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
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1.96969
•52985
1.88734
•55241
1.81025
•57541
I. 73788
S
S6
.48629
2.05637
.50806
1.96827
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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