teatise earing lARPE lENCl Practical Treatise on Gearing TWELFTH EDITION Brown & Sharpe Mfg. Co, Providence, R. I., U. S. A. 1920 25C-11-19 /<^-' /^l''-'^ i ^ :^,^'^ Copyright 1902, 1905, 1911, 1920 BY Brown & Sharpe Mfg. Co. m 24 iS20 g)C!.A559931 O-^-' PREFACE This book is made for men in practical life and deals with questions of Gear Cutting and the construction of Gear Wheels in a way to meet the needs of such men even when they may not have the time to acquire a technical knowledge of the subject. It is also specially adapted to the use of the student who desires to learn methods of approximating the theoretical forms of gear teeth. CONTENTS Chapter I Pitch Circle — Pitch — Tooth — Space — Addendum or Face — Flank — Clearance 9 Chapter II Classification — Sizing Blanks and Tooth Parts from Circular Pitch — Centre Distance — Pattern Gear 13 Chapter III Single-Curve Gears of 30 Teeth and more 17 Chapter IV Rack to Mesh with Single-Curve Gears having 30 Teeth and more 20 Chapter V Diametral Pitch — Sizing Blanks and Teeth of Spur Gears — Distance between the Centres of Wheels 24 Chapter VI A Single-Curve Gear having 12 Teeth and an Engag- ing Rack showing Interference — A Gear having 12 Teeth and an Engaging Rack without Inter- ference—Interchangeable Gears 29 Chapter VII Double-Curve Teeth — Gear of 15 Teeth — Rack 36 Chapter VIII Double-Curve Spur Gears, having More or Less than 15 Teeth — Annular Gears 41 Chapter IX Internal Gears 45 Chapter X Bevel Gear Blanks 48 Chapter XI Bevel Gears — Forms and Sizes of Teeth — Cutting Teeth 55 Chapter XII Curved Tooth or Spiral Bevel Gears 79 Chapter XIII Worm Gearing — Sizing Blanks of 32 Teeth and More 83 5 Chapter XIV. Sizing Gears When the Distance Between the Centres and the Ratios of Speeds are Fixed — General Remarks — Width of Face of Spur Gears — Speed of Gear Cutters 101 Chapter XV Spiral Gears — Calculations for Lead of Spirals 108 Chapter XVI Examples in Calculations of Lead of Spirals— Angle of Spiral — Circumference of Spiral Gears — A Few Hints on Cutting 114 Chapter XVII Normal Pitch of Spiral Gears — Curvature of Pitch Surface — Formation of Cutters 117 Chapter XVIII Cutting Spiral Gears in a Universal Milling Machine . . 124 Chapter XIX Spiral and Worm Gears — General Remarks 131 Chapter XX Strength of Gears 135 Chapter XXI Standard Proportions for Spur Gears 143 Chapter XXII Tangent of Arc and Angle 145 Chapter XXIII Sine and Cosine — Some of Their Applications in Machine Construction 151 Chapter XXIV Application of Circular Functions^ Whole Diameter of Bevel Gear Blanks — Angles of Bevel Gear Blanks 157 Chapter XXV Angle of Pressure 164 Chapter XXVI Continued Fractions — Some Applications in Machine Construction 166 Chapter XXVII Squares and Square Roots 172 Tables 174 Index 207 6 CHAPTER I Pitch Circle — Pitch — Tooth — Space — Addendum or Face — Flank — Clearance Let two cylinders, Fig. 1, touch each other, their 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 evidently move through the same distance in a given time. This surface speed is called linear velocity. Original Cyl- inders. Fig. 1 TANGENT CYLINDERS Linear Velocity is the distance a point moves along a line in a unit of time. The line 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. Linear Veloc- BROWN & 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 the angular unit, and 1 minute as the time unit, the loctty^'^^^'^ ^^" 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 Relative An- othcr makes. If one cylinder is twice as large as the guiar Velocity, q^j^^j.^ ^j^g Smaller will make two turns while 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. C/RCLE Fig. 3 Land. In the periphery of these two cylinders, as in Fig. 2, cut equidistant grooves. In any grooved piece the places between grooves are called lands. Upon the 10 BROWN & SHARPE MFG. CO. Addendum. lands add parts; these parts are called addenda. A land and its addendum is called a tooth. A toothed "^""^^ cylinder is called a gear. Two or more gears with teeth interlocking are called a train. A line c c', Fig. 2 or 3, between the centres of two wheels is called the line of ^vt^^^ ""^ ^^''' centres. A circle just touching the addenda is called the addendum circle. Gear. Train. Addendum Circle. Fig. 4 11 BROWN & SHARPE MFG. CO. Pitch Circle. Space. Linear or Cir- cular Pitch. Tooth Thick- ^ Abb re via- tions o f Parts for Teeth and Gears. To find the Circumference and Diameter of a Circle. The circumference of the cyHnders without teeth is called the pitch circle and exists geometrically in every gear. In the study of gear wheels, the problem is to so shape the teeth that the pitch circles will just touch each other without slipping. 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 centre of one tooth and the centre of the next tooth, measured along the pitch line, is the linear or circular pitch; that is, the linear or circular pitch is equal to a tooth and a space; hence, the thickness of a tooth at the pitch line is equal to one-half the linear or circular pitch. Let D = diameter of addendum circle. ** D' = diameter of pitch circle. '' P' = linear or circular pitch. " if = thickness of tooth at pitch line. ** 5 = addendum or face, also length of working part of tooth below pitch line or flank. *' 2s = T>'\ the working depth of tooth, or twice the addendum. ** /= 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 gear. '' 7r = 3.1416 or the circumference when diameter isl. P' is read 'T prime." D" is read ''D second." n is read ''pi." If we multiply the diameter of any circle by 7ty the product will be the circumference of this circle. If we divide the circumference of any circle by n, the quo- tient will be the diameter of this circle. 12 BROWN & SHARPE MFG. CO. 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 made Elements of the smallest possible, we ultimately come to the con- ^^^ Teeth. ception of teeth that are merely lines upon the original pitch surfaces. These lines are called elements 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). This book treats upon four kinds of gears. First — Spur Gears; those connecting parallel shafts spur Gears. and whose tooth elements are straight. Second — Bevel Gears; those connecting shafts whose Bevei Gears, 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 — Worm Gears; those connecting shafts that are worm Gears. not parallel and do not meet, and the elements of whose teeth are screw-like. Fourth — Spiral Gears; those connecting shafts that are either parallel or at an angle, but which do not meet, and the elements of whose teeth are helical. The circular pitch and number of teeth in a wheel being given, the diameter of the wheel and size of tooth BifS^Etc. parts are found as follows: Dividing by 3.1416 is the same as multiplying by 3^^. Now 3.1416 = .3183; hence, multiply the circum- ference of a circle by .3183 and the product will be the diameter of the circle. Multiply the circular pitch by .3183 and the product will be the same part of the 13 BROWN & SHARPE MFG. CO. A Diameter Pitch, or Mod- ule. The Module and the Adden- d u m measure the same, radi- ally. Diameter of Pitch Circle. Whole Diam- eter. Clearance. Example. Sizes of Blank and Tooth Parts for Gear of 30 teeth 13^ in. Circular Pitch. diameter of pitch circle that the circular pitch is of the circumference of pitch circle. This part is called the module of the pitch. There are as many m.odules con- tained in the diameter of a pitch circle as there are teeth in the wheel. Most mechanics make the addendum of teeth equal the module. Hence we can designate the module by the same letter as we do the addendum; that is, let s= the module. .3183 P'=s, or circular pitch multiplied by .3183=5, the addendum or the module. N5=D', or number of teeth in a wheel, multiplied by the module, equals diameter of pitch circle. (NH-2) s=D, or add 2 to the number of teeth, mul- tiply the sum by the addendum and the product will be the whole diameter. -Yq =/, or one-tenth of thickness of tooth at pitch line equals amount added to bottom of space for clearance. One-tenth of the thickness of tooth at pitch line is more than one-sixteenth of working depth, being .07854 Example— Whed 30 teeth, 1>^'' circular pitch. P'= 1.500''; then /=.750" or thickness of tooth equals y^". 5=1.500" X.3183=.4775=module for l^^''?'. (See Table of Tooth Parts, pages 178-181.) D'=30x.4775"=14.325"=diameter of pitch circle. D = (30+2)X.4775''=15.280"=diameter of addendum circle, or the diameter of the blank. /z=^ of .7500"=.0750''=clearance at bottom of space. D''=2X. 4775"=. 9549''= working depth of teeth. D"+/=2x.4775"+.0750"=1.0299"=whole depth of space. 5+/=.4775"+.0750"=.5525"=depth of space inside of pitch line. D"=25 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 14 BROWN & SHARPE MFG. CO. into the workshop, three places of decimals are usually sufficient. The distance between the centres of two wheels is 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 5 multiplied by one-half the number of teeth in the wheel. Distance be- tween centres of two Gears. Fig. 5. SPUR GEARING Hence, if we know the number of teeth in two wheels, in mesh, and the circular pitch, to obtain the distance between centres 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 centres. Example — What is the distance between the centres of two wheels 35 and 60 teeth, l}i" circular pitch? We 15 BROWN & SHARPE MFG. CO. first find s to be \yi" X .3183 = .3979'^ Multiplying by 47.5 (one-half the sum of 35 and 60 teeth) we obtain 18.9003'' as the distance between centres. shVinTagViS Pattern Gears should be made large enough to allow Gear Castings. ^^^ shrinkage in casting. In cast iron the shrinkage is about 1 inch in one foot. For gears one to two feet in diameter it is well enough to add simply -z^ of diameter of finished gear to the pattern. In gears about six inches diameter or less, the moulder will gener- ally 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 enough to make ternatfi.^^*' 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 abrasive 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 necessary and the blanks may be finished before they are cut. 16 BROWN & SHARPE MFG. CO. CHAPTER III Single-Curve Gears of 30 Teeth and More Single-curve teeth are so called because they have -peet? ^^"^"'^'^ 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 thirty 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 thirty 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 calculated the data of a gear of 30 teeth, Qg^r^Nfto'V^ yi inch circular pitch (as we did in Chapter II for Xyi" =^"- pitch), we proceed as follows: 1. Draw pitch circle and point it off into parts equal to one-half the circular pitch. 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 semicircle being equal to radius of pitch circle. Draw addendum, working depth and whole depth circles. 3. From the point B, where semicircle, pitch circle 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 as the chord BA. 4. Through this new point at A, upon the semicircle, draw a circle concentric to pitch circle. This last is called the base circle, and is the one for centres of tooth Base circle arcs. In the system of single-curve gears we have adopted, the diameter of this circle is .9682" of the diameter of 17 Method of laying out. BROWN & SHARPE MFG. CO. GEAR, 30 TEETH, K" CIRCULAR PITCH. P = %" or .75" N = 30 t--- .375" S:^ .2387" D"= .4775" 5+/= .2762" D"+/= .5150" i) '1=7.1618" 1D = 7.6392" Fig. 6 SINGLE -CURVE GEAR 18 BROWN & SHARPE MFG. CO. pitch circle. Thus the base circle of any gear 1 inch pitch diameter by this system is .9682''. If the pitch circle is 2" the base circle will be 1.9364''. 5. With dividers set to one-quarter of the radius of pitch circle, draw 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 centre through B. 6. With dividers set to one-seventh of the widest part of tooth space, draw 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 that invoiite^oea?- can run at varying distance of axes and transmit unvary- ^^s- ing angular velocity. This feature makes involute gears specially valuable for driving rolls or any rotating pieces, the distance of whose axes is likely to be changed. The assertion that gears crowd harder on bearings ^eSin^r^ °" when of involute than when of other forms of teeth, has not been proved in actual practice. Before taking the next chapter, the learner should make la^S'^ouueeth! several drawings of gears 30 teeth and more. Say make 35 and 70 teeth Yyi" P'. Then make 40 and 65 teeth %" P'. 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. 19 BROWN & SHARPE MFG. CO. CHAPTER IV Rack to Mesh with Single-Curve Gears Having 30 Teeth and More of a 'iaik. ^ "" This gear (Fig. 7) is made precisely the same as gear in Chapter III. 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 cor- respond 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 form 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. a Ra^ck!' '^"^ ° A Rack is a straight piece, having 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 of pit??Line °Jf circumference is a straight line. The pitch line of a ^^''^' rack, then, is merely a straight line just touching the pitch circle of a gear meshing with the rack. The thick- ness of teeth, addendum and depth of teeth below pitch line are calculated the same as for a wheel. (For pitches in common use, see Tables of Tooth Parts pages 178-181.) The term circular pitch when applied to racks can be more accurately replaced by the term linear pitch. Linear applies strictly to a line 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. 20 BROWN & SHARPE MFG. CO. 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 adden- dr^fng^Rack! dum line, working depth line and whole depth line, each parallel to the pitch line (see Fig. 7). Fig. 7 RACK TO MESH WITH SINGLE-CURVE GEAR HAVING 30 TEETH AND MORE 21 Angle _f o r sides of Teeth. BROWN & SHARPE MFG. CO. 2. Point off the pitch hne 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 t of the width of spaces between the rack teeth at the adden- dum line. Rack The sketch, Fig. 8, will show how to approximately obtain angle of sides of rack teeth, directly from pitch line of rack, without drawing a gear in mesh with the rack. Upon the pitch line hb', draw any semicircle — ba a'b'. From point b lay off upon the semicircle the distance ba^ equal to one-quarter of the diameter of semicircle, and draw a straight line through b and a. This line, ba, makes an angle of approximately 75^"^ with pitch line bb', 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 planing out spaces between rack teeth. Upon any line OB draw 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 approximately 29°, and are right for inclination of sides of rack tool. 22 BROWN & SHARPE MFG. CO. Make end of rack tool .3095 of circular pitch, and then round the corners of the tool to leave fillets at the bottom of rack teeth. Thus, if the circular pitch of a rack is lj4" and we multiply by .3095, the product .4642 will be the width of tool at end for rack of this pitch before corners are taken off. This width is shown at xy. B A Width of Rack Tool at end. ■ Fig. 9 A Worm is a screw that meshes with the teeth of a gear. This sketch and the foregoing rule are also right for a worm thread tool, but a worm thread tool is not usually rounded for fillet. In cutting worms, leave width of top of thread .3354 of the circular pitch. When this is done, the depth of thread will be right. Worm Thread Tool. 3354 P' SKETCH OF WORM THREAD 23 BROWN & SHARPE MFG. CO. CHAPTER V Diametral Pitch — Sizing Blanks and the Teeth of Spur Gears — Distance Between the Centres of Wheels ^^^clJt+^ In making drawings of gears, and in cutting racks, cuiaT Pitch ^''^' ^^ ^^ necessary to know the circular pitch, both on account of spacing teeth and calculating their strength. It would be more convenient to express the circular pitch in whole inches, and the most natural divisions of an inch, as VT\ %'T', }4'T' and so on. But as the ^'*f^ oJPi""! circumference of the pitch circle must contain the circular cumference ^ ciJcufa?''pitih' pitch some whole number of times, corresponding to the oSes''''"'^^'^ number of teeth in the gear, the 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 circumference multiplied by .3183. Terms^'^'f the ^^ practlcc, it Is better that the diameter should be Diameter. of somc slzc couvcnicntly measured. The same applies to the distance between centres. 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. and'a%1ImS ^^ ^^^ ^^^^ assumc thc module and pass to its equiv- 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 yi" \^ the module of this gear. In practice, the module is taken in some convenient part of an inch, as }4" module and so on. 24 BROWN & SHARPE MFG. CO. It is convenient in calculation to designate one of these ModSe°Diam- modules by s, as in Chapter II. Thus, for >^" module, ''''^^''^^■ s is equal to >^". Generally, in speaking of the module, the denominator of the fraction only is named. A module of \" is then called 2 diametral pitch. That is, it has been found more convenient to take the reciprocal of the module in making calculation. The reciprocal of a aN3er^ ""^ number is 1 divided by that number. Thus the reciprocal of ^ is 4, because X goes into 1 four times. Hence, we come to the common definition: Diametral Pitch is the number of teeth to one inch _ symboi f o r Diametral of diameter of pitch circle. Let this be denoted by P. Pitch. Thus, }i" diameter pitch we would call 4 diametral pitch or 4P, 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 Given, thepi- ^ ./ J. ametral to find is equal to the circular pitch. Thus to obtain the cir- the circular Pitch. cular pitch for 4 diametral pitch, we divide 3.1416 by 4 _ , . ^. •^ ^ "^ To obtain Cir- and obtam .7854 for the circular pitch, corresponding ? u i a r pitch ■^ r- o from D 1 a m e - to 4 diametral pitch. * trai pitch. In this case we would write P = 4, P' = .7854", s^yi". ^" = 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 equal to the module. -^ = P, or one inch divided by the module equals num- ber of teeth to one inch or the diametral pitch. ^ = t, or 1.5708 divided by the diametral pitch gives Given pi- thickness of tooth at pitch line. Thus, thickness of fjnd Thickness ■^ ot 1 o o t n at teeth along the pitch line for 4 diametral pitch is .3927". Pitch Line. f = D', or number of teeth in a gear divided by the be?'Jf^TeShTn diametral pitch equals diameter of the pitch circle, ametrafpuch^t^ Thus for a wheel, 60 teeth, 12 P, the diameter of pitch ofiitc^'arde' circle will be 5 inches. 25 BROWN & SHARPE MFG. CO. Given, Num- N+2 _-^ ii<-» i -i r ^ • ber of Teeth in ^r" = D, or add 2 to the number or teeth m a wheel wheel and Dia- ii-'ii ^• ■> • i metrai Pitch, to and divide the sum by the diametral pitch; and the find Whole Di- . „, , , , , ,. . , ameter. Quotient Will be the whole diameter of the gear or the diameter of the addendum circle. Thus, for 60 teeth, 12P, the diameter of gear blank will be SA inches. ^, = 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. -D~ = 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 3i\'' diameter, 14 teeth, the diametral pitch is 5. DT = 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. DP — 2 = 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. N+2 = 5, or divide the whole diameter of a spur gear by the number of teeth plus two, and the quotient will be the addendum or module. The Diame- Whcu wc Say the diametral pitch we shall mean the number of teeth to one inch of diameter of pitch cir- cle, or P, (-^''=P). anStraj'^pu^h Whcu thc clrcular pitch is given, to find the corre- sponding diametral pitch, divide 3.1416 by the circular pitch. Thus 1.5708P is the diametral pitch correspond- ing to 2-inch circular pitch, C-^ = P) • Example. What diametral pitch corresponds to }4" circular pitch? Remembering that to divide by a fraction we multiply by the denominator and divide by the numer- ator, we obtain 6.2832 as the quotient of 3.1416 divided by }4. 6.2832P, then, is the diametral pitch correspond- ing to yo circular pitch. This means that in a gear of }4 inch circular pitch there are six and two hundred and eighty-three one-thousandths teeth to every inch in the 26 from Circular Pitch BROWN & SHARPE MFG. CO. diameter of the pitch circle. In the table of tooth parts and for calculations, the diametral pitches corresponding to circular pitches are carried out to four places of decimals, but for use in the shop three places of decimals are usually enough. When two gears are in mesh, so that their pitch circles just touch, the distance between their axes or centres is equal to the sum of the radii of the two gears. The number of the modules between centres is equal to half the sum of the number of teeth in both gears. This principle is the same as given in Chapter II, page 13, but when the diametral pitch and numbers of teeth in two gears are given, add together the numbers of teeth in the two wheels and divide half the sum by the diametral pitch. The quotient is the centre distance. A gear of 20 teeth, 4P, meshes with a gear of 50 teeth; what is the distance between their axes or centres? Add- ing 50 to 20 and dividing half the sum by 4, we obtain ^}i" as the centre distance. The term diametral pitch is also applied to a rack. Thus, a rack 3P, means a rack that will mesh with a gear of 3 diametral pitch. It will be seen that if the expression for the module has any number except 1 for a numerator, we cannot express the diametral pitch by naming the denominator only. Thus, if the addendum or module is A" , the diametral pitch will be 2yi, because 1 divided by A equals 2}i. The term module is much used where gears are made to metric sizes, for the reason that, the millimetre being so short, the module is conveniently expressed in milli- metres. 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. Rule to find Distance be- tween Centres. Example. Fraction; Diametral Pitch. 27 BROWN & SHARPE MFG. CO. No. 5 AUTOMATIC GEAR CUTTING MACHINE Cuts spur gears to 60'' in diameter, W face. Cast iron, 2 diametral pitch; steel, 3 diametral pitch. This machine is representative of our line of Spur Gear Cutting Machines. 28 BROWN & SHARPE MFG. CO. CHAPTER VI A Single-Curve Gear Having 12 Teeth and an Engaging Rack Showing Interference — A Gear Having 12 Teeth and an Engaging Rack Without Inter- ference — Interchangeable Gears It has been customary to cut rack teeth with a cutter , construction ■I 1 1 r^r- AH i ' -, ^ lor Set of gears. shaped to cut a 135- tooth gear. All gears having 12 teeth or more shaped according to the data in Chapter III, interchange fairly well with one another and with such a rack, when the pitch is not coarser than ten to the inch diametral (lOP), but in coarser pitches there is an objectionable interference as indicated in Fig. 10. 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 the base circle out to the adden- dum circle, by the same method as the gear in Chapter III, but the spaces inside of the base circle are drawn as follows: In a gear of 12 teeth, the sides of the spaces inside of ^ f i a n k s of 11-1 1- 1 r 1- Gears in low the base circle are radial for a distance, ab, equal to Numbers of or leeth. 5fp or 3.5 divided by the product of the pitch by the number of teeth. With one leg of the dividers in the pitch circle in the , Method of drawing. centre of the next tooth, e, and the other leg just touching one of the radial lines at b, continue the tooth side into c, until it will touch a fillet arc, whose radius is one- seventh the width of the space at the addendum circle. The part b'c\ is an arc from the centre of the tooth g, etc. The flanks of the teeth or spaces in the 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 designer, to produce spaces as wide as practicable, just below 29 BROWN & SHARPE MFG. CO. BROWN & SHARPE MFG. CO. or inside of the base circle, and then strengthen the flank with as large a fillet as will clear the addenda of any gear. If the flanks in any gear will clear the addenda of a rack, they will clear the addenda of any other gear except internal gears. An internal gear is one internal having teeth upon the inner side of a rim or ring. See ^^'''■• Chapter IX. Now, it will be seen that the gear, Fig. 10, has teeth too much rounded at the points or at the adden- dum circle. In gears of pitch coarser than 10 to the inch (lOP), and having fewer than 30 teeth, this rounding A?de"nTa CS becomes objectionable. This rounding occurs, because '^''''*^- in these gears arcs of circles depart too far from the true involute curve : — it is so much that the points of the teeth get no bearing on the flanks of teeth in mating wheels. In the gear. Fig. 11, the teeth outside the base circle are made as nearly true involute as a workman can get tion Sf'^x rTe without special machinery. This is accomplished as follows: draw three or four tangents to the base circle, ii\ jj', kk', W, letting the points of tangency on base circle, i', j', k', V, be about one-third or one-quarter the circular pitch apart; the first point /', being distant from /, equal to one-quarter the radius of the pitch circle. With the dividers set to one-quarter the radius of the pitch circle, placing one leg in i', draw the arc a'ij; with one leg in j', and radius j'j, draw jk; with one leg in k' and radius k'k, draw kl. Should the addendum circle be outside of /, the tooth side can be completed with the last radius, I'l. The arcs, a'ij, jk and kU together form a very close approximation to a true involute from the base circle, i'j'k'V. 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 the base circle. The foregoing operation of drawing the tooth sides, although tedious in description, is very easy of practical application. It will also be seen that the addenda of the rack teeth Rounding of in Fig. 10, interfere with the gear- teeth flanks, as at Rack.^" 31 BROWN & SHARPE MFG. CO. BROWN & SHARPE MFG. CO. BROWN & SHARPE MFG. CO. Templets necessary for Rounding Points Teeth. off o f D iagrams for Cutters. set of m, n; to avoid this interference, the teeth of the rack, Fig. 11, are rounded at their points or addenda. It is also necessary to round off the points of the invo- lute teeth in all gears, when they are to interchange with low numbered gears. In interchangeable sets of gears the lowest numbered pinion is usually 12. Just how much to round off can be learned by making templets of a few teeth out of thin metal or cardboard, for the gear and rack, or for the two gears required, and fitting the addenda of the teeth to clear the flanks. However accurate we may make a diagram, it is quite as well to make templets in order to shape cutters accurately. Fig. IIA shows a pinion whose tooth faces have been corrected as in the foregoing. A rack engaging with this pinion is also shown. Ordinarily, in interchangeable sets it is best to make cutters to corrected diagrams, as in Fig. 11 A. When corrected diagrams are made, as in Fig. 11 A, take the following: For 135 to rack, diagram of 135 teeth. " 55 " 134 teeth, " " 55 '' 35 " 54 " " " 35 " 26 " 34 " " " 26 " 21 " 25 " " " 21 .. 17 - 20 " " " 17 - 14 " 16 " " ^L 14 '' 12 and 13 " " " 12 If greater accuracy is desired cutters can also be made for half numbers, in which case it is recommended that they be made as follows: For 80 to 134 teeth, diagram of 80 teeth. 42 30 25 19 15 13 " 42 " 54 '' '' 30 " 34 '' '' 23 " 25 " .. 19 " 20 " '' 15 " 16 " '' 13 34 BROWN & SHARPE MFG. CO. By making a cutter right for the lowest number of teeth for which it is to be used, the other teeth cut by this cutter will be more rounded off at the outer parts of the tooth faces. This rounding off is to facilitate easy running, the avoidance of interference and perhaps of noise. A SPUR GEAR TESTING MACHINE, WITH SPUR GEARS IN POSITION TO BE TESTED FOR CENTRE DISTANCE AND CONCENTRICITY OF THE TEETH 35 BROWN & SHARPE MFG. CO. CHAPTER VII Double-Curve Teeth — Gear of 15 Teeth — Rack Nature of III double-curve or epicycloidal teeth the formation of tooth sides changes at the pitch line. The outHne of the faces of the teeth may be traced by a point in a circle, rolling on the outside of the pitch circle of a gear, and the flanks by a point in a circle rolling on the inside of the pitch circle. In all gears the part of 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 interchangeable 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 Base of this system. Construction Wc wlll first draw a gear of fifteen teeth. This fifteen- teeth.'' ^ "'^'' tooth construction enters into gears of any number of teeth and also into racks. Let the gear be 3P. Having obtained data, we proceed as follows: 1. Draw pitch circle and point it off into parts equal to one-thirtieth of the circumference, or equal to thick- ness of tooth = t. 2. From the centre, 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 AK and O^. For any other double-curve gear of 3P, the radius of arcs, AK and Ok, will be the same as in this 15-tooth gear = 2>^". In a 15-tooth gear, the arc. Ok, passes through the centre O, but for a gear having any other number of teeth, this construction arc does not pass through centre of gear. 36 BROWN & SHARPE MFG. CO. GEAR, 3 P., 15 TEETH P= 3 N = 15 P'= 1.0472" t= .5236" .3333" D"= .(5666" .<+/= .3S57" D"+/= .7190" D'= 5.0000" D — 5.C666" \ Fig. 12 DOUBLE-CURVE GEAR 37 BROWN & SHARPE MFG. CO. Of course, the 15-tooth radius of arcs, AK and Ok, is always taken from the pitch we are working with. 4. Upon these arcs on opposite sides of Hne OTA, lay off tooth thickness, AK and Ok, and draw line KTk. 5. Perpendicular to KT^, draw line of pressure, LTP; also through O and A, draw lines AR and Or, perpendicular to KT^. The line of pressure is at an angle of 78° with the radius of gear. 6. From O, draw a line OR to intersection of AR with KT^. Through point c, where OR intersects LP, describe a circle about the centre, O. In this circle one leg of dividers is placed to describe tooth faces. 7. The radius, cd, of arc of tooth faces is the straight distance from c to tooth-thickness point, b, on the other side of radius, OT. With this radius, cb, describe both sides of tooth faces. 8. Draw flanks of all teeth radial, as Oe and Of. The base gear, 15 teeth only, has radial flanks. 9. With radius equal to one-seventh of the widest part of space, as gh, draw fillets at bottom of teeth. Approxima- Thc forcgolug Is a close approximation to epicycloidal ciddaiTeeth!'^' tccth. To gct cxact teeth, make two 15-tooth gears of thin metal. Made addenda long enough to come to a point, as at n and q. Make radial flanks, as at m and p, 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. standard Thcsc two tcmplct gears are exact, when the centres Templets. ^^^ ^^^ ^.-^j^^ distaucc apart and the teeth interlock without backlash. One of these templet 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-curve Rack. Let us draw a rack 3P. Hav- ing 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. 38 Double-curve Rack. BROWN & SHARPE MFG. CO. Fig. 13 DOUBLE-CURVE RACK 39 BROWN & SHARPE MFG. CO. 2. Through one of the points, as at T, Fig. 13, draw Hne OTA perpendicular to pitch Hne of rack. 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 centre T draw arcs Ok and AK; make Ok and AK equal to tooth thickness; draw KT^; draw Or, AR, and line of pressure, each perpendicu- lar to KT^. 4. Through R and r, draw lines parallel to OA. Through intersections c and c' of these lines, with pressure line LP, 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 cd of rack- tooth flanks, and is the straight distance from c to tooth-thickness point b on opposite side of line OA. 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. Fig. 14 Racks of 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. 40 BROWN & SHARPE MFG. CO. CHAPTER VIII Double-Curve Spur Gears, Having More or Less than 15 Teeth — Annular Gears Let us draw two gears, 12 and 24 teeth, 4P, in mesh, ^f s^t Sf douS In Fig. 15 the construction hnes of the lower or 24-tooth curve Gears. gear are full. The upper or 12- tooth gear construction lines are dotted. The line of pressure, LP, and the line KT^ answer for both gears. The arcs AK and Ok 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 AK and O^ is the tooth thickness for 4P. The line KT^ 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 KT^, and, perpendicular to KT^, the lines AR, Or and the line of pressure LTP, we proceed with the 24-tooth gear as follows: 1. From centre C, through r, draw line intersecting line of pressure in m. Also draw line from centre C to R, crossing the line of pressure LP at c. 2. Through m describe a 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, mn, of flanks is the straight distance from m to the first tooth-thickness point on other side of line of centres, CC, 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 draw tooth faces, the radius of tooth faces being cb. 41 BROWN & SHARPE MFG. CO. PINION, 12 TEETH, GEAR 24. TEETH, -4. P P=4 N=12and24 P'= .7854" t = .3927" S = .2500" D"= .5000" It/ = .2893" D"+/=.5393" D = 6.500 Fig. 15 DOUBLE-CURVE GEARS IN MESH 42 BROWN & SHARPE MFG. CO. 5. The radius of fillets at roots of teeth is equal to one-seventh the width of space at addendum circle. The constructions for flanks of 12, 13 and 14 teeth are Fianksfori2. 13 and 14 Teeth. similar to each other and as follows : 1. Through centre, C, draw line from R, 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 same side of CTC. 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 centre, 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 foregoing, the radius being wy] the arc is continued to x, to show construction. Annular Gears. Gears with teeth inside of a rim Annular Gears. 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 described, the pinion meshing with an annular gear, must differ 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 Chapter IX. Annular gear patterns require more clearance for moulding than external or spur gears. In speaking of different sized gears, the smaller of a Pinions. pair is often called a pinion. 43 BROWN & SHARPE MFG. CO. 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 only; in our example, it is 78° when one side of a tooth reaches the line of centres, 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 on pages 164-165. 44 BROWN & SHARPE MFG. CO. CHAPTER IX Internal Gears Special Cut- in Chapter VIII, it is stated that the space of an internal gear is the same as the tooth of a spur gear. This appHes to involute or single-curve gears as well as to double-curve gears. The sides of teeth in involute internal gears are hollow- ing. 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 pitch double-curve ters^Yor coarse . . . . Pitch. gears. In designmg mternal gears, 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 Chapters VII 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 necessary to resort to the method given at the end of Chapter 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 difference between , ^ase for in- ■'-' ternal Gear the internal gear and its pinion. The base can be smaller ^eeth. if desired. Let it be required to make an internal gear, and pinion 24 and 18 teeth, 3P. Here the base cannot be more than 6 teeth. In Fig. 16 the base is 6 teeth. The arcs AK and Oky drawn about T, have a radius equal to the radius of the pitch circle of a 6-tooth gear, 3P, instead of a 15-tooth gear, as in Chapter VIII. The outline of teeth of both gear and pinion is made ^.Description of ° ^ Fig. 16. similar to the gear in Chapter VIII. The same letters refer to similar parts. The clearance circle is, however, drawn on the outside for the internal gear. As before 45 BROWN & SHARPE MFG. CO. V- GEAR, 24 TEETH. PINION, 18 TEETH, 3 P, P = 3 N =24 and 18 P'= 1.0472" t=- 5236" S=^ .3333' D"= .6666" S+f= .3857" D"+/=. .7190' A .^it ■— — — ^^ZIL"T v\li ^ & \ // i 11 / A /; / / \ // ./ / \ // // if 1 1 1 1 1 1 1 j 1 j 1 I 1 I 1 1 1 1 C r \ Fig, 16 INTERNAL GEAR AND PINION IN MESH 46 BROWN & SHARPE MFG. CO. stated, the spaces of a spur wheel become the teeth of an internal wheel. 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. 16 the angle KTA is 30° instead of 12°, as in Fig. 12. This brings the line of pressure LP at an angle of 60° with the radius CT, 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 Chapter VIII. 47 BROWN & SHARPE MFG. CO. . Op, qr, and uv equal to the working depth of teeth, which in these gears is }4". The addendum of course is meas- ♦ ured perpendicularly from the cone pitch lines as at kr. 7. Draw lines Om, On, Op, Oo, Oq, Or. These lines give the height of teeth above the cone pitch lines as they approach 0, and would vanish entirely at 0. It is quite as well never to have the length of teeth, or face, mm' longer than one- third the apex distance mO, nor more than two and one-half times the circular pitch. 8. Having decided upon the length of face, draw limiting lines m'n' perpendicular to iO, q'r' perpendicular to kO, 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 through their axes. The distance mr is the whole diameter of the pinion. The Whole Diam- dlstaucc qo is the whole diameter of the gear. In practice eter of Bevel ^ o sr Gear Blanks ob- thcsc diamctcrs can be obtained by measuring the draw- tamed by Meas- . i- . . . uring Drawings, mg. Thc diamctcr of pmion is 3.4475" and of the gear 6.2225". We can find the angles also by measuring the drawing with a protractor. In the absence of a pro-. tractor, templets can be cut to the drawing. The angle formed by line mm' with ab is the angle of face of pinion, in this pinion 59° 10', or 59^°. The lines qq' and gh give us the angle of face of gear, for this gear 22° 18', or 22i° nearly. The angle formed by mn with ab is called the angle of edge of pinion, in our sketch 26° 34', or about 26>^°. The angle of edge of gear, line qr with gh, 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 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. 50 BROWN & SHARPE MFG. CO. BROWN & SHARPE MFG. CO. 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 equal. 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° 10'. ^u'^J* ^^^u"" The following method of obtaining the whole diam- method of ob- ° ^ taming Whole etcr of bcvcl gcars is sometimes preferred : Diameter of Blanks. From k, Fig. 18, lay off; upon the cone pitch line, a distance kw, equal to ten times the working depth of the teeth = 10D". Now add iV of the shortest distance of w from the line ghy which is the perpendicular dotted line wx, to the outside pitch diameter of gear, and the sum will be the whole diameter of gear. In the same manner tV of wy, 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. Chapter XXIV gives trigonometrical methods of figur- ing bevel gears. In our ''Formulas in Gearing" there arc trigonometrical formulas for bevel gears, and also tables for angles and sizes. bSSs^^ who^se A somewhat similar construction will do for bevel RSht^llies.^* gears whose axes are not at right angles. In Fig. 19 the axes are shown at OB and OD, the angle BOD being less than a right angle. 1. Parallel to OB, and at a distance from it equal to the radius of the gear, we draw the lines ab and cd. 2. Parallel to OD, and at a distance from it equal to the radius of the pinion, we draw the lines ef and gh. 52 BROWN & SHARPE MFG. CO. J ANGLE OF AXES MORE THAN 90° Fig. 20 ^ INSIDE BEVEL GEAR AND PINION Fig. 21 53 BROWN & SHARPE MFG. CO. 3. Now, through the point j at the intersection of cd and gh, we draw a Hne perpendicular to OB. This Hne kj, hmited by ab and cd, represents the largest pitch diameter of the gear. Through j we draw a line perpendicular to OD. This line jl, limited by ef and gh, represents the largest pitch diameter of the pinion. 4. Through the point k at the intersection of ab with kj, we draw a line to 0, a line from j to 0, and another from /, at the intersection jl and ef to 0. These lines Ok, Oj, and 01, represent the cone pitch lines, as in Fig. 18. 5. Perpendicular to the cone pitch lines we draw the lines uv, op, and qr. Upon these lines we lay off the addenda and working depth as in the previous figure, and then draw lines to the point as before. By a similar construction Figs. 20 and 21 can be drawn. STOCKING CUTTER 54 BROWN & SHARPE MFG. CO. CHAPTER XI Bevel Gears — Forms and Sizes of Teeth- Cutting Teeth To obtain the form of the teeth in a bevel gear we Form of bevc , . , , . gear teeth. do not lay them out upon a pitch circle, as we do m a 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. 22, let fgc be the base of a cone about the axis OA, the diameter of the cone being /c, and its radius gc. Now the radius of curvature of the surface, at c, is evidently longer than gc, as can be seen in the other view at C; the full line shows the curvature of the surface, and the dotted lines shows the curvature of a circle of the radius gc. 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 cA per- pendicular to Oc, letting cA reach the centre line OA, and take cA as the radius of a circle upon which to lay out the teeth. This is shown at cnm. Fig. 23. For con- venience the line cA is sometimes called the back cone radius. Let us take, for an example, a bevel gear and a pinion Example. 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. 23. 1. Draw the centre lines AO and BO, then the lines gh and cd, and the gear blank lines as described in Chapter X. Extend the lines o'p' and op until they meet the centre lines A'B' and AB. 2. With the radius Ac draw the arc cnm, which we take as the geometrical pitch circle upon which to 55 BROWN & SHARPE MFG. CO. lay out the teeth at the large end. This distance K'c' 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 Be for the geo- metrical pitch circle at the large end and B'c' for the small end: the distance B'c' is transferred to Be'''. 4. Upon the arc cnm 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 semicircle upon the radius Ac, 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 large end of the gear. Repeat this operation with the radius BC about B, and we have the form of the teeth at the 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. 23, the clearance, /, however, is usually the same at the small end as at the large, for conveni- ence 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 plrte. *^^ The sizes of the tooth parts at the small end are in the same proportion to those at the large end as the line 56 BROWN & SHARPE MFG. CO. • -^^ 57 BROWN & SHARPE MFG. CO. Oc' is to Oc. In our example Oc' is 2'\ and Oc is 3"; dividing Oc' by Oc we have i, or .666, as the ratio of the sizes at the small end to those at the large: f is .2095^' or f of .3142'', and so on. If the distance nm is equal to the outer tooth thickness, /, upon the arc cnm, the lines nA and mA will be a distance apart equal to the inner tooth thickness f 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. cJtting.^^'' ° ^ As an example of the cutting of bevel gears with rotary disk cutters, or common gear cutters, let us take a pair of 8 pitch, 12 and 24 teeth, shown in Fig. 25. Length of In making the drawing it is well to remember that tooth face. , . . ..,-,. nothmg IS gamed by havmg the face FE 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 apex distance Oc. To cut a bevel gear with a rotary cutter, as in Fig. 26, 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 Oc, Fig. 25. In the example, one-third of the apex distance is ^'Vbut FE 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 Thcsc data are needed before beginning to cut: cutting. ° 1. The pitch and the numbers of the teeth the same as for spur gears. 58 BROWN & SHARPE MFG. CO. P =5. N =1 8 and 24 p = .628" t' = .209" t = .314" S'= .133 8 = .200" D"= .266 D"= .400" s'+f = .165" 8+/ = .231" D"+/ =.298' 0"+/ = .431" Fig. 23 BEVEL GEARS, FORM AND SIZE OF TEETH 59 cutters BROWN & SHARPE MFG. CO. 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 the outside and inside ends; D''+/ 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 angle, as applied to bevel gears cut with a rotary cutter is the angle that the path of the cutter makes with the axes of the gears. In Fig. 25 the cutting angle for the gear cD is AOp, and that for the pinion is BOo. Thus the cutting angle of each gear equals the face angle of its mate. Selection of Thc form of the teeth in one of these gears differs 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. 23; we need merely measure the lines Ac and Be, Fig. 25, 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 Ac, in inches, multiplied by the diametral pitch, equals the number of teeth for which to select a cutter for 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, page 104. Twice Be, 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. 27 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 MM have been filed. In thus rounding the teeth they should be filed above the pitch line, being careful not to file them 60 BROWN & SHARPE MFG. CO. thinner at t' as in Fig. 24 where the dotted Unes FF show the tooth as it is left by the cutter and the full lines show it after being filed to shape. 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 Ac and Be as radii. In cutting a bevel gear, in the ordinary gear cutting machines, the finished spaces are not always of the same form as the cutter might be expected to make, Fig. 24 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. Thus, in Fig. 26, a gear and a cutter are set to have a space widened at the large end e\ and the last chip to be cut off by the left side of the cutter, the cutter Widening the space at the large end. 61 BROWN & SHARPE MFG. CO. BEVEL GEAR DIAGRAM FOR DIMENSIONS 62 BROWN & SHARPE MFG. CO. having been moved to the right, and the blank rotated in the direction ot 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 side wise, and the blank is then rotated by indexing the spindle to bring the large end of the tooth up against the cutter. This tends not only to cut ro Jed^mofe' the spaces wider at the large pitch circle, but also to atroot!^ ^^^"^ 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 BOo, Fig. 25, 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. 28 and 29. Fig. 28 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 different 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 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. 29, we shall cut the face F much thinner without materially changing the thick- ness of the root R. Almost all bevel gears are between the two extremes of Figs. 28 and 29, so that when the cutting angle BOo, Fig. 25, is smaller than about 30°, this change in the form of the spaces caused by the rotation of the blank may be so great as to necessitate the substitution of a cutter that is narrower at ee', Fig. 26, than is called for by the way of figuring that we have just given: thus 63 BROWN & SHARPE MFG. CO. LEFT RIGHT CUTTER MOVED IN THIS DIRECTION FOR THIS CUT Fig. 26 SETTING BEVEL GEAR CUTTER OUT OF CENTRE ON BEVEL GEAR CUTTING MACHINE BROWN & SHARPE MFG. CO. 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. 30 shows the same I Fig. 28 Fig. 29 gear as Fig. 27, but in this case the teeth have all been filed similar to MM, Fig. 27. Different workmen prefer different ways to com- niing the promise in the cutting of a bevel gear. When a blank smaii end. is rotated in adjusting to finish the large end of the teeth there need not be much filing of the small end, Fig. 30 FINISHED GEAR 65 BROWN & SHARPE MFG. CO. if the cutter is right, for a pitch circle of the radius Be, Fig. 25, which for our example is a No. 8 cutter, but the tooth faces may be rather thin at the large ends. This com- promise 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 cuuer°"whe°i g^ar cuttcrs, unless otherwise specified. This means teetji are to be ^^^^ ^^ should Send a No. 8, 8-pitch bevel gear cutter in reply to an order for a cutter to cut the 12- tooth pinion, Fig. 25; 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 Be in inches, Fig. 25, 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 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, at the large end. Fig. 25, are copied from the table of spur gear teeth, pages 178-181. gear cutTin°g Thc dlstancc Oe' is seven- tenths of the apex distanee order. Q^^ g^ ^j^^^ ^j^^ gj^gs of thc tooth parts at the small end, except /, are seven-tenths the large. The order for cutting these gears goes to the workmen in this form: Large Gear P = 8 N = 24 D''-f/-.2696'' D'''+/=.1946'' ^ = .1963'' f = .1374'' s = .1250'' 5' = .0875'' Cutting Angle = 59°10' = face angle of small gear. 66 BROWN & SHARPE MFG. CO. Small Gear. N = 12 Cutting Angle ==22°18' = face angle of large gear. Fig. 34 is a front view of a gear cutting machine. A bevel gear blank A is held by the work 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. 36 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 work 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. 34, 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 I- degree more than 59°. Mark the depth of the cut at the outside, as in Fig. 32: — it is also well enough to mark the depth at the inside as a check. The thickness of the teeth at the large end is conveniently determined by the solid gauge, Fig. 31. GEAR TOOTH GAUGE ■Fig. 32 GEAR TOOTH CALIPER Fig. 33 Setting machine. the 67 BROWN & SHARPE MFG. CO. The gear tooth vernier caliper, Fig. 33, 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. 31, to the thickness of the teeth at the small end. side°of*t°o''oth The index having been set to divide to the right number being finished. ^^ ^^^ ^^^ spaccs Central with the blank, leaving a tooth between that is a little too thick, as in the upper part of Fig. 27. 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 Fig. 26. 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-seventh to one-sixth of the thick- ness 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. ceSraf'Sts/"^ After 3. Uttlc practicc a workman can set his cutter 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. 27. The depth of cut is partly controlled by the hand elevating shaft H, Fig. 36, which determines the height of the work spindle, and partly by the position of the cutter spindle. if^of Ser*"" We now set the cutter out of centre the trial distance by means of the cutter spindle dial shaft, I, Fig. 36. 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 68 out of center. BROWN & SHARPE MFG. CO. Fig. 34 AUTOMATIC GEAR CUTTING MACHINE FRONT ELEVATION 69 Adjustments. BROWN & SHARPE MFG. CO. the large end than at the small is illustrated in Fig. 26, 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. The rotative adjustment of the work spindle is accom- plished 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 work 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 small end is in the same proportidn to the large end as Oc' is to Oc, Fig. 25, we can at once adjust the cutter 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. The amount of set-over may be calculated very closely from the accompanying table and formula: * TABLE FOR OBTAINING SET-OVER FOR CUTTING BEVEL GEARS o| Ratio of apex distance to width of face 6 "§ 1 1 1 33^ 1 3M 1 4 1 4M 1 4K 1 4M 1 5 1 5^ 1 6 1 7 1 8 1 1 2 3 4 5 6 7 8 .254 .266 .266 .275 .280 .311 .289 .275 .254 .268 .268 .280 .285 .318 .298 .286 .255 .271 .271 .285 .290 .323 .308 .296 .256 .272 .273 .287 .293 .328 .316 .309 .257 .273 .275 .291 .295 .330 .324 .319 .257 .274 .278 .293 .296 .334 .329 .331 .257 .274 .280 .296 .298 .337 .334 .338 .258 .275 .282 .298 .300 .340 .338 .344 .258 .277 .283 .298 .302 .343 .343 .352 .259 .279 .286 .302 .307 .348 .350 .361 .260 .280 .287 .305 .309 .352 .360 .368 .262 283 .290 .308 .313 .356 .370 .380 .264 .284 .292 .311 .315 .362 .376 .386 Set-over = Factor from Table P = diametral pitch of gear to be cut. Tc = thickness of cutter used, measured at pitch line. Given as a rule, this would read: — find the factor in the table corresponding to the number of the cutter used and *From an article in Machinery by Ralph E. Flanders, prepared and edited in collabora- tion with Brown & Sharpe Mfg. Co. 70 BROWN & SHARPE MFG. CO. to the ratio of the apex distance to the width of face; divide this factor by the diametral pitch, and subtract the quotient from half of the thickness of the cutter at the pitch line. Fig. 35 As an illustration of the use of this table in obtaining the set-over we will take the following example: a bevel gear of 24 teeth, 6 pitch, 30 degrees pitch cone angle and 1% face. These dimensions, by the ordinary calcula- tions for bevel gears call for a No. 4 cutter and an apex distance of 4 inches. In order to get our factor from the table, we have to know the ratio of the apex distance with the length of face. This ratio iSi|5 = ^or about ^. The factor in the table for this ratio with a No. 4 cutter is 0.280. We next measure the cutter at the proper depth of S+/ for 6 pitch, which is found in the column marked "depth of space below pitch line" in the Table of Tooth Parts, pages 178-181, or by dividing 1.157 by the diametral pitch. This gives S+/=.1928 inch. We find by measurement that the thickness of the cutter at this depth is .1745 inch. This dimension will vary with different cutters, and will vary in the same cutter as it is ground away, since formed bevel gear cutters are commonly provided with side relief. Substituting these values in the formula we have, set-over = required dimension. ' — 'f = .0406 inch, which is the 71 mes BROWN & SHARPE MFG. CO. Mnung^Maci I^ cutting bevel gears on milling machines the work must be set off centre on one side of the cutter by this amount, taking the usual precautions to avoid errors from backlash. In this position the cutter is run through the blank, the latter being indexed for each tooth space until it has been cut around. (If a central or roughing cut has been previously taken, it will be necessary to line up this cut at the small end of the tooth with the cutter. This is done by rotating the tooth space back toward the cutter, either by moving the index crank as many holes in the dial-plate as are necessary, or by means of such other special provisions as may be made for doing this in the index head, independently of the dial-plate.) Having thus cut one side of the tooth to proper dimen- sions, the work must be set-over by the same amount the other side of the position central with the cutter, taking the same precautions in relation to backlash as before, and rotating the blank to again line up the cutter with the tooth space at the small end of the tooth. With this setting, take a trial cut. This will be found to leave the tooth whose side is trimmed in this operation a little too thick, if the cutter is thin enough, as it ought to be, to pass through the small end of the tooth space of the completed gear. This trial tooth should now be brought to the proper thickness by rotating the blank toward the cutter, moving the crank around the dial for the rough adjustment, and bringing it to accurate thickness by such means as may be provided in the head. In the Brown & Sharpe head, this fine adjustment is effected by two thumbscrews near' the hub of the index crank, which turn the index worm with relation to the crank. It will evidently be wise to be sure we are right before going ahead, as the slight approximations involved in the derivation of the formula may bring the setting not quite right, so that the thickness of the tooth at the large and the small ends is not what it ought to be. 72 BROWN & SHARPE MFG. CO. This point may be tested by measuring the tooth at both the large and the small ends with the gear tooth vernier caliper as shown in Fig. 33, the caliper being set so that the addendum at the small end is in the proper proportion to the addendum at the large end — (that is to say, that it is in the ratio ^~^ Fig. 35.) In taking these measurements, if the thicknesses at both the large and the small ends, which should be in this same ratio, are too great, rotate the tooth toward the cutter and take another cut until the proper thickness at either the large or small end has been obtained. If the thickness is right at the large end and too thick at the small end, the set-over is too much. I f it is right at the small end and too thick at the large end, the set-over is not enough, and should be changed accordingly, as is done by the regular **cut-and-try" process. The formula and table given herewith, however, ought to bring it near enough right the first time, and in the general run of work it can be safely relied on. It may be said, in this connection, that nothing but a true running blank, with accurate angles and diameters, should be used in setting up the machine. If such a blank cannot be found in the lot of gears to be cut, it will be necessary to turn one up out of wood or other easily worked material. Otherwise the workman is inviting trouble, whatever his method of setting up. The directions for cutting bevel gears on the milHng , ^1^}!^^ A ^ machine apply in modified form to the automatic gear Mlchin?.''"'"^ cutting machine as well. The set-over is determined in the same way, but instead of moving the work off centre, the cutter spindle is adjusted axially by means provided for that purpose. Some machines are provided with dials for reading this movement. The cutter is first centred as in the milling machine, and then shifted — first to the right, and then to the left of this central position. The rotating of the work to obtain the proper thick- ness of the tooth is effected by unclamping the indexing 73 A second approximation. BROWN & SHARPE MFG. CO. worm from its shaft (means usually being provided for this purpose) and rotating the worm until the gear is brought to proper position. Otherwise the operations are the same as for the milling machine. 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 without cutting any central spaces at all. The cutter spindle stops, JJ, can now be set to control the out of centre of the cutter, without having to adjust it 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 atten- tion to the rotative adjustment of the work 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. 27 the tooth L is as the cutter left it, and is ready to be filed to the shape of the teeth MM, which have been filed. Fig. 37 is the pair of gears that we have been cutting; the teeth of the 12-tooth pinion have been filed. A second approximation in cutting with a rotary cutter is to widen the spaces at the large end by swing- ing either the work spindle or the cutter slide carriage, so as to pass the cutter through on an angle with the blank sideways, called the side-angle, and not rotate the blank at all to widen the spaces. It is available in 74 BROWN & SHARPE MFG. CO. 75 BROWN & SHARPE MFG. CO. the manufacture 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 ee' , Fig. 26, 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. pr^imation.^^" A third approximate method consists in cutting the teeth right at the large end by going 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 possibilities for considerable accuracy. proximation.^^" A fourth mcthod is to have a cutter fully as thick as the spaces at the small end, cut rather deeper than 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 is short 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, 5', at the small end by turning the blank down. This method is extensively employed by chuck manu- facturers. 76 BROWN & SHARPE MFG. CO. Fig. 37 FINISHED GEAR AND PINION 77 BROWN & SHARPE MFG. CO. the manufacture 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 ee\ Fig. 26, 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. pr^ximation.^^' A third approximate method consists in cutting the teeth right at the large end by going 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 possibilities for considerable accuracy. proximatbn.^^" A fourth mcthod is to have a cutter fully as thick as the spaces at the small end, cut rather deeper than 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 is short 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 manu- facturers. 76 BROWN & SHARPE MFG. CO. Fig. Zl FINISHED GEAR AND PINION 77 BROWN & SHARPE MFG. CO. Planing of bevel gears. Mounting gears. Angles and sizes of bevel gears. Mitre gears. 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. One form of gear planer is that in which the prin- ciple embodied is theoretically correct; this machine originates the tooth curves without a former and is more often called a gear generator for this reason. Another form of the same class of machines is that in which the tool is guided by a former. The gear generator is more often used on the smaller gears while the planer type, which uses a former for getting the shape of teeth, is used on the larger pitch gears. If gears are not correctly mounted in the place where they are to run, 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. i^ ^ *|^ ^ ^ The method of obtaining the sizes and angles pertaining to bevel gears by measuring a drawing is quite convenient, and with care is fairly accurate. Its accuracy depends, of course, upon the careful measuring of a good drawing. We may say, in general, that in measuring a diagram, while we can hardly obtain data mathematically exact, we are not likely to make wild mistakes. We, however, calcu- late the data without any measuring of a drawing. In the ''Formulas in Gearing" there are also tables pertain- ing to bevel gears. 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. 78 BROWN Si SHARPE MFG. CO. CHAPTER XII Curved Tooth or Spiral Bevel Gears These gears have been developed for use in automobile rear axles and are used to a certain extent for other purposes, requiring an especially smooth running drive. In these gears the axes of the pinion and the gear inter- sect as in regular bevel gears having radial teeth. The object in cutting curved teeth is to obtain a smoother and quieter drive and to increase the number of teeth in contact at a given instant. The bearing between the teeth, at any instant, instead of being along straight lines, as in bevel gears having radial teeth, runs from the base of the tooth at one end toward the top at the other end in a diagonal line as in herring- bone gears. This produces uniform wear and helps preserve the original tooth outline. Another advantage sometimes claimed is that the position of the pinion can be adjusted a greater amount than is possible with regular bevel gears without seriously affecting their running qualities. ^ The teeth of the gear are inclined, or curved, in the opposite direction to those of the pinion; one being right- hand and the other left. This inclination of the teeth causes the pinion to thrust in or out, according to the hand of the teeth and the direction of rotation, while the pitch angle causes the pinion to thrust out from its apex. Thus, when these forces act in opposite directions, the load on the thrust bearings is reduced and is equal to their difference, but when the direction of rotation of the gears is reversed, the thrust load on the bearing is increased and is equal to the sum of the above mentioned forces. No change is necessary in the general design of the gears when changing from the regular type to those 79 BROW^N & SHARPE MFG. CO. Fig. 38 CURVED TOOTH BEVEL GEARS 80 BROWN & SHARPE MFG. CO. having curved teeth. The same blanks can be used with the same number of teeth. The only difference in the tooth measurements is that the normal thickness is, of course, reduced. Fig. 38 shows a pair of curved tooth bevel gears. These are cut on a machine especially designed for this purpose and the cutting is done by means of an inserted tooth cutter the blades of which cut upon its face and thus give the tooth its curved shape. 81 BROWN & SHARPE MFG. CO. Fig. 39 WORM GEARING NUMBER OF TEETH, 54. THROAT DIAMETER, 44. 59' CIRCULAR PITCH, tVi" . OUTSIDE DIAMETER, 46' 82 BROWN & SHARPE MFG. CO. CHAPTER XIII Worm Gearing — Sizing Blanks of 32 Teeth and More A worm is a screw made to mesh with the teeth of ^'''"°^- a wheel called a worm wheel, Fig. 39. 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 a 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 in 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 usually given in a pitch of worm. 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 worm Lead of a 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 X" 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 one inch, we divide 83 BROWN & SHARPE MFG. CO. Fig. 40 WORM AND WORM WHEEL THE THREAD OF WORM IS LEFT-HAND; 'WORM IS SINGLE-THREADED 84 BROWN & SHARPE MFG. CO. 85 BROWN & SHARPE MFG. CO. V by the number of threads to one inch, and the quotient is the circular pitch. Thus, the wheel in Fig. 41 is }4" Linear Pitch, eircular 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 recipro- cal of the linear pitch. Thus, in the above example there are 2 threads to 1" as 2 is the reciprocal of }4" the linear pitch. We should say of a double- threaded worm advancing 1'' in 1 1/3 turns that: Lead - y^" or .75''. Linear Pitch or P' = 3/8'' or .375". Multiply 3.1416 by the number of threads to one inch, and the product will be the diametral pitch of the worm wheel. 11/3 turns per 1" double-threaded = 2 2/3 threads per inch. 2 2/3x3.1416-8.3776 times the diametral pitch or P. See Table of Tooth Parts. Drawing of To makc drawing of worm and wheel we obtain data Wormand • • ^ •, i Worm Wheel, as m circular pitch. 1. Draw centre line AO and upon it space off the distance ab equal to the diameter of pitch circle. 2. On each side of these two points lay off the dis- tance s, or the usual addendum = ^^^ as be and bd. 3. From c lay off the distance cO equal to the radius of the worm. The diameter of a worm is generally four or five times the circular pitch. 4. Lay off the distances eg and de 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 diameter at bottom of worm thread. 6. Draw hO and iO at an angle of 30° to 45° with AO. These lines give width of face of worm wheel. 7. Through g and d draw arcs about O, ending in„ hO and iO. 86 Teeth of Wheels fin- BROWN & SHARPE MFG. CO. 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. 41. Measure sketch for whole diameter of wheel blank. The foregoing instructions and sketch are for cases where the teeth of the wheels are finished with a hob. i^^ed with Hob A hob is shown in Fig. 42, being a steel piece threaded ^°^- with a tool of the same angle as the tool that threads the worm, the end of the tool being .3354 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/, or twice the clearance larger than the worm. In our relieved ^°^ hobs the diameter is made about .005'' to .010" larger for small sizes to allow for wear. The outer corners of the hob thread can be rounded down as far as the clearance distance. The width at top of the hob thread before rounding should be .3095 of the linear, or circular pitch = .3095P'. The whole depth of thread is thus the ordinary working depth plus the clearance = D'' +/• The diameter at the bottom of the hob thread should be 2/+ .005'' to .010" larger for small sizes than the diameter at bottom of worm thread. In both this diameter and the outside diameter an allowance up to .03" or .04" can be made when hobs are of large size. For thread tool and worm thread see end of Chapter IV. Proportions of 87 BROWN & SHARPE MFG. CO. How to use the Hob. Universal Milling Ma- chine used in Hobbing. Why a Wheel is Hobbed. Worm Wheel Blanks with Less than 30 Teeth. Interference of Thread and Flank. Example. Special Forms of Teeth. In the absence of a special worm gear cutting machine 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 machine is convenient for doing this work; with it the distance between axes of worm and wheel can be noted. In making wheels in quantities it is better to have a machine in which the work spindle is driven by gearing, so that the hob can cut the teeth from the solid without gashing. The object of hobbing a wheel is to get more bearing sur- face of the teeth upon worm thread. The worm wheels. Figs. 40 and 50, were hobbed. If we make the diameter of a worm wheel blank, that 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 undercut or hollowing. This is caused by the interference 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 (31^ geometrically), and interfere with nearly the whole flank in a wheel of 12 teeth. In Fig. 43 the blank for worm wheel of 12 teeth was sized by the same rule as given for Fig. 41. 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. In Fig. 44 the blank for wheel was sized so that pitch- circle comes midway between outermost part of teeth and innermost point obtained by worm thread. 88 BROWN & SHARPE MFG. CO. Fig. 43 ^vrcii^/Rc/.^ '^M^' ^..n-f Fig. 44 Fig. 45 89 BROWN & SHARPE MFG. CO. 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. 43. The pitch circle in Fig. 44 is the same diameter as the pitch circle in Fig. 43. The same . hob was used for both wheels. The flanks in this wheel are so much undercut as to materially lessen the bearing surface of teeth and worm thread. Avowtd/'''"'" In Chapter 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 of a worm, making them, like rack-teeth, to mesh with 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. 45 are not undercut, because the diameter of wheel is so large that there is hardly any tooth inside the pitch circle. The pitch circle in Fig. 45 is the same size as pitch circles in Figs. 43 and 44. This wheel was sized by the following rule: multiply the Th?oatto Avofd ^^^^^ diameter of the wheel by .937, and add to the product Interference. four timcs thc addcudum (45); 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 driven by independent mechanism and not by the hob. The diameter across the outermost parts of teeth, as at AB, 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 78 and Fig. 41, when the wheels are to be hobbed. The spaces can be cut the full depth, the cutter being dropped in. 90 BROWN & SHARPE MFG. CO. Fig. 46 shows a milling machine gashing the teeth of a worm wheel. In gashing the teeth the blank is dogged to the spiral head spindle, and the swivel table is swung to the required angle. The vertical feed is used and the teeth are indexed the same as in cutting a spur gear. Most of the stock is removed in gashing, enough only being left to allow the hob to take a light finishing cut. Fig. 47 shows the same wheel being hobbed. The work is set up practically the same as in the opera- tion of gashing the teeth, only the dog on the arbor is removed and the swivel table is set at zero. The worm wheel revolves freely on the centres, being rotated by the hob. The wheel can be hobbed to the right depth by using a steel rule at the back of the knee to measure a distance equal to the centre distance of the worm and wheel from a line marked "Centre", on the vertical slide to the top of the knee. This line on the vertical slide indicates the position of the top of the knee when the index centres are at the same height as the centre of the machine spindle. When worm wheels are not hobbed it is better to Blank Like a turn blanks like a spur wheel. Little is gained by having '^^'"' ^^^''^' wheels curved to fit worm unless teeth are finished 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. 48 and 51. In setting a cutter to gash a worm wheel. Figs. 49 and wheels for 52, the angle is measured from the axis of the worm Machines'! wheel and the angle of the worm thread is, in conse- quence, measured from the perpendicular to the axis of the worm. See Chapters XVI and XIX. Some mechanics prefer to make dividing wheels 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 91 BROWN & SHARPE MFG. CO. Fig. 46 GASHING TEETH IN WORM WHEEL 92 BROWN & SHARPE MFG. CO. Fig. 47 ROBBING TEETH IN WORM WHEEL 93 BROWN & SHARPE MFG. CO. an accurate cutting machine we have found wheels like Figs. 49 and 52, not hobbed, every way satisfactory. As to the different wheels, Figs 50, 51 and 52, when worm Diffe"renTstyies! IS in right position at the start, the lifetime of Fig. 50, under heavy and continuous work, will be the longest. Fig. 51 can be run in mesh with a gear or a rack as well as with a worm when made within the angular limits commonly required and is capable of lateral adjust- ment between the worm and wheel. Strictly, neither two gears made in this way, nor a gear and a rack would be mathematically exact, as they might bear at the sides of the gear or at the ends of the teeth only and not in the middle. At the start the contact of teeth in this wheel upon worm thread is in points only; yet such wheels have been many years successfully used in ele- vators. FiJfsMnJ'^ Rim^ Fig. 52 is a neat looking wheel. In gear cutting machines where the workman has occasion to turn the work spindle by hand, it is not so rough to take hold of as Figs. 50 and 51. The teeth are less liable to injury than the teeth of Figs. 50 and 51. 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 the diameter of the worm much less than four times the thread pitch or linear pitch but the worm can be of any larger diameter, five or ten times the linear pitch, if required. It is well to take off the outermost part of teeth in wheels (Figs. 40 and 50), as shown in these two figures. Limits for and not leave them sharp, as in Figs. 41 and 44. It is also well to round over the outer corners of the blanks for the wheels, Figs. 51 and 52. 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. 94 BROWN & SHARPE MFG. CO. Fig. 48 WORM WHEEL WITH TEETH CUT IN A STRAIGHT PATH DIAGONALLY ACROSS FACE. W^ORM IS DOUBLE-THREADED 95 BROWN & SHARPE MFG. CO. Fig. 49 / WORM AND WORM WHEEL FOR GEAR CUTTING MACHINE 96 BROWN & SHARPE MFG. CO. ^^ Fig. 50 Fig. 51 Fig. 52 97 BROWN & SHARPE MFG. CO. By stating 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. worm^and^of a Usually, in determining the length of a worm, the H°^' object is to make it just long enough to engage the teeth of the worm wheel in contact with it at one time. The length of the hob should be somewhat greater than that of the worm. The length of the worm varies with its diameter, the diameter and width of face of the worm wheel, pitch, pressure angle and helix angle, and to determine it for any particular case requires complicated calculations not within the scope of this treatise . For practical purposes the minimum length of worm can be determined by making a diagram (see Fig. 53) and measuring the length L of the addendum line repre- senting the outside diameter of the worm, at its inter- section with the throat circle of the worm wheel, or it may be calculated as follows: D = throat diameter of the worm wheel, D''= working depth of tooth, L = length of worm, V = length of hob. L = 2 Jd^'(D-D'0 V=L + If endwise movement of the worm relative to its wheel is required for adjustment or traverse the amount of such movement should also be added to the length of worm, but need not be added to the length of the hob. For a 30-tooth worm wheel of the form of Figs. 48 and 49, we can have only about three threads in con- tact and a hob four threads long, like Fig. 42, is long enough. From the diagram, Fig. 54, 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 98 BROWN & SHARPE MFG. CO. i Addendum Line Of Worm Fig. 53 PITCH LINE Fig. 54 99 BROWN & SHARPE MFG. CO. where the line CD 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 the worm. GASHING TEETH OF HOB Hobs with Relieved Teeth. Grinding Hobs for Accuracy. Hobs of any size are made with the teeth relieved the same as gear cutters, the faces of which may be ground without changing the form of the teeth. They are made with a precision screw so that the pitch of the thread is accurate before hardening. When assured accuracy is desired for hardened hobs it can be obtained by grinding. 100 BROWN & SHARPE MFG. CO. CHAPTER XIV 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 Cutters Let us suppose that we have two shafts 14'' apart, , centre d is - •^•^ r- 7 tance and Ratio centre to centre, and wish to connect them by gears so that they will have speed ratio 6 to 1. We add the 6 and 1 together, and divide W by the sum and get 2'' for a quotient; this 2'', multiplied by 6, ^ives us the radius of pitch circle of large wheel=12''. In the same manner we get 2" as radius of pitch circle 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 diameter of two wheels of a given ratio to connect shafts a given distance apart: Divide the centre distance by the sum of the terms of the ratio; find the product of twice the quotient by each term separately, and the two products will be the pitch diameters of the two 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 outlines. A blank l}4" 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 178-181. In computing it is well to take tt to more than four places, TT to nine places=3. 141592653. ^ to nineplaces= .318309886. 101 fixed. Rule for Di- ameter of Pitch Circles. BROWN & SHARPE MFG. CO. Gearing"" ' "" Thcrc is HO such thing as pure rolling contact in teeth of wheels; they always rub, and, in time, will wear them- selves 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 departure 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 (8P), 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. G™fS;es.^^'''' ^^^ width of facc 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 Thc spccd of gcar cuttcrs is subject to so many con- Cutters. . :: . „ . - . ditions that definite rules cannot be given. Carbon cutters can be run from 60 to 70 feet per minute in cast iron and from 30 to 40 feet per minute in machinery steel. High speed steel cutters can be run from 80 to 125 feet per minute in cast iron and from 65 to 100 feet per minute in machinery steel. Speed in In brass the speed of gear cutters can be twice as fast as in cast iron. Clockmakers and those making a specialty of brass gears exceed this rate even. A 12P cutter has been run 1200 turns a minute in bronze. A 32P cutter has been run 7000 turns a minute in soft brass. 102 BROWN & SHARPE MFG. CO. The most desirable rate of feed varies widely under different conditions, while slight changes have so marked an effect on the cost and quality of the product that no exact rules can be given. The best method is to start with a given feed, then increase it until the gear blank will stand no more or the economical limit of the cutter is reached, and then use this or a very slightly slower feed for all similar work. One way to increase the production when cutting cast iron is to use an exhaust back of the cutter to carry away the chips and to keep the cutter cool. This will allow for materially higher speed and for the cutting of a much greater number of teeth without re-sharpening. As most of the cutters used for manufacturing are of high speed steel, the following table will give useful data for the feed of these cutters in cutting cast iron and low carbon steel. Rate of Feed. Use of A: Exhaust. Feed of High Speed Steel Cutters Diametral Pch 2 2i 3 4 5 6 7 8 10 12 16 Feed, Inches Cast I. 3| 31 4 ^ ^ 5 6 6 7 8 9 per Minute Steel 11 n 2 2h 2h 3 4 4 4i 5 6 This table is based on finishing the gears in one cut. Whether this is permissible or not will depend on many things, such as the hardness of the material, size and shape or stiffness of the blank and the quality of finish desired. 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 form are rapidly destroyed if allowed to run after they are dull. Cutters are oftener Keep Cutters Sharp. 103 BROWN & SHARPE MFG. CO. 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 VI 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 4P, we would cut with it all 4P gears, having 135 teeth or more, and we would also cut with it a 4P rack. Now, instead of always referring to a cutter by the num- ber 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 4P, made to cut gears 135 teeth to a rack, inclusive. No. 1, 4P. We have adopted numbers for designating involute cutteS!"*^ ^^^'" S^^^ cutters as in the following table: No. 1 will cut wheels from 135 teeth to a rack inclusive. 2 '' 3 " 4 - 5 " 6 '' 7 " 8 " 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 4P gears having twelve teeth and more. It takes eight other cutters to cut all involute gears of 5P, having 12 teeth and more. A No. 8, 5P cutter cuts only 5P gears having 12 and 13 teeth. A No. 6, lOP cutter cuts only lOP 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. 55 ' ' 134 teeth 35 " 54 '' 26 ' ' 34 " 21 ' ' 25 " 17 ' ' 20 '' 14 ' ' 16 " 12 ' ' 13 " 104 BROWN & SHARPE MFG. CO. In ordering cutters for involute spur gears two things must be given : 1. Either the number of teeth to be cut in the gear or the number of the cutter, as given in the foregoing table. 2. Either the pitch of the gear or the diameter and number of teeth to be cut in the gear. li 2b teeth are to be cut in a 6P involute gear, the cutter will be No. 5, 6P, which cuts all 6P 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, 8P, No. 6, 8P, and No. 7, 8P. For each pitch of epicycloidal, or double-curve gears, 24 cutters are made. In coarse pitch gears, the varia- 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-curve teeth were formerly preferred in coarse pitch gears. The tendency now, however, is to use the involute form in all cases. 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 . 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 gear cutters: Cutter A cuts 12 teeth. Cutter M cuts - N O P " Q R B " 13 " " C - 14 " D " 15 " '' E " 16 - - F " 17 " ial or double-curve Its 27 to 29 teeth. " 30 '' 33 " " 34 .. 37 - " 38 .. 42 " " 43 - 49 " " 50 '' 59 " How to order Involute Cut- ters. Epicycloidal or D o u b 1 e - curve Cutters. Table of Epi- cycloidal or Double-curve Gear Cutters. 105 BROWN & SHARPE MFG. CO. Cutter G cuts 18 teeth. Cutter S cuts 60 to 74 teeth. " H ' ' 19 " T '' 75 " 99 " .. I ' ' 20 " U " 100 " 149 '^ '' J '' 21 to 22 V '' 150 '' 249 '' " K ' ' 23 to 24 w '' 250 '' Rack. '' L ' ' 24 to 26 X " Rack. 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. E^frj^^dofdli I^ ordering epicycloid al gear cutters designate the Cutters. letter of the cutter as in the foregoing table, also either give the pitch or give data that will enable us to deter- mine the pitch, the same as directed for involute cutters. More care is required in making and adjusting epi- cycloidal gears than in making involute gears. How to order In Ordering bevel gear cutters the following data Bevel Gear o o o Cutters. must be given: 1. The number of teeth in each gear. 2. Either the pitch of gears or the largest pitch diameter of each gear; see Fig. 18. 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 attempt should be made to cut epicycloidal tooth bevel gears with rotary disk cutters. How to order In ordcrlug worm wheel cutters, three things must Worm Gear ° Cutters. be given: 1. Number of teeth in the wheel. 2. Pitch of the worm; see Chapter XII I. 3. Whole diameter of worm. In any order connected with gears or gear cutters, when the word * 'diameter" occurs, we usually under- stand that the pitch diameter is meant. When the whole diameter of a gear is meant it should be plainly written. Care in giving an order often saves the delay 106 BROWN & SHARPE MFG. CO. of asking for further instructions. An order for one gear cutter to cut from 25 to 30 teeth cannot be filled, because it takes two cutters of involute form to cut from 25 to 30 teeth, and three cutters of epicycloidal form to cut from 25 to 30 teeth. In ordering, sheet zinc is convenient to sketch gears xempfetl ^""^ 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 mixture 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. ■nSS^gLvJ.'''' 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. When gears are cut too deep, which is more often the case rather than not deep enough, considerable noise results, especially if the driving gear is at fault. Cutting gears off centre may result in gea'rs being noisy in one direction when they may run quietly in the other direction. Another cause may be the centre distance, which if not right, allows the gears to mesh too tightly or run too loosely. Shafts that are not parallel or when the frame of the machine being of such a form as to give off sound vibrations are frequently found to be the cause of noisy gearing. There are numerous other causes for noisy gears and it is sometimes very difficult to tell what they may be, even after we have examined the gears in question. 107 BROWN & SHARPE MFG. CO. CHAPTER XV Spiral Gears^ — ^Calculations for Lead of Spirals spSl^Gear.°^ When the teeth of a gear are cut, not in a straight path, Hke a spur gear, but in a hehcal or screw-Hke 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. In screw gears there is an end movement, or slipping of the pitch sur- faces 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 path. 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 move- ment 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. 56. 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 spiral gears. Direction of Whcu two cxtcmal spiral gears run together, with eSn?i li* Axes' their axcs parallel, the teeth of the gears must have opposite hand spirals. Thus, in Fig. 56 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 108 BROWN & SHARPE MFG. CO. Fig. 55 RACKS AND GEARS ^ -^-iiMIMiiiiiiiiiiiilBliiiilliii^^ Fig. 56 SPIRAL GEARING 109 BROWN & SHARPE MFG. CO. must have the same hand spiral teeth. A and B, Fig. 56, have right-hand spiral teeth. If both gears A and B had left-hand spiral teeth, the relative direction in which they turn would be reversed. Fig. 57 shows in diagrama- tic form the hand and direction of revolution of spiral gears. Spiral Lead. ^^le Spiral Icad or lead of spiral is the distance the spiral advances in one turn on the pitch line. A cylinder or gear cut with spiral grooves is merely a screw 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 uni- versal 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 }4" lead is said to be 2 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. shewing "Jiatlre ^^^ ^^^^ papcr a right angle triangle, one side of the rii^^^''' °^ ^p^- right angle 6" and the other side of the right aingle 2" long. Make a cylinder 6" in circumference. It will be remembered (Chapter II) that the circumference of a cyUnder, multiplied by .3183, equals the diameter; 6''X.3183 = 1.9098'\ Wrap the paper triangle around the cyclinder, letting the 2" side be parallel to the axis, the 6" side perpendicular to the axis and reaching around the cylinder. The hypotheneuse now forms a helix or screw-like line, called a spiral. Fasten the paper triangle thus wrapped around. See Fig. 58. If we now turn this cylinder ABCD one turn in the direction of the arrow, the spiral will advance from O to E. This advance is the lead of the spiral. 110 BROWN & SHARPE MFG. CO. Fig. 57 LEFT-HAND SPIRAL GEAR DIAGRAMS SHOWING DIRECTIONS OF REVOLUTIONS OF SPIRAL GEARS RIGHT-HAND LEFT-HAND 111 BROWN & SHARPE MFG. CO. Fig. 58 Rules for cal- culating the parts of ral. spi- The angle EOF, which the spiral makes with the axis EO, is the angle of the spiral. This angle is found as in Chapter I. The circumference of the cylinder 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 : Divide the circumference of the cylinder or spiral by the number of inches of spiral to one turn, and the quotient will be the tangent of angle of spiral. For an explana- tion of the tangent and other functions of triangles, see Chapters XXII-XXIII. When the angle of spiral and circumference are given, to find the lead: Divide the circumference by the tangent of angle, and the quotient will be 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 by the lead, and the product will be the circumference. When applying calculations to spiral gears the angle is reckoned at the pitch circumference and not at the outer or addendum circle. 112 BROWN & SHARPE MFG. CO. It will be seen that when two spirals of different dia- meters have the same lead the spiral of less diahieter will have the smaller angle. Thu$ in Fig. 58 if the paper triangle had been 4'' long instead of &' the diameter of the cylinder would have been 1.27'' and the angle of the spiral would have been only 63^ degrees. 113 BROWN & SHARPE MFG. CO. CHAPTER XVI Examples in Calculation of the Lead of Spiral Angle of Spiral — Circumference of Spiral Gears — A Few Hints on Cutting The rules for calculating the circumference of spiral gears, angle and the lead of spiral are the same as in Chapter XXII, for the tangent and angle of a right angle triangle. In Chapter XV, the word ''circum- ference" is substituted for ''side opposite," and the words "lead of spiral" are substituted for "side adjacent." rait^'lith^reler- When two Spiral gears are in mesh the angle of spiral orihaFts.^"^'^ should be the same in one gear as in the other, 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. 56 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 rais^^of '"differ- tccth, aud thc othcr has 25 teeth, the lead of spiral in ent diameters. .^^^ 50-tooth gcar shouM bc twlcc as long as that of the 25-tooth gear. Of course, the diameter of pitch circle should be twice as large in the 50-tooth as in the 25-tooth gear. 114 BROWN & SHARPE MFG. CO. In spirals where the angle is 45° the circumference is the same as the spiral lead, because the tangent of 45° is 1. Sometimes the circumference is varied to suit a pitch ciTcim1e?ence that can be cut 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 con- siderable 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 may be 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'' (Si/g"), the axes to be parallel. By rule given in Chapter XIV, we find the diameters of pitch circles will be 5" and 1^4"- Let us take a spiral of 48'' lead for the large gear, and a spiral of 12" lead for the small gear. The circumference of the 5" pitch circle is 15.70796". Dividing the circumference by the lead of the spiral, we have '-^^ = .3272" for tangent of angle of spiral. In the table the nearest angle to tangent, .3272", is 18°7'. 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. . Now, this rule gives the angle at the pitch surface ^ difference only. Upon looking at a small screw of coarse pitch, ind"bittom*°Jf it will be seen that the angle at bottom of the thread ^p^^^^ Grooves, 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. 115 BROWN & SHARPE MFG. CO. Example i n calculation of Lead of Spiral. This will be seen in Fig. 59, where AO is the centre line; bj shows direction of bottom of thread, and dg shows direction of top of thread. The angle Afb is less than the angle Agd. The difference of angle being due to the warped nature of a screw thread. A cylinder 2'' diameter is to have spiral grooves 20° with the centre line of cylinder; what will be the lead of spiral? The circumference is 6.2832". The tangent of 20° is .36397. Dividing the circumference by the tangent of angle, we obtain Up = 17.26"+ for lead of spiral. Fig. 59 In Chapter XIII, 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 imakes with a plane perpendicular to the axis. 116 BROWN & SHARPE MFG. CO. CHAPTER XVII Normal Pitch of Spiral Gears — Curvature of Pitch Surface — Form of Cutters A Normal to a curve is a line perpendicular to the Normal to f. Curve. tangent at the point of tangency. Fig. 60 In Fig. 60, the line BC is tangent to the arc DEF, .and the line AEO, being perpendicular to the tangent at E the point of tangency, is a normal to the arc. Fig. 61 is a representation of the pitch surface of a spiral gear. A'D'C is the circular pitch. ADC is the same circular pitch seen upon the periphery of a wheel. Let AD be a tooth and DC a space. Now, to cut this space DC, the path of cutting is along the dotted line ab. By mere inspection, we can see that the shortest distance between two teeth along the pitch surface is not the dis- tance ADC. Let the line AEB 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 AEB, reaching over a tooth and a space along the normal helix, is called the normal pitch or the normal linear pitch. 117 BROWN & SHARPE MFG. CO. Fig. 61 118 mal Pitch. BROWN & SHARPE MFG. CO. The normal pitch of a spiral gear is then: — the shortest formal pitch. distance between the centres of two consecutive teeth measured along the pitch surface. In spur gears the normal pitch and circular pitch are alike. In the rack DD, Fig. 55, the linear pitch and normal pitch are alike. From the foregoing it will be seen that, if we should spSai^ceai.'' "^ cut the space DC 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 CAB is equal to the angle of the spiral. The line AEB corresponds to the cosine of the angle CAB. Hence the rule:— multiply the cosine of angle of spiral by the circular pitch and the product will be the to find ^ox- 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 by 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 subtracting 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 the circular pitch. Thus, if the norma] pitch is 12 diametral, the addendum should be //', the thickness .1309'', and so 119 BROWN & SHARPE MFG. CO. on. The diameter of pitch circle of a spiral gear is calculated from the diametral pitch. Thus a gear of 30 teeth lOP would be ?>" pitch diameter. But if the normal pitch is 12 diametral pitch, the blank will be 3 -—;" diameter instead of 3~'^ orma i c jt is cvldent that with a given pitch diameter and vanes, number of teeth the normal pitch varies with the angle of spiral. The cutter should be for the normal pitch. 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 fore- going; that is: To make An- First calculatc to the next finer pitch cutter than f onf^'irm^To wouM bc rcQulred for the diametral pitch. Cutters given. ^^^ ^^ ^^^^^ ^^^ cxamplc, two gcars 10 pitch and 30 teeth, spiral and axes parallel. Let the next finer cutter 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. 61) that the line ADC will be .3142" when AEB is .2618". Dividing .2618" by .3142" (see Chapter XV), we obtain the cosine of the angle CAB, which is also the angle of the spiral, '^^j^ = .8333. The same quotient comes by dividing 10 by 12, -\^ = .8333 + ; 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 .8333 is 33°34'. We now want to find the pitch of spiral that will give angle of 33°34' on the pitch surface of the wheel, 3" diameter. Dividing the circumference by the tangent of angle, we obtain the pitch of spiral (see Chapter XVI). The circumference is 9.4248". The tangent of 33°34' is .66356, ^|f|| = 14.20; and we have for our spiral 14.20" lead. 120 BROWN & SHARPE MFG. CO. When the machine is not arranged for the exact pitch of spiral wanted, it is generally well enough to 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 centre distances right. When two gears of the same size are in mesh with their axes parallel, a change of angle of teeth or spiral makes no difference in the correct meshing of the teeth. But when gears of different size are in mesh, due regard must be had to the spirals being in pitch, pro- portional to their angular velocities (see Chapter XVI). We come now to the curvature 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 to cut a spur 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 AEB, Fig. 61, 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. When exact Pitch cannot be cut. Spiral Gears of Different Sizes of Mesh, Shape of Cut- ter. BROWN & SHARPE MFG. CO. Thus, in Fig. 62, suppose the templet fits a circle 3}^" diameter, if the normal pitch is 12 to one inch, dia- metral, the cutter required is for 12P and 40 teeth. The curvature of the templet will not be quite circular, but is sufficiently near for practical purposes. Strictly, a flat templet cannot be made to coincide with the normal helix for any distance whatever, but any greater refine- ment than we have suggested can hardly be carried 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 number of the cutter required may be calculated by the formula: — number of teeth for which cutter is to be selected = the number of teeth in the gear -^ by the cube of the cosine of the angle of teeth with the axis. Thus, in the example given on page 120, 30, the number of teeth -^ .8333^ = 52, the number of teeth for which the cutter should be selected. Referring to the table, page 104, it will be seen that a No. 3 cutter will be required. The difference, between normal pitch and linear or circular pitch is plainly seen in Figs. 55 and 56. The rack DD, Fig. 55, is of regular form, the depth of teeth being W 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 spiral gears, looking at them upon the side of the gears, are thicker in proportion to their depth, as in Fig. 56. This difference is seen between the teeth of the two racks DD and EE, Fig. 55. In the rack DD we have 20 teeth, while in the rack EE we have but 14 teeth; yet each rack will run with each of the spiral gears A, B or C,^Fig. 56, but at different angles. 122 BROWN & SHARPE MFG. CO. The teeth of one rack will accurately 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. 123 BROWN & SHARPE MFG. CO. CHAPTER XVIII Cutting Spiral Gears in a Universal Milling Machine Machine. A rotary disk cutter 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. settinJ"of the Bcforc cutting into a blank it is well to make a slight 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. 56, 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 DD, Fig. 55. 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 consideration of the meshing of straight side rack teeth with a spiral gear, as in Fig. 55. Spiral gears that run correctly with a rack, as in Fig. 55, will run correctly with each other when their axes are parallel, as at BC, Fig. 56 ; 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 angles, as AB, Fig. 56, though they will run well enough practically. The operation of cutting spiral teeth with a planer tool is sometimes called planing the teeth. Planing is an accurate way of 124 BROWN & SHARPE MFG. CO. /> a c Fig. 63 ( > fi n /> <^ f) [| 11 J — 1 \ Fig. 64 125 Data. BROWN & SHARPE MFG. CO. 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 not been found available for general use. It is convenient to have the data of spiral gears arranged as in the following table: Gear. No. of Teeth . Pitch Diameter. Outside Diameter Circular Pitch . Angle of Teeth with Axis Normal Circular Pitch Pitch of Cutter Addendum 5 Thickness of Tooth t Whole Depth D'^+/ 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 Pinion. A spiral of any angle to 45° can generally be cut in a universal milling machine without special attachments, 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. 126 BROWN & SHARPE MFG. CO. 1^ .«^'<^;/>i^» Fig. 65 USE OF VERTICAL SPINDLE MILLING ATTACHMENT IN CUTTING SPIRAL GEARS 127 BROWN & SHARPE MFG. CO. ^_^centrai Set- p^^ ^^^^ accuratc work, it is safer to test the position of the centres after the table has been set to the angle. This can be done with a trial piece, Fig. 63, 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 BC drawn on the side of the arbor at the same height as the centres; 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. 65 and 66; 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 sqmetimes been taken out of the cutter, but this is not recommended. The safest plan is to drop the work away. thfnfs".^'^^^*^'' In cutting spiral gears of greater angle than 45°, a vertical spindle milling attachment is available, as shown in Figs. 65 and 66. 128 BROWN & SHARPE MFG. CO. Fig. 66 USE OF VERTICAL SPINDLE MILLING ATTACHMENT IN CUTTING SPIRAL GEARS 129 BROWN & SHARPE MFG. CO. In Fig. 65 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° spiral, we subtract 70° from 90°, and the remainder, 20°, is the angle to set the table. In cutting on the top, Fig. 65, the attachment is set to 0. In Fig. QQ 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. tinf "^^ ^°'' ^^*' In setting the cutter central it is convenient to have a small knee as at K, Fig. 64. 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. QQ is for a right- hand spiral. If the gear blank were brought in front of the cutter, and the reversing gear set between two change gears, the machine would be set for a left-hand spiral. For coarser pitches than about 12P diametral, it is well to cut more than once around, the finishing cut being light so as to produce a smooth cut. 130 BROWN & SHARPE MFG. CO. CHAPTER XIX Spiral and Worm Gears — General Remarks The working of spiral gears, when their axes are parallel, spTaTfers. °^ 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 epicycloidal and involute curves, however, 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 centres 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 variation of centre distance is sometimes necessary in gears for rolls. Relative to spur and bevel gears in Chapter XIV, it was stated that all gears finally wore themselves 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 there is end pressure upon the upon'^shafS^ol shafts, because of the 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 spiral. 131 Distinctive features of BROWN & SHARPE 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 worm gears the axes are generally at right angles, or nearly so. The distinctive features of worm gearing may be stated as follows: The relative angular velocities do no depend upon the diameters of pitch cylinders, as in Chapter I. Thus, the worm in Chapter XIII, Fig. 40, can be any diameter — Worm Gearing. ^^^ ^^^^ ^^ ^^^ inchcs— without afifcctiug thc velocity of the worm wheel. Conversely if the axes are not parallel we can have a pair of spiral or worm 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 worm 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 Chapter IV. Worms are sometimes cut in a milling machine. If the form of the teeth in a pair of worm gears is determined upon the normal helix, as in Chapter XVII, 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. The angle of a, worm thread can be calculated the same as the angle of teeth of a 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. 132 BROWN & SHARPE MFG. CO. When a multiple-threaded worm is cut in a milling machine and the angle of the thread is less than 72° with the axis of the worm, it may be desirable to work by the normal pitch. The normal pitch can be obtained by multiplying the thread-pitch by the sine of the angle of the thread with the axis. WORM AND ^VORM AVHEEL IN POSITION FOR TESTING ON A BEVEL GEAR TESTING MACHINE 133 BROWN & SHARPE MFG. CO. TABLE No. 1 No. OF Value OF "Y" Teeth For 14|° Involute* For 20° Involute 12 .067 ,.078 13 .070 .083 14 .072 .088 15 .075 .092 16 .077 .094 17 .080 .096 18 .083 .098 19 .087 .100 20 .090 .102 21 .092 .104 23 .094 .106 25 .097 .108 27 .100 .111 30 .102 .114 34 .104 .118 38 .107 .122 43 .110 .126 50 .112 .130 60 .114 .134 75 .116 .138 100 .118 .142 150 .120 .146 300 .122 .150 Rack .124 .154 * Originally given as 15° TABLE No. 2 Safe Working Stress {^^) For Different Speeds Velocity of Pitch Line in Feet per Minute Material 100 200 300 600 900 1200 1800 2400 Cast Iron Steel 8000 20000 6850 17000 6000 15000 5350 13300 4000 10000 3200 8000 2650 6650 2000 5000 1600 4000 134 BROWN & SHARPE MFG. CO. CHAPTER XX Strength of Gears There are in existence many rules for the strength of gear teeth, which have some merit when applied to the particular conditions for which they were designed. To establish a rule that would fit all conditions would be impracticable, and in adopting any rule we should see that as many factors as possible have been considered. The rule, or method, that has received the greatest recognition is that proposed by Wilfred Lewis and described by him in a paper read before the Engineers' Club of Philadelphia on October 15th, 1892, and published in the proceedings of the club, January, 1893. While the Lewis formula does not consider all the factors that enter into the problem, its almost universal acceptance proves quite conclusively its soundness for use in ordinary conditions of cut gearing. In this chapter we have adopted the Lewis formula with only such modifications as make it more adaptable for general use. The factors used are as follows : W = allowable load in pounds at the pitch line. S = allowable stress per sq. in. for static load (zero speed) for material of the gears and assumed to be 2/3 to ultimate strength for cast iron and 2/3 the elastic limit of steel. P' = circular pitch of gear; distance from centre to centre of teeth on pitch line. F = Width of face of gear. Y = a factor for strength depending on pressure angle and number of teeth. (Table No. 1, page 134.) V = Velocity of pitch line in feet per minute; is the same for both gears of a pair. 135 BROWN & SHARPE MFG. CO. = a factor to modify or reduce the factor *'S" as the velocity *'V" increases from zero speed. HP = Horsepower. The equation for ''W" is: 600XSPTY (1) W = and HP 600 +V WV (2) 33000 Equation (1) is appHcable to any material and speed, but for average conditions the values given in Table 2, page 134, can be used. When this is done the equation for *'W" is simplified and W-SPTY (3) WV and HP = 33000 (2) Example: What is the allowable load '*W" in pounds and what horsepower can be transmitted by a pair of cast iron spur gears having 30T 6P1^" face running at 500 R.P.M? ^^_ 5X3.1416X500 12 V = 654.5 ft. per minute. By referring to Table 2 under 600 F.P.M. we find the allow- able stress S = 4000 lbs. Now looking in Table 1 under 14>^° and opposite 30T we find that then Y = .102 W = 4000 X.5236X 1.25 X. 102 (3) = 267 lbs. 136 BROWN & SHARPE MFG. CO. , ,T^ 267x654.5 ^^ , aad HP = = 5.3 horsepower (2) 33000 Now if these same sizes were to be made of a steel with an elastic limit of 60000 lbs. per square inch, we would proceed as follows: I X 60000 = 40000 = S for static load. .u w 600 X40000X. 5236 X 1.25 X. 102 (1) thenW = ^ 600+654 = 1277 lbs. and HP = ^^^'^X^^^ = 25 Horsepower (2) 33000 This is a case where the gears would have to be case- hardened and be thoroughly lubricated to transmit this power and give satisfactory wear. The values in Table 2 for "S" at zero speed are based on the original ones as given by Mr. Lewis, and are lower than necessary when the better grades of materials are used. In assuming higher values, however, it should be borne in mind that a point may be reached where excessive wear occurs in spite of the fact that the teeth are amply strong to withstand fracture. In such a case the question of wear becomes the limiting consideration and particular attention should be given to the combination of material of which the gears are made. For drives of small and medium size it is probable that gears of a high-grade casehardening steel, properly case- hardened, will transmit more power than those of any other available material. Where the gear is too large to caseharden readily, it is good practice to use a casehardened pinion with a gear of unhardened steel of not less than .40% carbon, and if the pinion can not be hardened, a steel .80% or more carbon could be used with the lower carbon gear. Both members are often made of cast iron and this is satisfactory when the work to be done is comparatively light, but if the ratio between the gear and pinion is large 137 BROWN & SHARPE MFG. CO. it is often advisable to use a steel pinion in order to equalize the strength and the wear. Both members of a drive should not be of unhardened low carbon steel unless the service is very light, as they are liable to seize and rough up. In calculating the strength of a pair of gears, each made of the same material, it is only necessary to consider the strength of the pinion, which is always the weaker member; but when made of different materials the strength of each member should be figured and the lesser of the two used in determining the strength of the drive. When a considerable amount of power is to be trans- mitted, it is highly beneficial to lubricate the tooth surfaces of the gears. If the speeds are moderate this can be satisfactorily accomplished by arranging one member so that it will dip into an oil bath, but where the speed is very high the centrifugal force tends to throw the oil off the teeth and under this condition it is advisable to arrange an oil pump so that oil can be applied to the teeth at the point of engagement. The effect of the speed or pitch line velocity upon the breaking strength of cast iron gears was investigated several years ago by Prof. Guido H. Marx of Leland Stanford University, and the results of the tests were published in the transactions of the American Society of Mechanical Engineers — Volume 34 — Page 1323, and Volume 37— Page 503. The gears used in these tests were made of cast iron 10 diametral pitch, some of 14>^° pressure angle standard depth and others of 20° pressure angle and stub teeth. The general conclusions derived from these tests were that the Lewis formula, which assumes the entire load taken at approximately the end of a single tooth, does not give as great a strength- value as is warranted. This is because the contact between the teeth is near the pitch line when the load is carried on one tooth and when the contact is at the point of the tooth where there is a second tooth in contact thus dividing the load. 138 BROWN & SHARPE MFG. CO. A formula based on the results obtained from these tests gives lower values for the safe load at low speeds and higher values at high speeds than the Lewis formula. BEVEL GEARS To calculate the strength of bevel gears the following factors are used : W = allowable load at pitch line in pounds. S = allowable stress per sq. in. for static load (zero speed) for material of the gears and assumed to be 2/3 of the ultimate strength for cast iron and 2/3 the elastic limit for steel. Y = a factor for strength. P' = circular pitch. Distances from centre to centre of teeth at large end. F = face width of gears. V = velocity at pitch line (large end) in feet per minute; is the same for both gears of a pair. HP = horsepower. D' = pitch diameter at large end. d' = pitch diameter at small, end. N = actual number of teeth. n = formative number of teeth. The number of teeth for which the cutter is chosen when cutting bevel gears with a rotary cutter and for which the factor Y is chosen. See Fig. 23. a = edge angle of gear (pitch angle) see Fig. 77 Chapter XXIV. N (4) n = ^ COS. a W = SPFY§, (5) 33000 (2) when S is taken from the table (2) . When other values for ''S" based on better material are taken, then — 139 BROWN & SHARPE MFG. CO. W = 600SPTY'd' 600 +VD' HP= WV 33000 (6) (2) Before calculating the strength of a pair of bevel gears it is best to make use of a simple graphical layout as in Fig. 67 to get the diameter d' at the small end; this is easier than to calculate it. Of Gear Centreline Note. Symbols D'and A Have Sub Letters InThis Fig. Only In Using The Value For D'AudJl' Care MustBeTaken ToUseThe Proper One In The Formulas Pitch Diameter Of PmiON=D^ Fig. 67 140 BROWN & SHARPE MFG. CO. Example: What is the allowable load ''W" in pounds and what horsepower can be transmitted by a pair of bevel gears with 20 and 60T 6P lyi" face and 20^ pressure angle pinion running at 300 R.P.M. Pinion of steel and gear of cast iron. As **V" is the same for both gears we will base our calculation on this pinion. ,, ,j 3.333x3.1416x300 ^.^ ,^ . ^ then V = = 262 ft. per mmute 12 "S" from Table 2 equals 13300 for steel and 5350 for iron. 20 20 n for pinion = = m this case ^,^^^ = 22 nearly COS. a .94869 cjr\ pjr\ n for gear = = in this case ^^^^ =190 nearly COS. a .3162 now taking the inside diameter d'p for pinion from Fig. 67 we have W = 13300 X .5236X 1.5 X -104 X^|g ^^ W = 774 774 V 262 and HP = -^^^ = 6 Horsepower (2) For gear we have W= 5350 X. 5236 X 1.5 X. 146 X^^^^^ ® W = 440 440 X 262 then HP = -^355q- =3>^ Horsepower (2) 141 BROWN & SHARPE MFG. CO. 142 BROWN & SHARPS MFG. CO. CHAPTER XXI Standard Proportions for Spur Gears In order to ensure having gears of good proportion, and of sufficient strength without excessive weight with uniformity of appearance where the gears are used in sets, it is important to follow some regular system in proportioning the gears. The following formulas and tables have been used in our works for several years and are based on our experience in designing gears of this type. Standard Proportions for Spur Gears Formulas are for Gears 20P 4'' Diam. up to liP 12" Diam. A = .138D'(^^^+.28)+.67 a = 2A B = .4A .4a or ^? C = — D' = Pitch Diam, T=4 +.25 G = (D"+/)+t+.06 *K = 1.5H+it *L = 1.5H P = Diam. Pitch D" +/ = Full Depth of Tooth t = Thick, of Tooth Table Giving Value of (f- + .28) to Aid in Finding "A" Diam. Pitch 1 1.23 10 .38 IK .915 11 .37 2 .755 12 .36 2K .660 13 .354 3 .597 14 .350 4 .520 15 .345 5 .470 16 .340 6 .44 17 .337 7 .42 18 .334 8 .40 9 Value of(p +.28) .. 39 Diam. Pitch 19 .332 20 Value of (-^+.28) 330 *These proportions are suitable for general practice but must be varied to suit special cases: — Thus the face "F" should be less than given when used for light change gears and the hole "H" should vary to suit conditions but can usually be brought within one of the formulas (| +.5) ; (f +.5) or (4- +.5) . 143 BROWN & SHARPE MFG. CO. Rules and Table to Determine Whether Gears shall be Solid, Web or Arm It is specially important that, in a set to be used together, there should be consistency in making gears solid or with web or arms. The figures given are for the outside diameter of the largest gear in each style. Sizes larger than figures given in "Web" column are to be made with six arms, pro- portioned as per formulas above. The proportions for "Combination Gears" should be followed for ordinary cases, using the proportions for "Change Gears" only in cases where light service is required. Diam. Pitch. web ^^', Diam. Pitch . 4 5 6 7 8 9 10 5.50 4.80 4.50 4.30 4.25 4.00 3.80 6.00 5.00 4.75 4.57 4.25 4.00 3.80 9.00 7.50 7.30 6.86 6.75 6.67 5.80 9.50 8.00 7.75 7.00 6.88 6.75 6.00 12 14 16 18 20 3.67 3.29 3.13 3.00 2.80 3.67 3.29 5.67 5.30 4.15 4.55 4.40 5.75 5.50 Make gear rims with a draft of 5° on inside. Make hubs with a taper of 3>^° on a side, except where finished, in which case do not taper hubs. {Round the corners on spur gear teeth as follows: — ^" Rad. to 20 P. Inc., ^" Rad. to 14 P. Inc., Ye" Rad. to 8 P. Inc., A" Rad. to 4 P. Inc., i" Rad. for coarser pitches. fMake diameter of hub = 1>^X Diam. of hole+><" except where heavy service is required, or large key ways or set screws are used, in which case make hub = If X diam. of hole-J-j", or reinforce hub over key way or around set screw. 144 BROWN & SHARPE MFG. CO. CHAPTER XXII Tangent of Arc and Angle We shall now show how to calculate some of the func- tions of a right angle triangle from a table of circular functions and the application of these calculations in some problems of gearing such as figuring the angles of bevel gears and in sizing blanks and cutting teeth of spiral gears, the selection of cutters for spiral gears, etc. A Function 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 his wages per hour. In any right angle triangle, OAB, we shall, for con- venience, call the two lines that form the right angle OAB the sides, instead of base and perpendicular. Thus OAB, being the right angle we call the line OA a side, and the line AB a side also. When we speak of the angle AOB, we call the line OA the side adjacent and the line AB the side opposite. When we are speaking of the angle ABO we call the line AB the side adjacent and the line OA the side opposite. The line connecting points OB is called the hypothenuse. In the following pages the definitions of circular func- tions are for angles smaller than 90°, and not strictly applicable to the reasoning employed in analytical 145 Functions of Right angle Triangle. Function de- fined. Right angle Triangle. Side adjacent. Hypothenuse. Tangent. BROWN & SHARPE MFG. CO. trigonometry, where we find expressions for angles of 270°, 760°, etc. The Tangent of an arc is the line that touches it at one extremity and is terminated by a line drawn from the centre 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. 69 To find Degrees in Angle. the an Thus, in Fig. 69, the line AB is the tangent of the arc AC. The point of tangency is at A. An angle at the centre of a circle is measured by the arc intercepted by the sides of the angle. Hence the tangent AB of the arc AC is also the tangent of the angle AOB. 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 OA being 1", if we know the length of the line or tangent AB we can, by looking in a table of tangents, find the number of degrees in the angle AOB. Thus, if AB is 2.25'' long, we find the angle AOB is QQ° 2' very nearly. That is, having found that 2.24956 is the nearest number to 2.25 in the Table of Tangents 146 BROWN & SHARPE MFG. CO. at the end of this volume, we find the corresponding degrees of the angle in the column at the bottom of the table and the minutes to be added at the right-hand side of the table. Now, if we have a right angle triangle with an angle the same as OAB, but with OA two inches long, the line AB will also be twice as long as the tangent of angle AOB, as found in a table of tangents. Let us take a triangle with the side 0A = 5'' long, finl^'Thf" De- and the side AB = 8" long; what is the number of degrees ingfe" '" """ in the angle AOB? Dividing 8'' by 5 we find what would be the length of AB if OA was only V long. The quotient then would be the length of tangent when the radius is V 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 correspond- ing angle is 58°, which would be the angle AOB when AB is 8'' and the radius OA 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 required 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 angle triangle: — divide the side opposite the angle by the side adjacent the angle and the quotient will be the tangent of the angle (see page 182). 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 ofSnAngS!"* subtracting the angle from 90°. Thus 40° is the com- plement of 50°. The Cotangent of an angle is the tangent of the com- cotangent. plement of the angle. Thus, in Fig. 69, the line AB is the cotangent of AOE. In right angle triangles either acute angle is the complement of the other acute angle. Hence, if we know one acute angle, by subtracting this angle from 90° we get the other acute angle. As the arc 147 To find the Tangent. BROWN & SHARPE MFG. CO. approaches 90° the tangent becomes longer, and at 90° it is infinitely long. The sign of infinity is a. Tangent 90° = a. An^?e^^by''\he By a tablc of tangents, angles can be laid out upon JmSfe.'' Fig. ^o' sheet 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 .25862. Multiply- ing .25862 by 5 we obtain, in the enlarged triangle, 1.29310" 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. 70, draw the line AB 5'' long; perpendicular to this line at A draw the line AO 1.293'' long; now draw the line OB, and the angle ABO will be 14°30'. If special accuracy is required, the tangent can be multiplied by 10; the line AO will then be 2.586" long and the line AB 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 AOB. The reader will remember these angles as occurring in Chapter IV, and obtained approximately in a different way. A semicircle upon the line OB touching the extremi- ties O and B will just touch the right angle at A, and the line OB is four times as long as OA. Let it be required to turn a piece 4" long, \" 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? 148 BROWN & SHARPE MFG. CO. Fig. 70 Fig. 71 149 BROWN & SHARPE MFG. CO. Dilmeter^''pf^*f A sectioii, Fig. 71, through the axis of this piece is 5ece^ Mg! 7L ^he saiiie as if we added two right angle triangles, O AB and O'A'B', to a straight piece A'ABB^ V wide and 4'' long, the acute angles B and B^ being 5°, thus, making the sides OB and O'B'10° with each other. The tangent of 5° is .08749, which, multiplied by 4'', gives .34996" as the length of each line, AO and A'O', to be added to V at the large end. Taking twice .34996'' and adding to V, we obtain 1.69992'' 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 care- fully 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, problems fnTri- ^0 solvc problcms in right angle triangles see 'Table angles. £qj. ^^le Solutlou of Right Angle Triangles," page 182. 150 BROWN & SHARPE MFG. CO. CHAPTER XXIII Sine and Cosine — Some of their Applications in Machine Construction The Sine of an arc is the line drawn 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. 72, AC is the sine of the arc BC, and of the angle BOC. It will be seen that the sine is always inside of the arc, and can never be longer than the radius. As the arc approaches 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 AC, opposite the angle AOC, in any right angle triangle, is the sine of the angle AOC, when OC is the radius of the arc. Hence the rule: — in any right angle triangle, the side opposite either acute angle, divided by the hypothenuse, is equal to the sine of the angle. (See table, page 182). 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 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. 72, the chord FC is twice as long as the sine AC. Fig. 72 Sine of Arc and Angle. To find the Sine. Chord of an Arc. 151 BROWN & SHARPE MFG. CO. / 3 \ ,''''""/^ """v "s /* / s / / / / / / s \ \ 1 / \ \ \ / X \^ \ \ \ \ \) ■ 1 ' \ 1 \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / / ^v y \ y "-^. r ■^ ^^ .1. Fig. 73 Example to find the Chord. Polygon. Let there be four holes equidistant about a circle ?>" in diameter, Fig. 73; what is the shortest distance between two holes? This shortest distance is the chord AB, which is twice the sine of the angle COB. The angle AOB is one-quarter of the circle, and COB is one- eighth of the circle. 360°, divided by 8 = 45°, the angle COB. The sine of 45° is .70711, which multiplied by the radius 1.5", gives length CB in the circle, 3'' in diameter, as 1.06066''. Twice this length is the required distance AB = 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 by straight lines is called a polygon. 152 BROWN & SHARPE MFG. CO. Cosine. When the outside diameter and the number of sides of ierj?h of side. a regular polygon are given, to find the length of one of the sides: — divide 360° by twice the number of sides; multi- ply the sine of the quotient by the outer diameter, and the product will be the length of one of the sides. Multiplying by the diameter is the same as multi- plying by the radius, and that product again by 2. The Cosine of an angle is the sine of the complement of the angle. In Fig. 72, COD is the complement of the angle AOC; the line CE is the sine of COD, and hence is the cosine of BOC. The line OA is equal to CE. It is quite as well to remember the cosine as the part of the radius, from the centre that is cut off by the sine. Thus, the sine AC of the angle AOC cuts off the cosine OA. The line OA may be called the cosine because it is equal to the cosine CE. In any right angle triangle, the side adjacent either acute angle corresponds to the cosine when the hypothe- nuse is the radius of the arc that measures the angle; hence: — divide the side adjacent the acute 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 to obtain Length of the diameter across the sides. ^on"' °^ ^°^^^'' To find the Cosine. Rule for Di- ameter across "sides of a Poly- gon. BROWN & SHARPE MFG. CO. Let a cylinder, 2" diameter, Fig. 75, be cut six-sided; what is the diameter across the sides? The angle AOB, at the centre, occupied by one of these sides, is one-sixth of the circle, = 60°. The cosine of one-half this angle, 30°, is the line CO; twice this Hne is the diameter across the sides. The cosine of 30° is .86603, which, multiplied by 2, gives 1.73206^' 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 regular polygon, when the diameter across corners is given, will then be: — multiply the cosine of 360° divided by twice the number of sides, by the diameter across corners, and the product will be the diameter across sides. The Table of Sines and Cosines is arranged like the Table of Tangents and Cotangents as explained on page 146. C B Fig. 75 To find the Diameter across corners of a Polygon. A six-sided piece is to be 1}4" across the sides; how large must a blank be turned before cutting the sides? Dividing 360° by twice the number of sides or 180° by the number of sides, we have 30°, which is the angle COB, Fig. 75. 154 BROWN & SHARPE MFG. CO. The radius of the six-sided piece is .1^" . Dividing .75'' by the cosine of 30° which will be found by the table to be .86603 gives .8660 the hypothenuse OB. .8660 X 2 = 1.7320+ the required diameter of blank. Hence, in a regular polygon, when the diameter across sides and the number of sides are given, to find diameter across corners, divide the distance across sides by the cosine of 180 divided by the number of sides and the quotient will be the distance across corners. 155 BROWN & SHARPE MFG. CO. No. 13 AUTOMATIC GEAR CUTTING MACHINE FOR SPUR AND BEVEL GEARS Cuts Spur and bevel gears to 18^' diameter, 4^' face. Cast iron, 5 diametral pitch; steel, 6 diametral pitch. 156 BROWN & SHARPE MFG. CO. CHAPTER XXIV Application of Circular Functions — Whole Diameter of Bevel Gear Blanks — Angles of Bevel Gear Blanks The rules given in this chapter apply only to bevel gears having the centre angle c'Oi not greater than 90°. To avoid confusion we will illustrate one gear only. The same rules apply to all sizes of bevel gears. Fig. 76 is the outline of a pinion 4P, 20 teeth, to mesh with a gear 28 teeth, shafts at right angles. For making sketch of bevel gears see Chapter X. In Fig. 76, the line Om'm is continued to the line ab. The angle c'Oi that the cone pitch line makes with the centre line may be called the centre angle. The centre angle c'Oi is equal to the angle of edge c'ic, c'i is the mgi side opposite the centre angle c'Oi, and c'O is the side adjacent the centre angle. c'/ = 2.5"; c'0 = 3.5". Divid- ing 2.b" by 3.5" we obtain .71428''+ as the tangent of c'Oi. In the table we find .71417 to be the nearest tangent, the corresponding angle being 35°32'. 35°32^ then, is the centre angle c'Oi and the angle of edge c'in, very 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°32' from 90° we obtain 54°28' as the angle of edge of gear 28 teeth, to mesh with gear 20 teeth. Fig. 76, 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° and we have the centre angle of the gear. 157 Angle of BROWN & SHARPE MFG. CO. Fig. 76 BEVEL GEAR DIAGRAM 158 BROWN & SHARPE MFG. CO. 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. To obtain angle of face Om"c', the distance c'O becomes ^ngie of Face. the side opposite and the distance m"c' is the side adja- cent. 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 Om"c'. The nearest tangent in the table is 1.24079 and the corresponding angle is 51°8'. To obtain cutting angle c'On" we divide the distance c'n" by c'O. By measurement c'^" 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°9'. The largest pitch diameter, kj, of a bevel gear, as in Fig. 77, is known the same as the pitch diameter of any spur gear. Now, if we know the distance ho or its equal aq, we can obtain the whole diameter of bevel gear blank by adding twice the distance bo to the largest pitch diameter. Twice the distance bo, or what is the same thing, Diameter in- the sum of aq and bo is called the diameter increment, because it is the amount by which we increase the largest pitch diameter to obtain the whole or outside diameter of bevel gear blanks. The distance bo can be calculated without measuring the diagram. The angle boj 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 bo is equal to the cosine of angle of edge, multiplied by the distance jo. The distance jo is the addendum, as in previous chapters ( = 5). Hence the rule for obtaining the diameter increment of any bevel gear: — multiply the cosine of angle of edge 159 BROWN & SHARPE MFG. CO. Fig. 77 DIAGRAM— BEVEL GEAR AND PINION 160 Outside Diam- eter. BROWN & SHARPE MFG. CO. by the working depth of teeth (D"), cind the product will be the diameter increment. By the method given on page 157, we find the angle of edge of gear (Fig. 77) is 56° 19'. The cosine of 56°19' is .55436, which, multipHed by 2/3", or the depth of the 3P gear, gives the diameter increment of the bevel gear 18 teeth, 3P meshing with pinion of 12 teeth. 2/3 of .55436 = .3696" (or .37", nearly). Adding the diameter increment, .37", to the largest pitch diameter of gear, &\ we have 6.37" as the outside diameter. In the same manner, the distance cd is half the dia- meter increment of the pinion. The angle cdk is equal to the centre angle of pinion, and when axes are at right angles is the complement of centre angle of gear. The centre angle of pinion is 33°40^ The cosine, multiplied by the working depth, gives .555" for diameter incre- ment 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. The small angle oOj is called the angle increment. ^^^^^^ ^""^' When shafts are at right angles the face angle of one gear is equal to the centre angle of the other gear, minus the angle increment. Thus, the angle of face of gear (Fig. 77) is less than the centre angle DO^, or its equal Ojk by the angle oOj. That is, subtracting oOj from Ojk, the remainder will be the angle of face of gear. Subtracting the angle increment from the centre angle of gear, the remainder will be the cutting angle. The angle increment can be obtained by dividing oj, the side opposite, by O;', the side adjacent, thus finding the tangent as usual. The length of cone pitch line from the common centre, Length of O to j, can be found, without measuring diagram, by dividing the adjacent side OB of the triangle OB7 by the cosine of the centre angle; to find the length of cone pitch 161 BROWN & SHARPE MFG. CO. line, divide the side adjacent to the centre angle by the cosine of the centre angle. The length of the side adjacent equals the radius of the pinion 2" which divided by .55436 the cosine of 56°19' = 3.6045'' the length of the line Oj. Dividing oj by Oj, we have for tangent .0924, and for angle increment 5° 17'. The angle increment can also be obtained by the following rule: Divide the sine of centre angle by half the number of teeth, and the quotient will be the tangent of increment angle. Subtracting the angle increment from centre angles of gear and pinion, we have respectively: Cutting angle of gear, 51° 2'. Cutting angle of pinion, 28°25'. Remembering that when the shafts are at right angles, the face angle of a gear is equal to the cutting angle of its mate (Chapter XI), we have: Face angle of gear, 28°25'. Face angle of pinion, 51°2'. It will be seen that both the whole diameter and the angles of bevel gears can be obtained without making a diagram. "Formulas in Gearing," published by us, contains extensive tables for bevel gearing, including tables Sine. BROWN & SHARPE MFG. CO. for diameter increment, angle of face and edge, etc., for bevel gears. In laying out angles, the following method may be AnSfe^^by''\he preferred, as it does away with the necessity of making a right angle: — draw a circle, ABO (Fig. 78), 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 AB. From any point O in the cir- cumference, draw the lines OA and OB. The angle AOB is the angle required. Thus, let the required angle be 12°. The sine of 12° is .20791, which, multiplied by 10, gives 2.0791'', or 2^" nearly, for the distance AB. 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 OB passes through the centre then the two lines AB and AO will form a right angle at A. 163 BROWN & SHARPE MFG. CO. CHAPTER XXV Angle of Pressure In Fig. 79, let A be any flat disk lying upon a hori- zontal plane. Take any piece, B, with a square end, ab. Press against A with the piece B in the direction of the arrow. Fig. 79 Fig. 80 It is evident A will tend to move directly ahead of B in the normal line cd. Now (Fig. 80) let the piece B, at one corner /, touch the piece A. Move the piece B along the line de, in the direction of the arrow. It is evident that A will not now tend to move in the line de, but will tend to move in the direction of the normal cd. When one piece, not attached, presses against another, the tendency to move the second piece is in the direction of the normal, at the point of contact. Line of Press- TMs normal is called the line of pressure. The angle that this line makes with the path of the impelling piece, is called the angle of pressure. In Chapter IV, the lines BA and BA' are called lines of pressure. This means that if the gear drives the rack. 164 BROWN & SHARPE MFG. CO. the tendency to move the rack is not in the direction of pitch line of rack, but either in the direction BA or BA', 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 Hne; 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 press- ure, 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. The arc through which a wheel passes while one of its teeth is in contact is called the arc of action. Until within a few years, the base of a system of double- tem curve interchangeable gears was 12 teeth. It is now g 15 teeth in the best- practice. (See Chapter VII) . 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. Arc of Action. Base of Sys- of I nter- changeable ears. 165 BROWN & SHARPE MFG. CO. CHAPTER XXVI Continued Fractions — Some Applications in Machine Construction Fractions. a Conti*Jrue°d A continucd fraction is one that has unity for its numerator, and for its denominator an entire number plus a fraction, which fraction has also unity for its numerator, and for its denominator an entire number plus a fraction, and thus in order. The expression, ^^f- s+r ^ is called a continued fraction. By the use of continued fractions, we are enabled of^cpntinu'^d to find a fraction expressed in smaller numbers, 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 (4P), 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: 4P = SqP'- This example will be considered farther on. For illustration, we will take a simpler example. What fraction expressed in smaller numbers is near- est in value to ^g ? Dividing the numerator and the denominator of a fraction by the same number does not change the value of the fraction. Dividing both 1 by 29, we have qi or, what is the same 1 thing expressed as a continued fraction, 5+j, . The 29 1 29 continued fraction s+T is exactly equal to 145 • If 29 now, we reject the 29, the fraction -5 will be larger than Example r. ,. ^5 terms of Continued Fractions. 166 BROWN & SHARPE MFG. CO. 1 5+j^ , because the denominator has been diminished, 29 5 being less than 629. y is something near 146 expressed in smaller numbers than 29 for a numerator and 146 for a 1 29 denominator. Reducing -5 and 1^ to a common denominator, we have j = ^^ and ^ = ^o- Subtracting one from the other, we have 730, which is the difference between j and f|.. Thus, in thinking of §q as j, we have a pretty fair idea of its value. There are fourteen fractions with terms smaller than „ -^^^L^.^f °^ approximation. valent . 29 and 146, which are nearer i^e than j is, such as 76' si 28 and so on to yTi- In this case by continued frac- tions we obtain only one approximation, namely -5, and any other approximations, as ^, gi, etc., we find by trial. It will be noted that all these approximations 29 are smaller in value than j^. 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 millimetre is equal ^,„Metric equi- to .03937 inch; what fraction in smaller terms expresses .03937'' nearly? .03937, in a vulgar fraction, is i^. Dividing both numerator and denominator by 3937, 1 we have 0^1575 . Rejecting from the denominator '^'-'3937 of the new fraction, Iff, the fraction ^ gives us a pretty good idea of the value of .03937". If in the J expression, 2.5+1575, we divide both terms of the 3937 fraction ~ by 1575, the value will not be changed. Performing the division, we have l..^ 2+787 1575. We can now divide both terms of {§-^ by 787, without changing its value, and then substitute the new fraction for ^ in the continued fraction. 167 BROWN & SHARPE MFG. CO. Dividing again, and substituting, we have: 1 25+1 2+1 2 +1 787 as the continued fraction that is exactly equal to .03937. 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 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 2^ or .04 — a little larger than .03937. If, now, we take ^^» we shall come still nearer .03937. The expression 2^^ is merely stating that 1 is to be divided by 25 >^. To divide, we first reduce 25>^ to an improper fraction,^, and the expression becomes 51, or one divided by f. 2 To divide by a fraction, 'Invert the divisor, and proceed as in multiplication." We then have ^ as the next nearest fraction to .03937. ^ = .0392 + , which is smaller than .03937. To get still nearer, we take in the next part of the continued fraction, and have 25+1 2+1 2. We can bring the value of this expression into a frac- tion, 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 lyi, is equal 168 BROWN & SHARPE MFG. CO. to -f. Stopping here, the continued fraction would become .,Vt, 2o+l Now, 5^ equals f, and we have 25+2 . 25^ equals 2 5 127. „,.i ^:^..^: :„ i -\- t>v' •^.' _ 1 u,. 127 -5-; substituting again, we have 127^ . Dividing 1 by 5 » we have ^. ^j is the nearest fraction to .03937, unless we reduce the whole continued fraction ~ 25+1 2+1 787, which would give us back the .03937 itself. —=.03937007, which is only 100000000 larger .03937. It is not often that an approximation will come so near as this. This ratio, 5 to 127, is used in cutting millimetre Ppcticai use 01 the toregoing thread screws. If the leading screw of the lathe is Example. 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 millimetre threads near enough for practical purposes. The foregoing operations are more tedious in description 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, ^, ^, jlj, 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 = ^0 J 169 BROWN & SHARPE MFG. CO. reducing to lowest terms, we the operation for the greatest common divisor: have ig. 3927) 5000 (1 3927 1073) 3927 (3 3219 708) 1073 (1 708 365) 708 (1 365 343^ 365 (1 343 22) 343 (15 22 l23 110 13) 22 (1 13 9) 13 (1 4) 9 (2 _8 1) 4 (4 4 Applying Rule Bringing the various incomplete quotients as denomi- nators in a continued fraction as before, we have : 1 1 + 1 3 + 1 1 + 1 1 + 1 1 + 1 15 + 1 1 + 1 1 + 1 2 + i Now arrange each partial quotient in a line, thus: 1 1 1 15 3 4 7 11 172 183 355 893 3927 4 5 9 14 219 233 452 1137 5000 Now place under the first incomplete quotient the first reduction or convergent y, which, of course, is 1; put under the next partial quotient the next reduction or convergent 1^3/3 or 1%, which becomes % . 1 is larger than .7854, and ^ 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- 170 BROWN & SHARPE MFG. CO. tinned fraction, or the next incomplete quotient, and add to the product the numerator of the preceding convergent: the sum will he 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. Con- tinue until the last convergent is the original fraction. Under each incomplete quotient or denominator from the continued fraction arranged in line, will be seen the corresponding convergent or reduction. The con- vergent H is the one commonly used in cutting racks 4P. This is the same as calling the circumference of a circle f when the diameter is one (1); this is also the common ratio for cutting any rack. The equivalent decimal to ^ is .7857+, being about ^^ large. In three settings for rack teeth, this error would amount to about .OOr'. For a worm, this corresponds to n threads to V'\ now, with a leading screw of lathe 3 to I", 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 4P gear. In "Formulas in Gearing", tables of factors and prime numbers are given which are of assistance in problems requiring the use of continued fractions. 171 BROWN & SHARPE MFG. CO. CHAPTER XXVII Squares and Square Roots To square a number, multiply it by itself. The expression 7^ indicates that 7 is to be squared. 72 = 7 X 7 = 49, the square of 7. 234' = 234 X 234 = 54756, the squaie of 234. To obtain the square root of a number proceed as follows: (1) Space the number into groups of two figures each both ways from the decimal point. (2) Find the greatest square in the first group on the left, and set its root on the right. Subtract the square thus found from the first group and to the remainder annex the two figures of the next following group for a dividend. (3) Double the root first found for a divisor and find how many times it is contained in the dividend exclusive of the right hand figure of the latter and set that quotient figure both in the quotient and divisor. Multiply the whole divisor by the last quotient figure and subtract the product from the dividend, bringing down the next group for a new dividend. (4) Repeat the process under Rule 3 and so on through all the periods to the last. Let us find" the square root of 54756. 172 BROWN & SHARPE MFG. CO. The expression \l 54756 indicates that the square root of 54756 is to be found. 5 47 56 234 Ans. 2' ^ = 4 2X2=4— 1 47 Annex 3 43 1 29 23X2 = 46— 1856 \nnex 4 464 1856 Therefore x' 54756 = 234. Find the square root of .1968 4' I 19 68 16 .443+ Ans. 4x2=8— Annex 4 84 3 68 3 36 44x2 = 8^ Annex 3 320( 264^ ) SI B3 ) 551 Therefore \L 1986 = .443 + 173 BROWN & SHARPE MFG. CO. TABLES TABLE GIVING CHORDAL THICKNESS OF GEAR TEETH (t'O AND DISTANCE FROM CHORD TO TOP OF TOOTH (s") When accurate measure- ments of gear teeth are required, it is necessary to work to the chordal thickness of tooth. This thickness, also the distance from the chord to the top of tooth, varies from the figures given in the ''Table of Tooth Parts" being greatest for gears with the fewest teeth. The table gives the chordal thick- ness of teeth and the distance from the chord to the top of tooth for gears of 1 diametral pitch. To obtain V and s'' for any diametral pitch divide the figures given in the table opposite the required number of teeth by the required diametral pitch. Example: — Find f' and s'' for a ' gear 5 diametral pitch, 23 teeth 1.5696 -^5 = . 3139 = t'' 1.0268-^5 = .2054 = s'' To obtain V and s'' for any circular pitch multiply the figures given in the table, opposite the required number of teeth by s (taking ''s" from the ''Table of Tooth Parts", pages 178 and 179). Example: — Find V and s'' for a ^'' circular pitch gear, 15 teeth. 1. 5679 X. 2387 = . 3743 = t^ 1.0411 X. 2387 = . 2485 = s^ 174 BROWN & SHARPE MFG. CO. CHORDAL THICKNESS OF GEAR TEETH FOR 1 DIAMETRAL PITCH NUMBER OF TEETH t" s" NUMBER OF TEETH t" s* NUMBER OF TEETH t" s" 94 5707 1.0066 6 55^9 1. 1022 50 5705 1.0123 95 5707 1.0065 7 55^^^ I.0S73 51 5706 1.0121 96 5707 1.0064 8 5607 1.0769 52 5706 1.0119 97 5707 1.0064 9 5628 1,0684 53 5706 I.01I7 98 5707 1.0063 10 5643 I.0616 54 5706 I.01I4 99 5707 1.0062 1 1 5654 1-0559 55 5706 I.01I2 100 5707 1. 0061 12 S'^^j 1.0514 56 5706 1. 01 10 101 5707 1.0061 13 5670 1.0474 57 5706 1.0108 102 5707 1.0060 14 5675 1.0440 58 5706 1.0106 103 5707 1.0060 15 5679 1. 04 1 1 59 5706 I.OIO5 104 -5707 1.0059 16 56S3 1-0385 60 5706 1.0102 105 5707 1.0059 17 5686 1.0362 61 5706 I.OIOl 106 5707 1.0058 18 .5688 1.0342 62 5706 I.OIOO 107 5707 1.0058 19 5690 1.0324 63 5706 1.0098 108 5707 1.0057 20 5692 1.0308 64 5706 1.0097 109 5707 1.0057 21 5694 1.0294 65 5706 1.0095 1 10 5707 1.0056 22 5695 1.0281 66 5706 1.0094 1 i 1 5707 1.0056 23 5696 1.0268 67 5706 1.0092 1 12 5707 1-0055 24 5697 1.0257 68 5706 1.0091 1 13 5707 1-0055 25 5698 1.0247 69 5707 1.0090 1 14 5707 1.0054 26 5698 1.0237 70 5707 1.0088 1 15 5707 i.ooc;4 27 5699 1.0228 71 5707 1.0087 1 16 5707 1-0053 28 5700 1.0220 72 5707 1.0086 1 17 5707 1-0053 29 5700 1. 02 1 3 73 5707 1.0085 1 18 5707 1-0053 30 5701 1.0208 74 5707 1.0084 1 19 5707 1.0052 31 5701 1.0199 75 5707 1.0083 120 5707 1.0052 32 5702 1.0193 76 5707 1.0081 121 5707 1.0051 33 5702 1.0187 77 5707 1.0080 122 5707 1. 0051 34 5702 1.0181 78 5707 1.0079 123 5707 1.0050 35 5702 1.0176 79 5707 1.0078 124 5707 1.0050 36 5703 1. 0171 80 5707 1.0077 125 5707 1.0049 37 5703 1.0167 81 5707 1.0076 126 5707 1.0049 38 5703 1.0162 82 5707 1.0075 127 5707 1.0049 39 5704 1.0158 83 5707 1.0074 128 5707 1.0048 40 5704 1-0154 84 5707 1.0074 129 5707 1.0048 41 5704 1.0150 85 5707 1.0073 130 5707 1.0047 42 5704 1.0147 86 5707 1.0072 131 5708 1.0047 43 5705 1.0143 87 5707 1.0071 132 5708 1.0047 44 5705 1.0140 88 5707 1.0070 133 5708 1.0047 45 5705 1.0137 89 5707 1.0069 134 5708 1.0046 46 5705 1.0134 90 5707 1.0068 135 5708 1.0046 47 5705 1.0131 91 5707 1 .0068 150 5708 1.0045 48 5705 1.0129 92 5707 1.0067 250 5708 1.0025 49 1 5705 1.0126 93 5707 1.0067 Rack 5708 1.0000 175 BROWN & SHARPE MFG. CO. DIAMETRAL PITCH. "NUTTALL." Diametral Pitch is the Number of Teeth to Each Inch of the Pitch Diameter. To Get Havinj Rule. Formula. The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. Pitch Diameter. Pitch Diametei Pitch Diametei Pitch Diameter. Outside Diameter Outside Diameter Outside Diameter, Outside Diameter Number of Teeth. Number of Teeth. Thickness of Tooth Addendum. Root. Workins Depth. Whole Depth. Clearance. Clearance. The Circular Pitch. The Pitch Diameter and the Number of Teetii .... The Outside Diame- ter and theNunibe of Teeth .... The Number of Teetli and the Diametral Pitch The Number of Teeth and Outside Diam- eter The Outside Diame- ter and the Diam- etral 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 tlie Diametral Pitch The Outside Diame- ter and the Diame- tral Pitch . . . The Diametral Pitch The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. Thickness of Tooth. Divide 3.1416 by the Circular Pitch Divide Number of Teeth by Pitch Diameter Divide Number of Teeth plus 2 by Outside Diameter Divide Numl)er of Teeth by the Diametral Pitch Divide the i)roduct of Outside Diameter and Numl)er 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 ])lu 2 by the quotient of Number of Teeth and by the Pitch Diameter Multiply the Numl)er of Teeth plus 2 by Addendum .... Multiply Pitch Diameter by the Diametral Pitch Multiply Outside Diameter by the Diametral Pitch and subtract 2. Divide 1.5708 by the Diametral Pitch Divide 1 by the Diametral Pitch, D' or s = -^ Divide 1.157 by the Diametral Pitch Divide 2 by the Diametral Pitch. Divide 2.157 by the Diametral Pitch Divide .157 by the Diametral Pitch Divide Thickness of Tooth at pitch line by 10 .1416 P' = N D' N+2 D - N "P~ DN N+2 D'= D — D'= sN D^ N+2 P D = D = N+2 N . D'' D = (N+2) s N, = D'P N = DP — 1.5708 t = - S+f: D"= 1.157 2.157 f^ .157 176 BROWN & SHARPE MFG. CO. CIRCULAR PITCH. "NUTTALL." Circular Titch is the Distance from the Centre of One Tootli to the Centre of the Next Tooth, Measured along the Pitch Line. To Get The Circular Pitch, The Circular Pilch, The Circular Pitch, Pitch Diameter, Pitch Diameter, Pitch Diameter. Pitch Diameter, Outside Diameter Outside Diameter, Outside Diameter. Number of Teeth. Thickness of Tooth Addendum. Root. Workinjr Depth. Whole Depth. Clearance. Clearance. Havinf The Diametral Pitch. The Pitch Diameter and the Number of Teeth The Outside Diame- ter and the Number of Teeth „ . . . The Number of Teeth and the Cii'cular 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 Number of Teeth and the Circular Pitch The Pitch Diameter and the Circular Pitch ..... The llumber of Teeth r.nu t!ic Addendimi The Pitch Diameter and the Circular Pitch . . . The Circular Pitch. Tiie Circular Pitch. The Circular Pitch. The Circular Pit(^h. The Circular Pit(;h. The Circular Pitch. Thickness of Tooth. . Rule. Divide 3.1416 ))y the Diametral Pitch Divide Pitch Diameter by 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 product 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 plus 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 = -5-' N INFultiply the Circular Pit(;h bv .3683 " Multiply the Circular Pitch b\ .6366 ■ Multiply the Circular Pitch by .6866 iVfidtiply the Circular Pitch by .05 One-tenth the Thickness of Tooth at Pitch Line Formula. P'= 3.1416 P D^ .3183 N D .3183 N+2 D'=NP'.3183 D'= ND N+2 D'r=D— (P'.6366) D'=N8 D=:(N+2)P'.3183 D=D'-4-(P'.63G6) P' s = = P' 3 183 s + f = = P' .3683 I)' = P ' .()366 D ' + f = P'.6866 f = t 05 f = 177 BROWN & SHARPE MFG. CO. TABLE OF TOOTH PARTS CIRCULAR PITCH IN FIRST COLUMN Jl Threads or Teeth per inch Linear. 2 4 1^ Thickness of Tooth on Pitch Line. ^4 II t r Depth of Space below Pitch Line. « 1 > o Width of Thread-Tool at End. Width of Thread at Top. p' p' P t s D" s+f D"+/ P'X.3095 P'X.3354 2 1 2 1.5708 1.0000 .6366 1.2782 .7366 1.3732 .6190 .6707 If 8 15 1.6755 .9375 .5968 1.1937 .6906 1.2874 .5803 .6288 H 4 T 1.7952 .8750 .5570 1.1141 .6445 1.2016 .5416 .5869 If 8 1.9333 .8125 .5173 1.0345 „5985 L1158 .5029 .5450 li f 2.0944 .7500 .4775 .9549 .5525 1.0299 .4642 .5030 ih 16 -23 2.1855 .7187 .4576 .9151 .5294 .9870 .4449 .4821 11 -8. 11 2.2848 .6875 .4377 .8754 .5064 .9441 .4256 .4611 11 A. 4 2.3562 .6666 .4244 .8488 .4910 .9154 .4127 .4471 1^ 16 21 2.3936 .6562 .4178 .8356 .4834 .9012 .4062 .4402 H JL 5 2..5133 .6250 .3979 .7958 .4604 .8583 .3869 .4192 li 16 -19- 2.6456 .5937 .3780 .7560 .4374 .8154 .3675 .3982 li 8 9 2.7925 .5625 .3581 .7162 .4143 .7724 .3482 .3773 ifr 16 17 2.9568 .5312 .3382 .6764 .3913 .7295 .3288 -.3563 1 1 3.1416 .5000 .3183 .6366 .3683 .6866 .3095 .3354 15 16 1* 3.3510 .4687 .2984 .5968 .3453 .6437 .2902 .3144 7 8 li 3.5904 .4375 .2785 .5570 .3223 .6007 .2708 .2934 # If 3.8666 .4062 .2586 .5173 .2993 .5579 .2515 .2725 f 11 3.9270 .4000 .2546 .5092 ._2946 .5492 .2476 .2683 3. 4 11 4.1888 .3750 .2387 .4775 .2762 .5150 .2321 .2515 11 • 16 1^ 4.5696 .3437 .2189 .4377 .2532 .4720 ..2128 .2306 2 3 11 4.7124 .3333 .2122 .4244 .2455 .4577 .2063 .2236 5 8 11 5.0265 .3125 .1989 .3979 .2301 .4291 .1934 .2096 3 5 11 5.2360 .3000 .1910 .3820 .2210 .4120 .1857 .2012 -4- 7 11 5.4978 .2857 .1819 .3638 .2105 .3923 .1769 .1916 9 16- 11 5.5851 .2812 .1790 .3581 .2071 .3862 .1741 .1886 To obtain the size of any part of a multiply the corresponding part of 1' circular pitch not given in the ' pitch by the pitch required. table, 178 BROWN & SHARPE MFG. CO. TABLE OF TOOTH PARTS— Continued CIRCULAR PITCH IN FIRST COLUMN Threads or Teeth per inch Linear. TJiickness of Tooth on Pitch Line. t .a ^ r Depth of Space below Pitch Line. Width of Thread-Tool at End. o H SI ^ s P' 1" p' p t s D" s+f D'^/. P'X.30g5 P'X.3354 -|- 2 6.2832 .2500 .1592 .3183 .1842 .3433 .1547 .1677 ~ 21 7.0685 .2222 .1415 .2830 .1637 .3052 .1376 .1490 IT 2-f 7.1808 .2187 .1393 .2785 .1611 .3003 .1354 .1467 -f- 21- 7.3304 .2143 .1364 .2728 .1578 .2942 .1326 .1437 2 T 21- 7.8540 .2000 .1273 .2546 .1473 .2746 .1238 .1341 8 2f 8.3776 .1875 ,1194 .2387 .1381 .2575 .1161 .1258 i 11 2f 8.6394 .1818 .1158 .2316 .1340 .2498 .1125 .1219 1 3 9.4248 .1666 .1061 .2122 .1228 .2289 .1032 .1118 & 16 3i 10.0531 .1562 .0995 .1989 .1151 .2146 .0967 .1048 8 10 3i 10.4719 .1500 .0955 .1910 .1105 .2060 .0928 .1006 2 7 8i 10.9956 .1429 .0909 .1819 .1052 ,1962 .0884 .0958 1 T 4 12.6664 .1250 .0796 .1591 .0921 .1716 .0774 .0838 9 T 4i 14.1372 .1111 .0707 .1415 .0818 .1526 .0688 .0745 1 5 5 15.7080 .1000 .0637 .1273 .0737 .1373 .0619 .0671 3 10 51- 16.7552 .0937 .0597 .1194 .0690 .1287 .0580 .0629 11 5f 17.2788 .0909 .0579 .1158 .0670 .1249 .0563 .0610 1 6 6 18.8496 .0833 .0531 .1061 .0614 .1144 .0516 .0559 2 la 6i 20.4203 .0769 .0489 .0978 .0566 .1055 .0476 .0516 T 7 21.9911 .0714 .0455 .0910 .0526 .0981 .0442 .0479 2 15 7-1- 23.5619 .0666 .0425 .0850 .0492 .0917 .0413 .0447 1 T 8 25.1327 .0625 .0398 .0796 .0460 .0858 .0387 .0419 T 9 28.2743 .0555 .0354 .0707 .0409 .0763 .0344 .0373 JL 10 10 31.4159 .0500 .0318 .0637 .0368 .0687 .0309 .0335 1 16 16 50.2655 .0312 .0199 .0398 .0230 .0429 .0193 .0210 1 20 62.8318 .0250 .0159 .0318 .0184 .0343 .0155 .0168 To obtain the size of any part of a circular pitch multiply the corresponding part of 1" pitch by the not given in the table, pitch required. 179 BROWN & SHARPE MFG. CO. TABLE OF TOOTH PARTS DIAMETRAL PITCH IN FIRST COLUMN l-g 1^ II Thickness of Tooth on Pitch Line. 1^ <1 t 1 Depth of Space below Pitch Line. Whole Depth of Tooth. p P' t s D" s+f. D"+/. y2 G.2832 3.1416 2.0000 4.0000 2.3142 4.3142 M 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 IM 2.5133 1.2566 .8000 1.6000 .9257 1.7257 IK 2.0944 1.0472 .6666 1.3333 .7714 1.4381 IH 1.7952 .8976 .5714 1.1429 .6612 1.2326 2 1.5708 .7854 .5000 1.0000 .5785 1.0785 2M 1.3963 .6981 .4444 .8888 .5143 .9587 2K 1.2566 .6283 .4000 .8000 .4628 .8628 2M 1 . 1424 .5712 .3636 .7273 .4208 .7844 3 1.0472 .5236 .3333 .6666 .3857 .7190 33^ .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. 180 BROWN & SHARPE MFG. CO. TABLE OF TOOTH P ART S^Continued DIAMETRAL PITCH IN FIRST COLUMN 5^ Thickness of Tooth on Pitch Line. -^ or the Addendum or Module. Depth of Space below Pitch Line. Whole Depth of Tooth. P. P'. t. s. D". s+f. D"./. 15 .2094 .1047 .0866 .1333 .0771 .1438 16 .1963 .0982 .0625 .1250 .0723 .1348 17 .1848 .0924 .0588 .1176 .0681 .1269 18 .1745 .0873 .0555 .1111 .C643 .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 .0833 .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 GO .0524 .0262 .0166 .0333 .0193 .0360 To obtain the size of any part divide the corresponding part of of a diametral pitch not given in the table, 1 diametral pitch by the pitch required. 181 BROWN & SHARPE MFG. CO. TABLE FOR THE SOLUTION OF RIGHT ANGLED TRIANGLES SOLUTION OF TRIANGLES BY NATURAL LINES PARTS PARTS TO BE FOUND. GIVEN. Angle. Adj.- Side. Opp. Side. Hyp. Opp. Ang. ^^-°^. , Opp. and Hyp. jHyp.2-0pp.2 Co-il T-=^; , Opp. and Adj. J0pp.2+Adj.2 Co'- = i^: Adj. and Hyp. /-, Adj . Cos. = .^ Hyp. J Hyp.2-Adj.2 s'-ife Ang. and Opp. Opp.XCot. Opp. -^Sin. 90°— Ang. Ang. and Adj. Adj. X Tang. Adj. -^ Cos. 90°— Ang. Ang. and Hyp. Hyp. X Cos. Hyp. X Sin. 90°— Ang. ADJ. OPP. ABBREVIATIONS USED Opp. = Opposite side. Adj. = Adjacent side. Hyp. = Hypothenuse. Ang. = Angle. Sin. = Sine. Tan. = Tangent. Cos. = Cosine. Cot. = Cotangent. 182 Natural Sines and Cosines BROWN & SHARPE MFG. CO. NATURAL SINES AND COSINES / I 2° 3 D 4° r / Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine .00000 .01745 .99985 .03490 .99939 .05234 .99863 .06976 •99756 60 I ,00029 .01774 .99984 .03519 .99938 .05263 .99861 .07005 .99754 59 2 .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 5S 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 .997.36 50 II .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 .99725 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 .99915 .05873 .99827 .07614 .99710 38 23 .00669 .99998 .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 .99911 .05960 .99822 .07701 .99703 3S 26 .00756 .99997 .02501 .99969 .04246 .99910 .05989 .99821 .07730 .99701 34 27 .00785 .99997 .02530 .99968 .04275 .99909 .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 .998x5 .07817 .99694 31 30 .00873 .99996 .02618 .99966 .04362 .99905 .06105 .99813 .07846 .99692 30 31 .00902 .99996 .02647 .99965 .04391 .99904 .06134 .99812 .07875 .99689 29 32 .00931 .99996 .02676 .99964 .04420 .99902 .06163 .99810 .07904 .99687 28 33 .009O0 .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 2Z 38 .01105 .99994 .02850 .99959 .04594 .99894 .06337 .99799 .08078 .99673 22 39 .01134 .99994 .02879 .99959 .04623 .99893 .06366 .99797 .08107 .99671 21 40 .01164 .99993 .02908 .99958 .04653 .99892 .06395 •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 .99955 .04740 .99888 .06482 .99790 .08223 .99661 17 44 .01280 .99992 .03025 .99954 .04769 .99886 .06511 .99788 .08252 .99659 16 45 .01309 •99991 .03054 .99953 .04798 .99885 .06540 .99786 .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 .99951 .04885 .99881 .06627 .99780 .08368 .99649 12 49 .01425 .99990 .03170 .99950 .04914 .99879 .06656 .99778 .08397 .99647 II 50 .01454 .99989 .03199 .99949 .04943 .99878 .06685 •99776 .08426 .99644 10 51 .01483 .99989 .03228 .99948 .04972 .99876 .06714 .99774 .08455 .99642 9 52 .01513 .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 .9963s 6 .01600 .99987 .03345 .99944 .05088 .99870 .06831 .99766 .08571 .99632 S 56 .01629 .99987 .03374 .99943 .05117 .99869 .06860 •99764 .08600 .99630 4 n .01658 .99986 .03403 .99942 .05146 .99867 .06889 .99762 .08629 •99627 3 58 .01687 .99986 .03432 .99941 .05175 .99866 .06918 .99760 .08658 .99625 2 59 .01716 .99985 .03461 .99940 .05205 .99864 .06947 •99758 .08687 .99622 I 60 .01745 .99985 .03490 .99939 .05234 .99863 .06976 .99756 .08716 .99619 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 8c )° 8^ 5° 87 86 8^ -0 184 BROWN & SHARPE MFG. CO. NATURAL SINES AND COSINES 1 5 6 7 8° 9° / Sine Cosine Sine Cosine Sine Cosine Sine C Cosine Sine C 'osine .08716 .99619 .10453 •99452 .12187 .59255 • 13917 99027 .15643 98769 60 1 .08745 .99617 .10482 .99449 .12216 •99251 .13946 99023 .15672 98764 59 2 .08774 .99614 .10511 •99446 .12245 .99248 .13975 99019 .15701 98760 58 3 .08803 .99612 .10540 .99443 .12274 •99244 .14004 99015 .15730 98755 57 4 .08831 .99609 .10569 .99440 .12302 .99240 .14033 99011 .15758 98751 56 .08860 .99607 .10597 •99437 • 12331 .99237 .14061 99006 .15787 98746 55 6 .08889 .99604 .10626 •99434 .12360 .99233 .14090 99002 .15816 9874 X 54 7 .08918 .99602 • 10655 •99431 .12389 .99230 .14119 98998 .15845 98737 53 8 .08947 .99599 .10684 • 99428 .12418 .99226 .14148 98994 .15873 98732 52 9 .08976 .99596 .10713 •99424 .12447 .99222 .X4177 98990 .15902 98728 SX 10 .09005 .99594 .10742 .99421 .12476 .99219 .14205 98986 .15931 98723 50 II .09034 .99591 .10771 .99418 .12504 .992x5 .X4234 98982 .15959 98718 49 12 .09063 .99588 .10800 •99415 .12533 .9921 X .14263 98978 .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 9896s .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 9B676 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 25 .09440 .99553 .11176 •99374 .12908 .99163 .14637 98923 .16361 98652 35 26 .09469 .99551 • 11205 •99370 .12937 .99160 .14666 98919 .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 .09585 .99540 .11320 •99357 •13053 •99144 .14781 98902 .16505 98629 30 31 .09614 .99537 .11349 .99354 .13081 .99I4X .X4810 98897 .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 .99x29 .14896 98884 .16620 98609 26 35 .09729 .99526 .11465 .99341 •13197 .99125 .14925 98880 .16648 98604 25 36 .09758 •99523 •I1494 .99337 .13226 .99122 .14954 98876 .16677 98600 24 37 .09787 • 99520 ■ 11523 •99334 • 13254 .99118 .14982 98871 .16706 98595 23 38 .09816 .99517 •IISS2 •99331 •13283 .99114 .15011 98867 .16734 98590 22 39 .09845 .99514 .11580 •99327 • 13312 .99110 .15040 98863 .16763 98585 21 40 .09874 .99511 .11609 .99324 • 13341 .99106 .15069 98858 .16792 98580 20 41 .09903 .99508 .11638 .99320 •13370 .99102 .15097 98854 .16820 9857s 19 42 .09932 •99506 .11667 •99317 .13399 .99098 .15126 98849 .16849 98570 18 43 .09961 .99503 .11696 .99314 • 13427 .99094 • X515S 98845 .16878 98565 17 44 .09990 .99500 .11725 .99310 • 13456 •99091 .X5184 98841 .16906 98561 16 45 .10019 .99497 •II754 .99307 '3485 • 99087 .15212 98836 .16935 98556 IS 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 .10X35 .99485 .11869 .99293 .13600 •99071 • 15327 98818 .17050 98536 II SO .10164 .99482 .11898 .99290 .13629 .99067 • 15356 98814 .17078 98531 10 51 .10192 .99479 .11927 .99286 .13658 •99063 • 15385 98809 .17x07 98526 9 S2 .10221 .99476 .11956 .99283 .13687 .99059 .15414 98805 .17136 98521 8 S3 .10250 .99473 .11985 .99279 •13716 .99055 .15442 98800 .17164 98516 7 S4 .10279 .99470 .12014 .99276 • 13744 .99051 .15471 98796 .17193 98511 6 SS .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 .12.58 .99258 .13889 .99031 • IS615 98773 .17336 98486 I 60 .10453 .99452 .12187 .99255 .13917 ■99027 • 15643 98769 .17365 98481 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 8. ^° 8: 5° 8i 2° 81^ 3 80^ > 185 BROWN & SHARPE MFG. CO. NATURAL SINES AND COSINES / 10 II 12 13 M" 1 / Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine o •17365 .98481 .19081 .98163 .20791 .97815 .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 •97015 58 3 .I74SI .98466 .19167 .98146 .20877 .97797 .22580 .97417 •24277 .97008 57 4 .17479 .98461 .19195 .98140 .20905 .97791 .22608 •974 1 1 .24305 .97001 S6 S .17508 .98455 .19224 .98135 .20933 .97784 .22637 •97404 .24333 .96994 55 6 .17537 .98450 .19252 .98129 .20962 .97778 .22665 .97398 .24362 .96987 54 7 .17565 .98445 .19281 .98124 .20990 .97772 .22693 .97391 .24390 .96980 53 8 .17594 .98440 •19309 .98118 .21019 .97766 .22722 .97384 .24418 .96973 52 9 .17623 .98435 .19338 .98112 .21047 .97760 .22750 .97378 .24446 .96966 51 10 .17651 .98430 .19366 .98107 .21076 .97754 .22778 .97371 .24474 .96959 SO II .17680 .98425 .19395 .98101 .21104 .97748 .22807 .97365 .24503 .96952 49 12 .17708 .98420 •19423 .98096 .21132 .97742 .22835 .97358 .24531 .96945 48 13 .17737 .98414 •19452 .98090 .21161 .97735 .22863 .97351 .24559 .96937 47 14 .17766 .98409 .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 •97325 .24672 .96909 43 i8 .17880 .98389 •I9S9S .98061 .21303 .97705 .2300s •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 .98368 .19709 .98039 .21417 .97680 .23118 .97291 .24813 .96873 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 25 .18081 .98352 .19794 .98021 .21502 .97661 .23203 •97271 .24897 .96851 35 26 .i8iog .98347 .19823 .98016 .21530 •97655 .23231 •97264 .24925 .96844 34 21 .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 • 1996s •97987 .21672 •97623 .23373 .97230 .25066 .96807 ^l 32 .18281 .983 IS .19994 .97981 .21701 •97617 .23401 .97223 .25094 .96800 28 33 .18309 .98310 .20022 .97975 .21729 .97611 .23429 .97217 .25122 .96793 21 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 .18424 .20136 .97952 .21843 .97585 .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 ^l 44 .18624 .98250 .20336 .97910 .22041 •97541 .23740 .97141 .25432 .96712 16 45 .18652 .98245 .20364 •9790s .22070 .97534 .23769 .97134 .25460 .9670s IS 46 .18681 .98240 .20393 •97899 .22098 .97528 .23797 .97127 .25488 .96697 14 47 .18710 .98234 .20421 .22126 .97521 .23825 .97120 .25516 .96690 13 48 .18738 .98229 .20450 •97887 .22155 .97515 .23853 .97113 .25545 .96682 12 49 .18767 .98223 .20478 .97881 .^2183 .97508 .23882 .97106 .25573 .96675 11 SO .18795 .98218 .20507 .97875 .22212 .97502 .23910 .97100 .25601 .96667 10 51 .18824 .98212 .20535 .97869 .22240 .97496 .23938 .97093 .25629 .96660 9 S2 .18852 .98207 .20563 .97863 .22268 .97489 .23966 .97086 .25657 .96653 8 S3 .18881 .98201 .20592 .97857 .22297 .97483 .23995 .97079 .2568s .96645 7 54 .18910 .98196 .20620 .97851 .22325 .97476 .24023 .97072 .25713 .96638 6 55 .18938 .98190 .20649 .97845 .22353 .97470 .24051 .97065 .25741 .96630 5 56 .18967 .98185 .20677 .97839 .22382 .97463 •24079 .97058 .25769 .96623 4 57 .18995 .98179 .20706 .97833 .22410 .97457 .24108 .97051 .25798 .96615 3 S8 .19024 .98174 .•20734 .97827 .22438 .97450 .24136 .97044 .25826 .96608 2 59 .19052 .98168 .20763 .97821 .22467 .97444 .24164 •97037 ^5854 .96600 I 60 .19081 .98163 .20791 .97815 .22495 .97437 .24192 •97030 .25882 .96593 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine f 7< t 7^ ^° 7 f 7< 3 7 5° 186 BROWN & SHARPE MFG. CO. NATURAL SINES AND COSINES / 15° 16° 17° i8<^ 19° 1 Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine .25882 .96593 .27564 .96126 .29237 .95630 .30902 .95106 .32557 .94552 60 I .25910 .96585 .27592 .96118 .29265 .95622 .30929 .95097 .32584 .94542 59 2 .25938 .96578 .27620 .96110 .29293 .95613 .30957 .95088 .32612 .94533 58 3 .25966 .96570 .27648 .96102 .29321 .95605 .30985 .95079 .32639 .94523 57 4 .25994 .96562 .27676 .96094 .29348 .95596 .31012 .95070 .32667 .94514 56 S .26022 .96555 .27704 .96086 .29376 .95588 .31040 .95061 .32694 .94504 55 6 .26050 .96547 .27731 .96078 .29404 .95579 .31068 .95052 .32722 .94495 54 7 .26079 .96540 .27759 .96070 .29432 .95571 .31095 .95043 .32749 .94485 53 8 .26107 .96532 .27787 .96062 .29460 .95562 .31123 .95033 .32777 .94476 52 9 .26135 .96524 .27815 .96054 .29487 .95554 .31151 .95024 .32804 .94466 51 10 .26163 .96517 .27843 .96046 •29515 .95545 .31178 .95015 .32832 •94457 50 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 45 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 .9491S .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 il 24 .26556 .96410 .28234 .95931 .29904 .95424 .31565 .94888 .33216 .94322 36 25 .26584 .96402 .28262 .95923 .29932 .95415 .31593 .94878 .33244 .94313 35 26 .26612 .96394 .28290 .95915 .29960 .95407 .31620 .94869 .33271 .94303 34 27 .26640 .96386 .28318 .95907 .29987 .31648 .94860 .33298 .94293 33 28 .26668 .06379 .28346 .95898 .30015 .95389 .31675 .94851 .33326 .94284 32 29 .26696 .96371 .28374 .95890 .30043 .95380 .31703 .94842 .33353 .94274 31 30 .26724 .96363 .28402 .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 .26808 .96340 .28485 .95857 .30154 .95345 .31813 .94805 .33463 .94235 27 34 .26836 .96332 .28513 .95849 .30182 .35537 .31841 .94795 .33490 .94225 26 35 .26864 .96324 .^8s4i .95841 .30209 .95328 .31868 .94786 .33518 .94215 25 36 .26892 .96316 .28569 .95832 .30237 .95319 .3189:: .94777 .33545 .94206 24 37 .26920 .96308 .28597 .95824 .30265 .95310 .31923 .947C8 .33573 .94196 23 38 .26948 .96301 .28625 .95816 .30292 .95301 .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 .33655 .94167 20 41 .27032 .96277 .28708 .95791 .30376 .95275 .32034 .94730 .33682 •94157 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 .28792 .95766 .30459 .95248 .32116 .94702 .33764 .94127 16 45 .27144 .96246 .28820 .95757 .30486 .95240 .32144 .94693 .33792 .94118 15 46 .27172 .96238 .28847 .95749 .30514 .95231 .32171 .94684 .33819 .94108 14 ^Z .27200 .96230 .28875 .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 .28987 .95707 .30653 .95186 .32309 ■94637 .33956 .94058 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 54 .27396 .96174 .29070 .95681 .30736 ■95159 .32392 .94609 .34038 .94029 6 55 .27424 .96166 .29098 .95673 .30763 .95150 .32419 .94599 .34065 .94019 5 56 .27452 .96158 .29126 .95664 .30791 .95142 .32447 .94590 .34093 .94009 4 H .27480 .96150 .29154 .95656 .30819 .95133 .32474 .94580 .34120 .93999 3 S8 .27508 .96142 .29182 .95647 .30846 .95124 .32502 .94571 .34147 .93989 2 59 .27536 •96i3< ' .29209 .95639 .30874 .95115 .32529 .94561 •34175 .93979 I 60 .27564 .96126 .29237 .95630 .30902 .95106 .32557 .94552 .34202 .93969 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 7' 4° 7 5° 7' 2° 7 1° 7 0° 187 BROWN & SHARPE MFG. CO. NATURAL SINES AND COSINES ■ 20° 21 22^ 23° 24° 1 Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine o .34202 .93969 .35837 .93358 .37461 .92718 .39073 .920C0 .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 lO .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 .93211 •37838 .92565 .39448 .91891 .41045 .91188 46 IS .34612 .93819 .36244 .93201 •37865 .92554 .39474 .91879 .41072 .91176 45 i6 .34639 •93809 .36271 .93 J 90 •37892 .92543 .3950X .91868 .41098 .91164 44 17 .34666 .93799 .36298 .93180 .37919 .92532 .39528 .91856 .41125 .91152 43 i8 .34694 .93789 .36325 .93169 .37946 •92521 .39555 .91845 .41151 .91140 42 19 • 34721 .93779 .36352 .93159 .37973 •92510 .39581 .91833 .41178 .91128 41 20 .34748 .93769 .363'79 .93148 .37999 .92499 .39608 .91822 .41204 .91116 40 21 .34775 •93759 .36406 .93137 .38026 .92488 .39635 .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 .38161 .92432 .39768 .91752 .41363 .91044 34 27 .34939 .93698 .36569 .93074 .38188 .92421 .39795 .91741 .41390 .91032 33 28 .34966 .93688 .36596 .93063 .3821S .92410 .39822 .91729 .41416 .91020 32 29 .34993 .93677 .36623 .93052 .38241 .92399 .39848 .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 .90984 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 • 9358s • 36867 .92956 .38483 .92299 .40088 .91613 .41681 .90899 22 39 .35266 •93575 •36894 •92945 .38510 .92287 .40115 .91601 .41707 .90887 21 40 .35293 •93565 .36921 .92935 .38537 .92276 .40141 .91590 .41734 .90875 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 .92892 .38644 .92231 .40248 •91543 .41840 .90826 16 45 .35429 .93514 .37056 .92881 .38671 .92220 .40275 •91531 .41866 .90814 15 46 .35456 .93503 .37083 .92870 .3^698 .92209 .40301 •91519 .41892 .90802 14 47 .35484 .93493 .37110 .92859 .38725 .92198 .40328 .91508 .41919 .90790 13 48 .35511 .93483 .37137 .92849 .38752 .92186 .40355 .91496 .41945 .90778 12 49 .35538 .93472 .37164 .92838 .38778 .92175 .40381 .91484 .41972 .90766 11 SO .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 .38859 .92141 .40461 .91449 .42051 .90729 8 S3 .35647 •93431 .37272 .92794 .38886 .92130 .40488 .91437 .42077 .90717 7 54 .35674 •93420 •37299 .92784 .38912 .92119 .40514 .91425 .42104 .90704 6 55 .35701 .93410 .37326 .92773 •3S939 .92107 .40541 .91414 .42130 .90692 5 56 .35728 .93400 .37353 .92762 .38966 .92096 .40567 .91402 .42156 .90680 4 ^l .35755 •93389 .37380 .92751 .38993 .92085 .40594 •91390 .42183 .90668 3 58 .35782 .93379 .37407 .92740 .39020 .92073 .40621 .91378 .42209 .90655 2 59 .35810 .93368 .37434 .92729 .39046 .92062 .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( )° 6^ B° 6: 7" 6( 3° 6 5° 188 BROWN & SHARPE MFG. CO. NATURAL SINES AND COSINES / 25 26° 27 28 29° / Sine Cosine Sine Cosine Sine Cosine 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 .8:448 59 2 .42315 .90606 .43889 .89854 .45451 .89074 .46999 .88267 .48532 .87434 S8 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 51 10 .42525 .90507 .44098 .89752 .45658 .88968 .47204 .8815S •48735 .87321 50 II .42552 .90495 .44124 .89739 .45684 .88955 •47229 .88144 .48761 •87306 49 12 .42578 .90483 .44151 .89726 ■45710 .88942 •4725s .88130 .48786 .87292 48 13 .42604 .90470 ■44177 .89713 ■45736 .88928 .47281 .88117 .48811 .87278 47 14 .42631 .90458 ■44203 .89700 ■45762 .8891S .47306 .88103 .4S837 .87264 46 15 .42657 .90446 .44229 .89687 ■45787 .88902 .47332 .88089 .48862 .87250 45 i6 .42683 .90433 ■44255 .89674 .45813 .88888 .47358 .88075 .48888 •8723s 44 ly .42709 .90421 .44281 .89662 .45839 .88875 .47383 .88062 .48913 .87221 43 l8 .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 .903S3 .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 .49192 .87064 32 29 .43025 .90271 .44594 .89506 .46149 .88715 .47690 187896 .49217 .8701^0 31 30 .43051 .90259 .44620 .89493 .46175 .88701 .47716 .87882 .49242 .87036 30 31 •43077 .90246 .44646 .89480 .46201 .88688 .47741 .87868 .49268 .87021 ^9 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 .86578 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 II SO .43575 .90007 ■45140 .89232 .46690 .88431 .48226 .87603 .40748 .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 55 .43706 .8y943 .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 .43785 .89905 .45347 .89127 .46896 .88322 .48430 .87490 .49950 .86632 2 59 .43811 .89892 .45373 .89114 .46921 .88308 .48456 .87476 .49975 .86617 I 6o .43837 .89879 .45399 .89101 .46947 .88295 .48481 .87462 .50000 .86603 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 6. i° 6 3° 62 61 6 0° 189 BROWN & SHARPE MFG. CO. NATURAL SINES AND COSINES / 30° 3 1° 3- 2° 33° 34° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine .50000 .86603 .51504 .85717 .52992 .84805 .54464 .83867 .55919 .82904 60 I .50025 .86588 .51529 .85702 .53017 .84789 .54488 .83851 .55943 .82887 2 .50050 .86573 .51554 .85687 .53041 .84774 .54513 .83835 .55968 .82871 58 3 .50076 .86559 .51579 .85672 .53066 .84759 .54537 .83819 •55992 .82855 57 4 .50101 .86544 .51604 .85657 .53091 .84743 .54561 .83804 .56016 .82839 56 s .50126 .86530 .51628 .85642 .53115 .84728 .54586 .83788 .56040 .82822 55 6 .50151 .86515 .51653 .85627 .53140 .84712 .54610 .83772 .56064 .82806 54 7 .50176 .86501 .S1678 .856x2 .53164 .84697 .54635 .83756 .56088 .82790 53 8 .50201 .86486 .51703 .85597 .53189 .84681 .54659 .83740 .56112 •^2773 52 9 .50227 .86471 .51728 .85582 .53214 .84666 .54683 .83724 .56136 .82757 51 10 .50252 .86457 .51753 .85567 .53238 .84650 .54708 .83708 .56160 .82741 SO 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 .50403 .86369 .51902 .85476 .53386 .84557 .54854 .83613 .56305 .82643 44 17 .50428 .86354 .51927 .85461 .53411 .84542 .54878 .83597 .56329 .82626 43 i8 .50453 .86340 .51952 .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 .84495 .54951 .83549 .56401 .82577 40 21 .50528 .86295 .52026 .85401 .53509 .84480 .5497s .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 .55048 .83485 .56497 .82511 36 25 .50628 .86237 .52126 .85340 .53607 .84417 .55072 .83469 .56521 .82495 35 26 .50654 .86222 .52151 .85325 .53632 .84402 .55097 .83453 .56545 .82478 34 27 .50679 .86207 .52175 .85310 .53656 .84386 .55121 .83437 .56569 .82462 33 28 .50704 .86192 .52200 .85294 .53681 .84370 .55145 .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 .52275 .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 35 .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 .55388 .83260 .56832 .82281 22 39 .50979 .86030 .52473 .85127 .53951 .84198 .55412 .83244 .56856 .82264 21 40 .51004 .86015 .52498 .85112 .53975 .84182 .55436 .83228 -56880 .82248 20 41 .51029 .86000 .52522 .85096 .54000 .84167 .55460 .83212 .56904 .82231 19 42 .51054 .85985 .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 .55557 .83147 .57000 .82165 IS 46 .51154 .85926 .52646 .85020 .54122 .84088 .55581 .83131 .57024 .82148 14 47 .51179 .85911 .52671 .85005 .54146 .84072 .55605 .83115 .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 SO .51254 .85866 .52745 .84959 .54220 .84025 .55678 .83066 .57119 .82082 10 51 .51279 .85851 .52770 .84943 .54244 .84009 .55702 .83050 .57143 .82065 9 52 .51304 .85836 .52794 .84928 .54269 .83994 .55726 .83034 .57167 .82048 8 S3 .51329 .85821 .52819 .84913 .54293 .83978 .55750 .83017 .57191 .82032 7 54 .51354 .85806 .52844 .84897 .54317 .83962 .55775 .83001 .57215 .82015 6 SS .51379 .85792 .52869 .84882 .54342 .83946 .55799 .82985 .57238 .81999 S S6 .51404 .85777 .52893 .84866 .54366 .83930 .55823 .82969 .57262 .81982 4 57 .51429 .85762 .52918 .84851 .54391 .83915 .55847 .82953 .57286 .81965 3 58 .51454 .85747 .52943 .84836 .54415 .83899 .55871 .82936 .57310 .81949 2 59 .51479 .85732 .52967 .84820 .54440 .83883 .55895 .82920 .57334 .81932 I 60 .51504 .85717 .52992 .84805 .54464 .83867 .55919 .82904 .57358 .81915 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 59 )° 5^ 5° 5/ 7O si i° 5f .0 5 190 BROWN & SHARPE MFG. CO. NATURAL SINES AND COSINES f 35° 36 -2*7° 37 38° 39° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine o .57358 .81915 .58779 .80902 .60182 .79864 .61566 .78801 .62932 .77715 60 1 .57381 .81899 .58802 .80885 .60205 • 79846 .61589 .78783 .62955 .77696 59 2 .57405 .81882 .58826 .80867 .60228 .79829 .61612 .78765 .62977 .77678 58 3 • 57429 .81865 •58849 .80850 .60251 •79811 .61635 .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 •6304s .77623 55 6 .57501 .8181S .58920 •80799 .60321 •79758 .61704 .78694 .63068 .77605 54 7 .57524 .81708 .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 .63135 .77550 51 10 .57596 .81748 .59014 .80730 .60414 .79688 •6179s .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 15 .57715 .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 i8 .57786 .81614 .59201 •80593 .60599 .79547 .61978 .78478 .633.^8 .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 ■ 7^323 •63518 • 77236 34 27 .57999 .81462 •59412 .80438 .60807 .79388 .62183 .78315 •63540 .77218 33 28 .58023 .81445 •59436 .80420 .60830 .79371 .62206 .78297 •63563 .77199 32 29 .58047 .81428 •59459 .80403 .60853 .79353 .62229 .78279 •63585 •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 .80351 .60922 .79300 .62297 .78225 •63653 .77125 28 33 .58141 .81361 • 595*52 •80334 .60945 .79282 .62320 .78206 •63675 .77107 27 34 .58165 .81344 • 59576 .80316 .60968 .79264 .62342 .78188 .63698 .77088 26 35 .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 15 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 .58519 .81089 .59926 .80056 .61314 .78998 .62683 .77916 .64033 .76810 11 50 .58543 .81072 •59949 .80038 .61337 .78980 .62706 .77897 .64056 .76791 10 51 .58567 .810SS • 59972 .80021 .61360 .78962 .62728 .77879 .64078 .76772 9 52 .58590 .81038 •59995 .80003 .61383 .78944 .62751 .77861 .64100 .76754 8 53 .58614 .81021 .60019 .79986 .61406 .78926 .62774 .77843 .64123 .76735 7 54 •S^^F .81004 .60042 .79968 .61429 .78908 .62796 .77824 .64145 .76717 6 55 .58661 .80987 .60065 •79951 .61451 .78891 .62819 .77806 .64167 .76698 5 S6 .58684 .80970 .60089 • 79934 •61474 .78873 .62842 .77788 .64190 .76679 4 ^l .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 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 5^ t° . 53 5i >° 51 ° 5< 3° 191 BROWN & SHARPE MFG. CO. NATURAL SINES AND COSINES f 40 ° 41 42° 43° 44° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine .64279 .76604 .65606 .75471 .66^13 .74314 .68200 .73135 .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 .7^23,7 .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 .■7S337 .67064 .74178 .68349 .72996 .69612 .71792 S3 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 ■729Z7 •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 .67215 .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 .69:62 .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 25 .64834 .76135 .66153 .74992 .67452 .73826 .68730 .72637 .69987 .71427 35 26 .64856 .76116 .66175 .74973 .67473 .73806 .6C751 .72617 .70008 .71407 34 27 .64878 .76097 .66197 .74953 .67495 •73787 .6Gr72 .72597 .70029 .71386 33 28 .64901 .76078 .66218 .74934 .67516 •73767 .68793 .72577 .70049 .71366 32 29 .64923 .76059 .66240 .74915 .67538 .73747 .68C14 .72557 .70070 .71345 31 30 .64945 .76041 .66262 .74896 .67559 .73728 .C8C3S .72537 .70091 .71325 30 31 .64967 .76022 .66284 .74876 .67580 .73708 .68857 .72517 .70112 .71305 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 .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 .71100 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 .69x30 .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 11 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 .72095 .70546 .70875 8 53 .65452 .75604 .66762 .74451 .68051 .73274 •69319 .72075 .70567 .70855 7 54 .65474 .75585 .66783 .74431 .68072 .73254 .69340 .72055 .70587 .70834 6 55 .65496 .75566 .66805 .74412 .68093 .73234 .69361 .72035 .70608 .70813 s 56 .65518 .75547 .66827 .74392 .68115 .73215 .69382 .72015 .70628 .70793 4 57 .65540 .75528 .66848 .74373 .68136 .73195 .69403 .71995 .70649 .70772 3 58 .65562 .75509 .66870 .74353 .68157 .73175 •69424 .71974 .70670 .70752 2 59 .65584 .75490 .66891 .74334 .68179 .73155 .69445 .71954 .70690 .70731 I 60 .65606 •7S47I .66913 .74314 .68200 .73135 .69466 .71934 .70711 ,70711 f Cosine Sine Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 4< f 4^ ?° 4' 7° 4< 5° 4 5° 192 Natural Tangents and Cotangents BROWN & SHARPE MFG. CO. NATURAL TANGENTS AND COTANGENTS / 0° 1° 2 3 4° / Tang: Cotang: Tang: Cotang: Tang: Cotang: Tang: Cotang: Tang: Cotang: .00000 Infinite .01746 57.2900 .03492 28.6363 .05241 19.0811 .06993 14.3007 60 I .00029 3437.75 .01775 56.3506 .03521 28.3994 .05270 18.9755 .07022 14.2411 S9 2 .00058 1718.87 .01804 55.4415 .03550 28.1664 .05299 18.8711 .07051 14.1821 S8 3 .00087 1145.92 .01833 54.5613 -03579 27.9372 .05328 18.7678 .07080 14.123s 57 4 .00116 859.436 .01862 5'3.7o86 .03609 27.7117 .05357 18.6656 .07110 14.0655 S6 5 .00145 687.549 .01891 52.8821 .03638 27.4899 .05387 18.5645 .07139 14.0079 55 6 .00175 572.957 .01920 52.0807 .03667 ■27.271s .05416 18.4645 .07168 13.9507 54 7 .00204 491.106 .01949 S1.3032 .03696 27.0566 .05445 18.3655 .07197 13.8940 53 8 .00233 429.718 .01978 50.5485 .03725 26.8450 .05474 18.2677 .07227 13.8378 52 9 .00262 381.971 .02007 49.8157 .03754 26.6367 .05503 18.1708 .07256 13.7821 51 10 .00291 343.774 .02036 49.1039 .03783 26.4316 .05533 18.0750 .07285 13.7267 so 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 .05591 17.8863 .07344 13.6174 48 13 .00378 264.441 .02124 47.0853 .03871 25.8348 .05620 17.7934 ■OIZIZ 13.5634 47 14 -00407 245.552 .02IS3 46.4489 .03900 25.6418 .05649 17.7015 .07402 13.5098 46 IS .00436 229.182 .02182 45.8294 .03929 25-4517 .05678 17.6106 .07431 13.4566 45 i6 .00465 214.858 .02211 45.2261 .03958 25-2644 .05708 17.520S .07461 13.4039 44 17 .00495 202.219 .02240 44.6386 .03987 25-0798 .05737 17.4314 .07490 13.3515 43 i8 .00524 190.984 .02269 44.0661 .04016 24-8978 .05766 17.3432 .07519 13.2996 42 19 .00553 180.932 .02298 43.5081 .04046 24-7185 .05795 17.2558 .07548 13.2480 41 20 .00582 171.88s .02328 42.9641 .04075 24-5418 .05824 17.1693 .07578 13.1969 40 21 .00611 163.700 .02357 42.4335 .04104 24.3675 .05854 17.0837 .07607 13.1461 39 22 .00640 156.259 .02386 41.9158 .04133 24.1957 .05883 16.9990 .07636 13.0958 38 23 .00669 149-465 .0241S 41.4106 ,04162 24.0263 .05912 16.9150 .07665 13.0458 37 24 .00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 .07695 12.9962 36 25 .00727 137.507 .02473 40.4358 .04220 23.6945 .05970 16.7496 .07724 12.9469 35 26 .00756 132.219 .02502 39.9655 .04250 23-5321 .05999 16.6681 .07753 12.8981 34 21 .00785 127.321 .02531 39.5059 .04279 23-3718 .06029 16.5874 .07782 12.8496 33 28 .008 IS 122.774 .02560 39.0568 .04308 23-2137 .06058 16.5075 .07812 12.8014 32 29 .00844 118.540 .02589 38.6177 .04337 23-0577 .06087 16.4283 .07841 12.7536 31 30 .00873 114.589 ,02^19 38.188s .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 .06175 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 12.5199 26 35 .01018 98.2179 .02764 36-1776 .04512 22.1640 ,06262 15.9687 .08017 12.4742 25 36 .01047 95.4895 .02793 35.8006 .04541 22.0217 ,06291 15.8945 .08046 12.4288 24 37 .01076 92.908s .02822 35.4313 .04570 21.8813 .06321 15.8211 .08075 12.3838 23 38 .Olios 90.4633 .02851 35-0695 .04S99 21,7426 .06350 15.7483 .08104 12.3390 22 39 .01135 88.1436 .02881 34-7151 .04628 21.6056 .06379 15.6762 .08134 12.2946 21 40 .01164 85.9398 .02910 34.3678 .04658 21.4704 .06408 15.6048 .08163 12.250s 20 41 .01193 83.843s .02939 34.0273 .04687 21.3369 .06437 15.5340 .08192 12.2067 19 42 .01222 81.8470 .02968 33.6935 .04716 21.2049 .06467 1S.4638 .08221 12.1632 18 43 .01251 79.9434 .02997 33.3662 .04745 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 .03055 32.7303 .04803 20.8188 .06554 15.2571 .08309 12.0346 IS 46 .01338 74.7292 .03084 32.4213 .04833 20.6932 .06584 1S.1893 .08339 11,9923 14 47 .01367 73.1390 .03114 32.1181 .04862 20.5691 .06613 15.1222 .08368 IX. 9504 13 48 .01396 71.6151 .03143 31.8205 .04891 20.4465 ,06642 15.0557 .08397 11.9087 12 49 .01425 70.1533 .03172 31.5284 .04920 20.3253 .06671 14.9898 .08427 11.8673 II SO .OI4SS 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 ,08456 11.826-2 10 SI .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 ,08485 11.7853 9 52 .01513 66. loss .03259 30.6833 .05007 19.9702 .06759 14-7954 .08514 11.7448 8 S3 • .01542 64.8580 .03288 30.4116 .05037 19.8546 .06788 14.7317 .08544 11.7045 7 54 .01571 63.6567 •03317 30.1446 ,05066 19.7403 .06817 14.668s .08573 11.6645 6 SS .01600 62.4992 .03346 29.8823 .05095 19.6273 .06847 14.6059 .08602 11.6248 s S6 .01629 61.3829 .03376 29.6245 .05124 19.5156 .06876 14.5438 .08632 11,5853 4 S7 .01658 60.3058 .03405 29.3711 .05153 19.4051 .0690s 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 11.468s I 6o ,01746 57.2900 .03492 28.6363 .05241 19.0811 .06993 14.3007 .08749 11.4301 / Cotang Tang: Cotang: Tang: Cotang: Tang: Cotang: Tang: Cotang: Tang: ' 89° 88° 8: 7° 8( )° 8 -0 3 194 BROWN & SHARPE MFG. CO. NATURAL TANGENTS AND COTANGENTS 1 5° 6 ° 7° 8 9° / Tang: Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang .08749 11.4301 .10510 9.51436 .12278 8.14435 .14054 7.11537 .15838 6-31375 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 58 3 .08837 II. 3163 .10599 9.43515 .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 5 .08895 II. 2417 .10657 9-38307 .12426 8.04756 .14202 7^04105 .15988 6.25486 55 6 .08925 11.2048 .10687 9-35724 .12456 8.02848 .14232 7.02637 .16017 6.24321 54 7 .08954 11.1681 .10716 9-33155 .12485 8.00948 .14262 7.01174 .16047 6.23160 S3 8 .08983 11.1316 .10746 9-30599 • 12515 7.99058 .14291 6.99718 .16077 6.22003 52 9 .09013 11.0954 •I077S 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 SO 11 .09071 11.0237 .10834 9.23016 .12603 7.93438 .14381 6.9538s .\(^\(>'j 6.18559 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 6.92525 .16226 6.16283 47 14 .09159 10.9178 .10922 9- 1 5554 .12692 7-87895 .14470 6.91104 .16256 6.15151 46 IS .09189 10.8829 .10952 9-13093 .12722 7.86064 .14499 6.89688 .16286 6.14023 45 i6 .09218 10.8483 .10981 9.10646 .12751 7-84242 .14529 6.88278 .16316 6.12899 44 17 .09247 10.8139 .IIOII 9.08211 .12781 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 7-78825 .14618 6.84082 .16405 6.09552 41 20 .09335 10.7119 .11099 9.00983 .12869 7.7703s .14648 6.82694 .16435 6.08444 40 21 .09365 10.6783 .11128 8.98598 .12899 7.75254 .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 7.7171S .14737 6.78564 .16525 6.05143 37 24 .09453 10.5789 .11217 8.91520 7-69957 .14767 6.77199 .16555 6.04051 36 25 .09482 10.5462 .11246 8.89185 .13017 7.68208 .14796 6.75838 .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 Z2 29 .09600 10.4172 .11364 8.79964 .13136 7.61287 .14915 6.70450 .16704 5 -98646 31 30 .09629 10.3854 .11394 8.77689 .13165 7-59575 .14945 6.69116 .16734 5.97576 30 31 .09658 10.3538 .11423 8.75425 .13195 7.57872 .14975 6.67787 .16764 5^96510 29 32 .09688 10.3224 .11452 8.73172 .13224 7-56176 .15005 6.66463 .16794 S.95448 28 33 .09717 10.2913 .11482 8.70931 .13254 7-54487 .15034 6.65144 .16824 5^94390 27 34 .09746 10.2602 .iiSii 8.68701 .13284 7.52806 .15064 6.63831 .16854 5^93335 26 35 .09776 10.2294 -IIS4I 8.66482 .13313 7-51132 .15094 6.62523 .16884 5-92283 25 36 .09805 10.1988 .11570 8.6427s .13343 7-49465 .15124 6 61219 .16914 5-91236 24 37 .09834 10.1683 .11600 8.62078 •13372 7.47806 -15153 6.59921 .16944 5.90191 23 38 .09864 10.1381 .11629 8.59893 .13402 7.46154 .15183 6.58627 .16974 22 39 .09893 10.1080 .11659 8.57718 •13432 7-44509 • 15213 6.57339 .17004 5 88114 21 40 .09923 10.0780 .11688 8.55555 .13461 7.42871 .15243 6.560SS .17033 5-87080 20 41 .09952 10.0483 .11718 8.53402 .13491 7-41240 .15272 6.54777 .17063 5 -8605 1 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 5.84001 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 5.81966 15 46 .10099 9.90211 .11865 8.42795 • 13639 7-33190 .15421 6.48456 .17213 5-80953 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 •ISSII 6.44720 .17303 S.77936 11 50 .10216 9.78817 .11983 8.34496 .13758 7.26873 .15540 6.43484 .-i-llU 5-76937 10 SI .10246 9-76009 .12013 8.32446 .13787 7-25310 .15570 6.42253 .17363 5-75941 9 52 .10275 9-73217 .12042 8.30406 .13817 7-23754 .15600 6.41026 .17393 5-74949 8 S3 .10305 9-70441 .12072 8.28376 .13846 7.22204 .15630 6.39804 .17423 5-73960 7 54 .10334 9-67680 .12101 8.26355 .13876 7.20661 .15660 6.38587 .17453 5-72974 6 55 .10363 9.64935 .12131 8.24345 .13906 7-19125 .15689 6-37374 .17483 5-71992 S S6 .10393 9.62205 .12160 8.22344 .13935 7-17594 .15719 6.36165 .17513 5-7IOI3 4 S7 .10422 9.59490 .12190 8.20352 .13965 7-16071 • 15749 6.34961 • 17543 5-70037 3 58 .10452 9.56791 .12219 8.18370 .13995 7-I45S3 .15779 6.33761 •17573 5-69064 2 59 .10481 9-54106 .12249 8.16398 .14024 7-13042 .15809 6.32566 .17603 S-68094 I 60 .10510 9-51436 .12278 8.14435 .14054 7-II537 .15838 6.31375 .17633 5-67128 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 84° 8: i° 82° 8] 80° 195 BROWN & SHARPE MFG. CO. NATURAL TANGENTS AND COTANGENTS 1 10° 11° 12° 13° 14° / Tangf Cotangr Tang Cotang: Tangf Cotangf Tangf Cotangf Tangf Cotangf o .17633 S.67128 .19438 S-I445S .21256 4.70463 .23087 4.33148 -24933 4.01078 60 I .17663 5.66165 .19468 S-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 S8 3 .17723 S.64248 .19529 5.12069 .21347 4.68452 .23179 4.31430 .25026 3-99592 57 4 .17753 5.63295 -19559 5.11279 •21377 4-67786 .23209 4.30860 •25056 3.99099 56 S .17783 5.62344 -19589 5.10490 .21408 4.67121 .23240 4-30291 .25087 3.98607 55 6 .17813 5. 61397 .19619 5-09704 .21438 4.66458 .23271 4.29724 .25118 3.98117 54 7 .17843 S.60452 .19649 5-08921 .21469 4-65797 .23301 4.29159 •25149 3-97627 53 8 .17873 5-59511 .19680 5-08139 .21499 4-65138 .23332 4.28595 .25180 3-97139 52 9 .17903 5.58573 .19710 5-07360 .21529 4-64480 -23363 4.28032 .25211 3-96651 51 10 .17933 S.57638 .19740 5.06584 .21560 4.6382s • 222,92, 4.27471 •25242 3-96165 50 II .17963 5.56706 .19770 S-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 4.61868 .23485 4.25795 -25335 3-94713 47 14 •'^°^^ 5.53927 .19861 5-03499 .21682 4.61219 .23516 4.25239 .25366 3-94232 46 IS .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 5-51176 .19952 5.01210 .21773 4.59283 .23608 4-23580 .25459 3-92793 <3 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 .2212\ 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 25 .18384 5. 43966 .20194 4-95201 .22017 4-54196 •23854 4-19215 -25707 3-89004 35 26 .18414 5.43077 .20224 4.94460 .22047 4-53568 .23885 4-18675 .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 .2031S 4-92249 .22139 4-51693 •23977 4-17064 -25831 3.87136 31 30 .18534 S-39552 -20345 4-91516 .22169 4.51071 .24008 4-16530 .25862 3.86671 30 31 .18564 5-38677 .20376 4-90785 .22200 4-50451 .24039 4-15997 .25893 3.86208 29 32 .18594 S-37805 .20406 4-90056 .22231 4-49832 .24069 4-15465 .25924 3-85745 28 33 ..18624 5-36936 .20436 4-89330 .22261 4-49215 .24100 4-14934 .25955 3-85284 21 34 .18654 5-36070 .20466 4.88605 .22292 4.48600 .24131 4-14405 .25986 3.84824 ■ 26 35 .18684 S-35206 .20497 4-07882 .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 22 38 .18775 S-32631 .20588 4-85727 .22414 4-46155 -24254 4.12301 .26110 3-82992 22 39 .18805 5-31778 .20618 4-85013 .22444 4-45548 -24285 4-11778 .26141 3.82537 21 40 .18835 5-30928 .20648 4-84300 -22475 4-44942 -24316 4.II256 .26172 3-82083 20 41 .18865 5-30080 .20679 4-83590 .22505 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 4.82175 .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 S-26715 .20800 4.80769 .22628 4.41936 .24470 4.08666 .26328 3-79827 15 46 .19016 5-25880 .20830 4.80068 .22658 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 II SO .19136 5-22566 .20952 4-77286 .22781 4.38969 .24624 4.06107 .26483 3.77595 10 51 .19166 5-21744 .20982 4.76595 .22811 4.38381 -24655 4.05599 .26515 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 4.75219 .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 56 .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 2 58 .19378 5-16058 .21195 4.71813 .23026 4.34300 .24871 4-02074 -26733 3-74075 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 3-7320S / Cotang Tang Cotangf Tangf Cotangf Tang- Cotangf Tangf Cotangf Tangf / 79° ' 78° 77° 76° 7 196 BROWN & SHARPE MFG. CO. NATURAL TANGENTS AND COTANGENTS 1 15° 16° 17° 18° K P° / Tang: Cotang: Tang: Cotang: Tang: Cotang: Tang: Cotansr Tang: Cotang: .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 C.46837 .30732 3.25392 .32653 3.06252 -34596 2.890SS 55 6 .26982 3.70616 .28864 3.46458 .30764 3.25055 .3268s 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.OS049 .34726 2.87970 SI 10 .27107 3.68909 .28990 3.44951 .30891 3.23714 .32814 3.04749 .34758 2.87700 50 II .27138 3.6848s .29021 3.44576 .30923 3.23381 .32846 3-04450 .34791 2.87430 49 12 .27169 3.68061 .29053 3.44202 •30955 3.23048 .32878 3-04152 .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 IS .2726J 3.66796 .29147 3.43084 .31051 3.22053 .32975 3.03260 .34922 2.86356 45 i4 .27294 3.66376 .29179 3.42713 .31083 3.21722 .33007 3-02963 .34954 2.86089 44 I? .27326 3.65957 .29210 3.42343 .3111S 3.21392 .33040 3-02667 .34987 2.85822 43 i8 .27357 3.65538 .29242 3-41973 .31147 3-21063 .33072 3-02372 .35020 2-85555 42 19 .27388 3.65121 .29274 3.41604 .31178 3-20734 .33104 3-02077 .35052 2.85289 41 20 .27419 3.6470s .2930s 3-41236 .31210 3.20406 .33136 3.01783 .35085 2.85023 40 21 .274SI 3.64289 .29337 3.40869 .31242 3-20079 .33169 3.01489 .35118 2.84758 39 22 .274S2 3.63874 .20368 3.40502 .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 .33320 3.00028 .35281 2-83439 34 27 .27633 3.61814 .29526 3.38679 .31434 3.18127 .23363 2.99738 .35314 2.83176 33 28 .27670 3.614OS .29553 3.38317 .31466 3-17804 .33395 2.99447 .35346 2.82914 32 29 .27701 3.60996 .295QO 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 2.98580 .3S44S 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 3.53966 .23748 3.36158 .31658 3.15877 .33589 2.97717 .35543 2.81350 26 35 .27889 3.53562 .29780 3.35800 .31690 3-15558 .33621 2.97430 .35576 2.81091 25 36 .27921 3.53160 .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 .279S3 3.57357 .29875 3-34732 .31786 3.14605 .33718 2.96573 .35674 2.80316 22 39 .28015 3.56957 .29906 3-34377 .31818 3.14288 .33751 2.962S3 .35707 2.80059 21 40 .28046 3.56557 .29938 3.34023 .31850 3.13972 .33783 2.96004 .35740 2;798o2 20 41 .28077 3.56159 .29970 3-33670 .31882 3.13656 .33816 2.9S72I .35772 2.79545 19 42 .28109 3.55761 .30001 3-333-^7 .31914 3.13341 .33848 2.95437 .3580s 2.79289 18 43 .28140 3.55364 .30033 3.3296s .31946 3-13027 .33881 2.9515s .35838 2.79033 17 44 .28172 3.54968 .30065 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 15 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-II77S .34010 2.94028 .35969 2.78014 13 48 .28297 3-53353 .30192 3.31216 .32106 3.11464 .34043 2.93743 .36002 2.77761 12 49 .28329 3.53001 .30224 3.30868 .32139 3-I11S3 .34075 2.93468 .360,3s 2.77507 II SO .28360 3.52609 .30255 3-30521 •32171 3.10842 .34108 2.93189 .36068 2.77254 10 SI .28391 3.52219 .30287 3.30174 .32203 3-10532 .34140 2.92910 .36101 2.77002 9 52 .28423 3.51829 .30319 3.29829 .32235 3.10223 .34173 2.92632 .36134 2.76750 8 S3 .28454 3.51441 .30351 3-29483 .3^267 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 5S .28517 3.50666 .30414 3.28795 .32331 3.09298 .34270 2.91799 .36232 2.75996 S 56 .28549 3.50279 .30446 3.28452 .32363 S.08991 .34303 2.9i5.-:3 .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 2.90971 .36331 2.75246 2 59 .28643 3-49T2S .30541 3.27426 .32460 3.08073 .34400 2.90656 .36364 2.74997 I 6o .28675 3.48741 .30573 3.27085 .32492 3.07768 -34433 2.90421 .363Q7 2.;'4748 / Cotang Tang Cotang: Tang: CotanK Tan^ Cotang: Tangr Cotang: Tang: 74° 73° 72° 71° 7< )° 197 BROWN & SHARPE MFG. CO. NATURAL TANGENTS AND COTANGENTS / 20^ 21° 22° 23° 24° / Tang- Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang o .36397 2.74748 .38386 2.60509 .40403 2.47509 .42447 2.35585 .44523 2.24604 60 I .36430 2.74499 .3^420 2.60283 .40436 2.47302 .42482 2.35395 .44558 2.^44^ 59 2 .36463 2.74251 •3f4S3 2.60057 .40470 2.4709s .42516 2.35-205^ .44593 2.24252 58 ^ 3 .36496 2.74004 .38487 2.59831 .40504 2.46888 .42551 ^.35015 .44627 2.24077 57 4 .36529 2.73756 .38520 2.59606 .40538 2.46682 .4258^5 -2,3482s .44662 2.23902 56 5 .36562 2.73509 .38553 2.59381 .40572 2.46475 .42619 2.34636 .44697 2.23727 55 6 .36595 2.73263 .38587 2.59156 .40606 2.46270 .42654 2.34447 .44732 2.23553 54 7 .36628 2.73017 ,38620 2.58932 .40640 2.4606s .42688 2.34258 .44767 2.23378 S3 8 .36661 2.72771 .38654 2.58708 .40674 2.45860 .42722 2.34069 .44802 2.23204 52 9 .36694 2.72526 .38687 2.58484 .40707 2.45655 .42757 2.33881 .44837 2.23030 51 10 .36727 2.72281 .38721 2.58261 .40741 2.45451 .42791 2.33693 .44872 2.22857 SO II .36760 2.72036 .38754 2.58038 .40775 2.45246 .42826 2.3350s .44907 2.22683 49 12 .36793 2.71792 .38787 2.57815 .40809 2.45043 .42860 2.33317 .44942 2.22510 48 13 .36826 2.71548 .38821 2.57593 .40843 2.44839 .42894 2.33130 .44977 2.22337 47 14 .36859 2.71305 .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 2.70819 .38921 2.56928 .40945 2.44230 .42998 2.32570 .45082 2.21819 44 17 .36958 2.70577 .38955 2.56707 .40979 2.44027 .43032 2.32383 .45117 2.21647 43 i8 .36991 2.70335 .38988 2.56487 .41013 2.4382s .43067 2.32197 .45152 2.21475 42 19 .37024 2.70094 .39022 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 .45292 2.20790 38 23 .37157 2.69131 .39156 2.55389 .41183 2.42819 .43230 2.31271 .45327 2.20619 27 24 .37190 2.68892 .39190 2.55170 .41217 2.42618 .43274 2.31086 .45362 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 .45432 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 .43412 2.30351 .45502 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 2.5386s .41421 2.41421 .43481 2.29984 .45573 2.19430 30 31 .37422 2.67225 .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 .43585 2.29437 .45678 2.18923 27 34 .37521 2.66516 .39526 2.53001 .41558 2.^0629 .43620 2.29254 .45713 2.18755 26 35 .37554 2.66281 • 39559 2.52786 .41532 2.40432 .43654 2.29073 .45748 2.18587 25 36 .37588 2.66046 .39593 2.52571 .41626 2.4023s .43689 2.2S891 .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 2.39645 .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 2.64875 .39761 2.51502 .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 .39829 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 2.6394s .39896 2.50652 .41933 2.38473 .44001 2.27267 .46101 2.16917 15 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 2.1658s 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 2.16255 II SO .38053 2.62791 .40065 2.49597 .42105 2.37504 .44175 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 53 .38153 2.62103 .40166 2.48967 .42207 2.3692s .44279 2.25840 .46383 2.15596 7 54 .38186 2.61874 .40200 2.48758 .42242 2.36733 .44314 2.25663 .46418 2.15432 6 55 .38220 2.61646 .40234 2.48549 .42276 2.36541 .44349 2.25486 .46454 2.15268 s S6 .38253 2.61418 .40267 2.48340 .42310 2.36349 .44384 2.25309 .46489 2.15104 4 57 .38386 2.61190 .40301 2.48132 .42345 2.36158 .44418 2.25132 .46525 2.14940 3 58 .38320 2.60963 .40335 2.47924 .42379 2.35967 .44453 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.I44SI / Cotang Tang- Cotang Tang Cotang Tangr Cotang Tang Cotang Tang / 6c f 68° 67° 66° 65° 198 BROWN & SHARPE MFG. CO. NATURAL TANGENTvS AND COTANGENTS 1 25° 26° 27° 28° 29° / Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang .46631 2.14451 .48773 2.05030 .50953 1.96261 .53171 1.88073 .55431 1.80405 60 .46666 2.14288 .48809 2.04879 .50989 1.96120 .53208 1.87941 • 55469 1.80281 59 2 .46702 2.1412s .48845 2.04728 .51026 1^95979 .53246 1.87809 ■ 55507 1.80158 58 3 .46737 2.13963 .48881 2.04577 .51063 1^95838 .53283 1.87677 •55545 1.80034 57 4 .46772 2.13801 .48917 2.04426 .51099 1.95698 .53320 1.87546 •55583 1.79911 S6 s .46808 2.13639 .48953 2.04276 .51136 1^95557 .53358 1.87415 • 55621 1.79788 55 6 .46843 2.134-7 .48989 2.04125 .51173 1^95417 .53395 1.87283 •55659 1.79665 54 7 .46879 2.13316 .49026 2.03975 .51209 1^95277 .53432 1.87152 .55697 1.79542 53 8 .46914 2.13154 .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 51 10 .46985 2.12832 •49134 2.03526 .51319 1^94858 .53545 1.86760 .55812 1.79174 50 II .47021 2.12671 •49170 2.03376 •S1356 1^94718 .53582 1.86630 .55850 1. 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 .49315 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 1.93332 .53957 1.85333 •56232 1.77834 39 22 .47412 2.10916 .49568 2.01743 .51761 I.93195 .53995 1.85204 .56270 1.77713 •38 2i .47448 2.10758 .49604 2.01596 .51798 1.93057 .54032 1.85075 •56309 1.77592 37 24 .47483 2.10600 .49640 2.01449 .51835 1.92920 .54070 1.84946 • 56347 1.77471 36 25 .47519 2.10442 .49677 2.01302 .51872 1.92782 .54107 1.84818 •56385 1.77351 35 26 .47555 2.10284 .49713 2.01155 .51909 1.92645 .54145 1.84689 .56424 I 11210 34 27 .47590 2.10126 .49749 2.01008 .51946 1.92508 .54183 1.84561 .56462 I.771IO 33 28 .47626 2.09969 .49786 2.00862 .51983 1. 92371 .54220 1.84433 .56501 1.76990 32 29 .47662 2.09811 .49822 2.00715 .52020 1.92235 .54258 1.84305 .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 Z2 .47769 2.09341 .49931 2.00277 .52131 1. 91826 .54371 1.83922 .56654 1.76510 28 33 .47805 2.09184 .49967 2.00131 .52168 1. 91690 .54409 1.83794 .56693 1-76390 27 34 .47840 2.09028 .50004 1.99986 .52205 1. 91554 .54446 1.83667 .56731 1.76271 26 35 .47876 2.08872 .50040 1.99841 .52242 1.91418 .54484 1.83540 .56769 1.76151 25 36 .47912 2.08716 .50076 1.9969s -52279 1.91282 .54522 1.83413 .56808 1-76032 24 37 .47948 2.08560 .50113 1-99550 -52316 1.91147 .54560 1.83286 .56846 1. 75913 23 38 .47984 2.08405 .50149 1.99406 -52353 I. 91012 .54597 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 1.90741 .54673 1.82906 .56962 1.75556 20 41 .48091 2.07939 .50258 1.98972 -52464 1.90607 .S47II 1.82780 .57000 1-75437 19 42 .48127 2.07785 .50295 1.98828 .52501 1.90472 .54748 1.82654 .57039 1-75319 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 .52575 1.90203 .54824 1.82402 .57116 1.75082 16 45 .48234 2.07321 .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 1.74846 14 47 .48306 2.07014 •50477 1.98110 .52687 1.89801 .54938 1.82025 .S122,2 1.74728 13 48 .48342 2.06860 .50514 1.97966 •52724 1.89667 .54975 1.81899 .57271 1.74610 12 49 .48378 2.06706 .50550 1.97823 .52761 1.89533 .55013 1-81774 .57309 1-74492 II SO .48414 2.06553 .50587 1.97681 .52798 1.89400 .55051 1.81649 .57348 1.74375 10 51 .48450 2.06400 .50623 1.97538 .52836 1.89266 .55089 1.81524 .57386 1.74257 9 52 .48486 2.06247 .50660 1-97395 .52873 1.89133 .55127 1.81399 .57425 I. 74140 8 53 .48521 2.06094 .50696 1-97253 .52910 1.89000 .55165 1.81274 .57464 1.74022 7 54 .48557 2.05942 .50733 1.97111 .52947 1.88867 .55203 1.81150 .57503 1.7390S 6 55 .48593 2.05790 .50769 1.96969 -52985 1.88734 .55241 1. 81025 .57541 1-73788 5 56 .48629 2.05637 .50806 1.96827 .53022 1.88602 .55279 1.80901 .57580 1-73671 4 ^l .48665 2.05485 .50843 1.96685 .53059 1.88469 .55317 1.80777 .57619 1-73555 3 58 .48701 2.05333 .50879 1.96544 .53096 1.88337 .55355 1.80653 .57657 1-73438 2 V .48737 2.05182 .50916 1.96402 • 53134 1.8820s .55393 1.80529 .57696 I-73321 1 6o .48773 2.05030 •50953 1. 96261 •53171 1.88073 .55431 1.80405 .57735 1-73205 1 Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 64° 63° 6i 2° 6 [° 6 0° 199 BROWN & SHARPE MFG. CO. NATURAL TANGENTS AND COTANGENTS / 30° 31 32 33° 34° / Tang: Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang o •57735 1.73205 .60086 1.66428 .62487 1.60033 .64941 1.53986 .67451 1.48256 60 I •57774 i^73o89 .60126 1.66318 •62527 1-59930 .64982 1.53888 -67493 1.48163 59 2 .57813 1.72973 .60165 1.66209 .62568 1.59826 .65024 1^53791 -67536 1.48070 58 3 .57851 1.72857 .60205 1.66099 .62608 1.59723 .65065 1^53693 .67578 1.47977 57 4 .57890 1. 72741 .60245 1.65990 .62649 1.59620 .65106 1^53595 .67620 1.4788s 56 S .57929 1.7262s .60284 1.65881 .62689 1.59517 •65148 1^53497 .67663 1.47792 55 6 .57968 i^72S09 .60324 1.65772 .62730 1. 59414 •65189 K53400 .67705 1-47699 54 7 .58007 i^72393 .60364 1.65663 .62770 1.59311 .65231 1.53302 •67748 I -47607 S3 8 .58046 1.72278 .60403 1. 65554 .62811 1.59208 •65272 1.53205 .67790 1-47514 52 9 .58085 i^72i63 .60443 1^65445 .62852 1.59105 .65314 1.53107 .67832 1-47422 51 lO .58124 1.72047 .60483 i^6S337 .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 1-58695 .65480 1.52719 .68002 1-47053 47 14 .58279 1.71588 .60642 1.64903 .63055 i^58593 .65521 1.52622 .68045 1.46962 46 IS .58318 l^7i473 .60681 1.64795 .63095 1^58490 .65563 1^52525 .68088 1.46870 45 i6- .58357 i^7i358 .60721 1.64687 .63136 1-58388 .65604 1.52429 .68130 1.46778 44 17 .58396 1. 71244 .60761 1.64579 •63177 1.58286 .65646 1.52332 .68173 1.46686 43 i8 .58435 1.71129 .60801 1.64471 •63217 1.58184 .65688 1.5223s .6821S 1-46595 42 19 .58474 1.7101S .60841 1^64363 .63258 1-58083 .65729 1.52x39 .68258 1-46503 41 20 .58513 1. 70901 .60881 1.64256 .63299 1^57981 .65771 1.52043 .68301 I. 464 I I 40 21 .58552 1.70787 .60921 1.64148 .63340 i^57879 .65813 1.51946 .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 25 .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 1. 70106 .61160 1^63505 .63584 1.57271 .66063 1.51370 .68600 1.45773 33 28 .58826 1.69992 .61200 1^63398 .63625 1.57170 .6610s 1^51275 .68642 1.45682 32 29 .58865 1.69879 .61240 1.63292 .63666 1.57069 .66147 I^5il79 .68685 1-45592 31 30 .58905 1.69766 .61280 1.63185 .63707 1.56969 .66189 1.S1084 .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 1.50893 .68814 1-45320 28 33 .59022 1.69428 .61400 1.62866 .63830 1.56667 .66314 1.50797 .68857 1-45229 ,27 34 .59061 1.69316 .61440 1.62760 .63871 1.56566 .66356 1.50702 .68900 I-45139 . 26 35 .59101 1.69203 •.61480 1.62654 .63912 1.56466 .66398 1.50607 .68942 1-45049 25 36 .59149 1. 69091 .61520 1.62548 •63953 1.56366 .66440 1.50S12 .68985 1.44958 24 37 .59179 1.68979 .61561 1.62442 •63994 1.56265 .66482 1.50417 .69028 1.44868 23 38 .59218 1.68866 .61601 1.62336 •64035 1.56165 .66524 1.50322 .69071 1-44778 22 39 .59258 I.687S4 .61641 1.62230 .64076 1.56065 .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 i^6853i .61721 1.62019 •64158 I-SS866 .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 1.61703 .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 IS 46 .59533 1.67974 .61922 1.61493 •64363 1.55368 .66860 1.49566 .69416 1.44060 14 47 . .59573 1.67863 .61962 1.61388 .64404 1-55269 .66902 1.49472 .69459 1.43970 13 48 .59612 1.67752 .62003 1.61283 .64446 I-55170 .66944 1^49378 .69502 1.43881 12 49 .59651 1. 67641 .62043 1.61179 .64487 1.55071 .66986 1.49284 .69545 1-43792 11 SO .59691 1^67530 .62083 1. 61074 .64528 I.S4972 .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 1.6086s .64610 I-S4774 •67113 1.49003 .69675 1-43525 8 S3 .59809 1.67198 .62204 1.60761 .64652 1-54675 .67155 1.48909 .69718 1.43436 7 54 •59849 1.67088 .62245 1.60657 .64693 1-54576 .67197 1. 48816 .69761 1-43347 6 55 .59888 1.66978 .62285 1.60553 .64734 1.54478 .67239 1.48722 .69804 1-43258 5 56 .59928 1.66867 .62325 1.60449 .64775 1-54379 .67282 1.48629 -69847 1.43169 4 57 .59967 I.667S7 .62366 1.60345 .64817 1-54281 .67324 1-48536 .69891 1.43080 3^ S8 .60007 1.66647 .62406 1.60241 .64858 I-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 I.4281S / Cotang- Tang- Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 59° 5< 3° 5: 7° S6° 5 5° 200 BROWN & SHARPE MFG. CO. NATURAL TANGENTS AND COTANGENTS / 35° 36° 37° 38° 39° / Tang: Cotang Tang Cotang Tang Cotang Tang Cotang Tang Cotang .70021 1.42815 •72654 1.37638 •75355 1.32704 .78129 1.27994 .80978 1.23490 60 I .70064 1.42726 .72699 1.37554 .75401 1.32624 -78175 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 1.42462 .72832 1.37302 .75538 I •32384 -78316 1.27688 .81171 1.23196 S6 .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 1.27535 .81268 1.23050 54 7 .70325 1.42198 .72966 I.37CSO .75675 1-32144 .78457 1.27458 .81316 1.22977 53 8 .70368 1.42110 .73010 1.36967 .75721 1.32064 .78504 1.27382 .81364 1.22904 52 9 .70412 1.42022 .73055 1.36883 .75767 1-31984 .78551 1.27306 .81413 1.22831 SI 10 .70455 1.41934 .73100 I 36800 .75812 1.31904 .78598 1.27230 .81461 1.22758 SO II .70499 1. 41847 .73144 1.36716 .75858 1.31825 .7864s 1.27153 .81510 1.22685 49 12 .70542 1.41759 .73189 1.36633 .75904 1-31745 .78692 1.27077 31558 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 1.41497 ■73323 I 36383 .76042 1^31507 .78834 1.26849 .81703 1.22394 45 i6 •70717 1.41409 .73368 1.36300 .76088 1.31427 .78881 1.26774 .81752 I. 22321 44 17 .70760 1.41322 •73413 1.36217 .76134 1.31348 .78928 1.26698 .81800 1.22249 43 i8 .70804 1.41235 .73457 1.36134 .76180 1. 3 1 269 .78975 1.26622 .81849 I. 22176 42 19 .70848 1.41148 •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 I. 2203 I 40 21 .70935 1.40974 .73592 1.35885 .76318 1.31031 .79117 1.26395 .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 1-30795 .79259 1.26169 .82141 1.21742 36 25 .71110 1.40627 • 73771 1^35554 .76502 1.30716 .79306 1.26093 .82190 1.21670 35 26 .71154 1.40540 • 73816 I.3S472 .76548 1.30637 .79354 1.26018 .82238 1.21598 34 27 .71198 1.40454 .73861 1.35389 .76594 1.30558 .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 I 30244 .79591 1.25642 .82483 1.21238 29 32 .71417 1.40022 .74086 1.34978 .76825 1.30166 .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 .71505 1-39850 • 74176 1.34814 .76918 1.30009 .79734 1. 25417 .82629 I. 21023 26 35 .71549 1.39764 .74221 1.34732 .76964 1.29931 .79781 1.25343 .82678 1.20951 25 36 .71593 1.39679 .74267 1.34650 .77010 1.29853 .79829 1.25268 .82727 1.20879 24 37 .71637 1.39593 .74312 1.34568 .77057 1.29775 .79877 1.25193 .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 I -25044 .82874 1.20665 21 40 .71769 1.39336 .74447 1.34323 .77196 I. 29541 .80020 1.24969 .82923 1.20593 20 41 .71813 1.39250 .74492 1.34242 .77242 1.29463 .80067 1-24895 .82972 1.20522 19 42 .71857 1.39165 .74538 1.34160 .77289 1.29385 .80115 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 45 .71990 1.38909 .74674 1. 33916 .77428 1-29152 .80258 1-24597 .83169 1.20237 15 46 .72034 1.38824 • 74719 1.33835 .77475 1.29074 .80306 1-24523 .83218 1.20166 14 47 .72078 1.38738 .74764 1^33754 .77521 1.28997 .80354 1-24449 .83268 1.20095 13 48 .72122 1.38653 .74810 1^33673 .77568 1.28919 .80402 1.24375 .83317 1.20024 12 49 .72167 1.38568 .74855 1^33592 .77615 1.28842 .80450 1.24301 -83366 1.19953 II 50 .72211 1.38484 .74900 1.33511 .77661 1.28764 .80498 1.24227 .83415 1.19882 10 51 .72255 1.38399 .74946 1-33430 .77708 1.28687 .80546 1.24153 .8346s 1.19811 9 52 •72299 1. 383 1 4 .74991 1-33349 .77754 1,28610 .80594 1.24079 .83514 1.19740 8 S3 ■^''Hi 1.38229 • 75037 l^33268 .77801 1-28533 .80642 1.24005 .83564 1. 1 9669 7 54 .72388 1.38145 .75082 I-33187 .77848 1.28456 .80690 1-23931 .83613 1.19599 6 SS .72432 1.38060 • 75128 1-33107 .77895 1.28379 .80738 1-23858 .83662 1.19528 5 S6 ■72477 1.37976 •75173 1 -33026 .77941 1.28302 .80786 1-23784 .83712 1.19457 4 H .72521 1. 37891 •75219 1.32946 .77988 1.28225 .80834 1.23710 .83761 1. 19387 3 S8 •72565 1^37807 • 75264 1-32865 .78035 I. 28148 .80882 1.23637 .83811 1.19316 2 f^ .72610 1.37722 .75310 1-32785 .78082 1.28071 .80930 1.23563 .83860 1. 19246 I 6o •72654 1^37638 .75355 1-32704 .78129 1.27994 .80978 1.23490 .83910 1.19175 / Cotang Tangr Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 5^ 1° 52 ° 52 ,0 51 5C )° 201 BROWN & SHARPE MFG. CO. NATURAL TANGENTS AND COTANGENTS / 40° 41° 42- 43° 1 44° / Tang- Cotang Tang Cotang Tang Cotang Tang Cotang ' rang Cotang o 83910 1.19175 .86929 1.15037 .90040 1.11061 .93252 1.07237 96569 1.03553 60 I 83960 1. 1910s .86980 I. 14969 .90093 1. 10996 .93306 1.07174 96625 1.03493 59 2 84009 1.1903s .87031 1.14902 .90146 1. 10931 .93360 1. 0711.2 96681 1.03433 58 3 84059 1.18964 .87082 1.14834 .90199 1.10867 .93415 1.07049 96738 1.03372 57 4 84108 I. I 8804 .87133 1.14767 .90251 1.10802 .93469 1.06987 96794 1.03312 56 5 84158 I. 18824 .87184 I. 14699 .90304 1.10737 .93524 1.06925 96850 1.03252 55 6 84208 1.18754 .87236 1.14632 •90357 1. 10672 .93578 1.06862 96907 1. 03192 54 7 84258 1. 1 8684 .87287 1.14565 .90410 1. 10607 .93633 1.06800 96963 1.03132 53 8 84307 1.18614 .87338 1. 14498 .90463 1. 10543 .93688 1.06738 97020 1.03072 52 9 84357 1. 18544 .87389 1.14430 .90516 I. 10478 .93742 1.06676 97076 1. 03012 SI lO 84407 1. 18474 .87441 1.14363 .90569 1.10414 •93797 1. 066 1 3 97133 1.02952 50 II 84457 1. 1 8404 .87492 1. 14296 .90621 1.10349 .93852 1.06551 97189 1.02892 49 12 84507 1. 18334 .87543 1. 14229 .90674 1.10285 .93906 1.06489 97246 1.02832 48 13 84556 1. 18264 .87595 1. 14162 .90727 1.10220 .93961 1.06427 97302 1.02772 47 14 84606 1.18194 .87646 1.1409s .90781 1.10156 .94016 1.06365 97359 1.02713 46 IS 84656 1.18125 .87698 1. 14028 .90834 i.ioogi .94071 1.06303 97416 1.02653 45 i6 84706 1. 18055 .87749 1.13961 .90887 1. 10027 .94125 1.06241 97472 1.02593 44 17 84756 i.i79«6 .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 97586 1.02474 42 19 84856 1. 1 7846 .87904 1.13761 .91046 1.09834 .94290 1.06056 97643 1.02414 41 20 84906 1. 17777 .87955 1.13694 .91099 1.09770 .94345 1. 05994 97700 I.023SS 40 2X 84956 I. 17708 .88007 1. 13627 .91153 1.09706 .94400 1.05932 97756 1.02295 39 22 85006 1. 1 7638 .88059 1. 13561 .91206 1.09642 .94455 1.05870 97813 1.02236 38 23 85057 1. 1 7569 .88110 1.13494 .91259 1.09578 .94510 1.05809 97870 1.02176 37 24 85107 1.17500 .88162 1. 13428 .91313 1.09514 .94565 1.05747 97927 1.02117 36 25 85157 1. 1 7430 .88214 1.13361 .91366 1.09450 .94620 1.05685 97984 1.02057 35 26 85207 1.17361 .88265 1.1329s .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 I. 13162 .91526 1.09258 .94786 1.05501 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 .B8473 1. 13029 .91633 1.09131 .94896 1.05378 98270 1.01761 30 31 85458 1.17016 .88524 I. I 2963 .91687 1.09067 .94952 I.05317 98327 1.01702 29 32 85509 1.16947 .88576 1. 1 2897 .91740 1.09003 .95007 1.05255 98384 1.01642 28 33 85559 1.16878 .88628 1.12831 .91794 1.08940 .95062 1.05194 98441 1.01583 27 34 85609 1.16809 .88680 1.12765 .91847 1.08876 .95118 I.OS133 98499 1.01524 26 35 85660 1.16741 .88732 1.12699 .91901 1.08813 .95173 1.05072 98556 1. 01465 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 8581 1 1-16535 .88888 1.12501 .92062 1.08622 .95340 1.04888 98728 1.01288 22 39 85862 1. 16466 .88940 1.12435 .92116 1. 08559 •95395 1.04827 98786 1.01229 21 40 85912 1. 16398 .88992 1.12369 .92170 1.08496 .95451 1.04766 98843 1.01170 20 41 85963 I. 16329 .89045 1.12303 .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 86115 1. 16124 .89201 1.12106 .92385 1.08243 ■95673 1.04522 99073 1.00935 16 45 86166 1.16056 .89253 1.12041 .92439 1.08179 .95729 1. 04461 99131 1.00876 15 46 86216 1. 15987 .89306 1.11975 .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 1.15851 .89410 1.11844 .92601 1.07990 .95897 1.04279 99304 1. 00701 12 49 86368 I. 15783 .89463 1.11778 .92655 1.07927 .95952 1. 04218 99362 1.00642 II 50 86419 1.15715 .89515 1.11713 .92709 1.07864 .96008 1. 04158 99420 1.00583 10 51 86470 1.15647 .89567 1.11648 .92763 1. 07801 .96064 1.04097 99478 1.0052s 9 52 86521 1.ISS79 .89620 1.11582 .92817 1.07738 .96120 1.04036 99536 1.00467 8 53 86572 1.15511 .89672 1.11517 .92872 1.07676 .96176 1.03976 99594 1.00408 7 54 86623 1.15443 .89725 1.11452 .92926 1.07613 .96232 1.03915 99652 i.oqsso 6 55 86674 1.1S37S .89777 1. 11387 .92980 1.07550 .96288 1.03855 99710 1. 00291 S 56 86725 1.15308 .89830 1.11321 .93034 1.07487 .96344 1.03794 99768 1.00233 4 57 86776 I. 15240 .89883 1.11256 .93088 1.0742s .96400 1.03734 99826 I. 001 75 3 58 86827 1.15172 .89935 1. 11191 .93143 1.07362 .96457 1.03674 99884 1.00116 2 59 86878 1.IS104 .89988 1. 11126 .93197 1.07299 .96513 1.03613 99942 1.00058 I 60 86929 1.15037 .90040 1.11061 .93252 1.07237 .96569 1.03553 I 00000 1.00000 C otang Tang Cotang Tang Cotang Tang Cotang Tang C otang Tang / / "~ 4c )° 4^ r 4; 7° 4^ 5° 45° 202 BROWN & SHARPE MFG. CO. DECIMAL EQUIVALENTS OF PARTS OF AN INCH J^ ... .01563 A 03125 A ... .04688 i-i6 0625 ^5^ ... .07813 A- 09375 7- ... .10938 1-8 125 ^\ ... .14063 -352 15625 H ... .17188 3-16 1875 if ... .20313 3V 21875 if ... .23438 1-4 25 i| ... .26563 A 28125 if ... .29688 5-16 3125 IJ ... .32813 4i 34375 If ... .35938 3-8 375 II ... .39063 i| 40625 IJ ... .42188 7-16 4375 If ... .45313 If 46875 |i ... .48438 1-2 5 ff ... .51563 H 53125 M ... .54688 6 4 9-16 5625 ff ... .57813 if 59375 If ... .60938 5-8 .625 If ... .64063 21- ...... .65625 43 ... .67188 11-16 6875 45 6? 64 3-4 49 64 ,2 5 32 64 13-16 53 64 2.7. 32 55 6¥ 7-8 Al 64 29 32" 59. 64 15-16 31 '3 2- 13 64 . .70313 . .71875 . .73438 . 75 .76563 .78125 .79688 .8125 .82813 .84375 .85938 .875 . .89063 . .90625 . .92188 . .9375 , .95313 .96875 .98438 1.00000 203 BROWN & SHARPE MFG. CO. TABLE OF DECIMAL EQUIVALENTS OF MILLIMETRES AND FRACTIONS OF MILLIMETRES. mm. Inches. mm. Inches. mm. Inches. mm. Inches. , ^ = .00039 Wo = -01399 Wo = -03530 m = -03740 i§0 = .00079 fj = .01339 ^ = .03559 fo = .03780 il = -00118 Wo = -01378 Wo = -02598 fo = .03819 m = -00157 1) = -01417 ^ = .03638 fo = .03858 4 = .00197 #0 = -01457 m = -03677 ^ = .03898 4 = .00336 m = -01496 i = .03717 1 = .03937 ^ = .00376 ^ = .01535 fo = .03756 3 = .07874 4 = .00315 Wo = .01575 ^ = .03795 3 = .11811 4 = .00354 ^0 = -01614 fo = .03835 4 =- .15748 Wo = -00394 fo = -01654 fo = .02874 5 = .19685 ^ = .00433 f == .01693 fo = .02913 6 = .33633 i) = .00473 i) = .01733 ifo = .03953 7 = .37559 m = .00513 Wo = -01773 f = .03993 8 = .31496 ^ = .00551 f, = .01811 fo = .03033 9 = .35433 ij = .00591 f = .01850 f = -03071 10 = .39370 m = -00630 f = .01890 f = .03110 11 = .43307 1^ = .00669 ^ = .01939 i; == .03150 13 = .47344 ^ = .00709 j^ = .01969 fo = .03189 13 = .51181 Wo = -00748 iS) = .03008 fo = .03338 14 = .55118 Wo = .00787 Wo = .03047 fo = .03368 15 = .59055 ^ = .00837 Wo = .03087 fo = .03307 16 = .63993 il = -00866 fo = .03136 m = .03346 17 = .66939 H = .00906 ^ = .03165 fo = .03386 18 = .70866 ^ = .00945 i^ = .03305 fo = .03435 19 = .74803 ^ = .00984 fo = .03344 fo = .03465 30 = .78740 Wo = .01034 ^ = .03383 fj = .03504 31 = .83677 1^ = .01063 fo = .03333 fo = .03543 33 = .86614 fo = -01103 ^ = .03363 f = .03583 33 = .90551 m = -01143 ^ = .03403 fo = .03633 34 = .94488 fi - -01181 Wo - -O^il S = .03661 35 = .98435 ^ = .01330 ^ - .03480 Wo = .03701 36 =1.03363 Wo = .01360 10 mm. = 1 Centimeter : 10 cm. = 1 Decimeter = 0.3937 inches. J.937 inches. 10 dm. = 1 Meter = 39.37 inches. 25.4 mm. = 1 English Incli. 204 BROWN & SHARPE MFG. CO. OTHER PUBLICATIONS ISSUED BY THE BROWN & SHARPE MFG. CO. Practical Treatise on Milling and Milling Machines Edition of 1919 This work is a thorough treatise on MilHng and MiUing Machines. 332 pages, 210 illustrations. Sent on receipt of price. Cloth, $1.50; Cardboard, $1.00. Construction and Use of Automatic Screw Machines Edition of 1919 This book is published to assist those who are not familiar with the construction and use of the Automatic Screw Machine. Sent on receipt of price. Cardboard, 50 cents. Construction and Use of Universal Grinding Machines Edition of 1919 This work describes the construction and use of Universal Grinding Machines, as made by us. Fully illustrated. Sent on receipt of price. Cardboard, 25 cents. Construction and Use of Plain Grinding Machines Edition of 1920 This work describes the construction and use of Plain Grinding Machines, as made by us. Fully illustrated. Sent on receipt of price. Cardboard, 25 cents. Formulas in Gearing Edition of 1918 This work supplements the * 'Practical Treatise on Gearing" and contains formulas for solving the problems that occur in gearing. Sent on receipt of price. Cloth, $1.50. Hand Book for Apprenticed Machinists This book, illustrated, is for the Apprenticed Machinist. It is carefully written to assist the learner in the use of Machine Tools. Sent on receipt of price. Cloth, 75 cents. 205 BROWN & SHARPE MFG. CO. INDEX A PAGE Abbreviations of Parts of Teeth and Gears 12 Addendum 11 Angle, How to Lay Off an 146, 163 Angle Increment 161 Angle of Edge 157 Angle of Face 159 Angle of Pressure 164 Angle of Spiral 114 Angular Velocity 10 Annular Gears 43, 45 Arc of Action 165 Automatic Gear Cutting Machine 156 B Base Circle 17 Base of Epicycloidal System 36 Base of Internal Gears 45 Bevel Gear Blanks 48 Bevel Gear Cutting on Automatic Machine 73 Bevel Gear Angles by Diagram 50 Bevel Gear Angles by Calculation 157 Bevel Gear, Form of Teeth of 55 Bevel Gear, Spiral 79 Bevel Gear, Whole Diameter of 50, 161 C Centres, Line of 11 Chordal Thickness 174, 175 Circular Pitch, Linear or 12 Circular Pitch, "Nuttall" 177 Classification of Gearing 13 Clearance at Bottom of Space 14 Clearance in Pattern Gears 16 207 BROWN & SHARPE MFG. CO. INDEX PAGE Condition of Constant Velocity Ratio 10 Contact, Arc of 165 Continued Fractions 166 Coppering Solution 107 Cosine, Sine and 151 Cutters, How to Order 105 Cutters, Table of Epicycloidal 105 Cutters, Table of Involute 104 Cutters, Table of Feeds for 103 Cutters, Shape of 121 Cutting Bevel Gears on Automatic Machine 73 Cutting Spiral Gears in a Universal Milling Machine. 124 D Decimal Equivalents, Tables of 203, 204 Diameter Increment 159 Diameter of Pitch Circle 14 Diameter Pitch 14 Diametral Pitch 25 Diametral Pitch, "Nuttall" 176 Distance between Centres 15 E Elements of Gear Teeth 13 Epicycloidal Gears, with more or less than 15 Teeth 41 Epicycloidal Gears, with 15 Teeth 36 Epicycloidal Rack 38 F Face, Width of Spur Gear 102 Flanks of Teeth in Low Numbered Pinions 29 G Gear Cutters, How to Order 105 Gear Patterns 16 Gearing Classified 13 Gears, Bevel. : 48, 55, 157 Gears, Epicycloidal 36 208 BROWN & SHARPE MFG. CO. INDEX PAGE Gears, Involute 17 Gears, Spiral 108, 114, 117, 124 Gears, Spiral Bevel 79 Gears, Strength of 135 Gears, Worm 83 H Herring-bone Gears 132 Hobs 87 I Increment, Angle 161 Increment, Diameter 159 Interchangeable Gears 34 Internal or Annular Gears 43, 45 Involute Gears, 30 Teeth and over 17 Involute Gears, with Less than 30 Teeth 29 Involute Rack 20 L Land 10 Lead of a Worm 83 Length of a Worm and a Hob 98 Limiting Numbers of Tee thin Internal Gears 43 Line of Centres 11 Line of Pressure 20, 164 Linear or Circular Pitch 12 Linear Velocity 9 M Machine for Cutting Bevel Gears 69 Method for Obtaining Set-over for Bevel Gears 70 Method of Drawing a Rack 21 Method of Laying Out Single-curve Gears 17 Mitre Gears 78 Module 14 209 BROWN & SHARPE MFG. CO. INDEX N PAGE Normal 117 Normal Helix 117 Normal Pitch 119 O Original Cylinders 9 P Pattern Gears 16 Pinions '. 43 Pitch Circle 12 Pitch, Circular or Linear 12 Pitch, a Diameter 14 Pitch, Diametral 25 Pitch, Normal 117 Pitch of Spirals 112 Polygons, Calculations for Diameters of 153 R Rack 20 Rack for Epicycloidal Gears 38 Rack for Involute Gears 21- Rack for Spiral Gears .;.... 119 Relative Angular Velocity , .10 RolHng Contact of Pitch Circle 12 S Screw Gearing 108 Shape of Cutters 121 Sine and Cosine 151 Single-Curve Teeth 17 Speed of Gear Cutters 102 Spiral Bevel Gears 79 Spiral Gearing 108, 114, 117, 124 Standard Templets 38 Standard Proportions for Spur Gears 143 210 BROWN & SHARPE MFG. CO. INDEX PAGE Strength of Gears 135 Squares and Square Roots 172 T Table of Chordal Thickness of Gear Teeth 174, 175 Table of Decimal Equivalents 203, 204 Table of Feeds for Gear Cutters 103 Table of Gear Strength Factors 134 Table for Obtaining Set-over for Bevel Gears. . 70 Table for Right Angled Triangles 182 Table of Sines, etc 184-202 Table of Tooth Parts 178-181 Tangent of Arc and Angle 145 V Velocity, Angular 10 Velocity, Linear 9 Velocity, Relative 10 W Wear of Teeth 102, 131 Worm Gears 83 Worm, Length of 98 211