{"1": {"fulltext": "", "height": "4588", "width": "3052", "jp2-path": "practicaltreatis07beal_0001.jp2"}, "2": {"fulltext": "", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0002.jp2"}, "3": {"fulltext": "V\\nV\\noo\\ntf*\u00c2\u00ab\\nH -7*", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0003.jp2"}, "4": {"fulltext": "", "height": "4326", "width": "2475", "jp2-path": "practicaltreatis07beal_0004.jp2"}, "5": {"fulltext": "", "height": "4326", "width": "2475", "jp2-path": "practicaltreatis07beal_0005.jp2"}, "6": {"fulltext": "", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0006.jp2"}, "7": {"fulltext": "", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0007.jp2"}, "8": {"fulltext": "", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0008.jp2"}, "9": {"fulltext": "PRACTICAL TREATISE\\nON\\nGEARING\\noJ^Ls s JUQstS^\\no\\n1 4* 3 I\\n3\\nSIXTH EDITION.\\nPROVIDENCE, R. I.\\nBROWN SHARPE MANUFACTURING COMPANY.\\n1900.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0009.jp2"}, "10": {"fulltext": "TWO COPIES RECEIVED,\\nLibrary of C8Bgr\u00c2\u00a7\u00c2\u00ab%\\n(Jfflee of tb\u00c2\u00ae\\nAPR 1 1900\\ntegltter of Copyrights,\\nB3-7\\n)C[00\\n61040\\nEntered according to Act of Congress, in the year lyuO by\\nBROWN SHARPE MFG. CO.,\\nIn the Office of the Librarian of Congress at Washington.\\nRegistered at Stationers Hall, London, Eng.\\nAH rights reserved.\\nO0-2V9? P\\nSECOND GOPY,", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0010.jp2"}, "11": {"fulltext": "PREFACE.\\nThis Book is made for men in practical life for those that\\nwould like to know how to construct gear wheels, but whose\\nduties do not afford them sufficient leisure to acquire a technical\\nknowledge of the subject.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0011.jp2"}, "12": {"fulltext": "\\\\\\\\\u00c2\u00bb\\\\1\\nAf^.1^,1^", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0012.jp2"}, "13": {"fulltext": "CONTENTS\\nPAKT I\\nChapter I.\\nPAGE.\\nPitch Circle Pitch Tooth Space Addendum or Face\\nFlank Clearance 1\\nChapter II.\\nClassification Sizing Blanks and Tooth Parts from Circular\\nPitch Ce n ter D istan ce 5\\nChapter III.\\nSingle Curve Gears of 30 Teeth and more 9\\nChapter IV.\\nRack to Mesh with Single Curve Gears having 30 Teeth and\\nmore 12\\nChapter V.\\nDiametral Pitch Sizing Blanks and Teeth Distance be-\\ntween the Centers of Wheels 16\\nChapter VI.\\nSingle-Curve Gears, having Less than 30 Teeth Gears and\\nRacks to Mesh with Gears having Less than 30 Teeth... 20\\nChapter VII.\\nDouble-Curve Teeth\u00e2\u0080\u0094 Gear of 15 Teeth\u00e2\u0080\u0094 Rack 25\\nChapter VIII.\\nDouble-Curve Gears, having More and Less than 15 Teeth\\nAnnnlar Gears 30", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0013.jp2"}, "14": {"fulltext": "VI CONTENTS.\\nChapter IX.\\nPAGK.\\nBevel Gear Blanks 3-t\\nChapter X.\\nBevel Gears Form and Size of Teeth Cutting Teeth 41\\nChapter XI.\\nWorm Wheels Sizing Blanks of 32 Teeth and more G3\\nChapter XII.\\nSizing Gears when the Distance between the Centers and the\\nRatio of Speeds are fixed General Remarks Width of\\nFace of Spur Gears Speed of Gear Cutters Table of\\nTooth Parts 79\\nPART II.\\nChapter I.\\nTangent of Arc and Angle 8?\\nChapter II.\\nSine, Cosine and Secant Some of their Applications in\\nMachine Construction 93\\nChapter III.\\nApplication of Circular Functions Whole Diameter of Bevel\\nGear Blanks Angles of Bevel Gear Blanks 102\\nChapter IV.\\nSpiral Gears Calculations for Pitch of Spirals 109\\nChapter V.\\nExamples in Calculations of Pitch of Spirals Angle of\\nSpiral Circumference of Spiral Gears A few Hints\\non Cutting 113", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0014.jp2"}, "15": {"fulltext": "CONTENTS. VII\\nChapter VI.\\nPAGE.\\nNormal Pitch of Spiral Gears Curvature of Pitch Surface\\nFormation of Cotters 116\\nChapter VII.\\nCutting Spiral Gears in a Universal Milling Machine 122\\nChapter VIII.\\nScrew Gears and Spiral Gears General Remarks 129\\nChapter IX.\\nContinued Fractions Some Applications in Machine Con-\\nstruction 132\\nChapter X.\\nAngle of Pressure 137\\nChapter XI.\\nIn fcernal Gears Tables Index 139\\nChapter XII.\\nStrength of Gears Tables 142", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0015.jp2"}, "16": {"fulltext": "", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0016.jp2"}, "17": {"fulltext": "PART I.\\nCHAPTER I.\\nPITCH CIRCLE, PITCH, TOOTH, SPACE, ADDENDUM OR FACE, FLANK,\\nCLEARANCE.\\nLet two cylinders, Fig. 1, touch each other, their original Cyi\\naxes be para lei and the cylinders be on shalts, turning\\nfreely. If, now, we turn one cylinder, the adhesion of\\nits surface to the surface of the other cylinder will\\nmake that turn also. The surfaces touching each\\nother, without slipping one upon the other, will evi-\\ndently move through, the same distance in a given\\nLinear Veloci\\ntime. This surface speed is called linear velocity. ty.\\nTANGENT CYLINDERS.\\nLinear Velocity is the distance a point moves along\\na line in a unit of time.\\nThe line described by a point in the circumference\\nof either of these- cylinders, as it, rotates, may be called\\nan arc. The length of the arc (which may be greater\\nor less than the circumference of cylinder), described\\nin a unit of time, is the velocity. The length, expressed\\nin lineal* units, as inches, feet, etc., is the linear velocity.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0017.jp2"}, "18": {"fulltext": "EEOWS t SHAP.PE MEG. CO.\\nThe length, expressed in angular units, as degrees, is\\nthe angular Telocity.\\nIf now, instead of l 3 we take 360 or one turn, as\\nAngular v e- t le ann^ar nnit, and 1 minute as the time unit, the\\nangular velocity will be expressed in turns or revolu-\\ntions per minute.\\nIf these two cylinders are of the same size, one will\\nmake the same number of turns in a minute that the\\nRelative An- other makes. It cne cylinder is twice as large as the\\ngular elocitr. _\\nother, the smaller will make two turns while the larger\\nmakes one. but the linear velocity of the surface of\\neach cylinder remains the same.\\nThis combination would be very useful in mechan-\\nism if we could be sure that one cylinder would always\\nturn the other without slipping.\\n_-s-\\nLancL\\nAddendum.\\nTooth.\\nGear.\\nTrain.\\n-:::\u00e2\u0080\u00a2.:_\\n~z-\\nLINE\\n_~ t\\n=?:_\u00c2\u00a3\\nIFi~. 3\\nIn the periphery of these two cylinders, as in Fig.\\n2, cut equidistant grooves. In any grooved piece the\\nplaces between grooves are called lands. Upon the\\nlands add part-; these parts are called addenda. A\\nland and its al.lendum is called a tooth. A toothed\\ncylinder is called a gear. Two or more gears with\\nteeth interlocking are called a train. A Hue. c c\\\\ Fig.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0018.jp2"}, "19": {"fulltext": "PROVIDENCE, R. I\\nAddendum\\nCircle.\\n2 or 3, between the centers of two wheels is called the Line of Cen\\nline of centers. A circle just touching the addenda ters\\nis called the addendum circle.\\nThe circumference of the cylinders without teeth is\\ncalled the pitch circle. This circle exists geometri- Pltch Circle\\ncally in every gear and is still called the pitch circle pitcll circle\\nor the primitive circle. In the study of gear wheels, it th e alS PrinStive\\nis the problem to so shape the teeth that the pitch circle,\\ncircles will just touch each other without slipping.\\nOn two fixed centers there can turn only two circles,\\none circle on each center, in a given relative angular\\nyelocity and touch each other without slipping.\\nTHICKNESS OF\\nWIDTH OF SPACE j TOOTH\\nAT PITCH LINE AT PITCH LINE\\nFig. 4", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0019.jp2"}, "20": {"fulltext": "4 BROWN SHARPE MEG. CO.\\nSpace. The groove between two teeth is called a space.\\nIn cut gears the width of space at pitch line and thick-\\nness of tooth at pitch line are equal. The distance\\nbetween the center of one tooth and the center of the\\ncircular Pitch, next tooth, measured along the pitch line, is the cir-\\ncular pitch that is, the circular pitch is equal to a\\nTooth Thick- tooth and a space; hence, the thickness of a tooth at\\nness. x\\nthe pitch line is equal to one-half the circular pitch.\\n\u00e2\u0080\u00a2^P^^V Let D diameter of addendum circle.\\ntions of Parts\\nfor Teeth and t D pitch\\nP circular pitch.\\nt= thickness of tooth at pitch line.\\ns addendum or face, also length of working\\npart of tooth below pitch line or flank.\\n2s=D or twice the addendum, equals the\\nworking depth of teeth of two gears in\\nmesh.\\nf\u00e2\u0080\u0094 clearance or extra depth of space below\\nworking depth.\\ns +f= depth of space below pitch line.\\nD +f= whole depth of space.\\nN= number of teeth in one gear.\\n7r=3.1416 or the circumference wlien diameter\\nis 1.\\nP is read P prime/ D is read D second.\\nn is read, pi.\\nIf we multiply the diameter of any circle by tt, the\\nTo find the product will bs the circumference of this circle. If\\nand Diameter we divide the circumference of any circle by tt, the\\nquotient will be the diameter of this circle.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0020.jp2"}, "21": {"fulltext": "CHAPTER II.\\nCLASSIFICATION -SIZING BUNKS AND TOOTH PARTS FROM\\nCIRCULAR PITCE\u00e2\u0080\u0094 CENTRE DISTANCE\u00e2\u0080\u0094 PATTERN GEARS.\\nIf we conceive the pitch, of a pair of cears to be Elements of\\nx the Teeth.\\nmade the smallest possible, we ultimately come to the\\nconception of teetn that are merely lines upon the\\noriginal pitch surfaces. These lines are called ele-\\nments of the teeth. Gears may be classified with\\nreference to the elements of their teeth, and also with\\nreference to the relative position of their axes or -shafts.\\nIn most gears the elements of teeth are either straight\\nlines or helices (screw-like lines).\\nPart I. of this book, treats upon three kinds of\\ngears.\\nFirst Spur Gears those connecting parallel shafts tVX GearB\\nand whose tooth elements are straight.\\nSecond Bevel Gears; those connecting shafts Bevel Gears,\\nwhose axes meet when sufficiently prolonged, and the\\nelements of whose teeth are straight lines. In bevel\\ngears the surfaces that touch each other, without\\nslipping, are upon cones or parts of cones whose\\napexes are at the same point where axes of shafts meet.\\nThird Screw or Worm Gears; those connecting w e Jlars. r\\nshnfts that are not parallel and do not meet, and the\\nelements of whose teeth are helical or screw-like.\\nThe circular itcfi and number of teeth in a wheel giz n?\\nbeingf o-iven, the diameter of the wheel and size of Blanks,\\ntooth parts are found as follows\\nDividing by 3.1410 is the same as multiplying by\\nNow J\u00e2\u0080\u0094 T 3183; hence, multiply the cir-\\n3-14 16- 3-14 16\\ncumference of a circle by .3183 and the product will be\\nthe diameter of the circle. Mult ply the circular pitch\\nby .3183 and the product will be the same part of the", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0021.jp2"}, "22": {"fulltext": "G BBOWM SHARPE MFG. CO.\\ndiameter of pitch circle that the circular pitch is of the\\ncircumfeivnce of pitch circle. This part or module is\\na Diameter C; ,ii e d a diameter pitch. There are as manv diameter\\nPitch, or Mod-\\nuie. pitches contained in the diameter of a pitch circle as\\nthere are teeth in the wheel.\\nai^the^ddeH 6 Most mechanics make the addendum of teeth equal\\n?ili^^ a ^f to the module. Hence we can designate the module or\\nthe anie. raai- o\\nan y- diameter pitch by the same letter as we do the adden-\\ndum that is, let s=the module.\\n.3183 V or circular pitch multiplied by .3183=5.\\nor the module.\\nX =D or number of teeth in a wheel, multiplied by\\nDiameter of J\\nPitch circle, the module, equals diameter of pitch circle.\\n(N-j-2) s=D, or add 2 to the number of teeth, mul-\\nwiioieDiam-tinv the sum by s and product will be the whole di-\\neter. L r\\na meter.\\nor one tenth of thickness of tooth at pitch-line\\nclearance. equals amount added to bottom of space for clearance.\\nSome mechanics prefer to make/ 1 equal to T *g- of the\\nworking depth of teeth, or .0625 D One-tenth of\\nthe thickness of tooth at pitch-line is more than one-\\nsixteenth of working depth, being .07S54 D\\nExample. Example. Wheel 30 teeth. U circular pitch.\\nc _, P =1.5 then t\u00e2\u0080\u0094~o or thickness of tooth equals\\nSizes ol Blank J-\\nand Tooth s 1.5 x .31 83=.47 moduie for 1^ P (See\\nfor Gear\\nof so teeth. tables of tooth parts, pages 145 -14 s\\nphcA; 11 D 30X.4775 14.325 diaineter of pitch-circle.\\nD=(30+2)X.4775 =15.280 =diameter of adden-\\ndum circle.\\ntV \u00c2\u00b0f -75 .075 clearance at bottom of space.\\nD 2x. 4775 9549 working depth of teeth.\\nD -/-2x.4775 .075 1.0299 whole depth of\\nspace.\\ns+/=. 4775 -.075 5525 depth of space inside\\nof pitch-line.\\nD =-2s or the working depth of teeth is equal to\\ntwo modules.\\nIn making calculations it is well to retain the fourth\\nplace in the decimals, but when drawings are passed\\ninto the workshop, three places of decimals are suffi-\\ncient.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0022.jp2"}, "23": {"fulltext": "PROVIDENCE, K. I.\\nFig. 5, Spur Gearing.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0023.jp2"}, "24": {"fulltext": "8 BROWN SHARPE MFG-. CO.\\nDistance be- The distance between the centers of two wheels is\\nIween centers\\nof two Gears, evidently equal to the radius of pitch-circle of one wheel\\nadded to that of the other. The r.idius of pitch-circle\\nis equal to s multiplied by one-half the number of teeth\\nin the wheel.\\nHence, if we know the number of teeth in two wheels,\\nin mesh, and the circular pitch, to obtain the distance\\nbetween centers we first find s then multiply s by one-\\nhalf the sum of number of teeth in both wheels and the\\nproduct will be distance between centers.\\nExample What is the distance between the centers\\nof two wheels 35 and 60 teeth, 1J circular pitch. We\\nfirst find s to be 11 x .3183 3979 Multiplying by\\n47.5 (one-half the sum of 35 and 60 teeth) we obtain\\n18.899 as the distance between centers.\\nQ^T\u00e2\u0084\u00a2 f fJT Pattern Gears should be made large enough to\\ncnnnKage in o o\\nGear Castings, allow for shrinkage in casting. In cast iron the shrinkage\\nis about ijj inch in one foot. For gears one to two feet\\nin diameter it is well enough to add simply -j-j-g- of\\ndiameter of finished gear to the pattern. In gears\\nabout six inches diameter or less, the moulder will\\ngenerally rap the pattern in the sand enough to make\\nany allowance for shrinkage unnecessary. In pattern\\ngears the spaces between teeth should be cut wider\\nthan finished gear spaces to allow for rapping and to\\navoid having too much cleaning to do in order to have\\ngears run freely. In cut patterns of iron it is generally\\nMetal Pattern enough to make spaces .015 to .02 wider. This\\nmakes clearance .03 to .04 in the patterns. Some\\nmoulders might want .06 to .07 clearance.\\nMetal patterns should be cut straight they work\\nbetter with no draft. It is well to leave about .005 to\\nbe finished from side of patterns after teeth are cut\\nthis extra stock to be taken away from side where\\ncutter comes through so as to take out places where\\nstock is broken out. The finishing should be done\\nwith file or emery wheel, as turning in a lathe is likely\\nto break out stock as badly as a cutter might do.\\nIf cutters are kept sharp and care is taken when\\ncoming through the allowance for finishing is not nec-\\nessary and the blanks may be finished before they are\\ncut.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0024.jp2"}, "25": {"fulltext": "CHAPTER III.\\nSINGLE-CURVE GEARS OF 30 TEETH AND OVER.\\nSingle-curve teeth are so called because they have T g^ le Curve\\nbut one curve by theory, this curve forming both face\\nand flank of tooth sides. In any gear of thirty teeth\\nand more, this curve can be a single arc of a circle\\nwhose radius is one-fourth the radius of the pitch\\ncircle. In gears of thirty teeth and more, a fillet is\\nadded at bottom of tooth, to make it stronger, equal\\nin radius to one-seventh the widest part of tooth space.\\nA cutter formed to leave this fillet has the advantage\\nof wearing longer than it would if brought up to a\\ncorner.\\nIn gears less than thirty teeth this fillet is made the\\nsame as just given, and sides of teeth are formed with\\nmore than one arc, as will be shown in Chapter VI.\\nHaving calculated the data of a efear of 30 teeth, f Example of a\\nfo 4 Gear, N=30, P\\ninch circular pitch (as we did in Chapter II. for lh\\npitch), we proceed as follows\\n1. Draw pitch circle and point it off into parts equal Geometrical\\nConstruction.\\nto one-hall the circular pitch. Fig. 6.\\n2. From one of these points, as at B, Fig. 6, draw\\nradius to pitch circle, and upon this radius describe a\\nsemicircle; the diameter of this semicircle being equal\\nto radius of pitch circle. Draw addendum, working\\ndepth and whole depth circles.\\n3. From the point B, Fig. 6, where semicircle, pitch\\ncircle and outer end of radius to pitch circle meet, lay\\noff a distance upon semicircle equal to one-fourth the\\nradius of pitch circle, shown in the figure at BA, and\\nis laid off as a chord.\\n4. Through this new point at A, upon the semicircle,\\ndraw a circle concentric to pitch circle. This last is", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0025.jp2"}, "26": {"fulltext": "10\\nBROWS k SHAEPE MEG. CO.\\nFig. 6\\nr^^SWM\\nGEAR,\\n30 TEETH,\\nCIRCULAR PITCH,\\nP\\nfor .75\\nN\\n=30\\nP\\n=4.1888\\nt\\n.375\\nS\\n.2387*\\nD\\n.4775\\nS+/\\n.2762\\nD +f\\n.5150\\nD\\n7.1610\\nD\\n7.6384\\nSINGLE CURVE GEAR.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0026.jp2"}, "27": {"fulltext": "PROVIDENCE, R. I. 11\\ncalled tlie base circle, and is the one for centers of\\ntooth arcs. In the system of single cnrve gears, we\\nhave adopted the diameter of this circle is .968 of the\\ndiameter of pitch circle. Thus the base circle of any\\ngear 1 inch pitch diameter by this system is .968\\nIf the pitch circle is 2 the base circle will be 1.936.\\n5. With dividers set to one-quarter of the radius of\\npitch circle, draw arcs forming sides of teeth, placing\\none leg of the dividers in the base circle and letting\\nthe other leg describe an arc through a point in the\\npitch circle that was made in laying off the parts equal\\nto one-half the circular pitch. Thus an arc is drawn\\nabout A as center through B.\\n6. With dividers set to one-seventh of the widest part\\nof tooth space, draw the fillets for strengthening teeth\\nat their roots. These fillet arcs should just touch the\\nwhole depth circle and the sides of teeth already\\ndescribed.\\nSingle curve or involute gears are the only gears invori^Gear-\\nthat can run at varying distance of axes and transmit lng-\\nunvarying angular velocity. This peculiarity makes\\ninvolute gears specially valuable for driving rolls or\\nany rotating pieces, the distance of whose axes is\\nlikely to be changed.\\nThe assertion that gears crowd harder on bearings Pressure on\\nbearings.\\nwhen of involute than when of other forms of teeth,\\nhas not been proved in actual practice.\\nBefore taking next chapter, the learner should make Practice, De-\\nseveral drawings of gears 30 teeth and more. Say next chapter.\\nmake 35 and 70 teeth 1| P Then make 40 and 65\\nteeth F\\nAn excellent practice will be to make drawing on\\ncardboard or Bristol-board and cut teeth to lines, thus\\nmaking paper gears or, what is still better, make them\\nof sheet metal. By placing these in mesh the learner\\ncan test the accuracy of his work.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0027.jp2"}, "28": {"fulltext": "12 BKOWN SHAKPE MFG. CO.\\nCHAPTER IV.\\nRACK TO MESH WITH SINGLE-CURYE GEARS HAYING\\n30 TEETH AND OYER.\\nmade a g re ^ra This gear (Fig. 7) is made precisely the same as gear\\ntory to drawing i n Chapter III. It makes no difference in which direc-\\ntion the construction radius is drawn, so far as obtain-\\ning form of teeth and making gear are concerned.\\nHere the radius is drawn perpendicular to pitch line\\nof rack and through one of the tooth sides, B. A semi-\\ncircle is drawn on each side of the radius of the pitch\\ncircle.\\nThe points A and A are each distant from the point\\nB, equal to one-fourth the radius of pitch circle and\\ncorrespond to the point A in Fig. 6.\\nIn Fig. 7 add two lines, one passing through B and\\nA and one through B and A These two lines form\\nangles of 75- 10 (degrees) with radius BO. Lines BA\\nand BA are called lines of pressure. The sides of\\nrack teeth are made perpendicular to these lines.\\nRack. A Rack is a straight piece, having teeth to mesh\\nwith a gear. A rack may be considered as a gear of\\ninfinitely long radius. The circumference of a circle\\napproaches a straight line as the radius increases, and\\nwhen the radius is infinitely long any finite part of the\\nconstruction circumference is a straight line. The pitch line of a\\nof Pitch Line of\\nHack. rack, then, is merely a straight line just touching the\\npitch circle of a gear meshing with the rack. The\\nthickness of teeth, addendum and depth of teeth\\nbelow pitch line are calculated the same as for a wheel.\\n(For pitches in common use, see table of tooth parts.)\\nThe term circular pitch when applied to racks can be\\nmore accurately replaced by the term linear pitch.\\nLinear applies strictly to a line in general while circular\\npertains to a circle. Linear pitch means the distance\\nbetween the centres of two teeth on the pitch line\\nwhether the line is straight or curved.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0028.jp2"}, "29": {"fulltext": "PROVIDENCE, R. I.\\n13\\nA rack to mesh with a single-curve gear of 30 teeth\\nor more is drawn as follows\\n1. Draw straight pitch line of rack also draw ad-\\ndendum line, working depth line and whole depth line,\\neach parallel to the pitch line (see Fig. 7).\\nRack.\\nFig. 7.\\nRACK TO MESH WITH SINGLE CURVE GEAR\\nHAVING 30 TEETH AND OVER.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0029.jp2"}, "30": {"fulltext": "14 BROWN i: SHARPE MFG. CO.\\n2. Point off the pitch line into parts equal to one-\\nhalf the circular pitch, or t.\\n3. Through these points draw lines at an angle of\\n7-H with pitch lines, alternate lines slanting in oppo-\\nsite directions. The left-hand side of each rack tooth\\nis perpendicular to the line BA. The right-hand side\\nof each rack tooth is perpendicular to the line BA\\n4. Add fillets at bottom of teeth equal to 4- of the\\nwidth of spaces between the rack teeth at the adden-\\ndum line.\\nsid\\\\? g of e nick Tlie sketch, Fig. 8, will show how to obtain angle cf\\nTeeth. sides of rack teeth, directly from pitch line of rack,\\nwithout drawing a gear in mesh with the rack.\\nUpon the pitch line b b draw any semicircle\\nb a a 1/ From point b lay off upon the semicircle\\nthe distance b equal to one-quarter of the diameter\\nof semicircle, and draw a straight line through b and a.\\nThis line, b a, makes an angle of 75.}\u00c2\u00b0 with pitch line\\nbb and can be one side of rack tooth. The same\\nconstruction, b a will give the inclination 75|\u00c2\u00b0 in the\\nopposite direction for the other side of tooth.\\nThe sketch, Fig. 9, gives the angle of sides of a tool\\nfor planing out spaces between rack teeth. Upon any\\nline OB draw circle OABA From B lay off distance\\nBA and BA each equal to one-quarter of diameter of\\nthe circle.\\nDraw lines OA and OA These two lines form an\\nangle of 29\u00c2\u00b0, and are right for inclination of sides of\\nrack tool.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0030.jp2"}, "31": {"fulltext": "PROVIDENCE, R. I.\\n15\\nMake end of rack tool .31 of circular pitch, and then ^Y 11 1 of ack\\n\u00e2\u0096\u00a0t Tool at end.\\nround the corners of the tool to leave fillets at the\\nbottom of rack teeth.\\nThus, if the circular pitch of a rack is 1J and we\\nmultiply by .31, the product .465 will be the width of\\ntool at end for rack of this pitch before corners are\\ntaken off. This width is shown at x y.\\nA Worm is a screw that meshes with the teeth of a\\ngear.\\nThis sketch and the foregoing rule are also right for Worm Thread\\na worm-thread tool, but a worm-thread tool is not\\nusually rounded for fillet. In cutting worms, leave\\nwidth of top of thread .335 of the circular pitch.\\nWhen this is done, the depth of thread will be right.\\nSKETCH OF WORM THREAD", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0031.jp2"}, "32": {"fulltext": "16\\nCHAPTER V.\\nDIAMETRAL PITCH\u00e2\u0080\u0094 SIZING BLANKS AND THE TEETH OF SPUR GEARS\\n\u00e2\u0080\u0094DISTANCE BETWEEN THE CENTRES OF WHEELS.\\nIn making drawings of gears, and in cutting racks,\\nWhen it is 7\\nnecessary to it is necessary to know the circular pitch, both on\\nknow the Cir-\\ncuiar Pitch. account oi spacing teeth and calculating then* strength.\\nIt would be more convenient to express the circular\\npitch in whole inches, and the most natural divisions\\nin a complete of an inch, as 1 P f P P and so on. But as\\nPitch circum- the circumference of the pitch circle must contain the\\nference must i i i i m\\ncontain the cir- circular pitch soine whole number ot times, corre-\\nsome whole sponding to the number of teeth in the gear, the\\ntimes ber diameter of the pitch circle will often be of a size not\\nreadily measured with a common rule. This is because\\nthe circumference of a circle is equal to 3 1416 times\\nthe diameter, or the diameter is equal to the circum-\\nference multiplied by .3183.\\nIn practice, it is better that the diameter should be\\nPitch, j .n f some size conveniently measured. The same applies\\nTerms ot the L L\\nDiameter. fc the distance between centers. Hence it is generally\\nmore convenient to assume the pitch in terms of the\\ndiameter. In Chapter II. was given a definition of a\\ndiameter pitch, and also how to get a diameter pitch\\nfrom the circular pitch.\\nWe can also assume a diameter pitch and pass to its\\ncircular Pitch equivalent circular pitch. 11 the circumference of the\\nand a JJiame- x x\\nter Pitch. pitch circle is divided by the number of teeth in the\\ngear, the quotient will be the circular pitch. In the\\nsame manner, if the diameter of the pitch circle is\\ndivided by the number of teeth, the quotient will be a\\ndiameter pitch. Thus, if a gear is 12 inches pitch\\ndiameter and has 48 teeth, dividing 12 by 43, the\\nquotient J is a diameter pitch of this gear. In prac-", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0032.jp2"}, "33": {"fulltext": "PROVIDENCE, R. I. 17\\ntice, a diameter pitch is taken in some convenient part\\nof an inch, as A- diameter pitch, and so on. It ^Abbreviation\\nt of Diameter\\nis convenient in calculation to designate one of these Pitch,\\ndiameter pitches by s, as in Chapter II. Thus, for\\ndiameter pitch, s is equal to J Generally, in speak-\\ning of diameter pitch, the denominator of the fraction\\nonly is named. J diameter pitch is then called 3\\ndiametral pitch. That is, it has been found more con-\\nvenient to take the reciprocal of a diameter pitch in\\nmaking calculation. The reciprocal of a number is 1,\\ndivided by that number. Thus the reciprocal of J is a NuE?\\n4, because goes into 1 four times.\\nHence, we come to the common definition\\nDiametral Pitch is the number of teeth to one inch pitch.\\nof diameter of pitch circle. Let this be denoted by P.\\nThus, J diameter pitch we would call 4 diametral\\npitch or 4 P, because there would be 4 teeth to every\\ninch in the diameter of pitch circle. The circular\\npitch and the different parts of the teeth are derived\\nfrom the diametral pitch as follows.\\n^iifi P or 3.1416 divided by the diametral pitch Given the Di-\\nr i/ x ametral to find.\\nis equal to the circular pitch. Thus to obtain the cir-*\u00c2\u00a3 e Circular\\ncular for 4 diametral pitch, we divide 3.1416 by 4 and\\nget .7854 for the circular pitch, corresponding to 4 c ta pitoh\\ndiametral pitch. traiPit?n. ame\\nIn this case we would write P=4,P .7854 *=i\\ns, or one inch divided by the number of teeth to\\nan inch, gives distance on diameter of pitch circle\\noccupied by one tooth. The addendum or face of\\ntooth is the same distance as s.\\ni _\\ns\\n=P, or one inch divided by the distance occupied\\nby one tooth equals number of teeth to one inch.\\n\u00c2\u00b1*l\u00e2\u0080\u0094t t or 1.57 divided by the diametral pitch gives amJSprtchto\\nthickness of tooth at pitch line. Thus, thickness of JJes^of Tooth\\nteeth along the pitch line for 4 diametral pitch is .392 L* n e he Pitch\\n^=D or number of teeth in a gear divided by the N umber *o?\\ndiametral pitch equals diameter of the pitch circle, and the a Diam-\\nThus for a wheel, 60 teeth, 12 P, the diameter of^h^Diam-\\npitch circle will be 5 inches. gg^* Pitch\\ni^=D, or add 2 to the number of teeth in a wheelN umber of\\n_. -1-1 -i Teeth ina wheel\\nand divide the sum bv the diametral pitch, and the and the Diame-\\nJ r tral Pitch to\\nfind the Wholo\\nDiameter.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0033.jp2"}, "34": {"fulltext": "18 BKOWN SHARPE MFG. CO.\\nquotient will be the whole diameter of the gear or the\\ndiameter of the addendum circle. Thus, for 60 teeth,\\n12 P, the diameter of gear blank will be 5fv inches.\\n\u00c2\u00a3,=P, or number of teeth divided by diameter of\\npitch circle in inches, gives the diametral pitch or\\nnumber of teeth to one inch. Thus, in a wheel, 24\\nteeth, 3 inches pitch diameter, the diametral pitch is 8.\\njp P, or add 2 to the number of teeth; divide the\\nsum by the whole diameter of gear, and the quotient\\nwill be the diametral pitch. Thus, for a wheel 3^\\ndiameter, 14 teeth, the diametral pitch is 5.\\nD P=N, or diameter of pitch circle, multiplied by\\ndiametral pitch equals number of teeth in the gear.\\nThus, in a gear, 5 pitch, 8 pitch diameter, the num-\\nber of teeth is 40.\\nD P 2=N or multiply the whole diameter of the\\ngear by the diametral pitch,subtract 2, and the remain-\\nder will be the number of teeth.\\nj^p s, or divide the whole diameter of a spur gear\\nby the number of teeth plus two, and the quotient\\nwill be the addendum, or a diameter pitch.\\nPitch iameter n f u ure vvhen we speak of a diameter pitch, we\\nshall mean the addendum distance or s. If we speak\\nof so many diameter pitches, we shall mean so many\\nThe Diame- times s, (-3= s). When we say the diametral pitch we\\ntral Pitch. F J\\nshall mean the number of teeth to one inch of diameter\\nof pitch circle, or P, =P).\\ns\\nTo obtain Di-\\nPitc When the circular pitch is given, to find the corre-\\npitch Circular sponding diametral pitch, divide 3.1416 by the circular\\npitch. Thus 1.57 P is the diametral pitch correspond-\\ning to 2-inch circular pitch, (^l^i^=P).\\nExample. What diametral pitch corresponds to J- circular\\npitch Remembering that to divide by a fraction we\\nmultiply by the denominator and divide by the numer-\\nator, we obtain 6.28 as the quotient of 3.1416 divided by\\nJ 6.28 P, then, is the diametral pitch corresponding\\nto J circular pitch. This means that in a gear of\\ninch circular pitch there are six and twenty-eight one\\nhundredths teeth to every inch in the diameter of the\\npitch circle. In the table of tooth parts the diametral", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0034.jp2"}, "35": {"fulltext": "PROVIDENCE, R. I. 19\\npitches corresponding to circular pitches are carried\\nout to four places of decimals, but in practice three\\nplaces of decimals are enough.\\nWhen two gears are in mesh, so that their pitch\\ncircles just touch, the distance between their axes or\\ncenters is equal to the sum of the radii of the two\\ngears. The number of the diameter pitches between\\ncenters is equal to half the sum of number of teeth in\\nboth gears. This principle is the same as given in\\nChapter II., page 6, but when the diametral pitch an d -o^nce ^ht\\nnumbers of teeth in two gears are given, add together tween centers.\\nthe numbers of teeth in the two wheels and divide half\\nthe sum by the diametral pitch. The quotient is the\\ncenter distance.\\nA gear of 20 teeth, 4 P, meshes with a gear of 50 Example,\\nteeth what is the distance between their axes or\\ncenters Adding 50 to 20 and dividing half the sum\\nby 4, we obtain 8f as the center distance.\\nThe term diametral pitch is also applied to a rack.\\nThus, a rack 3 P, means a rack that will mesh with a\\ngear of 3 diametral pitch.\\nIt will be seen that if the expression for a diameter Fractional\\nDiametral\\npitch has any number except 1 for a numerator, we Pitch,\\ncannot express the diametral pitch by naming the\\ndenominator only. Thus, if the addendum or a diam-\\neter pitch is f^, the diametral pitch will be 2^, because\\n1 divided by T equals 2\u00c2\u00a3.\\nIn Chapter II, the term modtde is used in the same\\nsense as the term a diameter pitch. Modtde is much\\nused where gears are made to metric sizes, for the\\nreason that, the millimeter being so short, the module\\nis conveniently expressed in millimeters. If we know\\nthe module of a gear we can figure the other parts as\\neasily as we can if we know either the circular pitch\\nor the diametral pitch. The module is, in a sense, an\\nactual distance, while the diametral pitch, or the num-\\nber of teeth to an inch, is a relation or merely a ratio.\\nThe meaning of the module is not easily mistaken.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0035.jp2"}, "36": {"fulltext": "20 EEOWU i SHiEPE 5IT(i. CO.\\nCHAPTER VI.\\nSUGLE-GURYE GEARS HAYING LESS THAN 30 TEETH\u00e2\u0080\u0094 GEARS AND\\nRACKS TO MESH WITH GEARS HAVING LESS THAN 30 TEETH.\\nConstruction, Jq Yig. 10. tlie construction of the rack is the same\\nFig. 10.\\nas the construction of the rack in Chapter TV. Tne\\ngear in Fig. 10 is drawn from base circle out to adden-\\ndum circle, by the same method as the gear in Chapter\\nIII., but the spaces inside of base circle are drawn as\\nfollows\\nFlanks of T n gears, 12 to 19 t-eth, the sides of spaces inside\\nGears in low\\nnumbers of,,f the base circle are radial for a distance, a b, equal\\nTeeth. l\\nto or 3.5 divided by the product of the pitch by the\\nnumber of teeth. In gears with more than 10 teeth\\nthe radial construction is omitted.\\nconstruction Then, with one leg- of dividers in pitch circle in\\nof Fig. 10 con-\\ntinued. center of next tooth, e, and other leg just touching\\none of the radial lines at l continue the tooth side\\ninto c, until it will touch a fillet arc. whose radius is\\ni the width of space at the addendum circle. The\\npart, V c is an arc from center of tooth f/, etc. The\\nflanks of teeth or spaces in gear, Fig. 11, are made the\\nsame as those in Fig. 10.\\nThis rule is merely conventional or not founded\\nupon any principle other than the judgment of the de-\\nsigner, to efTe:-t the object to have spaces as wide as\\npracticable, just below or inside of base circle, and\\nthen strengthen flank with as large a fillet as will clear\\naddenda of any gear. If flanks in any gear will clear\\naddenda of a rack, they will clear addenda of any\\nInternal Gear, other gear, except internal gears. An internal gear is\\none having teeth upon the inner side of a rim or ring.\\nNow, it will be seen that the gear, Fig. 10, has teeth", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0036.jp2"}, "37": {"fulltext": "PROVIDENCE, R. I.\\n21\\nFig. 10", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0037.jp2"}, "38": {"fulltext": "22\\nre. :w i snAzrz irre.\\ntoo much rounded at the points or at the addendum\\narcle. In gears cf pitch coarser than 10 to r_ch (10\\nAddTSof an( having less than 30 teeth, this rounding\\n1 becomes obj ectionable. This rounding occur s, be a nse\\nin these gears arcs of en les depart too far from the\\ntrue involute carve, being so much that points of\\nteeth get no bearing on flanks of teeth in other wheels.\\nIn gear. Fir 11. r _e teetii outside of base circle are\\nmade as nearly true involute as a workman will be able\\nt:- ret vri:L: e: ;.l n; :_r_r.~. This is accomplished\\nAjproxtma-ag follows: draw three or four tan rents to the base\\nDon _rue jz.- o\\nvolute, circle, t jj\\\\ k k\\\\ 1 1 letting tie points of tangen sy\\non base circle i .j\\\\ k V be about or J the circular pitch\\napart the first point, being distant from i, equal to\\nthe radius of pitch circle. With dividers set to -J-\\nthe radius of pitch circle, placing one leg in draw\\nthe arc, a j/ with one leg in j 7 and radius j j,\\ndxawj U; with one leg in k and radius I: k draw J: 7.\\nShould the addendum circle be outside of 7. the I oth\\nside can be completed with the last radius, I The\\narcs, a ij, j k and k l. together form a very close\\napproximation to a true involute from the base circle,\\ni j k I The exact involute for gear teeth is the\\ncurve made by the end of a band when unwound from\\na cylinder of the same diameter as base circle.\\nThe foregoing operation of drawing tooth sides,\\nalthough te 1: in les iription. is very easy of practical\\napplication.\\nBounding of It will also be seen that the addenda of rack teeth\\nA i r z. _ i _\\nRack. in Tig. 10. interfere with the gear-teeth nanks, as\\nm n; to avoid this interference, the teeth of rack, Fig.\\n11. are rounded at points cr addenda-\\nIt is also necessary to round off the points of invo-\\nlute teeth in high numbered gears, when they are to\\ninterchange with low nunir. e: ed _ e In iniercshang\\nable sets of gears the lowest-numbered pinion is usual-\\nTemple:? iv 12. Just how much to round off can be 1 earned by\\nK::i::i. _ ----_\u00e2\u0096\u00a0- a Zzii: :n: :_:n met.-- cr\\nPoints of teeth. L\\n..dboard. for the gear and rack, or, two gears re-\\nquired, and fitting addenda of teeth to clear flar k\\nHowever accurate we may make a diagram, it is qu", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0038.jp2"}, "39": {"fulltext": "mOVIDENCE, E. I.\\n23\\nFig.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0039.jp2"}, "40": {"fulltext": "24\\nBROWN ft SHARPE MFG. CO.\\nas well to make templets in order to shape cutters\\naccurately.\\nIt is best to make cutters to corrected diagrams, as\\nin Fig. 11. TVhen corrected diagrams are made, as\\nin Fig. 11. take the following:\\nFor 12 and 13 teeth, diagram of 12 teeth.\\na Set of Cut-\\nters.\\n14\\nto 16\\n17\\n20\\n21\\n25\\n26\\n34\\n35\\n54\\n55\\n134\\n135\\nrack,\\ntc\\ni\\n14\\na\\nti\\na\\n17\\nic\\ntc\\ni. i\\n21\\nit\\n26\\ni i\\nIt\\na\\n35\\na\\nhi\\nt i\\n55\\ni\\n-135\\ni i\\nTemplets for large gears must be fitted to run with\\n12 teeth.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0040.jp2"}, "41": {"fulltext": "PROVIDENCE, R. I.\\n25\\nCHAPTER VII.\\nDOUBLE-CURYE TEETH\u00e2\u0080\u0094 GEAR, 15 TEETH\u00e2\u0080\u0094 RACK.\\nIn double-curve teeth the formation of tooth sides ah Double-\\ncurve Tooth\\nchanges at the pitch line. In all gears the part of Fa \u00c2\u00b0es are con-\\nteeth outside of pitch line is convex in some gears\\nthe sides of teeth inside pitch line are convex in some,\\nradial; in others, concave. Convex faces and concave\\nflanks are most familiar to mechanics. In interchange-\\nable sets of gears, one gear in each set, or of each\\npitch, has radial flanks. In the bast practice, this gear\\nhas fifteen teeth. Gears with more than fifteen teeth,\\nhave concave flanks gears with less than fifteen teeth,\\nhave convex flanks. Fifteen teeth is called the Base\\nof this system.\\nWe wid first draw a gear of fifteen teeth. This construction\\nof Fig. 12.\\nfifteen-tooth construction enters into gears of any\\nnumber of teeth and also into racks. Let the gear be\\n3 P. Having obtained data, we proceed as follows\\n1. Draw pitch circle and point it off into parts equal\\nto one-thirtieth of the circumference, or equal to thick-\\nness of tooth =i^.\\n2. From the center, through one of these points, as\\nat T, Fig. 12, draw line OTA. Draw addendum and\\nwhole-depth circles.\\n3. About this point, T, with same radius as 15-tooth\\npitch circle, describe arcs A K and k. For any other\\ndouble-curve gear of 3 P., the radius of arcs, A K and\\nO k, will be the same as in this 15-tooth gear=2J\\nIn a 15-tooth gear, the arc, O k, passes through the\\ncenter O, but for a gear having any other number of\\nteeth, this construction arc does not pass through\\ncenter of gear. Of course, the 15-tooth radius of arcs,\\nA K and O k, is always taken from the pitch we are\\nworking with.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0041.jp2"}, "42": {"fulltext": "26\\nBROWN k SHAEPE MTG. CO.\\nEiS- 12\\nDOUBLE CURVE GEAR.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0042.jp2"}, "43": {"fulltext": "PROVIDENCE, R. I. 27\\n4. Upon these arcs on opposite sides of line OTA,\\nlay off tooth thickness, A K and O k, and draw line\\nKT\\n5. Perpendicular to K T k, draw line of pressure,\\nL T P also through O and A, draw lines A R and O r,\\nperpendicular to K T The line of pressure is at\\nan angle of 78\u00c2\u00b0 with the radius of gear.\\n6. From 0, draw a line R to intersection of AR\\nwith KT^. Through point c, where R intersects\\nL P, describe a circle about the center, 0. In this\\ncircle one leg of dividers is placed to describe tooth\\nfaces\\n7. The radius, c d, of arc of tooth faces is th*.\\nstraight distance from c to tooth-thickness point, b,\\non the other side of radius, O T. With this radius, c b,\\ndescribe both sides of tooth faces.\\n8. Draw flanks of all teeth radial, as e and Of\\nThe base gear, 15 teeth only, has radial flanks.\\n9. With radius equal to one-seventh of the widest\\npart of space, as g h, draw fillets at bottom of teeth.\\nThe foresroiner is a close approximation to epicy- Approxima-\\nrr r J tion to Epicy-\\ncloidal teeth. To get exact teeth, make two 15-tootheioidai Teeth.\\ngears of thin metal. Make addenda long enough to\\ncome to a point, as at n and q. Make radial flanks, as\\nat m and p, deep enough to clear addenda when gears\\nare in mesh. First finish the flanks, then fit the long\\naddenda to the flanks when gears are in mesh.\\nWhen these two templet gears are alike, the centers standard\\nare the right distance apart and the teeth interlock\\nwithout backlash, they are exact. One of these tem-\\nplet gears can now be used to test any other templet\\ngear of the same pitch.\\nGears and racks will be right when they run cor-\\nrectly with one of these 15-tooth templet gears. Five\\nor six teeth are enough to make in a gear templet.\\nDouble- curve Rack. Let us draw a rack 3 P. R^^ig^ST 9\\nHaving obtained data of teeth we proceed as follows\\n1. Draw pitch line and point it off in parts equal\\nto one- half the circular pitch. Draw addendum and\\nwhole-depth lines.\\n2. Through one of the points, as at T, Fig. 13, draw\\nline OTA perpendicular to pitch line of rack.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0043.jp2"}, "44": {"fulltext": "28\\nBBOWS s 8HABPE MFG. CO.\\nZFi s 13\\nDOUBLE CURVE RACK.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0044.jp2"}, "45": {"fulltext": "PROVIDENCE, R. I.\\n29\\n3. About T make precisely the same construction as\\nwas made about T in Fig. 12. That is, with radius of\\n15-tooth pitch circle and center T draw arcs Jc and\\nA K make O k and A K equal to tooth thickness\\ndraw KT/;; draw r, A R, and line of pressure, each\\nperpendicular to K T k.\\n4. Through R and r, draw lines parallel to A.\\nThrough intersections c and c of these lines, with\\npressure line L P, draw lines parallel to pitch line.\\n5. In these last lines place leg of dividers, and draw\\nfaces and flanks of teeth as in sketch.\\n6. The radius c cV of rack-tooth faces is the same\\nlength as radius c d of rack-tooth flanks, and is the\\nstraight distance from c to tooth-thickness point b on\\nopposite side of line A.\\n7. The radius for fillet at bottom of rack teeth is\\nequal to i- of the widest part of tooth space. This\\nradius can be varied to suit the judgment of the\\ndesigner, so long as a fillet does not interfere with\\nteeth of engaging gear.\\nFig. 14\\nRacks of the same pitch, to mesh with interchange-\\nable gears, should be alike when placed side by side,\\nand fit each other when placed together as in Fig. 14.\\nIn Fig. 13, a few teeth of a 15-tooth wheel are shown\\nin mesh with the rack.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0045.jp2"}, "46": {"fulltext": "30\\nCHAPTER VIII.\\nDOUBLE-CURVE SPUR GEARS, HAYING MORE AND FEWER THAN\\n15 TEETH\u00e2\u0080\u0094 ANNULAR GEARS,\\nConsxracvion ^et us draw two gears, 12 and 24 teeth, 4 P, in\\nOl rig. Id. d\\nmesh. In Fig. 15 the construction lines of the lower\\nor 24-tooth gear are full. The upper or 12-tooth gear\\nconstruction lines are dotted. The line of pressure,\\nL P. and the line K T Jc answer for both gears. The\\narcs A K and O k are described about T. The radius\\nof these ares is the radius of pitch circle of a gear 15\\nteeth 4 pitch. The length of arcs A K and O k is the\\ntooth thickness for 4 P. The line K T k is obtained\\nthe same as in Chapter YH. for all double-curve gears,\\nthe distances only varying as the pitch. Having drawn\\nthe pitch circles, the line K T Jc. and, perpendicular to\\nK T Jc, the lines A Pi. r and the line of pressure\\nL T P. we proceed with the 24-tooth gear as follows\\n1. From center C. through r, draw line intersecting\\nline of pressure in m. Also draw line from center C\\nto E. crossing the line of pressure L P at\\n2. Through m describe circle concentric with pitch\\ncircle about C. This is the circle in which to place\\none leg of dividers to describe flanks of teeth.\\n3. The radius, m n. of flanks is the straight distance\\nfrom m to the first tooth-thickness point on other side\\nof hue of centers. C C at v. The arc is continued to\\nn. to show how constructed. This method of obtain-\\ning radius of double-curve tooth flanks applies to all\\ngear s having more than fifteen teeth.\\n4. The construction of tooth faces is similar to 15-\\ntooth wheel in Chapter VII. That is Draw a circle\\nthrough c concentric to pitch circle in this circle\\nplace one leg of dividers to draw tooth faces, the\\nradius of tooth faces being c b.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0046.jp2"}, "47": {"fulltext": "PROVIDENCE, R. I.\\n31\\n/v,\\nA\\nA\\nA y\\\\\\nPINION, 12 TEETH,\\nGEAR 24 TEETH, 4 P.\\nP=4\\nN=12 and 24.\\nP .7854\\nt .3927\\nS .2500\\nD 1 .5000\\ns+ .2893\\nD .5393\\nD 3\\nId=\\nD 6\\nC\\nDOUBLE CURVE GEARS IN MESH.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0047.jp2"}, "48": {"fulltext": "32 zzz^zs t =ez_zzz zzzz\\nT\\nrhc lira :z z_t:s it roots of ---~z is equal to\\nno-seventh the width of space at addendum circle.\\nThe constructions for flanks of 14\\nline pressure in u. Tizrzngh u draw circle about\\nC. Iz :zis circle :ze leg .-\u00e2\u0080\u0094_ lers zlzcei zdt\\niz zzz z.zil_:5.\\n2 __e rzzizz zzzizis is :ze 5zz:z:_:e zzzzz z\\n:Lt ziz: :zzz.- :z iczzzc s s r :izz ;:.:ir,- z z o:\\nCTC. Tzis zi _ r^ ::i _ r: zlzz/zs. Tiie zc c:z-\\n:i_.e ~z:~ czzszrzctizz.\\n3. This arc f:r zzzzzis i= miizzzei z z zr ~.z~\\nzzzter. :zlj a z: :ze siszz :ze zrlczzz zerzz\\nJ s.) the lower part of Hank is similar to flanks of\\ngear in Chapter VI.\\nCiizzter VII.. zzz :ze Ji-t::zz gear iz. zze fire-\\ngoing, Hie I lira zemg to y the arc is continued to a^\\nATJiz^r rirs. _Ajz-zz_z. Gziez Gears with teeth inside of a rim\\n:r zizg zze cailez r -zlar :r Iz.:einzl Grzrs. TJze\\nconstruction of tooth outlines is similar to the f ore-\\ng: iz.gr. c :ze z//::z5 z sjzzr ez:e:zizl zez: ::e::zze\\ntze :.z e_.r rizr gezz.\\nProf. MacCord has shown that in the system just\\nze=::r*:e~. :ze z:zi:z zirsizzrg _:z ez ar_zzziar .ez\\nzizz: :li\u00c2\u00b1er zrzrz i: cv :::tt1 Zr: _. Tlzz\\na gear teeth zz:: work with an annular gear\\nof 36 teeth, but it will work with annular gears of 39\\ntee:b and more. The n .ers at the roots of the teeth\\nmzs: re z I e s.s re ivs :z..z iz :riiz:-.:v srzr g:ars. .An\\naz:: z ge.-.r frrizz .zm :ts rz..:e cr i.-=s iz.z 15\\ntee: e z: r. This ^vill be shown in Part IL\\nAzzzziaz-gezz y.atreizis rerzzie zz:re jzzziz\\nzzz zizi^gtzazz ezrerzai or z z gears\\npzz:_=. In speaking linereni -siz ed gears, the smallest\\nTze zze of pressure iz. all z^:^^ except involute,\\neonstaiitlj :L i_ges. 78 is the pressure angle in\\ndouble-curve, or epicrcloidal gears foe an instant", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0048.jp2"}, "49": {"fulltext": "PROVIDENCE. R. I.\\n33\\nonly; in our example, it is 78\u00c2\u00b0 when one side of a\\nfcooth reaches the line of centers, and the pressure\\nagainst teeth is applied in the direction of the arrows.\\nThe pressure angle of involute gears does not\\nchange. An explanation of the term angle of pressure\\nis given in Part II.\\nWe obtain the forms for ejncycloidal gear cutters\\nby means of a machine called the Odontom Engine.\\nThis machine will cut original gears with theoretical\\naccuracy.\\nIt has been thought best to make 24 gear cutters 24 Double-\\ncurve Gear\\nfor each pitch. This enables us to fill any require- Cu era for\\nj i. each Pitcb.\\nment of gear-cutting very closely, as the range covered\\nby any one cutter is so small that it is exceedingly near\\nto the exact shape of all gears so covered.\\nOf course, a cutter can be exactly right for only one\\ngear. Special cutters can be made, if desired.\\n1 PITCH TOOTH CURVES\\nfrom the\\nODONTOM ENGINE.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0049.jp2"}, "50": {"fulltext": "34\\nCHAPTER IX.\\nBEVEL-GEAR BLANKS,\\nBevel Gears connect shafts whose axes meet when\\nBe J\u00c2\u00ab Qg ai f sufficiently prolonged. The teeth of bevel gears are\\ncones.\\nformed upon formed about the frustrums of cones whose apexes\\nfrustrums of l\\nare at the same point where the shafts meet. In Fig.\\n16 we have the axes A O and B 0, meeting at O, and\\nthe apexes of the cones also at O. These cones are\\ncalled the pitch cones, because they roll upon each\\nother, and because upon them the teeth are pitched.\\nIf, in any bevel gear, the teeth were sufficiently pro-\\nlonged toward the apex, they would become infinitely\\nsmall that is, the teeth would all end in a point, or\\nvanish at 0. We can also consider a bevel gear as\\nbeginning at the apex and becoming larger and larger\\nas we go away from the apex. Hence, as the bevel\\ngear teeth are tapering from end to end, we may say\\nBEVEL GEAR PITCH CONES.\\nFig. 16.\\nthat a bevel gear has a number of pitches and pitch\\ncircles, or diameters in speaking of the pitch of a\\nbevel gear, we mean always the pitch at the largest", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0050.jp2"}, "51": {"fulltext": "PROVIDENCE, R. I.\\n35\\npitch circle, or at the largest pitch diameter, as at\\nb d, Fig. 17.\\nFig. 17 is a section of three bevel gears, the gear\\no B q being twice as large as the two others. The\\nouter surface of a tooth as m m is called the face of Construction\\nof Bevel Gear\\nthe tooth. The distance m m is usually called the B1 anks.\\nlength of the face of the tooth, though the real length\\nis the distance that it occupies upon the line O i. The\\nouter part of a tooth at m n is called its large end, and\\nthe inner part m n the small end.\\nAlmost all bevel gears connect shafts that are at\\nright angles with each other, and unless stated other-\\nwise we always understand that they are so wanted.\\nThe directions given in connection with Fig. 17\\napply to gears with axes at right angles.\\nHaving decided upon the pitch and the numbers of\\nteeth\\n1. Draw centre lines of shafts, A O B and COD,\\nat right angles.\\n2. Parallel to A O B, draw lines a b and c d, each\\ndistant from A B, equal to half the largest pitch\\ndiameter of one gear. For 24 teeth, 4 pitch, this half\\nlargest pitch diameter is 3\\n3. Parallel to COD, draw lines e f and g h, dis-\\ntant from COD, equal to half the largest pitch\\ndiameter of the other gear. For a gear, 12 teeth, 4\\npitch, this half largest pitch diameter is 1 J ff\\n4 At the intersection of these four lines, draw\\nlines O i, O j, O k, and 1 these lines give the size\\nand shape of pitch cones. We call them Cone Pitch\\nLines.\\n5. Perpendicular to the cone- pitch lines and through\\nthe intersection of lines a b, c d, e f and g h, draw\\nlines m n, o p, q r. We have drawn also u v to show\\nthat another gear can be drawn from the same diagram.\\nFour gears, two of each size, can be drawn from this\\ndiagram.\\n6. Upon the lines m n, o p, q r, the addenda and\\ndepth of the teeth are laid off, these lines passing", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0051.jp2"}, "52": {"fulltext": "36\\nBROWN i- SHARPE MFG. CO.\\nthrough the largest pitch, circle the gears, Layoff\\nthe s id lentlnm. it beins; in these 2:ears 4- This sives\\ndistance m n. o p, q r. and a v equal to the working\\ndepth of teeth, which in these gears is J The\\naddendum of course is measured perpendicu .aily from\\nthe cone pitch lines as at k r.\\n7. Draw lines m, n, p, o. r.\\nThese lines _ the height of teeth above the cone-\\n:teh lines as they approach O, and would vanish\\nentirel It is quite us well never to have the\\nlength of teeth, or face, m n longer than one-third\\nth- apex distance m 0, nor more than two and one-\\nhalf times the circular pitch\\n8. Having decided upon the length of face, draw\\nlimiting lines in n perpendicular i 0. q r perpen-\\ndicular to k O. and so on.\\nTL- listance between the cone-pitch lines at the\\ninner ends of the teeth m n and q r is called the inner\\nor smaller pitch diameter, and the circle at these points\\nis called the smallest pitch circle. We now Lave the\\noutline a -rtion of the gears through their axes.\\nThe listance m i is the whole diameter of the pinion.\\n;f el hol f The distance q o is the whole diameter of the gear.\\nIn practice these diameters can be obtained bvmeasur-\\nobtained by i n2 r th c Irawing. The diameter of pinion is 3.45 and\\nMeasuring c x\\nDrawings. f r g eai __ We can find the angles also by\\nmeasuring the drawing with a protractor. In the\\nabsence of a protrack r, templetes can be cut to the\\ndrawing. The angle form line m m with a b is\\nthe angle of face of pinion, in this pinion o j c 11 or\\n59^\u00c2\u00b0 nearly. The lines q and g h give us angle of\\nface of gear, for this gear 22 z 19 or 2 2^ nea:~.\\nThe angle formed by m n with a b is called the angle\\nof edge of pinion, in our sketch 2\u00c2\u00bb3 34 or about 26J\\nThe angle of edge of gear, line q r with g h, is\\nor about 63J\u00c2\u00b0. In turning blanks to these angles we\\nplace one arm of the protractor or templet against the\\nend of the hub, when trying angles of a blank. Some\\ndesigners give the ausrJes from the axes of gears, bat", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0052.jp2"}, "53": {"fulltext": "PROVIDENCE, E. I.\\n37\\nFig. 17", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0053.jp2"}, "54": {"fulltext": "38 BE0WX SHARPE MFG. CO.,\\nit is not convenient to- try blanks in this way. The\\nmethod that we have given comes right also for angles\\nas figured in compound rests.\\nWhen axes are at right angles, the sum of angles\\nof edge in the two gears equals 90\u00c2\u00b0, and the sums of\\nangle of edge and face in each gear are alike.\\nThe angles of the axes remaining the same, all pairs\\nof bevel gears of the same ratio have the same angle\\nof edge all pairs of same ratio and of same numbers\\nof teeth have the same angles of both edges and faces\\nindependent of the pitch. Thus, in all pairs of bevel\\ngears having one gear twice as large as the other, with\\naxes at right angles, the angle of edge of large gear\\nis 63\u00c2\u00b0 26 and the angle of edge of small ffear is 26\u00c2\u00b0 34\\nDO G\\nIn all pairs of bevel gears with axes at right angles,\\none gear having 24 teeth and the other gear having 12\\nteeth, the angle of face of small gear is 59\u00c2\u00b0 11\\n4, n 9 t ie r The following method of obtaining the whole diam-\\nrnethod ot ob- c s\\ntaining Whole er f bevel gears is sometimes preferred\\nDiameter ot c l\\nBlanks. From k lay off upon the cone-pitch line, a distance\\nK w, equal to ten times the working depth of the\\nteetk=10D Xow add ts of the shortest distance\\nof w from the line g h, which is the perpendicular\\ndotted line w x. to the outside pitch diameter of gear,\\nand the sum will be the whole diameter of gear. In\\nthe same manner fu of w y. added to the outside pitch\\ndiameter of pinion, gives the whole diameter of pinion.\\nThe part added to the pitch diameter is called the\\ndiameter increment.\\nPart II gives trigonometrical methods of figuring\\nbevel gears in our Formulas in Gearing there arc\\ntrigonometrical formulas for bevel gears, and also\\ntables for angles and sizes.\\nConstruction somewhat similar construction will do for bevel\\nof Bevel-Gear\\nBlanks whose gears whose axes are not at right angles.\\nAxes are not ct\\nat Right An- J n Fig. 18 the axes are shown at O B and D, the\\nangle BOD being less than a right angle.\\n1. Parallel to B. and at a distance from it equal\\nto the radius of the gear, we draw the lines a b and c d.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0054.jp2"}, "55": {"fulltext": "PROVIDENCE, R. I.\\n39\\nINSIDE BEVEL GEAR\\nAND PINION. Fig.no", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0055.jp2"}, "56": {"fulltext": "40\\nBROWN SHARPE MFG. CO.\\n2. Parallel to O D, and at a distance from it equal\\nto the radius of the pinion, we draw the lines e f and g h-\\n3. Now, through the point j at the intersection of\\nc d and g h, we draw a line perpendicular to O B.\\nThis line k j, limited by a b and c d, represents the\\nlargest pitch diameter of the gear.\\nThrough j we draw a line perpendicular to D.\\nThis line j 1, limited by e f and g h, represents the\\nlargest pitch diameter of the pinion.\\n4. Through the point k at the intersection of a b\\nwith k j, we draw a line to O, a line from j to 0, and\\nanother from 1, at the intersection j 1 and e f to 0.\\nThese lines O k, O j, and O 1, represent the cone-\\npitch lines, as in Fig. 17.\\n5. Perpendicular to the cone-pitch lines we draw\\nthe lines u v, op, and q r. Upon these lines we h ry\\noff the addenda and working depth as in the previous\\nfigure, and then draw lines to the point O as before.\\nBy a similar construction Figs. 19 and 20 can be\\ndrawn.\\nSTOCKING CUTTER.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0056.jp2"}, "57": {"fulltext": "PROVIDENCE, K. I. 41\\nCHAPTER X.\\nBEVEL GEARS.\\nFORMS AND SIZES OF TEETH.\\nCUTTING TEETH.\\nTo obtain the form of the teeth in a bevel gear we Form of\\nbevel g e a r\\ndo not lay them out upon a pitch circle, as we do in a teeth,\\nspur gear, because the rolling pitch surface of a bevel\\ngear, at any point, is of a longer radius of curvature\\nthan the actual radius of a pitch circle that passes\\nthrough that point. Thus in Fig. 21, let f g c be a\\ncone about the axis O A, the diameter of the cone\\nbeing f c, and its radius g c. Now the radius of\\ncurvature of the surface, at c, is evidently longer than\\ng c, as can be seen in the other view at C the full\\nline shows the curvature of the surface, and the dotted\\nline shows the curvature of a circle of the radius g c.\\nIt is extremely difficult to represent the exact form of\\nbevel gear teeth upon a flat surface, because a bevel\\ngear is essentially spherical in its nature for practical\\npurposes we draw a line c A perpendicular to O c,\\nletting c A reach the centre line O A, and take c A\\nas the radius of a circle upon which to lay out the\\nteeth. This is shown at c n m, Fig. 22. For con-\\nvenience the line c A is sometimes called the back\\ncone radius.\\nLet us take, for an example, a bevel gear and a Fi Ex .4 mi le\\npinion 24 and 18 teeth, 5 pitch, shafts at right angles.\\nTo obtain the forms of the teeth and the data for\\ncutting, we need to draw a section of only a half of\\neach gear, as in Fig. 22.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0057.jp2"}, "58": {"fulltext": "42 BROWN SHAItPE MFG. CO.,\\n1. Draw the centre lines A and B 0. then the\\nlines g h and c d. and the gear blank lines as des-\\ncribed in Chapter IX. Extend the lines op and o p\\nuntil they meet the centre lines at A B and A B.\\n2. With the radius A c draw the arc c n m, which\\nwe take as the geometrical pitch circle upon which to\\nlay out the teeth at the large end. The distance Ac\\nis taken as the radius of the geometrical pitch circle\\nat the small end to avoid confusion an arc of this\\ncircle is drawn at c n m about A.\\n3. For the pinion we have the radius B c for the\\ngeometrical pitch circle at the large end and Be for\\nthe small end: the distance B c is transferred to\\nB c\\n4. Upon the arc cum lay off spaces equal to the\\ntooth thickness at the large pitch circle, which in our\\nexample is .314 Draw the outlines of the teeth as\\nin previous chapters for single curve teeth we draw a\\nsemi-circle upon the radius A c. and proceed as des-\\ncribed in chapter III. For all bevel gears that arc to\\nbe cut with a rotary disk cutter, or a common gear\\ncutter, single curve teeth are chosen and no attempt\\nshould be made to cut double curve teeth. Double\\ncurve teeth can be drawn by the directions given in\\nchapters VII aud VIII. We now have the form of\\nthe teeth at the large end of the gear. Eepeat this\\noperation with the radius B C about B, and we have\\nthe form of the teeth at the large end of the pinion.\\n5. The tooth parts at the small end are designated\\nby the same letters as at the large, with the addition\\nof an accent mark to each letter, as in the right hand\\ncolumn, Fig. 22, the clearance, f. however, is usually\\nthe same at the small end as at the large, for con-\\nvenience in cutting the teeth.\\nSizes of the The sizes of the tooth parts at the small end are in\\ntooth parts.\\nthe same proportion to those at the large end as\\nthe line c is to c. In our example O c is _\\nand O c is 3 dividing c by c we have -f. or\\n.666, as the ratio of the sizes at the small end to those", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0058.jp2"}, "59": {"fulltext": "PROVIDENCE, R. J.\\n43\\nCO\\nO\\nu\\nIE\\no\\nJH\\nh-\\ne*\\nQ.\\nLu\\nfe(\\no\\nUJ\\n\u00c2\u00a3E\\nz\\nh-\\nc\\nDC\\nID\\n(J", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0059.jp2"}, "60": {"fulltext": "44 BROWN A- SHARFE MFG. CO.\\nat the large t is .209 rf of .314 and so on. If\\nthe distance n m is equal to the outer tooth thickness,\\nt, upon the arc c n ni, the lines n A and m A will be a\\ndistance apart equal to the inner tooth thickness t\\nupon the arc c n m The addendum, s and the\\nworking depth. D are at o c and o p\\n6. Upon the arcs c n m and c we draw the forms\\nof the teeth of the gear and pinion at the inside.\\nExample of As an example of the cutting; of bevel sears with\\nCuttins. i\\nrotary disk cutters, or common gear cutters, let us\\ntake a pair of 8 pitch. 1 2 and 24 teeth, shown in\\nFig. 23.\\nLength of In making the drawing it is well to remember that\\nnothing is gained by having the face F E longer than\\nfive times the thickness of the teeth at the large\\npitch circle, and that even this is too long when it is\\nmore than a third of the apex distance c. To cut a\\nbevel gear with a rotary cutter, as in Fig. 24, is at\\nbest but a compromise, because the teeth change pitch\\nfrom end to end. so that the cutter, being of the right\\nform for the large ends of the teeth can not be right\\nfor the small ends, and the variation is too great when\\nthe length of face is greater than a third of the apex\\ndistance c. Fig. 23. In the example, one-third of\\nthe apex distance is T V but F E is drawn only a\\nhalf inch, which even though rather short, has changed\\nthe pitch from 8 at the outside to finer than 11 at the\\ninside. Frequently the teeth have to be rounded over\\nat the small ends by filing the longer the teeth the\\nmore we have to file. If there is any doubt about the\\nstrength of the teeth, it is better to lengthen at the\\nlarge end. and make the pitch coarser rather than to\\nlengthen at the small end.\\nData These data are needed before beginning to cut\\n1. The pitch and the numbers of the teeth the same\\nas for spur gears.\\n2. The data for the cutter, as to its form some-\\ntimes two cutters are needed for a pah of bevel gears.\\n3. The whole depth of the tooth spaces, both at\\n:-:Ti^r.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0060.jp2"}, "61": {"fulltext": "PEOVIDENCE, E. I.\\n45\\nS .200\\nD .400\\nS+/= .231\\nD .431\\nD .266\\nS +f =.165\\nD -h/ .298\\nFig. 22.\\nBEVEL GEARS, FORM AND SIZE OF TEETH,", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0061.jp2"}, "62": {"fulltext": "46 BROWN SHARPE MFG. CO.\\nthe outside and inside ends D f at the outside,\\nand D -J- f at the inside.\\n4. The thickness of the teeth at the outside and at\\nthe inside t and t\\n5. The height of the teeth above the pitch lines at\\nthe outside and inside s and s\\n6. The cutting angles, or the angles that the path\\nof the cutter makes with the axes of the gears. In\\nFig. 23 the cutting angle for the gear c D is A Op,\\nand the cutting angle for the pinion is B O o.\\nselection of The form of the teeth in one of these gears differs\\ncutters.\\nso much from that in the other gear that two cutters\\nare required. In determining these cutters we do not\\nhave to develop the forms of the gear teeth as in\\nFig. 22 we need merely measure the lines A c and\\nB c, Fig. 23, and calculate the cutter forms as if these\\ndistances were the radii of the pitch circles of the\\ngears to be cut. Twice the length Ac, in inches,\\nmultiplied by the diametral pitch, equals the number\\nof teeth for which to select a cutter tor the twenty-\\nfour-tooth gear this number is about 54, which calls\\nfor a number three bevel gear cutter in the list of\\nbevel gear cutters, page 61. Twice B c, multiplied\\nby 8, equals about 13, which indicates a No. 8 bevel\\ngear cutter for the pinion. This method of selecting\\ncutters is based upon the idea of shaping the teeth as\\nnearly right as practicable at the large end, and then\\nfiling the small ends where the cutter has not rounded\\nthem over enough.\\nIn Fig. 25 the tooth L has been cut to thickness at\\nboth the outer and inner pitch lines, but it must still\\nbe rounded at the inner end. The teeth M M have\\nbeen filed. In thus rounding the teeth ihey should not\\nbe filed thinner at the pitch lines.\\nThere are several things that affect the shape of the\\nteeth, so that the choice of cotters is not always so\\nsimple a matter as the taking of the lines A c and\\nB c as radii.\\nIn cutting a bevel gear, in the ordinary gear cutting", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0062.jp2"}, "63": {"fulltext": "PROVIDENCE, R. I.\\n47\\nBEVEL GEAR DIAGRAM FOR DIMENSIONS.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0063.jp2"}, "64": {"fulltext": "4* BROWS i SHARPE 3JTG. CO.\\ni_ii:_ines. the finished spaces are not always of the\\nsame form as the cutter might be expected to makr.\\nbecause of the changes in the positions of the eutter\\nand of the gear blank in order to cut the teeth of the\\nright thickness at both ends. The cutter must of\\ncourse be thin enough to pass through the small end of\\nthe spaces, so that the large end has to be cut to the\\nright width by adjusting either the cutter or the blank\\nside wise, then rotating the blank and cutting twice\\naround.\\nWidening T-T.15. Ill r \\\\Z- i 1 fiT :V_ 11 u C~I _ ;._r 5-rt Z~. _ V r\\nthe Urge a space widened at the large end e and the last chip\\nto be cut off by the right side of the eutter, the cutter\\nhaving been moved to the left, and the blank rotated\\nin the direction of the arrow in a Universal Milling\\n_\\nMachine the same result would be attained by moving\\nthe blank to the right and rotating it in the direction\\nof the arrow. It may be well to remember that in\\nsetting to finish the side of a tooth, the tooth and the\\ncutter are first separated side wise, and the blank is\\nthen rotated by indexing the spindle to bring the large\\nTteeth Mr _ end of the tooth up against the cutter. This trnds\\nt^re JJ^jj not only to cut the spaces wider at the large pitch\\nat root. circle, but also to cut off still more at .the face of the\\ntooth that is, the teeth may be cut rather thin at the\\nface and left rather thick at the root. This tendency\\nis greater as a cutting angle BOo. Fig. 23, is smaller,\\nor as a bevel gear approaches a spur gear, because\\nwhen the cutting angle is small the blank must be\\nrotated through a greater arc in order to set to cut the\\nright thickness at the outer pitch circle. This can be\\nunderstood by Figs. 26 and 27. Fig. i-i is a radial-\\ntoothed clutch, which for our present purpose can be\\nregarded as one extreme of a bevel gear in which the\\ntrrth are cut square with the axis: the dotted lines\\nindicate the different positions of the cutter, the side\\nof a tooth being finished by the side of the cutter that\\nis on the centre line. In setting to cut these\\nthere is the same side adjustment and rotation of the", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0064.jp2"}, "65": {"fulltext": "PROVIDENCE, R. I.\\nTig. 24\\nSETTING BEVEL GEAR CUTTER\\nOUT OF CENTRE.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0065.jp2"}, "66": {"fulltext": "50\\nBROWN SHAHl K MFG. CO.\\nspindle as in a bevel gear, but there is no tendency to\\nmake a tooth thinner at the face than at the root. On\\nthe other hand, if we apply these same adjustments to\\na spur gear and cutter, Fig. 27, we shall cut the face\\nF much thinner without materially changing the thick-\\nness of the root R.\\nFig* 26\\nAlmost all bevel gears are between the two extremes\\nof Figs. 26 and 27, so that when the cuttiug angle\\nB O o, Fig. 23, is smaller than about 30\u00c2\u00b0, this change\\nin the form of the spaces caused by the rotation of the\\nblank maybe so great as to necessitate the substitution\\nFig 1 28\\nFINISHED GEAR.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0066.jp2"}, "67": {"fulltext": "PROVIDENCE, R. I. 51\\nof a cutter that is narrower at e e Fig. 24, than is\\ncalled for by the way of figuring that we have just\\ngiven thus in our own gear cutting department we\\nmight cut the pinion with a No. G cutter, instead of a\\nNo. 8. The No. 6, being for 17 to 20 teeth, cuts the\\ntooth sides with a longer radius of curvature than the\\nNo 8, which may necessitate considerable filing at the\\nsmall ends of the teeth in order to round them over\\nenough. Fig. 28 shows the same gear as Fig. 25, but\\nin this case the teeth have all been filed similar to\\nM M, Fig. 25.\\nDifferent workmen prefer different ways to com- Filing the\\n,i \u00c2\u00a3\\\\-i Trri teeth at the\\npromise m the cutting or a bevel gear. When a email end.\\nblank is rotated in adjusting to finish the large end of\\nthe teeth there need not be much filing of the small\\nend, if the cutter is right, for a pitch circle of the\\nradius B c, Fig. 23, which for our example is a No. 8\\ncutter, but the tooth faces may be rather thin at the\\nlarge ends. This compromise is preferred by nearly\\nall workmen, because it does not require much filing\\nof the teeth it is the same as is in our catalogue by\\nwhich we fill any order for bevel gear cutters, unless\\notherwise specified. This means that we should send c 2SS? ti whcn\\na No. 8, 8-pitch bevel gear cutter in reply to an order J ee j*j are to\\nfor a cutter to cut the 12-tooth pinion, Fig. 23 while\\nin our own gear cutting department we might cut the\\nsame pinion with a No. 6, 8-pitch cutter, because we\\nprefer to file the teeth at the small end after cutting\\nthem to the right thickness at the faces of the large\\nend. We should take a No. G instead of a No. 8 only\\nfor a 12-tooth pinion that is to run with a gear two or\\nthree times as large. We generally step off to the\\nnext cutter for pinions fewer than twenty-five teeth,\\nwhen the number for the teeth has a fraction nearly\\nreaching the range of the next cutter thus, if twice\\nthe line B c in inches, Fig. 23, multiplied by the\\ndiametral pitch, equals 20.9, we should use a No. 5\\ncutter, which is for 21 to 25 teeth inclusive. In\\nfilling an order for a gear cutter, we do not consider", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0067.jp2"}, "68": {"fulltext": "52\\nx a- shap.fz\\nthe fraction but send the cotter indicated by the whole\\nnumber.\\nLater on ^c- will refer to other compromises that are\\nmadv :lr sotting f bevel gears.\\nThe sizes of the H-piteh tooth parts, Fig. 23. at the\\nlarge end. are copied from the table of spar _\\nteeth, pages 145 to 148.\\nThe distance Oc is seven-tenths of the apex\\ntance Oc. so that the sizes of the tooth y ts t the\\ngear cutting small end, rxieptf, are seven-tenths the larse. The\\norder for cutting these gears goes to the workmen in\\ntV-is :o::n\\nForm of\\nSenir _\\nir. ::.:re\\nLab 5b i.tear.\\nP 5\\nN 24\\ny f .270 D\\nf\\n.195\\nt\\nt\\n.137\\ns .12o\\n5\\n^7\\nCot: ins: Ansle 59\u00c2\u00b0 10\\nr\\nSmall Gkar.\\nX\\nC:::::: Angle l. IS\\nFig. B2 is aside view of a Geai Cutting Machine.\\nA evel gar blank A is held by the index spindle B.\\nThe cotter C is sanied y the cutter-slide D. The\\ncutter-slide- carriag E n be set to the cutting angle,\\nthe degrees being indicated on the quadrant F\\nFig l Z is a plan of the machine in this view the\\n::er-slide-carriage. in order to show the detaiis\\nlittle p ainer, is not set to an angle.\\nBefore bes^nnino; to cut the cutter is set central with\\nthe index spindle and the dial G is set zero, so\\nthat we can adjust the softer to any required distance\\nout of centre, in either direction. Set the c itternBlide-\\ncarriage E. Fig. 32, to the cutting angle of the gear.\\nwhich for 24-teeth is 59\u00c2\u00b0 10 the quadrant beii _\\ndivided to half -degrees, vre estimate thai l( or de-", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0068.jp2"}, "69": {"fulltext": "PROVIDKNCK, K. I\\n53\\ngree more than 59\u00c2\u00b0. Mark the depth of the cut at the\\noutside, as in Fig. 30 it is also well enough to mark\\nthe depth at he inside as a check. The thickness of\\nthe teeth at the large end is conveniently deter-\\nmined by the solid gauge, Fig. 29. The gear-tooth\\nJ GEAR TOOTH GAUGE.\\nDEPTH\\nGAUGE.\\n_Fi#. 30\\nGEAR TOOTH CALIPER.\\nJFig.31\\nvernier caliper, Fig. 31, will measure the thickness of\\nteeth up to 2 diametral pitch. In the absence of the\\nvernier caliper we can file a gauge, similar to Fig 29,\\nto the thickness of the teeth at the small end.\\nThe index having been set to divide to the right 8 i^.\u00c2\u00b0o? tooth\\nnumber we cut two spaces central with the blank, being finished\\nleaving a tooth between that is a little too thick, as in\\nthe upper part of Fig. 25. If the gear is of cast iron,\\nand the pitch is not coarser than about 5 diametral,\\nthis is as far as we go with the central cuts, and we\\nproceed to set the cutter and the blank to finish first\\none side of the teeth and then the other, going around\\nonly twice. The tooth has to be cut away more in\\nproportion from the large than from the small end,\\nwhich is the reason for setting the cutter out of centre,\\nas in Fig. 24.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0069.jp2"}, "70": {"fulltext": "54\\nBllOWN SHARPE MFG. CO.\\nFig. 32\\nAUTOMATIC GEAR CUTTING MACHINE.\\nSIDE ELEVATION.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0070.jp2"}, "71": {"fulltext": "rrsoviDENCE, r. i. 55\\nIt is important to remember that the part of the\\ncutter that is finishing one side of a tooth at the pitch\\nline should be central with the gear blank, in order to\\nknow at once in which direction to set the cutter out of\\ncentre. We can not readily tell how much out of\\ncentre to set the cutter until we have cut and tried,\\nbecause the same part of a cutter does not cut to the\\npitch line at both ends of a tooth. As a trial distance\\nout of centre we can take about one-tenth to one-\\neighth of the thickness of the teeth at the large end.\\nThe actual distance out of centre for the 12-tooth\\npinion is .021 for the 24-tooth gear, .030 when\\nusing cutters listed in our catalogue.\\nAfter a little practice a workman can set his blank Necessity of\\n1 central cuts.\\nthe trial distance out of centre, and take his first cuts,\\nwithout any central cuts at all but it is safer to take\\ncentral cuts like the upper ones in Fig. 25. The\\ndepth of cut is partly controlled by the index-spindle\\nraising-dial-shaft H, Fig. 33, which determines the\\nheight of the index spindle, and partly by the position\\nof the cutter spindle. We now set the cutter out of\\ncentre the trial distance by means of the cutter-spindle\\ndial-shaft, I, Fig. 33. The trial distance can be about\\none-tenth the thickness of the tooth at the large end\\nin a 12-tooth pinion, and from that to one-eighth the\\nthickness in a 24-tooth gear and larger. The principle\\nof trimming the teeth more at the large end than at\\nthe small is illustrated in Fig. 24, which is to move\\nthe cutter away from the tooth to be trimmed, and\\nthen to bring the tooth up against the cutter by\\nrotating the blank in the direction of the arrow. Adustments\\nThe rotative adjustment of the index spindle is\\naccomplished by loosening the connection between the\\nindex worm and the index drive, and turning the worm,:\\nthe connection is then fastened again. The cutter is\\nnow set the same distance out of centre in the other\\ndirection, the index spindle is adjusted to trim the\\nother side of the tooth until one end is down nearly\\nto the right thickness. If now the thickness of the", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0071.jp2"}, "72": {"fulltext": "D6 BROWN SHAKPE MFG. CO.\\nsmall end is in the same proportion to the large end as\\nOc is to Oc, Fig. 23, we can at once adjust to trim\\nthe tooth to the right thickness. But if we find that\\nthe large end is still going to be too thick when the\\nsmall end is right, the out of centre must be increased.\\nIt is well to remember this too much out of centre\\nleaves the small end proportionally too thick, and too\\nlittle out of centre leaves the small end too thin.\\nAfter the proper distance out of centre has been\\nlearned the teeth can be finish-cut by going around out\\nof centre first on one side and then on the other with-\\nout cutting any central spaces at all. The cutter\\nspindle stops, J J, can now be set to control the out\\nof centre of the cutter, without having to adjust by\\nthe dial G. If, however, a cast iron gear is 5-pitch\\nor coarser it is usually well to cut central spaces first\\nand then take the two out-of -centre cuts, going around\\nthree times in all. Steel gears should be cut three\\ntimes around.\\nBlanks are not always turned nearly enough alike to\\nbe cut without a different setting for different blanks.\\nIf the hubs vary in length the position of the cutter\\nspindle has to be varied. In thus varying, the same\\ndepth of cut or the exact J) -f f may not always be\\nreached. A slight difference in the depth is not so\\nobjectionable as the incorrect tooth thickness i hat it\\nmay cause. Hence, it is well, after cutting once\\naround and finishing one side of the teeth, to give\\ncareful attention to the rotative adjustment of the\\nindex spindle so as to cut the right thickness.\\nAfter a gear is cut, and before the teeth are filed, it\\nis not always a very satisfactory-looking piece of work.\\nIn Fig. 25 the tooth L is as the cutter left it, and is\\nready to be filed to the shipe of the teeth M M, which\\nhave been filed. Fig. 34 is the pair of gears that we\\nhave been cutting the teeth of the 12-tooth pinion\\nhave been filed.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0072.jp2"}, "73": {"fulltext": "PROVIDENCE, K. I.\\n57", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0073.jp2"}, "74": {"fulltext": "58\\nBROWN SHARPE MFG. CO.\\na second second approximation in cutting with a rotary\\napproxima- x l\\ntion- cutter is to widen the spaces atthe large end by swing-\\ning either the index spindle or the cutter-slide-carriage,\\nso as to pass the cutter through on an angle with the\\nblank side-ways, called the side-angle, and not rotate\\nthe blank at all to widen the spaces. This side-angle\\nmethod is employed in our No. 11 Automatic Bevel\\nGear Cutting Machines it is available in the manufac-\\nture of bevel gears in large quantities, because with\\nthe proper relative thickness of cutter, the tooth-\\nthickness comes right by merely adjusting for the\\nside-angle but for cutting a few gears it is not much\\nliked by workmen, because, in adjusting for the side-\\nangle, the central setting of the cutter is usually lost,\\nand has to be found by guiding into the central slot\\nalready cut. if the side-angle mechanism pivots about\\na line that passes very near the small end of the tooth\\nto be cut, the central setting of the cutter may not\\nbe lost. In widening the spaces at the large end,\\nthe teeth are narrowed practically the same amount at\\nthe root as at the face, so that this side-angle method\\nrequires a wider cutter at e e Fig. 24. than the first,\\nor rotative method. The amount of filing required\\nto correct the form of the teeth at the small end is\\nabout the same as in the first method.\\na third ap- A third approximate method consists in cutting\\nproximation.\\nthe teeth right at the large end by going around at\\nleast twice, and then to trim the teeth at the small end\\nand toward the large with another cutter, going around\\nat least four times in all. This method requires skill\\nand is necessarily a little slow, but it contains possi-\\nbilities for considerable accuracy.\\na fourth ap- A fourth method is to have a cutter fully as thick as\\nproximation.\\nthe spaces at the small end, cut rather deeper than\\nthe regular depth at the large end, and go only once\\naround. This is a quick method but more inaccurate\\nthan the three preceding it is available in the manu-\\nfacture of large numbers of gears when the tooth-face", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0074.jp2"}, "75": {"fulltext": "PROVIDENCE, K. I.\\n59\\nFig. 34\\nFINISHED GEAR AND PINION", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0075.jp2"}, "76": {"fulltext": "60\\nBROWN SIIARPE MFG. CO.\\nis sho.rt compared with the apex distance. It is little\\nliked, and seldom employed in cutting a few gears it\\nmay require some experimenting to determine the form\\nof cutter. Sometimes the teeth are not cut to the\\nregular depth at the small end in order to have them\\nthick enough, which may necessitate reducing the\\naddendum of the teeth, s, at the small end by turning\\nthe blank down. This method is extensively employed\\nby chuck manufacturers.\\nA machine that cuts bevel gears with a reciprocating\\nmotion and using a tool similar to a planer tool is\\ncalled a Gear Planer and the gears so cut are said to\\nbe planed.\\npianino- G f ne f\u00c2\u00b0 rm of Gear Planer is that in which the prin-\\nbevei gears. c [pi e embodied is theoretically correct this machine\\noriginates the tooth curves without a former. Another\\nform of the same class of machines is that in which the\\ntool is guided by a former.\\nUsually the time consumed in planing a bevel gear\\nis greater than the time necessary to cut the same gear\\nwith a rotary cutter, thus proportionately increasing\\nthe cost.\\nPitches coarser than 4 are more correct and some-\\ntimes less expensive when planed it is hardly prac-\\nticable, and certainly not economical, to cut a bevel\\nge-ir as coarse as 3P. with a rotary cutter. In gears as\\nfine as 1GP. planing affords no practical gain in quality.\\nWhile planing is theoretically correct, yet the wear-\\ning of the tool may cause more variation in the thick-\\nness of the teeth than the wearing of a rotary cutter,\\nand even a planed gear is sometimes improved by filing.\\nMounting of gears are not correctly mounted in the place where\\ngears. they are to run, they might as well not be planed. In\\nfact, after taking pains in the cutting of any gear,\\nwhen we come to the mounting of it we should keep\\nright on taking pains.\\nAngles and The method of obtaining the sizes and angles per-\\ngeais. taining to bevel gears by measuring a drawing is quite\\nconvenient, and with care is fairly accurate. Its", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0076.jp2"}, "77": {"fulltext": "PROVIDEXCK, K. I.\\n61\\naccuracy depends, of course, upon the careful measur-\\ning of a good drawing. We may say, in general, that\\nin measuring a diagram, while we can hardly obtain\\ndata mathematically exact, Ave are not likely to make\\nwild mistakes. Some years ago we depended almost\\nentirely upon measuring, but siuce the publication of\\nthis Treatise and our Formulas in Gearing we\\ncalculate the data without any measuring of a drawing.\\nIn the Formulas in Gearing there are also tables\\npertaining to bevel gears.\\nSeveral of the cuts and some of the matter in this\\nchapter are taken from an article by O. J. Beale, in\\nthe American Machinist, June 20, 1^95.\\nCutters for Mitre and Bevel\\nGears.\\nDiametral\\nDiameter of\\nHole in\\nPitch.\\nCutter.\\nCutter.\\n4\\n8 3-8\\n1 1-4\\n5\\n3 1-16\\n1 1\\n6\\n2 3-4\\n1 1-16\\n8\\n2 1-2\\n1 1\\n10\\n2 1-8\\n7-8\\n12\\n2\\nii\\n14\\n2\\ni\\n16\\n1 15-10\\n1 1\\n20\\n1 7-8\\nI c\\n24\\n1 3-4", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0077.jp2"}, "78": {"fulltext": "WORM WHEEL.\\nNumber of Teeth, 34. Circular Pitch, 2J/.j.\\nThroat Diameter, 44.39 Outside Diameter, 46", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0078.jp2"}, "79": {"fulltext": "C3\\nCHAPTER XI.\\nWORM WHEELS\u00e2\u0080\u0094 SIZING BLANKS OF 32 TEETH AND OVER.\\nA worm is a screw made to mesli with the teeth of Worm.\\na wheel called a worm-wheel. As implied at the end of\\nChapter IV., a section of a worm through its axis is, in\\noutline, the same as a rack of corresponding pitch.\\nThis outline can be made either to mesh with single or\\ndouble curve gear teeth but worms are usually made\\nfor single curve, because, the sides of involute rack\\nteeth being straight (see Chapter IV.), the tool for\\ncutting worm-thread is more easily made. The thread-\\ntool is not usually rounded for giving fillets at bottom\\nof worm-thread.\\nThe axis of a worm is usually at rigl t singles to the\\naxis of a worm wheel: no other angle of axis is treated\\nof in this book.\\nThe rules for circular pitch apply in the size of tooth\\nparts and diameter of pitch-circle of worm-wheel.\\nThe pitch of a worm or screw is sometimes given in Pitch of worm\\na way different from the pitch of a gear, viz. in num-\\nber of threads to one inch of the length of the worm or\\nscrew. Thus, to say a worm is 2 pitch may mean 2\\nthreads to the inch, or that the worm makes two turns\\nto advance the thread one inch. But a worm may be\\ndouble-threaded, triple-threaded, and so on; hence\\nto avoid misunderstanding, it is better always to call\\nthe advance of the worm thread the lead. Thus, a worm-Thread\\nworm-thread that advances one inch in one turn we\\ncall one-inch lead in one turn. A single-thread worm\\n4 to 1 is J lead. We apply the term pitch, that is\\nthe circular pitch, to the actual distance between the\\nthreads or teeth, as in previous chapters. In single-\\nthread worms the lead and the pitch are alike. If we\\nhave to make a worm and wheel so many threads to", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0079.jp2"}, "80": {"fulltext": "04\\nBROWN SHAEPE MFG. CO.\\nFIG. 35\u00e2\u0080\u0094 WORM AND WORM-WHEEL\\nThe thread of Worm is left-handed Worm is single-threaded", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0080.jp2"}, "81": {"fulltext": "PROVIDENCE, R. I.\\n05", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0081.jp2"}, "82": {"fulltext": "ir inehj we irs: ziride 1 hw lie number of threads Mm\\nne imdk 9 and the quotient iM-ihe circular ptidL Hence,\\n~_r- -f:- F .z. is .:::.::_. X j:::.\\nsxprcoBBa tI bat i3 mean: f ar pitch.\\nT. r:::l _e rf nz ei use.\\nri.-.:r ~:f: t m= i I :k=. T _ e\\n_~m.ljer of threads to one iceh linear, is the reciprocal\\ni r t i tie:\\nMultiply 3.1416 bj the number of threads to one\\ninch, and the product will be the diametral pitch of the\\nirr. r_~j. -r=i:~ -i iir .r-:*-: f..:r 3\\nworm advancing 1 in 1^ turns that:\\nDnvimg of Iiead=\u00c2\u00a7 or .7-5 Iane.xT ::P t r\\n-i^^^-AT ii-=-~-il; :_ F S ^rr :e. :*_-e- :_\\nZ: _ :t _E-~_zr :_ ~::z_ ezlI _-ee. ~r __\\nV ._ :_ :i :::n\\n1 iTi- :-:_:-: jzlt A J iz.1 u\\\\:_ i: :_e-\\n_:- t: ii_ ~_t ELLim-rT-EEr ::;_-:_;_\\n_ t. ._ .t -L-e-^t i :_;- Li; :z :_t ii=-\\n_ _ t _ _ t L ifzi 1 i_\\n3. From lay the itmtwocr c O equal to the\\nriiii= :_- ::_ T_t iiizi-E-Tr^ i :-_ is _ ri.-\\n-i L^~ :z :1t ii-:^z:\u00c2\u00bb5-~ evlI -e :_ e~:\\n_ _ t t _ e- :::_::::::_-;\u00e2\u0096\u00a0:: t\\nI_ ::__ i_:,- eh:! 5 j Xlr-s*\\nr-e~: TETz: tie ~_:1- Iie^-tEe: ~::_ :-._.*. :_r _-\\n:::_ -:::_-:_. e\\nI :_ :-_ .-.ei_ _t :e_\\nA r_r-T __-- r _ :i:- :l_--_tt..\\nZ_e :^_ ml ~_e.^ i i: rz.v~ _\\n:_:\\n~_i5 t.::::_ :_.-:t- :_r :^tln.e\\n::n-\u00e2\u0080\u0094 LettI. F:: :tt _ _ e :_- t_ _\\n_ _t:t: D. ~_:.^1 -z. i: :_t :_: :r\\n5i_:._tE-: L_e.l_t:t. jl-e-t! 7-_ JS/f---* f", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0082.jp2"}, "83": {"fulltext": "PROVIDENCE, R. I.\\n67\\nthreaded with a tool of the same angle as the tool that\\nthreads the worm, the end of the tool bein r .335 of\\nthe linear pitch the hob is then grooved to make teeth\\nfor cutting, and hardened.\\nThe whole diameter of hob should be at least 2 Proportionsof\\nHob.\\nor twice the clearance larger than the worm. In our\\nrelieved hobs the diameter is made about .005 to .010\\nlarger to allow for wear. The outer corners of hob-thread\\ncan be rounded down as far as the clearance distance.\\nThe width at top of the hob-thread before rounding\\nshould be .31 of the linear, or circular pitch .31P\\nThe whole depth of thread is thus the ordinary work-\\ning depth plus the cle:iranee=D The diameter\\nat bottom of hob-thread should he 2/+. 005 to .010\\nlarger than the diameter at bottom of worm-thread.\\nFig. 37\u00e2\u0080\u0094 HOB.\\nFor thread-tool and worm-thread see end of Chapter\\nIV.\\nIn the absence of a special worm gear cutting ma tn H( H r l) t0 use\\nchine, the teeth of the wheel are first cut as nearly to the\\nfinished form as practicable; the hob and worm-wheel\\nare mounted upon shafts and hob placed in mesh, it is\\nthen rotated and dropped deeper into the wheel until the\\nteeth are finished. The hob generally drives the worm-\\nwheel.during this operation. The Universal Milling Ma- universal\\nchine is convenient for doing this work with it the d is- chine used in\\nHobbing.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0083.jp2"}, "84": {"fulltext": "68\\nBROWN SHARPE MFG. CO.\\nmJf\\nFig. 38.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0084.jp2"}, "85": {"fulltext": "PROVIDENCE, R. I.\\nG9\\nH c,R s^\\nFig. 39.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0085.jp2"}, "86": {"fulltext": "~z.-.:.:-i i::}\\ntan ce be t ween axes of wo rm and wheel ea n be n oted In\\nmaking wheels in quantities it is be!ter to have a ma-\\nchine in which the work spindle is drive a y rearing,\\nso that the hob can cut the teeih from the solid with-\\n-va wheel OD gashing. The object of hobbing a wheel is to _\\niiH ivz-rt more bearing surface of the teeth upon worm-thread.\\nThe worm-wheels. Figs. 35 aud 43. I obbed.\\nwonn-wiieei If we make the diameter of a worm-wheel blank, that\\nleas than 30 is to have less than 30 teeth, by the common rules\\nfor sizing blanks, and finish the teeth with a hob.\\nshall find the flanks of teeth n sax the bottom to be un-\\ninierierenee dercut or hollowing. This is caused bv the interfer-\\ned __Lresi B.~i __ r v\\nFlank. ence spoken of m Chapter I. Inirty teeth was there\\ngiven as a limit, which will be right when teeth are\\nmade to circle arcs. With pressure an^ aid\\nrack-teeth with usual addendum, thU ::::e::erence of\\nrack-teeth with flanks of gear- teeth commences at 31\\nteeth (31^ geometrically), and interferes with nearly\\nthe whole flank in wheel of 12 re:h.\\nIn Fig 38 the blank for worm- wheel of 12 teeth w\\nsized by the same rule as given for Fig. 36. The wheel\\nand worm are sectioned to show shape of teeth at the\\nmid-plane of wheel. The flunks of teeth are undercut\\nby the hob. The worm-thread does not have a good\\nbearing on flanks inside of A. the bearing being that of\\na corner against a surf a e\\n^s- In Fig 39 the blank for wheel wae sized sc that pitch-\\ncircle comes midway between outermost part of tee:_\\n\u00e2\u0096\u00a0nd innermost point obtained by worm thread.\\nThis rule for sizing worm-wheel blanks has been in\\nuse me ri:ri:. The hob has cut away flanks of\\nteeth still more than in Fig. 3S. The pitch circle in\\nKg. 39 is the same diameter as the pitch-circle in Fig.\\n38. The same hob was used for both wheels. The\\nflanks in this wheel are so much undercut as to mate-\\nrially lessen the bearing surface of teeth and worm-\\nthread.\\ninterference In Chapter ~YX the interference of teeth in high-\\nnumbered g r ears and racks with flanks of 12 teeth vr\\nremedied by rounding off the addenda. Although it\\nw raid be move systematic to round off the threads ol\\na worm, making them, like rack-teeth, to mesh w::_", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0086.jp2"}, "87": {"fulltext": "PROVIDENCE, R. I. 71\\ninterchangeable gears, yet this has not generally been\\ndone, because it is easier to make a worm-thread tool\\nwith straight sides.\\nInstead of cutting away the addenda of worm-\\nthread, we can avoid the interference with flanks of\\nwheels having less than 30 teeth by making wheel\\nblanks larger.\\nThe flanks of wheel in Fig. 40 are not undercut, be- Fig. 40.\\ncause the diameter of wheel is so large that there is\\nhardly any tooth inside the pitch-circle. The\\npitch-circle in Fig. 40 is the same size as pitch-\\ncircles in Figs. 38 and 39. This wheel was sized\\nby the following rule Multiply the pitch diameter of Diameter at\\nThroat to Avoid\\nthe wheel by .937, and add to the product four times interference,\\nthe addendum (4 s) the sum will be the diameter for\\nthe blank at the throat or small part. To get the\\nwhole diameter, make a sketch with diameter of throat\\nto the foregoing rule and measure the sketch.\\nIt is impractical to hob a wheel of 12 to about 16 or\\n18 teeth when blank is sized by this rule, unless the\\nwheel is driven by independent mechanism and not by\\nthe hob. The diameter across the outermost parts of\\nteeth, as at A B, is considerably less than the largest\\ndiameter of wheel before it was bobbed.\\nIn general it is well to size all blanks, as by page 66\\nand Figs. 36 and 38, when the wheels are to be hobbed\\nof course the cutter should be thin enough to leave\\nstock for finishing. The spaces can be cut the full\\ndepth, the cutter being dropped in.\\nWhen worm-wheels are not hobbed it is better to\\nturn blanks like a spur-wheel. Little is gained by g ^J a n J h Li J e a\\nhaving wheels curved to fit worm unless teeth are fin-\\nished with a hob. The teeth can be cut in a straight\\npath diagonally across face of blank, to fit angle of\\nworm-thread, as in Figs. 41 and 44.\\nFor dividing wheels to gear-cutting engines the Wheels for\\nblanks are turned like a spur-wheel and a cutter about Machines.\\nT *g- larger diameter than the worm, is dropped in, as\\nin Figs. 42 and 45.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0087.jp2"}, "88": {"fulltext": "5:\\nJ _ v\\nFn 40", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0088.jp2"}, "89": {"fulltext": "PROVIDENCE, K. I. 73\\nSome mechanics prefer to make dividing wheels in\\ntwo parts, joined in a plane perpendicular to axis, hob\\nteeth then turn one part round upon the other, match\\nteeth and fasten parts together in the new position,\\nand hob again with a view to eliminate errors. With\\nan accurate cutting engine we have found wheels like\\nFigs. 42 and 45, not nobbed, every way satisfactory.\\nAs to the different wheels, Figs. 43, 44 and 45, when Figures 43, 44\\nand 45.\\nworm is in right position at the start, the life-time\\nof Fig. 43, under heavy and continuous work, will be\\nthe longest.\\nFig. 44 can be run in mesh with a gear or a rack as\\nwell as with a worm when made within the angular\\nlimits commonly required. Strictly, neither two gears\\nmade in this way, nor a gear and a rack would be\\nmathematically exact, as they might bear at the sides\\nof the gear or at the ends of the teeth only and not in\\nthe middle. At the start the contact of teeth in this\\nwheel upon worm-thread is in points only; yet such\\nwheels have been many years successfully used in ele-\\nvators.\\nFig. 45 is a neat-looking wheel. In gear cutting\\nengines where the workman has occasion to turn the\\nwork spindle by hand, it is not so rough to take hold\\nof as Figs. 43 and 44. The teeth are less liable to in-\\njury than the teeth of Figs. 43 and 44.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0089.jp2"}, "90": {"fulltext": "BROWN SHAUPE MFG. CO.\\nFig. 41.\\nWorm-wheel with teeth cut in a straight path diagonally across face.\\nWorm is double-threaded.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0090.jp2"}, "91": {"fulltext": "PROVIDENCE, E. I.\\n7a\\nFig. 42.\\nWorm and Worm- Wheel, for Gear-cutting Engine.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0091.jp2"}, "92": {"fulltext": "7G\\nBROWN SHAIirE MFG. CO.\\nUaaMjamik---\\nFig. 43.\\nFig. 44.\\nFig. 45.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0092.jp2"}, "93": {"fulltext": "PROVIDENCE, R. I. 77\\nSome designers prefer to take off the outermost part\\nof teeth in wheels (Figs. 35 and 43), as shown in these\\ntwo figures, and not leave them sharp, as in Figs. 36\\nand 80.\\nWe do not know that this serves any purpose except\\na matter of looks.\\nIn ordering worms and worm wheels the centre dis-\\ntances should be given.\\nIf there can be any limit allowed in the centre distance\\nit should be so stated.\\nFor instance, the distance from the centre of a worm\\nto the centre of a worm wheel might be calculated at\\n6 but 5 31-32 or 6 1-32 might answer.\\nBy stating all the limits that can be allowed there\\nmay be a saving in the cost of work because time need\\nnot be wasted in trying to make work within narrower\\nlimits than are necessary.\\nHOBS WITH RELIEVED TEETH.\\nWe are prepared to make hobs of any size with the\\nteeth relieved the same as our gear cutters. The teeth\\ncan be ground on their faces without changing their\\nform. The hobs are made with a precision screw so\\nthat the pitch of the thread is accurate before hard-\\nening.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0093.jp2"}, "94": {"fulltext": "BIIOTVX SHARFE MFG. CO.\\nGASHING TEETH OF HOB,\\nlO Inches Ou.tside Diameter.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0094.jp2"}, "95": {"fulltext": "PROVIDENCE, E. I,\\nCHAPTER XII.\\nSIZING GEARS WHEN THE DISTANCE BETWEEN CENTRES AND THE\\nRATIOS OF SPEEDS ARE FIXED\u00e2\u0080\u0094 GENERAL REMARKS\u00e2\u0080\u0094 WIDTH\\nOF FACE OF SPOR GEARS\u00e2\u0080\u0094 SPEED OF GEAR CUTTERS\u00e2\u0080\u0094 TABLE\\nOF TOOTH PARTS.\\nLet us suppose that we have two shafts 14 apart,\\ncenter to center, and wish to connect them by gears so, center dis-\\nJ tance and Ratio\\nthat they will have speed ratio 6 to 1. We add the 6 fixed\\nand 1 together, and divide 14 by the sum and get 2\\nfor a quotient; this 2 multiplied by 6, gives us the\\nradius of pitch circle of large wheel 12 In the same\\nmanner we get 2 as radius of pitch circle of small wheel.\\nDoubling the radius of each gear, we obtain 24 and 4\\nas the pitch diameters of the two wheels. The two num-\\nbers that form a ratio are called the tsrms of the ratio.\\nWe have now the rule for obtaining pitch-circle diame-\\nter of two wheels of a given ratio to connect shafts a\\ngiven distance apart:\\nDivide the center distance bii the sum of the terms of Rule f \u00c2\u00b0r r u\\nJ ameter of Pitch\\nthe ratio; find the product of tin ice the quotient by each circles.\\nterm sparately, and the two products ivill he the pitch\\ndiameters of the two ivheels.\\nIt is well to give special attention to learning the\\nrules for sizing blanks and teeth these are much\\noftener needed than the method of forming tooth out-\\nlines.\\nA blank 1-J- diameter is to have 16 teeth: what will\\nthe pitch be? What will be the diameter of the pitch\\ncircle See Chapter V.\\nA good practice will be to compute a table of tooth\\nparts. The work can be compared with the tables\\npages 86-89.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0095.jp2"}, "96": {"fulltext": "Sj BBovrs- jc szat.iz mtg.\\nIn compute:, i: is well to take re to more than four\\nrlaces. ,t to nine places 3.141592653. to nine\\nplaces .318309886.\\nThere is no such thing as pure rolling contact in\\nteeth of wheels: they always rub, and. in time, will\\nwear themselves out of snape and niay become noisy.\\nBevel gears, vriier. correctly ft-raieiL ran smoother\\nthan spur gears of same diameter and pitch, because\\nthe teeth continue in contact longer than the teeth of\\nspur gears. F r this reas n annular gears run smoother\\nthan either bevel or spur gears.\\nSometimes gears have to be cut a little deeper than\\ndesigned, in order to run easily on their shafts. If\\nany departure is made in ratio of pitch diameters it\\nbetter to have the driving gear the larger, that is, cut\\nthe follower smaller. For wheels coarser than eight\\ndiametral pitch (8 P it is generally better to cottwi x\\naround, when accurate wxk is wanted, also for lar _ -e\\nwheels, as the expansion of parts from heat often causes\\ninaccurate work when cut but once around. There is\\nnot so much trouble from heat in plain or we _ e is as\\nin arm gears.\\nw-. .-_:-; \u00e2\u0080\u009er Tiie t _ c cast-ii:n _ear :a:es r/ererai rtir-\\nposes can be made to the following rule\\nIt v:\\\\J.-i S :y -to the\\nquo* it sum toiU be width of face for the pitch\\nExample VThat width of face for gear 4 P Divid-\\ning 8 by -i and adding J we obtain 2^ for width of\\nface. For change gears on lathes, where it is desira-\\nble not to have face very wide, the following rule can\\nbe used:\\nDivide -ch and add J\\nBv the latter rule a 4 P change gear would have but\\nH race\\nSpeedof Gear The speed of gear cutters is subject t s many con-\\nditions that dehnite rules cannot be given. TVe append\\ntable of average speeds. A coarse pitch cutter for\\npinion. 12 teeth, would usually be fed slower than a\\ncutter for a large gear of same pitch.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0096.jp2"}, "97": {"fulltext": "PROVIDENCE, E. I.\\n81\\nTABLE OF AVERAGE SPEEDS FOR GEAR-CUTTERS.\\nA\\nbfi\\ntI) S\\nu\\nc\\nO\\n\u00c2\u00abw\\n^.S\\nC3^ A\\nS3 p\\no\\nS|\\nO\\n5S\\nftp g\\nw m\\nP.3 o\\nat\\nm d d\\nKg\\nt- d c\\nO.SM S3\\nS3^\u00c2\u00a3\\nDiametral\\nof Cut\\nu\\n5\\naa \u00c2\u00b0H\\nP S3 -u\\nJJ3 3\\nEr\\n3cm\\n7J CO\\nr\\n2 9^\\n^s5\\nr- S OQ\\n2\\n5 in.\\n24\\n18\\n.025 in\\n.011 in.\\n.60 in.\\n20 in.\\n2*\\n\u00c2\u00b1i\\n30\\n24\\n.028\\n.013\\n.84\\n.31\\n3\\n16\\n36\\n28\\n.031\\n.015\\n1.12\\n.42\\n4\\n3f\\n42\\n32\\n.034\\n.017\\n1.43\\n.54\\n5\\n3 T V\\n50\\n40\\n.037\\n.019\\n1.85\\n.76\\n6\\n2U-\\n16\\n75\\n55\\n.030\\n.016\\n2.25\\n.88\\n7\\n2-9-\\n1 6\\n85\\n65\\n.032\\n.018\\n2 72\\n1.17\\n8\\n9\u00c2\u00b1\\n~2\\n95\\n75\\n.034\\n020\\n3.23\\n1.50\\n10\\n01\\nZ S\\n125\\n90\\n.026\\n.014\\n3.25\\n1.26\\n12\\n2\\n135\\n100\\n.027\\n.017\\n3.64\\n1.70\\n20\\nH\\n145\\n115\\n.029\\n.021\\n4,20\\n2.41\\n32\\nH\\n160\\n135\\n.031\\n.025\\n4.96\\n3.37\\nIn brass the speed of gear-cutters can be twice as B g e e d s\\nfast as in cast iron. Clock-makers and those making a\\nspecialty of brass gears exceed this rate even. A 12 P\\ncutter has been run 1,200 (twelve hundred) turns a\\nminute in bronze. A 32 P cutter has been run 7,000\\n(seven thousand) turns a minute in soft brass.\\nIn cutting 5 P cast-iron gears, 75 teeth, a No. 1, BP^^g 9\\ncutter was run 136 (one hundred and thirty-six) turns\\na minute, roughing the spaces out the full 5 P depth\\nthe teeth were then finished with a 5 P cutter, running\\n208 (two hundred and eight) turns a minute, feeding\\nby hand. The cutter stood well, but, of course, the\\ncast iron was quite soft. A 4 P cutter has finished\\nteeth at one cut, in cast-iron gears, 86 teeth, running 48\\n(forty-eight) turns a minute and feeding T y at one\\nturn, or 3 in. in a minute.\\nHence, while it is generally safe to run cutters as in\\nthe table, yet when many gears are to be cut it is well to\\nsee if cutters will stand a higher speed and more feed.\\nIn gears coarser than 3 P it is more economical to\\ncut first the full depth with a stocking cutter and then\\nfinish with a gear cutter. This stocking cutter is made", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0097.jp2"}, "98": {"fulltext": "82 BROWN SHAEPE MEG. CO.\\non the jDrinciple of a circular splitting saw for wood.\\nThe teeth, however, are not set but side relief is ob-\\ntained by niakino- sides of cutter blank hollowing. The\\nshape of stocking cutter can be same as bottom of\\nspaces in a 1 --tooth gear, and the thickness of cutter\\ncan be J- of the circular pitch, see page 40.\\nKeep cutters The matter of keeping cutters sharp is so important\\nthat it has sometimes been found best to have the work-\\nman grind them at stated times, and not wait until he\\ncan see that the cutters are dull. Thus, have him\\nm.ind everv two hours or after cutting a stated number\\nof gears. Cutters of the style that can be ground\\nupon their tooth faces without changing form are rap-\\nidly destroyed if allowed to run after they are dull.\\nCutters are oftener wasted bv trying to cut with them\\nAvhen they are dull than by too much grinding. Grind\\nIhe faces radial with a free cutting wheel. Do not let\\nthe wheel become glazed, as this will draw the temper\\nof the cutter.\\nIn Chapter YI. was given a series of cutters for cut-\\nting gears having 12 teeth and more. Thus, it was\\nthere implied that any gear cf same pitch, having 135\\nteeth, 13G teeth, and so on up 1 the largest gears, and,\\nalso, a rack, could be cut with one cutter. If this cut-\\nter is 4 P, we would cut with it all 4 P gears, having\\n135 teeth or more, and we would also cut with it a 4P\\nrack. Now. instead of alwavs ref erring to a cutter bv\\nthe number of teeth in gears it is designed to cut, it\\nhas been found convenient to designate it by a letter\\nor by a number. Thus, we call a cutter of 4 P, made\\nto cut gears 135 teeth to a rack, inclusive, Nj. 1, 4 P.\\nWe have adopted numbers for designating involute\\ninvolute Gear orear-cutters a-s in the following table:\\nCutters.\\nNo. 1 will cut wheels from 135 teeth to a rack inclusive.\\n55\\nCI\\n134 teeth\\n35\\n54\\n26\\nu\\n34\\n21\\n25 ki\\n17\\n20\\n14\\nit\\n16\\n12\\n13", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0098.jp2"}, "99": {"fulltext": "PROVIDENCE, R. I. 83\\nBy this plan it takes eight cutters to cut all gears\\nhaving twelve teeth and over, of any one pitch.\\nThus it takes ei\u00c2\u00bbht cutters to cut all involute 4 P\\nO\\ngears having twelve teeth and more. It takes eight\\nother cutters to cut all involute gears of 5 P, having\\n12 teeth and more. A No. 8, 5 P cutter cuts only 5 P\\ngears having 12 and 13 teeth. A No. 6, 10 P cutter\\ncuts only 10 P gears having 17, 18, 19 and 20 teeth.\\nOn each cutter is stamped the number of teeth at the\\nlimits of its range, as well as the number of the cutter.\\nThe number of the cutter relates only to the number\\nof teeth in gears that the cutter is made for.\\nIn ordering cutters for involute spur-gears two things\\nmust be given\\n1. Either the number of teeth to be cut in the qear How to order\\nJ J Involute Cut-\\nor the number of the cutter, as given in the foregoing ters.\\ntable.\\n2. Either the pitch of the gear or the diameter and\\nnumber of teeth to be cut in the gear.\\nIf 25 teeth are to be cut in a 6 P involute gear, the\\ncutter will be No. 5, 6 P, which cuts all 6 P gears from\\n21 to 25 teeth inclusive. If it is desired to cut gears\\nfrom 15 to 25 teeth, three cutters will be needed, No.\\n5, No. 6 and No. 7 of the pitch required. If the pitch\\nis 8 and gears 15 to 25 teeth are to be cut, the cutters\\nshould be No. 5, 8 P, No. 6, 8 P, and No. 7, 8 P.\\nFor each pitch of epicycloidal, or double-curve gears, Epicycioidai\\n24 cutters are made. In coarse-pitch gears, the varia- cur e cutters,\\ntion in the shape of spaces between gears of consecu-\\ntive-numbered teeth is greater than in fine-pitch gears.\\nA set of cutters for each pitch, to consist of so large a\\nnumber as 24, has been established because double curve\\nteeth have generally been preferred in coarse-pitch gears,\\nthough the tendency of late years is toward the involute\\nform.\\nOur double curve cutters have a guide shoulder on each\\nside for the depth to cut. When this shoulder just reaches\\nthe periphery of the blank the depth is right. The marks\\nwhich these shoulders make on the blank, should be as nar-\\nrow as can be seen, when the blanks are sized right.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0099.jp2"}, "100": {"fulltext": "81 zz. _ y h terete z tee.-. :o.\\nDouble-curve gear-cutters are designate bv letters\\ninstead of by numbers this is to avoid confusion in\\n;~e:::i_\\nF allowing is the list of epicycloidal or double-c urve\\ngear-cutters\\n^-^vf 1 Cutter A :-tits 1*2 teeth. Cutter 31 :u: 27 t: 2? teeth\\nIknUe-curre -n -i t; v\\nC U Q 0\u00c2\u00b1\\nD 15 u P 3S 42\\nE 16 Q 43 49\\nu y 17 B 50 59\\nq is S 60 74\\nH 19 T 75 99\\nI 20 u T 100 149\\nJ 21 t: -_ V 150 24:9\\nK 23 t: 24 TV 250 Rack.\\nL 24 to 26 u X Rack.\\nA cutter that cuts more than one gear is made of\\nproper form for the smallest gear in its range. Thus,\\n:tttte: J ::t 21 t: 22 teeth is crrrect tor 21 teeth:\\ncutter S for 60 to 74 teeth is correct for 60 teeth,\\nand so on.\\nE r ~_ :;y In ordering epicycloidal gear-cutters designate the\\nOuters. letter of the cutter as in the foregoing table, also\\neither give the pitch or give data that will enable us\\nto determine the pitch, the same as directed for invo-\\nlute cutters.\\nAI ;.re care is required in making and adjusting epi-\\ncycloidal gears than in making involute gears.\\nIn lering bevel gear cutters three things must s\\nEe~el (.Tear c o\\nt:-\\n1 h er of teeth in each gear.\\n2 Either the pitch of gears or the largest pitch\\ndio gear; set Fig- -7\\n3. The length of tooth face.\\nIf the shafts are not to run at right angles, it\\nshould be so stated, and the angle given. Involute\\ncutters onlv are used for cutting bevel gears. Nc\\ntempt should be made to cut epicyclodial tooth bevel gears\\nwith rotarv di k cut:-", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0100.jp2"}, "101": {"fulltext": "PROVIDENCE, R. I. S5\\nIn ordering- worm-wheel cutters, three things must T How t0 order\\no Worm -gear\\nbe given Cutters.\\n1. Number of teeth in the toheel.\\n2. Pitch of the worm; see Chapter XI.\\n3. M^hole diameter of worm.\\nIn any order connected with gears or gear-cutters,\\nwhen the word Diameter occurs, we usually under-\\nstand that the pitch diameter is meant. When the\\nwhole diameter of a gear is meant it should be plainly\\nwritten. Care in giving an order often saves the delay\\nof asking further instructions. An order for one gear-\\ncutter to cut from 25 to 30 teeth cannot be filled, be-\\ncause it takes two cutters of involute form to cut from\\n25 to 30 teeth, and three cutters of epicycloidal form\\nto cut from 25 to 30 teeth.\\nSheet zinc is convenient to sketch gears upon, and\\nalso for making templets. Before making sketch, it is\\nwell to give the zinc a dark coating with the following\\nmixture Dissolve 1 ounce of sulphate of copper (blue\\nvitriol) in about 4 ounces of water, and add about one-\\nhalf teaspoonful of nitric acid. Apply a thin coating\\nwith a piece of waste.\\nThis mixture will give a thin coating of copper to\\niron or steel, but the work should then be rubbed dry.\\nCare should be taken not to leave the mixture where it\\nis not wanted, as it rusts iron and steel.\\nWe have sometimes been asked why gears are noisy.\\nNot many questions can be asked us to which we can\\ngive a less definite answer than to the question why\\ngears are noisy.\\nW T e can indicate only some of the causes that may\\nmake gears noisy, such as: depth of cutting not\\nright in this particular gears are oftener cut too deep\\nthan not deep enough (more noise may be caused\\nby cutting the driver too deep than by cutting the\\ndriven too deep;) cutting not central this may\\nmake gears noisy in one direction when they are quiet\\nwhile running in the other direction centre distance\\nnot right if too deep the outer corners of the\\nteeth in one gear may strike the fillets of the teeth\\nin the other gear shafts not parallel frame of the", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0101.jp2"}, "102": {"fulltext": "B\\nBE OWN i -HaF.FE MFG. CO.\\nmachine such a form as to give oft sound vibrat: as\\n;n when we examine a pair of gears e cannot\\nalways tell what is the matter.\\nN te For any pitch not in the folio wing tables\\nfind corresponding part multiply the tabular value\\nfor one inch by the circular pitch required, and the\\nwill be the value for the pitch given. Exam-\\nple What is the value of s for 4 inch circular pitch?\\n.3183 for 1 P and .3183 4= 1.2732=8 for 4\\nP.\\nThe expression Addendum and (addendum\\nand the modules mean- the distance of a tooth outside\\nof pitch line and also the distance occupied for every\\ntooth upon the diameter of pitch circle.\\n---i-z\\n:i z si", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0102.jp2"}, "103": {"fulltext": "PART II.\\nCHAPTER I.\\nTANGENT OF ARC AND ANGLE.\\nIn Pakt II. we shall show how to calculate some e |p2fned t0 be\\nof the functions of a right-angle triangle from a table\\nof circular functions, the application of these calcula-\\ntions in some chapters of Part I. and in sizing blanks\\nand cutting teeth of spiral gears, the selection of\\ncutters for spiral gears, the application of continued\\nfractions to some problems in gear wheels and cutting\\nodd screw-threads, etc., etc.\\nA. Function is a quantity that depends upon another\\nquantity for its value. Thus the amount a workman\\nearns is a function of the time he has worked and of fi nctl0u de\\nhis wages per hour.\\nIn any right-angle triangle, O A B, we shall, for Right angle\\nt i o Triangle.\\nconvenience, call the two lines that form the right\\nangle O A B the sides, instead of base and perpen-\\ndicular. Thus O A B, being the right angle we call\\nthe line O A a side, and the line A B a side also.\\nWhen we speak of the angle A O B, we call the line\\nO A the side adjacent. When we are speaking of the Skle ad J acent\\nangle ABO we call the liue A B the side adjacent.\\nThe line opposite the right angle is the hypothenuse. Hypothenuse.\\nIn the following pages the definitions of circular\\nfunctions are for angles smaller than 90\u00c2\u00b0, and not\\nstrictly applicable to the reasoning employed in ana-\\nlytical trigonometry, where we find expressions for\\nangles of 270\u00c2\u00b0, 760\u00c2\u00b0, etc.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0103.jp2"}, "104": {"fulltext": "88\\nTangent.\\nBROWN SHAEPE MFG. CO.\\nThe Tangent of an arc is the line that touches it at\\none extremity and is terminated by a line drawn from\\nthe center through the other extremity. The tangent\\nis always outside the arc and is also perpendicular to\\nthe radius which meets it at the point of tangency.\\n*iff ^7\\nThus, in Fig. 46, the line A B is the tangent of the arc\\nA C. The point of tangency is at A.\\nAn angle at the center of a circle is measured by the\\narc intercepted by the sides of the angle. Hence the\\ntangent A B of the arc A C is also the tangent of the\\nangle A O B.\\nIn the tables of circular functions the radius of the\\narc is unity, or. in common practice, we take it as one\\ninch. The radius O A being 1 if we know the length\\nof the line or tangent A B we can, by looking in a\\ntable of tangents, find the number of degrees in the\\nangle A O B.\\nTo find the Thus, if A B is 2.25 long, we find the angle A O B\\nDegrees in an\\nAngle. is 66\u00c2\u00b0 very nearly. That is, having found that 2.2460\\nis the nearest number to 2.25 in the table of tangents\\nat the end of this volume, we find the corresponding\\ndegrees of the angle in the column at the left hand of\\nthe table and the minutes to be added at the top of\\nthe column containing the 2.2460.\\nThe table gives angles for every 10 which is suf-\\nficient for most purposes.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0104.jp2"}, "105": {"fulltext": "PROVIDENCE, R. I- 89\\nNow, if we have a right-angle triangle with an angle\\nthe same as O A B, but with O A two inches long, the\\nline A B will also be twice as long as the tangent of\\nangle A O B, as found in a table of tangents.\\nLet us take a triangle with the side OA=5 long, fln J xa\\nand the side AB 8 long what is the number of e e f in au\\ndegrees in the angle A O B\\nDividing 8 by 5 we find what would be the length\\nof A B if O A was only 1 long. The quotient then\\nwould be the length of tangent when the radius is 1\\nlong, as in the table of tangents. 8 divided by 5 is\\n1.6. The nearest tangent in the table is 1.6003 and\\nthe corresponding angle is 58\u00c2\u00b0, which would be the\\nangle A O B when A B is 8 and the radius O A is 5\\nvery nearly. The difference in the angles for tangents\\n1.6003 and 1.6 could hardly be seen in practice. The\\nside opposite the required acute angle corresponds to\\nthe tangent and the side adjacent corresponds to the\\nradius. Hence the rule\\nTo find the tangent of either acute angle in a right- T JJ e ^J d the\\nangle triangle Divide the side opposite the angle by\\nthe side adjacent the angle and the quotient will be\\nthe tangent of the angle. This rule should be com-\\nmitted to memory. Having found the tangent of the\\nangle, the angle can be taken from the table of tan-\\ngents.\\nThe complement of an angle is the remainder after complement\\nsubtracting the angle from 90\u00c2\u00b0. Thus 40\u00c2\u00b0 is the com-\\nplement of 50\u00c2\u00b0.\\nThe Cotangent of an angle is the tangent of the cotangent,\\ncomplement of the angle. Thus, in Fig. 47, the line\\nA B is the cotangent of A O E. In right-angle tri-\\nangles either acute angle is the complement of the\\nother acute angle. Hence, if we know one acute angle,\\nby subtracting this angle from 90\u00c2\u00b0 we get the other\\nacute angle. As the arc approaches 90\u00c2\u00b0 the tangent\\nbecomes longer, and at 90\u00c2\u00b0 it is infinitely long.\\nThe sign of infinity is oo. Tangent 90\u00c2\u00b0 oo,", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0105.jp2"}, "106": {"fulltext": "AMi^by 1 ~^7 a ta e tangents, angles can be laid out npon\\nTangent Ex- 5neet zinc. ere. T_:s is often an advantage, as i: is not\\ne\\nconvenient to lay protractor flat down so as to mark\\nangles up to a sharp point. If we could lay off the\\nlength of a line e could take tangents dire\\nfrom table and obtain angle at moe. It. however, is\\ngenerally bet:-: multiply the tangent by 5 or 10\\nand make an riilarged triangle. If, then, there is a\\nslight error in laying off length of lines it will not\\nmake so much difference with the angle.\\nLet it be required to lay off an angle of 14 r 30 By\\nthe table we find the tangent to 1: e _ B 1 Multiply-\\ning .25861 hj 5 we obtain, in the enlarged triangle.\\n1.29305 as the length of side opposite the angle l-\u00c2\u00b1~\\n30 As we have made the side opposite five times as\\nlarge, we must make the sid^ \u00e2\u0080\u00a2int five times\\nlarge, in order to keep angle the same. Hence. F:_\\n48, draw* the line A B 5 long;; perpendicular to this\\nline at A Iraw ._e:ieA J 1.203 long; now draw the\\nline O B, and the angle A B O will be 14\u00c2\u00b0 30\\nIf special accuracy is required, the tangent can be\\nmultiplied by 10 the line AO will then be 2.5S6 long\\nand the line A B 10 long. Remembering that the\\nacute angles of a right-angle triangle are the comple-\\nments of each other, we subtract 11 3U from 90 and\\nbtain 75 30 as the angle of A O B.\\nThe reader will remember these angles as occurring\\nin Paj.i I.. Chapter IT., and obtained in a different\\nway. A Srimcirele upon the line O B touching the\\nextremitic ill just touch the right angle at\\nA. and the line 3:-: :ur times as long as O A\\nL: it be required to turn a piece 4 long. 1 diam-\\neter at small end. with a taper of 10 r one side with the\\nfcher what will be the diameter of the piece at th^\\nlarge end\\nA section. Fig. 49, through the axis of this piece is\\nTo .^lcuiate l le saine as if we added two riorht-anofle triangles.\\nDiameter of a -_\\nTapering^ B and A B to a straight piecr A A B B 1\\npiece. Fig. 50.\\nwide and 4 long,the a angles 1 an.i B bein_r 5\\nthus making th: sides B and OBI with -ach\\n_\\nther.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0106.jp2"}, "107": {"fulltext": "PROVIDENCE, R. I.\\n91\\n-1^293-+\\nFig. 4S.\\nFig. 49", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0107.jp2"}, "108": {"fulltext": "92\\nBROWN SHARPE MFG. CO.\\nThe tangent of 5\u00c2\u00b0 is .08748, which, multiplied by\\n4 gives 34992 as the length of each line, A O and\\nA O to be added to 1 at the large end. Taking\\ntwice .34992 and adding to 1 we obtain 1.69984 as\\nthe diameter of large end.\\nThis chapter must be thoroughly studied before\\ntaking up the next chapters. If once the memory\\nbecomes confused as to the tangent and sine of an\\nangle, it will take much longer to get righted than it\\nwill to first carefully learn to recognize the tangent\\nof an angle at once.\\nIf one knows what the tangent is, one can tell better\\nthe functions that are not tangents.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0108.jp2"}, "109": {"fulltext": "03\\nCHAPTER II.\\nSINE\u00e2\u0080\u0094 COSINE AND SECANT SOME OF THEIR APPLICATIONS IN\\nMACHINE CONSTRUCTION.\\nSine of Arc\\nand Angle.\\nThe Sine of an arc is the line drawn from one\\nextremity of the arc to the diameter passing through\\nthe other extremity, the line being perpendicular to\\nthe diameter.\\nAnother definition is The sine of an arc is the dis-\\ntance of one extremity of the arc from the diameter,\\nthrough the other extremity.\\nThe sine of an angle is the sine of the arc that\\nmeasures the angle.\\nIn Fig. 50 A C is the sine of the arc B C, and of\\nthe angle BOC. It will be seen that the sine is\\nalways inside of the arc, and can never be longer than\\nthe radius. As the arc ap-\\nproaches 90\u00c2\u00b0, the sine comes\\nnearer to the radius, and at 90\u00c2\u00b0\\nthe sine is equal to 1, or is the\\nradius itself. From the defini-\\ntion of a sine, the side A C,\\nopposite the angle A O C, in\\nany right-angle triangle, is the\\nsine of the angle A O C, when\\nO C is the radius of the arc.\\nHence the rule In any right-angle triangle, the side to find the\\nopposite either acute angle, divided by the hypothe-\\nnuse, is equal to the sine of the angle.\\nThe quotient thus obtained is the length of side\\nopposite the angle when the hypothenuse or radius is\\nunity. The rule should be carefully committed to\\nmemory.\\nFig. 50.\\nSine.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0109.jp2"}, "110": {"fulltext": "94\\nBROWN SHARPE MFG. CO.\\nChord of an Chord is a straight line joining the extremities of\\nan arc, and is twice as long as the sine of half the\\nangle measured by the arc. Thus, in Fig. 51, the\\nchord B C is twice as long as the sine A C.\\nFig. r l\\nLet there be four holes equidistant about a circle\\n3 in diameter Fig. 51 what is the shortest distance\\nbetween two holes This shortest distance is the\\nnnd X th\u00e2\u0084\u00a2chord\u00c2\u00b0 chord A B, which is twice the sine of the angle COB.\\nThe angle A O B is one quarter of the circle, and\\nC O B is one-eighth of the circle. 360\u00c2\u00b0, divided by\\n8=45\u00c2\u00b0, the angle COB. The sine of 45\u00c2\u00b0 is .70710,\\nwhich multiplied by the radius 1.5 gives length C B in tho\\ncircle, 3 in diameter, as 1.0G0G5 Twice this length is\\nthe required distance A B=2.1213\\nWhen a cylindrical piece is to be cut into any num-\\nber of sides, the foregoing operation can be applied to\\nobtain the w T idth of one side. A plane figure bounded\\nPolygon. by straight lines is called a polygon.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0110.jp2"}, "111": {"fulltext": "PROVIDENCE, E. I.\\n95\\nWhen the outside diameter and the number of sides of\\na regular polygon are given, to find the length of\\none of the sides: Divide 3G0\u00c2\u00b0 by twice the number of To fi tlie\\nJ J length of Side.\\nsides midtiply the sine of the quotient by the outer\\ndiameter, and the product will be the length of one of\\nthe sides.\\nMultiplying by the diameter is the same as multi-\\nplying by the radius, and that product again by 2.\\nThe Cosine of an angle is the sine of the comple- Cosine\\nment of the angle.\\nIn Fig. 50, C O D is the complement of the angle\\nA O C the line C E is the sine of COD, and hence\\nis the cosine of B O C. The line O A. is equal to C E.\\nIt is quite as well to remember the cosine as the part\\nof the radius, from the center that is cut off by the\\nsine. Thus the sine A C of the angle A O C cuts off\\nthe cosine O A. The line O A may be called the\\ncosine because it is equal to the cosine C E.\\nIn any right-angle triangle, the side adjacent either\\nacute angle corresponds to the cosine when the\\nhypothenuse is the radius of the arc that measures\\nthe angle hence: Divide the side adjacent the acute To find the\\nangle by the hypothenuse, and the quotient vnll be the\\ncosine of the angle.\\nWhen a cylindrical piece is cut into a polygon of\\nany number of sides, a table of cosines can be used tOg.^iverth of\\nobtain the diameter across the sides. g\u00c2\u00b0 n", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0111.jp2"}, "112": {"fulltext": "96 BROWN \u00c2\u00abs SHABPE MFG. CO.\\nLet a cylinder, 2 diameter. Fig. 53. be cut six-sided:\\nwhat is the diameter across the sides\\nThe angle A O B. at the center, occupied by one of\\nthese sides, is one-sixth of the circle, =60 c The\\ncosine of one-half this angle, 30 is the line C O;\\ntwice this line is the diameter across the sides. The\\ncosine of 30 is .86602. which, multiplied by 2, gives\\n1.7320-4 as the diameter across the sides.\\nOf course, if the radius is other than unity, the cosine\\nshould be multiplied by the radius, and the product\\nagain by 2. in order to get diameter across the sides\\nor what is the same thing, multiply the cosine by the\\nwhole diameter or the diameter across the corners,\\nameter f aoroi ^e ru e f or obtaining the diameter across sides of\\nsides of a Po1 regular polvgon. when the diameter across corners i\\ngiven, will then be: Multiply the cosine of 360 D\\ndivided l y twice the number of sides, ly the diameter\\nacross corners, and the product will be the diameter\\nacross sides.\\nLook at the right-hand column for degrees of the\\ncosine, and at bottom of page for minutes to add to\\nthe degrees.\\nThe Secant of an arc is a straight line drawn from\\nthe center through one end of an arc, and terminated\\nbv a tangent drawn from the other end of the arc.\\nThus, in Fig. 53, the line OB is the secant of the\\nangle. COB.\\nA C B\\nFig. J3.\\nTo cud the In a n v light -angle triangle, divide the hypothenuse\\nSecant. c\\nby the side adj icent either acute angle, and the quo-\\ntient will le the secant of that angle.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0112.jp2"}, "113": {"fulltext": "PROVIDENCE, R. I. 97\\nThat is, if we divide the distance OB by O C, in\\nthe right-angle triangle COB, the quotient will be\\nthe secant of the angle COB.\\nThe secant cannot be less than the radius it in-\\ncreases as the angle increases, and at 90\u00c2\u00b0 the secant is\\ninfinity =co\\nA six-sided piece is to be H across the sides towp^^^J\\nlarge must a blank be turned before cutting the sides JF\u00e2\u0084\u00a2 R)iygon! rS\\nDividing 360\u00c2\u00b0 by twice the number of sides, we have\\n30\u00c2\u00b0, which is the angle COB. The secant of 30\u00c2\u00b0 is\\n1.1547.\\nThe radius of the six-sided piece is .75\\nMultiplying the secant 1.1547 by .75 we obtain the\\nlength of radius of the blank O B multiplying again\\nby 2, we obtain the diameter 1.732\\nHence, in a regular polygon, when the diameter\\nacross sides and the number of sides are given, to find\\ndiameter across corners Multiply the secant of 360\u00c2\u00b0\\ndivided by tvnce the number of sides, by the diameter\\nacross sides, and the product will be the diameter\\nacross corners.\\nIt ^Till be seen that the side taken as a divisor has\\nbeen in each case the side corresponding to the radius\\nof the arc that subtends the angle.\\nThe versed sine of an acute angle is the part of\\nradius outside the sine, or it is the radius minus the\\ncosine. Thus, in Fig. 50, the versed sine of the arc\\nBC is AB. The versed sine is not given in the tables\\nof circular functions when it is wanted for any angle\\nless than 90\u00c2\u00b0 we subtract the cosine of that angle from\\nthe radius 1. Having it for the radius 1, we can\\nmultiply by the radius of any other arc of which we\\nmay wish to know the versed sine.\\nFig. 54 is a sketch of a gear tooth of IP. In\\nmeasuring gear teeth of coarse pitch it is sometimes a\\nconvenience to know the chordal thickness of the\\ntooth, as at ATB, because it may be enough shorter\\nthan the regular tooth-thickness AHB, or t, to require\\nattention. It may be also well to know the versed\\nsine of the angle 1J, or the distance II, in order to tell\\nwhere to measure the chordal thickness.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0113.jp2"}, "114": {"fulltext": "98\\nBliOWN SHAKPK MI G CO.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0114.jp2"}, "115": {"fulltext": "PROVIDENCE, R. I.\\n99\\nOn pages 104 and 105 are tables of data pertaining\\nto chordal thickness of IP. teeth. For any other\\ndiametral pitch, divide the number in the tabic by that\\npitch.\\nGEAR TOOTH CALIPER.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0115.jp2"}, "116": {"fulltext": "100\\nBROWN SHAEPE MFG. CO.\\nCHORDAL THICKNESS OF TEETH FOR GEARS AND CUTTERS,\\nON A BASTS OF 1 DIAMETRAL PITCH.\\nT D sin. ff\\nH R (1\u00e2\u0080\u0094 cos.\\nN Number of teeth in gears.\\nT Chorclal thickness of Tooth.\\nH Height of Arc.\\nD Pitch Diameter.\\nR Pitch Radius.\\nyS f 90\u00c2\u00b0 divided by the number of teeth.\\nNote. When the tooth of a gear is measured, add the height of arc to (S); and\\nwhen gear cutter is measured subtract the height of arc from (S f).\\nInvolute.\\nCutter.\\nT\\nH\\n.0047\\nCorrected\\nS+f forCutt.\\nCorrected\\nS for Gear.\\nNo.l 135 T-\\n-1P\\n1.5707\\n1.1524\\n1.0047\\n2 55 T-\\n-IP\\n1.5706\\n.0112\\n1.1459\\n1.0112\\nu g__ 35T-\\n-IP\\n1.5702\\n.0176\\n1.1395\\n1.0176\\n4\u00e2\u0080\u0094 26T-\\n-IP\\n1.5698\\n.0237\\n1.1334\\n1.0237\\n\u00c2\u00ab5\u00e2\u0080\u0094 21 T-\\n-IP\\n1.5694\\n.0294\\n1.1277\\n1.0294\\n6\u00e2\u0080\u0094 17 T-\\n-IP\\n1.5686\\n.0362\\n1.1209\\n1.0362\\n7\u00e2\u0080\u0094 14 T-\\n-IP\\n1.5675\\n.0440\\n1.1131\\n1.0440\\n\u00c2\u00ab8\u00e2\u0080\u0094 12 T-\\n-IP\\n1.5663\\n.0514\\n1.1057\\n1.0514\\n11 T-\\n-IP\\n1.5654\\n.0559\\n1.1011\\n1.0559\\n10T-\\n-IP\\n1.5643\\n.0616\\n1.0955\\n1.0616\\n9T-\\n-IP\\n1.5628 .0684\\n1.0887\\n1.0684\\n8T-\\n-IP\\n1.5607\\n.0769\\n1.0802\\n1.0769", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0116.jp2"}, "117": {"fulltext": "providence, r. i.\\nEpicycloidal.\\n101\\nCutter.\\nT\\nH\\nCorrected\\nS f f or Cutt.\\nCorrected\\nS for Gear.\\nA\\n12T-\\n-1P\\n1.5663\\n.0514\\n1.1057\\n1.0514\\nB\\n13 T-\\n-IP\\n1.5670\\n.0474\\n1.1097\\n1.0474\\nC\\n14 T-\\n-IP\\n1.5675\\n.0440\\n1.1131\\n1.0440\\nD\\n15T-\\n-IP\\n1.5679\\n.0411\\n1.1160\\n1.0411\\nE\\n16 T-\\n-IP\\n1.5683\\n.0385\\n1.1186\\n1.0385\\nF\\n17 T-\\n-IP\\n1.5686\\n.0362\\n1.1209\\n1.0362\\nG\\n18 T-\\n-IP\\n1.5688\\n.0342\\n1.1229\\n1.0342\\nH-\\n19T-\\n-IP\\n1.5690\\n.0324\\n1.1247\\n1.0324\\nI\\n20 T-\\n-IP\\n1.5692\\n.0308\\n1.1263\\n1.0308\\nJ\\n21 T-\\n-IP\\n1.5694\\n.0294\\n1.1277\\n1.0294\\nK\\n23T-\\n-IP\\n1.5696\\n.0268\\n1.1303\\n1.0268\\nL\\n25 T-\\n-IP\\n1.5698\\n.0247\\n1.1324\\n1.0247\\nM\\n27T-\\n-IP\\n1.5699\\n.0228\\n1.1343\\n1.0228\\nN-\\n30 T-\\n-IP\\n1.5701\\n.0208\\n1.1363\\n1.0208\\n34 T-\\n-IP\\n1.5703\\n.0181\\n1.1390\\n1.0181\\nP\\n38 T-\\n-IP\\n1.5703\\n.0162\\n1.1409\\n1.0162\\nQ\\n43 T-\\n-IP\\n1.5705\\n.0143\\n1.1428\\n1.0143\\nR\\n50 T-\\n-IP\\n1.5705\\n.0123\\n1.1448\\n1.0123\\nS\\n60T-\\n-IP\\n1.5706\\n.0102\\n1.1469\\n1.0102\\nT\\n75T-\\n-IP\\n1.5707\\n.0083\\n1.1488\\n1.0083\\nU-\\n100T-\\n-IP\\n1.5707\\n.0060\\n1.1511\\n1.0060\\nV-\\n150T-\\n-IP\\n1.5707\\n.0045\\n1.1526\\n1.0045\\nw\\n250T-\\n-IP\\n1.5708\\n.0025\\n1.1546\\n1.0025\\nSpecial.\\nNo. Teeth.\\n9T IP\\n10 T IP\\n11T IP\\n1.5628\\n1.5643\\n1.5654\\nH\\n.0684\\n.0616\\n.0559\\nCorrected\\nS f for Cutt.\\n1.0887\\n1.0955\\n1.1012\\nCorrected\\nS for Gear.\\n1.0684\\n1.0616\\n1.0559", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0117.jp2"}, "118": {"fulltext": "103\\nCHAPTER III.\\nAPPLICATION OF CIRCULAR FUNCTIONS\u00e2\u0080\u0094 WHOLE DIAMETER OF\\nBEVEL GEAR BLANKS\u00e2\u0080\u0094 ANGLES OF BEVEL GEAR BLANKS.\\nThe rules given in this chapter apply onl} T to bevel\\ngears having the center angle c O i not greater than 90\u00c2\u00b0.\\nTo avoid confusion we will illustrate one gear only.\\nThe same rules apply to all sizes of bevel gears. Fig.\\n55 is the outline of a pinion 4 P, 20 teeth, to mesh with\\na gear 28 teeth, shafts at right angles. For making\\nsketch, of bevel gears see Chapter IX.. Part I.\\nIn Fig. 55, the line O m m is continued to the line\\na b. The angle c O I that the cone pitch-line makes\\nwith the center line may be called the center angle.\\nAngle of The center angle c O i is equal to the angle of edge\\nc i c. c i is the side opposite the center angle c O\\ni, and c O is the side adjacent the center angle, c\\ni 2.5 O 3.5 Dividing 2.5 by 3.5 we\\nobtain .71128 as the tangent of c O i. In the table\\nwe find .71329 to be the nearest tangent, the corre-\\nsponding angle being 35\u00c2\u00b0 30 35 then, is the center\\nangle c O i and the angle of edge c i n, very nearly.\\nWhen the axes of bevel gears are at right angles the\\nangle of edge of one gear is the complement of angle\\nof edge of the other gear-. Subtracting, then, 35^\u00c2\u00b0\\nfrom 90\u00c2\u00b0 we obtain 54^\u00c2\u00b0 as the angle of edge of gear\\n28 teeth, to mesh with gear 20 teeth, Fig. 55. from which we\\nhave the rule for obtaining centre angles when the axes of\\ngears are at right angles.\\nDivide the radius of the pinion by the radius of the gear\\nand the quotient will be the tangent of centre angle of the\\npinion.\\nXow subtract this centre ande from 90 deer, and we have\\nthe centre angle of the gear.\\nThe same result is obtained by dividing the number of\\nteeth in the pinion by the number of teeth in the gear the\\nquotient is the tangent of the centre angle.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0118.jp2"}, "119": {"fulltext": "PROVIDENCE, R. I.\\n103\\nFig. 55.\\nBEVEL GEAR DIAGRAM,", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0119.jp2"}, "120": {"fulltext": "104 BROWN SHARPE MFG. CO.\\nAngle of Pace. To obtain angle of face O m c\\\\ the distance c O\\nbecomes the side opposite and the distance m c is\\nthe side adjacent.\\nThe distance c O is 3.5 the radius of the 28 tooth\\nbevel gear. The distance c m is by measurement\\n2.82\\nDividing 3.5 by 2.82 we obtain 1.2411 for tangent\\nof angle of face O m c. The nearest tangent in the\\ntable is 1.2422 and the corresponding angle is 51\u00c2\u00b0 10\\nTo obtain cutting angle c O n we divide the distance\\nc n by c O. By measurement c n is 2.2 Divid-\\ning 2.2 by 3.5 we obtain .62857 for tangent of cutting\\nangle. The nearest corresponding angle in the table\\nis 32\u00c2\u00b010\\nThe largest pitch diameter, kj, of a bevel gear, as in\\nFig. 56, is known the same as the pitch diameter of\\nany spur gear. Now, if we know the distance b o or\\nits equal a q, we can obtain the whole diameter of\\nbevel gear blank by adding twice the distance b o to\\nthe largest pitch diameter.\\ncr?ment ter Fig Twice the distance b o, or what is the same thing,\\n5b the sum of a q and b o is called the diameter incre-\\nment, because it is the amount by Avhich we increase\\nthe largest pitch diameter to obtain the whole or out-\\nside diameter of bevel gear blanks. The distance b o\\ncan be calculated without measuring the diagram.\\nThe angle b o j is equal to the angle of edge.\\nThe angle of edge, it will be remembered, is the\\nangle formed by outer edge of blank or ends of teeth\\nwith the end of hub or a plane perpendicular to the\\naxis of gear.\\nThe distance b o is equal to the cosine of angle of\\nedge, multiplied by the distance j o. The distance j o\\nis the addendum, as in previous chapters s).\\nHence the rule for obtaining the diameter increment\\nof any bevel gear: Multiply the cosine of angle of\\nedge by the working depth of teeth (D and the\\nproduct will be the diameter increment.\\nBy the method given on page 102 we find the angle\\nof edge of gear (Fig. 56) is 56\u00c2\u00b0 20 The cosine\\nof 56\u00c2\u00b0 20\u00c2\u00b0 is .55436, which, multiplied by or the\\ne? e U r Side Diam depth of the 3 P gear, gives the diameter increment of\\nthe bevel gear 18 teeth, 3 P meshing with pinion of 12", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0120.jp2"}, "121": {"fulltext": "PROVIDENCE, R. I.\\n105", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0121.jp2"}, "122": {"fulltext": "10G BROWN 6 SHABPE MFG. CO.\\nteeth, j of .55436 369. I or .37 nearly). Adding\\nthe diameter increment. .37 to the largest pitch\\ndiameter of gear, 6 we have 6.37 as the outside\\ndiameter.\\nIn the same manner, the distance c J is half the\\ndiameter increment of the pinion. The angle c J J: is\\nequal to the center angle of pinion, and when axes are\\nat right angles is the complement of center angle of\\ngear. The center angle of pinion is 33\u00c2\u00b0 40 The\\ncosine, multiplied by the working depth, gives .555\\nfor diameter increment of pinion, and we have 4-555\\nfor outside diameter of pinion.\\nIn turning bevel gear blanks, it is sufficiently accu-\\nrate to make the diameter to the nearest hundredth of\\nan inch.\\nAngle in-re The small angle o is called the angle increment.\\nmerit. f u\\nWhen shafts are at right angles the face angle of one\\ngear i 3 equal to the center angle of the other gear,\\nminus the angle increment.\\nThus the angle of face of gear Fig. 56) is less than\\nthe center angle D k\\\\ or its equal 0./ k by the angle\\no Oj. That is. subtracting o Oj from Oj A\\\\ the re-\\nmainder will be the angle of face of gear.\\nSubtracting the angle increment from the center\\nangle of gear, the remainder will be the cutting\\nangle.\\nThe angle increment can be obtained by dividing\\noj. the side opposite, by Oj. the side adjacent, thus\\nfinding the tangent as usual.\\nThe length of cone-pitch line from the common\\ncenter. t) j. can be found, without measuring dia-\\ngram, by multiplying the secant of angle 0,/ A or the\\ncenter angle of inion, by the radius of largest pitch\\ndiameter of gear.\\nThe secant of angle Oj k, 33 40 is 1.2015, which.\\nmultiplied by 3 the radius of gear, gives 3.6045 as\\nthe length of line\\nDividing oj by Oj, we have for tangent .092-4. and\\nfor angle increment 5 20\\nThe angle increment can also be obtained by the\\nfollowing rule", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0124.jp2"}, "123": {"fulltext": "PROVIDENCE, E. I. 107\\nDivide the sine of center angle by half the nun*\\nber of teeth, and the quotient will be the tangent of\\nincrement angle.\\nSubtracting the angle increment from, center angles\\nof gear and pinion, we have respectively\\nCutting angle of gear, 51\u00c2\u00b0.\\nCutting angle of pinion, 28\u00c2\u00b0 20\\n.Remembering that when the shafts are at right\\nangles, the face angle of a gear is equal to the cutting\\nangle of its mate (Chapter X. part 1), we have:\\nFace angle of gear, 28\u00c2\u00b0 20\\nFace angle of pinion, 51\u00c2\u00b0.\\nIt will be seen that both the whole diameter and the\\nangles of bevel gears can be obtained without making\\na diagram. Mr. George B. Grant has made a table of\\ndifferent pairs of gears from 1 to 1 up to 10 to 1, con-\\ntaining diameter increments, angle increments and\\ncentre angles, which is published in his Treatise on\\nGears. Formulas in Gearing, published by us, also\\ncontains extensive tables for bevel gearing. We have\\nadopted the terms diameter increment, angle incre-\\nment, and centre angle from him. lie uses the\\nterm (i back angle for what we have called angle of\\nedge, only he measures the angle from the axis of the\\ngear, instead of from the side of the gear, or from the t Tol a y\u00c2\u00b0 u n\\nAngle oy the\\nend of hub, as We have done that is, his back angle Sine.\\nis the complement of our angle of edge.\\nIn laying out angles, the following method may be\\n-t i j. 52.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0125.jp2"}, "124": {"fulltext": "108 BROWN SHARPE MFG. CO.\\npreferred, as it does awny with the necessity of making\\naright angle: Draw a circle, ABO (Pig. o t), ten\\ninches in diameter. Set the dividers to ten times the\\nsine of the required angle, and point off this distance\\nin the circumference as at A B. From any point O in\\nthe circumference, draw the lines O A and O B. The\\nangle AOB13 the angle required. Thus, let the re-\\nquired angle be 12\u00c2\u00b0. The sine of 12\u00c2\u00b0 is .20791, which,\\nmultiplied by 10, gives 2.0791 or Zj$q- nearly, for\\nthe distance A B.\\nAny diameter of circle can be taken if we multiply\\nthe sine by the diameter, but 10 is very convenient,\\nas- all we have to do with the sine is to move the\\ndecimal point one place to the right.\\nIf either of the lines pass through the centre, then the\\ntwo lines which clo not pass through the centre will form a\\nright angle. Thus, if O B passes through the centre then\\nthe two lines A B and A O will form a right angle at A.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0126.jp2"}, "125": {"fulltext": "109\\nCHAPTER IV.\\nSPIRAL GEARS CALCULATIOHS FOR PITCH OF SPIRALS.\\nWhen the teeth of a gear are cut, not in a straight s P iral Gear\\npath, like a spur gear, but in a helical or screw- like\\npath, the gear is called, t actinically, a twisted or screw\\ngear, but more generally among mechanics, a spiral\\ngear. A distinction is sometimes made between a\\nscrew gear and a twisted gear. In twisted gears the\\npitch surfaces roll upon each other, exactly like spur\\ngears, the axes being parallel, the same as in Fig. 1,\\nPart I. In screw gears there is an end movement,\\nor slipping of the pitch surfaces upon each other, the\\naxes not being parallel. In screw gearing the action\\nis analogous to a screw and nut, one gear driving\\nanother by the end movement of its tooth path. This\\nis readily seen in the case of a worm and worm-wheel,\\nwhen the axes are at right angles, as the movement of\\nwheel is then wholly due to the end movement of\\nworm thread. But, as we make the axes of gears more\\nnearly parallel, they may still be screw gears, but the\\ndistinction is not so readily seen.\\nWe can have two gears that are alike run together,\\nwith their axes at right angles, as at A B, Fig. 59.\\nThe same gear may be used in a train of screw gears\\nor in a train of twisted gears. Thus, B, as it relates to\\nA, may be called a screw gear; but in connection with\\nC, the same gear, B, may be called a twisted gear.\\nThese distinctions are not usually made, and we call\\nall helical or screw-like gears made on the Universal\\nMilling Machine spiral gears.\\nWhen two extarnal spiral gears run together, with Direction of.\\ntheir axes parallel, the teeth of the gears must have erence to Axes.\\nopposite hand spirals.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0127.jp2"}, "126": {"fulltext": "110\\nBROWN SHAKPE MFG. CO.\\nThus, in Fig. 59 the gear B has right hand spiral\\nteeth, and the gear C has left hand spiral teeth. When\\nthe axes of two spiral gears are at right angles, both\\ngears must have the same hand spiral teeth. A and\\nB, Fig. 59. have right hand spiral teeth. If both gears\\nA and B had left hand spiral teeth, the relative direc-\\ntion in which they turn would be reversed,\\nspiral Lead. ^he spiral lead or lead of spiral is the distance the\\nspiial advances in one turn. A cylinder or gear cut\\nwith spiral grooves is morel) a screw of coarse pitch or\\nlong lead that is, a spiral is a coarse lead screw, and\\na screw is a fine lead spiral.\\nSince the introduction and extensive use of the\\nUniversal Milling Machine, it lias become customary\\nto call any screw cut in the milling machine a spiral.\\nThe spiral lead is given as so many inches to one\\nturn. Thus, a cylinder having a spiral groove that\\nadvances six inches to one turn, is said to have a six\\ninch spiral.\\nIn screws the pitch is often given as so many turns\\nto one inch. Thus, a screw of -J- lead is said to be 2\\nturns to the inch. The reciprocal expression is not\\nmuch used with spirals. For example, it would not\\nbe convenient to speak of a spiral of 6 lead, as -J- turns\\nto one inch.\\nThe calculations for spirals are made from the func-\\ntions of a right angle triangle.\\nExample, Cut from paper a right angle triangle, one side of\\nshowing the r l\\nnature of a He- the right angle 6 long, and the other side of the\\nlix or Spiral. c c\\nright angle 2 Make a cylinder 6 in circumference.\\nIt will be remembered (Part I., Chapter II.) that the\\ncircumference of a cylinder, multiplied by .3183, equals\\nthe diameter 6 X .3183=1.9098 Wrap the paper\\ntriangle around the cylinder, letting the 2 side be\\nparallel to the axis, the 6 side perj)endicular to the\\naxis and reaching around the cylinder. The hypoth-\\neneuse now forms a helix or screw-like line, called\\na spiral. Fasten the paper triangle thus wrapped\\naround. See Fig. GO.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0128.jp2"}, "127": {"fulltext": "PROVIDENCE, R. I.\\nIll\\nFIG. 58 -RACKS AND GEARS.\\nFig. 59.-SPIRAL GEARING.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0129.jp2"}, "128": {"fulltext": "11*2\\ni\\nz vy-\\nWig SO\\nIf we dow turn this cylinder ABCD one lorn in\\nthe direction of the arrow, the spiral will adva nee from\\nto E. This advai.ee is the lead of the sp\\nThe angle E F, which the spiral makes with the\\naxis E 3 is the angle of the spiral. This ;mgle isfonnd\\nas in Chapter I. The circumference of the cylinder\\n:;s: :n::s t side opposite the angle. T pilch\\nof the spiral corresponds to the si seat the angle.\\nHence the role for angle of spiral:\\nelating tie Divide tlte circumference of the\\nparts of a spi -fry f] te numo er of inches of 5 the\\nquotient will he the tangent of angle\\nWhen the angk fspi and .inference are given,\\nto find the lead\\nDivU/e the circrr/i/ trence by the tangent of angle,\\nthe quotient will be the lead of the spi\\nWhen the angle of spiral and the lead or pitch of spiral\\nare sriven, to find, the ci rence\\nMuVij the t gent of angle by the lead, and the\\nproduct will be t ce.\\nWhen applying calculations to spiral gears the angle\\nis reckoned at the pitch circumference and not at the\\nouter or addendum circle.\\nIt will be seen that when two spirals of different\\ndiameters lave the b the spiral of less diame-\\nter will have the smaller angle. Thus in Fig. GO if ihe\\npaper triangle bid teen 4 lonsr ii s: of C the diam-\\neter of t e cylinder would have been 1.27 .d the\\nangle of the spiral would have been _ legrees.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0130.jp2"}, "129": {"fulltext": "113\\nCHAPTER V.\\nEXAMPLES IN CALCULATION OF THE LEAD OF SPIRAL\u00e2\u0080\u0094 ANGLE OF\\nSPIRAL\u00e2\u0080\u0094 CIRCUMFERENCE OF SPIRAL GEARS\u00e2\u0080\u0094\\nA FEW HINTS ON CUTTING.\\nIt will be seen that the rules for calculating the cir-\\ncumference of spiral gears, angle and the lead of spiral\\nare the same as in Chapter I., for the tangent and angle\\nof a right angle triangle. In Chapter IV., the word\\ncircumference is substituted for side opposite,\\nand the words lead of spiral are substituted for\\nside adjacent.\\nWhen two spiral gears are in mesh the angle of r 1 ff^ e f JjJ\\nspiral should be the same in one gear as in the other, *f C Q h l f t An s le\\nin order to have the shafts parallel and the teeth work\\nproperly together. When two gears both have right\\nhand spiral teeth, or both have left hand spiral teeth,\\nthe angle of their shafts will be equal to the sum of\\nthe angles of their spirals. But when two gears have\\ndifferent hand spirals the angle of then* shafts will be\\nequal to the difference of their angles of spirals.\\nThus, in Fig. 59 the gears A and B both have right\\nhand spirals. The angle of both spirals is 45\u00c2\u00b0, their\\nsum is 90\u00c2\u00b0, or their axes are at right angles. But C\\nhas a left hand spiral of 45\u00c2\u00b0. Hence, as the difference\\nbetween angles of spirals of B and C is 0, their axes\\nare parallel.\\nIf two 45\u00c2\u00b0 gears of the same diameter have the same\\nnumber of teeth the lead of the spiral will be alike in\\nboth gears: if one gear has more teeth than the other\\nthe lead of spiral in the larger gear should be longer\\nin the same ratio. Thus, if one of these gears has 50\\nteeth and the other has 25 teeth, the lead of spiral ra s ea d f ^lil^\\nin the 50 tooth gear should be twice as long as that of ent diameters.\\nthe 25 tooth gear. Of course, the diameter of pitch", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0131.jp2"}, "130": {"fulltext": "114: KBOWH SHARPE MFG. CD.\\ncircle should s a large in the 50 tooth as in the\\n25 tooth gear.\\nIn spirals where the angle is 45 the circumference\\nis the same as the spiral lead, because the tangent of\\n45\u00c2\u00b0 is 1.\\noiS2S? on Sometimes the circumference is varied to suit a pitch\\nv, 1 rPTl TO TPTPT1 Of* -I\\nSi nraL that can be cut on the machine and retain the angle\\nrequired. This would apply to catting rolls for mak-\\ning diamond-shaped impressions where the diameter\\nof the roll is not a matter of importance.\\nWhen two gears are to run together in a given\\nvelocity ratio, it is well first to select spirals that the\\nmachine will cut of the same ratio, and calculate the\\nnumbers of teeth and angle to correspond. This will\\noften save considerable time in fisfurinof.\\nThe calculations for spiral gears present no special\\ndifficulties, but sometimes a little ingenuity is required\\nto make work conform n the machine and to such\\ncutters as we may have in stock.\\nLet it be required to make two spiral gears to run\\nwith a ratio of 4 to 1, the distance between centres\\nbe 3.125 m the axes to be parallel.\\nBy rule given in Chapter XII.. Part I., we fiud the\\ndiameters of tch rclea will be 5 and 1\\\\ Let us\\ntake a spiral of 4S lead for the large gear, and a\\nspiral of 12 lead for the small gear. The circumfer-\\nence of the 5 pitch circle is 15.70796 Dividing\\nthe circumference by the lend of the spiral, we have\\n1 5 4g 7 9 6 =-32724 for tangent of angle of spiral. In\\nthe ta 3 the rest angle to tangent, .32724 is 1S\u00c2\u00b0 1 1\\nAs before stated, the angle of the teeth in the small\\ngear will be the same as the angle of teeth or spiral in\\nthe large gear.\\ninAiigtesafctop Now, this rule gives the angle at the pitch surface\\nspSjlGr^oave? on I^P 0U looking at a small screw of coarse pitch,\\nit will be seen that the angle at bottom of the thread\\nis not so great as the angle at top of thread; that is,\\nthe thread at bottom is nearer parallel to the centre\\nline than that at the top.\\nThis will be seen in Fig. 61, where A is the centre\\nline; i f shows lirectiou of bottom of thread, and d g", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0132.jp2"}, "131": {"fulltext": "PROVIDENCE, K. I.\\n115\\nshows direction of top of thread. The angle Afb is\\nless than the angle A g d. The difference of angle\\nbeing dne to the warped nature of a screw thread.\\nA cylinder 2 diameter is to have spiral grooves 20\u00c2\u00b0 Example in\\nJ ib calculation of\\nwith the centre line of cylinder; what will be the lead Leadof spiral.\\nof spiral? The circumference is 6.2832 The tan-\\ngent of 20\u00c2\u00b0 is .36397. Dividing the circumference by\\nthe tangent of angle, we obtain f 7 17.26 for\\nlead of spiral.\\nFig. 01.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0133.jp2"}, "132": {"fulltext": "116\\nCHAPTER VI.\\nNORMAL PITCH OF SPIRAL GEARS\u00e2\u0080\u0094 CURVATURE OF PITCH\\nSURFACE\u00e2\u0080\u0094 FORM OF CUTTERS.\\nNormal to a Normal to a curve is a line perpendicular to the\\ntangent at the point of tangency.\\nIn Fig. 62, the line B C is tangent to the arc DEF,\\nand the line A E O, being perpendicular to the tan-\\ngent at E the point of tangency, is a normal to the\\narc.\\nFig. 63 is a representation of the pitch surface of a\\nspiral gear. A D C is the circular pitch, as in Part\\nI. A D C is the same circular pitch seen upon the\\nperiphery of a wheel. Let A D be a tooth D and a\\nspace. Now, to cut this space D C, the path of cut-\\nting is along the dotted line a b. By mere inspection,\\nwe can see that the shortest distance between two\\nteeth along the pitch surface is not the distance\\nADC.\\nLet the line A E B be perpendicular to the sides of\\nteeth. upon the pitch surface. A continuation of this\\nline, perpendicular to all the teeth, is called the\\nNormal Helix. The line A E B, reaching over a\\ntooth and a space along the normal helix, is called the\\nNormal Pitch.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0134.jp2"}, "133": {"fulltext": "PUOVIDENCK, K. I.\\nn\\nFig. 63.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0135.jp2"}, "134": {"fulltext": "IIS -:~z\\nRoma: pitch. Xhe Xormal Fitch of a spiral gear is then: _\\nshortest distance betwee the centers of\\n:r.. u 1 p ;~h-\\\\\\nIn spur gears the normal pitch and circular pitch\\nare alike. In the rack D D. Fig 8, the linear pitch\\nand normal pitch are alike.\\nltter for From the foregoing it will be seen that, if we should\\nat the Bp:v:e D C with ::er 3 the thickness of which\\nat the pitch line is equal to one-half the circular pitch.\\na? in spur ^h\u00e2\u0080\u0094is. the space would be too ide, and\\nthe teeth would be too thin. Hence, spiral gears\\nshould be it with thinner cutters than spur gears of\\nthe same circular pitch.\\nThe aztgie CAB is equal to the angle of the spiral.\\nThe line AEBc rrresponds to the cosine of the angle\\nC A B. Hence the rule Mu It y lythe gle\\nmlTpPeL^ \u00c2\u00b0f *P* ra ty the t far pitch, t d he product will be\\nat pitch. One-half the normal pitch is the\\nproper thickness of cutter at the pitch line.\\nIf the normal pitch and the angle are kno~ L vide\\ne normal pitch by the cost f he angle and the quo-\\nbe Hie lint itch.\\nThis may be required iu a case of a spiral ion i on-\\nning in a rack. The perpendicular to the side of the\\nrack is taken as the line from which to calculate angle\\nof teeth. That is, this line would correspond to the\\naxial line in a spiral gear; and, when the a: ::s of the\\ngeai is at right angles t t .:e rack, the angle of\\n:e ;\u00e2\u0096\u00a0:_ .h ..e s:de of the ri.ck is rai r ysubtruit-\\niug this angle from 90\u00c2\u00b0.\\nThe angle of the r teeth with the side of the\\nrack can also be ained by remembering that the\\nsin of the angle of spiral is the sine of :_e angle of\\nhe teeth with the s i :1 e of the rack.\\nThe addendum and working dep:h of tooth sh:\\ncorrespond to the al pitch, and not to the circular\\ntch. Thus, if the normal pitch is 12 di the\\naddendum should be -jV the t hkness .13 hi\\non. The di of pitch circle of a spiral gear is\\ncalculated fr m the etral pitch. Thus a gear of\\n30 teeth 10 P w raid .itch diametr", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0136.jp2"}, "135": {"fulltext": "Normal Pitch\\nvaries.\\nPROVIDENCE, R. I. 119\\nBut il the normal pitch is 12 diametral pitch, the\\nblank will be 3 T y diameter instead of 3 r y.\\nIt is evident that the normal pitch varies with the\\nangle of spiral. The cutter should be for the normal\\npitch. In designing spiral gears, it is well first to look\\nover list of cutters on hand, and see whether there are\\ncutters to which the gears can be made to conform.\\nThis may avoid the necessity of getting a new cutter,\\nor of changing both drawing and gears after they are\\nunder way. To do this, the problem is worked the\\nreverse of the foregoing; that is:\\nFirst calculate to the next finer pitch cutter than gl e ?sp e irai\\nwould be required for the diametral pitch. cutter^^lven\\nLet us take, for example, two gears 10 pitch and 30\\nteeth, spiral and axes parallel. Let the next finer cut-\\nter be for 12 pitch gears. The first thing is to find the\\nangle that will make the normal pitch .2G18 when the\\ncircular pitch is .3142 See table of tooth parts.\\nThis means (Fig. 63) that the line A D C will be .3142\\nwhen A E B is .2018 Dividing .2015 by .3142 (see\\nOlnip. IV.), we obtain the cosine of the angle CAB,\\nwhich is also the angle of the spiral, ;\u00c2\u00a7f J\u00c2\u00a7=.833.\\nThe same quotient comes by dividing 10 by 12,\\nyf =.833 that is, divide one pitch by the other, the\\nlarger number being the divisor. Looking in the table,\\nwe find the angle corresponding to the cosine .833 is\\n33\u00c2\u00b0 30 We now want to find the pitch of spiral that\\nwill give angle of 33 1-\u00c2\u00b0 on the pitch sui face of the wheel,\\n3 diameter. Dividing the circumference by the tan-\\ngent of angle, we obtain the pitch of spiral (see Chap.\\nV.) The circumference is 9.4248 The tangent of\\n33\u00c2\u00b0 30 is .06188, \u00c2\u00a3;f 1^= 14.23 and we have for\\nour spiral 14.23 lead.\\nWhen the machine is not arranged for the exact When exact\\np ii Pitch cannot b\u00c2\u00a9\\npitch of spn*al wanted, it is generally well enough to cut.\\ntake the next nearest spiral. A half of an inch more\\nor less in a spiral 10 pitch or more would hardly be\\nnoticed in angle of teeth. It is generally better to\\ntake the next longer spiral and cut enough deeper to\\nbring center distances right. When two gears of the\\nsame size are in mesh with their axes parallel, a change", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0137.jp2"}, "136": {"fulltext": "120\\nBROWS .s SHAKPE MFG. CO.\\nof angle of teeth or spiral makes no difference in the\\ncorrect meshing f the teeth.\\nGears g nt when gears of different size are in mesh, clue\\nof Different\\nsizes of Mesh. re g ar j must be had to the spirals being in pitch, pro-\\nportional to then angular velocities (see Chapter V.\\nWe come now to the curvature of cutters for spiral\\ngears; that is, then shape as to whether a cutter is\\nmade to cut 12 teeth or 100 teeth. A cutter that is light,\\nShape of cut- to cut a spur gear 3 diameter, may not be right for a\\nspiral gear 3 diameter. To find the curvature of\\ncutter, fit a templet to the blank along the line of the\\nnormal helix, as A E B. letting the templet reach over\\nabout two or three normal pitches. The curvature of\\nthis templet will be nearer a straight line than an arc\\nof the addendum circle. Xow find the diameter of a\\ncircle that will approximately fit this templet, and con-\\neider this circle as the addendum circle of a gear for\\nwhich we are to s lect a cutter, reckoning the gear as\\nof a pitch the fame as the normal pitch.\\nFig 64\\nThus, in Fig. 64. suppose the templet fits a circle\\n3^- diameter, if the normal pitch is 12 to inch, dia-\\nmetral, the cutter required is for 12 P and 40 teeth.\\nThe curvature of the templet will not be quite circular,\\nbut is sufficiently near for practical purposes. Strictly,", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0138.jp2"}, "137": {"fulltext": "PROVIDENCE, R. I. 121\\na, flat templet cannot be made to coincide with the\\nnormal helix for any distance whatever, but any greater\\nrefinement than we have suggested can hardly be car-\\nried out in a workshop.\\nThis applies more to an end cutter, for a disk cutter\\nmay have the right shape for a tooth space and still\\nround off the teeth too much on account of the warped\\nnature of the teeth.\\nThe difference between normal pitch and linear or\\ncircular pitch is plainly seen in Figs 58 and 59.\\nThe rack T D, Fig. 58, is of regular form, the depth\\nof teeth being of the circular pitch, nearly (.6866 of\\nthe pitch, accurately). If a section of a tooth in either\\nof the gears be made square across the tooth, that is a\\nnormal section the depth of the tooth will have the\\nsame relation to the thickness of the tooth as in the\\nrack just named.\\nBut the teeth of spiral gears, looking at them upon\\nthe side of the gears, are thicker in proportion to their\\ndepth, as in Fig. 59 This difference is seen between\\nthe teeth of the two racks D D and E E, Fig. 58. In\\nthe rack D D we have 20 teeth, w T hile in the rack E E\\nwe have but 14 teeth yet each rack will run with each\\nof the spiral gears A, B or C, Fig. 59, but at different\\nangles.\\nThe teeth of one rack will accurately fit the teeth of\\nthe other rack face to face, but the sides of one rack\\nwill then be at an angle of 45\u00c2\u00b0 with the sides of the\\nother rack. At F is a guide for holding a rack m mesh\\nwith a gear.\\nThe reason the racks will each run with either of the\\nthree gears is because all the gears and racks have the\\nsame normal pitch. When the spiral gears are to run\\ntogether they must both have the same normal pitch.\\nHence, two spiral gears may run correctly together\\nthough the circular pitch of one gear is not like the\\ncircular pitch of the other gear.\\nIf a rack is to run at any angle other than 90\u00c2\u00b0 with\\nthe axis of the gear it is well to determine the data\\nfrom a diagram, as it is very difficult to figure the\\nangles and sizes of the teeth without a sketch or\\ndiagram.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0139.jp2"}, "138": {"fulltext": "122\\nCHAPTER VII.\\nCUTTING SPIRAL GEARS IN A UNIVERSAL MILLING MACHINE.\\nA rotary disk cutter is generally preferable to a shank\\ncotter or end mill on account of cutting faster and hold-\\ning its shape longer. In catting spiral grooves, it is\\nsometimes necessary to use an end mill on account of\\nthe warped character of the grooves, but it is very Sel-\\ndom necessary to use an end mill in cutting spiral gears.\\nProving the Before catting into a blank it is well to make a slight\\nSetting f the c\\nMachine. trace of the spiral with the cutter, after the change\\nsears are in place, to see whether the gears are correct.\\nIf the material of the gear blanks is quite expensive, it\\nis a safe plan to make trial blanks of cast iron in order\\nto prove the setting of the machine, before cutting into\\nthe expensive material.\\nThe cutting of spiral gears may develop some curi-\\nous facts to one that has not studied warped surfaces.\\nThe gears. Fig. 59, were cut with a planing tool in a\\nshaper, the spiral gear mechanism of a Universal Mill-\\ning Machine having been faste. el upon the shaper.\\nThe tool was of the same form as the spaces in tl\\nD D, Fig. 58. Ail spiral gears of the same pitch can be\\ncut in this manner with one tool. The nature of tl is\\ncutting operation can be understood from a considera-\\ntion of the meshing of straight side rack teeth with a\\nspiral gear, as in Fig. 58. Spiral gears that run cor-\\nrectly with a rack, as in Fig. 5S, will run correctly\\nwith each other when tin. ir axes are parallel, as at B C,\\nFig. 59 j but it is not considered that they are quite\\ncorrect, theoretically, to run together when the gears\\nhave the same hand spiral, and their axes are at right", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0140.jp2"}, "139": {"fulltext": "PKOVIDENCE, II. I.\\n12\\nQ\\nB\\nFig. 65\\nC\\nk\\nrrn-\\ntti\\n_Fi 7. ec", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0141.jp2"}, "140": {"fulltext": ":u\\n31. szi_?.ji ::e\\nangles, as A B Fig. thoush tbev will ran we\\n.It. Tie c^.^::.. c: c:::::g s i-\\na planer tool is sometimes caHlodplanimg the tee:\\nlog is an accurate way of shaping teeth th\\nwith rack teeth and for gears on parallel si\\nLr: .::lh:.si.7r j ez; i c s i: 1 ti\\nc. .re p ner i./o-6=, his r.:: .eer. :::m:\\ngeie use.\\nI: is c:Lve::t!i: live tie 5i:i :_ si\\nt s :i :ne ::^:~.ii i..b.e\\nh. p:.\\n_ __ j. _ __,\\n-ion.\\nNo. of Teeth\\nPitch Diam\\nEft 1\\niiside Lis\\ni_e:e:\\nCircular Pit\\n:ch\\nAlible of Teeth with Axis\\nN nnal Cin\\nc-nlar Pitch\\nP:::L Ci\\ntruer\\nA lie- ivr\\ns\\nTni?k-ess\\nTooth t\\nWh- I er\\n:LI\u00c2\u00bb\\ner\\nact Lead\\nof Spiral\\nA] _ rin\\n:e Lei i Si\\n1 rs n ~\\\\T\\nlhng Machine to Cut Spiral\\nGear on W\\n1st Gear on\\nr: :1\\n2nd Gear 01\\n*v\\nt-s.: :i_ .^;r\\ne 7\\nA spiral of any angle I can generally be cat in\\na Universal Milling Machine without special attach-\\nments, the cntter being at the top of the work. The\\ncutter is placed on the arbor in such position tha:\\ncan reach the work centrally after the table is s it :o\\nthe angle of the spiral. In order to cut central, it ia\\ngenerally well enough to place the table, before setting\\nit to the angle, so that the work centres will be central\\nwith the cutter, then swing the table and set it to the\\nui: t .e s ir;-/\\nFor very accurate work, i: is s\\\\fer to test the posi-^^ 1 1 Set\\ntion of the centres after the table has been set to the\\nangle.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0142.jp2"}, "141": {"fulltext": "PROVIDENCE, R. I.\\n125\\nFig. 67.\\nUSE OF VERTICAL SPINDLE MILLING ATTACHMENT IN CUTTING\\nSPIRAL GEARS.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0143.jp2"}, "142": {"fulltext": "120 BROWN SHARPE MFG. CO.\\nThis can be done with a trial piece, Fig. 65 which\\nis simply a round arbor with centre holes in the ends.\\nIt is mounted between the centres, and the knee is\\nraised until the cutter sinks a small gash, as at A.\\nThis gash shows the position of the cutter; ami if the\\ngash is central with the trial piece, the cutter will be\\ncentral with the work. If preferred, the arbor can be\\ndogged to the work spindle; and the line B C drawn\\non the side of the arbor at the same height as the cen-\\ntres the work spindle should then be turned quarter\\nway round in order to bring the line at the top. The\\ngash A can now be cut and its position determined with\\nthe line.\\nIn cutting small gears the arbor can be dogged to the\\nwork spindle the distance between the gear blank and\\nthe dog should be enough to let the dog pass the cutter\\narbor without striking.\\nA spiral gear is much more likely to slip in cutting\\nthan a spur gear.\\nFor gears more than three or four inches in diameter\\nit is well to have a taper shank arbor held directlv in\\nthe work spindle, as shown in Figs. 67 and GS and for\\nthe heaviest work, the arbor can be drawn into the spin-\\ndle with a screw in a threaded hole in the end of the\\nshank.\\nAfter cutting a space the work can be dropped away\\nfrom the cutter, in order to avoid scratching it when\\ncoming back for another cut. Some workmen prefer\\nnot to drop the work away, but to stop the cutter and\\nturn it to a position in which its teeth will not touch\\nthe work. To make sure of finding a place in the cut-\\nter that will not scratch, a tooth has sometimes been\\nt iken out of the cutter, but this is not recommended.\\nThe safest plan is to drop the work away.\\nAngie^reater In cutting spiral gears of greater angle than 45\u00c2\u00b0, a\\nvertical spindle milling attachment is available, as\\nshown in Figs. G7 and 6$.\\nIn Fig. G7 the cutter is at 90\u00c2\u00b0 with the work spindle\\nwhen the table is set to 0, so that the proper angle at\\nwhich the table should be set, is the difference between\\nthe angle of the spiral and 90\u00c2\u00b0. Thus, to cut a 70\u00c2\u00b0", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0144.jp2"}, "143": {"fulltext": "PROVIDENCE, R. I.\\n127\\nFig. 68.\\nUSE OF VERTICAL SPINDLE MILLING ATTACHMENT IN CUTTING\\nSPIRAL GEARS.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0145.jp2"}, "144": {"fulltext": "128 bbow fc ^:-:aspe mt\\nspiral, we subtract ?i from and the remainder,\\n20\u00c2\u00b0, is the angle to se: the In cutting on the\\ntop, Fig. 67, the attachment is set to 0.\\nIn Fig. 69 the cutter is at the side of the work; the\\nh ile is set to 0, and the attachment is set to the differ-\\ne between 9 1 an :I the r-: .ire 1 ^n_ e ci spiral.\\nIn setting the cotter central it is convenient have a\\nsmall knee as at K. Fig. 66. A line is drawn uponthc\\nknee at Lhe same height as the cent es. The cutter\\nart 9i is glit to tl. angle as just shown, and a gash\\nis cut in the knee. Wh n the g ish is central with the\\nlit r. the cutter wiD be central with the work.\\nThe cutter can be sir to act apon either side the\\ngear to be :::r ling as a right hand or a left hand\\nspiral is wanted. The setting in Fig. 68 is for a right\\nhand spiral.\\nIf the gear blank were bronght in front of the c\\nt:-r. and the reversing gen: s.t between two chance\\ngeai a, the machine w al .1 be 5t: f\u00c2\u00ab r a left hand spiral.\\nFor cc iraer pitches than abon: 12 IP diametral, it is\\nwell to cut more than once around, the finishing cut\\nbeing quite light so as to en: sn;::th.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0146.jp2"}, "145": {"fulltext": "129\\nCHAPTER VIII.\\nSCREW GEARS AND SPIRAL GEARS\u00e2\u0080\u0094 GENERAL REMARKS.\\nThe working of spiral gears, when their axes are working of\\nn iii Spiral Gears.\\npara. lei, is generally smoother than spur gears. A\\ntooth does not strike along its whole face or length at\\nonce. Tooth contact first takes place at one side of the\\ngear, passes across the face and ceases at the other\\nside of the gear. This action tends to cover defects in\\nshape of teeth and the adjustment of centres.\\nSince the invention of machines for producing accu-\\nrate epicyloidal and involute curves, it has not so often\\nbeen found necessary to resort to spiral gears for\\nsmoothness of action. A greater range can be had in\\nthe adjustment of centers in spiral gears than in spur\\ngears. The angle of the teeth should be enough, so\\nthat one pair of teeth will not part contact at one side\\nof the gears until the next pah of teeth have met on the\\nother side of the gears. When this is done the gears\\nwill be in mesh so long as the circumferences of their\\naddendum circles intersect each other. This is some-\\ntimes necessary in roll gears.\\nRelative to spur and bevel gears in Part I., Chapter\\nXII., it was stated that all gears finally wore them-\\nselves out of shape and might become noisy. Spiral\\ngears may be worn out of shape, but the smoothness\\nof action can hardly be impaired so long as there are\\nany teeth left. For every quantity of wear, of course,\\nthere will be an equal quantity of backlash, so that if\\ngears have to be reversed the lost motion in spiral\\ngears will be as much as in any gears, and may be\\nmore if there is end play of the shafts. In spiral gears End Pressure\\nthere is end pressure upon the shafts, because of the Spiral Gears,\\nscrew-like action of the teeth. This end pressure is\\nsometimes balanced by putting two gears upon each\\nshaft, one of right and one of left hand spiral.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0147.jp2"}, "146": {"fulltext": "130 BROWN SHARPE MFG. CO.\\nThe same result is obtained in solid cast gears by\\nmaking the pattern in two parts one right and one\\nleft-hand spiral. Such gears are colloquially called\\nherring-bone gears.\\nIn an internal spiral gear and its pinion, the spirals\\nof both wheels are either right-handed or left-handed.\\nSuch a combination would hardly be a mercantile\\nproduct, although interesting as mechanical feat.\\nIn screw or worm-gears the axes are generally at\\nright angles, or nearly so. The distinctive features of\\nscrew gearing may be stated as follows\\nThe relative angular velocities do not depend upon\\nthe diameters of pitch- cylinders, as in Chapter I.,\\nDistinctive Part I. Thus the worm in Chapter XL, Fig-. 35, can\\nteatures of x\\nScrew Gearing, be any diameter one inch or ten inches without\\naffecting the velocity of the worm-wheel. Conversely if the\\naxes are not parallel we can have a pair of spiral or screw\\ngears of the same diameter, but of different numbers of\\nteeth. The direction in which a worm-wheel turns depends\\nupon whether the worm has a right-hand or left-hand thread.\\nWhen angles of axes of worm and worm-wheel are\\noblique, there is a practical limit to the directional\\nrelation of the worm-wheel. The rotation of the\\nworm-wheel is made by the end movement of the\\nworm-thread.\\nThe term worm and worm-wheel, or worm -gearing,\\nis applied to cases where the worms are cut in a lathe,\\nand the shapes of the threads or teeth, in axial section,\\nare like a rack. The shape usually selected is like the\\nrack for a single curve or involute gear. See Chap.\\nIV., Parti. Worms are sometimes cut in a milling\\nmachine.\\nIf the form of the teeth in a pair of screw gears is\\ndetermined upon the normal helix, as in Chap. VI.,\\nthe gears are usually called Spiral Gears.\\nIf we let two cylinders touch each other, their axes\\nbeing at right angles, the rotation of one cylinder will\\nhave no tendency to turn the other cylinder, as in\\nChapter L, Part I.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0148.jp2"}, "147": {"fulltext": "PROVIDENCE, R. I. 13J\\nWe can now see why worms and worm-wheels wear why Wor\\nJ Wheels wear\\nout faster than other gearing. The length of worm- sofast\\nthread, equal to more than the entire circumference of\\nworm, comes in sliding contact with each tooth of the\\nwheel during one turn of the wheel.\\nThe angle of a worm-thread can be calculated the\\nsame as the angle of teeth of spiral gear only, the\\nangle of a worm thread is measured from a line or\\nplane that is perpendicular to the axis of the worm.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0149.jp2"}, "148": {"fulltext": "CHAPTER IX.\\nCONTINUED FRACTIONS\u00e2\u0080\u0094 SOME APPLICATIONS IN MACHINE\\nCONSTRUCTION.\\nDefinition of _^_ continued fraction is one which has unitv for its\\na Continued\\nFraction. numerator, and for its denominator an entire number\\nplus a fraction, which fraction has also unity for its\\nnumerator, and for its denominator an entire number\\nplus a fraction, and thus in order.\\nThe expression.\\n4 i_\\n3 1_\\n5 is called a continued frac-\\ntion. By the use of continued fractions, we are ena-\\nPracucai u=e D e d to nnd a fraction expressed in smaller numbers.\\nof Continued\\nFractions. that, for practical purposes, may be sufficiently near in\\nvalue to another fraction expressed in large numbers.\\nIf we were required to cut a worm that would mesh\\nwith a gear -4 diametral pitch -i P. m a lathe having\\n3 to 1-inch linear leading screw, we might, without\\ncontinued fractions, have trouble in finding- change\\ngears, because the circular pitch corresponding to\\n\u00e2\u0080\u00a24 diametral pitch is expressed in large numbers\\nThis example will be considered farther on. For\\nillustration, we will take a simpler example.\\nWhat fraction expressed in smaller numbers is near-\\nest in value to T Dividing the numerator and the\\ndenominator of a fraction by the same number does\\nnot change the value of the fraction. Dividing both\\nExample in terms f s;_ bv 29. we have ^T~ or. what is the\\nContinued u\u00c2\u00ab J T\\nsame thing expressed as a continued fraction, s-kju The\\ncontinued fraction s+j_ is exactly equal to -f\u00c2\u00a3j-\\nnow. we reject the v 7 the fraction 4- will be larger\\nthan 5\u00e2\u0080\u0094 :_. because the denominator has been diniin-\\n2 9\\nislied, 5 beii _ less than 5gV- i d something\\n^-_V expressed in smaller numbers than 29 for a", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0150.jp2"}, "149": {"fulltext": "PROVIDENCE, E. I. 133\\nnumerator and 146 for a denominator. Eeducing\\nand y 2 to a common denominator, we have ^=i|f\\nan d i 2_9_ \u00c2\u00b1|i-. Subtracting one from the other, we\\nhave t Jq, which is the difference between and -f\u00c2\u00a3-$.\\nThus, in thinking of T as we have a pretty fair\\nidea of its value.\\nThere are fourteen fractions with terms smaller than\\n29 and 146, which are nearer j 2 V than is, such as\\n\u00e2\u0096\u00a0if, -if and so on to f 2 f T In this case by continued frac-\\ntions we obtain only one approximation, namely -J, and\\nany other approximations, as T f, -J-f, c, we find by\\ntrial. It will be noted that all these approximations\\nare smaller in value than T 2 I There are cases, how-\\never, in which we can, by continued fractions, obtain\\napproximations both greater and less than the required\\nfraction, and these will be the nearest possible approxi-\\nmations that there can be in smaller terms than the\\ngiven fraction.\\nIn the French metric system, a millimetre is equal\\nto .03937 inch what fraction in smaller terms ex-\\npresses .03937 nearly? .03937, in a vulgar fraction,\\nis ToSood* Dividing both numerator and denominator\\nby 3937, we have 25II1JL, Rejecting from the de-\\nnominator of the new fraction, -Jf Jf the fraction -fa\\ngives us a pretty good idea of the value of .03937\\nIf in the expression, Tz+TJJA, we divide both terms of\\nthe fraction |jj|j by 1575, the value will not be changed.\\nPerforming the division, we have 1\\n25 1\\n2 787\\n1575\\nWe can now divide both terms of r 8 T 7 T by 787,\\nwithout changing its value, and then substitute the\\nnew fraction for 8 T n ne continued fraction.\\nDividing again, and substituting, we have\\n1\\n25 1\\n2 l_\\n2+ 1\\n787\\nas the continued fraction that is exactly equal to\\n.03937.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0151.jp2"}, "150": {"fulltext": "134 BROWN SHAEPE MFG. CO.\\nIn performing the divisions, the work stands thus\\n3937) 100000 (25\\n7874\\n21260\\n19685\\n1575) 3937 (2\\n3150\\n787) 1575 (2\\n1574\\n1) 787 (787\\n787\\n\u00e2\u0080\u00a2o-\\nThat is, dividing- the last divisor by the last remain-\\nder, as in finding the greatest common divisor. The\\nquotients become the denominators of the continued\\nfraction, with unity for numerators. The denominators\\n25, 2, and so on, are called incomplete quotients, since\\nthey are only the entire parts of each quotient. The\\nfirst expression in the continued fraction is or\\n.04 a little larger than .03937. If, now, we take\\n25 i we shall come still nearer .03937. The expres-\\nsion 25~+t is merely stating that 1 is to be divided by\\n25^. To divide, we first reduce 25J to an improper\\nfraction, and the expression becomes IT, or one\\ndivided by To divide by a fraction, Invert the\\ndivisor, and proceed as in multiplication. We\\nthen have -f T as the next nearest fraction to .03937.\\n-g2 T 0392 which is smaller than .03937. To get still\\nnearer, we take in the next part of the continued frac-\\ntion, and have 1\\n25 1\\n2 1\\n2\\nWe can bring the value of this expression into a\\nfraction, with only one number for its numerator and\\none number for its denominator, by performing the\\noperations indicated, step by step, commencing at the\\nlast part of the continued fraction. Thus, 2 J, or\\n2-|, is equal to Stopping here, the continued frac-\\ntion would become\\n25 -H_\\n2-\\n1 1__\\nNow, 5 equals and we have 25 +_\u00c2\u00bb_. 25f equals\\n2 5\\n1 3 substituting again, we have rhi. Dividing 1 by\\nJ-f- 1 w^e have T T T T is the nearest fraction to", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0152.jp2"}, "151": {"fulltext": "PROVIDENCE, E. I. 135\\n.03937, unless we reduce the whole continued fraction\\nl\\n25+1\\n2 1\\n2 L\u00e2\u0080\u009e, which would give us back the .03937 itself.\\ntot\\nT f T 03937007, which is only larger\\n.03937. It is not often that an approximation will\\ncome so near as this.\\nThis ratio, 5 to 127, is used in cutting millimeter Practical use\\nof the foregoing\\nthread screws. If the leading screw of the lathe isExampie.\\n1 to one inch, the change gears will have the ratio of\\n5 to 127; if 8 to one inch, the ratio will be 8 times\\nas large, or 40 to 127; so that with leading screw 8 to\\ninch, and change gears 40 and 127, we can cut milli-\\nmeter threads near enough for practical purposes.\\nThe foregoing operations are more tedious in de-\\nscription than in use. The steps have been carefully\\nnoted, so that the reason for each step can be seen\\nfrom rules of common arithmetic, the operations being\\nmerely reducing complex fractions. The reductions,\\n\u00e2\u0096\u00a0gig-, T 2 T T ^j, etc., are called conver gents, because they\\ncome nearer and nearer to the required .03937. The\\noperations can be shortened as follows:\\nLet us find the fractions converging towards .7854 Example.\\nthe circular pitch of 4 diametral pitch, .7854= T 8 Tr 5 (r\\nreducing to lowest terms, we have Applying\\nthe operation for the greatest common divisor:\\n3927) 5000 (1\\n3927\\n1073) 3927 (3\\n3219\\n708) 1073 (1\\n_708\\n365) 708 (1\\n365\\n843) 365 (1\\n343\\n22) 343 (15\\n22\\n123\\n110\\n13) 22 (1\\n13\\n9) 13 (1\\n9\\n4) 9 (2\\n8\\n1) 4 (4\\n4\\nBringing the various incomplete quotients as de-\\nnominators in a continued fraction as before, we have", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0153.jp2"}, "152": {"fulltext": "136 BROWN SHARPE MFG. CO.\\n1 1\\n3 1\\n1 1\\n1 1 _\\n1 1 _\\n15 1\\n1 1_\\ni i\\n2\\nNow arrange each partial quotient in a line, thus\\n13111 15 1 1 2 4\\n1 3. 4 1 J_l \u00c2\u00b1T_2 183 3 5 5 8_93_ 3 9 27\\nx 4 1 9 14 219 233 4~T2 T13 1 5 0\\nNow place under the first incomplete quotient the\\nfirst reduction or convergent -f, which, of course, is 1\\nput under the next partial quotient the next reduction or\\nconvergent 7 or which becomes f\\n1 is larger than .7854, and J is less than .7854.\\nHaving made two reductions, as previously shown,\\nwe can shorten the operations by the following rule for next\\neonvergents Multiply the numerator of the convergent\\njust found by the denominator of the next term of the con-\\ntinued fraction, or the next incomplete quotient, and add\\nto the product the numerator of the preceding convergent\\nthe sum will be the numerator of the next convergent.\\nProceed in the same way for the denominator, that\\nis multiply the denominator of the convergent just\\nfound by the next incomplete quotient and add to the\\nproduct the denominator of the preceding convergent\\nthe sum will be the denominator of the next convergent.\\nContinue until the last convergent is the original frac-\\ntion. Under each incomplete quotient or denominator\\nfrom the continued fraction arranged in line, will be\\nseen the corresponding convergent or reduction. The\\nconvergent is the one commonly used in cutting\\nracks 4 P. This is the same as calling the circumference of\\na circle 22-7 when the diameter is one (1) this is also the\\ncommon ratio for cutting any rack. The equivalent decimal\\nto is .7S57 X being about x g large. In three set-\\ntings for rack teeth, this error would amount to about .001\\nFor a worm, this corresponds to -if- threads to 1\\nnow, with a leading screw of lathe 3 to 1 we would\\nwant gears on the spindle and screw in a ratio of 33\\nto 14.\\nHence, a gear on the spindle with 66 teeth, and a\\ngear on the 3 thread screw of 28 teeth, would enable\\nus to cut a worm to fit a 4 P gear.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0154.jp2"}, "153": {"fulltext": "CHAPTER X.\\nANGLE OF PRESSURE.\\n137\\nIn Fig. 69, let A be any flat disk lying upon a hori-\\nzontal plane. Take any piece, B, with a square end,\\na b. Press against A with the piece B in the direction\\nof the arrow.\\nFig. 69.\\nFig.\\nIt is evident A will tend to move directly ahead of\\nB in the normal line c d. Now (Fig. 70) let the piece\\nB, at one corner f, touch the piece A. Move the piece\\nB along the line d e, in the direction of the arrow.\\nIt is evident that A will not now tend to move in\\nthe line d e, but will tend to move in the direction of\\nthe normal c d. When one piece, not attached, presses\\nagainst another, the tendency to move the second\\npiece is in the direction of the normal, at the point of\\ncontact. This normal is called the line of pressure. ^ineofPreas-\\nJ L ure.\\nThe angle that this line makes with the path of the\\nimpelling piece, is called the angle of pressure.\\nIn Part I., Chapter IV., the lines B A and B A are\\ncalled lines of pressure. This means that if the gear", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0155.jp2"}, "154": {"fulltext": "138\\nBE0WX k SHAEPE MFG. CO.\\ndrives the rack, the tendency to move the rack is not\\nin the direction of pitch line of rack, but either in the\\ndirection B A or B A as we turn the wheel to the left\\nor to the right.\\nThe same law holds if the rack is moved in the\\ndirection of the pitch line; the tendency to move the\\nwheel is not directly tangent to the pitch circle, as if\\ndriven by a belt, but in the direction of the line of\\npressure. Of course the rack and wheel do move in\\nthe paths prescribed by then connections with the\\nframework, the wheel turning about its axis and the\\nrack moving along its ways. This pressure, not in a\\ndirect path of the moving piece, causes extra friction\\nin all toothed gearing that cannot well be avoided.\\nAlthough this pressure works out by the diagram,\\nas we have shown, yet, in the actual gears, it is not at\\nall certain that they will follow the law as stated,\\nbecause of the friction of teeth among themselves. If\\nthe driver in a train of gears has no bearing upon its\\ntooth-flank, we apprehend there will be but little\\ntendency to press the shafts apart.\\nThe arc through which a wheel passes while one of\\nits teeth is in contact is called the arc of action.\\nBase of Sys- Until within a few vears, the base of a svstem of\\ntern of Inter- J\\nchange a bie c iible-cmve interchangeable gears was 12 teeth. It\\nGears. o fc\\nis now 15 teeth in the best practice (see Chapter VII.,\\nPart I.)\\nThe reason for this change was the base, 15 teeth,\\ngives less angle of pressure and longer arc of contact,\\nand hence longer lifetime of gears.\\nArc of Action.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0156.jp2"}, "155": {"fulltext": "139\\nCHAPTER XL\\nINTERNAL GEARS.\\nIn Part L, Chapter VIII., it is stated that the space\\nof an internal gear is the same as the tooth of a spur\\ngear. This applies to involute or single-curve gears as\\nwell os to double-curve gears.\\nThe sides of teeth in involute internal gears are\\nhollowing. It, however, has been customary to cut\\ninternal gears with spur gear-cutters, a No. 1 cutter\\ngenerally being used. This makes the teeth sides\\nconvex. Special cutters should be made for coarse Special Cut,\\nt -i ters for coarse\\npitch double-curve gears. In designing internal gears, Pitch.\\nit is sometimes necessary to depart from the system\\nwith 15-tooth base, so as to have the pinion differ from\\nthe wheel by less than 15 teeth. The rules given in\\nPart I., Chapters VII. and VIII., will apply in making\\ngears on any base besides 15 teeth. If the base is\\nlow-numbered and the pinion is small, it may be neces-\\nsary to resort to the method given at the end of Chap-\\nter VII., because the teeth may be too much rounded\\nat the points by following the approximate rules.\\nThe base must be as small as the difference between Base f or in-\\nt 6ril\u00c2\u00a3Ll Ct P ft t*\\nthe internal gear and its pinion. The base can be Teeth,\\nsmaller if desired.\\nLet it be required to make an internal gear, and\\npinion 24 and 18 teeth, 3 P. Here the base cannot\\nbe more than 6 teeth.\\nIn Fig. 71 the base is 6 teeth. The arcs A K and\\nO k, drawn about T, have a radius equal to the radius\\nof the pitch circle of a 6-tooth gear, 3 P, instead of a\\n15-tooth gear, as in Chapter VIII., Part I.\\nThe outline of teeth of both gears and pinion is pescription of\\nmade similar to the gear in Chapter VIII. The same", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0157.jp2"}, "156": {"fulltext": "140\\nBROWN SHARPE MFG. CO.\\nGEAR, 24 TEETH.\\nPINiON, 18 TEETH, 3P\\nP 3\\nN =24 and 18\\nP 1.0472\\nt\u00e2\u0080\u0094 5236\\nS= .3333\\nD .6666\\n-3857\\nP 7190\\nNTERNAL GEAR AND PINION IN MESH", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0158.jp2"}, "157": {"fulltext": "PROVIDENCE, R. I. 141\\nletters refer to similar parts. The clearance circle is,\\nhowever, drawn on the outside for the internal gear.\\nAs before stated, the spaces of a spur wheel become\\nthe teeth of an internal wheel. The teeth of internal\\ngears require but little for fillets at the roots they\\nare generally strong enough without fillets. The\\nteeth of the pinion are also similar to the gear in\\nChapter VIII., substituting 6-tooth for 15-tooth base.\\nTo avoid confusion, it is well to make a complete\\nsketch of one gear before making the other. The arc\\nof action is longer in internal gears than in external\\ngears. This property sometimes makes it necessary\\nto give less fillets than in external gears.\\nIn Fig. 71 the angle K T A is 30\u00c2\u00b0 instead of 12\u00c2\u00b0, as\\nin Fig. 12. This brings the line of pressure L P at\\nan angle of 60\u00c2\u00b0 with the radius C T, instead of 78\u00c2\u00b0.\\nA system of spur gears could be made upon this\\n6-tooth base. These gears would interchange, but no\\ngear of this 6-tooth system would mesh with a double-\\ncurve gear made upon the 15-tooth system in Part 1.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0159.jp2"}, "158": {"fulltext": "c:-:.-.?7z?.\\n:-Eii3\\n_ _- _\\n_ _ _ _ r-_\\n:k. 117 lri_ :-r _ .t _\\n:5r ;i-r:, -1 sZ-\\nz 1\\ni\\n.\u00c2\u00bb__ _\\n_\\n_\\n_ :-55~\\n7 T i 7 _U.~ JT\\nT\\nIirjr _ r\\n;iLS 4\\n7Z T 7Z\\n-tt.\\ni.\u00e2\u0080\u0094\\n1 1-4\\n40\\n1460\\n1 9-16\\n2220\\n5\\n24*\\n-,.._-_,\\n1 ;:r\u00c2\u00a3\\n1 t _\\n~-n\\nV^r 1 szir\\nrrS5\u00e2\u0080\u0094 r i: 1-\\n3 of the\\n:rr^o _i*\\nli T\\n10 Fitdb 353 1-3 Ubs. at the Pitch line.\\n8 _\\n6 40\\n823 1-3\\nhen it is 2} times the cJrealar pitch.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0160.jp2"}, "159": {"fulltext": "PROVIDENCE. II. I.\\n143\\nTABLE OF DECIMAL EQUIVALENTS\\nof 8ths, 16ths, 32nds and 64ths of an inch.\\n8ths.\\ni=.126\\ni=.250\\n#=.375\\ni .500\\nf=.750\\n\u00c2\u00a3=.875\\n16ths.\\nT V=.0625\\nA=-1875\\nf V=3125\\nT V=.4375\\nT 9 g .5625\\nU .6875\\n|f =.8125\\nif =.9375\\n32nds.\\nJ^ .03125\\nA=. 09375\\nA=. 15625\\nj\\n32\\n21875\\n.28125\\n.34375\\n.40625\\n.46875\\n.53125\\n.59375\\n.65625\\n.71875\\n.78125\\n.84375\\n.90625\\n.96875\\n64ths.\\n015625\\nA=- 046875\\n5\\n6 4\\n7\\n64\\n_9_\\n64\\n11\\n64\\n13\\n64\\n15.\\n64\\n\u00c2\u00b11\\n64\\n9_\\n32\\n11\\n32\\n1 Ji\\n32\\n15.\\n32\\nLI\\n32\\n19-\\n3 2\\n2 1_\\n32\\n2.3.\\n32\\n25\\n32\\n21-\\n3 2\\n_2_iL\\n32\\nill-\\n32\\n.078125\\n.109375\\n.140625\\n.171875\\n.203125\\n.234375\\n.265625\\nJL.9\\n64\\n2.1\\n64\\n2.3.\\n64\\n2.5.\\n64\\n2JT\\n64\\n29\\n64\\n_3_1\\n64\\n3.3.\\n64\\n3.5.\\n6 4\\n3.7\\n64\\n3.9.\\n64\\n41\\n64\\n43\\n64\\n15\\n64\\n11\\n64\\n19.\\n64\\n5_J_\\n64\\n5.3\\n64\\nA5_-\\n64\\n\u00c2\u00a31-\\n64\\nII-\\n64\\n6.1-\\n64\\n6.3-\\n64\\n296875\\n.328125\\n.359375\\n.390625\\n.421875\\n.453125\\n484375\\n.515625\\n546875\\n.578125\\n.609375\\n640625\\n.671875\\n703125\\n734375\\n765625\\n796875\\n.828125\\n859375\\n890625\\n921875\\n.953125\\n.984375", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0161.jp2"}, "160": {"fulltext": "144\\nBBOWH fc SHABPE MFG. CO.\\nGEAB WHEEL-.\\nTABLE OF TOOTH PAP.TS nBCUTAP. PIT 3H IX TTRST C 1 LUiiy.\\nn r\\n7\\nP\\nf YY +f p .31 p r\\no\\n3\\n1.5708\\ni o\\n0360\\n1.2732\\n.7360\\n1.3732\\n:_: 6700\\n1\\n1.6755\\n.9375\\n.596-\\nL.1937\\n90 i\\n1.2874\\n.5813 .6281\\n1-S-\\n4\\n1.7952\\n.8750\\n5570\\n1.1141\\n6445\\n1.201c\\n.5425 .3-::\\nIf\\n8\\n13\\n1.9333\\n.$125\\n.5173\\n1 0345\\n.5985\\n1.1158\\n5038 .5444\\n1:\\n2\\n3\\nD944\\n.7500\\n.4775\\n.9549\\n5525\\n1.0200\\n.4650 .5025\\n1-\\n1G\\n16\\n23\\n2.1855\\n.7187\\n.4576\\n9151\\n.5201\\n.9870\\n.4456 .4816\\n11\\n8\\n11\\n2.2848\\n875\\n.4377\\n-754\\n.5064\\n.9441\\n.4262 .41\\n1 I\\n16\\n\u00c2\u00bb1\\n2.31\\n.6562\\n.4178\\n.8\\n.4834\\n0-\\n4069 4397\\n1\\nn\\n4\\na\\n2.5133\\n.6252\\n3979\\n.7958\\n.4604\\n.8583\\n3875 .4188\\n1A\\n19\\n2.645C\\n.5937\\n.37-:\\n.7\\n.4374\\n.8156\\n.3681 .3978\\nH\\n9\\n2.7925\\n.5625\\n.35-1\\n.7162\\n.4143\\n.7724\\n.3488 37G9\\n1--\\nA 16\\nJ_6\\n2.9568\\n.5312\\n.3382\\n.6764\\n.3. 1;\\n.7295\\n.3294 .3559\\n1\\n1\\n3.1416\\n5000\\n3183\\n.3683\\n335C\\n15\\n16\\nItV\\n3.3510\\n.4687\\n.2984\\n5968\\n.3453\\n.6437\\n_ 6 3141\\n7\\n8\\nH\\n3.5904\\n.4375\\n.2785\\n.557\\n.3223\\n.6007\\n.2713 .2931\\n1 3\\n16\\n1*\\n3.866C\\n.4: 52\\n.2586\\n.5173\\n.2993\\n.5579\\n.2519 .2722\\n3.\\ni-\\n4.1858\\n.3750\\n.2387\\n.4775\\n.2762\\n5 1 5 _\\n.2325 .2513\\n1 1\\n16\\nx ll\\n4.5\\n.3437\\n.21 SO\\n.4377\\n2532\\n.4 2:\\n.2131 .2303\\n2\\n3\\nu\\n4.7124\\n.3333\\n2122\\n.4244\\n.2455\\n.4577\\n.2233", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0162.jp2"}, "161": {"fulltext": "PROVIDENCE, R. I.\\nHo\\nTABLE OF TOOTH PARTS.\u00e2\u0080\u0094 Continued.\\nCIRCULAR PITCH IN FIRST COLUMN.\\ne3\\n11\\nP\\n5\\n8\\nThreads or\\nTeeth per inch\\nLinear.\\nDiametral\\nPitch.\\nThickness of\\nTooth on\\nPitch Line.\\nAddendum\\nand\\nft\\nhflO\\n.g E-\\nG C\\nD\\nDepth of Space\\nbelow\\nPitch Line.\\nft\\nWidth of\\nThread-Tool\\nat End.\\nWidth of\\nThread at Top.\\n1\\nw\\n1 3\\nP\\nt\\ns\\n.4291\\nPx.31\\n.1938\\nP x.S35i\\n5.0265\\n.3125\\n.1989\\n.3979\\n.2301\\n.2094\\n9\\n1 G\\nIT\\nL 9\\n5.5851\\n.2812\\n.1790\\n.3581\\n.2071\\n.3862\\n.1744\\n.1884\\n1\\n2\\n2\\n6.2832\\n.2500\\n.1592\\n.3183\\n.1842 .3433\\n.1550\\n.1675\\n7\\n16\\n2f\\n7.1808\\n.2187\\n.1393\\n.2785\\n.1611 .3003\\n.1356\\n.1466\\n2\\n5\\n2i\\n7.8540\\n.2000\\n.1273\\n.2546\\n.1473 .2746\\n.1240\\n.1340\\n3\\n8\\n2|\\n8.3776\\n.1875\\n.1194\\n.2387\\n.1381\\n.2575\\n.1163\\n.1256\\n1\\n3\\n3\\n9.4248\\n.1666\\n.1061\\n.2122\\n.1228. 2289\\n.1033\\n.1117\\n5\\n1 G\\nH\\n10.0531\\n.1562\\n.0995\\n.1989\\n.1151\\n.2146\\n.0969\\n.1047\\n2\\n1\\n3}\\n10.9956\\n.1429\\n.0909\\n.1819\\n.1052\\n.1962\\n.0886\\n.0957\\n1\\n4\\n4\\n12.5664\\n.1250\\n.0796\\n.1591\\n.0921\\n.1716\\n.0775\\n.0838\\n2\\n9~\\n4-L\\n*2\\n14.1372\\n.1111\\n.0707\\n.1415\\n.0818\\n.1526\\n.0689\\n.0744\\n1\\n5\\n5\\n15.7080\\n1000\\n.0637\\n.1273\\n.0737\\n.1373\\n.0620\\n.0670\\n3\\n1 G\\n\u00c2\u00b03\\n16.7552\\n.0937\\n.0597\\n.1194\\n.0690\\n.1287\\n.0581\\n.0628\\nI\\nG\\n18.8496\\n.0833\\n.0531\\n.1061\\n.0614\\n.1144\\n.0517\\n.0558\\n1\\n1\\n7\\n21.9911\\n0714\\n.0455\\n0910\\n.0526\\n.0981\\n.0443\\n.0479\\n1\\n8\\n8\\n25.1327\\n.0625\\n.0398\\n.0796\\n.0460\\n.0858\\n.0388\\n.0419\\n1\\n9\\n9\\n28.2743\\n.0555\\n.0354\\n.0707\\n.0409\\n.0763\\n.0344\\n.0372\\n1\\n1\\n10\\n31.4159\\n.0500\\n.0318 .0637\\n.0368\\n.0687\\n.0310\\n.0335\\n1\\n1 6\\n1G\\n50.2655\\n.0312\\n.0199 .0398\\n.0230\\n.0429\\n.0194\\n.0209 i", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0163.jp2"}, "162": {"fulltext": "14 J\\nBROWN SHABTE ilFG. CO.\\nGEAR WHEELS.\\nTABLE OF TOOTH PART;\\n-DIAMETRAL PITCH EN FIRST COLUMN.\\n\u00e2\u0096\u00a0a\\nT- Z\\n71\\n5 C\\np\\nP\\nt\\ns\\nD\\nD\\n1\\n2\\n6.2832\\n3.1416\\n2 0000\\n4.0000\\n2.3142\\n4.3142\\na\\n4\\n4.1888\\n2 0944\\n1.3333\\n2.6666\\n1 5428\\n2.8761\\n1\\n3.1416\\n1 5708\\n1.0000\\n2 0000\\n1 .1571\\n2.1571\\nli\\n2.5133\\n1 2566\\n8000\\n1.6000\\n.9257\\n1.7257\\nH\\n2 0944\\n1.0472\\n.6666\\n1 3333\\n.7714\\n1.4381\\nIf\\n1.7952\\n.8976\\n.5714\\n1 1429\\n.6612\\n1.2326\\n2\\n1.5708\\n7854\\n.5000\\n1 0000\\n5785\\n1 0785\\n01\\n1.3963\\n.6981\\n.4444\\n.8888\\n.5143\\n9587\\n2i\\n1.2566\\n.6283\\n.4000\\n.8000\\n.4628\\n.8628\\n-4\\n1.1424\\n.5712\\n.3636\\n.7273\\n.4208\\n.7844\\n3\\n1 0472\\n5236\\n3333\\n.6666\\n3857\\n.7190\\n31\\n.S976\\n.4488\\n.2857\\n.5714\\n.3306\\n.6163\\n4\\n7854\\n.3927\\n.2500\\n5000\\n.2893\\n5393\\n5\\n.6283\\n.3142\\n.2000\\n.4000\\n.2314\\n.4314\\nG\\n.5236\\n.2618\\n.1666\\n.3333\\n.1928\\n.3595\\ni\\n.4488\\n.2244\\n1429\\n.2857\\n1653\\n.3081\\n8\\n.3927\\n.1963\\n.1250\\n2500\\n.1446\\n.2696\\n9\\n.3491\\n1745\\n.1111\\n2222\\n.1286\\n.2397\\n10\\n.3142\\n.1571\\n.1000\\n.2000\\n.1157\\n.2157\\n11\\n.2856\\n142S\\n0909\\n.1818\\n.1052\\n.1961\\n12\\n.2618\\n1309\\n0S33\\n.1666\\n0964\\n.1798\\n13\\n.2417\\n1208\\n.0769\\n.1538\\n.0890\\n.1659\\n14\\n.2244\\n.1122\\n.0714\\n1429\\n0S26\\n.1541", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0164.jp2"}, "163": {"fulltext": "PROVIDENCE. R. I.\\n117\\nTABLE OF TOOTH PARTS\u00e2\u0080\u0094 Continued.\\nDIAMETRAL PITCH IN FIRST COLUMN.\\nDiametral\\nPitch.\\nCircular\\nPitch.\\nThickness\\nof Tooth on\\nPitch Line.\\nAddendum\\nand J\\nWorking Depth\\nof Tooth.\\nDepth of Space\\nbelow\\nPitch Line.\\nWhole Depth\\nof Tooth.\\nP.\\nP\\nt.\\ns.\\nD\\n.0771\\nD\\n15\\n.2094\\n.1047\\n.0666\\n.1333\\n.1438\\n16\\n.1963\\n.0982\\n.0625\\n.1250\\n.0723\\n.1348\\n17\\n.1848\\n.0924\\n.05 8\\n.1176\\n.0681\\n.1269\\n18\\n.1745\\n.0873\\n.0555\\n.1111\\n.0643\\n.1198\\n19\\n.1653\\n.0827\\n.0526\\n.1053\\n.0609\\n.1135\\n20\\n.1571\\n.0785\\n.0500\\n.1000\\n.0579\\n.1079\\n22\\n.1428\\n.0714\\n.0455\\n.0909\\n.0526\\n.0980\\n24\\n.1309\\n.0654\\n.0417\\n.0833\\n.0482\\n.0898\\n26\\n.1208\\n.0604\\n.0385\\n.0769\\n.0445\\n.0829\\n28\\n.1122\\n0561\\n.0357\\n.0714\\n.0413\\n.0770\\n30\\n.1047\\n.0524\\n.0333\\n.0666\\n.0386\\n.0719\\n32\\n.0982\\n.0491\\n.0312\\n.0625\\n.0362\\n.0674\\n34\\n.0924\\n.0462\\n.0294\\n.0588\\n.0340\\n.0634\\n36\\n.0873\\n.0436\\n.0278\\n.0555\\n.0321\\n.0599\\n38\\n.0827\\n.0413\\n.0263\\n.0526\\n.0304\\n.0568\\n40\\n.0785\\n.0393\\n.0250\\n0500\\n.0289\\n.0539\\n42\\n.0748\\n.0374\\n.0238\\n.0476\\n.0275\\n.0514\\n44\\n.0714\\n.0357\\n.0227\\n.0455\\n.0263\\n.0490\\n46\\n.0683\\n.0341\\n.0217\\n.0435\\n.0252\\n.0469\\n48\\n.0654\\n.0327\\n.0208\\n.0417\\n.0241\\n.0449\\n50\\n.0628\\n.0314\\n.0200\\n.0400\\n.0231\\n.0431\\n56\\n.0561\\n.0280\\n.0178\\n.0357\\n.0207\\n.0385\\n60\\n.0524\\n.0262\\n.0166\\n.0333\\n.0193\\n0360", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0165.jp2"}, "164": {"fulltext": "us\\nBROWN SHARPE MFG. CO.\\nNATURAL SINE.\\nT eg.\\n0\\n10\\n20\\n33\\n40\\n50\\n60\\n.00000\\n.00291\\n.00581\\n.00872\\n.01163\\n.01454\\n.01745\\n89\\n1\\n.01745\\n.02036\\n.02326\\n.02617\\n.02908\\n.03199\\n.03489\\n88\\n2\\n.03489\\n.03780\\n04071\\n.04361\\n.04652\\n.04943\\n.05233\\ni 87\\n3\\n.052S3\\n.05524\\n.05814\\n.06104\\n.06395\\n.06685\\n.06975\\n1 80\\n4\\n.06975\\n.07265\\n.07555\\n.07845\\n.08135\\n.08425\\n.08715\\n85\\n5\\n.08715\\n.09005\\n.09295\\n.09584\\n.09874\\n.10163\\n10452\\nj 84\\n6\\n.10452\\n.10742\\n.11031\\n.11320\\n.11609\\n.11898\\n.12186\\n83\\n7\\n.12180\\n.12475\\n.12764\\n.13052\\n.13341\\n13629\\n.13917\\ni 82\\n8\\n.13917\\n.14205\\n.14493\\n14780\\n.15068\\n.15356\\n.15643\\n81\\n9\\n.15643\\n.15980\\n.16217\\n.16504\\n.16791\\n.17078\\n.17364\\ni 80\\n10\\n17364\\n.17651\\n.17937\\n.13223\\n.18509\\n.18795\\n.19080\\n79\\n11\\n.19080\\n.19366\\n.19651\\n.19936\\n.20221\\n.20506\\n.20791\\n78\\n12\\n.20791\\n.21075\\n.21359\\n.21644\\n.21927\\n.22211\\n.22495\\n77\\n13\\n.22495\\n.22778\\n.23061\\n.23344\\n.23627\\n.23909\\n.24192\\n7G\\n14\\n.24192\\n.24474\\n.24756\\n.25038\\n.25319\\n.25600\\n.25881\\n75\\n15\\n.25881\\n.26162\\n.26443\\n.26723\\n.27004\\n.27284\\n.27563\\n74\\n16\\n.27563\\n.27843\\n.28122\\n.28401\\n.28680\\n.28958\\n.29237\\n73\\n17\\n.29237\\n.29515\\n.29793\\n.30070\\n.30347\\n.30624\\n.30901\\n1 72\\n18\\n.30901\\n.31178\\n.31454\\n.31733\\n.32000\\n.32281\\n.32556\\n71\\n19\\n.32556\\n.3283L\\n.3310G\\n.33380\\n.33654\\n.83928\\n34202\\n70\\n20\\n.34202\\n.34475\\n.34748\\n.35020\\n.35293\\n.35565\\n.35836\\n69\\n21\\n.35836\\n.36103\\n.36379\\n.36650\\n.36920\\n.37190\\n.37460\\n68\\n22\\n.37460\\n.37780\\n.37999\\n.38268\\n.38536\\n.38805\\n.39073\\n67\\n23\\n.39073\\n.39340\\n.39607\\n.39874\\n.40141\\n.40407\\n.40673\\n66\\n24\\n.40673\\n.40989\\n.41204\\n.41469\\n.41733\\n.41998\\n.42261\\n65\\n25\\n.42261\\n.42525\\n.42788\\n.43051\\n.43313\\n.43575\\n.43887\\n64\\n26\\n.43837\\n.44098\\n.44359\\n.44619\\n.44879\\n.45139\\n.45399\\n63\\n27\\n.45399\\n.45658\\n.45916\\n.46174\\n.46482\\n.46690\\n.46947\\n62\\n28\\n.46947\\n.47203\\n.47460\\n.47715\\n.47971\\n.48226\\n.48481\\n61\\n29\\n.48481\\n.48735\\n.48989\\n.49242\\n.49495\\n.49747\\n.50000\\n60\\n30\\n.50000\\n.50251\\n.50503\\n.50753\\n.51004\\n.51254\\n.51503\\n59\\n31\\n.51503\\n51752\\n.52001\\n.52249\\n.52497\\n.52745\\n.52991\\n58\\n32\\n.52991\\n.53288\\n.53484\\n.53730\\n.53975\\n.54219\\n.54463\\n57\\n33\\n.54463\\n.54707\\n.54950\\n.55193\\n.55436\\n.55677\\n.55919\\n56\\n34\\n.55919\\n.56160\\n.56400\\n.56640\\n.56880\\n.57119\\n.57357\\n55\\n35\\n.57357\\n57595\\n.57833\\n58070\\n.58306\\n.58542\\n.58778\\n54\\n36\\n.58778\\n.59013\\n.59243\\n.59482\\n.59715\\n.59948\\n.60181\\n53\\n37\\n.60181\\n.60413\\n.60645\\n.60876\\n.61106\\n.61336\\n.61566\\n52\\n38\\n.61566\\n.61795\\n.62023\\n.62251\\n.62478\\n.62705\\n.62932\\n51\\n39\\n.62932\\n.63157\\n.63383\\n.63607\\n.63832\\n.64055\\n.64278\\n50\\n40\\n.64278\\n.64501\\n.64723\\n.64944\\n.65165\\n.65386\\n.65605\\n49\\n41\\n.65605\\n.65825\\n.66043\\n.66262\\n66479\\n.66696\\n.66913\\n48\\n42\\n.66913\\n.67128\\n.67344\\n.67559\\n.67773\\n.67986\\n.68199\\n47\\n43\\n.68199\\n.68412\\n.68624\\n68835\\n.69046\\n.69256\\n.69465\\n46\\n44\\n.69465\\n.69674\\n.69883\\n.70093\\n.70298\\n70504\\n70710\\n45\\n60\\n50\\n40\\n30\\n2Y\\n10\\nC\\ni\\n~Deg.\\nNATURAL COSINE.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0166.jp2"}, "165": {"fulltext": "PROVIDENCE. 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I.\\n149\\nNATURAL SINE.\\nBeg.\\n0\\n10\\n20\\n30\\n40\\n50\\n60\\nj\\n45\\n.70710\\n.70916\\n.71120\\n.71325\\n71528\\n.71731\\n.71934\\n44\\n46\\n.71934\\n72185\\n.72336\\n.72537\\n72737\\n72936\\n.73135\\n43\\n47\\n73185\\n73333\\n73530\\n.73727\\n73923\\n.74119\\n.74314\\n42\\n48\\n74314\\n.74508\\n.74702\\n.74895\\n.75088\\n.75279\\n.75471\\n41\\n49\\n.75471\\n.75661\\n75851\\n76040\\n.76229\\n.76417\\n.76604\\n40\\n50\\n.76U04\\n.76791\\n.76977\\n.77162\\n.77347\\n.77531\\n.77714\\n39\\n51\\n.77714\\n.77897\\n.78079\\n.78260\\n78441\\n.78621\\n78801\\n38\\n53\\n.78801\\n.78979\\n.79157\\n.79385\\n.79512\\n.79688\\n79863\\n37\\n53\\n.79863\\n.80038\\n.80212\\n.80385\\n.80558\\n.80730\\n.80901\\n36\\n54\\n.80901\\n.81072\\n.81242\\n.81411\\n.81580\\n.81748\\n.81915\\niiO\\n55\\n.81915\\n.82081\\n.82247\\n.82412\\n.82577\\n.82740\\n.82903\\n34\\n56\\n.82903\\n.83066\\n.83227\\n.83383\\n.83548\\n.83708\\n.83867\\n33\\n57\\n.83867\\n.84025\\n.84182\\n.84339\\n.84495\\n84650\\n.84804\\n32\\n53\\n.84804\\n.84958\\n.85111\\n.85264\\n.85415\\n.85566\\n.85716\\n31\\n59\\n.85716\\n.85866\\n.86014\\n.86162\\n.88310\\n.86456\\n86602\\n30\\nGO\\n.86602\\n.86747\\n.86892\\n.87035\\n.87178\\n.87320\\n.87462\\n29\\nGl\\n.87462\\n.87602\\n87742\\n.87881\\n.88020\\n.88157\\n.88294\\n28\\nG2\\n.88294\\n.88430\\n.88566\\n.88701\\n88835\\n.88968\\n.89100\\n27\\n03\\n.89100\\n.89232\\n.89363\\n.89493\\n.89622\\n.89751\\n.89879\\n\u00c2\u00a36\\n64\\n.89879\\n.90006\\n.90132\\n.90258\\n.90383\\n.90507\\n90630\\n25\\n65\\n.90630\\n.90753\\n.90875\\n.90996\\n.91116\\n.91235\\n.91354\\n24\\nG6\\n.91854\\n.91472\\n.91589\\n.91706\\n.91821\\n.91936\\n.92050\\n23\\n67\\n.92050\\n.92163\\n.92276\\n.92388\\n.92498\\n.92609\\n.92718\\n22\\nG8\\n.92718\\n92827\\n.92934\\n.93041\\n.93148\\n.93253\\n93358\\n21\\nG9\\n.93358\\n.93461\\n.93565\\n.93667\\n.93768\\n.93869\\n93969\\n20\\n70\\n.93969\\n.94068\\n.94166\\n.94264\\n.94360\\n.94456\\n.94551\\n19\\n71\\n.94551\\n.94646\\n.94789\\n.94832\\n.94924\\n.95015\\n.95105\\n18\\n72\\n.95105\\n.95195\\n.95283\\n.95371\\n.95458\\n.95545\\n.95630\\n17\\n73\\n.95630\\n.95715\\n.95799\\n.95882\\n.95964\\n.96045\\n.98126\\n16\\n74\\n.96126\\n.96205\\n.96284\\n96363\\n.96440\\n.96516\\n98592\\n15\\n75\\n.96592\\n.96667\\n.96741\\n.96814\\n.96887\\n.96958\\n.97029\\n14\\n76\\n.97029\\n.97099\\n.97168\\n.97237\\n.97304\\n.97371\\n.97437\\n13\\n77\\n.97437\\n.97502\\n.97566\\n.97623\\n.97692\\n.97753\\n.97814\\n12\\n78\\n.97814\\n.97874\\n.97934\\n.97992\\n.98050\\n.98106\\n.98162\\n11\\n79\\n.98162\\n.98217\\n.98272\\n.98325\\n.98378\\n.98429\\n.98480\\n10\\n80\\n.98480\\n.98530\\n.98580\\n.98628\\n.98676\\n98722\\n.98768\\n9\\n81\\n.98768\\n.98813\\n.98858\\n.98901\\n.98944\\n.98985\\n.99026\\n8\\n82\\n99026\\n.99066\\n.99106\\n.99144\\n.99182\\n.99218\\n.99254\\n7\\n83\\n.99254\\n.99289\\n.99323\\n.99357\\n.99889\\n.99421\\n.99452\\n6\\n84\\n.99452\\n.99482\\n.99511\\n.99539\\n.99567\\n.99593\\n.99619\\n5\\n85\\n.99619\\n.99644\\n.99668\\n.99691\\n.99714\\n99735\\n.99756\\n4\\n86\\n.99756\\n.99776\\n.99795\\n99813\\n.99880\\n.99847\\n.99863\\n3\\n87\\n.99863\\n.99877\\n.99891\\n.99904\\n.99917\\n.99928\\n99939\\n2\\n88\\n.99939\\n.99948\\n.99957\\n.99965\\n.99972\\n.99979\\n.99984\\n1\\n89\\n.99984\\n.99989\\n.99993\\n.99993\\n.99998\\n.99999\\n1.0000\\n60\\n50\\n40\\n30\\n\u00c2\u00a30\\nw\\n0\\nDeg.\\nNATURAL COSINE.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0167.jp2"}, "166": {"fulltext": "150\\nBUOWN SHARPK MFG. CO.\\nNATURAL TANGENT.\\nDej;.\\n0\\n10\\n20\\n80\\n40\\n50\\n60\\n.00000\\n.00290\\n.00581\\n00872\\n.01168\\n.01454\\n.01745\\n89\\n1\\n.01745\\n.02036\\n.02327\\n.02618\\n.02909\\n.03200\\n.03492\\n88\\n2\\n.03492\\n.03783\\n.04074\\n.04366\\n.04657\\n.04949\\n.05240\\n87\\n3\\n.05240\\n.05532\\n.05824\\n.06116\\n.06408\\n.06700\\n.06992\\n86\\n4\\n.06992\\n.07285\\n.07577\\n.07870\\n.08162\\n.08455\\n.08748\\n85\\n5\\n.08748\\n.09042\\n.09335\\n.09628\\n.09922\\n.10216\\n.10510\\n84\\n(5\\n.10510\\n.10804\\n.11099\\n.11393\\n.11688\\n.11983\\n12278\\n83\\n7\\n.12278\\n.12573\\n.12869\\n.13165\\n.13461\\n13757\\n.14054\\n82\\n8\\n.14054\\n.14350\\n.14647\\n14945\\n.15242\\n.15540\\n.15838\\n81\\n9\\n.15838\\n.16136\\n16435\\n16734\\n.17033\\n.17332\\n.17632\\n80\\n10\\n.17632\\n17932\\n18233\\n.18533\\n.18834\\n.19186\\n.19438\\n79\\n11\\n19438\\n19740\\n.20042\\n.20345\\n.20648\\n.20951\\n.21255\\n78\\n12\\n.21255\\n.21559\\n.21864\\n.22169\\n.22474\\n.22780\\n.23086\\n77\\n13\\n.23086\\n.23393\\n.23700\\n.24007\\n.24315\\n.24624\\n.24932\\n76\\n14\\n.24932\\n.25242\\n.25551\\n.25861\\n.26172\\n.26483\\n.26794\\n75\\n15\\n.26794\\n.27106\\n.27419\\n.27732\\n.28046\\n.28360\\n.28674\\n74\\n16\\n.28674\\n28989\\n.29305\\n.29621\\n.29938\\n.30255\\n.30578\\n73\\n17\\n.30573\\n.30891\\n.31210\\n.31529\\n.31850\\n.32170\\n.32492\\n72\\n18\\n.32492\\n.32813\\n.33136\\n.33459\\n.33783\\n.34107\\n.34432\\n71\\n19\\n.34482\\n34758\\n.35084\\n.35411\\n.85789\\n.36067\\n.36397\\n70\\n20\\n.36397\\n.3672(5\\n.37057\\n.37388\\n.37720\\n38053\\n.38386\\n69\\n21\\n.38386\\n.38720\\n.39055\\n.39391\\n.39727\\n.40064\\n.40402\\n68\\n22\\n.40402\\n.40741\\n.41080\\n.41421\\n.41762\\n.42104\\n.42447\\n67\\n23\\n.42447\\n.42791\\n.43135\\n.43481\\n.43827\\n.44174\\n.44522\\n60\\n24\\n.44522\\n.44871\\n.45221\\n45572\\n.45924\\n.46277\\n.46630\\n65\\n25\\n.46630\\n46985\\n.47341\\n.47697\\n.48055\\n.48413\\n.48773\\n64\\n26\\n.48773\\n.49133\\n.49495\\n.49858\\n.50221\\n.50586\\n.50952\\n63\\n27\\n.50952\\n.51319\\n.51687\\n52056\\n.52427\\n52798\\n.53170\\n62\\n28\\n.53170\\n.53544\\n.53919\\n54295\\n54672\\n.55051\\n.55430\\n61\\n29\\n.55480\\n.55811\\n.56195\\n.56577\\n.56961\\n57847\\n.57735\\n60\\n30\\n.57785\\n.58123\\n.58513\\n.58904\\n.59297\\n59690\\n60086\\n59\\n31\\n.60086\\n.60482\\n.60880\\n.61280\\n.61680\\n.62083\\n62486\\n58\\n32\\n62486\\n.62892\\n.63298\\n.63707\\n.64116\\n.64528\\n.64940\\n57\\n33\\n.64940\\n.65355\\n.65771\\n.66188\\n.66607\\n.67028\\n.67450\\n5,5\\n84\\n.67450\\n.67874\\n.68300\\n.68728\\n.69157\\n69588\\n.70020\\n55\\n35\\n.70020\\n.70455\\n.70891\\n71329\\n.71769\\n.72210\\n.72654\\n5-1\\n36\\n.72654\\n.73099\\n.73546\\n.73996\\n.74447\\n.74900\\n.75355\\n55\\n37\\n.75355\\n.75812\\n.76271\\n76732\\n77195\\n77661\\n.78128\\n52\\n38\\n.78128\\n.78598\\n.79069\\n.79543\\n.80019\\n.80497\\n80978\\n51\\n31)\\n.80978\\n.81461\\n.81946\\n82433\\n.82923\\n.83415\\n.83910\\n50\\n40\\n.83910\\n.84406\\n.84906\\n85408\\n.85912\\n.86419\\n86928\\n40\\n41\\n.86928\\n.87440\\n87955\\n.88472\\n88992\\n.89515\\n.90040\\n48\\n42\\n.90040\\n.90568\\n.91099\\n.91633\\n.921(59\\n.92709\\n.93251\\n47\\n4:5\\n.93251\\n.93796\\n.94845\\n.94896\\n.95450\\n.96008\\n.96568\\n46\\n44\\n.96568\\n.97132\\n.97699\\n.982(59\\n.118843\\n.99419\\n1.0000\\n45\\nCO\\n50\\n40\\n30\\n20\\n10\\n0\\nDeg.\\nNATURAL COTANGENT.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0168.jp2"}, "167": {"fulltext": "PROVIDENCE, R. I.\\n151\\nNATURAL TANGENT.\\nBeg.\\no\\n10\\n20\\n30\\n40\\n50\\n60\\n45\\n1.0000\\n1.0058\\n1.0117\\n1.0176\\n1.0235\\n1.0295\\n1.0355\\n44\\n46\\n1.0355\\n1.0415\\n1.0476\\n1.0537\\n1.0599\\n1.0661\\n1.0723\\n43\\n47\\n1.0723\\n1.0786\\n1.0849\\n1.0913\\n1.0977\\n1.1041\\n1.1106\\n42\\n48\\n1.1106\\ni.-;m\\n1 1236\\n1 1302\\n1.1369\\n1.1436\\n1.1503\\n41\\n49\\n1.1503\\n1.1571\\n1 1639\\n1.1708\\n1.1777\\n1.1847\\n1.1917\\n40\\n50\\n1.1917\\n1.1988\\n1.2059\\n1.2131\\n1.2203\\n1.2275\\n1 2349\\n39\\n51\\n1.2349\\n1.2422\\n1.2496\\n1.2571\\n1.2647\\n1.2723\\n1.2799\\n38\\n52\\n1.2799\\n1.2876\\n1.2954\\n1.3032\\n1.3111\\n1.3190\\n1.3270\\n37\\n53\\n1.3270\\n1.3351\\n1.3432\\n1.3514\\n1.3596\\n1.3680\\n1.3763\\n36\\n54\\n1.3763\\n1.3848\\n1.3933\\n1.4019\\n1.4106\\n1.4193\\n1.4281\\n35\\n55\\n1.4281\\n1.4370\\n1.4459\\n1.4550\\n1.4641\\n1.4733\\n1.4825\\n34\\n56\\n1 4825\\n1.4919\\n1.5013\\n1.5108\\n1.5204\\n1.5301\\n1.5398\\n33\\n57\\n1.5398\\n1.5497\\n1.5596\\n1.5696\\n1.5798\\n1.5900\\n1.6003\\n32\\n58\\n1.6003\\n1.6107\\n1.6212\\n1.6318\\n1.6425\\n1.6533\\n1.6642\\n31\\n59\\n1.6642\\n1.6753\\n1.6864\\n1.6976\\n1 7090\\n1.7204\\n1.7320\\n30\\n60\\n1.7320\\n1.7437\\n1.7555\\n1 7674\\n1.7795\\n1.7917\\n1.8040\\n29\\n61\\n1.8040\\n1.8164\\n1.8290\\n1.8417\\n1.8546\\n1.8676\\n1.8807\\n28\\n62\\n1.8807\\n.1.8940\\n1.9074\\n1.9209\\n1.9347\\n1.9485\\n1.9626\\n27\\n63\\n1.9626\\n1.9768\\n1.9911\\n2.0056\\n2.0203\\n2.0352\\n2.0503\\n26\\n64\\n2.0503\\n2.0655\\n2.0809\\n2.0965\\n2.1123\\n2.1283\\n2.1445\\n25\\n65\\n2.1445\\n2.1609\\n2.1774\\n2.1943\\n2.2113\\n2.2285\\n2.2460\\n24\\n66\\n2.2460\\n2.2637\\n2.2816\\n2.2998\\n2.3182\\n2.3369\\n2.3558\\n23\\n67\\n2.3558\\n2.3750\\n2.3944\\n2.4142\\n2.4342\\n2.4545\\n2.4750\\n22\\n68\\n2.4750\\n2.4959\\n2.5171\\n2.5386\\n2.5604\\n2.5826\\n2.6050\\n21\\n69\\n2.6050\\n2.6279\\n2.6510\\n2.6746\\n2.6985\\n2.7228\\n2.7474\\n20\\n70\\n2.7474\\n2.7725\\n2.7980\\n2.8239\\n2.8502\\n2.8770\\n2.9042\\n19\\n71\\n2.9042\\n2.9318\\n2.9600\\n2.9886\\n3.0178\\n3.0474\\n3.0776\\n18\\n72\\n3.0776\\n3.1084\\n3.1397\\n3.1715\\n3.2040\\n3.2371\\n3.2708\\n17\\n73\\n3.2708\\n3.3052\\n3.3402\\n3.3759\\n3.4123\\n3.4495\\n3.4874\\n16\\n74\\n3.4874\\n3.5260\\n3.5655\\n3.6058\\n3.6470\\n3.6890\\n3.7320\\n15\\n75\\n3.7320\\n3.7759\\n3.8208\\n3.8607\\n3.9136\\n3.9616\\n4.0107\\n14\\n76\\n4.0107\\n4.0610\\n4.1125\\n4.1653\\n4.2193\\n4.2747\\n4.3314\\n13\\n77\\n4.3B14\\n4.3896\\n4.4494\\n4.5107\\n4.5736\\n4.6382\\n4.7046\\n12\\n78\\n4.7046\\n4.7728\\n4.8430\\n4.9151\\n4.9894\\n5.0653\\n5.1445\\n11\\n79\\n5.1445\\n5.2256\\n5.3092\\n5.3955\\n5.4845\\n5.5763\\n5.6712\\n10\\n80\\n5.6712\\n5.7693\\n5.8708\\n5.9757\\n6.0844\\n6.1970\\n6.3137\\n9\\n81\\n6.3137\\n6.4348\\n6.5605\\n6.6911\\n6.8269\\n6.9682\\n7.1153\\n8\\n82\\n7.1153\\n7.2687\\n7.4287\\n7.5957\\n7.7703\\n7.9530\\n8.1443\\n7\\n83\\nS.1443\\n8.3449\\n8.5555\\n8.7768\\n9.0098\\n9.2553\\n9.5143\\n6\\n84\\n9.5143\\n9.7881\\n10.078\\n10.385\\n10.711\\n11.059\\n11.430\\n5\\n85\\n11.430\\n11.826\\n12.250\\n12.706\\n13.196\\n13.726\\n14.300\\n4\\n86\\n14.300\\n14.924\\n15.604\\n16.349\\n17.169\\n18.075\\n19.081\\n3\\n87\\n19.081\\n20.205\\n21.470\\n22.904\\n24.541\\n26.431\\n28.636\\n2\\n88\\n28.636\\n31.241\\n34.367\\n38.188\\n42.964\\n49.103\\n57.290\\n1\\n89\\n57.290\\n68.750\\n85.939\\n114.58\\n171.88\\n343 77\\n00\\n60\\n50\\nl\\n40\\n30\\n20\\n10\\n0\\nDeg.\\nNATURAL COTANGENT.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0169.jp2"}, "168": {"fulltext": "152\\nBROWN J: SHARPE MFG. CO.\\nNATURAL SECANT.\\n1\\nBeg.\\n0\\n10\\n20 30\\n4/\\n50\\n60\\n1.0000\\n1 0000\\n1.0000\\n1 0000\\n1 0000\\n1.0001\\n1 0001\\n89\\n1\\n1.000L\\n1 0002\\n1 0002\\n1.0003\\n1.0001\\n1 0005\\nl.OOOo\\n88\\n2\\n1 0006\\n1.0007\\n1 0008\\n1 0009\\n1.0010\\n1.0012\\n1.0013\\n87\\n3\\n1.0013\\n1 0015\\n1.0016\\n1.0018\\n1.0020\\n1 0022\\n1.0024\\n86 i\\n4\\n1 0024\\n1 002a\\n1.0028\\n1.0030\\n1.0033\\n1.0035\\n1.0033\\n85\\n5\\n1.0038\\n1 0040\\n1 0043\\n1.0016\\n1.0049\\n1.0052\\n1 0055\\n84\\n6\\n1.0055\\n1 0058\\n1.0031\\n1.0064\\n1.0068\\n1.0071\\n1.0075\\n83\\n7\\ni 1.0075\\n1.0078\\n1 0082\\n1 0086\\n1.0090\\n1 0094\\n1 0098\\n82\\n8\\n1.0098\\n1.0102\\n1.0103\\n1.0111\\n1.0115\\n1.0120\\n1.0124\\n1 81\\n9\\n1.0124\\n1.0129\\n1.0134\\n1 0139\\n1.0144\\n1 0149\\n1.0154\\ni 80\\n10\\n1.0154\\n1.0159\\n1.0164\\n1.0170\\n1.0175\\n1.0181\\n1.0187\\n79\\n11\\n1.0187\\n1 0192\\n1.0198\\n1.0204\\n1.0210\\n1.02L7\\n1 0223\\n1 78\\n12\\n1 0223\\n1.0229\\n1 02:36\\n1.0242\\n1 0249\\n1 0256\\n1.0263\\n77\\nu\\n1 0263\\n1.0269\\n1.0277\\n1 0284\\n1.0291\\n1 0298\\n1.0303 1\\n76\\n14\\n1.0303\\n1.0313\\n1.0321\\n1 0329\\n1.0336\\n1 0344\\n1.0352\\n75\\n15\\n1.0352\\n1.0380\\n1.0369\\n1.0377\\n1 0385\\n1 0394\\n1.0402\\n74\\n1(3\\n1.0402\\n1 0411\\n1 0420\\n1.0429\\n1.0438\\n1.0147\\n1 0456\\n73\\n17\\n1 0456\\n1.0436\\n1 0475\\n1.0485\\n1 0494\\n1.0504\\n1.0514\\n1 72\\n18\\n1.0514\\n1.0524\\n1 0534\\n1 0544\\n1 0555\\n1.0565\\n1.0576\\n71\\n19\\n1.0576\\n1.0586\\n1.0597\\n1.0:08\\n1.0319\\n1 0630\\n1.0641\\n70\\n20\\n1.0641\\n1 0853\\n1.06C4\\n1.0. i76\\ni.o::87\\n1 0699\\n1.0711\\nC9\\n21\\ni.07ii\\n1.0723\\n1 0735\\n1 0747\\n1.0760\\n1.0772\\n1.0785\\n68\\n23\\n1.0785\\n1.0798\\n1.0810\\n1.0823\\n1.0837\\n1 0850\\n1.0833\\n67\\n23\\n1 0863\\n1.0877\\n1 0890\\n1.0904\\n1.0918\\n1.0932\\n1.0946\\n66\\n1 24\\n1.0946\\n1.0960\\n1 0974\\n1.0989\\n1.1004\\n1.1018\\n1.1033\\n65\\n25\\n1 1033\\n1.1048\\n1.1063\\n1.1079\\n1.1094\\n1.1110\\n1.1126\\n64\\n26\\n1.1126\\n1.1141\\n1.1157\\n1.1174\\n1.1190\\n1.1206\\n1.1223\\n63\\n27\\n1.1223\\n1.1239\\n1.1256\\n1.1273\\n1.1290\\n1.1308\\n1.1125\\n62\\n28\\n1.1325\\n1 1343\\n1.1361\\n1.1378\\n1.1396\\n1.1415\\n1.1433\\n61\\n29\\n1.1433\\n1.1452\\n1.1470\\n1 1489\\n1.1508\\n1 1527\\n1 1547\\n60\\n30\\n1.1547\\n1 1566\\n1.1583\\n1.1605\\n1 1625\\n1.1646\\n1.1665\\n59\\n1 3L\\n1.1666\\n1.1686\\n1.1707\\n1.1723\\n1.1749\\n1.1770\\n1.1791\\n58\\n32\\n1.1791\\n1.1813\\n1.1835\\n1.1856\\n1.1878\\n1.1901\\n1.1923\\n57\\n33\\n1.1923\\n1.1946\\n1 1969\\n1.1992\\n1.2015\\n1.20-8\\n1 2062\\n56\\n34\\n1.2032\\n1.2085\\n1.2109\\n1.2134\\n1.2158\\n1.2182\\n1.2207\\n55\\n35\\n1.2207\\n1.2232\\n1.2257\\n1.2283\\n1.2308\\n1.2334\\n1.2360\\n54\\n3)\\n1.2360\\n1.2386\\n1.2413\\n1.2440\\n1 24 56\\n1.2494\\n1.2521\\n53\\n37\\n1.2521 i 1.2548\\n1.2576\\n1.2304\\n1.2632\\n1.2661\\n1.2690\\n52\\n33\\n1.26v 9 1 1.2719\\n1.2748\\n1.2777\\n1.2807\\n1.2837\\n1.2887\\n51\\n39\\n1.2867\\n1.2898\\n1.2328\\n1.2959\\n1.2990\\n1.3022\\n1.3054\\n50\\n49\\n1.3054\\n1.3086\\n1.3118\\n1.3150\\n1=3183\\n1.3216\\n1.3250\\n49\\n41\\n1.32-30\\n1.3283\\n1.3317 i\\n1.3351\\n1.3336\\n1.3421\\n1.3456\\n48\\n42\\n1 3456\\n1.3491\\n1.3527\\n1.3563\\n1.3599\\n1.3636\\n1.3673\\n47\\n43\\n1 3673\\n1.3710\\n1.3748\\n1.3785\\n1.3824\\n1.3862\\n1.3901\\n46\\n44\\n1.3901\\n1.3940\\n1.39S0\\n1.4020\\n1.40J0 1.4101\\n1.4142\\n45\\n1\\nGO\\n50\\n10\\n30\\n20 10\\nDeg.\\nNATURAL COSECANT.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0170.jp2"}, "169": {"fulltext": "PROVIDENCE, R. I.\\n153\\nNATURAL SECANT.\\n1\\nBeg.\\n0\\n10\\n20\\n30\\n40\\n50\\n60\\n45\\n1.4142\\n1.4183\\n1.422j\\n1.4267\\n1.4309\\n1.4352\\n1.4395\\n44\\n46\\n1.4395\\n1.4439\\n1.4483\\n1.4527\\n1.4572\\n1.4617\\n1.4662\\n43\\n47\\n1.4662\\n1.4708\\n1.4755\\n1.4801\\n1.4849\\n1.4896\\n1.4944\\n42\\n48\\n1.4944\\n1.4993\\n1 5042\\n1.5091\\n1.5141\\n1.5191\\n1.5242\\n41\\n49\\n1.5242\\n1.5293\\n1.5345\\n1.5397\\n1.5450\\n1 5503\\n1.5557\\n40\\n50\\n1.5557\\n1.5611\\n1.5666\\n1.5721\\n1.5777\\n1.5833\\n1.5890\\n39\\n51\\n1.5890\\n1.5947\\n1.6005\\n1.(3063\\n1.6122\\n1.6182\\n1.6242\\n38\\n52\\n1.6242\\n1.6303\\n1.6364\\n1.6426\\n1.6489\\n1.6552\\n1.6616\\n37\\n53\\n1.6616\\n1.6680\\n1 6745\\n1.6811\\n1.6878\\n1.6945\\n1.7013\\n36\\n54\\n1.7013\\n1.7081\\n1.7150\\n1.7220\\n1.7291\\n1.7362\\n1.7434\\n35\\n55\\n1.7434\\n1.7507\\n1.7580\\n1.7655\\n1.7730\\n1.7806\\n1.7882\\n34\\n56\\n1.7882\\n1.7960\\n1.8038\\n1.8118\\n1.8198\\n1.8278\\n1.8360\\n1 33\\n57\\n1.8360\\n1.8443\\n1.8527\\n1.8611\\n1.8697\\n1.8783\\n1.8870\\n1 32\\n58\\n1.8870\\n1.8959\\n1.9048\\n1.9138\\n1.9230\\n1.9322\\n1.9416\\n31\\n59\\n1.9416\\n1.9510\\n1.9603\\n1.9702\\n1.9800\\n1.9899\\n2.0000\\n30\\n60\\n2.0000\\n2.0101\\n2.0203\\n2 0307\\n2.0412\\n2.0519\\n2.0826\\n29\\n61\\n2.0623\\n2.0735\\n2.0845\\n2.0957\\n2.1070\\n2.1184\\n2.1300\\n28\\n62\\n2.1300\\n2.1417\\n2.1536\\n2.1656\\n2.1778\\n2.1901\\n2.2026\\n27\\n63\\n2.2026\\n2.2153\\n2.2281\\n2.2411\\n2.2543\\n2.2376\\n2.2811\\n26\\n64\\n2.2811\\n2.2948\\n2.3087\\n2.3228\\n2.3370\\n2.3515\\n2.3662\\n25\\n65\\n2.3662\\n2.3810\\n2.3961\\n2.4114\\n2.4239\\n2.4426\\n2.4585\\n24\\n66\\n2.4585\\n2.4747\\n2.4911\\n2.5078\\n2.5247\\n2.5418\\n2.5593\\n23\\n67\\n2.5593\\n2.5769\\n2.5949\\n2.6131\\n2.6316\\n2.6503\\n2.6694\\n22\\n68\\n2.6694\\n2.6883\\n2.7085\\n2.7285\\n2.7488\\n2.7694\\n2.7904\\n21\\n69\\n2.7904\\n2.8117\\n2.8334\\n2.8554\\n2.8778\\n2.9006\\n2.9238\\n20\\n70\\n2.9238\\n2.9473\\n2.9713\\n2.9957\\n3.0205\\n3.0458\\n3.0715\\n19\\n71\\n3.0715\\n3.0977\\n3.1243\\n3.1515\\n3.1791\\n3.2073\\n3.2360\\n18\\ni 72\\n3.2360\\n3.2853\\n3.2951\\n3.3255\\n3.3564\\n3.3880\\n3.4203\\n17\\n1 73\\n3.4203\\n3.4531\\n3.4867\\n3.5209\\n3.5558\\n3.5915\\n3.6279\\n16\\n74\\n3.6279\\n3.6651\\n3.7031\\n3.7419\\n3.7816\\n3 8222\\n3.8637\\n15\\n75\\n3.8337\\n3.9061\\n3.9495\\n3.9939\\n4.0393\\n4.0859\\n4.1335\\n14\\n76\\n4.1335\\n4.1823\\n4.2323\\n4.2836\\n4.3362\\n4.3901\\n4.4454\\n13\\n77\\n4.4454\\n4.5021\\n4.5604\\n4.6202\\n4.6816\\n4.7448\\n4.8097\\n12\\n78\\n4.8097\\n4.8764\\n4.9451\\n5 0158\\n5.0886\\n5.1635\\n5.2408\\n11\\n79\\n5.2408\\n5.3204\\n5.4026\\n5.4874\\n5.5749\\n5.6653\\n5.7587\\n10\\n80\\n5.7587\\n5.8553\\n5.9553\\n6.0588\\n6.16C0\\n6.2771\\n6.3924\\n9\\n81\\n6.3924\\n6.5120\\n6.6363\\n6.7654\\n6.8997\\n7.0396\\n7.1852\\n8\\n82\\n7.1852\\n7.3371\\n7.4957\\n7.6612\\n7.8344\\n8.0156\\n8.2055\\n17\\ni\\n83\\n8.2055\\n8.4046\\n8.6137\\n8.8336\\n9.0651\\n9.3091\\n9.5667\\n6\\n84\\n9.5667\\n9.8391\\n10.127\\n10.433\\n10.758\\n11.104\\n11.473\\n5\\n85\\n11.473\\n11.868\\n12.291\\n12.745\\n13.234\\n13.763\\n14.335\\n4\\n86\\n14.335\\n14.957\\n15.633\\n16.380\\n17.198\\n18.102\\n19.107\\n3\\n87\\n19.107\\n20.230\\n21.493\\n22.925\\n24.562\\n26.450\\n28 653\\n2\\n88\\n28.653\\n31.257\\n34.382\\n38.201\\n42 975\\n49.114\\n57.298\\n1\\n89\\n57.298\\n68.757\\n85.945\\n114.59\\n171.88\\n343.77\\n00\\nGO\\n50\\n43\\n80\\n20\\n10\\n0\\nDeg.\\nNATURAL COSECANT.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0171.jp2"}, "170": {"fulltext": ":.:.4\\nSHABPE MFG.\\n:o.\\nTABLE OF DECIMAL EQIIVALEXTS\\nIHILIMZTEES AXI FRACTIONS OF MILLrAEETEP.-.\\ninm. Inehi s.\\n1\\n5 0\\n_a_.\\n\u00e2\u0080\u00a2f\\n3\\n5\\n50\\n5\\n50\\n:_\\n9\\n50\\n1\\n1 1\\ni\\n1 3_-\\nJi\\nI\\n1 5\\n1 6-\\nLA\\n50\\n1 S\\nS\\ngo\\nf\\nt.\\nr\\n|S\\n5\\n.0007!\\n.00157\\n236\\n.00315\\n.00394\\n0551\\n.001\\n.007\\n.--7-7\\n00945\\n.01024\\n.01102\\n.01181\\n.012\\n01339\\n.01417\\n014\\n01575\\n-1654\\n.017;: 2\\n.01811\\n.01890\\n01\\nmm. Inches.\\n2047\\n02126\\n02205\\n02283\\n02362\\n02441\\n02520\\n02598\\n02677\\n02756\\n02835\\n02913\\n02992\\n03071\\n03150\\n03228\\n03307\\n033-6\\n03465\\n03543\\n03622\\n03701\\n03780\\n03858\\n03937\\na\\nLi-\\nI s\\n30\\n3J_\\n3A\\n3 5\\n6-\\nn\\na _\\nI 8.-\\n50\\n40-\\n1 2-\\n50\\n43-\\n50\\nS\\nti-\\n5\\n4 S-\\n4\\n1\\nmrn.\\nInches.\\n2\\n.07874\\n3\\n.11811\\n4\\n.15748\\n5=\\n.19685\\n6=\\n.2362-\\ni\\n.2755V\\n8\\n.31496\\n35433\\n10=\\n.39370\\n11\\n433-7\\n12=\\n.47244\\n13=\\n.51181\\n14=\\n.55118\\n15=\\n.59055\\n16=\\n.62992\\n17=\\nI\\n18=\\n19=\\n.74-\\n20=\\n.7874C\\n21\\n.-2677\\n22=\\n34 S14\\n23\\n.90551\\n24=\\n.94488\\n25=\\n-425\\n26 1\\n.02:;\\n10 mm. l Centimeter=0.3937 in es\\n=1 Decimeter =c 2\\n10 dm. =1 Meter =39.37\\n2~ A mm.=l Enirli h Inch.", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0172.jp2"}, "171": {"fulltext": "INDEX.\\nA.\\nPACK\\nAbbreviations of Parts of Teeth and Gears 4\\nAddendum 2\\nAngle, How to Lay OS. an SB, 107\\nAngle Increment 106\\nAngle of Edge 102\\nAngle of Face 104\\nAngle of Pressure 137\\nAngle of Spiral 113\\nAngular Velocity 2\\nAnnular Gears 32, 139\\nArc of Action 138\\nB.\\nBase Circle 11\\nBase of Epicycloidal System 25\\nBase of Internal Gears 130\\nBevel Gear Blanks 34\\nBevel Gear Cutting on B. S. Automatic Gear Cutter 52\\nBevel Gear Angles by Diagram 36\\nBevel Gear Angles by Calculation 102, 106\\nBevel Gear, Form of Teeth of 41\\nBevel Gear, Whole Diameter of 36, 104\\nC.\\nCenters, Line of 2\\nCircular Pitch 4\\nClassification of Gearing 5\\nClearance at Bottom of Space G\\nClearance in Pattern Gears 8\\nCondition of Constant Velocity Katio 2\\nContact, Arc of 138\\nContinued Fractions 132\\nCoppering Solution 85\\nCutters, How to Order 83\\nCutters, Table of Epicycloidal 8t", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0173.jp2"}, "172": {"fulltext": "15G INDEX.\\nPAGE.\\nCutters, Table of Involute 82\\nCutters, Table of Speeds for 81\\nCutting Bevel Gears on B. S. Automatic Gear Cutter 52\\nCutting Spiral Gears on a Universal Milling Machine 122\\nD.\\nDecimal Equivalents, Tables of 143, 154\\nDiameter Increment 101\\nDiameter of Pitch Circle G\\nDiameter Pitch 5\\nDiametral Pitch 17\\nDistance between Centers 8\\nE.\\nElements of Gear Teeth 5\\nEpicycloidal Gears, with more and. less than 15 Teeth 30\\nEpicycloidal Gears, with 15 Teeth 25\\nEpi cycl oid al Rack 27\\nF.\\nFace, Width of Spur Gear 80\\nFlanks of Teeth in Low-numbered. Pinions 20\\nG.\\nGear Cutters, How to Order 83\\nGear Patterns 8\\nGearing Classified 5\\nGears, Bevel 31, 41, 102\\nGears, Epicycloidal 25\\nGears, Involute 9\\nGears, Spiral 109, 122\\nGears, Worm G3\\nII.\\nHerring-bone Gears 130\\nI.\\nIncrement, Angle 100\\nIncrement, Diameter 104\\nInterchangeable Gears 24\\nInternal or Annular Gears 32, 139\\nInvolute Gears, 30 Teeth and over 9\\nInvolute Gears, with Less than 30 Teeth 20\\nInvolute Rack 12", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0174.jp2"}, "173": {"fulltext": "INDEX. 157\\nL\\nPAGE.\\nLead of a Worm 62\\nLimiting Numbers of Teeth in Internal Gears 32\\nLine of Centers 2\\nLine of Pressure 12, 137\\nLinear Velocity 1\\nM.\\nMachine, B. S., for Cutting Bevel Gears 52\\nN.\\nNormal 11G\\nNormal Helix 116\\nNormal Pitch 116\\n0.\\nOriginal Cylinders 1\\nP.\\nPattern Gears 8\\nPitch Circle 3\\nPitch, Circular or Linear -4\\nPitch, a Diameter 6\\nPitch, Diametral 17\\nPitch, Normal 116\\nPitch of Spirals 112\\nPolygons, Calculations for Diameters of 05\\n11\\nRack 12\\nPack for Epicycloidal Gears 27\\nPack for Involute Gears 12\\nPack for Spiral Gears 121\\nRelative Angular Velocity 2\\nRolling Contact of Pitch Circle 3\\nS.\\nScrew Gearing 109, 130\\nSingle-Curve Teeth 9\\nSpeed of Gear Cutters 81", "height": "4326", "width": "2624", "jp2-path": "practicaltreatis07beal_0175.jp2"}, "174": {"fulltext": "158 INDEX.\\nPAGE.\\nSpiral Gearing 109. 122\\nStandard Templets 27\\nStrength of Gears 142\\nTable of Decimal Equivalents 143, 154\\nTable of Sines, etc 148, 153\\nTable of Speeds for Gear Cutters 81\\nTable of Tooth Parts 144, 147\\nVelocity, Angular 2\\nVelocity, Linear 1\\nVelocity. 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