W ' . "? A HANDBOOK ON THE TEETH OF GEARS, THEIR CURVES, PROPERTIES AND PRACTICAL CONSTRUCTION. WITH ODONTOGRAPHS, FOR BOTH EPICYCLOIDAL AND INVOLUTE TEETH, RULES FOR THE STRENGTH OF TEETH, A TABLE OF PITCH DIAMETERS, AND MUCH OTHER GENERAL INFORMATION ON THE SUBJECT. GEORGE B. GRANT, ee BEVERLY STREET, BOSTON, MASS. PRICE, ONE DOLLAR. Copj-right, 1885, by Geo. B, Grant. LIBRARY OF CONGRESS. mpf — iijpri5¥ 1" -- SheK_.:..(i.'I.C tsi? CNITED STATES OF AMERICA. A HANDBOOK ON THE TEETH OF GEARS, THEIR CURVES, PROPERTIES AND PRACTICAL CONSTRUCTION. WITH ODONTOGRAPHS, FOR BOTH EPICYCLOIDAL AND INVOLUTE TEETH, RULES FOR THE STRENGTH OF TEETH, A TABLE OF PITCH DIAMETERS, AND MUCH OTHER GENERAL I N F O R M A T I _.. _ ^ ON THE SUBJECT. /^v? ^ '^f.aP^RHIMr. ^^T, GEORGE b/grant, \ MAR 11 188] 63 BEVERLY STREET, BOSTON, ^>^&By ^ ^^^ ^ ^^y^^^ PRICE, ONE DOLLAR. Copyright, 1885, by Geo. B. Grant. tK < fc &l. 1^ OV-357% THE TEETH OF GEAR WHEELS INTRODUCTION. Few meclianical subjects have attracted the attention of scientific men to such an extent, or are so intimately connected with mathematics, as tlie proper construction of tlie teetli of gear wheels, and, as a consequence, few can show such an advance as has here been made, from the rough cog wheel of not many years ago, to the neat cut gear of the present day. It is not apparent wherein much further improvement is needed in our knowledge of the theory of the subject, but it is evident that much remains to be done towards its practical application, and to induce the working mechanic to understand and use the improvements that have been developed by the mathematician and the inventor. The theory seems to be full and well nigh perfect, but the mill-wright and the machinist still clings to imperfect rules and clumsy devices that should have been forgotten years ago, and few workmen have a clear Jcnowledge of even the rudiments of the science which it is their business to apply to practical purposes. It is the mathematical and scientific character of the subject that makes it so difi&cult to the j^ractical man, who can understand but little of it as it is commonly presented in elaborate treatises or encyclopaedias, and who takes but little interest in the study of a matter that bristles with strange characters and technical terms. I have here undertaken to address the workman as well as the man of science, and have felt obliged to leave out nearly everything that cannot be treated in a i)lain, descriptive manner, to use language that any intelligent man can understand, and to refer to more pretentious works than this for demonstrations, or unessential details. A volume of a thousand pages would not properly present the whole subject, and this little pamphlet can deal only with the main principles and prominent points. It is not a treatise, it is a hand-book that does not pretend to cover the whole ground, and its principal object is to present the new odontographs, which I believe to be superior to those heretofore in use for the i)urpose of designing the teeth of gear wheels. FIRST PRINCIPLES. The original gear wheel had pins or projections for teeth, of any form that would serve the general purpose ar.d communicate an unsteady motion from one wheel to another. HG. 1. THE ORIGINAL GEAR WHEEL. The perfect gear wheel is the friction wheel, communicating a smooth, uniform, rolling motion, by means of the f rictional j contact of its surface. It is, in fact, a gear wheel with a great many very small, weak, and irregular teeth. The whole aim and object of the science of the teeth of gear wheels is to increase the size and strength of these teeth with- out destroying the uniformity of the motion they transmit, and this is accomplished by studying the shape of the teeth, and giving their bearing surfaces the curved outline that is found to produce the desired result. There are an infinite number of curves that will meet the requiremeht, but only two, the epicycloid and the involute, are of any practical impor- tance, or in actual use. FRICJION VVHEE-S. THE EPICYCLOIDAL TOOTH. The epicycloidal or double curve tooth has its bearing surface formed of two curves, meeting at the pitch line P, which corresponds to the working cir- cle of the perfect gear wheel of fig. 2. If a small circle5a,be rolled around on the outside of the pitch circle, p, a fixed ^ tracing point, a, in its edge, will trace • out the dotted line called an epicycloid, •and a small part of this curve near the I pitch line, usually one sixth of its full iiy/ height,forms the face of the tooth. Similarly, if a small circle, B, be rolled around on the inside of the pitch line, its tracing point, b, will describe the internal epicycloid, or hypocycloid, a small portion of which is used for the flank of the tooth. FIG. 3. THE EPICYCLOIDAL TOOTH. FIG .4 WHOLE EPICYCLOIDAL TEETH. If a projection be formed on the friction wheel fig. 4, the curved outline of which is a whole epicycloid E, and a depression be formed in the wheel N having a whole hypocycloid H for its outline, then, if both curves have been formed by the same describing circle B, it can be mathemati- cally demonstrated that the two curves will just touch and slide on each other, without separating or intersecting, while the two friction wheels roll together. The reverse of this fact is also true; that, if one wheel drives another by means of an epicycloidal projection on it working against a hypocycloidal depression in the other, both curves being formed by the same describing circle, tlie two wheels will roll together as uniformly as if driven by frictional contact, and it is this peculiar property of the epicycloid that gives it its value for the purpose in hand. The pressure acting between the two curves is in the direction of the line dg, is direct only at the start, and becomes more and more oblique, until, when the middle points, q q, come together, and beyond, there is no driving action at all. This defect forbids the use of the whole curve and we can use but a small portion of it near the pitch line. Another projection and depression must be formed so near the first that they will come into work- ing position before the first pair are out of contact, thus forming the theo- retically perfect but incomplete gears of fig. 5. Practical requirements still further modify the apparent shape of the tooth, for it is desirable that the wheels shall work in either direction, and that they shall be interchangea- ble, so that any one of a set of several shall work with any other of that set. This can be accomplished only *by making the curves face both ways. and by putting both projections and depressions on each gear, thus form- ing the familiar tooth of fig. 3. FIQ. 5. INCOMPLETE EPICYCLOIDAL TEETH. THE INTERCHANGEABLE SET. If all the curves of a set of several gears, both the faces and the flanks of each gear, are described by the same rolling circle, the set will be interchangeable, and any one vrill work perfectly with any other. This is a property of the greatest practical importance, and interchangea- ble sets should come into as universal use on heavy mill work as with cut p-ear- ing. It is the only system that will allow the use of a set of ready made cutters, and is therefore essential to the economical manufacture of cut gear wheels. The diameter of the rolling circle is usually made half the diameter of the smallest gear of the set, and that gear will have straight radial lines for flanks. The set in almost universal use and adopted for all the odontographs, has twelve teeth in its smallest gear, but there is a tendency to change this well established system, and create confusion for which the writer can see no adequate excuse, by the adoption of a pinion of fifteen teeth as the base or smallest gear. It may be admitted that as large a base as possible should be used, but the change from twelve to fifteen seems to be unwarranted in view of the confusion it creates by the abrupt change from an old and good rule to a new one that is a mere shade better, and the trouble it makes with small pinions of eight to twelve teeth. RADIAL FLANK TEETH. If the internal curves, or flanks, of a pair of gears that are to run together are on each radial straight lines described by a rolling circle of half its pitch diameter, and the rolling circle that describes the flanks of one gear is used to describe the faces of the other gear, then, the two gears will form a pair fitted to each other and not interchangeable with other gears. This style of gear is very often used under the erroneous impression that it is the best possible form, and will give the least possible friction and thrust on the bearings, but the saving in friction over the interchangeable form would be an exceedingly difficult thing to measure by any practicable method, although it can be mathematically demonstrated to be a fact, and the slender roots of such teeth make them weaker and much inferior to the others. The odontograph figures show both a pair of these gears, and the same pair on the interchangeable plan, also, by the dotted lines on the former figure, the shapes as they would be on the interchangeable plan. It is plainly seen that the interchangeable faces are but a shade more rounding, while their flanks are so curved that the teeth are much stronger at the roots. The larger the describing circle, the less the theoretical thrust and friction, and if the flanks Avere formed by a describing circle of more than half the diam- eter of the gear, the teeth would be undercurved, the friction less, and their strength less, than that of the radial flank tooth. In practical matters it is a good plan to give flrst place to practical points, and not to take too much notice of minute theoretical advantages, and there is no good reason, that will bear the test of experiment, for adopting the radial flank, non-interchangeable, and weak tooth, in preference to the strong tooth of the interchangeable system. THE PITCH. The pitch is a term used to designate the size of the tooth, and is either circular or diametral. THE CIRCULAR PITCH or more properly the circumferential pitch, is the actual distance from tooth to tooth measured along the curve of the pitch line, and is expressed in inches, as f inch pitch, 1| inch pitch, etc. The table gives the proper pitch diameter of a gear of any given number of teeth, and one inch circular pitch. The tabular numbers must be multi- plied by any other pitch that is in use. Formerly, the circular pitch was the only one known, but it has deser- vedly gone out of use on cut gears, and it is hoped may soon be abandoned altogether. It is a clumsy, awkward, and troublesome device on either large or small work, having its origin in the ignorance of the past, and owing its existence not to any perceptible merit, but to habit, and the natural per- sistence of an established custom. With the circular pitch the relation between the pitch diameter of the gear, and the number of teeth on it, is fractional. If the diameter is a convenient quantity, such as a whole number of inches, the pitch must be an inconvenient fraction, and if the pitch is a handy part of an inch, the diameter will contain an unhandy decimal. With the circular pitch there is no one length of tooth that is better than any other, and consequently there is no agreement upon that point. Each maker is at liberty to chose his own distance at random, and whatever he chose s is as good as any other. Its worst feature is that it leads to endless errors, for the average mechanic appreciates convenience more than accuracy, and will stretch his figures to suit his facts, with a botch as the common result. A millwright figures out a diameter of 22.29 inches for a gear of one inch pitch and 70 teeth, and failing to make such a clumsy figure fit his work or his foot rule, and thinking a quarter of an inch or so to be of no importance, he lets it go at 22 whole inches. The same process on its mate of 15 teeth gives a 5 inch gear instead of one of 4.78 inches diameter, and the pair will never run or wear together properly. His only alternatives are to adopt the clumsy true diameters, or else use the clumsy figure .988 inch for his pitch. Again, he is apt to apply a carpenter's rule directly to the teeth of the gear he is to repair or match, and naturally takes the nearest convenient fraction of an inch as his measurement, when the real pitch may be just enough different to spoil the job. There is no reason whatever for using the circular pitch, unless the work to be done is to match work already in use. THE DIAMETRAL PITCH is an immense improvement on the old fashioned circular pitch. It is not a measurement, but a number, or ratio. It is the number of teeth on the gear, for each inch of its pitch diameter, and its merit is that it establishes a convenient and manageable relation between these two principal elements, so that the calculations are of the simplest description and the results convenient and accurate. The product of the pitch and the pitch diameter is equal to the number of teeth, and the number of teeth divided by the pitch is equal to the pitch diame- ter. A gear of 15 inches diameter and 2 pitch has 30 teeth, and a gear of 27 teeth of 4 pitch has a pitch diameter of 6f inches. The rule that the length of the tooth is two pitch parts of an inch, | or i an inch for 4 pitch, f or 1 inch for 2 pitch, etc. is so simple and so much bet- ter than any other that it is never disputed, and is in universal use. The circular and diametral pitches are connected by the relation cXp=3.1416. or, the product of the circular and the diametral pitch is the number 3.1416. THE ADDENDUM. For reasons expressed above we can use but a small part of the epicy- cloidal curve near the pitch line, limiting it by a circle drawn at a distance inside or outside of the pitch line called the addendum. The outside limit need not be the same as the inside limit, but it is customary to make them equal. When the diametral pitch is used, the length of the addendum is always one pitch part of an inch, as Jthinch for 4 pitch, ^rd inch for 3 pitch, etc. If we use the same proportion for circular pitches the addendum will be -^ xir^ circular pitch, and the value ird of the circular pitch may be adopted as the most convenient for use. THE CLEARANCE. Theoretically, the depression formed inside the pitch line should be only as deep as the projection outside of it is high, but to allow for practical defects in the making or in the adjustment of the teeth, and to provide a place for dirt to lodge, the depression is always deeper than theory requires by an amount called the clearance. The amount of the clearance is arbitrary, but the sixteenth part of the depth of the tooth is a convenient and customary measure, or -i^th of the circular pitch, and 1 divided by 8 times the diametral pitch. The following tables will be convenient and save calculation : CLEARANCE FOR CIRCULAR PITCHES. Circular pitch. Clearance. .02 .03 f .03 .04 1 .04 .05 .05 If H .06 .06 If .07 2 .08 2i .09 .10 8 .12 CLEARANCE FOR DIAMETRAL PITCHES. Diametral pitch. Clearance. 6 .02 5 .03 4 .03 H .04 .04 3 .04 2f^ .05 2i .05 .06 2 .06 If .08 .09 H 1 .10 .12 THE BACKLASH. When wooden cogs or rough cast teeth are used, the inevitable irregular- ities require that the teeth should not pretend to fit closely, but that the spaces should be larger than the teeth by an amount called the backlash. The amount of the backlash is arbitrary, but it is customary to make it about equal to the clearance. Cut gears should have no allowance for backlash, and involute teeth need less backlash than epicycloidal teeth. PITCH DIAMETERS. TT-OR ONE HSrCH CIRCTJLA.R FITCH. For Any Other Pitch, Multiply by that Pitch. T. P.D. T. P.D. T. P.D. T. P.D. ]0 3.18 33 10.50 66 17.83 79 25.15 11 3.50 34 10.82 57 18.15 80 25.47 12 3 82 35 11.14 68 18.47 81 25.79 13 4.14 36 11.46 69 18.78 82 26.10 14 4 46 37 11.78 60 19.10 83 26.43 15 4.78 88 12.10 61 19.42 84 26.74 16 6.09 39 12.42 62 19.74 85 27.06 17 6.40 40 12.74 63 20.06 86 27.38 18 6.73 41 13.05 64 20.38 87 27.70 19 6.05 42 13.37 65 20.69 88 28.02 20 6.37 43 13.69 66 21.02 89 28.:^4 21 6 69 44 14.00 67 21.33 90 28.65 22 7.00 45 14.33 68 21.65 91 28.97 23 7.32 46 11.65 69 21.97 92 29.29 24 7.64 47 14.96 70 22.29 93 29.60 25 7.96 48 15.28 71 22^.60 94 29.93 26 8.28 49 15.60 72 22.92 9.i 30.25 27 8 60 50 15.92 73 23.24 96 30.56 28 8.90 51 16.24 74 23.56 97 30.88 2d 923 52 16 56 75 23.88 98 31.20 30 9.55 53 16.87 76 24.20 99 31.52 31 9.87 54 17.19 77 24.52 100 31.84 32 10.19 65 17.62 78 24.83 6 riG. 6. THE EPICYCLOID. THE EPICYCLOID. THEORETICAL FORMATION. The true epicycloid, shown by fig. 6, is perpendicular to the pitch line at the origin a, and forms an endless series of lobes about it, as in fig, 3. The most convenient and simple process for draAving it, is to step it off with the dividers. Several describing circles, M^ to M'^, are drawn at ran- dom ; steps are made, as shown by the figure, from the origin a^ to past each tangent point, a^ to a^, and then the same number back, around each circle, to locate the several points, b^ to b^, on the curve, which is then drawn by hand through the points, and is accurately in place if the steps are sinall. By the mechanical method for drawing the curve, the describing circle, B, is rolled around the pitch circle A, and a tracing point or pencil P, draws the curve. A steel ribbon s, is fastened to the templets at each end, and assists in keep- ing them in place. This process is the main principle of the epicy- cloidal engine, which carries a scribing tool, or a rotary cutter at p, to trace or cut out a tem- plet that is then used in forming gear teeth or gear cutters. It is, of course, the most accurate method known, but it is not available for ordinary pur- poses, for unless the templets are well made and skillfully handled, the resulting curve will be poorly drawn, and the method, although simple in principle, may be consid- ered difficult in its practical application. F^^ACTICAL FORMATION. Of course nothing but the perfect curve will answer its purpose with per- fect accuracy, but the epicycloid is a peculiar curve which cannot be accu- rately drawn by any simple process, or with common instruments, particu- larly when the teeth are small, and it is customary to use arcs of circles or other curves, which approximate as nearly as possible to the true curve. Such an arc can be made to agree with the curve so closely that it is a need- less refinement to be more particular for most practical purposes, such as drafting teeth, making wooden cogs or patterns for cast teeth, or even the templets for shaping gear cutters and planing bevel gear teeth. Some makers of rough cast or heavy planed gearing go to great expense to construct the (supposed to be) theoretically true epicvcloid, by means of rolling circles. This practice looks very much indeed like accuracy, but if he had an absolutely true curve as a templet, supposing he could make such a thing, the maker of this class of work could not produce from it a work- ing tooth more nearly perfect than if the templet was properly constructed of circular arcs. It is labor lost to lay out teeth to the thousandth of an inch, that must be constructed with ordinary hand or machine tools, or shaped with a chisel and mallet. Furthermore, it is a question if the delicate processes and epicycloidal engines used for the finest cut gear work, can serve practical purposes and construct templets to work from, better than intelligent and skillful hand-work. It is a fact that the best work in this line is made from tem- plets that are laid out by theory, but dressed into shape and perfected I y hand and eye processes. FIG. 7. THE EPICYCLOIDAL ENGINE. ODONTOGRAPHS. ^Nlaiiy arbitrary or " rule of thumb " methods for shapinor p^ear teeth have been proiX)secl, but they are p:enerally worthless, and reliance should be r>1aced only on such as are founded on the mathematical prniciples of the curve to be imitated. Of these only three are known to the writer. THE WILLIS ODONTOGRAPH is a method for finding the center m of the circle which is tangent to the epicycloid a b c, at the point b, where it is cut by a line bm, which passes through the adjacent pitch jjoint k, and makes the angle gkf=T5° with the radial line kf. The radius used, is not the line m b, 1-ut the more convenient line m a. The instrument is nothing Avhatever but a piece of card or sheet metal cut to the angle ef 75°, which is laid against the radial line kf, as a guide for drawing the line k m. The center distance k m, to be laid off along the line thus drawn is given by a table that accompanies the instrument. Xo instrument is necessary, for the line k m may be placed by drawing the arc f g with a radius of one inch, and laying off the chord fg=l. 22 inch. The tabular distance km can be readily computed from , c Ki nil — 2Q g t t+12 k^ m^ = 2^^3 t t-12 in which c is the circular pitch in inches, and t is the number of teeth in the gear. , The Willis odontograph, as found in use, is confined to the single case of an interchangeable series rmming from twelve teeth to a rack, but for any possible pair of gears the angle becomes g k f = 90° — i§i° and ki nil = k2 m2 = s c .628 sc t-f s t sill. sill. 180° s 180° .628 t — s s in which t is the number of teeth in the gear being drawn and s the number in the mate. The accuracy of the Willis circular arc will be examined further on. 8 THE IMPROVED WILLIS ODONTOGRAPH. EPICYCLOIDAL TEETH. TWELVE TO RACK. INTERCHANGEABLE SERIES. FOR ONE For one inch NUMBER OF DIAMETRAL PITCH. CIRCULAR PITCH. TEETH For any otliei L" pitch, divide For any other pitch, mul- IN THE GE^E. by that pitch. tiply by that pitch. Faces. Flanks. Faces. Flanks. 1 Exact. Intervals. Rad. Dis. Rad. Dis. Rad. .73 Dis. Rad. Dis. 12 12 2.30 .15 oo oo .05 OO OO m 13-14 2.35 .16 15.42 10,25 .75 .05 4.92 3.26 15i 15-16 2.40 .17 8.38 3.86 .77 .05 2.66 1.24 m 17-18 2.45 .18 6.43 2.35 .78 .06 2.05 .75 20 19-21 2.50 .19 5.38 1.62 .80 .06 1.72 .52 23 22-24 2.55 .21 4.75 1.23 .81 .07 1.52 .39 27 25-29 2.61 .23 4.31 .98 .83 .07 1.36 .31 33 30-36 2.68 .25 3.97 .79 .85 .08 1.26 .26 42 37^8 2.75 .27 3.69 .66 .88 .09 1.8 .21 58 49-72 2.83 .30 3.49 .57 .90 .10 3.0 .18 97 . 73-144 2.93 .33 3.30 .49 .93 .11 1.5 .15 290 145-rack. 3.04 .37 3.18 .42 .97 .12 1.2 ' .13 THE IMPROVED WILLIS ODONTOGRAPH. I have carefully calculated the distances nii n^ and nig n2 of the circles of centers from the pitch line, and also the radii aj m^ and a2 mg, and have arranged them in the table above, so that the data resulting from the usual process can be obtained without the usual labor. This improved Willis process will produce exactly the same circular arc as the usual method, with the same theoretical error, but its operation is simpler and less liable to errors of manipulation. By the usual process it is necessary to draw two radial lines, and to lay off a line at an angle with each. The tabular distances laid off on these lines, will locate the two centers. The two circles of centers are then drawn through them, and the dividers set to the radii to be used. By the new process the circles of centers are drawn at once without pre- liminary constructions, at the tabular distances from the pitch line, and the table also gives the radii to be taken on the dividers. No special instru- ment is required, no angles or special lines are drawn to locate the centers, and the chance of error is much less. This process, however, is not as correct, and is no simpler or more con- venient than the new odontographic process given further on. ROBINSON'S TEMPLET ODONTOGRAPH. This ingenious instrument, the invention of Prof. S. W. Robinson of the Ohio State University at Columbus, is based on the fact that some part of a certain curve of uniformly increasing curvature, called the logarithmic spiral, can be made to agree v^ith the true curve of a gear tooth with a degree of approximation that is very precise. It is a sheet metal templet having a graduated curved edge ac, shaped to a logarithmic spiral, and a hollow edge a b shaped to its evolute, an equal logarithmic spiral. To apply the instrument, draw a radial line from the pitch point d on the pitch line, and another from e, the center of the tooth, and then draw tangents d g and n e f , square with the radial lines. The instrument is then so placed that a certain graduation, given by accompanying tables, is at the point h on the tangent nef, while the grad- uated edge ac, is at the pitch point d, and the hollow edge ab, just touches the tangent line n e f at k, and then the face of the tooth is drawn with a pen along the graduated edge. The flank is similarly located by placing the instrument so that a certain other graduation is at the pitch point d, while its hollow edge touches the tangent line g d. The full theory of this instrument would be out of place here, but may be found in No. 24 of Yan Nostrand's Science Series, or in Yan Nostrand's Mag- azine for July, 1876. 10 A NEW ODONTOGRAPH. Having frequently to apply the "Willis Odontograph, it occurred to me that the process would be much simplified and much time and labor saved if the location of the circles of centers and the lengths of the radii were computed and tabulated, thus forming the improved Willis method already described. It was then evident that the process would be precisely the same, and the result much improved, if the centers tabulated were the centers of the near- est possible approximating circles, rather than of the Willis circles, and 1 have embodied this idea in the following tables. I have carefully computed, by accurate trigonometrical methods, and have tabulated the location of the center of the circular arc that passes through the three most important points on the curve, at the pitch line a, fig. 0, at the addendum line k, and the point e, half way between. The tables locate this center directly, giving its distance from the pitch line, and from the pitch point. The circles of centers are drawn at the tabular distances "dis" inside and outside the jjitch lines, and all the faces and flanks are drawn from centers on these circles, with the dividers set to the tabular radii "rad." The tables are arranged in an equidistant series of twelve intervals. For ordinary purposes the tabular value for any interval can be used for any tooth in that interval, but for greater precision it is exact only for the given "exact" number, and intermediate values must be taken for inter- mediate teeth. The tables are arranged for both the diametral and circular pitch sys- tems. The former is much the more manageable and should be used when the work is not to interchange with work already made on the latter system. The first table, giving an interchangeable set, from twelve teeth upwards, is the one for general use. The second, or radial flank table, is inserted because teeth are sometimes drawn that way, but, as before explained, they a:ft weak, not interchange- able, and but a mere shade more direct in their action than the interchange- able style. ACCURACY OF THE ODONTOGRAPH. The assertion is often made that no circular arc can be made to do duty for the epicycloid, except for rough work, but it can be shown that the state- ment is not true if applied to the new method, for few mechanical processes can be made to work closer to a given example, than this arc is close to the true curve. Figure 9 shows the true curve, and both the new and the Willis aj)proximating arcs, the actual proportions being exagerated to show the errors more clearly. The Willis arc runs altogether within the true curve, while the new arc crosses it twice. We will take, for an example, the case of a twelve tooth pinion, which will show the errors at their greatest, and calculate them with great care for a tooth of three inch circular pitch, which is twice the size of the figure on page 13, and may be considered a very large tooth. The distance from pitch line to addendum line '''°- '• is divided into eight equal spaces by parallel cir- cles, and the distance along each circle, in ten thousandths of an inch, from the true curve to each odontographic arc, is as follows : 11 GRANT. WILLIS. At a .0000 .0000 inches " b +.0088 +.0175 " •' c +.0091 +.0244 " " d +.0056 + .0283 " " e .0000 + .0288 '• " f -.0036 +.0297 " u -.0061 +.0308 " ^' h -.0046 +.0342 " " k .0000 +.0397 " Average, .0042 .0260 " It is seen that the new arc is in no place one hundredth of an inch in error, and that for a tooth of four pitch, a large size for cut work, its average error is one thousandth of an inch. A greater accuracy than this would be of no practical value. The twelve tooth gear, for which the errors of both arcs were com- puted, shows them at their maximum value, for, as the number of teeth in the gear increases, the errors diminish, and for several locations their values for the new arc at c, which is the point of greatest error, are as follows : r t == 12 = .009 inches " 20 a .008 " " 40 i. .006 '' " 100 a .004 '' " 300 a .002 '• and the errors of the Willis arc are subject to the same rule. The error of the Willis arc is plainly shown, at its greatest value, by the figure on page 13, where the dotted faces of the pinion teeth are correctly located by the Willis method. To further test the accuracy of the new method, construct the same tooth face several times by the same process, using either the method by points, or the usual Willis process. Unless the work is most carefully performed, it will be found that the several results will not agree with each other by amounts that are noticeable, while by the new method they will be sub- stantially the same curve. The new arc is most nearly correct at the most important point, the upper part of the curve, just where the Willis arc is most out of place, or where the true curve, unless drawn by some delicate and costly apparatus, in most likely to be out of place. CIRCULAR AND DIAMETRAL PITCHES COMPARED. CIR. P. DM. P. 6 .52 5i .58 5 .63 4i .70 4 .78 3i .90 3 1.05 21 1.15 2h 1.25 H 1.40 2 1.57 11 1.80 U 2.10 u 2.50 1 3.14 1 4.20 h 6.28 DM. P. CIR. P. h 6.28 1 4.20 1 3.14 u 2 50 u 2.10 11 180 2 1.57 2h 1.25 3 1 05 3i .90 4 .78 6 .63 6 .52 7 .45 8 .39 9 .35 10 .31 12 THE NEW ODONTOGRAPH. GENERAL DIRECTIONS. Draw the pitch line and divide it for the pitch points mag. Take from the tables, multiply or divide, as the case may require, by the pitch in use, and lay olf , the addendum a b and a c, the clearance e f , the backlash g- g', the face distance a d, and the flank distance a c. Draw the addendum line through b, the root line through e, the clearance line through f, the line of face centers through d, and the line of flank centers through c. Set the dividers to the face xadius, and draw all the faces ab from centers A. Set to the flank radius, and draw all the flanks a k from centers B. Round the flanks into the clearance line. The flanks of a gear of twelve teeth are straight radial lines. ODONTOGRAPH TABLE. EPICYCLOIDAL TEETH. INTERCHANGEABLE SERIES. From a Pinion of Twelve Teeth to a Rack. FOR ONE FOR ONE INCH J NUMBER OF TEETH DIAm For ar [ETRAL P ETCH. dde hy CIB Fors .CUI.AR PITCH. 1 ly other pitch, di my other pitch, multiply 1 IN THE GEAR. that pitch. 1 hy that pitch. | Faces. Flanks. Faces. Flanks. 1 Exact. Intervals. Rad. Dis. Rad. Dis. Rad. Dis. Rad. Dis. 12 12 2.01 .06 CO CO .64 .02 CO oo 134 13-U 2.04 .07 15.10 9.43 .65 .02 4.80 3.00 15i 15-16 2.10 .09 7.86 3.46 .67 .03 2.50 1.10 17i 17-18 2.14 .11 6.18 2.20 .68 .04 1.95 .70 20 19-21 2.20 .13 5.12 1.57 .70 .04 1.63 .50 23 22-24 2.26 .15 4.50 1.13 .72 .05 1.43 .36 27 25-29 2.33 .16 4.10 .96 .74 .05 1.30 .29 33 30-36 2.40 .19 3.80 .72 .76 .06 1.20 .23 42 37-48 2 48 .22 3.52 .63 .79 .07 1.12 .20 58 49-72 2 60 .25 3 33 .54 .83 .08 1.06 .17 97 73-144 2.83 .28 3.14 .44 .90 .09 1.00 .14 290 145-rack. 2.92 .31 3.00 .38 .93 .10 .95 .12 13 A PRACTICAL EXAMPLE OF THE WORK OF THE NEW ODONTOGRAPH, face dis, = .07. flank " = .54. face " = .03. flank " r= CO INTERCHANGEABLE SERIES. Example. — A gear of 24 teeth, and a gear of 12 teeth, of H circular pitch. Data. — Take from the table the numbers to be used, which are as follows when multiplied by 1^. For 24 teeth, face rad, = 1.08 " 24 '' flank " =2.15 '' 12 " face " — .96 " 12 " flank '' = cc Also take from the proper tables the pitch diameters 5.73 and 11.46 inches, the addendum, .5 inch, and clearance, .06 inch. Pkocess. — Draw the two pitch lines, and divide for the pitch points. Draw the addendum, root, and clearance lines of both gears. Draw the circles of centers, .03 inside the pitch line of the 12 tooth gear, and .07 inside of that of the other. Draw the circles of flank centers, .54 outside the pitch line of the 24 tooth gear, and draw straight radial flanks for the 12 tooth gear. Draw the faces of the 12 tooth gear with the radius. 96, and draw the faces of the 24 tooth gear with the radius, 1.08, and the flanks with the radius 2.15. Kound the flanks into the root line, and allow backlash by thinning the teeth according to judgement. The dotted faces of the 12 tooth gear show them as they would be laid out by the Willis odontograph, and the figure also shows the two centers in place. 14 RADIAL FLANK SYSTEM. TEETH NOT INTERCHANGEABLE. Gears on this system must work together in pairs, each gear being fitted to its mate and to no otner. See page 3. The process is the same that has been described on page 12 for the interchangeable set. RADIAL FLANK SYSTEM. ExPLANATioisr OF THE TABLE. — The Upper number in each square is tlie face radius, the lower is the center distance. The centers are mostly insid the pitch line, but some are on the line, and those having the negative sign are outside of it. The tabular numbers are for one inch circular pitch, and must be multi- plied by any other circular pitch in use. For the value for any diametral pitch, multiply the tabular number by 3.14, and then divide by the diame- tral pitch in use. Example. — A gear of 12 teeth, paired with a gear of 24 teeth. Circular pitch-l^ inches. Data. — Take from the table for 12 teeth into 24, face radius =.68 and cen- ter distance = 0, and for 24 teeth into 12. radius == 72, and distance = .05. These multiplied by 1^ give the values for use on the drawing, 12 rad. =1.02, 12 dis = 0, 24 rad. = 1.08, and 24 dis. = .07. The addendum is one third the pitch, = i inch, and the proper tables give the clearance =.06, and the pitch diameters = 5.73 and 11.46 inches. Process, — Draw the two pitch linos 5.73 and 11.46 inches in diameter and space them for the teeth. Lay off the addendum, .5 inch, and the clearance, .06 inch, and draw the addendum, root, and clearance lines. Draw all the faces of the twelve tooth gear, from centers on its pitch line, with the radius 1.02. Draw all the faces of the 24 tooth gear from centers on a line .07 inch inside its pitch line, with the radius 1.08 inches. Draw straight radial lines for the flanks of all tlie teeth. 15 ODONTOGRAPH TABLE. EPICYCIiOIDAIi TEETH. RADIAL FLANK TABLE. FOR ANY POSSIBLE PAIR OF GEARS, NOT INTERCHANGEABLE. One Inch Circitlar Pitch. For any other pitch, multiply by that pitch. Ntm TEETH BEING Exact. BER OF JN GEAR DRAWN. Intervals NUMBER OF TEETH IN THE MATE. .„ 13 15 17 19 22 25 30 37 49 73 145 ^" 11 16 18 21 24 29 36 48 72 144 rack 12 12 .64 .02 .65 .02 .64 .01 .65 .01 .66 .01 .67 .68 .69 -.01 .70 -.01 .71 -.02 .73 -.02 .74 -.03 .75 -.03 ISh 13-14 .66 .02 .67 .Oi .68 I .69 .01 .01 .70 .72 .74 -.01 .75 -.01 .76 -.02 .78 -.02 .79 -.03 15i 15-16 .67 .03 .68 .02 .69 .02 1 .70 1 .72 .01 .01 .74 .01 .75 ! .78 .79 -.01 .82 -.02 .84 -.02 .84 -.03 m 17-18 .68 .04 .70 .03 .71 .02 .73 .02 .75 .01 .77 .01 .78 .01 .82 .84 -.01 .87 -.01 .89 -.02 .90 -.03 20 19-21 .70 .04 .72 .04 .74 .03 .76 .02 .79 .02 .81 .01 .83 .01 .87 .90 .93 -.01 .96 -.02 .96 -.03 23 22-24 .72 .05 .74 .04 .76 .04 .79 .03 .82 .02 .85 .03 .84 .02 .87 .01 .91 .01 .94 .98 -.01 1.01 -.02 1.03 -.03 27 ' 23-29 .74 .05 .76 .05 .79 .04 .82 .04 .87 .02 .92 .02 .96 .01 .99 1.03 -.01 1.07 -.02 1.10 -.03 33 1 30-36 ! .76 .06 .79 .05 .83 .05 .86 .04 .90 .03 .94 .03 .98 .02 1.02 .01 1.06 1.11 1.17 -.01 1.23 -.02 42 37-48 .79 .07 .83 .06 .86 .05 .90 .05 .96 .04 .98 .04 1.03 .03 1.08 1.14 .03 .02 1.20 1.25 1.37 -.01 58 49-72 .83 .08 .87 .07 .91 .07 .96 .06 1.02 .06 1.05 .05 1.10 .04 1.17 1.24 .04: .03 1.30 .02 1.43 i 1.58 ! 97 73-144 .90 .09 .93 .08 .97 .08 1.01 .07 1.07 .07 1.11 .06 1.18 .06 1.28 1.34 .05 .04 1.47 .03 1.65 .02 2.03 290 145 rack .93 .10 .96 .09 1.00 .09 1.05 .09 1.10 .08 1.16 .08 1.24 .07 1.37 ' 1 50 .07 1 .06 1.70 .04 2.12 .03 2.90 .02 16 THE INVOLUTE TOOTH. With the exception of the epicycloid, the only curve in extensive use for the working face of a gear tooth, is the involute. THE INVOLUTE CURVE. As the rolling circle A of fig. 3 increases in size, it finally, when of infinite diameter, becomes the straight line d g of fig. 15, while the epicycloid traced by a fixed point in the circle becomes the involute. The involute is, therefore, not a new or sep- arate curve, but simply a particular case of the epicycloid. It is the infinite form of the epicy- cloid. As the rolling circle of infinite diameter is the same thing as a straight line, the involute can be formed by a fixed tracing point in a cord which is unwound from a circle, called its "base circle," which has been wrapped or " involved" FIG. 15. THE INVOLUTE. 1^ It, aud from this property it derives its name. ITS UNIFORM ACTION. If the two circles A and B, fig. 16, are separ- ated by the distance ab, and work together by means of two external epicycloids G and D, the motion communicated will be irregular, for the conditions of uniformity are that the two cir- cles shall touch, and that the external curve of one shall work with the internal curve of the other. See page 2 and figure 4. The amount of this irregularity will depend on the proportion between the separating dis- tance ab and the diameter of the rolling circle which describes the epicycloids. If the pro- portion is very small, the irregularity will be very small, and if the rolling circle has an in- finitely great diameter, the proportion and the irregularity will be infinitely small, that is, zero. Therefore, involutes will work together with perfect regularity and are suitable curves for gear teeth. ITS ADJUSTIBILITY. If tlie rolling circle is infinitely large, the proportion between the separat- ing distance and it will always be zero, and it will not be altered by any finite alteration of the former, and therefore the uniformity of the action of involute teeth is not in any way dependent upon, or affected by any change of the separating distance. The action will be perfect as long as the curves remain in contact, and this is a property of the greatest practical value, which gives the involute a great advantage over every other known or pos- sible curve. The curve of any gear tooth must of necessity be a "rolled curve " formed by a fixed object attached to the plane of or moving with some curve that rolls upon the base curve of the tooth, and, as the involute is the infinite form of any rolled curve, it is the only form that can possess this property of adjustibility. EXTERNAL EPICYCLOIDS. 17 ITS UNIFORM PRESSURE AND FRICTION. The point of contact oi the two involutes C and D will always be upon the sti'ai- 288 3.20 2;-.. 86 22.30 .96 8.26 7.3> 576 289-rack 9.60 73.95 70.10 2.88 23.65 22.30 INTERFERENCE FOR TWELVE TO RACK INTERCHANGEABLE SET. Teeth In the gear. 12 13 14 15 17 16 18 19 21 22 24 25 29 30 36 37 48 49 1 73 72 1 144 145 00 Pitch. Amount of the Interference. One in. cir. .003 .007 .007 .007 .007 .007 .010 .010 .010 .013 .017 .020 One diamet'll .01 .02 .02 .02 .02 .02 .03 .03 .03 .04' .05 .00 Interference always to commence at a point half way between pitch line and addendum line. 23 A PRACTICAL EXAMPLE. INVOLUTE TEETH. INTERCHANGEABLE SERIES. Example. — A rack, and a pinion of twelve teeth, of two diametral pitch. Data. — From the tables we have, after dividing by 2, the base distance .10'', face radius 1.35", and flank radius .42'^ The addendum is .5", the clearance .06", and the pitch diameter 6. inches. The limit of interference for the rack is .20", and for the gear .28", or may be assumed at .25" with small error. The amount of interference for the rack is .03", and for the gear .005". PiiOCESS. — Draw the pitch lines and divide for the pitch points, lay oft the addendum and the clearance, and draw the addendum, root, and clear- ance lines. Draw the base line .10" inside the pitch line of the gear. Draw the limit lines .20" and .28" from the pitch lines. With the face radius 3.35" and from the center d on the base line, draw the faces of the gear from pitch line to limit line, and, as the interference is imperceptible in this case, con- tinue it to the addendum line. With the flank radius .42" and from the center b on the base line, draw the flank from pitch line to base line. The flank, inside the base line, is a straight radial line. The face and flank of the rack is a straight line from root line to limit line, at an angle of 75° with the pitch line. From the limit line to the addendum linc^ the face of the rack curves inward, being .03 inch from the true lace at the point. The interference on the pinion tooth is here neglected, because it is very small, but on larger gears it must be accounted for if the gear is to belong- to the interchangeable set. At sixty teeth the root and base lines coincide, and there is no radial flank. 24 BEVEL GEARS. In laying out the teeth of a bevel gear but one new point needs to be con- sidered. The working pitch diameter a b c is not to be used, but the teeth are to be drawn on the conical pitch diameter adc, developed or rolled out as in fig. 25. The conical diameter adc may be found from a drawing, or if the gears are of some common proportion, from the following table by multiplying the true pitch diameters by the tabular numbers given for that proportion TABLE OF CONICi^L PITCH DIAMETERS OF BEVEL GEARS. Proportion. Larger Gear. 1.41 Smaller Gear. 1 to 1 1.41 2 » 1 2.24 1.12 3 " 2 1.80 1.20 3 " 1 3.16 1.05 4 " 3 1.67 1.25 4 " 1 4.12 1.03 5 " 4 1.60 1.28 5 " 3 1.94 1.17 5 " 2 2.69 1.08 5 " 1 5.10 1.02 6 " 5 1.56 1.30 6 « 1 6.08 1.01 7 " 1 7.07 1.01 8 " 1 8.06 1.01 9 " 1 9.06 1.01 10 '' 1 10.05 1.01 Examples. — A miter gear, proportion 1 to 1, of 4 pitch, 6" diameter, and 24 teeth, has a conical diameter of 6'^ x 1.41 = 8.46'^ and there are 24 x 1.41 = 33.8 teeth on the full circle of the developed cone. A pair of bevel gears of 3 to 1 proportion, 48^' and W diameters, 36 and 12 teeth, have conical diameters 48'' x 3.16 — 151.68". and 16" x'5rt6 = 16.80", and there are 36 x 3.16 — 113.76, and 12 x 1.05 = 12.60 teeth on thje full cir- cles of the developed cones. \^S 25 INTERNAL GEARS. The internal gear, sometimes called the "annular" gear, is drawn by the rules for spur gears, the teeth of a spur gear being the spaces between the teeth of an internal gear of the same pitch diameter, with the backlash and clearance reversed in position. Involute teeth should end at the base line, the radial part of the flank being omitted, or well rounded over if it is desirable to preserve the appear- ance of the full tooth. Internal teeth will interfere, even if properly drawn, unless the gear is considerably larger than the pinion running in it. If drawn for the common twelve to rack interchangeable set, there should be at least twelve more teeth in the gear than in the pinion, and if the difference is less, the teeth must be "doctored " or rounded over until they will pass, and there must be a difference of two teeth in any case. Involute teeth have a decided advantage over epicycloidal teeth for inter- nal gearing, their action being much more direct, with less sliding and friction. 26 STRENGTH AND HORSE-POWER OF GEARS. There are about as many different rules for this purpose, and contradictory re- sults, as there are writers upon the subject. I have preferred not to discuss the theory, but to adopt without quesiion the method given by Thomas Box in liis Prac- tical Treatise on Mill Gearing, because that engineer has most carefully considered the practical points in view, and because his formulze agree almost exactly with a great mauy cases in actual practice. STRENGTH OP A TOOTH. —For worm gears, crane gears, and slow-moving gears in general, we have to consider only the dead weight that the tooth can lift with safety. If we allow the iron to be subjected to but one tenth of its breaking strain, we can use the formula: — W = 350 c f, in which "W is the dead weight to be lifted, c is the circular pitch, and f the face, both in inches. For the wooden cogs of mortise wheels, use 120 instead of 350 as a factor in the formula. When the pinion is large enough to insure that two teeth shall always be in fair contact, the load, as found by this rule, may be doubled. Example. — A cast-iron gear of 3" circular pitch and G" face will lift W = 350 X 3 X 6 = 6300 lbs. HORSE-POWER OF A GEAR. — For very low speeds we can use the formula, HP for low speed = .0037 d n c f , in w^hich d is the pitch diameter, c the circular pitch, and f the face, all in inches, and n is the number of revolutions per minute. Example. — The horse-power of a gear of three feet diameter, three inch pitch, and ten inch face, at eight revolutions per minute, is, HP = .0037 X 36 X 8 X 3 X 10 = 32. For ordinary or high speeds, where impact has to be considered, it is found that the above formula gives too high results, and we must use the formula, HP at ordinary speeds = .012 c^ f \/du. Example. - A gear of three feet diameter, three inch pitch and ten inch face, at one hundred revolutions per minute, will carry but HP == .012 X 9 X 10' X VlOO X 36 = 65 horse-power, instead of the 400 horse-power found by the rule for low speeds. , At ordinary or high speeds a wooden cog, on account of its elasticity, will carry as much as or more power than a cast-iron tooth, and we can use .014 instead of .012 in the formula. "When in doubt as to whether a given speed is to be considered high or low, com- pute the horse-power by both formulae, and use the smallest result. For bevel gears the same rules will apply, if we use the pitch diameter and the pitch at the center of the face. Some ru!es in use take no account of the face of the gear, but assume that the tooth should be able to bear the whole strain upon one corner. A tooth that does not bear substantially along its whole face, at several points at least, is a very poor piece of work, and it would be better to straighten the tooih than to force the rule to follow it. 27 THE EQUIDISTANT SERIES. The shape of a tooth is not the same on two gears of different sizes, for its curvature continually decreases and the curve flattens as the number of teeth in the gear increases. When the teeth are formed by a rotary milling tool, we must use a cutter of fixed shape ; when formed by planing, a fixed guide is employed ; and when drawn by an odontograph, flxed tabular data are used ; and obviously, if we require the greatest possible accuracy we must have a different shape of cutter, or guide, or a separate tabular number, for each separate tooth, and at least two hundred in a set to cover the ordinary range of work. As this would be an expensive and clumsy system, it is customary to make one fixed shape do duty for several teeth, being just right for one tooth of a given interval, and approximately so for several teeth either way. This set of fixed intervals is known as the equidistant series, as it so dis- tributes the errors that the greatest error is the same in all the intervals. The equidistant series was invented by Willis, but he gives no rule for arranging it, and the example he gives was apparently found by some experimental method. In the American Machinist for Jan. 8th, 1881, I proposed the location of the dividing points of the series by the formula , a n n-s-4- — I 7. in which a is the first and z the last tooth, usually twelve and infinity, of a series of n intervals, s is the number, in the series, of any particular inter- val, and t is the last tooth in the interval s. This formula uniformly distributes, not the differences in form, but what is for all i^ractical purposes the same thing and much more easily handled, the differences in the lengths of the addendum arcs. It is general in its nature, and independent both of the form of the tooth and of its length, which have but a minute effect on the required series. Any method that recognizes these small differences must necessarily require more intricate and difficult trigonometrical work than the slightly increased accuracy will war- rant.* * In his treatise on Kinematics, Prof. C. W. MacCord has treated my formula in such a summary and unjust manner, that in replying to him I do not feel bound by the usual rules of courtesy, but am at liberty to state the facts in plain words, without fear or favor. He not only refers to my process with an evident attempt at ridicule, but he positively mangles the facts. He is careful to show its defects and to exagerate their importance, while he is equally careful to slight and conceal its real merit. His motive is evident when he next proposes as a substitute a "locus" method which he claims is "the perfect solution of the problem," which will give a series that is "exact to a single tooth," and the value of which he assumes but does not attempt to prove. It is, in fact, an arbitrary approximation, and so wonderfully intricate, clumsy, and inaccurate, that the result, determined by it with great care by its own expert inventor, does not divide the locus curve to a single tooth, or in some parts, within several teeth, or distnbute the errors of form any more unifonnly than does the method it was intended to displace. A full (and free) discussion of this matter may be found in several letters published in the American Machinist in 1884. The locus method gives a result almost identical (for cases in actual use) with the series found by my formula, and if the slight differ- ence can be proved to be in its favor, as has not been done, it is of imperceptible importance, and no offset whatever to the excessive intricacy of the method. I did not claim perfection for my formula, or imagine it worth the notice that has been taken of it, and I would not in ordinary cases criticise the work of any other writer, but as I have been used with unusual and unprovoked seventy, I find it necessary to publish this note in self defence. Both sides of the question are now accessible to any one who may be interested, and all I ask or expect is that my work shall be treated with ordinary fairness, and allowed whatever merit it really has. 28 For the ordinary series of eight intervals, to cover fit>m 12 to oo the form- ula becomes ^ 8-s and if we put s successively equal to 1, 2, 3, 4, 5, 6, 7, and 8, we get the series of last teeth 13f , 16, 191, 24, 32, 48, 96 and oo , the resulting equidistant series being 12 to 13, 25 to 32, 14 to 16, 33 to 48, 17 to 19, 49 to 96, 20 to 24, 97 to a rack. Similarly, if we apply the formula from a=24 to z=x ,for n=12, we get the series adopted above for the involute odontograph table, and it requires but a few figures and a simple operation to apply it to any other case. POSITION OF THE "PERFECT" TOOTH. The "perfect" tooth, whose shape does duty for the whole interval, can best be placed, not at the center of the interval, but by assuming the inter- val to be a short series of two intervals, and adopting the intermediate value, The proper fi-xed shape for the interval from c to d is that of the tooth found by the formula ^ c+d For the interval from 145 to 288 the perfect tooth is the 193rd, instead of the 216th at the center. MAXIMUM ERROR OF THE SERIES. The odontograph gives the correct position of the perfect tooth only, and the point of the tooth at either end of the interval is out of position by the very small amount found by the formula • .182 errors — pn in which p is the diametral pitch, and n is the number of intervals in the series. The odontograph table I have given for epicycloidal teeth has twelve intervals, and the greatest error in the position of the point of anj' tooth drawn by the table is error =— = '— inch. I2P p This becomes .015 inch for one diametral pitch, and .005 inch for one inch circular pitch and in direct proportion for other pitches. For involute teeth this formula becomes error = -2^ and for the given table having twenty-four intervals the greatest error is .006 inch for one diametral pitch, and .002 inch for one inch circular pitch. It is thus seen that the number of intervals used is sufficient for all prac- tical purposes, particularly if the error is still further reduced by adopting intermediate tabular numbers for intermediate numbers of teeth. 29 STANDARD FACES FOR GEAR WHEELS. It is desirable for the sake of uniformity and interchangeability, to have a regular system or law of fixed relation between the size of the teeth and the width of the face of a gear wheel. Such a law is recognized and in general use in a loose way, is a law of common sense, in fact, for it is almost invariably the custom to adopt a coarse tooth for a wide face, and although the practice is far from uniform, an examination of a great many cases, selected at random, will show that the "base," or product of the face and pitch, will average very near the number ten for cut iron gears. It is obvious that a fixed law should accommodate itself to actual practice as nearly as possible, and, adopting ten as a base as rigidly as a proper re- spect for standard pitches, and convenient fractions for the faces will permit, we can construct the following table for cut iron gears. Face Pitch Base i 20 10 f 16 10 f 12 9 1 10 10 li 8 10 If 6 lOi 2^ 4 10 For small cut gears, which are usually made of brass, the weaker metal requires a coarser base, and we can use the number six for the standard. Face Pitch Base i 48 6 i 24 6 A 20 6i f 16 6 A 14 6i i 12 6 In the same way for cast gears we can construct a system on the number three as a base, as follows : — Face n 2 3 4 6 e 7 8 8 9 Circular Pitch Base i 3 i 2f 1 3 li H H 3 If 2f 2 3 2i H 2i H 2i 2H 8 3 WATER MOTORS F^or Driving a.11 descriptions of Snaall IVIachin^ery t>y common City Water F*ressu.re. Send for Pamphlet, and enclose stamp for reply. Full particulars should be given of the work to be done and the water pressure at command, I am preparing to put on the market a line of GEAR CnTTING MACHINERY, ^vhieh I intend shall be equal to the best, and superior to most tools offered in this line. A large Universal Engine, to cut to six feet diameter, and entirely autoi"natic in its operations. A Medium Universal and Automatic Engine, to cut to three feet diameter. A Small Universal and Automatic Engine, to cut to one foot diameter. A cheap Gear-Cutting Attachment for common lathes. A Rack Cutter. A Rack-Cutting Attachment for ordinary gear cutters. Correspondence solicited with concerns that use and in- tend to purchase machinery of this description. THE CALCULATING MACHINE 19 AN INSTRUMENT FOR THE -^CCTJI^.-A-I'E], E-^S-S", and. I^-^^>I3D Performance of the usually tedious operations of ADDITION, SUBTRACTION, MCTLTIPLIOATION, AND DIVISION. IT HAS BEOJBIVED THE M:. C. M. -A.. GS-old MCedal, M. C. M:. J^, Silver IMedal, rVixG Centennial M!edal, and the Soott Xjegaoy and IVIedal, AND IS ACKNOWLEDGED TO BE THE BEST INSTRUMENT IN USE FOR ITS PURPOSE. SEND STAMP FOR ILLUSTRATED PAMPHLET, giving full particulars of its construction and operation, and from prominent scientific and practical experts. PRICE, ONE HUNDRED DOLLARS. NO DISCOUNTS. NO A$£NTS WANTED. All descriptions of Gear Wheels made or cut to order. Any kind, Spur, Bevel, Miter, Rack, Ratchet, Worm, In- ternal, etc. Any size from a quarter inch to six feet diameter. Any quantity from a Single Gear to thousands. Gears for Machine Work, Gears for Model Work, Gears for Light or Heavy Machinery. BRASS AND IRON GEAR WHEELS GEAR CUTTING OF ALL DESCRIPTIONS. Send for IHuBtr&ted and Descripi/ve Pamphlet and Prhe Liet GEORGE B. GRANT, 66 Beverly Street, BOSTON. Many sizes of ready-made Gears are kept in stock for immediate delivery. Brass Gears of all kinds for Models and light Machinery kept in stock and sent free by mail at low prices. Superior Spur, Bevel and Miter Gears with cast teeth. Cut Iron Gears, not ready-made, but that can be made to order at short notice from patterns and castings, always on hand. ODONTOGRAPHS. The new Odontographs, fully explained and discussed in the Handbook, are published in SEPARATE AND SIMPLE FORM, on heavy and durable paper, and, -with the tables of pitch diameters, clearances, etc., are arranged with special reference to PRACTICAL APPLICATION in the shop or drafting-room. IHor JN^srOUXJTJEl TKKXH, I'rloe, 25o. postpaid, mor EinCYCILiOIDAIj TEKTH, Prioe, S5o. postpaid. Both ODONTOG-RAFHS for 40o. I HAVE IN PREPARATION A SHOP IVIANXJAlv ON GEAR WHEELS. This will not interfere with the Handbook on the Teeth of Gears, or treat of Tooth Curves or Odontographs, but will deal "Brith the Gear Wheel as a whole and in connection with other Gears. It will fully describe and illustrate all the different Gears in common use, — the Spur, Miter, Bevel, Ratchet, Internal, Rack, and Worm Gear, showing what each one is, hovr it works, how it combines in pairs or trains with others, and how it is to be pro- portioned and shaped. Detailed examples with engravings to scale will illustrate each subject, and the object kept in view will not be to discuss or demonstrat>e, but to state facts in such plain tterms that they shall be readily understood by any intelligent mechanic. It will pay particular attention to the sizing and shaping of Miterand Bevel Gear Blanks, giving tables of angles and outside diameters, ready for shop use. Perfectly accurate dimensions and angles can be found at a glance, that usually require a careful 4lrawing and close measurements. It will be uniform with the Handbook in size and price. mm,^3!^l OF CONGRESS 021213^114 3 GEORGE B. GRANT, 66 BEVERLY. STREET, BOSTON, MASS. 3 GEAR CUTTING, Standard Gear Wheels, CALCULATING MACHINES, WATER MOTORS.