case the two interior epicycloids e h, b a, will be straight lines, tending to the centres B and A, and the labour of the mechanic will by this means be greatly abridged."

It is unnecessary to copy the engraving to which these letters of reference relate, as the demonstrations of M. Camus given above in the 6th Corollary (542), as deduced from the preceding numbers, commencing with 536, is so complete and satisfactory, that any one who will attentively study that portion of the work, and refer to the figures 178, 179, 180, 181, 182, and 183, plate 26, will see that the diameter of the generating circle of the epicycloid, must be equal to the RADIUS, and not to the diameter of the opposite WHEEL or PINION.

When, instead of the teeth of a pinion, the rungs of a lantern are treated of, and considered as of infinitely small dimensions, then, and then only, the generating circle of the epicycloid is defined to be equal to the DIAMETER, and not to the radius of the LANTERN; as is clearly shown by the 7th Corollary (543). And here appears to be the origin of the erroneous view of the editor of Imison's Elements ;' he has not distinguished between the epicycloid acting against a radial line, and that acting against a point: the radical line in the wheel or pinion demanding an epicycloid derived from the radius, while the point in the lantern requires an epicycloid derived from the diameter.

That the wheel or pinion requires the tooth of the driving-wheel or pinion to be an epicycloid generated by a circle equal to the radius of the wheel, or pinion, is again shown in the second solution (548), where these words occur, "the generating circle of which Y, has for diameter, the radius A B of the pinion," referring to fig. 171, where the generating circle Y is shown, by dotted lines, to be half the diameter of the primitive circle of the pinion.

And in the third part of the same solution (548), these words are found, "the generating circle of which V, has for diameter, the radius AF of the wheel," and this generating circle V is represented by dotted lines also in fig. 171.

There is therefore no obscurity in the language of M. Camus on this subject, and it would be superfluous to quote other passages to the same purport, from his descriptions and definitions of the modes of forming the teeth of bevel wheels, pinions, and lanterns, where he demonstrates that the same laws govern the genesis of the spherical epicycloid. (See 557 to 570.)

Although the above proofs must be sufficient for those who will go into the subject with the attention it deserves, yet as the false notion has obtained an extensive acquiescence on the part of many of our first-rate engine manufacturers, some of whom are pouring into the market multitudes of cast-iron wheels and pinions of various magnitudes, for cotton and other machinery, with teeth formed from the epicycloid of the diameter, instead of the radius of the opposite wheel, or of the opposite pinion; the following extracts from Rees's Cyclopædia (article "Clock movement," said to have been written by Mr. Thomas Reid, an eminent clock-maker of Edinburgh), are given in corroboration of the views before elicited, and in the hope of awakening those manufacturers to a sense of the injuries which they are occa

sioning to their customers, by supplying them with wheel-work that must wear out in a few years, instead of lasting the greater part of a century, which many of the wheels would do were the teeth formed on true principles.

"Camus, in his 'Cours de Mathématique,' liv. 10 and 11, has investigated the epicycloid as it forms a rule for the formation of teeth in wheel-work, which portion of the work has lately been translated into English, but the translator has added some practical directions respecting the shape of a tooth, taken from Imison's Elements of Science and Art,' the principles of which we think it necessary to correct; at the same time that we avail ourselves of the elucidation of our subject which Camus's masterly treatment of it affords."

*

"The translator of a part of Camus, and the editor of Imison's 'Elements of Science and Art,' have positively, though erroneously, asserted that the generating circle should, in all cases of wheels and pinions, be equal to the fellow of the wheel on which the curve is to be described, in which opinion some very respectable mechanicians agree; but others, on the contrary, assert, with equal confidence and more truth, that the said generating circle should have its diameter equal to only one half of the diameter of the said fellow. (See Camus, Dr. Young's 'Syllabus,' and Brewster's edition of Ferguson's 'Select Exercises,' &c.)

66 A careful examination of Camus's demonstrations would of itself have reconciled the disagreeing parties, which we trust a due attention to our elucidation, by means of the tracer and radial lever, will not fail to effect. The fact is, that where pins like our tracers or spindles, are used for teeth in any wheel or lantern, as is frequently the case in large works, the generating circle must be equal in diameter to the diameter of the acting wheel or lantern which it represents, in order to trace the epicycloidal teeth of its fellow;* but in clock movements, and in all other instances in wheel-work where both the wheels and pinions have the epicycloidal formation, the generating circle must be only one half in diameter to what is required when lanterns are used, for in this case, which is most frequent, the interior and exterior epicycloids impel each other alternately, the former being a portion of a radial lever, and the latter a portion of the epicycloidal curve."

"Olaus Roomer, the celebrated astronomer and mechanist of Denmark, according to Wolfius and Leibnitz, was the first who pointed out the utility of the epicycloidal curve, when applied to delineate the shape of a tooth; but De la Hire took up the subject after him, and demonstrated that if a tooth of either a wheel or pinion be formed by portions of an exterior epicycloid, described by a generating circle of any diameter whatever, the tooth of its fellow will be properly

*It should here be remembered that in this case the epicycloidal tooth must be first formed, as if working on a rung of infinitely small diameter, and then cut away to suit a rung of the given finite diameter; as clearly explained above by M. Camus (545, 546).

formed, by portions of an interior epicycloid, described by the same generating circle; which curious circumstance allows of an infinite variety in the two corresponding curves that form the teeth of the wheel and pinion, if they were practicable."

The epicycloidal curves are applicable to practice, only in proportion as they approximate to that generated by a circle, the diameter of which is equal to the radius of the opposite wheel or pinion.

The student, however, will be well rewarded for his trouble in going through the investigations of M. De la Hire, in the original, published in Paris in 1694, or in the translation of a considerable part of them into English, by Mandey, published in London in 1696.

M. De la Hire takes a different method of investigation, but arrives at the same conclusion as M. Camus; but M. Camus's process of reasoning is more concise, elegant, and clear.

It would be tedious to multiply quotations to enforce a point already so strongly proved. Should, however, any person have doubts still remaining, he would do well to pursue the inquiry, by referring to Dr. Thomas Young's Philosophy (published in London in 1807, 2 vols. 4to), page 176 of vol. i., plate 15, and page 55 of vol. ii.; Buchanan'sEssay on the Teeth of Wheels,' revised by Peter Nicholson (London, 1808, 8vo), pages 15 to 35; Ferguson's 'Lectures,' edited by Sir David Brewster (Edinburgh, 1806), pages 210 to 226; or indeed to any other author who has written on the teeth of wheels, except the editor of Imison's 'Elements;' all besides him having irrefutably proved that the epicycloidal part of a tooth, designed to act on another wheel or pinion, against a part of a tooth lying in a plane cutting the two axes, must, to ensure smooth and durable action, be generated by a circle equal to the radius of the wheel or pinion with which it is to be engaged; and not equal to the diameter, as contended for by the editor of Imison's 'Elements,' and unfortunately acted on by many of our most eminent engine manufacturers, some of whom have confessed that they adopted the erroneous practice without investigation, from faith in the very extensive mechanical knowledge possessed by the promulgator of the error.

The Editor appreciates that knowledge so highly, that he has been in the habit, for more than thirty years, of designating his friend a walking encyclopædia; and has referred to him hundreds of times for information, as to what has been discovered and performed in numerous branches of science and art. His personal regard, therefore, would have prompted him to screen the editor of Imison's 'Elements' from the exposure of his error, could he have forgotten the public duty which devolves upon him as editor of Camus; that duty imperiously calling on him to display the truth of the subject treated on, in the fullest and clearest manner, irrespective of the feelings or prejudices of any person whomsoever.

In order to address the understanding through the medium of the eye, for the information of those who either cannot or will not go into the mathematical investigation of this subject, two segments of the rims of spur wheels, the one of two feet, and the other of one foot radius, with epicycloidal teeth, are shown in plate 38. The shaded

teeth are of a proper length for smooth and durable action, either in wood or iron, requiring no play in the engagement, the entering corners of the teeth passing by each other without touching. Many millwrights, however, prefer the teeth somewhat longer, as represented by the dotted line a b; and some few, addicted to old fashions, make the teeth as long as is shown by the round dotted line c d.

Our forefathers made the teeth long for the sake of having several in action at the same time, in order that each tooth should have to bear only a part of the strain; but experience has been gradually shortening the teeth of wheels; and the best mechanicians now hold, that teeth longer than those represented by the shaded part of plates 38 and 39, are injurious.

It is much more advantageous to extend the bearing surfaces, by giving additional thickness to the wheels, and thereby additional breadth to the teeth; the division of the strain is thus effected without increasing the sliding of tooth upon tooth, which in long teeth is very considerable, but in short teeth is so small, that many good mechanicians have expressed an opinion that there is no sliding; because, say they, the truly-formed epicycloidal tooth rolls on the radial surface against which it acts; this is, however, an error, as will be obvious on inspecting the places of contact of the several teeth shown in plate 38, where, if we suppose the segment A B to be the driver, moving downwards and driving the segment C D, it is obvious, that the point of contact e of two teeth, when situated in the line of centres xy, will be at the place where the primitive circles z 1, 2 3, cut that line; but in descending the distance of a tooth and a space, the point of contact will be a little within the primitive circle of the segment C D, while the point g, which was in contact at e, will have receded from that primitive circle near f, by sliding four-tenths of an inch. If the teeth were elongated, as shown by the dotted lines h b i, i a k, and the motion continued the distance of another tooth and space, the point of contact i would be a little farther within the primitive circle of the segment C D, but the point l, which was in contact at e, when in the line of centres, would, by moving the distance of two teeth and two spaces, have slidden sixteen-tenths of an inch from that part of the primitive circle near i; a quantity of sliding too destructive to be compensated for by any advantage that could be derived from dividing the strain, by having two teeth always in contact instead of one. The quantity of sliding of any tooth on another, is always equal to the distance of the two primitive circles from each other, at the place nearest to the final contact of the two teeth.

If the other segment be considered as the driver, or both the segments be supposed to move upwards instead of downwards, the same results will become manifest, and it will be clearly seen that the sliding of the tooth increases in a direct ratio with the angular distance of the point of contact from the line of centres.

Short teeth, therefore, have a great advantage over long ones on the score of sliding; and they have a great advantage too in respect of strength; and hence equal strength may be obtained from thinner teeth; and a greater number may be given to a wheel, whereby the

strain may be beneficially divided, without occasioning the evil of excessive sliding, by which an injurious degree of friction would be created, to be overcome by a continual waste of power.

The dotted lines m n, no, show the elongation of a tooth of the segment CD, generated, as are the elongations of all the other teeth, by a circle equal to the radius of the opposite segment.

The dotted lines between and below the roots of the teeth, represent the necessary deepening of the spaces to allow the elongations free play. The dotted circular lines p q, show that it is not necessary to weaken the tooth, by making the space angular at the bottom.

It is obvious, on inspection, that the curves m n, no, h i, i k, are of the proper figure for acting on the radial lines, constituting the sides of the spaces of the opposite segments. Now these curves are accurately generated by circles, the diameters of which are equal to the radii of the primitive circles of the opposite segments; and are in strict accordance with the demonstrations of M. Camus.

The curve contended for by the editor of Imison's 'Elements,' is shown by the dotted lines mr, ro, which are generated by a circle equal in diameter with the diameter of the primitive circle of the opposite segment. It is clear from inspection, that a tooth thus bounded, could not act, except the sides of the space were made of a different figure from the line of radius. In fact, the part of the tooth within the primitive circle would have to be cut away, less or more, according to the length of the tooth; thus, if the tooth were of the length represented by the shaded part, an indentation should be made in the side of the tooth, equal to that shown by the short curved dotted line st; if the tooth were of the length indicated by the dotted line a b, the indentation would have to be longer and deeper, as shown by the curved dotted line su; and if the tooth were of the length shown by the dotted lines mr, ro, then the indentation ought to be as long and deep as that shown by the dotted line i v.

If these indentations are not made in the wheel by the millwright, the falsely-formed teeth will, in the course of working, excavate such indentations for themselves: hence we see too generally in old wheels, the teeth worn away into deep hollows within the primitive circles, while the points of the teeth have retained nearly their original false figure.

Many persons, from observing this form of old teeth, have adopted the erroneous notion that this is nature's form, and consequently must be the correct figure; they have, therefore, made the faces of their new teeth accurate copies of the old ones; merely taking care that the new teeth should be as strong as the old ones were before they began to wear away.

Now if that part of the teeth of wheels, projecting beyond the primitive circles, be made truly epicycloidal at the first, and so accurately geered, that the primitive circles shall always cut the line of centres in the same place; and if that part of the teeth within the primitive circles be made plane surfaces, lying in the direction of the radius, there appears to be no reason why, in wearing, they should ever change their figure.