Chapter VI: Gearing and Millwork

By 1830 the use of machine tools was becoming general; they were being regularly manufactured and their design was crystallizing. It was the period of architectural embellishment when no tool was complete without at least a pair of Doric columns, and planers were furnished in the Greek or Gothic style. As the first machine frames were made of wood, much of the work probably being done by cabinet makers, it was natural that they should show the same influence that furniture did. It took several generations of mechanics to work out the simpler lines of the later machines.

The application of scientific forms for gear teeth came at about this time with the general development of the machine tool. The suggestion of the use of epicyclic and involute curves is much older than most of us realize. The first idea of them is ascribed to Roemer, a Danish mathematician, who is said to have pointed out the advantages of the epicyclic curve in 1674. De la Hire, a Frenchman, suggested it a few years later, and went further, showing how the direction of motion might be changed by toothed wheels. On the basis of this, the invention of the bevel-gear has been attributed to him. Willis,[60] however, has pointed out that he missed the essential principle of _rolling_ cones, as the conical lantern wheel which he used was placed the wrong way, its apex pointing away from, instead of coinciding with, the intersection of the axes. De la Hire also investigated the involute and considered it equally suitable for tooth outlines. Euler, in 1760, and Kaestner, in 1771, improved the method of applying the involute, and Camus, a French mathematician, did much to crystallize the modern principles of gearing. The two who had the most influence were Camus and Robert Willis, a professor of natural philosophy in Cambridge, whose name still survives in his odontograph and tables. All of the later writers base their work on the latter’s essay on “The Teeth of Wheels,” which appeared originally in the second volume of the “Transactions of the Institution of Civil Engineers,” 1837. Willis’ “Principles of Mechanism,” published in 1841, which included the above, laid down the general principles of mechanical motion and transmission machinery. In fact, many of the figures used in his book are found almost unchanged in the text-books of today. Smeaton is said to have first introduced cast-iron gears in 1769 at the Carron Iron Works near Glasgow, and Arkwright used iron bevels in 1775. All of these, except the last two, were mathematicians; and no phase of modern machinery owes more to pure theory than the gearing practice of today.

[60] “Principles of Mechanism,” p. 49. London, 1841.

Camus gave lectures on mathematics in Paris when he was twelve years old. At an early age he had attained the highest academic honors in his own and foreign countries, and had become examiner of engines and professor in the Royal Academy of Architecture in Paris. He published a “Course of Mathematics,” in the second volume of which were two books, or sections, devoted to the consideration of the teeth of wheels, by far the fullest and clearest treatment of this subject then published. These were translated separately, the first English edition appearing in London in 1806, and the second in 1837.[61] In these the theory of spur-, bevel-, and pin-gearing is fully developed for epicycloidal teeth. In the edition of 1837, there is an appendix by John Hawkins, the translator, which is of unusual interest. He gives the result of an inquiry which he made in regard to the English gear practice at that time.[62] As the edition is long since out of print and to be found only in the larger libraries, we give his findings rather fully. His inquiries were addressed to the principal manufacturers of machinery in which gearing was used, and included, among others, Maudslay & Field, Rennie, Bramah, Clement, and Sharp, Roberts & Company. To quote Hawkins:

[61] “A Treatise on the Teeth of Wheels.” Translated from the French
of M. Camus by John Isaac Hawkins, C.E. London, 1837.

[62] _Ibid._, p. 175.

A painful task now presents itself, which the editor would gladly
avoid, if he could do so without a dereliction of duty; namely, to
declare that there is a lamentable deficiency of the knowledge of
principles, and of correct practice, in a majority of those most
respectable houses in forming the teeth of their wheel-work.

Some of the engineers and millwrights said that they followed
Camus, and formed their teeth from the epicycloid derived from the
_diameter_ of the _opposite wheel_....

One said, “We have no method but the rule of thumb;” another, “We
thumb out the figure;” by both which expressions may be understood
that they left their workmen to take their own course.

Some set one point of a pair of compasses in the center of a tooth,
at the primitive circle (pitch-circle), and with the other point
describe a segment of a circle for the off side of the next tooth....
Others set the point of the compasses at different distances from the
center of the tooth, nearer or farther off; also within or without
the line of centers, each according to some inexplicable notion
received from his grandfather or picked up by chance. It is said
inexplicable, because no tooth bounded at the sides by segments of
circles can work together without such friction as will cause an
unnecessary wearing away.

It is admitted that with a certain number of teeth of a certain
proportionate length as compared with the radii, there may be a
segment of a circle drawn from some center which would give “very
near” a true figure to the tooth; but “very near” ought to be
expunged from the vocabulary of engineers and millwrights; for that
“very near” will depend on the chance of hitting the right center and
right radius, according to the diameter of the wheel, and the number
of teeth; against which hitting, the odds are very great indeed.

Among the Mathematical Instrument Makers, Chronometer, Clock and
Watch Makers, the answers to the inquiries were, by some, “We have
no rule but the eye in the formation of the teeth of our wheels;” by
others, “We draw the tooth correctly on a large scale to assist the
eye in judging of the figure of the small teeth;” by another, “In
Lancashire, they make the teeth of watch wheels of what is called
the bay leaf pattern; they are formed altogether by the eye of the
workman; and they would stare at you for a simpleton to hear you talk
about the epicycloidal curve.” Again, “The astronomical instrument
makers hold the bay leaf pattern to be too pointed a form for smooth
action; they make the end of the tooth more rounding than the figure
of the bay leaf.”

It is curious to observe with what accuracy the practiced eye will
determine forms.... How important it is, then, that these Lancashire
bay leaf fanciers should be furnished with pattern teeth of large
dimensions cut accurately in metal or at least in cardboard; and
that they should frequently study them, and compare their work with
the patterns. These Lancashire workmen are called bay leaf fanciers,
because they cannot be bay leaf copiers; since it is notorious that
there are not two bay leaves of the same figure.

Hawkins then describes a method of generating correctly curved teeth, or rather of truing them after they had been roughly formed, devised by Mr. Saxton of Philadelphia, “who is justly celebrated for his excessively acute feeling of the nature and value of accuracy in mechanism; and who is reputed not to be excelled by any man in Europe or America for exquisite nicety of workmanship.” By this method the faces of the teeth were milled true by a cutter, the side of which lay in a plane through the axis of a describing circle which was rolled around a pitch circle clamped to the side of the gear being cut. It is by this general method that the most accurate gears and gear cutters are formed today.

While he by no means originated the system, Hawkins seems first to have grasped the practical advantages of the involute form of teeth. Breaking away from the influence of Camus, the very authority he was translating, who seems to have controlled the thought of everyone else, Hawkins writes the following rather remarkable words:[63]

[63] _Ibid._, pp. 160 _et seq._

Since M. Camus has treated of no other curve than the epicycloid,
it would appear that he considered it to supersede all others for
the figure of the teeth of wheels and pinions. And the editor must
candidly acknowledge that he entertained the same opinion until after
the greater part of the foregoing sheets were printed off; but on
critically examining the properties of the involute with a view to
the better explaining of its application to the formation of the
teeth of wheels and pinions, the editor has discovered advantages
which had before escaped his notice, owing, perhaps, to his prejudice
in favor of the epicycloid, from having, during a long life, heard
it extolled above all other curves; a prejudice strengthened too by
the supremacy given to it by De la Hire, Doctor Robison, Sir David
Brewster, Dr. Thomas Young, Mr. Thomas Reid, Mr. Buchannan, and
many others, who have, indeed, described the involute as a curve by
which equable motion _might_ be communicated from wheel to wheel,
but scarce any of whom have held it up as equally eligible with the
epicycloid; and owing also to his perfect conviction, resulting from
strict research, that a wheel and pinion, or two wheels, accurately
formed according to the epicycloidal curve, would work with the least
possible degree of friction, and with the greatest durability.

But the editor had not sufficiently adverted to the case where one
wheel or pinion drives, at the same time, two or more wheels or
pinions of different diameters, for which purpose the epicycloid
is not perfectly applicable, because the form of the tooth of the
driving wheel cannot be generated by a circle equal to the _radius_
of more than one of the driven wheels or pinions. In considering this
case, he found that the involute satisfies all the conditions of
perfect figure, for wheels of any sizes, to work smoothly in wheels
of any other sizes; although, perhaps, not equal to the epicycloid
for pinions of few leaves.

With Joseph Clement, he experimented somewhat to determine the relative end-thrust of involute and cycloidal teeth, deciding that the advantage, if any, lay with the former. He details methods of laying out involute teeth and concludes:

Before dismissing the involute it may be well to remark that what
has been said respecting that curve should be considered as a mere
sketch, there appearing to be many very interesting points in regard
to its application in the formation of the teeth of wheels which
require strict investigation and experiment.

It is the editor’s intention to pursue the inquiry and should he
discover a clear theory and systematic practice in the use of the
involute, he shall feel himself bound to give his views to the public
in a separate treatise. He thinks he perceives a wide field, but is
free to confess that his vision is as yet obscure. What he has given
on the involute is more than was due from him, as editor of Camus,
who treated only of the epicycloid, but the zeal of a new convert to
any doctrine is not easily restrained.

So far as the writer knows this is the first _real_ appreciation of the value of the involute curve for tooth outlines, and Hawkins should be given a credit which he has not received,[64] especially as he points the way, for the first time, to the possibility of a set of gears any one of which will gear correctly with any other of the set. It was thought at that time that there should be two diameters of describing circles used in each pair of gears, each equal to the pitch radius of the opposite wheel or pinion. This gave radial flanks for all teeth, but made the faces different for each pair. The use of a single size of describing circle throughout an entire set of cycloidal gears, whereby they could be made to gear together in any combination, was not known until a little later.

[64] John Isaac Hawkins was a member of the Institution of Civil
Engineers. He was the son of a watch and clock maker and was born
at Taunton, Somersetshire, in 1772. At an early age he went to the
United States and “entered college at Jersey, Pennsylvania, as
a student of medicine,” but did not follow it up. He was a fine
musician and had a marked aptitude for mechanics. He returned to
England, traveled a great deal on the Continent, and acquired a wide
experience. He was consulted frequently on all kinds of engineering
activities, one of them being the attempt, in 1808, to drive a tunnel
under the Thames. For many years he practiced in London as a patent
agent and consulting engineer. He went to the United States again in
the prosecution of some of his inventions, and died in Elizabeth, N.
J., in 1865. From a Memoir in the “Transactions of the Institution of
Civil Engineers,” Vol. XXV, p. 512. 1865.

Professor Willis seems to be the first to have pointed out the proper basis of this interchangeability in cycloidal gearing. With the clearness which characterized all his work he states: “If for a set of wheels of the same pitch a constant describing circle be taken and employed to trace those portions of the teeth which project beyond each pitch line by rolling on the exterior circumference, and those which lie within it by rolling on its interior circumference, then any two wheels of this set will work correctly together.... The diameter of the describing circle must not be made _greater_ than the radius of the pitch-circle of any of the wheels.... On the contrary, when the describing circle is _less_ in diameter than the radius of the pitch-circle, the root of the tooth spreads, and it acquires a very strong form.... The best rule appears to be that the diameter of the constant describing circle in a given set of wheels shall be made equal to the least radius of the set.”[65] This practice is standard for cycloidal gearing to this day. In his “Principles of Mechanism,” Willis did the work on involute gearing which Hawkins set before himself; and also describes “a different mode of sizing the teeth” which had “been adopted in Manchester,” for which he suggests the name “diametral pitch.”[66]

[65] Willis: “Principles of Mechanism,” Articles 114-116. London,
1841. See also “Transactions of the Institution of Civil Engineers,”
Vol. II, p. 91.

[66] Diametral pitch, which is credited to John George Bodmer, was
long known as “Manchester pitch.”