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Full text of "Practical treatise on gearing"

Class _rJil^l„ 

CopyrightE^ L^ 

COPYRIGHT DEPOSI-K 



/ 



//3 



Practical Treatise 

m^Gearing. 



EIGHTH EDITION. 



BROWN & SHARPK MANUFACTURING CO. 

PROVIDENCE, R. I., U. S. A. 
1905 



C^ -2.- 



THE LIBRARY OF 
CONGRESS. 

Two Copies Received ! 
MAR 80 1305 
Oopyrignt Entry 
m^i^ 36 f90S 
CLASS f^ XXs, Not 

©•IPY A. 



X 



A \ 



%' 






COPYRIGHT, 

188G, 1887, 1892, 1893, 1896, 1900, 1902, 1905, 

BY 

BROWN & SHARPE MFG. CO. 



0^-IOI(pO <u 



1^ 



PREFACE. 



This Book is made for men in practical life ; for those that 
would like to know how to construct gear wheels, but whose 
duties do not afford them suflficient leisure to acquire a technical 
knowledge of the subject. 



CONTENTS. 



P A K T I , 
Chapter I. 



PAGE. 

Pitch. Circle — Pitch — Tooth — Space — Addendum or Face — 

Flank — Clearance 1 

Chapter II. 

Classification — Sizing Blanks and Tooth Parts from Linear 

or Circular Pitch — Center Distance 5 

Chapter III. 
Single Curve Gears of 30 Teeth and more 9 

Chapter IV. 

Pack to Mesh with Single Curve Gears having 30 Teeth and 

more 12 

Chapter V. 

Diametral Pitch — Sizing Blanks and Teeth — Distance be- 
tween the Centers of Wheels 16 

Chapter VI. 

Single-Curve Gears, having Less than 30 Teeth — Gears and 

Packs to Mesh with Gears having Less than 30 Teeth... 20 

Chapter VIL 
Double-Curve Teeth— Gear of 15 Teeth— Pack 25 

Chapter VIII. 

Donble-Curve Gears, having More and Less than 15 Teeth 

— Annnlar Gears 30 



VI CONTENTS. 

Chapter IX. 

PAGE. 

Bevel Gear Blanks 34 

Chapter X. 
Bevel Gears — Form and Size of Teeth — Cutting Teeth 41 

Chapter XI. 
Worm Wheels — Sizing Blanks of 32 Teeth and more 63 

Chapter XII. 

Sizing Gears when the Distance between the Centers and the 
Katie of Speeds are fixed — General Eemarks — Width of 
Face of Spur Gears — Speed, of Gear Cutters — Table of 
Tooth Parts 79 



PART II, 



Chapter I. 
Tangent of Arc and Angle 87 

Chapter II. 

Sine, Cosine and Secant — Some of their Applications in 

Machine Construction 93 

Chapter III. 

Application of Circular Functions — Whole Diameter of Bevel 

Gear Blanks — Angles of Bevel Gear Blanks 100 

Chapter IV. 
Spiral Gears — Calculations for Pitch of Spirals 107 

Chapter V. 

Examples in Calculations of Pitch of Spirals — Angle of 
Spiral — Circumference of Spiral Gears — A few Hints 
on Cutting Ill 



CONTENTS. VII 

Chapter VI. 



PAGE. 



Normal Pitch of Spiral Gears — Curvature of Pitch Surface 

— Formation of Cutters 114 

Chapter VII. 
Cutting Spiral Gears in a Universal Milling Machine......... 120 

Chapter VIII. 
Screw Gears and Spiral Gears — General Eemarks 127 

Chapter IX. 

Continued Fractions — Some Applications in Machine Con- 
struction ' 130 

Chapter X. 
Angle of Pressure 135 

Chapteb XI. 
Internal Gears 187 

Chapter XII. 

Strength of Gears — Tables of Tooth Parts, Tables Sine, 

Cosine, etc., Index 140 



PART I. 

CHAPTER I. 

PITCH CIRCLE, PITCH, TOOTH, SPACE, ADDENDUM OR FACE, FLANK, 

CLEARANCE. 



Let two cylinders, Fie-. 1, touch each other, their Original Cyi- 

'^ ' ° ' ' Indera. 

axes be parallel and the cylinders be on shafts, turning 
freely. If, now, we turn one cylinder, the adhesion of 
its surface to the surface of the other cylinder will 
make that turn also. The surfaces touching each 
other, without slipping one upon the other, will evi- 
dently move through the same distance in a given , . 

'' ° ° Linear Veloci- 

time. This surface speed is called linear velocity. ty. 
TANGENT CYLINDERS. 



^ 




44. 







Linear Velocity is the distance a point moves along 
a line in a unit of time. 

The hne described by a point in the circumference 
of either of these cylinders, as it rotates, may be called 
an arc. The length of the arc (which may be greater 
or less than the circumference of cylinder), described 
in a unit of time, is the velocity. The length, expressed 
in linear units, as inches, feet, etc., is the linear velocity. 



Z BKOWN & SHARPE MFG. CO. 

The length, expressed in angular units, as degrees, is 
the angular velocity. 

If now, instead of 1° we take 360°, or one turn, as 
lo^i°y "^^^ ^^ ^^® angular unit, and 1 minute as the time unit, the 
angular velocity will be expressed in turns or revolu- 
tions per minute. 

If these two cylinders are of the same size, one will 
make the same number of turns in a minute that the 
other makes. If one cylinder is twice as large as the 
other, the smaller will make two turns whUe the larger 
makes one, but the linear velocity of the surface of 
each cylinder remains the same. 

This combination would be very useful in mechan- 
ism if we could be sure that one cylinder would always 
turn the other without slipping. 



Relative An 
gular Velocity 




T^ig. 3 




C/RCLE 



In the periphery of these two cylinders, as in Fig. 
2, cut equidistant grooves. In any grooved piece the 
Land. places between grooves are called lands. Upon the 

Addendum, lands add parts ; these parts are called addenda. A 
Tooth. land and its addendum is called a tooth. A toothed 

Gear. cylinder is called a gear. Two or more gears with 

Train. teeth interlocking are called a trai7i. A line, c c', Fig. 



PROVIDENCE, R. I. 



2 or 3, between the centers of two wheels is called the Line of cen- 

line of centers. A circle just touching the addenda *®'"®' 

is called the addendum circle. circle! ^"^^^^^ 

The circumference of the cylinders without teeth is 
called the pitch circle. This circle exists geometri- -^^^"^ Circle, 
cally in every gear and is still called the pitch circle p.^^j^ circle 
or the primitive circle. In the study of gear wheels, it t^ie*^^ Motive 
is the problem so to shape the teeth that the pitch Circle, 
circles will just touch each other without slipping. 

On two fixed centers there can tiu'n only two circles, 
one circle on each center, in a given relative angular 
velocity and touch each other without slipping. 




inig. 4 



4 BROWN & SHARPE MFG. CO. 

Space. The groove between two teeth is called a space. In 

cut gears the width of space at pitch line and thickness 

of tooth at pitch line are equal. The distance between 

the center of one tooth and the center of the next tooth, 

LinearorCir- measured aloug the pitch line, is the linear or circular 

cular Pitch. . . 

pitch; that is, the linear or circular pitch is equal to a 
Tooth Thick- tooth and a space; hence, the thickness of a tooth at 
the pitch line is equal to one-half the linear or circular 
pitch. 
tioiis^of*^parts ^^^ D=diameter of addendum circle. 
g>JJ«eth and .< D'= diameter of pitch circle. 
" P'= linear or circular pitch. 
" if = thickness of tooth at pitch line. 
** s = addendum or face, also length of working 

part of tooth below pitch line or flank. 
" 2s= D" or twice the addendum, equals the work- 
ing depth of teeth of two gears in mesh. 
*' /= clearance or extra depth of space below work- 
ing depth. 
" s+/= depth of space below pitch line. 
" D"+/= whole depth of space. 
" N^ number of teeth in one gear. 
** 7r=3.1416 or the circumference when diameter 
isl. 
P' is read "P prime." D" is read **D second." ttIs 
read "pi." 
To And the If we multiply the diameter of any circle by n, the 

Circumf ©rcriCG 

and Diameter product will be the circumfereucB of this circle. If we 

of 3i Oirclp 

divide the circumference of any circle by 7t, the quo- 
tient will be the diameter of this circle. 
Pitch Point. The pitch point of the side of a tooth is the point at 
which the pitch circle or line meets the side of the 
tooth. A gear tooth has two pitch points. 



CHAPTER II. 

CLASSIFICATION-SIZING BLANKS AND TOOTH PARTS FROM 
CIRCULAR PITCH— CENTRE DISTANCE— PATTERN GEARS. 



If we conceive the pitch of a pair of gears to be ^j^^iements of 
made the smallest possible, we ultimately come to the 
conception of teeth that are merely lines upon the 
original pitch surfaces. These lines are called ele- 
ments of the teeth. Gears may be classified with 
reference to the elements of their teeth, and also with 
reference to the relative position of their axes or -shafts. 
In most gears the elements of teeth are either straight 
lines or helices (screw-like lines). 

Part I. of this book, treats upon three kinds of 

GEARS. 

First — Spub Geaes ; those connecting parallel shafts ^p^^ Gears. 
and whose tooth elements are straight. 

Second — Bevel Gears ; those connecting shafts Bevei Gears, 
whose axes meet when sufficiently prolonged, and the 
elements of whose teeth are straight lines. In bevel 
gears the surfaces that touch each other, without 
slipping, are upon cones or parts of cones whose 
apexes are at the same point where axes of shafts meet. 

Third — Screw ok Worm Gears; those connecting ^1°^^ ^J^^^^.^' 
shafts that are not parallel and do not meet, and the 
elements of whose teeth are helical or screw-like. 

The circular pitch and number of teeth in a wheel g_^^ 
being given, the diameter of the wheel and size of Blanks, &c. 
tooth parts are found as follows : 

Dividing by 3.1416 is the same as multiplying by 
rrln- -^^"^ s.ilie —-^-*-^^' hence, multiply the cu'- 
cumference of a circle by .3183 and the product will be 
the diameter of the circle. Multiply the cu'cular pitch 
by .3183 and the product will be the same part of the 



6 BROWN & SHARPS MFG. CO. 

diameter of pitch circle that the circular pitch is of the 
^ j^.g^j^g^gj. circumference of pitch circle. This part is called the 

Pitch, or ^oa module of the pitch. There are as many modules con- 
tained in the diameter of a pitch circle as there are 
teeth in the wheel. 

andtheAdden- Most mechanics make the addendum of teeth equal 

dum measure,! n i tt j • i j.i ^ ^ ^ 

thesame,radi- toe module. Hence we can designate the module by 
^"^" the same letter as we do the addendum; that is, let 5 = 

the module. 

.3183 P — s, or circular pitch multiplied by .3183 =s, 
or the module. 
Diameter of Ns = D', or number of teeth in a wheel, multiplied 
Pitch Circle, -^y the modulc, equals diameter of pitch circle. 

(N+2) s = D, or add 2 to the number of teeth, mul- 
g^^'^^^^^^^^^'tiply the sum by the module and product will be the 
whole diameter. 
■—=/, or one tenth of thickness of tooth at pitch line 
Clearance. equals amount added to bottom of space for clearance. 
Some mechanics prefer to make / equal to j^g- of the 
working depth of teeth, or .0625 D". One-tenth of the 
thickness of tooth at pitch-line is more than one-six- 
teenth of working depth, being .07854 D". 
Example. Example.— Wheel 30 teeth, li" circular pitch. P'= 

Sizes of Blank ^•^"' ^^^^^ ^=.75" or thickness of tooth equals f". 5 = 
parfs for'aear ^'S" X '3183 =.4775 = module for 11" P'. (See table of 
9^^ 30 teeth^i3| tooth parts, pages 144-147. 

Pitch. D'=30x.4775"=14.325"=diameter of pitch-circle. 

D = (30+2) X. 4775"= 15.280"=diameter of adden- 
dum circle, or the diameter of the blank. 
f=j^ of .75"=.075"= clearance at bottom of space. 
"D"=r=2x.4775"=.9549"== working depth of teeth. 
D"-f/=2x. 4775"+. 075"=1.0299"= whole depth of 
space. 

s+/=.4775"+.075"=.5525"= depth of space inside 
of pitch-line. 

D"=2s or the working depth of teeth is equal to two 
modules. 

In making calculations it is well to retain the fourth 
place in the decimals, but when drawings are passed 
into the workshop, three places of decimals are suffi- 
cient. 



PROVIDENCE, K. I. 




Fig. 5, Spur Gearing. 



8 BEOWN & SHAKPE MFG. CO. 

tw^en^'centers ^^^ distance between the centers of two wheels is 

of two Gears, evidently equal to the radius of pitch-circle of one wheel 

added to that of the other. The radius of pitch-circle 

is equal to s multiplied by one-half the number of teeth 

in the wheel. 

Hence, if we know the number of teeth in two wheels, 
in mesh, and the circular pitch, to obtain the distance 
betAveen centers we first find s ; then multiply s by one- 
half the sum of number of teeth in both wheels and the 
product will be distance between centers. 

Example. — What is the distance between the centers 
of two wheels 35 and 60 teeth, 1^" circular pitch. "We 
first find s to be l^'x .3183=. 3979". Multiplying by 
47.5 (one-half the sum of 35 and 60 teeth) we obtain 
18.899" as the distance between centers. 
Shr\nTa°gl *fn Pattern GearS should be made large enough to 
Gear Castings, allow for shrinkage in casting. In cast iron the shrinkage 
is about ^ inch in one foot. For gears one to two feet 
in diameter it is well enough to add simply -j-oo" of 
diameter of finished gear to the pattern. In gears 
about six inches diameter or less, the moulder will 
generally rap the pattern in the sand enough to make 
any allowance for shrinkage unnecessary. In pattern 
gears the spaces between teeth should be cut wider 
than finished gear spaces to allow for rapping and to 
avoid having too much cleaning to do in order to have 
gears run freely. In cut patterns of iron it is generally 
Metal Pattern enough to make spaces .015" to .02' wider. This 
makes clearance .03" to .04" in the patterns. Some 
moulders might want .06" to .07" clearance. 

Metal patterns should be cut straight ; they work 
better with no draft. It is well to leave about .005" to 
be finished from side of patterns after teeth are cut ; 
this extra stock to be taken away from side where 
cutter comes through so as to take out places where 
stock is broken out. The finishing should be done 
with file or emery wheel, as turning in a lathe is likely 
to break out stock as badly as a cutter might do. 

If cutters are kept sharp and care is taken when 
coming through the allowance for finishing is not nec- 
essary and the blanks may be finished before they are 
cut. 



CHAPTER III. 
SINGLE-CURVE GEARS OF 30 TEETH AND MORE. 



Single-curve teeth are so called because they have rpgg°^^® ^^^^ 
but one curve by theory, this curve forming both face 
and flank of tooth sides. In any gear of thirty teeth 
and more, this curve can be a single arc of a circle 
whose radius is one-fourth the radius of the pitch 
circle. In gears of thhty teeth and more, a fillet is 
added at bottom of tooth, to make it stronger, equal 
in radius to one-seventh, the widest part of tooth space. 

A cutter formed to leave this fillet has the advantage 
of wearing longer than it would if brought up to a 
corner. 

In gears less than thu-ty teeth this fillet is made the 
same as just given, and sides of teeth are formed with 
more than one arc, as will be shown in Chapter VI. 

Having calciilated the data of a gear of 30 teeth, f Example of a 

. ° . . ^ ' * Gear, N=30, P 

inch circular pitch (as we did in Chapter II. for Ih" =%"■ 
pitch), we proceed as follows : 

1. Draw pitch cu^cle and point it off into parts equal Geometrical 

. 1 in ,1 • 1 •- 1 Construction. 

to one-halt the circular pitch. Fig. 6. 

2. From one of these points, as at B, Fig. 6, draw 
radius to pitch circle, and upon this radius describe a 
semicircle ; the diameter of this semichcle being equal 
to radius of pitch circle. Draw addendum, working 
depth and whole depth circles. 

3. From the point B, Fig. 6, where semichcle, pitch 
chcle and outer end of radius to pitch circle meet, lay 
off a distance upon semicircle equal to one-fourth the 
radius of pitch circle, shown in the figure at BA, and 
is laid off as a chord. 

4. Through this new point at A, upon the semichcle, 
draw a circle concentric to pitch cii'cie. This last is 



10 



BKOWN & SHAEPE MFG. CO. 



IT-ig. 6 




GEAR, 30 TEETH, 
CIRCULAR PITCH 
P'= %" or .75" 
N = 30 
t= .375" 
S= .2387" 
T>"= .4775" 
S+f= .2762" 
T>"+f= .5150" 
D'rr 7.1610" 
D = 7.6384" 



SINGLE CURVE GEAR. 



PROVIDENCE, R. I. 11 

called the base circle, and is the oue for centers of 
tooth arcs. In the system of single curve gears we 
have adopted, the diameter of this circle is .968 of the 
diameter of pitch circle. Thus the base circle of any 
gear 1 inch pitch diameter by this system is .968". 
If the pitch circle is 2" the base circle will be 1.936." 

5. With dividers set to one-quarter of the radius of 
pitch circle, di'aw arcs forming sides of teeth, placing 
one leg of the dividers in the base circle and letting 
the other leg describe an arc through a point in the 
pitch circle that was made in laying off the parts equal 
to one-half the circular pitch. Thus an arc is drawn 
about A as center through B. 

6. With dividers setto one-seventh of the widest part 
of tooth space, drav/ the fillets for strengthening teeth 
at their roots. These fillet arcs should just touch the 
whole depth circle and the sides of teeth already 
described. 

Single curve or involute gears are the only gears jj^^®j^^g^Qg^°^ 
that can run at varying distance of axes and transmit ''^^• 
unvarying angular velocity. This peculiarity makes 
involute gears specially valuable for driving rolls or 
any rotating pieces, the distance of whose axes is 
hkely to be changed. 

The assertion that gears crowd harder on bearings Pressure ou 

° ° toearmgs. 

when of involute than when of other forms of teeth, 
has not been proved in actual practice. 

Before taking next chapter, the learner should make Practice, be- 
several drawings of gears 30 teeth and more. Say next chapter. 
make 35 and 70 teeth 1^" P'. Then make 40 and 65 
teeth I" F. 

An excellent practice will be to make drawing on 
cardboard or Bristol-board and cut teeth to lines, thus 
making paper gears ; or, what is still better, make them 
of sheet metal. By placing these in mesh the learner 
can test the accuracy of his work. 



12 



CHAPTER IV. 

RACK TO MESH WITH SINGLE-CURVE GEARS HAYING 
30 TEETH AND MORE. 



made^prepara,- This gear (Fig. 7) is made precisely the same as gear 
a°ifack*^'^^^'°^ "^ Chapter III. It makes no difference in which direc- 
tion the construction radius is drawn, so far as obtain- 
ing form of teeth and making gear are concerned. 

Here the radius is drawn perpendicular to pitch line 
of rack and through one of the tooth sides, B. A semi- 
circle is drawn on each side of the radius of the pitch 
circle. 

The points A and A' are each distant from the point 
B, equal to one-fourth the radius of pitch circle and 
correspond to the point A in Fig. 6. 

In Fig. 7 add two lines, one passing through B and 

A and one through B and A'. These two lines forra 

angles of 75^° (degrees) with radius BO. Lines BA 

, and BA' are called lines of pressure. The sides of 

rack teeth are made perpendicular to these lines. 

Kack. A Rack is a straight j)iece, ha^dng teeth to mesh 

with a gear. A rack may be considered as a gear of 

infinitely long radius. The circumference of a circle 

approaches a straight line as the radius increases, and 

when the radius is infinitely long any finite part of the 

Construction circumference is a straisrht line. The pitch line of a 

of Pitoli Line of . .... 

Rack. rack, then, is merely a straight line just touching the 

pitch circle of a gear meshing with the rack. The 

thickness of teeth, addendum and depth of teeth 

below pitch line are calculated the same as for a wheel. 

(For pitches in common use, see table of tooth parts.) 

The term circular pitch when applied to racks can be 

more accurately replaced by the term linear pitch. 

Linear applies strictly to aline in general while circular 

pertains to a circle. Linear pitch means the distance 

between the centres of two teeth on the pitch line 

whether the line is straight or curved. 



PROVIDENCE, R. I. 



13 



A rack to mesh with a single-curve gear of 30 teeth 
or more is drawn as follows : 

1. Draw straight pitch line of rack ; also draw ad- 
dendum line, working depth line and whole depth line, 
each parallel to the pitch line (see Fig. 7). 



■Rack. 

Fig. 7. 




RACK TO MESH WITH SINGLE CURVE GEAR 
HAVING 30 TEETH AND MORE. 



14 BKOWN & SHARPE MFG. CO. 

2. Point off the pitch line into parts equal to one- 
half the circular pitch, or =^. 

3. Through these points draw lines at an angle of 
75^° with pitch lines, alternate lines slanting in oppo- 
site directions. The left-hand side of each rack tooth 
is perpendicular to the line BA. The right-hand side 
of each rack tooth is perpendicular to the line BA'. 

4. Add fillets at bottom of teeth equal to i of the 
width of spaces between the rack teeth at the adden- 
dum line. 

Bit^lt Rack "^^^ sketch, Fig. 8, wHl show how to obtain angle of 
Teeth. sides of rack teeth, directly from pitch line of rack, 

without drawing a gear in mesh with the rack. 




Upon the pitch line b b', draw any semicircle — 
baa' b'. From point h lay off upon the semicircle 
the distance b a, equal to one-quarter of the diameter 
of semicircle, and draw a straight line through b and a. 

This line, b a, makes an angle of 75 j° with pitch line 
b b', and can be one side of rack tooth. The same 
construction, b' a', will give the inclination 75^° in the 
opposite direction for the other side of tooth. 

The sketch, Fig. 9, gives the angle of sides of a tool 
for j)laning out spaces between rack teeth. Upon any 
line OB di"aw circle OABA'. From B lay off distance 
BA and BA', each equal to one-quarter of diameter of 
the circle. 

Draw lines OA and OA'. These two lines form an 
angle of 29°, and are light for inclination of sides of 
rack tool. 



PROVIDENCE, E. I. 



15 



Make end of rack tool .31 of circular pitch, and then J^'i'^*'* »* ^ac^ 

^ ' Tool at end. 

round the corners of the tool to leave fillets at the 
bottom of rack teeth. 

Thus, if the circular pitch of a rack is 1^" and we 
multiply by .31, the product .465" will be the width of 
tool at end for rack of this i^itch before corners are 
taken off. This width is shown at x y. 






A Worm is a screw that meshes with tlie teeth of a 
gear. 

This sketch and the foregoing rule are also right for worm Thread 
a worm-thread tool, but a worm-tln-ead tool is not 
usually rounded for fillet. In cutting worms, leave 
width of top of thread .335 of the circular pitch. 
When this is done, the depth of thi-ead will be right. 

:.335 P' 




16 



CHAPTER V. 

DIAMETRAL PITCH— SIZING BLANKS AND THE TEETH OF SPUR GEARS 
—DISTANCE BETWEEN THE CENTRES OF WHEELS. 



necessary to ^^ making drawings of gears, and in cutting racks, 
cuiaxPitchf"^^^ ^^ necessary to know the circular pitch, both on 
account of spacing teeth and calculating their strength. 
It would be niore convenient to express the circular 
pitch in whole inches, and the most natural divisions 
AVheer^'thl^f an inch, as 1" P', f" P', J" P', and so on. But as 
ference^'mu'st^'^® circumference of the pitch circle must contain the 
^'^"I'.^J.^pf^^'iy" circular j^itch some whole number of times, corre- 
n'\i"mblr°of ^P'^'^^^^^S ^^ ^^® number of teeth in the gear, the 
times. diameter of the pitch circle will often be of a size not 

readily measured with a common rule. This is because 
the circumference of a circle is equal to 3.1416 times 
the diameter, or the diameter is equal to the circumfer- 
ence multiplied by .3183. 
Pitch, in In practice, it is better that the diameter should be 

Terms of the ^ _ ' 

Diameter. of some size Conveniently measured. The same applies 
to the distance between centers. Hence it is generally 
more convenient to assume the pitch in terms of the 
diameter. In Chapter II. was given a definition of the 
module, and also how to obtain the module from the 
circular pitch. 
Circular Pitch ^e Can also assume the module and pass to its equiv- 

and a Diame- ^ ^ 

ter Pitch. alent circular pitch. If the circumference of the pitch 
circle is divided by the number of teeth in the gear, 
the quotient will be the circular pitch. In the 
same manner, if the diameter of the pitch circle is 
divided by the number of teeth, the quotient will 
be the module. Thus, if a gear is 12 inches pitch 
diameter and has 48 teeth, dividing 12" by 48, the 
quotient ^" is the module of this gear. In prac- 



PROVIDENCE, R. I. 17 

tice, the module is taken in some convenient part of 

an inch, as V module and so on. It is convenient in /''•^'i^V^ti?" 

' " of Module Diii- 

ealculation to designate one of these modules by s, as meter ritcb. 
in Chapter II. Thus, for ^" module, s is equal to ^". 
Generally, in speaking of the module, the denominator 
of the fraction only is named, i" module is then called 
3 diametral pitch. That is, it has been found more 
convenient to take the reciprocal of the module in mak- 
ing calculation. The reciprocal of a number is 1 divi- Reciprocal of 
ded by that number. Thus the reciprocal of \ is 4. 
because \ goes into 1 four times. 

Hence, we come to the common definition : 

Diametral Pitch is the number of teeth to one inch Diametral 

Pitch 

of diameter of pitch circle. Let this be denoted by P. 
Thus, 1" diameter pitch we would call 4 diam,etral pitch 
or 4 P, because there would be 4 teeth to every inch in 
the diameter of pitch circle. The circular pitch and 
the different parts of the teeth are derived from the 
diametral pitch as follows. 

^^^^ = P', or 3.1416 divided by the diametral pitch am Sto^flDd 
is equal to the circular pitch. Thus to obtain the cir- pitch? ^'^*'"^'^* 
cular for 4 diametral pitch, we divide 3 1416 by 4 and to obtain Cir- 
obtain .7854 for the circular pitch, corresponding to 4 ^j^^^jj^^'j)^^*^'^^ 
diametral pitch. trai Pitch. 

In this case we would write P=4, P'=: 7854", s=i". 
^ r=s, or one inch divided by the number of teeth to an 
inch, gives distance on diameter of pitch circle occupied 
by one tooth or the module. -The addendum or face of 
tooth is the same distance as the module. 

^ ::= P, or one inch divided by the module equals num- 
ber of teeth to one inch or the diametral pitch. 

^ _„ Given, the Di- 

-'^^ = t, or 1.57 divided by the diametral pitch gives ametraiPitehto 
P ' -^ i- & ^^^ jjjg Thick- 

thickness of tooth at pitch line. Thus, thickness of ness of Tooth 

fit the Pitch 

teeth along the pitch line for 4 diametral pitch is .392". Line. 

^=r:D', or number of teeth in a gear divided by the>,fi,mb°r *of 
diametral pitch equals diameter of the pitch circle. Jnd^tiie'^Di'am! 
Thus for a wheel, 60 teeth, 12 P, the diameter of pitch ^*^'5\ifJDi'am*? 
circle will be 5 inches. ^*frcie.* ^''''^ 

^^=:D, or add 2 to the number of teeth in a wheel Given, the 

P ' N II m b e r o f 

and divide the sum by the diametral pitch ; and the Teetiiinfuyheei 

'' ' and the Diame- 

tral Pitch to 
find the Whole 
Diameter. 



18 BKOWN & SHAKPE MFG. CO. 

quotient will be the whole diameter of the gear or the 
diameter of the addendum circle. Thus, for 60 teeth, 
12 P, the diameter of gear blank will be 5 j-^ inches. 

p,=P, or number of teeth divided by diameter of 
pitch circle in inches, gives the diametral pitch or 
number of teeth to one inch. Thus, in a wheel, 24 
teeth, 3 inches pitch diameter, the diametral pitch is 8. 

— ^ :=P, or add 2 to the number of teeth; divide the 
sum by the whole diameter of gear, and the quotient 
will be the diametral pitch. Thus, for a wheel 3^^^^" 
diameter, 14 teeth, the diametral pitch is 5. 

D' P=N, or diameter of pitch circle, multiplied by 
diametral pitch equals number of teeth in the gear. 
Thus, in a gear, 5 pitch, 8" pitch diameter, the num- 
ber of teeth is 40. 

D P — 3=N or multiply the whole diameter of the 
gear by the diametral pitch,subtract 2, and the remain- 
der will be the number of teeth. 

j^==s, or divide the whole diameter of a spur gear 

by the number of teeth plus two, and the quotient 
will be the module. 

+ '^1*^^-4.^**™®' When we say the diametral pitch we shall mean the 
number of teeth to one inch of diameter of pitch cir- 
cle, or P, (^'=P). 

ametrai^ Pitch When the circular pitch is given, to find the corre- 

Ht^ ^^^°'^^*'^ spending diametral pitch, divide 3.1416 by the circular 
pitch. Thus 1.57 P is the diametral pitch correspond- 
ing to 2-inch circular pitch, (?^/J-^=P). 

Example. What diametral pitch corresponds to ^" circular 

pitch % Remembering that to divide by a fraction we 
multiply by the denominator and divide by the numer- 
ator, we obtain 6.28 as the quotient of 3.1416 divided by 
\ . 6.28 P, then, is the diametral pitch corresponding 
to \ circular pitch. This means that in a gear of \ 
inch circular pitch there are six and twenty-eight one 
hundredths teeth to every inch in the diameter .of the 
pitch circle. In the table of tooth parts the diametral 
pitches corresponding to circular pitches are carried 
out to four places of decimals, but in practice three 
places of decimals are euough. 



PROVIDENCE, R. I. 19 

When two gears are in mesh, so that their pitch 
circles just touch, the distance between their axes or 
centers is equal to the sum of the radii of the two gears. 
The number of the modules between centers is equal to 
half the sum of number of teeth in both gears. This 
principle is the same as given in Chapter II., page 6, Rule to lind 

^ ^ ^ P » l^ 6 ' Distance l)e- 

but when the diametral pitch and numbers of teeth in twecn centers. 
two gears are given, add together the numbers of teeth in 
the two ioheels and divide half the stan by the diametral 
pitch. The quotient is the center distance. 

A gear of 20 teeth, 4 P, meshes with a gear of 50 Example, 
teeth ; what is the distance between their axes or cen- 
ters? Adding 50 to 20 and dividing half the sum by 4, 
we obtain 8|" as the center distance. 

The term diametral pitch is also applied to a rack. 
Thus, a rack 3 P, means a rack that will mesh with a 
gear of 3 diametral pitch. 

It will be seen that if the expression for the module Fractional 

, •■ i. -I i! A. .Diametral 

has any number except 1 for a numerator, we cannot pitcb. 
express the diametral pitch by naming the denominator 
only. Thus, if the addendum or module is y'^^, the 
diametral pitch will be 2^, because 1 divided by y^ 
equals 2^. 

The term module is much used where gears are made 
to metric sizes, for the reason that, the millimeter being 
so short, the module is conveniently expressed in milli- 
meters. If we know the module of a gear we can figure 
the other parts as easily as we can if we know either 
the circular pitch or the diametral pitch. The module 
is, in a sense, an actual distance, while the diametral 
pitch, or the number of teeth to an inch, is a relation or 
merely a ratio. The meaning of the module is not 
easily mistaken. 



20 



CHAPTER VI. 

SINGLE-CDRYE GEARS HAYING LESS THAN 30 TEETH— GEARS AHD 
RACKS TO MESH Y?ITH GEARS HAYING LESS THAN 30 TEETH. 



K^^io^'^*'*''^' In Fig. 10, the construction of the rack is the same 
as the construction of the rack in Chapter IV. The 
gear in Fig. 10 is drawn from base circle out to adden- 
dum circle, by the same method as the gear in Chapter 
III., but the spaces inside of base circle are drawn as 
follows : 
Flanks of In gears, 12 to 19 teeth, the sides of space inside 

Gears m low » , , ,. , 

Numbers of of the base circle are radial for a distance, a b, equal 

Teeth, ' t. 

to 1^, or 3.5 divided by the product of the pitch by the 
number of teeth. In gears with more than 19 teeth 
the radial construction is omitted. 
Construction Then, with one leg of dividers in pitch circle in 

of Fig. 10 con- ° . 

tinued. center of next tooth, e, and other leg just touching 

one of the radial lines at h, continue the tooth side 
into c, until it will touch a fillet arc, whose radius is 
1 the width of space at the addendum circle. The 
part, h' c\ is an arc from center of tooth g, etc. The 
flanks of teeth or spaces in gear, Fig. 11, are made the 
same as those in Fig. 10. 

This rule is merely conventional or not founded 
upon any principle other than the judgment of the de- 
signer, to effect the object to have spaces as wide as 
practicable, just below or inside of base circle, and 
then strengthen flank with as large a fillet as will clear 
addenda of any gear. If flanks in any gear will clear 
addenda of a rack, they will clear addenda of any 

Internal Gear, other gear, except internal gears. An internal gear is 
one having teeth upon the inner side of a rim or ring. 
Now, it will be seen that the gear. Fig. 10, has teeth 



PROVIDENCE, R. I 




Fig. 10 



22 



BKOWT^ & SHARPE MFG. CO. 



too much rounded at the points or at the addendum 
circle. In gears of pitch coarser than 10 to inch (10 
Add^^nd'a of ■^)' ^^^ having less than 30 teeth, this rounding 
Teeth. becomes objectionable. This I'ounding occurs, because 

in these gears arcs of circles depart too far from the 
true involute curve, being so much that points of 
teeth get no bearing on flanks of teeth in other wheels. 
In gear, Fig. 11, the teeth outside of base circle are 
made as nearly true involute as a workman will be able 
to get without special machinery. This is accomplished 
tiiOTpto^Tmeinl ^® f ollows : draw three or four tangents to the base 
volute. circle, i i', J J', k k', 1 1', letting the points of tangency 

on base circle i',j', k' , I' be about ^ or |- the circular pitch 
apart ; the first point, i', being distant from ^, equal to 
\ the radius of pitch circle. "With dividers set to ^ 
the radius of pitch circle, placing one leg in i', draw 
the arc, a' i j; with one leg in j', and radius j' j, 
draw J k; with one leg in k', and radius k' k draw k I. 
Should the addendum circle be outside of I, the tooth 
side can be completed with the last radius, I' I. The 
arcs, a' ij, j k and k I, together form a very close 
approximation to a true involute from the base circle, 
i' j' k' I'. The exact involute for gear teeth is the 
curve made by the end of a band when unwound from 
a cylinder of the same diameter as base circle. 

The foregoing operation of drawing tooth sides, 
although tedious in description, is very easy of practical 
application. 
Rounding of It will also be seen that the addenda of rack teeth 

Addenda of ^ n 

Rack. in Fig. 10, interfere with the gear-teeth flanks, as at 

m n; to avoid this interference, the teeth of rack, Fig. 
11, are rounded at points or addenda. 

It is also necessary to round off the points of invo- 
lute teeth in high-numbered gears, when they are to 
interchange with low-numbered gears. In interchange- 
able sets of gears the lowest-numbered pinion is usual- 
Tempietsly 12. Just how much to round off can be learned bv 

necessary for ^ i n i • i 

Rounding off makinef templets of a few teeth out of thin metal or 

Points of teeth. -, „ i -, ^ , 

cardboard, for the gear and rack, or, two gears re- 
quired, and fitting addenda of teeth to clear flanks. 
However accurate we may make a diagram, it is quite 



TROVIDENCE, K. I. 




Fig. 11 



24 



BROWN & SHARPE MFG. CO. 



Diagrams for 
a Set of Cut- 
ters. 



as well to make templets iu order to shape cutters 
accurately 

It is best to make cutters to corrected diagrams, as 
iu Fig 1 1 . When corrected diagrams are made, as 
in Fig. 1 1, take the following : 

For 12 and 13 teeth, diagram of 12 teeth. 



" 14 


to 10 ' 


c u u 14 u 


u 17 


' 20 ^ 


' " " 17 " 


" 21 


• 25 ' 


;t 21 " 


" 26 


' 34 ' 


" 26 " 


" 35 


' 54 ' 


" 35 " 


" 55 


' 134 ' 


' '' " 55 " 


" 135 


' rack, ' 


"135 " 



Templets for large gears must be fitted to run with 
12 teeth. 



25 



CHAPTER VII. 
DOUBLE-CDRYE TEETH— GEAR, 15 TEETH— RACK. 



In double-curve teeth the formation of tooth sides ^^^ ^ m^^l^J 

curve Tooth 

changes at the pitch Hne. In all gears the part of Faces are con- 
teeth outside of pitch line is convex ; in some gears 
the sides of teeth inside pitch line are convex ; in some, 
radial ; in others, concave. Convex faces and concave 
flanks are most familiar to mechanics. In interchange- 
able sets of gears, one gear in each set, or of each 
pitch, has radial flanks. In the best practice, this gear 
has fifteen teeth. Gears with more than fifteen teeth, 
have concave flanks; gears with less than fifteen teeth, 
have convex flanks. Fifteen teeth is called the Jiase 
of this system. 

We will fij.-st draw a a:ear of fifteen teeth. This , construction 

*^ , of Fig. 12. 

fifteen-tooth construction enters into gears of any 
number of teeth and also into racks. Let the gear be 
3 P. Having obtained data, we proceed as follows : 

1. Draw pitch cu'cle and point it off into pai'ts equal 
to one-thu'tieth of the circumference, or equal to thick- 
ness of tooth — ^. 

2. From the center, through one of these points, as 
at T, Fig. 12, draw line OTA. Draw addendum and 
whole-depth circles. 

3. About this point, T, with same radius as 15-tooth 
pitch circle, describe arcs A K and O k. For any other 
double-curve gear of 3 P., the radius of arcs, A K and 
O Jc, will be the same as in this 15-tooth gear =2^". 
In a 15-tooth gear, the arc, O k, passes through the 
center O, but for a gear having any other number of 
teeth, this construction arc does not pass through 
center of gear. Of course, the 15-tooth radius of arcs, 
A K and O k, is always taken from the pitch we are 
workinsf with. 



26 



BKOWN & SHAKPB MFG. CO. 






A^" 



GEAR, 3 P., 15 TEETH 
P= 3 
N = 15 
P'= 1.0472" 
t— .5236" 
S= .3833" 
D"=r .6666" 
S+f= .3857" 
D"+/= .7190" 
D'= 5.0000" 
D = 5.6666" 




V. 



DOUBLE CURVE GEAR. 



PROVIDENCE, K. I. 27 

4. Upon these arcs on opposite sides of line OTA, 
lay off tooth thickness, A K and O k, and draw line 
KT 7c. 

5. Perpendicular to K T Jc, draw line of pressvire, 
L T P ; also through O and A, draw lines A R and O r, 
perpendicular to K T k. The line of pressure is at 
an angle of 78° with the radius of gear. 

6. From O, draw a line O R to intersection of A R 
with K T Jc. Through point c, where O R intersects 
L P, describe a circle about the center, 0. In this 
circle one leg of dividers is placed to describe tooth 
faces 

7. The radius, c d, of arc of tooth faces is thv 
straight distance from c to tooth-thickness point, h, 
on the other side of radius, O T. With this radius, c h, 
describe both sides of tooth faces. 

8. Draw flanks of all teeth radial, as O e and O f 
The base gear, 15 teeth only, has radial flanks. 

9. With radius equal to one-seventh of the widest 
part of space, as g h, draw fillets at bottom of teeth. 

The foregoing is a close approximation to epicy- ^. ^'^'^'^^^^^^'_ 
cloidal teeth. To get exact teeth, make two 1 5 tooth cioidai Teeth, 
gears of thin metal. Make addenda long enough to 
come to a point, as at n and q. Make radial flanks, as 
at m and x>i deep enough to clear addenda when gears 
are in mesh. First finish the flanks, then fit the long 
addenda to the flanks when gears are in mesh. 

When these two templet gears are alike, the centers standard 
are the right distance apart and the teeth interlock 
without backlash, they are exact. One of these tem- 
plet gears can now be used to test any other templet 
gear of the same pitch. 

Gears and racks will be right when they run cor- 
rectly with one of these 15-tooth templet gears. Five 
or six teeth are enough to make in a gear templet. 

Double- cuEVE Rack. — Let us draw a rack 3 P. T,'^?^^-^^;f'Y7® 

Kack, Fig. la. 

Having obtained data of teeth we proceed as follows : 

1. Draw pitch line and point it off in parts equal 
to one-half the circular pitch. Draw addendum and 
whole-depth lines. 

2. Through one of the points, as at T, Fig. 13, draw 
line OTA perpendicular to pitch line of rack. 



28 



BROWN & SHARPE MFG. CO. 




I^ig. 13 

DOUBLE CURVE RACK. 



PROVIDENCE, R. I. 



2Q 



3. About T make precisely the same construction as 
■was made about T in Fig. 12. That is, with radius of 
15-tooth pitch circle and center T draw arcs k and 
A K ; make O k and A K equal to tooth thickness ; 
draw K T k ; draw r, A R, and line of pressui-e, each 
perpendicular to K T 7c. 

4. Through R and r, draw lines parallel to O A. 
Through intersections c and c' of these lines, with 
pressure line L P, draw lines parallel to pitch line. 

5. In these last lines place leg of dividers, and draw 
faces and flanks of teeth as in sketch. 

6. The radius c' d' of rack-tooth faces is the same 
length as radius c d of rack-tooth flanks, and is the 
straight distance from c to tooth-thickness point h on 
opposite side of line O A. 

7. The radius for fillet at bottom of rack teeth is 
equal to |- of the widest part of tooth space. This 
radius can be varied to suit the judgment of the 
designer, so long as a fillet does not interfere with 
teeth of engaging gear. 




Vig. 14: 



Racks 9f the same pitch, to mesh with interchange- 
able gears, should be alike when placed side by side, 
and fit each other when placed together as in Fig. 14. 

In Fig. 13, a few teeth of a 15-tooth wheel are shown 
in mesh with the rack. 



30 



CHAPTER VIII. 

DOUBLE-CURVE SPUR GEARS, HAYING MORE AND FEWER THAN 
15 TEETH— ANNULAR GEARS. 



^f Construction Let US di'Rw two gears, 12 and 24 teeth, 4 P, in 
mesh. In Fig. 15 the construction lines of the lower 
or 24-tooth gear are full. The upper or 12-tooth gear 
construction lines are dotted. The line of pressure, 
L P, and the line K T ^ answer for both gears. The 
arcs A K and O k are described about T. The radius 
of these arcs is the radius of pitch circle of a gear 15 
teeth 4 pitch. The length of arcs A K and O ^ is the 
tooth thickness for 4 P. The line K T /<; is obtained 
the same as in Chapter VII. for all double-curve gears, 
the distances only varying as the pitch. Having drawn 
the pitch circles, the line K T ^, and, perpendicular to 
K T h^ the lines A R, O r and the line of pressui'e 
L T P, we proceed with the 24-tooth gear as follows : 

1. From center C, through r, draw line intersecting 
line of pressure in w^. Also draw line from center C 
to R, crossing the line of pressxu-e L P at c. 

2. Through m describe circle concentric with pitch 
circle about C. This is the circle in which to place 
one leg of dividers to describe flanks of teeth. 

3. The radius, ?n «, of flanks is the straight distance 
from m to the first tooth-thickness point on other side 
of line of centers, C C', at v. The arc is continued to 
n, to show how constructed. This method of obtain- 
ing radius of double-curve tooth flanks applies to all 
gears having more than fifteen teeth. 

4. The construction of tooth faces is similar to 15- 
tooth wheel in Chapter VII. That is : Draw a circle 
through c concentric to pitch circle ; in this circle 
place one leg of dividers to di-aw tooth faces, the 
radius of tooth faces being c h. 



PROVIDENCE, R. I. 



31 




PINION, 12 TEETH, 
GEAR 24 TEETH, .4 P 

P=4 

N=12 and 24 
P'= .7854" 
t = .3927" 
S = .2500" 
D"= .5000' 
St/ = .2893' 
D"+/ = .5393' 



c<! 
DOUBLE CURVE GEARS IN 



lESH. 



32 BROWN & SHAEPE MFG. CO. 

Construction 5 Yhe radius of fillets at roots of teeth is equal to 

of Fig. 15 con- ^ 

linued. one-seventh the width of space at addendum circle. 

Flanks for 12 ^^® constructioiis for flanks of 12, 13 and 14 

13 and 14 Teeth, teeth are similar to each other and as follows : 

1. Through center, C, draw line from K, intersecting 
line of pressure in u. Through u draw circle about 
C. In this circle one leg of dividers is placed for 
drawing flanks. 

2. The radius of flanks is the distance from u to 
the first tooth-thickness point, e, on the scone side of 
C T C. This gives convex flanks. The arc is con- 
tinued to V, to show construction. 

3. This arc for flanks is continued in or toward the 
center, only about one- sixth of the working depth (or 
Is.) ; the lower part of flank is similar to flanks of 
gear in Chapter VI. 

4. The faces are similar to those in 15-tooth gear, 
Chapter VII., and to the 24-tooth gear in the fore- 
going, the radius being w y ; the arc is continued to tc, 
to show construction. 

Annular Gears. Anndlar Geaks. Gears with teeth inside of a rim 
or ring are called Annular or Internal Gears. The 
construction of tooth outlines is similar to the fore- 
going, but the spaces of a spur external gear become 
the teeth of an annular gear. 

It has been shown that in the system just de- 
scribed, the pinion meshing with an annular gear, 
must difi'er from it by at least fifteen teeth. Thus, 
a gear of 24 teeth cannot work with an annular gear 
of 36 teeth, but it will work with annular gears of 39 
teeth and more. The fillets at the roots of the teeth 
must be of less radius than in ordinary spur gears. An 
annular gear differing from its mate by less than 15 
teeth can be made. This will be shown in Part II. 

Annular-gear patterns requue more clearance for 
moulding than external or spur gears. 

Pinions. In speaking of different-sized gears, the smallest 

ones are often called " pinions." 

The angle of pressure in all gears except involute, 
constantly changes. 78° is the pressure angle in 
double-curve, or epicycloidal gears for an instant 



PROVIDENCE. R. I. 33 

only; in our example, it is 78° wheu one side of a 
tootli reaches the line of centers, and the pressure 
against teeth is applied in the direction of the arrows. 

The pressure angle of involute gears does not 
change. An explanation of the term angle of pressure 
is given in Part II. 

We obtain the forms for epicycloidal gear cutters 
by means of a machine called the Odontom Engine. 
This machine will cut original gears with theoretical 
accuracy. 

It has been thought best to make 24 gear cutters 24 Douuie- 

° ° curve Gear 

for each pitch. This enables us to fill any require- Cu tters for 

^ , ^ X eacli Pitch. 

ment of gear-cutting very closely, as the range covered 
by any one cutter is so small that it is exceedingly near 
to the exact shape of all gears so covered. 

Of course, a cutter can be exactly right for only one 
gear. Special cutters can be made, if desired. 



34 



CHAPTER IX. 

BEVEL-fiEAR BLANKS. 



Bevel Gears connect shafts whose axes meet when 
g''^|j®*-,'^^°^ sufficiently prolonged. The teeth of bevel gears are 
formed upon formed about the frustrums of cones whose apexes 

frustrums ol ^ 

cones. are at the same point where the shafts meet. In Fig. 

16 we have the axes A O and B O, meeting at O, and 
the apexes of the cones also at O. These cones are 
called the pitch cones, because they roll upon each 
other, and because upon them the teeth are pitched. 
If, in any bevel gear, the teeth were sufficiently pro- 
longed toward the apex, they would become infinitely 
small ; that is, the teeth would all end in a point, or 
vanish at O. We can also consider a bevel gear as 
beginning at the apex and becoming larger and larger 
as we go away from the apex. Hence, as the bevel 
gear teeth are tapering from end to end, we may say 




BEVEL GEAR PITCH CONES. 



Fig. 16. 



that a bevel gear has a number of pitches and pitch 
circles, or diameters : in speaking of the pitch of a 
bevel gear, we mean always the pitch at the largest 



PROVIDENCE, U. I. 



35 



pitch circle, or at the largest pitch diameter, as at 
bd, Fig. 17. 

Fig. 17 is a section of three bevel gears, the gear 
o B q being twice as large as the two others. The 
outer surface of a tooth as m m' is called the face of Construction 

ot Bevel Gem 

the tooth. The distance m m' is usually called the Blanks, 
length of the face of the tooth, though the real length 
is the distance that it occupies upon the line O i. The 
outer part of a tooth at m n is called its large end, and 
the inner part m' n' the small end. 

Almost all bevel gears connect shafts that are at 
right angles with each other, and unless stated other- 
wise we always understand that they are so wanted. 

The directions given in connection with Fig. 17 
apply to gears with axes at right angles. 

Having decided upon the pitch and the numbers of 
teeth : — 

1. Draw centre lines of shafts, A O B and COD, 
at right angles. 

2. Parallel to A O B, draw lines a b and c d, each 
distant from A O B, equal to half the largest pitch 
diameter of one gear. For 24 teeth, 4 pitch, this half 
largest pitch diameter is 3". 

3. Parallel to COD, draw lines e f and g h, dis- 
tant from COD, equal to half the largest pitch 
diameter of the other gear. For a gear, 12 teeth, 4 
pitch, this half largest pitch diameter is 1|". 

4 At the intersection of these four lines, draw 
lines O i, O j, O k, and O 1 ; these lines give the size 
and shape of pitch cones. We call them " Cone Pitch 
Lines." 

5. Perpendicular to the cone- i)itch lines and through 
the intersection of lines a b, c d, e f, and g h, draw 
lines m n, o p, q r. We have drawn also u v to show 
that another gear can be drawn from the same diagram. 
Four gears, two of each size, can be drawn from this 
diagram. 

6. Upon the lines m u, o p, q r, the addenda and 
depth of the teeth are laid off, these lines passing 



36 



BROWN & SHARPE MFG. CO. 



through the largest pitch circle of the gears. Lay off 
the addendum, it being in these gears |-". This gives 
distance m n, o p, q r, and u v equal to the Avorking 
depth of teeth, which in these gears is |". The 
addendum of course is measured perpendicularly from 
the cone pitch lines as at k r. 

7. Draw lines m, O n, O p, O o, O q, Or. 
These lines give the height of teeth above the cone- 
pitch lines as they approach O, and would vanish 
entirely at O. It is quite as well never to have the 
length of teeth, or face, m m' longer than one-third 
the apex distance m O, nor more than two and one- 
half times the circular pitch. 

8. Having decided upon the length of face, draw 
limiting lines m'u' pei-pendicular to i O, q' r' perpen- 
dicular to k O, and so on. 

The distance between the cone-pitch lines at the 
inner ends of the teeth m' n' and q' r' is called the inner 
or smaller pitch diameter, and the circle at these points 
is called the smallest pitch circle. We now have the 
outline of a section of the gears thx'ough their axes. 
The distance m r is the whole diameter of the pinion. 

Dian^eter'^o'f '^^^ distance q o is the whole diameter of the gear. 

Bevel- G ear j^ practice these diameters can be obtained by rueasur- 

Blanke can be ^ • -^ 

obtained by ing the drawing. The diameter of pinion is 3.45" and 

Measuring'^' ° '■ 

Drawings. of the gear 6.22". We can find the angles also by 
measuring the drawing with a protractor. In the 
absence of a protractor, templetes can be cut to the 
drawing. The angle formed by line m m' with a b is 
the angle of face of pinion, in this pinion 59° 11', or 
59^° nearly. The lines q q' and g h give us angle of 
face of gear, for this gear 22° 19', or 221° nearly 
The angle formed by m n with a b is called the angle 
of edge of pinion, in our sketch 26° 34', or about 26i°. 
The angle of edge of gear, line q r with g h, is 63° 26', 
or about 63|-°. In turning blanks to these angles we 
place one arm of the protractor or templet against the 
end of the hub, when trying angles of a blank. Some 
designers give the angles from the axes of gears, but 



PROVIDENCE, R. I. 



37 




FUj. 17 



38 BROWN & SHARPE MFG. CO., 

it is not convenient to try blanks in this way. The 
method that we have given comes right also for angles 
as figured in compound rests. 

When axes are at right angles, the sum of angles 
of edge in the two gears equals 90°, and the sums of 
angle of edge and face in each gear are alike. 

The angles of the axes remaining the same, all pairs 
of bevel gears of the same ratio have the same angle 
of edge ; all pairs of same ratio and of same numbers 
of teeth have the same angles of both edges and faces 
independent of the pitch. Thus, in all pairs of bevel 
gears having one gear twice as large as the other, with 
axes at right angles, the angle of edge of large gear 
is 63° 26', and the angle of edge of small gear is 26° 34'. 

In all pairs of bevel gears with axes at right angles, 
one gear having 24 teeth and the other gear having 12 
teeth, the angle of face of small gear is 59° 11'. 
Another -phe following method of obtaining the whole diam- 

method ot ob- - ° * 

taming Whole ter of bevel gears is sometimes preferred : 

Diameter o t ^ '■ 

Blanks. From k lay off ; upon the cone-pitch line, a distance 

K w, equal to ten times the working depth of the 
teeth = 10 D". Now add rV of the shortest distance 
of v^ from the line g h, which is the perpendicular 
dotted line w x, to the outside pitch diameter of gear, 
and the sum will be the whole diameter of gear. In 
the same manner xV of w y, added to the outside pitch 
diameter of pinion, gives the whole diameter of pinion. 
The part added to the pitch diameter is called the 
diameter increment. 

Part II gives trigonometrical methods of figuring 

bevel gears : in our Formulas in Gearing there are 

trigonometrical formulas for bevel gears, and also 

tables for angles and sizes, 

of^BOT*e?Gear ^ somewhat similar construction will do for bevel 

Blanks whose gears whose axes are not at right angles. 

Axes are not ° » & 

at Right An- In Fig. 18 the axes are shown at O B and O D, the 
angle BOD being less than a right angle. 

1. Parallel to O B, and at a distance from it equal 
to the radius of the gear, we draw the line^ a b and c do 



PROVIDKNCK, R. I. 



39 




Fig. 20 



INSIDE BEVEL GEAR 
AND PINION. 



40 BROWN & SHARPE MFG. CO. 

2. Parallel to O D, and at a distance from it equal 
to the radius of the pinion, we draw the lines e f and g h. 

3. Now, through the point j at the intersection of 
c d and g h, we draw a line perpendicular to O B. 
This line k j, limited by a b and c d, represents the 
largest pitch diameter of the gear. 

Through j we draw a line perpendicular to O D. 
This line j 1, limited by e f and g h, represents the 
largest pitch diameter of the pinion. 

4. Through the point k at the intersection of a b 
with k i, we draw a line to O, a line from j to O, and 
another from 1, at the intersection j 1 and e f to O. 
These lines O k, O j, and O 1, represent the cone- 
pitch lines, as in Fig. 17. 

5. Perpendicular to the' cone-pitch lines we draw 
the lines u v, o p, and q r. Upon these lines we lay 
off the addenda and working depth as in the previous 
figure, and then draw lines to the point O as before. 

By a similar construction Figs. 19 and 20 can be 
drawn. 




GEAR CUTTER. 



41 



CHAPTER X. 

BEVEL GEARS. 

FORMS AND SIZES OF TEETH. 

CUTTING TEETH. 

To obtain the form of the teeth iu a bevel gear we Form of 

IjgvgI "* g ti r 

do not lay them out upon a pitch circle, as we do iu a teeth, 
spur gear, because the rolling pitch surface of a bevel 
gear, at any point, is of a longer radius of curvature 
than the actual radius of a pitch circle that passes 
through that point. Thus in Fig. 21, let f g c be a 
cone about the axis O A, the diameter of the cone 
being f c, and its radius g c. Now the radius of 
curvature of the surface, at c, is evidently longer than 
g c, as can be seen in the other view at C ; the full 
line shows the curvature of the surface, and the dotted 
line shows the curvature of a circle of the radius g c. 
It is extremely difficult to represent the exact form of 
bevel gear teeth upon a flat surface, because a bevel 
gear is essentially spherical in its nature ; for practical 
purposes we draw a line c A perpendicular to c, 
letting c A reach the centre line A, and take c A 
as the radius of a circle upon which to lay out the 
teeth. This is shown at c n m. Fig. 22. For con- 
venience the line c A is sometimes called the back 
cone radius. 

Let us take, for an example, a bevel gear and a ^,.^^o^-,"^i^^®= 
pinion 24 and 18 teeth, 5 pitch, shafts at right angles. 
To obtain the forms of the teeth and the data for 
cutting, we need to draw a section of only a half of 
each gear, as in Fig. 22. 

1. Draw the centre lines A O and B O, then the 
lines g h and c d, and the gear blank lines as des- 
cribed in Chapter IX. Extend the lines o' p' and o p 
until they meet the centre lines at A' B' and A B. 

2. With the radius A c draw the arc c n m, which 
we take as the geometrical pitch circle upon which to 
lay out the teeth at the large end. The distance A' c' 



42 



BROWN & SHARPE MFG. CO., 



is taken as the radius of the geometrical pitch circle 
at the small end ; to avoid confusion an arc of this 
circle is drawn at c" n' m' about A. 

3. For the pinion we have the radius B c for the 
geometrical pitch circle at the large end and B' c' for 
the small end : the distance B' c' is transferred to 
B c'". 

4. Upon the arc c n m lay off spaces equal to the 
tooth thickness at the large pitch circle, which in our 
example is .314". Draw the outlines of the teeth as 
in previous chapters : for single curve teeth we draw a 
semi-circle upon the radius A c, and proceed as des- 
cribed in chapter III. For all bevel gears that are to 
be cut with a rotary disk cutter, or a common gear 
cutter, single curve teeth are chosen ; and no attempt 
should be made to cut double curve teeth. Double 
curve teeth can be drawn by the directions given in 
chapters VII and VIII. We now have the form of 
the teeth at the lai'ge end of the gear. Repeat this 
operation with the radius B C about B, and we have 
the form of the teeth at tlie large end of the pinion. 

5. The tooth parts at the small end are designated 
by the same letters as at the large, with the addition 
of an accent mark to each letter, as in the right hand 
column. Fig. 22, the clearance, f, however, is usually 
the same at the small end as at the large, for con- 
venience in cutting the teeth. 

When cutting bevel gears with rotary cutters, the 
cutting angle is the same as the working depth angle. 
This angle is used for two reasons : first, it is not neces- 
sary to figure the angle of the bottom; second, the 
inside of the teeth is rounded over a little more and 
this lessens the amount to be filed off at the point. 
When cut in this way, the line of the bottom of the 
tooth is parallel to the face of the mating gear and it 
does not pass through the cone apex or common point 
of the axes. 

tooth p\?tV."'' ^^^ ^^^^^ ^^ ^^® ^°^*^ P^^'^^ ^* *^^ ^^^^^ ®^*^^ ^^''^ ^^ 
the same proportion to those at the large end as 

the line O c' is to c. In our example O c' is 2", 

and O c is 3" ; dividing O c' by O c we have |, or 

.666, as the ratio of the sizes at the small end to those 



PROVIDENCE, R. J. 



43 





44 BROWN & SHARPE MFG. CO. 

at the large : t' is .209" or f of .314", and so on. If 
the distance n m is equal to the outer tooth thickness, 
t, upon the arc c n m, the lines n A and m A will be a 
distance apart equal to the inner tooth thickness t' 
upon the arc c" n' m'. The addendum, s', and the 
working depth, D'", are at o' c' and o' p'. 

6. Upon the arcs c" n' m' and c'" we draw the forms 
of the teeth' of the gear and pinion at the inside. 
Example of As an example of the cutting of bevel gears with 

Cutting. .-,.,, , 

rotary disk cutters, or common gear cutters, let us 
take a pair of 8 pitch, 12 and 24 teeth, shown in 
Fig. 23. 

Length of In making the drawing it is well to remember that 
nothing is gained by having the face F E longer than 
five times the thickness of the teeth at the large 
pitch circle, and that even this is too long when it is 
more than a third of the cy^ea; distance O c. To cut a 
bevel gear with a rotary cutter, as in Fig. 24, is at 
best but a compromise, because the teeth change pitch 
from end to end, so that the cutter, being of the right 
form for the large ends of the teeth can not be right 
for the small ends, and the variation is too great when 
the length of face is greater than a third of the apex 
distance O c. Fig. 23. In the example, one-third of 
the apex distance is -^^'\ but F E is drawn only a 
half inch, which even though rather short, has changed 
the pitch from 8 at the outside to finer than 11 at the 
inside. Frequently the teeth have to be rounded over 
at the small ends by filing ; the longer the teeth the 
more we have to file. If there is any doubt about the 
strength of the teeth, it is better to lengthen at the 
large end, and make the pitch coarser rather than to 
lengthen at the small end. 

Data for These data are needed before beginning to cut: 

1. The pitch and the numbers of the teeth the same 
as for spur gears. 

2. The data for the cutter, as to its form : some- 
times two cutters are needed for a pair of bevel gears. 

3. The whole depth of the tooth spaces, both at 



cutting. 



PROVIDENCE, R. I. 



45 




D"+/ = .431" 



Fig. 22. 

BEVEL GEARS, FORM AND SIZE OF TEETH. 



46 BROWN & SHARPE MFG. CO. 

the outside and inside ends ; D" + f at the outside, 
and D'" + f at the inside. 

4. The thickness of the teeth at the outside and at 
the inside ; t and t'. 

5. The height of the teeth above the pitch lines at 
the outside and inside ; s and s'. 

6. The cutting angles, or the angles that the path 
of the cutter makes with the axes of the gears. In 
Fig. 23 the cutting angle for the gear c D is A Op, 
and the cutting angle for the pinion is B O o. 

Selection of The form of the teeth in one of these gears differs 

cutters 

so much from that in the other gear that two cutters 
are required. In determining these cutters we do not 
have to develop the forms of the gear teeth as in 
Fig. 22 ; we need merely measure the lines A c and 
B c. Fig. 23, and calculate the cutter forms as if these 
distances were the radii of the pitch circles of the 
gears to be cut. Twice the length A c, in inches, 
multiplied by the diametral pitch, equals the number 
of teeth for which to select a cutter tor the twenty- 
four-tooth gear ; this number is about 54, which calls 
for a number three bevel gear cutter in accordance 
with the lists of gear cutters, pages 61 and 82. Twice 
B c, multiplied by 8, equals about 13, which indicates 
a No. 8 bevel gear cutter for the pinion. This method 
of selecting cutters is based upon the idea of shaping 
the teeth as nearly right as practicable at the large end, 
and then filing the small end where the cutter has not 
rounded them over enough. 

In Fig. 25 the tooth L has been cut to thickness at 
both the outer and inner pitch lines, but it must still 
be rounded at the inner end. The teeth M M have 
been filed. In thus rounding the teeth rhey should not 
be filed thinner at tlie pitch lines. 

There are several things that affect the shape of the 
teeth, so that the choice of cutters is not always so 
simple a matter as the taking of the lines A c and 
B c as radii. 

In cutting a bevel gear, in the ordinary gear cutting 



PROVIl>ENCE, H. I. 



47 




BEVEL GEAR DIAGRAM FOR DIMENSIONS. 



end. 



48 BROWN & SHAEPE MFG. CO. 

machines, the finished spaces are not always of the 
same form as the cutter might be expected to make, 
because of the changes in the positions of the cutter 
and of the gear blank in order to cut the teeth of the 
right thickness at both ends. The cutter must of 
course be thin enough to pass through the small end of 
the spaces, so that the large end has to be cut to the 
right width by adjusting either the cutter or the blank 
sidewise, then rotating the blank and cutting twice 
around. 
Widening Thus, in Fig. 24, a gear and a cutter are set to have 
th^^^a^rge a space widened at the large end e', and the last chip 
to be cut off by the right side of the cutter, the cutter 
having been moved to the left, and the blank rotated 
in the direction of the arrow : in a Universal Milling 
Machine the same result would be attained by moving 
the blank to the right and rotating it in the direction 
of the arrow. It may be well to remember that in 
setting to finish the side of a tooth, the tooth and the 
cutter are first separated sidewise, and the blank is 
then rotated by indexing the spindle to bring the large 
end of the tooth up against the cutter. This tends 
rowed iwe ^q(; Qy^\y iq ^qJ; the spaces wider at the large pitch 
at root. circle, but also to cut off still more at the face of the 

tooth ; that is, the teeth may be cut rather thin at the 
face and left rather thick at the root. This tendency 
is greater as a cutting angle B O o. Fig. 23, is smaller, 
or as a bevel gear approaches a spur gear, because 
when the cutting angle is small the blank must be 
rotated through a greater arc in order to set to cut the 
right thickness at the outer pitch circle. This can be 
understood by Figs. 26 and 27. Fig. 26 is a radial- 
toothed clutch, which for our present purpose can be 
regarded as one extreme of a bevel gear in which the 
teeth are cut square with the axis : the dotted lines 
indicate the diffei-ent positions of the cutter, the side 
of a tooth being finished by the side of the cutter that 
is on the centre line. In setting to cut these teeth 
there is the same side adjustment and rotation of the 



Teeth nar- 



PROVIDENCE, U. I. 



49 




Tig. 34 



SETTING BEVEL GEAR CUTTER 
OUT OF CENTRE. 




60 



BROWN & SHARPE MFG. CO. 



spindle as in a bevel gear, but there is no tendency to 
make a tooth thinner at the face than at the root. On 
the other hand, if we apply these same adjustments to 
a spur gear and cutter, Fig. 27, we shall cut the face 
F much thinner without materially changing the thick- 
ness of the root R. . 





Mg, 26 

Almost all bevel gears are between the two extremes 
of Figs. 26 and 27, so that when the cutting angle 
B O o, Fig. 23, is smaller than about 30°, this change 
in the form of the spaces caused by the rotation of the 
blank maybe so great as to necessitate the substitution 




Fig.. 28 

FINISHED GEAR. 



PROVIDENCE, R. I. 



51 



of a cutter that is narrower at e e', Fig. 24, than is 
called for by the way of figuring that we have just 
given : thus in our own gear cutting department we 
might cut the pinion with a No. 6 cutter, instead of a 
No. 8. The No. 6, being for 17 to 20 teeth, cuts the 
tooth sides with a longer radius of curvature than the 
No 8, which may necessitate considerable filing at the 
small ends of the teeth in order to round them over 
enough. Fig. 28 shows the same gear as Fig. 25, but 
in this case the teeth have all been filed similar to 
M M, Fig. 25. 

Difi'erent workmen prefer different ways to com- Filing the 
promise in the cutting of a bevel gear. When a gmaii end. ^^ 
blank is rotated in adjusting to finish the large end of 
the teeth there need not be much filing of the small 
end, if the cutter is right, for a pitch circle of the 
radius B c, Fig. 23, which for our example is a No. 8 
cutter, but the tooth faces may be rather thin at the 
large ends. This compromise is preferred by nearly 
all workmen, because it does not require much filing 
of the teeth : it is the same as is in our catalogue by 
which we fill any order for bevel gear cutters, unless 
otherwise specified. This means that we should send Selection of 

^ cutter when 

a No. 8, 8-pitch bevel gear cutter in reply to an order teeth are to 

for a cutter to cut the 12-tooth pinion, Fig. 23 ; while 

in our own gear cutting department we might cut the 

same pinion with a No. 6, 8-pitch cutter, because we 

prefer to file the teeth at the small end after cutting 

them to the right thickness at the faces of the large 

end. We should take a No. 6 instead of a No. 8 only 

for a 12-tooth pinion that is to run with a gear two or 

three times as large. We generally step off to the 

next cutter for pinions fewer than twenty-five teeth, 

when the number for the teeth has a fraction nearly 

reaching the range of the next cutter : thus, if twice 

the line B c in inches, Fig. 23, multiplied by the 

diametral pitch, equals 20.9, we should use a No. 5 

cutter, which is for 21 to 25 teeth inclusive. In 

filling an order for a gear cutter, we do not consider 



52 BROWN & SHARPE MFG. CO. 

the fraction but send the cutter indicated by the whole 
number. 

Later on we will refer to other compromises that are 
made in the cutting of bevel gears. 

The sizes of the 8-pitch tooth parts, Fig. 23, at the 
large end, are copied from the table of spur gear 
teeth, pages 146 to 149. 

The distance Oc' is seven-tenths of the apex dis- 
tance Oc, so that the sizes of the tooth parts at the 

Form of „ , ^ , , , r^,, 

gear cutting small end, except f , are seven-tenths the large. The 

order 

order for cutting these gears goes to the workmen in 
this form : 

Large Gear. 

P = 8 
N = 24 
D" + f := .270" D'" + f = .195" 

t --= .196" t' = .137" 

s = .125" s' = .087" 

Cutting Angle = 59° 10' 

Small Gear. 

N = 12 

Cutting Angle = 22° 18' 

Setting the ^^S- ^2 is a side view of a Gear Cutting Machine. 

machine. j^ bevel gear blank A is held by the index spindle B. 

The cutter C is carried by the cutter-slide D. The 

cutter-slide-carriage E can be set to the cutting angle, 

the degrees being indicated on the quadrant F. 

Fig. 33 is a plan of the machine : in this view the 
cutter-slide-carriage, in order to show the details a 
little plainer, is not set to an angle. 

Before beginning to cut the cutter is set central with 
the index spindle and the dial G is set to zero, so 
that we can adjust the cutter to any required distance 
out of centre, in either direction. Set the cutter-slide- 
carriage E, Fig. 32, to the cutting angle of the gear, 
which for 24-teeth is 59° 10' ; the quadrant being 
divided to half-degrees, we estimate that 10' or ^ de- 



PROVIDENCE, R. I 



53 



gree more than 59°. Mark the depth of the cut at the 
outside, as in Fig. 30 : it is also well enough to mark 
the depth at 'he inside as a cheek. The thickness of 
the teeth at the large end is conveniently deter- 
mined by the solid gauge, Fig. 29. The gear-tooth 



M^: 



mff.gd 



GEAR TOOTH GAUGE. 




DEPTH 
GAUGE. 



JFig. 30 



GEAR TOOTH CALIPER. 
mg.31 



vernier caliper, Fig. 31, will measure the thickness of 
teeth up to 2 diametral pitch. In the absence of the 
vernier caliper we can file a gauge, similar to Fig 29, 
to the thickness of the teeth at the small end. 

The index having been set to divide to the right side^of 'tooth 
number we cut two spaces central with the blank, hieing flnisiied. 
leaving a tooth between that is a little too thick, as in 
the upper part of Fig. "lb. If the gear is of cast iron, 
and the pitch is not coarser than about 5 diametral, 
this is as far as we go with the central cuts, and we 
proceed to set the cutter and the blank to finish first 
one side of the teeth and then the other, going around 
only twice. The tooth has to be cut away more in 
proportion from the large than from the small end, 
which is the reason for setting the cutter out of centre, 
as in Fiff. 24. 



54 



BROWN & yHARPE MFQ. CO. 




Fig. 32. 



AUTOMATIC GEAR CUTTING MACHINE. 



SIDE ELEVATION. 



PROVIDENCE, R. I. 65 

It is important to remember that the part of the 
cutter that is finishing one side of a tooth at the pitch 
line should be central with the gear blank, in order to 
know at once in which direction to set the cutter out of 
centre. We can not readily tell how much out of 
centre to set the cutter until we have cut and tried, 
because the same part of a cutter does not cut to the 
pitch line at both ends of a tooth. As a trial distance 
out of centre we can take about one-tenth to one- 
eighth of the thickness of the teeth at the large end. 
The actual distance out of centre for the 12-tooth 
pinion is .021" : for the *24-tooth gear, .030", when 
using cutters listed in our catalogue. 

After a little practice a workman can set his cutter ^^ecessityof 

'^ central cuts. 

the trial distance out of centre, and take his first cuts, 
without any central cuts at all ; but it is safer to take 
central cuts like the upper ones in Fig. 25. The 
depth of cut is partly controlled by the index-spindle 
raising-dial-shaft H, Fig. 33, which determines the 
height of the index spindle, and partly by the position 
of the cutter spindle. We now set the cutter out of 
centre the trial distance by means of the cutter-spindle 
dial-shaft, I, Fig. 33. The trial distance can be about 
one-seventh the thickness of the tooth at the large end 
in a 12-tooth pinion, and from that to one-sixth the 
thickness in a 24-tooth gear and larger. The principle 
of trimming the teeth more at the large end than at 
the small is illustrated in Fig. 24, which is to move 
the cutter away from the tooth to be trimmed, and 
then to bring the tooth up against the cutter by 

rotating the blank in the direction of the arrow. .^. ^ 

° Adjustments. 

The rotative adjustment of the index spindle is 
accomplished by loosening the connection between the 
index worm and the index drive, and turning the worm : 
the connection is then fastened again. The cutter is 
now set the same distance out of centre in the other 
direction, the index spindle is adjusted to trim the 
other side of the tooth until one end is down nearly 
to the right thickness. If now the thickness of the 



56 BROWN & SHARPK MFG. CO. 

small end is in the same proportion to the large end as 
Oc' is to Oc, Fig. 23, we can at once adjust to trim 
the tooth to the right thickness. But if we find that 
the large end is still going to be too thick when the 
small end is right, the out of centre must be increased. 

It is well to remember this : too much out of centre 
leaves the small end proportionally too thick, and too 
little out of centre leaves the small end too thin. 

After the proper distance out of centre has been 
learned the teeth can be finish-cut by going around out 
of centre first on one side and then on the other with- 
out cutting any central spaces at all. The cutter 
spindle stops, J J, can now be set to control the out 
of centre of the cutter, without having to adjust by 
the dial G. If, however, a cast iron gear is 5-pitch 
or coarser it is usually well to cut central spaces first 
and then take the two out-of-centre cuts, going around 
three times in all. Steel gears should be cut three 
times around. 

Blanks are not always turned nearly enough alike to 
be cut without a different setting for different blanks. 
If the hubs vary in length the position of the cutter 
spindle has to be varied. In thus varying, the same 
depth of cut or the exact D" -{- f may not always be 
reached. A slight difference in the depth is not so 
objectionable as the incorrect tooth thickness that it 
may cause. Hence, it is well, after cutting once 
around and finishing one side of the teeth, to give 
careful attention to the rotative adjustment of the 
index spindle so as to cut the right thickness. 

After a gear is cut, and before the teeth are filed, it 
is not always a very satisfactory-looking piece of work. 
In Fig. 25 the tooth L is as the cutter left it, and is 
ready to be filed to the sh ipe of the teeth M M , which 
have been filed. Fig. 3-t is the pair of gears that we 
have been cutting ; the teeth of the 12-tooth pinion 
have been filed. 



PROVIDENCE, R. I. 



57 







^P^^ 



UJ 



u 



(3 

p 
h- 

<: 

LU 

O 
O 

< 

O 

I- 

< 



58 



BROWN & SHARPE MFG. CO. 



A second ^ second approximation in cutting with a rotary 

approxima- '■ '^ *= -^ 

tion. cutter is to widen the spaces at the large end by swing- 

ing either the index spindle or the cutter-slide-carriage, 
so as to pass the cutter through on an angle with the 
blank side-ways, called the side-angle, and not rotate 
the blank at all to widen the spaces. This side-angle 
method is employed in our No. 11 Automatic Bevel 
Gear Cutting Machines : it is available in the manufac- 
ture of bevel gears in large quantities, because with 
the proper relative thickness of cutter, the tooth- 
thickness comes right by merely adjusting for the 
side- angle ; but for cutting a few gears it is not much 
liked by workmen, because, in adjusting for the side- 
angle, the central setting of the cutter is usually lost, 
and has to be found by guiding into the central slot 
already cut. If the side-angle mechanism pivots about 
a line that passes very near the small end of the tooth 
to be cut, the central setting of the cutter may not 
be lost. In widening the spaces at the large end, 
the teeth are narrowed practically the same amount at 
the root as at the face, so that this side-angle method 
requires a wider cutter at e e', Fig. 24, than the first, 
or rotative method. The amount of filing required 
to correct the form of the teeth at the small end is 
about the same as in the first method. 

A third ap- A third approximate method consists in cutting 

proximation. , , . , , i n i . -, . 

the teeth right at the large end by gomg around at 
least twice, and then to trim the teeth at the small end 
and toward the large with another cutter, going around 
at least four times in all. This method requires skill 
and is necessarily a little slow, but it contains possi- 
bilities for considerable accuracy. 
A fourth ap- A f ourth method is to have a cutter fully as thick as 
the spaces at the small end, cut rather deeper thuu 
the regular depth at the large end, and go only once 
around. This is a quick method but more inaccurate 
than the three preceding : it is available in the manu- 
facture of large numbers of gears when the tooth-face 



proximation. 



PROVIDENCE, R. I. 



59 




Fig. 34: 



FINISHED GEAR AND PINION, 



60 BROWN & SHARPE MFG. CO. 

is sho*-t compared with the apex distance. It is little 
liked, and seldom employed in cutting a few gears : it 
may require some experimenting to determine the form 
of cutter. Sometimes the teeth are not cut to the 
regular depth at the small end in order to have them 
thick enough, which may necessitate reducing the 
addendum of the teeth, s', at the small end by turning 
the blank down. This method is extensively employed 
by chuck manufacturers. 

A machine that cuts bevel gears with a reciprocating 

motion and using a tool similar to a planer tool is 

called a Gear Planer and the gears so cut are said to 

be planed. 

Planing of ^"^^ form of Gear Planer is that in which the prin- 

bevei gears. q\^\q embodied is theoretically correct ; this machine 
originates the tooth curves without a former. Another 
form of the same class of machines is that in which the 
tool is guided by a former. 

Usually the time consumed in planing a bevel gear 
is greater than the time necessary to cut the same gear 
with a rotary cutter, thus proportionately increasing 
the cost. 

Pitches coarser than 4 are more correct and some- 
times less expensive when planed ; it is hardly prac- 
ticable, and certainly not economical, to cut a bevel 
gear as coarse as 3P. with a rotary cutter. In gears as 
fine as 16P. planing affords no practical gain in quality. 
While planing is theoretically correct, yet the wear- 
ing of the tool may cause more variation in the thick- 
ness of the teeth than the wearing of a rotary cutter, 
and even a planed gear is sometimes improved by filing. 
Moiintino-of ^^ gears are not correctly mounted in the place where 

gears. ^j^gy ^^^ ^^ ^^^^ they might as well not be planed. In 

fact, after taking pains in the cutting of any gear, 

when we come to the mounting of it we should keep 

right on taking pains. 

Angles and The method of obtaining the sizes and angles per- 

gears.*' ^^^ taining to bevel gears by measuring a drawing is quite 
convenient, and with care is fairly accurate. Its 



PROVIDENCE, R. I. 

accuracy depends, of course, upon the careful measur- 
ing of a good drawing. We may say, in general, that 
in measuring a diagram, wliile we can hardly obtain 
data mathematically exact, we are not likely to make 
wild mistakes. Some years ago we depended almost 
entirely upon measuring, but since the publication of 
this "Treatise" and our " Formulas in Gearing " Ave 
calculate the data without any measuring of a drawing. 
In the " Formulas in Gearing" there are also tables 
pertaining to bevel gears. 

Several of the cuts and some of the matter in this 
chapter are taken from an article by O. J. Beale, in 
the "American Machinist," June 20, 1895. 

Cutters for 
.Mitre and Bevel Gears. 



61 



Diametral Pitch. 


Diameter of Cutter. 


Hole in Cutter. 


4 


3 1-2" 


1 1-4" 


5 


3 1-2 


1 1-4 


6 


3 1-2 


1 1-4 


7 


3 1-2 


1 1-4 


8 


3 1-4 


1 1-4 


10 


3 1-4 


7-8 


13 


3 


7-8 


14 


3 


7-8 


16 


2 3-4 


7-8 


20 


2 1-2 


7-8 


24 


2 1-4 


7-8 



When each gear of a pair of bevel gears is of the 
same size and the gears connect shafts that are at right 
angles, the gears are called "Mitre Gears'' and one 
cutter will answer for both. 



62 



BROWN & SHARPE MFG. CO. 




WORM WHEEL 



Number of Teeth, 54. 
Throat Diameter, 44.59". 



Circular Pitch, 2^. 
Outside Diameter 46". 



C3 



CHAPTER XI. 
WORM WHEELS— SIZING BLANKS OF 32 TEETH AND MORE. 



A WORM is a screw made to mesh with the teeth of Worm. 
a wheel called a worm-wheel. As implied at the end of 
Chapter IV., a section of a worm through its axis is, in 
outline, the same as a rack of corresponding pitch. 
This outline can be made either to mesh with single or 
double curve gear teeth ; but worms are usually made 
for single curve, because, the sides of involute rack 
teeth being straight (see Chapter IV.), the tool for 
cutting worm-thread is more easily made. The thread- 
tool is not usually rounded for giving fillets at bottom 
of worm-thread. 

The axis of a worm is usually at right angles to the 
axis of a worm wheel: no other angle of axis is treated 
of ill this book. 

The rules for circular pitch apply in the size of tooth 
parts and diameter of pitch-circle of worm-wheel. 

The pitch of a worm or screw is sometimes given inPitchof worm 
a way different from the pitch of a gear, viz. : in num- 
ber of threads to one inch of the length of the worm or 
screw. Thus, to say a worm is 2 pitch may mean 2 
threads to the inch, or that the worm makes two turns 
to advance the thread one inch. But a worm may be 
double- threaded, triple-threaded, and so on; hence 
to avoid misunderstanding, it is better always to call 
the advance of the worm thread the lead. Thus, a ^^ ^j^.^j^^^^^ 
worm-thread that advances one inch in one turn we 
call one-inch lead in one turn. A single-thread worm 
4 turns to 1" is |^" lead. We apply the term pitch, that is 
the circular pitch, to the actual distance between the 
threads or teeth, as in previous chapters. In single- 
thread worms the lead and the pitch are alike. In 
making a worm and wheel a given number of threads to 



64 



BKOWN & SHAKPE MFG. CO. 




FIG. 35 -WORM AND WORM-WHEEL 

The Thread of Worm is Left-handed; Worm is Single-threaded. 



PROVIDENCE, R. I. 



65 




OO BROWN & SHARPE MFO. CO. 

one inch, we divide 1 " by the number of threads to one 
inch, and the quotient is the circular pitch. Hence, 

Linear pitch, the wheel ill Fig. 36 is ^' circular pitch. Linear pitch 
expresses exactly what is meant by circular pitch. 
Linear pitch has the advantage of being an exact use 
of language when applied to worms and racks. The 
number of threads to one inch linear, is the reciprocal 
of the linear pitch. 

Multiply 3.1416 by the number of threads to one 
inch, and the product will be the diametral pitch of the 
worm-wheel. Thus, we should say of a double-threaded 
worm advancing 1" in \\ turns that: 

Drawing of Lead=f" or .75". Linear pitch or P'z=:f" or .375". 
Worm-wheel. Diametral pitch orP=8.377. See table of tooth parts. 
To make drawing of worm and wheel we obtain 
data as in circular pitch. 

1. Draw center line A O and upon it space off the 
distance a b equal to the diameter of pitch-circle. 

2. On each side of these two points lay off the dis- 
tance 5, or the usual addendum =^ , as 5 c and b d. 

3. From c lay off the distance c O equal to the 
radius of the woi'm. The diameter of a worm is gen- 
erally four or five times the circular pitch. 

4. Lay off the distances c g and d e each equal to /*, 
or the usual clearance at bottom of tooth space. 

5. Through c and e draw circles about O. These 
represent the whole diameter of worm and the diam- 
eter at bottom of worm-thread. 

6. Draw h O and i O at an angle of 30° to 45° with 
A O. These lines give width of face of worm-wheel. 

7. Through g and d draw arcs about O, ending in 
h O and i O. 

This operation repeated at a completes the outline 

of worm-wheel. For 32 teeth and more, the addendum 

diameter, or D, should be taken at the throat or 

smallest diameter of wheel, as in Fig. 36. Measure 

sketch for whole diameter of wheel-blank. 

Teeth of The foregoing instructions and sketch are for cases 

ished^^th Hob" where the teeth of the wheels are finished with a hob. 

Hob. "^ HOB is shown in Fig. 37, being a steel piece 



PEOVIDENCE, R. I. 67 

threaded with a tool of the same angle as the tool that 

threads the worm, the end of the tool being .335 of 

the linear pitch ; the hob is then grooved to make teeth 

for cutting, and hardened. 

The whole diameter of hob should be at least 2 f. Proportionsof 

■' ' Hob. 

or twice the clearance larger than the worm. In our 
relieved hobs the diameter is made about .005" to .010" 
larger to allow for wear. The outer corners of hob-thread 
can be rounded down as far as the clearance distance. 
The width at top of the hob-thread before rounding 
should be .31 of the linear, or circular pitch ^.31P'. 
The whole depth of thread is thus the ordinary work- 
ing depth plus the clearance ^D"-f/. The diameter 
at bottom of hob-thread should be 2/-f.005" to .010" 
larger than the diameter at bottom of worm-thread. 




Fig. 37— HOB. 

For thread-tool and worm-thread see end of Chapter 
IV. 

In the absence of a special worm gear cutting "^^-^^^q^^ ^®® 
chine, the teeth of the wheel are first cut as nearly to the 
finished form as practicable; the hob and worm-wheel 
are mounted upon shafts and hob placed in mesh, it is 
then rotated and dropped deeper into the wheel until the 
teeth are finished. The hob generally drives the worm- 
wheel during this operation. The Universal Milling Ma- universal 
chine is convenient for doing this work ; with it the dis- chW'used in 

Hobtoing. 



68 



BROWN & SHAEPE MFG. CO. 



^^■"^'^V^'-^ 




Fig. 38. 



PKOVIDENCE, K. I. 



69 



■CH ,CIR 




Fig. 39. 



70 



BKOWN & SHAKPE MFG. CO. 



tance between axes of worm and wheel can be noted. In 
making wheels in quantities it is better to have a ma- 
chine in which the work spindle is driven by gearing, 
so that the hob can cut the teeth from the solid with- 
whyawheei ^"* gashing. The object of bobbing a wheel is to get 
isHobtoed. more bearing surface of the teeth upon worm-thread. 

The worm-wheels, Figs. 35 and 43, were hobbed. 

Worm- Wheel If we make the diameter of a worm-wheel blank, that 

Less than 30 is to have less than 30 teeth, by the common rules 

for sizing blanks, and finish the teeth with a hob, we 

shall find the flanks of teeth near the bottom to be un- 

interference dercut OX hollowinsf. This is caused by the interfer- 

of Thread and ° '' 

Flank. ence spoken of in Chapter VI. Thirty teeth was there 

given as a limit, which will be right when teeth are 
made to circle arcs. With pressure angle 14^°, and 
rack-teeth with usual addendum, this interference of 
rack-teeth with flanks of gear-teeth begins at 31 teeth 
(31j2g. geometrically), and interferences with nearly the 
whole flank in wheel of 12 teeth. 

Fig. 38, In Fig 38 the blank for worm-wheel of 12 teeth was 

sized by the same rule as given for Fig. 36. The wheel 
and worm are sectioned to show shape of teeth at the 
mid-plane of wheel. The flanks of teeth are undercut 
by the hob. The worm-thread does not have a good 
bearing on flanks inside of A, the bearing being that of 
a corner against a surface. 

^is- 39. In Pig 39 the blank for wheel was sized so that pitch- 

circle comes midway between outermost part of teeth 
and innermost point obtained by worm-thread. 

This rule for sizing worm-wheel blanks has been in 
use to some extent. The hob has cut away flanks of 
teeth still more than in Fig. 38. The pitch circle in 
Fig. 39 is the same diameter as the pitch-circle in Fig. 
38. The same hob was used for both wheels. The 
flanks in this wheel are so much undercut as to mate- 
rially lessen the bearing surface of teeth and worm- 
thread. 
Interference In Cnapter VI. the interference of teeth in high- 
numbered gears and racks with flanks of 12 teeth was 
remedied by rounding off the addenda. Although it 
would be more systematic to round off the threads ol 
a worm, making them, like rack-teeth, to mesh with 



PROVIDENCE, II I. 71 

interchangeable gears, yet this has not generally been 
done, because it is easier to make a worm-thread tool 
with straight sides. 

Instead of cutting away the addenda of worm- 
thread, we can avoid the interference with flanks of 
wheels having less than 30 teeth by making wheel 
blanks larger. 

The flanks of wheel in Fig. 40 are not undercut, be- Fig. 40. 
cause the diameter of wheel is so large that there is 
hardly any tooth inside the pitch-circle. The 
pitch-circle in Fig. 40 is the same size as pitch- 
circles in Figs. 38 and 39. This wheel was sized 
by the following rule : Multiply the pitch diameter of Diameter at 

1 111 rvorr i i T , /i i ■ p • Throat to Avoid 

the wheel by .937, and add to the product four times interference, 
the addendum (4 s) ; the sum will be the diameter for 
the blank at the throat or small part. To get the 
whole diameter, make a sketch with diameter of throat 
to the foregoing rule and measure the sketch. 

It is impractical to hob a wheel of 12 to about 16 or 
18 teeth when blank is sized by this rule, unless the 
wheel is di'iven by independent mechanism and not by 
the hob. The diameter across the outermost parts of 
teeth, as at A B, is considerably less than the largest 
diameter of wheel before it was hobbed. 

In general it is well to size all blanks, as by page 66 
and Figs. 36 and 38, when the wheels are to be hobbed ; 
of course the cutter should be thin enough to leave 
stock for finishing. The spaces can be cut the full 
depth, the cutter being dropped in. 

When worm-wheels are not hobbed it is better to 
turn blanks like a spur-wheel. Little is gained by g ^^^ wh'^V * 
having wheels curved to fit worm unless teeth are fin- 
ished with a hob. The teeth can be cut in a straight 
path diagonally across face of blank, to fit angle of 
worm-thread, as in Figs. 41 and 44. 

Ill setting a cutter to gash a worm wheel, Figs. 42 and Gea*r-cutting 
45, the angle is measured from the axis of the worm- ^^'^'^"^^" 
wheel and the angle of the worm thread is, in conse- 
quence, measured from the perpendicular to the axis 
of the worm. See Chapters V and VIII, Part II. 



72 



BEOWN & SHAKPE MFG. CO. 



^\-3XH C/R 




Fig. 40. 



PROVIDENCE, K. 1. 73 

Some mechanics prefer to make dividing wlieels in 
two parts, joined in a plane perpendicular to axis, hob 
teeth , then turn one part round upon the other, match 
teeth and fasten parts together in the new position, 
and hob again with a view to eliminate errors. With 
an accurate cutting engine we have found wheels like 
Figs. 42 and 45, not bobbed, every way satisfactory. 
As to the different wheels, Figs. 43, 44 and 45, whenaJj^s""'^''^'''*' 
worm is in rigbt position at the start, tbe life-time 
of Fig. 43, under beavy and continuous work, will be 
the longest. 

Fig. 44 can be run in mesb with a gear or a rack as 
well as with a worm when made within tbe angular 
limits commonly required. Strictly, neither two gears 
made in this way, nor a gear and a rack would be 
matbematically exact, as they miglit bear at the sides 
of the gear or at the ends of the teeth only and not in 
the middle. At tbe start the contact of teeth in this 
wheel upon worm-tbread is in points only; yet such 
wheels have been many years successfully used in ele- 
vators. 

Fig 45 is a neat-looking wheel. In gear cutting 
engines where the workman has occasion to turn the 
work spindle by hand, it is not so rough to take hold 
of as Figs 43 and 44. The teeth are less liable to in- 
jury than the teeth of Figs. 43 and 44. 

The diameter of a worm has no necessary relation to 
the speed ratio of the worm to the worm wheel. The 
diameter of the worm can be chosen to suit any dis- 
tance between the worm shaft and the worm-wheel 
shaft. It is unusual to have tbe diameter of the worm 
much less than four times the thread-pitch or linear- 
pitch but the worm can be of anj^ larger diameter, five 
or ten times the linear-pitch, if required. 

It is well to take off the outermost j^art of teeth in 
wheels (Figs. 35 and 48), as shown in these two fig- 
ures, and not leave them sharp, as in Figs. 36 and 39. 
It is also well to round over the outer corners of the 
blanks for the wheels. Figs. 44 and 45. In ordering 
worms and worm wheels the centre distances should 
be given. If there can be any limit allowed in the 
centre distance it should be so stated. 



74 



BKOWN & SHARPE MFG. CO. 




Fig. 41. 



VVorm--wl:ieel with teeth cut in a straight path diagonally across face, 
^A''orm is double-threaded. 



PROVIDENCE, K. I. 



75 




Fig. 42. 

"Worm and Worm- Wheel, for Gear-cutting Engine. 



76 



BROWN & SHAKPE MFG. CO. 



' jjjgmiiiiiit'i^ 






^.«r^ 



Fig. 43. 



Fig. 44. 



Fig. 45. 



PROVIDENCE, K. I. 

For instance, the distance from the centre of a worm 
to the centre of a worm wheel might be calculated at 
6" but 5 31-32" or 6 1-32" might answer. 

By stating all the limits that can be allowed, there 
may be a saving in the cost of work because time need 
not be wasted in trying to make work within narrower 
limits than are necessary. 



11 




Fig. '46 A 

LENGTH OF A WORM AND OF A HOB. 

In worm-wheels, like Figs. 41 and 42, having 540 
teeth, worms can have bearings in ten places or along 
ten threads. Worms in wheels of 120 teeth bear on 
six threads. 

In order to hob a wheel of 540 teeth, the hob must 
be about eleven threads long, if the worm has ten 
threads. For the 120 tooth wheel, the hob should have 
about seven threads, if the worm has six threads. 

For a 80 tooth w^orm M^heel of the form of Figs. 41 
and 42, we can have only about three threads in con- 
tact and a hob four threads long, like Fig. 37, is long 
enough. 

From the diagram. Fig, 45 A, which is similar to 
Fig. 7, we can tell approximately the number of threads 
that can bear. Let the worm move to the right and 
the action begins at C and ends at A', C being the point 
where the line C D intersects the addendum circle of 
the gear and A' being the point where the line would 
intersect the addendum line of the worm. 

A short worm can be used in a large wheel by having 
the hob a little longer than tlie worm. 



78 



BKOWN & SHAKPE MFG. CO. 



GASHING TEETH OF HOB. 
10 Inches Outside Diameter. 







HOBS WITH RELIEVED TEETH. 

We make hobs of any size with the teeth telieved the 
same as our gear cutters. The teeth can be ground on 
their faces without changing tiieir form. The hobs are 
made with a precision screw so that the pitch of the 
thread is accurate before hardening. 



79 



CHAPTER XII. 



SIZING GEARS WHEN THE DISTANCE BETWEEN CENTRES AND THE 
RATIOS OF SPEEDS ARE FIXED— GENERAL REMARKS— WIDTH 
OF FACE OF SPUR GEARS— SPEED OF GEAR COTTERS— TABLE 
OF TOOTH PARTS. 



Let us suppose that we have two shafts 14" apart, 
center to center, and wish to connect them by sfears so, center dis- 

' "^ o tance and Ratio 

that they will have siDced ratio 6 to 1. We add the 6 ^^^d. 
and 1 together, and divide 14" by the sum and get 2" 
for a quotient; this 2", multiplied by 6, gives us the 
radius of pitch circle of large wheel = 12". In the same 
manner we get 2" as radius of pitch cu'cle of small wheel. 
Doubling the radius of each gear, we obtain 24" and 4" 
as the pitch diameters of the two wheels. The two num- 
bers that form a ratio are called the terms of the ratio. 
We have now the rule for obtaining pitch-circle diame- 
ter of two wheels of a given ratio to connect shafts a 
given distance apart: 

Divide the center distajice hy the sum of the terms of ^uie for 

if tf J ameterofPi 

the ratio; find the product of tiifice the quotient hy each circieB. 
term separately, and the t%oo products xoill he the pitch 
diameters of the tioo wheels. 

It is well to give special attention to learning the 
rules for sizing blanks and teeth ; these are much 
oftener needed than the method of forming tooth out- 
lines. 



Di- 
Pitch 



80 BROWN & SHARPE MFG. CO. 

A blank 1^" diameter is to have 16 teeth: what will 
the pitch be? "What will be the diameter of the pitch 
circle ? See Chapter V. 

A good practice will be to compute a table of tooth 
parts. The work can be compared with the tables 
pages 146-149. 

In computing it is well to take 7t to more than four 
places, Tt to nine places = 3.] 41592653. j^ to nine 
places = .318309886. 

There is no such thing as pure rolling contact in 
teeth of wheels ; they always rub, and, in time, will 
wear themselves out of shape and may become noisy. 

Bevel gears, when correctly formed, run smoother 
than spur gears of same diameter and pitch, because 
the teeth continue in contact longer than the teeth of 
spur gears. For this reason annular gears run smoother 
than either bevel or spur gears. 

Sometimes gears have to be cut a little deeper than 
designed, in order to run easily on their shafts. If 
any departui'e is made in ratio of pitch diameters it is 
better to have the driving gear the larger, that is, cut 
the follower smaller. For wheels coarser than eight 
diametral pitch (8 P), it is generally better to cut twice 
around, when accurate work is wanted, also for large 
wheels, as the expansion of parts from heat often causes 
inaccurate work when cut but once around. There is 
not so much trouble from heat in plain or web gears as 
in arm gears. 

£r*fkces.^"'^ The width of face of cast-iron gears can, for general 
use, be made 2| times the linear pitch. 
In small gears or pinions this width is often exceeded. 
The outer corners of spur gears may be rounded off 
for convenience in handling. This can be provided 
for when turning the blank. 
Speed of Gear The speed of gear cutters is subject to so many con- 
ditions that definite rules cannot be given. We append 
a table of average speeds. A coarse pitch cutter for 
pinion, 12 teeth, would usually be fed slower than a 
cutter for a large gear of same pitch. 



PKOVIDKNCE, E. I. 
TABLE OF AVERAGE SPEEDS FOR GEAR-CUTTERS. 



81 



^ 




bO 


bflrH 


^ 


tH 




a 




«4-l 
O 

&3 ^ 


P 

p. 3 g 


P. 3 o 






t- fl cj 


p.o«S 


s 




1 


2 "S^ 




oj -^ fH 5 


Td 

^ b6 


f^ag g 


2 


5 in. 


24 


18 


. 025 in. 


.011 in. 


. 60 in. 


. 20 in. 


2i 


41 u 


30 


24 


.028 " 


.013 " 


.84 " 


.31 " 


3 


3if " 


36 


28 


.031 " 


.015 " 


1.12 " 


.42 " 


4 


3| " 


42 


32 


.034 " 


.017 " 


1.43 " 


.54 " 


5 


3tV " 


50 


40 


.037 " 


.019 " 


1.85 " 


.76 " 


6 


2ji " 


75 


55 


.030 " 


.016 " 


2.25 " 


.88 " 


7 


2tV " 


85 


65 


.032 " 


.018 " 


2.72 " 


1.17 " 


8 


21 " 


95 


75 


.034 " 


.020 " 


3.23 " 


1.50 " 


10 


4 " 


125 


90 


.026 " 


.014 " 


3.25 " 


1.26 " 


12 


2 " 


135 


100 


.027 " 


.017 " 


3.64 " 


1.70 " 


20 


H " 


145 


115 


.029 " 


.021 " 


4.20 " 


2.41 " 


32 


If " 


160 


135 


.031 '^ 


.025 " 


4.96 " 


3.37 " 



In brass tlie speed of geai'-cutters can be twice as -^^^^^ ^'^ ^^ 
fast as in cast iron. Clock-makers and those making a 
specialty of brass gears exceed this rate even. A 12 P 
cutter has been run 1,200 (twelve hundred) tui-ns a 
minute in bronze. A 32 P cutter has been run 7,000 
(seven thousand) turns a minute in soft brass. 

In cutting 5 P cast-iron gears, 75 teeth, a No. 1, ^'P tromP^ctu^V 
cutter was run 136 (one hundred and thirty-six) turns 
a minute, roughing the spaces out the full 5 P depth ; 
the teeth were then finished with a 5 P cutter, running 
208 (two hundred and eight) turns a minute, feeding 
by hand. The cutter stood well, but, of course, the 
cast iron was quite soft. A 4 P cutter has finished 
teeth at one cut, in cast-iron gears, 86 teeth, running 48 
(forty-eight) turns a minute and feeding -^-^" at one 
turn, or 3 in. in a minute. 

Hence, while it is generally safe to run cutters as in 
the table, yet when many gears are to be cut it is well to 
see if cutters will stand a higher speed and more feed. 

In gears coarser than 3 P it is more economical to 
cut first the full depth with a stocking cutter and then 
finish with a gear cutter. This stocking cutter is made 



82 



BROWN & SHARPE MFG. CO. 



Keep Cutters 
sharp. 



on the principle of a circular splitting saw for wood. 
The teeth, however, are not set ; but side relief is ob- 
tained by making sides of cutter blank hollowing. The 
shape of stocking cutter can be same as bottom of 
spaces in a 12-tooth gear, and the thickness of cutter 
can be J of the circular pitch, see page 40. 

The matter of keeping cutters sharp is so important 
that it has sometimes been found best to have the work- 
man grind them at stated times, and not wait until he 
can see that the cutters are dull. Thus, have him 
grind every two hours or after cutting a stated number 
of gears. Cutters of the style that can be ground 
upon their tooth faces without changing foi'm are rap- 
idly destroyed if allowed to run after they are dull. 
Cutters are oftener wasted by trying to cut with them 
when they are dull than by too much grinding. Grind 
the faces radial with a free cutting wheel. Do not let 
the wheel become glazed, as this will draw the temper 
of the cutter. 

In Chapter YI. was given a series of cutters for cut- 
ting gears having 12 teeth and more. Thus, it was 
there implied that any gear of same pitch, having 135 
teeth, 136 teeth, and so on up to the largest gears, and, 
also, a rack, could be cut with one cutter. If this cut- 
ter is 4 P, we would cut with it all 4 P gears, having 
135 teeth or more, and we would also cut with it a 4 P 
rack. Now, instead of always referring to a cutter by 
the number of teeth in gears it is designed to cut, it 
has been found convenient to designate it by a letter 
or by a number. Thus, we call a cutter of 4 P, made 
to cut gears 135 teeth to a rack, inclusive, No. 1, 4 P. 

We have adopted numbers for designating involute 
Involute Gear ofear-cutters a-s in the following table : 

Cutters. 

No. 1 will cut wheels from 185 teeth to a rack inclusive. 



2 




55 




134 teeth 


o 
o 




35 




54 " 


4 




26 




34 " 


5 




21 




25 " 


6 




17 




20 " 


7 




14 




16 " 


8 




12 




13 " 



PROVIDENCE, R. I. 83 

By this plan it takes eight cutters to cut all gears 
having twelve teeth and over, of any one pitch. 

Thus it takes eight cutters to cut all involute 4 P 
gears having twelve teeth and more. It takes eight 
other cutters to cut all involute gears of 5 P, having 
12 teeth and more. A No. 8, 5 P cutter cuts only 5 P 
gears having 12 and 13 teeth. A No. 6, 10 P cutter 
cuts only 10 P gears having 17, 18, 19 and 20 teeth. 
On each cutter is stamped the number of teeth at the 
limits of its range, as well as the number of the cutter. 
The number of the cutter relates only to the number 
of teeth in gears that the cutter is made for. 

In ordering cutters for involute spur-gears two things 
must be given : 

1. Either the number of teeth to he cut in the gear ^ How to order 

■' _ "^ _ Involute Cut- 

or the number of the cutter, us given in the foregoing ters. 
table. 

2. Either the pitch of the gear or the diam^eter and 
number of teeth to be cut in the gear. 

If 25 teeth are to be cut in a 6 P involute gear, the 
cutter will be No. 5, 6 P, which cuts all 6 P gears from 
21 to 25 teeth inclusive. If it is desired to cut gears 
from 15 to 25 teeth, three cutters will be needed, No. 
5, No. 6 and No. 7 of the pitch required. If the pitch 
is 8 and gears 15 to 25 teeth are to be cut, the cutters 
should be No. 5, 8 P, No. 6, 8 P, and No. 7, 8 P. 

For each pitch of epicycloidal, or double-curve gears, Epicycioidai 
24 cutters are made. In coarse-pitch gears, the varia- curve cutters. 
tion in the shape of spaces between gears of consecu- 
tive-numbered teeth is greater than in fine-pitch gears. 

A set of cutters for each pitch to consist of so large 
a number as 24, was established for the reason that 
double carve teeth were formerly preferred in coarse 
pitch gears. The tendency now, however, is to use the 
involute form. 

Our double curve cutters have a guide shoulder on each 
side for the depth to cut. When this shoulder just reaches 
the periphery of the blank the depth is right. The marks 
which these shoulders make on the blank, should be as nar- 
row as can be seen, when the blanks are sized right. 



84 



BKOWN & SHAKPE MFG. CO. 



Double-curve gear-cutters are designated by letters 
instead of by numbers ; this is to avoid confusion in 
ordering. 

Following is the list of epicycloidal or double-curve 
gear-cutters : — 
cy^c^oida 1^ OT Cutter A cuts 12 teetb. Cutter M cuts 27 to 29 teeth. 



Double -curve 
Gear Cutters. 




B 

C 


' 13 " 

' 14 " 






D 


' 15 " 






E 


' 16 " 






F 


' 17 " 






G 


' 18 " 






H 


' 19 " 






I 


' 20 " 






J 


' 21 to 22 






K 


' 23 to 24 






L 


' 24 to 26 



N 
O 
P 

Q 
R 

S 
T 
U 
V 

w 

X 



30 " 33 " 
34 " 37 " 
42 ' 
49 " 
59 " 
74 " 
99 " 
149 " 
249 " 

250 " Eack. 

Rack. 



38 
43 

50 

60 

75 

100 

150 



A cutter that cuts more than one gear is made of 
proper form for the smallest gear in its range. Thus, 
cutter J for 21 to 22 teeth is correct for 21 teeth; 
cutter S for 60 to 74 teeth is correct for 60 teeth, 
and so on. 
Epicycloidal ^^ Ordering epicycloidal gear-cutters designate the 
Cutters. letter of the cutter as in the foregoing table, also 

either give the pitch or give data that v^ill enable us 
to determine the pitch, the same as directed for invo- 
lute cutters. 

More care is requii-ed in making and adjusting epi- 
cycloidal gears than in making involute gears. 
How to order j^ ordering bevel-gear cutters three thing's must be 

Bevel Gear & o t> 

Cutters. ffiven : 

1. The number of teeth in each gear. 

2. Either the pitch of gears or the largest pitch 

diameter of each gear; see Fig. 17. 

3. The length of tooth face. 

If the shafts are not to run at right angles, it 
should be so stated, and the angle given. Involute 
cutters only are used for cutting bevel gears. No at- 
tempt should be made to cut epicyclodial tooth bevel gears 
with rotary disk cutters. 



PROVIDENCE, K. I. 85 

In orderiuff worm-wheel cutters, three thinpfs must uow to order 

. ° » o Worm -gear 

be given : Cutters. 

1. N'umber of teeth in the loheel. 

2. Pitch of the worm; see Chapter XI. 

3. M^hole diameter of worm. 

In any order connected with gears or gear-cutters, 
when the word " Diameter " occurs, we usually under- 
stand that the intch diameter is meant. When the 
tohole diameter of a gear is meant it should be plainly 
written. Care in giving an order often saves the delay 
of asking further instructions. An order for one gear- 
cutter to cut from 25 to 30 teeth cannot be filled, be- 
cause it takes two cutters of involute form to cut from 
25 to 30 teeth, and thi-ee cutters of epicycloidal form 
to cut from 25 to 30 teeth. 

Sheet zinc is convenient to sketch gears upon, and 
also for making templets. Before making sketch, it is 
well to give the zinc a dark coating with the following 
mixture : Dissolve 1 ounce of sulphate of copper (blue 
vitriol) in about 4 ounces of water, and add about one- 
half teaspoonful of nitric acid. Apply a thin coating 
with a piece of waste. 

This mixtui'e will give a thin coating of copper to 
iron or steel, but the work should then be rubbed dry. 
Care should be taken not to leave the mixture where it 
is not wanted, as it rusts iron and steel. 

We have sometimes been asked why gears are noisy. 
Not many questions can be asked us to which we can 
give a less definite answer than to the question why 
gears are noisy. 

We can indicate only some of the causes that may 
make gears noisy, such as: — depth of cutting not 
right — in this particular gears are oftener cut too deep 
than not deep enough ; (more noise may be caused 
by cutting the driver too deep than by cutting the 
driven too deep;) cutting not central — this may 
make gears noisy iu one direction when they are quiet 
while running in the other direction ; centre distance 
not right — if too deep the outer corners of the 
teeth in one gear may strike the fillets of the teeth 
in the other gear ; shafts not parallel ; frame of the 



86 



BROWN & SHARPE MFG. CO. 



machine of such a form as to give off sound vibrations. 
Even when we examine a pair of gears we cannot 
always tell what is the matter. 



IMPROVED 29° SCREW THREAD TOOL GAUGE. 



'ACME STANDARD. 




DEPTH OF GEAR TOOTH GAUGES. 




Depth of Gear Tooth Gauges for all regular pitches, from 3 to 
48 pitch inclusive, are carried in stock. 

One Gauge answers for each pitch, and indicates the extreme 
depth to be cut. 



PART II. 

CHAPTER I. 
TANGENT OF ARC AND ANGLE. 



In Pakt II. we shall show how to calculate some g|"|^^?^^^_'^ ^® 
of the functions of a right-angle triangle from a table 
of circular functions, the application of these calcula- 
tions in some chapters of Part I. and in sizing blanks 
and cutting teeth of spiral gears, the selection of 
cutters for spii'al gears, the application of continued 
fractions to some problems in gear wheels and cutting 
odd screw-threads, etc., etc. 

A Functio7i is a quantity that depends upon another 
quantity for its value. Thus the amount a workman 
earns is a function of the time he has worked and of g^^^^*^'''^ *^®" 
his wages per hour. 




In any right- angle triangle, O A B, we shall, for Rigiit- angle 
convenience, call the two lines that form the right 
angle O A B the sides, instead of base and perpen- 
dicular. Thus O A B, being the right angle we call 
the line O A a side, and the line A B a side also. 

When we speak of the angle A O B, we call the line 
O A the side adjacent. "When we are speaking of the^^*^® *^'^^^°®^*- 
angle ABO we call the line A B the side adjacent. 
The line opposite the right angle is the hypothenuse. Hypothenuse. 

In the following pages the definitions of circular 
functions are for angles smaller than 90°, and not 
strictly applicable to the reasoning employed in ana- 
lytical trigonometry, where we find expressions for 
angles of 370°, 760°, etc. 



88 

Tangent. 



BEOWN & SHARPS MFG. CO. 

The Tangent of an arc is the line that touches it at 
one extremity and is terminated by a line drawn from 
the center through the other extremity. The tangent 
is always outside the arc and is also perpendicular to 
the radius which meets it at the point of tangency. 




Fig. 47. 

Thus, in Fig. 47, the line A B is the tangent of the arc 
A C. The point of tangency is at A. 

An angle at the center of a circle is measured by the 
arc intercepted by the sides of the angle. Hence the 
tangent A B of the arc A C is also the tangent of the 
angle A O B. 

In the tables of circular functions the radius of the 
arc is unity, or, in common practice, we take it as one 
inch. The radius O A being 1", if we know the length 
of the line or tangent A B we can, by looking in a 
table of tangents, find the number of degrees in the 
angle A O B. 
To find the Thus, if A B is 2.25" long, we find the angle A O B 

J_)©^r©6S In 3,11 

Angle. is 66 very nearly. That is, having found that 2.2460 

is the nearest number to 2.25 in the table of tangents 
at the end of this volume, we find the corresponding 
degrees of the angle in the column at the left hand of 
the table and the minutes to be added at the top of 
the column containing the 2.2460. 

The table gives angles for every 10', which is suf- 
ficient for most purposes. 



PROVIDENCE, R. I. 89 

Now, if we liave a right-angle triangle with an angle 
the same as O A B, but with O A two inches long, the 
line A B will also be twice as long as the tangent of 
angle A O B, as found in a table of tangents. 

Let us take a triangle with the side O A = 5" long, And^^hi'^l, e? 
and the side A B = 8" long; what is the number oflJ^®®^^ in an 
degrees in the angle A O B ? 

Dividing 8" by 5 we find what would be the length 
of A B if O A was only 1" long. The quotient then 
would be the length of tangent when the radius is 1" 
long, as in the table of tangents. 8 divided by 5 is 
1.6. The nearest tangent in the table is 1.6003 and 
the corresponding angle is 58°, which would be the 
angle A O B when A B is 8" and the radius O A is 5" 
very nearly. The difference in the angles for tangents 
1.6003 and 1.6 could hardly be seen in practice. The 
side opposite the requu'ed acute angle corresponds to 
the tangent and the side adjacent corresponds to the 
radius. Hence the rule : 

To find the tangent of either acute angle in a right- rpj° g^^*^ ^^^ 
angle triangle : Divide the side opposite the angle by 
the side adjacent the angle and the quotient toill be 
the tangent of the angle. This rule should be com- 
mitted to memory. Having found the tangent of the 
angle, the angle can be taken from the table of tan- 
gents. 

The complement of an angle is the remainder after complement 
subtracting the angle from 90°. Thus 40° is the com- 
plement of 50°. 

2'he Cotangent of an angle is the tangent of the Cotangent, 
complement of the angle. Thus, in Fig. 47, the line 
A B is the cotangent of A O E. In right-angle tri- 
angles either acute angle is the complement of the 
other acute angle. Hence, if we knoAv one acute angle, 
by subtracting this angle from 90° we get the other 
acute angle. As the arc approaches 90°, the tangent 
becomes longer, and at 90° it is infinitely long. 

The sign of infinity is oo. Tangent 90° = oo. 



90 BKOWN & SHAKPE MFG. CO. 

Angie^^by"\he ^J ^ table of tangents, angles can be laid out upon 
Tangent E^x- gi^eet zinc, etc. This is often an advantage, as it is not 
convenient to lay protractor flat down so as to mark 
angles up to a sharp point. If we could lay off the 
length of a line exactly we could take tangents direct 
from table and obtain angle at once. It, however, is 
generally better to multiply the tangent by 5 or 10 
and make an enlarged triangle. If, then, there is a 
slight error in laying off length of lines it will not 
make so much difference with the angle. 

Let it be required to lay off an angle of 14° 30'. By 
the table we find the tangent to be .25861. Multiply- 
ing .25861 by 5 we obtain, in the enlarged triangle, 
1.29305" as the length of side opposite the angle 14° 
30'. As we have made the side opposite five times as 
large, we must make the side adjacent five times as 
large, in order to keep angle the same. Hence, Fig. 
48, draw the line A B 5" long ; perpendicular to this 
line at A draw the line A O 1.293" long ; now draw the 
line O B, and the angle A B O wHl be 14° 30'. 

If special accuracy is required, the tangent can be 
multiplied by 10; the line A O will then be 2.586" long 
and the line A B 10" long. Remembering that the 
acute angles of a right-angle triangle are the comple- 
ments of each other, we subtract 14° 30' from 90' and 
obtain 75° 30' as the angle of A O B. 

The reader will remember these angles as occurring 
in Part I., Chapter IV., and obtained in a different 
way. A semicircle upon the line O B touching the 
extremities O and B will just touch the right angle at 
A, and the line O B is four times as long as O A. 

Let it be required to turn a piece 4" long, 1" diam- 
eter at small end, with a taper of 10° one side with the 
other ; what will be the diameter of the piece at the 
large end ? 

A section. Fig. 49, through the axis of this piece is 
To calculate ^;\^q same as if we added two right-angle triangles, O 

Ta pe r i n gA. B and O' A' B', to a straight piece A' A B B', 1" 

piece. Fig. 50. . ' to i , . n 

wide and 4 long, the acute angles B and B being 5 , 

thus making the sides O B and O' B' 10° with each 

other. 



PROVIDENCE, K. 1. 



91 




-h293-t- 
Fig. 48. 




Fig. 49. 



92 



BEOWN & SHAEPE MFG. CO. 

The tangent of 5° is .08748, which, multiplied by 
4 , gives .34992" as the length of each line, A O and 
A' O', to be added to 1" at the large end. Taking 
twice .34992" and adding to 1" we obtain 1.69984" as 
the diameter of large end. 

This chapter must be thoroughly studied before 
taking up the next chapters. If once the memory 
becomes confused as to the tangent and sine of an 
angle, it will take much longer to get righted than it 
will to first carefully learn to recognize the tangent 
of an angle at once. 

If one knows what the tangent is, one can tell better 
the functions that are not tangents. 




93 



CHAPTER II. 

SINE— COSINE AND SECANT : SOME OF THEIR APPLICATIONS IN 
MACHINE CONSTRUCTION. 



Sine of Arc 
' and Angle 



The Sine of an arc is the line di-awn from one 
extremity of the arc to the diameter passing through 
the other extremity, the line being perpendicular to 
the diameter. 

Another definition is : The sine of an arc is the dis- 
tance of one extremity of the arc from the diameter, 
through the other extremity. 

The sine of an angle is the sine of the arc that , 
measures the angle. 

In Fig. 50 , A C is the sine of the arc B C, and of 
the angle B O C. It will be seen that the sine is 
always inside of the arc, and can never be longer than 
the radius. As the arc ap- 
proaches 90°, the sine comes 
nearer to the radius, and at 90° 
the sine is equal to 1, or is the 
radius itself. From the defini- 
tion of a sine, the side A C, 
opposite the angle A O C, in 
any right-angle triangle, is the 
sine of the angle A O C, when 
O C is the radius of the arc. 
Hence the rule : I?i any right-angle triangle, the side To find the 
opposite either acute angle, divided hy the hypothe- 
nuse, is equal to the sine of the angle. 

The quotient thus obtained is the length of side 
opposite the angle when the hypothenuse or radius is 
unity. The rule should be carefully committed to 
memory. 



^^ 


A 


\ 


C 


'\ 


\ 




/ 


\ 


^ 


E 


") 



Fig. 50. 



94 



BROWN & SHARPE MFG. CO. 



Chord of an j^ Chord is a straight line joining the extremities of 
an arc, and is twice as long as the sine of half the 
angle measured by the arc. Thus, in Fig. 50, the 
chord F C is twice as long as the sine A C. 





f_B ^ 




^'-A 






— ^ 










^•^ 


\/ 


y 


^^ 






— -^ 




-^ / 






\ 








N 


y / 






N 


y / 






S 


/ / 






S 


/ / 
/ / 
/ / 

^/ 






\ 
\ 
\ 
\ 
\ 
\ 


' / \ 






\ 


1 / \ 






\ 








\ 

\ 


' / \ 


\ 




\ 


A V \ 


\ 




{ \ 


^ 1 ) 
1 / 

I 
\ 

\ 
\ 
\ 







/ 

/ 
/ 
/ 


\ 






/ 


\ 






/ 


\ 






/ 


\ 






/ 


\ 






/ 


\ 






/ 


\ 






/ 


\ 






/ 


\ 






/ 


s 






y 


V 






y 


s. 






y 










X 







^ 






^\ 
















\ 




"^--- 





1 — \ 


— -^ 



Fig. Rl. 

Let there be four holes equidistant about a circle 
3" in diameter — Fig. 51 ; what is the shortest distance 
between two holes ? This shortest distance is the 

flnd^th^chord! chord A B, which is twice the sine of the angle COB. 
The angle A O B is one-quarter of the circle, and 
C O B is one-eighth of the circle. 360^, divided by 
8=45°, the angle COB. The sine of 45° is .70710, 
which multiplied by the radius 1.5", gives length C B iu the 
circle, 8" in diameter, as 1.06065". Twice this length is 
the required distance A B=2.1213". 

When a cylindrical piece is to be cut into any num- 
ber of sides, the foregoing operation can be applied to 
obtain the width of one side. A plane figure bounded 

Polygon. by straight lines is called a polygon. 



PROVIDENCE, E. I. 95 

When the outside diameter and the number of sides of 

a regular polygon are given, to find tlie length of 

one of the sides: Divide 360° hy tioice the number of , To find the 
. -^ •' length of Side. 

Sides ; ')nidUply the sine of the quotient hy the outer 

diameter, and the product loill be the length of one of 

the sides. 

Multiplying by the diameter is the same as multi- 
plyitig by the radius, and that product again by 2. 

The Cosine of an angle is the sine of the comple- cosine. 
ment of the angle. 

In Fig. 50, C O D is the complement of the angle 
A O C ; the line C E is the sine of COD, and hence 
is the cosine of B O C. The line O A is equal to C E. 
It is quite as well to remember the cosine as the part 
of the radius, from the center that is cut off by the 
sine. Thus the sine A C of the angle A O C cuts off 
the cosine O A. The line A may be called the 
cosine because it is equal to the cosine C E. 

In any right-angle triangle, the side adjacent either 
acute angle corresponds to the cosine when the 
hypothenuse is the radius of the arc that measures 
the ans'le ; hence: Divide the side adjacent the ac^cte To find the 

Cosine. 

angle by the hypothenuse, and the quotient will be the 
cosine of the angle. 

When a cylindrical piece is cut into a polygon of 
any number of sides, a table of cosines can be used tOgj^^l^^^t^^j"^^^ 
obtain the diameter across the sides. s^"^- 




96 



BROWN & SHARPE MFG. CO. 



Let a cylinder, 2" diameter, Fig. 53, be cut six-sided : 
what is the diameter across the sides ? 

The angle A O B, at the center, occupied by one of 
these sides, is one-sixth of the circle, =60°. The 
cosine of one-half this angle, 30°, is the line C O; 
twice this line is the diameter across the sides. The 
cosine of 30° is .86602, which, multiplied by 2, gives 
1.73204" as the diameter across the sides. 

Of course, if the radius is other than unity, the cosine 
should be multiplied by the radius, and the product 
again by 2, in order to get diameter across the sides ; 
or what is the same thing, multiply the cosine by the 
whole diameter or the diameter across the corners. 

The rule for obtaining the diameter across sides of 
sidesof aPoiy-j.Qg.yjg_j. pQ;[yg.Q^^^ -^yjigj^ ^l^e ^i^™^®^®^ ^cross corners is 
given, will then be : Multiply the cosine of 360° 
divided by tvnce the number of sides, by the diameter 
across corners, and the product will be the diameter 
across sides. 

Look at the right-hand column for degrees of the 
cosine, and at bottom of page for minutes to add to 
the degrees. 

2^he Secant of an arc is a straight line di'awn from 
the center through one end of an arc, and terminated 
by a tangent drawn from the other end of the arc. 

Thus, in Fig. 53, the line O B is the secant of the 
angle COB. 

A C B 



Eule for Di- 
ameter across 



Secant. 




Fig. 63. 

To find the In any right-angle triangle, divide the hypothenuse 
by the side adjacent either acute angle, and the quo- 
tient will be the secant of that angle. 



PROVIDENCE, R. 1. 



07 



That is, if we divide the distauce OB by O C, in 
the right-angle triangle COB, the (jviotient will be 
the secant of the angle COB. 

The secant cannot be less than the radius ; it in- 
creases as the angle increases, and at 90° the secant is 
infinity =00 . 

A six-sided piece is to be l-j" across the sides ; liow^ j^^j^^J^^^'^ 
large mnst a blank be turned before cutting the sides ? ^f^°^^ comers 

o o ot a Polygon. 

Dividing 360° by twice the number of sides, we have 
30°, which is the angle COB. The secant of 30° is 
1.1547. 

The radius of the six-sided piece is .75". 

Multiplying the secant 1.1547 by .75", we obtain the 
length of radius of the blank O B ; multiplying again 
by 2, we obtain the diameter 1.732"-)-. 

Hence, in a regular polygon, when the diameter 
across sides and the number of sides are given, to find 
diameter across corners : Multiply the secant of 360° 
divided hy tvnce the number of sides, hy the diameter 
across sides, and the product toill he the diameter 
across corners. 

It will be seen that the side taken as a divisor has 
been in each case the side corresponding to the radius 
of the arc that subtends the angle. 

The versed sine of an acute angle is the part of 
radius outside the sine, or it is the radius minus the 
cosine. Thus, in Fig. 50, the versed sine of the arc 
BC is AB. The versed sine is not given in the tables 
of circular functions : when it is wanted for any angle 
less than 90° we subtract the cosine of that angle from 
the radius 1. Having it for the radius 1, we can 
multiply by the radius of any other arc of which we 
may wish to know the versed sine. 



98 



BROWN & SHARPE MFG. CO. 











NO. 13. AUTOMATIC GEAR CUTTING MACHINE. 
For Spur and Bevel Gears. 



PROVIDENCE, R. I. 



99 




FRONT VIEW. 




REAR VIEW. 



GEAR MODEL. 
Shows combination of six different kinds of gears. 



100 



CHAPTER III. 

APPLICATION OF CIRCULAR FUNCTIONS— WHOLE DIAMETER 
BEYEL GEAR BLANKS— ANGLES OF BEYEL GEAR BLANKS. 



The rules given in this chapter apply only to bevel 
gears having the center angle c' O i not greater than 90*^. 

To avoid confusion we will illustrate one gear only. 
The same rules apply to all sizes of bevel gears. Fig. 
55 is the outline of a pinion 4 P, 20 teeth, to mesh with 
a gear 28 teeth, shafts at right angles. For making 
sketch of bevel gears see Chapter IX., Pakt I. 

In Fig. 55, the line O in' m is continued to the line 
a b. The angle c' O i that the cone pitch-line makes 
with the center line may be called the center angle. 
Angle of The center angle c O ^ is equal to the angle of edge 
' c' i c. c' ^ is the side opposite the center angle c' O 
i, and c' O is the side adjacent the center angle, c' 
i = 2.5"; c' O = 3.5". Dividing 2.5" by 3.5" we 
obtain .71428" + as the tangent of c' O i. In the table 
we find .71329 to be the nearest tangent, the corre- 
sponding angle being 35° 30'. S5^°, then, is the center 
angle c' O i and the angle of edge c i n, yevj nearly. 

When the axes of bevel gears are at right angles the 
angle of edge of one gear is the complement of angle 
of edge of the other gear. Subtracting, then, 35^° 
from 90° we obtain 54^° as the angle of edge of gear 
28 teeth, to mesh with gear 20 teeth, Fig. 55, from which we 
have the rule for obtaining centre angles when the axes of 
gears are at right angles. 

Divide the radius of the pinion by the radius of the gear 
and the quotient will be the tangent of centre angle of the 
pinion. 

Now subtract this centre angle from 90 deg. and we have' 
the centre angle of the gear. 

The same result is obtained by dividing the number of 
teeth in the pinion by the number of teeth in the gear ; the 
quotient is the tangent of the centre angle. 



PBOVIDENCE, K. I. 



101 




Fig. 55. 

BEVEL GEAR DIAGRAM. 



102 BKOWN & SHAKPE MFG. CO. 

Angle Of Face. To obtain angle of face O m" c\ the distance c O 
becomes the side opposite and the distance m" c is 
the side adjacent. 

The distance c O is 3.5", the radius of the 28 tooth 
bevel gear. The distance c m" is by measurement 
2.82". 

Dividing 3.5 by 2.82 we obtain 1.2411 for tangent 
of angle of face O m" c . The nearest tangent in the 
table is 1.2422 and the corresponding angle is 51° 10'. 
To obtain cutting angle c O n" vre divide the distance 
c' n" by c O. By measurement c' n" is 2.2". Divid- 
ing 2.2 by 3.5 we obtain .62857 for tangent of cutting 
angle. The nearest corresponding angle in the table 
is 32°10'. 

The largest pitch diameter, kj, of a bevel gear, as in 
Fig. 56, is known the same as the pitch diameter of 
any spur geai*. Now, if we know the distance h o or 
its equal a q, we can obtain the whole diameter of 
bevel gear blank by adding twice the distance b o to 
the largest pitch diameter. 
crement.*^Vig' Twice the distance b o, or what is the same thing, 
^^- the sum oi a q and Z> o is called the diameter incre- 

ment, because it is the amount by Avhich we increase 
the largest pitch diameter to obtain the whole or out- 
side diameter of bevel gear blanks. The distance b o 
can be calculated without measuring the diagram. 

The angle b o j is equal to the angle of edge. 

The angle of edge, it will be remembered, is the 
angle formed by outer edge of blank or ends of teeth 
with the end of hub or a plane perpendicular to the 
axis of gear. 

The distance ^ o is equal to the cosine of angle of 
edge, multiplied by the distance j o. The distance j o 
is the addendum, as in previous chapters ( = s). 

Hence the rule for obtaining the diameter increment 
of any bevel gear: Multiply the cosine of angle of 
edge by the toorking depth of teeth (D"), and the 
product will be the diameter increment. 

By the method given on page 102 we find the angle 
of edge of gear (Fig. 56) is 56° 20'. The cosine 
of 56° 20° is .55436, which, multiplied by |", or the 
^Outside Diam- (Jepth of the 3 P gear, gives the diameter increment of 
the bevel gear 18 teeth, 3 P meshing with pinion of 12 



PROAaDENCE, R. I. 



103 




104 BKOWN & SHAEPE MFG. CO. 

teeth. I of .55436=.369"+ (or .37", nearly). Adding 
the diameter increment, .37", to the largest pitch 
diameter of gear, 6", we have 6.37" as the outside 
diameter. 

In the same manner, the distance c d is half the 
diameter increment of the pinion. The angle c d k is 
equal to the center angle of pinion, and when axes are 
at right angles is the complement of center angle of 
gear. The center angle of pinion is 33° 40'. The 
cosine, multiplied by the working depth, gives .555" 
for diameter increment of pinion, and we have 4.555" 
for outside diameter of pinion. 

In turning bevel gear blanks, it is sufficiently accu- 
rate to make the diameter to the nearest hundredth of 
an inch. 
Angle incre The Small angle o 0/ is called the angle increment. 
When shafts are at right angles the face angle of one 
gear is equal to the center angle of the other gear, 
minus the angle increment. 

Thus the angle of face of gear (Fig. 56) is less than 
the center angle D O ^, or its equal O^' ^ by the angle 
o 0/. That is, subtracting o O j from O j k, the re- 
mainder will be the angle of face of gear. 

Subtracting the angle increment from the center 
angle of gear, the remainder will be the cutting 
angle. 

The angle increment can be obtained by dividing 
o j, the side ojDposite, by Oj, the side adjacent, thus 
finding the tangent as usual. 

The length of cone-pitch line from the common 
center, O to j, can be found, without measuring dia- 
gram, by multiplying the secant of angle Oj k, or the 
center angle of pinion, by the radius of largest pitch 
diameter of gear. 

The secant of angle Oj k, 33° 40', is 1.2015, which, 
multiplied by 3", the radius of gear, gives 3.6045" as 
the length of line O j. 

Dividing oj by Oj, we have for tangent .0924, and 
for angle increment 5° 20'. 

The angle increment can also be obtained by the 
following rule : 



PROVIDENCE, E. I. 105 

Divide the sine of cerder angle hy half the nurn- 
her of teeth, and the quotient loill he the tangent of 
increment angle. 

Subtracting the angle increment from center angles 
of gear and pinion, we have respectively : 

Cutting angle of gear, 51°. 

Cutting angle of pinion, 28° 20'. 
Kemembering that when the shafts are at right 
angles, the face angle of a gear is equal to the cutting 
angle of its mate (Chapter X. part 1), we have : 

Face angle of gear, 28° 20'. 

Face angle of pinion, 51°. 
It will be seen that both the whole diameter and the 
angles of bevel gears can be obtained without making 
a diagram. Mr. George B. Grant has made a table of 
different pairs of gears from 1 to 1 up to 10 to 1, con- 
taining diameter increments, angle increments and 
centre angles, which is published in his "Treatise on 
Gears." "Formulas in Gearing," published by us, also 
contains extensive tables for bevel gearing. We have 
adopted the terms "diameter increment," "angle incre- 
ment," and "centre angle" from him. He uses the 
term "■'back angle" for what we have called angle of 
edge, only he measures the angle from the axis of tlie 

gear, instead of from the side of the gear, or from the . '•'^^i^y^^*?" 
° ' & ' Angle by the 

end of hub, as we have done ; that is, his *^back angle "sine, 
is the complement of our angle of edge. 

In laying out angles, the following method may be 




Ji"Kj. 57. 



106 



BROWN & SHARPE MFG. CO. 



Back 

Cone Radius. 



preferred, as it does away with the necessity of making 
aright angle: Draw a circle, ABO (Fig. 57), ten 
inches in diameter. Set the dividers to ten times the 
sine of the required angle, and point off this distance 
in the circumference as at A B. From any point O in 
the circumference, draw the lines O A and O B. The 
angle A O B is the angle required. Thus, let the re- 
quired angle be 12°. The sine of 12° is .20791, which, 
multiplied by 10, gives 2.0791", or 2^" nearly, for 
the distance A B. 

Any diameter of circle can be taken if we multiply 
the sine by the diameter, but 10" is very convenient, 
as all we have to do with the sine is to move the 
decimal point one place to the right. 

If either of the lines pass through the centre, then the 
two lines which do not pass through the centre will form a 
right angle. Thus, if B passes through the centre then 
the two lines A B and A will form a right angle at A. 




Na = No. of Teeth in Gear. 
Nb = No. of Teeth in Pinion. 
OC = Centre Angle of Gear. 



Measure the back cone radius a b for the gear, or 6 c for the pinion. 
This is equal to the radius of a spur gear, the nximber of teeth in which 
would determine the cutter to use. Hence twice a b times the diametral 
pitch equals the number of teeth for which the cutter should be selected 
for the gear. Looking in the list on page 240 the proper number for the 
cutter can be found. 

Thus, let the back cone radius a. b be 4" and the diameter pitch be 8. 
Twice four is 8 and 8 x 8 is 64, from which it can be seen that the cutter 
must be of shape No. 2, as 64 is between 55 and 134, the range covered by 
a No. 2 cutter. 

The number of teeth for which the cutter should be selected can also 
be found by tlie following formula : 

Na 
Nb 



Tan. OC 



No. of teeth to select cutter for gear =- 



Na 



for pinion =•; 



Nb 



"Cos.a "^ — Sin- a 

If the gears are mitres or are alike, only one cutter is needed; if one 
gear is larger than the other, two may be needed. 



J07 



CHAPTER IV. 
SPIRAL GEARS— CALCULATIONS FOR LEAD OF SPIRALS. 



When the teeth of a gear are cut, not in a straight Spiral Gear, 
path, like a spur gear, but in a hehcal or screw-like 
path, the gear is called, technically, a twisted or screw 
gear, but more generally among mechanics, a spiral 
gear. A distinction is sometimes made between a 
screw gear and a twisted gear. In twisted gears the 
pitch surfaces roll upon each other, exactly like spur 
gears, the axes being parallel, the same as in Fig. 1, 
Part I. In screw gears there is an end movement, 
or slipping of the pitch surfaces upon each other, the 
axes not being parallel. In screw gearing the action 
is analogous to a screw and nut, one gear driving 
another by the end movement of its tooth jDath. This 
is readily seen in the case of a worm and worm-wheel, 
when the axes are at right angles, as the movement of 
wheel is then wholly due to the end movement of 
worm thread. But, as we make the axes of gears more 
nearly parallel, they may still be screw gears, but the 
distinction is not so readily seen. 

Unless otherwise stated, the shafts of screw gears are 
at right angles, as at A and B, Fig. 59. 

The same gear may be used in a train of screw gears 
or in a train of twisted gears. Thus, B, as it relates to 
A, may be called a screw gear ; but in connection with 
C, the same gear, B, may be called a twisted gear. 
These distinctions are not usually made, and we call 
all helical or screw-like gears made on the Universal 
Milling Machine spiral gears. 

When two external spiral shears run together, with Direction of 

,1 • nT T .1 i ,1 ^ ji 1 Spiral with ref- 

their axes parallel, the teeth of the gears must have erence to Axes. 
opposite hand spirals. 



108 BROWN & SHAKPE MFG. CO. 

Thus, in Fig. 59 the gear B has right hand spiral 
teeth, and the gear C has left hand spiral teeth. "When 
the axes of two spiral gears are at right angles, both 
gears must have the same hand spiral teeth. A and 
B, Fig. 59, have right hand spiral teeth. If both gears 
A and B had left hand spiral teeth, the relative direc- 
tion in which they turn would be reversed. 

Spiral Lead. Ti^g spiral lead or lead of spiral is the distance the 
spiral advances in one turn. A cylinder or gear cut 
with spiral grooves is merely a scrcAV of coarse pitch or 
long lead ; that is, a spiral is a coarse lead screw, and 
a screw is a fine lead spiral. 

Since the introduction and extensive use of the 
Universal Milling Machine, it has become customary 
to call any screw cut in the milling machine a spiral. 
The spiral lead is given as so many inches to one 
turn. Thus, a cylinder having a spiral groove that 
advances six inches to one turn, is said to have a six 
inch spiral. 

In screws the pitch is often given as so many turns 
to one inch. Thus, a screw of y lead is said to be 3 
turns to the inch. The reciprocal expression is not 
much used with spirals. For example, it would not 
be convenient to speak of a spiral of 6'' lead, as \ turns 
to one inch. 

The calculations for spirals are made from the func- 
tions of a right angle triangle. 

Example, Cut from paper a right angle triangle, one side of 

showing the r r o o o ' ^ 

nature of a He- the right angle 6 long, and the other side of the 
right angle 2". Make a cylinder 6" in circumference. 
It will be remembered (Part I., Chapter II.) that the 
circumference of a cylinder, multiplied by .3183, equals 
the diameter — 6" X -3183=1.9098". Wrap the paper 
triangle around the cylinder, letting the 2" side be 
parallel to the axis, the 6" side perpendicular to the 
axis and reaching around the cylinder. The hypoth- 
eneuse now forms a helix or screw-like line, called 
a spiral. Fasten the paper triangle thus wrapped 
around. See Fig. 60. 



PROVIDENCE, It. I. 



109 




FIG, 58 -RACKS AND GEARS. 








Fig. 59.-SPIRAL GEARING. 



110 



BROWN & SHARPE MFG. CO. 




Fig. 60. 



If we now turn this cylinder A B C D one lurii in 
the direction of the arrow, the spiral will advance from 
to E. This advance is the lead of the spiral. 

The angle E E, which the spiral makes with the 
axis E 0, is the angle of the spiral. This angle is found 
as in Chapter I. The circumference of the cyhnder 
corresponds to the side opposite the angle. The pitch 
of the spiral corresponds to the side adjacent the angle. 
Hence the rule for angle of spiral: 
ci?/atin°gtiie Divide the circumference of the cylinder or spiral 
parts of a spi-j^ //^g number of inches of spiral to one turn, and the 
quotient will he the tangent of atigle of spiral. 

When the angle of spiral and circumference are given, 
to find, the lead : 

Divide the circumference hy the tangent of angle, and 
the quotient ivill he the lead of the spiral. 

When the angle of spiral and the lead or pitch of spiral 
are given, to find the circumference : 

Multiply the tangent of angle hy the lead, and the 
product luill he the circumference. 

When applying calculations to spiral gears the angle 
is reckoned at the pitch circumference and not at the 
outer or addendum circle. 

It will be seen that when two spirals of different 
diameters have the same lead the spiral of less diame- 
ter will have the smaller angle. Thus in Fig. 60 if the 
paper triangle had been 4" long instead of 6" the diam- 
eter of the cylinder would have been 1.37" and the 
angle of the spiral would have been only 63J degrees. 



Ill 



CHAPTER V. 

EXAMPLES m CALCULATION OF THE LEAD OF SPIRAL— ANGLE OF 

SPIRAL— CIRCUMFERENCE OF SPIRAL GEARS— 

A FEW HINTS ON CUTTING. 



It will be seen that the rules for calculating the cir- 
cumference of spiral gears, angle and the lead of spiral 
are the same as in Chapter I., for the tangent and angle 
of a right angle triangle. In Chapter IV., the word 
"circumference" is substituted for "side opposite," 
and the words "lead of spiral" are substituted for 
"side adjacent." 

When two spiral gears are in mesh the angle of raif^^itti^^ f ^'' 
spiral should be the same in one gear as in the other, ®^°e to Angle 
in order to have the shafts parallel and the teeth work 
properly together. When two gears both have right 
hand spiral teeth, or both have left hand spiral teeth, 
the angle of their shafts will be equal to the sum of 
the angles of their spirals. But when two gears have 
different hand spirals the angle of their shafts will be 
equal to the difference of their angles of spirals. 
Thus, in Fig. 59 the gears A and B both have right 
hand spirals. The angle of both spirals is 45°, their 
sum is 90°, or their axes are at right angles. But C 
has a left hand spiral of 45°. Hence, as the difference 
between angles of spirals of B and C is 0, their axes 
are parallel. 

If two 45° gears of the same diameter have the same 
number of teeth the lead of the spiral will be alike in 
both gears: if one gear has more teeth than the other 
the lead of spiral in the larger gear should be longer 
in the same ratio. Thus, if one of these gears has 50 
teeth, and the other has 25 teeth, the lead of spiral Lead in spi- 

' ' -t^ rals of diflfer- 

in the 50 tooth gear should be twice as long as that of ent diameters. 
the 25 tooth gear. Of course, the diameter of pitch 



IVZ BKOWN & SHAKPE MFG. CO. 

circle should be twice as large in the 50 tooth as in the 
25 tooth gear. 

In spirals where the angle is 45° the circumference 
is the same as the spiral lead, because the tangent of 
45° is 1. 
CircumPr'^n™ Sometimes the circumference is varied to suit a pitch 
tosuitaspirai. that can be cnt on the machine and retain the angle 
required. This would apply to cutting rolls for mak- 
ing diamond-shaped impressions where the diameter 
of the roll is not a matter of importance. 

When two gears are to run together in a given 
velocity ratio, it is well first to select spirals that the 
machine will cut of the same ratio, and calculate the 
numbers of teeth and angle to correspond. This will 
often save considerable time in figuring. 

The calculations for spiral gears present no special 
difficulties, but sometimes a little ingenuity is required 
to make work conform to the machine and to such 
cutters as we may have in stock. 

Let it be required to make two spiral gears to run 
with a ratio of 4 to 1, the distance between centres to 
be 3.125" (31"), the axes to be parallel. 

By rule given in Chapter XII., Part I., we find the 
diameters of pitch circles will be 5" and 1^". Let us 
take a spiral of 48" lead for the large gear, and a 
spiral of 12" lead for the small gear. The circumfer- 
ence of the 5" pitch circle is 15.70796". Dividing 
the circumference by the lead of the spiral, we have 
i^7_|.7_96 =,32724" for tangent of angle of spiral. In 
the table the nearest angle to tangent, .32724", is 18° 10'. 

As before stated, the angle of the teeth in the small 
gear will be the same as the angle of teeth or spiral in 
the large gear. 
iiiAngiesattop Now, this rule gives the angle at the pitch surface 
sph-aiGroOTe^s^o^'y* Upon looking at a small screw of coarse pitch, 
it will be seen that the angle at bottom of the thread 
is not so great as the angle at top of thread; that is, 
the thread at bottom is nearer parallel to the centre 
line than that at the top. 

This will be seen in Fig. 61, where A is the centre 
line; Z>/ shows direction of bottom of thread, and d g 



PKOVIDENCE, R. I. 



113 



shows direction of top of tbrciul. The angle A fb is 
less than the angle A y d. The diflferetice of angle 
being due to tlie warped nature of a screw thread. 

A cylinder 2" diameter is to have spiral grooves ^0° catofiat?on of 
with the centre line of cylinder; what will be the lead Lead of spiral, 
of spiral? The circumference is 6.2833". The tan- 
gent of 20° is .36397. Dividing the circumference by 
the tangent of angle, we obtain ^;||§f ^ = 17.26"-|-for 
lead of spiral. 




Fig. 61. 

In Chapter XI, part I, it is stated that, when gashing 
the teeth of a worm-wheel, the angle of the teeth 
across the face is measured from the line parallel to the 
axis of the wheel. 

To obtain this angle from the worm, divide the lead 
by the pitch circumference of the worm, and the quo- 
tient will be the tangent of the angle that the thread 
makes with a plane perpendicular to the axis. 



114 



CHAPTER VI. 

NORMAL PITCH OF SPIRAL GEARS— CURVATURE OF PITCH 
SURFACE— FORM OF CUTTERS. 



Curv™*^ ^° ^ ^ Normal to a curve is a line perpendicular to the 
tangent at the point of tangency. 




In Fig. 62, the line B C is tangent to the arc D E F, 
and the line A E O, being perpendicular to the tan- 
gent at E the point of tangency, is a normal to the 
arc. 

Fig. 63 is a representation of the pitch surface of a 
spiral gear. A' D' C is the cu'cular j)itch, as in Part 
I. A D C is the same circular pitch seen upon the 
periphery of a wheel. Let A D be a tooth D and a 
space. Now, to cut this space D C, the path of cut- 
ting is along the dotted line a h. By mere inspection, 
we can see that the shortest distance between two 
teeth along the pitch surface is not the distance 
ABC. 

Let the line A E B be perpendicular to the sides of 
teeth upon the pitch surface. A continuation of this 
line, perpendicular to all the teeth, is called the 
Normal Helix. The line A E B, reaching over a 
tooth and a space along the normal helix, is called the 
Normal Pitch, or the normal linear pitch. 



PKOVIDENCH, R. I. 



115 




Fig. 63. 



116 BROWN & SHAEPE MFG. CO. 

Normal Pitch. The Normol Pitch of a spiral gear is then : The 
shortest distance betioeen the centers of two consecutive 
teeth measured along the pitch surface. 

In spur gears the normal pitch and circular pitch 
are alike. In the rack D D, Fig. 58, the linear pitch 
and normal pitch are alike. 
Cutter for From the foregoing it will be seen that, if we should 

Spiral Gears. . . 

cut the space D C with a cutter, the thickness of which 
at the pitch line is equal to one-half the circular pitch, 
as in spur wheels, the space would be too wide, and 
the teeth would be too thin. Hence, spiral gears 
should be cut with thinner cutters than spur gears of 
the same circular pitch. 

The angle C A B is equal to the angle of the spiral. 
The line A E B corresponds to the cosine of the angle 
CAB. Hence the rule : Multiply the cosine of angle 
'^I'p^t'ii ^^^' ^-^ spiral by the circular pitch, and the product will he 
the normal pitch. One-half the normal pitch is the 
proper thickness of cutter at the pitch line. 

If the normal pitch and the angle are known, Divide 
the normal pitch hy the cosine of the angle and the quo- 
tient will be the circular pitch. 

This may be required in a case of a spiral pinion run- 
ning in a rack. The perpendicular to the side of the 
rack is taken as the line from which to calculate angle 
of teeth. That is, this line would correspond to the 
axial line in a spiral gear ; and, when the axis of the 
gear is at right angles to the rack, the angle of the 
teeth with the side of the rack is obtained by subtract- 
ing this angle from 90°. 

The angle of the rack teeth with the side of the 
rack can also be obtained by remembering that the 
cosine of the angle of spiral is the sine of the angle of 
the teeth with the side of the rack. 

The addendum and working depth of tooth should 
correspond to the normal pitch, and not to tiie circular 
pitch. Thus, if the normal pitch is 13 diametral, the 
addendum should be -j^'', the thickness .1309", and so 
on. The diameter of pitch circle of a spiral gear is 
calculated from the diametral pitch. Thus a gear of 
30 teeth 10 P would be 3" pitch diameter. 



±»ROVIDENCE, R. I. 117 

But if the normal pitch is 13 diametral pitch, the 
blank will be 3yV diameter instead of SjV'* 

It is evident that the normal pitch varies with the^^j.'?™^'^'^'^^ 
angle of spiral. The cutter should be for the normal 
pitcli. In designing spiral gears, it is well first to look 
over list of cutters on hand, and see whether there are 
cutters to which the gears can be made to conform. 
This may avoid the necessity of getting a new cutter, 
or of changing both drawing and gears after they are 
under way. To do this, the problem is worked the 
reverse of the foregoing; that is: 

First calculate to the next finer pitch cutter than gj^o^^l^^^^j^^j 
would be required for the diametral pitch. cutters ^glvea" 

Let us take, for example, two gears 10 pitch and 30 
teeth, spiral and axes parallel. Let the next finer cut- 
ter be for 12 pitch gears. The first thing is to find the 
angle that will make the normal pitch .2618", when the 
circular pitch is .3142". See table of tooth parts. 
This means (Fig. 63) that the line A D C will be .3142" 
when A E B is .2618". Dividing .2618" by .3142" (see 
Chap. IV.), we obtain the cosine of the angle CAB, 
which is also the angle of the spiral, iff if "=-833. 

The same quotient comes by dividing 10 by 12, 
■f^f =.833 4- ; that is, divide one pitch by the other, the 
larger number being the divisor. Looking in the table, 
we find the angle corresponding to the cosine .833 is 
33° 30'. We now want to find the pitch of spiral that 
will give angle of 33|^° on the pitch surface of the wheel, 
3" diameter. Dividing the circumference by the tan- 
gent of angle, we obtain the pitch of spiral (see Chap. 
V.) The circumference is 9.4248". The tangent of 
33° 30' is .66188, ^:fft|j=14.23 ; and we have for 
our spiral 14.23" lead. 

When the machine is not arrano-ed for the exact when exact 

° Pitch cannot be 

pitch of spnal wanted, it is generally well enough to cut. 
take the next nearest spiral. A half of an inch more 
or less in a spiral 10" pitch or more would hardly be 
noticed in angle of teeth. It is generally better to 
take the next longer spiral and cut enough deeper to 
bring center distances right. Wlien two gears of the 
same size are in mesh with their axes parallel, a change 



118 



BROWN & SHAKPE MFG. CO. 



of angle of teeth or spiral makes no difference in the 
correct meshing of the teeth. 
Spiral Gears B^t when gears of different size are in mesh, due 

of Different ° . i , • • • i 

Sizes of Mesii. regard must be had to the spirals being in pitch, pro- 
portional to their angular velocities (see Chapter V. ) 
"VVe come now to the curvatui'e of cutters for spiral 
gears; that is, their shape as to whether a cutter is 
made to cut 12 teeth or 100 teeth. A cutter that is right, 
Shape of Cut- to cut a spuT gear 3" diameter, may not be right for a 
spiral gear 3" diameter. To find the curvature of 
cutter, fit a templet to the blank along the line of the 
normal helix, as A E B, letting the templet reach over 
about one normal pitch. The curvature of this templet 
will be nearer a straight line than an arc of the adden- 
dum circle. Now find the diameter of a circle that will 
approximately fit this templet, and consider this circle 
as the addendum circle of a gear for which we are to 
select a cutter, reckoning the gear as of a pitch the 
same as the normal pitch. 




Fig. 64. 



Thus, in Fig. 64, suppose the templet fits a circle 
3^" diameter, if the normal pitch is 12 to inch, dia- 
metral, the cutter required is for 12 P and 40 teeth. 
The curvature of the templet will not be quite circular, 
but is sufiiciently near for practical purposes. Strictly, 



PROVIDENCE, R. I. 119 

a flat templet cannot be made to coincide with the 
normal helix for any distance whatever, but any greater 
refinement than we have suggested can hardly be car- 
ried out in a workshop. 

This applies more to an end cutter, for a disk cutter 
may have the right shape for a tooth space and still 
round off the teeth too much on account of the warped 
nature of the teeth. 

The difference between normal pitch and linear or 
cii'cular pitch is plainly seen in Figs 58 and 59. 

The rack T> D, Fig. 58, is of regular form, the depth 
of teeth being J-|- of the circular pitch, nearly (.6866 of 
the pitch, accurately). If a section of a tooth in either 
of the gears be made square across the tooth, that is a 
normal section , the depth of the tooth will have the 
same relation to the thickness of the tooth as in the 
rack just named. • 

But the teeth of spu'al gears, looking at them upon 
the side of the gears, are thicker in proportion to their 
depth, as in Fig. 59 This difference is seen between 
the teeth of the two racks D D and E E, Fig. 58. In 
the rack D D we have 20 teeth, while in the rack E E 
we have but 14 teeth ; yet each rack will run with each 
of the spiral gears A, B or C, Fig. 59, but at different 
angles. 

The teeth of one rack will accui-ately fit the teeth of 
the other rack face to face, but the sides of one rack 
will then be at an angle of 45° with the sides of the 
other rack. At F is a guide for holding a rack in mesh 
with a gear. 

The reason the racks will each run with either of the 
three gears is because all the gears and racks have the 
same normal pitch. When the spiral gears are to run 
together they must both have the same normal pitch. 
Hence, two spiral gears may run correctly together 
though the circular pitch of one gear is not like the 
circular pitch of the other gear. 

If a rack is to run at any angle other than 90° with 
the axis of the gear it is well to determine the data 
from a diagram, as it is very difficult to figure the 
angles and sizes of the teeth without a sketch or 
diagram. 



120 



CHAPTER VII. 
CUTTING SPIRAL GEARS IN A UNIVERSAL MILLING MACHINE. 



A rotary disk cntter is generally preferable to a shank 
cutter or end mill on account of cutting faster and hold- 
ing its shape longer. In cutting spiral grooves, it is 
sometimes necessary to use an end mill on account of 
the warped character of the grooves, but it is very sel- 
dom necessary to use an end mill in cutting spiral gears. 
se^tST^'^of the ^^fore Cutting into a blank it is well to make a slight 
Machine. trace of the spiral with the cutter, after the change 
gears are in place, to see whether the gears are correct. 
If the material of the gear blanks is quite expensive, it 
is a safe plan to make trial blanks of cast iron in order 
to prove the setting of the machine, before cutting into 
the expensive material. 

The cutting of spiral gears may develop some curi- 
ous facts to one that has not studied warped surfaces. 
The gears. Fig. 59, were cut with a planing tool in a 
shaper, the spiral gear mechanism of a Universal Mill- 
ing Machine having been fastened upon the shaper. 
The tool was of the same form as the spaces in the rack 
D D, Fig. 58. All spiral gears of the same pitch can be 
cut in this manner with one tool. The nature of this 
cutting operation can be understood from a considera- 
tion of the meshing of straight side rack teeth with a 
spiral gear, as in Fig. 58. Spiral gears that run cor- 
rectly with a rack, as in Fig. 58, will run correctly 
with each other when their axes are pai*allel, as at B C, 
Fig. 59; but it is not considered that they are quite 
correct, theoretically, to run together when the gears 
have the same hand spiral, and their axes are at right 



PROVIDENCE, K. I. 



121 





/> 






Ua 


c 


" 



Fig. 65 



( 


> 

fl 11 


/> 


^ () 


(] 


K 


[[ 11 


_J 












— 




— 




{ 








\ 



Fig. 66 



122 



BKOWN & SHABPE MFG. CO. 



angles, as AB, Fig. 59, though they will run well enough 
practically. The operation of cutting spiral teeth with 
a planer tool is sometimescalled^/fl!;^m^ the teeth. Plan- 
ing is an accurate way of shaping teeth that are to mesh 
with rack teeth and for gears on parallel shafts; this 
method has been employed to cut spiral pinions that 
drive planer tables, but has noc been found available 
for general use. 

It is convenient to have the data of spiral gears 
arranged as in the following table : 



Data. 



No. of Teeth 
Pitch Diameter . 
Outside Diameter 
Circular Pitch 
Angle of Teeth with Axis 
Normal Circular Pitch 
Pitch of Cutter . 
Addendum s 
Thickness of Tooth t 
Whole Depth D"+f . 
No. of Cutter 
Exact Lead of Spiral 
Approximate Lead of Spiral 



Gears on Milling Machine to Cut Spiral 

Gear on Worm .... 

1st Gear on Stud 

2nd Gear on Stud 

Gear on Screw .... 



Gear. 



Pinion. 



A spiral of any angle to 45° can generally be cut in 
a Universal Milling Machine without special attach- 
ments, the cutter being at the top of the work. The 
cutter is placed on the arbor in such position that it 
can reach the work centrally after the table is set to 
the angle of the spiral. In order to cut central, it is 
generally well enough to place the table, before setting 
it to the angle, so that the work centres will be central 
with the cutter, then swing the table and set it to the 
angle of the spiral. 

For very accurate work, it is safer to test the T)osi-..*^^"*^^^ ^®*' 

. , ^ ting. 

tion of the centres after the table has been set to the 
angle. 



PROVIDENCE, R. I., U. S. A. 



123 




Fig. 67. 



USE OF VERTICAL SPINDLE MILLING ATTACHMENT 
IN CUTTING SPIRAL GEARS. 



124 BROWN & SHAKPE MFG. CO. 

This can be done with a trial piece. Fig. 65, which 
is simply a round arbor with centre holes in the ends. 
It is mounted between the centres, and the knee is 
raised until the cutter sinks a small gash, as at A. 
This gash shows the position of the cutter; and if the 
gash is central with the trial piece, the cutter will be 
central with the work. If preferred, the arbor can be 
dogged to the work spindle ; and the line B drawn 
on the side of the arbor at the same height as the cen- 
tres ; the work spindle should then be turned quarter 
way round in order to bring the line at the top. The 
gash A can now be cut and its position determined with 
the line. 

In cutting small gears the arbor can be dogged to the 
work spindle; the distance between the gear blank and 
the dog should be enough to let the dog pass the cutter 
arbor without striking. 

A spiral gear is much more likely to slip in cutting 
than a spur gear. 

For gears more than three or four inches in diameter 
it is well to have a taper shank arbor held directly in 
the work spindle, as shown in Figs. 67 and 68; and for 
the heaviest work, the arbor can be drawn into the spin- 
dle with a screw in a threaded hole in the end of the 
shank. 

After cutting a space the work can be dropped away 
from the cutter, in order to avoid scratching it when 
coming back for another cut. Some workmen prefer 
not to drop the work away, but to stop the cutter and 
turn it to a position in which its teeth will not touch 
the work. To make sure of finding a place in the cut- 
ter that will not scratch, a tooth has sometimes been 
taken out of the cutter, but this is not recommended. 
The safest plan is to drop the work away. 
Angle greater In cutting spiral gears of greater angle than 45°, a 
than 45° vertical spindle milling attachment is available, as 
shown in Figs. 67 and 68. 

In Fig. 67 the cutter is at 90° with the work spindle 
when the table is set to 0, so that the proper angle at 
which the table should be set, is the difference between 
the angle of the spiral and 90°. Thus, to cut a 70° 



PROVIDENCK, K. I., U. S. A. 



125 




Fig. 68. 



USE OF VERTICAL SPINDLE MILLING ATTACHMENT 
IN CUTTING SPIRAL GEARS. 



126 BROWN & SHARPE MFG. CO. 

spiral, we subtract 70° from 90°, and the remainder, 
20°, is the angle to set the table. In cutting on the 
top. Fig. 67, the attachment is set to 0. 

In Fig. 68 the cutter is at the side of the work ; the 
table is set to 0, and the attachment is set to the differ- 
ence between 90° and the required angle of spiral. 

In setting the cutter central it is convenient to have a 
small knee as at K, Fig. 66. A line is drawn upon the 
knee at the same height as at the centres. The cutter 
arbor is brought to the angle as just shown, and a gash 
is cut in the knee. When the gash is central with the 
line, the cutter will be central with the work. 

The cutter can be set to act upon either side of the 
gear to be cut, according as a right hand or a left hand 
spiral is wanted. The setting in Fig. 68 is for a right 
hand spiral. 

If the gear blank were brought in front of the cut- 
ter, and the reversing gear set between two change 
gears, the machine would be set for a left hand spiral. 

For coarser pitches than about 12 P diametral, it is 
well to cut more than once around, the finishing cut 
being quite light so as to cut smooth. 



127 



CHAPTER VIII. 
SCREW GEARS AND SPIRAL GEARS— GENERAL REMARKS. 



The working of spiral gears, when their axes are working of 

., , . n J.I ii A Spiral Gears. 

para.lel, is generally smoother than spur gears. A 

tooth does not strike along its whole face or length at 

once. Tooth contact first takes place at one side of the 

gear, passes across the face and ceases at the other 

side of the gear. This action tends to cover defects in 

shape of teeth and the adjustment of centres. 

Since the invention of machines for producing accu- 
rate epicyloidal and involute curves, it has not so often 
been found necessary to resort to spiral gears for 
smoothness of action. A greater range can be had in 
the adjustment of centers in spiral gears than in spur 
gears. The angle of the teeth should be enough, so 
that one pair of teeth will not part contact at one side 
of the gears until the next pair of teeth have met on the 
other side of the gears. When this is done the gears 
will be in mesh so long as the circumferences of their 
addendum circles intersect each other. This is some- 
times necessary in gears for rolls. 

Relative to spur and bevel gears in Part I., Chapter 
XII., it was stated that all gears finally wore them- 
selves out of shape and might become noisy. Spiral 
gears may be worn out of shape, but the smoothness 
of action can hardly be impaired so long as there are 
any teeth left. For every quantity of wear, of course, 
there will be an equal quantity of backlash, so that if 
gears have to be reversed the lost motion in spiral 
gears will be as much as in any gears, and may be 
more if there is end play of the shafts. In spiral gears End Pressure 
there is end pressure upon the shafts, because of the Spiral Gears. 
screw-like action of the teeth. This end pressure is 
sometimes balanced by putting two gears upon each 
shaft, one of right and one of left hand spu-al. 



128 BBOWK & SHAKPE MFG. CO. 

The same result is obtained in solid cast gears by 
making the pattern in two parts — one right and one 
left-hand spiral. Such gears are colloquially called 
"herring-bone gears." 

In an internal spiral gear and its pinion, the spirals 
of both wheels are either right-handed or left-handed. 
Such a combination would hardly be a mercantile 
product, although interesting as a mechanical feat. 

In screw or worm-gears the axes are generally at 
right angles, or nearly so. The distinctive features of 
screw gearing may be stated as follows : 

The relative angular velocities do not depend upon 

the diameters of pitch- cylinders, as in Chapter I., 

Distinctive Part I. Thus the worm in ChaDter XL, Fisf. 35, can 

features of . j. ' o ' 

Screw Gearing, be any diameter — one inch or ten inches — without 
affecting the velocity of the worm-wheel. Conversely if the 
axes are not parallel we can have a pair of spiral or screw 
gears of the same diameter, but of different numbers of 
teeth. The direction in which a worm-wheel turns depends 
upon whether the worm has a right-hand or left-hand thread. 
When angles of axes of worra and worm-wheel are 
oblique, there is a practical limit to the directional 
relation of the worm-wheel. The rotation of the 
worm-wheel is made by the end movement of the 
worm-thread. 

The term worm and worm-wheel, or worm-gearing, 
is applied to cases where the worms are cut in a lathe, 
and the shapes of the threads or teeth, in axial section, 
are like a rack and the pitch is measured on a line 
parallel to the axis. The shape usually selected is like 
the rack for a single curve or involute gear. See 
Chap. IV, Part I. Worms are sometimes cut in a 
milling machine. 

If the form of the teeth in a pair of screw gears is 
determined upon the normal helix, as in Chap. VI,, 
the gears are usually called Spiral Gears. 

If we let two cylinders touch each other, their axes 
being at right angles, the rotation of one cylinder will 
have no tendency to turn the other cylinder, as in 
Chapter I., Part I. 



PROVIDENCE, R. I. 129 

We can now see why worms and worm-wheels wear wiiy worm 

•' Wheels wear 

ont faster than other gearing. The length of worm-sof^^s'^ 

thread, equal to more than the entire circumference of 

worm, comes in sliding contact with each tooth of the 

wheel during one turn of the wheel. 

The angle of a worm-thread can be calculated the 
same as the angle of teeth of spiral gear ; only, the 
angle of a worm thread is measured from a line or 
plane that is perpendicular to the axis of the worm. 

When a multiple threaded worm is cut in a milling 
machine and the angle of the thread is less than 75° 
with the axis of the worm, it may be desirable to work 
by the normal pitch. The normal pitch can he obtained 
by multiplying the thread-pitch by the sine of the angle 
of the thread with the axis. 



130 



CHAPTER IX. 

CONTINUED FRACTIONS— SOME APPLICATIONS IN MACHINE 
CONSTRUCTION. 



Definition of ^ continued fraction is one that has unity for its 

a Continued •' 

Fraction. numerator, and for its denominator an entire number 

plus a fraction, which fraction has also unity for its 
numerator, and for its denominator an entu'e number 
plus a fraction, and thus in order. 



The expression, 



4 + 1 

^ is called a continued frac- 
tion. By the use of continued fractions, we are ena- 
Practicai use ^jed to find a fraction expressed in smaller numbers, 

of Continued -'■ ' 

Fractions. that, for practical purposes, may be sufficiently near in 
value to another fraction expressed in large numbers. 
If we were required to cut a worm that would mesh 
with a gear 4 diametral pitch (4 P.), in a lathe having 

3 to 1-inch linear leading screw, we might, without 
continued fractions, have trouble in finding change 
gears, because the circular pitch corresponding to 

4 diametral pitch is expressed in large numbers : 

^-^ 10000 -^ • 

This example will be considered farther on. For 
illustration, we will take a simpler example. 

What fraction expressed in smaller nulnbers is near- 
est in value to jVe ^ Dividing the numerator and the 
denominator of a fraction by the same number does 
not change the value of the fraction. Dividing both 
con^u^ue^d^®^"'^^ ^^ TIT ^J 2^' ^^ ^^"^^ 5J~ ^^'' what is the 
same thing expressed as a continued fraction, 5-t- i . The 
continued fraction s+gT is exactly equal to -^^j. If 
now, we reject the -^q, the fraction ^ will be larger 
than 5-i- 1 , because the denominator has been dimin- 
ished, 5 being less than 5-^-^. ^ is something near 
y?j9g- expressed in smaller numbers than 29 for a 



PROVIDENCE, R. I. 131 

numerator and 146 for a denominator. Reducing -J 
and y^j"^ to a common denominator, we have ^ = ^|^ 
and i%V=Y3 0". Subtracting one from the other, we 
have Tj-^y, which is the difference between l and j^?"- 
Thus, in thinking of ^^V ''^ i» ^''^ have a pretty fair 
idea of its vakie. 

There are fourteen fractions with terms smaller than 
29 and 146, which are nearer j^-a^. than ^ is, such as 
■^, If- and so on to /^y. In this case by continued frac- 
tions we obtain only one approximation, namely -^, and 
any other approximations, as \f, -|^f-, &c., we find by 
trial. It will be noted that all these approximations 
are smaller in value than ^W- There are cases, how- 
ever, in which we can, by continued fractions, obtain 
approximations both greater and less than the required 
fraction, and these "will be the nearest possible approxi- 
mations that there can be in smaller terms than the 
given fraction. 

In the French metric system, a milHmetre is equal 
to .03937 inch; what fraction in smaller terms ex- 
presses .03987" nearly? .03937, in a vulgar fraction, 
^^ To o 00 - Dividing both numerator and denominator 
by 3937, we have 25i5?5- Rejecting from the de- 
nominator of the new fraction, ^ifr? the fraction -^^ 
gives us a pretty good idea of the value of .03937". 
If in the expression, "ai+TIIi, we divide both terms of 
the fraction -jfir ^7 1575, the value will not be changed. 
Performing the division, we have ^ 

° 25 + 1 



2 + 787 
1575 • 

We can now divide both terms of fW? ^J "^^"^j 
without changing its value, and then substitute the 
new fraction for ^W? ^^ ^^^ continued fraction. 

Dividing again, and substituting, we have : 
1 

25 +J^ 

2 + 1 

8+1 

787 

as the continued fraction that is exactly equal to 
.03937. 



132 BROWN & SHARPE MFG. CO. 

In performing the divisions, the work stands thus : 

3937) 100000 (25 
7874 
21260 
19685 
1575) 3937 (2 
3150 

787) 1575 (2 
1574 

1) 787 (787 
787 

•0- 
That is, dividing the last divisor by the last remain- 
der, as in finding the greatest common divisor. The 
quotients become the denominators of the continued 
fraction, with unity for numerators. The denominators 
25, 2, and so on, are called incomplete quotients, since 
they are only the entire parts of each quotient. The 
first expression in the continued fraction is -^-^ or 
.04 — a little larger than .03937. If, now, we take 
gg-qri, we shall come still nearer .03937. The expres- 
sion 25'x-f is merely stating that 1 is to be divided by 
25|-. To divide, we first reduce 1h\ to an improper 
fraction, ^, and the expression becomes ST, or one 
divided by -^. To divide by a fraction, "Invert "the 
divisor, and proceed as in multiplication." We 
then have -^-^ as the next nearest fraction to .03937. 
-g2j-=.0392 + , which is smaller than .03937. To get still 
nearer, we take in the next part of the continued frac- 
tion, and have i 



2 + 1 
2' 

We can bring the value of this expression into a 
fraction, with only one number for its numerator and 
one number for its denominator, by performing the 
operations indicated, step by step, commencing at the 
last part of the continued fraction. Thus, 2-|-^, or 
2^, is equal to |, Stopping here, the continued frac- 
tion would become i 

25+J_ 
5 

2- 

1 \ 

Now, ^ equals f , and we have 25 +^. 25f equals 

2 5 

^\^ ; substituting again, we have li^. Dividing 1 by 
J-|^, we have yl^- yf y is the nearest fraction to 



PROVIDENCE, R. I. 133 

.03937, unless we reduce the whole continued fraction 

_i 

25 + 1 

2 + 1 

^ + 1_, which would afive us back the .03937 itself. 

787 

y|y=. 03937007, which is only ^^^^^ larger 
.03937. It is not often that an approximation will 
come so near as this. 

This ratio, 5 to 127, is used in cutting millimeter Practical use 

Trt IT fill' of the foregoing 

thread screws. If the leading screw of the lathe isExampie. 
1 to one inch, the change gears will* have the ratio of 
5 to 127; if 8 to one inch, the ratio will be 8 times 
as large, or 40 to 127; so that with leading screw 8 to 
inch, and change gears 40 and 127, we can cut milli- 
meter threads near enough for practical purposes. 

The foregoing operations are more tedious in de- 
scription than in use. The steps have been carefully 
noted, so that the reason for each step can be seen 
from rules of common arithmetic, the operations being 
merely reducing complex fractions. The reductions, 
^, ■fj, yIy, etc., are called conver gents, because they 
come nearer and nearer to the required .03937. The 
operations can be shortened as follows: 

Let us find the fractions converging towards .7854", Example, 
the circular pitch of 4 diametral pitch, .7854=Yyg%\; 
reducing to lowest terms, we have \\\l . Applying 
the operation for the greatest common divisor: 



392; 


r) 5000 (1 

3927 






1073) 3927 <3 




3219 






708) 


1073 (1 
708 

365) 708 (1 
365 

343) 365 (1 
343 

22) 343 (15 
23 
123 
110 

13) 22 (1 
13 

9) 13 (1 
9 

4) 9 (2 
8 

Y) 4 (4 
4 




Bringing the various incomplete quotients as de- 
nominators in a continued fraction as before, we have : 



134 BROWN & SHAEPE MFG. CO. 

1 

1 + 1 

3 + 1 

i+1 

1+1 

1+1__ 

15 + 1 

1 + 1 

1+1 

2 + |- 

Now arrange each partial quotient in a line, thus : 
13111 15 1 1 2 4 

1 3 4 1 ±1. 112 18 3 35 g 8 9 3 3927 
-L i T i it ST-g" T3 3" Ttt TT3T TO 

Now place under the first incomplete quotient the 
first reduction or convergent ^, which, of course, is 1 ; 
put under the next partial quotient the next reduction or 
convergent ^-rr or ^, which becomes f . 

'-^ 1 + 3 ■'3 

1 is larger than .7854, and f is less than .7854. 
Having made two reductions, as previously shown, 
we can shorten the operations by the following rule for next 
convergents: Multiply the numerator of the convergent 
just found by the denominator of the next term of the con- 
tinned fraction, or the next incomplete quotient^ and add 
to the product the numerator of the 2oreceding convergent ; 
the sum ivill be the numerator of the next convergent. 

Proceed in the same way for the denominator, that 
is multiply the denominator of the convergent just 
found by the next incomplete quotient and add to the 
product the denominator of the preceding convergent ; 
the sum will be the denominator of the next convergent. 
Continue until the last convergent is the original frac- 
tion. Under each incomplete quotient or denominator 
from the continued fraction arranged in line, will be 
seen the corresponding convergent or reduction. The 
convergent ^l is the one commonly used in cutting 
racks 4 P. This is the same as calling the circumference of 
a circle 22-7 when the diameter is one (1) ; this is also the 
common ratio for cutting any raclv. The equivalent decimal 
to li is .7857 X, being about 1 tf ^arge. In three set- 
tings for rack teeth, this error would amount to about .001" 
For a worm, this corresponds to ^f threads to 1" ; 
now, with a leading screw of lathe 3 to 1", we would 
want gears on the spindle and screw in a ratio of 33 
to 14. 

Hence, a gear on the spindle with 66 teeth, and a 
gear on the 3 thread screw of 28 teeth, would enable 
us to cut a worm to fit a 4 P gear. 



135 



CHAPTER X. 
ANGLE OF PRESSURE. 



In Fig. 69, let A be any flat disk lying upon a hori- 
zontal plane. Take any piece, B, with a square end, 
a b. Press against A with the piece B in the direction 
of the arrow. 




Fig. 69. 




Fig. 



It is evident A will tend to move directly ahead of 
B in the normal line c d. Now (Fig. 70) let the piece 
B, at one corner^/", touch the piece A. Move the piece 
B along the line d e, in the direction of the arrow. 

It is evident that A will not now tend to move in 
the line d e, but will tend to move in the direction of 
the normal c d. When one piece, not attached, presses 
against another, the tendency to move the second 
piece is in the du'ection of the normal, at the point of 
contact. This normal is called the line of pressu7'e. Line of Press- 

«' ^ lire. 

The angle that this line makes with the path of the 
impelling piece, is called the atigle of pressure. 

In Part I., Chapter IV., the lines B A and B A' are 
called lines of pressure. This means that if the gear 



136 BEOWN & SHAKPE MFG. CO. 

drives the rack, the tendency to move the rack is not 
in the direction of pitch line of rack, but either in the 
direction B A or B A', as we turn the wheel to the left 
or to the right. 

The same law holds if the rack is moved in the 
direction of the pitch line ; the tendency to move the 
wheel is not directly tangent to the pitch circle, as if 
driven by a belt, but in the direction of the line of 
pressure. Of course the rack and wheel do move in 
the paths prescribed by their connections with the 
framework, the wheel turning about its axis and the 
rack moving along its ways. This pressure, not in a 
direct path of the moving piece, causes extra friction 
in all toothed gearing that cannot well be avoided. 

Although this pressure works out by the diagram, 
as we have shown, yet, in the actual gears, it is not at 
all certain that they will follow the law as stated, 
because of the friction of teeth among themselves. If 
the driver in a train of gears has no bearing upon its 
tooth-flank, we apprehend there will be but little 
tendency to press the shafts apart. 
Arc of Action. rpj^g ^^^ through which a wheel passes while one of 

its teeth is in contact is called the arc of action. 
tenf^of "^ Inter- ^^^il within a few years, the base of a system of 
^^^^^seabie^jo^l^le.curye interchangeable gears was 12 teeth. It 
is now 15 teeth in the best practice (see Chapter VII., 
Part I.) 

The reason for this change was : the base, 15 teeth, 
gives less angle of pressure and longer arc of contact, 
and hence longer lifetime to gears. 



137 



CHAPTER XI. 
INTERNAL GEARS. 



In Part I., Chapter YIII., it is stated that the space 
of an internal gear is the same as the tooth of a spur 
gear. This applies to involute or single-curve gears as 
well as to double-curve gears. 

The sides of teeth in involute internal gears are 
hollowing. It, however, has been customary to cut 
internal gears with spur gear-cutters, a No. 1 cutter 
generally being used. This makes the teeth sides 
convex. Special cutters should be made for coarse Special cut- 

r" -_-... , ters for coarse 

pitch double-curve gears, in designing internal gears. Pitch, 
it is sometimes necessary to depart from the system 
with 15-tooth base, so as to have the pinion differ from 
the wheel by less than 15 teeth. The rules given in 
Part I., Chapters YII. and VIII., will apply in making 
gears on any base besides 15 teeth. If the base is 
low-numbered and the pinion is small, it may be neces- 
sary to resort to the method given at the end of Chap- 
ter VII., because the teeth may be too much rounded 
at the points by following the approximate rules. 
The base must be as small as the diiference between Base for in> 

^ ternal Gear 

the internal gear and its pinion. The base can be Teeth, 
smaller if desired. 

Let it be requii-ed to make an internal gear, and 
pinion 24 and 18 teeth, 3 P. Here the base cannot 
be more than 6 teeth. 

In Fig. 71 the base is 6 teeth. The arcs A K and 
O k, drawn about T, have a radius equal to the radius 
of the pitch cu'cle of a 6-tooth gear, 8 P, instead of a 
15-tooth geai", as in Chaj^ter VIII., Part I. 

The outline of teeth of both gears and pinion is Description of 

Fig. 67. 

made similar to the gear in Chapter VIII. The same 



138 



BEOWN & SHAKPE MFG. CO. 




GEAR, 24 TEETH. 
PINION, 18 TEETH, 3 P. 

P = 3 

N =24 and 18 
P'= 1.0472" 
t=- 5236" 
S= .3333' 
D= .6666" 
S+/= .3857" 
(}"+/= .7190" 



INTERNAL GEAR AND PINION IN MESH. 



PROVIDENCE, E. I. 

lettei's refer to similar parts. The clearance circle is, 
however, drawn on the outside for the internal gear. 
As before stated, the spaces of a spur wheel become 
the teeth of an internal vrheel. The teeth of internal 
gears require but little for fillets at the roots ; they 
are generally strong enough without fillets. The 
teeth of the pinion are also similar to the gear in 
Chapter VIII., substituting 6-tooth for 15-tooth base. 
To avoid confusion, it is well to make a complete 
sketch of one gear before making the other. The arc 
of action is longer in internal gears than in external 
gears. This property sometimes makes it necessary 
to give less fillets than in external gears. 

In Fig. 71 the angle K T A is 30° instead of 12°, as 
in Fig. 12. This brings the line of pressure L P at 
an angle of 60° with the radius C T, instead of 78°. 
A system of spur gears could be made upon this 
6-tooth base. These gears would interchange, but no 
gear of this 6-tooth system would mesh with a double- 
curve gear made upon the 15-tooth system in Part 1. 



139 



140 



CHAPTER XIL 



STRENGTH OF GEARING. 



We have been unable to derive from oar own experi- 
ence, any definite rule on this subject but would refer 
those interested to "Kent's Mechanical Engineers' 
Pocket Book/' where a good treatment of the subject 
can be found. 

We give a few examples of average breaking strain 
of our Combination Gears, as determined by dyna- 
mometer, the pressure being measured at the pitch line. 
These gears are of cast iron, with cut teeth. 



Diametral Pitch. 


No. Teeth. 


Revolutions 
per 

Minute. 


Pressure at 




Face. 




10 
8 
6 
5 


1 1-16 
1 1-4 
1 9-16 

1 7-8 


110 

72 
72 
90 


27 
40 
27 
18 


1060 
1460 
2220 

2470 



These are the actual pressures for the particular 
widths given. 

If we take a safe pressure at 1-3 of the foregoing 
breaking strain, we shall have for 



10 Pitch 353 1-3 Lbs. at the Pitch Line. 

8 '' 486 2-3 " 

6 " 740 *' " 

5 " 823 1-3 

The width of the face of a gear is in good proportion 
when it is 2^ times the circular pitch. 



PROVIDENCE, K. I. 



141 



TOOTH PARTS. 




Fig. 73. 

GEAR TOOTH 1 P 



142 BROWN & SHARPE MFG. CO. 

The dimensions of tooth parts as given in the tables, 
pages 144 to 147, are correct according to the definition 
of tooth parts, pages 4 and 16 ; but, as the pitch line 
of gears is curved, the thickness of a tooth will not be 
measured on the pitch line if the caliper is set to the 
figures given in the tables mentioned. To measure the 
teeth accurately on the pitch line, the caliper must be 
set to the chordal thickness and the depth setting to the 
pitch line must be to the corrected s, as explained and 
tabulated. If the gear blank is not of the correct 
diameter, the proper allowance must be made in setting 
the caliper for the depth. It is utterly useless to be 
guided by the outside of a gear blank when the outside 
diameter is not right. The measuring of the tooth 
thickness is well enough, as a check, but it is oftentimes 
as well first to make sure that the spaces are cut to the 
right depth. 

Fig. 73 is a sketch of a gear tooth of 1 P. In meas- 
uring gear teeth of coarse pitch accurately the chordal 
thickness of the tooth, ATB, must be known, because 
it may be enough shorter than the regular tooth-thick- 
ness AHB, or t, to require attention. It may be also 
well to know the versed sine of the angle /?', or the dis- 
tance H, in order to tell where to measure the chordal 
thickness. 

Chordal Thicknesses of Teeth of Gears, on a 
Basis of 1 Diametral Pitch. 

N = Number of teeth in gears. 

T = Chordal thickness of Tooth. T = D' sin. /?' 

H = Height of Arc. H = R (1— cos. ^') 

D' = Pitch Diameter. 

R = Pitch Radius. 

= 90° divided by the number of teeth. 

Note. — For any pitch not in the following tables to find 
corresponding part : — Multiply the tabular value for one inch 
by the circular pitch required, and the product will be the 
value for the pitch given. 

Exafnple : What is the value of s for 4 inch circular pitch ? 
.3183 = s for 1" P' and .3183 X 4 = 1 .2732 = s for 4" P^ 

The expression "Addendum and ^" (addendum and the 
module) means the distance of a tooth outside of pitch line 
and also the distance occupied for every tooth upon the diam- 
eter of pitch circle. 



PROVIDENCE, R. I. 



143 



CHORDAL THICKNESSES OF TEETH OF GEARS. 

INVOLUTE. 



Cutter. 


T 


H 


Corrected 
S for Gear. 


No. I —135 T — 


P 


1-5707 


.0047 


1 .0047 


No. 2 — 55 T — 


[ P 


1.5706 


.0112 


1.0112 


No. 3—35 '1' — 


[ P 


1.5702 


.0176 


1.0176 


No. 4—26 'J' — 


[ P 


1.5698 


.0237 


1.0237 


No. 5 — 21 T — 


I P 


1.5694 


.0294 


1.0294 


No. 6—17 T — 


[ P 


1.5686 


.0362 


1.0362 


No. 7—14 T — 


I P 


1-5675 


.0440 


1.0440 


No. 8— 12 T — 


I P 


1-5663 


.0514 


1.05 14 


II T — 


I P 


1-5654 


■0559 


1-0559 


loT — 


I P 


1-5643 


.0616 


I.0616 


9T- 


I P 


1.5628 


.0684 


1.0684 


8T — 


I P 


1.5607 


.0769 


1.0769 



EPICYCLOIDAL. 



SPECIAL. 



Cutter. 


T 


H 


Corrected 
S for Gear. 


A— 12T— I 


P 


1-5663 


.0514 


1.0514 


B - 13 T _ I 


P 


1.5670 


-0474 


1.0474 


C — 14 T — I 


P 


1-5675 


.0440 


1 .0440 


D— 15 T — 1 


P 


1-5679 


.0411 


1. 041 1 


E_ 16T — ] 


P 


1-5683 


-0385 


1-0385 


F — 17 T — 1 


P 


1.5686 


.0362 


1.0362 


G — 18 T — ] 


P 


1.5688 


.0342 


1.0342 


H— 19T— ] 


P 


1.5690 


.0324 


1.0324 


I — 20 T — ] 


P 


1.5692 


.0308 


1.0308 


J — 21 T — ] 


P 


1-5694 


.0294 


1.0294 


K — 23 T — 


[ P 


1.5696 


.0268 


1.0268 


L — 25 T — 


t P 


1.5698 


.0247 


1.0247 


M— 27 T — 


[ P 


1-5699 


.0228 


1.0228 


N — 30 T — 


[ P 


1.5701 


.0208 


1 .0208 


- 34 T - 


[ P 


1-5703 


.0181 


1.0181 


P - 38 T — 


[ P 


1-5703 


.0162 


I.0162 


Q - 43 T - 


[ P 


1-5705 


-0143 


1.0143 


R _ 50 T — 


[ P 


1-5705 


.0123 


1.0123 


S _ 60 T — 


[ P 


1.5706 


.0102 


1. 01 02 


T-75T- 


[ P 


1-5707 


.0083 


1 .0083 


U —100 T — 


[ P 


1-5707 


.0060 


1.0060 


V —150 T — 


I P 


1-5707 


.0045 


1 .0045 


W— 250 T — 


I P 


1.5708 


.0025 


1.0025 



No. Teeth. 


T 


H 


Corrected 
S for Gear. 


9T— I P 

10 T — I P 

11 T— I P 


1.5628 
1-5643 
1-5654 


.0684 
.0616 
-0559 


1.0684 
1. 061 6 
I-OS59 



144 



BROWN & SHARPE MFG. CO. 



DIAMETRAL PITCH. 

"NUTTALL." 
Diametral Pitch is tlie Number of Teetli to Each Incli of the Pitch Diameter. 



To Get 



The Diametral 
Pitch. 



The Diametral 
Pitch. 



The Diametral 
Pitoli. 



Pitch 

Diameter 



Pitch 

Diameter 



Pitch 

Diameter. 



Pitch 

Diameter. 



Outside 

Diameter. 



Outside 

Diameter. 



Outside 

Diameter. 



Outside 

Diameter 



Number of 

Teeth. 



Number of 

Teeth. 

Thickness 

of Tooth. 

Addendum. 

Root. 

Working 

Depth. 

Whole Depth. 

Clearance. 

Clearance. 



Havina 



The Circular Pitch. 

The Pitch Diameter 
and the Nunil)er of 
Teeth 

The Outside Diame- 
ter and the Numl)ei 
of Teeth .... 

The Number of Teetli 
and the Diametral 
Pitch .... 

The Number of Teeth 
and Outside Diam 
eter 

The Outside Dinme 
ter and the Diam 
etial Pitch . . . 

Addendum and the 
Number of Teeth. 

The Number of Teeth 
and the Diametral 
Pitch 

The Pitch Diameter 
and the Diametral 
Pitch 

The Pitch Diameter 
and the Number of 
Teeth .... 

The Number of Teeth 
and Addendum . 

The Pitch Diameter 
and the Diametral 
Pitch 

The Outside Diame- 
ter and the Diame- 
tral Pitch . . . 



The Diametral Pitch. 
Tire Diametral Pitch. 

The Diametral Pitch. 
The Diametral Pitch. 
The Diametral Pitch. 
The Diametral Pitch. 
Thickness of Tooth. 



Rule. 



Divide 3.1416 by the Circular Pitch 

Divide Number of Teeth by Pitch 
Diameter 



Divide Number of Teeth plrrs 2 by 
Outside Diameter . . . . . 



Divide Number of Teeth by the 
Diametral Pitch ....". 

Divide the" product of Outside 
Diameter and Number of Teeth 
by Number of Teeth plus 2 

Subtract from the Outside Diame- 
ter the quotient of 2 divided by 
the Diametral Pitch .... 

Multiply Addendum by the Num- 
ber of Teeth 



Divide Number of Teeth plus 2 by 
the Diametral Pitch .... 



Add to the Pitch Diameter the 
quotient of 2 divided by the 
Diametral Pitch 

Divide the Number of Teeth plus 
2 by the quotient of Number of 
Teeth and by the Pitch Diameter 

Multiply the Number of Teeth 
plus 2 by Addendum .... 

Multiply Pitch Diameter by the 
Diametral Pitch 



Multiply Outside Diameter by the 
Diametral Pitch and subtract 2. 

Divide 1.570S by the Diametral 
Pitch . 

Divide 1 by the Diametral Pitch, 
D' 

°'"^="-N 

Divide 1.157 bythe Diametral Pitch 



Divide 2 Ijy the Diametral Pitch. 
Divide 2.157 bythe Diametral Pitch 

Divide .157 bythe Diametral Pitch 

Divide Thickness of Tooth at 
pitch line by 10 



Formula. 



3.1416 
P' 
N 
' D' 

" D 

\ 



D'= 



n;_ 
p 

DN 

N+2 



D'=D_- 
D'=sN 

y+2 



D = - 



D = D'+^ 



D = 


N+2 
= N 




D~ 


D = 


= (N+2) s 


N = 


= DP 


N = 


='DP — 2 


t = 


1.570S 
P 


s = 


1 
P 


s + 


„ 1.157 
^- P 


D"= 


2 

"^ P" 


D"- 


f '-l'' 



.1.57 



PROVIDENCE, R. I. 



145 



CIRCULAR PITCH. 



"NUTTALL." 

Circular Pitch is tlic Distance from the Centre of One Tooth to the Centre of the 
Next Tootli, Measured alonj:? tlie Pitch Line. 



To Get 



The Circular 
I'itch, 



Tlie Circular 
Pitcli, 



The Circular 
Pitch, 



Pitch 

Diameter, 



Pitch 

Diameter, 



Pitch 

Diameter, 



Pitch 

Diameter 



Outside 

Diameter. 



Outside 

Diameter. 



Outside 

Diameter. 



Number of 

. Teeth. 



Thicliness 

of Tooth, 



Addendum. 

Root. 

Working 

Depth, 

Whole Depth. 

Clearance. 

Clearance. 



Havini! 



The Diametral Pitch. 

The Pitch Diameter 
and the Numl)er of 
Teeth 

The Outside Diame- 
ter and the Number 
of Teeth .... 

The Number of Teeth 
and the Circular 
Pitch 

The Number of Teeth 
and the Outside Di- 
ameter .... 

The Outside Diame- 
ter and the Circular 
Pitch 

Addendum and the 
Number of Teeth. 

The Numljer of Teeth 
and the Circuhir 
Pitch 

The Pitch Diameter 
and the Circular 
Pitch 

The Number of Teeth 
and the Addendum 

The Pitch Diameter 
and the Circular 
Pitch . . „ . . 

The Circular Pitch. 
The Circular Pitch. 



The Circular Pitch. 

Tlie Circular Pitch. 

The Circular Pitch. 
The Circular Pitch. 
Thickness of Tooth. 



Rule. 



Divide 3.1416 by the Diametral 
Pitch 

Divide Pitch ' Diameter liy the 
product of .3183 and Number of 
Teeth 

Divide Outside Diameter by the 
product of .3183 and Number of 
Teeth plus 2 

The continued product of the 
Number of Teeth, the Circular 
Pitch and .3183 

Divide the pi'oduct of Number of 
Teeth and Outside Diameter by 
Number of Teeth plus 2 . . . 

Subtract from the Outside Diame- 
ter the product of the Circular 
Pitch and .6366 

Multiply the Number of Teeth by 
the Addendum 

The continued product of the 
Number of Teeth ])lus 2, the 
Circular Pitch and .3183 . . . 

Add to the Pitch Diameter the 
product of the Circular Pitch 
and .6366 

Multiply Addendum by Number 
of Teeth plus 2 

Divide the product of Pitch Diam- 
eter and 3.1416 by the Circular 
Pitch 

One-half the Circular Pitch . . 

Multiply the Circular Pitch by 
.3183, or 8 = -^' 

Multiply the Circular Pitch by 
.3683 

Multiply the Circular Pitch bv 
.6366 ' 

Multiply the Circular Pitch by 
.6866 

Multiply the Circular Pitch by .05 

One-tenth the Thickness of Tooth 
at Pitch Line 



Formula. 



3.1416 
P 

D' 

.3183 N 

D 



^ .3183 N-f 2 
D'=NP'.3183 



D'= 



ND 



N+2 
D=:D— (P'.6366) 
D'= N 8 

D:=(N+2)P'.31SS 
D=D'-(-(P'.6366) 




s = P' .3183 

s + f = P' .3683 

D"= P' .6366 

D"= P' .6866 

f =P.05 
t 



f = 



10 



146 



BROWN & SHARPE MFG. CO. 



GEAR WHEELS. 



TABLE OF TOOTH PAKTS- — CIKCULAK PITCH IN FIKST COLUMN. 



■6^ 


Threads or 

Teeth per inch 

Linear . 




Thickness of 

Tooth on 
Pitch Line. 




Working Depth 
of Tooth. 


Depth of Space 

below 

Pitch Line. 


^ 8 


Width of 

Thread-Tool 

at End. 


Width of 
Thread at Top. 


P' 


p' 


p 


t 


. S 


D" 


«+/ 


D"+/ 


P'X.31 


P'X.335 


2 


1 
a 


1.5708 


1.0000 


.6366 


1.27S2 


.7366 


1.3732 


.6200 


.6700 


If 


8 
15 


1.6755 


.9375 


.5968 


1.1937 


.6906 


1.2874 


.5813 


.6281 


11 


i 

7 


1.7952 


.8750 


.5570 


1.1141 


.6445 


1.2016 


.5425 


.5863 


li 


8 
13 


1.9333 


.8125 


.5173 


1.0345 


.5985 


L1158 


.5038 


.5444 


li 


2 
3 


2.0944 


.7500 


.4775 


.9549 


.5525 


1.0299 


.4650 


.5025 


ih 


16 
23 


2.1855 


.7187 


.4576 


.9151 


.5294 


.9870 


.4456 


.4816 


1-1- 


b 
11 


2.2848 


.6875 


.4377 


.8754 


.5064 


.9441 


.4262 


.4606 


li 


3 

i 


2.3562 


.6666 


.4244 


.8488 


.4910 


.9154 


.4133 


.4466 


1^ 


16 
21 


2.3936 


.6562 


.4178 


.8356 


.4834 


.9012 


.4069 


.4397 


li 


i 
5 


2.5133 


.6250 


.3979 


.7958 


.4604 


.8588 


.3875 


.4188 


1^ 


16 

ir 


2.6456 


.5937 


.3780 


V.7560 


.4374 


.8156 


.3681 


.3978 


if 


8 
9 


2.7925 


.5625 


.3581 


.7162 


.4143 


.7724 


.3488 


.3769 


1^ 


16 

17 


2.9568 


.5312 


.3382 


.6764 


.3913 


.7295 


.3294 


.3559 


1 


1 


3.1416 


.5000 


.3183 


.6366 


.3683 


.6866 


.3100 


.3350 


15 
16" 


11 


3.3510 


.4687 


.2984 


.5968 


.3453 


.6437 


.2906 


.3141 


7 
8 


li 


3.5904 


.4375 


.2785 


.5570 


.3223 


.6007 


.2713 


.2931 


13 

16" 


1^ 


3.8666 


.4062 


.2586 


.5173 


.2993 


.5579 


.2519 


.2722 


1. 
5 


If 


3.9270 


.4000 


.2546 


.5092 


.2946 


.5492 


.2480 


.2680 


3 
4 


If 


4.1888 


.3750 


.2387 


.4775 


.2762 


.5150 


.2325 


.2513 


11 
16 


li 


4.5696 


.3437 


.2189 


.4377 


.2532 


.4720 


.2131 


.2303 


2 
3 


If 


4.7124 


.3333 


.2122 


.4244 


.2455 


.4577 


.2066 


.2233 


5 
8 


If 


5.0265 


.3125 


.1989 


.3979 


.2301 


.4291 


.1938 


.2094 


3 
5 


11 


5.2360 


.3000 


.1910 


.3820 


.2210 


.4120 


.1860 


.2010 


.7 


If 


5.4978 


.2857 


.1819 


.3638 


.2105 


.3923 


.1771 


.1914 


^ 


If 


5.5851 


.2812 


.1790 


.3581 


.2071 


.3862 


.1744 


.1884 



To obtain the size 
table, multiply the 
required. 



of any part of a circular pitch not given in the 
corresponding part of 1" pitch by the pitch 



PROVIDENCE, R. I, 

TABLE OF TOOTH TAUTB.— Contimteil 



147 



CIRCULAR riTCIl IN FIRST COLUMN. 





Threads or 

Teeth per inch 

Linear. 


"3 


Thickness of 

Tooth on 
Pitch Line. 


§1 

n 


- ^ 

bO o 

.an 

o 


Depth of Space 

below 

Pitch Line. 


ft o 

CD P, 


Width of 

Thread-Tool 

at End. 


o H 
R 


P' 


1" 


p 


t 


s yi 


D" 


s-t-/ 


D'^f. 


Pk.3i 


PX.335 


1 

2 


2 


6.2832 


.2500 


.1592 


.3183 


.1842 


.3433 


.1550 


.1675 


i 
» 


21 


7.0685 


.2222 


.1415 


.2830 


.1637 


.3052 


.1378 


.1489 


7 
IG 


2f 


7.1808 


.2187 


.1393 


.2786 


.1611 


.3003 


.1356 


.1466 


8 

7 


^8 


7.3304 


.2143 


.1364 


.2728 


.1578 


.2942 


.1328 


.1436 


2 
5 


^2 


7.8540 


.2000 


.1273 


.2546 


.1473 


.2746 


.1240 


.1340 


8 
8 


2f 


8.3776 


.1875 


.1194 


.2387 


.1381 


.2575 


.1163 


.1256 


i 
11 


2f 


8.6394 


.1818 


.1158 


.2316 


.1340 


.2498 


.1127 


.1218 


1 
3 


3> 


9.4248 


.1666 


.1061 


.2122 


.1228 


.2289 


.1033 


.1117 


6 
IG 


Si- 


10.0531 


.1562 


.0995 


.1989 


.1151 


.2146 


.0969 


.1047 


3 
10 


Si 


10.4719 


.1500 


.0955 


.1910 


.1105 


.2060 


.0930 


.1005 


2_ 


3i 


10.9956 


.1429 


.0909 


.1819 


.1052 


.1962 


.0886 


.0957 


1 
i 


4 


12.5664 


.1250 


.0796 


.1591 


.0921 


.1716 


.0775 


.0838 


9' 


4i 


14.1372 


.1111 


.0707 


.1415 


.0818 


.1526 


.0689 


.0744 


1 

5 


5 


15.7080 


.1000 


.0637 


.1273 


.0737 


.1373 


.0620 


.0670 


8 

16 


5f 


16.7652 


.0937 


.0597 


.1194 


.0690 


.1287 


.0581 


.0628 


11 


51- 


17.2788 


.0909 


.0579 


.1158 


.0670 


.1249 


.0564 


.0609 


1 
6 


6 


18.8496 


.0833 


.0531 


.1061 


.0614 


.1144 


.0517 


.0558 


2 
13 


6i 


20.4203 


.0769 


.0489 


.0978 


.0566 


.1055 


.0477 


.0515 


1 

7 


7 


21.9911 


.0714 


.0455 


.0910 


.0526 


.0981 


.0443 


.0479 


2 
15 


7i 


23.5619 


.0666 


.0425 


.0850 


.0492 


.0917 


.0414 


.0446 


1 
8 


8 


25.1327 


.0625 


.0398 


.0796 


.0460 


.0858 


.0388 


.0419 


1 

9 


9 


28.2743 


.0555 


.0354 


.0707 


.0409 


.0763 


.0344 


.0372 


1 
10 


10 


31.4159 


.0500 


.0318 


.0637 


.0368 


.0687 


.0310 


.0335 


1 
16 


16 


50.2655 


.0312 


.0199 


.0398 


.0230 


.0429 


.0194 


.0209 


1 

20 


20 


62.8318 


.0250 


.0159 


.0318 


.0184 


.0343 


.0155 


.0167 



To obtain the 
table, multiply 
required. 



size of any part of 
the corresponding 



a circular pitch not given in the 
part of 1" pitch by the pitch 



148 



BROWN & SHARPE MFG. CO. 



GEAE WHEELS. 



TABLE OF TOOTH PARTS DIAMETRAL PITCH IN FIRST COLUMN. 



Diametral 
Pitch. 




Thickness 
of Tooth on 
Pitch Line. 


3= 1 

re a 
< 


ft 
o 


Depth of Space 

below 

Pitch Line. 


Whole Depth 
of Tooth. 


P 


P' 


t 


s 


D" 


s+f. 


D"+/. 


i 


6.2832 


3.1416 


2.0000 


4.0000 


2.3142 


4.3142 


I 


4.1888 


2.0944 


1.3333 


2.6666 


1.5428 


2.8761 


1 


3.1416 


1 . 5708 


1.0000 


2.0000 


1.1571 


2.1571 


li 


2.5133 


1.2566 


.8000 


1.6000 


.9257 


1.7257 


n 


2.0944 


1.0472 


.6666 


1.3333 


.7714 


1.4381 


If 


1.7952 


.8976 


.5714 


1 1429 


.6612 


1.2326 


2 


1.5708 


.7854 


.5000 


1.0000 


.5785 


1.0785 


2i 


1.3963 


.6981 


.4444 


.8888 


.5143 


.9587 


2i 


1.2566 


.6283 


.4000 


.8000 


.4628 


.8628 


2f - 


1.1424 


.5712 


.3636 


.7273 


.4208 


.7844 


3 


1.0472 


.5236 


.3333 


.6666 


.3857 


.7190 


3^ 


.8976 


.4488 


.2857 


.5714 


.3306 


.6163 


4 


.7854 


.3927 


.2500 


.5000 


.2893 


.5393 


5 


.6283 


.3142 


.2000 


.4000 


.2314 


.4314 


6 


.5236 


.2618 


.1666 


.3333 


.1928 


.3595 


7 


.4488 


.2244 


.1429 


.2857 


.1653 


.3081 


8 


.3927 


.1963 


.1250 


.2500 


.1446 


.2696 


9 


.3491 


.1745 


.1111 


.2222 


.1286 


.2397 


10 


.3142 


.1571 


.1000 


.2000 


.1157 


.2157 


11 


.2856 


.1428 


.0909 


.1818 


.1052 


.1961 


12 


.2618 


.1309 


0833 


.1666 


.0964 


.1798 


13 


.2417 


.1208 


.0769 


.1538 


.0890 


.1659 


14 


.2244 


.1122 


.0714 


.1429 


.0826 


.1541 



To obtain the size of any part of a diametral pitch not given in the 
table, divide the corresponding part of 1 diametral pitch by the pitch 
required. 



PROVIDENCE, n. I, 



149 



TABLE OF TOOTH TARTS— Contmuecl 



DIAMETRAL PITCH IN FIRST COLUMN. 



u ■ 

II 

o3CL| 
5 


1.1 
O 


Thickness 
of Tooth on 
Pitch Line. 


< 


fcCo 
o ° 


Depth of Space 

below 

Pitch Line. 


Is 


P. 


P'. 


t. 


s. 


D". 


s+f. 
.0771 


D' + /. 


15 


.2094 


.1047 


.0666 


.1333 


.1438 


16 


.1963 


.0982 


.0625 


.1250 


.0723 


.1348 


17 


.1848 


.0924 


.0588 


.1176 


.0681 


. 1269 


18 


.1745 


.0873 


.0555 


.1111 


.0643 


.1198 


19 


.1653 


.0827 


.0526 


.1053 


.0609 


.1135 


20 


.1571 


.0785 


.0500 


.1000 


.0579 


.1079 


22 


.1428 


.0714 


.0455 


.0909 


.0526 


.0980 


24 


.1309 


.0654 


.0417 


.0830 


.0482 


.0898 


26 


.1208 


.0604 


.0385 


.0769 


.0445 


.0829 


28 


.1122 


.0561 


.0357 


.0714 


.0413 


.0770 


30 


.1047 


.0524 


.0333 


.0666 


.0386 


.0719 


32 


.0982 


.0491 


.0312 


.0625 


.0362 


.0674 


34 


.0924 


.0462 


.0294 


.0588 


.0340 


.0634 


36 


.0873 


.0436 


.0278 


.0555 


.0321 


.0599 


38 


.0827 


.0413 


.0263 


.0526 


.0304 


.0568 


40 


.0785 


.0393 


.0250 


. 0500 


.0289 


.0539 


42 


.0748 


.0374 


.0238 


.0476 


.0275 


.0514 


44 


.0714 


.0357 


.0227 


.0455 


.0263 


.0490 


46 


.0683 


.0341 


.0217 


.0435 


.0252 


.0469 


48 


.0654 


.0327 


.0208 


.0417 


.0241 


.0449 


50 


.0628 


.0314 


.0200 


.0400 


.0231 


.0431 


56 


.0561 


.0280 


.0178 


.0357 


.0207 


.0385 


60 


.0524 


.0262 


.0166 


.0333 


.03 93 


. 0360 



To obtain the size of any part of a diametral pitch not given in the 
table, divide the corresponding part of 1 diametral pitch by the pitch 
required. 



Natural Sines and Cosines, 



International Correspondence Schools. 



NATURAL SINES AND COSINES 



151 



/ 


0° 


1° 


2° 


3° 


4° 


f 


Sine ( 


Cosine 


Sine C 


Cosine 


Sine C 


:osine 


Sine ( 


"osine 


Sine C 


Cosine 





.00000 




.01745 


9998s 


.03490 


99939 


•05234 


99863 


.06976 


99756 


60 


I 


.00029 






.01774 


99984 


.03519 


99938 


•05263 


99861 


.07005 


99754 


59 


a 


.00058 






.01803 


99984 


.03548 


99937 


•05292 


99860 


.07034 


99752 


58 


3 


.00087 






.01832 


99983 


.03577 


.99936 


•05321 


99858 


.07063 


99750 


57 


4 


.00116 






.01862 


99983 


.03606 


99935 


•05350 


99857 


.07092 


99748 


S6 


5 


.00145 






.01891 


99982 


.03635 


99934 


•05379 


99855 


.07121 


99746 


55 


6 


.00175 






.01920 


99982 


.03664 


99933 


.05408 


99854 


.07150 


99744 


54 


7 


.00204 






.01949 


99981 


•03693 


99932 


•05437 


99852 


.07179 


99742 


S3 


8 


.00233 






.01978 


99980 


.03723 


99931 


•05466 


99851 


.07208 


99740 


52 


9 


.00262 






.02007 


99980 


.03752 


99930 


•05495 


99849 


.07237 


99738 


51 


10 


.00291 






.02036 


99979 


.03781 


99929 


•05524 


99847 


.07266 


99736 


50 


11 


.00320 


99999 


.02065 


99979 


.03810 


99927 


•05553 


99846 


.07295 


99734 


49 


12 


.00349 


99999 


.02094 


99978 


.03839 


99926 


•05582 


99844 


.07324 


99731 


48 


13 


.00378 


99999 


.02123 


99977 


.03868 


99925 


.05611 


99842 


.07353 


99729 


47 


14 


.00407 


99999 


.02152 


99977 


.03897 


99924 


.05640 


99841 


.07382 


99727 


46 


IS 


.00436 


99999 


.02181 


99976 


.03926 


99923 


.05669 


99839 


.07411 


9972s 


45 


i6 


.00465 


99999 


.02211 


99976 


.03955 


99922 


.05698 


99838 


.07440 


99723 


44 


17 


.00495 


99999 


.02240 


99975 


.03984 


99921 


•05727 


99836 


.07469 


99721 


43 


i8 


.00524 


99999 


.02269 


99974 


.04013 


99919 


•05756 


99834 


.07498 


99719 


42 


19 


.00553 


99998 


.02298 


99974 


.04042 


99918 


•05785 


99833 


.07527 


99716 


41 


20 


.00582 


99998 


.02327 


99973 


.04071 


99917 


.05814 


99831 


.07556 


99714 


40 


21 


.00611 


99998 


.02356 


99972 


.04100 


99916 


.05844 


99829 


.07585 


99712 


39 


22 


.00640 


99998 


.02385 


99972 


.04129 


999 IS 


.05873 


99827 


.07614 


99710 


38 


23 


.00669 


Cr998 


.02414 


99971 


.04159 


99913 


.05902 


99826 


.07643 


99708 


37 


24 


.00698 


99998 


.02443 


99970 


.04188 


99912 


•05931 


99824 


.07672 


99705 


36 


25 


.00727 


99997 


.02472 


99969 


.04217 


9991 1 


•05960 


99822 


.07701 


99703 


35 


26 


.00756 


99997 


.02501 


99969 


.04246 


99910 


•05989 


99821 


.07730 


99701 


34 


27 


.00785 


99997 


.02530 


99968 


.04275 


9S909 


.06018 


99819 


■07759 


99699 


33 


28 


.00814 


99997 


.02560 


99967 


.04304 


99907 


.06047 


99817 


.07788 


99696 


32 


29 


.00844 


99996 


.02589 


99966 


.04333 


99906 


.06076 


9981s 


.07817 


99694 


31 


30 


.00873 


99996 


.02618 


99966 


.04362 


99905 


.06105 


99813 


.07846 


99692 


30 


31 


.00902 


99996 


.02647 


9996s 


.04391 


99904 


.06134 


99812 


•07875 


99689 


29 


32 


■00931 


99996 


.02676 


99964 


.04420 


99902 


.06163 


99810 


•07904 


99687 


28 


33 


.00960 


99995 


.02705 


99963 


.04449 


99901 


.06192 


99808 


•07933 


99685 


27 


34 


.00989 


99995 


.02734 


99963 


.04478 


99900 


.06221 


99806 


.07962 


99683 


26 


35 


.01018 


99995 


.02763 


99962 


.04507 


99898 


.06250 


99804 


.07991 


99680 


25 


36 


.01047 


99995 


.02792 


99961 


.04536 


99897 


.06279 


99803 


.08020 


99678 


24 


37 


.01076 


99994 


.02821 


99960 


.04565 


99896 


.06308 


99801 


.08049 


99676 


23 


38 


.Olios 


99994 


.02850 


999S9 


.04594 


99894 


•06337 


99799 


.08078 


99673 


22 


39 


.01134 


99994 


.02879 


99959 


.04623 


99893 


•06366 


99707 


.08107 


99671 


21 


40 


.01164 


99993 


.02908 


99958 


.04653 


99892 


.0639s 


99795 


.08136 


99668 


20 


41 


.01193 


99993 


.02938 


99957 


.04682 


99890 


.06424 


99793 


.08165 


99666 


19 


42 


.01222 


99993 


.02967 


99956 


.04711 


99889 


.06453 


99792 


.08194 


99664 


18 


43 


.01251 


99992 


.02996 


9995S 


.04740 


99888 


.06482 


99790 


.08223 


99661 


17 


44 


.01280 


99992 


.03025 


99954 


.04769 


99886 


.06511 


99788 


.08252 


99659 


16 


4S 


.01309 


99991 


.03054 


99953 


.04798 


99885 


.06540 


997^6 


.08281 


99657 


IS 


46 


.01338 


99991 


.03083 


99952 


.04827 


99883 


.06569 


99784 


.08310 


99654 


14 


47 


.01367 


99991 


.03112 


99952 


.04856 


99882 


.06598 


99782 


.08339 


99652 


13 


48 


.01396 


99990 


.03141 


9995 1 


.04885 


99881 


.06627 


99780 


.08368 


99649 


12 


49 


.01425 


99990 


.03170 


99950 


.04914 


99879 


.06656 


99778 


•08397 


99647 


11 


SO 


.01454 


99989 


.03199 


99949 


.04943 


99878 


.06685 


99776 


.08426 


99644 


10 


SI 


.01483 


99989 


.03228 


99948 


.04972 


99876 


.06714 


99774 


.08455 


99642 


9 


52 


•OIS13 


99989 


•03257 


99947 


.05001 


99875 


.06743 


99772 


.08484 


99639 


8 


53 


.01542 


99988 


.03286 


99946 


.05030 


99873 


.06773 


99770 


.08513 


99637 


7 


54 


.01571 


99988 


.03316 


99945 


.05059 


99872 


.06802 


99768 


.08542 


99635 


6 


55 


.01600 


99987 


.03345 


99944 


.05088 


99870 


.06831 


99766 


.08571 


99632 


S 


56 


.01629 


99987 


.03374 


99943 


.05117 


99869 


.06860 


99764 


.08600 


99630 


4 


£7 


.01658 


99986 


•03403 


99942 


.05146 


99867 


.06889 


99762 


.08629 


99627 


3 


S8 


.01687 


99986 


.03432 


99941 


•05175 


99866 


.06918 


99760 


.08658 


99625 


2 


S9 


.01716 


9998s 


.03461 


99940 


•05205 


99864 


.06947 


997.';8 


.08687 


99622 


I 


60 


.OI74S 


99985 


.03490 


99939 


•05234 


99863 


.06976 


99756 


.08716 


99619 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


89° ^ 


88° 


87° 


86° 


85° 



152 



NATURAL SINES AND COSINES 



/ 


5 





6 





7° 


8° 


9° 




/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine C 


-osine 





.08716 


•99619 


.10453 


.99452 


.12187 


•99255 


.13917 


.99027 


■15643 


98769 


60 


I 


.08745 


.99617 


.10482 


.99449 


.12216 


.99251 


.13946 


■99023 


■ 15672 


98764 


59 


2 


.08774 


.99614 


.10511 


.99446 


.12245 


.99248 


.13975 


■99019 


■15701 


98760 


S8 


3 


.08803 


.99612 


.10540 


.99443 


.12274 


•99244 


.14004 


■99015 


■15730 


9875s 


57 


4 


.08831 


.99609 


.10569 


.99440 


.12302 


.99240 


.14033 


.99011 


■15758 


98751 


56 


S 


.08860 


.99607 


.10597 


■99437 


•12331 


•99237 


.14061 


.99006 


.15787 


98746 


55 


6 


.o888g 


.99604 


.10626 


.99434 


.12360 


•99233 


.14090 


.99002 


.15816 


98741 


54 


7 


.08918 


.99602 


.10655 


.99431 


.12389 


.99230 


.14119 


.98998 


.15845 


98737 


S3 


8 


.08947 


.99599 


.10684 


.99428 


.12418 


.99226 


.14148 


■9^994 


■15873 


98732 


52 


9 


.08976 


.99596 


.10713 


.99424 


.12447 


.99222 


.14177 


.98990 


■15902 


98728 


SI 


10 


.09005 


.99594 


.10742 


•99421 


.12476 


.99219 


.14205 


.98986 


.15931 


98723 


SO 


II 


.09034 


.99591 


.10771 


.99418 


.12504 


.99215 


.14234 


.98982 


.15959 


98718 


49 


12 


.09063 


.99588 


.loSoo 


•99415 


.12533 


.99211 


.14263 


.93978 


■15988 


98714 


48 


13 


.09092 


.99586 


.10829 


•99412 


.12562 


.99208 


.14292 


.98973 


.16017 


98709 


47 


14 


.09121 


.99583 


.10858 


•99409 


.12591 


.99204 


.14320 


.98969 


.16046 


98704 


46 


IS 


.09150 


.99580 


.10887 


.99406 


.12620 


.99200 


.14349 


.98965 


.16074 


98700 


45 


i6 


.09179 


.99578 


.10916 


.99402 


.12649 


.99197 


.14378 


.98961 


.16103 


98695 


44 


17 


.09208 


.99575 


.10945 


•99399 


.12678 


•99193 


.14407 


■98957 


.16132 


98690 


43 


i8 


.09237 


•99572 


.10973 


•99396 


.12706 


.99189 


.14436 


■98953 


.16160 


98686 


42 


19 


.09266 


.99570 


.11002 


.99393 


■12735 


.99186 


.14464 


.98948 


.16189 


98681 


41 


20 


.09295 


■99567 


.11031 


.99390 


.12764 


.99182 


■ 14493 


.98944 


.16218 


98676 


40 


21 


.09324 


.99564 


.11060 


.99386 


•12793 


•99178 


.14522 


.98940 


.16246 


98671 


39 


22 


.09353 


.99562 


.11089 


■99383 


.12822 


•99175 


.14551 


.98936 


.16275 


98667 


38 


23 


.09382 


.99559 


.11118 


.99380 


.12851 


.99171 


.14580 


•98931 


.16304 


98662 


37 


24 


.09411 


.99556 


.11147 


■99377 


.12880 


.99167 


.14608 


•98927 


.16333 


98657 


36 


2S 


.09440 


•99553 


.11176 


•99374 


.12908 


.99163 


.14637 


.98923 


.16361 


98652 


35 


26 


.09469 


•99551 


.11205 


•99370 


.12937 


.99160 


.14666 


.9S919 


.16390 


98648 


34 


27 


.09498 


•99548 


.11234 


•99367 


.12966 


.99156 


■14695 


.98914 


.16419 


98643 


33 


28 


.09527 


•99545 


.11263 


•99364 


.12995 


.99152 


■14723 


.98910 


.16447 


98638 


32 


29 


.09556 


•99542 


.11291 


.99360 


.13024 


.99148 


■14752 


.98906 


.16476 


98633 


31 


30 


.0958s 


.99540 


.11320 


.99357 


■13053 


.99144 


.14781 


.98902 


.16505 


98629 


30 


31 


.09614 


.99537 


.11349 


.99354 


.13081 


•99141 


.14810 


.9S897 


.16533 


98624 


29 


32 


.09642 


•99534 


.11378 


.99351 


.13110 


•99137 


.14838 


.98893 


.16562 


98619 


28 


33 


.09671 


•99531 


.11407 


.99347 


.13139 


•99133 


.14867 


.98889 


.16591 


98614 


27 


34 


.09700 


.99528 


.11436 


.99344 


.13168 


.99129 


.14896 


.98884 


.16620 


98609 


26 


35 


.09729 


.99526 


.11465 


•99341 


.13197 


.99125 


■14925 


.98880 


.16648 


98604 


25 


36 


.09758 


.99523 


.11494 


.99337 


.13226 


.99122 


■14954 


.98876 


.16677 


98600 


24 


37 


.09787 


.99520 


•11523 


•99334 


.13254 


.99118 


.14982 


.98871 


.16706 


9859s 


23 


38 


.09816 


.99517 


.11552 


.99331 


.13283 


.99114 


.15011 


.98S67 


.16734 


98590 


22 


39 


.09845 


.99514 


.11580 


.99327 


.13312 


.99110 


.15040 


.98863 


.16763 


9858s 


21 


40 


.09874 


.99511 


.11609 


.99324 


.13341 


.99106 


.15069 


.98858 


.16792 


98580 


20 


41 


.09903 


.99508 


.11638 


.99320 


.13370 


.99102 


.15097 


.98854 


.16820 


98575 


19 


42 


.09932 


.99506 


.11667 


■99317 


.13399 


.99098 


.15126 


.98849 


.16849 


98570 


18 


43 


.09961 


.99503 


.11696 


.99314 


.13427 


.99094 


.15155 


.98845 


.16878 


98565 


17 


44 


.09990 


.99500 


•11725 


.99310 


.13456 


.99091 


.15184 


.98841 


.16906 


98561 


16 


4S 


.10019 


.99497 


•11754 


.99307 


.13485 


.99087 


.15212 


.98836 


.16935 


98556 


15 


46 


.10048 


.99494 


.11783 


•99303 


.13514 


.99083 


.15241 


.98832 


.16964 


98551 


14 


47 


.10077 


.99491 


.11812 


•99300 


.13543 


.99079 


.15270 


.98827 


.16992 


98546 


13 


48 


.10106 


.99488 


.11840 


•99297 


.13572 


.99075 


■15299 


.98823 


.17021 


98541 


12 


49 


.10135 


.99485 


.11869 


.99293 


.13600 


.99071 


■15327 


.98818 


-17050 


98536 


11 


50 


.10164 


.99482 


.11898 


.99290 


.13629 


.99067 


.15356 


.98814 


.17078 


98531 


10 


SI 


.10192 


.99479 


.11927 


.99286 


.13658 


.99063 


.15385 


.98809 


.17107 


98526 


9 


S2 


.10221 


•99476 


.11956 


.99283 


.13687 


.99059 


.15414 


.98805 


.17136 


98521 


8 


53 


.10250 


•99473 


.11985 


.99279 


.13716 


■9905s 


.15442 


.98800 


.17164 


98516 


7 


54 


.TO279 


•99470 


.12014 


•99276 


.13744 


.99051 


■15471 


.98796 


.17193 


98511 


6 


55 


.10308 


•99467 


.12043 


.99272 


.13773 


.99047 


■15500 


.98791 


.17222 


98506 


5 


56 


.10337 


.99464 


.12071 


.99269 


.13802 


.99043 


.15529 


■98787 


.17250 


98501 


4 


57 


.10366 


.99461 


.12100 


.99265 


.13831 


.99039 


.15557 


■98782 


.17279 


98496 


3 


58 


.10395 


.99458 


.12129 


.99262 


.13860 


.99035 


.15586 


■98778 


.17308 


98491 


2 


59 


.10424 


•99455 


.12158 


.99258 


.13S89 


.99031 


.15615 


.98773 


.17336 


98486 


I 


60 


.10453 


.99452 


.12187 


.99255 


.13917 


.99027 


.15643 


.98769 


.17365 


08481 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


8 


4° 


8 


?° 


82 





81 





80^ 


3 



NATURAL SINES AND COSINES 



153 



/ 


10 





I I 





12° 


13° 


14° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine C 


;osine 





.1736s 


.98481 


.19081 


.98163 


.20791 


.9781s 


•22495 


.97437 


.24192 


97030 


60 


I 


.17393 


.98476 


.19109 


.98157 


.20820 


.97809 


•22523 


.97430 


.24220 


97023 


59 


2 


.17422 


•98471 


.19138 


.98152 


.20848 


.97803 


•22552 


.97424 


.24249 


9701s 


S8 


3 


.17451 


.98466 


.19167 


.98146 


.20877 


•97797 


.22580 


•97417 


.24277 


97008 


57 


4 


.17479 


.98461 


.19195 


.98140 


.20905 


•97791 


.22608 


•97411 


.24305 


97001 


56 


5 


.17508 


.98455 


•19224 


.98135 


.20933 


•97-84 


■22637 


.97404 


.243.33 


96994 


55 


6 


•17537 


.98450 


.19252 


.98129 


.20962 


•97778 


.22665 


•97398 


.24362 


96987 


S4 


7 


■1756s 


.98445 


.19281 


.98124 


.20990 


•97772 


.22693 


•97391 


.24390 


96980 


S3 


8 


.17594 


.98440 


.19309 


.98118 


.21019 


•97766 


.22722 


•97384 


.24418 


56973 


52 


9 


.17623 


•9S435 


.19338 


.98112 


.21047 


•97760 


.22750 


•9737S 


.24446 


96966 


SI 


10 


.17651 


.98430 


.19366 


.98107 


.21076 


•97754 


.22778 


.97371 


•24474 


96959 


SO 


II 


.17680 


.98425 


.19395 


.98101 


.21104 


•97748 


.22807 


•9736s 


•24503 


96952 


49 


12 


.17708 


.98420 


.19423 


.98096 


.21132 


•97742 


.22835 


•97358 


•24531 


96945 


48 


13 


.17737 


.98414 


.19452 


.98090 


.21161 


•9773S 


.22863 


•97351 


•24559 


96937 


47 


14 


.17766 


.9S409 


.19481 


.98084 


.21189 


•97729 


.22892 


•97345 


.24587 


96930 


46 


IS 


.17794 


.98404 


■19509 


.98079 


.21218 


•97723 


.22920 


•97338 


.24615 


96923 


45 


i6 


.17823 


.98399 


•19538 


.98073 


.21246 


•97717 


.22948 


•97331 


.24644 


96916 


44 


17 


.17852 


.98394 


■19566 


.98067 


.21275 


•97711 


.22977 


•9732s 


.24672 


96909 


43 


i8 


.17880 


.98389 


■19S9S 


.98061 


.21303 


.97705 


.23005 


•97318 


.24700 


96902 


42 


19 


.17909 


•98383 


■19623 


.98056 


.21331 


.97698 


.23033 


•97311 


.24728 


96894 


41 


20 


.17937 


.98378 


.19652 


.98050 


.21360 


.97692 


.23062 


.97304 


.24756 


96887 


40 


21 


.17966 


.98373 


.19680 


.98044 


.21388 


.97686 


.23090 


.97298 


.24784 


96880 


39 


22 


.17995 


.98363 


■19709 


.98039 


.21417 


.97680 


.23118 


.97291 


.24813 


96S73 


38 


23 


.18023 


.98362 


• 19737 


.98033 


.21445 


•97673 


.23146 


.97284 


.24841 


96866 


37 


24 


.18052 


.98357 


.19766 


.98027 


.21474 


•97667 


■23175 


.97278 


.24869 


96858 


36 


2S 


.18081 


.98352 


■19794 


.98021 


.21502 


.97661 


■23203 


.97271 


.24897 


96851 


35 


26 


.18109 


■98347 


.19823 


.98016 


.21530 


•97655 


■23231 


.97264 


.24925 


96844 


34 


27 


.18138 


.98341 


.19851 


.98010 


.21559 


.97648 


.23260 


.97257 


.24954 


96837 


33 


28 


.18166 


.98336 


.19880 


.98004 


.21587 


.97642 


.23288 


.97251 


.24982 


96829 


32 


29 


.18195 


.98331 


.19908 


.97998 


.21616 


.97636 


.23316 


.97244 


.25010 


96822 


31 


30 


.18224 


.98325 


.19937 


.97992 


.21644 


.97630 


.23345 


.97237 


.25038 


96815 


30 


31 


.18252 


.98320 


.19965 


.97987 


.21672 


•97623 


.23373 


.97230 


.25066 


96807 


29 


32 


.18281 


.98315 


.19994 


■97981 


.21701 


.97617 


.23401 


.97223 


.25094 


96800 


28 


33 


.18309 


.98310 


.20022 


■97975 


.21729 


.97611 


.23429 


.97217 


.25122 


96793 


27 


34 


.18338 


.98304 


.20051 


■97969 


.21758 


.97604 


.23458 


.97210 


.25151 


96786 


26 


35 


.18367 


.98299 


.20079 


■97963 


.21786 


.97598 


.23486 


.97203 


.25179 


96778 


25 


36 


.18395 


.98294 


.20108 


■97958 


.21814 


.97592 


.23514 


.97196 


.25207 


96771 


24 


37 


.18424 


.98288 


.20136 


■97952 


.21843 


.975S5 


•23542 


.97189 


.25235 


96764 


23 


38 


.18452 


.98283 


.20165 


■97946 


.21871 


.97579 


.23571 


.97182 


.25263 


96756 


22 


39 


.18481 


.98277 


.20193 


■97940 


.21899 


•97573 


.23599 


.97176 


.25291 


96749 


21 


40 


.18509 


.98272 


.20222 


■97934 


.21928 


.97566 


.23627 


.97169 


.25320 


96742 


20 


41 


.18538 


.98267 


.20250 


.97928 


.21956 


.97560 


.23656 


.97162 


.25348 


96734 


19 


42 


.18567 


.98261 


.20279 


.97922 


.21985 


.97553 


.23684 


.97155 


.25376 


96727 


18 


43 


.18595 


.98256 


.20307 


.97916 


.22013 


.97547 


.23712 


.97148 


.25404 


96719 


17 


44 


.18624 


.98250 


.20336 


.97910 


.22041 


.97541 


.23740 


.97141 


.25432 


96712 


16 


45 


.18652 


.98245 


.20364 


.97905 


.22070 


.97534 


.23769 


•97134 


.25460 


96705 


IS 


46 


.18681 


.98240 


.20393 


.97899 


.22098 


.97528 


.23797 


.97127 


.25488 


96697 


14 


47 


.18710 


■98234 


.20421 


.97893 


.22126 


.97521 


.23825 


»97I20 


.25516 


96690 


13 


48 


.18738 


.98229 


.20450 


.97887 


.22155 


.97515 


.23853 


.97113 


.25545 


96682 


12 


49 


.18767 


.98223 


.20478 


.97881 


..22183 


.97508 


.23882 


.97106 


.25573 


96675 


II 


SO 


.18795 


.98218 


.20507 


•9787s 


.22212 


.97502 


.23910 


.97100 


.25601 


96667 


10 


SI 


.18824 


.98212 


•20S3S 


.97869 


.22240 


.97496 


.23938 


.97093 


.25629 


96660 


9 


52 


.18852 


.98207 


.20563 


.97863 


.22268 


,97489 


.23966 


.97086 


.25657 


96653 


8 


S3 


.18881 


.98201 


.20592 


.97857 


.22297 


.97483 


.23995 


.97079 


.25685 


9664s 


7 


54 


.18910 


.98196 


.20620 


.97851 


.22325 


.97476 


.24023 


.97072 


.25713 


96638 


6 


55 


.18938 


.98190 


.20649 


.97845 


.22353 


.97470 


.24051 


•97065 


.25741 


96630 


S 


56 


.18967 


.98185 


.20677 


.97839 


.22382 


.97463 


.24079 


.97058 


.25769 


96623 


4 


57 


.18995 


.98179 


.20706 


.97833 


.22410 


.97457 


.24108 


.97051 


.25798 


96615 


3 


58 


.19024 


.98174 


..20734 


.97827 


.22438 


.97450 


.24136 


.97044 


.25826 


96608 


2 


59 


.19052 


.98168 


.20763 


.97821 


.22467 


.97444 


.24164 


.97037 


^5854 


96600 


I 


6o 


.19081 


.98163 


.20791 


■97815 


.22495 


.97437 


.24192 


.97030 


.25882 


96593 





f 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


1 


79 





78 





7' 


7° 


7< 


^0 


75' 


> 



154 



NATURAL SINES AND COSINES' 



/ 


15 





16° 


17° 


18° 


19° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine C 


'osine 


Sine C 


-osine 





.25882 


.96593 


.27564 


.96126 


.29237 


.95630 


.30902 


95106 


.32557 


94SS2 




60 


I 


.25910 


.96585 


.27592 


.96118 


.29265 


.95622 


.30929 


95097 


.32584 


94542 


59 


2 


.25938 


.96578 


.27620 


.96110 


.29293 


.95613 


.30957 


95088 


.32612 


94533 


S8 


3 


.25966 


.96570 


.27648 


.96102 


.29321 


.95605 


.30985 


95079 


.32639 


94523 


57 


4 


.25994 


.96562 


.27676 


.96094 


•29348 


.95596 


.31012 


95070 


.32667 


94S14 


S6 


S 


.26022 


.96555 


.27704 


.96086 


.29376 


.95588 


.31040 


95061 


.32694 


94504 


SS 


6 


.26050 


.96547 


.27731 


.96078 


.29404 


.95579 


.31068 


95052 


.32722 


94495 


S4 


7 


.26079 


.96540 


■27759 


.96070 


.29432 


.95571 


.31095 


95043 


■32749 


9448s 


S3 


8 


.26107 


.96532 


.27787 


.96062 


.29460 


.95562 


.31123 


95033 


.32777 


94476 


52 


9 


.26135 


.96524 


.2781S 


.96054 


.29487 


.95554 


.31151 


95024 


.32804 


94466 


SI 


10 


.26x63 


.96517 


.27843 


.96046 


.29515 


.95545 


.31178 


95015 


.32832 


94457 


SO 


II 


.26191 


.96509 


.27871 


.96037 


.29543 


.95536 


.31206 


95006 


.32859 


94447 


49 


12 


.26219 


.96502 


.27899 


.96029 


.29571 


.95528 


.31233 


94997 


.32887 


94438 


48 


13 


.26247 


.96494 


.27927 


.96021 


.29599 


.95519 


.31261 


94988 


.32914 


94428 


47 


14 


.26275 


.96486 


.27955 


.96013 


.29626 


.95511 


.31289 


94979 


.32942 


94418 


46 


IS 


.26303 


.96479 


.27983 


.96005 


.29654 


•95502 


.31316 


94970 


.32969 


94409 


4S 


i6 


.26331 


.96471 


.28011 


•95997 


.29682 


•95493 


.31344 


94961 


.32997 


94399 


44 


17 


.26359 


.96463 


.28039 


.95989 


.29710 


.95485 


•31372 


94952 


.33024 


94390 


43 


i8 


.26387 


.96456 


.28067 


•95981 


.29737 


.95476 


•31399 


94943 


.33051 


94380 


42 


19 


.26415 


.96448 


.28095 


.95972 


.29765 


.95467 


.31427 


94933 


.33079 


94370 


41 


20 


.26443 


.96440 


.28123 


.95964 


•29793 


.95459 


.31454 


94924 


.33106 


94361 


40 


21 


.26471 


.96433 


.28150 


.95956 


.29821 


■95450 


.31482 


94915 


.33134 


94351 


39 


22 


.26500 


.96425 


.28178 


.95948 


.29849 


.95441 


.31510 


94906 


.33161 


94342 


38 


23 


.26528 


.96417 


.28206 


•95940 


.29876 


.95433 


.31537 


94897 


.33189 


94332 


37 


24 


.26556 


.96410 


.28234 


.95931 


.29904 


.95424 


.31565 


94888 


.33216 


94322 


36 


2S 


.26584 


.96402 


.28262 


.95923 


.29932 


.95415 


•31593 


94878 


.33244 


94313 


35 


26 


.26612 


.96394 


.28290 


.95915 


.29960 


.95.107 


.31620 


94869 


.33271 


94303 


34 


27 


.26640 


.96386 


.28318 


.95907 


.29987 


.95398 


.31648 


94860 


.33298 


94293 


33 


28 


.2666S 


.96379 


.28346 


•95898 


.30015 


.95389 


•31675 


94851 


.33326 


94284 


32 


29 


.26696 


.96371 


.28374 


•95890 


.30043 


.95380 


•31703 


94842 


-33353 


94274 


31 


30 


.26724 


.96363 


.28403 


.95882 


.30071 


.95372 


•31730 


94832 


.33381 


94264 


30 


31 


.26752 


.96355 


.28429 


.95874 


.30098 


•95363 


.31758 


94823 


.33408 


94254 


29 


32 


.26780 


.96347 


.28457 


.95865 


.30126 


•95354 


.31786 


94814 


.33436 


94245 


28 


33 


.26B08 


.96340 


.28485 


.95857 


.30154 


•95345 


•31813 


9480s 


.33463 


94235 


27 


34 


.26836 


.96332 


.28513 


•95849 


.30182 


.05337 


.31841 


94795 


.33490 


94225 


26 


35 


.26864 


.96324 


..28541 


.95841 


.30209 


.95328 


.31868 


94786 


•33518 


94215 


25 


36 


.26892 


.96316 


.28569 


•95832 


.30237 


.95319 


.31895 


94777 


•33545 


94206 


24 


37 


.26920 


.96308 


.28597 


•95824 


.30265 


•9S3IO 


.31923 


947C8 


■33573 


94196 


23 


38 


.26948 


.96301 


.28625 


•95816 


.30^9:; 


•9S30I 


.31951 


94753 


■33600 


94186 


22 


39 


.26976 


.96293 


.28652 


.95807 


.30320 


•95293 


■31979 


94749 


■33627 


94176 


21 


40 


.27004 


.96285 


.28680 


.95799 


.30348 


.95284 


.32006 


94740 


■3365s 


94167 


20 


41 


.27032 


.96277 


.28708 ■ 


.95791 


.30376 


.95275 


■32034 


94730 


■33682 


941S7 


19 


42 


.27060 


.96269 


.28736 


.95782 


.30403 


.95266 


.32061 


94721 


•33710 


94147 


18 


43 


.27088 


.96261 


.28764 


.,95774 


.30431 


.95257 


.32089 


94712 


•33737 


94137 


17 


44 


.27116 


.96253 


J28792 


.95766 


•30459 


.95248 


.32116 


94702 


•33764 


94127 


16 


45 


.27144 


.96246 


.,28820 


.95757 


.30486 


.95240 


.32144 


94693 


•33792 


94118 


IS 


46 


.27172 


.96238 


.28847 


.95749 


.30514 


.95231 


.32171 


94684 


•33819 


94108 


14 


47 


.27200 


.96230 


.128S75 


.95740 


.30542 


.95222 


.32199 


94674 


•33846 


94098 


13 


48 


.27228 


.96222 


.28903 


.95732 


-30570 


.95213 


.32227 


94665 


•33874 


94088 


12 


49 


.27256 


.96214 


.28931 


.95724 


•30597 


.95204 


.32254 


94656 


•33901 


94078 


II 


SO 


.27284 


.96206 


.28959 


.95715 


.30625 


.95195 


,32282 


94646 


•33929 


94068 


10 


SI 


.27312 


.96198 


128987 


.95707 


.30653 


.95186 


.32309 


94637 


.33956 


940S8 


9 


52 


.27340 


.96190 


.29015 


.95698 


.30680 


.95177 


■32337 


94627 


•33983 


94049 


8 


S3 


.27368 


.96182 


.29042 


.-95690 


.30708 


.95168 


.32364 


94618 


•34011 


94039 


7 


S4 


.27396 


.96174 


.29070 


.95681 


.30736 


.95159 


•32392 


94609 


•34038 


94029 


6 


SS 


.27424 


.96166 


.29098 


.95673 


.30763 


.95150 


.32419 


94599 


•3406s 


94019 


5 


S6 


.27452 


.961^ 


.29126 


.95664 


.30791 


.95142 


•32447 


94590 


•34093 


94009 


4 


57 


.27480 


.96150 


.29154 


.95656 


.30819 


.95133 


•32474 


94580 


•34120 


93999 


3 


58 


.27508 


.96142 


.29182 


.95647 


.30846 


.95124 


•32502 


94571 


.34147 


93989 


2 


59 


.27536 


.96134 


.29209 


.95639 


.30874 


.95115 


•32529 


94561 


•3417s 


93979 


I 


60 


•27564 


.96126 


.29237 


.95630 


.30902 


.95106 


•32557 


94552 


.34202 


93969 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


74 





7, 


3° 


7' 


2° 


71' 


) 


70° 



NATURAL SINES AND COSINES 



155 



/ 


20° 


2] 





22° 


23° 


24° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





.34202 


.93969 


•35837 


.93358 


.37461 


.92718 


.39073 


.92050 


•40674 


.91355 


60 


I 


.34229 


.93959 


■35864 


•93348 


.37488 


.92707 


.39100 


•92039 


.40700 


.91343 


59 


2 


.34257 


.93949 


•35891 


•93337 


.37515 


.92697 


.39127 


.92028 


.40727 


.91331 


58 


3 


.34284 


.93939 


•35918 


•93327 


.37542 


.92686 


.39153 


.92016 


.40753 


.91319 


57 


4 


.34311 


.93929 


• 35945 


•93316 


.37569 


.92675 


.39180 


.92005 


.40780 


.91307 


56 


S 


.34339 


•93919 


•35973 


•93306 


•37595 


.92664 


.39207 


.91994 


.40806 


•91295 


55 


6 


.34366 


.93909 


.36000 


•93295 


.37622 


.92653 


.39234 


.91982 


.40833 


•91283 


54 


7 


.34393 


•93899 


•36027 


•93285 


•37649 


.92642 


.39260 


.91971 


.40860 


•91272 


53 


8 


.34421 


.93889 


•36054 


•93274 


•37676 


.92631 


.39287 


.91959 


.40886 


.91260 


52 


9 


.34448 


.93879 


.36081 


•93264 


.37703 


.92620 


.39314 


.91948 


•40913 


.91248 


51 


10 


.34475 


.93869 


.36108 


.93253 


.37730 


.92609 


•39341 


.91936 


•40939 


.91236 


50 


II 


.34503 


•93859 


.36135 


.93243 


•37757 


.92598 


■39367 


.91925 


.40966 


.91224 


49 


12 


.34530 


•93849 


.36162 


.93232 


•37784 


.92587 


.39394 


.91914 


.40992 


.91212 


48 


13 


.34557 


•93839 


•36190 


•93222 


•37811 


•92576 


■39421 


.91902 


.41019 


.91200 


47 


14 


.34584 


•93829 


•36217 


•9321 1 


•37838 


.92565 


•39448 


.91891 


.41045 


.91188 


46 


IS 


.34612 


•93819 


.36244 


•93201 


•37S65 


•92554 


•39474 


.91879 


.41072 


.91176 


45 


i6 


.34639 


•93809 


•36271 


•93190 


.37892 


•92543 


•39501 


.91868 


.41098 


.91164 


44 


17 


.34666 


•93799 


.36298 


.93180 


.37919 


.92532 


•39528 


.91856 


.41125 


.91152 


43 


i8 


.34694 


•93789 


■3632s 


.93169 


.37946 


•92521 


.39555 


.91845 


.41151 


.91140 


42 


19 


.34721 


•93779 


.36352 


.93159 


.37973 


•92510 


.39581 


•91833 


.41178 


•91128 


41 


20 


.34748 


.93769 


•36379 


.93148 


.37999 


•92499 


.39608 


.91822 


.41204 


.91116 


40 


21 


.34775 


•93759 


.36406 


.93137 


.38026 


.92488 


•3963s 


.91810 


.41231 


.91104 


39 


22 


.34803 


•93748 


•36434 


.93127 


.38053 


.92477 


.39661 


.91799 


.41257 


.91092 


38 


23 


.34830 


•93738 


•36461 


.93116 


.38080 


.92466 


.39688 


.91787 


.41284 


.91080 


37 


24 


.34857 


•93728 


.36488 


.93106 


.38107 


■92455 


.39715 


.91775 


.41310 


.91068 


36 


25 


.34884 


•93718 


•36515 


.93095 


.38134 


■92444 


.39741 


.91764 


.41337 


.91056 


35 


26 


.34912 


■93708 


•36542 


•93084 


.3S161 


.92432 


.39768 


.91752 


.41363 


.91044 


34 


27 


.34939 


•93698 


•36569 


•93074 


.38188 


.92421 


■39795 


.91741 


.41390 


.91032 


33 


28 


.34966 


.93688 


•36596 


•93063 


.3S21S 


.92410 


•39822 


.91729 


.41416 


.91020 


32 


29 


.34993 


•93677 


•36623 


•93052 


.38241 


.92399 


•39S48 


.91718 


.41443 


.91008 


31 


30 


.35021 


•93667 


.36650 


.93042 


.38268 


.92388 


.39875 


.91706 


•41469 


.90996 


30 


31 


.35048 


•93657 


•36677 


.93031 


.38295 


.92377 


.39902 


.91694 


•41496 


.909S4 


29 


32 


•35075 


•93647 


.36704 


.93020 


.38322 


.92366 


.39928 


.91683 


.41522 


.90972 


28 


33 


.35102 


•93637 


■36731 


.93010 


.38349 


.92355 


.39955 


.91671 


.41549 


.90960 


27 


34 


.35130 


•93626 


.36758 


•92999 


.38376 


.92343 


.39982 


.91660 


.41575 


.90948 


26 


35 


.35157 


•93616 


.36785 


.92988 


.38403 


.92332 


.40008 


.91648 


.41602 


.90936 


25 


36 


.35184 


•93606 


.36812 


.92978 


.38430 


.92321 


.40035 


.91636 


.41628 


.90924 


24 


37 


.35211 


•93596 


.36839 


.92967 


.38456 


.92310 


.40062 


.91625 


.41655 


.90911 


23 


38 


.35239 


•93585 


.36867 


.92956 


.38483 


.92299 


.40088 


.91613 


.41681 


.90899 


22 


39 


.35266 


•93575 


.36894 


.92945 


.38510 


.92287 


.40115 


.91601 


.41707 


.90887 


21 


40 


.35293 


•9356s 


.36921 


.92935 


.38537 


.92276 


.40141 


.91590 


.41734 


.9087s 


20 


41 


•35320 


•93555 


.36948 


.92924 


•38564 


.92265 


.40168 


.91578 


.41760 


.90863 


19 


42 


.35347 


•93544 


.36975 


.92913 


•38591 


.92254 


.40195 


.91566 


.41787 


.90851 


18 


43 


.35375 


•93534 


.37002 


.92902 


•38617 


.92243 


.40221 


.91555 


.41813 


.90839 


17 


44 


.35402 


•93524 


.37029 


.92S92 


•38644 


.92231 


.40248 


.91543 


.41840 


.90826 


16 


45 


•35429 


•93514 


.37056 


.92881 


•38671 


.92220 


•40275 


.91531 


.41866 


.90814 


IS 


46 


.35456 


•93503 


•37083 


.92870 


.3e698 


.92209 


.40301 


.91519 


.41892 


.90802 


14 


47 


.35484 


•93493 


•37110 


.92859 


.38-25 


.92198 


.40328 


.91508 


•41919 


.90790 


13 


48 


.35511 


•93483 


•37137 


.92849 


.38752 


.92186 


.4035s 


.91496 


•41945 


.90778 


12 


49 


.35538 


•93472 


•37164 


.9283S 


.38778 


.92175 


.40381 


.91484 


•41972 


.90766 


11 


50 


.35565 


•93462 


•37191 


.92827 


.38805 


.92164 


.40408 


.91472 


.41998 


■90753 


10 


51 


.35592 


■93452 


.37218 


.92816 


.38832 


.92152 


.40434 


.91461 


.42024 


•90741 


9 


52 


.35619 


•93441 


.37245 


.92805 


.3S859 


.92141 


.40461 


.91449 


.42051 


.90729 


8 


53 


.35647 


•93431 


.37272 


.92794 


.38S86 


.92130 


.40488 


.91437 


.42077 


.90717 


7 


54 


.35674 


•93420 


.37299 


.92784 


.3S912 


.92119 


.40514 


.91425 


.42104 


.90704 


6 


55 


.35701 


•93410 


■37326 


.92773 


.3S939 


.92107 


.40541 


.91414 


.42130 


.90692 


5 


56 


•35728 


.93400 


.37353 


.92762 


.3S966 


.92096 


.40567 


.91402 


.42156 


.90680 


4 


57 


•35755 


•93389 


.37380 


.92751 


•3S993 


.920S5 


.40594 


•91390 


.42183 


.90668 


3 


58 


•35782 


•93379 


.37407 


.92740 


.39020 


.92073 


.40621 


•91378 


.42209 


•9065s 


2 


59 


.35810 


.93368 


•37434 


.92729 


.39046 


.9::o63 


.40647 


•91366 


.42235 


.90643 


I 


60 


.35837 


•93358 


•37461 


.92718 


.39073 


.92050 


.40674 


•91355 


•42262 


.90631 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


6( 


3° 


6 


3° 


6 


7° 


6 


5° 


6 


5° 



156' 



NATURAL SINES AND COSINES 



/ 


25 


D 


26 





27 


D 


28 





29 





1 


Sine ( 


"osine 


Sine ( 


I^osine 


Sine ( 


l^^osine 


Sine 


Cosine 


Sine 


Cosine 





.42262 


.90631 


.43837 


.89879 


.45399 


.89101 


.46947 


.88295 


.48481 


.87462 


60 


I 


.42288 


.90618 


.43863 


.89867 


.45425 


.89087 


.46973 


.88281 


.48506 


.87448 


59 


2 


.42315 


.90606 


.43889 


.89854 


.45451 


.89074 


.46999 


.88267 


.48532 


•87434 


58 


3 


.42341 


.90594 


.43916 


.89841 


•45477 


.89061 


.47024 


.88254 


•48557 


.87420 


57 


4 


.42367 


.90582 


.43942 


.89828 


•45503 


.89048 


.47050 


.88240 


•48583 


.87406 


56 


S 


.42394 


.90569 


.43968 


.89816 


■45529 


.89035 


.47076 


.88226 


.48608 


.87391 


55 


6 


.42420 


.90557 


.43994 


.89803 


■45554 


.89021 


.47101 


.88213 


.48634 


.87377 


54 


7 


.42446 


•90545 


.44020 


.89790 


■45580 


.89008 


.47127 


.88199 


•48659 


.87363 


53 


8 


.42473 


.90532 


.44046 


.89777 


.45606 


.88995 


•47153 


.88185 


.48684 


.87349 


52 


9 


.42499 


.90520 


.44072 


.89764 


.45632 


.88981 


•47178 


.88172 


.48710 


.87335 


SI 


10 


.42525 


.90507 


.44098 


•89752 


.45658 


.88968 


•47204 


.88158 


.48735 


.87321 


50 


II 


.42552 


•90495 


.44124 


•89739 


.45684 


.8895s 


.47229 


.88144 


.48761 


.87306 


49 


12 


.42578 


•90483 


.44151 


.89726 


.45710 


.88942 


.47255 


.88130 


.48786 


.87292 


48 


13 


.42604 


.90470 


.44177 


•89713 


.45736 


.88928 


.47281 


.88117 


.48811 


.87278 


47 


14 


.42631 


•90458 


.44203 


.89700 


.45762 


.88915 


.47306 


.88103 


.48837 


.87264 


46 


IS 


.42657 


.90446 


.44229 


.89687 


.45787 


.88902 


.47332 


.88089 


.48862 


.87250 


45 


i6 


.42683 


.90433 


.44255 


.89674 


.45813 


.88888 


.47358 


.88075 


.48888 


.87235 


44 


17 


.42709 


.90421 


.44281 


.89662 


.45839 


.88875 


•47383 


.88062 


.48913 


.87221 


43 


i8 


.42736 


.90408 


.44307 


.89649 


.45865 


.88862 


.47409 


.88048 


.48938 


.87207 


42 


19 


.42762 


.90396 


.44333 


.89636 


.45891 


.88848 


.47434 


.88034 


.48964 


•87193 


41 


20 


.42788 


.90383 


.44359 


.89623 


•45917 


.88835 


.47460 


.88020 


.48989 


.87178 


40 


21 


.42815 


.90371 


.44385 


.89610 


•45942 


.88822 


.47486 


.88006 


.49014 


.87164 


39 


22 


.42841 


.90358 


.44411 


.89597 


•45968 


.88808 


.47511 


.87993 


.49040 


.87150 


38 


23 


.42867 


■90346 


•44437 


.89584 


•45994 


.88795 


•47537 


.87979 


.49065 


.87136 


37 


24 


.42894 


■90334 


.44464 


•89571 


.460-J 


.88782 


•47562 


.87965 


.49090 


.87121 


36 


25 


.42920 


.90321 


.44490 


•89558 


.46046 


.88768 


•47588 


■87951 


.49116 


.87107 


35 


26 


.42946 


.90309 


.44516 


•89545 


.46072 


•88755 


•47614 


■87937 


.49141 


.87093 


34 


27 


.42972 


.90296 


.44542 


•89532 


.46097 


.88741 


•47639 


.87923 


.49166 


.87079 


33 


28 


.42999 


.90284 


•44568 


•89519 


.46123 


.88728 


.47665 


■87909 


.49192 


.87064 


32 


29 


.43025 


.90271 


•44594 


•89506 


.46149 


.88715 


.47690 


.87896 


.49217 


.87050 


31 


30 


.43051 


.90259 


.44620 


•89493 


.4617s 


.88701 


.47716 


.87882 


.49242 


.87036 


30 


31 


.43077 


.90246 


.44646 


.89480 


.46201 


.88688 


.47741 


.87868 


.49268 


.87021 


29 


32 


.43104 


.90233 


• 44672 


.89467 


.46226 


.88674 


.47767 


.87854 


.49293 


.87007 


28 


33 


.43130 


.90221 


.44698 


.89454 


.46252 


.88661 


.47793 


.87840 


.49318 


.86993 


27 


34 


.43156 


.90208 


.44724 


.89441 


.46278 


.88647 


.47818 


.87826 


.49344 


.86978 


26 


35 


.43182 


.90196 


•44750 


.89428 


.46304 


.88634 


.47844 


.87812 


.49369 


.86964 


25 


36 


.43209 


.90183 


.44776 


.89415 


.46330 


.88620 


.47869 


.87798 


.49394 


.86949 


24 


37 


.43235 


.90171 


.44802 


.89402 


.46355 


.88607 


.47895 


.87784 


.49419 


.86935 


23 


38 


.43261 


.90158 


.44828 


.89389 


•46381 


.88593 


•47920 


.87770 


•49445 


.86921 


22 


39 


.43287 


.90146 


.44854 


.89376 


.46407 


.88580 


•47946 


.87756 


.49470 


.86906 


21 


40 


.43313 


.90133 


.44880 


.89363 


.46433 


.88566 


•47971 


.87743 


.49495 


.86892 


20 


41 


.43340 


.90120 


.44906 


.89350 


.46458 


•88553 


.47997 


.87729 


.49521 


.86878 


19 


42 


.43366 


.90108 


.44932 


.89337 


.46484 


•88539 


.48022 


.87715 


.49546 


.86863 


18 


43 


.43392 


.90095 


.44958 


.89324 


.46510 


.88526 


.48048 


.87701 


.49571 


.86849 


17 


44 


.43418 


.90082 


•44984 


.89311 


•46536 


.88512 


.48073 


.87687 


.49596 


.86834 


16 


45 


.43445 


.90070 


•45010 


.89298 


.46561 


.88499 


.48099 


.87673 


.49622 


.86820 


15 


46 


.43471 


■90057 


•45036 


•89285 


.46587 


.88485 


.48124 


.87659 


.49647 


.86805 


14 


47 


.43497 


■90045 


•45062 


.89272 


.46613 


.88472 


.48150 


.87645 


.49672 


.86791 


13 


48 


.43523 


■90032 


.45088 


.89259 


.46639 


.88458 


.48175 


.87631 


.49697 


.86777 


12 


49 


.43549 


.90019 


•45114 


.89245 


.46664 


.88445 


.48201 


.87617 


.49723 


.86762 


11 


SO 


.43575 


.90007 


.45140 


.89232 


.46690 


.88431 


.48226 


.87603 


.49748 


.86748 


10 


51 


.43602 


.89994 


.45166 


.89219 


.46716 


.88417 


.48252 


.87589 


.49773 


.86733 


9 


52 


.43628 


.89981 


•45192 


.89206 


.46742 


.88404 


.48277 


•87575 


.49798 


.86719 


8 


53 


.43654 


.89968 


•45218 


.89193 


.46767 


.88390 


•48303 


•87561 


.49824 


.86704 


7 


54 


.43680 


.89956 


•45243 


.89180 


.46793 


.88377 


.48328 


•87546 


.49849 


.86690 


6 


SS 


.43706 


.89943 


•45269 


.89167 


.46819 


.88363 


■48354 


•87532 


.49874 


.86675 


S 


S6 


•43733 


.89930 


.45295 


•89153 


.46844 


.88349 


■48379 


•87518 


.49899 


.86661 


4 


57 


.43759 


.89918 


.45321 


.89140 


.46870 


.88336 


■48405 


.87504 


.49924 


.86646 


3 


58 


•4378s 


.89905 


.45347 


.89127 


.46896 


.88322 


■48430 


•87490 


.49950 


.86632 


2 


59 


.43811 


.89892 


.45373 


.89114 


.46921 


.88308 


■48456 


.87476 


.49975 


.86617 


I 


60 


.43837 


.89879 


•45399 


.89101 


.46947 


.88295 


.48481 


.87462 


.50000 


.86603 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


6z 


\° 


62 





62 





6] 





6c 


° 



NATURAL vSINES AND COSINES 



157 



/ 


30 





31 


3 


32 





33 





34 


3 


/ 


Sine 


Cosine 


Sine C 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine ( 


.Cosine 





.50000 


.86603 


.51504 


85717 


.52992 


.84805 


•S4464 


•83867 


•55919 


82904 


60 


I 


.50025 


.86588 


•51529 


85702 


•53017 


.84789 


•54488 


•83851 


•55943 


82887 


59 


2 


.50050 


.86573 


•51554 


85687 


•53041 


•84774 


•S45I3 


•83835 


•55968 


82871 


58 


3 


.50076 


■86559 


• 5 1 579 


85672 


•53066 


•84759 


•54537 


•83819 


•5.S992 


82855 


57 


4 


.50101 


.86544 


•51604 


85657 


•53091 


•84743 


•54561 


.83804 


•56016 


82839 


56 


S 


.50126 


.86530 


.51628 


85642 


•S3II5 


.84728 


.5-!586 


.83788 


.56040 


82822 


55 


6 


.S0151 


.86515 


•51653 


85627 


•53140 


.84712 


.54610 


.83772 


.56064 


82806 


54 


7 


.50176 


.86501 


.51678 


85612 


•53164 


.84697 


.54635 


•83756 


.56088 


82790 


53 


8 


.50201 


.86486 


•51703 


85597 


•53189 


.84681 


.54659 


.83740 


.56112 


82773 


52 


9 


.50227 


.86471 


•51728 


85582 


.53214 


.84666 


.54683 


•83724 


.56136 


82757 


51 


10 


.50252 


.86457 


•S1753 


85567 


•53238 


.84650 


•54708 


•83708 


.56160 


82741 


50 


II 


■50277 


.86442 


•51778 


85551 


.53263 


.84635 


• 54732 


•83692 


.56184 


82724 


49 


12 


.50302 


.86427 


•51803 


85536 


•53288 


.84619 


.54756 


•83676 


.56208 


82708 


48 


13 


■50327 


.86413 


.51828 


85521 


•53312 


.84604 


.54781 


.83660 


.56232 


82692 


47 


14 


.50352 


.86398 


.51852 


85506 


•53337 


.84588 


•54805 


•83645 


.56256 


82675 


46 


IS 


.50377 


.86384 


.51877 


85491 


•53361 


.84573 


•54829 


•83629 


.56280 


82659 


45 


i6 


•S0403 


.86369 


.51902 


85476 


•53386 


.84557 


•54854 


•83613 


•56305 


82643 


44 


17 


.50428 


■86354 


.51927 


85461 


■5341 1 


.84542 


•54878 


•83597 


•56329 


82626 


43 


i8 


.50453 


.86340 


.51953 


85446 


•53435 


.84526 


-54902 


.83581 


•56353 


82610 


42 


19 


.50478 


.86325 


•51977 


85431 


.53460 


.84511 


.54927 


•83565 


•56377 


82593 


41 


20 


.50503 


.86310 


.52002 


85416 


•53484 


•8449s 


.54951 


•83549 


•56401 


8257f 


40 


21 


.50528 


.86295 


.52026 


85401 


•53509 


.84480 


•54975 


•83533 


•56425 


82561 


39 


22 


.50553 


.86281 


.52051 


85385 


•53534 


.84464 


•54999 


•83517 


■56449 


82544 


38 


23 


•50578 


.86266 


.52076 


85370 


•53558 


.84448 


•55024 


.83501 


•56473 


82528 


37 


24 


.50603 


.86251 


.52101 


85355 


•53583 


.84433 


•SS048 


.83485 


• 56497 


82511 


36 


2S 


.50628 


.86237 


.52126 


85340 


•S3607 


.84417 


•SS072 


.83469 


•56521 


82495 


35 


26 


.50654 


.86222 


.52151 


85325 


•53632 


.84402 


•55097 


.83453 


•56545 


82478 


34 


27 


.50679 


.86207 


.5217s 


85310 


•53656 


.84386 


•55121 


.83437 


•56569 


82462 


33 


28 


.50704 


.86192 


.52200 


85294 


•53681 


.84370 


•S5I45 


.83421 


•56593 


82446 


32 


29 


.50729 


.86178 


.52225 


85279 


•53705 


.84355 


•55169 


•83405 


■56617 


82429 


31 


30 


.50754 


.86163 


.52250 


85264 


•53730 


.84339 


•55194 


■83389 


.56641 


82413 


30 


31 


■50779 


.86148 


•S2275 


85249 


•53754 


.84324 


•55218 


.83373 


.56665 


82396 


29 


32 


.50804 


.86133 


•52299 


85234 


•53779 


.84308 


•55242 


.83356 


.56689 


82380 


28 


33 


.50829 


.86119 


•52324 


85218 


•53804 


.84292 


•55266 


.83340 


.56713 


82363 


27 


34 


.50854 


.86104 


•52349 


85203 


.53828 


.84277 


•55291 


.83324 


.56736 


82347 


26 


3S 


.50879 


.86089 


•52374 


85188 


•53853 


.84261 


.55315 


.83308 


.56760 


82330 


25 


36 


.50904 


.86074 


•52399 


85173 


.53877 


.84245 


■55339 


■83292 


.56784 


82314 


24 


37 


.50929 


.86059 


•52423 


85157 


.53902 


•84230 


.55363 


■83276 


.56808 


82297 


23 


38 


.50954 


.86045 


•52448 


85142 


.53926 


.84214 


.553S8 


.83260 


.56832 


82281 


22 


39 


.50979 


.86030 


•52473 


85127 


.53951 


•84198 


.55412 


.83244 


.56856 


82264 


21 


40 


.51004 


.86015 


•52458 


85112 


•53975 


.84182 


.55436 


.83228 


.56880 


82248 


20 


41 


■.51029 


.86000 


•52522 


85096 


.54000 


.84167 


.55460 


.83212 


.56904 


82231 


19 


42 


.51054 


.859S5 


•52547 


85081 


.54024 


•84151 


.55484 


.83195 


.56928 


82214 


18 


43 


.51079 


.85970 


•52572 


85066 


•54049 


•84135 


.55509 


.83179 


•56952 


82198 


17 


44 


.51104 


•85956 


•52597 


85051 


•54073 


.84120 


.55533 


.83163 


•56976 


82181 


16 


45 


.51129 


•85941 


•52621 


85035 


•54097 


.84104 


•5S5S7 


■83147 


•57000 


8216s 


15 


46 


.51154 


.85926 


.52646 


85020 


•54122 


.840S8 


•5SS8i 


■83131 


•57024 


82148 


14 


47 


■51179 


•8591 1 


.52671 


85005 


•54146 


.84072 


•55605 


■8311S 


•57047 


82132 


13 


48 


.51204 


.85896 


.52696 


84989 


•54171 


.84057 


•55630 


.83098 


•57071 


82115 


12 


49 


.51229 


.85881 


.52720 


84974 


•54195 


.84041 


•55654 


.83082 


■57095 


82098 


II 


50 


■5I2S4 


.85866 


.52745 


84959 


.54220 


.84025 


•55678 


.83066 


.57119 


S2082 


10 


SI 


■51279 


.85851 


•52770 


84943 


.54244 


.84009 


,55702 


.83050 


•57143 


82065 


9 


52 


■51304 


.85836 


•52794 


84928 


.54269 


.83994 


•55726 


•83034 


■57167 


82048 


8 


S3 


•SI 329 


.85821 


.52819 


84913 


.54293 


.83978 


•55750 


.83017 


.57191 


82032 


7 


54 


•S1354 


.85806 


.52844 


84897 


.54317 


.83962 


•55775 


.83001 


■57215 


82015 


6 


55 


.51379 


.85792 


.52869 


84882 


.54342 


.83946 


•55799 


.82985 


.57238 


81999 


5 


S6 


■51404 


•85777 


.52893 


84866 


.54366 


.83930 


•55823 


.82969 


.57262 


81982 


4 


57 


•51429 


•85762 


.52918 


84851 


•54391 


.83915 


■55847 


.82953 


.57286 


8196s 


3 


S8 


.51454 


•85747 


.52943 


84836 


-5441S 


.83899 


•55871 


.82936 


■57310 


81949 


2 


59 


•S1479 


•85732 


.52967 


84820 


■54440 


.83883 


•55895 


.82920 


■57334 


81932 


I 


60 


■S1504 


.85717 


.52992 


84805 


.54464 


.83867 


.55919 


.82904 


■57358 


81915 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


59 





58° 


57 





S6 





55° 



158 



NATURAL SINES AND COSINES 



/ 


35 





36 





37° 


38° 


39° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





.57358 


.81915 


.58779 


.80902 


.60182 


.79864 


.61566 


.78801 


.62932 


.77715 


60 


I 


.57381 


.81899 


.5 802 


.80885 


.60205 


.79846 


.61589 


.78783 


■62955 


.77696 


59 


2 


.57405 


.81882 


■ 5 826 


.80867 


.60228 


.79829 


.61612 


.78765 


.62977 


.77678 


S8 


3 


.57429 


.81865 


■58C49 


.80850 


.60251 


.79811 


.61J35 


.78747 


.63000 


.77660 


57 


4 


.57453 


.81848 


.58873 


.80833 


.60274 


.79793 


.61658 


.78729 


.63022 


.77641 


56 


S 


.57477 


.81832 


.58896 


.80816 


.60298 


.79776 


.61681 


.78711 


.63045 


.77623 


55 


6 


.57501 


.8181S 


.58920 


.80799 


.60321 


.79758 


.61704 


.78694 


.63068 


.77605 


54 


7 


.57524 


.81798 


.58943 


.80782 


.60344 


.79741 


.61726 


.78676 


.63090 


.77586 


53 


8 


.57548 


.81782 


.58967 


.80765 


.60367 


.79723 


.61749 


.78658 


.63113 


.77568 


52 


9 


.57572 


.81765 


.58990 


.80748 


.60390 


.79706 


.61772 


.78640 


.6313s 


.77550 


51 


10 


.57596 


.81748 


.59014 


.80730 


.60414 


.79688 


.61795 


.78622 


.63158 


.77531 


50 


II 


.57619 


.81731 


.59037 


.80713 


.60437 


.79671 


.61818 


.78604 


.63180 


.77513 


49 


12 


.57643 


.81714 


.59061 


.80696 


.60460 


.79653 


.61841 


.78586 


.63203 


.77494 


48 


13 


.57667 


.81698 


.59084 


.80679 


.60483 


.79635 


.61864 


.78568 


.63225 


.77476 


47 


14 


.57691 


.81681 


.59108 


.80662 


.60506 


.79618 


.61887 


.78550 


.63248 


.77458 


46 


IS 


•5771S 


.81664 


■59131 


.80644 


.60529 


.79600 


.61909 


.78532 


.63271 


.77439 


45 


i6 


.57738 


.81647 


.59154 


.80627 


.60553 


•79583 


.61932 


.78514 


■63293 


.77421 


44 


17 


.57762 


.81631 


.59178 


.80610 


.60576 


.79565 


.61955 


.78496 


.63316 


.77402 


43 


l8 


.57786 


.81614 


.59201 


.80593 


.60599 


-79547 


.61978 


.78478 


.63338 


.77384 


42 


19 


.57810 


.81597 


.59225 


.80576 


.60622 


.79530 


.62001 


.78460 


.63361 


.77366 


41 


20 


■57833 


.81580 


■59248 


.80558 


.60645 


.79512 


.62024 


.78442 


.63383 


.77347 


40 


21 


.57857 


.81563 


.59272 


.80541 


.60668 


.79494 


.62046 


.78424 


.63406 


.77329 


39 


22 


.57881 


.81546 


.59295 


.80524 


.60691 


.79477 


.62069 


.78405 


.63428 


.77310 


38 


23 


.57904 


.81530 


.59318 


.80507 


.60714 


.79459 


.62092 


.78387 


.63451 


.77292 


37 


24 


.57928 


.81513 


.59342 


.80489 


.60738 


.79441 


.62115 


.78369 


.63473 


.77273 


36 


25 


■57952 


.81496 


.59365 


.80472 


.60761 


.79424 


.62138 


.78351 


.63496 


.77255 


35 


26 


.57976 


.81479 


■59389 


.80455 


.60784 


.79406 


.62160 


.78333 


.63518 


.77236 


34 


27 


.57999 


.81462 


.59412 


.80438 


.60807 


.79388 


.62183 


.7831S 


.63540 


.77218 


2Z 


28 


.58023 


.81445 


.59436 


.80420 


.60830 


.79371 


.62206 


.78297 


.63563 


.77199 


32 


29 


.58047 


.81428 


.59459 


.80403 


.60853 


.79353 


.62229 


.78279 


.6358s 


.77181 


31 


30 


.58070 


.81412 


.59482 


.80386 


.60876 


.79335 


.62251 


.78261 


.63608 


.77162 


30 


31 


.58094 


.81395 


.59506 


.80368 


.60899 


.79318 


.62274 


.78243 


.63630 


.77144 


29 


32 


.58118 


.81378 


.59529 


.83351 


.60922 


.79300 


.62297 


.78225 


.63653 


.77125 


28 


33 


.58141 


.81361 


.59S.S2 


.80334 


.60945 


.79282 


.62320 


.78206 


.63675 


.77107 


27 


34 


.58165 


.81344 


.59576 


.80316 


.60968 


.79264 


.62342 


.78188 


.63698 


.77088 


26 


3S 


.58189 


.81327 


.59599 


.80299 


.60991 


.79247 


.62365 


.78170 


.63720 


.77070 


25 


36 


.58212 


.81310 


.59622 


.80282 


.61015 


.79229 


.62388 


.78152 


.63742 


.77051 


24 


37 


.58236 


.81293 


.59646 


.80264 


.61038 


.79211 


.62411 


.78134 


.63765 


.77033 


23 


38 


.58260 


.81276 


.59669 


.80247 


.61061 


.79193 


.62433 


.78116 


.63787 


.77014 


22 


39 


.58283 


.81259 


.59693 


.80230 


.61084 


.79176 


.62456 


.78098 


.63810 


.76996 


21 


40 


.58307 


.81242 


.59716 


.80212 


.61107 


.79158 


.62479 


.78079 


.63832 


.76977 


20 


41 


■58330 


.81225 


.59739 


.80195 


.61130 


.79140 


.62502 


.78061 


.63854 


.76959 


19 


42 


•58354 


.81208 


.59763 


.80178 


.61153 


.79122 


.62524 


.78043 


.63877 


.76940 


18 


43 


.58378 


.81191 


.59786 


.80160 


.61176 


.79105 


.62547 


.78025 


.63899 


.76921 


17 


44 


.58401 


.81174 


.59809 


.80143 


.61199 


.79087 


■62570 


.78007 


.63922 


.76903 


16 


45 


.58425 


.81157 


.59832 


.80125 


.61222 


.79069 


.62592 


.77988 


.63944 


.76884 


IS 


46 


.58449 


.81140 


.59856 


.80108 


.61245 


.79051 


.62615 


.77970 


.63966 


.76866 


14 


47 


.58472 


.81123 


■59879 


.80091 


.61268 


.79033 


.62638 


.77952 


.63989 


.76847 


13 


48 


.58496 


.81106 


.59902 


.80073 


.61291 


.79016 


.62660 


.77934 


.64011 


.76828 


12 


49 


.5S519 


.81089 


.59926 


.80056 


.61314 


.78998 


.62683 


.77916 


.64033 


.76810 


11 


50 


.38543 


.81072 


.59949 


.80038 


.61337 


.78980 


.62706 


.77897 


.64056 


.76791 


10 


51 


.58567 


.810SS 


.59972 


.80021 


.61360 


.78962 


.62728 


.77879 


.64078 


.76772 


9 


S2 


.58590 


.81038 


.59995 


.80003 


.61383 


.78944 


.62751 


.77861 


.64100 


.76754 


8 


S3 


.58614 


.81021 


.60019 


.79986 


.61406 


.78926 


.62774 


.77843 


.64123 


.76735 


7 


54 


.58637 


.81004 


.60042 


.79968 


.61429 


.78908 


.62796 


.77824 


.64145 


.76717 


6 


55 


.36661 


.80987 


.60065 


•799SI 


.61451 


.78891 


.62819 


.77806 


.64167 


.76698 


5 


S6 


.58684 


.80970 


.60089 


.79934 


.61474 


■78873 


.62842 


.77788 


.64190 


.76679 


4 


57 


.58708 


■80953 


.60112 


.79916 


.61497 


.78855 


.62864 


.77769 


.64212 


.76661 


3 


S8 


.58731 


.80936 


.60135 


.79899 


.61520 


.78837 


.62887 


.77751 


.64234 


.76642 


2 


59 


.58755 


.80919 


.60158 


.79881 


.61543 


.78819 


.62909 


.77733 


.64256 


.76623 


I 


60 


.58779 


.80902 


.60182 


.79864 


.61566 


.78801 


.62932 


.77715 


.64279 


.76604 





t 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


54 





53 





5- 


2° 


5 


[° 


5 


3° 



NATURAL SINES AND COSINES 



159 



/ 


40° 


41 





42 





43° 


44° 


1 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine C 


osine 





.64279 


.76604 


.65606 


.75471 


.66913 


.74314 


.68200 


•7313s 


.69466 


71934 


60 


I 


.64301 


.76586 


.65628 


.75452 


.66935 


.74295 


.68221 


.73116 


.69487 


71914 


59 


2 


.64323 


.76567 


.65650 


.75433 


.66956 


•74276 


.68242 


.73096 


.69508 


71894 


58 


3 


.64346 


.76548 


.65672 


.75414 


.66978 


.74256 


.68264 


.73076 


.69529 


71873 


57 


4 


.64368 


•76530 


.65694 


.75395 


.66999 


.74237 


.68285 


.73056 


.69549 


71853 


56 


S 


.64390 


.76511 


.65716 


.75375 


.67021 


.74217 


.68306 


.73036 


.69570 


71833 


55 


6 


.64412 


.76492 


.65738 


.75356 


•67043 


.74198 


.68327 


.73016 


.69591 


71813 


54 


7 


.64435 


.76473 


■65759 


.75337 


.67064 


.74178 


.68349 


.72996 


.69612 


71792 


53 


8 


.64457 


•76455 


.65781 


.75318 


.67086 


.74159 


.68370 


.72976 


.69633 


71772 


52 


9 


.64479 


.76436 


.65803 


.75299 


.67107 


.74139 


.68391 


.72957 


■69654 


71752 


51 


10 


.64501 


•76417 


.65825 


.75280 


.67129 


.74120 


.68412 


.72937 


.6967s 


71732 


50 


II 


.64524 


.76398 


.65847 


.75261 


.67151 


.74100 


.68434 


.72917 


.69696 


71711 


49 


12 


.64546 


.76380 


.65869 


.75241 


.67172 


.74080 


.68455 


.72897 


.69717 


71691 


48 


13 


.64568 


.76361 


.65891 


.75222 


.67194 


.74061 


.68476 


.72877 


.69737 


71671 


47 


14 


.64590 


.76342 


.65913 


.75203 


.6721S 


.74041 


.68497 


.72857 


.69758 


71650 


46 


IS 


.64612 


•76323 


.65935 


.75184 


.67237 


.74022 


.68518 


.72837 


.69779 


71630 


45 


i6 


.64635 


.76304 


.65956 


.75165 


.67258 


.74002 


.68539 


.72817 


.69800 


71610 


44 


17 


.64657 


.76286 


.65978 


•75146 


.67280 


•73983 


.68561 


.72797 


.69821 


71590 


43 


i8 


.64679 


.76267 


.66000 


.75126 


.67301 


•73963 


.68582 


•72777 


.6:842 


71569 


42 


19 


.64701 


.76248 


.66022 


•75107 


.67323 


.73944 


.68603 


.72757 


.69C62 


71549 


41 


20 


.64723 


.76229 


.66044 


.75088 


.67344 


.73924 


.68624 


.72737 


.69883 


71529 


40 


21 


.64746 


.76210 


.66066 


.75069 


.67366 


.73904 


.68645 


.72717 


.69904 


71508 


39 


22 


.64768 


.76192 


.66088 


.75050 


.67387 


.73885 


.68666 


.72697 


.69925 


71488 


38 


23 


•64790 


.76173 


.66109 


.75030 


.67409 


.7386s 


.68688 


■72677 


.69946 


71468 


37 


24 


.64812 


.76154 


.66131 


.75011 


.67430 


.73846 


.68709 


.72657 


.69966 


71447 


36 


2S 


.64834 


.76135 


.66153 


.74992 


.67452 


.73826 


.68730 


.72637 


.69987 


71427 


35 


26 


.64856 


.76116 


.66175 


.74973 


.67473 


.73806 


.68751 


.72617 


.70008 


71407 


34 


27 


.64878 


.76097 


.66197 


•74953 


.67495 


.73787 


.6G772 


.72597 


.70029 


71386 


33 


28 


.64901 


.76078 


.66218 


.74934 


.67516 


•73767 


.68793 


.72577 


.70049 


71366 


32 


29 


.64^23 


•76059 


.66240 


.74915 


.67538 


•73747 


.683 1 4 


.72557 


.70070 


7134s 


31 


30 


.64945 


.76041 


.66262 


.74896 


.67559 


.73728 


.6GG3S 


.72:37 


.70091 


71325 


30 


31 


.64967 


.76022 


.66284 


.74876 


.67580 


.73708 


.68857 


.72517 


.70112 


7 1 30s 


29 


32 


.64989 


.76003 


.66306 


.74857 


.67602 


.73688 


.68878 


.72497 


.70132 


71284 


28 


33 


.65011 


.75984 


.66327 


.74838 


.67623 


.73669 


.68899 


.72477 


.70153 


71264 


27 


34 


.65033 


■75965 


.66349 


.74818 


.67645 


.73649 


.68920 


.72457 


.70174 


71243 


26 


35 


•65055 


.75946 


.66371 


.74799 


.67666 


.73629 


.68941 


.72437 


.70195 


71223 


25 


36 


.65077 


.75927 


.66393 


.74780 


.67688 


.73610 


.68962 


.72417 


.70215 


71203 


24 


37 


.65100 


.75908 


.66414 


.74760 


.67709 


.73590 


.68983 


.72397 


.70236 


71182 


23 


38 


.65122 


.75889 


.66436 


.74741 


.67730 


.73570 


.69004 


.72377 


.70257 


71162 


22 


39 


.65144 


.75870 


.66458 


.74722 


.67752 


.73551 


.69025 


.72357 


.70277 


71141 


21 


40 


.65166 


.75851 


.66480 


•74703 


.67773 


.73531 


.69046 


.72337 


.70298 


71121 


20 


41 


.65188 


.75832 


.66501 


.74683 


.67795 


.73511 


.69067 


.72317 


.70319 


71 100 


19 


42 


.65210 


.75813 


.66523 


.74664 


.67816 


.73491 


.69088 


.72297 


•70339 


71080 


18 


43 


.65232 


.75794 


•66545 


.74644 


.67837 


.73472 


.69109 


.72277 


.70360 


71059 


17 


44 


.65254 


.75775 


.66566 


.74625 


.67859 


.73452 


.69130 


.72257 


.70381 


71039 


16 


45 


.65276 


.75756 


.66588 


.74606 


.67880 


.73432 


.69151 


.72236 


.70401 


71019 


IS 


46 


.65298 


.75738 


66610 


.74586 


.67901 


.73413 


.69172 


.72216 


.70422 


70998 


14 


47 


.65320 


•75719 


.66632 


•74567 


.67923 


.73393 


.69193 


.72196 


.70443 


70978 


13 


48 


.65342 


•75700 


.66653 


■74548 


.67944 


.73373 


.69214 


.72176 


.70463 


70957 


12 


49 


•65364 


.75680 


.66675 


.74528 


.67965 


.73353 


.69235 


.72156 


.70484 


70937 


II 


SO 


.65386 


.75661 


.66697 


.74509 


.67987 


.73333 


.69256 


.72136 


.70505 


70916 


10 


51 


.65408 


.75642 


.66718 


.74489 


.68008 


.73314 


.69277 


.72116 


.70525 


70896 


9 


52 


.65430 


.75623 


.66740 


.74470 


.68029 


.73294 


.69298 


.7209s 


.70546 


7087s 


8 


S3 


•65452 


.75604 


.66762 


.7445r 


.68051 


.73274 


.69319 


.72075 


.70567 


70855 


7 


54 


.65474 


.755S5 


.66783 


.74431 


.68072 


.73254 


.69340 


.72055 


.70587 


70B34 


6 


55 


.65496 


.75566 


.66805 


.74412 


.68093 


.73234 


.69361 


.72035 


.70608 


70813 


S 


S6 


.65518 


•75547 


.66827 


.74392 


.68115 


.73215 


.69382 


.72015 


.70628 


70793 


4 


S7 


.65540 


.75528 


.66848 


.74373 


.68136 


.73195 


.69403 


.71995 


.70649 


70772 


3 


58 


.65562 


•75509 


.66870 


.74353 


.68157 


.'7317s 


.69424 


.71974 


.70670 


70752 


2 


59 


.65584 


•75490 


.66891 


.74334 


.68179 


.73155 


.69445 


.71954 


.70690 


70731 


I 


60 


.65606 


.75471 


.66913 


.74314 


.68200 


.73135 


.69466 


.71934 


.70711 


7071 1 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


4 


3° 


4i 


i° 


4 


7° 


4< 


5° 


45^ 


3 



Natural Tangents and Cotangents. 



International Correspondence Schools. 



NATURAL TANGENTS AND COTANGENTS 



IGl 



/ 


O 







I 


2 





3 





4° 


/ 


Tangr 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 





.00000 


Infinite 


.01746 


37.2900 


.03492 


28.6363 


.05241 


19.0811 


.06993 


14.3007 


60 


I 


.ooo^g 


3437.75 


.01775 


36.3306 


.03521 


28.3994 


.05270 


18.9755 


.07022 


14.2411 


S9 


2 


.00058 


1718.87 


.01804 


53.4415 


.03550 


28.1664 


.05299 


18.8711 


.07051 


14.1821 


58 


3 


.00087 


1143.92 


.01833 


54.3613 


.03579 


27.9372 


.05328 


18.7678 


.07080 


14.123s 


57 


4 


.00116 


859.436 


.01862 


3'3.7o86 


.03609 


27.7117 


.05357 


18.6656 


.07110 


14.0653 


56 


S 


.00143 


687.549 


.01891 


52.8821 


.03638 


27.4899 


.05387 


18.5643 


.07139 


14.0079 


35 


6 


.0017s 


572.957 


.01920 


52.0807 


.03667 


27.271S 


.05416 


18.4643 


.07168 


13.9507 


34 


7 


.00204 


491.106 


.01949 


31.3032 


.03696 


27.0566 


.05445 


18.3655 


.07197 


13.8940 


53 


8 


.00233 


429.718 


.01978 


50.5485 


.03725 


26. 84 50 


.03474 


18.2677 


.07227 


13.8378 


52 


9 


.00262 


381.971 


.02007 


49.8157 


.03754 


26.6367 


.03503 


18.1708 


.07256 


13.7821 


31 


10 


.00291 


343-774 


.02036 


49.1039 


.03783 


26.4316 


.05333 


18.0750 


.07285 


13.7267 


50 


II 


.00320 


312.521 


.02066 


48.4121 


.03812 


26.2296 


.05562 


17.9802 


.07314 


13.6719 


49 


12 


.00349 


286.478 


.02095 


47-7395 


.03842 


26.0307 


.05391 


17.8863 


.07344 


13.6174 


48 


13 


.00378 


264.441 


.02124 


47.0853 


.03871- 


25.8348 


.05620 


17.7934 


.07373 


13.5634 


47 


14 


.00407 


245.552 


.02153 


46.4489 


.03900 


25.6418 


.05649 


17.701S 


.07402 


13.5098 


46 


IS 


.00436 


229.182 


.02182 


45.8294 


.03929 


25.4517 


.05678 


17.6106 


.07431 


13.4566 


43 


i6 


.00465 


214.858 


.02211 


45-2261 


.03958 


25.2644 


.05708 


17.5205 


.07461 


13-4039 


44 


17 


.00495 


202.219 


.02240 


44-6386 


.03987 


25.0798 


.05737 


17.4314 


.07490 


13-351S 


43 


i8 


.00524 


190.984 


.02269 


44-0661 


.04016 


24.8978 


.05766 


17.3432 


.07319 


13-2096 


42 


19 


.005S3 


180.932 


.02298 


43-5081 


.04046 


24.7185 


.0579s 


17.2558 


.07548 


13-2480 


41 


20 


.00582 


171.885 


.02328 


42.9641 


.04073 


24.5418 


.05824 


17.1693 


•07578 


13.1969 


40 


21 


.00611 


163.700 


.02337 


42.4335 


.04104 


24.367s- 


.05854 


17.0837 


.07607 


13.1461 


39 


23 


.00640 


136.259 


.02386 


41.9158 


.04133 


24.1957 


.05883 


16.9990 


.07636 


13.0958 


38 


23 


.00669 


149-465 


.02415 


41.4106 


.04162 


24.0263 


.05912 


16.9150 


.07665 


13.0458 


37 


24 


.00698 


143-237 


.02444 


40.9174 


.04191 


23-8393 


.05941 


16.8319 


.07695 


12.9962 


36 


25 


,00727 


137.507 


.02473 


40.4338 


.04220 


23.6943 


.05970 


16.7496 


-07724 


12.9469 


33 


26 


.00756 


132.219 


.02502 


39.965s 


.04230 


23-5321 


.05999 


16.6681 


.07753 


12.8981 


34 


2/ 


.0078s 


127.321 


.02531 


39.5059 


.04279 


23.3718 


.06029 


16.5874 


.07782 


12.8496 


33 


28 


.0081 S 


122.774 


.02560 


39.0568 


.04308 


23.2137 


.06058 


16-5075 


.07812 


12.8014 


32 


29 


.00844 


218.340 


.02589 


38.6177 


.04337 


23.0577 


.06087 


16.4283 


.07841 


12-7536 


31 


30 


.00873 


114.589 


.02^19 


38.1885 


.04366 


22.9038 


.06116 


16.3499 


-07870 


12.7062 


30 


31 


.00902 


110.892 


.02648 


37.7686 


.04395 


22.7519 


.06145 


16.2722 


.07899 


12.6591 


29 


32 


.00931 


107.426 


.02677 


37.3579 


.04424 


22.6020 


.06173 


16.1952 


.07929 


12.6124 


28 


33 


.00960 


104.171 


.02706 


36.9560 


.04454 


22.4541 


.06204 


16.1190 


.07958 


12.5660 


27 


34 


.00989 


101.107 


.02735 


36.5627 


.04483 


22.3081 


.06233 


16.0435 


.07987 


12.3199 


26 


33 


.01018 


98.2179 


.02764 


36.1776 


.04512 


22.1640 


.06262 


15.9687 


.08017 


12.4742 


25 


36 


.01047 


93.4895 


.02793 


35.8006 


.04341 


22.0217 


.06291 


15-8945 


.08046 


12.4288 


24 


37 


.01076 


92.9085 


.02822 


35.4313 


.04370 


21.8813 


.06321 


15.8211 


.0807s 


12.3838 


23 


33 


.01103 


go.4633 


.02831 


33.0693 


.04599 


i 1.7426 


.06350 


iS-7483 


.08104 


12.3390 


22 


39 


.01133 


88.1436 


.02881 


34.7151 


.04628 


21.6056 


.06379 


15.6762 


.08134 


12.2946 


21 


40 


.01164 


85.9398 


.02910 


34-3678 


.04638 


21.4704 


.06408 


15.6048 


.08163- 


12.250S 


20 


41 


.01193 


83.8435 


.02939 


34.0273 


.04687 


21.3369 


•06437 


IS. 3340 


.08192 


12.2067 


19 


42 


.01222 


81.8470 


.02968 


33.6933 


.04716 


21.2049 


.06467 


13.4638 


.08221 


12.1632 


18 


43 


.01231 


79.9434 


.02997 


33.3662 


.04743 


21.0747 


.06496 


15.3943 


.08251 


12.1201 


17 


44 


.01280 


78.1263 


.03026 


33.0452 


.04774 


20.9460 


.06525 


15.3254 


.08280 


12.0772 


16 


45 


.01309 


76.3900 


.03033 


32.7303 


.04803 


20.8188 


•06554 


15-2571 


.08309 


12.0346 


15 


46 


.01338 


74.7292 


.03084 


32.4213 


■04833 


20.6932 


.06584 


15.1893 


.08339 


11.9923 


14 


47 


.01367 


73.1390 


.03114 


32.1181 


.04862 


20.5691 


.06613 


15.1222 


.08368 


11.9504 


13 


48 


.01396 


71.6151 


.03143 


31.8205 


.04891 


20.4463 


.06642 


15.0557 


.08397 


11.9087 


12 


49 


.014:3 


70.1533 


.03172 


31.3284 


.04920 


20.3253 


.06671 


14-9898 


.08427 


11.8673 


11 


SO 


.01453 


68.7501 


.03201 


31.2416 


.04949 


20.2056 


.06700 


14.9244 


.08456 


11.8262 


10 


SI 


.01484 


67.4019 


.03230 


30.9599 


.04978 


20.0872 


.06730 


14.8596 


.08485 


11.7853 


9 


52 


.01513 


66.1055 


.03259 


30.6833 


.05007 


19.9702 


.06759 


14-7954 


.08514 


11.7448 


8 


53 


.01542 


64.8580 


.03288 


30.4116 


.05037 


19.8546 


.06788 


14-7317 


.08544 


11.7043 


7 


54 


.01571 


63.6567 


•03317 


30.1446 


.05066 


19.7403 


.06817 


14.6685 


.08573 


11.664s 


6 


55 


.01600 


62.4992 


.03346 


29.8825 


.05095 


19.6273 


.06847 


14.6059 


.08602 


11.6248 


5 


S6 


.01629 


61.3829 


.03376 


29.6245 


.05124 


19.3136 


.06876 


14.5438 


.08632 


11-5833 


4 


^ 


.01658 


60.3058 


.03405 


29.3711 


.05153 


19.4051 


.06905 


14.4823 


.08661 


11.5461 


3 


58 


.01687 


59.2659 


.03434 


29.1220 


.05182 


19.2959 


.06934 


14.4212 


.08690 


11.5072 


2 


59 


.01716 


58.2612 


.03463 


28.8771 


.05212 


19.1879 


.06963 


14.3607 


.08720 


11.4685 


I 


6o 


.01746 


57.2900 


.03492 


28.6363 


.05241 


19.0811 


.06993 


14.3007 


.08749 


11.4301 
Tang 





f 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


/ 


8( 


3° 


88° 


8; 


7° 


8( 


5° 


8 


-0 



162 



NATURAL TANGENTS AND COTANGENTS 



/ 


5 





6 





7° 


8° 


5 


° 


/ 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 





.08749 


II. 4301 


.10310 


9.51436 


.12278 


8.14435 


•14054 


7.I1S37 


.15838 


6-3I37S 


60 


I 


.08778 


11.3919 


.10540 


9.48781 


.12308 


8.12481 


.14084 


7.10038 


.15868 


6.30189 


59 


2 


.08807 


11.3540 


.10569 


9.46141 


.12338 


8.10536 


.14113 


7.08546 


.15898 


6.29007 


S8 


3 


.08837 


11.3163 


.10599 


9.4351s 


.12367 


8.08600 


.14143 


7.07059 


.15928 


6.27829 


57 


4 


.08866 


11.2789 


.10628 


9.40904 


.12397 


8.06674 


•14173 


7.05579 


.15958 


6.26655 


56 


S 


.08895 


11.2417 


.10657 


9.38307 


.12426 


8.04756 


.14202 


7. 04 1 OS 


.15988 


6.25486 


55 


6 


.0892s 


11.2048 


.10687 


9.35724 


.12456 


8.02848 


.14232 


7.02637 


.16017 


6.24321 


54 


7 


.08954 


II. 1681 


.10716 


9.33155 


.12485 


8.00948 


.14262 


7.01174 


.16047 


6.23160 


S3 


8 


.08983 


11.1316 


.10746 


9.30599 


•1251S 


7.99058 


.14291 


6.997x8 


.16077 


6.22003 


52 


9 


.09013 


11.0954 


.10775 


9.28058 


•12544 


7.97176 


.14321 


6-98268 


.16107 


6.20851 


51 


10 


.09042 


11.0594 


.10805 


9.25530 


•12574 


7-95302 


.14351 


6.96823 


.16137 


6.19703 


50 


II 


.09071 


11. 0237 


.10834 


9.23016 


.12603 


7-93438 


.14381 


6.9538s 


.16167 


6.i8sS9 


49 


12 


.09101 


10.9882 


.10863 


9.20516 


.12633 


7-91582 


.14410 


6.93952 


.16196 


6.17419 


48 


13 


.09130 


10.9529 


.10893 


9.18028 


.12662 


7-89734 


.14440 


6.9252s 


.16226 


6.16283 


47 


14 


.09159 


10.9178 


.10922 


9.1 5554 


.12692 


7-87895 


.14470 


6.91 104 


.16256 


6.1S151 


46 


15 


.09189 


10.8829 


.10952 


9.13093 


.12722 


7.86064 


.14499 


6-89688 


.16286 


6.14023 


4S 


l6 


.09218 


10.8483 


.10981 


9.10646 


.12751 


7.84242 


.14529 


6.88278 


.16316 


6.12899 


44 


17 


.09247 


10.8139 


.IIOII 


9.0821 1 


.12-81 


7.82428 


.14559 


6.86874 


.16346 


6.11779 


43 


i8 


.09277 


10.7797 


.11040 


9.05789 


.12810 


7.80622 


.14588 


6.85475 


.16376 


6.10664 


42 


19 


.09306 


10.7457 


.11070 


9-03379 


.12840 


7.7882s 


.14618 


6-84082 


.1640s 


6-09552 


41 


20 


•0933s 


10.7119 


.11099 


9.00983 


.12869 


7-77035 


.14648 


6.82694 


■1643s 


6.08444 


40 


21 


.09365 


10.6783 


.11128 


8.g85g8 


.12899 


7.7S2S4 


.14678 


6-81312 


•16465 


6.07340 


39 


22 


.09394 


10.6450 


.11158 


8.96227 


.12929 


7.73480 


.14707 


6.79936 


•16495 


6.06240 


38 


23 


.09423 


10.6118 


.11187 


8.93867 


.12958 


7-71715 


.14737 


6.78564 


•16525 


6.05143 


37 


24 


•09453 


10.5789 


.11217 


8.91520 


.12988 


7.69957 


.14767 


6.77199 


.16555 


6.04051 


36 


25 


.09482 


10.5462 


.11246 


8.89185 


.13017 


7.68208 


.14796 


6.758^8 


.16585 


6.02962 


35 


26 


.09511 


10.5136 


.11276 


8.86862 


•13047 


7.66466 


.14826 


6.74483 


.16615 


6.01878 


34 


27 


.09541 


10.4813 


.11305 


8.84551 


.13076 


7.64732 


.14856 


6.73133 


.16645 


6.00797 


33 


28 


•09570 


10.4491 


.11335 


8.82252 


.13106 


7-63005 


.14886 


6.71789 


.16674 


5-99720 


32 


29 


.09600 


10.4172 


.11364 


8.79964 


•13136 


7.61287 


.14915 


6.70450 


.16704 


5-98646 


31 


30 


.09629 


10.3854 


.11394 


8.77689 


•1316s 


7-59575 


.14945 


6.69116 


.16734 


S-97576 


30 


31 


.09658 


10.3538 


.11423 


8.7S42S 


.13195 


7-57872 


.14975 


6.67787 


.16764 


5.96510 


29 


32 


.096S8 


10.3224 


.11452 


8.73172 


.13224 


7.56176 


.15005 


6.66463 


.16794 


5-95448 


28 


33 


.09717 


10.2913 


.11482 


8.70931 


.13254 


7-54487 


.15034 


6.65144 


.16824 


5-94390 


27 


34 


.09746 


10.2602 


.11511 


8.68701 


.13284 


7.52806 


.15064 


6.63831 


.16854 


S-93335 


26 


35 


.09776 


10.2294 


.11541 


8.66482 


.13313 


7-51132 


.15094 


6.62523 


.16884 


S-92283 


25 


36 


.09805 


10.1988 


.11570 


8.64275 


.13343 


7-49465 


.15124 


6.61219 


.16914 


S-91236 


24 


37 


.09834 


10.1683 


.11600 


8.62078 


.13372 


7.47806 


.15153 


6.59921 


.16944 


S.90191 


23 


38 


.09864 


10.1381 


.11629 


8-59893 


.13402 


7.46154 


.15183 


6.58627 


.16974 


5-89151 


22 


39 


.09893 


10.1080 


.11659 


8-57718 


.13432 


7.44509 


.15213 


6-57339 


.17004 


S-88114 


21 


40 


.09923 


10.0780 


.11688 


8-55555 


.13461 


7.42871 


•15243 


6.56055 


.17033 


5.87080 


20 


41 


.09952 


10.0483 


.11718 


8.53402 


.13491 


7.41240 


•15272 


6-54777 


.17063 


5-86051 


19 


42 


.09981 


10.0187 


.11747 


8.51259 


.13521 


7.39616 


• 15302 


6.53503 


.17093 


5-85024 


18 


43 


.10011 


9.98931 


.11777 


8.49128 


.13550 


7-37999 


• 15332 


6.52234 


.17123 


S- 8400 1 


17 


44 


.10040 


9.96007 


.11806 


8.47007 


.13580 


7-36389 


.15362 


6.50970 


.17153 


5-82982 


16 


45 


.10069 


9-93101 


.11836 


8.44896 


.13609 


7-34786 


•15391 


6.49710 


.17183 


S-81966 


IS 


46 


.10099 


g. 90211 


.11865 


8-42795 


.13639 


7-33190 


.15421 


6-48456 


.17213 


S-80953 


14 


47 


.10128 


9.87338 


.11895 


8.40705 


.13669 


7.31600 


.15451 


6.47206 


.17243 


5-79944 


13 


48 


.10158 


9.84482 


.11924 


8.3862s 


.13698 


7.30018 


.15481 


6.45961 


.17273 


S.78938 


12 


49 


.10187 


9.81641 


.11954 


8.36555 


.13728 


7.28442 


.155" 


6.44720 


.17303 


5-77936 


II 


50 


.10216 


9.78817 


.11983 


8-34496 


.13758 


7.26873 


.15540 


6.43484 


.17333 


S.76937 


10 


51 


.10246 


9.76009 


.12013 


8.32446 


.13787 


7-25310 


.15570 


6.42253 


.17363 


S.75941 


9 


52 


.10275 


9.73217 


.12042 


8.30406 


.13817 


7-23754 


.15600 


6.41026 


.17393 


5-74949 


8 


S3 


.10305 


g. 70441 


.12072 


8.28376 


•13S46 


7.22204 


.15630 


6.39804 


.17423 


5-73960 


7 


54 


•10334 


g. 67680 


.12101 


8.2635s 


•13876 


7.20661 


.15660 


6.38587 


.17453 


S-72974 


6 


55 


.10363 


9-64935 


.12131 


8.24345 


.13906 


7-19125 


.15689 


6.37374 


.17483 


5-71992 


S 


56 


•10393 


9.62205 


.12160 


8.22344 


.13935 


7-17594 


.15719 


6.36165 


.17513 


5-71013 


4 


57 


.10422 


9-59490 


.12190 


8-20352 


.13965 


7.16071 


.15749 


6.34961 


.17543 


S-70037 


3 


58 


.10452 


9.56791 


.12219 


8.18370 


.13995 


7-14553 


.15779 


6.33761 


.17573 


S -69064 


2 


59 


.10481 


9-54106 


.12249 


8.16398 


.14024 


7.13042 


.15809 


6.32566 


.17603 


5 -68094 


1 


60 


.10510 


9^51436 


.12278 


8.1443S 


.14054 


7-11537 


.15838 


6.3137s 


.17633 


5-67128 





1 
1 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


/ 


8. 


t° 


8: 


5° 


82° 


81° 


8( 


3° 



NATURAL TANGENTS AND COTANGENTS 



163 



1 


10° 


11° 


12° 


13° 


I 


4° 


/ 


Tang: 


Cotang 


Tangr 


Cotangr 


Tans: 


Cotang: 


Tangr 


Cotang 


Tang 


Cotang 





.17633 


5.67128 


.19438 


5. 14455 


.21256 


4-70463 


.23087 


4.33148 


.24933 


4.01078 


60 


I 


.17663 


S^66i6s 


.19468 


5.13658 


.21286 


4-69791 


.23117 


4.32573 


.24964 


4.00582 


59 


2 


.17693 


5^65205 


-19498 


5.12862 


.21316 


4.69121 


.23148 


4.32001 


.24995 


4.00086 


58 


3 


.17723 


5.64248 


■19529 


5^12069 


.21347 


4.68452 


.23179 


4.31430 


.2%oz(> 


3.99592 


57 


4 


.17753 


5^63295 


-19559 


5-11279 


•21377 


4.67786 


.23209 


4^30860 


.25056 


3.99099 


S6 


S 


.17783 


5^62344 


-19589 


5^10490 


.21408 


4.67121 


.23240 


4^3029i 


.25087 


3.98607 


55 


6 


.17813 


5. 61397 


.19619 


5.09704 


.21438 


4.66458 


.23271 


4.29724 


.25118 


3.98117 


54 


7 


-17843 


5-60452 


.19649 


5.08921 


.21469 


4.65797 


.23301 


4.29159 


.25149 


3-97627 


53 


8 


.17873 


S.S95II 


.196S0 


5.08139 


.21499 


4.65138 


.23332 


4.28595 


.25180 


3.97139 


52 


9 


.17903 


5^58S73 


.19710 


5^07360 


.21529 


4.64480 


.23363 


4-28032 


.25211 


3.96651 


51 


10 


.17933 


5^57638 


.19740 


5-06534 


.21560 


4.6382s 


■2Zi<ii 


4-27471 


.25242 


3.9616s 


50 


u 


•17563 


5.56706 


.19770 


5^05809 


.21590 


4.63171 


.23424 


4-26911 


.25273 


3^95680 


49 


12 


•17993 


5^55777 


.19801 


5.05037 


.21621 


4-62518 


.23455 


4^26352 


.25304 


3.95196 


48 


13 


.18023 


5.54851 


.19831 


5.04267 


.21651 


4^6i868 


.23485 


4^25795 


.25335 


3.94713 


47 


14 


.18053 


5.53927 


.19861 


5.03499 


.21682 


4^61219 


.23516 


4^25239 


.25366 


3.94232 


46 


15 


.18083 


5-53007 


.19891 


5.02734 


.21712 


4^60572 


.23547 


4^24685 


.25397 


3-93751 


45 


i6 


.18113 


5.52090 


.19921 


5^01971 


.21743 


4.59927 


.23578 


4^24132 


.25428 


3^93271 


44 


17 


.18143 


5.SI176 


.19952 


5.01210 


.21773 


4-59283 


.23608 


4^23580 


.25459 


3-92793 


CI 


i8 


.18173 


5.50264 


.19982 


5.00451 


.21804 


4-58641 


.23639 


4^23030 


.25490 


3.92316 


42 


19 


.18203 


5.49356 


.20012 


4.99695 


.21834 


4-58001 


.23670 


4.22481 


.25521 


3.91839 


41 


20 


.18233 


5^48451 


.20042 


4.98940 


.21864 


4.57363 


.23700 


4.21933 


.25552 


3.91364 


40 


21 


.18263 


5^47548 


.20073 


4.98188 


.21895 


4-56726 


.23731 


4.21387 


.25583 


3.90890 


39 


22 


.18293 


5.46648 


.20103 


4.97438 


.21925 


4-56091 


.23762 


4^20842 


.25614 


3.90417 


38 


23 


.18323 


5.45751 


.20133 


4.96690 


.21956 


4.55458 


.23793 


4.20298 


.25645 


3.89945 


37 


24 


.18353 


5^44857 


.20164 


4-95945 


.21986 


4.54826 


.23823 


4.19756 


.25676 


3.89474 


36 


2S 


.18384 


5.43966 


.20194 


4.95201 


.22017 


4.54196 


.23854 


4.1921s 


.25707 


3.89004 


35 


26 


.18414 


5.43077 


.20224 


4.94460 


.22047 


4.53568 


.23885 


4.1867s 


.25738 


3.88536 


34 


27 


.18444 


5. 42192 


.20254 


4-93721 


.22078 


4-52941 


.23916 


4.18137 


.25769 


3.88068 


33 


28 


.18474 


5-41309 


.20285 


4-92984 


.22108 


4-52316 


.23946 


4.17600 


.25800 


3.87601 


32 


29 


.18504 


5.40429 


.20315 


4-92249 


.22139 


4-S1693 


.23977 


4.17064 


.25831 


3.871.36 


31 


30 


.18534 


5-39552 


.20345 


4.91516 


.22169 


4.51071 


.24008 


4.16530 


.25862 


3.86671 


30 


31 


.18564 


S-38677 


-20376 


4.9078s 


.22200 


4.50451 


.24039 


4.15997 


.25893 


3.86208 


29 


32 


.18594 


S-37S0S 


.20406 


4.90056 


.22231 


4.49832 


.24069 


4.15465 


.25924 


3-85745 


28 


33 


•18624 


5-36956 


.20436 


4-89330 


.22261 


4.49215 


.24100 


4.14934 


.25955 


3.85284 


27 


34 


•18654 


S.36070 


.20466 


4.88605 


.22292 


4.48600 


.24131 


4^14405 


.25986 


3 •84824 


26 


35 


.18684 


5.35206 


.20497 


4.87882 


.22322 


4.47986 


.24162 


4^13877 


.26017 


3^84364 


25 


36 


.18714 


5-34345 


.20527 


4-87162 


.22353 


4-47374 


.24193 


4^13350 


.26048 


3.83906 


24 


37 


.18745 


S-33487 


.20557 


4-86444 


.22383 


4.46764 


.24223 


4.12825 


.26079 


3-83449 


23 


38 


.18775 


5-32631 


.20588 


4.85727 


.22414 


4.4615s 


.24254 


4.12301 


.26110 


3-82992 


22 


39 


.18805 


5-31778 


.20618 


4.85013 


.22444 


4.45548 


.2428s 


4.11778 


.26141 


3.82537 


21 


40 


.18835 


5.30928 


.20648 


4.84300 


.22475 


4.44942 


.24316 


4.11256 


.26172 


3^82083 


20 


41 


.18865 


S-30080 


.20679 


4-83590 


.2250s 


4-44338 


.24347 


4.10736 


.26203 


3.81630 


19 


42 


.18895 


5-29235 


.20709 


4.82882 


.22536 


4.43735 


.24377 


4.10216 


.26235 


3.81177 


18 


43 


.18925 


5.28393 


.20739 


4.8217s 


.22567 


4.43134 


.24408 


4.09699. 


•26266 


3.80726 


17 


44 


.18955 


5^27553 


.20770 


4.81471 


.22597 


4-42534 


.24439 


4.09182 


•26297 


3.80276 


16 


45 


.18986 


5.26715 


•20800 


4.80769 


.22628 


4.41936 


.24470 


4.08666 


.26328 


3-79827 


15 


46 


.19016 


5^25880 


.20830 


4.80068 


.2265S 


4-41340 


.24501 


4.08152 


.26359 


3.79378 


14 


47 


.19046 


5.25048 


.20861 


4.79370 


•22689 


4.40745 


.24532 


4.07639 


.26390 


3.78931 


13 


48 


.19076 


5.24218 


•20891 


4^78673 


.22719 


4.40152 


.24562 


4.07127 


.26421 


3.78485 


12 


49 


.19106 


5.23391 


.20921 


4.77978 


•22750 


4.39560 


.24593 


4.06616 


.26452 


3-78040 


11 


50 


.19136 


5^22566 


.20952 


4.77286 


•22781 


4.38969 


.24624 


4.06107 


.26483 


3.77595 


10 


SI 


.19166 


S^2I744 


.20982 


4.76595 


.22811 


4.38381 


.24655 


4.05599 


.2651S 


3.77152 


9 


52 


.19197 


5^20925 


.21013 


4^75906 


.22842 


4.37793 


.24686 


4.05092 


.26546 


3.76709 


8 


S3 


.19227 


5^20107 


.21043 


4^752I9 


.22872 


4-37207 


.24717 


4.04586 


.26577 


3.76268 


7 


54 


.19257 


5^ 19293 


.21073 


4^74534 


.22903 


4-36623 


.24747 


4.04081 


.26608 


3.75828 


6 


55 


.19287 


5.18480 


.21104 


4^73851 


.22934 


4.36040 


.24778 


4.03578 


.26639 


3.75388 


5 


S6 


.19317 


5.17671 


.21134 


4^73170 


.22964 


4-35459 


.24809 


4.03076 


.26670 


3.74950 


4 


57 


.19347 


5.16863 


.21164 


4.72490 


.22995 


4^34879 


•24840 


4.02574 


.26701 


3-74512 


3 


S8 


.19378 


S^i6os8 


.21195 


4.71813 


.23026 


4.34300 


•24871 


4.02074 


.26733 


3.7407s 


2 


59 


.19408 


5^15256 


.21225 


4-71137 


.23056 


4.33723 


•24902 


4.01576 


.26764 


3.73640 


I 


60 


.19438 


5^14455 


.21256 


4.70463 


-23087 


4.33148 


•24933 


4.01078 


.26795 


3.73205 





/ 


Cotang: 


Tang: 


Cotang- 


Tang: 


Cotang: 


Tang- 


Cotang 


Tang 


Cotang 


Tang 


/ 


79° 


78° 


yi"" 


76° 


7. 


-0 
5 



164 



NATURAL TANGENTS AND COTANGENTS 



1 


15° 


16° 


17° 


18° 


I 



9 


1 


Tang: 


Cotangr 


Tang 


Cotang 


Tang 


Cotang 


Tang 


CotansT 


Tang 


Cotang 


o 


.2679s 


3-73205 


.28675 


3.48741 


.30573 


3.27085 


.32492 


3.07768 


•34433 


2.90421 


60 


I 


.26826 


3.72771 


.28706 


3-48359 


.30605 


3.26745 


.32524 


3-07464 


•34465 


2.90147 


59 


2 


.26857 


3.72338 


.28738 


3.47977 


.30637 


3.26406 


.32556 


3.07160 


•34498 


2.89873 


58 


3 


.26888 


3.71907 


.28769 


3.47596 


.30669 


3-26067 


.32588 


3-06857 


.34530 


2.89600 


57 


4 


.26920 


3-71476 


.28800 


3-47216 


.30700 


3-25729 


.32621 


3.06554 


.34563 


2.89327 


56 


S 


.26951 


3.71046 


.28832 


3-46837 


.30732 


3.25392 


.32653 


3-06252 


.34596 


2.8905s 


55 


6 


.26982 


3.70616 


.28864 


3-46458 


.30764 


3-25055 


.32685 


3-05950 


.34628 


2.88783 


54 


7 


.27013 


3.70188 


.28895 


3.46080 


.30796 


3.24719 


.32717 


3-05649 


.34661 


2.88511 


S3 


8 


.27044 


3-69761 


.28927 


3-45703 


.30828 


3-24383 


.32749 


3-05349 


.34693 


2.88240 


52 


9 


.27076 


3-69335 


.28958 


3-45327 


.30860 


3-24049 


.32782 


3-05049 


.34726 


2.87970 


51 


10 


.27107 


3.68909 


.28990 


3.44951 


.30891 


3.23714 


.32814 


3.04749 


.34758 


2.87700 


SO 


II 


.27138 


3-68485 


.29021 


3.44576 


.30923 


3.23381 


.32846 


3-04450 


.34791 


2.87430 


49 


12 


.27169 


3.68061 


.29053 


3.44202 


.30955 


3.23048 


.32878 


3-04IS2 


.34824 


2.87161 


48 


13 


.27201 


3.67638 


.29084 


3.43829 


.30987 


3.22715 


.32911 


3-03854 


.34856 


2.86892 


47 


14 


.27232 


3.67217 


.29116 


3.43456 


.31019 


3.22384 


.32943 


3.03556 


.34889 


2.86624 


46 


15 


.27263 


3.66796 


.29147 


3.43084 


.31051 


3.22053 


.3297s 


3.03260 


.34922 


2.86356 


45 


i6 


.27294 


3.66376 


.29179 


3.42713 


.31083 


3.21722 


.33007 


3.02963 


•34954 


2.86089 


44 


17 


.27326 


3.65957 


.29210 


3.42343 


.31115 


3.21392 


.33040 


3.02667 


.34987 


2.85822 


43 


i8 


.27357 


3.65538 


.29242 


3.41973 


.31147 


3.21063 


.33072 


3-02372 


.35020 


2.8555s 


42 


19 


.27388 


3.65121 


.29274 


3.41604 


.31178 


3-20734 


.33104 


3.02077 


.35052 


2.85289 


41 


20 


.27419 


3.6470s 


.29305 


3.41236 


.31210 


3.20406 


.33136 


3.01783 


.3508s 


2.85023 


40 


21 


.27451 


3.64289 


.29337 


3.40869 


.31242 


3.20079 


.33169 


3.01489 


.35118 


2.84758 


39 


22 


.27482 


3.63874 


.29368 


3.40S02 


.31274 


3-19752 


.33201 


3.01196 


.35150 


2.84494 


38 


23 


.27513 


3.63461 


.29400 


3.40136 


.31306 


3.19426 


.33233 


3.00903 


.35183 


2.84229 


37 


24 


.27545 


3.63048 


.29432 


3.39771 


.31338 


3.19100 


.33266 


3.00611 


.35216 


2.8396s 


36 


25 


.27576 


3.62636 


.29463 


3.39406 


.31370 


3-18775 


.33298 


3.00319 


.35248 


2.83702 


35 


26 


.27607 


3.62224 


.29495 


3.39042 


.31402 


3-18451 


.33330 


3.00028 


• 35281 


2.83439 


34 


27 


.27638 


3.61814 


.29526 


3.38679 


.31434 


3.18127 


.33363 


2.99738 


•35314 


2.83176 


33 


28 


.27670 


3.61405 


.29558 


3.38317 


.31466 


3.17804 


.33395 


2.99447 


•35346 


2.82914 


32 


29 


.27701 


3.60996 


.29590 


3-37955 


.31498 


3.17481 


.33427 


2.99158 


•35379 


2.82653 


31 


30 


.27732 


3.60588 


.29621 


3.37594 


•31530 


3.17159 


.33460 


2.98868 


•35412 


2.82391 


30 


31 


.27764 


3.60181 


.29653 


3.37234 


.31562 


3.16838 


.33492 


Z.98580 


•35445 


2.82130 


29 


32 


.27795 


3-59775 


.29685 


3.36875 


.31594 


3.16517 


.33524 


2.98292 


•35477 


2.81870 


28 


33 


.27826 


3-59370 


.29716 


3.36516 


.31626 


3.16197 


.33557 


2.98004 


•35510 


2.81610 


27 


34 


.27858 


3.58966 


.29748 


3.36158 


.31658 


3.15877 


.33589 


2.97717 


•35543 


2.81350 


26 


35 


.27889 


3.58562 


.29780 


3.35800 


.31690 


3.15558 


.33621 


2.97430 


•35576 


2.8iogi 


25 


36 


.27921 


3.58160 


.29811 


3.35443 


.31722 


3.15240 


.33654 


2.97144 


.35608 


2.80833 


24 


37 


.27952 


3-57758 


.29843 


3-35087 


•31754 


3.14922 


.33686 


2.96858 


•35641 


2.80574 


23 


38 


.27983 


3-57357 


.29875 


3.34732 


.31786 


3-14605 


•33718 


2.96573 


•35674 


2.80316 


22 


39 


.2801S 


3.56957 


.29906 


3.34377 


.31818 


3.14288 


•33751 


2.96288 


•35707 


2.80059 


21 


40 


.28046 


3.56557 


.29938 


3.34023 


.31850 


3.13972 


•33783 


2.96004 


•3S740 


2-.79802 


20 


41 


.28077 


3.56159 


.29970 


3.33670 


.31882 


3-13656 


.33816 


2.95721 


•35772 


2.79S4S 


19 


42 


.28109 


3.55761 


.30001 


3.33317 


.31914 


3.13341 


.33848 


2.95437 


•3580s 


2.79289 


18 


43 


.28140 


3.55364 


.30033 


3-32965 


.31946 


3-13027 


.33881 


2.9515s 


.35838 


2.79033 


17 


44 


.28172 


3.54968 


.3006s 


3-32614 


.31978 


3.12713 


.33913 


2.94872 


.35871 


2.78778 


16 


45 


.28203 


3.54573 


.30097 


3.32264 


.32010 


3.12400 


.33945 


2.94591 


.35904 


2.78523 


IS 


46 


.28234 


3-54179 


.30128 


3.31914 


.32042 


3.12087 


.33978 


2-94309 


.35937 


2.78269 


14 


47 


.28266 


3-53785 


.30160 


3-31565 


.32074 


3.1177s 


.34010 


2.94028 


.35969 


2.78014 


13 


48 


.28297 


3.53393 


.30192 


3.31216 


.32106 


3.11464 


.34043 


2.93748 


.36002 


2.77761 


12 


49 


.28329 


3.53001 


.30224 


3.30868 


.32139 


3.11153 


■3407s 


2.93468 


.36035 


2.77507 


II 


SO 


.28360 


3.52609 


■30255 


3.30521 


.32171 


3.10842 


.34108 


2.93189 


.36068 


2.77254 


JO 


51 


.28391 


3.52219 


.30287 


3.30174 


.32203 


3.10532 


.34140 


2.92910 


.36101 


2.77002 


9 


52 


.28423 


3.S1829 


.30319 


3.29829 


.32235 


3.10223 


.34173 


2.92632 


.36134 


2.76750 


8 


53 


.28454 


3.S1441 


.30351 


3.29483 


.32267 


3.09914 


.34205 


2.92354 


.36167 


2.76498 


7 


54 


.28486 


3-51053 


.30382 


3.29139 


.32299 


3.09606 


.34238 


2.92076 


.36199 


2.76247 


6 


55 


.28517 


3.50666 


.30414 


3.28795 


.32331 


3.09298 


.34270 


2.91799 


.36232 


2.75996 


5 


56 


.28549 


3.50279 


.30446 


3.28452 


.32363 


3.08991 


.34303 


2.91523 


.3626s 


2.75746 


4 


57 


.28580 


3-49894 


.30478 


3.28109 


.32396 


3.08685 


•34335 


2.91246 


.36298 


2.75496" 


3 


58 


.28612 


3.49509 


.30509 


3.27767 


.32428 


3.08379 


.34368 


Z.90971 


■36331 


2.75246 


2 


59 


.28643 


3.49125 


.30541 


3.27426 


.32460 


3.08073 


•34400 


2.90696 


■36364 


2.74997 


I 


60 


.28675 


3.48741 


.30573 


3.2708s 


.32492 


3.07768 


•34433 


2.90421 


■36397 


2.74748 





/ 


Cotang 


Tangr 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


/ 


7^ 


J° 


73° 


7i 


)0 


7J 





7( 


J° 



NATURAL TANGENTS AND COTANGENTS 



1G5 



/ 


20° 


2] 





22° 


2; 


° 


24° 


/ 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 





.36397 


2.74748 


.38386 


2.60509 


.40403 


2.47509 


.42447 


2.35585 


•44S23 


2.24604 


60 


I 


.36430 


2.74499 


.38420 


2.60283 


.40436 


2.47302 


•42482 


2.35395 


•44558 


2.24428 


59 


2 


.36463 


2.74251 


.38453 


2.60057 


.40470 


2.47095 


.42516 


2.35205 


• 44.193 


2.24252 


S8 


3 


.36456 


2.74004 


.38487 


2.59831 


.40504 


2.468S8 


.42551 


2.35015 


•44627 


2.24077 


57 


4 


.36329 


2.73756 


.38520 


2.59606 


.40538 


2.46682 


.42585 


2.34825 


.44662 


2.23902 


56 


S 


.36562 


2.73509 


.38553 


2.59381 


.40572 


2.46476 


.42619 


2.34636 


.44697 


2.23727 


55 


6 


.36595 


2.73263 


.38587 


2.59156 


.40606 


2.46270 


.42654 


2.34447 


.44732 


2.235.53 


54 


7 


.36628 


2.73017 


.38620 


2.58932 


.40640 


2.46065 


.42688 


2.34258 


.44767 


2.2,^378 


S3 


g 


.36661 


2.72771 


.38654 


2.58708 


.40674 


2.45S60 


.42722 


2.34069 


.44S02 


2.23204 


52 


9 


.36694 


2.72526 


.38687 


2.58484 


.40707 


2.45655 


•42757 


2.33881 


.44837 


2.23030 


SI 


10 


.36727 


2.72281 


.38721 


2.58261 


.40741 


2.45451 


•42791 


2.33693 


.44872 


2.22857 


50 


II 


.36760 


2.72036 


.38754 


2.58038 


.4077s 


2.45246 


.42826 


2.33505 


.44907 


2.22683 


49 


12 


.36793 


2.71792 


.38787 


2.57S1S 


.40S09 


2.45043 


.42860 


2.33317 


.44942 


2.22510 


48 


13 


.368.:6 


2.71548 


.38821 


2.57593 


.40843 


2.44839 


.42894 


2.33130 


.44977 


2.22337 


47 


14 


.36859 


2.7130s 


.38854 


2.57371 


.40877 


2.44636 


.42929 


2.32943 


.45012 


2.22164 


46 


IS 


.36892 


2.71062 


.38888 


2.57150 


•40911 


2.44433 


.42963 


2.32756 


.45047 


2.21992 


45 


i6 


.36925 


2.70813 


•38921 


2.56928 


.40945 


2.44230 


.42998 


2.32570 


.45083 


2.2i8ig 


44 


17 


.36958 


2.70577 


.38955 


2.56707 


.40979 


2.44027 


.43032 


2.32383 


.45117 


2.21647 


43 


i8 


.36991 


2.7033s 


.38988 


2.56487 


.41013 


2.43825 


.43067 


2.32197 


.45152 


2.21475 


42 


19 


.37024 


2.70094 


.39023 


2.56266 


.41047 


2.43623 


.43101 


2.32012 


.45187 


2.21304 


41 


20 


■37057 


2.69853 


.39055 


2.56046 


.41081 


2.43422 


.43136 


2.31826 


.45222 


2.21132 


40 


21 


.37090 


2.69612 


.39089 


2.55827 


.41115 


2.43220 


.43170 


2.31641' 


.45257 


2.20961 


39 


22 


.37123 


2.69371 


.39122 


2.55608 


•41149 


2.43019 


.43205 


2.31456 


.45293 


2.20790 


38 


23 


.37157 


2.69131 


.39156 


2.55389 


.41183 


2.42819 


.43230 


2.31271 


.45327 


2.20619 


37 


24 


.37190 


2.68892 


.39190 


2.55170 


.41217 


2.42618 


.43274 


2.31086 


.45363 


2.20449 


36 


25 


.37223 


2.68653 


.39223 


2.54952 


.41251 


2.42418 


.43308 


2.30902 


•45397 


2.20278 


35 


26 


.37256 


2.68414 


•39257 


2.54734 


.41285 


2.42218 


.43343 


2.30718 


•45433 


2.20108 


34 


27 


.37289 


2.68175 


•39290 


2.54516 


■41319 


2.42019 


.43378 


2.30534 


.45467 


2.19938 


33 


28 


.37322 


2.67937 


•39324 


2.54299 


.41353 


2.41819 


.43413 


2.30351 


•45503 


2.19769 


32 


29 


.37355 


2.67700 


•39357 


2.54082 


.41387 


2.41620 


.43447 


2.30167 


•45538 


2.19599 


31 


30 


.37388 


2.67462 


.39391 


2.53865 


.41421 


2.41421 


.43481 


2.29984 


•45573 


2.19430 


30 


31 


.37422 


2.6722s 


.39425 


2.53648 


.41455 


2.41223 


.43516 


2.29801 


.45608 


2.19261 


29 


32 


.37455 


2.66989 


.39458 


2.53432 


.41490 


2.41025 


.43550 


2.29619 


.45643 


2.19092 


28 


33 


.37488 


2.66752 


.39492 


2.53217 


.41524 


2.40827 


.4358s 


2.29437 


.45678 


2.18923 


27 


34 


.37521 


2.66516 


.39526 


2.53001 


.41558 


2.40629 


•43620 


2.29254 


.45713 


2.18755 


26 


35 


•37554 


2.66281 


.39559 


2.52786 


.41592 


2.40432 


•43654 


2.29073 


.45748 


2.18587 


25 


36 


.37588 


2.66046 


.39593 


2.52571 


.41626 


2.4023s 


.43689 


2.28891 


.45784 


2.18419 


24 


37 


.37621 


2.65811 


.39626 


2.52357 


.41660 


2.40038 


•43724 


2.28710 


.45819 


2.18251 


23 


38 


.37654 


2.65576 


.39660 


2.52142 


.41694 


2.39841 


.43758 


2.28528 


.45854 


2.18084 


22 


39 


.37687 


2.65342 


.39694 


2.51929 


.41728 


2.3964s 


•43793 


2.28348 


.45889 


2.17916 


21 


40 


.37720 


2.65109 


.39727 


2.51715 


.41763 


2.39449 


•43828 


2^28167 


.45924 


2.17749 


20 


41 


.37754 


2.6487s 


.39761 


2.S1502 


.41797 


2.39253 


.43862 


2^27987 


.45960 


2.17582 


19 


42 


.37787 


2.64642 


.39795 


2.51289 


.41831 


2.39058 


.43897 


2.27806 


.45995 


2.17416 


18 


43 


.37820 


2.64410 


.39S29 


2.51076 


.41865 


2.38863 


.43932 


2.27626 


.46030 


2.17249 


17 


44 


.37853 


2.64177 


.39862 


2.50864 


.41899 


2.38668 


.43966 


2.27447 


.46065 


2.17083 


16 


45 


.37887 


2.63945 


.39896 


2.50652 


.41933 


2.38473 


.44001 


2.27267 


.46101 


2.16917 


IS 


46 


.37920 


2.63714 


.39930 


2.50440 


.41968 


2.38279 


.44036 


2.27088 


.46136 


2.16751 


14 


47 


•37953 


2.63483 


.39963 


2.50229 


•42002 


2.38084 


.44071 


2.26909 


.46171 


2.16585 


13 


48 


.37986 


2.63252 


.39997 


2.50018 


•42036 


2.37891 


.44105 


2.26730 


•46206 


2.16420 


12 


49 


.38020 


2.63021 


.40031 


2.49807 


.42070 


2.37697 


.44140 


2.26552 


.46242 


2.1625s 


11 


50 


.38053 


2.62791 


.40065 


2.49597 


.42105 


2.37504 


.4417s 


2.26374 


•46277 


2.16090 


10 


SI 


.38086 


2.62561 


.40098 


2.49386 


.42139 


2.37311 


•44210 


2.26196 


•46312 


2.15925 


9 


52 


.38120 


2.62332 


.40132 


2.49177 


.42173 


2.37118 


.44244 


2.26018 


•46348 


2.15760 


8 


S3 


.38153 


2.62103 


.40166 


2.48967 


.42207 


2.36925 


.44279 


2.25840 


•46383 


2.15596 


7 


54 


.38186 


2.61874 


.40200 


2.48758 


.42242 


2.36733 


.44314 


2.25663 


.46418 


2.1S432 


6 


SS 


.38220 


2.61646 


.40234 


2.48549 


.42276 


2.36541 


.44349 


2.25486 


•46454 


2.15268 


5 


S6 


.38253 


2.61418 


.40267 


2.48340 


.42310 


2.36349 


.44384 


2.25309 


.46489 


2.I5I04 


4 


57 


.38286 


2. 61190 


.40301 


2.48132 


.42345 


2.36158 


.44418 


2.25132 


.46525 


2.14940 


3 


S8 


.38320 


2.60963 


.40335 


2.47924 


.42379 


2.35967 


.444S3 


2.24956 


.46560 


2.14777 


2 


59 


.38353 


2.60736 


.40369 


2.47716 


•42413 


2.35776 


.44488 


2.24780 


.46595 


2.14614 


I 


60 


.38386 


2.60509 


.40403 


2.47509 


■42447 


2.35585 


.44523 


2.24604 


.46631 


2.I445I 





/ 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


( 


6( 


?° 


6i 


i° 


6; 


7° 


6( 


5° 


6 


5° 



166 



NATURAL TANGENTS AND COTANGENTS 



/ 


2 = 





26° 


27° 


28° 


29° 


1 


Tang 


Cotang: 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 





.46631 


2.I44SI 


.48773 


2.05030 


•S09S3 


1. 96261 


•53171 


1.88073 


•55431 


1.8040S 


60 


I 


.46666 


2.14288 


.48809 


2.04879 


.50989 


1.96120 


.53208 


1.87941 


•55469 


1.80281 


59 


3 


.46702 


2.14125 


.48845 


2.047:8 


.51026 


1.95979 


-53246 


1.87809 


•55507 


1.80158 


S8 


3 


.46737 


2.13963 


.488S1 


2.04577 


.51063 


1.95838 


.53283 


1-87677 


•5S545 


1.80034 


57 


4 


.46772 


2.13801 


•48917 


2.04426 


.51099 


1.95698 


•53320 


1-87546 


•55583 


1.79911 


56 


S 


.46808 


2.13639 


.48953 


2.04276 


.51136 


1-95557 


•53358 


1.87415 


•55621 


1.79788 


ss 


6 


.46843 


2.13477 


.48989 


2.04125 


•51 173 


1.95417 


•53395 


1.87283 


•55659 


1.7966s 


54 


7 


.46879 


2.13316 


.49026 


2.03975 


.51209 


l^95277 


•53432 


1.87152 


.55697 


1.79542 


S3 


8 


.46914 


2.I3IS4 


.49062 


2.0382s 


.51246 


1^95137 


•53470 


1. 87021 


.55736 


1.79419 


52 


9 


.46950 


2.12993 


.49098 


2.03675 


.51283 


1.94997 


•53507 


1.86891 


.55774 


1.79296 


SI 


10 


.4698s 


2.12832 


.49134 


2.03526 


•51319 


1.94858 


•S3545 


1.86760 


.55812 


1.79174 


SO 


:i 


.47021 


2.12671 


.49170 


2.03376 


.51356 


1.94718 


•S3582 


1.86630 


•55850 


I.79051 


49 


12 


.47056 


2.12511 


.49206 


2.03227 


.51393 


1.94579 


•53620 


1.86499 


.55888 


1.78929 


48 


13 


.47092 


2.12350 


.49242 


2.03078 


.51430 


1.94440 


•53657 


1.86369 


•55926 


1.78807 


47 


14 


.47128 


2. 12190 


.49278 


2.02929 


.51467 


1.94301 


•53694 


1.86239 


•55964 


1.78685 


46 


IS 


.47163 


2.12030 


.4931s 


2.02780 


.51503 


1. 94162 


•53732 


1.86109 


.56003 


1.78563 


45 


i6 


.47199 


2.11871 


.49351 


2.02631 


•51540 


1.94023 


•53769 


1-85979 


.56041 


1^78441 


44 


17 


.47234 


2.11711 


.49387 


2.02483 


•51577 


1.93885 


•53807 


1.85850 


.56079 


1.78319 


43 


i8 


.47270 


2.11552 


.49423 


2.02335 


.51614 


1^93746 


.53844 


1.85720 


.56117 


1-78198 


42 


19 


.4730s 


2.11392 


.49459 


2.02187 


•51651 


1.93608 


.53882 


1-85591 


.56156 


1.78077 


41 


20 


.47341 


2.11233 


■49495 


2.02039 


.51688 


1.93470 


•53920 


1.85462 


•56194 


1-77955 


40 


21 


.47377 


2.11075 


.49532 


2.01891 


.51724 


I^93332 


•53957 


1.85333 


.56232 


1-77834 


39 


22 


.47412 


2.10916 


.49568 


2.01743 


.51761 


1-93195 


.53995 


1.85204 


.56270 


I-77713 


38 


23 


.47448 


2.10758 


.49604 


2.01596 


.51798 


1-93057 


•54032 


1.85075 


.56309 


1.77592 


37 


24 


.47483 


2.10600 


.49640 


2.01449 


.S183S 


1.92920 


.54070 


1.84946 


•56347 


1-77471 


36 


25 


.47519 


2.10443 


.49677 


2.01302 


.51872 


1.92782 


•54107 


1.84818 


.56385 


I-77351 


35 


26 


.47555 


2.10284 


.49713 


2.0II5S 


.51909 


1.92645 


.54145 


1.84689 


.56424 


1.77230 


34 


27 


.47590 


2.10126 


.49749 


2.01008 


■51946 


1.92508 


.54183 


1.84561 


.56462 


1.77110 


33 


28 


.47626 


2.09969 


.49786 


2.00862 


•51983 


1-92371 


.54220 


1-84433 


.56501 


1.76990 


32 


29 


.47663 


2.09811 


.49822 


2.00715 


•52020 


1^92235 


.54258 


I-8430S 


.56539 


1.76869 


31 


30 


.47698 


2.09654 


.49858 


2.00569 


•52057 


1.92098 


.54296 


1.84177 


•56577 


1.76749 


30 


31 


.47733 


2.09498 


.49894 


2.00423 


.52094 


I. 91962 


.54333 


1.84049 


•56616 


1.76629 


29 


32 


.47769 


2.09341 


.49931 


2.00277 


•52131 


I. 91826 


.54371 


1.83922 


.56654 


1.76510 


28 


33 


.4780s 


2.09184 


.49967 


2.00131 


.52168 


1.91690 


.54409 


1.83794 


.56693 


1.76390 


27 


34 


.47840 


2.09023 


.50004 


1.99986 


.52205 


I.9I5S4 


.54446 


1.83667 


.56731 


1.76271 


26 


3S 


.47876 


2.08873 


.50040 


1.99841 


.52242 


I.91418 


.54484 


1.83540 


.56769 


1.76151 


25 


36 


.47912 


2.08716 


.50076 


1.99695 


.52279 


1-91282 


•54522 


1-83413 


.56808 


1.76032 


24 


37 


.47948 


2.08560 


.50113 


1.99550 


.52316 


1-91147 


•54560 


1.83286 


.56846 


I-75913 


23 


38 


.47984 


2.0840s 


.50149 


1.99406 


.52353 


I.91012 


•S4597 


1.83159 


.56885 


1-75794 


22 


39 


.48019 


2.08250 


.50185 


1. 99261 


.52390 


1.90876 


•54635 


1.83033 


.56923 


1.75675 


21 


40 


.48055 


2.08094 


.50222 


1.99116 


.52427 


1. 9074 1 


•54673 


1.82906 


.56962 


1-75556 


20 


41 


.48091 


2.07939 


.50258 


1.98972 


•52464 


1.90607 


.54711 


1.82780 


.57000 


1-75437 


19 


42 


.48127 


2.0778;; 


.50295 


1.98828 


.52501 


1.90472 


.54748 


1.82654 


.57039 


I-75319 


18 


43 


.48163 


2.07630 


.50331 


1.98684 


•52538 


1-90337 


.54786 


1.82528 


.57078 


1.75200 


17 


44 


.48198 


2.07476 


.50368 


1.98540 


•S2S75 


1.90203 


•54824 


1.82402 


.57116 


1.75082 


16 


4S 


.48234 


2.073-'2i 


.50404 


1.98396 


.52613 


1.90069 


.54862 


1.82276 


•57155 


1.74964 


15 


46 


.48270 


2.07167 


.50441 


1.98253 


.52650 


1-89935 


.54900 


1.82150 


•57193 


I-74846 


14 


47 


.48306 


2.07014 


.50477 


1.98110 


.52687 


1.89801 


.54938 


1.82025 


•57232 


1-74728 


13 


48 


.48342 


2.06860 


.50514 


1.97966 


.52724 


1.89667 


•54975 


1.81899 


•57271 


I-74610 


12 


49 


.4S378 


2.06706 


.50550 


1.97823 


.52761 


1.89533 


•55013 


1.81774 


.57309 


1-74492 


11 


SO 


.48414 


2.06553 


.50587 


1. 97681 


.52798 


1.89400 


•55051 


1.81649 


•57348 


1.74375 


10 


51 


.48450 


2.06400 


.S0623 


1.97538 


.52836 


1.89266 


•55089 


I. 81524 


.57386 


1.74257 


9 


32 


.484B6 


2.06247 


.50660 


1.97395 


.52873 


1-89133 


•55127 


1.81399 


.57425 


1.74140 


8 


53 


.48521 


2.06094 


.50696 


1.97253 


.52910 


1.89000 


•55165 


1.81274 


•57464 


1.74022 


7 


S4 


.48557 


2.05942 


.50733 


1.97111 


.52947 


1.8S867 


•55203 


1.81150 


•57503 


1.73905 


6 


SS 


.48593 


2.05790 


.50769 


1.96969 


•52985 


1.88734 


•55241 


1.81025 


•57541 


I. 73788 


S 


S6 


.48629 


2.05637 


.50806 


1.96827 


.53022 


I. 886c 2 


•55279 


1.80901 


•57580 


1.73671 


4 


S7 


.48665 


2.0548s 


.50843 


1.96685 


•53059 


1.88469 


•55317 


1.80777 


•57619 


1.73555 


3 


S8 


.48701 


2.05333 


.50879 


1.96544 


.53096 


1.88337 


•55355 


1.80653 


•57657 


1.73438 


2 


S9 


.48737 


2.05182 


.50916 


1.96402 


.53134 


1.88205 


.55393 


1.80529 


•57696 


1.73321 


I 


60 


.48773 


2.05030 


.50953 


1.96261 


.53171 


1.88073 


•S543I 


I. 80405 


•5773S 


1.7320s 





/ 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


/ 


6. 


i° 


6 


3° 


6: 


2° 


6 


[° 


6 


3° 



NATURAL TANGENTS AND COTANGENTS 



167 



1 


30° 


3] 





3- 





33° 


34° 


/ 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 





.57735 


1.73205 


.60086 


1.66428 


.62487 


1.60033 


.64941 


1.53986 


.67451 


1.48256 


60 


I 


.57774 


1.73089 


.60126 


1.66318 


.62527 


1.59930 


.64982 


1.53888 


.67493 


1.48163 


59 


2 


.57813 


1.72973 


.60165 


1.66209 


.62568 


1.59826 


.65024 


1.S379I 


.67536 


1.48070 


58 


3 


.57851 


1.72857 


.60205 


1.66099 


.62608 


1.59723 


-6506s 


1.53693 


.67578 


1.47977 


57 


4 


.57890 


1. 72741 


.60245 


1.65990 


.62649 


1.59620 


.65106 


I.S359S 


.67620 


1.4788s 


56 


S 


.57929 


1.7262s 


.60284 


1. 6588 1 


.62689 


1.59517 


.65148 


1.53497 


.67663 


1.47792 


55 


6 


.57968 


1.72509 


.60324 


1.65772 


.62730 


1.59414 


.65189 


1-53400 


.67705 


1.47699 


54 


7 


.58007 


1-72393 


.60364 


1.65663 


.62770 


1.5931 1 


.65231 


1.53302 


.67748 


1.47607 


S3 


8 


.58046 


1.72278 


.60403 


1.65554 


.62811 


1.59208 


.65272 


1-53205 


.67790 


I-47514 


52 


9 


.58085 


I. 72163 


.60443 


1-65445 


.62852 


I-S9I0S 


.65314 


1-53107 


.67832 


1.47422 


51 


10 


.58124 


1.72047 


.60483 


1.65337 


.62892 


1.59002 


.65355 


1.53010 


.67875 


1.47330 


50 


II 


.58162 


1. 71932 


.60522 


1.65228 


.62933 


1.58900 


.65397 


1.52913 


.67917 


1.47238 


49 


12 


.58201 


1.71817 


.60562 


1.65120 


.62973 


1-58797 


.65438 


1.52816 


.67960 


1.47146 


48 


13 


.58240 


1.71702 


.60602 


1. 65011 


.63014 


i^586gs 


.65480 


1. 52719 


.68002 


1.47053 


47 


14 


.58279 


1. 71588 


.60642 


1.64903 


.63055 


I^58S93 


.65521 


1.52622 


.68045 


1.46962 


46 


IS 


.58318 


1. 71473 


.60681 


1.64795 


.63095 


1.58490 


.65563 


1.52525 


.68088 


1.46870 


45 


l6 


.58357 


I. 71358 


.60721 


1.64687 


.63136 


1.58388 


■65604 


1.52429 


.68130 


1.46778 


44 


17 


.5S396 


I. 71244 


.60761 


1.64579 


.63177 


1.58286 


.65646 


1.52332 


.68173 


I.466E6 


43 


i8 


.58435 


1.71129 


.60801 


1.64471 


.63217 


1.58184 


.65688 


1.5223s 


.68215 


1.46595 


42 


19 


.58474 


1.71015 


.60841 


1.64363 


.63258 


1.58083 


.65729 


1.52139 


.68258 


1.46503 


41 


20 


.58513 


1.70901 


.60881 


1.64256 


.63299 


1.57981 


.65771 


1.52043 


.68301 


1.4641 1 


40 


21 


.58552 


1.70787 


.60921 


1.64148 


.63340 


1.57879 


.65813 


I.51946 


.68343 


1.46320 


39 


22 


.58591 


1.70673 


.60960 


1.64041 


.63380 


1^57778 


.65854 


1. 51850 


.68386 


1.46229 


38 


23 


.58631 


1.70560 


.61000 


1.63934 


.63421 


1.57676 


.65896 


1.51754 


.68429 


1.46137 


37 


24 


.58670 


1.70446 


.61040 


1.63826 


.63462 


1.57575 


.65938 


1.51658 


.68471 


1.46046 


36 


2S 


.58709 


1.70332 


.61080 


1.63719 


■63503 


1.57474 


.65980 


1.51562 


.68514 


1.45955 


35 


26 


.58748 


1. 70219 


.61120 


1. 63612 


.63544 


1.57372 


.66021 


1.51466 


.68557 


1.45864 


34 


27 


.58787 


I. 70106 


.61160 


1.63505 


-63584 


1-57271 


.66063 


1-51370 


.68600 


1.45773 


33 


28 


.58826 


1.69992 


.61200 


1.63398 


.6362s 


I-57170 


.66105 


1-51275 


.68642 


1.45682 


32 


29 


.58865 


1.69879 


.61240 


1.63292 


.63666 


1.57069 


.66147 


1.51179 


.68685 


1-45592 


31 


30 


.58905 


1.69766 


.61280 


1.63185 


.63707 


1.56969 


.66189 


1.51084 


.68728 


I.45501 


30 


31 


.58944 


1.69653 


.61320 


1.63079 


.63748 


1.56868 


.66230 


1.50988 


.68771 


1.45410 


29 


32 


.58983 


1.69541 


.61360 


1.62972 


.63789 


1.56767 


.66272 


1.S0893 


.68814 


1.45320 


28 


33 


.59022 


1.69428 


.61400 


1.62866 


.63830 


1.56667 


.66314 


1.S0797 


.68857 


1.45229 


27 


34 


.59061 


1. 69316 


.61440 


1.62760 


.63871 


1.56566 


.66356 


1.50702 


.68900 


1.45139 


26 


35 


.59101 


1.69203 


.61480 


1.62654 


.63912 


1.56466 


.66398 


1.50607 


.68942 


1.45049 


25 


36 


.59149 


1. 6909 1 


.61520 


1.62548 


■63953 


1.56366 


.66440 


1.50S12 


.68985 


1.44958 


24 


37 


.59179 


1.68979 


.61561 


1.62442 


■63994 


1.5626s 


.66482 


1. 50417 


.69028 


1.44868 


23 


38 


.59218 


1.68866 


.61601 


1.62336 


■6403s 


1.56165 


.66524 


1.50322 


.69071 


1.44778 


22 


39 


.59258 


1.68754 


.61641 


1.62230 


.64076 


1.5606s 


.66566 


1.50228 


.69114 


1.44688 


21 


40 


.59297 


1.68643 


.61681 


1. 62125 


.64117 


1.55966 


.66608 


1.50133 


.69157 


1.44598 


20 


41 


.59336 


1.68531 


.61721 


1.62019 


.64158 


1.55866 


.66650 


1.S0038 


.69200 


1.44508 


19 


42 


.59376 


1. 68419 


.61761 


1. 61914 


.64199 


1.55766 


.66692 


1.49944 


.69243 


1.44418 


18 


43 


.59415 


1.68308 


.61801 


1.61808 


.64240 


1.55666 


.66734 


1.49849 


.69286 


1.44329 


17 


44 


•59454 


1. 68196 


.61842 


I. 61703 


.64281 


1.55567 


.66776 


1-49755 


.69329 


1.44239 


16 


45 


.59494 


1.6808s 


.61882 


1.61598 


.64322 


1.55467 


.66818 


1.49661 


.69372 


1.44149 


15 


46 


.59533 


1.67974 


.61922 


1.61493 


.64363 


1.55368 


.66860 


1.49566 


.69416 


1.44060 


14 


47 


.59573 


1.67863 


.61962 


1.6138S 


.64404 


1.55269 


.66902 


1.49472 


.69459 


1.43970 


13 


48 


.59612 


1.67752 


.62003 


1.61283 


.64446 


1^55170 


.66944 


1.49378 


.69502 


1.43881 


12 


49 


.59651 


1. 67641 


.62043 


1.61179 


.64487 


1.SS071 


.66986 


1.49284 


.69545 


1.43792 


11 


50 


.59691 


1.67530 


.62083 


1. 61074 


.64528 


1-54972 


.67028 


1.49190 


.69588 


1.43703 


10 


SI 


.59730 


1.67419 


.62124 


1.60970 


.64569 


1.54873 


.67071 


1.49097 


.69631 


1.43614 


9 


52 


.59770 


1.67309 


.62164 


1.60865 


.64610 


1.54774 


.67113 


1.49003 


.69675 


1.43525 


8 


53 


.59809 


1.67198 


.62204 


1.60761 


.64652 


l^S467S 


.67155 


1.48909 


.69718 


1.43436 


7 


54 


.59849 


1.67088 


.62245 


1.60657 


.64693 


1.54576 


.67197 


1.48816 


.69761 


1.43347 


6 


SS 


.59888 


1.66978 


.62285 


1-60553 


.64734 


1.54478 


.67239 


1.48722 


.69804 


1.43258 


5 


S6 


.59938 


1.66867 


.62325 


1.60449 


.64775 


1.54379 


.67282 


1.48629 


.69847 


1.43169 


4 


57 


.59967 


1.66757 


.62366 


1.6034s 


.64817 


1. 54281 


.67324 


1.48536 


.69891 


1.43080 


3 


58 


.60007 


1.66647 


.62406 


1.60241 


.64858 


1. 54183 


.67366 


1.48442 


.69934 


1.42992 


2 


59 


.60046 


1.66538 


.62446 


1. 60137 


.64899 


1.54085 


.67409 


1.48349 


.69977 


1-42903 


I 


6o 


.60086 


1.66428 


.62487 


1.60033 


.64941 


1.53986 


.67451 


1.48256 


.70021 


1.4281s 





/ 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


/ 


5 


?° 


5 


3° 


5 


7° 


5< 


5° 


5 


5° 



168 NATURAL TANGENTS AND COTANGENTS 



/ 


35° 


36° 


37° 


38° 


39° 


/ 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 





.70021 


I •4281s 


.72654 


1^37638 


.75355 


1.32704 


.78129 


1.27994 


.80978 


1.23490 


60 


I 


.70064 


1.42726 


.72699 


1^37554 


.75401 


1.32624 


■7817s 


1.27917 


.81027 


1.23416 


59 


2 


.70107 


1.42638 


.72743 


1.37470 


•75447 


1.32544 


.78222 


1.27841 


.81075 


1.23343 


S8 


3 


.70151 


1.42550 


.72788 


1.37386 


•75492 


1.32464 


.78269 


1.27764 


.81123 


1.23270 


57 


4 


.70194 


I^42462 


.72832 


1.37302 


•75538 


1.32384 


.78316 


1.27688 


.81171 


I. 23196 


S6 


S 


.70238 


1^42374 


.72877 


1.37218 


•75584 


1.32304 


.78363 


1.27611 


.81220 


1.23123 


55 


6 


.70281 


1.42286 


.72921 


1.37134 


•75629 


1.32224 


.78410 


1.2753s 


.81268 


1.23050 


54 


7 


.70325 


1. 42198 


.72966 


I. 37050 


•75675 


1.32144 


.78457 


1.27458 


.81316 


1.22977 


53 


8 


.70368 


I.42IIO 


.73010 


1.36967 


•75721 


1.32064 


.78504 


1.27382 


.81364 


1.22904 


52 


9 


.70412 


1.42022 


.73055 


1.36883 


•75767 


I. 31984 


.78551 


1.27306 


.81413 


1.22831 


SI 


10 


.70455 


1^41934 


.73100 


1.36800 


•75812 


1.31904 


.78598 


1.27230 


.81461 


1.22758 


SO 


II 


.70499 


1. 41847 


.73144 


1.36716 


.75858 


1.3182s 


.7864s 


1.27153 


.81510 


1.2268s 


49 


12 


.70542 


1^41759 


.73189 


1.36633 


.75904 


1.31745 


.78692 


1.27077 


.81558 


1.22612 


48 


13 


.70586 


1.41672 


.73234 


1^36549 


.75950 


1. 31666 


.78739 


1. 27001 


.81606 


1.22539 


47 


14 


.70629 


1.41584 


.73278 


1.36466 


.75996 


1.31586 


.78786 


1.26925 


.81655 


1.22467 


46 


IS 


.70673 


I.41497 


.73323 


1.36383 


.76042 


1.31507 


.78834 


1.26849 


.81703 


1.22394 


45 


i6 


.70717 


1. 41409 


.73368 


i^36300 


.76088 


1.31427 


.78881 


1.26774 


.81752 


1.22321 


44 


17 


.70760 


1.41322 


.73413 


I^362I7 


.76134 


1.31348 


.78928 


1.26698 


.81800 


1.22249 


43 


i8 


.70804 


1.4123s 


.73457 


1^36134 


.76180 


1.31269 


.7897s 


1.26622 


.81849 


1.22176 


42 


19 


.70848 


I.41148 


■73502 


1^36051 


.76226 


1.31190 


.79022 


1.26546 


.81898 


1.22104 


41 


20 


.70891 


1.41061 


■73547 


1.35968 


.76272 


1. 31110 


.79070 


1.26471 


.81946 


1.22031 


40 


21 


.7093s 


1.40974 


•73592 


1.35885 


.76318 


1.31031 


.79117 


1.2639s 


.81995 


1.21959 


39 


22 


.70979 


1.40887 


•73637 


1.35802 


.76364 


1.30952 


.79164 


1.26319 


.82044 


1.21886 


38 


23 


.71023 


1.40800 


•73681 


1.35719 


.76410 


1.30873 


.79212 


1.26244 


.82092 


1.21814 


37 


24 


.71066 


1.40714 


.73726 


1.35637 


.76456 


1 ■3079s 


■79259 


1.26169 


.82141 


1.21742 


36 


2S 


.71110 


1.40627 


■73771 


1 .35554 


.76502 


1. 30716 


.79306 


1.26093 


.82190 


I. 21670 


35 


26 


.71154 


1.40540 


•73816 


1^35472 


.76548 


1.30637 


.79354 


1.26018 


.82238 


1.21598 


34 


27 


.71198 


1.40454 


.73861 


i^35389 


.76594 


l^30S58 


■79401 


1.25943 


.82287 


1.21526 


33 


28 


.71242 


1.40367 


■73906 


1^35307 


.76640 


1.30480 


■79449 


1.25867 


.82336 


1.21454 


32 


29 


.71285 


1.40281 


■73951 


1^35224 


.76686 


1.30401 


.79496 


1.25792 


.82385 


1.21382 


31 


30 


.71329 


1.40195 


■73996 


1.35142 


•76733 


1.30323 


■79544 


1.25717 


.82434 


1. 21310 


30 


31 


•71373 


1.40109 


■74041 


1.35060 


•76779 


1.30244 


■79591 


1.25642 


.82483 


1.21238 


29 


32 


.71417 


1.40022 


.74086 


1.34978 


.76825 


I. 30166 


■79639 


1.25567 


•82531 


1. 21166 


28 


33 


.71461 


1.39936 


■74131 


1.34896 


.76871 


1.30087 


.79686 


1.25492 


.82580 


1.21094 


27 


34 


.7150s 


i^398so 


.74176 


1.34814 


.76918 


1.30009 


.79734 


1.25417 


.82629 


1.21023 


26 


35 


•71549 


1^39764 


.74221 


1^34732 


.76964 


1.29931 


.79781 


1.25343 


.82678 


1.209s I 


25 


36 


■71593 


i^39679 


.74267 


1.34650 


.77010 


1.29853 


.79829 


1.25268 


.82727 


1.20879 


24 


37 


.71637 


1^39593 


■74312 


1.34568 


•77057 


1.2977S 


.79877 


I.25193 


.82776 


1.20808 


23 


38 


.71681 


1.39507 


•74357 


1.34487 


•77103 


1.29696 


.79924 


1.25118 


.82825 


1.20736 


22 


39 


.71725 


1.39421 


•74402 


1 .34405 


•77149 


1. 29618 


.79972 


1.25044 


.82874 


1.2066s 


21 


40 


.71769 


1.39336 


•74447 


1^34323 


.77196 


1.29541 


.80020 


1.24969 


.82923 


1.20593 


20 


41 


.71813 


1.39250 


.74492 


1.34242 


.77242 


1.29463 


.80067 


1.2489s 


.82972 


1. 20522 


19 


42 


•71857 


1.39165 


•74538 


1.34160 


.77289 


1.29385 


.8011S 


1.24820 


.83022 


1.20451 


18 


43 


.71901 


1.39079 


•74583 


1.34079 


.77335 


1^29307 


.80163 


1.24746 


.83071 


1.20379 


17 


44 


•71946 


1.38994 


.74628 


1^33998 


•77382 


1.29229 


.80211 


1.24672 


.83120 


1.20308 


16 


4S 


.71990 


1.38909 


.74674 


I^339i6 


.77428 


1.29152 


.80258 


1.24597 


.83169 


1.20237 


15 


46 


•72034 


1.38824 


.74719 


i^3383S 


•77475 


1.29074 


.80306 


1 ■24523 


.83218 


1.20166 


14 


47 


.72078 


1.38738 


.74764 


1 ■33754 


■77521 


1.28997 


.80354 


1.24449 


.83268 


1. 2009s 


13 


48 


.72122 


1.38653 


.74810 


1-33673 


■77568 


1.28919 


.80402 


1.24375 


.83317 


1.20024 


12 


49 


•72167 


1.38568 


■74855 


I ■33592 


■77615 


1.28842 


.80450 


1.24301 


.83366 


1.19953 


II 


SO 


.72511 


1.38484 


.74900 


l^33Sii 


.77661 


1.28764 


.80498 


1.24227 


.8341S 


1.19882 


10 


SI 


•72255 


1.38399 


■74946 


l^33430 


.77708 


1.28687 


.80546 


I.24IS3 


.8346s 


1.19811 


9 


S2 


.72299 


i^383i4 


■74991 


i^33349 


■77754 


1.28610 


.80594 


1.24079 


.83514 


1.19740 


8 


S3 


.72344 


1.38229 


■75037 


1.33268 


.77801 


1.28533 


.80642 


1.24005 


.83564 


1. 19669 


7 


S4 


.72388 


1.3814s 


■75083 


1.33187 


.77848 


1.28456 


.80690 


1.23931 


.83613 


I. I 9599 


6 


SS 


.72432 


1.38060 


.75128 


I^33I07 


.7789s 


1.28379 


.80738 


1.23858 


.83662 


1. 19528 


S 


S6 


■72477 


1.37976 


•75173 


1^33026 


.77941 


1.28302 


.80786 


1.23784 


.83712 


1.19457 


4 


57 


.72521 


1.37891 


•75219 


1.32946 


.77988 


1.28225 


.80834 


1.23710 


.83761 


1. 19387 


3 


58 


■72565 


1.37807 


■75264 


i^3286s 


■7803s 


1.28148 


.80882 


1.23637 


.83811 


1.19316 


2 


59 


.72610 


1.37722 


■75310 


1.3278s 


.78082 


1.28071 


.80930 


1.23563 


.83860 


1. 19246 


1 


6o 


•72654 


1.37638 


■75355 


1.32704 


.78129 


1.27994 


.80978 


1.23490 


.83910 


1.1917s 





/ 


Cotang 


Tang 


Cotang 


Tang ( 


I^otang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


1 


5^ 


t° 


5: 


i° 


52 


>° 


53 





5< 


)° 



NATURAL TANGENTS AND COTANGENTS 



1G9 



/ 


40° 


4 


° 


42° 


43° 


44° 


/ 




Tangf 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang ' 


rang 


Cotang 


o 


.83910 


1.19175 


.86929 


1. 15037 


.00040 


1.11061 


.93252 


1.07237 


96569 


1-03553 


60 


I 


.83960 


1.19105 


.86980 


1.14969 


.90093 


1. 1 0996 


.93306 


1.07174 


9662s 


1.03493 


59 


a 


.84009 


1. 1 9035 


.87031 


1.14902 


.90146 


1.10931 


.93360 


1. 07112 


96681 


1.03433 


S8 


3 


.84059 


1. 1 8964 


.87082 


1. 14834 


.90199 


1.10867 


.93415 


1.07049 


96738 


1.03372 


57 


4 


.84108 


1. 1 8894 


.87133 


1.14767 


.90251 


1. 10802 


■93469 


1.06987 


96794 


1. 033 1 2 


56 


5 


.84158 


1. 1 8824 


.87184 


1.14699 


.90304 


1. 10737 


■93524 


1.0692s 


96850 


1.03252 


55 


fi 


.84208 


1.18754 


.87236 


1.14632 


.90357 


1.10672 


■93578 


1.06862 


96907 


1.03192 


54 


I 


.84258 


1.18684 


.87287 


1.1456s 


.90410 


1.10607 


■93633 


1 .06800 


96963 


1.03132 


S3 


.84307 


1.18614 


.87338 


1.14498 


.90463 


I.IOS43 


.93688 


1.06738 


97020 


1.03072 


52 


9 


.84357 


1. 18544 


.87389 


1.14430 


.90516 


1.10478 


.93742 


1.06676 


97076 


1.03012 


SI 


10 


.84407 


1.18474 


.87441 


1. 14363 


.90569 


1.10414 


.93797 


I. 06613 


97133 


1.02952 


SO 


II 


■84457 


1. 18404 


.87492 


1.14296 


.90621 


1.10349 


.93852 


1.06551 


97189 


1.02892 


49 


12 


.84507 


1. 1 8334 


.87543 


1.14229 


.90674 


1.10285 


.93906 


1.06489 


97246 


1.02832 


48 


13 


.84556 


1. 1 8264 


.87595 


1.14162 


.90727 


1.10220 


.93961 


1.06427 


97302 


1.02772 


47 


14 


.84606 


1. 18194 


.87646 


1.14095 


.90781 


1.10156 


.94016 


1.06365 


97359 


I. 02713 


46 


IS 


.84656 


1.1812s 


.87698 


1.14028 


.90834 


1.10091 


.94071 


1.06303 


97416 


1.02653 


45 


l6 


.84706 


1.180SS 


.87749 


1.13961 


.90887 


1.10027 


.94125 


I. 06 24 I 


97472 


1.02593 


44 


i7 


.84756 


1.17986 


.87801 


1.13894 


.90940 


1.09963 


.94180 


1.06179 


97529 


1.02533 


43 


i8 


.84806 


1.17916 


.87852 


1.13828 


.90993 


1.09899 


.94235 


1.06117 


975S6 


1.02474 


42 


J9 


.84856 


1.17846 


.87904 


1.13761 


.91046 


1.09834 


.94290 


1.06056 


97643 


1.02414 


41 


20 


.84906 


1.17777 


■87955 


I. 13694 


.91099 


1.09770 


.94345 


I. 05994 


97700 


1.02355 


40 


ai 


.84956 


1.17708 


.88007 


I. 13627 


.91153 


1.09706 


.94400 


1.05932 


97756 


1.0229s 


39 


23 


.85006 


1.17638 


.88059 


1.13561 


.91206 


1.09642 


.94455 


1.05870 


97813 


1.02236 


38 


as 


.85057 


1-17569 


.88110 


1.13494 


.91259 


1.09578 


.94510 


1.05809 


97870 


1.02176 


37 


24 


.85107 


1.17500 


.88162 


1.13428 


.91313 


1.09514 


.94565 


I.OS747 


97927 


1.02117 


36 


2S 


.85157 


1. 1 7430 


.88214 


1.13361 


.91366 


1.09450 


.94620 


1.0568s 


97984 


1.02057 


35 


26 


.85207 


1.17361 


.88265 


1.13295 


.91419 


1.09386 


.94676 


1.05624 


98041 


1. 01998 


34 


27 


.85257 


1.17292 


.88317 


1.13228 


.91473 


1.09322 


.94731 


1.05562 


98098 


1.01939 


33 


28 


.85308 


1.17223 


.88369 


1.13162 


.91526 


1.09258 


.94786 


l.OSSOI 


98155 


1.01879 


32 


29 


.85358 


1.17154 


.88421 


1.13096 


.91580 


1.09195 


.94841 


1.05439 


98213 


1.01820 


31 


30 


.85408 


1.1708s 


.88473 


1. 13029 


•91633 


1.09131 


.94896 


1.05378 


98270 


1.01761 


30 


31 


.85458 


1.17016 


.88524 


1.12963 


.91687 


1.09067 


.94952 


1.05317 


98327 


1.01702 


29 


32 


.85509 


1.16947 


.88576 


1. 1 2897 


.91740 


1.09003 


.95007 


1.0525s 


98384 


1.01642 


28 


33 


.85559 


1.16878 


.88628 


1.12831 


.91794 


1.08940 


.95062 


1.OS194 


98441 


1.01583 


27 


34 


.85609 


1.16809 


.88680 


1.1276s 


.91847 


1.08876 


.95118 


I.05133 


98499 


1.01524 


26 


35 


.85660 


1.16741 


.88732 


1. 1 2699 


.91901 


1. 088 1 3 


.95173 


1.05072 


98556 


1.0146s 


25 


36 


.85710 


1.16672 


.88784 


1.12633 


.91955 


1.08749 


.95229 


1. 05010 


98613 


1.01406 


24 


37 


.85761 


1.16603 


.88836 


1.12567 


.92008 


1.08686 


.95284 


1.04949 


98671 


1.01347 


23 


38 


.85811 


1.16535 


.88888 


1.12501 


.92062 


1.08622 


.95340 


1.04888 


98728 


1.01288 


22 


39 


.85862 


1.16466 


.88940 


1. 1 2435 


.92116 


1.08559 


.95395 


1.04827 


98786 


1. 01 229 


21 


40 


.85912 


I. 16398 


.88992 


1.12369 


.92170 


1.08496 


.95451 


1.04766 


98843 


1. 01 1 70 


20 


41 


.85963 


1.16329 


.89045 


1. 1 2303 


.92224 


1.08432 


.95506 


1.0470s 


98901 


1.01112 


19 


42 


.86014 


1.16261 


.89097 


1. 12238 


.92277 


1.08369 


.95562 


1.04644 


98958 


1-01053 


18 


43 


.86064 


1.16192 


.89149 


1. 12172 


.92331 


1.08306 


.95618 


1.04583 


99016 


1.00994 


17 


44 


.86ns 


1.16124 


.89201 


1.12106 


.9238s 


1.08243 


.95673 


1.04522 


99073 


1.0093s 


16 


45 


.86166 


I. 16056 


.89253 


I. 12041 


.92439 


1. 08 1 79 


.95729 


I.04461 


99131 


1.00876 


15 


46 


.86216 


1.15987 


.89306 


I.11975 


.92493 


1.08116 


.95785 


1.04401 


99189 


1.00818 


14 


47 


.86267 


1.15919 


.89358 


1.11909 


.92547 


1.08053 


.95841 


1.04340 


99247 


1.00759 


13 


48 


.86318 


1.158S1 


.89410 


1.11844 


.92601 


1.07990 


.95897 


1.04279 


99304 


I. 00701 


12 


49 


.86368 


1. 15783 


.89463 


1.11778 


.92655 


1.07927 


.95952 


1.04218 


99362 


1.00642 


II 


SO 


.86419 


1.15715 


.89515 


1.11713 


.92709 


1.07864 


.96008 


1.04158 


99420 


1.00583 


10 


SI 


.86470 


1.15647 


.89567 


1.11648 


.92763 


1. 07801 


.96064 


1.04097 


99478 


1.00525 


9 


52 


.86521 


1.15579 


.89620 


1.11582 


.92817 


1.07738 


.96120 


1.04036 


99536 


1.00467 


8 


53 


.86572 


I.I5SII 


.89672 


1.11517 


.92872 


1.07676 


.96176 


1.03976 


99594 


1.00408 


7 


54 


.86623 


1. 15443 


.89725 


1.11452 


•.92926 


1.07613 


.96232 


1.0391s 


99652 


i.otyso 


6 


55 


.86674 


1. 15375 


.89777 


1.11387 


.92980 


I.07S50 


.96288 


1.03855 


99710 


1.00291 


5 


56 


.86725 


1.15308 


.89830 


1.11321 


.93034 


I.074S7 


.96344 


1-03794 


99768 


1.00233 


4 


57 


.86776 


1.15240 


.89883 


1.11256 


.93088 


1.07425 


.96400 


1-03734 


99826 


1.0017s 


3 


58 


.86827 


1.15172 


.89935 


1.11191 


.93143 


1.07362 


.96457 


1-03674 


99884 


1.00116 


2 


59 


.86878 


1. 15104 


.89988 


1.11126 


.93197 


1.07299 


.96513 


I-03613 


99942 


1.00058 


I 


60 


.86929 


1.ISO37 


.90040 


1.11061 


.93252 


1.07237 


.96569 


1.03553 I 


00000 


I. 00000 





1 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang C 


otang 


Tang 


1 


4( 


?° 


4^ 


?° 


4: 


7° 


4( 


5° 


4 


5° 



170 



BROWN & SHARPE MFG. CO. 



NATUEAL SECAl^T. 



Deg. 


0' 


10' 


20' 


30' 


43' 


50 


60' 







1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0001 


1.0001 


89 


1 


1.0001 


1 . 0002 


1.0002 


1.0003 


1.0004 


1.0005 


1.0006 


88 


3 


1.0006 


1.0007 


1 . 0008 


1.0009 


1.0010 


1.0013 


1.0013 


87 


3 


1.0013 


1.0015 


1.0016 


1.0018 


1.0020 


1.0033 


1.0034 


86 


4 


1.0024 


1.0036 


1.0028 


1.0030 


1.0083 


1.0035 


1.0038 


85 


5 


1.0088 


1.0040 


1.0043 


1.0046 


1.0049 


1.0053 


1.0055 


84 


6 


1.0055 


1.0058 


1.0081 


1.0064 


1.0068 


1.0071 


1.0075 


83 


7 


1.0075 


1.0078 


1.0082 


1.0086 


1.0090 


1.0094 


1.0098 


83 


8 


1.0098 


1.0103 


1.0108 


1.0111 


1.0115 


1.0130 


1.0124 


81 


9 


1.0124 


1.0139 


1.0134 


1.0139 


1.0144 


1 . 0149 


1.0154 


80 


10 


1.0154 


1.0159 


1.0164 


1.0170 


1.0175 


1.0181 


1.0187 


79 


11 


1.0187 


1.0193 


1.0198 


1.0204 


1.0210 


1.0317 


J. 0333 


76 


13 


1.0223 


1.0329 


1.0236 


1.0242 


1.0249 


1.0856 


1.0363 


77 


13 


1.0263 


1.0269 


1.0277 


1.0384 


1.0291 


1.021)8 


1.0303 


76 


14 


1.0308 


1.0313 


1.0321 


1.0329 


1.0336 


1.0344 


1.0353 


75 


15 


1.0352 


1.0360 


1 . 0369 


1.0377 


1.0385 


1.0394 


1.0403 


74 


16 


1.0403 


1.0411 


1.0420 


1.0429 


1.0438 


1.0447 


1.0456 


73 


17 


1.0456 


1.0466 


1.0475 


1.0485 


1.0494 


1.0504 


1.0514 


73 


18 


1.0514 


1.0524 


1.0534 


1.0544 


1.0555 


1.0565 


1.0576 


71 


19 


1.0576 


1.0586 


1.0597 


1.0608 


1.0619 


1.0630 


1.0641 


70 


20 


1.0641 


1.0653 


1.0664 


1.0876 


1.0887 


1.0699 


1.0711 


69 


31 


1.0711 


1.0723 


1.0735 


1.0747 


1.0760 


1.0773 


1.0785 


68 


23 


1.0785 


1.0798 


1.0810 


1.0823 


1.0837 


1.0850 


1.0863 


67 


23 


1.0863 


1.0877 


1.0890 


1.0904 


1.0918 


1.0933 


1.0946 


66 


24 


1.0946 


1.0960 


1.0974 


1.0989 


1.1004 


1.1018 


1.1033 


65 


35 


1.1033 


1.1048 


1.1063 


1.1079 


1.1094 


1.1110 


1.1126 


64 


36 


1.1126 


1.1141 


1.1157 


1.1174 


1.1190 


1.1206 


1.1233 


63 


27 


1.1223 


1.1239 


1.1256 


1.1273 


1.1290 


1.1308 


1.1335 


63 


28 


1.1325 


1.1343 


1.1361 


1.1378 


1.1396 


1.1415 


1.1433 


61 


29 


1 . 1433 


1.1452 


1.1470 


1.1489 


1.1508 


1.1537 


1.1547 


60 


30 


1.1547 


1.1566 


1.1586 


1.1605 


1.1625 


1.1646 


1.1666 


59 


31 


1.1666 


1.1686 


1.1707 


1.1738 


1.1749 


1.1770 


1.1791 


58 


32 


1.1791 


1.1813 


1.1835 


1.1856 


1.1878 


1.1901 


1.1923 


57 


33 


1.1923 


1.1946 


1.1969 


1.1993 


1.2015 


1.2038 


1.3062 


56 


34 


1.2082 


1.2085 


1.2109 


1.3134 


1.3158 


1.2182 


1.3307 


55 


35 


1.2207 


1.3333 


1.2257 


1.3383 


1.3308 


1.2334 


1.3360 


54 


36 


1.2360 


1.2386 


1.3413 


1.3440 


1.8466 


1.2494 


1.3531 


53 


37 


1.2531 


1.2548 


1.3576 


1.3804 


1.2633 


1.2661 


1.3690 


53 


38 


1.36i?0 


1.2719 


1.2748 


1.3777 


1.3807 


1.3837 


1.3887 


51 


39 


1.2867 


1.2898 


1.3928 


1.3959 


1.2990 


1.3033 


1.3054 


50 


40 


1.3054 


1.3086 


1.3118 


1.3150 


1.3183 


1.3216 


1.3350 


49 


41 


1.3350 


1.3283 


1.3317 


1.3351 


1.3386 


1.3421 


1.3456 


48 


43 


1.3456 


1.3491 


1.3527 


1.3563 


1.3599 


1.3636 


1.3673 


47 


43 


1.3673 


1.3710 


1.3748 


1.3785 


1.3824 


1.3863 


1.3901 


46 


44 


1.3901 


1.3940 


1.3980 


1.4030 


1.4060 


1.4101 


1.4143 


45 




GO' 


50' 


40' 


30' 


.20' 


10' 


0- 


Deg. 



NATURAL COSECANT. 



TROVIDENCE, R. I. 



171 



NATUEAL SECANT. 



1 
Deg. 


0' 


10' 


20' 


30' 


40' 


50' 


GO' 




45 


1.4143 


1.4183 


1.4325 


1.4267 


1.4309 


1.4352 


1.4395 


44 


46 


1.4395 


1.4439 


1.4483 


1.4537 


1.4573 


1.4617 


1.46G3 


43 


47 


1.4663 


1.4708 


1.4755 


1.4801 


1.4849 


1.4896 


1.4944 


43 


48 


1.4944 


1.4993 


1.5043 


1.5091 


1.5141 


1.5191 


1.5343 


41 


49 


1.5343 


1.5393 


1.5345 


1.5397 


1.5450 


1.5503 


1.5557 


40 


50 


1.5557 


1.5611 


1.5666 


1.5731 


1.5777 


1.5833 


1.5890 


39 


51 


1.5890 


1.5947 


1.6005 


1.6063 


1.6122 


1.6182 


1.6343 


38 


52 


1.6342 


1.6303 


1.6364 


1.6436 


1.6489 


1.6552 


1.6616 


37 


53 


1.6616 


1.6680 


1.6745 


1.6811 


1.6878 


1,6945 


1.7013 


36 


54 


1.7013 


1.7081 


1.7150 


1.7220 


1.7391 


1.7362 


1.7434 


35 


55 


1.7434 


1.7507 


1.7580 


1.7655 


1.7780 


1.7806 


1.7883 


34 


56 


1.7883 


1.7960 


1.8038 


1.8118 


1.8198 


1.8278 


1.8360 


33 


57 


1.8360 


1.8443 


1.8537 


1.8611 


1.8697 


1.8783 


1.8870 


33 


58 


1.8870 


1.8.959 


1.9048 


1.9138 


1.9330 


1.9322 


1.9416 


31 


59 


1.9416 


1.9510 


1.9608 


1.9702 


1.9800 


1.9899 


3.0000 


30 


60 


2.0000 


2.0101 


3.0303 


2.0307 


3.0413 


2.0519 


3.0836 


89 


61 


2.0636 


2.0735 


3.0845 


2.0957 


3.1070 


2.1184 


3.1300 


28 


63 


3.1300 


2.1417 


3.1536 


3.1656 


3.1778 


3.1901 


3.8026 


37 


63 


3.3036 


2.2153 


3.3381 


2.2411 


3.3543 


2.2676 


3.3811 


86 


64 


3.3811 


3.3948 


3.3087 


2.3338 


3.3370 


3.3515 


2.3668 


25 


65 


3.3662 


3.3810 


3.3961 


3.4114 


2.4289 


3.4436 


8.4585 


34 


66 


2.4585 


3.4747 


3.4911 


3.5078 


3.5347 


2.5418 


3.5593 


23 


67 


2.5593 


3.5769 


3.5949 


3.6131 


3.6316 


2.6503 


3.6694 


33 


68 


3.6694 


3.6883 


3.7085 


3.7385 


3.7488 


2.7694 


3.7904 


31 


69 


3.7904 


3.8117 


3.8334 


3.8554 


3.8778 


2.9006 


3.9838 


30 


70 


2.9338 


3.9473 


3.9713 


3.9957 


3.0305 


3.04.58 


3.0715 


19 


71 


3.0715 


3.0977 


3.1343 


3.1515 


3.1791 


3.3073 


3.3360 


18 


73 


3.2360 


3.3653 


3.3951 


3.3355 


3.3564 


3.3880 


3.4803 


I 17 


73 


3.4303 


3.4531 


3.4867 


3.5309 


3 5558 


3.5915 


3.6379 


; 16 


74 


3.6379 


3.6651 


3.7031 


3.7419 


3.7816 


3.8338 


3.8637 


15 


75 


3.8837 


3.9061 


3 9495 


3.9939 


4.0393 


4.0859 


4.1335 


14 


76 


4.1335 


4.1833 


4.3333 


4.3836 


4.3363 


4.3901 


4.4454 


13 


77 


4.4454 


4.5031 


4.5604 


4.6303 


4.6816 


4.7448 


4.8097 


13 


78 


4.8097 


4.8764 


4.9451 


5 0158 


5.0886 


5.1635 


5.8408 


li 


79 


5.8408 


5.3304 


5.4036 


5.4874 


5.5749 


5.6653 


5.7587 


10 


80 


5.7587 


5.8553 


5.9553 


6.0588 


6.1660 


6.3771 


6.3934 


9 


81 


6.3934 


6.5120 


0.6363 


6.7654 


6.8997 


7.0396 


7.1853 


8 


83 


7.1853 


7.3371 


7.4957 


7.6613 


7.8344 


8.0156 


8.3055 


7 


83 


8.3055 


8.4046 


8.6137 


8.8336 


9.0651 


9.3091 


9.5667 


6 


84 


9.5667 


9.8391 


10.137 


10.433 


10.758 


11.104 


11.473 


5 


85 


11.473 


11.868 


13.391 


13.745 


13.234 


13.763 


14.335 


4 


86 


14.335 


14.957 


15.636 


16.380 


17.198 


18.103 


19.107 


3 


87 


19.107 


30.230 


31.493 


33.935 


34.563 


36.450 


88 653 


3 


88 


38.653 


31.357 


34.382 


38.201 


43 975 


49.114 


57.398 


1 


89 


57.398 


68.757 


85.945 


114.59 


171.88 


343.77 


00 







60' 


50' 


40' 


30' 


20' 


10' 


0' 


Deg. 



NATUEAL COSECANT. 



172 



BROWN & SHARPE MFG. CO. 



DECIMAL EQUIVALENTS OF PARTS OF AN INCH, 



eV ... .01563 

sV - 03125 

-i-^ ... .04688 
i-i6 0625 

^\ ... .07813 

3^ 09375 

/^ ... .10938 
1-8 125 

-i^ ... .14063 

3^ 15625 

^ ... .17188 
3-i6 1875 

if ... .20313 

^2 21875 

If ... .23438 
1-4 25 

1| ... .26563 

A 28125 

if ... .29688 
5-i6 3125 



If ... .32813 

^ 34375 

If ... .35938 
3-8 375 

2 5 ... ..39063 

if 40625 

If ... .42188 
7-i6 4375 

If ... .45313 

-If 46875 

If ... .48438 
i-2 5 

If ... .51563 
■II 53125 

fl ... .54688 

6 4 

9-i6 5625 

If ... .57813 

if 59375 

If ... .60938 
5-8 625 

If ... .64063 
If ...... .65625 

If ... .67188 
11-16 6875 



If ... .70313 

If 71875 

If ... .73438 
3-4 75 

If ... .76563 

If 78125 

If ... .79688 
13 16 8125 

If ... .82813 

If 84375 

If ... .85938 
7-8 875 

If ... .89063 

If 90625 

If ... .92188 
15-16 9375 

If ... .95313 

If 96875 

If ... .98438 
1 1.00000 



BROWN & SHARPE MFG. CO. 



173 



TABLE OF DECIMAL EQUIVALENTS 



MILLIMETRES AND FRACTIONS OF MILLIMETRES. 



mm. Inches. 


mm. Inches. 


mm. Inches. 


mm. Inches. 


jf5 = .00039 


Wo = -01399 


m = -03530 


Wo = -03740 


iio = .00079 


fj = .01339 


jlo = .03C59 


fo = .03780 


m = -00118 


^0 = -01373 


^ = .03598 


fo = -03819 


■joo = .00157 


tl = .01417 


^ = .03633 


fo = .03853 


Wo = -00107 


Wo = -01457 


Wo = -03677 


-^ = .03898 


ilo = -00330 


ij = .01496 


1 = -03717 


1 = .03937 


j^ = .0037a 


1^ = .01535 


70 „„_.„ 

100 ~ -0-J'o6 


3 = .07874 


ilo = .00315 


100 = -01575 


m = -03795 


3 = .11811 


i5o = -00354 


^0 = -01614 


Joo = .03835 


4 = .15748 


^ = .00394 


fo = .01654 


fo = -03874 


5 = .19685 


^ == .00433 


fo = .01693 


fo = -03913 


6 = .23633 


m = -00473 


Wo = -01733 


if, = .03953 


7 = .37559 


^ = .00513 


#0 = .01773 


^0 = -03993 


8 = .31496 


m = -00551 


M = -01811 


m = -03033 


9 = .354^3 


ij = .00591 


^ = .01850 


^0 = -03071 


10 == .39370 


^ = .00630 


^ = .01890 


f - -03110 


11 = .4a307 


^ = .00009 


m = -01939 


^; = .03150 


13 = .47344 


^ = .00709 


Wo = -01969 


fo = -03189 


13 = .51181 


m = -00748 


^ = .03008 


fo = -03338 


14 = .55118 


Z = -00787 


M = -03047 


fo = .03368 


15 = .59055 


if = .00837 


^ = -03087 


fo = .03307 


16 = .63993 


^ = .00866 


H = .03L36 


fo = .03346 


17 = .66939 


"jflQ ^ .00906 


M = -03165 


fo = .03386 


18 = .70866 


^ = .00945 


-g = .03305 


fo = -03435 


19 = .74803 


H = .00984 


fo = -03244 


fo = .03465 


30 = .78740 


^ = .01024 


^ = .03383 


■^0 = -03504 


21 = .83677 


^ = .01063 


fo = -03333 


fo = .03543 


22 = .86614 


Oft 

^0 = -01103 


^ = .03363 


fo = .03583 


33 = .90551 


M = -01143 


Wo = -03103 


fo = .03633 


34 = .94488 


Wo = -01181 


Wo = -03441 


ff = .03661 


35 = .98435 


m = -01330 


M = -03480 


^ = .03701 


26 =1.03363 


> fj = .01360 









10 mm. = 1 Centimeter = 0.3f)37 inches. 
10 cm. = 1 Decimeter = 3.937 inches. 



10 Im. = 1 Meter = 39.37 inches. 
25.4 mm. = 1 English Inch. 



INDEX 



A. 

PAGE 

Abbreviations of Parts of Teeth and Gears 4 

Addendum 2 

Angle, How to Lay Off an 88, 105 

Angle Increment 104 

Angle of Edge 100 

Angle of Face 103 

Angle of Pressure 135 

Angle of Spiral Ill 

Angular Velocity 3 

Annnlar Gears 32, 137 

Arc of Action 136 

B. 

Base Circle 11 

Base of Epicycloidal System 25 

Base of Internal Gears 137 

Bevel Gear Blanks 34 

Bevel Gear Cutting on B. & S. Automatic Gear Cutter 53 

Bevel Gear Angles by Diagram 36 

Bevel Gear Angles by Calculation 100, 104 

Bevel Gear, Form of Teeth of 41 

Bevel Gear, Whole Diameter of 36, 103 

C. 

Centers, Line of 2 

Chordal Thickness 142, 148 

Circular Pitch, Linear or 4 

Classification of Gearing 5 

Clearance at Bottom of Space 6 

Clearance in Pattern Gears 8 

Condition of Constant Velocity Eatio 2 

Contact, Arc of 136 

Continued Fractions 130 

Coppering Solution 85 

Cutters, How to Order 83 

Cutters, Table of Epicycloidal 84 



176 INDEX. 

PAGE. 

Cutters, Table of Involute 8:^ 

Cutters, Table of Speeds for 81 

Cuttiog Bevel Gears on B. & S. Automatic Gear Cutter 52 

Cutting Spiral Gears in a Universal Milling Machine 120 

D. 

Decimal Equivalents, Tables of 172 

Diameter Increment 102 

Diameter of Pitch Circle 6 

Diameter Pitch 5 

Diametral Pitch 17 

Distance between Centers 8 

E. 

Elements of Gear Teeth 5 

Epicycloidal Gears, with, more and less than 15 Teeth 30 

Epicycloidal Gears, with 15 Teeth 25 

Epicycloidal Rack 27 

F. 

Face, Width of Spur Gear 80 

Flanks of Teeth in Low-numbered Pinions 20 

G. 

Gear Cutters, How to Order 83 

Gear Patterns 8 

Gearing Classified 5 

Gears, Bevel 34, 41, 100 

Gears, Epicycloidal 25 

Gears, Involute 9 

Gears, Spiral ..107, 120 

Gears, Worm 63 

H. 
Herring-bone Gears 128 

I. 

Increment, Angle 104 

Increment, Diameter 102 

Interchangeable Gears 24 

Internal or Annular Gears 32, 137 

Involute Gears, 30 Teeth and over 9 

Involute Gears, with Less than 30 Teeth 20 

Involute Rack ,„ „. 12 



INDEX. 177 

L 

PAOE. 

Lead of a Worm 62 

Limiting Numbers of Teeth in Internal Gears 32 

Line of Centers 2 

Line of Pressure 12, 135 

Linear or Circular Pitch 4 

Linear Velocity 1 

M. 

Machine, B. & S., for Cutting Bevel Gears 52 

Module 6 

N. 

Normal 114 

Normal Helix 114 

Normal Pitch 114 

0. 

Original Cylinders 1 

P. 

Pattern Gears .,.... = 8 

Pitch Circle 3 

Pitch, Circular or Linear 4 

Pitch, a Diameter 6 

Pitch, Diametral 17 

Pitch, Normal 114 

Pitch of Spirals 110 

Polygons, Calculations for Diameters of 95 

B, 

Pack 12 

Back for Epicycloidal Gears 27 

Back for Involute Gears 12 

Back for Spiral Gears 119 

Belative Angular Velocity 2 

Boiling Contact of Pitch Circle 3 

S. 

Screw Gearing 107, 128 

Single-Curve Teeth 9 

Speed of Gear Cutters 81 



178 INDEX. 

PAGE. 

Spiral Gearing 107, 120 

Standard Templets 27 

Strength of Gears 140 

T. 

Table of Decimal Equivalents , 172 

Table of Sines, etc . 150-171 

Table of Speeds for Gear Cutters 81 

Table of Tooth Parts 146-149 

V. 

Velocity, Angular 2 

Velocity, Linear 1 

Velocity, Kelative 2* 

Wear of Teeth.. ..., ., 80, 127 

Worm Gears, ..,0... , .,....o..,. 63 



LE N '09 



■.^*r 



Gearing 




PROVIDENCE. R. L, U» S. A