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'^^6tiAV^^!0''
1
STATISTICAL METHODS
WITH SPECIAL BEFERENCB TO
BIOLOGICAL VARIATION.
• •
C. b; 2AVBNP0BT,
Director of Departrnent of Experimental EvoliUwnt
Carnegie Institution of Waehington.
THIRD, REVISED EDITION.
FIRST THOUSAKD.
NEW YORK
JOHN WILEY & SONS, Inc.
London: CHAPMAN & HALL, Limited
1914
Oopyrlglit, 1899, 1904,
BY
C. B. DAVENPORT.
THE .SCIENTIFIC PRESS
MBCirr ORUMMOND AND COMPANT
BROOKLYN, N. V.
?>
■^
-=) PKEFACE.
This book has been issued in answer to a repeated call for a
simple presentation of the newer statistical methods in their
application to biology. The immediate need which has called
it forth is that of a handbook containing the working formuin
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 tbe 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 tbe cordial cooperation
which the publishers have given in making the book ser-
viceable.
C. B. Davbnpobt,
Biological Laboratory of the Brooklyn Institutb,
Cold Spring Harbor, Long Island,
June 29, 1899.
iii
288912.
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 Kari 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 por ExperimbntaIj Evolution
Carnegie Institution op Washington.
Cold Spring Harbor,
March 27, 1904.
iv
CONTENTS.
CHAPTER I.
On Methods of Measuring Organisms.
PAGE
Preliminary definitions 1
Methods of collecting individuals for measurement 2
Processes preliminary to measuring characters 2
The determination of integral variates — Methods of counting 3
The determination of graduated variates — Method of measurement. 4
Straight lines on a plane surface 4
Distances through solid bodies or cavities 4
Area of plane surfaces 4
Area of a curved surface 6
Characters occupying three dimensions of i^pace 6
Characters having weight 6
Color characters 6
Marking-characters 7
^ds in calculating 7
Precautions in arithmetical work 8
CHAPTER II.
On thb Seriation and Plotting op 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 CliAsses of Frequency Poltqons.
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 abscissse 25
The normal curve as a binomial curve 25
Example of a normal curve 26
To find the average difference between the pth and the (p + l)th
individual in any seriation 27
To find the best fitting normal frequency distribution when only a
portion of an empirical distribution is given t 28
Other unimodal frequency polygons 30
The range of the curve ^0
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 30
CHAPTER IV.
Correlated Variabilftt.
General principles 42
Methods of determining coefficient of correlation, 44
CONTENTS. VH
PAOll
Galton's graphic method. 44
Pearson's method. 44
Brief method 45
Probable error of r 45
Example 45
Coefficient of regression 47
The quantitative treatment of characters not quantitatively meas-
urable. 47
The correlation of non-quantitative qualities 49
Example 51
Quick methods of roughly determining the coefficient of correlation. 54
Spurious correlation in indices 54
Heredity 55
Uniparental inheritance 55
Biparental inheritance 55
To find the coefficient of correlation between brethren from the
means of the arrays 56
Galton's law of ancestral heredity. 57
Mendel's law of inheritance in hybrids 57
A dissymmetry index 60
CHAPTER V.
SoMB Rsaniyrs of STATisncAii Biolooioal Studt.
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
MoUusca 67
Echinodermata 68
Coelenterata. 68
Protista. 69
Plants 69
Some types of biological distributions 71
Type I. 71
Type IV. 72
Type V 72
Normal 72
Skewness 72
Complex distributions 73
• • •
TUl COISTTEKTS.
PAOB
Correlation 73
General 73
Man 73
Lower animaln 76
Plants 78
Heredity 78
General 78
Parental 79
Grandparental SO
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
BlBLIOGRAPHT 85
Explanation of Tables. 105
List of Tables;
The Greek alphabet 114
Index to the principal letters used in the formulse of this book. . . 115
Table I. Formulas 116
** II. Certain constants and their logarithms 117
•• III. Table of ordinates of normal curve, or values of —
Vo
corresponding to values of — 118
o
** lY. Table of half-class index values (^) or the values
of the normal probability integral corresponding to
values of — ; or the fraction of the area of the curve
o
X X
between the limits and -1 — or and — 119
a a
•• V. Table of Log T functions of p 126
** Vl. Tal>le of reduction of linear dimensiona from common
to metric sjrstem 128
** \ VII. Minutes and seconds in decimals of a degree 128
•• / VIII. First to sixth powers of integers from 1 to 60. . . * . . . 129
•• ' IX. Probable errors of the coefficient of correlation 130
** X. Squares, cubes, square-roots, cube-roots, and recip-
rocals 131
** XI. Logarithms of numbers: 149
** XII. Logarithmic sines, cosines, tangents, and cotangents. 176
STATISTICAL METHODS
WITH SPECIAL REFERENCE TO
BIOLOGICAL VAKIATION.
CHAPTER I.
On Methods of Measuring Organisms.
Preliminary Definitions.
An individual is a segregated mass of living matter, capable
of independent existence. Individuals are either simple or
compound, i.e., stocks or corms. In the case of a compound
individual the morphological unit may be called a person!
A multiple organ is one that is repeated many times on the
same individual. Example, the leaves on a tree, the scales
on a fish.
A chara/^ter 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.
Graduxited variates are magnitude-determinations of char-
acters which do not exist as integers and which may c-onse-
quently differ in different variates by any degree of magni-
tude however small; e.gr., the stature of man.
A variant^ among integral variates, is a single number-con-
dition, e.gr., 5 (flowers), 13 (ray-flowers), etc.
A clasSf 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.
I
>M
2 STATISTICAL METHODS.
Methods of Collecting ludlviduals 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:
Hating settled upon the general conditions, of race, sex,
locality, age, which the individuals to be measured must fulfil,
take Vie individuals methodically at random and toithout possible
selection of individuals on the basis of the magnitude of the
eha/racter to be measured. If the iudividuals are simply not
consciously selected on the basis of niagnilude of the character
they will often be taken sufficiently at random.
The number of variates to be obtamed 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 reproduiing 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 dramnga 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 Yariates.—
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.
Straight lines on a plane sur&ce are easilj meas-
ured by meana of a measuring- scale of some sort. The meas-
uremeut should always be metric because
this t3 the universal scientific system. Vari-
ous kinds of Bcalea may l>e obtained of
Optical companies and hardware dealers,—
such as steel measuritig tapes, graduated to
millimetres (about |1,00), and steel rules
(6 cm. to 16 cm. ) graduated to } of a millU
metre. Steel " spdog-bon " dividers with
milled-head screw are useful for getting
distances which may be laid off on a scale.
Tortuous Hues, e.g., the contour of the
seriated margiu of a leaf or Ibe outer
margin of the wing of a sphinx raolb, may
be measured by a map-measurer (" Eiitfer-
nungsmesser, " Fig. 1), supplied at artist's
i and engineer's supply stores at about (^.50.
Dtatances through solid bodies
' or cavities are measured by ciilipcrs of
Home sort. Calipers for measuring diumtters
of solid bodies are made in various styles.
Micrometer screw calipers (" speeded ")
reading to oue-bundredtbs of a. millimetre
Fid. 1. and sold by dealers in pliysicitl apparatus for
about f 5.00 are excellent for determining diameters of bonea,
birds' eggs, gastropod shells, etc. Leg calipers for rougher
work can be obtained for from 30 cents to $4.00. The
micrometer " caiiper-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 colloldia scratched
In millimetre squares. By rubbing in a little carmine the
MEASUREMENT OF ORGAlflSMS. 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
Fio. 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 quautllatively 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 Cliaracters. Color may be qualitatively ex-
pressed by reference to named standard color samples. Such
standard color samples are given in Ridgeway's book,
** Nomenclature of Color/* and also in a set of samples manu-
factured by the Milton Bradley Co., Springfield, Mass. , costing
6 cents. The best way of designating a color character is by
means of the color wheel, a cheap form of which (costing 6
cents) is made by the Milton Bradley Co. The colors of this
"top" are standard and are of known wave-length as follows:
Bed, 656 to 661 Green, 514 to 519
Orange, 606 to 611 Blue, 467 to 472
Yellow, 577 to 582 Violet, 419 to 424.
It is desirable to use Milton Bradley's color top as a standard.
Any color character can be matched by using the elementary
colors and white and black in certain proportions. The pro-
portions are given in percents. In practice the fewest possible
colors necessary to give the color character should be employed
and two or three independent determinations of each should
be made at different times and the results averaged. So far
as my experience goes any color character is given by only
one least combination of elementary colors. (See Science,
July 16, 1897.)
When there is a complex color pattern the color of the
different patches must be determined separately. In case of
a close intermingling of colors, the colored area may be rapidly
rotated on a turntable so that the colors blend and the result-
MEASUREMENT OF ORGANISMS. 7
ant may then be compared with the color -wheel. By this
means also the total raelunism or albinism, viridescence, etc.,
may be measured.
Markiug-cbaracters. 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 vaiiation of the elements can usually
be treated under one of the preceding categories. Find in how
far the variation of the color pattern is due to the variation of
some number or other magnitude, and express the variation in
terms of that magnitude. Remember that it is rarely a ques-
tion whether the variation of the character can be expressed
quantitatively but rather what is the best method of express-
ing it quantitatively.
Aids in Calculating^. An indispensable aid in multi-
plying and dividing is a book of reckoning tables of which
Crelle's Rechnungstafeln (Berlin: Geo. Reimer) is the best.
This work enables us to get directly any product to 999X999
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 numbera 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 Manufactiuing Co.,
Chicago ($225), and foimd 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 ^ xvii, 706). It
is sold by the Spectator Company, 95 WiUiam Street, New
York, price $250. The same firm is agent for Tate's Im-
8 STATISTICAL METHODS.
proved Arithometer ($300 to $400). The "Brunsviga" csiU
culating machine (Herm 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 J xvii, 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., quadniple ruled,
with about three squares to the centimetre, so that each
figure may occupy a distinct square. I like to work with a
pencil, of 2H grade, so that slight errors may be erased and
rectified. In case of larger errors running through several
steps of the work, the erroneous calculations should not be
erased but cancelled.
In using logarithms with the six-place table given in this
book, it is ordinarily necessary to write the entire mantissa
to six places, and to determine the number corresponding to
any logarithm to at least six places by use of the table of
proportional parts given at the bottom of the page. Upon
the completion of the calculation the number of decimal
places to be recorded will depend upon the probable error of
I
V
10 STATISTICAL METHODS.
CHAPTER II.
On the Sbriatiok and Plotting op Data and the
Frequency Polygon.
The data obtained by measuring any character in a lot of
individuals consists either of amass of numbers for the charac-
ter in each individual ; or, perhaps, two numbers which are to
be united to form a ratio ; or, finally, a series of numbers such
as are obtained by the color wheel, of the order : W40%, I^
(Black) 88^, 7 12^, Q 10^. 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 X\iq frequency
of the class. Each class has a central valve, 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 :
8.2
4.5
5.2
5.6
6.0
8.8
4.7
5.2
5.7
6.2
4.1
4.9
5.3
5.8
6.4
4.3
6.0
5.3
5.8
6.7
4.3
5.1
5.4
5.9
7.3
In this case it is clear that our magnitudes are not exact, but are merely
approximations of the real (forever unknowable) value. The question
SERIATION" XISTD PLOTTIiq^G OF DATA. 11
arises eoncerning the inclusiveuess of a class— the class range. An
approximate rule is : Make the classes only just large enough to haTO
no or very few vacant classes in the series. Following this rule we get
r 8.0-8.4; 8.5-8.9; 4.0-4.4; 4.6-4.9; 5.0-6.4;
aasses.... \ 8.2 8.7 4.2 4.7 5.2
( 1 2 8 4 5
Frequency 118 8 7
r 6.6-5.9; 6.0-6.4; 6.6-6.9; 7.0-7.4;
aasses.... •< 6.7 6.2 6.7 7.2
(67 8 9
Frequency 6 8 11
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 6 or 10, each class should include one
and only one of these round numbers.
As Fechner (*97) has pointed out, the frequency of the classes and all
the data to be calculated from the series will vary according to the
point at which we begin our seriation. Thus if, instead of beginning the
series with 8.0 as in our example, we begin with 3.1 we get the series :
riu^sg^ i 3.1-3.6; 8.6-4.0; 4.1-4.6; 4.6-5.0; 5.1-6.6;
v,ii««w....^ 3.8' 8.8 4.8 4.8 8.6
Frequency 114 8 6
Classes j 5-8-«-0; «.l-«.5; 6.6-7.0; 7.1-7.6;
^^ *** ( 5.8 6.8 6.8 7.8
Frequency 6 2 11
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 Une 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.
.a^t ri— h .... .k.T .
12
BTATisrrrcAL methods.
This method of drawing the frequency polygon is known aa
the method of rectangles.
When the number of classes is large the frequencies may be
represented by ordinates as follows: At equal intervals along
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 metliod of loaded ordiuates.
2»00 LE*WE5 NOHMiL CUflV!
Fia. 4.— Veins in Beech Leaveb. attek Pbaiibon, '02'.
The r^ection of extreme variatcs 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 (ko) is calculated, k is the argument in Table IV cor-
responding to a tabular entry equal to
2n-I
SERIATION AND PLOTTING OF DATA. 13
Example. — In 1000 minnows from one lake there are found the
following frequencies of anal fin-rays:
7 8 9 ' 10 11 12 13
1 2 15 279 554 144 6
A = 10.835 ; o = .7 28 fin-rays.
1999 .„__
''=40()0=-^^^^^-
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 clasa
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 averagfe or mean (A) is the abscissa of the centre of
gravity of the frequency polygon. It is found by the formula
n '
in which V is the magnitude of any class; / its frequency;
2 indicates that the sum of the products for all classes into
frequency is to be got, and n is the number of variates.
Thus in the example on p. 10:
A =(3.2X1+3.7X1+4.2X3+4.7X3 + 5.2X7+5.7X5+6.2X3
+6.7Xl+7.2Xl)+25=-5.24,
or
ili==(lX 1+2X1+3X3+4X3+5X7 +6X5+7X3+8X1+ 9X1)
+ 25=5.08,
A =6.2* + .08(5.7 -5.2) = 5.24.
A still shorter method of finding A i^ 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 1902^, 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 luagnitiide 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 lota: those less than
the middle doss, i.e., the one that contains the median magnitude, of
which the total number is a; those of the middle doss, b; and those
greater, c. Then a+b+c^n^the total number of variates". Let r=-
the lower limiting value of the middle doss, and I" =the upper limiting
value, and let x^the absdsspl distance of the median ordinate above the
lower limit QP^elow the upper limit of the median doss according as x
is positive or negative. Then ^n — a : b='X : I" — V when x is positive,
or ^— c : b'^x : I" — V when x is negative.
Thus in the last example: (12.5—8): 7==a; : 0.5; a; = .32; the median
magnitude = 5.0 + .32 = 5.32. Or (12.5-10): 7= -a; : 0.5; a;=-.18;
the median magnitude =5.5 — .18=5.32. (Of. 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 foimd. 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:*
db2Eare 4.5:1 ±5E are 1,310:1
±3Eare21 :1 ±6E are 19,200:1
±4E are 142 : 1 ± IE are 420,000 : 1
±8E are 17,000,000:1
±9E are about a billion to 1.
The probable error should be found to two significant
* These values are easily deduced from Table IV.
SERIATION 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 diflPerence between two averages (A^ and
A 2) of which the probable errors (E^ and E^) are known is
the square root of the sum of the squared probable errors, or
(Pearson, '02):
Probable Difference of A^-A^ is \/eJ+E^,
The probable error of the mean is given by the
formula
■^n fi74Svg*!ggg L d deviatio n [see belo w]^ ±0.6745-^.
Vnumber 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<t
-s-Vn" (Sheppard, '98).
The geometric mean of a series of values (v) is the
nimiber corresponding to the average of the logarithms of
the values. Thus,
n
The index of the variability, <y, 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
nimiber of variates, and extracting the square root of the
quotient, thus:
V
sum of [(deviation of class from mean)'
X frequency o f class]
number of variates
«/
^'■•"XX;
n
where X is the number of imits 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
• /MyPTAg standard deviation , r^ «pt^p <r
j:0.6745 . == ±0.6745-7::^.
V 2Xnimiber of variates V 2n
Otlier Indices of Variation. Tlie average deviation,
or average departure, is found thus:
. -J ^ Bum of [deviatioDB of class from mean X frequency]
number of variates
The average deviation is equal to .7979 X standard deviation, or
-0.7979ff.
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 measiu^d in other units. It has been pro-
posed to reduce the index of variation to an abstract number,
independent of any particular unit, by dividing the index of
variation of any variates by the mean; the quotient multi-
plied by 100 is called the coefficient of variability* In
a formula, C=-|-XlOO% (Pearson, '96; Brewster, '97).
The probable error of the coefficient of vari-
ability is given by Pearson as:
SERIATION 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
= <T.
3/
(.43 is the value of i — , at which the area of the curve
included between these limits of x equals one-third of the
whole).
Or, 2. Select roughly two specimens that seem to be about
one-third of the distance from the two extremes j»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^j and the number from the
middle to the larger n^^ Also, the x distance to the lower
division point \ and to the upper division point K. Then
(Ji^-\-h^ = 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 /la on the other (percentage of individuals between
h h
the middle and h. and hz), we can find — and — from Table IV,
(J a
X
since they are the values — corresponding to the percentage
♦ See Macaoqell, X902.
18 STATISTICAL METHODS.
areas determined. But — +— =■ ; thus a is deter-
a a a
mined. Knowing a we can get \ or h^y 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.
!i2!=M» 2105- VlL^lk^ 1974
From Table IV:
At
Therefore — = .555.
a
hi
a
% area.
.56
.55
.21226
.20884
Similarly ^=-.517;
o
Ai+fe2 ^Qy 0^ 2.22-1.78 ^
.44
•*• """ 1:072 ^•'*^°^'
— .555; 4inc'^'517;
.4105 • .4105
Ai = .2278, ^2=2122.
Mean is at 1.78 + .2278 = 2.01.
THE CLASSES OF FREQUENCY POLYGONS. 19
CHAPTER III.
The Classes op Frequency Polygons.
The plotted curve may fall into one of the folio wing classes:
A. Unimodal.
I. Simple.
1. Range unlimited in both directions:
a. Symmetrical. The normal curve.
b, Unsymmetrical (Pearson's Type IV).
2. Range limited in one direction, together with
skewness (Types III, V, and VI).
8. 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 foilows :
1. Composed of two curves, whose modes are different but so near that
the component curves biend Into one ; such curves are usually unsym-
metrical.
2. The sum of two curves having the same mode but differing varia-
bility.
8. 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 Vq)
80 STATISTICAL METHODS.
as a zero point ; then the departure of all the other classes
will be - 1, - 2, - 3, etc.. and + 1, + 2, + 3, etc.
Add the products of all these departures multiplied by the
frequency of the corresponding class and divide by n; call
the quotient rj.
Add the products of the squares of all the departures multi-
plied by the frequency of the corresponding class and divide '\
by 71 ; call the quotient ^a.
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 v^.
Add the products of \h^ fourth powers of all the departures
multiplied by the frequency of the corresponding class and
divide by n\ call the quotient v^. Or,
^i = -^ = departure of V^ from mean. Vq being
n
known, A may be found [A= Vo + yJ;*
_i'(7-7oV
^ n
n *
^4=
2iv-v,y
n
The values vi, v^, vs, v^, are called respectively the first,
second, third, and fourth moments of the curve about Kow
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 ^hort method of £lii(iiiig A referred to on page 13.
THE CLASSES OP FftEQttElfCT tOtTQOKS. 21
r n
A«5- V5— 6yiV4 + lOvi^^a - lOvi'va + ^^ ;
(B) To find moments in case of graduated variates:
/'5'=K-5v,y4+10y>3-10y,«y2 + 4yj'^-f;«3]A*;
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:
E;£2= .67449(72 j/-; ^^^^ 57449 ^24.
E/£3= .67449(T«|/| ; ^^vJi = .67449>|/^;
E/£,= .67449a^|/- ; I^d = .67449|/|^<r (p. 31) ;
^ of Skewness= .67449|/2^. (See page 30.)
(From Pearson, 1903°).
The classification of any empirical frequency polygon
depends upon the value of its "critical function," F* (Pear-
son, 1901<i).
F^
^0^2+3)'
4(4/?,-3/?,)(2^,-3A-6)-
* This value of F is ganeral. For the special case of Tsi>e8 I-IV
tha foUowing oritioal fonotioii iras given by Pearaon imm*
-■■ .'^-iA
STATiaTICA.L METHODS.
CorreBpondins Frequenoy Curve
P>Oand <1
j;'_0,^,=0,a,=3
F=Q,B,-0,3,not'
F<0
Type III, Transitional between Type
I and Type VI.
Type VI.
Type V. TraoMtional between Typo
IV and Type II.
Type IV.
Normal curve.
Type II.
Type I.
An important relation to be referred to later U
05
The Normai. Cubvb.
The normal curve is Bymmetiical about the mode; ooo-
sequently the mode and the median and mean coindde.
The mathematical formula of the normal curve, a formula
u ^ven 08 foUovn:
■e ia of Type I.
IB of Typo n. •
Tin u „ J ) fll^". P3^0, curvB IB ot Typn ni.
WhwiP-Oand 1ft=0.ft-3,«ir™i»nonn»I.
When F ia pomtive and ^i>0, ^>3, ourve la oTTVpe TV.
much used, F|-a»j-Mi-
Wben F is negative »nd J g_f.' "g'^X '
lft>o!fc>3,'oi;
J
THE CLASSES OF FREQUENCY POLYGOKS. 23
of which one does not have to understand the development
in order to make use of it, is
n 1
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-
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, y^ being taken as
equal to 1, and <r is the standard deviation.
For the case of a polygon built up of rectangles represent-
ing the relative frequency of the variates, Table IV gives
immediately the theoretical number of individuals occurring
between the values a;=0 and x— ±— . By looking up the
X
given values of — the corresponding theoretical percentage
of variates between the limits x=0 and a;= i— will be found
a
directly. The ratio — may be called the Index of Abmodality.
The normal curve may preferably be employed even when
^1 is not exactly equal to 0, nor ^g exactly equal to 3, nor F
exactly equal to 0. Use the normal curve when
FX/i,'<±l and ?^^^^^'=1±.2;
L...
24 STATISTICAL METHODS.
also the skewness (p. 30) should be less than twice the value
.67449 |/|.
To determine the closeness of fit of a theoreti-
cal polygon to the observed polygon. Find for
each class the difference (S^) 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
quotients is the index of closeness of fit (J). Or, J=y 2—*
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 frequency.
_.-i^2
P=e
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 {%) 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
number of variates; or, in a formula, -r— ^; where I,^f means
add all the frequencies from the median value to Xi ^^d n
is the number of variates. Next find for each class the sum
of A-\-ax. This should equal L. The difference is the
actiuil discrepancy. The probable discrepancy should next be
calculated for all but the extreme values. It is calculated
by use of the formula
0«4V I 'Ji^- (.+1) I KV^
THE CLASSES OF FREQUEKCY POLYGOKS. "Zb
where the value of z corresponding to ;f is got from Table III,
or from the formula
e^ix^ =
e
ix^
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) oi ratios found, according
to the following table :
Ai
P.L.
yli
P.L.
Ai
P.L
Ai
P.L.
1
1.000
6
2.376
11
2.777
16
3.009
2
1.559
7
2.481
12
2.832
17
3.046
3
1.874
8
2.570
13
2.882
18
3.080
4
2.088
9
2.648
14
2 928
19
3.112
5
2.248
10
2.716
15
2 970
20
3.142
The foregoing method is from Sheppard (1898).
The probable raiijye of abscissae (2xi) 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 <7, in accordance with the following formula derived
by the transposition of y= ^— - 6~^Vg<y' by putting y=l:
r\/25
2xi=2ay
log
n
!Es:ample. For the ventri^cosity of 1000 shells of Lit-
tomea littorea from Tenby, Wales, A = 90.964% and a=
2.3775%. What is the probable range of ventricosity
expressed in per cent.?
/;
2xi=2x2.S775y .460517xlog
1000
-15.2.
2.506628X2.3775
The observed range was 15 (Duncker, '98). See also the
criterion of Chauvenet (^88) for the rejection of extreme
variates (page 12).
The Normal Curve of Frequency as a Binomiaij
Curve.
The normal curve may also be expressed by the binomial
formula (pXq)'^, where p'^i, q^^^h 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
-4=4X (Standard Deviation) '=4<t'.
Linple of Norma,! Carre* — Number of rays in lower valve
of Pecten opercularis from Firth of Forth:
v-Vo nv-Vo) f(v-Voy nv-Vo^ kv-Vo)*
V
/
14
1
15
8
16
63
17
154
18
164
19
96
20
20
21
2
3
-3
9
-27
81
2
-16
32
-64
128
1
-63
63
63
63
1
164
164
164
164
2
192
384 •
768
1536
3
60
180
540
1620
4
8
32
128
512
n-508
342
864
1446
4104
^^-ii--'^732; v,= |^-1.7008;»^-^^-2.8465; .,-:^-8.0787.
yl - Fo+ VI =- 17 + .6732 = 17.6732.
/AS -1.7008 -0.6732* -1.2475; a =» VJ^ =1.1169.
,jLt -2.8465 -3 X0.6732 X1.7008 +2 X0.6732' =0.0217.
114 -8.0787 -4 X0.6732 X2.8465 +6X0.6732* X 1.7008 -3 X0.6732«
0.0217« _ooQQg. o_i:4223 ^
ril69« "•"^^' ^* 1.1169»^'^^*-
0.0002 X 5^^1^^^ _ - 0.00047 ; Fm2^ = 0.0009.
4.4223.
4X11.3650
3v8g-2i;i< 3(1.7008)2-2X^.7059^
V4 " ,8.0787
= 1.011.
n
Theoretical maximum frequency, yo= — r— "■
508
a\/2jt 1.1169\/2«
-181.5.
The probable discrepancy, based on the five larger values
of y, is found as follows, the Xi values being taken from a
table like Table IV ;
•
Ratio of
Actual
Probable
Actual to
L
a
Xi
A+axi
Dis-
crepancy.
Dis-
crepancy.
Probable
Dis-
14.5
-0.99606
crepancy.
15.5
-0.96457
-2.11
15.34
+ 0.17
.083
, 2.05
16.5
-0.71654
-1.07
16.51
-0.01
.032
0.31
17.5
-0.11023
—0.138
17.55
-0.05
.025
2.00
18.5
+ 0.53543
0.73
18.51
—0.01
.027
0.37
19.5
+ 0.91439
1.72
19. G2
-0.12
.054
2.22
20.5
+ 90213
THE CLASSES OF FREQUENCY POLYGONS. 27
The extreme values are not calculated for the relations
indicated by the formula do not hold well there where the
frequencies are small and the proportionate values of y are
changing rapidly for small changes of x. For the five values
considered the actual discrepancy is less than the probable
discrepancy in three cases and less than the probable limit
in alL
To find the average difference between the
pt\\ and the (|>+ l)th individual in any seriation
(Galton*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:
^/2np pPe"^ 1 ,,
*p=^ jp •^U+Ci+Ca+C3+ ...},
where 2/„= — t^hc"*^'.
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 | -\ .
c,= -.75 ^7^^'+/' + 1-5^ -
-'^^('-^.)(fJ"
28 STATISTICAL METHODS.
* ' nXn-p)V n\n-p)p ' ym
The solution of the equations for Cj, Cj, and Cj will be facili-
tated by finding, once for all, the logarithms of n, (n— p),
171
(n-2p), (n-p)p, and --.
(2). When n and p are both large and not nearly equal:
iffym
(3). When n is small the unsimplified form of the equa-
tion must be used.
^ n-p p n^ ^ .^^ y n
X— (I+C1 + C2+C3+ • • •)•
|n means the products of all integers from 1 to n. The
series Cj, Cg, C3 is not complete, but the values of c with higher
subscripts are so small that they may be neglected.
Let IpY' be the difference measured in units of a between
the p'th and the p"th individual, then
The foregoing method is that of Pearson (1902^^) based
upon some considerations of Galton (1902).
To fiud 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) y=2/o|«o + «iy+«2(j) if
where I is the half range and
€i=3/i;
€,=3.75(3^2- Ao);
also,
To find niQ arrange the frequencies in the usual manner
(p. 26) and find the logarithm of each; their sum is equal
to rriQ. Making the class situated at the middle of the
range 0, find the deviation of each of the other classes from
this class. The algebraic sum of the product of the loga-
rithms by the deviations gives m^. The second moment
about the same zero point gives Wj. Or,
mo-i-log/^i-y; m,=^I[Y{V-VJ]; m,= IlY{V-V,n
Substituting in (1) we get a numerical quadratic equation
which can be put in the form
»'-»•!-[ (T)"+f;T+(ft)>--(ft)"!
If the normal curve be y=z^ 2o^ i
(3) r= log 2/= log 2o - 2g2 log e;
whence, by comparison of right-hand expressions in equa-
tions (2) and (3),
yoX«a
; Then the required normal curve is
(Pearson, 1902".)
30 STATISTICAL METHODS.
Oi-HER Unimodal Frequency Polygons.
The formulas of Pearson's Types I to VI are as follows:
m
X
Type 11. y=yoyl~j
Type III. 2/=2/o(l+j)^«"^/^
Type IV. y=yocoa6^^e~'^, where tan 6=^^.
Type V. y=yox'~Pe~^^''.
Type VI. y=yo(x-l)^^/aflK
In these formulas:
X, abscissae;
^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 y^;
I, a part of the abscissa-axis XX' expressed in units of the
classes;
e, the base of the Naperian system of logarithms, 2.71828.
The other letters stand for relations that are explained in
the sections below treating of each type separately.
The raiig^e 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 and 1. The ratio of carapace
length to abdominal length in various crustaceans may, how-
ever, conceivably take any value from + oo to 0. In the ratio
of dorsoventral to antero-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 8kewness (a) is found 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. Asynmietry 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 T3rpes I and III skewness is measured also by the
rato a=iVA:-||. where '>=^^^E^- When
5^2—6^1—9 is positive, a has the sign of pt^; if negative,
a has the opposite sign to fi^ (Duncker, *00**).
In Type I. „=iV?. l±|(=iVA-^^^>
III, a=i\/Pi= ^=, where the sign is the
+ 2V fh same as that of /£,.
IV. a=iVK '~^
it t(
It it
It It
V, a=
V
sinc« p— 4 is the positive root of the quadratic:
(p_4).-^(p_4)-g=0.
p is readily found.
LiType VI, a ^ (gi + ?.)/(?!-?. -3)
(gi-^2)\/K^i-l)fe+l)|
where (1— gO and (ga+l) are the two roots of the equation
^'-^^+4-fi^,(s+2)V(8+l)^^-
To compare any observed frequency polyg^on
of Type I with its corresponding^ theoretical
curve.
/. , X \ wi / x\m2
32 STATISTICAL METHODS.
To find l„ ^2, mi, m^, y^.
The total rangre, Z, of the curve (along the abscissa axis]
is found by the equation
2| and Z, are the ranges to the one side and the other of 2/bf
7
mi=y(s— 2); mi+ma=«— 2;
^' i ' (mi+m^)'"^-^'"^' r(mi4-l)r(m2 4-l)*
To solve this equation it will be necessary to determine
the value of each parenthetical quantity following the P
sign and find the corresponding value of P from Table V.
It is, however, sometimes easier to calculate the value of t/^
from the following approximate formula:
Jl / I 1 l\
__n (mi+in2+l)\/mi+mz 12\mi+in2 mi m2/
* V2;rmim2
With these data the theoretical curve of Type I maybe
drawn. Frequency polygons of Type I are often found in
biological measurements.
To compare any observed frequency polygon
of Type II with its corresponding theoretical
curve.
This equation is only a special form of the equation of Type
I in which li=l2 and mi=m2.
As from page 22, /?i=0 in Type II, l=2a\/7+T; since the
curve is S5mmietrical, D=0, and
iZ\/;rr(m-f 1)*
The P values will be found from Table V.
THE CLASSES OF FREQUENCY POLYGOIfS. 33
An approximate formula for i/q is given by Dimcker as fol-
lows:
n «— 1
Vo = — 7= y ^=0 4(« - 2).
a\/27r\/(« +!)(«- 2)
To compare any observed frequeucy polygron
of Type III with its corresponding theoretical
curve*
y=y.(i+|)
p —x/d
e
The range at one side of the mode is infinite; at the other
is found by the formula
l,^ a ^^= a^—^ (for Type III).
' 2\/A " y^ ^
Also, P=j\ = -r:.» Vo
D aa* l^ 'ePPip+l)
The value of F corresponding to p+1 can be got from
Table V, Appendix.
To compare any observed frequency polygron
of Type lY with its corresponding theoretical
curve.
This is the conmionest type of biological skew curves.
^ is a variable, dependent upon x as shown in the equation
x= I tan 6,
The factor (cos^)'"* following y^ indicates that the curve
is not calculated from the mean ordinate (A), or the mode
(A^D)f but that the zero ordinate is at A —mD; or at a dis-
t&nce mXD from the mean.
^=^Vl6(«-l)-A(«-2)'; m=i(«+2);
- ^^i-V^Sl' ^^=i-V^(3-2);
xssB Aj — —t with the opposite sign to ft^;
34 STATISTICAL METHODS.
(arc of circle) =
A^2 1
•7y
180^'
(cob «^)2 1
n,/Te 3.
128
I ^ 2;r (cos^)'+^ *
^— angle whose tangent is — .
To compare any obseryed frequency polyg^on
of Type Y with its corresponding theoretical
curve.
To find p solve the quadratic equation
(p-4)'-^(p-4)-15=0,
and take the positive root.
To compare any observed frequency polygon
of Type YI with its corresponding theoretical
curve.
\—qi and ^2+ 1 ^^e the two roots of the equation
^i=» Vtt^-cW-^ — N» where (1-gi) and «are negative;
Kqi+q^)
i>=
(gi-92)(gi-g2-2)'
* 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
Vo-J.
/*'(sin d)'e
^^dd
the formula for reducing which is to be gained from the integral eai-
cuius.
1 2
8 4 5 6
7
8
9
10
209 365
482 414 277 134
72
22
8
2
THE CLASSES OF FREQUBKCY POLYGONS. 35
Bxample of calcnlatlnff tlie tlieoretlcal curve corre«
■pondlns uritli observed data* (Fig. 6.)
Distribution of frequency of glands in the right fore leg of 2000 female
swine (integral variates):
Number of glands
Frequency 16
Assume the axis yy' ( Vm) to pass through ordinate 4, then:
V V-Vm f /(r— Fm) /{V^Vm)* f{V-^Vm)* fiV—Vm)*
—4 15 _> 60 240
1 —3 209 —627 1881
2 ~2 365 —780 1460
8 _1 482 —482 482
4 414
6 1 277 277 277
6 2 134 268 636
7 8 72 216 648
8 4 22 88 852
9 6 8 40 200
10 6 2 12 72
2 2000 —"998 6148 —8872 48568
r J = — 998-4- 2000 = — .499.
r, = 6148 -4- 2000 = 3.074.
V, = — 3872 -♦■ 2000 = — 1.936.
V4 =: 48568 -H 2000 = 24.284.
pi=0; A -4-. 499 -3.501.
fit = 3.074 — (— .499)* = 2.824999.
M, = - 1.936 - 3(- .499 X 3.074) + 2(- .499)« = 2.417278.
fi4 = 24.284-4(-.499x- 1.936) + 6(.249001 X 3.074) - 3(- 499)« = 24.826297,
_ (2.417278)« _ 6.848282929 _
^* ■* (2.824999)« " 22.545241688 " "•'*^^^^'
24.826897 _ 24.826297 _
'^' "" (2.«24999)« "" 7.98061985 ~ ^•^***»*^«
.259 X (6.111)2 .
^ 4(12.443 - .778)(6.222 - 6.778) - " '^ " ^^^ '"
^ 6(8.11082 - 0.25918 - 1) ,„ „„
• == :55589 = ^^•®®^'^-
960
3840
5648
16929
2920
6840
482
482
277
277
1072
2144
1944
5832
1406
5632
1000
5000
432
2592
a- H ^-^^l^S j^^l^ = .31116.
D- 1.680774 X .3111 = .5230.
D.»- .5230 X 19.9857 = 10.4519.
i- .840887 Vl6 X 20.9857 + 0.25918 X (21.9857)a = 18.0448.
,^, 18.0448 -10.4519 ^3^
36 STATISTICAL METHODS.
^-18.0448-3.7965-14.2483;
^ 3.7965X17.9857 ^y^^.
•^ 18:0448 ^^^^^ •
^ 14.248 3X17.9857 ,.«nnA.
m2 ig-o|4g 14.2006;
2000 (18.9846)^^17.9846 ^- ,»,j,„.0833(. 0556 -.2643 -.0704)
18.0448 V2,rX 3.7840X14.2006
—475.24, the frequency of the modal class.
Position of the mode, i/o— A —2>— 3.501 — .523— 2.978. The close-
ness of fit to the theoretical curve is calculated below by Pearson's
method (page 24).
/ Theoretical (i/) 9 a«
V
-1 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 0.2 - 0.2 .04
12 0.0
9^
.-3.09; P-2.71828-««-^>(l+2f + (5.^V(2^+(^»)_ ,8
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 op Logarithms in CuRVB-FimNQ.
Most of the statistical operations can be greatly facilitated
by the use of logarithms. In curve-fitting their use becomes
\\
,8H)
axi
—
—
1
111
J_
—
I
—
—
—
360
SOO
;j60
100
]
/
—
—
—
\
\
_
—
£0
n
A
r
—
—
—
—
—
\
is.
;!a^
—
• '—; pcdygoD of theoretloftl Ireqaeocy n3\A^>
38 STATISTICAL METHODS.
necessary. The following paradigm will be found of asost-
ance:
GENERAL.
log Vi=log I(y-Vo)-\ogn, A^Vm+v^.
log 1^2= log I(V- Vo)' -log n. log <T= J log fJLy
log 1^3= log liy—VoY -log n. log C= J log /«2-log A.
log v,=log 2X^- V-o)* -log n.
log E. A = 9.828982 + log <t - J log n.
log E.a = logE.^ -0.150515.
log E.c=log E.<, —log A.
log 2= 0.301030 tV = 08333 Find 2 log Vj
log 3= 0.477121 5|^= .02916 3 log v^
log 4= 0.602060 ,f ^= .0125 4 log v^
log 6=0.778151 log J=9. 98970
/!,= iV(log vjj) -iV(2 log V,) -[.0833]. Find: log fi^; 2 log ^;
3 log fJL2.
/£3=iV(log V3) -iVOog 3+log vi + log V2) 4-iVaog 2 + 3 log vj
Find: log ^3; 2 log fi.^.
fi^= NQog v^) - Ni\og 4 + log Vi + log V3)
+ iV(log 6 + 2 log V, + log V.,) - NClog 3 + 4 log y,)
- iSr[9.698970 + log ft^] - ^jj. Find log fi^,
log ^1=2 log )t£3 - 3 log fL^.
\ogP^=\ogfi^-2\ogfi^.
ti;=5/32-6/3i-9 (Types I, IV).
Skewness:
Typeu I : log a = i .og fi^ + log mj - log O?, + 3) - 0.301030.
Type III: log a=} log ^1-0.301030.
THE CLASSES OF FREQUENCY POLYGOKS. 39
Type IV: log a= J log p^ + log OffjH- 3) - log w; - 0.301030.
Type V: log a=log 2+ J log (p— 3) —log p.
Type VI: log a=log (g^-f g2) + ilog (gi-g2-3)-log (gi-gj
-Jlog(gi-l)-ilog(g2+l).
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 j= i log ^i+log (s-2). log A;=log /+i log /£,.
log a=log/-log (8 + 2) -0.301030.
log Z=i log j«2+i log(iV[log (s-l) + 1.204120]
-iV[log ^1 + 2 log(s-2)]} -0.602060.
5 + 2
logD=log a + i log n^) m=— 2-.
log mD = log A; - 0.602060.
log T=log fc+log 5-0.602060 -log I.
log tan ^==log T— log 5.
log (? =8.241877+ log ^°.*
log 2/o= log n + J log 5 + AT j log [iV (2 log cos <j> — log 3»)
-iV(8.920819-log 5)-iSr(log r+log ^)]+9.637784{
- 0.399090 - log Z - (s + 1) log cos 4k
log t/= log Vo + -^ [log (s + 2) + log log cos d]
+ iV[7.8796612 + log 0° * +log x\
Multimodal Curves.
Multimodal curves are given when the frequency in the
different classes exhibits more than one mode. False mul-
timodal cvuves 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.
* In degrees and fractions of a degree; see Table VII*
40
STATISTICAI. METHODS.
Muttimodal currea differ in degree. The modes may be so
close that only a single mode (usually lu an asymmetrical
curve) appears In tlie result; or one of the modes may appear
as a hump on the other; or the two modes may eveu be far
apart and separated by a deep sinus (Figs. 7 to 10).
^
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.
Fio. e.
The Index of diTei^ence of two modes of a multi-
modal curve is the distance between the modes expressed In
THE CLASSES OF FREQUENCY POLTOONS. 41
terms of the Blaadard deviatloii of the jaon variable of Ibe
The Index of Isolation of two maaBea of Tariatea
grouped about adjaceot modes Is the ratio of the deptesdon
between the modes to the height of the shorter mode.
The meaning of multimodal curves Is direrse. Sometimes
L7!
A-
Fia.9.
they indlcatea polymorphic condition of the species, the modes
representing the different type forms. This is the case wilh
il~
=7V
:^=
Fia. 10.
the number of niy flowers of the white dalay which has modes
at 8, 18, 21, S4, etc. Sometimes they indicate a splitting of a
species into two or more varieties.
rergen
e ina
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 ab modality 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 _ _
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.
Heace we must deal with masses and with averages.
II
=;■ 7 f" ^ ' =^ " " " " "
u
1 M 1 II § 1 1 5 1
n
i i ! • 1 S § 3 1 E §
=
s
-
s
= ■"""-
.
«
■
-
3
i i M - =
S U - " 1
.
5
i ;- -ss
S S " " :
.
^
1 1 " S C 3 !
S S " " :
•
'.
: " B S E a !
3 - " : :
-
^
: - S g S S °
"
3
« S 3 S a »- i : : ; ;
-
^
« K 12 3 ^ - : i i : :
•
■f
- - " j ! i i i i ; i
s
s
■s
1
|i S S S 1 1 5 5 5 3 M
IK .
It
I
14 STATISTICAL METHODS.
In studying correlation one (either one) of the characters is
regarded as subject and the other as relative. A correlation
table is then arranged as in the example on page 43, which
gives data for determining the correlation between the num-
ber of 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 Dbtermikikg Coefficient of Correlation.
Galton's g^raphic method* On co-ordinate paper
draw perpendicular axes X and T ; locate a series of points
from the pairs of indices of abmodality of the relative and sub-
ject corresponding to each subject class. The indices of the
subjects are laid off as abscisssB ; 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 eacli assoc.
rel. class X no. of cases in both)
total no. of iudivs. X Staud. Dev. of subject x Stand. Dev.
of relative ;
or, expressed in a formula :
^(dev. X X dev. y X/)
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:
(-3.640X8
- 8.547 X -^ - 2.540 X 5
(-1.540X3
r- 3.540 X 4
- 2.647 X \ - 2.540 X 151
- 1.540 X 58
etc*
COBBELATED VABIABILITY. 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 ihe 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 a/ and 1/ 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, v^ (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 v^ 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, '97*», pp. 12-17.)
The probable error of the determination of r is
„ 0.6745(1 -r^)
\/n
fPearson and Filon, '98, p. 242.)
[Example. Correlation in number of Miillerian glands
on right and left legs of 2000 male swine. (See table on next
page.)
For + quadrants 2'(a/^)= 5243
5125_ J(xy)
46
STATISTICAL METHODS.
«« O lO M b-
^ ^ M »-• cJ
^ a a ^ ^
b. o e^ CO Q «c>i>c
0» C^ O to O CO C4
C^ to t* C4 CO CO
'to
3
eo
H
D
»o
coo
aoa»
CO
>o
CO
D
M
J?
oco
iHlO
O'^.
^
O
lO
SS
t^
00
^
CO
CO
o
SS
00
v-i
o
1
>►
1
-^
CO
w
«H
1
1
1
1
1
t> O ^ ^ Q
S ,-1 CO CO S
2^ CO M
cc(«-i 00
a> 00
»o CO t^
^ U ^
^ '*
M
»H d CO ^ »0 CO
_, N> U) 00 CO C4 iH
- S 15 ^ - -^
aw
Sf-4
.H '•OJ oot^ S20» S<N 8(N
eo «'-' «<2 a^ 2co SCO
fH '"' '"'
fiH ?«o CO wW *Q0 «00 *>0 gCO
' ' W lO ^ r-l
il^ rOO N. »-• ~* «0 "*C0 "'fH
COOOOO COMCOOO'H
W M »o o» »-'
I
CO nm5
lo cop
^ fHW
0)10
I 'I ,*w
^ ' I I I I
* I v^«. g 5-
CO
096
9 81
a>o 8c^ a<N S'H S g 09
Uf ozi
Se 691
§2 818
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§ I 56Z
§ I88I-
s
•-•CO cqO) wCO CO O r-"^ WOO eoiH
« l-08f'-
lO ■»>
CO bc
I
-s
?" '*<
s
Soo
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•«0 "«0 rH
a^ «N
<N 1
»o Y'^
-^roMi-i Or-icico'^ioco
I I I I
I Z-Z19-
^ 8-e2Z-
SB
»H
I
I
s
»HMC0"^»0Ob»00OO (^|
CORRELATED VARIABILITY. 47
r^ fl^^-v,'vA ^= (2.5525-. 4535 X. 4605)
^ L7T95X 1.730 '='^''^^^^'
6745[1 -07919)-]^^^^^
\/2000
The average variability of an array is = (r\/l— r*.
The coefiicieiit of reg^ressiou marks the proportional
change of the relative organ for a unit's change of the sub-
ject organ. It is given by the equation /o=r — , where a^ is
the standard deviation of the subject, o^ that of the relative.
The Quantitative Treatment op 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.gr., in eye-color we may
have black, brown and gray, and blue), and let the frequency
of these classes be n,, n^, rij, in which rii and n^ are each less
than Jn, so that rig contains the median. Let Lj, L^ be the
(unknown) distances of the mean from the two boundaries
of rij. Call Li/a=\ and LJa=^\^ then
rii — n« —
and
ftT phi
n Y kJo
48
STATISTICAL METHODS.
Now the left-hand side in these equations is known; it is Ja
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 ^j and h^ 1 = — ) are found, and hence L^/a
and L^/tJ and the entire range
of the middle class.
in terms of <t, is known. Call the range in absolute units I.
Then 1=^Lq+Li and I /a is known and for a second series Z/</
can be similarly determined. Hence <t/</, the ratio of the
variabilities of the two series, is determined.
Again, since Li/a and — ^ are known, Li/{L^+Lj) is
(T
knojni, and this gives us the ratio in which the mean divides
the true range of the central class. (Pearson and Jjce, 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 h^ + h2= — is known and hence
<j= -^ — r^, the standard deviation, is found. Since L, = h^a —
the distance of the mean from the left-hand boundary of n^
the position of the mean is known.
The probable error of a is
E.,= .67449(^
Li + L3 J n, (n — til) , rigCn — rig)
+
n'H,
where
\/2n
Hi»
and ^3=
1 -iA3»
\/^
COBRELATED VARIABILITY,
49
The values of the last two equations may be obtamed
directly from Table III.
The probable vcrror of Lp or of the mean, is
E.^ = . 67449
{^^-A.}'.
where J^= ( ^--.^ 1 ,1 . = \r, , , and J. = \tt « «
«* \.67749/ ^1 n^/fj' ' ^s n»/^,'
The Correlation op Non-Quantitative Qualities.
•
Pearson (1900°) has ingeniously discovered a method of ex-
pressing correlation quantitatively when the variables cannot
be so expressed, as, for example, in the case of effectiveness
of vaccination. Strictly, this method assumes normal vari-
/ ation in variables, but it can be employed generally, in
default of a better method, with fairly accurate results.
The prime requisite is that the qualities to be compared
shall be separable into two grades, an upper and a lower.
For example, in the case of the result of vaccination: on
the one hand, either presence or absence of a scar; on
the other, either recovery or death. As either of the
second pair may occur with either of the first pair, four
classes, a, 6, c, d, will be formed altogether and a correlation
surface like the following may be made:
—y
—a;
a
b
a + b
c
d
c+d
a+c
b+d
n
The axes y,— t/ and x^—x probably do not coincide with the
axes y and x passing through the *' origin" of the correlation
60 STATISTICAL METHODS.
surface, but may be regarded as situated from those axes at
the respective distances h and A;. These values may be
found from the formulae
n f n Jq *"
n r ;: Jo
a, 6, c, and d being known, h and k are found from Table IV.
Then
H ^e - H* . and X= -^-^^\
of which the values may be looked up in Table III, or, better,
their product may be calculated by logarithms as follows:
log //ii:=9.201820-JVriog^^±^V 9.637784].
Find also log hk^ h^, and A;*. To find r solv« the following
equation to as many terms as may be necessary:
+ ^{h* - Qh^ + 3) (fc* - C/b> + 3)r»
j^ ^hk{h* -10h^ + 15)ik*-lQk^+ 15)r« + etc.
This gives us a numerical equation of the nth degree which
can be solved by ordinary algebraic methods, using Sturm's
functions and Homer'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 ax^ + 6x + c » 0,
CORRELATED VARIABILITY. 61
z— ^T " ft 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 f{x) is obtained by
multiplying each term in f{x) by the exponent of x in that
term and diminishing the exponent of a; by 1). The correc-
f(r)
tion 777-T should be added to the value of r used in substi-
/ W
tuting. Repeat this process as often as the correction affects
the fourth place of decimals, and go to r* if necessary.
The probable error of r as thus determined is
h~~rk
found as follows: First calculate the relations fi^-
\/r^
and /9a = . — =. Also find
dt.^—r^l *6 *^'dy and ib^=—==f^e ^^^dr
from Table TV. Moreover,
(h^+k^-2rhk)
Then,
A744q
Prob. error of r^'^^^[iia+d)ic + h) + (a+c)(d+6)^2'
which can be easily solved by substitution. In using the
foregoing formula, it must be noted that " o is the quadrant
in which the mean falls, so that h and k are both positive."
In other words, a+c>h+d and a+b>c + d. (Pearson, '00°.)
Example* The eye-colors of a certain set of people (see Bio-
metrika, II. 2 pp 237-240) and of their great-grandparents were
found to be distributed as follows:
STATISTICAL UBTHODB.
I
1
s
2
I
3
i
1
1
5
1
i
s
s
S
8
1
^
,
i
1
i
IS
1
g
i
30
15
17
21
448
20
S
3
1
4
1
140
51
a
»■
269
213
17
103
'"
"
1113
It WM dflfflred to de(erm[ne the oorrelation between the eye-oolor
of the offspring and that of their (raat-grand parents. Claariy ths
Tinges of the claaaea given above are not quantitatively equal nor
determinabla Consequently a fourfold table wbb formed by dividini
the population into those having eyefl whose color was gray blue-groen,
or lighter, and those having dark gray, haiel. or darker eyea. Tfaia
gives a good basis fnr calculation If the dark gray and hsiel eyea
had been grouped with the lighter eyes it would have made quadrant
iQ large; and there is nothing in the natun of the dat*
that si
)ngly (a
^ 7aB-388 __
635-478
u
1-3
4-8
Total.
1-3
4-8
460
276
203
725
3SS
Totals.
636
478
U18
"s
*
:\l
! 17637
01
.01275
CORRELATED VARIABILITY. 63
Log A-9.5900512 Los As=» 9.1801024 A^-. 151392
Log A;"9.2497412 Log A;2» 8.4994824 A;S_.03i585
Log A* -8.8397924 A* +A;2 .182977
A* -.069150 HJk -.034576 ^~^*-. 091489
Log (450X203 -275X185) = 4.6071869
Log HK - - log 2 jr - .091489 log e
-9.2018201 -M8.9613689+ 9.63778428]
-9.2018201-0.0397332-9.1620869
Log ^^^-4.607 1869 -(9. 1620869 + 2 log 1113) -9.3521096
.224962-r + .034755r2 + i(A2-l)(A:2-l)r«+siAifc(A2-3)(A;«-3)r*+eto.
Solving .034575r2 +r - .224962 = 0,
1 ±\/l +4( .034575 X .22496Y) oooooc * ,*
*• 2003457^) ^^-.223225 to Ist approx.
A3 - 1 - - .848608 Jfc2 _ 1 _ _ .968415 Coeff . r« - .136967
Coeff. ^_+069150X2.848608X2.96841S_o^3g3
24
.024363H + .136967r3 + .034575r2 +r - .224962 - 0.
Applying Newton's approximation, we reach the result
r-.2217.
E.^ - '^^^(75095 + 303530^2* + 2813OO0i2 + 8095001^^2
n««o -8619502-27425<^i)i
Log wo - log * JT - ^ log( 1 - r2) - Mlog log e + log( A* + *» - 2rAA:)
-Iog(l-r2)-log2]
A2+A;2-2rAA;= 0.152315, l-r2= 0.950850.
Logwo-9.20182-9.989056-iV[9.637784+9.18274-9.978112-0.30103]
-9.1779797
«
, .67449 .^ „ „«„« .«„.«.„
».0^0»/iJ— «.00»/^
to — ».i# / wou=^.uox^c»o.
A-
0.358614
^2=0.093794
Table IV:
^1
^
^2 02
.358
.13983
22.2
.4
.093 .03705
27.3
3.5
0i - .14006 02 = .03736
Log E.r = 4.0812530 + Uog 74426.858
£.,.-0.03289
64 STATISTICAL METHODS.
QmcK Methods of Roughly Determining the Coeffi-
cient OF Correlation.
The method just described may be used m lieu of the rela-
tion r= — ^-^ whenever the distributions of frequencies of
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 C00*») as
ad— he
and this may be used as a rough approximation to the coeffi-
cient of correlation.
But Pearson ('00<') h£is shown that this simple relation ia
not nearly as close to the true r as the following:
. n 1
ra==smy — _ ,
2 Vl + Aja
where
, Aahcd . n'
"2
{(id-hc)\a + d){h + cy
The superiority of the value r^ as an approximation to r^
justifies the additional work its determination demands.
Spurious Correlation in Indices.
When two characters a and h are measured in each indi-
vidual of a series of individuals, and each absolute magnitude
is transformed into an index by dividing it by the magnitude
of a third character c as found in the same individuals, a
spurious correlation will be found to exist between the indices
of — and — (Pearson, '97).
c c
Let C|=the coefficient of variability of a;
C^= '' '' '' '' '' 6;
<< tt it i( (I ^.
3 — *^f
Tq— '* " *' spurious correlation.
''0 =
C,'
VcJTclWc^'+c/
CORRELATED VARIABILITY. 66
The precise method of using Tq in modifying any determi-
nation of r is uncertain. Pearson recommends using r— Tq
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 uuipareutal inlieritaiicey 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, a^, may (as a result of selection or of environ-
mental change) be different from the variability of the char-
acter in the first generation, <Ji, the index should be taken as
r— , called the coefficient of regression.
The probable error of this determiuation is
' -i/ ^—, in which r,, 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— r^ is to 1.
In biparental iuheritaiice, if there is no evidence of
assortative mating, or correlation between the two parents in
the character in question, the mean abmodality of any frater*
66 STATISTICAL METHODS,
nity will be
where /ii= average abmodality of fraternity;
/i2= average abmodality of male parent;
/i3= average abmodality of female paient;
r2= correlation coefficient between fraternity and
female parent;
r^= correlation coefficient between fraternity and male
parent;
^1= standard deviation of fraternity;
<j2= standard deviation of male parent;
a^— standard deviation of female parent.
When assortative mating occurs, as is usually the ca^e, the
abmodality of a fraternity is given by
'"'" l-r,i '7p^ 1-r,^ '7,'^'
where ri= correlation between male and female parents.
The other letters have the same signification as before.
The strength of heredity in assortative mating is measuved
by the formula
To find the coefficient of correlation between
brethren from the means of the arrays.
This is given by the formula
2[in,(n,-l)A,]/n'-A,'
o^
where rii is the number of the brethren in an array [and there-
fore iniin^ — l) is the number of possible pairs of brothers in
that array]; A^ is the mean value of the array; o is the
standard deviation of the character in the brethren taken
all together, n is the total number of variates, and A 2 is the
average of the brethren. This method will be found useful
where to take all possible pairs of brethren would be found
a work of too great magnitude (Pearson, Lee, etc./99, p. 271).
CORBELATBD VARIABILITY, 57
Gallon ('97) has shown that an individual inherits l)ot only
from his parents, but also from his grandparents, great-grand-
parents, and so on. The heritage from his 2 parents together
is, on the average, 50^ or ^ of the whole ; from the 4 grand-
parents 25^ ot\; from the 8 great-grandparents 12.6^ or |;
from the nth ancestral generation ~ of the whole ; the total
heritage adding up 100^. This law has been generalized b/
Pearson (*98) as follows :
- 1 0*0- , 1 0*0, , 1 (To, , 1 Coj .
where hi = average abmodality of fraternity.
0*0 = standard deviation of fraternity.
(Ti, (Ti , , , or, = standard deviation of mid-parent of
1st, 2d . . . «th ancestral generation.
ki = abmodality of mid-parent of 1st ancestral genera«
tion.
kt, kt , , , kg = abmodality of mid-parent of 2d, 8d
• . . «th 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 op Alternative Inheritance. ^
In 1865 Gregor 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.
68 STATISTICAL METHODS.
Of the two antagonistic peculiarities united in the cross,
that which becomes visible in the soma is called by Mendel the
dominating, that which lies latent is called the recessive char-
acter. What determines which character shall be dominating
is still unknown, and the determination of this point offers an
enticing field of inquiry. In some cases the dominating form
is the systematically higher, in others it is the older or ances-
tral form.
The law of dichotomy may now be developed. When a
mongrel (monohybrid) fertilization takes place the zygote con-
tains both the dominant quality (abbreviated d) and the re-
cessive quality (r). In the early cleavages d and r are both
passed over into both the daughter-cells; but apparently, at
the time of segregation of the germ-cells, the somatic cells
are provided with d only, while the germ-cells retain both
qualities. In the ripening of these germ-cells, probably in
the' reduction division, d and r come to reside in distinct cells,
so that we have
of the female cells 50%d+50%r, and
of the male cells 50%d + 50%r.
If now mongrels are crossed haphazard, a male d cell may
unite with either a female d cell or with a female r cell; like-
wise a male r cell may unite with a female d or a female r celL
Consequently in the long run we shall have of all the zygotes
25%d, d+50%d, r+25%r, r,
or 50% of the zygotes hybrid and 50% of pure blood, and of
the latter half exclusively maternal and half paternal. But
since the soma developed from the hybrid germ-cell has the
dominant character, we shall have
75% of the cases with the dominant character;
25% '* '' '' *' '' recessive *'
and this agrees with various empirical results, of which the
following from Correns is instructive. A cross W£is 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.
Gen. 2.
Gen. 3.
31 ]/ (hybrid) peas produced 12 plants;
these bore:
775 y (hybrid +2/) peas ( =75.8%)
21 plants were produced:
7 (33%) pure- 14 (66%) hybrids,
blooded y, because the
because they
blooded
;ause
bord:
292 2/ peas
because they
bore:
I
247 g (pure-blooded)
peas (=24.2%).
20 plants bore:
462 y 149 g 670 ffreen 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) ({i+r)r=={i, 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. 6.25%d, d', 12.5%d, r; 6.25%r, r.
A. 60%d, r
, ' »
B, \2.b%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. +5. 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 polyhybrids may be likewise
predicted, by extending the principles already laid down.
Measure op Dissymmetry in Organisms.
A Dissyinmetry-IiideXy a, measuring the average de-
gree of asynmietry in the right and left organs of bilateral
organisms, has been proposed by Dimcker (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
Od
= ^ a^ + a^ — 2ra^a
ii»
in which the subscriptsi refer to the bilateral series of which
the asymmetry is to be found, and r is the coefficient of cor-
relation between the two sides.
Let d' represent any positive differences in the series, and
d" any negative differences; and let //, //, etc., represent*
the frequencies of the negative-difference classes, and /j",
//', etc., the frequencies of the positive-difference classes.
Then the asymmetry-index
"^ 5^2(^0 + ^(d'O] "
COBBELATED VABIABILITY. 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-U: -10 12 3
/: 1 63 310 23 3
i'(d0 = 310Xl + 23X2+3x3=365, 2'(/0«33a
2'(d") = l, i'(/'0 = li n=400.
„ 336X365-1X1 122639
400X366 146400
0.8377a
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, '00^, '01, '02; Davenport, '00, '00*, 01»>;
Duncker, '99^; Eigenmann, '96; Galton, '01; Gallardo,
'00, '01, '01b; Ludwig, '00, '03; Redeke, '00; Volterre,
'01.
Text-books: Galton, '89; Bateson, '94; Dimcker, '00;
Pearson, '00; Vernon, '03.
Method: Camerano, '00; Engberg, '03; Fechner, '97;
Galton, '89, '02; Heincke, '97; Johannsen, '03; Pear-
son, '94, '95, '96, '97, '97^, '98, '00«, '01^, '02«, '02«, '02ar,
'02"», '02'^, '03®; 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; ShuU, '02; Yule, '02; Johannsen, '03.
Proper value of ratio of first to second prizes: Galton, '02;
Pearson, '02'^,
8'^ATISTICAL BIOLOGICAL STUDY. 63
Coefficient of variability; significance : Pearson, '96; Brew-
ster, '97; Duncker, '00^; Davenport, '00«.
Mutations: Bateson, '94; Howe, '98; deVries, '01-'03;
Weldon, '02«.
Individual vs. specific variation: Brewster, '97, '99; Field,
'98; Mayer, '02; Davenport '03b.
Variability independent of sexual reproduction: Warren,
'99, '02; Pearson and others, 'Ol^, pp. 359-362.
Relative variability of the sexes: — in man, Pearson, '97*';
Brewster, '99; Pearl, '03; in crabs, Schuster, '03.
Relative variability of primitive and modem ra^es: — in man,
primitive races less variable: Pearson, '96, p. 281; Pearson
(and others), 'Ol*', p. 362.
Man.
Stature. — Seriation for adults of different races: Bavari-
ans, Anunon, '99; United States, recruits, Baxter, '75, Pear-
son, '95, p. 385; various, Macdonell, '02; English middle
upper classes, Galton, '^9, Pearson, '96, p. 270; Germans,
Pearson, '96, p. 278; French, Pearson, '96, p. 281; Cam-
bridge University students, Pearson, '99.
Lot. n A C
Engl, upper middle class 4 683 69.215'' ±.066 2.592'' d: .047
do. husbands. 200 69.135"±.126 2.628"±.089 3.66
Cambridge Univ. students 68.863" ±.054 2 522" ±.048
cm. cm.
English fathers 1078 171.95 6.81 3.99
English sons 1078 174.40 6.94 3.98
U.S. recruits 25878 170.94 6.56 3.84
N.S.Wales, criminals.... 2862 169.88 6.58 3.80
Frenchmen 284 166.80 6.47 3.88
English criminals 3000 166.46 6.45 3.88
French, Lyons 166.26±.53 6.50±.37
Germans 390 156.93 6.68 4.02
in. in.
Engl, upper middle class 9 652 64.043 ±.061 2.325 ±043
(<o. wive.s 200 63.869±.110 2.303±.078
Cambridge Un. students? 63.883 ±.130 2.361 ±.092 3.69
French, Lyons 9 154.02 cm. ± .52 5.45 ± .37
Seriation at different ages: British infant at birth, Pearson,
'99; school children, Bowditch, '91; St. Louis schoolgirfe,
Porter, '94, Pearson, '95, p. 386; Australian adult whites,
Powys, '01.
64
STATISTICAL METHODS.
Lot. Average.
New-bom infant, British S. 20 . 503 ± . 028 in.
9. 20.124±.025 "
««
1.332±.020
1.117 ±.018
C
6.500
5.840
Ob. JJUJIXIB B
Australian
A>lX\J%JHm IB-
whites:
' XJ
.O .^1 X MUH,
«. 1 f VI
t
Age.
Years
Average.
S 9
i
a
9
i
C
9
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
3.76
(
1.144±
.017
15
.66%
1.006±
.015
14
.23
16.547 ±
.25
10
.83
14.030±.
.57
11,
.17
Weight. — Seriations at different ages, British: Infants,
Pearson, '99; University students, Pearson, '99; 5552 Eng-
lishmen, Sheppard, *98.
Lot. • Average.
New-bom infants, i 7 .301 ± .024 lb.
9 7.073±.021
Cambridge Univ. students, S 152 .783 ± . 35
9 125. 605 ±.77
Sktdl. — Cephalic index: Bavarians, Ranke, '83; 6800 20-
year old Badeners, working cla,ss, Ammon, '99, p. 85; various
races, Pearson, '96, p. 280, Macdonell, '02.
Lot. n
Bavarian peasants 100
Baden recruits 6748
Modem Parisians
French peasants 56
Cambridge students 1000
Criminals (British) 100
Brahmans of Bengal 100
Whitechapel English 107
Maquada race
A
a
C
83.41
3.58
4.29
81.15
3.63
4.48
79.82
3.79
4.74
79.79
3.84
4.81
78.33
2.90
3.70
76.86
3.65.
4.75
75.77
3.37
4.44
74.73
3.31
4.43
72.94
2.98
3.95
Skull capacity:
Lee, '02.
Lot.
Andamanese
Ainos
Negroes
liow-caste Punjabs . .
Parisian French
17th Century English.
coefficients of variability. Fawcett and
S 9 Lot. S 9
5.04 5.59 Naquadas 7.72 6.92
6.89 6.82 Germans 7.74 8.19
7.07 6.90 Egyptian mummies. . 8.13 8.29
7.24 8.99 Polynesians 8.20 5.65
7.36 7.10 Italians 8.34 8.99
7.37 6.68 Modem Egyptians. . . 8.59 7.17
7.68 8.15 Etmscans 9.58 8.54
J
STATISTICAL BIOLOGICAL STUDY. 65
Various cranial dimensions, Lee and Pearson, '01.
Other Organs. — Coefl&cient of variability of bones of skele-
ton of French and Naquada (C. of limb-tones, 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 niunber of Miillerian glands in Sus scrofa, n, 2000;
A, 3.501 ±.025; a, 1.680±.018; C, 48.0, Davenport and Bul-
lard, '96.
Avcs.
Seriations of various proportions of N. A. birds, Allen, '71 ;
characters of Lanius (** shrike") and its races, Strong, '01;
Lot. n A o C
Shrike, length L. wins J 168 99.06 mm. 2.74 mm. 2.81
*• ? 112 97.98 2.64 2.69
tail lengths 141 101.57 3.48 3.43
^ ? 95 99 55 3.63 3.65
bill length, i 164 12.01 0.71 5 89
•• •• 9 112 11.71 0.63 5.35
" depth, i 126 9.27 0.42 4.57
9 85 8.95 0.41 4.61
*• melanism of crown, i 144 83.57% 3.0% 3 58
5 99 83.66 3.19 3.81
" upper tail-coverts i 142 53.13 15.42 29 02
•* , 9 104 47.98 18.99 39.58
Curvature of culmen 29.94** 2.74** 9.15
Eggs, proportions: Passer domesticus, Bumpus, '97, Pear-
son, '02*; various species, Latter, '02.
Av.
Length, length, mm. Breadth, mm.
Species. Bird, n A a C A a C
in.
Cuckoo 14 243 22.40 1.059 4.72 16.54 .650 3.93
Blackbird 10 114 29.44 1.357 4.61 21.73 .787 3 62
Song-thrush 9 151 27.44 0.999 3.64 20.69 .516 2 50
Starling 8-8.5 27 29.78 1.097 3.68 21.76 .423 1.94
Yellowhammer. . 7 32 21.55 0.682 3.17 16.04 .405 2 53
Tree-pipit 6.5 27 20.01 0.698 3.49 15.09 .449 2 97
Meadow-pipet . . 6 74 19.72 1.250 6.37 14.56 .561 3.84
H ouae-sparro w
(English) 6 687 21.82 1.195 5.47 15.51 .525 3.38
House-sparrow
(American)... 6 868 21.32 1.05 4.92 15.34
Hedge-sparrow.. 6 26 20.12 0.810 4.02 14.73 .415 2 81
Robin 6 57 20.22 0.8.57 4.24 15.43 .477 309
Linnet 5.5-6 65 17.14 0.598 3.49 13.33 .358 2^69
66 STATISTICAL METHODS.
Amphibia.
Sedations 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,
B3mie, '02; dimensions of 141 Petromyzon; Lonnberg, '93.
Tracheata.
Leindoptera. — Seriations of wing dimensions of Thyreus
abbotti. Field, '98; nimiber of *' eye-spots" on wing of Epi-
nephele, Bachmetjew, '03; number of spots on different
species of the genus PapiUo, Mayer, '02; breadth of wing,
98 i Strenia clathrata C=4.57, Warren, '02.
Aphides. — Asexually produced offspring show an average
variability of 60% that of the race, Warren, '02, p. 144;
seriation of fertility, empirical mode =7 young, Warren, '02,
p. 133; reduced variability of the earlier generations, because
they include only such as can produce fertile offspring, War-
ren, '02.
Dimension. Grandmothers. Children.
o C o C
Frontal breadth 2.28mm. 6.07% 2.96mm. 8.26
Length R. antenna 7.36 8.77 10.94 12.97
Ratio : ^°^^, T^Tl X 10 . . 1 . 23% 5.67 1.84 7.82
Frontal breadth
Myriapoda. — Lithobius: seriations of length of adults,
C, for 3 's= 10.97; «'s= 11.25; number of prostemal teeth;
of antennal joints; of coxal pores in which C varies from 9.9
to 15.4, Williams, '03.
Crustacea.
Podophthalmata. — Seriations of 12 dimensions of right-
handed and left-handed * 'fiddler-crabs," Gelasimus pugilator,
C varies from 7.0 to 11.1, Yerkes, '01; relative variability of
male and female Eupagurus prideauxi from deep and from
shallow water, Schuster, '03; forehead breadths of Carcinus
STATISTICAL BIOLOGICAL STUDY. 67
moenas, Weldon, '93, Pearson, '94; various dimensions, Cran-
gon, Weldon, '90; length of rostrum, Palaemon serratus,
Thompson, '94, Pearson, '94; number of rostral teeth of
Palaemonetes, Weldon, '92^, Pearson, '95, Duncker, '00.
Lot. A, mm. a, mm. C, %
Eupagurus, short edge of R. chela:
i deep water 9.708±.085 2.76 28.5
i shallow water 10.272±.075 2.69 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:
i deep water 17. 97 ±.14 4.73 27.8
i shallow water 18. 68+. 13 4.38 23.5
« deep water 14.14±.06 1.67 11.9
9 shallow water 13. 97 ±.05 1.82 13.0
Eupagurus, carapace length:
i deep water 8.59±.05 1.67 19.4
i shallow water 7.54±.03 0.94 12.5
9 deep water 7.12±.03 0.86 12.1
Pabemonetes 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 . 48 28 . 26
•« tt
Amphipoda. — Seriations of lengths of body, of second
antennae, and of ratio of second antennae to body-length,
Smallwood, '03.
Annelida.
ChoBtopoda. — Teeth on jaws of Nereis virens. Right: A —
10.055 ±.045, (7=1. 339 ±.032, C=13.3%; Left: A = 10.00 ±
.044, a=1.306±.031, C=13.1%, Hefferan, '00.
Bracliiopoda.
Seriation of width -t- 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.
i4 = 13.782±.031, <j= 1.318 ±.022, C=9.57±.16, Davenport,
'00«.
Molliisca,
Gastropoda. — Frequency polygons of ventricosity, weight,
and index of Littorina littorea for 3 British and 10 American
localities — ^greater variability in America. Index: ap=2.3%,
68 STATISTICAL METHODS.
<7^=2.7%, CB=2,e%, C^ = 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-
diimi. Baker, '03; Pecten; ray-frequency, Lutz, '00, Daven-
port, '00, '03, '03**; change in proportions with age, acquisi-
tion of new synametry about transverse axis; definition of
form units from different locaUties, Davenport, '03, '03*>.
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
Cutchoicue, 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 20^12±.030 0.991 ±.021 4.83±.10
Pecten ventricosus:
San Diego, Cal., R. valve .... 19.495 ± .087 0.885 ± .019 4.55 ± .10
Hcliiii Oder iiiata.
Seriation of ray-frequency in starfish, Crossaster papposus:
i4 = 12.391, C= 0.788, i;=6.36%, Ludwig, '93^
Coelenterata.
ScyphomediiscB. — Seriation of number of tentaculocysts of
Aurelia aurita: n=3000, empirical range 4-15; empirical
mode=8, genital sacs, Af'=4, range, 2-10, Browne, '95, '01.
Hydromedusas, — Seriation of number of radial canals,
gonads, gastric lobes, and tentacles of Gonionemus, Hargitt,
'01 ; radial canals and lips of Pseudoclytia pentata, Mayer, '01,
Davenport, '02; radial canals, etc., of Eucope, Agassiz and
Woodworth, '96.
o
• C
19.15
8.36%
9.16
13.44
4.03
5.73±.12
10.27±.22
2. 17 ±.05
13.66±.30
STATISTICAL BIOLOGICAL STUDY. 6.9
Lot. A a C
Paeudocyltia, num. radial canals 5 . 004 ± . 094 . 441 8.81
lips 4.868±.0l2 0.556 11.4
Protista.
Paramecium recently divided, Simpson, *02; seriation of
diameter of Actinospherimn and number of cysts and nuclei
in body, Smith, '03; outer and inner diameters of shell of
502 Arcella vulgaris, Pearl and Dunbar, '03; various diatoms,
Schrdter and Vogler, '01.
Lot. A
Paramecium, length n 229.05
breadth 68.13
index 29.91
Arcella, outer diameter 55 . 79 ± . 17
** inner diameter .... . 15.91±.07
Plants.
General. — ^Multimodal polygons especially frequent in
plants, Ludwig, '97; crlticaJ, Lee, '02; Pearscn, '02*'.
Ray-flowers in CoMPOsiT-fi. — Seriation of ray-frequency
of Coreopsis, de Vries, '94; of Senecio nemorensis, S. fuchsii,
Centurea cyanus, C. jacea, Solidago virga aurea, Achilla mille-
foUum, Ludwig, '96; ray-frequency in Chrysanthemum,
Ludwig, '97°, Lucas, '98, Tower, '02, Pearson and Yule, '02;
Helianthus, Wilcox, '02; Bellis perennis, Ludwig, '98*>; Soli-
dago serotina, Ludwig, '00*»;' Arnica montana, Ludwig, '01;
Aster, ShuU, '02.
Nimi. Ray-flowers. A
Aster shortii 14.000± .068
A. novsB-angli£B 42.874 ± . 302
A. punicens 36.672± .107
A. prenanthoides 28.080± . 107
Other Seriations op Floral Organs: Ranunctilacece. —
Petals, Ranunculus bulbosus, de Vries, '94, Pearson, '95;
calyx, coralla, stamens, and pistils of Ficaria vema, Ludwig,
'01; number of Ficaria pistils, early flowers, A = 17.448, <j=
3.89; late flowers, A = 12.147, <t=3.88; number of stamens,
early, A = 26.731, <7= 3.761 and late, -4 = 17.863, <7= 3.298,
e
C
1.526±
.048
10.90
6.308i:
.213
14.71
4.480i:.
.076
12.22
4.070±
.077
14.52
.»:;
itttti
70 STATISTICAL METHODS.
MacLeod, '99, Weldon, '01; number of petals of Caltha
palustris, de Vries, '94; number of cal}^ parts and petals
of Trollius europseus and number of fruits per head of Ranun-
culus acris, Ludwig, '93*>, '00*>; 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.
Pa'paveracece. — 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.
CaryophyllacecB. — Nimiber 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, Reindhl,
'03.
Sapidaceas. — Number of compartments in fruit of Acer
pseudoplatanus, de Vries, '94.
Leguminosoe. — Niunber of blossoms in clover plants, Type I:
cT= 2.788, de Vries, '94, Pearson, '95, p. 402; number of ele-
vated flowers in blossoms of Trifoliiun repens perumbellatiun,
de Vries, '94; floral organs of Lotus uliginosus, L. comicu-
latus, Medicago saliva, M. falcata, Ludwig, '97; flowers per
head of Lathyrus, Ludwig, '00^.
RosaceoB. — Number of stamens of Prunus spinofa and Cra-
taegus, Ludwig, '01; sepals of 1000 Potentilla tormentilla
and petals of 4097 Potentilla anserina, de Vries, '94.
ComacecB. — Number of flowers in head of Comus mas and
C. sanguinea, not in Fibonacci series, Vogler, '03.
CaprifoliacecB. — 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.
DipsacoB. — Number of flowers per head in Knautia arven-
sis, maximum at 64, Vogler, '03.
Compositce. — Number of male and female flowers in lunbel
of Homogyne, Ludwig, '01.
Primvlaceoe. — Number of flowers per umbel. Primula,
multimodal, Ludwig, '97, '98**, '00; rays in Primula farinosa,
Vogler, '01,
STATISTICAL BIOLOGICAL STUDY. 71
ScrophuIariaceoB, — Number of parts in peloria of Lenaria
spuria, Yost, '99; number of stamens, Digitalis, Gallardo, '00.
OrchidacecB. — Extremes in variability of niunber of spots
on flower, Chodat, '01.
Leaves. — Seriation of niunbers of paired leaflets of Pirus
aucuparia, Fraxinus excelsior, Senecio nemorencis, and Pole-
monium, Ludwig, '97, '98*^. 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 sicjfe 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, '98; pine needles. Ludwig, '01;
from various branches of Pinus silvestris, Lee, '02.
Lot. length of pine needles A mm. a mm. C
Pinus silv. , lower branches . . 22 . 163 ± . 048 4 . 474 ± . 034 20 . 19
•* middle branches. 26. 524 ±.055 5. 167 ±.039 19.48
" upper branches . 25. 949 ±.062 5. 858 ±.044 22.69
Fruit. — Number of ears in head of Agropyrum repens and
Brachypodium, Ludwig, '01; of the grass Lolium, Ludwig,
'00^; fruits per head of Ranunculus acris Ludwig, '00**; niun-
ber of seeds per capsule-compartment, Helleboriis, 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, '00**.
SOME TYPES OF BIOLOGICAL DISTRIBUTIONS.
General. — Pearson, '95 '01<^. a modified by selection,
Reindhl, '03.
Type I.
Petals of 222 flowers of Ranunculus bulbosus, de Vries, '94,
Pearson, '95, p. 401.
Number of glands of fore legs of swine, Davenport and
Bullard, '96, Pearson, '96, p. 291: a=, 311 ±.016.
72 STATISTICAL METHODS.
Fertility (percentage of births with one year of marriage)
of wives at different ages, Powys, '01.
Rays in dorsal fin of Pleuronectes i , Duncker, '00.
*' ''anal '' '' '* ?, " "
Type IV.
Stature of St. Louis schoolgirls, Pearson, '95, p. 386.
a= -0.489.
Number of teeth, Palaemonetes varians Plymouth, Pear-
son, '95, p. 404. a= 0.134.
Stature of Australian whites, Powys, '01.
Rays in dorsal fin of Pleuronectes, « , Duncker, '00.
'' '' anal '' '* '' i '* "
" '' pectoral " " " « " "
Type V.
Nimiber of lips of medusa, P. pentata, Mayer, '01, Pearson^
'Old. a=.549.
Normal.
Stature, U. S. recruits, Baxter, 75, Pearson, 95, p. 385.
Ray frequency, Pectens, Davenport, '00, '03^.
Skcwness.
General. — Mathematical Analysis. — Pearson, '95, '01*, '02',
'028r, »02«». Biological /nterpre^a/icm.— Davenport, '01»>, 'Olc.
QuarUitaiive Results,
Numerous cranial characters, Naquada race, Fawoett, '02.
Nearly always +.
Num. lips of medusa, P. pentata (Mayer, '01 ; Pearson, '01**) + .649
Num. MliUerian glands, legs of swine (Pearson and Filon, '98). . . + .311
Num. dorsal teeth, Palasmonetes varians (Pearson, '95). ...*.... + . 130
Num. rays, Pecten of)ercularis, Irish Sea (Davenport, 'OS"*) + .087
Eddy stone (Davenport, '03'») .... + .080
" hooks on statoblasts, Pectinatella (Davenport, *00') + .077
Weldon's crab measurements, "No. 4 " (Pearson, '95) + .077
Num. rays lower valve, Pecten irradians, L. I (Davenport, '00* )+ .023
•• " •• ** P. opercularis, F. of Forth + .007
" " upper valve, P. irradians (Davenport, '00«) ± .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
2l-rayed Chrysanthemum (de Vries, '99) .'— . 13
13- •• •* " *• •* + .12
Selected 12- (and 13-) rayed Chrysanthemum (de Vries, '99) +1.9
Raysof Pectenirradians, fossil, Va oldest (Davenport, '01^) .... — .22
• •• •• youngest... -.16
recent. N C - .09
" •• •• •• recent, LI +.023
Length of wings of long-winged chinch-bug (Davenport, '01^). . . — .43
•* •• •* short-winged chinch- bug " ** ...+ .44
Length horns rhinoceros-beetle, long-homed (Davenport, '01^). . — .03
•• short-homed " " .. + .48
Complex Distributions.
Bimodal Polygons. — Discontinuity in hairiness of Biscu-
tella, Saunders, '97; of Lychnis, Bateson and Saunders, '02,
Weldon, 'Q2\
Length of cephalic horns of rhinoceros-beetle, anc^ forceps
length of male earwigs, Bateson, '94; explanation of di-
morphism, Giard, '94.
Multimodal Polygons. — Mode^ fall in Fibonacci series, Lud-
wig, '96, '96*', '96°, '97, '97^, '97«.
Modes of Chrysanthemum segetum at 13, 21, de Vries,'95.
Opposed to Fibonacci series, complex polygon due to lack
of homogeneity, Lucas, '98, ShuU, '02, Pearson, '02»», Lee, '02,
Reinohl, '03, Vogler, '03.
CORRELATION.
General and Metliod. — Galton, '88, '89, Pearson,
'96, Yule, '97, '97*»; spurious correlation, Pearson,'97; non-
quantitative characters, Pearson, 'OO**, Pearson and Lee, '00,
Yule, '00, '00*», '02; index not constant in related races,
Weldon, '92, Pearson, '96, '98^ p. 175, '02»» p. 2, Daven-
port, '03^
Man.
General. — Galton, '88; British criminals, various dimen-
sions, r=.13 to .84, Macdonell, '02.
SkvU. — Correlated with cranial capacity in living persons,
Lee and Pearson, '01; breadth -and length, Naquada, Bavari-
ans, French, Pearson, '96, p. 280; N. A. Indians, Boas, '99;
74 STATISTICAL METHODS.
various dimensions, Aino and German, Lee and Pearson, '01 ;
Naquadas, Fawcett and Lee, '02. With civilization woman's
correlation tends to gain on man's, Lee and Pearson, '01,
Pearson, '02»».
Lot. r
Breadth and Length:
German, « 49± .05
Smith Sound Eskimo 47
Aino, i 43±.06
Aino, 9 37db .07
German, i 29±.06
Modem Bavarian peasants 28db .06
Naquada race 27
Sioux Indians 24
Modem French peasants 13 ± . 09
British Columbian Indians i 08
Modem French (Parisians) 05db .06
Shuswap Indians 04
Lot. rS r?
Aino:
Capacity and length 89± .01 .66db .05
'' '** breadth 56±.05 .50±.07
*' '* height 54±.05 .52±.07
Length and height 50± .05 .35± .07
Breadth and height 35± .06, .18db .08
Cap. and ceph. index - .31 ± .07 — . 25db .09
German:
Capacity and breadth 67 ± . 04 . 70 db . 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. — Rollet, '89; stature correlated with length of
long bones, reconstruction of stature of extinct races, Pear-
son, '98*>; various coefficients of correlation, Pearson, '99, '00,
p. 402; in hand-bones, Whiteley and Pearson, '99, Lewena
and Whiteley, '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 80(9) to .81(i)
Stature and humerus 77( « ) to .81( « )
Stature and tibia 78(0 to .80(«)
Humerus and ulna 75 to .86
Humerus and radius 74 to . 84
Radius and stature .67 («) to 70(«)
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-bom infant S 644d: .012
? 622±.013
Weight and stature of Cambridge (Engl.) students, S . . . .486± .016
" " ? 721 ±.026
Breadth of head (reduced to* 12th yr.) and intelligence,
youth 084± .024
Length of head (reduced to 12th yr.) and intelligence,
youth 044± .024
Cephalic index and intelligence, youth 005 ± .024
Breadth of head and ability, adults 045± .032
Cephalic index and ability, University men 031 ± .035
•• length of head. University men -.086 ±.033
Vdccination and Recovery. — Pearson, 'OO*'; Macdonell, *02,
'03. r= .23 to .91.
Assortative Mating. — Pearson, '96, '99**, '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.
liower Animals.
Antimerically symmetrical orgaxs:
Paired organs. — Number of Miillerian glands on R. and L.
fore legs of swine, Daiiyiport 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, Heflferan, '00;
breadth of R. and L. valves of Pecten, Davenport, '03^;
skeletal spicules on R. and L. half of Echinus larva.
Subject and Relative. r
Leng^th R. and L. sides of carapace, Gielasimus 947 ± .003
*' " *' *• meropodite, first walking leg 948 ± .005
Breadth R. and L. valve of Pecten opercularis, Irish Sea. . . .858 ± .006
Num. of teeth R. and L jaws of Nereis .820± .008
** ** fin-rays R. and L. pectoral, Acerina 710
** coxal pores R. and L. 14th pair legs, Lithobius • .69 ± .02
13th pair legs, Lithobius 686±.029
12th pair legs, Lithobius 58 ±.04
anal pair legs, Lithobius 575 ±.039
Other antimeric organs:
r
Num of dorsal and ventral spines, Palsemonetes vulgaris
(Duncker, '00b) 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 dorsaUand anal fin -rays in flounder, $ 651
•* *' " '* " " '* *' 2 . .690
Length antero-poaterior 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. chelse of Galasimus 754 ± . 014
" carpopodite, R. and L. chelse of Gelasimus 698 ± .017
" propodite, R. and L. chelse of Gelasimus 473 ± .026
Num. rays R. and L. ijectoral fin, flounder, Pleuronectes, i . .594
•• tt it it •« •• << •• •• Q 582
" of dorsal fin-rays at which lateral line ends, R. and L.
Pleuronectes, J 467
Num. rays R. and L. ventral fin, Pleuronectes. i 243
t<
««
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 lie
" 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 rostnun, 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 chelse
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 Actinosphaerium Echomi
and the number of cysts and of nuclei. Smith, '03; inner and
outer diameters and color of the shell of Arcella, Pea^l and
Dunbar, '03.
Organs. r
Carapace length and chela length, Eupagurus, i .9389::t 0036
" ? 8626±.0080
Diameter of body of Actinospherian and num. of nuclei .854 ± . 017
Inner and outer diameter shell of Arcella 836 ± .007
Diam. of body of Actinosphaerium and num. of cysts. . .769 ± .026
Wing leng^th and tail length, Lanius 569
Diam. of cell and body length, Daphnia, hatching to
3d molt 551
Diam.. of cell aqd body length, Daphnia, 3.1 to 4th
molt 393
Diam. of cell and body length, Daphnia, after 4th molt.. . 248
Num. coxal pores, R. anal and L. 12th seg., Lithobius. . . .427 ± .046
Frontal breadth and antennal length (Warren, '02) 320 ± .032
Ccxal pores, R. 14tb leg and body length, Lithobius.. . .308 ± .059
Num. rays dorsal fin and end-point of L. lateral line,
Pleuronectes, i 208
Outer diameter and color Arcella 012
Num. dorsal spines and L. pectoral rays, Pleuronectes. .004
t:>-t
78 STATISTICAL METHODS.
Organs. r
Body length and number antennal joints — .013± .087
Circumference of statoblast and number spines.
Pectinatella — .092±.006
Num. R. definite teeth and L. indefinite. Nereis — .524 ± .023.
Carappce leng^th 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 Astor, Shull, '02; various organs
on Lesser Celandine, Pearison and others, '03.
Organs. ' r
Num. rayB and bracts. Aster 856 to .799
" stamens and pistils Ficaria ranunculoides, early. . . . 507 ± .031
•• late 749± .016
' rays and disc florets, Aster 574 to .353
•• petals and sepals Ficaria verna + . 34 to — . 18
* ' stamens and pistils, Celandine 43 to .75
•• petals. Celandine 38 to .22
*' pistils and petals, Celandine 35 to .19
•• sepals, Celandine 25 to .03
" stamens and sepals, Celandine 06 to .02
Other parts. — Size of leaves of same rosette of Bellis peren-
nis, Verschaffelt, '99; various pairs of dimensions of fruits
and leaves, Harshberger, '01; parts of Syndesmon, Keller-
man, '01.
HEREDITY.
Geueral.
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.
Ineqtuility in parental transmission. — Father prepotent m
sons; mother in daughters, Pearson and Lee, '00, p. 115;
heredity weakened by change of sex, Pearson and Lee, '00,
p. 115, Lutz, '03.
i
STATISTICAL BIOLOGICAL STUDY.
79
Inheritance of Eye-color, Homo.
No. of Changes of Sex.
«, son; d, daughter; /.father; m, mother.
1
2
3
( Averaoe of r.^ and f ^^
.530
.459
.300
.296
.145
Parental .r^..,*'„ar"
I sm ■■**" 'df
Grand- j .. ..^ ^ ^^"^ ,
.370
parental j 8fm*'ajp amp'gmm''
^ tmf* 'afm
Great-grand-parental inheritance, average . . .
.347
.222
.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, '00«, pp. 38, 47.
X y
Longevity:
Father and son (Beeton and Pearson, '99)
'* " adult son (Beeton and Pearson, '01)
•• adult dau.
Mother and adult son "
«. jj^^ .. .. .. ..
Eye-color (Pearson and Lee, '00) 55 to .44
Stature, English middle class:
Father and son (Pearson, '96, p. 270)
" dau. *
Mother ana fion " " '*
" dau. " :** "
Head index. N. Amer. Indian:
Mother and son (Pearson, '00, p. 458) '.
•• dau. •• •• ••
Coat-color, thoroughbred horses:
Sire, foal (Pearson, '00, p. 458)
Dam, foal **
Fertility:
Mother and daughter, British upper class
Father and son, " " *'
Mother and daughter, British peerage
Father and son, ** "
Mother and daughter, landed gentry
Father and son " "
Frontal breadth, Hyaloptenis (Warren, '02)
Length R. antenna, Hyaloptenis ** **
Ratio: R. antenna + frontal breadth (Warren, '02) . . .
Ratio: Length protopodite-»- length body, Daphniii
(Warren, '02)
*.,!*
Cor.
r
.12
Reg.
.135
.10
.130
.08
.131
.12
.149
.12
JO .44
.396
.352
.360
.419
.302
.269
.284
.275
.370
.300
.517
.527
.042±
.010
.051 ±
.009
.210
.066
.105
.116
r
P
.335
.359
.427
.507
.439
.539
.466
.619
80 STATISTICAL METHODS.
GrandparentaL
r p
Coat color, thoroughbred race-horses 339
•* Basset hounds 113
Frontal breadth, Hyalopterus, Aphidse (Warren, '02) 321 .269
Length, R. antenna, Aphidse (Warren, '02) 177 . 192
Ratio. R. antenna + frontal breadth, Aphidse (Warren, '02) .231 .295
Ratio Length protopodite + length body. Daphnia (War-
ren, '02) [.27 .5]
Stature .
Gr'dson and gr'df.. homo male line (Pearson, '96) I99
• female line (Pearson, '96), 089
Grtgr'dson and grtgr'df.. homo i line " " \ 105
•• ..I. ». Q «. «t 4* ftai
Eye-color, homo, f., grandfather, and son (Blanchard, '03) .421
Coat
horse, **
»»
«» t. '
it
• ,i '
.324
Eye
homo, **
'* dau.
•«
«*
.380
Coat
horse, **
•« it
««
«*
.360
Eye
homo, m.«
*' son
««
»*
.372
Coat
horse, '*
*• »»
»•
*•
.359
Eye
homo, **
•* dau.
«•
•»
.297
Coat
horse, **
•« «*
««
*«
.311
Eye
homo, f., grandmother, and son
••
••
.272
Coat
horse, **
it II
»•
••
.309
Eye
homo, **
" dau.
«•
M
.221
Coat
horse, **
tt tt
«•
• «
.204
E.ye
homo, m.,
" son
••
• »
.262
Coat
horse, "
*t *(
»«
• «
.261
Eye
homo, *•
** dau.
«•
••
.318
Coat
horse, *'
•• •»
••
••
.239
Fraternal.
Daphnia, length of spine
(Warren,
'99; Pearson
.*01«).
• • • .
r
693
Aphis, antennal length (Warren, '02) 679
frontal breadth (Warren, 02) 666
Parameciiun, index of just separated fission pairs (Simpson, '02). .664
Horse, coat-color (Pearson, Lee, and Moored, average of 3 sets. . .633
Man, forearm, English (Pearson, '01*=) 542
Hound, coat-color, Bassett (Pearson and Lee, '00) 526
Man, eye-color, English (Pearson, '01*=). 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, '01") 317
** longevity, British peerage (Pearson, '01) 260
Quakers " " " 197
Average of 23 sets 476
Mean of 42 fraternal correlations (Pearson, '02*') 496
Some mental characteristics, inherited exactly like physical
characters (Pearson, '01*):
Conscientiousness 593 Popularity 504
Self-consciousne8s 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)^
Brothers 4000 .4 to 1 .0
Half-brothers 2000 .2 to 0.5
Uncle and nephew . 1500 . 250
First cousins 0750
First cousins once removed 0344
Second cousins 0172
Third cousins 0041
Homotyposis.
Correlation in non-sexual reproduction, as in production of
homologous undifferentiated physiologically independent
parts, Pearson, '01^; criticism, Bateson, '01; reply, Pearson,
'02^; rejoinder, Bateson, '03; correlation between differen-
tiated homologous organs, Pearson, '02*.
% Var. to f^^^^
Velt. of P°rro-
Race. ^**^°^-
Ceteract, Somersetshire Lobes on fronds 78 .631
Hartstongue, Somersetshire Sori on fronds 78 .630
Shirley poppy, Chelsea Stigmatic bands 79 .615
English onion. Hampden Veins in tunics 79 .611
HoUy Dorsetshire Prickles on leaves ... 80 .699
Spanish chestnut, mixed Veins in leaves . .. . i . 81 .591
Beech, Buckinghamshire Veins in leaves 82 .570
Papaver rhceas, Hampden Stigmatic bands 83 .562
Mushroom, Hampden Gill indices 84 .549
Papaver rhncas, Quantocks Stigmatic bands 85 .533
Shirley poppy. Hampden Stigmatic bands 85 .624
Spanish chestnut, Buckinghamshire . Veins in leaves 89 .466
Broom. Yorkshire Seeds in pods 91 .416
Ash. Monmouthshire Leaflets on leaves. ... 91 .405
Papaver rhoeas. Lower Chiltems Stigmatic bands 92 .400
Ash, Dorsetshire Leaflets on leaves. , . . 92 .396
Ash Buckinghamshire Leaflets on leaves. ... 93 .374
Holly. Somersetshire Prickles on leaves. ... 93 .355
Wild ivy, mixed localities Leaf indices 96 .273
Nigella hispanica, Slough Seg of seed -capsules. 98 . 190
Malva rotundi folia, Hampden Seg. of seed-vessels. . . 98 .183
Woodruff, Buckinghamshire Members of whorls . . 98 .173
Lot. Character. Var, of
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 ctUM of homotypons (Ftan»n, '02*) .499
82 STATISTICAL METHODS.
Mendelism.
General Statement.— Mendel, '66, de Vries, '00, '00^, '00^,
'03, Correns, '00, Davenport, '01, Bateson, '02, Castle, '03;
critical, Weldon, '02, '03, Pearson, '03*'.
Plants.— Correns, '00, '00t>, '01, '02-'02c, '03-'03S de Vries,
'02, '01-'03, Bateson and Saunders, '02.
Animals. — Echinoids, Doncaster, '03; poultry, Bateson
and Saunders, '02; mice, Darbishire, '02, '03, '03t», Castle, '03b,
Bateson, '03*»; 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 unsjnnmetrical as 3 to 4, Mayer, '01.
SELECTION.
General. — Intensity of selection connotes a lessening cf
correlation, Pearson, '02^, p. 23; mediocre individuals not
the fittest to survive, Pearson, '02**, 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 bom brood
(r= —.188 ±.084); in Carcinus moenas, Weldon, '95, '99;
in Clausilia, Weldon, '01.
Plants. — Transformation of skew frequency curve to a 83rm-
metrical one by selection, de Vries, '94, '98; shifting of the
mode by selection, de Vries, '99.
Sexual. — Pearson, '96: A a
Stature of husbands, inches 69 . 136 ± . 126 2 . 628 ± . 089
•• males in general 69.215±.066 2.592d:.047
"wives 63.869±.110 2. 303d:. 078
" adult females in general . . 64 . 043 ± . 061 2 . 325 ^ . 043
See also Correlation: Assortative mating (p. 75).
DISSYMMETRY.
Ihe following values for 3 have been determined by
Duncker, '00 and '03:
STATISTICAL BIOLOGICAL STUDY. 83
Pleuronectes flesus L., 1060 R.-eyed and 60 L.^yed: 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-cr£.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 chelffi, all 1 .00 1 .00
Num.of rays on R. and L. pectoral fins, Acerina —0.111
glands on wrists of swine ".0053
•« ti
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. . ^1 = 37 . 56 33 . 93
Length of R. antenna. ... A = 83 . 91 76 . 59
Ratio 1^' A== 22.46 22.57
R. 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 HoU. Waquoit. Bristol, R. I.
Dorsal fin-iays. .. il = 66.1 65.2 64.9
Anal ** ... A = 49.7 48.6 48.7
_Ui-V •«...-* .-
84 STATISTICAL METHODS,
Number of fin-rays of Pleuronectes flesus from Western
Baltic, M'=39, southern North Sea 41i, 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.
InvertebrcUn. — 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, '03*>.
Plants, — Lesser celandine, Pearson and others, '03.
USEFUL TABLES.
Probability Integral. — Area and ordinate of normal curve
in terms of abscissa, Sheppard, '9S, '03; abscissa of normal
curve in terms of ordinate, Sheppard, '9S; abscissa and ordi-
nate in terms of difference of area, Sheppard, '03; abscissa
of normal curve in t^rms of class index, Sheppard, '98.
Probability of fitted curve being the true one:
_ Elderton, '02.
Values of log ] Xv—^~^^^ \ ^^^ various values of x^-
^ ^ "^ ^ Elderton, '02.
Table of log —, ^r^- jr . Elderton, '02.
° n(n — 2)(n — 4) ...
Table ofy—j e~^^^dx, for different values of Xj Elder-
ton, '02.
Table of log,o (1+x)— a;log,oe for various values of x, for
use with curves of Type III.
Tables for calculating probable error, Sheppard, '98.
Table of values of 1— r^ and \/l— r^ for all values of r
from to 1 proceeding by hundredths. Yule, '97.
Probable errors of r for all values of n. Yule, '97.
BIBLIOGRAPHY^ 85
BIBLIOGRAPHY.
Note. — An effort has been made to include all recent
works containing usable quantitative data in botany and
zoology; but the literature on the mathematical treatment
of statistics and that affording data in anthropology are
by no means completely listed.
ABBREVIATIONS.
The following names of journals often referred to have
been much abbreviated:
Amer. Nat. = American Naturalist.
Ber. d. deutsch. bot. Ges. = Berichte der deutschen botanischen
Gesellschaft.
Biom. = Biometrika.
Bot. Centralbl. = Botanisches Centralblatt.
Phil. Trans. = Philosophical Transactions of the Royal
Society of London.
Proc. Roy. Soc.= Proceedings of the Royal Society of Lon-
don.
The references are scattered through fifty-seven periodi-
cals.
Adams, C. C. *00. Variation in lo. Proc. Amer. Assoc, for
the Adv. of Sci., XLIX, 18 pp., 27 plates.
Agassiz, a., and W. McM. Woodworth, '96. Some varia-
tions in the Genus Eucope. Bull. Mus. Comp. Zool.,
XXX, 123-150. Plates I-IX. Nov.
Allen, J. A., '71. On the Mammals and Winter Birds of
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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 ordi nates 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; a=: 1.117; y^=lSlA.
(See page 26.)
Entries in Table
V — M corresponding to
V
V-M
a
V-M
Vo
V
/
14
-3.673
3.29
o
.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; <7 = 1.1169. (See page 26.)
Deviation
£i
(i-
~ia)XlO(
]!lass.
z
Per
Class
from
1
\f<X>
a
cent.
Limits.
A='Xi
a
less^^j^.^
14
-3.29
.2
14.5
-3.173
-2.841
.225
15
-2.39
1.6
15.5
-2.173
-1.945
2.364
16
-1.50
12.4
16.5
-1.173
-1.050
12.097
17
- .60
30.3
17.5
-0.173
-0.155
29.155
18
.29
32.3
18.5
0.827
0.740
33.194
19
1.19
18.9
19.5
1.827
1.636
17.873
20
2.08
3.9
20.5
2.827
2.531
4.524
21
2.98
0.4'
.568
100.0 100.000
106 STATISTICAL METHODS.
In the example, the data of which are given on p. 26, the
frequ^icy between the limits is given in % column. The — of
o
each limit (as an inner class limit) is found and the entries
in Table IV corresponding to the limits are taken. Each
such entry is subtracted from 0.50000, is multiplied by
100, and from the product is subtracted the total theoreticsJ
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.
y. Table of log T fuuctioiis of p. This table
will enable one to solve the equations for yo 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 f unc«
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(4.378). The value cannot be found
directly because the value of p is larger than the numbers la
the table (1 to 2). The solution is made by aid of the equation
r(p + l)=prlp),ihus:
logr(1.273) = 9.955185
log 1.278 =0.104828
log r(2.273) = 0.060018
log 2.273 =0.356599
log r(3.273) = 0.416612
log 3.273 =0.514946
log r(4.273) = 0.981558
or, more briefly, log r(1.278) = 9.955185
log 1.273 = .104828
log 2.273 = .356599
log 3.273 = .514946
log r(4.273) = 0.931558 = log 8.543
EXPLANATION OF TABLES. 107
VI. Table of reduction ft*oin the common to
the metric system. This is given first for whole iDches
from 1 to 99 excepting even tens, wliich may be got from the
first line of figures by shifting the decimal point one place
to the right. The table may be used for hundredths of an
inch by shifting the decimal point two places to the left.
Other fractions than decimals are given in the lower tables.
Vn. Table of minutes and seconds of arc in
decimals of a deforce. This table will be foimd of use
in the fitting of ciures of Type IV (p. 33).
VIII. First to sixth powers of integers ft'oin 1
to 30. 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
6745(1— r^)
being — ^= — -, a table for the varying values of n and r
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 1054. The use of
this table can be extended by using the principle that if any
number be multiplied by n, its square is multipUed by n*. its
cube by n', and its reciprocal by — .
XI. liOg^arithms 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].
Tlie 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.
ia the number. It is always one less thaa 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
verm. 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 characteristio
is negative. We may avoid the use of a negative characteristic
by arbitrarily adding ID, which may be neglected at the close
of the calculation. By this rule we have
Number, Logarithm,
1.384 0.141136
.1384 9.141136
.01384 8.141136
.001384 7.141136
etc. etc.
No confusion need arise from this method in finding a number
from its logarithm; for although the logarithm 6.141136 repre-
sents either the number 1,384,000, or the decimal .0001384, yet
these are so diverse in their values that we can never be uncer-
tain in a given problem which to adopt.
Table XI, contains the mantissas of logarithms, car-
ried to six places of decimals, for numbers between 1 and 9d99,
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 colimui
contains the average difference of consecutive logarithms ou
EXPLANATION OF TABLES. 109
the same line, but for a given case the difference needs to be
verified by lactual 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
fig^ures :— 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.
^(jmpfe.— Find the log of 23.4275.
For 2342 mantissa is 369587
" diff. 185col. 7 129.5
" ** ** " 5 9.2
Ana. For 23.4275 log is 1.369726
The decimals of the corrections are added together to deter-
mine the nearest value of the sixth figure of the mantissa.
To find the number corresponding^ to a given
logarithm. — If the given mantissa is not in the table find the
one next less, and take out the four figures corresponding to it;
divide the difference between the two mantissas by the tabu-
lar difference in that part of the table, and annex the figures of
the quotient to the four figures already taken out. Finally,
place the decimal point according to the rule for characteristics,
prefixing or annexing ciphers if necessary. The division re*
quired is facilitated by the table of proportional parts, •which
furnishes by inspection the figures of the quotient.
Example, — ^Find the nimiber of which the logarithm is
8.263927 8.263927
First 4 figures 1836 from 263873
Diff.
54.0
Tabular diff. = 236 .-. 5th fig. = 2
47.2
6.80
6th fig. = 8
7.08
An$. No. = .0183623 or 183,623,000.
. ..../, .
110 STATISTICAL METHODS.
Tlie number derived from a six-place logarithm is not
reliable beyond the sixth figure. i
At the end of Table XI is a small table of logarithms of
nmnbers 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 siue, 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
nvimber 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.
iSmwpfe.— Find the log sin of 9° 28' 20'.
Log sin of %" 28' is 9.216097
Add correction 20 X 12.62 252
Ann. 9.216849
J&fe.— Find the log cot of 9* 28' 20*.
Log cotan of 9" 28' is 10.777948
Subtract correction 20 X 12.97 259
-4n«.lo 777689
EXPLANATION OF TABLES. Ill
To find the angle or arc eorrespondingr to a
g^ven lograrithmic sine, tang^ent, 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.
JShBample,^9.d8S7Sl is the log tan of what angle?
Next le88 9.383682 gives 13' 36'
Diff. 49.00 -*- 9.20 = 05'.3
Ana, 13" 36' 05".8
Example. — ^9.249348 is the log cos of what angle?
Next ffreater 683 gives 79' 46'
Diff. 235 -*- 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) bccau.se
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^r^^ 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 the fourth
are the last three figures of a logarithm which is the difference
between the log sin and the logarithm of the number of sec-
onds m 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° griven
to seconds* — Reduce the given arc to seconds, and take the
logarithm of the number of seconds from the table of loga-
rithms, and (idd to this the logarithm from the fourth column
opposite the same number of seconds. The Bum 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 tlM
given number of seconds.
Mcample.— Find the log sin of 1" 39' 14'.4.
1** 80' 14'.4 = 5954'.4 log 3.774838
log 3.
(2-0 4.
add ((7 - 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 (g — Q from the fifth column.
Jfcampfo.— Find the log tan of 0* 52' 35*.
52' 35" = (3120" + 35*) = 3155" log i?.498999
log 8.
(?-0 4.
add (o - 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-\- Q
the logarithm corresponding to the given angle, interpolating
for the last figure if necessary, and from this mbtract the loga-
rithm of the niunber of seconds in the given angle.
.Ecampfo.— Find the log cotan of 1** 44' 22'. 5.
q + I 15.314293
6240" + 22". 5 = 6262.5 log 3.796748
Am. 11.517544
These two pages may be used in the same way when the
given angle lies between 88" and 92°, or between 178° and 180°;
but if the number of degrees be found at the bottom of the page,
the title of each column will be found there also; and if the
number of degrees be found on the right hand side of the page,
the number of minutes must be found in the right hand col-
unm, and since here the minutes increase upward; the number
of seconds on the same line In the first column must be dtmin-
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.
Mcampte.^Fiad the log cos of 88° 41' 12'. 5
(2-0 4.685537
4740" - 12'.5 = 4727.6 log 8.674631
Ans. 8.860168
EXPLAKATION OF TABLES. 113
Bxcmple.— Find the log tan of 90" 30' 50'.
q + 1 15.314413
1800" + 50' = 1850" log 3.267172
Ans, 12;04724i
To find the arc corresponding: to a given log
sin, coSy tan, or cotan wliich falls vrithin 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 tho
given function is found at the top of that column read the
degrees from the top of the page; if at the hottom read from
the bottom.
Find the value ot{q — l) or (q + 0, as the case may require,
corresponding to the given log (interpolating for the last figure
if necessary). Then if g = given log and I = log of number of
seconds, ti, in the required arc, we have at once l=q — (q —J)
or ^ = (g'+O — g, 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 tho
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 tho number adopted and the num-
ber 71.
-S«aw8^^6.— 11.734268 is the log cot of what arc?
<r + 1 15.314376
q 11 .734268
.-. «= 3802.8 3.580108
For V adopt 3780. givmg 03'
J)ifference 22'. 8
Ans, r 03' 22\8 or 178" 56' 37".2.
Mkample. — 8.201795 is the log cos of what arc?
q - I 4.685556
q a201795
.•. 71=1 3282\8 3.51623d
For 89° adopt 3300. giving 05'
Difference 17".2
Ans. 8r 05' 17".2 or 90** 54' 42\8.
114 STATISTICAL METHODS,
THE GREEK ALPHABET.
A a Alpha
/ t
Iota
Pp
Rho
B /J Beta
Kk
Kappa
2orS
Sigma
r y Gamma
A X
Tiamba
T r
Tau
J (5 Delta
Mm
Mu
Tv
UpsiloQ
E € Epsilon
N V
Nu
^ 4>
Phi
Z C Zeta
s$
Xi
Xx
Chi
Hr; Eta
Oo
Omicron
Wtl)
Psi
e ^ Theta
Iln
Pi
£1 QO
Omeira
.;^>
EXPLANATION OF TABLES.
115
INDEX TO THE PRINCIPAL LETTERS USED IN THE
FORMULAE OF THIS BOOK.
At average, mean,
a, class index (p. 24); also upper
left-hand quadrant (p. 40).
a, skewness index.
b, the frequency of the upper
right quadrant (p. 49).
fi, ratio of moments.
C, coefficient of variability.
c, the frequency of the lower left
quadrant (p. 49).
D, distance from mean to mode.
dt a difference; differential; the
frequency of lower right quad-
rant (p. 49).
Jt index of closeness of fit.
d, difference between y and /.
E, probable error.
e, base of Naperian logarithms,
» 2.7 18282.
Ft critical function.
/, class frequency.
O, geometric mean.
Ht a function of h.
h, a fixed value of x; also, index of
heredity.
/, interval between the p'th and
p"th individual.
i, interval between the pth and
(p + l)th individual (p. 27).
Kt a function of k.
kt a fixed value of x.
L, limiting value of class.
It range of curve along x.
lit ht portions of the curve range.
A f number of classes.
A, class range.
Af, abscissal value of the mode
(theoretical).
M't abscissal value of the mode
(empirical).
fit moment about A.
Nt 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.
B , index of dissynunetry.
P, probability-
p, ordinal rank of a particular
individual or case (p. 27); a
root or power.
Kt circumference in units of diame-
ter, 3.14159.
Qt a root or power.
r, coefficient of correlation.
p, coefficient of regression.
a, a relation of 0*b (p. 22).
It summation sign.
o, standard deviation; index of
variability.
T, transmuting factor, o into E,
.67449.
T, in Type IV.
' > angles.
Vt magnitude of any class.
Vo, magnitude of central class.
V, any variate or value.
ti>-6j92-6;?i-9 (p. 31).
Xt the horizontal axis or base of
polygon.
X, a varying abscissal value.
Xi, X2t etc., definite values of x.
X
"•7-
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.
.-..1
>.
116 STATISTICAL METHODS.
I. FORMULAS.
ji^^iZJ) :^Vo+vi. -E. -±0.6745—^ «-F-il
c-\/^^=v^FTr«=v;:;. ^0-0.6746 -''_..
\/2n
C = j-X100%.
''-'■''''M:^'iWy-
A, D. = — — =0.7979o. Jg: , ^ -0.6745O.
Vl-= =A— Ko. »*2 = .
n n
VB** . ^4= •
n n
2.111 Kx^.f) . J 1 J
/i3 — V8 - 3viV2 + 2vj' —
71
_iXx^/) j i-(ar«./) ■ 7 I
n < 2n "^240 »•
Zfl! ft =Z± * 6(/?2-igi-l)
;/2*' ^^ M2^' 3/?, -2^8+6*
i?l(-92 + 3)2
2) = o.A.
4(4^2-3/9,)(2^2-3;?,-6)*
a = i V^ * ^1 (Types I. IV). a ^^^ilEl (Type V).
Probable discrepancy, ^-^f^ j | . ^l-^^f^ - (l +|^ I *
Kdev.xXdey.yXf) I{x^rd) „ 0.6745(1 -r»)
fi, 01,02 fio\02 \/n
■ Ca^
ro (spurious correlation) = — . .
\/Ci2+C32\/C'22+C8*
A ( uniparental ) = r—^ ; Ai ( biparen t al ) = rz^ A2 + rj — Aa.
02 0} 03
E ^_ 674501 A^/ r^gg
* 02 n '
To solve any equation of the second degree,
ax^ + bx + c='0; x=-
2a
CERTAIN CONSTAKTS AKD THEIR LOGARITHMS. 117
II.— CERTAIN CONSTANTS AND THEIR LOGARITHMS.
Title.
Ratio of circumference to diameter
Reciprocal of same
Square root of same
Reciprocal of square root of same
Square root of 27t
Reciprocal of same
Reciprocal of 2n
Square root of 2
Reciprocal of same
Square root of —
It
Base of hyp>erbolic logarithms
Reciprocal of square root of same .
Modulus of common system of logs = log e
Reciprocal of same — hyp. log 10
Factor to reduce a to probable error.
Com. logx==mXhyp. logx, or
Com. log (com. log x)
= 9.6377843+ com. log (hyp. logar)
Hyp. log x = com. log xX— , or
m
Com. log(hyp. logx)
=com. log (com. log) x +0.3622157
Circumference of circle
Area of circle
Area of sector (length of arc = Z)
Area of sector (angle of arc=a°)
Symbol
It
V7
1
1
\/2n
J_
27r
vT
1
VT
m
J_
m
2ffr
7rr2
Nimiber.
3.1416927
0.3183099
*
1.7724538
0.5641896
2.506628
0.3989422
0.159155
1.4142136
0.707105
0.797816
2.7182818
0.606530
0.4342945
2.3025851
0.67449
Log.
0.4971499
9.602€5C1
0.2485749
9.7514251
0.399090
9.6009101
9.201820
0.150515
9.8494849
9.9019401
0.4342945
9.7828528
9.6377843
0.3622157
9.828976
Eccentricity of an ellipse, « = \/^i-A^, where o = semi.maj
major axis; b =
118 STATISTICAL METHODS.
TABLE III.— TABLE OF ORDINATES (i) OF NORMAL CUEVK
OH VALUES OF -^ CORRESPONDING TO VALUES OF —,
)1 99.196 W
»l 97Slfl 9;
ro 77721
15 72033
tS 06097
H607«
48875 46
42056 4S
91SSBSI
S7353 S(
77167 7f
71448 7(
65494 64
7 59440 .«
17 53400 6!
1247511 4(
r9.41S4S4]
'1 354S9
W 30550
57623 5-
45783 4;
40202 3(
34950 34
1 12740 i:
>B 78817
10 73193
16 6729S
IS 61259
lOGS20g
IB 40360
t9 38056
17 32054
iS 28702 28251
15 23078
1 2014S
4 17081|167B2
4 14083 If-- "
..'11496 11259
)5 00200 OBOgo
,7 03232(13
■--2474 02.
1876 01;
19 01408 01;
■3 0377S Of
<6 02908 0^
<0 02218 0!
a 01874 01
{9 01252 01
19 0Ul.'i3 0(
\7 020S8 02040
17 01681 0153B
5 01179 0114-
TALUKS OP NOEMAL PEOBABIMTY IHTEORAL, 119
ING TO VALUES OF -i; OR THE FRACTION OF THE AREA
OF THE CURVE BETWEEN THE LIMITS AND +~. OR
K amantd Id b> lOO.M
-
—
1 1
J^^_
-
—
-
—
tSt
UOOOo' 40
SO
4
J_L.
9| 199
6
7
27!
^
9
^
039 438
47
en
9 591
75!
07B8 838
87
on
9
7' 997
1037
is;
03
lie 123;
6 396
47f
131f
a
04
<5 795
21
>3 1U3
27;
00
239 2431
247
Z5li
25
1 259
2631
2671
2711
2751
07
2700 2831
2910
9 29SI
3020
Ofl)
310)
326
7 338;
4 784
406
4102
41
1 181
4201
4301
441'
44M
449^
45
i 4S7I
466;
460i
4736
4855
5131
£2B
5409
560
5646
5685
5 76.
dS04
5841
6922
15
800
604
ensc
9 159
6198
623(
63171
639
643
3 SS3
6592
8710,
' 6941
6985
7103
714
71S
757
761
7652
79
776
78091 7848
7887,
20
702
796S
S356
804;
sa
t 12.
8t6
81991 823*
8278
! 890'
oog
34
94&
9.W
B6
* 967;
975'
2fi
987
0000
004;
100.
5 0064
10141
0180
0295
1 0441
10520
0665
« 083.
10911
094C
2S
103
9 ISr
133E
29
140
1447
1148
1524
115
2 1600
I67(
i7i;
17K
1820 1867
190(
119
3 1081
206(
20W
2131
22101 Z24S
228(
t 236!
3S
2689 2S27
266;
280!
33
293
2968' 3005
3269
34
330
3344 3&
342C
134
T 349i
353S
1357(
360*
3646
13683
13720 3758
379S
138
a 3870
^
3083
14020
1 .
f™,
™
^
AI,P.
™_
4.0
2
8,0
—
-
B.O
20.0
24.0
~28.0
32
36,0
3.0
T.8
7
'■?
19,5
23,4
27,3
2
37
3,7
7.4
"
'
4.8
18.5
22. 2
26,9
29
*
33:3
STATISTICAL METHODS-
TABLE JV.—ConHntied.
T/«
o'
.Uh
.1.
7
s
jL
.mL
424^1 -H
a -17
''«
4 t'
fi 7 (.«
fiO
"^
47fi
3-
4
36
46
24
47
MIS2
81
S4 4
50
5
3S
2l»e0 2U 2a
20 60
54
'*«
S0436 204
208SC
S'i
2 9"
i 98
SB
60
1 J
64
65
68
,-;■■■'-■-, '■'■;■'
30
29
PhOJ OHTlllSAL PAUtS.
J
1
!
3
^
B
_i_
g
11. 1
4.S
.fl
22.2
33.3
34
is
■!
9
i
i
1
Q
S
1
1
1
8
1
3
2E
:f
S
22
2
8
1
24
2B
s e
5 H
_^
7
^
s
17
4
ao
3
^
2€
:i
VALUES OF NORMAL PROBABILITY INTEGRAL. 181
TABLE IV.— ConKniierf.
. 3BI(lV^I32|2ei6n ''QIS9 3S21 {zaZ^epB"?) 20303 29J32 20JbO
541 S63 SS6
SI" 5J1 S51 57U|
414 464 471
STATISTICAL MBTHODa,
TABLE IV.—CoMinjud
./. 1
1
2 1 3 1 4
5
6
^!^
9
J
3S6S6
~70;
781
810' 838
857
971
991
008: 02:
.23
39065
0*
lOi
121
131
453; 471
4SE
662
581
508
.26
eS4 052
670
688
70.
72;
74;
76.
77t
18
.27
BIS 831
MS
866
884
.SB
; 008
025
060
095
112
m
.29
40147
16fi IB:
IBE
21f
23;
251
268
28:
.30
320
43)
17
64(
641
692
808
:33
825
84!
857
906
022
038
055
071
1.34
087
004
020
165
ISI
20s
16
:afl
3oa
324
340
355
38:
40;
It
434
.37
466
SIS
52;
543
558
74
500
.38
60t
Ti;
2S
74.
;«i
024
off
^t
S
981
90)
8fl4
90<
15
013
028
04;
OS
42073
131
161
17f
24 1 263
30 40T
47)
507
521
53 540
80;
619
83'
-45
647
6B1
67 ; 688
70:
711
730
75!
77
.46
022
03S
94 ; B62
97;
981
~002
018
~M0
043
1,48
43056
060
096
1«
122
13E
149
!6:
17
1.49
228
2S.
26;
28C
13
ii
460
49i
574
587
39i
eiH
624
849
86!
87.
i.sa
609
72!
736
748
761
773
78f
70;
810
1.64
822
870
943
090
002
014
026
038
I M
074
osi
097
108
120
144
158
1:57
20^
214
22f
23;
1
26(
271
1.5B
406
419
430
442
453
404
486
408
Pro™
"^
L Pai
'^
1
i
1
2
3
4
B
6
7
8
?
'■7
1
s
*
i
1
'■5
lo^a
11^9
1.
.3
I's
\
2
1
1
*
11
96
11.2
■
i'3
I
I
I
I
;
'2
7^8
1
■7
4
3
-8
i'-o
7.2
b:4
.8
1.1
J_
2
'■*
3
^
__
IL
6.6
7.7
!
.9
VALUES OP NORMAL PROBABILITY INTEGRAL. 133
TABLE IV — CotUtnuaJ.
«/»
0_
1
2
3
4
,
6
7
S 1 9
TeiT
M52(
~6s;
Bfi;
67;
681
597
m
en
1.61
70j
1.02
k
1^
B7(
s
i;
g07
908
91(
021
03(
04:
45053
063
073
09.1
3-
[S4
164
174
19'
20-
2s;
26;
27'
3o;
31}
:
34s
1.68
1.6
44!
481
54;
662i 671
fiSl
601
i
631
665 064
6s:
701
711
7461 75S
1,7
845
85'
1.7
_™
OIB
BZ-
933
942
05U
059
068
7
086
oo;
028
03:
06
06!
071
.76
113
.77
24(
25:
261
271
1
28;
29!
3U
311
31)
32j
33;
343
37!
38
391
391
.SO
23
47;
.SI
500
501
.62
57{
77
SSI
592
6oi
60:
645
6S2
66.;
67:
es:
69;
7a
711
26
77;
.85
784
98
sm
SSI
834
,86
70
88'
801
80(
0U6
939
94i
953
960
967
97
081
088
028
1,89
a7J
OS'.
081
005
10!
108
i.ao
141
148
15.
161
167
174
1B(
87
20b
21s
21'
225
231
238
244
51
257
2T(
28)
301
3o;
13
34
7!
I.Oi
381
38;
iV
**!
S?
4 i
47C
485
48i
l^M
SO
sot
s'li
52:
521
641
641
65;
55S
Sfii
aoi
1.B8
l.M
6Si
602
70::
70(
1(
735
75;
757
702
7flS
778
781
791
70
sit
Sit
M20
at
2.(12
831
S(i'.
87:
77
2,03
9(12
907
92;
S32
837
042
B47
...
057
J*!
967
972
Phoh.
.„,«
[T^
STB.
1
J
~V
2
3
4
fi
6 1 7
s
g
"l
^-i
f;
i'?
2:4
1
a'.o
6.0
i-i
i|
11
STATISTICAL METHODS.
TABLE IV.-
,'.
JL
_^
2
jj^_
5
_^
^_
_^
_9_
_-_
3.IU
479S3
eei
990
OH
01!
02f
02!
3.IA
48U30
036
039
OTT
OSi
08T
001
m
UK
if^
:
lis
13*
1»
133
137
142
146
19
Wi
178
182
18;
191
31E
221
227
236
240
244
2(7
201
?''-
■27t,
2H,i
287
■i
300
3<H
:!3<i
3
3T8
I
i
i
i'm 463
4i;
4
i
496
«
EOC
S37
570
671
577
500
602
606
iii
P
SS
62
58
66
ea'i'
634
609
638
-3
68
08
n
29
89
30
00
-5
77
3
06
800
27
8
58
90
96
20
809
Of
3
25
31
28
TO
94.
9fie
06
004
9fiB
998
77
980
01
004
°33
2
U3S
00
1)64
08
B
10
2
USB
08
.
1
00
»
sa
3
12
20
W
08
S
3
^24
286
OS
aw
29
M
305
34
54
K
t6
58
02
sa
■»[ 358
2<
-
P
BIB
•
e
VALUES OF NORMAL PROBABILITY INTEGRAL.
TABLE IV— CminiMrf
„
2 1 a
.
. ] 6 7 8
3
P
JBB 3
9B7
1
1
1
i
1
!?
1
99
99b M6l
?
8
W
r.o oraoK r ■ .
1
,
1
s
3
.
I 1 e
'
8
Q
3
O.B
oil
81
1:2
Q.i
(J.S
3.0
o'.a
f1
0:7
1-6
;;i
126 STATISTICAL METHODS.
v.— TABLE OF LOG r FUNCTIONS OF p (mow
p
.
a
8
.
.
•
'
S
•
JIM
0T.1I)
0.W0
9261
BOOS
s;sfl
S.-.09
BJ03
8017
7773
;ais
63«B
8026
ilos
^iTo
1S83
luijii
li!05
0981
l.N
ossa
OBI I
00«»
<i888
9fl47
94S7
§■.■08
8-989
a:7a
§56*
1.05
8.988.118
8639
8119
1.06
s-'fa
6876
6lfiB
4963
47S8
4.563
3943
3741
saa
0403
K09
0212
OlhM
0;:;8
9W1
lias
8365
soes
saw
B710
sws
8157
7068
BBIfl
I'.U
NlOO
*««
1.18
J78a
4871
4101
a.»!
S764
m:
3931
K7«6
10,51
1790
0331*
1.15
B.sBsnoi
S7iT
9694
914a
D290
9139
8988
RSsa
8668
BSSfl
ma
809U
rm
TUftB
7513
7369
7U25
7082
6374
sooa
H08,
60[fc;
4387
lilB
4-WS
8944
ssia
aesa
35B7
MM
3308
3175
8W8
]f,
IB9S
Hra
UW
a^
Ta?S
m?
M«l
M67
OKI
WM
0067
S95a
SS43
973'J
9621
9511
liaa
B-Jfli
siai
noTB
8968
RW1
S;65
8S4B
8430
»ai
7650
i.ai
TMl
Te£3
71 m
T0»
6930
BHM
8738
6842
6547
64SS
1.2S
S359
BiM7
mi
B808
ftHOT
B716
6637
S360
4842
4757
4MS
4':a»
4097
I'.W
a;*)
BMl
354T
3470
saw
3213
3168
3094
1.30
awr
S730
flSSS
2448
!J379
1907
i:»a
leis
16R1
1450
1397
1211
1.S8
0977
flBIH
0H61
0747
flow
0179
0584
Ml«
OMi
4)309
1.3a
fl. BIOSSI
Booa
nsM
980S
9767
9710
9063
9S17
9571
B52S
l.M
B^a^
9391
9348
9ai9
9118
9180
9095
901.-)
M7S
6S^
8«M
BSTB
8.171
8503
lisg
B8U
8811
a»o
tBM
Siil
aioi
fil6»
EL35
8107
S080
BD»
8000
V9TS
7877
T8M
7831
i!ii
7e83
i.«
T608
fsoo
W73
raw
7340
7S24
7600
7491
746i(
TJS8
74-M
^^^
7378
7368
73,58
7S4B
«!■»
SMl
!3l4
7*78
IS73
l.(S
reas
7804
-469
7S55
7M1
7548
7S1II
7214
7£43
7911
1.4S
■^JM
7S4S
7*48
7261
*Wia
72- »
7308
Tsm
7317
7336
7303
7396
.
i.it
7«I7
7419
:444
7457
7471
748S
7499
7610
i
TABLE OP LOG r FUNCTIONS.
127
v.— TABLE OF LOG r FUNCTIONS OF p (see pages 32-34).
p
1
2
8
•
4
6
6
7
8
9
1.50
9.947545
7661
7577
7694
7612
7629
7647
7666
76S5
7704
1.51
7724
7744
7764
7785
7806
7828
7850
7873
7896
7919
1.53
7943
7967
7991
8016
8011
8067
8093
8120
8146
8174
1.63
8201
8229
8>.')8
8287
a316
.a346
a376
8406
8487
8468
1.54
8500
8532
8564
8597
8630
8664
8698
8732
8767
8808
1.55
8837
8873
8910
8946
8988
9021
9a59
9097
9185
9174
1.56
9-nj
9254
9294
9334
9:^75
9417
9458
9500
9543
9586
1.57
9Vn
9672
9716
9761
9806
9851
9896
9942
9989
5085
1.58
9.96U082
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
1880
1487
1494
1.562
1610
1.61
1668
1727
1786
1846
1905
1965
2025
2086
2147
2209
1.62
2271
28-W
2396
2459
2522
2586
26.50
2715
2780
2845
1.68
2911
2977
8043
3110
3177
3244
a312
3880
3449
&517
1.64
8687
3656
3726
3797
8807
3938
4010
4081
4164
4226
1.65
4299
4372
4446
4619
4594
4668
4743
4819
4894
4970
1.66
»V47
5124
5201
5278
5356
5434
6513
5692
.5671
676U
1.67
5880
5911
5991
6072
6154
6235
6317
6400
6482
6666
1.68
6649
6733
6817
6901
6986
7072
7157
7243
7329
7416
1.69
7503
7590
7678
7766
7854
7943
8082
8122
8211
8801
1.70
8391
8482
8673
8664
8756
8848
8941
9034
9127
9820
1.71
9814
9409
9502
9598
9t.93
9788
9884
9980
6077
5174
1.75J
9.960271
0869
0467
0565
0664
0763
0862
0961
1061
1162
1.78
1262
1863
1464
1566
166S
1770
1873
1976
2079
2188
1.74
2287
2391
2496
2601
2706
2812
2918
8024
3131
8288
1.75
8345
8453
3561
3669
3778
8887
8996
4105
4215
4820
1.76
4436
4547
4669
4770
4882
4994
6107
6220
6388
5447
1 77
5561
5675
5789
6904
6019
6135
6261
6367
6484
6600
1.78
6718
6885
6953
7071
7189
7308
7427
7547
76^6
7787
1.79
7907
8028
8149
8270
8392
8514
8636
8769
8882
9005
1.80
9129
9258
9377
9501
9626
9761
9877
5006
5189
5265
1.81
9.970:)83
0509
0637
0765
0893
1021
1150
1279
1408
1588
1.82
1668
1798
1929
2060
2191
2322
2464
2586
2719
8862
1.83
2985
8118
3252
3:^
3520
8665
3790
8925
4061
4197
1.84
4833
4470
4606
4744
4881
5019
6167
6295
5434
6578
1.85
6712
6852
6992
6132
6273
6414
6.556
6697
6888
6980
1.86
7128
7266
7408
76.52
7696
7840
7984
8128
8278
8419
1.87
a564
8710
8856
9002
9149
9296
9443
9591
9739
9887
1.88
9.980036
0184
0:383
0483
0633
0783
0938
1084
1284
1886
1.89
1537
1689
1841
1994
2147
2299
2468
2607
2761
8915
1.90
8069
3224
8379
35.35
3690
3846
4003
4169
4316
4474
1.91
4631
4789
4947
5105
52(S4
5423
6682
5742
5902
6062
1.92
6223
6383
6.544
6706
6867
7029
7192
7a54
7517
7680
1.93
7844
8007
8171
8386
8500
8665
8a30
8996
9161
9327
1.94
9494
9660
9827
9995
5162
oa30
5498
5G66
5885
1004
1.95
9.991173
1848
1512
1683
18.58
2024
2196
2366
2537
2709
1.96
2881
3054
8227
3399
3573
3746
3920
4094
4269
4448
1.97
4618
4794
4969
5145
6821
6498
.5674
5861
6029
6206
1.98
6884
6562
6740
6919
7098
7277
7457
7637
7817
7997
1.99
8178
8369
8640
8722
8908
9085
9268
9450
9638
9818
128
STATISTICAL METHODS.
VI.— TABLE OF REDUCTION FROM COMMON TO METRIC SYSTEM.
• • • •
Inches to Millimeters.
•
1
2
8
•
4
6
6
7
. 8
9
25.40
50.80
76.20
101.60
127.00
152.40 177.80
203.20
228.60
10
279.40
304.80
330.19
355.59
380 99
406.39 431.79
457.19
482.59
20
533.39
558.79
584.19
609.59
634.99
660.39 6&5.79
711.19
r<J6.59
30
787.39
812.79
8:i8 19
863.59
888.99
914.39 939.78
965.18
990.58
40
1041.4
1066.8
1092.2
1117.6
1143.0
1168.4
1193.8
1219.2
1244.6
50
1295.4
1320.8
1346.2
1371.6
1397.0
1422.4
1447.8
1473.2
1498.6
60
1549.4
1574.8
1600.2
16-25.6
1651.0
1676.4
1701.8
1727.2
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
2133.6
2159.0
2184.4
2209.8
2235.2
2260.6
90
2311.4
2336.8
2362.2
2387.6
2413.0
2438.4
2463.8
2489.2
2514.6
■ 1
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
yn
4.23
8/12
16.93
1/8
3.17
3/8
9.52
5/8
15
.87
7/8
22.22
8/12
6.35
9/12
19.05
3/16
4.76
7/16
11.11
11/16
17
.46
15/16
23.81
4/12
8.47
10/12
21.17
1/4
6.35
1/2
12.70
3/4
19.0-
1
25.40
5/12
10.58
11/12
23.28
6/13
12.70
12/12
25.40
TABLE VII.— MINUTES AND SECONDS IN DECIMALS OF A DEGREE.
1
o
21
o
4?
o
n
1
o
21
o
41
o
.016666
.350000
.683333
.000278*
.006833
.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
46
.012500
6
.100000
26
.433333
146
.766666
6
.001667
26
.007222
46
.012778
7
.1166661
27
.450000, 47
.783333
7
.001944
27
.007500
47
.013056
8
. 1333331
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
.806660
12
.003333
32
.008889
52
.014444
13
.216666
33
.550000
53
.883333
13
.003611
33
.009167
63
.014722
14
. 233333
34
.566666
54
.9(K)00()
14
.003889
34
.009444
54
.016000
15
.250000
35
.583333
55
.916666
15
.004167
35
.009722
66
.016278
16
.266666
36
.600000
56
.933333
; 16
.004444
36
.010000
66
.016656
17
.283333
37
.616666
57
.950000
i 17
.004722
37
.010278
67
.016833
18 .300000
38
.633333
58
.966666
18
.005000
38
.010656
68
.016111
19;. 316666
39
.650000,
59
.983333
19
.005278
39
.010833
69
.016389
20 .333333
40
.666666
60
1.000000
20
.005556
40
.011111
60
.016667
* .0002777778.
FIRST TO SIXTH POWERS OF INTEGERS.
129
TABLE VIII.— FIRST TO SIXTH POWERS OF INTEGERS FROM 1 TO 50.
•
Powers.
First.
Gecond.
Third.
Fourth.
Fifth.
Sixth.
1
1
1
1
1
1
2
4
8
16
82
64
8
9
27
81
248
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
7-29
6561
59049
531441
10
100
1000
10000
100000
1000000
11
121
1331
14641
161051
1771561
12
144
1728
20736
248882
2985984
13
169
2197
28561
371-493
4826809
14
196
2744
38416
537824
75.i9536
15
225
3375
50625
759375
11390625
16
256
4096
65536
1048576
16777216
17
289
4913
83521
1419857
24137569
18
824
58:32
104976
1889568
34012224
19
861
6859
1303'J1
2476099
47045881
20
400
8000
160000
8200000
64000000
21
441
9-^61
194481
4084101
85766121
22
484
10648
284256
5153632
113379904
23
529
12167
279841
6436343
1480:^6889
24
576
13824
33I?76
79t)Si624
191102976
25
625
156..'5
390625
9765625
24414U625
26
676
17576
456976
11881376
808915776
27
7:^:9
19683
531441
14348907
387420489
28
7H4
21952
614656
1?210368
481890304
29
841
24389
707281
20511149
694823321
90
900
■27000
810000
24300000
729900000
• 31
961
29791
923521
28629151
887508681
32
105J4
32768
104H576
33554432
1073741824
33
ias9
35937
1185921
391.35:^93
129146706B
34
1156
3a»4
1386336
454.35424
1544804416
35
1225
42875
1500625
52521875
1838265625
36
1296
46656
1679616
60464176
2176782336
37
13b^
50653
1874161
69843957
25a5726409
38
1444
54872
2085136
79235168
3010986384
39
1521
59319
2313441
90224199
3518743761
40
1600
64000
2560000
102400000
4096000000
41
1681
68921
2825761
115856201
4750104241
42
1764
74088
3111696
1306912;«
54S9031744
43
1849
7950r
3418801
147008443
6321363049
44
1936
&5184
3748096
164916224
7256313K56
45
2025
91125
4100625
184528125
8303765625
46
2116
97336
4477456
205962976
9474296896
47
2209
103S23
4879681
229:^45007
10779215329
48
2304
110592
5308416
254803968
12230590464
40
2401
117649
5764801
282475249
13841287201
50
2500
125000
6250000
312500000
15625000000
130
STATISTICAL METHODS.
TABLE IX.— PROBABLE ERRORS OF THE COEFFICIENT OF COR-
RELATION FOR VARIOUS NUMBERS OF OBSERVATIONS OR
VARIATES in) AND FOR VARIOUS VALUES OF r.
Decimal point, properly preceding each entry, is omitted. (Specially Cal-
.^ culated.)
Number
of Obser-
yations-
Correlation Coefficient r
•
0.0
0.1
0.2
0.3
0.4
0.5
0.6
20
30
40
60
60
1508
1231
1067
0954
0871
1493
1219
1056
0944
0862
1448
1182
1024
0915
0836
1373
1121
0971
0868
0793
1267
1035
0896
0801
0731
1131
0924
0800
0716
0653
0966
0788
0683
0610
0667
70
80
90
100
150
0806
0754
0711
0674
0551
0798
0747
0704
0668
0546
0774
0724
0683
0648
0529
0734
0686
0647
0614
0501
0677
0633
0597
0567
0463
0606
0566
0533
0506
0413
0616
0483
0465
0432
0362
200
250
300
400
500
0477
0420
0389
0337
0302
0472
0421
0386
0334
0299
0458
0409
0374
0324
0290
0434
0387
0354
0307
0274
0401
0358
0327
0283
0253
0358
' 0319
0292
0253
0226
0305
0272
0249
0216
0193
600
700
800
900
1000
0275
0255
0239
0225
0213
0272
0252
0236
0222
0211
0264
0245
0229
0216
0205
0251
0232
0217
0205
0194
0232
0214
0200
0189
0179
0207
0191
0179
0169
0160
0176
0163
0163
0144
0137
2000
5000
20
30
40
50
60
0151
0095
0149
» 0094
0145
0092
0137
0087
0127
0080
0113
0072
0007
0061
■ 1
0.65
0.7
0.75
0.8
0.85
0.9
o.gs
0871
0711
0616
0551
0503
0769
0628
0544
0486
0444
0660
0539
0467
0417
0381
0543
0444
0384
0343
0313
0419
0342
0296
0265
0241
0287
0234
0203
0181
0165
0147
0120
0104
0003
0085
70
80
90
100
150
0466
0436
0411
0391
0318
0411
0385
0363
0345
0281
0353
0330
0311
0294
0241
0290
0271
0256
0242
0198
0224
0209
0197
0187
0153
0153
0143
0135
0128
0106
0079
0074
0069
0066
0064
200
250
300
400
500
0275
0246
0225
0195
0174
0243
0218
0199
0172
0154
0209 s
0187
0170
0148
0132
0172
0154
0140
0122
0109
0133
0118
0108
0094
0084
0091
0081
0074
0064
i)067
0047
0042
0038
0083
0029
600
700
800
900
1000
0159
0147
0138
0130
0123
0140
0130
0122
0114
0109
0121
0112
0105
0098
0093
0099
0092
0086
0081
0077
0076
0071
0066
0062
0059
0052
0049
0046
0043
0041
0027
0025
0023
0022
0021
2000
5000 1
0087
0055
0077
0049
0066
0042
0054
0034
0042
0026
1
0029
0018
0014
0009
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
8
9
27
1.7320508
1.4422496
.833333333
4
16
64
2.0000000
1.5874011
.250000000
5
25
125
2.2360680
1.7099759
.200000000
6
36
216
2 4494897
1.8171206
.166666667
7
49
343
2.6457513
1.9129312
.14285n48
8
64
512
2.8284271
2.0000000
.125000000
9
81
729
8.UO0O00O
2.0600837
.111111111
10
100
1000
8.1622777
2.1544347
.100000000
11
121
1331
8.3166218
2.2288801
.090909091
12
144
17%
8.4041016
2.2894286
.063333388
13
169
2197
8.6065513
S.3513347
.076023077
14
196
2744
8.7416574
2.4101422
.071428571
15
225
8875
8.8729838
2.4662121
.066666667
16
256
4096
4.0000000
2.5196421
.062600000
17
289
4913
4.1231056
2.5712816
.058823529
18
824
5832
4.2426407
S.6207414
.055556566
19
861
6859
4.3588989
S.6684016
.062631579
20
400
8000
4.4721360
2.7144177
.050000000
21
441
9261
4.5825757
2.7589248
.047619048
22
484
10648
4.6904158
2.8020393
.045454545
28
620
12167
4.7958315
2.8438670
.048478261
24
576
13824
4.8969795
2.8844991
.041666667
25
625
15625
50000000
2.9240177
.010000000
26
676
17576
5.0990195
2.9624960
.088461588
27
729
19683
5.1961524
8.0000000
.087037087
28
784
21952
5.2916026
8.0365889
.085714286
29
841
24389
5.3851648
8.0?^168
.034482759
ao
900
27000
5.4VV22b6
8.1072325
.083338888
81
961
29791
5.6677614
8.1413806
.032258066
82
1024
82768
5.6568542
8.1748021
.031250000
88
1089
85937
5.7445626
8.2075843
080308080
84
1156
89304
5.8809519
8.2396118
.029411765
85
1225
42875
5.9160798
8.2710663
.028571429
86
1296
46656
6.0000000
8.3019272
.0277:7778
87
1860
50653
6.0827625
8.3322218
.027027027
88
1444
54872
6.1644140
8.8619754
.026315780
89
1521
50319
6.2449980
8.3912114
.025641026
40
1600
64000
6.3245553
8.4199519
.025000000
41
1681
68921
6.4031342
8.4482172
.024890244
42
1764
74088
6.4807407
8 4760266
.023809624
48
1849
79507
6.5574385
8.503:)981
.028256814
44
1936
a5184
6.63%M96
8.5303483
.02272?^78
45
2025
91125
6.7082039
8.5568933
.022222222
46
2116
97im
6.ffl23a00
8.5830479
.021739180
47
2209
103823
6.8556546
8.6088261
.021276600
48
2304
llOiOa
•6.0282032
8.6342411
.(^0838388
49
2101
117649
7.0000000
3.6593057
.020408168
50
'2500
125000
7.0710678
3.6840814
.020000000
51
2601
132651
7.1414284
8.7064298
.019607848
52
2704
140608
7.2111026
8.7325111
.019230769
53
2809
148877
7.2801099
8.7562858
.018867925
54
2916
157464
7.3484692
8.7797631
.018518519
55
8025
166375
7.4161985
8.8029525
.018181818
56
8186
175616
7.4833148
8.8258624
.01786n48
57
3249
18510;}
7.&i98344
8.84&y)ll
.017543860
58
33()4
195112
7.615rr31
8.8708766
.017241379
59
3481
205379
7.6811457
8.8929965
.016949153
60
8600
216000
7.7459667
8.9148676
.016666667
61
3721
226981
7.8102497
8.9364972
.016893443
62
3844
2rW328
7.8740079
a.^'t^vs
^ 5SS&'^fint&
\z\
TABLE X. — SQUARES, CUBES, SQUARE ROOTS,
No.
Squares.
63
3969
64
4096
65
4225
66
4356
67
4489
68
4624
69
4761
70
4900
71
5011
72
5184
73
6329
74
5476
75
5625
76
5776
77
5929
78
6084
79
6241
80
6400
81
6561
82
6724
83
6889
84
7056
85
7225
86
7396-
87
7569
88
89
90
91
92
93
91
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
lt>2
/
7744
7921
8100
8281
&464
8649
8836
9025
9216
9109
9604
9801
10000
10201
10404
10609
10816
11025
11236
11449
11004
11881
12100
12321
12544
127G9
12996
1322,5
13156
13689
13924
14161
14400
14611
14884
15129
15.376
Cubes.
76 I
250047
262144
274625
287496
300763
314432
328509
843000
a>7911
373248
389017
405224
42187'5
438976
45653:i
474552
493039
B12000
5;il441
551368
571787
592704
61 1125
636056
6585(«
681472
704969
729000
75:3571
77S688
804:357
&30584
85?375
884736
912673
941192
97ai99
1000000
10:30:301
10()1208
1092727
1124804
1157625
1191016
1225013
125!)712
1295029
1331000
1:307631
1404928
1142897
1481544
1520875
15(50896
1001613
161:3032
1685159
1728000
1771561
181 5818
1S()0867
Square
Roots.
7.9372539
8.0000000
8.0622577
8.1240384
8.1853528
8.2462113
8.3066239
8.3666003
8.4261498
8.4852814
8.54400:37
8.6023253
8.6602M0
8.7177979
8.7749644
8.8317609
8.8881944
8.9442719
9.0000000
9.055:3a51
9.11043:36
9.1651514
9.2195415
9.2r36185
9.3273791
9.3808:315
9.4:339811
9.4868330
9.5393920
9.5916630
9.6436508
9.695a597
9.7467S>43
9.7979590
9.848S578
9.8994949
9.9498744
10.0000000
10.0198756
10.0995049
10.1488916
10.1980390
10.2469508
10.2950:301
10.3440804
10.:3923048
10.4403065
10.4880885
10.5356538
10.5830052
10.6301458
10.(770783
10.7238053
10.7703296
10.81665:38
10.8627805
10.9087121
10.9544512
11.00 0000
11.0453610
ll.(i9053(r)
11.1355287
Cube Roots.
3.9790571
4.0000000
4.020?256
4.0412401
4.061&480
4.0816551
4.1015661
4.1212853
4.1408178
4.1601676
4.1793390
4.1983364
4.2171633
4.2358236
4.2543210
4.2726586
4.2908404
4.3088695
4.3267487
3444815
3620707
3795191
3968296
4140049
4.4310476
4.4479602
4.4647451
4.4814047
4.4979414
4.5143574
4.5306&49
4.5468359
4.5629026
4.5788570
4.5947009
4.6104363
4.6260650
4.6415888
4.6570095
4.6723287
4
4
4
4
4
4
6875482
7026694
7176940
7326235
7474594
7622032
4.7768562
4.
4.
4.
4.
4.
4.
4.
.7914199
.8058955
.8202845
.8345881
.&488076
.8629443
.8769990
4.8909732
4.9048681
4.9186847
4.9324343
4.9460874
4.9596757
4.9731898
4.9860310
Reciprocals.
.015873016
.015625000
.015384615
.015151515
.014925373
.014705883
.014492754
.014285714
.014064507
.013888889
018696G30
.013513514
.013333333
.013157893
.012967013
.012830513
.013658223
.013500000
.013345679
.013195132
.012048193
.011904762
.011764706
.011627907
.011494253
.011363636
.011235955
.011111111
.010989011
.010869565
.010753688
.010638396
.010536310
.010416667
.010309378
.010204063
.010101010
.010000000
.009900990
.009803933
.009706738
.009615385
.009533810
.009433963
.009345794
.00.^359359
.009174313
.009090909
.009009009
.008928571
.008849558
.006771930
.006695653
.006630690
.006547009
.008474576
.006103361
.006388333
.008364463
.008196?21
.008130061
.008064516
1
\^%
CUBE ROOTS, AKD RECIPROCALS.
No.
Squares.
Cubes.
Square
Roots.
Cube Roots.
Reciprocals.
125
15685
1953125
11.1803.399
5.0000000
.008000000
12G
15^6
2000376
11.2249722
5.01.32979
.007936508
ir
lffl29
2048383
11.2694277
5.0265257
.0U7874016
1-28
16384
2097152
11.3137085
5.0:390842
.007812500
129
16641
2140089
11.357^167
5.0527743
.007751938
130
16900
2197000
11.4017543
5.0657970
.007692308
131
17161
2248091
11.4455231
5.0787531
.00763.3588
332
17424
2299968
11.4891253
5.09164.34
.007575758
133
17689
2352637
11.5325626
5.1044687
.007518797
1^
17956
2406104
11.5758369
5.1172299
.007462687
135
18225
2460375
11.6189500
5.12992:3
.007407407
136
18496
2515456
11. 66190:38
5.142.5632
.007352941
137
18769
2571353
11.7046999
5.1.551.367
.00?299270
138
19044
2628072
11.74r^01
5.1676493
.007246.377
139
19321
2685619
11.7898261
5.1801015
.007194i^45
140
19600
2744000
11.8321596
5.1924941
.007142857
la
19881
28032^1
11.874:3421
5.2048279
.007092199
142
20164
2863288
11.910.3753
5.2171034
.007042254
143
20449
5924207
11.9582607
5.2293215
.006993007
144
' 20786
2965984
12.0000000
5.2414828
.006944444
145
21025
8048625
12.0415946
5.25.35879
.0068965.52
146
21316
8112136
12.0830460
5.2656374
.006849315
147
21609
8176528
12.1243557
5.2776321
.006802721
148
21904
8241792
12. 1655251
5.28a5r25
.006756757
149
82201
8307949
12.2065556
5.3014592
.000ni409
150
22500
8375000
12.s;474487
5.3132928
.006666667
151
22801
&442951
12.2882057
5.3250740
.006622517
152
23104
8511808
12.3288280
5.3308083
.006578947
153
23409
8581577
12.3693169
5.3484812
.006535948
154
23716
8652264
12.40967:36
5.8601084
.006493506
155
24025
3723875
12. -4498996
5.3716854
.006451613
156
24336
8796416
12.4899960
5.3832126
.0064102.56 •
157
24649
8869893
12.. 5299041
6 8946907
.006369427
158
1^964
3944312
12.. 5698051
5.4061202
.006.329114
159
25281
4019679
12.6095202
5.4175015
.006289308
160
25600
4096000
12.W91106
• 5.4288a53
.006250000
161
25921
4173281
12.688.5775
5.4401218
.006211180
162
26244
4251528
12.?279221
5.4513(U8
.006172840
163
26569
43:30747
12.76n453
5.4625.5.56
.006134969
164
26896
4410944
12.8062485
5.47370:37
.006097561
165
2?225
4492125
12.8452326
5.4848066
.006060606
166
27556
4574296
12.8840987
6.4958647
.006024096
167
27889
4657403
12 9228480
6.5068784
.005988024
168
28224
4741632
12.9614814
6.5178484
.005952381
169
28561
4826809
13.0000000
5.5287748
.005917160
170
88900
4913000
13 03S4048
5.5896.583
.005882353
171
29241
5000211
13.0760968
5.5504991
.005847953
172
29584
5088448
13 1148770
5.. 5612978
.005818953
173
29929
5177717
13.1529464
5.5?20546
.005780347
174
80276
5208024
13 1909060
5.5827702
.005747126
175
30625
6350375
13 2287506
5.59.34447
.005714286
176
80976
5451776
13.2604992
5.6040787 "
.006681818
177
31329
5545233
13..304K347
5.6146724
.005649718
178
81684
5639752
13, .3416641
6.6252263
.005617978
179
32041
5735.339
13.3790882
5.6357408
.005586592
180
3:^400
5832000
13.4164079
6.6462162
.005555.5.56
181
82761
5929741
13.45.36240
5.&56(}528
.005524862
182
33124
6028568
13.4907376
5.6670511
.005494505
183
83489
612W87
13.5277493
6 6774114
.0a5464481
184
83856
6229504
13.5646600
5.6877340
.0054.34783
185
^4225
63.31625
13.6014705
5.6980192
.005406405
186
a4596
64.34856
13.6381817
6.708a(S7^
.<i«£{ia'«dA^.
Vi&
TABLE X. -SQUARES, CUBES, SQUARE ROOTS,
No.
Squares.
Cubes.
Square
Roots.
Cube Roots.
Reciprocals.
187
84969
6539203
13.6747943
6.7184791
.005347B04
188
a5344
6&14672
13.7113092
6.7286543
.005319149
189
35?^1
6751269
13.747r271
5.7387936
.005291005
190
86100
6869000
13.7840488
6.7488971
.005268168
191
S(U81
6967871
13.8202750
5.7589652
.00528560e
19S
36864
7077888
13.8564065
6.7689982
.006208883
193
37249
7189057
13.8924440
5.7789966
.005181847
194
376;«
7301384
13.9283883
6.7889604
.005154689
195
38025
7414875
13.9642400
6.7988900
.005128206
196
88416
7529536
14.0000000
6.8067857
.005102041
397
88809
7645373
14.0a56688
6.8186479
.005076142
198
89204
7762392
14.0712473
6.8284767
..00506050i5
199
89601
7880599
14.1067360
6.8882?25
.005025126
200
40000
8000000
14.1421356
5.8480865
.006000000
201
40101
8120601
14.1774469
5.8577660
.004976124
202
40604
8242408
14.2126704
6.8674643
.004950496
208
41209
8365427
14.2478068
6.8771307
.004926108
204
41616
8189664
14.2828569
6.6867653
.004901961
205
42025
8615125
14.3178211
6.8963685
.004878049
206
4iM36
8741810
14.8627001
6.9059406
.004854869
207
42849
8869743
14.3874946
6.9164817
.004830918
208
43264
8998912
14.4222061
6.9249921
.004807698
209
43681
9129829
14.4668328
6.9344721
.004784689
210
44100
9261000
14.4913767
6.9489220
.004761906
211
44521
9393931
14.6258390
6.9533418
.004780886
212
44944
9528128
14.6602198
6.9627320
.004716981
213
45369
9663597
14.5945195
6.9720926
.004694836
214
45796
9600344
14.6287388
6.9814240
.004672897
215
46225
9938875
14.6628783
6.9907264
.004661168
216
46656
10077696
14.6969385
6.0000000
.004629680
217
47089
10218313
14.7809199
6.0092450
.004606296
218
47524
10360232
14.7648231
6.0184617
.004587166
219
47961
10503459
14.7966486
6.0276502
.004666210
220
48100
10648000
14.8323970
6.0368107
.004646455
221
48841
10793861
14.8660687
6.0459435
.004624887
232
40284
10941048
14.8996644
6.0660489
.004504506
223
49729
11089567
14.9331845
6.0641270
.004484801
224
50176
112394^
14.9666295
6.0781779
.004464286
225
50625
11390625
15.0000000
6.0822020
.004444444
226
5107&
11M3176
15.0332964
6.0911994
.004424779
227
51529
11697083
15.0665192
6.1001702
.004406286
228
51984
11852:i52
15.0996689
6.1091147
.004886066
229
52441
12008989
15.1327460
6.11808S2
.004366812
230
52900
12167000
15.1657509
6.1269257
.004847826
231
53:361
12:326391
15.1986842
6.1357924
.004829004
232
5;i824
l:;M87168
15.2315462
6.1446387
.004810346
233
54289
12649:337
15.2643376
6.1584496
.004291845
234
54756
12812904
15.2970585
6.1622401
.004273604
235
55225
12977875
15.3297097
6.1710058
.004265319
VS^
55696
13144256
15.3622915
6.1797466
.004287288
237
56169
ia312053
15.3948048
6.1884628
.004219409
2m
50644
1»481272
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
.004149678
242
58564
14172488
! 15.5563492
6.2316797
.004182281
243
69049
14^48907
1 15.5884673
6.2402515
.004116226
244
59536
14526784
15.6204994
6.2487998
.004098861
2i5
60025
14706125
15.6524758
6.2573248
.004061688
246
60516
148869:36
15.6843871
6.2658266
.004066041
J
347
61009
15069223
15.7162336
6.2743054
.004048688
L
J0^ 1
61504
15252992
15.7480157
6.2827618
.004082266
134
CUBE ROOTS, AND RECIPROCALS.
No.
Squares.
Cubes.
Square
Boots.
Cube Boots.
Beciprocals.
fU9
62001
15438249
15.7797338
6.2911946
.004016064
250
62500
15625000
15.8113883
6.2996053
.004000000
251
63001
15813251
15.8429795
6.3079935
.003984064
252
63604
16003008
15.8745079
6.3168596
.003968254
253
64009
16194277
15.9059737
6.3;U7035
.003952569
254
64516
16387064
15.93r3775
6:3330256
.003937008
255
65025
16581375
15.9687194
6.3413257
.003921569
256
65536
16777216
16.0000000
6.3496042
.003906250
257
66049
16974593
16.a3l2195
6.3578611
.003891051
25d
66564
17173512
16.0623784
6.3660968
.003875969
259
67081
17373979
16.09347U9
6.3743111
.003861004
260
67600
17576000
16.1245155
6.3825043
.003846154
261
68121
17779581
16.1554944
6.3906765
.003831418
262
68644
17984723
16.1864141
6.3988279
.008816794
263
69169
18191447
16.2172747
6.4069585
.003802281
264
69696
18399744
16.2480768
6.4150687
.003787879
265
70225
18609625
16.2788206
6.4231583
.003778685
266
70756
18821096
16.3095064
6.4312276
.003759398
267
71289
19034163
16.3401346
6.4392767
.003745318
268
71824
19^48832
16.3707055
6.4473057
.003731343
269
72361
19465109
16.4012195
6.4553148
.003717472
270
72900
19683000
16.4316767
6.4633(U1
.003703704
2n
73441
19902511
16.4620776
6.4712736
.003690037
2?2
73984
20123648
16.4924225
6.4792236
.003676471
273
7452J
20346417
16.5227116
6.4871541
.003663004
274
75073
20570824
16.5529454
6.4950653
.003649635
275
75625
20796875
16.5831240
6.5029572
.003636364
276
76176
21024576
16.6132477
6.5106300
.003623188
277
76729
21258933
16.6433170
6.5186839
.003610108
278
77284
21484952
16.678.3320
6.5265189
.003597122
279
77841
21717639
16.7032931
6.5343351
.003584229
280
78400
21952000
16.7332005
6.5421326
.003571429
281
78961
22188041
16.7630546
6.5499116
.003558719
282
79524
2^425768
16.7928556
6.5576722
.003546099
283
80089
22065187
16.8226038
6.5654144
.008588569
284
80656
22906304
16.8522995
6.5731385
.003521127
285
81225
23149125
16.8819430
6.5806443
.003608772
286
81796
23393656
16.9115345
6.5885323
.003496503
287
82369
23639903
16.9410743
6.5962023
.003484321
288
82944
23887872
16.9705627
6.6038545
.0aS472222
289
83521
24137569
17.0000000
6.6114890
.003460206
290
84100
24389000
17.0293864
6.6191060
.003448276
291
84681
24642171
17.0587221
6.6267054
.003436426
292
85264
24897088
17.0880075
6.6342874
.003424658
293
85849
25153757
17.1172428
6.6418522
.003412969
294
864.36
25412181
17.1464282
6.6493998
.003401361
295
87025
25672375
17.1756646
6.6569302
.003389831
296
87616
25934a36
17.2046505
6.6644437
.003378378
297
88209
26198073
17.2336879
6.6719403
.003367003
298
88804
26403592
17.262G765
6.6794200
.003.355705
299
89401
26r30899
17.2916165
6.6868831
.008344482
300
90000
27000000
17.3205081
6.6043295
.003333333
301
90601
27270901
17.3493516
6.7017593
.00:3322259
302
91204
27543608
17.3781472
6.7091729
.00a311258
303
91809
27818127
17.4068952
6.n66700
.003300330
304
92418
28094464
17.4355958
6.7239508
.003289474
305
93025
283?2625
17.4642492
6.7313155
.00:3278689
306
930:36
28652616
17.4928557
6.7386641
.003867974
307
94249
28934443
17.5214155
6.7459967
.003257329
308
94864
29218112
17.5499288
6.7533134
.003246753
309
954«1
2950:3629
17.5783958
6.7606143
.003236246
310
96100
29791000
17.6068169
6.7678995
.003225806
1^5
TABLE X. — SQUARES, CUBES, SQUARE ROOTS,
L
No.
Squares.
Cubes.
Square
Kuuts.
Cube Roots.
Reciprocals.
311
90721
30080231
17.0351921
0.7751090
.0032154ai
31«>
97344
30371:328
17.00:35217
0.7824229*
.0032U5128
313
97909
30004297
17.0918000
6.7890013
.0031»4HK8
314
98590
30959144
17.7200451
6.7908844
.003l84n3
315
99225
31255875
17.7482:393
6.8040921
.003174608
310
99850
31554490
17.7703888
6.8112847
.0081W557
317
100189
31855013
17.80449:38
0.8184020
.003154674
SIS
101124
321574:32
17.8325545
0.8250242
.003144654
319
101701
35^01759
17.8005711
0.8327714
.003134796
320
102400
32708000
17.8885438
6.a^9fl037
.003125000
3-^1
103041
33070101
17.9104729
0.8470213
.003ll5si65
"^i
103084
33380248
17.9443584
6.8541240
.003105590
3:i}
104329
33098207
17.9722008
0.8012120
.003005975
3:;i4
104970
340122-^4
18.0000000
6.8082855
.003086420
3.J5
105025
34328125
18.0277504
6.8753443
.0p30769i33
3;>0
100270
"34045970
18.0554701
6.8823888
.003067485
3;2r
100929
34905783
18.0831413
6.8894188
.008058104
3:i8
107584
352S7552
18.1107703
6.8904:345
.003048780
'ixHi
108;W1
35011289
18.1383571
6.9034359
.003039514
330
108900
a*i937000
18.1059021
6.9104233
.003030303
3^31
109501
30204091
18.1934054
6.9173904
.003021148
33:3
1102;i4
3»i594:308
18.2208072
6.9--^3550
.003012048
333
110889
30J20037
18.:i4b2870
6.9:313008
.003003003
:«4
111550
37259704
18.2750009
6.9382321
.002994012
335
112225
37595375
18.:3030052
6.9451496
.002985075
33G
112890
379:33050
18.3303028
6.9520533
.002976190
2A1
113509
38272753
18.3575598
6.9589431
.002967359
338
1145J44
88014472
18.384r.03
6.9058198
.002958580
339
114921
38958219
18.4119520
6.9720826
.002949853
aio
115000
39304000
18.4390889
6.9795321
.002941178
341
110281
39051821
18.400185:3
6.980:3081
.002982551
342
110904
40001088
18.4932420
6.9931900
.002023977
343
117049
40:35:j(i07
18.5202592
7.0000000
.00291545^
344
1183J30
40707584
18.54?2;370
7.0(MJ7902
.002906977
345
119025
4100;J025
18.5741750
7 0135791
.002898551
310
119710
414217:30
18.0010752
7.0203490
.002890173
347
120409
41781923
18.0279:300
7.0271058
.002881814
348
121104
42144102
18.0547581
7.03:3W97
.002878563
349
121801
42508549
18.0815417
7.0405806
.002865330
350
12251X)
4287r)000
18.7082809
7.04'^2987
.002857143
351
12:i2()l
4:324:3551
18.7:349940
7.0540041
.002819003
352
12:3904
43011208
18.7010(>3O
7.0000907
.002840909
353
124009
4398(5977
18.7882942
7.0073707
.002832801
a54
125310
44301804
18.8148877
7.0740440
.002824859
355
120025
44738875
18. 84144: J7
7.0800988
.002816901
aMi
12(>7;i0
45118010
18.8079023
7.08753411
.002808989
357
127449
45499293
18.89444:30
7.0939709
.002801120
358
128104
45882712
18.9208879
7.1005885
.002798296
359
128881
40208279
18.9^472953
J ^^ ^L ^bdtf^ f^ ^^ ^% j^
7.1071937
.002785515
300
129000
40050000
18.9730000
7.1137866
.002777778
301
130:321
47045?iSl
19 0000000
7.1203074
.002770088
302
131044
474:37928
19.0202970
7.1209360
.002762431
3o;3
131709
478:32147
19.05255vS9
7.1334925
.002764821
304
13^490
48228544
19.0787840
7.1400370
.002747258
. 305
133225
48027125
19.10497:32
7.1405095
.0027397MS
300
13:3950
4JK)27890
19.1311205
7.1530901
.002732240
307
134089
494:30803
19.1572441
7.1595988
.002724796
308
1:35424
498:30032
19.1^3:3201
7.1060957
.002717391
309
130101
5024:3409
19.2093727
7.1725809
.002n0087
370
130900
50053000
19.2353^41
7.1790544
.002702708
371
1:37041
51004811
19.2013003
7.1855102
.002695418
sr2 1
imm
51478t^8
19.28r3015
7.1919663
.002688178
1
\%^
CUBE ROOTS, AND RECIPROCALS.
No.
Squares.
Cubes.
Square
Roots.
Cube Roots.
Reciprocals.
373
139129
51895117
19.3132079
7.1984050
.002680965
374
139876
52:^1:3624
19.3390796
7.2048322
.002673797
375
140625
52734:375
19:3649167
7.2112479
.002666667
376
141376
53157376
19.3907194
7.2176522
.002669574
877
142129
5:3582633
19.4164878
7.2240450
.002662520
378
142884
54010152
19.4422221
7.2304268
.002645603
379
143U41
54439939
19.4679223
7.2367972
.002688622
380
144400
54872000
19.4935887
7.2431565
.002681679
381
145161
55306341
19.5192213
7.2495045
.002624673
S82
145924
55742968
19.5448203
7.2558415
.002617801
383
146089
56181887
19.5703a')8
7.2621675
.002610966
8^
147456
56623104
19.5959179
7.2684824
.002604167
385
118225
67060625
19.6214169
7.2747864
.002597408
386
148996
57512456
19.6468827
7.2810794
.002590674
387
149769
57960603
19.6723156
7.2873617
.002588979
388
150544
58411072
19.6977156
7.2936830
.002577880
389
151321
58863869
19.'n)i30629
7.2996986
.002670604
390
152100
59319000
19.7484177
7.8061486
.002564108
391
152881
59776471
19.7737199
7.3128828
.002567645
392
153GM
60236288
19.7989899
7.3186114
.002561020
393
154449
60698457
19.8242276
7.S248295
.002544629
894
155236
61162984
19.8494332
7.8810869
.002688071
895
156025
61629875
19.8746069
7.38?-J889
.002681646
396
150816
62099136
19.8997487
7.3484205
.002625258
397
157609
62570773
19.9248588
7.8496966
.002518892
398
158404
63044792
19.9499378
7.8557624
.002612668
399
159201
63521199
19.9749644
7.861917>J
.002506266
400
160000
64000000
20.0000000
7.8680630
.002500000
401
160801
64481201
20.0249844
7.8741979
.002498766
402
161004
64964808
20.0499377
7. £808227
.002487662
403
162409
65450827
20.0748599
7.38643r3
.002481890
404
163216
65939264
20.0997512
7.3925418
.002475248
405
164025
66430125
20.1246118
7.3986863
.002469186
406
1&1836
66923416
20.1494417
7.4047206
.002468064
407
165649
67419143
20.174^10
7.4107950
.002467002
408
1664(>4
67917:312
20.1990099
7.4168595
.002450960
409
Wi)iSl
6W17929
20.2237484
7.4229142
.002444988
410
168100
68921000
20.2484567
7.4289589
.002489024
411
168921
60420531
20.2731349
7.4349938
.002438090
412
169744
6i)934528
20.2977831
7.4410189
.002427184
413
170569
70444997
20.3224014
7.4470842
.002421806
414
1713.06
70957944
20.3469899
7.45:30399
.002416460
415
172225
71473375
20.3715488
7.4590359
.002409689
416
173056
71991296
20.3960781
7.4650228
.002408846
417
173889
72511713
20.4205779
7.4709991
.002898062
418
174?^
7:3034632
20.4450483
7.4769664
.002892844
419
175561
73560059
20.4694895
7.4829242
.002386686
420
176400
74088000
20.4939015
7.4888724
.002880962
421
177241
74618461
20.5182845
7.4948113
.002875297
422
178064
75151448
20.5426386
7.5007406
.002869668
423
178929
75686967
20.5669638
7.5066607
.002364066
424
179776
;6225024
20 5912603
7.5125715
.002358491
425
180625
76765625
20.6155281
7.5184730
.002352941
426
181476
77308776
20.6397674
7.5248652
.002347418
427
182329
77854483
20.6639783
7.5302482
.002341920
428
ia3184
78402752
20.6881609
7.5361221
.002836449
429
184041
78953589
20.7123152
7.5419867
.002381002
430
184900
79507000
20.7364414
7.5478428
.002825581
431
185761
8(K)62991
20.7605395
7.5536888
.002320186
432
186624
80621568
20.7846097
7.5595263
.002814815
433
187489
81182737
20.8086520
7.5653548
.002809469
4M
188356
81746504
20.8326667
7.571174a
ly .<5ffiSfc^^s\ \
rdi
TABLE X. — SQUARES, CUBES, SQUARE ROOTS,
No.
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
435
436
437
438
439
440
441
442
448
444
445
416
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467 '■
468 :
469 i
/
Squares.
Cubes.
Square
Boota
Cube Roots.
Reciprooals.
189225
82312875
20.8566536
7.5769849
.008298851
190096
82881856
20.8806130
7.5827865
.002293578
190969
83453453
20.9045450
7.5885798
.00S2288SaO
191844
&40276?2
20.92tm95
7.5943638
.002388105
192721
84604519
20.9523268
7.6001385
.0Q2S779O1
193600
85184000
20.9781770
7.6059049
.Utt272787
194481
85766121
21.0000000
7.6116626
.002267574
195364
86350(588
21.0237960
7.6174110
.002262443
196249
86938307
21.0475652
7.6231519
.002257830
197138
87528384
21.0718075
7.6288837
198025
88121125
21.0950231
7.6346067
.002Mn91
198916
88716536
21.1187121
7.6403213
.002242152
199809
89314623
21.1423745
7.6460272
.002287136
200704
89915392
21.1660105
7.6517247
.008232143
201601
90518849
21.1896201
7.657418J
.002227171
202500
91125000
21.2132034
21.2367606
7.6630943
.0Q22228S8
20:^1
91733851
7.6687665
.0a2217i95
204304
92345408
21.2602916
7.6744303
.002212889
205209
92959677
21.2837967
7.6800857
.00S2075O6
206116
93576661
21.3072758
7.6a57328
.002209643
207025
94196375
21.3307290
7.6918717
.002107808
207ai6
94818816
21.3541565
7.6970023
.002198968
208849
95443993
21.3775583
7.7026246
.002188184
209764
96071912
21.4009346
7.7082388
.002188406
210681
96702579
21.4242853
7.7138443
.002178640
211600
97336000
21.4476106
7.7194426
.008178918
212521
97972181
21.4709106
7.72D0325
.002160197
213444
98611128
21.4941853
7.7306141
.002104508
214369
99252847
21.5174348
7.7361877
.002169637
215296
99897344
21.5406592
7.7417532
.002165178
216225
100544625
21.5638587
7.7473109
.002150588
217156
101194696
21.58703:31
7.7328606
.002145088
218089
10184750:3
21.6101828
7.7584023
.002141828
210024
1(K5032:32
21.633:3077
7.70:19361
.002186758
219961
103161709
21.65&4078
7.7694620
.002182196
220900
103823000
21.6794834
7.7749801
.002187860
221841
104487111
21.7025344
7.7804904
.002123148
222784
105154048
21 .7255610
7.7839928
.002118844
223?29
105823817
21.7485632
7.7914875
.002114165
224676
106490424
21.7715411
7.7969745
.002109705
225625
107171875
21.7944947
7.8024538
.002106268
226576
107850176
21.8174242
7.8079254
.002100840
227529
10853133:3
21.840:3297
7.8133892
.002096486
228484
109215352
21 8632111
7.818^456
.002092050
229441
109902239
21.8860686
7.8242942
.002067668
230400
110592000
21.9089023
7.8297353
.002068388
231:^1
1112&4641
21.9317122
7.8:351688
.002078008
232:^24
111980168
21.9:>44984
7.a405949
.002074689
2:33289
112678587
21.9772610
7.a460134
.002070893
2:^56
113379904
22.0000000
7.8514^44
.002066116
235225
J 14084125
22.0227155
7.8568281
.002061856
236196
114791256
22.0454077
7.8622242
.002057618
237169
115501303
22.0680765
7.8676130
.002068888
2:38144
116214272
22.0907220
7.8729944
.002049180
239121
1169:30169
22.11:33444
78783684
.002014990
240100
117649000
22.1.359436
7.8837a'i8
.0OXM0616
241081
118370771
22.15ail98
7.8890946
.008080060
2420(54
119095488
22.1810730
7.8944468
.002088580
24:3049
119823157
22.2036033
7.8997917
.008028896
2440:36
120553784
22.2261108
7.9051294
.002081891
^5025
121287375
22.2485955
7.9104599
.008080808
240016 \
12^2:39:36
22.2710575
7.9157832
.002016199
1^8
CUBE ROOTS, AND RECIPROCALS.
1
No.
Squares.
Cubes.
Square
Roots.
Cub© Roots.
Reciprocals.
497
247009
122763473
22.2034968
7.9210994
.00201207^
498
248004
123505992
22.3159136
7.9264085
.002008032
499
219001
1;^4251499
22.3383079
7.9317104
.002004006
500
250000
125000000
22.3606798
7.9370053
.002000000
501
251001
125751501
22.3830293
7.9422931
.001996008
603
252lX)4
126506008
22.4053565
7.9475739
.001992062
503
253009
127263527
22.4276615
7.9528477
.001968072
504
254016
128024061
22.4499448
7.9581144
.001984127
505
255025
128787625
22.4722051
7.9638743
.001980198
506
256036
129554216
:^.4944438
7.9686271
.001976285
507
257049
130823ai3
22.5166605
7.9738781
.001972887
508
258064
131096512
22.5388553
7.9791122
.001968504
509
259081
1318?.2229
22.5610283
7.9843444
.001964637
610
260100
182651000
22.5831796
7.9895697
.001960784
611
261121
133432831
22.6058091
7.9947883
.001956947
512
262144
134217728
22.6274170
8.0000000
.001958125
513
263169
135005697
22.6495038
8.0052049
.001949318
514
261196
135796744
22.6715681
8.0104082
.001945525
515
265225
136590875
22.6936114
8.0155946
.001941748
516
266256
137888096
22.7156334
8.0207794
.001937964
517
267289
138188413
22.7376310
8.0259574
.001934236
518
268:i24
138991832
22.7596134
8.0311287
.001930502
519
269361
139796359
22.7815715
8.0362935
.001926782
530
270400
140608000
22.8035085
8.0414515
.001923077
521
271441
141420761
22.8254244
8.0466030
.001919386
522
27SM84
142236648
22.8473193
8.0517479
.0019157X)9
523
2r3529
148055667
22.8691933
8.0568862
.001912046
524
274576
143877824
22.8910463
8.0620180
.001906397
525
275625
144703125
22.9128785
8.0671482
.001904762
526
276676
145531576
22.9346899
8.0722620
.001901141
627
277729
146363183
22.9564806
8.0778743
.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
23.04343;^
8.0977589
.001883289
532
28:^^4
150568768
23.0651252
8.1028390
.001879699
533
284089
151419437
23.0667928
8.1079128
.001876178
534
285156
152273304
23.1084400
8.1129803
.001872659
535
286225
153180375
23.1300670
8.1180414
.001869159
536
287296
153990656
23.1516738
8.1230962
.001865672
537
288369
154854153
aj. 1782605
8.1281447
.001862197
538
289444
155720872
23.1948270
8.1331870
.001858786
539
290521
156590819
23.2163735
8.1382230
.001855288
540
291600
157464000
23.2379001
8.1432529
.001851858
541
292681
158340421
2;^. 2594067
8.1482765
.001848429
542
293764
159220068
23.2808935
8.1532939
.001845018
543
294849
16C103007
213.3023604
8.1583051
.001841621
544
295936
160989184
23.3238076
8.16:33102
.001838235
645
297025
161878625
23.^452351
8.1083092
.001834862
546
298116
162771336
23.3666429
8.17^3020
.001831502
547
299209
163667323
23.3880311
8.1782888
.001828154
548
300304
164566592
2:3.4093998
8.1832695
.001824818
649
801401
165169149
23.4307490
8.1882441
.001821494
550
802500
166375000
23.4520788
8.1932127
.001818182
551
303601
16?284151
2:3. 47*3892
8.1981753
.001814882
552
304704
168196608
2:3.4946802
8.2031319
.001811594
553
305809
169112377
23.5159520
8.2080625
.001808318
554
306916
170031464
2:3.5372046
8.2130271
.001805054
555
308025
170953875
23.5584880
8.2179657
.001801803
556
809136
171879616
23.6796522
8.2228985
.001796561
657
310249
172808693
23.6008474
, 8.22r:«KA
558
811364
173741112
[ sa.esaojjafe
V ^.^asaAsa
1^^
TABLE X. — SQUARES, CUBES, SQUARE ROOTS,
No.
Squares.
Cubes.
559
312481
174676879
560
313600
175616000
5(il
314721
176558481
502
315844
177504328
503
316969
17845:3547
564
318096
179406144
5(>5
319225
180362125
566
320356
181321496
567
321489
182284263
668
322624
18;J2504J«
509
323761
lt^4220000
570
324900
185193000
571
326041
186169411
572
327184
187149248
573
328329
188132517
674
329476
189119224
575
a30625
19010*375
676
331776
191102976
577
332929
192100033
678
334084
19310a'>52
579
335241
194104539
580
a3(>400
195113000
581
337561
196122941
582
338?24
197137:^68
583
339889
198155287
584
341056
199176704
585
342225
200201625
586
34*196
201230056
587
344509
202262003
588
345744
20:i297472
689
340921
204336469
590
348100
205379000
• 591
349281
200425071
592
3504(>4
207474688
593
351649
203527857
594
,3528.36
20(i5&4584
595
JJM025
210044875
596
3-)521G
211708736
597
3r)()4()9
2127701 7:i
598
3-)7604
213847192
599
358801
214921799
360000
361201
362404
363609
304816
366025
367236
368449
369664
370881
3721()0
37*321
374.M4
375769
3769{)6
378225
379456
380689
3MJ024
3H3161
384400
216000000
217081801
218167208
219256227
220348864
221445125
222545016
223648543
2SM755712
225866529
226981000
228099131
229220928
230346:397
2:31475544
2:32608:375
2:<:37448{M5
2348K5n3
236029032
237170659
238328000
Square
Koots.
Cube Roots.
23.6431806
8.2376614
23.6643191
8.2425706
23.6854:386
8.2474740
23.7065392
8.2523715
2:3.?276210
8.2572633
2:3.7486842
8.2621492
23.7697286
8.2670294
23.7907545
8.2719039
23.8117618
8.2767788
23.8:327506
8.2816365
^o-tibbT-Mi
8.2864928
23.8746728
8.2918444
23.895606:3
8.2961903
23.9165215
8.3010804
23.9374184
8.3058661
23.9582971
8.3106941
23.9791576
8.3165175
a4. 0000000
8.3203368
^.0208243
8.3251475
24.0416806
8.3299542
24.0624188
8.3347563
24.0831891
8.3395509
24.10:39416
8.3443410
24.1246762
8.3491256 -
24.1453929
8.3539047
^.1660919
8.3580784
SM. 1867732
8.3634466
24.2074369
8.3682095
24.2280829
8.3729668
24.:^487n3
8.3777188
24.2693222
8.3824653
24.2899156
8.3872065
24.:3104916
8.:391942:3
24.3310501
8.:i966729
24.:3515913
8.4018981
24.:3721152
8.4061180
24.:3926218
8.4108326
24.4131112
8.41,55419
24.4:3:358:34
8.4202460
24.4r>40:385
8.4249448
24.4744765
8.4296383
24.4948074
8.4348267
24.5153013
8.4;3iK)098
24.53.56883
8.4436877
24.5,560583
8.4483605
24.5764115
8.4530281
24.5967478
8.4576906
»4. 6170673
8.4623479
24.6:373^00
8.4670001
JM. 6.576.560
8.4716471
24.6779254
8.4762892
24.6981781
6.4809261
24.7184142
8.4855579
24.7380338
8.4901848
24.758a3(>8
8.4948065
»4. 7790234
8.4994233
^.7991935
8.5040350
24.8193473
8.5086417
24.KJ94847
8., 5132435
iM. 8596058
8.5178403
24.87U7106
a4.8W?.m
Reciprocals.
\
001788809
.001T95714
.001768531
0017r9859
001776109
.OOITTSOSO
.001709912
.001766784
.001768668
.001760668
.001757469
.001754886
.001751318
.001748252
.001745201
.001742160
.001789180
.001736111
.001788102
.001780104
.60]72ni6
.001T84188
.00178X170
.001718218
.001715266
.001712829
.001709402
001706485
.001703578
.001700680
.001697793
.001694915
.001692047
.001689189
.001686341
.001683502
001680672
.001677852
.001675042
.001678241
.001669449
.001666667
.001663894
.001661130
.001668875
.001C65629
.001662898
.001650165
.001647446
.001644737
.001642086
.001639844
.001686661
001688987
001631821
.001628664
.001626016
.001628377
.001620746
.001618128
.001615509
140
A
CUBE ROOTS, AND RECIPROCALS.
C30
631
(m
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
653
653
654
655
end
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
67r
678
079
680
681
682
Squares.
Cubes.
Square
IvUOtS.
1
3a5&ll
2:J9483061
24.9198716
386881
210641848
24.9399278
388129
241804867
24.9599679
389376
^12970624
24.9799920
390625
244140625
25.0000000
391876
245314876
25.0199920
893129
246491883
25.0399681
894384
247673162
25.0599282
395641
248858189
25.0798724
896900
250047000
25.0998006
398161
251239591
25.1197181
399421
252435968
25.1396102
4lK)689
253636137
25.1594913
401956
254840101
25.1793566
403>?25
256047875
25.1992063
404496
257259456
25.2190404
405769
258474853
25.2388589
407044
259694072
25.2586619
406321
260917119
25.2784493
409600
262144000
25.2982213
410881
263374721
25.3179778
412164
264609288
25.3377189
413449
265847r07
25.3574447
414736
267069984
25.3771551
416025
268336125
25.3968502
417316
2695861:%
25.4165301
418609
270840023
25.4361947
419904
272097792
25.4558441
421201
273359449
25.4754784
422500
^7462'5000
25.4950976
423801
275894451
25.5147016
425104
277167808
25.5842907
426409
278445077
25.5538W7
427716
279?26264
25.5734237
429025
281011375
25.5929678
430336
282800416
25.6124969
431649
28359;«93
25.6820112
432964
284890312
25.&515107
4S1281
286191179
25.6709953
4a5600
287496000
25.6904652
436921
288804781
25.7099203
438244
290117528
25.?298607
43ft'>69
2914*4247
25.7487864
440896
292754941
25.7681975
442225
294079625
25.7875939
443556
295406296
25.8069758
444889
296740963
25.8263431
446224
298077632
25.8456960
447561
299418309
25.8650343
448900
300763000
25.8848582
450241
302111711
25.9036677
451584
30*464448
25.9229628
452929
304821217
25.9422435
454276
306182024
25.9615100
455625
807546875
25.9807621
456978
808915776
26.0000000
458329
810288733
26.0192237
459684
311665752
26.0384331
461041
813046839
J:6. 0676281
462400
314432000
26.0768096
463761
315821241
26.0959767
465124
317214568
26.1151297
Cube Roots.
8.5316009
8.5361780
8.5407501
8.5453173
8.5498797
8. >541372
8.5589899
8.5635377
8.5680807
8.5726189
8.5771523
8.5816809
8.5862047
8.5907238
8.5952380
8.5997476
6.6042525
8.6087526
8.6132480
8.617r388
8.6222248
6.6267063
8.6311830
8.6356551
8.6401226
6.6445855
8.6490437
8.6584974
8.6579465
8.6623911
8.6668810
8.6712665
8.6756974
8.6801237
8.6845456
8.6889630
8.6933759
8.6977843
8.7021882
8.7065877
8.7109827
8.7153784
8.7197596
8.7241414
8.7285187
8.7328918
6.7372604
8.7416246
8.7459846
6.7503401
8.7546913
8.7590383
8.7633809
8.767ri92
6.7720532
8.7763830
6.7807084
8.7850296
8.7898466
8.7936.593
8.7979679
8.8022721
Reciprocals.
.001610306
.001607717
.001605136
.001602564
.001600000
.001597444
.001594896
.001592357
.001589825
.001587^02
.001584786
.001582278
.001579779
.001577287
.0015748a3
.001572327
.001669869
.001667898
.0016&4945
.001562500
.001560062
.001657632
.001555210
.001552795
.001550388
.001547988
.001545695
.001543210
.001540832
.001588462
.001536098
.001533742
.001531394
.001529062
.001526718
.001521390
.001522070
.001519767
.001517451
.001515152
.001612860
.001510674
.001506296
.001506024
.001508769
.001601602
.001499260
.001497006
.001494768
.001492687
.001490818
.001488095
.001486884
.001483680
.001481481
.001479290
.001477105
.001474926
.001472754
.00147a'588
.001468429
.001466278
l^i
TABLE- X. — SQUARES, CUBES, SQUARE ROOTS,
No.
Squares.
Cubt's.
Square
lioots.
Cube Roots.
.Reciprocals.
683
466489
318611987
26.1342687
8.8066722
.001464139
6&i
467856
32001:3504
26.15:38937
8.8108681
.001461968
fy85
469225
321419125
26.1725047
8.8151598
.001459654
686
470596
322828856
26.1916017
8.8194474
.001457726
687
471969
324242703
26.2106848
8.8237307
.001456604
688
473314
325660672
26.2297541
8.8280099
.001453488
689
474721
327082769
26.2488095
8.8322850
.001451379
690
476100
828509000
26.2678511
8.8365559
.001449275
691
477481
329939371
26.2868789
8.8408227
.00144n78
692
478864
331373888
26.3058929
8.8460854
.001445067
693
480249
332812557
26.3248932
8.8493440
.001443001
694
481636
334255384
26.3438797
8.85:35985
.0014400BS3
695
483025
335702375
26.3628527
8.8578489
.001438849
696
484416
38715S536
26.3818119
8.8620952
.001436788
697
485809
338606873
26.4007676
8.8668375
.0014847^
698
48?204
340068392
26.4196896
8.8705757
.001432665
699
488601
341532099
26.4386081
8.8748099
.001430615
TOO
490000
343000000
26.4575131
8.8790400
.001428571
701
491401
344472101
26.4764046
8.8832661
.001426584
* 708
492804
34594&108
26.4952820
8.8874882
.001424501
703
494209
347428927
26.51414?2
8.8917063
.001422475
704
495616
ai8913664
26.5329983
8.8969204
.001420455
705
497025
350402625
26.6518361
8.9001304
.001418440
706
498436
351895816
20.5706605
8.9043366
.001416431
707
499849
353393243
26.5894716
8.9086387
.001414427
'
708
501264
354804912
26.0082694
8.9127369
.001412429
709
502681
356400829
26 0270539
8.9169311
.001410437
710
504100
357911000
26.6458252
8.9211214
.001406451
711
605521
359425431
26.0645833
8.925:3078
.001406470
712
506944
360944128
26.6833281
8.92949P2
.001404404
713
608:369
362467097
26.7020598
8.93:36687
.001402525
714
609796
363994344
26.7207784
8.9378433
.001400560
715
511225
365525875
26.7394839
8.9420140
.001396601
716
512656
367061696
26.7581763
8.9461809
.001896648
717
514089
368601813
26.7768557
8.95031:38
.001894700
718
515524
370146232
26.7955220
8.9546029
.001:392758
I-
719
516901
371694959
26.8141754
8.9586581
.001390621
i.
720
518100
373248000
26.8328157
8.9628095
.001888889
1 ■
721
519841
374805:301
26.K514432
8.9609570
.lK)1386968
i!-
7*22
5212H4
376367048
26.8700577
8.9711007
.001386042
!■
723
522729
3779:3:3067
26.8886593
8.9752406
.001383126
*
TO4
524176
37950:^24
20.9072481
8.9793766
.001381215
1
725
525625
381078r25
26.9258240
8.9835089
.001379810
•h
726
527076
382657176
26.9443872
8.9876373
.001377410
1 '
■^27
528529
3H424()58:3
26.9629375
8.9917620
.001375516
1' '
■I
72S
5299H4
:385828:352
2(i.lW14751
8.9958829
.001373626
■ ,
729
531441
38742W89
27.0000000
9.0000000
.001371742
\[
730
5^2900
389017000
27.0185122
9.0041134
.001369863
731
534361
390617-891
27.0370117
9.0082229
.001367989
!,
732
5:35824
39222:3108
27.0554985
9.0123288
.001366120
,-
783
53?289
3938:3:^:37
27.0739727
9.0164309
.001364256
."
734
5:38756
395446904
27.0924344
9.0205293
.001362396
1
735
540-225
397065375
27.1108834
9.0246239
.001360544
736
541696
398688256
27.129:3199
9.0287149
.001358696
.-
737
54:3169
400:315553
27.1477439
9.0328021
.001356862
738
544644
40194^272
27.1(K)1554
9.0368857
.001355014
739
54G121
403583419
27.1*45544
9.0409655
.001353180
■
740
547000
4a5224000
27.202<)410
9.0450419
.001351851
,
741
549081
4068()1K)21
27.221:31.52
9.0491142
.001349526
■■
742
550564
408518488
27.2'39G7«9
9.05:31831
.001347/09
743
552049
410172407
27.258026:3
9.067.M82
.001845805
. r^ /
053536 1
411830784
27.2703634
9.0613098
.001344066
142
CUBE ROOTS, AND RECIPROCALS.
No.
Squares.
Cubes.
Square
Roots.
Cube Roots.
Reciprocals.
•
745
555025
413493625
27.2946861
9.0653677
.001342282
746
556516
415160936
27.3130006
9.0694220
.001340483
747
558009
416832723
27.3313007
9.0734726
.001338668
748
559504
418506992
27.3495867
9.0775197
.001336696
749
561001
420189749
27.3678644
9.0615631
.001.S35113
750
562500
421875000
27.3861279
9.0656030
.001833388
751
564001
423564751
27.4043792
9.0696392
.001831558
752
565504
425259008
27.4226184
9.0936719
.0013297^
753
567009
42695VVT/
27.4408455
9.0977010
.001826021
754
568516
428661064
27.4590604
9.1017265
.001326260
755
570025
430368875
27.4772633
9.1057485
.001324503
756
571536
432081216
27.4954542
9.1097669
.001822751
757
573049
433798093
27.5136330
9.1137818
.001321004
758
574564
485519512
27.5317998
9.1177931
.001319261
759
576061
437245479
27.5499546
9.1218010
.001317528
760
577600
438976000
27.5680975
9:1258053
.001315789
761
579121
440711081
27.5862264
9.1298061
.001314060
762
580644
442450^8
27.6043475
9.1338084
.001312386
763
582169
444194947
27.62^4546
9.1377971
.001810616
764
583696
445943744
27.6405499
9.1417674
.001808901
765
585225
447697125
27.0566334
9.1457742
.001307190
766
586756
449455096
27.6767050
9.1497576
.001805483
767
588289
451217663
27.6947648
9.1537375
.001303761
768
589824
452984832
27.7128129
9.1577189
.001802063
769
591361
454756609
27.7306492
9.1616669
.001800890
770
592900
456533000
27.7486739
9.1656565
.001296701
771
594441
458314011
27.7666868
9.1696225
.001297017
772
595984
460099648
27.7848860
9.1735652
.001295887
773
597529
461889917
27.6026775
9.1775445
.001298661
774
599076
463684824
27.8206555
9.1615003
.001291990
775
600625
465484375
27.8388218
9.1654527
.001290328
776
602176
467288576
27.8567766
9.1694018
.001286660
777
603729
469097433
27.6747197
9.1988474
.001267001
778
605284
470910952
27.6926514
9.1972697
.001286847
779
606841
472729139
27.9105715
9.2012286
.001288697
780
606400
474552000
27.9284601
9.2061641
.001282061
781
609961
476379541
27.9468772
9.2090962
.001280410
782
611524
478211768
27.9642629
9.2130250
.001278772
783
613069
480048687
27.9821373
9.2169505
.001277189
784
614656
461890804
26.0000000
9.2206726
.001275510
785
616225
483736625
28.0178515
9.2247914
.001273866
786
617796
485587656
28.0356915
9.2267068
.001272265
787
619369
487443403
26.0535203
9.2826189
.001270648
788
620944
489303872
26.07ia377
0.2865277
.001269036
789
622521
491169069
28.0691436
9.2404383
.001267427
790
624100
493039000
26.1069366
9.2448865
.001265828
791
625681
494913671
28.1247222
9.2482844
.001264228
792
627264
496793068
28.1^946
28.1602557
9.2521300
.001262626
793
628849
498677257
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
6.35209
506261573
26.2311684
9.2715592
.001254705
798
636804
508169592
26.2468938
9.2754352
.001253183
799
638401
510082399
28.2665661
9.2798081
.001251664
800
640000
512000000
28.2842712
9.2881777
.001250000
801
641601
513922401
28.3019434
9.2870440
.001248439
802
643204
515849608
28.3196045
9.2909072
.001246668
803
644809
517781627
28.3372546
9.2947671
.001245830
804
646416
519718464
28.3548938
9.2986289
.001243761
805
648025
521660125
28.3725219
9.8024775
, .C»V?A799&
806
649636
623606616
i 28.3att\^\
V "^.^fSRSSn^
\ .<($S^Sf«K«:k
14^
TABLE X. — SQUARES, CUBES, SQUARE ROOTS,
No.
Squares.
Cubes.
Square
Roots.
Cube Roots.
Reciprocals.
807
651249
525557943
28.4077454
9.3101750
.001239157
806
652864
527514112
28.4253408
9.8140190
.0012876^
809
654481
529475129
28.4429253
9.8178599
.001236094
810
656100
531441000
28.4604989
9.3216975
.001234568
811
657721
533411731
28.4780617
9.3255320
.001233046
813
659344
535387328
28.4956137
9.3293634
.001231627
813
660969
537367797
28.5131549
9.3331916
.001230012
814
662596
539353144
28.5306852
9.3370167
.001228501
815
664225
541343375
28.5482048
9.3408386
.001226994
816
665856
543338496
28.5657137
9.3446575
.001^5490
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
830
672400
551368000
28.6350421
9.3599016
.001219512
821
674041
553387601
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
.001218592
825
680625
561515625
28.7228132
9.3788873
.001212121
826
682276
563559976
28.7402157
9.3826752
.001210664
827
683929
565609283
28.7576077
9.3864600
.001209190
828
685584
567663552
28.7749891
9.3902419
.001207729
829
687241
569722789
28.7923601
9.3940206
.001206278
830
688900
571787000
28.8097206
9.8977964
.001204819
831
690561
573856191
28.8270706
9.4015691
.001208869
832
692221
575930368
28.8444102
9.4053887
.001201928
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
.001191896
840
705600
592704000
28.9827535
9.4353880
.001190476
841
70?281
594823321
29.0000000
9.4391307
.001189061
842
708964
596947688
29.0172363
9.4428704
.001187648
843
710649
599077107
29.0344023
9.4466072
.001186240
844
712336
601211584
29.0516781
9.4503410
.001184834
845
714025
603351125
29.0088837
9.4540719
.001183432
846
715716
605495736
29.0800791
9.45?r999
.001182083
847
717409
607045423
29.1032044
9.4615249
.001180638
848
719104
609800192
29.1204390
9.4052470
.001179245
819
720801
611960049
29.1376040
9.4689661
.001177856
a50
722500
614125000
29.1547595
9.4726824
.001176471
851
724201
616295051
29.1719043
9.4763957
.001175088
852
725904
618470208
29.1890390
9.4801061
.001178P?09
853
727609
620650477
29.2001037
9.4838136
.001172888
854
729316
622835804
29.2232784
9.4875182
.001170960
855
731025
625020375
29.2403830
9.4912200
.001169691
856
732736
627222016
29.2574777
9.4949188
.001168224
857
734449
629422793
29.2745623
9.4986147
.001166861
858
736104
631628712
29.2910370
9.5023078
.001166501
859
737881
633839779
29.3087018
9.5059980
.001164144
860
739600
636056000
29.3257566
9.5096854
.001162791
861
741321
638277381
29.3428015
9.5133699
.001161440
862
743044
6405a3928
29.a5983(»
9.5170515
.001160093
863
744769
6427:^5047
29.3768010
9.5207303
.001158749
864
746496
644972544
29.3938709
9.5244063
.001157407
865
748225
647214625
29.410882;i
9.5280794
.001156069
866
749956
649461896
29.4278779
9.5317497
.001164784
867 J
808 /
751689
753424
m\n4sm
29.44486:37
9.5:^172
.001158408
653972032
29.4618aV!r[
i %.5a9(»ia
.001152074
U4
CUBE ROOTS, AND RECIPROCALS.
No.
Squares.
Cubes.
Square
Roots.
Cube Roots.
Reciprocals.
8C9
755161
656234909
29.4788059
9.5427437
.001150748
870
756900
658503000
29.4957624
9.5464027
.001149425
871
758641
060776311
29.5127091
9.5500589
.001148106
872
760384
663054848
29.5296461
9.5537123
.001146789
873
762129
665338617
29.5465734
9.5573630
.00U45475
874
763876
667627624
29.5634910
9.5610106
.001144165
875
765625
669921875
29.5803989
9.5646559
.001142857
876
767376
672221378
29.5972972
9.5682982
.001141563
877
769129
674526138
29.6141858
9.5719377
.001140251
878
770884
6768ii6152
29.6310648
9.5755745
.001138952
879
772641
679151439
29.6479342
9.5792065
.001137656
880
774400
681472000
29.6M7939
9.5828397
.001136364
881
776161
683797841
29.6816442
9.5864682
.001135074
882
77/924
686128968
29.6984848
9.5900939
.001133787
883
779689
68&465387
29.7153159
9.5937169
.001132503
884
781456
690807104
29.7321375
9.5973378
.001131222
885
783225
693154125
29.7489496
9.6009548
.001129944
886
784996
695506456
29.7657521
9.6045696
.001128668
887
786769
697864103
29.7825452
9.6081817
.001127896
888
788544
700227072
29.7993289
9.6117911
.001126126
889
790821
702595369
29.8161030
9.6153977
.001124859
890
792100
704969000
29.8328678
9.6190017
.001123596
891
793881
707347971
29.8496231
9.6226080
.001122334
892
795664
709732288
29.8663690
9.6262016
.001121076
893
797449
712121957
29.8831056
9.6297975
.001119821
891
799236
714516984
29.8998328
9.6333907
.001118568
895
801025
716917375
29.9165506
9.6369812
.001117818
896
802816
719323136
29.9332591
9.6405690
.001116071
897
804609
721734273
29.9499583
9.6441542
.001114827
898
806404
724150792
29.9666481
9.6477367
.001118586
899
808201
726572699
29.9633287
9.6513166
.001112347
900
810000
^9000000
80.0000000
9.6548938
.001111111
901
811801
731432701
30.0166620
9.6584684
.001109878
902
813604
733870808
30.0333148
9.6620403
.001108647
903
815409
786314327
30.0499584
9.6656096
.001107420
904
817216
738763264
30.0665928
9.6601762
.001106195
905
819025
741217625
30.0882179
9.6727403
.001104972
906
820836
743677416
30.0998339
9.6763017
.001108753
907
822649
746142643
30.1164407
9.6798604
.001102536
906
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
80.1827765
9.6940694
.001097695
912
831744
758650528
80.1993377
9.6976151
.001096491
913
83a569
761048497
80.2158899
9.7011583
.001095290
914
835396
763551944
80.2324329
9.7046989
.001094092
915
83^225
766060875
80.2489669
9.7082369
.001092896
916
a39056
768575296
80.2654919
9.7117723
.001091703
917
840689
771095213
80.2820079
9.7153051
.001090513 •
918
842724
773620632
80.2985148
9.7188854*
.001089325
919
844561
776151559
80.8150128
9.7223631
.001088139
920
846400
77B688000
30.3315018
9.7258888
.001086957
921
848241
781229961
80.3479818
9.7294109
.001085776
922
850084
783'i77448
80.3644529
9.7829309
.001084599
923
851929
786330467
80.8809151
9.7^64484
.001068423
924
853776
788889024
80.3973683
9.7899634
.001062251
925
855625
791453125
30.4138127
9.7434758
.001081061
926
857476
7!M(fc>2rr6
30.4302481
9.7469867
.001079914
927
R')9329
7965*17983
30.4466747
9.7504930
.001078749
928
801184
7n9i7.sr:>2
30.4630924
9.7539979
.001077586
929
863041
801765089
80.4795013
9.7575QCW
930
864900
804357000
aO. 495^14
^ 'i.';^v;«*sv
U5
TABLE X. — SQUARES, CUBES, SQUARE ROOTS,
No.
Squares.
Cubes.
Square
Koots.
Cube Roots.
Reciprocals
931
866761
806954491
30.5122926
9.7644974
.001074114
932
868624
809557568
30.5286750
0.7679922
.001072961
033
870489
812166237
30.5450487
9.7714645
.001071811
»ai
8^356
814780504
30.5614136
9.7749748
.001070664
035
874225
817400375
80.5777607
9.7784616
.001060619
936 -
876096
820025856
80.5941171
9.7819466
.001068376
037
877969
822656953
80.6104557
9.78M288
.001067286
938
879844
825293672
80.6267857
9.7889087
.001066096
S39
881721
827936019
80.6431069
0.7923861
.001064968
940
883600
830584000
30.6594194
9.7958611
.0010688»0
941
885481
833237621
30.6757233
9.7993386
.001062690
942
887364
835896888
30.6020185
9.8026036
.001061671
943
889249
838561807
30.7083051
9.6062711
.001060445
944
801136
841232384
J.0.7245830
9.8097362
.001060322
945
893025
843906625
30.7408523
9.6181969
.001068201
946
894916
846590536
80.7571130
0.8166591
.001057062
947
896809
849278123
80.7733651
0.6201169
.001066066
948
896704
851971302
30.7896086
9.6236723
.001064862
949
900601
854670349
80.8056436
9.8270262
.001063741
950
902500
857375000
80.8220700
9.8804757
.001062882
951
904401
860085351
80.8362679
9.8839288
.001061685
953
906304
862801406
80.8544972
9.8373695
.001060420
953
908209
865623177
30.8706981
9.6406127
.001049818
954
910116
868250664
80.6868904
9.6442586
.001048218
955
012025
870088875
80.0030743
9.6476920
.001047120
956
913936
873722816
80.0102407
9.6511280
.001046026
957
915849
876«>7403
30.0354166
9.8545617
.001044982
958
917764
879217012
30.0515751
9.6570029
.001048841
959
910681
881974079
30.0677251
9.6614218
.001042768
060
021600
884736000
30.0838666
0.8648488
.001041607
961
923521
887503681
31.0000000
0.8662724
.001040688
962
925444
800277128
31.0161248
9.6716041
.001089601
063
927369
803056347
31.0322413
0.8751135
.001068422
964
'929296
605841344
31.0483404
0.8785805
.001087344
965
931225
808632125
31.0644491
9.6619451
.001086269
066
933156
001428606
31.0605405
9.6868674
.001085107
067
935089
0042:^1063
31.0066236
9.8887673
.001084126
068
937024
007030232
31.1126064
9.6921749
.001088058
069
938961
000853209
81.1267648
9.6965801
.001081902
970
940900
912678000
31.1448280
0.8989880
.001080928
971
942841
915406611
31.1606729
0.0028836
.001029666
972
944784
918330048
31.1760145
0.9a57«17
.001028807
073
046729
021167317
31.1029479
0.0001776
.001027749
074
048676
024010424
81.2089731
0.0125712
.001026694
075
050625
026850375
31.2249900
0.0150624
.001025641
076
052576
020714176
31.2409987
0.0108518
.001024690
077
054529
032574833
31.2569992
0.0227870
.001028641
078
956484
035441352
31.2729915
0.0261222
.001022495
. 079
958441
038313739
81.2889757
0.0296042
.001021450
080
960400
941192000
81.3049517
9.9328889
.001020406.
081
962361
944076141
31.3209195 .
9.9362618
.001019868
082
064324
046066168
81.3366792
9.9396363
.001018880
083
066289
040862087
81.8528308
9.9430092
.001017294
084
968256
052763004
31.3687743
9.9463797
.001016260
985
970225
055671625
31.3847097
9.9497479
.001016226
986
05^196
058585256
31.4006369
9.9531138
.001014199
987
074169
061504803
31.4165561
9.9564775
.001018171
988
976144
064430272
31.4324673
0.9698889
.001012146
989
978121
067361669
31.4463704
9.9681981
.001011182
fffft? /
901
992 1
0fiOJOO
970200000
31.4642654
9.9666649
.001010101
9H2081
97^242271
81.4801525
9.0699095
.001009068
984064 1
976191488
81.4960^15
^ .^»V»»M6
146
CUBE ROOTS, AKD RECIPROCALS.
No.
Squares.
Cubes.
Square
Roots.
Cube Roots.
Reciprocals.
993
986049
979146657
81.5119025
9.9766120
.001007049
/
994
988036
982107784
31.5277655
9.9799599
.001006036
995
990025
985074875
31.5436206
9.9633055
.001005025
996
992016
988047936
31.5594677
9.9866488
.001004016
997
994009
991026973
81.5753068
9.9699900
.001003009
998
996004
994011992
31.5911380
9.9933289
.001002004
999
998001
997002999
81 .6069613
9.9966656
.001001001
1000
1000000
1000000000
31.6227766
10.0000000
.001000000
1001
1002001
1003003001
81.6385840
10.0038322
.0009990010
1003
1004004
1006012008
81. 65438:^6
10.0066622
.0009980040
1003
1006009
1009027027
81.6701752
10.0099699
.0009970090
1004
1006016
1012J48064
81.6859590
10.0138155
.0009960150
1005
1010025
1015075125
81.7017349
10.0166869
.0009950249
1006
1012038
1018106216
31.n75030
10.0199601
.0009940358
1007
1014049
1(^1147343
81.7332633
10.0282791
.0009980187
1008
1016064
1024192512 ,
81.7490157
10.0266058
.0009920686
1009
1018081
1027243729
81.7647603
10.0299104
.0009910606
1010
1020100
1030301C00
31.78049?2
10.0332228
.0009900990
1011
1022121
1033364331
31.7962262
10.0865330
.0009691107
1012
1(K4144
1036433?28
31.811W74
10.0896410
.0009881428
1013
1026169
1039509197
81.8276609
10.0481469
.00098n668
1014
1028196
1042590744
31.6433666
10.0464506
.0009661933
1015
1030225
1045678:^75
81.8590646
10.0497521
.0009662217
1016
1032256
1048772096
81.8747549
10.0680514
.0009842520
1017
ia34280
1051871913
31.8904374
10.0563485
.0009632842
1018
10:^6324
1054977*32
81.9061123
10.0596435
.0009828183
1019
1038361
1058089859
81.9217794
10.0629864
.0009818548
10;J0
1040400
1061208000
81.9374388
10.0662271
.0009608922
1021
1042441
1064332261
81.9530906
10.0695156
.0009794819
1022
1044484
1067462648
81.9687347
10.0?28020
.0009784786
1023
1046529
1070599167
81.9843712
10.0760863
.0009775171
1024
1048576
1073741824
82.(1000000
10.0798684
.0009786625
1025
1050625
10r6890625
32.0156212
10.06264S4
.0009756006
1026
1052676
1080045576
32.0312848
10.0859262
.0009746580
1027
1054729
1083206683
82.0468407
10.0692019
.0009787096
1028
1056784
1086373952
82.0624391
10.0924755
.0009727826
1029
1058841
10S9547389
82.0780296
10.0957469
.0009718173
1030
1060900
1092727000
32.0936131
10.0990163
.0009706788
1031
1063961
1095912791
32.1091887
10.1022835
.0009699321
1032
1065024
1099104768
82.1247568
10.1056487
.00 9689922
1033
1067089
1102302937
82.1403173
10.1088117
.0009680542
1034
1069156
11(6507304
82.1556704
10.1120726
.0009671180
1035
1071225
1108717875
82.1714159
10.1153314
.0009661686
1036
1073-296
1111934656
32.1869539
10.11*5862
.0009652510
1037
1075:369
1115157653
32.2024844
10.1218428
.0009643202
1038
1077444
1118386872
82.2180074
10.1250953
.0009683911
ia39
1079.521
1121622319
82.2335229
10 1283457
.0009624680
1040
1081000
1124864000
32.2490310
10.1815941
.0009615386
1041
1083681
1128111921
32.2645316
10.1348408
.0009606148
1042
1085764
1131366088
82.2800248
10.1380645
.0009596929
1043
1087849
lia4()26507
32.2955105
10.1413266
.0009587788
1044
1089936
1137803184
32.3100888
10.1445667
.0009578544
1045
1092025
1141106125
32.3264598
10.1478047
.0009569378
1046
1094116
1144445336
32.3419233
10.1510406
.0009560229
1047
1096209
1147730823
32.a573794
10.1542744
.0009551098
1048
1098304
1151022592
32.3728'«1
10.1575002
.0009541986
1049
1100401
1154320649
32.3882695
10.1607359
1050
1102500
1157625000
32.4037035
10.1639636
.0009623810
1051
1104601
1160935651
82.4191301
10.1671893
.0009514748
1052
1106704
1164252606
32.4345495
10.1704129
.0009505703
1053
1106809
1167575877
32.4499615
10.1786844
^ .<M»Ma«S»i \
1054
1110916
1170905464
32.46536«ISi
,■ \^.Yi^«aft
\ 5W5W«WRfe ^
14T
TABLE XI. — I.OnAIHTIIMS OF NDMBBK8.
No. 110 L. Ml.]
WlSftl 17S7
06069S ims
^
PBOPOItTIOH.lL PaBTS.
I asa j
/ asr .
71:
TABLE XI. — LOGARITHMS OF NUMBERS.
No.
130 L. 079.]
[No. 184 L. 180.
N.
1
2
8
4
6
6
7
8
9
Diff.
079181
^ 9543
9904
120
0266
3861
7426
0626
4219
7781
0987
4576
8136
1347
4934
8190
1707
5291
8845
2067 2126
360
1
2
8
088785
6360
9905
3144
6716
3503
7071
5647
9198
6004
9552
357
355
0358
3773
7257
0611
4133
7R0d
0963
4471
1315
4820
8398
1667
5169
8644
3018
5518
8990
2370
6866
9335
2721
6215
9681
8071
6563
852
4
5
093432
6910
849
1 lAr» t wx
0036
3463
6871
346
343
341
6
7
8
100371
3801
/ 7210
0715
4146
7549
1059
4487
7888
1403
4838
8227
1747
5169
8565
2091
, 5510
' 8903
2434
5851
9241
2777
6191
9579
8119
6531
9916
0253
3609
6910
388
885
338
9
130
1
110590
3943
7271
0926
4277
7603
1263
4611
7934
1599
4944
8265
1934
5378
•8595
2270
6611
8926
1
2605
5943
9256
2940
6276
9586
8375
6608
0915
0215
3525
6781
380
338
825
2
3
4
120574
3862
7105
IS
0^
4178
7439
1331
4504
7753
1560
4830
8076
1888
5156
8399
' 2316
6481
8733
2544
6806
9045
2871
6131
9368
8198
6156
9690
0012
328
Proportional Parts.
Diflf.
355
354
353
352
351
350
349
848
317
346
345
344
343
312
341
340
839
338
337
336
335
334
333
333
331
330
839
338
337
336
325
321
323
322
1
2
8
4
6
35.6
71.0
106.5
142.0
177.5
35.4
70.8
106.2
141.6
ITT.O
35.3
70.6
105.9
141.2
176.5
35.2
70.4
105.6
140.8
178.0
35.1
70.3
105.8
140.4
175.5
35.0
70.0
105.0
140.0
175.0
34.9
69.8
101.7
139.6
174.5
34.8
69.6
104.4
139.2
174.0
34.7
69.4
101.1
138.8
173.5
34.6
69.3
108.8
138.4
173.0
34.5
69.0
108.5
138.0
172.5
34.4
68.8
108.2
137.6
172.0
34.3
68.6
102.9
137.2
171.5
34.2
68.4
102.6
136.8
171.0
34.1
68.2
102.3
136.4
170.5
34.0
68.0
102.0
136.0
170.0
33.9
67.8
101.7
135.6
169.5
33.8
67.6
101.4
135.2
169.0
33.7
67.4
101.1
131.8
168.5
33.6
67.2
100.8
134.4
168.0
83.5
67.0
100.5
134.0
167.5
33.4
66.8
100.2
133.6
167.0
33.3
66.6
99.9
133.2
166.5
33.2
66.4
99.6
132.8
166.0
33.1
66.2
99.3
132.4
165.5
33.0
66.0
99.0
132.0
165.0
32.9
65.8
98.7
131.6
164.5
32.8
65.6
98.4
131.2
164.0
33.7
65.4
98.1
130.8
163.5
33.6
65.2
97.8
130.4
163.0
33.5
65.0
97.5
130.0
163.6
33.4
64.8
97.2
129.6
163.0
33.3
64.6
96.9
129.2
161.5
33.3
64.4
96.6
128.8
161.0 [
6
.0
.4
313.0
313.4
211.8
211.2
210.6
210,
209
208.8
206.2
207.6
2or.o
206.4
205.8
205.2
204.6
204.0
203.4
202.8
202.2
201.6
201.0
200.4
199.8
199.2
198.6
198.0
197.4
196.8
196.2
195.6
195.0
194.4
193.8
218.5
247.8
247.1
216.4
245.7
245.0
244.3
243.6
242.9
242.3
241.5
240.8
240.1
239.4
238.7
238.0
237.3
236.6
235.9
235.2
234.5
233.8
233.1
8
.4
.7
.0
232.
231.
231
230.3
239.6
228.9
227.5
236.8
336
384.0
283.2
282.4
281.6
280.8
280.0
279.2
278.4
277.6
276.8
276.0
275.2
274.4
273.6
272.8
272.0
271.2
270.4
269.6
268.8
268.0
267.2
266.4
265.6
264.8
264.0
263.2
262.4
261.6
260.8
260.0
259.2
9
319.5
318.6
317.7
316.8
315.9
315.0
314.1
318.2
312.8
811.4
810.5
309.6
808.7
807.8
806.9
806.0
805.1
804.2
803.3
803.4
801.5
800.6
299.7
296.8
297.9
297.0
296.1
295.2
294.8
293.4
292.5
291.6
228.1 I 2B&AV'»«5i!V\
151
TABLE XI. — LOGARITHMS OF NUMBERS.
No. 135 L. 130.]
[No. 149 L. 175.
N.
1
2
8
4
•
6
7
8
9
Diff.
6
7
8
i3a3.'«
a539
6721
9879
0655
3858
7037
0977
4177
7:3m
1298
4496
7671
1619
4^14
7987
1939
5i:«
83U3
2260
fr45l
8618
2580
5769
8934
2900
6086
9249
3219
640!)
9564
321
318
316
0194
3327
64;«
9527
a-ios
3639
6748
9835
0H22
3951
70)8
1136
4263
7367
1450
4574
7676
17m
4885
7985
2076
5196
82&t
2:J89
5507
8603
2702
5818
8911
314
311
309
9
140
1
143015
6128
9219
0142
3205
6246
9266
0449
3510
6549
9567
0756
381 5
6853
9868
1063
4120
7154
1370
4424
7457
1676
4728
7759
19^
5032
8061
307
805
303
-2
3
4
152288
5336
8362
2594
5640
8664
fum
5943
8965
0168
3161
6131
9086
0469
3460
6430
9380
0769
3758
6726
9674
1068
4055
7022
9968
301
299
297
295
5
6
7
161368
4353
7317
1667
4650
7613
1967
4947
7908
2266
5^14
8203
2564
5541
&497
2863
5a38
8792
8
9
170262
3186
0555
8478
0848
8769
1141
4060
1434
4351
1726
4641
2019
4932
2311
5222
^iS03
5512
28%
5808
293
291
Proportionaij Parts.
DiflP.
821
320
319
318
317
316
315
314
313
312
301
300
299
298
297
296
295
294
293
292
30.5
64.2
64.0
63.8
63.6
63.4
63.2
6:^.0
62.8
62.6
62.4
02.2
62.0
61.8
61
61
61
61
60
60.6
60.4
60.2
60.0
59.8
59.6
59.4
59.2
59.0
58.8
58.6
58.4
58.2
58.0
57.8
57.6
rt7.4
57.2
8
96.3
96.0
95.7
95.4
95.1
94.8
94.
94.
.5
.2
9:3 9
93.6
93.3
93.0
9:^.7
92.4
92.1
91.8
91.5
91.2
90.9
90.6
90.3
90.0
89.7
89.4
89.1
88.8
88.5
88.2
87.9
87.6
87.3
87.0
86.7
86.4
86.1
85.8
4
6
6
7
8
128.4
160.5
192.6
234.7
256.8
128.0
160.0
192.0
221.0
256.0
127.6
159.5
191.4
223.3
255.2
127.2
159.0
190.8
222.6
25'!. 4
126.8
158.5
190.2
221.9
253.6
126.4
158.0
189.6
221.2
252.8
126.0
157.5
189.0
220.5
252.0
125.6
157.0
188.4
219.8
251.2
125.2
156.5
187.8
219.1
250.4
VZ\.^
156.0
187.2
218.4
219.6
124.4
155.5
186.6
217.7
248.8
124.0
155.0
186.0
217.0
248.0
123.6
154.5
ia5.4
216.3
247.2
123.2
154.0
184.8
215.C
216.4
l.':2.8
153.5
184.2
214.9
2466
122.4
153.0
1&3.6
214.2
244.8
122.0
152.5
183.0
213.5
244.0
121.6
152.0
182.4
212.8
243.2
121.2
151.5
181.8
212.1
242.4
120.8
151.0
181.2
211.4
241.6
120.4
150.5
180.6
210.7
240.8
120.0
150.0
180.0
210.0
210.0
119.6
149.6
179.4
209.3
239.2
119.2
149.0
178.8
208.6
238.4
118.8
148.5
178.2
207.9
237.6
118.4
148.0
177.6
207.2
236.8
118.0
147.5
177.0
206.5
236.0
117.6
147.0
176.4
205.8
235.2
117.2
146.5
175.8
205.1
234.4
116.8
146.0
175.2
204.4
233.6
116.4
145.5
174.6
203.7
232.8
116.0
145.0
174.0
203.0
232.0
115.6
144.5
173.4
202.3
231.2
115.2
144.0
172.8
201.6
2:)0.4
114.8
143.5
\ 14a .0
172.2
200.9
229.6
114.4
\ «».^
I %».8
9
288.9
288.0
287.1
286.2
285.3
284.4
283.5
282.6
281.7
280.8
279.9
279.0
278.1
277.2
276.8
275.4
274.5
273.6
272.7
271.8
270.9
270.0
269.1
268.2
267.8
266.4
265.5
264.6
283.7
262.8
261.9
261.0
260.1
259.2
258.8
8OT.4
152
TABLE XI.— LOOAMTHMS OF NUMBERS.
"i sis
liif
^ I "3? I BOW
ill
SI™
11
if
If
a s
B s
m
r
11
"Ira"
TABLE XI. — LOGARITHI^S OF NUMBERS.
No. 170 L. 230.]
[No. 189 L. 278.
N.
170
1
2
8
4
5
6
7
8
9
180
1
3
3
4
5
6
7
8
9
230449 : 0704
2996
5528
8016
32-)0
5781
8297
240549
3038
5513
7973
0799
3286
5759
8219
250420
2853
5273
7679
8
8
4
09i>0
1215
1470
3,-MJ4
3757
4011
(50:«
6285
65.37
8548
8799
9019 '■
1048
1297
1546
3534
3782
40.*iO
0006
6252
6499
i^64
8709
8954
6
1724
42(U
6789
1979
4517
7041
8
9
Diff.
9299 i 9550
22;m
4?70
7292
9800
1795
4277
6745
9198
06(;4
3096
5514
7918
0908
33;i8
5755
8158
1151
3580
5996
8398
vm
3822
6237
86;J7
260071
2451
4818
7172
9513
0310
2688
5a>i
7406
9746
0548
2925
5290
7641
9980
0787
3162
5525
7875
0213
25:38
4850
7151
271842
4158
6462
2074
4:i89
6t)92
2:306
4620
6921
1025
3399
5761
8110
0446
2770
6081
7:380
1638
4064
6477
8877
2044
4525
6991
9443
1263
3636
5996
8344
0679
3001
5311
7609
1881
4306
6718
9116
2293
4772
7237
9687
2125
4548
6958
9355
2488
5023
7544
0050
2541
5019
7482
9932
1501
3873
6232
8578
0912
82:33
5542
78:iS
1739
4109
6467
8812
1144
3464
5772
8067
2368
4790
7198
9594
2742 I
5276 !
7795
0300
2790
5266
772a
1976
4346
6702
9046
1377
3696
6002
8296
0176
2610
5031
7439
9833
2214
4582
6937
9279
1609
8927
6232
a525
255
253
252
250
249
248
246
245
243
242
241
239
238
237
235
234
233
232
230
229
Proportional Parts.
DiflP.
1
2
8
4
5
6
7
8
9
255
25.5
51.0
76.5
102.0
127.5,
153.0
178.5
204.0
229.6
254
25.4
50.8
76.2
101.6
127.0
152.4
177.8
203.2
228.6
2V3
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
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
12:3.0
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
198.6
217.8
241
^.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
2:38
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
2:36
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
161.5
188.0
211.5
234
23.4
46.8
70.2
93.6
117.0
140.4
163.8
187.2
210.6
2:3:3
2:3.3
46.6
69.9
93.2
116.5
139.8
163.1
186.4
209.7
2:32
2:3.2
40.4
09.6
92.8
116.0
139.2
162.4
185.6
206.8
2:31
23.1
46.2
69.3
92.4
115.5
i:38.6
161.7
184.8
207.9
2:30
23.0
46.0
69.0
92.0
115.0
1:38.0
161.0
184.0
207.0
229
22.9
45.8
68.7
91.6
114.5
137.4
160.3
183.2
206.1
228
22.8
45.6
68.4
91.2
114.0
136.8
159.6
182.4
205.2
227 J
22.7
2n. 1
45.4
68.1
90.8
113.5
136.2
158.9
181.6
204.8
45.2
67.8
90.4
113.0
135.6
158 2
180.8
203.4
154
TABLE XI, — LOGARITHMS OF NCMBEHB.
ritoroBTioNAi. Fakts,
TABLE XI. — LOGARITHMS OF NUMBERS.
No. 215 L. 332.]
[No. 1339 L. 880.
N.
1
9
8
4
6
6
7
8
9
Diff,
215
6
7
8
832438
4454
6460
8456
2640
4655
6660
8656
2842
4856
6860
6855
3044
5067
7060
9U54
3246
5257
7260
9253
a447
5458
7459
9451
8649
5658
7659
9650
3850
5859
7858
9849
4051
6059
8058
4253
6260
8257
202
201
200
0047
2028
8999
5962
791&
9860
0246
2225
4196
6157
8110
199
198
197
196
195
9
220
1
2
A
340444
2423
4392
6353
8305
0642
S620
4589
6549
8500
0841
2817
4785
6744
8694
1039
8014
4981
6939
8889
1237
8212
5178
7135 '
9083 '
1435
8409
5:^74
7330
9278
1632
3606
5570
7525
9472
1830
8802
6766
7?20
9666
0054
1969
8916
5634
7744
9646
194
193
193
192
191
190
4
5
6
7
8
9
850248
2183
4108
6026
7935
9835
0442
2375
4301
6217
8125
0636
2568
4493
6408
8316
0829
2761
4685
6599
8506
1023
2954
4876
6790
8696 i
1216
3147
5068
6981
8886
1410
8339
5260
7172
9076
1603
8532
5452
7^63
9266
1796
8724
5643
7554
9456
0025
1917
8800
5675
7542
9401
0215
2105
3988
5862
7729
9587
0404
2294
4176
6049
7915
9772
0593
2482
4363
6236
8101
9958
0783
26n
4551
6423
, 8287
0972
2859
4?39
6610
8473
1161
3048
4926
6796
8659
1350
8286
6113
6963
8845
1539
8424
6301
7169
OORO
189
188
188
187
186
230
1
2
3
4
861728
3612
5488
7356
9216
i 0143
1991
8831
5664
7488
9306
0328
2175
4015
6816
7670
9487
0518
2360
4198
6029
7852
9668
0698
2544
4882
6212
8034
9649
0683
2728
4565
6894
8216
185
184
184
183
182
6
6
7
8
9
871068
2912
4748
6577
8398
88
1253
8096
4932
6759
8580
1437
8280
5115
6942
8761
1622
8464
5298
7124
8943
1806
8647
5481
7306
9124
0080
181
Proportional Parts.
Diflf.
1
2
3
4
5
6
7
8
9
202
20.2
40.4
60.6
80.8
101.0
121.2
141.4
161.6
181.8
201
^.1
40.2
60.3
80.4
100.5
120.6
140.7
160.8
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
1T9.1
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
177.8
196
19 6
392
58.8
78.4
98
117.6
137.2
156.8
176.4
195
19.5
39.0
58.5
78.0
97.5
1170
136.5
156.0
175.6
194
19.4
388
58.2
77.6
97.0
116.4
135.8
156.2
174.6
193
19 3
38.6
67.9
77.2
96.5
115.8
135.1
164.4
173.7
192
19 2
38.4
57.6
76.8
96.0
115.2
184.4
158.6'
172.8
1>1
19 1
38.2
57 3
76.4
95.5
114.6
133.7
152.8
m.9
190
19
38.0
57
76.0
95.0
114
133.0
168.0
171.0
189
18.9
37 8
56.7
75.6
945
113.4
132«3
151.2
170.1
188
18 8
37.6
56.4
75.2
94.0
112.8
131.6
150.4
169.2
187
18 7
37 4
56 1
74.8
93.5
112.2
130.9
149.6
168.3
186
18.6
37 2
55.8
74 4
93.0
111.6
130.2
148.8
167.4
185
18 5
37
55 5
740
925
111
129.5
148.0
166.5
184
18 4
36 8 *
55 2
73.6
92.0
110.4
128.8
147.8
165.6
183
18 3
366
54.9
73 2
91 5
109 8
128.1
1464
164.7
182 ^
18 2
36.4
54 6
72 8
91
109 2
127.4
145.6
168.8
' 181
18 1
362
54 3
72 4
90.5
106 6
126.7
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TABLE SI. — LOGARITHMS OF NUMBERS.
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TABLE ir.—LOOABITHMS OP NUHBERS.
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TABLE XI. — L0GAKITHM8 OF NUUBERS.
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LOOABITHIES 01
N.
Loff.
N.
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N.
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J^
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SI ]
ssaaio
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xir
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SINB8,
ITS
J^
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TABLE
XII
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ITS
Ll
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lane.
y+'l
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4.4Sfi
5.314
i
1
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1
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6.463^ 13
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LOOARITHUIC
SINKS
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I,.,-.
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COSINES, TANGENTS, AND COTANGENTS.
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11.232583
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11.211446
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11.191283
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11.172008
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2
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1
11.155356
Tnno'
1 /
Sine.
D. r.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
a5
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
8.718800
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8.744536
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8.766675
768828
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8.787736
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.791828
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.797894
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.801892
.803876
.805852
8.807819
.809777
.811726
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.817522
.819436
.821343
.823240
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8.827011
.8288^4
.a30749
.832007
.8:«456
.a36297
.838130
.a39956
.W1774
8.843585
Cosine.
40.07
39.85
39.62
39.42
39.18
38.98
38.78
38.55
38.37
38.17
37.95
87.77
37.55
37.37
37.18
86.98
36.80
86.60
36.43
36.23
36.07
35.88
35.70
85.52
35.37
35.17
35.02
34.83
34.68
34.50
a4.35
84.18
34.02
83.85
83.70
33.55
33.38
33.25
33.07
32.93
32.78
82.63
82.48
32.35
32.20
32.05
31.90
31.78
31.62
31.50
81.35
31.22
31.08
30.97
30.82
30.68
30.55
30.43
30.30
80.18
Cosine.
D. r.
9.999404
.999398
.999391
.999384
.999378
.999371
.999364
.999357
.999350
.999343
.999336
9.999329
.999322
.999315
.999308
.999301
.999294
.999287
.999279
.999272
.999265
9.999257
.999250
.999242
.999235
.999227
.999220
.999212
.999205
.999197
.999189
0.999181
.999174
.999166
.999158
.999150
.999142
.999134
.999126
.999118
.999110
9.099102
.999004
.999086
.999077
.999069
.999061
.999053
.999044
.999036
.999027
9.099019
.999010
.999002
.998093
.998984
.908976
.908067
.0080r)8
.008050
0.008041
Sine.
.10
.12
.12
.10
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.13
.12
.12
.13
.12
.13
.12
.13
.12
.13
.12
.13
.13
.13
.12
.13
.13
.13
.13
.13
.13
.13
.13
.13
.13
.13
.15
.13
.13
.13
.15
.13
.15
.13
.15
.13
.15
.15
.13
.15
.15
.13
.15
D. r.
Tang.
D. r.
8.710306
.721806
.724204
.726588
.728050
.731317
.733663
.736006
.738317
.740626
.742022
8.745207
.747479
.749740
.751969
.7M227
.75W53
.758668
.760872
.763065
.765246
8.767417
.769578
.771727
.773866
.775995
.778114
.780-222
.782320
.784408
.786486
8.78a554
.790613
.792662
.794701
.796731
.798f52
.800763
.802705
.804758
.806742
8.808717
.810683
.812641
.814589
.816529
.818461
.820384
.822208
.834205
.820103
8.827002
.820874
.831748
.833613
.835471
.837321
.830103
.840008
.842825
8.844644
Cotang.
40.17
30.07
30.73
30.52
30.30
30.10
88.88
38.68
88.48
38.27
38.06
37.87
37.68
37.48
87.30
37.10
36.92
36.73
36.55
86.35
36.18
36.02
85.82
85.65
35.48
35.32
35.13
84.97
84.80
34.63
84.47
34.32
34.15
as. 96
33.83
83.68
&3.52
83.37
a3.22
as. 07
82.92
32. C"
82.63
32.47
82. as
82.20
82.05
31.90
31.78
31.63
31.48
81.37
31.23
31.08
30.97
30.83
30 70
30.58
30. 4'.
30.32
D. r.
98<
nsi
TABLE XII. — LOGARITHMIC SINES,
174*
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
60
51
52
53
54
55
56
57
58
59
60
Sine.
8.843585
.846387
.847183
.848971
.850751
.852525
.854291
.856049
.857801
.859546
.861283
8.863014
.864738
.866455
.868165
.869868
.871565
.873255
.874938
.876615
.878285
8.879949
.881607
.883258
.884903
.886542
.888174
.889801
.891421
.893036
.894643
8.896246
.897842
.899432
.901017
.902596
.904169
.9a5736
.907297
.908853
.910404
8.911949
.913488
.915022
.916550
.918073
.919591
.921103
.922610
.924112
.925609
8.927100
.928587
.930068
.931544
.933015
.9:34481
.a35942
.937398
.938850
8.940296
D. r.
30.03
29.93
29.80
29.67
29.57
29.43
29.30
29.20
29.08
28.95
28.85
28.73
28.62
28.50
28.38
28.28
28.17
28.05
27.95
27.^
27.73
27.63
27.52
27.42
27.32
27.20
27.12
27.00
26.90
26.80
26.72
26.60
26.50
26.42
26.32
26.22
26.12
26.02
25.93
25. »>
25.75
25.65
25.57
25.47
25.38
25.30
25.20
25.12
25. a3
24.95
24.85
24.78
24.68
24.60
21.52
24.43
24.35
24.27
24.20
24.10
Cosine. / D. 1'.
Cosine.
9.998941
.998932
.998923
.998914
.998905
.998896
.998887
.998878
.998869
.998860
.998851
9.996841
.998832
.998823
.998813
.998804
.998795
.998785
.998776
.998766
.998757
9.998747
.998738
.998728
.998718
.998708
.998699
.998689
.998679
.998669
.998659
9.998649
.998639
.998629
.998619
.998609
.998599
.998589
.998578
.998568
.998558
9.998548
.998537
.998527
.998516
.99a506
.998495
.998485
.998474
.998464
.998453
9 998442
.998431
.998421
.998410
.998399
.998388
.998377
.998366
.99Ki55
9.998344
Sine.
D. r.
15
15
15
15
15
15
15
15
15
15
17
15
15
17
15
15
17
15
17
15
17
15
17
17
17
15
17
17
17
17
17
17
17
17
17
17
17
18
17
17
17
18
17
18
17
18
17
18
17
18
18
18
17
18
18
18
18
18
18
18
Tang.
8.844644
.846455
.848260
.850057
.851846
.853628
.855403
.857171
.858932
.860686
.862433
8.864173
.865906
.867632
.869351
.871064
.872770
.874469
.876162
.877849
.879529
8.88120S
.882869
.884530
.886185
.887833
.889476
.891112
.892742
.894366
.895984
8.897596
.899203
.900803
.902398
.903987
.905570
.907147
.908719
.910285
.911846
8.913401
.914961
.916495
.918034
.919568
.921096
.922619
.924136
.925649
.927156
8.928658
.930155
.931647
.933134
.934616
.936093
.937565
.939a32
.940494
8.941952
D. r.
Cotang.
30.18
30.08
29.95
29.82
29.70
29.58
29.47
29.35
29.23
29.12
29.00
28.88
28.77
28.66
28.65
28.43
28.32
28.22
28.12
28.00
27.88
27.78
27.68
27.68
27.47
27.38
27.27
27.17
27.07
26.97
26.87
26.78
26.67
26.68
26.48
26.88
26.28
26.20
26.10
26.02
25.92
25.83
25.73
26.63
26.67
25.47
25.38
26.28
25.22
25.12
26.03
24.95
24.87
24.78
24.70
24.62
24.53
24.45
24.37
24.30
94*
I). 1'. li Cotang. [ D. 1'.
180
i:.:fi.')356
.153545
.151740
.149943
.148154
.146372
.144597
.142829
.141068
.189814
.187667
11.185827
.184004
.182368
.180649
.128986
.127280
.125681
.128888
.122151
.120471
11.118798 V
.117181 38
.116470 ! 37
.118815
.112167
.110624
.106888
.107268
.106684
.104016
11.102404
.100797
.099197
.097808
.006018
.004430
.092863
.091281
.089715
.068164
11.066599
.066049
.068606
.061966
.060488
.076904
.077881
.076664
.074851
.072644
11.071848
.069646
.066858
.066666
.066864
.068907
.062486
.060968
.069606
11.068048
60
69
58
67
56
66
64
63
58
51
50
49
48
47
46
45
44
48
48
41
40
86
85
34
33
82
81
80
Tang.
29
S8
27
S6
26
24
23
22
21
80
19
18
17
16
15
14
18
18
11
10
8
7
6
6
4.
i
8
1
^iW'
6^
TABLE XII. — LOGARITHMIC SINES,
173*
/
Sine.
9.019235
1
. 0204^55
2
.021632
3
.022825
4
.024016
5
.02520:^
6
.026:386
7
.027567
8
.028744
9
.029918
10
.031089
11
9.032257
12
.0S:U21
13
.034582
14
.035741
15
.036896
16
.038048
17
.039197
18
.040342
19
.041485
20
.042625
21
9.043762
22
.044vS95
23
.046026
24
.047154
25
.048279
26
.049400
27
.050519
28
.051635
29
.052749
30
.053859
31
9.054966
32
.056071
33
.057172
34
.058271
35
.059367
3<j
.060460
87
.061551
38
.0626:39
39
.063724
40
.064806
41
9.065885
42
.066962
43
.0680:36
44
.069107
45
.070176
4G
.071242
47
.072:U16
48
.073:366
49
.074424
50
.075480
51
9.07^533
52
.07758:3
5:3
.0786:31
54
.079676
55
.080719
50
.081759
57
.082797
58
.08:3832
59
.084864
60
9.085894
D. r.
Cosine.
20.00
9.95
9.88
9.85
9.78
9.72
9.68
9.62
9.57
9 52
9.47
9.40
9.35
9.32
9.25
9.20
9.15
9.08
9.05
9.00
8.95
8.88
8.85
8.80
8.75
8.68
8.65
8.60
8.57
8.50
8.45
8.42
8.35
8.:32
8.27
8.22
8.18
8.13
8.08
8.0:3
7.98
7.95
7.90
7.85
7.82
7.77
7.73
7.67
7.63
7.60
7.55
7.50
7.47
7.42
7.38
7.. 33
7.30
7.25
7.20
7.17
9.997614
.91J7601
.iwr588
.997574
.997561
.997547
.997534
.997520
.997507
.997493
.997480
9.997466
.997452
.i)97439
.997425
.997411
.997397
.997383
.997369
.997355
.997341
9.997327
.997313
.997299
.997285
.997271
.99^257
.997242
.997228
.99?214
.997199
9.907185
.997170
.997156
.997141
.997127
.997112
.997098
.997083
.9970()H
.99705;]
9.9970.39
.997024
.997009
.99(5994
.JK)6979
.99(5964
.1KK5949
.9'M59:34
.99(5919
.996904
9.996889
.99(5874
.99(58.58
.9%843
.996828
.9%812
.996797
.996782
.9967(56
9.91K5751
/ ' I Cosine. I I), v. / Sine.
D. r.
.22
.22
.2:3
.22
.23
!23
.22
.23
.23
.23
.22
.2:3
.23
.2.3
.23
.23
.23
.23
.2:3
.23
.23
.23
.2:3
.23
.25
. ^^
.25
.23
.25
.23
.25
.23
*"»
OK
.25
.25
.25
.25
or;
or:
.25
.27
, «••!
. «•!
07
. -^>
. ^.j
.27
.25
Tang.
D. r.
CotADg.
9.021620
.022834
.024044
.025251
.026455
.027655
.028852
.030046
.031237
.0:32425
.033609
9.034791
.0:i5969
.0:37144
.0:38:316
.039485
.040651
.041813
.042973
.044130
.045284
9.046434
.047582
.048727
.049869
.051008
.052144
.053277
.054407
.055535
.056659
9.057781
.(X58900
.060016
.061130
.062240
.063348
.06445:3
.065556
.066655
.067752
9.068846
.069^38
.071027
.072113
.073197
.074278
.075356
.076432
.077505
.078576
9.079r>44
.080710
.081773
.082833
.083891
.084947
.086000
.087a")0
.088098
9.089144
20.23
20.17
20.12
20.07
20.00
9.95
9.90
9.85
9.80
9.73
9.70
9.63
9.58
9.53
9.48
9.43
9.37
9.3:3
9.28
9.23
9.17
19.13
19.08
I9.a3
18.98
18.93
18.88
18 83
.80
73
.70
8.65
8.60
8.57
8.50
8.47
8.42
8.38
8.32
8.28
8.25
20
15
10
8.07
8.02
7.97
7.93
7.88
7.a5
7.80
7.77
7.72
7.67
7.63
7.60
7.55
7.50
7.47
7.43
10.978380
.977166
.075956
.974749
.973545
.972345
.971148
.969954
.968763
.967575
.966891
10.965209
.964031
.962856
.961684
.960515
.959349
.958187
.957027
.955870
.954716
10.953566
.952418
.951273
.950131
.948992
.947856
.946723
.945593
944465
.943341
10.942219
.941100
.939984
.938870
.937760
.936652
.935547
.933345
.932248
I0.a31154
.930062
.928973
.927887
.926803
.925722
.924644
.923568
.922495
.921424
10.920356
.919290
.918227
.917167
.916109
.915053
.914000
.912950
.911902
10.910656
60
59
58
57
56
55
54
53
52
51
60
49
48
47
46
45
44
43
42
41
40
S9
88
37
86
85
34
83
32
81
80
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
8
2
1
I). V. V CoUing. \ D. V. [ Tang.
182
^^
7'
COSINES,
TAfiaENTS, AND COTANQBNTS.
172'
~
s,„.
„..-.
«...
D r
T«DB.
D.1-.
Co,a...
0O85BW
S.9967BI
'2;
9.0B9144
IT. 38
II
17)3
10 910858
80
1
17,00
.OIWT:ia
,UW1BT
59
2
OHtOT)
;oBaM»
S
S?
I
.mm
liNwiss
.0»33(W
Be
'.(mim
:SIM633
£4
.09303r
'0tfU»41
:27
!o963ua
.903609
88
.094047
.9B6«fi!5
.«K4ai
62
a
10
:K
.96WilU
:!S^
1
I
:099065
IOCS
11
9,996578
!27
9-l(KMBT
II
ibIm
10.899513
i
i
,10006a
!m7
48
19
'. oa048
.m6H
U0648B
;i04&ia
.10B550
:»fl*M58
46
1
. 0303/
43
, 04025
4a
'«■!
.9H440
,WM33
iiuBsaa
:8B0141
w
1
fl.iosirta
II
ioSMOD
:w
"IS
18.60
Ifl.'lO
10.889444
38
:i09B01
.306384
.BHBM
: 113133
!««467
s
SO
.110878
.9S63S1
35
34
3S
: 113774
: 11747!;
'seasm
SI
ao
,B«M«
SI
.8B0B71
3i
9ii«»a
1:3 ft.!
15 Jr
IS.TO
9.m6B.'a
,28
1
e.is!M04
.1!I13TT
16 aa
"';S
33
18. (W
27
.996902
W
se
^Ss
ii
iiawMii
«74l'.l
28
a?
i'i
23
,28
sa
isius
15. «3
15 .%5
1BB3
21
«
: ittisr
mioo
JSWIBT
;870»13
20
O.iHMOBS
9 13DM1
II
15.63
15 62
10 mm
1
.1370110
.8U9UU9
43
.187903
:b«wb
1
.KttHOSfi
Boeim
:i3a893
.W7107
!l:NHM
ia3H39
1
46
130781
inoauBS
«
131708
ISM
.mm
48
laiiao
.1SB66:
8633,13
49
60
.134470
Si
.^'^
30
137605
.S6IOB6
I
BB
9 1S5387
IS 27
11
iSOSBll
996894
sa
9 133478
14O109
15 S9
15 53
ii
10 880524
.8511591
,30
.868800
S4
.13)51»(
limaeo
.867731
139037
13VM4
flISWS
14»I9((
146W4
:8.M95a
68
.141TM
liiOi
i)!wrH8
8640S4
GO
3 14^
.30
9,147803
IS. 30
10(e>^197
~
Coetne. ' 1>, V. \ Sine,
D.r.
Cul«ns.
D.r.
Tans,
8*
TABLE
XII.—
BIHIS,
171
. SU.e.
i..,-.l|„«...
D. 1'.
Tang.
D. r.
Cotang.
to
e-i«^
14, B7
14.93
14. eo
14,M8
14 '73
.BttirBS
8.HT803
11
10.85S197
.148718
.asiasa
,149833
'.M(M3
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B
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19.19
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COSINES TANGENTS AND COTANGENTS
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Sine.
I,.,-.! c™,,-..
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Tang.
D. r.
C0t«Dg. 1
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81787B
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13'
COSINES, TANGENTS, AND COTANGENTS.
166'
Sine.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
IG
17
18
19
20
21
22
2:3
24
25
20
27
28
29
30
31
32
3;^
U
3.-)
36
37
38 i
39 I
40
41
42
43
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9.352088
.352635
.353181
.;i53726
.354271
.354815
.355358
.355901
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9
9
358064
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360215
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,362889
36.^422
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.365016
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.366604
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9.368711
.3()9236
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9.373933
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9 379089
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.380113
.3806^4
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.381643
.:W2152
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.;383168
9.383675
D. r.
Cosine.
9.12
9.10
9.08
9.08
9.07
9.05
9.05
9.0:i
9.02
9.00
9.00
8.98
8.97
8.95
8.95
8.95
8.92
8.92
8.90
8.88
8.88
8.87
8.85
8.K5
8.82
8.82
8.78
8.80
8
. < t
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8.75
8.75
8.72
8.72
8.70
8.70
8.68
8.68
8.67
8.65
8.65
8.(53
8.62
8.60
8.60
8.60
8.57
8.57
8.57
8.53
8.53
8..'>;^
8.52
8.50
8.48
8.48
8.48
8.45
8.45
Cosine.
D. r
9.988724
.988695
.988666
.988636
.988607
.988578
.988548
.988519
.988489
.988460
.988430
9.988401
.988371
.988342
.988312
.988282
.988252
.988223
.988193
.988163
.988133
9.988103
.988073
.988043
.988013
.98798:j
.987953
.987922
.987892
.987862
.987832
9.987801
.987771
.987740
.987710
.987679
.987649
.987618
.J)87588
.987557
.987526
9.987496
.987465
.987434
.987403
.987372
.987341
.987310
.987279
.987248
.987217
9.987186
.987155
.987124
.987092
.987061
.987ft30
.986998
.986967
.986936
9.986904
D. 1*.
Sine.
.48
.48
.50
.48
.48
.50
.48
.50
.48
.50
.48
.50
.48
.50
.50
.50
.48
.50
.50
.50
.50
.50
.50
.50
.50
.50
.50
.50
.50
.52
.50
.52
.50
.52
.52
.50
.52
.52
.50
.52
.52
.52
.52
.52
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.52
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.52
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.52
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Tang.
9.363364
.363940
.364515
.365090
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.366237
.366810
.367382
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.368524
.369094
9.369663
.370232
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.371367
.371933
.372499
.373064
.373629
.374193
.374756
9.375319
.375881
.376442
.377003
.377563
.378122
.378681
.379239
.379797
.380354
9.380910
.381466
.382020
.382575
.383129
.383682
.384234
.384786
.385:337
.385888
9.386438
.386987
.387536
.3880^4
.388631
.389178
.389724
.390270
.390815
.391360
9.391903
.392447
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.393,531
.394073
.394614
.395154
.395694
.396233
9.396771
109*
D. r. ! Cotang.
D. r.
9.60
9.58
9.58
9.57
9.55
9.55
9.53
9.52
9.52
9.50
9.48
9.48
9.45
9.47
9.43
9.43
9.42
9.42
9.40
9.38
9.38
9.37
9.35
9.85
9.3:3
9.32
9.32
9.30
9.30
9.28
9.27
9.27
9.23
9.25
9.23
9.22
9.20
9.20
9.18
9.18
9.17
9.15
9.15
9.13
9.12
9.12
9.10
9.10
9.08
9.08
9.05
9.07
9.03
9.a3
9.03
9.02
9.00
9.00
8.98
8.97
D. r.
Cotang.
/
10.636636
60
.636060
59
.635485
58
.634910
57
.634336
66
.633763
65
.633190
64
.632618
63
.632047
62
.631476
61
.630906
60
10.630837
49
.629768
48
.629201
47
.628633
46
.628067
45
.627601
44
.6269:36
43
.626371
42
.625807
41
.625244
40
10.624681
S9
.624119
38
.623558
37
.622997
36
.622437
35
.621878
84
.621319
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82
.620203
31
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10.619090
29
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.617980
27
.617425
26
.616871
25
.616318
24
.615766
23
.615214
22
.614663
21
.614112
20
10.613562
19
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.612464 ; 17
.611916 ' 16
.(511369 15
.610822 14
.610276 18
.6097:30 12
.609185 11
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10.608097
9
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8
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7
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.605927
5
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4
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2
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1
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ang. "
.
TABLE xn. — LOGABTTH&nC SINIS,
D. I". Cosine, D. 1
Tang. D, 1", OMang
9.389SII
.saoiio
'.mtm
! 303191
.VXTM?
m
.'Man
'.aueaia
.wan
9.4ceas6
.M)3187
.■WKIS
.405004
! 9.W7M5
' .408471
.40ei«6
!41«M5
! 41 109*
.411G15
,mm
!4i4at9
.4147^
;widi7
.'enti&ts
!39»I09
Ui.SK3*t
.59e£ft2 .
.595^1 ,
.59532^
.591092
.Aftiiet
^sasioe '.
.587343
[o.seeaei
.siwaai
.£8^074
ia.5SIB4Z I IS
lO.STSSKI
.STCOtrr
u*
COSINES
TANGENTS, AND COTANQilMTS.
lU
Sine.
D.r.
CcBine. I>. 1-. II
^
D. 1-. C
tang.
~
S.41£9S«
7.85
9.ttl4M4
.1? ' 8
438063
's^
"^
,»(491U
571443
3
ilHW
S7
Sw
8:33
570938
«:
7:so
7'.7T
56
;41B347
;»84774
4aB73
W9437
.981740
431075
168935
8
:«»i7
!9Kie3a
.984603
1
43jo;«
433S80
433080
560930
oa',
50
11
S.3SU69
58
^1
Aim-S
7!73
7.7»
7,68
.984533
48
.4180T9
.9W500
431.179
sfiwai
.984460
1
4.^^78
!4axi(>7
,W443i
46
18
5U3937
44
:*!0933
436570
503)30
43
.«i3aa
:9aiaia
43706-
663933
42
b:S
20
!4£i3LS
:984a51(
438U39
40
3.41Kmi
i'.rti
D.9a«M
i ■
1
4385M
si
BB1148
■39
S2
38
K
!4ia607
:98408r.
4405S9
M94T1
ST
30
35
X
:^0T^
411032
658978
.tffiSW
R.'SO
iSSttS
33
ss
435SS7
33
»
.438*13
7^00
: 883940
44^497
31
30
.4Mi3y»
.Bsagii
44S988
6B70ia
9.427iW
t'.'j^
9 WWTB
B8 "
39
4139US
38
33
1
414458
S.W.513
444W7
65.1053
36
3$
i
SS^-iM
35
ai
3T
B:ia
5535^
33
38
44«898
653103
23
39
7:!so
7,50
4473«
SO
B.983Se3
1 '
GO
1 '"
.433328
.083487
5,M1S0
.4aS7TB
.iXOK
7;48
:oo
:4a4ia3
:083S45
4.10777
.4*4589
.983309
45IS80
548740
1
.9S3S73
3S 1
51Ka57
'.VSMSi
&I7775 1 11
SO
.tssaos
:9B3ao«
453703
5i73M
10
61
fl,«S3»3
B,9831M
30 *
XI
6(1
4531B7
Z >o
H68ia
7
U
! 43768(1
!983(»8
iMa>»
7:98
t:95
7.95 10
e
.438139
.9B3IHa
4551U7
M489a
.063981)
S14414
W39B8
ffl
:e829i4
450543
U
,439897
,993878
45701B
1
~^~
9.440338
9.982812
M3504
C»L,.e.
D.r.
BioB. 1 D
Tr.T
,taus.
D.V. \ 1
--*■■ ^ '
\
16<
TABLE XII. — LOGARITHMIC SINES,
163'
1
2
8
4
6
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
80
81
82
88
84
85
86
87
38
89
40
41
42
43
44
45
46
47
48
49
60
51
52
53
54
55
56
57
58
59
60
Sine.
-/
9.440338
.440778
.441218
.441658
.442096
.442535
.442973
.448410
.448847
.444284
.444720
9.445155
.445590
.446025
.446459
.446893
.447326
.44rr59
.448191
.448623
.449054
9.449485
.449915
.460345
.450775
.451204
.461632
.452060
.452488
.462915
.458342
9.453768
.454194
.454619
.465044
.465469
.455893
.466316
.456739
.467162
.467584
9.468006
.468427
.468848
.469268
.469688
.460108
.460527
.460946
.461364
.461782
9.462199
.462616
.463032
.463448
.463864
.464279
.464694
.466108
.466522
9.465935
D.r.
7.33
7.33
7.33
7.30
7.32
7.80
7.28
7.28
7.28
7.27
7.25
7.25
7.25
7.23
7.23
7.22
7.22
7.20
7.20
7.18
7.18
7.17
7.17
7.17
7.15
7.13
7.13
7.13
7.12
7.12
7.10
7.10
.08
.08
08
.07
.05
.05
05
.03
7
7
7
7
7
7
7
7
7.03
7.02
02
00
00
00
6.98
6.98
6.97
6.97
6.95
6.95
6.93
6.93
6.93
6.92
6.92
6.90
6.90
6.88
h
Cosine.
9.982842
.982805
.982769
.982733
.982696
.982660
.982624
.982587
.982561
.982514
.982477
9.982441
.982404
.982367
.962331
.982294
.982257
.982220
.962183
.962146
.982109
9.982072
.982035
.981998
.981961
.981921
.981886
.981849
.981812
.981774
.981737
9.981700
.981662
.981625
.981587
.981649
.981512
.981474
.981436
.981399
.981361
9.981323
.981285
.981247
.981209
.981171
.981133
.981095
.981057
.981019
.980981
9.980942
.980904
.980866
.980827
.980789
.980750
.980712
.980673
.980636
9.980596
Cosine, I D.V. jl Sine.
D. r.
.62
.60
.60
.62
.60
.60
.62
.60
.62
.62
.60
.62
.62
.60
.62
.62
.62
.62
.62
.62
.62
.62
.62
.62
.62
.63
.62
.62
.63
.62
.62
.63
.62
.63
.63
.62
.63
.63
.62
.03
.63
.63
.63
.63
.63
.63
.63
.63
.63
.63
.65
.63
.63
.65
.63
.65
.63
.&5
.63
.65
Tang.
D. r.
192
9.457496
.457973
.458149
.458925
.459400
.459675
.460349
.460623
.461297
.461770
.462242
9.462715
.463186
.463668
.464128
.464599
.465069
.465539
.466008
.466477
.466945
9.467413
.467880
.468347
.468814
.469280
.469746
.470211
.470676
.4ni41
.471605
9.472069
.472532
.472995
.473457
.478919
.474381
.474842
.475303
.475763
.476223
9.4766&S
.477142
.477601
.478059
.478517
.478975
.479432
.479889
.480345
.480801
9.481257
.481712
.482167
.482621
.483075
.483529
.483982
.484435
.4^1887
9.485339
Cotaiig.
D. r
7.95
7.93
7.93
7.92
7.92
7.90
7.90
7.90
7.88
7.87
7.88
7.85
7.87
7.83
7.85
7.83
7.83
7.82
7.82
7.80
7.80
7.78
i . to
7.78
7.77
7.77
75
75
75
73
7.72
7.72
7.70
7.70
7.70
7.68
7.68
7.67
7.67
7.67
7.65
7.65
7.63
7.63
7.63
7.62
7.62
7.60
7.60
7.60
7.58
7.58
7.57
7.57
7.57
7.55
7.56
7.53
7.53
I D. r.
Cotang.
10.527931
527468
.527006
.526543
.526081
.525619
.525158
.524697
.524237
.523777
10.523317
.522858
.522399
.521941
.521483
.521025
.520568
.520111
.519655
.519199
10.518743
.518288
.517833
.617379
.516925
.516471
.516018
.515565
.515113
10.514661
10.542504 60
.542027 59
.541551 58
.541075 57
.540600 56
.540125 55
..539651 54
.539177 53
..538703 52
.538230 51
.537758 50
10.537285 ' 49
.536814 ' 48
.536342 I 47
.535872 46
.535401 45
.534931 44
.534461 43
.533992 I 42
.533523 41
.533055 40
10.532587 89
.532120 I 38
.531653 I 37
.531186 ! 36
..530720 35
.5.30254 34
.529789 33
.529324 32
.528859 31
.528395 SO
Tang.
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
5
4
3
2
1
nv
COSINBfi, TANGENT8, AND COTANGENTS.
IS I
.4il78tl
.SOUISD
■xmeo
soihieT I
.soesn !
.60S3T3
18°
TABLE
XII. — LOGARITHMIC SINES,
161
,-| Bin. D
r. j t
□Elne. D
1 '
ang I
. 1 Coiflng-
-_
a 9
48U38S g
^"s^
*7820e
8 "
5 ~a
"T io"
4KBSS4
1
a ssob
4H77M
a
190758 f.
47
lisiai
6 J«S5
487385
S;
4M147 ^
r:sm
-0
n
-0
■ TIM
1
4I»I9£3 ^
m
4BS30S I
intm
48M51
49afi95 J
9-7919
14-7
4B30fll S
o-TBrr
493400 S
9 TttSi
4BSS51 J
0-TTB4
5 8057
4BSB43-
4B4S8B (
1
wma
^ ■
4S351B
49
K
977869
1-336
483680
18
4fl57T3 I
jfraso
^
i898
r
4S054S
491301
30
977419
5 1SS3
480118
49T6Si!
977377
530305
21 s
33 ^
3a
aa
3a
33
isorea
B 10
33
03
03
03
473973
39
49R8ai 1
sHlBl
i
TO
i.ift73 ;
471«1H
47S437
ii
ao
49»ge3 J
BTTIBB
3t
as
500731
07T0B3
S ;
4T87J1
»
[101099 J
471IHIO
30
475480
30
SI 9
B01R54 ,
WBOtJ
™ •
mm 1
08 M
475080
sx
S3
fiOittSl J
nrusra.
i=S360 1
38
ar
%
35
eo
i
73
97
an
30
473385
36
BB
6037*1 [
876708
M7033. \
4^887
47S133
471716
1
40
wsaaj •
97<I5K
K
issToa. 1
471398
30
605608 ,
97BIBB
18 '"
4S
Botmw. I
33
9711301.
1
ra
•aWflfl. }
i
4fl!W84
17
BoroM 1
ii
JSOTBI I
489318
4fll«04
n
4S
97fllS0
BSaKB
608.'>85 I
97(1148
i3M39 J
m
487501
re
10
El 9
iso«se ,
17 1"
9^080
re 8
73
78
79
re
73
re
SSSEOS ,
i
4887S4
9
maao [
07IKII7
533079
406321
8
5U40113 )
485908
is
K
67
ii •
s
M ''
484361
1
H
IS
i 10
59
TO S
filBW!!
12 g
9750711
saoKu ''
463038
T
'' /c
w™. I'd
TT.'^
'iinc^. I'd"
olang, 1 n
TTt'^
"^r*
19^
COSINES, TANGENTS, AND COTANGENTS.
160'
/
Sine.
9.512642
1
.513009
.513:i75
3
.51:^741
4
.514107
5
.514472
6
.5148.'C
7
.515202
8
.515066
9
.515930
10
.516294
11
9.516657
12
.517020
13
.517382
14
.517745
15
.518107
16
.518468
17
.518829
18
.519190
19
.519551
20
.519911
21
9.520271
22
.5200:31
23
.520990
24
.521»19
25
.521707
26
.522066
27
.522424
28
.522781
29
.523138
80
.523495
31
9.523862
32
.524208
83
.524561
84
.524920
a5
.525275
86
.525630
37
.525984
38
.526339
30
.526698
40
.627046
41
9.527400
42
.527r53
43
.528105
41
.528458
45
.528810
46
.529161
47
.529513
48
.529864
49
.530215
50
.530505
51
9.530915
52
.531265
53
.531614
54
.531963
55
.532312
56
.532661
57
.533009
58
.,533357
59
.533701
60
9.534052
D. 1'.
Cosine.
Cosine.
6.12
6.10
6.10
6.10
6.«'8
6.08
6.08
6.07
6.07
6.0Z
6.05
6.05
6.03
6.05
6.03
6.02
6.02
6.02
6.02
6.00
6.00
6.00
5.98
5.98
.97
.98
.97
.95
.95
.95
.95
5.
5.
5.
5.
5.
5.
5.
5.93
5.93
5.93
5.92
5.
5
5,
5
5.
5.
92
90
92
90
88
90
5.88
5.87
5.88
5.87
5.85
5.87
5.85
5.85
5.83
5.83
5.83
5.82
5.82
5.82
5.82
5.80
5.80
5.78
5.80
D. V.
9.975670
.975627
.975583
.975539
.975496
.975452
.975406
.975365
.975321
.975277
.975233
9.975189
.975145
.975101
.975057
.975013
.974969
.974925
.974880
.974836
.974792
9.974748
.974703
.974659
.974614
.974570
.974525
.974481
.974436
.974391
.974347
9.974302
.974257
.974212
.974167
.974122
.974077
.974032
.973987
.973942
.973807
9.973858
.973807
.973761
.973716
.97;B671
.973625
.97a580
.9735.35
.973489
.97344^1
9.973398
.973352
.973307
.973261
.973215
.9731(59
.973124
.973078
.973032
9.972986
Sine.
D. r.
.72
.73
.73
.72
.73
.73
.72
.73
.73
.78
.73
.73
.73
.73
.73
.73
.73
.75
.73
.73
.73
.75
.73
.75
.73
.75
.73
.75
.75
.73
.75
.75
.75
.75
.75
.75
.75
.75
.75
.75
.75
.75
.77
.75
.75
.77
.75
.75
.77
.75
.77
.77
.75
.77
.77
.77
.75
.77
.77
.77
D. r.
Tang.
D. r.
9.536972
.537382
.537792
.538202
.588611
.539020
.539429
.539837
.540245
.540653
.541061
9.541468
.541875
.542281
.542688
.543094
.543499
.543905
.544310
.544715
.545119
9.545524
.545928
.546381
.5467;«
.517138
.547540
.547943
.548845
.548747
.549149
9.549550
.549951
.550352
.550752
.551153
.551552
.551952
.552351
.552750
.553149
9.553548
.553946
.554344
.554741
.555139
.55 536
.555933
.55(5329
.556725
.557121
9.557517
.557913
.55a308
.558703
.559097
.559401
.559885
.560279
.56067:3
9.561066
Cotang.
6.83
6.83
6.83
6.82
6.82
6.82
6.80
6.80
6.80
6.80
6.78
6.78
6.77
6.78
6.77
6.75
6.77
6.75
6.75
6.73
6.75
6.73
6.72
6.73
6.72
6.70
6.72
6.70
6.70
6.70
6.68
6.68
6.68
6.67
6.68
6.65
6.67
6.65
6.65
6.65
6.65
6.63
6.63
6.62
6.63
6.62
6.62
6.60
6.60
6.60
6.60
6.60
6.58
6.58
6.57
6.57
6.57
6.57
6.57
6.55
i D.V.
Cotang.
109*
10.463028
.462618
.462208
.461798
.461389
.460980
.460571
.460163
.459755
.459347
.458939
10.45&532
.458125
.457719
.457312
.456906
.456501
.456095
.455690
.455285
.454881
10.454476
.454072
453669
.453265
.452862
.452460
.452057
.451055
.451253
.450851
10.450450
.450049
.449648
.449248
.448847
.448448
.448048
.447649
.447250
.446851
10.446452
.446054
.445656
.445259
.444861
.444464
.444067
.443671
.443275
.442879
10.442483
.442087
.441692
.441297
.440903
.440509
.440115
.439721
.439327
10.438934
\^5
«•
TABLE
xn-
-LOGARITHMK
SINES,
169
^
Bine.
D.,-.
ToBlDe.
D.1-.
' Tang.
D. V. C
».„.
—
B.5a((isa
D7S99G
77
'77
■ 9 5GlW
666 '"
488931
,W«93
IfWWO
.5*1745
58
11
67
.ftaXifA
437B64
'saana
436972
I
.M«IM
fi!73
WJ«e8
.ra
WWII
11
43S»SI
64
O'^IT
435788
63
.5
:637;fl7
Mssm
OWWS
4S54ro
B.M7S51
5.73 ^
S.TO
BTU4T8
.78
9.B65f!73
650 "*
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9:45
0,47
434687
434a87
3
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.bcislss
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s.^88eo
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arsaa]
483068
ttrasM
4a!»80
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5677(19
43S91
43
5'^
■64051W
BT^IOS
't8
431514
UTSoaa
481 lOT
40
s °
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4-W739
89
22
:mi6[s
Kl
.641051
ire
189965
^
.Miues
1171870
1 !5704SB
871888
35
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9-1578
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84
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.543310
in;a9
428119
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S
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11
4»«B3
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;-7:;73s
.7B
.W)
.7H
.SO
,78
.BO
u.,s70iai
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4369?7
X!
071493
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a?
Moon
ICISBS
sf-o^iV)
435n;4
»
3G
6:38
6.38
0.3S
G.38
.54ao.s:i
.a;r.ia7
484578
?f
iiTiaw
4!M190
s •
483807
4eS4S4
80
o.;4fl(i3J
D710nf!
6 37 ^''
423041
1171018
.077311
432659
«
MHIMJ
s.w 1
.STTTM
1
45
MMdO
:78
411
.MOBBS
3
WOHST
578MT
481138
4T
070778
.'bo
tl
480752
117073!
480371
40
.580009
419991
»
ISSIOM
seossm
419611
e.55135S
.SI
9 wnTTfl
6.38 1"
i:i „
GS
.5SI0BJ
Biiia
BS
.SESnS
B04!«0
.ss-jais
J704JB
fmi
i
iiS™
11
S
!to
31
Sii
,
60
g.SWJ:.-!!
i70i.'.a
416833
L
1^
~a^^!
», V. ' ~
~>^
ITiT-i
CoVaiiE.
B.V, T
.HE.
'
CONINES, TANGENTS, AND COTANGENTS.
.6ffit14fl
B.MraST
.0DMID9
.eou)T5
.1WT977
MTKV
.WHWIl
.5HW40
.5HW14
.HMIHS
D.Ggt!0.->4
.51.7317
.59T018
9.003137
.I3034US
AVJKa
io.4iieS4
.41I30»
10.4OIS33 I
.VJSMOt I
.40H49U !
io.4«aMi
.3eHTD4
23"
TABLE
XII.-
-LOGARITHinc
SINES,
1S7
Siiio.
D. r. c
OBine.
a,-.
Tang.
D.V.
Cotane,
—
9
KBSTB
5,as B
fi.ao
s.po
9B71Be
9,606410
6.0S
10 8035B0
m3888
wniB
.BOBTTB
:so3aj7
59
I
!8fi
.007131
.tiOTMO
am
eioa
^
1I169B1
S7B136
968910
:m
;B0H2i!5
:a9ir75
958861)
.811WJK8
.mm
:87
5780119
»
b:!?
;«
:B9oa9j
51
10
i'lma
,389964
60
1 '
S7JB0B
877618
i:iS "
tmim
.87
.8S
'.mm
,389341
,388880
Is
.611480
b:^
:8T
IB
17
B7H8W
;3874,W
43
18
BToioa
mm
1
.6iaei
BttTUJO
5'l3
Booisa
6:ro
40
M 9
BBOOW
906065
9,614000
5.98
698
697
10,388000
.39
5803H2
,87
1
.87
.B7
.8S
.BM3B!)
38
!3ffi28a
081 DOS
384923
iie
20
B81313
384S65
■a
M
581618
SflSftM
.BISTM
381SOT
iH
b:«
,383849
BflS
S9
imss
slOH
BBSflOH
"b1M7
iaeaiaa
:ll
,B8OT78
30
31 9
»»14S
r..07 ^
o'ofl
5^03
B.9B
593
G.9a
:(. 388418
ss
K
S8344S
606611
ina
.3K061
V8
9UM68
,381705
.881348
r>»i3ei
1
.380902
■a,
36
■^m
9GB301
.3806.'*
37
JOBUia
-6]»7aO
.380280
:^
!S7U668
40
586817
965090
.37«aia
h
BOB "
965087
B,8ail49
11
4a
:88
44
tSSTOWi
0fum9
is
,H82S07
;S77793
5ST3ftB
JBWM
.aaajBi
.377439
4a
s!oa
6,ce
5M
W82W
;B70377
4B
588590
WAm
.628978
ti
.376034
50
.6E4B30
6S
589789
ii "
MiMO
;9o
':S
il
10,376317
Is
590981
if
iwim
!8i«T97
11
JjSre
BS
5912^
91M133
:m
.375851
nam)
b:85
,873199
(SO B
iuis™
9(Moai
0.6!!78sa
10.373148
-^
^cv
s»je. /
D.I-.
Bin«.
D.1-.
Cotang.
■»,v.
Taut.
»-
COSINES, TANGENTS, AND COTANGENTS.
1S6
—
«... B
V. 1 C
osine. D
r. !l Ta...
D. 1'. Cc
tone.
9 501S78
8
xuoaa
w
»
90
99
»
687853
B85 *"
173148
60
69
BS
3
!b«t™ 1
ll«»05
07
.SO306T 1
•03811
JTW45
.B933OT 1
li
.flOSftW J
Msoao
63
B
9US59S
^:83
5.B3
63
!b94B17 J
W3543
!i3im»
.891812 I
668046
a.MSlST ,
^ »
K *
fl317M
5.W
sosass
49
.HK43a I
903879
367W7
48
»
103385
i
16^59
633099
36690!
45
'.meiM *
WilfiS
J3
633147
i
360653
(03108
63.^795
360305
i
W
36S857
:B97Jfllll ^
SO
.MTTiH J
go^ftis
fl34BaS
3fii516a
40
9.B080TB ,
»2890
M "
635155
B.TS 10
si
B^TT
b:77
SflWlB
Kl
.B9B3A8 \
ff
38
B7
aj
'.bsmi i
17
92
036^
363774
66
le
.5S0S44 3
903673
636573
363438
8B
2a
.ii»95aa ^
902817
636B19
m
ieoons 1
« 1
037611
Beams
302380
63
S3
29
.iHKWn ]
WiSiSa
637950
saxni
81
so
ftia»e8
638803
361098
30
SI
0.6009BO ^
^ 9
9Bi»iS
W *
ai8047
B^ra "
36iaV!
39
wwsss
301008
28
03
oa
»3
630337
b'.n
6.73
B^TS
360663
»7
ieoiKso 1
630U33
360318
26
85
.BOilW *
IS
89
llfl««3
369073
£5
3U
.B034SB
goaooT
640371
a-iiKag
.flOSTW ^
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iflosaofi 2
36I9*K
B3
3581506
21
40
.floai»4 3
BBiaie
W1747
35»ffi3
ao
M
B. 003882 ^
:6(wii7 ;
80 ^
i
961080
9B ^
99
93
92
042777
B.ra
B.75
H
ii
.BMTis ;
ftl3130
awsM
10
78
77
356537
IB
46
■fi^ '
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M114B
355803
13
48
:60589a ;
Mnoa
93
6H190
B^TO
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to
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Boiiiao
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B.eoftrai .
b
961 2^
03
i
93
^ l9
645510
6.'es w
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.florasK ,
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353119
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fwrna
352778
St
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980999
fi.M JO
GB
n J
»2097
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60
B.ewBia *
9M730
648583
Coame. I-
!■-
Blao. 1 D
r. 1, c
o«Dg.
D.f. 1
UJSf,
k
TABLE XII. — LOGARITHMIC SINES,
>. 615333
B.OSOTia
B«
COBINBS
TANGENTS, AND COTANQBNTS.
164'
Sloe. D
■■■
c..-.
1
^ng.
D. r.
Colang.
~o^T
easMS ,
S3
1
9 0573™
1
0666^7
5.48
6.50
10.331327
60
«ab2i9 j
UHKim
HM90 5
Iflsnes
mm
:S30668
BS
B
,330839
-m
.330000
07039)
.320689
56
'.wman
'-m
670H0
.320351
M
1
BTSW 3
.9BUSia
670977
6^48
.assess
63
3
.956803
671306
1338366
10
ioseeei
!SS316 ^
4B
47
i
43
48
0.056633
lioo
ojsasi
10.327700
40
.956666
48
.9IW00
«™847
829721 1
SS0980 3
:0B6S87
«ms
:32e39e
45
.966387
e^«£ia
.aason
.xam
IS
.0360)8
19
831069 1
:9B
674911
b;43
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41
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:9560^
876437
.BW763
40
0.956039
lioo
11
10.324430
89
£3
aa
63S125 1
.955900
076217
6323113 \
43
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43
42
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670M8
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ae
87
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b:43
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%
638154 3
67:&40
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sasTiB 1
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B78171
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6339BI 5
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e»124» ^
42
4a
40
40
1
09SM2R
i:o3
6ffiH21
It
10.821170
29
E8
S8
584778 J
,'955307
079171
M
B8Rm 1
.BMiMT
B797BB
6.iSi
.^0205
as
easaoa 1
6aoiso
.319880
es
eSUTD \
b'M
bIw
636880 \
681416
.318684
21
40
636623 J
!l»18B3
.818260
20
636886 .
87
i
87
9054823
ii '
10 317937
M
63T1-1S 3
.B547(a
^1
.317618
i:o3
.317290
17
.816967
18
1951570
46
.9W518
683679
47
63S4.W
S7
681001
.3 6999
48
638T20 ^
lOS ■
.8 5676
12
1
I95J3S5
*-^
.954274
«849fi8
! 316032
61 S
639i)03 ,
ss
33
S
33
OOMSia
6.37
b!37
10 3 4710
.8 4388
84066
7
lift!
6863r>6
.813746
B5
:05WB8
686577
.313423
fiiOSW ^
i:o3 J
6H6S08
B..15
.813102
,812781
3
687M0
GO
iofflTSS
6»78fll
60 S
BJlka *
9.963660
io:3iiBia
(Mine. 1 D
■?r
Sine. 1 Ur. He
oLaii^.
o.v.
V ■^^.
iL,
TABLE XII. — ^LOGARITHMIC SINES,
1«'
Sine.
/
16*
1
2
8
4
5
6
7
8
10
11
n
18
14
15
16
17
18
10
SO
21
22
23
24
25
26
27
28
29
80
81
82
83
81
85
86
87
88
89
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9
9
9
9
9
641842
642101
642360
642618
642877
643135
643308
648660
648906
644165
644428
644680
644936
645198
645450
645706
645962
646S18
646474
646739
646964
647240
647494
647749
648004
648258
648512
646766
&19020
649274
649527
649781
650034
650287
650539
650792
651044
651297
651549
651800
652052
652304
652555
652806
653057
653308
653558
653808
654059
654309
654558
654806
655058
655307
655556
655805
656051
656302
656551
656799
fi57047
^ / Cosine. I
D. 1".
4.82
4.32
4
4
4
4
4
4.
4
4
4.
30
32
80
80
26
80
28
30
26
4.27
4.28
4.28
4.27
4.27
4.27
4.27
4.25
4.25
4.27
4.
4.
4.
4.
4.
4.
.23
.25
.25
.23
.23
.23
4.23
4.23
4.22
4.28
4.22
4.22
4.20
4.22
4.20
4.22
4.20
4.18
4.20
4.20
4.18
4.18
4.18
4.
4.
4.
4.
4.
4.
4.
18
17
17
18
17
15
17
4.17
4.15
4.15
4.15
4.15
4.13
4.15
4.13
4.13
Ck>sine.
9.953660
.953599
.953537
.953475
.953413
.953852
.953290
.953226
.953166
.953104
.958042
9.952980
.952918
.962856
.952798
.952781
.952669
.952606
.952544
.952481
.952419
9.952856
.952294
.952231
.952168
.952106
.952043
.951960
.951917
.951851
.951791
9.951728
.951665
.951602
.951589
.951476
.951412
.951819
.951280
' .951222
.951159
9.951096
.951032
.950968
.950905
.950841
.950778
.950n4
.950650
.950586
.950522
0.950458
.950394
.950330
.950266
.950202
.950138
.950074
.950010
.949945
9.949881
D. r.
.02
.08
.03
.03
.02
.03
.03
.03
.08
.08
.03
.08
.05
.08
.08
.08
.05
.08
.05
.03
.05
.03
.05
.05
.03
.05
.05
.05
.05
.05
.05
.05
.05
.05
.05
.07
.05
.05
.07
.05
.05
.07
.07
.05
.07
.05
.07
.07
.07
.07
.07
.07
.07
.07
.07
.07
.07
.07
.08
.07
Tang.
0.688182
.688502
.688828
.689143
.689463
.689788
.690103
.690428
.690742
.691062
.691881
0.691700
.692019
.6923^
.692656
.692975
.693293
.693612
.694248
.694566
9.094883
.695201
.695518
.695836
.696153
.696470
.696787
.697103
.697420
.697786
0.698053
.696369
.698685
.699001
.699316
.699632
.699947
.700203
.700578
.700693
0.701208
.701523
.701887
.702152
.702466
.702781
.703095
.703409
.703722
.704036
0.704350
.704663
.704976
.705290
.705603
.705916
.706228
.706541
.706854
9.707166
D. 1'.
5.38
5.32
5.33
5.33
5.33
5.33
5.33
5.82
6.38
6.82
5.32
5.82
5.82
5.30
5.32
5.30
5.32
5.30
5.30
5.30
6.26
5.30
5.26
5.30
5.26
5.26
5.28
5.27
5.26
5.27
5.26
5.27
5.27
5.27
5.25
5.27
5.25
5.27
5.25
5.25
6.25
6.25
5.23
5.25
5.23
5.25
5.23
5.28
5.22
5.23
5.23
5.22
5.22
5.23
5.22
5.22
5.20
5.22
6.22
5.20
Cotang.
10.811816 ' 60
.811496 69
.311177
.310657
.310687
.310217
.309897
.309577
.809256
.306936
.306619
10.308300
.307981
.307062
.307344
.307025
.306707
.306386
.306070
.805752
.806484
10.806117
.304799
.804462
.804164
.808647
.303530
.803218
.302897
.802580
.802264
10.801947
.801631
.801315
.300999
.300664
.300366
.300053
.299737
.299422
.299107
10.296792
.29647:
.298163
.297848
.297534
.297219
.296906
.296591
.296278
.295964
10.29.5650 ,
.2953S7 '
.29,'5021
.294710
.294397
.294064
.293772
.293459
.293146
10.292634
D.V, 1 1 Sine. I D.V. \^ Cotaiie. \ "D.V. I Tang.
202
58
67
66
55
64
53
52
61
60
49
46
47
46
45
44
43
42
41
40
39
88
37
86
85
84
83
82
81
80
29
26
27
26
25
21
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
6
7
6
5
4
3
2
1
^»
«•
COSINES, TANGENTS, AND COTANGENTS.
IW
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
23
27
28
29
80
81
82
83
34
35
86
37
88
89
40
41
42
43
44
45
46
47
48
49
60
51
62
63
54
55
56
57
68
59
60
Sine.
D. r.
9.657047
.657295
.657542
.657790
.658037
.658284
.658531
.658778
.659025
.659271
.658517
9.659763
.660009
.660255
660501
660746
.660991
.661236
.661481
.661726
.661970
9.662214
.662^9
.062703
662946
.663190
.663433
.663677
.663920
.664168
.664406
9.664648
.664891
.665133
.665375
.665617
.665859
.666100
.666312
.666583
.666824
9.667066
.667305
.667546
.667786
.668027
.668867
.668506
.668746
.668986
.669225
9.669464
669703
669942
670181
670419
.670658
670896
.671134
.671372
9.671609
Cosine.
4.13
4.12
4.13
4.12
4.12
4.12
4.
4.
4.
4.
4.
12
12
10
10
10
4.
4.
4.
4.
4.
.10
.10
.10
.08
.06
4.06
4.06
4.06
4.07
4.07
4.08
4.07
4.05
4.07
4.05
4.07
4.05
4.05
4.05
4.03
4.05
4.
4.
4.
4.
4
4.
4.
4.
03
03
03
03
02
08
02
02
4.02
00
02
00
02
00
3.98
4 00
4.00
3.98
3.98
8 98
3.98
8.98
3.97
3.98
3.97
3.97
3.97
3.95
D. r.
Cosine.
9.949881
.949816
.949752
.949688
.949623
.949558
.949494
.949429
.949364
.949300
.949235
9.949170
.949105
949040
.948975
.948910
.948^5
948780
.948715
.948650
.918564
9.948519
946454
.948388
.946323
.948257
.948192
.948126
.948060
.947995
.947929
9.947863
947797
.947731
947605
.947600
.947533
.947467
.947401
.947335
.947269
9 947203
.947136
.947070
.947004
.946937
.946871
&46804
.946738
.940671
.946604
9.946538
.5)46471
.946404
946337
.94(5270
9462(^3
.946136
.&46009
940002
9.9459135
D. 1'.
.08
.07
.07
.08
.06
.07
.08
.06
.07
.08
.08
.08
.06
.06
.06
.06
.06
.06
.06
.10
.06
.08
.10
.06
.10
.08
.10
.10
.08
.10
.10
10
.10
.10
.08
.12
.10
.10
.10
.10
.10
12
.10
.10
.12
.10
.12
.10
.12
.12
.10
.12
.12
.12
.12
.12
. i/W
.12
.12
.13
Tang.
9.707166
.707478
.'rC7790
.708102
.706414
.708726
.709037
.709349
.709660
.709971
.710282
9.710593
.710904
711215
.711525
.711836
.712146
.712456
.712766
.718076
.718386
9.713696
.714005
714314
.714624
.714983
.715242
.715551
715860
.716168
.716477
9.716785
.717093
. 717401
717709
.718017
718325
.718633
.718940
.719248
.719555
9.719862
.720169
.720476
.720783
.721089
.721396
.721702
.722009
.722315
.722621
9.722927
7232.^
723538
.723844
724149
.724454
724700
725065
?^5370
9.725074
D. 1'.
5.20
5.20
5.
5.
5,
5
5.
5
5
5
6
20
20
20
18
20
18
18
18
18
5.18
5.18
6.17
5.18
6.17
6.17
5.17
5.17
5.17
6.17
5.
5.
6.
6.
6.
5.
5.
6.
15
15
17
15
15
15
15
13
6.15
5.13
13
13
13
13
13
18
12
13
12
12
5.12
5.12
6.12
5.10
6.12
5.10
5.12
5.10
5.10
5.10
5.08
5.10
5.10
5.08
5.08
5.10
5.08
5.08
5.07
Cotang.
f 1
60
10.292834
.292522
69
.292210
68
.291898
67
.291586
56
.291274
55
.290963
54
.290651
53
.290840
52
.290029
61
.289718
50
10.289407
49
.289096
48
.288785
47
.288475
46
.288164
45
.287854
44
.287544
48
.287234
42
.286924
41
.286614
40
10.286304
89
.285995
88
.286686
87
285876
86
.285067
85
.284758
84
.284449
83
.284140
82
.288832
81
.283523
80
10.283215
29
.282907
28
.282599
27
.282291
26
281983
25
.281675
24
.281367
23
281060
22
280752
21
.280445
20
10.280138
19
279831
18
.279524
17
.279217
16
.278911
15
.278604
14
.278298
13
.277991
12
277085
11
.277379
10
10.277078
9
.276768
8
276462
7
276156
6
275851
6
.275546
4
275240
3
274935
2
274630
1
10.274326
Sine. \ D. V . \\ ^vi\,a.\x^.\ \>.V A '^^^^^ \
117-
I^Z
28'
TABLE XII. — LOGARITHMIC SINES,
IBV
Sine.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
82
a3
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
SO
9.671609
.671847
.672084
.672321
.672558
.672795
.673032
.673268
.673505
.673741
.6?3977
9.674213
.674448
.674684
.674919
.675155
.675390
.675624
.675859
.676094
.676328
9.676562
.676796
.677030
-/■
.677264
.677498
.677731
.677964
.678197
.678430
.678663
9.678895
.679128
.679300
.079592
.6798i.'4
.080050
.080288
.080519
.080750
.680982
9.681213
.081443
.681074
.681905
.682135
.082365
.682595
.682825
.6&3055
.683284
9.683514
.683743
.683972
.684201
.684430
.684658
.684887
.685115
.685:^43
P. 685571
D. r.
3.97
3.95
3.95
3.95
3.95
3.95
3.93
3.95
3.93
3.93
3.93
3.
3
3.
3
3.
92
93
92
93
92
3.90
3.92
3.92
3.90
3.90
3.90
3.90
3.90
3.90
3.88
3.88
3.88
3.88
3.88
3.87
3.88
3.87
3.87
3.87
3.87
3.87
3.85
3.87
3.85
3. as
3.85
3.85
3 83
3.83
3.83
3.83
3.83
3.82
8.83
3.82
3.82
3.82
3.82
3.80
3.82
3.80
3.80
3.80
Cosine. I D. 1'.
i
Cosine.
9.945935
.945868
.945800
.94573:^
.945666
.945598
.9455:^1
.945404
.945396
.945328
.945201
9.945193
.945125
.9450.58
.944990
.944922
.944854
.944786
.944718
.944650
.944582
9.944514
.944446
.944377
.944309
.944241
.944172
.944104
.944036
.943907
.943899
9.943830
.943701
.943093
.943024
.943555
.943480
.94.^17
.{M.3348
.943279
.943210
9.943141
.94:^072
.943003
.(M2934
.942804
.942795
.942726
.9426,56
.942.587
.942517
9.942448
.942378
.942308
.942239
.942109
.942099
.942029
.9419.')9
.941889
9.941819
Sine.
D. r.
,12
,13
.12
,12
,13
,12
.12
.13
.13
.12
.13
.13
.12
.13
.13
.13
.13
.13
.13
.13
.13
.13
.15
.13
.13
.15
.13
.13
.15
.13
.15
.15
.13
.15
.15
.15
.15
.15
.15
.15
,15
,15
.15
.15
,17
,15
,15
,17
.15
.17
,15
,17
,17
15
,17
,17
,17
.17
.17
.17
Tang.
9.725674
.725979
.726284
.720588
.726892
.727197
.7'27501
.7'
27805
.728109
.728412
.728716
9.729020
.729323
.729626
.729929
.730233
.730535
.r30838
.731141
.731444
.731746
9.732048
.732351
.732653
.732955
.733257
.733558
.733860
.734162
.734463
.734704
9.7a')066
.735307
.735608
.7a5969
.736269
.730570
.736870
.737171
• .737471
.737771
9.738071
.738371
.738671
.738971
.739271
.739570
.739870
.740169
.740468
.740767
9.741066
.741365
.741664
.741902
.742261
.7425.59
.742a58
.743156
.743454
9.743752
D. 1'.
Cotang.
5.08
5.08
5.07
5.07
5.05
5.07
5.07
5.07
5.05
5.07
5.07
05
05
05
07
03
05
5.05
5.05
5.03
5.03
5.05
5.03
5.03
5.03
5.02
5.03
5.03
5.02
5.02
5.03
5.02
5.02
5.02
5.00
5.02
5.00
5.02
5.00
5.00
5.00
5.
5.
5.
5.
4.
5.
.00
.00
.00
.00
.98
.00
4.98
4.98
4.98
4.98
4.98
4.98
4.97
4.98
4.97
4.98
4.97
4.97
4.97
10.274326
.274021
.273716
.273412
.273108
.272803
.272499
.272195
.271891
.271588
10.270980
.270677
.270374
.270071
.269767
.269465
.269162
.268859
.268556
.268254
10.267952
.267649
.267347
.267046
.266743
.266442
.266140
.265838
.265537
.265236
10.264934
.264633
.264332
.264031
.263731
.26*130
.26:3130
.262829
.262529
10.258934
.258635
.258336
.258038
.257739
.257441
.257142
.256844
.256546
10.256248
U0*
D. r. \ Cotai\6.\ T>.V. \ Twig
204
60
59
58
57
56
55
54
53
52
51
.271284 50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
.262229 i 20
10.261929 i 19
.261629 18
.261329 17
.261029 , 16
.260729 ; 15
.260430 14
.260130 13
.259831 12
.259532 : 11
.259233 ! 10
9
8
7
6
5
4
3
2
1
^V
29'
COSINES, TANGENTS, AND COTANGENTS.
16a'
Sine.
1
2
3
4
6
6
7
8
9
10
11
13
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
81
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9.0a5571
.6«5r99
.086027
.686254
.68(>I82
.686709
.686936
.687163
.687389
.687616
.687843
9.668069
.688295
.688521
.688747
.688972
.689198
.689423
.689648
.689873
.690098
9.690323
.69a>48
.690772
.690996
.691220
,691444
.691668
.691892
.692115
.692339
9.692562
.692785
.693008
.693231
.69;«53
.693676
.693898
.694120
.694;i42
.6945(>4
9.694786
.695007
.695229
.695450
.695071
.69.5892
.6901 13
.090334
.696554
,690775
9.696995
.697215
.6974;^
.697&'>4
.697874
.698094
.698313
.6985:^
.698751
9.698970
' I Cosine.
D. r
3.80
3.80
3.78
3.80
3.78
3.78
3.78
3.77
3.78
3.78
3.77
3.77
3.77
3.77
3.75
3.77
3.75
3.75
3.75
3.75
8.75
8.75
3.73
3.73
8.73
3.73
3.7:3
3.73
3.72
3.7:i
3.72
3.73
3.72
8.72
3.70
3.72
8.70
8.70
3.70
3.70
3.70
3.68
3 70
3 68
8 OS
3.68
3.08
3.68
3.67
s.m
3. 07
3.67
3.67
3.65
3.67
3.67
3.65
3.65
3.&5
3.65
Cosine.
X !i
D. 1"
D. r.
9.941819
.941749
.941679
.941609
.911533
.941 40 J
.9U31)S
.941328
.941258
.9111^^7
.941117
9.941046
.940975
.9409a5
.940834
.910763
.940693
.940622
.94a551
.940480
.940409
9.940338
.940267
.940196
.940125
.940054
.93:)982
.939911
.939840
.939768
.939697
9.939625
.939554
.9:39482
.9:B9410
.9393:39
.9:39207
.939195
.9:30123
.939052
.938980
9.938908
.9:388:30
.9:38703
.938691
.9:38619
.938547
.938475
.938402
.938:330
.938258
9.938185
.938113
.938040
.9:379()7
.9:37895
.937822
.9:37749
.9:37676
.9:37004
9.937531
Sine.
l.li
1.17
1.17
1.17
1.17
1.18
1.17
1.17
1.18
1.17
1.18
1.18
1.17
1.18
1.18
1.17
1.18
1.18
1.18
1.18
1.18
1.18
1.18
1.18
1.18
1.20
1.18
1.18
1.20
1.18
1.20
1.18
1.20
1 20
1.18
1.20
1.20
1.20
1.18
1.20
1.20
1.20
1.22
1.20
1.20
1.20
1.20
1.22
1.20
1.20
1 00
1.20
1 ^»
1 . 'SfA*
1 i»
A • 'V^
1.20
1 kV>
1.22
1.20
1 o.>
Taiif
r» I
9.74:J7r>2
.744050
.744:«8
.744645
.744iM3
.745240
.7455:38
.745835
.740132
.746429
.746726
9.747023
.747319
.747616
.747913
.748209
.74&505
.748801
.749097
.749393
.749689
9.749985
.750281
.750576
.750872
.751107
.751462
.751757
.752052
.752347
.752642
9.7.-)2937
.7.'):32:31
.5:3528
75
.7.'):3820
.754115
.754409
.754703
.751997
.755291
.755585
9.7.')5878
.75()172
.756405
.7.56759
.757052
.757:W5
.757(J:38
.757931
.758224
.75a517
9.758810
.759102
.759;W5
.759087
.759979
.7002?2
.760564
.760a56
.701148
9.701439
D. r.
4.97
4.97
4.95
4 97
4.95
4.97
4.95
4.95
4.95
4.95
4.95
4.93
4.95
4
4
4
4
4
4
4
4
a5
93
93
93
93
93
93
93
4. as
4.92
4.93
4.92
4.92
4.92
4.92
4.92
4.92
4.92
90
92
90
92
90
4.90
4.90
4.90
4 90
4.88
4.90
4.88
4.90
4.88
4.88
4.88
4.88
4.88
4.88
4.88
4 87
4.88
4.87
4.87
4.88
4.87
4.87
4.87
4.85
Cotang.
10.256248
.255950
.255652
.25.5:355
.255057
.254760
.25^4402
.254105
.2.53868
.25:3571
.253274
10.252977
.252081
.2523W
.252087
.251791
.251495
.251199
.250903
.250607
.250311
10 250015
.249719
.2494-^
.249128
.24883:3
.248538
.248243
.{M7948
.247053
.217358
10.247063
.240709
.246474
.246180
.5M5885
.245591
245297
.245003
.244709
.244415
10 2441 '22
.24:3828
.2435:35
.24:3241
.iM2948
.242655
.242:362
.^42009
.241776
.5M1483
10.241190
.24081W
.240005
240313
.240021
.2:39?28
.2:39436
.239144
.238a52
10.238561
19
18
17
16
1 15
14
13
13
11
10
1>.V. i*^ C^oVaxv^A \i.V A '^^^'^^ ^
119'
^^h
80'
TABLE XII. — ^LOGARITHMIC SINES,
149'
L
ISO'
/
Sine.
9.698970
1
.699189
2
.699407
3
.699626
4
.699844
6
.700062
6
.700280
7
.700498
8
.700716
9
.7001133
10
.701151
11
9.701308
12
.711585
13
.701802
14
.7oroi9
15
.70 «6
16
.702452
17
.702669
18
.702885
19
.703101
20
.703317
21
9.703533
22
.703749
23
.703964
24
.704179
25
.704^395
26
.704610
27
.704825
28
.705040
29
.705254
30
.705469
31
9.705683
32
.705898
33
.706112
34
.706326
35
.706539
36
.706753
37
.706967
38
.707180
39
.707393
40
.707606
41
9.707819
42
.708032
43
.708245
44
.708458
45
.708670
46
.708882
47
.709094
48
.709306
49
.709518
60
.709730
51
9.709941
62
.710153
63
.710364
64
.710575
65
.710786
66
.710997
67
.711208
68
.711419
69
.711629
60
9.711839
D. r.
^ / Cosine. '
3.65
8.63
8.65
8.68
3.63
3.63
3.63
8.63
3.62
3.63
3.62
3.62
3.62
3.62
3.62
3.60
3.62
8.60
3.60
8.60
3.60
8.60
8.58
8.68
8.60
3.68
3.58
3.68
3.57
3.58
3.57
3.58
3.57
3.57
3.55
3.57
3.57
3.55
3-55
3.55
3.55
3.
3.
3.
3.
.55
.55
3.55
3.53
.53
.53
3.53
3.53
3.53
3.52
3.53
3.52
3.52
3.52
3.52
3.52
3.52
3.50
3.50
D.J'
Cosine.
9.937.531
.937458
,937385
.937318
.937238
.937165
.937092
.937019
.936946
,936872
.936799
9.936725
.936652
.936678
.936605
.936431
.936357
.936284
.936210
.936136
.936062
9.935988
,935914
.935&40
.935766
.935692
.935618
,935543
.935469
.935395
.935320
9.935246
.9a5171
.935097
.935022
.934948
.934873
.a^798
.934723
.934649
.934574
9.934499
.9134424
.934349
.934274
.934199
.934123
.934048
.93:3073
.a33898
.933822
9.933747
.9a3671
.9a3596
.933520
933445
.9a3369
.93'32<)3
.933217
.9a'J141
9.9a3066
D. r.
Sine.
1.22
22
22
23
22
22
22
1.22
1.23
1.22
1.23
1.22
1.23
1.22
1.23
1.23
1.22
1.23
1.23
1.23
1.23
1.23
1.23
1.23
.23
.23
25
1.23
1.23
l.£5
1.23
1.25
1.23
1.25
1.23
1.25
1.25
1.25
1.23
1.25
1.25
1.25
25
25
25
27
25
1.25
1.25
1.27
1.25
1.27
1.25
1.27
1.25
1.27
1.27
1.27
1.27
1.25
Tang.
D.V.
20^
9.761439
.761731
.762023
.762314
,762606
.762897
.763188
.763479
.763770
,764061
.764862
9.764ft43
.764933
.766824
.766514
.765805
.766095
.766385
.766675
.766965
.767255
9.767546
.767834
.768124
.768414
.768703
.768992
.769281
.769571
.769860
.770148
9
770437
,770726
.771015
.771303
.771592
,771880
.772168
.772457
.772745
.773033
9. 773321
.773608
.773896
.774184
.774471
.774759
.775046
.775333
.775621
.775908
9.776195
.776482
.776768
.777055
.777342
.777628
.777915
.778201
.778488
9.778774
. Cotaivg.
D. 1".
4.87
4.87
4.85
4.87
4.85
4.85
4.
4
4.
4.
4.
85
85
85
85
85
4.83
4.85
4.83
4.85
4.83
483
4.83
4.83
4.83
4.83
82
83
83
82
82
82
83
82
4.80'
4.82
4
4
4
4
4
4
82
82
80
82
80
80
4.82
4.80
4.80
4.80
,78
.80
.80
.78
.80
,78
,78
.80
,78
,78
4.78
4.77
4.78
4.78
4.77
4.78
4.77
4.78
4.77
D.r
Cotanif.
10.238561
.238269
,237977
.237686
.237894
.237103
.236813
.286621
.286230
.235939
.286648
10.236357
.235067
.234776
.234486
.284195
.233905
.233615
.233325
.283035
.232745
10.232455
.232166
.231876
.231686
.231297
.231008
.230719
.230429
.230140
.229852
10.229563
.229274
,228986
.228697
.228408
.228120
.227832
.227543
.227255
.226967
10.226679
.226392
.226104
.225816
.225629
.225241
.224954
.224667
.224379
.224092
10.223805
.223618
.223232
.222946
.228658
.222372
.222085
.221799
.221512
10.221226
Tang.
60
69
68
67
56
55
54
53
52
51
60
49
48
47
46
45
44
43
42
41
40
89
38
87
36
35
34
33
32
31
30
29
28
27
28
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
tsr
11"
COSINES, TANGSNTB, AND COTANQSNTS.
lU"
8lno. D
r. Cosinp.
D. 1-.
Tang.
D. 1".
taOB.
so
~"b
711839
>9 "
i
48
48
iS
nsMia
1.37
1.37
9,7W774
4.77 '0
aaiaae
miM
.7790S0
TISdeo
58
8
»d898
11
1.27
!779t)32
aaosuH
57
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.nujia
i.ti
220088
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9!BIWS
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aiB7B7
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ai7799
48
93*076
.res48a
4^75
4-7S
4. 75
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HIWW
48
15
4B
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7&ISB 5
47
muta
aieusa
ffllTIW
.7Ii3«!W
48
eiaso
4S
47
41
SO
aiacei
40
21 9
eam
45
45
45
[Si4ai
l.lffl
9.784TM
4.73
4.78
4!78
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4.7a
m
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214MB
87
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;;8Mi8
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7878
.TSflTtffl
aissB
as
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IUUSI3
.7W088
81
BO
7180S5
.781319
aii!U8i
SO
ai s
71BKH
48 °
48
48
411
4U
mass
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I.3U
I 30
1-^
4.7a '"
718197
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4.7a
4'.7S
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afl
2iun8
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i7S9585
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St0132
^ '
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aooew
»
730345 ,
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j-ii 10
eoniioa
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4a
7ao&i9 I
Itwtio
1;S
4.70
4.70
SOKSi
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il
.rao9iio
20Haoi
.raiasi
IS
45
mm I
20M37
5
«
781389 ;
naossi
1:*!
:79184e
4:70
4170
4.70
2081M
4
7*1570 I
.■wsisa
20XX
8
i3
a
19
inssse
GO
raaiBi ^
949207
.798974
amoae
1
1.32
1 aa
lias
~.i~
g.TSSMB
4^70
4.70
4,68
4.70
4,eB
3:^ „
«
S0li4S2
s
72S7B1 i
iiaSBTi
:7fl38l9
Tism \
938893
.79J101
9!«NI5
5
TsMdo ;
723003 ;
9SWA7
58
7S3a» ;
93«.™
'.Tinm!
TMom ;
.7B6608
aoMoa
1
HI B
9:»i«)
201211
~^~(
OHine, D
IT i~
sine.
Cutaiig.
D.1-.
Can?,.
TABLE XII. — LOGARITHMIC SINES,
147*
1
2
8
4
6
6
7
8
9
10
11
12
18
14
15
16
17
18
19
20
21
22
28
24
26
26
27
28
29
80
81
82
88
84
8S
86
37
38
39
40
41
42
48
44
45
46
47
48
49
60
51
62
68
54
65
66
67
58
69
60
zzz
Sine.
9.724210
.724412
.724614
.724816
.725017
.723219
.7^420
.725622
.725823
.726024
.786225
9.726426
.726626
.726827
.72r027
.727228
.727428
.727628
.727828
.728027
.728227
9.728427
.728626
.728825
.729024
.729223
.729422
.729621
.729620
.780018
.780217
9,780415
.730613
.780811
.781009
.781206
.73J404
.781602
.781799
.781996
.732193
9.732390
.732587
.732784
.732980
.733177
.733373
.733569
.783765
.783961
.784157
9.734853
.7^549
.784744
.734939
.785135
.735330
.785525
.785719
.785914
9.7S6109
D. 1'.
3.37
3.37
8.37
8.35
3.37
3.35
3.87
3.35
3.35
3.85
3.35
3.33
3.35
3.33
8.35
3.33
333
3.33
3.32
3.33
3.33
3.32
3.32
3.32
3.32
3.32
3.32
8.32
3.80
8.32
8.30
3.80
8.30
3.30
3.28
8.30
3.30
3.28
3.28
3.28
3.28
8.28
3.28
3.27
3.28
8.27
3.27
3.27
3.27
3.27
8.27
3.27
3.25
3.25
3.27
3.25
8.25
8.23
8.25
3.25
Coalne. j D, 1\ 1 1 Sine.
Cosine.
D. 1'. , Tang.
D. r.
Cotang.
9.928120
.928342
.928263
.923183
.928104
.928025
.927946
.927867
.927787
.927708
.927623
9.927549
.927470
.927390
.927310
.927231
.927151
.927071
.926991
.926911
.926831
9.926751
.926671
.926591
.926511
.92&ldl
.926351
.926270
.926190
.926110
.926029
9.925949
.925868
.925788
.925707
.925626
.925545
.925465
.925384
.925303
.925222
9.925141
.925060
.924979
.924897
.924816
.924735
.924654
.924572
.924491
.924409
9.924328
.924246
.924164
.924083
.924001
.923919
.923a37
.923755
.923673
9. 923591
1.30
1.32
i.m
1.32
1.32
1.32
1.32
1.33
1.32
1.32
1.33
1.32
1.3:5
1.33
1.32
1.33
1.33
1.33
1.33
1.33
1.33
1.33
.33
.33
.33
.33
1.35
1.33
1.33
1.35
1.33
1.35
1.33
1.35
1.35
1.35
1.33
1.35
1.35
1.35
1.35
1.35
1.35
1-37
1.35
1.35
1.35
1.37
1.35
1.37
1.35
1.37
1.37
1.35
1.37
1.37
1.37
1 37
1.37
1.37
D. r. 1
208
9.795789
.796070
.796351
.796632
.796913
.797194
.797474
.797755
.798036
.798316
,798596
9.798877
.799157
.799437
.799717
.799997
.800277
.800557
.800836
.801116
.801396
9.801675
.801955
.802234
.802513
.802792
,803072
.803351
.803630
.803909
.804187
9.804466
.804745
.805023
.805302
.805580
.805859
.806137
.806415
.806693
.806971
9.807249
.807527
.807805
.808083
.806361
.808638
.808916
.809193
.809471
.809748
9.810025
.810302
.810580
.8lOa57
.811134
.811410
.811687
.811964
.812241
9.812517
Cotang. I D. V
4.68
4.68
4.68
4.68
4.68
4.67
4.68
4.68
4.67
4.67
4.68
4.67
4.67
4.67
4.67
4.67
4.67
4.65
4.67
4.67
4.65
4.67
4 65
4.05
4.65
4.67
4.65
4.65
4.65
4.63
4.65
4.65
4.63
4 <)5
4.03
4.65
4 63
4.63
4.63
4.63
4.63
4.63
4.63
4.63
4.63
4.62
4
4
4
4
4
63
62
63
62
62
4.62
4.6:}
4.62
4
4.
4.
4.
4
4.
62
60
62
62
62
60
10.204211 ; 00
.20:i9:« : 59
.203649 58
.203368 67
.203087 56
.202806 55
.202526 54
,202245 53
.201964 52
.201684 51
.201404 50
10.201123
.200843
.200563
.200283
.200003
.199723
.199443
.199104
.198884
.198604
10.198325
.198045
.197766
.197487
.197208
.196928
.196649
.196370
,196091
.195813
10.195534
.195255
.194977
.194698
.194420
.194141
.193863
.19;i585
.193307
.193029
10.192751
.192473
.192195
.191917
.191639
.191362
.191084
.190807
.190529
.190252
10.189975
.189698
.189420
.189143
.188866
.188590
.188313
.188036
.187759
10.187483
Taug.
49
48
47
46
45
44
43
42
41
40
89
38
37
36
35
84
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
IS
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Vl'
B"
COSINES,
TANGENTS, AND COTANGENTS.
146
1
s^
D.r.
C...
11
Tang.
D, 1-.
Colang. '
!^3aixi3
3.!3
3,ai
!|
3:23
I'M
3.23
"■S
.37
D.fil»l7
4.1)3
10.1S74B3 \ 80
s
.:%40B
■jKi^
Isibfi^
4:113
4.W
4.30
4:eo
!:!8
4, BO
jeagso
: 186377
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inToso
:o.ffli''i
.ISblOl
.ra7S7*
7
a
:^7S55
,ftt,tK!.i
.B14T«
;is
i
10
i
!738820
a,23
a. 22
1
3.29
IS
>i2272
1
1
^38
!»172W
4. BO
4.00
4:uo
4.58
4. DO
4^00
4.98
'. 83007
4B
48
47
4S
44
43
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!hi7T,'iw
'. mm
I
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B.TWIBT
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89
ss
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1
8. IB
1
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1740748
: 80805 38
25
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■ wifai
'.WMia
! 80318
38
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en
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(at
ran
;:|
:i79tBa
31
.lttlH)7
30
9-7Jsmi
140
O.R^IWT
4.W
10,178343
aa
.T4«tri
Vm
ATSm
38
33
'.t-aeoi
ifleOTTi
IS
1.40
1.40
1,40
r4D
'.W^l'KKi
.ireaw 27
.74K1Z
i:i?
1'^
.T430B3
It-'Wiat
24
.743a«3
.W03a^
:ir7-.97
23
as
il
.177023
3»
! 743809
21
to
.743702
a^n
iieoaa
:i7M70
20
49
43
B.743Wa
:744550
3.IS
:bibo3i
^40
1
^40
:ssiins
II
10,1761MB
.173928
.175855
.175381
17
.744739
lis
.uioew
.ttMM
.175107
.Ktt;iifl
4.a
13
48
:re-,7i:4
:i74S87
13
49
.743 «1
l!a
.Biar*B
.KiMKU
.174014
.74Baa
1:^
lO.iraiBB
g
ta
>4flD00
8.13
Ididisi
.1731U5
8
S3
.7«S«
.(iioiflo
;S27U78
.172923
.740436
ii
,K735I
4^55
4..^3
.172849
SB
se
:BlS9t5
iiTBioa
67
.7«a!r3
s:i3
.B1W30
1:43
;^^
eo
a! 747502
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8:ai»Bt(7
10; 171013
Cosine.
D.r.
Bl.e.
".'■■
I^Q^B.V.t,
■o.v.
S -^.^-t-
^
\
M'
TABLE XII. — ^LOGARITHMIC SINES^
IW
Sine.
1
2
8
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
S4
35
36
87
38
89
40
41
42
43
44
45
46
47
48
49
60
61
62
63
64
6i
66
67
68
59
60
9.747562
.747749
.747936
.748123
.748310
.748497
.748683
.748870
.749056
.749243
.749429
9.749615
.749801
.749987
.750172
.750358
.750543
.750729
.750914
.751099
.751284
9.751469
.751654
.751839
.752023
.752208
.752392
.752576
.752760
.752944
.753128
9.753312
.758495
.753679
.753862
.754046
.754229
.754412
.754595
.754778
.754960
9.755143
.755326
.755508
.755690
.755872
.756054
.756236
.756418
.756600
.756782
9.756963
.757144
.757326
.757507
.757688
.757869
.758050
.758230
.758411
P. 758591
D. r.
Cosine.
D. r.
Tang.
D. r.
/ou / ».7«©yi
^ /Cosine. I
3.12
3.12
8.12
3.12
3.12
3.10
3.12
3.10
3.12
3.10
3.10
3.10
3.10
3.08
8.10
8.08
3.10
3.08
3.08
3.08
3.08
3.08
3.08
3.07
3.08
3.07
3.07
3.07
8.07
8.07
3.07
3.05
3.07
8.07
3.07
3.05
3.05
3.05
3.05
3.a3
3.05
3.05
3.03
3.03
3.03
3.03
3.03
8.03
3.03
3.03
3.02
3.02
303
3.02
3.02
3.02
3.02
3.00
3.02
3.00
D.l\ I Sine.
9.918574
.918489
.918404
.918318
.918233
.918147
.918062
.917976
.917891
.917805
.917719
9.917634
.917548
.917462
.917376
.917290
.917204
.917118
.917032
.916946
.916859
9.916773
.916687
.916600
.916514
.916427
.910841
.916254
.910167
.916081
.915994
9.915907
.915820
.915733
.915646
.915559
.915472
.915385
.915297
.915210
.915123
9.915a35
.914948
.914860
.914773
.914685
.914598
.914510
.914422
.914334
.914246
9.914158
.914070
.913982
.913894
.913806
.913718
.913630
913541
.913453
9.913365
1.42
1.42
1.4;}
42
43
42
1.43
1.42
1.43
1.43
1.42
1.43
1.43
1.43
1.^3
1.43
1.43
1.43
1.45
1.43
1.43
1.45
1.43
1.45
1.43
1.45
1.45
1.4:3
1.45
1.45
1.45
1.45
1.45
1.45
1.45
1.45
47
1.45
1.45
1.47
1
1
taa*
1.45
1.47
1.45
1.47
1.45
47
1.47
1.47
1.47
1.47
1.47
1.47
1.47
1.47
1.47
1.47
1.48
1.47
1.47
D. r. \
210
9.828987
.829260
.829532
.829805
.830077
.830349
.830621
.830893
.831165
.8314:37
.831709
9.831981
.832253
.832525
.832796
.833068
.833339
.833611
.833882
.834154
.834425
9.834696
.8^907
.8352:38
.835509
.835780
.836051
.836322
.836593
.836864
.837134
9.837405
.837675
.837946
.838216
.83.3487
.838757
.839027
.839297
.839568
.839838
9.840108
.840378
.840648
.U0917
.841187
.841457
.841727
.841996
.842266
.842535
9.842805
.843074
.843343
.843612
.843882
.844151
.844420
.844689
.844958
9.845227
4.55
4.53
4.
4.
4.
4.
4.
4.
4.
55
53
53
53
53
53
53
4.53
4.53
4.53
4.53
4. .52
4.
4
.4.
4.
4.
4.
53
52
53
52
53
52
4.52
4.52
4.52
4.
4.
4.
4.
4.
.52
.52
.52
.52
.52
4.52
4 50
4.52
4.50
4.52
4.50
4.52
4.50
4.50
4.50
4.52
4.50
4.50
4.50
4.50
4.48
4.
4
4.
4,
4
4.
50
50
50
48
50
48
4.50
4.48
4.48
4.48
4.50
4.48
4.48
4.48
4.48
4.48
Cotaiig. \\>.V.
Cotang.
60
10.171013
.170740
69
.170468
68
.170196
57
.169923
66
.169651
65
.169379
64
.169107
63
.168835
62
.168563
51
.168291
60
10.168019
49
.167747
48
.167475
47
.167204
46
.166932
45
.166661
44
.166389
43
.166118
42
.165846
41
.165575
40
10.165304
39
.165033
38
.164762
87
.164491
86
.164220
36
.163949
34
.163678
83
.163407
32
.163136
31
.162866
30
10.162595
29
.162325
28
.162054
27
.161784
26
.161513
25
.161243
24
.160973
28
.160703
22
.160432
21
.160162
20
10.159892
19
.159622
18
.159352
17
.159083
16
.158813
15
.158543
14
.158273
13
.158004
12
.157734
11
.157465
10
10.r)7195
9
156926
8
.156657
7
.156388
6
.156118
5
.155849
4
.155580
8
.155311
2
.155042
1
10.154773
Tang.
W^
K'
COSINES, TANGENTS, AN
D COTANGENTS.
14i
•
Sine. ri
!iii
(U ' 1. 1- '
rang.
D.r. c
jtuiB.
~
9 7Smi
09 »
00
00
Bisaiw
1,48 "
^^
4.48
4.47
!:|
4!48
4.47
1M773
60
siai!:ii
omsr
8l57fil
154230
i:i8
SI80M
1B3907
67
!75i«is :
153038
6
.T5B1W '
w
JO
30
Bi3g2i
11
.TfiaSTS \
nias33
8W839
153181
64
S47108
153893
63
l!4B
152024
g
152350
:t«suo I
BI7B13
B.TBOfiTO g
Iff
m
918388
IS '
11
W8181
1BI819
49
4B
4:48
iraiKW ;
848980
151014
46
.Miaas 1
DlliOSI
819S5I
til
lS074fl
45
MflSM
II
IS0178
1
42
Itbimb 1
I
911074
850325
14SC75
SI1G84
S505B3
149407
17
S1I4!»
4:45
SS
ivess^ ;
B1140J
1.B0
8511^
88
S3
.T8!(7i3 ;
BliaiB
851396
14860*
gj
.Teaasa *
Bitaao
851BM
*-*5
148336
!S
»7
B51B31
X
4:45
W
ilso
1.60
14JB34
na
iisaeoo I
91080U
14^67
29
.76*777 ;
BlOTTfl
30
.7831IH 1
Biosas
8&3a68
146733
31
D.T«13( „
IS "
eiOMB
150 ^
853535
i-i '"
146405
31
.7M308 J
853802
140198
33
^9
34
!7(We«-2 5
n
ilw
35
.TWS38 I
Bioasa
^54003
33
<3
0101J4
854870
145130
I'.m
W5137
it
144863
38
1
at
'.TVSMi r
USl
.766720 1
iBBTSa
855BS8
144002
4a
a.TSKoa J
Si
IM *
i!5a
s
4A5 >«
4ST96
43629
1
43
;7(18W7 5
43203
.T6ft«3
i
90B419
a7004
4:43
42990
.TBeaos
joBsaft
«r270
12730
46
.TB8774
:w
47
.TaiWIB
I»0I4B
1
48
.7B71M
13
noBOM
419B1
49
.707300
noewM
4:43
4I6M
SO
906373
413SB
92 '
« 1
1 B2 "
443 '"
4,43
tss
:7eiKi
xmo
,7fiT9W
XBSSB
!:S
859400
8 i
Gam
!7B88l8
M
,708^2
i
»fl331
I '.13 1
443
39802
4
S
■7fli71
mess3
ia0464
39530
3.
to
!709IMB ;
« 1 fl
1.53 9
1
eo
B.msaa
B07U58
(Giasi
138739
Conine. D
1-. 11 Sine.
D. r. O
^i^
D,V. "
*».t-
^
36'
TABLE XII. — LOGARITHMIC SINES,
143<
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Sine.
r>. 1".
9.769219
.769393
.769566
.769740
.769913
.770087
.770260
.7704*3
.770606
.770779
.770952
9.771125
.771298
.771470
.771643
.771815
.771987
.772159
.772331
.772503
'^'r2675
to
9.772847
.773018
.773190
.773361
.773533
.773704
.773875
.774046
.774217
.774388
9.774558
.774729
.774899
.775070
.775240
.775410
.775580
not
50
.775920
.776090
9.776259
.776429
.776598
.776768
.776937
.777106
.777275
.777444
.777613
.777781
7
/
9.777950
.778119
.778287
.778455
.778624
.778792
.778960
.779128
.779295
9.779463
2.90
2.88
2.90
2.88
2.90
88
88
88
88
2.88
2.88
2.88
2.87
2.88
2.87
2.87
2.87
2.87
2.87
2.87
2.87
2.85
2.87
2.85
2.87
2.85
2.85
85
,85
83
2.85
2.83
2.85
2.83
2.83
2.83
2.83
2.83
2.83
2.82
2.
2.
2.
2.
2.
2.
.83
.82
.83
.82
.82
.82
2.82
2.82
2.80
2.82
2.82
2.80
2.80
2.82
2.80
2.80
2.80
2.78
2.80
r/
Cosine. I D. V.
Cosine.
D. r.
9.907C58
.9078(J6
.907774
.907682
.907590
.907498
.907406
.907314
.907222
.907129
.907037
9.906945
.906852
.906760
.906667
.906575
.906482
.906389
.906296
.906204
.906111
9.906018
.905925
.905832
.905739
.905645
.905552
.905459
.905366
.905272
.905179
9.905085
.904992
.904898
.904804
.904711
.904617
.904523
.904429
.904335
.904241
9.904147
.904053
.903959
.903864
.903770
.903676
.903581
.903487
.903392
.903298
9.903203
.903108
.903014
.902919
.902824
.902729
.902634
.902539
.902444
9.902349
Sine.
1.53
1.53
1.53
1.53
1.53
1.53
1.53
1.53
1.55
1.53
1.53
1.55
1
1
1,
53
55
53
1.55
1
1.
1
.55
.55
.53
1.55
1.55
1.55
1.55
1.55
1.57
1.55
1.55
1.55
1.57
1.55
1.57
1.55
1.57
1.57
1.55
1.57
1.57
1.57
1.57
1.57
1.57
1.57
57
58
57
57
58
57
1.58
1.57
1.58
1.58
1.57
1.58
1.58
1.58
1.58
1.58
1.58
1.58
Tang.
9.861261
.861527
.861792
.862058
.862323
.862589
.862854
.863119
.86.33a5
.863650
.863915
9.864180
.864445
.864710
.864975
.865240
.865505
.865770
.866035
.866300
.866564
9.866829
.867094
.867358
.867623
.867887
.868152
.868416
.868680
.808945
.869209
9.869473
.86'J737
.870001
.870265
.870529
.870793
.871057
.871321
.871585
.871849
9.872112
.872376
.8?2640
.872903
.873167
.87*430
.87"3094
.873957
.874220
.874484
9.874747
.875010
.875273
.875537
.875800
.870063
.8715:526
.87'6589
.876852
9.877114
D. r.
Cotang.
4.43
4.42
4.43
4.42
4.43
4.42
4.42
4.43
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.40
4.42
4.42
4.40
4.42
4.40
4.42
4.40
4.40
4 42
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.38
4.40
4.40
4
4
4
4
4
4
4
.38
,40
.38
,40
,38
.38
,40
4.38
4.38
4.38
4.40
4.38
4.38
4.38
4.38
4.38
4.37
10.138739
.138473
.138208
.137942
.137677
.137411
.137146
.136881
.136615
.136350
.136085
10.135820
.135555
.135290
.135025
.134760
.134495
.134230
.133965
.133700
.133436
10.133171
.132906
.132642
.132377
.132113
.131848
.131584
.131320
.131055
.130791
10.130527
.130263
.129999
.129735
.129471
.129207
.128943
.128679
.128415
.128151
10.127888
.127624
.127360
.127097
.126833
.126570
.126306
.126043
.125780
.125516
10.125253
.124990
.124727
.124463
.124200
.123937
.123674
.123411
.123148
10.122886
J26'
D. 1* . \\ Cotang. V D. V . \ '^a.w^.
212
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
S6
85
34
33
38
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
37*
COSINES, TANGENTS, AND COTANGENTS.
142'
1
2
3
4
5
6
7
S
10
11
12
13
14
15
19
17
18
19
20
21
22
23
24
25
26
27
23
29
30
31
S'2
33
34
35
3<>
37
a?
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
eo
Sine.
9.779463
.779631
.779798
.779966
.Tsoias
.780300
.780467
.780634
.780801
.780968
.781134
9.781301
.781468
.781684
.781800
.781966
.782132
.782298
.782464
.782630
.782796
9.782961
.783127
.783292
.783458
.783623
.7a3788
.783953
.784118
.784282
.784447
9.784612
.784776
.784941
.785105
.785269
.7854*3
.785597
.785761
.785925
.786089
9.786252
.786416
.786579
.786742
.786906
.787069
.787232
.787395
.787557
.787720
9.787883
."788045
.788208
.788370
.788532
.788694
.788856
.789018
.789180
9.78a342
Cosine.
D. 1".
2.80
2.78
2.
2.
2.
2.
80
78
78
78
2.78
2.78
2.78
2.77
2.78
2.
2,
2.
2.
2.
2.
2.
78
77
77
77
77
77
77
2.77
2.77
2.
75
2.77
2.75
2.77
2.75
2.
2.
2.75
2.73
2.
2.
75
75
75
75
2.73
2.75
2.73
2.73
2.73
2.73
2.73
2.73
2.73
2.72
2.73
2.72
2.72
2.73
2.72
2.72
2.72
2.70
2.72
2.72
2.70
2.72
2.70
2.70
2.70
2.70
2.70
2.70
2.70
D r.
Cosine.
9.902349
.902253
.902158
.902063
.901967
.901872
.901776
.901681
.901585
.901490
.901394
9.901298
.901202
.901106
.901010
.900914
.900818
.900722
.900626
.900529
.900433
9.900337
.900240
.900144
.900047
.899951
.899854
.899757
.899660
.899564
.899467
9.899370
.899273
.899176
.899078
.898981
.898884
.898787
.898689
.898592
.898494
9.898397
. .898299
.898202
.898104
.898006
.897908
.81)7810
.897712
.897614
.897516
9.897418
.897320
.897222
.897123
.897025
.896926
.896828
.896729
.896631
9.896532
Sine.
D. 1".
1.60
1.58
1.58
1.60
1.58
1.60
1.58
1.60
1.58
1.60
1.60
1.60
1.60
1.60
1.60
1.60
1.60
1.60
1.62
1.60
1.60
1.62
1.60
1.62
1.60
1.62
1.62
1.62
1.60
1.G2
1.02
1.62
1.62
1.63
1.62
1.62
1.G2
1.63
1.62
1.03
1.02
l.a3
1.G2
1.63
1.63
1.63
1.63
1.63
1.03
1.63
1.63
1.6:3
1.63
1.65
1.63
1.65
1.63
1.65
1.&3
1.65
I D. r
Tang.
D. 1".
Cotang.
/
9.877114
4.38
4.38
4.38
4.37
4.38
4.38
4.37
4.38
4.37
4.38
4.37
10.122886
60
.877377
.122623
59
.877640
.122360
58
.877903
.122097
57
.878165
.121835
56
.878428
.121572
55
.878691
.121309
54
.8:'8953
.121047
53
.879216
.120784
52
.879478
• .120522
51
.879741
.120259
50
9.880003
4.37
4.38
4M7
4.37
4.37
4.38
4.37
4.37
4.37
4.37
10.119997
49
.880265
.119735
48
.880528
.119472
47
.880790
.119210
46
.881052
.118948
45
.881314
.118686
44
.881577
.118423
43
.8818:39
.118x61
42
.882101
.117899
41
.882363
.117637
40
9.882625
4.37
4.35
4.37
4.37
4.37
4.37
4.35
4.37
4.35
4.37
10.117375
S9
.882887
.117113
38
.883148
.116a52
37
.883410
.116590
36
.883672
.116328
35
.8839*4
.116066
34
.884196
.115804
83
.884457
.11554:3
32
.884719
.115281
31
.884980
.115020
30
9.885242
4.37
4.:35
4.35
4.37
4.35
4:37
4.35
4.35
4.. 35
4.35
10.114758
29
.885504
.114496
28
.885765
.1142:35
27
.886026
.113974
26
.886288
.113712
25
.886549
.11:3451
iU
.886811
.113189
23
.887072
.112928
22
.887*33
.112667
21
.887594
.112406
20
9.887855
4.35
4.37
4.35
4.35
4.35
4.33
4.35
4.35
4.'i5
4.35
10.112145
19
.888116
.111884
18
.888378
.111622
17
.888639
.111361
16
.888900
.111100
15
.889161
.110839
14
.889421
.110579
13
.889682
.110318
12
.889943
.110057
11
.890204
.109796
10
9.8904&5
4.33
4.35
4.35
4. as
4.a5
4.33
4.35
4.33
4.35
10.109535
9
.890725
.109275
8
.890986
.109014
7
.891247
.108753
6
.891507
.108493
6
.891768
.108232
4
.892028
.107972
8
.892289
.107711
2
.892549
.107451
1
; 9.892810
10.107190
A Cotaxv^.
•^liA'.
k "^VCCSJ^.
\ »
i27'
1\Z
TABLE xn. — LOOASITHHIC SIKBS,
»°
COaiKES, TANGBNTB, AND COTANGENTB.
MO*
Sloe.
D. I-. C
□sine.
"■'•■i
fli.8.
D. r. c
ilang.
80
a
TOMSTS
|80 8
800508
1.73 "
ooesro
4.M '«
4:90
091M1
ttMoa
iioBeat
(91873 se
lira
mem
ea
Tnatsg
TSKMitt
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lira
mm
0911698
XW600
)U0»»
K
TSaWG
if
tetme
leseas
Ruon?
asikws
11
oiwas
UHB585
BB
9
SOihra
iwnsTfl
01 0898
MW07
61
eiosei
ii
889874
i;«
911309
4.IM
0887^
IS
B»737
880371
H1487
)88S3S
48
13
9i)IM1B
mm
388981
1BT780
48
le
eoiaoB
JDMB8
»7M8
IT
8D15I1
3887H
49
18
48
801813
;8KM8
W
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88SU4
BiaSiM
ffli8471
40
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sceiffl
fisr 8
888911
3.-3 ^
l.T
1.-3
9137m
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914302
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176188
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TABLE Xll, — LOGARITHMIC SINES,
Bine.
„^..
o..-^
ane.
D. r. oc
itang.
a
1mm
II
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in •
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4.87 '«
4.28
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4.28
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4.!J7
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4,88
4.27
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2:50
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15
45
18
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11061
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011060
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070804
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38 7l»
11
Ds»ra8
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929904
070038
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S1IW4
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930987
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TABLE XII. — LOGAHITHUIC SINES,
;
- I a-u,. Iv.i:
«" COSINES, TANGENTS, AND COTANGENTS. 186?
1
iS
H"
TABLE
xn-
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a I-. 5
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4*'
INDEX.
AnoBslraJ heredity 78
Amieiida. corjeLatioD. - , . - . ^ ,,-.-.,-,-,.-,-.-. - ^ ,-.-.-., . 76
. variability 67
ApbidiB, pec Honapnda 66
AriUi
Avenss 13. 17
deviation. 16
Avea, oorrelation 77
, variability 65
o[ MtB. as
Bimodal trequenoy poTyBDM 73
Birtis, «e Avaa.
Braohiopoda, variahility 87
BruiiKV^ca calculator .,.......,.., S
Bryophyta, variability 71
Bryovw. correlation 77
, heroriitv. . SO
, variability 67
CaloulatinE maohiae^ --.-..,.-.- ,.,....., 7
lablos. ... 7
c»ryophyuaoe», variability.'."'.!!!!;!!!!;!:!!;!:;:;!!!;;;;;;:;;; 67
Character defined 1
Cbauvensfs oritarioQ 12
Cl«83, defined I
cioaenesB of fit: ; 1 '.;:;:;::;:;::;:;::;::;;;;;;;;:::!!!!:;;;;:;; 24
CoeffieioDt ol torrolation 44
variability ;;;;:;!;;:;: a, as
C<Blenl«ra(a. eae Hydrt.nie.iu*a,
■Color. meaMirement of 6
Compoailw, com^lation 78
, variability 69. 70
Comptometer 7
dated van
■CoroaeeK, variability. , ,
-Criminali, akuU iadeit. .
Oridcal functioa
CnioiCarK. vaiiaJuIity, . ,
■££%
CrusUoea, Amphipoda, Tariatrility .
, Daphnia, oornJatioD.
, Eupasunu,
liereditv. 79, 80
, variabiUty 63. 86
Beeimal places, number to en —
DipBBBte. VHTiability.
to employ -..-.-
FibIihi, Bee Pisoes.
FreqiKQcy notygon , ...-...---^-.--.-.-..-^-.-..-.-•-. ,.-,,*.< 63
Fnut, variability of 71
Galton's differeniw prsblem .' 27
Gastropoiia. correlatioa. ,..-., 77
, variability 67
GenmBtrio mean IS
Heredity 6S, 78
, BDceetral TO
Beiapoda. correlation 77
, variability 66
Homo, oorrelation 73
, eye-color, heredity of ,.■'-■'-■'■■ - ...-.-,--.,-... 70
, rertilitv. heredity of. 70, 80
linherilince 70
, head Indei, heredity of 70
, mental characters, heredity of. SO
. akelBlal, correlation 74
I sUtiire. correlation; .' .".'.!! [i !!!!!!!!!;!!! i ! i ! I!!!! I !!!!! 79
, weight, variability. --..-,...-,-.-....,.,-.-.,,.-. 63
.varfability 6* .
ISoe alBO Naquada ra™) 64, 6S, 74
HoraotypoBiB . . . 81
iscdatioo «
variability. 16, 17
n. epecifie vari
Integral vr~—
iL^aves, variability.
LBgumiiioHB. vartaL,ij.,j. .. .
Lepidoptera, variability. . . .
Loeded ordiaatee, metliad ol
Local raoea
Longevity, iDberitaace af. . .
, vaiiatHlity. . . .
irrelation. 76
. Jial races 84
variability 68, 71
INDEX. 226
PAGE
MendeUsm 67, 82
Mid-departure 16
Mode 13
Multimodal polygons 39. 73
Multii>le organ 1
Mutations 63
Myriapoda, correlation 76
, variability 66
Naquada race, skeletal variability 64, 65, 74
Normal curve of frequency 22
Nmnber of variates to employ -2
Oroliidacefle, variability 71
Organ variation 1
Pi4>averace8B, variability 70
Partial variation. 1
Person 1
Pisces, correlation 76, 77
, local races 83
, variability 66
Plants, correlation 78
, homotyposis. 81
, variability. 60
Prepotency 78
Primulacese, variability 70
Probable departure. Id*-
dinerenoe 16 1-
error. 14
in uniparratal heredity 56
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 60
Range of variability 25
Ranunculacese, variability. 60
Recessive characters 58
Rectangles, method of, in platting frequency distributions 11
Rejection of extreme variates 12
Relative variability of the sexes 63
Rosaces, variability 70
Siq>idaceflB, variability 70
Bcrophulanacee, 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.
gyrmmetry in frequency distribution 10
Telegony 82
Types of frequency distribution 10, 71. 72
Variability. 15, 17, 62-71
Variant 1
Variate. I
Weight, variability, see Homo.
FEB 8 - 1916
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