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 THE FATHER OF AMERICAN COTTON MANUFACTURE 
 
 TJ"cw Yoi-lr, D, AzddI e ton ,<;:■ C ° 
 
A SHORT TREATISE 
 
 ON THE 
 
 DESIGNING AND CONSTEUCTION 
 
 OF 
 
 GEEROG AND MILL-WORK. 
 
 ACCOMPANIED BY AN EXPLANATION OF THE CONSTRUCTION AND USE OP 
 
 THE ODONTOGEAPH OP PEOF. WILLIS. 
 
 IREPUBLISEED FROM APPLETONS* DICTIONARY OF MACHINES, MECHANICS, ENGINE-WORK, AND ENGINEERING, 
 FOR MESSRS. REED & TENNE7, PROVIDENCE, R. L] 
 
 1 
 
 ISTEW YOEK: 
 D. APPLETON AND COMPANY, 
 
 549 & 551 BROADWAY. 
 1873. 
 
 /..,v- COPYRIGHT '^^ 
 
 

 Entebed, according to Act of Congress, in the year 1873, 
 
 By D. APPLETON & COMPAI^, 
 
 in the Office of the Librarian of Congress, at Washington. 
 
 4 
 
 0(o -2^^30 
 
PEEFAOE. 
 
 There are few subjects of greater importance and interest to the mechanic 
 than that of designing and constructing the various kinds of Geering. Every 
 mechanic feels the necessity of obtaining a knowledge of this branch of his busi- 
 ness, yet few know where to look for the desired information, and very few pos- 
 sess a knowledge of even the rudiments of the art. 
 
 Nothing has hitherto been published treating exclusively of the subject, to 
 which the ordinary workman can readily turn, and in which the information that 
 he needs is stated plainly and concisely. A treatise on Geering, low priced, com- 
 pact in form, and containing no extraneous matter, is evidently much needed, and 
 our brother mechanics will, we hope, appreciate the attempt which we have here 
 made to furnish them the best treatise on the subject yet published. 
 
 We take this work, by the kind permission of the publishers, Messrs. D. Ap- 
 pleton & Co., from " Appletons' Dictionary of Mechanics and Engineering," re- 
 printing the entire article with all its illustrations. The latter are one hundred 
 and twenty-five in number, and include all ordinary forms of geering, shafting, 
 and connections. Our readers will find here a complete description of the invalu- 
 able Odontograph of Prof. Willis. By the use of this instrument any intelligent 
 mechanic can lay out his own geers, whatever their size or the number and pitch 
 of their teeth. The geering thus designed will invariably run well with any 
 geers of the same pitch, whatever their size, provided they have been laid out by 
 the same method. A vast amount of time and labor is saved by its use, as the 
 calculations have all been previously made, and the results embodied in the tables 
 accompanying the instrument. These figures being taken from the tables and 
 set off from graduations on the instrument, liability to error, as well as loss of 
 valuable time, is avoided. 
 
 This work is published for and sold by the undersigned, who will also furnish 
 Odontographs well made and graduated with perfect accuracy. 
 
 REED & TENNEY, 
 
 JProvide7ice, It. I, 
 
Professor E. H. Thurston, of tlie Stevens Institute of Technology, 
 
 Hoboken, l!T. J., who has nsed the Odontograph, in designing some of 
 
 the heaviest geering in the country, permits us to refer to him for 
 
 testiniony in regard to the value of the instrument, as well as to our 
 
 own personal standing, and our character as mechanics and makers of 
 
 geering. n 
 
 EEED & TENNEY. 
 
A SHORT TREATISE 
 
 ON THE 
 
 DESIGNING AND CONSTEUCTION 
 
 OF 
 
 GEERING AND MILL-WORK. 
 
 ACCOMPANIED BY AN EXPLANATION OP THE CONSTEUCTION AND USE OF 
 
 THE ODONTOGEAPH OP PEOF. WILLIS. 
 
 GEERING is the general term employed to denote a combination of mechanical organs, interposed 
 between the prime mover and the working parts of machinery. Frequently, however, the signification 
 is restricted to the series of toothed wheels by which the motion is conducted from one revolving 
 axis to another, independently of the shafts and bearings by which they are supported. Two toothed 
 wheels are also said to geer when they have their teeth engaged together, and to be out of geer when 
 separate and consequently out of action. 
 
 In the transfer of motion from one axis to another in a system of mechanism, wheels are the most 
 important organs. Of these there are two principal varieties with several modifications, distinguished 
 by peculiarities of form, construction, and adaptation. 
 
 When motion is to be transferred from one axis to another which is parallel to it, the peripheries of 
 |the wheels act upon each other tangentially, and the teeth are disposed round their cylindrical surfaces 
 ^ the direction of radii from their centres. Wheels of this sort take the general name of spur geer. In 
 
GEERING. 
 
 the ordinary cast-iron wheels, the teeth are made in one piece, with the rim ; and commonly the wheel 
 
 consists of one entire piece, though sometimes the wheel is cast in segments when of very large size, 
 
 and especially if intended for shipment, for convenience of packing and transport. Cast-iron wheels of all 
 
 sizes above, and frequently below 18 inches diameter, are made with arms ; but in 
 
 very small-sized wheels the arms are commonly omitted, and the rim and teeth are 
 
 united to the central boss by a thin continuous plate, as represented in elevation and 
 
 section, by the annexed cuts. Wheels of this sort are usually denominated plate- 
 
 loheels, to distinguish them from those of similar size having arms, and which are 
 
 therefore described, especially by clock-makers, as being crossed out. In very 
 
 small machinery the wheels are sometimes formed out of plain disks by cutting 
 
 out a series of equidistant notches round the circumference. 
 
 In wooden wheels, which were in general use for all species of millwork till towards the end of the last 
 century, and of which we have still some examples, the teeth are formed of separate pieces, and fixed 
 into equidistant mortises pierced through the rim of the wheel. The rim is formed of segments of hard 
 wood iii-mly bound together by iron straps and bolts, and is connected to the shaft by a wooden framing, 
 consisting of bars set at right angles. The square opening, thus left at the centre to receive the shaft, 
 also of wood, and square at that part, is purposely made larger than the section of the shaft, to admit 
 of adjustment and fixing by wedges. The arms are, however, very frequently mortised into the shaft ; 
 but such a mode of fixing weakens the shaft, and at the same time renders it difficult to get the wheel 
 off, should this be required in consequence of the failure of the shaft. 
 
 Wheels of this kind are technically known as cog-wheels, and the teeth take the name of cogs. These 
 are made of some well-seasoned hard wood, as mountain-beech, plane-tree, hornbeam, hickory, and the 
 like, with the grain disposed in the direction of their length, which being the radical direction, is the 
 most favorable to transverse strength. 
 
 A modification of this construction of toothed wheels is still very commonly employed in millwork 
 under the name of mortise-wheels. In these, the body of the wheel is of cast-iron, and the teeth of 
 wood, fixed into mortises made in the rim, as in the old cog-wheel. The individual teeth, or cogs, are 
 often formed so that the part which projects without the rim is .the tooth ; and the shank, tail, or tenon, 
 is made to fit its mortise in the rim of the wheel very tightly, and is left sufficiently long to project on 
 the inside, so that being driven into the mortise up to the shoulders, it is secured in its place by an 
 h'on pin inserted into a hole bored through the tenon, closely under the rim of the wheel. 
 
 There is, however, another mode of fixing the cogs, which is more commonly practised than that 
 described. This mode will be understood by reference to Fig. 1990, from which it will be observed 
 that the cogs, instead of being made fast in their places by pins, are retained in the mortises of the rim 
 by dovetail keys vv, driven between every contiguous pair inside of the rim. The tenons, when the 
 cogs are intended to be fitted in this manner, are made with an expanding dovetail at their extremities, 
 to receive one side of the key, which being driven tightly into the V-shaped space, thus formed between 
 the ends of each pair inside of the rim of the wheel, retains the cogs in their places. The keys, which 
 are made of well-seasoned hard wood, usually of the same kind as that used for the cogs, are made 
 long enough to project some way beyond the side of the rim, to 1991. 
 
 allow of their being driven in more tightly, should the cogs in the 
 course of working become loose. The ends of the tenons are 
 also very commonly vandejlush with the surface of the keys, but 
 the mode of finishing shown in Fig. 1991 has some advantage in 
 point of strength. 
 
 Spur mortise-^oheel. — Figs. 1989 and 1990 give two views in 
 elevation, and Fig. 1991a section of a mortise spur-wheel of 34:^ 
 inches diameter, and containing 48 cogs ; pitch 2^ inches. 
 
 Fig. 1991 is a section of the wheel where a a represents the 
 cogs ; r r the ring ; c c the face arms ; h h the feathers of the arms, 
 and S S the socket or eye of the wheel 
 
 Fig. 1989 is a view parallel to the axis of the wheel, and Fig. 
 1990 is a plan of the same wheel in the line of its centre. 
 
 These figures are intended to illustrate the mode of drawing 
 the wheel. The pitch and number of cogs being given, the first 
 part of this operation is to determine the diameter of the pitch 
 circle. This may be done by multiplying the pitch of the teeth 
 by their number, and dividing the product by 31416 ; or it may 
 be directly found from the rule and table given in page 795 ; or 
 the compasses may be at once set to draw the pitch circle from 
 scale of Fig. 2022, (see pages 199 and 800.) The pitch being 
 2i inches, the radius of the pitch circle will be found in the line 
 wliich runs parallel with A D, and which is marked 2\ ; and the 
 distance C to 48 in that line being the radius for a 1\ inch pitch 
 with 48 teeth, one point of the compasses placed in the intersec- 
 tion of the line 2i by C B, and the other in the intersection of the 
 line terminating in 48, will be that radius. The other required 
 
 dimensions will also be found in the line 2\ ; the distance A = the pitch \ ac = the length ; 6 c =. the 
 (liickness \ ac'ss. the length from the pitch line to the point, and twice this is the working depth ; the 
 width of space is A h, and their lateral clearance is A S — Qh, and the bottom clearance a C — 2 a c. 
 
 The same scale may be used in finding the proportions of parts in making out reduced drawings of 
 wheels to different scales. If the compasses be set to the required pitch, and moved up the line B, 
 keeping one point in C B until the other point meets the fine A B, the points running parallel to A C 
 
GEERING. 
 
 then in a line drawn parallel to A D, and passing through the points in which the compasses meet the 
 converging lines, will be found all the other required dunensions in accordance with that scale. 
 
 Fig. 1989 shows the method of finding the shape and position of the teeth, by the application of tht 
 T square to Fig. 1990, as indicated by the dotted lines eeee at 2, 2, 2. The hnes forming the outline 
 of the shaft are found in a similar maimer from Fig. 1990. 
 
 The mode of drawing the curves of the top and bottom of the teeth is shown by the dotted circles, 
 mai-ked ffff, and which are di-awn from the centres of the teeth a a. This curve is, however, very 
 variable, and depends much upon the size of the pinion intended to work in the wheel. 
 
 The mortise-wheel is preferred, wliere a high velocity with smoothness of motion is required. It is 
 usually made to work into a wheel with cast-hon teeth, a pair of this sort being found to work together 
 with less vibration, less noise, and less wear, than when both wheels of the pair have hon teeth. For 
 these reasons it is very common, in good millwork, to make one wheel of every large-sized pair with 
 wooden cogs, especially when the speed is high, and subject to variations of velocity. But it may be 
 here remarked, that in cases where a very large wheel is reqmi'ed to geer with a smaller one, the former 
 is commonly made the mortise-wheel. 
 
 It may be here well to observe, that when two wheels geer together, the one which communicates 
 the motion to the other is called the driver or leader ; and the wheel impelled is the follower. If the 
 two wheels be of very different sizes, the smaller one very commonly takes the name of pinion, which 
 may be regarded as the diminutive of toothed-wheel. For the sake of further distinction the teeth of 
 pinions are often termed leaves. In geering of the ordinary kinds, the pinions are commonly of the 
 same form of construction as the wheels with which they act ; but in the old wooden machinery the 
 pinion is commonly formed by inserting the extremities of a number of wooden cylinders into equidistant 
 holes, in two parallel disks, (technically called heads and cheeks,) upon a square shaft. A pinion of 
 this description is denominated a trundle, lantern, or wallower, and its cyhndrical teeth are termed 
 staves, sometimes rounds and roundles, very commonly pronounced rungs. 
 
 In early machinery the toothed wheels were often cut out of thin metal plates, which rendered it im- 
 possible to make a pair thus formed to work together, for the slightest deviation of one of the wheels 
 from the plane of rotation of the pair, would allow the teeth to lose hold of each other sidewise. To 
 obviate this, one of the wheels of a pair was always made either in the lantern form, as just described, 
 or more commonly with pins inserted at one end only, into a disk. A modification of this method also 
 in use, was to form the teetli on the edge of a hoop ; by these arrangements, the thin wheel was enabled 
 to retain its hold of the teeth of the wheel with which it geered, notwithstanding a little deviation from 
 the plane of rotation. This form of wheel is still in common use in watch and clock work, under the 
 names oi croivn and contrate wheel. <^^'^ 
 
 Fig. 1992 represents the case of a wheel working into a rack, commonly called 5/^\^1992. 
 
 a rack and pinion, t^C^'^ 
 
 If the rack be curved upon the pinion it becomes an internal or annular r'^-^'''^'^^^^''^'^>^wvv\A^ 
 wheel, that is, a wheel having the teeth inside of the rim, as represented by Fig. 1993. In this the 
 axes of the wheel and pinion are parallel, and moreover, revolve in the same direction. The arms o! 
 the internal wheel are necessarily situated behind the rim to prevent their interference with the pinion, 
 and the latter must overhang its bearing, that is, be fixed on the 
 extremity of its shaft, to avoid interference with the arms of the 
 wheel 
 
 TMien the pinion is made half the diameter, that is, equal in 
 diameter to the radius of the annular wheel, the arrangement 
 admits, in small steam-engines, of an application, known as 
 White's parallel motion. In this the annular wheel is fixed, and 
 the pinion is attached upon a crank-arm. The rod being at- 
 tached at the circumference of revolution of the pinion, it is 
 thereby made to describe a right line, coinciding with a diameter 
 of the annular wheel, which is therefore equal to the length of 
 the stroke of the engine. 
 
 Similar combinations to that of the crown-wheel and pinion 
 were early introduced into mechanism. In examples of this con- 
 struction, whicli are still to be found along with cog-wheels and trundles, the cogs are simply disposed 
 on the rim of the wheel, from which they project in lines parallel to the axis, so as to geer with those 
 of an ordinary cog-wheel, having the cogs disposed roimd the circumference, or with a trundle when 
 
 1993. 
 
 1994, 
 
 one of the axes revolves much quicker than the other. This is exemplified in the hand-mill still to bo 
 found in some parts of Germany and the north of Europe, as depicted in Fig, 1995, in wliich we have 
 \he face-wheel upon the crank-axle, geering with the trundle upon the millstone axle, which forms an 
 angle of 90° with the former. 
 
GEERING. 
 
 In several parts of Scotland, the combination sho-svn in Fig. 1996, as a first pair in the geering of a 
 thrashing-mill, is still common. 
 
 In spur-geer, the principle depends upon two cylindrical surfaces Tjeing made to act on each otliei 
 tangentially ; in bevel-geer the cylinders are replaced by thin frusta of cones, \vhich have their smooth 
 Burfaces exchanged for a regular series of equidistant teeth directed to the apex of the cone ; so that a 
 right line passing through the apex, if brought into contact mth any part of the side, or top of a tooth, 
 shall touch it throughout its whole breadth ; consequently in any pair adapted to work together, the 
 apices of their cones must meet in the same point, and thus the contact of one tooth with another will 
 take place along their sides. This is illustrated in Figs. 1997, 1998, and 1999. 
 
 Detailed drawings of a mitre-ivheel. Fig. 1997 is a section. Fig. 1998 a side view, and Fig. 1999 a 
 face view of a mitre-wheel. This is a kind of bevel- wheel that is used when it is required to change 
 the direction of motion 90° without altering its angular velocity — a case which occurs very frequently 
 in practice. It is obvious that only one pattern is required for the pair of wheels of this kind, and the 
 method of drawing any one of them will apply to both. The following is the process : 
 
 Draw a line A B equal to the diameter of the wheel, (diameter of pitch circle ;) bisect A B, and pro- 
 duce a line at right angles to it, which will represent the centre line of the shaft. The points C and D 
 in this line, Figs. 1997 and 1998, where the produced centre lines of the two sliafts intersect each other, 
 is the point to which all the lines nmning in the direction of the breadth of the teeth are drawn. TKe 
 distance of the point D or from the line A B is equal to half the diameter of the pitch circle. This 
 will be evident if B C be drawn from the point B at right angles to A B : the line B C being equal to 
 A B, will represent the proper position of the other wheel, and if bisected in E, the line E D produced 
 perpendicular to it will be the centre line of the shaft. The point D is therefore the position of the 
 common apex of the two cones. 
 
 Join A I) and B D, and A and B 0, and at right angles to these lines draw e m and r i, Fig. 1997, 
 passing through the points A and B. Make AJc,Bn equal to the breadth of the teeth, and draw kg 
 and n h j)arallel to A w, B I. Set off upon A wz, B I the length of the teeth, as ef,rs; also the thick- 
 ness of rim, as fm, s I ; and from these points let lines be drawn to the apex of the cones, as shown. 
 Join g h, and make q v equal to the size of the eye ; P q and V w are each to be equal to the intended 
 thickness of the eye, and P t equal to the depth. Draw pt, qu,vx,wy, at right angles to g h and 
 join t m and y I, and thus a section of the wheel is determined, from which all the lines and points re- 
 quisite for drawing the plan can be found. 
 
 Fig. 1998 is a face view, the mode of proceeding with which is to transfer the points *1, 2, 3, 4, 5, 
 from Fig. 1997 to it, by means of the T square, which determines the extremities of the teeth and rim ; 
 circles are then described passing through these points, and that passing through the point 4 is divided 
 into the number of teeth intended to be in the wheel : the thickness of the teeth is then set off, and 
 drawn in a similar manner to that described in Fig. 1990. The lines forming the edges of the top and 
 bottom of the teeth are radial lines, and of course are drawn in the direction of the centre of the wheel. 
 
 Fig. 1999 the side view. The elementary lines for this drawing, viz. 6, 7, 8, 9, 10, correspond to 
 the hnes in the section having the same figures, and the points forming the teeth are derived irom Fig. 
 1998, by the application of the square. For example, the two points a b, forming the top of the out- 
 side of the tooth F, are transferred to the line marked 8, on Fig. 1999, and the points cd to the line 
 marked 7 ; these points being connected by the hnes a c and h d, form the end of the tooth : straight 
 lines are then drawn from a 6 in the direction of the point D, until they meet the fine marked 10, and 
 a fine in the same direction from c meeting the line marked 9. The terminations of these hnes mark 
 out the extremities of the inside of the tooth, which being joined, complete the tooth F. The shape and 
 direction of the other teeth in the wheel are found by a similar operation to that used for finding F, 
 keeping in view that the slope of all the teeth tends to the point D. 
 
 The eye of this wheel is intended for a round boss, on which it may be keyed in the usual manner. 
 The thickness of the web is equal to that of a tooth of the wheel ; the arms have a thickness somewhat 
 less, and the thickness of the metal of the eye is equal to the pitch of the teeth. 
 
 The radius representing the circle of action of wheels of the kind described, is found by produdrg 
 the hne B I till it meets the centre hne of the shaft. 
 
GEERING. 
 
 :^evel-wheel and pinion. — This pair of wheels, shown in Fig. 2000, differs from that in the preceding 
 HI beino- of unequal size ; and hence come under the denomination of bevel-wheels. Both wheels of thit 
 pair are supposed to be placed on horizontal shafts, in which they differ from the pair of bevels in Fig 
 2001, of which the pmion is placed on an upright or vertical shaft, having its bottom bearing in a foot 
 step, 'while the wheel with which it is in geer is placed on a horizontal shaft. The mode of drawing a 
 pair'of wheels of this kind wiU be understood from what follows. 
 
 Fig. 2000 is a side view of the wheel and pinion. 
 
 2000. 
 
 When only the side view of a bevel-wheel and pinion is required, it is not necessary that the whole 
 section, as in Fig. 1997, should be drawn, farther than to determine the position of lines abode and f^ 
 and the position of the apex of the two cones, as at 0, Fig. 1997. It should also be observed, in the 
 example given in Fig. 1997, that the view in Fig. 1998 was requisite for drawing the side view, only 
 60 far as finding the position of the four points of the teeth abed was concerned ; but for common 
 purposes, these points can be found with suffi- 
 cient exactness by merely drawing circles of the 
 same diameter as the wheel and pinion, and 
 dividing them into the given number of teeth, 
 and marking off their tliickness. This will be 
 readily understood, by examining Fig. 2000, in 
 which circles, so divided, are laid down equal 
 in diameter to the wheel and pinion ; and 6, 7, 
 8, (fcc, show the thickness of the teeth. Now 
 by applying the T square, as at 6 and 7,* and 
 marking these points upon the pitch line c, the 
 lines forming the sides of the teeth at 3, 4, are 
 found by drawing them parallel to the centre 
 line, which is drawn to the point where the 
 lines 1 and 2, forming the bevel of the ends of 
 the teeth, meet. This point is not shown in the 
 drawing, being at too great a distance for the 
 size of the plate. The lines forming the face of 
 the teeth are all drawn to the point where 
 the cones meet, as shown by the lines gg ; and 
 exactly the same mode is adopted in determin- 
 ing the shape and position of the teeth of the 
 pinion C. 
 
 'V\Tien the wheels are di-awn to a small scale, 
 as in Fig. 2001, the two lines forming the sides 
 of the teeth may both be drawn to the same 
 point, as shown at a, as in such cases they will 
 not differ sensibly from the parallel. 
 
 A bevelled jnortise-wheel. — Figs. 2002, 2003, 
 and 200-4, are plan, section, and side view of a 
 bevel-wheel, having wooden teeth, commonly 
 termed cogs. The ratio which the wheel and 
 pinion bear to each other is four to one, as will be observed from the section, Fig. 2003, in which A B 
 
 * The scale is too small in the cut to show these figures, but the process will readily be followed by a glance at the 
 method pursued in Fig. 1997. 
 
10 GEERING. 
 
 represent the diameter of the wheel, and B C that of the pinion. By following the instructions given m 
 Figs. 1997, 1998, and 1999, the point D will be found to be the junction of the two cones, and conse- 
 quently that point to which all the lines of the teeth are drawn. 
 
 The manner of finding the points for drawing the plan. Fig. 2002, is by applying the square to Fig 
 2003, as shown by the dotted lines 1, 2, 3, 4, 5, 6, 1, (fee, and the shape of tooth in Fig. 2004, is pre- 
 cisely the same as described in Fig. 1999. 
 
 In these examples, motion is led off at an angle of 90° ; but bevel-geer may be employed to change 
 the direction of the motion to any angle required. This will readily appear from the following dia- 
 grams, in which, for the sake of simplicity, we suppose the cones continued till their apices meet. 
 
 In Fig. 2005 we have an illustration of the principle of those cases already described. Here a is tlis 
 point at which the apices meet, and supposing the motion to be conveyed in the axis A a, it is trans- 
 ferred to the axis a B in a right angle to its first direction. 
 
 In Fig. 2006 the change of direction is no longer a right angle ; for the angle mcluded between the 
 axes is obviously less than 90°, and might evidently be made stiU less by diminishing the angles which 
 the slanting sides of the cones make with their respective axes. Wheels of this kind — rather wheels 
 of which the angle included by^the axes is less or greater than a right angle — are usually distinguished 
 by the name of cordcal-wheels. 
 
 2005. 2007. ^^?^- 
 
 In Fig. 2007 the apices of the two cones meet in a point, so that one of them becomes a plane sur- 
 face ; its teeth therefore become radial as referred to its apex ; in other words, they diverge in the 
 manner of radii, drawn from the centre a. The change in the direction of the motion is in this case 
 greater than 90°, and might be further increased, if consistent with the relative velocity of the pair, by 
 increasing tlie angle of the cone. 
 
 In Fig. 2008 we have an expression of a mode of changing the angle of direction differing from those 
 shown in the preceding figures, inasmuch as one of the cones has become hollow, and the teeth are dis- 
 posed upon its interior surface, thereby forming an internal bevel-wheel analogous to the internal spur- 
 geer previously noticed ; wheels are, however, very rarely made in this manner. 2009. 
 
 When the direction approaches a right line, the cones may be made to roll upon 
 each other, as in Fig. 2009. 
 
 In cases where the axes do not meet in a point as hitherto considered, the geer- 
 ing becomes somewhat more complicated. 
 
 The direct mode of arrangemen|; in this case is to employ an intermediate dou- 
 ble bevel-wheel. 
 
 To avoid the multiplication of parts, the object is more commonly attained by a 
 modification in the direction of the teeth, by which they are made to come into 
 contact in an oblique position, answering to the obliquity of the position of tlie 
 cones upon which they are formed. This artifice is often practised when it is ne- 
 cessary to cross the two axes, — which it must be observed are not tlieoretical lines but actual shafts of 
 diameters proportionate to the power they are required to transmit. This arrangement is represented 
 by Fig. 2010, and illustrates that variety of toothed-geer, known as sTcew-wheels. Wheels of this char- 
 acter are commonly avoided in practice ; as, on account of the oblique form of the teeth, they are not 
 only of more difficult construction, and therefore rarely possess the same degree of accuracy which we 
 find in bevels of the ordinary sort, but the strain upon the shafts is likewise oblique, and therefore more 
 severe upon the bearings. 
 
 Another form of toothed- wheel, known as " Hooke's Geering," from the name of the inventor, Dr. 
 Hooke, consists in disposing the teeth upon the peripheries of the wheels in equidistant steps, or in 
 such a manner as to form a continuous slope. This remarkable contrivance, which has several times 
 since been re-invented and patented, was intended, to use the words of the learned inventor, " First, to 
 make a piece of wheel-work so that, both the wheel and pinion, though of never so small a size, shall 
 have as great a number of teeth as shall be desired, and yet neither weaken the work nor make the 
 teeth so small as not to be practicable by any ordinary workman. Next, that the motion shall be so 
 equally communicated from the wheel to the pinion,'that the work being well made, there can be no 
 inequality of force or motion communicated. Thirdly, that the point of touching and bearing shall be 
 always in the line that joins the two centres together. Fourthly, that it shall have no manner of rub- 
 bing, nor be more difficult to be made than the common way of wheel-work, save only that workmen 
 have not been accustomed to make it." 
 
 The mode of construction contemplated by Dr. Hooke was to make the wheel and pinion of several 
 plates, laid one beside the other, and to bolt them together. These plates are individually cut into 
 wheels, which are fitted together in the manner described, but in such order that the teeth of the suc- 
 cessive plates follow each other in regular gradation, so that the last tooth of each group may, within 
 one step, answer to the first tooth of the next group. The pinion being constructed in the same manner, 
 and of the same number and thickness of plates, it is obvious that the inequalities in the touching sur- 
 faces are reduced in proportion to the number of thicknesses ; and the amount of rubbing friction is 
 reduced to what it would be between the two next teeth of one of the sets. 
 
 This is still further reduced by reducing the steps formed by the ends of the teeth to a straight, or 
 rather a spiral edge. A wheel and pinion of this kind, observes Dr. Hooke, are equivalent to the same 
 having an infinite number of teeth, and are best made each of a single plate of convenient thickness, 
 
GEERING. 
 
 11 
 
 ** which thickness must be more or less, according to the bigness of the sloped tooth. And this is always 
 to be observed in the cutting of the teeth, that the end of one sloped tooth on the one side be full a& 
 forward as the beginning of the next tooth on the other side ;" in other words, that the end b c{ ono 
 tooth on the right side be fuU as low as c, the beginning of the next tooth on the left side 
 
 In a pair of wheels of this construction it is easy to perceive that the contact of the teeth will beai 
 arery instant at a single point, which, as the wheels revolve, will travel from one side to the other, a 
 fresh contact always beginning on the first side immediately before the last contact has ceased on the 
 
 opposite side. The contact, moreover, being always in the plane of the centres of the pair, the action 
 h reduced to that of rolling ; and as there is no sliding motion, there is consequently no rubbing friction 
 between the teeth. 
 
12 
 
 GEERTNG. 
 
 The motion of wheel work of this description is remarkably smooth and free of vibration, but is liable 
 to the objectiofi of introducing endlong pressure upon the axes, in consequence of the obliquity of the sur- 
 faces of contact to the planes of rotation. To obviate this objection, the wheels have been made with dou- 
 ble sloped teeth, in the form represented in Fig. 2011. This wheel may be conceived to be composed oi 
 two thicknesses of metal, each having teeth formed upon its periphery in the same manner as the wheel 
 of Dr. Hooke just described, and that these plates are fitted together with the direction of the teeth reverse 
 to each other. More strictly, the wheel may be supposed to consist of two equal and concentric plates, 
 which being, in the first place, made as separate wheels, have their teeth of equal size, and cut at equal 
 but contrary angles with the axis of rotation, are fitted together so that the teeth of the one plate meet 
 those of the other plate in the plane of contact. The slopes of the teeth on the two sides being thu3 
 reverse, and projected at equal but contrary angles with the axis of the wheel, and the pinion being 
 formed in the same manner, the action of the respective sides of the V-sliaped teeth on each other will 
 be as before described ; but the endlong pressure arising from the obliquity of contact on one side, will 
 be exactly equal and opposite to that resulting from the obliquity of action on the contrary side ; these 
 pressures must, therefore, neutralize each other, and relieve the journals of an amount of friction which 
 IS necessarily involved in the mode of action contemplated in the original invention. 
 
 A further modification of Dr. Hooke's geering has of late been somewhat extensively adopted, espe- 
 cially in the newer cotton-spinning machines. This consists, when the direction of the motion is simply 
 to be changed to an angle of 90°, in forming the teeth upon the periphery of the pair at an angle of 45° 
 to the respective axes of the wheels ; it will then be perceived that if the sloped teeth be presented to 
 each other, in such a way as to have exactly the same horizontal angle, the wheels will geer together, 
 and motion being communicated to one axis, the same will be transmitted to the other at a right angle 
 to it, as in a common bevel pair. Thus, if the wheel A, Fig. 2012, upon a horizontal shaft have the 
 teeth formed upon its circumference with an angle of 45° to the plane of the axis, it can geer with a 
 Bimilar wheel B upon a vertical axis. Let it be upon the driving-shaft ; then the motion transmitted 
 will be changed in its direction, as if A and B were a pair of bevels of the ordinary kind, and as with 
 bevels generally, the direction of motion will be changed through an angle equal to the sum of the 
 angles which the teeth of the wheels of the pair form with their respective axes. 
 
 The contrivance, as stated, is directly a modification of Dr. Hooke's geering ; but, in effect, it may b»3 
 described as a modification of the tangent screw and screw-wheel. In this last arrangement, repre- 
 sented by Fig. 2013, in its common form, it will be observed that the plane of the thread coinciding so 
 nearly with that of the axis of the wheel, renders it necessary that the screw be uniformly the driver ; 
 and as the screw, by one revolution, passes only a single tooth of the wheel, the motion is necessarily 
 Blow. Both of these circumstances are manifest advantages in a large class of instances ; but for very 
 many purposes it is desirable to retain the principle of the screw, with an increase of velocity and a 
 diminution of its rubbing friction. This is accomplished by diminishing the inclination of the thread oi 
 the screw to the axis — more correctly by increasing the number of separate spiral threads upon the 
 surface of its cylinder; for, as every one of those spirals will pass its own wheel-tooth across the line of 
 centres in a revolution of the screw, it follows that as many teeth of the wheel will pass that line dur- 
 ing one revolution of the screw as this last has threads. If, therefore, we suppose the number of threads 
 to be increased until they equal in number the teeth of the wheel, then the screw and wheel may be 
 made exactly alike. This is precisely the case exemplified by Fig. 2012, in which we suppose A and 
 B to be the same size, and to have the same number of spn-al teeth — which might manifestly be con 
 .tinned round the axes of the wheels without affecting in the least their mode of action. 
 
GEERENG. 
 
 13 
 
 2014. 
 
 ■N^ 
 
 Relative direction and velocity of rotation. — A wheel describes a cii'cle because its axis is fixed; bu^ 
 the direction and the velocity of its rotation depend upon its connection with the next wheel of the 
 train, which lies between it and the moving power ; and it is evident that the relative velocities of any 
 Cfiven pair, must depend upon the relative magnitudes of the circles which they respectively describe. 
 Thus, supposing the circumference of the wheel be double that of its pinion, and, therefore, to have 
 double the number of teeth upon it, every revolution of the wheel must of necessity eifect two revolu- 
 tions of the pinion about its axis. And similarly knowing the respective circumferences of any pair 
 acting together, it is easy to assign the ratio of their velocities. This form of the question, however, 
 more rarely comes into use than that requii'ing the determination of those quantities from the diameters, 
 which is slightly more involved, and renders it necessary, before entering further upon the consideration 
 of the subject, to examine the conditions a little more closely. 
 
 Supposing that A and B, Fig. 2014, are two axes 
 of motion referred to the same perpendicular plane, 
 and that A B is a right line connecting them. Fm-ther, 
 let it be supposed that the moving force is transmit- 
 ted through A, and that it is required to connect it 
 with B in such a way — that is, by a pair of spm*- 
 wheels of such sizes — that while A makes two revo- 
 lutions B shaU make three revolutions, the distance 
 between A and B being six feet. 
 
 It is manifest that the circumferences of the wheels, 
 the axes being fixed and the circumferences in geer, 
 must of necessity have the same velocity. Thus, 
 supposing for an instant that the question is already 
 resolved, and that the circles described about A and 
 B, having their point of contact at c, then if a con- 
 stant force, Pi, act tangentially to the circle described 
 about A with the radius Ac, it is evident that the 
 rotation of this circle being transmitted to that de- 
 scribed about B with the radius Be, will cause it to 
 describe an equal length of arc to that through which 
 itself passes, in any unit of time. But although the 
 arcs be of the same absolute length, it is easy to per- 
 ceive that they fonn very different portions of their 
 respective circles. In fact the arcs described by the 
 two points of a and h, taken as portions of the circle 
 to which they belong, are in the inverse ratio of the radii Ac and Be by which the circles arc 
 described. 
 
 Now, if this be true for a part of a revolution, it is equally true for a whole revolution of either oi 
 the wheels, and consequently for any number of revolutions. Hence, to obtain the radii of the wheels, 
 it only remains to divide the distance between the centres — that is, the line A B into parts inversely 
 proportional to the number of revolutions which the wheels respectively make in the same unit ot 
 time. These rules are expressed as follows : — 
 
 I. To find the radius of the wheel, the relative number of revolutions and the distance between the 
 centres of the pairs being given, multiply the distance between the centres by the numbers expressing 
 the velocity of the pinion, and divide the product by the sum of the numbers expressing the relative 
 velocities of the pair — understanding by velocities the number of revolutions which the wheels make in 
 the same time. 
 
 IL To find the radius of the pinion or other wheel, multiply the distance between the centres by the 
 number expressing the velocity of the wheel, and divide the result by the sum of the velocities ; or sub- 
 tract the radius of the wheel from the distance between the centres : the remainder is manifestly the 
 radius of the pinion. 
 
 Now in the arithmetical question proposed, the distance d between the centres is six feet, the velocity 
 of the wheel two, and that of the pinion three ; hence, 
 
 6 feet X3 18 feet „ ^ , h, • i. 
 — — I = — - — = 3 feet 7^ mchea 
 
 6 feet X 2 12 feet 
 
 2-f 3 5 
 
 Sum of radii c/ = R + r = 6 feet. 
 
 Having the radii or diameters of the wheels given, it is not necessary to find their circumferences m 
 order to determine the ratio of the velocities of their axes. The relative number of revolutions which 
 they will make in a given unit of time is ascertained by dividing the measure of the one by that of the 
 pther, the quotient being the number of revolutions, or parts of a revolution, the axis of that wheel will 
 nahe which is made the divisor, while the axis of the wheel whose diameter or radius is divided ma\es 
 one revolution. 
 
 Questions of this kind may be solved very readily by means of the compasses and a scale of equal 
 parts. Thus let A and B, Fig. 2014, be the given centres, the ratio of their velocities being respectively 
 two and three, if the line joining the centres A and B be divided into 2 -j- 3 = 5 equal parts, that is, into 
 as many equal parts as there are units in the terms of the given ratio, the radius of the wheel upon A 
 vill contain three of those parts, and the radius of the pinion on B will contain the remaining two parts, 
 
14 GEERING. 
 
 and the point of contact of the wheels will be at c. This method is very convenient in practice when 
 the terms of the ratios are small. ' 
 
 In the preceding examples, only a single pair has been taken ; but it frequently occurs in practice 
 that motion is to be transferred through a system of shafts, with varying degrees of intensity, according 
 to the purposes to be attained. It Ukewise not unfrequently happens that a pair of shafts are to be 
 connected at such distances as to require the interposition of carrier wheels, that is, wheels intended 
 simply to connect the two, without being subservient to any other purpose. 
 
 Let there be two motion-axes, A and B, of which the angular velocities are as 2*7 to 343, and let it 
 be required to connect them by four wheels on two intermediate shafts, the wheels to have a common 
 velocity ratio throughout. Let A be the driver ; then the intensity of the motion is required to increase 
 from 27 towards B, in which it is expressed by 343. 
 
 Now, it has been shown that if a pair of wheels be in geer, their radii are to each other as the angular 
 velocities, that is, the relative number of revolutions which they make upon their axes. In order, there- 
 fore, that the geering pairs in the question proposed may have a common velocity ratio, it is obvious 
 that the same must apply to the axes upon which they are fixed. These velocities ought, therefore, to 
 form a geometrical progression, of which the first and last terms, (A = 27 and B = 343,) and also the 
 number of terms, namely, the four axes, are given. Now, the formula for all such questions, m being 
 the number of geometrical means to be inserted, p the ratio, is for an increasing series. 
 
 m+\ 
 
 'B 
 
 j3 = </ -7 which becomes p = \/- 
 
 7 
 Hence the angular velocity of C = 27 X ^ = 63 
 
 3 
 
 of D==63 X ^-=147 
 
 o 
 
 Also, A win be connected with C, and C with D, and D with B by wheels of radii 7 and 3 respectively. 
 
 These exaniples illustrate the whole process of calculation of the transfer of angular velocity from 
 one axis to another, and include every case of the kind to be met with in practice. 
 
 In the case taken, we have supposed the motions of the pair to be in the same plane, and the axes of 
 rotation of the circles parallel. But it is often found necessary to change the du'ection of motion through 
 all conceivable angles, as in case the axes of rotation .meet in a point by common bevels and face wheels, 
 and when the axes are neither parallel nor do they meet in direction by screw-wheels. These cases 
 require to be separately noticed. 
 
 Let us, in the first place, suppose that the two axes of rotation B A and C A intersect in A, and in- 
 clude the angle BAG. If we suppose a cone to be generated by the revolution of the line A E about 
 A B, and another by the revolution of the line A E about A 0, then these cones 
 being made to revolve in contact about their respective axes A B and A C, their 
 surfiices will roll upon one another along their whole length of contact A E. 
 Eor, as already shown, if n times the circumference of a circle D e be equal to 
 m times that of a circle F e, and these circles be conceived to revolve in contact 
 about their centres b and c, and to carry the cones with them, then it is evident 
 that whilst the cone E A G- makes n revolutions the cone E A H will make m 
 revolutions. But n times the circumference of any other circle E G- of the one 
 cone, is equal to m times the circumference of the corresponding circle E H of 
 the other cone ; for the diameters of these circles, and, therefore, their circum- 
 ferences are to one another in the same proportion as the diameters and circum- 
 ferences of the circles e D and e F. It is therefore obvious, that whilst the cones 
 make n and m revolutions respectively, the circles E G- and E H are carried 
 through n and m revolutions respectively, and that n times the circumference 
 of E G- being equal to m times the circumference of E H, it follows that these 
 circles roll in contact through the whole of their path. And the same is equally 
 
 true of any other corresponding circles in the cones, and therefore of their whole surfaces, so that the 
 rotation of one axis being communicated to the other, by the rolling together of the two cones, the sur- 
 face of the one cone will carry with it the surface of the other along the whole line of contact A E with 
 equal perimetral velocities, and with angular velocities inversely proportional to the circumferences, 
 diameters, or radii, of the corresponding circles. 
 
 In practice, thin frusta only of the cones, with teeth upon their perimeters, are employed ; but in 
 this there is no new consideration involved, as respects the angular velocities of the axes upon which 
 the wheels are carried. In determining the size of a pair of bevels, we are not, however, limited to any 
 particular diameters as when the axes are parallel ; the wheels may be made of any convenient sizes, 
 and the teeth consequently of any breadth, according to the stress they are intended to bear. 
 
 The question, however, which presents itself here, is the mode of determining the relative sizes of the 
 conical frusta of a pair ; and this resolves itself into a division of the angle included between the two 
 axes inversely as the ratio of their angular velocities. Let B and C be the position of the two given 
 axes, and let them be prolonged till they meet in a point A. Further, let it be required that C make 
 seven revolutions while B makes four. From any points D and E in the lines A B, A 0, and perpendicular 
 to them, draw D d and E e of lengths (from a scale of equal parts) inversely as the number of revolu- 
 tions which the axes are severally required to make in the same unit of time. Thus, the angular velocity 
 of axis B being 4, Fig. 2016, and that of the axis C being 7, the line D d must be drawn = 7, and the hna 
 E c = 4. Then through d and e parallel with the axes A B and A C draw dc and e c till they meet in 
 
GEERING. 
 
 15 
 
 «. A straight line dra\ni from A through c will then make the required division of the angle B A Q 
 and define the hne of contact of the two cones, by means of which the two rolling frusta may be pro 
 jected at any convenient distance from A. 
 
 Otherwise, having determined the relative perimeters, 
 diameters, or radii, of tlie pair, then the lines D d and E e 
 are to each other directly as these quantities. 
 
 The point c may also be found more directly thus : Prom 
 A towards C in the axis A C, set off from a scale as many 
 equal parts (A/) as there are units in the number (1) ex- 
 pressing the velocity of that axis ; from the point / draw 
 fc parallel to AB, and set off from the same scale as 
 many parts {fc) as there are units in the number (4) ex- 
 pressing the velocity of the axis A B ; then a line drawn 
 from A through c, as before, will divide the angle, as required. 
 
 By one or other of these methods the division of the 
 angle of inclination of the axes may always be determined, 
 the ratio of the angular velocities of the pair* being known. 
 
 The case in which the axes are neither parallel nor inter- 
 secting, admits of solution by means of a pair of bevels 
 upon an intermediate axis, so situated as to meet the others 
 in any convenient points. 
 
 Thus, if D F and E G, Fig. 201 Y, be the two given axes, 
 they may be connected by a third axis F G intersecting them 
 in the points F and G ; and if a shaft be mounted in the posi- 
 tion of this third axis with a pair of bevels upon it, geering 
 with the bevels on the main axes, and having their apices 
 in the points F and G of these axes, the motion of the driving- 
 shaft will be communicated, as if the two bevel- wheels C and M were in immediate contact ; the ratic 
 of their velocities remaining the same, provided the bevels H and K be of the same size. If it be re- 
 quired to bring up the speed or reduce it, between the two shafts, in a higher ratio than is convenient by 
 that arrangement, the interposed bevels will afford additional facility of accomplishing the purpose. 
 
 When the contiguity of the shafts is such as to per- , 2017. 
 
 mit of their being connected by a single pair, skewed- 
 bevels are frequently employed as shown in Fig. 2010 ; 
 and as respects the relative velocities of a pair of this 
 kind, it is evident that the same law obtains as in the 
 preceding cages. 
 
 When the axes are at right angles to each other, and 
 do not intersect, the wheel and screw may be em- 
 ployed to connect them. The velocity of angular 
 motion is in this arrangement immediately deduced 
 from that of the screw, its number of threads, and the 
 number of teeth in its geering wheel. Thus, if it be 
 required to transmit the motion of one shaft to another 
 contiguous, and at right angles to it — the angular mo- 
 tions being as 20 to 1 ; then, if the screw be a single- 
 threaded one, the wheel must have 20 teeth ; but if 
 double-threaded the number of teeth will be increased 
 to 40, for 2 teeth will be passed at every revolution. 
 If the velocities be as 2 to 1, the condition is, that the 
 ecrew have half as many threads upon its barrel, as 
 there are teeth on the wheel; and if 1 to 1, the wheel 
 and screw lose their distinctive characters : both be- 
 come many-threaded screws under the form of wheels, 
 as represented by Fig. 2011. Wheels of this sort may 
 often be applied with peculiar advantage, especially in light geering ; and when so applied it is not 
 essentially necessary that the axes be at right angles to each other any more than it is in bevel-geer. 
 
 If the screw^ have few threads compared with the number of teeth of the wheel, it must always 
 assume the position of driver on account of the obliquity of the thread to the axis ; and in this respect 
 its action is analogous to that of a travelling rack, moving endwise one tooth, whilst the screw makes 
 one revolution on its axis. • 
 
 On the pitch of wheels.— The primary object aimed at in the construction of toothed-geer is the uni- 
 form transmission of the power, supposing that to be constant and equal. This implies that the one 
 wheel ought to conduct the other, as if they simply touched in the plane, passing through both their 
 centres. This plane is denoted by the line A B, in Fig. 2018. 
 
 When this line— which is usually denommated the line of centres— is divided into two parts, A c and 
 B c, proportional to the number of teeth formed upon the perimeters of the pinion and wheel, these two 
 parts are proportional or primitive radii of the pan: ; and a circle C being described from each centre 
 passing through the common point c, limits what is called the pitch line or circle ; that is, a circle de- 
 Bcribed from the centre A, and another from the centre B, through the same point, are called, the first, 
 the pitch circle or pitch line of the pinion, and the other of the wheel. They are also sometimes called 
 the primitive and proportional circles. If the pitch circle be divided into as many equal parts as there 
 are teeth to be given to the wheel the length of one of these parts is termed the pitch of the teeth 
 
16 
 
 GEERING. 
 
 One of these arcs, as that intercepted hj ppm Fig. 2018, comprehends a complete tooth and space, 
 meaning bj space the hollow opening between two contiguous teeth. 
 
 ^ W " 
 
 By the pitch lines of a geering pair is, therefore, to be 
 understood the proportional circles in which they would 
 revolve upon each other if they were simply cylinders 
 without teeth ; and the pitch of the teeth is the length 
 of arc of the pitch circles, measured from the centre of 
 one tooth to the centre of the contiguous one. Any right 
 lines, P and P, drawn from the centre of the wheels till 
 they meet the pitch circles or lines, are termed propor- 
 tional radii, because they determine the relations of their 
 angular velocities ; and any similar radial lines, P' and 
 P', continued to the extremities of the teeth, are called 
 the true radii of the wheel and pinion — for no very ob- 
 vious reason. In bevel and conical wheels the pitch cir- 
 cle is the base of the frustum, as AB of Fig. 1997. 
 
 Rules. — I. To find the pitch of the teeth of a wheel, 
 the diameter and number of teeth being given, divide 
 the diameter D, (in inches,) by the number of teeth N", 
 and multiply the quotient by 3-1416 : the product is the 
 pitch in inches or parts of an inch. 
 
 II. To find the diameter of a wheel, the number of 
 teeth and pitch being given, divide the pitch by 3-1416, 
 and multiply the quotient by the number of teeth. 
 
 III. To find the number of teeth, the diameter and 
 pitch being given, divide 3-1416 by the pitch, and multi- 
 ply the result by the diameter in inches. 
 
 In ordinary geering the pitches most commonly in use 
 range from 1 inch to 4 inches, increasing up to 2 inches 
 by eighths, and beyond by fourths of an inch. Below 
 inch, the pitches decrease by eighths down to \ inch. 
 
 The pitches being few and definite, the rules given 
 above may be greatly simplified by the use of the an- 
 nexed table, which will be found very convenient when the 
 diameter D is to be determined, the pitch P and number 
 of teeth IST being given ; and conversely, when the diame- 
 ter and pitch are given to find the number of teeth. 
 
 The use of this table may be rendered obvious by the 
 following examples : — ■ 
 
 _ 1. Given a wheel of 88 teeth, 2^-inch pitch, to find the 
 diameter of the pitch circle. Here the tabular number in 
 
 Pitcli in inches 
 
 TT 
 
 N = ^XD 
 
 Rule.— To And the 
 
 Rule.— To find the 
 
 and parts of an 
 
 diameter in inches, 
 
 number of teeth, mul- 
 
 'inch. 
 
 multiply the number 
 
 tiply the ^ivenxliam. 
 
 
 of teeth by the tabu- 
 
 eter in inches by th« 
 
 
 lar number answer- 
 
 tabular number an- 
 
 
 ing to the g'iven pitch. 
 
 swering to the giren 
 pitch. 
 
 Values of P 
 
 P 
 
 Values of — 
 
 Values of 1 ! 
 
 6 
 
 1-9095 
 
 •5236 
 
 5 
 
 1-5915 
 
 •6283 
 
 41 
 
 1-4210 
 
 •6981 
 
 4 
 
 1-2732 
 
 •7854 
 
 Si 
 
 11141 
 
 •8976 
 
 3 
 
 •9547 
 
 1-0472 
 
 2| 
 
 •8754 
 
 1-1333 
 
 2i 
 
 •7958 
 
 1 2566 
 
 2^ 
 
 •7135 
 
 1-3963 
 
 2 
 
 •6366 
 
 1-5708 
 
 1| 
 
 •5937 
 
 1-6755 
 
 
 •5570 
 
 1-7952 
 
 1^ 
 
 •5141 
 
 1-9264 
 
 H 
 
 •4774 
 
 2-0944 
 
 If 
 
 •4377 
 
 2-2848 
 
 U 
 
 •3979 
 
 2-5132 
 
 i| 
 
 •3568 
 
 2-7926 
 
 
 •3183 
 
 3-1416 
 
 1 
 
 '2785 
 
 3-5904 
 
 1 
 
 •2387 
 
 4-1888 
 
 1 
 
 •1989 
 
 5-0266 
 
 i 
 
 •1592 
 
 6-2832 
 
 
 •1194 
 
 8-3776 
 
 ^ 
 
 •0796 
 
 12-5664 
 
GEERING. 
 
 17 
 
 ilie second column answering to the given pitch is -7958, which, multiphed by 88, gives 70-03 for the 
 diameter required. 
 
 2. Given a wheel 33 inches diameter, l|-inch pitch, to find the number of teeth. The corresponding 
 factor is 1-7952, which multiphed by 33 gives 59-242 for the number of teeth, that is, 59^ teeth nearly. 
 Now, 59 would here be the nearest whole number ; but as a wheel of 60 teeth may be prefeiTed for 
 convenience of calculation of speeds, we may adopt that number and find the diameter corresponding 
 The factor in the second column answering to 1| pitch is '557, and this multiplied by 60 gives 33-4 
 inches as the diameter which the wheel ought to have. 
 
 A mode of sizing wheels in relation to their pitches, diameters, and numbers of teeth, is adopted in 
 some engineering factories as a simplification of tliat explained above. 
 
 Suppose the diameter of the pitch circle to be divided into as many equal parts as there are teeth to 
 be given to the wheel ; let one of these parts be called the diametral pitch, to distinguish it from the 
 circular pitch hitherto employed, and let a few definite values (in terms of the inch) be assigned to it ; 
 
 then it is clear, that calling M the diametral pitch, we have the relation - = M. And as M is always 
 a simple fraction of an inch, let M = — , then we have the general expressions, 
 
 N = ?»XD 
 
 
 To illustrate this by an arithmetical example, let it be assumed that a wheel of 20 inches diameter is 
 required to have 40 teeth ; then the diametral pitch, 
 
 ^x D 20 1 ... 
 
 M = - = — = -.= i mch ; 
 N 40 m 
 
 that is, the diameter being divided into equal parts corresponding in number to the number of teeth in 
 the circumference of the wheel, the length of each of these parts is ^ an inch, consequently w = 2 ; and 
 according to the phraseology of the workshop, the wheel is said to be one of tivo pitch. The circular 
 pitch corresponding to this diametral pitch is by the properties of the circle ^ X 3-1416 = 1-5708 
 inch. 
 
 In this mode of sizing wheels a few determined values are given to m, as 20, 16, 14, 12, 10, 9, 8, 7, 
 6, 5, 4, 3, 2, 1, which includes a variety of pitches from § inch up to 3 inches, according to the following 
 table, which shows the value of the circular pitches corresponding to the assigned values of m. 
 
 Values of m. 
 
 1 
 
 2 
 
 3 
 
 4 
 •785 
 
 5 
 
 •628 
 
 6 
 
 7 
 
 8 
 
 9 
 
 •349 
 
 10 
 
 12 
 •262 
 
 14 
 
 16 
 •196 
 
 20 
 •157 
 
 Corresponding circular ) 
 pitch in decimals of an in. \ 
 
 3-142 
 
 1-571 
 
 1-047 
 
 •524 
 
 •449 
 
 -393 
 
 •314 
 
 •224 
 
 From this table, having the valuo of m given, the corresponding circular pitch is found ; and from the 
 rules given above, if the number of teeth and value of m be known the diameter of the wheel is also 
 known, for D=N-f-«t. Thus if the number of teeth be 80 and w = 10, then the diameter D = 8 
 inches, and the circular pitch is ^314 inch, that is, 5 -16th inch very nearly. Generally, the diameter and 
 value of m being given, the number of teeth is found from the rule N" = ?n X D. Thus the value of m 
 being 10, and the diameter 20 inches, the number of teeth is 200. 
 
 From these remarks, it is easy to perceive that this mode of sizing wheels differs from that before 
 
 explained simply in tliis, th«.t it expresses in small whole numbers the quantity ^ instead of the quan- 
 tity P, and therefore affords a ready way of calculating the diameter and number of teeth of any re- 
 quired wheel. This method, however, has not hitherto been introduced into millwright work ; but has 
 been confined to the sizing of small wheels of spinning and like machinery. 
 
 It lias long been regarded as a rule among millwrights that the number of teeth in a wheel should 
 be prime to the number of teeth in its pinion; in other words, that the number of teeth of the wheel 
 should not be divisible by the number of teeth in the pinion without a remainder ; and that the best 
 possible relation of the numbers is such, that in effecting the division the remainder be 1. Tliis one is 
 kermed a hicnting tooth, and effects the purpose of preventing the same pair of teeth of the wheel and 
 pinion from coming together until the former has made as many revolutions as it has teeth. By such 
 an arrangement it is supposed the wear would be less uniform ; and it may be observed, that if the 
 teeth be at first incorrectly made, there is some advantage to be gained by taking prime numbers. 
 But in the practice of the present day, when millwrights are fully ahve to the method and advantages 
 of giving to the teeth, in the construction of their wheel-patterns, the proper geometrical form, and do 
 not trust to the wheels wearmg themselves into shape, the precaution of making the numbers of the 
 "wheel and pinion prime to each other, is less required, and may, in fact, be disregarded in proportion 
 fts accuracy of construction is attained. 
 
 In respect of the relative sizes of the pairs which geer together, the main purpose to be accomplished 
 is the modification of the contemporary velocities of the parts to such extent that their respective speeds 
 shall be adapted to the work to be performed at the several points. 
 
 To exhibit the method of applying the principles of angular velocity to the computation of the num 
 bers of a system of toothed geering, we shall consider, in the first place, the action of a single pair. 
 The fundamental proposition may be stated thus : — If there be an equal pair in geer, then whether the 
 pinion drive the wheel or be driven by it, the number of turns of the wheel multiplied by the number 
 of its teeth is equal to the number of turns which the pinion makes in the same time, multiplied hj 
 
18 GEERIN'G. 
 
 the number of its teeth, so that the number of the contemporary turns of the wheel anu pinion are re- 
 ciprocally proportional to the numbers of their teeth. 
 
 Applying this principle to a system of geering, we deduce the following rules : To find the number 
 of revolutions of the last pinion of the system, multiply the number of revolutions of the first wheel by 
 the quotient which is found by dividing the continued product of the numbers of teeth of all the wheels, 
 by the continued product of the numbers of teeth of all the pinions. 
 
 It thence also follows that the number of revolutions of the last pinion, for one revolution of the first 
 wheel, is equal to the product of all the wheels divided by the product of all the pinions. 
 
 And conversely for one revolution of the last pinion, the first wheel will make that portion of a revo- 
 lution expressed by a fraction, having for its numerator the product formed by multiplying together the 
 numbers of the teeth of all the pinions, and for its denominator the product formed by multiplying to- 
 gether the numbers of teeth of all the wheels. 
 
 From these rules it immediately follows, that whether a system of geering contains one wheel ana 
 pinion, or any greater number of wheels and a like number of pinions, if we designate the product of all 
 the wheels by W, and the product of all the pinions by P, and ifp represent the number of revolutions 
 made by the last pinion during one revolution of the first wheel, we shall have 
 
 W 
 W=^XPand^=— . 
 
 In reference to the strength of the teeth of wheels, the first subject of inquiry is the stress they aro 
 severally required to sustain when in action. 
 
 The fundamental principle involved in the consideration of this subject, is expressed by the condition 
 that the pitch being the same, the stress is inversely as the velocity ; and this is obviously true, since the 
 teeth which act with superior force must be proportionally stronger ; and the momentum of the power 
 remaining constant, the higher the velocity becomes, the more is the weight of the power diminished, 
 so that in any 'combination of wlieels, the stress upon the teeth is reciprocally proportional to the ve- 
 locity at a given point of the train. Thus, the strength which is sufficient to transmit a given amount 
 of horse power, when the velocity is one foot per second, will be equal to the transmission of double 
 that amount when the velocity is two feet a second, three times the amount when the velocity is tripled. 
 Elnowing therefore the strength of teeth necessary to transmit a given amount of horse power, the same 
 strength of wheel will be sufficient under any other circumstances of increase, or diminution of velocity, 
 when the horse power of the first mover in both cases, divided by the velocity in feet per second, pro- 
 duces the same quotient. Thus assuming as the standard the received mechanical unit of a horse 
 power, namely, 83,000 pounds raised one foot high in a minute, or 550 pounds raised one foot in a 
 second, then if H be the number of horses' power of any first mover, and v the velocity of the pitch 
 circle (in feet per second) of any wheel in the system of geering moved by it, then the stress will be 
 
 ^ ^ 550 X H 
 expressed m pounds by . 
 
 For example, if the pitch line of a wheel move with a velocity of 11 feet per second, and the power 
 
 550 X 20 
 of the prime mover be twenty-horse power, the stress will be — = 1000 pounds. 
 
 Otherwise, if P be the power of the prime mover in pounds, and V be the velocity of that power h 
 feet per second, the stress on the teeth of a wheel through which the power is transmitted, and of which 
 
 P X V 
 
 the velocity of the pitch circle is v, will be expressed in pounds as before, by . 
 
 It is necessary, however, to be observed, that the absolute power of the prime mover must only be 
 considered at those points of the geering through which it is wholly transmitted ; for if the power be 
 taken off at different points, it is obvious that the stress will be successively diminished as these points 
 are passed. For instance, the power of a steam-engine being employed to drive a cotton factory : if the 
 first geering be so arranged that the whole power of the engine is transmitted from the fly-wheel shaft 
 to a vertical shaft, which ascends from the bottom to the top-flat of the building, by a bevel pair, and if 
 the geering of the several flats be successively connected with this upright shaft, it is clear that, in esti- 
 mating the stress at the several points, with a view to ascertaining the requisite strength of the several 
 pairs, the whole power of the engine ought to be taken only at the first point ; that is, at the point 
 where it is connected with the vertical shaft. In estimating the strength of the bevel pair there placed, 
 H, in our formula, wiU be equivalent to the whole power given off by the engine ; but at the successive 
 points, where the power is taken off to drive the machinery of the several flats, H will represent only 
 the amount of power requisite to do the work at these points. 
 
 A difference is besides very properly made in practice, in the strength of those wheels of a system of 
 geering which are placed near the first mover, to compensate for irregularities in the motion ; for we»e 
 the strength exactly limited to the resistance to be overcome under constant action, a sudden accelera- 
 tion of the speed would tend to stripe the wheels, in other words, to break the teeth. Also in operations 
 of an irregular kind, the strength ought to be greatly more than is requisite in such geering as that of 
 a cotton factory. Thus the geering in iron-works, and the like, is greatly beyond the strength which a 
 calculation of the power of the prime mover would indicate, and this is required to counteract the sud- 
 den shocks which result from the chocking of rolls and the like. 
 
 It may also be necessary to remark, that, in estimating the strain upon a system of geering, it is the 
 actual power required to do the work which is to be taken as an element of the data — that is, the horse 
 power at which the resistance is valued. 
 
GEERING. 
 
 19 
 
 In the teeth of wheels, it is of importance that the -whole be made of such strength as to sustain un- 
 injured the greatest stress that is likely to come upon it in the course of "working in the worst possibldi 
 position ; that is to say, in the direction in which the 
 structure is capable of offering the least effective resist- 
 ance to fracture. Now supposing the .strain to act with 
 its whole energy upon the extreme corner of a tooth, it 
 Is easy to perceive that it will there more readily pro- 
 duce fracture than if it acted along the whole line of the 
 breadth ; for supposing it to act along the whole line of 
 breadth, if fracture of the tooth should take place, it 
 must traverse the whole line A B, or some line parallel 
 to it ; but acting at D, should the force be sufficiently 
 great to break the tooth, the fracture will take place, 
 not along the root of the tooth A B, but in the line B b, 
 or in some line parallel to it — these being the lines of 
 least resistance. Rather, under tlie circumstances of 
 the force being applied at D, the strain will be greatest along the line B 6, which is defined from 
 BD = D6. 
 
 To show that this is strictly true, let it be borne in mind that the strength of beams of equal thickness 
 is directly as their breadths, and inversely as their lengths ; consequently, if the proportion of the length 
 to the breadth be preserved, the strength will remain unaltered whatever difference is made in the 
 actual dimensions. But this being true, it necessarily follows that the line of fracture, under the cir- 
 cmnstances presumed, will be along the line which, as a base, bears the least proportion to the perpen- 
 dicular height from that base to the point where the pressure is applied. This least proportion is when 
 B D = D 6, for the base B 6 is then only d^ouble the effective length of the beam, that is, double the 
 perpendicular <? D. If a line be drawn from B to d, it will manifestly be more than double the altitude 
 of the triangle B c? D, of which it is the base ; and, similarly, the line B c is more than double the alti 
 tude of its triangle B c D. 
 
 This might be directly proved by application of the rules of maxima and minima ; rather, it might be 
 shown that tlie strain is greatest under a force acting at D, in a line determined by D B = D 6, which 
 makes B 6 ^ 2 e D. 
 
 In effect, therefore, the line B h (including all lines parallel to it) is the line of least resistance of the 
 tooth, and consequently the line in which fracture would be produced by a force sufficiently great ap- 
 phed at D. Presuming, then, that wheels, in consequence of inaccuracy of workmanship, unequal 
 wearing of brasses, vibration of shafts, and other circumstances incidental to the action of a system of 
 geering, are liable to stress acting upon them in the least efficient position of the teeth, it would appear 
 that the effective proportion of the breadth of the tooth, assuming the thickness to be uniform, does not 
 exceed twice the length. Whatever may be the force, the principle informs us that there is a limit be- 
 yond which no strength is practically gained, and this limit will be found in general not to differ 
 materially from double the length of the tooth. 
 
 It must not, however, be inferred from this that it is useless to make the breadth of any proportion 
 greater than that stated ; for although no additional practical strength be gained by increase of breadth, 
 it is still highly advisable that the dimensions named should bear a much higher ratio, than is given by 
 the consideration of the merely mathematical principle. This is accordingly followed in practice, and 
 the advantage is, that the wearing action, by being distributed over a larger surface, does not so soon 
 reduce the thickness, and thereby render the wheel too weak for the work it has to perform. Moreover, 
 there is the additional advantage in giving more breadth than is indicated above, that the surfaces oi 
 contact being longer, vibration is to some extent diminished ; the centres are accordingly better pre- 
 served, and the wearing of the tooth becomes greatly more uniform. An error of workmanship, and 
 of unequal contraction of the casting, becomes likewise more apparent, and may possibly admit t.f 
 correction. 
 
 2020. 
 
 As it is convenient to express all the dimensions in terms of the same unit, and the pitch being an 
 appropriate quantity, is nearly universally adopted as the term of comparison. 
 
 These proportions differ, though slightly, in different works and in different localities ; but they are 
 the most commonly employed, and are besides the most consistent with good and accurate workman* 
 ship. For the sake of more easy reference, we collect them into a table, which the annexed Fig. 2080 
 wiU serve fully to explain. They stand thus : 
 
20 GEERIKG, 
 
 a 6 =-: Piteh of teeth, =; 1 pitch. 
 
 acrs:^ Depth to ptch line, P P, sa yq " 
 A a X a c =^ Working depth of tooth, sss jf- 
 C c — A a s= Bottom clearance, =r jV 
 
 C a =: Whole depth to root, =s j-g- 
 C 6 := Thickness of tooth, =s jx 
 
 A 6 =s: Width of space, -s^ xT " 
 
 These proportions, as remarked, are found to work very advantageously^ and are those adopted io 
 several workshops ; but the following are preferred by some engineers of experience. Thus supposing 
 the pitch divided into 15 equal parts: then the 
 
 Depth from point to pitch line, :ss 5^ 
 
 Depth from pitch line to root of tooth, ss 6^ 
 Whole length of tooth, =!l2 
 
 Working depth, ssll " 
 
 Thickness of tooth, (also of arms and rim,) :== 1 " 
 Width of space, 3=s 8 " 
 
 In practice, these proportions are usually laid off in lines for the convenieiisce of the Workmen fn the 
 pattern shop, so that for any given pitch the other dimensions may at once be determined by means of 
 the compasses, and without having recourse to calculation. In Figs. 2021 and 2022 two diagrams of that 
 sort are given. Fig. 2021 contains the proportions last enumerated, in which the pitch is supposed to be 
 divided into 15 equal parts ; and Fig. 2022 is constructed nearly according to the proportions first given, 
 but embraces the recognized principle that the relative amount of clearance ought to vary inversely as the 
 pitch, wheels of small pitch requiring more clearance relatively than those in which the pitch is greater. 
 Accordingly, in this scale, the clearance in a wheel of |-inch pitch is 1-lOth the pitch, whereas were the 
 scale sufficiently extended, it would show a clearance of only 1-1 8th for a pitch of six inches. 
 
 The construction of these scales is very simple. Thus in Fig. 2021, and to six inches pitch, let A B 
 be divided into 15 equal parts, and draw BC perpendicular to it; and again divide BC into a 
 determinate number of parts from B, actual measures of the pitches for which the scale is intended 
 to be used ; that is, B a = ^ inch ; B-6 = 1 inch ; B c = 2 inches, and so on, and join a and A, h and A, 
 and A, and so on. To complete the scale, di'aw 15 parallels to B C irom the points numbered in the 
 line A B, numbering their intersections (if thought proper) with the line A C in the same order ; and 
 also the two parallels T and U, (which are full hues in the diagram,) equidistant from the parallels on 
 each side of them. 
 
 The scale is thus ready for use, and its principle is self-evident. To get from it the several propor- 
 tions for a given pitch, say of 3 inches == B c?, let the compasses be extended from the intersection of the 
 parallel marked T, with the line A B, to the point where it intersects the line A d ; this will be the pari 
 of the tooth from the pitch line to the point, and equivalent to 5^ parts of the pitch, (viz. of B </ ;) sim- 
 ilarly the compasses being extended from the intersection of the parallel U, with the hne A B, to its 
 point of intersection of the line A c?, will give the part of the length of the tooth from the pitch line to 
 the root, and equivalent to 6^ parts of the pitch. For the whole length of the tooth (if wanted in one 
 measurement) set the compasses to the point where the parallel marked 1 2 meets the line A B, and 
 extend to its point of intersection of the line A c? at s, the length is 12 parts of the pitch Be?; the work- 
 ing depth is in like manner found from the parallel marked 1 1 ; the thickness from that marked 1 ; and 
 the width of space from that marked 8. 
 
 The proportions for any other given pitch comprised in the scale are found in precisely the same way, 
 and if the scale be well constructed they may be measured off with the utmost accuracy and readiness. 
 To save confusion it is, however, better in practice to insert in the diagram only those parallels, namely, 
 T, U, 12, 11, 8, ^, which are required ; the others are not requisite, and by inattention may lead to error. 
 Both this scale and that marked 2 are commonly drawn on hard-wood boards ; but sometimes, for the 
 sake of greater accuracy, on plates of polished brass. 
 
 The description of the scale as here given supposes that the lateral clearance is constantly l-15th ol 
 the pitch ; but as it is commonly desired and desirable that this should vary slightly with the pitch, 
 relatively increasing as the pitch decreases, two other lines, m n and p q, have been introduced into the 
 scale, to enable such modification to be adopted, should it be required. These lines are drawn at such 
 angles as to give a clearance at 6 inches pitch of 1-1 8th, wliich is increased at |-inch pitch to 1-lOth. 
 From these hues the thickness and space are to be taken, instead of using tlie hnes marked 7 and 8, 
 setting the compasses in the points of intersection with the pitch lines, and extending perpendicularly 
 to the line A B ; in other words, the shortest distance from the point of intersection with the pitch line 
 to the hne A B, is the required measure of the space when the line pqis taken, and of the thickness of 
 tooth when the Hne m n is taken. 
 
 Fig. 2022 is more complete than the one described, and when well cons|;ructed insures, with moderate 
 care, a degree of accuracy and uniformity, in the construction of the various sizes of wheels for which it 
 is employed, that can hardly be otherwise attained. The principle of its construction is in effect the 
 same as that described, but its use is more extended ; the diameter of the wheel being found from it 
 simultaneously with the length and thickness of tooth, width of space, and clearances. The scale is 
 adapted to wheels of all the pitches enumerated in the table, p. 195, from ^ inch up to 3 inches. The 
 mode of construction is this : having drawn the line A D of any convenient length, raise the perpendic- 
 ular C B to it, also of any convenient length. On the line A D lay off the greatest pitch of the scale 
 from C to A ; then from C towards D lay off seven times the pitch once or twice, according to the sizes 
 €f wheels to which the scale is intended to be applied. In the scale given, double of seven time« tha 
 
 52 
 
GEERING. 
 
 21 
 
 pitcli is laid off, namely, 42 inches ; then each of these great divisions being subdivided into 11 equal 
 parts, one of these parts Avill be equal to four teeth upon the radius of the wheel, so that the whole line 
 C D will be divided into 88 radial pitches. Next on the line C B set off the pitches which may be re- 
 quired in the scale, and through these points draw the 24 parallels to A D, terminating in the lines A B 
 and D B. Then each parallel measured from the line B C to its point of termination in B D, is the 
 radius of a wheel of 88 teeth of the particular pitch marked against it on the line A B. They also 
 express the radii of wheels having less than 88 teeth when measured only to the corresponding point 
 in the line joining B, and the divisional on C D, against which the number of teeth is marked. Thus 
 the radius of a wheel of 52 teeth and l|-inch pitch, is rs := 15 7-16tIi iaches very nearly. (The trua 
 answer by the table, p. 795, is 308724 ~ 2 = 15-4362 inches.) 
 
 tJ02J. 
 
 a 
 
 PROPORTION SCALES FOR GEERING. 
 
 c • d 
 
 554 
 
 UZ 21 all Jfl 18 U 16 15 14 . 13 12 IL 10 
 
 u i:< 12 li. 10 
 
 ft 
 
 The scale may also be used when the number of teeth exceeds 88 ; for example, to find the radius of 
 a wheel having 100 teeth. Thus having found the radius answering to 88 teeth, upon the same parallel 
 take off the measure answering to the difference 100 — 88 = 12 teeth ; and the two measures together 
 will be the radius required. 
 
 To adapt the scale to odd numbers of teeth, the first division on the right of C is divided into single 
 radial pitches, so that the radius of any wheel may be measured off without having recourse to calcu- 
 lation of any kind. Thus, for example, if the wheel is intended to contain 50 teeth, the compasses being 
 
22 
 
 GEERING. 
 
 exitended from 52 to the intersection of the parallel answering to the particular pitch to -where it meets 
 the line joining Q and B, will give the radius required, that is, a radius answering to 52 — 2 = 50 teeth 
 and any other number of teeth "when not marked against the base may be found in the same "way, ob 
 serving that it is more convenient to subtract than to add in this use of the scale. 
 
 For the proportions of the teeth, set off a = 1-tentks of the pitch, then -will A a = Z-tenths of the 
 pitch, -which corresponds to the depth from the point of the tooth to the pitch line. Again, set of! 
 C 6 = ^-fifteenths of the 3-inch pitch, and ^-elevenths on the parallel against the 1-inch pitch ; this -wiD 
 be the thickness of the tooth, allowing from a fifteenth for clearance on the largest pitch, to a tenth on 
 those from |-inch and under ; and A b will be the width of space, including the clearance. Lines being 
 flrawn from those points to B complete the diagram, which will be found to contain all the proportions 
 enumerated in the preceding table. 
 
 To use the scale, lay off the addendum of the tooth ; that is, the length beyond the pitch line, equal 
 to A a = y^^ pitch, and the same length marked off within the pitch line will give the whole working 
 depth of the tooth, namely, 6-lOths pitch. Then with the measure Ca=z^^ pitch in the compasses, 
 mai-k off the whole length of the tooth, and this will allow 1-1 Oth at bottom for clearance. Again, set 
 off the thickness of tooth = C &, and the space = A 6 which will contain the clearance for the particular 
 pitch, varying from l-15th to fully 1-lOth on the small pitches. It is hardly necessary to observe that 
 these measurements must be taken upon the parallel corresponding to the particular pitch under con- 
 sideration at the time. 
 
 The amount of bottom clearance is here presumed to be uniformly 1-lOth of the pitch; but if it be 
 thought advisable to make this vary as in the case of the lateral clearance, it will then be necessary to 
 insert a third line c B in the scale, and so related to a B that the space a c shall be throughout equal to 
 the depth of tooth from the pitch circle to the root, and giving any bottom clearance that may be desired. 
 
 In relation to the strength of wheels, M. Morin, in his Aide Memoire de Mecanique Pratique, gives it 
 as a rule that when the velocity of the pitch circle does not exceed five feet per second, the breadth ol 
 the tooth measured parallel to the axis ought to be equal to four times the thickness ; but when the velo- 
 city is higher the breadth ought to be equal to five thicknesses, the teeth being constantly greased. If the 
 teeth be constantly wet, he recommends the breadth to be made equal to six thicknesses at all velocities. 
 
 With respect to the thickness of the tooth, it is plain that it must be dependent on the pressure which 
 the tooth is required to sustain. This relation may be conveniently expressed for all cases by the formula, 
 t=.c y/ W. where t is the thickness of the tooth, "W the pressure upon it in pounds, and c a constant 
 multiplier depending upon the nature of the material of which the tooth is formed. 
 
 Therefore for cast-iron c =-025 ; and reasoning in the same way for brass, we find it = -035 for hard 
 wood, = -038 ; so that for the thickness of teeth of these materials, we have, 
 
 IZ wf """' / -'.of. ^/ w \ ^l^icl^ g^^e i i^ i^el^es or P^rts of an inch. 
 
 For brass, ^ = -035 -^ W 
 
 For hard wood, t = -038 V W 
 
 f W being taken in pounds. 
 
 As an example of the application of these formulae, let it be required to find the thickness of a tooth 
 (cast-u-on) which is to sustain a pressure of 4000 poimds at the pitch circle, the breadth being double 
 the length. Here W = 4000 ; therefore v' W = 63-25. 
 
 Hence t = -025 v' W = -025 X 63-25 = 1-58 inch. 
 
 The same calculatioo applied with the formula for brass, would give t = 2'21 inches ; aiid for wood 
 it gives t = 2-4 inches. 
 
 The thickness of tooth giveo by our rules is intended to make allowance for wear at a velocity o^ 
 three feet per second, and has been found to be sufficient in practice. It is, however, less by a small 
 fraction than would be found by application of Mr. Tredgold's rule ; that is, divide the stress at the 
 pitch circle in pounds by 1500, and the square root of the result is the thickness of the tooth in inches. 
 
 To compare this rule with that given above we subjoin the following table, which will likewise be 
 found useful in calculating the strength of wheels. 
 
 
 Thickness of teeth. 
 
 
 
 Stress in lbs. at 
 
 
 
 Actual pitches to 
 
 Corresponding thickness^ al- 
 
 
 
 the pitch circle. 
 
 By Authors 
 rule. 
 
 By Tredgold's 
 Rule. 
 
 which the wheels 
 would be made. 
 
 lowing 1-lOtk for clearance. 
 
 lbs. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 400 
 
 0-50 
 
 0-52 
 
 1| to li 
 
 0-536 to 0-593 
 
 809 
 
 0-71 
 
 0-75 
 
 li — If 
 
 0-714 — 0-774 
 
 1200 
 
 0-87 
 
 0-90 
 
 1| — 2 
 
 0-893 — 0-952 
 
 1600 
 
 1-00 
 
 1-03 
 
 2 — 2J 
 
 0-952 — 1-012 
 
 2000 
 
 1-12 
 
 1-15 
 
 2i — 2| 
 
 1-031 — 1-132 
 
 2400 
 
 1-22 
 
 1-26 
 
 n — 2f 
 
 1-190 — 1-250 
 
 2800 
 
 1-32 
 
 1-36 
 
 2f — 2| 
 
 1-250 -- 1-309 
 
 3200 
 
 141 
 
 1-43 
 
 2| — 3 
 
 1-369 — 1-429 
 
 3600 
 
 1-50 
 
 1-56 
 
 H — H 
 
 1-488 — 1-548 
 
 4000 
 
 1-58 
 
 1-63 
 
 H — H 
 
 1-548 — 1-607 
 
 4400 
 
 1-66 
 
 1-70 
 
 3| — Si 
 
 1-607 — 1-667 
 
 4800 
 
 1-73 
 
 1-78 
 
 3^- 3S 
 
 1-667 — 1-726 
 
 5200 
 
 1-80 
 
 1-86 
 
 3| — 3| 
 
 1-726 — 1-786 
 
 5600 
 
 1-87 
 
 1-93 
 
 3i-4 
 
 1-786 — 1-904 
 
 6000 
 
 1-94 
 
 2-00 
 
 4 — 4i 
 
 1-904 — 2-024 
 
GEERING. 23 
 
 The last column of this table is calculated from the expression pitch ^. = (2 -|- ^L) = 2. 1 1 the clear* 
 ance being a tenth. The thickness, allowing l-15th for cleai-ance, may in like manner be calculated 
 trom the expression pitch =zt. (2 -f" tV^ ^^ 2*067 t. 
 
 The formula reduced to expressions giving the pitch p in the same manner as the thicknesses are 
 ^iven above, will stand thus : — 
 
 Clearance a tenth. Clearance a fifteenth. 
 
 For cast-iron teeth ^ = '0525 y/ W and p = -0517 y/ W. 
 
 For brass teeth p = -0735 y/ W and p = -0723 ^ ^• 
 
 For wooden teeth p = '0798 ^ W and jo = -0785 -/ W. 
 
 By means of these rules the pitch may be directly calculated. Thus, for a pressure of 15,000 pounds 
 
 we have, 
 
 ^W = ^ 15,000 = 122-5. 
 Consequently, the pitch of a cast-iron wheel capable of sustaining that pressure at the pitch circle, 
 allowing 1-1 5th for clearance, will be 
 
 •0517 ^/ W = -0517 X 122-5 = 6-333 inches. 
 TJie wheel from whicli this example is taken has been several years in action, and has an actual pitch 
 of six inches, and the mean pressure at the pitch cii-cle is 14-786 pounds. 
 
 It may be here observed that it is common, in calculations relative to the strength of the teeth of 
 wheels, to make additional allowance for wear of the pinion ; for if the pinion make double the number 
 of revolutions made by the wheel with which it is engaged, its teeth will manifestly be subject to double 
 the amount of wear by friction ; consequently, to proportion the teeth of the pair so that they shall 
 v/ear equally long, it is necessary to give an allowance of tliickness on those of the pinion equivalent 
 to the increase of abrasion to whicli they are subject. If we assume one-third the thickness as the 
 
 (2 -{- n) t 
 proper allowance, this will give for the thickness of the pinion the expression — — , in which n is 
 
 the number of revolutions which the pinion makes for one revolution of the wheel, and t the thickness 
 of the teeth of the wheel. Thus, for example, if the pinion make 2^ times as many revolutions as the 
 wheel, of whicli the thickness of the teeth is 1-12 inch, then the thickness of those of the pinion will be 
 ti2 + n) ^m2(2±^ 
 3 3 
 
 This amount of difference, is not, however, commendable in practice, at least in spur-geer, and it is 
 therefore rarely adopted, as wheels would in that case require to be constantly made in pairs, which 
 would lead to an endless accumulation of wheel patterns. Instead of making the allowance spoken of, 
 the common practice is to adopt a larger pitch — rather, indeed, to use wheels somewhat beyond the 
 strength which is requisite for the work. 
 
 It has already been shown that by a horse power is meant a pressure of 33,000 pounds moved with 
 a velocity of one foot in a minute. But this is the mean of the force exerted, and as most prime movers 
 are more or less variable in their motion, any wheel required to transmit tliat motion should be strong 
 enougli to bear the maximum force with safety. For ordinary and general purposes we may assume, 
 as a very safe approximation, that it exceeds the mean of the whole force exerted by the fraction 
 3-llths. Making that allowance, we shall have, as the practical strain of a horse power, 550 lbs. X 
 ly^y = 700 lbs. raised one foot per second. By substitution of this value of the horse power, in the rule 
 
 formerly given, it will become — X H = the stress on the teeth of the wheel in pounds. 
 
 As an example, let a steam-engine of 12-horse power be applied to di-ive the machinery of a factory, 
 and let it be required to find the strength of the teeth of a first wheel on the main shaft, which wiU 
 have a velocity of four feet a second, at the pitch circle. 
 
 Here H = 12 and w = 4 ; therefore X I? = — — • X 12 = 2100 lbs., the pressure at the pitch cir- 
 cle = "W in the rule for the tliickness of teeth, 
 
 Now the square root of 2100 is 45-826, and supposing the wheel to be of cast-iron, then we have 
 t = -025 y/W = -025 X 45-826 = 1-1456 inch, 
 the thickness of the teeth of the wheel; consequently, if 1-lOth be allowed for clearance, the pitch wil 
 be 1-1456 X 2*1 =2-52 inches. The actual pitch would therefore be made from 2^ to 2f inches. 
 
 If the wheel have wooden teeth, then the rule t = 038 ^/W must be used, which gives 1-74 inch aa 
 the thickness of tooth, and 3-66 inches as the pitch ; the pitch to be adopted would therefore be 3| 
 inches. 
 
 When cast-iron and wooden teeth work together, their action upon each other tends, in consequence 
 of the elasticity of the wood, to maintain a more uniform distribution of the strain, and being at first 
 commonly more accurately dressed, to prevent abrasion of the wood by the iron, they work with much 
 less friction, are less liable to shocks, and nearly exempt from accident by hard particles coming be- 
 tween the teeth. 
 
 The best practice, when a mortise and iron wheel are to work together, is to make both of the same 
 pitch, and in tlie first instance, of the same thickness of tooth — the pitch being of course calculated for 
 the wooden wheel by the rule jo = -0785 \/W, which allows l-15th for clearance, (and with good work- 
 manship this is amply sufficient in cases of the kind proposed ;) afterwards to dress down the teeth ol 
 the iron wheel, by the chipping tool, or the wheel-cutting machine, and file, to the exact form and thick- 
 ness, as given by the formula i =■ -025 ^Z W ; that is, to a thickness in relation to the thickness of the 
 
 ooden teeth, which shaU have the ratio of 25 to 38. Both of the wheels will then be of the same 
 
24 GEERING. 
 
 strength, and from their superior finish, will work with much less friction, and, consequently, less weai 
 than if both wheels were of iron. 
 
 If m be the number of revolutions to be made by the wheel in a minute, and n the number of teeth 
 to be cut on it, and W the pressure upon each tooth in lbs., we shall have the following values of the 
 pitchy in inches and parts of an inch, agreeable to the method adopted in the preceding rules, namely, 
 
 For cast-iron » = 12 v/ — or a / ^ > = P 
 
 ^ y/ mn y/ i mn S 
 
 T? -U no /^^ /^2197H) 
 
 For Brass » = 13a/ — or a/ { > = P 
 
 ^ \/ mn W \ mn S ^ 
 
 For hard wood j9 = 14 a /— or a / ^ — — \ =p 
 y/ mn W i mn S 
 
 Conversely, the power which the teeth of a wheel of given pitch are capable of transmitting may be 
 readily calculated from the following rules, which are immediately deduced from the above : 
 
 For cast-iron . H = ~ 
 
 12' 1728 
 
 For brass . . h = !^^' or ^^i^' 
 13^ 2197 
 
 „ , , , T-T mnp^ mnp^ 
 
 For hard wood . H = i- or ^ » 
 
 14^ 2744 
 
 Thus supposing the pitch j? = 3 inches, the number of teeth n = 60, and the number of revolutions 
 n which the wheel is designed to make per minute = 50 ; then the wheel being cast-iron, the power H 
 which it is capable of transmitting will be 
 
 60 X 5 X 3^ ,H, , 
 
 H = ---— = 47 horses power nearly. 
 
 This wheel is employed to transmit 45 horses' power at the given velocity, and has been at work for 
 several years. 
 
 Every writer on the teeth of wheels has thought it necessary to adduce rules for finding the proper 
 breadth of the teeth. As respects strength, such a calculation has been shown to be in a manner un- 
 necessary beyond merely doubling the length, which is immediately deducible from the pitch ; and as. 
 respects durability there seems to be no theoretical limit to the breadth, for the more the pressure and 
 rubbing action is diffused, the less rapidly will tlie teeth be worn. The breadth to be assigned in prac- 
 tice must, therefore, be always a quantity to be determined by circumstances, and modified by the par- 
 ticular opinions of those concerned ; if the motion be particularly uniform and free of vibration, the 
 breadth may be extended even to four times the pitch with advantage ; but if the contrary circum- 
 stances obtain, this great breadth will in like manner have a contrary effect, the teeth becoming fre- 
 quently locked together, will more speedily wear out of shape if they be made too strong to twist and 
 break one another out at the ends. When the shafting is light, tliis is a frequent occurrence, although 
 the reason does not seem to be always understood ; and accordingly, the remark is not micommon, that 
 the wheel ought on account of its great breadth to have been more than sufficient for the work. How- 
 ever contradictory it may appear, strength in this sense is not an unfrequent cause of weakness and 
 failure. 
 
 Assuming that the teeth of wheels follow the same law of strength by increase of breadth as in thick- 
 ness, and referring back to the general formula for the strength of a beam of given dimensions, we 
 arrive at the conclusion that the breadth b and the length I are in the relation of 6 = | Z ; and suppos- 
 ing with some engineers of experience that a breadth of 6 inches is sufficient for a power equivalent to 
 9 horses, when the pitch line moves with a velocity of 3 feet per second, then we have the following 
 rule : Double the number of horses' power of the prime mover, and divide the result by the velocity of 
 the pitch ch'cle in feet per second : the result is the breadth of the teeth in inches. 
 
 Thus the power transmitted by a wheel moving at 5^ feet velocity per second, transmits 16 horses* 
 power ; the breadth of the wheel will therefore be 
 
 2X16 S-2 ,„„. , 
 
 — — — = — = 5-82 mches nearly. 
 
 Again, on the same principle, having the breadth and velocity of teeth given, multiply the velocity 
 of the pitch circle in feet per second by the breadth of the teeth in inches, and half the product will be 
 the number of horses' power which the wheel is capable of transmitting. 
 
 Thus, let the breadth be 12 inches, and the velocity 21 feet per second; then 12 X 21 =252, half 
 of which is 126, the number of horses' power required. 
 
 It is easy to see, however, that unless the breadth be a function of the pitch, any calculation of this 
 kind cannot be satisfactory ; and from the remarks already made, there cannot exist such difficulty in 
 fixing upon the proper breadth the wheel ought to have for the particular purpose intended. 
 
 The following table may be useful in determining the relation of the dimensions of the teeth of 
 wheels of the given pitches, and the power which they are capable of transmitting safely at the various 
 
 2 p^ X V 
 speeds named. The table was originally constructed from the formula -^^-— = H, and has sinc^ 
 
 Deen extensively used. 
 
GEERING. 
 
 25 
 
 
 Thickness 
 of teeth. 
 
 Length of 
 teeth. 
 
 Least 
 
 Velocity of the wheel at the pitch circle. j 
 
 Pitch. 
 
 breadth of 
 teeth. 
 
 3 feet per 
 
 4 feet per 
 
 5 feet per 
 
 7 feet per 
 
 11 feet per 
 
 
 
 
 
 second. 
 
 second. 
 
 second. 
 
 second. 
 
 second. 
 
 i 
 
 Inches. 
 
 Inches. 
 
 Incheff. 
 
 Inches. 
 
 H. p. 
 
 H. P. 
 
 H. P. 
 
 H. P. 
 
 H. P. 
 
 6 
 
 2-9 
 
 4-2 
 
 8-4 
 
 43i 
 
 57| 
 
 72 
 
 100^ 
 
 158f 
 
 6| 
 
 4 
 
 2-6 
 
 3-85 
 
 7-7 
 
 36A 
 
 48f 
 
 60i 
 
 84tV 
 
 133^V 
 
 1-9 
 
 2-8 
 
 6-6 
 
 19 
 
 251 
 19| 
 
 32^ 
 
 45 
 
 701 
 
 f 
 
 1-6 
 
 2-45 
 
 4-9 
 
 14| 
 
 241 
 
 341- 
 
 54 
 
 1-4 
 
 2-1 
 
 4-2 
 
 11 
 
 l4 
 
 18 
 
 25 
 
 391 
 
 f 
 
 1-2 
 
 1-75 
 
 3-5 
 
 74 
 
 10"" 
 
 121 
 
 171 
 
 271 
 
 0-95 
 
 1-4 
 
 2-8 
 
 4| 
 
 6} 
 
 8 
 
 iT 
 
 171 
 
 1| 
 
 0-83 
 
 1-225 
 
 2-45 
 
 34 
 
 5 
 
 6} 
 
 81 
 
 13^ 
 
 ^ 
 
 0-71 
 
 1-05 
 
 2-1 
 
 2| 
 
 34 
 
 2|- 
 
 41 
 
 6| 
 
 10 
 
 0-59 
 
 0-875 
 
 1-75 
 
 2 
 
 3| 
 
 4^ 
 
 6f 
 
 li 
 
 0-53 
 
 0-7875 
 
 1-575 
 
 1| 
 
 2| 
 
 n 
 
 3i 
 
 51 
 
 1 
 
 0-48 
 
 0-7 
 
 1-4 
 
 U 
 
 If 
 
 2 
 
 2j 
 
 4| 
 
 1 
 
 0-41 
 
 0-6125 
 
 1-225 
 
 1 
 
 If 
 
 If 
 
 21 
 
 ^ 
 
 I 
 
 0-36 
 
 0-525 
 
 1-05 
 
 A 
 
 A 
 
 H 
 
 H 
 
 21 
 
 5 
 
 0-33 
 
 0-4375 
 
 0-875 
 
 i 
 
 
 3 
 
 1 
 
 1?^ 
 
 J 
 
 0-24 
 
 0-35 
 
 0-7 
 
 ^3 
 
 r 
 
 t 
 
 2 
 
 A 
 
 11 
 
 i: 
 
 40 
 
 1 
 
 0-18 
 
 0-2625 
 
 0-525 
 
 1 
 
 fV 
 
 1 
 
 T 
 
 0-12 
 
 0-175 
 
 0-35 
 
 3 
 
 .1 
 
 To 
 
 k 
 
 .^. 
 
 To find the power which a wheel is capable of transmitting for other velocities than those in the 
 table -. — For 6 feet per second, double the result given for 3 feet ; for 8 feet double the result at 4 feet, 
 and so on ; and for lower velocities than those given, divide the tabular number by the ratio which they 
 bear to those enumerated. Thus, for 2^ feet velocity, take half the result at 5 feet, and so of other 
 velocities. 
 
 When a wheel and pinion, which differ very much in size, work together, the teeth of the latter, on 
 account of their unequal thickness, are capable of sustaining much less pressure than the teeth of the 
 wheel : they are in effect, if not in fact, much reduced in thickness ; and, in applying rules to the cal- 
 culation of the strength of wheels, the difference of size of the pair ought not to be overlooked, unless, 
 aS is indeed very common in practice, the deficiency of strength be made up to the pinion by a flange 
 cast on one or both sides of the rim, of the same depth as the teeth, and binding these together like 
 the staves of a trundle. In this case the pinion is commonly the stronger wheel of the pair. 
 
 In the construction of wheels, the problem which presents itself relative to the shape of the teeth is 
 this, that the surfaces of mutual contact shall be so formed that the wheels shall be made to turn by 
 the intervention of the teeth, precisely as they would by the friction of their circumferences, 
 2023. 
 
 Thufi, if we take Fig. 2023 to represent two teeth of a pair of wheels of which that marked A is the 
 driver ; then it is plain that the faces of the teeth may be of such curvature in relation to each other, 
 that as the circle A revolves and carries with it the circle B by then: mutual contact at C, the teeth 
 may continually touch one another throughout their lines of curvature, continually altering their rela- 
 tive positions and their point of contact t, as the primitive circles change their point of contact ; and 
 this being true, it evidently follows that the two circles would be made to revolve by the contact o/ 
 taeth, whose surfaces are thus formed, precisely as they would by the friction of their circumferences 
 
26 
 
 GEERING. 
 
 at the point c. For observing the action of the circles upon each other, in the former case we find a 
 series of points of contact passing through c, and bringing about a corresponding rotation of points of 
 contact of the teeth at t ; and in the latter case, the action being transferred to the teeth, the same 
 transposition of the points of contact at t is followed by the same transposition of points, as formely, 
 through c. 
 
 To form teeth whose surfaces of contact shall possess the property here assigned to the curves ol 
 the teeth m m and n n, is the problem to be solved ; and the solution is dependent on the following 
 
 Fundamental principle. — In order that two circles A and B may be made to revolve by the contact 
 of the surfaces of the curves m m and nn of their teeth precisely as they would by the friction of thei; 
 cii-cumferences, it is necessary and sufficient that a line drawn from the point of contact t of the teeth 
 to the point of contact c of the circumferences, (pitch circles,) should in every position of the point t be 
 perpendicular to the surfaces of contact at that point ; that is, in the language of mathematicians, that 
 the straight line be a normal to both the curves m m and n n. 
 
 In proceeding to establish this principle, we must in the first place recur to a proposition, which in 
 effect is that employed by the mechanic in the use of his templets for drawing in the curves of the 
 teeth. Thus if we have two circles, as in Fig. 2024, described about A and B as fixed axes, and these 
 circles be free to revolve by their mutual contact at T ; then supposing A B the line of centres divided 
 as usual in T, in the inverse ratio of the angular velocities of the circles ; if the circle B be provided 
 with a fine tracing point fixed into its circumference, and it be made to roll upon A, the point will 
 describe the curve a b. Again, if the curve a h continued to c be cut out of thin plate and caused to 
 turn round the centre A, and the pin at b be carried by an arm round the centre B, the motion commu- 
 nicated by the pin to the curve will fulfil the required conditions. For at the beginning of the motion 
 let T e be the position of the curve, then the pin will coincide with T ; and if the curve move into any 
 other position abc, driving the pin to b, the arc T a will be equal to T b, and the path described by the 
 pin in its motion will be that indicated by a b. But the arcs T a and T b are also those described by the 
 two pitch circles respectively in moving from T to the second position, and since these equal arcs are 
 described in the same unit of time, the angular velocity is not charged, but remains constant as if the 
 motion had been produced by the rolling contact of the two pitch circles. 
 
 A further illustration of the same principle is afforded by Fig. 2025, in which A and B are centres of 
 motion as before, and T the point of contact of the pitch circles of the wheels. In this case let. the 
 curve ibche described by a point b fixed in the circumference of a circle T b B, haying for its diameter 
 the radius B T of the pitch circle of the wheel B. From the centre B, let a radial Ime B6 be drawn, 
 touching b and meeting the pitch circle in d. Further, let motion be communicated by contact trom 
 the edge abcoi the curve which revolves about the centre A to the radial fine B b d, which revolves 
 about B and let the beginning of the motion be reckoned from the position m which a coincides with 1, 
 and therefore (^ with a; then in moving to any other position of contact, the arcs simultaneously 
 described by the two pitch circles will be T « and T d Now T B 6 is an angle at B, the circumference 
 of the roUbi circle, and T B (^ an angle at the same point, which is the centre of the pitch circle ; there- 
 fore T b measures an angle double of T d. Also the radius T b is half that of T c?, consequently the arc 
 rj. J _ rp ^ _ rp ^ . t^^t is, thc arcs of the pitch circles measured from the beginning ot motion are equal, 
 and therefore the angular velocity ratio, as before, is constant and the same as would be obtained by 
 the rolling contact of the pitch circles. . •, , . x, . -a i 
 
 To exhibit this principle in another point of view, let there be two circles havmg their axes in A and 
 B, and their point of contact c as shown in Fig. 2026 • then supposing that the circle B is made to roU 
 
GEERING. 27 
 
 upon the circumference of A, a point in the circumference of the former will describe a curve a h ; thai 
 they are applied by their axes to a thii'd plane M M, into which their axes are fixed, and which has« 
 also an axis of motion coinciding with A ; then it is evident if the plane M M be made to move while 
 the chcle on A is kept at rest, the chcle upon B will be made to revolve upon the circumference of this 
 last, and a point b fixed into its circumference will trace out a curve a h upon the plane M M, precisely 
 the same as would have been described by that point, if the latter plane had remained at rest, and the 
 centre of the circle B had been set free from its axis, and been made to roll by its circumference on the 
 circumference of A. This is obvious, and it is also obvious that, both circles being fixed by their axes 
 to the plane M M, and the circle on A being made to revolve with an equal, but opposite angular 
 velocity to M M, and which communicates its angular velocity to the circles A and B, these revolving 
 meantime in respect to one another, and by the mutual contact of their circumferences precisely as 
 they would if the plane M M were at rest, then the circle A being carried round by its own proper 
 motion in one direction, and by the motion common to it, and the plane M M with the same angular 
 velocity in the opposite direction, will in point of fact rest in space ; and at the same time the circle B 
 having no motion proper to itself, will revolve with the angular velocity of the plane M M, and all the 
 points of that circle will have angular velocities compounded of their proper velocities, and the veloci- 
 ties they receive in common with the plane M M ; but these velocities being equal and opposite at the 
 point c, will there neutralize one another. ITow this point c is the point of contact of the two circles ; 
 so that while B revolves about A, the point c at which it is in contact with the latter is at rest ; yet 
 this quiescent point of the circle B is continually varying its position on the circumferences of the two 
 circles, so that the circle B is thus made to roll on the circle A, which, in consequence of its own proper 
 motion in one direction being neutralized by the equal and opposite motion it receives from the plane 
 M M, is in reahty at rest. 
 
 It therefore appears that by communicating a certain common angular velocity to both the circles A 
 and B about the centre A, the former is made to roll upon the latter at rest, and moreover that this 
 common angular velocity does not alter the form of the curve a b, which a point b in the circumference 
 of B describes upon the plane M, that is, in effect upon the plane of the circle A ; in other words, that 
 the curve traced under these circumstances is the same whether the circles revolve about fixed centres 
 by their mutual contact, or whether the centre of one circle be released and it be made to roll upon the 
 circumference of the other at rest. And this shown to be true, the principle announced becomes 
 evident ; for if B roU on the circumference of A, it is evident that a point b will at any instant be de- 
 scribing a circle about their point of contact c, and this being true, it likewise follows that the point b 
 is at every instant of the revolution describing during that instant an exceedingly small circular arc 
 about c, and therefore is 6 c always perpendicular to the curve a 6 at the point b ; in other words, it is 
 always a normal to it. 
 
 Now returning to Fig. 2023, let P be a point exceedingly near to t in the curve n n, which is fixed 
 upon the circle B ; it is evident from what has been shown, that as that point passes through the point 
 of contact t of the two curves, it will be made to describe on the plane of the circle A an exceedingly 
 small portion of the cxurve m m. But the curve which under these circumstances it describes, has been 
 shown to be always perpendicular to the line t c, now the curve m m being perpendicular to that line 
 at t, the point of contact with n n, that curve must likewise be perpendicular to it at the same instant, 
 and consequently we have tc; a normal to both curves at the point t This is the characteristic prop- 
 erty of the two curves m m and n n by which they satisfy the condition of a continual contact with each 
 other at the same time that the circles revolve by the contact of their circumferences at e ; and con- 
 versely, supposing the motion to be induced by the mutual contact of the curves, they will communi- 
 cate the same motion to the circles, as these would receive by the mutual contact of their circum- 
 ferences. 
 
 The principle here announced, exhibits a special application of one particular property of that curve 
 known to mathematicians as the epicycloid. The mode of generating it is that described, and that upon 
 which a mathematical definition of it is founded. Thus generally when two circles are in contact at 
 their circumferences, and the one is made to roll upon the other, any point beyond the centre in the 
 moving circle describes during its revolution the particular curve named. This curve, as already stated, 
 may be traced upon an immovable plane, against which a point in the moving circle is made to bfjar. 
 For the sake of distinctness this is termed the generating circle, and the circle upon which it roll>3 is 
 called the fundamental circle, and the portion of it on which the epicycloid rests is called the base. 
 
 The definition of the epicycloid is rendered obvious by reference to Fig. 2026, in which B is the gen- 
 erating circle, and A the fundamental circle of the epicycloidal curve a b. If the generating circle, in- 
 stead of roUing on the outside, roll within the base circle, the curve is usually called an interior epicy- 
 cloid ; that generated when the circle rolls on the convex circumference, being termed for distinction an 
 exterior epicycloid. 
 
 An essential condition of the epicycloid is that the generating circle, in revolving from its first posi- 
 tion to dififerent other situations, as shown in Fig. 2026, applies successively all the parts of its circum- 
 ference to those of the base ; it is therefore evident that the base of the complete epicycloid is equal to 
 the circumference of the generating circle, and each portion of its base as c a is equal to the corres 
 ponding part c 6 of the generating circle by the rolling of which it is traced. Hence we have a method 
 of drawing the epicycloid by describing a series of circles which have all the same diameters as the 
 generating circle B, and which all touch the base A ; then making the lengths of the arcs c b taken from 
 the points of contact with the base equal to the arcs c a of the base, we can readily determine the 
 points of the curve, and consequently the curve itself. 
 
 The epicycloid may be described by the compasses in the following manner. Let us in the first place 
 take the exterior curve. 
 
 Having divided the circumference ABD, Fig, 2027, into a series of equal parts 1, 2, 3, ... . begin- 
 liing from the point A ; set off in the same manner, upon the circle Ax Ay, the divisions 1', 2', 8', . . 
 
28 
 
 GEERING. 
 
 points 1, 2, 3, they will intersect each other successively 2027. 
 
 at the points cde, Take now the distance 1 to c, and 
 
 set it oif on the same arc from the point of intersection of the 
 radius AC to t\ fti like manner, set off the distance 2 to d, 
 from h to w, and the distance 3 to e from a to v, and so on. 
 Then the points A.tuv, . . . will be so many points in the 
 epicycloid ; and their frequency may be increased at pleas- 
 ure by shortening the divisions of the circular arcs. Thus the 
 form of the curve may be determined to any amount of accu- 
 racy, and completed by tracing a line through the points 
 found. 
 
 As the distances 1 to c, ... . which are near the commence- 
 ment of the curve, must be very short, it may, in some in- 
 stances, be more convenient to set oflf the whole distance t to 
 1 from c to t\ and in the same way the distance 6 to 2 from 
 d to u^, and so on. In this manner the form of the curve is 
 the more likely to be accurately defined. 
 
 A second method of finding the points in the curve is, in 
 the first place, to find the positions m n, of the centre of the 
 rolling circle corresponding to the points of contact 1', 2', 3', 
 {fee, which may be readily done by producing the radii from 
 the centre O, to cut the circle E F. From these centres de- 
 scribe arcs of a cii'cle with the radius of C A, cutting the cor- 
 responding arcs described from the centre 0, and passing 
 through the points tuv, as before. 
 
 Two distinct portions of the curves are represented in combination so as to form the faces of a tooth 
 of a wheel of which the primitive circumference is A a; A y. 
 
 When the moving circle A B D is made to roll on the interior of the circumference Ax Ay, as shown 
 in the under part of Fig. 2027, the curve described by the point A is called an interior epicycloid. It 
 may be constructed in the same way as in the preceding case. The first operation is to divide the two 
 circles into equal parts, at the points 1, 2, 3, and 1', 2', 3', &c. Draw radii from the points 1', 2', S', 
 (fee, to the centre O, and also arcs of circles through the points 1, 2, 3, (fee, from the same centre 0, 
 meeting the corresponding radii at the points cde. Then by transferring the distances c to 2, c? to 3, 
 to the axis A D, as in the other case, we find a series of points t it v, which may be increased in number 
 to any extent, and are points in the curve, through which, if a line be traced, the epicycloid will be 
 formed. 
 
 To determine the relation of the epicycloid and hypocycloid in 
 connection with the form of the teeth of wheels, let mp andp n, Fig. 
 2028, be respectively portions of these curves, having the same gener- 
 ating circle c Vp and having for their bases the pitch circles c D c 
 and c E c of two wheels. If the faces of the teeth upon the cir- 
 cumferences of these wheels coincide with these curves, it may be 
 shown that tliey will work truly together ; for let them be in con- 
 tact at p, and let their common generating circle be in contact 
 with both pitch circles at c, then will its circumference manifestly 
 pass through the point of contact of the two teeth : and if it were 
 made to roll through an exceedingly small angle upon the point c, 
 rolling there upon the circumferences of both circles, its generating 
 point would traverse exceedingly small portions of both curves. 
 But since a given point in the circumference of the generating cir- 
 cle is thus at the same instant in the perimeters of both the curves, 
 that point must of necessity be the point p of the curves ; and 
 since moreover the generating circle rolls upon the point of con- 
 tact c, its generating point traverses a small portion of the perim- 
 eter of each of the curves at p, it follows that the line cp is a 
 normal to both curves at that point ; for whilst the generating 
 circle cPp is rolUng through an exceedingly small angle upon c 
 the point p, it is describing a circle whose radius is cp. If the 
 
 teeth of a pair of wheels have their edges formed to these curves, they will therefore satisfy the condi- 
 tion that the line cp, drawn from the point of contact of the two pitch circles to any point of contact ol 
 the teeth, is a normal to the surfaces of both at that point, and this condition has been shown to be 
 necessary and sufficient to the correct working of the teeth. 
 
 From this then it appears that if an epicycloid be described on the plane of one of the wheels with 
 any generating circle, and with the pitch circle of that wheel for its base ; and if a hypocycloid be de- 
 scribed on the plane of the other wheel with the pitch circle of that wheel as its base, and if the acting 
 eurfaces of the teeth on the two wheels be cut so as to coincide with these curves, they will be driven 
 by the intervention of these teeth in the same manner as they would by simple contact of their pitch 
 circles. 
 
 ' It might be shown in exactly the same manner that the curves mp and np may be generated by the 
 rolling of any other curve than a circle upon the pitch circles of the wheels ; they would still poosess 
 
 2028. 
 
GEERING. 
 
 29 
 
 this property, that a line drawn from any point of their contact to the point of contact of the pitch 
 circles is a normal to both, "wliich, as already shown, is the one necessary and sufficient conditioa 
 Further, it can be shown that a tooth of any form whatever being cut upon a wheel, it is possible to 
 find a curve which, rolHng upon the pitch circle of that wheel, shall, by a certain generating point, 
 traverse -the edge of the given tooth ; and the curve thus found being made to roll on the circuraferenc*' 
 of a second wheel, will trace out the form of tooth which will work truly with the first. 
 
 Let aM6, Fig. 2029, be any curve whatever intended to 
 form the condition of the acting surfaces of teeth of the 
 wheel, and let c M be the pitch of the wheel ; take c m =s 
 c M, it is required to find the curve which passes through the 
 point m, and which being continually in contact with the 
 curve a M 6 will be impelled by the latter according to the 
 required condition. Now, as the common normal must al- 
 ways pass through the point c, if we draw the perpendic- 
 ular c i to the curve aMb, wliich may be done by describ- 
 ing an ai-c of a circle from the point c as a centre, with a 
 radius such that that arc will cut the curve aMb in two 
 points infinitely near and joining the point c to the middle 
 of the distance of these two points, the point t will be the 
 point of contact of the two curves. Supposing now that 
 we divide the pitch c M and c m into the same number of 
 equal parts at the points 1, 2 and V 2\ when these points 
 arrive successively in contact in the course of the revolution 
 of the two circles, they will coincide with the point c ; and 
 the normal to the curve a M b, passing through the point 1, 
 will also be the normal to the cm've sought, and will pass 
 thi-ough its point of contact with a M 6. If, then, from the 
 point 1' as a centre, with a radius equal to a normal to the 
 curve a M b, we describe an arc of a ckcle, it will be a tan- 
 gent to the curve sought. And doing the same with the other points of division, we find a numbe? 
 of points through which, if a curve be traced tangential to the small arcs described, it will fulfil the 
 conditions of the problem, with more or lesa accm-acy, according as the points taken have been more 
 or less numerous. 
 
 From this it therefore appears that, having the curve given of the teeth of one wheel, the curve to 
 which the teeth of the second wheel ought to be formed may be readily found ; which is the problem 
 in its most general form, and may be stated summarily thus : Given the form of the teeth of one wheel, 
 to find out the form of those of another that may work with it correctly. Solution : Describe the pitch 
 circles of the required wheels, and find the curve which, revolving upon the one, will describe the given 
 tooth ; make the same curve revolve within the other, and with the same describing point it will gen- 
 erate the tooth required. 
 
 2031. 
 
 2030. 
 
 
 \ I'ii 
 
 It is havvever to be observed, that these curves, to be applicable in practice, imply the condition that 
 the curvature of the concavity of one tooth should be greater than the convexity of the other, or else 
 that both should be convex. The practical solution may be conducted in the following simple manner : 
 Let A and B, Fig. 2030, be two boards, whose edges are formed into arcs of the given pitch circles. 
 Attach to one of them the shape of the proposed tooth C, and to the other a piece of strong paper D, 
 the tooth being slightly raised to allow the paper to pass under it. Keep the circular edges of the 
 board in contact, and make them roll together. Draw upon D a number of successive positions of the 
 outhno of the edge of C, which, touching one another in a corresponding number of points, a curve /e, 
 passing through these points, will be the corresponding tooth required for B. For it is obvious, from the 
 method it is obtained, that if the tooth be cut out, and be made to work upon C, it will touch that tooth 
 in every position ; and therefore the contact of these curves will exactly fulfil the required condition, 
 by replacing the rolUng action of the two pitch circles. 
 
 To illustrate the preceding process more particularly; by means of Fig. 2031, let o be the centre of 
 the piece B, supposed to be fixed ; and suppose the piece A rolling upon it over the arc c c' of sufiicient 
 ^jxtent to include the extreme positions of the acting surfaces. Then c and c', being the extreme points 
 
30 
 
 GEERING. 
 
 2040. 
 
 fi contact of the pieces, draw the radii oc, oc', and from the centre o, with a conyenient radius, dratv 
 an arc d e, meeting o c and o c', produced at d and e ; divide this arc into a number of equal parts, cor- 
 responding to the number of positions of the tooth C intended to be taken ; the radii drawn to the pointa 
 of division will give corresponding points of contact at the circumference of the piece B, from which the 
 relative positions of the tooth C may be severally described. In Fig. 2039, the extreme positions of the 
 piece A and the tooth C are indicated by dot lines, though, with the exception of two other positions of 
 the tooth, the rest is omitted to prevent confusion. A curve n n, being traced so as to touch, in succes- 
 sion, the different positions of the tooth C, will indicate the form of tooth required for the piece B, so as 
 to geer with the piece A. 
 
 It is, however, true, that many forms of C may be tried which will prove impracticable ; for some of 
 the successive portions of its edge may come up and interfere with parts of a b, previously drawn ; 
 thereby showing, that although it may be geometrically possible to assign a form of a b, which shall 
 work with any given form of C, it by no means foUows that tliis is practically true ; and, indeed, it does 
 not appear that any new forms of curves deduced from this general principle are likely to adapt them- 
 selves in practice so well as those commonly adopted. 
 
 There is also one curve highly applicable in practice, the importance of which requires that it be 
 noticed as a separate principle, which may be thus stated : 
 
 The teeth of two wheels will also work truly together, if their acting surfaces coincide with curves 
 traced out by the extremity of a flexible Hne, unwinding from the circuniference of a circle, and called 
 the involute of that circle, provided the circles, of which the bounding curves of the teeth are respect- 
 ively portions, be concentric with the pitch circles of the wheels, and have their radii in the same ratio 
 as the pitch circles. 
 
 Thus, let there be two centres of rotation, A and B, 
 Fig. 2040, of which A T and B T are the primitive radii ; 
 through the point T draw any straight line D E ; through 
 the points A and B draw the perpendiculars A D and B E, 
 and describe circles from the points A and B as centres, 
 with these lines as radii. Again, let K T H be the invo- 
 lute of a portion of the circle A D, traced by the (flexible) 
 line D T, and this line is manifestly a normal to that curve 
 at T ; this point will also be that at which the second 
 curve, traced by E T, the line touches the first. But sup- 
 pose the circle A D to turn on its centre, and taking with 
 it the curve K T H, the line D E will be a constant normal 
 to that curve, in all its positions; and consequently also / 
 to the conducting curve in all its positions, and at the / 
 same point. Now this property of a curve, of having its ' 
 normal tangential to a circle, belongs only to the involute 
 of that circle ; the conducted curve must therefore be the 
 involute of the circle B E. 
 
 As an illustration of this principle, let D E be conceived 
 to represent a band passing round the two circles drawn 
 with the radii D E and A D ; the wheels will evidently be 
 driven by this band precisely as they would by the con- 
 tact of their pitch circles, since the radii of the involute 
 circles have the same ratio as the radii of the pitch circles. 
 
 Let it also be supposed, that the projected circles carry 
 round with them their planes as they revolve, and that 
 a tracing point is fixed at any point T of the band ; it 
 will trace at the same instant the two curves, and upon both planes, as they revolve beneath it ; it is 
 then obvious that these curves, being traced by the same point, must be in contact in all positions of 
 the circles when driven by the band, and therefore when driven by their mutual contact. The wheels 
 would therefore be driven by the contact of teeth of the forms of the curves thus traced by the point T 
 of the band, precisely as they would by the contact of their pitch circles ; and it is easy to see that the 
 curves H T K and F T Gr, so described, are involutes of the cu-cles drawn with the radii A D and B E, 
 from the given centres. 
 
 A particular property of teeth formed according to the principle here pointed out, is that they trans- 
 mit the pressure without altering its intensity — this being the same when we suppose the force con- 
 stantly applied at the pitch circles, but not at the points of contact of the teeth. In epicycloidal teeth 
 the normal to the curves is variable, as may be seen from Fig. 2029. Thus the pressure at the pitch 
 circles being constant, may be expressed by A c, while that upon the points of contact of the teeth will 
 be expressed by A t, w^hich manifestly varies with the direction of the normal c t, and consequently 
 with the position of the points of contact of the teeth. Such variation does not occur when the teeth 
 are of the involute form ; for then the relation of A D to B E, Fig. 2040, is invariable, since the constant 
 line D E is always normal to the curves in contact ; and hence it is obvious that the pressure transmit- 
 ted from the one circle to the other at D and E has always the same value. 
 
 An advantage of this form of teeth is, that if the distance of the centres A and B be altered, but so 
 that the involutes still remain in contact, the velocity of the pitch circles of the wheels will not be 
 affected, and therefore the angular motion of the curves in contact will remain unaltered. _ Tliis is a 
 property which distinguishes the involute from the other curves that have been given, and is of some 
 practical importance ; for, in a pair of wheels of which the teeth are of that form, it is not only unne- 
 cessary to fix the centres at a precise distance, but a derangement of the centres, from wearing of the 
 journals and brasses, or settlement of the framework, does not impair the action of the teeth. 
 
 \ 
 
GEERINCr. 31 
 
 It may also be observed, that for every pair of pitch circles an infinite number of pairs of involutes 
 may be assigned that will answer the required conditions ; for the inclination of E T D to the line ol 
 centres is manifestly arbitrary, and every change of inclination niani- 2041. 
 
 festly produces a new pair of base-circles, and of involutes to these 
 circles. 
 
 Of epicycloidal teeth. — The simplest illustration of the action of epi- 
 cycloidal teeth, is when they are employed to drive a trundle, as rep- 
 resented in Fig. 2041. In the first place, let it be assumed, that the 
 staves of the trundle have no sensible thickness ; that the distance of 
 their centres apart, that is, their pitch, and also their distance from the 
 centre of the trundle, that is, their pitch circle, are known. The pitch 
 circles of the trundle and wheel being then drawn from their respec- 
 tive centres B and A, set off the pitches upon these circumferences, 
 corresponding to tlie number of teeth in the wheel and number of 
 staves in the trundle ; let five pins, ah c, Ac, be fixed into the pitch 
 circle of the trundle to represent the staves, and let a series of epicy- 
 cloidal arcs be traced with a describing circle, equal in diameter to 
 the radius of the pitch circle of the trundle, and meeting in the points 
 klmn, (fee, alternately from right and left. If new motion be given to \ \ j / / 
 
 the wheel in the direction of the arrow, then the curved face m r will \ 1 j / / 
 
 press against the pin b, and move it in the same direction ; but as the \ • i / / 
 
 motion continues, the pin will slide upwards till it reaches m, when \ \\! / 
 
 the tooth and pin will quite contact. Before this happens, the next \;]// 
 
 pin a will have come into contact with the face a I of the next tooth, '%■' 
 
 which repeating the same action will bring the succeeding pair into A 
 
 contact ; and so on continually. 
 
 "We have here assumed the pins to represent the staves of the trundle and to be without sensible 
 thickness, which is not true in practice. To allow, therefore, of the required thickness of staves, it is 
 manifestly necessary and sufficient to diminish the size of the teeth of the wheel, by a quantity equal 
 to the radius of the staves, sometimes increased by a certain fraction of the pitch for clearance, in this 
 case termed backlash. This may readily be done by drawing within the primary epicycloids at the 
 required distance, another series of curves parallel to these. For example, let it be required to draw 
 the proper tooth to impel a stave of the radius b t ; with that radius plus the requisite amount of clear- 
 ance in the compasses, draw a series of small arcs from and within the original curve, then a curve 
 touching all the arcs so drawn will be the epicycloid required ; this curve acting upon the stave will 
 fulfil the condition stated, for being symmetrical with the first, which passes through the axis of the 
 Btave, it will act similarly at its circumference. 
 
 In practice, a portion must be cut from the points of the teeth, and also a space must be cut out 
 within the pitch circle of the driver, to allow the staves to pass ; but as the sides of these parts never 
 come into contact with the staves, no particular form is requisite ; the condition to be attended to is 
 simply to allow of sufiicient space for' the staves to pass without contact. 
 
 We have here supposed the wheel to be the driver, and this being the case it is evident that the 
 staves being indefinitely small, the contact of the tooth will begin at the instant its base reaches the 
 line of centres at a ; and during the action of the tooth the point of contact will gradually slide upwards, 
 remaining always in the pitch circle of the trundle, at the same time it recedes from the line of centres, 
 until the contact is finally terminated at the point of the tooth. If the trundle be made the driver, the 
 pitch lines of the pair will still move with the same uniform angular velocity ; but the reverse of the 
 preceding action will take place, for in that case the contact would commence at the top of the teeth 
 and cease at their base ; it would moreover commence before the line of centres, and terminate when 
 it had reached the point of intersection of that line. Now the friction which takes place between teeth 
 whose point of contact is approaching the line of centres, is of a much more injurious character than that 
 which happens while the points of contact are receding from it. Not only does it cause much more 
 friction and vibration, in consequence of the inequalities of the surfaces in contact — and the surfaces 
 even of the most highly-polished bodies have some inequalities, which, when pressed together, interlace 
 — but the teeth at the same time tend to force the axes of the wheels outwards, and very speedily 
 induces injurious eflfects upon the journals, and also upon the planes of the teeth. When the action 
 is receding from the line of centres the friction is less intense, and its effects less injurious ; it tends to 
 draw the axes together, and induces much less vibratory action of the geering. 
 
 For these reasons it is studied in practice to avoid as much as possible the kind of contact which 
 takes place before the line of centres, by making the wheel the driver and the trundle the follower. 
 rhe diameter of the staves is also commonly made equal to that of the teeth, with an allowance of a 
 teeth of the pitch for clearance ; the radius is therefore rather less than a quarter of the pitch, conse- 
 quently the contact wiU begin, the wheel being the driver, when the centre of the stave reaches the line 
 of centres, and therefore at a distance before that line equal to the radius of the stave, or rather less 
 than a quarter of the pitch. 
 
 It is also evident that since one tooth must not quit contact before the succeeding tooth is engaged, 
 that when the point a has reached the line of centres the tooth T must not have quitted contact with 
 the stave b, and the point at which contact ceases must therefore be at an angular distance from tho 
 line of centres equal at least to half the distance a r, that is, to half the pitch. In a pair of this kind, 
 the action which takes place before the line of centres is less than a half of that which takes place after 
 passing it. 
 
 The action of a wheel and tnmdle being understood, it is easy to comprehend that of the teeth of a 
 pair of wheels of the ordinary construction. Let A and B of Fig. 2042 be respectively the centres of a 
 
GEERING. 
 
 2043. 
 A 
 
 XII 
 
 wheel and pinion of -whicii the teeth are intended to be of the epicycloidal form, and A c and B c their 
 
 primitive radii. To lay off the teeth of this pair, having determined tlie pitch and number of teeth ii» 
 
 the wheel and pinion, let the pitch lines be divided 
 
 into as many equal parts, setting out from the point 
 
 of contact c, as there are teeth in them respectively. 
 
 Let the thickness of the teeth be next set off, taking 
 
 ca for the thickness of a tooth of the wheel, and ch 
 
 for that of a tooth of the pinion. Upon the radii A c 
 
 and B c as diameters, describe two circles, having also 
 
 their point of contact at c and their centres X and Y. 
 
 Now let the circle Y be made to roll upon the pitch 
 
 line of the wheel, and a point io its circumference at 
 
 c will describe the epicycloidal arc c m, and this curve 
 
 determines the form of the point of the tooth of the 
 
 wheel. In the same way describe the epicycloidal arc 
 
 c n by making the circle X to roll upon the pitch circle 
 
 of the pinion, and this curve will determine the form 
 
 of the part of the tooth of the pinion beyond the pitch 
 
 line. 
 
 Again, the teeth must have the same form on both 
 sides ; in other words, they must be symmetrical in 
 reference to the radius which passes through the mid- 
 dle of their thickness, in order that the wheel and 
 pinion may be turned in either direction. If, then, 
 through a point equidistant from c and a and from the 
 centre A of the wheel, a right line be drawn as the 
 axis of the tooth, it hmits at m the outline of the tooth ; 
 to complete the form of the addendum, it therefore 
 only remams to describe the opposite epicycloidal arc 
 
 am equal to cwi. A similar process completes the \ \ / 
 
 form of the addendum of the pinion. ^^^-^-.1.-'"'' 
 
 The curve c m of the tooth of the wheel is constant- ^ 
 
 iy in contact with ths radius Be, and its point of 
 
 contact is at the same time situated in the circumference of the circle X ; the contact will therefore 
 cease when the extremity ni of the tooth becomes the point of contact ; and this occurs when the point 
 m has arrived at the circumference of Y. If, then, an arc of a circle be described from the centre A 
 with the radius A wi, its point / of intersection with the circumference of Y is that at which the tooth 
 ought to cease to act, to secure uniformity of motion, and at the same instant that another new tooth 
 advances into geer. Tlie determination of the point /limits also the useful length of the flank; for il 
 from B as a centre, with radius B/ an arc be described, the part c s of the radius c B is that which is in 
 contact with the tooth till it arrives at the position B/; and consequently this is the useful length of 
 the flank of the pinion. 
 
 Reasoning in the same manner with respect to the teeth of the wheel, it may be determined, that the 
 useful length of the flank of the tooth c a is the portion c g. 
 
 We have yet to find the form of the portions of the teeth within their respective pitch circles, or 
 more properly of the spaces between them. The first condition which presents itself is manifestly that 
 the spaces ought to be sufficiently large to allow the projecting portions of the teeth of the opposite 
 wheel to enter freely between them. To resolve this problem in its most general form, it is necessary 
 to find the curve described by the point m upon the movable plane of the circle of B, the pinion being 
 driven by the wheel A. For practical purposes this is not reckoned a matter of much importance ; it 
 is usually considered enough that sufficient space for play be allowed whilst the tooth remains strong 
 enough for its work. As every tooth moves between two flanks, and touches only one of them, the 
 space may be bounded literally by radial lines prolonged towards the centre of the wheel. The bottom 
 of the space being sufficiently removed from the pitch circumference to allow the tops of the teeth of 
 the pinion to pass round without touching them, may be described by arcs of a circle drawn from the 
 centre. 
 
 By this arrangement, then, the flanks of the teeth of both wheels of the pair, that is, the portions of 
 tlie teeth which lie within the respective pitch circles, are radial lines drawn from the centres to the 
 pitch circles, and the faces of the teeth, or those portions which lie without the pitcli circles, are arcs of 
 epicycloids traced in each wheel with a describing circle equal in diameter to the pitch radius of the 
 otlier wheel. Accordingly each flank and face will act in contact to produce a constant angular velocity 
 ratio of the pitch circles of the pair, and the action of each pair of teeth will be confined to its own side 
 of the line of centres. 
 
 In practice, and especially "when the teeth of the wheels are small, it is not usually considered 
 necessary to apply strictly the form of the epicycloid to find curves of the teeth; but to define them 
 approximately by some such mode as the following. 
 
 Referring to Fig. 2043, after having divided the pitch circles into the proper number of parts cor- 
 responding to the number of teeth, and set off upon the pitch lines the thicknesses of the teeth, through 
 tlie extremity h of the pitch a 6 of the pinion, draw a radius B h cutting the circumference of the epicy- 
 cloidal circle Y in d. Joining the point d with the first division h' of the wheel, draw a perpendicular 
 upon the middle of the line dh' meeting the circumference of A at the point h ; an arc described from 
 this point as a centre, and with /i 6 as a radius, will define the curve of the teeth of the wheel with 
 Bufi&cient exactness. 
 
GEERINa. 
 
 33 
 
 
 2043. 
 
 i!\ 
 
 2<f 
 
 The length of the teeth is limited at d where the curve of the tooth meets the chcumference of tliG 
 epicycloidal circle Y ; for when the tooth has arrived at that point, another pair ought to come in*-' 
 geer at the line of centres. The flank of the tooth being 
 radial, may be defined by a right line from the termination 
 of the curve of the face of the tooth in the pitch circle 
 drawn towards the centre A. 
 
 As the tooth is symmetrical about tlie axis A k, its op- 
 posite face may be described by taking off the form of the 
 side found upon a mould ; and the two sides being thus 
 found, a pattern tooth may be formed and used in draw- 
 ing in the other teeth of the wheel. 
 
 A similar process gives the form of the teeth of the pin- 
 ion. Having taken an arc a e equal to the pitch, draw A e 
 till it meets the circumference of X in c;' ; with this join the 
 division c' of the pinion, and tlirough the middle of the line 
 diaw a perpendicular, meeting the circumference of Y in 
 h' \ from this point as a centre, and with radius h' e', de- 
 scribe the curve g e\ which defines the face of the tooth. 
 A radial line meeting the curve at e' and drawn towards 
 B as a centre, will in like manner define the flank of the 
 tooth. 
 
 To determine the bottom of the space between two 
 teeth, it is to be remarked that in the pinion this must be 
 at a distance from the centre A, at least equal to A a ; if, 
 therefore, from the point B as a centre, with radius B ?/?, a 
 circle be described, it will define the bottoms of the teeth 
 of the pinion ; and in like manner, if with radius A ?i a 
 circle be described from the centre of the wheel, it will 
 define the bottoms of its teeth. The angles at bottom are ^-^^ \ ,y 
 
 commonly rounded, which, at the same time that it im- 1b 
 
 proves the apjDcarance, strengthens the teeth. 
 
 The forms of the teeth are also occasionally described by arcs, of which the radii are equal to the 
 pitch, and with the centres taken upon the pitch lines. 
 
 In this case the centre.3 of these arcs must fall upon the sides of the teeth, which being already 
 marked off, it becomes a simple process to describe the upper portions. This method, which differs 
 little from the preceding, may be conveniently substituted for it, when the diameters of the wheels are 
 not very unequal, and the teeth not thick. But for small pinions, with larger teeth, the first method 
 must be adopted. 
 
 In many cases the curves of the teeth are described by arcs drawn from the centres of the adjacent 
 teeth upon the pitch line. This gives a radius equal to the pitch, 'plus half the thickness of a tooth 
 In hke manner, the sides of the teetli also are sometimes described by arcs from the centres of the ad- 
 acent teeth giving a radius equal to the pitch, minus half the thickness of a tooth. 
 
 This form of the tooth may be defective, in the case of a very small pinion having to transmit great 
 pi-essure, as the extremities of the teeth may be too much reduced. In this case, the curves or faces of 
 the teeth may be described with radii equal to three-fourths of the pitch ; and if this be not suflBcient 
 curvature, radii equal to some smaller fraction of the pitch may be used. 
 
 When, on the contrary, the pitch is large, and the pressure comparatively small, the teeth may be 
 too short ; this will be remedied by employing arcs of which the radius is one and a half or twice the 
 pitch. A general method of determining these radii will presently be given. 
 
 Another condition which it is necessary to advert to, is the point in the circumference at which con 
 tact of tlie teeth commences. This point is necessarily that in which the two pitch circles cut the line 
 of centres, as may be rendered obvious by reference to Fig. 2042. Here it is plain that the face c m of 
 the tooth of the wheel ought to act upon the side of the tooth of the pinion, until the extremity m of the 
 tooth arrives at the extremity of that side, as shown by the position of the tooth/. But were this to 
 take place before the acting surface of the succeeding tooth of the pinion had arrived at the line of 
 centres, the succeeding tooth of the wheel would of necessity act, during the first part of its contact, 
 upon the face of that of the pinion, and continue so to do until it had arrived at the hne of centres. 
 
 On the other hand, should the extremity of the tooth on arriving at / not immediately cease contact, 
 but only be disengaged wlien the acting surface of the next tooth of the pinion had arrived at or passed 
 the line of centres, the /ace of that tooth would not be at all acted upon by the tooth of the wheel, and 
 the contact with the flank would be entirely beyond the line of centres. In this case, were the wheel 
 constantly to drive the pinion, the curved faces of the teeth of the latter might be altogether dispensed 
 with, without in any way injuring the equality of action. But the condition assumed does not occur in 
 practice ; for, on account of the inequalities which take place in the motion of the wheels, whether 
 arising from inequahty in the resistance or of speed in the moving power, the wheel and pinion act upon 
 each other alternately, even when the general movement is continuously in the same direction, a cir- 
 cumstance which cannot be' safely neglected in the construction of any system of geering. 
 
 When the teeth fall into action before the line of centres, they must obviously slide upon each other 
 as they approach that line, and their friction, as in the case of the wheel and trundle, is then much 
 greater than it is after they have passed it, on account of their sliding inwards upon each other instead 
 of outvcards, and against a pressure which is augmented in proportion to the curvature of the acting 
 Burfaces. And, moreover, in the contact after the line of centres, although there exist inequalities on 
 the acting surfaces, they offer comparatively little obstruction to the sliding of the teeth upon each 
 
34 
 
 GEERINa. 
 
 other; whereas, -when the contact is in advance, every inequality becomes an abutting obstacle to the 
 closing in of the teeth, and which it is necessary to surmount by a species of backward motion at great 
 expense of motive power, and greatly to the detriment of smoothness of motion among the parts of 
 the geering. 
 
 But although it is thus clear that an advantage is gained by making the teeth engage after the line 
 of centres, yet this is as clearly desirable only within certain limits. For, as already observed, it is 
 necessary that the tooth of the pinion continue to geer with the wheel until the succeeding tooth has 
 arrived at the line of centres — a condition in which is involved the problem of the minimum number of 
 teeth that can be given to a pinion to work with a wheel of given size, and which may be determined 
 when the pitch and radii are given. From these data, the angle of the sides of the teeth is immediately 
 deduced. The length and thickness of teeth and width of space must be modified to answer the con- 
 dition imposed, and the smaller the number of teeth in the pair, the more difficulty there is in accom- 
 f)lishing the solution. To resolve the conditions of the question into the simplest form which they admit, 
 et us take the imaginary case of a pinion of seven teeth, without visible thickness, to geer with a wheel 
 ol fifty epicycloidal teeth, as represented in Fig. 2044. In this the line A B is the line of centres \ cm.h 
 one of the teeth of the wheel, and B m a one of the radial teeth of the pinion, with which the tooth cmh 
 has been in contact during its motion from c to h. JSTow since the pinion has seven radial teeth, the value 
 of the angle cBa is 3 6io__5|o 25/ 43'/; and making the radius Bc=:l, the right-angled triangle 
 B m c, completed by drawing the right line c m, gives, 
 
 Bm = cos aBc = cos 51° 25' 43" = 0-62349. 
 If radius B c = 1, then A c = \" and A B = 5_7, Thus we have given in the triangle A m B the two 
 sides AB andBm and the angle aBc; and therefore we have also the angle B Am = 3° 35' 50". 
 Now if the wheel have 50 teeth, we have the angle c A 6, containing a tooth and a space, = '-^-^^ ° = 
 7° 12' ; and subtracting from this the angle c Km, there remains the angle 6 A m = 3° 36' 10" ; but 
 the angles h Km and mKc are equal, and by consequence the angle 6 A c is double the former of thosG; 
 that is, 7° 12' 20", the tooth being symmetrical in reference to A m. Now we found the angular value 
 of the pitch c6 = 7° 12', which ought to contain the tooth and the space. It therefore follows that 
 even when the tooth of the pinion has no thickness, it is impossible to place the tooth of the wheel in 
 the pitch interval— for this is shown to be less than the tooth of the wheel would occupy ; and it is 
 easy to understand that a wheel of less than 50 teeth could still less be made to drive a pinion of seven 
 teeth. And although in a wheel which has more than 50 teeth the pitch angle would be found gi-eater 
 than the thickness of a tooth, still it would be less than that which should contain the thickness and the 
 space together; hence the angle which would be found due to the space would be so small as to be 
 insufficient to receive the tooth of the pinion, allowing for lateral clearance, and in consequence the pair 
 vould necessarily geer before the line of centres. 2^^^ 
 
 Ai- / 
 
 I ! 
 
 \ i 
 
 ii 
 
 This problem is often more conveniently solved organically, when the teeth of the wheel and pimon 
 are formed in the usual manner. Thus in Fig. 2045, let c B m be the angle through which it is desired 
 that the contact of the tooth a m should continue after passing the line of centres ; then as the contact ends 
 at m, the point of contact will be at the extremity m of the tooth. Join c m by a right line which will 
 4be a perpendicular to the radius B r ; also join A m. Then, since a was in contact with r at the line oi 
 
GEERIVG. 
 
 35 
 
 centres, the arc carrrcr is given, and is that portion of the pitch through which the contact of the 
 tooth is required to continue. Then a 6 is half the thickness of the tooth, minus half the base of the 
 portion cut off; and eais equal to the pitch, and must contain one tooth and the space between. Sup- 
 posing these qualities to be equal, then since a b contains less than half a tooth, and cannot contain 
 more, e h must contain half a tooth and space at least, and in this case as much more as a 6 is less 
 than half a tooth. Tlierefore for every given value of c B m, a value of c A may be assigned which 
 shall make e b exactly equal to a space and half a tooth, when the tooth is pointed, and with a corre- 
 sponding and known increment when the tooth is blunted. 
 
 If a greater radius c A' be taken, the point b will manifestly fall still nearer a and the tooth may be 
 still more blunted ; but if a less radius c A" be assumed, then the point b will fall nearer to c, and e b 
 will become too small to contain the space and remaining half tooth ; and in this case, if the wheel 
 were set out, it would be found that the epicycloidal arcs on the two sides of the tooth would intersect 
 between m and b, and therefore the tooth would be too short to continue in action through the required 
 arc c a. 
 
 This mode of illustration is due to Professor Willis, to whom we are also indebted for the following 
 tables derived organically in the manner described, for spur and annular wheels : — 
 
 Table I.- 
 
 -For spur wheels, showing the least num- 
 
 Table 11.- 
 
 -For annular wheels, showing the great- 
 
 Ijere of teeth that wiU work with given pinions— Tooth 
 
 est numbers of teeth that will work with given pinions | 
 
 space. 
 
 
 
 —Tooth ■= space. 
 
 
 
 
 Least number of teeth in 
 
 
 
 Greatest number of teeth in 
 
 
 No. of teeth in 
 given pinion. 
 
 the wheel. 
 
 
 No. of teeth in 
 pinion. 
 
 annular wheel. 
 
 
 Wheel driving. 
 
 Pinion driving. 
 
 
 Wheel driving. 
 
 Pinion driving. 
 
 
 5 
 
 impossible. 
 
 impossible. 
 
 
 2 
 
 impossible. 
 
 5 
 
 i 
 
 6 
 
 
 
 176 
 
 i 
 
 3 
 
 — 
 
 12 
 
 *Pk 
 
 7 
 
 
 
 52 
 
 'p-, 
 
 4 
 
 — 
 
 26 
 
 II 
 
 8 
 
 — 
 
 35 
 
 11 
 
 5 
 
 — 
 
 85 
 
 9 
 
 — 
 
 27. 
 
 7 
 
 14 
 
 any number 
 
 « 
 
 10 
 
 rack. 
 
 23 
 
 s 
 
 8 
 
 25 
 
 
 
 § 
 
 11 
 
 54 
 
 21 
 
 .1 
 
 9 
 
 60 
 
 
 
 2 
 
 12 
 
 30 
 
 19 
 
 
 
 
 
 
 13 
 
 20 
 
 18 
 
 
 
 
 
 O 
 
 14 
 
 24 
 
 17 
 
 O 
 
 
 
 
 ^ 
 
 15 
 16 
 
 17 
 15 
 
 16 
 
 
 
 
 
 
 3 
 
 impossible. 
 
 impossible. 
 
 
 2 
 
 impossible. 
 
 10 
 
 ^ 
 
 4 
 
 — 
 
 35 
 
 ^ 
 
 3 
 
 — 
 
 77 
 
 "a. 
 
 5 
 
 — 
 
 19 
 
 o 
 
 4 
 
 5 
 
 any number 
 
 6 
 
 — 
 
 14 
 
 'fr. 
 
 5 
 
 12 
 
 
 
 II 
 
 7 
 
 31 
 
 12 
 
 II 
 
 6 
 
 77 
 
 — 
 
 8 
 
 16 
 
 10 
 
 
 
 
 « 
 
 9 
 10 
 
 12 
 10 
 
 10 
 10 
 
 
 
 
 
 
 2 
 
 impossible. 
 
 impossible 
 
 
 2 
 
 impossible. 
 
 14 
 
 1 
 
 3 
 
 — 
 
 36 
 
 i 
 
 4 
 
 8 
 
 any number 
 
 Ph 
 
 4 
 
 — 
 
 15 
 
 ■&. 
 
 5 
 
 64 
 
 — 
 
 evrs 
 
 5 
 
 
 
 13 
 
 WICO 
 
 
 
 
 II 
 
 6 
 
 20 
 
 10 
 
 II 
 
 
 
 
 «^ 
 
 7 
 
 11 
 
 9 
 
 V. 
 
 
 
 
 V> 
 
 8 
 
 8 
 
 8 
 
 « 
 
 
 
 
 These tables contain all the limiting cases under the three suppositions of the arc of action being 
 equal to the pitch, to three-fo'urths of the pitch, and to two-thirds of the same. When the arc of action 
 and pitch are the same, the teeth of the follower may be reduced in length to the pitch circle, and the 
 contact of the teeth confined to their recess from the hne of centres ; when the arc of action is reduced, 
 as in the other two cases given, the reduction will of course indicate the portion of contact which must 
 take place in advance of the line of centres. Thus when c r = | and § of the pitch, then it is obvious 
 that the contact of the teeth must begin at least \ and J of the pitch respectively before they reach 
 the line of centres. 
 
 It also appears from the table, that a smaller pinion may be employed to drive than to follow. 
 Thus when the action begins at the line of centres, the least wheel that can drive a pinion of eleven 
 teeth i^ fifty-four ; but the same pinion can drive a wheel of twenty-one and upwards. Again, nothing 
 less than a rack can drive a pinion of ten, but this pinion can drive a wheel of twenty-three and up- 
 wards. No pinion of less than ten leaves can be driven, but pinions as low as six may be employed 
 to drive any number above those in the table. And lastly, the least pair of equal pinions that will 
 work together is 16. These limits being geometrically exact, and making no allowance for wear, it is 
 better, indeed necessary, in practice, to allow more teeth than the table assigns. 
 
36 
 
 GEERING. 
 
 It often occurs in practice that a wheel is required to drive more than one pinion, and vice versa, 
 and those of different diameters. In this case the usual mode of obtaining the epicycloidal form oi 
 the face of the driving teeth is inapplicable, as that curve is made to depend upon the diameters ol 
 the pair when uniformity of motion is studied. But before adverting to the methods by which this 
 condition may be fulfilled, it will relieve us of some of the difficulties of the question to examine the 
 ordinary mode in which epicycloidal teeth are set out. As the workman cannot be expected to be in 
 possession of that accurate mathematical knowledge which would enable him to determine the neces- 
 sary forms by purely theoretical means, he has recourse to the mechanical method of forming the 
 pattern teeth by templets. We give the process in detail. 
 
 204G. 
 
 Having determined the pitch of the teeth and the radius of the pitch circle by the methods de- 
 scribed, a thin slip of wood — say | thick — is provided, and on this an arc of the pitch circle is struck. 
 The shp is then cut to the circumference of the circular arc described upon it. Another similar slip is 
 now provided, and on this an arc of a circle equal to the diameter of the wheel to the points of the 
 teeth is in like manner struck, and the edge worked off to the circumference of the circular arc as be- 
 fore. These two pieces so prepared are laid together as in the Fig. 2046 ; the piece A, whose 
 
 2047. 
 
 edge s/is an arc of the pitch circle, is fixing upon B, whose edge is an arc of the extreme circumfer- 
 ence of the wheel, the space s s between those edges being in breadth equal to the length of the teetb 
 from the pitch circle to the points. The slips are held firmly in their relative position by two screws c c. 
 This done, a like templet is prepared for the pinion, Fig. 2047 ; the piece C, whose edge s^ is an arc of the 
 pitch circle, is similarly fixed on the piece D, whose edge is an arc of a circle the diameter of which is 
 the true diameter of the pinion, the breadth of space s s being, as before, equal to the length of 
 addenda of the teeth. 
 
 2049. 
 
 The pair of templets being thus prepared, two tracing points m m are inserted obliquely, and from 
 Dohind into the piece D of the pinion templet. One of these points passes out at the edge of the 
 piece C, and the other at the edge of the piece D, and the templets are then placed upon each other, 
 as Bhown in Figs. 2048 and 2049 annexed, so that the cu-cumference of the piece C, that is, the pitch 
 
GEERING. 
 
 37 
 
 circumference of the pinion, shall meet Ihe circumference of the piece A, that is, the pitch circumference 
 «f the wheel. If in this position the templets be made to roll upon each other through a certain arc, 
 pressing them at the same time slightly together, the tracing points will mark two epicycloidal curves 
 upon the pieces A and B, as v and r ; of these two curves, that marked v, which is traced on the face 
 of the piece A, will be the curve of the lower portion or flank, and that marked r will be the curve of 
 the upper portion, or face of a tooth of the wheel. If now the thickness of the tooth be marked off on 
 the edge of the piece 0, that is, on the pitch circle, and the corresponding tracing point be made to 
 coincide with that point, the curves of the opposite side of the tooth will be formed by making the 
 templets roll together in the contrary direction. A complete outline of a tooth of the wheel is thus 
 described, to which a pattern tooth may be cut and used to shape the teeth by in making the wheel 
 pattern. 
 
 Let now the tracing points be taken out of the pinion templet, and inserted in exactly the same way 
 into the templet of the wheel, and the templets applied to each other as above shown, the operation 
 described will give the outline of a tooth of the pinion upon the parts C and D of the pinion templet. 
 
 Having thus found the cui'ves of the teeth of the pair, pat- 2050. 
 
 tern teeth may be made and used in setting them out; but s 
 
 this mode is not uniformly adoj^ted. Instead of forming 
 pattern teeth, many prefer to lay off the teeth by circular 
 arcs coinciding approximately with the epicycloidal arcs 
 found by the templets. This is readily done by taking any 
 tliree points in the curve, and finding the radius with which 
 a circle may be drawn whose circumference will pass through 
 the points so taken, and which will thus of necessity coincide 
 approximately with the normal curve. Radii being found 
 for the curves of the top and bottom of the teeth of the wheel 
 and pinion, their centres are transferred to the circumference of the pitch circles, and circles drawn 
 within these last. Figs. 2050 and 2051 illustrate the process. The arcs B and D, drawn with the 
 radii AB and CD, are those of the .face and flank of the wheel tooth; and the arcs F and H, drawn 
 with the radii E F and G- H, those of the corresponding off teeth, are drawn from the pitch circles c e, 
 Fig. 2051, and parts of the tooth. The pinion curves of the faces in setting those of the flank are drawn 
 from the circles n n within the others. 
 
 2051. 
 
 B»- 
 
 E»- 
 
 «D 
 
 • H 
 
 Tlie preceding is a very common mode of obtaining the curves of the teeth of a pair of wheels which 
 tre to geer together ; but it is faulty, in as far as it gives a form of teeth smaller at the root than at the 
 pitch circles, and also in the cii-cumstauce that a pair of wheels set off with templets formed in the 
 manner described, will only work correctly with each other, and not with wheels of any other numbers, 
 although of the same pitch. Thus, a wheel of 30, set off as described, to work with another of 50, wiU 
 not work correctly with a wheel of 40 or of 60, although the pitch be the same. To effect this it is 
 necessary to employ the proposition already announced and illustrated, and which shows that if there 
 be two pitch circles touching each other, then an epicycloidal tooth formed by causing a given describing 
 circle to roll on the exterior circumference of the one, will work correctly with an interior epicycloid 
 formed by causing the sa77ic generating circle to roll on the interior circumference of the other. This 
 proposition having been already established, it is unnecessary to dwell upon it further than to observe, 
 that although it has been admitted into the workshop as a fundamental truth, the practice founded upon 
 it does not afford the full advantage which it is adapted to afford, and which might be derived from it 
 hy a slight modification of the ordinary practice. 
 
 To describe the practical application of the principle, let a thin slip of wood be provided, and let an 
 arc of the pitch circle ^be struck upon it ; divide the slip into two portions through the line of this arc 
 tvith a fine saw ; one part. A, will have a concave, and the other, B, a corresponding convex circular edge. 
 (Or the same slip may be made with a convex and concave edge of the same radius.) Describe an arc 
 dd of the pitch circle upon a second board C D, upon which the pattern tooth is to be di-awn. Fix the 
 piece B upon the board, so that its circular edge may accurately coincide with the circumference of the 
 arc dd. A portion of a circular plate D is next provided, of the same radius which it is proposed to give 
 fen the generating cii-cle ; this plate has a fine tracing point at p inserted into it, and projecting slightly 
 
38 
 
 GEERI5G. 
 
 from its under surface, and accurately coinciding with its circumference. Having set oif the thickness o^ 
 the tooth a c upon the pitch circle d d, so that twice this width increased by the clearance which it is 
 desired the teeth should have, may be equal to 
 
 205f. 
 
 C \o.i / 
 
 
 
 
 the pitch, the generating circle D is made to roll 
 upon the convex edge of B ; meantime the point 
 at ip will trace upon the board the curve of the fa- 
 ces of the tooth, having caused the point to coin- 
 cide successively with the two points a and c, and 
 the circle to roll from right to left, and vice versa. 
 Let the piece B be now removed, and the 
 piece A applied and fixed, so that its concave 
 edge may accurately coincide with the circular 
 arc dd; then, with the same circular plate D 
 pressed against the concave edge of A, and 
 made to roll upon it^ the point at p, which is 
 made as before to coincide successively with the 
 
 goints a and c, will trace upon the surface of the 
 oard C D the two hypocycloidal arcs ab, cb, 
 which form the flanks of the tooth. The com- 
 plete tooth, thus formed, has this advantage 
 over the form of tooth found by the former 
 mode, that it will work correctly with the teeth 
 similarly described upon any other wheel, pro- 
 vided the pitch of the teeth be the same, and 
 provided the same generating circle D be used 
 to strike the curves upon the two wheels. 
 
 In this manner the general forms of the teeth of the pair are determined. And it only remains to 
 cut them oflf at such lengths that they shall come into contact in the act of passing through the line of 
 centres. Thus, in Fig. 2042, let Y be the centre of the generating circle of the teeth, then the points 
 a h, where the circle intersects the edges of the driving teeth, are the points of contact of the teeth. 
 Now as the teeth ought to come into action only at the line of centres, it is clear that the tooth/ must 
 have been driven by the tooth of the wheel from the time their contact was at c, to which another 
 tooth of the wheel is shown to have advanced, the former being about to quit contact ; and since the 
 tooth now advanced to c is about to take up the task of impelling the driven wheel, and the other to 
 yield it, it is clear that all the part of the tooth falling within the circumference of the generating circle 
 might be removed ; this done, it is evident that all the driving strain would immediately be transferred 
 to the succeeding pair of teeth as required. In order, therefore, to obtain a form of tooth which shall 
 satisfy this condition, by the mode of setting out which has just been described, it is only necessary to 
 take ac, in Fig. 2052, equal to the pitch of the tooth, and to bring the circumference of the generating 
 circle D to touch the convex circumference of the asrc d d in that point : the point of intersection o ot 
 this circle, with the face t a, will be the last acting part of the tooth ; and if a circle passing through 
 that point be struck from the centre of the pitch circle, all that portion of the faces of the teeth whicli 
 lie beyond it may be cut off. The length of the tooth of the wheel intended to act with this may be 
 determined in exactly the same manner. It may, however, happen that the point o will fall beyond 
 the point t of the tooth ; when tliis occurs, it is then impossible to cut the teeth of such lei^th as to 
 satisfy the condition that they shall drive only after the line of centres. This may be proved organi- 
 cally with a pair of 15 to 15, and other numbers which the preceding table of cases will suggest. 
 
 We have here supposed the same generating circle to be used in striking the entire surfaces of the 
 teeth of both wheels. Tliis is convenient, but not essentially necessary ; for if the same generating, 
 circle be used in striking those parts of the teeth which act together, the required condition will be 
 satisfied. Thus the flank of the driving tooth and the /ace of the driven tooth come into actual contact ; 
 it is therefore requisite that these curves be respectively an epicycloid and hypocycloid, struck with the 
 same generating circle. And, again, the face of the driving tooth and the Jlank of the driven tooth being 
 in contact, these curves must in like manner be struck with the same generating circle. But it is evi- 
 dently unnecessary that the same generating circle be used in the first and second cases ; any generating 
 circle will satisfy the proposed conditions in either case, provided it be the same for the epicycloid and 
 the hypocycloid, which are to act together ; observing, however, that as the diameter of the generating 
 circle is increased, the thickness of the root of the teeth vi^ill be diminished ; and conversely, when 
 made less than the radius of the pitch circle, the root of the teeth will spread out, and the curvature of 
 the epicycloidal faces will be proportionally increased, and the teeth shortened. From these observa- 
 tions it is also clear that the forms of the teeth of a 
 pair of wheels which are required to drive continuously in 
 one direction may be so modified as to give additional 
 strength to the teeth without impairing their delicacy of ac- 
 tion. Thus, if the wheels represented here, Fig. 2053, the 
 lower A B being the driver, move always in the direction of 
 the arrows, then, instead of the teeth being formed symmet- 
 rically, the back of each tooth may be filled up, either by 
 striking the curves by a very small generating circle, or bet- 
 ter by involutes, which being proportional to the pitch circles 
 will be sure to clear during the working of the wheels. By 
 this means the strength of the teeth is greatly augmented, _ 
 
 as must be obvious, and their action is in no way impaired, since the acting surfaces are not altered 
 
 2853 
 
GEERING. 
 
 39 
 
 Ileturning now to the condition often required, though seldom fulfilled in practice, of making wheela 
 of different numbers, when of the same pitch, to work correctly together, it is clear from what has been 
 advanced in connection with the preceding mode of setting out, that it is not absolutely necessary to 
 assume the generating circle of any precise radius ; it may be taken greater or less within certain limits 
 without affecting the correctness of the working sm-faces of the teeth. 
 
 Proposition. — " If for a set of wheels of the same pitch, a constant describing circle be taken and 
 emplo3^ed to trace those portions of the teeth which project beyond each pitch line by rolling on the 
 exterior circumference, and those which lie within by rolling on the interior circumference ; then any 
 two wheels of this set will work correctly together." 
 
 In illustration of this principle, first announced by Professor Willis, he adduces the established and 
 well-known condition, that the portion of each tooth within the pitch line of a driving-wheel works only 
 with the portion which lies beyond the pitch line of the follower, as above shown, and that its action ia 
 confined to the approach of the point of contact to the line of centres. After the point of contact of the 
 teeth has passed that line, then the case is reversed, and the portion of the driving tooth which lies 
 beyond the pitch line is in contact only with some part of the follower's tooth, which lies within its pitch 
 line. Consequently, if a constant describing circle be used for the whole set, it is clear that the propo- 
 sition will apply to any pair of the wheels, both before and after the teeth have passed the line of 
 centres ; for in each case we have an exterior epicycloid working with an interior epicycloid, and both 
 have been drawn by the same describing circle ; that is, by the constant circle of the set. 
 
 To carry this scheme into practice, it only remains to settle the proper diameter to be given to the describ- 
 ing circle. Let the external circle B C, Fig. 2054, be a pitch circle whose centre is ; then upon this system 
 the flank of the tooth — that is, the portion of it which 2054. 
 
 lies within the circle — will be an arc of an interior 
 epicycloid (or hypocycloid) mn or m' n. Now, if the 
 describing circle be half the diameter of the pitch cii-- 
 cle, the flank will become a straight line, coinciding 
 with the radius n, as shown in Figs. 2042 and 2043. 
 But if the describing circle be of less than half the 
 diameter of tlie pitch circle, the flank m n will be con- 
 cave, and the base of the tooth will spread ; on the 
 contrary, if the describing circle be more than half the 
 diameter, the flank m' n will be convex, and the base 
 of the tooth will be contracted inwards — a form man- 
 ifestly unpractical and useless. It is therefore clear, 
 that the describing circle must not be greater than 
 half the diameter of the pitch line ; nor must it be too 
 small, for then the base of the teeth would spread 
 inconveniently, and the curvature of the exterior epi- 
 cycloids would be injuriously increased, and the teeth 
 become too short. The circle selected should there- 
 fore be as large as it can, consistently with the Umita- 
 tion just stated, so that we finally arrive at the prac- 
 tical rule : Make it equal in diameter to the radius of 
 the least pitch circle of the set. And, as pinions should never have less than 12, and if possible not less 
 than 14 teeth, it would be weU to establish one of these numbers for that least pitch circle. 
 
 Thus, then, by the use of the same generating circle for all the wheels of the same pitch, they will 
 all work correctly together. 
 
 It has already been observed, that, having found the epicycloidal curves of the teeth, by means of the 
 templets, a common method of proceeding is to find, by trial, a centre and small radius, by which the 
 arc of a circle can be described that will coincide nearly with the templet-traced curve. Having found 
 these, as in Fig. 2052, and having struck upon the ends of the rough cogs of his pattern, a circle, con- 
 centric with the pitch circle, and whose distance from it is equal to that of the centre of this arc, the 
 workman adjusts his compasses to the small radius, and always keeping one point in the circle just 
 described, he steps with the other to gach cog in succession, the cogs having been previously divided 
 into two parts corresponding to the pitch and thickness of the teeth. Upon each cog he describes two 
 arcs, one to the right and the other to the left, which serve him as guides in shaping and finishing the 
 acfing surfaces of the teeth. The practical convenience of this method is evinced by the extent of its 
 adoption, and requires only to be aided by a more commodious and certain method of determining the 
 centre and radius of the approximate arc, which has been supplied by Prof. Willis, 
 
 But before entering upon the geometrical consideration of the conditions involved, it may simplify 
 the investigation to explain, in the first place, the practical application which Mr. Willis has made of it 
 in the construction of his Odontagraph, an instrument which deserves to be extensively known and un- 
 derstood. The fundamental principle which it involves may be thus exhibited : — 
 
 Let A, Fig. 2055, be the centre and A T the radius of the pitch circle of a proposed wheel ; draw T 3, 
 making an angle A T 3 = 75° 30', with the radius, and drop a perpendicular A 3 upon T 3. Or draw 
 A T, and upon it describe a semicircle, and cut off T 3 = ^ of A T, that is, equal to a quarter of the 
 radius, then will 3 be the centre from which, if an arc o^ be described through T, that arc will be the 
 side of the tooth required. 
 
 This principle being established, it may be adapted conveniently to practice by constructing a bevel 
 of brass, of the requisite form, the angle at T being 75^ degrees : the side T 3 may for convenience bo 
 graduated into a scale of quarters of inches as in the Figure ; and these divisions may be further sub- 
 divided if thought proper. If the bevel so formed be laid upon the radius A T of the proposed wheel 
 and its point T to the pitch circle, the centre point 3 will be found at once by reading off the length ol 
 
40 GEERING. 
 
 the radius of the "wheel in inches upon the reduced scale. Thus in the figure the radius A T is three 
 inches long, and the point 3 is found at 3 of the scale. 
 
 To draw the curves of the other teeth, describe from centre A with radius A 3 the circle p q, dotted 
 in Fig. 2055, and falling within the pitch circle ; this will be the locus of the centres of the teeth. Thus 
 having divided the pitch circle into arcs corresponding to the pitch of the teeth, take the constant radius 
 3 T in the compasses, and keeping one point in the dotted circle as at q, step from tooth to tooth, de- 
 scribing successively the arcs of the faces first to the right and then to the left. For example, m n is 
 described fi-om centre q, and p o from P. 
 
 2055. 
 
 The sides of teeth thus formed, consisting each of a single arc, bear considerable analogy to involute 
 teeth, and like these have the fault of acting together with rather too much obliquity ; but the mode of 
 describing them is exceedingly simple, and they have this advantage, that they will work correctly with 
 wheels of any number of teeth, with any amount of clearance, and without any great nicety of adjust- 
 ment of the pitch circles. For wheels of a small pitch — as | inch and under — this mode of setting out 
 may be adopted without risk ; but when the sides of the teeth cctfisist of single arcs, there can only be 
 one position of action in which the angular velocity will be strictly constant — and that is when the 
 point of contact is on the line of centres. This objection is obviated by making the side of each tooth 
 to consist of two arcs, joined at the pitch line, and so struck that the point of action of the one shall be a 
 little before the line of centres — say at the distance of half the pitch ; and the exact point of the other 
 at the same distance beyond that line. 
 
 The mode of drawing these arcs is very fully described by Professor Willis, but it is much too com- 
 plex to be rendered available in the workshop ; besides, it has been rendered superfluous for practical 
 purposes by his Odontagraph, in which it is embodied. This instrument, which is fast coming into use, 
 is usually formed of card-paper, sheet-brass, or plane-tree. Fig. 2056, as constructed by Messrs. Holt- 
 zapffel and Co., of London, contains within it the requisite tables of numbers. 
 
 One side is graduated into a scale of half inches, each half inch being again subdivided into ten equal 
 parts. The line t D, which corresponds to the radius of the wheel, makes an angle of 75° with the side. 
 The mode of using the instrument is thus explained in its application to the setting out of a wheel oj 
 29 teeth of 3-inch pitch. 
 
 Describe an arc of the required pitch circle, and upon it set off the pitch T t. Fig. 2057 ; bisect this 
 in e, Rnd draw radial lines B T and B t. For the arc within the pitch circle apply the slant edge of the 
 scale to the radial line B t, placing its extremity t on the pitch circle as in the Fig. In the table 
 headed Centres for the Flanks of the Teeth, look down the column of 3-inch pitch, and opposite to 30 
 teeth — which is the nearest number to that required — will be found the number 49. The point r indi- 
 cated on the drawing board by the position of this number on the scale of parts marked " Scale of 
 centres of Teeth within the Pitch Circle^'' is the centre required, from which the arc ef must be drawn 
 with a radius r e. 
 
 The centre for the arc de, which lies outside the pitch, is found in a manner precisely similar, by 
 aj^plying the slant edge of the scale to the radial line B T. The number 21 obtained from the table of 
 Centres for the Faces of the Teeth, will indicate the position of this centre upon the scale of Centres f&j 
 I'eeth outside the Pitch Circle, namely, at R. 
 
GEERING. 
 
 41 
 
 8 K 
 
 <2p 
 
 
 2056. 
 THE ODONTAGRAPH. 
 
 TABLE SHOWING THE PLACE OF THE CENTRES 
 
 
 UPON THE SCALE. 
 
 CENTRES FOR THE FLANKS OF THE TEETH. 
 
 PITCH IN INCHES AND PARTS. 
 
 Num- 
 ber of 
 
 i 
 
 H 
 
 i 
 
 1 
 
 1 
 
 1 
 
 n 
 
 Hi| 
 
 2 
 
 2i2i 
 
 3 
 
 H 
 
 teeth. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 13 
 
 32 
 
 48 
 
 64 
 
 80 
 
 96 
 
 129 
 
 160 
 
 193 
 
 225 
 
 257'289'32l'386 
 
 450 
 
 14 
 
 17 
 
 26 
 
 35 
 
 43 
 
 52 
 
 69 
 
 87 
 
 104 
 
 121 
 
 139156173 208-242 
 
 15 
 
 12 
 
 18 
 
 25 
 
 31 
 
 37 
 
 49 
 
 62 
 
 74 
 
 86 
 
 99,111123 1481173 
 
 16 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 4(] 
 
 50 
 
 59 
 
 69 
 
 79j 89: 99 191 
 
 138 
 
 17 
 
 8 
 
 13 
 
 17 
 
 21 
 
 25 
 
 34 
 
 42 
 
 50 
 
 59 
 
 67 
 
 75 
 
 84 101 
 
 117 
 
 18 
 
 7 
 
 11 
 
 15 
 
 19 
 
 22 
 
 30 
 
 37 
 
 45 
 
 52 
 
 59 
 
 67 
 
 74 
 
 89 
 
 104 
 
 19 
 
 
 10 
 
 13 
 
 17 
 
 20 
 
 27 
 
 35 
 
 40 
 
 47 
 
 54 
 
 60 
 
 67 
 
 80 
 
 94 
 
 20 
 
 6 
 
 9 
 
 12 
 
 16 
 
 19 
 
 25 
 
 31 
 
 37 
 
 43 
 
 49 
 
 56i 62 
 
 74 
 
 86 
 
 22 
 
 5 
 
 8 
 
 11 
 
 14 
 
 16 
 
 22 
 
 27 
 
 33 
 
 39 
 
 43 
 
 491 54 
 
 65 
 
 76 
 
 24 
 
 
 7 
 
 10 
 
 J2 
 
 15 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 49 
 
 59 
 
 69 
 
 26 
 
 
 
 9 
 
 11 
 
 14 
 
 18 
 
 23 
 
 27 
 
 32 
 
 37 
 
 41 
 
 46 
 
 55 
 
 64 
 
 28 
 
 4 
 
 6 
 
 
 .. 
 
 13 
 
 
 22 
 
 26 
 
 30 
 
 35 
 
 40 
 
 43 
 
 52 
 
 60 
 
 30 
 
 
 
 8 
 
 10 
 
 12 
 
 17 
 
 21 
 
 25 
 
 29 
 
 33 
 
 371 41 
 
 49 
 
 58 
 
 35 
 
 
 .. 
 
 
 9 
 
 11 
 
 16 
 
 19 
 
 23 
 
 26 
 
 30 
 
 34 
 
 38 
 
 45 
 
 53 
 
 40 
 
 •• 
 
 5 
 
 7 
 
 •• 
 
 •• 
 
 15 
 
 18 
 
 21 
 
 25 
 
 28 
 
 32 
 
 35 
 
 42 
 
 49 
 
 60 
 
 3 
 
 
 fi 
 
 8 
 
 9 
 
 13 
 
 15 
 
 19 
 
 22 
 
 25 
 
 28 
 
 31 
 
 37 
 
 43 
 
 80 
 
 
 4 
 
 
 7 
 
 .. 
 
 12 
 
 
 17 
 
 20 
 
 23 
 
 26 
 
 29 
 
 35 
 
 41 
 
 100 
 
 
 
 
 
 8 
 
 11 
 
 14 
 
 .. 
 
 
 22 
 
 25 
 
 28 
 
 34 
 
 39 
 
 150 
 
 
 .. 
 
 5 
 
 .. 
 
 
 
 13 
 
 16 
 
 19 
 
 21 
 
 24 
 
 27 
 
 32 
 
 38 
 
 Rack. 
 
 2 
 
 •• 
 
 
 6 
 
 7 
 
 io 
 
 12 
 
 15 
 
 17 
 
 20i22 
 
 25 
 
 30 
 
 34 
 
 
 TABLE SHOWING THE PLACE OF THE CENTRES UPON THE SCALE. 
 
 
 CENTRES FOR THE FACES OF THE TEETH. 
 
 
 
 
 PITCH IN INCHES AND PARTS. 
 
 Number 
 of teeth. 
 
 i 
 
 1 
 
 i 
 
 1 
 
 1 
 
 1 
 
 U 
 
 n 
 
 1| 
 
 2 
 
 H 
 
 2-^ 
 
 3 
 
 Si 
 
 12 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 1 
 
 9 
 
 10 
 
 U 
 
 12 
 
 15 
 
 17 
 
 15 
 
 
 
 3 
 
 
 
 
 1 
 
 8 
 
 10 
 
 11 
 
 12 
 
 14 
 
 17 
 
 19 
 
 20 
 
 2 
 
 
 ... 
 
 4 
 
 5 
 
 6 
 
 8 
 
 9 
 
 11 
 
 12 
 
 14 
 
 15 
 
 18 
 
 21 
 
 30 
 
 ... 
 
 3 
 
 4 
 
 ... 
 
 
 7 
 
 9 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 21 
 
 25 
 
 40 
 
 ... 
 
 ... 
 
 
 
 6 
 
 8 
 
 
 11 
 
 13 
 
 15 
 
 17 
 
 19 
 
 23 
 
 26 
 
 60 
 
 
 
 
 5 
 
 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 25 
 
 29 
 
 80 
 
 ... 
 
 ... 
 
 
 
 
 9 
 
 11 
 
 13 
 
 15 
 
 17 
 
 19 
 
 21 
 
 26 
 
 30 
 
 100 
 
 ... 
 
 
 ... 
 
 
 7 
 
 
 
 
 
 18 
 
 20 
 
 22 
 
 
 31 
 
 150 
 
 
 ... 
 
 5 
 
 6 
 
 ... 
 
 ... 
 
 ... 
 
 14 
 
 16 
 
 19 
 
 21 
 
 23 
 
 27 
 
 32 
 
 Rack. 
 
 ... 
 
 4 
 
 
 
 ... 
 
 10 
 
 12 
 
 15 
 
 17 
 
 20 
 
 22 
 
 25 
 
 30 
 
 34 
 
42 
 
 GEERING. 
 
 The double curve df is alsD true for an annular wheel of the same number of teeth, / becoming ol 
 com-se the point of the tooth and d its root. For a rack the pitch hne T t will be a straight line, and 
 B ^, B T, drawn perpendicular to it at a distance from each other, equal to the pitch 
 
 2057. 
 
 The numbers for pitches not stated in the table, may be obtained from the column of some other 
 pitch by direct proportion. Thus for 4-inch pitch, by doubling the number of the column of the 2-inch 
 pitch ; for 4^ by doubling 2^, and so on ; or if the difference be small, the column belonging to th« 
 
 nearest pitch may be employed without a serious error ; or more accurately, a number may be taken 
 half-way between those given in the two nearest columns. Also no tabular numbers are given for 12 
 teeth in the upper table, because witliin the pitch circle their teeth are radial lines. 
 
GEERING 
 
 43 
 
 In reference to the numbers given in the tables, the inventor observes that it is unnecessary to extend 
 these to every wheel, as the error produced by taking those belonging to the nearest, as above directed, 
 is so small as to be unajspreciable in practice. 
 
 The difference in form between the tooth of one wheel and of another is due to two causes ; the first 
 is the difference of curvature, which is provided for in the Odontagraph, by placing the compasses at 
 the different points of the scale of equal parts ; and secondly, the form depends on the variation of the 
 angle TBt, Fig. 2057, which is met by placing the instrument upon the two radii in succession. 
 
 The first is the only cause with which these calculations are concerned. Now in three-inch pitch, tho 
 greatest difference of form produced by mere curvature in the portion of tooth which lies beyond the 
 pitch circle, is only l-25th of an inch between the extreme cases of a pinion of 12 teeth and of a rack; 
 and in the acting part of the arc within the pitch circle it is 1-1 0th of an inch ; so that as all the other 
 forms lie betAveen those two, it is clear that if we select only four or five examples for the outer side of 
 the tooth, and ten or twelve for the inner side, we can never incur a greater error than the -^^o" P^^^ °^ 
 an inch, in a three-inch pitch, by always taking the nearest number in the manner directed, and a pro- 
 portionally smaller error in smaller pitches. But to insure this, the numbers ought to be so taken, that 
 the corresponding form shall lie equally between the extremes. This is more necessary with wheels of 
 small numbers of teeth, since the variation of form is much greater among the teeth of low than among 
 those of larger number. 
 
 Fig. 2058 shows the relation of the curves of the teeth of a rack and pinion of 12 teeth, both adapted 
 to work into the same wheel. They are drawn with accuracy, and on a large scale, so as to afford a 
 graphical comparison of the two extreme cases. 
 
 The Odontagraph is likewise applied to determine the correct form for cutters used in the wheel- 
 cutting machine, for shaping the teeth of metal and cog wheels. The form of the cutter is manifestly 
 that of the space between two contiguous teeth ; and may therefore be determined by drawing a pair 
 of teeth in any particular case. But in making a set of cutters especially for small pitches, it is not by 
 any means necessary to make one for every number. The forms for numbers of teeth that lie closely 
 together are, as already remarked, very nearly alike — so nearly alike indeed, that the errors of work- 
 manship would entirely destroy the difference, particularly when the numbers are high. When the num- 
 bers are very low, there is less room for deviation from the strict rule — not more, for example, between 
 the form of a cutter for 16 and 17, than there is for a cutter for a wheel of 150 teeth and another for 
 one of 300. This being the case, it appeared to Mr. Willis to be worth while to investigate some rule 
 by which the necessary cutters could be determined for a set of wheels, so as to incur the least chance 
 of error ; and to this effect he has calculated, by a method sufficiently accurate for the purpose, the 
 following series of what may be termed equidistant values of cutters ; that is, a table of cutters so 
 arranged that the same difference of form exists between any two consecutive numbers. 
 
 
 
 TABLE OF EQUIDISTANT VALUES FOR CUTTERS 
 
 • 
 
 
 
 
 
 
 
 1 
 
 2 
 300 
 
 3 
 
 4 
 
 5 
 
 7-6 
 
 6 
 60 
 
 7 
 50 
 
 8 
 43 
 
 9 10 
 38 34 
 
 11 
 
 12 
 
 27 
 
 13 
 
 25 
 
 14 
 23 
 
 15 
 21 
 
 16 
 
 20 
 
 17 
 T9 
 
 18 
 17 
 
 19 
 16 
 
 20 
 >< 
 
 21 
 15 
 
 22 
 U 
 
 23 
 13 
 
 24 
 X 
 
 25 
 12 
 
 No. of 
 teeth. 
 
 Rack. 
 
 This table affords the requisite information in the selection of the wheel to which each cutter shall be 
 accurately adapted, after it has been determined how many are necessary in a set. For example, if a 
 single cutter were thought sufficient for a set of very small wheels, it had better be accurately adapted 
 to teeth of 25, for that value is intermediate between the two extremes. If three cutters are to su05ce 
 for the whole set, then 76, 25, and 15 ought to be selected — of which the cutter 76 may be used for all 
 teeth from a rack to 38 ; the cutter 25 from 38 to 19, and the cutter 15 from 19 to 12 ; and so on. In 
 cutters, the greatest difference of form is at the apex of the tooth, that is, at the base of the cutter, and 
 this amounts to ^ of an inch in 2-inch pitch, when the teeth have the usual addendum. From this the 
 difference for any smaller pitch may be ascertained, and as many cutters interposed as the workman's 
 notion of his own powers of accuracy may induce him to think necessary. Thus, if the hundredth of an 
 inch be his limit of accm-acy in forming his cutters, and he proposes to make a set for half-inch pitch, 
 where the difference of form is \ X i= ys, that is, yf^ nearly, then half a dozen cutters will be 
 sufficient, and these must be made as nearly as possible to suit the wheels 150, 50, 30, 21, 16, 13. 
 
 Tlie following table contains a selection of numbers for different cases which may save trouble r — 
 
 TABLE OF CUTTERS. 
 
 
 Numbers of Teeth to be selected. 
 
 2 
 
 50 
 
 16 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 4 
 
 75 
 100 
 150 
 
 25 
 34 
 
 -50 
 
 15 
 
 20 
 30 
 
 -13 
 
 21 
 
 Te" 
 
 13 
 
 — 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 8 
 10 
 12 
 
 Is" 
 
 24 
 
 200 
 200 
 300 
 300 
 
 rack 
 
 77 
 100 
 150 
 300 
 
 40 
 50 
 GO 
 100 
 150 
 
 29 
 -35 
 "43 
 
 70 
 100 
 
 27" 
 34 
 
 50 
 76 
 
 18 
 22" 
 27 
 
 0(7 
 
 15 
 
 23 
 30 
 50 
 
 13 
 
 1q 
 20" 
 
 26 
 43 
 
 IT 
 
 24" 
 
 '38 
 
 u 
 
 15 
 22 
 34 
 
 H 
 
 20 
 30- 
 
 13 
 
 18 
 
 27' 
 
 To" 
 
 25 
 
 IF 
 23- 
 
 IT 
 21 
 
 13 
 
 20 
 
 z 
 
 12 
 
 19 
 
 l8 
 
 _ 
 
 IK 
 
 I5 
 
 _ 
 
 H 
 
 I2 
 
44 
 
 GEERING. 
 
 When the numbers have been selected, the Odontagraph may be employed to draw the figure of th« 
 cutter corresponding to each wheel, either on the same scale, or better, on a larger scale, which may be 
 afterwards reduced proportionally to the size required. 
 
 To explain the principle of the Odontagraph, let us, in the first place, observe the action of the two 
 pieces in the annexed diagram, Fig. 2059, of which P and Q are the centres of curvature at the points 
 
 2059. 
 
 ^- / 
 
 of contact, and which are capable of revolving round A and B. It is plain that the line P Q is, in every 
 position of contact of these curves, equal to the sum of their radii, and must therefore be constant during 
 such motion ; and hence, if for the circular arcs, a pair of rods A P, B Q, connected by a link P Q, be 
 substituted, their angular velocity will be to each other as the segmental arcs which they respectively 
 represent, and P Q will be a constant and common normal to their arcs of motion. It is also clear that 
 a change in the actual lengths of the radii will not affect the motion, the distances of the centres being 
 constant. Now let the rod A P be moved into a new position A.p, as shown in Fig. 2060; its extremity 
 
 will manifestly carry with it the end of the link P Q, and communicate through it a motion to the arm 
 B Q, causing it to assume likewise a new position B q. It is necessary to know the relative value ol 
 this motion to that of A P which produced it, which, though constantly changing, may be thus deter- 
 mmed at any instant. 
 
 When the rod P Q changes its position, it may be considered during its motion to turn about some 
 centre in space, although the relative position of that centre be perpetually shifting. Tliis centre must 
 of necessity be the point in which the arms A P, B Q would intersect if produced. Thus the momen- 
 tary centre, when the motion begins, is K. For as the extremity P moves round the centre A, the 
 direction of its motion at starting must be perpendicular to A P, and therefore the momentary centre 
 must be somewhere in the line A P produced. In like manner the initial motion of the extremity Q 
 must be perpendicular to B Q, and the momentary centre must lie somewhere in the direction B Q : 
 but these directions intersect in K, which is therefore the centre in space about which the rod P Q on 
 starting into motion may be conceived to turn. And since the rod P Q turns about K, the direction of 
 motion of P and Q are to each other at any given instant as their radial distances from K ; that is, as 
 P K and Q K, which is true whether we consider them as the extremities of the rod P Q or of the radii 
 A P, B Q; also tlie angular motions of the latter will be found by dividing those direct motions by their 
 radii ; that is, 
 
 Angular motioi of 
 
 P K OK 
 
 P round A : angular motion of Q round B : : -r-^- : =r— -- 
 
 A P B (4 
 
GEERING. 45 
 
 If now we draw K L, A R, B S, then we have 
 
 PK : AP : : KL : A R by simUar triangles KPL ; APR 
 
 BQ:QK::BS:KL BQS;KLQ 
 
 AT:BT::AR:BS ARTjBTS 
 
 And compounding these three proportions we obtain 
 
 — .^^..BT.AT 
 
 that is to say, the anguhir motion of the arms are to each at any moment inversely as the segments intfl 
 which the link divides the line A B, joining the centres of motion or line of centres. If now the lini 
 move into a new position p q near the first, and it happen that this second position intersects the first 
 in a pomt L above or below the line of centres, as shown in the Figure, then the ratio AT: B T will 
 be changed into At \'Bt, and consequently the angular motion will be an increasing or decreasing ratio. 
 But if the point L coincide with the line of centres, this ratio for the moment will remain constant. 
 
 Now, a little consideration will show that the point of intersection between two successive positions 
 P Q, jD 5' of the link must be at the place where the perpendicular from K falls upon it. For as K is 
 the momentary centre of motion, the extremity L of the perpendicular will begin to move in line at 
 right angles with it, and consequently will remain in the dii-ection of the first position when the link has 
 passed into the second position ; in other words, it wiU be in the point of intersection of the two posi- 
 tions. When, therefore, the rods are in such a position that the perpendicular from K meets the link 
 P Q in the line of centres, the ratio of the angular motions of A P and B Q is constant ; and if in this 
 state the points P and Q be employed as centres whence short arcs are drawn through any common 
 point M', as in Fig. 2059, and applied as teeth, these arcs will manifestly drive each other correctly 
 when in the exact relative position described, and very nearly so when removed to a short distance on 
 either side of it, which is the thing required. 2061 
 
 Now these relative positions of P and Q may be determined by a ^ ' ^ 
 
 simple construction founded upon the necessary coincidence of L and ^^^s::^ 7^— -V— ->^^ 
 
 T. Thus let A and B, Fig. 2061, be the centres of a pair of wheels, ^^~""*^~-^/^v\\ /^, 
 and A B the Hne of centres divided at T, the point of contact of the ^\ ^i^'-^-^l^^fX 
 
 pitch circles. Draw P T Q, making any angle with the line of centres, ""n^ / ^^^^^^^^\\. 
 
 and upon it assume P as a centre from whence the circular side is to '^ / ^/l \,^~~~~-^^i^ 
 
 be described of a tooth of the wheel whose centre of motion is A. To / 7^ I ^"^ ''' 
 
 find tl\e corresponding centre for a tooth of the wheel which turns upon y^ ' 7^'\ 
 
 B, draw T K perpendicular to P Q, produce A P to meet it in K ; join y^ *^ 
 
 K B and produce it to meet P Q in Q, then will Q be the required ^ 
 
 centre — as will appear by comparing Fig. 2061 with Fig. 2060. If 
 
 then a small arc m n struck from P be taken as the face of a tooth of the wheel on A, it will work cor- 
 rectly with an arc mp struck from Q through ?n, and employed as a tooth of the wheel on B. 
 
 Were B Q so taken as to make the angle K B T acute — for example at B', then would Q fall at Q' 
 on the same side of T as P, but beyond it, and the effect would be that the tooth m p would be concave 
 instead of convex. And if the angle K B T = P T A, then will K B become parallel to P T, and the 
 point Q being thus removed to an infinite distance, the arc mp oi the tooth of the wheel on B will be 
 a right line perpendicular to P T. 
 
 The angle A T P is arbitrary, and its value may, therefore, be determined from other conditions than 
 those stated. It may be reuiarked, however, that if made a right angle, the points P and Q will vanish 
 by coinciding with T ; and if it be made a little less than a right angle, the points P and Q will be 
 thrown so near to T, that the radii by which the arcs aro struck become too short, and the points of the 
 teeth too much rounded ofl". On the other hand, if the angle A T P be made too acute, the action of the 
 teeth upon each other at the moment of passing the hne of centres and elsewhere becomes very oblique, 
 and an injurious pressure is thereby thrown upon the axis of the wheels. By various trials Mr. Willis 
 has adopted 75° as the value of the angle which appears to avoid these extremes, and he has accord- 
 ingly employed it in the construction of his Odontagraph. 
 
 Again, the position of the point 7n, through which the arcs are to be struck, is also arbitrary, and 
 must be determined by considering which point of the action we wish to make the correct point. ^ II 
 the teeth consist of a single arc each, the correct point may be fixed at the moment of passing the line 
 of centres, and therefore the arcs must be struck through the point T ; but if the side of the tooth be 
 formed of two arcs joined, one lying within and the other beyond the pitch line, then the action of one 
 of them will be confined to the approach of the point of contact of the teeth and the line of centres, and 
 the action of the other to its recess from that line, and m must be assumed upon such a principle that 
 the correct point of each arc shall fall nearly in the middle of its action. 
 
 The distance between the centres may also be found by calculation, thus : A R being perpendiculaJ 
 to P T, we have by the similar triangles A R P ; P T K 
 
 ^_ PTX AR __ PTX AR 
 PR TR — PT 
 
 PT . AT . sin ATP 
 
 AT. cos ATP — PT 
 
 ,^ ^ ^^ AT . KT . cos ATP ^ 
 
 therefore P T = ^, „, , -. -— = D 
 
 K T -f A T . sm A T P 
 
46 
 
 GEERING. 
 
 vrhich is the (Jentre for the arc m n. And similarly drawing B S perpendicular to T Q we have for tha 
 corresponding arc m p. 
 
 BT . KT . cos ATP _ 
 ^ ""■ KT-f B"T sin ATP~ 
 
 Now the point P being obtained from the point K on the opposite point of T to A, these formulae be- 
 long to that part of the tooth of the lower wheel beyond the pitch line and the flank of the tooth of the 
 upper wheel. The corresponding face of the upper and flank of the lower wheel may be determined 
 in a similar manner, by joining B k and following the course of reasoning stated for the point K. 
 
 In these formulse, the values of P T and Q T may be determined for any forms of teeth, and also for 
 a set of wheels which shall work correctly together. But the greatest value which in this last case can 
 be correctly assigned to K T must be one which, if r be the smallest radius, shall make 
 
 KT = rsin ATP 
 
 Consequently, putting R, for the radius of the wheel, and angle A T P = the values of D and d, that 
 is, the distances of the centre points of the arcs measured from T, will become ; 
 
 _ R . r , cos 6 , R . r . cos 9 
 
 R+r R— r 
 
 By assuming constant values of r and in a set of wheels, the values of D and d which coiTespond 
 to different numbers and pitches may be calculated and arranged in tables for use. In this way the 
 tables which accompany the diagram of the Odontagraph were obtained by taking the least number 
 of teeth to be given to a wheel = 12 and 6 = 75°, the numbers being expressed in twentieths of an inch. 
 
 In this way these tables may be extended at pleasure, and for any values of r and 6 ; and perhaps it 
 would be preferable in the construction of such tables to express the numbers in inches and 32ds of an 
 inch, which would better adapt them to the common foot-rule. 
 
 Involute teeth.— luYolnte teeth have the disadvantage, already stated, of being, when in contact, too 
 much inclined to the radius, by which an undue pressure is transferred to their axes. Their mutual 
 friction is thereby little affected, but that of the axes is increased, and their journals are more speedily 
 worn. But they have at the same time the advantage of working with more accuracy under derange- 
 ment and incorrectness of fitting, and any pairs of them will work truly to- 
 gether in sets within certain limits, however diff'erent in diameters, the pitch ^^g^ 
 
 being the same. Until Professor Willis had developed the mode of adapting ' v 
 
 the epicycloid to the condition in question, the involute was the only form ,-''" """^^^ 
 
 known in practice by which the conditions of perfect figure for wheels of any / 
 
 sizes to work smoothly in wheels of any other sizes, could be satisfied. ' ■" 
 
 To describe this curve for the teeth of a pair, of which the radii of the pitch 
 circles, and pitch of the teeth are determined, we may employ the mode 
 illustrated by Fig. 2062. Let A and B be the centres of the pair, and e 6 be 
 their pitch lines ; join A and B by a right line passing through c ; from this -'^'J 
 last point draw cd, cd perpendicular to the radials Bd,Ad, and cutting them '■■ 
 in d and d; this line dd is then a common normal to the teeth in contact, and 
 the perpendiculars Ad,Bd, are the radii of the involute circles which form 
 the acting faces of the teeth. 
 
 The involute curve may be described mechanically in several ways. Thus 
 let A, Fig. 2063, be the centre of a wheel, for which the form of involute teeth is 
 to be 'found. Let m.w a be a thread lapped round its circumference, having a 
 loop-hole at its extremity a ; in this fix a pin, with which describe the curve or involute ah — K\yj 
 unwinding the thread gradually from the circumference, and this curve will be the proper form for tJw 
 teeth of a wheel of the given diameter. 
 
 2064. 
 
 At- 
 
 o)i 
 
 2063. 
 
 ■ ' " ■ *$ 
 
 The involute ab---h may also be produced by an epicycloidai motion ; for since the circumference 
 of a generating circle, whose centre is infinitely distant, must be a straight line, we may form the invo- 
 
GEERING. 47 
 
 lute by making a straight ruler roll upon the circumference of the circle to be evolved. Thus let R, ii? 
 ^ig. 2003, be a straight ruler, at whose extremity a pin jd is fixed with the point resting upon the poini 
 q of the circle ; then by rolling the straight ruler upon the circumference, so that the point at -which it 
 touches the circle may move gradually from q towards r, the curve qp will be generated exactly simi- 
 lar to the involute ab h, and the method is perhaps easier and more accurate than by unlapping a 
 
 thread from the circumference of the evolute, as above described. 
 
 Mr. Hawking, in his Appendix to Camus' Treatise on the Teeth of Wheels, recommends a very simple 
 instrument for striking involute teeth. Two views of this instrument are given in Figs. 2064 and 2065. 
 In these a 6 is a piece of watch-spring, with two teeth c c, formed upon its edges ; at a is a small screw 
 by which the spring is attached to a templet formed of thin board, and equal to a sector of the circle 
 upon which the involute is intended to be generated. At 6 a small bit of wire is inserted into the 
 spring, to form two knees as a handle by which the spring can be kept stretched. The templet is equal 
 iji the thickness to the width of the spring. 
 
 In using the instrument, the templet is to be fixed upon the drawing board or surface upon which the 
 involute is to be traced ; the centre / coinciding with the axis of the wheel, or its representative on the 
 drawing board ; then by holding the knobs 6 as a handle, and keeping the spring thereby tightly 
 stretched, so that it remains a correct tangent to the circle, and lapping the spring upon the arc of the 
 templet, one of the points c will trace upon the surface over which it passes a curve which, when the 
 point has arrived at the circumference of the pitch circle represented by the templet, will be a true 
 involute of that circle. 
 
 If the tracing instrument be now turned over, the opposite tooth c on the side of the spring now un- 
 dermost, will, by repeating the same operation, trace the counter involute for the other side of the tooth 
 of the wheel. 
 
 Mr. Hawkins' mode of proceeding to draw the outlines of a pair of involute wheels with his tracer is 
 simple, and may be explained by reference to Fig. 2062. Having drawn the line of centres A B, and 
 divided it at c into parts Ac, Be, proportionate to the intended velocities of the two wheels, next draw 
 the right hne dd cutting the line of centres at c, and making with it an angle deviating not more than 
 20 degrees from a right angle. From centre A describe a circle touching the line dd a.t d within the 
 pitch circle of that wheel, and which will thus have dd as a tangent at d. In like manner from B as a 
 centre describe a circle touching the same straight line at d within the pitch circle of that wheel, and 
 also having that straight line as a tangent. Then the distance between the two circles is the proper 
 length of the teeth, from which, however, the necessary clearance ought to be deducted. The two cir- 
 cles last drawn are likewise the bases upon which the involutes for forming the boundaries of the teeth 
 are to be generated. 
 
 Next, having provided two involute traces made in the way above described, each with a templet 
 corresponding to a sector of the base circle of its wheel, the centre / of the templet is to be applied to 
 the centres A or B of the wheel of the pair to which it belongs, and meeting the points d ; then sup- 
 posing the templet for A to be first used, let it be adjusted xmtil the tracing point c of the spring coin- 
 cides with some point as d of the base circle of the wheel B, by moving it until it meets the circum- 
 ference of its own base circle at b, the side db of the tooth deb will be formed, and of the proper 
 length, by moving the templet spring over the space included between the exterior of one base circle 
 to the other. By turning over the templet, and adjusting it for the required thickness, the outline of 
 the opposite side of the tooth may be traced in the same manner. Repetition of the operation with the 
 templet of B will give the form of the teeth of that wheel. 
 
 If thought desirable all the teeth of the pair may be drawn in the way indicated, or a single tootli 
 of each wheel may be accurately drawn, and two pattern teeth cut to the forms thus found. 
 
 Mr. Hawkins states that teeth formed according to this method will communicate equable motion 
 from either wheel to the other, without possessing any tendency to press the shafts outwards — which 
 tendency would exist if the angle of the common tangent [dd) of the two base circles deviated much 
 more than twenty degrees from a right angle with the line of centres. 
 
 If the teeth of both wheels are to be of equal thickness at the base, which they ought to be when of 
 the same material for the sake of equal strength, the part df of the common tangent ought to be bi- 
 sected in the middle c of its length, and the involutes drawn in the way described will give the roots 
 of the teeth so nearly equal, that a very slight correction only will be required — the amoimt depending 
 on the relative diameters of the wheels. But when the teeth of one wheel are required to be thicker 
 than those of the other, the part df of the common normal d d must be unequally bisected in propor- 
 tion to the required difference. 
 
 It is finally to be observed that the teeth may have their bases in any other circles than those given, 
 since the radii of the generating circles are entirely arbitrary ; but the proportion taken is found to 
 satisfy the conditions required with very little outward action. In respect to the character of the in- 
 volute curve as compared with the epicycloid, it is obvious that the former possesses the advantag;e 
 that a greater number of teeth of equal strength may be given to a wheel by this than the epicycloidal 
 form ; for with the latter the space must be at least equal to the tooth, while not much more than half 
 the space is required to be marked off on the base circle for an involute tooth of the same length. There 
 are also more teeth engaged at the same time, and thus the strain is divided. 
 
 Teeth of a rack and pinion. — A rack, as before observed, may be regarded as a wheel of infinite 
 diameter. The teeth may, therefore, be defined by the epicycloidal method explained for spur-wheels, 
 by making their planks straight, and the teeth of the pinion involutes. Now, the properties which have 
 been shown to belong to involute teeth manifestly obtain, however great may be the dimensions of the 
 pitch circle of their wheels, or whatever disproportion may exist between them. Thus of the two wheels 
 A and B with involute teeth, which work together, let the radius of the pitch circle A c, Fig. 2066, of ona 
 become infinite ; its circumference will then become a straight line represented by the face of a rack 
 But whilst the radius A c becomes infinite, the radius A D of the circle from which its involute teeth ar 
 
48 
 
 GEERING. 
 
 Btruck, and which bears a constant ratio to the first, 'will also become infinite, and therefore the invo- 
 lute m a will become a straight line, perpendicular to the pitch line at a. For it is evident that the 
 extremity of a line of infinite length, unwinding itself from the circumference of a circle of infinite 
 diameter, will describe through a finite space a straight line perpendicular to the circumference of the 
 cii-cle. The involute teeth of the rack will, 
 
 therefore, have their faces perpendicular to 2060. 
 
 the pitch line; and this line will be deter- 
 mined by drawing a tangent to the circle on B 
 through the point of contact from which the 
 involute teeth of the pinion are to be struck. If 
 the radius B m, with which the involute circle 
 of the pinion is described, be equal to the 
 radius of its pitch circle, the line D m will 
 become parallel to the face of the rack, and 
 the sides of the teeth of the rack perpendicu- 
 lar to it. But in this, while the teeth of one 
 wheel, B, have remained unaltered, and the 
 accuracy of their action uninfluenced by the 
 change in the dimensions of the pitch circle 
 of the other, which has converted it into a 
 rack, and its curved into straight teeth; it 
 therefore follows that straight teeth upon a 
 rack work truly with involute teeth upon a 
 pinion. 
 
 But as straight teeth are not the form of 
 greatest strength which may be attained, and 
 as strength is of importance as well as uni- 
 formity of action, the conditions of the problem 
 
 may be examined with a view to the best modi- \ 1/ 
 
 fication which the form admits with respect to \ ^ 
 
 this test. If with the pitch radius B c of the Ai 
 
 pinion. Fig. 206*7, we describe the pitch circle, 
 and draw M N a tangent to it at c, we next 
 
 make this right line the pitch line of the rack ; and, further, describe the circle whose radius is B h, 
 making this last the base of an involute h h ; then if the tooth of the rack be bounded by a right line 
 fe D E, making an angle B D M with the pitch line equal to B c A, and the involute A 6 be moved into 
 the position h' h', it will drive the sloped tooth to the position P m, always touching it in the line A c H ; 
 and the velocity of the circumference of the pitch circle will always be equal to that of the pitch line. 
 
 A wheel with involute teeth will work truly 
 with a rack whose teeth are straight-sided, and 20G7. 
 
 inclined to the pitch line at an angle d, pro- 
 vided only 
 
 \ 
 
 \ 
 
 radius of base 
 
 B/i 
 
 B 
 
 : sm 9. 
 
 radius of pitch circle 
 
 In a rack of this kind, the locus of contact of 
 the teeth being the tangent line h H, the 
 sloping teeth will be pressed downwards by a 
 certain portion of the working pressure, which 
 in some cases may be an advantage in itself 
 by neutralizing vibration, and the advantage 
 of a greatly increased form of teeth is at the 
 same time secured. 
 
 The teeth of the pinion being of the regular 
 epicycloidal form, the faces of the teeth of the 
 rack will have a cycloidal form. But accord- 
 ing to the usual method of setting out racks 
 and pinions, the pitch line of the rack is the 
 locus of contact, and the action on one side of 
 the line of centres is confined to a constant 
 point in the rack tooth, which is thus subjected . . 
 
 to rapid wear. This disadvantage may be abridged by reducing the length of the teeth of the pinion, 
 but it may also be entirely avoided by taking a constant generating circle, and employing it to describe 
 cycloidal flanks for the rack teeth by rolling on its pitch line, and then by describing the faces of the 
 teeth of the pinion with the same generating circle, in the manner before described. By this simple 
 modification, the contact will no longer be confined to the pitch line of the rack, but will be distributed 
 over the flanks of its teeth ; and these, it may be observed, may be made of any length desired, by 
 regulating the diameter of the generating circle accordingly. If the generating circle have a diameter 
 equal to the radius of the pinion, the flanks of the pinion-teeth will then be radial, and this form, may 
 also be modified by taking a circle of greater or less diameter. 
 
 To determine the pitch circle of the pinion, let D represent the distance through which the rack is to 
 be moved by each tooth of the pinion, and let these teeth be in number —N; then will the rack be 
 moved through the space N X D during one complete revolution of the pinion. 
 
 Now the rack and 
 
GEERING. 
 
 49 
 
 pinion are to be driven by the action of their teeth, as they would by the contact of the circumference of 
 the pitch circle of the pinion with the plane face of the rack, so that the space moved through by the 
 rack during one complete revolution of the pinion must precisely equal the circumference of the revolv- 
 ing pitch circle ; consequently, calling R the radius of the pitch circle of the pinion, we will have 
 
 Two of these quantities being given, tlie other immediately becomes known. Thus, let D 
 and N = 13, then 
 
 2 inches 
 
 R = 
 
 2 X 3-1416 
 
 :4:-138 inches. 
 
 Annular wheel and pinion. — An annular wheel is one having its teeth formed within its periphery 
 and consequently the pinion works internally. In this arrangement, the form of the teeth may be de- 
 termined according to the principles already explained, with very slight modification in their practical 
 application. 
 
 Thus, in Fig. 2068, let A B be the centres of the wheel and pinion ; and let c be the point of contact 
 of the pitch circles. The epicycloid c m described by the circle having a diameter = B c, gives the 
 lateral form of the tooth of the wheel, Hmited as before by a radius drawn to the middle of the thickness 
 of the tooth ; and au arc described from the centre B, with radius A n, cuts the circumference B c at a 
 point which determines the side of the tooth of the pinion at n. But if we attempt to apply the same 
 process to finding the face of the tooth of the pinion as we have employed in determining the flank ot 
 the wheel-teeth, we shall find that the construction will either not give the side of the tooth as the arcs 
 do not meet, or it will give it a position upon the radius Ac on the same side as the face cm. From 
 this, then, it appears that the teeth of the wheel will have no flanks, and that their outline will consist 
 simply effaces. From this it follows that the pinion would not drive the wheel uniformly according to 
 the required condition, and in consequence we must find another curve for the faces of the teeth of this 
 last. Now if we trace the epicycloid by a point c of the circle to centre A in the figure, rolling it exter- 
 nally upon the circumference of the circle on B, it will give the outline of a tooth capable of fulfilling 
 the condition of uniformity of action at any point of the circumference of the circle on A. Should the 
 teeth engage before the line of centres, the tooth of the wheel would necessarily act constantly at that 
 point and cause great wear, particularly as this species of geering is usually employed in conveying the 
 motion of water-wheels where it is exposed to incessant wetting and vibration. 
 
 In practice it is found sufficiently accurate to substitute for the epicycloidal method above described 
 for finding the form of the face of the tooth of the pinion, the more convenient mode of defining it by an 
 arc of a circle drawn from the root of the tooth, with a radius equal to the chord of the pitch. 
 
 "When the teeth do not engage until they arrive at the line of centres, the faces of the teeth of the 
 pinion may be entirely suppressed, leaving the flanks only, which are alone acted upon by the teeth oi 
 the wheel. This may be uniformly done, at least when the pitch of the pinion is small, and when the 
 wheels are more than strong enough for the power which they are required to transmit. In such cases 
 the following method of setting out the teeth of the pair may be adopted with safety. 
 
 In Fig. 2068, let A be the centre of the wheel, and B the centre of the pinion as before. From tne 
 point c of contact of the two pitch circles mark off the arcs ca, cb, equal respectively to the pitch of the 
 pinion and wheel ; join a B and h a, and bisect this last ; from the point of bisection raise a perpendicu- 
 lar, and the point d, where this meets the pitch circle of the wheel, will be the centre from which the 
 arc a b, defining the side of the tooth of the wheel, may be struck. The flank of the tooth of the pinion 
 is determined by the two radii which limit its thickness and meet in B. To define the length of the 
 eeth, it is sufficient to describe a circumference with the radii A r, B w, leaving sufficient bottom clear 
 4 
 
50 
 
 GEERING. 
 
 auce ; and the extremities of the teeth of the pinion, and the bottoms of those of the wheel, ought to be 
 rounded off by arcs drawn with a radius equal to the chord of the pitch taken upon the pitch circle of 
 the pinion. 
 
 By this method it will be seen that the strength of the pinion-teeth is greatly reduced, and ought 
 therefore only to be applied when the power to be transmitted is comparatively light, and when the 
 wheel drives the pinion. It is consequently not recommended for general practice. The epicycloidal 
 method, explained for external wheels, may, on the contrary, be applied in all cases, observing that in 
 the case considered, the sjpace answers to the tooth, and conversely, the tooth to the space ; so that 
 keeping this distinction in view, the forms of the teeth may be determined with as much readiness as if 
 the pair were of the ordinary kind of spur-geer. The teeth may also be set out with the Odontagrapb, 
 if that instrument be in use in the pattern shop. 
 
 There is one property of the internal pinion which may be here pointed out, namely, the diminished 
 friction which attends its working, as compared with an external pinion of the same diameter. To 
 render this obvious, let P P, Fig. 2069, be the pitch circle of the wheel, Q Q the pitch circle of an internal 
 pinion, and Q' Q' that of an external pinion of equal diameter. Then supposing the wheel to drive, and 
 that it moves until the point a arrives at h ; by this advance, the same point of the pinions will have 
 arrived respectively at d and c of the circles Q Q and Q' Q', having both moved over a space equal to 
 ah. Now it is manifest that the distance from & to c is less than the distance from 6 to c?; but these 
 distances may be assumed as the measure of the amount of slide of the teeth of the respective pinions 
 in moving to their new position ; for whatever may be the actual amount of slide, it must evidently be 
 in the ratio of bciohd, since these quantities measure the velocity of the sliding contact of the pair ; 
 and since, when the circumstances are otherwise identical, the friction is as the space passed over. It 
 is, therefore, evident that the internal pinion has less friction than the external one ; and the same figure 
 may be employed to show that it has a larger arc of contact with the wheel, and is, therefore, other 
 things being equal, stronger than the external pinion. 
 
 Teeth of bevel-wheels. — When a cone is made to revolve upon the surface of another fixed cone, in 
 Buch a manner that their summits always coincide, the curve which is generated by a point in the cir- 
 cumference of the rolhng cone is a kind of epicycloid, which will plainly lie on the surface of a sphere 
 whose centre is the common apex of the cones, and whose radius is equal to the slant side of the rolling 
 cone. From this property the curve is termed the spherical epicycloid; and if the cone roll on the 
 eoncave surface of the base, the curve produced is denominated the spherical hypocycloid. 
 
 To apply this definition, let there be two cones in contact along their slant sides A c. Fig. 2070 ; and 
 ' let c A a be any other cone, having its line of contact coinciding with A c, and having its apex at A ; the 
 raxes of these three cones will then be in the same plane A R B. The cii-cumferences will also be at the 
 ~«ame distance A c from A, and wiU lie on the surface of a sphere whose centre is A and radius A c. 
 
 Now, suppose the three cones to revolve above their respective axes with the same relative velocity 
 
GEERING. 
 
 61 
 
 as -vrould be produced by the rolling contact of their surfaces, it is clear that the line of contact will 
 oonstantlj be Ac, and calling a Ac, Fig. 2070, the describing cone, a line db upon the circumference of 
 the surface of this directed to the common apex A, will generate one surface / 6 c?^ on the outside of the 
 cone A cm, and another surface dehb on the inside of the cone c Ar. These surfaces will touch along 
 the describing line db; for since ft db is generated by the rolling of the generating cone on the cone 
 ?n A c, the motion of db is at every instant perpendicular to the hne of contact A c, and therefore the 
 iiornijil plane at c? 6 to the surface generated by that line, must pass through the line of contact A c. 
 In like manner the normal plane to the surface hedb will pass through the line of contact of the cones, 
 and consequently the surfaces must touch along db. 
 
 Now, under these conditions, it is clear that if the rotative motion of the cone m- A c be communicated 
 by teeth to the cone c A r by their contact action, the teeth being generated by the describing cone a A c, 
 the motion will be the same as that produced by the rolling contact of the conical surfaces, for at the 
 beginning of the motion/^ and eh coincide with the line of contact Ac; and since the arcs cf, ce, de- 
 scribed by the bases of the cones, are, respectively, equal to c d, and therefore to each other. 
 
 In practice the portion of spherical surfaces occupied by these arcs, when employed as teeth, are 
 narrow belts of the cones extending only a small distance within the circle of the bases, and therefore 
 cones tangent to the sphere along c m and c r may be substituted for the sphere itself without 
 practical error. Thus draw E, c B perpendicular to the line of contact A c, and intersecting the axes of 
 the two cones at B and E, ; then B R revolving about A B will generate a conical surface tangent to 
 the sphere along the base c m ; and the same line R B revolving about A R wiU generate a conical 
 surface tangent to the sphere along the base c r of the cone c A r. And since the arc df, which really 
 lies in the spherical surface, is very short, it may be supposed, without sensible error, to lie in the sur- 
 face of the tangent cone c B m, and to be described with a circle whose diameter is equal to that of the 
 base of the describing cone. And, in hke manner, the arc d e may be supposed to be similarly de- 
 Bcribed upon the surface of the tangent cone c A r. 
 
 These principles furnish us with a ready mode of obtaining the form of the teeth of a pair of bevei 
 
52 GEERING. 
 
 wheels. Thus let A P and A Q be the axes of the shafts between which the motion is to be trans" 
 mitted. The angular velocities with which the two shafts are driven being determined, draw A D sucla 
 that it divides the angle P A Q into two parts, proportional to the numbers of teeth in the wheels 
 respectively, or inversely proportional to tlie angular velocities or numbers of turns per minute of the 
 wheels on the shafts. If we conceive the line A D to revolve about the axis A P, at the same angle 
 PA D, the conical surface A D E will evidently be described, a portion of which, D E e c?, is represented 
 in Fig. 2071. Similarly, by the revolution of the line A D about the axis A Q, the conical frustum 
 DF/c? will be described. It is obvious, therefore, that these two cones, communicating the motion by 
 simple contact, will transmit motion in the given ratio, and hence they are termed the primitive cones, 
 and have an obvious analogy to the primitive circumferences or pitch circles of spur-wheels. In the 
 interior of their mass the flanks and spaces are conceived to be formed, and upon their surfaces are 
 placed the projecting faces of the teeth — the returning surfaces E m and F w of the cones, or the crowns, 
 as they are termed, are defined by inverted conical surfaces formed upon the bases D E and D F ; they 
 are short frusta of the cones B D E and CDF, the common apex of which is determined by the line 
 BD C drawn at right angles to AD. To perceive the propriety of this form of the crown, it is only 
 necessary to consider that the ends of the teeth E m and F n will thus always be square with the sur- 
 face of the primitive cones, and they will geer most completely into each other. The other ends of the 
 teeth are defined in like manner by the conical recesses ehd and dcf, which are parallel to the exterior 
 surfaces of the crowns ; the line bdc being, like the exterior line, at right angles to A D. 
 
 It appears, then, that the true forms of the teeth may be described with facility upon the surfaces 
 of the cones DEB and D F C ; and for our present purpose of representing the forms of these teeth 
 upon a plane surface, it is only necessary to conceive those surfaces spread out or developed upon a 
 plane. Now, the development of a cone, that is, the form assumed by the superficies when laid out 
 flat, is a sector of a circle, of which the radius is equal to the slant height of the cone, and the arc equal 
 to the circumference of the cone at the base. In the case before us, therefore, if a circular arc D G- be 
 described from the centre C with the radius C D, it will be the development of a portion of the base, 
 and the sector C D G will likewise be the development of a portion of the cone ; and by describing an 
 arc kg with the radius C y>: or C^, the annular space contained between kg and D G- will be the de- 
 velopment of the crown of the wheel. It is upon this annular segment, then, that the forms of the 
 teeth may be drawn precisely as if it was a part of a spur-wheel ; and if we suppose a piece of sheet 
 copper or other suitable material to be cut into the form of the teeth, and wrapped upon the crown of 
 the wheel, the outline which could thus be traced off would represent the true forms of the teeth of the 
 bevel-wheel. 
 
 The breadth D d being settled by the same rules as for spur-wheels, it is obvious that the teeth must 
 follow the taper of the cone toAvards the point A. Draw d k parallel to A C, and from the centre C 
 describe the arc k g ; then, as before, the segment Gkg will be the development of a portion of the 
 cone dcf, and upon this segment the forms of the upper ends of the teeth may be described. Lines 
 drawn from the teeth on the arc D G to the centre 0, will determine the magnitude of the teeth on the 
 arc kg, and the teeth may thence be described, as in the figure. 
 
 The largest diameter D F is reckoned the diameter of the wheel, and, similarly, D E is said to be 
 that of the pinion. The process which we have detailed for describing the teeth of the wheel is pre- 
 cisely the same as that for describing the teeth of the pinion. It is unnecessary, therefore, to particu- 
 larize further, than that Fig. 2071 shows the teeth of the pinion drawn similarly to those of the wheel, 
 upon the arcs D H and i h. 
 
 That the teeth thus formed will work truly together, becomes obvious, when we reflect that if any 
 two exceedingly thin wheels, with the form of teeth described, had been taken in a plane* perpendicu- 
 lar to A D passing through the point D and having their centres in the axes of the given wheels, they 
 would work truly together, and their angular velocities would be in the given ratio. Now it is evident 
 that the portion oi each of the conical surfaces which is at any instant passing the line ^ i is at that 
 instant revolving in the plane perpendicular to A D which passes through the point D, the one surface 
 revolving in that plane about the axis A C, and the other about the axis A B. Those portions of the 
 teeth of the wheels which lie in these two conical surfaces will, therefore, drive one another truly, at 
 the instant when they are passing through the line k i, if they be cut of the form which they must have 
 had, in order that they might drive one another truly and with the required angular velocity, had they 
 acted entirely in the plane perpendicular to A D and about the given axes. But this is precisely the 
 forms in which the teeth are supposed to be cut, and therefore those portions of the bevel teeth which 
 lie in the conical surfaces will drive one another truly at the instant they are passing through the line 
 k i ; and, therefore, they will drive one another truly through a small arc on either side of that line, 
 which is the condition required, since it is only through an exceedingly small distance on either side of 
 that line that any two given teeth remain in contact. It therefore follows, that those portions of the 
 teeth which lie in the conical surfaces Df, D e, work truly with one another throughout the whole 
 breadth of the conical belt. 
 
 If the radius of the base of the frustum be called R, and the radius of its developed circle be r ; also if 
 
 the semi-angle CAD of the rolling cone = d ; then r = -. Hence the action of the teeth in any bevel- 
 
 ^ ^ cos 
 
 ■p 
 
 wheel is equal to that of a spur-wheel whose radius = ; and if N be the number of teeth in the 
 
 ^ ^ cos 
 
 bevel-wheel, will be the number of those in the spur-wheel of equal action. Thus if = 50°, 
 
 cos r ^ 
 
 •p -] 
 
 then = -—=: 1*556, and therefore the action of a bevel-wheel of 50*^ is fully equivalent to 
 
 cos 9 -64279 "^ ^ 
 
 that of a spur-wheel having a radius one-half greater, and consequently a half more teeth 
 
 54 
 
GEERING. 
 
 53 
 
 Tliis, then, is a reason for the superior action of bevel-wheels, as compared with spur-wheels of the 
 same number of teeth ; for spur-wheels always act the better the more teeth they have, and it appears 
 that the bevel-wheel is always equivalent in its action to a spur-wheel of a greater radius, and conse 
 quently a greater number of teeth. 
 
 When a pair of bevel-wheels have equal numbers of teeth, and have their axes at right angles to 
 each other, they ai-e termed mitre-wheels. In this case d =45° and cos = '707 11 ; therefore taking 
 
 radius equal unity, we have = 1-4:; in words, the action of a meter- wheel is nearly equivalent to 
 
 cos Q 
 
 tliat of a spur-wheel, with half as many teeth. 
 
 Skew-bevels. — When the axes ax-e inclined to each other, and yet do not meet in direction, and it is 
 proposed to connect them by a single pair of bevels, the teeth must be inclined to the base of the frusta 
 to allow them to come into contact. 2072. 
 
 To find the line of contact upon a given frustum of the tangent- 
 cone; let the Fig. 2072 be the plane of the frustum; a the centre. 
 Set oS ae equal to the shortest distance between the axes, (called the 
 eccentricity,) and divide it in c, so that a c is to e c as the mean radius 
 of the f>-ustum to the mean radius of that with which it is to work ; 
 draw cp perpendicular to a e, and meeting the circumference of the 
 conical surface at m ; perform a similar operation on the base of the 
 frustum by drawing a line parallel to cm and at the same distance a c 
 from the centre, meeting the circumference in p. 
 
 The line pc'is then plainly the line of direction of the teeth. We 
 are also at liberty to employ the equally inclined line cq in the op- 
 posite direction, observing only that, in laying out the two wheels, 
 the pair of directions be taken, of which the inclinations corres- 
 pond. 
 
 Fig. 2073 renders tlds mode of laying off the outlines of the wheels at once obvious, 
 figure the line ae corresponds to the line marked by the same letters in Fig. 2072; and the division of 
 it at e is determined in the manner directed. The line c m being thus found in direction, it is drawn 
 
 In this 
 
 aidetmitely to d. Parallel to this line and from the point c draw e to e, and in this line take the centra 
 of the second wheel. The line cmd gives the direction of the teeth ; and if from the centre a with 
 rstdius at a circle be described, the direction of any tooth of the wheel will be a tangent to it as at c 
 
54 
 
 GEERING, 
 
 And similarly if a centre e be taken in the line e d, ar.d with radins ed=:c e a circlo be drawn, th« 
 direction of the teeth of the second wheel will be tangents to this last, as at d. 
 
 Having thus found the direction of the teeth, their outlines may be described exactl}) as in the case oi 
 ordinary bevel-wheels, and with equal exactness and facility. 
 
 Wheel and tangent screw. — This is often a very convenient and ready mode of reducing a high 
 velocity. 
 
 To determine the form af the teeth of the wheel and thread of the screw, it may be remarked that, 
 from the nature of the screw, the section of its thread, made by a plane passing through its axis, is 
 eveiywhere the same ; and that if a series of such sections of the entire screw be made by planes at 
 equal angular distances round the circle, a set of figures, resembling a double rack, will be obtained 
 alike in the number and form of their teeth, but in which the teeth will approach nearer and nearer to 
 the extremity of the screw. Now, while revolving, the screw successively brings these sections into 
 operation upon the whole teeth, producing exactly the same effect as if the screw were pushed endlong 
 without rotation, in the manner of a rack. But this supposition furnishes a ready mode of obtaining the 
 forms of the wheel-teeth and thread of the screw, upon the principle of the rack and pinion. 
 
 In Fig. 2074 a c is the line of centres of the wheel and screw. The screw is shown in section by 
 a plane cutting it in the line of its axis. Now, the screw being considered a driving rack, it passes a 
 tooth of the wheel during each revolution, and therefore the point a will, at the end of a revolution of 
 the screw, be the point b ; but the thread must necessarily have continued in contact with the tootli 
 while passing from a to b. Now, if it be required that the thread of the screw sliali be in contact with 
 the tooth at a point only, the tooth will be straight, and its obliquity equal to the pitch of the screw. 
 But if it be desired that the thread shall be in contact with the entire side of the tooth, the outline oi 
 the wheel-teeth must be different in every section perpendicular to the axis of the wheel. The required 
 form will be found by making such series of sections of the screw as proposed, and adapting the portioE 
 of the side of the tooth to that particular section with which it is intended to work. And since in every 
 eection two or even three teeth may be in simultaneous contact, the scrow may be in contact along the 
 entire side of those teeth. 
 
 2074 
 
 In practice, then, the form of the tooth must be deternamed for the respective sections of the screw, 
 as in the case of the rack ; and the thread of the screw may obviously take the same sloped form i? 
 great strength be required. 
 
 To find the diameter of a wheel, driven by a tangent-screw, which is required to make one revolution 
 for a given number of turns of the screw, it is obvious, in the first place, that when the screw is single- 
 threaded, the number of teeth in the wheel must be equal to the number of turns of the screw. Conse- 
 quently, the pitch being also given, the radius of the wheel will be found by multiplying the pitch by 
 the number of turns of the screw during one turn of the wheel, and dividing the product by 6-28. 
 
 When a wheel-pattern is to be made, the first consideration is the determination of the diameter to 
 Buit the required speed ; the next is the pitch which the teeth ought to have so that the wheel may be 
 in accordance with the power which it is intended to transmit ; the next, the number of the teeth in 
 relation to the pitch and diameter ; and, lastly, the proportions of the teeth, the clearance, length, and 
 breadth. 
 
 The size and proportions of the wheel being thus settled, the operation of constructing the pattern ia 
 ready to be undertaken. Let us, in the first place, assume that the wheel is ta be a radial gels with sia 
 
GEERING. 
 
 55 
 
 arms ; and here it must be premised, that the number of arms increases according to the diameter ol 
 the wheel — thus, with occasional deviation : 
 
 Wheels from IJ to 3 J feet diameter have 4 arms. 
 
 " from 3^ to 5 feet diameter have 5 " 
 
 from 5 to 8^ feet diameter have 6 " 
 
 " from 8:^ to 16 feet diameter have 8 " 
 
 " from 16 to 24 feet diameter have 10 " 
 
 It IS assumed in the numbers of this table, that the wheels are cast complete, the boss supporting thft 
 arms, and these last the rim with the teeth cast upon it. The cases of deviation are those in which the 
 wheels are very small, and cast with a continuous web, instead of radial arms, and where the wheels are 
 required to be particularly hght, and to possess great accuracy : it is in this case preferable to increase 
 the number of arms, so that they may be made thinner, and thereby incur less risk of the rim being 
 drawn out of form dm-ing the cooling of the metal. 
 
 In arm- wheels the rim is always built of segmental pieces, cut out of plank of convenient thickness, 
 to an external radius equal to the pitch radius of ihe wheel, which allows a depth equal to the flanks of 
 the teeth for di-essing. A templet or mould is, in the first place, made of thin, well-seasoned and dry 
 board, and by this tiie form of the building segments is drawn upon the wood from which they are to 
 be cut. Suppose that the rim is of such breadth that six thicknesses of wood are required to make it 
 up ; then four segments are cut of the requisite breadth and to the same internal radius, and two of 
 them to an internal radius less than the former by at least as much as the intended thickness of the rim. 
 With these segments the rim is to be built in the following manner : — 
 
 A circular plate of wood, with an iron centre on the back of it, usually with an internal screwed boss 
 Dn the centre of it, having the same pitch of thread as the thread of the screw on the projecting end of 
 the lathe-spiudle, is prepared and turned up truly on the face. A circle is also described upon it to 
 indicate the pitch line of the wheel, and to this circle the exterior arcs of the rough segments are to be 
 adjusted. 
 
 Thus prepai-ed, the first course of segments are planed true on one side, and fitted with sprigs upon 
 the plate, their ends accurately jointed together, glued, and sprigged. When the glue is in some de- 
 gree set, the plate is put upon the spindle of the lathe, and the segmental ring is turned true on the 
 flat side ; but the interior and exterior circumferences are left in their rough state. The plate is then 
 removed from the lathe-spindle — usually a simple spindle with not more than two speeds upon it. 
 Another course of the segments is now dressed with the plane on one side, and built upon the first. 
 This course is bedded with glue and sprigged to the first course, observing that the joints of this course 
 do not coincide with those of the first, but that all the former joints are covered by the whole wood o* 
 the last-laid course. The glue being allowed time to set, the plate is again put upon the lathe-spindle, 
 and the face of the ring is dressed as before, stiU leaving the circumferences in their rough state. 
 
 The segments of these two courses are of the same breadth, but the next course to be laid must have 
 the increased breadth necessary to form the web or feather of the rim ; this, as already stated, is usually 
 made equal in breadth to the thickness of the rim. This course avlII, therefore, project inwards over the 
 ring formed by the two courses already laid. In other respects, however, it is not different, and it is 
 laid and fastened in exactly the same manner as the second course. 
 
 Two courses are laid of these broader segments ; afterwards two others of segments of the same 
 breadth as the two courses first laid. This done, the building of the rim is complete, and a cross sec- 
 tion of it would present the appeai-ance shown by the part adbc. Fig. 2075, and an arc of it in elevation 
 
 2075. 
 
 2076. 
 
 Tilt ^ 
 
 c cl 
 
 is represented by the part A h, B «, Fig. 2076. In Fig. 2075 the courses are numbered in the order in 
 which they are supposed to be laid, the layers 3 and 4 being those which are intended to form the web 
 of the rim. 
 
 This process beiag completed, and the face-plate placed on the lathe-spindle, the exterior edge of the 
 rim and half tlie interior circumference, that is, the interior of the three courses rriarked 4, 5, 6, are 
 turned true to the required form. The rim is then taken off the face-plate and reversed ; but is now 
 fastened to the plate by screws passed from the back of the plate into the edge of the rim. In this 
 position the interior of the other half of the breadth of the rim is dressed, that is, the inner circumfer- 
 ences of the layers 1, 2, 3, when the rim is ready to have the arms fitted into it. 
 
 In being dressed, the web or feather formed by the two middle layers, 3 and 4, is diminished in 
 thickness, till at their edge they are together exactly of the same thickness as the arms which are to 
 be put into the pattern ; and a good practical rule is, to make these equal in thickness to seven-eig/ith^ 
 of that of tlie rim of the wheel. In breadth they ought to be equal, at least, to the breadth of the 
 rim. when this is calculated according to the rule of maximum strcngtli already pointed out, ana 
 
56 GEERING. 
 
 towards the centre should be increased to ^-ths of that breadth. More commonly they are made eqtiaj 
 in breadth to the rim, and tapered to -f ths of that breadth at their extremities. 
 
 The breadth of the arms being determined, they are to be fitted into the rim by a half-s^ieck upon 
 the web, which is done by checking out at six points, answering to the number of the arms, the breadth 
 of an arm in the web layer marked 3. The extremities of the arms are then checked to the same 
 depth, and fitted into their places with a little glue, and two short but strong screw-nails at each end. 
 
 The mode of fitting the arms together is dififerent in diff'erent shops ; but the easiest mode is to fit a 
 double arm in the manner described — that is, an arm extending through the diameter, and having a 
 half-check at the middle of its length cut to the proper angle to receive the second double arm extend- 
 ing, like the first, through the complete diameter. These being fitted in, two half length arms are then 
 inserted, butting into the wide angles left at the centre between the arms already fitted. Fillets are 
 next inserted to fill up and round off the acute angles, both at the centre and extremities of the arms ; 
 these are intended to give additional strength to the wheel, and to facilitate the extraction of the 
 pattern from the sand in the process of moulding. "Were the acute angles left open, the sand would 
 be ready to break down and give the wheel an uncouth and rough appearance, a result which is to be 
 guarded against in all kinds of moulding, and more especially in wheels. 
 
 The flat arms being thus fitted in, the next operation is to insert a centre, upon which the boss or 
 eye of the pattern may be at any time built according to the size necessary for the particular wheel to 
 be cast off the pattern. The centre is formed of two hexagons, one laid on each side, and built of six 
 separate pieces, or made of single solid blocks of half the depth of the rim, so that on each side the 
 centre projects beyond the plane of the rim by half the thickness of the flat arms. The centre is firmly 
 screwed together, and to the arms of the pattern ; and on one side — the side which is intended to be 
 uppermost in the process of moulding — a flat plate of iron, usually of the same form as the centre 
 itself, is indented and fastened with screws to the wood. This plate has a hole in the centre, intended 
 to admit an iron prong for starting and Hfting the pattern out of the sand. 
 
 The next operation is to attach the feathers of the arms. These are placed along the middle of the 
 flat arms, the planes of the two being at right angles to each other, so that their cross section at any 
 point is represented by -f"- The breadth of the feathers diminishes from the centre towards their 
 extremities, in exactly the same ratio as the flat arms ; at the centre they are half the width of the 
 rim, and consequently stand flush with the depth of the centre, but at the extremities wliich meet the 
 rim the width is reduced a sixth. I'iUets are fitted in to fill up the angles which the ends thus form 
 with the inside of the rim, for the reasons already stated. Small running fillets are likewise fitted 
 into the angles which they form with the flat arms. 
 
 The interior of the pattern may now be considered as finished ; and it may be remarked that it is 
 always desirable to have the arms completely fitted and secure, before commencing to turn tlie exte- 
 rior of the rim ; for should the rim be dressed on the outside previously to the arms being put in — 
 which is practised by some makers — it is hardly possible to prevent its springing, and being thrown 
 out of truth. By the process described, the arms are fitted in while the exterior of the rim is in its 
 rough state, and consequently, although it should slightly spring, the external and most iinportant cir- 
 cumference will be brought to the truth in the after process of turning. The rim is also stronger and 
 less liable to be thrown from the truth, while in the rough state, than it is after being finished to the 
 thickness. 
 
 After the arms are fitted into the interior of the wheel, and finished in the manner described, it is 
 taken from the face-plate ; a hard-wood plate, with a centre for the lathe-spindle, is screwed upon the 
 eye of the pattern ; it is then placed in the lathe, and the exterior of the rim is turned to the required 
 thickness. In this process it receives a very slight bevel — usually from an eighth to a tenth of an inch 
 in a foot — to allow it to draw cleanly from the sand in the process of moulding. About the same frac- 
 tion is allowed upon the diameter of the pattern for contraction of the metal in cooling. 
 
 The next part of the operation is to cut the dovetail seats of the teeth. The form of these recesses 
 is shown in Fig. 20^6, beneath the teeth. They extend completely across the breadtli of the rim, and are 
 intended to receive the dovetail pieces {hd, Fig. 20^75) on which the teeth are fixed. The dovetail 
 pieces are themselves made of hardwood, and besides being otherwise convenient, they serve to bind 
 the several layers of the rim firmly together. 
 
 These pieces are, in the first place, ^accurately fitted into the seats and marked, so that every piece 
 may be replaced inserted into the proper groove to which it was fitted ; the rough pieces of which the 
 teeth are to be formed — these being cleaned with the plane on one side — are then fitted, each with 
 a dovetail fastened by screws to its dressed side ; and being cut nearly to the proper length, they are 
 fitted into their places by the dovetails, which are made with a slight taper, so that they can be the more 
 easily driven in its seat, and taken ouf: again when required. The blocked teeth being thus all placed 
 and partially fixed upon the circumference of the pattern, this is again placed on the lathe for the pur- 
 pose of dressing off the ends and points of the teeth. This done, the pitch circle of the wheel is finally 
 marked off upon the ends of the teeth with a fine point. 
 
 The pattern is now ready to have the form of the teeth di-awn in, as represented in Fig. 3, by an 
 arc of the rim, at this stage, with the outhnes of the teeth also drawn. In this figure the pitch line is 
 denoted by the dotted circle pp, and the form of the teeth is denoted by the shaded portions of the 
 rectangular blocks. The various methods of obtaining the outline have already been fully explained. 
 
 To finish the pattern, it therefore only remains to dress off the teeth to the forms indicated, and to 
 fix the dovetails permanently in their beds. This last is easily effected by a film of glue ; and if the 
 teeth are to be finished by the wheel-cutting machine, it ought to be performed previous to placing 
 the pattern in the lathe for the purpose of dressing the ends and points of the teeth, and consequently 
 before pitching them. But when the teeth are to be finished by hand, the order of the operation is 
 that described and the most commonly practised. 
 
 In finishing the teeth by hand, they are removed in succession, by driving out the dovetails, one caJj 
 
GEERING. 
 
 57 
 
 at a time ; the unshaded portions of the block are then accurately dressed off by concav e and con ve:a 
 gage-planes adapted to the purpose ; the dovetail is then fitted into its bed with a film of glue ; and 
 the tooth is complete. Every tooth is treated in exactly the same manner, until the whole number be 
 dressed and fastened upon the circumference of the rim. But unless the operation of fixing the teeth 
 be performed with care, there is danger of twisting the rim and throwing it out of truth. — To guard 
 against this, the safest practice is to take always in succession the pair of teeth opposite to each othev 
 on the circumference : that is, having fitted a tooth on one side of the pattern, the tooth opposite to it 
 on the other side ought to be the next taken, so that any tendency of the glueing to twist the rim at 
 one point will be neutralized by an opposite tendency in fitting the next tooth. — The wheel pattern 
 may now be considered complete. 
 
 It is to be remarked, that in making the pattern of a cast-iron wheel, it is necessary to take into 
 account the nature of the material : the pattern must not only be of such a proportion of parts as to 
 be sufficiently strong, calculated by the series of the parts ; it must also be so proportioned that the 
 fluid metal, when poured mto the mould, shall set in every part as nearly as possible at the same 
 instant. For if the parts contain disproportionate quantities of metal, it is plain that the thinner parts 
 will cool more quickly than the others ; and as cooling is attended with contraction, the parts of the 
 casting will be put upon strain, from their contracting irregularly and faster at one point than at 
 another ; and if the irregularity be carried to a certain extent — and the limits are not wide — the cast- 
 ing, on being removed from the mould, will be found fractured through some of the thinner and first 
 cooled and fii-st contracted parts. Thus, one of the most common errors is to put too much thickness 
 into the boss of the wheel, and in consequence it not unfrequently happens that the arms are drawn 
 away from the rim. To avoid this, the metal round the eye ought never to be more, in arm-wheels, 
 than twice the thickness of the rim ; that is, equal to the pitch of the teeth, and this is quite sufficient 
 in strength to resist the driving of the keys in fitting the wheel on the shaft, provided this operation 
 be done with ordinary skill and care. 
 
 To prevent the bad effects of unequal contraction, the arms are sometimes made of a curved form. 
 - When the wheel is very light in proportion to its diameter, the arms are advantageously made with 
 a double curve like the old letter/, which admits extension to a certain extent, depending on the 
 elasticity of the material, in the direction of its length, without fracture. 
 
 The rules and directions wliich apply 
 to spur-wheels, likewise apply, with 
 slight technical modifications, to bevel- 
 wheels. The mode of laying down 
 the working drawings of a pair of this 
 kind has been already explained and 
 figured in relation to the manner of 
 obtaining the form of the teeth ; and 
 the proportion of parts may be depicted 
 in connection with the same drawing, 
 by substituting for the primitive cones 
 there given, a section of the pair as 
 represented in Fig. 2011. 
 
 Here A P is the common pitch line 
 of the pair, determined as before de- 
 scribed ; and B P, C P are the radii of 
 the wheels. The breadth of the rim 
 of the wheel is marked off from P 
 towards A, and through that point the 
 perpendicular b b' is drawn. Another 
 perpendicular to the same line is 
 drawn through the point P, and of 
 course parallel to b b'. Upon these 
 perpendiculars to the pitch line AP, 
 the lengths of the teeth are set off, and also the thickness of the rims, as indicated by the lines a o, 
 a'b'. The distance a a' will be the length of the tooth plus the bottom clearance ; the space inclosed 
 between a or a' and the pitch fine the length of the flank of the tooth ; and from the pitch line to the 
 dotted lines at a and a' is the measure of the addendum of the teeth. The thicknesses of the rim 
 and arms are in the same relation as for spur-wheels ; but in this case they are not placed opposite the 
 middle of the rim and boss as before, but entering on one side of the smallest diameter of tiie wheel, 
 with the feather entire and projecting outwards, so that a cross section of the arm complete would 
 present the appearance T, the horizontal line indicating the face-arm, or web, and the vertical piece 
 the feather of the same. The boss or centre of the wheel is built in the manner already described, and 
 of the same relative proportions, but altogether on the side which is furthest from the apex of the ideal 
 cone, of which the wheel is a frustum. The feather of the arm is inserted upon the same side, and is 
 of equal breadth to the face-arm. This is commonly sweeped by a double curve, as in the diagram. 
 Fig. 2077, but often it is simply tapered from the Vim till it meets the central boss, which, in well- 
 proportioned wheels, is usually equal to the breadth of the rim plus the thickness of the face-arm. 
 
 The mode of building the rim, which, with the exception of the teeth, forms the only practical differ- 
 ence between wheels of tliis class and spur-wheels, will be understood from Figs. 2078 and 2079, which 
 show the order of position of the segmental rings in plan and section. The segments of the ring 
 marked 4 being cut to the proper breadth, according to the thickness of the wood used, they aro 
 sprigged to the face-plate, and dressed on their exterior surface, as before explained. But in the oper- 
 ;ition of dressing, their internal edge is commonly cut away to the diameter at which the next course 
 
 B\- 
 
58 
 
 GEERING. 
 
 marked 2 is to be laid. This operation is not indeed essentially necessary, but it affords to the workman 
 a better guide than a mere surface line described with the required radius, and is done without trouble 
 
 The succeeding courses are sprigged and glued on in succession, every succeeding course being 
 increased in diameter according to the 
 angle which the side of the cone forms 
 with the axis ; in other words, accord- 
 ing as the bevel of the wheel is more 
 or less. Usually, the first two courses 
 we flush on their inside circumference, 
 IS shown in the sectional diagram of 
 Fig. 20^79. 
 
 The rim being thus built, is next to 
 be dressed on its inner circumference 
 to the diameter required. This done, 
 the face-arms are half-checked into the 
 radial part of the ring marked 3, ex- 
 actly as in the case of the spur-wheel 
 pattern. The centre and feathers of 
 the arms are inserted in succession, and 
 before the rim is removed from the 
 face-plate. But as soon as these oper- 
 ations are completed, a centre-plate is 
 fitted to the pattern, and concentric 
 with it ; the faceplate is then removed, 
 and the pattern is fitted on the lathe 
 
 by the centre-plate, for the purpose of having the face of the rim turned. This must necessarily 
 be true to the required bevel, and therefore implies additional care on the part of the workman 
 whose usual method is quite exact. In the first place, he turns down the edge of the course marked 
 1 to the required diameter; then having fixed for the apex of the imaginary cone a point or edge, 
 he applies a straight-edge from time to time during the process of turning, when it is evident that 
 so soon as his straight-edge coincides with the whole breadth of the rim, and at the same time touches 
 the apex and base of the cone, the operation of turning is complete, and not before this coincidence 
 is attained. 
 
 An easier method is, however, to turn the greatest and least diameters in succession — with which 
 the drawings furnish him — and afterwards to dress down the face of the rim until a short straight-edge 
 touches the two diameters at the same time, and consequently coincides with the face of the rim 
 throughout its whole breadth. If this be not deemed sufficiently exact, a mould may be formed from 
 the drawing of two slips of wood so joined together as to coincide at the same time with the face-arm 
 and face of the rim ; the greater of the two diameters being struck, as in the method first proposed, 
 it is evident that the mould will, on being applied, coincide with the lines on which it was found only 
 when these agree with the drawing. 
 
 The rim being finished by one or other of the methods described, the next part of the operation is 
 to make the dovetail-grooves which are to receive the dovetails of the teeth. These are cut with the 
 same attention to accuracy of taper as was used to determine the bevel of the rim ; for as the teeth 
 ought manifestly to diminish in thickness as they extend towards the apex of the cone, where, if con- 
 tinued to that point, they would vanish, it is consistent with the general character of the work, although 
 not essentially necessary, that the dovetails upon which they are to be fastened take the same tapered 
 form. As in the spur-wheel patterns, those dovetails ought to pass through and take hold of each of 
 the several courses of segments of which the rim is built, and thereby act as keys to bind them per- 
 manently together. 
 
 When the dovetails are fitted and the teeth fastened to them, they are firmly driven into their seats ; 
 and the teeth are dressed in the lathe on the ends and points to the required size. The external and 
 internal pitch lines, that is, lines coinciding with the surface of the imaginary cone of which the wheel 
 is a frustum, are drawn upon the ends of the teeth. Upon these lines the centres of the teeth are 
 marked off, next their thickness, and lastly their form is described either by the method of curves, or 
 by pattern teeth shaped according to the mode of development already explained. The dovetails 
 are then driven out, and the body of the teeth accurately dressed to the curves traced on the ends. 
 But this part of the operation ought to be performed in the successive manner directed for spur- 
 wheels, and with the same attention and precaution to prevent the twisting of the rim, which is almost 
 invariably the result when the teeth are fitted consecutively. 
 
 It very seldom happens that two castings are to be made from the same pattern with the same' sizo 
 and form of eye. It becomes therefore . necessary to have a ready mode of altering the size and form 
 of the centre according to the size and form of the eye wanted. This is provided for by making the 
 original and solid centre smaller than any boss likely to be required for a wheel of the particular pitch. 
 This allows of a temporary centre being built around the original one, of the particular size wanted ; 
 and as the eye is always taken out by a core, a. print of corresponding diameter is sprigged on to guide 
 the founder in the size and also in the setting of his core. 
 
 It is also necessary often to have the boss and arms more on one side of the wheel than the other 
 a case which cannot be provided for by making these loose. To modify the pattern for a casting of this 
 kind, the difference is made up temporarily, on the side to which the increase is desired, by pieces 
 simply sprigged to the proper arras and centre of the pattern ; the quantity to be removed from the 
 other side is then carefully marked off by chalk lines drawn upon the feathers of the arms and round 
 the centre ; and Avhen the pattern is moulded, the parts so chalked off are filled up in the sand. li 
 
GEERING. 59 
 
 the modification, liowever, be of such extent as to alter the position of the face-arms, it then becomes 
 necessary enl irely to inclose the arms with thin board, forming a species of box, of a depth determined 
 on the one side by the position of the arms and centre of the pattern, and on the other by the position 
 which thev are to occupy in the casting. The pattern being moulded in this condition, there is left 
 only large open spaces for the arms and centre ; and into these the new arms and centre are built by 
 cores formed by core-boxes made to the particular shape and size desired. This is perhaps one of the 
 greatest niceties of green sand moulding. 
 
 For wheels of small diameter and pitch, it is now becoming a general practice to use metal patterns, 
 of cast-iron and brass. In works where small machinery is made, and which are commonly provided 
 with wheel-cutting machines, the rim is usually made of sufficient thickness to allow the teeth to be 
 cut upon it, and yet to leave the proper thickness of metal for strength between the bottoms of the 
 spaces, and the wcb-plate which connects the rim with the eye-boss of the wheel. Sometimes also the 
 same method is adopted with wheels which are crossed-out. This method admits of greater accuracy 
 than that described for large wheels ; for before proceeding to cut the teeth of the wheel, the casting 
 may be accurately adjusted to a centre in the lathe, and have the strength of its parts at the same 
 time exactly proportioned to the pitch of the teeth. The centre wheel thus admits of being polished, 
 a process which not only miproves the appearance, but likev'se insures correctness of the work. 
 
 The mode of constructing the pattern of a mortise-wheel, diflers from the ordinary routine for toothed 
 patterns in httle more than the omission of the teeth. The rim is however made wider than that of a 
 toothed wheel of the same pitch, by twice the thickness of tooth ; and its thickness is double that ol 
 the ordinary toothed rim. 
 
 The manner of construction is this : The rim is built and dressed, and has the arms and centre fittpd 
 into it as aheady described. The dovetails are also fitted into the corresponding grooves in the rim ; 
 but these, instead of having teeth fixed upon them, carry simply their pieces to serve as core-prints to 
 guide the founder in laying in his cores to make the mortises for receiving the tails or tenons of the 
 teeth. These prints are made in length equal to the breadth of the teeth, and in breadth equal to 
 their thickness. When the pattern is moulded the prints leave their impressions in the sand, and these 
 are filled by cores of coiTesponding size and of a depth equal to the thickness of the rim. 
 
 The rim of the casting being carefully faced up, or dressed in the lathe, is ready to be cogged. The 
 cog-tenons are in the first place fitted tightly into the mortises of the rim ; the wheel is then put into 
 the lathe and the points of the teeth and ends of the tenons are dressed off in precisely the same way 
 as the wooden teeth of a wheel-pattern : the pitch hues of the wheel are then run upon them with a 
 point, and the curves of the teeth being described, the cogs are ready to be taken out and finished by 
 hand ; or the wheel may be put into the wheel-cutting machine, and cut to the form and pitch required. 
 
 Wooden cogs, it will be seen from the rules already given for the strength of the teeth of wheels, 
 ought to be thicker than iron teeth in the proportion of the cube of 14 to the cube of 12 ; the difference 
 is commonly made greater, and perhaps correctly, as the strain falls principally upon the necks of the 
 tenons of the teeth. This allows of the iron teeth of the wheel with which the cogs are to geer, to be 
 dressed to the exact form and thickness, and polished on their acting surfaces, to prevent injurious 
 abrasion of the wooden cogs. By dressing, the iron teeth are however diminished in strength, and 
 consequently a pair of wheels of the kind here described ought to be made fully more strong than 
 wheels of the ordinary kind. 
 
 W^hen wheels are beyond a certain size, it becomes necessary to cast them in segments, which are 
 afterwards united t^ form the complete wheel. The pattern of the rim of a wheel of this kind consists 
 of a single segment of which the required number of castings are made. The segment is provided at 
 the ends with the requisite means of connecting the parts together, and to the arms which are also 
 commonly cast separately, and bolted to the rim and centre. As the rim in this case may be cast ol 
 any degree of thickness without risk of injury to the wheel, it is commonly made with recesses to re- 
 ceive the ends of the arms, which are fixed in their places by keys, bolts, or both, according to the mode 
 of construction preferred by the maker. The segments are also sometimes, and advantageously, dove- 
 tailed into each other on the inside at the ends, and the arms are fitted also by dovetail-checks and 
 bolts to the middle of their length. This mode of fitting requires great accuracy of workmanship, but 
 when well executed it possesses a degree of neatness which cannot be otherwise attained. The arms 
 are fitted in the same manner to the centre. 
 
 Another mode is to cast each segment with an arm attached to the middle of its length, and to fit 
 these upon the centre in the way described. Thus if the wheel have eight arms, the rim will be com- 
 posed of eight segments, and the centre will be cast with an equal number of recesses to receive the 
 ends of the arms. 
 
 When the wheels again are of an intermediate size, too large to be cast in one piece, yet not larger 
 than will admit of their being cast in two parts, they are then cast in halves, each half complete in itself 
 from the circumference to the centre. These halves are then fitted together to form the complete 
 wheel. The manner of fitting is various. The common way is to cast each half with strong flanges 
 throughout the whole diameter, and to bolt these together. Wheels constructed in this way, must of 
 necessity have an even number of arms, so that the division may be effected through the middle of an 
 arm running through the diameter of the wheel. The two flanges, that is, the flange of each segment, 
 will form the feather of that arm, but double the thickness of the ordinary feather. 
 
 We have seen wheels put together in three pieces in this way very satisfactorily, but the difficulty of 
 fitting is then very considerably increased. It is also not uncommon to cast the segments with strong 
 flanges only at the rim and centre, and to trust to these alone to hold the segments of the wheel to 
 gether. This greatly reduces the work of fitting, and we do not see that sufficient strength could not 
 thus be attained. 
 
 Witli respect to the material of the patterns of wheels, it is only necessary to observe that the coni- 
 vam practice of using wliite pine for the arms and rim, and hard wood fur the teeth, and 'heir dovetails 
 
60 
 
 GEERING 
 
 is both economical in point of time and expense. But in all cases where the teeth are expected to 
 require no dressing after leaving the sand, it is of importance that those of the pattern be formed with 
 the utmost accuracy, and have a smooth surface. And it may be here observed, that unless in the 
 circumstances referred to, it is not desirable, in point of economy of wear, to dress the teeth, at least to 
 the extent of removing the cast surface, which is by far the hardiest and most durable part, and after 
 working some time, it takes likewise a smoother surface than can be given to the softer metal beneath. 
 The surfaces ought, however, to be cleaned of any imbedded sand, and excrescences which may be found 
 adhering to them when taken out of the sand ; and it is also of advantage to run-up, or clean the points 
 of the teeth in the lathe when the wheel is small. 
 
 The key-seats of the eye are usually cut by the slotting machine, and of a breadth proportioned to the 
 size of the eye, and not to the pitch or diameter of the wheel. 
 
 Shafting is a department of mill- work which embraces some of the most important considerationa 
 within the compass of practical dynamics. If on the one hand shafting be too light, it will be of little 
 importance that the wheels are accurately made, and proportioned to the power conveyed ; the tremor 
 and hobbling gait to which an overburdened condition of the shaft-geer invariably gives rise, will 
 speedily destroy their adjustment by irregular wearing of the teeth. The bush-brasses at the same 
 time suffer, and the evil is aggravated till ultimately a sudden failure at some important point sets the 
 whole at rest. On the other hand, if the shafting be made too heavy, an unnecessary expense is in- 
 curred in the construction ; and, what is usually of more importance, and in most cases of more serious 
 consequence, a waste of power is occasioned by the unnecessary friction and wear produced by the 
 superfluous weight at the journals and footsteps. Both extremes are thus to be avoided with equal care. 
 
 One important feature of our modern practice is the higher velocities at which the main shafts are 
 impelled. By this means the geering can be made lighter for the same power, and therefore more 
 durable, since the friction and inertia are diminished, and the impulsion thereby rendered more steady 
 and uniform. The power of the prime mover is therefore economized ; and if that power be derived 
 from steam, a corresponding saving is effected in the maintenance of the engine ; and if water power be 
 employed, a proportionally greater quantity of machinery can be impelled by the same waterfall. 
 Hence the principles which ought to guide the engineer in the construction of a system of shaft and 
 wheel geering is, to regulate the connections in such a manner that the inertia of the mass and the 
 friction of the motion may be reduced to a minimum, and to effect this purpose the velocities and 
 strength of the parts must be adjusted to one another, and to the speeds of the machines in which the 
 power is to be expended. 
 
 But before entering upon the investigation of the practical bearings of this general principle, it may 
 be necessary to glance briefly at the various kinds of shafting employed by the millwright and engineer. 
 As respects material, there is only the choice between wood and iron ; and the forms are chiefly cylin- 
 drical and square, but sometimes octagonal. When large wooden shafts are employed, as they some- 
 times still are, especially as axles of wooden water-wheels, they are commonly made of one solid log, 
 with gudgeons inserted into their ends, according to a method to be subsequently described. But the 
 scarcity of large timber has not unfrequently led to the substitution of shafts built of planks, and these, 
 when properly made and of sufficient strength, have been found little less durable, and much less expensive. 
 
 Iron shafts are of two varieties — forged and cast. Those of large size are commonly of cast-iron, 
 while smaller shafts and spindles are formed of malleable iron. The form most common to both is that 
 of a cylindrical solid ; but often they are square, sometimes octagonal, and occasionally hexagonal. 
 Cast-iron shafts are also not unfrequently made with ribs upon their peripheries, and then they are 
 ca\^Qdi feathered shafts — probably from the slight resemblance which they bear to the feathered part ol 
 an arrow. Cylindrical shafts of large size are also sometimes cast tubular, and they are then termed 
 hollow shafts. 
 
 The subjoined figures represent the forms of cross section most commonly adopted in shafts of the 
 respective materials, iron, malleable and cast, and wood. 
 
 2080. 
 
 Malleable iron, cast-iron, wood. 
 2081, 
 
 Cast-iron. 
 
 In respect of these forms, the cylindrical is in general to be preferred, not that it possesses greater 
 Btrength, as is commonly supposed, for the same weight of materials, but because it is the simplest and 
 the most elegant in appearance when mounted. Feathered shafts, as they are commonly made, although 
 their strength to withstand lateral pressure be augmented by the breadth of the feathers, are very rarely 
 free of tremor from want of substance between the feathers, and unequal distribution of the material 
 around their axes. It is, however, an error to assume, with Tredgold and others, that the circle is the 
 only form of section which gives to the axes tlie property of offering in every direction the same resist- 
 ance to flexure ; for it might readily be proved geometrically that the square section must offer the 
 
GEERING. 
 
 61 
 
 same resistance to flexure in the direction of its sides and diagonals ; and what is true of the square, is 
 equally true of a great number of other figures which may be formed by combining symmetrically tlio 
 circle and square. The hollow cylindrical form admits of being adopted only in shafts of the very 
 largest class, and for the same weight of material is greatly the strongest form that can be employed, 
 and the best adapted to resist the combined action of torsion and cross strain. It is difficult to make a 
 casting of a tubular form below a certain diameter with the necessary accm-acy ; and, therefore, in shafts 
 of small diameter it is much greater economy to make them solid. 
 
 The sections represented above. Figs. 2080 and 2081, must be imderstood to be those of the body- 
 parts of the shafts ; but the shaft is very seldom uniform throughout its length even when cylindrical, 
 and never when of any other form ; for whatever be the figure of the body-part, the journals must 
 always have a circular section, and every shaft must — as will be subsequently seen — have at least one 
 journal or gudgeon on wliich it revolves. If the shaft be one of a line leading the power to some dis- 
 tant point from the source of motion, it will hkewise have its extremities adapted to coitj^le with the 
 shafts between which it is placed ; and if intended to carry a wheel or belt-pulley, it is commonly pro- 
 vided with a boss on which the same is keyed. In very large shafts the boss may be replaced by four 
 snugs projecting from the periphery to receive the keys, and when the shaft is of cast-iron and square, 
 as shown by the section marked a, the part to receive the wheel takes frequently the form of section 
 marked b. 
 
 The following cut. Fig. 2082, illustrates the usual-form of a cylindrical iron shaft, on which a wheel la 
 to be keyed at the enlarged part 6, denominated the wheel-boss, and which is to be coupled to another 
 shaft at the part c, by what is termed a half-lap coupling, to be afterwards described. The journal is 
 marked a, and has the same diameter as the body-part of the shaft marked d. 
 
 2082. 
 
 In ordinary shading, the gudgeons are 7iow always formed of the same piece as the shaft, and turned. 
 But in ordinary water-wheels the axles are formed of cast-iron, usually hollow, with independent gud- 
 geons fitted into their extremities ; and when made of wood, the gudgeons must of necessity be inserted. 
 
 The drawings in Fig. 2083 show the general form of a hollow cast-iron axle for a water-wheel, with 
 two modes of fitthig the gudgeons. The form of gudgeon fitted into the end marked A is peculiarly 
 adapted to axles of this kind, on account of its simplicity. The metal of the shaft is here thickened 
 
 2083. 
 
 mternally to a distance equal to the interior diameter, and three ribs cast in it have key-seats cut in 
 thsir faces. Tlie gudgeon marked a has three arms cast or forged upon it, (according as it is made ol 
 cast or wrought iron,) of the same length as the ribs on the interior of the shaft ; and these being dressed 
 in the lathe at the same time that the gudgeon is turned, are correspondingly key-seated. A wrought- 
 iron hoop being driven red-hot upon the end of the shaft to prevent it from splitting, the arms of the 
 gudgeon are adjusted in their places and keyed tight, the keys having as little taper as can conveniently 
 be allowed for fitting. They are thus less likely to become loose in their seats. 
 
 Sometimes the gudgeon is provided with four arms, but it is commonly admitted that the mode de- 
 scribed is preferable, because with three arms the keys must all bear equally, whereas when four are 
 employed it is difficult to obtain a uniform tightness of the set. 
 
 The form of gudgeon shown at B is more applicable to a vertical than to a horizcftital shaft, although 
 employed indifferently with both. The shaft is in this case cast with a projecting flange round its ex- 
 tremity, of the same diameter as a similar flange cast upon the gudgeon. The interior of the end of the 
 aliaft is, in the first place, bored out truly cylindrical, and the flange is at the same time faced. The 
 flange of the gudgeon is likewise faced, and a piece cast on the back of it is turned to fit into the turned 
 part of the end of the shaft ; the two flanges are then brought together and secured in tight met.illic 
 contact by bolts and nuts. 
 
 In this form of gudgeon the pivot is sometimes formed of wrought-iron or steel, and fitted into the 
 flange and socket, which must of necessity be cast. This allows of the pivot part being replaced when 
 worn out, at less expense than if the whole consisted of one piece. 
 
 To render this a good and lasting arrangemcjnt, it would be necessary to make the socket of the 
 
62 
 
 GEERING. 
 
 gudgeon of considerable length, and to bind the t"WL flanges together by a hoop of malleable iron, em- 
 bracing both their circumferences, and put on red-hot. 
 
 When the shafts are of small diameter these contrivances are not usually employed; the part of the 
 gudgeon which enters the shaft is formed as a solid cylinder, in the surface of which the key-seats are 
 formed ; the cylinder fits loosely into the end of the shaft, and is made tight hv the keys, as in the first 
 method described. The shaft, with its gudgeons fitted, is then put into the mthe, and the pivots are 
 turned truly concentric with the body-part, and of the strength desired, the original diameters being 
 usually much greater to allow for any inaccuracy in the fitting. This is also sometimes practised with 
 shafts of the larger class ; but more commonly the gudgeon part alone is dressed in the lathe, and the 
 adjustment intrusted to the fitter, who can adapt his keys to obviate any small inequalities due to 
 the casting. 
 
 The form of gudgeon most approved of for wooden water-wheel axles is that known as the cross-tail 
 gudgeon, which consists of a pivot piece projecting from the centre of a cross, roughly represented by 
 f- in side elevation, and by 4" on end. The vertical pieces forming the tails are let into mortises cut 
 in the end of the shaft, and the pivot coinciding with the axis of the shaft, the mortises are filled up on 
 each side of the neck by pieces of wood. A strong malleable-iron hoop is then wedged tightly on the 
 end of the shaft, in order to fasten and retain the pivot of the gudgeon in its parallel position. 
 
 This gudgeon has also been made with only two cross-tails ; and in well-finished work the tails, 
 whether two or four, pass within a malleable-iron ring, which embraces the shaft at the point where 
 they project to the circumference of the shaft : this ring being made slightly less in diameter when cold, 
 is fitted in its place at a temperature of red heat, and contracting, embraces the tails and shaft tightly. 
 The second ring at the end is made of full size, and fixed by wedges. As an additional security, the 
 arms of tlie cross are sometimes fixed by screws, which pass through into the wood of the shaft. The 
 arms are in this case made thicker, and do not enter so far into the shaft. The screws by which they 
 are fastened are made of considerable length, and the distance being noted to which they will pass into 
 the timber, mortises are cut at these points large enough to receive nuts suitable to the bolts. The nuts 
 being dropped into their places, opposite to the screwed ends of the bolts, they are screwed up by a 
 spanner, and thereby hold the arms tightly in their place. 
 
 By these . and similar means the gudgeon can be made for a time quite fast ; but when it is consid- 
 ered that the direction of the stress which tends to loosen it is continually changing, and that such stress 
 being exerted on wood, a material which is comparatively very easy and permanently compressed, it is 
 not difficult to perceive that the fastening of gudgeons must have been a matter of much importance to 
 the older millwrights, and that with all their care and ingenuity they seldom succeeded in rendering 
 them firm, and lasting. Accordingly several modifications of the schemes described have from time to 
 time been resorted to with indifferent success. The best of these is undoubtedly that of Robert 
 Hughes, described in the Transactions of the Society of Arts, (vol. xxxi. p. 223.) " It consists in casting 
 the gudgeon with cross-arms, which fit into notches in an octangular box of cast-iron which has been 
 previously fixed on the end of the shaft." The contrivance is represented by Figs. 2084 and 2085, in 
 which " A represents (in section) a portion 
 of the end of a wooden shaft of an octangu- 2085. 
 
 lar form ; it is long enough to reach across 
 the pit in which the water-wheel works, and, 
 having a gudgeon at each end, is supported 
 and revolves upon them in proper bearings. 
 B B is the cast-iron box accurately fitted on 
 the end of the shaft, and being wedged tight, 
 prevents the wood from splitting as effectu- 
 ally as any hoops can do. Upon the end of 
 the box is a projecting flange a a, with four 
 notches to receive the cross-arms hh, dd of 
 the gudgeon C. This cross is firmly attached 
 to the box by four screw-bolts, which pass 
 through the flange and the ends of the cross-arms. The figure in section (in which the cross and box 
 are represented separately to show more clearly the mode of putting them together) explains a further 
 precaution which is necessary for strength ; and which consists in the inside face of the cross having 
 projections e e, which enter the end of the box, and keep the pivot truly in the centre, and prevent any 
 lateral strain on the bolts, which have thus merely to hold the gudgeon fast to the end of the box. 
 When a gudgeon, fitted according to this method, becomes worn out so as to require replacement, it 
 can be removed by takujg off the four nuts, and a new one applied ; the gudgeon also being of small 
 dimensions, the cylindrical part admits of being conveniently turned in a lathe, which is a considerable 
 advantage." 
 
 This gudgeon involves nevertheless a considerable deal of workmanship, and the real practical advan- 
 tage which it possesses over the common cross-tail gudgeon is simply, that it does not require the end 
 of the shaft to be impaired by mortising, and affords the means of renewing the gudgeon when the pivot 
 becomes worn out, with more facility and without injury to the shaft. 
 
 The stress to which water-wheel gudgeons are subject is generally of a lateral and simple character. 
 The gudgeons have manifestly all the weight of the shaft, wheel, and sometimes the water, to sustain, 
 and ought, therefore, to be made sufficiently strong for that purpose ; but they ought obviously to be as 
 little as possible increased in diameter beyond the required strength, and a proper allowance for wear 
 to insure durability. 
 
 In practice it is common to make the length of the gudgeon equal to the diameter. Establishing this 
 as a principle, from experiments on the strength of iron, we deduce the following practical rules : — 
 
 Find the weight in pounds to be supported by each gudgeon, extract the square root of the 
 
 2084. 
 
GEEKING. G3 
 
 number, and tliat root divided by 26 Tvhen the gudgeon is cast-iron, and by 2G ^'hen it is wrought- 
 ii-on, will give the diameter and length of the gudgeon from the shoulder to the extremity ei'pressed ia 
 inches. 
 
 Example. — The gross weight on the gudgeons of a water-wheel is 86,800 lbs. ; required their diameter, 
 the -gudgeons being cast-ii'on ? The gross weight being 86,800 lbs., that upon each gudgeon will be 
 43,400 lbs., and V^ 43400 = 208-3266'7, which divided by 25 gives 8*3336 inches for the diameter of the 
 gudgeon. The actual diameter of the gudgeons is in this case 8*5 inches, and they have worked in 
 cast-iron bearings for upwards of twelve years, the bushes being only once and lately renewed during 
 that period. 
 
 . \Yith resprct to the strength of shafts generally, there are different circumstances to be considered in 
 the calculation. It frequently happens, in examining the conditions under which the shaft is to be 
 placed, even when the stress is manifestly compounded of lateral pressure and torsion, that one of these 
 may be neglected in the calculation of the strength. Thus the eliect of lateral pressure is to produce 
 deflection, which must be provided against, and ought not to exceed y^ ^th of an inch in one foot of 
 length ; if, therefore, the dimension requisite to give this degree of transverse stiffness be greater than is 
 required for the twisting power brought on at the circumference, this latter strain may obviously be dis- 
 regarded ; and conversely, when the torsion is great and the lateral pressure insignificant, as in the case 
 of vertical shafts, the effect of the former only requires to be considered. And again, when the shafts 
 are long, and have the power and resistance acting at their extremities, it is not enough that the body 
 part of the shaft be sufficiently stiff to prevent deflection beyond the proper limit; it must have 
 sufficient dimensions to prevent twisting beyond a certain quantity ; and in estimating the effect of 
 torsion, it is not so much the shaft itself which ought to be considered as the journal, which is constantly 
 exposed to wear, and which is thereby rendered more liable to rupture. 
 
 In examining successively the amount and effect of these species of stress, and the circumstances 
 under which their intensity is developed separately and in combination, the first part of the inquiry 
 which natm-ally claims attention is the degree of strength requisite to withstand lateral stress. In this 
 inquiry it is not the ultimate transverse strength, but the stitihess of the shaft which claims attention ; 
 in other words, the resistance which the material offers to bending by its own and any superadded 
 pressure tending to deflect it from the straight line. Now, the shaft being square, if 
 
 d = the depth or side of the square in inches, 
 
 L = the length of the shaft in feet, 
 
 W = the weight or pressure upon it in lbs., 
 
 I z= the deflection at the middle of the shaft in inches, 
 
 M = the modules of elasticity ; then 
 1®. When the shaft is supported at both ends, and the stress is intermediate, • 
 
 432 L^ W 
 : or 
 
 ^=vv-^\ 
 
 Tiat is, in words — Multiply the weight upon the shaft in lbs. by the cube of the length expressed in 
 feet, and by 432, and divide the product by the value of M, multiplied by the assumed amount of 
 deflection 6 to be allowed : the fourth root will express in inches the depth of the side of the shaft wheo 
 the transverse section is square. Or the square root wiU express the area in square inches of the cross 
 section when the shaft has any other form. 
 
 This rule answers for any material of which the weight of the modulus of elasticity M is known. In 
 its general form it is, however, laborious, from the magnitude of the numerical quantities involved : fixing 
 upon particular values of M for the different materials to be employed, and also fixing the maximum 
 value of S, it may be greatly simplified. In shafting, as already remarked, the deflection should not 
 exceed y^th of an inch for every foot of length ; hence, 
 
 L For wood. — Taking M generally = 1,500,000 lbs. and S = inches. 
 
 Then for square shafts, d being the depth of the side of the square in inches, 
 
 '=-jr (^) 
 
 A.nd for round shafts, d being the diameter in inches, 
 
 '11 
 
 20 
 
 ^' = -^?r (B) 
 
 II. For cast-iron.— Taking M = 18,000,000 lbs. and 6= inches. 
 
 When the section is square, d being the depth of side, 
 
 ^=-412- (^) 
 
 vVhen the section is round, d being the diameter, 
 
 ^ = 1^ (D) 
 
 240 ^^ 
 
 IIL For loro^/5'A^^Vo7i.— Taking M = 24,500,000 lbs. and ^ = y^ inches as before. 
 
64 GEERING. 
 
 When the section is square and d the side, 
 
 -^^ (^) 
 
 \^en the section is round and d the diameter, 
 
 ^=-334- (^ 
 
 2°. — If the stress be uniformly distributed over the length of a square shaft it will produce the same 
 eflection of the shaft in the middle as fths of that -weight applied at that point. And if W be the 
 weight of the shaft itself, and it be otherwise unloaded, the deflection produced by its own weight 
 will be, 
 
 I. — For cast-iron shafts when square, 
 _ L* 
 
 ^ ~" 256000 d» ^^^ 
 
 And when the shaft is cylindrical and solid, 
 
 L* 
 
 ~ 150000 d"" ^ ^ 
 
 II. — In wrought-iron shafts when square, 
 
 i i^ (I) 
 
 336000 d" ^ ' 
 
 And when the shaft is cylindrical and solid, 
 
 _ ^* 
 
 ■" 198000 d"" ; ^ ^ 
 
 As an example of the application of these rules, let it be required to determine the deflection at the 
 middle of the length of a 3 inch wrought-iron round shaft of 15 feet length betwixt its bearings : 
 
 Here L = 15, and therefore L* = 50625, 
 c?=: 3, and therefore c?^ = 9 ; 
 
 L* 50625 ^ . , 
 
 Hence, h = ^ ^^^^ 70 = 7777; - = •03151 mch. 
 
 198000 c?^ 1782000 
 
 The converse of this case frequently occurs ; that is, having determined the diameter of the shaft, 
 and assumed a maximum amount of deflection to be allowed, what may be the distance between the 
 bearings ? 
 
 Let the maximum deflection be y|oth of an inch for every foot of length, that is, h = — — inches on 
 
 <^he whole length ; then from equation, 
 
 (G) ; L = V2560^ (K) 
 
 (H) L = Vr5007^ (L) 
 
 (I) L = V3360T^ (M) 
 
 (J) L = Vl980^ (N) 
 
 I'hus, let the diameter c? = 3 inches, as in the last example, the shaft being of wrought-iron and a solid 
 rylinder ; then from equation (N) we have 
 
 L = n/1980 X 9 = 26-12 feet, 
 
 the distance between the bearings, on the condition that the deflection of the shaft by its own weight 
 shall not exceed y^o^th of an inch for each foot of the length. This being 26 feet in all, the deflection 
 at the middle will be "26 •.'=.\ inch nearly, which is a safe allowance, in ordinary cases, on this length 
 of shaft. 
 
 Again, if the torsion be very small, and it be more convenient, and possibly more economical, to 
 adjust the diameter of the shaft to the limit of deflection than the length of the shaft, this being fixed, 
 then, assuming the value of I as before, we obtain from equation 
 
 (K) 
 
 A^^, <°^ 
 
 '^^ ■■ '-A^\ ^^^ 
 
 w '-^A^\ ^'^^ 
 
 m • '^^^'ImoI («' 
 
 Thiis, the length between tbf 
 
GEERING. 65 
 
 shaft in which the deflection shall not exceed 30 X '01 = -3 inch at the middle of the length ? In thia 
 case the rule is equation, (0,) whence, 
 
 d=y/< ■^—- I = N^ 10-59 = 3-2642 inches, 
 
 the side of the square, the area of the cross section of the shaft being 10*59 square inches. 
 
 The distances between the bearings given in these examples are not, however, such as would be 
 adopted in practice ; but simply show the limits in reference to the deflection of the shaft by its own 
 weight. The spring to which a shaft is always liable in working, from irregularities in the power and 
 resistance, even when its deflection is far within the limit prescribed, renders bearers in general not 
 more than 10 feet apart necessary to prevent vibration ; these are usually of a less substantial charac- 
 ter than those in the immediate vicinity of the working points. 
 
 III. It is sometimes found unavoidably necessary in a system of geering to overhang a wheel or belt 
 pulley, leaving only one end of the shaft supported,, while the stress falls upon the other. The equa- 
 tion for the stiffness witlxin the elastic limit is in this case, 
 
 _ 4:PW __ 6912 L» W 
 
 "SVhen L = the length of the shaft from the point of pressure to the ber».ring expressed in feet, and M 
 the weight of the modulus in lbs., d and 5 have the same values as before. 
 
 Now, in cast-iron, the value of M was before assumed = 18,000,000 lbs. ; therefore, when the shaft 
 is cylindrical, (the only case whicli it is necessary to consider) we shall have 
 
 I^ I^ (S) 
 
 1532 a 16 ^ ^ 
 
 \Mien 5 = inches. This equation determines the least diameter of the journal, which is never 
 
 greater than that of the shaft 
 
 Again, the value of M for wrought-iron is 24,600,000 lbs. ; hence, for round shafts of that material, 
 
 d< = lLJL = inL ' (T) 
 
 "* 2086 a 21 ^ ' 
 
 "When 5 = — -- inches as before. 
 
 As an example of this case, let it be required to determine the diameter of the journal of a mallea- 
 ble iron shaft on which a belt pulley is overhung, at a distance of 2^ feet from the jommal ? Let the 
 weight of the pulley be 180 lbs. and the tension of the belt, when driving, 320 lbs. 
 
 Here L = 2-5 feet, and W = (180 -f 320) lbs. = 600 lbs. 
 
 Hence, <i'=(^^^ = 148-81, 
 
 And d= y/ v'14881 = ^12-21 = 3*49 inches, the diameter required. 
 
 In this equation it is presumed that no flexure of the shaft takes place beyond the journal, and, to 
 fulfil that condition — which is necessary to save the bearing, and avoid undue friction — the shaft must 
 either be made very strong or have only a short distance between the first and second bearings. Thus, 
 suppose a pulley overhung on the shaft, to prevent the shaft from springing between the proper bear- 
 ings, an intermediate bearing is placed at a short distance behind the bearing nearest the pulley, which 
 prevents upward flexure, and thereby saves the pillows adjoining the stress. It may also be here ob- 
 served that the distance between the bearings which our rules seem to allow are greater than is com- 
 monly practised in modern mill- work ; steadiness of motion is much desired, and, as friction is not 
 increased by the number of bearings, provided these be properly made and fixed, it is reckoned good 
 economy rather to multiply the bearings than to risk even the small amount of deflection which we 
 have taken as a safe limit 
 
 We have hitherto referred only to two forms of shafts — square and round ; and these solid. But the 
 same rules may be considered to apply generally to any other forms of solid section ; for knowing the 
 side of a square shaft of a given strength, the cross sectional area of any shaft of equivalent strength is 
 thereby approximately determined, however different in form, 
 
 Mr, Tredgold's rule for the strength of hollow shafts within the elastic limit, the cylinder being sup- 
 ported at the ends, is 
 
 500 (1 — N^) f ' 
 
 and is thus expressed verbally : Fix on some proportion between the diameters ; let the external diam- 
 eter be to the internal as 1 is to IS" ; the number N will always be a decimal, " and ought not to ex- 
 ceed 0-8." Then multiply the length L in feet by the weight W to be supported in lbs. Also, multi- 
 ply 500 by the difference between 1 and the fourth power of N", and divide the product of the length 
 and the weight by the last product, and the cube root of the quotient will be the exterior diameter d 
 of the shaft in inches. The interior diameter will be the number N multiphed by the exterior diameter, 
 and half the difference of the diameters will be the thickness of metal. 
 
 As an example of the application of this rule, let it be required to find the dimensions of a hollow 
 shaft for a water-wheel, which, including the weight of the water in the buckets, weighs 44,800 lbs, 
 the whole length of the shaft is 8 feet, from which, deducting 6 feet, the width of the wheel, leaves 3 
 feet for the length of tlie bearing : required the exterior and interior diameters of the shaft ? 
 5 
 
 ^i 
 
66 GEERING. 
 
 Making N" = -1, its fourth power is -2401 ; and 1 — •24 = -76. Therefore, we have by the rule 
 
 — ^- — -— — =354 in the nearest whole number; and the cube root of 354 is T, the exterior diameter 
 
 n inches ; and 7 inches X '7 = 4-9 inches, the interior diameter. 
 
 When the shaft is supported at the ends, and the stress is not in the middle of the length, but at 
 distances p and g from the respective ends, the rule becomes. 
 
 V ( 500 L X (1 — N*) j 
 
 Thus, let the weight of a wheel and other pressure on the shaft be 36,000 lbs. = "W ; the distance of 
 the point of stress from the bearing at one end being 3 feet =p, and the distance from the other bear- 
 ing 1-5 foot = q ; and let IS" = -8 ; required the exterior and interior diameters of the shaft ? 
 The fourth power of -8 is -409 ; and 1 — N* = -591. Therefore, by the rule 
 _ 4 X 36,000 X 3 X 1-5 _ 
 
 500 X 4-5 X -591 ' 
 
 and the cube root of 485 is 1-SQ=d, the exterior diameter in inches; and 1-86 inches X •8 = 6-3 
 mches, the interior diameter. These rules give the strength of hollow shafts within the elastic limit, 
 but when the deflection is restricted to a given amount, the diameter must be determined from the rule 
 
 d = diameter of solid shaft 
 
 the diameter of a hollow shaft of the same stiffness as the shaft of diameter d. 
 
 The second species of stress to which shafts are liable is torsion. The question of torsion divides 
 itself into two cases as apphed to shafts ; for, if the shaft be very short, the amount of twist will only 
 be limited by the ultimate strength of the material, but in ranges of shafting of considerable length, 
 the angle of torsion must necessarily be confined within certain limits, depending upon the degree of 
 accuracy of motion, requisite in the particular instance. In the first case it is usually the strength of 
 journal which is to be determined, for the journal being of suflficient diameter to resist the force applied 
 to it tangentially, the body part of the shaft will be in no danger of rupture from that cause, since it 
 has at least equal strength, and is not subject to wear ; and, moreover, the ultimate strength must ob- 
 viously, as respects torsion, be independent of the length — provided, indeed, the length be not less than 
 the diameter of the shaft. For shafts of which the lengths are small, and in which the angle of torsion 
 may consequently be neglected, we have the following rules ; 
 
 I. For solid shafts to resist torsion. — Equations (S) and (T). Find the pressure in lbs. acting upon 
 the shaft at the circumference of the wheel or pulley receiving or transmitting the power : multiply 
 that pressure by the radius of the wheel or pulley, that is, by the leverage at which it acts ; if the shaft 
 be cast-iron, divide that product by 125, but, if malleable iron, divide it by 140 ; and extract the cube 
 root of the quotient ; this root expresses in inches the diameter of shaft corresponding to the given 
 pressure and leverage. 
 
 II. JPor hollow cast-iron shafts, the thickness of metal being l-5th of the diameter — Equation (W). 
 Find, as before, the product of the pressure and leverage: divide that product by 109, and take the 
 cube root of the quotient as the required diameter in inches. 
 
 For particular cases in which a different thickness of metal is employed, the equation, of which the 
 rule is only another form of expression, must be reduced to find the proper divisor. Thus, supposing 
 the thickness of metal to be fixed at l-'Zth of the diameter, then 
 
 d—2t 5 , ^ 625 
 = - ; and n* = 
 
 d 7' 2401' 
 
 1776 
 therefore, 125 (1 — n") = 125 X tj^ = 92-5 ; 
 
 which must be taken as the divisor instead of 109 when the thickness of metal is reduced from l-6tli 
 to l-7th of the diameter of the shaft. 
 
 These rules are necessary and sufficient to determine the strength of short shafts to resist twisting, 
 and, consequently, ought to be employed in calculating the strength of journals ; but in shafts of great 
 length in comparison with their diameters the angle of torsion becomes an important element in the 
 investigation. 
 
 N'ow, if the extension which the material will bear by twisting without injury when the length is 
 taken as unity be assumed, 
 
 For cast-iron . = — — — 
 1200 
 
 For wrought-iron = 
 
 then the value of in our fundamental equation, that is, the angle of torsion, will be, 
 
 T. .. - 2284 L 
 
 For cast-iron . 
 
 For wrought-iron i 
 
 1000 d 
 1965 L 
 
 1000 d 
 
 in which L is the length of the shaft in feet; d the diameter in inches; and d the angle of tcrsion in 
 degrees of a circle. 
 
GEERING. 67 
 
 The rule, moreoTer, indicates the condition that the angle of torsion is as the length directly, and the 
 diameter inversely; and may, therefore, be adjusted with precision to the degree of accuracy with 
 which the motion of the fii'st mover must be transmitted. 
 
 Taking the modulus of elasticity of the two varieties of iron at the same values as before, our funda- 
 mental rule becomes, for round shafts of 
 
 cast-iron and sohd d* = 
 " and hollow d* = 
 
 LR W 
 53-0 d 
 LR\Y 
 
 wrouffht-iron and solid d* = 
 
 53-5 0(1—71*) 
 LR W 
 
 and hollow d* = 
 
 71-3 
 LR W 
 
 (S) 
 
 (T) 
 
 71-3 0(1 — 71*) 
 
 The data usually attainable in practice are the power which the shaft is required to transmit, the lever- 
 age at which the power acts, and the length of the shaft ; it then remains to fix the degree of torsion 
 which may be permitted without injuriously affecting the regularity of the motion : these quantities 
 being settled, the rules expressed by (S) and (T) for sohd cylindrical shafts are thus applied : 
 
 Multiply tiae power in lbs. by the leverage at which it acts, and by the length of the shaft, both in 
 feet; divide the product obtained by 53-5 or 71-3 times the number of degrees in the angle of torsion 
 allowed, according as the shaft is of cast or wrought hon ; and the fourth root of the quotient will be 
 the diameter of the shaft in inches. 
 
 Thus, by a line of shafting of 40 feet length, the power transmitted is 500 lbs. acting at the circum- 
 ference of pulleys of 1 foot radius : required the diameter so that the angle of torsion shall not exceed 
 2 degrees at the extremity, the shafts being wrought-iron ? 
 
 Here 500 X 40 X 1 = 20000 ) , 20000 _ 
 
 and 71-3 X 2 = 142-6 f ^^^^ 142-6 —^^^-^^ 
 And the fourth root of 140-25 is 3-44, so that the shafting would be made 3^ inches diameter — which 
 is the actual case. 
 
 The rule for hollow shafts differs from that given only in this, that the thickness of metal in i elation 
 to the diameter must be assigned as before explained ; and for square shafts, the rule differs only in 
 having different coefi&cients of the angle of torsion. 
 
 These rules are sufficient to meet all the cases of calculation which occur in practice relative to the 
 strength of shafts ; but the mode of expression often causes considerable previous computation to 
 determine the value of the power transmitted in lbs., as here requked. The more common and conve- 
 nient mode is, therefore, to estimate the strength and sizes of the shafting by the horse-power trans- 
 mitted, and the velocity. According to this measure, from what has already been explained in respect 
 of velocity, it wiU be perceived that the resistance must be estimated as the horse-power directly, and 
 as the number of revolutions inversely, since, with a given power, the velocity must be greater, as the 
 resistance is less. Thus, the resistance due to 32 horse-power on a journal making 50 revolutions 
 per minute will strain it to the same, and only to the same extent, as the resistance due to 16 
 
 horse-power acting on a journal which makes only 25 revolutions in the same unit of time ; for 
 09 1 g 
 
 — = — ='64. And, in all cases, when the horse-power, divided by the velocity, gives the same quo- 
 tient, the stress is the same. 
 
 If the rule contained in equation (S) for cast-iron shafts be reduced to the notation corresponding to 
 the dynamical imit of power, it assumes the form, 
 
 <i'=^x^^ (U) 
 
 71 
 
 in which H represents the number of horse -power transmitted by the shaft, and n the number of turns 
 which it makes in a minute ; L and are respectively the length and angle of torsion expressed in feet 
 and degrees as before. From the rule in equation (T) for -wrought-iron shafts we have the form, 
 
 -=!x^ (VO) 
 
 Tliese rules -vvill apply to the ordinary kinds of shafting, and where it is necessary to keep the amount 
 of torsion in view ; but, in cases where less exactness is required, the simpler rules furnished by 
 Robertson Buchanan may be adopted. These rules are intended to comprehend three classes, as — 1st 
 Steam-engine fly-wheel shafts ; 2d. Shafts in immediate connection with water-wheels ; and 3d. Shafts 
 for ordinary internal mill-geering. The following are the rules : 
 
 1st class c7= Y/ j - X 400 
 
 2d class d= \/ S 5 X 200 ? 
 y/ I n ) 
 
 3d class d— vV - X 100 
 V I n 
 
 These rules are stated to be derived from " observation '^f shafts in actual use, and of ackno-vrledged 
 
68 
 
 GEERING. 
 
 good proportions ;" and for short shafts, they express pretty nearly the practice of some of our best 
 millwrights ; but, in his last edition of the Essays on MQI-work, the rule for cylindrical cast-iron shafts i3 
 
 </\lx 
 
 240 
 
 Couplings. — Couplings are of necessity employed in a line of shafting when of greater length than ia 
 found practicable to cast or forge in one continued shaft. They aie also frequently required in cases where 
 one length of the shaft would suffice, for the purpose of occasionally disconnecting parts of the geering 
 beyond a certain point of the line ; and likewise for connecting and disconnecting particular machines. 
 
 The most simple species of connection is the square coupling. In this the end of the shafts to be 
 connected are made square, and are embraced by a square couplmg-hox, the internal surface of which 
 is fitted exactly to the squares of the shafts, A box is divided into two parts, which close together 
 diagonally upon the shaft, and are provided with flanges at their junction, by means of which they are 
 bolted together, and to secure the shafts, so that one cannot turn without driving the other. The 
 coupling is sometimes made quite plain ; embracing the shafts like the one now described ; and, when 
 occasion requires it, the box may be slipped back on one shaft, to leave it clear of the other, thereby 
 admitting of the motion being discontinued, or of one or both being removed for repairs or alterations. 
 
 An objection to this arrangement of a rigid square couphng at once suggests itself, when we reflect 
 that, although the motion would go on all very smoothly so long as the shafts remained mathemati- 
 cally true to each other, when the one of them wears down in its bearings faster than the other, or 
 when the wearing is in diff'erent directions, it must follow, that in some part of the revolution the shaft 
 is lifted off its bearings, where there are two bearings, one on each side, and unsteady motion is pro- 
 duced, together with further straining and wearing of the couplings. This objection, it is true, applies 
 principally to the shafts of heavy mill- work ; but it is only for such purposes almost exclusively that 
 this form of coupling is employed ; in small machinery it is only occasionally employed. 
 
 A variety of the square coupling consists in having fitting-strips, or projections along the comers of 
 the square parts of the shafts. The square form of the shaft is in this case virtually preserved, and 
 there is the advantage of the coupling being fitted with greater facility, while the strain is concentrated 
 upon the corners of the shaft. 
 
 The round coupling is, in some respects, open to the same objections as the square coupling. It will 
 be understood from Fig, 2086, which represents it in section. In this the ends of the shafts are made 
 cyhndrical, and meet together, with flat 
 ends, under a round coupling fitted truly 
 upon them, and secured to them by pins 
 a and h at right angles to one another. 
 This joint may be made with greater pre- 
 cision and much less expensively than the 
 other, as the parts may all be accurately 
 turned and fitted together. It is clear, 
 however, that, as the strain is concen- 
 trated on the pins and holes, these parts 
 must wear out soon ; at the same time, 
 it is easy to renew the pins, though, obviously, they cannot be made to fit the holes so accurately as 
 they did at first. This coupUng is not often used for heavy shafting, on account of the objection just 
 stated, though we think it might readily be applied to lighter shafts. 
 
 Fig. 2087^ is a rigid sleeve coupling for a cast 
 iron shaft ; it consists of a solid hub or ring of cast 
 iron, hooped with wrought iron ; the shafts are made 
 with bosses, the coupling is slipped on to one of the 
 shafts, the ends of the two are then brought to- 
 gether ; the coupling is now slipped back over the 
 joint, and firmly keyed. Ihis is an extremely rigid 
 connection. Fig. 2087^ is a screw coupling, a very 
 neat and excellent rigid coupling, for the connecting 
 of wrought iron shafts, more especially the lighter 
 
 kinds. It win be observed that this coupling admits of rotation but in one direction, the one tending to bring 
 the ends of the shafts towards each other, the reverse motion tends to unscrew and throw them apart, 
 and uncouple them. 
 
 Fig. 2087" is a clamp coupling for a square shaft. 
 
 2087' 
 
 V:^ 
 
 2037' 
 
 The coupling shown in figs. 2088 and 2089 is constructed of two cast-uron plates, a a and 6 &, keyed 
 one on the end°of each of the shafts. The plate a is foraied with two segmental openings in it, which 
 will be better understood from ^.g. 2089 at c c ; these openings are intended to receive correspondmg 
 
 55 
 
GEERING. 
 
 69 
 
 proiecti^ns on the face of the plate h, as represented in the section • and thus the shafts become 
 engaged. The rim of the plate a over- 
 laps and embraces the circumference 
 of the othei- plate, and thus they are 
 preserved exactly concentric. 
 
 This coupling, applicable to shafts 
 with two bearings, one on each side of 
 the coupling, is simple and durable. 
 It is easly adjusted, and may be dis- 
 connected without difficulty. 
 
 Fig. 2090 is another combination of 
 disks suitable for couplings with only 
 one bearing. The disk 6 is keyed on 
 one shaft, and is recessed on the face, 
 to receive the smaller disk c ; this disk 
 is sunk flush with the face of the other, 
 and is screwed tightly up to it by 
 means of the ring a, wliich is bolted 
 to the disk b, and secures that marked 
 c. Between the three plates, a, 5, and 
 c, annular pieces of leather are interposed, which bring them all to a proper bearing. 
 
 This combination, termed a friction coupling, is useful for preventing breakage of the connections in 
 case of a sudden stoppage or reversal of the motion. It 
 is plain that the holding power of the coupling depends 
 smiply upon the tightness with which the disks are screwed 
 together, and the consequent frictional force of the surfaces 
 of leather and metal. 
 
 Of late years, in this country, turned wrought-iron shafts 
 have been very generally adopted in manufactories and 
 workshops : the coupling in most common use for connection 
 is the faced couphng. Figs. 2091 and 2092. This coupling 
 consists of two parts, one of which being firmly keyed to 
 the end of each of the shafts to be connected, the faces are 
 then brought together, and securely united by bolts. In 
 fitting this coupling, considerable care is requisite. Each 
 part of the coupling is first turned and drilled, then driven 
 hot on to its shaft, reduced a little in diameter to receive 
 it, or forced on by screws. It is then strongly secured by 
 a key or keys to the shaft. The shaft is now put into a 
 lathe, and the coupling is faced ; that is, the faces of the 
 couphng which are to be bolted together are turned, so that they are perfect planes, at right angles 
 to the shaft. The bolt-holes on the two portions should correspond exactly. They are therefore drilled 
 on a chuck, with an index ; the holes are made slightly tapering, and the bolts, fitting tightly, are 
 driven in by smart blows of a hammer, and secured by nuts. Faced couplings, thus fitted, afford com- 
 
 2090. 
 
 2091. 
 
 plete and firm connections. The chief objection to their employment lies in their stiflfoess or rigidity 
 They are, therefore, mostly used to connect wrought-iron shafts, when the elasticity of the shafts 
 obviates the inconveniences resulting from little, and sometimes unavoidable, settlements or inequalities 
 of the bearin-gs. 
 
 Another form of permanent coupling is that known as Ilooke's Universal Joint, from the name of 
 
GEERING. 
 
 the inventor, Dr. Robert Hooke. The object of this coupling is to unite shafts which are inchned 
 to each other in the line of direction, and which do not, ti'.erefore, admit of being rigidly con- 
 nected, as in ordinary cases. This coupling is very commonlj' employed in light machinery, as in 
 steeple clocks, for taking off the index-motion, and is then usually constructed by forming an are 
 
 a 
 
 on the two extremities which it is intended to connect, and to form the joint by a central cross '+6 • 
 
 a 
 
 the extremities of the ark on the end of 
 
 one shaft being jointed to the arms a a, 
 and the extremities of the arc on the other 
 shaft on the arms b b, at right angles to 
 the former. But this simple mode of con 
 struction is not adequate to the purposes 
 for which the coupling is required in a 
 line of shaft-geering ; in this case, al- 
 though the principle is not in any way 
 changed, the construction is much more 
 substantial. Figs. 2093 and 2094 rep- 
 resent a form of it adapted to heavy 
 strains. A is a strong disk keyed on the 
 end of each shaft carrying a pair of 
 bearings for the reception of the gud- 
 geons formed on the extremities of the 
 cross. Fig. 2094 is a face view of one 
 of the disks, showing the cross in its 
 place, with its alternate joui'nals disen- 
 
 2093. 
 
 2094. 
 
 In this illustration the shafts are 
 shown at nearly the limit of the angle 
 to which the single joint ought to be 
 applied. This angle ought not exceed 
 tifteen degrees ; when a higher angle 
 is introduced, the rotatory motion be- 
 comes very sensibly irregular, and the 
 friction is greatly increased. This defect may be obviated by using a double joint. 
 
 With the view of admitting the disengagement of the connection, in cases of sudden steppage oi 
 reversal, the coupling. Fig. 2095, is sometimes employed. In this instance, the shaft is supposed to be 
 continuous, and the coupling may be termed 
 a disengaging coupling ; a and b are the two 
 parts of the coupling, formed on the acting 
 faces into alternate projections and recesses, 
 such as they correspond with, and exactly fit 
 into each other when in goer. The part a is, 
 in this example, cast on a. spur-wheel, from 
 which the motion of the ehaft is supposed to 
 be taken off. Both of the parts a and b ai-e, 
 to a certain extent, loose on the shaft ; the 
 former being capable oi moving round on it, 
 though deprived of lodgiiadinal motion by 
 washers and pins marked e, and the latter 
 being free to slide on the shaft, though pre- 
 vented from turning on it by a sunk key, 
 which slides in a slit inside the clutch or slid- 
 ing piece b. The mechanism is put into geer 
 by means of the handle d, which terminates 
 in 2^ fork with cylindrical extremities c, and 
 it is obvious, that by the contact of the flat 
 faces of a and 6, the latter will immediately 
 carry with it the other part at the same speed 
 as the shaft. Supposing, now, that the mo- 
 tion of the wheel a is suddenly accelerated, the oblique faces of the couplings immediately fall out of 
 contact, and slide free of each other, leaving the couplings clear, and the shaft free to continue in 
 motion. 
 
 In the old form of this contrivance, known as the sliding bayonet clutch, the part h, instead of the 
 tooth-like projections on the face, had two or more prongs which laid hold of corresjDonding snugs cast 
 on the face of the part a — which, moreover, was usually a broad belt pulley introduced with a view to 
 modify the shock on the geering on throwing the clutch into action. In an older form still the pulley 
 was made to slide end-long on the shaft. A form analogous to this was known as the " lock pulley." 
 Instead of the end-long motion common to the other modes, the parts were " locked" together by a bolt 
 fixed upon the side of the pulley, and which, when shifted towards the axis, engaged with an arm of a 
 cross, of which the part 6, in Fig. 2095, is the modern representative. The iDolt was wrought by means 
 of a key and stop, the turning of the key throwing back the bolt, and thereby unlocking and disengaging 
 the pulley. 
 
 The form of coupling represented by Fir. 2095 is particularly applicable when the impelling powei 
 
GEERING. 
 
 71 
 
 Is derived from two sources — a circumstance which frequently occurs in locahties affording water powei 
 to some extent, and yet not in sufficient abundance for the demands of the work. The deficiency it 
 usually supplied by a steam-engine ; and the two powers are concentrated in the main line of shafting 
 by a coupling of the kind depicted. In cases of this kind, the speed of the shafting being fixed, and the 
 supply of water inconstant, tiie power of the water-wheel ought to be thrown upon the wheel a a, and 
 that of the engine upon the shaft at another point. By tliis arrangement, the speed of the line can be 
 exactly regulated by >vorking the engine to a greater or less power, according to the supply of water. 
 The proper speed of the water-wheel will likewise be maintained, which is of importance in economizing 
 the water power. 
 
 The same form of coupling is also used occasionally for engaging and disengaging portions of tlie 
 machinery. But for this purpose the object is to obtain a mode of connection by which the motion may 
 be commenced without shock ; for it is a law in mechanics that when a body is struck by another in 
 motion, some time elapses before it is diffused from the point struck through the other parts ; conse- 
 quently, if the parts receiving the blow have not sufficient elasticity and cohesive force to absorb the 
 whole momentum of the striking body till the motion be transmitted to the centre of rotation, fracture 
 of the body struck must necessarily ensue. Hence, in a system of mechanism, any parts intended to 
 be acted upon suddenly by others in full motion ought not only to be strong, but they ought to be ca 
 pable of yielding on the first impulse of the impclhng force with as little resistance as possible, and 
 gradually bring the whole weight into motion. The common mode of driving by belts and pulleys 
 accomphshes this object very satisfactorily. In this the elasticity of the belt comes into action ; and 
 should this be inadequate, it has the liberty of slipping on the pulleys, till, by the friction between the 
 eliding surfaces, the belt gradually brings the quiescent pulley into full motion. This mode of connection 
 is unexceptionable when the power to be transferred is not great ; but its application to large machinery 
 is attended with inconvenience. 
 
 When the belt connection is employed, the provision of the fast and loose pulley is the most simple 
 and effective form yet devised for the purpose. It consists simply of two pulleys in juxtaposition on 
 the same axis — the one fast, and the other loose, so that the belt which transmits the motion may be 
 shifted at pleasure upon one pulley or the other, by that means putting in or out of motion the axis 
 upon which they are placed. The driving pulley — that is, the pulley on the shaft from which the mo- 
 tion is derived — ought to be equal in breadth to the two on the second shaft; and these last ought to 
 have their rims shghtly rounded or swelled in the middle to prevent the belt slipping off — which it is 
 apt to do when the rims of the pulleys are flat. This curious property is of great practical importance, 
 and has obviated all those clumsy appendages formerly required to keep the belt from being thrown. 
 
 This mode of driving is not, however, always convenient ; and, accordingly, many attempts have been 
 caade \o accomplish the same with wheels. Perhaps the best of these is the method of friction cones, 
 
 represented in Fig. 2096. The two parts of this coupling, c and d, are arranged on the shaft b, in the 
 same manner as we have described under the preceding figure, a is a shaft driven by means of bevel- 
 geer off the main shaft b, its motion being derived from the latter shaft through the coupling, c is an 
 interior cone east upon the back of the bevel-wheel ; d is an exterior cone having the same taper as the 
 cone c, such that by means of the handle e it may be moved into contact with the interior cone. The 
 surfaces being supposed to be well fitted to each other, the cone d will, by its friction, drive the cone c, 
 and thereby also the upright shaft. When either of the shafts a or 6 is accidentally stopped, the cone» 
 immediately fall out of geer, and the connection is broken. They are held in geer by means of a screw, 
 or more commonly, and perhaps better, by a weight. 
 
 This coupling works very well, if properly adjusted at first; but requires some nicety in communi- 
 cating the exact degree of taper to the cones ; for if, on the one hand, the taper be too small, they art 
 liable to adhere into each other too firmly, and on the other hand, if the taper be too great, they do not 
 possess sufficient frictional force to keep them in contact. 
 
 Another mode of accomplishing the same purpose in small machinery, by means of an epicyclic train, 
 is represented by Fig. 2097. In this the shaft A A is continuous, and supposed to be that through 
 which the motive power is transmitted. The wheel a is fast, but those marked b and c run loose on 
 
72 
 
 GEERING. 
 
 Ihis shaft. The two pinions c?c? have their bearings in the wheel c c, and geer with the tAVO opposite 
 bevel- wheels, a and b. (One of these pinions only is requisite to complete the motion ; the second being 
 introduced merely to maintain the equipoise of the system.) If now motion be given to the shaft A A, 
 it is clear that the wheel b, which is loose, will be made to revolve in the contrary direction to the wheel 
 a which is fixed, by means of the carriers dd; but no motion of the wheel c, if slightly opposed, wili 
 ensue ; and so long as this last remains at rest, the wheels a and b will have the same angular velocity 
 in opposite directions. But if the motion of the wheel b be opposed by means of a friction gland e, 
 which can be tightened by means of the T-screw marked / to any degree required, the teeth of that 
 wheel will serve as fulcra to the carrier pinions dd, which, becoming levers of the second kind, with the 
 resistance at their axes, they will carry round the wheel c with half the velocity of the prime mover a ; 
 and geering with the wheel h, on the main spindle of the machine to be impelled, will transfer to it tlie 
 motion which itself receives. We have supposed the wheel 6 to be held absolutely still ; but it is ob- 
 vious that it may be brought gradually to rest by means of the friction gland ; and as the wheel c can 
 attain motion orSy as the motion of the wheel b is reduced, and can attain its full speed only when b is 
 brought to rest, it is clear that the wheel h, and, consequently, the machine, may be brought into action 
 without the slightest degree of shock ; and, moreover, may be driven at any velocity less than the max- 
 imum, that may be desired. 
 
 2098. 
 
 LI^ 
 
 Another mode of obviating shock in starting machinery, which has been long in use, is represented by 
 Fig. 2098. On the shaft B is fixed a drum or pulley wliich is embraced by a friction-band a as tightly 
 as may be found necessary ; this band is provided with projecting ears, with which the prongs & 6 of a 
 fixed cross on the driving-shaft A can be shifted into contact. This cross can be shifted end-long on its 
 shaft A, but is connected to it by a sunk key, so that being thrown into geer with the ears of the friction- 
 band, the shaft being in motion, the band slips round on its pulley until the friction becomes equal to 
 the resistance, and the pulley gradually attains the motion of the clutch. The arms and sockets c c, 
 which are keyed firmly on the shaft A, are used to steady the prongs and to remove the strain from the 
 shifting part. 
 
 Bnt of all slide couplings to engage and -^^^^^ 2098* 
 
 disengage with the least shock, and at any 
 speed, ^Q friction cone coupling, (fig. 2098^,) 
 is by far the best. It consists of an exterior 
 and interior cone, ah; a is fastened to the 
 shaft A, whilst 6 slides in the usual way on 
 the feather /of the shaft B ; pressing h for- 
 ward, its exterior surface is brought in con- 
 tact with the interior conical surface of a ; 
 this should he done gradually ; the surfaces 
 of the two cones slip on each other till the 
 friction overcomes the resistance, and motion 
 is transmitted comparatively gradually, and 
 
 without danger to the machinery. It must be observed that the longer 
 the taper of the cones, the more difficult the disengagement ; but the 
 more blunt the cones, the more difficult to keep the surfaces in contact. 
 The limiting angle of resistance for surfaces of cast-iron upon cast-iron 
 is 8° 39', and this angle with the line of shaft will give a very good an- 
 gle for the surfaces of the cones of this material. "When thrown into 
 geer, the handle of the lever or sUj)per is slipped into a notch, that it 
 may not be thrown out by accident. 
 
 For the engaging and disengaging of a machine, that is, for putting 
 into or out of motion, the arrangement o£sifast-and-loose pulley is adopt- 
 ed as simpler and better than the clutches before given ; it is almost in- 
 variably used in connection with single machines which may be required 
 to be thrown in and out of motion. It consists merely of two pulleys in 
 
GEEEING. 
 
 73 
 
 juxtaposition on the same axis, one fast, the other loose, so that the belt which transmits the motion ma.} 
 be shifted from one to the other. The face of the driving pulley, that is, the one on the driving shaft, onght 
 to be equal in width to that of both the fast and loose pulleys. By making the face of the pulleys slightly 
 convex, the belt is prevented from shpping off, as the tendency of a belt is always to the larger diameter. 
 
 Beamigs. — Another important feature in a system of mill-geering is the mode of supporting the shaft- 
 ing — the form, arrangement, and general adaptation of the plumber-blocks and steps, and their supporting 
 apparatus. These are manifold in their proportion and construction — depending, as they do, in a great 
 measm-e, on the taste of the engineer and the circumstances in wliich they are employed. Even the 
 length of journal in relation to its diameter is far from being a fixed quantity; and as the journals de 
 termine the length of the piUotvs or brasses employed, these last arc equally indefinite in then" propor- 
 tions. The range of variation is gradually becoming less as the principle of friction is becoming better 
 known ; but still we find journals made equal in length and diameter ; and others having a length 
 equal to their cii'cumference. These proportions, except for particular cases, are extremes to be 
 avoided. 
 
 "Without attempting to establish a general rule for the length to be given to bearings, under all cir- 
 cumstances, it may be observed, that the pressm-e upon the pillows or steps ought in no case to exceed 
 1,000 lbs. on the square inch of rubbing surfaces. Within this limit the wear of the brasses is moder- 
 ate ; but, when a less intense pressure can conveniently be attained by increasing the amount of sur- 
 face, without a countervailing evil, it ought to be adopted, even to half the pressure stated. To calcu- 
 late every particular case which occurs in practice, and to adapt the patterns to it, would, however, be 
 attended with an amount of labor and expense which it is found necessary to avoid ; and, accordingly, 
 the mode commonly adopted is to fix upon two extreme cases, and to proportion the intermediate 
 lengths to these. Thus, taking a journal of 2 inches diameter, and another 12 inches; and suppose 4 
 inches to be taken as a proper length for the former and 19 inches for the latter; then the lengths for 
 intermediate diameters may be assigned by arithmetical progression. Thus, between 2 inches and 12 
 inches there are 40 quarters, and between 4 inches and 19 inches there are 120 eighths ; consequently, 
 for every increment of \ inch of diameter, we have a corresponding increment of f inch of length of 
 journal 
 
 The common material employed for pillows is a composition of copper and tin, in the proportion of 
 ] 2 to 1. This alloy is much 'harder than common brass, and is supposed to work with cast and wrought 
 iron with less friction, and to be more durable than most other compounds. Cast-iron pillows, espe- 
 cially when cast in iron moulds, and when the rubbing surfaces are made large in proportion to the 
 pressure upon them, are not inferior in any respect to the bush-metal commonly employed — the fric- 
 tion is, indeed, rather less than greater, and being harder, they are still more lasting. But when the 
 pressure is so great as to produce abrasion of the surfaces when left occasionally without a proper sup- 
 T)ly of oil, the wearing process, thus commenced, goes on increasing, and the surfaces never afterwards 
 acquire the necessary polish. In this the bush-metal has an advantage, for although abrasion has pro- 
 ceeded a certain way, when the lubrication is again perfect, the pillows take a new polish as perfect 
 as before, and the abrasion ceases. Wooden pillows are also sometimes employed with advantage. 
 Box-tree and lignum vitas were long in use, but those woods, besides being much more expensive, are 
 far inferior for the purpose to home-grown beech. We have known journals which, with ordinary 
 metal pillows, were ever liable to heat, run for many years on beech- wood without manifesting any 
 tendency of that kind. 
 
 Pure tin is, perhaps, of all other substances, that which produces the least friction with iron ; but its 
 softness has prevented it being very extensively employed. When the pressure upon the rubbing sur- 
 faces exceeds very moderate Kmits, the tin yields to it, and becomes extended in the direction of the 
 length of the pillow. A mode of obviating this difiiculty has been patented in the United States by 
 Mr. Babbitt ; it consists in placing the soft metal in a species of casing of cast-iron, of the common form 
 of the half pillow, but with ledges round its concave surface, which thus presents a recess of a depth 
 corresponding to the size of the pillow — from ith inch to an inch in very large diameters. This recess 
 is filled with the soft metal, which is retained by the ledging of the casing, and prevented from yielding 
 and becoming extended by the pressure on its surface. 
 
 2101. 
 
 2099. 
 
 2102. 
 
 The material used Vjy Mr. liabbitt is not pure tin, but a soft alloy, in which lead predominates, anq 
 which, besides answering all the purposes of tin, has the advantage of being greatly cheaper. 
 
 A patent for a composition-metal has lately been obtained by Mr. Fenton ; it is harder than Mr 
 Babbitt's metal equally fusible, and still cheaper, and has been found very efficient as bush-metal in 
 
V4 
 
 GEERING 
 
 railway engines. It is simply a compound of tin and zinc, with a little copper to harden it ; and a very 
 excellent composition of the sort may be made of equal parts of the two former metals and a sixteenth 
 by weight of antimony to give hardness. 
 
 Forms of plumber-blocks and pillows, very commonly adopted for heavy shafts, are shown by Figa 
 2099, 2100, 2] 01, and 2102. In this the pillows are made octagonal to prevent their turning round with 
 the shaft, and are cast with flanges on their ends to prevent longitudinal displacement. This form has, 
 however, been departed from by some of our best millwrights. Instead of casting the brasses with 
 flanges and plane faces, they are cast of a cylindrical form, and with two snugs at the middle of their 
 length, which, entering corresponding recesses in the plumber-block, retain them securely in their places 
 when the cover is fixed. By this arrangement they are more easily fitted, contain less useless metal, 
 and have a lighter appearance. The plumber-block itself, also, admits of being bored out to the size 
 of the pillows ; whereas, with the octagonal form, the faces require to be planed to the required 
 angles — a much more expensive process. 
 
 2104. 
 
 2103. 
 
 The plumber-block consists of a sole and cover, which, when practicable, are independently bolted 
 together, the heads of the cover-bolts being countersunk in the sole. The cover is made to check 
 accurately within the cheeks of the block, the surfaces being planed true to each other, so that the 
 stress upon the bolts is reduced to a minimum. The sole has two projecting ends by which to bolt it 
 in its place, as shown in Fig. 2103, and elsewhere. The figure specified is an example of a plumber- 
 block with what is denominated a false or shell cover — a form often adopted for economy when the 
 pressure is entirely down. In this case there is only one brass, the cover part being cast with side pro- 
 jections resembling the ends of a brass, and with projections on its upper surface resembling the nuts 
 and ends of bolts of the regular form represented by Fig. 2099. This cover is not fitted, but is simply 
 laid loosely in its place, and serves merely to preserve the journal from sand and similar injurious 
 matters. 2105. 2106. 
 
 The form of pedestal represented under the footstep-bridge by Fig. 2107, is one of the best examples, 
 combining strength, elegance, and simplicity of make. It is in every way appropriate for its purpose. 
 The base and cover fit together with a degree of symmetry and character which are not found in the 
 examples above referred to ; and the amount of fitting is much reduced. The base and cap being 
 checked together by planing, they are fixed in their relative position by the cover-bolts, and bored out 
 to the extreme diameter of the brasses. But instead of a uniform concave surface, the interior of the 
 base and cap have a circular recess of small depth occupying about a half of their breadth, so that the 
 bored parts occupy about a fourth of the breadth of the concave surface on each side of this recess 
 the object of this is to avoid the necessity of turning the exterior of the brasses for an equal extent at 
 the mnidle of their length, and, therefore, allows of their being provided with snugs as already noticed. 
 
GEERING. 
 
 15 
 
 to secure them from tm-niiig round in their places with the journaL The brasses being tm-ned to fit 
 their position in the pedestal, they are bored out concentrical with the outer circle; and the points oi 
 the cap-bolts and the nuts being dressed, the plumber- block is ready to be bolted in its place. 
 
 The foundation-plate, upon which the sole of the pedestal rests, is usually provided with snugs, set 
 HO much apart as to allow of a wooden key being driven between them and the ends of the pedestal- 
 sole. The bolt-holes in the sole and foundation are also made oblong, to allow of a small amount ol 
 adjustment in setting the pedestal; and, to obtain this latitude of adjustment laterally, and also in the 
 direction of the axis of the brasses, the holes are lengthened in the sole and foundation in the corres- 
 ponding directions. Adjustment is attained vertically by allowing for a given thickness of wood to be 
 placed under the sole of the pedestal, and which can be increased or diminished at pleasure to obviate 
 any small inaccuracy of workmanship. 
 
 In ordinary mill-work pedestals are usually pro- 
 vided with some form of cast-iron foundation-plate 
 upon which they are fixed. Tliis is exemplified by 
 Figs. 2103, 2104, 2106, 2108, and 2107 ; and it fre- 
 quently happens that much ingenuity is necessary 
 to compose what is technically denominated a loall- 
 box ox fixing for the proper reception of the number 
 of pedestals required by a confluence of shafting to 
 a point. Fig. 210-i aftbrds an example of one of 
 the most common and simplest forms of wall-box, 
 being intended to carry only one pedestal for a shaft 
 which may be supposed either to pass through the 
 waU into which the frame is inserted at that point, 
 or to terminate there. If intended to carry the end 
 of a shaft, it is usually made in the form of a rect- 
 angular box with the bottom into the wall, and with 
 a shelve at a convenient height to receive the sole of 
 the pedestal, the breadth of which determines the 
 depth of the box. But when the shaft passes com- 
 pletely through the wall, the box is simply a rect- 
 angular frame, corresponding in depth with the 
 
 thickness of the wall into which it is inserted and with the cross-shelve, for the purpose before stated. 
 This makes a neat, convenient, and substantial fixing, and one which is in constant requisition. 
 
 Fig. 2108 gives an example of a pedestal wall-plate, and which serves the same purpose as the wall- 
 box above described, when the shaft terminates at the wall and does not pass through it. This species 
 of fixing is simply a bracket bolted to the wall by three bolts which pass completely through and are 
 secured on the outside by wall-washers, bonnets, or stars. These are merely plates of cast-iron, some- 
 times with radial arms to increase the amount of bearing surface, with holes through their centres, tc 
 receive the ends of the bolts, which being screwed, are retained by nuts passed over them in the usual 
 maimer. Commonly the bolts are entered from the outside, which is generally advisable for conve- 
 
 nience and safety. The wall-washers serve the purpose of increasing the amount of surface of the wall 
 acted upon, and thereby allow the bolts to be screwed up more tightly ; and the more friable the 
 material of the wall is, the larger these ought manifestly to be made. For brick walls a very common 
 proportion between the size of bolt and bonnet, estimated by diameter, is one to six. For dressed 
 Btone walls, the proportion may be somewhat less ; but for rubble work the stellar form ought to be 
 preferred. 
 Tlie bracket is provided with a projecting sole upon which the pedestal rests, and which is supported 
 
76 
 
 GEERING. 
 
 at the middle of its length by a rib cast upon it. The sole has a stiuff at each end, between which the 
 pedestal is keyed, and holes through it to receive the pedestal-bolts. 
 
 Fig. 2109 represents another forni of wall-br&cj^et. This form is employed for carrying a shaft along 
 a wall, and projects sufficiently to allow jspvice for any wheels and pulleys which may be upon the shaft 
 to work clear of the wall. The bearing is in this case formed in the bracket itself, and the cover is hela 
 in its place by two bolts secured in the cheeks of the block by cotterals. 
 
 As this species of bracket does not allow of any adjustment at the bearing, and as it would be diffi- 
 cult to adjust, if fitted directly upon the wall, it is usually provided with a wall or ground-plate, as 
 shown in the figure. This plate is, in the first place, secured to the wall by wall-bolts and washers, in 
 the manner above described ; and the bracket is then adjusted and fixed upon it by bolts and keys, in 
 the same manner that the simple pedestal is secured to its seat. But sometimes, and judiciously, 
 its bolts also pass Entirely through the wall, and serve to maintain the stability of both foundation and 
 bracket. 
 
 Figs, 2110 to 2112 are forms of pendent brackets intended to carry lines of shafting along the ceiling 
 of a building. Fig. 2111 is the simplest form, and is often used to carry the weight of the shaft afc 
 points intermediate to those at which the power is taken off, but sometimes also it is the only species of 
 bracket employed when the shafting is light. It is easily fitted, and has very commonly only an under 
 pillow for the shaft, and no cover ; but when the shaft has an upward pressure — which it must at any 
 point where motion is transferred to or from it by toothed geer, whose line of contact is in the horizon- 
 tal plane, — a species of block-brass is fitted into it and retained by a cotter, as shown in the figm-e. Tht 
 ends of this block are checked, and the back of it has a groove to receive the edge of the cotter ; it is 
 thus effectually prevented from moving on end, and may be forced downwards upon the shaft by ad- 
 vancing the cotter, which is made of a tapering form for that purpose 
 
 2112. 2111. 
 
 This form of gallows has, however, no other recommendation than cheapness. It is inelegant in the 
 last degree, and deficient in stability, is easily twisted out of position in con-sequence of the oblique 
 action of the weight upon it ; and, moreover, the hook into which the brasses are fitted being usually 
 narrow — another feature of economy always associated with it — the brasses soon become loose and 
 allow a hobbling motion of the shaft, not more unpleasant to witness than it is injurious to all the COQ- 
 nections of the ]i2.e, and to the accurate transfer of the power at the working points. 
 
 2114. 2113. 
 
 Fig. 2112 represents a gallows of a more substantial character, but still inelegant in appearance, and, 
 m some respects, inconvenient. The weight being equally and directly borne by the two supporting 
 points, it possesses all the advantages of strength ; it may be provided with a cover and cover-brass, 
 which are necessary when the bracket is placed in the vicinity of a pair of wheels which geer together. 
 The form of bracket has the manifest inconvenience that the shaft cannot be taken out of its place 
 except by moving it end-long. 
 
 Figs. 2110 and 2110^ are two views of a form of pendent bracket, which combines to a considerable 
 extent the two requisites of strength and simphcity. The pillar and expanded base of this bracket are 
 cast hollow, as indicated by the dotted lines in Fig. 21 10^. The cover is fitted to the pillar by planed 
 faces which check together, leaving space to receive a bevel-edged key which holds the cover in its 
 place when the stress is entirely downwards; but when the bracket is intended to be placed in the 
 vicinity of two jgeering-wheels, the cover and projection on which the under pillow rests are provided 
 
GEERING. 
 
 11 
 
 with snugs through which a nib-bolt, as it is technically called, passes and assists the key in retaining, 
 the cover in its place. 
 
 This form of pendent is also convenient to the mill-wright as a pattern. In the two forms above 
 described, any difference of length is attended with extensive alteration of the pattern, but in this, tha 
 ptalkof the pattern is made at least as long as there is any probabihty of its ever being required ; a 
 series of ferules are then turned to fit upon the stalk, and of the requisite outside diameter; the 
 necessary number of these are put on to make up the length less the length of the octagonal part which 
 forms the bearing ; this part is formed of cast-iron, and has also a hole through it to receive the stalk, 
 and when put on and fixed in the proper position the bracket is ready to be moulded, any part of the 
 stalk which may be projecting tlirough being cut off in the sand. The process of moulding is likewise 
 simple, and no core-box is required, except when the bracket has a double bearing, which is always the 
 case when it is intended for a point at which a bevel pair meet. 
 
 The only objection to this form of bracket is the oblique direction in which the strain falls upon the 
 holding bolts, and which might be obviated by making the cover one piece with the stem, and checking 
 the under bearing to it by a strong dovetail and single holding-bolt, the office of which would simply be 
 to prevent end-long motion. 
 
 Pendent brackets such as those here described, are sometimes cast with vertical soles, prepared to 
 bolt to beams ; but more frequently the soles are horizontal ; and in that case are bolted to grounds of 
 planking fixed between two adjoining beams by wood-screws. When the brackets can be attached to 
 beams, that mode of fixing is, however, to be preferred ; and in that case the sole ought to be formed 
 with a projecting rib to bear against the under side of the beam, and relieve the bolts as much aa 
 possible of strain. 
 
 The kinds of bearings described are all designed to support shafts placed horizontally ; but in almost 
 every system of geering there is one large vertical shaft from which the minor horizontal shafts derive 
 their motion. The bottom extremity of the vertical shaft is supported on a species of bearing techni- 
 cally named a foot-step, of which Fig. 2107 represents a specimen. In this case, the motion is usually 
 transferred by a pair of bevel-wheels, as represented by Fig. 2001, and more completely still by the 
 arrangement shown in Fig. 2115. In this last the first shaft is marked H; on this is the spur pinion Q 
 
 2115. 
 
 which geers with the equally pitched wheel F on the horizontal shaft K. On the other extremity ol 
 this shaft is the bevel-wheel C, which geers with a similar wheel on the vertical shaft ; and through 
 this last wheel motion is communicated to the equal wheel B and the geering on the right. The 
 arrangement represented by Fig. 2001 is strictly analogous — the horizontal shaft being the driver and 
 the vertical shaft the carrier of the motion to the superior parts of the factory. 
 
 In these arrangements the foot-step of the vertical shaft is carried upon an arch, technically a foot- 
 bridge, of which Fig. 2107 gives a front view, and Fig. 2115 a side view: Fig. 2001 shows a section of 
 the same through the foot-step box. This bridge rests on a foundation-plate of cast-iron, which in turn 
 is fixed upon a stone foundation more or less substantial, according to the weight and stress which are 
 to be resisted. The pedestal of the horizontal shaft ought to be supported on the same foundation 
 sole ; for, if supported on a separate sole, the foundation bolts, and possibly the foundation itself, will 
 in all probability be soon destroyed by the twisting action of the wheels at the fixed points. The foot- 
 step box occupies the summit of the arch ; three sides of it are cast upon the bridge, but the fourth side, 
 which forms the door, is a separate casting, attached by four bolts which pass right through from back 
 to front of the box, hollow rolls being cast on the two opposite sides for that purpose. In Fig. 2107 
 the door is supposed to be removed to show the interior arrangement; Fig. 2001 gives a transverse 
 section through the door, showing its thickness ; and Fig. 2115 shows the door in its place and fixed by 
 its bolts. This door is fitted with great exactness, and the whole interior of the box — at least a 
 sufficient bearing surface of it at the angles — is chipped and filed true: a foot-plate as in Fig. 2001, 
 and sometimes a species of box solid above, and sometimes both, is fitted into the bottom of the box. 
 This plate or pillow is first planed ; and in fitting, its superior surface is set truly square with the sides 
 of the box. The surface of the plate is frequently steeled by case-hardening when the end of the shaft 
 is intended to be rested immediately upon it, as shown in the drawings referred to ; but more commonly 
 it is simply a plate of cast-iron on which the bottom of the cup-formed brass is supported. In these 
 drawings the brass is supposed to be without bottom, and to embrace the cylindrical end of the shaft, 
 and support it laterally. This anangement has its advantages, in so far as the foot-plate can easily be 
 
78 GEERING. 
 
 replaced when wora out ; and the brass may be formed in halves, which will allow it also to be replaced 
 with little difficulty. But more commonly the brass has the form of a deep cup, interiorly ; while, 
 exteriorly, it is made in size exactly to fit the box. On one side — that from which the wheel acts — a 
 recess is left in the interior to receive and retain oil in considerable quantity, so that the lubrication may 
 be abundant and prevent heating. With every attention to lubrication it is, however, very difficult, 
 when the weight upon the step is great, to prevent its becoming hot, evaporating the oil, and destroying 
 the rubbing surfaces. Various modes of remedying the evil have been attempted ; but the best expe- 
 dient hitherto tried is to enlarge the sui-face as much as convenient. Formerly, the end of the shaft was 
 greatly reduced to form the foot — often, indeed, a pin was inserted and made to serve the functions of a 
 foot ; but at present the foot is formed on the same principle as any other bearing — namely, with aa 
 much bearing surface as the diameter will allow. But even this is often found too little, and the next 
 resource is to continue the wheel boss and give the foot a conical form, thereby increasing the supporting 
 surface as much as circumstances wiU admit. 
 
 We have also seen attempts to line the brass step with ribs of steel to increase its durability ; but 
 without success. A step made on Babbitt's principle, and lined with soft metal, would, in our opinion, 
 offer a better chance of success than any scheme which has hitherto been attempted. But the experi- 
 ment remains to be made on a proper scale, and with sufficiently good workmanship, to test the principle 
 thoroughly before its being adopted very extensively in mill-work. 
 
 Among the more ingenious of the older contrivances for rendering foot-bearings durable was that of 
 attaching a hardened steel plate to the extremity of the foot by a square tang, which was driven hard 
 into a corresponding hole in the foot. On the under side of this disk a groove was cut across, the better 
 to distribute the oil which found its way to the bottom of the step by vertical grooves cut in it. 
 
 This was found to work very well when there was little lateral stress upon the shaft ; but ultimately 
 gave place to an arrangement which is sometimes adopted in modem work, and which consists in 
 inserting a steel pivot into the lower end of the shaft. When this pivot is bound in its place by a 
 malleable-iron ring put on hot, the arrangement is one of the best which can be adopted ; but the 
 original mode of simply boring a hole in the end of the shaft and inserting the pivot, without any 
 further lateral support, left the foot very weak and inadequate to sustain any considerable amount of 
 lateral stress. 
 
 Another scheme, also sometimes put in practice in modern work, when the shaft is very large, is to 
 make the foot turn on conical rollers, on the same principle as horizontal shafts are sometimes made tc 
 turn on parallel friction rollers to diminish the amount of friction, by converting a rubbing into a rolling 
 motion, with only the sliding action of the small axes which retain the rollers in their places. The 
 cones, however, require very accurate workmanship, and therefore become expensive, and are, moreover, 
 very liable to get out of order ; they are, in consequence, very seldom employed. A modification of the 
 conical rollers, consisting of a single ball placed under the foot of the shaft, although very rarely tried 
 in mill-work, seems to offer advantages more than equivalent to the additional workmanship which it 
 would entail, by dividing the motion, in some measure, between the shaft and itself, and leaving an 
 open space to receive oil, which would tend materially to keep the foot cool. This arrangement is 
 almost universally adopted in vertical sugar-mills, and is found to answer, although in process of time 
 the ball does become flattened, and, accordingly, requires to be replaced. 
 
 The scheme of the late Mr. Bramah, of supporting the pivot entirely on water, the pressure of which 
 was to be maintained by a small forcing-pump and stop-cock, would, if all tear and wear could be 
 avoided, be perfect ; but unfortunately it holds out little prospect of being successfully applied to the 
 end in question, although there appears little doubt that it will ultimately be found both practicable 
 and efficient as applied to railway turn-tables, in which the bearing sm-face is large, and the only prob- 
 lem to be resolved is the diminution of friction. 
 
 Water lias been successfully applied to keep the common foot-step cool, by simply perforating the foot- 
 box and running a stream constantly through it. An example of this may be seen at the Deanston mills. 
 
 Sizes and proportions of bolts. — The bolts employed in securing together the pedestals and parts of 
 the fixings of shaft-geering constitute another important element in mill-work. Bolts may be generally 
 described as pins having a head formed on one end, and the other end screwed externally to enter an 
 internally screwed ring, technically named a nut. When the bolt is passed through holes placed oppo- 
 sitely in two pieces, a thin ring or washer being first placed on the screwed end of the bolt, if the nut 
 be then put on and screwed tightly down upon the washer the two pieces will be held firmly together 
 between this last and the head, the body of the bolt, or part between, serving as the mediunl through 
 which the force is maintained by one surface upon that opposite. 
 
 Bolts are denominated according to their diameters of the body part, ^-inch, |-inch, |-inch, |-inch, 
 1-inch, 1^-inch, &c., bolts ; the particular diameter being determined in any case by the supposed tension 
 to which the bolt will be exposed. It is rarely that this force can be estimated even approximately ; 
 but assuming it to be known, the tension ought not to exceed 1 J ton on the square inch of section. 
 Therefore, if we designate by T the tension in tons, that is, the eftbrt tending to separate the head and 
 nut by tearing the body part of the bolt asunder, and the diameter in inches by D, we have the simple 
 formula, D = | V T. 
 
 Thus, supposing the known tension to be nine tons, then, 
 
 D = I ^/ 9 = 2i inches, 
 the diameter of the bolt necessary to resist that pressvure. 
 
 The number of consecutive threads in an inch, that is, the pitch of the screw, is regulated by the di- 
 ameter of the bolt. For small bolts of about -^-inch the pitch is usually about a sixth of the diameter, but 
 relatively decreases inversely with the increase of diameter. 
 
 The following table is that of the dies and taps of an extensive millwright and engineering establish 
 ment in Glasgow, and it does not differ sensibly from the numbers adopted in some other works. 
 
GEERINCr. 
 
 79 
 
 The depth of the thread is also a matter of importance. The section of thread most approved of fof 
 strength imd easy motion of the nut is an equilateral triangle, thus a, the bevelled sides being equal bo- 
 tween themselves and to the base or pitch. 
 
 Bolts of larger diameter than those 
 named in the annexed table are usually 
 chased in the screw-cutting lathe, and in 
 that case the thread is conmionly made 
 of a rectangular section ; but such bolts 
 are very rarely requued in mill-work, 
 and when they do liappen to be employ- 
 ed, it is under such circmnstances as re- 
 move them from the ordinary class of bolts 
 and the proportions therein recognized. 
 
 The length of the body of the bolt 
 must, of course, depend upon cu'cum- 
 stances ; but a certain proportion is ob- 
 served between the diameter and the 
 sizes of head and nut. For common bolts the head is usually square, and the nut six-paned, that is, 
 has six focets on its periphery. Sometimes, also, the head has the same form, and occasionally it is 
 made round, and for particular purposes it is conical, but oftener pyramidal. When of this shape, it is 
 intended to he flush or even with the surface of the piece into which the bolt is inserted, and it is then 
 said to be countermnk. The cover-bolts of pedestals are of this form, being necessarily flush with the 
 sole of the pedestal. The thickness of the head when of the usual form is from two-thirds to three- 
 fourths of the diameter of the bolt ; and the diameter of its circumscribed circle is double of the primary 
 diameter. The tliickness of the nat is usually made sufficient to contain from eight to ten threads, the 
 smaller having proportionally the greater depth. The part of the body towards the head, equal in 
 length to a third of the whole, is usually made square, and the remaining part or two-thirds is round; 
 and of this a portion equal at least to double the thickness of the nut is screwed. Sometimes, however, 
 as in pedestal cover-bolts, the whole length of the body is cylindrical, the bolt being prevented from 
 turning round when the nut is being screwed up by the pyramidal countersunk head. 
 
 Although the length of bolts generally can only be assigned for particular cases, those adapted for 
 particular sizes of pedestals may be given. These, it is true, will vary with the particular form of pe- 
 destal used ; but the sizes given in the following table will be found very generally applicable for those 
 of good proportions. 
 
 Diameter of 
 
 Threads in an 
 
 Diameter of 
 
 Threads in an 
 
 screw in inches. 
 
 inch. 
 
 screw in inches. 
 
 inch. 
 
 i 
 
 12 
 
 1* 
 
 5 
 
 1 
 
 11 
 
 1| 
 
 4i 
 
 1 
 
 10 
 
 2 
 
 H 
 
 i 
 
 8| 
 
 2^ 
 
 4 
 
 1 
 
 8 
 
 2* 
 
 4 
 
 n 
 
 H 
 
 2| 
 
 H 
 
 u 
 
 7 
 
 2i 
 
 3i 
 
 If 
 
 H 
 
 2| 
 
 3 
 
 H 
 
 6 
 
 3 
 
 3 
 
 11 
 
 H 
 
 
 
 
 
 gmi 
 
 p^ 
 
 Q^. 
 
 ^^^ 
 
 > * 
 
 Diameter of 
 journal. 
 
 Diameter of 
 bolts. 
 
 Holding. 
 
 Cover. 
 
 Double-ended. 
 
 Inches. 
 
 Inche-j, 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. i 
 
 li 
 
 ■i» 
 
 3^ 
 
 4i 
 
 n 
 
 5 
 
 1| 
 
 1 
 
 3i 
 
 H 
 
 H 
 
 5 
 
 o 
 
 f 
 
 3| 
 
 5 
 
 24 
 
 6 
 
 H 
 
 1 
 
 4 
 
 51 
 
 OJ 
 
 64 
 
 2i 
 
 1 
 
 4 
 
 51 
 
 2| 
 
 6| 
 
 2| 
 
 1 
 
 4i 
 
 6i 
 
 3 
 
 Vi 
 
 3 
 
 1 
 
 H 
 
 61 
 
 3 
 
 '71 
 
 H 
 
 i 
 
 4f 
 
 n 
 
 H 
 
 8i 
 
 H 
 
 i 
 
 4| 
 
 8 
 
 Si 
 
 8| 
 
 H 
 
 1 
 
 4| 
 
 8i 
 
 31 
 
 9 
 
 4 
 
 1 
 
 5 
 
 9 
 
 3| 
 
 94 
 
 4i 
 
 n 
 
 H 
 
 n 
 
 4 
 
 lOi 
 
 4i 
 
 u 
 
 6 
 
 10^ 
 
 4i 
 
 10| 
 
 4| 
 
 H 
 
 H 
 
 101 
 
 41 
 
 Hi 
 
 o 
 
 It 
 
 H 
 
 11 
 
 4| 
 
 111 
 
 5i 
 
 If 
 
 6i 
 
 Hi 
 
 4| 
 
 12i 
 
 H 
 
 li 
 
 6| 
 
 12 
 
 5 
 
 121 
 
 5| 
 
 H 
 
 6f 
 
 12| 
 
 H 
 
 13 
 
 6 
 
 H 
 
 1 
 
 m 
 
 H 
 
 13| 
 
 6i 
 
 If 
 
 7i 
 
 13| 
 
 54 
 
 141 
 
 H 
 
 If 
 
 n 
 
 14 
 
 5| 
 
 15 
 
 H 
 
 H 
 
 8i 
 
 141 
 
 6 
 
 16 
 
 7 
 
 n 
 
 H 
 
 15 
 
 6i 
 
 161 
 
 H 
 
 2 
 
 8| 
 
 15| 
 
 61 
 
 17 
 
 ^ 
 
 2 
 
 9 
 
 16i 
 
 6| 
 
 m 
 
 n 
 
 2| 
 
 H 
 
 16| 
 
 7 
 
 18 
 
 8 
 
 2i 
 
 H 
 
 m 
 
 7i 
 
 181 
 
 8i 
 
 n 
 
 9| 
 
 181 
 
 71 
 
 194 
 
 Si 
 
 2i 
 
 10 
 
 19| 
 
 8 
 
 20i 
 
 9 
 
 2| 
 
 lOi 
 
 20 
 
 8i 
 
 21 
 
 91 
 
 21 
 
 11 
 
 20| 
 
 81 
 
 22 
 
 1 
 
80 ' GEERING. 
 
 The heads of bolts intended for wood are commonly segments of a sphere ; and the bolt is not often 
 provided with a nut, the holding being in the wood itself. But sometimes, when the wood is thin, the 
 screw is passed sufficiently far through to take a broad wash and nut above it, an arrangement which 
 gives strength in proportion to the acting surface. 
 
 Guide pulleys. — It frequently happens, when the power is to be conveyed by a belt, that the connec- 
 tion cannot be obtained dh-ectly, in consequence of the relative positions of the shafts, which may be 
 placed obliquely to one another ; and sometimes other parts of the geering, machinery, cross-beams oi 
 the building, and the like, come between the points to be connected, and by their intervention prevent 
 the belt from passing immediately from one point to the other. Under these circumstances, it is ne- 
 cessary to pass the belt over guide pulleys. 
 
 An example of the common guide-pulley frame is given in Figs. 2113 and 2114. The axes of the 
 pulleys are adjustable in the frame B to any requii-ed angle within a certain range in a common plane, 
 which is parallel with a plane passing through the driving axis and perpendicular to the plane of the 
 driven shaft. "When the belt, therefore, passes from the driver over one of the guide pulleys and is 
 returned upon the other, the bight will be thrown into a plane at right angles to the normal plane, and 
 a puUey placed in that plane, its axis coinciding with it, wiH be driven by the belt without any ten- 
 dency to change its plane of action. 
 
 We have supposed the beams to be parallel and perpendicular to one another ; but the arrangement 
 may be adapted to other positions by placing the guide pulleys in different planes ; and in order to 
 aUow of further accommodation, the frame may be made adjustable at the point of attachment to the 
 beam at D. 
 
 The above treatise on geering has been taken from the Engineer and Machinist's Assistant, and ia 
 generally admitted to be the best yet published. 
 
GEER-CUTTING MACHINE, BEVEL. 
 
 81 
 
 GEER-CUTTING MACHINE, BEVEL. For the cut- 
 ting of bevel geer an admirable machine has been recently- 
 invented and patented by George H. Corliss, of Provi- 
 dence, K. I., in whose shop (CorHss, Nightingale & Co.) it 
 is in successful operation. From the specitications of the 
 patentee we extract the following full and clear descrip- 
 tion of the machine, its distinguishing characteristic, con- 
 struction, and operation. 
 
 Fig. 1973, plan of the machine. 
 
 Fig. 197-4, a side elevation. 
 
 Figs. 1975 and 1976, enlarged cross sections, taken at 
 the line A a of Fig. 1973 ; the former with certain parts 
 removed. 
 
 Fig. 1977, another cross section at the line B6 of Fig. 
 1974. 
 
 Fig. 1978, a vertical cross section at the line C c of Fig, 
 1976. 
 
 Figs. 1979 and 1980, enlarged end and side views of the 
 cutter and carriage. 
 
 Fig. 1981, an enlarged separate view of the hinge which 
 forms the axis of the guide-bar on which the cutter carriage 
 slides. 
 
 Fig. 1982, an enlarged separate view of the end of the 
 guide-bar, with a stem connected therewith. 
 
 The same letters indicate like parts in all the figures. 
 
 In the machines heretofore employed for cutting cogs 
 of toothed wheels, a rotating burr cutter has been used ; 
 and although this is, to a certain extent, effective for cut- 
 ting spin geer, yet in cutting the cogs of bevel geer it is, 
 from the very nature of the case, defective. The cogs, 
 when cut with a rotating cutter, must be defective for the 
 following reasons : if the sides of the cutter-wheel be 
 parallel, the space cut out between the cogs will also be 
 parallel, wliilst in bevel-wheels they should be in the lines 
 of the radii, that is, farther apart at the outer than at the 
 inner periphery ; and if to avoid this the sides of the cut- 
 ter-wheel are bevelled, to make the spaces wider by cut- 
 ting deeper towards the outer periphery, then the spaces 
 will be wedge-formed in their section, which is at variance 
 with the proper formation of cogs, for the spaces below 
 ths pitch-Line should be vertical, or curved inwards, and 
 from the pitch-line upwards curved outwards ; and these 
 curves should be sections of cones, which cannot be formed 
 by a rotating cutter, whicli, from 
 the very nature of its operation, 
 will make the same curve from 
 end to end of the cog. 
 
 The object of this invention is to 
 avoid this objection ; and for tliis 
 purpose the first part of the inven- 
 tion consists in the use of a recip- 
 rocating cutter, governed by a 
 guide-bar on which the cutter 
 carriage slides, and which has its 
 axes of vibration to adapt the cut- 
 ter to the required depth of cogs 
 at the apex of a cone correspond- 
 ing with the bevel of the wheel 
 to be cut, whether such axes be 
 fixed or adjustable to wheels of 
 different sizes, that all the cuts 
 may be in the direction of the 
 radii. 
 
 The second part of the invention 
 consists m combining the guide-bar, 
 on which the cutter carriage runs, 
 with a secondary frame hinged to 
 the main frame outside of the circle 
 of the largest wheel intended to 
 
82 
 
 GEER-CUTTING MACHINE, BEVEL. 
 
 be cut in the machine, that the axis of vibration of the guide-bar may be elevated or depressed tc 
 adapt the machine to different bevels ; and that the main driving shaft, which communicates motion tc 
 tlie operative parts of the machinery placed at the hinged end of the said frame, may be in the line oi 
 the axis of vibration of the said frame, that the vibration thereof may not change the relative position 
 of the driving shaft and the parts receiving motion therefrom. 
 
 The third part of the invention consists in combining with the guide-bar a guide-plate, against whicl 
 it bears by means of a .weight, spring, or the equivalent thereof, so that as the guide-bar descends to 
 give the proper depth to the cogs, the said guide-bar shall follow the curve of the guide, and thus de- 
 termine the form of the face of the cogs. 
 
 And the last part of the invention consists in making that part of the rear end of the guide-bar which 
 rests against the guide, movable, so as to have an endwise motion, in or on the said bar, in the direction 
 of its length ; the said movable part dr stem being bevelled back of where it rests against the guide, 
 and so connected, either with the guide-bar or some other part of the machinery, that at the time of the 
 cutting motion it will move forward, that its bevelled surface may be brought in contact with the guide 
 and give a lateral motion to the guide-bar, to relieve the cutter from the surface of the cog that is being 
 cut, to admit of its moving back clear of the cog ; and then, at the end of the return motion, a reversed 
 motion to bring the cutter in the proper line for cutting. 
 
 In the accompanying figures, a represents the main frame of the machine, to the inside of which is 
 adapted a frame h that carries the spindle c of an index-plate, made in the usual manner. This frame 
 h is secured to the main frame by the bolts dd that pass through elongated holes eem the side pieces 
 of the main frame ; and as there are several of these holes along the frame, the index-plate can be 
 moved to any place required on the main frame, to adapt the machiue to wheels of various sizes. The 
 upper end of the spindle c is adapted, like other cutting engines, to receive the wheel / to be cut. A 
 secondary inclined frame h is provided near one end with a shaft i, the journals of which run in boxes 
 jj at the end of the main frame, so that the secondary frame can be inclined to any desired extent with 
 the axis of the index-wheel to determine the bevel of the cogs to be cut. And the extreme rear end of 
 the secondary frame is provided with a bolt and temper-screw that pass through a segment mortice 
 in the main frame, by means of which the secondary frame may be secured and held in place at any 
 inclination required. The shaft i of the secondary frame extends out sufficiently on one side to receive 
 one loose pulley k, and two fast pulleys / m, one on each side of the loose pulley. The loose pulley k 
 turns freely between and on the barrel of the other two in a manner well known to machinists. And 
 the inner end of the fast pulleys I mis provided with a pinion n, which engages and carries a cog-wheel 
 that turns on a stud-pin p, and the arbor of the wheel carries a pinion (see dotted hne in Fig. 1914) 
 that engages and carries a sector-rack q on the end of a rock-shaft r, provided with a pendulous arm s, 
 to the lower end of which is joined a connecting rod t that takes hold of the rear end of a carriage w, 
 'as seen in Fig* 1980,) to which is secured the cutter v, in any appropriate manner. The carriage slides 
 
GEER-CUTTING MACHINE, BEVEL. 
 
 83 
 
 on a guide-bar iv properly formed for this purpose, as shown in the figures, and in turn this guide-bar 
 slides inways x, connected by a socket y with a stud z that projects from an arbor a' which turns on 
 pivot screws b' b', that pass tlu'ough the two cars c' c' of a plate d', secured by bolts or screws to the side 
 pieces of the secondary frame, so that by shifting these screws the plate can be moved along the sec- 
 ondary frame, as the index-plate and its spindle can be moved along the main frame to adapt the 
 machine to the cutting of wheels of various diameters. The guide-bar is thus connected with the sec- 
 ondary frame by a universal joint, and the connection of the universal joint with the secondary frame 
 can be shifted to adapt the machine to the cutting of wheels of different sizes, and as the axis of the 
 vertical vibration of the guide-bar must always be in the hne of the axis of the index-plate, the mode of 
 shifting either the one or the other must be such as will admit of accurate adjustment. For this purpose 
 the holes in the main frame, through which the seeming bolts pass, are elongated. The machine is 
 driven by two belts e'f, one being crossed, and the two governed by a double belt-shipper g', so formed 
 that when the direct belt e' runs on the pulley ?n, to give the cutting motion to the cutter carriage by 
 the connection of parts from the pinion n, the crossed belt f runs on the loose pulley Jc, and when the 
 belts are shifted at the end of the cutting motion, to reverse the motion, the crossed belt runs on and 
 
 197 
 
 parries the pulley I, and with it the cutter carriage by the same connections, and the direct belt e' thea 
 runs on the loose pulley ; or, if desired, this arrangement of belt may be reversed. In this way the 
 desired motions are given, and the shifting of the belts is effected in the following manner : The belt- 
 shipper is attached to the outer end of two rods i' i' that slide in a plate h' attached to the frame, and 
 these way-rods i' i' arc connected by a cross-bar i^ with one arm of a right-angle lever,/', the other arm 
 of which passes to the inside of the main frame, and is there jointed to a rod k' which passes through a 
 thimble I, jointed to the pendulous arm s that communicates motion to the carriage. The rod k' is pro- 
 vided with two adjustable stops m' m\ on each side of the thimble, and at such distance apart that 
 when the pendulous arm s moves forward to effect the cutting motion, towards the end of this motion 
 the thimble strikes one of the stops m', and shifts the belts to give the return motion, towards the end 
 of which the other stop m' is struck to re-sliift the belts. In this way, by simply varying the distance 
 between the two stops in', any desired range of motion can be given to the cutter carriage, to adapt it 
 to the length of the cogs to be cut. 
 
 As before described, by reason of the universal joint connection, the guide-bar is free to move either 
 vertically or horizontally, and with it the cutter carriage which slides on it. Its rear end is suspended to 
 a cord o' which passes over a pulley p' with a counter weight q' attached to it, by which it is held up 
 against the end of a set screw r', the turning of which will therefore determine the depth of cut to be 
 made by the cutter. This guide-bar is also borne laterally by means of a bent lever s', (see also Fig. 
 1977,) one arm of which acts against it, and the other attached to a cord t' that passes over a pulley u' 
 4nd is provided with a weight v'. This weight always tends to bear the guide-bar in one direction, 
 norizontally, and against a guide-plate v/, one edge of which is formed so as to determine the form to bo 
 
84 
 
 GEER-CUTTING MACHINE, BEVEL. 
 
 given to the face of the cog, and as this plate can be removed, others may be substituted to suit ths 
 various and desired forms of cogs. The rear end of the guide-bar, however, does not bear against this 
 guide-plate, but, instead of this, there is a stem x' "with a socket in its forward end that slides accu- 
 rately, but freely, on a projection y'. Fig. 1982, from the rear end of the guide-bar, so that one can slide 
 on the other longitudinally ; and this stem it is that bears against the guide-plate. The rear end of the 
 stem is looped to receive the arm z' of a slide a^. ■ It will be observed that whilst the stem is in the 
 position represented in the drawings, as the rear end of the guide-bar is moved up and down to cut the 
 depth of the cog, the stem x' follows the curvature of the guide w', and therefore communicates a cor- 
 responding motion to the point of the cutter in a direction converging to the centre of the universal 
 joint, on which the guide-bar w vibrates, and that, therefore, any curve to be given to the cross section 
 of the cog will be gradually reduced as it approaches the axis of the wheel. But when the cutter is to be 
 moved back, it is necessary that it should run clear of the metal, and for this purpose the stem x', back 
 of tlie part which is represented as bearing against the guide-plate, is curved inwards, or bevelled as 
 at b"^, so that when this part is brought in contact with the guide-plate a slight lateral motion is given 
 to the guide-bar sufficient to relieve the cutter. The required endwise motion for this purpose is given 
 to the stem by the operation of shifting the belts to reverse the motion of the cutter. The inner arm 
 of the lever/, of the belt, carries a box c\ that slides freely on the slide a^, and towards the end of the 
 forward motion of the cutter carriage ; this box strikes against a stop d'\ from which the arm z\ con- 
 nected with the stem x', projects and forces forward the stem to the distance required to bring the 
 bevel b"^ against the guide-plate w' to relieve the cutter. The cutter carriage then runs back and 
 towards the end of this back motion ; the box on the lever of the belt -shipper strikes another stop e'^, 
 on the slide a^, and moves back the stem to bring the cutter in the proper line for making its cut. In 
 this way at each operation the cutter is relieved and returned to its proper position for cutting. The 
 cutter is fitted in any desired manner in a socket in the carriage, and when it is desired to cut cogs ol 
 the form represented in the enlarged Fig. 1983, the cutting edge of the cutter must be bent forward, as 
 shown in that figure. 
 
 1982. 
 
 1380. 
 
 
 ■vf 
 
 y 
 
 
 ^ 
 
 
 
 ■} 
 
 1 
 
 197a. 
 
 J — 
 
 o 
 
 O 
 
 O 
 
 i 
 
 ) 
 
 
 / 
 
 ^SSSB^ 
 
 7Z 
 
 a o 
 
 o o 7 
 
 
 w 
 
 1981. 
 
 In a machine constructed and operating on the principle of this invention, every cut converges to the 
 dpex of a cone that represents the bevel of the wheel, and therefore the cogs, and the spaces between 
 them, will gradually and in the true proportion diminish from the outer to the inner periphery. Thus far 
 as to the mode of operation of the machine for cutting the faces of the cogs on one side ; but as each cog 
 has two faces on opposite sides, so soon as one face of all the cogs have been cut, the machine must be 
 reversed to cut the other side, and for this purpose it is simply required to reverse the cutter^ V, the 
 guide-plate w', and the lever s. The reversed position of these parts is shown in dotted lines in Figs. 
 1977 and 1979. By the shifting of these parts, it will be observed that the machine will cut the re- 
 versed side of the cogs. The lower end of the cutter should be properly formed to give the required 
 shape to the bottom of the cogs. 
 
 The patentee claims, first, The method of cutting the cogs of bevelled wheels by means of a recipro- 
 cating cutter that moves in or on a slide, or slides, that vibrates on an axis that coincides, or nearly so, 
 ^rirtithe apex of a cone representing the bevel of the wheel to be cut, substantially as herein described, 
 
GEER-CUTTING ENGINE. 
 
 85 
 
 by which vibration the depth of the cut is determined, and irrespective of the adjustment of the axes 
 of vibration as described. 
 
 Secondly, The guide-bar, (or its equivalent,) on -which the cutter carriage runs, and having its axis ot 
 vibration for the depth of cut, as above described, when combined with a secondary frame jointed to the 
 main frame at some point outside of the circumference of the wheel to be cut, that the machinery may 
 be adapted to the cutting of cogs on various bevels. 
 
 Tliirdly, In combination with the guide-bar, having a universal joint, or the equivalent thereof, and 
 operated substantially as described in combination with the guide-plate, to guide the cutter and deter 
 mine the form of the face of the cogs, as described. 
 
 And lastly. Making that part of the guide-bar which rests against the guide-plate to determine the 
 form of the face of the cogs separate from, and movable on, the guide-bar, and properly bevelled to 
 relieve and clear the cutter for its back movement, substantially as described. 
 
 GEER-CUTTIXG EXG^E. Figs. 1984 and 1985 are front and side elevations of a geer-cutting 
 engine, as built by the Lowell Machine Shop, sufficiently large to cut a geer 1 feet in diameter and 8 
 
 inches face. This is a hea\7 and strong-built machine, adapted to cutting spiral spur and bevel geers, 
 without any additional fixtures. 
 
 M is the spindle to which the geers are fastened when being cut, and to which is attached the dial 
 
88 
 
 GEER-CUTTING ENGINE. 
 
 plate B, on the upper surface of which are drilled circular rows of holes very accurately spaced off; by 
 means of this, and the spring and point N, Fig. 1984, the geer is divided into the required number of 
 teeth. 
 
 A, hand-wheel for raising and lowering spindle M when cutting small geers. For cutting large spur 
 or bevel geers, the geer remains stationary, and the head to which the cutter is attached is traversed 
 up and down by means of the crank D, which moves the pinion d, geering into a larger geer H, on chain 
 pinion-shaft._ An endless chain running up over a pulley K is attached to the plate, to which the cutter 
 shaft stand is attached. This stand can be turned around at any angle for spiral or worm geers. 
 
 ]pe4. 
 
 R, guide for spindle when cutting worm, or slightly spiralled geers. When spiral geers of greater 
 angle are to be cut, there is a circular inclined plane of plate steel screwed to the cylindrical part of the 
 yoke, (imder the dial,) which bears on a friction roll attached to the main frame. It is kept against this 
 roll by means of the weight G, at the end of the string passing over the grooved pulley F. 
 
 P, weight to balance spindle M and the geer, being cut. This weight has a pinion playing into the 
 rack on the top of the long arm of the lever 0, by means of which it can be moved forth and back to 
 balance geers of different weight. 
 
 G', crank and pinion geering into segment, for setting the head at an angle for cutting bevel geers. 
 
 C, driving pulley coupled to cutter shaft. 
 
 L, geer on nut for raising the cutting head. It is worked by a pinion L' on the end of the upright 
 shaft, to which a hand- wheel is attached. 
 
 E, crank and bevel geers which communicate with the screw inside of frame, by means of the geers 
 E'. Turning this crank moves the head stock to different positions, for different sized geers. 
 
 b b, arms to which the cord and weight are attached when cutting spiral geers. 
 
 This machine is a very convenient one to work at, as the operator can stand in one position and com- 
 mand everv part of it. 
 
GEER-CUTTING ENGINE. 
 
 87 
 
 Fig3. 1986, 1987, another form of geer-cutting engine for cutting bevel, spiral, and spu/ ^eers, also 
 made by the Lowell Machine Shop. 
 
 Fig. 198G, side elevation. 
 
 Fig. 1987, front or end elevation. 
 
 E E are the end standards which form part of the frame. 
 
 D, swing shde, in which the cutter head slides when cutting bevel geers. 
 
 B, hand-wheel on pinion shaft passing through the axis of the swing slide D, wliich has a pinio< 
 geering into a rack on the back side of the cutter head, for traversing the cutter. 
 
 C, handle nuts for fastening the swing slide in the circular slot at any angle required. 
 
 A, driving pulley on the shaft that drives the cutter shaft. This pulley and shaft are fixed. The 
 cutter shaft can be taken out without throwing off the belt, as the two shafts are connected together 
 with a clutch coupling. 
 
 N, hand-wheel on screw for traversing cutter head to suit geers of different diameters, or in cutting 
 worm geers, &c. ; to the left of N are seen the screw and lever for fastening the head. 
 
 51 
 
GEERING. 
 
 G, dial or index, having rows of holes of different numbers drilled on its face, which can be divided 
 by means of the point and spring I, so as to cut any number of teeth wanted. 
 
 K is a hand lever for raising spindle F and dial with the geer while cutting. This lever is on one 
 end of a short shaft passing through a stand, and on the other end there is a quadrant, to which a chain 
 is attached, that connects with the long lever L, on which the spindle stands. 
 
 1987 
 
 L" is a balance lever weight to counterbalance the spindle and dial, and the weight of the geer 
 which is being cut. The weight has a movable cover for the purpose of putting in more or less weight, 
 as the heft of the geer being cut may require. 
 
 To the extreme left of the upper part of the frame, Fig. 1986, is a thumb-nut for tightening up the 
 spindle in case it should wear and become loose. 
 
 H, pulley over which the cord passes to support the weight P, used in cutting spiral geers. 
 

 A SHORT TREATISE 
 
 ON THE 
 
 DESIGNING AND CONSTRUCTION 
 
 OF 
 
 GEERING AND MILL-WORK. 
 
 ACCOMPANIED BY AN EXPLANATION OF THE CONSTEUCTION AND USE OP 
 
 THE ODONTOGRAPH OF PEOF. WILLIS. 
 
 \_REPUDLISIIED FROM APPLETOXS' DICTIONARY OF MACHINES, MECHANICS, ENGINE-WORE, AND ENGINEERING, 
 FOR MESSRS. REED <& TENNEY, PROVIDENCE, R. /.] 
 
 NEW YOEK: 
 D. APPLETON AND COMPANY, 
 
 549 & 551 BROADWAY. 
 1873. 
 
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 of the Exact Scierices explained to the Non-Mathe- 
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