which some teeth have been thus constructed with great skill and success. But as it is often difficult to describe this compound curve, and sometimes impossible to discover its nature, we shall endeavour to select such a form for the teeth as may be easily described by the practical mechanic, while it ensures a uniformity of pressure and velocity In order to avoid circumlocution and obscurity, we shall call, as is customary with practical men, the small wheel (which is supposed always to be driven by a greater one) the pinion, and its teeth, the leaves of the pinion. The line which joins the centres of the wheel and pinion may be called the line of centres. Now there are three different ways in which the teeth of one wheel may act upon the teeth of another: and cach of these modes of action requires a different form for the teeth.

1. When the teeth of the wheel begin to act upon the leaves of the pinion just as they arrive at the line of centres; and, when their mutual action is carried on after they have passed this line. II. When the teeth of the wheel begin to act upon the leaves of the pinion, before they arrive at the line of centres, and conduct them either to this line, or a very little beyond it.

III. When the teeth of the wheel begin to act upon the leaves of the pinion, before they arrive at the line of the centres, and continue to act after they have passed this line.

I. The first of these modes of action is recommended by Ca mus and De la Hire, the latter of whom has investigated the form of the teeth solely for this particular case. When this mode of action is adopted, the acting faces of the leaves of the pinion should be parts of an interior epicycloid generated by a circle of any diameter rolling upon the concave super bicies of the pinion, and the acting faces of the teeth of the wheel should be portions of an exterior epicycloid formed by the same generating circle rolling upon the convex superficies of the wheel.

Now it is demonstrable (see the article CYCLOID, Supp. English Encyclo.) and has before been mentioned in our article PARALLEL MOtions, that when one circle rolls within another whose diameter is double that of the rolling circle, the line generated by any point of the latter will be a straight line, tending to the centre of the larger circle. If the generating circle, therefore, mentioned above, should bet ken with its diameter equal to the radius of the pinion, and be made to roll upon the concave superficies of the pinion, it will generate a straight line tending to the pinion's centre, which will be the form of the acting faces of its leaves; and the teeth of the wheel will, in this case, be exterior epicycloids, formed by a generating circle, whose diameter is equal to the radius of the pinion, rolling upon the convex superficies of the wheel This form of the teeth, viz. when the acting faces of the pinion's leaves are right lines tending to its centre, is exhibited in fig.. 14. pl. XXXII. and is perhaps the most advantageous, as it requires less trouble, and may be executed with greater accuracy than if the epicycloidal form had been employed: it is justly recommended both By De la Hire and Camus as particularly advantageous in clock and watch work.

The attentive reader will perceive that, in order to prevent the teeth of the wheel from acting upon the leaves of the piùion, befor: REV. MARCH, 1807.

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they

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they reach the line of centres, and that one tooth of the wheel may not quit the leaf of the pinion till the succeeding tooth begins to act upon the succeeding leaf, there must be a certain proportion between the number of leaves in the pinion and the number of teeth in the wheel, or between the radius of the pinion and the radius of the wheel, when the distance of the leaves is given. But in machinery the number of leaves and teeth are always known from the velocity which is required at the working point of the machine: it becomes a matter, therefore, of great importance, to determine with accuracy the relative radii of the wheel and pinion.'

"The author next determines the ratio that ought to obtain between the radius of the wheel and of the pinion; and then he considers what ought to be the form of the teeth of the wheel, when the teeth of the pinion or small wheel are cylindrical staves, fastened between two circular boards or plates parallel to each other. The method which he states may be found in Camus's Dissertation, and is merely graphical. The equation of the curve for the form of the teeth is not given either by the present author or M. Camus.

If the teeth of wheels (says Mr. G.) and the leaves of pinions be formed according to the directions already given, they will act upon each other, not only with uniform force, but also without friction. The one tooth rolls upon the other, and neither slides nor rubs to such a degree as to retard the wheels, or wear their teeth. But as it is impossible in practice to give that perfect curvature to the acting faces of the teeth which theory requires, a certain quantity of friction will remain after every precaution has been taken in the formation of the communicating parts.'

It appears to us that some error must lurk in this passage. If an uniformity of action be produced, friction will ensue : the author has not proved that no friction will ensue; and if we doubt our own speculations and researches on this point, we have great authority to produce against the assertions in the passage just quoted. After having stated the only Values of certain quantities, (x, y,) that satisfy an equation involving the two conditions, uniform motion and no friction, Euler says, sicque prodirent due rota dentibus destituta: ac propter ea fieri nequit, ut utrique conditioni præscripte satisfiat."-Again; "Quoniam autem fieri nequit, ut motus atriusque rota reddetur uniformis, simulque attritus in contactu dentium mutuo evitetur, videndum est utri barum duarum conditionum potius satisfieri conveniat, altera neglecta," &c.

If we examine the form of the teeth of which the author speaks in the first volume, (that is, when they are involutes of circles,) it will readily appear that friction must take place; and indeed, on general grounds, without entering into Euler's intricate equations, it might be shewn that, when equable motion is produced, friction cannot be avoided.

Mr.

Mr. Gregory has spoken of Euler's memoir; and in dark language he has commended it. We plainly declare that there are parts in it, the meaning of which we do not comprehend: we refer not to the symbolical operations, but to an observation. (p.307,) that is made subsequently to the deduction of the equation for the form of the teeth when they act without friction, and when the contact takes place in the line joining the centres of the two wheels. The "Verum thus commences; passage hic ingens incommodum occurrit quo hujusmodi dentes ad praxin planè inutiles redduntur," &c. and the reason which he assigns is that, in the tooth of the wheel A urged by the teeth of another wheel B, a line drawn from the centre of the wheel A must make with the curve of its tooth an obtuse angle: A M.C. is that angle: he then says, " Cum igitur dentium natura non permittat, ut angulus A.M.C. ubique sit obtusus, evidens est, fieri non posse, ut hoc modo rota alia ab alia ad motum incitetur." We apprehend that the angle A.M.C. may be obtuse, and consequently that motion can be produced without friction.

The motion of bodies towards fixed centres, of bodies revolving in curves, and of bodies moving in constrained paths, are treated in the division or book intitled Dynamics. The laws of the motions of the first and second kinds, at least in the cases usually given, are assigned without difficulty; and the author has neatly and exactly demonstrated the theorems relating to central forces, the laws of the motions of projectiles, of the oscillations of bodies in Cycloids, &c.

In treating of motion of the third kind, under which is comprehended the rotation of bodies round a fixed Axis, he has availed himself of the labours and inventions of the geometricians of the continent, but the matter might have been better arranged and woven together. In the rotation of bodies round a fixed Axis, it is the fundamental proposition that occasions the difficulty of demonstration: that being once established, and an expression for the force acting on a particle at a given distance from the centre of motion being obtained, all the ordinary propositions follow with the greatest ease; and expressisions for the centre of Oscillation and Gyration flow as corollaries from the original propositions. To this subject belongs almost all that Mr. Atwood has included within his 6th section. There are two methods,-related methods, however,-of treating this part of Mechanics: if A, B, C, &c. be the parts of a system, the force acting on any particle (as A) may be found by dividing the moving force by the quantity of matter moved, taking into consideration the difference of the velocities with which the several parts are moved: from such accelerating force, the velocity may be found, and the time. This is

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nearly the plan pursued by Mr. Atwood in his sixth section; and it is sufficiently plain in those cases in which the parts of a system revolve round a fixed Axis. If, however, the veloci ties of the parts of the system do not vary as their distances from the centre of motion, (which case happens when the parts of the system are connected with flexible strings,) although the accelerating force may be found, yet the determination of the velocity will frequently depend on a difficult Integral. A confirmation of the truth of this remark may be obtained from the problem given by Mr. A. towards the end of his sixth section; which problem, in consequence of its erroneous solution, has been several times discussed in a valuable work intitled Leybourn's Mathematical Repository. Here we may remark that this problem was solved (and exactly solved) seventy years ago by Bernoulli. If we recollect rightly, for we cannot immediately refer to the proper documents, neither Thomas Simpson nor Atwood, nor yet the subsequent demonstrators of the problem, have noticed this circumstance. The determination of the accelerating force, then, is one method of solving this class of problems. Foreign mathematicians employ the theorem of the Conservatio Virium Vivarum. In Mr. Leybourn's publication, just mentioned, a partial demonstration of this theorem has been given by Mr. Dawson; and the employment of it undoubtedly leads to the solution of problems with greater facility and conciseness than the former method.

This theorem was employed by Bernoulli as a principle for the solution of various problems in Dynamics; and it is indeed most fruitful in the consequences to which it leads. The merit of this mathematician, and of Leibnitz, is not always in this country fairly and sufficiently appreciated. They were emulous, perhaps envious, of Newton; and therefore Englishmen, zealous in opposing their claims to mathematical distinction and pre-eminence, depressed their real merit beyond its just level. We still in some measure retain, and our books communicate to us, this prejudice: but it would be corrected if we now examined their writings; and we should then probably confess that, although Newton was the greatest mathematician and philosopher, Bernoulli and Leibnitz were certainly very great mathematicians and philosophers.

At p. 265. cor. 5. a wrong inference is made concerning the convertibility (if we may so term it) of the centres of suspension and oscillation. If S, O, G, be the centres of suspension, oscillation, and gravity, respectively, it does not follow immediately, because SO. SG=md2+m'd+ &c. that, if S be transferred to O, O will be transferred to S: the thing is true, but a process or proposition is omitted: it ought to be shewn that SG=

, g', g", &c. being the respective distances of the particles m, m', m', &c. from the centre of gravity.

In the first volume of this work are contained, besides Mechanics properly so called, Hydrostatics, Hydrodynamics, Pneumatics, &c. We approve the latter part of the volume less than the first. The subjects, indeed, are in their nature rather vague, and bear not easily the strictness and precision of mathematical discussion: but the author has not laboured

with felicity; his article on Hydrostatics is rather meagre, and to our taste much too wordy.-Over the second volume, which contains the account of Machines, we have whiled away many an hour, and have gained some instruction. The drawings are well executed, and, which is seldom the case, adequately present to the eye the construction of the machines. Mr. Gregory, however, sometimes writes too much or not enough for instance, in Archimedes' screw, the mathematical processes, which do not go one-tenth of the way to explain its theory, should have been omitted, and a plain description of the uses and mode of action of the machine alone retained :—while to aid the explanation of the machine next described, (the shoe-maker's implement to enable him to work in an upright posture,) a diagram ought to have been added.

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To the second volume, and the accompanying volume of plates, this publication is principally indebted for its claim to distinction and patronage. It is our duty and our wish to state and enforce that claim; for although, in the nicety or fastidiousness of criticism, we should reject some parts and alter others, yet the performance on the whole is useful and valuable, creditable equally to the talents and the industry of its author.

ART. III. New Observations on the Natural History of Bees, by Francis Huber. Translated from the Original. 12mo. pp. 310. 5s. 6d. Boards. Longman and Co. 18c6.

SINCE the experiments reported in these pages appear to have been conducted with great accuracy, and especially since they lead in several instances to curious and very unexpected results, we cannot refrain from expressing our surprize that, during a term of fifteen years, they should have remained inaccessible to the mere English reader. This singular fact may, perhaps, be partly ascribed to the general diffusion of the French language among our men of science, and partly to a defect of zeal in the prosecution of entomological studies.