614
.810763
.810912,
.811061

2^48
2.48
2.48
2.48
2.48
2.48
2.48

.882764
.882657
.882550
.882443
.882336
.882229
.882121

1 .78
1.78
1.78
1.78
.78
.78
.80
.78

.927403
.927659
.927915
.928171

.928427
.928684
.928940

4.2<
4.27
4.27

4.27
4.27
4.28

4.27
4.27

.072597
.072341
.072085
.071829
.071573
.071316
.071060

46
45
44
43
42
41
40

21
22
23
24

9.811210
.811358
.811507
.811655

2.47
2.48
2.47

9.882014
.881907
.881799
.881692

.78
.80
.78

9.929196
.929452
.929708
.929964

4.27
4.27
4.27

10.070804
.070548
.070292
.070036

39

38
37
36

25

26

.811804
.811952

2.48
2.47

9 47

.881584

.881477

.80
.78

• pn

.930220
.930475

4.27
4'2£

.069780
.069525

35
34

27

.812100

/c.4i

2 47

.881369

.OU

PA

.930731

97

.069269

33

28

.812248

.881261

.OU

.930987

.XI

.069013

32

29
30

.812396
.812544

2.47
2.47
2.47

.881153
.881046

.80
.78
.80

.931243
.931499

.27
.27
.27

.068757
.068501

31
30

31

9.812692

247

9.880938

DA

9.931755

10.068245

29

32

.812840

.44

2 47

.880830

.OU
PA

.932010

27

.067990

28

33
34
35

.812988
.813135
.813283

2^45

2.47

.880722
.880613
.880505

.OU

.82
.80
on

.932266
.932522
.932778

4^27
4.27

.067734
.067478
.067222

27
26
25

36
37

.813430
.813578

2.45
2.47

24K.

.880397
.880289

.ou
.80

.933033
.933289

4.25

4.27

497

.066967
.066711

24

23

38

.813725

.40

2 AC

.880180

PA

.933545

,Xt

4 OK

.066455

22

39
40

.813872
.814019

.40

2.45
2.45

.880072
.879963

.OU

.82
.80

.933800
.934056

:.O9

4.27
4.25

.066200
.065944

21
20

41
42
43

9.814166
.814313
.814460

2.45
2.45

9.879855
.879746
.879637

.82

.82

PA

9.934311
.934567
.934822

4.27
4.25

497

10.065689
.065433
.065178

19

18
17

44

.814607

?•• j5

.879529

.oU

.935078

.«<

.064922

16

45

.814753

2.4o

.879420

.82

89

.935333

4.25

.064667

15

46

47

.814900
.815046

2^43

.879311
.879202

.0,*

.82

89

.935589
.935844

4^25

497

.064411
.064156

14
13

48
49

.815193
.815339

2^43

.879093

.878984

.0.*

.82

.936100
.936355

•./ft

4.25

.063900
.063645

12
11

50

.815485

2.43
2.45

.878875

.82

.82

.936611

4.27
4.25

.063389

10

51
52
53
54

9.815632

.815778
.815924
.816069

2.43
2.43
2.42

9.878766
.878656
.878547
.878438

.as

.82
.82

9.936866
.937121
.937377
.937632

4.25
4.27
4.25

10.063134
.062879
.062623
.062368

9

8
7
6

55
56

.816215
.816361

2.43
2.43

.878328
.878219

.83

.82

.937887
.938142

4.25
4.25

.062113
.061858

5
4

57

.816507

2.43

.878109

.83

.938398

4.27

.061602

3

58
59

.816652
.816798

2.42
2.43

.877999

.877890

.83

.82

.938653
.938908

4.25
4.25

.061347
.061092

2
1

60

9.816943

2.42

9.877780

1 .83

9.939163

4.25

10.060837

0

'

Cosine.

D. 1".

Sine.

D. r.

Cotang.

D. r.

Tang.

'

130°

216

49'

41°

COSINES, TANGENTS, AND COTANGENTS.

138°

,.

i |

'

Sine.

D. 1'.

Cosine.

D. r. '

Tang.

D. r.

/

1

o

9.816943

o to 9.877780

0.030103

1

.817088

; .8776:0 i {•'':; .'.MM is

4M-.

2

[817233

,) | }•*•*

3

.817379

.877450 .o:500-JS

4 OR

4

.817524

040 ! •S77"1° l"S

5

.817008

242 .877230 ,22 .!M»M:i'.)

4 or

6
7
8
9
10

.817813
.817953
.818103
.818247
.81839:2

2'.42
2.43

2.40
2.42

2.40

.877120
.877010
.870800
.876789
.876078

I >:{

1.85

1.83
1.85
1.83

.940694

.010010

.941804
.941459

.941713

4.25

4.86
4.88
4.85

.050:

.066641

.058287

64

68
51

50

11
12

9.818536

818081

2.42

9.8705G8
.870457

1.85

9.941968
.942223

4.25

10.058032

49

13
14
15
10

17
18
19
20

.818825
.818909
.819113
.819257
.819401
.819545
.819689
.819832

2.40
2.40
2.40

2.40
2.40
2.40
2.40
2.38
2 40 i

.876347
.876236
.876125
.876014
.875904
.875793
.875682
.875571

1.85

1.85
1.85
1.83 .
1.85
1.85
1.85
1.87

.942478
.942788

.942988
.943243
.943498
.943752
.944007
.94-4202

4.26
4.86

4.25
4.25
4.23
4.25
4.25
4 25

.057012
.056757
.066608
.066248

.055993
.055738

47
46
45
41
43
42
41
40

21
22
23

24
25

26
27
28
29
30

9.819976
.820120
.820263
.820406
.820550
.820693
.820836
.820979
.821122
.821265

2.40
2.38
2.38
2.40
2.38
2.38
2.38
2.38
2.38
2 37

9.875459

.875348
.875237
.875126
.875014
.874903
.874791
.874680
.874568
.874456

1.85
1.85
1.85

1.87
1.85

1.87
1.85
1.87
1.87
1 87

9.944517
.944771
.945026
.945281
.945535
' .945790
.946045
.946299
.946554
! .946808

4.23
4.25
4.25
4.23
4.25
4.25
4.23
4.25
4.23
4.25

10.055483

.054974
.054719
.054465
.054210
.053955
.053701
.068446
.053192

39

as

87
86

35
84

32
81

SO

31
32
33
34
35
36
37
38
39
40

9.821407
821550
.821093
.821835
.821977
.822120
.822202
.822404
.822546
.822088

2.38
2.38
2.37
2.37
2.38
2.37
2.37
2.37
2.37
9 37

9.874341
.874232
.874121
.874009
.873896
.873784
.873672
.873560
.873448
.873335

1.87
1.85
1.87
1.88
1.87
1.87
1.87
1.87
1.88
1 87

9.947063
.947318
.947572
.947827
1 .948081
.948885
.948590
.948844
.949099
.949353

4.25
4.23
4.25
4.23
4.23
4.25
4.23
4.25
4.23
4.25

10.052937
.052682
.052428
.052173
.051919
.051665
.051410
.051156
.060901
.050647

29
28
27

25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

51

52

9.822830
.822972
.823114
.823255
.823397
.823539
.823080
.823821
.8-2:5003
.824104

9.824245

.824380

2.37
2.37
2.35
2.37
2.37
2.35
2.35
2.37
2.35
2.35

2.35

! 9.873223
.873110
1 .872998
.872885
.872772
.872659
.872547
.872434
.872321
.872208

9.872095
I .871981

1 88
1.87

1.88
1.88
1.88
1.87
1.88
1.88
1.88
1.88

1.90
1 88

9.949608
.949862
.950116
.950371
.950625
.950879
.951133
.951388
.951642
.951896

9.952150
.952405

4.23
4.23
4.25
4.23
4.23
4.23
4.25
4.23
4.23
4.23

4.23

10.050392
.060188
[048884
.048689

.048181
.048867
.048618

10.047850
.047595
047341

In

18
17
16
15
14
18
18
11
10

9

8

7

53
54
55
56
57
58
59
60

.824527

.824668
.834808
.824949
.825090
.825230
.825371
9.825511

2.35
2.33
2.35
2.35
2.33
2.35
2.33

! .871868
.871755
.871641
.871528
.871414
.871301
.871187
9.871073

1.88
1.90
1.88
1.90

i.as

1.90
1.90

.952913
.953167
; 958481
.868878
.968981
.954183
9.954487

4.23
4.23
4.23

4.23
4.23

.047087
.046888
.046579
I046886

1 045817

6

5

4
8
2

1
0

1 Cosine.

D. 1'.

Sine.

D. 1". I Cotnntf

D. r

Tang. 1 '

4R«

131°

217

42°

TABLE XII. — LOGARITHMIC SINES,

i

'

Sine.

D. 1'.

Cosine.

I). 1".

Tang.

D. 1".

Cotang.

'

0

9.825511

9.871073

1 ftQ

9.954437

4 90

10.045563

60

1

2
3

.825651
.825791
.825931

2! 33
2.33

: .870960
.870846
.870732

1 .00

1.90
1.90

.954691
.954946
.955200

.In)

4.25

4.23

.045309
.045054
.044800

59
58
57

4
5
6

.826071
.826211
.826&51

2.33
2.33
2.33

200

.870618
.870504
.870390

1.90
1.90
1.90
1 on

.955454
.955708
.955961

4.23
4.23
4.22
490

.044546
.044292
.044039

56
55
54

8

. 9

10

.826491
.826631

.826770
.826910

. OO

2.33
2.32
2.33
2.32

.870276
.870161
.870047
.869933

i .yu
1.92
1.90
1.90
1.92

.956215
.956469
.956723
.956977

./OO

4.23
4.23
4.23
4.23

.043785
.043531
.043277
.043023

53

52
51
50

11
12
13
14
15
16

9.827049

.827189
.827328
.827467
.827606
.827745

2.33

2.32
2.32
2.32
2.32

9.869818
.869704
.869589
.869474
.869360
.869245

1.90
1.92
1.93

1.90
92

9.957231
.957485
.957739
.957993

.958247
.958500

4.23
4.23
4.23
4.23

4.22

10.042769
.042515
.042261
.042007
.041753
.041500

49
48
47
46
45
44

17

18
19

20

.827884
.828023
.828162
.828301

2.32
2.32
2.32
2.32
2.30

.869130
.869015
.868900
.868785

.92
.92
.92
.92
1.92

.958754
.959008
.959262
.959516

4.23
4.23
4.23
4.23
4.22

.041246
.040992
.0407.8
.040484

43

42
41
40

21

9.828439

9.868670

9.959769

4cyf>

10.040231

39

22
23
24
25
26
27
28
29
30

.828578
.828716
.828855
.828993
.829131
.829269
.829407
.829545
.829683

2! 30
2.32
2.30
2.30
2.30
2.30
2.30
2.30
2.30

.868555
.868440
.868321
.868209
.868093
.867978
.867862
.867747
.867631

1.'92
1.93
1.92
1.93
1.92
1.93
1.92
1.93
1.93

.960023
.960277
.960530
.960784
.961038
.961292
.961545
.961799
.962052

./OO

4.23

4.22
4.23
4.23
4.23
4.22
4.23
4.22
4.23

.039977
.039723
.039470
.039216
.038962
.038708
.038455
.038201
.037948

38
37
36
35
34
33
32
31
30

31
88

0.829821
.829959

2.30

9.867515
.867399

1.93

9.962306
.962560

4.23

10.037694
.037440

29

28

33
34
35

.830097
.830234
.830372

2 '.28
2.30

2Oft

.867283
.867167
.867051

1.93
.93
.93

no

.962813
.963067
.963320

4.22
4.23

4.22

.037187
.036933
.036680

27
26
25

36

.830509

.«o

.866935

.Jo

no

.963574

49°.

.036426

24

37

.830646

6.40

.866819

.yo

.963828

.JSO

.036172

23

38
39

.830784
.830921

2.30

2.28

.866703
.866586

.93
.95

.964081
.964335

4.22
4.23

.035919
.035665

22

21

40

.831058

2.28
2.28

.866470

.93
.95

.964588

4^23

.035412

20

41
42
43
44
45

9.831195
.831332
.831469
.831606
.831742

2.28
2.28
2.28
2.27

9.866353
.866237
.866120
.866004

.865887

.93
.95
.93
.95

(\K

9.964842
.965095
.965349
.965602
.965855

4.22
4.23
4.22
4.22
490

10.035158
.034905
.034651
.034398
.034145

19
18
17
16
15

46
47
48
49
50

.831879
.832015
.832152
.832288
.832425

2. '27
2.28
2.27
2.28
2.27

.865770
.865653
.865536
.865419
.865302

.yo
.95
.95
.95
.95
.95

.966109
.966362
.966616
.966869
.967123

./OO

4.22
4.23
4.22
4.23
4.22

.033891
.033638
.033384
.033131
.032877

14
13
12
11
10

51

9-832561

9 97

9.865185

Qt

9.96737'6

99

10.032624

9

52
53
54
55

56
57
58
59

.832697
.832833
..832969
.833105
.833241
.833377
.833512
.833648

Z.-KY

2.27
2.27
2 27
2^27
2.27
2.25
2.27

.865068
.864950
.864833
.864716
.864598
.864481
.864363
.864245

.yo
.97
.95
.95
.97
.95
.97 i

:997

.967629
.967883
.968136
.968389
! .968643
.968896
.969149
.969403

4. .23
4.22
4.22
4.23

4.22
4.22
4.23

.032371
.032117
.031864
.031611
.031357
.031104
.030851
.030597

8
7
6
5
4
3

1

60

9.833783

2.25

9.864127

1 9.969656

4.22

10.030344

0

1

Cosine.

D.I".

Sine.

D. 1".

Cotang.

D. 1".

Tang.

'

132'

47-

43°

COSINES, TANGENTS, AND COTANGENTS.

136°

'

Sine.

I), r.

Cosine.

D. r.

Tang.

D. r.

Cotang.

'

0

9.833783

o o~

9.864127

1 95

9.969056

10.030344

60

1

.833919

2 25 '

.804010

.969909

.080091

59

o

.834054

2 ^5

.8C.3S-92 '

,')!

.970162

, T

58

g

.834189

o »7

.863774

1 97

.970410

.029584

57

4

.834325

2 °5

.86365IJ

1 97

.970669

'T^t

56

5

6

8

.834460
.834595
.834730
.834865

2.25
2.25
2.25

.863538 !
.863419
.863301
.863183

1.98
1.97 i
1.97

.9709^
.971175
.971429 i
.971682

4.22
4.23
4.22

.088885

.028318

55
54
53

52

9

10

.834999
.835134

2.25
2.25

.863064
.862946

1.97
1.98

.971935
.972188

4.22
4.22

.027812

51
50

11
12
13

9.835269
.835403
.835538

2.23
2.25

9.862827
.862709
.862590

1.97
1.98

1 QS '

9.972441
.972695

.972948

4.23
4.88

10.027559
.027052

49

48
47

14

.835672

99

.862471

.973201

.026799

46

15
16

.835807
.835941

2.23

.862353
.862234

1.98 I

.973454
.973707 i

4.22

499

.026293

45
44

17
18

.836075
.836209

2.23
290

1 .862115
.861996

1.98

.973960 i
.974213

4.22

499

.026040
.025787

43
42

19

20

.836343
.836477

2.23
2 23

.861877

.861758

1.98 ;
2.00

.974466
.974720

4.23

4.22

.025534
.025280

41
40

21

9.836611

2 ^3

9.861638

9.974973

4 °2

10.025027

39

22

.836745

! .861519

' 0

.975226

.024774

38

23
24
25

26
27
28
29
30

.836878
.837012
.837146
.837279
.837412
.837546
.837679
.837812

2.23
2.23
2.22
2.22
2.23
2.22
2.22
2 22

.861400
.861280
; .861161
.861041
.860922
.860802
.860682
i .860562

2.00
1.98
2.00
1.98
2.00
2.00
2.00
2.00

.975479
.975732
.9759a5
.976238
.976491
.976744
.976997
.977250

4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22

.024521
.084868
.084015

.023702
.023509
.088856
.023003
.022750

37
36
35
34

as

32
31

30

31
32
33
34
35
36
37
38
39
40

9.837945

.838078
.838211
.838344
.838477
.838610
.838742
.838875
.839007
.839140

2.22
2.22
2.22
2.22
2.22
2.20
2.22
2.20
8.28
2 k)0

9.860442
.860322
.860202
.860082
.859962
.859842
.859721
; .859001
i .859480
i .859360

2.00
2.00
2.00
2.00
2.00
2.02
2.00
2.02
2.00
2 02

9.977503
.971756
.978009
.978262
.978515
.978768
.979021
.979274
.979527
.979780

4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22

10.022497
.022244
.021991
.021738
.U:214S5
.021 •,>:!•,!
.080979

.080478

.020220

90

28
27
26
25
24
23
22
21
80

41
42
43
44
45
46
47
48
49
50

51
52
53
54
55
56
57
58
59
60

9.839272
.839404

".839668
.839800
.839932
.840064
.840196
.840328
.840459

9.840591
.840722
.840854
.840985
.841116
.841247
.841378
.841509
.841640
9.841771

2.20
2.20
2.20
2.20
2.20
2.20
2.20
2.20
2.18
2.20

2.18
2.20
2.18
2.18
2.18
2.18
2.18
2.18
2.18

9.859239
.859119
.858998
.858877
.858756
.858635
i .858514
.858393
.858272
.858151

9.858029
.857908
.857786
.857665
.857543
.857422
.857300
.857178
.867066
9.856934

2.00

2.02
2.02
2.02
2.02
2.02
2.02
2.02
2.02
2.03

2.02
2.03
2.02
2.03
2.02
2.03
2.03
2.03
2.03

9.980033
.980286
.980538
.980791
.981044
.981297
.! IS 1550
.981803
.982056
.982309

9.982562
.982814
.983067
.983320

.984079

9.984837

4 22
4^20
4.22
i 4.22
4.88
4.88
4.22
4.22
4.22
4.22

4.20
4.22

4.22

4.22

4.22
4.20
4.22

10.019967
.019714

! 019809

.018703
.018450
.018197
.017944
.017691

10.017438
.010088

.016081
!015668

10.015163

19
18
17
16
15
14
13
12
11
10

y

8
7
6
5
4
3
2
1
0

'

1 Cosine.

D. 1".

Sine.

i D. r.

Cotang.

D.r.

Tang.

1 '

AM*

133°

213

44°

TABLE XII. — LOGARITHMIC SINES,

135°

'

Sine.

D. 1'.

Cosine.

D. I'.

Tang.

D. r.

Cotang.

'

0

9.841771

9.856934

2 no

9.984837

10.015163

60

1

.841902

2.18

.850812

.Uo
o no

.985090

499

.014910

59

2

.842033

91^

.856690

.985343

,TC6

.014657

58

3
4
5

.842163
.842294
.842424

2.18
2.17
21ft

.856568
.856446
.856323

2! 08

2.05

2 no

.985596
.985848
.986101

4.20
4.22

.014404
.014152
.013899

57
56
55

6

7
8
9

.842555

.842685
.842815
.842946

. lo
2.17
2.17

2.18

.856201
.856078
.855956
.855833

.Uo

2.05
2.03
2.05

2 no

.986354
.986607
.986860
.987112

4 '.22
4.22

4.20

.013646
.013393
.013140

.012888

54
53
52
51

10

.843076

2.17

.855711

.Uo

2.05

.987365

4.22

.012635

50

11
12
13
14
15
16
17

9.843206
.843336
.843466
.843595
.843725
.843855
.843984

2.17
2.17
2.15
2.17
2.17
2.15

917

9.855588
.855465
.855342
.855219
.855096
.854973
.854850

2.05
2.05
2.05
2.05
2.05
2.05

9.987618
.987871
.988123
.988376
.988629
.988882
.989134

4.22
4.20
4.22
4.22
4.22
4.20

10.012382
.012129
.011877
.011624
.011371
.011118
.010866

49
48
47
46
45
44
43

18
19

.844114
.844243

J6.lt

2.15

21 K

.854727
.854603

2.05
2.07

.989387
.389640

4.22
4.22

.010613
.010360

42
41

20

.844372

. JO

2.17

.854480

2.05
2.07

.989893

4^0

.010107

40

21
22
23
24
25
26
27
28
29
30

9.844502
.844631
.844760
.844889
.845018
.845147
.845276
.845405
.845533
.845662

2.15
2.15
2.15
2.15
2.15
2.15
2.15
2.13
2.15
2.13

9.854356
.854233
.854109
.853986
.853862
.853738
.853614
.85349^
.853366
.853242

2.05
2.07
2.05
2.07
2.07
2.07
2.07
2.07
2.07
2.07

9.990145
.990398
.990651
.990903
.991156
.991409
.991662
.991914
.992167
.992420

4.22
4 22

4^20

4 90

4^22

4 90

4. '20
4.22
4.22
4.20

10.009855
.009602
.009349
.009097
.008844
.008591
.008338
.008086
.007833
.007580

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9.845790
.845919
.846047
.846175
.846304
.846432
.846560
.846688
.846816
.846944

2.15
2.13
2.13
2.15
2.13
2.13
2.13
2.13
2.13
2.12

9.853118
.852994
.852869
.852745
.852620
.852496
.852371
.852247
.852122
.851997

2.07
2.08
2.07
2.08
2.07
2.08
2.07
2.08
2.08
2.08

9.992672
.992925
.993178
.993431
.993683
.993936
.994189
.994441
.994694
.994947

4.22
4.22
4.22
4.20
4.22
4.22
4.20
4.22
4.22
4.20

10.007328
.007075
.006822
.006569
.006317
.006064
.005811
.005559
.005306
.005053

29
28
27
26
25
24
23
oo

21
20

41

9.847071

9 1Q

9.851872

9.995199

10.004801

19

42

.847199

21 °.

.851747

2.08

.995452

4.22

.004548

18

43

44
45
46
47
48
49
50

.847327
.847454

.847582
.847709
.847836
.847964
.848091
.848218

. lu

2.12
2.13
2.12
2.12
2.13
2.12
2.12
2.12

.851622
.851497
.851372
.851246
.851121
.850998
.850870
.850745

2.08
2.08
2.08
2.10
2.08
2.08
2.10
2.08
2.10

.995705
.995957
.996210
.996463
.996715
.996968
.997221
.997473

4.20
4.22
4.22
4.20
4.22
4.22
4.20
4.22

.004295
.004043
.003790
.003537
.003285
.003032
.002779
.002527

17
16
15
14
13
12
11
10

51

9.848345

219

9.850619

9.997726

49O

10.002274

9

52
53

.848472
.848599

. l~

2 12

219

.850493
.850368

2 . 10

2.08

.997979
.998231

. 4£
4.20

.002021
.001769

8

7

54
55
56
57
58
59
60

.848726
.848852
.848979
.849106
.849232
.849359
9.849485

. 14

2.10
2.12
2.12
2.10
2.12
2.10

.850242
.850116
.849990
.849864
.849738
.849611
9 84^485

2. 10
2.10
2.10
210 i
2.10 i
2.12 i
2.10

.998484
.998737
.998989
.999242
,999495
.999747
0.000000

4. 22
4.22
4.20
4.22
4.22
4.20
4.22

.001516
.001263
.001011
.000758
.000505
.000253
10.000000

6
5
4
3
o

1

0

'

Cosine.

D. r.

Sine.

D. 1' . j i Cotang.

D. r.

Tang.

9

134°

220

45'

INDEX.

(Names of animals are to be looked for under their class name.)

Amphibia, variability 66

Amphipoda, see Crustacea 67

Ancestral heredity 7g

Annelida, correlation 75

, variability ... 67

Aphidae, see Hexapoda 66

Area, measurement of 5

Arithmetical work, precautions in g

Arithometer 7

Assortative mating 75

Average 13, 17

deviation. m 16

Aves, correlation ' 77

, variability 65

of eggs 65

Bimodal frequency polygons 73

Birds, see Aves.

Brachiqpoda, variability 67

Brunsviga calculator. 8

Bryophyta, variability 71

Bryozoa, correlation 77

, heredity 80

, variability 67

Calculating machines 7

tables 7

Caprifoliaceae, variability 70

Caryophyllacese, variability 67

Character denned 1

Chauvenet 's criterion 12

Class, denned

range

Closeness of fit 2

Coefficient of correlation. . .. 4

regression 47

variability 16, 63

Crelenterata, see Hydromedusa.

Color, measurement of _'•

Compositae, correlation 'i

, variability 69, 70

Comptometer ~

Coordinate paper. J*

Cornaceae, variability JJ

Correlated variability *j

Counting, methods of ;

Crabs f

Criminals, skull index »•

Critical function z \

Cruciferse, variability

223

224 IKDEX,

PAGE

Crustacea, Amphipoda, variability 67

, correlation 76

, Daphnia, correlation 77

, heredity 79, 80

, Eupagurus, correlation 77

, local races 84

, variability 63, 66

Decimal places, number to employ 8

Dipsacse, variability 70

Discontinuous variates 1

Dissymmetrical animals, bilateral correlation of 76

Dissymmetry 82

index 60

Dominating characters 58

Echinodermata, correlation 76

, variability 68

Environment, direct effect of . 83

Fertility, heredity of 82

Fishes, see Pisces.

Frequency polygon 62

Fruit, variability of 71

Galton 's difference problem 27

Gastropoda, correlation 77

, variability 67

Geometric mean -. 15

Graduated variates 1

Heredity 55, 78

, ancestral 78

Hexapoda, correlation 77

, variability 66

Homo correlation 73

eye-color, heredity of 79

fertility, heredity of 79, 80

inheritance 79

head index, heredity of 79

mental characters, heredity of 80

skeletal, correlation 74

skull, variability of 64

stature, correlation 79

weight, variability 63

variability 64

(See also Naquada race) 64, 65, 74

Homotyposis 81

Hydromedusae, variability 68

Index of abmodality 23

dissymmetry 6

divergence 40

isolation 41

variability 15, 17

Individual 1

variation 1

vs. specific variation 63

Integral variate 1

Lamellibranchiata, correlation 76

, local races 84

, variability 68, 71

Leaves, variability 71

Leguminosse, variability 70

Lepidoptera, variability 66

Loaded ordinates, method of 12

Local races 83

Longevity, inheritance of 79

Mammalia, correlation. 76

, variability 65

Mean 13

Median. . . . 14

INDEX. 225

Mendelism ........................................... &G£

Mid-departure .................

Mode ............................. :::::::::;:

Multimodal polygons .............................. '39 73

Multiple organ ............................... ' '

Mutations ................................ ........ .63

Myriapoda, correlation ........................... .

, variability ........................ ........ ! ! 66

Naquada race, skeletal variability .................... 64, 65, 74

Normal curve of frequency .........................

Number of variates to employ ................. .........

Orchidacese, variability .............................. ' [ \\ 71

Organ variation ................................. . . .

Papaveracese, variability ............................. 70

Partial variation .................................. ........ 1

Person ................................................... ' i

Pisces correlation .................................... ..... 76, 77

local races . . ......................................... 83

variability ........................................ . . 66

Plants correlation ........................................... 7g

homotyposis .......................................... 81

variability ....................................... . .„ . 69

Prepotency .................................................. 78

Primulacese, variability ...................................... 70

Probable departure ........................................... 16

difference ........................................... 15

error. .............................................. 14

in uniparental heredity ........................... 55

of coefficient of correlation ....................... 44

of variability ........ * .............. 16

of mean ....................................... 15

of median ...................................... 15

of standard deviation ............................ 16

Probability of normality of a given distribution .................. 24

Protista, correlation .......................................... 77

, variability .......................................... 69

Range of variability .......................................... 25

Ranunculaceae, variability ..................................... 69

Recessive characters .............................. ............ 58

Rectangles, method of, in platting frequency distributions ......... 11

Rejection of extreme variates .................................. 12

Relative variability of the sexes ................................ 63

Rosacese, variability .......................................... 70

Sapidacese, variability. . . ..................................... 70

Scrophulariaceae, variability ........ . .......................... 71

Selection ....... .... ......................................... 82

Sex, relative variability ....................................... 63

Sedation .................................................... 10

Skewness ................. ............................. 30, 71, 72

Skull, see Homo.

Spurious correlation ..... . .................................... 54

Standard deviation ........................................... 16

Stature, see Homo.

Symmetry in frequency distribution ............................ 1

Telegony .................................................... 82

Types of frequencydistribution .......................... 19, 71, 72

Variability ......................................... 15, 17, 62-71

Variant .......................................... ...........

Variate ..................................................... 1

Weight, variability, see Homo.

UNIVERSITY OF CALIFORNIA
BRANCH OF THE COLLEGE OF AGRICULTURE

THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW

6 1937

FEB 5 1946
£ 8 1947

194?'

38395

UNIVERSITY OF CALIFORNIA LIBRARY