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Class _rJil^l„ CopyrightE^ L^ COPYRIGHT DEPOSI-K / //3 Practical Treatise m^Gearing. EIGHTH EDITION. BROWN & SHARPK MANUFACTURING CO. PROVIDENCE, R. I., U. S. A. 1905 C^ -2.- THE LIBRARY OF CONGRESS. Two Copies Received ! MAR 80 1305 Oopyrignt Entry m^i^ 36 f90S CLASS f^ XXs, Not ©•IPY A. X A \ %' COPYRIGHT, 188G, 1887, 1892, 1893, 1896, 1900, 1902, 1905, BY BROWN & SHARPE MFG. CO. 0^-IOI(pO <u 1^ PREFACE. This Book is made for men in practical life ; for those that would like to know how to construct gear wheels, but whose duties do not afford them suflficient leisure to acquire a technical knowledge of the subject. CONTENTS. P A K T I , Chapter I. PAGE. Pitch. Circle — Pitch — Tooth — Space — Addendum or Face — Flank — Clearance 1 Chapter II. Classification — Sizing Blanks and Tooth Parts from Linear or Circular Pitch — Center Distance 5 Chapter III. Single Curve Gears of 30 Teeth and more 9 Chapter IV. Pack to Mesh with Single Curve Gears having 30 Teeth and more 12 Chapter V. Diametral Pitch — Sizing Blanks and Teeth — Distance be- tween the Centers of Wheels 16 Chapter VI. Single-Curve Gears, having Less than 30 Teeth — Gears and Packs to Mesh with Gears having Less than 30 Teeth... 20 Chapter VIL Double-Curve Teeth— Gear of 15 Teeth— Pack 25 Chapter VIII. Donble-Curve Gears, having More and Less than 15 Teeth — Annnlar Gears 30 VI CONTENTS. Chapter IX. PAGE. Bevel Gear Blanks 34 Chapter X. Bevel Gears — Form and Size of Teeth — Cutting Teeth 41 Chapter XI. Worm Wheels — Sizing Blanks of 32 Teeth and more 63 Chapter XII. Sizing Gears when the Distance between the Centers and the Katie of Speeds are fixed — General Eemarks — Width of Face of Spur Gears — Speed, of Gear Cutters — Table of Tooth Parts 79 PART II, Chapter I. Tangent of Arc and Angle 87 Chapter II. Sine, Cosine and Secant — Some of their Applications in Machine Construction 93 Chapter III. Application of Circular Functions — Whole Diameter of Bevel Gear Blanks — Angles of Bevel Gear Blanks 100 Chapter IV. Spiral Gears — Calculations for Pitch of Spirals 107 Chapter V. Examples in Calculations of Pitch of Spirals — Angle of Spiral — Circumference of Spiral Gears — A few Hints on Cutting Ill CONTENTS. VII Chapter VI. PAGE. Normal Pitch of Spiral Gears — Curvature of Pitch Surface — Formation of Cutters 114 Chapter VII. Cutting Spiral Gears in a Universal Milling Machine......... 120 Chapter VIII. Screw Gears and Spiral Gears — General Eemarks 127 Chapter IX. Continued Fractions — Some Applications in Machine Con- struction ' 130 Chapter X. Angle of Pressure 135 Chapteb XI. Internal Gears 187 Chapter XII. Strength of Gears — Tables of Tooth Parts, Tables Sine, Cosine, etc., Index 140 PART I. CHAPTER I. PITCH CIRCLE, PITCH, TOOTH, SPACE, ADDENDUM OR FACE, FLANK, CLEARANCE. Let two cylinders, Fie-. 1, touch each other, their Original Cyi- '^ ' ° ' ' Indera. axes be parallel and the cylinders be on shafts, turning freely. If, now, we turn one cylinder, the adhesion of its surface to the surface of the other cylinder will make that turn also. The surfaces touching each other, without slipping one upon the other, will evi- dently move through the same distance in a given , . '' ° ° Linear Veloci- time. This surface speed is called linear velocity. ty. TANGENT CYLINDERS. ^ 44. Linear Velocity is the distance a point moves along a line in a unit of time. The hne described by a point in the circumference of either of these cylinders, as it rotates, may be called an arc. The length of the arc (which may be greater or less than the circumference of cylinder), described in a unit of time, is the velocity. The length, expressed in linear units, as inches, feet, etc., is the linear velocity. Z BKOWN & SHARPE MFG. CO. The length, expressed in angular units, as degrees, is the angular velocity. If now, instead of 1° we take 360°, or one turn, as lo^i°y "^^^ ^^ ^^® angular unit, and 1 minute as the time unit, the angular velocity will be expressed in turns or revolu- tions per minute. If these two cylinders are of the same size, one will make the same number of turns in a minute that the other makes. If one cylinder is twice as large as the other, the smaller will make two turns whUe the larger makes one, but the linear velocity of the surface of each cylinder remains the same. This combination would be very useful in mechan- ism if we could be sure that one cylinder would always turn the other without slipping. Relative An gular Velocity T^ig. 3 C/RCLE In the periphery of these two cylinders, as in Fig. 2, cut equidistant grooves. In any grooved piece the Land. places between grooves are called lands. Upon the Addendum, lands add parts ; these parts are called addenda. A Tooth. land and its addendum is called a tooth. A toothed Gear. cylinder is called a gear. Two or more gears with Train. teeth interlocking are called a trai7i. A line, c c', Fig. PROVIDENCE, R. I. 2 or 3, between the centers of two wheels is called the Line of cen- line of centers. A circle just touching the addenda *®'"®' is called the addendum circle. circle! ^"^^^^^ The circumference of the cylinders without teeth is called the pitch circle. This circle exists geometri- -^^^"^ Circle, cally in every gear and is still called the pitch circle p.^^j^ circle or the primitive circle. In the study of gear wheels, it t^ie*^^ Motive is the problem so to shape the teeth that the pitch Circle, circles will just touch each other without slipping. On two fixed centers there can tiu'n only two circles, one circle on each center, in a given relative angular velocity and touch each other without slipping. inig. 4 4 BROWN & SHARPE MFG. CO. Space. The groove between two teeth is called a space. In cut gears the width of space at pitch line and thickness of tooth at pitch line are equal. The distance between the center of one tooth and the center of the next tooth, LinearorCir- measured aloug the pitch line, is the linear or circular cular Pitch. . . pitch; that is, the linear or circular pitch is equal to a Tooth Thick- tooth and a space; hence, the thickness of a tooth at the pitch line is equal to one-half the linear or circular pitch. tioiis^of*^parts ^^^ D=diameter of addendum circle. g>JJ«eth and .< D'= diameter of pitch circle. " P'= linear or circular pitch. " if = thickness of tooth at pitch line. ** s = addendum or face, also length of working part of tooth below pitch line or flank. " 2s= D" or twice the addendum, equals the work- ing depth of teeth of two gears in mesh. *' /= clearance or extra depth of space below work- ing depth. " s+/= depth of space below pitch line. " D"+/= whole depth of space. " N^ number of teeth in one gear. ** 7r=3.1416 or the circumference when diameter isl. P' is read "P prime." D" is read **D second." ttIs read "pi." To And the If we multiply the diameter of any circle by n, the Circumf ©rcriCG and Diameter product will be the circumfereucB of this circle. If we of 3i Oirclp divide the circumference of any circle by 7t, the quo- tient will be the diameter of this circle. Pitch Point. The pitch point of the side of a tooth is the point at which the pitch circle or line meets the side of the tooth. A gear tooth has two pitch points. CHAPTER II. CLASSIFICATION-SIZING BLANKS AND TOOTH PARTS FROM CIRCULAR PITCH— CENTRE DISTANCE— PATTERN GEARS. If we conceive the pitch of a pair of gears to be ^j^^iements of made the smallest possible, we ultimately come to the conception of teeth that are merely lines upon the original pitch surfaces. These lines are called ele- ments of the teeth. Gears may be classified with reference to the elements of their teeth, and also with reference to the relative position of their axes or -shafts. In most gears the elements of teeth are either straight lines or helices (screw-like lines). Part I. of this book, treats upon three kinds of GEARS. First — Spub Geaes ; those connecting parallel shafts ^p^^ Gears. and whose tooth elements are straight. Second — Bevel Gears ; those connecting shafts Bevei Gears, whose axes meet when sufficiently prolonged, and the elements of whose teeth are straight lines. In bevel gears the surfaces that touch each other, without slipping, are upon cones or parts of cones whose apexes are at the same point where axes of shafts meet. Third — Screw ok Worm Gears; those connecting ^1°^^ ^J^^^^.^' shafts that are not parallel and do not meet, and the elements of whose teeth are helical or screw-like. The circular pitch and number of teeth in a wheel g_^^ being given, the diameter of the wheel and size of Blanks, &c. tooth parts are found as follows : Dividing by 3.1416 is the same as multiplying by rrln- -^^"^ s.ilie —-^-*-^^' hence, multiply the cu'- cumference of a circle by .3183 and the product will be the diameter of the circle. Multiply the cu'cular pitch by .3183 and the product will be the same part of the 6 BROWN & SHARPS MFG. CO. diameter of pitch circle that the circular pitch is of the ^ j^.g^j^g^gj. circumference of pitch circle. This part is called the Pitch, or ^oa module of the pitch. There are as many modules con- tained in the diameter of a pitch circle as there are teeth in the wheel. andtheAdden- Most mechanics make the addendum of teeth equal dum measure,! n i tt j • i j.i ^ ^ ^ thesame,radi- toe module. Hence we can designate the module by ^"^" the same letter as we do the addendum; that is, let 5 = the module. .3183 P — s, or circular pitch multiplied by .3183 =s, or the module. Diameter of Ns = D', or number of teeth in a wheel, multiplied Pitch Circle, -^y the modulc, equals diameter of pitch circle. (N+2) s = D, or add 2 to the number of teeth, mul- g^^'^^^^^^^^^'tiply the sum by the module and product will be the whole diameter. ■—=/, or one tenth of thickness of tooth at pitch line Clearance. equals amount added to bottom of space for clearance. Some mechanics prefer to make / equal to j^g- of the working depth of teeth, or .0625 D". One-tenth of the thickness of tooth at pitch-line is more than one-six- teenth of working depth, being .07854 D". Example. Example.— Wheel 30 teeth, li" circular pitch. P'= Sizes of Blank ^•^"' ^^^^^ ^=.75" or thickness of tooth equals f". 5 = parfs for'aear ^'S" X '3183 =.4775 = module for 11" P'. (See table of 9^^ 30 teeth^i3| tooth parts, pages 144-147. Pitch. D'=30x.4775"=14.325"=diameter of pitch-circle. D = (30+2) X. 4775"= 15.280"=diameter of adden- dum circle, or the diameter of the blank. f=j^ of .75"=.075"= clearance at bottom of space. "D"=r=2x.4775"=.9549"== working depth of teeth. D"-f/=2x. 4775"+. 075"=1.0299"= whole depth of space. s+/=.4775"+.075"=.5525"= depth of space inside of pitch-line. D"=2s or the working depth of teeth is equal to two modules. In making calculations it is well to retain the fourth place in the decimals, but when drawings are passed into the workshop, three places of decimals are suffi- cient. PROVIDENCE, K. I. Fig. 5, Spur Gearing. 8 BEOWN & SHAKPE MFG. CO. tw^en^'centers ^^^ distance between the centers of two wheels is of two Gears, evidently equal to the radius of pitch-circle of one wheel added to that of the other. The radius of pitch-circle is equal to s multiplied by one-half the number of teeth in the wheel. Hence, if we know the number of teeth in two wheels, in mesh, and the circular pitch, to obtain the distance betAveen centers we first find s ; then multiply s by one- half the sum of number of teeth in both wheels and the product will be distance between centers. Example. — What is the distance between the centers of two wheels 35 and 60 teeth, 1^" circular pitch. "We first find s to be l^'x .3183=. 3979". Multiplying by 47.5 (one-half the sum of 35 and 60 teeth) we obtain 18.899" as the distance between centers. Shr\nTa°gl *fn Pattern GearS should be made large enough to Gear Castings, allow for shrinkage in casting. In cast iron the shrinkage is about ^ inch in one foot. For gears one to two feet in diameter it is well enough to add simply -j-oo" of diameter of finished gear to the pattern. In gears about six inches diameter or less, the moulder will generally rap the pattern in the sand enough to make any allowance for shrinkage unnecessary. In pattern gears the spaces between teeth should be cut wider than finished gear spaces to allow for rapping and to avoid having too much cleaning to do in order to have gears run freely. In cut patterns of iron it is generally Metal Pattern enough to make spaces .015" to .02' wider. This makes clearance .03" to .04" in the patterns. Some moulders might want .06" to .07" clearance. Metal patterns should be cut straight ; they work better with no draft. It is well to leave about .005" to be finished from side of patterns after teeth are cut ; this extra stock to be taken away from side where cutter comes through so as to take out places where stock is broken out. The finishing should be done with file or emery wheel, as turning in a lathe is likely to break out stock as badly as a cutter might do. If cutters are kept sharp and care is taken when coming through the allowance for finishing is not nec- essary and the blanks may be finished before they are cut. CHAPTER III. SINGLE-CURVE GEARS OF 30 TEETH AND MORE. Single-curve teeth are so called because they have rpgg°^^® ^^^^ but one curve by theory, this curve forming both face and flank of tooth sides. In any gear of thirty teeth and more, this curve can be a single arc of a circle whose radius is one-fourth the radius of the pitch circle. In gears of thhty teeth and more, a fillet is added at bottom of tooth, to make it stronger, equal in radius to one-seventh, the widest part of tooth space. A cutter formed to leave this fillet has the advantage of wearing longer than it would if brought up to a corner. In gears less than thu-ty teeth this fillet is made the same as just given, and sides of teeth are formed with more than one arc, as will be shown in Chapter VI. Having calciilated the data of a gear of 30 teeth, f Example of a . ° . . ^ ' * Gear, N=30, P inch circular pitch (as we did in Chapter II. for Ih" =%"■ pitch), we proceed as follows : 1. Draw pitch cu^cle and point it off into parts equal Geometrical . 1 in ,1 • 1 •- 1 Construction. to one-halt the circular pitch. Fig. 6. 2. From one of these points, as at B, Fig. 6, draw radius to pitch circle, and upon this radius describe a semicircle ; the diameter of this semichcle being equal to radius of pitch circle. Draw addendum, working depth and whole depth circles. 3. From the point B, Fig. 6, where semichcle, pitch chcle and outer end of radius to pitch circle meet, lay off a distance upon semicircle equal to one-fourth the radius of pitch circle, shown in the figure at BA, and is laid off as a chord. 4. Through this new point at A, upon the semichcle, draw a circle concentric to pitch cii'cie. This last is 10 BKOWN & SHAEPE MFG. CO. IT-ig. 6 GEAR, 30 TEETH, CIRCULAR PITCH P'= %" or .75" N = 30 t= .375" S= .2387" T>"= .4775" S+f= .2762" T>"+f= .5150" D'rr 7.1610" D = 7.6384" SINGLE CURVE GEAR. PROVIDENCE, R. I. 11 called the base circle, and is the oue for centers of tooth arcs. In the system of single curve gears we have adopted, the diameter of this circle is .968 of the diameter of pitch circle. Thus the base circle of any gear 1 inch pitch diameter by this system is .968". If the pitch circle is 2" the base circle will be 1.936." 5. With dividers set to one-quarter of the radius of pitch circle, di'aw arcs forming sides of teeth, placing one leg of the dividers in the base circle and letting the other leg describe an arc through a point in the pitch circle that was made in laying off the parts equal to one-half the circular pitch. Thus an arc is drawn about A as center through B. 6. With dividers setto one-seventh of the widest part of tooth space, drav/ the fillets for strengthening teeth at their roots. These fillet arcs should just touch the whole depth circle and the sides of teeth already described. Single curve or involute gears are the only gears jj^^®j^^g^Qg^°^ that can run at varying distance of axes and transmit ''^^• unvarying angular velocity. This peculiarity makes involute gears specially valuable for driving rolls or any rotating pieces, the distance of whose axes is hkely to be changed. The assertion that gears crowd harder on bearings Pressure ou ° ° toearmgs. when of involute than when of other forms of teeth, has not been proved in actual practice. Before taking next chapter, the learner should make Practice, be- several drawings of gears 30 teeth and more. Say next chapter. make 35 and 70 teeth 1^" P'. Then make 40 and 65 teeth I" F. An excellent practice will be to make drawing on cardboard or Bristol-board and cut teeth to lines, thus making paper gears ; or, what is still better, make them of sheet metal. By placing these in mesh the learner can test the accuracy of his work. 12 CHAPTER IV. RACK TO MESH WITH SINGLE-CURVE GEARS HAYING 30 TEETH AND MORE. made^prepara,- This gear (Fig. 7) is made precisely the same as gear a°ifack*^'^^^'°^ "^ Chapter III. It makes no difference in which direc- tion the construction radius is drawn, so far as obtain- ing form of teeth and making gear are concerned. Here the radius is drawn perpendicular to pitch line of rack and through one of the tooth sides, B. A semi- circle is drawn on each side of the radius of the pitch circle. The points A and A' are each distant from the point B, equal to one-fourth the radius of pitch circle and correspond to the point A in Fig. 6. In Fig. 7 add two lines, one passing through B and A and one through B and A'. These two lines forra angles of 75^° (degrees) with radius BO. Lines BA , and BA' are called lines of pressure. The sides of rack teeth are made perpendicular to these lines. Kack. A Rack is a straight j)iece, ha^dng teeth to mesh with a gear. A rack may be considered as a gear of infinitely long radius. The circumference of a circle approaches a straight line as the radius increases, and when the radius is infinitely long any finite part of the Construction circumference is a straisrht line. The pitch line of a of Pitoli Line of . .... Rack. rack, then, is merely a straight line just touching the pitch circle of a gear meshing with the rack. The thickness of teeth, addendum and depth of teeth below pitch line are calculated the same as for a wheel. (For pitches in common use, see table of tooth parts.) The term circular pitch when applied to racks can be more accurately replaced by the term linear pitch. Linear applies strictly to aline in general while circular pertains to a circle. Linear pitch means the distance between the centres of two teeth on the pitch line whether the line is straight or curved. PROVIDENCE, R. I. 13 A rack to mesh with a single-curve gear of 30 teeth or more is drawn as follows : 1. Draw straight pitch line of rack ; also draw ad- dendum line, working depth line and whole depth line, each parallel to the pitch line (see Fig. 7). ■Rack. Fig. 7. RACK TO MESH WITH SINGLE CURVE GEAR HAVING 30 TEETH AND MORE. 14 BKOWN & SHARPE MFG. CO. 2. Point off the pitch line into parts equal to one- half the circular pitch, or =^. 3. Through these points draw lines at an angle of 75^° with pitch lines, alternate lines slanting in oppo- site directions. The left-hand side of each rack tooth is perpendicular to the line BA. The right-hand side of each rack tooth is perpendicular to the line BA'. 4. Add fillets at bottom of teeth equal to i of the width of spaces between the rack teeth at the adden- dum line. Bit^lt Rack "^^^ sketch, Fig. 8, wHl show how to obtain angle of Teeth. sides of rack teeth, directly from pitch line of rack, without drawing a gear in mesh with the rack. Upon the pitch line b b', draw any semicircle — baa' b'. From point h lay off upon the semicircle the distance b a, equal to one-quarter of the diameter of semicircle, and draw a straight line through b and a. This line, b a, makes an angle of 75 j° with pitch line b b', and can be one side of rack tooth. The same construction, b' a', will give the inclination 75^° in the opposite direction for the other side of tooth. The sketch, Fig. 9, gives the angle of sides of a tool for j)laning out spaces between rack teeth. Upon any line OB di"aw circle OABA'. From B lay off distance BA and BA', each equal to one-quarter of diameter of the circle. Draw lines OA and OA'. These two lines form an angle of 29°, and are light for inclination of sides of rack tool. PROVIDENCE, E. I. 15 Make end of rack tool .31 of circular pitch, and then J^'i'^*'* »* ^ac^ ^ ' Tool at end. round the corners of the tool to leave fillets at the bottom of rack teeth. Thus, if the circular pitch of a rack is 1^" and we multiply by .31, the product .465" will be the width of tool at end for rack of this i^itch before corners are taken off. This width is shown at x y. A Worm is a screw that meshes with tlie teeth of a gear. This sketch and the foregoing rule are also right for worm Thread a worm-thread tool, but a worm-tln-ead tool is not usually rounded for fillet. In cutting worms, leave width of top of thread .335 of the circular pitch. When this is done, the depth of thi-ead will be right. :.335 P' 16 CHAPTER V. DIAMETRAL PITCH— SIZING BLANKS AND THE TEETH OF SPUR GEARS —DISTANCE BETWEEN THE CENTRES OF WHEELS. necessary to ^^ making drawings of gears, and in cutting racks, cuiaxPitchf"^^^ ^^ necessary to know the circular pitch, both on account of spacing teeth and calculating their strength. It would be niore convenient to express the circular pitch in whole inches, and the most natural divisions AVheer^'thl^f an inch, as 1" P', f" P', J" P', and so on. But as ference^'mu'st^'^® circumference of the pitch circle must contain the ^'^"I'.^J.^pf^^'iy" circular j^itch some whole number of times, corre- n'\i"mblr°of ^P'^'^^^^^S ^^ ^^® number of teeth in the gear, the times. diameter of the pitch circle will often be of a size not readily measured with a common rule. This is because the circumference of a circle is equal to 3.1416 times the diameter, or the diameter is equal to the circumfer- ence multiplied by .3183. Pitch, in In practice, it is better that the diameter should be Terms of the ^ _ ' Diameter. of some size Conveniently measured. The same applies to the distance between centers. Hence it is generally more convenient to assume the pitch in terms of the diameter. In Chapter II. was given a definition of the module, and also how to obtain the module from the circular pitch. Circular Pitch ^e Can also assume the module and pass to its equiv- and a Diame- ^ ^ ter Pitch. alent circular pitch. If the circumference of the pitch circle is divided by the number of teeth in the gear, the quotient will be the circular pitch. In the same manner, if the diameter of the pitch circle is divided by the number of teeth, the quotient will be the module. Thus, if a gear is 12 inches pitch diameter and has 48 teeth, dividing 12" by 48, the quotient ^" is the module of this gear. In prac- PROVIDENCE, R. I. 17 tice, the module is taken in some convenient part of an inch, as V module and so on. It is convenient in /''•^'i^V^ti?" ' " of Module Diii- ealculation to designate one of these modules by s, as meter ritcb. in Chapter II. Thus, for ^" module, s is equal to ^". Generally, in speaking of the module, the denominator of the fraction only is named, i" module is then called 3 diametral pitch. That is, it has been found more convenient to take the reciprocal of the module in mak- ing calculation. The reciprocal of a number is 1 divi- Reciprocal of ded by that number. Thus the reciprocal of \ is 4. because \ goes into 1 four times. Hence, we come to the common definition : Diametral Pitch is the number of teeth to one inch Diametral Pitch of diameter of pitch circle. Let this be denoted by P. Thus, 1" diameter pitch we would call 4 diam,etral pitch or 4 P, because there would be 4 teeth to every inch in the diameter of pitch circle. The circular pitch and the different parts of the teeth are derived from the diametral pitch as follows. ^^^^ = P', or 3.1416 divided by the diametral pitch am Sto^flDd is equal to the circular pitch. Thus to obtain the cir- pitch? ^'^*'"^'^* cular for 4 diametral pitch, we divide 3 1416 by 4 and to obtain Cir- obtain .7854 for the circular pitch, corresponding to 4 ^j^^^jj^^'j)^^*^'^^ diametral pitch. trai Pitch. In this case we would write P=4, P'=: 7854", s=i". ^ r=s, or one inch divided by the number of teeth to an inch, gives distance on diameter of pitch circle occupied by one tooth or the module. -The addendum or face of tooth is the same distance as the module. ^ ::= P, or one inch divided by the module equals num- ber of teeth to one inch or the diametral pitch. ^ _„ Given, the Di- -'^^ = t, or 1.57 divided by the diametral pitch gives ametraiPitehto P ' -^ i- & ^^^ jjjg Thick- thickness of tooth at pitch line. Thus, thickness of ness of Tooth fit the Pitch teeth along the pitch line for 4 diametral pitch is .392". Line. ^=r:D', or number of teeth in a gear divided by the>,fi,mb°r *of diametral pitch equals diameter of the pitch circle. Jnd^tiie'^Di'am! Thus for a wheel, 60 teeth, 12 P, the diameter of pitch ^*^'5\ifJDi'am*? circle will be 5 inches. ^*frcie.* ^''''^ ^^=:D, or add 2 to the number of teeth in a wheel Given, the P ' N II m b e r o f and divide the sum by the diametral pitch ; and the Teetiiinfuyheei '' ' and the Diame- tral Pitch to find the Whole Diameter. 18 BKOWN & SHAKPE MFG. CO. quotient will be the whole diameter of the gear or the diameter of the addendum circle. Thus, for 60 teeth, 12 P, the diameter of gear blank will be 5 j-^ inches. p,=P, or number of teeth divided by diameter of pitch circle in inches, gives the diametral pitch or number of teeth to one inch. Thus, in a wheel, 24 teeth, 3 inches pitch diameter, the diametral pitch is 8. — ^ :=P, or add 2 to the number of teeth; divide the sum by the whole diameter of gear, and the quotient will be the diametral pitch. Thus, for a wheel 3^^^^" diameter, 14 teeth, the diametral pitch is 5. D' P=N, or diameter of pitch circle, multiplied by diametral pitch equals number of teeth in the gear. Thus, in a gear, 5 pitch, 8" pitch diameter, the num- ber of teeth is 40. D P — 3=N or multiply the whole diameter of the gear by the diametral pitch,subtract 2, and the remain- der will be the number of teeth. j^==s, or divide the whole diameter of a spur gear by the number of teeth plus two, and the quotient will be the module. + '^1*^^-4.^**™®' When we say the diametral pitch we shall mean the number of teeth to one inch of diameter of pitch cir- cle, or P, (^'=P). ametrai^ Pitch When the circular pitch is given, to find the corre- Ht^ ^^^°'^^*'^ spending diametral pitch, divide 3.1416 by the circular pitch. Thus 1.57 P is the diametral pitch correspond- ing to 2-inch circular pitch, (?^/J-^=P). Example. What diametral pitch corresponds to ^" circular pitch % Remembering that to divide by a fraction we multiply by the denominator and divide by the numer- ator, we obtain 6.28 as the quotient of 3.1416 divided by \ . 6.28 P, then, is the diametral pitch corresponding to \ circular pitch. This means that in a gear of \ inch circular pitch there are six and twenty-eight one hundredths teeth to every inch in the diameter .of the pitch circle. In the table of tooth parts the diametral pitches corresponding to circular pitches are carried out to four places of decimals, but in practice three places of decimals are euough. PROVIDENCE, R. I. 19 When two gears are in mesh, so that their pitch circles just touch, the distance between their axes or centers is equal to the sum of the radii of the two gears. The number of the modules between centers is equal to half the sum of number of teeth in both gears. This principle is the same as given in Chapter II., page 6, Rule to lind ^ ^ ^ P » l^ 6 ' Distance l)e- but when the diametral pitch and numbers of teeth in twecn centers. two gears are given, add together the numbers of teeth in the two ioheels and divide half the stan by the diametral pitch. The quotient is the center distance. A gear of 20 teeth, 4 P, meshes with a gear of 50 Example, teeth ; what is the distance between their axes or cen- ters? Adding 50 to 20 and dividing half the sum by 4, we obtain 8|" as the center distance. The term diametral pitch is also applied to a rack. Thus, a rack 3 P, means a rack that will mesh with a gear of 3 diametral pitch. It will be seen that if the expression for the module Fractional , •■ i. -I i! A. .Diametral has any number except 1 for a numerator, we cannot pitcb. express the diametral pitch by naming the denominator only. Thus, if the addendum or module is y'^^, the diametral pitch will be 2^, because 1 divided by y^ equals 2^. The term module is much used where gears are made to metric sizes, for the reason that, the millimeter being so short, the module is conveniently expressed in milli- meters. If we know the module of a gear we can figure the other parts as easily as we can if we know either the circular pitch or the diametral pitch. The module is, in a sense, an actual distance, while the diametral pitch, or the number of teeth to an inch, is a relation or merely a ratio. The meaning of the module is not easily mistaken. 20 CHAPTER VI. SINGLE-CDRYE GEARS HAYING LESS THAN 30 TEETH— GEARS AHD RACKS TO MESH Y?ITH GEARS HAYING LESS THAN 30 TEETH. K^^io^'^*'*''^' In Fig. 10, the construction of the rack is the same as the construction of the rack in Chapter IV. The gear in Fig. 10 is drawn from base circle out to adden- dum circle, by the same method as the gear in Chapter III., but the spaces inside of base circle are drawn as follows : Flanks of In gears, 12 to 19 teeth, the sides of space inside Gears m low » , , ,. , Numbers of of the base circle are radial for a distance, a b, equal Teeth, ' t. to 1^, or 3.5 divided by the product of the pitch by the number of teeth. In gears with more than 19 teeth the radial construction is omitted. Construction Then, with one leg of dividers in pitch circle in of Fig. 10 con- ° . tinued. center of next tooth, e, and other leg just touching one of the radial lines at h, continue the tooth side into c, until it will touch a fillet arc, whose radius is 1 the width of space at the addendum circle. The part, h' c\ is an arc from center of tooth g, etc. The flanks of teeth or spaces in gear, Fig. 11, are made the same as those in Fig. 10. This rule is merely conventional or not founded upon any principle other than the judgment of the de- signer, to effect the object to have spaces as wide as practicable, just below or inside of base circle, and then strengthen flank with as large a fillet as will clear addenda of any gear. If flanks in any gear will clear addenda of a rack, they will clear addenda of any Internal Gear, other gear, except internal gears. An internal gear is one having teeth upon the inner side of a rim or ring. Now, it will be seen that the gear. Fig. 10, has teeth PROVIDENCE, R. I Fig. 10 22 BKOWT^ & SHARPE MFG. CO. too much rounded at the points or at the addendum circle. In gears of pitch coarser than 10 to inch (10 Add^^nd'a of ■^)' ^^^ having less than 30 teeth, this rounding Teeth. becomes objectionable. This I'ounding occurs, because in these gears arcs of circles depart too far from the true involute curve, being so much that points of teeth get no bearing on flanks of teeth in other wheels. In gear, Fig. 11, the teeth outside of base circle are made as nearly true involute as a workman will be able to get without special machinery. This is accomplished tiiOTpto^Tmeinl ^® f ollows : draw three or four tangents to the base volute. circle, i i', J J', k k', 1 1', letting the points of tangency on base circle i',j', k' , I' be about ^ or |- the circular pitch apart ; the first point, i', being distant from ^, equal to \ the radius of pitch circle. "With dividers set to ^ the radius of pitch circle, placing one leg in i', draw the arc, a' i j; with one leg in j', and radius j' j, draw J k; with one leg in k', and radius k' k draw k I. Should the addendum circle be outside of I, the tooth side can be completed with the last radius, I' I. The arcs, a' ij, j k and k I, together form a very close approximation to a true involute from the base circle, i' j' k' I'. The exact involute for gear teeth is the curve made by the end of a band when unwound from a cylinder of the same diameter as base circle. The foregoing operation of drawing tooth sides, although tedious in description, is very easy of practical application. Rounding of It will also be seen that the addenda of rack teeth Addenda of ^ n Rack. in Fig. 10, interfere with the gear-teeth flanks, as at m n; to avoid this interference, the teeth of rack, Fig. 11, are rounded at points or addenda. It is also necessary to round off the points of invo- lute teeth in high-numbered gears, when they are to interchange with low-numbered gears. In interchange- able sets of gears the lowest-numbered pinion is usual- Tempietsly 12. Just how much to round off can be learned bv necessary for ^ i n i • i Rounding off makinef templets of a few teeth out of thin metal or Points of teeth. -, „ i -, ^ , cardboard, for the gear and rack, or, two gears re- quired, and fitting addenda of teeth to clear flanks. However accurate we may make a diagram, it is quite TROVIDENCE, K. I. Fig. 11 24 BROWN & SHARPE MFG. CO. Diagrams for a Set of Cut- ters. as well to make templets iu order to shape cutters accurately It is best to make cutters to corrected diagrams, as iu Fig 1 1 . When corrected diagrams are made, as in Fig. 1 1, take the following : For 12 and 13 teeth, diagram of 12 teeth. " 14 to 10 ' c u u 14 u u 17 ' 20 ^ ' " " 17 " " 21 • 25 ' ;t 21 " " 26 ' 34 ' " 26 " " 35 ' 54 ' " 35 " " 55 ' 134 ' ' '' " 55 " " 135 ' rack, ' "135 " Templets for large gears must be fitted to run with 12 teeth. 25 CHAPTER VII. DOUBLE-CDRYE TEETH— GEAR, 15 TEETH— RACK. In double-curve teeth the formation of tooth sides ^^^ ^ m^^l^J curve Tooth changes at the pitch Hne. In all gears the part of Faces are con- teeth outside of pitch line is convex ; in some gears the sides of teeth inside pitch line are convex ; in some, radial ; in others, concave. Convex faces and concave flanks are most familiar to mechanics. In interchange- able sets of gears, one gear in each set, or of each pitch, has radial flanks. In the best practice, this gear has fifteen teeth. Gears with more than fifteen teeth, have concave flanks; gears with less than fifteen teeth, have convex flanks. Fifteen teeth is called the Jiase of this system. We will fij.-st draw a a:ear of fifteen teeth. This , construction *^ , of Fig. 12. fifteen-tooth construction enters into gears of any number of teeth and also into racks. Let the gear be 3 P. Having obtained data, we proceed as follows : 1. Draw pitch cu'cle and point it off into pai'ts equal to one-thu'tieth of the circumference, or equal to thick- ness of tooth — ^. 2. From the center, through one of these points, as at T, Fig. 12, draw line OTA. Draw addendum and whole-depth circles. 3. About this point, T, with same radius as 15-tooth pitch circle, describe arcs A K and O k. For any other double-curve gear of 3 P., the radius of arcs, A K and O Jc, will be the same as in this 15-tooth gear =2^". In a 15-tooth gear, the arc, O k, passes through the center O, but for a gear having any other number of teeth, this construction arc does not pass through center of gear. Of course, the 15-tooth radius of arcs, A K and O k, is always taken from the pitch we are workinsf with. 26 BKOWN & SHAKPB MFG. CO. A^" GEAR, 3 P., 15 TEETH P= 3 N = 15 P'= 1.0472" t— .5236" S= .3833" D"=r .6666" S+f= .3857" D"+/= .7190" D'= 5.0000" D = 5.6666" V. DOUBLE CURVE GEAR. PROVIDENCE, K. I. 27 4. Upon these arcs on opposite sides of line OTA, lay off tooth thickness, A K and O k, and draw line KT 7c. 5. Perpendicular to K T Jc, draw line of pressvire, L T P ; also through O and A, draw lines A R and O r, perpendicular to K T k. The line of pressure is at an angle of 78° with the radius of gear. 6. From O, draw a line O R to intersection of A R with K T Jc. Through point c, where O R intersects L P, describe a circle about the center, 0. In this circle one leg of dividers is placed to describe tooth faces 7. The radius, c d, of arc of tooth faces is thv straight distance from c to tooth-thickness point, h, on the other side of radius, O T. With this radius, c h, describe both sides of tooth faces. 8. Draw flanks of all teeth radial, as O e and O f The base gear, 15 teeth only, has radial flanks. 9. With radius equal to one-seventh of the widest part of space, as g h, draw fillets at bottom of teeth. The foregoing is a close approximation to epicy- ^. ^'^'^'^^^^^^'_ cloidal teeth. To get exact teeth, make two 1 5 tooth cioidai Teeth, gears of thin metal. Make addenda long enough to come to a point, as at n and q. Make radial flanks, as at m and x>i deep enough to clear addenda when gears are in mesh. First finish the flanks, then fit the long addenda to the flanks when gears are in mesh. When these two templet gears are alike, the centers standard are the right distance apart and the teeth interlock without backlash, they are exact. One of these tem- plet gears can now be used to test any other templet gear of the same pitch. Gears and racks will be right when they run cor- rectly with one of these 15-tooth templet gears. Five or six teeth are enough to make in a gear templet. Double- cuEVE Rack. — Let us draw a rack 3 P. T,'^?^^-^^;f'Y7® Kack, Fig. la. Having obtained data of teeth we proceed as follows : 1. Draw pitch line and point it off in parts equal to one-half the circular pitch. Draw addendum and whole-depth lines. 2. Through one of the points, as at T, Fig. 13, draw line OTA perpendicular to pitch line of rack. 28 BROWN & SHARPE MFG. CO. I^ig. 13 DOUBLE CURVE RACK. PROVIDENCE, R. I. 2Q 3. About T make precisely the same construction as ■was made about T in Fig. 12. That is, with radius of 15-tooth pitch circle and center T draw arcs k and A K ; make O k and A K equal to tooth thickness ; draw K T k ; draw r, A R, and line of pressui-e, each perpendicular to K T 7c. 4. Through R and r, draw lines parallel to O A. Through intersections c and c' of these lines, with pressure line L P, draw lines parallel to pitch line. 5. In these last lines place leg of dividers, and draw faces and flanks of teeth as in sketch. 6. The radius c' d' of rack-tooth faces is the same length as radius c d of rack-tooth flanks, and is the straight distance from c to tooth-thickness point h on opposite side of line O A. 7. The radius for fillet at bottom of rack teeth is equal to |- of the widest part of tooth space. This radius can be varied to suit the judgment of the designer, so long as a fillet does not interfere with teeth of engaging gear. Vig. 14: Racks 9f the same pitch, to mesh with interchange- able gears, should be alike when placed side by side, and fit each other when placed together as in Fig. 14. In Fig. 13, a few teeth of a 15-tooth wheel are shown in mesh with the rack. 30 CHAPTER VIII. DOUBLE-CURVE SPUR GEARS, HAYING MORE AND FEWER THAN 15 TEETH— ANNULAR GEARS. ^f Construction Let US di'Rw two gears, 12 and 24 teeth, 4 P, in mesh. In Fig. 15 the construction lines of the lower or 24-tooth gear are full. The upper or 12-tooth gear construction lines are dotted. The line of pressure, L P, and the line K T ^ answer for both gears. The arcs A K and O k are described about T. The radius of these arcs is the radius of pitch circle of a gear 15 teeth 4 pitch. The length of arcs A K and O ^ is the tooth thickness for 4 P. The line K T /<; is obtained the same as in Chapter VII. for all double-curve gears, the distances only varying as the pitch. Having drawn the pitch circles, the line K T ^, and, perpendicular to K T h^ the lines A R, O r and the line of pressui'e L T P, we proceed with the 24-tooth gear as follows : 1. From center C, through r, draw line intersecting line of pressure in w^. Also draw line from center C to R, crossing the line of pressxu-e L P at c. 2. Through m describe circle concentric with pitch circle about C. This is the circle in which to place one leg of dividers to describe flanks of teeth. 3. The radius, ?n «, of flanks is the straight distance from m to the first tooth-thickness point on other side of line of centers, C C', at v. The arc is continued to n, to show how constructed. This method of obtain- ing radius of double-curve tooth flanks applies to all gears having more than fifteen teeth. 4. The construction of tooth faces is similar to 15- tooth wheel in Chapter VII. That is : Draw a circle through c concentric to pitch circle ; in this circle place one leg of dividers to di-aw tooth faces, the radius of tooth faces being c h. PROVIDENCE, R. I. 31 PINION, 12 TEETH, GEAR 24 TEETH, .4 P P=4 N=12 and 24 P'= .7854" t = .3927" S = .2500" D"= .5000' St/ = .2893' D"+/ = .5393' c<! DOUBLE CURVE GEARS IN lESH. 32 BROWN & SHAEPE MFG. CO. Construction 5 Yhe radius of fillets at roots of teeth is equal to of Fig. 15 con- ^ linued. one-seventh the width of space at addendum circle. Flanks for 12 ^^® constructioiis for flanks of 12, 13 and 14 13 and 14 Teeth, teeth are similar to each other and as follows : 1. Through center, C, draw line from K, intersecting line of pressure in u. Through u draw circle about C. In this circle one leg of dividers is placed for drawing flanks. 2. The radius of flanks is the distance from u to the first tooth-thickness point, e, on the scone side of C T C. This gives convex flanks. The arc is con- tinued to V, to show construction. 3. This arc for flanks is continued in or toward the center, only about one- sixth of the working depth (or Is.) ; the lower part of flank is similar to flanks of gear in Chapter VI. 4. The faces are similar to those in 15-tooth gear, Chapter VII., and to the 24-tooth gear in the fore- going, the radius being w y ; the arc is continued to tc, to show construction. Annular Gears. Anndlar Geaks. Gears with teeth inside of a rim or ring are called Annular or Internal Gears. The construction of tooth outlines is similar to the fore- going, but the spaces of a spur external gear become the teeth of an annular gear. It has been shown that in the system just de- scribed, the pinion meshing with an annular gear, must difi'er from it by at least fifteen teeth. Thus, a gear of 24 teeth cannot work with an annular gear of 36 teeth, but it will work with annular gears of 39 teeth and more. The fillets at the roots of the teeth must be of less radius than in ordinary spur gears. An annular gear differing from its mate by less than 15 teeth can be made. This will be shown in Part II. Annular-gear patterns requue more clearance for moulding than external or spur gears. Pinions. In speaking of different-sized gears, the smallest ones are often called " pinions." The angle of pressure in all gears except involute, constantly changes. 78° is the pressure angle in double-curve, or epicycloidal gears for an instant PROVIDENCE. R. I. 33 only; in our example, it is 78° wheu one side of a tootli reaches the line of centers, and the pressure against teeth is applied in the direction of the arrows. The pressure angle of involute gears does not change. An explanation of the term angle of pressure is given in Part II. We obtain the forms for epicycloidal gear cutters by means of a machine called the Odontom Engine. This machine will cut original gears with theoretical accuracy. It has been thought best to make 24 gear cutters 24 Douuie- ° ° curve Gear for each pitch. This enables us to fill any require- Cu tters for ^ , ^ X eacli Pitch. ment of gear-cutting very closely, as the range covered by any one cutter is so small that it is exceedingly near to the exact shape of all gears so covered. Of course, a cutter can be exactly right for only one gear. Special cutters can be made, if desired. 34 CHAPTER IX. BEVEL-fiEAR BLANKS. Bevel Gears connect shafts whose axes meet when g''^|j®*-,'^^°^ sufficiently prolonged. The teeth of bevel gears are formed upon formed about the frustrums of cones whose apexes frustrums ol ^ cones. are at the same point where the shafts meet. In Fig. 16 we have the axes A O and B O, meeting at O, and the apexes of the cones also at O. These cones are called the pitch cones, because they roll upon each other, and because upon them the teeth are pitched. If, in any bevel gear, the teeth were sufficiently pro- longed toward the apex, they would become infinitely small ; that is, the teeth would all end in a point, or vanish at O. We can also consider a bevel gear as beginning at the apex and becoming larger and larger as we go away from the apex. Hence, as the bevel gear teeth are tapering from end to end, we may say BEVEL GEAR PITCH CONES. Fig. 16. that a bevel gear has a number of pitches and pitch circles, or diameters : in speaking of the pitch of a bevel gear, we mean always the pitch at the largest PROVIDENCE, U. I. 35 pitch circle, or at the largest pitch diameter, as at bd, Fig. 17. Fig. 17 is a section of three bevel gears, the gear o B q being twice as large as the two others. The outer surface of a tooth as m m' is called the face of Construction ot Bevel Gem the tooth. The distance m m' is usually called the Blanks, length of the face of the tooth, though the real length is the distance that it occupies upon the line O i. The outer part of a tooth at m n is called its large end, and the inner part m' n' the small end. Almost all bevel gears connect shafts that are at right angles with each other, and unless stated other- wise we always understand that they are so wanted. The directions given in connection with Fig. 17 apply to gears with axes at right angles. Having decided upon the pitch and the numbers of teeth : — 1. Draw centre lines of shafts, A O B and COD, at right angles. 2. Parallel to A O B, draw lines a b and c d, each distant from A O B, equal to half the largest pitch diameter of one gear. For 24 teeth, 4 pitch, this half largest pitch diameter is 3". 3. Parallel to COD, draw lines e f and g h, dis- tant from COD, equal to half the largest pitch diameter of the other gear. For a gear, 12 teeth, 4 pitch, this half largest pitch diameter is 1|". 4 At the intersection of these four lines, draw lines O i, O j, O k, and O 1 ; these lines give the size and shape of pitch cones. We call them " Cone Pitch Lines." 5. Perpendicular to the cone- i)itch lines and through the intersection of lines a b, c d, e f, and g h, draw lines m n, o p, q r. We have drawn also u v to show that another gear can be drawn from the same diagram. Four gears, two of each size, can be drawn from this diagram. 6. Upon the lines m u, o p, q r, the addenda and depth of the teeth are laid off, these lines passing 36 BROWN & SHARPE MFG. CO. through the largest pitch circle of the gears. Lay off the addendum, it being in these gears |-". This gives distance m n, o p, q r, and u v equal to the Avorking depth of teeth, which in these gears is |". The addendum of course is measured perpendicularly from the cone pitch lines as at k r. 7. Draw lines m, O n, O p, O o, O q, Or. These lines give the height of teeth above the cone- pitch lines as they approach O, and would vanish entirely at O. It is quite as well never to have the length of teeth, or face, m m' longer than one-third the apex distance m O, nor more than two and one- half times the circular pitch. 8. Having decided upon the length of face, draw limiting lines m'u' pei-pendicular to i O, q' r' perpen- dicular to k O, and so on. The distance between the cone-pitch lines at the inner ends of the teeth m' n' and q' r' is called the inner or smaller pitch diameter, and the circle at these points is called the smallest pitch circle. We now have the outline of a section of the gears thx'ough their axes. The distance m r is the whole diameter of the pinion. Dian^eter'^o'f '^^^ distance q o is the whole diameter of the gear. Bevel- G ear j^ practice these diameters can be obtained by rueasur- Blanke can be ^ • -^ obtained by ing the drawing. The diameter of pinion is 3.45" and Measuring'^' ° '■ Drawings. of the gear 6.22". We can find the angles also by measuring the drawing with a protractor. In the absence of a protractor, templetes can be cut to the drawing. The angle formed by line m m' with a b is the angle of face of pinion, in this pinion 59° 11', or 59^° nearly. The lines q q' and g h give us angle of face of gear, for this gear 22° 19', or 221° nearly The angle formed by m n with a b is called the angle of edge of pinion, in our sketch 26° 34', or about 26i°. The angle of edge of gear, line q r with g h, is 63° 26', or about 63|-°. In turning blanks to these angles we place one arm of the protractor or templet against the end of the hub, when trying angles of a blank. Some designers give the angles from the axes of gears, but PROVIDENCE, R. I. 37 FUj. 17 38 BROWN & SHARPE MFG. CO., it is not convenient to try blanks in this way. The method that we have given comes right also for angles as figured in compound rests. When axes are at right angles, the sum of angles of edge in the two gears equals 90°, and the sums of angle of edge and face in each gear are alike. The angles of the axes remaining the same, all pairs of bevel gears of the same ratio have the same angle of edge ; all pairs of same ratio and of same numbers of teeth have the same angles of both edges and faces independent of the pitch. Thus, in all pairs of bevel gears having one gear twice as large as the other, with axes at right angles, the angle of edge of large gear is 63° 26', and the angle of edge of small gear is 26° 34'. In all pairs of bevel gears with axes at right angles, one gear having 24 teeth and the other gear having 12 teeth, the angle of face of small gear is 59° 11'. Another -phe following method of obtaining the whole diam- method ot ob- - ° * taming Whole ter of bevel gears is sometimes preferred : Diameter o t ^ '■ Blanks. From k lay off ; upon the cone-pitch line, a distance K w, equal to ten times the working depth of the teeth = 10 D". Now add rV of the shortest distance of v^ from the line g h, which is the perpendicular dotted line w x, to the outside pitch diameter of gear, and the sum will be the whole diameter of gear. In the same manner xV of w y, added to the outside pitch diameter of pinion, gives the whole diameter of pinion. The part added to the pitch diameter is called the diameter increment. Part II gives trigonometrical methods of figuring bevel gears : in our Formulas in Gearing there are trigonometrical formulas for bevel gears, and also tables for angles and sizes, of^BOT*e?Gear ^ somewhat similar construction will do for bevel Blanks whose gears whose axes are not at right angles. Axes are not ° » & at Right An- In Fig. 18 the axes are shown at O B and O D, the angle BOD being less than a right angle. 1. Parallel to O B, and at a distance from it equal to the radius of the gear, we draw the line^ a b and c do PROVIDKNCK, R. I. 39 Fig. 20 INSIDE BEVEL GEAR AND PINION. 40 BROWN & SHARPE MFG. CO. 2. Parallel to O D, and at a distance from it equal to the radius of the pinion, we draw the lines e f and g h. 3. Now, through the point j at the intersection of c d and g h, we draw a line perpendicular to O B. This line k j, limited by a b and c d, represents the largest pitch diameter of the gear. Through j we draw a line perpendicular to O D. This line j 1, limited by e f and g h, represents the largest pitch diameter of the pinion. 4. Through the point k at the intersection of a b with k i, we draw a line to O, a line from j to O, and another from 1, at the intersection j 1 and e f to O. These lines O k, O j, and O 1, represent the cone- pitch lines, as in Fig. 17. 5. Perpendicular to the' cone-pitch lines we draw the lines u v, o p, and q r. Upon these lines we lay off the addenda and working depth as in the previous figure, and then draw lines to the point O as before. By a similar construction Figs. 19 and 20 can be drawn. GEAR CUTTER. 41 CHAPTER X. BEVEL GEARS. FORMS AND SIZES OF TEETH. CUTTING TEETH. To obtain the form of the teeth iu a bevel gear we Form of IjgvgI "* g ti r do not lay them out upon a pitch circle, as we do iu a teeth, spur gear, because the rolling pitch surface of a bevel gear, at any point, is of a longer radius of curvature than the actual radius of a pitch circle that passes through that point. Thus in Fig. 21, let f g c be a cone about the axis O A, the diameter of the cone being f c, and its radius g c. Now the radius of curvature of the surface, at c, is evidently longer than g c, as can be seen in the other view at C ; the full line shows the curvature of the surface, and the dotted line shows the curvature of a circle of the radius g c. It is extremely difficult to represent the exact form of bevel gear teeth upon a flat surface, because a bevel gear is essentially spherical in its nature ; for practical purposes we draw a line c A perpendicular to c, letting c A reach the centre line A, and take c A as the radius of a circle upon which to lay out the teeth. This is shown at c n m. Fig. 22. For con- venience the line c A is sometimes called the back cone radius. Let us take, for an example, a bevel gear and a ^,.^^o^-,"^i^^®= pinion 24 and 18 teeth, 5 pitch, shafts at right angles. To obtain the forms of the teeth and the data for cutting, we need to draw a section of only a half of each gear, as in Fig. 22. 1. Draw the centre lines A O and B O, then the lines g h and c d, and the gear blank lines as des- cribed in Chapter IX. Extend the lines o' p' and o p until they meet the centre lines at A' B' and A B. 2. With the radius A c draw the arc c n m, which we take as the geometrical pitch circle upon which to lay out the teeth at the large end. The distance A' c' 42 BROWN & SHARPE MFG. CO., is taken as the radius of the geometrical pitch circle at the small end ; to avoid confusion an arc of this circle is drawn at c" n' m' about A. 3. For the pinion we have the radius B c for the geometrical pitch circle at the large end and B' c' for the small end : the distance B' c' is transferred to B c'". 4. Upon the arc c n m lay off spaces equal to the tooth thickness at the large pitch circle, which in our example is .314". Draw the outlines of the teeth as in previous chapters : for single curve teeth we draw a semi-circle upon the radius A c, and proceed as des- cribed in chapter III. For all bevel gears that are to be cut with a rotary disk cutter, or a common gear cutter, single curve teeth are chosen ; and no attempt should be made to cut double curve teeth. Double curve teeth can be drawn by the directions given in chapters VII and VIII. We now have the form of the teeth at the lai'ge end of the gear. Repeat this operation with the radius B C about B, and we have the form of the teeth at tlie large end of the pinion. 5. The tooth parts at the small end are designated by the same letters as at the large, with the addition of an accent mark to each letter, as in the right hand column. Fig. 22, the clearance, f, however, is usually the same at the small end as at the large, for con- venience in cutting the teeth. When cutting bevel gears with rotary cutters, the cutting angle is the same as the working depth angle. This angle is used for two reasons : first, it is not neces- sary to figure the angle of the bottom; second, the inside of the teeth is rounded over a little more and this lessens the amount to be filed off at the point. When cut in this way, the line of the bottom of the tooth is parallel to the face of the mating gear and it does not pass through the cone apex or common point of the axes. tooth p\?tV."'' ^^^ ^^^^^ ^^ ^^® ^°^*^ P^^'^^ ^* *^^ ^^^^^ ®^*^^ ^^''^ ^^ the same proportion to those at the large end as the line O c' is to c. In our example O c' is 2", and O c is 3" ; dividing O c' by O c we have |, or .666, as the ratio of the sizes at the small end to those PROVIDENCE, R. J. 43 44 BROWN & SHARPE MFG. CO. at the large : t' is .209" or f of .314", and so on. If the distance n m is equal to the outer tooth thickness, t, upon the arc c n m, the lines n A and m A will be a distance apart equal to the inner tooth thickness t' upon the arc c" n' m'. The addendum, s', and the working depth, D'", are at o' c' and o' p'. 6. Upon the arcs c" n' m' and c'" we draw the forms of the teeth' of the gear and pinion at the inside. Example of As an example of the cutting of bevel gears with Cutting. .-,.,, , rotary disk cutters, or common gear cutters, let us take a pair of 8 pitch, 12 and 24 teeth, shown in Fig. 23. Length of In making the drawing it is well to remember that nothing is gained by having the face F E longer than five times the thickness of the teeth at the large pitch circle, and that even this is too long when it is more than a third of the cy^ea; distance O c. To cut a bevel gear with a rotary cutter, as in Fig. 24, is at best but a compromise, because the teeth change pitch from end to end, so that the cutter, being of the right form for the large ends of the teeth can not be right for the small ends, and the variation is too great when the length of face is greater than a third of the apex distance O c. Fig. 23. In the example, one-third of the apex distance is -^^'\ but F E is drawn only a half inch, which even though rather short, has changed the pitch from 8 at the outside to finer than 11 at the inside. Frequently the teeth have to be rounded over at the small ends by filing ; the longer the teeth the more we have to file. If there is any doubt about the strength of the teeth, it is better to lengthen at the large end, and make the pitch coarser rather than to lengthen at the small end. Data for These data are needed before beginning to cut: 1. The pitch and the numbers of the teeth the same as for spur gears. 2. The data for the cutter, as to its form : some- times two cutters are needed for a pair of bevel gears. 3. The whole depth of the tooth spaces, both at cutting. PROVIDENCE, R. I. 45 D"+/ = .431" Fig. 22. BEVEL GEARS, FORM AND SIZE OF TEETH. 46 BROWN & SHARPE MFG. CO. the outside and inside ends ; D" + f at the outside, and D'" + f at the inside. 4. The thickness of the teeth at the outside and at the inside ; t and t'. 5. The height of the teeth above the pitch lines at the outside and inside ; s and s'. 6. The cutting angles, or the angles that the path of the cutter makes with the axes of the gears. In Fig. 23 the cutting angle for the gear c D is A Op, and the cutting angle for the pinion is B O o. Selection of The form of the teeth in one of these gears differs cutters so much from that in the other gear that two cutters are required. In determining these cutters we do not have to develop the forms of the gear teeth as in Fig. 22 ; we need merely measure the lines A c and B c. Fig. 23, and calculate the cutter forms as if these distances were the radii of the pitch circles of the gears to be cut. Twice the length A c, in inches, multiplied by the diametral pitch, equals the number of teeth for which to select a cutter tor the twenty- four-tooth gear ; this number is about 54, which calls for a number three bevel gear cutter in accordance with the lists of gear cutters, pages 61 and 82. Twice B c, multiplied by 8, equals about 13, which indicates a No. 8 bevel gear cutter for the pinion. This method of selecting cutters is based upon the idea of shaping the teeth as nearly right as practicable at the large end, and then filing the small end where the cutter has not rounded them over enough. In Fig. 25 the tooth L has been cut to thickness at both the outer and inner pitch lines, but it must still be rounded at the inner end. The teeth M M have been filed. In thus rounding the teeth rhey should not be filed thinner at tlie pitch lines. There are several things that affect the shape of the teeth, so that the choice of cutters is not always so simple a matter as the taking of the lines A c and B c as radii. In cutting a bevel gear, in the ordinary gear cutting PROVIl>ENCE, H. I. 47 BEVEL GEAR DIAGRAM FOR DIMENSIONS. end. 48 BROWN & SHAEPE MFG. CO. machines, the finished spaces are not always of the same form as the cutter might be expected to make, because of the changes in the positions of the cutter and of the gear blank in order to cut the teeth of the right thickness at both ends. The cutter must of course be thin enough to pass through the small end of the spaces, so that the large end has to be cut to the right width by adjusting either the cutter or the blank sidewise, then rotating the blank and cutting twice around. Widening Thus, in Fig. 24, a gear and a cutter are set to have th^^^a^rge a space widened at the large end e', and the last chip to be cut off by the right side of the cutter, the cutter having been moved to the left, and the blank rotated in the direction of the arrow : in a Universal Milling Machine the same result would be attained by moving the blank to the right and rotating it in the direction of the arrow. It may be well to remember that in setting to finish the side of a tooth, the tooth and the cutter are first separated sidewise, and the blank is then rotated by indexing the spindle to bring the large end of the tooth up against the cutter. This tends rowed iwe ^q(; Qy^\y iq ^qJ; the spaces wider at the large pitch at root. circle, but also to cut off still more at the face of the tooth ; that is, the teeth may be cut rather thin at the face and left rather thick at the root. This tendency is greater as a cutting angle B O o. Fig. 23, is smaller, or as a bevel gear approaches a spur gear, because when the cutting angle is small the blank must be rotated through a greater arc in order to set to cut the right thickness at the outer pitch circle. This can be understood by Figs. 26 and 27. Fig. 26 is a radial- toothed clutch, which for our present purpose can be regarded as one extreme of a bevel gear in which the teeth are cut square with the axis : the dotted lines indicate the diffei-ent positions of the cutter, the side of a tooth being finished by the side of the cutter that is on the centre line. In setting to cut these teeth there is the same side adjustment and rotation of the Teeth nar- PROVIDENCE, U. I. 49 Tig. 34 SETTING BEVEL GEAR CUTTER OUT OF CENTRE. 60 BROWN & SHARPE MFG. CO. spindle as in a bevel gear, but there is no tendency to make a tooth thinner at the face than at the root. On the other hand, if we apply these same adjustments to a spur gear and cutter, Fig. 27, we shall cut the face F much thinner without materially changing the thick- ness of the root R. . Mg, 26 Almost all bevel gears are between the two extremes of Figs. 26 and 27, so that when the cutting angle B O o, Fig. 23, is smaller than about 30°, this change in the form of the spaces caused by the rotation of the blank maybe so great as to necessitate the substitution Fig.. 28 FINISHED GEAR. PROVIDENCE, R. I. 51 of a cutter that is narrower at e e', Fig. 24, than is called for by the way of figuring that we have just given : thus in our own gear cutting department we might cut the pinion with a No. 6 cutter, instead of a No. 8. The No. 6, being for 17 to 20 teeth, cuts the tooth sides with a longer radius of curvature than the No 8, which may necessitate considerable filing at the small ends of the teeth in order to round them over enough. Fig. 28 shows the same gear as Fig. 25, but in this case the teeth have all been filed similar to M M, Fig. 25. Difi'erent workmen prefer different ways to com- Filing the promise in the cutting of a bevel gear. When a gmaii end. ^^ blank is rotated in adjusting to finish the large end of the teeth there need not be much filing of the small end, if the cutter is right, for a pitch circle of the radius B c, Fig. 23, which for our example is a No. 8 cutter, but the tooth faces may be rather thin at the large ends. This compromise is preferred by nearly all workmen, because it does not require much filing of the teeth : it is the same as is in our catalogue by which we fill any order for bevel gear cutters, unless otherwise specified. This means that we should send Selection of ^ cutter when a No. 8, 8-pitch bevel gear cutter in reply to an order teeth are to for a cutter to cut the 12-tooth pinion, Fig. 23 ; while in our own gear cutting department we might cut the same pinion with a No. 6, 8-pitch cutter, because we prefer to file the teeth at the small end after cutting them to the right thickness at the faces of the large end. We should take a No. 6 instead of a No. 8 only for a 12-tooth pinion that is to run with a gear two or three times as large. We generally step off to the next cutter for pinions fewer than twenty-five teeth, when the number for the teeth has a fraction nearly reaching the range of the next cutter : thus, if twice the line B c in inches, Fig. 23, multiplied by the diametral pitch, equals 20.9, we should use a No. 5 cutter, which is for 21 to 25 teeth inclusive. In filling an order for a gear cutter, we do not consider 52 BROWN & SHARPE MFG. CO. the fraction but send the cutter indicated by the whole number. Later on we will refer to other compromises that are made in the cutting of bevel gears. The sizes of the 8-pitch tooth parts, Fig. 23, at the large end, are copied from the table of spur gear teeth, pages 146 to 149. The distance Oc' is seven-tenths of the apex dis- tance Oc, so that the sizes of the tooth parts at the Form of „ , ^ , , , r^,, gear cutting small end, except f , are seven-tenths the large. The order order for cutting these gears goes to the workmen in this form : Large Gear. P = 8 N = 24 D" + f := .270" D'" + f = .195" t --= .196" t' = .137" s = .125" s' = .087" Cutting Angle = 59° 10' Small Gear. N = 12 Cutting Angle = 22° 18' Setting the ^^S- ^2 is a side view of a Gear Cutting Machine. machine. j^ bevel gear blank A is held by the index spindle B. The cutter C is carried by the cutter-slide D. The cutter-slide-carriage E can be set to the cutting angle, the degrees being indicated on the quadrant F. Fig. 33 is a plan of the machine : in this view the cutter-slide-carriage, in order to show the details a little plainer, is not set to an angle. Before beginning to cut the cutter is set central with the index spindle and the dial G is set to zero, so that we can adjust the cutter to any required distance out of centre, in either direction. Set the cutter-slide- carriage E, Fig. 32, to the cutting angle of the gear, which for 24-teeth is 59° 10' ; the quadrant being divided to half-degrees, we estimate that 10' or ^ de- PROVIDENCE, R. I 53 gree more than 59°. Mark the depth of the cut at the outside, as in Fig. 30 : it is also well enough to mark the depth at 'he inside as a cheek. The thickness of the teeth at the large end is conveniently deter- mined by the solid gauge, Fig. 29. The gear-tooth M^: mff.gd GEAR TOOTH GAUGE. DEPTH GAUGE. JFig. 30 GEAR TOOTH CALIPER. mg.31 vernier caliper, Fig. 31, will measure the thickness of teeth up to 2 diametral pitch. In the absence of the vernier caliper we can file a gauge, similar to Fig 29, to the thickness of the teeth at the small end. The index having been set to divide to the right side^of 'tooth number we cut two spaces central with the blank, hieing flnisiied. leaving a tooth between that is a little too thick, as in the upper part of Fig. "lb. If the gear is of cast iron, and the pitch is not coarser than about 5 diametral, this is as far as we go with the central cuts, and we proceed to set the cutter and the blank to finish first one side of the teeth and then the other, going around only twice. The tooth has to be cut away more in proportion from the large than from the small end, which is the reason for setting the cutter out of centre, as in Fiff. 24. 54 BROWN & yHARPE MFQ. CO. Fig. 32. AUTOMATIC GEAR CUTTING MACHINE. SIDE ELEVATION. PROVIDENCE, R. I. 65 It is important to remember that the part of the cutter that is finishing one side of a tooth at the pitch line should be central with the gear blank, in order to know at once in which direction to set the cutter out of centre. We can not readily tell how much out of centre to set the cutter until we have cut and tried, because the same part of a cutter does not cut to the pitch line at both ends of a tooth. As a trial distance out of centre we can take about one-tenth to one- eighth of the thickness of the teeth at the large end. The actual distance out of centre for the 12-tooth pinion is .021" : for the *24-tooth gear, .030", when using cutters listed in our catalogue. After a little practice a workman can set his cutter ^^ecessityof '^ central cuts. the trial distance out of centre, and take his first cuts, without any central cuts at all ; but it is safer to take central cuts like the upper ones in Fig. 25. The depth of cut is partly controlled by the index-spindle raising-dial-shaft H, Fig. 33, which determines the height of the index spindle, and partly by the position of the cutter spindle. We now set the cutter out of centre the trial distance by means of the cutter-spindle dial-shaft, I, Fig. 33. The trial distance can be about one-seventh the thickness of the tooth at the large end in a 12-tooth pinion, and from that to one-sixth the thickness in a 24-tooth gear and larger. The principle of trimming the teeth more at the large end than at the small is illustrated in Fig. 24, which is to move the cutter away from the tooth to be trimmed, and then to bring the tooth up against the cutter by rotating the blank in the direction of the arrow. .^. ^ ° Adjustments. The rotative adjustment of the index spindle is accomplished by loosening the connection between the index worm and the index drive, and turning the worm : the connection is then fastened again. The cutter is now set the same distance out of centre in the other direction, the index spindle is adjusted to trim the other side of the tooth until one end is down nearly to the right thickness. If now the thickness of the 56 BROWN & SHARPK MFG. CO. small end is in the same proportion to the large end as Oc' is to Oc, Fig. 23, we can at once adjust to trim the tooth to the right thickness. But if we find that the large end is still going to be too thick when the small end is right, the out of centre must be increased. It is well to remember this : too much out of centre leaves the small end proportionally too thick, and too little out of centre leaves the small end too thin. After the proper distance out of centre has been learned the teeth can be finish-cut by going around out of centre first on one side and then on the other with- out cutting any central spaces at all. The cutter spindle stops, J J, can now be set to control the out of centre of the cutter, without having to adjust by the dial G. If, however, a cast iron gear is 5-pitch or coarser it is usually well to cut central spaces first and then take the two out-of-centre cuts, going around three times in all. Steel gears should be cut three times around. Blanks are not always turned nearly enough alike to be cut without a different setting for different blanks. If the hubs vary in length the position of the cutter spindle has to be varied. In thus varying, the same depth of cut or the exact D" -{- f may not always be reached. A slight difference in the depth is not so objectionable as the incorrect tooth thickness that it may cause. Hence, it is well, after cutting once around and finishing one side of the teeth, to give careful attention to the rotative adjustment of the index spindle so as to cut the right thickness. After a gear is cut, and before the teeth are filed, it is not always a very satisfactory-looking piece of work. In Fig. 25 the tooth L is as the cutter left it, and is ready to be filed to the sh ipe of the teeth M M , which have been filed. Fig. 3-t is the pair of gears that we have been cutting ; the teeth of the 12-tooth pinion have been filed. PROVIDENCE, R. I. 57 ^P^^ UJ u (3 p h- <: LU O O < O I- < 58 BROWN & SHARPE MFG. CO. A second ^ second approximation in cutting with a rotary approxima- '■ '^ *= -^ tion. cutter is to widen the spaces at the large end by swing- ing either the index spindle or the cutter-slide-carriage, so as to pass the cutter through on an angle with the blank side-ways, called the side-angle, and not rotate the blank at all to widen the spaces. This side-angle method is employed in our No. 11 Automatic Bevel Gear Cutting Machines : it is available in the manufac- ture of bevel gears in large quantities, because with the proper relative thickness of cutter, the tooth- thickness comes right by merely adjusting for the side- angle ; but for cutting a few gears it is not much liked by workmen, because, in adjusting for the side- angle, the central setting of the cutter is usually lost, and has to be found by guiding into the central slot already cut. If the side-angle mechanism pivots about a line that passes very near the small end of the tooth to be cut, the central setting of the cutter may not be lost. In widening the spaces at the large end, the teeth are narrowed practically the same amount at the root as at the face, so that this side-angle method requires a wider cutter at e e', Fig. 24, than the first, or rotative method. The amount of filing required to correct the form of the teeth at the small end is about the same as in the first method. A third ap- A third approximate method consists in cutting proximation. , , . , , i n i . -, . the teeth right at the large end by gomg around at least twice, and then to trim the teeth at the small end and toward the large with another cutter, going around at least four times in all. This method requires skill and is necessarily a little slow, but it contains possi- bilities for considerable accuracy. A fourth ap- A f ourth method is to have a cutter fully as thick as the spaces at the small end, cut rather deeper thuu the regular depth at the large end, and go only once around. This is a quick method but more inaccurate than the three preceding : it is available in the manu- facture of large numbers of gears when the tooth-face proximation. PROVIDENCE, R. I. 59 Fig. 34: FINISHED GEAR AND PINION, 60 BROWN & SHARPE MFG. CO. is sho*-t compared with the apex distance. It is little liked, and seldom employed in cutting a few gears : it may require some experimenting to determine the form of cutter. Sometimes the teeth are not cut to the regular depth at the small end in order to have them thick enough, which may necessitate reducing the addendum of the teeth, s', at the small end by turning the blank down. This method is extensively employed by chuck manufacturers. A machine that cuts bevel gears with a reciprocating motion and using a tool similar to a planer tool is called a Gear Planer and the gears so cut are said to be planed. Planing of ^"^^ form of Gear Planer is that in which the prin- bevei gears. q\^\q embodied is theoretically correct ; this machine originates the tooth curves without a former. Another form of the same class of machines is that in which the tool is guided by a former. Usually the time consumed in planing a bevel gear is greater than the time necessary to cut the same gear with a rotary cutter, thus proportionately increasing the cost. Pitches coarser than 4 are more correct and some- times less expensive when planed ; it is hardly prac- ticable, and certainly not economical, to cut a bevel gear as coarse as 3P. with a rotary cutter. In gears as fine as 16P. planing affords no practical gain in quality. While planing is theoretically correct, yet the wear- ing of the tool may cause more variation in the thick- ness of the teeth than the wearing of a rotary cutter, and even a planed gear is sometimes improved by filing. Moiintino-of ^^ gears are not correctly mounted in the place where gears. ^j^gy ^^^ ^^ ^^^^ they might as well not be planed. In fact, after taking pains in the cutting of any gear, when we come to the mounting of it we should keep right on taking pains. Angles and The method of obtaining the sizes and angles per- gears.*' ^^^ taining to bevel gears by measuring a drawing is quite convenient, and with care is fairly accurate. Its PROVIDENCE, R. I. accuracy depends, of course, upon the careful measur- ing of a good drawing. We may say, in general, that in measuring a diagram, wliile we can hardly obtain data mathematically exact, we are not likely to make wild mistakes. Some years ago we depended almost entirely upon measuring, but since the publication of this "Treatise" and our " Formulas in Gearing " Ave calculate the data without any measuring of a drawing. In the " Formulas in Gearing" there are also tables pertaining to bevel gears. Several of the cuts and some of the matter in this chapter are taken from an article by O. J. Beale, in the "American Machinist," June 20, 1895. Cutters for .Mitre and Bevel Gears. 61 Diametral Pitch. Diameter of Cutter. Hole in Cutter. 4 3 1-2" 1 1-4" 5 3 1-2 1 1-4 6 3 1-2 1 1-4 7 3 1-2 1 1-4 8 3 1-4 1 1-4 10 3 1-4 7-8 13 3 7-8 14 3 7-8 16 2 3-4 7-8 20 2 1-2 7-8 24 2 1-4 7-8 When each gear of a pair of bevel gears is of the same size and the gears connect shafts that are at right angles, the gears are called "Mitre Gears'' and one cutter will answer for both. 62 BROWN & SHARPE MFG. CO. WORM WHEEL Number of Teeth, 54. Throat Diameter, 44.59". Circular Pitch, 2^. Outside Diameter 46". C3 CHAPTER XI. WORM WHEELS— SIZING BLANKS OF 32 TEETH AND MORE. A WORM is a screw made to mesh with the teeth of Worm. a wheel called a worm-wheel. As implied at the end of Chapter IV., a section of a worm through its axis is, in outline, the same as a rack of corresponding pitch. This outline can be made either to mesh with single or double curve gear teeth ; but worms are usually made for single curve, because, the sides of involute rack teeth being straight (see Chapter IV.), the tool for cutting worm-thread is more easily made. The thread- tool is not usually rounded for giving fillets at bottom of worm-thread. The axis of a worm is usually at right angles to the axis of a worm wheel: no other angle of axis is treated of ill this book. The rules for circular pitch apply in the size of tooth parts and diameter of pitch-circle of worm-wheel. The pitch of a worm or screw is sometimes given inPitchof worm a way different from the pitch of a gear, viz. : in num- ber of threads to one inch of the length of the worm or screw. Thus, to say a worm is 2 pitch may mean 2 threads to the inch, or that the worm makes two turns to advance the thread one inch. But a worm may be double- threaded, triple-threaded, and so on; hence to avoid misunderstanding, it is better always to call the advance of the worm thread the lead. Thus, a ^^ ^j^.^j^^^^^ worm-thread that advances one inch in one turn we call one-inch lead in one turn. A single-thread worm 4 turns to 1" is |^" lead. We apply the term pitch, that is the circular pitch, to the actual distance between the threads or teeth, as in previous chapters. In single- thread worms the lead and the pitch are alike. In making a worm and wheel a given number of threads to 64 BKOWN & SHAKPE MFG. CO. FIG. 35 -WORM AND WORM-WHEEL The Thread of Worm is Left-handed; Worm is Single-threaded. PROVIDENCE, R. I. 65 OO BROWN & SHARPE MFO. CO. one inch, we divide 1 " by the number of threads to one inch, and the quotient is the circular pitch. Hence, Linear pitch, the wheel ill Fig. 36 is ^' circular pitch. Linear pitch expresses exactly what is meant by circular pitch. Linear pitch has the advantage of being an exact use of language when applied to worms and racks. The number of threads to one inch linear, is the reciprocal of the linear pitch. Multiply 3.1416 by the number of threads to one inch, and the product will be the diametral pitch of the worm-wheel. Thus, we should say of a double-threaded worm advancing 1" in \\ turns that: Drawing of Lead=f" or .75". Linear pitch or P'z=:f" or .375". Worm-wheel. Diametral pitch orP=8.377. See table of tooth parts. To make drawing of worm and wheel we obtain data as in circular pitch. 1. Draw center line A O and upon it space off the distance a b equal to the diameter of pitch-circle. 2. On each side of these two points lay off the dis- tance 5, or the usual addendum =^ , as 5 c and b d. 3. From c lay off the distance c O equal to the radius of the woi'm. The diameter of a worm is gen- erally four or five times the circular pitch. 4. Lay off the distances c g and d e each equal to /*, or the usual clearance at bottom of tooth space. 5. Through c and e draw circles about O. These represent the whole diameter of worm and the diam- eter at bottom of worm-thread. 6. Draw h O and i O at an angle of 30° to 45° with A O. These lines give width of face of worm-wheel. 7. Through g and d draw arcs about O, ending in h O and i O. This operation repeated at a completes the outline of worm-wheel. For 32 teeth and more, the addendum diameter, or D, should be taken at the throat or smallest diameter of wheel, as in Fig. 36. Measure sketch for whole diameter of wheel-blank. Teeth of The foregoing instructions and sketch are for cases ished^^th Hob" where the teeth of the wheels are finished with a hob. Hob. "^ HOB is shown in Fig. 37, being a steel piece PEOVIDENCE, R. I. 67 threaded with a tool of the same angle as the tool that threads the worm, the end of the tool being .335 of the linear pitch ; the hob is then grooved to make teeth for cutting, and hardened. The whole diameter of hob should be at least 2 f. Proportionsof ■' ' Hob. or twice the clearance larger than the worm. In our relieved hobs the diameter is made about .005" to .010" larger to allow for wear. The outer corners of hob-thread can be rounded down as far as the clearance distance. The width at top of the hob-thread before rounding should be .31 of the linear, or circular pitch ^.31P'. The whole depth of thread is thus the ordinary work- ing depth plus the clearance ^D"-f/. The diameter at bottom of hob-thread should be 2/-f.005" to .010" larger than the diameter at bottom of worm-thread. Fig. 37— HOB. For thread-tool and worm-thread see end of Chapter IV. In the absence of a special worm gear cutting "^^-^^^q^^ ^®® chine, the teeth of the wheel are first cut as nearly to the finished form as practicable; the hob and worm-wheel are mounted upon shafts and hob placed in mesh, it is then rotated and dropped deeper into the wheel until the teeth are finished. The hob generally drives the worm- wheel during this operation. The Universal Milling Ma- universal chine is convenient for doing this work ; with it the dis- chW'used in Hobtoing. 68 BROWN & SHAEPE MFG. CO. ^^■"^'^V^'-^ Fig. 38. PKOVIDENCE, K. I. 69 ■CH ,CIR Fig. 39. 70 BKOWN & SHAKPE MFG. CO. tance between axes of worm and wheel can be noted. In making wheels in quantities it is better to have a ma- chine in which the work spindle is driven by gearing, so that the hob can cut the teeth from the solid with- whyawheei ^"* gashing. The object of bobbing a wheel is to get isHobtoed. more bearing surface of the teeth upon worm-thread. The worm-wheels, Figs. 35 and 43, were hobbed. Worm- Wheel If we make the diameter of a worm-wheel blank, that Less than 30 is to have less than 30 teeth, by the common rules for sizing blanks, and finish the teeth with a hob, we shall find the flanks of teeth near the bottom to be un- interference dercut OX hollowinsf. This is caused by the interfer- of Thread and ° '' Flank. ence spoken of in Chapter VI. Thirty teeth was there given as a limit, which will be right when teeth are made to circle arcs. With pressure angle 14^°, and rack-teeth with usual addendum, this interference of rack-teeth with flanks of gear-teeth begins at 31 teeth (31j2g. geometrically), and interferences with nearly the whole flank in wheel of 12 teeth. Fig. 38, In Fig 38 the blank for worm-wheel of 12 teeth was sized by the same rule as given for Fig. 36. The wheel and worm are sectioned to show shape of teeth at the mid-plane of wheel. The flanks of teeth are undercut by the hob. The worm-thread does not have a good bearing on flanks inside of A, the bearing being that of a corner against a surface. ^is- 39. In Pig 39 the blank for wheel was sized so that pitch- circle comes midway between outermost part of teeth and innermost point obtained by worm-thread. This rule for sizing worm-wheel blanks has been in use to some extent. The hob has cut away flanks of teeth still more than in Fig. 38. The pitch circle in Fig. 39 is the same diameter as the pitch-circle in Fig. 38. The same hob was used for both wheels. The flanks in this wheel are so much undercut as to mate- rially lessen the bearing surface of teeth and worm- thread. Interference In Cnapter VI. the interference of teeth in high- numbered gears and racks with flanks of 12 teeth was remedied by rounding off the addenda. Although it would be more systematic to round off the threads ol a worm, making them, like rack-teeth, to mesh with PROVIDENCE, II I. 71 interchangeable gears, yet this has not generally been done, because it is easier to make a worm-thread tool with straight sides. Instead of cutting away the addenda of worm- thread, we can avoid the interference with flanks of wheels having less than 30 teeth by making wheel blanks larger. The flanks of wheel in Fig. 40 are not undercut, be- Fig. 40. cause the diameter of wheel is so large that there is hardly any tooth inside the pitch-circle. The pitch-circle in Fig. 40 is the same size as pitch- circles in Figs. 38 and 39. This wheel was sized by the following rule : Multiply the pitch diameter of Diameter at 1 111 rvorr i i T , /i i ■ p • Throat to Avoid the wheel by .937, and add to the product four times interference, the addendum (4 s) ; the sum will be the diameter for the blank at the throat or small part. To get the whole diameter, make a sketch with diameter of throat to the foregoing rule and measure the sketch. It is impractical to hob a wheel of 12 to about 16 or 18 teeth when blank is sized by this rule, unless the wheel is di'iven by independent mechanism and not by the hob. The diameter across the outermost parts of teeth, as at A B, is considerably less than the largest diameter of wheel before it was hobbed. In general it is well to size all blanks, as by page 66 and Figs. 36 and 38, when the wheels are to be hobbed ; of course the cutter should be thin enough to leave stock for finishing. The spaces can be cut the full depth, the cutter being dropped in. When worm-wheels are not hobbed it is better to turn blanks like a spur-wheel. Little is gained by g ^^^ wh'^V * having wheels curved to fit worm unless teeth are fin- ished with a hob. The teeth can be cut in a straight path diagonally across face of blank, to fit angle of worm-thread, as in Figs. 41 and 44. Ill setting a cutter to gash a worm wheel, Figs. 42 and Gea*r-cutting 45, the angle is measured from the axis of the worm- ^^'^'^"^^" wheel and the angle of the worm thread is, in conse- quence, measured from the perpendicular to the axis of the worm. See Chapters V and VIII, Part II. 72 BEOWN & SHAKPE MFG. CO. ^\-3XH C/R Fig. 40. PROVIDENCE, K. 1. 73 Some mechanics prefer to make dividing wlieels in two parts, joined in a plane perpendicular to axis, hob teeth , then turn one part round upon the other, match teeth and fasten parts together in the new position, and hob again with a view to eliminate errors. With an accurate cutting engine we have found wheels like Figs. 42 and 45, not bobbed, every way satisfactory. As to the different wheels, Figs. 43, 44 and 45, whenaJj^s""'^''^'''*' worm is in rigbt position at the start, tbe life-time of Fig. 43, under beavy and continuous work, will be the longest. Fig. 44 can be run in mesb with a gear or a rack as well as with a worm when made within tbe angular limits commonly required. Strictly, neither two gears made in this way, nor a gear and a rack would be matbematically exact, as they miglit bear at the sides of the gear or at the ends of the teeth only and not in the middle. At tbe start the contact of teeth in this wheel upon worm-tbread is in points only; yet such wheels have been many years successfully used in ele- vators. Fig 45 is a neat-looking wheel. In gear cutting engines where the workman has occasion to turn the work spindle by hand, it is not so rough to take hold of as Figs 43 and 44. The teeth are less liable to in- jury than the teeth of Figs. 43 and 44. The diameter of a worm has no necessary relation to the speed ratio of the worm to the worm wheel. The diameter of the worm can be chosen to suit any dis- tance between the worm shaft and the worm-wheel shaft. It is unusual to have tbe diameter of the worm much less than four times the thread-pitch or linear- pitch but the worm can be of anj^ larger diameter, five or ten times the linear-pitch, if required. It is well to take off the outermost j^art of teeth in wheels (Figs. 35 and 48), as shown in these two fig- ures, and not leave them sharp, as in Figs. 36 and 39. It is also well to round over the outer corners of the blanks for the wheels. Figs. 44 and 45. In ordering worms and worm wheels the centre distances should be given. If there can be any limit allowed in the centre distance it should be so stated. 74 BKOWN & SHARPE MFG. CO. Fig. 41. VVorm--wl:ieel with teeth cut in a straight path diagonally across face, ^A''orm is double-threaded. PROVIDENCE, K. I. 75 Fig. 42. "Worm and Worm- Wheel, for Gear-cutting Engine. 76 BROWN & SHAKPE MFG. CO. ' jjjgmiiiiiit'i^ ^.«r^ Fig. 43. Fig. 44. Fig. 45. PROVIDENCE, K. I. For instance, the distance from the centre of a worm to the centre of a worm wheel might be calculated at 6" but 5 31-32" or 6 1-32" might answer. By stating all the limits that can be allowed, there may be a saving in the cost of work because time need not be wasted in trying to make work within narrower limits than are necessary. 11 Fig. '46 A LENGTH OF A WORM AND OF A HOB. In worm-wheels, like Figs. 41 and 42, having 540 teeth, worms can have bearings in ten places or along ten threads. Worms in wheels of 120 teeth bear on six threads. In order to hob a wheel of 540 teeth, the hob must be about eleven threads long, if the worm has ten threads. For the 120 tooth wheel, the hob should have about seven threads, if the worm has six threads. For a 80 tooth w^orm M^heel of the form of Figs. 41 and 42, we can have only about three threads in con- tact and a hob four threads long, like Fig. 37, is long enough. From the diagram. Fig, 45 A, which is similar to Fig. 7, we can tell approximately the number of threads that can bear. Let the worm move to the right and the action begins at C and ends at A', C being the point where the line C D intersects the addendum circle of the gear and A' being the point where the line would intersect the addendum line of the worm. A short worm can be used in a large wheel by having the hob a little longer than tlie worm. 78 BKOWN & SHAKPE MFG. CO. GASHING TEETH OF HOB. 10 Inches Outside Diameter. HOBS WITH RELIEVED TEETH. We make hobs of any size with the teeth telieved the same as our gear cutters. The teeth can be ground on their faces without changing tiieir form. The hobs are made with a precision screw so that the pitch of the thread is accurate before hardening. 79 CHAPTER XII. SIZING GEARS WHEN THE DISTANCE BETWEEN CENTRES AND THE RATIOS OF SPEEDS ARE FIXED— GENERAL REMARKS— WIDTH OF FACE OF SPUR GEARS— SPEED OF GEAR COTTERS— TABLE OF TOOTH PARTS. Let us suppose that we have two shafts 14" apart, center to center, and wish to connect them by sfears so, center dis- ' "^ o tance and Ratio that they will have siDced ratio 6 to 1. We add the 6 ^^^d. and 1 together, and divide 14" by the sum and get 2" for a quotient; this 2", multiplied by 6, gives us the radius of pitch circle of large wheel = 12". In the same manner we get 2" as radius of pitch cu'cle of small wheel. Doubling the radius of each gear, we obtain 24" and 4" as the pitch diameters of the two wheels. The two num- bers that form a ratio are called the terms of the ratio. We have now the rule for obtaining pitch-circle diame- ter of two wheels of a given ratio to connect shafts a given distance apart: Divide the center distajice hy the sum of the terms of ^uie for if tf J ameterofPi the ratio; find the product of tiifice the quotient hy each circieB. term separately, and the t%oo products xoill he the pitch diameters of the tioo wheels. It is well to give special attention to learning the rules for sizing blanks and teeth ; these are much oftener needed than the method of forming tooth out- lines. Di- Pitch 80 BROWN & SHARPE MFG. CO. A blank 1^" diameter is to have 16 teeth: what will the pitch be? "What will be the diameter of the pitch circle ? See Chapter V. A good practice will be to compute a table of tooth parts. The work can be compared with the tables pages 146-149. In computing it is well to take 7t to more than four places, Tt to nine places = 3.] 41592653. j^ to nine places = .318309886. There is no such thing as pure rolling contact in teeth of wheels ; they always rub, and, in time, will wear themselves out of shape and may become noisy. Bevel gears, when correctly formed, run smoother than spur gears of same diameter and pitch, because the teeth continue in contact longer than the teeth of spur gears. For this reason annular gears run smoother than either bevel or spur gears. Sometimes gears have to be cut a little deeper than designed, in order to run easily on their shafts. If any departui'e is made in ratio of pitch diameters it is better to have the driving gear the larger, that is, cut the follower smaller. For wheels coarser than eight diametral pitch (8 P), it is generally better to cut twice around, when accurate work is wanted, also for large wheels, as the expansion of parts from heat often causes inaccurate work when cut but once around. There is not so much trouble from heat in plain or web gears as in arm gears. £r*fkces.^"'^ The width of face of cast-iron gears can, for general use, be made 2| times the linear pitch. In small gears or pinions this width is often exceeded. The outer corners of spur gears may be rounded off for convenience in handling. This can be provided for when turning the blank. Speed of Gear The speed of gear cutters is subject to so many con- ditions that definite rules cannot be given. We append a table of average speeds. A coarse pitch cutter for pinion, 12 teeth, would usually be fed slower than a cutter for a large gear of same pitch. PKOVIDKNCE, E. I. TABLE OF AVERAGE SPEEDS FOR GEAR-CUTTERS. 81 ^ bO bflrH ^ tH a «4-l O &3 ^ P p. 3 g P. 3 o t- fl cj p.o«S s 1 2 "S^ oj -^ fH 5 Td ^ b6 f^ag g 2 5 in. 24 18 . 025 in. .011 in. . 60 in. . 20 in. 2i 41 u 30 24 .028 " .013 " .84 " .31 " 3 3if " 36 28 .031 " .015 " 1.12 " .42 " 4 3| " 42 32 .034 " .017 " 1.43 " .54 " 5 3tV " 50 40 .037 " .019 " 1.85 " .76 " 6 2ji " 75 55 .030 " .016 " 2.25 " .88 " 7 2tV " 85 65 .032 " .018 " 2.72 " 1.17 " 8 21 " 95 75 .034 " .020 " 3.23 " 1.50 " 10 4 " 125 90 .026 " .014 " 3.25 " 1.26 " 12 2 " 135 100 .027 " .017 " 3.64 " 1.70 " 20 H " 145 115 .029 " .021 " 4.20 " 2.41 " 32 If " 160 135 .031 '^ .025 " 4.96 " 3.37 " In brass tlie speed of geai'-cutters can be twice as -^^^^^ ^'^ ^^ fast as in cast iron. Clock-makers and those making a specialty of brass gears exceed this rate even. A 12 P cutter has been run 1,200 (twelve hundred) tui-ns a minute in bronze. A 32 P cutter has been run 7,000 (seven thousand) turns a minute in soft brass. In cutting 5 P cast-iron gears, 75 teeth, a No. 1, ^'P tromP^ctu^V cutter was run 136 (one hundred and thirty-six) turns a minute, roughing the spaces out the full 5 P depth ; the teeth were then finished with a 5 P cutter, running 208 (two hundred and eight) turns a minute, feeding by hand. The cutter stood well, but, of course, the cast iron was quite soft. A 4 P cutter has finished teeth at one cut, in cast-iron gears, 86 teeth, running 48 (forty-eight) turns a minute and feeding -^-^" at one turn, or 3 in. in a minute. Hence, while it is generally safe to run cutters as in the table, yet when many gears are to be cut it is well to see if cutters will stand a higher speed and more feed. In gears coarser than 3 P it is more economical to cut first the full depth with a stocking cutter and then finish with a gear cutter. This stocking cutter is made 82 BROWN & SHARPE MFG. CO. Keep Cutters sharp. on the principle of a circular splitting saw for wood. The teeth, however, are not set ; but side relief is ob- tained by making sides of cutter blank hollowing. The shape of stocking cutter can be same as bottom of spaces in a 12-tooth gear, and the thickness of cutter can be J of the circular pitch, see page 40. The matter of keeping cutters sharp is so important that it has sometimes been found best to have the work- man grind them at stated times, and not wait until he can see that the cutters are dull. Thus, have him grind every two hours or after cutting a stated number of gears. Cutters of the style that can be ground upon their tooth faces without changing foi'm are rap- idly destroyed if allowed to run after they are dull. Cutters are oftener wasted by trying to cut with them when they are dull than by too much grinding. Grind the faces radial with a free cutting wheel. Do not let the wheel become glazed, as this will draw the temper of the cutter. In Chapter YI. was given a series of cutters for cut- ting gears having 12 teeth and more. Thus, it was there implied that any gear of same pitch, having 135 teeth, 136 teeth, and so on up to the largest gears, and, also, a rack, could be cut with one cutter. If this cut- ter is 4 P, we would cut with it all 4 P gears, having 135 teeth or more, and we would also cut with it a 4 P rack. Now, instead of always referring to a cutter by the number of teeth in gears it is designed to cut, it has been found convenient to designate it by a letter or by a number. Thus, we call a cutter of 4 P, made to cut gears 135 teeth to a rack, inclusive, No. 1, 4 P. We have adopted numbers for designating involute Involute Gear ofear-cutters a-s in the following table : Cutters. No. 1 will cut wheels from 185 teeth to a rack inclusive. 2 55 134 teeth o o 35 54 " 4 26 34 " 5 21 25 " 6 17 20 " 7 14 16 " 8 12 13 " PROVIDENCE, R. I. 83 By this plan it takes eight cutters to cut all gears having twelve teeth and over, of any one pitch. Thus it takes eight cutters to cut all involute 4 P gears having twelve teeth and more. It takes eight other cutters to cut all involute gears of 5 P, having 12 teeth and more. A No. 8, 5 P cutter cuts only 5 P gears having 12 and 13 teeth. A No. 6, 10 P cutter cuts only 10 P gears having 17, 18, 19 and 20 teeth. On each cutter is stamped the number of teeth at the limits of its range, as well as the number of the cutter. The number of the cutter relates only to the number of teeth in gears that the cutter is made for. In ordering cutters for involute spur-gears two things must be given : 1. Either the number of teeth to he cut in the gear ^ How to order ■' _ "^ _ Involute Cut- or the number of the cutter, us given in the foregoing ters. table. 2. Either the pitch of the gear or the diam^eter and number of teeth to be cut in the gear. If 25 teeth are to be cut in a 6 P involute gear, the cutter will be No. 5, 6 P, which cuts all 6 P gears from 21 to 25 teeth inclusive. If it is desired to cut gears from 15 to 25 teeth, three cutters will be needed, No. 5, No. 6 and No. 7 of the pitch required. If the pitch is 8 and gears 15 to 25 teeth are to be cut, the cutters should be No. 5, 8 P, No. 6, 8 P, and No. 7, 8 P. For each pitch of epicycloidal, or double-curve gears, Epicycioidai 24 cutters are made. In coarse-pitch gears, the varia- curve cutters. tion in the shape of spaces between gears of consecu- tive-numbered teeth is greater than in fine-pitch gears. A set of cutters for each pitch to consist of so large a number as 24, was established for the reason that double carve teeth were formerly preferred in coarse pitch gears. The tendency now, however, is to use the involute form. Our double curve cutters have a guide shoulder on each side for the depth to cut. When this shoulder just reaches the periphery of the blank the depth is right. The marks which these shoulders make on the blank, should be as nar- row as can be seen, when the blanks are sized right. 84 BKOWN & SHAKPE MFG. CO. Double-curve gear-cutters are designated by letters instead of by numbers ; this is to avoid confusion in ordering. Following is the list of epicycloidal or double-curve gear-cutters : — cy^c^oida 1^ OT Cutter A cuts 12 teetb. Cutter M cuts 27 to 29 teeth. Double -curve Gear Cutters. B C ' 13 " ' 14 " D ' 15 " E ' 16 " F ' 17 " G ' 18 " H ' 19 " I ' 20 " J ' 21 to 22 K ' 23 to 24 L ' 24 to 26 N O P Q R S T U V w X 30 " 33 " 34 " 37 " 42 ' 49 " 59 " 74 " 99 " 149 " 249 " 250 " Eack. Rack. 38 43 50 60 75 100 150 A cutter that cuts more than one gear is made of proper form for the smallest gear in its range. Thus, cutter J for 21 to 22 teeth is correct for 21 teeth; cutter S for 60 to 74 teeth is correct for 60 teeth, and so on. Epicycloidal ^^ Ordering epicycloidal gear-cutters designate the Cutters. letter of the cutter as in the foregoing table, also either give the pitch or give data that v^ill enable us to determine the pitch, the same as directed for invo- lute cutters. More care is requii-ed in making and adjusting epi- cycloidal gears than in making involute gears. How to order j^ ordering bevel-gear cutters three thing's must be Bevel Gear & o t> Cutters. ffiven : 1. The number of teeth in each gear. 2. Either the pitch of gears or the largest pitch diameter of each gear; see Fig. 17. 3. The length of tooth face. If the shafts are not to run at right angles, it should be so stated, and the angle given. Involute cutters only are used for cutting bevel gears. No at- tempt should be made to cut epicyclodial tooth bevel gears with rotary disk cutters. PROVIDENCE, K. I. 85 In orderiuff worm-wheel cutters, three thinpfs must uow to order . ° » o Worm -gear be given : Cutters. 1. N'umber of teeth in the loheel. 2. Pitch of the worm; see Chapter XI. 3. M^hole diameter of worm. In any order connected with gears or gear-cutters, when the word " Diameter " occurs, we usually under- stand that the intch diameter is meant. When the tohole diameter of a gear is meant it should be plainly written. Care in giving an order often saves the delay of asking further instructions. An order for one gear- cutter to cut from 25 to 30 teeth cannot be filled, be- cause it takes two cutters of involute form to cut from 25 to 30 teeth, and thi-ee cutters of epicycloidal form to cut from 25 to 30 teeth. Sheet zinc is convenient to sketch gears upon, and also for making templets. Before making sketch, it is well to give the zinc a dark coating with the following mixture : Dissolve 1 ounce of sulphate of copper (blue vitriol) in about 4 ounces of water, and add about one- half teaspoonful of nitric acid. Apply a thin coating with a piece of waste. This mixtui'e will give a thin coating of copper to iron or steel, but the work should then be rubbed dry. Care should be taken not to leave the mixture where it is not wanted, as it rusts iron and steel. We have sometimes been asked why gears are noisy. Not many questions can be asked us to which we can give a less definite answer than to the question why gears are noisy. We can indicate only some of the causes that may make gears noisy, such as: — depth of cutting not right — in this particular gears are oftener cut too deep than not deep enough ; (more noise may be caused by cutting the driver too deep than by cutting the driven too deep;) cutting not central — this may make gears noisy iu one direction when they are quiet while running in the other direction ; centre distance not right — if too deep the outer corners of the teeth in one gear may strike the fillets of the teeth in the other gear ; shafts not parallel ; frame of the 86 BROWN & SHARPE MFG. CO. machine of such a form as to give off sound vibrations. Even when we examine a pair of gears we cannot always tell what is the matter. IMPROVED 29° SCREW THREAD TOOL GAUGE. 'ACME STANDARD. DEPTH OF GEAR TOOTH GAUGES. Depth of Gear Tooth Gauges for all regular pitches, from 3 to 48 pitch inclusive, are carried in stock. One Gauge answers for each pitch, and indicates the extreme depth to be cut. PART II. CHAPTER I. TANGENT OF ARC AND ANGLE. In Pakt II. we shall show how to calculate some g|"|^^?^^^_'^ ^® of the functions of a right-angle triangle from a table of circular functions, the application of these calcula- tions in some chapters of Part I. and in sizing blanks and cutting teeth of spiral gears, the selection of cutters for spii'al gears, the application of continued fractions to some problems in gear wheels and cutting odd screw-threads, etc., etc. A Functio7i is a quantity that depends upon another quantity for its value. Thus the amount a workman earns is a function of the time he has worked and of g^^^^*^'''^ *^®" his wages per hour. In any right- angle triangle, O A B, we shall, for Rigiit- angle convenience, call the two lines that form the right angle O A B the sides, instead of base and perpen- dicular. Thus O A B, being the right angle we call the line O A a side, and the line A B a side also. When we speak of the angle A O B, we call the line O A the side adjacent. "When we are speaking of the^^*^® *^'^^^°®^*- angle ABO we call the line A B the side adjacent. The line opposite the right angle is the hypothenuse. Hypothenuse. In the following pages the definitions of circular functions are for angles smaller than 90°, and not strictly applicable to the reasoning employed in ana- lytical trigonometry, where we find expressions for angles of 370°, 760°, etc. 88 Tangent. BEOWN & SHARPS MFG. CO. The Tangent of an arc is the line that touches it at one extremity and is terminated by a line drawn from the center through the other extremity. The tangent is always outside the arc and is also perpendicular to the radius which meets it at the point of tangency. Fig. 47. Thus, in Fig. 47, the line A B is the tangent of the arc A C. The point of tangency is at A. An angle at the center of a circle is measured by the arc intercepted by the sides of the angle. Hence the tangent A B of the arc A C is also the tangent of the angle A O B. In the tables of circular functions the radius of the arc is unity, or, in common practice, we take it as one inch. The radius O A being 1", if we know the length of the line or tangent A B we can, by looking in a table of tangents, find the number of degrees in the angle A O B. To find the Thus, if A B is 2.25" long, we find the angle A O B J_)©^r©6S In 3,11 Angle. is 66 very nearly. That is, having found that 2.2460 is the nearest number to 2.25 in the table of tangents at the end of this volume, we find the corresponding degrees of the angle in the column at the left hand of the table and the minutes to be added at the top of the column containing the 2.2460. The table gives angles for every 10', which is suf- ficient for most purposes. PROVIDENCE, R. I. 89 Now, if we liave a right-angle triangle with an angle the same as O A B, but with O A two inches long, the line A B will also be twice as long as the tangent of angle A O B, as found in a table of tangents. Let us take a triangle with the side O A = 5" long, And^^hi'^l, e? and the side A B = 8" long; what is the number oflJ^®®^^ in an degrees in the angle A O B ? Dividing 8" by 5 we find what would be the length of A B if O A was only 1" long. The quotient then would be the length of tangent when the radius is 1" long, as in the table of tangents. 8 divided by 5 is 1.6. The nearest tangent in the table is 1.6003 and the corresponding angle is 58°, which would be the angle A O B when A B is 8" and the radius O A is 5" very nearly. The difference in the angles for tangents 1.6003 and 1.6 could hardly be seen in practice. The side opposite the requu'ed acute angle corresponds to the tangent and the side adjacent corresponds to the radius. Hence the rule : To find the tangent of either acute angle in a right- rpj° g^^*^ ^^^ angle triangle : Divide the side opposite the angle by the side adjacent the angle and the quotient toill be the tangent of the angle. This rule should be com- mitted to memory. Having found the tangent of the angle, the angle can be taken from the table of tan- gents. The complement of an angle is the remainder after complement subtracting the angle from 90°. Thus 40° is the com- plement of 50°. 2'he Cotangent of an angle is the tangent of the Cotangent, complement of the angle. Thus, in Fig. 47, the line A B is the cotangent of A O E. In right-angle tri- angles either acute angle is the complement of the other acute angle. Hence, if we knoAv one acute angle, by subtracting this angle from 90° we get the other acute angle. As the arc approaches 90°, the tangent becomes longer, and at 90° it is infinitely long. The sign of infinity is oo. Tangent 90° = oo. 90 BKOWN & SHAKPE MFG. CO. Angie^^by"\he ^J ^ table of tangents, angles can be laid out upon Tangent E^x- gi^eet zinc, etc. This is often an advantage, as it is not convenient to lay protractor flat down so as to mark angles up to a sharp point. If we could lay off the length of a line exactly we could take tangents direct from table and obtain angle at once. It, however, is generally better to multiply the tangent by 5 or 10 and make an enlarged triangle. If, then, there is a slight error in laying off length of lines it will not make so much difference with the angle. Let it be required to lay off an angle of 14° 30'. By the table we find the tangent to be .25861. Multiply- ing .25861 by 5 we obtain, in the enlarged triangle, 1.29305" as the length of side opposite the angle 14° 30'. As we have made the side opposite five times as large, we must make the side adjacent five times as large, in order to keep angle the same. Hence, Fig. 48, draw the line A B 5" long ; perpendicular to this line at A draw the line A O 1.293" long ; now draw the line O B, and the angle A B O wHl be 14° 30'. If special accuracy is required, the tangent can be multiplied by 10; the line A O will then be 2.586" long and the line A B 10" long. Remembering that the acute angles of a right-angle triangle are the comple- ments of each other, we subtract 14° 30' from 90' and obtain 75° 30' as the angle of A O B. The reader will remember these angles as occurring in Part I., Chapter IV., and obtained in a different way. A semicircle upon the line O B touching the extremities O and B will just touch the right angle at A, and the line O B is four times as long as O A. Let it be required to turn a piece 4" long, 1" diam- eter at small end, with a taper of 10° one side with the other ; what will be the diameter of the piece at the large end ? A section. Fig. 49, through the axis of this piece is To calculate ^;\^q same as if we added two right-angle triangles, O Ta pe r i n gA. B and O' A' B', to a straight piece A' A B B', 1" piece. Fig. 50. . ' to i , . n wide and 4 long, the acute angles B and B being 5 , thus making the sides O B and O' B' 10° with each other. PROVIDENCE, K. 1. 91 -h293-t- Fig. 48. Fig. 49. 92 BEOWN & SHAEPE MFG. CO. The tangent of 5° is .08748, which, multiplied by 4 , gives .34992" as the length of each line, A O and A' O', to be added to 1" at the large end. Taking twice .34992" and adding to 1" we obtain 1.69984" as the diameter of large end. This chapter must be thoroughly studied before taking up the next chapters. If once the memory becomes confused as to the tangent and sine of an angle, it will take much longer to get righted than it will to first carefully learn to recognize the tangent of an angle at once. If one knows what the tangent is, one can tell better the functions that are not tangents. 93 CHAPTER II. SINE— COSINE AND SECANT : SOME OF THEIR APPLICATIONS IN MACHINE CONSTRUCTION. Sine of Arc ' and Angle The Sine of an arc is the line di-awn from one extremity of the arc to the diameter passing through the other extremity, the line being perpendicular to the diameter. Another definition is : The sine of an arc is the dis- tance of one extremity of the arc from the diameter, through the other extremity. The sine of an angle is the sine of the arc that , measures the angle. In Fig. 50 , A C is the sine of the arc B C, and of the angle B O C. It will be seen that the sine is always inside of the arc, and can never be longer than the radius. As the arc ap- proaches 90°, the sine comes nearer to the radius, and at 90° the sine is equal to 1, or is the radius itself. From the defini- tion of a sine, the side A C, opposite the angle A O C, in any right-angle triangle, is the sine of the angle A O C, when O C is the radius of the arc. Hence the rule : I?i any right-angle triangle, the side To find the opposite either acute angle, divided hy the hypothe- nuse, is equal to the sine of the angle. The quotient thus obtained is the length of side opposite the angle when the hypothenuse or radius is unity. The rule should be carefully committed to memory. ^^ A \ C '\ \ / \ ^ E ") Fig. 50. 94 BROWN & SHARPE MFG. CO. Chord of an j^ Chord is a straight line joining the extremities of an arc, and is twice as long as the sine of half the angle measured by the arc. Thus, in Fig. 50, the chord F C is twice as long as the sine A C. f_B ^ ^'-A — ^ ^•^ \/ y ^^ — -^ -^ / \ N y / N y / S / / S / / / / / / ^/ \ \ \ \ \ \ ' / \ \ 1 / \ \ \ \ ' / \ \ \ A V \ \ { \ ^ 1 ) 1 / I \ \ \ \ / / / / \ / \ / \ / \ / \ / \ / \ / \ / \ / s y V y s. y X ^ ^\ \ "^--- 1 — \ — -^ Fig. Rl. Let there be four holes equidistant about a circle 3" in diameter — Fig. 51 ; what is the shortest distance between two holes ? This shortest distance is the flnd^th^chord! chord A B, which is twice the sine of the angle COB. The angle A O B is one-quarter of the circle, and C O B is one-eighth of the circle. 360^, divided by 8=45°, the angle COB. The sine of 45° is .70710, which multiplied by the radius 1.5", gives length C B iu the circle, 8" in diameter, as 1.06065". Twice this length is the required distance A B=2.1213". When a cylindrical piece is to be cut into any num- ber of sides, the foregoing operation can be applied to obtain the width of one side. A plane figure bounded Polygon. by straight lines is called a polygon. PROVIDENCE, E. I. 95 When the outside diameter and the number of sides of a regular polygon are given, to find tlie length of one of the sides: Divide 360° hy tioice the number of , To find the . -^ •' length of Side. Sides ; ')nidUply the sine of the quotient hy the outer diameter, and the product loill be the length of one of the sides. Multiplying by the diameter is the same as multi- plyitig by the radius, and that product again by 2. The Cosine of an angle is the sine of the comple- cosine. ment of the angle. In Fig. 50, C O D is the complement of the angle A O C ; the line C E is the sine of COD, and hence is the cosine of B O C. The line O A is equal to C E. It is quite as well to remember the cosine as the part of the radius, from the center that is cut off by the sine. Thus the sine A C of the angle A O C cuts off the cosine O A. The line A may be called the cosine because it is equal to the cosine C E. In any right-angle triangle, the side adjacent either acute angle corresponds to the cosine when the hypothenuse is the radius of the arc that measures the ans'le ; hence: Divide the side adjacent the ac^cte To find the Cosine. angle by the hypothenuse, and the quotient will be the cosine of the angle. When a cylindrical piece is cut into a polygon of any number of sides, a table of cosines can be used tOgj^^l^^^t^^j"^^^ obtain the diameter across the sides. s^"^- 96 BROWN & SHARPE MFG. CO. Let a cylinder, 2" diameter, Fig. 53, be cut six-sided : what is the diameter across the sides ? The angle A O B, at the center, occupied by one of these sides, is one-sixth of the circle, =60°. The cosine of one-half this angle, 30°, is the line C O; twice this line is the diameter across the sides. The cosine of 30° is .86602, which, multiplied by 2, gives 1.73204" as the diameter across the sides. Of course, if the radius is other than unity, the cosine should be multiplied by the radius, and the product again by 2, in order to get diameter across the sides ; or what is the same thing, multiply the cosine by the whole diameter or the diameter across the corners. The rule for obtaining the diameter across sides of sidesof aPoiy-j.Qg.yjg_j. pQ;[yg.Q^^^ -^yjigj^ ^l^e ^i^™^®^®^ ^cross corners is given, will then be : Multiply the cosine of 360° divided by tvnce the number of sides, by the diameter across corners, and the product will be the diameter across sides. Look at the right-hand column for degrees of the cosine, and at bottom of page for minutes to add to the degrees. 2^he Secant of an arc is a straight line di'awn from the center through one end of an arc, and terminated by a tangent drawn from the other end of the arc. Thus, in Fig. 53, the line O B is the secant of the angle COB. A C B Eule for Di- ameter across Secant. Fig. 63. To find the In any right-angle triangle, divide the hypothenuse by the side adjacent either acute angle, and the quo- tient will be the secant of that angle. PROVIDENCE, R. 1. 07 That is, if we divide the distauce OB by O C, in the right-angle triangle COB, the (jviotient will be the secant of the angle COB. The secant cannot be less than the radius ; it in- creases as the angle increases, and at 90° the secant is infinity =00 . A six-sided piece is to be l-j" across the sides ; liow^ j^^j^^J^^^'^ large mnst a blank be turned before cutting the sides ? ^f^°^^ comers o o ot a Polygon. Dividing 360° by twice the number of sides, we have 30°, which is the angle COB. The secant of 30° is 1.1547. The radius of the six-sided piece is .75". Multiplying the secant 1.1547 by .75", we obtain the length of radius of the blank O B ; multiplying again by 2, we obtain the diameter 1.732"-)-. Hence, in a regular polygon, when the diameter across sides and the number of sides are given, to find diameter across corners : Multiply the secant of 360° divided hy tvnce the number of sides, hy the diameter across sides, and the product toill he the diameter across corners. It will be seen that the side taken as a divisor has been in each case the side corresponding to the radius of the arc that subtends the angle. The versed sine of an acute angle is the part of radius outside the sine, or it is the radius minus the cosine. Thus, in Fig. 50, the versed sine of the arc BC is AB. The versed sine is not given in the tables of circular functions : when it is wanted for any angle less than 90° we subtract the cosine of that angle from the radius 1. Having it for the radius 1, we can multiply by the radius of any other arc of which we may wish to know the versed sine. 98 BROWN & SHARPE MFG. CO. NO. 13. AUTOMATIC GEAR CUTTING MACHINE. For Spur and Bevel Gears. PROVIDENCE, R. I. 99 FRONT VIEW. REAR VIEW. GEAR MODEL. Shows combination of six different kinds of gears. 100 CHAPTER III. APPLICATION OF CIRCULAR FUNCTIONS— WHOLE DIAMETER BEYEL GEAR BLANKS— ANGLES OF BEYEL GEAR BLANKS. The rules given in this chapter apply only to bevel gears having the center angle c' O i not greater than 90*^. To avoid confusion we will illustrate one gear only. The same rules apply to all sizes of bevel gears. Fig. 55 is the outline of a pinion 4 P, 20 teeth, to mesh with a gear 28 teeth, shafts at right angles. For making sketch of bevel gears see Chapter IX., Pakt I. In Fig. 55, the line O in' m is continued to the line a b. The angle c' O i that the cone pitch-line makes with the center line may be called the center angle. Angle of The center angle c O ^ is equal to the angle of edge ' c' i c. c' ^ is the side opposite the center angle c' O i, and c' O is the side adjacent the center angle, c' i = 2.5"; c' O = 3.5". Dividing 2.5" by 3.5" we obtain .71428" + as the tangent of c' O i. In the table we find .71329 to be the nearest tangent, the corre- sponding angle being 35° 30'. S5^°, then, is the center angle c' O i and the angle of edge c i n, yevj nearly. When the axes of bevel gears are at right angles the angle of edge of one gear is the complement of angle of edge of the other gear. Subtracting, then, 35^° from 90° we obtain 54^° as the angle of edge of gear 28 teeth, to mesh with gear 20 teeth, Fig. 55, from which we have the rule for obtaining centre angles when the axes of gears are at right angles. Divide the radius of the pinion by the radius of the gear and the quotient will be the tangent of centre angle of the pinion. Now subtract this centre angle from 90 deg. and we have' the centre angle of the gear. The same result is obtained by dividing the number of teeth in the pinion by the number of teeth in the gear ; the quotient is the tangent of the centre angle. PBOVIDENCE, K. I. 101 Fig. 55. BEVEL GEAR DIAGRAM. 102 BKOWN & SHAKPE MFG. CO. Angle Of Face. To obtain angle of face O m" c\ the distance c O becomes the side opposite and the distance m" c is the side adjacent. The distance c O is 3.5", the radius of the 28 tooth bevel gear. The distance c m" is by measurement 2.82". Dividing 3.5 by 2.82 we obtain 1.2411 for tangent of angle of face O m" c . The nearest tangent in the table is 1.2422 and the corresponding angle is 51° 10'. To obtain cutting angle c O n" vre divide the distance c' n" by c O. By measurement c' n" is 2.2". Divid- ing 2.2 by 3.5 we obtain .62857 for tangent of cutting angle. The nearest corresponding angle in the table is 32°10'. The largest pitch diameter, kj, of a bevel gear, as in Fig. 56, is known the same as the pitch diameter of any spur geai*. Now, if we know the distance h o or its equal a q, we can obtain the whole diameter of bevel gear blank by adding twice the distance b o to the largest pitch diameter. crement.*^Vig' Twice the distance b o, or what is the same thing, ^^- the sum oi a q and Z> o is called the diameter incre- ment, because it is the amount by Avhich we increase the largest pitch diameter to obtain the whole or out- side diameter of bevel gear blanks. The distance b o can be calculated without measuring the diagram. The angle b o j is equal to the angle of edge. The angle of edge, it will be remembered, is the angle formed by outer edge of blank or ends of teeth with the end of hub or a plane perpendicular to the axis of gear. The distance ^ o is equal to the cosine of angle of edge, multiplied by the distance j o. The distance j o is the addendum, as in previous chapters ( = s). Hence the rule for obtaining the diameter increment of any bevel gear: Multiply the cosine of angle of edge by the toorking depth of teeth (D"), and the product will be the diameter increment. By the method given on page 102 we find the angle of edge of gear (Fig. 56) is 56° 20'. The cosine of 56° 20° is .55436, which, multiplied by |", or the ^Outside Diam- (Jepth of the 3 P gear, gives the diameter increment of the bevel gear 18 teeth, 3 P meshing with pinion of 12 PROAaDENCE, R. I. 103 104 BKOWN & SHAEPE MFG. CO. teeth. I of .55436=.369"+ (or .37", nearly). Adding the diameter increment, .37", to the largest pitch diameter of gear, 6", we have 6.37" as the outside diameter. In the same manner, the distance c d is half the diameter increment of the pinion. The angle c d k is equal to the center angle of pinion, and when axes are at right angles is the complement of center angle of gear. The center angle of pinion is 33° 40'. The cosine, multiplied by the working depth, gives .555" for diameter increment of pinion, and we have 4.555" for outside diameter of pinion. In turning bevel gear blanks, it is sufficiently accu- rate to make the diameter to the nearest hundredth of an inch. Angle incre The Small angle o 0/ is called the angle increment. When shafts are at right angles the face angle of one gear is equal to the center angle of the other gear, minus the angle increment. Thus the angle of face of gear (Fig. 56) is less than the center angle D O ^, or its equal O^' ^ by the angle o 0/. That is, subtracting o O j from O j k, the re- mainder will be the angle of face of gear. Subtracting the angle increment from the center angle of gear, the remainder will be the cutting angle. The angle increment can be obtained by dividing o j, the side ojDposite, by Oj, the side adjacent, thus finding the tangent as usual. The length of cone-pitch line from the common center, O to j, can be found, without measuring dia- gram, by multiplying the secant of angle Oj k, or the center angle of pinion, by the radius of largest pitch diameter of gear. The secant of angle Oj k, 33° 40', is 1.2015, which, multiplied by 3", the radius of gear, gives 3.6045" as the length of line O j. Dividing oj by Oj, we have for tangent .0924, and for angle increment 5° 20'. The angle increment can also be obtained by the following rule : PROVIDENCE, E. I. 105 Divide the sine of cerder angle hy half the nurn- her of teeth, and the quotient loill he the tangent of increment angle. Subtracting the angle increment from center angles of gear and pinion, we have respectively : Cutting angle of gear, 51°. Cutting angle of pinion, 28° 20'. Kemembering that when the shafts are at right angles, the face angle of a gear is equal to the cutting angle of its mate (Chapter X. part 1), we have : Face angle of gear, 28° 20'. Face angle of pinion, 51°. It will be seen that both the whole diameter and the angles of bevel gears can be obtained without making a diagram. Mr. George B. Grant has made a table of different pairs of gears from 1 to 1 up to 10 to 1, con- taining diameter increments, angle increments and centre angles, which is published in his "Treatise on Gears." "Formulas in Gearing," published by us, also contains extensive tables for bevel gearing. We have adopted the terms "diameter increment," "angle incre- ment," and "centre angle" from him. He uses the term "■'back angle" for what we have called angle of edge, only he measures the angle from the axis of tlie gear, instead of from the side of the gear, or from the . '•'^^i^y^^*?" ° ' & ' Angle by the end of hub, as we have done ; that is, his *^back angle "sine, is the complement of our angle of edge. In laying out angles, the following method may be Ji"Kj. 57. 106 BROWN & SHARPE MFG. CO. Back Cone Radius. preferred, as it does away with the necessity of making aright angle: Draw a circle, ABO (Fig. 57), ten inches in diameter. Set the dividers to ten times the sine of the required angle, and point off this distance in the circumference as at A B. From any point O in the circumference, draw the lines O A and O B. The angle A O B is the angle required. Thus, let the re- quired angle be 12°. The sine of 12° is .20791, which, multiplied by 10, gives 2.0791", or 2^" nearly, for the distance A B. Any diameter of circle can be taken if we multiply the sine by the diameter, but 10" is very convenient, as all we have to do with the sine is to move the decimal point one place to the right. If either of the lines pass through the centre, then the two lines which do not pass through the centre will form a right angle. Thus, if B passes through the centre then the two lines A B and A will form a right angle at A. Na = No. of Teeth in Gear. Nb = No. of Teeth in Pinion. OC = Centre Angle of Gear. Measure the back cone radius a b for the gear, or 6 c for the pinion. This is equal to the radius of a spur gear, the nximber of teeth in which would determine the cutter to use. Hence twice a b times the diametral pitch equals the number of teeth for which the cutter should be selected for the gear. Looking in the list on page 240 the proper number for the cutter can be found. Thus, let the back cone radius a. b be 4" and the diameter pitch be 8. Twice four is 8 and 8 x 8 is 64, from which it can be seen that the cutter must be of shape No. 2, as 64 is between 55 and 134, the range covered by a No. 2 cutter. The number of teeth for which the cutter should be selected can also be found by tlie following formula : Na Nb Tan. OC No. of teeth to select cutter for gear =- Na for pinion =•; Nb "Cos.a "^ — Sin- a If the gears are mitres or are alike, only one cutter is needed; if one gear is larger than the other, two may be needed. J07 CHAPTER IV. SPIRAL GEARS— CALCULATIONS FOR LEAD OF SPIRALS. When the teeth of a gear are cut, not in a straight Spiral Gear, path, like a spur gear, but in a hehcal or screw-like path, the gear is called, technically, a twisted or screw gear, but more generally among mechanics, a spiral gear. A distinction is sometimes made between a screw gear and a twisted gear. In twisted gears the pitch surfaces roll upon each other, exactly like spur gears, the axes being parallel, the same as in Fig. 1, Part I. In screw gears there is an end movement, or slipping of the pitch surfaces upon each other, the axes not being parallel. In screw gearing the action is analogous to a screw and nut, one gear driving another by the end movement of its tooth jDath. This is readily seen in the case of a worm and worm-wheel, when the axes are at right angles, as the movement of wheel is then wholly due to the end movement of worm thread. But, as we make the axes of gears more nearly parallel, they may still be screw gears, but the distinction is not so readily seen. Unless otherwise stated, the shafts of screw gears are at right angles, as at A and B, Fig. 59. The same gear may be used in a train of screw gears or in a train of twisted gears. Thus, B, as it relates to A, may be called a screw gear ; but in connection with C, the same gear, B, may be called a twisted gear. These distinctions are not usually made, and we call all helical or screw-like gears made on the Universal Milling Machine spiral gears. When two external spiral shears run together, with Direction of ,1 • nT T .1 i ,1 ^ ji 1 Spiral with ref- their axes parallel, the teeth of the gears must have erence to Axes. opposite hand spirals. 108 BROWN & SHAKPE MFG. CO. Thus, in Fig. 59 the gear B has right hand spiral teeth, and the gear C has left hand spiral teeth. "When the axes of two spiral gears are at right angles, both gears must have the same hand spiral teeth. A and B, Fig. 59, have right hand spiral teeth. If both gears A and B had left hand spiral teeth, the relative direc- tion in which they turn would be reversed. Spiral Lead. Ti^g spiral lead or lead of spiral is the distance the spiral advances in one turn. A cylinder or gear cut with spiral grooves is merely a scrcAV of coarse pitch or long lead ; that is, a spiral is a coarse lead screw, and a screw is a fine lead spiral. Since the introduction and extensive use of the Universal Milling Machine, it has become customary to call any screw cut in the milling machine a spiral. The spiral lead is given as so many inches to one turn. Thus, a cylinder having a spiral groove that advances six inches to one turn, is said to have a six inch spiral. In screws the pitch is often given as so many turns to one inch. Thus, a screw of y lead is said to be 3 turns to the inch. The reciprocal expression is not much used with spirals. For example, it would not be convenient to speak of a spiral of 6'' lead, as \ turns to one inch. The calculations for spirals are made from the func- tions of a right angle triangle. Example, Cut from paper a right angle triangle, one side of showing the r r o o o ' ^ nature of a He- the right angle 6 long, and the other side of the right angle 2". Make a cylinder 6" in circumference. It will be remembered (Part I., Chapter II.) that the circumference of a cylinder, multiplied by .3183, equals the diameter — 6" X -3183=1.9098". Wrap the paper triangle around the cylinder, letting the 2" side be parallel to the axis, the 6" side perpendicular to the axis and reaching around the cylinder. The hypoth- eneuse now forms a helix or screw-like line, called a spiral. Fasten the paper triangle thus wrapped around. See Fig. 60. PROVIDENCE, It. I. 109 FIG, 58 -RACKS AND GEARS. Fig. 59.-SPIRAL GEARING. 110 BROWN & SHARPE MFG. CO. Fig. 60. If we now turn this cylinder A B C D one lurii in the direction of the arrow, the spiral will advance from to E. This advance is the lead of the spiral. The angle E E, which the spiral makes with the axis E 0, is the angle of the spiral. This angle is found as in Chapter I. The circumference of the cyhnder corresponds to the side opposite the angle. The pitch of the spiral corresponds to the side adjacent the angle. Hence the rule for angle of spiral: ci?/atin°gtiie Divide the circumference of the cylinder or spiral parts of a spi-j^ //^g number of inches of spiral to one turn, and the quotient will he the tangent of atigle of spiral. When the angle of spiral and circumference are given, to find, the lead : Divide the circumference hy the tangent of angle, and the quotient ivill he the lead of the spiral. When the angle of spiral and the lead or pitch of spiral are given, to find the circumference : Multiply the tangent of angle hy the lead, and the product luill he the circumference. When applying calculations to spiral gears the angle is reckoned at the pitch circumference and not at the outer or addendum circle. It will be seen that when two spirals of different diameters have the same lead the spiral of less diame- ter will have the smaller angle. Thus in Fig. 60 if the paper triangle had been 4" long instead of 6" the diam- eter of the cylinder would have been 1.37" and the angle of the spiral would have been only 63J degrees. Ill CHAPTER V. EXAMPLES m CALCULATION OF THE LEAD OF SPIRAL— ANGLE OF SPIRAL— CIRCUMFERENCE OF SPIRAL GEARS— A FEW HINTS ON CUTTING. It will be seen that the rules for calculating the cir- cumference of spiral gears, angle and the lead of spiral are the same as in Chapter I., for the tangent and angle of a right angle triangle. In Chapter IV., the word "circumference" is substituted for "side opposite," and the words "lead of spiral" are substituted for "side adjacent." When two spiral gears are in mesh the angle of raif^^itti^^ f ^'' spiral should be the same in one gear as in the other, ®^°e to Angle in order to have the shafts parallel and the teeth work properly together. When two gears both have right hand spiral teeth, or both have left hand spiral teeth, the angle of their shafts will be equal to the sum of the angles of their spirals. But when two gears have different hand spirals the angle of their shafts will be equal to the difference of their angles of spirals. Thus, in Fig. 59 the gears A and B both have right hand spirals. The angle of both spirals is 45°, their sum is 90°, or their axes are at right angles. But C has a left hand spiral of 45°. Hence, as the difference between angles of spirals of B and C is 0, their axes are parallel. If two 45° gears of the same diameter have the same number of teeth the lead of the spiral will be alike in both gears: if one gear has more teeth than the other the lead of spiral in the larger gear should be longer in the same ratio. Thus, if one of these gears has 50 teeth, and the other has 25 teeth, the lead of spiral Lead in spi- ' ' -t^ rals of diflfer- in the 50 tooth gear should be twice as long as that of ent diameters. the 25 tooth gear. Of course, the diameter of pitch IVZ BKOWN & SHAKPE MFG. CO. circle should be twice as large in the 50 tooth as in the 25 tooth gear. In spirals where the angle is 45° the circumference is the same as the spiral lead, because the tangent of 45° is 1. CircumPr'^n™ Sometimes the circumference is varied to suit a pitch tosuitaspirai. that can be cnt on the machine and retain the angle required. This would apply to cutting rolls for mak- ing diamond-shaped impressions where the diameter of the roll is not a matter of importance. When two gears are to run together in a given velocity ratio, it is well first to select spirals that the machine will cut of the same ratio, and calculate the numbers of teeth and angle to correspond. This will often save considerable time in figuring. The calculations for spiral gears present no special difficulties, but sometimes a little ingenuity is required to make work conform to the machine and to such cutters as we may have in stock. Let it be required to make two spiral gears to run with a ratio of 4 to 1, the distance between centres to be 3.125" (31"), the axes to be parallel. By rule given in Chapter XII., Part I., we find the diameters of pitch circles will be 5" and 1^". Let us take a spiral of 48" lead for the large gear, and a spiral of 12" lead for the small gear. The circumfer- ence of the 5" pitch circle is 15.70796". Dividing the circumference by the lead of the spiral, we have i^7_|.7_96 =,32724" for tangent of angle of spiral. In the table the nearest angle to tangent, .32724", is 18° 10'. As before stated, the angle of the teeth in the small gear will be the same as the angle of teeth or spiral in the large gear. iiiAngiesattop Now, this rule gives the angle at the pitch surface sph-aiGroOTe^s^o^'y* Upon looking at a small screw of coarse pitch, it will be seen that the angle at bottom of the thread is not so great as the angle at top of thread; that is, the thread at bottom is nearer parallel to the centre line than that at the top. This will be seen in Fig. 61, where A is the centre line; Z>/ shows direction of bottom of thread, and d g PKOVIDENCE, R. I. 113 shows direction of top of tbrciul. The angle A fb is less than the angle A y d. The diflferetice of angle being due to tlie warped nature of a screw thread. A cylinder 2" diameter is to have spiral grooves ^0° catofiat?on of with the centre line of cylinder; what will be the lead Lead of spiral, of spiral? The circumference is 6.2833". The tan- gent of 20° is .36397. Dividing the circumference by the tangent of angle, we obtain ^;||§f ^ = 17.26"-|-for lead of spiral. Fig. 61. In Chapter XI, part I, it is stated that, when gashing the teeth of a worm-wheel, the angle of the teeth across the face is measured from the line parallel to the axis of the wheel. To obtain this angle from the worm, divide the lead by the pitch circumference of the worm, and the quo- tient will be the tangent of the angle that the thread makes with a plane perpendicular to the axis. 114 CHAPTER VI. NORMAL PITCH OF SPIRAL GEARS— CURVATURE OF PITCH SURFACE— FORM OF CUTTERS. Curv™*^ ^° ^ ^ Normal to a curve is a line perpendicular to the tangent at the point of tangency. In Fig. 62, the line B C is tangent to the arc D E F, and the line A E O, being perpendicular to the tan- gent at E the point of tangency, is a normal to the arc. Fig. 63 is a representation of the pitch surface of a spiral gear. A' D' C is the cu'cular j)itch, as in Part I. A D C is the same circular pitch seen upon the periphery of a wheel. Let A D be a tooth D and a space. Now, to cut this space D C, the path of cut- ting is along the dotted line a h. By mere inspection, we can see that the shortest distance between two teeth along the pitch surface is not the distance ABC. Let the line A E B be perpendicular to the sides of teeth upon the pitch surface. A continuation of this line, perpendicular to all the teeth, is called the Normal Helix. The line A E B, reaching over a tooth and a space along the normal helix, is called the Normal Pitch, or the normal linear pitch. PKOVIDENCH, R. I. 115 Fig. 63. 116 BROWN & SHAEPE MFG. CO. Normal Pitch. The Normol Pitch of a spiral gear is then : The shortest distance betioeen the centers of two consecutive teeth measured along the pitch surface. In spur gears the normal pitch and circular pitch are alike. In the rack D D, Fig. 58, the linear pitch and normal pitch are alike. Cutter for From the foregoing it will be seen that, if we should Spiral Gears. . . cut the space D C with a cutter, the thickness of which at the pitch line is equal to one-half the circular pitch, as in spur wheels, the space would be too wide, and the teeth would be too thin. Hence, spiral gears should be cut with thinner cutters than spur gears of the same circular pitch. The angle C A B is equal to the angle of the spiral. The line A E B corresponds to the cosine of the angle CAB. Hence the rule : Multiply the cosine of angle '^I'p^t'ii ^^^' ^-^ spiral by the circular pitch, and the product will he the normal pitch. One-half the normal pitch is the proper thickness of cutter at the pitch line. If the normal pitch and the angle are known, Divide the normal pitch hy the cosine of the angle and the quo- tient will be the circular pitch. This may be required in a case of a spiral pinion run- ning in a rack. The perpendicular to the side of the rack is taken as the line from which to calculate angle of teeth. That is, this line would correspond to the axial line in a spiral gear ; and, when the axis of the gear is at right angles to the rack, the angle of the teeth with the side of the rack is obtained by subtract- ing this angle from 90°. The angle of the rack teeth with the side of the rack can also be obtained by remembering that the cosine of the angle of spiral is the sine of the angle of the teeth with the side of the rack. The addendum and working depth of tooth should correspond to the normal pitch, and not to tiie circular pitch. Thus, if the normal pitch is 13 diametral, the addendum should be -j^'', the thickness .1309", and so on. The diameter of pitch circle of a spiral gear is calculated from the diametral pitch. Thus a gear of 30 teeth 10 P would be 3" pitch diameter. ±»ROVIDENCE, R. I. 117 But if the normal pitch is 13 diametral pitch, the blank will be 3yV diameter instead of SjV'* It is evident that the normal pitch varies with the^^j.'?™^'^'^'^^ angle of spiral. The cutter should be for the normal pitcli. In designing spiral gears, it is well first to look over list of cutters on hand, and see whether there are cutters to which the gears can be made to conform. This may avoid the necessity of getting a new cutter, or of changing both drawing and gears after they are under way. To do this, the problem is worked the reverse of the foregoing; that is: First calculate to the next finer pitch cutter than gj^o^^l^^^^j^^j would be required for the diametral pitch. cutters ^glvea" Let us take, for example, two gears 10 pitch and 30 teeth, spiral and axes parallel. Let the next finer cut- ter be for 12 pitch gears. The first thing is to find the angle that will make the normal pitch .2618", when the circular pitch is .3142". See table of tooth parts. This means (Fig. 63) that the line A D C will be .3142" when A E B is .2618". Dividing .2618" by .3142" (see Chap. IV.), we obtain the cosine of the angle CAB, which is also the angle of the spiral, iff if "=-833. The same quotient comes by dividing 10 by 12, ■f^f =.833 4- ; that is, divide one pitch by the other, the larger number being the divisor. Looking in the table, we find the angle corresponding to the cosine .833 is 33° 30'. We now want to find the pitch of spiral that will give angle of 33|^° on the pitch surface of the wheel, 3" diameter. Dividing the circumference by the tan- gent of angle, we obtain the pitch of spiral (see Chap. V.) The circumference is 9.4248". The tangent of 33° 30' is .66188, ^:fft|j=14.23 ; and we have for our spiral 14.23" lead. When the machine is not arrano-ed for the exact when exact ° Pitch cannot be pitch of spnal wanted, it is generally well enough to cut. take the next nearest spiral. A half of an inch more or less in a spiral 10" pitch or more would hardly be noticed in angle of teeth. It is generally better to take the next longer spiral and cut enough deeper to bring center distances right. Wlien two gears of the same size are in mesh with their axes parallel, a change 118 BROWN & SHAKPE MFG. CO. of angle of teeth or spiral makes no difference in the correct meshing of the teeth. Spiral Gears B^t when gears of different size are in mesh, due of Different ° . i , • • • i Sizes of Mesii. regard must be had to the spirals being in pitch, pro- portional to their angular velocities (see Chapter V. ) "VVe come now to the curvatui'e of cutters for spiral gears; that is, their shape as to whether a cutter is made to cut 12 teeth or 100 teeth. A cutter that is right, Shape of Cut- to cut a spuT gear 3" diameter, may not be right for a spiral gear 3" diameter. To find the curvature of cutter, fit a templet to the blank along the line of the normal helix, as A E B, letting the templet reach over about one normal pitch. The curvature of this templet will be nearer a straight line than an arc of the adden- dum circle. Now find the diameter of a circle that will approximately fit this templet, and consider this circle as the addendum circle of a gear for which we are to select a cutter, reckoning the gear as of a pitch the same as the normal pitch. Fig. 64. Thus, in Fig. 64, suppose the templet fits a circle 3^" diameter, if the normal pitch is 12 to inch, dia- metral, the cutter required is for 12 P and 40 teeth. The curvature of the templet will not be quite circular, but is sufiiciently near for practical purposes. Strictly, PROVIDENCE, R. I. 119 a flat templet cannot be made to coincide with the normal helix for any distance whatever, but any greater refinement than we have suggested can hardly be car- ried out in a workshop. This applies more to an end cutter, for a disk cutter may have the right shape for a tooth space and still round off the teeth too much on account of the warped nature of the teeth. The difference between normal pitch and linear or cii'cular pitch is plainly seen in Figs 58 and 59. The rack T> D, Fig. 58, is of regular form, the depth of teeth being J-|- of the circular pitch, nearly (.6866 of the pitch, accurately). If a section of a tooth in either of the gears be made square across the tooth, that is a normal section , the depth of the tooth will have the same relation to the thickness of the tooth as in the rack just named. • But the teeth of spu'al gears, looking at them upon the side of the gears, are thicker in proportion to their depth, as in Fig. 59 This difference is seen between the teeth of the two racks D D and E E, Fig. 58. In the rack D D we have 20 teeth, while in the rack E E we have but 14 teeth ; yet each rack will run with each of the spiral gears A, B or C, Fig. 59, but at different angles. The teeth of one rack will accui-ately fit the teeth of the other rack face to face, but the sides of one rack will then be at an angle of 45° with the sides of the other rack. At F is a guide for holding a rack in mesh with a gear. The reason the racks will each run with either of the three gears is because all the gears and racks have the same normal pitch. When the spiral gears are to run together they must both have the same normal pitch. Hence, two spiral gears may run correctly together though the circular pitch of one gear is not like the circular pitch of the other gear. If a rack is to run at any angle other than 90° with the axis of the gear it is well to determine the data from a diagram, as it is very difficult to figure the angles and sizes of the teeth without a sketch or diagram. 120 CHAPTER VII. CUTTING SPIRAL GEARS IN A UNIVERSAL MILLING MACHINE. A rotary disk cntter is generally preferable to a shank cutter or end mill on account of cutting faster and hold- ing its shape longer. In cutting spiral grooves, it is sometimes necessary to use an end mill on account of the warped character of the grooves, but it is very sel- dom necessary to use an end mill in cutting spiral gears. se^tST^'^of the ^^fore Cutting into a blank it is well to make a slight Machine. trace of the spiral with the cutter, after the change gears are in place, to see whether the gears are correct. If the material of the gear blanks is quite expensive, it is a safe plan to make trial blanks of cast iron in order to prove the setting of the machine, before cutting into the expensive material. The cutting of spiral gears may develop some curi- ous facts to one that has not studied warped surfaces. The gears. Fig. 59, were cut with a planing tool in a shaper, the spiral gear mechanism of a Universal Mill- ing Machine having been fastened upon the shaper. The tool was of the same form as the spaces in the rack D D, Fig. 58. All spiral gears of the same pitch can be cut in this manner with one tool. The nature of this cutting operation can be understood from a considera- tion of the meshing of straight side rack teeth with a spiral gear, as in Fig. 58. Spiral gears that run cor- rectly with a rack, as in Fig. 58, will run correctly with each other when their axes are pai*allel, as at B C, Fig. 59; but it is not considered that they are quite correct, theoretically, to run together when the gears have the same hand spiral, and their axes are at right PROVIDENCE, K. I. 121 /> Ua c " Fig. 65 ( > fl 11 /> ^ () (] K [[ 11 _J — — { \ Fig. 66 122 BKOWN & SHABPE MFG. CO. angles, as AB, Fig. 59, though they will run well enough practically. The operation of cutting spiral teeth with a planer tool is sometimescalled^/fl!;^m^ the teeth. Plan- ing is an accurate way of shaping teeth that are to mesh with rack teeth and for gears on parallel shafts; this method has been employed to cut spiral pinions that drive planer tables, but has noc been found available for general use. It is convenient to have the data of spiral gears arranged as in the following table : Data. No. of Teeth Pitch Diameter . Outside Diameter Circular Pitch Angle of Teeth with Axis Normal Circular Pitch Pitch of Cutter . Addendum s Thickness of Tooth t Whole Depth D"+f . No. of Cutter Exact Lead of Spiral Approximate Lead of Spiral Gears on Milling Machine to Cut Spiral Gear on Worm .... 1st Gear on Stud 2nd Gear on Stud Gear on Screw .... Gear. Pinion. A spiral of any angle to 45° can generally be cut in a Universal Milling Machine without special attach- ments, the cutter being at the top of the work. The cutter is placed on the arbor in such position that it can reach the work centrally after the table is set to the angle of the spiral. In order to cut central, it is generally well enough to place the table, before setting it to the angle, so that the work centres will be central with the cutter, then swing the table and set it to the angle of the spiral. For very accurate work, it is safer to test the T)osi-..*^^"*^^^ ^®*' . , ^ ting. tion of the centres after the table has been set to the angle. PROVIDENCE, R. I., U. S. A. 123 Fig. 67. USE OF VERTICAL SPINDLE MILLING ATTACHMENT IN CUTTING SPIRAL GEARS. 124 BROWN & SHAKPE MFG. CO. This can be done with a trial piece. Fig. 65, which is simply a round arbor with centre holes in the ends. It is mounted between the centres, and the knee is raised until the cutter sinks a small gash, as at A. This gash shows the position of the cutter; and if the gash is central with the trial piece, the cutter will be central with the work. If preferred, the arbor can be dogged to the work spindle ; and the line B drawn on the side of the arbor at the same height as the cen- tres ; the work spindle should then be turned quarter way round in order to bring the line at the top. The gash A can now be cut and its position determined with the line. In cutting small gears the arbor can be dogged to the work spindle; the distance between the gear blank and the dog should be enough to let the dog pass the cutter arbor without striking. A spiral gear is much more likely to slip in cutting than a spur gear. For gears more than three or four inches in diameter it is well to have a taper shank arbor held directly in the work spindle, as shown in Figs. 67 and 68; and for the heaviest work, the arbor can be drawn into the spin- dle with a screw in a threaded hole in the end of the shank. After cutting a space the work can be dropped away from the cutter, in order to avoid scratching it when coming back for another cut. Some workmen prefer not to drop the work away, but to stop the cutter and turn it to a position in which its teeth will not touch the work. To make sure of finding a place in the cut- ter that will not scratch, a tooth has sometimes been taken out of the cutter, but this is not recommended. The safest plan is to drop the work away. Angle greater In cutting spiral gears of greater angle than 45°, a than 45° vertical spindle milling attachment is available, as shown in Figs. 67 and 68. In Fig. 67 the cutter is at 90° with the work spindle when the table is set to 0, so that the proper angle at which the table should be set, is the difference between the angle of the spiral and 90°. Thus, to cut a 70° PROVIDENCK, K. I., U. S. A. 125 Fig. 68. USE OF VERTICAL SPINDLE MILLING ATTACHMENT IN CUTTING SPIRAL GEARS. 126 BROWN & SHARPE MFG. CO. spiral, we subtract 70° from 90°, and the remainder, 20°, is the angle to set the table. In cutting on the top. Fig. 67, the attachment is set to 0. In Fig. 68 the cutter is at the side of the work ; the table is set to 0, and the attachment is set to the differ- ence between 90° and the required angle of spiral. In setting the cutter central it is convenient to have a small knee as at K, Fig. 66. A line is drawn upon the knee at the same height as at the centres. The cutter arbor is brought to the angle as just shown, and a gash is cut in the knee. When the gash is central with the line, the cutter will be central with the work. The cutter can be set to act upon either side of the gear to be cut, according as a right hand or a left hand spiral is wanted. The setting in Fig. 68 is for a right hand spiral. If the gear blank were brought in front of the cut- ter, and the reversing gear set between two change gears, the machine would be set for a left hand spiral. For coarser pitches than about 12 P diametral, it is well to cut more than once around, the finishing cut being quite light so as to cut smooth. 127 CHAPTER VIII. SCREW GEARS AND SPIRAL GEARS— GENERAL REMARKS. The working of spiral gears, when their axes are working of ., , . n J.I ii A Spiral Gears. para.lel, is generally smoother than spur gears. A tooth does not strike along its whole face or length at once. Tooth contact first takes place at one side of the gear, passes across the face and ceases at the other side of the gear. This action tends to cover defects in shape of teeth and the adjustment of centres. Since the invention of machines for producing accu- rate epicyloidal and involute curves, it has not so often been found necessary to resort to spiral gears for smoothness of action. A greater range can be had in the adjustment of centers in spiral gears than in spur gears. The angle of the teeth should be enough, so that one pair of teeth will not part contact at one side of the gears until the next pair of teeth have met on the other side of the gears. When this is done the gears will be in mesh so long as the circumferences of their addendum circles intersect each other. This is some- times necessary in gears for rolls. Relative to spur and bevel gears in Part I., Chapter XII., it was stated that all gears finally wore them- selves out of shape and might become noisy. Spiral gears may be worn out of shape, but the smoothness of action can hardly be impaired so long as there are any teeth left. For every quantity of wear, of course, there will be an equal quantity of backlash, so that if gears have to be reversed the lost motion in spiral gears will be as much as in any gears, and may be more if there is end play of the shafts. In spiral gears End Pressure there is end pressure upon the shafts, because of the Spiral Gears. screw-like action of the teeth. This end pressure is sometimes balanced by putting two gears upon each shaft, one of right and one of left hand spu-al. 128 BBOWK & SHAKPE MFG. CO. The same result is obtained in solid cast gears by making the pattern in two parts — one right and one left-hand spiral. Such gears are colloquially called "herring-bone gears." In an internal spiral gear and its pinion, the spirals of both wheels are either right-handed or left-handed. Such a combination would hardly be a mercantile product, although interesting as a mechanical feat. In screw or worm-gears the axes are generally at right angles, or nearly so. The distinctive features of screw gearing may be stated as follows : The relative angular velocities do not depend upon the diameters of pitch- cylinders, as in Chapter I., Distinctive Part I. Thus the worm in ChaDter XL, Fisf. 35, can features of . j. ' o ' Screw Gearing, be any diameter — one inch or ten inches — without affecting the velocity of the worm-wheel. Conversely if the axes are not parallel we can have a pair of spiral or screw gears of the same diameter, but of different numbers of teeth. The direction in which a worm-wheel turns depends upon whether the worm has a right-hand or left-hand thread. When angles of axes of worra and worm-wheel are oblique, there is a practical limit to the directional relation of the worm-wheel. The rotation of the worm-wheel is made by the end movement of the worm-thread. The term worm and worm-wheel, or worm-gearing, is applied to cases where the worms are cut in a lathe, and the shapes of the threads or teeth, in axial section, are like a rack and the pitch is measured on a line parallel to the axis. The shape usually selected is like the rack for a single curve or involute gear. See Chap. IV, Part I. Worms are sometimes cut in a milling machine. If the form of the teeth in a pair of screw gears is determined upon the normal helix, as in Chap. VI,, the gears are usually called Spiral Gears. If we let two cylinders touch each other, their axes being at right angles, the rotation of one cylinder will have no tendency to turn the other cylinder, as in Chapter I., Part I. PROVIDENCE, R. I. 129 We can now see why worms and worm-wheels wear wiiy worm •' Wheels wear ont faster than other gearing. The length of worm-sof^^s'^ thread, equal to more than the entire circumference of worm, comes in sliding contact with each tooth of the wheel during one turn of the wheel. The angle of a worm-thread can be calculated the same as the angle of teeth of spiral gear ; only, the angle of a worm thread is measured from a line or plane that is perpendicular to the axis of the worm. When a multiple threaded worm is cut in a milling machine and the angle of the thread is less than 75° with the axis of the worm, it may be desirable to work by the normal pitch. The normal pitch can he obtained by multiplying the thread-pitch by the sine of the angle of the thread with the axis. 130 CHAPTER IX. CONTINUED FRACTIONS— SOME APPLICATIONS IN MACHINE CONSTRUCTION. Definition of ^ continued fraction is one that has unity for its a Continued •' Fraction. numerator, and for its denominator an entire number plus a fraction, which fraction has also unity for its numerator, and for its denominator an entu'e number plus a fraction, and thus in order. The expression, 4 + 1 ^ is called a continued frac- tion. By the use of continued fractions, we are ena- Practicai use ^jed to find a fraction expressed in smaller numbers, of Continued -'■ ' Fractions. that, for practical purposes, may be sufficiently near in value to another fraction expressed in large numbers. If we were required to cut a worm that would mesh with a gear 4 diametral pitch (4 P.), in a lathe having 3 to 1-inch linear leading screw, we might, without continued fractions, have trouble in finding change gears, because the circular pitch corresponding to 4 diametral pitch is expressed in large numbers : ^-^ 10000 -^ • This example will be considered farther on. For illustration, we will take a simpler example. What fraction expressed in smaller nulnbers is near- est in value to jVe ^ Dividing the numerator and the denominator of a fraction by the same number does not change the value of the fraction. Dividing both con^u^ue^d^®^"'^^ ^^ TIT ^J 2^' ^^ ^^"^^ 5J~ ^^'' what is the same thing expressed as a continued fraction, 5-t- i . The continued fraction s+gT is exactly equal to -^^j. If now, we reject the -^q, the fraction ^ will be larger than 5-i- 1 , because the denominator has been dimin- ished, 5 being less than 5-^-^. ^ is something near y?j9g- expressed in smaller numbers than 29 for a PROVIDENCE, R. I. 131 numerator and 146 for a denominator. Reducing -J and y^j"^ to a common denominator, we have ^ = ^|^ and i%V=Y3 0". Subtracting one from the other, we have Tj-^y, which is the difference between l and j^?"- Thus, in thinking of ^^V ''^ i» ^''^ have a pretty fair idea of its vakie. There are fourteen fractions with terms smaller than 29 and 146, which are nearer j^-a^. than ^ is, such as ■^, If- and so on to /^y. In this case by continued frac- tions we obtain only one approximation, namely -^, and any other approximations, as \f, -|^f-, &c., we find by trial. It will be noted that all these approximations are smaller in value than ^W- There are cases, how- ever, in which we can, by continued fractions, obtain approximations both greater and less than the required fraction, and these "will be the nearest possible approxi- mations that there can be in smaller terms than the given fraction. In the French metric system, a milHmetre is equal to .03937 inch; what fraction in smaller terms ex- presses .03987" nearly? .03937, in a vulgar fraction, ^^ To o 00 - Dividing both numerator and denominator by 3937, we have 25i5?5- Rejecting from the de- nominator of the new fraction, ^ifr? the fraction -^^ gives us a pretty good idea of the value of .03937". If in the expression, "ai+TIIi, we divide both terms of the fraction -jfir ^7 1575, the value will not be changed. Performing the division, we have ^ ° 25 + 1 2 + 787 1575 • We can now divide both terms of fW? ^J "^^"^j without changing its value, and then substitute the new fraction for ^W? ^^ ^^^ continued fraction. Dividing again, and substituting, we have : 1 25 +J^ 2 + 1 8+1 787 as the continued fraction that is exactly equal to .03937. 132 BROWN & SHARPE MFG. CO. In performing the divisions, the work stands thus : 3937) 100000 (25 7874 21260 19685 1575) 3937 (2 3150 787) 1575 (2 1574 1) 787 (787 787 •0- That is, dividing the last divisor by the last remain- der, as in finding the greatest common divisor. The quotients become the denominators of the continued fraction, with unity for numerators. The denominators 25, 2, and so on, are called incomplete quotients, since they are only the entire parts of each quotient. The first expression in the continued fraction is -^-^ or .04 — a little larger than .03937. If, now, we take gg-qri, we shall come still nearer .03937. The expres- sion 25'x-f is merely stating that 1 is to be divided by 25|-. To divide, we first reduce 1h\ to an improper fraction, ^, and the expression becomes ST, or one divided by -^. To divide by a fraction, "Invert "the divisor, and proceed as in multiplication." We then have -^-^ as the next nearest fraction to .03937. -g2j-=.0392 + , which is smaller than .03937. To get still nearer, we take in the next part of the continued frac- tion, and have i 2 + 1 2' We can bring the value of this expression into a fraction, with only one number for its numerator and one number for its denominator, by performing the operations indicated, step by step, commencing at the last part of the continued fraction. Thus, 2-|-^, or 2^, is equal to |, Stopping here, the continued frac- tion would become i 25+J_ 5 2- 1 \ Now, ^ equals f , and we have 25 +^. 25f equals 2 5 ^\^ ; substituting again, we have li^. Dividing 1 by J-|^, we have yl^- yf y is the nearest fraction to PROVIDENCE, R. I. 133 .03937, unless we reduce the whole continued fraction _i 25 + 1 2 + 1 ^ + 1_, which would afive us back the .03937 itself. 787 y|y=. 03937007, which is only ^^^^^ larger .03937. It is not often that an approximation will come so near as this. This ratio, 5 to 127, is used in cutting millimeter Practical use Trt IT fill' of the foregoing thread screws. If the leading screw of the lathe isExampie. 1 to one inch, the change gears will* have the ratio of 5 to 127; if 8 to one inch, the ratio will be 8 times as large, or 40 to 127; so that with leading screw 8 to inch, and change gears 40 and 127, we can cut milli- meter threads near enough for practical purposes. The foregoing operations are more tedious in de- scription than in use. The steps have been carefully noted, so that the reason for each step can be seen from rules of common arithmetic, the operations being merely reducing complex fractions. The reductions, ^, ■fj, yIy, etc., are called conver gents, because they come nearer and nearer to the required .03937. The operations can be shortened as follows: Let us find the fractions converging towards .7854", Example, the circular pitch of 4 diametral pitch, .7854=Yyg%\; reducing to lowest terms, we have \\\l . Applying the operation for the greatest common divisor: 392; r) 5000 (1 3927 1073) 3927 <3 3219 708) 1073 (1 708 365) 708 (1 365 343) 365 (1 343 22) 343 (15 23 123 110 13) 22 (1 13 9) 13 (1 9 4) 9 (2 8 Y) 4 (4 4 Bringing the various incomplete quotients as de- nominators in a continued fraction as before, we have : 134 BROWN & SHAEPE MFG. CO. 1 1 + 1 3 + 1 i+1 1+1 1+1__ 15 + 1 1 + 1 1+1 2 + |- Now arrange each partial quotient in a line, thus : 13111 15 1 1 2 4 1 3 4 1 ±1. 112 18 3 35 g 8 9 3 3927 -L i T i it ST-g" T3 3" Ttt TT3T TO Now place under the first incomplete quotient the first reduction or convergent ^, which, of course, is 1 ; put under the next partial quotient the next reduction or convergent ^-rr or ^, which becomes f . '-^ 1 + 3 ■'3 1 is larger than .7854, and f is less than .7854. Having made two reductions, as previously shown, we can shorten the operations by the following rule for next convergents: Multiply the numerator of the convergent just found by the denominator of the next term of the con- tinned fraction, or the next incomplete quotient^ and add to the product the numerator of the 2oreceding convergent ; the sum ivill be the numerator of the next convergent. Proceed in the same way for the denominator, that is multiply the denominator of the convergent just found by the next incomplete quotient and add to the product the denominator of the preceding convergent ; the sum will be the denominator of the next convergent. Continue until the last convergent is the original frac- tion. Under each incomplete quotient or denominator from the continued fraction arranged in line, will be seen the corresponding convergent or reduction. The convergent ^l is the one commonly used in cutting racks 4 P. This is the same as calling the circumference of a circle 22-7 when the diameter is one (1) ; this is also the common ratio for cutting any raclv. The equivalent decimal to li is .7857 X, being about 1 tf ^arge. In three set- tings for rack teeth, this error would amount to about .001" For a worm, this corresponds to ^f threads to 1" ; now, with a leading screw of lathe 3 to 1", we would want gears on the spindle and screw in a ratio of 33 to 14. Hence, a gear on the spindle with 66 teeth, and a gear on the 3 thread screw of 28 teeth, would enable us to cut a worm to fit a 4 P gear. 135 CHAPTER X. ANGLE OF PRESSURE. In Fig. 69, let A be any flat disk lying upon a hori- zontal plane. Take any piece, B, with a square end, a b. Press against A with the piece B in the direction of the arrow. Fig. 69. Fig. It is evident A will tend to move directly ahead of B in the normal line c d. Now (Fig. 70) let the piece B, at one corner^/", touch the piece A. Move the piece B along the line d e, in the direction of the arrow. It is evident that A will not now tend to move in the line d e, but will tend to move in the direction of the normal c d. When one piece, not attached, presses against another, the tendency to move the second piece is in the du'ection of the normal, at the point of contact. This normal is called the line of pressu7'e. Line of Press- «' ^ lire. The angle that this line makes with the path of the impelling piece, is called the atigle of pressure. In Part I., Chapter IV., the lines B A and B A' are called lines of pressure. This means that if the gear 136 BEOWN & SHAKPE MFG. CO. drives the rack, the tendency to move the rack is not in the direction of pitch line of rack, but either in the direction B A or B A', as we turn the wheel to the left or to the right. The same law holds if the rack is moved in the direction of the pitch line ; the tendency to move the wheel is not directly tangent to the pitch circle, as if driven by a belt, but in the direction of the line of pressure. Of course the rack and wheel do move in the paths prescribed by their connections with the framework, the wheel turning about its axis and the rack moving along its ways. This pressure, not in a direct path of the moving piece, causes extra friction in all toothed gearing that cannot well be avoided. Although this pressure works out by the diagram, as we have shown, yet, in the actual gears, it is not at all certain that they will follow the law as stated, because of the friction of teeth among themselves. If the driver in a train of gears has no bearing upon its tooth-flank, we apprehend there will be but little tendency to press the shafts apart. Arc of Action. rpj^g ^^^ through which a wheel passes while one of its teeth is in contact is called the arc of action. tenf^of "^ Inter- ^^^il within a few years, the base of a system of ^^^^^seabie^jo^l^le.curye interchangeable gears was 12 teeth. It is now 15 teeth in the best practice (see Chapter VII., Part I.) The reason for this change was : the base, 15 teeth, gives less angle of pressure and longer arc of contact, and hence longer lifetime to gears. 137 CHAPTER XI. INTERNAL GEARS. In Part I., Chapter YIII., it is stated that the space of an internal gear is the same as the tooth of a spur gear. This applies to involute or single-curve gears as well as to double-curve gears. The sides of teeth in involute internal gears are hollowing. It, however, has been customary to cut internal gears with spur gear-cutters, a No. 1 cutter generally being used. This makes the teeth sides convex. Special cutters should be made for coarse Special cut- r" -_-... , ters for coarse pitch double-curve gears, in designing internal gears. Pitch, it is sometimes necessary to depart from the system with 15-tooth base, so as to have the pinion differ from the wheel by less than 15 teeth. The rules given in Part I., Chapters YII. and VIII., will apply in making gears on any base besides 15 teeth. If the base is low-numbered and the pinion is small, it may be neces- sary to resort to the method given at the end of Chap- ter VII., because the teeth may be too much rounded at the points by following the approximate rules. The base must be as small as the diiference between Base for in> ^ ternal Gear the internal gear and its pinion. The base can be Teeth, smaller if desired. Let it be requii-ed to make an internal gear, and pinion 24 and 18 teeth, 3 P. Here the base cannot be more than 6 teeth. In Fig. 71 the base is 6 teeth. The arcs A K and O k, drawn about T, have a radius equal to the radius of the pitch cu'cle of a 6-tooth gear, 8 P, instead of a 15-tooth geai", as in Chaj^ter VIII., Part I. The outline of teeth of both gears and pinion is Description of Fig. 67. made similar to the gear in Chapter VIII. The same 138 BEOWN & SHAKPE MFG. CO. GEAR, 24 TEETH. PINION, 18 TEETH, 3 P. P = 3 N =24 and 18 P'= 1.0472" t=- 5236" S= .3333' D= .6666" S+/= .3857" (}"+/= .7190" INTERNAL GEAR AND PINION IN MESH. PROVIDENCE, E. I. lettei's refer to similar parts. The clearance circle is, however, drawn on the outside for the internal gear. As before stated, the spaces of a spur wheel become the teeth of an internal vrheel. The teeth of internal gears require but little for fillets at the roots ; they are generally strong enough without fillets. The teeth of the pinion are also similar to the gear in Chapter VIII., substituting 6-tooth for 15-tooth base. To avoid confusion, it is well to make a complete sketch of one gear before making the other. The arc of action is longer in internal gears than in external gears. This property sometimes makes it necessary to give less fillets than in external gears. In Fig. 71 the angle K T A is 30° instead of 12°, as in Fig. 12. This brings the line of pressure L P at an angle of 60° with the radius C T, instead of 78°. A system of spur gears could be made upon this 6-tooth base. These gears would interchange, but no gear of this 6-tooth system would mesh with a double- curve gear made upon the 15-tooth system in Part 1. 139 140 CHAPTER XIL STRENGTH OF GEARING. We have been unable to derive from oar own experi- ence, any definite rule on this subject but would refer those interested to "Kent's Mechanical Engineers' Pocket Book/' where a good treatment of the subject can be found. We give a few examples of average breaking strain of our Combination Gears, as determined by dyna- mometer, the pressure being measured at the pitch line. These gears are of cast iron, with cut teeth. Diametral Pitch. No. Teeth. Revolutions per Minute. Pressure at Face. 10 8 6 5 1 1-16 1 1-4 1 9-16 1 7-8 110 72 72 90 27 40 27 18 1060 1460 2220 2470 These are the actual pressures for the particular widths given. If we take a safe pressure at 1-3 of the foregoing breaking strain, we shall have for 10 Pitch 353 1-3 Lbs. at the Pitch Line. 8 '' 486 2-3 " 6 " 740 *' " 5 " 823 1-3 The width of the face of a gear is in good proportion when it is 2^ times the circular pitch. PROVIDENCE, K. I. 141 TOOTH PARTS. Fig. 73. GEAR TOOTH 1 P 142 BROWN & SHARPE MFG. CO. The dimensions of tooth parts as given in the tables, pages 144 to 147, are correct according to the definition of tooth parts, pages 4 and 16 ; but, as the pitch line of gears is curved, the thickness of a tooth will not be measured on the pitch line if the caliper is set to the figures given in the tables mentioned. To measure the teeth accurately on the pitch line, the caliper must be set to the chordal thickness and the depth setting to the pitch line must be to the corrected s, as explained and tabulated. If the gear blank is not of the correct diameter, the proper allowance must be made in setting the caliper for the depth. It is utterly useless to be guided by the outside of a gear blank when the outside diameter is not right. The measuring of the tooth thickness is well enough, as a check, but it is oftentimes as well first to make sure that the spaces are cut to the right depth. Fig. 73 is a sketch of a gear tooth of 1 P. In meas- uring gear teeth of coarse pitch accurately the chordal thickness of the tooth, ATB, must be known, because it may be enough shorter than the regular tooth-thick- ness AHB, or t, to require attention. It may be also well to know the versed sine of the angle /?', or the dis- tance H, in order to tell where to measure the chordal thickness. Chordal Thicknesses of Teeth of Gears, on a Basis of 1 Diametral Pitch. N = Number of teeth in gears. T = Chordal thickness of Tooth. T = D' sin. /?' H = Height of Arc. H = R (1— cos. ^') D' = Pitch Diameter. R = Pitch Radius. = 90° divided by the number of teeth. Note. — For any pitch not in the following tables to find corresponding part : — Multiply the tabular value for one inch by the circular pitch required, and the product will be the value for the pitch given. Exafnple : What is the value of s for 4 inch circular pitch ? .3183 = s for 1" P' and .3183 X 4 = 1 .2732 = s for 4" P^ The expression "Addendum and ^" (addendum and the module) means the distance of a tooth outside of pitch line and also the distance occupied for every tooth upon the diam- eter of pitch circle. PROVIDENCE, R. I. 143 CHORDAL THICKNESSES OF TEETH OF GEARS. INVOLUTE. Cutter. T H Corrected S for Gear. No. I —135 T — P 1-5707 .0047 1 .0047 No. 2 — 55 T — [ P 1.5706 .0112 1.0112 No. 3—35 '1' — [ P 1.5702 .0176 1.0176 No. 4—26 'J' — [ P 1.5698 .0237 1.0237 No. 5 — 21 T — I P 1.5694 .0294 1.0294 No. 6—17 T — [ P 1.5686 .0362 1.0362 No. 7—14 T — I P 1-5675 .0440 1.0440 No. 8— 12 T — I P 1-5663 .0514 1.05 14 II T — I P 1-5654 ■0559 1-0559 loT — I P 1-5643 .0616 I.0616 9T- I P 1.5628 .0684 1.0684 8T — I P 1.5607 .0769 1.0769 EPICYCLOIDAL. SPECIAL. Cutter. T H Corrected S for Gear. A— 12T— I P 1-5663 .0514 1.0514 B - 13 T _ I P 1.5670 -0474 1.0474 C — 14 T — I P 1-5675 .0440 1 .0440 D— 15 T — 1 P 1-5679 .0411 1. 041 1 E_ 16T — ] P 1-5683 -0385 1-0385 F — 17 T — 1 P 1.5686 .0362 1.0362 G — 18 T — ] P 1.5688 .0342 1.0342 H— 19T— ] P 1.5690 .0324 1.0324 I — 20 T — ] P 1.5692 .0308 1.0308 J — 21 T — ] P 1-5694 .0294 1.0294 K — 23 T — [ P 1.5696 .0268 1.0268 L — 25 T — t P 1.5698 .0247 1.0247 M— 27 T — [ P 1-5699 .0228 1.0228 N — 30 T — [ P 1.5701 .0208 1 .0208 - 34 T - [ P 1-5703 .0181 1.0181 P - 38 T — [ P 1-5703 .0162 I.0162 Q - 43 T - [ P 1-5705 -0143 1.0143 R _ 50 T — [ P 1-5705 .0123 1.0123 S _ 60 T — [ P 1.5706 .0102 1. 01 02 T-75T- [ P 1-5707 .0083 1 .0083 U —100 T — [ P 1-5707 .0060 1.0060 V —150 T — I P 1-5707 .0045 1 .0045 W— 250 T — I P 1.5708 .0025 1.0025 No. Teeth. T H Corrected S for Gear. 9T— I P 10 T — I P 11 T— I P 1.5628 1-5643 1-5654 .0684 .0616 -0559 1.0684 1. 061 6 I-OS59 144 BROWN & SHARPE MFG. CO. DIAMETRAL PITCH. "NUTTALL." Diametral Pitch is tlie Number of Teetli to Each Incli of the Pitch Diameter. To Get The Diametral Pitch. The Diametral Pitch. The Diametral Pitoli. Pitch Diameter Pitch Diameter Pitch Diameter. Pitch Diameter. Outside Diameter. Outside Diameter. Outside Diameter. Outside Diameter Number of Teeth. Number of Teeth. Thickness of Tooth. Addendum. Root. Working Depth. Whole Depth. Clearance. Clearance. Havina The Circular Pitch. The Pitch Diameter and the Nunil)er of Teeth The Outside Diame- ter and the Numl)ei of Teeth .... The Number of Teetli and the Diametral Pitch .... The Number of Teeth and Outside Diam eter The Outside Dinme ter and the Diam etial Pitch . . . Addendum and the Number of Teeth. The Number of Teeth and the Diametral Pitch The Pitch Diameter and the Diametral Pitch The Pitch Diameter and the Number of Teeth .... The Number of Teeth and Addendum . The Pitch Diameter and the Diametral Pitch The Outside Diame- ter and the Diame- tral Pitch . . . The Diametral Pitch. Tire Diametral Pitch. The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. Thickness of Tooth. Rule. Divide 3.1416 by the Circular Pitch Divide Number of Teeth by Pitch Diameter Divide Number of Teeth plrrs 2 by Outside Diameter . . . . . Divide Number of Teeth by the Diametral Pitch ....". Divide the" product of Outside Diameter and Number of Teeth by Number of Teeth plus 2 Subtract from the Outside Diame- ter the quotient of 2 divided by the Diametral Pitch .... Multiply Addendum by the Num- ber of Teeth Divide Number of Teeth plus 2 by the Diametral Pitch .... Add to the Pitch Diameter the quotient of 2 divided by the Diametral Pitch Divide the Number of Teeth plus 2 by the quotient of Number of Teeth and by the Pitch Diameter Multiply the Number of Teeth plus 2 by Addendum .... Multiply Pitch Diameter by the Diametral Pitch Multiply Outside Diameter by the Diametral Pitch and subtract 2. Divide 1.570S by the Diametral Pitch . Divide 1 by the Diametral Pitch, D' °'"^="-N Divide 1.157 bythe Diametral Pitch Divide 2 Ijy the Diametral Pitch. Divide 2.157 bythe Diametral Pitch Divide .157 bythe Diametral Pitch Divide Thickness of Tooth at pitch line by 10 Formula. 3.1416 P' N ' D' " D \ D'= n;_ p DN N+2 D'=D_- D'=sN y+2 D = - D = D'+^ D = N+2 = N D~ D = = (N+2) s N = = DP N = ='DP — 2 t = 1.570S P s = 1 P s + „ 1.157 ^- P D"= 2 "^ P" D"- f '-l'' .1.57 PROVIDENCE, R. I. 145 CIRCULAR PITCH. "NUTTALL." Circular Pitch is tlic Distance from the Centre of One Tooth to the Centre of the Next Tootli, Measured alonj:? tlie Pitch Line. To Get The Circular I'itch, Tlie Circular Pitcli, The Circular Pitch, Pitch Diameter, Pitch Diameter, Pitch Diameter, Pitch Diameter Outside Diameter. Outside Diameter. Outside Diameter. Number of . Teeth. Thicliness of Tooth, Addendum. Root. Working Depth, Whole Depth. Clearance. Clearance. Havini! The Diametral Pitch. The Pitch Diameter and the Numl)er of Teeth The Outside Diame- ter and the Number of Teeth .... The Number of Teeth and the Circular Pitch The Number of Teeth and the Outside Di- ameter .... The Outside Diame- ter and the Circular Pitch Addendum and the Number of Teeth. The Numljer of Teeth and the Circuhir Pitch The Pitch Diameter and the Circular Pitch The Number of Teeth and the Addendum The Pitch Diameter and the Circular Pitch . . „ . . The Circular Pitch. The Circular Pitch. The Circular Pitch. Tlie Circular Pitch. The Circular Pitch. The Circular Pitch. Thickness of Tooth. Rule. Divide 3.1416 by the Diametral Pitch Divide Pitch ' Diameter liy the product of .3183 and Number of Teeth Divide Outside Diameter by the product of .3183 and Number of Teeth plus 2 The continued product of the Number of Teeth, the Circular Pitch and .3183 Divide the pi'oduct of Number of Teeth and Outside Diameter by Number of Teeth plus 2 . . . Subtract from the Outside Diame- ter the product of the Circular Pitch and .6366 Multiply the Number of Teeth by the Addendum The continued product of the Number of Teeth ])lus 2, the Circular Pitch and .3183 . . . Add to the Pitch Diameter the product of the Circular Pitch and .6366 Multiply Addendum by Number of Teeth plus 2 Divide the product of Pitch Diam- eter and 3.1416 by the Circular Pitch One-half the Circular Pitch . . Multiply the Circular Pitch by .3183, or 8 = -^' Multiply the Circular Pitch by .3683 Multiply the Circular Pitch bv .6366 ' Multiply the Circular Pitch by .6866 Multiply the Circular Pitch by .05 One-tenth the Thickness of Tooth at Pitch Line Formula. 3.1416 P D' .3183 N D ^ .3183 N-f 2 D'=NP'.3183 D'= ND N+2 D=:D— (P'.6366) D'= N 8 D:=(N+2)P'.31SS D=D'-(-(P'.6366) s = P' .3183 s + f = P' .3683 D"= P' .6366 D"= P' .6866 f =P.05 t f = 10 146 BROWN & SHARPE MFG. CO. GEAR WHEELS. TABLE OF TOOTH PAKTS- — CIKCULAK PITCH IN FIKST COLUMN. ■6^ Threads or Teeth per inch Linear . Thickness of Tooth on Pitch Line. Working Depth of Tooth. Depth of Space below Pitch Line. ^ 8 Width of Thread-Tool at End. Width of Thread at Top. P' p' p t . S D" «+/ D"+/ P'X.31 P'X.335 2 1 a 1.5708 1.0000 .6366 1.27S2 .7366 1.3732 .6200 .6700 If 8 15 1.6755 .9375 .5968 1.1937 .6906 1.2874 .5813 .6281 11 i 7 1.7952 .8750 .5570 1.1141 .6445 1.2016 .5425 .5863 li 8 13 1.9333 .8125 .5173 1.0345 .5985 L1158 .5038 .5444 li 2 3 2.0944 .7500 .4775 .9549 .5525 1.0299 .4650 .5025 ih 16 23 2.1855 .7187 .4576 .9151 .5294 .9870 .4456 .4816 1-1- b 11 2.2848 .6875 .4377 .8754 .5064 .9441 .4262 .4606 li 3 i 2.3562 .6666 .4244 .8488 .4910 .9154 .4133 .4466 1^ 16 21 2.3936 .6562 .4178 .8356 .4834 .9012 .4069 .4397 li i 5 2.5133 .6250 .3979 .7958 .4604 .8588 .3875 .4188 1^ 16 ir 2.6456 .5937 .3780 V.7560 .4374 .8156 .3681 .3978 if 8 9 2.7925 .5625 .3581 .7162 .4143 .7724 .3488 .3769 1^ 16 17 2.9568 .5312 .3382 .6764 .3913 .7295 .3294 .3559 1 1 3.1416 .5000 .3183 .6366 .3683 .6866 .3100 .3350 15 16" 11 3.3510 .4687 .2984 .5968 .3453 .6437 .2906 .3141 7 8 li 3.5904 .4375 .2785 .5570 .3223 .6007 .2713 .2931 13 16" 1^ 3.8666 .4062 .2586 .5173 .2993 .5579 .2519 .2722 1. 5 If 3.9270 .4000 .2546 .5092 .2946 .5492 .2480 .2680 3 4 If 4.1888 .3750 .2387 .4775 .2762 .5150 .2325 .2513 11 16 li 4.5696 .3437 .2189 .4377 .2532 .4720 .2131 .2303 2 3 If 4.7124 .3333 .2122 .4244 .2455 .4577 .2066 .2233 5 8 If 5.0265 .3125 .1989 .3979 .2301 .4291 .1938 .2094 3 5 11 5.2360 .3000 .1910 .3820 .2210 .4120 .1860 .2010 .7 If 5.4978 .2857 .1819 .3638 .2105 .3923 .1771 .1914 ^ If 5.5851 .2812 .1790 .3581 .2071 .3862 .1744 .1884 To obtain the size table, multiply the required. of any part of a circular pitch not given in the corresponding part of 1" pitch by the pitch PROVIDENCE, R. I, TABLE OF TOOTH TAUTB.— Contimteil 147 CIRCULAR riTCIl IN FIRST COLUMN. Threads or Teeth per inch Linear. "3 Thickness of Tooth on Pitch Line. §1 n - ^ bO o .an o Depth of Space below Pitch Line. ft o CD P, Width of Thread-Tool at End. o H R P' 1" p t s yi D" s-t-/ D'^f. Pk.3i PX.335 1 2 2 6.2832 .2500 .1592 .3183 .1842 .3433 .1550 .1675 i » 21 7.0685 .2222 .1415 .2830 .1637 .3052 .1378 .1489 7 IG 2f 7.1808 .2187 .1393 .2786 .1611 .3003 .1356 .1466 8 7 ^8 7.3304 .2143 .1364 .2728 .1578 .2942 .1328 .1436 2 5 ^2 7.8540 .2000 .1273 .2546 .1473 .2746 .1240 .1340 8 8 2f 8.3776 .1875 .1194 .2387 .1381 .2575 .1163 .1256 i 11 2f 8.6394 .1818 .1158 .2316 .1340 .2498 .1127 .1218 1 3 3> 9.4248 .1666 .1061 .2122 .1228 .2289 .1033 .1117 6 IG Si- 10.0531 .1562 .0995 .1989 .1151 .2146 .0969 .1047 3 10 Si 10.4719 .1500 .0955 .1910 .1105 .2060 .0930 .1005 2_ 3i 10.9956 .1429 .0909 .1819 .1052 .1962 .0886 .0957 1 i 4 12.5664 .1250 .0796 .1591 .0921 .1716 .0775 .0838 9' 4i 14.1372 .1111 .0707 .1415 .0818 .1526 .0689 .0744 1 5 5 15.7080 .1000 .0637 .1273 .0737 .1373 .0620 .0670 8 16 5f 16.7652 .0937 .0597 .1194 .0690 .1287 .0581 .0628 11 51- 17.2788 .0909 .0579 .1158 .0670 .1249 .0564 .0609 1 6 6 18.8496 .0833 .0531 .1061 .0614 .1144 .0517 .0558 2 13 6i 20.4203 .0769 .0489 .0978 .0566 .1055 .0477 .0515 1 7 7 21.9911 .0714 .0455 .0910 .0526 .0981 .0443 .0479 2 15 7i 23.5619 .0666 .0425 .0850 .0492 .0917 .0414 .0446 1 8 8 25.1327 .0625 .0398 .0796 .0460 .0858 .0388 .0419 1 9 9 28.2743 .0555 .0354 .0707 .0409 .0763 .0344 .0372 1 10 10 31.4159 .0500 .0318 .0637 .0368 .0687 .0310 .0335 1 16 16 50.2655 .0312 .0199 .0398 .0230 .0429 .0194 .0209 1 20 20 62.8318 .0250 .0159 .0318 .0184 .0343 .0155 .0167 To obtain the table, multiply required. size of any part of the corresponding a circular pitch not given in the part of 1" pitch by the pitch 148 BROWN & SHARPE MFG. CO. GEAE WHEELS. TABLE OF TOOTH PARTS DIAMETRAL PITCH IN FIRST COLUMN. Diametral Pitch. Thickness of Tooth on Pitch Line. 3= 1 re a < ft o Depth of Space below Pitch Line. Whole Depth of Tooth. P P' t s D" s+f. D"+/. i 6.2832 3.1416 2.0000 4.0000 2.3142 4.3142 I 4.1888 2.0944 1.3333 2.6666 1.5428 2.8761 1 3.1416 1 . 5708 1.0000 2.0000 1.1571 2.1571 li 2.5133 1.2566 .8000 1.6000 .9257 1.7257 n 2.0944 1.0472 .6666 1.3333 .7714 1.4381 If 1.7952 .8976 .5714 1 1429 .6612 1.2326 2 1.5708 .7854 .5000 1.0000 .5785 1.0785 2i 1.3963 .6981 .4444 .8888 .5143 .9587 2i 1.2566 .6283 .4000 .8000 .4628 .8628 2f - 1.1424 .5712 .3636 .7273 .4208 .7844 3 1.0472 .5236 .3333 .6666 .3857 .7190 3^ .8976 .4488 .2857 .5714 .3306 .6163 4 .7854 .3927 .2500 .5000 .2893 .5393 5 .6283 .3142 .2000 .4000 .2314 .4314 6 .5236 .2618 .1666 .3333 .1928 .3595 7 .4488 .2244 .1429 .2857 .1653 .3081 8 .3927 .1963 .1250 .2500 .1446 .2696 9 .3491 .1745 .1111 .2222 .1286 .2397 10 .3142 .1571 .1000 .2000 .1157 .2157 11 .2856 .1428 .0909 .1818 .1052 .1961 12 .2618 .1309 0833 .1666 .0964 .1798 13 .2417 .1208 .0769 .1538 .0890 .1659 14 .2244 .1122 .0714 .1429 .0826 .1541 To obtain the size of any part of a diametral pitch not given in the table, divide the corresponding part of 1 diametral pitch by the pitch required. PROVIDENCE, n. I, 149 TABLE OF TOOTH TARTS— Contmuecl DIAMETRAL PITCH IN FIRST COLUMN. u ■ II o3CL| 5 1.1 O Thickness of Tooth on Pitch Line. < fcCo o ° Depth of Space below Pitch Line. Is P. P'. t. s. D". s+f. .0771 D' + /. 15 .2094 .1047 .0666 .1333 .1438 16 .1963 .0982 .0625 .1250 .0723 .1348 17 .1848 .0924 .0588 .1176 .0681 . 1269 18 .1745 .0873 .0555 .1111 .0643 .1198 19 .1653 .0827 .0526 .1053 .0609 .1135 20 .1571 .0785 .0500 .1000 .0579 .1079 22 .1428 .0714 .0455 .0909 .0526 .0980 24 .1309 .0654 .0417 .0830 .0482 .0898 26 .1208 .0604 .0385 .0769 .0445 .0829 28 .1122 .0561 .0357 .0714 .0413 .0770 30 .1047 .0524 .0333 .0666 .0386 .0719 32 .0982 .0491 .0312 .0625 .0362 .0674 34 .0924 .0462 .0294 .0588 .0340 .0634 36 .0873 .0436 .0278 .0555 .0321 .0599 38 .0827 .0413 .0263 .0526 .0304 .0568 40 .0785 .0393 .0250 . 0500 .0289 .0539 42 .0748 .0374 .0238 .0476 .0275 .0514 44 .0714 .0357 .0227 .0455 .0263 .0490 46 .0683 .0341 .0217 .0435 .0252 .0469 48 .0654 .0327 .0208 .0417 .0241 .0449 50 .0628 .0314 .0200 .0400 .0231 .0431 56 .0561 .0280 .0178 .0357 .0207 .0385 60 .0524 .0262 .0166 .0333 .03 93 . 0360 To obtain the size of any part of a diametral pitch not given in the table, divide the corresponding part of 1 diametral pitch by the pitch required. Natural Sines and Cosines, International Correspondence Schools. NATURAL SINES AND COSINES 151 / 0° 1° 2° 3° 4° f Sine ( Cosine Sine C Cosine Sine C :osine Sine ( "osine Sine C Cosine .00000 .01745 9998s .03490 99939 •05234 99863 .06976 99756 60 I .00029 .01774 99984 .03519 99938 •05263 99861 .07005 99754 59 a .00058 .01803 99984 .03548 99937 •05292 99860 .07034 99752 58 3 .00087 .01832 99983 .03577 .99936 •05321 99858 .07063 99750 57 4 .00116 .01862 99983 .03606 99935 •05350 99857 .07092 99748 S6 5 .00145 .01891 99982 .03635 99934 •05379 99855 .07121 99746 55 6 .00175 .01920 99982 .03664 99933 .05408 99854 .07150 99744 54 7 .00204 .01949 99981 •03693 99932 •05437 99852 .07179 99742 S3 8 .00233 .01978 99980 .03723 99931 •05466 99851 .07208 99740 52 9 .00262 .02007 99980 .03752 99930 •05495 99849 .07237 99738 51 10 .00291 .02036 99979 .03781 99929 •05524 99847 .07266 99736 50 11 .00320 99999 .02065 99979 .03810 99927 •05553 99846 .07295 99734 49 12 .00349 99999 .02094 99978 .03839 99926 •05582 99844 .07324 99731 48 13 .00378 99999 .02123 99977 .03868 99925 .05611 99842 .07353 99729 47 14 .00407 99999 .02152 99977 .03897 99924 .05640 99841 .07382 99727 46 IS .00436 99999 .02181 99976 .03926 99923 .05669 99839 .07411 9972s 45 i6 .00465 99999 .02211 99976 .03955 99922 .05698 99838 .07440 99723 44 17 .00495 99999 .02240 99975 .03984 99921 •05727 99836 .07469 99721 43 i8 .00524 99999 .02269 99974 .04013 99919 •05756 99834 .07498 99719 42 19 .00553 99998 .02298 99974 .04042 99918 •05785 99833 .07527 99716 41 20 .00582 99998 .02327 99973 .04071 99917 .05814 99831 .07556 99714 40 21 .00611 99998 .02356 99972 .04100 99916 .05844 99829 .07585 99712 39 22 .00640 99998 .02385 99972 .04129 999 IS .05873 99827 .07614 99710 38 23 .00669 Cr998 .02414 99971 .04159 99913 .05902 99826 .07643 99708 37 24 .00698 99998 .02443 99970 .04188 99912 •05931 99824 .07672 99705 36 25 .00727 99997 .02472 99969 .04217 9991 1 •05960 99822 .07701 99703 35 26 .00756 99997 .02501 99969 .04246 99910 •05989 99821 .07730 99701 34 27 .00785 99997 .02530 99968 .04275 9S909 .06018 99819 ■07759 99699 33 28 .00814 99997 .02560 99967 .04304 99907 .06047 99817 .07788 99696 32 29 .00844 99996 .02589 99966 .04333 99906 .06076 9981s .07817 99694 31 30 .00873 99996 .02618 99966 .04362 99905 .06105 99813 .07846 99692 30 31 .00902 99996 .02647 9996s .04391 99904 .06134 99812 •07875 99689 29 32 ■00931 99996 .02676 99964 .04420 99902 .06163 99810 •07904 99687 28 33 .00960 99995 .02705 99963 .04449 99901 .06192 99808 •07933 99685 27 34 .00989 99995 .02734 99963 .04478 99900 .06221 99806 .07962 99683 26 35 .01018 99995 .02763 99962 .04507 99898 .06250 99804 .07991 99680 25 36 .01047 99995 .02792 99961 .04536 99897 .06279 99803 .08020 99678 24 37 .01076 99994 .02821 99960 .04565 99896 .06308 99801 .08049 99676 23 38 .Olios 99994 .02850 999S9 .04594 99894 •06337 99799 .08078 99673 22 39 .01134 99994 .02879 99959 .04623 99893 •06366 99707 .08107 99671 21 40 .01164 99993 .02908 99958 .04653 99892 .0639s 99795 .08136 99668 20 41 .01193 99993 .02938 99957 .04682 99890 .06424 99793 .08165 99666 19 42 .01222 99993 .02967 99956 .04711 99889 .06453 99792 .08194 99664 18 43 .01251 99992 .02996 9995S .04740 99888 .06482 99790 .08223 99661 17 44 .01280 99992 .03025 99954 .04769 99886 .06511 99788 .08252 99659 16 4S .01309 99991 .03054 99953 .04798 99885 .06540 997^6 .08281 99657 IS 46 .01338 99991 .03083 99952 .04827 99883 .06569 99784 .08310 99654 14 47 .01367 99991 .03112 99952 .04856 99882 .06598 99782 .08339 99652 13 48 .01396 99990 .03141 9995 1 .04885 99881 .06627 99780 .08368 99649 12 49 .01425 99990 .03170 99950 .04914 99879 .06656 99778 •08397 99647 11 SO .01454 99989 .03199 99949 .04943 99878 .06685 99776 .08426 99644 10 SI .01483 99989 .03228 99948 .04972 99876 .06714 99774 .08455 99642 9 52 •OIS13 99989 •03257 99947 .05001 99875 .06743 99772 .08484 99639 8 53 .01542 99988 .03286 99946 .05030 99873 .06773 99770 .08513 99637 7 54 .01571 99988 .03316 99945 .05059 99872 .06802 99768 .08542 99635 6 55 .01600 99987 .03345 99944 .05088 99870 .06831 99766 .08571 99632 S 56 .01629 99987 .03374 99943 .05117 99869 .06860 99764 .08600 99630 4 £7 .01658 99986 •03403 99942 .05146 99867 .06889 99762 .08629 99627 3 S8 .01687 99986 .03432 99941 •05175 99866 .06918 99760 .08658 99625 2 S9 .01716 9998s .03461 99940 •05205 99864 .06947 997.';8 .08687 99622 I 60 .OI74S 99985 .03490 99939 •05234 99863 .06976 99756 .08716 99619 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 89° ^ 88° 87° 86° 85° 152 NATURAL SINES AND COSINES / 5 6 7° 8° 9° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine C -osine .08716 •99619 .10453 .99452 .12187 •99255 .13917 .99027 ■15643 98769 60 I .08745 .99617 .10482 .99449 .12216 .99251 .13946 ■99023 ■ 15672 98764 59 2 .08774 .99614 .10511 .99446 .12245 .99248 .13975 ■99019 ■15701 98760 S8 3 .08803 .99612 .10540 .99443 .12274 •99244 .14004 ■99015 ■15730 9875s 57 4 .08831 .99609 .10569 .99440 .12302 .99240 .14033 .99011 ■15758 98751 56 S .08860 .99607 .10597 ■99437 •12331 •99237 .14061 .99006 .15787 98746 55 6 .o888g .99604 .10626 .99434 .12360 •99233 .14090 .99002 .15816 98741 54 7 .08918 .99602 .10655 .99431 .12389 .99230 .14119 .98998 .15845 98737 S3 8 .08947 .99599 .10684 .99428 .12418 .99226 .14148 ■9^994 ■15873 98732 52 9 .08976 .99596 .10713 .99424 .12447 .99222 .14177 .98990 ■15902 98728 SI 10 .09005 .99594 .10742 •99421 .12476 .99219 .14205 .98986 .15931 98723 SO II .09034 .99591 .10771 .99418 .12504 .99215 .14234 .98982 .15959 98718 49 12 .09063 .99588 .loSoo •99415 .12533 .99211 .14263 .93978 ■15988 98714 48 13 .09092 .99586 .10829 •99412 .12562 .99208 .14292 .98973 .16017 98709 47 14 .09121 .99583 .10858 •99409 .12591 .99204 .14320 .98969 .16046 98704 46 IS .09150 .99580 .10887 .99406 .12620 .99200 .14349 .98965 .16074 98700 45 i6 .09179 .99578 .10916 .99402 .12649 .99197 .14378 .98961 .16103 98695 44 17 .09208 .99575 .10945 •99399 .12678 •99193 .14407 ■98957 .16132 98690 43 i8 .09237 •99572 .10973 •99396 .12706 .99189 .14436 ■98953 .16160 98686 42 19 .09266 .99570 .11002 .99393 ■12735 .99186 .14464 .98948 .16189 98681 41 20 .09295 ■99567 .11031 .99390 .12764 .99182 ■ 14493 .98944 .16218 98676 40 21 .09324 .99564 .11060 .99386 •12793 •99178 .14522 .98940 .16246 98671 39 22 .09353 .99562 .11089 ■99383 .12822 •99175 .14551 .98936 .16275 98667 38 23 .09382 .99559 .11118 .99380 .12851 .99171 .14580 •98931 .16304 98662 37 24 .09411 .99556 .11147 ■99377 .12880 .99167 .14608 •98927 .16333 98657 36 2S .09440 •99553 .11176 •99374 .12908 .99163 .14637 .98923 .16361 98652 35 26 .09469 •99551 .11205 •99370 .12937 .99160 .14666 .9S919 .16390 98648 34 27 .09498 •99548 .11234 •99367 .12966 .99156 ■14695 .98914 .16419 98643 33 28 .09527 •99545 .11263 •99364 .12995 .99152 ■14723 .98910 .16447 98638 32 29 .09556 •99542 .11291 .99360 .13024 .99148 ■14752 .98906 .16476 98633 31 30 .0958s .99540 .11320 .99357 ■13053 .99144 .14781 .98902 .16505 98629 30 31 .09614 .99537 .11349 .99354 .13081 •99141 .14810 .9S897 .16533 98624 29 32 .09642 •99534 .11378 .99351 .13110 •99137 .14838 .98893 .16562 98619 28 33 .09671 •99531 .11407 .99347 .13139 •99133 .14867 .98889 .16591 98614 27 34 .09700 .99528 .11436 .99344 .13168 .99129 .14896 .98884 .16620 98609 26 35 .09729 .99526 .11465 •99341 .13197 .99125 ■14925 .98880 .16648 98604 25 36 .09758 .99523 .11494 .99337 .13226 .99122 ■14954 .98876 .16677 98600 24 37 .09787 .99520 •11523 •99334 .13254 .99118 .14982 .98871 .16706 9859s 23 38 .09816 .99517 .11552 .99331 .13283 .99114 .15011 .98S67 .16734 98590 22 39 .09845 .99514 .11580 .99327 .13312 .99110 .15040 .98863 .16763 9858s 21 40 .09874 .99511 .11609 .99324 .13341 .99106 .15069 .98858 .16792 98580 20 41 .09903 .99508 .11638 .99320 .13370 .99102 .15097 .98854 .16820 98575 19 42 .09932 .99506 .11667 ■99317 .13399 .99098 .15126 .98849 .16849 98570 18 43 .09961 .99503 .11696 .99314 .13427 .99094 .15155 .98845 .16878 98565 17 44 .09990 .99500 •11725 .99310 .13456 .99091 .15184 .98841 .16906 98561 16 4S .10019 .99497 •11754 .99307 .13485 .99087 .15212 .98836 .16935 98556 15 46 .10048 .99494 .11783 •99303 .13514 .99083 .15241 .98832 .16964 98551 14 47 .10077 .99491 .11812 •99300 .13543 .99079 .15270 .98827 .16992 98546 13 48 .10106 .99488 .11840 •99297 .13572 .99075 ■15299 .98823 .17021 98541 12 49 .10135 .99485 .11869 .99293 .13600 .99071 ■15327 .98818 -17050 98536 11 50 .10164 .99482 .11898 .99290 .13629 .99067 .15356 .98814 .17078 98531 10 SI .10192 .99479 .11927 .99286 .13658 .99063 .15385 .98809 .17107 98526 9 S2 .10221 •99476 .11956 .99283 .13687 .99059 .15414 .98805 .17136 98521 8 53 .10250 •99473 .11985 .99279 .13716 ■9905s .15442 .98800 .17164 98516 7 54 .TO279 •99470 .12014 •99276 .13744 .99051 ■15471 .98796 .17193 98511 6 55 .10308 •99467 .12043 .99272 .13773 .99047 ■15500 .98791 .17222 98506 5 56 .10337 .99464 .12071 .99269 .13802 .99043 .15529 ■98787 .17250 98501 4 57 .10366 .99461 .12100 .99265 .13831 .99039 .15557 ■98782 .17279 98496 3 58 .10395 .99458 .12129 .99262 .13860 .99035 .15586 ■98778 .17308 98491 2 59 .10424 •99455 .12158 .99258 .13S89 .99031 .15615 .98773 .17336 98486 I 60 .10453 .99452 .12187 .99255 .13917 .99027 .15643 .98769 .17365 08481 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 8 4° 8 ?° 82 81 80^ 3 NATURAL SINES AND COSINES 153 / 10 I I 12° 13° 14° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine C ;osine .1736s .98481 .19081 .98163 .20791 .9781s •22495 .97437 .24192 97030 60 I .17393 .98476 .19109 .98157 .20820 .97809 •22523 .97430 .24220 97023 59 2 .17422 •98471 .19138 .98152 .20848 .97803 •22552 .97424 .24249 9701s S8 3 .17451 .98466 .19167 .98146 .20877 •97797 .22580 •97417 .24277 97008 57 4 .17479 .98461 .19195 .98140 .20905 •97791 .22608 •97411 .24305 97001 56 5 .17508 .98455 •19224 .98135 .20933 •97-84 ■22637 .97404 .243.33 96994 55 6 •17537 .98450 .19252 .98129 .20962 •97778 .22665 •97398 .24362 96987 S4 7 ■1756s .98445 .19281 .98124 .20990 •97772 .22693 •97391 .24390 96980 S3 8 .17594 .98440 .19309 .98118 .21019 •97766 .22722 •97384 .24418 56973 52 9 .17623 •9S435 .19338 .98112 .21047 •97760 .22750 •9737S .24446 96966 SI 10 .17651 .98430 .19366 .98107 .21076 •97754 .22778 .97371 •24474 96959 SO II .17680 .98425 .19395 .98101 .21104 •97748 .22807 •9736s •24503 96952 49 12 .17708 .98420 .19423 .98096 .21132 •97742 .22835 •97358 •24531 96945 48 13 .17737 .98414 .19452 .98090 .21161 •9773S .22863 •97351 •24559 96937 47 14 .17766 .9S409 .19481 .98084 .21189 •97729 .22892 •97345 .24587 96930 46 IS .17794 .98404 ■19509 .98079 .21218 •97723 .22920 •97338 .24615 96923 45 i6 .17823 .98399 •19538 .98073 .21246 •97717 .22948 •97331 .24644 96916 44 17 .17852 .98394 ■19566 .98067 .21275 •97711 .22977 •9732s .24672 96909 43 i8 .17880 .98389 ■19S9S .98061 .21303 .97705 .23005 •97318 .24700 96902 42 19 .17909 •98383 ■19623 .98056 .21331 .97698 .23033 •97311 .24728 96894 41 20 .17937 .98378 .19652 .98050 .21360 .97692 .23062 .97304 .24756 96887 40 21 .17966 .98373 .19680 .98044 .21388 .97686 .23090 .97298 .24784 96880 39 22 .17995 .98363 ■19709 .98039 .21417 .97680 .23118 .97291 .24813 96S73 38 23 .18023 .98362 • 19737 .98033 .21445 •97673 .23146 .97284 .24841 96866 37 24 .18052 .98357 .19766 .98027 .21474 •97667 ■23175 .97278 .24869 96858 36 2S .18081 .98352 ■19794 .98021 .21502 .97661 ■23203 .97271 .24897 96851 35 26 .18109 ■98347 .19823 .98016 .21530 •97655 ■23231 .97264 .24925 96844 34 27 .18138 .98341 .19851 .98010 .21559 .97648 .23260 .97257 .24954 96837 33 28 .18166 .98336 .19880 .98004 .21587 .97642 .23288 .97251 .24982 96829 32 29 .18195 .98331 .19908 .97998 .21616 .97636 .23316 .97244 .25010 96822 31 30 .18224 .98325 .19937 .97992 .21644 .97630 .23345 .97237 .25038 96815 30 31 .18252 .98320 .19965 .97987 .21672 •97623 .23373 .97230 .25066 96807 29 32 .18281 .98315 .19994 ■97981 .21701 .97617 .23401 .97223 .25094 96800 28 33 .18309 .98310 .20022 ■97975 .21729 .97611 .23429 .97217 .25122 96793 27 34 .18338 .98304 .20051 ■97969 .21758 .97604 .23458 .97210 .25151 96786 26 35 .18367 .98299 .20079 ■97963 .21786 .97598 .23486 .97203 .25179 96778 25 36 .18395 .98294 .20108 ■97958 .21814 .97592 .23514 .97196 .25207 96771 24 37 .18424 .98288 .20136 ■97952 .21843 .975S5 •23542 .97189 .25235 96764 23 38 .18452 .98283 .20165 ■97946 .21871 .97579 .23571 .97182 .25263 96756 22 39 .18481 .98277 .20193 ■97940 .21899 •97573 .23599 .97176 .25291 96749 21 40 .18509 .98272 .20222 ■97934 .21928 .97566 .23627 .97169 .25320 96742 20 41 .18538 .98267 .20250 .97928 .21956 .97560 .23656 .97162 .25348 96734 19 42 .18567 .98261 .20279 .97922 .21985 .97553 .23684 .97155 .25376 96727 18 43 .18595 .98256 .20307 .97916 .22013 .97547 .23712 .97148 .25404 96719 17 44 .18624 .98250 .20336 .97910 .22041 .97541 .23740 .97141 .25432 96712 16 45 .18652 .98245 .20364 .97905 .22070 .97534 .23769 •97134 .25460 96705 IS 46 .18681 .98240 .20393 .97899 .22098 .97528 .23797 .97127 .25488 96697 14 47 .18710 ■98234 .20421 .97893 .22126 .97521 .23825 »97I20 .25516 96690 13 48 .18738 .98229 .20450 .97887 .22155 .97515 .23853 .97113 .25545 96682 12 49 .18767 .98223 .20478 .97881 ..22183 .97508 .23882 .97106 .25573 96675 II SO .18795 .98218 .20507 •9787s .22212 .97502 .23910 .97100 .25601 96667 10 SI .18824 .98212 •20S3S .97869 .22240 .97496 .23938 .97093 .25629 96660 9 52 .18852 .98207 .20563 .97863 .22268 ,97489 .23966 .97086 .25657 96653 8 S3 .18881 .98201 .20592 .97857 .22297 .97483 .23995 .97079 .25685 9664s 7 54 .18910 .98196 .20620 .97851 .22325 .97476 .24023 .97072 .25713 96638 6 55 .18938 .98190 .20649 .97845 .22353 .97470 .24051 •97065 .25741 96630 S 56 .18967 .98185 .20677 .97839 .22382 .97463 .24079 .97058 .25769 96623 4 57 .18995 .98179 .20706 .97833 .22410 .97457 .24108 .97051 .25798 96615 3 58 .19024 .98174 ..20734 .97827 .22438 .97450 .24136 .97044 .25826 96608 2 59 .19052 .98168 .20763 .97821 .22467 .97444 .24164 .97037 ^5854 96600 I 6o .19081 .98163 .20791 ■97815 .22495 .97437 .24192 .97030 .25882 96593 f Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine 1 79 78 7' 7° 7< ^0 75' > 154 NATURAL SINES AND COSINES' / 15 16° 17° 18° 19° / Sine Cosine Sine Cosine Sine Cosine Sine C 'osine Sine C -osine .25882 .96593 .27564 .96126 .29237 .95630 .30902 95106 .32557 94SS2 60 I .25910 .96585 .27592 .96118 .29265 .95622 .30929 95097 .32584 94542 59 2 .25938 .96578 .27620 .96110 .29293 .95613 .30957 95088 .32612 94533 S8 3 .25966 .96570 .27648 .96102 .29321 .95605 .30985 95079 .32639 94523 57 4 .25994 .96562 .27676 .96094 •29348 .95596 .31012 95070 .32667 94S14 S6 S .26022 .96555 .27704 .96086 .29376 .95588 .31040 95061 .32694 94504 SS 6 .26050 .96547 .27731 .96078 .29404 .95579 .31068 95052 .32722 94495 S4 7 .26079 .96540 ■27759 .96070 .29432 .95571 .31095 95043 ■32749 9448s S3 8 .26107 .96532 .27787 .96062 .29460 .95562 .31123 95033 .32777 94476 52 9 .26135 .96524 .2781S .96054 .29487 .95554 .31151 95024 .32804 94466 SI 10 .26x63 .96517 .27843 .96046 .29515 .95545 .31178 95015 .32832 94457 SO II .26191 .96509 .27871 .96037 .29543 .95536 .31206 95006 .32859 94447 49 12 .26219 .96502 .27899 .96029 .29571 .95528 .31233 94997 .32887 94438 48 13 .26247 .96494 .27927 .96021 .29599 .95519 .31261 94988 .32914 94428 47 14 .26275 .96486 .27955 .96013 .29626 .95511 .31289 94979 .32942 94418 46 IS .26303 .96479 .27983 .96005 .29654 •95502 .31316 94970 .32969 94409 4S i6 .26331 .96471 .28011 •95997 .29682 •95493 .31344 94961 .32997 94399 44 17 .26359 .96463 .28039 .95989 .29710 .95485 •31372 94952 .33024 94390 43 i8 .26387 .96456 .28067 •95981 .29737 .95476 •31399 94943 .33051 94380 42 19 .26415 .96448 .28095 .95972 .29765 .95467 .31427 94933 .33079 94370 41 20 .26443 .96440 .28123 .95964 •29793 .95459 .31454 94924 .33106 94361 40 21 .26471 .96433 .28150 .95956 .29821 ■95450 .31482 94915 .33134 94351 39 22 .26500 .96425 .28178 .95948 .29849 .95441 .31510 94906 .33161 94342 38 23 .26528 .96417 .28206 •95940 .29876 .95433 .31537 94897 .33189 94332 37 24 .26556 .96410 .28234 .95931 .29904 .95424 .31565 94888 .33216 94322 36 2S .26584 .96402 .28262 .95923 .29932 .95415 •31593 94878 .33244 94313 35 26 .26612 .96394 .28290 .95915 .29960 .95.107 .31620 94869 .33271 94303 34 27 .26640 .96386 .28318 .95907 .29987 .95398 .31648 94860 .33298 94293 33 28 .2666S .96379 .28346 •95898 .30015 .95389 •31675 94851 .33326 94284 32 29 .26696 .96371 .28374 •95890 .30043 .95380 •31703 94842 -33353 94274 31 30 .26724 .96363 .28403 .95882 .30071 .95372 •31730 94832 .33381 94264 30 31 .26752 .96355 .28429 .95874 .30098 •95363 .31758 94823 .33408 94254 29 32 .26780 .96347 .28457 .95865 .30126 •95354 .31786 94814 .33436 94245 28 33 .26B08 .96340 .28485 .95857 .30154 •95345 •31813 9480s .33463 94235 27 34 .26836 .96332 .28513 •95849 .30182 .05337 .31841 94795 .33490 94225 26 35 .26864 .96324 ..28541 .95841 .30209 .95328 .31868 94786 •33518 94215 25 36 .26892 .96316 .28569 •95832 .30237 .95319 .31895 94777 •33545 94206 24 37 .26920 .96308 .28597 •95824 .30265 •9S3IO .31923 947C8 ■33573 94196 23 38 .26948 .96301 .28625 •95816 .30^9:; •9S30I .31951 94753 ■33600 94186 22 39 .26976 .96293 .28652 .95807 .30320 •95293 ■31979 94749 ■33627 94176 21 40 .27004 .96285 .28680 .95799 .30348 .95284 .32006 94740 ■3365s 94167 20 41 .27032 .96277 .28708 ■ .95791 .30376 .95275 ■32034 94730 ■33682 941S7 19 42 .27060 .96269 .28736 .95782 .30403 .95266 .32061 94721 •33710 94147 18 43 .27088 .96261 .28764 .,95774 .30431 .95257 .32089 94712 •33737 94137 17 44 .27116 .96253 J28792 .95766 •30459 .95248 .32116 94702 •33764 94127 16 45 .27144 .96246 .,28820 .95757 .30486 .95240 .32144 94693 •33792 94118 IS 46 .27172 .96238 .28847 .95749 .30514 .95231 .32171 94684 •33819 94108 14 47 .27200 .96230 .128S75 .95740 .30542 .95222 .32199 94674 •33846 94098 13 48 .27228 .96222 .28903 .95732 -30570 .95213 .32227 94665 •33874 94088 12 49 .27256 .96214 .28931 .95724 •30597 .95204 .32254 94656 •33901 94078 II SO .27284 .96206 .28959 .95715 .30625 .95195 ,32282 94646 •33929 94068 10 SI .27312 .96198 128987 .95707 .30653 .95186 .32309 94637 .33956 940S8 9 52 .27340 .96190 .29015 .95698 .30680 .95177 ■32337 94627 •33983 94049 8 S3 .27368 .96182 .29042 .-95690 .30708 .95168 .32364 94618 •34011 94039 7 S4 .27396 .96174 .29070 .95681 .30736 .95159 •32392 94609 •34038 94029 6 SS .27424 .96166 .29098 .95673 .30763 .95150 .32419 94599 •3406s 94019 5 S6 .27452 .961^ .29126 .95664 .30791 .95142 •32447 94590 •34093 94009 4 57 .27480 .96150 .29154 .95656 .30819 .95133 •32474 94580 •34120 93999 3 58 .27508 .96142 .29182 .95647 .30846 .95124 •32502 94571 .34147 93989 2 59 .27536 .96134 .29209 .95639 .30874 .95115 •32529 94561 •3417s 93979 I 60 •27564 .96126 .29237 .95630 .30902 .95106 •32557 94552 .34202 93969 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 74 7, 3° 7' 2° 71' ) 70° NATURAL SINES AND COSINES 155 / 20° 2] 22° 23° 24° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine .34202 .93969 •35837 .93358 .37461 .92718 .39073 .92050 •40674 .91355 60 I .34229 .93959 ■35864 •93348 .37488 .92707 .39100 •92039 .40700 .91343 59 2 .34257 .93949 •35891 •93337 .37515 .92697 .39127 .92028 .40727 .91331 58 3 .34284 .93939 •35918 •93327 .37542 .92686 .39153 .92016 .40753 .91319 57 4 .34311 .93929 • 35945 •93316 .37569 .92675 .39180 .92005 .40780 .91307 56 S .34339 •93919 •35973 •93306 •37595 .92664 .39207 .91994 .40806 •91295 55 6 .34366 .93909 .36000 •93295 .37622 .92653 .39234 .91982 .40833 •91283 54 7 .34393 •93899 •36027 •93285 •37649 .92642 .39260 .91971 .40860 •91272 53 8 .34421 .93889 •36054 •93274 •37676 .92631 .39287 .91959 .40886 .91260 52 9 .34448 .93879 .36081 •93264 .37703 .92620 .39314 .91948 •40913 .91248 51 10 .34475 .93869 .36108 .93253 .37730 .92609 •39341 .91936 •40939 .91236 50 II .34503 •93859 .36135 .93243 •37757 .92598 ■39367 .91925 .40966 .91224 49 12 .34530 •93849 .36162 .93232 •37784 .92587 .39394 .91914 .40992 .91212 48 13 .34557 •93839 •36190 •93222 •37811 •92576 ■39421 .91902 .41019 .91200 47 14 .34584 •93829 •36217 •9321 1 •37838 .92565 •39448 .91891 .41045 .91188 46 IS .34612 •93819 .36244 •93201 •37S65 •92554 •39474 .91879 .41072 .91176 45 i6 .34639 •93809 •36271 •93190 .37892 •92543 •39501 .91868 .41098 .91164 44 17 .34666 •93799 .36298 .93180 .37919 .92532 •39528 .91856 .41125 .91152 43 i8 .34694 •93789 ■3632s .93169 .37946 •92521 .39555 .91845 .41151 .91140 42 19 .34721 •93779 .36352 .93159 .37973 •92510 .39581 •91833 .41178 •91128 41 20 .34748 .93769 •36379 .93148 .37999 •92499 .39608 .91822 .41204 .91116 40 21 .34775 •93759 .36406 .93137 .38026 .92488 •3963s .91810 .41231 .91104 39 22 .34803 •93748 •36434 .93127 .38053 .92477 .39661 .91799 .41257 .91092 38 23 .34830 •93738 •36461 .93116 .38080 .92466 .39688 .91787 .41284 .91080 37 24 .34857 •93728 .36488 .93106 .38107 ■92455 .39715 .91775 .41310 .91068 36 25 .34884 •93718 •36515 .93095 .38134 ■92444 .39741 .91764 .41337 .91056 35 26 .34912 ■93708 •36542 •93084 .3S161 .92432 .39768 .91752 .41363 .91044 34 27 .34939 •93698 •36569 •93074 .38188 .92421 ■39795 .91741 .41390 .91032 33 28 .34966 .93688 •36596 •93063 .3S21S .92410 •39822 .91729 .41416 .91020 32 29 .34993 •93677 •36623 •93052 .38241 .92399 •39S48 .91718 .41443 .91008 31 30 .35021 •93667 .36650 .93042 .38268 .92388 .39875 .91706 •41469 .90996 30 31 .35048 •93657 •36677 .93031 .38295 .92377 .39902 .91694 •41496 .909S4 29 32 •35075 •93647 .36704 .93020 .38322 .92366 .39928 .91683 .41522 .90972 28 33 .35102 •93637 ■36731 .93010 .38349 .92355 .39955 .91671 .41549 .90960 27 34 .35130 •93626 .36758 •92999 .38376 .92343 .39982 .91660 .41575 .90948 26 35 .35157 •93616 .36785 .92988 .38403 .92332 .40008 .91648 .41602 .90936 25 36 .35184 •93606 .36812 .92978 .38430 .92321 .40035 .91636 .41628 .90924 24 37 .35211 •93596 .36839 .92967 .38456 .92310 .40062 .91625 .41655 .90911 23 38 .35239 •93585 .36867 .92956 .38483 .92299 .40088 .91613 .41681 .90899 22 39 .35266 •93575 .36894 .92945 .38510 .92287 .40115 .91601 .41707 .90887 21 40 .35293 •9356s .36921 .92935 .38537 .92276 .40141 .91590 .41734 .9087s 20 41 •35320 •93555 .36948 .92924 •38564 .92265 .40168 .91578 .41760 .90863 19 42 .35347 •93544 .36975 .92913 •38591 .92254 .40195 .91566 .41787 .90851 18 43 .35375 •93534 .37002 .92902 •38617 .92243 .40221 .91555 .41813 .90839 17 44 .35402 •93524 .37029 .92S92 •38644 .92231 .40248 .91543 .41840 .90826 16 45 •35429 •93514 .37056 .92881 •38671 .92220 •40275 .91531 .41866 .90814 IS 46 .35456 •93503 •37083 .92870 .3e698 .92209 .40301 .91519 .41892 .90802 14 47 .35484 •93493 •37110 .92859 .38-25 .92198 .40328 .91508 •41919 .90790 13 48 .35511 •93483 •37137 .92849 .38752 .92186 .4035s .91496 •41945 .90778 12 49 .35538 •93472 •37164 .9283S .38778 .92175 .40381 .91484 •41972 .90766 11 50 .35565 •93462 •37191 .92827 .38805 .92164 .40408 .91472 .41998 ■90753 10 51 .35592 ■93452 .37218 .92816 .38832 .92152 .40434 .91461 .42024 •90741 9 52 .35619 •93441 .37245 .92805 .3S859 .92141 .40461 .91449 .42051 .90729 8 53 .35647 •93431 .37272 .92794 .38S86 .92130 .40488 .91437 .42077 .90717 7 54 .35674 •93420 .37299 .92784 .3S912 .92119 .40514 .91425 .42104 .90704 6 55 .35701 •93410 ■37326 .92773 .3S939 .92107 .40541 .91414 .42130 .90692 5 56 •35728 .93400 .37353 .92762 .3S966 .92096 .40567 .91402 .42156 .90680 4 57 •35755 •93389 .37380 .92751 •3S993 .920S5 .40594 •91390 .42183 .90668 3 58 •35782 •93379 .37407 .92740 .39020 .92073 .40621 •91378 .42209 •9065s 2 59 .35810 .93368 •37434 .92729 .39046 .9::o63 .40647 •91366 .42235 .90643 I 60 .35837 •93358 •37461 .92718 .39073 .92050 .40674 •91355 •42262 .90631 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 6( 3° 6 3° 6 7° 6 5° 6 5° 156' NATURAL SINES AND COSINES / 25 D 26 27 D 28 29 1 Sine ( "osine Sine ( I^osine Sine ( l^^osine Sine Cosine Sine Cosine .42262 .90631 .43837 .89879 .45399 .89101 .46947 .88295 .48481 .87462 60 I .42288 .90618 .43863 .89867 .45425 .89087 .46973 .88281 .48506 .87448 59 2 .42315 .90606 .43889 .89854 .45451 .89074 .46999 .88267 .48532 •87434 58 3 .42341 .90594 .43916 .89841 •45477 .89061 .47024 .88254 •48557 .87420 57 4 .42367 .90582 .43942 .89828 •45503 .89048 .47050 .88240 •48583 .87406 56 S .42394 .90569 .43968 .89816 ■45529 .89035 .47076 .88226 .48608 .87391 55 6 .42420 .90557 .43994 .89803 ■45554 .89021 .47101 .88213 .48634 .87377 54 7 .42446 •90545 .44020 .89790 ■45580 .89008 .47127 .88199 •48659 .87363 53 8 .42473 .90532 .44046 .89777 .45606 .88995 •47153 .88185 .48684 .87349 52 9 .42499 .90520 .44072 .89764 .45632 .88981 •47178 .88172 .48710 .87335 SI 10 .42525 .90507 .44098 •89752 .45658 .88968 •47204 .88158 .48735 .87321 50 II .42552 •90495 .44124 •89739 .45684 .8895s .47229 .88144 .48761 .87306 49 12 .42578 •90483 .44151 .89726 .45710 .88942 .47255 .88130 .48786 .87292 48 13 .42604 .90470 .44177 •89713 .45736 .88928 .47281 .88117 .48811 .87278 47 14 .42631 •90458 .44203 .89700 .45762 .88915 .47306 .88103 .48837 .87264 46 IS .42657 .90446 .44229 .89687 .45787 .88902 .47332 .88089 .48862 .87250 45 i6 .42683 .90433 .44255 .89674 .45813 .88888 .47358 .88075 .48888 .87235 44 17 .42709 .90421 .44281 .89662 .45839 .88875 •47383 .88062 .48913 .87221 43 i8 .42736 .90408 .44307 .89649 .45865 .88862 .47409 .88048 .48938 .87207 42 19 .42762 .90396 .44333 .89636 .45891 .88848 .47434 .88034 .48964 •87193 41 20 .42788 .90383 .44359 .89623 •45917 .88835 .47460 .88020 .48989 .87178 40 21 .42815 .90371 .44385 .89610 •45942 .88822 .47486 .88006 .49014 .87164 39 22 .42841 .90358 .44411 .89597 •45968 .88808 .47511 .87993 .49040 .87150 38 23 .42867 ■90346 •44437 .89584 •45994 .88795 •47537 .87979 .49065 .87136 37 24 .42894 ■90334 .44464 •89571 .460-J .88782 •47562 .87965 .49090 .87121 36 25 .42920 .90321 .44490 •89558 .46046 .88768 •47588 ■87951 .49116 .87107 35 26 .42946 .90309 .44516 •89545 .46072 •88755 •47614 ■87937 .49141 .87093 34 27 .42972 .90296 .44542 •89532 .46097 .88741 •47639 .87923 .49166 .87079 33 28 .42999 .90284 •44568 •89519 .46123 .88728 .47665 ■87909 .49192 .87064 32 29 .43025 .90271 •44594 •89506 .46149 .88715 .47690 .87896 .49217 .87050 31 30 .43051 .90259 .44620 •89493 .4617s .88701 .47716 .87882 .49242 .87036 30 31 .43077 .90246 .44646 .89480 .46201 .88688 .47741 .87868 .49268 .87021 29 32 .43104 .90233 • 44672 .89467 .46226 .88674 .47767 .87854 .49293 .87007 28 33 .43130 .90221 .44698 .89454 .46252 .88661 .47793 .87840 .49318 .86993 27 34 .43156 .90208 .44724 .89441 .46278 .88647 .47818 .87826 .49344 .86978 26 35 .43182 .90196 •44750 .89428 .46304 .88634 .47844 .87812 .49369 .86964 25 36 .43209 .90183 .44776 .89415 .46330 .88620 .47869 .87798 .49394 .86949 24 37 .43235 .90171 .44802 .89402 .46355 .88607 .47895 .87784 .49419 .86935 23 38 .43261 .90158 .44828 .89389 •46381 .88593 •47920 .87770 •49445 .86921 22 39 .43287 .90146 .44854 .89376 .46407 .88580 •47946 .87756 .49470 .86906 21 40 .43313 .90133 .44880 .89363 .46433 .88566 •47971 .87743 .49495 .86892 20 41 .43340 .90120 .44906 .89350 .46458 •88553 .47997 .87729 .49521 .86878 19 42 .43366 .90108 .44932 .89337 .46484 •88539 .48022 .87715 .49546 .86863 18 43 .43392 .90095 .44958 .89324 .46510 .88526 .48048 .87701 .49571 .86849 17 44 .43418 .90082 •44984 .89311 •46536 .88512 .48073 .87687 .49596 .86834 16 45 .43445 .90070 •45010 .89298 .46561 .88499 .48099 .87673 .49622 .86820 15 46 .43471 ■90057 •45036 •89285 .46587 .88485 .48124 .87659 .49647 .86805 14 47 .43497 ■90045 •45062 .89272 .46613 .88472 .48150 .87645 .49672 .86791 13 48 .43523 ■90032 .45088 .89259 .46639 .88458 .48175 .87631 .49697 .86777 12 49 .43549 .90019 •45114 .89245 .46664 .88445 .48201 .87617 .49723 .86762 11 SO .43575 .90007 .45140 .89232 .46690 .88431 .48226 .87603 .49748 .86748 10 51 .43602 .89994 .45166 .89219 .46716 .88417 .48252 .87589 .49773 .86733 9 52 .43628 .89981 •45192 .89206 .46742 .88404 .48277 •87575 .49798 .86719 8 53 .43654 .89968 •45218 .89193 .46767 .88390 •48303 •87561 .49824 .86704 7 54 .43680 .89956 •45243 .89180 .46793 .88377 .48328 •87546 .49849 .86690 6 SS .43706 .89943 •45269 .89167 .46819 .88363 ■48354 •87532 .49874 .86675 S S6 •43733 .89930 .45295 •89153 .46844 .88349 ■48379 •87518 .49899 .86661 4 57 .43759 .89918 .45321 .89140 .46870 .88336 ■48405 .87504 .49924 .86646 3 58 •4378s .89905 .45347 .89127 .46896 .88322 ■48430 •87490 .49950 .86632 2 59 .43811 .89892 .45373 .89114 .46921 .88308 ■48456 .87476 .49975 .86617 I 60 .43837 .89879 •45399 .89101 .46947 .88295 .48481 .87462 .50000 .86603 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 6z \° 62 62 6] 6c ° NATURAL vSINES AND COSINES 157 / 30 31 3 32 33 34 3 / Sine Cosine Sine C Cosine Sine Cosine Sine Cosine Sine ( .Cosine .50000 .86603 .51504 85717 .52992 .84805 •S4464 •83867 •55919 82904 60 I .50025 .86588 •51529 85702 •53017 .84789 •54488 •83851 •55943 82887 59 2 .50050 .86573 •51554 85687 •53041 •84774 •S45I3 •83835 •55968 82871 58 3 .50076 ■86559 • 5 1 579 85672 •53066 •84759 •54537 •83819 •5.S992 82855 57 4 .50101 .86544 •51604 85657 •53091 •84743 •54561 .83804 •56016 82839 56 S .50126 .86530 .51628 85642 •S3II5 .84728 .5-!586 .83788 .56040 82822 55 6 .S0151 .86515 •51653 85627 •53140 .84712 .54610 .83772 .56064 82806 54 7 .50176 .86501 .51678 85612 •53164 .84697 .54635 •83756 .56088 82790 53 8 .50201 .86486 •51703 85597 •53189 .84681 .54659 .83740 .56112 82773 52 9 .50227 .86471 •51728 85582 .53214 .84666 .54683 •83724 .56136 82757 51 10 .50252 .86457 •S1753 85567 •53238 .84650 •54708 •83708 .56160 82741 50 II ■50277 .86442 •51778 85551 .53263 .84635 • 54732 •83692 .56184 82724 49 12 .50302 .86427 •51803 85536 •53288 .84619 .54756 •83676 .56208 82708 48 13 ■50327 .86413 .51828 85521 •53312 .84604 .54781 .83660 .56232 82692 47 14 .50352 .86398 .51852 85506 •53337 .84588 •54805 •83645 .56256 82675 46 IS .50377 .86384 .51877 85491 •53361 .84573 •54829 •83629 .56280 82659 45 i6 •S0403 .86369 .51902 85476 •53386 .84557 •54854 •83613 •56305 82643 44 17 .50428 ■86354 .51927 85461 ■5341 1 .84542 •54878 •83597 •56329 82626 43 i8 .50453 .86340 .51953 85446 •53435 .84526 -54902 .83581 •56353 82610 42 19 .50478 .86325 •51977 85431 .53460 .84511 .54927 •83565 •56377 82593 41 20 .50503 .86310 .52002 85416 •53484 •8449s .54951 •83549 •56401 8257f 40 21 .50528 .86295 .52026 85401 •53509 .84480 •54975 •83533 •56425 82561 39 22 .50553 .86281 .52051 85385 •53534 .84464 •54999 •83517 ■56449 82544 38 23 •50578 .86266 .52076 85370 •53558 .84448 •55024 .83501 •56473 82528 37 24 .50603 .86251 .52101 85355 •53583 .84433 •SS048 .83485 • 56497 82511 36 2S .50628 .86237 .52126 85340 •S3607 .84417 •SS072 .83469 •56521 82495 35 26 .50654 .86222 .52151 85325 •53632 .84402 •55097 .83453 •56545 82478 34 27 .50679 .86207 .5217s 85310 •53656 .84386 •55121 .83437 •56569 82462 33 28 .50704 .86192 .52200 85294 •53681 .84370 •S5I45 .83421 •56593 82446 32 29 .50729 .86178 .52225 85279 •53705 .84355 •55169 •83405 ■56617 82429 31 30 .50754 .86163 .52250 85264 •53730 .84339 •55194 ■83389 .56641 82413 30 31 ■50779 .86148 •S2275 85249 •53754 .84324 •55218 .83373 .56665 82396 29 32 .50804 .86133 •52299 85234 •53779 .84308 •55242 .83356 .56689 82380 28 33 .50829 .86119 •52324 85218 •53804 .84292 •55266 .83340 .56713 82363 27 34 .50854 .86104 •52349 85203 .53828 .84277 •55291 .83324 .56736 82347 26 3S .50879 .86089 •52374 85188 •53853 .84261 .55315 .83308 .56760 82330 25 36 .50904 .86074 •52399 85173 .53877 .84245 ■55339 ■83292 .56784 82314 24 37 .50929 .86059 •52423 85157 .53902 •84230 .55363 ■83276 .56808 82297 23 38 .50954 .86045 •52448 85142 .53926 .84214 .553S8 .83260 .56832 82281 22 39 .50979 .86030 •52473 85127 .53951 •84198 .55412 .83244 .56856 82264 21 40 .51004 .86015 •52458 85112 •53975 .84182 .55436 .83228 .56880 82248 20 41 ■.51029 .86000 •52522 85096 .54000 .84167 .55460 .83212 .56904 82231 19 42 .51054 .859S5 •52547 85081 .54024 •84151 .55484 .83195 .56928 82214 18 43 .51079 .85970 •52572 85066 •54049 •84135 .55509 .83179 •56952 82198 17 44 .51104 •85956 •52597 85051 •54073 .84120 .55533 .83163 •56976 82181 16 45 .51129 •85941 •52621 85035 •54097 .84104 •5S5S7 ■83147 •57000 8216s 15 46 .51154 .85926 .52646 85020 •54122 .840S8 •5SS8i ■83131 •57024 82148 14 47 ■51179 •8591 1 .52671 85005 •54146 .84072 •55605 ■8311S •57047 82132 13 48 .51204 .85896 .52696 84989 •54171 .84057 •55630 .83098 •57071 82115 12 49 .51229 .85881 .52720 84974 •54195 .84041 •55654 .83082 ■57095 82098 II 50 ■5I2S4 .85866 .52745 84959 .54220 .84025 •55678 .83066 .57119 S2082 10 SI ■51279 .85851 •52770 84943 .54244 .84009 ,55702 .83050 •57143 82065 9 52 ■51304 .85836 •52794 84928 .54269 .83994 •55726 •83034 ■57167 82048 8 S3 •SI 329 .85821 .52819 84913 .54293 .83978 •55750 .83017 .57191 82032 7 54 •S1354 .85806 .52844 84897 .54317 .83962 •55775 .83001 ■57215 82015 6 55 .51379 .85792 .52869 84882 .54342 .83946 •55799 .82985 .57238 81999 5 S6 ■51404 •85777 .52893 84866 .54366 .83930 •55823 .82969 .57262 81982 4 57 •51429 •85762 .52918 84851 •54391 .83915 ■55847 .82953 .57286 8196s 3 S8 .51454 •85747 .52943 84836 -5441S .83899 •55871 .82936 ■57310 81949 2 59 •S1479 •85732 .52967 84820 ■54440 .83883 •55895 .82920 ■57334 81932 I 60 ■S1504 .85717 .52992 84805 .54464 .83867 .55919 .82904 ■57358 81915 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 59 58° 57 S6 55° 158 NATURAL SINES AND COSINES / 35 36 37° 38° 39° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine .57358 .81915 .58779 .80902 .60182 .79864 .61566 .78801 .62932 .77715 60 I .57381 .81899 .5 802 .80885 .60205 .79846 .61589 .78783 ■62955 .77696 59 2 .57405 .81882 ■ 5 826 .80867 .60228 .79829 .61612 .78765 .62977 .77678 S8 3 .57429 .81865 ■58C49 .80850 .60251 .79811 .61J35 .78747 .63000 .77660 57 4 .57453 .81848 .58873 .80833 .60274 .79793 .61658 .78729 .63022 .77641 56 S .57477 .81832 .58896 .80816 .60298 .79776 .61681 .78711 .63045 .77623 55 6 .57501 .8181S .58920 .80799 .60321 .79758 .61704 .78694 .63068 .77605 54 7 .57524 .81798 .58943 .80782 .60344 .79741 .61726 .78676 .63090 .77586 53 8 .57548 .81782 .58967 .80765 .60367 .79723 .61749 .78658 .63113 .77568 52 9 .57572 .81765 .58990 .80748 .60390 .79706 .61772 .78640 .6313s .77550 51 10 .57596 .81748 .59014 .80730 .60414 .79688 .61795 .78622 .63158 .77531 50 II .57619 .81731 .59037 .80713 .60437 .79671 .61818 .78604 .63180 .77513 49 12 .57643 .81714 .59061 .80696 .60460 .79653 .61841 .78586 .63203 .77494 48 13 .57667 .81698 .59084 .80679 .60483 .79635 .61864 .78568 .63225 .77476 47 14 .57691 .81681 .59108 .80662 .60506 .79618 .61887 .78550 .63248 .77458 46 IS •5771S .81664 ■59131 .80644 .60529 .79600 .61909 .78532 .63271 .77439 45 i6 .57738 .81647 .59154 .80627 .60553 •79583 .61932 .78514 ■63293 .77421 44 17 .57762 .81631 .59178 .80610 .60576 .79565 .61955 .78496 .63316 .77402 43 l8 .57786 .81614 .59201 .80593 .60599 -79547 .61978 .78478 .63338 .77384 42 19 .57810 .81597 .59225 .80576 .60622 .79530 .62001 .78460 .63361 .77366 41 20 ■57833 .81580 ■59248 .80558 .60645 .79512 .62024 .78442 .63383 .77347 40 21 .57857 .81563 .59272 .80541 .60668 .79494 .62046 .78424 .63406 .77329 39 22 .57881 .81546 .59295 .80524 .60691 .79477 .62069 .78405 .63428 .77310 38 23 .57904 .81530 .59318 .80507 .60714 .79459 .62092 .78387 .63451 .77292 37 24 .57928 .81513 .59342 .80489 .60738 .79441 .62115 .78369 .63473 .77273 36 25 ■57952 .81496 .59365 .80472 .60761 .79424 .62138 .78351 .63496 .77255 35 26 .57976 .81479 ■59389 .80455 .60784 .79406 .62160 .78333 .63518 .77236 34 27 .57999 .81462 .59412 .80438 .60807 .79388 .62183 .7831S .63540 .77218 2Z 28 .58023 .81445 .59436 .80420 .60830 .79371 .62206 .78297 .63563 .77199 32 29 .58047 .81428 .59459 .80403 .60853 .79353 .62229 .78279 .6358s .77181 31 30 .58070 .81412 .59482 .80386 .60876 .79335 .62251 .78261 .63608 .77162 30 31 .58094 .81395 .59506 .80368 .60899 .79318 .62274 .78243 .63630 .77144 29 32 .58118 .81378 .59529 .83351 .60922 .79300 .62297 .78225 .63653 .77125 28 33 .58141 .81361 .59S.S2 .80334 .60945 .79282 .62320 .78206 .63675 .77107 27 34 .58165 .81344 .59576 .80316 .60968 .79264 .62342 .78188 .63698 .77088 26 3S .58189 .81327 .59599 .80299 .60991 .79247 .62365 .78170 .63720 .77070 25 36 .58212 .81310 .59622 .80282 .61015 .79229 .62388 .78152 .63742 .77051 24 37 .58236 .81293 .59646 .80264 .61038 .79211 .62411 .78134 .63765 .77033 23 38 .58260 .81276 .59669 .80247 .61061 .79193 .62433 .78116 .63787 .77014 22 39 .58283 .81259 .59693 .80230 .61084 .79176 .62456 .78098 .63810 .76996 21 40 .58307 .81242 .59716 .80212 .61107 .79158 .62479 .78079 .63832 .76977 20 41 ■58330 .81225 .59739 .80195 .61130 .79140 .62502 .78061 .63854 .76959 19 42 •58354 .81208 .59763 .80178 .61153 .79122 .62524 .78043 .63877 .76940 18 43 .58378 .81191 .59786 .80160 .61176 .79105 .62547 .78025 .63899 .76921 17 44 .58401 .81174 .59809 .80143 .61199 .79087 ■62570 .78007 .63922 .76903 16 45 .58425 .81157 .59832 .80125 .61222 .79069 .62592 .77988 .63944 .76884 IS 46 .58449 .81140 .59856 .80108 .61245 .79051 .62615 .77970 .63966 .76866 14 47 .58472 .81123 ■59879 .80091 .61268 .79033 .62638 .77952 .63989 .76847 13 48 .58496 .81106 .59902 .80073 .61291 .79016 .62660 .77934 .64011 .76828 12 49 .5S519 .81089 .59926 .80056 .61314 .78998 .62683 .77916 .64033 .76810 11 50 .38543 .81072 .59949 .80038 .61337 .78980 .62706 .77897 .64056 .76791 10 51 .58567 .810SS .59972 .80021 .61360 .78962 .62728 .77879 .64078 .76772 9 S2 .58590 .81038 .59995 .80003 .61383 .78944 .62751 .77861 .64100 .76754 8 S3 .58614 .81021 .60019 .79986 .61406 .78926 .62774 .77843 .64123 .76735 7 54 .58637 .81004 .60042 .79968 .61429 .78908 .62796 .77824 .64145 .76717 6 55 .36661 .80987 .60065 •799SI .61451 .78891 .62819 .77806 .64167 .76698 5 S6 .58684 .80970 .60089 .79934 .61474 ■78873 .62842 .77788 .64190 .76679 4 57 .58708 ■80953 .60112 .79916 .61497 .78855 .62864 .77769 .64212 .76661 3 S8 .58731 .80936 .60135 .79899 .61520 .78837 .62887 .77751 .64234 .76642 2 59 .58755 .80919 .60158 .79881 .61543 .78819 .62909 .77733 .64256 .76623 I 60 .58779 .80902 .60182 .79864 .61566 .78801 .62932 .77715 .64279 .76604 t Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 54 53 5- 2° 5 [° 5 3° NATURAL SINES AND COSINES 159 / 40° 41 42 43° 44° 1 Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine C osine .64279 .76604 .65606 .75471 .66913 .74314 .68200 •7313s .69466 71934 60 I .64301 .76586 .65628 .75452 .66935 .74295 .68221 .73116 .69487 71914 59 2 .64323 .76567 .65650 .75433 .66956 •74276 .68242 .73096 .69508 71894 58 3 .64346 .76548 .65672 .75414 .66978 .74256 .68264 .73076 .69529 71873 57 4 .64368 •76530 .65694 .75395 .66999 .74237 .68285 .73056 .69549 71853 56 S .64390 .76511 .65716 .75375 .67021 .74217 .68306 .73036 .69570 71833 55 6 .64412 .76492 .65738 .75356 •67043 .74198 .68327 .73016 .69591 71813 54 7 .64435 .76473 ■65759 .75337 .67064 .74178 .68349 .72996 .69612 71792 53 8 .64457 •76455 .65781 .75318 .67086 .74159 .68370 .72976 .69633 71772 52 9 .64479 .76436 .65803 .75299 .67107 .74139 .68391 .72957 ■69654 71752 51 10 .64501 •76417 .65825 .75280 .67129 .74120 .68412 .72937 .6967s 71732 50 II .64524 .76398 .65847 .75261 .67151 .74100 .68434 .72917 .69696 71711 49 12 .64546 .76380 .65869 .75241 .67172 .74080 .68455 .72897 .69717 71691 48 13 .64568 .76361 .65891 .75222 .67194 .74061 .68476 .72877 .69737 71671 47 14 .64590 .76342 .65913 .75203 .6721S .74041 .68497 .72857 .69758 71650 46 IS .64612 •76323 .65935 .75184 .67237 .74022 .68518 .72837 .69779 71630 45 i6 .64635 .76304 .65956 .75165 .67258 .74002 .68539 .72817 .69800 71610 44 17 .64657 .76286 .65978 •75146 .67280 •73983 .68561 .72797 .69821 71590 43 i8 .64679 .76267 .66000 .75126 .67301 •73963 .68582 •72777 .6:842 71569 42 19 .64701 .76248 .66022 •75107 .67323 .73944 .68603 .72757 .69C62 71549 41 20 .64723 .76229 .66044 .75088 .67344 .73924 .68624 .72737 .69883 71529 40 21 .64746 .76210 .66066 .75069 .67366 .73904 .68645 .72717 .69904 71508 39 22 .64768 .76192 .66088 .75050 .67387 .73885 .68666 .72697 .69925 71488 38 23 •64790 .76173 .66109 .75030 .67409 .7386s .68688 ■72677 .69946 71468 37 24 .64812 .76154 .66131 .75011 .67430 .73846 .68709 .72657 .69966 71447 36 2S .64834 .76135 .66153 .74992 .67452 .73826 .68730 .72637 .69987 71427 35 26 .64856 .76116 .66175 .74973 .67473 .73806 .68751 .72617 .70008 71407 34 27 .64878 .76097 .66197 •74953 .67495 .73787 .6G772 .72597 .70029 71386 33 28 .64901 .76078 .66218 .74934 .67516 •73767 .68793 .72577 .70049 71366 32 29 .64^23 •76059 .66240 .74915 .67538 •73747 .683 1 4 .72557 .70070 7134s 31 30 .64945 .76041 .66262 .74896 .67559 .73728 .6GG3S .72:37 .70091 71325 30 31 .64967 .76022 .66284 .74876 .67580 .73708 .68857 .72517 .70112 7 1 30s 29 32 .64989 .76003 .66306 .74857 .67602 .73688 .68878 .72497 .70132 71284 28 33 .65011 .75984 .66327 .74838 .67623 .73669 .68899 .72477 .70153 71264 27 34 .65033 ■75965 .66349 .74818 .67645 .73649 .68920 .72457 .70174 71243 26 35 •65055 .75946 .66371 .74799 .67666 .73629 .68941 .72437 .70195 71223 25 36 .65077 .75927 .66393 .74780 .67688 .73610 .68962 .72417 .70215 71203 24 37 .65100 .75908 .66414 .74760 .67709 .73590 .68983 .72397 .70236 71182 23 38 .65122 .75889 .66436 .74741 .67730 .73570 .69004 .72377 .70257 71162 22 39 .65144 .75870 .66458 .74722 .67752 .73551 .69025 .72357 .70277 71141 21 40 .65166 .75851 .66480 •74703 .67773 .73531 .69046 .72337 .70298 71121 20 41 .65188 .75832 .66501 .74683 .67795 .73511 .69067 .72317 .70319 71 100 19 42 .65210 .75813 .66523 .74664 .67816 .73491 .69088 .72297 •70339 71080 18 43 .65232 .75794 •66545 .74644 .67837 .73472 .69109 .72277 .70360 71059 17 44 .65254 .75775 .66566 .74625 .67859 .73452 .69130 .72257 .70381 71039 16 45 .65276 .75756 .66588 .74606 .67880 .73432 .69151 .72236 .70401 71019 IS 46 .65298 .75738 66610 .74586 .67901 .73413 .69172 .72216 .70422 70998 14 47 .65320 •75719 .66632 •74567 .67923 .73393 .69193 .72196 .70443 70978 13 48 .65342 •75700 .66653 ■74548 .67944 .73373 .69214 .72176 .70463 70957 12 49 •65364 .75680 .66675 .74528 .67965 .73353 .69235 .72156 .70484 70937 II SO .65386 .75661 .66697 .74509 .67987 .73333 .69256 .72136 .70505 70916 10 51 .65408 .75642 .66718 .74489 .68008 .73314 .69277 .72116 .70525 70896 9 52 .65430 .75623 .66740 .74470 .68029 .73294 .69298 .7209s .70546 7087s 8 S3 •65452 .75604 .66762 .7445r .68051 .73274 .69319 .72075 .70567 70855 7 54 .65474 .755S5 .66783 .74431 .68072 .73254 .69340 .72055 .70587 70B34 6 55 .65496 .75566 .66805 .74412 .68093 .73234 .69361 .72035 .70608 70813 S S6 .65518 •75547 .66827 .74392 .68115 .73215 .69382 .72015 .70628 70793 4 S7 .65540 .75528 .66848 .74373 .68136 .73195 .69403 .71995 .70649 70772 3 58 .65562 •75509 .66870 .74353 .68157 .'7317s .69424 .71974 .70670 70752 2 59 .65584 •75490 .66891 .74334 .68179 .73155 .69445 .71954 .70690 70731 I 60 .65606 .75471 .66913 .74314 .68200 .73135 .69466 .71934 .70711 7071 1 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 4 3° 4i i° 4 7° 4< 5° 45^ 3 Natural Tangents and Cotangents. International Correspondence Schools. NATURAL TANGENTS AND COTANGENTS IGl / O I 2 3 4° / Tangr Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang .00000 Infinite .01746 37.2900 .03492 28.6363 .05241 19.0811 .06993 14.3007 60 I .ooo^g 3437.75 .01775 36.3306 .03521 28.3994 .05270 18.9755 .07022 14.2411 S9 2 .00058 1718.87 .01804 53.4415 .03550 28.1664 .05299 18.8711 .07051 14.1821 58 3 .00087 1143.92 .01833 54.3613 .03579 27.9372 .05328 18.7678 .07080 14.123s 57 4 .00116 859.436 .01862 3'3.7o86 .03609 27.7117 .05357 18.6656 .07110 14.0653 56 S .00143 687.549 .01891 52.8821 .03638 27.4899 .05387 18.5643 .07139 14.0079 35 6 .0017s 572.957 .01920 52.0807 .03667 27.271S .05416 18.4643 .07168 13.9507 34 7 .00204 491.106 .01949 31.3032 .03696 27.0566 .05445 18.3655 .07197 13.8940 53 8 .00233 429.718 .01978 50.5485 .03725 26. 84 50 .03474 18.2677 .07227 13.8378 52 9 .00262 381.971 .02007 49.8157 .03754 26.6367 .03503 18.1708 .07256 13.7821 31 10 .00291 343-774 .02036 49.1039 .03783 26.4316 .05333 18.0750 .07285 13.7267 50 II .00320 312.521 .02066 48.4121 .03812 26.2296 .05562 17.9802 .07314 13.6719 49 12 .00349 286.478 .02095 47-7395 .03842 26.0307 .05391 17.8863 .07344 13.6174 48 13 .00378 264.441 .02124 47.0853 .03871- 25.8348 .05620 17.7934 .07373 13.5634 47 14 .00407 245.552 .02153 46.4489 .03900 25.6418 .05649 17.701S .07402 13.5098 46 IS .00436 229.182 .02182 45.8294 .03929 25.4517 .05678 17.6106 .07431 13.4566 43 i6 .00465 214.858 .02211 45-2261 .03958 25.2644 .05708 17.5205 .07461 13-4039 44 17 .00495 202.219 .02240 44-6386 .03987 25.0798 .05737 17.4314 .07490 13-351S 43 i8 .00524 190.984 .02269 44-0661 .04016 24.8978 .05766 17.3432 .07319 13-2096 42 19 .005S3 180.932 .02298 43-5081 .04046 24.7185 .0579s 17.2558 .07548 13-2480 41 20 .00582 171.885 .02328 42.9641 .04073 24.5418 .05824 17.1693 •07578 13.1969 40 21 .00611 163.700 .02337 42.4335 .04104 24.367s- .05854 17.0837 .07607 13.1461 39 23 .00640 136.259 .02386 41.9158 .04133 24.1957 .05883 16.9990 .07636 13.0958 38 23 .00669 149-465 .02415 41.4106 .04162 24.0263 .05912 16.9150 .07665 13.0458 37 24 .00698 143-237 .02444 40.9174 .04191 23-8393 .05941 16.8319 .07695 12.9962 36 25 ,00727 137.507 .02473 40.4338 .04220 23.6943 .05970 16.7496 -07724 12.9469 33 26 .00756 132.219 .02502 39.965s .04230 23-5321 .05999 16.6681 .07753 12.8981 34 2/ .0078s 127.321 .02531 39.5059 .04279 23.3718 .06029 16.5874 .07782 12.8496 33 28 .0081 S 122.774 .02560 39.0568 .04308 23.2137 .06058 16-5075 .07812 12.8014 32 29 .00844 218.340 .02589 38.6177 .04337 23.0577 .06087 16.4283 .07841 12-7536 31 30 .00873 114.589 .02^19 38.1885 .04366 22.9038 .06116 16.3499 -07870 12.7062 30 31 .00902 110.892 .02648 37.7686 .04395 22.7519 .06145 16.2722 .07899 12.6591 29 32 .00931 107.426 .02677 37.3579 .04424 22.6020 .06173 16.1952 .07929 12.6124 28 33 .00960 104.171 .02706 36.9560 .04454 22.4541 .06204 16.1190 .07958 12.5660 27 34 .00989 101.107 .02735 36.5627 .04483 22.3081 .06233 16.0435 .07987 12.3199 26 33 .01018 98.2179 .02764 36.1776 .04512 22.1640 .06262 15.9687 .08017 12.4742 25 36 .01047 93.4895 .02793 35.8006 .04341 22.0217 .06291 15-8945 .08046 12.4288 24 37 .01076 92.9085 .02822 35.4313 .04370 21.8813 .06321 15.8211 .0807s 12.3838 23 33 .01103 go.4633 .02831 33.0693 .04599 i 1.7426 .06350 iS-7483 .08104 12.3390 22 39 .01133 88.1436 .02881 34.7151 .04628 21.6056 .06379 15.6762 .08134 12.2946 21 40 .01164 85.9398 .02910 34-3678 .04638 21.4704 .06408 15.6048 .08163- 12.250S 20 41 .01193 83.8435 .02939 34.0273 .04687 21.3369 •06437 IS. 3340 .08192 12.2067 19 42 .01222 81.8470 .02968 33.6933 .04716 21.2049 .06467 13.4638 .08221 12.1632 18 43 .01231 79.9434 .02997 33.3662 .04743 21.0747 .06496 15.3943 .08251 12.1201 17 44 .01280 78.1263 .03026 33.0452 .04774 20.9460 .06525 15.3254 .08280 12.0772 16 45 .01309 76.3900 .03033 32.7303 .04803 20.8188 •06554 15-2571 .08309 12.0346 15 46 .01338 74.7292 .03084 32.4213 ■04833 20.6932 .06584 15.1893 .08339 11.9923 14 47 .01367 73.1390 .03114 32.1181 .04862 20.5691 .06613 15.1222 .08368 11.9504 13 48 .01396 71.6151 .03143 31.8205 .04891 20.4463 .06642 15.0557 .08397 11.9087 12 49 .014:3 70.1533 .03172 31.3284 .04920 20.3253 .06671 14-9898 .08427 11.8673 11 SO .01453 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 .08456 11.8262 10 SI .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 .08485 11.7853 9 52 .01513 66.1055 .03259 30.6833 .05007 19.9702 .06759 14-7954 .08514 11.7448 8 53 .01542 64.8580 .03288 30.4116 .05037 19.8546 .06788 14-7317 .08544 11.7043 7 54 .01571 63.6567 •03317 30.1446 .05066 19.7403 .06817 14.6685 .08573 11.664s 6 55 .01600 62.4992 .03346 29.8825 .05095 19.6273 .06847 14.6059 .08602 11.6248 5 S6 .01629 61.3829 .03376 29.6245 .05124 19.3136 .06876 14.5438 .08632 11-5833 4 ^ .01658 60.3058 .03405 29.3711 .05153 19.4051 .06905 14.4823 .08661 11.5461 3 58 .01687 59.2659 .03434 29.1220 .05182 19.2959 .06934 14.4212 .08690 11.5072 2 59 .01716 58.2612 .03463 28.8771 .05212 19.1879 .06963 14.3607 .08720 11.4685 I 6o .01746 57.2900 .03492 28.6363 .05241 19.0811 .06993 14.3007 .08749 11.4301 Tang f Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang / 8( 3° 88° 8; 7° 8( 5° 8 -0 162 NATURAL TANGENTS AND COTANGENTS / 5 6 7° 8° 5 ° / Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang .08749 II. 4301 .10310 9.51436 .12278 8.14435 •14054 7.I1S37 .15838 6-3I37S 60 I .08778 11.3919 .10540 9.48781 .12308 8.12481 .14084 7.10038 .15868 6.30189 59 2 .08807 11.3540 .10569 9.46141 .12338 8.10536 .14113 7.08546 .15898 6.29007 S8 3 .08837 11.3163 .10599 9.4351s .12367 8.08600 .14143 7.07059 .15928 6.27829 57 4 .08866 11.2789 .10628 9.40904 .12397 8.06674 •14173 7.05579 .15958 6.26655 56 S .08895 11.2417 .10657 9.38307 .12426 8.04756 .14202 7. 04 1 OS .15988 6.25486 55 6 .0892s 11.2048 .10687 9.35724 .12456 8.02848 .14232 7.02637 .16017 6.24321 54 7 .08954 II. 1681 .10716 9.33155 .12485 8.00948 .14262 7.01174 .16047 6.23160 S3 8 .08983 11.1316 .10746 9.30599 •1251S 7.99058 .14291 6.997x8 .16077 6.22003 52 9 .09013 11.0954 .10775 9.28058 •12544 7.97176 .14321 6-98268 .16107 6.20851 51 10 .09042 11.0594 .10805 9.25530 •12574 7-95302 .14351 6.96823 .16137 6.19703 50 II .09071 11. 0237 .10834 9.23016 .12603 7-93438 .14381 6.9538s .16167 6.i8sS9 49 12 .09101 10.9882 .10863 9.20516 .12633 7-91582 .14410 6.93952 .16196 6.17419 48 13 .09130 10.9529 .10893 9.18028 .12662 7-89734 .14440 6.9252s .16226 6.16283 47 14 .09159 10.9178 .10922 9.1 5554 .12692 7-87895 .14470 6.91 104 .16256 6.1S151 46 15 .09189 10.8829 .10952 9.13093 .12722 7.86064 .14499 6-89688 .16286 6.14023 4S l6 .09218 10.8483 .10981 9.10646 .12751 7.84242 .14529 6.88278 .16316 6.12899 44 17 .09247 10.8139 .IIOII 9.0821 1 .12-81 7.82428 .14559 6.86874 .16346 6.11779 43 i8 .09277 10.7797 .11040 9.05789 .12810 7.80622 .14588 6.85475 .16376 6.10664 42 19 .09306 10.7457 .11070 9-03379 .12840 7.7882s .14618 6-84082 .1640s 6-09552 41 20 •0933s 10.7119 .11099 9.00983 .12869 7-77035 .14648 6.82694 ■1643s 6.08444 40 21 .09365 10.6783 .11128 8.g85g8 .12899 7.7S2S4 .14678 6-81312 •16465 6.07340 39 22 .09394 10.6450 .11158 8.96227 .12929 7.73480 .14707 6.79936 •16495 6.06240 38 23 .09423 10.6118 .11187 8.93867 .12958 7-71715 .14737 6.78564 •16525 6.05143 37 24 •09453 10.5789 .11217 8.91520 .12988 7.69957 .14767 6.77199 .16555 6.04051 36 25 .09482 10.5462 .11246 8.89185 .13017 7.68208 .14796 6.758^8 .16585 6.02962 35 26 .09511 10.5136 .11276 8.86862 •13047 7.66466 .14826 6.74483 .16615 6.01878 34 27 .09541 10.4813 .11305 8.84551 .13076 7.64732 .14856 6.73133 .16645 6.00797 33 28 •09570 10.4491 .11335 8.82252 .13106 7-63005 .14886 6.71789 .16674 5-99720 32 29 .09600 10.4172 .11364 8.79964 •13136 7.61287 .14915 6.70450 .16704 5-98646 31 30 .09629 10.3854 .11394 8.77689 •1316s 7-59575 .14945 6.69116 .16734 S-97576 30 31 .09658 10.3538 .11423 8.7S42S .13195 7-57872 .14975 6.67787 .16764 5.96510 29 32 .096S8 10.3224 .11452 8.73172 .13224 7.56176 .15005 6.66463 .16794 5-95448 28 33 .09717 10.2913 .11482 8.70931 .13254 7-54487 .15034 6.65144 .16824 5-94390 27 34 .09746 10.2602 .11511 8.68701 .13284 7.52806 .15064 6.63831 .16854 S-93335 26 35 .09776 10.2294 .11541 8.66482 .13313 7-51132 .15094 6.62523 .16884 S-92283 25 36 .09805 10.1988 .11570 8.64275 .13343 7-49465 .15124 6.61219 .16914 S-91236 24 37 .09834 10.1683 .11600 8.62078 .13372 7.47806 .15153 6.59921 .16944 S.90191 23 38 .09864 10.1381 .11629 8-59893 .13402 7.46154 .15183 6.58627 .16974 5-89151 22 39 .09893 10.1080 .11659 8-57718 .13432 7.44509 .15213 6-57339 .17004 S-88114 21 40 .09923 10.0780 .11688 8-55555 .13461 7.42871 •15243 6.56055 .17033 5.87080 20 41 .09952 10.0483 .11718 8.53402 .13491 7.41240 •15272 6-54777 .17063 5-86051 19 42 .09981 10.0187 .11747 8.51259 .13521 7.39616 • 15302 6.53503 .17093 5-85024 18 43 .10011 9.98931 .11777 8.49128 .13550 7-37999 • 15332 6.52234 .17123 S- 8400 1 17 44 .10040 9.96007 .11806 8.47007 .13580 7-36389 .15362 6.50970 .17153 5-82982 16 45 .10069 9-93101 .11836 8.44896 .13609 7-34786 •15391 6.49710 .17183 S-81966 IS 46 .10099 g. 90211 .11865 8-42795 .13639 7-33190 .15421 6-48456 .17213 S-80953 14 47 .10128 9.87338 .11895 8.40705 .13669 7.31600 .15451 6.47206 .17243 5-79944 13 48 .10158 9.84482 .11924 8.3862s .13698 7.30018 .15481 6.45961 .17273 S.78938 12 49 .10187 9.81641 .11954 8.36555 .13728 7.28442 .155" 6.44720 .17303 5-77936 II 50 .10216 9.78817 .11983 8-34496 .13758 7.26873 .15540 6.43484 .17333 S.76937 10 51 .10246 9.76009 .12013 8.32446 .13787 7-25310 .15570 6.42253 .17363 S.75941 9 52 .10275 9.73217 .12042 8.30406 .13817 7-23754 .15600 6.41026 .17393 5-74949 8 S3 .10305 g. 70441 .12072 8.28376 •13S46 7.22204 .15630 6.39804 .17423 5-73960 7 54 •10334 g. 67680 .12101 8.2635s •13876 7.20661 .15660 6.38587 .17453 S-72974 6 55 .10363 9-64935 .12131 8.24345 .13906 7-19125 .15689 6.37374 .17483 5-71992 S 56 •10393 9.62205 .12160 8.22344 .13935 7-17594 .15719 6.36165 .17513 5-71013 4 57 .10422 9-59490 .12190 8-20352 .13965 7.16071 .15749 6.34961 .17543 S-70037 3 58 .10452 9.56791 .12219 8.18370 .13995 7-14553 .15779 6.33761 .17573 S -69064 2 59 .10481 9-54106 .12249 8.16398 .14024 7.13042 .15809 6.32566 .17603 5 -68094 1 60 .10510 9^51436 .12278 8.1443S .14054 7-11537 .15838 6.3137s .17633 5-67128 1 1 Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 8. t° 8: 5° 82° 81° 8( 3° NATURAL TANGENTS AND COTANGENTS 163 1 10° 11° 12° 13° I 4° / Tang: Cotang Tangr Cotangr Tans: Cotang: Tangr Cotang Tang Cotang .17633 5.67128 .19438 5. 14455 .21256 4-70463 .23087 4.33148 .24933 4.01078 60 I .17663 S^66i6s .19468 5.13658 .21286 4-69791 .23117 4.32573 .24964 4.00582 59 2 .17693 5^65205 -19498 5.12862 .21316 4.69121 .23148 4.32001 .24995 4.00086 58 3 .17723 5.64248 ■19529 5^12069 .21347 4.68452 .23179 4.31430 .2%oz(> 3.99592 57 4 .17753 5^63295 -19559 5-11279 •21377 4.67786 .23209 4^30860 .25056 3.99099 S6 S .17783 5^62344 -19589 5^10490 .21408 4.67121 .23240 4^3029i .25087 3.98607 55 6 .17813 5. 61397 .19619 5.09704 .21438 4.66458 .23271 4.29724 .25118 3.98117 54 7 -17843 5-60452 .19649 5.08921 .21469 4.65797 .23301 4.29159 .25149 3-97627 53 8 .17873 S.S95II .196S0 5.08139 .21499 4.65138 .23332 4.28595 .25180 3.97139 52 9 .17903 5^58S73 .19710 5^07360 .21529 4.64480 .23363 4-28032 .25211 3.96651 51 10 .17933 5^57638 .19740 5-06534 .21560 4.6382s ■2Zi<ii 4-27471 .25242 3.9616s 50 u •17563 5.56706 .19770 5^05809 .21590 4.63171 .23424 4-26911 .25273 3^95680 49 12 •17993 5^55777 .19801 5.05037 .21621 4-62518 .23455 4^26352 .25304 3.95196 48 13 .18023 5.54851 .19831 5.04267 .21651 4^6i868 .23485 4^25795 .25335 3.94713 47 14 .18053 5.53927 .19861 5.03499 .21682 4^61219 .23516 4^25239 .25366 3.94232 46 15 .18083 5-53007 .19891 5.02734 .21712 4^60572 .23547 4^24685 .25397 3-93751 45 i6 .18113 5.52090 .19921 5^01971 .21743 4.59927 .23578 4^24132 .25428 3^93271 44 17 .18143 5.SI176 .19952 5.01210 .21773 4-59283 .23608 4^23580 .25459 3-92793 CI i8 .18173 5.50264 .19982 5.00451 .21804 4-58641 .23639 4^23030 .25490 3.92316 42 19 .18203 5.49356 .20012 4.99695 .21834 4-58001 .23670 4.22481 .25521 3.91839 41 20 .18233 5^48451 .20042 4.98940 .21864 4.57363 .23700 4.21933 .25552 3.91364 40 21 .18263 5^47548 .20073 4.98188 .21895 4-56726 .23731 4.21387 .25583 3.90890 39 22 .18293 5.46648 .20103 4.97438 .21925 4-56091 .23762 4^20842 .25614 3.90417 38 23 .18323 5.45751 .20133 4.96690 .21956 4.55458 .23793 4.20298 .25645 3.89945 37 24 .18353 5^44857 .20164 4-95945 .21986 4.54826 .23823 4.19756 .25676 3.89474 36 2S .18384 5.43966 .20194 4.95201 .22017 4.54196 .23854 4.1921s .25707 3.89004 35 26 .18414 5.43077 .20224 4.94460 .22047 4.53568 .23885 4.1867s .25738 3.88536 34 27 .18444 5. 42192 .20254 4-93721 .22078 4-52941 .23916 4.18137 .25769 3.88068 33 28 .18474 5-41309 .20285 4-92984 .22108 4-52316 .23946 4.17600 .25800 3.87601 32 29 .18504 5.40429 .20315 4-92249 .22139 4-S1693 .23977 4.17064 .25831 3.871.36 31 30 .18534 5-39552 .20345 4.91516 .22169 4.51071 .24008 4.16530 .25862 3.86671 30 31 .18564 S-38677 -20376 4.9078s .22200 4.50451 .24039 4.15997 .25893 3.86208 29 32 .18594 S-37S0S .20406 4.90056 .22231 4.49832 .24069 4.15465 .25924 3-85745 28 33 •18624 5-36956 .20436 4-89330 .22261 4.49215 .24100 4.14934 .25955 3.85284 27 34 •18654 S.36070 .20466 4.88605 .22292 4.48600 .24131 4^14405 .25986 3 •84824 26 35 .18684 5.35206 .20497 4.87882 .22322 4.47986 .24162 4^13877 .26017 3^84364 25 36 .18714 5-34345 .20527 4-87162 .22353 4-47374 .24193 4^13350 .26048 3.83906 24 37 .18745 S-33487 .20557 4-86444 .22383 4.46764 .24223 4.12825 .26079 3-83449 23 38 .18775 5-32631 .20588 4.85727 .22414 4.4615s .24254 4.12301 .26110 3-82992 22 39 .18805 5-31778 .20618 4.85013 .22444 4.45548 .2428s 4.11778 .26141 3.82537 21 40 .18835 5.30928 .20648 4.84300 .22475 4.44942 .24316 4.11256 .26172 3^82083 20 41 .18865 S-30080 .20679 4-83590 .2250s 4-44338 .24347 4.10736 .26203 3.81630 19 42 .18895 5-29235 .20709 4.82882 .22536 4.43735 .24377 4.10216 .26235 3.81177 18 43 .18925 5.28393 .20739 4.8217s .22567 4.43134 .24408 4.09699. •26266 3.80726 17 44 .18955 5^27553 .20770 4.81471 .22597 4-42534 .24439 4.09182 •26297 3.80276 16 45 .18986 5.26715 •20800 4.80769 .22628 4.41936 .24470 4.08666 .26328 3-79827 15 46 .19016 5^25880 .20830 4.80068 .2265S 4-41340 .24501 4.08152 .26359 3.79378 14 47 .19046 5.25048 .20861 4.79370 •22689 4.40745 .24532 4.07639 .26390 3.78931 13 48 .19076 5.24218 •20891 4^78673 .22719 4.40152 .24562 4.07127 .26421 3.78485 12 49 .19106 5.23391 .20921 4.77978 •22750 4.39560 .24593 4.06616 .26452 3-78040 11 50 .19136 5^22566 .20952 4.77286 •22781 4.38969 .24624 4.06107 .26483 3.77595 10 SI .19166 S^2I744 .20982 4.76595 .22811 4.38381 .24655 4.05599 .2651S 3.77152 9 52 .19197 5^20925 .21013 4^75906 .22842 4.37793 .24686 4.05092 .26546 3.76709 8 S3 .19227 5^20107 .21043 4^752I9 .22872 4-37207 .24717 4.04586 .26577 3.76268 7 54 .19257 5^ 19293 .21073 4^74534 .22903 4-36623 .24747 4.04081 .26608 3.75828 6 55 .19287 5.18480 .21104 4^73851 .22934 4.36040 .24778 4.03578 .26639 3.75388 5 S6 .19317 5.17671 .21134 4^73170 .22964 4-35459 .24809 4.03076 .26670 3.74950 4 57 .19347 5.16863 .21164 4.72490 .22995 4^34879 •24840 4.02574 .26701 3-74512 3 S8 .19378 S^i6os8 .21195 4.71813 .23026 4.34300 •24871 4.02074 .26733 3.7407s 2 59 .19408 5^15256 .21225 4-71137 .23056 4.33723 •24902 4.01576 .26764 3.73640 I 60 .19438 5^14455 .21256 4.70463 -23087 4.33148 •24933 4.01078 .26795 3.73205 / Cotang: Tang: Cotang- Tang: Cotang: Tang- Cotang Tang Cotang Tang / 79° 78° yi"" 76° 7. -0 5 164 NATURAL TANGENTS AND COTANGENTS 1 15° 16° 17° 18° I 9 1 Tang: Cotangr Tang Cotang Tang Cotang Tang CotansT Tang Cotang o .2679s 3-73205 .28675 3.48741 .30573 3.27085 .32492 3.07768 •34433 2.90421 60 I .26826 3.72771 .28706 3-48359 .30605 3.26745 .32524 3-07464 •34465 2.90147 59 2 .26857 3.72338 .28738 3.47977 .30637 3.26406 .32556 3.07160 •34498 2.89873 58 3 .26888 3.71907 .28769 3.47596 .30669 3-26067 .32588 3-06857 .34530 2.89600 57 4 .26920 3-71476 .28800 3-47216 .30700 3-25729 .32621 3.06554 .34563 2.89327 56 S .26951 3.71046 .28832 3-46837 .30732 3.25392 .32653 3-06252 .34596 2.8905s 55 6 .26982 3.70616 .28864 3-46458 .30764 3-25055 .32685 3-05950 .34628 2.88783 54 7 .27013 3.70188 .28895 3.46080 .30796 3.24719 .32717 3-05649 .34661 2.88511 S3 8 .27044 3-69761 .28927 3-45703 .30828 3-24383 .32749 3-05349 .34693 2.88240 52 9 .27076 3-69335 .28958 3-45327 .30860 3-24049 .32782 3-05049 .34726 2.87970 51 10 .27107 3.68909 .28990 3.44951 .30891 3.23714 .32814 3.04749 .34758 2.87700 SO II .27138 3-68485 .29021 3.44576 .30923 3.23381 .32846 3-04450 .34791 2.87430 49 12 .27169 3.68061 .29053 3.44202 .30955 3.23048 .32878 3-04IS2 .34824 2.87161 48 13 .27201 3.67638 .29084 3.43829 .30987 3.22715 .32911 3-03854 .34856 2.86892 47 14 .27232 3.67217 .29116 3.43456 .31019 3.22384 .32943 3.03556 .34889 2.86624 46 15 .27263 3.66796 .29147 3.43084 .31051 3.22053 .3297s 3.03260 .34922 2.86356 45 i6 .27294 3.66376 .29179 3.42713 .31083 3.21722 .33007 3.02963 •34954 2.86089 44 17 .27326 3.65957 .29210 3.42343 .31115 3.21392 .33040 3.02667 .34987 2.85822 43 i8 .27357 3.65538 .29242 3.41973 .31147 3.21063 .33072 3-02372 .35020 2.8555s 42 19 .27388 3.65121 .29274 3.41604 .31178 3-20734 .33104 3.02077 .35052 2.85289 41 20 .27419 3.6470s .29305 3.41236 .31210 3.20406 .33136 3.01783 .3508s 2.85023 40 21 .27451 3.64289 .29337 3.40869 .31242 3.20079 .33169 3.01489 .35118 2.84758 39 22 .27482 3.63874 .29368 3.40S02 .31274 3-19752 .33201 3.01196 .35150 2.84494 38 23 .27513 3.63461 .29400 3.40136 .31306 3.19426 .33233 3.00903 .35183 2.84229 37 24 .27545 3.63048 .29432 3.39771 .31338 3.19100 .33266 3.00611 .35216 2.8396s 36 25 .27576 3.62636 .29463 3.39406 .31370 3-18775 .33298 3.00319 .35248 2.83702 35 26 .27607 3.62224 .29495 3.39042 .31402 3-18451 .33330 3.00028 • 35281 2.83439 34 27 .27638 3.61814 .29526 3.38679 .31434 3.18127 .33363 2.99738 •35314 2.83176 33 28 .27670 3.61405 .29558 3.38317 .31466 3.17804 .33395 2.99447 •35346 2.82914 32 29 .27701 3.60996 .29590 3-37955 .31498 3.17481 .33427 2.99158 •35379 2.82653 31 30 .27732 3.60588 .29621 3.37594 •31530 3.17159 .33460 2.98868 •35412 2.82391 30 31 .27764 3.60181 .29653 3.37234 .31562 3.16838 .33492 Z.98580 •35445 2.82130 29 32 .27795 3-59775 .29685 3.36875 .31594 3.16517 .33524 2.98292 •35477 2.81870 28 33 .27826 3-59370 .29716 3.36516 .31626 3.16197 .33557 2.98004 •35510 2.81610 27 34 .27858 3.58966 .29748 3.36158 .31658 3.15877 .33589 2.97717 •35543 2.81350 26 35 .27889 3.58562 .29780 3.35800 .31690 3.15558 .33621 2.97430 •35576 2.8iogi 25 36 .27921 3.58160 .29811 3.35443 .31722 3.15240 .33654 2.97144 .35608 2.80833 24 37 .27952 3-57758 .29843 3-35087 •31754 3.14922 .33686 2.96858 •35641 2.80574 23 38 .27983 3-57357 .29875 3.34732 .31786 3-14605 •33718 2.96573 •35674 2.80316 22 39 .2801S 3.56957 .29906 3.34377 .31818 3.14288 •33751 2.96288 •35707 2.80059 21 40 .28046 3.56557 .29938 3.34023 .31850 3.13972 •33783 2.96004 •3S740 2-.79802 20 41 .28077 3.56159 .29970 3.33670 .31882 3-13656 .33816 2.95721 •35772 2.79S4S 19 42 .28109 3.55761 .30001 3.33317 .31914 3.13341 .33848 2.95437 •3580s 2.79289 18 43 .28140 3.55364 .30033 3-32965 .31946 3-13027 .33881 2.9515s .35838 2.79033 17 44 .28172 3.54968 .3006s 3-32614 .31978 3.12713 .33913 2.94872 .35871 2.78778 16 45 .28203 3.54573 .30097 3.32264 .32010 3.12400 .33945 2.94591 .35904 2.78523 IS 46 .28234 3-54179 .30128 3.31914 .32042 3.12087 .33978 2-94309 .35937 2.78269 14 47 .28266 3-53785 .30160 3-31565 .32074 3.1177s .34010 2.94028 .35969 2.78014 13 48 .28297 3.53393 .30192 3.31216 .32106 3.11464 .34043 2.93748 .36002 2.77761 12 49 .28329 3.53001 .30224 3.30868 .32139 3.11153 ■3407s 2.93468 .36035 2.77507 II SO .28360 3.52609 ■30255 3.30521 .32171 3.10842 .34108 2.93189 .36068 2.77254 JO 51 .28391 3.52219 .30287 3.30174 .32203 3.10532 .34140 2.92910 .36101 2.77002 9 52 .28423 3.S1829 .30319 3.29829 .32235 3.10223 .34173 2.92632 .36134 2.76750 8 53 .28454 3.S1441 .30351 3.29483 .32267 3.09914 .34205 2.92354 .36167 2.76498 7 54 .28486 3-51053 .30382 3.29139 .32299 3.09606 .34238 2.92076 .36199 2.76247 6 55 .28517 3.50666 .30414 3.28795 .32331 3.09298 .34270 2.91799 .36232 2.75996 5 56 .28549 3.50279 .30446 3.28452 .32363 3.08991 .34303 2.91523 .3626s 2.75746 4 57 .28580 3-49894 .30478 3.28109 .32396 3.08685 •34335 2.91246 .36298 2.75496" 3 58 .28612 3.49509 .30509 3.27767 .32428 3.08379 .34368 Z.90971 ■36331 2.75246 2 59 .28643 3.49125 .30541 3.27426 .32460 3.08073 •34400 2.90696 ■36364 2.74997 I 60 .28675 3.48741 .30573 3.2708s .32492 3.07768 •34433 2.90421 ■36397 2.74748 / Cotang Tangr Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 7^ J° 73° 7i )0 7J 7( J° NATURAL TANGENTS AND COTANGENTS 1G5 / 20° 2] 22° 2; ° 24° / Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang .36397 2.74748 .38386 2.60509 .40403 2.47509 .42447 2.35585 •44S23 2.24604 60 I .36430 2.74499 .38420 2.60283 .40436 2.47302 •42482 2.35395 •44558 2.24428 59 2 .36463 2.74251 .38453 2.60057 .40470 2.47095 .42516 2.35205 • 44.193 2.24252 S8 3 .36456 2.74004 .38487 2.59831 .40504 2.468S8 .42551 2.35015 •44627 2.24077 57 4 .36329 2.73756 .38520 2.59606 .40538 2.46682 .42585 2.34825 .44662 2.23902 56 S .36562 2.73509 .38553 2.59381 .40572 2.46476 .42619 2.34636 .44697 2.23727 55 6 .36595 2.73263 .38587 2.59156 .40606 2.46270 .42654 2.34447 .44732 2.235.53 54 7 .36628 2.73017 .38620 2.58932 .40640 2.46065 .42688 2.34258 .44767 2.2,^378 S3 g .36661 2.72771 .38654 2.58708 .40674 2.45S60 .42722 2.34069 .44S02 2.23204 52 9 .36694 2.72526 .38687 2.58484 .40707 2.45655 •42757 2.33881 .44837 2.23030 SI 10 .36727 2.72281 .38721 2.58261 .40741 2.45451 •42791 2.33693 .44872 2.22857 50 II .36760 2.72036 .38754 2.58038 .4077s 2.45246 .42826 2.33505 .44907 2.22683 49 12 .36793 2.71792 .38787 2.57S1S .40S09 2.45043 .42860 2.33317 .44942 2.22510 48 13 .368.:6 2.71548 .38821 2.57593 .40843 2.44839 .42894 2.33130 .44977 2.22337 47 14 .36859 2.7130s .38854 2.57371 .40877 2.44636 .42929 2.32943 .45012 2.22164 46 IS .36892 2.71062 .38888 2.57150 •40911 2.44433 .42963 2.32756 .45047 2.21992 45 i6 .36925 2.70813 •38921 2.56928 .40945 2.44230 .42998 2.32570 .45083 2.2i8ig 44 17 .36958 2.70577 .38955 2.56707 .40979 2.44027 .43032 2.32383 .45117 2.21647 43 i8 .36991 2.7033s .38988 2.56487 .41013 2.43825 .43067 2.32197 .45152 2.21475 42 19 .37024 2.70094 .39023 2.56266 .41047 2.43623 .43101 2.32012 .45187 2.21304 41 20 ■37057 2.69853 .39055 2.56046 .41081 2.43422 .43136 2.31826 .45222 2.21132 40 21 .37090 2.69612 .39089 2.55827 .41115 2.43220 .43170 2.31641' .45257 2.20961 39 22 .37123 2.69371 .39122 2.55608 •41149 2.43019 .43205 2.31456 .45293 2.20790 38 23 .37157 2.69131 .39156 2.55389 .41183 2.42819 .43230 2.31271 .45327 2.20619 37 24 .37190 2.68892 .39190 2.55170 .41217 2.42618 .43274 2.31086 .45363 2.20449 36 25 .37223 2.68653 .39223 2.54952 .41251 2.42418 .43308 2.30902 •45397 2.20278 35 26 .37256 2.68414 •39257 2.54734 .41285 2.42218 .43343 2.30718 •45433 2.20108 34 27 .37289 2.68175 •39290 2.54516 ■41319 2.42019 .43378 2.30534 .45467 2.19938 33 28 .37322 2.67937 •39324 2.54299 .41353 2.41819 .43413 2.30351 •45503 2.19769 32 29 .37355 2.67700 •39357 2.54082 .41387 2.41620 .43447 2.30167 •45538 2.19599 31 30 .37388 2.67462 .39391 2.53865 .41421 2.41421 .43481 2.29984 •45573 2.19430 30 31 .37422 2.6722s .39425 2.53648 .41455 2.41223 .43516 2.29801 .45608 2.19261 29 32 .37455 2.66989 .39458 2.53432 .41490 2.41025 .43550 2.29619 .45643 2.19092 28 33 .37488 2.66752 .39492 2.53217 .41524 2.40827 .4358s 2.29437 .45678 2.18923 27 34 .37521 2.66516 .39526 2.53001 .41558 2.40629 •43620 2.29254 .45713 2.18755 26 35 •37554 2.66281 .39559 2.52786 .41592 2.40432 •43654 2.29073 .45748 2.18587 25 36 .37588 2.66046 .39593 2.52571 .41626 2.4023s .43689 2.28891 .45784 2.18419 24 37 .37621 2.65811 .39626 2.52357 .41660 2.40038 •43724 2.28710 .45819 2.18251 23 38 .37654 2.65576 .39660 2.52142 .41694 2.39841 .43758 2.28528 .45854 2.18084 22 39 .37687 2.65342 .39694 2.51929 .41728 2.3964s •43793 2.28348 .45889 2.17916 21 40 .37720 2.65109 .39727 2.51715 .41763 2.39449 •43828 2^28167 .45924 2.17749 20 41 .37754 2.6487s .39761 2.S1502 .41797 2.39253 .43862 2^27987 .45960 2.17582 19 42 .37787 2.64642 .39795 2.51289 .41831 2.39058 .43897 2.27806 .45995 2.17416 18 43 .37820 2.64410 .39S29 2.51076 .41865 2.38863 .43932 2.27626 .46030 2.17249 17 44 .37853 2.64177 .39862 2.50864 .41899 2.38668 .43966 2.27447 .46065 2.17083 16 45 .37887 2.63945 .39896 2.50652 .41933 2.38473 .44001 2.27267 .46101 2.16917 IS 46 .37920 2.63714 .39930 2.50440 .41968 2.38279 .44036 2.27088 .46136 2.16751 14 47 •37953 2.63483 .39963 2.50229 •42002 2.38084 .44071 2.26909 .46171 2.16585 13 48 .37986 2.63252 .39997 2.50018 •42036 2.37891 .44105 2.26730 •46206 2.16420 12 49 .38020 2.63021 .40031 2.49807 .42070 2.37697 .44140 2.26552 .46242 2.1625s 11 50 .38053 2.62791 .40065 2.49597 .42105 2.37504 .4417s 2.26374 •46277 2.16090 10 SI .38086 2.62561 .40098 2.49386 .42139 2.37311 •44210 2.26196 •46312 2.15925 9 52 .38120 2.62332 .40132 2.49177 .42173 2.37118 .44244 2.26018 •46348 2.15760 8 S3 .38153 2.62103 .40166 2.48967 .42207 2.36925 .44279 2.25840 •46383 2.15596 7 54 .38186 2.61874 .40200 2.48758 .42242 2.36733 .44314 2.25663 .46418 2.1S432 6 SS .38220 2.61646 .40234 2.48549 .42276 2.36541 .44349 2.25486 •46454 2.15268 5 S6 .38253 2.61418 .40267 2.48340 .42310 2.36349 .44384 2.25309 .46489 2.I5I04 4 57 .38286 2. 61190 .40301 2.48132 .42345 2.36158 .44418 2.25132 .46525 2.14940 3 S8 .38320 2.60963 .40335 2.47924 .42379 2.35967 .444S3 2.24956 .46560 2.14777 2 59 .38353 2.60736 .40369 2.47716 •42413 2.35776 .44488 2.24780 .46595 2.14614 I 60 .38386 2.60509 .40403 2.47509 ■42447 2.35585 .44523 2.24604 .46631 2.I445I / Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang ( 6( ?° 6i i° 6; 7° 6( 5° 6 5° 166 NATURAL TANGENTS AND COTANGENTS / 2 = 26° 27° 28° 29° 1 Tang Cotang: Tang Cotang Tang Cotang Tang Cotang Tang Cotang .46631 2.I44SI .48773 2.05030 •S09S3 1. 96261 •53171 1.88073 •55431 1.8040S 60 I .46666 2.14288 .48809 2.04879 .50989 1.96120 .53208 1.87941 •55469 1.80281 59 3 .46702 2.14125 .48845 2.047:8 .51026 1.95979 -53246 1.87809 •55507 1.80158 S8 3 .46737 2.13963 .488S1 2.04577 .51063 1.95838 .53283 1-87677 •5S545 1.80034 57 4 .46772 2.13801 •48917 2.04426 .51099 1.95698 •53320 1-87546 •55583 1.79911 56 S .46808 2.13639 .48953 2.04276 .51136 1-95557 •53358 1.87415 •55621 1.79788 ss 6 .46843 2.13477 .48989 2.04125 •51 173 1.95417 •53395 1.87283 •55659 1.7966s 54 7 .46879 2.13316 .49026 2.03975 .51209 l^95277 •53432 1.87152 .55697 1.79542 S3 8 .46914 2.I3IS4 .49062 2.0382s .51246 1^95137 •53470 1. 87021 .55736 1.79419 52 9 .46950 2.12993 .49098 2.03675 .51283 1.94997 •53507 1.86891 .55774 1.79296 SI 10 .4698s 2.12832 .49134 2.03526 •51319 1.94858 •S3545 1.86760 .55812 1.79174 SO :i .47021 2.12671 .49170 2.03376 .51356 1.94718 •S3582 1.86630 •55850 I.79051 49 12 .47056 2.12511 .49206 2.03227 .51393 1.94579 •53620 1.86499 .55888 1.78929 48 13 .47092 2.12350 .49242 2.03078 .51430 1.94440 •53657 1.86369 •55926 1.78807 47 14 .47128 2. 12190 .49278 2.02929 .51467 1.94301 •53694 1.86239 •55964 1.78685 46 IS .47163 2.12030 .4931s 2.02780 .51503 1. 94162 •53732 1.86109 .56003 1.78563 45 i6 .47199 2.11871 .49351 2.02631 •51540 1.94023 •53769 1-85979 .56041 1^78441 44 17 .47234 2.11711 .49387 2.02483 •51577 1.93885 •53807 1.85850 .56079 1.78319 43 i8 .47270 2.11552 .49423 2.02335 .51614 1^93746 .53844 1.85720 .56117 1-78198 42 19 .4730s 2.11392 .49459 2.02187 •51651 1.93608 .53882 1-85591 .56156 1.78077 41 20 .47341 2.11233 ■49495 2.02039 .51688 1.93470 •53920 1.85462 •56194 1-77955 40 21 .47377 2.11075 .49532 2.01891 .51724 I^93332 •53957 1.85333 .56232 1-77834 39 22 .47412 2.10916 .49568 2.01743 .51761 1-93195 .53995 1.85204 .56270 I-77713 38 23 .47448 2.10758 .49604 2.01596 .51798 1-93057 •54032 1.85075 .56309 1.77592 37 24 .47483 2.10600 .49640 2.01449 .S183S 1.92920 .54070 1.84946 •56347 1-77471 36 25 .47519 2.10443 .49677 2.01302 .51872 1.92782 •54107 1.84818 .56385 I-77351 35 26 .47555 2.10284 .49713 2.0II5S .51909 1.92645 .54145 1.84689 .56424 1.77230 34 27 .47590 2.10126 .49749 2.01008 ■51946 1.92508 .54183 1.84561 .56462 1.77110 33 28 .47626 2.09969 .49786 2.00862 •51983 1-92371 .54220 1-84433 .56501 1.76990 32 29 .47663 2.09811 .49822 2.00715 •52020 1^92235 .54258 I-8430S .56539 1.76869 31 30 .47698 2.09654 .49858 2.00569 •52057 1.92098 .54296 1.84177 •56577 1.76749 30 31 .47733 2.09498 .49894 2.00423 .52094 I. 91962 .54333 1.84049 •56616 1.76629 29 32 .47769 2.09341 .49931 2.00277 •52131 I. 91826 .54371 1.83922 .56654 1.76510 28 33 .4780s 2.09184 .49967 2.00131 .52168 1.91690 .54409 1.83794 .56693 1.76390 27 34 .47840 2.09023 .50004 1.99986 .52205 I.9I5S4 .54446 1.83667 .56731 1.76271 26 3S .47876 2.08873 .50040 1.99841 .52242 I.91418 .54484 1.83540 .56769 1.76151 25 36 .47912 2.08716 .50076 1.99695 .52279 1-91282 •54522 1-83413 .56808 1.76032 24 37 .47948 2.08560 .50113 1.99550 .52316 1-91147 •54560 1.83286 .56846 I-75913 23 38 .47984 2.0840s .50149 1.99406 .52353 I.91012 •S4597 1.83159 .56885 1-75794 22 39 .48019 2.08250 .50185 1. 99261 .52390 1.90876 •54635 1.83033 .56923 1.75675 21 40 .48055 2.08094 .50222 1.99116 .52427 1. 9074 1 •54673 1.82906 .56962 1-75556 20 41 .48091 2.07939 .50258 1.98972 •52464 1.90607 .54711 1.82780 .57000 1-75437 19 42 .48127 2.0778;; .50295 1.98828 .52501 1.90472 .54748 1.82654 .57039 I-75319 18 43 .48163 2.07630 .50331 1.98684 •52538 1-90337 .54786 1.82528 .57078 1.75200 17 44 .48198 2.07476 .50368 1.98540 •S2S75 1.90203 •54824 1.82402 .57116 1.75082 16 4S .48234 2.073-'2i .50404 1.98396 .52613 1.90069 .54862 1.82276 •57155 1.74964 15 46 .48270 2.07167 .50441 1.98253 .52650 1-89935 .54900 1.82150 •57193 I-74846 14 47 .48306 2.07014 .50477 1.98110 .52687 1.89801 .54938 1.82025 •57232 1-74728 13 48 .48342 2.06860 .50514 1.97966 .52724 1.89667 •54975 1.81899 •57271 I-74610 12 49 .4S378 2.06706 .50550 1.97823 .52761 1.89533 •55013 1.81774 .57309 1-74492 11 SO .48414 2.06553 .50587 1. 97681 .52798 1.89400 •55051 1.81649 •57348 1.74375 10 51 .48450 2.06400 .S0623 1.97538 .52836 1.89266 •55089 I. 81524 .57386 1.74257 9 32 .484B6 2.06247 .50660 1.97395 .52873 1-89133 •55127 1.81399 .57425 1.74140 8 53 .48521 2.06094 .50696 1.97253 .52910 1.89000 •55165 1.81274 •57464 1.74022 7 S4 .48557 2.05942 .50733 1.97111 .52947 1.8S867 •55203 1.81150 •57503 1.73905 6 SS .48593 2.05790 .50769 1.96969 •52985 1.88734 •55241 1.81025 •57541 I. 73788 S S6 .48629 2.05637 .50806 1.96827 .53022 I. 886c 2 •55279 1.80901 •57580 1.73671 4 S7 .48665 2.0548s .50843 1.96685 •53059 1.88469 •55317 1.80777 •57619 1.73555 3 S8 .48701 2.05333 .50879 1.96544 .53096 1.88337 •55355 1.80653 •57657 1.73438 2 S9 .48737 2.05182 .50916 1.96402 .53134 1.88205 .55393 1.80529 •57696 1.73321 I 60 .48773 2.05030 .50953 1.96261 .53171 1.88073 •S543I I. 80405 •5773S 1.7320s / Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 6. i° 6 3° 6: 2° 6 [° 6 3° NATURAL TANGENTS AND COTANGENTS 167 1 30° 3] 3- 33° 34° / Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang .57735 1.73205 .60086 1.66428 .62487 1.60033 .64941 1.53986 .67451 1.48256 60 I .57774 1.73089 .60126 1.66318 .62527 1.59930 .64982 1.53888 .67493 1.48163 59 2 .57813 1.72973 .60165 1.66209 .62568 1.59826 .65024 1.S379I .67536 1.48070 58 3 .57851 1.72857 .60205 1.66099 .62608 1.59723 -6506s 1.53693 .67578 1.47977 57 4 .57890 1. 72741 .60245 1.65990 .62649 1.59620 .65106 I.S359S .67620 1.4788s 56 S .57929 1.7262s .60284 1. 6588 1 .62689 1.59517 .65148 1.53497 .67663 1.47792 55 6 .57968 1.72509 .60324 1.65772 .62730 1.59414 .65189 1-53400 .67705 1.47699 54 7 .58007 1-72393 .60364 1.65663 .62770 1.5931 1 .65231 1.53302 .67748 1.47607 S3 8 .58046 1.72278 .60403 1.65554 .62811 1.59208 .65272 1-53205 .67790 I-47514 52 9 .58085 I. 72163 .60443 1-65445 .62852 I-S9I0S .65314 1-53107 .67832 1.47422 51 10 .58124 1.72047 .60483 1.65337 .62892 1.59002 .65355 1.53010 .67875 1.47330 50 II .58162 1. 71932 .60522 1.65228 .62933 1.58900 .65397 1.52913 .67917 1.47238 49 12 .58201 1.71817 .60562 1.65120 .62973 1-58797 .65438 1.52816 .67960 1.47146 48 13 .58240 1.71702 .60602 1. 65011 .63014 i^586gs .65480 1. 52719 .68002 1.47053 47 14 .58279 1. 71588 .60642 1.64903 .63055 I^58S93 .65521 1.52622 .68045 1.46962 46 IS .58318 1. 71473 .60681 1.64795 .63095 1.58490 .65563 1.52525 .68088 1.46870 45 l6 .58357 I. 71358 .60721 1.64687 .63136 1.58388 ■65604 1.52429 .68130 1.46778 44 17 .5S396 I. 71244 .60761 1.64579 .63177 1.58286 .65646 1.52332 .68173 I.466E6 43 i8 .58435 1.71129 .60801 1.64471 .63217 1.58184 .65688 1.5223s .68215 1.46595 42 19 .58474 1.71015 .60841 1.64363 .63258 1.58083 .65729 1.52139 .68258 1.46503 41 20 .58513 1.70901 .60881 1.64256 .63299 1.57981 .65771 1.52043 .68301 1.4641 1 40 21 .58552 1.70787 .60921 1.64148 .63340 1.57879 .65813 I.51946 .68343 1.46320 39 22 .58591 1.70673 .60960 1.64041 .63380 1^57778 .65854 1. 51850 .68386 1.46229 38 23 .58631 1.70560 .61000 1.63934 .63421 1.57676 .65896 1.51754 .68429 1.46137 37 24 .58670 1.70446 .61040 1.63826 .63462 1.57575 .65938 1.51658 .68471 1.46046 36 2S .58709 1.70332 .61080 1.63719 ■63503 1.57474 .65980 1.51562 .68514 1.45955 35 26 .58748 1. 70219 .61120 1. 63612 .63544 1.57372 .66021 1.51466 .68557 1.45864 34 27 .58787 I. 70106 .61160 1.63505 -63584 1-57271 .66063 1-51370 .68600 1.45773 33 28 .58826 1.69992 .61200 1.63398 .6362s I-57170 .66105 1-51275 .68642 1.45682 32 29 .58865 1.69879 .61240 1.63292 .63666 1.57069 .66147 1.51179 .68685 1-45592 31 30 .58905 1.69766 .61280 1.63185 .63707 1.56969 .66189 1.51084 .68728 I.45501 30 31 .58944 1.69653 .61320 1.63079 .63748 1.56868 .66230 1.50988 .68771 1.45410 29 32 .58983 1.69541 .61360 1.62972 .63789 1.56767 .66272 1.S0893 .68814 1.45320 28 33 .59022 1.69428 .61400 1.62866 .63830 1.56667 .66314 1.S0797 .68857 1.45229 27 34 .59061 1. 69316 .61440 1.62760 .63871 1.56566 .66356 1.50702 .68900 1.45139 26 35 .59101 1.69203 .61480 1.62654 .63912 1.56466 .66398 1.50607 .68942 1.45049 25 36 .59149 1. 6909 1 .61520 1.62548 ■63953 1.56366 .66440 1.50S12 .68985 1.44958 24 37 .59179 1.68979 .61561 1.62442 ■63994 1.5626s .66482 1. 50417 .69028 1.44868 23 38 .59218 1.68866 .61601 1.62336 ■6403s 1.56165 .66524 1.50322 .69071 1.44778 22 39 .59258 1.68754 .61641 1.62230 .64076 1.5606s .66566 1.50228 .69114 1.44688 21 40 .59297 1.68643 .61681 1. 62125 .64117 1.55966 .66608 1.50133 .69157 1.44598 20 41 .59336 1.68531 .61721 1.62019 .64158 1.55866 .66650 1.S0038 .69200 1.44508 19 42 .59376 1. 68419 .61761 1. 61914 .64199 1.55766 .66692 1.49944 .69243 1.44418 18 43 .59415 1.68308 .61801 1.61808 .64240 1.55666 .66734 1.49849 .69286 1.44329 17 44 •59454 1. 68196 .61842 I. 61703 .64281 1.55567 .66776 1-49755 .69329 1.44239 16 45 .59494 1.6808s .61882 1.61598 .64322 1.55467 .66818 1.49661 .69372 1.44149 15 46 .59533 1.67974 .61922 1.61493 .64363 1.55368 .66860 1.49566 .69416 1.44060 14 47 .59573 1.67863 .61962 1.6138S .64404 1.55269 .66902 1.49472 .69459 1.43970 13 48 .59612 1.67752 .62003 1.61283 .64446 1^55170 .66944 1.49378 .69502 1.43881 12 49 .59651 1. 67641 .62043 1.61179 .64487 1.SS071 .66986 1.49284 .69545 1.43792 11 50 .59691 1.67530 .62083 1. 61074 .64528 1-54972 .67028 1.49190 .69588 1.43703 10 SI .59730 1.67419 .62124 1.60970 .64569 1.54873 .67071 1.49097 .69631 1.43614 9 52 .59770 1.67309 .62164 1.60865 .64610 1.54774 .67113 1.49003 .69675 1.43525 8 53 .59809 1.67198 .62204 1.60761 .64652 l^S467S .67155 1.48909 .69718 1.43436 7 54 .59849 1.67088 .62245 1.60657 .64693 1.54576 .67197 1.48816 .69761 1.43347 6 SS .59888 1.66978 .62285 1-60553 .64734 1.54478 .67239 1.48722 .69804 1.43258 5 S6 .59938 1.66867 .62325 1.60449 .64775 1.54379 .67282 1.48629 .69847 1.43169 4 57 .59967 1.66757 .62366 1.6034s .64817 1. 54281 .67324 1.48536 .69891 1.43080 3 58 .60007 1.66647 .62406 1.60241 .64858 1. 54183 .67366 1.48442 .69934 1.42992 2 59 .60046 1.66538 .62446 1. 60137 .64899 1.54085 .67409 1.48349 .69977 1-42903 I 6o .60086 1.66428 .62487 1.60033 .64941 1.53986 .67451 1.48256 .70021 1.4281s / Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 5 ?° 5 3° 5 7° 5< 5° 5 5° 168 NATURAL TANGENTS AND COTANGENTS / 35° 36° 37° 38° 39° / Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang .70021 I •4281s .72654 1^37638 .75355 1.32704 .78129 1.27994 .80978 1.23490 60 I .70064 1.42726 .72699 1^37554 .75401 1.32624 ■7817s 1.27917 .81027 1.23416 59 2 .70107 1.42638 .72743 1.37470 •75447 1.32544 .78222 1.27841 .81075 1.23343 S8 3 .70151 1.42550 .72788 1.37386 •75492 1.32464 .78269 1.27764 .81123 1.23270 57 4 .70194 I^42462 .72832 1.37302 •75538 1.32384 .78316 1.27688 .81171 I. 23196 S6 S .70238 1^42374 .72877 1.37218 •75584 1.32304 .78363 1.27611 .81220 1.23123 55 6 .70281 1.42286 .72921 1.37134 •75629 1.32224 .78410 1.2753s .81268 1.23050 54 7 .70325 1. 42198 .72966 I. 37050 •75675 1.32144 .78457 1.27458 .81316 1.22977 53 8 .70368 I.42IIO .73010 1.36967 •75721 1.32064 .78504 1.27382 .81364 1.22904 52 9 .70412 1.42022 .73055 1.36883 •75767 I. 31984 .78551 1.27306 .81413 1.22831 SI 10 .70455 1^41934 .73100 1.36800 •75812 1.31904 .78598 1.27230 .81461 1.22758 SO II .70499 1. 41847 .73144 1.36716 .75858 1.3182s .7864s 1.27153 .81510 1.2268s 49 12 .70542 1^41759 .73189 1.36633 .75904 1.31745 .78692 1.27077 .81558 1.22612 48 13 .70586 1.41672 .73234 1^36549 .75950 1. 31666 .78739 1. 27001 .81606 1.22539 47 14 .70629 1.41584 .73278 1.36466 .75996 1.31586 .78786 1.26925 .81655 1.22467 46 IS .70673 I.41497 .73323 1.36383 .76042 1.31507 .78834 1.26849 .81703 1.22394 45 i6 .70717 1. 41409 .73368 i^36300 .76088 1.31427 .78881 1.26774 .81752 1.22321 44 17 .70760 1.41322 .73413 I^362I7 .76134 1.31348 .78928 1.26698 .81800 1.22249 43 i8 .70804 1.4123s .73457 1^36134 .76180 1.31269 .7897s 1.26622 .81849 1.22176 42 19 .70848 I.41148 ■73502 1^36051 .76226 1.31190 .79022 1.26546 .81898 1.22104 41 20 .70891 1.41061 ■73547 1.35968 .76272 1. 31110 .79070 1.26471 .81946 1.22031 40 21 .7093s 1.40974 •73592 1.35885 .76318 1.31031 .79117 1.2639s .81995 1.21959 39 22 .70979 1.40887 •73637 1.35802 .76364 1.30952 .79164 1.26319 .82044 1.21886 38 23 .71023 1.40800 •73681 1.35719 .76410 1.30873 .79212 1.26244 .82092 1.21814 37 24 .71066 1.40714 .73726 1.35637 .76456 1 ■3079s ■79259 1.26169 .82141 1.21742 36 2S .71110 1.40627 ■73771 1 .35554 .76502 1. 30716 .79306 1.26093 .82190 I. 21670 35 26 .71154 1.40540 •73816 1^35472 .76548 1.30637 .79354 1.26018 .82238 1.21598 34 27 .71198 1.40454 .73861 i^35389 .76594 l^30S58 ■79401 1.25943 .82287 1.21526 33 28 .71242 1.40367 ■73906 1^35307 .76640 1.30480 ■79449 1.25867 .82336 1.21454 32 29 .71285 1.40281 ■73951 1^35224 .76686 1.30401 .79496 1.25792 .82385 1.21382 31 30 .71329 1.40195 ■73996 1.35142 •76733 1.30323 ■79544 1.25717 .82434 1. 21310 30 31 •71373 1.40109 ■74041 1.35060 •76779 1.30244 ■79591 1.25642 .82483 1.21238 29 32 .71417 1.40022 .74086 1.34978 .76825 I. 30166 ■79639 1.25567 •82531 1. 21166 28 33 .71461 1.39936 ■74131 1.34896 .76871 1.30087 .79686 1.25492 .82580 1.21094 27 34 .7150s i^398so .74176 1.34814 .76918 1.30009 .79734 1.25417 .82629 1.21023 26 35 •71549 1^39764 .74221 1^34732 .76964 1.29931 .79781 1.25343 .82678 1.209s I 25 36 ■71593 i^39679 .74267 1.34650 .77010 1.29853 .79829 1.25268 .82727 1.20879 24 37 .71637 1^39593 ■74312 1.34568 •77057 1.2977S .79877 I.25193 .82776 1.20808 23 38 .71681 1.39507 •74357 1.34487 •77103 1.29696 .79924 1.25118 .82825 1.20736 22 39 .71725 1.39421 •74402 1 .34405 •77149 1. 29618 .79972 1.25044 .82874 1.2066s 21 40 .71769 1.39336 •74447 1^34323 .77196 1.29541 .80020 1.24969 .82923 1.20593 20 41 .71813 1.39250 .74492 1.34242 .77242 1.29463 .80067 1.2489s .82972 1. 20522 19 42 •71857 1.39165 •74538 1.34160 .77289 1.29385 .8011S 1.24820 .83022 1.20451 18 43 .71901 1.39079 •74583 1.34079 .77335 1^29307 .80163 1.24746 .83071 1.20379 17 44 •71946 1.38994 .74628 1^33998 •77382 1.29229 .80211 1.24672 .83120 1.20308 16 4S .71990 1.38909 .74674 I^339i6 .77428 1.29152 .80258 1.24597 .83169 1.20237 15 46 •72034 1.38824 .74719 i^3383S •77475 1.29074 .80306 1 ■24523 .83218 1.20166 14 47 .72078 1.38738 .74764 1 ■33754 ■77521 1.28997 .80354 1.24449 .83268 1. 2009s 13 48 .72122 1.38653 .74810 1-33673 ■77568 1.28919 .80402 1.24375 .83317 1.20024 12 49 •72167 1.38568 ■74855 I ■33592 ■77615 1.28842 .80450 1.24301 .83366 1.19953 II SO .72511 1.38484 .74900 l^33Sii .77661 1.28764 .80498 1.24227 .8341S 1.19882 10 SI •72255 1.38399 ■74946 l^33430 .77708 1.28687 .80546 I.24IS3 .8346s 1.19811 9 S2 .72299 i^383i4 ■74991 i^33349 ■77754 1.28610 .80594 1.24079 .83514 1.19740 8 S3 .72344 1.38229 ■75037 1.33268 .77801 1.28533 .80642 1.24005 .83564 1. 19669 7 S4 .72388 1.3814s ■75083 1.33187 .77848 1.28456 .80690 1.23931 .83613 I. I 9599 6 SS .72432 1.38060 .75128 I^33I07 .7789s 1.28379 .80738 1.23858 .83662 1. 19528 S S6 ■72477 1.37976 •75173 1^33026 .77941 1.28302 .80786 1.23784 .83712 1.19457 4 57 .72521 1.37891 •75219 1.32946 .77988 1.28225 .80834 1.23710 .83761 1. 19387 3 58 ■72565 1.37807 ■75264 i^3286s ■7803s 1.28148 .80882 1.23637 .83811 1.19316 2 59 .72610 1.37722 ■75310 1.3278s .78082 1.28071 .80930 1.23563 .83860 1. 19246 1 6o •72654 1.37638 ■75355 1.32704 .78129 1.27994 .80978 1.23490 .83910 1.1917s / Cotang Tang Cotang Tang ( I^otang Tang Cotang Tang Cotang Tang 1 5^ t° 5: i° 52 >° 53 5< )° NATURAL TANGENTS AND COTANGENTS 1G9 / 40° 4 ° 42° 43° 44° / Tangf Cotang Tang Cotang Tang Cotang Tang Cotang ' rang Cotang o .83910 1.19175 .86929 1. 15037 .00040 1.11061 .93252 1.07237 96569 1-03553 60 I .83960 1.19105 .86980 1.14969 .90093 1. 1 0996 .93306 1.07174 9662s 1.03493 59 a .84009 1. 1 9035 .87031 1.14902 .90146 1.10931 .93360 1. 07112 96681 1.03433 S8 3 .84059 1. 1 8964 .87082 1. 14834 .90199 1.10867 .93415 1.07049 96738 1.03372 57 4 .84108 1. 1 8894 .87133 1.14767 .90251 1. 10802 ■93469 1.06987 96794 1. 033 1 2 56 5 .84158 1. 1 8824 .87184 1.14699 .90304 1. 10737 ■93524 1.0692s 96850 1.03252 55 fi .84208 1.18754 .87236 1.14632 .90357 1.10672 ■93578 1.06862 96907 1.03192 54 I .84258 1.18684 .87287 1.1456s .90410 1.10607 ■93633 1 .06800 96963 1.03132 S3 .84307 1.18614 .87338 1.14498 .90463 I.IOS43 .93688 1.06738 97020 1.03072 52 9 .84357 1. 18544 .87389 1.14430 .90516 1.10478 .93742 1.06676 97076 1.03012 SI 10 .84407 1.18474 .87441 1. 14363 .90569 1.10414 .93797 I. 06613 97133 1.02952 SO II ■84457 1. 18404 .87492 1.14296 .90621 1.10349 .93852 1.06551 97189 1.02892 49 12 .84507 1. 1 8334 .87543 1.14229 .90674 1.10285 .93906 1.06489 97246 1.02832 48 13 .84556 1. 1 8264 .87595 1.14162 .90727 1.10220 .93961 1.06427 97302 1.02772 47 14 .84606 1. 18194 .87646 1.14095 .90781 1.10156 .94016 1.06365 97359 I. 02713 46 IS .84656 1.1812s .87698 1.14028 .90834 1.10091 .94071 1.06303 97416 1.02653 45 l6 .84706 1.180SS .87749 1.13961 .90887 1.10027 .94125 I. 06 24 I 97472 1.02593 44 i7 .84756 1.17986 .87801 1.13894 .90940 1.09963 .94180 1.06179 97529 1.02533 43 i8 .84806 1.17916 .87852 1.13828 .90993 1.09899 .94235 1.06117 975S6 1.02474 42 J9 .84856 1.17846 .87904 1.13761 .91046 1.09834 .94290 1.06056 97643 1.02414 41 20 .84906 1.17777 ■87955 I. 13694 .91099 1.09770 .94345 I. 05994 97700 1.02355 40 ai .84956 1.17708 .88007 I. 13627 .91153 1.09706 .94400 1.05932 97756 1.0229s 39 23 .85006 1.17638 .88059 1.13561 .91206 1.09642 .94455 1.05870 97813 1.02236 38 as .85057 1-17569 .88110 1.13494 .91259 1.09578 .94510 1.05809 97870 1.02176 37 24 .85107 1.17500 .88162 1.13428 .91313 1.09514 .94565 I.OS747 97927 1.02117 36 2S .85157 1. 1 7430 .88214 1.13361 .91366 1.09450 .94620 1.0568s 97984 1.02057 35 26 .85207 1.17361 .88265 1.13295 .91419 1.09386 .94676 1.05624 98041 1. 01998 34 27 .85257 1.17292 .88317 1.13228 .91473 1.09322 .94731 1.05562 98098 1.01939 33 28 .85308 1.17223 .88369 1.13162 .91526 1.09258 .94786 l.OSSOI 98155 1.01879 32 29 .85358 1.17154 .88421 1.13096 .91580 1.09195 .94841 1.05439 98213 1.01820 31 30 .85408 1.1708s .88473 1. 13029 •91633 1.09131 .94896 1.05378 98270 1.01761 30 31 .85458 1.17016 .88524 1.12963 .91687 1.09067 .94952 1.05317 98327 1.01702 29 32 .85509 1.16947 .88576 1. 1 2897 .91740 1.09003 .95007 1.0525s 98384 1.01642 28 33 .85559 1.16878 .88628 1.12831 .91794 1.08940 .95062 1.OS194 98441 1.01583 27 34 .85609 1.16809 .88680 1.1276s .91847 1.08876 .95118 I.05133 98499 1.01524 26 35 .85660 1.16741 .88732 1. 1 2699 .91901 1. 088 1 3 .95173 1.05072 98556 1.0146s 25 36 .85710 1.16672 .88784 1.12633 .91955 1.08749 .95229 1. 05010 98613 1.01406 24 37 .85761 1.16603 .88836 1.12567 .92008 1.08686 .95284 1.04949 98671 1.01347 23 38 .85811 1.16535 .88888 1.12501 .92062 1.08622 .95340 1.04888 98728 1.01288 22 39 .85862 1.16466 .88940 1. 1 2435 .92116 1.08559 .95395 1.04827 98786 1. 01 229 21 40 .85912 I. 16398 .88992 1.12369 .92170 1.08496 .95451 1.04766 98843 1. 01 1 70 20 41 .85963 1.16329 .89045 1. 1 2303 .92224 1.08432 .95506 1.0470s 98901 1.01112 19 42 .86014 1.16261 .89097 1. 12238 .92277 1.08369 .95562 1.04644 98958 1-01053 18 43 .86064 1.16192 .89149 1. 12172 .92331 1.08306 .95618 1.04583 99016 1.00994 17 44 .86ns 1.16124 .89201 1.12106 .9238s 1.08243 .95673 1.04522 99073 1.0093s 16 45 .86166 I. 16056 .89253 I. 12041 .92439 1. 08 1 79 .95729 I.04461 99131 1.00876 15 46 .86216 1.15987 .89306 I.11975 .92493 1.08116 .95785 1.04401 99189 1.00818 14 47 .86267 1.15919 .89358 1.11909 .92547 1.08053 .95841 1.04340 99247 1.00759 13 48 .86318 1.158S1 .89410 1.11844 .92601 1.07990 .95897 1.04279 99304 I. 00701 12 49 .86368 1. 15783 .89463 1.11778 .92655 1.07927 .95952 1.04218 99362 1.00642 II SO .86419 1.15715 .89515 1.11713 .92709 1.07864 .96008 1.04158 99420 1.00583 10 SI .86470 1.15647 .89567 1.11648 .92763 1. 07801 .96064 1.04097 99478 1.00525 9 52 .86521 1.15579 .89620 1.11582 .92817 1.07738 .96120 1.04036 99536 1.00467 8 53 .86572 I.I5SII .89672 1.11517 .92872 1.07676 .96176 1.03976 99594 1.00408 7 54 .86623 1. 15443 .89725 1.11452 •.92926 1.07613 .96232 1.0391s 99652 i.otyso 6 55 .86674 1. 15375 .89777 1.11387 .92980 I.07S50 .96288 1.03855 99710 1.00291 5 56 .86725 1.15308 .89830 1.11321 .93034 I.074S7 .96344 1-03794 99768 1.00233 4 57 .86776 1.15240 .89883 1.11256 .93088 1.07425 .96400 1-03734 99826 1.0017s 3 58 .86827 1.15172 .89935 1.11191 .93143 1.07362 .96457 1-03674 99884 1.00116 2 59 .86878 1. 15104 .89988 1.11126 .93197 1.07299 .96513 I-03613 99942 1.00058 I 60 .86929 1.ISO37 .90040 1.11061 .93252 1.07237 .96569 1.03553 I 00000 I. 00000 1 Cotang Tang Cotang Tang Cotang Tang Cotang Tang C otang Tang 1 4( ?° 4^ ?° 4: 7° 4( 5° 4 5° 170 BROWN & SHARPE MFG. CO. NATUEAL SECAl^T. Deg. 0' 10' 20' 30' 43' 50 60' 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1.0001 89 1 1.0001 1 . 0002 1.0002 1.0003 1.0004 1.0005 1.0006 88 3 1.0006 1.0007 1 . 0008 1.0009 1.0010 1.0013 1.0013 87 3 1.0013 1.0015 1.0016 1.0018 1.0020 1.0033 1.0034 86 4 1.0024 1.0036 1.0028 1.0030 1.0083 1.0035 1.0038 85 5 1.0088 1.0040 1.0043 1.0046 1.0049 1.0053 1.0055 84 6 1.0055 1.0058 1.0081 1.0064 1.0068 1.0071 1.0075 83 7 1.0075 1.0078 1.0082 1.0086 1.0090 1.0094 1.0098 83 8 1.0098 1.0103 1.0108 1.0111 1.0115 1.0130 1.0124 81 9 1.0124 1.0139 1.0134 1.0139 1.0144 1 . 0149 1.0154 80 10 1.0154 1.0159 1.0164 1.0170 1.0175 1.0181 1.0187 79 11 1.0187 1.0193 1.0198 1.0204 1.0210 1.0317 J. 0333 76 13 1.0223 1.0329 1.0236 1.0242 1.0249 1.0856 1.0363 77 13 1.0263 1.0269 1.0277 1.0384 1.0291 1.021)8 1.0303 76 14 1.0308 1.0313 1.0321 1.0329 1.0336 1.0344 1.0353 75 15 1.0352 1.0360 1 . 0369 1.0377 1.0385 1.0394 1.0403 74 16 1.0403 1.0411 1.0420 1.0429 1.0438 1.0447 1.0456 73 17 1.0456 1.0466 1.0475 1.0485 1.0494 1.0504 1.0514 73 18 1.0514 1.0524 1.0534 1.0544 1.0555 1.0565 1.0576 71 19 1.0576 1.0586 1.0597 1.0608 1.0619 1.0630 1.0641 70 20 1.0641 1.0653 1.0664 1.0876 1.0887 1.0699 1.0711 69 31 1.0711 1.0723 1.0735 1.0747 1.0760 1.0773 1.0785 68 23 1.0785 1.0798 1.0810 1.0823 1.0837 1.0850 1.0863 67 23 1.0863 1.0877 1.0890 1.0904 1.0918 1.0933 1.0946 66 24 1.0946 1.0960 1.0974 1.0989 1.1004 1.1018 1.1033 65 35 1.1033 1.1048 1.1063 1.1079 1.1094 1.1110 1.1126 64 36 1.1126 1.1141 1.1157 1.1174 1.1190 1.1206 1.1233 63 27 1.1223 1.1239 1.1256 1.1273 1.1290 1.1308 1.1335 63 28 1.1325 1.1343 1.1361 1.1378 1.1396 1.1415 1.1433 61 29 1 . 1433 1.1452 1.1470 1.1489 1.1508 1.1537 1.1547 60 30 1.1547 1.1566 1.1586 1.1605 1.1625 1.1646 1.1666 59 31 1.1666 1.1686 1.1707 1.1738 1.1749 1.1770 1.1791 58 32 1.1791 1.1813 1.1835 1.1856 1.1878 1.1901 1.1923 57 33 1.1923 1.1946 1.1969 1.1993 1.2015 1.2038 1.3062 56 34 1.2082 1.2085 1.2109 1.3134 1.3158 1.2182 1.3307 55 35 1.2207 1.3333 1.2257 1.3383 1.3308 1.2334 1.3360 54 36 1.2360 1.2386 1.3413 1.3440 1.8466 1.2494 1.3531 53 37 1.2531 1.2548 1.3576 1.3804 1.2633 1.2661 1.3690 53 38 1.36i?0 1.2719 1.2748 1.3777 1.3807 1.3837 1.3887 51 39 1.2867 1.2898 1.3928 1.3959 1.2990 1.3033 1.3054 50 40 1.3054 1.3086 1.3118 1.3150 1.3183 1.3216 1.3350 49 41 1.3350 1.3283 1.3317 1.3351 1.3386 1.3421 1.3456 48 43 1.3456 1.3491 1.3527 1.3563 1.3599 1.3636 1.3673 47 43 1.3673 1.3710 1.3748 1.3785 1.3824 1.3863 1.3901 46 44 1.3901 1.3940 1.3980 1.4030 1.4060 1.4101 1.4143 45 GO' 50' 40' 30' .20' 10' 0- Deg. NATURAL COSECANT. TROVIDENCE, R. I. 171 NATUEAL SECANT. 1 Deg. 0' 10' 20' 30' 40' 50' GO' 45 1.4143 1.4183 1.4325 1.4267 1.4309 1.4352 1.4395 44 46 1.4395 1.4439 1.4483 1.4537 1.4573 1.4617 1.46G3 43 47 1.4663 1.4708 1.4755 1.4801 1.4849 1.4896 1.4944 43 48 1.4944 1.4993 1.5043 1.5091 1.5141 1.5191 1.5343 41 49 1.5343 1.5393 1.5345 1.5397 1.5450 1.5503 1.5557 40 50 1.5557 1.5611 1.5666 1.5731 1.5777 1.5833 1.5890 39 51 1.5890 1.5947 1.6005 1.6063 1.6122 1.6182 1.6343 38 52 1.6342 1.6303 1.6364 1.6436 1.6489 1.6552 1.6616 37 53 1.6616 1.6680 1.6745 1.6811 1.6878 1,6945 1.7013 36 54 1.7013 1.7081 1.7150 1.7220 1.7391 1.7362 1.7434 35 55 1.7434 1.7507 1.7580 1.7655 1.7780 1.7806 1.7883 34 56 1.7883 1.7960 1.8038 1.8118 1.8198 1.8278 1.8360 33 57 1.8360 1.8443 1.8537 1.8611 1.8697 1.8783 1.8870 33 58 1.8870 1.8.959 1.9048 1.9138 1.9330 1.9322 1.9416 31 59 1.9416 1.9510 1.9608 1.9702 1.9800 1.9899 3.0000 30 60 2.0000 2.0101 3.0303 2.0307 3.0413 2.0519 3.0836 89 61 2.0636 2.0735 3.0845 2.0957 3.1070 2.1184 3.1300 28 63 3.1300 2.1417 3.1536 3.1656 3.1778 3.1901 3.8026 37 63 3.3036 2.2153 3.3381 2.2411 3.3543 2.2676 3.3811 86 64 3.3811 3.3948 3.3087 2.3338 3.3370 3.3515 2.3668 25 65 3.3662 3.3810 3.3961 3.4114 2.4289 3.4436 8.4585 34 66 2.4585 3.4747 3.4911 3.5078 3.5347 2.5418 3.5593 23 67 2.5593 3.5769 3.5949 3.6131 3.6316 2.6503 3.6694 33 68 3.6694 3.6883 3.7085 3.7385 3.7488 2.7694 3.7904 31 69 3.7904 3.8117 3.8334 3.8554 3.8778 2.9006 3.9838 30 70 2.9338 3.9473 3.9713 3.9957 3.0305 3.04.58 3.0715 19 71 3.0715 3.0977 3.1343 3.1515 3.1791 3.3073 3.3360 18 73 3.2360 3.3653 3.3951 3.3355 3.3564 3.3880 3.4803 I 17 73 3.4303 3.4531 3.4867 3.5309 3 5558 3.5915 3.6379 ; 16 74 3.6379 3.6651 3.7031 3.7419 3.7816 3.8338 3.8637 15 75 3.8837 3.9061 3 9495 3.9939 4.0393 4.0859 4.1335 14 76 4.1335 4.1833 4.3333 4.3836 4.3363 4.3901 4.4454 13 77 4.4454 4.5031 4.5604 4.6303 4.6816 4.7448 4.8097 13 78 4.8097 4.8764 4.9451 5 0158 5.0886 5.1635 5.8408 li 79 5.8408 5.3304 5.4036 5.4874 5.5749 5.6653 5.7587 10 80 5.7587 5.8553 5.9553 6.0588 6.1660 6.3771 6.3934 9 81 6.3934 6.5120 0.6363 6.7654 6.8997 7.0396 7.1853 8 83 7.1853 7.3371 7.4957 7.6613 7.8344 8.0156 8.3055 7 83 8.3055 8.4046 8.6137 8.8336 9.0651 9.3091 9.5667 6 84 9.5667 9.8391 10.137 10.433 10.758 11.104 11.473 5 85 11.473 11.868 13.391 13.745 13.234 13.763 14.335 4 86 14.335 14.957 15.636 16.380 17.198 18.103 19.107 3 87 19.107 30.230 31.493 33.935 34.563 36.450 88 653 3 88 38.653 31.357 34.382 38.201 43 975 49.114 57.398 1 89 57.398 68.757 85.945 114.59 171.88 343.77 00 60' 50' 40' 30' 20' 10' 0' Deg. NATUEAL COSECANT. 172 BROWN & SHARPE MFG. CO. DECIMAL EQUIVALENTS OF PARTS OF AN INCH, eV ... .01563 sV - 03125 -i-^ ... .04688 i-i6 0625 ^\ ... .07813 3^ 09375 /^ ... .10938 1-8 125 -i^ ... .14063 3^ 15625 ^ ... .17188 3-i6 1875 if ... .20313 ^2 21875 If ... .23438 1-4 25 1| ... .26563 A 28125 if ... .29688 5-i6 3125 If ... .32813 ^ 34375 If ... .35938 3-8 375 2 5 ... ..39063 if 40625 If ... .42188 7-i6 4375 If ... .45313 -If 46875 If ... .48438 i-2 5 If ... .51563 ■II 53125 fl ... .54688 6 4 9-i6 5625 If ... .57813 if 59375 If ... .60938 5-8 625 If ... .64063 If ...... .65625 If ... .67188 11-16 6875 If ... .70313 If 71875 If ... .73438 3-4 75 If ... .76563 If 78125 If ... .79688 13 16 8125 If ... .82813 If 84375 If ... .85938 7-8 875 If ... .89063 If 90625 If ... .92188 15-16 9375 If ... .95313 If 96875 If ... .98438 1 1.00000 BROWN & SHARPE MFG. CO. 173 TABLE OF DECIMAL EQUIVALENTS MILLIMETRES AND FRACTIONS OF MILLIMETRES. mm. Inches. mm. Inches. mm. Inches. mm. Inches. jf5 = .00039 Wo = -01399 m = -03530 Wo = -03740 iio = .00079 fj = .01339 jlo = .03C59 fo = .03780 m = -00118 ^0 = -01373 ^ = .03598 fo = -03819 ■joo = .00157 tl = .01417 ^ = .03633 fo = .03853 Wo = -00107 Wo = -01457 Wo = -03677 -^ = .03898 ilo = -00330 ij = .01496 1 = -03717 1 = .03937 j^ = .0037a 1^ = .01535 70 „„_.„ 100 ~ -0-J'o6 3 = .07874 ilo = .00315 100 = -01575 m = -03795 3 = .11811 i5o = -00354 ^0 = -01614 Joo = .03835 4 = .15748 ^ = .00394 fo = .01654 fo = -03874 5 = .19685 ^ == .00433 fo = .01693 fo = -03913 6 = .23633 m = -00473 Wo = -01733 if, = .03953 7 = .37559 ^ = .00513 #0 = .01773 ^0 = -03993 8 = .31496 m = -00551 M = -01811 m = -03033 9 = .354^3 ij = .00591 ^ = .01850 ^0 = -03071 10 == .39370 ^ = .00630 ^ = .01890 f - -03110 11 = .4a307 ^ = .00009 m = -01939 ^; = .03150 13 = .47344 ^ = .00709 Wo = -01969 fo = -03189 13 = .51181 m = -00748 ^ = .03008 fo = -03338 14 = .55118 Z = -00787 M = -03047 fo = .03368 15 = .59055 if = .00837 ^ = -03087 fo = .03307 16 = .63993 ^ = .00866 H = .03L36 fo = .03346 17 = .66939 "jflQ ^ .00906 M = -03165 fo = .03386 18 = .70866 ^ = .00945 -g = .03305 fo = -03435 19 = .74803 H = .00984 fo = -03244 fo = .03465 30 = .78740 ^ = .01024 ^ = .03383 ■^0 = -03504 21 = .83677 ^ = .01063 fo = -03333 fo = .03543 22 = .86614 Oft ^0 = -01103 ^ = .03363 fo = .03583 33 = .90551 M = -01143 Wo = -03103 fo = .03633 34 = .94488 Wo = -01181 Wo = -03441 ff = .03661 35 = .98435 m = -01330 M = -03480 ^ = .03701 26 =1.03363 > fj = .01360 10 mm. = 1 Centimeter = 0.3f)37 inches. 10 cm. = 1 Decimeter = 3.937 inches. 10 Im. = 1 Meter = 39.37 inches. 25.4 mm. = 1 English Inch. INDEX A. PAGE Abbreviations of Parts of Teeth and Gears 4 Addendum 2 Angle, How to Lay Off an 88, 105 Angle Increment 104 Angle of Edge 100 Angle of Face 103 Angle of Pressure 135 Angle of Spiral Ill Angular Velocity 3 Annnlar Gears 32, 137 Arc of Action 136 B. Base Circle 11 Base of Epicycloidal System 25 Base of Internal Gears 137 Bevel Gear Blanks 34 Bevel Gear Cutting on B. & S. Automatic Gear Cutter 53 Bevel Gear Angles by Diagram 36 Bevel Gear Angles by Calculation 100, 104 Bevel Gear, Form of Teeth of 41 Bevel Gear, Whole Diameter of 36, 103 C. Centers, Line of 2 Chordal Thickness 142, 148 Circular Pitch, Linear or 4 Classification of Gearing 5 Clearance at Bottom of Space 6 Clearance in Pattern Gears 8 Condition of Constant Velocity Eatio 2 Contact, Arc of 136 Continued Fractions 130 Coppering Solution 85 Cutters, How to Order 83 Cutters, Table of Epicycloidal 84 176 INDEX. PAGE. Cutters, Table of Involute 8:^ Cutters, Table of Speeds for 81 Cuttiog Bevel Gears on B. & S. Automatic Gear Cutter 52 Cutting Spiral Gears in a Universal Milling Machine 120 D. Decimal Equivalents, Tables of 172 Diameter Increment 102 Diameter of Pitch Circle 6 Diameter Pitch 5 Diametral Pitch 17 Distance between Centers 8 E. Elements of Gear Teeth 5 Epicycloidal Gears, with, more and less than 15 Teeth 30 Epicycloidal Gears, with 15 Teeth 25 Epicycloidal Rack 27 F. Face, Width of Spur Gear 80 Flanks of Teeth in Low-numbered Pinions 20 G. Gear Cutters, How to Order 83 Gear Patterns 8 Gearing Classified 5 Gears, Bevel 34, 41, 100 Gears, Epicycloidal 25 Gears, Involute 9 Gears, Spiral ..107, 120 Gears, Worm 63 H. Herring-bone Gears 128 I. Increment, Angle 104 Increment, Diameter 102 Interchangeable Gears 24 Internal or Annular Gears 32, 137 Involute Gears, 30 Teeth and over 9 Involute Gears, with Less than 30 Teeth 20 Involute Rack ,„ „. 12 INDEX. 177 L PAOE. Lead of a Worm 62 Limiting Numbers of Teeth in Internal Gears 32 Line of Centers 2 Line of Pressure 12, 135 Linear or Circular Pitch 4 Linear Velocity 1 M. Machine, B. & S., for Cutting Bevel Gears 52 Module 6 N. Normal 114 Normal Helix 114 Normal Pitch 114 0. Original Cylinders 1 P. Pattern Gears .,.... = 8 Pitch Circle 3 Pitch, Circular or Linear 4 Pitch, a Diameter 6 Pitch, Diametral 17 Pitch, Normal 114 Pitch of Spirals 110 Polygons, Calculations for Diameters of 95 B, Pack 12 Back for Epicycloidal Gears 27 Back for Involute Gears 12 Back for Spiral Gears 119 Belative Angular Velocity 2 Boiling Contact of Pitch Circle 3 S. Screw Gearing 107, 128 Single-Curve Teeth 9 Speed of Gear Cutters 81 178 INDEX. PAGE. Spiral Gearing 107, 120 Standard Templets 27 Strength of Gears 140 T. Table of Decimal Equivalents , 172 Table of Sines, etc . 150-171 Table of Speeds for Gear Cutters 81 Table of Tooth Parts 146-149 V. Velocity, Angular 2 Velocity, Linear 1 Velocity, Kelative 2* Wear of Teeth.. ..., ., 80, 127 Worm Gears, ..,0... , .,....o..,. 63 LE N '09 ■.^*r Gearing PROVIDENCE. R. L, U» S. A