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Science Library 

62 H 



STATISTICAL METHODS 



WITH SPECIAL REFERENCE TO 

BIOLOGICAL VARIATION. 



tty^'^ BY 

0. B: DAVENPOET, Ph.D., 

Instrttctor in Zoology at Harvard University, 



FIRST EDITION. 
FIRST THOUSAND. 



NEW YORK: 

JOHN WILEY & SONS. 
London: CHAPMAN & HALL, Limited. 

1899. 



Copyright, 1899, 

BY 

C. B. DAVENPORT. 



ROBERT DRUMMOND, PBINTBR, NEW TOBK. 



I 
I 

I i 



PREFACE. 



-J 

I 





V 



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

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

C. B. Davenport. 

BioLoaicAL Laboratory of the Brooklyn Institute, 

Cold Sprino Harbor, LoNa Island, 

June 29, 1899. 

ill 



CONTENTS. 



CHAPTER I. 
On Methods of Mbasubino OBOAinsMS. 

PAQB 

Preliminary DeflnitioDS 1 

Methods of Collecting Individuals for Measurement 2 

Processes Preliminary to Measuring Characters 2 

The Dermination of Integral Variates—Methods of Counting 8 

The Detwmination of Graduated Variates— Method of Measurement.. 

Straight lines on a plane«urface 

Distances through solid bodies or cavities 

Area of plane surfaces 

Area of a curved surface 

Form of a plane figure 

Characters occupying three dimensions of space 

Characters having weight 9 

Color characters 9 

Marking-characters 10 

CHAPTER n. 

On the Serution and Plotting of Data and the Frequenot 

Polygon. 

Seriation 11 

Plotting i 12 

Method of loaded ordinates 12 

Method of rectangles 18 

Certain constants of the Frequency Polygon 18 

The mean 18 

The mode 14 

The median magnitude 14 

The probable error of the mean 14 

The index of variability 15 

The probable error of the standard deviation 15 

Average deviation and probable error 15 

CoefQcient of variability 15 

V 



yi CONTENTS. 

CHAPTER III. 
The Classes of Frequency Polygons. 

PAGE 

Classiflcation 16 

To classify a simple frequency polygon 16 

The normal curve 18 

To compare any observed curve with the theoretical normal 

curve 19 

The index of abmodality 19 

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

observed polygon 19 

The normal curve as a binomial curve 20 

Example of a nearly normal curve. 20 

Other unimodal frequency polygons . 21 

Curves of limited range 93 

Asymmetry or skewness 22 

To compare any observed frequency polygon of Type I with its 

corresponding theoretical curve 22 

To compare any observed frequency polygon of Type II with its 

corresponding theoretical curve 23 

To compare any observed frequency polygon of Type III with its 

corresponding theoretical curve 23 

To compare any observed frequency curve of Type IV with its 

corresponding theoretical curve 24 

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

Multimodal curves 26 

CHAPTER IV. 

COBBBLATED VARIABILITY. 

General principles 80 

Methods of determining coefficient of correlation 82 

Oalton^s graphic method 82 

Pearson *s method 32 

Duncker^s brief method 88 

Spurious Correlation in Indices 35 

Heredity 85 

Uniparental inheritance 86 

Biparental inheritance 86 

Galton's law of ancestral heredity 87 

CHAPTER V. 

Soke Applications of Statistical Biological Study. 

The laws of variation. ... 88 

The causes of variation 38 

Selection 38 

The origin of species 38 



CONTENTS. VU 

PAGE 

The definition of species 3d 

Distinction between species and varieties S9 

Criterion for homology * 39 

Prepotency 39 

Selected Bibliooraphy of Works on the Quantitative Study of 

Organism 40 

expi^anation of tables 43 

List of Tables. 

Table I. Formulas. 53 

** II. Certain constants and their lofcarithms 54 

" m. Table of ordinates of normal curve, or values of -^ 

Vo 

corresponding to values of — 55 

" IV. Table of values of the normal probability int^ral corre- 
sponding to values of — ; or the fraction of the area of 

the curve between the limits and -i or and 56 

** V. Table of Log r functions of p 57 

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

metric system 50 

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

Vm. Squares, cubes, square-roots, cube-roots and reciprocals, 60 
IX. Logarithms of numbers 77 



(4 



Z. Logarithmic sines, cosines, tangents and cotangents 104 



STATISTICAL METHODS 

WITH SPECIAL BBFEBBNGE TO 

BIOLOGICAL VAEIATION 



CHAPTER I. 
On Methods of Mbasurino Organisms. 

Preliminary I>efinitions. 

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

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

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

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

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

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

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



2 STATISTICAL METHODS. 

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

Methods of Collecting Individuals for Meas- 
urement. 

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

Having settled upon the general eondiUans, of race, sex, 
locality, etc., toliieh the individuals to he measured must fulfil^ 
take Vie individuals metliodieally at random and vfithout possible 
selection of individuals on the basis of the magnitude of the 
character to be measured. If the individuals are simply not 
consciously selected on the basis of magnitude of the character 
they will often be taken sufficiently at random. 

Processes Preliminary to Measuring 

Characters. 

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

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

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



MEASUREMENT OF ORGANISMS. 3 

gmphic 'paper. More or less transpareut organs, such as 
leaves, petals, iDsect-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 dramngs are a convenient although slow method of 
reproducing on paper greatly enlarged outlines of microscopic 
characters, such as tbe 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 Determinatiou 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 
small fields each containing only a few organs. Counting 
under the microscope, e.g,, the number of spines, scales or 
plant-hairs per square millimetre, may be aided by cross-hair 
rectangles in the eyepiece. Tbe 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 HETHODS. 



Straight lltiea on a plane surface are easily meas- 
ured by means ot a measurlng-acnle of some sort. The meas- 
urement should always be metric because 
thU is the unirei-Btil scientific system. Vari- 
ous kluda of scales roay be obtained of 
optical companies and Lai'dware dealers, — 
BUCb as sleel measuring tapes, graduated to 
millimetres {about $1.00), and steel rules' 
(6 cm. to 15 cm.) graduated to | of a milli- 
metre. Steel "sprlDg-bow" dividers \rltb 
mllled-bead screw are useful for getting 
distances which may be laid off on ft scale. 
Tortuous lines, e.g.. the couiour of the 
serrated margin of a leaf or Ibe outer 
margin of the wing of a sphinx motb, may 
be measured by a map-measurer ("Eiitfer- 
Duugsmesaer," Fig. 1), supplied at artist's 
and engineer's supply stores nt about ta.SO. 
Distances through solitl bodies 
or cavities are niensured by cidlpcrs of 
some sort. Calipers for measuriog diameters 
of solid bodies are made iu various slylcs. 
Micrometer screw cnliptrs (" speeded") 
reading to one-bundredtlis of a millimetre 
Titi. I. and sold by dealers in pliysic:i1 apparatus for 

about 15.00 are excellent for determining diaiiictcrs of bones, 
birds' Gggs, gastropod shells, eic. Leg calipers for rougher 
work can be obtained for from 30 cents to |4,00, The 
micrometer "caliper-square," availnblc for inside or outside 
measurements and measuring to hundredths of a millimetre, 
la a useful instrument.* 

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



MEASUREMENT OF ORGANISMS. 5 

scratches may be made clearer. The number of squares 
covered by the surface is counted (fractional squares being 
mentally summated) and the required area is at oucc 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 oS. 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. 



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

Linear, more than thrice longer than wide and of nearly the 
same breadth throughout (Fig. 8). 

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

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

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

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

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

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



8 



e 



\]) 





FiGB. 3-7. 



Ouneaie or cuneiform, wedge-shaped. 

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

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

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

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

2'runcaie, terminating as though cut off (Fig. 11). 

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

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

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

Crenate, rounded teeth (Fig. 17). 

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



MEASUREMENT OF ORGANISMS. 




8 9 10 



1 12 

Fios. 8-14. 



13 



U 



Sinuate, still stroDger waves, height equals or exceeds 
breadth (Fig. 19). 
Inmed, with sharp, deep incisions (Fig. 20). 




15 



16 



17 18 19 

Fios. 15-SO. 



90 



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

greatest length 



Index of Linearness, 



greatest breadth* 



•* " Lanceolateness, — — - — --r: — ^-r, also angle ode. 

greatest breadth ° 

, ^, , greatest length , area 

" " Oblongness, ___,_^ , ^^,^, , also 



greatest breadth' 



breadth* 



.. >■ ElHpticity, (S^t««t;gth.)-(greate8t brdth.)^ 
'^ "^ (greatest length) 

for values from 1 to .50. 



8 STATISTICAL VETHODEL 

Index of Ovalness, (g^e»'e«t ie°gth) - (greatest breadth) 

(greatest length) 
for values from .50 to .1. 



it 



it 



it 



f < 

(I 
(( 
<< 

ft 

<< 
(( 

41 



l< 



• Orbicularneas. (gre»te9tdiam.)-(greate8tbrdth.) 

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



'* Ovateness or obovateuess, 



radius of curvature of 
larger end 

radius of smaller end 



diameter at ^ 

** Cuneateness, -j-. : rr* or angle abc (line Or-e 

diameter at | 

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

" Spatulateness, ■— ^. -^ — ^ • 

*^ transverse diameter of narrow 

part of organ 

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

*' Acuteness, angle ctbc at apex. 

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

** Truncatedness, angle <ibc at apex and radius of curva- 
ture. 

*' Retuseness, ; — of i angle abc, 

2 X sine 

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

average angle of tooth. 
** Dentateness, number of teeth, average angle of tooth, 
*' Crenateness, number of waves, average radius of 

curvature of waves. 

«* T» J depth of waves ,, . 

" Bepandness, ; — ^--r — ; , average radius of cur- 

*^ length of waves ° 

vature of waves. 

** di X depth of waves ,. . 

'* Binuateness, z — =--r — z , average radms of cur- 
length of waves ° 

vature of waves. 

depth of incision 



" " Incisedness, 



opening of incision 



MEASUREMENT OF ORGANISMS. 9 

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

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

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

Red, 656 to 661 .Green, 514 to 519 

Orange, 606 to 611 Blue, 467 to 473 

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- 



10 STATISTICAL METHODS. 

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

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



SERIATION AND PLOTTING OF DATA. 11 



CHAPTER II. 

On the Sbbiation and Plotting of Data and the 

Fkbquency Polygon. 

The data obtaiued by measuring any character iixalotof 
individuals consists either of amass of numbers for the charac- 
ter in each individual ; or, perhaps, two numbers which are to 
be united to form a ratio ; or, finally, a series of numbers such 
as are obtained by the color wheel, of the order : W 40%, 2^ 
(Black) 38jf, 7 12^, G 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 vadates occurring in each class is determined. 
The number of variates in the class determines ih% freqiLerusy 
of the class. 

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

Example 1. The magnitude of 21 integral variates are found to be as 
follows : 12, 14, 11, 18, 12, 12, 14, 13, 12, 11, 13, 12, 11, 12, 10, 11, 12, 13, 12, 
13, 12, 12. In seriation they are arranged as follows : 

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

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



3.2 


4.5 


6.8 


5.6 


6.0 


8.8 


4.7 


5.2 


5.7 


6.2 


4.1 


4.9 


5.3 


6.8 


6.4 


4.8 


5.0 


5.3 


5.8 


6.7 


4.3 


• 5.1 


6.4 


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 



12 STATISTICAL METHODS. 

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

C 3.0-8.4; 3.5-3.9; 4.0-4.4; 4.6-4.9; 5.0-6.4; 
Classes ... -n 8.2 3.7 4.2 4.7 5.2 



\ 



1 -S 3 4 5 

Frequency 118 3 7 

C 5.5-5.9; 6.0-6.4; 6.5-6.9; V.0-7.4; 
Classes.... -| 5.7 6.2 6.7 7.2 

(6 7 8 9 • 

Frequency 6 3 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 5 or 10, each class should include one 
and only one of these round numbers. 

As Fechner C9T) has pointed out, the frequency of the classes and all 
the data to be calculated from the series will vary according to the 
point at which we begin our seriation. Thus if, instead of beginning the 
series with 3.0 as in our example, we begin with 3.1 we g^t the series : 

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

' *"j 8.3 3.8 4.8 4.8 3.6 

Frequency 114 3 6 

Classes i 5.0-6.0; 6.1-6.5; 6.6-7.0; 7.1-7.5; 

V**( 5.8 6.3 6.8 7.3 

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. 

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

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

» ■ ■■ ■- I ■■ 1 .1. ■■■■■.■ ■ I ^— ^M»» 

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



SEBIATION AND PLOTTING OF DATA. 



13 



In the case of graduated variates proceed as follows : Lay 
off along a horizontal line equal conliguous spaces each of 
which shall represeut one class, number the spaces iu order 



10 
6 




























^ 


I 






.-_^^ 



d 



10 



u 



12 
Fig. 21. 



13 



14 



15 



from left to right with the class magnitudes iu succession, 
and erect upon these bases rectangles proportionate in height 
to the frequency of the respective clashes (Fig. 32). 





























* 


t 








—n 


_T 



8.0 



3.5. 



4.0 4.6 



6.0 6.6 
Fio. 83. 



e.o 



6.6 



7.0 7,6 



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

CsBTAiN Constants of the Frequency Poltqon. 

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

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



Jf= 



2iV.f) 



n 



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



14 STATISTICAL METHODS. 

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

M = (3.3 X 1 + 3.7 X-1 +4.2X3 4- 4.7X3 + 5.2X7 + 6.7X6 + 6.2X8 

+ 6.7 X 1 + 7.2 X 1) -*- 25 = 6.24, 
or 

Ml = (1X1+2X1+3X3+4X8+5X7+6X5+7X3+8X1+9X1)-*- 25 = 6.08, 

M = 5.2* + .08 (5.7 - 5.2) = 5.24 

A still shortef method of finding Jf is given on page 17. 

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

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

To find its position: Divide the variates into three lots: those less than 
the middle class, of which the total number is a; those of the middle 
clasSy h; and those greater^ c. Then a-\-h -\- c = n = the total number 
of variates. Let V = the lower limiting value of the. middle class^ and 
V = the upper limiting value^ and let x = the abscissal distance of the 
median ordinate above the lower limit or below the upper limit of the 
median class according as x is positive or negative. Then in -a:b = 
x;l" — V when xis pos^itive^ or ^n — c '. b = x : I" — V when x is negative. 

Thus in the last example : 12.5 — 8 : 7 = or : 0.5; a; = .32; the median 
magnitude = 5,0 + .32 = 5.32. Or 12:5 - 10 : 7 = -x-. 0.5; a? = - .18; the 
median magnitude = 5.5 — .18 = 5.32. (Cf. p. 11.) 

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

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

i^ ,*^A^ Standard deviation [see below] ^ ^_, <t 

± 0.6745 X , *• . = ± 0.6745-7=^. 

y number of variates yn 

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

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



S 



SERIATION AND PLOTTING OF DATA. 15 

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

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



-W 



8um of [(deviation of class from mean)' 
X frequency of class] 



number of variates 



=/ 



2{xKf) 



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

/x/»«^i^ standard deviation ^ /.w^e. <^ 

± 0.6745 — . = ± 0.6745 



^2 X number of variates |^2n 

Olher Indices of Variation are the aw^era^e deTtatlon, or aver- 

age departure^ which is found thus: 

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

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

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

the coefficient of variability. In a formula, CV=z — 

(Pearson. *96 ; Brewster, '97 ) 



16 STATISTICAL METHODS. 



CHAPTER III. 

The Clabses of Frequency Polygons. 

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

A. Unimodal. 

I. Simple. 

1. Range unlimited iu both directions: 

a. Symmetrical. The normal curve. 
h, Unsymmetrical (Pearson's Type IV). 

2. Range limited in one direction, together with 

skewness (Type III). 
8. Range limited in both directions : 
a. Symmetrical, Type II. 
6. Unsymmetrical, Type I. 
II. Complex. 

B. Multimodal. 

The classification of any given curve is not always an ea«y 
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 a« follows : 

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

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

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

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

carried further. 

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



THE CLASSES OF EKEQUENCT POLYGOi^S. 17 

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

Add the products of all these departures multiplied by the 
frequency of the corresponding class and divide by n; call 
the quotient Vi, 

Add the products of the squarea of all the departures multi- 
plied by the frequency of the corresponding class and divide 
by n; call the quotient v^. 

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

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

"SIV — Vm) 

yj = _b .' =3 departure of Vm from mean. Vm being 

n 

known, M may be found [if = Fm + ^i]; * 

2{v- Vmy. 



r» = 



r« = 



n 
n 
n 



The values vi, r^, vs, ^'4, are called respectively the first, 
second, third, and fourth moments of the curve about Vm> 

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

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

Ui = 0; 

>"a = Va — Vi^; 

Mt = y» — 3yiVa + 2vi*; 

Ma = ^4 — ^yiVi + Syj^va — Svi^, 

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

* This is the^short mettiod of finding M referred to on page 14. 



18 



STATISTICAL METHODS. 



//> = ^s ~ 3j^iVa + Sj'i'; 

f^A = y* — 4:riVt + 6viVa — S^'i* + ra — ^1* + ^. 



Also, 






»» 



yUa* /^a 

i?* = 6 + 3/?i - 2/?a = the '* critical function.' 
Now the classification of any empirical curve depends upon 
the value of its critical function, F. 



■nn-i XT" •*• ^ ( A > 0, curve IS of Type I. 
When jP* IS positive and J ' a ,7 ^ o - trr 

{ ytf 1 = 0, /?a < 3, curve is of 1 



(( 



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

curve is normal. 
F is negative and /?i > 0, /?a > 3, curve is of Type IV. 
An important relation to be referred to later is 

_ 6(/?a - /?! - 1 ) 

in which s is an unknown, positive number. 

M 











f 


\ 


















f 


\ 
















/ 






\ 














/ 






\ 














/ 






\ 














/ 






\ 












y 










y 










/ 










\ 








J 


/ 










\ 


k 






/ 














\ 


fc- 


1 


1 


i : 


i 


L ( 


) J 


> 1 


6 « 


1 


15 



Fig. 28. 

The Normal Curve. 

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

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



y = 



a 



0-4/2^ ■ «^/'" 



THE CLASSES OF FREQUENCY POLYGONS. 19 

This formula gives the value of any ordinate y (or any 
class) at any distance x (measured along the base, X, X!^ of 
Fig. 28) from the mode, d is a constant number, 2.71828, the 
base of the Naperian system of logarithms, a is the total are a 
of the curve or nu mber of variajLes, and or 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 iutegral variates the theoretical fre- 
quency of any class at a deviation — from the mean can be 

taken directly from Table III. Here a; is the actual deviation 
from the nlean expressed in the unit of the maximum, and a 
is the standard deviation. 

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

X 

number of individuals occurring between the values -\ and 

X X 

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

XX X 

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

called the Index of Ahmodality, 

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

FX //a* < ± 1 and ^^«' - ^^^^ = 1 ± .2 

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

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



20 STATISTICAL METHODS. 

(B) Subtract in order each theoretical value of y from the 
corresponding observed value, regarding signs. Call the dif- 
ference 8i, "Whenever in the successive values of 8x there is 
a change of sign, divide the product of these successive values 
of ^1, in pail's, by their sum. Call this value ^s; make its 
sign always minus. Then the difference between the two 
polygons in per cent of one of them is given by the equation 

where <^i is summated without regard to sign, and n equals 

the total number of variates. This is the method of Duncker, 

'98. It may be considered a sufficient agreement between 

100 
observation and calculation when A < — ^. 

yn 

The Normal Curve op Frequency as a Binomial 

Curve. 

The normal curve may also be expressed by the binomial 
formula (p + qY, where p = i,? = J, and I is the number 
of terms, less 1, iu the expansion of the binomial ; hence 
approximately the number of classes into which the magni- 
tudes of the variates should fall. If the standard deviation be 
known, I may be found by the equation 

^ = 4 X (Standard Deviation)* = 4o-«. 



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

V V-Vm f fiV-Vm) /{V-Vm)^ AV-Vm)^ f{V-Vm)* 

11 -8 1 - 3 9 - 27 81 

12-2 2 -4 8 -16 32 

13 - 1 189 - 189 189 - 189 189 

14 1234 

15 1 454 454 454 464 454 

16 2 20 40 80 160 320 

S 1900 298 740 "m 1076 

r, = ^=0.1668; ..= — = 0.8895; ^.= i^ = 0.2041; .,= |^»0.M68. 

M^Vm-\-Vi = 14 + 0.1668 =14.1568. 
fi, = 0.8896 ~ 0.1568* = 0.3650. 
M, = 0.2011 - 3 X 0.1568 X 0.3895 + 2 X 0.1668» = 0.0267. 



THE CLASSES OF FREQUEKCY POLYGONS. 21 

|44 = 0.B668 - 4X0.1568 X 0.2011 + « X 0.1668« X 0.8895 - 3 X 0.1568* = 0.4929. 

^» = 03650^ = ^•^'^' ^» = 0:3650i = ^■^^' 
F=6+ .04074 - 7.8996 = - 1.3689. J* . /»a» = 1.3589 X 0.365» = .066. 
Si',* - 2v,* 8 X 0.3895' - 2 X .1568< „. .,— ^. 

-^, — oise^i = •^^- " = Vm.= .fl04i. 

n 1900 
Maximum frequency = =: = = = 1255. 

<r 4/2ir -6041 X V2» 

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

The difference between it and the normal is found below to be 1 .39^. 

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

V-M f / y «i «« 

-8.157 5.23 1 0.0 +1 

-2.167 8.67 2 2.1 - 0.1 - 0.09 

-1.157 1.92 189 200.4 -10.6 

-0.167 .26 1284 1218.0 +21.0 - 7.04 

+ 0.848 . 1.40 454 474.0 -20.0 -10.24 

+ 1.848 3.60 20 11.9 + 8.1 - 5.75 

1901.5 00.8 23.1 

m 

The values of y in the table above are calculated from the formula 
y =z Pq . e~^^*. The sum of the theoretical y values should equal the 
total number of variates. 

Other TJnimodal Frequency Polygons. 

The formulas of the remaining four types of unimodal simple f r& 
quency polygons have a family resemblance with the formula 



y=V ^' 



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

Unsymmetrical, y = yo(i+ —)^^ (l - ^)^\ Type L 

Symmetrical, y = yo\} 5) » Type II, 

Curve of range limited on one side: 

Unsymmetrical, l/=yo(lH — /^« " » Tyi>e III, 



22 STATISTICAL METHODS. 



Curves of unlimited range on both sides: 

Unsymmetrical, y—Vo cos e , where tan ^ = -, Type IV. 



a 



x^ 



2(7-3 

[Symmetrical, y ^ y^e , the normal curve.] 

In these formulas : 

2/o = modal ordinate, to be especially reckoned for each type. 

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

the distance x from 2/0 • 
a = a part of the abscissa-axis XX' expressed in units of the 

classes. 
e = the base of the Naperian system of logarithms, S.718*^. 

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

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

Antero-Post. Diam. - , • 1 i. « » 1-1 m a* 

'TT = =r: of a bivalve shell may conceivably range from to 

Doreo-Veut. Diam. "^ j ^ 

c». The forms of Che molluscan genera Pinna (or Malleus) and Solen 

approach such extremes. 

Asyminetry or skei^ness is found in Type I (of which Type II la 

the symmetrical limit), Type III and Type IV. In skew curves the mode 

and the mean are separated from each other by a certain distance, d. 

Asymmetry is measured by a factor 

A^±-yzA/a-t±± -here , - 51^3_:iij^ll). 
^ - ^ - ^ y^i^ ^ 2 , Where « - .^^ _ 33, _ e • 

the redult has the sanie sign as /A|. 
In Type l, A = %Vj[j^. 



(I It 



«t <( 






To compare any obMerved frequency polygon of Type 
I i¥Uli Its correspondlns tlieoretleal carre. 



» = ».(! + -) \\--^ '. 



THE CLASSES OF FREQUENCY POLYGONS. 23 

To find Oi, a,, mj« m,, ?v 

The total ranffe, 6, of the curve (along the abscissa axis) is found by 
the equation 



Oj and a^ are the ranges to the one side and the other of yo; 

Oi = J^(6 — d«); d = or^ = V/uia . ^; 

Oj = 6 — Oj; 

wij = -^*(« - 2); m, -f ma = a - 2; 

-^ wti"** . w,^* r(fflt + ma4-2) 

*'•".', , ,wii 4- »»a * r(nii 4- l)(w, + 1)* 

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



1(— ! -L_-L) 

12\m. 4- m. tn, m./ 



^ (wij 4- Wa + 1) Vmi 4- Wi g I2\mi 4- »»« w*i w»i 

Vo — r • — . <^ 

** V2ir»ijnia 

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

To compare any observed fyeqnency polygon of Type 
II ivltli Its eorrespondlns theoretical curve. 






This equation is only a special form of the equation of Type I in which 
Oi = tta and wii = m^. 

As from page 17, 0i = in Tyi)e II, 6 = 2<r Va 4- 1 ; since the curve is 
symmetrical, d = 0, and 

6 , „ _, or T(m 4- 1.5) 

a = — ; m = ^(a - 2); y^ = — — ^ 



'-^ « yirVm 4- 1) 

The r values will be found from Table V. 

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

1 



a 8-1 4(8-2) 

<r V2ir Vis 4- !)(» - 2) 

To compare any observed frequency curve of Type III 
iprltli Its corresponding tbeoretlcal curve. 



»=».(n-^) 



p -.x/d 

^ m 



24 STATISTICAL METHODS. 

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

a = <r * ~ ^JL = <r ^ "/' (for Type IB). 

a a a. pP + ^ 

Also, p = — =-— ; yQ = — . -—£- . 

^ d or^' ^« tt ePr(i) + l) 

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

To compare any obnerTed flrequency cnrre of Type IV 
witli Its correspondlns theoretical curve. 

This is the commonest type of biological skew curves. 

y = i/o(cos «)^"* . c~^. 

9 is a variable, dependent upon x as shown in the equation 

X =i a tan 6. 

The factor (cos B)^^ following Vo indicates that the curve is not calcu 
lated from the^ean ordinate (3f), or the mode {M— d), but that the 
2sero ordinate is at 3f — md\ or at a distance vi x d from the mean. 

a = -^ V 16(s - 1) -/3i(« - i)*; m = yi(8-h2); 

^ _ Vfi,8(s-2) VPt ^ ^.^j^ ^^^ opposite sign to f*,; 
• (arc of circle) = ^; 



(C08»)^__2,^,^., 



=^/I" "' '''' '^' 



« '^ (COS0)*^^ 



V 

= angle whose tangent is — . 



* 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 Pear- 
son as 



»o = — 



a giVTT 



y (sin tf)* 



e^«d« 



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



THE CLASSES OF FBEQUEN^CY P0LYG02STS. 25 

Bxample of calculating tlie tlieoretlcal cnrre corre- 
■pondlns ^v^ltli obserTcd data, (Fig. 2i.) 

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

Number of glands 0123456780 10 
Frequency 15 209 365 482 414 277 134 72 22 8 2 

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

V V-Vm f f{V—Vm) /(F— F»n)« f{V—Vfn)i fiV—VmY 






4 


15 


1 


— 8 


209 


2 


— 2 


365 


8 


-1 


482 


4 





414 


5 


1 


277 


6 


2 


134 


7 


3 


72 


8 


4 


22 


9 


5 


8 


10 


6 


2 



60 


240 


— 960 


3840 


627 


1881 


— 6643 


16929 


780 


1460 


— 2920 


6840 


482 


482 


— 482 


482 














277 


277 


277 


277 


268 


686 


1072 


2144 


216 


648 


1944 


5832 


88 


352 


1408 


6633 


40 


200 


1000 


5000 


12 


72 


432, 


2592 



2 2000 —998 6148 —8872 48568 

Kj = — 998 -H 2000 = — .499. 

K, = 6148 -♦-2000= 3.074. 

V, = — 3872 -♦• 2000 = — 1.936. 

1/4 = 48568 -»- 2000 = 24.284. 

fii = If = 4 — .499 = 3.501. 

Ma = 8.074 — (— .499)9 = 2.824999. 

Ms = - 1.936 - 8(- .499 X 3.074) -f 2(- .499)* = 2.417278. 

fi4 = 24.284-4(-.499x - 1.936) + 6(.249001 X 3.074) - 3(- 499)* =s 24.826297. 

(2.417278)a 5.848232929 _ 
^^ " (2.824999)* " 22.545241688 " ''"^^^'^' 

24.826297 24.826297 _ 
^^ ~" (2.824999)9 "~ 7.98061985 ~ **-'^^***^- 

F = 6 -}- 8 X 0.259178 - 2 X 3.110828 = + D.556888 (Type I). 
. = ■'<'"'^^»«-^> = 19.9857. 

01 OAK? 

A = M ^-259178^^^= .31115. 

d = 1.680774 X .8111 = .5280. 
d. «= .6230 X 19.9857 = 10.4519. 

h = .840887 1^16 X 20.9857 + 0.25918 X (21.9657)* = 18.0448. 
^ 18.0448 ~ 10.4519 ^ _^ 

CI] = s ~ o.TwJO. 



26 STATISTICAL METHODS. 



a, = 18.0448 - S.7965 r= 14.2488. 
3.7965 X 17.9867 _ 

"*» = — i8:oi48 — = ^•'^^• 

^ 14.248^ X 17.9857 . . .^ 
*"« = 18:0448 = ^^•^^• 

2000 (18.9846) VrTMiQ ^ « i m ^ •<>^3(.0556 - .2648 - .0704) 
18.0448 ^2w X 3.7840 X 14.2008 
= 475.24, the number of cases in the modal class. 
The equation of the theoretical curve is thus 



V = 475.24(1 + 3-.^) (i-jj:^ 



S'784 / « \ l4-a01 

t 



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

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

The mean percentage deviation of the theoretical ordinates from the 
observed ordinates is 11.4^* (Method A). .This is calculated as follows: 



V 


observed 


y 

theoretical 


a 


1 





0.0 


0.0 





15 


21.1 


— 6.1 


1 


209 


185.8 


+ 23.2 


2 


865 


395.1 


— 30.1 


3 


482 


475.2 


+ 6.8 


4 


414 


405.6 


+ 8.4 


5 


277 


272.1 


+ 4.9 


6 


134 


147.6 


— 13.6 


7 


72 


65.9 


+ 6.1 


8 


23 


24.1 


— 2.1 


9 


8 


7.0 


+ 1.0 


10 


2 


1.6 


+ 0.4 


11 





0.2 




12 





0.0 





40.7 

11.1 

8.2 

1.4 

2.0 

1.8 

10.2 

8.5 

9.5 

12.5 

20.0 

11.4j( 

Multimodal Cukvks. 

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

<*■ The mean percentage deviation by Duncker^s determination witk 
method B using the same data is 1.78^ of area. 



" THE CLAeSES OF FREQUENCY POLYGONS. 



Fia.M. 
DtstTJbutlon of f i«queDci' Id glaoIlK of sWine. 

I polygon of ohBer^ed frequency. 

— , polygon of theorellcal frequency <Typo I). 

- - - -, normal frequency polygon. 



28 



STATISTICAL METHODS. 



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




R 



6.6 L6 8.6 2.5 1.6 J 0.6 1.6 2.6 8i> i.6 6.6 6.6 7.6 

Fig. 26. 

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 veiy tedious and rarely 
applicable. 







/ 


\ 














/ 


\ 
















\ 
















\ 


















\ 
















V 


















\ 
















\ 








/ 










\ 






/ 










\ 






/ 










\ 


^ 


i 

4 


: 


L 1 


) ] 




4 
« 


) i 


; 



Fig. 26. 

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



THE CLASSES OF FREQUEIifCY POLYGONS. 



29 



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

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

The meaning of multimodal curves is diverse. Sometimes 







/ 


\ 




















/ 














































\ 






















\ 
























I 
















I 




V. 


/ 


\ 






















\ 






















\ 










/ 














\ 






ri 


/ 


L ( 


) 


L i 


! 2 


) i 


. 


\ 


i 


^ 



Fia. 27. 



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







/ 


\ 




















/ 


\ 




















/ 


\ 




















/ 










































1 


1 






/ 


\ 










J 




1 




/ 


^ 
















\ 


/ 






V 








/ 






V 


/ 






\ 






> 


/ 






v. 


/ 






\ 


^ 




8 i 




I ( 


) 


L . 


9 * 


( 


1 ] 


L i 


I \ 


\ i 



Fig. 28. 
the number of ray flowers of the white daisy which has modes 
at 8, 13, 21, 34, etc. Sometimes they indicate a splitting of a 
species into two or more varieties. 

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



30 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 accompained by a corresponding abmodality of the 
other or others. 

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

TThe principles upon which the measure of correlated varia- 
tion rest are these. When we take individuals at random we 
find that the mean magnitude of any character is equal to the 
mean magnitude of this character in the whole population. 
Deviation from the mean of the whole population in any lot of 
individuals implies a selection. If we select individuals on the 
basis of one character {A, called the subject) we select also any 
closely correlated character (B, called the relaiive) (e.g. leg- 
length and stature). If perfectly correlated, the index of 
abmodality of 5 will be as great as that of A or 

Index abmodality of relative _ ^ 
Index abmodality of subject ~~ 

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

Index of abmodality of relative _ ___ ^ 
Index of abmodality of subject "" m "" ' 

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



CORRELATED VARIABILITY. 



31 



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32 STATISTICAL METHODS. 

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

Methods op Determining Coefficient op Correlation. 

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

A more precise method is given by Pearson as follows: 
Sum of products (deviation subj. class X deviation each assoc. 

rel. class X no. of cases in both) 

total no. oFiudivs. X Stand. Dev. ( f subject x Stand. Dev. 

of relative ; 

or, expressed in a formula : 

^(dev. X X dev. y X :f) 

p = 

nO'iO'a 

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 
is indicated below ; 

(- 3.540 X 8 

- 8.547 X \ - 2.540 X 5 

(-1.540X2 

f- 3.540 X 4 

- 2.540 X 151 

- 2.547 X ^ - 1.540 X 58 

- 0.540 X 9 

- 0.460 X 3 
etc. 



CORRELATED VARIABILITY, 33 

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

A *, f ^ ^(dev. gXde v. yX/) , ^ 
p IS composed of two factors: and 



To find 



^(dev. X X dev. y Xf) 
n 



Separate the deviation from the mean of each class into its 
integral and its fractional parts ; the fractional parts for all 
classes below the mean will be equal to the fractional part of 
the mean ; of all classes above the mean, to the complement 
of that number. Designate the integral parts of the variants 
of the subject by ± X ; of the relatives by ± Xa, and the frac- 
tional complement parts of the means of subject or relative by 
^1, $1. Let/ equal the frequency of any deviation in the com- 
bination XjXa, as shown in the correlation table. Draw rect- 
angular co-ordinates as shown on page 84 through the zero- 
poiut of the correlation table. Number the N". W. quadrant, 
which should include negative deviations of both subject and 
relative variants, I ; the N. E. quadrant, II ; the S. W. 
quadrant containing solely positive deviations III; and the 
S. E. quadrant, IV. Then if 2i, 2u, etc., indicate a summa- 
tion for the quadrant I, II, etc., and having regard to signs : 

:S'i«if(/X,XO - 2i{fX,) - 2i(fX,) 
2X,Xif + 2i(f) - ^iiifX,) - 2TJi{fX,) , , 

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

positive: 2i(fX^X,l 2nifX,X,), 2i{f), 2n{fX,\ 
negative : S^^(/XaX,), ^niifX.X,), 2i(fX^), 2i(fX,). 

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

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



34 



STATISTICAL METHODS. 



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

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

he third quadrant multiplied by their frequency. 

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

To get /o, divide — ^— ^ by the product of o"i and 



n 



0*a. 



The probable error of the determination of p is 



P,E»a = 



0.6745(1 - /o») 

Vn{l + P) ' 



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



Mean, right leg, = 3.5465 ; Mean, left leg, = 



0-, = 1.7195 

Right leg, subject : 



o-a 
Left leg, relative. 



3.5395 
1.7304 



Xa - 


3- 


-2- 


-1 








1 


2 


3 


4 


5 


6 


Rel. class 





1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


Sub. class (I) 






1 














(II) 


X 
























0-3 


8 


5 


2 


















1 - 2 


4 151 


58 


9 


3 














2- 1 


2 


65 154 


96 


28 


7 


1 










3 




14 


88 173 


128 


28 


6 








- 


4 




5 


27 119 


153 


77 


26 


3 


1 






5 1 




1 


7 


24 


92 101 


52 


11 


9 






6 2 








8 


16 


58 


48 


16 


7 




2 


7 3 








1 


8 


20 


18 


17 


9 


5 




8 4 










1 


3 


5 


3 


2 


2 




9 5 












1 


3 


8 


2 


2 


1 


10 6 (III) 


















1 


(IV) 



CORRELATED VARIABILITY. 35 



2/ - irK/XiXa) = 1142-9—9+1652 

- :sj(/Xi) - :s(/Xa) + :sj{f) 

= -f 806 + 814 + 829 

- 2//(/xo - :Sjxi(/Xt) 

= - 49 _ 51 



. 5125 

■^ -T- ?i = = 2 5625 



-«.«. = - 4535 X. 4605= -^ 

0-,0-a = 1.7195 X 1.7304 = 2.9754; p = -|^^ = .7919 

«.97d4 



.6745[1 - (.7919)«] 
4/2000 X 1.627 



p.^.^ ^-::ilZ^ y. >«..« .; J ^ ± 0.0044 



Spurious Correlation in Indices. 

When two characters A and ^are measured in each individual 
of a series of individuals, and each absolute magnitude is trans- 
formed into an index by dividing it by the magnitude of a 
third character C7as found in the same individual, a spurious 
correlation will be found to exist between the indices of 

A , B 

-7= and ■= 
C C 

Let ®i = the coefficient of variation of A 
tja = •* ** •• " " B 

t% = ** " " ** •* G 

po = '* ** " spurious correlation. 



4/ui* + «8*4/V + V 

The precise method of using po in modifying any determina- 
tion of p is uncertain. Pearson recommends usiug p — po 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. Tlie statistical laws of 
heredity deal not with relations between one descendant and 



36 STATISTICAL METHODS. 

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

In uniparental inheritanoe, 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, cr^, may (as a result of selection or of environ- 
mental change) be different from the variability of the char- 
acter in the first generation, cr,, the index should be taken as 

The probable error of this determination is 

^— , in which pn means the correlation coeffi- 

n 

cieut 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 — p* is to 1. 

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

hi = Pa ~-h% + Pa — n%, 

where Ai = average abmodality of fraternity ; 
Aa = average abmodality of male parent ; 
As = average abmodality of female parent ; 
p, = correlation coefficient between fraternity and 

female parent ; 
Pt = coiTelation coefficient between fraternity and male 

parent ; 
(Ti = standard deviation of fraternity ; 
0*3 = standard deviation of male parent ; 
CTs = standard deviation of female parent. 



CORRELATED VARIABILITY. 37 

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

1 — Pi* 0-. 1 — Pi* era 

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

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

Pt — Pi Pa Oj 
1 — pi* * cr,* 

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

from the nth ancestral generation r- of the whole ; the total 

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

1 (To, . 1 CTo 1 Co, , 1 (To 

a (Ti 4 (79 O (Tt 10 <T4 

where hi = average abmodality of fraternity. 
(To = standard deviation of fraternity. 
cTi , cTa . . . Cf = standard deviation of mid-parent of 

1st, 2d . . . «th ancestral generation. 
ki = abmodality of mid-parent of 1st ancestral genera- 
tion. 
kt, ks , . , k, = abmodality of mid-parent of 2d, 3d 

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



1 



38 STATISTICAL METHODS. 



CHAPTER V. 
So]£B AppLiCATioirs OP Statistical Biological Studt. 

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

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

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

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

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



STATISTICAL BIOLOGICAL STUDY. 39 

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

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

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

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



40 STATISTICAL METHODS. 

SELECTED BIBLIOGRAPHY 

OP WORKS ON THE QUANTITATIVB STUDY OF ORGANISMS. 

Amann, J., '96. Application du calcul des probabilites ft 
Tetiide de la variatioa d'lin type vegetal. Bull, de 
I'Herb. Bossier. Geneve et B^le. IV, 578-590. 

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

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

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

Davenport, C. B., and J. W. Blankinbhip, *98. A Precise 
Criterion of Species. Science, VII, 685-695. 

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

DuNCKER, G., '97. Correlation Studien an den Strahlzahlen 
einiger Flossen von Acerina cernua L. Biol. Centralbl., 
XVII. 785h-794 ; 815-831. 

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

DuNCKER, G., *99. Die Methode der Variation s-Statistik. 
Arch. f. Entwickelungs-Mcchan. d. Organismen, VIII, 
112-183. [The most important elementary presentation 
of the subject ; extensive, nearly complete bibliogmphy. ] 

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

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

Fechner, G. T.,*97. Kollektivmasslehre. Im Auftrage der 
KOnlglich Sachsischen Gesellschaft der Wissenschaften 
herausgegeben von Gottl. Friedr. Lipps. Leipzig : Engel 
mann. 483 pp. [Important but too much neglected work.] 



SELECTED BIBLIOGRAPHY. " 41 

Field, W. L. W., '98. A Contribution to the Study of Indi- 
vidual Variation in the Wiugs of Lepidoplera. Proc. 

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

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

XLV, 136-145. 
Galton, F., '89. Natural Inheritance. London : Macmillan. 
Galton, F. '97. The Average Contribution of each several 

Ancestor to the total Heritage of the Offspring. Proc. 

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

theWhiteDaisy. Amer. Naturalist, XXXII, 509-511. 2figs. 
LuDWiG, F., '95. Ueber Variationskurven und Variations- 

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

2 Tafn. 
LxJDWiG, F., '96. Weiteres tlber Fibonacci-Kurven und die 

numerische Variation der gesammten Bliithenstd.nde der 

Kompositen. Bot. Centralbl. LXVIII, 1 et folg. 1 Taf. 
LxJDWiG, F., *96. Eine ftinfgipfplige Variations- Kurve. Ber. 

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

Gauss'sche Wahrscheinlichkeitskurve. Bot. Centralbl., 

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

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

rodes Rafinesque in Turkey lake and Tippecanoe lake. 

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

of Evolution. [I. On the Dissection of Frequency 

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

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

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

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

of Evolution, IIL Regression, Heredity, and Panmixia. 

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

Form of Spurious Correlation, which may Arise when 



42 STATISTICAL METHODS. 

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

Pbarbon, K. '98. Mathematical Contributions, etc. On the 
Law of Ancestral Heredity. Proc. Roy. Soc. London, 
LXII, 886-412. 

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

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

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

DB Ybies, H., '94. Ueber halbe Galton-Eurven als Zeichnen 
diskontinuirlichen Variation. Ber. deutsch. Bot. Ges., 
Xn, 197-207. Taf. X. 

dbVkies, H., '95. Eine zweigipfelige Variations-Eurve. 
Arch. f. Entwickelungsmecbanik, II, 52-65. 1 Taf. 

Warren, E., '96. Variation in Portuntu depuraior, Proc. 
Roy. Soc. London, LX, 221-243. 

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

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

Weldon, W. F. R., '92. Certain Correlated Variations in 
Ch^angon vulgaris. Proc. Roy. Soc. London, LI, 2-21. 

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

Weldon, W. F. R., '95. Report of the Committee for Con- 
ducting Statistical Inquiries into the measurable Char- 
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to Measure the Death-rate due to Selective Destruction of 
Gardnus maenas with respect to a Particular Dimension. 
Proc. Roy. Soc. London, LVII, 860-879. 



EXPLANATIOlSr OF TABLES. 



43 



EXPLANATION OF TABLES. 

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

II. Certain constants and their logarithms. 

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

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

Example: Jf = 14.157 ; o- = 0.604 ; ly. = 1256. 

(See page 19.) 

Entries In Table 
^ ^ „ V - M correspondine to 



r 




0" 


F- Jf 




<r 


11 


- 8.157 


5.2 


.000004 


12 


- 2.157 


8.6 


.0015 


18 


- 1.157 


1.9 


.164 



Vo 



1 

2 

189 



X 1255 = 0.0 
X 1255 = 1.8 
X 1255 = 201«8 

lY. Table of values of probability integral. 

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

Example: Jf. 5.24 ; a = 0.987. (See page 12). 





% 


blast 
limits 


Deviation 
from 


X 


X 

^0 




4 


8.0 


-2.84 


-2.27 


48.8 




4 


8.6 


-1.74 


-1.76 


46.1 




13 


4.0 


-1.24 


-1.26 


89.6 




12 


4.5 


-0.74 


-0.76 


27.8 




88 


6.0 


-0.24 


-0.25 


9.9 




20 


6.5 


0.26 


0.27 


10.6 




12 


6.0 


0.76 


0.77 


27.9 




4 


6»5 


1.26 


1.28 


40.0 




4 


7.0 


1.76 


1.78 


46.8 


» 


100 


7.5 


2.26 


2.20 


48.9 



..••.......-----^ 








fre 
02.4 

tre 


.... ■ 

The 
iO.5 

Ob 

* * * * < 


oret 
55.2 

bo 
serv 


ioal 
79.6 

ed 






...**...••.... .-^ 





WT" 



100 

qtiency 



44 STATISTICAL METHODS. 

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

quency reduced to percents in column headed %. The — of 

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

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

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

log r(1.273) = 9.955185 
log 1.273 =0.104828 

log r(2.273) = 0.060013 
log 2.273 =0.356599 

log r(3.273) = 0.416612 
log 3.273 =0.514946 

logr(4.27a) = 0.931558 

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

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

log r(4.273) = 0.931558 = log 8.542 



EXPLANATIOK OF TABLES. 45 

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

VII. First to sixth povirers of integers from 

1 to 30. This table is useful in calculating moments. 

YIII. Squares, cubes, square roots, and re- 
ciprocals 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 multiplied by n*, its 

cube by n', and its reciprocal by — . 

IX. Lograrithms 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]. 

Appendix IX. — The logarithm of a number consists of 
two parts, a whole number called the cha/robcteristiCf and a deci- 
mal called the mantissa. All numbers which consist of the 
same figures standing in the same order have the same man- 
tissa, regardless of the x>osition 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 
in the number. It is always one less than the number of 
figures in the number to the left of the decimal point. The 
value is therefore diminished by one every time the decimal 
point of the number is removed one place to the left, and vice 
versa. Thus 



Number, 


Logarithm. 


13840. 


4.141136 


1384.0 


8.141136 


138.40 


2.141136 


13.84 


1.141136 


1.384 


0.141136 


.1384 


—1.141136 


.01384 


—2.141136 


.001384 


--3.141136 


etc. 


etc. 



46 



STATISTICAL METHODS. 



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

Number, Logarithm. 

1.384 0.141136 

.1384 9.141136 

.01384 8.141136 

.001384 7.141136 

etc. etc. 

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

The table IX. contains the mantissas of logarithms, car- 
ried to six places of decimals, for nxmibers between 1 and 0999, 
inclusive. The first three figures of a number are given in the 
first colunm, the fourth at the top of the other colunms. 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 oft the ciphers at either end) con- 
sists of not more than four figures, the mantissa may be taken 
direct from the table; but by interpolation the logarithm of a 
number having six figures may be obtained. The last column 
contains the average difference of consecutive logarithms <m 
the same line, but for a given case the difference needs to he 
verified by actual subtraction, at least so far as the last figure 
is concerned. The lower part of the page contains a complete 
list of differences, with their multiples divided by 10. 

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



EXPLANATION OP TABLES, 47 

MeampU.—Fmd the log of 23.4275. 

For 2342 mantissa is 869587 

" diff. 185col. 7 129.5 

" " *' " 5 9.2 



An8, 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, "vvhich 
furnishes by inspection the figures of the quotient. 

Ex€M7iple, — ^Find the number of which the logarithm is 

8.268927 8.263927 

First 4 figures 1836 from 263873 

Biff. Slo 

Tabular diff .= 286 .-. 5thfig.=2 47.2 



6.80 
6th fig. =3 J\08 

Am, No. = .0188628 or 183,623,000. 

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

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

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



48 STATISTICAL METHODS. 

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

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

Example.— Yin^ the log sin of 9' 28' 20'. 

Log sin of r 28' is 9.216097 

Add correction 20 X 12.62 252 

Ans. 9.216849 

Bccmple.— Find the log cot of 9° 28' 20*. 

Log cotan of Q" 28' is 10.777948 

Subtract correction 20 X 12.97 259 

^?M.lo .777689 

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

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



EXPLANATIOK OF TABLES. 49 

Example.— 9.dSS7^1 is the log tan of ^liat angle? 
Next less 9.883682 gives l^"* 86' 

Diff. 4^.00 -¥• 9.20 = 05'.3 



Ans. 13** 86' OST.Z 
Example,— 9.^9B^ is the log cos of what angle? 
Next ffreater 688 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) because 
here the differences vary so rapidly that interpolation made by 
them in the usual way will not give exact results. 

On the first two pages, the first column contains the number 
of seconds for every minute from 1' to 2° ; the minutes are 
given in the second, the log. sin. in the third, and in the fourtli 
are the last three figures of a logarithm which is the difference 
between the log sin and the logarithm of the number of sec- 
onds in the first column. The first three figures' and the char- 
acteristic of this logarithm are placed, once for all, at the head 
of the column. 

To find the logr 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 Ioga> 
rithms, and add to this the logarithm from the fourth column 
opposite the same number of seconds. The simi is the log sin 
required. 

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

Mnample,— Find the log sin of 1** 39' 14'. 4. 

V 89' 14'.4 = 5954'.4 log 8.774838 

add (g - 4.685515 

Ana. log sin 8.460868 

Log tangents of small arcs are found in the same way, only 
taking the last four figures ot{q^l) from the fifth coliunn. 



60 STATISTICAL METHODS. 

£itample.— Find the log tan of 0* 52' 85'. 

62' 85" = (8120' + 85') = 8155* log ^.498999 

add (9 - 4.685609 

Ans, log tan 8.184608 

To find the log^ cotangrent of an angrle less than 
2" ^ven to seconds.— Take from the column headed (q+l) 
the logarithm corresponding to the given angle, interpolating 
for the last figure if necessary, and from this svJbtraci the loga- 
rithm of the number of seconds in the given angle. 

EaDomple.—Tmd the log cotan of 1** 44' 22'.5. 

q + I 15.814292 
6240* + 22*. 5 = 6262.5 log 8.796748 

An9, 11.517544 

These two pages may be used in the same way when the 
given angle lies between 88" and 92% or between 178** and 180**; 
but if the number of degrees be found at the bottom of the page, 
the title of each column will be found there also; and if the 
number of degrees be found on the right hand side of the page, 
the number of minutes must be found in the right hand col- 
umn, and since here the minutes increase upward, the number 
of seconds on the same line in the first column must be 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 

BkcampU.^Fmd the log cos of 88" 41' 12'.5 

fe - 4.685507 
4740* - 12'.5 = 4727.5 log 8.674631 

Ans. 8.860168 

JBircmple.—Tmd the log tan of 90** 80' 50'. 

q + 1 15.814413 
1800' + 50' = 1850' log 8.267172 

Ans. 12.047241 

To find the arc corresponding: to a g^ven log 
sin, cos, tan, or cotan which falls within the 
limits of the first two pages of Table X, 

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



EXPLANATIOK OF TABLES. 51 

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

Find the value of {q-r- I) or (g + 0, as the case may require, 
corresponding to the given log (interpolating for the last figure 
if necessary). Then if ^ = given log and I = log of number of 
seconds, n, in the required arc, we have at once l=q — (g — I) 
or I =: {q -\-l) — q, whence n is easily found. 

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

Iikamg>le,-^ll,7d42Q8 is the log cot of what arc? 

q + 1 15.314376 

q ' 11.734268 

.-. «= 8803.8 8.580108 

For V adopt 3780. giving 03' 

Difference 22'. 8 

Ans. V 08' 22'.8 or 178" 56' 37'^. 

Mkample» — 8.201795 is the log cos of what arc? 

q - 1 4.685556 

q 8.201795 

.-. n=L 3282".8 8.516289 

For 89° adopt 8300. giving 05' 

Difference 17'. 2 

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



52 STATISTICAL METHODS. 

I. -FORMULAS. 
->^=^^p^= T^m-"!- P.J2?.j|f=± 0.6745-^. aj=F-lC 

' = ^/^^^= v;^^^ = Viir. p.i?.^ = 0.6745-JL^. 

^D = ^%^ = 0.7979<r. P.iZ?. = g = 0.6745cr. 

"1- n -^-^m- »'« = ;j . 

"•- n • •'*- n • 

Ma = »'« - «'!«(+ i) = ^^^'•■^> (+ J). 

w 

n 

'^^ ft,*' ^' " ;:?■ 



j» 



A% (for graduated variates) = * o * . KW. 



2n 



d 



A^ (for integral variates) = S^-j^ , lOOjf, where fc equals the number of classes. 
_ S(dev.a;.dev. I/./) _ SXiXg/ 

yXiXa _ SJ_Jp</X^Xa)-SJ(/X^)-2J{/Xa)+SJ</)-2Jy(/Xa)~SJ^</3^^) 

> Vn(l + p) 
Po (spurious correlation) = — ^ * 

7i (index of heredity, uniparental inheritance) = p^. 

^1 = f»8~^« + Pi^^i [biparental inheritance; unassortative mating]. 

*i = \ _ \' • ^h^ + ][ ^ ' a* • ^ • '*» [biparental inheritance; assortatiTO 

mating]* 



CERTAIN CONSTANTS AND THEIR LOGARITHMS. 53 



n.— CERTAIN CONSTANTS AND THEIR LOGARITHMS. 



Title. 



Ratio of circumference to diameter. 
Reciprocal of sfune 



Square root of same. 



Reciprocal of square root of same. 



Square root of 2v . 



Reciprocal of same 

Base of hjHPerboIic logarithms 

Modulus of common system of logs = log c 



Reciprocal of same = hyp. log 10 . 



Com. log X =»m x hyp. log a:, or 
Com.log(com.log x)=9.6377B43+com.log(hyp.log x) 

Hyp. log « = com. log x X — , or 

m. 

Com.]og(hyp.logj;)=oom.log(com.loga;)+0.3622157 

Circumference of circle = — 

Area of circle 

Area of sector (length of arc = 

Area of sector (angle of arc = a**) 

Eccentricity of an ellipse, e = V 5 — , whei 



Symbol 


Number. 


Log. 


IT 


3.1415927 


0.4971499 




0.318S099 


9.6028501 


V^ 


1.7724538 


0.2485749 


1 


0.5641896 


9.7514251 


^^ 


2.506628 


0.399090 


1 

V2» 


0.3969428 


9.6009100 


c 


2.7182818 


0.4842945 


fii 


0.4342945 


9.6877843 


m 


2.3025851 


0.3622157 


2irr 






irr« 






Jtfr 






860"' 






x=semi-majorazl8; &=semi- 


minoi 


r axis of el 


lipse. 



54 



STATISTICAL METHODS. 



1 



ra.-TABLE OP ORDINATES OF NORMAL CURVE, OR YALUES.OF 

^ CORRESPONDINQ TO VALUES OF -. 
Vo «" 

X = deyiatlon from mean, y = frequency. 



c = standard deviation. 



y© = =. = maximum f requeiM^. 



x/<r 


y/Vo 


x/tr 


y/Vo 


x/v 


y/y^ 


X/9 


y/y^ 





1. 


0.8 


.7262 


1.6 


.2780 


2.8 


.0198 


0.1 


.9950 


0.9 


.6670 


1.7 


.2857 


8.0 


.0111 


0.2 


.9602 


1.0 


.6065 


1.8 


.1979 


8.2 


.0060 


0.3 


.9560 


1.1 


.6467 


1.9 


.1645 


8.4 


.0081 


0.4 


.9281 


1.2 


.4868 


2.0 


.1368 


8.6 


.0015 


0.5 


.8825 


1.3 


.4286 


2.2 


.0889 


8.8 


.0007 


0.6 


.8353 


1.4 


.3758 


2.4 


.0561 


4.0 


.0003 


0.7 


.7827 


1.5 


.3246 


2.6 


.0840 


5.0 


.000004 



VALUES OF THE NORMAL PBOBABILITY UiTTBGRAL. 55 



IV.— TABLE OP VALUES OF THE NORMAL PROBABILITY INTEGRAL 



X 



COBRESPONDma TO VALUES OF- ; OR THE FRACTION OF THE 



ABSA OP THE CURVE BETWEEN THE LIMITS AND + - OR 

AND-?. 

TotcU area of curve assumed to be 10 009. 

X = deviation from mean, 
o- = standard deviation. 



X 

9 





1 


2 


8 


4 


6 


6 


7 


8 


9 


A 


0.0 


0000 


0040 


0060 


0120 


0160 


0200 


0239 


0279 


0319 


0369 


40 


0.1 


0399 


0438 


0478 


0617 


0567 


0597 


0686 


0676 


0715 


0754 


40 


0.2 


0793 


083:2 


0671 


0910 


0948 


0987 


1026 


1064 


1103 


1141 


39 


0.8 


1179 


1217 


1256 


1293 


1380 


1368 


1406 


1443 


1480 


1517 


36 


0.4 


1554 


1591 


1628 


1664 


1700 


1737 


1773 


1808 


1844 


1879 


36 


0.; 


1915 


1950 


1985 


2020 


2054 


2069 


2124 


2157 


2191 


2225 


84 


0.6 


2268 


2291 


2824 


2857 


2;J89 


2422 


2464 


2486 


2516 


2549 


32 


0.7 


2581 


2612 


2648 


2672 


2704 


2734 


2764 


2794 


2623 


2653 


80 


0.8 


2882 


2910 


2939 


2967 


2995 


8028 


3051 


3078 


8106 


3133 


26 


0.9 


8160 


8186 


8212 


3238 


3264 


8290 


3315 


3340 


8865 


3369 


;e6 


1.0 


8414 


8438 


8461 


3485 


3509 


3532 


3555 


3677 


3600 


3622 


23 


1.1 


8644 


8665 


8686 


3708 


8729 


8750 


3770 


3791 


3811 


3830 


21 


1.2 


8850 


3869 


8888 


8906 


3925 


8944 


8962 


3960 


3997 


4016 


19 


1.8 


4082 


4049 


4066 


4068 


4099 


4115 


4132 


4147 


4162 


4178 


17 


1.4 


4193 


4206 


4322 


4237 


4261 


4265 


4279 


4292 


4306 


4319 


14 


1.6 


4882 


4845 


4358 


4370 


4388 


4396 


4406 


4418 


4429 


4441 


12 


1.6 


4462 


4463 


4474 


4485 


4496 


4506 


4516 


4526 


4536 


4646 


10 


1.7 


4654 


4564 


4578 


4582 


4591 


4600 


4606 


4617 


4625 


4633 


9 


1.8 


4641 


4648 


4656 


4664 


4671 


4678 


4686 


4693 


4700 


4706 


7 


1.0 


4n3 


4720 


47;26 


4732 


4788 


4744 


4750 


4756 


4762 


4767 


6 


8.0 


4778 


4778 


4783 


4788 


4794 


4799 


4804 


4806 


4813 


4817 


6 


2.1 


4822 


4826 


4830 


4834 


4888 


4842 


4846 


4850 


4864 


4866 


4 


2.2 


4861 


4865 


4868 


4872 


4875 


4878 


4881 


4884 


4887 


4890 


3 


2.8 


4898 


4896 


4899 


4901 


4904 


4906 


4909 


4911 


4914 


4916 


3 


2.4 


4918 


4921 


4923 


4926 


4927 


4929 


4931 


4933 


4935 


4936 


2 


2.5 


4988 


4940 


4942 


4943 


4945 


4946 


4947 


4949 


4961 


4952 


2 


2.6 


4958 


4955 


4056 


4958 


4959 


4960 


4961 


4962 


4964 


4966 


1 


2.7 


4966 


4967 


4968 


4969 


4970 


4970 


4971 


4972 


4973 


4974 


1 


2.8 


4976 


4975 


4976 


4977 


4978 


4978 


4979 


4960 


4981 


4961 


0.5 


2.9 


4962 


4982 


4983 


4983 


4984 


4964 


4985 


4965 


4986 


4986 


0.6 


8 


4967 


4991 


4993 


4995 


4997 


4996 


4999 


4999 


4999 


6000 


. • • • 


00 


5000 


« • • • 


• • • • 


■ • • • 


• • • • 


• • • • 


• • • • 


• • . • 


. . . • 


• ■ • • 


.... 



56 



STATISTICAL METHODS. 



v.— TABLE OF LOG T FUNCTIONS OF p. 



V 





1 


2 


8 


4 


6 


6 


7 


8 





i 


1.00 




9750 


9500 


J25I 
^801 


90a3 


8756 


a509 


8263 


8017 


7778 




1.01 


"9!997B29' 


7286 


7043 


6660 


6320 


6080 


5841 


5602 


6866 




1.02 


6128 


4892 


4656 


4421 


4187 


3953 


3721 


3489 


3257 


8Q26 




1.03 


2796 


2567 


2388 


2110 


1883 


1656 


1430 


1205 


0981 


0775 




1.04 


0538 


0611 


0089 


9868 


§647 


§427 


§208 


§989 


§772 


§554 




1.05 


9.988338 


8122 


7907 


7692 


7478 


7265 


7052 


6841 


6629 


6419 




1.06 


6209 


6000 


5791 


5583 


5.378 


5169 


4963 


4758 


4553 


4349 




1.07 


4145 


3943 


3741 


3539 


3338 


3138 


2939 


2740 


2541 


2344 




1.06 


2147 


1951 


1755 


1560 


1365 


1172 


0978 


0786 


0594 


0403 




1.09 


0212 


0022 


§838 


§644 


§456 


§269 


§062 


§900 


§710 


§525 




1.10 


9.978341 


8157 


7974 


7791 


7610 


7438 


7248 


7068 


6888 


6709 




1.11 


6531 


6354 


6177 


6000 


5825 


6650 


5475 


5301 


5128 


49&5 




1.12 


4783 


4612 


4441 


4271 


4101 


3982 


8764 


3596 


8429 


8262 




1.13 


3096 


2931 


2766 


2602 


2438 


2275 


2113 


1951 


1790 


1629 




1.14 


1469 


1309 


1150 


0992 


0835 


0677 


0521 


0365 


0210 


0055 




l.i.5 


9.969901 


9747 


9594 


9442 


9290 


9139 


8988 


8838 


8688 


8539 




1.16 


8390 


8243 


8096 


7949 


7803 


7658 


7513 


7869 


7225 


7062 




1.17 


6039 


6797 


6655 


6614 


6374 


6234 


6095 


6957 


5818 


6681 




1.18 


5544 


6406 


6272 


6137 


5002 


4868 


4734 


4601 


4469 


4337 




1.19 


4205 


4075 


8944 


8815 


3686 


8567 


3429 


3802 


3175 


8048 




l.SO 


2922 


2797 


2672 


2548 


2425 


2302 


2179 


2057 


1936 


1815 




i.ai 


1695 


1575 


1456 


1337 


1219 


1101 


0964 


0667 


0751 


0686 




1.22 


0521 


0407 


0293 


0180 


0067 


9955 


6643 


9782 


9621 


9511 




1.23 


9.959401 


9292 


9184 


9076 


8968 


8861 


8755 


8649 


8544 


8439 




1.24 


8335 


8231 


8128 


8025 


7923 


7821 


7720 


7620 


7520 


7420 




1.25 


7321 


7223 


7125 


7027 


6930 


6834 


6738 


6642 


6547 


6453 




1.26 


6359 


6267 


6173 


6081 


5989 


5898 


5807 


5716 


5627 


6587 




1.27 


5449 


5360 


6278 


6185 


6099 


5013 


4927 


4842 


4757 


4678 




1.28 


4589 


4506 


4428 


4341 


4'i59 


4178 


4097 


4017 


8938 


8868 




1.29 


8780 


8702 


8624 


8547 


8470 


8394 


8318 


3243 


3168 


8094 




1.30 


8020 


2947 


2874 


2802 


2730 


2659 


2668 


2518 


2448 


2379 




1.31 


2310 


2242 


2174 


2106 


2040 


1973 


1907 


1842 


1777 


1712 




1.32 


1648 


1586 


1522 


1459 


1397 


1386 


1275 


1214 


1154 


1094 




1.33 


1035 


0977 


0918 


0661 


0603 


0747 


0690 


0634 


0579 


0524 




1.34 


0470 


0416 


0362 


0309 


0257 


0205 


0153 


0102 


0051 


0001 




1.35 


9.949951 


9902 


9853 


9805 


9757 


9710 


9663 


9617 


9571 


9525 




1.36 


94S0 


9435 


9391 


9348 


9304 


9262 


9219 


9178 


9186 


9095 




1.37 


9054 


9015 


8975 


8936 


8898 


8859 


8822 


8785 


8748 


8711 




1.38 


8676 


8640 


8605 


8571 


8587 


85a3 


8470 


8437 


8405 


8:i78 




1.39 


8342 


8311 


8280 


8250 


8221 


8192 


8163 


8135 


8107 


8060 




1.40 


8053 


8026 


8000 


7975 


7950 


7925 


7901 


7877 


7854 


7831 




1.41 


7808 


7786 


7765 


7744 


7723 


7703 


7683 


7664 


7645 


7626 




1.42 


7608 


7590 


7573 


7556 


7540 


7534 


7509 


7494 


7479 


7465 




1.43 


7451 


7488 


7425 


7413 


7401 


7389 


7378 


7368 


7358 


7348 




1.44 


7338 


7329 


7321 


7312 


7305 


7298 


7291 


7284 


7278 


7273 




1.45 


7268 


7268 


7259 


7255 


7251 


7246 


7246 


7244 


7242 


7241 




1.46 


7240 


7239 


7239 


7240 


7341 


7242 


7243 


7245 


7248 


7251 




1.47 


7254 


7258 


7262 


7266 


7271 


7277 


7282 


7289 


7295 


7802 




1.48 


7310 


7317 


7326 


7334 


7343 


7353 


7363 


7373 


7384 


7896 




1.49 


7407 


7419 


7431 


7444 


7457 


7471 


7485- 


7499 


7515 


7529 





TABLE OF LOG T FUNCTIONS. 



57 



v.— TABLE OF LOG r FUNCTIONS OF p. 



p 





1 


2 


8 


4 


6 


6 


7 


8 


1 
9 


1.50 


0.047545 


7661 


7577 


7694 


7612 


7629 


7647 


7666 


76S6 


7704 


1.51 


7724 


7744 


7764 


7785 


7806 


7828 


7850 


7873 


7896 


7919 


1.63 


7948 


7967 


7991 


8016 


8041 


8067 


8093 


8120 


8146 


8174 


1.53 


8201 


8229 


8258 


8287 


8316 


8346 


8376 


8406 


8437 


816R 


1.54 


8500 


8532 


8664 


8597 


8630 


8664 


8698 


8732 


8767 


8802 


1.55 


8837 


8873 


8910 


8946 


8968 


9021 


9059 


9097 


9185 


9174 


1.58 


0214 


9254 


9294 


9334 


9375 


9417 


9458 


9500 


9543 


9580 


1.57 


06 iO 


9672 


9716 


9761 


9806 


9851 


9896 


9942 


9989 


6035 


1.58 


0.050082 


0130 


0177 


0225 


0274 


0323 


0372 


0422 


0472 


0522 


1.50 


0578 


0624 


0676 


0728 


0780 


0833 


0886 


0939 


0993 


1047 


1.60 


1102 


1157 


1212 


1268 


1324 


1380 


1487 


1494 ■ 


1552 


1610 


1.61 


1668 


1727 


1786 


1846 


1905 


1965 


2025 


2086 


2147 


2209 


1,68 


2271 


2833 


2396 


2459 


2522 


2586 


2650 


2715 


2780 


2845 


1.68 


2011 


2977 


8043 


3110 


3177 


8244 


a312 


3880 


3449 


3517 


1.64 


8587 


3656 


3726 


3797 


8867 


8938 


4010 


4081 


4154 


4226 


1.65 


4200 


4372 


4446 


4519 


4594 


4668 


4748 


4819 


4894 


4970 


1.66 


5047 


5124 


5201 


5278 


5356 


5484 


5513 


5592 


5671 


5740 


1.67 


5880 


5911 


5991 


6072 


6154 


6285 


6317 


6400 


6482 


6566 


1.68 


6640 


6783 


6817 


6901 


6986 


7072 


7157 


7243 


7322 


7416 


1.60 


7608 


7690 


7678 


7766 


7854 


7943 


8032 


8122 


8211 


8301 


1.70 


8301 


8482 


8673 


8664 


8756 


8848 


8941 


9034 


9127 


9220 


1.71 


9814 


9409 


9502 


9598 


9»93 


9788 


9884 


9980 


6077 


6174 


1.7a 


0.060271 


0869 


0467 


0565 


0664 


0763 


0662 


0961 


1061 


1162 


1.78 


1902 


1363 


1464 


1566 


1668 


1770 


1873 


1976 


2079 


2ia3 


1.74 


2287 


2391 


2496 


2601 


27*06 


2812 


2918 


8024 


3131 


3238 


1.7» 


8845 


3458 


8561 


8669 


3778 


3887 


3996 


4105 


4215 


4326 


1.78 


4486 


4547 


4659 


4770 


48S2 


4994 


5107 


5220 


5338 


5447 


1.77 


6561 


6675 


5789 


5904 


6019 


6135 


6251 


6367 


6484 


6600 


1.78 


6718 


6885 


6953 


7071 


7189 


7308 


7427 


7547 


76C6 


7787 


1.70 


7007 


8023 


8149 


8270 


8392 


8514 


8636 


8759 


8882 


9005 


1.80 


0120 


9258 


9377 


9501 


9626 


9751 


9877 


d006 


6129 


6255 


1.81 


0.070388 


0509 


0637 


0765 


0693 


1021 


1150 


1279 


1408 


1538 


1.88 


1668 


1798 


1929 


2060 


2191 


2322 


2454 


2586 


2719 


2852 


1.83 


2985 


8118 


3252 


3386 


3520 


3655 


3790 


8925 


4061 


4197 


1.84 


4383 


4470 


4606 


4744 


4881 


5019 


5157 


5295 


5434 


5573 


1.85 


5712 


5852 


6992 


6132 


6273 


6414 


6555 


6607 


C83S 


6980 


1.86 


7128 


7966 


7408 


7552 


7696 


7840 


7984 


8128 


8273 


8419 


1.87 


a564 


8710 


8856 


9002 


9149 


9296 


9443 


9591 


9739 


9887 


1.88 


0.080036 


0184 


0383 


0483 


06.33 


0783 


0933 


1084 


1234 


1386 


1.89 


1537 


1689 


1841 


1994 


2147 


2299 


2463 


2607 


2761 


2915 


1.00 


' 8060 


8224 


3379 


3535 


3690 


3846 


4003 


4150 


4316 


4474 


1.01 


4631 


4789 


4947 


5105 


5264 


5423 


5562 


5742 


5902 


6062 


1.02 


6223 


6888 


6544 


6706 


6867 


7029 


7192 


7354 


7517 


7680 


1.03 


7844 


8007 


8171 


8386 


8500 


8665 


8830 


8996 


9161 


9327 


1.04 


0404 


9660 


9627 


9995 


6162 


5830 


5498 


6666 


6835 


1004 


1.05 


0.001178 


1848 


1512 


1683 


1858 


2024 


2195 


2366 


2537 


2709 


1.06 


2881 


3054 


3227 


3399 


3573 


3746 


3920 


4094 


4269 


4443 


1.07 


4618 


4794 


4960 


5145 


53>1 


5498 


5674 


5851 


6029 


6206 


1.08 


6884 


6562 


6740 


6919 


^ 


7277 


7457 


7637 


7817 


7997 


1.00 


8178 


8869 


8640 


8722 


9085 

\ 


9268 


9450 


9633 


9816 



; 



58 



STATISTICAL METHODS. 



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



• • • • 


Inches to Millimeters. 


1 


2 


8 


4 


6 


6 


7 


8 


9 


2S.40 


50.80 


76.20 


101.60 


137.00 


152.40 177.80 


.203.20 


228.60 


10 


279 40 


804.80 


330.19 


855.59 


380.99 


406.89 481.79 


457 19 


482.59 


20 


538.39 


558.79 


584.19 


609.59 


634.99 


060.89 685.79 


711.19 


rd6.59 


30 


787.39 


812.79 


838 19 


863.59 


888.99 


914.89 939.76 


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 


1820.8 


1346.2 


1371.6 


1397.0 


1422.4 


1447.8 


1473.2 


1498.6 


t)0 


1549.4 


1574.8 


1600.2 


16».6 


1651.0 


1676.4 


1701.8 


1727.2 


1752.6 


70 


1803.4 


1828.8 


1854.2 


1»79.6 


1906.0 


1930.4 


1955.8 


1981.2 


2006.6 


80 


2057.4 


2082.8 


2108.2 


2133.6 


2159.0 


2181.4 


2209.8 


2285.2 


2260.6 


90 


2311.4 


2336.8 


2:362.2 


2387.6 


2413.0 


2438.4 


2463.8 


2489.2 


2514.6 


Twelfths. 


Sixteenths. 


1/12 


2.12 


7/12 


14.82 


1/16 


1.59 


5/16 


7.94 


9/16 


14.29 


18/16 


20.64 


•2/12 


4.28 


8/12 


16.9:) 


.1/8 


3.17 


8/8 


9.62 


6/8 


15.87 


7/8 


22.22 


8/12 


6.35 


9/12 


19.05 


8/16 


4.76 


7/16 


11.11 


11/16 


17.46 


16/16 


28.81 


4/12 


8.47 


10/12 


21.17 


1/4 


6.35 


1/2 


12.70 


8/4 


19.05 


25.4( 


5/12 


10.58 


11/12 


23.28 


















6/12 


12.70 


12/12 


25.40 



















FIBST TO SIXTH POWERS OF INTEGERS. 



59 



TABLE Vn.— FIBST TO SIXTH POWERS OF INTEGERS FROM 1 TO 80. 



Powers. 


First. 


Gecond. 


Third. 


Fourth. 


Fifth. 


Sixth. 


1 


1 


1 


1 


1 


1 


8 


4 


8 


16 


32 


64 


8 


9 


27 


81 


248 


729 


4 


16 


64 


266 


1024 


4096 


5 


25 


125 


625 


8125 


15625 


6 


96 


216 


1296 


7776 


46656 


7 


49 


343 


2401 


16807 


117649 


8 


64 


512 


4096 


S2r68 


262144 


9 


81 


739 


6661 


59049 


631441 


10 


100 


1000 


10000 


100000 


1000000 


11 


121 


1831 


14641 


161051 


1771561 


12 


144 


1788 


20786 


248832 


2985984 


18 


169 


2197 


28561 


371;^ 


4826809 


14 


196 


2744 


3S416 


537824 


75^9536 


16 


225 


8875 


50625 


769375 


11390625 


16 


256 


4096 


65536 


1048676 


16777216 


17 


289 


4918 


83521 


1419857 


24137569 


18 


824 


5832 


104976 


1889568 


34012224 


19 


861 


' 6859 


1803:21 


2476099 


47045881 


90 


400 


8000 


160000 


3200000 


64000000 


21 


441 


9261 


194481 


4084101 


86766121 


28 


484 


10648 


284256 


5153632 


113379904 


28 


529 


12167 


279841 


6436343 


1480:^5889 


24 


576 


13824 


331776 


7962624 


191102976 


25 


625 


156-^5 


390625 


9765625 


24414U625 


26 


676 


17576 


456976 


11881376 


806915776 


27 


729 


19663 


531441 


14348907 


387420489 


28 


784 


21952 


614656 


17210868 


481890804 


29 


841 


24389 


707281 


20511149 


594828821 


90 


900 


27000 


810000 


24800000 


729000000 



TABLE VIII, — SQUARES, CUBES, SQUARE ROOTS. 



No. 


Squares. 


Cubes. 


Square 
RootR. 


Cube Roots. 


Reciprocals. 


1 


1 


1 


1.0000000 


1.0000000 


l.OOOOOOOOO 


2 


4 


8 


1.4142136 


1.2599210 


.500000000 


3 


9 


27 


1.7320508 


1.4422496 


.333333333 


4 


16 


64 


2.0000000 


1.6874011 


.250000000 


5 


25 


126 


2.2360680 


1.7099759 


.200000000 


6 


36 


216 


24494897 


1.8171206 


.166666667 


7 


49 


343 


2.6457513 


1.9129312 


.142857143 


8 


64 


612 


2.8284271 


2.0000000 


.125000000 


9 


81 


729 


3.0000000 


2.0600637 


.111111111 


10 


100 


1000 


3.1622777 


2.1544347 


.100000000 


11 


121 


1331 


8.3166248 


2.2239601 


.090909091 


12 


144 


1728 


3.4641016 


2.2804286 


.063333333 


13 


169 


2197 


3.6065513 


S.3513347 


.076923077 


14 


196 


2744 


8.7416674 


2.4101422 


.071428571 


15 


225 


3376 


8.8729633 


2.4662121 


.066666667 


16 


256 


4096 


4.O0O0O0O 


2.6196421 


.062500000 


17 


289 


4913 


4.1231056 


2.5712616 


.058823529 


18 


324 


5832 


4.2426407 


2.6207414 


.055556556 


19 


361 


6859 


4.3588089 


2.6684016 


.052631579 


20 


400 


8000 


4.4721360 


2.7144177 


.050000000 


21 


441 


9861 


4.5825757 


2.7589243 


.047619048 


22 


484 


10648 


4.6904158 


2.8020393 


.045454545 


23 


529 


12167 r 


4.7968315 


2.8438670 


.043478261 


24 


676 


13824 


4.8969795 


2.8844991 


.041666667 


26 


626 


15625 


5.0000000 


2.9240177 


.040000000 


26 


676 


17576 


5.0990195 


2.9624960 


.038461538 


27 


729 


19683 


5.1961524 


3.0000000 


.037037037 


28 


784 


21952 


5.2915026 


3.0365889 


.035714286 


29 


841 


24389 


5.3851^i48 


8.0?23168 


.034482759 


30 


900 


27000 


5.477^256 


8.1072325 


.033333333 


31 


961 


29791 


5.5677644 


8.1413806 


.032258065 


82 


1024 


32768 


5.6568542 


8.1748021 


.031250000 


83 


1089 


35937 


5.7445626 


8.2075343 


030309030 


34 


1156 


39304 


5.8309519 


3.2396118 


.029411765 


35 


1225 


42875 


5.9160798 


8.2710663 


.028571429 


36 


1296 


46656 


6.0000000 


8.3019272 


.027777778 


37 


1369 


50653 


6.0827625 


3.3322218 


.027027027 


38 


1441 


54873 


6.1644140 


8.3619754 


.026315780 


39 


1521 


59319 


6.2449960 


8.3912114 


.025641026 


40 


1600 


64000 


6.8245553 


8.4199619 


.025000000 


41 


1681 


68921 


6.4081242 


8.4482172 


.024390244 


42 


1764 


74088 


6.4807407 


8.4760266 


.023809624 


43 


1849 


79507 


6.55743R5 


3.5033981 


.023255814 


44 


1936 


85184 


6.6332496 


8.5303483 


.022'«>72ra 


45 


2025 


91125 


6.7082U39 


3.5568933 


.022222222 


46 


2116 


97:«6 


6.7823300 


8.5830479 


.021739130 


47 


2209 


103823 


6.8556546 


3.6068261 


.021276600 


48 


2304 


110592 


6.9282032 


3.6342411 


.0208333:^3 


49 


2401 


117X549 


7.0000000 


3.6593057 


.020406163 


50 


2600 


125000 


7.0710678 


3.6840314 


.020000000 


61 


2601 


132651 


7.1414284 


8.7064298 


.019607843 


52 


2704 


140608 


7.2111026 


8.7325111 


.019280769 


53 


2809 


148877 


7.2801099 


8.7562858 


.018867925 


64 


2916 


157464 


7.3484692 


8.7797631 


.018518519 


55 


3025 


166375 


7.4161965 


8.8029525 


.018181818 


56 


8136 


175616 


7.4833148 


8.8258624 


.017857148 


57 


3249 


185193 


7.5498344 


3.8485011 


.017543860 


58 


3364 


195112 


7.6157731 


8.8708766 


.017241879 


69 


3481 


206379 


7.6811457 


3.8929965 


.016949158 


60 


3600 


216000 


7.7459667 


8.9148676 


.016666667 


61 


8T21 


226981 


7.6102497 


3.9364978 


.016398443 


62 


3844 


238328 


7.8740079 


8.9578915 


.016129082 



60 



i 



11 M 



TABLE VIII. — SQUARES, CUBES, SQUARE ROOTS. 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Roots. 


Reciprocals. 


125 


15625 


1953125 


11.1803399 


5.0000000 


.008000000 


126 


15876 


2000376 


11.2249722 


5.0132979 


.007936608 


rr 


16129 


2048388 


11.2694277 


5.0265257 


.007874016 


128 


16384 


2097152 


11.3137085 


5.0396842 


.007812500 


129 


16641 


2146689 


11.3578167 


5.0627743 


.007751938 


130 


16900 


2197000 


11.4017548 


5.0657970 


.007692308 


131 


17161 


2248091 


11.4455231 


5.0787531 


.007633588 


132 


17424 


2299968 


11.4891253 


5.0916434 


.007575758 


133 


17689 


2352637 


11.5325626 


5.1044687 


.007518797 


134 


17956 


2406104 


11.5758369 


5.1172299 


.007462687 


135 


18225 


2460875 


11.6189500 


5.1299278 


.007407407 


136 


18496 


2515456 


11.6619038 


5.1425632 


.007352941 


137 


18769 


2571353 


11.7046999 


5.1551367 


.007299270 


138 


19044 


2628072 


11.7473401 


5.1676493 


.007246377 


139 


19321 


2685619 


11.7898261 


5.1801015 


.007194245 


140 


19600 


2744000 


11.8321596 


6.1924941 


.007142857 


141 


19681 


2803221 


11.8743421 


5.2048279 


.007092199 


142 


20164 


2863288 


11.9163753 


5.2171034 


.007042254 


143 


20449 


2924207 


11.9582607 


5.2293215 


.006998007 


144 


20736 


2985984 


12.0000000 


5.2414828 


.006944444 


145 


21025 


3048625 


12.0415946 


5.2536879 


.006896552 


146 


21316 


8112136 


12.0830460 


5.2656374 


.006849315 


147 


21609 


8176628 


12.1243557 


5.2rr6321 


.006802721 


148 


21904 


8241792 


12.1655251 


5.2895725 


.006756757 


149 


22201 


8307949 


12.2065556 


5.8014592 


.006711409 


150 


22500 


8375000 


12.:i474487 


5.3132928 


.006666667 


151 


22801 


3442951 


12.2882057 


5.3250740 


.006622517 


152 


23104 


3511806 


12.3288280 


5.3368038 


.006578947 
.006535948 


158 


23409 


8581577 


12.3693169 


5.3484812 


154 


23716 


8652264 


12.4096736 


6.3601084 


.006493606 


155 


24025 


if/28875 


12.4496996 


5.3716854 


.006451613 


156 


24336 


8796416 


12.4899960 


5.3832126 


.006410256 


157 


24649 


8869698 


12.5299641 


6 3946907 


.006369427 


158 


24964 


3944312 


12.5698051 


5.4061202 


.006329114 


159 


26281 


4019679 


12.6095202 


5.4175015 


.006289808 


160 


25600 


4096000 


12.6491106 


5.4288852 


.006250000 


161 


25921 


4173281 


12.6885775 


5.4401218 


.006211180 


162 


26244 


4251528 


12.7279221 


5.4518618 


.006172840 


163 


26569 


4330747 


12.7671453 


5.4625556 


.006184969 


164 


26896 


4410944 


12.8062485 


5.4737037 


.006097561 


165 


27225 


4492125 


12.8452326 


5.4848066 


.006060606 


166 


27556 


4574296 


12.8840987 


6.4958647 


.006024096 


167 


27889 


4657463 


12.9228480 


6.5068784 


.005968024 


168 


28224 


4741632 


12.9614814 


6.5178484 


.005952381 


169 


28561 


4826809 


13.0000000. 


6.5287748 


.oocoinoo 


170 


28900 


4918000 


18.08S4048 


5.5396583 


.006882353 


171 


29241 


5000211 


18.0766968 


5.5504991 


.005847953 


172 


29584 


5088448 


13.1148770 


5.5612978 


.005813953 


173 


29929 


5177717 


13.1529464 


5.57^0546 


.006780347 


174 


30276 


5268024 


13.1909060 


5.5827702 


.00574n28 


175 


30625 


5359875 


13.2287566 


5.5934447 


.005714286 


176 


80976 


5451776 


18.2664992 


5.6040787 


.006681818 


177 


31329 


5545233 


18.3041347 


5.6146724 


.005649718 


178 


31684 


5639752 


18.3416641 


6.62.52263 


.005617978 


179 


82041 


5735339 


13.3790882 


6.6357408 


.005586592 


180 


32400 


5832000 


13.4164079 


5.6462162 


.005555556 


181 


82761 


5929741 


13.4536240 


5.6566528 


.006624862 


182 


33124 


6028568 


18.4907376 


6.6670511 


.005494505 


183 


83489 


6128487 


13.5277493 


6.6774114 


.006464481 


184 


33856 


6229504 


13.5646600 


5.6877340 


.0054S4'n3 


185 


34225 


6331625 


13.6014705 


5.6960192 


.005405405 i 
.005376344 


186 


34596 


6434856 


13.6381817 


6.7082675 

1 



62 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Roots. 


Reciprocals. 


187 


84969 


6639203 


13.6747943 


5.7184791 


.005347594 


188 


85344 


6644672 


13.7113092 


6.7286543 


.005319149 


189 


85721 


6751269 


13.7477271 


6.7887986 


.005291005 


190 


86100 


6859000 


13.7840488 


6.7488971 


.005263158 


191 


86481 


0967871 


18.8202750 


6.7589652 


.005235602 


198 


36864 


7077688 


13.8664065 


6.7689962 


.005208833 


198 


87»I9 


7189057 


13.8924440 


5.7789966 


.005181347 


194 


37636 


7301384 


13.9288883 


6.7889604 


.005154639 


196 


38025 


7414875 


13.9642400 


6.7988900 


.005128205 


196 


88416 


7520536 


14.0000000 


6.8087857 


.005102041 


197 


88809 


7645373 


14.0356688 


5.8186479 


.005076142 


196 


39204 


7762392 


14.0712473 


5.8284767 


.005050505 


199 


39601 


7880699 


14.1067360 


5.8382725 


.005025126 


aoo 


40000 


6000000 


14.1421356 


6.8480855 


.005000000 


• aoi 


-40401 


8120601 


14.1774469 


6.8577660 


.004975124 


aoe 


40604 


8242406 


14.2126704 


6.8674643 


.004950496 


ao6 


41209 


8365427 


14.2478068 


6.8771307 


.004926106 


204 


41616 


8489664 


14.2828569 


6.8867653 


.004901961 


906 


42025 


8615125 


14.3178211 


5.89fi86R'> 


.004878049 


ao6 


42436 


8741816 


14.3627001 


6.9059400 


.004854369 


207 


42849 


8869743 


14.3874946 


6.9154817 


.004830918 


206 


43264 


8996912 


14.4222051 


5.9249921 


.004807692 


200 


43681 


9120829 


14.4568323 


6.9344721 


.004784689 


210 


44100 


9261000 


14.4913767 


5.9439220 


.004761905 


211 


44521 


9893931 


14.5258390 


5.9583418 


.004739336 


212 


44944 


0528128 


14.5602198 


5.9627320 


.004716981 


213 


45369 


9663597 


14.5945195 


5.9720926 


.004694836 


214 


45796 


9600344 


14.6287388 


5.9814240 


.004672897 


215 


46225 


99HR875 


14.6628783 


5.9907264 


.004651168 


216 


46666 


10077696 


14.6969385 


6.0000000 


.004629630 


217 


47069 


10218313 


14.7309199 


6.0092450 


.004608296 


218 


47524 


10860232 


14.7648281 


6.0184617 


.004587156 


219 


47961 


10506459 


14.7966486 


6.0276502 


.004566210 


220 


48400 


10048000 


14.8323970 


6.0868107 


.004545455 


221 


46841 


10793861 


14.8660687 


6.0459485 


.004524887 


222 


49284 


10941048 


14.8996644 


6.0550489 


.004504505 


228 


49729 


11069567 


14.9331845 


6.0641270 


.004484301 


224 


50176 


11239424 


14.9666295 


6.0731779 


.004464286 


225 


50625 


11390625 


15.0000000 


6.0822020 


.004444444 


226 


51076 


11543176 


15.0332964 


6.0911994 


.004424779 


227 


51529 


11697063 


15.0665192 


6.1001702 


.004405286 


228 


61964 


11852852 


15.0996689 


6.1091147 


.004385966 


229 


52441 


12006069 


15.1327460 


6.1180332 


.004366812 


230 


62900 


12167000 


15.1657509 


6.1269257 


.004347826 


231 


63361 


12326391 


15.1966842 


6.1357924 


.004329004 


2Stt 


63824 


12487166 


15.2315462 


6.1446337 


.004310346 


233 


54289 


12649337 


15.2643375 


6.1534495 


.004291845 


284 


54756 


12812904 


15.2970585 


6.1622401 


.004273504 


235 


65225 


12977875 


15.3297097 


6.1710058 


.004255319 


236 


55696 


13144250 


15.3622915 


6.1797466 


.004237288 


237 


56169 


13312063 


15.3948043 


6.1884628 


.004219409 


288 


56644 


13481272 


16.4272486 


6.1971544 


.004201681 


239 


67121 


13651919 


15.4596248 


6.2058218 


.004184100 


240 


67600 


13824000 


15.4919334 


6.2144650 


.004166667 


241 


68061 


13997531 


15.5241747 


6.2230843 


.004149378 


242 


58504 


14172488 


15.5563492 


6.2316797 


.004132281 


243 


59049 


14348907 


15.5884573 


6.2402515 


.004116226 


244 


59536 


14526784 


15.62(Vt994 


6.2487998 


.004098861 


245 


60025 


14706125 


15.6524758 


6.2573248 


.004081633 


246 


60516 


14866936 


15.6848871 


6.2658266 


.004065041 


247 


61009 


15069223 


15.7162336 


6.2743054 


.004048688 


248 


61504 


15252992 


15.7480167 


6.2827613 


.004082268 



63 



TABLE VIII. — SQUARES, CUBES, SQUARE ROOTS. 



No, 


Squares. 


Cubes. 


Square 
Roots. 


Cube Boots. 


Reciprocals. 


249 


62001 


15438249 


15.7797338 


6.2911946 


.0MO16064 


250 


62500 


15625000 


15.8113883 


6.2990068 


.OOMOOOOO 


251 


63001 


15813251 


15.8429796 


6.8079986 


.008964064 


252 


63694 


16003008 


15.8745079 


6.3163596 


.003968254 


253 


61009 


16194277 


15.9059737 


6.8247036 


.008068560 


254 


64516 


16387064 


15.93ra773 


6.3390256 


.008087006 


255 


65025 


16581375 


15.9687194 


6.3413257 


.008921669 


266 


65536 


16777216 


16.0000000 


6.3496042 


.008906250 


257 


66049 


16974593 


16.0312195 


6.3578611 


.008891051 


258 


66564 


17173512 


16.0623784 


6.8660968 


.003875969 


259 


67081 


17373979 


16.0934769 


6.3748111 


.003861004 


260 


67600 


17576000 


16.1245155 


6.8825048 


.003846164 


261 


68121 


17779581 


16.1564944 


6.3906765 


.008831418 


262 


68644 


17984728 


16.1864141 


6.3988279 


.008816794 


263 


69169 


18191447 


16.2172747 


6.4069585 


.-003802281 • 


264 


69696 


18399744 


16.2480768 


6.4150687 


.008787879 


265 


70225 


18609625 


16.2788206 


6.4231683 


.003778686 


266 


70756 


18821096 


16.3096064 


6.4812276 


.003769896^ 


267 


71289 


19034163 


16.3401346 


6.4392767 


.003745818 


268 


71824 


19248832 


16.3707055 


6.4473057 


.008731843 


269 


72361 


19465109 


16.4012195 


6.4653148 


.008717472 


270 


72900 


19683000 


16.4316767 


6.4683041 


.008708704 


271 


73441 


19902511 


16.4620776 


6.4712786 


.008690087 


272 


73984 


20123648 


16.4924225 


6.4792286 


.008676471 


273 


7452J 


20346417 


16.5227116 


6.48n641 


.008668004 


274 


75076 


20570824 


16.6529454 


6.4960668 


.008649686 


275 


75623 


20796875 


16.5831240 


6.5029672 


.008686864 


276 


76176 


21024576 


16.6132477 


6.6108300 


.008628188 


2rr 


76729 


21253933 


16.6433170 


6.6186839 


.008610106 


278 


T7284 


21484952 


16.6783320 


6.5266189 


.00359n8S 


279 


77841 


21717639 


16.7032931 


6.6848361 


.008584220 


280 


78400 


21952000 


16.7332006 


6.6421326 


.008671429 


281 


78961 


22188041 


16.7680546 


6.6499116 


.008568710 


282 


79524 


22425768 


16.7928656 


6.6676722 


.008646090 


283 


80089 


22665187 


16.8226038 


6.6664144 


.003688660 . 


284 


80656 


22906304 , 


^16.8522995 
Ni6.8819430 


6.6731885 


.008621127 


285 


81225 


23149125 


6.5808443 


.003606772 


286 


^1796 


23393656 


16^116346 


6.6885323 


.003496608 


287 


82369 


23639903 


16]m^43 


6.6962023 


.006484321 


288 


82944 


23887872 


16.970S9I(27 


6.6038545 


.0034';^282 


289 


83521 


24137569 


17.0298864 


6.6114890 


.003460206 


290 


84100 


24389000 


"^ .6.6191060 


.003448276 


291 


84681 


24643171 


17.0587221 


6.6267054 


.003436426 


292 


85264 


24897068 


17.0880076 


6.6842874 


.003424668 


293 


85849 


23153757 


17.1172428 


6.6418522 


.003412960 


294 


86406 


25412184 


17.1464282 


6.6493996 


.008401861 


295 


87025 


25672375 


17.1755640 


6.6569802 


.003880681 


296 


87616 


23934a36 


17.2046506 


6.6644437 


.008378878 


297 


88209 


26198073 


17.2336879 


6.671940S 


.003367008 


298 


88804 


26463592 


17.2626765 


6.6794200 


.003356706- 


299 

n/\/\ 


89401 


26730699 


17.2916165 

4 jk <V\Afe^A04 


6.6868831 


.003344482 


300 


90000 


27000000 


17.3205081 


6.6943295 


.003888888 


301 


90601 


27270901 


17.3493516 


6.7017593 


.003322250 


302 


91204 


27543608 


17.3781472 


6.7091729 


.003311256 


303 


91809 


27818127 


17.4068952 


6.7166700 


.003800880 


304 


92416 


28094464 


17.4356958 


6.7289508 


.008280474 


305 


93025 


28372625 


17.4642492 


6.7318156 


.003278680 


306 


93636 


28652616 


17.4928557 


6.7886641 


.003267074 


807 


94249 


28934443 


17.6214156 


6.7469967 


.003267820 


308 


94864 


29218112 


17.6499288 


6.7688184 


.008246788 


809 


95481 


2a503629 


17.5783968 


6.7606148 


.008288246 


310 


9C100 


29791000 


17.6068169 


6.7678995 


.008226806 



64 



CUBE ROOTS, AND RECIPKOOALS. 



No. 


Sqpiarea. 


Cubes. 


Squai-e 
Roots. 


Cube Roots. 


Reciprocals. 


311 


96721 


80080231 


17.6351921 


6.7751690 


.003215434 


812 


97344 


80371328 


17.6635217 


6.7824229 


.003205128 


813 


97969 


80664297 


17.6918060 


6.7896618 


.008194888 


814 


96696 


80959144 


17.7200451 


6.7968844 


.008184713 


315 


99225 


81255875 


17.7482398 


6.8040921 


.008174603 


316 


99656 


81554496 


17.7763888 


6.8112847 


.003164557 


317 


100489 


81855013 


17.8044938 


6.8184620 


.003154574 


818 


101124 


32157432 


17.8325545 


6.8256242 


.003144654 


319 


101761 


82461759 


17.8605711 


6.8827714 


.003134796 


330 


102400 


32768000 


17.8885438 


6.8399087 


.008125000 


m 


108Q41 


33076161 


17.9164729 


6.8470213 


.003115265 


3^ 


103684 


83386248 


17.9443584 


6.8541240 


.003105590 


333 


104829 


33696267 


17.9722008 


6.8612120 


.003095975 


3^ 


104976 


84012224 


18.0000000 


6.6682855 


.003066420 


325 


105625 


843281^ 


18.0277564 


6.8753443 


.003076923 


326 


106276 


84645976 


18.0554701 


6.8823888 


.003067485 


327 


100929 


84965783 


18.0631413 


6.8894188 


.003058104 


328 


107584 


85287552 


18.1107703 


6.8964345 


.003048780 


829 


108241 


35611289 


18.1383571 


6.9034359 


.003039514 


830 


108900 


85937000 


18.1059021 


6.9104232 


.003030303 


331 


109561 


36264691 


18.1934054 


6.9173964 


.003021148 


332 


110224 


36594368 


18.22066?2 


6.9243556 


.003012048 


333 


110689 


36U26037 


18.2482876 


6.9313006 


.003003008 


334 


111556 


3?259701 


18.2756669 


6.9382321 


.002994012 


335 


112225 


37595375 


18.3030062 


6.9451496 


.002985075 


336 


112896 


87933056 


18.3808028 


6.9520533 


.002976190 


337 


113669 


88272758 


18.3575598 


6.9589484 


.002967359 


338 


114214 


88614472 


18.3847763 


6.9658198 


.002958580 


839 


114921 


88958219 


18.4119526 


6.9?26826 


.002949853 


840 


115600 


89304000 


18.4390889 


6.9795321 


.002941176 


341 


116281 


89651821 


18.4661858 


6.9863681 


.002932551 


342 


116964 


40001688 


18.4932420 


6.9961906 


.002923977 


343 


117649 


40353607 


18.5202592 


7.0000000 


.002915462 


. 844 


118336 


40707584 


18.5472370 


7.0067962 


.002906977 


845 


119025 


41063625 


18.5741756 


7 0135791 


.002898551 


846 


119n6 


41421736 


18.6010752 


7.0203490 


.002890173 


847 


120409 


41781928 


18.6279360 


7.02n058 


.002881844 


848 


121104 


42144192 


18.6547581 


7.0338497 


.002873568 


849 


121801 


42508549 


18.6815417 


7.0405806 


.002865330 


ffiiO 


122500 


42875000 


18.7062869 


7.0472987 


.002857143 


851 


123201 


43243551 • 


18.7349940 


7.0540041 


.002849003 


852 


•123904 


43614208 


18.7616630 


7.0606967 


.002840909 


868 


124609 


43986977 


18.7882942 


7.0673767 


.002832861 


854 


125816 


44361864 


18.8148877 


7.0740440 


.002824859 


355 


126025 


44788875 


18.8414437 


7.0806968 


.002816901 


856 


126736 


45118016 


18.8679628 


7.0873411 


.002808989 


857 


127449 


45499298 


18.8»i4436 


7.0939709 


.002801120 


858 


128164 


45882712 


18.9208879 


7.1005885 


.002793290 


859 


128881 


46268279 


18.9472953 


7.1071937 


.002785515 


.860 


129600 


46656000 


18.9736660 


7.1137866 


.002777778 


361 


130321 


47045881 


19.0000000 


7.1203674 


.002770063 


862 


131044 


47437928 


19.0262976 


7.1269360 


.002762431 


863 


131769 


47832147 


19.0525589 


7.1334925 


.002754821 


864 


132496 


48228544 


19.0787840 


7.1400370 


.002747253 


365 


133225 


48627125 


19.1049732 


7.1465695 


.002739726 


366 


138056 


49027896 


19.1311265 


7.1530901 


.002732240 


867 


134689 


49430863 


19.1572441 


7.1595988 


.002^4796 


868 


135424 


49636082 


19.1838261 


7.1660957 


.002n7391 


869 


186161 


60243409 


19.2093727 


7.1725809 


.002n0027 


870 


136900 


60658000 


19.2353841 


7.1790544 


.002702703 


8n 


137641 


51064811 


19.2613603 


7.1855162 


.002695418 


872 


138384 


61478848 


19.2873015 


7.1910663 


.002688172 



bo 



TABLE VIII. — SQUABE8, CUBES, SQUARE ROOTS. 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Roots. 


Reciprocals. 


878 


139129 


51895117 


19.3132079 


7.1984050 


.002680965 


874 


139876 


52313624 


19.3390796 


7.2046322 


.002673797 


875 


140625 


52734375 


19.3649167 


7.2112479 


.002666667 


876 


141376 


53157376 


19.3907194 


7.2176522 


.002660674 


W7 


142129 


53582638 


19.4164878 


7.2240450 


.002662520 


878 


142884 


54010152 


19.4422221 


7.2304268 


.002645603 


879 


143641 


54439939 


19.4679228 


7.2367972 


.002638522 


880 


144400 


54872000 


19.4935887 


7.2431565 


.002681579 


881 


145161 


55306841 


19.5192213 


7.2496045 


.002624672 


dS2 


145924 


55742968 


19.5448208 


7.2558415 


.002617801 


888 


146680 


56181887 


19.5703858 


7.2621675 • 


.002610966 


884 


147456 


56623104 


19.5959179 


7.2684824 


.002604167 


385 


148225 


57066625 


19.6214169 


7.2747864 


.002697403 


886 


148996 


67512456 


19.6468827 


7.2810794 


.002690674 


887 


149769 


57960603. 


19.6723156 


7.2873617 


.002688979 


888 


150544 


58411072 


19.6977156 


7.2936330 


.0Q2677SM) 


889 


151821 


58863869 


19.7230629 


7.2996936 


.002670694 


890 


152100 


59319000 


19.748417;L 
19.778719lr 


7.3061486 


.002664103 


891 


152881 


59776471 


7.3123628 


.002657545 


892 


153664 


60236288 


19.7969899 


7.3186114 


.002651020 


898 


154449 


60696457 


19.8^^276 


7.3248296 


.002544629 


894 


155236 


61162984 


19 8404832 


7.3310960 


.002536071 


895 


156025 


61629875 


19'.a?46069 
19.i997487 


7.387:2339 


.002631646 


896 


156816 


62099186 


7.3484206 


.00S625253 


897 


157609 


62570773 


19.9248588 


7.3496966 


.002518692 


896 


158404 


63044792 


19.9499873 


7.8557«24 


.002612663 


899 


159201 


63521199 


19.9749644 


7.3619178 


.002506266 


400 


160000 


64000000 


20.0000000 


7.8680630 


.008600000 


401 


160601 


64481201 


20.0249644 


7.8741979 


.008496766 


400 


161604 


64964806 


20.0499377 


7.8808227 


.008487662 


408 


162409 


65450627 


20.0748599 


7.88643ra 


.002481390 


404 


168216 


65939264 


20.0997512 


7.8926418 


.008476848 


406 


164025 


66430125 


20.1246118 


7.8966363 


.008469136 


406 


164836 


66923416 


20.1494417 


7.4047206 


.002468064 


407 


165649 


67419143 


20.1742410 


7.4107960 


.002467008 


408 


166464 


67917312 


20.1990099 


7.4166696 


.008460960 


409 


167281 


68417929 


20.2237484 


7.4229142 


.008444988 


410 


168100 


68921000 


20.2484567 


7.4269589 


.008489084 


411 


168921 


69426531 


20.2781349 


7.4349938 


.008483090 


412 


169744 


69934528 


20.2977831 


7.4410189 


.00842n84 


413 


170569 


70444997 


20.8224014 


7.4470842 


.008421808 


414 


171396 


70967944 


20.3469699 


7 4530399 


.008416459 


415 


172225 


71473375 


20.3715488 


7.459UU59 


.008409639 


416 


173056 


71991296 


20.3960rBl 


7.4650223 


.008406846 


417 


178889 


72511713 


20.4205779 


7.4709991 


.002396062 


418 


174ra4 


. 73034632 


20.4450483 


7.4769664 


.002392844 


419 


175561 


73560059 


20.4694695 


7.4829242 


.002886636 


420 


176400 


74088000 


20.4939015 


7.4888724 


.002380958 


421 


17^41 


74618461 


20.5182845 


7.4946113 


.002375297 


422 


178084 


75151448 


20.5426386 


7.5007406 


.002360668 


423 


178929 


76686967 


20.5669636 


7.5066607 


.008364066 


424 


179776 


.6225024 


20 5912603 


7.5125715 


.008868401 


425 


180625 


76765625 


20.6156281 


7.5184730 


.002358041 


426 


181476 


77306776 


20.6397674 


7.5243652 


.008847418 


427 


182329 


77854483 


20.6639783 


■ 7.5302482 


.008841920 


428 


183184 


78402758 


20.6881609 


7.5361221 


.002336449 


429 


184041 


Tsa'sa'^o 


20.7123152 


7.5419667 


.008331002 


430 


184900 


79507000 


20.7364414 


7.5478423 


.00232S681 


431 


186761 


80062991 


20.7605395 


t .OOSOotx} 


.008320186 


432 


186624 


80621568 


20.7846097 


7.5595263 


.002814816 


438 


187489 


81182737 


20.8086520 


7.5653548 


.008309469 


434 


188356 


81746604 


20.8326667 


7.5711743 


.0'J8a04147 



06 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Roots. 


Reciprocals. 


435 


189225 


82312875 


20.8566536 


7.5769849 


.002298851 


436 


190096 


82881856 


20.8806130 ■ 


7.5827865 


.002293578 


437 


190969 


83453453 


20.9045450 


7.5885793 


.0022d8330 


438 


191844 


84027672 


20.9284495 


7.5943633 


.002283105 


439 


192721 


84604519 


20.9623268 


7.6001385 


.002277904 


440 


193600 


85184000 


20.976irr0 


7.6059049 


.002272727 


441 


191481 


85766121 


21.0000000 


7.6116626 


.002267574 


442 


195364 


86350888 


21.0237960 


7.6174116 


.002262143 


443 


196249 


86938307 


21.0475652 


7.6231519 


.002257336 


444 


197138 


87528384 


21.0713075 


7.6288837 


.002252252 


445 


198025 


88121125 


21.0950231 


7.6346067 


.002247191 


446 


198916 


88716536 


21.1187121 


7.6403213 


.002242152 


447 


199809 


89314623 


21.1423745 


7.6160272 


.002237136 


448 


200704 


89915392 


21.1660105 


7.6517247 


.002232143 


449 


201601 


90518849 


21.1896201 


7. 657413 J 


.002227171 


450 


202500 


911^^000 


21.2132034 

21.2367606 


7.6830913 


.002222222 


451 


203^1 


91733851 


7.6687665 


.002217295 


45@ 


204304 


92345408 


21.2602916 


7.6744303 


.002212389 


453 


205209 


92959677 


21.2837967 


7.6800857 


.002207506 


454 


206116 


93576664 


21.30f/«'«''58 


7.6857328 


.002202643 


455 


207025 


94196375 


21.3307290 


7.6913717 


.002197802 . 


456 


207936 


91818816 


21.3541565 


7.6970023 


.002192982 


457 


208819 


95443993 


21.3775583 


7.7026246 


.002188184 


458 


209764 


96071912 


21.4009316 


7.7082388 


.002183406 


459 


210681 


96702579 


21.4242853 


7.7138448 


.002178649 


460 


2116db 


97336000 


21.4476106 


7.7194426 


.002173913 


461 


212521 


97972181 


21.4709106 


7.7250325 


.002169197 


462 


213444 


98611128 


21.4941853 


7.7306141 


.002164502 


463 


214369 


99252847 


21.5174348 


7.7361877 


- .002159827 


464 


215296 


99897344 


21.5406592 


7.7417532 


.002155172 


465 


216225 


100644625 


21.5638587 


7.7473109 


.002150538 


466 


217156 


101194696 


21.5870331 


7.7528606 


.002145923 


467 


218089 


101847563 


21.6101828 


7.7581023 


.002141328 


468 


219024 


102503232 


21.6333077 


7.T639361 


.002136752 


469 


219961 


103161709 


21.6564078 


7.7694620 


.002132196 


470 


220900 


103823000 


21.6791834 


7.7749801 


.002127660 


471 


221841 


104487111 


21.7025314 


7.7801904 


.002123142 


472 


222784 


105154048 


21.';^55610 


7.7859928 


.002118644 


473 


223729 


105823817 


21.7485632 


7.7914875 


.002114165 


474 


224676 


106496424 


21.7715411 


7.7969745 


.002109705 


475 


225625 


107171875 


21.7944947 


7.8024538 


.002105263 


476 


226576 


107850176 


21.8174242 


7.8079254 


.002100840 


477 


227529 


108531333 


21.8403297 


7.8133892 


.002096436 


478 


228484 


109215352 


21 8632111 


7.8188456 


.002092050 


479 


229441 


109902239 


21.8860686 


7.8242942 


.002087683 


480 


230100 


110592000 


21.9089023 


7.8297353 


.002088333 


481 


231361 


111284641 


21.9317122 


7.8351688 


.002079002 


482 


232324 


111980168 


21.9544981 


7.8405919 


.002074689 


483 


233289 


112678587 


21.9772610 


7.8160134 


.002070393 


484 


234256 


113379904 


22.0000000 


7.8514244 


.002066116 


485 


23522!$ 


114064125 


22.0227155 


7.8568281 


.002061856 


486 


236196 


114791256 


22.0451077 


7.8622242 


.002057613 


487 


237169 


115501303 


22.0680765 


7.8676130 


.002053388 


488 


238144 


11C214272 


22.0907220 


7.8729944 


.002049180 


489 


239121 


116930169 


22.1133144 


7 8783684 


.002044990 


490 


240100 


117649000 


22.1359136 


7.8837352 


.002040816 


491 


241081 


lia370771 


22.15a5198 


7.8890916 


.002036660 


492 


242064 


119095488 


22.1810730 


7.8914468 


.002032520 


493 


243049 


119823157 


22.2036033 


7.8997917 


.002028398 


494 


241036 


120553784 


22.2261108 


7.9051294 


.002021291 


495 


245025 


121287375 


22.24*5955 


7.9101599 


.002020202 


496 


246016 


122023936 


22.2710575 


7.9157832 


.002016129 



67 



TABLE VIII. — SQUABES, CUBES, SQUAEB ROOTS. 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Boots. 


Reciprocals. 


497 


247009 


122763473 


22.2934968 


7.9210994 


.002012072 


496 


248004 


123505992 


22.3159136 


7.9264085 


.002006032 


499 


249001 


124251499 


2S2.3383a79 


7.9317104 


.002001006 


.600 


250000 


125000000 


22.3606798 


7.9370053 


.002000000 


601 


251001 


125751501 


22.3830293 


7.9422931 


.001996008 


603 


2S20O4 


126506006 


22.4053565 


7.9475739 


.001992U»2 


608 


253009 


127263527 


22.4276615 


7.9528477 


.001988073 


604 


254016 


128024064 


22.4499448 


7.9581144 


.001964127 


605 


255025 


128787625 


22.4722051 


7.968*743 


.001960196 


606 


256036 


129554216 


22.4944488 


7.9686271 


.001970285 


5or 


257049 


130323843 


22.5166605 


7.9738781 


.001972387 


608 


258064 


131096512 


22.5388553 


7.9791128 


.001968604 


609 


259081 


131872229 


22.5610283 


7.9843444 


.001964637 


510 


260100 


132651000 


22.5881796 


7.9895697 


.001960784 


611 


261121 


133432831 


22.6053091 


7.9947883 


.001956947 


512 


262144 


134217728 


22.6274170 


8.0000000 


.001953125 


613 


263169 


185005697 


22.6495033 


8.0052049 


.001949818 


514 


264196 


185796744 


22.6715681 


8.0104032 


.001945625 


515 


265225 


186590875 


22.6936114 


8.0155946 


.001941748 


616 


266256 


13738S096 


22.7156834 


8.0207794 


.001937964 


617 


26?289 


188188413 


22.7376340 


8.0259574 


.001934286 


518 


268324 


18899ia32 


22.7596134 


8.0311287 


.001980502 


519 


269361 


189798359 


22.7815715 


8.0862935 


.001926782 


620 


270400 


140608000 


22.8035065 


8.0414515 


.001983077 


521 


271441 


141420761 


22.8254244 


8.0466030 


.001919386 


522 


272484 


142236648 


22.8473193 


8.0517479 


.001915709 


523 


273529 


143055667 


22.8691933 


8.0568862 


.001912046 


624 


274576 


143877824 


22.8910463 


8.0620180 


.001906897 


625 


275625 


144703125 


22.9128785 


8.0671432 


.001904762 


626 


276676 


145531576 


22.9846899 


8.0722620 


.001901141 


627 


2^.7'i'29 


146363183 


22.9564806 


8.0778743 


.001897533 


528 


278784 


147197952 


22.9782506 


8.0824800 


.001898939 


529 


279841 


148035889 


23.0000000 


8.0875794 


.001890359 


530 


280900 


148877000 


23.0217289 


8.0926723 


.001886792 


531 


281961 


149?21291 


23.0434372 


8.0977589 


.001883239 


532 


28:^024 


150568768 


23.0651252 


8.1028390 


.001879699 


533 


284069 


151419437 


23.0867928 


8.1079128 


.001876178 


534 


285156 


152273304 


23.1084400 


8.1129803 


.001872659 


635 


286225 


153130375 


23.1300670 


8.1180414 


.001869159 


536 


287296 


153990656 


23.1516738 


8.1230962 


.001865672 


537 


288369 


154854158 


23.1732605 


8.1281447 


.001862197 


638 


289444 


155720872 


23.1948270 


8.1331870 


.001858736 


639 


290521 


156590619 


23.2163785 


8.1382230 


.001855288 


640 


291600 


157464000 


23.2379001 


8.1432529 


.001851852 


641 


292681 


158340421 


23.2594067 


8.1482765 


.001848429 


542 


293764 


159220088 


2:3.2808935 


8.1532939 


.001845018 


543 


294849 


16C103007 


2:3.3028604 


8.1588051 


.001841621 


544 


295936 


160989184 


23.3238076 


8.1633102 


.001838235 


545 


297025 


161878625 


23.3452351 


8.1683092 


.001834862 


546 


298116 


162771336 


23.3666429 


8.1|-33020 


.001831502 


547 


299209 


163667323 


23.3880311 


8.1782888 


.001828154 


648 


300304 


164566592 


23.4093998 


8.1832695 


.001834818 


549 


801401 


165469149 


23.43(K'490 


8.1882441 


.001821494 


550 


802500 


166375000 


28.4520788 


8.1932127 


.001818183 


551 


803601 


107284151 


23.4733892 


8.1981753 


.001814883 


552 


304704 


1G819CC08 


23.4946802 


8.2031319 


.001811594 


553 


305809 


] 60112377 


23.5159520 


8.2080825 


.001808318 


554 


806916 


170031464 


2:3.5373046 


8.2130271 


.001805054 


655 


308025 


170958875 


23.5584380 


8.2179657 


.001801808 


656 


309136 


171879616 


23.5796522 


8.2228985 


.001796561 


667 


810249 


172808693 


2:3.6008474 


8.2278254 


.001795888 


668 


811864 


173741112 


23.6220236 


8.2327463 


.001792115 



68 



CUBE ROOTS, AND RECIPR0GAIJ3. 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Roots. 


Reciprocals. 


559 


812481 


174676879 


' 28.6481808 


8.2376614 

• 


001788909 


seo 


313600 


175616000 


23.6643191 


8.2425706 


.001785714 


561 


314721 


176558481 


23.6854886 


8.2474740 


.001782631 


563 


315844 


177604328 


28.7065392 


8.2523716 


001779359 


563 


316960 


178453547 


23.7276210 


8.2672633 


001776199 


564 


318090 


179406144 


83.7486842 


8.2621492 


.001778050 


565 


319226 


180862126 


23.7697288 


8.2670294 


.001769912 


566 


820856 


181321498 


28.7907545 


8.2719089 


.001766784 


567 


321489 


182284263 


23.8117618 


8.2767726 


.001763668 


568 


322624 


183260432 


23.8327506 


8.2816856 


.001760668 


569 


823761 


184220000 


2b.853?209 


8.2864928 


.001757469 


570 


324900 


185193000 


23.8746728 


8.2918444 


.001754386 


571 


826041 


186169411 


23.8956063 


8.2961903 


.001751813 


5ra 


327184 


187149248 


23.9166215 


8.3010804 


.001748252 


578 


828329 


188132617 


23.9374184 


8.3068661 


.001746201 


574 


329470 


189119224 


23.9682971 


8.8106941 


.001742160 


575 


330625 


190109375 


23.9791576 


8.3165176 


. .001739180 


576 


iXSl'm 


191102976 


24.0000000 


8.3203353 


.001736111 


577 


332929 


192100033 


24.0206243 


6.3251476 


.001783102 


678 


834064 


193100552 


24.0416306 


8.3299642 


.001730104 


579 


836241 


194104639 


24.0624188 


8.3347663 


.eor/2'7116 


580 


836400 


195112000 


24.0831891 


8.3395609 


.001724138 


581 


837561 


196122941 


24.1039416 


6.3443410 


.001721170 


583 


338724 


197187368 


24.1246762 


8.3491256 


.001718213 


583 


839889 


198155287 


24.1453929 


8.3539047 


.001715266 


584 


841050 


199176704 


24.1660919 


6.3586784 


.001712329 


585 


342226 


200201625 


24.1867732 


8.3634466 


.001709402 


580 


343390 


201230056 


24.2074869 


8.3682096 


001706486 


587 


344569 


2022620a3 


24.2280829 


8.3729668 


.001703678 


588 


345744 


203297472 


24.2487113 


8.3V77188 


.001700680 


589 


346921 


204336469 


24.2693222 


8.3824663 


.001697793 


590 


348100 


205879000 


24.2899156 


6.3872065 


.001694915 


591 


349281 


206425071 


24.3104916 


8.3919423 


.001692047 


592 


350464 


207474688 


24.3310501 


6.3966729 


.001689189 


598 


351649 


208527857 


24.3515913 


8.4013981 


.001686341 


594 


^ 352836 


200584584 


24.3721152 


8.4061180 


.001683502 


596 


354025 


210644875 


24.3926218 


8.4108326 


001680672 


596 


355216 


211706786 


24.4181112 


6.4155419 


• .001677852 


597 


356409 


212776173 


24.4335834 


6.4202460 


.001676042 


696 


857604 


218847192 


24.4540385 


6.4249448 


.001672241 


599 


858801 


214921799 


24.4744765 


8.4296383 


.001669449 


600 


860000 


216000000 


24.4948974 


8.4348267 


.001666667 


601 


361201 


217061801 


24.6153013 


6.4390098 


.001663894 


602 


362404 


218167203 


24.5356883 


8.4436877 


.001661130 


603 


363609 


219256227 


24.5560683 


6.4483605 


.001658375 


604 


364816 


220348864 


24.5764116 


6.4530281 


.001G55629 


605 


366025 


221446125 


24.5967478 


6.4576900 


.001652893 


606 


367236 


222545016 


24.6170678 


6.4623479 


.001650165 


607 


368449 


228648543 


24.6378,00 


8.4670001 


.001647446 


608 


369664 


224755712 


»4. 6576560 


6.4716471 


.001644787 


609 


370881 


2258665^ 


^.6779254 


8.4762892 


.001642086 


610 


8?2100 


226981000 


24.6981781 


8.4809261 


.001639344 


611 


373;i21 


228099131 


24.7184142 


8.4856579 


.001036661 


612 


374544 


229220928 


24.7386338 


8.4901848 


001633987 


618 


375769 


230846397 


24.7588368 


8.4948066 


001631821 


614 


876996 


231475544 


24.7790834 


8.4994233 


.001628664 


615 


378225 


232606375 


24.7991936 


8.5040850 


.001626016 


616 


379450 


233744896 


24.8198473 


8.5086417 


.001628877 


617 


880689 


234885113 


24.8394847 


8.5182435 


.001620746 


CIS 


381924 


236029032 


24.8596068 


8.5178403 . 


.001618123 


619 


383161 


237176659 


24.8797106 


8.5224821 


.001615509 


620 


884400 


238828000 i 


24.8997992 


8.5270189 


.001612908 



69 



TABLE VIII.— SQnARES, CHBEB, SQUAEE EO0T8. 



CUBE BOOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


683 


466489 


818611967 


681 


467856 


820013504 


685 


469225 


821419125 


686 


470696 


822828856 


687 


471969 


324242703 


688 


473344 


325660672 


689 


474721 


827088769 


690 


476100 


328509000 


691 


477481 


329939371 


692 


478864 


331373888 


693 


480249 


332812557 


694 


481636 


3&1255384 


695 


483025 


a35702375 


696 


4&1416 


33715JW36 


697 


485809 


338608873 


698 


487204 


340068892 


699 


488(301 


341532099 


700 


490000 


843000000 


701 


491401 


344472101 


702 


492804 


345948408 


703 


494209 


347428927 


704 


495616 


348913664 


705 


497025 


350402625 


706 


496436 


351805816 


707 


499849 


a')8393243 


708 


501264 


854894912 


709 


502681 


856400829 


710 


504100 


357911000 


711 


505521 


859425431 


712 


506944 


360944128 


713 


506369 


362467097 


714 . 


509796 


863994344 


715 


511225 


365525875 


716 


512656 


867061696 


717 


514089 


868601813 


718 


515524 


370146232 


719 


516061 


371694959 


730 


518400 


373248000 


721 


519841 


374805361 


722 


521284 


376367048 


723 


622729 


877933067 


724 


524176 


879503424 


725 


625625 


381078125 


726 


627076 


882657176 


727 


628529 


884240583 


728 


529984 


385^8352 


729 


531441 


387420489 


730 


' 632900 


389017000 


731 


534361 


390617891 


732 


635824 


392223168 


733 


637289 


39»83S837 


784 


638756 


895446904 


735 


540^225 


397065376 


786 


641696 


898688256 


787 


643169 


400315653 


738 


644944 


401947272 


789 


646121 


403588419 


740 


647600 


406224000 


741 


649061 


406869021 


742 


650564 


406518488 


743 


562049 


410173407 


744 


553536 


411880784 



Square 
Hoots. 


Cube Roots. 


Reciprocals. 


26.1342687 


8.8065722 


.001464129 


26.1533937 


8.8108681 


.001461968 


26.1725047 


8.8151596 


.001459854 


26.1916017 


8.8194474 


.001457726 


26.2106848 


8.8237307 


.001455604 


26.2297541 


8.8280099 


.001453488 


26.2488095 


8.8322850 


.001451379 


26.2678511 


8.8865559 


.001449275 


26.2868789 


8.8408227 


.001447178 


26.3058929 


8.8450854 


.001445087 


26.3248982 


8.8493440 


.001443001 


26.3438797 


8.8535985 


.001440922 


26.3628527 


8.8578489 


.001438849 


26.3818119 


8.8620952 


,001436782 


26.4007576 


8.8663375 


.001434720 


26.4196896 


8.8705757 


.001432665 


26.4386061 


8.8748099 


.001430615 


26.4575131 


8.8790400 


.001428571 


26.4764046 


8.8832661 


.001426534 


26.4952826 


8.8874882 


.001424501 


26.5141472 


8.8917063 


.001422475 


26.5329963 


8.89592M 


.001420455 


26.5518361 


8.9001304 


.001418440 


26.5706605 


8.9043366 


.001416431 


26.5894716 


8.9085387 


.001414427 


26.6082694 


8.9127369 


.001412429 


26 6270539 


8.9169311 


.001410437 


26.6458252 


8.9211214 


.001408451 


26.6645833 


8.9253078 


.001406470 


26.6833281 


8.9294902 


.001404494 


26.7020598 


8.9336687 


.001402525 


26.7207784 


8.9378433 


.001400560 


26.7394839 


8.9420140 


.001398601 


26.7581763 


8.9461809 


.001396648 


26.7768557 


8.9503438 


.001394700 


26.7955220 


8.9545029 


.001:392758 


26.8141754 


8.9586581 


.001390621 


26.8328157 


8.9628095 


.001388889 


26.8514432 


8.9669570 


.U01386963 


26.8700577 


8.9711007 


.001385042 


26.8886593 


8.9752406 


.0013a3126 


26.9073481 


8.9793766 


.001381215 


26.9258240 


8.9835089 


.001379310 


26.9443872 


8.9676373 


.001377410 


26.9629375 


8.9917620 


.001375516 


26.9814751 


8.9958829 


.001373626 


27.0000000 


9.0000000 


.001371742 


27.0185122 


9.0041134 


.001369863 


27.0370117 


9.0082229 


.001367989 


27.0554985 


9.0123288 


.001366120 


27.0739;^; 


9.0164309 


.001364256 


27.0924344 


9.0205293 


.001362398 


27.1108834 


9.0246239 


.001360544 


27.1293199 


9.0287149 


.001358696 


27.1477439 


9.0328021 


.001356862 


27.1661554 


9.0868857 


.001355014 


27.1845544 


9.0409655 


.001353180 


27.2029410 


9.0450419 


.001351351 


27.2213152 


9.0491142 


.001349528 


27.2396769 


9.0531831 


.001347709 


27.2580263 


9.0572482 


.001345895 


27.2763634 


9.0613096 


.001344066 



71 



TABLE VIII. — SQUARES, CUBES, SQUARE ROOTS. 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Roots. 


Reciprocals. 


745 


555025 


413498625 


27.2946881 


9.0663677 


.001342282 


746 


556516 


415160936 


27.3180006 


9.0694220 


.001340483 


747 


558009 


416832723 


27.8813007 


9.0784726 


.001338688 


748 


559504 


418506992 


27.3495667 


9.0'/75197 


.001336896 


749 


561001 


420189749 


27.3678644 


9.0615631 


.001886113 


7S0 


562500 


421876000 


27.3861279 


9.0666030 


.001838338 


751 


564001 


428564751 


27.4043792 


9.0H96392 


.001831558 


752 


565504 


425259006 


27.4226164 


9.0936719 


.001829787 


753 


567009 


426957V77 


27.4406455 


9.0977010 


.001328021 


754 


568516 


428661064 


27.4690604 


9.1017265 


.001826260 


755 


570025 


430368875 


27.4772633 


9.1057486 


.001324608 


756 


571536 


482061216 


27.4954542 


9.1097669 


.001322751 


757 


573049 


483798093 


27.5136330 


9.1137818 


.001821004 


758 


574564 


435519512 


27.5817998 


9.1177931 


.001319261 


759 


576061 


437245479 


27.5499546 


9.1218010 


.001317523 


760 


577600 


438976000 


27.6680975 


9.1258053 


.001316789 


761 


579121 


440711081 


27.5862284 


9.1296061 


.001314060 


762 


580644 


442450728 


27.6043475 


9.1338084 


.001312886 


768 


582169 


444194947 


27.62:^4646 


9.13r7971 


.001310616 


764 


683696 


445943744 


27.6405499 


9.1417874 


.001306901 


765 


585225 


447697125 


27.6566334 


9.1457742 


.001807190 


766 


686756 


449455096 


27.6767050 


0.1497576 


.001305483 


767 


688289 


451217663 


27.6947648 


9.1537875 


.001803781 


768 


689624 


452984832 


27.7126129 


9.1577189 


.0018U2U83 


769 


691861 


454756609 


27.7306492 


9.1616869 


.001800890 


770 


592900 


456683000 


27.7488739 


9.1666665 


.001296701 


771 


594441 


458314011 


27.7668868 


9.1696225 


.001297017 


772 


595984 


460099648 


27.7848880 


9.1736862 


.001295337 


773 


597529 


461889917 


27.8028775 


9.1776445 


.001298661 


774 


599076 


463684624 


27.82(18555 


9.1815003 


.001291990 


775 


600625 


465484375 


27.8388218 


9.1854627 


.001290328 


rre 


602176 


467288676 


27.6567766 


9.1894018 


.001286660 


777 


603729 


469097433 


27.8747197 


9.1963474 


.001267001 


778 


605284 


470910952 


27.6926514 


9.1972897 


.001286847 


779 


606641 


472729139 


27.9106715 


9.2012286 


.001288697 


780 


606400 


474562000 


27.9284801 


9.2051641 


.001282051 


781 


609961 


476379641 


27.9468rnj 


9.2090962 


.001280410 


782 


611524 


478211768 


27.9642629 


9.2130250 


.001278778 


783 


618069 


480048687 


27.9821372 


9.2169505 


.001277180 


784 


614656 


481890304 


26.0000000 


9.2206726 


.001276610 


785 


616225 


483736625 


28.0178515 


9.2247914 


.001273885 


786 


617796 


485687666 


28.0356915 


9.2267068 


.001272265 


787 


619369 


487443403 


28.0536203 


9.2826189 


.001270648 


788 


620944 


489303872 


28.0n8377 


9.2365277 


.001269086 


789 


622521 


491169069 


28.0691438 


9.2404833 


.001267427 


790 


624100 


498039000 


28.1069386 


9.2443855 


.001266823 


791 


625681 


494913671 


28.1247222 


9.2482344 


.001264223 


792 


627264 


496793068 


28.1424946 


9.2521300 


.001262626 


793 


628649 


498677257 


26.1602567 


9.2560224 


.001261084 


794 


630436 


500566184 


28.1780066 


9.2509114 


.001259446 


795 


632025 


602459675 


28.1957444 


9.2637973 


.001257862 


796 


633616 


604358836 


28.2134720 


9.2676798 


.001256281 


797 


635209 


606261573 


28.2311884 


9.2715602 


.001264705 


798 


636804 


606169592 


28.2488988 


9.2754862 


.001258133 


799 


638401 


510062399 


28.2666881 


9.2799061 


.001251664 


800 


640000 


612000000 


28.284^n2 


9.2881777 


.001250000 


801 


641601 


618922401 


28.3019434 


9.2870440 


.001248489 


802 


643204 


615849608 


28.3196045 


9.2909072 


.001246868 


803 


644809 


617781627 


28.3372546 


9.2947671 


.001245830 


804 


646416 


519718464 


28.3548938 


9.2966289 


.001243781 


805 


648025 


521660125 


28.3725219 


9.8024775 


.001242288 


806 


649636 


523606616 


28.3901391 


9.8063278 


.001240005 



T4 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Roots. 


Reciprocals. 


807 


661249 


625657948 


28.4077454 


0.3101750 


.001239157 


806 


662864 


627514112 


28.4258408 


0.3140190 


.001237624 


809 


664481 


629475129 


28.4420258 


0.3178599 


.001236094 


810 


666100 


681441000 


28.4604060 


9.8216975 


.001234568 


811 


657721 


688411781 


28.4780617 


9.8255320 


.001233046 


812 


660844 


6S6887828 


28.4956187 


9.3298634 


- .001231527 


818 


660960 


687367797 


28.5131549 


9.8831916 


.001230012 


814 


662606 


630868144 


28.5306852 


9.8370167 


.001228501 


816 


664225 


541848975 


28.5482048 


9.3408386 


.001226994 


816 


666866 


64888R496 


28.6657137 


9.3446575 


.001225490 


817 


667480 


646888R18 


28.6832119 


9.3484731 


.001223990 


818 


660124 


547843482 


28.6006098 


9.8522857 


.001222494 


819 


670761 


649363259 


28.6181760 


9.3£;60952 


.001221001 


890 


672400 


661868000 


28.6866421 


9.3599016 


.001219612 


821 


674041 


663887661 


28.6630976 


9.3637049 


.001218027 


822 


676684 


666412248 


28.6706424 


9.3675051 


.001216545 


828 


677829 


667441767 


28.6879766 


9.3713(^ 


.001215067 


824 


678976 


660470224 


28.7064002 


9.3750963 


.001213592 


825 


680625 


661616625 


28.7228132 


9.8786873 


.001212121 


826 


682276 


568660076 


28.7402167 


9.3826752 


.001210664 


827 


6R8829 


666609288 


28.7576077 


9.3864600 


.001209190 


828 


686684 


667668662 


28.7740601 


9.3902419 


.001207729 


829 


687241 


660722789 


28.7923601 


9.3940206 


.001206273 


880 


688000 


671787000 


28.8007206 


9.3977964 


.001204819 


881 


690661 


573866191 


28.8270706 


9.4015691 


.001203369 


882 


6U2224 


57&0»n868 


28.8444102 


9.4053887 


.001201923 


888 


606889 


578UU0687 


28.8617394 


9.4091054 


.001200480 


884 


606666 


660093704 


28.6790582 


9.4128690 


.001199041 


885 


697225 


682182875 


28.6068666 


9.4166297 


.001197605 


886 


608806 


684277U56 


26.0136646 


9.4^03873 


.001196172 


887 


700669 


686876258 


26.0309528 


9.4241420 


.001194743 


838 


702244 


688480472 


26.9482297 


9.4278936 


.001193817 


889 


700021 


600689719 


26.9664967 


9.4316423 


.001191895 


840 


706600 


592704000 


28.0627636 


9.4353880 


.001190476 


841 


707881 


594823821 


20.0000000 


9.4391307 


.001189061 


842 


708964 


590947888 


20.0172363 


9.4428704 


.001187648 


848 


710649 


699077107 


20.0344628 


9.4466072 


.001186240 


844 


712886 


601211684 


20.0616781 


9.4508410 


.001184834 


845 


714025 


608361125 


20.0688837 


9.4540719 


.001183432 


846 


715716 


605406786 


29.0660791 


9.4577999 


.001182038 


847 


717409 


607646428 


29.1062644 


9.4615249 


.001180638 


848 


719104 


600800102 


29.1204896 


9.4662470 


.001179245 


849 


720801 


611060049 


29.1376046 


9.4689661 


.001177856 


850 


722500 


614126000 


29.1547595 


9.4726624 


.001176471 


851 


724201 


616295061 


29.1710043 


9.4763957 


.001175088 


862 


726004 


618470208 


29.1890890 


9.4801061 


.001178709 


868 


727609 


620660477 


29.2061637 


9.4838136 


.001172338 


851 


729B16 


022886864 


29.2282764 


9.4675182 


.001170960 


855 


781025 


626026376 


29.240R830 


9.4912200 


.001169591 


866 


782786 


627222016 


29.2574777 


9.4949166 


.001168224 


867 


784440 


029422708 


29.2745623 


9.4966147 


.001166861 


868 


786164 


681628712 


29.2916370 


9.5023078 


.001166501 


809 


787881 


6338807:0 


29.3087016 


9.5069960 


.001164144 


860 


780600 


686066000 


29.3257566 


9.6096854 


.001162791 


861 


741821 


" 688277881 


29.8428016 


9.6133699 


.001161440 


802 


748044 


640606028 


20.3596365 


9.5170515 


.001160098 


868 


744780 


642786647 


20.3768616 


9.6207308 


.001168749 


864 


746496 


644072644 


20.8038760 


9.5244063 


.001157407 


866 


748226 


647214625 


20.4106623 


9.5260794 


.001166069 


866 


749966 


649461896 


20.4278779 


9.5817497 


.001164784 


867 


751689 


651714868 


20.4448637 


9.6354172 


.001168408 


868 


758424 


658972082 


20.4616307 


9.5390616 


.001162074 



7a 



TABLE VIII.— SQUARES, CUBES, SQUARE ROOTS. 



Na 


Squares. 


Cubes. 


Square 
Roots. 


Cube Boots. 


869 


756161 


666234900 


29.4788060 


9.6427437 


870 


756900 


668608000 


29.4967624 


9.6464027 


871 


758641 


660778311 


29.5127091 


9.5500589 


878 


760884 


663064848 


29.6296461 


9.5637128 


873 


762129 


666888617 


29.5465734 


9.5573630 


874 


768876 


667tti7624 


29.5634910 


9.5610106 


875 


766625 


669921875 


29.6808060 


9.5646569 


876 


76rd76 


672221376 


29.5972972 


9.5662962 


877 


769129 


674526133 


29.6141868 


9.5719877 


878 


770884 


676886152 


29.6310648 


9.5755745 


87» 


773641 


679151439 


29.6479342 


9.6792066 


880 


774400 


681472000 


29.6647939 


9.5826897 


881 


776161 


683797841 


29.6816442 


9.6864682 


88!sS 


777924 


686128968 


29.6964848 


9.6900989 


883 


779680 


688466387 


29.7163159 


9.6937169 


884 


781456 


69080n04 


29.7321875 


9.5973378 


886 


783225 


696154126 


29.7489496 


9.6009548 


886 


784996 


696606466 


29.7667621 


9.6046696 


887 


786769 


697864108 


29.7826452 


9.6081817 


888 


788644 


700227072 


29.7998289 


9.6117911 


889 


790621 


700606369 


29.8161080 


9.6168977 


890 


792100 


704969000 


29.8328678 


9.6190017 


891 


798881 


707847971 


29.8496281 


0.6226090 


890 


796664 


709782288 


29.8663690 


9.6262016 


898 


797449 


712121967 


29.8831066 


9.6297975 


804 


799236 


714616984 


29.8998328 


9.6338907 


895 


801025 


716917375 


29.9165506 


9.6869612 


896 


802810 


719323136 


29.9332691 


9.6406690 


897 


804609 


721734273 


29.9499688 


9.6441642 


698 


806401 


724160792 


29.9666481 


9.6477367 


899 


806201 


726672699 


29.9883287 


9.6613166 


900 


810000 


729000000 


30.0000000 


9.6548968 


901 


811801 


781432701 


30.0166620 


9.6564664 


90S 


818604 


788870606 


30.0333148 


9.6620406 


908 


815409 


786314827 


80.0499684 


9.6666096 


904 


8irai6 


738763264 


30.0665928 


9.6691762 


905 


819025 


741217625 


30.0832179 


9.6727408 


906 


820886 


748677416 


30.0996339 


9.6768017 


907 


822649 


746142648 


30.1164407 


9.6796604 


908 


824464 


748618812 


30.1380883 


9.6884166 


900 


826281 


751069429 


30.1496269 


9.6869701 


910 


828100 


753671000 


30.1662063 


9.6906211 


911 


829921 


766066081 


80.1827766 


9.6940694 


912 


831744 


758550528 


30.1998377 


9.6976151 


913 


888569 


761048497 


30.2168899 


9.7011688 


914 


886396 


763661944 


30.2324329 


9.7046969 


916 


887226 


766060875 


30.2489660 


9.7062360 


916 


839066 


768675296 


30.2654919 


O.TllTTOS 


917 


840689 


771096213 


30.2620079 


9.7158051 


918 


842724 


773620632 


80.2966148 


9.7188864 


919 


844661 


776151559 


30.3160128 


9.7228631 


920 


846400 


778688000 


80.3316018 


9.7258883 


921 


848241 


781229961 


30.3479618 


9.7294109 


922 


860064 


788777448 


30.3644629 


9.7329809 


923 


861929 


786330497 


30.8809151 


9.7864484 


924 


863776 


788889024 


80.3973683 


9.7899634 


926 


866625 


791453125 


30.4188127 


9.7434768 


926 


867476 


794022776 


90.4302481 


9.7469667 


927 


869829 


796697988 


30.4460747 


9.7504980 


928 


861184 


799178752 


30.4630924 


9.7539979 


909 


868041 


801765089 


80.4795013 


9.7675002 


980 


864900 


804357000 


30.4959014 


9.7610001 



Reciprocals. 



.001160748 

.001149425 
.001146106 
.001146789 
.001145475 
.001144165 
.001142867 
.001141663 
.001140251 
.001138062 
.001137666 

.001136864 
.001135074 
.001138787 
.001132508 
.001131222 
.001129944 
.001128668 
.001127396 
.001126126 
.001124860 

.001123606 
.001122384 
.001121076 
.001119821 
.001118666 
.001117318 

.ooiiioon 

.001114827 
.001118686 
.001112847 

.001111111 
.001109678 
.001106647 
.001107420 
.001106196 
.001104972 
.001109768 
.001102686 
.001101822 
.001100110 

.001096901 
.001097696 
.001096491 
.001096290 
.001094092 
.001092896 
.001001708 
.001090618 
.001069826 
.001068189 

.001066967 
.001066770 
.001064609 
.001068428 
.001062251 
.001061061 
.001079914 
.001076749 
.001077586 
.001078406 
.001075060 



74 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square 
Boots. 


Cube Roots. 


Reciprocals. 


931 


8697V1 


806954491 


30.5122926 


9.7644974 


.001074114 


932 


868624 


800687568 


30.6286750 


9.7679922 


.00107T2961 


983 


870489 


812166S87 


80.6450487 


9.7714845 


.001071811 


9S4 


872356 


814780504 


80.5614136 


9.7749743 


.001070664 


966 


874225 


817400375 


30.5777697 


9.7784616 


.001069519 


986 


876096 


820025866 


80.5941171 


, 9.7819466 


.001068376 


967 


877969 


822656953 


30.6104557 


9.7854288 


.001067286 


988 


879644 


825293672 


30.6267857 


9.7889087 


.001066098 


939 


881721 


827936019 


30.6431069 


,9.7928861 


.001064963 


940 


883600 


630584000 


80.6594194 


9.7968611 


.001068880 


941 


885481 


833237621 


80.6757283 


9.7998386 


.001062699 


942 


887864 


885896888 


80.6920185 


9.8028036 


.001061571 


943 


889249 


MH8561807 


80.7083051 


9.8062711 


.001060445 


944 


891136 


841282884 


JiO.7245880 


9.8097862 


.001059322 


945 


893025 


843906625 


80.7408523 


9.8181989 


.001058201 


946 


894916 


846590536 


80.7571130 


9.8166591 


.001057082 


947 


ovuOUv 


849278123 


80.7733651 


9.8201169 


.001065966 


948 


898704 


851971392 


80.7896066 


9.8285723 


.001054852 


949 


900601 


854670349 


80.8058486 


9.8270252 


.001053741 


950 


902500 


857375000 


80.8220700 


9.8804757 


.001062682 


951 


904401 


860085851 


80.8882879 


9.8889288 


.001051625 


952 


906304 


862801408 


80.8644972 


9.8373695 


.001060420 


953 


906209 


865523177 


80.8706961 


9.8406127 


.001049818 


954 


010116 


866250664 


80.8868904 


9.8442586 


.001048218 


965 


912025 


870983875 


80.9030743 


9.8476920 


.001047120 


966 


918986 


878722816 


80.9102497 


9.8511280 


.001046025 


95r 


915849 


876467493 


30.9864166 


9.8545617 


.001044982 


958 


917764 


879217912 


80.9515751 


9.8579929 


.001048841 


959 


919681 


881974079 


80.9677251 


9.8614218 


.001042758 


960 


921600 


884736000 


80.9888668 


9.8648483 


.001041667 


961 


923521 


887603681 


81.0000000 


9.8682724 


.001040583 


962 


925444 


890277128 


81.0161248 


9.8716941 


.001039601 


968 


927369 


898066347 


81.0822413 


9.8751185 


.001088422 


964 


929296 


606641344 


81.0483494 


9.8785805 


.001087844 


065 


961225 


896682125 


81.0644491 


9.8819451 


.001036269 


966 


963156 


901428696 


81.0605406 


9.8858574 


.001085197 


967 


986069 


904231068 


81.0966236 


9.8887673 


.001034126 


968 


987024 


907039232 


31.1126964 


9.8921749 


.001038068 


969 


988961 


909858209 


81.1267648 


9.8956601 


.001031992 


970 


940900 


912678000 


81.1448280 


0.8989680 


.001030928 


971 


942841 


915496611 


81.1608729 


9.9028885 


.001029666 


972 


944784 


918380048 


81.1769145 


9.9057817 


.001028807 


973 


946729 


921167317 


81.1929479 


9.9091776 


.001027749 


974 


948676 


924010424 


81.2069731 


9.9125712 


.001026694 


975 


950625 


926859375 


81.2249900 


9.9159624 


.001025641 


976 


952576 


929714176 


81.2409987 


9.9198513 


.001024590 


977 


054529 


932674833 


81.2669992 


9.9227879 


.001023541 


978 


. 956484 


985441362 


81.2729915 


9.9261222 


.001022496 


979 


968441 


988313789 


81.2889757 


9.9296042 


.001021450 


980 


960400 


941192000 


81.8049617 


9.9828889 


.001020408 


981 


062861 


944076141 


81.8209195 


9.9862618 


.001019868 


962 


964824 


946966168 


81.8868792 


9.9396363 


.001018330 


988 


966289 


949862087 


81.3626308 


9.9480092 


.001017294 


984 


968266 


962763904 


81.8687743 


9.9463797 


.001016260 


985 


971)226 


966671625 


31.8847097 


9.9497479 


.001015228 


986 


972196 


968685266 


31.4006369 


9.9531138 


.001014199 


987 


974169 


961604803 


31.4166661 


9.9664775 


.001013171 


988 


976144 


964430272 


31.4324678 


9.9598889 


.001012146 


980 


978121 


967861669 


31.4488704 


9.9681981 


.001011122 


990 


960100 


970299000 


81.4642654 


9.9665549 


.001010101 


991 


962081 


973242271 


31.4801526 


9.9699096 


.001009062 


902 


984064 


976191488 


31.4960815 


9.9732619 


.001006065 



75 



TABLE VIII. — SQUARES, CUBES, ETC. 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Roots. 


Reciprocals. 


993 


986049 


979146657 


81.5119025 


9.9766120 


.001007049 


994 


988086 


982107784 


31.5277655 


9.9799599 


.001006036 


995 


990025 


965074875 


31.5436206 


9.9688065 


.001006025 


996 


992016 


968047986 


31.5594677 


9.9666488 


.001004016 


997 


994009 


991026973 


81.5753068 


9.9699900 


.001009009 


998 


996004 


994011992 


81.5011380 


9.9988289 


.001002004 


999 


996001 


997002999 


31.6069613 


9.9966656 


.001001001 


1000 


1000000 


1000000000 


31.62557766 


10.0000000 


.001000000 


1001 


1002001 


1003003001 


81.6385840 


10.0088322 


.0009990010 


1002 


1004004 


1006012008 


81.6643836 


10.0066622 


.0009960040 


1003 


1006009 


1009027027 


81.6/01752 


10.0099699 


.0009970090 


1004 


1006016 


1012.148064 


31.6859590 


10.0188155 


.0009960159 


1005 


1010025 


1015075125 


31.7017349 


10.0166389 


.0009960249 


1006 


1012036 


1018106216 


31.n75080 


10.0199601 


.0009940858 


1007 


1014049 


1021147848 


81.7332633 


10.0282791 


.0009930487 


1006 


1016064 


1024192512 


31.7490157 


10.0265956 


.0009920635 


1009 


1018061 


1027243729 


81.7647603 


10.0299104 


.0009910603 


1010 


1020100 


1080301C00 


81.7804972 


10.0632226 


.0009900990 


1011 


1022121 


1033364331 


31.7962262 


10.0365330 


.0009601197 


1012 


1024144 


1036433726 


31.8119474 


10.0896410 


.0009681423 


1013 


1026169 


1039509197 


81.8276609 


10.0481469 


.0009671668 


1014 


1(^196 


1042590744 


31.8433666 


10.0464506 


.0009661933 


1015 


1030225 . 


1046678375 


31.8590646 


10.0497521 


.0009662217 


1016 


1032256 


1048772096 


81.8747549 


10.0680514 


.0009642^0 


1017 


1084280 


1051871913 


81.8904374 


10.0568485 


.0009632842 


1018 


1086324 


1054977832 


81.9061123 


10.0696435 


.0009823183 


1019 


1088361 


1058069859 


81.9217794 


10.0629364 


.0009813548 


1020 


1040400 


1061206000 


31.9374388 


10.0662271 


.0009603922 


1021 


1042441 


1064882261 


81.9530906 


10.0695156 


.0009794819 


1022 


1044484 


1067462648 


31.9687847 


10.0728020 


.0009784786 


1023 


1046529 


1070599167 


81.9643712 


10.0760668 


.0009775171 


1024 


1048576 


1078741824 


32.(K)00000 


10.0798684 


.0009766625 


1(^ 


1050625 


1076890625 


32.0156212 


10.0626484 


.0009756096 


i(m 


1052676 


1060045576 


32.0312348 


10.0650262 


.0009746589 


1087 


1054729 


1083206683 


32.0468407 


10.0692019 


.0009787098 


1028 


1056784 


1066373952 


32.0624891 


10.09247S5 


.0009727026 


1029 


1058841 


1089547389 


32.0780296 


10.0967469 


.0009718173 


loao 


1060900 


1092727000 


82.0936131 


10.0990163 


.0009706738 


1031 


1062961 


1095012791 


32.1091887 


10.1022835 


.0009699321 


1032 


1065024 


1099104768 


32.1247568 


10.1055487 


.0a)9689922 


1033 


1067089 


1102302937 


82.1403173 


10.1088117 


.0009680642 


1034 


1069156 


1105507304 


32.1558704 


10.1120726 


.0009671180 


1035 


1071225 


1108717875 


82.1714159 


10.1158814 


.0009661836 


1086 


1073296 


1111934656 


82.1869539 


10.118)882 


.0009652510 


1087 


1075369 


1115157853 


82.2024844 


10.1218428 


.0009643202 


1038 


107r444 


1118386672 


32.2180074 


10,1250958 


.0009633911 


1U89 


1079521 


1121622319 


82.2335229 


10 .1283457 


.0009624639 


1040 


1061600 


1124864000 


32.2490310 


10.1815941 


.0009616885 


1041 


1063681 


1128111921 


82.2645316 


10.1348408 


.0009606148 


1042 


1065764 


1131366068 


82.2800248 


10.1880645 


.0009506929 


1043 


1067849 


1134626507 


82.2955106 


10.1413266 


.0000667738 


1044 


1089936 


1137803184 


32.3109888 


10.1445667 


.0009678544 


1045 


1092025 


1141166125 


32.3264596 


10.1478047 


.0009609378 


1046 


1094116 


1144445336 


32.3419233 


10.1510406 


.0009660229 


1047 


1096209 


1147780623 


32.8578794 


10.1542744 


.0009661006 


1048 


1096804 


1151022592 


82.3728281 


10.1575062 


.0009541966 


1049 


1100401 


1154820649 


82.8882695 


10.1607859 


.0009532888 


1050 


1102500 


1157625000 


32.4037085 


10.1689636 


.0009623810 


1051 


1104601 


1160936651 


82.4191301 


10.1671893 


.0009514748 


1062 


1106704 


1164262606 


82.4345495 


10.1704129 


.0009506703 


1058 


1106809 


1167575877 


82.4499615 


10.1786344 


.0009496676 


1054 


1110916 


1170905464 


82.4653662 


10.1768639 


.0009487066 



76 



TABLE IX. — LOGARITHMS- OF NUMBERS. 



No. 100 L. 000.] 



No. 109 L. 040. 



N. 





1 


2 


« 


4 


6 


6 


7 


8 


9 


Diff. 


100 
1 
2 


000000 

4321 
8600 


0434 
4751 
9026 


0668 
5181 
9451 


1801 
6609 
9676 


1784 
6038 


2166 
6466 


2598 
6894 


8029 
7321 


8461 
7748 


8891 
8174 


482 

426 


0300 
4521 

8700 


0724 
4940 
9116 


1147 
5360 
9532 


1570 
5779 
9947 


1993 
6197 


2415 
6616 


424 
420 


8 

4 


012837 
7033 


8259 
7451 


3680 
7868 


4100 
8284 


0861 
4486 
8571 


4806 
8978 


416 
412 
408 


6 
6 

7 


021189 
5306 
9884 


1603 

5715 

,9789 


2016 
6125 


2428 
6533 


2841 
6942 


8252 
7350 


3664 
7757 


4075 
8164 


1 


0195 
4227 
8223 


0600 
4628 
8620 


1004 
5029 
9017 


1408 
5430 
9414 


1812 
5830 
9611 


2216 
6230 


2619 
6629 


8021 
7028 


404 
400 


8 
9 


083424 
7426 
01 


3826 
7825 


0207 


0602 


0998 


897 



Proportional Parts. 



Diff. 


1 


2 


8 


4 


6 


6 


7 


8 


9 


484 


48.4 


86.8 


130.2 


178.6 


217.0 


260.4 


803.8 


347.2 


890.6 


488 


48.8 


86.6 


129.9 


178.2 


216.5 


259.8 


803.1 


846.4 


389.7 


432 


48.2 


86.4 


129.6 


172.8 


216.0 


269.2 


802.4 


846.6 


888.8 


431 


43.1 


86.2 


129.3 


172.4 


215.5 


258.6 


301.7 


844.8 


887.9 


430 


48.0 


86.0 


129.0 


172.0 


215.0 


258.0 


801.0 


844.0 


887.0 


429 


42.9 


85.8 


128.7 


171.6 


214.5 


257.4 


300.8 


848.2 


386.1 


428 


42.8 


85.6 


128.4 


171.2 


214.0 


206.8 


299.6 


842.4 


386.2 


427 


42.7 


85.4 


128.1 


170.8 


213.5 


266.2 


298.9 


841.6 


884.3 


426 


42.6 


85.2 


127.8 


170.4 


213.0 


255.6 


29S.2 


840.8 


888.4 


425 


42.5 


85.0 


127.5 


170.0 


212.5 


255.0 


297.6 


840.0 


382.5 


424 


42.4 


84.8 


127.2 


169.6 


212.0 


254.4 


296.8 


889.2 


381.6 


423 


42.3 


84.6 


126.9 


169.2 


211.5 


253.8 


296.1 


838.4 


880.7 


422 


42.2 


84.4 


126.6 


168.8 


211.0 


253.2 


296.4 


887.6 


879.8 


421 


42.1 


84.2 


126.3 


168.4 


210.6 


252.6 


294.7 


886.8 


878.9 


420 


42.0 


84.0 


126.0 


168.0 


210.0 


252.0 


294.0 


836.0 


878.0 


419 


41.9 


83.8 


126.7 


167.6 


209.5 


251.4 


293.3 


835.2 


877.1 


418 


41.8 


83.6 


125.4 


167.2 


209.0 


250.8 


292.6 


834.4 


876.2 


417 


41.7 


83.4 


126.1 


166.8 


208.5 


260.2 


291.9 


833.6 


875.3 


416 


41.6 


83.2 


124.8 


166.4 


208.0 


249.6 


291.2 


832.8 


874.4 


415 


41.5 


83.0 


124.5 


166.0 


207.5 


249.0 


290;6 


832.0 


878.5 


414 


41.4 


82.8 


124.2 


165.6 


207.0 


248.4 


289.8 


881.2 


872.6 


413 


41.3 


82.6 


123.9 


165.2 


206.5 


247.8 


289.1 


880.4 


871.7 


412 


41.2 


82.4 


123.6 


164.8 


206.0 


247.2 


288.4 


829.6 


870.8 


411. 


41.1 


82.2 


123.3 


164.4 


205.5 


246.6 


287.7 


828.8 


869.9 


410 


41.0 


82.0 


123.0 


164.0 


205.0 


246.0 


287.0 


828.0 


369.0 


409 


40.9 


81.8 


122.7 


163.6 


204.5 


245.4 


286.3 


827.2 


368.1 


408 


40.8 


81.6 


122.4 


163.2 


204.0 


244.8 


286.6 


326.4 


367.2 


407 


40.7 


81.4 


122.1 


162.8 


203.5 


244.2 


284.9 


825.6 


866.3 


406 


40.6 


81.2 


121.8 


162.4 


208.0 


243 6 


284.2 


824.8 


365.4 


406 


40.5 


81.0 


121.5 


162.0 


202.6 


243.0 


283.5 


824.0 


364.5 


404 


40.4 


80.8 


121.2 


161.6 


202.0 


242.4 


282.8 


823.2 


868.6 


406 


40.3 


80.6 


120.9 


161.2 


201.5 


241.8 


282.1 


822.4 


362.7 


402 


40.2 


80.4 


120.6 


160.8 


201.0 


241 2 


281.4 


821.6 


361.8 


401 


40.1 


80.2 


120.3 


160.4 


200.6 


240.6 


280.7 


320.8 


360.9 


400 


40.0 


800 


120.0 


160.0 


200.0 


240.0 


280.0 


820.0 


360.0 


899 


39.9 


79.8 


119.7 


159.6 


199.5 


239.4 


279.3 


319.2 


359.1 


896 


39.8 


79.6 


1M.4 


159.2 


199.0 


238.8 


278.6 


318.4 


368.2 


897 


39.7 


: 79.4 


119.1 


158.8 


196.5 


238.2 


277.9 


317.6 


867.8 


896 


39.fi 


79.2 


118.8 


158.4 


198.0 


237.6 


277.2 


316.8 


856.4 


806 


39.5 


79.0 


118.6 


168.0 


197.6 


237.0 


276.6 


816 1 855.6 



77 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 110 L. 041.] 



[No. 119 L. 078. 



N. 





1 


8 


8 


4 


6 


6 


7 


8 


9 


Diff. 


110 

1 
2 


041393 
5323 
9218 


1787 
5714 
9606 


2182 
6105 
9993 


2576 
6495 


2969 
6885 


3362 
•ri?75 


3755 
7664 


4148 
8068 


4540 
8442 


4982 
8830 


898 
880 




0380 
4230 
8046 


0766 
4613 
8426 


1153 
4996 
8805 


1588 
5378 
9186 


1924 
5760 
9563 


2309 
6142 
9942 


2694 
6524 


886 


S 

4 


053078 
6905 


3463 
7286 


8846 
7666 


888 


0820 
4088 
7815 


879 
876 
878 


6 
6 

7 


060696 
4458 
8186 


1075 
4832 

8567 


1452 
5206 
8928 


1829 
5580 
9298 


2206 
5953 
9668 


2582 

6326 


2958 
6609 


8833 
7071 


3709 
7443 


0038 
3718 
7368 


0407 
4086 
7781 


0776 
4451 
8094 


1145 
4816 

8457 


1514 
5182 
8819 


870 
866 
868 


8 
9 


071883 
6547 


2250 
5912 


2617 
6276 


2965 
6640 


3352 
7004 



Diff. 



895 
894 
898 
892 
891 
890 
880 
388 
887 
886 
885 

884 
883 
882 
881 
880 
879 
878 
377 
376 
875 

374 
378 
372 
371 
870 
369 
368 
867 
366 
866 

364 
868 
362 
861 
360 
369 
858 
867 
856 



39.6 
89.4 
89.3 
39.2 
89.1 
89.0 
88.9 
38.8 
38.7 
38.6 
88.5 

88.4 
88.3 
88.2 
88.1 
88.0 
87.9 
87.8 
87.7 
87.6 
87.5 

87.4 
87.3 
87.2 
87.1 
37.0 
36.9 
86.8 
86.7 
86.6 
86.5 

86.4 
86.3 
86.2 
86.1 
86.0 
85.9 
35.8 
85.7 
86.6 



8 



79.0 
78.8 
78.6 
78.4 
78.2 
78.0 
77.8 
77.6 
77.4 
77.2 
77.0 

76.8 
76.6 
76.4 
76.2 
76.0 
75.8 
75.6 
75.4 
75.2 
75.0 

74.8 

74.6 

74. 

74. 

74.0 

78.8 

73.6 

73.4 

73.2 

78.0 

72.8 
72.6 
72.4- 
72.2 
72.0 
71.8 
71.6 
71.4 
71.2 



.4 
.2 



Proportional Parts. 



8 



.0 

.7 
.4 
.1 



118.5 

118.2 

117.9 

117.6 

117.3 

117 

116 

116 

116 

115.8 

116.6 

115.2 
114.9 
114.6 
114.8 
114.0 
118.7 
113.4 
118.1 
112.8 
112.6 

112.2 
111.9 
111.6 
111.8 
111.0 
110.7 
110.4 
110.1 
109.8 
109.5 

109.2 
108.9 
108.6 
106.8 
106.0 
107.7 
107.4 
107.1 
106.8 



158.0 
157.6 
157.2 
156.8 
156.4 
156.0 
155.6 
156.2 
154.8 
154.4 
154.0 

153.6 
153.2 
152.8 
152.4 
152.0 
151.6 
151.2 
150.8 
150.4 
150.0 

149.6 
149.2 
148.8 
148.4 
148.0 
147.6 
147.2 
146.8 
146.4 
146.0 

145.6 
146.2 
144.8 
144.4 
144.0 
143.6 
143.2 
142 8 
142.4 



197.5 
197.0 
196.5 
196.0 
195.5 
195.0 
194.5 
194.0 
193.5 
193.0 
192.6 

192.0 
191.5 
191.0 
190.5 
190.0 
189.5 
189.0 
188.5 
188.0 
187.5 

187.0 
186.5 
186.0 
185.5 
185.0 
184.5 
184.0 
183.5 
183.0 
182.5 

182.0 
181.5 
181.0 
180.5 
180.0 
179.5 
179.0 
178.5 
178.0 



6 



237.0 
236.4 
235.8 
235.2 
234.6 
234.0 
233.4 
232.8 
282.2 
231.6 
231.0 

230.4 
229.8 
229.2 
228.6 
228.0 
227.4 
226.8 
226.2 
225.6 
225.0 

224.4 
223.8 
228.2 
222.6 
222.0 
221.4 
220.8 
220.2 
219.6 
219.0 

218.4 
217.8 
217.2 
216.6 
216.0 
215.4 
214.8 
214.2 
213.6 



276.6 
275.8 
276.1 
274.4 
278.7 
273.0 
272.3 
271.6 
270.9 
270.2 
260.6 

268.8 
268.1 
267.4 
266.7 
266.0 
266.8 
264.6 
268.9 
268.2 
262.6 

261.8 
261.1 
260.4 
259.7 
259.0 
258.8 
257.6 
256.9 
266.2 
255.7 

254.8 
254.1 
258.4 
252.7 
252.0 
251.8 
250.6 
249.9 
249.2 



8 



816.0 
815.2 
814.4 
818.6 
812.8 
812.0 
811.2 
310.4 
809.6 
808.8 
806.0 

807.2 
806.4 
805.6 
804.8 
804.0 
808.2 
802.4 
801.6 
800.8 
800.0 

299.2 
296.4 
297.6 
296.8 
296.0 
296.2 
294.4 
298.6 
292.8 
292.0 

291.2 
290.4 
289.6 
288.8 
288.0 
287.2 
286.4 
886.6 
884.8 



866.6 
854.6 
853.7 
862.8 
851.9 
851.0 
850.1 
849.2 
848.8 
847.4 
846.5 

845.6 
844.7 
848.8 
342.9 
842.0 
841.1 
840.2 
880.8 
888.4 
887.6 

886.6 
885.7 
884.8 
888.9 
888.0 
882.1 
881.2 
880.8 
829.4 
828.6 

827.6 
826.7 
826.8 
884.9 
824.0 
888.1 
888.8 
321.8 
830.4 



78 



TABLE IX. — LOGARITHMS 0^ NUMBERS. 



No. 190 L. 079.] 



[No. 184 L. 180. 



N. 





1 


2 


8 


4 


5 


6 


7 


8 


9 


Diff. 




079181 


9648 


9904 
















ISO 


0200 

8861 
7426 


0626 

4219 
7781 


0967 

4576 
8136 


1847 

4984 
8490 


1707 

6291 
8845 


2067 

6647 
9196 


2426 

6004 
9662 


860 


1 
8 
8 


080786 
6860 
9905 


8144 
6716 


8508 

7071 


857 
866 


0258 
8772 
7867 


0611 
4122 
760A 


0963 
4471 


1315 
4820 
8208 


1667 

6169 

, 8644 


2018 
6518 
8990 


2370 
6866 
9335 


2721 
6216 
9681 


8071 
6662 


862 
849 


4 
5 


008422 
6010 




0026 
8462 
6871 


846 
843 
341 


6 

7 
8 


100871 
8804 
7210 


0716 
4146 
7549 


1059 
4487 
7888 


1403 
4828 
8227 


1747 
6169 
8666 


2091 
5610 
8908 


2434 
5851 
9241 


2777 
6191 
9579 


8119 
6581 
9916 


0258 
8609 

6940 


888 

885 

333 


9 

180 

1 


110690 

8948 
7W1, 


0926 

4277 
760B 


1268 

4611 
7984 


1599 

4944 
8266 


1984 

5278 
8696 


2270 

6611 
8926 


2606 

6943 
9256 


2940 

6276 
9586 


8275 

6608 
9916 


0245 
8526 
6781 


830 
828 
825 


2 

8 
4 


120674 
8862 
7106 

18 


0906 
4178 
7429 


1281 
4504 
7758 


1560 
4880 
8076 


1888 
6156 
8399 


2216 
6451 
8722 


2544 
6806 
9046 


2871 
6131 
9368 


8196 
6466 
9690 


0012 


823 



Paopo&TioNAb Parts. 



Wff. 


t 


855 


86:6 


864 


86.4 


863 


86.8 


852 


86.2 


861 


86.1 


860 


86.0 


849 


84.9 


848 


84.8 


847 


84.7 


846 


84.6 


846 


84.5 


844 


84.4 


848 


84.8 


842 


84.2 


841 


84.1 


840 


84.0 


889 


88.9 


888 


88.8 


887 


88.7 


886 


88.6 


836 


88.5 


834 


88.4 


838 


88.8 


882 


88.2 


881 


88.1 


880 


88.0 


829 


82.9 


828 


88.8 


827 


82.7 


826 


82.6 


825 


82.1S 


824 


82.4 


888 


92.S 


802 


82.2 



71.0 
70.8 
70.6 
70.4 
70.2 
70.0 
69.8 
69.6 
69.4 
69.2 

60.0 
68.8 
68.6 
68.4 
68.2 
68.0 
67.8 
67.6 
67.4 
67.2 

67.0 
66.8 
66.6 
66.4 
66.2 
66.0 
66.8 
66.6 
66.4 
66.2 

65.0 
64.8 
64.6 
64.4 



106.6 
106.2 
106.9 
106.6 
106.8 
106.0 
104.7 
104.4 
104.1 
108.8 

106.6 
103.2 
102.9 
102.6 
102.3 
102.0 
101.7 
101.4 
101.1 
100.8 

100.6 
100.2 
99.9 
99.6 
99.8 
99.0 
96.7 
06.4 
06.1 
97.8 

97.6 
97.2 
96.9 
96.6 



142.0 
141.6 
141.2 
140.8 
140.4 
140.0 
189.6 
189.2 
188.8 
188.4 

138.0 
137.6 
187.2 
186.8 
186.4 
186.0 
186.6 
186.2 
134.8 
184.4 

134.0 
138.6 
188.2 
182.8 
182.4 
132.0 
181.6 
131.2 
180.8 
180.4 

180.0 
129.6 
129.2 
128.8 



6 



177.5 

irr.o 

176.6 
176.0 
175.5 
175.0 
174.6 
174.0 
178.6 
178.0 

172.6 
172.0 
171.6 
171.0 
170.6 
170.0 
169.6 
169.0 
168.6 
168.0 

167.6 
167.0 
166.6 
166.0 
165.6 
165.0 
164.5 
164.0 
168.6 
168.0 

162.6 
162.0 
161.5 
161.0 



.0 

.4 



213.0 

212.4 

211.8 

211.2 

210.6 

210 

209 

208.8 

208.2 

207.6 

207.0 
206.4 
206.8 
206.2 
204.6 
204.0 
208.4 
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 

196.0 
194.4 
198.8 
193.2 



248.6 
247.8 
247.1 
246.4 
245.7 
245.0 
244.8 
243.6 
242.9 
242.2 

241.5 
240.8 
240.1 
239.4 
238.7 
238.0 
287.3 
236.6 
236.9 
235.2 

284.5 
233.8 
233.1 
232.4 
281.7 
231.0 
230.3 
229.6 
228.9 
228.2 

227.6 
226.8 
226.1 
226.4 



8 



284.0 
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 
250.2 
258.4 
257.6 



9 



319.5 
318.6 
817.7 
816.8 
315.9 
316.0 
314.1 
813.2 
312.3 
811.4 

310.6 
309.6 
808.7 
307.8 
306.9 
306.0 
305.1 
304.2 
303.3 
802.4 

801.5 
300.6 
299.7 
298.8 
297.9 
297.0 
296.1 
296.2 
294.8 
293.4 

292.5 
291.6 
290.7 
289.8 






i 



"i 



s 



S 



"S s 



1 



S 



S 



is 



E 



= 



S 



a 



Biffi 



i 



a 



s 



s 



I 



5 



a 



i 



I 



I 



TABLE IX. — LOa&RITEHS OP NCMBBH8. 



Si'S 



11 



i 



i 



ii 



11 



s 



i 



B 



a'_s_s!^ 



"% S SiS 



li 



i 



i 



ii 



i 



i 



a 



TABLE IX.— LOGARITHMS OF KtJMBERS. 



No. 


170 L. 230.] 














[No. 189 L. 278. 


N. 





1 


t 


9 


4 


« 


• 


7 


8 


9 


Ittff. 


170 
1 
2 
3 


230449 
2996 
6528 
8046 


0704 

asiso 

5781 
8297 


0960 
8504 
6083 
8548 


1215 
3757 
6285 
8799 


1470 
4011 
6537 
9049 


ITM 
4264 
6789 
9299 


1979 
4517 
7041 
95R0 


2234 

4770 
7292 
9600 


2488 
5023 
7544 


2742 
5276 
7795 


255 
253 

252 


0060 
2541 
5019 
7482 
9982 


O80O 
2790 
ocsoo 
7728 


250 
249 
248 
246 


4 
5 
6 

7 


240549 
3088 
^M8 

7978 


0799 
^86 
5759 
8219 


1048 
8534 
6006 
8464 


1297 

3782 
6252 
8709 


1546 
4030 
6499 
8054 


1796 
4277 
6745 
9196 


2044 
4S25 
6991 
9443 


2293 
4772 
7237 
9687 


0178 
2610 
5081 

7439 
9633 


245 
248 
242 

241 
239 


8 
9 

180 

1 


2S0420 
2853 

5273 
7679 


0664 
8096 

5514 
7918 


0906 
8336 

5755 
8158 


1151 
3580 

5996 
8396 

0787 
8162 
5525 

7875 


1305 
3822 

6237 
8637 


1638 
4064 

6477 
8877 


1881 
4306 

6718 
9116 


2125 
4548 

6958 
9356 


2868 

4790 

7198 
9594 


2 
3 

4 
5 
G 


260071 
2451 
4818 
7172 
9518 


0810 
2688 
5054 

7406 
9746 


0548 
2925 
5290 
7841 
9960 


1025 
8399 
5761 
6110 


1263 
3636 
5996 
8344 


1501 
8873 
6232 

8678 


1739 
4109 
6467 
8812 


1976 
4346 
6702 
9046 


S»4 
4582 
6037 
9279 


288 
287 
235 
284 


0218 
2538 
4800 
7151 


0446 
2770 
5081 
7380 


0679 
3001 
5311 
7609 


0912 
8238 
6542 

7838 


1144 
8464 
6772 

8067 


1877 
8696 
6002 
8396 


1609 
8927 
6232 
8625 


233 
232 

230 
229 


7 
8 
9 


271842 
4156 
6462 


2074 
4889 
6692 


2306 

4620 
6921 



Proportional Parts. 



Diff. 


1 


2 


8 


4 


6 


6 


7 


8 


9 


255 


25.5 


51.0 


76.6 


102.0 


127.6 


158.0 


178.6 


204.0 


220.5 


254 


25.4 


50.8 


78.2 


101.6 


127.0 


162.4 


177.« 


206.2 


228.6 


253 


25.3 


50.6 


75.9 


101.2 


126.6 


151.8 


177.1 


202.4 


227.7 


252 


25.2 


60.4 


76.6 


100.8 


126.0 


151.2 


178.4 


201.6 


226.8 


251 


25.1 


60.2 


75.8 


100.4 


125.5 


150.6 


176.7 


200.8 


225.9 


250 


250 


60.0 


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 


174.8 


190.2 


224.1 


248 


24.8 


49.6 


74.4 


90.2 


124.0 


148.8 


178.6 


196.4 


228.2 


247 


24.7 


49.4 


74.1 


96.8 


123.5 


148.2 


172.9 


197.6 


222.3 


246 


24.6 


49.2 


7S.8 


96.4 


123.0 


147.6 


172.2 


196.8 


221.4 


245 


24.5 


49.0 


78.5 


96.0 


122.5 


147.0 


171.5 


196.0 


220.5 


244 


24.4 


48.8 


73.2 


97.6 


122.0 


146.4 


170.8 


195.2 


219.6 


243 


24.3 


48.6 


72.9 


97.2 


121.5 


145.8 


170.1 


194.4 


218.7 


242 


24.2 


48.4 


72.6 


96.8 


121.0 


145.2 


169.4 


193.6 


217.8 


241 


24.1 


48.2 


72.3 


96.4 


120.6 


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 


^.9 


47.8 


n.7 


95.6 


119.5 


143.4 


167.8 


191.2 


216.1 


238 


28.8 


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 


180.6 


213.3 


236 


23.6 


47.2 


70.8 


94.4 


118.0 


141.6 


165.3 


188.8 


212.4 


235 


28.6 


47.0 


70.5 


94.0 


117.5 


141.0 


164.5 


188.0 


211.5 


234 


23.4 


46.8 


70.2 


98.6 


117.0 


140.4 


163.8 


187.2 


210.6 


283 


23.3 


46.6 


69.9 


93.2 


116.5 


189.8 


163.1 


186.4 


209.7 


232 


23.2 


46.4 


69.6 


92.8 


116.0 


139.2 


162.4 


185.6 


206.8 


231 


23.1 


46.2 


69.3 


92.4 


115.5 


138.6 


161.7 


184.8 


207.9 


230 


28.0 


46.0 


69.0 


92.0 


115.0 


138.0 


161.0 


184.0 


207.0 


229 


22.9 


45.8 


68.7 


91.6 


114.5 


187.4 


160.3 


183.2 


206.1 


228 


^.8 


45.6 


68.4 


91.2 


114.0 


136.8 


159.6 


182.4 


206.2 


227 


se.7 


45.4 


66.1 


90.8 


113.5 


136.2 


156.9 


181.6 


204.8 


220 


22.6 


45.2 


67.8 


90.4 


113.0 


135.6 


1582 


180.8 


208.4 



32 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 190 L. 278.] 



[No. 214 L. 332. 



N. 





1 


8 


S 


4 


6 


6 


7 


8 


9 


Diff. 




278754 


8082 


9211 


9430 


0667 


9696 












19U 


0J23 
2o96 

4056 
6905 
9143 


0351 
2622 
4882 
7130 
9306 


0578 
2849 
5107 
7354 
9589 


0606 
8075 
5332 

7578 
9812 


228 


1 
2 
3 
4 


281033 
3301 
5557 
7802 


1261 
3627 

5782 
8026 


1488 
3753 
6007 
8249 


1716 
3979 
6232 
8473 


1942 
4205 
6456 
8696 


2169 

4431 

6681 

1 8920 


227 
226 
225 
223 


5 
6 
7 
8 
9 


290035 
2256 
4466 
6666 
8853 


0857 
2478 
4687 
6884 
9071 


0480 
2690 
4907 
7104 
9289 


0702 
2920 
6127 
7323 
9507 


0925 
3141 
6847 
7542 
9726 


1147 
3363 
6567 
7761 
9943 


1360 
8584 
6787 
7979 


1591 
3804 
6007 
8196 


1818 
4025 
6226 
8416 


2034 
4246 
6446 
8635 


222 
221 
220 
219 


0161 

2331 
4491 
6639 
8778 

0906 
8023 
5130 
7227 
9314 


0878 

2547 
4706 
6854 
8991 


0606 

S764 
4921 
7068 
9204 


0818 

2960 
6136 
7282 
9417 


218 


200 

1 
2 
8 

4 


301030 
8196 
5351 
7496 
9630 


1247 
3412 
5566 
7710 
9843 


1464 
3628 
6781 
7924 


1681 
3844 
5996 
8137 


1896 
4059 
6211 
8.S51 


' 2114 

1 4276 

&425 

8564 


217 
216 
215 
213 


0056 

2177 
4289 
6390 
8481 


0268 
2389 
4499 
6599 
8689 


0481 
2600 
4710 
6809 
8898 


0693 
2812 
■ 4920 
7018 
9106 


1118 
8234 
5340 
7436 
9522 


1330 
3446 
6551 
7646 
9730 


1642 
8656 
6760 
7864 
9938 


212 
211 
210 
209 
206 


6 
6 

7 
8 


8117M 
8867 
5970 
8063 


1966 
4078 
6180 
8272 


9 

210 
1 
2 
3 


820146 

2219 
4282 
6336 
8380 


0354 

2426 
4488 
6541 
8583 


0562 

2633 
4694 
6745 

8787 


0769 

2839 
4899 
6950 
8991 


0977 

8046 
5105 
7155 
9194 


1184 

3252 
5810 
7359 
9398 


1391 

8458 
6516 
7563 
9601 


1598 

8665 
6?21 

7767 
9806 


1805 

3871 
5926 
7972 


2012 

4077 
6131 
8176 


207 

206 
205 
204 


0006 
2034 


0211 
2286 


aos 


4 


330414 


0617 


0619 


1023 


1225 


1 1427 


1630 


1832 


202 



Pbopo&tionjll Parts. 



Diff. 


1 


2 


8 


4 


5 


6 


7 


8 


9 


225 


22.6 


45.0 


67.6 


90.0 


112.5 


135.0 


157.5 


180.0 


202.5 


224 


22.4 


44.^8 


67.2 


89.6 


112.0 


134.4 


156.8 


179.2 


201.6 


223 


22.3 


44.6 


66.9 


89.2 


111.6 


133.8 


156.1 


178.4 


200.7 


222 


22.2 


44.4 


66.6 


88.8 


111.0 


133.2 


155.4 


177.6 


199.8 


221 


22.1 


44.2 


66.3 


88.4 


110.5 


132.6 


154.7 


176.8 


198.9 


220 


82.0 


44.0 


66.0 


88.0 


110.0 


132.0 


154.0 


176.0 


198.0 


219 


21.9 


43.8 


65.7 


87.6 


109.5 


131.4 


153.3 


175.2 


197.1 


218 


21.8 


43.6 


66.4 


87.2 


109.0 


130.8 


152.6 


174.4 


196.2 


217 


21.7 


43.4 


65.1 


86.8 


108.5 


130.2 


151.9 


173.6 


195.3 


216 


21.6 


43.2 


64.8 


86.4 


108.0 


129.6 


151.2 


172.8 


194.4 


215 


21.5 


43.0 


64.5 


86.0 


107.5 


129.0 


150.5 


ira.o 


198.5 


214 


21.4 


42.8 


64.2 


85.6 


107.0 


128.4 


149.8 


171.2 


192.6 


213 


21.3 


42.6 


63.9 


85.2 


106.5 


127.8 


149.1 


170.4 


191.7 


212 


21.2 


42.4 


63.6 


84.8 


106.0 


127.2 


148.4 


169.6 


190.8 


211 


21.1 


42.2 


63.3 


84.4 


105.5 


126.6 


147.7 


168.8 


189.9 


210 


21.0 


42.0 


63.0 


84.0 


105.0 


126.0 


147.0 


168.0 


189.0 


209 


20.9 


41.8 


62.7 


88.6 


104.5 


125.4 


146.3 


167.2 


188.1 


208 


20.8 


41.6 


62.4 


83.2 


104.0 


124.8 


145.6 


166 4 


187.2 


207 


20.7 


41.4 


62.1 


82.8 


108.5 


124.2 


144.9 


165.6 


186.3 


206 


20.6 


41.2 


61.8 


82.4 


108.0 


123.6 


144.2 


1&1.8 


185.4 


296 


20.5 


41.0 


CI .5 


82.0 


102.5 


133.0 


143.5 


164.0 


184.5 


204 


20.4 


40.8 


61.3 


81.6 


102.0 


122.4 


142.8 


163.2 


183.6 


203 


20.3 


40.6 


60.9 


81.2 


101.5 


121.8 


142.1 


162.4 


182.7 


202 


20.2 


40.4 


60.6 


'».8 


101.0 


121.2 


141.4 


161.6 


181.8 



83 



TABLE IX. — LOGARITHMS OF NUMBEBS. 



No. 215 L. 833.] 



[No. 239 L. 880. 



N. 



til5 
6 
7 
8 

9 

220 
1 
2 
8 

4 

5 
6 
7 
8 
9 

230 
1 
2 
8 
4 

5 
6 

7 
8 
9 



832438 
4454 
6460 
8456 



840444 
2423 



6358 

8305 



850248 
2188 
4106 
6026 
7935 
9635 



861728 
8612 
5488 
7856 
9216 



871068 
2912 
4748 
6577 
8398 

88 



2640 
4666 
6660 
8656 



0642 

2620 
4589 
6549 
8500 



0442 
2375 
4301 
6217 
81-^5 



2842 
4856. 
6860 
8855 



8044 
5067 
7060 
9054 



0641 

2817 
4785 
6744 
8694 



1039 

8014 
4961 
6089 
8889 



0636 
2568 
4493 
6406 
8316 





1 


0025 


0215 


1917 


2105 


8800 


3988 


5675 


5862 


7542 


7729 


9401 


9587 


1253 


1437 


8096 


8280 


4932 


6115 


6759 


6942 


8580 


8761 



0629 
2761 
4686 
6599 
6506 



0404 

2294 
4176 
6049 
7916 
9772 



4 


5 


3246 


34<7 


5257 


5458 


7260 


7459 


9253 


9451 


1237 


1435 


8212 


8409 


5178 


5374 


7136 


7330 


9083 


9278 



1023 
2954 
4876 
6790 
8696 



0593 



6236 
8101 
9958 



1622 


1806 


8464 


8647 


5296 


6481 


7124 


7306 


8943 


9124 



1216 
3147 
5068 
6961 
8886 



0783 

26n 
4551 
6423 
8287 



0143 
1991 
8831 
5664 
7486 
9306 



6 


7 


8649 


3650 


5656 


5869 


7659 


7858 


9650 


9849 


1632 


1830 


3606 


8802 


5570 


6766 


7525 


7720 


9472 


9666 


1410 


1603 


aS39 


3532 


5260 


5452 


7172 


7363 


9076 


9266 


0972 


1161 


2869 


3048 


4739 


4926 


6610 


6796 


8473 


8659 


0328 


0513 


2175 


2360 


4015 


4198 


5846 


6029 


7670 


7862 


9467 


9668 



8 



4061 
6059 
8058 



0047 
2026 

8999 
6962 
7916 
9860 



4258 
6260 
8257 



0246 
2225 

4196 
6157 
8110 



1796 
8724 
5643 
7564 
9466 



0054 
1969 
8916 
6834 
7744 
9646 



1350 

8236 
6113 
6963 

8846 



0698 
2544 
4383 
6212 
8084 
9649 



1639 

8424 
6801 
7169 
9030 



0683 
2788 
4566 
6394 
8216 



0080 



Diff. 



208 
201 
200 

199 
198 

197 
196 
196 

194 
198 
198 
192 
191 
190 

189 

188 
186 
187 
186 



186 
184 
184 
183 
182 

181 



Proportional Parts. 



Diff. 


1 


2 


8 


4 


6 


6 


7 


8 


9 


202 


20.2 


40.4 


60.6 


80.8 


101.0 


121.2 


141.4 


161.6 


181.8 


201 


"20.1 


40.2 


60.8 


80.4 


100.6 


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 


69.7 


79.6 


99.6 


119.4 


139.3 


160.2 


179.1 


196 


19 8 


39.6 


69.4 


79.2 


99.0 


118.8 


138.6 


168.4 


178.2 


197 


19.7 


39.4 


69.1 


78.8 


98.6 


118.2 


137.9 


167.6 


177.8 


196 


19 6 


39.2 


68.8 


78.4 


960 


117.6 


137.2 


166.8 


176.4 


195 


19.5 


39.0 


68.5 


78.0 


97.5 


117.0 


136.5 


166.0 


175.6 


194 


19.4 


388 


682 


77.6 


97.0 


116.4 


136.8 


156.2 


174.6 


193 


19 3 


38.6 


67.9 


77.2 


96.5 


115.8 


135.1 


164.4 


178.7 


192 


19 2 


38.4 


57.6 


76.8 


96.0 


115.2 


134.4 


163.6 


i'ra.8 


lul 


19.1 


38.2 


67.3 


76.4 


95.6 


114.6 


133.7 


152.8 


171.9 


190 


19.0 


38.0 


57.0 


76.0 


95.0 


114 


133.0 


152.0 


171.0 


189 


18.9 


37.6 


56.7 


76.6 


946 


113.4 


132.3 


161.2 


170.1 


1HH 


18.8 


37.6 


56.4 


76.2 


94.0 


112.8 


131.6 


160.4 


169.2 


187 


18.7 


37 4 


56.1 


74.8 


93.6 


112.3 


130.9 


149.6 


168.8 


186 


18.6 


37.2 


65.8 


74.4 


93.0 


111.6 


130.2 


148.8 


167.4 


186 


186 


87.0 


655 


74.0 


92.5 


111.0 


129.6 


148.0 


166.6 


184 


18 4 


368 


55.2 


73.6 


92.0 


110.4 


126.8 


147.3 


165.6 


1H3 


18 8 


366 


54.9 


73.2 


91.6 


109 8 


128.1 


1464 


164.7 


182 


18 2 


364 


64 6 


728 


91 


109 2 


127.4 


146.6 


168.8 


181 


18 1 


362 


64 3 


724 


90.5 


108 6 


126.7 


144.8 


162.9 


180 


18 


360 


54 


720 


900 


106 


1260 


144.0 


162.0 


179 


17 9 


85 8 


53.7 


71.6 


895 


107.4 


126.3 


143.3 


161.1 



84 



TABLE IX. — LOGARITHMS OF KUMBER8. 



No. 240 L. 380.] LNo. 269 L. 431. 


N. 

240 
1 
2 
8 
4 
5 

6 

7 
8 
9 

250 

1 

2 

3 
4 
6 
6 

7 

8 
9 

260 
1 
2 
8 

4 
5 
6 
7 
8 
9 





1 


8 


8 


4 


ft 





7 


8 


9 


Dm. 


380211 
2017 
3815 
5606 
7390 
9160 


0892 
2197 
8995 
5785 
7568 
9343 


0673 
2877 

4174 
6964 
7746 
9520 


0754 
2557 
4353 
6142 
7924 
9698 


0934 
2737 
4533 
6321 
8101 
9875 


1115 
2917 
4712 
6499 
82T9 


1296 
3097 
4891 
6677 
8456 


1476 
3277 
6070 
6856 
8634 


1656 
8456 
5249 
7034 
8811 


1837 
8636 
5428 
7212 
8989 


181 
180 
179 
178 
178 


0051 
1817 
3575 
5326 
7071 

8806 


0228 
1993 
3751 
5501 
7245 

8981 


0405 
2169 
8926 
6676 
7419 

9154 


0582 
2345 
4101 
5850 
7592 

9326 


0759 
2521 
4277 
6025 
7766 

9501 


177 


390935 
2697 
4452 
6199 

7940 
9674 


1112 
2873 
4627 
6374 

8114 
9847 


12KK 
8048 
4802 
6548 

8287 


14&4 
3224 
4977 
6722 

8461 


1641 
3400 
5152 
6896 

8634 


176 
176 

175 
174 

173 


0020 
1745 
8464 
5176 
6881 
8579 


0192 
1917 
3635 
5346 
7051 
8749 


0365 
2089 
.3807 
5517 
7221 
8918 


0538 
2261 
3978 
5688 
7391 
9087 


0711 
2433 
4149 
5858 
7561 
9257 


0888 
2605 
4320 
6029 
7731 
9426 


1056 
27?/ 
4492 
6199 
7901 
9595 


1228 
2949 
4663 
6370 
8070 
9764 


173 


401401 
3121 
4834 
6540 
8240 
9938 


1573 
3292 
5005 
6710 
8410 


172 
171 
171 
170 
160 


0102 
1788 
3467 

5140 
6807 
8467 


0271 
1956 
3635 

5307 
6973 
8638 


0140 
2124 
3803 

5474 
7139 
8796 


0609 
2293 
8970 

6641 
7306 
8964 


0777 
2461 
4137 

5806 
7473 
9129 


0046 
2629 
4305 

5974 
7638 
9295 


1114 
2796 
44?3 

6141 

7804 
WOO 


1283 
2964 
4639 

6306 
7970 
9625 


1451 
8132 
4806 

6474 
8135 
9791 


160 


411620 
3300 

4973 
6641 
8301 
9956 


168 
167 

167 
166 
165 


0121 
1768 
3410 
5045 
6674 
8297 
9914 


0286 
1933 
8574 
5206 
6836 
8459 


0451 
2097 
3737 
5371 
6999 
8621 


0616 
2261 
8901 
5534 
7161 
8783 


1 0781 
2426 
4065 
6697 
7324 

; 8944 


0945 
2590 
4228 
5860 
7486 
9106 


1110 
2754 
4392 
6023 
7648 
9268 


1275 
2918 
4555 
6186 
7811 
9429 


1439 
8062 
4718 
6349 
7978 
9591 


165 


421604 
3246 
4882 
6511 
8135 
97B8 

48 


164 
164 
163 
162 
162 


0075 0236 


0398 


' 0559 


0720 0681 1 1042 


1203 


161 








• 


Pbo 


PORTIO 


NAL Pi 


ATS. 











Diff. 


1 


2 


8 


4 


5 


6 

106.8 


7 


8 


9 


178 


17.8 


85.6 


53.4 


71.2 


89.0 


124.6 


142.4 


160.2 


177 


17.7 


35.4 


53.1 


. 70.8 


88.5 


106.2 


123.9 


141.6. 


159.3 


178 


17.6 


85.2 


62.8 


70.4 


88.0 


105.6 


123.2 


140.8 


158.4 


175 


17.5 


85.0 


62.5 


70.0 


87.5 


105.0 


122.5 


140.0 


157.5 


174 


17.4 


84.8 


62.2 


69.6 


87.0 


104.4 


121.8 


189.2 


156.6 


173 


17.3 


84.6 


61.9 


69.2 


86.5 


103.8 


121.1 


138.4 


155.7 


172 


17.2 


34.4 


61.6 


68.8 


86.0 


103.2 


120.4 


137.6 


154.8 


171 


17.1 


84.2 


61.8 


68.4 


85.5 


102.6 


119.7 


186.8 


153.9 


170 


17.0 


84.0 


51.0 


68.0 


85.0 


102.0 


119.0 


136.0 


153.0 


109 


16.9 


33.8 


60.7 


67.6 


84.5 


101.4 


118.3 


135.2 


152.1 


168 


16.8 


83.6 


60.4 


67.2 


84.0 


100.8 


117.6 


134.4 


151.2 


167 


16.7 


33.4 


60.1 


66.8 


83.5 


100.2 


116.9 


133.6 


150.3 


166 


16.6 


83.2 


49.8 


66.4 


83.0 


99.6 


116.2 


132.8 


149.4 


165 


16.5 


88.0 


49.5 


66.0 


82.5 


99.0 


115.5 


132.0 


148.5 


164 


16.4 


82.8 


49.2 


65.6 


82.0 


98.4 


114.8 


131.2 


147.6 


163 


10.3 


82.6 


48.9 


65.2 


81.5 


97.8 


114.1 


130.4 


146.7 


162 


16.2 


82.4 


48.5 


64.8 


81.0 


97.2 


113.4 


129.6 


145.8 


161 


16.1 


82.2 


48.8 


MA 


80.5 


96.6 


112.7 


128.8 


144.9 



85 



XABLFl IX. — LOGARITHMS OF NUMBERS. 



1 



No. 270 L. 431.] 



[No. 290 L. 4T«i 



N. 



270 
1 
2 
8 

4 
6 

6 
7 
8 
9 

280 
1 

• 2 
8 

4 
5 
6 

7 
8 

9 

290 
1 
2 
8 
4 
6 

6 

7 
8 
9 



4S1864 



4560 
6163 
7751 
9333 



440900 
2480 
4045 
6604 

7158 
8706 



450249 
1786 
3318 
4845 
6366 
7882 
9392 



460898 
2398 



1S2S 
8130 
4720 
6322 
7900 
9401 



1066 
2637 
4201 
5760 

7313 
8861 



8 



1686 
8290 
4898 
6481 
8067 
9648 



8 



1224 
2798 

4357 
6015 

7468 

9015 



0408 
1940 
8471 
4997 
6518 
8033 
0548 



5383 
6868 
•8347 
0822 



471292 
2756 
4216 
56n 



1048 

2548 
4042 
5532 
7016 
8495 
9969 



1438 
2903 
4362 
5816 



0557 
2093 
8624 
5150 
6670 
8184 
9694 



1108 

269? 
4191 
6680 
7164 
8643 



0116 
1585 
3049 
4506 
5962 



1846 
8450 
5048 
6640 
8226 
9806 



1381 
2950 
4513 
6071 

7628 
9170 



2007 
3610 
6207 
6799 
8384 
9964 



0711 
2247 
8777 
5302 
6821 
8336 
9846 



1348 

2847 
4340 
6829 
7312 
8790 



03G3 
1732 
3195 
4658 
6107 



1538 
8106 
4660 
6226 

7778 
9324 



0865 
2400 
3930 
5454 
6973 
8487 
9995 



1499 

2997 
4490 
5977 
7460 
8938 



0410 
1878 
8341 
4799 
6252 



2167 
8770 
5367 
6957 
8542 



0122 
1695 
3263 
4825 
6382 

7983 
9478 



1018 
2553 
4082 
6606 
7125 
8638 



0146 
1649 

8146 

4630 
6126 

7608 
9065 



2328 
8930 
5626 

7116 
8701 

0279 
1852 
^19 
4981 
6587 

8088 
9633 



2488 
4090 
6685 
7275 
8859 



0437 
2009 
8576 
5137 
6692 

82«S 

9787 



1172 
2706 
4235 
5758 
7276 
8789 



1826 
2860 
4387 
6010 
7428 
8040 



0296 
1799 

8296 
4788 
6274 
7766 
9233 



0557 
2025 
3487 
4944 
6397 



0704 
2171 
3633 
6000 
6542 



0447 
1948 

3446 

4936 
6423 
7904 
0380 



2640 
4240 
6844 

7488 
0017 



0604 
2166 
8732 
6208 
6848 

8397 
9041 



1470 
3012 
4540 
6062 
7670 
0001 



2800 
4400 
6004 

7S92 
9175 



0752 
2323 
3889 
5449 
7003 

8552 



0661 
2318 
8779 
5235 
6687 



0597 
2006 

8594 
6086 
6671 
80G2 
9627 



0908 
2464 
8925 
5881 



0095 
1633 
8165 
4692 
6214 
7731 
9242 



0748 
2248 

8744 
6234 
6719 
8200 
9675 



1145 
2610 
4071 
6626 
6976 



Difl. 



161 
160 
160 
150 
158 

158 
157 
157 
166 
155 

155 

154 
164 
168 
168 
162 
152 
151 

151 
160 

160 
140 
140 

148 
148 

147 
146 
146 
146 
145 






Pbopobtional Parts. 



Diflf. 


1 


2 


8 


>4 


5 


6 


7 


8 





161 


16.1 


32.2 


48.3 


64.4 


80.5 


96.6 


112.7 


128.8 


144.0 


160 


16.0 


32.0 


48.0 


64.0 


80.0 


96.0 


112.0 


128.0 


144.0 


159 


15.9 


31.8 


47.7 


63.6 


79.5 


95.4 


111.3 


127.2 


143.1 


158 


15.8 


31.6 


47.4 


63.2 


79.0 


94.8 


110.6 


126.4 


142.2 


157 


15.7 


81.4 


47.1 


62.8 


78.5 


94.2 


109.0 


125.6 


141.8 


156 


15.6 


31.2 


46.8 


62.4 


78.0 


93.6 


109.2 


124.8 


140.4 


155 


15.5 


81.0 


46.5 


62.0 


77.5 


93.0 


108.5 


124.0 


180.5 


154 


15.4 


30.8 


46.2 


61.6 


77.0 


92.4 


107.8 


123.2 


138.6 


153 


16.3 


80.6 


45.9 


61.2 


76.5 


91.8 


107.1 


122.4 


137.7 


152 


16.2 


80.4 


45.6 


60.8 


76.0 


91.2 


106.4 


121.6 


186.8 


151 


15.1 


80.2 


45.3 


60.4 


75.5 


90.6 


105.7 


120.8 


185.0 


150 


15.0 


80.0 


45.0 


60.0 


75.0 


90.0 


105.0 


120.0 


185 


149 


14.9 


29.8 


44.7 


59.6 


74.5 


89.4 


104.3 


119.2 


184.1 


148 


14.8 


29.6 


44.4 


59.2 


74.0 


88.8 


103.6 


118.4 


183.2 


147 


14.7 


29.4 


44.1 


58.8 


73.5 


88.2 


102.9 


117.6 


182.3 


146 


14.6 


29.2 


43.8 


58.4 


73.0 


87.6 


102.2 


116.8 


181.4 


145 


14.5 


29.0 


43.5 


58.0 


72.5 


87.0 


101.5 


116.0 


180.5 


144 


14.4 


28.8 


43.2 


57.6 


72.0 


86.4 


100.8 


115.2 


120.6 


143 


14.3 


28.6 


42.9 


57.2 


71.5 


85.8 


100.1 


114.4 


128.7 


142 


14.2 


28.4 


42.6 


56.8 


71.0 


852 


99.4 


118.6 


127.8 


141 


14.1 


28.2 


42.8 


56.4 


70.5 


84.6 


08.7 


112.8 


126.0 


140 


14.0 


28.0 


42.0 


56.0 


70.0 


81.0 


98.0 


112.0 


126.0 



86 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. aOO L. 477.] 



N. 



300 
1 

2 
3 
4 

6 
6 

7 
8 
9 

810 
1 
2 
3 
4 
5 
6 

7 
8 

320 
1 
2 

4 
5 
6 
7 
8 
9 

830 
1 

2 
3 
4 
6 
6 
7 
8 



477121 
8666 



480007 
1443 
2874 
4300 
5721 
7138 
8S51 
9058 



491962 
2760 
4155 
6544 



8811 
9687 



501059 
2427 
3791 

6150 
6605 
7856 
9203 



510545 
1888 
S218 
4548 
5874 
7196 

8514 
9828 



6^138 
2444 

8746 

6045 



7630 
8917 



5a(K»0 



7266 
8711 



0151 
1586 
3016 
4442 
5863 
"TSdO 
8692 



0099 

1502 
2900 
4294 
5683 
7068 
8448 
9824 



1196 
2564 
3927 

5286 
6640 
7991 
9337 



0679 
2017 
3351 
4681 
6006 
7828 

8646 
9959 



8 



7411 
8855 



0294 
1729 
8159 
4585 
6005 
7421 
8833 



0239 

1642 
8040 
4433 
5822 
7206 
8586 
9962 



1333 
2700 
4063 

5421 
6776 
8126 
9471 



0813 
2151 
3484 
4813 
6139 
7460 

8777 



1269 
2575 
8876 
5174 
6469 
7759 
9045 



0828 



0090 
1400 
2705 
4006 
5304 
6598 
7888 
9174 



0450 



7555 
8999 



04:38 
1872 
3302 
4727 
6147 
7563 
8974 



0380 

1782 
3179 
4572 
5960 
7344 
8724 



0099 
1470 
2837 
4199 

6557 
6911 
8260 
9606 



7700 
9143 



0582 
2016 
3445 
4869 
6289 
7704 
9114 



0520 

1922 
8319 
4711 
6099 
7483 
8862 



7844 
9287 



0725 
2159 
3587 
6011 
6430 
7845 
9255 



7989 
9431 



0947 
2284 
3617 
4946 
6271 
7592 

8909 



0221 
1530 
2835 
4136 
5434 
6727 
8016 
9302 



0584 



0236 
1607 
2973 
4335 

6693 
7046 
8395 
9740 



1081 
2418 
3750 
6079 
6403 
7724 

9040 



0661 

2062 
8458 
4850 
6238 
7621 
8999 



0374 
1744 
3109 
4471 

5828 
7181 
8530 

9874 



0353 
1661 
2966 
4266 
6563 
6856 
8145 
9430 



0713 



1215 
2551 
8883 
6211 
6535 
7855 

9171 



0869 
2302 
3730 
5153 
6572 
7986 
9396 



8133 
9575 



[No. 338 L. 531. 



8 



8278 
9719 



1012 
2445 
3872 
5295 
6714 
8127 
9537 



1156 
2588 
4015 
5437 
€855 
8269 
9677 



0601 

2201 
8597 
4989 
6376 
7759 
9137 



0511 
1880 
8246 
4607 

6964 
7816 
8664 



0941 

2341 
8787 
5128 
6515 
7897 
9275 



0648 
2017 
8882 
4748 

6099 
7451 
8799 



1081 

2481 
8876 
6267 
6658 
8035 
9412 



0785 
2154 
8518 
48T8 

6234 
7586 
8934 



0484 
1792 
8096 
4396 
5693 
6985 
8274 
9559 



0009 
1349 
2684 
4016 
6344 
6668 
7987 

9303 



0615 
1928 
3226 
4526 
6822 
7114 
8402 
9687 



0840 I 0968 



0143 


0277 


1482 


1616 


2818 


2951 


4149 


4282 


6476 


6609 


6800 


6932 


8119 


8251 


9434 


9566 


0745 


0876 


2053 


2188 


8356 


8486 


4656 


4785 


5951 


6081 


7243 


7372 


8531 


86C0 


9815 


9943 



iC96 i 1223 



0922 
2291 
3656 
5014 

6370 
7721 

9068 



0411 
1760 
8084 
4415 
5741 
7064 
8382 

9697 



1007 
2314 
3616 
4915 
6210 
7f01 
8788 



0072 
1351 



9 


Diff. 


8422 


145 


9863 


144 
144 


1299 


2781 


143 


4157 


143 


6579 


142 


6997 


142 


8410 


141 


9818 


141 
140 


1222 


2621 


140 


4015 


139 


5406 


139 


6791 


189 


8173 


138 


9560 


188 



137 
137 
186 
136 

186 
185 
135 

134 
134 
133 
138 
138 
132 
182 

181 

131 
181 
180 
180 
129 
129 
129 

128 
128 



PROPOBTiONAii Parts. 



Diff. 


1 


8 


8 


4 


5 


6 


7 


8 


9 


139 


13.9 


27.8 


4J.7 


55.6 


69.5 


83.4 


97.3 


111.2 


125.1 


138 


13.8 


27.6 


41.4 


55.2 


69.0 


82.8 


96.6 


110.4 


124.2 


137 


13.7 


27.4 


41.1 


64.8 


68.5 


82.2 


95.9 


109.6 


123.3 


136 


18.6 


87.8 


40.8 


54.4 


68.0 


81.6 


95.2 


106.8 


122.4 


135 


13.5 


27.0 


40.5 


64.0 


67.5 


81.0 


94.5 


108.0 


121.5 


134 


13.4 


26.8 


40.2 


63.6 


67.0 


80.4 


93.8 


107.2 


120.6 


133 


13.8 


26.6 


89.9 


53.2 


66.5 


79.8 


93.1 


106.4 


119.7 


182 


18.2 


26.4 


89.6 


52.8 


66.0 


79.2 


92.4 


105.6 


118.8 


181 


18.1 


26.2 


89.3 


62.4 


65.5 


78.6 


91.7 


104.8 


117.9 


180 


18.0 


26.0 


89.0 


62.0 


65.0 


78.0 


91.0 


104.0 


117.0 


129 


12.9 


25.8 


88.7 


61.6 J 


64.5 


77.4 


90.3 


108.2 


116.1 


128 


12.8 


25.6 


88.4 


51.2 


64.0 


76.8 


80.6 


102.4 115.2 


187 


127 


25.4 


88.1 


50.8 


63.5 


76.2 


88.9 


101.6 


114.3 



Q»r 



TABLE IX. — LOGARITHMS OF NUMBERS. 



1 



No. 340 L. 631.] 



LNo. 379 L. 679. 



N. 





1 


2 


8 


4 


6 


6 


7 


8 


9 


Diff. 


340 

1 
2 
3 
4 
5 
6 


531479 
2754 
4026 
5294 
6568 
7819 
9076 


1607 
288d 
4158 
5421 
6685 
7945 
9202 


1734 
8009 
4280 
6547 
6811 
8071 
9327 


1862 
8136 
4407 
6674 
6937 
8197 
9452 


1990 
8264 
4534 
6800 
7063 
8322 
9578 


2117 
3391 
4661 
6927 
7189 
8448 
9r03 


2245 
8518 
4787 
6058 
7315 
8574 
9829 


23r2 
3645 
4914 
6160 
7441 
8699 
9954 


2500 
3772 
5041 
6806 
7567 
8625 


2627 
8899 
6167 
6482 
7693 
8951 


128 
127 
127 
126 
126 
126 


0079 
1330 
2576 
8820 

6060 
6296 
7529 
8758 
9984 

1206 
2425 
8640 

4852 
6061 

7267 
8469 
96G7 


0204 
1454 
2701 
8944 

5188 
6419 
7652 
8RR1 


125 
125 
125 
124 

124 
124 
128 
128 


7 
8 
9 

350 
1 
2 
3 
4 


540829 
1579 
2825 

4068 
5307 
6548 
7775 
9003 


0455 
1704 
2950 

4192 
5431 
6666 
7898 
9126 


0580 

3074 

4316 
5655 

6789 
8021 
9249 


oro5 

1953 
8199 

4440 
5678 
6918 
8144 

93ri 


0830 
2078 
3323 

4364 
5802 
7036 
8267 
9494 


0955 

1 2203 

8447 

4688 
6925 
7159 
8389 


1060 
2327 
8571 

4812 
6049 
7282 
8512 
9739 


1205 
2452 
8096 

4936 
6172 
7405 
8685 
9661 


»*»» 1 


0106 
1828 
8547 
8762 
4973 
6182 

7387 
8589 
9787 


123 
122 
122 
121 
121 
1^ 

120 
120 
120 


5- 
6 

7 
8 
9 

360 
1 

a 

8 


560228 
1450 
2668 
8883 
6091 

6308 
7507 
8709 
9907 


0351 
1572 
2790 
4004 
6215 

6428 

7627 

8829 


0473 
1694 
2911 
4126 
6386 

6544 
7748 
8948 


0595 
1810 
3033 
4247 
6457 

6664 

7868 
9068 


0717 
1938 
3155 
4368 
6578 

6785 
7988 
9188 


; 0640 
2060 
3276 
4489 
6699 

6906 
8106 
9308 


0962 
2181 
3396 
4610 
6&» 

7026 
8228 
9428 


1064 
2308 
3519 
4r31 
6940 

7146 
8349 
9548 


0026 
1221 
2412 
8600 
4784 
5966 
7144 

8819 
9491 


0146 
1340 
2531 
3718 
4908 
6084 
7262 

8436 
9606 


0265 
1459 
2650 
3837 
6021 
6202 
7879 

8554 
9725 

0698 
2058 
8220 
4879 
6534 
6687 
7836 
8983 


0385 
1578 
2769 
3955 
6189 
6320 
7497 

8671 
9842 


1 0504 
1696 
2887 
4074 
6257 
6437 
7614 

8788 
9959 


0624 
1817 
3006 
4192 
6376 
6556 

7r<« 

8905 


0743 
1936 
3125 
4311 
6494 
6678 
7849 

9023, 


0863 
2055 
8244 
4429 
6612 
6791 
7967 

9140 


0982 

2174 
3362 
4548 
6730 
6909 
8064 

9257 


119 
119 
119 
119 
118 
118 

lis 

117 


4 
6 
6 
7 
8 


870 

1 


661101 
2293 
8181 
4666 
5848 
7026 

8202 
8374 


0076 

vm 

2407 
8568 
4726 
6880 
7082 
8181 
9326 


0198 
1859 
2528 

3684 
4841 
5096 
7147 
8295 
9441 


0309 
1476 
2639 
8800 
4957 
6111 
7262 
&410 
9555 


0426 
1592 
8755 
8915 
6072 
6226 
7377 
8525 
9669 


J17 

117 

116- 

116 

116 

115 

115 

115 

114 


2 
3 
4 
6 
6 
7 
8 
9 


570548 
1709 
2878 
4081 
5188 
6341 
7492 
8639 


0660 
1825 
2988 
4147 
6303 
6457 
7607 
8754 


0778 
1942 
8104 
4263 
5419 
6573 
7722 
8868 


1010 
2174 
8336 
4494 
6650 
6802 
7951 
9097 


1126 
2291 
3452 
4610 
6765 
6917 
8066 
9212 



Pboportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


128 


12.8 


25.6 


38.4 


51.2 


64.0 


76.8 


89.6 


102.4 


115.2 


127 


12 7 


254 


88.1 


50.8 


63.5 


76.2 


88.9 


101.6 


114.3 


126 


12 6 


25.2 


87.8 


50.4 


63.0 


75.6 


88.2 


100.8 


113.4 


125 


12.5 


25.0 


87.5 


50.0 


62.5 


75.0 


87.5 


100.0 


112.5 


124 


12.4 


24.8 


87.2 


49.6 


62.0 


74.4 


868 


99.2 


111.6 


123 


12.3 


24.6 


36.9 


49.2 


61.5 


73.8 


86.1 


96.4 


110.7 


122 


12.2 


24 4 


86.6 


48.8 


61.0 


7:5.2 


85.4 


97.6 


109.8 


121 


12.1 


24.2 


86.3 


48.4 


60.5 


72.6 


84.7 


96.8 


108.9 


120 


12.0 


IM 


860 


48.0 


60.0 


72.0 


84.0 


96.0 


108.0 


119 


11 9 


23.8 


85.7 


47.6 


59.5 


71.4 


83.8 


95.2 


107.1 



88 



TABLE IX. — L06ARITHUS ( 



Mo. »8a L. B78.] [No. 414 L, BIT. | 


N. 





1 


s 


S 


4 


t 


e 


7 


8 


» 


Diff. 


3D0 

4 

8 

400 

i 
S 
1 


S^ 


sese 




















0019 
IIM 

4557 
9060 


DISS 

4870 
6799 
B%!5 

8107 


0S41 

1S81 

3S62 
4T8S 

B378 


OSSB 

1495 
2881 

i 


M69 

leoe 

1 


0688 
17S2 

6250 
7371 


1S36 

8362 

8608 
S7-M 


1950 

4218 
5348 
WIS 


ll<f 

108 
107 

106 


Gaosa 
ttxs 

'£ 

fi9S7 

4398 

i 

tlB83 

saw 

8144 

4ex 

G381 
T45S 

aan 


SBH 


0061 
117S 

6606 

SIBO 
M53 

64RB 

s 


i 

«»17 


oas4 

a 

B%7 

son 


1510 

i 


2782 
6047 

S 


OfllB 
E81S 

7256 
8558 


29S1 

S 


i 

7476 


8175 
lae 

7586 
8681 
9774 


0101 

8361 
4412 

T669 


osio 

i 

i 


a 

S4M 

1 


W2« 
36H8 

■mi 


TOM 
80»S 
0167 


s 

i 

8205 


07M 
4010 


3036 
4119 

8419 


ooai 


SJ'S 

2451 2360 
4870 4475 

7W5 -ma 


0341 
■teSl 


4686 


is 
1 


010660 

s 

7000 


(»76T 
8647 


0873 

29M 
40S3 

B180 
7210 


09711 
4)M 


pBOPosTiotcAi. Fism 


IB 

1 

is 

108 

1 


11 

11 

5 
I 

10 




«3.« 
£34 

Is 

SUA 

Is 

11 


«S4 

1 

33!i 

1 

1 


40^4 
4B.0 

45!s 
44 8 

43^0 


i 




70.2 
6S.6 
69.0 
«S,4 

84.8 


B 

7 
7E 
7f 

r 

7 
7 


:? 


91 

i 




100 
10! 

101 

m 

s 

i 





TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 415 L. 618.) 



[No. 459 L. 662 



N. 


• 


415 


618048 


6 


9093 


7 


620186 


8 


1176 


9 


2214 


420 


8249 


1 


4282 


2 


5312 


8 


6340 


4 


7366 


6 


8389 


6 


9410 


7 


630428 


8 


1444 


9 


2457 


490 


8468 


1 


4477 


2 


6484 


8 


6488 


4 


7490 


5 


8489 


6 


9486 


7 


640481 


8 


1474 


9 


2465 


440 


8453 


1 


4439 


2 


5422 


8 


6404 


4 


7383 


5 


8360 


6 


9835 


7 


650308 


8 


1278 


9 


2246 


450 


8213 


1 


4177 


2 


5138 


8 


6098 


4 


7056 


6 


8011 


6 


8965 


7 


9916 


8 


660665 


9 


1813 



8153 
9198 



0240 
1280 
2318 

3358 
4385 
5415 
6143 
7468 
8491 
9512 



0530 
1545 
2559 

8569 
4578 
5584 
6588 
7590 
8589 
9586 



0581 
1578 
2568 

8551 
4587 
5521 
6502 
7481 
8458 
9482 



0405 
1375 
2348 

3309 
4273 
5235 
6194 
7152 
8107 
9060 



0011 
0960 
1907 



a 


t 


8257 


8362 


9302 


9406 


0344 


0448 


1384 


1488 


2421 


2525 


8456 


8559 


4488 


4591 


6518 


5621 


6546 


6648 


7571 


7673 


8593 


8695 


9613 


9715 



8466 
9611 



0552 
1592 

2628 

8668 
4695 
5724 
6751 
7775 
8797 
9817 



0631 


0733 


1647 


1748 


2660 


2761 


8670 


8771 


4679 


4779 


5685 


5785 


6688 


6789 


7690 


7790 


8689 


8789 


9686 


9785 


0680 


0779 


1672 


1771 


2662 


2761 


8650 


8749 


4686 


4734 


5619 


5717 


6600 


6698 


7579 


7676 


8555 


8658 


9530 


9627 


0502 


0599 


1472 


1569 


2440 


2536 


8405 


8502 


4369 


4465 


5331 


5427 


6290 


6386 


7247 


7843 


8202 


8298 


9155 


9250 


0106 


0201 


1055 


1150 


2002 


2096 



0835 
1849 
2862 

8872 

4880 
5886 
6889 
7890 
8888 
9885 



0879 
1871 
2860 

8847 
4832 
5815 
6796 
7774 
8750 
9724 



0696 
1666 
2633 

8598 
4562 
6523 
6482 
7438 
8393 
9346 



0296 
1245 
2191 



8571 
9615 



0656 
1695 
2732 

8766 
4798 
6827 
6853 
78re 
8900 
9919 



8676 
9719 



0760 
1799 
2886 

8869 
4901 
6929 
6956 
7980 
9002 



7 


8 


8780 


8884 


9824 


9928 


0664 


0968 


1903 


2007 


2939 


8042 


8978 


4076 


5004 


5107 


6082 


6135 


7058 


7161 


8082 


8185 


9104 


9206 



8969 



0082 
1072 
2110 
8146 

4179 
5210 
6238 
7268 
8287 
9308 



0936 
1951 
2963 

I 8973 
: 4981 
' 5986 
6969 
7990 
8988 
9984 



0021 


0123 


0224 


1088 


1139 


1241 


2052 


2158 


2256 


8064 


8165 


8266 


4074 


4175 


4276 


5081 


6182 


6283 


6087 


6187 


6287 


7089 


7189 


7290 


8090 


8190 


8290 


9088 


9188 


9287 



0978 
1970 
2959 

8946 
4931 
5913 
6894 
7872 
8848 
9821 



0798 
1762 
2730 

8695 
4658 
5619 
6577 
7534 
8488 
9441 



0326 
1342 
2356 
3367 

4876 
5388 
6388 
7890 
8389 
9887 



0064 

lorr 

2069 
8058 

4044 
6029 
6011 
6992 
7969 
8945 
0919 



0890 
1859 
2826 

8791 
4754 
5715 
6673 
7629 
8584 
9636 



0391 
1339 
2286 



0486 
1434 
2380 



0183 
1177 
2168 
8156 

4148 
6127 
6110 
7089 
8067 
9043 



0288 
1276 
2267 
8255 

4242 
6226 
6208 
7187 
8165 
9140 



0016 
0987 
1956 
2928 

8888 
4850 
5810 
6769 
7725 
8679 
9631 



0581 
1529 
2475 



0118 
1084 
2058 
8019 

8964 
4946 
6906 
6864 
7820 
8774 
9726 



0678 
1628 
2569 



0382 
1875 
2366 
8864 

4840 
6824 
6806 
7286 
8262 
9287 



0210 
1181 
2160 
8116 

4060 
6042 
6002 
6960 
7916 
8870 
9621 



0771 

in8 

2663 



Diir. 



106 
104 

K» 
102 



101 



100 



99 



96 



97 



90 



95 



Proportional Part& 



Diflf. 

106 
1U4 
103 
102 
101 
100 
99 



1 


2 


3 


4 


6 


6 


7 


8 


10 5 


21.0 


81 5 


42.0 


52 5 


630 


73.6 


840 


10 4 


208 


31 2 


41 6 


B2.0 


624 


728 


832 


10 3 


206 


80 9 


41.2 


51 5 


61 8 


721 


82.4 


10 2 


204 


30 6 


408 


51 


61 2 


71 4 


81 6 


10 1 


202 


303 


40.4 


505 


606 


707 


808 


10.0 


200 


30.0 


40 


500 


60.0 


70 


800 


99 


19 8 


29 7 


39 6 


49 6 


59 4 


69 8 


792 



9 



94.S 

98.6 

82 7 

91 8 

90.9 

90 

89 



t] 



90 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 460 L. 662.1 



[No. 499 L. 69a 



N 





1 


8 


t 


4 


6 


C 


7 


8 


• 


Diff. 


460 
1 
2. 
8 

4 
5 
6 

7 


662758 
3701 
4642 
5581 
6518 
7453 
8386 
9317 


2852 
3795 
4736 
5675 
6612 
7540 
8479 
9410 


2947 
3889 
4830 
5760 
6706 
7640 
8572 
9503 


3041 
3983 
4924 
5862 
6799 
7733 
8665 
9596 


3135 
4078 
5018 
5956 
6892 
7826 
8759 
9689 


3280 
4172 
5112 
6050 
6966 
7920 
8852 
9782 


3824 
4266 
5206 
6143 
7079 
8013 
8945 
9875 


3418 
4360 
5299 
6237 
7173 
8106 
9038 
9967 


8512 
4454 
5393 
6331 
7266 
8199 
9131 


3607 
4548 
5487 
6424 
7360 
8293 
9224 


94 


0060 
0988 
1913 

2836 
3758 
4677 
5595 
6611 
7424 
8336 
9246 


0153 
1080 
2006 

2929 
3850 
4769 
5687 
6602 
7516 
8427 
9337 


98 


8 
9 

470 
1 
2 
8 
4 
6 
6 
7 
8 


670246 
1173 

2098 
8021 
8942 
4861 
5778 
6694 
7607 
8518 
9428 


0339 
1265 

2190 
3113 
4034 
4953 
5870 
6785 
7698 
8609 
9519 


0431 
1358 

2283 
3205 
4126 
5045 
5962 
6876 
7789 
8700 
9610 


0524 
1451 

2375 
3297 
4218 
5137 
6053 
6968 
7881 
8791 
9700 


0617 
1543 

2467 
8390 
4810 
5228 
6145 
7059 
7972 
8882 
9791 


0710 
1636 

2560 
3482 
4402 
5320 
6236 
7151 
8068 
8973 
9682 


0602 
1728 

2652 
8574 
4494 
5412 
6328 
7242 
8154 
9064 
9973 


0895 
1821 

2744 
3666 
4586 
5503 
6419 
7333 
8215 
9155 


92 
91 


0063 
0970 

1874 
2777 
3677 
4576 
6473 
6368 
7261 
8153 
9042 
9930 


0154 
1060 

1964 
2867 
8767 
4666 
6563 
6458 
7351 
8242 
9131 


6245 
1151 

2055 
2957 
3857 
4756 
5652 
6547 
7440 
8331 
9220 




9 

480 

1 

1 
4 
5 
6 
7 
8 
9 


680336 

1241 
2145 
8047 
8947 
4845 
6742 
6636 
7529 
8420 
9909 


0426 

1332 
2235 
8137 
4037 
4935 
5831 
6736 
7618 
8509 
9396 


0517 

1422 
2326 
8227 

4127 
5025 
5021 
6815 
7707 
8596 
9486 


0607 

1513 
2416 
8817 
4217 
5114 
6010 
6904 
7796 
8687 
9575 


0696 

1603 

2506 

3407 

4307 

5204 

6100 

6994 

7886 1 

8776 ; 

9664 


0789 

1693 

2596 

3497 

4396' 

5294 

6189 

7083 

7975 

8865 

9753 


0879 

1784 
2686 
3587 
4486 
5383 
6279 
7172 
8064 
8953 
9641 


90 
80 


0019 

0906 
1789 
2671 
8551 
4430 
6307 
6182 
7055 
7926 
8796 


0107 

0993 
1877 
2759 
3639 
4517 
5394 
6269 
7142 
8014 
8883 




490 
1 
2 
8 
4 
5 
6 
7 
8 
9 


090196 
1061 
1965 
2847 
3?»7 
4606 
6482 
6866 
7229 
8100 


0285 
1170 
2053 
2935 
3815 
4693 
5669 
6444 
7317 
8168 


0373 
12^8 
2142 
8023 
8903 
4781 
6667 
6531 
7404 
8275 


0462 
1347 
2230 
3111 
3991 
4868 
6744 
6618 
7491 
8362 


0550 1 

1435 ' 

2318 

3199 

4078 

4956 

5832 

6706 

7578 

8449 


0639 
1524 
2406 
3267 
4160 
5044 
5919 
6798 
7665 
8535 


0728 
1612 
2494 
3375 
4254 
5131 
6007 
6880 
7752 
8622 


0816 
1700 
2583 
3463 
4342 
5219 
6094 
6968 
7839 
8709 


88 

87 



Proportional Parts. 



Diff. 


1 


2 


8 


4 


6 


6 


7 


8 


96 


9.8 


19.6 


29.4 


39.2 


49.0 


68.8 


68.6 


78.4 


97 


9.7 


19.4 


29.1 


88.8 


48.5 


68.2 


67.9 


77.6 


96 


0.6 


19.2 


28.8 


38.4 


48.0 


67.6 


67.2 


76.8 


95 


9.5 


19.0 


28.5 


38.0 


47.5 


67.0 


66.5 


76.0 


94 


9.4 


18.8 


28.2 


87.6 


47.0 


66.4 


66.8 


76.2 


98 


9.3 


18.6 


27.9 


87.2 


46.5 


66.8 


66.1 


74.4 


92 


9.2 


18.4 


27.6 


36.8 


46.0 


55.2 


MA 


73.6 


91 


9.1 


18.2 


27.8 


86.4 


46.6 


64.6 


63.7 


72.8 


90 


9.0 


18.0 


27.0 


36.0 


45.0 


64.0 


68.0 


T2.0 


89 


8.9 


17.8 


26.7 


86.6 


44.5 


68.4 


62.8 


71.2 


88 


8.8 


17.6 


26.4 


§5J 


44.0 


62.8 


61.6 


70.4 


87 


8.7 


17.4 


96.1 


84.-8 


48.6 


52.2 


60.9 


09.6 


86 


8.6 


17.2 


25.8 


84.4 


48.0 


51.6 


60.2 


68.8 



88.2 
87.8 
86.4 
86.5 
84.6 
83.7 
82.8 
81.9 
81.0 
80.1 
79.2 
TH.TJ 
77.4 



oi 



TABLE IX. — LOGARITHMS OF NUMBERS. 



1 



No. 500 L. 698.1 



[No. 644 L. 786. 



N. 



500 
1 

2 
3 

4 
5 
6 

7 
8 
9 

510 
1 
)t 

3 

4 
5 
6 
7 
8 
9 

520 
1 
2 
8 

4 

5 
6 

7 
8 
9 

580 

1 
2 
3 
4 
5 
6 
7 

8 
9 

540 
1 
2 



698070 
9638 



700704 
1568 
2431 
8291 
4151 
6006 
5864 
6718 

7570 
8421 
9270 

710117 
0963 
1807 
2650 
3491 
4330 
5167 

6008 
6838 
7671 
8502 
9331 



720159 
0966 
1811 
2634 
3456 

4276 
5095 
5912 
6727 
7541 
8354 
9165 
9974 



730782 
1589 

2394 
8197 
8999 
4800 
6599 



1 


t 


9057 


9144 


9924 




0011 

0677 


0790 


1654 


1741 


2517 


2608 


3377 


3463 


4286 


4322 


5094 


5lV9 


5949 


6085 


6808 


6888 


7655 


7740 


8606 


8691 


9365 


9440 


0202 


0287 


1048 


1132 


1892 


1976 


2734 


2818 


3576 


3659 


4414 


4497 


5251 


5335 


6087 


6170 


6921 


7004 


7754 


7837 


8585 


8668 


9414 


9497 


0242 


0325 


1068 


1151 


1803 


1975 


2716 


2798 


3538 


3620 


4358 


4440 


5176 


5258 


5993 


6075 


6809 


6800 


7623 


7704 


8435 


8516 


9246 


9827 


0055 


0136 


0663 


0944 


1669 


1750 


2474 


2555 


3278 


3358 


4079 


4160 


4880 


4960 


5679 


6759 



9281 



0098 
0963 
1827 
2689 
8549 
4406 
5265 
6120 
6974 

7826 
8676 
9524 



9817 



0371 
1217 
2060 
2902 
8742 
4581 
5418 

6254 
7088 
7920 
8751 
9580 



0407 
1233 
2058 
2881 
8702 

4522 
5340 
6166 
6972 
7785 
8597 
9406 



0217 
1024 
1830 

2636 
8438 
4240 
5040 
6838 



0184 
1060 
1913 
2775 
8635 
4494 
5850 
6206 
7059 

7911 
8761 
9609 



0456 
1301 
2144 
2986 
3826 
4665 
5502 

6337 
7171 

8003 
8834 
9663 



0490 
1316 
2140 
2963 
8784 

4604 
6422 
6238 
7058 
7866 
8678 
9489 



0296 
1105 
1011 

2715 
8518 
4320 
5120 
5918 



6 


e 


7 


8 


» 


9404 


9491 


9578 


9664 


9751 


0271 
1136 
1999 
2861 
8721 
4579 
5436 
6291 
7144 

7996 
8846 
9694 


0858 
1222 
2066 
2947 
8807 
4665 
6522 
6376 
7229 

8061 
8931 
9779 


0444 
1309 
2172 
8083 
3893 
4751 
6607 
6462 
7315 

8166 
9015 
9663 


0531 
1396 
2268 
3119 
3979 
4887 
5693 
6547 
7400 

8251 
9100 
9946 


0617 
1482 
2844 
8205 
4066 
4922 
6778 
6632 
7485 

8836 
9185 


0083 
0679 
1728 

3407 
4246 
6064 
6920 

6754 
7687 
8419 
9248 


0540 
1385 
2229 
8070 
3910 
4749 
5686 

6421 
7254 
8086 
8017 
9745 


0625 
1470 
2313 
8154 
3994 
4833 
5669 

6504 
7338 
8169 
9000 
9628 


0710 
1554 
2397 
3238 
4078 
4916 
5753 

6588 
7421 
8253 
9063 
9911 


0794 
1639 
2481 
8323 
4162 
6000 
5886 

6671 
7504 
8836 
9166 
9994 


0077 
0903 
1728 
2562 
3874 
4194 

5018 
5830 

6646 
7460 
8278 
9084 
9693 


0573 
1896 
2222 
8045 
8866 

4685 
5508 
6320 
7134 
7948 
8759 
9570 


0655' 

1481 

2305 

3127 

8948 

4767 
5685 
6401 
7216 
8029 
8841 
9661 


0738 
1563 
2387 
3209 
4080 

4849 
5667 
6483 
7297 
8110 
8922 
9782 


0621 
1646 
2469 
3291 
4112 

49bl 
5748 
6564 
7879 
8191 
9003 
9613 


0878 
1186 
1991 

2796 
3596 
4400 
5200 
6996 


0459 
1266 
2072 

2876 
3679 
4480 
5279 
6078 


0640 
1347 
2152 

2956 
3759 
4560 
5359 
6157 


0621 
1428 
2238 

8037 
3839 
4640 
5439 
6287 


0702 
1606 
2313 

8117 
8919 
4720 
6619 
6817 



Diflf. 



86 



86 



84 



88 



88 



81 



80 



Proportional Parts. 



Biff. 


1 


2 


8 


4 


6 


6 


7 


8 


87 


8.7 


17.4 


26 1 


84.8 


485 


622 


60.9 


69 6 


86 


8.6 


17.2 


25.8 


34.4 


430 


51 6 


602 


688 


85 


8.5 


17.0 


255 


84.0 


426 


61.0 


596 


680 


84 


8.4 


16.8 


252 


88.6 


420 


60.4 


688 


67.S 



783 
774 
765 
75.6 



92 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 545 L. 786.] 



LNo. 584 L. mt. 



N. 



545 
6 

7 
8 


560 
1 
2 
3 
4 
5 
6 
7 
8 
9 

560 
1 
2 

3 

4 
6 
6 
7 
8 
9 

570 
1 
2 
3 
4 
5 

6 

7 
8 
9 

580 
1 
2 
8 

4 



750508 
1279 
2048 
2816 
3683 
4348 
5112 

6875 
6636 
7396 
8155 
8912 
9668 



760422 
1176 
1928 
2079 

8428 
4176 
4923 
5660 
6418 






1 


8 


736897 


6476 


6666 


7198 


7272 


7362 


7987 


8067 


8146 


8781 


8860 


8939 


9572 


9651 


9781 


740863 


0442 


0521 


1152 


1280 


1309 


1089 


2018 


2096 


2725 


2804 


2882 


3510 


8588 


3667 


4293 


4871 


4449 


5075 


5153 


5281 


5665 


5933 


6011 


6634 


6712 


6790 


7412 


7480 


7567 


8188 


8266 


8343 


8963 


9040 


9118 


9786 


9614 


9891 



6686 
7481 
8225 
9018 
9610 

0600 
1388 
2175 
2961 
3745 
4528 
5309 
6069 
6868 
7645 

8421 
9195 
9968 



0686 
1356 
2125 



8660 
4426 
5189 

5961 
6712 
7472 
8280 

flORR 
OuOO 

9748 



0498 
1261 
2003 
2754 

8608 
4251 
4998 
5743 
0487 



0663 
1438 
2202 
2970 
3736 
4501 
5265 

6027 
6788 
7548 
8806 
9063 
9619 



0740 
1510 
2279 
3047 
8813 
4578 
5341 

6108 
6864 
7624 
8882 
9189 
9694 



6715 I 
7511 I 
8305 
9097 
9689 



6796 
7590 
8384 
9177 
9968 



0678 
1467 
2254 
8089 
3823 
4606 
6387 
6167 
6945 
7722 

8496 
9272 



0573 
1326 
2078 
2829 

8678 
4326 
5072 
5818 
6562 



0649 
1402 
2153 
2904 

8658 
4400 
5147 
5892 
6686 



0045 
0817 
1587 
2356 
3123 
3889 
4654 
5417 

6180 
6940 
7700 
8458 
9214 
9970 



0724 
1477 
2228 
2978 

8727 
4475 
5221 
5966 
6710 



0757 
1546 
2332 
3118 
3902 
4684 
6465 
6245 
7023 
7800 

' 8576 
9350 

0123 

0894 

16(34 

2483 

i 8200 

3966 

■■ 4730 

I 6494 

6256 
7016 

i 7775 
8533 

I 9290 

I 0045 
0799 
I 1552 
' 2308 
I 3053 

I 8802 
4550 
5296 
1 6041 
! 6786 



6874 
7670 
8468 
9256 



0047 

0636 
1624 
2411 
3196 
3980 
4762 
5643 



7101 
7878 

8653 
9427 



0200 
09n 
1741 
2509 
3277 
4042 
4807 
6570 

6332 
7092 
7851 
8609 
9366 



0121 
0875 
1627 
2878 
3128 

8877 
4624 
5370 
6115 
6859 



7 


8 


6964 


7064 


7749 


7829 


8643 


8622 


9385 


9414 


0126 


0206 


0915 


0994 


1708 


1782 


2489 


2568 


8275 


8353 


4058 


4136 


4840 


4919 


5621 


5699 


6401 


6479 


7179 


7256 


7955 


8083 


8781 


8808 


9504 


9582 



Diff. 



7118 
7908 
8701 
9498 



0284 

1073 
1860 
2647 
3431 
4215 
4997 
5777 
6556 
7334 
8110 

8885 
9659 



0277 


0354 


1048 


1125 


1818 


1896 


2686 


2663 


3»i8 


8480 


4119 


4196 


4888 


4960 


6646 


5722 


6408 


6484 


7168 


':^44 


7927 


8003 


8685 


8761 


9441 


9617 


0196 


0272 


0950 


1025 


1702 


1778 


2458 


2529 


3203 


3278 


8952 


4027 


4699 


4774 


5445 


5620 


6190 


6264 


6938 


70O7 



0431 
1202 
1972 
2740 
3506 
4272 
5086 
5799 

6560 
7320 
8079 
8836 
9592 



0847 
1101 
1868 
2604 
3363 

4101 
4848 
6604 
6888 
7062 



79 



78 



77 



76 



75 



Proportional Partsl 



Diff, 


1 


2 


8 


4 


5 


6 


7 


8 


9 


88 


8.8 


16.6 


24.9 


83.2 


41.5 


49.8 


58.1 


66.4 


74.7 


82 


8.2 


16.4 


24.6 


32.8 


41.0 


49.2 


67.4 


66.6 


73.8 


81 


8.1 


16.2 


24.3 


82.4 


40.5 


48.6 


56.7 


64.8 


72.9 


80 


8.0 


16.0 


24.0 


82.0 


40.0 


48.0 


56.0 


64.0 


72.0 


79 


7.9 


15.8 


23.7 


31.6 


39.5 


47.4 


65.3 


63.2 


71.1 


78 


7.8 


15.6 


28.4 


31.2 


39.0 


46.8 


54.6 


62.4 


70.2 


77 


7.7 


15.4 


23.1 


80.8 


88.5 


46.2 


68.9 


61.6 


69.8 


76 


76 


15.2 


23.8 


80.4 


88.0 


45.6 


68.2 


60.8 


68.4 


75 


7.5 


15.0 


22.5 


80.0 


87.5 


45.0 


62.5 


60.0 


67.5 


74 


7.4 


14.8 


23.2 


29.6 


87.0 


44.4 


51.8 


60.2 


66.6 



93 



TABLE IX.—LOGARITHMS OF NUMBERS. 



r 



No. 585 L. 767.1 



[No. 629 L. 799. 



N. 





1 


8 


8 


4 


585 


767156 


7230 


7804 


7879 


7453 


6 


7896 


7972 


8046 


8120 


8194 


7 


8638 


8712 


8786 


8860 


8934 


6 


9877 


9461 


9625 


9599 


9673 


9 


T70115 


0189 


0263 


0386 


0410 


590 


0862 


0926 


0999 


1073 


1146 


1 


1587 


1661 


1734 


1808 


1881 


2 


2322 


2395 


2468 


2542 


2615 


8 


3055 


3128 


3201 


8274 


8348 


4 


8786 


3860 


3933 


4006 


4079 


5 


- 4517 


4590 


4663 


4786 


4809 


6 


5246 


5819 


5892 


5465 


5536 


7 


5974 


6047 


6120 


6193 


6265 


8 


6701 


6774 


6S46 


6919 


6992 


9 


7427 


7499 


7572 


7644 


7717 


600 


8151 


8224 


8296 


8368 


8441 


1 


8874 


8947 


9019 


9091 


9163 


2 


9596 


9669 


9741 


9613 


9685 


3 


780317 


0380 


0461 


0583 


0605 


4 


1037 


1109 


1181 


1253 


1324 


5 


1755 


1827 


1899 


1971 


2042 


6 


2473 


2544 


2616 


2688 


2759 


7 


8189 


8260 


3332 


3403 


8475 


8 


3904 


8975 


4046 


4118 


4189 


9 


4617 


4689 


4760 


4831 


4902 


610 


5330 


5401 


5472 


5548 


5615 


1 


6041 


6112 


6183 


6254 


6325 


2 


6751 


6822 


6893 


6964 


7035 


8 


7460 


7531 


7602 


7678 


7744 


4 


8168 


8239 


8310 


8881 


8451 


5 


8875 


8946 


9016 


9087 


9157 


6 


9581 


9651 


9722 


9792 


9863 


7 


790285 


0356 


0426 


0496 


0567 


8 


0988 


1069 


1129 


1199 


1269 


9 


1691 


1761 


1831 


1901 


1971 


620 


2392 


2462 


2532 


2602 


2672 


1 


3092 


8162 


3281 


8301 


3371 


2 


3790 


8860 


8930 


4000 


4070 


8 


4488 


4568 


4627 


4697 


4767 


4 


5185 


5254 


5824 


5893 


5463 


5 


5880 


5949 


6019 


6068 


6158 


6 


6574 


6644 


6713 


6782 


6852 


7 


7268 


7837 


7406 


7475 


7545 


8 


7960 


8029 


8096 


8167 


8236 


9 


8651 


8720 


8789 


8858 


8927 



6 



7527 


7601 


8268 


8342 


9006 


9082 


9746 


9820 



0484 

1220 
1955 
2688 
3421 
4152 
4882 
5610 
6338 
7064 
7789 

8513 
9236 
9957 



0677 
1396 
2114 
2831 
8546 
4261 
4974 

5686 
6396 
7106 
7815 
8622 
9228 
9933 



0637 
1340 
2041 

2742 
3441 
4139 
4836 
5532 
6227 
6921 
7614 
I 8305 
8996 



0557 

1293 
2028 
2762 
3494 
4225 
4955 
5683 
6411 
7137 
7862 

8585 
9306 



0029 
0749 
1466 
2186 
2902 
3618 
4332 
5045 

5757 
6467 
7177 
7885 
8593 
9299 



00O4 
0707 
1410 
2111 

2812 
3511 
4209 
4906 
5602 
6297 
6990 
7683 
8374 
9065 



7 


8 


7676 


7749 


8416 


6490 


9156 


9230 


9694 


9966 


0631 


0706 


1367 


1440 


2102 


2175 


28a5 


2906 


3567 


3640 


4298 


4371 


5028 


5100 


5756 


5829 


6488 


6556 


7209 


7282 


7934 


8006 


8658 


8730 


ftS80 


9452 


0101 


0173 


0621 


0698 


1540 


1612 


2258 


2329 


2974 


3046 


8669 


8761 


4403 


4475 


5116 


5187 


5828 


5899 


6536 


6609 


7248 


7819 


7956 


8027 


8663 


8784 


9369 


9440 


0074 


0144 


0778 


0648 


1480 


1550 


2181 


2252 


2882 


2952 


3581 


3651 


4279 


4349 


4976 


5045 


5672 


5741 


6366 


6486 


7060 


7(29 


7752 


7821 


8443 


&518 


9134 


9203 



7823 

8564 
9303 



0042 
0778 

1514 
2246 
2981 
8718 

5173 
5902 
6629 
7354 
8079 

8802 
9524 



0245 
0965 
1684 
2401 
8117 
8832 
4546 
5250 

5970 
6680 
7890 
8096 
8804 
9510 



Diflf. 



74 



78 



0215 
0916 
1620 
2322 

3022 
8721 
4418 
5115 
5811 
6605 
7198 
7890 
8582 
9272 



72 



71 



70 



69 



Proportional. Parts. 



Diff. 


1 


2 


8 


4 


5 


6 


7 


8 


75 


7.5 


15.0 


22.5 


80.0 


37.5 


45.0 


62.6 


60.0 


74 


7.4 


14.8 


22.2 


29.6 


37.0 


44.4 


51.8 


59.2 


78 


7.8 


14.6 


21.9 


29.2 


36.6 


43.8 


61.1 


58.4 


72 


7.2 


14.4 


21.6 


28.8 


86.0 


43.2 


50.4 


57.6 


71 


7.1 


14.2 


21.8 


28.4 


35.5 


42.6 


49.7 


56.8 


70 


7.0 


14.0 


21.0 


28.0 


35.0 


42.0 


49.0 


56.0 


69 


6.9 


18.8 


20.7 


27.6 


84.5 


41.4 


48.8 


55.2 







67.5 
66.6 
65.7 
64.8 
68.9 
63.0 
62.1 



94 



TABLE IX. — LOGARITHMS OF KUMBERS. 



r 



K(V 630 L. 799.] 



[No. 674 L. 829. 



1 



-•N. 





1 


2 


8 


4 


6 


6 


7 


8 


9 


Diff. 


'630 

1 
2 
3 
4 
5 
6 
7 
8 
9 

640 

1 

2 

3 

4 
5 


799341 


9409 


9478 


9547 


9616 


9685 


9754 


9823 


9892 


9961 




800029 
0717 
1404 
2069 
2774 
3457 
4139 
4821 
5501 

806180 
6858 
7535 
8211 
co86 
9560 


0098 
0786 
1472 
2168 
2842 
3526 
4206 
4889 
6669 

6248 
6926 
7603 
8279 
8953 
9627 


0167 
0654 
1641 
2226 
2910 
3694 
4276 
4957 
5637 

6316 
6994 
7670 
8346 
9021 
9694 


0236 
0923 
1609 
2295 
2979 
3662 
4344 
5025 
5705 

6384 
7061 
7736 
8414 
9068 
9762 


0805 
0992 
1678 
2363 
3047 
3730 
4412 
6093 
5773 

6451 
7129 
7806 
8481 
9156 
9629 


0873 
1061 
1747 
2482 
3116 
3798 
4480 
6161 
6841 

6519 
7197 
7873 
8549 
9223 
9896 


0142 
1129 
1815 
2500 
8184 
8867 
4548 
6229 
5908 

6587 
7264 
7941 
8616 
9290 
9964 


0511 
1198 
1884 
2568 
3252 
3935 
4616 
6297 
5976 

6655 
7832 

8008 
8684 
9358 


0580 
1266 
1952 
2637 
8321 
4003 
4685 
5866 
6044 

6723 
7400 
8076 
8751 
9425 


0648 
1835 
2021 
2706 
8889 
4071 
4753 
5433 
6112 

6790 
7467 
8143 
8818 
9492 


» 

68 




0031 
0703 
1374 
2044 
2713 

3881 
4048 
4714 
5378 
6042 
6705 
7367 
8028 
8688 
9346 


0098 
0770 
1441 
2111 

2780 

3448 
4114 
4780 
5445 
6109 
6771 
7483 
8094 
8754 
9412 


0165 
0837 
1506 
2178 
2847 

8514 
4181 
4847 
6611 
6175 
6838 
7499 
8160 
8820 
9478 






7 
8 
9 

650 
1 
2 
8 

4 
5 
6 
7 
8 
9 


810233 
0904 
1576 
2245 

2913 
3681 
4248 
4913 
5578 
6241 
6904 
7565 
8226 
8686 

9644 


0300 
09n 
1642 
2312 

2960 
3648 
4314 
4960 
6644 
6306 
6970 
7631 
8292 
8961 

9610 


0367 
1039 
1709 
2379 

3047 
3714 
4381 
5046 
5711 
6374 
7066 
7696 
8358 
9017 

9676 


0434 
1106 
1776 
2445 

3114 
3781 
4447 
6113 
5777 
6440 
7102 
7764 
8424 
9063 

9741 


0501 
1173 
1843 
2512 

3181 
3848 
4514 
5179 
6843 
6506 
7169 
7830 
8490 
9149 

9807 


0569 
1240 
1910 
2579 

8247 
3914 
4581 
.5246 
5910 
6573 
7235 
7896 
8556 
9215 

9873 


0636 
1307 
1977 
2646 

3314 
3981 
4647 
5312 
5976 
6639 
7301 
7962 
8622 
9281 

9939 


67 
66 


660 


0004 
0661 
1317 
1972 
2626 
3279 
3930 
4581 
5231 
5880 

6528 
7175 
7821 
8467 
9111 


0070 
0727 
1382 
2037 
2691 
3344 
3996 
4646 
5296 
5945 

6593 
7340 
7886 
8531 
9175 


0136 
0792 
1448 
2103 
2756 
3409 
4061 
4711 
6361 
6010 

6658 
7306 
7951 
8595 
9239 




1 
2 
3 
4 
5 
6 
7 
8 
9 

670 
1 
2 
8 

4 


820201 
0656 
1614 
2168 
2622 
8474 
4126 
4778 
6426 

6075 
6723 
7369 
8016 
8660 


0267 
0924 
1579 
2233 
2887 
3639 
4191 
4841 
6491 

6140 
6787 
7434 
8060 
8724 


0838 
0969 
1645 
2299 
2952 
3606 
4256 
4906 
6556 

6204 
6852 
7499 
8144 
8789 


0899 
1055 
1710 
2364 
3018 
3670 
4321 
4971 
5621 

6269 
6917 
7563 
8209 
8863 


0464 
1120 
1775 
2430 
3083 
3735 
4386 
5036 
5686 

6834 

6961 
7628 
8273 
8918 


0530 
1186 
1841 
2495 
3148 
3800 
4451 
5101 
6751 

6399 
7046 
7692 
8338 
8982 


0595 
1251 
1906 
2560 
3213 
3865 
4516 
5166 
5815 

6464 
7M1 
7757 
8402 
9046 


65 



Pbofobtional Pabt& 



.Diff 



68 
67 
66 
66 
64 



1 


2 


8 


4 


6 


6 


7 


8 


9 


68 


13 6 


204 


272 


34 


408 


47 6 


54 4 


61 2 


67 


13 4 


20.1 


268 


33 5 


40 2 


46 9 


53 6 


603 


66 


13.2 


19 8 


26.4 


830 


39 


4« 2 


528 


59 4 


65 


13 


19 6 


260 


82.5 


39 


45 5 


52 


585 


6.4 


1£ 8 


19.2 


25 6 


320 


38.1 


44 8 


51 2 


57.6 



95 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 675 L. 829.] 



[No. 719 L. 867. 



N. 





1 


2 


8 


4 


6 


6 


7 


8 


9 


Diff. 


675 

A 


829304 
9947 


9368 


9432 


9497 


9561 


9625 


9690 


9754 


9618 


9682 






0011 
0653 
1294 
1934 

2578 
3211 

3848 
4484 
5120 
5754 
6387 
7020 
7652 
8282 

8912 
9541 


0075 
0717 
1358 
1996 

2637 
3275 
3912 
4548 
5183 
5817 
6451 
7083 
7715 
8345 

8976 
9604 


0139 
0781 
1422 
2062 

270O 
3338 
8975 
4611 
6247 
5881 
6514 
7146 
7778 
8406 

9038 
9667 


0204 
0645 
1486 
2126 

2764 
3402 
4039 
4675 
6310 
6944 
6577 
7210 
7841 
84n 

9101 
9729 


0268 
0909 
1550 
2189 

2828 
8466 
4108 
4789 
6878 
6007 
6641 
7273 
7904 
8534 

9164 
9792 


0882 
0973 
1614 
2258 

2892 
8530 
4166 
4802 
6487 
6071 
6704 
7386 
7967 
8597 

9227 
9866 


0396 
1037 
1678 
2317 

2956 
8593 
4280 
4866 
6600 
6134 
6767 
7399 
8030 
8660 

9289 
9918 


0460 
1102 
1742 
2881 

8020 
8657 
4294 
4929 
6664 
6197 
6880 
7462 
8093 
8728 

9352 
9981 


0685 
1166 
1806 
2445 

8068 
8721 
4857 
4993 
5627 
6261 
6894 
7626 
8166 
8786 

9415 




7 
8 
9 

680 
1 
2 
8 
4 
6 
6 
7 
8 
9 

690 

1 


830589 
1230 
1870 

2609 
' 3147 
3784 
4421 
5056 
5691 
6324 
6957 
7588 
8219 

8849 
9478 


64 
68 


0043 

oen 

1297 
1922 
2547 
8170 
8793 
4415 
6086 

6666 

6275 
6894 
7511 
8126 
8743 
9368 
9972 




2 
3 

4 
6 
6 
7 
8 
9 

700 
1 
2 
8 
4 
5 
6 
7 


840106 
0733 
1359 
1985 
2609 
8238 
8865 
4477 

6098 
6718 
6387 
6955 
7578 
8189 
8805 
9419 


0169 
0796 
1422 
2047 
2672 
8295 
8918 
4539 

6160 
5780 
6899 
7017 
7634 
8251 
8866 
9481 


0232 
0859 
1485 
2110 
2784 
3857 
3980 
4601 

6222 

6842 
6461 
7079 
7696 
8312 
8928 
9542 


0294 
0921 
1547 
2172 
2796 
8420 
4042 
4664 

6284 
5904 
6523 
7141 
7768 
8874 
8989 
9604 


0357 
0964 
1610 
2235 
2859 
3482 
4104 
4726 

6346 
5966 
6585 
7202 
7819 
8435 
9051 
9665 


0420 
1046 
1672 
2297 
2921 
8544 
4166 
4788 

5406 
6028 
6646 
7264 
7881 
8497 
9112 
9726 


1109 
1736 
2360 
2983 
8606 
4229 
4860 

6470 
6090 
6708 
7326 
7948 
8559 
9174 
9788 


0545 
1172 
1797 
2422 
3046 
3669 
4291 
4912 

6532 
6151 
6770 
7888 
8004 
8620 
9236 
9649 


0606 
1234 
1860 
2484 
8106 
8781 
4868 
4974 

6694 
6218 
6882 
7449 
8066 
8682 
9297 
9911 


m 


8 
9 

710 
1 
2 
8 
4 
5 
6 
7 
8 
9 


850033 
0646 

1258 
1870 
2480 
3090 
8698 
4306 
4913 
6519 
6124 
6729 


0095 
0707 

1320 
1931 
2541 
3150 
8759 
4367 
4974 
5580 
6185 
6789 


0166 
0769 

1381 
1992 
2602 
3211 
3820 
4428 
5034 
6640 
6245 
6850 


0217 
0830 

1442 
2053 
2663 
3272 
3881 
4488 
6095 
6701 
6306 
6910 


0279 
0891 

1503 
2114 
2724 
8333 
8941 
4549 
5156 
6761 
6866 
6970 


0340 
0952 

1564 
2175 
2786 
8394 
4002 
4610 
5216 
6822 
6427 
7081 


0401 
1014 

1625 
2236 
2846 
8455 
4063 
4670 
5277 
6882 
6487 
7091 


0462 
1075 

1686 
2297 
2907 
8616 
4124 
4781 
6387 
6948 
6548 
7152 


0624 
1186 

1747 
2868 
2968 
8577 
4185 
4792 
6896 
6003 
6608 
7212 


0685 
1197 

1809 
2419 
8029 
8637 
4246 
4862 
5450 
6064 
6668 
7272 


01 



Proportional Parts. 



Diflf. 


1 


2 


8 


4 


6 


6 


7 


8 


9 


65 


6.5 


13.0 


19.5 


26.0 


32.5 


39.0 


45.5 


62.0 


58.5 


64 


6.4 


12.8 


19.2 


25.6 


32.0 


86.4 


44.8 


51.2 


57.6 


68 


6.8 


12.6 


18.9 


26.2 


81.6 


37.8 


44.1 


60.4 


66.7 


62 


6.2 


12.4 


18.6 


24.8 


81.0 


87.2 


43.4 


40.6 


65 8 


61 


6.1 


12.2 


18.3 


24.4 


80.5 


86.6 


42.7 


48.6 


64.9 


60 


6.0 


12.0 


18.0 


24.0 


80.0 


86.0 


42.0 


48.0 


54.0 



96 



TABLE IX. — LOGARITHMS OF KUMBEBS. 



No. 730 L. 867.1 



[No. 764 L. 883. 



N. 



790 
1 
2 
8 

4 

6 
6 

7 
8 
9 

730 
1 
2 
3 
4 
6 
6 
7 
8 
9 

740 
1 

2 
8 

4 
6 
6 
7 
8 
9 

750 
1 
2 
8 
4 
5 
6 
7 
8 



760 
1 
2 
8 

4 



867332 
7985 
8687 
9188 
9789 



800838 
0087 
1584 
2181 
2728 

8823 
8917 
4511 
6104 
6606 
6287 
6878 
7467 
8066 
8644 

9282 

9818 



870404 
0989 
1578 
2166 
2789 
8821 
8902 
4482 

6061 
6640 
6218 
6796 
7371 
7947 



7893 
7995 
8697 
9198 
9799 



0808 
0996 
1694 
2191 
2787 



7458 

8066 
8067 
9258 
9869 



0458 
1066 
1664 
2251 

2847 



7518 
8116 
8718 
9818 
9918 



0518 
1116 
1714 
2310 
2906 



7574 
8176 
8778 
9379 
9978 



0578 
1176 
1778 
2370 
2966 



9096 
9669 



880242 

0814 
1886 
1966 
2625 
8008 




7884 
8286 

8838 
9489 



7694 
8297 
8898 
9499 



0088 
0637 
1236 
1838 
2480 
8025 

8620 
4214 
4808 
5400 
6992 
6583 
7178 
7762 
8350 
8988 

9525 



6119 
5698 
6276 
6853 
7429 
8004 
8679 
9158 
9726 



0299 

0671 
1442 
2012 
2681 
8160 



1042 
1618 
2183 
2762 
8821 



0111 
0696 
1281 
1865 
2448 
3080 
8611 
4192 
4772 

6851 
5929 
6607 
7088 
7659 
8234 
8809 
9388 
9956 



0098 
0607 
1295 
1803 
2489 
8065 

8680 
4274 
4867 
6459 
6051 
C642 
7232 
7B21 
8409 
8997 

9584 



0528 

1099 
1670 
2240 
2809 
8877 



0170 
0756 
1839 
1923 
2506 
8068 
8669 
4250 
4880 

5409 
6987 
6564 
7141 
7717 
8292 
8866 
9440 



7765 
6857 
6958 
9559 



0158 
0767 
1355 
1962 
2549 
8114 

8739 
4333 
4926 
5619 
6110 
6701 
7291 
7880 
8468 
9066 

9642 



0228 
0613 
1396 
1961 
2664 
8146 
3787 
4806 
4866 

6466 
6045 



0013 
0585 

1156 
1727 
2297 
2866 
8484 



7199 
7774 
8849 
8924 
9497 



8 



7815 
6417 
9018 
9619 



0216 
0617 
1415 
2012 
2606 
8204 

3709 
4392 
4965 
6678 
6169 
6760 
7860 
7939 
8627 
9114 

9701 



0070 
0642 

1213 
1784 
2354 
2923 
8491 



0287 
0672 
1456 
2040 
2622 
3204 
8785 
4866 
4945 

6624 
6102 
6660 
7256 
7832 
6407 
8961 
9666 



7875 
8477 
9078 
9679 



0278 
0677 
1475 
2072 
2666 
3268 

8868 
4452 
6045 
5687 
6228 
6819 
7409 
7996 
8566 
9I48 

9760 



0127 
0699 

1271 
1841 
2411 
2960 
8648 



0845 
0980 
1515 
2006 
2681 
3262 
3844 
4424 
5008 

6682 

6160 
6787 
7314 
7880 
8464 
0080 
0612 



Dlff. 



60 



60 



0185 
0756 

1328 
1898 
2468 
8087 
8606 



68 



67 









Propobtional Parts. 








• 


Dlff. 


1 


2 


8 


4 


6 


6 


7 


8 





• 


60 
68 
W 
66 


6.0 
6.8 
6.7 
5.6 


11.8 
11.6 
11.4 
11.2 


17.7 
17.4 
17.1 
16.8 


23.6 
23.2 
22.8 
22.4 


20.5 
20.0 
28.6 
28.0 


35.4 
84.8 
84.2 
83.6 


41.8 
40.6 
30.0 
30.2 


47.2 
46.4 
45.6 
44.8 


58.1 
52.2 
51.8 
60.4 



97 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 765 L. 883.] 



[No. 809 L. 906. 



N. 





1 


2 


S 


4 


6 


e 


7 


8 


• 


Diff. 


765 
6 
7 
8 
9 

770 

1 
2 
3 
4 
6 
6 


888661 
4229 
4'?96 
6361 
6026 

6491 
7054 
7617 
8179 
8741 
9302 
9662 


8718 
4285 
4862 
5418 
6963 

6547 
7111 
7674 
8236 
8797 
9358 
9918 


3775 
4342 
4909 
5474 
6039 

6604 
7167 
7730 
8292 
8653 
»I14 
9974 


8832 
4809 
4965 
6581 
6096 

6660 
7223 
7786 
8348 
8909 
9470 


8888 
4456 
6022 
6687 
6152 

6716 
7280 
7842 
8404 
8966 
9526 


8945 
4512 
6078 
6644 
6209 

6778 
7386 
7898 
8460 
9021 
9682 


4002 
4669 
6185 
6700 
6265 

6829 
7892 
7955 
8516 
9077 
9(>88 


4059 
4625 
6192 
5757 
6321 

6865 
7449 
8011 
8578 
9184 
9694 


4116 
4682 
6248 
6813 
6378 

6M2 
7505 
8067 
8629 
9190 
9750 


4172 
4739 
5805 
6870 
6484 

6996 
7561 
8128 
8685 
Won 
9806 


56 


0090 
0589 
1147 
1705 

2818 
3878 
8928 
4482 
6036 
5588 
6140 
6692 
7242 

7792 
8341 
8890 
9437 
9986 


0066 
0645 
1203 
1760 

2317 
2873 
3429 
3964 
4536 
6091 
5644 
6195 
6747 
7297 

7847 
8896 
8944 
9492 


0141 
0700 
1259 
1816 

2873 
2929 
3484 
4039 
4598 
6146 
6699 
6251 
6802 
7352 

7902 
8451 
8999 
9647 


0197 

o;^ 

1314 
1872 

2429 
2985 
3640 
4094 
4648 
5201 
6754 
6306 
6857 
7407 

7957 
8906 
9054 
9602 


0253 
0612 
1370 
1928 

2484 
3040 
3595 
4150 
4704 
6257 
5809 
6361 
6912 
7462 

8012 
8561 
0109 
9656 


0809 
0868 
1426 
1968 

2540 
8096 
8661 
4206 
4759 
5812 
6864 
6416 
6967 
7617 

8067 
8615 
9164 
9711 


0365 
0924 
1482 
2039 

2595 
8151 
8706 
4261 
4814 
5867 
5920 
6471 
7022 
7672 

8122 
8670 
9218 
9766 




7 
8 
9 

780 
1 
2 
8 
4 
5 
6 
7 
8 
9 

790 
1 
2 
3 
4 


890421 
0980 
1537 

2095 
2651 
8207 
8762 
4816 
4870 
5423 
6975 
6526 
7077 

7627 
8176 
8725 
9273 
9621 


0477 
1035 
1593 

2150 
2707 
8262 
3817 
4371 
4925 
6478 
6080 
6581 
7132 

7682 
8231 
8780 
9328 
9875 


0533 
1091 
1649 

2206 

2762 
3318 
3873 
4427 
4960 
5533 
6085 
6636 
7187 

7737 
8286 
8835 
9383 
9980 


55 


0039 
0586 
1131 
1676 
2221 
2764 

8807 
3849 
4391 
4932 
5472 
6012 
6551 
7089 
7626 
8163 


0094 
0640 
1186 
1731 
2275 
2618 

8361 
8904 
4445 
4986 
5526 
6066 
6604 
7143 
7680 
8217 


0149 
0605 
1240 
1785 
2329 
2873 

8416 
8958 
4499 
5040 
5580 
6119 
6658 
7196 
7784 
8270 


0203 
0749 
1295 
1840 
2364 
2927 

8470 
4012 
4558 
6094 
6634 
6178 
6712 
7250 
7787 
8324 


0258 
0804 
1849 
1894 
2488 
2981 

8624 
4066 
4607 
5148 
6688 
6227 
6766 
7304 
7841 
8378 


0812 
0650 
1404 
1948 
2492 
8086 

3678 
4120 
4661 
6202 
6742 
6281 
6820 
7858 
7696 
8481 




5 
6 

7 
8 
9 

800 
1 
2 
3 
4 
5 
6 
7 
8 
9 


900367 
0913 
1458 
2003 
2547 

8090 
8633 
4174 
4716 
5256 
6796 
6335 
6874 
7411 
7949 


0422 
0968 
1513 
2057 
2601 

8144 
8687 
4229 
4770 
6310 
5850 
6389 
6927 
7465 
8002 


0476 
1022 
1667 
2112 
2665 

8199 
8741 
4283 
4824 
6364 
5904 
6443 
6981 
-7519 
8056 


0581 

lorr 

1622 
2166 
2710 

3258 
8795 
4337 
4878 
5418 
6958 
6497 
7035 
7578 
8110 


54 

* 



Proportional Parts. 



Diff. 


1 


2 


8 


4 


5 


6 


7 


8 
45.6 





57 


5.7 


11.4 


17.1 


22.8 


28.5 


84.2 


89.9 


51.3 


56 


5.6 


11.2 


16.8 


22.4 


28.0 


38.6 


39.2 


44.8 


60.4 


55 


5.5 


11.0 


16.6 


22.0 


27.6 


33.0 


38.5 


44.0 


49.5 


54 


6.4 


10.8 


16.2 


21.6 


27.0 


32.4 


37.8 


43.2 


48.6 



98 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 810 L. 90a] 



[No. 854 L. 981. 



N. 





1 


8 


8 


4 


6 


6 


7 


8 


9 


Diff. 


810 

1 
2 


906485 
9021 
9556 


8B89 
9074 
9610 


8692 
9128 
9668 


8646 
9181 
9716 


8699 
9235 
9770 


8768 
9289 
9828 


8807 
9342 

9877 


8860 
9396 
9980 


8914 
9449 

9984 


8967 
9503 






0037 
0571 
1104 
1637 
2169 
2700 
3231 
8761 

4290 
4819 
5347 
5875 
6401 
6927 
7453 
7978 
8602 
9026 

9549 




8 

4 
5 
6 

7 
8 
9 

820 
1 
2 
8 
4 
5 
6 
7 
8 
9 

830 
1 


910091 
0624 
1158 
1690 
2222 
2753 
3284 

8814 
4848 
4872 
6400 
5927 
6454 
6960 
7506 
8080 
8565 

9078 
9601 

920123 
0645 
1166 
1686 
2206 
2725 
8244 
8762 

4279 
4796 
6812 
6828 
6842 
6857 
7370 
7883 
8896 
8906 

0419 
9980 


0144 
0678 
1211 
1748 
2275 
2806 
8387 

8867 
4396 
4925 
5458 
5960 
6507 
7083 
7558 
8088 
8607 

9130 
9653 


0197 
0731 
1264 
1797 
2328 
2859 
8890 

8920 
4449 
4977 
6505 
6033 
6559 
7085 
7611 
8135 
8659 

9183 
9706 


0251 
0784 
1817 
1850 
2381 
2918 
3448 

3973 
4502 
5080 
5558 
6086 
6612 
7188 
7663 
8188 
8712 

9235 
9758 


0304 
0838 
1371 
1908 
2435 
2966 
8496 

4026 
4555 
5063 
5611 
6138 
6664 
7190 
7n6 
8240 
8764 

9287 
9810 


0358 
0891 
1424 
1956 
2488 
8ul9 
3549 

4079 
4606 
5136 
5664 
6191 
6717 
7243 
7768 
8293 
8816 

9340 
9862 


0411 
0944 
1477 
2009 
2541 
8072 
3602 

4132 
4660 
5189 
57H) 
6243 
6770 
7295 
7820 
8345 
8869 

9392 
9914 


0464 
0998 
1530 
2063 
2594 
8125 
3655 

4184 

4718 1 

5241 

5769 

6296 

C822 

7348 

'«73 

8397 

8921 

9444 
9967 


0518 
1051 
1584 
2116 
2647 
8178 
3708 

4237 
4766 
5294 
5822 
6349 
6875 
7400 
7925 
8450 
8973 

9496 


53 


0019 
0541 
1062 
1582 
2102 
2622 
3140 
3658 
4176 

4693 
6209 
5725 
6240 
6754 
7268 
7781 
8293 
8805 
9317 

9827 


0071 
0593 
1114 
1684 
2154 
2674 
3192 
8710 
4228 

4744 
6261 
5776 
6291 
6805 
7819 
7882 
8345 
8857 
9868 

9879 




2 
8 
4 
5 
6 
7 
8 
9 

840 
1 
2 
8 
4 
5 
6 
7 
8 
9 

860 

1 


0176 
0697 
1218 
1788 
2258 
2777 
8296 
8814 

4881 
4848 
5364 
6879 
6394 
6906 
7422 
7986 
8447 
8969 

9470 
9981 


0228 
0749 
1270 
1790 
2310 
2829 
8348 
8865 

4883 

4899 
6415 
6081 
6445 
6969 
7473 
7986 
8496 
9010 

9521 


0280 
0801 
1322 
1842 
2362 
2881 
8399 
3917 

4434 
4951 
6467 
6962 
6497 
7011 
7624 
8087 
8649 
9061 

9572 

0088 
0592 
1102 
1610 


0332 
0853 
1374 
1894 
2414 
2938 
8451 
8969 

4486 
5003 
5518 
6034 
6548 
7062 
7576 
8088 
8601 
9112 

9623 


0884 
0906 
1426 
1916 
2466 
2985 
8508 
4021 

4538 
5054 
5570 
6065 
6600 
7114 
7627 
8140 
8652 
9163 

9674 


0436 
0958 
1478 
1998 
2518 
3037 
3555 
4072 

4589 
5106 
5621 
6137 
6651 
7165 
7678 
8191 
8703 
9215 

9725 


0489 
1010 
1530 
2050 
2570 
3089 
3607 
4124 

4641 
5157 
5673 
6188 
6702 
TO16 
7780 
8242 
8754 
9266 

9776 


62 
61 


0032 
0542 
1051 
1560 


0134 
0643 
1153 
1661 


0185 
0694 
1204 
1712 


0236 
0745 
1254 
1768 


0287 
0796 
1305 
1814 


0338 
0847 
1356 
1865 


0389 
0898 
1407 
1915 




2 
8 
4 


980440 
0949 
1466 


0491 
1000 
1509 





Proportional Parts. 



Diff. 


1 


2 


8 


4 


5 


6 


7 


8 


9 


58 


6.3 


10.6 


15.9 


21.2 


26.5 


31.8 


87.1 


42.4 


47.7 


62 


6.2 


10.4 


15.6 


20.8 


26.0 


31.2 


86.4 


41.6 


46.8 


51 


6.1 


10.2 


15.3 


20.4 


25.5 


30.6 


85.7 


40.8 


45.9 


fiO 


5.0 


10.0 


15.0 


20.0 


25.0 


30.0 


35.0 


40.0 


45.0 



99 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 865 L. 931.1 iNa 899 L. 964. ] 


N. 

.855 





1 


S 


% 


4 


S 


6 


7 


8 


• 


Dlflf. 


981966 


2017 


2068 


2118 


2169 


2220 


2271 1 2322 


2372 


2428 




6 


2474 


2524 


2575 


2626 


2677 


2727 


2rr8 


2829 


2679 


2930 




7 


2961 


3031 


3062 


8133 


3183 


3234 


3285 


3335 


8366 


3437 




8 


3487 


3638 


3589 


3639 


3690 


3740 


3791 


3841 


3892 


3943 




9 


8998 


4014 


4094 


4145 


4195 


4246 


4296 


4347 


4397 


4448 




860 


4498 


4549 


4599 


4650 


4700 


4751 


4601 


4852 


4902 


4968 




1 


6003 


6054 


5104 


5154 


5205 


5255 


5306 


5356 


5406 


5457 


• 


2 


6507 


5668' 


5608 


5658 


5709 


6759 


5809 


5860 


5910 


6960 




8 


6011 


6061 


6111 


6162 


6212 


6262 


6313 


6363 


6413 


6468 




4 


6514 


6664 


6614 


6666 


6715 


6786 


6815 


6866 


6916 


6966 




6 


7016 


7066 


7116 


7167 


7217 


7267 


7317 


7367 


7418 


7468 




6 


7518 


7568 


7618 


7668 


7718 


7769 


7819 


7869 


7919 


7969 

sffio 


50 


7 


8019 


8069 


6119 


6169 


8219 


8269 


8320 


8370 


8420 


8 


8520 


8570 


8620 


8670 


8720 


8770 


8820 


8870 


8920 


8970 




9 


9020 


9070 


9120 


9170 


9220 


9270 


9320 


9369 


9419 


9469 




870 
1 


9519 


9569 


9619 


9660 


9719 


9769 


9619 


0869 


9918 


9968 




940018 


0068 


0118 


0168 


0218 


0667 


0817 


0367 


0417 


0467 


2 


0516 


0666 


0616 


0666 


0716 


0765 


0815 


0665 


0915 


0964 




8 


1014 


1064 


1114 


1163 


1213 


1263 


1313 


1362 


1412 


1462 




4 


1511 


1561 


1611 


1660 


1710 


1760 


1809 


1669 


1909 


1956 




5 


2008 


2058 


2107 


2157 


220r 


2256 


2306 


2365 


2405 


2456 




6 


2504 


2!>54 


2603 


2658 


2702 


2752 


2801 


2861 


2901 


2950 




7 


8000 


8049 


8099 


3148 


3196 


3247 


3297 


8346 


8396 


8446 




8 


8496 


3544 


3593 


3648 


3692 


3742 


3791 


8841 


8690 


3939 




9 


8969 


4088 


4068 


4137 


4186 


4236 


4285 


4336 


4384 


4438 




880 


4483 


4532 


4581 


4631 


4680 


4729 


4779 


4828 


4877 


4927 




1 


4976 


5025 


5074 


5124 


5173 


fi?SJ^ 


6272 


6321 


5870 


6419 




8 


6469 


6518 


5567 


5616 


5666 


5716 


6764 


6813 


5862 


6912 




3 


6961 


6010 


6059 


6108 


6157 


6207 


6266 


6306 


6364 


6408 




4 


6452 


6601 


6551 


6600 


6649 


6698 


6747 


6796 


6845 


6894 




5 


6943 


6992 


7041 


7090 


7139 


7189 


7238 


7287 


7336 


7885 


49 


6 


7434 


7483 


7582 


7581 


7630 


7679 


7728 


7777 


7826 


7875 


7 


7924 


7973 


8022 


6070 


8119 


8166 


8217 


8266 


8316 


8364 




8 


8413 


8462 


8511 


8560 


8606 


8657 


8706 


8765 


8604 


8858 




9 


8902 


8951 


8999 


9048 


9097 


9146 


9195 


9244 


9292 


9841 




890 


9890 


9439 


9468 


0536 


9565 


9684 


9683 


9781 


9780 


9829 




1 


9878 


9926 


9975 
















0(124 


0073 
0560 


0121 
0606 


0170 


0219 
0706 


0267 
0754 


(Ktlfi 




2 


950365 


0414 


0462 


0511 


0657 


0608 




8 


0651 


0900 


0949 


0997 


1046 


1096 


1143 


1192 


1240 


1289 




4 


1388 


1886 


1436 


1483 


1532 


1580 


1629 


1677 


1728 


1776 




6 


1823 


1872 


1920 


1969 


2017 


2066 


2114 


2168 


2211 


8S60 




6 


2306 


2356 


2406 


2453 


2502 


2560 


2599 


2647 


2696 


2744 




7 


2792 


2841 


2689 


2938 


2966 


3034 


8063 


3131 


3180 


3228 




8 


8276 


3326 


3378 


3421 


3470 


8518 


8566 


3615 


3663 


3711 




9 


8760 


3806 


8866 


8905 8953 


4001 


4049 


4098 


4146 


4194 




Proportional Parts. 


Diff. 


1 


2 


8 


4 


6 


6 


7 


8 





61 


6.1 


10.2 


15.8 


20.4 


25.6 


80.6 


85.7 


40.8 


45.0 


50 


6.0 


10.0 


15.0 


20.0 


25.0 


80.0 


35.0 


40.0 


45.0 


49 

4A 


4.9 


9.8 


14.7 


19.6 


24.5 


29.4 


84.8 


89.2 


44.1 


48 1 4.8 1 ».o 1 14.4 


19.2 


24.0 


28.8 


88.6 


88.4 


48.S 



100 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No 900 L. 964.1 








[No. 944 L. 975. 


N. 
900 





1 


f 


t 


4 


« 


8 


7 


8 


9 


Diff. 


964948 


4291 


4339 


4887 


4436 


4484 


4532 


46R0 


4626 


4677 




1 


4725 


4773 


4821 


4860 


4918 


4966 


5014 


5062 


6110 


5158 




2 


6207 


6255 


5303 


5361 


5899 


5447 


6496 


5543 


5592 


6640 




8 


6688 


6736 


5784 


5832 


5880 


5928 


6976 


6024 


6072 


6120 




4 


6168 


6216 


6265 


6313 


6361 


6409 


6467 


6506 


6568 


6601 


48 


5 


6649 


6697 


6745 


6793 


6840 


6888 


6936 


6964 


7082 


7080 


6 


7128 


7176 


7224 


7272 


ra20 


7368 


7416 


7464 


7512 


7669 




7 


7607 


7655 


7703 


7751 


7799 


7847 


7804 


7942 


7990 


8088 




8 


8086 


8134 


8181 


8229 


8277 


8325 


8378 


8421 


8468 


8516 




9 


8664 


8612 


8650 


8707 


8755 


8803 


8660 


ooiro 


8946 


8994 




910 


9041 


9069 


9137 


9185 


9282 


9260 


9328 


9375 


9428 


9471 




1 


9518 


9666 


9614 


9661 


9709 


9757 


9804 


9662 


9900 


9947 




2 


9996 






















0042 
0518 


0090 
0566 


0138 
0613 


0185 
0661 


0233 
0709 


0280 
0756 


0828 
0604 


0376 
0861 


0428 
0R99 




8 


960471 




4 


0946 


0994 


1041 


1069 


1136 


1184 


1281 


1279 


1326 


1874 




6 


1421 


1469 


1616 


1663 


1611 


1668 


1706 


1763 


1801 


1848 




6 


1895 


1948 


1990 


2038 


2086 


2132 


2180 


2227 


2276 


2822 




7 


2369 


2417 


2464 


2511 


2659 


2606 


2668 


2701 


2748 


2796 




8< 


2848 


2890 


2937 


2966 


8082 


8079 


8126 


8174 


3221 


8268 




9 


8816 


8863 


8410 


3457 


8604 


8552 


8599 


8646 


8698 


8741 




920 


8788 


8835 


3882 


8929 


8077 


4024 


4071 


4118 


4166 


421S 




1 


4260 


4807 


4354 


4401 


4448 


4496 


4542 


4600 


4687 


4684 




2 


4731 


4778 


4825 


4872 


4919 


4966 


5018 


5061 


6108 


5166 




8 


6202 


6249 


6206 


6343 


5390 


5437 


5484 


6581 


6678 


6626 




4 


5672 


6719 


6766 


6813 


5860 


5907 


5054 


6001 


6048 


6096 


< 


5 


6142 


6189 


6236 


6263 


6329 


6876 


6428 


647t) 


6517 


6664 




6 


6611 


6658 


6706 


6752 


6799 


6845 


6892 


6939 


6986 


7088 




7 


7080 


7127 


7173 


7220 


7267 


7814 


7361 


7408 


7464 


7501 




8 


7548 


7596 


7642 


7688 


7735 


7782 


7829 


7875 


7922 


7969 




9 


8016 


8062 


8109 


8156 


8208 


8249 


8296 


8343 


8390 


8486 




990 


8483 


8530 


8576 


8623 


8670 


8716 


876S 


8810 


8856 


8908 




I 


8950 


8996 


9043 


9090 


9136 


9183 


9229 


9276 


0323 


9869 




2 


9416 


9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9636 




a 


9882 


9928 


9975 




















0021 
0486 


0068 
0533 


0114 
0579 


0161 
0626 


0207 


0254 


0300 




4 


970347 


0393 


0440 


0672 


0719 


0765 




5 


0812 


0658 


0904 


0951 


0997 


1044 


1090 


1137 


1183 


1229 




6 


1276 


1322 


1369 


1415 


1461 


1508 


1554 


1601 


1647 


1693 




7 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


2064 


2110 


2157 




8 


2208 


2249 


2296 


2342 


2388 


2484 


2481 


2527 


2678 


2619 




9 


2666 


2712 


2758 


2804 


2861 


2897 


2948 


2969 


8086 


8062 




940 


8128 


8174 


8220 


8266 


%13 


8369 


8405 


8451 


8497 


8643 




1 


8590 


8636 


3682 


8728 


8774 


8820 


8666 


8918 


8959 


4006 




2 


4061 


4097 


4143 


4189 


4235 


4281 


4327 


4374 


4420 


4466 




8 


4512 


4568 


4604 


4660 


4696 


4742 


4788 


4834 


4880 


4926 




4 


4972 


6018 


6064 


5110 


6166 


6202 


6248 


6294 


5840 


6886 


46 






Pbo] 


POBTIONAL PAJ 


RT8. 




Diff. 


1 


2 


3 


4 


6 


• 6 


7 


8 


9 • 


47 


4.7 


9.4 


14 


.1 


] 


18.8 


28.6 


28.2 


8S 


S.9 


87.6 


42.8 


46 


4.6 


9.2 


18 


8 


] 


18.4 


23.0 


27.6 


8S 


5.2 


86.8 


41.4 



101 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 945 L. 975.] 








[No. 969 L. 995. 1 


N. 
945 





1 


t 


8 


4 


6 


6 


7 


8 


9 


Diff. 


975432 


5478 


5524 


6570 


5616 


6662 


5707 


5753 


5799 


6845 




6 


5891 


5937 


5963 


6029 


6075 


6121 


6167 


6212 


6268 


6304 




7 


6350 


6396 


6442 


6488 


6533 


6579 


6625 


6671 


6717 


6763 




8 


6806 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 




9 


T266 


7312 


7358 


7403 


7449 


7495 


7541 


7586 


7632 


7678 




950 


7724 


7769 


7815 


7861 


7906 


7952 


7998 


8043 


8089 


8135 




1 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


8546 


8591 




2 


8637 


8683 


8728 


8774 


8819 


8865 


8911 


8956 


9002 


9047 


* 


3 


9093 


9138 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


9503 




4 
5 


9548 


9594 


9639 


9685 


9730 


9776 


9821 


9867 


9912 


9958 




960003 


0049 


0094 


0140 


0185 


0231 


0276 


0322 


0367 


0412 


6 


(458 


0503 


0549 


0591 


0640 


0685 


0730 


0776 


0821 


0867 




7 


0912 


0957 


1003 


1048 


1093 


1139 


1184 


1229 


1275 


1320 




8 


1366 


1411 


1456 


1501 


1547 


1562 


1637 


1683 


1728 


1773 




9 


1819 


1864 


1909 


1954 


2000 


2045 


2090 


2135 


2181 


2226 




960 


2271 


2316 


2362 


2407 


2452 


2497 


2543 


2588 


2633 


2678 




1 


2723 


2769 


2814 


2859 


2904 


2949 


2994 


3040 


3066 


3130 




2 


3175 


3220 


3265 


3310 


3356 


3401 


3446 


3491 


3536 


3681 




3 


3626 


3671 


3716 


3762 


3807 


3852 


3897 


3942 


3987 


4032 




4 


4077 


4122 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 


45 


6 


4527 


4572 


4617 


4662 


4707 


4752 


4t97 


4842 


4887 


4982 


6 


4977 


5022 


5067 


5112 


5157 


5202 


6247 


5292. 


533? 


6382 




7 


5426 


5471 


5516 


5561 


5606 


5651 


5696 


5741 


5786 


6830 




8 


5875 


5920 


5965 


6010 


6055 


6100 


6144 


6189 


6234 


6279 




9 


6324 


6369 


6413 


6458 


6503 


6548 


6593 


6637 


6682 


6727 




970 


6772 


6817 


6861 


6906 


6951 


6996 


7040 


7065 


7130 


7175 




1 


7219 


7264 


7309 


7353 


7398 


7443 


7488 


7532 


7577 


7622 




2 


7666 


7711 


7756 


7800 


7845 


7890 


7984 


7979 


8024 


8068 




3 


8113 


8157 


8202 


8247 


8291 


8336 


8381 


8425 


8470 


8614 




4 


8559 


8604 


8648 


8693 


8737 


8782 


8826 


8871 


8916 


8960 




5 


9005 


9049 


9094 


9138 


9183 


9227 


9272 


9316 


9361 


9405 




6 


9450 


9494 


9539 


9583 


9628 


9672 


9717 


9761 


9806 


9850 




7 


9895 


9939 


9983 


















0028 
0472 


0072 


0117 


0161 


0206 


0250 


0294 




8 


990339 


0383 


0428 


0516 


0561 


0605 


0650 


0694 


0738 




9 


0783 


0627 


0871 


0916 


0960 


1004 


1049 


1098 


1137 


1182 




980 


1226 


1270 


1315 


1359 


1403 


1448 


1492 


1536 


1580 


1625 




1 


1669 


1713 


1758 


1802 


1846 


1890 


1935 


1979 


2023 


2067 




2 


2111 


2156 


2200 


2244 


2288 


2833 


2377 


2421 


2466 


2609 




3 


2554 


2598 


2642 


2686 


2730 


2774 


2819 


2863 


2907 


2961 




4 


2995 


3039 


3083 


3127 


3172 


3216 


3260 


3804 


3348 


3392 




5 


3436 


3480 


3524 


3568 


8613 


3657 


3701 


3745 


3789 


3833 




6 


3877 


3921 


3965 


4009 


4053 


4097 


4141 


4185 


4229 


4273 




7 


4317 


4361 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


4713 


44 


8 


4757 


4801 


4845 


4889 


49») 


4977 


5021 


5065 


6106 


5152 




9 


5196 


5240 


5284 


5328 


5372 


5416 


5460 


5504 


6547 


5591 






• 


Phoi 


PORTIONS L Pa 


RTa. 




Diflf. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


46 


4.6 


9.2 


13 


8 


] 


18.4 


23.0 


27.6 


3S 


1.2 


36.8 


41.4 


45 


4.5 


9.0 


13 


5 


] 


18.0 


22.5 


27.0 


81 


.5 


36.0 


40.5 


44 


4.4 


8.8 


13 


2 


' 


17.6 


22.0 


26.4 


80 


1.8 


35.2 


39.6 


43 


4.3 


8.6 


12 


.9 


: 


[7.2 


21.5 


25.8 


30 


M 


84.4 


88.7 



102 



TABLE IX. — LOGARITHMS OF NUMBERS. 



No. 990 L. 905.] 














[No. 999 L. 999. 


N. 





1 


8 


8 


4 


6 


6 


7 


8 


9 


Difl. 


990 


996635 


5679 


6723 


6767 


5811 


5854 


5898 


5942 


5986 


6030 




1 


6074 


6117 


6161 


6205 


6249 


6293 


6:S7 


6880 


6424 


6468 


44 


2 


6513 


6555 


6599 


6643 


6687 


6731 


6774 


6818 


6862 


6906 


^ 


3 


6949 


6993 


7037 


7080 


7124 


7168 


7212 


7255 


7299 


7343 




4 


7386 


7430 


7474 


7517 


7561 


7606 


7648 


7692 


7736 


7779 




5 


7823 


7867 


7910 


7964 


Two 


8041 


8085 


8129 


8172 


8216 




6 


8250 


8308 


8347 


8390 


8434 


8477 


8521 


8564 


8606 


8652 




7 


8695 


8739 


8782 


8826 


8869 


8913 


8956 


9000 


9043 


9087 




R 


9181 


9174 


9218 


9261 


9305 


9348 


9392 


9435 


9479 


9522 




9 


9565 


9609 


9652 


9696 


9789 


9783 


9626 


9870 


9913 


9957 


43 



Logarithms of Nuvbers from 1 to 100. 



N. 



1 
2 
3 

4 
6 

6 
7 
8 
9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 



Log. 



0.000000 
0.301030 
0.477121 
0.602060 
0.698970 

0.778151 
0.845098 
0.903090 
0.954243 
1.000000 

1.041393 
1.079181 
1.113943 
1.146128 
1.176091 

1.204120 
1.230449 
1.255278 
1.278754 
1.301060 



N. 
21 


Log. 


1.822219 


22 


1.342423 


23 


1.861728 


24 


1.380211 ' 


25 


1.897940 


26 


1.414973 


27 


1.431364 ! 


28 


1.447158 ' 


29 


1.462398 


30 


1.477121 


31 


1.491362 


82 


1.506150 


33 


1.518514 


34 


1.531479 


85 


1.544068 


86 


1.556303 


37 


1.568202 


88 


1.579784 


89 


1.591065 


40 


1.602060 



N. 



41 
42 

43 
44 
45 

46 
47 
48 
49 
60 

51 
52 
63 
54 
55 

56 
57 
58 
59 
60 



Log. 


N. 
61 


1.612784 


1.628249 


62 


1.633468 


68 


1.643453 


64 


1.653213 


65 


1.662758 


66 


1.672098 


67 


1.681241 


68 


1.690196 


69 


1.696070 


70 


1.707570 


71 


1.716003 


72 


1.724276 


73 


1.732394 


74 


1.740363 


75 


1.748188 


78 


1.755875 


r? 


1.763428 


78 


1.770852 


79 


1.778151 


80 



Log. 


N. 
81 


1.785830 


1.792392 


82 


1.799341 


83 


1.806180 


84 


1.812913 


85 


1.819544 


86 


1.826076 


87 


1.832509 


88 


1.888849 


89 


1.845098 


90 


1.851258 


91 


1.857332 


92 


1.863323 


93 


1.869232 


94 


1.876061 


95 


1.880814 


96 


1.886491 


97 


1.898095 


96 


1.897627 


99 


1.903090 


100 



Log. 



1.906485 
1.913814 
1.919078 
1.924279 
1.929419 

l.a34498 
1.939519 
1.9444a3 
1.949390 
1.954243 

1.959041 
1.963788 
1.968483 
1.973128 
1.9r;724 

1.982271 
1.966772 
1.991226 
1.995635 
2.000000 



Value 

ato*. 



8in.«.. 
Tan... 
Seo... 
Vwsin 
Ckw... 
Cot.... 
Coflec. 



O 

g 

O 
R 

00 



Sign 
inlst 
Quad. 



Value 
atOO". 



R 

00 

oo 
R 
O 
O 
R 



Sign 
in 2d 
Quad. 



4- 



Value 


Sign 


Value 


Sign 


at 


in8d 


at 


in 4th 


180°. 


Quad. 


270« 


Quad. 


O 


, ^ 


R 




o 


-1- 


00 


^_ 


R 


_ 


00 


.. . 


2R 


-4- 


R 




R 


_ 





— . 


00 


4- 





~ 


00 




R 


"- 



Value 

at 

860<». 



O 
O 
R 
O 
R 

00 
00 



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

108 



TABLE X. — LOGARITHMIC 8INBB, 



I 
1 

1 
1 



TABLE X. — LOGARITHMIC SINES, 



178* 



tr 



moo 

8660 
8330 
8780 
8840 
8000 
8060 
40S0 
4060 
4140 
4900 

4860 
4890 
4880 
4440 
4500 
4660 
4680 
4660 
4740 
4800 

4860 
4800 
4860 
S040 
6100 
6160 



S06O 
6840 
5400 

6460 
6680 
6660 
6640 
6700 
6780 



6880 
6040 
6000 

6060 
6180 
6160 
6840 
6800 



6480 
6460 
6640 
6600 



6790 
6780 
6840 
6000 
6860 
7080 
7060 
7140 
7900 



// 



C 
1 
8 
8 
4 
6 
6 
7 
8 

10 

11 
19 
18 
14 
15 
16 
17 
18 
10 
80 

81 
88 
88 
84 
86 
88 
87 
88 
89 
80 

81 
88 

^ 
84 

85 

86 

87 

88 

89 

40 

41 
49 
48 
44 
46 
46 
47 
48 
49 
60 

61 
69 
68 
64 
66 
66 
67 
66 
60 
60 



Sine. 



8.941655 
.949068 
.956094 
.968048 
.809881 
.876614 



.869773 
.896807 
.80854ft 

.806794 

6.814954 
.881097 
.887016 
.888984 
.888758 
.844504 
.860181 
.866788 
.861815 
.866777 

8.879171 
.877499 
.862769 
.887969 
.888101 
.886179 
.408199 
.406161 
.418066 
.417919 

6.499717 
.497469 
.489156 
.486600 
.441894 
.446941 
.460440 
.464698 
.450801 
.468666 

6.467965 
.472968 
.476496 
.480608 
.484848 
.488968 
.498040 
.497078 
.501060 
.506045 

8.606974 
.519867 
.616796 
.690551 
.694848 
.696109 
.681886 
.686698 
.689186 

8.549819 



«-« 



4.686 



553 
669 
551 
561 
550 
549 
548 
547 
546 
646 
545 

544 
543 
649 
641 
640 
680 
680 
688 
587 
586 

585 
584 
583 



539 
581 
580 
599 
587 
586 
586 

584 
693 
598 

581 
690 
616 
617 
516 
515 
614 

519 
511 
510 
509 
69/ 
606 
505 
508 
609 
601 

499 
496 
497 
496 
494 
499 
491 
490 
468 
487 
4. 



619 
690 
699 



695 
687 
698 
680 
639 
638 



685 



687 
686 
640 
649 
644 
646 
648 
649 
651 
668 

656 
667 
668 
661 
663 
666 
666 
670 
679 
674 

676 
679 
681 
668 
685 
688 
690 
698 
695 
697 

700 
709 
705 
707 
710 
718 
715 
716 
780 
788 

796 
799 
781 
734 
737 
740 
748 
746 
748 
•?51 



Tang. 



685 



8.841981 
.849108 
.866166 
.968115 
.969966 
.976691 
.983393 
.989856 



.309634 
.806884 

8.816046 
.891199 
.887114 
.883085 
.388666 
.844610 
.860869 
.865695 
.861430 
.366895 

8.379298 
.377628 



.888098 
.888284 
.896815 
.408888 
.408804 
.413918 
.418068 

6.499869 
.497618 
.489815 
.486969 
.441560 
.446110 
.450618 
.456070 
.459481 
.468849 

6.468172 
.479454 
.476698 
.480699 
.485050 
.489170 
.498950 
.497998 
.601296 
.606967 

6.600900 
.513096 
.516061 
.580790 
524586 
.528349 
.588060 
.536779 
.539447 

8.548064 



Cotang. 


« + « 




15.814 


11.758079 


381 


.750698 


880 


.748885 


378 


.736885 


srr 


.780044 


375 


.■793809 


373 


.716677 


372 


.710144 


370 


.708708 


368 


.697366 


367 


.691116 


866 


11.684954 


863 


.678878 


862 


.679886 


860 


.666975 


868 


.661144 


856 


.666890 


354 


.649711 


359 


.644105 


351 


.688570 


849 


.688105 


347 


11.627708 


345 


.622878 


343 


.617111 


841 


.611908 


889 


.606766 


337 


.601685 


834 


.596662 


832 


. .591696 


830 


.586787 


828 


.581982 


826 


11.577181 


824 


.572862 


821 


.667685 


319 


.668086 


317 


.658440 


315 


.658890 


312 


.549887 


810 


.644980 


807 


.640519 


805 


.536151 


803 


11.531826 


800 


.527546 


898 


.528807 


295 


.519106 


293 


.614950 


290 


.610880 


287 


.506750 


285 


.508707 


282 


.496709 


280 


.494738 


277 


11.490600 


874 


.486909 


871 


.488089 


269 


.479910 


866 


.475414 


263 


.471661 


960 


.467990 


957 


.464921 


955 


.460563 


959 


11.466916 


949 




16.814 


Tang. 


5 + 1 



Dl' 



Cosine. 



0& 
05 
03 
03 
05 
08 
08 
06 
08 
05 

05 
03 
05 
06 
03 
05 
05 
05 
05 
05 

05 
06 
05 
05 
05 
05 
05 
05 
07 
05 

05 
07 
05 
05 
07 
05 
07 
05 
07 
07 

05 
07 
07 
07 
07 
05 
07 
07 
07 
07 

07 
06 
07^ 
07 
07 
07 
08 
07 
07 



9 



9 



9 



9 



9 



9 



999934 
999982 
999999 
999997 
999025 
999989 
999920 
999918 
999915 
999913 
999910 

999007 
999905 
999908 

999897 

999691 
999888 
999885 
999882 

999679 
999676 
999878 
999870 
999667 
999864 
999861 
999858 
999854 
999851 

999648 
999844 
999641 
999888 
999834 
999831 
999827 
999824 
999820 
999816 

99981 3 
999809 
999805 
999801 
999797 
999794 
999700 
999786 
999782 
999778 

999774 
999769 
999765 
999761 
999757 
999753 
999748 
999744 
999740 
999736 



60 
59 
58 
57 
56 
55 
54 
58 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
37 
36 
85 
34 
33 
32 
81 
SO 

29 
28 
97 
26 
25 
24 
28 
22 
21 
90 

19 
18 
17 
16 
15 
14 
13 
19 
11 
10 

9 
8 
7 
6 
5 
4 
8 
9 
1 




91' 



lAfS 



2- 



COSmKS, TANGENTS, AND COTANGENTS. 



IT?* 




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 

81 
82 
83 
34 
35 
36 
37 
88 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
65 
66 
57 
58 
59 
60 



Sine. 



8 542319 
.546422 
.649995 
.653539 
.557054 
.560540 
.563999 
.567431 
.670636 
.574214 
.577566 

8.580692 
.584193 
.587469 
.590721 
.593948 
.59n52 
.600332 
.603489 
.606623 
.609734 

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

8.642563 
.645428 
.648274 
.651102 
.653911 
.656702 
.669475 
.662230 
.664968 
.667689 

8.670393 
.673080 
.675751 
.678405 
.681048 
.683665 
.686272 
.688863 
.6914-« 
.693996 

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

8.718800 



D.r. 



60.06 
69.65 
69.07 
68.58 
68.10 
67.65 
67.20 
56.75 
66.30 
65.87 
65.43 

55.02 
54.60 
64.20 
63.78 
58.40 
58.00 
52.62 
52.23 
61.85 
51.48 

51.13 

50.77 

50.42 

60.05 

49.72 

49.38 

49 

48 

48.40 

48.05 



?^ 



47.75 
47.43 
47.18 
46.82 
46.52 
46.22 
45.92 
45.63 
45.35 
45.07 

44.78 
44.52 
44.23 
43.97 
43.70 
43.45 
43.18 
42.92 
42.67 
42.42 

42.17 
41.93 
41.68 
41.45 
41.20 
40.97 
40.75 
40.52 
40.28 



' I Cog*-^e. D. 1". 



Cosine. 



9.999735 
999731 
999*:^ 
999722 
999717 
999713 
999706 
099704 
999699 
999694 
999689 



9 



9 



9 



9 



9 



9 



999685 
999680 
999675 
999670 
999665 
999660 
999655 
999650 
999645 
999640 

999635 
999629 
999624 
999619 
999614 
999606 
999603 
999597 
999592 
999586 

999581 
999575 
999570 
999564 
999558 
999553 
999547 
999541 
999535 
999529 

999524 
999518 
999512 
999506 
999500 
999493 
999487 
999481 
999475 
999469 

999463 
099456 
999450 
999443 
999437 
999431 
999424 
999418 
999411 
999404 



Sine. 



D. 1- 



07 
08 
07 
06 
07 
06 
07 
08 
06 
08 
07 

06 
06 
06 
06 
06 
06 
06 
06 
06 
06 

10 
06 
08 
06 
10 
06 
10 
06 
10 
06 

10 
06 
10 
10 
06 
10 
10 
10 
10 
06 

10 
10 
10 
10 
12 
10 
10 
10 
10 
10 

12 
10 
12 
10 
10 
12 
10 
12 
12 



i 



92< 



D. 1". 



106 



Tang. 



8.5480S4 
.546691 
.650268 
.553817 
.557336 
.560628 
.564291 
.567^7 
.571137 
.674520 
.577877 

8.581206 
.584514 
.587795 
.591051 
.594283 
.697492 
.600677 
.603839 



.610094 

8.618189 
.616262 
.619313 
.622348 
.625352 
.628340 
.631306 
.634256 
.687184 
.640098 

8.642982 
.645858 
.648704 
.651537 
.654852 
.65n49 
.659928 
.662689 
.665483 
.668160 

8.670670 
.673663 
.676239 
.678900 
.681544 
.684172 
.686784 
.689381 
.691968 
.694529 

8.697061 
.699617 
.702139 
.704646 
.707140 
.709618 
.712068 
.714534 
.716078 

8.719896 



D. 1'. 



Ootaog. 



.. 



00.12 
69.62 
69.15 
68.66 
68.20 
67.72 
67.27 
66.83 
66.88 
66.96 
65.62 

55.10 
54.68 
64.27 
53.87 
53.48 
53.06 
62.70 
62.82 
61.93 
61.58 

61.22 
60.85 
60.60 
60.15 
49.80 
49.47 
49.18 
48.80 
48.48 
48.15 

47.85 
47.52 
47.22 
46.92 
46.62 
46.82 
46.02 
45.73 
45.45 
45.17 

44.88 
44.60 
44.85 
44.07 
43.80 
43.58 
43.26 
43.08 
42.77 
42.58 

42.27 
42.03 
41.78 
41.57 
41.80 
41.08 
40.86 
40.68 
40.40 



Cotang. I D. 1'. 



11.456016 
.453309 
.449732 
.446188 
.442664 
.489172 

. .435709 
.432273 
.428868 
.425480 
.422123 



11 



11 



11 



11 



11 



11 



418792 
415486 
412205 
406049 
405n7 
402508 
899823 
396161 
393022 
389906 

886811 
383738 
380687 
877667 
874648 
871660 
368692 
365744 
862816 
859907 

857018 
854147 
861296 
848463 
846648 
842861 
840072 
837311 
334567 
331840 

329130 
326487 
32S761 
321100 
318456 
316828 
313216 
310619 
308087 
805471 

302919 
300683 

297861 
295354 
292860 
290682 

287917 
286466 
283028 
280604 



Tang. 



60 
59 
68 
57 
56 
55 
54 
58 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
81 
90 

20 
23 
27 
26 
25 
24 
28 
22 
21 
20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 


8 
7 
6 
5 
4 
3 
2 
1 




•7* 



TABLE X. — LOGARITHMIC SINES, 



176- 





1 

2 
8 
4 
6 
6 
7 
8 
9 
10 

11 
12 
18 
14 
15 
16 
17 
18 
19 
90 

21 
22 
28 
24 
25 
26 
27 
28 
29 
80 

81 
82 
88 
84 
35 
86 
87 
86 
89 
40 

41 
42 
48 
44 
45 
46 
47 
48 
49 
50 

61 
52 
58 
54 
55 
56 
57 
56 
SO 
60 



Sine. 



8.718800 
.721204 
.728696 
.789078 
.7S8387 
.780688 
.788027 
.785854 
.787667 
.739969 
.742259 

8.744536 
.746802 
.749055 
.751297 
.758528 
.755747 
.767955 
.760151 
.702837 
.764511 

8.766675 
.7688S8 
.770970 
.778101 
.775228 
.777388 
.779484 
.781524 
.783605 
.785675 

8.787786 
.789787 
.791828 
.798859 
.796881 
.797894 
.799697 
.801802 
.808876 

8.807819 
.809777 
.811726 
.818667 
.815599 
.817522 
.819486 
.821348 
.828240 
.825130 

8.827011 
.888884 
.830749 
.832607 
.834456 
.836297 
.838130 
.639966 
.841774 

8.848665 



D.r. 



Cosine. 



40.07 
89.85 
39.62 
89.42 
80.18 
38.96 
88.78 
38.55 
38.87 
88.17 
87.96 

37.77 
87.55 
87.97 
87.18 
86.98 
86.80 
86.60 
86.48 
86.23 
86.07 

35.88 
35.70 
85.52 
85.87 
85.17 
85.02 
84.83 
84.68 
84.50 
84.85 

84.18 
84.02 
83.85 
83.70 
83.55 
83.38 
88.26 
88.07 
82.93 
82.78 

82.63 
82.48 
82.85 
82.20 
82.05 
81.90 



81 
81 
81 
81 



.78 
.02 
.50 
.85 



81.22 
81.06 
80.97 
30.82 
30.68 
30.55 
80.48 
80.80 
80.18 



D r. 



Ck)6in6. 



D.r. 



9.999404 
.999391 

QQQSfti. 
.•fW004 

.999878 
.999871 
.999864 
.990657 
.999360 
.999848 
.999836 

9.999329 
.999322 
.999815 
.9ov9Uo 
.999801 
.999294 
.999287 
.999279 
.999272 
.999265 

9.990257 
.999250 
.999242 
.999235 
.999227 
.999220 
.999212 
.999205 
.999197 
.999189 

0.999181 
.999174 
.999166 
.999158 
.999150 
.999142 
.999134 
.999126 
.999118 
.999110 

9.999102 
.999094 
.999080 
.999077 
.999069 
.999061 
.999053 

.99903C 
.999027 

9.999019 
.999010 
.999002 
.996993 
.998084 
.998976 
.996967 
.996958 
.998950 

9.996941 



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 
.l.') 
.13 

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



Tang. 



8.719396 
.721806 
.724204 
.796588 
.728959 
.781317 
.793663 
.785996 
.738317 
.740626 
.742922 

8.745207 
.747479 
.749740 
.751969 
.754227 
.756453 
.758668 
.760672 
.763065 
.766246 

8.767417 
.769578 
.771727 
.773866 
.775995 
.778114 
.780222 
.782320 
.784408 
.786486 

8.788554 
.790613 
.792662 
.794701 
.796731 
.798^2 
.800763 
.802765 
.804758 
.806742 

8.808717 
.810683 
.812641 
.814589 
.816529 
.818461 
.820884 
.822298 
.834305 
.826103 

8.827992 
.829874 
.831748 
.833813 
.835471 
.837321 
.839163 
.840998 
.842825 

6.844644 



D.r. 



D. r. I Cotang. 



40.17 
39.97 
39.73 
80.62 
89.80 
89.10 
88.88 
88.68 
88.48 
88.27 
88.06 

87.87 
87.68 
87.48 
87.80 
87.10 
86.92 
86.78 
36.55 
86.35 
36.18 

86.02 
85.82 
85.65 
35.48 
35.32 
35.18 
84.97 
84.80 
84.63 
84.47 

84.82 
84.15 
83.98 
88.83 
83.68 
83.52 
83.87 
33.22 
a3.07 
82.92 

82. 

82. 

32.47 

82.33 

82.20 

32.05 

31.90 

81.78 

31.63 

31.48 

31.37 
31.23 
31.06 
30.97 
30.83 
80.70 
30.58 
80.45 
80.32 



Ootang. 



11.280604 
.276194 
.275796 
.278412 
.271041 
.268683 
.266337 
.264004 
.261683 
.259374 
.257078 

11.254793 
.252521 
.250260 
.248011 
.245773 
.243547 
.241332 
.239128 
.236935 
.234754 

11.282563 
.230422 
.228273 
.226134 
.224005 
.221886 
.219778 
.217680 
.215692 
.213514 

11.211446 
.209887 
.207838 
.205299 
.203269 
.201248 
.199237 
.197236 
.195242 
.193258 

11.191283 
.189317 
.187359 
.185411 
.183471 
.181539 
.179616 
.177702 
.175795 
.173897 

11.172008 
.170126 
.168252 
.166387 
.164529 
.162679 
.160837 
.159002 
.157175 

11.155856 



D. 1'. 



Tang. 



60 
69 
68 
67 
66 
55 
64 
68 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

89 

36 

87^ 

86 2 

86 

84 

38 

82 

81 

80 

29 
26 
27 
26 
26 
24 
28 
22 
21 
20 

19 
18 
17 
16 
16 
14 
13 
12 
11 
10 

9 
8 
7 
6 
6 
4 
3 
2 
1 




w 



107 



COSINES, TANGENTS, AND COTANGENTS. 



175* 



Sine. 






8.848586 


1 


.845387 


2 


.847183 


3 


.848971 


4 


.850751 


5 


.85:2525 


6 


.854291 


7 


.856049 


8 


.857801 


9 


.859546 


10 


.861283 


11 


8.863014 


12 


.864738 


13 


.866455 


14 


.868165 


15 


.869668 


16 


.871565 


17 


.878255 


18 


.874938 


19 


.876615 


20 


.878285 


21 


8.879949 


22 


.881607 


23 


.883258 


24 


.884903 


25 


.886542 


26 


.888174 


27 


.889801 


28 


.891421 


29 


.893035 


ao 


.894643 


31 


8.896246 


82 


.897842 


38 


.899432 


34 


.901017 


35 


.902506 


86 


.904169 


37 


.905736 


38 


.907297 


88 


.906853 


40 


.910404 


41 


8.911949 


42 


.913488 


43 


.915022 


44 


.916550 


45 


.918073 


46 


.919591 


47 


.921103 


48 


.922610 


49 


.924112 


60 


.925609 


51 


8.927100 


52 


.928687 


58 


.980068 


54 


.931544 


55 


.938015 


56 


.934481 


57 


.935942 


58 


.937398 


59 


.938860 


60 


8.940296 



Cosine. 



D. r. 



80.03 
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 
28.06 
27.95 
27/83 
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 
25.85 
25.75 

25.65 
25.57 
25.47 
25.38 
25.30 
25.20 
25.12 

25. as 

24.95 
24.85 

24.78 
24.68 
24.60 
24.52 
24.43 
24.35 
24.27 
24.20 
24.10 



D. r. 



Cosine. 



<7. WVO(l4l 

.996932 
.996923 
.998914 
.99dvII5 
.998896 
.998887 
.996878 
.996869 
.998860 
.996851 

9.996841 
.996832 
.996623 
.998813 
.996804 
.996795 
.998785 
.998776 
.996766 
.996757 

9.998747 
.996738 
.998728 
.998718 
.996706 
.99ou99 
.996689 
.996679 
.99ooov 
.998659 

9.998649 
.998639 
.996629 
.998619 
.996609 
.996599 
.996589 
.996578 
.996568 
.996558 

0.998648 
.998537 
.998527 
.996516 
.998606 

.998485 
.996474 
.998464 
.998453 

9.998442 
.996431 
.998421 
.998410 
.996899 

QQftStflA 

.OVOOOO 

.998377 

.996366 

.996355 

9.998344 



Sine. 



D. r. 



.15 
.16 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.17 

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

.15 
.17 
.17 
.17 
.15 
.it 
.17 
.17 

.11 

.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 



D. r. 



Tang. 



8.844644 
.846455 
.848260 
.850057 
.851846 
.863628 
.655403 
.857171 
.858932 
.860686 
.862433 

8.864173 
.865906 
.867632 
.869351 
.871064 
.872770 
.874469 
.876162 
.877849 
.879529 

8.881202 
.882869 
.884530 
.886185 
.887833 
.889476 
.891112 
.892742 
.894866 
.895984 

8.897596 
.899203 
.900603 
.902896 
.903967 
.906570 
.907147 
.908719 
.910285 
.911846 

8.913401 
.914951 
.916495 
.918084 
.919568 
.021096 
.922619 
.924136 
.925649 
.927156 

8.928658 
.930155 
.931647 
.933134 
.934616 
.936093 
.987565 
.939032 
.9404.94 

8.941952 



D. r. 



80.18 
30.06 
29.95 
29.82 
29.70 
29.58 
29.47 
29.35 
29.23 
29.12 
29.00 

28.88 
28.77 
28.65 
28.55 
28.43 
28.32 
28.22 
28.12 
28.00 
27.88 

27.78 
27.68 
27.58 
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 
25.63 
25.57 
25.47 
25.38 
25.28 
25.22 
25.12 
25.03 

24.95 
24.87 
24.78 
24.70 
24.62 
24.53 
24.45 
24.37 
24.30 



Cotang. 



Cotang. I D. 1". 



ll.:.')5356 


60 


.153545 


50 


.151740 


68 


.149943 


67 


.148154 


66 


.146372 


65 


.144697 


64 


.142829 


58 


.141068 


62 


.139314 


51 


.137567 


.*)0 


11.135827 


49 


.134094 


48 


.132368 


47 


.180649 


46 


.128936 


45 


.127230 


44 


.125631 


43 


.128888 


42 


.122151 


41 


.120471 


40 


11.118798 


19 


.117131 


38 


.116470 


37 


.118815 


36 


.112167 


85 


.110524 


34 


.106888 


38 


.107258 


82 


.105684 


31 


.104016 


30 


11.102404 


29 


.100797 


28 


.099197 


27 


.097602 


26 


.096013 


25 


.094430 


24 


.092853 


23 


.091281 


22 


.089715 


21 


.088154 


20 


11.066599 


19 


.065040 


18 


.088505 


17 


.061966 


16 


.060432 


15 


.078904 


14 


.077381 


13 


.075664 


12 


.074351 


11 


.072844 


10 


11.071842 


9 


.069645 


8 


.068853 


7 


.066866 


6 


.066884 


5 


.068907 


^ 


.062485 


3 


.060968 


2 


.069606 


1 


11.058048 






Tang. 



k 



M* 



108 



TkSLE X. — LOOABITHUIC SINES, 



e- 



COSmiCS, TANGENTS. AND COTANGENTS. 



178* 





1 

2 
8 

4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
10 

ao 

21 
23 
23 
24 
25 
26 
27 
28 
29 
SO 

81 
32 
33 
34 
35 
36 
37 
88 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 

51 
62 
58 
54 
65 
56 
67 
68 
69 
60 



Slna 



9.019235 
.020435 
.021632 
.022825 
.0S4016 
.025208 
.026386 
.087567 
.028744 
.029918 
.031069 

9.082257 
.068421 
.084582 
.035741 
.036806 
.038048 
.030197 
.040342 
.041485 
.042625 

9.043782 
.044895 
.046026 
.047154 
.048279 
.049400 
.050519 
.051635 
.052749 
.058850 

9.064966 
.056071 
.057172 
.068271 
.059367 
.060460 
.061551 
.068689 
.063724 
.064806 

9.066885 
.066962 
.068086 
.069107 
.070176 
.071242 
.072806 
.078366 
.074424 
.0754B0 

9.076633 
.077588 
.078631 
.079676 
.080719 
.081759 
.062797 



.064864 
9.065804 



Cosine. 



D. r. 



20.00 
19.96 
19.88 
1&.85 
19.78 
19.72 
19.68 
19.62 
19.57 
19 52 
19.47 

19.40 
19.35 
19.32 
19.25 
19.20 
19.15 
19.06 
19.05 
19.00 
18.95 

18.88 
18.85 
18.80 
18.75 
18.68 
18.65 
18.60 
18.57 
18.50 
18.45 

18.42 
18.35 
18.32 
18.27 
18.S22 
18.18 
18.13 
18.06 
18.03 
17.98 

17.98 
17.90 
17.85 
17.82 
17.77 
17.78 
17.67 
17.68 
17.60 
17.55 

17.50 
17.47 
17.42 
17.88 
17.38 
17.30 
17.25 
17.20 
17.17 



D. r. 



Cosine. 



9.997614 
.997601 
.997588 
.997574 
.997561 
.997547 
.997534 
.997520 
.997607 
.997493 
.997480 

9.997466 
.997452 
.997439 
.997425 
.997411 
.997897 
.097883 
.997369 
.99^955 
.99*^341 

9.997827 
.997313 
.997299 
.997285 
.997271 
.997257 
.997242 
.997228 
.997214 
.997199 

9.997185 
.997170 
.997156 
.997141 
.997127 
.997112 
.997098 
.997063 
.907068 
.997063 

9.997089 
.997024 
.997009 
.996994 
.996979 
.996964 
.990949 
.996934 
.996919 
.996904 

9.996889 
.996674 
.996a58 
.996843 
.996828 
.996812 
.996797 
.990782 
.996766 

9.996751 



Sine. 



D. r. 



.22 
.22 
.23 

!23 
.22 
.23 
.22 
.23 

!23 

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

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

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

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

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



D. r. 



Tang. 



9.021620 
.022834 
.024044 
.025251 
.026455 
.027655 
.028852 
.030046 
.081237 
.032425 
.033609 

9.034791 
.035969 
.037144 
.038316 
.039485 
.040651 
.041818 
.042978 
.044130 
.045284 

9.046484 
.047582 
.048727 
.049669 
.051006 
.052144 
.053277 
.054407 
.055585 
.056659 

9.057781 
.058900 
.060016 
.061130 
.062240 
.063348 
.064453 
.066556 
.066655 
.067762 

9.068846 
.009936 
.071027 
.072113 
.078197 
.074278 
.075356 
.076432 
.077505 
.078576 

9.079644 
.060710 
.061773 
.062883 
.083691 
.064947 
.066000 
.067050 

0.069144 



Cotang. 



D. r. 



Cotang. 



20.23 

20.17 

20.12 

20.07 

20.00 

9.95 

9.90 

9.85 

9.80 

9.78 

9.70 

9.63 
9.68 
9.63 
9.48 
9.43 
9.37 
9.33 
9.28 
9.23 
9.17 

9.13 
9.06 
9.03 
8.96 
8.93 
8.88 
883 
8.80 
8.73 
8.70 

8.65 
8.60 
8.57 
8.60 
8.47 
8.42 
8.88 
8.82 
8.28 
8.25 

8.20 

8.15 
8.10 
8.07 
8.t» 
7.97 
7.93 
7.88 
7.86 
7.80 

7.77 



72 
67 
68 
60 
56 
60 
47 



7.43 



D. r. 



10.978880 
.977166 
.975966 
.974749 
.973545 
.972345 
.971148 
.969954 
.966763 
.067575 
.966891 

10.066200 
.064031 
.962866 
.061684 
.060515 
.060840 
.058187 
.057027 
.066870 
.054716 

10.058666 
.062418 
.051273 
.050131 
.048002 
.047866 
.046723 
.045503 
.044465 
.043341 

10.042210 
.041100 
.989064 
.038870 
.037760 



.035647 

OStAAAd. 

.033845 
.082248 

10.931154 
.080062 
.028078 
.927887 
.926808 
.925722 
&24A44 
.028668 
.922496 
.921424 

10.920856 
.919290 
.918227 
.917167 
.916109 
.916053 
.914000 
.912960 
.011902 

10.910666 



Tang. 



60 
69 
58 
67 
66 
65 
64 
53 
52 
61 
60 

49 
48 
47 
46 
46 
44 
43 
42 
41 
40 

89 
38 
87 
36 
35 
34 
88 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

19 
18 
17 
16 
16 
14 
18 
12 
11 
10 

9 

8 
7 
6 
6 
4 
8 
2 
1 




8e^ 



110 



TABLE X. — LOGARITHMIC SIKES, 



172* 



Sine. 




1 
8 
3 
4 
5 
6 
7 
8 
9 
10 

11 
12 
18 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
80 

81 
82 
88 
84 
85 
86 
87 
88 
89 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 

51 
52 
63 
64 
65 
66 
67 
68 
69 
60 



9065894 
.086922 
.067947 
.068970 
.060990 
.091006 
.002024 
.009087 
.094047 
.096066 
.096062 

9.097065 
.096066 
.099065 
.100062 
.101056 
.102048 
.108037 
.104025 
.106010 
.105992 

9.106973 
.107951 
.108927 
.100901 
.110678 
.111842 
.112809 
.113774 
.114737 
.115698 

9116656 
.117613 
118667 
.119519 
.120460 
.121417 
.122862 
.128306 
.124248 
.125187 

0.126125 
.127060 
.127993 
.128925 
.129654 
130781 
.131706 
.132630 
.133551 
.134470 

9 135387 
.136303 
.187216 
.138128 
.189037 
.139944 
140660 
.141754 
.142655 

9143555 



D. 1'. 



17.13 
17.08 
17.05 
17.00 
16.97 
16.93 
16.88 
16.83 
16.82 
16.77 
16.72 

16.68 
16.65 
16.62 
16.57 
16.53 
16.48 
16.47 
16.42 
16.37 
16.35 

16.30 
16.27 
16.23 
16.20 
16.15 
16.12 
16.06 
16.05 
16.02 
15.97 

15.95 
15.90 
15.87 
15.83 
15.80 
15.75 
15.73 
15.70 
15.65 
15.63 

15 58 
15.55 
15 53 
15.48 
15 45 
15.42 
15.40 
15 35 
15.32 
15.28 

15.27 
15 22 
15.20 
15 15 
15.12 
15.10 
15.07 
15.02 
15.00 



Cosine. 



Cosine. 



D. r. 



9.996751 
.996735 
.996720 
.996704 
.996688 
.996673 
.996657 
.996641 
.996625 
.996610 
.996594 

9.996578 
.996562 
.996546 
.996530 
.996514 
996498 
.996482 
.996465 

.996433 

9.996417 
.996400 
.996384 
.996368 
.996351 
.996335 
.996318 
.996302 
.996285 
.906269 

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

0.996063 
.096066 
.996049 
.996032 
.996015 
.995998 
.995960 
.995963 
.995946 
.995928 

d 995911 
.995894 
.995876 
.995859 
.995841 
.995823 
.995806 
.995788 
996771 

9.995758 



D. r. 



Sine. 



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

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

.28 
.27 
.27 
.•28 
.27 
.28 
.27 
.28 
.27 
.28 

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

.28 
.28 
.26 
.28 
.28 
.30 
.28 
.28 
.80 
.28 

.28 
.30 
.28 
80 
.80 
.28 
.30 
28 
.30 



Tang. 



D. r 



D. r. 



9.069144 
.090167 
.091228 
.092266 
.093802 
.094386 
.096367 
.096395 
.09r422 
.098446 
.099468 

9.100487 
.101504 
.102519 
.108532 
.104542 
.105560 
.106556 
.107559 
.106660 
.109559 

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

9.120404 
.121377 
.122348 
.123317 
.124284 
.125249 
.126211 
.127172 
.128130 
.129067 

9.130041 
.130994 
.131944 
.132893 
.133839 
.134784 
.136726 
.136667 
.137605 
.138542 

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

9.147903 



Cotang. 



17.38 
17.35 
17.30 
17.27 
17.23 
17.18 
17.13 
17.12 
17.07 
17.03 
16.98 

16.95 
16.92 
16.88 
16.83 
16.80 
16.77 
16.72 
16.68 
16.65 
16.62 

16.58 

16.53 

16.50 

16.47 

16.48 

16.40. 

16.35 

16.33 

16.28 

16.25 

16.22 
16.18 
16.15 
16.12 
16.06 
16.03 
1602 
15.97 
15.95 
15.90 

15.88 
15.83 
15.82 
15.77 
15.75 
15.70 
15.68 
15.63 
15 62 
15.57 

15.55 
15.52 
15.48 
15.45 
15.42 
15.38 
15 37 
15.32 
15.30 



Cotang. 



D. r. 



10.910666 
.9U9ol3 
.908772 
.907784 
.906698 
.905664 
.904633 
.903605 
.902578 
.901554 
.900532 

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

10.889444 
.888449 
.887457 
.886467 
.885479 
.884493 
.883509 
.882528 
.881548 
.880671 

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

10.869059 
.869006 
.868056 
.867107 
.866161 
.865216 
.864274 
863333 
.862305 
.861458 

10 860524 
.859591 
.868660 
.857731 
.856804 
.855679 
.854056 
.864034 
.853115 

10.852197 



Tang. ' 



60 
50 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
37 
36 
35 
34 
38 
32 
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 
8 
2 
1 




VI* 



111 



Ji 



9* 



TABLE X. — LOGARITHMIC SINES, 



iro« 



Sine. 






9.194382 


1 


.195129 


2 


.195825 


3 


.196719 


4 


.197511 


5 


.198802 


6 


.199091 


7 


.199679 


8 


.200666 


9 


.201451 


10 


• M^/WWOrB 


11 


9.203017 


12 


.2U8797 


18 


.204577 


14 


.205354 


15 


.206131 


16 


.206906 


17 


.20r679 


18 


.206452 


19 


.209222 


ao 


.209998 


21 


9.210760 


22 


.211526 


23 


.212291 


24 


.218065 


25 


.213818 


26 


.214579 


27 


.215338 


28 


.216097 


29 


.216854 


80 


.217609 


81 


9.218368 


82 


.219116 


88 


.219868 


84 


.220618 


35 


.221867 


86 


.222115 


87 


.222861 


88 


.228606 


39 


.224849 


40 


.225002 


41 


9.225838 


42 


.226578 


48 


.227311 


44 


.228048 


45 


.228784 


46 


.229518 


47 


.280252 


48 


.230984 


49 


.23in5 


50 


232444 


61 


9.233172 


52 


.233899 


58 


.234625 


54 


.235849 


55 


.236073 


66 


.286795 


57 


.287615 


58 


.238235 


60 


.238958 


60 


9.239670 



Cosine. 



D. r. 



13.28 
13.27 
13.23 
13.20 
13.18 
18.15 
18.13 
13.12 
13.06 
18.05 
18.05 

13.00 
13.00 
12.95 
12.95 
12.92 
12.88 
12.88 
12.83 
12.88 
12.80 

12.77 
12.75 
12.73 
12.72 
12.68 
12.65 
12.65 
12.62 
12.58 
12.57 

12.55 
12.53 
12.50 
12.48 
12.47 
12.43 
12.42 
12.38 
12.38 
12.85 

12.33 
12.30 
12.28 
12.27 
12.23 
12.23 
12.20 
12.18 
12.15 
12.13 

12.12 
12.10 
12.07 
12.07 
12.03 
12.00 
12.00 
11.97 
11.95 

D. r. 



Cosine. 



9.994620 
.994600 
.994580 
.994560 
.994540 
.994519 
.994499 
.994479 
.994459 
.994438 
.994418 

9.994398 
.994877 
.994357 
.904886 
.994316 
.994205 
.994274 
.994254 
.994233 
.994212 

9.994191 
.994171 
.994150 
.994129 
.994108 
.994067 
.994066 

.994024 
.994003 

9.993962 
.993960 
.993939 
.993918 
.998897 
.998875 
.998854 
.993832 
.993811 
.993789 

9.993768 
.993746 
.9937^ 
.993708 
.998681 
.993660 
.998638 
.993616 
.993594 
.993572 

9.993550 
.993528 
.993506 
.998484 
.998462 

.993418 

.998896 

.993374 

9.993351 

Sine. 



D. r. 



09<» 



.33 
.33 
.33 
.33 
.35 
.33 
.33 
.33 
.86 
.33 
.33 

.35 
.33 
.35 
.88 
.85 
.85 
.83 
.85 
.85 
.35 

.83 
.85 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 

.87 
.35 
.35 
.35 
.87 
.35 
.37 
.35 
.37 
.35 

.37 
.35 
.87 
.87 
.35 
.37 
.87 
.37 
.87 
.37 

.87 
.37 
.37 
.87 
.87 
.37 
.37 
.37 
.88 

D. r. 



118 



Tang. 



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

9.206619 
.209420 
.210220 
.211018 
.211815 
.212611 
.213405 
.214198 
.214969 
.215780 

9.216568 
.217356 
.218142 
.218926 
.219710 
.220492 
.221272 

.222830 
.228607 

9.224882 
.225156 
.225929 
.226700 
.227471 
.228239 
.229007 
.229778 
.230589 
.231302 

9.232065 
.232826 
.238586 
.234345 
.235108 
.235859 
.236614 
.237368 
.238120 
.238872 

9.239622 
.240371 
.241118 
.241865 
.242610 
.243354 
.244097 
.244839 
.245579 

9.246319 

Cotang. 



D. r. 



13.60 
13.60 
13.57 
13.53 
13.52 
18.50 
13.47 
13.45 
13.43 
13.40 
13.87 

13.35 
13.88 
13.30 
13.28 
18.27 
18.28 
18.22 
13.18 
13.18 
13.18 

13.18 
13.10 
13.07 
13.07 
18.08 
18.00 
18.00 
12.97 
12.95 
12.92 

12.90 
12.88 
12.85 
12.85 
12.80 
12.80 
12.77 
12.77 
12.72 
12.72 

12.68 
12.67 
12.65 
12.68 
12.60 
12.58 
12.57 
12.53 
12.58 
12.50 

12.48 
12.45 
12.45 
12.42 
12.40 
12.88 
12.87 
12.83 
12.33 

D. r. 



Cotang. 



10.800287 
.799471 
.798655 
.797841 
.797029 
.796218 
.795408 
.794600 
.798798 
.792987 
.792183 

10.791381 
.790580 
.789780 
.788982 
.788185 
.787389 
.786595 
.785802 
.785011 
.784220 

10.783432 
.782644 

.781858 
.781074 
.780200 
.779508 
.778728 
.777948 
.777170 
.776393 

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

10.767935 
.767174 
.766414 
.765655 
.764897 
.764141 
.768386 
.762632 
.761880 
.761128 

10.760378 
.759629 
.758882 
.758135 
.767390 
.756646 
.755903 
.755161 
.754421 

10.753681 

Tang. 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41, 
40 

39 
38 
37 
36 
35 
34 
88 
82 
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 

a 

5 
4 

8 
2 
1 




80« 



il 






1 



I 



12- 



COSINES, TANGENTS. AND COTANGENTS. 



167* 



Sine. 




1 
2 
8 

4 
6 
6 
7 
8 
9 
10 

11 
12 
18 
14 
16 
16 
17 
18 
19 
20 

21 
22 
28 
24 
25 
26 
27 
28 
29 
80 

81 
82 
38 
84 
86 
86 
87 
88 
89 
40 

41 
42 
48 
44 
45 
46 
47 
48 
49 
50 

61 
62 
68 
64 
66 
66 
67 
68 
69 
60 



108* 



9.817879 
.818473 
.819066 
.819668 
.820349 
.820640 
.821430 
.822019 
.822607 
.828194 
.823780 

9.324866 
.824960 
.325634 
.826117 
.826700 
.327281 
.327862 
.328442 
.329021 
.329599 

9.330176 
.880758 
.831329 
.831908 
.382478 
.888051 
.883624 
.384195 
.384767 
.836337 

9.385906 
.836475 
.337048 
.837610 
.338176 
.338742 
.339307 
.889871 
.840434 
.340996 

9.341558 
.342119 
.342679 
.843289 
.843797 
.844355 
.344912 
.345469 
.846024 
.846579 

9.347134 
.347687 
.848240 
.848792 
.849343 
.849893 
.850443 
.350092 
.&51540 

9.852068 



D. r. 



9.90 
9.88 
9.87 
9.85 
9.85 
9.88 
9.82 
9.80 
9.78 
9.77 
9.77 

9.73 
9.78 
9 72 
9.72 
9.68 
9.68 
9.67 
9.65 
9.68 
9.62 

9.62 
9.60 
9.57 
9.68 
9.66 
9.56 
9.52 
9.58 
9.60 
9.48 

9.48 
9.47 
9.45 
9.43 
9.48 
9.42 
9.40 
9.38 
9.37 
9.37 

9.85 
9.38 
9.38 
9.80 
9.80 
9.28 
9.28 
9.25 
9.25 
9.25 

9.22 
9.22 
9.20 
9.18 
9.17 
9.17 
9.15 
9.18 
9.18 



Cosine. D. 1 



Cosine. 



D. r. 



9.990404 
.990378 
.990851 
.990324 
.990297 
.990270 
.990243 
.990216 
.990188 
.990161 
.990134 

9.990107 
.990079 
.990062 
.990025 
.989997 
.989970 
.989942 
.969915 

.969860 

9.960632 
.969604 
.969777 
.969749 
.969721 
.969693 
.969666 
.969637 
.969610 
.969582 

9.960553 
.969525 
.960497 
.969469 
.969441 
.969418 
.969686 
.989856 
.989328 
.969800 

9.969271 
.969248 
.969214 
.969186 
.969157 
.969128 
.969100 
.969071 
.969042 
.969014 

9.9nJoOO 
.9oo960 
.968927 
.968898 

.9tKX}Q9 

.ddbs40 
.968811 
.968782 
.968758 
9.988724 



Sine, 



.43 
.45 
.45 
.45 
.45 
.46 
.47 
.45 
.45 
.45 
.45 

.47 
.46 
.45 
.47 
.45 
.47 
.45 
.47 
.45 
.47 

.47 
.45 
.47 
.47 
.47 
.47 
.47 
.45 
.47 
.48 

.47 
.47 
.47 
.47 
.47 
.47 
.48 
.47 
.47 
.48 

.47 
.48 
.47 
.48 
.48 
.47 
.48 
.48 
.47 
.48 

.48 
.48 
.48 
.48 
.48 
.48 
.48 
.48 
.48 



D. r. 



116 



Tang. 



9.327475 
.328095 
.328715 
.329334 
.329953 
.830570 
.831187 
.331808 
.332418 
.333083 
.833646 

9.334259 
.834871 
.335482 
.336093 
.836702 
.887811 
.387919 
.338627 
.839138 
.339789 

9.340844 
.340948 
.341562 
.342165 
.342757 
.343358 
.843958 
.344568 
.845167 
.346756 

9.346363 
.346949 
.347545 
.848141 
.348735 
.349329 
.849922 
.850514 
.851106 
.351697 

9.862287 
.352876 
.853466 
.354058 
.354640 
.856227 
.365818 
.856898 
.356962 
.357566 

9.858149 
.858731 
.359813 
.359698 
.360474 
.861058 
.861632 
.362210 
.862787 

9.803364 



Cotang. 



D. r. 



10.38 
10.38 
10.32 
10.32 
10.28 
10.28 
10.27 
10.25 
10.26 
10.22 
10.22 

10.20 
10.18 
10.18 
10.15 
10.15 
10.13 
10.13 
10.10 
10.10 
10.08 

10.07 
10.07 
10.05 
10.08 
10.02 
10.00 
10.00 

V. oO 

9.97 
9.97 

9.93 
9.93 
9.93 
9.90 
9.90 
9.88 
9.87 
9.87 
9.86 
9.88 

9.82 
9.82 
9.80 
9.78 
9.78 
9.77 
9.75 
9.73 
9.73 
9.72 

9.70 
9.70 
9.67 
9.68 
9.66 
9.66 
9.63 
9.62 
9.62 



Cotang. 



10.672525 
.671906 
.671285 
.670666 
.670047 
.669480 
.668818 
.668197 
.667582 
.666967 
.666864 

10.666741 
.665129 
.664518 
.668907 
.668296 
.662689 
.662061 
.661478 
.660867 
.660261 

10.659666 
.659052 
.668448 
.667845 
.657248 
.666642 
.656042 
.655442 
.654843 
.654245 

10.653647 
.653061 
.662455 
.651869 
.651266 
.660671 
.660078 
.649486 
.648894 
.648803 

10.647718 
.647124 
.646685 
.646947 
.646860 
.644778 
.644187 
.648602 
.648018 
.642484 

10.641861 
.641269 
.640687 
.640107 
.639526 
.688947 
.636868 
.637790 
.687218 

10.( 



D. 1'. I Tang. 



60 
50 
58 
67 
56 
55 
54 
58 
68 
51 
50 

49 
48 
47 
46 
45 
44 
48 
42 
41 
40 

89 
88 
87 
86 
85 
84 
83 
82 
81 
80 

29 
28 
27 
26 
25 
24 
28 
22 
21 
20 

19 
18 
17 
16 
16 
14 
18 
18 
11 
10 

9 
8 
7 
6 
6 
4 
8 
2 
1 




77* 



1 

I 
1 



1 
1 



14- 



COSINES, TANGENTS, AND COTANGENTB. 



169< 



i 





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 
80 

31 
32 
33 
34 
35 
36 
37 
38 
89 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
66 
57 
58 
59 
60 

10? 



Sine. 



9.383675 
.384182 
.384687 
.385192 
.385697 
.386201 
.386704 
.387207 
.387709 
.388210 
.888711 

9.389211 
.389711 
.390210 
.390706 
.891206 
.391703 
.392199 
.392695 
.393191 
.893685 

9.394179 
.394673 
.395166 
.895658 
.396150 
.896641 
.89n32 
.897621 
.398111 
.898600 

9.899068 
.899575 
.400062 
.400549 
.401035 
.401520 
.402005 
.402489 
.4029?2 
.408155 

9.403938 
.404420 
.404901 
.405383 
.405862 
.406341 
.406820 
.407299 
.407777 
.406254 

9.408731 
.409207 
.409682 
.410157 
.410632 
.411106 
.411579 
.412052 
.412524 

9.412996 



Cosine. 



D. r. 



8.45 
8.42 
8.42 
8.42 
8.40 
8.38 
8.88 
8.37 
8.35 
8.35 
8.33 

8.38 
8.32 
8.30 
8.30 
8. 28 
8.27 
8.27 
8.27 
8.23 
8.23 

8.23 
S.22 
8.20 
8.20 
8.18 
8.18 
8.15 



8. 
8. 



17 

15 



8.13 

8.12 
8.12 
8.12 
8.10 
8.08 
8.08 
8.07 
8.05 
8.05 
8.05 

8.03 
8.02 
8.02 
8.00 



7, 
7, 
7, 
7. 
7. 
7. 



t 

7 

4 

7 
7 

7, 

7, 



98 
98 
98 
97 
95 
95 

.93 
.93 
.92 

.92 
.90 
,00 
88 
87 
87 



D. r 



Cosine. 



9.966904 
.966873 

. Vouo41 

.986809 
.966778 
.966746 
.966714 
.986683 
.986651 
.966619 
.986587 

9.966655 
.966523 
.966491 

.9te427 
.966395 
.966368 
.966331 
.986299 
.986266 

9.086234 
.966202 
.966169 
.966137 
.966104 
.96607^ 
.966039 
.986007 
.985974 
.985942 

9.965909 
.985876 
.985843 
.985811 
.985778 
.985745 
.985712 
.965679 
.985646 
.985613 



9 



985580 
985547 
.965514 
.985480 
.985447 
.985114 
.985381 
.985347 
.985314 
.985280 

9.965247 
.985213 
.985180 
.965146 
.965113 
.985079 
.965045 
.985011 
.964978 

9.964944 



Sine. 



D. r 



.52 
.53 
.53 
.62 
.53 
.58 
.52 
.58 
.53 
.53 
.53 

.53 
.53 
.58 
.53 
.53 
.53 
.53 
.53 
.55 
.53 

.58 
.55 
.53 
.55 
.53 
.55 
.53 
.55 
.53 
.55 

.55 
.55 
.53 
.55 
.55 
.5) 
.55 
.55 
.55 
.55 

.55 
.55 
.57 
.55 
.55 
.55 
.57 
.55 
.57 
.55 

.57 
.55 
.57 
.55 
.57 
.57 
.57 
.55 
.57 



Tang. 



9 



9 



9 



9 



9 



9 



9 



996771 
897809 
397846 
896383 
396919 
899455 
399990 
400524 
401068 
401591 
402124 

402656 
408187 
408718 
404249 
404778 
405306 
405836 
406364 
406892 
407419 

407945 
406471 
406996 
409521 
410045 
410569 
411092 
411615 
412137 
412658 

413179 
413699 
414219 
414738 
415257 
415775 
416293 
416810 
417326 
417842 

418358 
418873 
419387 
419901 
420415 
420927 
421440 
421952 
422463 
422974 

428484 
423993 
424503 
425011 
425519 
426027 
426534 
427041 
427547 
428052 



D. r. 



8.97 

8.96 
8.96 
8.98 
8.98 
8.92 
8.90 
8.90 
8.88 
8.88 
8.87 

8.85 
8.86 
8.86 
8.82 
8.88 
8.80 
8.80 
8.80 
8.78 
8.77 

8.77 
8.76 
8.75 
8.78 
8.78 
8.72 
8.72 
8.70 
8.68 
8.68 

8.67 
8.67 
8.66 
8.66 
8.63 
8.68 
8.62 
8.60 
8.60 
8.60 

8.58 
8.57 
8.67 
8.57 
8.56 
8.65 
6.58 
8.52 
8.52 
8.50 

8.48 
8.50 
8.47 
8.47 
8.47 
8.45 
8.46 
8.48 
8.42 



D. 1-. Cotang. D. 1'. 



118 



Cotang. 



10.608229 
.602691 
.602154 
.601617 
.601061 
.600545 
.600010 
.599476 
.596942 
.598409 
.697876 



10 



10 



10 



10 



10 



10 



597344 
696813 
696282 
595751 
595222 
691692 
694164 
698636 
503106 
592581 

692065 
691529 
601004 
500479 
580955 
589431 
588906 
688385 
687863 
587342 

686821 
586801 
585781 
585262 
584748 
584225 
583707 
583190 
582674 



581642 
581127 
580613 
580099 
679585 
579073 
578560 
578048 
577537 
677026 

576616 
576007 
575497 
674969 
574481 
578973 
673466 
572959 
572453 
571948 



Tang. 



60 

50 

58 

57 

56 

55 

54 

53 

52 

51 

60 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

89 
88 
87 
36 
35 
34 
33 
83 
81 
80 

29 
88 
27 
26 
26 
24 
28 
22 
21 



582168 20 



19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 
7 
6 
5 
4 
3 
2 
1 




76' 



15* 



TABLE X. — LOGARITHMIC BINES, 



184' 



Sine. 



D. r. 




1 
8 
8 

4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
SO 

81 
22 
23 
24 
25 
26 
27 
28 
29 
80 

81 
82 
S3 
34 
85 
86 
87 
88 
89 
40 

41 
42 
48 
44 
45 
46 
47 
48 
49 
60 

51 
58 
53 
54 
55 
56 
57 
68 
59 
60 



9.412996 
.413467 
.418938 
.414408 
.414878 
.416847 
.415815 
.416283 
.416751 
.417217 
.417884 

9.418150 
.418615 
.419079 
.419544 
.420007 
.420470 
.420933 
.421895 
.421857 
.422818 

9.422778 
.423238 
.423697 
.424156 
.424615 
.425073 
.425530 
42S987 
.426443 
.426899 

9.427354 

.427809 
.428263 
.428717 
.429170 
.429623 
.430075 
.430527 
.430978 
.431429 

9.431879 

.'432778 
.433226 
.433675 
.434122 
.434569 
.435016 
.435462 
.485906 

9.486353 
.436798 
.487242 
.437686 
.488129 
.488572 
>I38014 
.430456 
.430697 

9.440638 



' I CcMBine. 



7.85 
7.85 
7.83 
7.83 

7.82 
7.80 
7.80 
7.80 
7.77 
7.78 
7.77 

7.75 
7.73 
7.75 
7.72 
7.72 
7.72 
7.70 
7.70 
7.68 
7.67 

7.67 
7.65 
7.65 
7.65 
7.63 
7.62 
7.62 
7.60 
7.60 
7.58 



58 
57 
57 
55 
55 
S3 
53 
52 
7.52 
7.50 

7.50 
7.48 
7.47 
7.48 
7.45 
7.45 
7.45 
7.43 
7.43 
7.42 

7.48 
7.40 
7:40 
7.38 
7.38 
7.87 
7.37 
7.85 
7.85 



D. r. 



Cosine. 



D. 1'. 



Tang. 



9tiMtkAA 

.984910 
.984876 
.084842 
.084806 
.984774 
.984740 
.984706 
.984672 
.984638 
.984603 

9.964569 
.984535 
.964500 
.984466 
.984432 
.964397 
.984368 
.964328 
.984294 
.984259 

9.964224 
.984190 
.984155 
.984120 
.984085 
.964050 
.984015 
.OoSool 
.988946 
.968911 

9.96S875 
.968840 
.068805 
.988770 
.963735 
.983700 
.988664 
.968629 
.983594 
.983558 

9.988523 
.963487 
.988452 
.963416 
.983381 
.9833i5 
.983309 
.968273 
.968238 
.968202 

9.968166 
.968130 

• WSWv4 

.968058 
.968022 
.962986 
.982950 
.962914 
.982878 
9.982842 



Sine. 



.67 
.67 
.57 
.57 
.57 
.57 
.67 
.57 
.67 
.68 
.57 

.67 
.58 
.57 
.57 
.58 
.67 
.58 
.57 
.58 
.58 

.57 
.58 
.58 
.58 
.68 
.68 
.57 
.58 
.58 
.60 

.58 
.58 
.68 
.68 
.58 
.60 
.58 
.68 
.60 
.58 

.60 
.58 
.60 
.68 
.60 
.60 
.60 
.68 
.60 
.60 

.60 
.60 
.60 
.60 
.60 
.60 
.60 
.60 
.60 



9.428052 
.428558 
.429062 
.429566 
.430070 
.430573 
.431075 
.431577 
.432079 
.482580 
.433060 

9.433680 
.434060 
.434679 
.435078 
.435576 
.436073 
.436670 
.437067 
.437663 
.488059 

9.488554 
.439048 
.439543 
.440036 
.440529 
.441022 
.441514 
.442006 
.442497 
.442988 

9.448479 
.448968 
.444458 
.444947 
.445435 
.445923 
.446411 
.446898 
.447384 
.447870 

9.448356 
.448841 
.449326 
.449610 
.450294 
.450777 
.451260 
.451743 
.452225 
.452706 

9.453187 
.453668 
.454148 
.454628 
.455107 
.456586 
.456064 
.456542 
.457019 

9.457496 



105- 



D. 1'. 1 1 Cotang. 



119 



D. r. 



8.43 
8.40 
8.40 
8.40 
8.38 
8.37 
8.87 
8.87 
8.85 
8.83 
8.83 

8.88 
8.32 
8.32 
8.30 
8.28 
8.28 
8.28 
8.27 
8.27 
8.25 

8.23 
8.25 
8.22 
8.22 
8.22 
8.20 
8.20 
8.18 
8.18 
8.18 

8.15 
8.17 
8.15 
8.13 
8.13 
8.13 
8.18 
8.10 
8.10 
8.10 

8.08 
8.08 
8.07 
8.07 
8.05 
8.05 
8.06 
8.08 
8.02 
8.02 

8.02 
8.00 
8.00 
7.98 
7.98 
7.97 
7.97 
7.95 
7.95 



Cotang. 



D. r. 



10.571948 
.571442 
.570938 
.570434 
.669930 
.569427 
.668025 
.568423 
.567921 
.567420 
.566920 

10.566420 
.666920 
.665421 
.664922 
.664424 
.563927 
.563430 
.662938 
.662437 
.561941 

10.561446 
.560952 
.560457 
.569964 
.569471 
.558978 
.568486 
.557994 
.657603 
.557012 

10.556521 
.556032 
.655542 
.555053 
.554565 
.554077 
.563589 
.658102 
.552616 
.552130 

10.551644 
.551159 
.550674 
.650190 
.549706 
.649223 
.548740 
.548257 
.547775 
.517294 

10.546813 
.546332 
.545852 
.545872 
.544893 
.544414 
.543936 
.543458 
.542981 

10.542504 



Tang. 



60 

59 
58 
57 
56 
55 
54 
53 
52 
61 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

89 
38 
87 
36 
35 
34 
38 
82 
81 
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 
8 
2 
1 




74 



16« 



COSINES, TANGENTS, AND COTANGENTS. 



163< 



Sine. 




1 
2 
8 
4 
6 
6 
7 
8 
9 
10 

11 
12 
18 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
80 

81 
82 
88 
84 
86 
86 
87 
88 
89 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

61 
52 
53 
54 
55 
56 
57 
58 
59 
60 



9.440888 
.440778 
.441218 
.441658 
.412096 
.442535 
.442973 
.448410 
.448847 
.444284 
.444720 

9.445155 
.445590 
.446025 
.446459 
.446898 
.447826 
.447159 
.448191 
.448623 
.449054 

9.449485 
.449915 
.450345 
.450775 
.451204 
.451682 
.452060 
.452488 
.452915 
.453342 

9.453768 
.454194 
.454619 
.456044 
.455469 
.455898 
.456316 
.456739 
.457162 
.457584 

9.458006 
.458427 
.458848 
.459268 
.459688 
.460108 
.460527 
.460946 
.461364 
.461782 

9.462199 
.462616 
.463082 
.463448 
.463864 
.464279 
.464604 
.465108 
.465522 

9.465935 



D. r. 



' I Cosine. 



7.83 
7.83 
7.33 
7.30 
7.82 
7.80 
7.28 
7.28 
7.28 
7.27 
7.25 

7.25 
7.25 
7.23 
7.23 
7.28 
7.28 
7.20 
7.20 
7.18 
7.18 

7.17 
7.17 
7.17 



15 
13 
13 
13 

12 
12 



7. 
7. 

7. 
7. 
7. 



7.10 

7.10 
.06 
.06 
.08 
.07 
.06 
.06 
7.05 
7.03 
7.03 

7.02 
7.02 
7.00 
7.30 
7.00 
6.98 
6.96 
6.97 
6.97 
6.95 

6.95 
6.93 
6.93 
6.98 
6.92 
6.-92 
6.90 
6.90 
6.88 



Cosine. 



D. 1" 



9.982842 
.962805 
.982769 
.962733 
.982696 
.982660 
.982624 
.982587 
.982551 
.982514 
.968477 

9.982441 
.982404 
.982367 
.982331 
.982294 
.968257 
.988820 
.982188 
.982146 
.982109 

9.982072 
.988085 
.981996 
.981961 
.981924 
.981886 
.981849 
.961812 
.981774 
.981737 

9.981700 
.981662 
.981625 
.981587 
.981549 
.981512 
.981474 
.981436 
.981399 
.981361 

0.981388 
.981885 
.981847 
.981809 
.981171 
.981183 
.981095 
.961057 
.981019 
.980961 

9.960948 
.960904 

. vOUODO 

.980827 
.960789 
.980760 
.980712 
.960673 
.960685 
9.980596 



D. r. 



.62 
.60 
.60 
.62 
.60 
.60 
.62 
.60 
.62 
.62 
.00 

.62 
.62 
.60 
.62 
.62 
.68 
.62 
.62 
.68 
.62 

.62 
.62 
.62 
.62 
.63 
.62 
.62 
.63 
.62 
.62 

.63 
.62 
.63 
.63 
.62 
.63 
.63 
.62 
.63 
.63 

.63 
.68 
.68 
.63 
.63 
.63 
.63 
.63 
.63 
.65 

.63 
.63 
.65 
.63 
.65 
.68 
.65 
.63 
.65 



Tang. 



IW 



Sine. D. 1". 



120 



9.457496 
.457978 
.458449 
.458925 
.459400 
.459675 
.460849 
.460823 
.461297 
.461770 
.462242 

9.462715 
.468186 
.463658 
.464128 
.464599 
.466069 
.465530 
.466006 
.466477 
.466946 

9.467418 
.467880 
.468847 
.468814 
.469280 
.469746 
.470811 
.470676 
.471141 
.471605 

9. 4^^2069 
.472532 
.472995 
.478457 
.473919 
.474881 
.474848 
.475808 
.475763 
.476223 

9.476688 
.477142 
.477601 
.478059 
.478517 
.478975 
.479432 
.479889 
.480345 
.480601 

9.481257 
.481718 
.488167 
.488681 
.488075 
.488689 
.488968 
.484486 
.484887 

9.485889 



D. r. 



Cotang. 



7.95 
7.93 
7.93 
7.98 
7.98 
7.90 



7. 
7. 
7, 

7. 

7. 



90 
90 

88 
87 
88 



7.85 

7.87 



83 
85 
83 
83 
88 
7.88 
7.80 
7.80 



i 

7. 
7. 
7. 
7. 



.78 
.78 
.78 
.77 
.77 
7.75 
7.75 
7.75 
7.73 
7.73 

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 
7.68 
7.68 
7.68 
7.68 
7.60 
7.60 
7.60 

7.58 
7.58 
7.57 
7.57 



7, 
7, 
7, 
7, 

7. 



57 
55 
55 
58 
53 



Cotang. 



10.542504 
.548087 
.541551 
.541076 
.540600 
.540125 
.539651 
.539177 
.538703 
.538830 
.587758 

10.637285 
.536814 
.536848 
.535878 
.585401 
.534931 
.584461 
.583998 
.533583 
.533056 

10.588587 
.588180 
.581658 
.581186 
.580780 
.580854 
.680789 
.589384 
.588859 
.588895 

10.527981 
587468 
.587005 
.526548 
.586061 
.585619 
.525158 
.584697 
.584887 
.583777 

10.688817 
.582858 



.521041 
.581488 
.681025 
.580568 
.680111 
.510666 
.510100 

10.518748 
.518888 
.517888 
.517870 
.516985 
.516471 
.516018 
.515Sfi6 
.515118 

10.614661 



60 
50 
58 
57 
56 
55 
54 
53 
58 
51 
50 

40 
48 
47 
46 
45 
44 
43 
48 
41 
40 

80 
88 
87 
86 
35 
84 
83 
88 
31 
80 

80 
88 
27 
86 
25 
84 
28 
28 
81 
80 

19 
18 
17 
16 
15 
14 
18 
18 
11 
10 


8 
7 
6 
5 
4 
8 
8 
1 




D. r. i Tang. I ' 



78« 



TABLE X. — LOOABITHMIC 8IKBB, 



1 



0. r. I Cosine. D. Y 



I 
I 



18< 



COSINES, TANGENTS, AND COTANGENTS. 



161< 



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 

81 
82 
83 
34 
85 
86 
87 
88 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 

51 
52 
53 
54 
66 
56 
57 
68 
69 
60 



lOS- 



9.489982 
.490371 
.490759 
.491147 
.491535 
.491922 
.492308 
.492695 
.493081 
.493406 
.493851 

9.494236 
.494621 
.495005 
.495388 
.495772 
.496154 
.496537 
.496919 
.497301 
.497682 

9.498064 
.498444 
.498825 
.499204 
.499584 
.499963 
.50aW2 
.500721 
.501099 
.501476 

9.501854 
.502231 
.502607 
.502984 
.503360 
.503735 
.504110 
.504485 
.504860 
.505234 

9.5a5608 
.505981 
.506354 
.506727 
.507099 
.507471 
.507843 
.508214 
.508585 
.606956 

9.509326 
.509696 
.510065 
.510434 
.510603 
.511172 
.511640 
.611907 
.612276 

9.512642 



D. 1". 



Cosine. 



Cosine. 



6.48 
6.47 
6.47 
6.47 
6.45 
6.43 
6.45 
6.43 
6.42 
6.42 
6.42 

6.42 
6.40 
6.38 
6.40 
6.37 
6.38 
6.37 
6.87 
6.35 
6.35 

6.33 
6.36 
6.32 
6.38 
6.82 
6.82 
6.32 
6.80 
6.28 
6.30 

6.28 
6.27 
6.28 
6.27 
6.25 
6.25 
6.25 
6.26 
6.28 
6.28 

6.22 
6.22 
6.22 
6.20 
6.20 
6.20 
6.18 
6.18 
6.18 
6.17 

6.17 
6.16 
6.16 
6.16 
6.15 
6.13 
6.12 
6.13 
6.12 



D. r 



9.978206 
.978165 
.978124 
.978083 
.978042 
.978001 
.977959 
.977918 
.97r877 
.977885 
.977794 

9.977758 
.977711 
.977669 
.977628 
.977586 
.97r544 
.977508 
.977461 
.977419 
.977377 

9.977335 
.977293 

.977251 
.977209 
.977167 
.977125 
.977083 
.977041 
.976990 
.976957 

9.976914 
.976872 
.976830 

' .976787 
.976745 
.976702 
.976660 
.976617 
.976574 
.976582 

9.976489 
.976446 
.976404 
.976861 
.976318 
.976275 
.976282 
.976189 
.976146 
.976108 

9.976060 
.976017 
.975974 
.975980 
.975867 
.975844 
.975800 
.975757 
.975714 

9.975670 



D. r. 



Tang. 



.68 
.68 
.68 
.68 
.68 
.70 
.68 
.68 
.70 
.68 
.70 

.68 
.70 
.68 
.70 
.70 
.68 
.70 
.70 
.70 
.70 

.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.72 

.70 
.70 
.72 
.70 
.72 
.70 

.70 
.72 

.72 
.70 
.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 

.72 
.72 
.78 
.72 
.72 
.73 
.72 
.72 
.78 



Sine. 



D. r 



9.511776 
.512206 
.512635 
.513064 
.518493 
.513921 
.514349 
.514777 
.515204 
.515631 
.516057 

9.516484 
.516910 
.517385 
.517761 
.518186 
.518610 
.519064 
.519458 
.519882 
.520305 

9.620728 
.521151 
.521578 
.521995 
.522417 
.522838 
.523209 
.523680 
.524100 
.524520 

9.524940 
.525359 
.525778 
.526197 
.526615 
.527088 
.527461 
.527868 
.528285 
.528702 

9.529119 
.529535 
.529951 
.530866 
.530781 
.581196 
.531611 
.632025 
.532489 
.532858 

9.538266 
.533679 
.534092 
.584604 
.634916 
.536828 
.635739 
.536160 
.536561 

9.536972 



D. 1'. 



Cotang. 



7.17 
7.15 
7.15 
7.15 



7. 
7. 
7. 
7. 
7. 
7, 
7. 



13 
13 
13 
12 
12 
10 
12 



7.10 
7.08 
7.10 
7.08 
7.07 
07 
07 
07 
06 
7.05 

7.05 
7.08 
7.03 
7.08 
7.02 
02 
,02 
00 
00 
00 



6.98 
6.98 
6.98 
6.97 
6.97 
6.97 
6.96 
6 95 
6.96 
6.95 

6.93 
6.98 
6.92 
6.92 
6.92 
6.92 
6.90 
6.90 
6.90 
6.88 

6.88 
6.88 
6.87 
6.87 
6.87 
6.86 
6.86 
6.85 
6.85 



Cotang. I D. 1'. 



10.488224 
.487794 
.487365 
.486936 
.486607 
.486079 
.485651 
.485223 
.484796 
.484369 
.483943 

10.483516 
.488090 
.482665 
.482239 
.481814 
.481890 
.480966 
.480542 
.480118 
.479695 

10.479272 
.478849 
.478427 
.478005 
.477583 
.477162 
.476741 
.476820 
.475900 
.475480 

10.475060 
.474641 
.474222 
.478803 
.473385 
.472967 
.472649 
.472182 
.471716 
.471296 

10.470681 
.470466 
.470049 
.469684 
.469219 
.468804 
.468389 
.467975 
.467561 
.467147 

10.466784 
.466821 
.466906 
.466496 
.466064 
.464673 
.464261 
.468860 
.468488 

10.468028 

Tang. 



122 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
48 
42 
41 
40 

39 
88 
37 
36 
85 
84 
33 
32 
31 
90 

29 
2S 
27 
26 
25 
24 
23 
22 
21 
20 

19 
18 
17 
16 
16 
14 
18 
12 
11 
10 

9 
8 
7 
6 
6 
4 
8 
2 
1 




71* 



19« 



TABLE X. — LOGARITHMIC SINES, 



160- 



Sine. 




1 
2 
3 
4 
5 

7 
8 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
80 

81 
82 
33 
84 
85 
86 
87 
88 
89 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 

61 
52 
63 
54 
65 
66 
67 
68 
69 
60 



WW 



9.512642 
.513009 
.513375 
.513741 
.514107 
.514472 
.514837 
.515202 
.515566 
.515930 
.516294 

9.5166.57 
.5l70a0 
.517382 
.517745 
.518107 
.518468 
.518829 
.519190 
.519551 
.519911 

9.520271 
.520631 
.520990 
.521349 
.521707 
.522066 
.622424 
.522781 
.523138 
.523495 

9.523852 
.624206 
.624664 
.524920 
.625275 
.625630 
.525984 
.526389 



D. 1-. 



.6S7046 

9.527400 
.527758 
.528105 
.628468 
.528810 
.529161 
.529518 
.529664 
.530215 
.530565 

9.530915 
.531265 
.531614 
.531963 
.532312 
.532661 
.533009 
.533857 
.533704 

9.684052 

Cosine. 



6.12 
6.10 
6.10 
6.10 
6.U8 
6.08 
6.08 
6.07 
6.07 
6.07 
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 
6.98 
6.97 
6.98 
.97 
.95 
.95 
.95 
.95 



6. 
6. 
5. 
5. 
5. 



Cosine. 



6.93 
6.93 



5. 
5. 
6. 
5. 
5. 



.93 
.92 
.92 
.90 
.92 
5.90 
6.88 
6.90 

6.88 
5.87 
5.88 
5.87 
5.85 
6.87 
6.85 
5.85 
5.83 
5.83 

5.83 
5.82 
5.82 
5.82 
5.82 
6.80 
6.80 
6.78 
6.80 

D. r. 



9 



9 



9 



9 



9 



9 



9 



975670 
975627 
975583 
975539 
975496 
975452 
975406 
975365 
975321 
975277 
975233 

975189 
975145 
975101 
975057 
975013 
974969 
974925 
974880 
974836 
974792 

974748 
974703 
974659 
974614 
974570 
974525 
974481 
974436 
974391 
974347 

974302 
97425? 
974212 
974167 
974122 
974077 
974032 
973967 
973942 
973897 

973862 
973807 
973761 
973716 
973671 
973625 
973580 
973535 
973489 
973444 

973396 
973352 
973307 
973261 
978215 
973169 
973124 
973078 
978032 
972966 

Sine. 



D. r. 



.72 
.73 
.73 
.72 
.73 
.73 
.72 
.73 
.73 
.73 
.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. 



9.536972 
.587382 
.537792' 
.538202 
.538611 
.539020 
.539429 
.539837 
.540245 
.540653 
.541061 

9.541468 
.5418';5 
.542281 
.542688 
.543094 
.543499 
.543905 
.544310 
.544715 
.545119 

9.545524 
.545928 
.546331 
.546735 
.547138 
.547540 
.547943 
.548346 
.548747 
.549149 

9!549550 
.549951 
.550352 
.550752 
.551153 
.55f552 
.551952 
.652351 
.552750 
.653149 

9.553548 
.553946 
.554344 
.554741 
.555139 
.55-536 
.555933 
.556329 
.556725 
.657121 

9.557517 
.557913 
.558308 
.558703 
.559097 
.559491 

.560279 

.560673 

9.561066 

Cotang. 



D. r. 



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 



D. r. 



10.463028 
.462618 
.462206 
.461798 
.461389 
.460980 
.460571 
.460163 
.459756 
.459347 
.458939 

10.45a532 
.458126 
.457719 
.457312 
.456906 
.456501 
.456095 
.455690 
.455285 
.454881 

10.454476 
.454072 
.453669 
.453265 
.452862 
.452460 
.452057 
.451655 
.451253 
.450861 

10.450450 
.450049 
.449648 
.449248 
.448847 
.448448 
.448048 
.447649 
.447250 
.446851 

10.446452 
.446064 
.445656 
.445259 
.444861 
.444464 
.444067 
.443671 
.443275 
.442879 

10.442483 
.442087 
.441692 
.441297 
.440903 
.440509 
.440115 
.439721 
.439327 

10.438934 

Tang. 



123 



60 
59 
68 
67 
66 
55 
54 
53 
62 
51 
50 

49 
48 
47 
46 
46 
44 
43 
42 
41 
40 

89 
38 
37 
86 
35 
34 
S3 
32 
81 
30 

29 
28 
27 
26 
26 
24 
23 
22 
21 
20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 
8 
7 
6 
6 
4 
3 
2 
1 




70- 



20* 



COSINES, TANGENTS. AND COTANGENTS. 



159* 





1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

n 

12 
13 
14 
15 
16 
17 
18 
19 
SO 

21 
22 
23 
21 

25 
2G 
27 
2H 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
4G 
47 
48 
49 
50 

61 
52 
&S 
54 
55 
5G 
57 
58 
59 
60 



HO* 



Sine. 



9.534052 
.534399 
.534745 
.5:i5092 
.535438 
5:i57B3 
.536129 
.536474 
.5:36818 
.537163 
.537507 

9.537851 
538194 
.538538 
.538880 
.&39223 
5.39565 
.539907 
.540249 
.540590 
.540931 

9.541272 
.541613 
.541953 
.542293 
.542632 
.542971 
.543310 
543649 
.543987 
.544325 

9.544663 
.545000 
.545338 
.545GW 
.546011 
.546347 
.546683 
.547019 
.547354 
.547689 

9.548024 
.548359 
..548693 
.549027 
.549360 
..•)49693 
.550026 
.550a')9 

.551024 

9.551356 
.551087 
.552t'.l8 
.55^^9 
.552080 
.ri.'iaOJO 
.55334! 

.55;«ro 

.5.'>4000 
9.554329 



D. r. 



Cosine. D. 1'. 



6. 

5 

6. 

5. 

6. 

5. 

5 

5. 

5. 



.78 
.77 
.78 
.77 
.75 
.77 
75 
.78 
.75 
6.78 

6.ra 

6.72 
6.78 
6.70 
6.72 
6.70 
6.70 
6.68 
6.68 
5.68 
6.68 

6.68 
6.67 
6.67 
6.65 
5.65 
5.65 
6.65 
5.68 
5.63 
5.63 

6.62 
5.63 
5.60 
5.62 
5.60 
6.60 
6.60 
5.58 
6.58 
6.58 

5.58 
6 57 
5.57 
6.55 
5.55 
5.. 55 
6.55 
5.55 
5.53 
5.53 

5. 53 
5.52 
5.52 
6.. 52 
6.50 
5.52 
6.'18 
5.50 
5.48 



Cosine. 



9.972966 
• .972940 

.972894 
972848 
97^2802 
972756 

.972700 

.97^9668 
9^TW17 
972570 

.972624 

9.972478 

972481 

972886 

972888 

.972291 

.972245 

.972196 

.972151 

.978105 

.972068 

9972011 
.971964 
971917 
971870 
.971828 
.971776 
.971729 
.971682 
.971685 
.971588 

9.971540 
.971498 
.971446 
.971898 
.971851 
.971808 
.971256 
.971208 
.971161 
.971118 

9.971066 
.971018 
.970970 
.970922 
.970874 
.970827 
.970779 
.970781 
.970683 
970635 

9.970586 
.970538 
970490 
.970442 
.970804 
.970845 
.970297 
.970249 
.970200 

9.970152 



D. r. 



Slue. 



.77 

.77 

77 

77 

78 

.77 

.77 

.77 

.78 

.77 

.77 

.78 

.77 
78 
.78 
77 
.78 
.78 
.77 
.78 
.78 

.78 
.78 
.78 
.78 
.78 
.78 
.78 
.78 
.78 
.80 

.78 
.78 
.80 
.78 
.80 
.78 
.80 
.78 
80 
.78 

.80 
.80 
.80 
.80 
.78 
.80 
.80 
80 

.eo 

.82 

.80 

.80 
.60 
.80 
.82 
.80 
.80 
.82 
.80 



Tan^:. 



D. r. 



124 



9 661066 

.661469 

661851 

662244 

.662636 

.668028 

668419 

668811 

.664202 

.664608 

.664968 

9.666878 
666768 
.566168 
666642 
.666962 
.567320 
.667709 

.ODoUvS 

.668486 
.668873 

9.669261 
.569648 
.670085 
.670422 
.670809 
.671196 
.571581 
.571967 
.672862 
672738 

9.678128 
.678607 
.578892 
.674276 
.674660 
.676044 
.676427 
.676810 
.676196 
.676676 

9.676960 
.677841 
.677728 
.678104 
.678486 
.678867 
.579248 
579629 
.580009 
.580389 

9.680769 
.681149 
681528 
581907 
.582286 
.582665 
.583014 
.583422 
.588800 

9.584177 



Cotau^. 



D. r. 



656 
6 68 
6.66 
6.68 
6.58 
652 
658 
652 
6.52 
650 
6.50 

6.60 
6.60 
6.48 
6.60 
6.47 
6.48 
6.48 
6.47 
6.45 
6.47 

6.45 
6.45 
6.45 
6.45 
6.48 
6 48 
6.48 
6.42 
6.48 
6.42 

6.40 
6.42 
6.40 
6.40 
6.40 
6.88 
638 
6.88 
6.38 
6.38 

6.87 
6.87 
6.85 
6.87 
6.86 
6.86 
6.35 
6.38 
6.38 
6.33 

6.88 
6.82 
6.32 
6.82 
6.32 
6.32 
6.30 
6.30 
6.28 



Cotang^. 



D. 1'. 



10.488034 
.488641 
.488149 
.487756 
.487364 
.486072 
.436661 
.486180 
.485798 
.485407 
.486017 

10.484627 
.484237 
.438847 
.483468 
.488068 
.482680 
.482291 
.481902 
.481614 
.481127 

10.4807^ 
.480662 
.429966 
.429578 
.429191 
.428805 
.428419 
.428063 
.427648 
.427262 

10.426877 
.426498 
.426108 
.426724 
.425340 
.424956 
.424573 
.424190 
.428807 
.423424 

10.428041 
.422659 

!421896 
.421614 
.421188 
.420752 
.420871 
.419991 
.419611 

10.419231 
.418851 
.418472 
.418093 
.417714 
.417885 
.416956 
.416578 
.416200 

10.415823 



60 
59 
58 
67 
56 
55 
64 
63 
S2 
51 
60 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 



TODg. 



87 
86 
85 
34 
38 
82 
31 
30 

29 
28 
27 
26 
25 
24 
23 
23 
21 
20 

19 
18 
17 
16 
15 
14 
18 
12 
11 
10 

9 
8 
7 
6 
6 
4 
8 
2 
1 




69* 



««• 



COSINES. TANGENTS, AND COTANGENTS. 





167« 






Cotang. 


60 


10.893690 


.898227 


59 


.892863 


68 


.892500 


57 


.392187 


56 


.891775 


55 


.891412 


54 


.891050 


53 


.890688 


52 


.890326 


51 


.889964 


60 


10.889603 


49 


.889241 


48 


.888880 


47 


.8886iiiU 


46 


.8HKI59 


45 


.887799 


44 


.887489 


48 


.887079 


42 


.886719 


41 


.886360 


40 


10.886000 


39 


.886641 


38 


.386282 


37 


.884928 


36 


.384665 


85 


.384207 


34 


.883849 


33 


.883491 


32 


.383188 


31 


.882'/r/6 


30 


10 882418 


20 


.882061 


28 


.881705 


27 


.881348 


26 


.880992 


25 


.880636 


24 


.380280 


28 


.879924 


22 


.379568 


21 


.879218 


ao 


10.878868 


19 


.878608 


18 


.878148 


17 


.877798 


16 


.877480 


15 


.877086 


14 


.878781 


18 


.876877 


12 


.876024 


11 


.875670 


10 


10.376817 


9 


.374964 


8 


.374612 


7 


.374259 


6 


.378907 


5 


.878665 


4 


.878208 


8 


.872861 


2 


.872499 


1 


10.872148 




/ 


Tang. 





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 
80 

31 
82 
83 
34 
35 
86 
37 
38 
39 
40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
50 

51 

52 
53 
54 
65 
66 
57 
68 
59 
60 



Sine. 



9.578575 
.673888 
.674200 
.574512 
.574824 
.575136 
.575447 
.676758 
.676069 
.676879 
.576680 

9.676999 
.577309 
.577618 
.577927 
.578236 
.578545 
.578853 
.579162 
.579470 
.579777 

9.680065 
.580892 
.580699 
.681006 
.681312 
.681618 
.681924 
.582229 
.582535 
.582840 

9.588146 
.583449 
583754 
.584068 
.584361 
.684665 
.584968 
.585272 
.586674 
.585877 

9.586179 
.586482 
586783 
.687085 
.587886 
.687688 
.587989 
.588289 
.588590 
.588890 

9.589190 
.589489 
.589789 
.590068 
.590387 
.590686 
.690984 
691282 
.591580 

9.691878 

Cosine. 



D. r. 



22 
20 
20 
20 
20 
18 
18 
18 
17 
5.17 
6.17 



17 
15 
15 
15 
15 
18 
15 
13 
12 
13 



6.12 
5>12 



.10 
.12 
10 
.10 
.08 
5.10 
5.08 
6.06 



5. 

5. 

5 

5. 

6. 



5.07 



5 
5 
5 
5 
5 
5 
5 
5, 



06 
07 
05 
07 
05 
07 
03 
05 



5.03 



.05 
.02 
5.03 
5.02 
.03 
.02 
.00 
5.02 
5.00 
5.00 



5. 
5. 



5. 
5. 
5. 



4.98 
5.00 
4.96 



4 

4. 
4. 
4. 
4. 

4. 



96 
96 
97 
97 
97 
97 



D. 1". 



Cosine. 



9.967166 
.967115 
.967064 
.967018 
.966961 
.966910 
.966859 
.966806 
.966756 
.966705 
.966653 

9.966602 
.966550 
.966499 
.966447 
.966895 
.966344 
.966292 
.966240 
.966188 
.966136 

9.966065 
.966038 
.965961 
.965929 
.965876 
.965824 
.965772 
.965720 
.965668 
.965615 

9966668 

965511 

966456 

965406 

965353 

965301 

965248 

.965195 

.966148 

.966090 

9.966037 

.964964 

.964931 

.964879 

.964826 

964778 

.964720 

964666 

964613 

.964560 

9.964507 
.964454 
.964400 
.964347 
.964294 
.964240 
.964187 
.964183 
964080 

9.964026 

Sine. 



D. 1' 



.85 
.85 
.85 
.87 
.85 
.65 
.85 
.87 
.85 
.87 
.85 

.87 
.85 
.87 
.67 
.85 
.87 
.87 
.87 
.87 
.85 

.87 
.87 
.87 
.oo 
.87 
87 
.87 
.87 
.oo 
.87 

.87 
.86 

87 
.86 
.67 
.88 

88 
.87 
.86 
.88 

.86 
.88 
.87 
.88 
.88 
.88 
.90 
.88 
.68 
.68 

.86 
.90 
.66 
.86 
.90 

.Oo 

.90 
.66 
.90 

D. 1'. 



Tang. 



9.606410 
.606778 
.607137 
.607500 
.607863 
.606225 
.606586 
.606950 
.609312 
.609674 
.610036 

9.610897 
.610769 
.611120 
.611480 
.611841 
.612201 
.612561 
.612921 
.613281 
.613641 

9.614000 
.614359 
.614716 
.616077 
.615435 
.615798 
.616151 
.616509 
616867 
.617224 

9.617582 
617939 
.618296 
.618662 
619008 
.619364 
.619720 
.620076 
.620432 
.620787 

9.621142 
.621497 
.621662 
.622207 
.622661 
.622915 
.623269 
.623623 
.623976 
.624330 

9.624663 
.625086 
.625368 
.625741 
.626093 
.626446 
.626797 
.627149 
.627501 

9.627852 

Cotang. 



D. 1'. 



6.05 
6.07 
6.05 
6.05 
6.08 
6.05 
6.03 
6.03 
6.03 
6.03 
6.02 

6.03 
6.02 
6.00 
6.02 
6.00 
6.00 
6.00 
6.(0 
6.00 
5.98 

5.96 
'5.98 
598 
5.97 
6.97 
97 
97 
97 
95 
97 



6. 
6. 
5. 
6. 



5.95 
5.98 
.96 
.93 
.93 
.93 
6.93 
598 
5.92 
6.92 

5.92 
5.92 
5.92 
5.90 



6. 
5. 



90 
90 
5.90 
6.88 
6.90 
5.88 



5.88 
5.87 
.88 
.67 
.87 
.87 
5.87 
5.87 
5.66 



5. 
5. 
6. 
5. 



D. r. 



112< 



126 



67* 



i^ 



TABLE X. — LOGARITHMIC SINES, 



156« 



.' 


Sine. 





9.591878 


1 


.592176 


2 


.592473 


8 


.592770 


4 


.593067 


5 


.593363 


6 


.593659 


7 


.598955 


6 


.594251 


9 


.594547 


10 


.694842 


11 


9.595137 


12 


.595432 


13 


.595727 


14 


.596021 


15 


.596315 


16 


.596609 


17 


.596903 1 


18 


.597196 


19 


.597490 1 


20 


.597783 


21 


9.596075 . 


22 


.596368 


23 


.596660 


24 


.596952 


25 


.599244 


26 


.599536 


27 


.599627 


28 


.600118 


29 


.600409 


30 


.600700 


81 


9.600990 


32 


.601280 


83 


.601570 



D. r. 



84 
85 
36 
87 
88 
89 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 

51 
52 
58 
54 
65 
66 
57 
58 
69 
60 



118< 



.601860 
.602150 
.602139 
.602728 
.603017 
.603305 
.603604 

9.603882 
.604170 
.604457 
.604745 
.606032 

.605319 

AARiina 
.uUOoUD 

.605892 
.606179 
.606465 

9.606751 
.607036 
.607322 
.607607 
.607892 
.608177 
.606461 
.606745 
.609029 

9.609313 



Cosine. 



4.97 



4. 
4. 
4. 
4. 
4. 
4. 
4. 
4. 
4. 
4. 

4. 
4. 
4. 
4. 
4. 
4 
4, 
4 
4 



95 
95 
95 
93 
93 
93 
93 
93 
92 
92 

92 
92 
90 
90 
90 
90 
88 
90 
.88 



4.87 

4.88 
4.87 
4.87 



87 

,87 
.85 
.85 
.85 
.85 
.83 



4.83 
4.83 



4. 
4. 
4. 
4 
4 
4. 
4 



83 
83 
82 
82 
82 
80 
82 



4.80 

4.80 
4 
4 
4 

4 

4 



78 
80 
78 
78 

78 
4.77 
4.78 
4.77 
4.77 

4.75 
4.77 
4.75 
4.75 
4.75 
4.73 
4.73 
4.73 
4.73 



Cosine. 



D.r. 



D. r 



9.964026 
.963972 
.963919 
.963865 
.963811 
.963757 
.963704 
.963650 
.963596 
.963542 
.963488 

9.963434 
.963379 
.963325 
.963271 
.963217 
.963163 
.963106 
.963054 
.962999 
.962945 

9.962890 
.962836 
.962781 
.962727 
.962672 
.962617 
.962562 
.962508 
.962453 
.962398 

9.962843 
.962288 
.962233 
.962178 
.962123 
.962067 
.962012 
.961957 
.961902 
.961846 

9.961791 
.961735 
.961680 
.961624 
.961569 
.961513 
.961458 
.961402 
.961846 
.961290 

9.961235 
.961179 
.961123 
.961067 
.961011 
.960055 
.960699 
.960643 
.960786 

9.960730 



Sine. 



.90 

.oo 

.90 
.90 
.90 
.88 
.90 
.90 
.90 
.90 
.90 

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

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

.92 
.92 
.92 
.92 
.98 
.92 
.92 
.92 
.93 
.92 

.93 
.92 
.93 
.92 
.98 
.92 
.93 
.93 
.93 
.92 

.99 
.93 
.93 
.93 
.93 
.93 
.93 
.95 
.93 



Tang. 



9.637852 



.628554 
.628905 
.629255 
.629606 
.629956 
.630806 
.680656 
.631005 
.631366 

9.681704 
.632053 
.632402 
.632750 
.633099 
.633447 
.633795 
.634143 
.634490 
.634838 

9.635185 
.635532 
.635879 
.636226 
.636572 
.636919 
.637265 
.637611 
.637956 
.638302 

9.638647 
.638992 
.639337 
.639682 
.640027 
.640871 
.640716 
.641060 
.641404 
.641747 

9.642091 
.642434 
.642777 
.643120 
.643468 
.643806 
.644148 
.644490 
.644832 
.645174 

9.645516 
.646857 
.646199 
.646540 
.646881 
.647222 
.647562 
.647903 
.648243 

9.648583 



D. r. 



Cotang. 



D. 1". II Cotang. 



127 



6.85 

5.85 
5.85 
5.83 
.85 
.83 



5. 
5. 
6. 
5. 
5. 
5. 



88 

83 
82 
88 



5.82 

5.82 
6.82 
5.80 
5.82 
5.80 
5.80 
5.80 
5.78 
5.80 
5.78 

5.78 
6.78 
78 
77 
78 
77 
77 
75 
77 
75 



6 
6 
5 

6 
5 
5 
5 



5.75 
5.75 



5. 
5. 
5. 
5. 
5. 
6. 
5. 
5. 

5. 
6. 
6. 
5. 
5. 
5. 
5, 
5, 
5, 
6. 

5.68 
5.70 
5.68 
5.68 
5.68 
6.67 
6.68 
5.67 
5.67 



75 
75 
73 
75 
73 
73 
72 
73 

72 

72 
72 
72 
72 
70 
70 
70 
70 
70 



D. r. 



10.372148 
.371797 
.371446 
.371095 
.370745 
.370394 
.370044 
.369694 
.369844 
.368995 
.868645 

10.868296 
.367947 
.367598 
.367250 
.366901 
.866553 
.366205 
.365857 
.865510 
.865162 

10.364815 
.364468 
.364121 
.363774 
.363428 
.363061 
.862735 
.362389 
.362044 
.361698 

10.361858 
.361006 
.360663 
.360318 
.359973 
.359C29 
.359284 
.358940 
.358596 
.358253 

10.857909 
.357566 
.357223 
.356880 
.356537 
.356194 
.355852 
.355510 
.355168 
.354826 

10.354484 
.354143 
.353801 
.353460 
.353119 
.352778 
.352438 
.352097 
.351757 

10.351417 



Tang. 



60 
59 
58 
57 
56 
55 
54 
63 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
88 
37 
36 
35 
34 
33 
82 
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 




68* — 



1 
1 

J 



26* 



TABLE X. — LOGABITHMIC SINES, 



154« 



Sine. 



D. r. 




1 
2 
8 
4 
6 
6 
7 

I 

10 

11 
13 
13 
14 
15 
16 
17 
18 
19 
SO 

21 
82 
28 
24 
25 
26 
27 
28 
29 
80 

81 
82 
88 
84 
85 
86 
87 
88 
89 
40 

41 
42 
48 
44 
45 
46 
47 
48 
40 
60 

51 
68 
58 
54 
65 
56 
57 
56 
60 



9.626048 
.626219 
.686490 
.626760 
.627060 
.627800 
.627570 
.627840 
.628109 
.628878 
.628647 

9.6S8916 
.629185 
.629453 
.629721 



.680257 
.680624 
.680792 
.681059 
.681826 

9.681593 
.631869 
.682125 



.682658 



.638189 
.633454 
.638719 
.688984 

9.684249 
.684514 
.684778 

686042 
.686806 

686570 
.686884 
.636097 

686360 



9.636886 
68n48 
. 6^411 
.687878 
.637985 
.688197 
.688458 
.6383120 
.638961 



9.689503 
.639764 
.640024 
.640264 
.640544 
.640604 
.641064 
.641824 
.641688 

9.641842 



Oosine. 



4.52 
4.62 
4.50 
4.50 
4.60 
4.50 
4.50 
4.48 
4.48 
4.48 
4.48 

4.48 
4.47 
4.47 
4.47 
4.47 
4.45 
4.47 
4.45 
4.45 
4.45 

4.43 
4.43 
4.45 
4.48 
4.42 
4.43 
4.42 
4.42 
4.42 
4.42 

4.42 

4.40 
4.40 
4.40 
4.40 
4.40 
4.38 
438 
4.38 
4.88 

4.37 
4.88 
4.37 
4.87 
4.87 
4.35 
4.87 
4.85 
4.35 
4.85 

4.85 
4.83 
4.83 
4.88 
4.88 
4.83 
4.38 
4.82 
4.82 



Cogine. 



D. r. 



D. r. 



9.9572T6 
.967217 
.957158 
.957099 
.957040 
.966961 
.956021 
.956862 
.956808 
.956744 
.956684 

9.956625 
.966566 
.956606 
.956447 
.966387 
.966827 
.956268 
.956206 
.956148 
.956069 

9.956029 
.956969 
.955009 
.955649 
.956789 
.955729 
.956669 
.965609 
.965548 
.965488 

9955428 
.965368 
.956807 
.966247 
.965186 
.965126 
.966065 
.965005 
.964944 
.954888 

9954828 
.954762 
.954701 
.954640 
.954579 
.954516 
.954457 
.954896 
.954885 
.954274 

9954218 
.954162 
.954090 
.954029 
.953966 
.968906 
.958646 
.953788 
.968722 

9.958660 



Sine. 



.96 
.98 

.98 

.DO 

1.00 
.96 
.96 
.96 

1.00 
.96 

1.00 

.96 

1.00 

1.00 

.DO 

1.00 

1.00 

.96 

1.00 



1. 
1. 
1, 
1. 



.00 
.00 
.00 
.00 
1.00 
1.00 
1.00 
.96 
1 00 
1.00 

1.00 



1. 
1 
1. 
1. 
1. 



.02 
.00 
.02 
.00 
.02 
1.00 
1.02 
1.02 
1.00 

1 02 
1.02 
1.02 
1.02 
1.02 
1.02 
1 02 
1 02 
1.02 
1.02- 

1.02 
1.03 



02 
02 
03 
02 
03 
02 



i.oa 



115' 



D. 1". 



129 



Tang. 



D. 1". 



Cotang. 



9.668673 
.669002 
.669332 
.669661 
.669991 
.670620 
.670649 
.670977 
.671806 
.671635 
.671968 

9.672291 
.672619 
.672947 
.673274 
.673602 
.678929 
.674257 
.674564 
.674911 
.675287 

9.676664 
.675890 
.676217 
.676548 
.676869 
.677194 
.677520 
.677846 
.678171 
.678496 

9.678821 
679146 
.679471 
.679795 
.660120 
.680444 
.660766 
.661082 
.661416 
.661740 

9.682068 
.682887 
.682710 
.668033 
.663366 
.668679 
.684001 
.664324 
.684646 
.684968 

9.685290 
.686612 
.C85934 
686266 
.686577 
.686896 
.687219 
.687540 
.687861 

9.688182 



Ck>tang. 



5.48 
5.60 
5.46 
5.60 
5.48 
5.48 
5.47 
5.48 
5.48 
5.47 
5.47 

5 47 
547 
5.45 
5.47 
5.45 
5.47 
5.45 
5.45 
5.43 
5.45 

543 
5.45 
5.43 
5.43 
5.42 
5.43 
5.43 
5.42 
5.42 
5.42 

5.42 
5 42 
5.40 
5.42 
5.40 
5.40 
5.40 
5.40 
5.40 
5.88 

5.40 
5.38 
5.86 
86 
36 



37 
86 
87 
87 
87 



87 
87 
35 
87 
35 
5.85 
5 85 
5.85 
5.85 



10.331827 
.880996 
.380666 
.330339 
.380009 
.829680 
.829851 
.329023 
.828694 
.828865 
.326087 

10.327709 
.327881 
.327058 
.826726 
.826396 
.826071 
.325743 
.325416 
.825069 
.824763 

10.324436 
.824110 
.823788 
.823457 
.828131 
.822806 
.322480 
.822154 
.821829 
.821504 

10.821179 
.820664 
.320529 
.320205 
.819680 
.819556 
.819282 
.818806 
.818584 
.818260 

10 817987 
.817618 
.817290 
.816967 
.816644 
.816321 
.315099 
.315676 
.815364 
.316032 

10 814710 
.314386 
314066 
818745 
.313428 
.313102 
.812781 
.312460 
.312189 

10.311616 



D. 1'. 1 Tang. 



60 
59 
58 
57 
56 
55 
54 
53 
62 
51 
50 

49 
46 
47 
46 
45 
44 
43 
42 
41 
40 

39 
36 
37 
86 
35 
84 
88 
82 
31 
30 

29 
26 
27 
26 
25 
24 
28 
22 
21 
20 

19 
16 
17 
16 
15 
14 
13 
12 
11 
10 

9 

8 
7 
6 
5 
4 
8 
2 
1 




w 



: 



•: 



: 



; 



r^ri.-c 



27* 



TABLE X. — LOGARITHMIC SINES. 



162« 



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 

ao 

81 
32 
33 
34 
86 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 

51 
62 
&3 
64 
55 
56 
57 
58 
69 
60 



117' 



9.667047 
.657395 
.657542 
.657790 
.658087 
.668284 
.658531 
.658778 
.669025 
.669271 
.659517 

9.669768 
.660009 
.660255 
.660601 
660746 
.660991 
.661236 
.661481 
.661726 
.661970 

9.662214 
.662459 
.662703 
662946 
.663190 
.663433 
.663677 
.663920 
.664163 
.664406 

9.664648 
.664891 
.665133 
.666375 
.665617 
.665869 
.666100 
.666342 
.666583 
.666824 

9.667066 
.667306 
.667546 
.667786 
.668027 

668267 
.668606 
.668746 
.668966 

669225 

9.669164 

669703 

669942 

670181 

670419 

.670658 

670896 

.671134 

.671872 

9.671609 



Cosine. 



4. 
4. 
4. 
4. 
4. 



13 
12 
13 
12 
12 



4.12 
4.12 
4.12 
4.10 
4.10 
4.10 



4 
4 
4 
4 
4 
4 
4 
4, 
4 
4. 



10 
10 
10 
08 
08 
08 
08 
08 
07 
07 



4.06 



4 
4 
4 
4. 
4 
4 
4 
4 
4 



07 
05 
07 
05 
07 
05 
05 
05 
03 



4.05 



4 
4 
4 
4 
4 
4 
4 
4 
4 

4 
4 
4 
4 
4 



03 
03 
03 
03 
02 
03 
02 
02 
02 

00 
02 
00 
02 
00 



3.98 



4 00 
4.00 

3.98 

898 
3.98 
3.96 
3.97 
3.98 
3.97 
3.97 
3.97 
8.95 



D. r. 



Cosine. 



9.949881 
.949816 
.949752 
.949688 
.949623 
.949658 
.949494 
.949429 

.949300 
.949235 

9.949170 
.949105 
949040 
.948975 
.948910 
.948845 
.948780 
.948715 
.948650 
.948684 

9.948519 
.948454 
.948388 
.948323 
.948257 
.948192 
.948126 
.948060 
.947995 
.947929 

9.947868 
.947797 
.947731 
947665 
.947600 
.947533 
.947467 
.947401 
.947^35 
.947269 

9 947203 
.947186 
.947070 
.947004 
.946937 
.946871 
946804 
.946788 
.946671 

9.946538 

.946471 

.946404 

946337 

.946270 

946203 

.946136 

.946069 

946002 

9.945935 



Sine. 



D. 1". 



.06 
.07 
.07 
.06 
.06 
.07 
.08 
.06 
.07 
.08 
.08 

.08 
.08 
.06 
.08 
.08 
.06 
.06 
.06 
.10 
,06 

.06 
.10 
.06 
.10 
.06 
.10 
.10 
.06 
.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 
.12 
.12 
,12 
.13 



D. 1". 



131 



Tang. 



9.707166 
.707478 
.707790 
.706102 
.708414 
.708726 
.709037 
.709349 
.709660 
.709971 
.710282 

9.710593 
.710904 
.711215 
.711526- 
.711836 
.712146 
.712466 
.712766 
.713076 
.713886 

9.713696 
.714005 
.714314 
.714624 
.714933 
.715242 
.715651 
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 
728282 
723538 
.723844 
784149 
.724464 
724760 
725065 
725370 

9.725674 



Cotang. 



D. r. 



Cotang. 



5.20 
6.20 
5.20 
5.20 
5.20 
.18 
.20 
.18 
.18 
5.18 
6.18 



6. 
5. 
5. 
5. 



5.18 
5.18 



5 

5. 

5. 

5. 

6 

5 

5 

5. 

5. 
6 
5, 
6. 
5 
6 



17 
18 
17 
17 
17 
17 
17 
17 

15 
15 
17 
15 
15 
15 



6.15 
6.13 
5.15 
6.13 



5 
5 
5. 
5. 
5, 
5. 
6. 
5. 
6. 



13 
13 
13 
13 
18 
13 
12 
13 
12 



5.12 



5. 
5. 
6. 
6. 
5. 
5. 



.12 
.12 
.12 
.10 
.12 
.10 
5.12 
5.10 
5.10 
6.10 



06 
10 
10 
06 
08 
5.10 
5.06 
5.08 
5.07 



D. r 



10.292834 
.292522 
.292210 
.291898 
.291586 
.291274 
.290963 
.290651 
.290340 
.290029 
.289718 

10.289407 
.289096 
.288785 
.288475 
.288164 
.287854 
.287544 
.287234 
.286924 
.286614 

10.286804 
.285995 
.285686 
285376 
.285067 
.284758 
.284449 
.284140 
.283832 

. .283523 

10.283215 
.282907 
.282699 
.282201 
281983 
.281675 
.281367 
.281060 
.280752 
.280445 

10.280188 
.279831 
.279524 
.279217 
.278911 
.278604 
.278298 
.277991 
.277685 
.277379 

10.277073 

.276768 

276462 

276156 

.275861 

.275646 

275240 

274935 

.274630 

10.274326 



Tang. 



60 
59 
68 
57 
56 
65 
54 
63 
62 
61 
60 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
88 
87 
36 
35 
84 
33 
82 
31 
80 

29 
28 
27 
26 
26 
24 
23 
22 
21 
20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 
8 
7 
6 
5 
4 
3 
2 
1 




62'' 



S8« 



COSINES, TANGENTS, AND COTANGENTS. 



I5r 



/ 


Sine. 





9.671609 


1 


.671847 


2 


.672084 


3 


.672321 


4 


.672558 


5 


.672795 


6 


.673032 


7 


.673268 


8 


.673506 


9 


.673741 


10 


.673977 


11 


9.674213 


12 


.674448 


13 


.674684 


14 


.674919 


15 


.675155 


16 


.675390 


17 


.675624 


18 


.675859 


19 


.676094 


20 


.676328 


21 


9.676562 


22 


.676796 


23 


.677030 


24 


.677264 


25 


.677498 


26 


.677731 


27 


.6rr964 


28 


.678197 


29 


.678430 


80 


.678663 


81 


9.678895 


82 


.679128 


S3 


.679860 


34 


.679592 


85 


.679824 


86 


.680056 


87 


.680288 


88 


.680519 


89 


.680750 


40 


.680962 


41 


9.681213 


42 


.681443 


43 


.681674 


44 


.681905 


45 


.682135 


46 


.682865 


47 


.682595 


48 


.682825 


49 


.683055 


50 


.683284 


51 


9.683514 


52 


.68:3743 


53 


.683972 


54 


.684201 


55 


.684430 


56 


.684658 


57 


.684887 


58 


.685115 


59 


.685343 


60 


9.685571 



Cosine. 



D. 1". 



3.97 
8.95 
3.95 
3.95 
8.95 
3.95 
8.93 
3.95 
3.93 
3.93 
3.93 

3.92 
3.93 
3.92 
3.93 
3.92 
3.90 
8.92 
8.92 
3.90 
3.90 

3.90 
3.90 
3.90 
3.90 
3.88 
3.88 
8.88 
8.88 
3.88 
8.87 

8.88 
3.87 
3.87 
3.87 
3.87 
8.87 
3.85 
3.85 
3.87 
8.85 

8.83 
8.85 
3.85 
8.83 
3.83 
3.83 
3.88 
3.83 
3.82 
8.83 

3.82 
3.82 
3.82 
3.82 
3.80 
3.82 
3.80 
3.80 
3.80 

D. 1'. 



Cosine. 



9.945935 
.945868 
.945800 
.945733 
.945666 
.945598 
.945531 
.945464 
.945396 
.945328 
.945261 

9.945193 
.945125 
.945058 
.944990 
.944922 
.944854 
.944786 
.944n8 
.944660 
.944582 

9.944514 
.944446 
.944377 
.944309 
.944241 
.944172 
.944104 
.944036 
.943967 
.948899 

9.943830 
.943761 
.948693 
.943624 
.943555 
.948486 
.943417 
.943348 
.943279 
.943210 

9.943141 
.94:^072 
.943003 
.942934 
.942864 
.942795 
.942726 
.942656 
.942587 
.942517 

9.942448 
.942378 
.942308 
.942239 
.942169 
.942099 
.942029 
.941959 
.941889 

9.941819 

Sine. 



D. 1". 



.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 
.18 
.18 
.15 
.18 
.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 



D. r. 



Tang. 



9.725674 
.725979 
.■^6284 
.726588 
.726892 
.727197 
.727501 
.727805 
.728109 
.728412 
.728716 

9.7S9020 
.729823 
.729628 
.729929 
.780283 
.730535 
.730638 
.731141 
.781444 
.731746 

9.732048 
.782861 
.732653 
.732955 
.733257 
.733668 
.733860 
.734162 
.734468 
.784764 

9.785066 
.735367 
.736668 
.735969 
.736269 
.786570 
.736870 
.787171 
.737471 
.737771 

9.738071 
.788371 
.738671 
.788971 
.739271 
.789570 
.789870 
.740169 
.740468 
.740767 

9.741066 
.741366 
.741664 
.741962 
.742261 
.742559 
.742858 
.743156 
.743454 

9.743752 

Cotang. 



D. r. 



5.06 
5.08 
5.07 
5.07 
6.05 
5.07 
6.07 
6.07 
5.05 
5.07 
5.07 

5.05 
5.05 
5.05 
6.07 
5.03 
5.05 
5.05 
5.05 
5.03 
5.03 

5.05 
5.03 
6.03 
6.03 
02 
03 
03 
02 
02 
03 



5 
6 
6 
6 
6 
5 

5 
5 
6 
6 
5 
5 
5 
5 
5 
5 

6.00 
6.00 
6.00 
5.00 
4.96 
5.00 
4.96 
4.98 
4.98 
4.98 



.02 
02 
.02 
00 
02 
00 
02 
00 
00 
00 



.98 
.98 
.97 

.97 
.96 
.97 
97 



4.97 



D. r. 



Cotang. 



10.274326 
.274021 
.278716 
.278412 
.278108 
.272803 
.272499 
.272196 
.2n891 
.271588 
.271264 

10.270960 
.270677 
.270874 
.270071 
.269767 
.269465 
.269162 
.268859 
.268556 
.268254 

10.267952 
.267649 
.267347 
.267045 
.266743 
.266442 
.266140 
.266838 
.266637 
.266236 

10.264934 
.264688 
.264832 
.264031 
.263731 
.263480 
.263130 
.262829 
.262529 



10.261929 
.261629 
.261329 
.261029 
.260729 
.260430 
.260130 
.259831 
.259532 
.269288 

10.268984 
.258635 
.258886 
.258038 
.257789 
.257441 
.257142 
.266844 
.256646 

10.256248 

Tang. 



118' 



132 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
37 
36 
35 
34 
88 
32 
31 
30 

29 
28 
27 
26 
25 
24 
28 
22 
21 
20 

19 
18 
17 
16 
15 
14 
18 
12 
11 
10 

9 
8 
7 
6 
5 
4 
8 
8 
1 




9V 



39* 



TABLE X. — LOGARITHMIC SINES, 



150< 



Sine. 




1 
2 
8 
4 
6 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

ao 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

81 
82 
83 
34 
85 
36 
87 
88 
89 
40 

41 
42 

43 
44 
45 
46 
47 
48 
49 
CO 

61 
52 
53 
54 
55 
56 
67 
58 
59 
60 



119« 



9.«a5571 
.685799 
.686027 
.686254 
.686482 
.686709 



.68n63 
.687389 
.687616 
.687843 

9.688009 
.688295 
.688521 
.688747 
.688972 
.689196 
.689123 
.689648 
.689673 
.690096 

9.600823 
.600548 
.690772 
.690996 
.691220 
.691444 
.691668 
.691892 
.692115 
.692339 

9.692562 
.692785 
.693008 
.693231 
.693453 
.693676 
.693896 
.694120 
.694342 
.694564 

9.694786 
.695007 
.695229 
.695450 
.695671 
.695892 
.696113 
.696334 
.696554 
.696775 

9.^96995 
.697215 
.697435 
.697654 
.697874 
.696094 
.606318 
.696533 
.698751 

9.606970 



Cosine. 



D. r. 



3.80 
3.80 
3.78 
3.80 
3.78 
3.78 
3.78 
3.77 
3.78 
3.78 
8.77 

8.77 
3.77 
3.77 
8.75 
3.77 
8.75 
3.75 
3.75 
3.75 
8.75 

3.75 
3.73 
3.73 
3.73 
3.73 
3.r3 
3.73 
3.72 
3.73 
3.72 

3.72 
3. 72 
3.72 
3.70 
3.72 
3.70 
8.70 
8.70 
8.T0 
3.70 



68 
70 



368 
3 6S 
3.68 
3.68 
68 
67 
68 
07 



3.67 
3.67 
8.65 
3.67 
3.67 
3.65 
8.65 
8.65 
3.65 



D. r. 



Cosine. 



9.941819 
.941749 
.941679 
.941609 
.941589 
.941469 
.941398 
.941328 
.941258 
.941187 
.941117 

9.941046 
.940975 
.940905 
.940634 
.940763 
.940693 
.940622 
.940551 
.940480 
.940409 

9.940388 
.940267 
.940196 
.940125 
.940054 
.989962 
.939911 
.939640 
.939768 
.939697 

9.939625 
.939554 
.939482 
.939410 
.939339 
.939267 
.939195 
.939123 
.939052 
.938980 

9.988906 
.938886 
.986768 
.938691 
.938619 
.936547 
.936175 
.936402 
.938330 
.938258 

9.938185 
.938113 
.938040 
.937907 
.ft37BJ)5 
.937822 
.937749 
.937676 
.937604 

9.937531 



Sine. 



D.V, 



.17 
.17 
.17 
,17 
,17 
,18 
.17 
,17 
.18 
.17 
.18 

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

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

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

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

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



Tang. 



D. r. 



188 



9.743752 
.744050 
.744348 
.744645 
.744943 
.745240 
.745586 
.74'«35 
.74ol32 
.746429 
.746726 

9.747023 
,747819 
.747616 
.747913 
.748209 
.748605 
.748801 
.749097 
.749393 
.749689 

9.749985 
.750281 
.750576 
.750672 
.751167 
.751462 
.751757 
.752052 
.752347 
.752612 

9.752937 
.753231 
.75;^26 
.753820 
.754115 
.754409 
.754703 
.754997 
.755291 
.755585 

9.755878 
.756172 
.756465 
.756759 
.757052 
.757345 
.757688 
.757931 
.758224 
.758517 

9.758810 
.759102 
.759395 
.759687 
.759979 
.760272 
.760561 
.760856 
.761148 

9.761439 



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 
4.95 
4.93 
4.93 
4.93 
4.93 
4.93 
4.93 
4.93 



4. 
4. 
4. 
4. 
4. 



.93 
.92 
.93 
.92 
.92 
4.92 
4.92 
4.92 
4.92 
4.92 

4.90 
4 92 
4.90 
4.92 
4.90 
4.90 
4.90 
4.90 
4.90 
4.88 

4.90 



4 
4 
4 
4 
4 
4 
4 
4 



88 
90 
88 
88 
88 
88 
88 
88 



4.88 

4.87 
4.88 
4.87 
4.87 
4.88 
4.87 
4.87 
4.87 
4.85 



Cotang. I D. r. 



Cotang. 



10.356248 
.255950 
.255652 
.255355 
.255057 
.254760 
.254462 
.254165 
.253868 
* .253571 
.253274 

10.252977 
.252681 
.252384 
.252087 
.251791 
.251495 
.251199 
.250903 
.250607 
.250311 

10.250015 
.249719 
.24941^ 
.249128 
.248833 
.248538 
.248243 
.247948 
.247653 
.247358 

10.247063 
.24^69 
.246474 
.246180 
.215885 
.245591 
24529? 
.2450a3 
.244709 
.244415 

10.244122 
.243828 
.24353r) 
.243241 
.242948 
.242655 
.242362 
.242069 
.241776 
.241483 

10.241190 
.240898 
.240605 
240313 
.240021 
.239728 
.239436 
.239144 
.238a52 

10.238561 

Tang. 



GO' 

69 

58 

57 

56 

65 

54 

53 

52 

51 

50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
37 
36 
35 
84 
33 
32 
31 
30 

29 
26 
27 
26 
23 
24 
23 
22 
21 
90 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 
8 
7 
6 
5 
4 
3 
2 
1 




60* 



5 ' 



i 



SI* 



TABLE X. — LOGABITHMIC SINES, 



148- 



Sine. 




1 
2 
8 
4 
6 
6 
7 
8 
9 
10 

11 
12 
18 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
80 

81 
82 
83 
84 
86 
86 
87 



40 

41 
42 
48 
44 
45 
46 
47 
48 
49 
50 

61 
52 
58 
54 
65 
56 
67 
58 
59 
60 



9.711839 
.712050 
.712260 
.712469 
.712679 
.712889 
.713098 
.718808 
.718517 
.713726 
.713935 

9.714144 
.714352 
.714561 
.714769 
.714978 
.715186 
.715394 
.715602 
.715809 
.716017 

9.716a^ 
.716432 
.716639 
.716846 
.717053 
.717259 
.717466 
.717678 
.717879 
.718085 

9.718291 
.718497 
.718708 
.718909 
.719114 
.719320 
.719525 
.719730 
.719935 
.720140 

9.720845 
.720549 
.720754 
.720958 
.721162 
.721366 
.721570 
.721774 
.721978 
.723181 

9.722385 
.722588 
.722791 
.722994 
.728197 
.723400 
.723608 
.728805 
.724007 

9.724210 

Ck»ine. 



D. r. 



3.52 
3.60 
3.48 
8.60 
8.50 
8.48 
3.60 
8.48 
8.48 
8.48 
8.48 

3.47 
3.48 
3.47 
3.48 
3.47 
3.47 
3.47 
3.45 
3.47 
3.45 

3.47 
3.45 
3.45 
3.45 
3.43 
3.45 
3.45 
3.43 
3.43 
8.43 

8.43 
3.48 
8.43 
8.42 
3.43 
8.42 
3.42 
3.42 
3.42 
3.42 

8.40 
8.42 
8.40 
3.40 
8.40 
8.40 
8.40 
8.40 
8.38 
3.40 

8.88 
3.38 
8.38 
8.38 
8.38 
8.38 
3.87 
8.87 
3.88 

D. 1'. 



OoBine. 



9.933066 
.932990 
.932914 
.932838 
.982762 
.932685 



.932533 
.932457 
.932380 
.932304 

9.982228 
.932151 
.932075 
.931998 
.931921 
.931845 
.931768 
.931691 
.931614 
.931537 

9.931460 
.931383 
.931306 
.931229 
.931152 
.931075 
.930998 
.930921 
.930643 
.930766 

v.VnjUuCXS 

.930611 
.930533 
.980466 
.930378 
.930600 
.930223 
.980145 
.930067 
.929989 

9.929911 
.929833 
.929755 
.929677 
.929599 
.929521 
.929442 
.929364 
.929286 
.929207 

9.929129 
.929060 
.928972 
.928893 
.928815 
.928786 
.928657 
.928578 
.928499 

9.928420 

Sine. 



D. r. 



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

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

.28 
.28 
.28 
.28 
.28 
.28 
.28 
.80 
.28 
.80 

.28 
.80 
.28 
.80 
.80 
.28 
.30 
.30 
.30 
.80 

.80 
.30 
.30 
.30 
.80 
.32 
.30 
.30 
.32 
.30 

.32 
.80 
.32 
.30 
.82 
.82 
.82 
.32 
.32 



D. r. 



Tang. 



9.778774 
.779060 
.779346 
.779682 
.779918 
.780203 
.780489 
.780775 
.781060 
.781346 
.781631 

9.781916 
.782201 
.782486 
.782771 
.783066 
.783341 
.783626 
.783910 
.784195 
.784479 

9.784764 
.786048 
.785332 
.786616 
.785900 
.786184 
.786468 
.786752 
.787036 
.787319 

9.787603 
.787886 
.788170 
.788463 
.788736 
.789019 
.789302 
.789685 

.790151 

9.790434 
.790716 
.790999 
.791281 
.791563 
.791846 
.792128 
.792410 
,792692 
.792974 

9.793256 
.793588 
.793819 
.794101 
.794383 
.794664 
.7949(6 
.796227 
.795508 

9.79578iJ 

Cotang. 



D. r. 



4. 

4. 

4. 

4 

4. 

4. 

4 

4 

4 

4 

4 



77 
77 
77 
77 
75 
77 
77 
75 
77 
76 
75 



4.75 
4.75 
4.75 
4.75 
4.75 
4.75 
4.73 
4.76 
4.73 
4.75 

4.73 
4.73 



73 
73 
73 
73 
73 
78 
4.72 
4.78 

4.72 



.73 
.72 
.72 
72 
72 



4.72 
4.72 
4.72 
4.72 



70 
72 
70 
70 
72 
70 
70 
70 
4.70 
4.70 



.70 
,68 
70 
70 
68 



4.70 
4.68 
4.68 
4.68 

D. 1". 



Cotang. 



10.221226 
.220940 
.220654 
.220868 
.220062 
.219797 
.219511 
.219225 
.218040 
.218654 
.218869 

10.218064 
.217799 
.217514 
.217229 
.216944 
.216659 
.216374 
.216090 
.215805 
.215521 

10.216236 
.214952 
.214668 
.214384 
.214100 
.213816 
.213532 
.213248 
.212964 
.212681 

10.212397 
.212114 
.211880 
.211547 
.211264 
.210961 
.210696 
.210415 
.210132 
.200649 

10.200666 
.200284 
.209001 
.206719 
.208437 
.206154 
.207872 
.207690 
.207808 
.207026 

10.206744 
.206462 
.206181 
.206899 
.205617 
.206336 
.206064 
.204778 
.204492 

10.204211 

Tang. 



60 
59 
68 
57 
56 
65 
54 
53 
62 
61 
60 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
88 
37 
36 
85 
34 
S3 
82 
81 
80 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 


8 
7 
6 
5 
4 
8 
2 
1 




12l< 



185 



58 



82- 



COSINES. TANGENTS, AND COTANGENTS, 



147* 





1 

2 
8 
4 
5 
6 
7 
8 

10 

n 

12 
13 

u 

15 
16 
17 
18 
19 
SO 

21 
22 
23 
24 
26 
26 
27 
28 
29 
30 

31 
82 
33 
34 
85 
36 
37 
88 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
62 
63 
54 
56 
56 
67 
58 
50 
60 



Sine. 



D. r. 



9.724210 
.724413 
.724614 
.724816 
.725017 
.725219 
.725420 
.735622 
.725823 
.726021 
.726225 

.726626 
.726827 
.72;027 
.727228 
.727428 
.727628 
.727828 
.728027 
.728227 

9.728427 
.788626 
.728825 
.729024 
.729223 
.729422 
.729621 
.729620 
.730018 
.780217 

9.730415 
.730613 
.780811 
.781009 
.781206 
.78^404 
.781602 
.731799 
.731996 



9.782890 
.782587 
.732784 
.7ai»80 
.738irr 
.738378 
.733569 
.783785 
.783961 
.784157 

0.784353 
.734549 
.784744 
.734989 
.735135 
.735880 
.785525 
.735n9 
.785014 

0.786109 



I Cosine. 



8.37 
8.37 
3.87 
8.35 
3.87 
3.35 
8.37 
3.35 
3.85 
3.35 
3.35 

8.33 
3.85 
3.33 
8.85 
8.33 
883 
8.38 
8.32 
8.83 
3.83 

3.82 
3.82 
8.32 
8.82 
3.82 
8.82 
8.32 
8.30 
8.32 
8.30 

8.80 
3.30 
8.30 
8.28 
3.80 
3.80 
3.28 
3.28 
8.28 
8.28 

8.28 
8.28 
8.27 
8.28 
8.27 
3.27 
3.27 
8.27 
3.27 
8.27 

3.27 
8.25 
3.28 
8.27 
8.25 
8.25 
8.28 
8.25 
8.25 



Cosine. 



D. 1' 



D. 1". 



9.928420 
.928342 
.928263 
.988183 
.928104 
.928025 
.927946 
.927867 
.927787 
.927708 
.927623 

9.927549 
.927470 
.927890 
.927810 
.927281 
.927151 
.927071 
.926991 
.926911 
.926831 

9.926751 

,9ami 

.926601 
.926511 
.926481 
.926861 
.926270 
.926190 
.926110 
.920029 

9.926040 
.025868 
.926788 
.925707 
.925628 
.925545 
.925466 
.926884 
.925803 
.925222 

9.925141 
.925060 



.924897 
.924816 
.924785 
.924654 
.924573 
.924491 
.924409 

9.924828 
.924246 
.924164 
.924068 
.924001 
.923919 
.928887 
.928755 
.928673 

9928591 



1.80 
1.82 
1.83 
1.82 
1.82 
1.82 
1.82 
1.83 
1.82 
1.32 
1.83 

1.32 
1.33 
1.33 
1.32 
1.83 
1.83 
1.83 
1.83 
1.83 
1.33 

1 

1 

1 

1.33 

1.33 

1.85 

1.83 

1.33 

1.36 

1.33 

1.85 
1.83 
1.85 



83 



85 
85 
33 



1.85 
1.35 
1.35 
1.35 

1.35 

1.35 

1 

1 

1 

1.35 

1 

1 



37 
86 
35 



87 
35 
1.87 

1.37 
1.87 
1.35 
1.87 
1.87 
1.87 
1 87 
1.37 
1.37 



122« 



Sine. I D. 1'. 



186 



Tang. 



D. r. 



9.795789 
.796070 
.796851 
.796682 
.796918 
.797194 
.797474 
.797766 
.796036 
.796816 
.796666 

9.798877 
.799157 
.799437 
.799717 
.799997 
.800277 
.800667 
.800636 
.801116 
.801896 

9.801675 
.801965 
.802234 
.802518 
.802792 
.803072 
.806351 
.808630 
.806909 
.804187 

9.804466 

.804745 
.806028 
.806802 
.805680 
.805869 
.806187 
.806415 
.806603 
.806071 

0.807240 
.807527 
.807805 
.808068 
.808361 
.806688 
.806916 
.809193 
.809471 
.809748 

9.810025 
.810602 
.810680 
.810857 
.811134 
.811410 
.811687 
.811964 
.812241 

0.812517 



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. 
4. 
4. 
4. 
4. 



.67 
.67 
.67 
.67 
.67 
4.65 
4.67 
4.67 
4.65 

4.67 
4.65 
4.65 
4.65 
4.67 
4.65 
4.65 
4.65 
4.63 
4.65 



4 
4. 

4 

4. 

4. 

4 

4. 



63 
63 
65 
68 
65 
63 
63 



4.68 
4.63 
4.63 

4.68 
4.63 
4.68 
4.63 
4.62 
4.63 
4.62 
4.63 
4.62 
4.62 

4.62 
4.63 
4.62 
4.62 
4.60 
4.62 
4.62 
4.62 
4.60 



■f 



Cotang. I D. 1\ 



Cotang. 



10.204211 
.208930 
.208649 
.208868 
.908067 
.202806 
.202526 
.202245 
.201964 
.201684 
.201404 

10.201128 
.200848 
.200668 
.200283 
.200003 
.199?28 
.199448 
.199164 
.196884 
.196604 

10.196325 
.196045 
.197766 
.197487 
.197208 
.196928 
.196649 
.196870 
.196091 
.195818 

10.196584 
.196255 
.194077 
.194096 
.194420 
.194141 
.193863 
.198585 
.193807 
.198029 

10.192751 
.192478 
.192195 
.191917 
.191689 
.191862 
.191064 
.190607 
.190629 
.190262 

10.189075 
.180696 
.180420 
.180143 
.188866 
.188680 
.188818 
.188086 
.187750 

10.167488 



Tang. 



60 
59 
68 
67 
66 
55 
64 
63 
52 
51 
60 

49 
48 
47 
46 
45 
44 
43 
43 
41 
40 

89 
88 
87 
86 
85 
84 
83 
83 
81 
80 

20 
28 
27 
26 
25 
24 
28 
22 
21 
20 

10 
IS 
17 
16 
15 
14 
13 
12 
11 
10 


8 
7 
6 
5 
4 
8 
3 
1 




57« 



S3* 



TABLE X. — LOGARITHMIC SINES, 



146< 




1 
S 
8 

4 
5 

7 

8 

9 

10 

11 
12 
18 
14 
16 
16 
17 
18 
19 

ao 

21 

22 

28 

24^ 

26 

26 

27 

28 

29 

80 

81 
82 
88 
84 
86 
86 
87 
88 
89 
40 

41 
42 
48 
44 
45 
46 
47 
48 
49 
60 

61 
62 
68 
64 
66 
66 
67 
58 
69 
60 



Ll 

188< 



Sin«. 



9.796109 
.796303 
.736498 
.786602 
.796886 
.787080 
.737274 
.787467 
.737661 
.787855 
.788048 

9.788241 
.798434 
.738627 
.788820 
.739013 
.739206 



.739600 
.730788 
.7890r5 

9.740167 
.740659 
.740660 
.740742 
.740984 
.741125 
.741816 
.741506 
.741699 
.741889 

9.742080 
.742271 
.742462 
.742652 
.742842 
.748083 
.743323 
.748418 
.743602 
.748792 

9.743963 
.744m 
.744361 
.744650 
.744789 
.744988 
.745117 
.745806 
.745494 
.745683 

9.745671 
.746060 
.746248 
.746436 
.746624 
.746812 
.746999 
.747187 
.747874 

9.747S62 

Cosine. 



D. r. 



8.28 
3.25 
8.28 
3.28 
8.28 
3.28 
8.22 
3.28 
8.28 
3.22 
3.22 

3.22 
3.22 
8.22 
8.22 
8.22 
8.20 
8.20 
3.22 
8.20 
8.20 

8.20 
8.18 
8.20 
8.20 
8.18 
8.18 
8.20 
3.18 
3.17 
8.18 

3.18 
8.18 
3.17 
8.17 
3.18 
3.17 
8.17 
3.15 
8.17 
8.17 

8.15 
8.17 
8.15 
3.15 
3.15 
8.15 
8.15 
3.13 
8.15 
8.18 

8.15 
8.13 
3.13 
3.13 
3.18 
8.12 
8.18 
3.12 
3.18 

D. r. 



Cosine. 



9.923591 
.928509 
.923427 
.998345 
.923263 
.923181 
.928098 
.928016 
.922983 
.922851 
.922768 

9.922686 
.922608 
.922620 
.922488 
.922355 
.922272 
.922189 
.922106 
.922028 
.921940 

9.921867 
.921774 
.921691 
.921607 
.921524 
.921441 
.921857 
.921274 
.921190 
.921107 

9.921028 
.920989 
.920656 
.920r72 
.920688 
.990604 
.920620 
.920486 
.920362 



9.920184 
.920099 
.920015 
.919981 
.919646 
.919762 
.919677 
.919598 
.919606 
.919424 

9.919839 
.910254 
.919169 
.919065 
.919000 
.918915 
.918830 
.918745 
.918659 

9.918574 

Sine. 



X , 



1.87 
1.37 
1.37 
1.87 
1.87 
1.88 
1.87 
1.88 
1.37 
1.88 
1.87 



88 
88 
87 
88 
88 



1.88 



,88 
.86 
88 



1.86 

1.88 
1.88 
1.40 
1.36 
1.86 
1.40 
1.36 



.40 
.88 



1.40 



.40 
.86 
.40 



1.40 



1, 
1. 



.40 
.40 
1.40 
1.40 
1.40 
1.40 

1.42 



.40 
.40 



1.42 
1.40 
1.42 
1.40 
1.42 
1.40 
1.42 

1.42 
1.42 
1.40 



42 
42 
42 
42 

1.43 
1.42 



D. r. 



137 



Tang. 



9.612517 
.612794 
.613070 
.613347 
.613623 
.813899 
.614176 
.614452 
.814728 
.615004 
.815280 

9.615555 
.615831 
.616107 
.616382 
.616656 
.616938 
.617209 
.617484 
.617759 
.618035 

9.818810 
.818585 
.818860 
.819185 
.619410 
.619684 
.619959 
.820234 
.820506 
.820783 

9.821057 
.821832 
.821606 
.821880 
.822154 
.822429 
.622703 
.882977 
.623251 
.823524 

9.823796 
.824072 
.624345 
.824619 
.624693 
.825166 
.825439 
.825713 
.825986 
.826259 

9.826532 
.826805 
.827078 
.827351 
.827624 
.827897 
.828170 
.826442 
.628715 

9.826967 

Cotang. 



D. r. 



4.62 
4.60 
4.62 
4.60 



4. 
4. 
4. 
4. 
4. 
4. 
4. 

4. 
4. 
4. 

4. 
4. 
4. 
4. 
4. 
4. 



60 
62 
60 
60 
60 
60 
58 

60 
60 
58 
60 
58 
60 
56 
56 
60 



4.56 



.56 
.58 
.58 
.58 
.57 
.58 
4.56 
4.57 
4.58 
4.57 



4. 
4. 
4. 
4. 
4. 
4. 



4. 
4. 
4. 
4. 
4. 
4. 
4. 



.58 
.57 
.57 
.57 
.58 
.57 
.57 
4.57 
4.55 
4.67 

4.67 
4.55 
4.57 
4.57 
4.55 
4.55 
4.57 
4.55 
4.55 
4.55 



4. 
4. 
4. 
4. 
4. 
4. 
4. 



.55 
.55 
.55 
.65 
.55 
.55 
.53 
4.55 
4.53 



D. r. 



Cotang. 



10.167463 
.187206 
.186930 
.186653 
.186877 
.186101 
.185824 
.185648 
.186272 
.184990 
.184720 

10.184445 
.184169 
.188893 
.183618 
.188842 
.188067 
.182791 
.182516 
.182241 
.181965 

10.181690 
.181415 
.181140 
.180665 
.180590 
.180316 
.180041 
.179766 
.179492 
.179217 

10.178943 
.179668 
.178394 
.178120 
.177846 
.177571 
.irr297 
.177023 
.176749 
.176476 

10.176202 
.175928 
.175655 
.175361 
.175107 
.174834 
.174561 
.174267 
.174014 
.173741 

10.173468 
.173195 
.172922 
.172649 
.172876 
.172103 
.171830 
.171558 
.171285 

10.171013 

Tang. 



60 
59 
56 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
88 
87 
36 
35 
34 
33 
32 
31 
30 

29 
26 
27 
26 
25 
24 
23 
22 
21 
20 

19 
16 
17 
16 
15 
14 
13 
12 
11 
10 

9 
8 
7 
6 
5 
4 
3 
2 
1 




66- 



84* 



CX)SINES, TANGENTS. AND COTANGENTS. 



146* 





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 

61 
52 
53 
54 
6S 
66 
57 
58 
59 
60 



1940 



Sine. 



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 
.753495 
.753679 
.753862 
.754046 
.754229 
.754412 
.754595 
.754778 
.754960 

9.755143 
.755326 
.755508 
.755690 
.755872 
.756054 
.756236 
.756418 
.756600 
.756782 

9.756963 
.757144 
.757328 
.757507 
.757688 
.757869 
.758050 
.758230 
.758411 

9.758591 

Cosine. 



D. r. 



3.12 
3.12 
3.12 
3.12 
3.12 
3.10 
3.12 
3.10 
3.12 
3.10 
3.10 

3.10 
3.10 
3.06 
3.10 
8.06 
3.10 
3.06 
3.08 
3.08 
3.06 

3.08 
3.08 
3.07 
3.06 
3.07 
3.07 
3.07 
3.07 
3.07 
3.07 

3.05 
3.07 
3.07 
3.07 
3.05 
3.05 
3.05 
3.05 
3.03 
3.05 

3.05 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.02 

3.02 
303 
3.02 
3.02 
3.02 
3.02 
3.00 
3.02 
3.00 



D. r. 



Cosine. 



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 
.916341 
.916254 
.916167 
.916081 
.915994 

9.915907 
.915820 
.915733 
.915646 
.915559 
.915472 
.915385 
.915297 
.915210 
.915123 

9.915035 
.914948 
.914860 
.914773 
.914685 
.914598 
.914510 
.914422 
.914334 
.914246 

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

9.913365 



Sine. 



D. 1". 



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

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

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

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

.45 

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

.47 

.47 
.47 
.47 
.47 
.47 
.48 
.47 
.47 



D. r. i 



138 



Tang. 



9.828967 
.829260 
.829532 
.829805 
.880077 
.830349 
.830621 
.830693 
.831165 
.831437 
.831709 

9.831981 
.832253 
.832525 
.832796 
.833068 
.833339 
.833611 
.833882 
.834154 
.834425 

9.834696 
.834967 
.835238 
.835509 
.835780 
.836051 
.836322 
.836593 
.836864 
.837134 

9.837405 
.837675 
.a37946 

.838216 
.838487 
.838757 
.839027 
.839297 
.839568 
.839838 

9.840106 
.840378 
.840648 
.U0917 
.841187 
.841457 
.841727 
.841996 
.842266 
.842535 

9.842805 
.843074 
.843343 
.843612 
.813882 
.844151 
.844420 
.844689 
.844958 

9.845227 



D. r. 



Cotang. 



4.55 
4.53 
4.55 
4.53 
4.53 
4.53 
4.53 
4.53 
4.53 
4.53 
4.53 

4.53 
4.33 
4.52 



4. 

4 

4, 

4. 
4. 
4. 
4, 



53 
52 
53 
52 
53 
52 
52 



4.52 
4.52 
4.52 
4.52 
4.52 
4.52 
4.52 
4.52 
4.50 
4.52 

4.50 
4.52 



4 

4. 
4. 
4. 
4. 
4. 
4, 



50 
52 
50 
50 
50 
52 
50 



4.50 



4. 

4. 

4. 

4. 

4. 

4 

4. 

4. 

4. 



50 
50 
48 
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 



Cotang. 



D.l'. 



10.171018 
.170740 
.170468 
.170195 
.169923 
.169661 
.169379 
.169107 
.168835 
.168563 
.168291 

10.168019 
.167747 
.167475 
.167204 
.166932 
.166661 
.166389 
.166118 
.165846 
: 165575 

10.165304 
.165033 
.164762 
.164491 
.164220 
.163949 
.163678 
.163407 
.163136 
.162866 

10.162595 
.162325 
.162054 
.161784 
.161513 
.161243 
.160973 
.160703 
.160432 
.160162 

10.159892 
.159622 
.159352 
.159063 
.158813 
.158543 
.158273 
.158004 
.157734 
.157465 

10.157195 
156926 
.156657 
.156388 
.156118 
.155849 
.155680 
.165811. 
.155042 

10.154773 



Tang. 



60 
50 
58 
57 
56 
55 
54 
58 
52 
51 
50 

49 
48 
47 
46 
45 
44 
48 
42 
41 
40 

S9 
38 
87 
86 
85 
34 
33 
8d 
31 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

19 

18 
17 
16 
15 
14 
18 
12 
11 
10 

9 

8 
7 
6 
5 
4 
8 
2 
1 




W 



85< 



TABLE X. — LOGARITHMIC SINES, 



144< 



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 

81 
82 
88 
84 
85 
86 
87 
38 
39 
40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
60 

61 
52 
63 
54 
55 
56 
57 
58 
SO 
60 



9.758691 
.758772 
.758952 
.760132 
.760312 
.759492 
.759672 
.759852 
.760031 
.780211 
.760390 

9.780560 
.760748 
.760927 
.761106 
.761285 
.761464 
.761642 
.761821 
.761999 
.762177 

9.762356 
.762534 
.768712 
.762889 
.763067 
.763245 
.763422 
.768600 
.763777 
.763954 

9.764181 
.764306 
.764485 
.764662 
.764888 
.766015 
.765191 
.765867 
.766544 
.766720 

9.765896 
.766072 
.766247 
.766423 
.766698 
.766774 
.766949 
.767124 
.767800 
.767476 

9.767649 
.767824 
.767999 
.768178 
.768848 
.7686^3 
.768697 
.768871 
.760045 

9.709219 

Cosine. 



D. r. 



3.02 
3.00 
3.00 
3.00 
8.00 
8.00 
3.00 
2.96 
3.00 
2.96 
2.98 

2.96 
2.96 
2.96 
2.98 
2.98 
2.97 
2.98 
2.97 
2.97 
2.98 

2.97 
2.97 
2.95 
2.97 
2.97 
2.95 
2.97 
2.95 
2.95 
2.95 

2.95 
2.95 
2.95 
2.93 
2.95 
2.93 
2.93 
2.95 
2.93 
2.93 

2.98 
2.92 
2.93 
2.92 
2.93 
2.92 
2.92 
2.93 
2.92 
2.90 

2.92 
2.92 
2.90 
2.92 
2.90 
2.92 
2.90 
2.90 
2.90 

D. 1'. 



Cosine. 



D.V. 



9.918865 
.913376 
.918187 
.918099 
.918010 
.912922 
.912833 
.912744 
.912655 
.912566 
.912477 

9.912388 
.912299 
.912210 
.912121 
.912081 
.911942 
.911858 
.911763 
.911674 
.911684 

9.911496 
.911405 
.911815 
.911226 
.911186 
.911046 
.910956 
.910866 
.910776 
.910686 

9.910696 
.910506 
.910416 
.910625 
.910285 
.910144 
.910054 
.909963 
.909673 
.909782 

9.909691 
.909601 
.909510 
.909419 
.909828 
.909287 
.909146 
.909055 
.908964 
.908873 

9.906781 
.906690 
.906599 
.906507 
.906416 
.906824 
.906233 
.906141 
.906049 

9.907958 

Sine. 



1.48 
1.48 



.47 
.48 
.47 
.48 
.48 
.48 
.48 
.48 
.48 



1.48 
1.48 
1.48 
1.50 
1.48 
1.48 



50 
48 
50 
48 



1.50 
1.60 
1.48 
1.50 
1.50 



.50 
.50 
.50 
.50 



1.60 

1.50 

1 

1 

1 

1 



52 
50 
50 
52 
1.50 



52 
50 
52 
52 



1.50 
1.52 
1.52 
1.52 



52 
62 
52 



1.52 
1.52 
1.53 

1.52 
1.52 
1.53 
1.52 
1.53 
1.52 
1.53 



.53 
.52 



D. r. 



Tang. 



D. r. 



Cotang. 



9.845227 
.845496 
.845764 
.846033 
.846302 
.846570 
.846839 
.847108 
.847376 
.817644 
.847913 

9.848181 
.848449 
.848717 
.848966 
.849254 
.849522 
.849790 
.850057 
.850325 
.850593 

9.850661 
.851129 
.851396 
.851664 
.851931 
.852199 
.852466 
.8527a3 
.853001 
.853268 

9.853535 
.853802 
.854069 
.854336 
.854603 
.854870 
.855137 
.855404 
.855671 
.855938 

9.856204 
.856471 
.856737 
.857004 
.857270 
.857637 
.857803 
.858069 
.858336 
.858602 

9.858868 
.859134 
.859400 
.859666 
.859982 
.860198 
.860464 
.860780 
.860995 

9.861261 

Cotang. 



4.48 
4.47 
4.48 
4.48 
4.47 
4.48 
4.48 
4.47 
4.47 
4.48 
4.47 

4.47 
4.47 
4.48 
4.47 
4.47 
4.47 
4.45 
4.47 
4.47 
4.47 



.47 
45 
47 
45 
,47 
45 
45 
,47 
45 
,45 



4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.43 

4.45 
4.43 
4.45 
4.43 
4.45 
4.43 
4.43 
4.45 
4.43 
4.43 

4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.42 
4.43 

D. r. 



10.154773 
.154604 
.154286 
.153967 
.153696 
.163480 
.153161 
.162892 
.162624 
.162356 
.162087 

10.161819 
.161661 
.151283 
.151014 
.160746 
.160478 
.150210 
.149943 
.149675 
.149407 

10.149139 
.148871 
.148604 
.148336 
.148069 
.147801 
.147534 
.147267 
.146999 
.146732 

10.146465 
.146198 
.146931 
.145664 
.145397 
.145180 
.144868 
.144596 
.144329 
.144062 

10.143796 
.143529 
.143263 
.142996 
.142780 
.142463 
.142197 
.141981 
.141664 
.141398 

10.141132 
.140666 
.140600 
.140334 
.140068 
.139802 
.139536 
.139270 
.139005 

10.138739 

Tang. 



60 
59 
58 
67 
56 
56 
64 
53 
62 
51 
60 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

89 
88 
87 
86 
35 
34 
83 
32 
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 




125' 



1QO 



W 



86' 



COSINES, TANGENTS, AND COTANGENTS. 



143* 



Sine. 




1 
2 
8 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

2t 
SSi 
23 
24 
25 
28 
27 
28 
29 
80 

81 
82 
83 
84 
85 
86 
37 
88 
89 
40 

41 
42 
48 
44 
45 
46 
47 
48 
49 
60 

51 
52 
63 
64 
65 
66 
57 
68 
69 
60 



126' 



9.760219 
.769893 
.769566 
.769740 
.769918 
.770087 
.770260 
.770433 
.770606 
.770779 
.770952 

9.771125 
.771298 
.771470 
.771643 
.771815 
.771987 
.772159 
.772381 
.772503 
.772675 

9.772847 
.773018 
.773190 
.773861 
.773533 
.773704 
.773875 
.774046 
.774217 
.774388 

9.774558 
.774729 
.774899 
.775070 
.775240 
.775410 
.775580 
.775750 
.775920 
.776090 

9.776259 
.776429 
.776598 
.776768 
.776937 
.777106 
.777275 
.777444 
.777618 
.777781 

9.777950 
.778119 
.778287 
.778455 
.778624 
.778792 
.778960 
.779128 
.779295 

9.779463 



Cosine. 



D. r. 



2.90 
2.88 
2.90 
2.88 
2.90 
2.88 
2.88 
2.88 
2.88 
2.88 
2.88 

2.88 

2.87 
2.88 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 



86 

,87 



2.85 

2.87 



2.85 
2.86 
2.86 
2.86 
2.86 
2.83 

2.85 
2.83 
2.85 
2.83 
2.83 
2.83 
2.83 
2.83 
2.83 
2.82 

2.83 
2.82 
2.88 
2.82 
2.82 
2.82 
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 



D. r. 



Cosine. 



9.907958 
.907866 
.907774 
.907682 
.907590 
.907498 
.907406 
.907314 
.907222 
.907129 
.907037 

9.906945 
.906852 
.906760 
.906667 
.906575 
.906482 
.906389 
.906296 
.906204 
.906111 

9.906018 
.905925 
.905882 
.905739 
.905645 
.906652 
.905459 
.905366 
.905272 
.905179 

9.905065 
.904992 
.904898 

.904711 
.904617 
.904523 
.904429 
.904835 
.904241 

9.904147 
.904053 
.903959 
.903864 
.903770 
.903676 
.903581 
.903487 
.908392 
.903298 

9.903203 
.903108 
.903014 
.902919 
.9028^ 
.902729 
.902634 
.902539 
.902444 

9.902349 



Sine. 



D. r. 



.53 
.53 
.53 
.53 
.53 
.53 
.53 
.53 
.55 
.53 
.53 

.55 
.53 
.55 
.53 
.55 
.55 
.55 
.53 
.65 
.56 

.65 
.55 
.55 
.57 
.55 
.55 
.55 
.57 
.55 
.57 

.55 
.57 
.57 
.55 
.57 
.57 
.57 
.57 
.57 
.57 

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

.58 
.57 
.58 
.58 
.58 
.58 
.58 
.5(5 
.58 



D. r. 



140 



Tang. 



9.861261 
.861527 
.861792 
.862058 
.862323 
.862589 
.862854 
.863119 
.863385 
.863650 
.863915 

9.864180 
.864445 
.864710 
.864975 
.865240 
.865505 
.865770 
.866035 
.866300 
.866564 

9.866829 
.867004 
.867358 
.867623 
.867887 
.868152 
.868416 
.868680 
.868945 
.869209 

9.869478 
.869737 
.870001 
.870265 
.870629 
.870793 
.871057 
.871321 
.871585 
.871849 

9.872112 
.872376 
.872640 
.872903 
.873167 
.873480 
.873694 
.873957 
.874220 
.874484 

9.874747 
.875010 
.875273 
.875587 
.875800 
.876063 
.876326 
.876589 
.876852 

9.877114 



Cotang. 



D. 1". 



4. 
4. 
4. 
4. 
4. 
4. 



.43 
.42 
.43 
.42 
.43 
.42 
4.42 
4.43 
4.42 
4.42 
4.42 

4.42 
4.42 
4.42 



42 
42 
.42 
42 



4.42 



.40 
.42 



4.42 

4.40 
4.42 
4.40 
4.42 
4.40 
4.40 
4.42 
4.40 
4.40 



4. 
4. 
4. 
4. 
4. 



.40 
.40 
.40 
.40 
.40 
4.40 
4.40 
4.40 
4.40 
4.38 



4. 
4. 
4. 
4. 
4. 



.40 
.40 
.88 
.40 
.88 
4.40 
4.88 
4.88 
4.40 
4.88 



4. 
4. 
4. 
4. 
4. 



.88 
.38 
.40 
.88 
.38 
4.88 
4.88 
4.88 
4.37 



D. r. 



Cotang. 



10.138730 
.138473 
.138206 
.187942 
.137677 
.137411 
.137146 
.136881 
.186615 
.136350 
.136065 

10.135620 
.135555 
.135290 
.135025 
.134760 
.134495 
.134280 
.133965 
.133700 
.133436 

10.133171 
.132906 
.132642 
.132377 
.132113 
.131848 
.131684 
.131320 
.131065 
.130791 

10.130527 
.130263 
.129999 
.129735 
.129171 
.129207 
.128948 
.128679 
.128415 
.128151 

10.127888 
.127624 
.127360 
.127097 
.126888 
.126570 
.126806 
.126043 
.125780 
.125516 

10.125253 
.124990 
.124727 
.124463 
.124200 
.123987 
.123674 
.123411 
.123148 

10.122886 

Tang. 



60 
69 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
87 
86 
85 
34 
33 
32 
31 
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 
6 
4 
8 
2 
1 




5S< 



87« 



TABLE X.— LOGAniTttMIC SINES, 



142« 





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 

81 
32 
33 
34 
35 
36 
87 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 



9.779463 
.779631 
.779798 
.779966 
.780133 
.780900 
.780467 
.780634 
.780801 
.780968 
.'^^1134 

9.781301 
.781468 
.781634 
.781800 
.781966 
.782132 
.78S296 
.782464 
.782630 
.782796 

9.782961 
.783127 
.783292 
.783458 
.783623 
.783788 
.783953 
.784118 
.784282 
.784447 

9.784612 
.784776 
.784941 
.785106 
.785269 
.785433 
.785597 
.785761 
.786925 
.786089 

9.786252 
.786416 
.786579 
.786742 
.786906 
.787069 
.787332 
.787395 
.787557 
.787720 

9.787888 
.788045 
.788206 
.788370 
.788532 
.788694 
.788866 
.789018 
.789180 

9.789842 

Cosine. 



D. r. 



2.80 
2.78 
2.80 
2.78 
2.78 
•2.78 
2.78 
2.78 
2.78 
2.77 
2.78 

2.78 
2.77 
2.77 
2.77 
2.77 
2.77 
2.77 
2.77 
2.77 
2.75 

2.77 
2.75 
2.77 
2.75 
2.75 
2.75 
2.75 
2.73 
2.75 
2.75 

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 
2.78 
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 
.9018?2 
.901776 
.901681 
.901585 
.901490 
.901394 

9.901298 
.901202 
.901106 
.901010 
.900914 
.900618 
.900722 
.900626 
.900629 
.900433 

9.900337 
.900240 
.900144 
.900047 
.899951 
.899654 
.899757 
.899660 
.899564 
.899467 

9.899370 
.899873 
.899176 
.899078 
.896961 
.896884 
.898787 
.898689 
.896592 
.896494 

9.896397 
.896299 
.896202 
.896104 
.898006 
.897906 
.897810 
.897712 
.897614 
.897516 

9.897418 
.897320 
.897222 
.897123 
.897026 
.896926 
.896828 
.896729 
.896631 

9.896532 

Sine. 



D. 1". 



.60 
,58 
,58 
60 
,58 
,60 
,58 
.60 
.58 
.60 
,60 

.60 
,60 
,60 
,60 
,60 
.60 
.60 
.62 
.60 
60 

.62 
.60 
.62 
.60 
.62 
.62 
.62 
.60 
.62 
.62 

.62 
,62 
,63 
.62 
,62 
.62 
,63 
.62 
.63 
,62 

,63 
,62 
63 
,63 
,63 
,63 
.63 
.63 
,63 
,63 

63 
63 
65 
,63 
65 
63 
,65 
,63 
,65 



D. r. 



Tang. 



9.67ni4 
.877377 
.877640 
.877903 
.878165 
.878428 
.878691 
.878963 
.879216 
.879478 
.879741 

9.880008 
.880265 
.880628 
.880790 
.881052 
.881814 
.881577 
.881839 
.882101 
.882868 

9.882626 
.88S887 
.883148 
.883410 
.883672 
.883934 
.884196 
.884457 
.884719 
.884960 

9.886842 
.886504 
.885765 
.886026 
.886288 
.886649 
.886811 
.887072 
.887383 
.887594 

9.887856 
.888116 
.888378 
.888639 
.888900 
.889161 
.889421 
.889682 
.889943 
.890204 

9.890466 
.890726 
.890966 
.891247 
.891507 
.891768 
.892088 



.892649 
9.892810 

Cotang. 



D. r. 



.38 
.38 
.88 
.87 
.38 
.38 
.87 
.38 
.87 
.38 
.87 

.37 
.38 
.37 
.87 
.87 
.88 
.87 
.87 
.87 
.87 

.87 
.35 
.37 
.87 
.87 
.37 
.36 
.37 
.35 
.37 

.37 
.35 
.35 
.37 
.85 
.37 
.35 
.35 
.35 
.35 

.86 
.87 
.36 
.86 
.85 
.33 
.86 
.36 
.85 
.86 

.83 
.36 
.36 
.83 
.86 
.88 
.86 
.33 
.85 



D. r. 



Cotang. 



10.122886 
.122623 
.122360 
.122097 
.121835 
.121672 
.121309 
.121047 
.120784 
.120522 
.120259 

10.119997 
.119735 
.119472 
.119210 
.118948 
.118686 
.118423 
.118161 
.117899 
.117887 

10.117875 
.117113 
.116852 
.116590 
.116328 
.116066 
.115804 
.115543 
.116281 
.115020 

10.114758 
.114496 
.114236 
.118974 
.113712 
.113451 
.113189 
.112928 
.112667 
.112406 

10.112146 
.111884 
.111622 
.111861 
.111100 
.110689 
.110579 
.110818 
.110067 
.109796 

10.109635 
.109275 
.109014 
.106753 
.106493 
.106282 
.107972 
.107711 
.107451 

10.107190 

Tang. 



60 
59 
58 
57 
66 
65 
64 
63 
52 
51 
50 

49 
48 
47 
46 
46 
44 
43 
42 
41 
40 

89 
88 
87 
86 
86 
84 
83 
82 
81 
80 

29 
28 
27 
26 
25 
24 
28 
22 
21 
20 

19 
18 
17 
16 
16 
14 
IS 
12 
11 
10 

9 
8 
7 
6 
5 
4 
8 
2 
1 




127' 



141 



62» 



Xr. Oorino. D. r. 



..[h^. 



I 



40* 



COSINES, TANaENTS. AND COTANaENTS. 





139* 


Ootang. 


60 


10.076186 


.O7B09O 


SO 


.075678 


58 


.076417 


57 


.075160 


56 


.074904 


55 


.074648 


54 


.074881 


5» 


.074185 


60 


.073878 


51 


.078682 


50 


10.078866 


49 


.078110 


48 


.078868 


47 


.072607 


46 


.072841 


45 


.072086 


44 


.071889 


48 


.071678 


4S 


.071816 


41 


.071060 


40 


10.070804 


80 


: .070648 


88 


.070892 


87 


.070086 


86 


.069780 


85 


.069685 


84 


.069269 


38 


.069018 


33 


.068767 


31 


.068601 


80 


10.068246 


20 


.067990 


28 


.067734 


27 


.067478 


26 


.067282 


25 


.066067 


24 


.066711 


23 


.066466 


22 


.066200 


21 


.066044 


20 


10.066689 


19 


.066483 


18 


.066178 


17 


.064922 


16 


.064667 


15 


.064411 


14 


.064166 


18 


.068900 


12 


.068646 


11 


.068889 


10 


10.068184 


9 


.068879 


8 


.068688 


7 


.068868 


6 


.062118 


5 


.061868 


4 


.061602 


8 


.061847 


2 


.061082 


1 


10.060887 




1 


Tang. 





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 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 

61 
62 
63 
54 
56 
56 
67 
58 
59 
60 



Bine. 



9.808067 
.808218 
.806368 
.808619 
.806669 
.808819 
.808969 
.809119 
.809269 
.809419 
.809569 

9.809718 
.809668 
.810017 
.810167 
.810316 
.810465 
.810614 
.810763 
.810912 
.811061 

9.811210 
.811358 
.811507 
.811655 
.811804 
.811952 
.812100 
.812248 
.812396 
.812544 

9.812692 
.812840 
.812988 
.813185 
.813283 
.813430 
.813578 
.813725 
.813872 
.814019 

9.814166 
.814313 
.814460 
.814607 
.814753 
.814900 
.815046 
.815193 
.815339 
.815485 

9.815632 
.815778 
.815924 
.816069 
.816215 
.816361 
.816507 
.816652 
.816798 

9.816943 



D. r. 



Cosine. 



2.52 
2.50 
2.52 
2.50 
2.60 
2.50 
2.50 
2.50 
2.50 
2.50 
2.48 

2.50 
2.48 
2.50 
2.48 
2.48 
2.48 
2.48 
2.48 
2.48 
2.48 

2.47 
2.48 
2.47 
2.48 
2.47 
2.47 
2.47 
2.47 
2.47 
2.47 



2. 
2. 
2. 



2.47 
2.47 
.45 
.47 
.45 
2.47 
2.45 
2.45 
2.45 
2.45 

2.45 
2.45 
2.45 
2.43 
2.45 
2.43 
.45 
.43 
.43 
.45 



2. 
2. 
2. 
2. 



2.43 
2.43 
2.42 
43 
43 
43 
42 
2.43 
2.42 



Cosine. 



D. r 



9.884264 
.884148 
.884042 
.888036 
.888889 
.883723 
.883617 
.883510 
.883404 
.883297 
.883191 

9.883084 
.882977 
.8888n 
.888764 
.882667 
.882660 
.882443 
.882336 
.888229 
.888121 

9.882014 
.881907 
.881799 
.881692 
.881584 
.881477 
.881369 
.881261 
.881153 
.881046 

9.880988 
.880830 
.880722 
.880613 
.880505 
.880397 
.880289 
.880180 
.880072 
.879963 

9.879855 
.879746 
.879637 
.879629 
.879420 
.879311 
.879802 
.879093 
.878984 
.878875 

9.878766 
.878656 
.878547 
.878438 
.878388 
.878219 
.878109 
.877999 
.877890 

9.877780 



D. r. 



Sine. 



1.77 
1.77 
1.77 
1.78 
1.77 
1.77 
1.78 
1.77 
1.78 
1.77 
1.78 

1.78 
1.77 
1.78 
1.78 
1.78 



.78 
.78 
.78 
.80 
.78 



1.78 
1.80 
1.78 



.80 
.78 
.80 
.80 
.80 



1.78 
1.80 



80 
80 
82 
80 
80 
80 
82 
80 
82 



1.80 

1.82 
1.82 
1.80 
1.82 
1.82 
1.88 
1.82 
1.82 
1.88 
1.82 

1.88 
1.82 
1.82 
1.83 
1.82 
1.83 
1.83 
1.82 
1.83 



D. r. 



Tang. 



9.928814 
.924070 
.924887 
.924588 
.924840 
.925096 
.925852 
.ussotxw 
.926865 
.926122 
.926378 

9.926634 
.926890 
.927147 
.927403 
.927669 
.927915 
.988171 
.988427 
.988684 
.988940 

9.929196 
.929462 
.929708 
.929964 
.930280 
.990475 
.980781 
.930987 
.931243 
.931499 

9.931765 
.982010 
.938266 
.938588 
.988778 
.938088 
.938889 
.988646 
.988800 
.934066 

9.934811 
.934567 
.934888 
.986078 
.985888 

.wluOOv 

.986844 
.986100 
.986865 
.986611 

0.986866 
.937121 
.987877 
.937682 
.937887 
.988142 
.988898 
.988658 
.988906 

9.989168 



D. 1'. 



Cotang. 



4.27 
4.28 
4.27 
4.28 
4.27 
4.27 
4.28 
4.27 
4.28 
4.27 
4.27 

4.27 
4.28 
4.27 
4.27 



27 
27 

,27 
.28 
27 



4.27 

4.27 
4.27 
4.27 
4.27 
4.25 
4.27 
4.27 
4.27 
4.27 
4.27 



4 
4 
4 
4 
4 
4 
4 
4 
4 
4 



25 
27 
27 
27 
25 
27 
27 
25 
27 
25 



4. 
4. 
4. 
4. 
4. 



.27 
.25 
.27 
.26 
.27 
4.25 
4.27 
4.2(» 
4.27 
4.25 

4.25 
4.27 
4.25 
4.25 
4.25 
4.27 
4.25 
4.25 
4.25 



D. r. 



180« 



144 



41* 



TABLE X.— -LOGARITHMIC SINES, 



138' 



Sine. 




1 
2 
S 
4 
5 
6 
7 
8 
9 
10 

11 
13 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
21 
25 
26 
27 
28 
29 
30 

31 
32 
33 
84 
35 
86 
87 
88 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 

51 
62 
68 
54 
55 
56 
57 
68 
60 
00 



lar 



9.816943 
.817088 
.817233 
.817379 
.817524 
.817668 
.817813 
.817958 
.818103 
.818247 
.818302 

9.818536 
.818681 
.818825 
.818969 
.819113 
.819257 
.819401 
.819545 
.819689 
.819832 

9.819976 
.820120 
.820263 
.820i06 
.820550 
.820693 
.820836 
.820979 
.821122 
.821265 

9.821407 
.821550 
.821693 
.821835 
.821977 
.822120 



.822401 
.822546 
.822688 

9.822830 
.822972 
,823114 
.823255 
.823397 
.823539 
.823680 
.823821 
.823963 
.824104 

9.824245 
.824386 
.824527 
.824668 
.824806 
.824949 
.825090 
.825230 
.825371 

9.825511 

Cosine. 



D. r. 



2.42 
2.42 
2.43 
2.42 
2.40 
2.42 
2.42 
2.42 
2.40 
2.42 
2.40 

2.42 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 
2.38 
2.40 

2.40 
2.38 
2.38 
2.40 
2.38 
2.88 
2.38 
2.38 
2.38 
2.37 

2.38 
2.38 
2.37 
2.87 
2.38 
2.37 
2.37 
2.37 
2.37 
2.37 

2.37 
2.37 
2.35 
2.37 
2.37 
2.35 
2.85 
2.37 
2.35 
2.35 

2.35 
2.35 
2.35 
2.33 
2.35 
2.35 
2.33 
2.35 
2.33 

D. 1'. 



Cosine. 



D. r. 



9.877780 
.877070 
.877560 
.877450 
.877340 
.877230 
.8T?120 
.877010 
.876899 
.876789 
.876678 

9.876568 
.876457 
.876347 
.876236 
.876125 
.876014 
.875901 
.875793 
.875682 
.875571 

9.875459 
.875348 
.875237 
.B75126 
.875014 
.874903 
.874791 
.874680 
.874568 
.874456 

9.874344 
.874232 
.874121 
.874009 
.873896 
.873784 
.878672 
.873560 
.873448 
.873335 

9.873223 
.873110 
.872998 
.872885 
.872772 
.878659 
.8ra547 
.872434 
.872321 
.872208 

9.872095 
.871981 
.871868 
.871755 
.871641 
.871528 
.871414 
.871301 
.871187 

9.871078 

Sine. 



.1.83 
1.83 
1.83 
1.83 
1.83 
1.83 
1.83 
1.85 
1.83 
1.85 
1.83 

1.85 
1.83 
1.85 
1.85 
1.85 
1.83 
1.85 
1.85 
1.85 
1.87 



.85 
.85 
.&) 
.87 
.85 
.87 
.85 
.87 
.87 



1.87 



87 
85 
87 
88 
87 
87 
87 
87 
1.88 
1.87 



.87 
.88 
.88 
.88 
.87 
1.88 
1.88 
1.88 
1.88 

1.90 



1, 

1, 

1. 

1 

1. 

1. 



.88 
.88 
.90 



1.88 
1.90 
1.88 
1.90 
1.90 



D. r 



Tang. 



9.939163 
.939418 
.939673 
.939928 
.940183 
.940439 
.940694 
.940949 
.941204 
.941459 
.941713 

9.941968 
.942223 
.942478 
.942733 
.942988 
.943243 
.948498 
.943752 
.914007 
.944262 

9.944517 
.944771 
.945026 
.945281 
.945535 
.945790 
.946045 
.946299 
.946654 
.946808 

9.947063 
.947818 
.947572 
.947827 
.948061 
.948835 
.948590 
.948644 
.949099 
.949353 

9.949608 
.949862 
.950116 
.950371 
.950625 
.960879 
.951133 
.951368 
.951642 
.951896 

9.952150 
.952405 
.952659 
.952913 
.953167 
.953421 
.953675 
.958929 
.954183 

9.954437 

Cotang. 



D. r. 



4.25 
4.25 
25 
25 
27 
25 
25 
25 
4.25 
4.28 
4.25 



4.25 
4.25 
4.25 



4. 
4. 
.4. 
4. 
4. 



.25 
.25 
.25 
.23 
.25 
4.25 
4.25 



4.23 
4.25 



4. 
4. 
4 
4 
4 
4, 
4 
4, 



25 
28 
25 
25 
23 
25 
23 
25 



4. 
4. 
4. 
4. 
4. 



.25 
.23 
.25 
.23 
.23 
4.25 
4.23 
4.25 
4.23 
4.25 



.23 
23 
25 
23 
23 
23 
25 
23 
23 
23 



4. 
4. 
4. 
4. 
4. 
4. 



.25 
.28 
.28 
.23 
.23 
.23 
4.23 
4.23 
4.23 



Cotang. 



D. r. 



10.060887 
.060682 
.06082^/ 
.060072 
.050617 
.059561 
.059306 
.059051 
.056796 
.056541 
.058287 

10.058032 
.057777 
.057522 
.057267 
.057012 
.056757 
.066502 
.056248 
.055993 
.055738 

10.055483 
.055229 
.054974 
.054719 
.054165 
.054210 
.063955 
.053701 
.058446 
.053192 

10.052987 
.052682 
.052428 
.052178 
.051919 
.051665 
.051410 
.051156 
.060901 
.060647 

10.050892 
.060138 
.049684 
.049629 
.049375 
.049121 
.048867 
.048612 
.046358 
.048104 

10.047860 
.047596 
.047841 
.047067 
.046633 
.046579 
.046325 
.046071 
.015817 

10.045668 



60 
59 
58 
57 
56 
65 
64 
53 
52 
61 
tJ 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

89 
S3 
87 
36 
35 
84 
83 
82 
81 
SO 

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 




Tang. I ' 



145 



W 



I 



] 



I 



1 



"Tl 




^^I 081. 






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