GIFT OF

BIOLOGY

LIBRARY

G

STATISTICAL METHODS

WITH SPECIAL REFERENCE TO

BIOLOGICAL VARIATION.

BY

C. B. DAVENPORT,

Head of Department of Experimental Biology and Director of Station
for Experimental Evolution of the Carnegie Institution.

SECOND, REVISED EDITION.
FIRST THOUSAND.

NEW YORK :

JOHN WILEY & SONS.

LONDON : CHAPMAN & HALL, LIMITED.

1904.

Copyright, 1889, 1904,

BY
C. B. DAVENPORT.

| i

> 3

BIOLOGY

LIBRARY

G

ROBERT DRUMMOND, PRINTER, NEW YORK.

PREFACE.

THIS book has been issued in answer to a repeated call fora
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

250822

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 Biometricians 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

OP THE CARNEGIE INSTITUTION,

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

V

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.

Classification 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-fl)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 polygon 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 39

Multimodal curves 39

CHAPTER IV.
CORRELATED VARIABILITY.

General principles » . . . . 42

Methods of determining coefficient of correlation 44

CONTENTS. Vii

PAGE

Gallon'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

Gallon'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

Coelenterata 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 CONTENTS.

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 formulae 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

o

" 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 30 129

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

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

" XI. Logarithms of numbers 148

** XII. Logarithmic sines, cosines, tangents, and cotangents. 175

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.

Methods of Collecting Individuals for Meas-
urement.

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

Having settled upon 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 the
character to be measured. If the individuals are simply not
consciously selected on the basis of magnitude of the character
they will often be taken sufficiently at random.

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

The Determination of Integral Variates.—
Methods of Counting.

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

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

STATISTICAL METHODS.

The Determination of Graduated Variates,—
Methods of Measurement.

Straight lines on a plane surface are easily meas-
ured by means of a measuring-scale of some sort. The meas-
urement should always be metric because
this is the universal scientific system. Vari-
ous kinds of scales may be obtained of
optical companies and hardware dealers, —
such as steel measuring tapes, graduated to
millimetres (about $1.00), and steel rules
(6 cm. to 15 cm. ) graduated to £ of a milli-
metre. Steel "spring-bow" dividers with
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 ("Entfer-
nungsmesser," Fig. 1), supplied at artist's
and engineer's supply stores at about $3.50.
Distances through solid bodies
or cavities are measured by calipers of
some sort. Calipers for measuring diameters
of solid bodies are made in various styles.
Micrometer screw calipers (" speeded")
reading to one-hundredths of a millimetre
and sold by dealers in physical apparatus for
about $5.00 are excellent for determining diameters of bones,
birds' eggs, gastropod shells, etc. Leg calipers for rougher
work can be obtained for from 30 cents to $4.00. The
micrometer " caliper-square," available for inside or outside
measurements and measuring to 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 ORGANISMS. 5

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

FIG. 2.

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

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

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. Sucl}
standard color samples are given in Ridgeway's book,
" Nomenclature of Color," and also in a set of samples manu-
factured by the Milton Bradley Co., Springfield, Mass., costing
6 cents. The best way of designating a color character is by
means of the color wheel, a cheap form of which (costing 6
cents) is made by the Milton Bradley Co. The colors of this
"top" are standard and are of known wave-length as follows:
Red, 656 to 661 Green, 514 to 519

Orange, 606 to 611 Blue, 467 to 472

Yellow, 577 to 582 Violet, 419 to 424.

It is desirable to use Milton Bradley's color top as a standard.
Any color character can be matched by using the elementary
colors and white and black in certain proportions. The pro-
portions are given in 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 Calculating1. 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 ($125), 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, Natalis & Co., Brunswick,
Germany, Manufacturers; price $140 to & 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 work.
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 both 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 cO be recorded will depend upon the probable* error of

MEASUREMENT OF ORGANISMS. 9

each constant. It will ordinarily suffice if the probable error
contain two significant figures, e.g., ±0.17 or ±0.0089; then
the constant will be carried out to the same number of places
and not farther.

10 STATISTICAL METHODS.

CHAPTER II.

ON THE SERIATION AND PLOTTING OP DATA ANJD 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 : TF40#, N
(Black) 38#, 7 12$, Q 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, 12,
13, 12, 12. In seriation they are arranged as follows :
Classes: 10,11,12,13,14.
Frequency : 1, 4, 11, 4, 2.

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

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

4.3 5.0 5.3 5.8 6.7
4.3 5.1 5.4 5.9 7.3

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

SERIATIO^ AND PLOTTING OF DATA. 11

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

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 Fechuer ('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 :

3.0-3.4;

3.5-3.9;

4.0-4.4;

4.5-4.9;

5.0-5.4;

3.2

3.7

4.2

4.7

5.2

1

Q

3

4

5

1

1

3

3

7

5.5-5.9;

6.0-6.4;

6.5-6.9;

V.0-7.4;

5.7

6.2

6.7

7.2

6

7

8

9

5

3

1

1

Classes —

( 3.1-3.5;
"j 3.3

3.6-4.0;
3.8

4.1-4.5;
4.3

4.6-5.0;

4.8

5.1-5.
3.5

Frequency

1

1

4

3

6

Classes

( 5.6-6.0;

6.1-6.5;

6.6-7.0;

7.1-7.5;

( 5.8

6.3

6.8

7.3

Frequency

6

2

1

1

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

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

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

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

Q280
HI

E

NUMBER OF V£!NS
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 — - — •

SERIATION AND PLOTTING OF DATA. 13

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

7
1

8
2

9 10 11
15 279 554

A = 10.835 ; o = .728 fin-rays.

1999 ._n__
"= 77^ =.49975.

12
144

13
5

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;
I 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+4.2X3-1-4.7X3 + 5.2X7+5.7X5+6.2X3

+ 6.7 XI +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 V =
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" — I' 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; x=-.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

iCEare 17,000,000:1

are about a billion to 1.

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

SERIATIOK 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 (A1 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 ot A,-A2 is \/E* + E*.

The probable error of the menu is given by the
formula

+ 0.fi74Svstal^rd action [see below]= +a<t74fi*
A/number of variates Vn

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(7
-7-Vn (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,

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:

sum of [(deviation of class from mean)2

X frequency of class]

number of variates

whore A 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

+O.B74K standard deviation ^ ±(m46_»
V 2 X number of variates v2n

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

. j. ^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
-0.7979*.

The probable (or mid) departure is the distance from the mode
of that ordinate which exactly bisects the half curve OMX or OMX} ,
Fig. 5, it is equal to 0.6745 X standard deviation = 0.6745*7. 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=-j-XlOO% (Pearson, '96; Brewster, '97).
A.

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

SERIATIOK 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 is the value of - ± — , at which the area of the curve
o

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 a
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 n/, and the number from the
middle to the larger n2". Also, the x distance to the lower
division point /^ and to the upper division point h2. Then
(/ij-|-/&2) = 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 /^ on the one
hand and h2 on the other (percentage of individuals between

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

o o

since they are the values — corresponding to the percentage
* See Macdonell, 1902.

18 STATISTICAL METHODS.

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

O (J O

mined. Knowing a w,e can get ^ 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 laaves from the middle, and
the larger 15.

n 76

From Table IV:
>H

.56
.55

.21223

.20884

Therefore — = .555.

Similarly = .5

44

71! = . 2278, h2=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 F0)

20 STATISTICAL METHODS.

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

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

Add the products of the squares of all the departures multi-
plied by the f-requency 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 r3.

Add the products of ih&fourtfi powers of all the departures
multiplied by the frequency of the corresponding class and
divide by ?i; call the quotient r4. Or,

2(y— V )
••', = — — = departure of VQ from mean. V0 being

known, A may be found [A = 1

The values rlf r9t ra, v4, 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 FREQUEKCY POLYGONS. 21

Q,,,

/£3— V3 — cSv1v2

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

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:

fi2= .67449<r2 \~] #/?2= .67449 ^— ;

//3= .67449<734/-; ^\/7i= .67749i/-;

'71 '71

; JFj)=. 67749 /

E of Skewness= .67749 i- . (Sre page 30.)
' 2ifi

(From Pearson, 1903°).

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

F=

4(4/?2-3/91)(2/?2-3/?1-6)*

* 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

22

STATISTICAL METHODS.

Value of F.

Corresponding Frequency Curve.

F>1 and <oo
^=1

F>0and <1
^=0,ft = 0,ft=3
=0, ft = 0, /?2not

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
M

A 5^3210

Fio. 5.

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. jFi = 2/?2 — 3ft — 6. The classification was given as follows:

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

Pi =0, /?2<3, curve is of Type II.

/?i> 0, /?2> 3, curve is of Type III.

ft =0, ,52 = 3, curve is normal.
When F is positive and ft > 0, 32> 3, curve is of Type IV.

When F is negative and
When F = Q 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 a 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-

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

taken directly from Table III. Here — is the actual devia-
tion from the mean expressed in units of the standard devia-

tion, and — the corresponding ordinate, y0 being taken as

2/o
equal to 1, and a 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 x=Q and x= ±—. By looking up the
given values of — the corresponding theoretical percentage

of variates between the limits x=0 and x= db— 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 4~.
V 2n

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

/~fl~2

quotients is the index of closeness of fit (J). Or, J= y I—-

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.

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 xf

number of variates; or, in a formula, -p, where Jf0x/ 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

THE CLASSES OF FREQUENCY POLYGONS.

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

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.

1
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 (n) as well
as with a, in accordance with the following formula derived

by the transposition of y=

by putting y=l:

2^=2*7

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

2zj=2X2.3775|/ .46051 7 X log x-^;

1000

= 15.2.

'2.506628X2.3775
The observed range was 15 (Duncker, '98). See also the

criterion of Chauvenet ('£
variates (page 12).

for the rejection of extreme

THE NORMAL CURVE OF FREQUENCY AS A BINOMIAL
CURVE.

The normal curve may also be expressed by the binomial
formula (pXgH where p=i, #=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

A=4X (Standard Deviation) 2=4<72.

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

V } F-Fo /(F-F0) /(F-F0)2 /(F-F0)3 /(F-F0)4

14 1-3-3 9 -27 81

15 8 -2 -16 32 -64 128

16 63 -1 -63 63 63 63

17 154 0 0 0 0 0

18 164 1 164 164 164 164

19 96 2 192 384 768 1536

20 20 3 60 180 540 1620

21 2 4 8 32 128 512

n = 508 342 864 1446 4104

.4 = Fo+ vi = 17 +.6732 = 17. 6732.

/£2= 1.7008 -0.70592- 1.2475; <r=\///2=1.1169.

//3 = 2.8467 -3 X 0.7059 XI. 7008 +2 X0.70593 = 0.0217.

/t4=8.0787 -4 X0.7059X2.8465+6X0.70592X 1.7008-3X0.70594=4.4223

v22-2v14 3( 1 .7008 )2 - 2 X -7_05_94 _
v4 8.0787

Theoretical maximum frequency, 7/0 = — — -j= = 181.5.

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

L
14.5

a
-0.99606

Xi

,*i,

Actual
Dis-
crepancy.

Probable
Dis-
crepancy.

Ratio of
Actual to
Probable
Dis-
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

2.92

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
ptli and the (p + l)th individual in any seriation
(Galtoii's difference problem). Let xp be the aver-
age interval between the pth and (p+l)th individual; n the
total number of variates; and a their standard deviation.

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

ppe~p
~

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 ( -- -\ .

. 833

_. ..

n(n-p)p n2 ym n3 \ym

2) \ym

28 STATISTICAL METHODS.

»o 57 -- r~^ — 41 • -O — 77 -- \ — ' •

n\n-p)2p2 n\n-p)p y

2

2/ \ym

The solution of the equations for clt c2, and c3 will be facili-
tated by finding, once for all, the logarithms of n, (n — p),

(n — 2p), (n—p)p, and — .

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

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

|n means the products of all integers from 1 to n. The
series clf 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 m units of a 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

(1) 2/=:

where I is the half range and

£2=3.75(3A2->10);
also,

= =-

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 frcm
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 2<?2
(3) y

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

Then the required normal curve is

^y^v

(Pearson. 1902m.)

30 STATISTICAL METHODS.

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

Type I. ,=

Type II. y=

TypellL y=y0 l + j */<*.

Type IV. y=y0cos62me-*e, where tan 0=j.

Type V. y=yQx-pe-r/x.

Type VI. y=y0(x-l)^/x^.

In these formulas:
x, abscissae;
2/0, 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;
<?, 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 -f oo to 0. In the ratio
of dorsoventral to aiitero-posterior diameter the forms of the
molluscan genera Pinna or Malleus on the one hand and
Solen on the other approach such extremes.

Asymmetry or Skewuess (a) is tound in Types I, III,
IV, V, and VI, In skew curves "the mode and the mean are

THE CLASSES OF FREQUENCY POLYGONS. 31

separated from each other by a certain distance D; or D =
mean — mode. Asymmetry is measured by the ratio 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

raVo «-WA- where

5/?2 — 6ft — 9 is positive, a has the sign of //3; if negative,
a has the opposite sign to /z3 (Duncker, '00b).

In Type I, .-Jv*

" " III, a=%\/Jl=— 3 -, where the sign is the
+ 2\/>23 same as that of p9.

" " IV, a=

,,
P

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

p is readily found.

in Type VI, a=

,

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

s2
«2 _ o« I .

"h

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

32 STATISTICAL METHODS.

To find 119 12, ml9 m2, yQ.

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

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

1=(s — 2); m1 + m2=5 — 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
2/o—

I '

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,

y == 2/0 \ •*• 172

\ ¥ >

This equation is only a special form of the equation of Type
I in which Z1 = Z2 and m1 = m2.

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

— if _ov _n r(m+1.5)

yVfriXm+l)'

The F values will be found from Table V.

THE CLASSES OF FREQUENCY POLYGONS. 33

An approximate formula for y0 is given by Duncker as fol-
lows:

n s-1 - 1

2/o = -

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

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

_ iP -*/d
-T-] e

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

(for Type III).
pp+l

I, I, n

T>-Tai *-^

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.

0 is a variable, dependent upon x as shown in the equation
x=l tan0.

The factor (cos 0)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.

-l)-A(s-2)2; m=i(H-2);

T—' ^S ~ — ?t with the opposite sign to u~;

34 STATISTICAL METHODS.

6 (arc of circle) = rr^™,*

f—^_l^

<f)= angle whose tangent is — .
s

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

y=y0x~~pe~r/x.
To find p solve the quadratic equation

and take the positive root.

D— ^.

T(P-iy JKF1*)'

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

1 — q1 and q2+ 1 are the two roots of the equation

s2

' where U— 9i) and sare negative;

* 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

jf

(sin 0fered0

the formula for reducinc which is to be gained from the integral oaJ-
culus.

THE CLASSES OF FREQUENCY POLYGONS. 35

Example of calculating the theoretical curve corre-
sponding with observed data. (Fig. 6.)

Distribution of frequency of glands in the right fore leg of 2COO 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' ( Vm) to pass through ordinate 4, then:

f f(V—Vni) f(V—V™)* f(V— Fm)3 f(V— Fm)«

0

— 4

15

— GO

240

— 960

3840

1

— 3

209

— 627

1881

— 5643

16929

2

— 2

365

— 730

1460

— 2920

5840

3

— 1

482

— 482

482

— 482

482

4

0

414

0

0

0

0

5

1

277

277

277

277

277

6

2

134

268

536

1072

2144

7

3

72

216

648

1944

5832

,8

4

22

88

352

1408

5633

9

5

8

40

200

1000

5000

10

6

2

12

72

432

2592

2 2000 —998 6148 —3872 48568

= _ 998 -v- 2000 = — .499.
, = 6148 -f- 2000 = 3.074.
, = — 3872 -*• 2000 = — 1.936.
i = 48568 -*- 2000 = 24.284.

= 0; A = 4-. 499 =3.501.
, = 3.074 — (— .499)2 _ 2.824999.

, = - 1.936 - 8(- .499 X 3.074) + 2(- .499)» = 2.417278.
i = 24.284 -4(-.499 X - 1.936) + 6('.249001 X 3.074) - 3(- 499)* = 24.826297
(2.417278)2 5.843232929

(2.H24999)3 22.545241683
!?4. 826297 ?4. 826297

= 0.259178.

' <xv!.824999ja 7.98061935

= 3.110823.

.259 X (6.111)2 _ _ .

4(12.443 - .778)(6.222 - 6.778)

.55589
7^r,21.P857

D- 1.680774 X .3111 = .5230.
D a=» .5230 X 19.9857 = 10.4519.

J- .840887 1/16 X 20.9857 -f C.25918 X (21/J857)a = 18.0448.
I. -I'"*8-""8", 3.7965.

36 STATISTICAL METHODS.

*a= 18.0448 - 3.7965 = 14.2483 ;
3.7965X17.9857

. ,__..,
=3.78401;

14.24317.98

18:0448

14.2483X17.9857
18.0448

2000 (18.9846)^17.9846 „„ „ __.0833(. 0556 -.2643 -.0704)

4/0 - 10 f\ A A O / _ A ^.1« JO'SO

18.0448 V27rX 3.7840 X 14.2006

=475.24, the frequency of the modal class.

Position of the mode, yQ=A -D= 3.501 -.523 = 2.978. The close-
ness of fit to the theoretical curve is calculated below by Pearson's
method (page 24).

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

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

Classes. ...0 1 3

8

4 5

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

• -i polygon of observed frequency.

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

- - - -, normal frequency polygon.

10

38 STATISTICAL METHODS.

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

GENERAL.

log v^log I(V- T0) -log n. A = Vm + vv

log v2= log I(V- F0)2 -log n. log a= \ log /*2.

log v3= log 2(V- F0)3 -log n. log <7= \ log /*2-log A.

Iogv4=log 2'(F-F0)4 -logn.

log E.4 = 9.828982 + log o - \ log n.

logE.a = logE.^ -0.150515.

log E.e=log E.tf —log A.

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

log 3= 0.477121 ^= .02916 3 log ^

log 4= 0.602060 •5^G= .0125 4 log vl

log 6=0.778151 log i=9. 98970

/z2= AT (log v2) -JV(2 log yL) -[.0333]. Find: log ^ 2 log ^;

31og'//3.
//3=]V(log v3) - N(log 3 + log ^4-log y2) +N(log 2 + 3 log vx)

Find: log /*3; 2 log /i3.
/£4=^(log^v4) -JV (log 4 + log Vi + log v8)

+ 7V(log 6 + 2 log Vi + log v,) -AT (log 3 + 4 log v,)
-Ar[9.698970 + log ^J-^lj. Find log /*4.
log ft =2 log /£3-3 log ,«2.
log/?2=log//4-21og /(,.
^=5^-6^-9 (Types I, III).

Skewness:

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

THE CLASSES OF FKEQUEXCiT POLYGONS. 39

Type IV: log «=£ log ft + log (&+ 3) -log w-0.301030.

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

Type VI : log a = log (& + g2) + i log (^ - ?2 - 3) - log (q, - q2)

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 y

log a=log/-log (« + 2) -0.301030.

log Z=i log /£24- J log{#[log (8-1)4-1.204120]

-N[\og ft 4-2 log(s-2)J! -0.602060.

s + 2

log mD = log k - 0.602060.

log T= log k 4- log s - 0.602060 - log I

log tan <£=log T — log s.

g cos 0 — log 3s)
-.¥(8.920319 -log s)-JV(log r + log ^)] + 9.637784}
-0.399090-log Z- (s4- 1) log cos <£.
log y = log 2/0 + A^ [log (s + 2) + log log cos 0]

T AT7.S796612 + log 0°* + log 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).

5.5 4.5 3.5 2.5 1.5 .5 0.5 1.5 2.5 3.5 4.5 5.5 6.5 75

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.

FIG. 8.

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

THE CLASSES OF FREQUENCY POLYGONS.

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

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

The meaning of multimodal curves is diverse. Sometimes

\

Fio.9.

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

32101231

01234
FlQ. 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=3XStandard Deviation, which has certain advantages in species
study.

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.

43

8 $ S

g 3 8 S S

O O O TH TH TH O*

P r? P

O rH TH

Sd

1 0)

"

I- tfi TH

eo *-• «o

20* «0 00 TH
Oi TH

$ £ TH 3

s a

2 25

> a eo c» TH o o
£o I I I I

~

a-e,

O'E

44 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 Miillerian 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 OP 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 ypt 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 iissoc.

rel. class X no. of cases in both)

total no. oFludivs. X Stand. Dey. of subject x Stand. Dev.
of relative ;

or, expressed in a formula :

^(dev. x X dev. y X /)
n<Ti<r2

This method requires finding many products in the numera-
tor, as many sets of products as there are entries in the body of
the correlation table. A portion of the 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

"t.540 X 151

,540 X 58
etc.

- 2.547 X 4 - 2.540 X 151
I - l.£

COEKELATED 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 x' and y' 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, i^ (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

= 0.6745(1 -r2)

v^

(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 -f quadrants 2(x'y')= 5243

—

46

STATISTICAL METHODS.

•c ^

*l

w § S

g e §

«o t>.

r}< CO tN

I I I

O O CO
»O O CO
<N CO

O CC|»H (

co t- i

096
W9 sT

2S 09
w ^ OSI

§ 2 818

• >

III!

CORRELATED VARIABILITY. 47

»i V ) — = (2.5625 - .4535 X .4605)

1. 7195+1. 73J

r

\/2000

The average variability of an array is =o\/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 al is

°2

the standard deviation of the subject, cr2 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, n3, in which rat and n3 are each less
than %n, so that n2 contains the median. Let Llt L.+ be the
(unknown) distances of the mean from the two boundaries
of n2. Call Li/a=ht and Ls/a=/i3, then

n, — n.2 — n ,

•AT'-"1*

r 7T "0

and

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 /ij and 7t3 I = — J are found, and hence L^/a

and Lz/a and the entire range — of the middle class,

in terms of a, is known. Call the range in absolute units I.
Then Z=Z/3 + I/1 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 — — -1 are known, ^/(Z/g + Z/j) 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 \ 4- h2 =

is known and hence

-=J — r2
"-i ~\~n3

, the standard deviation, is found. Since L, = h^ =

the distance of the mean from the left-hand boundary of n2,
the position of the mean is known.
The probable error of a is

E. =tfV7^o A + £8 Wn-Q , n3(n-n3)

where

-

V 2*

and

CORRELATED VARIABILITY.

49

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

The probable error of Llt or of the mean, is

where
6

( E° \ * r
-

V67749/

and ? _
' a * 2~

n,(n-n,)

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, b, 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 x1 — x probably do not coincide with the
axes y and 2 -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

_e_tlf
TT •/()

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

and

-= -

V27T V27T

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:

(/i4 - 67i2 + 3) (A;4 - GF + 3)r5

+-_Lhk(h*- lOh2 + "15) (k4 - lOfc2 + 15)r6 + etc.

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

n I , and 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 /(x) by the exponent of x in that
term and diminishing the exponent of x by 1). The correc-
tion 7774 should be added to the value of r used in substi-
f (r)

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 &= — '-

fr — rh

and /?2 = — ^=. Also find
V 1— r2

7= f le
2^«/0

and ^2= — 7=

from Table IV. Moreover,

l

2(1 -r)'

7TVl_r

Then,

A744Q
Prob. error of r=—,?te(a

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 + b > c + d. (Pearson, J00C.)

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

g

0

3 •

>>

rown.

d

0

PQ

r^'c)

PQ

PQ

II

>>£

44 *

-g
*

i

44

44

J2

3

3

O

£W

^

PQ

1

«

1

1. Light blue

4

3

8

5

1

21

2 Blue — dark blue . .

8

177

95

76

5

39

31

17

448

3 Gray — blue-green.

1

69

85

52

2

20

26

1

256

4. Dark gray — hazel.

6

30

21

27

2

7

15

1

109

5 Light brown.

4

4

6. Brown

2

37

27

17

3

30

20

4

140

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

1113
From the tables.

Offspring

Great -grand-
parents.

1-3

4-8

Totals.

1-3

4-8

450
185

275
203

725

388

Totals.

635

478

1113

.31
.30

.01

. 39886
.38532

01354

.15
.14

01

.18912
. 17637

.01275

h - .38532 + ( 1 354 X .002785) = .389091
A; -. 17637 4- (1. 275 X. 001060) = .177722

CORRELATED VARIABILITY. 53

Log ^=9.5900512 Log A2 = 9.1801024 /i2-. 151392

Log A;=9.2497412 Log A;2 = 8. 4994824 &2 = .031585

Log fcfc =«8.8397924 A2+fc2 .182977

fc2+fc2
hk = . 069150 }Afc = .034575 J =.091489

Log (450 X 203 - 275 X 185) = 4.607 1869

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

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

log 11 13) = 9.3521096

Solving .03457 5r2+r-. 224962 = 0,
1 ± V/l +40034575 X .224962)

-^034575)-
2 _ x = _ .848608 fc2 - 1 = - .968415 Coeff . r3 = .136967

Coeff.

24

.024363r4 + . 136967r3 + .03457 5r2 + r - .224962 = 0.
Applying Newton's approximation, we reach the result
r = .2217.

fi744.Q

E.r= ^^(75095 + 303530&52 + 281300012

n5w0

Log w0 = l

-Iog(l-r2)-log2]

}l2 + jc2-2rhk =0.152315, l-r2 = 0.950850.

Log ^o =9.20182 - 9.989056 - M9.637784 + 9.18274 - 9.978112 - 0.30103]
= 9.1779797

.
Log ^^ = 9.828975 -4.569743 -9.177980 =4.081253.

n'a>o

^=0.358614 ^2=0.093794

From Table IV:

<f>2

.358 .13983 .093 .03705
22.2 27.3

.4 3.5

<!>i = .14006 ^2 - .03736

Log E.r = 4. 0812530 + ilog 74426. 858
E.r = 0.03289

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 y
tion r= 1 l whenever the distributions of frequencies of

fUFjtfj

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

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

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, J97).
c c

Let C1 = the coefficient of variability of a;
C2= " " " " " b;

ri _ (t (( (t (t tf r.

U3 — c,

r0= " " " SDurious correlation.

CORRELATED 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 miiparental 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, <r2, may (as a result of selection or of environ-
mental change) be different from the variability of the char-
acter in the first generation, alt the index should be taken as

r— , called the coefficient of regression.

The probable error of this determination is

.6745(7,. /I -r122 .

~\/ *-) m which r12 means the correlation coeffi-
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
fissortative mating, or rorrolation 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;

^3= average abmodality of female parent;

r2= correlation coefficient between fraternity and

female parent;
rz= correlation coefficient between fraternity and male

parent;

GI= standard deviation of fraternity;
<72= 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 ^ = 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 nv is the number of the brethren in an array [and there-
fore JWiCwj — 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 founc?
a work of too great magnitude (Pearson, Lee, etc./99, p. 271).

CORRELATED VARIABILITY. 57

Galtoii ('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, 011 the average, 50$ or i of the whole ; from the 4 grand-
parents 25# or J ; from the 8 great-grandparents 12. 5# or £;

from the nth ancestral generation — of the whole ; the total

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

where hi = average abmodality of fraternity.
CTO = standard deviation of fraternity.
<TI , cr, . . . <T8 = standard deviation of mid-parent of
1st, 2d ... «th ancestral generation.
A?i = abmodality of mid-parent of 1st ancestral genera-

tion.
&a, k* . , . ka = abmodality of mid-parent of 2d, 3d

. . . sth ancestral generation.

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

MENDEL'S LAW OF ALTERNATIVE INHERITANCE.

In 1865 Georg Mendel 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, dnd 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% " " " " M 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 (g) germ and one having
a yellow (y) germ. Yellow is dominating.

CORRELATED VARIABILITY. 59

Gen. 1.

3en. 2

Gen. 3.

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

77.5 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 plants bore :

because they bore:

bore : |

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. 25%d,d

B. G.25%d, d; 12.5%J, r; 6.25%r, r.
A.

B. I2.5%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. + B. 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 df 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-L8: -10123
/: 1 63 310 23 3

2X<f) = 310X 1 + 23X2 + 3X3=365,

J(<T) = 1, ^ (/") = !; n=400.

w_ 336X365-X1

^
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.

MHTHOD: Camerano, '00; Engberg, '03; Fechner, '97;
Galton, '89, '02; Heincke, '97; Johannsen, '03; Pear-
son, '94, '95, '96, '97, '97b, '98, '00°, 'Old, '02C, '02*, '02*,
'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, '02C.

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', 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.
Engl. upper middle class 1
do. husbands .
Cambridge Univ. students

English fathers.

n
683
200

1078

A
69.215" ±.066 5
69. 135" ±.126 2
68 .863" ±.054 I
cm.
171.95

a
5.592" ±.047
5.628" ±.089
2 522" ± .048
cm.
6.81

C
3.66

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.. . .
Frenchmen

2862
284

169.88
166 . 80

6.58
6.47

3.80

3.88

English criminals

3000

166 . 46

6.45

3.88

166. 26 ±.53

5.50± .37

Germans

390

156 . 93

6.68

4.02

Engl . upper middle class ?

do. wives. . . .
Cambridge Un. students ?

French. Lvons S . .

652

200

in.
64.043 ±.061
63.869 ±.110
63. 883 ±.130
154.02 cm. ±.52

in.
2.325 ±.043
2. 303 ±.078
2.361 ±.092
5. 45 ±.37

3.69

Seriation at different ages: British infant at birth, Pearson,
'99; school children, Bowditch, '91; St. Louis schoolgirls,
Porter, '94, Pearson, '95, p. 336; Australian adult whites,
Powys, '01.

64 STATISTICAL METHODS.

Lot. Average. a C

New-born infant , British 3 . 20 . 503 ± . 028 in . 1 . 332 ± . 020 6 . 500

?. 20.124±.025 " 1.117±.018 5.849

St. Louis schoolgirls 118.271 cm. 2.776

Australian whites:

Average. a C

Year's * ? * ? * ?

20-25 66.95 62.50 2.475 2.365 3.70 3.79

25-30 67.30 62.76 2.562 2.432 3.81 3.87

30-40 67.15 62.44 2.587 2.303 3.86 3.69

40-50 66.91 62.96 2.618 2.555 3.91 4.06

50-60 66.74 62.22 2.633 2.591 3.95 4.16

60 & over 66.26 61.31 2.682 2.300 4.04 3.75

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

Lot. Average. a 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. 280, Macdonell, '02.

Lot.

n A

a

C

Bavarian peasants . .

, 100 83.

41

3

,58

4.

29

Baden recruits

. 6748 81

.15

3.63

4

.48

Modern Parisians

79

.82

3

.79

4

.74

French peasants ....

. 56 79.

79

3

.84

4.

81

Cambridge students . .

. 1000 78

.33

2

.90

3

.70

Criminals (British) . ..

. 100 76

.86

3

.65

4.75

Brahmans of Bengal ,

. 100 75.

,77

3,

,37

4,

,44

Whitechapel English. .

107 74.73

3

.31

4

.43

Maquada race

72.94

2

.98

3

.95

Skull capacity:

coefficients

of variability. Fawcett

and

Lee, '02.

Lot.

5

?

Lot.

i

\

?

Andamanese

5

.04

5.59

7.72

6.

.92

Ainos

6.89

6

.82

Germans . . .

7.

74

8

.19

Negroes

7

.07

G

.90

Egyptian mummies. .

8.

13

8

.29

Low-caste Pun jabs . .

7

.24

8

.99

Polynesians .

8.

20

5

.55

Parisian French

7

.36

7

.10

Italians ....

8.

34

8

.99

Kanakas

7

.37

6

.68

Modern Egyptians. . .

8.

59

7

.17

17th Century English.

7

.68

8

.15

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, 4.58-5.57),
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, 1.680±.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

Shrike, length L. wing $ ......... 168

" ? ......... 112

tail length $ ............. 141

"

bill length, $

?

95
164
112

............

" depth, $ ............ 126

? ............ 85

melanism of crown, $ ..... 144

" " ? ..... 99

" upper tail-coverts $ 142

" ? 104
Curvature of culmen ..............

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

A

a

C

99.06mm.

2.74mm.

2.81

97.98

2.64

2.69

101.57

3.48

3.43

99.55

3.63

3.65

12.01

0.71

5.89

11.71

0.63

5.35

9.27

0.42

4.57

8.95

0.41

4.61

83.57%

3.0%

3.58

83.66

3.19

3.81

53.13

15.42

29.02

47.98

18.99

39.58

29.94°

2.74°

9.15

Species.
Cuckoo

Av.
Length,
Bird,
in.
14

243

Blackbird
Song-thrush . . .
Starling ...

. 10
. 9

. 8-8.5

114
151

?7

Yellowhammer.
Tree-pipit

7
6.5

32
?7

Meadow-pipet .
House-sparrow
(English) ....
House-sparrow
(American) . .
Hedge-sparrow .
Robin
Linnet . .

. 6
6

6
6
. 6
. 5.5-6

74
687

868
26
57
65

Length, mm. Breadth, mm

A a C A a C

22 .40
29.44
27.44
29.78
21.55
20.01
19.72

21.82 1.195 5.47 15.51 .525 3.38

1
1

0

1

0
0

1

.059
.357
.999
.097
.682
.698
.250

4.72
4.61
3.64
3.68
3.17
3.49
6.37

Ifi

21
20
21
1(3
15
14

.54
.73
.69
.76
.04
.09
.56

.650
.787
.516
.423
.405
.449
.561

3
3
2

1
2
2
3

.93

.62
.50
.94
.53
97
.84

21.32
20.12
20.22
17.14

1.05
0.810
0.857
0.598

4.92
4.02
4.24
3.49

15.34
14.73
15.43
13.33

.415 2.81
.477 3.09
.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, Moenkhaus, '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 Petromyzon, 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,
98 * 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 offsprmg,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 x 23% 5.67 1>84 7.82

Frontal breadth

Myriapoda. — Lithobius: seriations of length of adults,
C, for <$'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.

Podophihalmata. — Seriations of 12 dimensions of right-
handed and left-handed " 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, Palsemon serratus,
Thompson, '94, Pearson, '94; number of rostral teeth of
Pakemonetes, Weldon, '92b, Pearson, '95, Duncker, '00.

Lot. A, mm. a, mm. C,%

Eupagurus, short edge of R. chela:

$ deep water 9.708±.085 2.76 28.5

t shallow water 10.272±.075 2.59 25.2

? deep water 7.400±.033 1.06 14.3

5 shallow water 7.485±.029 1.02 13.6

Eupagurus, long edge of R. chela:

t deep water 17. 97 ±.14 4.73 27.8

* shallow water 18. 68+. 13 4.38 23.5

9 deep water 14.14±.06 1.67 11.9

? shallow water 13.97±.05 1.82 13.0

Eupagurus, carapace length:

$ deep water 8.59db.05 1.67 19.4

t shallow water 7.54±.03 0.94 12.5

$ deep water 7. 12 ±.03 0.86 12.1

PalsBmonetes vulgaris, 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
antennae, and of ratio of second antennae to body-length,
Smallwood, '03.
Annelida.

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

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, <r= 1.318 ±.022, C= 9.57 ±.16, Davenport,
'0CK
Mollusca.

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.

<J4=2.7%, Cj5=2.6%, (7.4 = 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, '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

Echiiiodermata.

Seriation of ray-frequency in starfish, Crossaster papposus:
A = 12.391, C=0.788, v=6.36%, Ludwig, '98b.
Coelenterata.

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
Wood worth, '96.

STATISTICAL BIOLOGICAL STUDY. 69

Lot. A a C

Pseudocyltia, 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; the diatom Syndes-
mon, polygon varies with season and year, Kellerman, '01;
other diatoms, Schroter and Vogler, '01.

Lot. A a C

Paramecium, length /i 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; Pearscn, '02h.

RAY-FLOWERS IN COMPOSITE.- -Seriation of ray-frequency
of Coreopsis, de Vries, '94; of Senecio nemorensis, S. fuchsii,
Centurea 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 a C

Aster shortii 14.000±.068 1.526±.048 10.90

A. novje-angliae 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, o=
3.89; late flowers, A = 12.147, <r=3.88; number of stamens,
early, A = 26.731, a= 3.761 and late, A = 17.863, G= 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.

Cruciferw. — 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:
a= 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.

Rosacece. — Number of stamens of Prunus spino: a and Cra-
tsegus, Ludwig, '01; sepals of 1000 Potentilla tormentilla
and petals of 4097 Potentilla anserina, de Vries, '94.

Cornacece. — Number of flowers in head of Cornus mas and
C. samguinea, 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.

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

Compositce. — 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. ?1

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, '93; pine needles. Ludwig, '01;
from various branches of Pinus silvestris, Lee, '02.

Lot, length of pine needles A mm. a mmf C

Pinus silv. , lower branches . . 22 . 163 ± . 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
Billiard, '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 s , 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.
" tl anal " " " s " "

il " pectoral " " " S " "

Type V.

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

Norigal.

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

Skewness.

GENERAL.— Mathematical Analysis. — Pearson, '95, 'Old, '02f,
'02&, '02m. Biological Interpretation. — Davenport, 'Olb, 'Olc.

Quantitative Results.

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

Nearly always +.

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

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

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

Num. rays, Pecten opercularis, Irish Sea (Davenport, '031') + .087

Eddystone (Davenport, '03b) .... + .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)4- .023

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

" " upper valve, P. irradians (Davenport, '00C) ± .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. 73

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

Index of Littorina Newport (Bumpus. '98). + .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, N C -.09

44 4t " " recent, L.I + .023

Length of wings of long-winged chinch-bug (Davenport, '01*>). . . — .43
44 " " " short-winged chinch-bug " ...+ .44

Length horns rhinoceros-beetle, long-horned (Davenport, 'Olb). . — .03
44 " " '* short-horned 4* .. + .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, '98, 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. 2SO; N. A. Indians, Boas, '99;

74 STATISTICAL METHODS.

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

Naquadas, Fawoett 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, $ 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

" 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 db . 04 . 70 ± . 03

" length 51±.05 .69±.04

" " height 24±.06 .45±.05

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

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

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

Skeletal— RoUet, '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 Whitelev, '02.

STATISTICAL BIOLOGICAL STUDY. 75

Lot. r

Right and left femur 96

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 S0(*) to .81(5)

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

Stature and tibia 78(«) to .80(«)

Humerus and ulna 75 to .86

Humerus and radius 74 to .84

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

Clavicle and humerus*. 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 3 644 ± . 012

' " " " 9 622±.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

41 " " 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

" " " *• 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

44 4> 4< " " " 13th pair legs, Lithobius 686 ±.029

44 *4 " " " " 12th pair legs, Lithobius 58 ±.04

44 44 " " " " anal pair legs, Lithobius 575±.039

Other antimeric organs:

r
Num. of dorsal and ventral spines, Palsemonetes vulgaris

(Duncker, '00*>) 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 dorsaLand 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 7 54 ±.014

44 '* carpopodite, R. and L. chelae of Gelasimus 698 ± .017

44 4< propodite, R. and L. chelae of Gelasimus 473 ± .026

Num. rays R. and L. pectoral fin, flounder, Pleuronectes, $ . .594
44 " " " " " " " ?. .582

44 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

" " " Cottus 110

" 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 chelae
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, '00*.
diameter of body of the Heliozoan Actinosphserium 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

" 9 .8626±.0080

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

Inner and outer diameter shell of Arc'ella 836 ± .007

Diam. of body of Actinosphaerium 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

Diaro. of cell an/1 body length, Daphnia, 3d 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. .001

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

Carapace 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 desmid, Syndesmon,
KeUeraian, '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, '03t

STATISTICAL BIOLOGICAL STUDY.

Inheritance of Eye-color, Homo.
«, son; d, daughter? /.father; m, mother.

Parental I AVrge?f^and;^ -
I rsmandrdf..

rdf

rsjjr anc* rdmm

rs/m' rd& rdmf> rsmm ' •

r«m/' rdfm •

Great-grand-parental inheritance, average , . .

Grand-
parental

No. of Changes of Sex.

.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) .517

Dam, foal " -527

Fertility :

Mother and daughter, British upper class .042 ± . 010

Father and son , . . 05 1 ± . 009

Mother and daughter, British peerage .210

Father and son, -066

Mother and daughter, landed gentry . 105

Father and son -116

r f>

Frontal breadth, Hyalopterus (Warren, '02) .335 .359

Length R. antenna, Hyalopterus " .427 .507

Ratio: R. antenna -^frontal breadth (Warren, ''02) . . . .439 .539

Ratio: Length protopodite -*- length body, Daphnia

(Warren, '02) , 466 .619

80

STATISTICAL METHODS.

Graiidparental.

Coat color, thoroughbred race-horses

" " Basset hounds

Frontal breadth, Hyalopterus, Aphidse (Warren, '02)

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

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

Stature .

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

female line (Pearson, '96)..

Ortgr'dson and grtgr'df. . homo $ 1;«~

339
113
321
177
231

.269
.192
.295

27 .5]

line

Eye-color, homo, f., grandfather, and son (Blanchard, '03)
Coat horse. *' " ' " " "

.199
.089
.105
.031

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

horse,

homo, " ' dau.

horse, "

homo, m.,

horse, "

homo, " ' dau.

horse, "

homo, f ., grandmother, and son

horse, "

homo, " ' dau.

horse, "

homo, m.,

horse, "

homo, " ' dau.

horse, '*

.421
.324
.380
.360
.372
.359
.297
.311
.272
.309
.221
.204
.262
.261
.318
.239

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°) 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):

'Consciousness 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

Homoty posts.

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

Lot.
Ceteract Somersetshire

%^
Character. V

Lobes on fronds .

/ar. to
ar. of
Race.
. 78
78
79
79
80
81
82
83
84
85
85
89
91
91
92
92
93
93
96
98
98
98

Corre-
lation.

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

Hartstongue, Somersetshire
Shirley poppy Chelsea . ...

.Sori on fronds
Stigmatic bands

English onion. Hampden
Holly Dorsetshire

.Veins in tunics
Prickles on leaves .

Spanish chestnut, mixed
Boech Buckinghamshire .

. Veins in leaves
Veins in leaves .

Papaver rhoeas, Hampden
Mushroom Hampden . . .

. Stigmatic bands ....
Gill indices .

Papaver rhoras, Quantocks
Shirley poppy. Hampden
Spanish chestnut, Buckinghamshire
Broom Yorkshire

. Stigmatic bands ....
.Stigmatic bands. . . .
Veins in leaves .....
Seeds in pods

Ash, Monmouthshire
Papaver rhoeas. Lower Chilterns. . .
Ash, Dorsetshire
Ash Buckinghamshire
Holly. Somersetshire
Wild ivy, mixed localities
Nigella hispanica, Slough

.Leaflets on leaves. . .
.Stigmatic bands
. Leaflets on leaves. . .
. Leaflets on leaves
. Prickles on leaves. . . .
. Leaf indices.

Seg of seed -capsules

Malva rotundi folia, Hampden. . . .
Woodruff, Buckinghamshire

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

Mean of 22 cases 87 .4 .457

Bands of capsules of Shirley poppies, mean 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, '031'.

Plants.— Correns, '00, '00b, '01, '02-' 02°, '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 of
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

44 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 & 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-crt,b). 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 chelae, all 1 .00 1 .00

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

" " glands on 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

.TV. A.

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

Plants. — Conditions of life affect number of floral parts in
poppy, de Vries, '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 Holl. Waquoit. Brfstol, R. I.

Dorsal fin-iays. . . A = 66 , 1 65 . 2 64 . 9
Anal " ... 4 = 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
Eddy stone, 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, '98, '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:

Elderton, '02.

Values of log ] ^i/— e~^2 [ for various values of f.

Elderton, '02.

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

& n(n — 2)(n — 4) . . .

/2 7*00 i o

Table of A/ — I e ~ *x d%, for different values of 7, Elder-

ton, '02.

Table of Iog10 (l+x)— x log,0 e for various values of x, for
use with curves of Type III.

Tables for calculating probable error, Sheppard, 798.

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-
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The references are scattered through fifty-seven periodi-
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XXX, 123-150. Plates I-IX. Nov.
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86 STATISTICAL METHODS.

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BATESON, W., '89. On some Variations of Cardium edule
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BATESON, W., '94. Materials for the Study of Variation.
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BATESON, W., and E. R. SAUNDERS, '02. Reports to the
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BATESON, W., '02. Mendel's Principles of Heredity. Cam-
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BATESON, W., '03. Variation and Differentiation in Parts
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BATESON, W., '03b. The Present State of our Knowledge of
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BEETON, MARY, and KARL PEARSON, '01. On the Inheritance
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BIBLIOGRAPHY. 87

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BLANCHARD, NORMAN, '03. On Inheritance (Grandparent
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BOAZ, FRANZ, '99. The Cephalic Index. American An-
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BOWDITCH, H. P., '01. The Growth of Children, Studied by
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BUMPUS, H. C., '98b. On the Identification of Fish Arti-
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BYRNE, L. W., '02. On the Number and Arrangement of
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88 STATISTICAL METHODS.

CAMERANO, L., '00. Lo studio quantitative degli organismi

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CAMERANO, L., '00b. Lo studio quantitative degli organ-
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Accademia delle sci. di Torino, XXXV, 19 pp.
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Atti d. R. Accademia delle sci. di Torino, XXXVI,

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BIBLIOQEAPHY. 89

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

Example: A = 17.673; o= 1.117; 2/0= 181.4.
(See page 26.)

Entries in Table
V — M corresponding to

V V-M —7" V-M 2/0 y f

a

14 -3.673 3.29 .00449 X181.4 = 0.8 1

15 -2.673 2.39 .05750 X 181.4 = 10.4 8

16 -1.673 1.50 .32465 X 181.4 = 58.9 63

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; d = 1.1169. (See page 26.)

Class.

X
a

Per
cent.

Class
Limits.

Deviation
from

Xi
a

(*-*a)XK
. v°°

less 2 Xl+(

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 METHODS.

j .

In the example, the data of which are given on p. 26, the
frequency between the limits is given in % column. The — of

G

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, *Js 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
"Percent." 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 r(4.273). The value cannot be found
directly because the value of p is larger than the numbers in
the table (1 to 2). The solution is made by aid of the equation
r(p + l)=pF(p), thus:

log r(1.273) = 9.955185
log 1.273 =0.104828

log T(3.273) = 0.060013
log 2.273 =0.356599

log r(3.273) = 0.416612
log 3.273 =0.514946

log T(4. 273) = 0.931558

or, more briefly, log r(1.273) = 9.955185
log 1.273 = .104828
log 2.273 = .356599
log 3.273 = .514946

log T(4.273) = 0.931558 = log 8.542

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

. VII. 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 (n) and for various values

of r. The probable error of the coefficient of correlation

, . 0.6745(1 -r2) , , . , ^

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-
tissa, regardless of the position of the decimal point in the
number, or of the number of ciphers which precede or follow
the significant figures of the number. The value of the char-
acteristic depends entirely on the position of the decimal point

108 STATISTICAL METHODS.

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

Number. Logarithm.

13840. 4.141136

1384.0 3.141136

138.40 2.141136

13.84 1.141136

1.384 0.141136

.1384 —1.141136

.01384 —2.141136

.001384 —3.141136

etc. etc.

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

Number. Logarithm.

1.384 0.141136

.1384 9.141136

.01384 8.141136

.001384 7.141136

etc, etc.

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

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 ^ 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

Am. For 23.4275 log is 1.369726

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

To find the number corresponding to a given
logarithm. — If the given mantissa is not in the table find the
one next less, and take out the four figures corresponding to it ;
divide the difference between the two mantissas by the tabu-
lar difference in that part of the table, and annex the figures of
the quotient to the four figures already taken out. Finally,
place the decimal point according to the rule for characteristics,
prefixing or annexing ciphers if necessary. The division re-
quired is facilitated by the table of proportional parts, \vhich
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. 540
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.

JSxample.—Yind 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
Mcample.—"Fm<l the log cot of 9° 28' 20".

Log cotan of 9° 28' is 10.777948

Subtract correction 20 X 12.97 259

Ans. 10777689

EXPLANATION OF TABLES. Ill

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

An*. 13° 36' 05".3

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

Diff. 235 -5- 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 the 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 third, and in thefourfo
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.

112 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 sm of 1° 39' 14". 4.

1° 39' 14".4 = 5954".4 log 3.774838

add (g-Z) 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 - 1) from the fifth column.

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

52' 35" = (3120" + 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-\- 1}
the logarithm corresponding to the given angle, interpolating
for the last figure if necessary, and from this subtract the loga-
rithm of the number of seconds in the given angle.

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

q + 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.

k— Find the log cos of 88° 41' 12". 5

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

Ans. $.360168

EXPLANATION OF TABLES. 113

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

q + I 15.314413

1800" + 50" = 1850' log ^267172

Ans. 12.047241

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

Find in the proper column two consecutive logarithms be-
tween which the given logarithm falls. If the title of the
given function is found at the top of that column read the
degrees from the top of the page; if at the "bottom read from
the bottom.

Find the value of (q — 1) or (g + 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, nt in the required arc, we have at once I = q — (q — 1}
or I = (q + 1) — q, whence n is easily found.

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

E*am&le.— 11.734268 is the log cot of what arc?

a + I 15.314376

q 11.734268

•. n^ 3802.8 3.580108

Fo? 1° adopt 3780. giving 03'

difference 22". 8

Ans. 1° 03' 22".8 or 178° 56' 37".2.

Example. — 8.201795 is the log cos of what arc?
q - I 4.685556

q 8.201795

/', w=. 3282".8 3.51623d

For 89° adopt 3300. giving 05'

Difference 17". 2

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

114

STATISTICAL METHODS.

THE GREEK ALPHABET.

A a

Alpha

I i

Iota

Pp

Rho

B/S

Beta

KK

Kappa

2 (75

Sigma

^ y

Gamma

A A

Lamba

T r

Tau

A d

Delta

MJLI

Mu

TV

Upsilon

E e

Epsilon

N v

Nu

$ (J)

Phi

z c

Zeta

S 1

Xi

X x

Chi

Hrj

Eta

Oo

Omicron

Wil>

Psi

@0fl

> Theta

IlTt

Pi

Q, GO

Omega

EXPLANATION OF TABLES.

115

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

A, average, mean,
a, class index (p. 24); also upper
left-hand quadrant (p. 49).

a, skewness index.

b, 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).

J, index of closeness of fit.
8, difference between y and /.

E, pfobable error.

e, base of Naperian logarithms,
= 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

p"th individual,
i, 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.
I, range of curve along x.
li, lz, portions of the curve range.
A , number of classes.
A, class range.
M, abscissal value of the mode

(theoretical).
Mr , abscissal value of the mode

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

a log.

n, number of variates; area of
polygon; any, not specified,
number.

\n, .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.

TT, circumference in units of diame-
ter, 2.14159.

q, a root or power.

r, coefficient of correlation.

P, coefficient of regression.

s, a relation of /?'s (p. 22).

I, summation sign.

a, standard deviation; index of
variability.

T, transmuting factor, a into E,
.67449.

T, in Type IV.

' j- angles.
<f>, )

V, magnitude of any class.

V0, magnitude of central class.

v, any variate or value.

w; = 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.

yo, value of the ordinate at the
origin.

z, ordinate value.

116 STATISTICAL METHODS.

!

I. FORMULAS.

. = ± 0.6745—^=. x = V-A

V n

.=L =0.7979*7. W -0.6745*.

n ^4. D.

2(£J) , Jl&J) , _7_t
n t 2n 240 »*

= D = o.A.

4(402- 30i)(2& *• 3ft-6)

£|| (Types I, IV). a== p~3 C^6 V>-

0.6745o ( TT (1-a2) /, , X2\ )t
Probable discrepancy, — — - j — . 3— - ( 1 + ^ } V 3.

r _ J(dev. x X dev. y X /) = ^(xiX2/) ^ ^0.6745(l-r2)

n. 01.02 noia-2 \/~n

r0 (spurious correlation) =

02 o2 03

—, 6745oi A / 1 — ri22

h~ o2 n

To solve any equation of the second degree,

-b±Vb2-4ac
2a

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

Title.

Symbol.

Number.

Log.

Ratio of circumference to diameter • .

•K
1

x

VT
i
VT

\/2ic

1
\/2~n
1

2*

VT
i

V2~

VI

*

e
1

vT

ra

1

m

T

3.1415927
0.3183099

1.7724538
0.5641896
2.506628
0 . 3989422

0.159155
1.4142136
0.707106
0.797816

2.7182818
0.606530
0.4342945
2.3025851
0.67449

0.4971499
9.502S5C1

0.2485749
9.7514251
0.399090
9.6009100
9.201820
0.150515
9.8494849
9.9019401

0.4342945
9.78285281
9.6377843
0.3622157
9.828976

Reciprocal of same

Souare root of same

Reciprocal of souare root of same

Square root of 2?r

Reciprocal of same

Reciprocal of 2?r . .

Reciprocal of same ..

2
Scruare root of —

7T

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. logz = raXhyp. logo;, or

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

Hyp. log x = com. log xX — , or
w

Com. log(hyp. log x)
=com. log (com. log) x + 0.3622157

Circumference of circle

2nr
nr*

y*ir

a •>
360^

ivhere a =
semi-n

= semi-major
unor axis of

axis : b =
ellipse.

Area of circle

Area of sector (length of arc =Z)

Area of sector (angle of arc =a°)
Eccentricity of an ellipse c ^v • -\

o2 '

118

STATISTICAL METHODS.

TABLE III.— TABLE OF ORDINATES (2) OF NORMAL CURVE,
OR VALUES OF -^ CORRESPONDING TO VALUES OF — .

2/0 a

x = deviation from mean.
o = standard deviation.

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

X/a

0

1

2

3

4

99920
99025
97161
94387
90774

5

6

7

8

9

0.0
0.1
0.2
0.3
0.4

1000QO
99501
98020
95600
92312

99995
99396
97819
95309
91939

99980
99283
97609
95010
91558

99955
99158
97390
94702
91169

99875
98881
96923
94055
90371

99820 199755
98728 [ 98565
96676 96420
9372393382
89961 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

1.0
1.1
1.2
1.3
1.4

60653
54607
48675
42956
37531

60047
54007
48092
42399
37007

59440
53409
47511
41845
36487

58834
52812
46933
41294
35971

58228
52214
46857
40747
35459

57623
51620
45783
40202
34950

57017
51027
45212
39661
34445

56414
50437
44644
39123
33944

55810 55209
49848'49260
4407863516
3858938058
33447 j 32954

1.5
1.6
1.7
1.8
1.9

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 28251
24385 23978
20511:20148
17081J16762
14083 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 11259
09290 09090
07433 07265
05888 05750
04618 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 03494
02757 02684
02098 02040
0158101536
01179,01145

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
OR 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

0

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

259

40

0.01

03991 439

479

519

559

598

638

678

718

758

0.02

0798J 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

45381 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

6804

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

7926

7965

8004

8043

8082

8121

8160

8199

8238

82781

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

102181

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

12552

12589

12627

12665

12703

12741

12778

12816

12854

12892

0.33

12930

12968113005

13043

13081

13118

13156

13194

13232

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

23 4 5

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.— Continued.

x/a

0

1

2

3

4

5

6

7

8

9

J

0.36

14058

14095

14132 14169

14207

14244

14281

14319

14356

14393

0.37

14431

14468

14505 145

IL'

1457

9

14617

14654

14691

1'

1728

147

'65

0.38

14803

14840

14877| 14914

14951

14988

15025

15062

15099

15136

37

0.39

15173

15210

15247

152

84

1532

1

15357

15394

15431

1,

3468

1«

>05

0.40

15542

15579

15616

156

52

1568

9

15726

15763

15799

1,

3836

15*

573

0.41

15910

15946

15983

16019

16056

16093

16129

16166

16202

16239

0.42

16276

16312

16348

163

85

1642

1

16458

16494

16531

1(

3567

16(

,i)l

0.43

16640

16676

16713

16749

16785

16821

16858

16894

16930

16967

0.44

17003

17039

17075

171

11

1714

7

17184

17220

17256

r

^292

17^

528

0.45

17364

17400

17436

17472

17508

17544

17580

17616

17652

17688

36

0.46

17724

17760

17796

178

31

1786

7

17903

17939

17975

1*

3011

18(

)46

0.47

18082

18118

18153

181

89

1822

5

18260

18296

18332

1*

3367

1#

HI:;

0.48

18439

18474

18509

18545

18580

18616

18651

18687

18722

18758

0.49

18793

18829

18864

188

99

1893

4

18969

19005

19040

1<

3075

19

ill

0.50

19146

19181

19216

19251

19287

19322

19357

19392

19427

19462

0.51"
0.52

19497
19847

19532
19881

19567
19916

19602 19637
1995 ij 19986

19672
20020

19707
20055

19742
20090

19777
20125

19812
20160

35

0.53

20194

20229

20263

202

98 2033

20367

20402

20436

2(

3471

20,

305

0.54

20540

20574

20609

20643

20678

20712

20746

20781

20815

20850

0.55

20884

20918

20952

20986

21021

21055

21089

21123

21158

21192

0.56

21226

21260

21294

213

28

2136

2

21396

21430

21464

2

1498

21,

332

34

0.57

21566

21600

21634

216

67

2170

1

21735

21769

21803

2

1836

21*

370

0.58

21904

21938

21971

22005

22039

22072

22106

22139

22173

22207

0.59

22240

22274

22307

223

41

2237

4

22407

22441

22474

2

2508

22,

341

0.60

22575

22608

22641

22674

22707

22741

22774

22807

22840

22874

0.61

22907

22940

22973

230

06

2303

9

23072

23105

23138

2

3171

23.

204

33

0.62

23237 23270

23303

23335

23368

23401

23434

23467

23499

23532

0.63

23565 23598

23630

236

63

2369

5

23728

23761

23793

2

3826

23*

359

0.64

23891

23924

23956

23988

24021

24053

24085

24118

24150

24183

0.65

24215

24247

24280

243

12

2434

4

24376 24408

24441

2

4473

24,

305

0.66

24537

24569

24601

246

33

2466

5

24697 24729

24761

2

4793

24*

325

32

0.67

24857

24889

24920

24952

24984

2501625048

25079

25111

25143

0.68

25175

25206

25238

252

G9

2530

1

25332 25364

25395

2

5427

25<

459

0.69

25490

25521

25553

25584

25615

25647

25678

25709

25741

25772

0.70

25804

25835

25866

258

97

2592

s

25959

25990

26021

2

5052

26(

384

0.71

26115

26146

26176

26207

26238

26269

26300

26331

26362

26393

31

0.72

26424

26454

26485

265

16

2654

6

26577

26608

26638

2

5669

26

roo

0.73

26730

26761

26791

268

22

2685

•2

26883

26913

26943

2

5974

27(

304

0.74

27035

27065

27095

27125

27156

27186

27216

27246

27277

27307

0.75

27337

27367

27397

274

27

2745

7

27437

27517

27547

2

7577

27

307

30

0.76

27637

276671 27697

27726

27756

27786

27816

27845

27875

27905

0.77

27935

27964

27994

280

23

2805

3

28082

28112

28142

2

8171

28

201

0.78

28230

28260

28289

283

18

2834

7

28377

28406

28435

2

8465

28

494

0.79

28524

28553

28582

286

11

2864

0

28669

28698

28727

2

8756

28

785

0.80

28814

28843

28872

28901

28930

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

VALUES OF NORMAL PROBABILITY INTEGRAL.

TABLE IV.— Continued.

X/a

0.

1

2

3

4

5

6

7

8

9

4

0.81

29103

29132

29160

29189 29217

29246

29274

29303

29332

29360

0.82

29389

29417

29446

294

74 29502

29531

29559

29588

2C

1616

296

>4/5

0.83
0.84

29673
29954

29701

J'.MISL'

29729
30010

29757
30038

29785
30066

29814129842
30094 30122

29870
30150

29898
30178

29926
30206

28

0.85

30234 30261

30289

303

17

30344

30372

30400

30427

3C

)455

304

US3

0.86

30510

30538

30565

30593

30620

30648

30675

30702

30730

30757

0.87

30785

30812

30839

30866

30894

30921

30948

30975

31002

31030

0.88

31057

31084

31111

311

;s

31165

31192

31219

31246

31

273

31C

500

27

0.89

31327

31353

31380

314

)7

31433

31460

31487

31514

31

540

3U

>67

0.90

31594

31620

31647

31673

31700

31726

31753

31780

31806

31832

0.91

31859

31885

31911

319

•57

31964

31990

32016

32042

32

5069

32C

)95

0.92

32121

32147

32173

32199

32225

32251

32277

32303

32329

32355

26

0.93
0.94

32381
32639

32407
32665

32433
32690

32459
32715

32484
32741

32510
32766

32536
32792

32562132587
3281832843

32613
32869

0.95

32894 32919

32945

329

70

32995

33021

33046

3307]

31

5096

331

22

0.96

3314733172

33197

33222

33247

33272

33297

33322

33347

33373

25

0.97

3339833422

33447

334

72

33497

33521 33546

33571

3C

5596

33(

521

0.98

33646 '33670

33695

33719

33744 S 33768 33793

33817

33842

33867

0.99

33891

33915 33940

339

(H

33988

34013

34037

34061

3^

L086

34]

L10

1.00

34134

34158(34182

342

06

34230

34255

34279

34303

3^

L327

341

551

24

1.01

34375

34399 34423

34446

34470

34494

34518

34542

34566

34590

1.02

34613

34637

34661

346

M

34708

34731

34755

34778

3^

1802

34*

>26

1.03

34849

34873

34896

34919

34943

34966

34989

35013

35036

35059

1.04

35083

35106

35129

351

52

35175

35198

35221

35245

3,

5268

35S

291

23

1.05

35314

35337

35360

353

SL>

35405

35428

35451

35474

3^

5497

35,

520

1.06

35543

35565

35588

35610

35633

35656

35678

35701

35724

35746

1.07

35769

35791

35814

35836

35858

35881

35903

35926

35948

35970

1 .08

35993

m K

037

Q

£0

nsi

1 n^

1 or

1 Aft

no

'I'.OQ'

36214 236

258

280

UoJ

302

lUo

324

1ZO

345

14o

367

389

1U4

411

22

1.10

433 455

477

4

98

520

541

563

585

607

(

328

1.11

650: 671

693

714

735

757

778

800

821

843

1.12

864 885

906

C]

28

94C

970

991

^c r

1.13

37176 097

118

139

160

181

202

012
223

Oo4
244

Ooo
265

21

1.14

286 306

327

348

368

389

410

430

451

472

1.15

493 513

534

5

54

574

595

615

636

656

(

377

1.16

697

718

738

758

778

798

819

839

859

880

1. 17

900

920

940

c

60

98C

000

ft9O

flAO

n/»n

i

\or\

1.18

38100

120

139

159

179

199

\jZ\J

218

U4U

238

uou

258

Uou

278

20

1.19

298

317

337

3

50

376

395

415

434

454

173

1.20

493

512

531

551

57G

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

7

8

9

857

d

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

1.35
1.36
1.37
1.38
1.39
1.40

1.41
1.42
1.43
1.44
1.45
1.46
1.47

.48
.49
.50
.51
.52
.53
.54
1.55

1.56
1.57
1.58
1.59

38686

877

705
895

724
914

743
933

762
952

781
971

800
990

819

838

19

18
17

16
15

14

13
12

008
195
380
562
742
920

027
214
398
580
760
937

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

139
324
507
688
866

158
343
525
706

884

177
361
544
724

902

008
182
354
524
692
857

025
199
371
540
709
873

042
216
388
557
725
899

060
233
405
574
742
906

077
251
422
591
758
922

095
268
439
608
775
938

112
285
456
625
792
955

130
303
473
641
808
971

40147
320
490
658
825
987

41149
308
466
621
774
924

165
337
507
676
840

004
165
324
481
637
789
939

020
181
340
497
652
804
954

036
197
355
512
667
819
969

052
213
371

527
683
834
984

068
229
387
543
698
849
998

084
245
403
558
713
864

101
261

418
574
728
879

117

277
434
590
744
894

133
292
450
605
759
909

013
161
306
449
591
730
867

028
175
321
464
605
744
881

016
149

280
409
536
662
785
906

043
190
335
478
619
758
895

058
205
350
492
633
772
908

42073
220
364
507
647
785
922

088
234
378
521
661
799
935

102
248
393
535
675
813
949

117
263
407
549
688
826
962

131
277
421
563
702
840
975

146
292
435
577
716
854
989

002
136
267
396
524
649
773
894

029
162
293
422
549
674
797
919

043
175
306
435
562
687
810
931

43056
189
319
448
574
699
822
943

069
202
332
460
587
711
834
955

083
215
345
473
599
724
846
967

096
228
358
486
612
736
858
978

109
241
371

498
624
748
870
990

122
254
383
511
637
760
882

002
120
237
351
464

014
132
248
363
475

026
144
260
374
486

038
156
271
385
498

050
167
283
397
509

44062
179
295
408

074
191
306
419

085
202
317
430

097
214
329
442

109
225
340
453

PROPORTIONAL PARTS.

J

19
18
17
16
15
14
13
12
11

1

1.9

1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1

234

5

6

7

8 9

3.8 5.7 7.6
3.6 5.4 7.2
3.4 5.1 6.8
3.2 4.8 6.4
3.0 4.5 6.0
2:$ 4.2 5.6
2.6 3.9 5.2
2.4 3.6 4.8
2.2 3.3 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
12.6
11.9
11.2
10.5
9.8
9.1
8.4
7.7

15.2 17.1
14.4 16.2
13.6 15.3
12.8 14.4
12.0 13.5
11.2 12.6
10.4 11.7
9.6 10.8
8.8 9.9

VALUES OF NORMAL PROBABILITY INTEGRAL. 123

TABLE IV —Continued.

X'O

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

Q^n

Qfin

Q7n

osn

QO1

I

you

you

y/u

yoLi

yy i

nni

ol 1

f\99

r\00

njo

1.65

45053

063

073

083

093

UU 1

103

U-l 1

114

\J££
124

UoZ

134

{.)**£
144

1.06

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

593

599

609

6181 627

1.71

637

646

655

664

673

682

692

701

710

7i9

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

9071 916

924

933

942

950

959

968

977

985

1 75

QQJL '

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

.78

246

254

262

270

279

287

295

303

311

319

.79

327

335

343

351

359

367

375

383

391

399

8

.80

407

415

423

430

438

446

454 462

469

477

.81

485

493

500

508

516

523

531! 539

547

554

.82

562

570

577

585

592

600

607

615

622

630

.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
1 .88

926

QQC

933

939

946

953

960

967

974

981

988

yyo

001

008

015

021

028

035

042

049

055

1.89

47062

069

075

" 082

088

095

102

108

115

122

1.90

128

135

141

148

154

161

167

174

180

187

.91

193

200

206

212

219

225

231

238

244

251

.92

257

263

270

276

282

288

294

301

-307

313

.93

320

326

332

338

344

350

356

362

369

375

.94

381

387

393

399

405

411

417

423

429

435

6

.95

441

447

453

459

465

471

476

482

488

494

.96

500

506

512

517

523

529

535

541

546

552

.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

97,2

977

PROPORTIONAL, PARTS.

J

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

5.6 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

4

2 05

47982

987

99]

996

nm

nnf

m i

me

non

2.06

48030

035

039

044

UUJ

049

uuo
054

UA l

058

o

063

\jt\j
068

02o
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

214

218

222

227

231

235

240

244

248

253

2.11

257

261

266

270

274

278

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

204

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

855

658

662

665

669

672

676

2.22

679

682

686

689

692

696

699

702

706

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

830

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

828

930

933

936

939

942

944

947

950

953

2.31

956

958

961

964

966

969

972

975

977

980

200

983

986

988

991

994

996

999

. O4

On9

f\f\A

on 7

2.33

49010

012

015

017

020

023

025

uu*
028

uu^
031

uu«
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

489

506

520

16

2.6

534

547

560

573

585

598

609

621

632

643

12

2.7

653

664

674

683

693

702

711

720

728

736

9

2.8

744

752

760

767

774

781

788

795

801

807

7

PROPORTIONAL PARTS.

A

1

2

345

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

A

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

98£

989

1

3.7

989

990

990

990

991

991

992

992

991

992

0

3.8

993

993

993

994

994

994

994

995

99f,

995

0

3.9

995

995

996

996

996

996

996

996

99.

997

0

4

997

998

999

999

999

000

000

000

00

000

0

PROPORTIONAL PARTS.

J

1

234

5 6

7

8 9

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 J

.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 T FUNCTIONS OF p (see pages 32-34).

p

0

1

2

3

4

5

6

7

8

9

1.00

9750

9500

9251

9003

8755

8^09

8263

8017

7773

1.01

9.997529

7285

7043

6801

6560

0320

6080

5841

5602

5365

1 .0:2

5128

4892

4656

4421

4187

3953

3721

3489

3257

3026

1.03

2796

2567

2338

2110

1883

1G5G

1430

1205

0981

0757

1.04

0533

0311

0089

9868

§647

§427

§208

§989

8-T2

§554

1.05

9.988338

8122

7907

7692

7478

7265

7052

6841

6629

6419

1.06

6209

6000

5791

5583

5376

5169

4963

4758

4553

4349

1.07

4145

3943

3741

3539

3338

3138

2939

2740

2541

2344

1.08

2147

1951

1755

1560

1365

1172

0978

0786

0594

0403

1.09

0212

0022

§833

§644

§456

§269

§082

8896

§710

§525

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

3764

3596

3429

3262

1.13

3096

2931

2766

2602

2438

2275

2113

1951

1790

1629

1.14

1469

1309

1150

0992

0835

0677

0521

0365

0210

0055

1.15

9.969901

9747

9594

9442

9290

9139

8988

8838

8688

8539

1.16

8390

8243

8096

7949

7803

7658

7513

7369

7225

7082

1.17

6939

6797

6655

6514

6374

6234

6095

5957

5818

5681

1.18

5544

5408

5272

5137

5002

4868

4734

4601

4469

4337

1.19

4205

4075

3944

3815

3686

3557

3429

3302

3175

3048

1.20

2922

2797

2672

2548

2425

2302

2179

2057

1936

1815

1.21

1695

1575

1456

1337

1219

1101

0984

0867

0751

0636

1.22

0521

0407

0293

0180

0067

9955

8843

9732

9621

9511

1.23

9.959401

9292

9184

9076

8968

8861

8755

8649

8544

8439

1.24

8335

8231

8128

8025

7923

7821

7720

7620

7520

7420

1.25

7321

7223

7125

7027

6930

6834

6738

6642

6547

6453

1.26

6359

6267

6173

6081

5989

5898

5807

5716

5627

5537

1.27

5449

5360

5273

5185

5099

5013

4927

4842

4757

4673

1.28

4589

4506

4423

4341

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

0690

0634

0579

0524

1.34

0470

0416

0362

0309

0257

0205

0153

0102

0051

0001

1.35

9.949951

9902

9853

9805

9757

9710

9663

9617

9571

9525

1.36

9480

9435

9391

9348

9304

9262

9219

9178

9136

9095

1.37

9054

9015

8975

8936

8898

8859

8822

8785

8748

8711

1.38

8676

8640

8605

8571

8537

8503

8470

8437

8405

8373

1.39

8342

8311

8280

8250

8221

8192

8163

8135

8107

8080

1.40

8053

8026

8000

7975

7950

7925

7901

7877

7854

7831

1.41

7808

7786

7765

7744

7723

7703

7683

7664

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

7289

7295

7302

1.48

7310

7317

7326

7334

7343

7353

7363

7373

7384

7395

1.49

7407

7419

7431

7444

7457

7471

7485

7499

7515

7529

TABLE OF LOG r FUXCTIOKS. 127

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

P

0

1

2

3

4

5

6

7

8

9

1.50

9.947545

7561

7577

7594

7612

7629

7647

7666

76S5

7704

1.51

7724

7744

7764

7785

7806

7828

7850

7873

7896

7919

1.52

7943

7967

7991

8016

8041

8067

8093

8120

8146

8174

1.53

8201

8229

8258

8287

8316

8346

8376

8406

8437

8-168

1.54

8500

8532

8564

8597

8630

8664

8698

8732

8767

8802

1.55

8837

8873

8910

8946

8983

9021

9059

9097

9135

9174

1.56

9-214

9254

9294

9334

9375

9417

9458

9500

9543

9586

1.57

93 .'9

9672

9716

9761

9806

9851

9896

9942

9989

5035

1.58

9.950082

0130

0177

0225

0274

0323

0372

0422

0472

0522

1.59

0573

0624

0676

0728

0780

0833

0886

0939

0993

1047

1.60

1102

1157

1212

1268

1324

1380

1437

1494

1552

1610

1.61

1668

1727

1786

1845

1905

1965

2025

2086

2147

2209

1.62

2271

2333

2396

2459

2522

2586

2650

2715

2780

2845

1.63

2911

2977

3043

3110

3177

3244

3312

3380

3449

3517

1.64

3587

3656

3726

3797

3867

3938

4010

4081

4154

4226

1.66

4299

4372

4446

4519

4594

4668

4743

4819

4894

4970

1.66

5047

5124

5201

5278

5356

5434

5513

5592

5671

5740

1.67

5830

5911

5991

6072

6154

6235

6317

6400

6482

6566

1.68

6649

6733

6817

6901

6986

7072

7157

7243

7322

7416

1.69

7503

7590

7678

7766

7854

7943

8032

8122

8211

8301

1.70

8391

8482

8573

8664

8756

8848

8941

9034

9127

9220

1.71

9314

9409

9502

9598

9«93

9788

9884

9980

5077

5174

1.72

9.960271

0369

0467

0565

0664

0763

0862

0961

1061

1162

1.73

1262

1363

14G4

1566

1668

1770

1873

1976

2079

2183

1.74

2287

2391

2496

2601

2706

2812

2918

3024

3131

3238

1.75

3345

3453

3561

3669

3778

3887

3996

4105

4215

4326

1.76

4436

4547

4659

4770

4882

4994

5107

5220

5333

5447

1.77

5561

5675

5789

5904

6019

6135

6251

6367

6484

6600

1.78

6718

6835

6953

7071

7189

7308

7427

7547

7666

7787

1.79

7907

8028

8149

8270

8392

8514

8636

8759

8882

9005

1.80

9129

9253

9377

9501

9626

9751

9877

5003

5129

5255

1.81

9.970383

0509

0637

0765

0893

1021

1150

1279

1408

1538

1.82

1668

1798

1929

2060

2191

2322

2454

2586

2719

2852

1.83

2985

3118

3252

3386

3520

3655

3790

3925

4061

4197

1.84

4333

4470

4606

4744

4881

5019

5157

5295

5434

5573

1.85

5712

5852

5992

6132

6273

6414

6555

6697

6838

6980

1.86

7123

7266

7408

7552

7696

7840

7984

8128

8273

8419

1.87

8564

8710

8856

9002

9149

9296

9443

9591

9739

9887

1.88

9.980036

0184

0333

0483

0633

0783

0933

1084

1234

1386

1.89

1537

1689

1841

1994

2147

2299

2453

2607

2761

2915

1.90

3069

3224

3379

3535

3690

3846

4003

4159

4316

4474

1.91

4631

4789

4947

5105

5264

5423

5582

5742

5902

6062

1.92

6223

6383

6544

6706

6867

7029

7192

7354

7517

7680

1.93

7844

8007

8171

8336

8500

8665

8830

8996

9161

9327

1.94

9494

9660

9827

9995

5162

5330

5498

5666

5835

1004

1.95

9.991173

1343

1512

1683

1853

2024

2195

2366

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

6384

6562

6740

6919

7078

7277

7457

7637

7817

7997

1.99

8178

8359

8540

8722

8903

9085

9268

9450

9633

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.

80

76.20

lor.eo

127.00

152.40

177

.RO

2(

)3.20

228.60

10

279.40

304.

80

330.19

355.59

380.99

406.39

431

.79

457.19

482.59

20

533.3$

558.

79

584.19

6

09.59

634.99

660.39

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

132J).

8

1346.2

1371.6

1397.0

1422.4

1447

.8

14

?3.2

1498.6

(i()

1549.4

1574.

8

1600.2

16

<!5.6

1651.0

1676.4

1701

.8

17

27.2

1752.6

70

1803.4

1828.

8

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,

}5.2

2260.6

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

7/12

14.82

1/16

1.59

5/16

7.94

9/16

14.29

13/16 20.64

2/12

4.23

8/12

16.93

1/8

3.17

3/8

9.52

5/8

15.87

7/8 22.22

3/12
4/12

6.35
8.47

9/12
10/12

19.05
21.17

3/16
1/4

4.76
6.35

7/16

1/3

11.11

12.70

11/16

3/4

17.46
19.05

15/16 23.81
1 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.

'

0

'

0

'

0

"

0

"

o

"

0

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

.006,667

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

51

.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

1

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.

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

Powers.

First.

Second.

Third.

Fourth.

Fifth.

Sixth.

1

1

1

1

1

1

2

4

8

16

32

64

3

9

27

81

243

729

4

16

64

256

1024

4096

5

25

125

625

3125

15625

6

36

216

1296

7776

46656

7

49

343

2401

16807

117649

8

64

512

4096

32768

262144

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

537824

7529536

15

225

3375

50625

759375

11390625

16

256

4096

65536

1048576

16777216

17

289

4913

83521

1419857

24137569

18

324

5832

104976

1889568

54012224

19

361

6859

130321

2476099

47045881

20

400

8000

160000

3200000

64000000

21

441

9261

194481

4084101

85766121

22

484

10648

234256

5153632

113379904

23

529

12167

279841

6436343

148035889

24

576

13824

331776

7962624

191102976

25

625

15625

390625

9765625

244140625

26

676

17576

456976

11881376

308915776

27

729

19683

531441

14348907

387420489

98

784

21952

614656

17210368

481890304

29

841

24389

707281

30511149

594823321

30

900

27000

810000

24300000

729000000

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.
(Specially Calculated.)

Number

t /~VU

Correlation Coefficient r.

OI <JD-

servations.

0.0

-0.1

0.2

0.3

0.4

0.5

25

0.1349

0.1336

0.1295

0.1228

0.1133

0 1012

50

0.0954

0.0944

0.0916

0.0868

0.0801

0.0715

75

0.0779

0.0771

0.0748

0.0709

0.0654

0.0584

100

0.0674

0.0668

0.0648

0.0614

0.0567

0.0506

200

0.0477

0.0472

0.0458

0.0434

0.0401

0.0358

300

0.0389

0.0386

0.0374

0.0354

0.0327

0.0292

400

0.0337

0.0334

0.0324

0.0308

0.0283

0.0253

500

0.0302

0.0299

0.0290

0.0274

0.0253

0.0226

600

0.0275

0.0273

0.0264

0.0251

0 . 0231

0.0207

700

0.0255

0.0252

0.0245

0.0232

0.0214

0.0191

800

0.0239

0.0236

0.0229

0.0217

0.0200

0.0179

900

0.0225

0.0221

0.0216

0.0205

0.0189

0.0169

1000

0.0213

0.0211

0.0205

0.0194

0.0179

0.0160

0.6

0.7

Ys

0.9

1.0

25

0.0863

0.0688

0.0486

0.0256

0

50

0.0611

0.0487

0.0343

0.0181

0

75

0.0498

0.0397

0.0280

0.0148

0

100

0.0432

0.0344

0.0243

0.0128

0

200

0.0305

0.0243

0.0172

0.0091

0

300

0.0249

0.0199

0.0140

0.0074

0

400

0.0216

0.0172

0.0121

0.0064

0

500

0.0193

0.0154

0.0109

0.00a7

0

600

0.0176

0.0140

0.0099

0.0052

0

700

0.0163

0.0130

0.0092

0.0048

0

800

0.0153

0.0122

0.0086

0.0045

0

900

0.0144

0.0115

0.0081

0.0043

0

1000

0.0137

0.0109

0.0077

0.0041

0

TABLE X. — SQUARES, CUBES, ETC.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

1

1

1

1.0000000

1.0000000

1.000000000

2

4

8

1.4142136

1.2599210

.500000000

3

9

27

1.7320508

1.4422496

.333333333

4

16

64

2.0000000

1.5874011

.250000000

5

25

125

2'. 2360680

1.7099759

.200000000

6

36

216

2 4494897

1.8171206

.166666667

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

.066666667

16

256

4096

4.0000000

2.5198421

.062500000

17

289

4913

4.1231056

2.5712816

.058823529

18

324

5832

4.2426407

2.6207414

.055555556

19

361

6859

4.3588989

2.6684016

.052631579

20

400

8000

4.4721360

2.7144177

.050000000

21

441

9261

4.5825757

2.7589243

.047619048

22

484

10648

4.6904158

2.8020393

.045454545

23

529

12167

4.7958315

2.8438670

.043478261

24-

576

13824

4.8989795

2.8844991

.041666667

25

625

15625

' 5.0000000

2.9240177

.040000000

26

676

17576

5.0990195

2.9624960

.038461538

27

729

19683

5.1961524

3.0000000

.037037037

28

784

21952

5.2915026

3.0365889

.035714286

29

841

24389

5.3851648

3.0723168

.034482759

30

900

27000

5.4772256

3.1072325

.033333333

31

961

29791

5.5677644

3.1413806

.032258065

32

1024

32768

5.6568542

3.1748021

.031250000

33

1089

35937

5.7445626

3.2075843

030303030

34

1156

39304

5.8309519

3.2396118

.029411765

35

1225

42875

5.9160798

3.2710663

' .028571429

36

1296

46656

6.0000000

3.3019272

.027777778

37

1369

50653

6.0827625

3.3322218

.027027027

38

1444

54872

6.1644140

3.3619754

.026315789

39

1521

59319

6.2449980

3.3912114

.025641026

40

1600

64000

6.3245553

3.4199519

.025000000

41

1681

68921

6.4031242

3.4482172

.024390244

42

1764

74088

6.4807407

3.4760266

.023809524

43

1849

79507

6.5574385

3.5033981

.023255814

44

1936

85184

6.6332496

3.5303483

.022727273

45

2025

91125

6.7082039

3.5568933

.022222222

46

2116

97336

6.7823300

3.5830479

.021739130

47

2209

103823

6.8556546

3.6088261

.021276600

48

2304

110592

6.9282032

3.6342411

.020833333

49

2401

117649

7.0000000

3.6593057

.020408163

50

2500

125000

7.0710678

3.6840314

.020000000

51

2601

132651

7.1414284

3.7084298

.019607843

52

2704

140608

7.2111026

3.7325111

.019230769

53

2809 -

148877

7.2801099

3.7562858

.018867925

54

2916

157464

7.3484692

3.7797631

.018518519

55

3025

166375

7.4161985

3.8029525

.018181818

56

3136

175616

7.4833148

3.8258624

.017857143

57

3249

185193

7.5498344

3.8485011

.017543860

58

.3364

195112

7.6157731

3.8708766

.017241379

59

3481

205379

7.6811457

3.8929965

.016949153

60

3600

216000

7.7459667

3.9148676

.016666667

61

3721

226981 7.8102497

3.9364972

.016393443

62

3844

238328 7.8740079

3.9578915

.016129032

131

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

I
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 ! 246211 3

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

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

82

6724

551368

9.0553851

4.3444815

.012195122

83

6889

571787

9.1104336

4.3620707

.012048193

84

7056

592704

9.1651514

4.3795191

.011904762

85

7225

614125

9.2195445

4 3968296

.011764706

86

7396

" 636056

9.2736185

4.4140049

.011627907

87

7569

658503

9.3273791

4.4310476

.011494253

88

7744

681472

9.3808315

4.4479602

.011363636

89

7921

704969

9.4339811

4.4647451

.011235955

90

8100

729000

9.4868330

4.4814047

.011111111

91

8281

753571

9.5393920

4.4979414

.010989011

92

8464

778688

9.5910030

4.5143574

.010869565

93

8649

804357

9.643C508

4.5306549

.010752688

94

8836

830584

9.6953597

4.5468359

.010638298

95

9025

857375

9.7467943

4.5629026

.010526316

96

9216

884736

9.7979590

4.5788570

.010416667

97

9409

912673

9.8488578

4.5947009

.010309278

98

9604

941192

9.8994949

4.6104363

.010204082

99

9801

970299

9.9498744

4.6260650

.010101010

100

10000

1000000

10.0000000

4.6415888

.010000000

101

10201

1030301

10.0498756

4.6570095

.009900990

102

10404

1061208

10.0995049

4.6723287

.009803922

103

10609

1092727

10.1488916

4.6875482

.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

.003259259

109

11881

1295029

10.4403065

4.7768562

.009174312

110

12100

1331000

10.4880885

4.7914199

.009090909

111

12321

1367631

10.5356538

4.8058955

.009009009

112

12544

1404928

10.5830052

4.8202845

.008928571

113

12769

1442897

10.6301458

4.8345881

.008849558

114

12996

1481544

iO. 6770783

. 4.8488076

.008771930

115

13825

1520875

10.7238053

4.8629442

.008695652

116

13456

1560896

10.7703296

4.8769990

.008620690

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

4.9324242

.008333333

121

14641

1771561

11.00 0000

4.9460874

.00820-4463

1£2

14884

1815848

11.0453610

4.9596757

.008190721

123

15129

1860867

11.0905365

4.9731898

.008130081

124

15376

1906624

11.1355287

4.9866310

.008064516

132

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Sqirare
Roots.

Cube Roots.

Reciprocals.

125

15625

1953125

11.1803399

5.0000000

.008000000

126

15876"

2000376

11.2249722

5.0132979

.007930508

127

16129

2048383

11.2694277

5.0265257

.007874016

128

16384

2097152

11.3137085

5.0396842

.007812500

129

16641

2146689

11.3578167

5.0527743

.007751938

130

16900

2197000

11.4017543

5.0657970

.007692308

131

17161

2248091

11.4455231

5.0787531

.007633588

132

17424

2299968

11.4891253

5.0916434

.007575758

133

17689

2352637

11.5325626

5.1044687

.007518797

134

17956

2406104

11.5758369

5.1172299

.007462687

135

18225

2460375

11.6189500

5.1299278

.007407407

136

18496

2515456

11.6619038

5.1425632

.007352941

137

18769

2571353

11.7046999

5.1551367

.007299270

138

19044

2628072

11.7473401

5.1676493

.007246377

139

19321

2685619

11.7898261

5.1801015

.007194245

140

19600

2744000

11.&321596

5.1924941

.007142857

141

19881

2803221

11.8743421

5.2048279

.007092199

142

20164

2863288

11.9163753

5.2171034

.007042254

143

20449

2924207

11.9582607

5.2293215

.006993007

144

20736

2985984

12.0000000

5.2414828

.006944444

145

21025

3048625

12.0415946

5.2535879

.006896552

146

21316

3112136

12.0830460

5.2656374

.006849315

147

21609

3176523

12.1243557

5.2776321

.006802721

148

21904

3241792

12.1655251

5.2895725

.006756757

149

22201

3307949

12.2065556

5.3014592

.006711409

150

22500

'3375000

12.^474487

5.3132928

.006666667

151

22801

3442951

12.2882057

5.3250740

.006622517

152

23104

3511808

12.3288280

5.3368033

.006578947

153

23409

3581577

12.3693169

5.3484812

.006535948

154

23716

3652264

12.4096736

5.3601084

.006493506

155

24025

3723875

12.4498996

5.3716854

.006451613

156

24336

3796416

12.4899960

5.3832126

.006410256

157

24649

3869893

12.5299641

5 3946907

.006369427

158

24964

3944312

12.5698051

5.4061202

.006329114

159

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

.006134969

164

26896

4410944

12.8062485

5.4737037

.006097561

165

27225

4492125

12.8452326

5.4848066

.006060606

166

27556

4574296

12.8840987

5.4958647

.006024096

167

27889

4657463

12.9228480

5.5068784

.005988024

168

28224

4741632

12.9614814

5.5178484

.005952381

169

28561

4826809

13.0000000

5.5287748

.005917160

170

28900

4913000

13:0384048

5.5396583

.005882353

171

29241

5000211

13.0766968

5.5504991

.005847953

172

29584

5088448

13.1148770

5.5612978

.005813953

173

29929

5177717

13.1529464

5.5720546

.005780347

174

30276

5268024

13.1909060

5.5827702

.005747126

175

30625

5359375

13.2287566

5 5934447

.005714286

176

30976

5451776

13.2664992

5.6040787

.005681816

177

31329

5545233

13.3041347

5.6146724

.005649718

178

31684

5639752

13.3416641

5.6252263

.005617978

179

32041

5735.339

13.3790882

5.6357408

.005586592

180

32400

5832000

13.4164079

5.6462162

.005555556

181

82761

5929741

13.4536240

5.6566528

.005524862

182

33124

6028568

13.4907376

5.6670511

.005494505

183

33489

6128487

13.5277493

5 6774114

.005464481

184

33856

6229504

13.5646600

5.6877:340

.005434783

185

34225

6331625

13.6014705

5.6980192

.005405405 i

186

34596

6434856

13.6381817

5.7082675

.005376344

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

5.7689982

.005208333

193

37249

7189057

13.8924440

5.7789966

.005181347

194

37636

7301384

13.9283883

5.7889604

.005154639

195

38025

7414875

13.9642400

5.7988900

.005128205

196

38416

7529536

14.0000000

5.8087857

.005102041

197

38809

7645373

14.0356688

5.8186479

.005076142

198

39204

7762392

14.0712473

5.8284767

.005050505

199

39601

7880599

14.1067360

5.8382725

.005025126

200

40000

8000000

14.1421356

5.8480355

.005000000

201

40401

8120601

14.1774469

5.8577660

.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

.004830918

208

43264

8998912

14.4222051

5.9249921

.004807692

209

43681

. 9129329

14.4568323

5.9344721

.004784689

210

44100

9261000

14.4913767

5.9439220

.004761905

211

44521

9393931

14.5258390

5.9533418

.004739336

212

44944

9528128

14.5602198

5.9627320

.004716981

213

45369

9663597

14.5945195

5.9720926

.004694836

214

45796

9800344

14.6287388

5.9814240

.004672897

215

46225

9938375

14.6628783

5.9907264

.004651163

216

46656

10077696

14.6969385

6.0000000

.004629630

217

47089

10218313

14.7309199

6.0092450

.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

231

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

.004098861

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

.004016064

250

62500

15625000

15.8113883

6.2996053

.004000000

251

63001

15813251

15.8429795

6.3079935

.0039840(54

252

63504

16003008

15.8745079

6.3163596

.003968254

253

64009

16194277

15.9059737

6.3247035

.00:3952569

254

64516

16387064

15.9373775

6.3330256

.003937008

255

65025

16581375

15.9687194

6.3413257

.003921509

256

65536

16777216

16.0000000

6.3496042

.003906250

257

66049

16974593

16.0312195

6.3578611

.003891051

258

66564

17178612

16.0623784

6.3660968

.003875969

259

67081

17373979

16.0934769

6.3743111

.003861004

260

67600

17576000

16.1245155

6.3825043

.003846154

261

68121

17779581

16.1554944

6.3906765

.003831418

262

68644

17984728

16.1864141

6.3988279

.003816794

263

69169

18191447

16.2172747

6.4069585

.003802281

264

69696

18399744

16.2480768

6.4150687

.003787879

265

70225

18609625

16.2788206

6.4231583

.003773585

266

70756

18821096

16.3095064

6.4312276

.003759398

267

71289

19034163

16.3401346

6.4392767

.003745318

268

71824

19248832

16.3707055

6.4473057

.003731343

269

72361

19465109

16.4012195

6.4553148

.003717472

270

72900

19683000

16.4316767

6.4633041

.003703704

271

73441

19902511

16.4620776

6.4712736

.003690037

272

73981

20123648

16.4924225

6.4792236

.003676471

273

74523

20346417

16.5227116

6.4871541

.003663004

274

75076

20570824

16.5529454

6.4950653

.003649635

275

75625

20796875

16.5831240

6.5029572

.003636364

276

76176

21024576

16.6132477

6.5108300

.003623188

277

76729

21253933

16.6433170

6.5186839

.003610108

278

77284

21484952

16.6733320

6.5265189

.003597122

279

77841

21717639

16.7032931

6.534:3351

.003584229

280

78400

21952000

16.7332005

6.5421326

.003571429

281

78961

22188041

16.76130546

6.5499116

.003558719

282

79524

22425768

16.7928556

6.5576722

.00a546099

283

80089

22665187

16.8226038

6.5654144

.003533569

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

6.5962023

.003484321

288

82944

23887872

16.9705627

6.6038545

.003472222

289

83521

24137569

17.0000000

6.6114890

.003460208

290

84100

24389000

17.0293864

6.6191000

.003448276

291

84081

24642171

17.0587221

6.6207054

.003436426

292

85264

24897088

17.0880075

6.6342874

.003424658

293

85849

25153757

17.1172428

6.6418522

.003412969

294

86433

25412184

17.1464282

6.6493998

.003401361

295

87025

25672375

17.1755640

6.6569302

.003389831

296

87616

25934.336

7.2046505

6.6644437

.003378378

297

88209

26198073

7.2336879

6.6719403

.003367003

298

88804

26463592

7.2626765

6.G794200

.003355705

299

89401

26730899

7.2916165

6.6808831

.003344482

300

90000

27000000

7.3205081

6.6943295

.003333333

301

90G01

27270901

7.3493516

6.701751)3

.00:3322259

302

91204

27543608

7.378147'2

6.7091729

.00:3311258

303

91809

27818127

7.4068952

6.7165700

.003300330

304

92416

28094464

17.4355958

6.7239508

.003289474

305

93025

28372625

17.4642492

6.7313155

.003278689

306

93636

28652616

17.4928557

6.7386641

.003267974

307

94249

28934443

17.5214155

6.7459967

.003257329

308

94864

29218112

17.5499288

6.7533134

.003246753

309

95481

29503629

17.5783958

6.7006143

.003236246

310

96100

29791000

17.6008169

6,7678995

.003225800

135

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

No.

Squares.

Cubes. '

Square
Roots.

Cube Roots.

Reciprocals.

311

96721

30080231

17.6351921

6.7751690

.003215434

312

97344

30371328

17.6635217

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

.003115265

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

326

106276

34645976

18.0554701

6.8823888

.003067485

327

106929

34965783

18.0831413

6.8894188

.003058104

328

107584

35287552

18.1107703

6.8964345

.003048780

329

108241

35611289

18.1383571

6.9034359

.003039514

330

108900

35937000

18.1659021

6.9104232

.003030303

331

109561

36264691

18.1934054

6.9173964

.003021148

332

110224

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

336

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

18.5741756

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

7.0472987

.002857143

351

123201

43243551

18.7349940

7.0540041

.002849003

352

123904

43614208

18.7616630

7.0606967

.002840909

353

124609

43986977

18.7882942

7.0673767

.002832861

354

125316

44361864

18.8148877

7.0740440

.002824859

355

126025

44738875

18.8414437

7.0806988

.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

129600

46656000

18.9736660

7.1137866

.002777778

361

130321

47045881

19 0000000

7.1203674

.002770083

362

131044

47437928

19.0262976

7.1269360

.002762431

363

131769

47832147

19.0525589

7.1334925

.002754821

364

132496

48228544

19.0787840

7.1400370

.002747253

365

133225

48627125

19.1049732

7.1465695

.002739726

366

133956

49027896

19.1311265

7.1530901

.002732240

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

CUBE ROOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Hoots.

Reciprocals.

373

139129

51895117

19.3132079

7.1984050

.002680965

374

139876

52313024

19.3390796

7.2048322

.002673797

375

140625

52734375

19.3049167

7.2112479

.002666667

376

141376

53157376

19.3907194

7.2176522

.002659574

377

142129

53582633

19.4104878

7.2240450

.002652520

378

142884

54010152

19.4422221

7.2304268

.002645503

379

143641

54439U3U

19.4679223

7.2367972

.002638522

380

144400

54872000

19.4935887

7.2431565

.002631579

381

145161

55306*41

19.5192213

7.2495045

.002624672

382

145924

55742968

19.5448203

7.2558415

.002617801

383

146689

56181887

19.5703858

7.2621675

.002610966

884

147456

56623104

19.5959179

7.2684824

.002604167

385

118225

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

58863809

19.7230829

7.2998936

.002570694

390

152100

59319000

19.7484177

7.3061436

.002564103

391

152881

59776471

19.7737199

7.3123828

.002557545

392

153664

60236288

19.7989899

7.3186114

.002551020 .

393

154449

60698457

19.8242276

7.3248295

.002544529

394

155236

61162984

19.8494332

7.3310369

.002538071

395

156025

61629875

19.8746069

7.3372339

.002531646

396

156816

62099136

19.8997487

7.3434205

.002525253

397

157609

62570773

19.9248588

7.3495966

.002518892

398

158404

63044792

19.9499373

7.3557624

.002512563

399

159201

63521199

19.9749844

7.3619178

.002506266

400

160000

64000000

20.0000000

7.3680630

.002500000

401

160801

64481201

20.0249844

7.3741979

.002493766

402

161604

64964808

20.0499377

7.3803227

.002487562

403

162409

65450827

20.0748599

7.3864373

.002481390

404

163216

65939264

20.0997512

7.3925418

.002475248

405

164025

66430125

20.1246118

f.3986363

.002469136

406

164836

66923416

20.1494417

7.4047206

.002463054

407

165649

67419143

20.1742410

7.4107950

.002457002

408

166464

67917312

20.1990099

7.4168595

.002450980

409

167281

68417929

20.2237484

7.4229142

.002444988

410

168100

68921000

20.2484567

7.4289589

.002439024

411

168921

69426531

20.2731349

7.4349938

.002433090

412

169744

69934528

20.2977831

7.4410189

.002427184

413

170569

70444997

20.3224014

7.4470342

.002421308

414

171396

70957944

20.3469899

7.4530399

.002415459

415

172225

71473375

20.3715488

7.4590359

.002409639

416

173056

71991296

20.3960781

7.4650223

.002403846

417

173889

72511713

20.4205779

7.4709991

.002398082

418

174724

73034632

20.4450483

7.4769664

.002392344

419

175561

73560059

20.4694895

7.4829242

.002386635

420

176400

74088000

20.4939015

7.4888724

.002380952

421

177241

74618461

20.5182845

7.4948113

.002375297

422

178084

75151448

20.5426386

7.5007406

.002369668

423

178929

75686967

20.5669638

7.5066607

.002364066

424

179776

#225024

20 5912603

7.5125715

.002358491

425

180625

76765625

20.6155281

7.5184730

.002352941

426

181476

77308776

20.6397674

7.5243652

.002347418

427

182329

77854483

20.6639783

7.5302482

.002341920

428

183184

784027'52

20.6881609

7.5361221

.002336449

429

184041

78953589

20.7123152

7.5419867

.002331002

430

184900

79507000

20.7364414

7.5478423

.002325581

431

185761

80002991

20.7605395

7.5536888

.002320186

432

186624

80621568

20.7846097

7.5595263

.002314815

433

187489

81182737

20.8086520

7.5653548

.002309469

434

188356

81746504

20.&326667

7.5711743

.002304147

137

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

435

189225

82312875

20.8566536

7.57'69849

.002298851

436

19009G

82881856

20.8806130

7.5827865

.00229-3578

437 ! 190969

83453453

20.9045450

7.5885793

.002288330

438

191844

84027672

20.9284495

7.5943633

.002283105

439

192721

84G04519

20.9523268

7.G001385

.002277904

440

193600

85184000

20.9761770

7.6059049

.002272727

441

194481

8576G121

21.0000000

7.6116626

.002267574

442

195364

86350888

21.0237960

7.6174116

.002262443

443

196249

86938307

21.0475652

7.6231519

.002257336

444

197136

87528384

21.0713075

7.6288837

.002252252

445

198025

88121125

21.0950231

7.6346067

.002247191

446

198916

88716536

21.1187121

7.6403213

.002242152

447

199809

89314G23

21.1423745

7.6460272

.002237136

448

200704

89915392

21.1660103

7.6517'247

.002232143

449

201601

90518849

21.1896201

7.657'4133

.002227171

450
451

202500
203401

91125000
91733851

21.2132034
21.2367600

7.6630943
7.6687665

.002222222
.002217295

452

204304

92345408

21.2602916

7.6744303

.002212389

453

205209

92959677

21.2837967

7.6800857

.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

.002183406

459

210681

96702579

21.4242853

7.7138443

.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

7.736187?

.002159827

464

215296

99897344

21.5406592

7.7417533

.002155172

465

216225

100544625

21.5638587

7.7473100

.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.C564078

7.7694620

.002132196

470

220900

103823000

21.6794834

7.7749801

.002127660

471

221841

104487111

21.7025344

7.780-4904

.002123142

472

222784

105154048

21.7'255610

7.7859923

.002118644

473

223729

105823817

21.7485G32

7.7914875

.002114165

474

224676

106496424

21.7715411

7.7969745

.002109705

475

225625

107171875

21.7944947

7.8024538

.002105263

476

226576

107850176

21.8174242

7.8079254

.002100840

477

227529

108531333

21.8403297

7.8133892

.002096436

478

228484

109215352

21 8632111

7.8188456

.002092050

479

229441

109902239

21.8800686

7.8242942

.002087683

480

230400

110592000

21.9089023

7.8.297353

.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

23.0000000

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

7.8944468

.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 KOOTS, AND RECIPROCALS.

No.

Squares.

Cubes.

Square
Koots.

Cube Hoots.

Kec'procals.

497

247009

122763473

22.2934968

7.9210994

.002012072

. 498

248004

123505UU3

22.3159136

7.92640&5

.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

252.104

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

130323843

22.5166605

7.9738731

.001972387

508

258064

131096512

22.5388553

7.9791122

.001968504

509

259081

131872229

22.5610283

7.9843444

.001964637

510

260100

132651000

22.5a31796

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

8.0104032

.001945525

515

265225

136590875

22.6936114

8.0155946

.001941748

516

26G256

137388096

22.7156334

8.0207794

.001937984

517

267289

138188413

22.7376340

8.0259574

.001934236

518

268:324

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

.001919386

522

272484

142236648

22.8473193

8.0517479

.001915709

523

27.3529

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

.001872659

535

286225

153130375

23.1300670

8.1180414

.001869159

536

287296

153990656

23.1516738

8.1230962

.001865672

537

288369

154854153

23.1732605

.8.1281447

.001862197

538

289444

155720872

23.1948270

8.1&31870

.001858736

539

290521

156590819

23.2163735

8.1382230

.001855288

540

291600

157464000

23.2379001

8.1432529

.001851852

541

292681

158340421

23.2594067

8.1482765

.001848429

542

293764

159220088

23.2808935

8.1532939

.001845018

543

294849

160103007

23.3023604

8.1583051

.001841621

544

295936

160989184

23.3238076

8.1633102

.001838235

545

297025

161878625

23.3452351

8.1683092

.001834862

546

298116

162771336

23.3666429

8.1733020

,00ia31502

547

299209

163667323

23.3880311

8.1782888

.001828154

548

300304

164566592

23.4003998

8.ia32695

.001824818

549

301401

165469149

23.4307490

8.1882441

.001821494

550

302500

166375000

23.4520788

8.1932127

.001818182

551

303601

167284151

23.4733892

8.1981753

.001814882

552

304704

168196608

23.4946802

8.2031319

.001811594

553

305809

160112377

23.5159520

8.2080825

.001808318

554

306916

170031464

23.5372046

8.2130271

.001805054

555

308025

170953875

23.5584380

8.2179657

.001801802

556

309136

171879616

23.5796522

8.2228985

.001798561

557

310249

172808693

23.6008474

8.2278254

.001705332

558

311364

173741112

23.6220236

8.2327463

.001792115

139

TABLE X. — SQUARES, CUBES, SQUARE KOOTS,

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

559 .

312481

174676879

23.6431808

8.2376614

.001788909

5GO

313600

175616000

23.6643191

8.2425706

.001785714

5G1

314721

176558481

23.6854386

8.247'4740

.001782531

502

315844

177504328

23.7065392

8.2523715

001779359

503

316969

178453547

23.7276210

8.2572633

001776199

504

318096

179406144

23.7486842

8.2621492

.001773050

505

319225

180362125

23.7697286

8.2670294

.001709912

506

320356

181321496

23.7907545

8.2719039

.001700784

507

321489

1828843368

23.8117618

8.2767726

.001763068

508

322024

183250432

23.8327500

8.2816355

.001760563

509

323761

184220000

£>. 8537209

8.2864928

.001757469

570

324900

185193000

23.8746728

8.2913444

.001754386

571

320041

180169411

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

24.0024188

8.3347553

.001727116

580

336400

195112000

24.0831891

8.3395509

.001724138

581

337561

196122941

24.1039416

8.3443410

.001721170

582

338724

197137368

24.1246702

8.3491256

.001718213

583

339889

198155287

24.1453929

8.3539047

.001715266

584

341056

199176704

24.1000919

8.3586784

.001712329

585

342225

200201625

24.1807732

8.3634466

.001709402

586

343396

201230056

24.2074309

8.3682095

001706485

587

344569

202262003

24.2280829

8.3729008

.001703578

588

345744

203297472

24.2487113

8.3777188

.001700680

589

346921

204336409

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

209584584

24.3721152

8.4061180

.001683502

595

a54025

210644875

24.3926218

8.4108326

001680672

596

355216

211708736

24.4131112

8.4155419

.001677852

597

356409

212776173

24.4335834

8.4202460

.001675042

598

357604

213847192

24.4540385

8.4249448

.001072241

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

363009

219256227

24.5560583

8.4483605

.001658375

604

304816

220348864

24.5764115

8.4530281

.001C55629

605

360025

221445125

24.5967478

8.4576906

.001652893

606

307236

222545016

24.6170673

8.4623479

.001650165

607

308449

223648543

24.6373/00

8.4670001

.001047446

608

309004

224755712

24.6576560

8.4716471

.001644737

609

310881

225866529

24.6779254

8.4762892

.001642036

610

37'2100

226981000

24.6981781

8.4809261

.001639344

611

373321

228099131

24.7184142

8.4855579

.001036061

612

374544

229220928

24.7386338

8.4901848

001033987

613

375709

230346397

24.7588368

8.4948065

001031321

614

370996

231475544

24.7790234

8.4994233

.001628664

615

378225

23200&375

24.7991935

8.5040350

.001626016

616

379456

233744896

24.8193473

8.5086417

.001623377

617

380689

234885113

24.8394847

8.5132435

.001020746

618

381924

230029032

24.&596058

8.5178403

.001618123

619

383161

237176659 ! 24.8797106

8.5224321

.001015509

620

384400

238328000 24.8!)97r;92 8.5270189

.001612903

140

CUBE HOOTS, AND HKCIPHOCALS.

No.

Squares.

Cubes.

Square
Hoots.

Cube Hoots.

Reciprocals.

621
622
623
624
625
626
627
628
629

385641
386884
388129
389376
390625
391876
393129
394384
395641

239483061
240641848
241804867
242970624
244140625
245314376
246491883
247673152
248858189

24.9198716
24.9399278
24.9599679
24.9799920
25.0000000
25.0199920
25.03iK)681

20 '.0798724

8.5316009
8.5361780
8.5407501
8.5453173
8.5498797
8.^544372
8.5589899
8.5635377
8.5680807

.001610306
.001607717
.001605136
.001602564
.001600000
.001597444
.001594896
.001592357
.001589825

630
631
632
633
634
635
636
637
638
639

396900
398161
399424
41)0689
401956
403225
404496
405769
407044
408321

250047000
251239591
252435968
253636137
254840104
256047875
257259456
258474853
259694072
260917119

25.0998008
25.1197134
25.1396102
25.1594913
25.1793566
25.1992063
25.2190404
25.2388589
25.2586619
25.2784493

8.5726189
8.5771523
8.5816809
8.5862047
8.5907238
8.5952380
8.5997476
8.6042525
8.6087526
8.6132480

.001587302
.001584786
.001582278
.001579779
.001577287
.001574803
.001572327
.001569859
.001567398
.001564945

640
641
642
643
644
645
646
647
648
649

409600
410881
412164
413449
414736
416025
417316
418609
419904
421201

262144000
263374721
264609288
265847707
267089984
268336125
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
8.6401226
8.6445855
8.6490437
8.6534974
8.6579465

.001562500
.001560062
.001557632
.001555210
.001552795
.001550388
.001547988
.001545595
.001543210
.001540832

650

651
652
653
654
655

657
658
659

422500
423801
425104
426409
427716
429025
430336
431649
432964
434281

274625000
275894451
277167808
278445077
279726264
281011375
282800416
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
667
668
669

4&5600
436921
438244
439569
440896
442225
443556
444889
446224
447561

287496000
288804781
290117528
291434247
292754944
294079625
295408296
296740963
298077632
299418309

25.6904652
25.7099203
25.7293607
25.7487864
25.7681975
25.7875939
25.8069758
25.8263431
25.8456960
25.8650343

8.7065877
8.7109827
8.7153734
8.7197596
8.7241414
8.7285187
8.7328918
8.7372604
8.7416246
8.7459846

.001515152
.001512859
.001510574
.001508296
.001506024
.001503759
.001501502
.001499250
.001497006
.001494768

670
671
672
673
674
675
676
677
678
679

448900
450241
451584
452929
454276
455625
456976
458329
459684
461041

300763000
302111711
303464448
304821217
306182024
307546875
308915776
310288733
311665752
313046839

25.8843582
25.9036677
25.9229628
25.9422435
25.9615100
25.9807621
26.0000000
26.0192237
26.0384331
£6. 0576284

8.7503401
8.7546913
8.7590383
8.7633809
8.7677192
8.7720532
8.7763830
8.7807084
8.7a50296
8.7893466

.001492537
.001490313
.001488095
.001485884
.001483680
.001481481
.001479290
.001477105
.001474926
.001472754

680
681
682

462400
463761
465124

314432000
315821241
317214568

26.0768096
26.0959767
26.1151297

8.7936593
8.7979679
8.8022721

.001470588
.001468429
.001466276

141

TABLE X.— SQUARES, CUBES, SQUAttK ROOTS,

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

683

466489

318611987

26.1342667

8.8065722

.001464129

684

467856

320013504

26.1533937

8.8108681

.001401988

685

469225

321419125

26.1725047

8.8151598

.001459854

686

470596

322828856

26.1916017

8.8194474

.001457720

687

471969

324242703

26.2106848

8.8237307

.001455604

688

473344

325660672

26.2297541

8.8280099

.001453488

689

474721

327082769

26.2488095

8.8322850

.001451379

690

476100

328509000

26.2678511

8.8365559

.001449275

691

477481

329939371

26.2868789

8.8408227

.001447178

692

478864

331373888

26.3058929

8.8450854

.001445087

693

480249

332812557

26.3248932

8.8493440

.001443001

694

481636

334255384

26.3438797

8.8535985

.001440922

695

483025

335702375

26.3628527

8.8578489

.001438849

696

484416

337153536

26.3818119

8.8620952

.001436782

697

485809

338608873

26.4007576

8.8663375

.001434720

698

487204

340068392

26.4196896

8.8705757

.001432665

699

488601

• 341532099

26.4386081

8.8748099

.001430615

700

490000

343000000

26.4575131

8.8790400

.001428571

701

491401

344472101

26.4764046

8.8832661

.001426534

702

492804

345948408

26.4952826

8.8874882

.001424501

703

494209

347428927

26.5141472

8.8917063

.001422475

704

495616

348913664

26.5329983

8.8959204

.001420455

705

497025

350402625

26.5518361

8.9001304

.001418440

706

498436

351895816

26.5706605

8.9043306

.001416431

707

499849

353393243

26.5894716

8.9085387

.001414427

708

501264

354894912

26.0082694

8.9127369

.001412429

709

502681

356400829

26 6270539

8.9169311

.001410437

710

504100

357911000

26.6458252

8.9211214

.001408451

711

505521

359425431

26.6645833

8.9253078

.001406470

712

506944

360944128

26.6833281

8.9294902

.001404494

713

508369

362467097

26.7020598

8.9336687

.001402525

714

509796

363994344

26.7207784

8.9378433

.001400560

715

511225

365525875

26.7394839

8.9420140

.001398601

716

512656

367061696

26.7581763

8.9461809

.001396648

717

514089

368601813

26.7768557

8.9503438

.001394700

718

515524

370146232

26.7955220

8.9545029

.001392758

719

516961

371694959

26.8141754

8.9586581

.001390821

720

518400

373248000

26.8328157

8.9628095

.001388889

721

519841

374805361

26.8514432

8.9009570

001386903

722

521284

376367048

26.870057:

8.9711007

.001385042

723

522729

37793:3067

26.8886593

8.9752406

001383126

724

524176

379503424

26.9072481

8.9793766

.001381215

725

525625

381078125

26.9258240

8.9835089

.001379310

726

527076

382657176

26.9443872

8.9870373

.001377410

727

528529

384240583

26.9629375

8.9917020

.001375516

728

529984

385828352

26.9814751

8.9958829

.001373026

729

531441

387420489

27.0000000

9.0000000

.001371742

730

532900

389017000

27.0185122

9.0041134

.001369803

731

534361

390617891

27.0370117

9.0082229

.001307989

732

535824

392223168

27.05549&5

9.0123288

.001306120

733

537'289

393832837

27.0739727

9.0164309

.001364256

734

538756

395446904

27.0924344

9.0205293

.001302398

735

540225

397065375

27.1108834

9.0246239

.001360544

736

541696

398688256

27.1293199

9.0287149

.001358696

737

543169

400315553

27.1477439

9.0328021

.001356852

738

544644

401947272

27.1601554

9.0368857

.001355014

739

546121

403583419

27.1845544

9.0409655

.001353180

740

547600

405224000

27.2029410

9.0450419

.001351351

741

549081

4068(39021

27.2213152

9.0491142

.001349528

742

550564

408518488

27.2390709

9.0531831

.001347709

743

552049

410172407

27.2580203

9.0572482

.001345895

744

553536

411830784

27.27'63034 | 9-0613098

.001344086

142

CUBE HOOTS, AKD

No.

Squares.

Cubes.

Square
Hoots.

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

450189749

27.3678644

9.0815631

.001335113

750

562500

421875000

27.3861279

9.0856030

.001333333

751

564001

423564751

27.4043792

9.0896392

.Oi 1331558

752

565504

425259008

27.4226184

9.0936719

-.001329787

753

567009

426957777

27.4408455

9.0977010

.001328021

754

568516

428661064

27.4590604

9.1017265

.001326260

755

570025

430368875

27.4772633

9.1057485

.001324503

756

571536

432081216

27.4954542

9.1097669

.001322751

757

573049

433798093

27.5136330

9.1137818

.001321004

758

574564

435519512

27.5317998

9.1177931

.001319261

759

576081

437245479

27.5499546

9.1218010

.001317523

7GO

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

27.7668868

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

47'0910952

27.8926514

9.1972897

.001285347

779

606841

472729139

27.9105715

9.2012286

.001283697

780

608400

474552000

27.9284801

9.2051641

.001282051

781

609961

476379541

27.9463772

9.2090962

.001280410

782

611524

478211768

27.9642629

9.2130250

.001278772

783

613089

480048687

27.9821372

9.2169505

.001277139

784

614656

481890304

28.0000000

9.2208726

.001275510

785

616225

483736625

28.0178515

9.2247914

.001273885

786

617796

485587656

28.0356915

9.2287068

.001272265

787

619369

487443403

28.0535203

9.2326189

.001270648

788

620944

489303872

28.0713377

9.2365277

.001269036

789

622521

491169069

28.0891438

9.2404333

.001267427

790

624100

493039000

28.1069386

9.2443355

.001265823

791

625681

494913671

28.1247222

9.2482344

.001264223

792

627264

496793088

28.1424946

9.2521300

.001262626

793

628849

498677257

28.1602557

9.2560224

.001261034

794

630436

500566184

28.1780056

9.2599114

.001259446

795

632025

502459875

28.1957444

9.2637973

.001257862

796

633616

504358336

28.2134720

9.2676798

.001256281

797

635209

506261573

28.2311884

9.2715592

.001254705

798

636804

508169592

28.2488938

9.2754352

.001253133

799

638401

510082399

28.2665881

9.2793081

.001251564

800

640000

512000000

28.2842712

9.2831777

.001250000

801

641601

513922401

28.3019434

9.2870440

.001248439

802

643204

515849608

28.3196045

9.2909072

.001246883

803

644809

517781627

28.3372546

9.2947671

.001245330

804

646416

519718464 28.3548938

9.2986239

.001243781

805

648025

521660125 28.3725219

9.3024775

.001242236

806

649636

523606616 28.3901391

9.3063278

.001240695

143

TABLE X. — SQUARES, CUBES, SQUARE ROOTS,

No.

Squares.

Cubes.

Square
Roots.

Cube Boots.

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

813

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

i 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

9.3£60952

.001221001

820

672400

551368000

28.6356421

9.3599016

.001219512

821

674041

553387661

28.6530976

9.3637049

.001218027

822

675684

555412248

28.6705424

9.3675051

.001216545

823

677329

557441767

28.6879766

9.3713022

.001215067

824

678976

559476224

28.7054002

9.3750963

.001213592

825

680625

561515625

28.7228132

9.3788873

.001212121

826

682276

563559976

28.7402157

9.3826752

.001210654

827

683929

565609283

28.7576077

9.3864600

.001209190

828

685584

567663552

28.7749891

9.3902419

.001207729

829

687241

569722789

28.7923601

9.3940206

.001206273

830

688900

571787000

28.8097206

9.3977964

.001204819

831

690561

573856191

28.8270706

9.4015691

.001203369

832

692224

5759303G8

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

9.5ia3699

.001161440

862

743044

640503928

29.3598365

9.5170515

.001160093

863

744769

642735647

29.3768616

9.5207303

.001158749

864

746496

644972544

29.3938709

9. 5244063

.001157407

865

748225

647214625

29.4108823

9 5280794

.001156069

866

749956

649461896

29.4278779

9.5317497

.001154734

867

751689

651714363

29.4448637

9.5354172

.001153403

868

753424

653972032

29.4618397

9.5390818

,001152074

144

CUBE ROOTS, AN!) RECIPROCALS.

No.

Squares.

Cubes.

Square
Roots.

Cube Roots.

Reciprocals.

869

755161

656234909 29.4788059

9.5427437

.001150748

870

756900

658503000 29.4957624

9.5464027

.001149425

871

758641

660776311 1 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

29.7657521

9.6045696

.001128668

887

786769

697864103

29.7825452

9.6081817

.001127396

888

788544

700227072

29.7993289

9.6117911

.001126126

889

790321

702595369

29.8161030

9.6153977

.001124859

890

792100

704969000

29.8328678

9.6190017

.001123596

891

793881

707347971

29.8496231

9.6226030

.001122334

892

795664

709732288

29.8663690

9.6262016

.001121076

893

797449

712121957

29.8831056

9.6297975

.001119821

894

799236

714516984

29.8998328

9.6333907

.001118568

895

801025

716917375

29.9165506

9.6369812

.001117318

896

802816

719323136

29.9332591

9.6405690

.001116071

897

804609

721734273

29.9499583

9.6441542

.001114827

898

806404

724150792

29.9666481

9.6477367

.001113586

899

808201

726572699

29.9833287

9.6513166

.001112347

900

810000

729000000

30.0000000

9.6548938

.001111111

901

811801

731432701

30.0166620

9.6584684

.001109878

902

813604

733870808

30.0333148

9.6620403

.001108647

903

815409

736314327

30.0499584

9.6656096

.001107420

904

817216

738763264

30.0665928

9.6691762

.001106195

905

819025

741217625

30.0832179

9.6727403

.001104972

906

820836

743677416

30.0998339

9.6763017

.001103753

907

822649

746142643

30.1164407

9.6798604

.001102536

908

824464

748613312

30.1330383

9.6834166

.001101322

909

826281

751089429

30.1496269

9.6869701

.001100110

910

828100

753571000

30.1662063

9.6905211

.001098901

911

829921

756058031

30.1827765

9.6940694

.001097695

912

831744

758550528

30.1993377

9.6976151

.001096491

913

833569

761048497

30.2158899

9.7011583

.001095290

914

835396

763551944

30.2324329

9.7046989

.001094092

915

837225

766060875

30.2489669

9.7082369

.001092896

916

839056

768575296

30.2654919

9.7117723

.001091703

917

840889

771095213

30.2820079

9.7153051

.001090513

918

842724

773620632

30.2985148

9.7188354

.001089336

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

9.7399634

.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

799178752

30.4630924

9.7539979

.001077586

929

863041

801765089

•30.4795013

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 .001072961

933

870489

812166237

30.5450487

9.7714845 .001071811

934

872356

814780504

30.5614136

9.7749743 .001070664

935

874225

817400375

30.5777697

9.7784616 ! .001069519

936

876096

820025856

30.5941171

9.7819466 .001068376

937

877969

822656953

30.6104557

9.7854288 I .001067236

938

879844

825293672 •

30.6267857

9.7889087 ! .001060)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

30.7083051

9.8062711

.001060445

944

891136

841232384

tt). 7245830

9.8097362

.001059322

945

893025

843908625

30.7408523

9.8131989

.001058201

946

894916

846590536

30.7571130

9.8166591

.001057082

947

896809

849278123

30.7733651

9.8201169

.001055966

948

898704

851971392

30.7896086

9.8235723

.001054852

949

900601

854670349

30.8058436

9.8270252

.001053741

950

602500

857375000

30.8220700

9.8304757

1001052632

951

904401

860085351

30.8382879

9.8339238

.001051525

952

906304

862801408

30.8544972

9.8373695

.001050420

953

908209

865523177

30.8706981

9.8408127

.001049318

954

910116

868250664

30.8868904

9.8442536

.001048218

955

912025

870983875

80.9030743

9.8476920

.001047120

956

913936

873722816

30.9192497

9.8511280

.001046025

957

915849

876467493

30.9354166

9.8545617

.001044932

958

917764

879217912

30.9515751

9.8579929

.001043841

959

919681

881974079

30.9677251

9.8614218

.001042753

960

921600

884736000

30.9838668

9.8648483

.001041667

961

923521

887503681

31.0000000

9.8682724

.001040583

962

925444

890277128

31.0161248

9.8716941

.001039501

963

927369

893056347

31.0322413

9.8751135

.001038422

964

929296

695841344

31.0483494

9.8785305

.001037344

965

931225

898632125

31.0644491

9.8819451

.001036269

966

933156

901428696

31.0805405

9.8853574

.001035197

967

935089

904231063

31.0966236

9.8887673

.001034126

968

937024

907039232

31.1126984

9.8921749

.001033058

969

938961

909853209

31.1287648

9.8955801

.001031992

970

940900

912673000

31.1448230

6. 8989830

.001030928

971

942841

915498611

31.1608729

9.9023835

.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

97'5

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

9498G2087

31.3528308

9.9430092

.001017294

984

968256

952763904

31.3687743

9.9463797

.001016260

985

970225

955671625

31.3847097

9.9497479

.001015228

986

972196

958585256

31.4006369

9.9531138

.001014199

987

974169

961504803

31.4165561

9.9564775

.001013171

988

976144

964430272

31.4324673

9.9598389

.001012146

989

978121

967361669

31.4483704

9.9631981

.001011122

900

980100

970299000

31.4642654

9.9665549

.001010101

991

982081

973242271

31.4801525

9.9699095

.001009082

992

984064

976191488

31.4960315

9.9732619

.001008065

I

146

CUBE ROOTS, AND 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

991026973

31.5753068

9.9899900

.001003009

998

996004

994011992

31.5911380

9.9933289

.001002004

999

998001

997002999

31 .6069613

9.9966656

.001001001

1000

1000000

1000000000

31.6227766

10.0000000

.001000000

1001

1002001

1003003001

31.6385840

10.0033322

.0009990010

1002

1004004

1006012008

31.6543836

10.0066622

.0009980040

1003

1006009

1009027027

31.6701752

10.0099899

.0009970090

1004

1008016

101248064

31.6859590

10.0133155

.0009960159

1005

1010025

1015075125 31.7017349

10.0166389

.0009950249

1006

1012036

1018108216

31.7175030

10.0199601

.0009940358

1007

1014049

1021147'343

31.7332633

10.0232791

.0009930487

1003

1016064

1024192512 j 31.7490157

10.0265958

- .0009920635

1009

1018081

1 027243729 31 . 7647603

10.0299104

.0009910803

1010

1020100

1030301COO 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

31.9687347

10.0728020

.0009784736

1023

1046529

1070599167

31.9843712

10.0760863

.0009775171

1024

1048576

1073741824

32.0000000

10.0793684

.0009765625

1025

1050625

1076890625

32.0156212

10.0826484

.0009756098

1026

1052676

1080045576

32.0312348

10.0859262

.0009746589

1027

1054729

1083206683

32.0468407

10.0892019

.0009737098

1028

1056784

1086373952 •

32.0624391

10.0924755

.0009727626

1029

1058841

10S9547389

32.0780298

10.0957469

.0009718173

1030

1060900

1092727000

32.0936131

10.0990163

.0009708738

1031

1062961

1095912791

32.1091887

10.1022835

.0009699321

1032

1065024

1099104768

32.1247568

10.1055487

.00 9689922

1033

1067089

1102302937

32.1403173

10.1088117

.0009680542

1034

1069156

1105507304

32.1558704

10.1120726

.0009671180

1035

1071225

1108717875

32.1714159

10.1153314

.0009661836

1036

1073296

1111934656

32.1869539

10.1185882

.0009652510

1037

1075369

1115157653

32.2024844

10.1218428

.0009643202

1038

1077444

1118386872

32.2180074

10.1250953

.0009633911

1039

1079521

1121622319

32.2335229

10 .1283457

.0009624639

1040

1081600

1124864000

32.2490310

10.1315941

.0009615385

1041

108:3681

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

32.3728281

10.1575062

.0009541985

1049

1100401

1154320649

32.3882695

10.1607359

.0009532888

1050

1102500

1157625000

32.4037035

10.1639636

.0009523810

1051

1104601

11609:35651

32.4191301

10.1671893

.0009514748

1052

1106704

1164252608

32.4345495

10.1704129

.0009505703

1053

1108809

1167575877

32.4499615

10.1736344

.0009496676

1054

1110916 i 1170905464

32.4653662

10.1768539

.0009487666

147

TABLE XI. — LOGARITHMS OF NUMBERS.

No.

100 L. 000.]

.No. 109 L. 040.

N.

0

1

2

8 4

6

6

7

8

9

Diff.

100

000000

0434

0868

1301 1734

2166

2598

3029

3461

3891

432

1

4321

4751

5181

5609 6038

6466

6894

7321

7748

8174

426

2

8600

9026

9451

9876

0300

0724

1147

1570

1993

2415

A£>A

3

012837

3259

3680

4100 4521

4940

5360

5779

6197

6616

420

4

7033

7451

7868

8284 8700

9116

9532

9947

0361

0775

416

5

021189

1603

2016

2428 2841

3252

3664

4075

4486

4896

412

6

5306

5715

6125

6533 6942

7350

7757

,8164

8571

8978

408

9384

9789

0195

0600 1004

1408

1812

2216

2619

3021

404

8

033424

3826

4227

4628 5029

5430

5830

6230

6629

7028

400

7426

7825

8223

8620 9017

9414

981 1

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

12

9.9

173.2

216.5

259

8

H

)3.1

346.4

389.7

432

43.2

86.4

14

9.6

172.8

216.0

259

2

ft

)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

43.0

86.0

12

9.0

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

2i

)9.6

342.4

385.2

427

42.7

85.4

12

8.1

170.8

213.5

256

2

21

)8.9

341.6

384.3

426

42.6

85.2

127.8

170.4

213.0

255.6

298.2

340.8

383.4

425

42.5

85.0

127:5

170.0

212.5

255.0

297.5

340.0

382.5

424

42.4

84.8

127.2

169.6

212.0

254.4

2<

J6.8

339.2

381.6

423

42.3

84.6

12

6.9

169.2

211.5

253

8

2<

)6.1

338.4

380.7

422

42.2

84.4

126.6

168.8

211*0

253

2

295.4

337.6

379.8

421

42.1

84.2

12

6.3

168.4

210.5

252

6

2<

W.7

336.8

378.9

420

42.0

84.0

126.0

168.0

210.0

252.0

294.0

336.0

378.0

419

41.9

83.8

12

5.7

167.6

209.5

251

4

2<

)3.3

335.2

377.1

418

41.8

83.6

125.4

167.2

209.0

250.8

292.6

334.4

376.2

417

41.7

83.4

12

5.1

166.8

208.5

250

2

2<

W.9

333.6

375.3

416

41.6

83.2

"124.8

166.4

208.0

249.6

2<

W.2

332.8

374.4

415

41.5

83.0

124.5

166.0

207.5

249.0

290.5

332.0

373.5

414

41.4

82.8

124.2

165.6

207.0

248.4

2\

39.8

331.2

372.6

413

41.3

82.6

12

3.9

165.2

206.5

247

8

2

39.1

330.4

371.7

412 1 41.2

82.4

123.6

164.8

206.0

247

2

a

38.4

329.6 370.8

411

41.1

82.2

12

3.3

164.4

205.5

246

6

2!

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

12

2.7

163.6

204.5

245

4

2!

36.3

327.2 368.1

408 40 8 81.6

122.4

163.2

204.0

244

8

2!

35.6

326.4 367.2

407

40.7 81.4

12

2.1

162.8

203.5

244

2

21

34.9

325.6 366.3

406 40.6 81.2

121.8

162.4

203.0 243

6

21

34.2

324.8 365.4

405 i 40.5 81.0

121.5

162.0

202.5

243

0

283.5

324.0 | 364.5

404

40.4

80.8

121.2

161.6

202.0

242.4

2!

32.8

323.2 363.6

40?

40.3 80.6

12

SO. 9

161.2

201.5 241

8

23

32.1

322.4 362.7

402

40.2

80.4

120.6

160.8

201.0

241

2

281.4

321.6 i 361.8

401

40.1

80.2

IS

X).3

160.4

200.5

240

6

a

30.7

320.8 ! 360.9

40(

) i 40.0

80-C

IS

0.0

160.0

200.0

240

.0

21

30.0

320.0 360.0

391

) 39.9 79.8

119.7

159.6

199.5

239

4

279.3

319.2 359.1

39*

5

39. H ! 79. C

11

9.4

159.2

199.0

238

8

g

78.6

318.4 ! 358.2

397 39.7 79.4

119.1

158.8

198.5

238.2

277.9

317.6 - 357.3

39(

> 3».<i 79.*

1]

8,8

158.4

198.0

287

6

2

77.2

316.8 356.4

395 i 39.5 79.0 118.5

158.0 197.5 237.0 276.5 3160 355.5

148

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

HKM

OOJTQ

9MA

OOfi

3

053078

3463

3846

4230 4613

4996

5378

5760

6142

6524

OoO

383

4

6905

7286

7666

8046 8426

8805

9185

9563

9942

0320

379

5

060698

1075

1452

1829 2206

2582

2958

3333

3709

4083

376

6

4458

4832

5206

5580 5953

6326

6699

7071

7443

7815

373

7

8186

8557

8928

9298 9668

AAOQ

H4O7

ftrr'fi

.

8

071882

2250

2617

2985 3352

uuoo
3718

4085

4451

1145
4816

5182

366

9

5547

5912

6276

6640 7'004

7'368

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

237.0
236.4

276.5
275.8

316.0
315.2

356.5

354.6

393

39.3

78.6

11

7.9

157.2

196.5

235

.8

275.1

314.4

353.7

392

39.2

78.4

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

273.7

312.8

351.9

390

39.0

78.0

11

7.0

156.0

195.0

234

.0

273.0

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

116.4

155.2

194.0

232

.8

271.6

310.4

349.2

387

38.7

77.4

11

6.1

154.8

193.5

232

.2

270.9

309.6

348.3

386

38.6

77.2

115.8

154.4

193.0

231

.6

270.2

308.8

347.4

385

38.5

77.0

115.5

154.0

192.5

231

.0

269.5

308.0

346.5

384

38.4

76.8

115.2

153.6

192.0

230

.4

268.8

307.2

345.6

383

38.3

76. (

11

4.9

153.2

191.5

229

.8

268.1

306.4

344.7

382

38.2

76.4

114.6

152.8

191.0

229.2

267.4

305.6

343.8

381

38.1

76.2

11

4.3

152.4

190.5

228

.6

266.7

304.8

342.9

38C

38.0

76.0

11

4.0

152.0

190.0

228

.0

266.0

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

264.6

302.4

340.2

377

37.7

75.4

113.1

150.8

188.5

226.2

263.9

301.6

339.3

376

37.6

75.2

11

2.8

150.4

188.0

225

.6

263.2

300.8

338.4

375

37.5

75.0

112.5

150.0

187.5

225.0

262.5

300.0

337.5

374

37.4

74.8

112.2

149.6

187.0

224

.4

261.8

299.2

336.6

373

37.3

74.6

111.9

149.2

186.5

223.8

261.1

298.4

335.7

37$

1

37.2

74.4

11

1.6

148.8

186.0

223

.2

260.4

297.6

334.8

371

37.1

74.2

111.3

148.4

185.5

222.6

259.7

296.8

333.9

37C

1

37.0

74. C

11

1.0

' 148.0

185.0

222

.0

259.0

296.0

333.0

36<

1

36.9

73.8

110.7

147.6

184.5

221

.4

258.3

295.2

332.1

36£

;

36.8

73.6

11

0.4

147.2

184.0

220

.8

257.6

294.4

331.2

S67

36.7

73.4

110.1

146.8

183.5

220.2

256.9

293.6

330.3

36(

>

36.6

73.2

1(

)9.8

146.4

183.0

219

.6

256.2

292.8

329.4

365

36.5

73.0

109.5

146.0

182.5

219.0

255.7

292.0

328.5

364

(

36.4

72.8

109.2

145.6

182.0

218.4

254.8

291.2

327.6

36£

5

36.3

72. e

1(

)8.9

145.2

181.5

217

.8

254.1

290.4

326.7

362

36.2

72.4

108.6

144.8

181.0

217.2

253.4

289.6

325.8

361

36.1

72.2

1(

)8.3

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. e

1

1(

)7.7

143.6

179.5

215

.4

251.3 287.2

••j23 1

35?

]

35.8

71. e

1(

W.4 143.2 179.0

214

.8

250.6 286.4

322^2

357

35.7

71.4

107.1 ! 142 8 178.5

214

.2

249.9

285.6

321.3

35(

)

35.6

71.2

106.8 142.4

178.0

213

.6

249.2 284.8

320.4

149

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 120 L. 079.]

[No. 134 L. 130.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

120

079181

9543

9904

2067

1

0266

0626

I 0987

1347

1707

2426

360

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

son

352

4

093422

3772

4122

4471

4820

5169

5518

5866

6215

6562

349

5

6910

7257

7'604

7951

8298

8644

8990

9335

9681

0026

346

6

100371

0715

1059

1403

1747

2091

2434

2777

3119

3462

343

7

3804

4146

4487

4828

5169

5510

5851

6191

6531

6871

341

8 7210

7549

7888

8227

8565

8903

9241

957'9

9916

0253

338

9

110590

0926

1363

1599

1934

2270

2605

2940

3275

3009

335

130

3943

4277

4611

4944

5278

5611

5943

6276

6608

6940

333

7271

7603

7934

8265

8595

8926

9256

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

93G8

9690

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

a5.4

70.8

ioe

.2

141.6

177.0

212.4 24

7.8

283.2

318.6

353

35.3

70.6

105

.9

14}. j

176.5

211.8 24

7.1

282.4

317.7

35.2

70.4

105.6

140.8

176.0

211.2 246.4

281.6

316.8

351

35.1

70.2

105

.3

140.4

175.5

210.6 24

5.7

280.8

315.9

ar>o

35.0

70.0

105.0

140.0

175.0

210.0 245.0

280.0

315.0

349

34.9

69.8

104

.7

139.6

174.5

209.4 24

4.3

279.2

314.1

348

34.8

69.6

104.4

139.2

174.0

208.8 243.6

278.4

313.2

347

34.7

69.4

104

.1

138.8

173.5

208.2 24

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 i 241.5

276.0

310.5

344

34.4

68.8

103.2

137.6

172.0

206.4

240.8

275.2

309.6

343

34.3

68.6

102

.9

137.2

171.5

205.8

24

[).l

274.4

308.7

342

34.2

68.4

102.6

136.8

171.0

205.2

239.4

-273.6

307.8

341

34.1

68.2

102

.3

136.4

170.5

204.6

3.7

272.8

306.9

310

34.0

68.0

102.0

136.0

170.0

204.0

2a

3.0

272.0

300. 0

339

as. 9

67.8

101

.7

135.6

169.5

203.4

23

r.3

271.2

305.1

888

33.8

67.6

101

.4

135.2

169.0

202.8

236.6

270.4

304.2

337

33.7

67.4

101

.1

134.8

168.5

202.2

23

3.9

269.6

303.3

ase

33.6

67.2

100.8

134.4

168.0

201.6

235.2

268.8

302.4

ass

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

LS3.6

167.0

200.4

233.8

267.2

300.6

333

33.3

66.6

99

.9 133.2

166.5

199.8

2a

5.1

266.4

299.7

332

33.2

66.4

99

.6 132.8

166.0

199.2

23,

2.4

265.6

298.8

asi

33.1

66.2

99.3

132.4

165.5

198.6

231.7

264.8

297.9

330

33.0

66.0

99

.0

132.0

165.0

198.0

23

1.0

264.0

297.0

329

32.9

65.8

98.7

131.6

164.5

197.4

230.3

263.2

296.1

328

32.8

65.6

98

.4

131.2

164.0

196.8

221

).6

262.4

295.2

327

32.7

65.4

98

.1

130.8

163.5

196.2

22*

261.6

294.3

326

32.6

65.2

97

.8

130.4

163.0

195.6

228.2

260.8

293.4

325

32.5

65.0

97.5

130.0

162.5

195.0

227.5

260.0

292.5

324

32.4

64.8

97.2

129.6

162.0

194.4

226.8

259.2

291.6

323

32.3

64.6

96

.9

129.2

161.5 193.8

22(

1. 1

s

258.4

290.7

322

32.2

64.4

96

.6

128.8 161.0 193.2

225.4

257.6

289.8

150

TAtfLK XI.— LOGARITHMS OF NUMBERS.

No. 135 L. 130.]

[No. 149 I,. 175.

N.

0

1

2

8

4

5

6

7

8 9

Diff.

135

130.334

0655

0977

1298

1619

1939 i 2260

2580

2900 3219

321

6

3539

3858

4177

4190

4814

5133 j 5451

5709

0080 6403

318

7

6721

7037

7354

7671

7987

8303 1 8018

8934

9249 9564

316

g

9879

0194 |

0508

0822

1136

1450 1703

2076

2389 2702

314

9

143015

3327

3039

3951

4263

4574

4885

5196

5507 5818

311

140

6128

6438

6748

7058

7367

7676 7985

8294

8003 8911

309

j

9219

9527

9835

0142

0449

0756

1063

1370

1076 1982

307

0

152288

2594

2900

3205

3510

3815

4120

4424

4728 5032

305

3

5336

5640

5943

6246

6549

6853

7154

7457

7759 8001

303

4

8362

8664

8965

9266

9567

9868

0168

0469

0769 1068

OA1

5

161368

1667

1967

2266

2564

2863

3161

3460

3758 4055

oul

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.

1

2

3

4

5

6

7

8

9

an | 33.1 64. a

96.3

128.4

160.5

192.

6

224.7

256.8

288.9

;;•-'>> 3-j.o 64.0

96

0

128.0

160.0

192.

0

224.0

256.0

288.0

319 31.9 63.8

95

7

]

27.6

159.5

191.

4

22

3.3

255.2

287.1

31R 31.8 ! 63.6

95

4

•

127.2

159.0

190.

8

22

2.6

251.4

286.2

317

31.7 63.4

95

1

126.8

158.5

190.

2

221.9

253.6

285.3

316 31.6 63.2

94

8

]

126.4

158.0

189.

0

22

1.2

252.8

284.4

315 I 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

]

[25.6

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 i 62.2

93

3

124.4

155.5

186.

6

217.7

248.8.

279 9

310

31.0 62.0

93

0

]

24.0

155.0

186.

0

21

7.0

248.0

279 !o

309

30.9 61.8

92

7

123.6

154.5

185.

4

216.3

247.2

278.1

308 j 30.8 61.6

92.4

123.2

154.0

184.

8

215.6

240,4

277.2

307 l 30.7 61.4

92

1

]

122.8

153.5

184.

2

21

4.9

245.6

276.3

306 30.6 ! 61.2

91

8

]

22.4

153.0

183.

6

21

4.2

244.8

275.4

305

30.5 61.0

91

5

122.0

152.5

183.

0

213.5

244.0

274 ,5

304

30.4 60.8

91

2

]

21.6

152.0

182.

4

21

2.8

243.2

273.6

303

30.3 j ,60.6

i 90

9

121.2

151.5

181.

8

212.1

242.4

27\J 7

302

30.2 60.4

! 90

6

120.8

151.0

181.

2

211.4

241.0 271.8

301

30.1 60.2

90.3

120.4

150.5

180.

6

210.7 j 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

]

[19.6

149.5

179.

4

2f

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

1

18.8

148.5

178.

0

2C

7.9

237.6 207.3

296

29.6 59.2

88

8

]

18.4

148.0

177.

6

2C

7.2

236.8 200.4

295

29.5 59.0

I 88

5

118.0

147.5

177.

0

206.5

236.0 205.5

294

29.4 , 58.8

88

2

17.6

147.0

176.

4

2C

5.8 235.2 264.6

293

29.3 58.6

i 87

9

117.2

146.5

175.

8

205.1 234.4 263.7

292

29.2 : 58.4

87

6

116.8

146.0

175.

0

24.4 233.6 262.8

291

29.1

58.2

87

3

116.4

145.5

174.

6

203.7 232.8 261.9

290

29.0 58.0

i 87

0

116.0

145.0

174.

0

203.0 232.0 261.0

289

2S 9 57.8

. 86

7

15.6

144.5

173.

4

2(

2.3 231.2 260.1

288

28.8 57.6

8fi

4

]

15.2

144.0

172.

8

2(1

1.6 2M0.4 259.2

287

28.7

57.4

86

1

114.8

143.5

172.

2

200.9 229.6 258.3

286

28.6

57.2

85

8

114.4

143.0

171.

6

200.2 228.8 257.4

J51

TABLE XI. — LOGAKITHMS OF NUMBERS.

No. 150 L. 176.]

[No. 169 L. 230.

N.

0

*

2

3

4

5

6

7

8

9

Diff.

150

176091 6381

6G70

6959

7248

7536

7825

8113

8401

8689

289

8977 9264

9552 9839

0126

0413

0699

no««

1272

1 ViS

007

2

181844

2129

2415

2700

2985

3270

3555 ! 3839

4123

4407

mot

285

3

4691

4975

5259

5542

5825

6108

6391 6674

6956

1 7239

283

7521 7803

8084

8366

8647

8928

9209

9490

9771

0051

9Q1

5

'190332 0612

0892

1171

1451

1730

2010

2289

2567

! 2846

279

6

3125 3403

3681

3959

4237

4514

4792

5069

5346

5623

378

r<

5900 6176

6453

6729

7005

7281

7556

7832

8107

8382

276

g

8657

8932

9206

9481

9755

0029

0303

fW"""'

nOfrA

9

201397

1670

1943

2216

2488

2761

3033

3305

UOOU

3577

i 11*4

3848

#74

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 1 •! <M

1^&£

Ifi'vi

1Q91

9R7

3

212188

2454

2720

2986

3252

3518

3783

4049

1004

4314

IMSl

4579

DPOif

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 i 250.2

277

27.7

55.4

83.1

110.8

138.5

166.2

193.9

221.6 I 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

2-15.7

272

27.2

54.4

81.6

108.8

136.0

163.2

190.4

217.6

244.8

271

27.1

54.2

81.3

108.4

135.5

162.6

189.7

216.8

243.9

270

27.0

54.0

81.0

108.0

135.0

162.0

189.0

216.0 ! 243.0

269

26.9

53.8

80.7

107.6

134.5

161.4

188.3

215.2 ! 242.1

268

26.8

53.6

80.4

107.2

134.0

160.8

187.6

214.4 2-11.2

267

26.7

53.4

80.1

106.8

133.5

160.2

186.9

213.6 240.3

266

26.6

53.2

79.8

106.4

133.0

159.6

186.2

212.8

239.4

265

26.5

53.0

79.5

106.0

132.5

159.0

185.5

212.0

238.5

264

26.4

52.8

79.2

105.6

132.0

158.4

184.8

211.2 237.6

263

26.3

52.6

78.9

105.2

131.5

157.8

184.1 !

210.4

236.7

262

26.2

52.4

78.6

104.8

131.0

157.2

209.6

235.8

261

26.1

52.2

78.3

104.4

130.5

156.6

182 '.7

208.8

234.9

260

26.0

52.0

78.0

104.0

130.0

156.0

182.0

208.0

234.0

259

25.9

51.8

77.7

103.6

129.5

155.4

181.3

207.2

233.1

! 258

25.8

51.6

77.4

103.2

129.0

154.8

180.6 ;

206.4

232.2

257

25.7

51.4

77.1

102.8 128.5 154.2

179.9

205.6

231.3

256

25.6

51.2

76.8

102.4 128.0 153.6

179.2 i

204.8

230.4

255

i — ,. ,

25.5

51.0

76.5

102.0 1£7.5 153.0

178.5 !

204.0

229.5

152

TABLE XI. — LOGARITHMS Otf

Xo. 170 L. 230.1 [No. 189 L. 278.

N.

0

1

2

3

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 i 253

2

5528

5781

(5033

6285

6537

6789

7041

7292

7544

7795

252

3

8046

8297

8548

8799

9049

9299

9550

Q«nn

4

2541

2790

249

240549

0799

1043

1297

1546

1795

2044

2293

5

3038

3286

3534

3782

4030

4277

4525

4772

5019

5266

248

6

5513

5759

6006

6252

6499

6745

6991

7237

7482

7728

246

7

7973

8219

8464

8709

8954

9198

9443

9687

9932

0176

245

8

250420

OGG4

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

9513

9746

9980

0213

0446

0679

0912"

1144

1377

IftftQ

ooq

7

271842

2074

2306

2538

2770

3001

3233

3464

3696

3927

!B9Q

232

' 8

4158

4389

4620

4850

5081

5311

5542

5772

6002

6232

230

9

6462

6692

6921

7151

7380

7609

7838

8067

8296

8525

229

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

7

8

9

255

25.5

51.0

76.5

102.0

127.5

153.0

178.5

204.0

229 .'5

254

25.4

50.8

76.2

101.6

127.0

152.4

177.8

203.2

228.6

253 j 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

1&5.6

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

1&3.2

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

i-)..

TABLE XI. — LOGARITHMS OF NUMBERS.

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

N.

0

1

2

3

4

5

6

7

8

9

Diff.

190

278754

8982

9211

9439

9667

9895

0133

0351

0578

0806

228

1

281033

1261

1488

1715

1942

2109

2390

2033

2849

3075

237

2

3301

3527

3753

3979

4305

4431

4050

4882

5107

5332

236

3

5557

5782

6007

63.33

6456

6681

6UJ5

7130

7354

7578

225

4

7802

8036

8349

8473

8096

8930

9143

9366

9589

9812

223

5

390035

0357

0480

0703

0935

1147

1369

1591

1813

2034

222

6

3356

2478

2699

2920

3141

3303

3584

3804

4025

4346

221

7

44G6

4687

4907

5127

5347

5507

5787

6007

6330

6446

220

8

6665

6884

7104

7333

7542

7761

7979

8198

8416

8635

219

g

8853

9071

9389

9507

9735

9943

0161

0378

0595

0813

218

200

301030

1247

1464

1681

1898

2114

2331

2547

2764

2980

217

1

3196

3412

3638

3844

4059

4375

4491

4706

4931

5136

216

2

5351

5566

5781

5990

6311

6435 0039

6854

7068

7383

215

3

7496

7710

7934

8137

8351

8504 8778

8991

9304

9417

213

9630

9843

0056

0308

0481

0093

0906

1118

1330

1542

212

5

311754

1900

2177

3-389

2GOO

2813

3033

3334

3445

3656

211

6

3867

4078

4389

4499

4710

1 4930

5130

5340

5551

5760

210

7

5970

6180

6390

0599

6809

i 7018

7337

7436

7646

7854

209

8

8003

8372

8481

8069

8898

! 9106

9314

9532

9730

9938

208

9

330140

0354

0503

07CO

0077

! 1184

1391

1598

1805

2012

207

210

2319

2426

2033

28H9

3046

1 3353

3458

3665

3871

4077

206

1

4382

4488

4094

4899

5105

! 5310

5516

5731

5936

6131

205

2

6336

6541

6745

6950

7155

! 7359

7563

7707 7973

8176

204

3

8380

8583

8787

8991

9194

9398

9001

9805

0008

0211

203

4

330414

0617

0819

1033

1335

1437

1030

1833

3034

3336

202

PROPORTIONAL PARTS.

Diff.

1

2

3

4

5

6

7

8

9

225

22.5

45.0

67.5

90.0

113.5

135.0

157.5

180.0

202.5

224

22.4

44,8

67.2

89.6

113.0

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

300.7

222

22.2

44.4

66.6

88.8

111.0

133.2

155.4

177.6

199.8

221

23.1

44.2

66.3

88.4

110.5

132.6

154.7

176.8

198.9

220

23.0

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

153.6

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

139.0

150.5

172.0

193.5

214

21.4

43.8

64.2

85.6

107.0

138.4

149.8

171.2

193.6

213

21.3

43.6

63.9

85.2

100.5

137.8

149.1

170.4

191.7

212

21.2

43.4

63.6

84.8

100.0

137.2

148.4

169.6

190.8

211

21 ..1

43 3

63.3

84.4

105.5

126.6

147.7

168.8

189.9

210

21.0

43'.0

63.0

84.0

105.0

13G.O

147.0

168.0

189.0

209

20.9

41.8

62.7

83.6

104.5

125.4

146.3

107.2

188.1

208

20.8

41.6

63.4

83.2

104.0

124.8

145.6

160 4

187.3

307

20.7

41.4

63.1

82.8

103.5

134.2

144.9

165.0

186.3

306

20.6

41.3

G1.8

83.4

103.0

133.6

144.3 104.8

185.4

305

20.5

44.0

C1.5

83.0

103.5

133.0

143.5

164.0

184.5

304

20.4

40.8

61.2

81.6 103.0

123.4

143.8 163.2

183.0

303 j 20.3

40.6

60.9

81.3 101.5

131.8

142.1 163.4

183.7

203 i 20.2

40.4

GO. 6

<0.8 101.0

131.3

141.4 161.6

181.8

154

TABLE XI. — LOGAK1THMS OF NUMBERS.

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

N.

0

1

2

8

4

6

6

7

8

9

Diff.

215

332438

2640

2842

3044

3246

3447

3649

3850

4051

4253

202

6

4454

4055

4856

50f,7

5257

5458

5658

5859

6059

6~>00

201

7

6460

6660

6860

7060

7260

7459

7059

7858

8058

8257

200

8

8456

8656

8855

9054

9253

9451

9650

9849

0047

0246

199

9

340444

0642

0841

1039

1237 J ,:>5

1632

1830

2028

2225

198

220

2423

2620

2817

3014

321.3

3^09

3606

3802

3999

4196

197

1

4392

4589

4785

4981

5118

53',4

5570

6766

5962

61"57

196

2

6353

6549

6744

6939

7135

•jaao

7525

7720

7915

8110

195

3

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

3532

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

8125

8316

8506

8696

8886

9076

9266

9456

9646

190

g

9835

0025

0215

0404

0593

0783

0972

1161

1350

1539

1GQ

230

361728

1917

2105

2294

2482

2671

2859

3048

3236

3424

ioy
188

1

3612

3800

3988

4176

4363

4551

4739

4926

6113

5301

188

2

5488

5675

5862

6049

6236

6423

6610

6796

6983

7169

187

3

7356

7542

7729

7915

8101

8287

8473

8659

8845

9030

186

4

9216

9401

9587

9772

9958

0143

0328

0513

0698

0883

185

5

371068

1253

1437

1622

1806

11)91

2175

2360

2544

2728

184

6

2912

3096

3280

3404

3047

3831

4015

4198

4382

4565

184

7

4748

4932

5115

5298

5481

5664

5846

6029

6212

6394

183

8

6577

6759

6942

7124

7306

7488

7670

7852

8034

8216

182

9

8398

8580

8761

8943

9124

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

185

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

1086

126 7

144.8

102.9

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

101 1

155

TABLE XX. — LOGARITHMS OK NUMBERS.

No. 240 L. 380.]

LNo. 269 L. 431.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

240
1
2
3
4
5

6

8
9

250
1

2

3

4
5

6

7

8
9

260
1
2
3

4
5
6

7
8
9

380211
2017
3815
5606
7390
9166

0392
2197
3995
5785
7568
9343

0573
2377
4174
5964
7746
9520

0754
2557
4353
6142
7924
9698

0934
2737
4533
6321
8101
9875

1115
2917
4712
6499
8279

1296
3097
4891
6677
8456

1476
3277
5070
6856
8634

1656
3456
5249
7034
8811

1837
3636
5428

7212
8989

181
180
179

178

178

177
176
176
175
174
173

173
172
171
171
170
169

169
168
167

167
166
165

165
164
164
163
102
102

161

0051
1817
3575
5326
7071

8808

0228
1993
3751
5501
7245

8981

0405
2169
3926
5676
7419

9154

0582
2345
4101
5850
7592

9328

0759
2521
4277
6025
7766

9501

390935
2697
4452
6199

7940
9674

1112
2873
4627
6374

8114
9847

1288
3048
4802
6548

8287

1464
3224
4977
6722

8461

1641
3400
5152
6896

8634

0020
1745
3464
5176

6881
8579

0192
1917
3635
5346
7051
8749

0365
2089
.3807
5517
7221
8918

0538
2261
3978

5688
7391
9087

0711
2433
4149
5858
7561
9257

0883
2605
4320
6029
7731
9426

1056
2777
4493
6199
7901
9595

1228
2949
4663
6370
8070
9764

401401
3121
4834
6540
8240
9933

411620
3300

4973
6641
8301
9956

1573
3292
5005
6710
8410

0102
1788
3467

5140

6807
8467

0271
1956
3635

5307
6973
8633

0440
2124
3803

5474
7139

8798

0609
2293
3970

5641
7306
8964

\ 0777
: 2461
4137

5808
7472
9129

0946
2629
4305

5974
7638
9295

1114
2796
4472

6141

7804
9460

1283
2964
4639

6308
7970
9625

1451
3132

4806

6474
8135
9791

0121
1768
3410
5045
6674
8297
9914

0286
1933
3574
5208
6836
8459

0451
2097
3737
5371
6999
8621

0616
2261
3901
5534
7161
8783

0781
i 2426
4065
5697
7324
1 8944

0945
2590
4228
5860
7486
9106

1110
2754
4392
6023
7648
9268

1275

2918
4555
6186
7811
9429

1439

3082
4718
6349
7973
9591

421604
3246
4882
6511
8135
9752
43

0075 0236 0398

0559

0720

0881 | 1042 | 1203

PROPORTIONAL PARTS.

Diff. 1

2 3

4

5

6

7 8

9

78

76
75
74
73

71

ro

109
168
167
166
1G5
164
163
162
161

17.8
7.7
7.6
7.5
7.4
7.3
7.2
7.1
7.0

16.9
16.8
16.7
16.6
16.5
16.4
16.3
16.2
16.1

a5.6 53.4
35.4 53.1
.35.2 52.8
35.0 52.5
34.8 52.2
34.6 51.9
34.4 51.6
34.2 51.3
34.0 51.0

33.8 50.7
33.6 50.4
33.4 50.1
33.2 49.8
33.0 49.5
32.8 49.2
32.6 48.9
32.4 48.5
32.2 48.3

71.2
70.8
70.4
70.0
69.6
69.2
68.8
68.4
68.0

67.6
67.2
66.8
66.4
66.0
65.6
65.2
64.8
64.4

89.0

as. 5

88.0
87.5
87.0
86.5
86 0
85.5
85.0

84.5
84.0
83.5
83.0
82.5
82.0
81.5
81.0
80.5

106.8
106.2
105.6
105.0
104.4
103.8
103.2
102.6
102.0

101.4
100.8
100.2
99.6
99.0
98.4
97.8
97.2
96.6

124.6 142.4
123.9 141.6
123.2 140.8
122.5 140.0
121.8 139.2
121.1 138.4
120.4 137.6
119.7 136.8
119.0 136.0

118.3 135.2
117.6 134.4
116.9 133.6
116.2 132.8
115.5 132.0
114.8 131.2
114.1 130.4
113.4 129.6
112.7 128.8

160.2
159.3
158.4
157.o
156.6
155.7
154.8
153.9
153.0

152.1
151.2
150.3
149.4
148.5
147.6
146.7
145.8
144.9

156

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 270 L. 431.] [No. 209 L. 476.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

270

431364

1525

1685

1846

2007

2167

2328

2488

2619 2809

161

1

2969

3130

8290

3450

3610

3770

3930

4090

4249

4409

160

2

4569

4729

4888

5048

5207

5367

5526

5685

5844

6004

159

3

6163

6322

6481

6640

6799

6957

7116

7275

7433

7592

159

4

7751

7909

8067

8226

8384

8542

8701

8859

9017 9175

158

5

9333

9491

9648

9806

9964

0122

0279

0437

0594 flTKo

158

6

440909

1066

1224

1381

1538

1695

1852

2009

2166

2323

157

7

2480

2637

2793

2950

3106

3263

3419

3576

3732

3889

157

8

4045

4201

4357

4513

4669

4825

4981

5137

5293

5449

156

9

5604

5760

5915

6071

6226

6382

6537

6692

6848

7003

155

280

7158

7313

7468

7623

7778

7933

8088

8242

8397

8552

155

j

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

2859

3012

3165

153

4

3318

3471

3624

3777

3930

4082

4235

4387

4540

4692

153

5

4845

4997

5150

5302

5454

5606

5758

5910

6062

6214

152

6

6366

6518

6670

6821

6973

7125

7276

7428

757?

7731

152

7

7882

8033

8184

8336

8487

8638

8789

8940

9091

9242

151

g

9392

9543

9694

9845

9995

0146

0296

0447

0597

0748

151

9

460898

1048

1198

1348

1499

1649

1799

1948

2098

2248

150

290

2398

2548

2697

2847

2997

3146

3296

3445

3594

3744

150

1

3893

4042

4191

4340

4490

4639

4788

4936

5085

5234

149

2

5383

5532

5680

5829

5977

6126

6274

6423

6571

6719

149

3

6868

7016

7164

7312

7460

7608

7756

7904

8052

8200

148

4
g

8347
9822

8495
9969

8643

8790

8938

9085 9233

9380

9527

9675

148

0116

Of63

0410

0557

0704

0851

0998

1145

147

6

471292

1438

1585

1732

1878

2025

2171

2318

2464

2610

146

7

2756

2903

3049

3195

as4i

3487

3633

3779

3925

4071

146

8

4216

4362

4508

4653

4799

4944

5090

5235

5381

5526

146

9

5671

5816

5962

6107

6252

6397 6542

6687

6832

6976

145

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

80.4

104.3

119.2

134.1

148

14.8

29.6

44.4

59.2

74.0

88.8

103.6

118.4

ias.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

2s!o

42.0

56.0

70.0

84.0

98.0

112.0

126.0

157

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 300 L. 477.1

[No. 339 L. 5C1.

K.

0

1

2

S

4

5

6

7

8

9

Diff.

800

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

8
9

330

1

2
3
4
5
6

8
9

477121
8566

7266
8711

7411

8855

0294
1729
3159
4585
6005
7421
8833

7555

8999

7700
9143

7844
9287

7989
9431

8133
9575

8278
9719

1156
2588
4015
5437
C855
6269
9677

8422
9663

145
144

144
143
143
142
142
141
141

140

140
139
139
139
138
138

137
137
136
136

136
135
135

134
134
133
133
133
132
132

131

131
131
130
130
129
129
129

128
128

480007
1443
2874
4300
5721
7138
8551
9958

491362
2760
4155
5544
6930
8311
9687

0151
1586
3016
4442
5863
7280
8692

0438
1872
330;J
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
(5714
8127
9537

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

0099
1470
2837
4109

5557
6911
8260
9606

0520

1922
3319
4711

6099
7483
8862

0236
1607
2973
4335

5693
7046
8395
9740

1081
2418
P.750
5079
6403
7724

9040

0661

2062
j 3458
4850
6238
7621
1 8999

0374
1744

8109
4471

5828
I 7181
1 8530
9874

0801

2201
3597
4989
6376
7759
9137

0511
1880
3246
4C07

5964
7316
8G64

0941

2341
3737
5128
6545
7897
9275

0648
2017
3382
4743

6009
7451
8799

1061

2481
3876
5267
6653
6035
9412

0785
2154
3518
4878

6234
7586
6934

1222

2621
4015
5406
C791
6173
C550

C922
2291
£655
C014

6370
7721
S068

501059
2427
3791

5150
6505
7856
9203

510545
1883
3218
4548
5874
7196

8514
9828

1196
2564
3927

5286
6640
7991
9&37

1333
2700
4063

5421
6776
8126
9471

0009
1349
2084
40*6
6344
6C68
7987

9303

0615
1922
3226
4526
5822
7114
8402
9687

0968

0143
1462
2818
4149
5476
6800
8119

9434

0745
2053
3356
4656
5951
7243
8531
9815

1096

0277
1616
2951
4282
5609
6932
8251

9566

0876
2163
3486
4765
608*
7372
8GGO
9943

1223

0411
1750

£084
4415

5741
7064
6S82

9697

1C07
2314
3616
4915
6210

7:01

6768

0679
2017
3351
4681
6006
7328

8646
9959

0813
2151
3484
4813
6139
7460

8777

0947
2284
3617
4946
6271
7592

8909

I 1215
2551
3883
5211
6535
7855

9171

0484
1792
3096
! 4396
: 5693
; 69a5
8274
9559

0840

0090
1400
2705
400(5
5304
f>598
7H88
9174

0221

1530
2835
4136
5434
6727
6016
<):X)2

0353
1661
2966
4266
5563
6856
8145
9430

521138
2444
3746
5045
6a39
7630
8917

1269
2575
3876
5174
6469
7759
9045

0072
1351

530200 ! 0328

0456

0584

0712

PROPORTIONAL PARTS.

Biff. 1

2 3

4

5

6

7 8

9

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

27.8 41.7
27.6 41.4
27.4 41.1
27.2 40.8
27.0 40.5
• 26.8 40.2
26.6 39.9
26.4 39.6
26.2 89.3
26.0 89.0
25.8 38.7
25.6 38.4
25.4 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

97.3 111.2
96.6 110.4
95.9 109.6
95.2 108.8
94.5 108.0
93.8 107.2
93.1 106.4
92.4 105.6
91.7 104.8
91.0 104.0
90.3 103.2
89.6 1 102.4
88.9 101.6 |

125.1
124.2
123.3
122.4
121.5
120.6
119.7
118.8
117.9
317.0
116.1
115.2
114.3

TABLE XI.-— LOGARITHMS OF NUMBERS.

No. 34 j L. L:A.]

[No. 379 L. 579.

N.

0

1

o

3

4 5

6

7

8

9

Diff.

340
1
2
3
4
5
6

8

9

350
1
2
3
4

5
6

7
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
2bb2
4153
5421
6685
7945
9202

1734
3009
4280
5547
6811
8071
9327

1S02
3130
4407
5074
6937
8197
9452

1990
3204
4534
5800
7003
8322
9578

2117
3391
4001
5927
7189
8448
9703

2245
3518

47'87
6053
7315

8574
9829

2372
3645
4914
6180
7441
8699
9954

1205
2452
3096

4936
6172
7405
86:35
9861

2500
3772
5041
0306
7567
8825

2627
3899
5167
6432
7693
8951

128
127
127
126
120
126

125
125
125
124

124
124
123
123

123
122
122
121
121
121

120
120
120

119
119
119
119
118
118
118

117

117
117
116
116
116
115
115
115
114

0079
1330
2576
3820

5060
6296
7529
8758
9984

0204
1454
2701
3944

5183
6419
7652
8881

540329
1579
2825

4068
5307
6543
7775

9003

0455
1704
2950

4192
5431
6666
7898
9126

0580
1829
3074

4316
5555
6789
8021
9249

0705
1953
3199

4440
5078
6913
8144
9371

0830
20,78
3323

4504
5802
7036
8207
9494

0955

2203
1 3447

I 4688
! 5925
! 7159
! 8389
9016

1080
2327
3571

4812
0049
7282
8512
97'39

0106
1328
2547
3762
4973
6182

7387
8589
9787

550228
1450
2668
3883
5094

6303
7507
8709
9907

01551
1572
2790
4004
5215

6423
7627

8829

0473
1694
2911
4126
5336

6544
7748
8948

0595
1816
3033
4247
5457

6664

7'808
9008

0717
1938
3155

4308
5578

6785

7988
9188

0&40
2060
3276
4489
i 5099

6905
8108
9308

0962
2181
3398
4010
6820

7026
8228
9428

1084
2303
3519
4731
5940

7146

8349
9548

1206
2425
3640
4852
6061

7267
8469
9067

0026
1221
2412
3600
4784
5966
7144

8319
9491

0146
1340
2531
3718
4903
6084
7262

8436
9008

0205
1459
2050
3837
5021
6202
7379

8554

9725

0385
1578
2709
3955
5139
6320
7497

8671
9S42

i 0504
1098
2887
4074
5257
6437
7014

8788
9959/

0024
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
4606
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
9669

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
21.8 37.2
24.6 36.9
24 4 30.0
24.2 36.3
24 0 30 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 89.6
76.2 88.9
75.6 88.2
75.0 87.5
74.4 86 8
73.8 86.1
73.2 &5.4
72.6 84.7
72.0 84.0
71.4 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

159

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 380. L. 579.]

[No. 414 L. 617.

N.

0

1

2

3

4

5

G

7

8

9

Diff.

380

579784

9898

0012

0126

0241

0355

0469 0583

0697

0811

114

1

580925

1039

1153

1267

1381

1495 1608 1'

22

1836

1950

2,

2063

2177

2291

2404

2518

[ 2031

2745

2858

2972

3085

3

3199

3312

3426

9

3052

! 3705

3879

3<

>92

4105

4218

4

4331

4444

4557

46'

0

4783

4896

5009 5

122

5235

5348

113

5

5461

5574

5080

5799

5912

6024

6137 6250

6302

6475

6

6587

6700

0812

69X

5

7037

7149

7262 7

JT4

7480

7599

7

7711

7823

7935

8047

8160

8272

8384

8490

8008

8720

112

8

8832

8944

9050

91C

7

9279

9391

9503

9

$15

9720

9838

g

9950

0061

0173

0284

0396

0507

0619

0730

0812

0953

390

591065

1176

1287

1399

1510

1621

1732

1843

1955

2060

1

2177

2288

2399

25]

0

2021

2732

2843

2*

)54

3004

3175

111

2

3286

3397

3508

3618

3729

3840

3950

4001

4171

4282

3

4393

4503

4014

47i

^4

4834

4945

5055

5

105

5270

5380

4

5496

5000

5717

5&

5937

6047

0157

6

,»67

6377

6187

5
6

6597
7695

6707
7805

6817
7914

6927
8024

7037
8134

7146
8243

8853

,7306
8402

7470
8572

7580
8081

110

7

8791

8900

9009

91]

9

9228

9337

9446

9

350

9005

9774

g

9883

9992

0101

0210

0319

0428

0537 1 0640

0755

0804

109

9

600973

1082

1191

1.299

1408

1517

1025

1

734

1843

1951

400

2060

2169

2277

2386

2494

2603

2711

2819

2928

3036

1

3144

3253

3301

3469

3577

3086

3794

3902

4010

4118

108

0

4226

4334

4442

45.

>0

4058

4766

4874

4

J82

SON)

5197

3

5305

5413

5521

5028

5730

5844

5951

6059

6100

0274

4

6381

6489

6596

67(

)4 6811

6919

7026

7

133

7241

7348

5

7455

7562

7009

77

7 i 7884

7991

8098

8205

8312

8419

107

6

8526

8633

8740

88-

17 ' 8954

9001

9107

9

474

9381

9188

7

9594

9701

9808

9014 ,
nnoi

0128

0234

O

} 11 n,i i<7

0554

8

610660

0767

0873

0979

1080

1192

1298

1405

1511

1017

9

1723

1829

1936

2042

2148

2254

2300

2

'•Go

2572

2078

106

410

2784

2890

2996

31(

>2

3207

3313

3419 3525

3(330

3736

1

3842

3947

4053

41

)9

4264

4370

4475 4

581

4080

4792

2

4897

5003

5108

52

3

5319

5424

5529 5

>34

5740

5845

3

5950

6055

6100

6265

0370

6476

6581 6080

6790

0895

105

4

7000

7105

7210

7315

7420

7525

7029 7

734

783J

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 1 94.4

106.2

117 11.7

23.4

35.1

46.8

58.5

70.2

81.9 93.6

105.3

110 11.0

23.2

34.8

46.4

58.0

69.6

81.2 92.8

104.4

115 11.5

23.0

34.5

46.0

57.5

69.0

80.5 92.0

103.5

114 11.4

22.8

34.2

45.6

57.0

68.4

79.8 91.2

102.0

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

66.0

77.7 88.8

99.9

110 11.0

ao

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

G5.4

70.3 87.2

98.1

108 10.8

21.6

32.4

43.2 54.0

64.8

75.0 80.4

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

95.4

105 10.5

21 .0

31.5

42.0 52.5 63.0

73.5 84.0

94.5

105 1 10.5

21.0

31.5

42.0 5X1.5

03.0

73.5 84.0

94.5

104 1 10.4

20.8

31.2

41.6

52.0

62.4

72.8 83.2

93.6

TABLE XT. — LOGARITHMS OF NUMBERS.

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

N.

0

1

2

3

4

6

6

7

8

9

Diff.

415
6

8
9

420
1
2
3
4
5
6

8
9

430
1
2
3
4
5
6

7
8
9

440
1

4
5

6

7
8
9

450
1
2
3
4
5
6
7

8
9

618048
9093

020130
1176
2214

3249

4282
5312
0340
7306
8389
9410

6:30428
1414
2457

3468

4477
5484
6488
7490
8489
9486

8153
9198

8257
9302

8302
9406

8466
9511

8571
9615

8676
9719

8780
9824

8884
9928

8989

0032
1072
2110
3146

4179
5210

G238
7203

8287
9308

105

104

103
102

101
100

99

98

97
96

95

0240

1280
2318

3353
4385
5415
6443
7468
8491
9512

0530
1545
2559

3569
4578
5584
6588
7590
8589
9586

0344
1384
2421

3456

4488
5518
6546
7571
8593
9013

0631
1647
2660

3670
4679

5685
6688
7690
8689
9686

0448
1488
2525

3559
4591
5021
OG48
7073
8695
9715

0733
1748
2701

3771
4779

5785
0789
7790
8789^
9785

0552
1592

2028

3003
4095
5724
0751
7775
8797
9817

0&35
18-19
2862

3872
4880
5886
0889
7890
8888
9885

0656
1695
2732

3766
4798
5827
6853
7878
8900
9919

~0936~
1951
2963

3973

4981
5986
6989
7990
8988
9984

0760
1799

2835

3869
4901
5929
6956
7980
9002

0021
1038
2052
3064

4074

5081
6087
7089
8090
9088

0864
1903
2939

3973
5004
6032
7058
8082
9104

0968
2007
3042

4076
5107
6135
7161
8185
9206

0123
1139
2153
3165

4175
5182
6187
7189
8190
9188

0224
1241
2255
3266

4276

5283
6287
7290
8290
9287

0326
1342
2356
3367

4376

5383
6388
7390
8389
9387

0084
1077
2069
3058

4044
5029
6011
6992
7969
8945
9919

0890
1859
2826

3791
4754
5715
6673
7629
8584
9536

0486
1434
2380

0183
1177
2168
3156

4143
5127
6110
7089
8067
9043

0016
0987
1956
2923

3888
4850
5810
6769
7725
8679
9631

0581
1529
2475

0283
1276
2UI7
3255

4242
5226
6208
7187
8165
9140

0113
1084
2053
3019

3984
4946
5906
6864
7820
8774
9726

0076
1623
2509

0382
1375
2366
3354

4340
5324
6306
7285
8262
9237

0210
1181
2150
3116

4080
5042
6002
6960
7916
8870
9821

0771
1718
2663

640481
1474

2405

3453
4439
5422
0404
7383
8300
9335

0581
1573
2563

3551
4537
5521
6502
7481
8458
9432

0405
1375
2343

3309
4273
5235
6194
7152
8107
9000

0011
0900
1907

0680
1672
2602

3050
4030
5619
6000
7579
8555
9530

0502
147'2
2440

3405
4369
5331
6290
7247
8202
9155

0106

1055
2002

0779
1771
27G1

3749
4734
5717
6698
7'676
8653
9627

0879
1871
2860

3847

4832
5815
6796
7774
8750
9724

0096
1666
2033

3598
4562
5523
6482
7438
8393
9346

0978
1970
2959

3946
4931
5913
6894
7872
8848
9821

0793
1762
2730

3695
4658
5619
6577
7534
8488
9441

G50308
1278
2246

3213
4177
5138
6098
7056
8011
8965
9916

600805
1813

0599
1509
2536

3502
4465
5427
6386
7343
8298
9250

0201
1150
2096

0296
1245
2191

0391
1339

2286

PROPORTIONAL PARTS.

Diff 1

234

5

6 7 8

9

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

21.0 315. 42.0
20 8 31 2 1 41 6
20 6 30 9 41.2
20 4 30 i 40.8
20 2 ?!0 3 40.4
20 0 30.0 40 0
19 8 29 7 39 6

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

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

94.5
93.0
92 7
91 8
90.9
90 0
89.1

161

TABLE XI. — LOGAKITHMS OF NUMBERS.

No. 460 L. 662.]

[No. 499 L. 698.

N

0

1

2

8

4

5

6

7

8

9

Diff.

460

662758 2852

2947

3041 3135

3230

3324 3418 3512

3607

1

3701 3795

3889

39

S3

4078

4172

4206 ; 4300 445

4

4548

2

4642 4736

4830

4924

5018

5112

5206 i 5299 5393

5487

94

3

5581 5675

5769

58

32

5956

6050

6143 ! 6237 j 633

1

6424

4

6518 ; 6612

6705

6799

6892

6986

7079 1 7173 7200

7360

5

7453 i 7546

7640

77

33

7826

7920

8013 ' 8106 | 819

9

8293

6

8386 8479

8572

86

35

8759

8852

8945 9038 ! 913

1

9224

9317

9410

9503

95

30

9689

9782

9875 9967

006

n

0153

93

8

670246 0339

0431

0524

0617

0710

0802

0895 0988

1080

9

1173

1265

1358

1451

1543

1636

1728

1821

1913

2005

470

2098

2190

2283

2375

2467

2560

2652

2744 i 2836

2929

1

3021

3113

3205

32

97

3390

3482

3574

3666 ! 375

8

385C

2

3942

4034

4126

4218

4310

4402

4494

4586 467

7

4769

92

3

4861

49.53

5045

51

37

5228

5320

5412

5503 559

5

5687

4

5778

5870

5962

60

53

6145

6236

6328

6419

651

1

6002

5

6694

6785

6876

69

6S

7059

7151

7242

7333

7424

7516

6

7607

7698

7789

78

SI

7972

i 8063

8154

8245

833

0

8427

7

8518

8609

8700

8791

8882

8973

9064

9155

9246

9337

91

8

9428

9519

9610

97

00

9791

: 9882

9973 !

! 0063

OlS-i

i

0245

9

680336

0426

0517

0607

0698

0789

0879 0970

1060

1151

480

1241

1332

1422

1513

1603

1693

1784

1874

1964

2055

1

2145

2235

2326

24

16

2506

i 2596

2686

2777

28t

7

2957

2

3047

3137

3227

3317

3407

3497

3587

3677

3767

3857

90

3

3947

4037

4127

42

17

4307

4396

4486

4576

466

6

4756

4

4845

49:35

5025

5114

5204

5294

5383

5473

5563

5652

5

5742

5831

5921

60

10

6100

6189

6279

6368

645

8

6547

6

6636

6726

6815

6904

6994

7083

7172

7261

7351

7440

7

7529

7618

7707

77

JO

7886

7975

8064

8153

824

2

8331

89

8 8420

8509

8598

8B

37

8776

8865

8953

9042

9131

9220

9 ! Q2HQ

9398

9486

95

75

9664

9753

9841

9930

001

n

0107

490

690196

0285

0373

0462

0550

0639

0728

0816

0905

0993

1

1081

1170

1258

1347

1435

1524

1612

1700

178

9

1877

2

1965

2053

2142

22

30

2318

2406

2494

2583

267

1

2759

3

2847

2935

3023

3111

3199

3287

3375

3463

3551

3039

88

4

3727

3815

3903

39

31

4078

4166

4254

4342

443

0

4517

5

4605

4693

4781

48

38

4956

5044

5131

5219

5307

5394

6

5482

5569

5657

57

14

5832

5919

6007

6094

618

2

6269

7

6356

6444

6531

66

18

6706

6793

6880

6968

705

5

7142

8

7229

7317

7404

74

n

7578

7665

7752

7839

792

6

8014

9

8100

8188

8275

8362

8449

8535

8622

8709

8796

8883

87

PROPORTIONAL PARTS.

Diff

1

2 3

4

5

6 7

8

9

98

9.8

19.6 29.4

39.2

49.0

58.8 68.6

78.4

88.2

97

9.7

19.4 29.1

38.8

48.5

58.2 67.9

77.6

87.3

96

9.6

19.2 28.8

38.4

48.0

57.6 67.2

76.8

86.4

95

9.5

19.0 28.5

38.0

47.5

57.0 66.5

76.0

85-. 5

94

9.4

18.8 28.2

37.6

47.0

56.4 65.8

75.2

84.6

93

9.3

18.6 27.9

37.2

46.5

55.8 65.1

74.4

83.7

92

9.2

18.4 27.6

36.8

46.0

55.2 64.4

73.6

82.8

91

9.1

18.2 27.3

36.4

45.5

54.6 63.7

72.8

81.9

90

9.0 18.0 27.0

36.0

45.0

54.0 63.0

72.0

81.0

89

8.9 17.8 26.7

35 6

44.5

53.4 62.3

71.2

80.1

88

8.8 1 17.6 26.4

35.2

44.0 52.8 61.6

70.4

79.2

8.7 1 17.4 26.1

34.8

43.5 i 52.2 00. 9

69.6

78:3

«fl 8.6 | 17.2 25.8

34.4

43.0 51.6 I 60.2

68.8

77.4

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
1

2
3

4
5
6

7
8
9

510

1
2

3
4
5
6
7
8
9

520
1
2
3
4

5
6

8
9

530
1
2
3

4
5
6

7

8
9

540
1
2
3
4

698970
9838

700704
1568
2431
3291
4151
5008
5864
6718

7570
8421
9270

9057
9924

9144

9231

9317 1

9404

9491

9578

9664

9751

0011
0877
1741
2603
3463
4322
5179
6035
6888

7740

8591
9440

0098
0963
1827
2689
3549
4408
5265
6120
6974

7826
8676
9524

0184 !
1050 i
1913
2775
3635
4494
5350 !
6206
7059

7911
8761 i
9609 ;

0271
1136
1999
2861
3721
4579
5436
6291
7144

7996
8846
9694

0358
1222
2086
2947
3807
4665
5522
6376
7229

8081
8931
9779

0444
1309
2172
3033
3893
4751
5607
6462
7315

8166
9015
9863

0531
1395
2258
3119
3979
4837
5693
6547
7400

8251
9100
9948

0617
1482
2344

3205
4065
4922

5778
6632
7485

8336
9185

86
85

84
83

82
81

80

0790
1654
2517
3377
4236
5094
5949
6803

7655
8506
9355

0033
0879
1723
2566
3407
4246
5084
5920

6754
7587
8419
9248

710117
0963
1807
2650
3491
4330
5167

6003
6838
7671
8502
9:331

720159
0986
1811
2634
3456

4276
5095
5912
6727
7541
8354
9165
9974

0202
1048
1892
2734
3575
4414
5251

6087
6921
7754

8585
9414

0287
1132
1976
2818
3659
4497
5335

6170

7004
7837
8668
9497

0371
1217
2060
2902
3742
4581
5418

6254
7088
7920
8751
9580

0456 '
1301 I
2144 !
2986 1
3826 i
4665
5502

6337
7171 |
8003 !

8834 i
9663

0490
1316
2140
2963

3784

4604
5422

6238
7053
7866
8678
9489

0540
1385
2229
3070
3910
4749
5586

6421
7254
8086
8917
9745

0625
1470
2313
3154
3994
4833
5669

6504
7338
8169
9000
9828

0710
1554
2397
3238
4078
4916
5753

6588
7421
8253
9083
9911

0794
1639

4162
5000
5836

6671
7504
8336
9165
9994

0077
0903
1728
2552
3374
4194

5013
5830
6646
7460
8273
9084
9893

0242
1068
1893
2716
3538

4358
5176
5993
6809
7625
8435
9246

0325
1151
1975
2798
3620

4440
5258
6075
6890
7704
8516
9327

0407
1233
2058
2881
3702

4522
5340
6156
6972
7785
8597
9408

0573
1398
2222
3045
3866

4685
5503
6320
7134
7948
8759
9570

0655
1481
2305
3127
3948

4767
5585
6401
7216
8029
8841
9651

0738
1563
2387
3209
4030

4849
5667
6483
7297
8110
8922
9732

0821
1646
2469
3291
4112

49bl

5748
6564
7379
8191
9003
9813

0055
0863
1669

2474
3278
4079
4880
5679

0136
0944
1750

2555
3358
4160
4960
5759

0217
1024
1830

2635
3438
4240
5040
5838

0298
1105
1911

2715
3518
4320
5120
5918

0378
1186
1991

2796
8598
4400
5200
5998

0459
1266

2072

2876
3679
4480
5279
6078

0540
1347
2152

2956
3759
4560
5359
6157

0621
1428
2233

3037
3839
4640
5439
6237

0702
1508
2313

3117
3919
4720
5519
6317

730782
1589

2394
8197
3999
4800
5599

PROPORTIONAL PARTS.

Diff. 1

234

5

678

9

87 8.7
86 8.6
85 ! 8.5
84 | 8.4

17.4 26 1 34 8
17.2 258 34.4
17.0 25 5 34.0
16.8 25 2 33.6

43 5 52 2 60.9 69 6
43 0 ! 51 6 60 2 i 68 8
425 i 510 59.5 680
42 0 60.4 58 8 67.2

78 3
77 4
76 5
75 6

163

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 545 L. 736.]

[No. 584 L. 767.

N.

0

1

o

8

4

5

v'

7

8

i 9 Diff.

545 736397

6476 6556

6635

6715 6795

6874

6954

7034 7113

6 i 7193

7272 7352

74C

51

7511 7590

767'0

7749

782<

) 7908

7 ! 7987

8067 8146

82$

{5

8305 8384

8463

8543

862

2 8701 !

8 I 8781

8860 8939

9018

9097 9177

9256

9335

941<

I 9493

9 9572

9651 9731

981

o

9889 ^ w>8

0047

0126

0205

0284

79

550 740363

0442 0521

0600

0678 0757

0836

0915

0994

1073

1

1152

1230 i 1309

13*

^8

1467 1546

1624

1703

178$

3

1860

2

1939

2018 1 2096

21"

'5

2254 2332

2411

2489

256*

3

2647

3

2725

2804 ' 2882

29f

>1

3039

3118

3196

3275

3353 3431

4

3510

3588 '• 3667

374

5

3823

3902

3980

4058

4131

i 4215

5

4293

4371 i 4449

4528

4606

4684

4762

4840

4919

4997

6

5075

5153 | 5231

53(

)9

5387

5465

5543

5621

569

1

5777

78

y

5*5

5933 , 6011

60*

«)

6167

6245

6323

6401

647

1

6556

8
9

6634
7412

6712

7489

6790
7567

6868
7645

6945

7722

; 7023 7101

7800 7878

7179
7955

7256
8033

7334
8110

560

8188

8266 8343

8421

8498

1 8576 8653

8731

8808

8885

1

8963

9040 9118

9195 9272

9350 9427

9504

9582

9659

2

9736

9814 9891

99(

jg

rwxis;

0123 0200

0277

4 fMO-1

3

750508

0586 i 0663

0740

0817

C894 0971

1048

IKXXt

1125

V*'J.l

1202

4

1279

1356

1433

15]

0

1587

16(J4 1741

1818

189

1972

^

5

2048

2125

2202

2279

2356

2433 2509

2586

2663

2740

<7

6

2816

2893

2970 3fr

17

3123

3200 3277

3353

343

)

3506

7

3583

3660

3736 3813

3889

3966 4042

4119

4195

4272

8

4348

4425

4501 45'

'8

4654

4730 4807

4883

496<

)

5036

9

5112

5189

5265 5341

5417

. 5494 , 5570

5646

5722

5799

570

5875

5951

6027 6103

6180

! 6256 6332

6408

6484

6560

1

6636

6712

6788 68<

)4

6940

7016 7092 7168

724

i

7320

76

2

7396

7472

7548 76*

.'4

7700

7775 7851 7927

800

3

8079

3

8155

8230

8306

8382

8458

8533 8609

8685

8761

4

8912

8988

9063

91.'

K

9214

9290 9366

9441

951

9592

5

9668

9743

9819

98'

H

QOTft

0045 0121

0196

027

->

0347

6

760422

0498

0573

0649

0724

0799 0875

0950

1025

1101

7

1176

1251

1326

14(

)2

1477

1552 1627

1703

177

J

1853

8

1928

2003

2078

21,

>3

2228

2303 2378

2453

252

1

2604

9

2679

2754

2829

2904 | 2978

3053 3128

3203

3278

3353

75

580

3428

3503

3578

3653 3727

3802 3877

3952

402

4101

1

4176

4251

4326 44(

X)

4475

4550 4624

4699

477

1

4848

2

4923

4998

5072 5147

5221

5296 5370

5445

5520

5594

3

5669

5743

5818

5892

5966

! 6041 6115

6190

6264

6338

4

•6413

6487

6562

6636

6710

6785 6859

6933

7007

7082

i

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 5'

r.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 5(

5.0

64.0

72.0

; 79 ! -.9

15.8 23.7

31.6

39.5

47.4 5J

>.3

63.2

71.1

"88

15.6 23.4

31.2

39.0

46.8 54.6

62.4

70.2

77 '"7

15.4 23.1

30.8

38.5

46.2 5.f

5.9

61.6

69.3

76 ~ 6

15.2 22.8

30.4

38.0

45.6 53.2

60.8

68.4

75 T5

15.0 225

30.0

37.5

45.0 5S

5.5

60.0

67.5

74 "".4

14 8 22.2

29.6 37.0 44.4 51.8

59.2

66.6

164

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 585 L. 767.] [No. 629 L. 799.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

585

767156

7230

7304

7379

7453

7627

7601

7675

7749

7823

6

7898

7972

8046

8120

8194

8268

8342

8416

8490

8564

74

8638

8712

8786

8860

8934

9008

9082

9156

9230

9303

g

9377

9451

9525

9599

9673

9746

9820

9894

9968

9

0778

770115

0189

0283

0336

0410

0484

0557

0631

0705

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

6

5246

5319

5392

5465

5538

5610

5683

5756

5829

5902

7

5974

6047

6120

6193

6265

6338

6411

6483

6556

6629

8

6701

6774

6846

6919

6992

7064

7137

7209

7282

7354

9

7427

7499

7572

7644

7717

7789

7862

7934

8006

8079

COO

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

O94^

3

780377

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

2759

2831

2902

2974

3046

3117

7

3189

3260

3332

3403

3475

3546

3618

3689

3761

3832

8

3904

3975

4046

4118

4189

4261

4332

4403

4475

4546

9

4617

4689

4760

4831

4902

4974

5045

5116

5187

5259

G10

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

Q

9581

9651

9722

9792

9863

9933

0004

0074

0144

0215

7

790285

0356

0426

0496

0567

0637

0707

0778

0848

0918

8

0988

1059 i 1129

1199

1269

1340

1410

1480

1550

1620

9

1691

1761

1831

1901

1971

2041

2111

2181

2252

2322

620

2392

2462

2532

2602

2672

2742

2812

2882

2952

3022

70

1

3092

3162

3231

3301

3371

3441

3511

3581

3651

3721

o

3790

3860

3930

4000

4070

I 4139

4209

4279

4349

4418

3

4488

4558

4627

4697

4767

1 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

7129

7198

7 7268 7337

7406

7475

7545

7614

7683

7752

7821

7890

8 7960 8029

8098

8167

8236

8305

8374

8443

8513

8582

9

8651

8720

8789

8858

8927

8996

9065

9134

9203

9272

69

PROPORTIONAL PARTS.

Diff. 1

224

5

678

9

75 ""5

15.0 22.5 30.0

37.5

45.0 52.5 60.0

67.5

74 ^.4

14.8 22.2 29.6

37.0

44.4 51.8 59.2

66.6

14.6 21.9 29.2

36.5

43.8 51.1 58.4

65.7

72 "'.2

14.4 21.6 28.8

36.0

43.2 50.4 57.6

64.8

71 l~. 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

165

TABLE XI. — LOGARITHMS OF NUMBERS.

, —
I To. G30 L. 799.] [No. 674 L. 829.

?•» .

o

1

2

3

4

6

6

7

8

9

Diff.

!<>30

799341

9409

9478

9547

9616

9685

9754

9823

9892

9961

1

800029

0098

0167

0236

0305

0373

0442

0511

0580

0648

2

0717

0786

0854-

0923

0992

1061

1129

1198

1266

1335

3

1404

1472

1541

1609

1678

1747

1815

1884

1952

2021

4

2089

2158

2226

2295

2363

2432

2500

2568

2637

2705

5

2774

2842

2910

2979

3047

3116

3184

3252

3321

3389

6

3457

3525

3594

3662

3730

3798

3867

3935

4003

4071

7

4139

4208

4276

4344

4412

4480

4548

4616

4685

4753

8

4821

4889

4957

5025

5093

5161

5229

5297

5365

5433

68

9

5501

5569

5637

5705

5773

5841

5908

5976

6044

6112

640

806180

6248

6316

6384

6451

6519

6587

6655

6723

6790

1

6858

6926

6994

7061

7129

7197

7264

7332

7400

7467

2

7535

7603

7670

7738

7806

7873

7941

8008

8076

8143

3

8211

8279

8346

8414

8481

8549

8616

8684

8751

8818

4

8886

8953

9021

9088

9156

9223

9290

9358

9425

9492

5

9560

9627

9694

9762

9829

9896

9964

0031

0098

0165

6

810233

0300

0367

0434

0501

0569

0636

0703

0770

0837

7

0904

0971

1039

1106

1173

1240

1307

1374

1441

1508

67

8

1575

1642

1709

1776

1843

1910

1977

2044

2111

2178

9

2245

2312

2379

2445

2512

2579

2646

2713

2780

2847

650

2913

2980

3047

3114

3181

3247

3314

3381

3448

3514

1

3581

3648

3714

3781

3848

3914

3981

4048

4114

4181

2

4248

4314

4381

4447

4514

4581

4647

4714

4780

4847

3

4913

4980

5046

5113

5179

5246

5312

5378

5445

5511

4

5578

5644

5711

5777

5843

5910

5976

6042

6109

6175

5

6241

6308

6374

6440

6506

6573

6639

6705

6771

6838

6

6904

6970

7036

7102

7169

7235

7301

7367

7433

7499

7

7565

7631

7698

7764

7830

7896

7962

8028

8094

8160

8

8226

8292

8358

8424

8490

8556

8622

8688

8754

8820

(\K

9

8885

8951

9017

9083

9149

9215

9281

9346

9412

9478

DO

660

9544

9610

9676

9741

9807

9873

9939

0004

0070

0136

1

820201

0267

0338

0399

0464

0530

0595

0661

0727

0792

2

0858

0924

0989

1055

1120

1186

1251

1317

1382

1448

3

1514

1579

1645

1710

1775

1841

1906

1972

2037

2103

4

2168

2233

2299

2364

2430

2495

2560

2626

2691

2756

5

2822

2887

2952

3018

3083

3148

3213

3279

3344

3409

6

3474

3539

3605

3670

3735

3800

3865

3930

3996

4061

7

4126

4191

4256

4321

4386

4451

4516

4581

4646

4711

ftR

8

4776

4841

4906

4971

5036

5101

5166

5231

5296

5361

OO

9

5426

5491

5556

5621

5686

5751

5815

5880

5945

6010

670

6075

6140

6204

6269

6334

6399

6464

6528

6593

6658

1

6723

6787

6852

6917

6981

7046

7111

7175

7240

7305

2

7369

7434

7499

7563

7628

7692

7757

7821

7886

7951

3

8015

8080

8144

8209

8273

8338

8402

8467

8531

8595

4

8660

8724

8789

8853

8918

8982

9046

9111

9175

9239

PROPORTIONAL PARTS.

!Diff 1

234

5

678

9

68 68

13 6 20 4 27 2

34 0

40 8 47 6 54 4

61 2

67 67

13 4 20.1 26 8

33 5

40 2 46 9 53 6

60 3

66 66

13.2 19 8 26.4

330

39 6 46 2 52 8

59 4

65 65

130 195 260

32.5

39 0 45 5 52 0

58 5

64 6.4

1£ 8 19.2 25 6

32.0

38.4 44 8 51 2

57 6

166

TABLE XI. — LOGARITHMS OF NUMBERS.

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

N.

0

1

2

8 4

6

6

7

8

9

Diff.

675

829304

9368

9432

9497

9561

9625

9690

9754

9818

9882

g

9947

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

27&4

2828

2892

2956

3020

3083

1

3147

3211

3275

3338

3402

3466

3530

3593

3657

3721

2

3784

3848

3912

3975

4039

4103

4166

4230

4294

4357

3

4421

4484

4548

4611

4675

4739

4802

4866

4929

4993

4

5056

5120

5183

5247

5310

5373

5437

5500

5564

5627

5

5691

5754

5817

5881

5944

6007

6071

6134

6197

6261

6

6324

6387

6451

6514

6577

6641

6704

6767

6830

6894

7

6957

7020

7083

7146

7210

7273

7336

7399

7462

7525

8

7588

7652

7715

7778

7841

7904

7967

8030

8093

8156

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

4106

4229

4291

4353

4415

9

4477

4539

4601

4664

4726

4788

4850

4912

4974

5036

700

5098

5160

5222

5284

5346

5408

5470

5532

5594

5656

62

1

5718

5780

5842

5904

5966

6028

6090

6151

6213

6275

2

6337

6399

6461

6523

6585

6646

6708

6770

6832

6894

3

6955

7017

7079

7141

7202

7264

7326

7388

7449

7511

4

7573

7634

7696

7758

7819

7881

7943

8004 8066

8128

5

8189

8251

8312

8374

8435

8497

8559

8620 8682

8743

6

8805

8866

8928

8989

9051

9112

9174

9235 9297

9358

7

9419

9481

9542

9604

9665

9726

9788

9849 9911

9972

8

850033

0095

0156

0217

0279

0340

0401

0462

0524

0585

9

0646

0707

0769

0830

0891

0952

1014

1075

1136

1197

710

1258

1320

1381

1442

1503 i

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 I 3577

3637

4

3698

3759

3820

3881

3941

4002

4063

4124

4185

4245

5

4306

4367

4428

4488

4549

4610

4670

4731

4792

4852

6

4913

4974

5034

5095

5156

5216

5277

5337

5398

5459

7

5519

5580

5640

5701

5761

5822

5882

5943

6t)03

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 1 48.0

54.0

167

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 720 L. 857.] [No. 7fo L. 883.

N.

0

1

2

3

4

6

8

7

S

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

9739

9799

9859

9918

9978

4

0038

0098

0158

0218

0278

5

860338 1 0398

0458

0518

0578

0637

0697

0757

0817

0877

6

0937

0996

1056

1116

1176

123<?

1295

1355

1415

1475

7

1534

1594

1654

1714

1773

1833

1893

1952

2012

2072

8

2131

2191

2251'

2310

2370

2430

2489

2549

2608

2668

9

2728

2787

2847

2906

2966

3025

3085

3114

3204

3263

730

3323

3382

3442

3501

3561

3620

3680

3739

3799

3858

1

3917

3977

4036

4096

4155

4214

4274

4333

4392

4452

2

4511

4570

4630

4689

4748

4808

4867

4926

4985

5045

3

5104

5163

5222

5282

5341

5400

5459

5519

5578

5637

4

5696

5755

5814

5874

5933

5992

6051

6110

6169

6228

5

6287

6346

6405

6465

6524

6583

C642

6701

6760

6819

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

8702

8821

8879

8938

8997

9056

9114

91V3

740

9232

9290

9349

9408

9466

9525

9584

9642

9701

9760

Q81R

9877

9935

9994

1

yolo

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

9096

9153

9211

92(58

9325

9383

9440

9497

9555

9612

9669

9726

9784

9841

9898

9956

0013

0070

0127

0185

9

880242

0299

0356

0413

0471

0528

0585

0642

0699

0756

760

0814

0871

0928

0985

1042

1099

1156

1213

1271

1328

1

1385

1442

1499

1556

1613

1670

1727

1784

1841

1898

1955

2012

2069

2126

2183

2240

2297

2354

2411

2468

57

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

234

5

678

9

59 5.9 11.8 17.7 23.6

29.5

35.4 41.3 47.2

53.1

58 5.8

11.6 17.4 23.2

29.0

34.8 40.6 46.4 52.2

57 5.7

11.4 17.1 22.8 -

28.5

34.2 39.9 45.6 51.3

56 5.6

11.2 16.8 22.4

28.0

33.6 39.2 44.8 | 50.4

108

TABLK XI. — LOGARITHMS OP LUMBERS.

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 «T(J5

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

S%5

9021

9077

9134

9190

9246

5

9302

9358

9414

9470

9526

9582

9638

9694

9750

9806

56

Q

9862

9918

9974

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

287'3

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

498£

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

8835

8890

8944

8999

9054

9109

9164

9218

3

92T3

9328

9:383

9437

9492

9547

9602

9656

9711

9766

4

9821

9875

9930

9985

0039

0094

0149

0203

0258

0312

5

9003G7

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

j 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

G

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

7787

7841

7895

9

7949

8002

8056

8110

8163

8217

8270

8324

8378

8431

PROPORTIONAL PARTS.

DiflP. 1

234

5

678

9

57 5.7

11.4 17.1 22.8

28.5

34.2 39.9 45.6

51.3

56 5.6

11.2 16.8 22.4

28.0

33.6 39.2 44.8

50.4

55 5.5

11.0 1 16.5 22.0

27.5

33.0 38.5 44.0

49.5

54 5.4

10.8 1 16.2 21.6

27.0

32.4 37.8 432

48.6

169

TABLE XI. — LOGARITHMS OF NUMBERS.

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

N.

0

1

2

3

4' 5

6

7

8

9

Diff.

810

908485

8539

8592

8646

8699

8753

8807

8860

8914

8967

1

9021

9074

9128

9181

9235

9289

9342 j 9396

9449

9503

9556

9610

9663

9716

9770

9823

9877 9930

9984

HAST

3

910091

0144

0197

0251

0304 !! 0358

0411 04C4

0518

0571

4

0624

0678

0731

0784

0838

0891

0944

0998

1051

1104

5

1158

1211

1264

1317

1371

1424

1477

1530

1584

1637

6

1690

1743

1797

1850

1903

1956

2009

2063

2116

2169

7

2222

2275

2328

2381

2435

2488

2541

2594

2647

2700

8

2753

2806

2859

2913

2966

3U19

3072

3125

3178

3231

9

3284

3337

3390

3443

3496

3549

3602

3655

"3708

3761

53

820

3814

3867

3920

3973

4026

4079

4132

4184

4237

4290

1

4343

4396

4449

4502

4555

4608

4660

4713

4766

4819

2

4872

4925

4977

5030

5083

5136

5189

5241 5294

5347

3

5400

5453

5505

5558

5611

5664

571t>

5769 5822

5875

4

5927

5980

6033

6085

6138

6191

6243

6296 6349

6401

5

6454

6507

6559

6612

6664

6717

6770

C822

6875

6927

6

6980

7033

7085

7138

7190

7243

7295

7348

7400

7453

7

7506

7558

7611

7G63

7716

7768

7820

7873

7925

7978

8

8030

8083

8135

8188

8240

8293

8345

8397

8450

8502

9

8555

8607

8659

8712

8764

8816

8869

8921

8973

9026

830

9078

9130

9183

9235

9287

9340

9392

9444

9496

9549

1

9601

9653

9706

9758

9810

9862

9914

9967

0019

0071

2

920123

0176

0228

0280

0332

0384

0436

0489 i 0541

0593

3

0645

0697

0749

0801

0853

0906

0958

1010 i 1062

1114

g»

4

1166

1218

1270

1322

1374

1426

1478

1530

1582

1634

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

6415

5467

5518

5570

5621

5673 5725

577'6

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

a549

8601

8652

8703

8754 8805

8857

9

8908

8959

9010

9061

9112

9163

9215

9266

9317

9368

850
j

9419
noon

9470
9981

9521

9572

9623

; 9674

9725

9776

9827

9879

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

234

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

170

TABLE XI. — LOGARITHMS OF NUMBERS.

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

N1

.
355

DMT.

931966

2017

2068

2118

2169

2220

2271

2322

2372

2423

6

2474

2524

2575

2626

2677

! 2727

2778

2829

2879

2930

7

2981

3031

3082

3133

3183

3234

3285

3335

3386

3437

8

3487

3538

3589

3639

3690

i 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

2

5507

5558

5608

5658

5709

5759

5809

5860

5910

5960

3

601*

6061

6111

6162

6212

6262

6313

6363

6413

6463

4

6514

6564

6614

6665

6715

6765

6815

6865

6916

6966

5

7016

7066

7116

7167

7217

i 7267

7317

7367

7418

7468

6

7518

7568

7618

7668

7718

7769

7819 7869

7919

7969

8019

8069

8119

8169

8219 ! 8269

8320 i 8370

8420

&170

50

8

8520

8570

8620

8670

8720 8770

8820 i 8870

8920

8970

9

9020

9070

9120

9170

92fO

9270

9320 9369

9419

9469

870

9519

9569

9615

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

2i>54

2603

2653

2702

2752

2301

2851

2901

2950

7

3000

3049

3090

3148

3198

1 3247

3297

3346

3396

3445

8 1 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

I 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
6

6943
7434

6992
7483

7041
7532

7090

7581

7139
7630

7189
7679

7238
7T28

8£

7336

7826

7385
7875

49

7

7924

7973

8022

8070

8119

8168

8217

8266

8315

8364

8

8413

8462

8511

8560

8608

8657

8706

8755

8804

8853

9

8902

8951

8999

9048

9097

9146

9195

9244

9292

9341

890
I

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

0851

0900

0949

0997

1046

1095

1143

1192

1240

1289

4

1338

1386

1435

1483

1532

1580

1629

1677

1736

1775

5

1823

1872

1920

1969

2017

2066

2114

2163

2211

2260

6

2308

2356

2405

24o3

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 0

50

5.0

10.0

15.0 20.0

25.0

30.0 35.0 40.0 45 0

49 4.y

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

171

TABLE 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

4966

5014

5062

5110 | 5158

2

5207

5255

5303

5351

5399

5447

5495

5543

5592

5640

3

5688

5736

5784

5832

5880

5928

5976

6024

6072

6120

4

6168

6216

6265

6313

6361

6409

6457

6505

6553

6601

5

6649

6697

6745

6793

6840

6888

6936

6984

7032

7080

48

6

7128

7176

7224

7272

7320

7368

7416

7464

7512

7559

7

7607

7655

7703

7751

7799

7847

7894

7942

7990

8038

8

8086

8134

8181

8229

8277

8325

8373

8421

8468

8516

9

8564

8612

8659

8707

8755

8803

8850

8898

8946

8994

910

9041

9089

9137

9185

9232

9280

9328

9375

9423

9471

1
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

0661

0709

0756

0804

0851

0899

4

0946

0994

1041

1089

1136

1184

1231

1279

1326

1374

5

1421

1469

1516

1563

1611

1658

1706

1753

1801

1848

6

1895

1943

1990

2038

2085

2132

2180

2227

2275

2322

7

2369

2417

2464

2511

2559

2606

2653

2701

2748

2795

8

2843

2890

2937

2985

3032

3079

3126

3174

3221

3268

9

3316

3363

3410

3457

3504

3552

3599

3646

3693

3741

920

3788

3835

3882

3929

3977

4024

4071

4118

4165

4212

1

4260

4307

4354

4401

4448

4495

4542

4590

4637

4684

2

4731

4778

4825

4872

4919

4966

5013

5061 5108

5155

3

5202

5249

5296

5343

5390

5437

5484

5531

5578

5625

4

5672

5719

5766

5813

5860

5907

5954

6001

6048

6095

f"

5

6142

6189

6236

6283

6329

6376

6423

6470

6517

6564

6

6611

6658

6705

6752

6799

6845

6892

6939

6986

7033

7

7080

7127

7173

7220

7267

7314

7361

7408

7454

7501

8

7548

7595

7642

7688

7735

7782

7829

7875

7922

7969

9

8016

8062

8109

8156

8203

8249

8296

8343

8390

8436

930

8483

8530

8576

8623

8670

8716

8763

8810

8856

8903

1

8950

8996

9043

9090

9136

9183

9229

9276

9323

9369

2
3

9416

9882

9463
9928

9509
9975

9556

9602

9649

9695

9742

9789

9835

0021

0068

0114

0161

0207

0254

0300

4

970347

0393

0440

0486

0533

0579

0626

0672

0719

0765

5

0812

0858

0904

0951

0997

1044

1090

1137

1183

1229

6

1276

1322

1369

1415

1461

1508

1554

1601

1647

1693

1740

1786

1832

1879

1925

1971

2018

2064

2110

2157

8

2203

2249

2295

2342

2388

2434

2481

2527

2573

2619

9

2666

2712

2758

2804

2851

2897

2943

2989

3035

3082

940

3128

3174

3220

3266

3313

3359

3405

3451

3497

3543

1

3590

3636

3682

3728

3774

3820

3866

3913

3959

4005

2

4051

4097

4143

4189

4235

4281

4327

4374

4420

4466

3

4512

4558

4604

4650

4696

4742

4788

4834

4880

4926

4

4972

5018

5064

5110

5156

5202

5248

5294

5340

5386

46

PROPORTIONAL PARTS.

Diff. 1

234

5

678

9

47 4.7

9.4 14.1 18.8

23.5

28.2 32.9 37.6

42.3

46 4.6

9.2 13.8 18.4

23.0

27.6 32.2 36.8

41.4

172

TABLE XI. — LOGARITHMS OF n I' M 15KRS.

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

N.

0

1

2

3

4

5

6

7

8

9

Diff.

945

975432

5478

5524

5570

5616

5662

5707

5753

5799

5845

6

5891

£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

1456

1501

1547

1502

1637

1683

1728

1773

9

1819

1864

1909

1954

2000

2045

2090

2135

2181

2226

960

2271

2316

2362

2407

2452

2497

2543

2588

2633

2678

1

2723

2769

2814

2859

2904

2949

2994

3040

3085

3130

2

3175

3220

3265

3310

3356

3401

3446

3491

3536

3581

3

3626

3671

3716

3762

3807

3852

3897

3942

3987

4032

4

4077

4122

4167

4212

4257

4302

4347

4392

4437

4482

5-

4527

4572

4617

4662

4707

4752

4797

4842

4887

4932

45

6

4977

5022

5067

5112

5157

5202

5247

5292

5337

5382

7

5426

5471

5516

5561

5606

5651

5696

5741

5786

5830

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

8336

8381

8425

8470

8514

4

8559

8604

8648

8693

8737

8782

8826

8871

8916

8960

5

9005

9049

9094

9138

9183

9227

9272

9316

9361

9405-

6

9450
9895

9494
9939

9539

QQftQ

9583

9628

9672

9717

9761

9806

9850

0028

0072

0117

0161

0206

0250

0294

8

990339

0383

0428

0472

0516

0561

0605

0650

0694

0738

9

0783

0827

0871

0916

0960

1004

1049

1093

1137

1182

980

1226

1270

1315

1359

1403

1448

1492

1536

1580

1625

1

1669

1713

1758

1802

1846

1890

1935

1979

2023

2067

2

2111

2156

2200

2244

2288

2333

2377

2421

2465

2509

3

2554

2598

2642

2686

2730

2774

2819

2863

2907

2951

4

2995

3039

3083

3127

3172

3216

3260

3304

3348

3392

5

3436

3480

&524

3568

3613

3657

3701

3745

3789

3833

6

3877

3921

3965

4009

4053

4097

4141

4185

4229

4273

7

4317

4361

4405

4449

4493

4537

4581

4625

4669

4713

44

8

4757

4801

4845

4889

4933

4977

5021

5065

5108

5152

9

5196

5240

5284

5328

5372

5416

5460

5504

5547

5591

PROPORTIONAL FARTS.

Diff . 1

234

5

678

9

46 4.6

9.2 13.8 18.4

23.0

27.6 32.2 36.8

41.4

45 4.5

9.0 13.5 18.0

22.5

27.0 31.5 36.0

40.5

44 4.4

8.8 13.2 17.6

22.0

26.4 30.8 35.2

39.6

43 4.3

8.6 12.9 17.2

21.5

25.8 30.1 34.4

38.7

173

TABLE XI. — LOGARITHMS OF NUMBERS.

No. 990 L. 995.]

[No. 999 L. 999.

N.

0

1

2

3

4

5

6

7

8

9

Diff.

990

995635

5679

5723

5767

5811

5854

5898

5942

5986

6030

1

6074

6117

6161

6205

6249

6

293

6337

638

0

6424

6468

44

2

6512

6555

6599

6643

6687

6731

6774

6818

6862

6906

3

6949

6993

7037

7080

7124

7

168

7212

725

5

7293

7343

4

7386

7430

7474

7517

7561

7

605

7648

769

2

7736

7779

5

7823

7867

7910

7954

7998

8041

8085

8129

8172

8216

6

8259

8303

8347

8390

8434

8

477

8521

856

4

860*

\ 8652

7

8695

8739

8782

8826

8869

8913

8956

9000

904?

I 9087

8

9131

9174

9218

9261

9305

9

348

9392

942

0

9471

> 9522

9

9565

9609

9652

9696

9739

9783

9826

9870

991J

J 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

1.908485

2

0.301030

22

1.342423

42

1

.6

23241

)

62

-95]

392

82

1.913814

3

0.477121

23

1.361728

43

1

.6,

334$

)

63

-9S

341

as

1.919078

4

0.602060

24

1.380211

44

1

.643453

64

1806180

84

1.924279

5

0.698970

25

1.397940

45

1

.653213

65

.812913

85

1.929419

6

0.778151

26

1,414973

46

1

.662758

66

.819544

86

1.934498

7

0.845098

27

.431364

47

1

.6

7209*

J

67

.

321

.075

87

1.939519

8

0.903090

28

.447158

48

1.681241

68

.

«{-,

509

88

1.944483

9

0.954243

29

.462398

49

]

.6

9011X

)

69

.,

0

£49

89

1.949390

10

1.000000

30

.477121

50

3

.6

9897(

)

70

.845098

IK)

1.954243

11

1.041393

31

.491362

51

1

.707570

71

.851258

91

1.959041

12

1.079181

32

.505150

52

1

leoo;

J

72

.

357

332

92

1.963788

13

1.113943

33

.518514

53

1

1724276

73

.86.3323

93

1.968483

14

1.146128

34

.531479

54

1

7

3239^

1 \

74

.

m

232

94

1.973128

15

1 176091

35

.544068

55

]

17

403fc

i i

75

.875061

95

1.977724

16

1.204120

36

.556303

56

1

.748188 !

76

.880814

96

1.982271

17

1.230449

37

.568202

57

1

.755875 j

77

.886491

97

1.986772

18

1.255273

38

.579784

58

1

7

6342*

i !

78

.

m

095

98

1.991226

19

1.278754

39

.591065

59

1

i-

70851

5

79

.]

627

99

1.995635

20

1.301030

40

.602060

60

M. 778151 I

80

.903090

100

2.000000

Value

at 0°.

Sign
in 1st
Quad.

Value
at 90°

• Quad.

Value
at

180°.

Sign
in3d
Quad.

Value
at

270°

Sign
in 4th
Quad.

Value
at
360°.

Sin

'

R

4.

o

R

o

Tan

r\

oo

o

i

00

o

Sec

R

oo

R

00

R

Versin....

O

__

R

4-

2R

4-

R

0

Cos

R

o

R

o

R

Cot .. .

00

__

o

00

j

o

Cosec

00

--

R

4-

CO

-

R

-

00

R signifies equal to rad; oo signifies infinite ; O signifies evanescent.

174

TABLE XII. — LOGARITHMIC SINES, ETC.

I

"

'

Sine.

q-l

Tang.

Cotang.

ff + *|

Dl"

Cosine.

/

4.685

15.314J

0

0

Inf. neg.

575 1 575

Inf. neg.

Inf. pos.

425 1

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

1575

.162696

.837304

425

ten

55

300

6

.241877

575

575

.241878

.758122

425

.02

9.999999

54

420

7

.308824

575

575

.308825

.691175

425

.00

.999999

53

480

8

.366816

574

'576

.366817

.633183

424

.00

.999999

52

540

9

.417968

574

576

.417970

.582030

424

.00

.999999

51

GOO

10

.463726

574

•576

.463727

.536273

424

.02

.999998

50

660

11

7.505118

574

,576

7.505120

12.494880

424

.00

9.999998

49

720

12

.542906

574

577

.542909

.457091

423

.02

.999997

48

780

13

.577668

574

1 577

.577672

.422328

423

.00

.999997

47

840

14

.609853

574

577

.609857

.390143

423

.02

.999996

46

900

15

.639816

573

!578

.639820

.360180

422

.00

.999996

45

960

16

.667845

573

;578

.667849

.332151

422

.02

.999995

44

1020

17

.694173

573

;578 .694179

.305821

422

.00

.999995

43

1080

18

.718997

573 T 579 i .719003

.280997

421

.02

.999994

42

1140

19

.742478

573 '579 .742484

.257516

421

.02

.999993

41

1200

20

.764754

572

| 580 .764761

.235239

420

.00

.999993

40

1260

21

7.785943

572

!580: 7.785951

12.214049

420

.02

9.999992

39

1320

22

.806146

572 ; 581 : .800155

.193845

419

.02

.999991

38

1380

23

.825451

572 i 581

.825460

.174540

419

.02

.999990

37

1440

24

.843934

571

582

.843944

.156056

418

.02

.999989

36

1500

25

.861662

571

583

.861674

.138:326

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

584

.910894

.089106

416

.02

.999986

32

1740

29

.926119

570

585

.926134

.073866

415

.02

.999985

31

1800

30

.940842

569

• W

.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

2*340

34

7.995198

568

589

7.995219

12.004781

411

.02

.999979

26

2100

35

8.007787

567 590

8.007809

11.992191

410

.03

.999977

25

2160

36

.020021

567 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

1595

8.076531

11.923469

405

.03

9.999969

19

2520 ! 42

.086965

564

:596

.086997

.913003

404

.02

.999968

18

2580

43

.097183

564

598

.09721J

.902783

402

.03

.999966

17

2640

44

.107167

563

!599

.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 i 602

.135851

,864149

398

.03

.999959

13

2880

48

.144953

561 1603

.144996

.855004

397

.02

.999958

12

2940

49

.153907

560 |! 604

.153952

.846048

396

.03

.999956

11

3000

5C

.162681

560 j 605

.162727

.837273

395

.03

.999954

10

3060

51

8.171280

559.! 607

8.171328

11.828672

393

.03

9.999952

9

3120

52

.179713

558 608

.179763

.820237

392

.03

rvq

.999950

8

3180

53

.187985

558 i 609

.188036

.811964

391

.{JO

.999948

7

3240

54

.196102

557 611

.196156

.803844

389

.03

.999946

6

3300

55.

.204070

556! 612

.204126

.795874

388

.03

.999944

5

3160

56

.211895

556 1613

.211953

.788047

387

.03

.999942

4

3^0

57

.219581

555 ; 615

.219641

.780359

385

.03

.999940

3

3180

58

.227134

554 !616

.227195

.772805

384

.03

.999938

2

3540

59

.234557

554

618

.234621

.765379

382

.03

.999936

1

3000

60

8.241855

553

1619

8.241921

11.758079

381

.03

9.999934

0

4.065

15.314

//

/

Cosine.

q-l

Cotang.

Tang.

<z + z

Dl'

Sine.

'

90°

175

TABLE Xli. — LOlxAKiTIIMIC SIN^ES,

'•

>

Sine.

q - I

Tang.

Cotang.

q + l

Dl" Cosine.

t

4.685

15.314

3600

c

8.241855

553

619

8.241921 !

11.758079

381

9.999934

60

3660

1

.249033

552

620

.249102

.750898

380 i •«*

.999932

59

3720

g

.256094

551

622

.256165

.743835

378 ! -J5 .999929

58

3780

3

.263042

551

623

.263115

.736885

377 ! -JS .999927

57

3840

4

.269881

550,

625

.269956

.730044

375 i -J5 .999925

56

3900

5

.276614

549

627

.276691

.723309

373 'JJg . .999922

.55

3960

6

.283243

548

628

.283323

.716677

.999920

54

4020

7

.289773

547

630

.289856

.710144

370 '

.999918

53

4080

8

.296207

546

632

.296292

.703708

368 Q

.999915

52

4140

9

.302546

546

633

.302634

.697366

367 'Jg

.999913

51

4200

10

.308794

545

635

.308884

.691116

.999910

50

4260

11

8.314954

544

637

8.315046

11.684954

363 -°5

9.999907

49

4320

12

.321027

543

638

.321122

.678878

362 I -JS

.999905

48

4380

13

.327016

542

640

.327114

.672886

3GO -J5 .999902

47

4440

14

.332924

541

642

.333025

.666975

358 -Xq .999899

46

4500

15

.338753

540

644

.338856

.661144

356 '}S .999897

45

4560

16

.344504

539

646

.344610

.655390

354 'J£

.999894

44

4620

17

.350181

539

648

.350289

.649711

352 'J2

.999891

43

4680

18

.355783

538

649

.355895

.644105

351 'J2

.999888

42

4740

19

.361315

537

651

.361430

.638570

349 *}5

.999885

41

4800

20

.366777

536

653

.366895

.633105

347 ; -05

.999882

40

4860

21

8 372171

535

655

8.372292

11.627708

345 •

9.999879

39

4920

22

.377499

534

657

.377622

.622378

343- -2

.999876

38

4980

23

.382762

533

659

.382889

.617111

341 i •!£

.999873

37

5040

24

.387962

532

661

.388092

.611908

339

°/-vt

.999870

36

5100

25

.393101

531

663

.393234

.606766

337

, .uo

.999867

35

5160

26

.398179

530

666

.398315

.601685

334

.05

/-it

.999864

34

5220

27

.403199

529

668

.403338

.596662

332

1 .UO

.999861

33

5280

28

.408161

527'

670

.408304

.591696

330

1 .05

O7

.999858

32

5340

29

.413068

526 !

672

.413213

.586787

328

.Ul

.998854

31

5400

30

.417919

525

674

.418068

.581932

326

\ .05

.999851 i 30

5460

31

8.422717

524

676

8.422869

11.577131

324

.05

9.990848

29

5520

32

.427462

523

679

.427618

.572382

321

*nn

.999844

28

5580

33

.432156

522

681

.432315

.567685

319

"/•(t

.999841 27

5640

34

.436800

521

683

.436962

.563038

317

.05

IY7

.999838 26

5700

35

.441394

520

685

.441560

.558440

315

.Ul

.999834 25

5760

36

.445941

518

688

.446110

.553890

312

.05

.999831 24

5820

37

.450440

517

690

.450613

.549387

310

•JJs

.999827 23

5880

38

.454893

516

693

.455070

.544930

307

.05

.999824 i 22

5940

39

.459301

515

695

.459481

.540519

305

*H7

.999820 21

6000

40

.463665

514

697

.463849

.536151

303

.UY

.999816 20

6060

41

8.467985

512

700

8.468172

11.531828

300

.05

9.999813 19

6120

42

.472263

511

702

.472454

.527546

298

.07

O7

.999809 18

6180

43

.476498

510

705

.476693

.523307

295

.Ul

O7

.999805 17

6240

44

.480693

509

707

.480892

.519108

293

.Ul
O7

.999801 ! 16

6300

45

.484848

50Y:

710

.485050

.514950

290

.Ul

.919797 15

6360

46

.488963

506;

713

.489170

.510830

287

g*f

.999794 i 14

6420

47

.493040

505

715

.493250

.506750

285

' r.n

.999790 ! 13

&480

48

.49707-8

503

718

.497293

.502707

282

.VI

H7

.999786 i 12

6540

49

.501080

502

720

.501298

.498702

280

.Ul

07

.999782 11

6600

50

.505045

501

723

.505267

.494733

277

.Ul

.999778 10

6660

51

8.508974

499

726

8.509200

11.490800

274

.07

OS

9.999774

9

6720

52

.512867

498

729

.513098

.486902

271

.Uo

.999769

8

6780

53

.516726

497

731

.516961

.483039

269

'07

.999765

7

6840

54

.520551

495

734

.520790

.479210

266

.Ul
O7

.999761

6

6900

55

.524343

494

737

524586

.475414

263

.Ul

07

.999757

5

6960

56

.528102

492

740

.528349

.471651

260

'AQ

.999753

4

7020

57

.531828

491

743

.532080

.467920

257

.UO
CV7

.999748

3

7080

58

.535523

490

745

.535779

.464221

255

.Ul

07

.999744

2

7140

59

,539186

488

748

.539447

.460553

252

Oft

.999740

1

7200

60

8.542819

487'

751

8.543084

11.456916

249

•uo

9.999735

0

4.685

15.314

H

'

Cosine.

q-l

Cotang.

Tang.

q + l

PI"

Sine,

'.J

COSINES, TANGENTS, AND COTANGENTS.

'

Sine.

D.r.

Cosine.

D. 1".

Tang.

D. r.

Cotang.

'

0

1

2
3
4
5

8 54S319
.546422
.549995
.553539
.557054
.560540

60.05
59.55
59.07
58.58
58.10

9.999735
.999731
.999726
.999722
.999717
.999713

.07
.08
.07
.08
.07

AO

8.543084
.546691
.550268
.553817
.557336
.560828

60.12
59.62
59.15
58.65
58.20

11.456916
.453309
.449732
.446183
.442664
.439172

60
59
58
57
56
55

6

7

.563999
.567431

57.65
57.20

.999708
.999704

.Uo

.07

AO

.564291
.567727

57.72
57.27

.435709
.432273

54
53

8
9
10

.570836
.574214
.577566

56.75
56.30
55.87
55.43

.999699
.999694
.999689

.Uo

.08
.08
.07

.571137
.574520

.577877

56 . 83
56.38
55.95
55.52

.428863
.425480
.422123

52
51
50

11
12

8.580892
.584193

55.02

9.999685
.999680

.08

AQ

8.581208
.584514

55.10

11.418792
.415486

49

48

13
14
15
16
17
18
19
20

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

54.60
54.20
53.78
53.40
53.00
52.62
52.23
51.85
51.48

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

.Uo

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

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

54.68
54.27
53.87
53.48
53.08
52.70
£2.32
51.93
51.58

.412205
.408949
.405717
.402508
.399323
.396161
.393022
.389906

47
46
45
44
43
42
41
40

21
22

8.612823
.615891

51.13

9.999635
.999629

.10

8.613189
.616262

51.22

11.386811
.383738

39

38

23
24
25

.618937
.621962
.624965

50.77
50.42
50.05

.999624
.999619
.999614

.08
.08
.08

-| A

.619313
.622343
.625352

50.85
50.50
50.15

4Q 8fl

.380687
.377657
.374648

37
36
35

26

.627948

49.72

.999608

. JU
AO

.628340

4y .ou
49.47

.371660

34

27
28
29
30

.630911
.633854
.636776
.639680

49.38
49.05
48.7(f
48.40
48.05

.999603
.999597
.999592
.999586

.Uo

.10
.08
.10
.08

.631308
.634256
.637184
.640093

49.13

48.80
48.48
48.15

.368692
.365744
.362816
.359907

33

32
31
30

31
32
33
34
35
36
37
38
39
40

8.642563
.645428
.64827*4
.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
.665433
.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
2S
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49

8.670393
.673080
.675751
.678405
.681043
.683665
.686272
.688863
.6914^8

44.78
44.52
44.23
43.97
43.70
43.45
43.18
42.92

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

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

8.670870
.673563
.676239
.67'8900
.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

11

50

.693998

42.67
42.42

.999469

.10
.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
.295,354
.292860
.290382
.287917
.285466
.283028
11.280604

9
8
7
6
5
4
3
2
1
0

'

Cor-e.

D r.

Sine.

D. 1". 1

Cotang. 1 P. 1". i Tang. '

177

TABLE XII. — LOGARITHMIC SINES,

'

Sine.

D. r.

Cosine.

D.I".

Tang.

D.I*.

Cotang.

'

0

1

2

8.718800
.721204
.723595

40.07
39.85

qQ Of)

9.999404
.999398
.999391

.10
.12

8.719396
.721806
.724204

40.17
39.97

11.280604
.278194
.275796

60
59
58

3

4
5

.725972
.728337
.730688

oy .0/6
39.42
39.18

qo QQ

.999384
.999378
.999371

.12
.10
.12

19

.726588
.728959
.731317

39.73
39.52
39.30

qQ -if\

.273412
.271041

.2686aS

57
56
55

6

.733027

oo.yo

qo r»Q

.999364

. 1/4

.7aS663

O» . 1U

qo 00

.266337

54

7
8

.735354
.737667

OO. <O

38.55

qo qrf

.999357
.999350

!l2

.735996
.738317

OO.OO

38.68

qo -jo

.264004
.261683

53

52

9

.739969

OO.OY
qo -jrv

.999343

•*J

.740626

OO.4o

.259374

51

10

.742259

OO.1<

37.95

.999336

.12
.12

.742922

38.27
38.08

.257078

50

11
12
13
14
15

8.744536
.746802
.749055
.751297
.753528

37.77
37.55
37.37
37.18

qc QQ

9.999329
.999322
.999315
.999308
.999301

.12
.12
.12
.12

19

8.745207
.747479
.749740
.751989
.754227

37.87
37.68
37.48
37.30

q« in

11.254793
.252521
.250260
.248011
.245773

49

48
47
46
45

16
17

18
19
20

.755747
.757955
.760151
.762337
.764511

oo.yo
36.80
36.60
36.43
36.23
36.07

,999294
.999287
.999279
.999272
.999265

. 1 -w

.12
.13
.12
.12

.756453
.758668
.760872
.763065
.765246

O< . 1U

36.92
36.73
36.55
36. a5
36.18

.243547
.24iaS2
.239128
.236935
.234754

44
43
42
41
40

21

8.766675

qt QO

9.999257

19

8.767417

11.232583

39

22
23
24
25
26
27
28
29
30

.768828
.770970
.773101
.775223
.777333
.779434
.781524
.783605
.785675

OO.oo

35.70
35.52
35.37
35.17
35.02
34.83
34.68
34.50
34.35

.999250
.999242
.999235
.999227
.999220
.999212
.999205
.999197
.999189

. 1/6

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

.769578
.771727
.773866
.775995

.778114
.780222
.782320,
.784408
.786486

35^82
35.65
a5.48
35.32
35.13
34.97
34.80
34.63
34.47

.230422
.228273
.226134
.224005
.221886
.219778
.217680
.215592
.213514

38
37!
36
35
34
33
32
31
30

31
32
33

8.787736
.789787
.791828

34.18
34.02

qq or

9.999181
.999174
.999166

.12
.13

•f q

8.788554
.790613
.792662

34.32
34.15

qq QQ

11.211446
.209387
.207aS8

29
28
27

34
35

36
37
38
39
40

.793859
.795881
.797894
.799897
.801892
.803876
.805852

OO.OO

33.70
33.55
33.38
33.25
33.07
32.93
32.78

.999158
.999150
.999142
.999134
.999126
.999118
.999110

. lo
.13
.13
.13
.13
.13
.13
.13

.794701
.796731
.798752
.800763
.802765
.804758
.806742

oo.yo

as. as
as. 68
as. 52
as. 37
as. 22

33.07
32.92

.205299
.203269
.201248
.199237
.197235
.195242
.193258

26
25
24
23

22
21
20

41
42
43

8.807819
.809777
.811726

32.63
32.48

9.999102
.999094
.999086

.13
.13

8.808717

.8io6as

.812641

32.7.7
32.63

qo Af*

11.191283
.189317

.187359

19
18
17

44
45
46
47
48
49
50

.813667
.815599
.817522
.819436
.821343
.823240
.825130

32.35
32.20
32.05
31.90
31.78
31.62
31.50
31.35

.999077
.999069
.999061
.999053
.999044
.999036
.999027

.15
.13
.13
.13
.15
.13
.15
.13

.814589
.816529

.818461
.820384
.822298
.824205
.826103

• >~ . 4 1

32. as

32.20
32.05
31.90
31.78
31.63
31.48

.185411
.183471
.181539
.179616
.177702
.175795
.173897

16
15
14
13
12
11
10

51
52
53
54
55
56
57
58

8.827011
.828884
.830749
.832607
.834456
.836297
.838130
.839956

31.22
31.08
30.97
30.82
30.68
30.55
30.43

on qn

9.999019
.999010
.999002
.998993
.998984
.998976
.998967
.998958

.15
.13
.15
.15
.13
.15
.15

•JO

8.827092
.829874
.831748
.833613
.835471
.837321
.839163
.840998

31.37
31.23
31.08
30.97
30.83
30.70
30.58

Of) AK.

11.172008
.170126
.168252
.166387
.164529
.162679
.160837
.159002

9
8
7
6
5
4
3
2

59
60

.841774
8.843585

oU . oU

30.18

.998950
9.998941

. Jo

.15

.842825
8.844644

OU . 40

30.32

.157175
11.155356

1

0

' Cosine.

D r.

Sine.

D. IV

Cotang.

D. 1*. ! Tang.

'

178

COSINES, TANGENTS, AND COTANGENTS.

175*

'

Sine.

D. 1".

Cosine.

D. i".

Tang.

D. r.

Cotang.

•

0

8.843585

on no

9.998941

8.844644

11.166866

60

1

2
3
4
5
6
7
8
9
10

.845387
.847183
.848971
.850751
.852525
.854291
.856049
.857801
.859546
.861283

29.93
29.80
29.67
29.57
29.43
29.30
29.20
29.08
28.95
28.85

.998932
.998923
.998914
.998905
.998896
.998887
.998878
.998869
.998860
.998851

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

.846455
.848260
.850057
.851846
.853628
.855403
.857171
.858932
.860686
.862433

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

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

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

11.135827
.134094
. 132368
.130649
.128936
.127230

49
48
47
46
45
44

17

.873255

.998785

.874469

.125531

43

18
19
20

.874938
.876615

.878285

27.95

27,83
87.73

.998776
.998766
.998757

.17
.15
.17

.876162
.877849
.879529

28.12
28.00
27.88

.123838
.122151
.120471

42
41
40

21
22
23
24

8.879949
.881607
.883258
.884903

27.63
27.52
27.42

9.998747
.998738
.998728
.998718

.15
.17
.17

8.881202
.882869
.884530
.886185

27.78
27.68
27.58

11.118798
.117131
.115470
.113815

39

38
37
36

25
26
27
28
29
30

.886542
.888174
.889801
.891421
.893035
.894643

27.20
27.12
27.00
26.90
26.80
26.72

.998708
.998699
.998689
.998679
.998669
.998659

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

.887833
.889476
.891112
.892742
.894366
.895984

27.38
27.27
27.17
27.07
26.97
26.87

.112167
.110524
.108888
.107258
.105634
.104016

35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

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

26.60
26.50
26.42
26.32
26.22
26.12
26.02
25.93
25.85
25.75

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

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

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

26.78
26.67
26.58
26.48
26.38
26.28
26.20
26.10
26.02
25.92

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

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

8.911949
.913488
.915022
.916550
.918073
.919591
.921103
.922610
.924112
.925609

25.65
25.57
25.47
25.38
25.30
25.20
25.12
25.03
24.95
24.85

9.998548
.998537
.998527

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

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

8.913401
.914951
.916495
.918034
.919568
.921096
.922619
.924136
.925649
.927156

25.83
25.73
25.63
25.57
25.47
25.38
25.28
25.22
25.12
25.03

11.086599
.085049
.083505
.081966
.080432
.078904
.077381
.075864
.074351
.072844

19
18
17
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
.99&S66
.99a355
; 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.

s Sine.

D. r.

Cotang.

D. 1".

Tang. '

94-

17Q

TABLE XII. — LOGARITHMIC SINES,

174°

'

Sin*.

D. 1'.

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

1

/

0

1

2
3

4
5
6

7

8.940296
.941738
.943174
.944606
.946034
.947456
.948874
.950287

24.03
23.93
23.87
23.80
23.70
23.63
23.55

9.998344
.998333
.998322
.998311
.998300
.998289
.998277
.998266

.18
.18
.18
.18
.18
.20
.18
1 n

8.941952
.943404
.944852
.946295
.947734
.949168
.950597
.952021

24.20
24.13
24.05
23.98
23.90
23.82
23.73

11.058048
.056596
.055148
.053705
.052266
.050832
.049403
.047979

60
59
58
57
56
55
54
53

8
9
10

.951696
.953100
.954499

23.40
23.32
23.25

.998255
.998243
.998232

.20

.18

.953441
.954856
.956267

23.58
23.52
23.45

.046559
.045144
.043733

52
51
50

11
12
13
14
15
16
17
18
19
20

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

23.17
23.10
23.03
22.95
22.87
22.82
22.73
22.65
22.60
22.52

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

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

8.957674
.959075
.960473
.961866
.963255
I .964639
.966019
.967394
1 ,968766
.970133

23.35
23.30
23.22
23.15
23.07
23.00
22.92
22.87
22.78
22.72

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

49
48
47
46
45
44
43
42
41
40

21
22
23
24
25
26
27

8.969600
.970947
.972289
.973628
.974962
.976293
.977619

22.45
22.37
22.32
22.23
22.18
22.10
99 m

9.998104
.998092
.998080
.998068
.998056
.998044
.998032

.20
.20
.20
.20
.20
.20
20

8.971496
.972855
.974209
.975560
.976906
.978248
.979586

22.65
22.57
22.52
22.43
22.37
22.30
22 25

11.028504
.027145
.025791
.024440
.023094
.021752
.020414

39
38
37
36
35
34
33

28

.978941

.998020

.980921

99 1"

.019079

32

29
30

.980259
.981573

21.90
21.83

.998008
.997996

.20
.20

.982251
.983577

22.10
22.03

.017749
.016423

31

30

31
32
33

8.982883
.984189
.985491

21.77
21.72

9.997984
.997972
.997959

.20
.22

8.984899
.986217
.987532

21.97
21.92
21 83

11.015101

.013783
.012468

29
28

27

34

.986789

~**jj?

.997947

on

.988842

91 ^A

.011158

26

35
36
37
38
39
40

.988083
.989374
.990660
.991943
.993222
.994497

21.52
21.43
21.38
21.32
21.25
21.18

.997935
.997922
.997910
.997897
.997885
.997872

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

.990149
.991451
.992750
.994045
.995337
.996624

21.70
21.65
21.58
21.53
21.45
21.40

.009851
.008549
.007250
.005955
.004663
.003376

25
24
23
22
21
20

41
42
43

44
45
46
47
48
49
50

8.995768
.997036
.998299
8.999560
9.000816
.002069
.003318
.004563
.005805
.007044

21.13
21.05
21.02
20.93
20 88
20.82
20.75
20.70
20.65
20.57

9.997860
.997847
.997835
.997822
.997809
.997797
.997784
.997771
.997758
.997745

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

'.22
.22
.22

8.997908
8.999188
9.000465
.001738
.003007
.004272
.005534
.006792
.008047
.009298

21. as

21.28
21.22
21.15
21.08
21.03
20.97
20.92
20.85
20.80

11.002092
11.000812
10.999535
.998262
.996993
.995728
.994466
.993208
.991953
.990702

19
18
17
16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

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

20 53
20.45
20.42
20.33
20.30
20.22
20.18
20.12
20.07

9.997732
.997719
.997706
.997693
.997680
.P97667
.997654
.997641
.997628
9.997614

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

9.010546
.011790
.013031
.014268
.015502
.016732
.017959

.0191 as

.020403
9.021620

20.73
20.68
20.62
20 57
20.50
20.45
20.40
20. 33
20.28

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

9

8
7
6

1

3
2

1
0

'

Cosine.

D. 1'.

Sine.

D. 1*.

Cotang.

D. 1".

Tang.

'

180

COSINES, TANGENTS, AND COTANGENTS.

173°

•

Sine.

D. 1*.

Cosine.

D. r.

Tang.

D. r.

Cotang.

'

0

1

2

9.019235
.0204a5
.021632

20.00
19.95

9.997614
.997601

.997588

.22

22

9.021620
.022834
.024044

20.23
20.17

10.978380
.977166
.975956

60

59
58

3
4
5
6

8
9
10

.022825
.024016
.025203
.026386
.027567
.028744
.029918
.031089

19.88
19.85
19.78
19.72
19.68
19.62
19.57
19 52 I
19.47 I

.997574
.997561
.997547
.997534
.997520
.997507
.997493
.997480

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

.025251
.026455
.027655
.028852
.030046
.031237
.032425
.033609

20. 12
20.07
20.00
19.95
19.90
19.85
19.80
19.73
19.70

.974749
.973545
.972345
.971148
.969954
.968763
.967575
.966391

57
56
55
54
53
52
51
50

11
12
13
14
15
16
17
18
19
20

9.032257
.0&3421
.034582
.035741
.036896
.038048
.039197
.040342
.041485
.042625

19.40
19.35 j
19.32
19.25
19.20 1
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
.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

ift ^

9.997327
.997313
.997299
.997285

.23
.23
.23

9.046434
.047582
.048727
.049869

19.13
19.08
19.03

1ft Qft

10.953566
.952418
.951273
.950131

39
38
37
36

25
26

.048279
.049400

Jo. <o

18.68

.997271
.997257

'.23

.051008
.052144

Jo.yo
18.93

ift ftft

.948992
.947856

35
34

27
28
29
30

.050519
.051635
.052749
.053859

18.65
18.60
18.57
18.50
18.45

.997242
.997228
.997214
.997199

.25
.23

iss
jsa

.053277
.054407
.055535
.056659

Jo.oo

1883
18.80
18.73
18.70

.946723
.945593
.944465
.943341

33
32
31
30

31

32
33

9.054966
.056071
.057172

18.42
18,35

9.997185
.997170
.997156

.25
.23

9.057781
.058900
.060016

18.65
18.60

1ft f\7

10.942219
.941100
.939984

29

28
27

34

35
36
87
38
39
40

.058271
.059367
.060460
.061551
.062639
.063724
.064806

18'.27
18.22
18.18
18.13
18.08
18.03
17.98

.997141
.997127
.997112
.997098
.997083
.997068
.997053

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

.061130
.062240
.063348
.064453
.065556
.066655
.067752

Jo. o<

18.50
18.47
18.42
18.38
18.32
18.28
18.25

.938870
.937760
.936652
.935547
.934444
.933345
.932248

26
25
24
23
22
21
20

41
42
43
44
45
46
47
48

9.065885
.066962
.068036
.069107
.070176
.071242
.072306
.073366

17.95
17.90
17.85
17.82
17.77
17.73
17.67

9.997039
.997024
.997009
.996994
.996979
.996964
.996949
.996934

.25
.25
.25
.25
.25
.25
.25

9.068846
.069938
.071027
.072113
.073197
.074278
.075356
.076432

18.20
18.15
18.10
18.07
18.02
17.97
17.93

10.931154
.930062
.928973
.927887
.926803
.925722
.924644
.923568

19
18
17
16
15
14
13
12

49

.074424

17.63

.996919

.25

.077505

17.88

.922495

11

50

.075480

17.60
17.55

.996904

.25

.25

1 .078576

17.85
17.80

.921424

10

51
52
53
54

9.076533
.077583
.078631
.079676

17.50
17.47
17.42

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

.080719

17.38

.996828

.27

.083891

17.63

.916109

5

56
57
58
59
60

.081759
.082797
.083832
.084864
9.085894

17.33
17.30
17.25
17.20
17.17

.996812
.996797
.996782
.996766
9.996751

.27
.25
25

'.25

.084947
.086000
.087050
.088098
9.089144

17.60
17.55
17.50
17.47
17.43

.915053
.914000
.912950
.911902
10.910856

4
3
2
1
0

'

Cosine.

D. 1'.

Sine.

D. r.

Cotang. | D. 1*.

Tang.

'

1S1

TABLE XII. — LOGARITHMIC SINES,

'

Sine.

D. r.

Cosine.

D. r.

Tang.

D. 1\

Cotang.

'

0

1

9 085894
.086922

17.13

9.996751
.996735

27

9.089144
.090187

17.38

10.910856
909813

60

59

2

3
4
5
6
7
8
9
10

.087947
.088970
.089990
.091008
092024
.093037
.094047
.095056
.096062

17.08
17.05
17.00
16.97
16.93 i
16.88 !
16.83
16.82
16.77
16.72

.996720
.996704
.996688
.996673
.996657
.996641
.996625
.996610
.996594

.25
.27
.27
.25
.27
.27
.27
.25
.27
.27

.091228
.092266
.093302
.094336
.095367
.096395
.097422
.098446
.099468

17.35
17.30
17.27
17.23
17.18
17.13
17.12
17.07
17.03
16.98

.908772
.907734
.906698
.905664
.904633
.903605
.902578
.901554
.900532

58
57
56
55
54
53
52
51
50

11
12
13
14
15
16
17
18
19
20

9 097065
.098066
.099065
.100062
. 101056
.102048
.103037
.104025
.105010
.105992

16.68
16.65
16.62
16.57
16.53
16.48
16.47
16.42
16.37
16.35

9.996578
.996562
.996546
.996530
.996514
996498
.996482
.996465
.996449
,996433

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

9.100487
.101504
.102519
.103532
.104542
.105550
.106556
.107559
.108560
.109559

16.95
16.92
16.88
16.83
16.80
16.77
16.72
16.68
16.65
16.62

10.899513
.898496
.897481
.896468
.895458
.894450
.893444
.892441
.891440
.890441

49
48
47
46
45
44
43
42
41
40

21
22
23
24

25
26
27

28

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

16.30
16.27
16.23 !
16.20
16.15
16.12
16.08
16 05

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

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

Oft

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

16.58
16.53
16.50
16.47
16.43
16.40
16.35

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

39
38
37
36
35
34
33
32

29
30

.114737
. 115698

16^02
15.97

.996285
.996369

. .Co

.27
.28

.118452
.119429

16.33
16.28
16.25

.881548
.880571

31
30

31
32
33
34
35

9 116656
.117613
118567
.119519
.120469

15 95
15.90
15 87
15.83

•if Qf\

9.996252
.996235
.996219
.996202
.996185

.28
.27
.28
.28

9.120404
.121377
.122348
.123317

.124284

16 22
16.18
16.15
16.12

10.879596
.878623
.877652
.876683
.875716

29
28
27

2e

25

36
37
38
39
40

.121417
.122362
.123306
.124248
.125187

ID.oU

15.75
15.73
15.70
15.65
15.63

.996168
.996151
.996134
.996117
.996100

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

.125249
.126211
.127172
.128130
.129087

16.08
16.03
16 02
15.97
15.95
15.90

.874751
.873789
.872828
.871870
.870913

24
23
22
21
20

41
42
43

9.126125
.127060
.127993

15.58
15.55

9.996083
.996066
.996049

.28
.28

9.130041
.130994
. 131944

15.88
15.83

10.869959
.869006
.868056

19

18
17

44

.128925

15 53

1 ^ 4ft

.996032

.28

9ft

.132893

15.82

-JR iyr»

.867107

16

45

46

47

48

129854
130781
131706
132630

ID. 40

15 45
15.42
15.40

•JE qc

.996015
.995998
.995980
.995963

,«0

.28
.30
.28

Oft

133839
.134784
.135726
.136667

ID. t i

15.75
15.70
15.68

•JK CO

.866161
.865216
.864274
863333

15
14
13
12

49
50

133551
134470

JO o.)

15.32
15.28

.995946
.995928

. 4a

30
.28

.137605
.138542

1O . Do

15 62
15.57

.862395 | 11
.861458 ! 10

51

52
53
54
55
56
57
58
59
60

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

15 27
15 22
15.20
15 15
15 12
15 10
15 07
15.02
15 00

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

28

:30

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

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

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

10 860524
.859591
.858660
.857731
.856804
.855879
.854956
854034
853115
10 852197

9
8
7
6
5
4
3
2
1
0

'

Cosine. D. 1'.

Sine.

D.r.

Cotang.

D. 1'.

Tang.

1

97°

182

COSINES, TANGENTS, AND COTANGENTS.

/

Sine.

D. r.

Cosine.

D. i'.

Tang.

D. 1'.

Cotang.

'

0

1

2
3
4

9.143555
.144453
.145349
.146243
.147136

14.97
14.93
14.90
14.88

14 ft3

9.995753
.995735
.995717
.995699
.995681

.30
.30
.30
.30

28

9.147803
.148718
.149632
.150544
.151454

15.25
15.23
15.20
15.17
i^ 1 ^

10.852197
.851282
.850368
.849456
.848546

60
59
58
57
56

5

6
7
8
9

.148026
.148915
.149802
.150686
.151569

14. OO

14.82
14.78
14.73
14.72

.995664
.995646
.995628
.995610
.995591

!30
.30
.30
.32
30

.152363
.153269
.154174
.155077
.155978

ID. ID

15.10
15.08
15.05
15.02

HQQ

.847637
.846731
.845826
.844923
.844022

55
54
53
52
51

10

.152451

14.70
14.65

.995573

.'30

.156877

.yo
14.97

.843123

50

11
12

9.153330
.154208

14.63

14 f\A

9.995555
.995537

.30

on

9.157775
.158671

14.93

14 Qfl

10.842225
.841329

49

48

13
14
15
16

.155083
.155957
.156830
.157700

14. Oo

14.57
14.55
14.50

.995519
.995501
.995482
.995464

.oU

.30
.32
• .30

on

.159565
.160457
.161347
.162236

14. yu

14.87
14.83
14.82

14 7R

.840435
.839543
.838653
.837764

47
46
45
44

17
18
19
20

.158569
.159435
.160301
.161164

14.48
14.43
14.43
14.38
14.35

.995446
.995427
.995409
.995390

.OU

.32
.30

.32
.30

.163123
.164008
.164892.
.165774

14. tO

14.75
14.73
14.70
14.67

.836877
.835992
.835108
.834226

43
42

41
40

21
22

9.162025
.162885

14.33

9.995372
.995353

.32

qo

9.166654
.167532

14.63

14 RO

10.833346
.832468

39

38

23
24

.163743
.164600

14.30
14.28

.995334
.995316

,«M

.30

oo

.168409
.169284

14. IM

14.58

14 f\f\

.831591
.830716

37
36

25

.165454

14.23

14 9')

.995297

.38

qo

.170157

14. OD
14 ^3

.829843

35

26

.166307

14. •£«

.995278

.«M

.171029

14. OO

.828971

84

27

.167159

14.20

.995260

.30

qO

.171899

14.50

.828101

33

28
29

.168008
.168856

14.15
14.13

.995241
.995222

.0-6

.32

••>•»

.172767
.173634

14.47
14.45

.827233
.826366

32
31

30

.169702

14.10
14.08

.995203

.0^
.32

.174499

14.42
14.38

.825501

30

31
32

9.170547
.171389

14.03

9.995184
.995165

.32

OO

9.175362
.176224

14.37

14 33

10.824638
.823776

29

28

33
34

.172230
.173070

14.02
14.00

.995146
.995127

.00

.32

.177084
.177942

14. OO

14.30

.822916
.822058

27
26

35

. 173908

13.97

•JO Qq

.995108

.32

qo

.178799

14.28

14 9*7

.821201

25

36
37
38
39
40

.174744
.175578
.176411
.177242
.178072

lo. yd

13.90
13.88
13 85
13.83
13.80

.995089
.995070
.995051
.995032
.995013

.058

.32
.32
.32
.32
.33

.179655
.180508
.181360
.182211
.183059

14 tii

14.22
14.20
14.18
14.13
14.13

.820345
.819492
.818640
.817789
.816941

24
23
22
21
20

41
42
43
44
45
46

9 178900
.179726
.180551
.181374
.182196
.183016

13.77
13.75
13.72
13.70
13.67

9.994993
.994974
.994955
.994935
.994916
.994896

.32
.32
.33
.32
.33

oo

9.183907
.184752
.185597
.186439
.187280
.188120

14.08
14.08
14.03
14.02
14.00

10.816093
.815248
.814403
.813561
.812720
.811880

19
18
17
16
15
14

47

.183834

13.63

13 fiO

.994877

.0-6

qq

.188958

13.97

13 Q3

.811042

13

48
49

.184651
.185466

Id. O*

13.58

13 W

.994857
.994838

.OO

.32

qq

.189794
.190629

lo.yo
13.92

13 Rft

.810206
.809371

12
11

50

.186280

Id. Of

13.53

.994818

.Oo

.33

.191462

1O.OO

13.87

.808538

10

51

9.187092

9.994798

qo

9.192294

13 R51

10.807706

9

52

.187903

13.52

.994779

.O<*

.193124

10. OO

13 QO

.806876

8

53
54
55
56

57
58
59
60

.188712
.189519
.190325
.191130
.191933
.192734
.193534
9.194332

13.48
13.45
13.43
13.42
13.38
13 35
13 33

1330

.994759
.994739
.994720
.994700
.994680
.994660
.994640
9.994620

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

.as

.33

.193953
.194780
.195606
.196430
.197253
.198074
.198894
9.199713

lo.oXJ
13.78
13.77
13.73
13.72
13.68
13.67
13.65

.806047
.805220
.804394
.803570
.802747
.801926
.801106
10.800287

7
6
5
4
3
2
1
0

/

Cosine.

D. r. ii Sine.

D. r.

Cotang.

D. r.

Tang.

i

98°

TABLE XII. LOGARITHMIC SINES,

1.70°

'

Sine.

D. 1\

Cosine.

D. 1".

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
.994600
.994580
.994560
.994540
.994519
.994499
.994479
.994459
.994438
.994418

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

9.199713
.200529
.201345
.202159
.202971
.203782
.204592
.205400
.206207
.207013
.207817

13.60
13.60
13.57
13.53
13.52
13.50
13.47
13.45
13.43
13.40
13.37

10.800287
.799471
.798655
.797841
.797029
.796218
. .795408
.794600
.703793
.792987
.792183

60
59
58
57
56
55
54
53
52
51
50

11
12
13
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
1 .994212

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

9.208619
.209420
.210220
.211018
.211815
1 .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
.789780
.788982
.788185
.787389
.786595
.785802
.785011
.784220

49
48
47
46
45
44
43
42
41
40

21
22
23
24

25
26
27
28
29
30

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

12.77
12.75
12.73
12.72
12.68
12.65
12.65
12.62
12.58
12.57

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

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

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

13.13
13.10
13.07
13.07
13.03
13.00
13.00
12.97
12.95
12.92

10.783432
.782644
.781858
.781074
.780290
.779508
.778728
.777948
.777170
.776393

39
38
37
36
35
34
33
32
31
30

31
32
33
34
35
36
37
38
39
40

9.218363
.219116
.219868
.220618
.221367
.222115
.222861
.223«06
.224349
.225092

12.55
12.53
12.50
12.48
12.47
12.43
12.42
12.38
12.38
12.35

9.993982
.993960
.993939
.993918
.9938&T
.993875
.993854
.993832
.993811
.993789

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

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

12.90

12.88
12.85
12.85
12.80
12.80
12.77
12.77
12.72
12.72

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

29
28
27
26
25
24
23
22
21
20

41

42
43
44

45
46
47
48
49
50

9.225833
.226573
.227311

.228048
.228784
.229518
.230252
.230984
.231715
.232444

12.33
12.30
12.28
12.27
12.23
12.23
12.20
12.18
12.15
12.13

9.993768
.993746
.993725
.993703
.993681
.993660
.993638
.993616
.993594
.993572

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

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

12.68
12.67
12.65
12.63
12.60
12.58
12.57
12.53
12.53
12.50

10.767935
.767174
.766414
.76565*
.764897
.764141
.763386
.762632
.761880
.761128

19
18
17
16
15
14
13
12
11
10

51

52
53
54
55

9.233172
.233899
.234625
.235349

.236073

12.12
12.10
12.07
12.07

9.993550
.993528
.993506
.993484
.993462

.37
.37
.37
.37

9.239622
.240371
.241118
.241865
.242610

12.48
12.45
12.45
12.42

10.760378
.759629

.758882
.758135
.757390

9
8
7
6
5

56

57
58

.236795
.237515
.238235

12.00
12.00

nQ7

.993440
.993418
.993396

.37

.37

q7

.243354
.244097
.244839

12.38
12.37

.756646
.755903
.755161

4
3
2

59

.238953

.993374

qo

.245579

.754421

1

60

9.239670

9.993351

9.246319

12.33

10.753681

0

'

Cosine. D. 1".

Sine.

D. r. i

Cotang.

D. 1".

Tang.

'

184

80°

COSINES, TANGENTS, AND COTANGENTS.

169"

•

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. r.

Cotang.

'

0

1

9.239670
.240386

11.93
11 .92

9.993351
.99:3329

.37

9.246319
.247057

12.30

10.753681
.752943

60

59

2

.241101

.993307

'net

.247794

12. «8

.752206

58

3
4

.241814
.242526

11.88
11.87

.993284
.993262

.38
.37

.248530
.249264

12.27
12.23

.751470
.750736

57
56

5
6

.243237
.243947

11.85
11.83

.993240

.993217

.37
.38

.249998
.250730

12.23
12.20

.750002
.749270

55
54

7
8

.244656
.245363

11 .82
11.78

.993195
.993172

''<£ .

.251461
.252191

12.18
12.17

.748539
.747809

53
52

9

10

.246069
.246775

11.77
11.77
11.72

.993149
.993127

.38
.37 !
.38

.252920
.253648

12.15
12.13
12.10

.747080
.746352

51
50

11
12
13
14
15
16
17
18
19

9.247478
.248181
.248883
.249583
.250282
.250980
.251677
.252373
.253067

11.72
11.70
11.67
11.65
11.63
11.62
11.60
11.57

9.993104
.993081
.993059
.993036
.993013
.992990
.992967
.992944
.992921

.38
.37
.38
.38
.38
.38
.38
.38

9.254374
.255100
.2f>824
.256547
.257269
.257990
.258710
.259429
.260146

12.10
12.07
12.05
12.03
12.02
12.00
11.98
11.95

10.745626
.744900
.744176
.743453
.742731
.742010
.741290
.740571
.739854

49
48
47
46
45
44
43
42
41

20

.253761

11.57
11.53

.992898

.38
.38

.260863

11.95
11.92

.739137

40

21
22

9.254453
.255144

11.52

9.992875
.992852

.38

9.261578
.262292

11.90

10.738422
.737708

39

38

23

.255834

11 .50

.992829

.38

.263005

11.88

.736995

37

24

.256523

11.48

.992806

.38

.263717

11.87

.736283

36

25

.257211

11 .47

.992783

.38

.264428

11.85

.735572

35

26

.257898

11.45

.992759

.40

.265138

11.83

.734862

34

27

.258583

11.42

.992736

.38

.265847

11.82

.734153

33

28

.259268

11.42

.992713

.38

.266555

11.80

.733445

32

29
30

.259951
.260633

11.38
11.37

.992690
.992666

.38
.40

.267261
.267967

11.77
11.77

.732739
.732033

31
30

11.35

.38

11.73

31

32

9.261314
.261994

11.33

9.992643
.992619

.40

9.268671
.269375

11.73

10.731329

.730625

29
28

33

.262673

11.32

.992596

.38

.270077

11.70

.729923

27

34

.263351

11.30

.992572

.40

.270779

11.70

.729221

26

35
36
37

.264027
.264703
.265377

11.27
11.27
11.23
11 .23

.992549
.992525
.992501

.38
.40
.40

OQ

.271479
.272178

.272876

11.67
11.65
11.63

nfiO

.728521

.727822
.727124

25
24
23

38

.266051

.992478

.OO

.273573

.001

.726427

22

39
40

.266723
.267395

11.20
11.20
11.17

.992454
.992430

.40
.40
.40

.274269
.274964

11 .60
11.58
11.57

.725731
.725036

21
20

41
42
43
44
45
46
47
48
49
50

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

11.15
11.13
11.12
11.10
11.08
11.07
11.03
11.03
11.02
10.98

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

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

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

11.55
11.53
11.52
11.50
11.48
11.47
11.45
11.43
11.40
11.40

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

19
18
17
16
15
14
13
12
11
10

51
52
53
54

9.274708
.275367
.276025
.276681

10.98
10.97
10.93

9.992166
.992142
.992118
.992093

.40
.40
.42

Af\

9.282542

.283225
.283907
.284588

11.38
11.37
11.35

10.717458
.716775
.710093
.715412

9

8

6

55
56
57
58
59
.60

.277337
.277991
.278045
.279297
.279948
9.280599

10.93
10.90
10.90
10.87
10.85
10.85

.992069
.992044
.992020
.991996
.991971
9.991947

.40 i
.42
.40
.40
.42
.40

.285268
.285947
.286624
.287301
.287977
9.288652

11.33
11.32
11.28
11.28
11.27
11.25

.714732
.714053
.71&376
.712699
.712023
10.711348

5
4
3
2

0

'

Cosine.

D. r.

i Sine.

D. 1'.

Cotang.

D.I'.

Tang.

'

100°

185

11*

TABLE XII.— LOGARITHMIC SINES,

3G3

'

Sine.

D. 1'.

Cosine.

D. 1*.

Tang.

D. 1".

Cotang.

'

0

1

2
3

9.280599

.281248
.281897
.282544

10.82
10.82
10.78

9.991947
.991922
.991897
.991873

.42
.42

.40

9.288652
.289326
.289999
.290671

11.23
11.22
11.20

10.711348
.710674
.710001
.709329

60
59
58
57

4
5
6
7
8
9
10

.283190
.283836
.284480
.285124
.285766
.286408
.287048

10.77
10.77
10.73
10.73
10.70
10.70
10.67
10.67

.991848
.991823
.991799
.991774
.991749
.991724
.991699

.42
.42
.40
.42
.42
.42
.42
.42

.2913-12
.292013
.292682
.293350
.294017
.294684
.295349

11.18
11.18
11.15
11.13
11.12
11.12
11.08
11.07

.708658
.707987
.707318
.706650
.705983
.705316
.704651

56
55
54
53
52
51
50

11
12

9.287688
.288326

10.63

1ft A3

9.991674
.991649

.42

9.296013
.296677

11.07

10.703987
.70:3323

49

48

13

.288964

1U.DO

.991624

•~*

.297339

11 .03

.702661

47

14
15
16
17
18
19

.289600
.290236
.290870
.291504
.292137
.292768

10.60
10.60
10.57
10.57
10.55
10.52
1ft x&

.991599
.991574
.991549
.991524
.991498
.991473

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

A9

.298001
.298662
.299322
.299980
.300638
.301295

11.03
11.02
11.00
10.97
10.97
10.95

1ft QQ

.701999
.701338
.700678
.700020
.699362
.698705

46
45
44
43
42
41

20

.293399

1U.O*

10.50

.991448

.4/0

.43

.301951

lu.yo
10.93

.698049

40

21
22
23
24

25
26
27
28
29
30

9.294029
.294658
.295286
.295913
.296539
.297164
.297788
.298412
.299034
.299655

10.48
10.47
10.45
10.43
10.42
10.40
10.40
10.37
10.35
10.35

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

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

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

10.90
10.88
10.88
10.85
10.85
10.83
10.82
10.80
10.78
10.77

10.697393
.696739
.696086
.6954aS
.694782
.694131
.693481
.692832
.692184
.691537

39
38
37
36
35
34
33
32
31
30

31

32

as

34

9.300276
.300895
.301514
.302132

10.32
10.32
10.30

1ft O"

9.991167
.991141
.991115
.991090

.43
.43
.42

9.309109
.309754
.310399
.311042

10.75
10.75
10.72

10.690891
.690246
.689601
.688958

29

28
27
26

35

36

.302748
.303364

1U .tit

10.27

.991064
.991038

!43

.311685
.312327

10. "70

.688315
.687673

25
24

37
38

! 304593

10.25
10.23

1ft OQ

.991012
.990986

.43
.43

xo

.312968
.313608

10.68
10.67
ift fifi

.687032
.686392

23

22

39
40

.305207
.305819

1U .M

10.20
10.18

.990960
.990934

.**o
.43
.43

.314247
.314885

JU. DO

10.63
10.63

.685753
.685115

21
20

41
42
43

44
45

40
47
43
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
.682570
.681936
.681303
.680670
.680039
.679408
.678778

19
18
17
16
15
14
13
12
11
10

51

9.312495

1ft ftQ

9.990645

9.321851

10.678149

9

52
53
54
65
56
57
58
59

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

IU.UO

10.02
9.98
10.00
9.97
9.95
9.95
9.92

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

.45
.45
.43
.45
.45
.43
.45 :
.45 |

.322479
.323106
.323733
.324358
.3249&3
.325607
.326231
.326853

10^45
10.45
10.42
10.42
10.40
10.40
10.37

.677521
.676894
.676267
.675642
.675017
.674393
.673769
.673147

8
7
6
5
4
3
2
1

60

9.317879

9.92

9.990404

.45 j

9.327475

10.37

10.672525

0

1

Cosine.

D. r.

Sine.

D. 1". i

Cotang. D. 1'.

Tang.

'

101*

186

COSINES, TANGENTS, AND COTANGENTS.

167*

•

Sine.

D. r.

Cosine.

D. r.

Tang.

D. 1'.

Cotang.

•

0

1

2
3
4
5
6

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

9.90

9.88 :
9.87
9.85
9.85
9.83

9.990404
.990378
.990351
.990324
.990297
.990270
.990243

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

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

10.33
10.33
10.32
10.32
10.28
10.28

1ft 97

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

60
59
58
57
56
55
54

.322019

U'JH

.990215

.47

.331803

IV. Iff

.668197

53

8
9
10

.322607
.323194

.323780

9.80
9.78
9.77
9.77

.990188
.990161
.990134

.45
.45
.45
.45

.332418
.333033
.333646

10 '.25
10.22
10.22

.667582
.666967
.666354

52
51
50

11
12
13
14

9.324366
.324950
.325534

.326117

9.73
9.73 1
9.72 1

979

9.990107
.990079
.990052
.990025

.47
.45
.45

47

9.334259
.334871
.335482
.336093

10.20
10.18
10.18

10 1 ^

10.665741
.665129
.664518
.663907

49
48
47
46

15

16

.326700
.327281

. < <v

9.68

.989997
.989970

.44

.45

.336702
.337311

JU. 1O

10.15

.663298
.662689

45
44

17
18
19
20

.327862
.328442
.329021
.329599

9.68
9.67
9.65
9.63
9.62

.989942
.989915
.989887
.989860

'.45
.47
.45

.47

.337919
.338527
.339133
.339739

10. 13
10.13
10.10
10.10
10.08

.662081
.661473
.660867
.660261

43
42
41
40

21
22
23
24
25

9.330176
.330753
.331329
.331903
.332478

9.62
9.60
9.57
9.58

9KK

9.989832
.989804
.989777
.989749
.989721

.47
.45

.47

.47

9.340344
.340948
.341552
.342155
.342757

10.07
10.07
10.05
10.03

10 O9

10.659656
.659052
.658448
.657845
.657243

39
38
37
36
35

26

27
28

.333051
.333624
.334195

,OO

9.55
9.52

.989693
.989665
.989637

.47
.47
.47

.343358
.343958
i .344558

lU.UJs

10.00
10.00

9QQ

.656642
.656042
.655442

34
33

32

29
30

.334767
.335337

9.53
9.50
9.48

.989610

.989582

.45

.47

.48

.345157
.345755

.00

9.97
9.97

.654843
.654245

31
30

31
32

9.335906
.336475

9.48

9.989553
.989525

.47

9.346353
.346949

9.93

10.653647
.653051

29
28

33
34

.337043
.337610

9.47
9.45

.989497
.989469

.47

.47

.347545
.348141

9!93

.652455
.651859

27
26

35

.338176

9.43

.989441

.47

.348735

9 Oft

.651265

25

36

.338742

9.43

.989413

.47

.349329

.yu

9 Oft

.650671

24

37
38
39
40

.339307
.339871
.340434
.340996

9.42
9.40
9.38
9.37
9.37

.989385
.989356
.989328
.989300

.47
.48
.47
.47
.48

.349922
.350514
.351106
.351697

.00

9.87
9.87
9.85
9.83

.650078
.649486
.648894
.648303

23

22
21
20

41
42
43

9.341558
.342119
.342679

9.35
9.33

9.989271
.989243
.989214

.47

.48

9.352287
.352876
.353465

9.82
9.82

9ftO

10.647713
.647124
.646535

19
18
17

44
45
46
47

.343239
.343797
.344355
.344912

9.33
9.30
9.30
9.28

99ft

.989186
.989157
.989128
.989100

.47
.48
.48
.47

AQ.

.354053
.354640
.355227
.355813

.oU

9.78
9.78
9.77
9 75

.645947
.645360
.644773
.644187

16
15
14
13

48
49

.345469
.346024

.ico

9.25

.989071
.989042

.4o
.48

.356398
.356982

9>3

97°.

.643602
.643018

12
11

50

.346579

9.25
9.25

! .989014

.47
.48

.357566

. i O

9.72

.642434

10

51

52
53

9.347134
.347687
.348240

9.22
9.22

! 9.988985
.988956
.988927

.48
.48

9.358149
.358731
.359313

9.70
9.70
9 67

10.641851
.641269
.640687

9
8

7

54

.348792

9.20

.988898

'AO

.359893

9fift

.640107

6

55

.349343

9.18

.988869

.48

.360474

.DO

9 65

.639526

5

56

.349893

9.17

.988840

*^40

.361053

9CK

.638947

4

57

.350443

9.17

.988811

.48

.361632

.DO
9 A0.

.638368

3

58
59

.350992
.351540

9.15
9.13

.988782
! .988753

.48
.48

.362210
.362787

.Do

9.62

91-.)

.637790
.637213

2
1

60

9.352088

9.13

I 9.988724

.48

9.3C3364

. 0*

10.636636

0

'

Cosine.

D. 1".

i Sine,

D. 1".

Cotang.

D. r.

Tang.

'

IBS'

187

TABLE XII. — LOGARITHMIC SINES,

166°

'

Sine.

D. r.

Cosine.

D. 1'.

Tang.

D. 1".

Cotang.

/

0

9.352088

919

9.988724

48

9.363364

9rn

10.636636

CO

I

3
4

5
6
7
8
9
10

.352635
.353181
.353726
.354271
.354815
.355358
.355901
.356443
.356984
.357524

. 1.-*

9.10
9.08
9.08
9.07
9.05
9.05
9.03
9.02
9.00
9.00

.988695
.988666
.988636
.988607
.988578
.988548
.988519
.988489
.988460
.988430

.48

.48
.50
.48
.48
.50
.48
.50
.48
.50
.48

.363940
.364515
.365090
.365664
.366237
.366810
.367382
.367953
.368524
.369094

.uU

9.58
9.58
9.57
9.55
9.55
9.53
9.55
9.52
9.50
9.48

.G36060
.635485
.634910
.634336
.633763
.033190
.632618
.632047
.631476
.630906

59
58
57
56
55
54
53
52
51
50

11

9.358064

8QQ

9.988401

r A

9.369663

10.630337

49

12

.358603

.«7O

8Q7

.988371

.OU

.370232

9.48

.629768

48

13

.359141

.•ft

.988342

" tA

.370799

9.45

.629201

47

14

.359678

8.95

8QK

.988312

.50

t A

.371367

9.47

.628633

46

15
16
17

18
19

.360215
.360752
.361287
.361822
.362356

.yo
8.95
8.92

8.92
8.90

8 88

.988282
.988252
.988223
.988193
.988163

.OU

.50
.48
.50
.50
fc/'i

.3719&3
.372499
.373064
.373629
.374193

9.43
9.43
9.42
9.42
9.40

9QQ

.628067
.627501
.626936
.626371
.625807

45
44
43
42
41

20

.362889

8^88

.988133

.ou
.50

.374756

.00

9.38

.625244

40

21
22
23

9.363422
.363954
.364485

8.87
8.85

SDK

9.988103
.988073
.988043

.50
.50

9.375319
.375881
.376442

9.37
9.35

10.624681
.624119
.623558

39

38
37

24

.365016

,oO

8 QO

.988013

.50

tA

.377003

9.35

90°,

.622997

36

25
26

.365546
.366075

.00

8.82

.987983
.987953

.OU

.50
iy>

.377563
.378122

.00
9.32

.622437
.621878

35
34

27
28

.366604
.367131

8>8

8 DA

.987922

.987892

!o5

tA

.378681
.379239

9! 30

90A

.621319
.620761

33

32

29

.367659

.OU

8r->-»

.987862

. OU

frft

.379797

.OU

.620203

31

30

.368185

. t 1

8.77

.987832

.OU

.52

.380354

9.28
9.27

.619646

30

31
32

9.368711
.369236

8.75

8r*c

9.987801
.987771

.50

9.380910
.381466

9.27

10.619090
.618534

29

28

33
34

.369761
.370285

. < •>

8.72

.987740
.987710

!M

.382020
.382575

9l25

.617980
.617425

27
26

35

.370808

8f~o

.987679

.52

tA

.383129

Q 99

.616871

25

36
37

.371&30
.371852

. lO

8.70

SAP,

.987649

.987618

.OU

.52

tA

.383682
.384234

9^20

.616318
.615766

24

23

38

.372373

.Do

8 CO

.987588

.OU

.384786

918

.615214

22

39

.372894

.Do

.987557

.O-v

.385337

. 1O

.614663

21

40

.373414

8.67

8.65

.987526

.52
.50

.385888

9.18
9.17

.614112

20

41
42
43
. 44

9.373933
.374452
.374970
.375487

8.65
8.63
8.62

9.987496
.987465
.987434
.987403

.52
.52
.52

9.386438
.386987
.387536
.388084

9.15
9.15
9.13

919

10.613562
.613013
.612464
.611916

19

18
17
16

45

.376003

8.60
8 fin

.987372

.52

.388631

. lii
919

.611369

15

46
47

.376519
.377035

,ou
8.60

.987341
.987310

!52

.389178
.389724

. I ~

9.10

.610822
.610276

14
13

48

.377549

8.57

.987279

.52

.390270

9.10

9AQ

.609730

12

49

.378063

8.57

.987248

.52

.390815

.Uo

9 08

.609185

11

50

.378577

8l53

.987217

!52

.391360

9^05

.608640

10

51

52

9-379089
.379601

8.53

9.987186
.987155

.52

9.391903
.392447

9.07

9AO

10.608097
.607553

9

8

53
54

55
56
57
58
59
60

.380113
.380624
.381134
.381643
.382152
.382661
.383168
9.383675

8.53
8.52
8.50
8.48
8.48
8.48
8.45
8.45

.987124
.987092
.987061
.987030
.986998
.986967
.986936
9.986904

.52
.53

.52
.52
.53

.52
.52
.53

.392989
.393531
.394073
.394614
.395154
.395694
.396233
9.396771

.Uo

9.03
9.03
9.02
9.00
9.00
8.98
8.97

.607011
.606469
.005927
.005386
.604846
.604306
.803767
10.603229

6
5
4
3
2
1
0

'

! Cosine.

D. 1s.

Sine.

D. 1*.

Cotang.

D. r.

Tang.

'

188

7.6"

14°

COSINES, TANGENTS, AND COTANGENTS.

165°

'

Sine.

D. r.

Cosine. D. r.

Tang.

D. 1'.

Cotang.

>

0

1

2

3

9.383675
.384182
.384687
.385192

8.45
8.42
8.42

849

9.986904
.986873
.986841
.986809

.52
.53
.53

9.396771
.397309
.397846
.398383

8.97
8.95
8.95

10.603229
.602691
.602154
.601617

60
50
58
57

4

.385697

.•*«

8 40

.986778

53

.398919

8 QO

.601081

56

5

.386201

8OQ

.986746

KQ

.399455

.yo

8Q9

.600545

55

6

.386704

.OO
800

.986714

.Do
to

.399990

• MB

Son

.600010

54

7
8
9

.387207
.387709
.388210

. OO

8.37
8.35

8 OK

.986683
.986651
.986619

tiXg

.53
.53

KQ

.400524
.401058
.401591

.yu
8.90

8.88

Q 00

.599476
.598942
.598409

53
52
51

10

.388711

.OO

8.33

.986587

.DO

.53

.402124

o.OO

8.87

.597876

50

11

12

9.389211
.389711

8.33

8 Of)

9.986555
.986523

.53

to

9.402656
.403187

8.85

10.597344
.596813

49

48

13
14

.390210
.390708

.oa

8.30

8O.fl

.986491
.986459

.DO

.53

tq

.403718
.404249

8.85
8.85

8P.9

.596282
.595751

47
46

15

.391206

.OU

8 28

.986427

.Do

KO

.404778

.0/6

8QQ

.595222

45

16

.391703

897

.986395

.Do

to

.405308

.Oo
Sftrt

.594692

44

17
18
19
20

.392199
.392695
.393191
.393685

. <,(

8.27
8.27
8.23
8.23

.986363
.986331
.986299
.986266

.Do

.53
.53
.55
.53

.405836
.406364
.406892
.407419

.OU

8.80
8.80
8.78
8.77

.594164
.593636
.593108
.592581

43
42
41
40

21
22
23

9.394179
.394673
.395166

8.23

8.22

9.986234
.986202
.986169

.53
.55

to

9.407945
.408471
.408996

8.77
8.75

8r»K

10.592055
.591529
.591004

39

38
37

24
25
26

27

.395658
.396150
.396641
.397132

8^20
8.18
8.18

.986137
.986104
.986072
.986039

.Do

.55
.53
.55

.409521
.410045
.410569
.411092

. to
8.73
8.73
8.72

.590479
.589955
.589431
.588908

36
35
34
33

28

.397621

8.15

81 •"*

.986007

tt

.411615

8rvj

.588385

32

29

.398111

. \(

8 -1C

.985974

.DO

KO

.412137

. lU

SCO

.587863

31

30

.398600

. ID

8.13

.985942

.Do

.55

.412658

. OO

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
34
35

.400062
.400549
.401035

8.12
8.12
8.10
Srtft

.985843
.985811
.985778

.55
.53
.55

KK

.414219
.414738
.415257

8. '65
8.65

Q CO

.585781
.585262
.584743

27
26
25

36
37
38
39
40

.401520
.402005
.402489
.402972
.403455

.Uo

8.08
8.07
8.05
8.05
8.05

.985745
.985712
.985679
.985646
.985613

.DO

.55
.55
.55
.55
.55

.415775
.416293
.416810
.417326
.417842

O.OO

8.63
8.62
8.60
8.60
8.60

.584225
.583707
.583190
.582674
.582158

24
23
22
21
20

41
42
43
44

9.403938
.404420
.404901
.405382

8.03

8.02
8.02

9.985580
.985547
.985514
.985480

.55
.55
.57

9.418358
.418873
.419387
.419901

8.58
8.57
8.57

10.581642
.581127
.580613
.580099

19

18
17
16

45

.405862

8.00

7QQ

.985447

.55
tt

.420415

8.57

Q KK

.579585

15

46

.406341

.yo

.985414

.DO

.420927

O.OO

8tt

.579073

14

47

.406820

7.98

7 Qft

.985381

.55

.421440

.00

8KO

.578560

13

48

.407299

t .yo

.985347

.57

.421952

.DO

.578048

12

49

.407777

7.97

7Qt

.985314

.55

K7

.422463

8.52

.577537

11

50

.408254

.yo
7.95

.985280

.Of

.55

.422974

8.50

.577026

10

51

8.408731

9.985247

9.423484

10.576516

9

52

.409207

7.93

.985213

.57

.423993

o.4o

8Krt

.576007

8

53

54
55

.409682
.410157
.410632

7.93
7.92
7.92

f on

.985180
.985146
.985113

.55
.57
.55

K7

.424503
.425011
.425519

. DU

8.47
8.47

847

.575497
.574989
.574481

7
6
5

56

.411106

i . yu

.985079

.O<

.426027

.4<

8AK

.573973

4

57

58

.411579
.412052

7.88
7.88

.985045
.985011

.57

.57

.426534
.427041

.40

8.45

.573466
.572959

3
2

59

.412524

7.87

.984978

.55

.427547

8.43
840

.572453

1

60 9.412996

7.87

9.984944

.57

9.428052

.4/v

10.571948

0

'gl Cosine.

D. r.

Sine.

D. r.

Cotang. D. 1'.

Tang.

'

189

75°

15°

TABLE XII. — LOGARITHMIC SIKES,

164°

•

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

9.412996

7 Of

9.984944

9.428052

10.571948

60

1

.413467

.00

7 8?«

.984910

R-»

.428558

8.43

84ft

.571442

59

2
3

.413938
.414408

I .oO

7.83

.984876
.984842

•91

.57

.429062
.429566

.4U

8.40

.570938
.570434

58
57

4
5
6

8
9
10

.414878
.415347
.415815
.416283
.416751
.417217
.417684

7.83
7.82
7.80
7.80
7.80
7.77
7.78
7.77

.984808
.984774
.984740
.984706
.984672
.984638
.984603

.57
.57
.57
.57
.57
.57
.58
.57

.430070
.430573
.431075
.431577
.432079
.432580
.433080

8.40
8.38
8.37
8.37
8.37
8.35
8.33
8.33

.569930
.569427
.568925
.568423
.567921
.567420
.566920

56
55
54
53
52
51
50

11

12
13

9.418150
.418615
.419079

7.75
7.73

9.984569
.984535
.984500

.57

.58

9.433580
.434080
.434579

8.33

8.32

10.566420
.565920
.565421

49

48
47

14
15
16

.419544
.420007
.420470

7.75

7.72
7.72

.984466
.984432
.984397

.'57
.58

.435078
.435576
.436073

8.32
8.30

8.28

.564922
.564424
.563927

46
45
44

17
18
19
20

.420933
.421395
.421857
.422318

7.72
7.70
7.70
7.68
7.67

.984363
.984328
.984294
.984259

.57
.58
.57
.58
.58

.436570
.437067
.437563
.438059

8.28
8.28
8.27
8.27
8.25

.563430
.562933
.562437
.561941

43
42
41

40

31
22
23

9.422778
.423238
.423697

7.67
7.65

9.984224
.984190
.984155

.57
.58

9.438554
.439048
.439543

8.23
8.25

10.561446
.560952
.560457

39

38
37

24
25

.424156
.424615

7.65
7.65

.984120
.984085

.58
.58

.440036
.440529

8.22
8.22

.559964
.559471

36
35

26
27

.425073
.425530

7.63
7.62

.984050
.984015

.58
.58

.441022
.441514

8.22
8.20

.558978
.558486

34

as

28
29

425987
.426443

7.62
7.60

r» (*n

.983981
.983946

.57
.58

KO

.442006
.442497

8.20
8.18

.557994
.557503

32
31

30

.426899

i . OU

7.58

.983911

.OO

.60

.442988

8J8

.557012

30

31

9.427354

9.983875

9.443479

10.556521

29

32

.427809

7.58

.983840

n

.443968

8.15

.556032

28

33

.428263

7.57

.983805

.58

.444458

8.17

.555542

27

34
35

.428717
.429170

7.57
7.55

7KK

.983770
.983735

'.58

KO

.444947-
.445435

8.15
8.13

.555053
.354565

26
25

36

.429623

.OO

.983700

,OO

.445923

® • *£

.554077

24

37

.430075

7.53

.983664

.60

.446411

8.13

.553589

23

38
39
40

.430527
.430978
.431429

7 53
7.52
7.52
7.50

.983629
.983594
.983558

.58
.58
.60

.58

.446898
.447384
.447870

8.12
8.10
8.10
8.10

.553102
.552616
.552130

22
21
20

41
42

43

9.431879
.432329
.432778

7.50

7.48

9.983523
.983487
.983452

.60

.58

9.448356
.448841
.449326

8.08
8.08

10.551644
.551159
.550674

19
18
17

44

45
46
47

48
49
50

.433226
.433675
.434122
.434569
.435016
.435462
.435908

7.47
7.48
7.45
7.45
7.45
7.43
7.43
7.42

.983416
.983381
.983345
.983309
.983273
.983238
.983202

.60
.58
.60
.60
.60
.58
.60
.60

.449810
.450294
.450777
.451260
.451743
.452225
.452706

8.07
8.07
8.05
8.05
8.05
8.03
8.02
8.02

.550190
.549706
.549223
.548740
.548257
.547775
.547294

16
15
14
13
12
11
10

51

9.436353

9.983166

9.453187

10.546813

9

52
53

.436798
.437242

7.42
7.40

.983130
.983094

.60
.60

.453668
.454148

8 02
8.00

.546332
.545852

8

7

54

.437685

7.40

.983058

.60

.454628

8.00

.545372

6

55

.438129

7.38

.983022

.60

.455107

".98

.544893

5

56

.438572

7.38

.982986

.60

.455586

".98

.544414

4

57

.439014

7.37

.982950

.60

.456064

f'tft

.543936

3

58

.439456

7.37

.982914

.60

.456542

.9«

.543458

2

59
60

.439897
9.440338

7.35
7.35

.982878
9.982842

.60
.60

.457019
9.457496

.95

-.95

.542981
10.542504

1
0

t

Cosine.

D. 1*.

Sine.

D. r.

Cotang.

D. r.

Tang.

'

190

74°

16"

COSINES, TANGENTS, AND COTANGENTS.

'

Sine.

D. 1".

Cosine.

D. r.

Tang.

D.I".

Cotang.

'

0

9.440338

r* OO

9.982842

9.457496

10.542504

60

1

.440778

< .OO
r- qq

.982805

'fift

.457973

r- i o

.542027

59

2

.441218

1 .OO

7 qo.

.982769

.ou
fift

.458449

. ;)o
•~ QQ

.541551

58

3

.441658

i .OO
7 °.ft

.982733

.ou

fi9

.458925

. yo

1" QS)

.541075

57

4

.442096

i .OU
7 %>

.982696

.0/6

fift

.459400

.4MB

.540600

56

5

442535

i .&£

7OA

.982660

.OU
fift

.459875

r» QA

.540125

55

6

.442973

.OU

7 28

.982624

.ou
62

.460349

. yu

.539651

54

7

.443410

79tt

.982587

.460823

i*'JH(

539177

53

8

.443847

.-60

7 9ft

.982551

.60

.461297

.90

r» QQ

.538703

52

9

.444284

i . -6O

7 27

.982514

*62

.461770

.Oo

.538230

51

10

.444720

7i25

.982477

!eo

.462242

^88

.537758

50

11

9.445155

r» OK.

9.982441

9.462715

fr OK

10.537285

49

12

.445590

-•''OR

.982404

fi9

.463186

.oO

.536814

48

13
14

.446025
.446459

7l23

.982367
.982331

.0/6

.60

.463658
.464128

-\83

rr OK

.536342
.535872

47
46

15

.446893

7 22

.982294

*R9

.464599

.oO
r- oo

.535401

45

16
17

18

.447326
.447759
.448191

7^22
7.20

.982257
.982220
.982183

.'62
.62

.465069
.465539
.466008

.OO

7.83
7.82

7 QO

.534931
.534461
.533992

44
43
42

19

.448623

7 18

.982146

'62

.466477

< .0/6

7 80

.533523

41

20

.449054

.982109

.466945

7^80

.533055

40

21

9.449485

7 17

9.982072

fi9

9.467413

7 7fi

10.532587

39

22

.449915

1 .11

.982035

.04
fi9

.467880

i . to
7 '"ft

.532120

38

23

.450345

717

.981998

.UM

.468347

4 . iO
7 7ft

.531653

37

24

.450775

. li
r» -IK

.981961

fi9

.468814

^ . <o

777

.531186

36

25

.451204

i . 1O
7 1°.

.981924

.tw

fiP.

.469280

. < <

7 77

.530720

35

26
27

28
29

.451632
.452060
.452488
'.452915

< . 1O

7.13
7.13

7.12

719

.981886
.981849
.981812
.981774

.DO

.62
.62
.63

.469746
.470211
.470676
.471141

7^75
7.75

7.75

.530254
.529789
.529324
.528859

34
33
32
31

30

.453342

. i~

7.10

.981737

!62

.471605

7! 73

.528395

30

31

9.453768

f -tn

9.981700

9.472069

7 79

10.527931

29

32

.454194

i . 1U
*' ftft

.981662

* on

.472532

i .tX

7 79

527468

28

33
34
35

36

.454619
.455044
.455469
.455893

i .Uo

7.08
7.08
7.07

r? AK

.981625
.981587
.981549
.981512

. o2
.63
.63
.62

.472995
.473457
.473919
.474381

1 . ( -V

7.70
7.70
7.70

7 fift

.527005
.526543
.526081
.525619

27
26
25
24

37

38

.456316
.456739

I .UO

7.05

.981474
.981436

.63
.63

.474842
.475303

< .DO

7.68
i .67

.525158
.524697

23
22

39
40

.457162
.457584

7^03
7.03

.981399
.981361

.62
.63
.63

.475763
.476223

7.67
7.67

.524237
.523777

21
20

41
42

9.458006

.458427

7.02

9.981323
.981285

.63

9.476683
.477142

7.65

10.523317

.522858

19

18

43
44

.458848
.459268

7.02
7.00

7 OA

.981247
.981209

!«3

on

.477601
.478059

7! 63

7 fi^

.522399
.521941

17
16

45

.459688

i .JO

.981171

.DO

.478517

< .DO

rt on

.521483

15

46

.460108

7.00

.981133

.63

.478975

t .DO

.521025

14

47

.460527

n 'r\o

.981095

»52

.479432

7 62

.520568

13

48

.460946

O.98

6Q7

.981057

.DO
on

.479889

7 60

.520111

12

49
50

.461364
.461782

.»<

6.97
6.95

.981019
.980981

.DO

.63
.65

.480345
.480801

7^60
7.60

.519655
.519199

11
10

51

9.462199

6AR

9.980942

on

9.481257

7 58

10.518743

9

52

.462616

.yo

.980904

.DO

pq

.481712

7 58

.518288

8

53

54
55
56
57
58
59
60

.463032
.463448
.463864
.464279
.464694
.465108
.465522
9.465935

6.93
6.93
6.93
6.92
6.92
6.90
6.90
6.88

.980866
.980827
.980789
.980750
980712
.980673
.980635
9.980596

.O.3

.65
.63
.65
.63
.65
.63

.482167
.482621
.483075
.483529
.483982
.484435
.484887
9.485339

7^57
7.57
7.57
7.55
7.55
7.53
7.53

.517833
.517379
.516925
.516471
.516018
.515565
.515113
10.514661

7
6
5
4
3
2

1
0

~7~

Cosine.

D. 1'.

Sine.

D. 1".

Cotang.

D. r.

Tang.

*

191

73*

17°

TABLE XII. — LOGARITHMIC SINES,

162°

'

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

9.485935

600

9.980596

no

9.485339

7KO

10.514661

60

1

.466348

.00

6QQ

.980558

.Oo
A5;

.485791

.Oo

.514209

59

g

3

.466761
.467173

.OO

6.87
607

.980519
.980480

.00

.65

A°.

.486242
.486693

7.52
7.52

7 fUl

.513758
.513307

58
57

4

.467585

.o<

6QK

.980442

.Oo

65

.487143

f .OU
7 en

.512857

56

5

.467996

.OO

6 OK

.980403

p~

.487593

.OU

.512407

55

6

7

.468407

.468817

.00

6.83

6QQ

.980364
.980325

.OO

.65 i

.488043
.488492

7.50

7.48

7 4ft

.511957
.511508

54
53

8

.469227

.OO
6QQ

.980286

(•e

.488941

f .4O

.511059

52

9

.469637

.OO

.980247

.00

.489390

JL*55

.510610

51

10

.470046

6.82
6.82

.980208

.65
.65

.489838

i .47
7.47

.510162

50

11
12

9.470455

.470863

6.80

6 on

9.980169
.980130

.65

9.490286
.4JW733

7.45

10.509714
.509267

49

48

13
14

.471271
.471679

.OU

6.80

6 '"ft

.980091
.980052

'.65

A7

.491180
.491627

7.45
7.45

7 4°.

.508820
.508373

47
46

15
16

.472086
.472492

. tty
6.77

677

.980012
.979973

.0*

.65

A^

.492073
.492519

f .<*O

7.43

.507927
.507481

45
44

17
18
19
20

.472898
.473304
.473710
.474115

. < i

6.77
6.77
6.75 I
6.73

.979934
.979895
.979855
.979816

.OO

.65
.67
.65
.67

.492965
.493410
.493854
.494299

7^42
7.40

7.42
7.40

.507035
.506590
.506146
.505701

43

42
41
40

21
22

9.474519
.474923

6.73

9.979776
.979737

.65

9.494743
.495186

7.38

10.505257

.504814

39
38

23

.475327

6 . 73 j

079

.979697

.67

.495630

7.40
700

.504370

37'

24
25

.475730
.476133

. < •- i

6.72 j

679

.9?0658
.979618

!67

BK

.496073
.496515

.Oo

7.37
707

.503927
.503485

36
35

26

.476536

. lA ••
67ft

.979579

.OO

.496957

.of
707

.503043

34

27

.476938

• ' ^
67ft

.979539

.67

A7

.497399

.01

7.37

.502601

33

28

.477340

. <U
6 Aft

.979499

.Of

.497841

7OK

.502159

32

29

.477741

.Oo

.979459

' ct

.498282

.oO

.501718

31

30

.478142

6.68
6.67

.979420

.05
.67

.498722

7!35

.501278

30

31

32

9.478542
.478942

6.67

9.979380
.979340

.67

9.499163
.499603

7.33

10.500837
.500397

29

28

33
34
35
36
37

.479342
.479741
.480140
.480539
.480937

6 '.65
6.65
6.65
6.63

.979300
.979260
.979220
.979180
.979140

!67
.67
.67
.67

A7

.500042
.500481
.500920
.501359
.501797

7^32
7.32
7.32

Z'30

.499958
.499519
.499080
.498641
.498203

27
26
25
24
23

38

.481334

« R9

.979100

.Of

Aft

.502235

7 9ft

.497765

22

39

40

.481731
.482128

6 '.62
6.62

.979059
.979019

.Oo

.67
.67

.502672
.503109

f .*o

7.28
7.28

.497328
.496891

21
20

41

9.482525

6 Aft

9.978979

A7

9.503546

797

10.496454

19

42

.482921

.OU

.978939

.0*

.503982

.<£<

.496018

18

43

.483316

6.58

6 Aft

.978898

.68

A7

.504418

7.27

797

.495582

17

41
45

.483712
.484107

.OU

6.58

.978858
.978817

.Of

.68

.504854
.505289

.lit

7.25

.495146
.494711

16
15

46
47

.484501
.484895

6.57
6.57

6K7

.978777
.978737

.67
.67

Aft

.505724
.506159

7.25
7.25

7 9°.

.494276
.493841

14
13

48
49
50

.485289
.485682
.486075

.01

6.55
6.55
6.53

.978696
.978655
.978615

.Oo

.68
.67
.68

.506593
.507027
.507460

1 .<CO

7.23
7.23
7.22

.493407
.492973
.492540

12
11
10

51

»9.486467

6KK

9.978574

Aft

9.507893

10.492107

9

52
53

.486860
.487251

.OO

6.52

6 CO

.978533
.978493

.Oo

.67

Aft

.508326
.508759

7! 28

7 20

.491674
.491241

8

7

54

.487643

.00

6KO

.978452

.Oo
Aft

.509191

71ft

.490809

6

55

.488034

.OSB

6ff\

.978411

.Oo
Aft

.509622

. lo

. 490378

5

56

.488424

.OU
6 en

.978370

.Oo
AS

.510054

7 1ft

.489946

4

57

.488814

.OU
6 en

978329

.Oo
Aft

.510485

f . Jo
71ft

.489515

3

58

.489204

.OU

.978288

.Oo

.510916

. lo

.489084

2

59

.489593

6.48
6.48

.978247

.68

Aft

.511346

7.17

717

.488654

1

60

9.489982

9.978206

.Oo

9.511776

i . 1 f

10.488224

0

'

Cosine.

D r.

Sine.

D. i".

Cotang.

D.r.

Tang.

'

107'

192

72*

18°

COSINES, TANGENTS, AND COTANGENTS.

161°

•

Sine.

D. 1'.

Cosine.

D. 1".

Tang.

D. 1'.

Cotang.

'

0 ! 9.489982

6 48

9.978206

C.Q

9.511776

r. «r. 10.488224

60

1 .490371

647

.978165

.Do
Aft

.512206

< . J i
7 Ifi

.487794

59

2 .490759

.'*<

.978124

.DO
Aft

.512635

.487365

58

3
4

.491147
.491535

6.47
6.47

6AK

.978083
.978042

.Do

.68

Aft

.513064
.513493

7.15
7.15

.486936
.486507

57
56

5
6

.491922
.492308

.40

6.43

64s!

.978001
.977959

.Do

.70

Aft

.513921
.514349

. 7.13
7.13

f -JO

.486079
.485651

55
54

7

.492695

.-SO

.977918

.Do

.514777

< .Jo

.485223

53

8

.493081

6 42

.977877

^n

.515204

7.12

719

.484796

9

.493466

649

.977835

Aft

.515631

. J -V

.484369

51

10

.493851

.•*«

6.42

.977794

.Do

.70

.516057

7.10
7.12

.483943

50

11

9.494236

6 42

9.977752

fift

9.516484

7 1ft

10.483516

49

12

.494621

.977711

.Do
7ft

.516910

< . 1U
7 ftft

.483090

48

13

.495005

6OO

.977669

. lU

Aft

.517335

< .Uo

.482665

47

14

.495388

.QO

6 40

.977628

.DO

.517761

7.10

r» no

.482239

46

15

.495772

607

.977586

'rv!

.518186

i .Uo

.481814

45

16

.496154

.01

6 38

.977544

. <0
Aft

.518610

7.07

r» fff

.481390

44

17

.496537

6 '07

.977503

.Do

tan

.519034

i . U<

.480966

43

18
19

.496919
.497301

.Of

6.37

6 WE

.977461
.977419

. iU
'1°

.519458
.519882

7!07

r» ne

.480542
.480118

42
41

20

.497682

. oo
6.35

.977377

'.70

.520305

t .UD

7.05

.479695

40

21
22
23
24
25
26
27
28

9.498064
.498444
.498825
.499204
.499584
.499963
.500342
.500721

6.33
6.35
6.32
6.33
6.32
6.32
6.32

6 on

9.977335
.977293
.977251
.977209
.977167
.977125
.977083
.977041

.70
.70
.70
.70
.70
.70
.70

9.520728
.521151
.521573
.521995
.522417
.522838
.523259
.523680

7.05
7.03
7.03
7.03
7.02
7.02
7.02

10.479272
.478849
.478427
.478005
.477583
.477162
.476741
.476320

39
38
37
36
35
34
33
32

29

.501099

.oU

.976999

.70

.524100

7.00

7 ftft

.475900

31

30

.501476

6^30

.976957

!72

.524520

I . UU

7.00

.475480

30

31
32
33

9.501854
.502231
.502607

6.28
6.27
6 28

9.976914
.976872
.976830

.70
.70

r»o

9.524940
.525359
.525778

6.98
6.98

6QQ

10.475060
.474641
.474222

29

28
27

34
35
36
37
38
39

.502984
.503360
.503735
.504110
.504485
.504860

6^27
6.25
6.25
6.25
6.25

.976787
.976745
.976702
.976660
.976617
.97&574

!70
.72
.70
.72

.72

.526197
.526615
.527033
.527451
.527868
.528285

.yo
6.97
6.97
6.97
6.95
6.95

.473803
.473385
.472967
.472549
.472132
.471715

26
25
24
23
22
21

40

.505234

6^23

.976532

.70
.72

.528702

6.95
6.95

.471298

20

41

9.505608

699

9.976489

r»o

9.529119

10.470881

19

42

.505981

.168
699

.976446

*7ft

.529535

A QQ

.470465

18

43

.506&54

,TSm

699

.976404

. i U

.529951

A Q9

.470049

17

44 .506727

.A8
69ft

.976361

*79

.530366

6Q9

.469634

16

45 .507099

.<DU

.976318

. i --

.530781

.MB

6Q9

.469219

15

46
47
48

.507471
.507843
.508214

6^20
6.18
61ft

.976275
.976232
.976189

.72
.72
.72

79

.531196
.531611
.532025

.We

6.92
6.90

6 Oft

.468804
.46&S89
.467975

14
13
12

49 ' .508585
50 j .508956

. Jo
6.18
6.17

.976146
.976103

. 1 -v

.72

.72

.532439
.532853

.yu
6.90
6.88

.467561
.467147

11
10

51

9.509326

9.976060

9.533266

6 Oft

10.466734

9

52

.509696

O.17

6-jC

.976017

. fm

79

.533679

.00
6 ftft

.466321

8

53

.510065

. JD

.975974

. i -

.534092

.oO

.465908

7

54

.510434

6.15

.975930

.73

.534504

6.87

6ft7

.465496

6

55

56

.510803
.511172

6.15
6.15
61°.

.975887
.975844

.72
.72

.534916
.535328

.of

6.87

.465084
.464672

5

4

57
58

.511540

.511907

.Jo

6.12

.975800
.975757

.73

.72

.535739
.536150

6^85

.464261
.463850

3
2

59
60

.512275
9.512642

6.13
6.12

.975714
9.975670

.72
.73

.536561
9.536972

6.85
6.85

.463439
10.463028

0

'

Cosine. I D. 1*. | i Sine. D. 1'.

Cotang. D. 1'.

Tang. I '

108"

193

71°

TABLE XII. — LOGARITHMIC SIXES,

1GO-

/

Sine.

D. 1". Cosine.

D. 1".

Tang.

D. 1".

Cotang.

/

i

0 9.512642
1 i .5J3009
2 i .513375

6.12
6.10

9.975670
.975627
.9755a3

.72
.73

9.536972
.537382
.537792

6.83

e. as

10.463028
.462618
.462208

CO
58

3

.513741

975539

.13
70

.538202

6.83

.461798

57

4

.514107

6. 10

.975496

.538611

6.82

.461389

56

5
0
7

.514472
.514837
.515202

6. ('8
6.08
6.08

.975452
.975408
.975365

.73
.73

.72

.539020
.539429
.539837

6.82
6.82
6.80

.460980
.460571
.460163

55

54
53

8

.515566

6.07

.975321

.73

.540245

6.80

.459755

52

9

.515930

6.07

.975277

. <O

.540053

6.80

.459347

51

10

.516294

6.07
6.05

.975233

.73

.73

.541061

6.80
6.78

.458939

50

11

9.516657

9.975189

9.541468

6r-o

10.458532

49

12
13

.517020
.517382

D.05
6.03

.975145
.975101

. <o
.73

.541875
.542281

. <0

6.77

.458125
.457719

48
47

14

.517745

6.05

6Aq

.975057

.73

.542688

6.78

677

.457312

46

15

.518107

.UQ

.975013

'r-q

.543094

. < (

.456906

45

16

.518468

O.OSS

.974969

. «<J

.543499

6. <5

.456501

44

17
18

.518829
.519190

6.02
6.02

.974925

.974880

.73

.75

.543905
.544310

6.77
6.75

.456095
.455690

43
42

19
20

.519551
.519911

6.02
6.00

.974836
.974792

.73
.73

.544715
.545119

6.75
6.73

.455285
.454881

41
40

6.00

.73

6.75

21
22
23

9.520271
.520631
.520990

6.00
5.98

9.974748
.974703
.974659

.75
.73

9.545524
.545928
.546&31

6.73

6.72

10.454476
.454072
.453669

39

38
37

24

.521349

5.98

.974614

.75

.546735

6.73

.453265

36

25

.521707

5.97

.974570

.73

.547138

6.72

.452862

35

26

.522066

5.98

.974525

.75

.547540

6.70

.452460

34

27

28

.522424
.522781

5.97
5.95

.974481
.974436

.73
.75

.547943
.548345

6.72
6.70

.452057
.451655

33
32

29

.523138

5.95

.974391

.75

.548747

6.70

.451253

31

30

.523495

5.95
5.95

.974347

.73
.75

.549149

6.70
6.68

.450851

30

31

9.523852

9.974302

9.549550

10.450450

29

32

.524208

5.9o

.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

.525275

5.92

.974122

.75

.551153

6.68

.448847

25

36

.525630

5.92

.974077

.75

.551552

6.65

.448448

24

37

.525984

5.90

.974032

.75

.551952

6.67

.448048

23

38

.526339

5.92

.973987

.75

.552351

6.65

.447649

22

39

.526693

5.90

.973943

.75

.552750

6.65

.447250

21

40

.527046

5.88
5.90

.973897

.75
.75

.553149

6.65
6.65

.446851

20

41

9.527400

9.973852

9.553548

10.446452

19

42
43

.527753
.528105

5.88
5.87

.973807
.973761

.75

.77

.553946
.554344

6.63
6.03

.446054
.445656

18
17

44

.528458

5.88

.973716

.75

.554741

6.62

.445259

16

45

46

.528810
.529161

5.87
5.85

.973671
.973625

.75

.77

.555139
.55' 536

6. 63
6.62

.444861
.444464

15
14

47

.529513

5.87

.973580

.75

.555933

6.62

.444067

13

48

.529864

5.85

.973535

.75

.550329

6.60

.443671

12

49

.530215

5.85

.973489

*m

.556725

6.60

.443275

11

50

.530565

5.83
5.83

.973444

'.77

.557121

6.60
6.60

.442879

10

51
52
53
54
55
56
57
58
59

9.530915
.531265
.531614
.531963
.532312
.532661
.533009
.533357
.533704

5.83

5.82
5.82
5.82
5.82
5.80
5.80
5.78

9.973398
.973352
.97.3307
.973261
.973215
.973169
.973124
.973078
.973032

.77
75
.77
.77
.77
.75
.77
.77

9.557517
.557913
.558308
.558703
.559097
.559491
.559885
.560279
.560673

6.60
6.58
6.58
6.57
6.57
6.57
6.57
6.57

10.4424aS
.442087
.441692
.441297
.440903
.440509
.440115
.439721
.439327

9
8
7
6
5
4
3
2
1

60

9.534052

5.80

9.972986

.77

9.561066

6.55

10.438934

0

'

Cosine.

D. 1".

Sine. D. 1".

Cotang.

D. r.

Tang. '

194

70-

20'

COSINES, TAN-GENTS, AND COTANGENTS.

'

Sine.

D. r.

Cosine.

D.I'.

Tang.

D. r.

Cotang.

'

0

1

2
3

9.534052
.534399
.534745
.535092

5.78
5.77

5.78
5.77

9.972986
.972940
.972894
.972848

77
.77
77
77

9 561066
.561459
561851
562244

6 55
6 53
6.55

6fcO

10.438934
.438541
438149
437756

60
59
58
57

4

.5354J38

972802

70

.562636

.DO

6 CO

.437364

56

5
6

535783
.536129

5.75
5.77

972755
.972709

<o

.77

77

.563028
563419

DO

6 52

6 tq

.436972
.436581

55
54

7 . 536474
8 .536818
9 537163

5 75
5.73
5.75

.972663
972617
972570

. i t

77
.78
77

563811
564202
564593

DO

6 52
6 52

6 en

436180
.4357'98
.436407

53
52
51

10

.537507

5.7'3
5.73

.972524

'.77

564983

OU

6.50

.435017

50

11

9.537851

9.972478

70

9.565373

6 en

10.434627

49

12
13
14
15
16
17
18

538194
538538
538880
.539223
539565
.539907
.540249

5.72
5.73
5.70

5.72
5.70
5.70

5.68

972431
972385
972338
.972291
.972245
.972198
.972151

. lO

.77
.78
.78
77
.78
.78

565763
.566153
566542
.566932
.567320
567709
.568098

.OU

6 50
6.48
6.50
6.47
6.48
6.48

.434237
.4433847
433458
.433068
.432680
.432291
.431902

48
47
46
45
44
43
42

19

20

.540590
.540931

5.68
5.68
5.68

.972105
.972058

.77
.78
.78

568486
566873

6^45
6.47

.431514
.431127

41
40

21

9.541272

5/>Q

9.972011

r-o

9.569261

6AK

10.430739

39

23

.541613
.541953

.DO

5.67

.971964
971917

. 40

.78
70

.569648
'• .570035

.40

6.45

6AK

.430352
.429965

38
37

24

25

. 542293
.542632

5.67
5.65

971870
.971823

. to

.78
70

.570422

.570809

.40

6.45

6AQ

.429578
.429191

36
35

26
27
20

29

.542971
.543310
543649
.543987

5.65
5.65
5.65
5.63

5t>n

.971776
.971729
.971682
.971635

. <o
.78
.78

.78

70

.571195
.571581
.571967
.572352

.40

6 43
6.43
6.42
6 43

.428805
.428419
.428033
.427648

34
33

32
31

30

.544325

.DO

5.63

.971588

. iO

.80

572738

6^42

.427262

30

31
32
33

9.544663
.545000
.545&3S

5.62
5.63

9.971540
.971493
.971446

.78
.78

Rft

9.573123
.573507
.573892

6.40

6.42

10.426877
.426493
.426108

29

28

27

34
35
36
37

.545674
.546011
. 546347
.546683

5.60
5.62
5.60
5.60

.971398
.971351
.971303
.971256

.OU

.78
.80

.78

.574276
.574660
.575044
.575427

e!40

6.40
6.38
600

.425724
.425340
.424956
.424573

26
25
24
23

38

.547019

5.60

.971208

.80

70

.575810

.00

60Q

.424190

22

39

.547354

5.58

.971161

. i O

.576193

.OO
6qO

.423807

21

40

.547689

5.58
5.58

.971113

80
.78

.576576

.OO

6.38

.423424

20

41

9.548024

K KQ

9.971066

on

9.576959

6 37

10.423041

19

42
43
44

. 548359
.548693
.549027

D.DO

5 57
5.57

.971018
.970970
.970922

.OU

.80
.80
on

.577341
.577723
.578104

6. '37
6.35
607

.422659

.422277
.421896

18
17
16

46

.549360

5.55

.970874

.oU

.578486

.o<

6 OK

.421514

15

46

.549693

5.55

.970827

.78

on

.578867

.00

6 OK

.421133

14

47

.550026

5.55

.970779

.OU

.579248

.OO
6 OK

.420752

13

48
49

.550359
.550692

5.55
5.55

5 to

.970731

.970683

.80
.80

on

.579629
.580009

.00

6.33
6 33

.420371
.419991

12
11

50

.551024

.DO

5.53

.970635

. oU

.82

.580389

6^33

.419611

10

51

9.551356

9.970586

8.580769

6oq

10.419231

9

52

.551087

5.52

.970538

.80

.581149

.00

6QO

.418851

8

53

552* -18

5.52

5 tf>

.970490

.80

c/n

581528

.cte

6 32

.418472

7

54

! 552349

,9m

.970442

.oU
on

581907

6qo

.418093

6

55

.552680

5.52

K KA

.970394

.oU

QO

.582286

.On

6 32

.417714

5

56

.553010

D. DU

.970345

.0.*

.582665

.417335

4

.553341

5.52

.970297

.80

.583044

6.32

.416956

3

58

.553070

5.48

.970249

.80

.583422

ft tm

.416578

2

59

.554000

5 . 50

.970200

.82

.583800

5'2j

.416200

1

60

9.554329

5.48

9.970152

.80

9.584177

o.2o

10.415823

0

'

Cosine.

D. 1".

Sine.

D. r.

Cotang.

D. r.

Tang.

'

110°

195

'TABLE XII. — LOGARITHMIC SINES,

158°

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

1

9.554329
.554658

5.48
t 40

9.97'0152
.970103

.82

9.584177
.584555

r on 1 10.415823
.415445

60
59

2
3
4
5
6

.554987
.555315
.555643
.555971
.556299

O.4o

5.47
5.47
5.47
5.47

5*K

.970055
.97'0006
.969957
.969909
.969860

.80
.82
.82 j
.80 !
.82

.584932
.585309
.585686
.586062
.586439

t).!«5

6.28
6.28
6.27
6.28

.415068
.414691
.414314
.413938
.413561

58
57
56
55
54

.556626

.4D

.969811

.82

.586815

6.27

.413185 53

8
9

.556953

.557'280

5.45
5.45

5 An

.969762
.969714

.82
.80

.587190
.587566

6.25
6.27

.412810
.412434

52
51

10

.557606

.4o
5.43

.969665

.82

.82

.50,941

6.25
6.25

.412059

50

11

9.557932

5 JO

9.969616

CO

9.588316

6 Of?

10.411684

49

12

.558258

.4o

.969567

.0,4

.588691

.<&)

.411309

48

13

.558583

5.42

.969518

.82

.589066

6.25

.410934

47

14
15

.558909
.559234

5.43
5.42

.969469
.969420

.82
.82

.589440
.589814

6.23
6.23

.410560 ! 46
.410186 ! 45

16

.559558

5.40

.969370

.83

.590188

6.23

.409812

44

17

18
19

.559883
.560207
.560531

5.42
5.40
5.40

.969321
.969272
.969223

.82
.82
.82

.590562
.5909a5
.591308

6.23
6.22
6.22

.409438
.409065
.408692

43
42
41

20

.560855

5.40
5.38

i .969173

.83
.82

.591681

6.22
6.22

.408319

40

21

9.561178

9.969124

9.592054

10.407946

39

22

.561501

5.o8

5OQ

.969075

.82

.592426

6.20

.407574

38

23

.561824

.00
507

.969025

.83

CO

.592799

6.22

.407201

37

24
25
26

.562146
.562468
.562790

.01

5.37
5.37

.968976
.968926
.968877

.04

.83

.82

.593171
.593542
.593914

6^18
6.20

.406829
.406458
.406086

36
35
34

27

.563112

5.37

.968827

.83

.594285

6.18

.405715

33

28

.563433

5.35

.968777

.83

.594656

6.18

.4051344

32

29

.563755

5.37

5OO

.968728

.82

.595027

6.18

.404973

31

30

.564075

.66

5.35

.968678

.83
.83

.595398

6.18
6.17

.404602

30

31

9.564396

5qq

9.968628

9.595768

10.404232

29

32

.564716

.00

.968578

.83

.596138

6.17

.403862 ! 28

33

.565036

5.33

.968528

.83

.596508

6.17

.40:3492 27

34
35

.565356
.565676

5.33
5.33

.968479
.968429

.82
.83

.596878
.567247

6.17
6.15

.403122 26
.402753 25

36

.565995

5.32

.968379

.83

.597616

6.15

.402384 24

37
38
39

.566314
.566632
.566951

5.32
5.30
5.32

.968329
.968278
.968228

.83
.85
.83

.597985
.598354
.598722

6.15
6.15
6.13

61 K.

.402015
.401646
.401278

23
22
21

40

.567269

5.30
5.30

.968178

.83
.83

.599C91

.lO

6.13

.400909

20

41

9.567587

9.968128

9.599459

10.400541

19

42
43

.567904
.568222

5^30

5 Oft

.968078
.968027

.83
.85

.599827
.600194

6.13
6.12

.400173 18
.399806 ! 17

44

.568539

.40

.967977

.83

.600562

6.13

.399438 ! 16

45
46

.568856
.569172

5.28
5.27

.967927
.967876

.83
.85

.600929
.601296

6.12
6.12

.399071 15
.398704 i 14

47

.569488

5.27

.967826

.83

.601663

6.12

.398337

13

48

.569804

5.27

.967775

.85

.602029

6.10

.397971

12

49
50

.570120
.570435

5.27
5.25

5.27

.967725
.967674

.83
.85
.83

.602395
.602761

6.10
6.10
6.10

.397605
.397239

11
10

51

52

9.570751
.571066

5.25

9.967624
.967573

.85

9.603127
.603493

6.10

10.396873
.396507

9
8

53
54

.571380
.571695

5.23
5.25

.967522
.967471

.85
.85

.603858
.604223

6.08
6.08

.396142
.395777

6

55

.572009

5.23

.967421

.83

.604588

6.08

.395412

5

56

.572323

5.23

.967370

.85

.604953

6.08

.395047

4

57

.572636

5.22

.967319

.85

.605317

6.07

.394683

3

58
59
60

.572950
.573263
9.573575

5.23
5.22
5.20

.967268
.967217
9.967166

.85
.85

.85

.605682
.606046
9.606410

6.08
6.07
6.07

.394318
.393954
10.393590

2
1
0

'

Cosine.

D. 1".

Sine.

D. 1". 1 1 Cotang.

D. 1". 1 Tang.

111°

196

68°

COSINES, TANGENTS, AND COTANGENTS.

157*

'

Sine.

D. 1".

Cosine.

D. 1'.

Tang.

D. 1'.

Cotang.

•

0

9.573575

9.967166

QK

9.606410

10.393590

60

1

2

3
4
5
6

8

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

5^20
5.20
5.20
5.20
5.18
5.18
5.18

517

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

.00

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

.606773
.607137
.607500
.607803
.608225
.608588
.608950
.609312

6.05
6.07
6.05
6.05
6.03
6.05
6.03
6.03

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

59
58
57
56
55
54
53
52

9

.576379

. li

517

.966705

.85

87

.609674

6.03
6 no

.390326

51

10

.576689

. 1 <

5.17

.966653

• Ol

.85

.610036

.Uo

6.02

.389964

50

11
12
13

14
15

9.576999
.577309
.577618
,577927
.578236

5.17
5.15
5.15
5 15

9.966602
.966550
.966499
.966447
.966395

.87
.85
.87
.87

9.610397
.610759
.611120
.611480
.611841

6.03
6.02
6.00
6.02

10 389603
.389241
.888880
.388520
.388159

49

48
47
46
45

16
17
18
19
20

.578545
.578853
.579162
.579470
.579777

5.15
5.13
5.15
5.13
5.12
5.13

.966344
.966292
.966240
.966188
.966136

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

.612201
.612561
.612921
.613281
.613641

6.00
6.00
6.00
6.(-0
6.00
5.98

.387799
.387439
.387079
.386719
.386359

44
43
42
41
40

21

9.580085

519

9.966085

Q.7

9.614000

10.386000

on

22
23

580392
.580699

. IJ

5.12

966033
.965981

.oY

87

.614359
.614718

5.98
5.98

.385641

.385282

38
37

24

25

26
27
28
29

.581005
.581312
581618
.581924
.582229
.582535

5.10
5.12
5 10
5.10
5.08
5.10

.965929
.965876
.965824
.965772
.965720
.965668

.87
.88
.87
87
.87
.87

.615077
.615435
.615793
.616151
.616509
616867

5 98
5.97
5.97
5.97
5.97
5.97

384923
384565
384207
.383849
.383491
.383133

36
5

•34

32
il

30

.582840

5.08
5.08

.965615

.88
87

.617224

5 95
5 97

.382776

30

31
32

9.583145
.583449

5.07

K no

9 965563
965511

.87

QQ

9 617582
617939

5.95

10382418
.382061

20
28

33
34

583754
.584058

O.Uo
5.07

5nK

965458
965406

.OO

87

QQ

.618295
.618652

5 '.95

.381705
.381348

27

35

.584361

.Uo

965353

.OO

619008

5 'QQ

.380992

05

36

.584665

5.07

965301

.87

.619364

.yo

.380636

24

37

.584968

5.05

965248

.88

.619720

5.93

5QO

.380280

23

38
39

.585272
.585574

5.07
5.03

.965195
.965143

88
.87

QQ

.620076
.620432

.yo
5 93

5QO

.379924
.379568

22
21

40

.585877

5.05
5.03

.965090

.OO

.88

.620787

.WE

5.92

.379213

20

41

9.586179

5AK

9.965037

QQ.

9.621142

K 92

10.378858

19

42

.586482

.UO

.964984

.00

.621497

.378503

18

43
44

586783
.587085

5.02
5.03
5 no

.964931
.964879

.88
.87

QQ

.621852
.622207

5.92
5.92
5 on

.378148
.377793

17
16

45

.587386

.UK

.964826

.OO

QQ

.622561

.yu
Son

.377439

15

46

.587688

5.03
5 no

.964773

.OO

QQ

.622915

.yu
Son

.377085

14

47
48

.587989
.588289

•Us

5.00
5 no

.964720
964660

.OO

.90

QQ

.623269
.623623

.yu
5.90

5QQ

.376731
.376377

13

12

49

.588590

.UB

964613

.OO

.623976

.Oo

.376024

11

50

.588890

5.00
5.00

.964560

.88
.88

.624330

5.90
5.88

.375670

10

51

9.589190

9.964507

QQ

9.624683

500

10.375317

9

52
53

.589489
.589789

5!oo

4QQ

.964454
.964400

.OO

.90

QQ

.625036
.625388

.00

5.87

500

.374964
.374612

8

54
55
56
57

58

.590088
.590387
.590686
590984
591282

.yo
4.98
4.98
4.97
4.97

.964347
.964294
.964240
.964187
.964133

.OO

.88
.90
.88
.90

.625741
.626093
.626445
.626797
.627149

.00

5.87
5.87
5.87
5.87

.374259
.373907
.373555
.373203
.372851

6
5
4
8

2

59
60

.591580
9.591878

4.97
4.97

.964080
9.964026

.88
.90

.627501
9.627852

5.87
5.85

.372499
10.372148

1
0

'

Cosine.

D. 1".

Sine.

D. 1'.

Cotang.

D. 1*.

Tang. | '

197

TABLE XII. — LOGA1UTHMIC SINES,

156*

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

1

2

9.591878
.592176
.592473

4.97
4.95

4 QIC

9.964026
.963972
.963919

on 9.627852
•15 .628203
•2 ; .628554

5.85

5.85

5 OK

10.372148
.371797
.371446

60
59

58

3
4
5

.592770
.593067
.593363

. yo
4.95
4.93

4QQ

.963865
.963811
.963757

.yu
.90
.90

00

.628905
.629255
.629606

.OO

5.83
5.85

5QQ

.371095
.370745
.370394

57
56

55

6

.593659

.yo

4 no

.963704

.00

.629956

.OO
5QO

.370044

54

7

.593955

.yo

.963650

Qrt

.630306

.OO
5 DO

.369694

53

8
9
10

.594251
.594547
.594842

4!93
4.92
4.92

.963596
.963542
.963488

.yu
.90
.90
.90

.630656
.631005
.631355

.OO

5.82
5.83
5.82

.369344
.368995
.368645

52
51
50

11

9.595137

.

9.963434

9.631704

569

10.368296

49

12

.595432

4Q9

.963379

Qrt

.632053

.O/s

50.)

.367947

48

13
14
15

.595727
.596021
.596315

. \)6

4.90
4.90

A f\f\

.963325
.963271
.963217

.yu
.90
.90

Qrt

.632402
.632750
.633099

.o&

5.80
5.82

Son

.367598
.367250
.366901

47
46
45

16
17

.596609 |-jg
.596903 1 ?-£X

.963163
.963108

.yu
.92

Qrt

.633447
.633795

.oU
5.80
5 PA

.366553
.366205

44
43

18

.597196 T'SS

.963054

.yu

no

.634143

.oU

57ft

.365857

42

19

.597490 1 J'S

.962999

,y*

.634490

. <O

.365510

41

20

.597783 J'gJ

.962945

.90
.92

.634838

5.80
5.78

.365162

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
no

9.635185
.635532
.635879
.636226
.636572

5.78
5.78
5.78

5.77

K 70

10.364815
.364468
.364121
.363774
.363428

39
38
37
36
35

26
27

.599536
.599827

.o<
4.85

4QC

.962617
.962562

,\)£
.92

Qrt

.636919
.637265

o . to
5.77
5 77

.363081
.362735

34
33

28
29

.600118
.600409

.oD

4.85

4 OK

.962508
.962453

.yu
.92

Q9

.637611
.637956

5>5

.362389
.362044

32
31

30

.600700

.oO

4.83

.962398

.tt/4

.92

.638302

5^75

.361698

30

31
32

9.600990
.601280

4.83

4 DO

9.962343

.962288

.92

9.638647
.638992

5.75

57K

10.361353
.361008

29

28

33
34

.601570
.601860

.OO

4.83

.962233
.962178

'.92

.639337
.639682

. <O

5.75

.360663
.360318

27
26

35
36
37

38

.602150
.602439
.602728
.603017

4.83
4.82
4.82
4.82

4 on

.962123
.962067
.962012
.961957

.92
.93
.92
.92

.640027
.640371
.640716
.641060

5.75
5.73
5.75
5.73
5 73

.359973
.&59C29
.359284
.358940

25
24
23

22

39
40

.603305
.603594

.oU

4.82
4.80

.961902
.961846

!93
.93

.641404
.641747

5 '.72
5.73

.358596
.358253

21
20

41

9.603882

9.961791

9.642091

579

10.357909

19

42
43

.604170
.604457

4.80
4.78

.961735
.961680

192

no

.642434
.642777

. < <«

5.72

579

.357566
.357223

18
17

44
45

.604745
.605032

4.80
4.78

.961624
.961569

.yo
.92

.643120
.643463

. i *

5.72'

.356880
.356537

16
15

46
47

.605319
.605606

4.78

4.78

477

.961513
.961458

.93

.92

.643806
.644148

5.72
5.70

57rt

.356194
.355852

14
13

48

.605892

. I i

4lfQ

.961402

•|5

.644490

. <U

.355510

12

49
50

.606179
.606465

. to
4.77
4.77

.961346
.961290

!93

.92

.644832
.645174

5>0
5.70

.355168
.354826

11
10

51

9.606751

9.961235

9.645516

10.354484

9

52

.607036

4.75

.961179

"no

.645857

0.08

.354143

8

53

.607322

4.77

.961123

.yo

.646199

t oo

.353801

7

54

.607607

4.75

.9610(57

.93

.646540

5.68

.353460

6

55
56
57
58

.607892
.608177
.608461
.608745

4.75
4.75
4.73
4.73

.961011
.960955
.960899
.960843

.93
.93
.93
.93

.646881
.647222
1 .647562
.647903

5.68
5.68
5.67
5.68

.353119

.352778
.352438
.352097

5
4
3

2

59

.609029

4.73

.960786

•?5 .648243

5.67

.351757

1

60

9.609313

4.73

9.960730

•Jd 1 9.648583

5.67

10.351417

0

' ! Cosine.

I). 1'.

Sine.

D. 1'. ll Cotang.

D. 1".

Tang.

'

198

66"

COSINES, TANGENTS, AND COTANGENTS.

155-

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1'.

Cotang.

t

0

1

9.609313
.609597

4.73

9.960730
.960674

.93

9.648583
.648923

5.67

5 Off

10.351417
.351077

60
59

2
3
4

.609880
.610164
.610447

4 . 72 '
4.73
4.72

47ft

.960618
.960561
.960505

!95
.93

QK

.649263
.649602
.649942

.Dl

5.65
5.67

.350737
.350398
.350058

58
57
56

5

.610729

. <U

479

.960448

.yo

.650281

5 OK

.349719

55

6
7
8
9
10

.611012
.611294
.611576
.611858
.612140

. i ~

4.70
4.70
4.70
4.70
4.68

.960392
.960335
.960279
.960222
.960165

'.95
.93
.95
.95
.93

.650620
.650959
.651297
.651636
.651974

.DO

5.65
5.63
5.65
5.63
5.63

.349380
.349041
.348703
.348364
.348026

54
53
52
51
50

11

9.612421

4 no

9.960109

QK

9.652312

5pn

10.347688

49

12
13
14

.612702
.612983
.613264

.DO

4.68
4.68

.960052
.959995
.959938

.yo
.95
.95

.652650
.652988
.653326

.DO

5.63
5.63

.347350
.347012
.346674

48
47
46

15

.613545

4.68

.959882

.93

QK

.653663

5.62

.346337

45

16
17
18

.613825
.614105
.614385

4 '.67
4.67

.959825
.959768
.959711

.yo
.95
.95

.654000
.654337
.654674

5.62
5.62
5.62

.346000
.345663
.345326

44
43
42

19
20

.614665
.614944

4.67
4.65
4.65

.959654
.959596

.95
.97
.95

.655011
.655348

5.62
5.62
5.60

.344989
.344652

41
40

21

9.615223

9.959539

QK

9.655684

Sftft

10.344316

39

22

23

.615502
.615781

4.65
4.65

.959482
.959425

.yo
.95

QK

.656020
.656356

.ou
5.60

50ft

.343980
.343644

38
37

24
25

.616060
.616338

4.65
4.63

.959368
.959310

.yo
.97

QK

.656692
.657028

.OU

5.60

.343308
.342972

36

26

.616616

4.63

.959253

.yo

.657364

5 fro

.342636

34

27

28
29
30

.616894
.617172
.617450
.617727

4.63
4.63
4.63
4.62
4.62

.959195
.959138
.959080
.959023

!95
.97
.95

.97

.657699
.658034
.658369
.658704

.OO

5.58
5.58
5.58
5.58

.342301
.341966
.341631
.341296

33
32
31

30

31
33
33

9.618004
.618&81
.618558

4.62
4.62

9.958965
.958908
.958850

.95

.97

Q7

9.659039
.659373
.659708

5.57
5.58

10.340961
.340627
.340292

29
28
27

34

.618834

on

.958792

.y«

.660042

5K<y

.339958

26

35
36
37
38
39
40

.619110
.619386
.619662
.619938
.620213
.620488

4.60
4.60
4.60
4.60
4.58
4.58
4.58

.958734
.958677
.958619
.958561
.958503
.958445

'.95
.97
.97
.97
.97
.97

.660376
.660710
.661043
.661377
.661710
- .662043

.O<

5.57
5.55
5.57
5.55
5.55
5.55

.339624
.a39290
.338957
.338623
.338290
.337957

25
24
23
22
21
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

46
47

.621861
.022135
.622409

4^57
4.57

4KK

.958154
.958096
.958038

!97
.97

QQ

.663707
.664039
.664371

5^53
5.53

K KQ

.336293
.335961
.335629

15
14
13

48
49
50

.622682
.622956
.623229

.OO

4.57
4.55
4.55

.957979
.957921
.957863

.yo
.97
.97
.98

.664703
.665035
.665366

o.oo
5.53
5.52
5.53

.335297
.334965
.334634

12
11
10

51

! 623774

4.53

4K-

9.957804
T957746

.97

QO

9.665698
.666029

5.52
5 52

10.334302
.333971

9
8

53
54
55
56

.624047
.624319
, .624591
.624863

.OO

4.53
4.53
4.53

.957687
.957628
.957570
I .957.M1

.yo
.98
.97
.98

.666360
.666691
.667021
.667352

5^52
5.50
5.52

.333640
.333309
.a32979
.332648

7
6
5
4

57
58
59

.6251,35
.625406
.625677

4.53
4.52

4.52

.957452
.957393
.957385

.98
.98
.97

.667682
.668013
.668343

5.50
5.52
5.50

.332318
.331987
.331657

3
2
1

CO

9.625948

4.52

9.957276

.98

9.668673

5.50

10.331327

0

'

Cosine.

D. 1".

Sine.

D.I".

Cotang.

D. r.

Tang.

'

114°

65*

TABLE XII. — LOGARITHMIC SINES,

154°

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1*.

Cotang.

'

0

9.625948

4 fro

9.957276

OP

9.668673

10.331327

60

1

2
3
4
5
6
7

.626219
.626490
.626760
.627030
.627300
.627570
.627840

.OR

4.52
4.50
4.50
4.50
4.50
4.50

.957217
.957158
.957099
.957040
.956981
.956921
.956862

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

QO

.669002
.669332
.669661
.669991
.670320
.670649
.670977

5.48
5.50
5.48
5.50
5.48
5.48
5.47

.330998
.330668
.330339
.330009
.329680
.329351
.329023

59
58
57
56
55
54
53

8

.628109

4.48

.956803

.yo

QO

.671306

5.48

.328694

52

9
10

.628378
.628647

4.'48
4.48

.956744
.956684

.yo
1.00
.98

.671635
.671963

5.48
5.47
5.47

.328365
.328037

51

50

11

9.628916

9.956625

9.672291

10.327709

49

12
13

.629185
.629453

4.48
4.47

.956566
.956506

.98
1.00

.672619
.672947

5.47
5.47

.327381
.327053

48

47

14
15
16

.629721
.629989
.630257

4.47
4.47
4.47

.956447
.956387
.956327

.98
1.00
1.00

.673274
.673602
.673929

5.45
5.47
5.45

.326726
.326398
.326071

46
45
44

17
18
19

.630524
.630792
.631059

4.45
4.47
4.45

.956268
.956208
.956148

.98
1.00
1.00

GO.

.674257
.674584
.674911

5.47
5.45
5.45

.325743
.325416
.325089

43
42
41

20

.631326

4.45
4.45

.956089

.yo
1.00

.675237

5.43
5.45

.324763

40

21
22
23
24
25
26
27
28

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

4.43
4.43
4.45
4.43
4.42
4.43
4.42

9.956029-
.955969
.955909
.955849
.955789
.955729
.955669
.955609

1.00
1.00
1.00
1.00
1.00
1.00
1.00

QQ

9.675564
.675890
.676217
.676543
.676869
.677194
.677520
.677846

5 43
5.45
5.43
5.43
5.42
5.43
5.43

5A9

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

39
38
37
36
35
34
33
32

29
30

.633719
.633984

4!42
4.42

.955548
.955488

.yo
1 00
1.00

.678171
.678496

.4/3

5.42
5.42

.321829
.321504

31
30

31
32
33
34

9.634249
.634514
.634778
.635042

4.42
4.40
4.40

9955428
.955368
.955307
.955247

1.00
1.02
1.00

9.678821
679146
.679471
.679795

5.42
5 42
5.40

10.321179
.320854
.320529
.320205

29
28
27
26

35
36
37
38
39
40

.635306
.635570
.635834
.636097
636360
.636623

4.40
4.40
4.40
4.38
4 38
4.38
4.38

.955186
.955126
.955065
.955005
.954944
.954883

1.02
1.00
1.02
1.00
1.02
1.02
1.00

.680120
.680444
.680768
.681092
.681416
.681740

5.42
5.40
5.40
5.40
5.40
5.40
5.38

.319880
.319556
.319232
.318908
.318584
.318260

25
24
23
22
21
20

41
42

9.636886
637148

4.37

4QQ

9 954823
.954762

1 02

9.682063

.682387

5.40

500

10.317937
.317613

19

18

43
44
45

.637411
.637673
.637935

.00
4.37
4.37

.954701
.954640
.954579

1.02
1.02
1.02

.682710
.683033
.683356

.00

5.38
5 38

50Q

.317290
.316967
.316644

17
16
15

46

.638197

4.37

.954518

1.02

.683679

.OO

.316321

14

47

.638458

4.35

.954457

1.02

.684001

5.37

500

.315999

13

48
49

.638720
.638981

4.37
4.35

.954396
.954335

1 .02
1 02

.684324
.684646

.00
5.37

.315676
.315354

12
11

50

.639243

4.35
4.35

.954274

1.02
1.02

.684968

5.37
5.37

.315032

10

51
52
53
54
55
56

9.639503
.639764
.640024
.640284
.640544
.640804

4.35
4.33
4.33
4.33
4.33

9.954213
.954152
.954090
.954029
.953968
.953906

1.02

1.03
1.02
1.02
03

9.685290
.6856*2
.685934
686255
.686577
.686898

5.37
5.37
5.35
5.37

5 35
5 OR

10 314710
.314388
.314066
.313745
.313423
.313102

9

8
7
6
5

4

57
58
59

.641064
.641324
.641583

4.33
4.33
4.32

.953845
.953783
.953722

.02
.03

: .02

.687219
.687540
.687861

.00
5 35
5.35

.312781
.312460
.312139

3
2

1

60

9.641842

4.32

9.953660

.03

9.688182

5.35

10.311818

0

'

Cosine.

D. 1%

Sine.

D. 1". I

Cotang.

D. 1'.

Tang.

'

200

26'

COSINES, TANGENTS, AND COTANGENTS.

'

Sine.

D. r.

Cosine.

D. 1'.

Tang.

D. r.

Cotang.

-

0

1

2

9.641842
.642101
.642360

4.32
4.32

9.953660
.953599
.953537

1.02
1.03
1 na

9.688182
.688502
.688823

5.33
5.32

10.311818
.311498
.311177

60
59
58

3
4

.642618

.642877

4^32
4 30

.953475
.953413

1 .Uu

1.03
1 02

.689143
.689463

5.33
5.33

5OO

.310857
.310537

57
56

5 .643135

4 30

.953352

l'03

.689783

.OO

5 33

.310217

55

6

.6433i#
.643650

4.28

.953290
| .953228

1.03

.690103
.690423

5.'33

.309897
.309577

54
53

8
9
10

.643908
.644165
.644423

4.'28
4.30
4.28

.953166
.953104
.953042

1.'03
1.03
1.03

.690742
.691062
j .691381

5.32
5.33
5.32
5.32

.309258
.308938
.308619

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
1 .692019
.692338
.692656
.692975
.693293
.693612
.693930
.694248
.694566

5.32
5.32
5.30
5.32
5.30
5.32
5.30
5.30
5.30
6.28

10.308300
.307981
.307662
.307344
.307025
.306707
.306388
.306070
.305752
.305434

49
48
47
46
45
44
43
42
41
40

21
22
23
24

9.647240
.647494
.647749
.648004

4.23
4.25
4.25

9.952356
.952294
.952231
.952168

1.03
1.05
1.05

9.694883
.695201
.695518
.695836

5.30
5.28
5.30

10.305117
.304799
.304482
.304164

39
38
37
36

25

26
27
28
29
30

.648258
.648512
.648766
.649020
.649274
.649527

4.'23
4.23
4.23
4.23
4.22
4.23

.952106
.952043
.951980
.951917
.951854
.951791

l!05
1.05
1 05
1.05
1.05
1.05

.696153
.696470
.696787
.697103
.697420
.697736

5.28
5.28
5.28
5.27
5.28
5.27
5.28

.303847
.303530
.303213
.302897
.302580
.302264

35
34
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
1 05
1.07
1 05
1.05

9.698053
.698369
.698685
.699001
.699316
.699632
.699947
.700263
.700578
.700893

5.27
5.27
5.27
5.25
5.27
5.25
5.27
5.25
5.25
6.25

10.301947
.301631
.301315
.300999
.300684
.300368
.300053
.299737
.299422
.299107

29
28
27
26
25
24
23
22
21
20

41
42
43

9.652304
.652555
.652806

4.18
4.18

41 Q

9.951096
.951032
.950968

1.07
1.07

9.701208
.701523
.701837

5.25
5.23

10.298792
.298477
.298163

19
18
17

44

.653057

.10

41 ft

.950905

1.05
1 07

.702152

5.25
500

.297848 16

45
46
47
48
49

653308
.653558
.653808
.654059
.654309

.10
4.17
4.17
4.18
4.17

.950841
.950778
.950714
.950650
.950586

I .VI

1.05
1.07
1.07
1.07

1A7

.702466
.702781
.703095
.703409
.703722

.HI

5.25
5.23
5.23
5.22

500

.297534
.297219
.296905 |
.296591
.296278

15
14
13
12
11

50

.654558

4.15
4.17

.950522

.VI

1.07

.704036

.to
5.23

.295964

10

51
52
53
54
55
56
57
58
59
60

9.654808
.655058
.655307
.655556
.655805
.656054
656302
.656551
.656799
9.657047

4.17
4.15
4.15
4.15
4.15
4.13
4.15
4.13
4.13

0.950458
-.950394
.950330
.950266
.950202
.950138
.950074
.950010
.949945
9.949881

1.07
1.07
1.07
1.07
1.07
1.07
1.07
1.08
1.07

9.704350
.704663
.704976
.705290
.705603
.705916
.706228
.706541
.706854
9.707166

5.22
5.22
5.23
5.22
5.22
5.20
5.22
5.22
5.20

10.295650
.295337
.295024
.294710
.294397
.294084
.293772
.293459
.293146
10.292834

9

8
7
6
5
4
3
2
1
0

'

Cosine.

D. r.

Sine. D. 1". > Cotang.

D. r.

Tang.

'

201

TABLE XII. — LOGAKTTHMEC SINES,

152°

'

Sine.

D. 1".

Cosine.

D. r.

Tang.

D. 1".

Cotang.

'

0

9.657047

4 -JO

9.949881

9.707166

10.292834

60

1

.657295

:. lO

.949816

.08 1

.707478

5. 20

.292522

59

2
3

.657542
.657790

4.12
4.13

.949752
.949688

.07 ;
: .07 !

.707790
.708102

5.20
5.20

.292210
.291898

58
57

4
5

.658037
.658284

4.12
4.12

.949623
.949558

: .08 '

.08

.708414
.708726

5.20
5.20

.291586
.291274

56
55

6

.658531

4.12

.949494

.07

.709037

5.18

.290963

54

8

.658778
.659025

4.12
4.12

.949429
.949364

.08
.08

.709349
.709660

5.20
5.18

.290651
.290340

53
52

9

.659271

4.10

.949300

.07

.709971

5.18

.290029

51

10

.659517

4.10

.949235

.08

.710282

5.18

.289718

50

4.10

.08

5.18

11

9.659763

4 10

9.949170

AQ

9.710593

5 18

10.289407

49

12
13

.660009
• .660255

4^10

.949105
.949040

.UO

•: .08

.710904
711215

5^18

f 17

.289096
.288785

48

47

14

660501

T'lJJ

.948975

'/\0

.711525

0. 1 i
51ft

.288475

46

15

16
17

18
19
20

660746
.660991
.661236
.661481
.661726
.661970

4.08
4.08
4.08
4.08
4.08
4.07
4.07

.948910
.948845
.948780
.948715
.948650
.948584

! .08
.08
.08
1.08
1.08
1.10
1.08

.711836
.712146
.712456
.712766
.713076
.713386

.15
5.17
5.17
5.17
5.17
5.17
5.17

.288164
.287854
.287544
.287234
.286924
.286614

45
44
43
42
41
40

21
22
23

9.662214
.662459
.662703

4.08
4.07

9.948519
.948454
.948388

1.08
1.10

9.713696
.714005
. 714314

5.15
5.15

10.286304
.285995

.285686

39

38
37

24
25

26

27 .

.662946
.663190
.663433
.663677

4.05
4.07
4.05
4.07

.948323
.948257
.948192
.948126

1.08
1.10
1.08
1.10

.714624
.714933
.715242
.715551

5.17
5.15
5.15
5.15

285376
.285067
.284758
.284449

36
35
34
33

28
29
30

.663920
.664163
.664406

4.05
4.05
4.05
4.03

.948060
.947995
.947929

1.10
1.08
1.10
1.10

715860
.716168
.716477

5.15
5.13
5.15
5.13

.284140
.283832
.283523

32
31
30

31
32
33

9.664648
.664891
.665133

4.05
4.03

9.947863
.947797
.947731

1 10
1.10

9.716785
.717093
.717401

5 13

5.13

10.283215

.282907
.282599

29

28
27

34
35
36
37

38
39
40

.665375
.665617
.665859
.666100
.666342
.666583
.666824

4.03
4.03
4.03
4.02
4.03
4.02
4.02
4.02

947665
.947600
.947533
.947467
.947401
.947335
.947269

1.10
1.08
1.12
1.10
1.10
1.10
1.10
1.10

717709
.718017
.718325
.718633
.718940
.719248
.719555

5.13
5.13
5.13
5.13
5.12
5.13
5.12
5.12

.282291
281983
.281675
.281367
.281060
280752
.280445

26
25
24
23
22
21
20

41
42
43
44
45

43
49
50

9.667065
.667305
.667546
.667786
.668027
668267
.668506
.668746
.668986
669225

4.00
4.02
4.00
4.02
4.00
3.98
4.00
4.00
3.98
3.98

9.947203
.947136
.947070
.947004
.946937
.946871
946804
.946738
.946671
.946604

1 12
1.10
1.10
1.12
1.10
1.12
1.10
1.12
1.12
1.10

9.719862
.720169
.720476
.720783
.721089
.721396
.721702
.722009
.722315
.722621

5.12
5.12
5.12
5.10
5.12
5.10
5.12
5.10
5.10
5.10

10.280138
.279831
.279524
.279217
.278911
.278604
.278298
.277991
277685
.277379

19
18
17
16
15
14
13
12
11
10

51
52
53

9.669464
669703
669942

3 98
3.98

9.946538
.946471
.946404

1.12
1.12

9.722927
723232
723538

5.08
5.10

10.277073

.276768
276462

9
8

7

54

55

670181
670419

3.98
3.97

C46337
.946270

1.12
1.12

,7'23844
.724149

5^08

.276156
.275851

6
5

56
57
58
59
60

.670658
670896
.671134
.67137'2
9.671609

3.98
3.97
3.97
3.97
3.95

946203
.946136
.946069
946002
9.945935

1.12
1.12
1.12
1.12
1.12

.724454
724760
725065
725370
9.725674

5.08
5.10
5.08
5.08
5.07

.275546
275240
274935
274630
10.274326

4
3
o

1
0

1

Cosine.

D. r.

Sine.

D. 1".

Cotang.

D. 1'.

Tang.

'

202

62°

28°

COSINES, TANGENTS, AND COTANGENTS.

151*

'

Sine.

D. 1'.

Cosine.

D. 1'.

Tang.

D. r.

Cotang.

'

0

1

2
3

4
5
6

7
8

9.671609
.671847
.672084
.67^1
.672558
.672795
.673032
.673268
.673505

3.97
3.95
3.95
3.95
3.95
3.95
3.93
3.95

9.945935
.945868
.945800
.945733
.945666
.945598
.945531
.945464
.945396

1.12
1.13
1.12
1.12
1.13
1.12
1.12
.13

9.725674
.725979
.726284
.726588
.726892
.727197
.727501
.727805
.728109

5.08
5.08
5.07
5.07
5.05
5.07
5.07
5.07

10.274326
.274021
.273716
.273412
.273108
.272803
.2?2499
.272195
.271891

60
59
58
57
56
55
54
53
52

9

.673741

3.93

.945328

.13

.728412

5.05

.271588

51

10

.673977

3.93

.945261

: .12

.728716

5.07

.271284

50

3.93

.13

5.07

11

12

9.674213

.674448

3.92

9.945193
.945125

.13

9.729020
.729323

5.05

10.270980
.270677

49

48

13

.674684

3.93

.945058

.12

.729626

5.05

.270374

47

14

.674919

3.92

.944990

.13

.729929

5.05

.270071

46

15

.675155

3.93

.944922

.13

.730233

5.07

.269767

45

16
17

.675390
.675624

3.92
3.90

.944854
.944786

.13
.13

.730535
.730838

5.03
5.05

.269465
.269162

44
43

18
19

.675859
.676094

3.92
3.92

.944718
.944650

: .13

.13

.731141
.731444

5.05
5.05

.268859
.268556

42
41

20

.676328

3.90
3.90

.944582

.13

.13

.731746

5.03
5.03

.268254

40

21
22

23

9.676562
.676796
.677030

3.90
3.90

9.944514
.944446
.944377

.13
.15

9.732048
.732351
.732653

5.05
5.03

5nn

10.267952
.267649
.267347

39
38
37

24
25

.677264
.677498

3.90
3.90

.944309
.944241

! .13
.13

.732955
.733257

.Uo
5.03

.267045
.266743

36
35

26

.677731

3.88

.944172

.15

.733558

5.02

.266442

34

27

.677964

3.88

.944104

.13

.733860

5.03

5(\n

.266140

33

28

.678197

3.88

.944036

.13

.734162

.UO

.265838

32

29

. 678430

3.88

.943967

.15

.734463

5.02

.265537

31

30

.678663

3.88

.943899

: .13

.734764

5.02

5nn

.265236

30

3.87

'. .15

.Uo

31
•32
33
34
35
36
37
38

9.678895
. 679128
.679360
.679592
.679824
.680056
.680288
.680519

3.88
3.87
3.87
3.87
3.87
3.87
3.85

9.943830
.943761
.943693
.943624
.943555
.943486
.943417
.943348

.15
.13
.15
.15
.15
.15
.15

9.735066
.735367
.735668
.735969
.736269
.736570
.736870
.737171

5.02
5.02
5.02
5.00
5.02
5.00
5.02

10.264934
.264633
.264332
.264031
.263731
.263430
.263130
.262829

29
28
27
26
25
24
23
22

39
40

.680750
.680982

3.85
3.87
3.85

.943279
.943210

1.15
1.15
1.15

.737471
.737771

5!(X)
5.00

.262529
.262229

21
20

41

42
43
44
45
46
47
48

9.681213
.681443
.681674
.681905
.682135
.682365
.682595
.682825

3.83
3.85
3.85
3.83
3.83
3.83
3.83

9.943141
.948072
.943003
.942934
.942864
.942795
.942726
.942656

1.15 '
1.15 i
1.15 i
1.17
1.15
1.15
1.17

9.738071
.738371
.738671
.738971
.739271
.739570
.739870
.740169

5.00
5.00
5.00
5.00
4.98
5.00
4.98

4 Oft

10.261929
.261629
.261329
.261029
.260729
.260430
.260130
.259831

19
18
17
16
15
14
13
12

49
50

.683055

.683284

3.83
3.82
3.83

.942587
.942517

.15

.17
.15

.740468

. 740767

.yo
4.98
4.98

.259532
.259233

11
10

51

52

9.683514
.683743

3.82

9.942448
.942378

.17

9.741066
.741365

4.98
4 98

10.258934
.258635

9

8

53

54
55

r~

58

59
60

.683972
.684201
.684430
.684658
.684887
.685115
.685343
9.685571

3.82
3.82
3.82
3.80
3.82
3.80
3.80
3.80

.942308
.942239
.942169
.942099
.942029
.941959
.941889
9.941819

.17
.15
.17
1.17
1.17
1.17
1.17
1.17

.741664
.741962
.742261
.742559
.742858
.743156
.743454
9.743752

4i97
4.98
4.97
4.98
4.97
4.97
4.97

.258336
.258038
.257739
.257441
.257142
.256844
.256546
10.256248

7
6
5
4
3
2
1
0

'

I Cosine.

D. 1'.

1 Sine.

D. r.

Cotang.

D. 1'.

Tang.

1

118°

29°

TABLE XII. — LOGAKITHMIC SINES,

150°

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

•

0

9.ea557i

3 Of)

9.941819

1 17

9.743752

4 97

10.256248

60^

1

2

.685799
.686027

.OU

3.80

.941749
.941679

1 . i i
1.17

11"

.744050
.744348

4^97

4QK

.255950
.255652

59

58

3
4

.686254
.686482

3.78
3.80

3r>o

.941609
.941539

. 1 1
1.17

.744645
.744943

.yo
4.97

4QK

.255355
.255057

57
56

5
6

7

.686709
.686936
.687163

. <0

3.78
3.78

.941469
.941398
.941328

. 17
.18
.17
17

.745240
.745538

.745835

.yo
4.97
4.95

.254760
.254462
.254165

55
54
53

8
9

.687389
.687616

3.77

3.78

.941258
.941187

ilfl

.74ol32
.746429

4^95

4Q^

.253868
.253571

52
51

10

.687843

3.78
3.77

.941117

. 17
.18

.746726

. yo
4.95

.253274

50

11
12
13
14
15

9.688069

.688295
.688521
.688747
.688972

3.77
3.77
3.77
3.75

9.941046
.940975
.940905
.940834
.940763

.18
.17
.18
.18

9.747023

.747319
.747616
.747913

.748209

4.93
4.95
4.95
4.93

4 no

10.252977
.252681
.252384
.252087
.251791

49

48
47
46
45

16
17
18
19
20

.689198
.689423
.689648
.689873
.690098

3.77
3.75
3.75
3.75
3.75
3.75

.940693
.940622
.940551
.940480
.940409

. 17

.18
.18
.18
.18
.18

.748505
.748801
.749097
.749393
.749689

.yo
4.93
4.93
4.93
4.93
4.93

.251495
.251199
.250903
.250607
.250311

44
43
42
41
40

21

9.690323

9.940338

9.749985

4QO

10.250015

39

22

.690548

3r>q

.940267

•JO

.750281

. yo

.249719

38

23
24
25

26

27
28
29
30

.690772
.690996
.691220
.691444
.691668
.691892
.692115
.692339

. «<J

3.73
3.73
3.73
3.73
3.73
3.72
3.73
3.72

.940196
.940125
.940054
.939982
.939911
.939840.
.939768
.939697

. lo
.18
.18
.20
1.18
1.18
1.20
1.18
1.20

.750576
.750872
.751167
.751462
.751757
.752052
.752347
.752642

4^93
4.92
4.92
4.92
4.92
4.92
4.92
4.92

.249424
.249128
.248833
.248538
.248243
.247948
.247653
.247358

37
36
35
34
33
32
31
30

31
32
33
34
35
36

9.692562

.692785
.693008
.693231
.693453
.693676

3.72
3.72
3.72
3.70
3.72

3r»/\

9.939625
.939554
.939482
.939410
.939339
.939267

1.18
1.20
1 20
1.18

1.20

9.752937
.753231
.753526
.753820
.754115
.754409

4.90
4.92
4.90
4.92
4.90
4 on

10.247063
.246769
.246474
.246180
.245885
.245591

29
28
27
26
25
2-1

37
38
39
40

.693898
.694120
.694342
.694564

. tO
3.70
3.70
3.70
3.70

.939195
.939123
.939052
.938980

1 .20
1.20
1.18
1.20
1.20

.754703
.754997
.755291
.755585

. yu
4.90
4.90
4.90
4.88

245297
.245003
.244709
.244415

23
22
21
20

41

9.694786

3CQ

9.938908

9.755878

4AA

10.244122

19

42
43
44

.695007
.695229
.695450

.Do

3.70
3.68

3 CO

.938836
.938763
.938691

1^22
1.20

.756172
.756465
.756759

. yu
4.88
4.90

4QQ

.243828
.243535
.243241

18
17
16

45
46
47

.695671
.695892
.696113

. Do

3.68
3.68

300

.938619
.938547
.938475

1^20
1.20

.757052
.757345
.757638

.OO

4.88
4.88

400

.242948
.242655
.242362

15
14
13

48
49
50

.696334
.696554
.696775

.Do

3.67
3.68
3.67

.938402
.938330
.938258

l!20
1.20
1.22

.757931
.758224
.758517

.00

4.88
4.88
4.88

.242069
.241776
.241483

12
11
10

51
52
53
54
55
56
57
58
59

9.696995
.697215
.697435
.697654
.697874
.698094
.698313
.698532
.698751

3.67
3.67
3.65
3.67
3.67
3.65
3.65
3.65

3«e

9.938185
.938113
.938040
.937967
.937895
.937822
.937749
.937676
.937604

1.20
1.22
1.22
.20
.22
.22
.22
.20
• oo

9.758810
.759102
.759395
.759687
.759979
.760272
.760564
.760856
.761148

4.87
4.88
4.87
4.87
4.88
4.87
4.87
4.87

4 OK.

10.241190
.240898
.240605
240313
.240021
.239728
.239436
.239144
.238852

9
8
7
6
5
4
3

0

1

60

9.698970

.DO

9.937531

9.761439

.OO

10.238561

0

'

Cosine.

D. 1".

Sine.

D. 1'. i

Cotang.

D. r.

Tang.

'

204

60°

30°

COSINES, TANGENTS, AND COTANGENTS.

149-

'

Sine.

D. r.

Cosine.

D. r.

Tang.

1

D. r.

Cotang.

'

0

1

9.698970
.699189

3.65

3 pa

9.937531
.937458

1.22

9.761439
.761731

4.87

10.238561
.238269

60
59

2
3
4
5
6
7

.699407
.699626
.699844
.700062
.700280
.700498

3.65
3.63
3.63
3.63
3.63

o po

.937385
.937312
.937238
.937165
.937092
.937019

1.22
1.23
1.22
1.22
1.22

199

.762023
.762314
.762606
.762897
.763188
.763479

4.85
4.87
4.85
4.85
4.85

4 OK

.237977
.237686
.237394
.237103
.236812
.236521

58
57
56
55
54
53

8
9

.700716
.700y33

3.62

.936946
.936872

1.2*

199

.763770
.764061

4.85

4 OK

.236230
.235939

52
51

10

.701151

3.62

.936799

1.23

.764352

4.85

.235648

50

11

9.701368

3R9

9.936725

19Q

9.764643

4QQ

10.235357

49

12
13
14

.7-J1585
.701802
.702019

3.62
3.62

.936652
.936578
.936505

1.23
1.22

19°.

.764933
; 765224
.765514

4.85
4.83

4 OK

.235067
.234776
.234486

48
47
46

15

.702236

.936431

190

.765805

400

.234195

45

16

.702452

3fi9

.936357

1 k)2

.766095

A QQ

.233905

44

17

.702669

3 fin

.936284

190

.766385

4OQ

.233615

43

18
19
20

.702885
.703101
.703317

3.60
3.60
3.60

.936210
.936136
.936062

1.23
1.23
1.23

.766675
.766965
.767255

4.83

4.83
4.83

.233325
.233035
.232745

42
41
40

21
22
23
24

9.703533
.703749
.703964
.704179

3.60
3.58
3.58

9.935988
.935914
.935840
.935766

1.23
1.23
1.23

19°.

9.767545
.767834
.768124
.768414

4.82

4.83
4.83

4Q9

10.232455
.232166
.231876
.231586

39
38
37
36

25
26
27

28
29
30

.704395
.704610
.704825
.705040
.705254
.705469

3.58
3.58
3.58
3.57
3.58
3.57

.935692
.935618
.935543
.935469
.935395
.935320

1.23
1.25
1.23
1.23
1.25
1.23

.768703
.768992
.769281
.769571
.769860
.770148

4.82
4.82
4.83
4.82
4.80
4.82

.231297
.231008
.230719
.230429
.230140
.229852

35
34
33
32
31
30

31

32
33
34
35
36
37
38
39
40

9.705683
.705898
.706112
.706326
.706539
.706753
.706967
.707180
.707393
.707606

3.58
3.57
3.57
3.55
3.57
3.57
3.55
3-55
3.55
3 55

9.935246
.935171
.935097
.935022
.934948
.934873
.934798
.934723
.934649
.934574

1.25
1.23
1.25
1.23
1.25
1.25
1.25
1.23
1.25
1.25

9.770437
.770726
.771015
.771303
.771592
.771880
.772168
.772457
.772745
.773033

4.82
4.82
4.80
4.82
4.80
4.80
4.82
4.80
4.80
4.80

10.229563
.229274
.228985
.228697
.228408
.228120
.227832
.227543
.227255
.226967

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

9.707819
.708032
.708245
.708458
.708670
.708882
.709094
.709306
.709518
.709730

3.55
3.55
3.55
3.53
3.53
3.53
3.53
3.53
3.53
3 52

9.934499
.934424
.934349
.934274
.934199
.934123
.934048
.933973
.9&S898
.933823

1.25
1.25
1.25
1.25
1.27
1.25
1.25
1.25
1.27
1 25

9.773321

.773608
.773896
• .774184
. 774471
.774759
.775046
.775333
.775621
.775908

4.78
4.80
4.80
4.78
4.80
4.78
4.78
4.80
4.78
4.78

10.226679
.226392
.226104
.225816
.225529
.225241
.224954
.224667
.224379
.224092

19
18
17
16
15
14
13
12
11
10

51
52
53
54
55
56
57
58
59
60

9.709941
.710153
.710364
.710575
.710786
.710997
.711208
.711419
.711629
9.711839

3.53
3.52
3.52
3.52
3.52
3.52
3.52
3.50
3.50

9.933747
.933671
.933596
.933520
933445
.933369
.933293
.933217
.9.33141
9.933066

1.27
1.25
1.27
1.25
1.27
1.27
1.27
1.27
1.25

9.776195
.776482
.776768
.777055
.777342
.777628
.777915
.778201
.778488
9.778774

4.78
4.77
4.78
4.78
4.77
4.78
4.77
4.78
4.77

10.223805
.223518
.223232
.222945
.222658
.222372
.222085
.221799
.221512
10.221226

9

8

6
5

4
3
2
1
0

'

Cosine.

D. 1".

Sine.

D. 1'.

! Cotang.

D. 1'.

Tang.

-j

120°

205

TABLE XIT. — LOGARITHMIC SIXES,

148*

'

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

1

2

9.711839
.712050
.712260

3.52
3.50

9.933066
.932990
.932914

1
1.27
1.27 1

1*>7 1

9.778774
.779060
.779346

4.77
4.77

10.221226
.220940
.220654

60
59

58

3
4
5
6

7
8
9
10

.712469
.712679
.712889
.713098
.713308
.713517
.713726
.713935

3^50
3.50
3.48
3.50
3.48
3.48
3.48
3.48

.932838
.932762
.932685
.932609
.932533
.932457
.932380
.932304

,V4 l

1.27
1.28 !
1.27
1.27
1.27
1.28
1.27
1.27

.779632
.779918

.780203
.780489
.780775
.781060
.781346
.781631

4.77
4.77
4.75
4.77
4.77
4.75
4.77
4.75
4.75

.220368
.220082
.219797
.219511
.219225
.218940
.218654
.218369

57
56
55
54
53
52
51
50

11
12
' 13
14

9.714144
.714352
.714561
.714769

3.47
3.48
3.47

q AQ

9.932228
.932151
.932075
.931998

1.28
1.27

1.28

1 9ft

9.781916
.782201

.782486

.782771

4.75
4.75
4.75

10.218084
.217799
.217514
.217229

49

48
47
46

15
16

.714978
.715186

O.4O

3.47
3 47

.931921
.931845

1 ./so

1.27

.783056
.783341

4.75
4.75

.216944
.216659

45
44

17

.715394

3 Aft

.931768

1 9ft

.783626

4.75

.216374

43

18
19
20

.715602
.715809
.716017

.4<

3.45
3.47
3.45

.931691
.931614
.931537

1^28
1.28
1.28

.783910
.784195
.784479

4.73
4.75
4.73
4.75

.216090
.215805
.215521

42
41
40

21
22

9.716224
.716432

3.47
3 45

9.931460
.931383

1.28

19ft

9.784764
.785048

4.73

10.215236
.214952

39
38

23
24

.716639
.716846

3^45

.931306
.931229

.^o

1.28

.785332
.7'85616

4^73

.214668
.214384

37
36

25

.717053

3.45
340

.931152

1 9ft

.785900

4.73

.214100

35

26

27

.717259
.717466

.4o

3.45

3AK

.931075
.930998

1 . *o

1.28

.786184
.786468

4.73
4.73

.213816
.213532

34
33

28
29

.717673

.717879

.40

3.43
340

.930921
.930843

1^30

.786752
.787036

4.73
4.73

.213248
.212964

32
31

30

.718085

.40

3.43

.930766

l!30

.787319

4.72
4.73

.212681

30

31
32
33
34
35
36
37

9.718291
.718497
.718703
.718909
.719114
.719320
.719525

3.43
3.43
3.43
3.42
3.43
3.42

9.930688
.930611
.930533
.930456
.930378
.930300
.930223

1.28
1.30
1.28
1.30
1.30
1.28

9.787603
.787886
.788170
.788453
.788736
.789019
.789302

4.72
4.73
4.72

4.72
4.72
4.72

10.212397
.212114
.211830
.211547
.211264
.210981
.210698

29
28
27
26
25
24
23

38
39
40

.719730
.719935
.720140

3^42
3.42
3.42

.930145
.930067
.929989

liw

1.30
1.30

.789585
.789868
.790151

4i 72
4.72

4.72

.210415
.210132
.209849

22
21
20

41

9.720345

9.929911

Qfi

9.790434

10.209566

19

42

.720549

A 4<!

.929833

1 'on

.790716

A' i~r»

.209284

18

43
44
45
46

47
48
49
50

.720754
.720958
.721162
.721366
.721570
.721774
.721978
.722181

o.42
3.40
3.40
3.40
3.40
3.40
3.40
3.38
3.40

.929755
.929677
.929599
.929521
.929442
.929364
.929286
.929207

l.ou
1.30
1.30
1.30
1.32
1.30
1.30
1.32
1.30

.790999
.791281
.791563
.791846
.792128
.792410
.792692
.792974

4. i£

4.70
4.70
4.72
4.70
4.70
4.70
4.70
4.70

.209001
.208719
.208437
.208154
.207872
.207590
.207308
.207026

17
16
15
14
13
12
11
10

51
52

9.722385

.722588

3.38

9.929129
.929050

1.32
Ion

9.793256
.793538

4.70

4 pa

10.206744
.206462

9
8

53

.722791

3.38

.928972

.OU

.793819

.DO

.206181

7

54
55

.722994
.723197

3.38
3.38

3 no

.928893
.928815

i.'ao

.794101
.794383

4. '70

4 CO

.205899
.205617

6
5

56

.723400

.OO

.928736

'QO

.794664

.Do

.205336

4

57
58
59
60

.723603
.723805
.724007
9.724210

3.38
3.37
3.37
3.38,

.928657
.928578
.928499
9.928420

i!aa

1 32

.794946
.795227
.795508
9.795789

4.70

4.68
4.68
4.68

.205054
.204773
.204492
10.204211

3
2
1
0

'

Cosine.

D. r.

Sine.

p. r.

Cotang.

D. r.

Tang.

'

88-

32°

COSINES, TANGENTS, AND COTANGENTS.

147°

'

Sine.

D. 1'.

Cosine.

D. 1'.

Tang.

D. 1".

Cotang.

'

0

1

9.724210
.724412

3.37

39.7

9.928420
.928342

1.30

IqO

9.795789
.796070

4.68

4 AS

10.204211
.203930

60
59

2

.724614

.Oi

.928263

.ISO

.796351

.Do

.203649

58

3
4
5

.724816
.725017
.725219

3.37
3.35
3.37

39.K

.938183
.928104
.928025

lisa

1.32

Iqo

.796632
.796913
.797194

4.68
4.68
4.68

.203308
.203087
.202806

57
56
55

6

.725420

.00

39.7

.927946

.06

1 9.9

.797474

4 Aft

.202526

54

7

.725622

. Of

.927867

1 .«KB

.797755

.00
A Aft

.202245

53

8

.725823

3.35

3qK

.927787

1 99

.798036

<J. Oo

.201964

52

9

10

.726024
.726225

.00

3.35
3.35

.927708
.927623

I'.m

1.33

.798316
.798596

4^67
4.68

.201684
.201404

51
50

11

9.726426

300

9.927549

1 32

9.798877

4 67

10.201123

49

12

.726626

.00

3qK.

.927470

1 9*)

.799157

4A7

.200843

48

13
14

.726827
.72i 027

. GO

3.33

.927390
.927310

1 .00

1.33

1 9.9

.799437
.799717

.O<

4.67

.200563
- .200283

47

46

15

.727228

3 .35

.927231

1 .<M

1 9.9.

.799997

4A7

.200003

45

16
17
18

.727428
.727628

.727828

3.33
3.33
3.33

3qo

.927151
.927071
.926991

1 .08

1.33
1.33

1 99

.800277
.800557
.800836

.0*

4.67
4.65
4 67

.199723
.199443
.199164

44
43
42

19

.728027

.<>£

3qq

.926911

1 .OO

Iqo

.801116

4A7

.198884

41

20

.728227

.OO

3.33

.926831

.00

1.33

.801396

. Of

4.65

.198604

40

21

fi. 728427

39.9

9.926751

Iqq

9.801675

4A1"*

10.198325

39

22 .728626
23 .728825
24 .729024
25 .729223
26 .729422

.64

3.32
3.32
3.32
3.32

.926671
.926591
.926511
.926431
.926351

.00

1.33
1.33
1.33
1.33

1QK

.801955
.802234
.802513
.802792
.803072

.vt

4.65
4.65
4.65
4.67

.198045
.197766
.197487
.197208
.196928

38
37
36
35
34

27 .729621

3.32

3qO

926270

.00
Iqo

.803351

4. 65
4c-

.196649

33

28
29
30

.729820
.730018
.730217

.OSa

3.30
3.32
3.30

.926190
.926110
.926029

.00

1.33
1.35
1.33

.803630
.803909
.804187

. DO

4.65
4.63
4.65

.196370
.196091
.195813

32
31
30

31

9 730415

3OA

9.925949

19.K

9.804466

10.195534

29

32

.730613

.oO

.925868

.00

1 9.9.

.804745

4A9.

.195255

28

33
34
35
36

.730811
.731009
.731206
.73J404

3.30
3.30
3.28
3.30

.925788
.925707
.925626
.925545

1 .OO

1.35
1.35
1.35

.805023
.805302
.805580
.805859

.OO

4 65
4.63
4.65

4A9.

.194977
.194698
.194420
.194141

27
26
25
24

37

.731602

0 rtO

.925465

1'qjr

.806137

DO
4A9.

.193863

23

38

.731799

3.2o

.925384

.OO

.806415

_ . Do
4A9.

.193585

22

39

40

. 731996
.732193

3.28
3.28
3.28

.925303
.925222

1 .35

1.35
1.35

.806693
.806971

.Do
4.63
4.63

.193307
.193029

21
20

41
42
43
44
45
46
47
48
49

9.732390
.732587
.732784
.732980
.733177
.733373
.733569
.733765
.733961

3.28
3.28
3.27
3.28
3.27
3.27
3.27
3.27

9.925141
.925060
.924979
.924897
.924816
.924735
.924654
.924572
.924491

1.35
1.35
1-3?
1.85
1.35 j
1.35
1.37
1.35

9.807249
.807527
.807805
.808083
.808361
.808638
.808916
.809193
.809471

4.63
4.63

4.'63
4.62
4.63
4.62
4.63

10.192751
.192473
.192195
.191917
.191639
.191362
.191084
.190807
.190529

19
18
17
16
15
14
13
12
11

50

.734157

3.27
3.27

.924403

1 .37
1.35

.809748

4!62

.190252

10

51
52
53
54
55
56

9.734353
.734549
.734744
.734939
.735135
.735330

3.2?
3.25
3.25
3 27
3.25

9.924328
.924246
.924164
.924083
.924001
.923919

1.37
1.37
1.35
1.37
1.37

9.810025
.810302
.810580
.810857
.811134
.811410

4.62
4.63
4.62
4.62
4.60

10.189975
.189698
.189420
.189143
.188866
.188590

9
8
7
6
5
4

57

.735525

3.25

.923837

1.37

.811687

4 62

.188313

3

58

.735719

3.23

.923755

1w

.811964

4 62

.188036

2

59

.735914

3.25

.923673

,01

.812241

.187759

1

60 • 9.736109

3.25

9 923591

1 .37

9.812517

'

10.187483

0

Cosine.

D. r.

Sine.

D. 1'.

Cotang.

D. i".

Tang.

'

307

$7-

TABLE XII. LOGARITHMIC SINES,

'

Sine.

D. r.

Cosine.

D. r.

Tang.

D.I".

Cotang.

'

0

1

2

9.736109
.736303
.736498

3.23
3.25

9.923591
.923509
.923427

1.37

.37

9.812517
.812794
.813070

4.62
4.60

10.187483

.187206
.186930

60
59
58

3

4
5

.736692
.736886
.737080

3.23
3.23
3.23

.923345
.923263
.923181

.37
.37
.37

OO

.813347
.813623
.813899

4.62
4.60
4.60

4R9

.186653
.186377
.186101

57
56

55

6

7

.737274
.737467

3.23
3.22

.923098
.923016

.OO

.37

OO

.814176
.814452

.DxJ
4.60
4 fin

.185824
.185548

54
53

8

.737661

3.23

.922933

.00

.814728

.ou
4 fin

.185272

52

9

.737855

«

.922851

'no

.815004

.ou

.184996

51

10

.738048

3.22
3.22

.922768

.08

.37

.815280

4.60
4.58

.184720

50

11

9.738241

9.922686

OQ

9.815555

4 Art

10.184445

49

12

.738434

ctct

.922603

.OO

.815831

.OU

.184169

48

13
14

.738627
.738820

p.9H

3.22

.922520
.922438

.38
.37

oo

.816107
.816382

4.60
4.58
4 fin

.183893
.183618

47
46

15

.739013

3.22

399

.922355

.00

OQ

.816658

. ou

4 CO

.183342

45

16
17

.739206
.739398

.£&

3.20

.922272
.922189

.OO

.38

.816933
.817209

.00

4.60

4KQ

.183067
.182791

44
43

18
19

.739590
.739783

3.20
3.22

.922106
.922023

.38
1.38

1QQ

.817484
.817759

.00
4.58

.182516
.182241

42

41

20

.739975

3.20
3.20

.921940

.OO

1.38

.818035

4.60
4.58

.181965

40

21
22

9.740167
.740359

3.20

31ft

9.921857
.921774

1.38

100

9.818310
.818585

4.58

4KQ

10.181690
.181415

39

38

23
24

.740550
.740742

. lo
3.20

3 on

.921691
.921607

.00
1.40
1 °j}

.818860
.819135

.Oo

4.58

4 CO

.181140

.180865

37
36

25

.740934

.*u

3f 0

.921524

1 .00
1 00

.819410

.Oo

.180590

35

26
27

.741125
.741316

.lo

3.18

.921441
.921357

1 .00
1.40

.819684
.819959

4.57
4.58

4 to

.180316
.180041

34
33

28
29

.741508
.741699

3.20
3.18

.921274
.921190

l."40

.820234
.820508

.Oo

4.57

.179766
.179492

32
31

30

.741889

3.17
3.18

.921107

1.38
1.40

.820783

4.58
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
r .920436
.920352
.920268

.40

.38
.40
.40
.40
.40
.40
.40
.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
.178394
.178120
.177846
.177571
.177297
.177023
.176749
.176476

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46
47
48
49
50

9.743982
.744171
.744361
.744550
.744739
.744928
.745117
.745306
.745494
.745683

3.15
3.17
3.15
3.15
3.15
3.15
3.15
3.13
3.15
3.13

9.920184
.920099
.920015
.919931
.919846
.919762
.919677
.919593
.919508
.919424

1.42
1.40
1.40
1.42
.40
.42
.40
.42
.40
.42

9.823798
.824072
.824345
.824619
.824893
.825166
.825439
.825713
.825986
.826259

4.57
4.55
4.57
4.57
4.55
4.55
4.57
4.55
4.55
4.55

10.176202
.175928
.175655
.175381
.175107
.174834
.174561
.174287
.174014
.173741

19
18
17
16
15
14
13
12
11
10

51
52
53

9.745871
.746060
.746248

3.15
3.13

9.919339
.919254
.919169

.42
.42

Af\

9.826532
.826805
.827078

4.55
4.55

4CK

10.173468
.173195
.172922

9
8

7

54
55
56
57
58
59
60

.746436
.746624
.746812
.746999
.747187
.747374
9.747562

3. 13
3.13
3.13
3.12
3.13
3.12
3.13

.919085
.919000
.918915
.918830
.918745
.918659
9.918574

.4U

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

.827351
.827624

.827897
.828170
.828442
.828715
9.828987

.OD

4.55
4.55
4.55
4.53
4.55
4.53

.172649
.172376
.172103
.171830
.171558
.171285
10.171013

6
5
4
3

1
0

'

Cosine.

D. r.

Sine.

D. r. I

! Cotang.

D. 1".

Tang.

'

56*

34*

COSINES, TANGENTS, AND COTANGENTS.

145-

,

Sine.

D. r.

Cosine.

D. r. <

Tang.

D. 1".

Cotang.

,

1

0

1

2
3
4
5
6

9.747562
.747749
.747936
.748123
.748310
.748497
.748683

3.12
3.12
3.12
3.12
3.12
3.10

319

9.918574
.918489
.918404
.918318
.918233
.918147
.918062

1.42
1.42
.43
.42
.43
.42

9.828987
.829260
.829532
.829805
.830077
.830349
.830621

4.55
4.58

4.55
4.53
4.53
4.53

10.171013
.170740
.170468
.170195
.169923
.169651
.169379

CO
59
58
57
56
55
54

8
9
10

.748870
.749056
.749243
.749429

.19

3.10
3.12
3.10
3.10

.917976
.917891
.917805
.917719

.43
.42
.43
.43
.42

.830893
.831165
.831437
.831709

4.53
4.53
4.53
4.53
4.53

.169107
.168835
.168563
.168291

53
52
51
50

11

9.749615

31ft

9.917634

9.831981

4 to

10.168019

49

12
13
14
15
16

.749801
.749987
.750172
.750358
.750543

. 1U

3.10
3.08
3.10
3.08

.917548
.917462
.917376
.917290
.917204

!43
.43
.43
.43

.832253
.832525
.832796
.833068
.833339

.Oo

4.53
4.52
4.53
4.52

.167747
.167475
.167204
.166932
.166661

48
47
46
45
44

17

18

.750729
.750914

3. 10
3.08

3 no

.917118
.917032

.43
.43

Af>

.833611

.833882

4.53
4.52

4 CO

.166389
.166118

43
42

19

20

.751099
.751284

.UO

3.08
3.08

.916946
.916859

.<*')

.45
.43

.834154
.834425

.Do

4.52
4.52

.165846
.165575

41
40

21
22

9.751469
.751654

3.08

3AQ

9.916773
.916687

.43

AK.

9.834696
.834967

4.52

4 co

10.165304
.165033

39

38

23
24
25
26
27
28
29
30

.751839
.752023
.752208
.752392
.752576
.752760
.752944
.753128

.Uo
3.07
3.08
3.07
3.07
3.07
3.07
3.07
3.07

.916600
.916514
.916427
.916341
.916254
.916167
.916081
.915994

.40

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

.835238
.835509
.835780
.836051
.836322
.836593
.836864
.837134

,*M

4.52
4.52
4.52
4.52
4.52
4.52
4.50
4.52

.164762
.164491
.164220
.163949
.163678
.163407
.163136
.162866

37
36
35
34
33
32
31
30

31
32
33
34

9.753312
.753495
.753679
.753862

3.05
3.07
3.07

9.915907.
.915820
.915733
.915646

.45
.45
.45

9.837405
.837675
.837946
.838216

4.50
4.52
4.50

10.162595
.162325
.162054
.161784

29
28
27
26

35
36

.754046
.754229

3.07
3.05

.915559
.915472

.45
.45

.838487
.838757

4.52
4.50

.161513
.161243

25
24

37

38

.754412
.754595

3.05
3.05

.915385
.915297

.45
.47

.K39027
.839297

4.50
4.50

.160973
.160703

23

22

39
40

.754778
.754960

3.05
3.03
3.05

.915210
.915123

.45
.45
.47

.839568
.839838

4.52
4.50
4.50

.160432
.160162

21
20

41
42

9.755143
.755326

3.05

9.915035
.914948

1.45

9.840108
.840378

4.50

10.159892
.159622

19

18

43

44
45
46
47
48
49
50

.755508
.755690
.755872
.756054
.756236
.756418
.756600
.756782

3.03
3.03
3.03
3.03
3.03
3.03
3.03
3.03
3.02

.914860
.914773
.914685
.914598
.914510
.914422
.914334
.914246

1.47
1.45
1.47
1.45
1.47
1.47
1.47
1.47
1.47

.840648
.U0917
.841187
.841457
.841727
.841996
.842266
.842535

4.50
4.48
4.50
4.50
4.50
4.48
4.50
4.48
4.50

.159352
.159083
.158813
.158543
.158273
.158004
.157734
.157465

17
16
15
14
13
12
11
10

51
52
53
54

5«
56
57
58
59
60

9.756963
.757144
.757326
.757507
.757688
.757869
.758050
.758230
.758411
9.758591

3.02
303
3.02
3.02
3.02
3.02
3.00
3.02
3.00

9.914158
.914070
.913982
.913894
.913806
.913718
.913630
913541
.913453
9.913365

1.47
1.47
1.47
1.47
1.47
1.47
1.48
1.47 j
1.47 |

9.842805
.843074
.843343
.843612
.843882
.844151
.844420
.844689
.844958
9.845227

4.48
4.48
4.48
4.50
4.48
4.48
4.48
4.48
4.48

10.T7195
156926
.156657
.156388
.156118
.155849
.155580
.155311
.155042
10.154773

9
8
7
6
5
4
3
* 2
1
0

i

Cosine.

r>. r.

Sine.

D. r. *

Cotang.

D.r.

Tang.

'

134°

209

35'

TABLE XII.— LOGARITHMIC SINES,

144*

'

Sine.

D. r.

Cosine.

D. 1".

Tang.

D. I'.

Cotang.

'

0

1

9.758591

.758772

3.02

9.913365
.913276

1.48

9.845227
.$45496

4.48

10.154773
.154504

60
59

2

.758952

3.00

.913187

1 .48

.845764

4.47

.154236

58

3
4

.759132
.759312

3.00
3.00

.913099
.913010

1.47

1.48

.846033
.846302

4.48
4.48

.153967
.153698

57
56

5
6
7
8
9
10

.759492
.759672
.759852
.760031
.760211
.760390

3.00
3.00
3.00
2.98
3.00
2.98
2.98

.912922
.912833
.912744
.912655
.912566
.912477

1.47
1.48
1.48
1.48
1.48
1.48
1.48

.846570
.846&S9
.847108
.847376
.847644
.847913

4.47
4.48
4.48
4.47
4.47
4.48
4.47

.153430
.153161

.152892
.152624
.152356
.152087

55
54
53
52
51
50

11
12
13
14
15

9.760569
.760748
.760927
.761106
.761285

2.98
2.98
2.98
2.98

9.912388
.912299
.912210
.912121
.912031

1.48
1.48
1.48
1.50

9.848181
.848449
.848717
.848986
.849254

4.47
4.47
4.48
4.47

10.151819
.151551
.151283
.151014
.150746

49
48
47
46
45

16
17

.761464
.761642

2.98
2.97

.911942
.911853

1 .48
1.48

ItA

.849522
.849790

4.47
4.47

4JK

.150478
.150210

44
43

18

.761821

2.98

.911763

.OU

.850057

.40

.149943

42

19
20

.761999
.762177

2.97
2.97
2.98

.911674
.911584

1 .48
1.50
1.48

.850325
.850593

4.47
4.47
4.47

.149675
.149407

41
40

21

9.762356

2 97

9.911495

Itrv

9.850861

10.149139

39

22
23
24
25
26
27
28

.762534
.762712
.762889
.763067
.763245
.763422
.763600

2^97
2.95
2.97
2.97
2.95
2.97

2ne

.911405
.911315
.911226
.911136
.911046
.910956
.910866

.OU

1.50
1.48
1.50
1.50
1.50
1.50

.851129
.851396
.851664
.851931
.852199
.852466
.852733

4.4<
4.45
4.47
4.45
4.47
4.45
4.45

447

.148871
.148604
.148336
.148069
.147801
.147534
.147267

38
37
36
35
34
33
32

29
30

.763777
.763954

.yo
2.95
2.95

.910776
.910686

i!so

1.50

.853001
.853268

.4<
4.45
4.45

.146999
.146732

31
30

31
32
33
34
35
36

9.764131
.764308
.764485
.764662
.764838
.765015

2.95
2.95
2.95
2.93
2.95

9.910596
.910506
.910415
.910325
.910235
.910144

1.50
1.52
1.50
1.50
1.52

9.853535
• .853802
.854069
.854336
.854603
.854870

4.45
4.45
4.45
4.45
4.45

4AK

10.146465
.146198
.145931
.145664
.145397
.145130

29
28
27
26
25
24

37
38
39
40

.765191
.765367
.765544
.765720

2.93
2.93
2.95
2.93
2.93

.910054
.909963
.909873
.909782

l'.52
1.50
1.52
1.52

.855137
.855404
.855671
.855938

.40

4.45
4.45
4.45
4.43

.144863
.144596
.144329
.144062

23
22
21
20

41
42
43
44
45

9.765896
.766072
.766247
.766423
.766598

2.93
2.92
2.93
2.92

9.909691
.909601
.909510
.909419
.909328

1.50
1.52
1/52
1.52

9.856204
.856471
.856737
.857004
.857270

4.45
4.43
4.45
4.43

10.143796
.143529
.143263
.142996
• .142730

19

18
17
16
15

46

.766774

4.&6

.909237

1 .52

.857537

4.45

.142463

14

47
48
49
50

.766949
.767124
.767300
.767475

2.92
2.92
2.93
2.92
2.90

.909146
.909055
.908964
.908873

1 .52
1.52
1.52
1.52
1.53

.857803
.&58069
.858336
.858602

4^43
4.45
4.43
4.43

.142197
.141931
.141664
.141398

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
.908324
.908233
.908141
.908049
9.907958

1.52
1.52
1.53
1.52
1.53
1.32
1.53
1.53
1.52

9.858868
.859134
.859400
.859666
.859932
.860198
.860464
.860730
.860995
9.861261

4.43
4.43
4.43
4.43
4.43
4.43
4.43
4.42
4.43

10.141132
.140866
.140600
.140334
.140068
.139802
.139536
.139270
. 139005
10.138739

9

8

6
5
4
3
2
1
0

'

Cosine.

D. r.

Sine.

D. 1".

Cotang.

D. 1".

Tang.

'

36°

COSINES, TANGENTS, AND COTANGENTS.

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

1

9.769219
.769393

2.90

2QQ

9.907C58
.907866

1.53

Itq

9.861261
.861527

4.43

10.138739
.138473

60
59

2
3
4
5
6

.769566
.769740
.769913
.770087
.770260

.OO

2.90

2.88
2.90
2.88

2QQ

.907774
.907682
.907590
.907498
.907406

.Do

1.53
1.53
1.53
1.53

1KO

.861792
.862058
.862323
.862589
.862854

4.42
4.43
4.42
4.43
4.42

.138208
.137942
.137677
.137411
.137146

58
57
56
55
54

7
8
9
10

.770433
.770606
.770779
.770952

. Oo

2.88
2.88
2.88
2.88

.907314
.907222
.907129
.907037

.DO

1.53
1.55
1.53
1.53

.863119
.863385
.863650
.863915

4.42
4.43
4.42
4.42
4.42

.136881
.136615
.136350
. 130085

53
52
51
50

11

12
13
14
15
16
17
18
19
20

9.771125
.771298
.771470
.771643
.771815
.771987
.772159
.772331
.772503
.772675

2.88
2.87
2.88
2.87
2.87
2.87
2.87
2.87
2.87
2.87

9.906945
.906852
.906760
.906667
.906575
.906482
.906389
.906296
.906204
.906111

1.55
1.53
1.55
1.53
1.55
1.55
1.55
1.53
1.55
1.55

9.864180
.864445
.864710
.864975
.865240
.865505
.865770
.866035
.866300
.866564

4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.40
4.42

10.135820
.135555
.135290
.135025
.134760
.134495
.134230
.133965
.133700
.133436

49
48
47
46
45
44
43
42
41
40

21
22
23
24

9.772847
.773018
.773190
.773361

2.85
2.87
2.85

9.906018
.905925
.905832
.905739

1.55
1.55
1.55

9.866829
.867094
.867358
.867623

4.42
4.40
4.42

10.133171
. 132906
.132642
.132377

39

38
37
36

25
26

27
28
29

.773533
.773704
.773875
.774046
.774217

2.87
2.85
2.85
2.85
2.85

.905645
.905552
.905459
.905366
.905272

1.57
1.55
1.55
1.55
1.57

.867887
.868152
.868416
.868680
.868945

4.40
4.42
4.40
4.40
4 42

.132113
. 131848
.131584
. 131320
. 131055

35
34
33

31

30

.774388

2.85
2.83

.905179

1.55
1.57

.869209

4.40
4.40

.130791

30

31

32

9.774558
.774729

2.85

9.905085
.904992

.55

9.809473
.860737

4.40

10.130527
.130263

29

28

33

.774899

2.83

.904898

'. .57

.870001

4.40
4dfi

.129999

27

34

.775070

2.85

.904804

.57 j

fti

.870265

.w

.129735

26

35

.775240

2.83

200

.904711

. .DO
5rf

.870529

4.40

.129471

25

36
37
38

.775410
.775580
.775750

.00
2.83
2.83

.904617
.904523
.904429

(

.57
.57

.870793
.871057
.871321

4.40
4.40
4.40

.129207
.128943

.128679

24
23

22

39
40

.775920
.776090

2.83
2.83
2.82

.904335
.904241

.57
.57
.57

.871585
.871849

4.40
4.40
4.38

.128415
.128151

21
20

41

9.776259

2QO

9.904147

• K(->

9.872112

10.127888

19

42

.776429

.OO

.904053

.Ol

.872376

4.40

.127624

18

43

.776598

2.82

20°.

.903959

'. .57

• CO

.872640

4.40

400

.127360

17

44

.776768

.00

.903864

. .Do

.872903

.00

.127097

16

45

.776937

2.82

.903770

'. .57

.873167

4.40

.126833

15

46
47

48

.777106
.777275
.777444

2.82
2.82
2.82

.903676
.903581
.903487

: '.SB

.57

pro

.873430
.873694
.873957

4^40
4.38

4QQ

.126570
.126306
.126043

14
13
12

49

.777613

2.82

.903392

.Do

.874220

.OO

.125780

11

50

.777781

2.80
2.82

i .903298

.57
.58

.874484

4.40
4.38

.125516

10

51
52

9.777950
.778119

2.82

9.903203
.903108

: .58

9.874747
.875010

4.38

400

10.125253
.124990

9
8

53
54

.778287
.778455

2.80
2.80

.903014
.902919

.57

.58

KQ

.875273
.875537

.00
4.40

400

.124727
.124463

7

6

55

.778624

2.82

.902824

.Do

.875800

.00

40Q

.124200

5

56
57

.778792
.778960

2.80
2.80

.902729
.902634

.58
.58

.876063
.876326

.00
4.38

400

.123937
.123674

4
3

58

.779128

2.80

.902539

! .58

KQ

.876589

.00
4OQ

.123411

2

59
60

.779295
9.779463

2.78
2.80

.902444
9.902349

. Do

1.58

.876852
9.877114

.00

4.37

.123148
10.122886

1
0

'

Cosine.

D. 1". i

1 Sine.

D. r.

Cotang.

v.r.

Tang.

'

126°

211

TABLE XII. — LOGARITHMIC SINES,

142°

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

9.779463

9.902349

9.S77114

10.122886

60

2
3

5
6

8
9

.779631
.779798
.779966
.780133
.780300
.780467
.780634
.730801
.780968

2.80
2.78
2.80
2.78
2.78
2.78
2.78
2.78
2.78

277

.902253
.902158
.902063
.901967
.901872
.901776
.901681
.901585
.901490

1.60
1.58
1.58
.60
.58
.60
.58
.60
.58

an

.877377
.877640
.877903
.878165
.878428
.878691
.878953
.879216
.879478

4.38
4.38
4.38
4.37
4.38
4.38
4.37
4.38
4.37

4OQ

.122623
.122360
.122097
.121835
.121572
.121309
.121047
.120784
.120522

59
58
57
56
55
54
53
52
51

10

.781134

. < i

2.78

.901394

.OU

1.60

.879741

.OO

4.37

.120259

50

11
12

9.781301
.781468

2.78

9.901298
.901202

1.60

9.880003
.880265

4.37

4OQ

10.119997
.119735

49

48

13
14
15
16
17
18
19
20

.781634
.781800
.781966
.782132
.782298
.782464
.782630
.782796

2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.75

.901106
.901010
.900914
.900818
.900722
.900626
.900529
.900433

i!eo

1.60
1.60
1.60
1.60
1.62
1.60
1.60

.880528
.880790
.881052
.881314
.881577
.881839
.882101
.882363

.OO

4.37
4.37
4.37
4.38
4.37
4.37
4.37
4.37

.119472
.119210
.118948
.118686
.118423
.118161
.117899
.117637

47
46
45
44
43
42
41
40

21
22

23

24
25
26

27
28
29
30

9.782961
.783127
.783292
.783458
.783623
.783788
.783953
.784118
.784282
.784447

2.77
2.75
2.77
2.75
2.75
2.75
2.75
2.73
2.75
2.75

9.900337
.900240
.900144
.900047
.899951
.899854
.899757
.899660
.899564
.899467

1.62
1.60
1.62
1.60
1.62 '
1.62
1.62
1.60
1.62
1.62

9.882625
.882887
.883148
.883410
.883672
.883934
.884196
.884457
.884719
.884980

4.37
4.35
4.37
4.37
4.37
4.37
4.35
4.37
4.35
4.37

10.117375
.117113
.116852
.116590
.116328
.116066
.115804
.115543
.115281
.115020

39
38
37
36
35
34

as

32
31
30

31
32
33
34
35
36
37
38
39
40

9.784612
.784776
.784941
.785105
.785269
.785433
.785597
.785761
.785925
.786089

2.73
2.75
2.73
2.73
2.73
2.73
2.73
2.73
2.73
2.72

9.899370
.899273
.899176
.899078
.898981
.898884
.898787
.898689
.898592
.898494

1.62
1.62
1.63
1.62
1.62
1.62
1.63
1.62
1.63
1.62

9.885242
.885504
.885765
.886026
.886288
.886549
.886811
.887072
.887333
.887594

4.37
4.35
4.35
4.37
4.35
4.37
4.35
4.35
4.35
4.35

10.114758
.114496
.114235
.113974
.113712
.113451
.113189
.112928
.112667
.112406

29
28
27
26
25
24
23
22
21
20

41
42
43
44
45
46

9.786252
.786416
.786579
.786742
.786906
.787069

2.73
2.72
2.72
2.73

2.72

9.898397
.898299
.898202
.898104
.898006
.897908

1.63
1.62
1.63
1.63
1.63

9.887855
.888116
.888378
.888639
.888900
.889161

4.35
4.37
4.35
4.35
4.35

10.112145
.111884
.111622
.111361
.111100
.110839

19
18
17
16
15
14

47

48

.787232
.787395

2.72
2.72

.897810
.897712

1.63
1.63

.889421
.889682

4.33
4.35

.110579
.110318

13

12

49
50

.787557
.787720

2.70

2.72
2.72

.897614
.897516

1.63
1.63
1.63

.889943
.890204

4.35
4.35
4.35

.110057
.109796

11
10

51
52

9.787883
.788045

2.70

9.897418
.897320

1.63

9.890465
.890725

4.33

10.109535
.109275

9

8

53
54

.788208
.788370

2.72
2.70

.897222
.897123

1.63
1.65

.890986
.891247

4.35
4.35

.109014
.108753

7
6

55

56
57

58

.788532
.788694
.788856
.789018

2.70
2.70
2 70
2.70

.897025
.896926
.896828
.896729

1 .63
1.65
1.63
1.65

.891507
.891768
.892028
.892289

4.33
4.35

4. as

4.35

.108493
.108232
.107972
.107711

5
4
3
2

59
60

.789180
9.789342

2.70
2.70

.896631
9.896532

1 .63
1.65

.892549
9.892810

4.33
4.35

.107451
10.107190

1
0

'

Cosine.

D r.

Sine.

D. 1".

Cotang.

D. 1".

Tang.

'

212

38°

COSINES, TANGENTS, AND COTANGENTS.

141-

'

Sine.

D. 1*.

Cosine.

D. 1'.

Tang.

D. 1'.

Cotang.

•

0

1

9.789342
.789504

2.70

9.896532
.896433

1.65

9.892810
.893070

4.33

10.107190
.106930

60
59

2
3

.789665
.789827

2>0

.896335
.896236

1 .63
1.65

.893331
.893591

4.35
4.33

4qq

.106669
.106409

58
57

4
5
6

8
9
10

.789988
.790149
.790310
.790471
.790632
.790793
.790954

2^68
2.68
2.68
2.68
2.68
2.68
2.68

.896137
.896038
.895939
.895840
.895741
.895641
.895542

l'.65
1.65
1.65
1.65
1.67
1.65
1.65

.893851
.894111
.894372
.894632
.894892
.895152
.895412

.00

4.33
4.35
4.33
4.33
4.33
4.33
4.33

.106149
.105889
.105628
.105368
.105108
.104848
.104588

50
55
54
53
52
51
50

11
12

9.791115
.791275

2.67

2 Aft

9.895443
.895343

1.67

9.895672
.895932

4.33

10.104328
.104068

49

48

13

.791436

.Do
2 AT

.895244

J*22

.896192

4'qq

.103808

47

14
15
16
17
18

.791596
.791757
.791917
.792077
.792237

.Ol

2.68
2.67
2.67
2.67
2 67

.895145
.895045
.894945
.894846
.894746

1^67
1.67
1.65
1.67

1 A7

.896452
.896712
.896971
.897231
.897491

.00
4.33
4.32
4.33
4.33

4OQ

.103548
.103288
.103029
.102769
.102509

46
45
44
43
42

19
20

.792397
.792557

2^67
2.65

.894346
.894546

1 .Oi

1.67
1.67 i

.897751
.898010

.OO

4.32
4.33

.102249
.101990

41
40

21

9.792716

2R7

9.894446

1 R7

9.898270

A 00

10.101730

39

22
23
24
25
26
27

.792876
.793035
.793195
.793354
.793514
.793673

.O<

2.65
2.67
2.65
2.67
2.65

.894346
.894246
.894146
.894046
.893946
.893846

1 .Ol i

1.67
1.67
1.67
1.67
1.67

IAft

.898530
.898789
.899049
.899308
.899568
.899827

Ht.OO

4.32
4.33
4.32

4. as

4.32

4qq

.101470
.101211
.100951
.100692
.100432
.100173

38
37
36
35
34
33

28
29
30

.793832
.793991
.794150

2.65
2.65
2.65
2.63

.893745
.893645
.893544

.Oo

1.67
1.68
1.67

.900087
.900346
.900605

.OO

4.32
4.32
4.32

.099913
.099654
.099395

32
31
30

31

9.794308

2 OK

9.893444

1 Aft

9.900864

400

10.099136

29

32
33
34

.794467
.794628
.794784

. OO

2.65
2.63

.893343
.893243
.893142

1 .Oo

1.67
1.68

.901124
.901383
.901642

.00

4.32
4.32

4 mat

.098876
.098617
.098358

23
27
26

35
36
37
38
39
40

.794942
.795101
.795259
.795417
.795575
.795733

2^65
2.63
2.63
2.63
2.63
2.63

.893041
.892940
.892839
.892739
.892638
.892536

lies

1.68
1.67
1.68
1.70
1.68

.901901
.902160
.902420
.902679
.902938
.903197

.98

4.32
4.33
4.32
4.32
4.32
4.32

.098099
.097840
.097580
.097321
.097062
.096803

25
24
23
22
21
20

41
42
43

9.795891
.796049
.796206

2.63
2.62

9.892435
.892334
.892233

1.68
1.68

9.903456
.903714
.903973

4.30
4.32

10.096544
.096286
.096027

19
18
17

44

45

.796364
.796521

2.63
2.62

.892132
.892030

1.68
1.70

.904232
.904491

4^32

.095768
.095?09

16
15

46

.796679

2.63

.891929

1.68

.904750

4'qrt

.095250

14

47
48
49
50

.796836
.796993
.797150
.797307

2.62
2.62
2.62
2.62
2.62

.891827
.891726
.891624
.891523

1.70
1.68
1.70
1.68
1.70

.905008
.905267
.905526
.905785

.OU
4.32
4.32
4.32
4.30

.094992
.094733
.004474
.094215

13
12
11
10

51

52
53

54

9.797464
.797621
.797777
.797934

2.62
2.60
2.62

9.891421
.891319
.891217
.891115

1.70
1.70
.70

9.906043
.906302
.906560
.906819

4.32
4.30
4.32

10.093957
.093698
.093440
.093181

9

8

7
6

55
56

.798091
.798247

2.62
2.60

.891013
.890911

!70

.907077
.907336

4. '32
4 on

.092923
.092664

5

4

57

.798403

2.60

.890809

.70

.907594

.oU

.092406

3

58
59
60

.798560
.798716
9.798872

2.62
2.60
2.60

.890707
.890605
9.890503

'.70 i
.70

I

.907853
.908111
9.9083C9

.4.'30
4.30

.092147
.091889
10.091631

2

1
0

t

Cosine.

D. 1'.

Sine.

D. r. i

Cotang.

D. r.

Tang.

'

128°

213

TABLE XII. — LOGARITHMIC SINES,

140«

'

Sine.

D. 1".

Cosine.

D. r.

Tang.

D. 1".

Cotang.

'

0

1

2
3
4
5
6

8

9.798872
.799028
.799184
.799339
.799495
.799651
.799806
.799962
.800117

2.60
2.60
2.58
2.60
2.60
2.58
2.60

9.890503
.890400
.890298
.890195
.890093
.889990
.889888
.889785
.889682

1.72
1.70
1.72
1.70
1.72
1.70
1.72
1.72

9.908369
.908628
.908886
.909144
.909402
.909660
.909918
.910177
.910435

4.32
4.30
4.30
4.30
4.30
4.30
4.32
4.30

10.091631
.091372
.091114
.090856
.090598
.090340
.090082
.089823
.089565

60
59
58
57
56
55
54
53
52

9
10

.800272
.800427

2. 58
2.58

.889579
.889477

1.72
1.70
1.72

.910693
.910951

4.30
4.30
4.30

.089307
.089049

51
50

11

12
13
14

15
16
17
18

9.800582
.800737
.800892
.801047
.801201
.801356
.801511
.801665

2.58
2.58
2.58
2.57
2.58
2.58
2.57

9.889374
.889271
.889168
.889064
.888961
.888858
.888755
.888651

1.72

1.72
1.73

.72
.72
.72

: .73

9.911209
.911467
.911725
.911982
.912240
.912498
.912756
.913014

4.30
4.30
4.28
4.30
4.30
4.30
4.30

10.088791
.088533
.088275
.088018
.087760
.087502
.087244
.086986

49

48
47
46
45
44
43
42

19
20

.801819
.801973

2.57
2.57
2.58

.888548
.888444

.72
.73

.913271
.913529

4.28
4.30
4.30

.086729
.086471

41
40

21
22
23
24
25
26
27

9.802128
.802282
.802436
.802589
.802743
.802897
.803050

2.57
2.57
2.55
2.57
2.57
2.55

2K7

9.888341

.888237
• .888134
.888030
.887926
.887822
.887718

.73

.72
.73
.73
1.73
1.73

1r»O

9.913787
.914044
.914302
.914560
.914817
.915075
.915332

4.28
4.30
4.30

4.28
4.30
4.28

40A

10.086213
.085956
.085698
.085440
.085183
.084925
.084668

39
38
371
36
35
34
33

28
29

.803204
.803357

.Ol

2.55

.887614
.887510

. 1 •)

1.73

1r»q

.915590
.915847

.OU

4.28

A OQ

.084410
.084153

32
31

30

.803511

2.-r'7

.887406

. IO

1.73

.916104

4. <ff>

4.30

.083896

30

31
32
33
34

9.803664
.803817
.803970
.804123

2.55
2.55
2.55

2ee

9.887302
.887198
.887093
.886989

1.73
1.75
1.73
170

9.916362
.916619
.916877
.917134

4.28
4.30
4.28

A OQ

10.083638
.083381
.083123
.082866

29

28
27
26

35
36
37
38
39
40

.804276
.804428
.804581
.804734
.804886
.805039

.OO

2.55
2.55
2.55
2.53
2.55
2.53

.886885
.886780
.886676
.886571
.886466
.886362

. 10

1.75
1.73
1.75
1.75
1.73
1.75

.917391
.917648
.917906
.918163
.918420
.918677

1.6O

4.28
4.30
4.28
4.28
4.28
4.28

.082609
.082352
.082094
.081837
.081580
.081323

25
24
23
22

21
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
.080552
.080295
.080038
.079781
.079524
.079267
.079010
.078753

19
18
17
16
15
14
13
12
11
10

51

9.806709

9.885205

9.921503'

10.078497

9

52
53

.806860
.807011

2.52
2.52

.885100
.884994

'.77

.921760
.922017

4^28

4 no

.078240
.077983

8

7

54
55

56
57
58
59
60

.807163
.807314
.807465
.807615
.807766
.807917
9.808067

2.53
2.52
2.52
2.50
2.52
2.52
2.50

.884889
.884783
.884677
.884572
.884466
.884360
9.884254

. 75
.77
.77
.75
.77
1.77
1.77

.922274
.922530
.922787
.923044
.923300
.923557
9.923814

,4X5

4.27
4.28
4.28
4.27
4.28
4.28

.077726
.077470
.077813
.076956
- .076700
.076443
10.076186

6
5
4
3
2
1
0

'

Cosine.

D. r.

Sine.

D.I".

Cotang.

D. r.

Tang.

'

40°

COSINES, TANGENTS, AND COTANGENTS.

139°

>

Sine.

D. 1".

Cosine.

D. r.

Tang.

D. r.

Cotang.

'

0

1

9.808067
.808218

2.52
2 50

9.884254
.884148

.77
77

9.923814
.924070

4.27

10.076186
.075930

60
59

2
3

4
5
6

7
8

.808368
.808519
.808669
.808819
.808960
.809119
.809269

2. 52
2.50
2.50
2.50
2.50
2.50
2 50

.884042
.883936
.883829
.883723
.883617
.883510
.883404

'.77
.78

.77
.77
.78
.77

7ft

.924327
.924583
.924840
.925096
.925352
.925609
.925865

4'.27
4.28
4.27
4.27
4.28
4.27

4 Oft

.075673
.075417
.075160
.074904
.074648
.074391
.074135

58
57
56
55
54
53
52

9
10

.809419
.809569

2^50
2.48

.883297
.883191

. to
.77
.78

.926122
.926378

.40

4.27
4.27

.073878
.073622

51
50

11
12
13
14
15
16
17
18
19
20

9.809718
.809808
.810017
.810167
.810316
.810465
.810614
.810763
.810912
.811061

2.50
2.48
2.50
2.48
2.48
2.48
2.48
2.48
2.48
2.48

9.883084
.882977
.882871
.882764
.882657
.882550
.882443
.882336
.882229
.882121

.78
.77
.78
.78
.78
.78
.78
.78
.80
.78

9.926634
.926890
.927147
.927403
.927659
.927915
.928171
.928427
.928684
.928940

4.27

4.28
4.27
4.27
4.27
4.27
4.27
4.28
4.27
4.27

10.07&366
.073110
.072853
. .072597
.072341
.072085
.071829
.071573
.071316
.071060

49
48
47
46
45
44
43
42
41
40

21
22

9.811210
.811358

2.47

24ft.

9.882014
.881907

.78

Oft

9.929196
.929452

4.27

10.070804
.070548

39
38

23

.811507

.40

O Aff

.881799

.O(J
r-0

.929708

f'ctrt

.070292

37

24
25
26

.811655
.811804
.811952

6 .4<

2.48
2.47

247

.881692
.881584
.881477

. <O

.80
.78

Qf\

.929964
.930220
.930475

4.27
4.27
4.25
407

.070036
.069780
.069525

36
35
34

27
*•
29
30

.812100
.812248
.812396
.812544

.41

2.47
2.47
2.47
2.47

.881369
.881261
.881153
.881046

.oU

.80
.80
.78
.80

.930731
.930987
.931243
.931499

.*•

4.27
4.27
4.27
4.27

.069269
.069013
.068757
.068501

33
32
31
30

31

9.812692

247

9.880938

OA

9.931755

4 OR

10.068245

29

32
33

.812840
.812988

.4l

2.47

.880830
.880722

.oU

.80

Co

.932010
.932266

>2SD

4.27

.067990
.067734

28
27

34
35

.813135
.813283

2.45

2.47

.880613
.880505

.04

.80
on

.932522
.932778

4.27
4.27

.067478
.067222

26
25

36

37

.813430
.813578

2^47

.880397
.880289

.oU

.80

.933033
.933289

4.25

4.27

.066967
.066711

24
23

38

.813725

2.45

.880180

.82

Qf\

.933545

4.27

.066455

22

39

.813872

2.45

2AK

.880072

.oU

.933800

4.25

.066200

21

40

.814019

.40

2.45

.879963

!so

.934056

4.27
4.25

.065944

20

41

9.814166

9.879855

CO

9.934311

10.065689

19

42
43

.814313
.814460

2.45
2.45

.879746
.879637

.(Sis

.82

.934567
.934822

4.27
4.25

.065433
.065178

18
17

44
45

.814607
.814753

2.45
2.43

.879529
.879420

.80
.82

.935078
.935333

4.27
4.25
407

.064922
.064667

16
15

46
47

.814900
.815046

2.45
2.43

.879311

.879202

.82
.82

.935589
.935844

.41

4.25

497

.064411
.064156

14
13

48

.815193

2.45

.879093

Ort

.936100

.41

.063900

12

49
50

.815339

.815485

2.43
2.43
2.45

.878984
.878875

.82
.82
.82

.936355
.936611

4.25
4.27
4.25

.063645
.063389

11
10

51

9.815632

9.878766

QQ

9.936866

10.063134

9

52

.815778

2.43

.878656

.OO

.937121

I'rt^

.062879

8

53
54

.815924
.816069

2.43
2.42

.878547
.878438

.82
.82

.937377
.937632

< .27
.25

.062623
.062368

7
6

55

.816215

2.43

.878328

.83

.937887

' .25

.062113

5

56

.816361

2.43

.878219

.82

.938142

' .25

.061858

4

57

.816507

2.43

.878109

.83

.938398

'c\L

.061602

3

58

.816652

2.42

.877999

.83

.938653

' .25

.061347

2

59
60

.816798
9.816943

2.43
2.42

.877890
9.877780

.82
.83

.938908
9.939163

4^25

.061092
10.060837

1
0

'

Cosine.

D. r.

Sine.

D. r. 1

Cotang.

D. r.

Tang.

'

130°

215

41°

TABLE XII. — LOGARITHMIC SIXES,

138°

•'

Sine.

D. 1'.

Cosine.

P.I'.

Tang.

D. 1".

Cotang.

'

0

1

2

9.81694-5
.817'088
.817233

2.42

2.42

9.877780
.877670
.877560

1.83

.83

9.939163
.939418
.939673

4.25
4.25

10.060837
.060582
.060327

60
59

58

3

4

.817379
.817524

2.43
2.42

.877450
.877340

.83
.83

.939928
.940183

4.25
4.25

.060072
.059817

57

56

5
6

7
8
9

.817668
.817813
.817953
.818103
.818247

2.40
2.42
2.42
2.42
2.40

Q At)

.877230
.877120
.877010
.876899
.876789

.83
.83
.83
.85
.83

QK

.940439
.940694
.940949
.941204
.941459

4.27
4.25
4.25
4.25
4.25

.059561
.059306
.059051
.058796
.058541

55
54
53

52
51

10

.818392

S5.Ce

2.40

.876678

.00

.83

.941713

4^25

.058287

50

11
12

9.818536
.818681

2.42

9.876568
.876457

.85

9.941968
.942223

4.25

10.058032
.057777

49

48

13

.818825

2.40

.876347

.83

QK

.942478

4.25

.057522

47

14

.818969 ,

2.40

.876236

.OO

.942733

4.25

.057267

46

15
16
17
18
19

.819113
.819257
.819401
.819545
.819689

2.40
2.40
2.40
2.40
2.40

.876125
.876014
.875904
.875793
.875682

.85
.85
.83
.85
.85
1 ft'%

.942988
.943243
.943498
.943752
.944007

4.25
4.25
4.25
4.23
4.25

.057012
.056757
.056502
.056248
.055993

45
44
43
42
41

20

.819832

2^40

.875571

1 . oo
1.87

.944262

4.25
4.25

.055738

40

21
22
23
24
25

9.819976
.820120
.820263
.820406
.820550

2.40
2.38
2.38
2.40

9.875459
.875348
.875237
.875126
.875014

1.85

1.85
1.85
1.87

9.944517
.944771
; .945026
.945281
.945535

4.23
4.25
4.25
4.23

4 OK

10.055483
.055229
.054974
.054719
.054465

39

38
37
36
35

26

27

.820693
.820836

2^38

.874903
.874791

l."87

; .945790
.946045

./CO

4.25

.054210
.053955

34
33

28
29

.820979
.82112-3

2^38

2qo

.874680
.874568

til?

.946299
.946554

4^25

4oq

.053701
.058446

32

31

30

.821265

.00

2.37

.874456

l."87

.946808

,xo

4.25

.053192

30

31

9.821407

9.874344

9.947063

4 OK

10.052937

29

32
33

.821550
.821693

5! 38

o q"

.874232
.874121

l!$5

.947318
.947572

./£)

4.23

4 OK

.052682
.052428

28

27

34
35

.821835
.821977

/O.OI

2.37

.874009
.873896

l!88

.947827
.948081

.28)

4.23
400

.052173
.051919

26
25

36
37
38
39

.822120
.822262
.822404
.822546

2^37
2.37
2.37
2 37

.873784
.873672
.873560
.873448

1.S7
1.87
1.87

1DQ

.9483ar>
.948590
.948844
.949099

.Ou

4.25
4.23
4.25

.051665
.051410
.051156
.050901

24
23
22
21

40

.822688

2!37

.873335

.OO

1.87

.949353

4^25

.050647

20

41

9.822830

9.873223

9.949608

10.050392

19

42
43

.822972
.823114

2.3<
2.37

2q>r

.873110
.872998

1.88
1.87

.949862
.950116

4.23
4.23

4 OK

.050138
.049884

18
17

44

.823255

.OD

.872885

i ft«

.950371

.3D

.049629

16

45

.823397

2.37

.872772

1QQ

.950625

A OQ

.049375

15

46
47

.823539
.823680

2.35

.872659

.872547

. .OB

1.87

Ipo

.950879
.951133

4. '23

.049121

.048867

14
13

48

.823831

2.35

.872434

.00

.951388

4.25

.048612

12

49
50

.823963
.824104

2.37
2.35
2.35

.872321
.872208

1.88
1.88
1.88

.951642
.951896

4.23
4.23
4.23

.048358
.048104

11
10

51
52
53

9.824245
.824386

.824527

2.35
2.35

9.872095
.871981
.871868

1.90
1.88

1QQ

9.952150
.952405
.952659

4.25
4.23

10.047850
.047595
.047341

9
8
7

54
55

.824668
.824808

2.35
2.33

.871755
.871641

.00

1.90

1QQ

.952913
.953167

4i23

.047087
.046833

6
5

56
57

.824949
.825090

2.35
2.35

.871528
.871414

.OO

1.90

.953421
.953675

4^23

.046579
.046325

4
3

58
59

.825230
.825371

2.33
2.35

.871301

.871187

1.88
1.90

.953929
.954183

4.23
4.23
400

.046071
.045817

2

1

60

9.825511

2.33

9.871073

1 .90

9.954437

.60

10.045563

0

'

Cosine.

D. 1'.

Sine.

D. 1".

Cotang.

D. 1".

Tang.

'

131°

216

48°

42°

COSINES, TANGENTS, AND COTANGENTS.

137°

'

Sine.

D. r.

Cosine.

D. 1'.

Tang.

D. 1".

Cotang.

•

0

1

9.825511
.825651

2.33

9.871073
.870960

.88

9.954437
.954691

4.23

10.045503
.045:309

60

59

2
3

.825791

.8251)31

2.33
2.33

.870846
.870732

.1)0
.90

.1)541)1(5
.95521)0

4.25
4.23

.045054
.044800

58

57

4
5

.826071
.826211

2.33
2.33

O OQ

.870618
.870504

.1)0
.90
on

.956454

.955708

4.23
4.23

499

.044546
.044292

56

55

6

7

.826351
.826491

/v. OO

2.133

.870390
.870276

.j\)
.90

QO

.955961
.956215

,iCO

4.23

4OQ

.044039
.043785

51
53

8
9
10

.826631

.826770
.826910

2^32
2.33
2.32

.870161

.870047
.869933

. J/w

.90
.90
.92

.956409
.956723
.956977

..OO

4.23
4.23
4.23

.043531
.043277
.043023

53
51
50

11

9.827049

9.869818

9.957231

10.012709

49

12
13
14
15
16
17
18
19

.827189
.827328
.827467
.827606
.827745
; 827884
.828023
.828162

2.3o
2.32
2.32
2.32
2.32
2.32
2.32
2.32

.809704
.869589
.869474
.869360
.869245
.869130
.869015
.868900

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

.957485
.957739
.957993
.958247
.958500
.958754
.959008
.959262

4 . 2o
4.23
4.23
4.23
4.22
4.23
4.23
4.23

.042515
.042261
.042007
.041753
.041500
.041246
.040992
.0407.8

48
47
40
45
44
43
42
41

20

.828301

2.32
2.30

.868785

.92
.92

.959516

4^22

.040484

40

21
23
23
24

9.828439

.828573
.828716
.828855

2.32
2.30
2.32

9.868670
.868555
.868440
.868324

.92

.92
.93

9.959769
.900023
.960277
.960530

4.23
4.23
4.22
4 23

10.040231
.039977
.039723
.039470

39

38
37
36

25
26
27

28
29
30

.828993
.829131
.829269
.821)407
.829545
.829683

2.30
2.30
2.30
2.30
2.30
2.30
2.30

.868209
.868093
8G7978
.867862
.867747
.867631

.92
.93
.92
.93
.93
.93
.93

.960784
.961038
.961292
.961545
.96179'J
.962052

4. 23
4.23
4.22
4.23
4.23
4.23

.039216
.038962
.038708
.038455
.038201
.037948

35
34
33
32
31
30

31

0.829821

9.867515

9.962306

400

10.037694

29

33
33

.829959
.830097

2.30
2.30

.867399
.867283

.9o
.93

.962560
.962813

.Too

4.22

.037440
.037187

28
27

34
35

36
37
38
39

.830234
.830372
.830509
.&30646
.830784
.&30921

2.28
2.30
2.28
2.28
2.30
2.28

.867167
.867051
.866935
.866819
.866703
.866586

.93
.93
.93
.93
.93
.95

.963067
.963320
.963574
.963828
.964081
.964335

4^22
4.23
4.23
4.22
4.23

499

.036933
.036680
.036426
.036172
.035919
.0,35665

26
25
24
23
22
21

40

.831058

2.28
2-. 28

.866470

.93
.95

.964588

:.Ol9

4.23

.035412

20

41

42
'43
44
45

46
47
48
49
00

9.831195
.831332
.831469
.831606
.831742
.831879
.832015
.832152
.832288
.832425

2.28

2.28
2.28
2.27
2.28
2.27
2.28
2.27
2.28
2.27

9.866353
.866237
.866120
.866004
.865887
.865770
.865653
.865536
.865419
.865302

.93
.95
.93
.95
.95
.95

'.95
.95
.95

9.964842
.965095
.965349
.965602
.965855
.966109
.966362
.966616
.966869
.967123

4.22
4.23

4.22
4.22
4.23
4.22
4.23
4.22
4.23
4.22

10.035158
.034905
.094061
.034398
.034145
.033891
.033638
.0333*4
.033131
.032877

19
18
17
16
15
14
18
12
11
10

51
52
53
54
55
56
57
58
59
60

9-832561
.832697
.832833
. 832969
.833105
.833241
.833377
.833512
.833648
9.833783

2.27
2.27
2.27
2.27
2.27
2.27
2.25
2.27
2.25

9.865185
.865068
.864950
.864833
.864716
.864598
.864481
.864363
.864245
9.864127

.95
.97
.95
.95
.97
.95
.97
.97
.97

9.967376
.967629
.967883
.968136
.968:389
.968643
.968896
.969149
.969403
9.969G56

4.22
4.23
4.22

4.22
4.23
4.22
4.22
4.23
4.22

10.032624
.088871
.03&17
.031804
.031011
.031357
.031101
.030851
.030597
10.030344

9
8
7
6
5
4
3

1

0

'

Cosine.

D. 1".

Sine.

1 D. r.

Cotang.

D. r.

Tang.

'

132*

217

43°

TABLE XII. — LOGARITHMIC SINES,

186°

•

Sine.

D. 1'.

Cosine.

D. 1".

Tang.

D. 1".

Cotang.

'

0

9.833783

o 07

9.864127

9.969656

10.030344

60

1

.833919

•» . AM

2 25

.864010

07

.969909

4.22

.030091

59

2

3

.834054
.834189

2>>5

9 97

.863892
.863774

. y<

.97

.97'0162
.970416

4. 22
4.23

.029838
.029584

58
57

4

.834325

16, <Cf

2 OK

.863656

.97

07

.970669

4.2-

.029331

56

5

.834460

. «O
2r>K

.863538

. VI

Oft

.970923

A 'rirt

.029078

55

6

8

.834595
.834730
.834865

. «0

2.25
2.25

.863419
.863301
.863183

.yo
.97
.97

.971175
.971429

.971682

4.22
4.23
4.22

.028825
.028571
.028318

54

53
52

9

10

.834999
.835134

2. 25
2.25

.863064
.862946

.98
1.97
1.98

.971935
.972188

4.22
4.22
4.22

.028065

.027812

51
50

11
12

9.835269
.835403

2.23

9.862827
.862709

1.97

9.972441
.972695

4.23

10.027559
.027305

49

48

13

.835538

2.25
2 23

.862590

1.98

1 Oft

.972948

4.22
4 22

.027052

47

14

.835672

.862471

1 . yo

.973201

.026799

46

15

.835807

2.25

2 23

.862353

1.97

1QQ

.973454

4.22

499

.026546

45

16

.835941

.862234

.yo

.973707

.3BH

.026293

44

17

.836075

2.23

.862115

1.98

.973960

4.22

.026040

43

18

.836209

2.23

.861996

.98

.974213

4.22

.025787

42

19

.836343

2.23

.861877

.98

.974466

4.22

.025534

41

20

.836477

2.23
2.23

.861758

.98
2.00

.974720

4.23

4.22

.025280

40

21

9.836611

200

9.861638

QO

9.974973

499

10.025027

39

22
23
24

.836745

.836878
.837012

.AO

2.22
,2.23

.861519
.861400
.861280

. yo
.98
2.00

.975226
.97'5479
.975732

.353

4.22
4.22

.024774
.024521
.024268

38
37
36

25

26

.837146
.837279

2.23
2.22

.861161
.861041

1.98
2.00

.975985
.97(5238

4.22
4.22

.024015
.023762

35
34

27
28
29
30

.837412
.837546
.837679
.837'812

2.22
2.23
2.22
2.22

2.22

.860922
.860802
.860682
.860562

1.98
2.00
2.00
2.00
2.00

.976491
.976744
.976997
.977'250

4.22
4.22
4.22
4.22
4.22

.023509
.023256
.023003
.022750

33
32
31

30

31

32
313
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

9 22
2.20
2.22
2.20
2.22
2.20

9.860442
.860322
.860202
.860082
.859962
.859842
.859721
.859601
.859480
.859360

2.00
2.00
2.00
2.00
2.00
2.02
2.00
2.02
2.00
2.02

9.977'503
.977756
.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
.021485
.021232
.020979
.020726
.020473
.020220

29
28
27
26
25
24
23
22
21
20

41

9.839272

2 on

9.859239

9.980033

10.019967

19

42
43
44

45
46

47
48
49
50

.839404
.839536
.839668
.839&X)
.839932
.840064
.840196
.840328
.840459

.«u

2.20
2.20
2.20
2.20
2.20
2.20
2.20
2.18
2.20

.859119

.858998
.858877
.858756
.858635
.858514
.858393
.858272
.858151

2.00
2.02
2.02
2.02
2.02
2.02
2.02
2.02
2.02
2.03

.980286
.980538
.980791
.981044
.981297
.981550
.981803
.982056
.982309

4.2,4
4.20
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22

.019714
.019462
.019209
.018956
.018703
.018450
.018197
.017944
.017691

18
17
16
15
14
13
12
11
10

51

52
53
54
55
56
57
58
59
60

9.840591

.840722
.84085-4
.840985
.841116
.841247
.841378
.841509
.841640
9.841771

2.18
2.20
2.18
2.18
2.18
2.18
2.18
2.18
2.18

9.858029
.857908

'. 857665
.857543
.857422
.857300
.857178
.857056
9.856934

2.02
2.03
2.02
2.03
2.02
2.03
2.03
2.03
2.03

9.982562
.982814
.983067
.983320
.983573
.983826
.984079
.984332
.984584
9.984837

4.20
4.22
4.22
4.22
4.22
4.22
4.22
4.20
4.22

10.017438
.017186
.016933
.016680
.016427
.016174
.015921
.015668
.015416
10.015163

9
8
7
6
5
4
3
2
1
0

'

Cosine.

D. 1".

Sine.

T^iT

Cotang.

D. r.

Tang.

'

133°

218

44°

COSINES, TANGENTS, AND COTANGENTS.

'

Sine.

D. 1".

Cosine.

D. 1".

Tang.

D. 1'.

Cotang.

'

0

9.841771

218

9.856934

9.984837

10.015163

60

1

.841902

. JO

O -JQ

.856812

41 1 kO

.985090

4.22

.014910

59

2
3
4
5
6

.842033
.842163
.842294
.842424
.842555

& . Jo

2.17
2.18
2.17
2.18
21'"'

.856690
.856568
.856446
.856323
.856201

2 . Oo
2.03
2.03
2.05
2.03

! 985596
.Q85848
.986101
.986354

4.22
4 22
4^20
.22
.22

.014657
.014404
.014152
.013899
.013646

58
57
56
55
54

7
8
9

.842685
.842815
.842946

2.17
2.18
2 I'"'

.856078
.855956

2.05
2.03
2.05
'•* 03

.986607
.986860
.987112

'.22
.20

.013393
.013140

.012888

63

52
51

10

.843076

2^17

.'855711

a! 05

.987365

4! 22

.012635

50

11
12
13
14

9.843206
.843338

.843466
.843595

2.17
2.17
2.15

217

9.855588
.a55465
.855342
.855219

2.05
2.05
2.05

9.987618
.987871
.988123
.988376

4.22
4.20
4.22

10.012382
.012129
.011877
.011624

49
48
47
46

15

.843725

. J i
9 17

.855096

2.05

.988629

4.22

.011371

45

16
17

.843855
.843984

6 . I i

2.15

9 17

.854973
.854850

2.05
2.05

.988882
.989134

4.22
4.20

.011118
.010866

44
43

18

.844114

Z. if

2 IK

.854727

2.05

.989387

4.22

.010613

42

19
20

.844243
.844372

. ID

2.15
2.17

.854603

.854480

2 07
2 .05

2.07

.989640
.989893

4.22
4 22
4^20

. 010360
.010107

41
40

21

9.844502

9 1ft;

9.854356

9.990145

10.009855

39

22

.844631

16. JD

2 15

.85423:1

2.05

.990398

4.22

.009602

38

23

.844760

21 ^

.854109

2.07

.990651

4 .22

.009349

37

24
25

.844889
.845018

. JD

2.15

2-tK

.853986
.853862

2.05
2 07

.990903
.991156

4.20
4.22

.009097

.008844

36
35

26

.845147

. ID

.853738

2 O/

.991409

4 22

ooar}9i

.34

27

28
29
30

.845276
.845405
.845533
.845662

2.15
2.15
2.13
2.15
2.13

.853614
.853490
.a53366
.853242

2.07
2.07
2.07
207
2.07

.991662
.991914
.992167
.992420

4 .22
4.20
4 22
4 22
4^20

! 008338

.008086
.007833
.007580

33

32
31
30

31

9.845790

21 fi

9.853118

9.992672

10.007328

29

32
33
34
35
36
37
38
39

.845919
.846047
.846175
.846304
.846432
.846560
.846688
.846816

. ID

2.13
2.13
2.15
2.13
2.13
2.13
2.13
21°.

.852994
.852869
.852745
.852620
" .852496
.852371
.852247
.852122

2^08
2.07
2.08
2.07
2.08
2.07
2.08

O Aft

.992925
.993178
.993431
.993683
.993936
.994189
.994441
.994694

4.22
4 22
4.22

4.20
4 22
4 22
4.20
4.22
4 2s*

.007075
.006822
.006569
.006317
.006064
.005811
.005559
.005306

28
27
26
25
24
23
22
ST

40

.846944

. Jo
2.12

.851997

s.Uo

2.08

.994947

4^20

.005053

20

41
42
43
44

9.847071
.847199
.847327
.847454

2.13
2.13
2.12

9.851872
.a51747
.851622
.851497

2 08

2.08
2 08

9.995199
.995452
.995705
.995957

4.22

4.22

4.20

10.004801
.004548
.004295
.004043

19
18
17
16

45
46
47

.847582
.847709
.847836

2.13
2.12
2.12

.851372
.851246
.851121

2.08
2 10

2.08

.996210
.996463
.996715

4.22
4.22
4.20

499

.003790
.003537
.003285

15
14
13

48
49
50

.847964
.848091
.848218

2. 13
2.12
2.12
2.12

.850998
.830870
.850745

2.08
2 10
2 .08
2 10

.996968
.997221
.997473

.££

4 22
4.20
4.22

.003032
.002779
.002527

12
11
10

51

9.84a345

9.a50619

9 997726

400

10.002274

9

52
53

.848472
.848599

2 12
2 12

.850493
.850368

2 10

2.08

.997979
.998231

. <r<*
4.20

.002021
.0017'69

8

7

54

.848726

2. 12

.a50242

2 10

.99848-4

4 oo

.001516

6

55
56
57

.848852
.848979
.849106

2.10
2 12
2.12

.ar>0116
.849990

849864

2.10
2.10
2 10

.998737
998989
.999242

4"20
4.22

.001263
.001011

.000758

5
4
3

58

.849232

2. 10

2.10

999495

4 .22

.000505

2

59

.849359

2.12

! 849611

2. 12

.9911747

4.20
40.)

.000253

1

60

9.849485

2.10

9 &:>485

2. 10

0 000000

.<i«.

10.000000

0

'

Cosine.

D. r.

Sine.

D. r. i

Cotang. 1

D. r.

Tang,

'

134°

219

INDEX.

(Names of animals are to be looked for under their class name.)

PAGE

Amphibia, variability 66

Amphipoda, see Crustacea 67

Ancestral heredity 78

Annelida, correlation 76

, variability 67

Aphidac, see Hexapoda 66

Area, measurement of 5

Arithmetical work, precautions in 8

Arithometer 7

Assortative. mating 75

Average 13, 17

deviation. 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

i variability 67

Calculating machines 7

tables 7

Caprifoliacese, variability 70

Caryophyllaceae, variability 67

Character defined 1

Chauvenet 's criterion 12

Class, denned

range

Closeness of fit 24

Coefficient of correlation 44

regression 47

variability 16, 63

Coelenterata, see Hydromedusa.

Color, measurement of 6

Compositae, correlation 78

, variability 69, 70

Comptometer

Coordinate paper. 11

Cornaceae, variability 70

Correlated variability 42

Counting, methods of 3

Crabs 63

Criminals, skull index 64

Critical function 21

Cruciferse, variability 70

221

j »

222 INDEX.

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

Dipsacae, 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. 223

PAGE

Mendehsm 57, 82

Mid-departure 16

Mode 13

Multimodal polygons 39, 73

Multiple organ 1

Mutations 63

Myriapoda, correlation 76

, variability 66

Naquada race, skeletal variability 64, 65, 74

Normal curve of frequency 22

Number of variates to employ 2

Orchidaceae, variability 71

Organ variation 1

Papaveracese, variability 70

Partial variation 1

Person 1

Pisces, correlation 76, 77

, local races 83

, variability 66

Plants, correlation „ 78

, homotyposis 81

, variability 69

Prepotency 78

Primulacece, 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

Ranunculacese, variability 69

Recessive characters 58

Rectangles, method of, in platting frequency distributions . 11

Rejection of extreme variates 12

Relative variability of the sexes 63

Rosacece, variability 70

Sapidacea), variability 70

Scrophulariacese, variability 71

Selection 82

Sex, relative variability 63

Seriation 10

Skewness 30, 71, 72

Skull, see Homo.

Spurious correlation 54

Standard deviation 16

Stature, see Homo.

Symmetry in frequency distribution 19

Telegony 82

Types of frequencydistribution 19, 71, 72

Variability 15, 17, 02-71

Variant 1

Variate 1

Weight, variability, see Homo.

^

14 DAY USE

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