Mechanics, from μηχανή, a machine, is the science which inquires into the laws of the equilibrium and motion of solid bodies; into the forces by which bodies, whether animate or inanimate, may be made to act upon one another; and into the means by which these may be increased so as to overcome such as are more powerful. The term mechanics was originally applied to the doctrine of equilibrium. It has by some late writers been extended to the motion and equilibrium of all bodies, whether solid, fluid, or aeriform; and has been employed to comprehend the sciences of hydrodynamics and pneumatics.

HISTORY.

As the science of mechanics is intimately connected with the arts of life, and particularly with those which exist even in the rudest ages of society, the construction of machines must have arrived at considerable perfection before the theory of equilibrium, or the simplest properties of the mechanical powers, had engaged the attention of philosophers. We accordingly find that the lever, the pulley, the crane, the capstan, and other simple machines, were employed by the ancient architects in elevating the materials of their buildings, long before the dawn of mechanical science; and the military engines of the Greeks and Romans, such as the catapulte and balista, exhibit an extensive acquaintance with the construction of compound machinery. In the splendid remains of Egyptian architecture, which in every age have excited the admiration of the world, we perceive the most surprising marks of mechanical genius. The elevation of immense masses of stone to the tops of these stupendous fabrics, must have required an accumulation of mechanical power which is not in the possession of modern architects.

The earliest traces of any thing like the theory of mechanics are to be found in the writings of Aristotle. In some of his works (Questions Mechanicae) we discover a few erroneous and obscure opinions respecting the doctrine of motion, and the nature of equilibrium; and in his 28th mechanical question he gives some vague observations on the force of impulse, tending to point out the difference between that species of force and pressure. He maintained that there cannot be two circular motions opposite to one another; that heavy bodies descended to the centre of the universe, and that the velocities of their descent were proportional to their weights. In answering the question, Why "small powers move great weights by means of the lever, when they have, besides, the weight of the lever itself to move?" Aristotle states, that the power moves the weight more easily in proportion as it is more distant from the fulcrum; and the reason which he assigns for this is, "that the end farthest from the centre describes a greater circle, so that the power which moves the weight will, by the same force, be transferred through a larger space. In discussing the same question, he remarks, that the shorter arm of the lever is moved more against nature than the other; a distinction which shows the inaccurate notions which he entertained on the subject.

The views of Aristotle, however, were so confused and erroneous, that the honour of laying the foundation of theoretical mechanics is exclusively due to the celebrated Archimedes, who, in addition to his inventions in geometry, discovered the general principles of hydrostatics. In his two books entitled Isoporricea, or De Equiponderantibus, he has demonstrated that when a balance with unequal arms is in equilibrium, by means of two weights in its lever, opposite scales, these weights must be reciprocally proportional to the arms of the balance. From this general principle all the other properties of the lever, and of machines referrible to the lever, might have been deduced as corollaries; but Archimedes did not pursue the discovery through all its consequences. In demonstrating the leading property of the lever, he lays it down as an axiom, that if the two arms of the balance are equal, the two weights must also be equal when an equilibrium takes place; and then shows, that if one of the arms be increased, and the equilibrium still continue, the weight appended to that arm must be proportionally diminished. This important discovery conducted the Syracusan philosopher to another equally useful in mechanics. Reflecting on the construction of his balance, which moved upon a fulcrum, he perceived that the two weights exerted the same pressure on the fulcrum as if they had both rested upon it. He then considered the sum of these two weights as combined with a third, and the sum of these three as combined with a fourth; and he thus saw, that in every such combination the fulcrum must support their united weight; and therefore that there is in every combination of bodies, and in every single body and the which may be conceived as made up of a number of lesser centre of bodies, a centre of pressure or gravity. This discovery gravity. Archimedes applied to particular cases, and pointed out the method of finding the centre of gravity of plane surfaces, whether bounded by a parallelogram, a triangle, a trapezium, or a parabola. It appears also from Plutarch and other ancient authors, that a great number of machines which have not reached our times were invented by this philosopher. The military engines which he employed in the siege of Syracuse, against those of the Roman engineer Appius, are said to have displayed the greatest mechanical genius, and to have retarded the capture of his native city.

In his mathematical collections Pappus has published a theory of the inclined plane, the pulley, the wheel and axle, the screw, and the wedge; but, for reasons which we do not know, these theories have been ascribed to Archimedes. In his theory of the inclined plane, Pappus thus announces the question: "Given a weight, and the power which will move it along a horizontal plane, to find the power which will move it along a given inclined plane." In the erroneous solution which he gives of this question, he supposes the weight to be formed into a sphere, and placed on the inclined plane; and he regards the weight of this sphere as supported by a lever, the fulcrum being the point of contact of the sphere with the plane, and the power being applied at the extremity of the horizontal radius. "No reasonable ground," says Mr. Whewell, "is or can be assigned for identifying the effects of such a lever with those of the inclined plane, for which it is thus substituted."

Among the various inventions which we have received from antiquity, that of water-mills is entitled to the highest place, whether we consider the ingenuity which they mills and wind-mills. display, or the useful purposes to which they are subservient. In the infancy of the Roman republic, the corn was ground by hand-mills, *mola versatiles*, consisting of two millstones, one of which was moveable and the other at rest; and at one period these mills were driven by bondsmen. Shafts were subsequently added; and when cattle were yoked to them, they were called *mola frumentariae*. The upper millstone was made to revolve either by the hand applied directly to a winch, or by means of a rope winding round a capstan. The precise time when the impulse or the weight of water was substituted in the place of animal labour is not exactly known. From an epigram in the *Anthologia Graeca*, and from Vitruvius's description of them, it is evident that water-mills were invented previously to the reign of Augustus. The invention of wind-mills is of a later date. According to some authors, they were first used in France in the sixth century; while others maintain that they were brought to Europe in the time of the crusades, and that they had long been employed in the East, where the scarcity of water precluded the application of that agent to machinery. In the twelfth century they had come into use both in France and in England.

The problem of the inclined plane assumed a more distinct shape in the hands of Cardan, though he was unsuccessful in his solution of it. By considering that the force necessary for supporting a weight upon an inclined plane was nothing when the plane was horizontal, or its inclination nothing, and that a force equal to the whole weight of the body was necessary to support it when the plane was vertical, or its inclination 90°, he concluded that the force increased uniformly from nothing to the weight of the body, and was therefore proportional to the angle of the plane's inclination.

The Marquis Guido Ubaldi, the pupil of Commandine, and descended of the illustrious house of Del Monte, devoted much of his time to mechanical statics. His principal works are, *Mechanicarum Libri sex*, Pisa, 1577; *De Archimedem de Equiponderantibus Paraphrasis*; and a treatise on the Screw (*De Cochlea*), published after his death by his son in 1616, in which he gives a full account of Archimedes's forces.

Several mechanical topics were discussed about this time by T. B. Benedetti, in a work entitled *Diversarum Speculationum Math. et Phys. Liber*, which were published at Turin in 1585. He demonstrates that the forces in the bent lever are as the perpendiculars drawn from the fulcrum to the lines of direction in which they act; and he ascribes the centrifugal force of moving bodies to their tendency to move in straight lines. Benedetti was among the small number of philosophers who supported the Copernican system.

Simon Stevinus of Bruges, a Dutch mathematician, contributed greatly to the progress of mechanical science. He discovered the parallelogram of forces, and has demonstrated, in his work on statics, published in 1586, that if a body is urged by two forces in the direction of the sides of a parallelogram, and proportional to these sides, the combined action of these two forces is equivalent to a third force acting in the direction of the diagonal of the parallelogram, and having its intensity proportional to that diagonal. This important discovery, which has been of such service in the different departments of physics, should have conferred upon its author a greater degree of celebrity than he has actually enjoyed. His name has scarcely been enrolled in the temple of fame; but justice may yet be done to the memory of such an ingenious man. He had likewise the merit of illustrating other parts of statics; and he appears to have been the first who, without the aid of the properties of the lever, discovered the law of equilibrium in bodies placed on an inclined plane, namely, that a body resting upon such a plane will be in equilibrio, when the power acting parallel to the plane is to the weight as the height of the plane is to its length. Mr Drinkwater, in his excellent life of Galileo, has stated that this problem had received an earlier solution from Jordanus in the thirteenth century, and that the solution was published by Tartalea in 1565. Mr Whewell, however, is of opinion "that this (Jordanus's) solution, even if it be interpreted so as to be right in the result, was mixed up with many of the usual Aristotelian errors on such subjects, and was not connected, so far as we know, with any consistent and tenable train of mechanical reasoning. We may still, it would seem, consider Stevin to be the father of modern statics." His works were reprinted in the Dutch language in 1605. They were translated into Latin in 1608, and into French in 1634; and in these editions of his works, his Statics were enlarged by an appendix, in which he treats of the rope machine, and on pulleys acting obliquely.

The doctrine of the centre of gravity, which had been applied by Archimedes only to plane surfaces, was now extended to solid bodies by Lucas Valerius, professor of mathematics at Rome. In his work entitled *De Centro Gravitatis Solidorum*, published at Bologna in 1661, he has discussed this subject with such ability, as to receive from Galileo the honourable appellation of the *Novus natra etatis Archimedes*, the Archimedes of the present age.

In the hands of Galileo the science of mechanics assumed a new form. In 1572 he wrote a small treatise on statics, which he reduced to this principle, that it requires an equal power to raise two different bodies to altitudes in the inverse ratio of their weights, or that the same power is requisite to raise 10 pounds to the height of 100 feet, and 20 pounds to the height of 50 feet. This fertile principle was not pursued by Galileo to its different consequences. It was left to Descartes to apply it to the determination of the equilibrium of machines, which he did in his explanation of machines and engines, without acknowledging his obligations to the Tuscan philosopher. In addition to this new principle, Galileo enriched mechanics with his theory of local motion. This great discovery has immortalized its author; and whether we consider its intrinsic value, or the change which it produced on the physical sciences, we are led to regard it as nearly of equal importance with the theory of universal gravitation, to which it paved the way. The first hints of this new theory were given in his *Systema Cosmicum*, dialogues ii. The subject was afterwards fully discussed in another, entitled *Discursus et Demonstrations Mathematicae circa duas novas Scientias pertinentes ad Mechanicam et Motum Localem*, and published in 1638. This work is divided into four dialogues; the first of which treats of the resistance of solid bodies before they are broken; the second points out the cause of the cohesion of solids; in the third he discusses his theory of local motions, comprehending those which are equable, and those which are uniformly accelerated; in the fourth he treats of violent motion, or the motion of projectiles; and in an appendix to the work, he demonstrates several propositions relative to the centre of gravity of solid bodies. In the first of these dialogues he has founded his reasoning on principles which are far from being correct; but he has been more successful in the other three. In the third dialogue, which contains his celebrated theory, he discusses the doctrine of equable motions in six theorems, containing

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1 De Ponderibus et Mensuris. 2 The First Principles of Mechanics, p. 43. the different relations between the velocity of the moving body, the space which it describes, and the time employed in its description. In the second part of the dialogue, which treats of accelerated motion, he considers all bodies as heavy, and composed of a number of parts which are also heavy. Hence he concludes, that the total weight of the body is proportional to the number of the material particles of which it is composed, and then reasons in the following manner: As the weight of a body is a power always the same in quantity, and as it constantly acts without interruption, the body must be continually receiving from it equal impulses in equal and successive instants of time. When the body is prevented from falling, by being placed on a table, its weight is incessantly impelling it downwards; but these impulses are as incessantly destroyed by the resistance of the table, which prevents it from yielding to them. But when the body falls freely, the impulses which it perpetually receives are perpetually accumulating, and remain in the body unchanged in every respect excepting the diminution which they experience from the resistance of the air. It therefore follows, that a body falling freely is uniformly accelerated, or receives equal increments of velocity in equal times. Having established this preliminary proposition, he then demonstrates, that the time in which any space is described by a motion uniformly accelerated from rest, is equal to the time in which the same space would be described by an uniform equable motion with half the final velocity of the accelerated motion; and that in every motion uniformly accelerated from rest, the spaces described are in the duplicate ratio of the times of description. After having proved these theorems, he applies the doctrine with great success to the ascent and descent of bodies moving on inclined planes.

The theory of Galileo was embraced by his pupil Torricelli, who illustrated and extended it in his excellent work entitled De motu gravium naturaliter accelerato, published in 1644. In his treatise De motu projectorum, published in the Florentine edition of his works in 1664, he has added several new and important propositions to those which were given by his master on the motion of projectiles; and he appears to have been the first who established the principle applicable to all statical problems, "that if two weights are so connected together, that when placed in any position, their common centre of gravity neither ascends nor descends, they are in equilibrium in all these positions."

It was about this time that steam began to be employed as the first mover of machinery. This great discovery has been ascribed by the English to the Marquis of Worcester, and to Papin by the French. About thirty-four years before the date of the marquis's invention, and about sixty-one years before the construction of Papin's digester, steam was employed as the impelling power of a stamping engine by one Brancas an Italian, who published an account of his invention in 1629; but in this case it acted merely by its impulse, and could not even have suggested the idea of its acting by its expansive force. The advantages of steam as an impelling power being thus known, the ingenious Captain Savary invented an engine which raised water by the expansion and condensation of steam. Several engines of this construction were actually erected in England and France, but they were incapable of raising water from depths which exceeded thirty-five feet. The steam-engine received great improvements from our countrymen Newcomen, Beighton, and Blackley; but it was brought to its present state of perfection by Mr Watt of Birmingham, one of the most accomplished engineers of his age. Hitherto it had been employed merely as a hydraulic machine for draining mines or raising water, but in consequence of Mr Watt's improvements it has long been used as the impelling power of almost every species of machinery. It is a curious circumstance, that History, the steam-engine was not only invented, but has received all its improvements, in our own country.

The success of Galileo in investigating the doctrine of Descartes' rectilinear motion, induced the illustrious Huygens to turn his attention to curvilinear motion. In his celebrated work Huygens, De Horologio Oscillatorio, published in 1673, he has shown that the velocity of a heavy body descending along any curve, is the same at every instant in the direction of the tangent, as it would have been if it had fallen through a height equal to the corresponding vertical abscissa; and from the application of this principle to the reversed cycloid with its axis vertical, he discovered the isochronism of the cycloid, or that a heavy body, from whatever part of the cycloid it begins to fall, always arrives at the lower point of the curve in the same space of time. By these discussions, Huygens was gradually led to his beautiful theory of central forces in the circle. This theory may be applied to the motion of a body in any curve, by considering all curves as composed of an infinite number of small arcs of circles of different radii, which Huygens had already done in his theory of evolutes. The theorems of Huygens concerning the centrifugal force and circular motions were published without demonstrations. They were first demonstrated by Dr Keill, at the end of his Introduction to Natural Philosophy. The demonstrations of Huygens, however, which were more prolix than those of the English philosopher, were afterwards given in his posthumous works.

About this time the true laws of collision or percussion were separately discovered by Wallis, Huygens, and Sir Christopher Wren, in 1661, without having the least communication with each other. They were transmitted to the Royal Society of London in 1788, and appeared in the forty-third and forty-sixth numbers of their Transactions. The solution of Wallis was first sent, that of Wren next, and that of Huygens last of all, in consequence of his living at a greater distance. The rules given by Wallis and Wren are published in No. 43, p. 864 and 867, and those of Huygens in No. 46, p. 927. The foundation of all their solutions is, that in the mutual collision of bodies, the absolute quantity of motion of the centre of gravity is the same after impact as before it, and that when the bodies are elastic, the relative velocity is the same after as before the shock. We are indebted likewise to Sir Christopher Wren for an ingenious method of demonstrating the laws of impulsion by experiment. He suspended the impinging bodies by threads of equal length, so that they might touch each other when at rest. When the two bodies were separated from one another, and then allowed to approach by their own gravity, they impinged against each other when they arrived at the positions which they had when at rest, and their velocities were proportional to the chords of the arches through which they had fallen. Their velocities after impact were also measured by the chords of the arches through which the stroke had forced them to ascend, and the results of the experiments coincided exactly with the deductions of theory. The laws of percussion were afterwards more fully investigated by Huygens in his posthumous work De Motu Corporum ex Percussione, and by Wallis in his Mechanica, published in 1670.

The attention of philosophers was at this time directed to the two mechanical problems proposed by Mersenne in 1635. The first of these problems was to determine the centre of oscillation in a compound pendulum; and the second, to find the centre of percussion of a single body, or a system of bodies turning round a fixed axis. The centre of oscillation is that point in a compound pendulum, or a system of bodies moving round a centre, in which, if a small body were placed and made to move round the same centre, it would perform its oscillations in the same time. as the system of bodies. The centre of percussion, which is situated in the same point of the system as the centre of oscillation, is that point of a body revolving or vibrating about an axis, which being struck by an immovable obstacle, the whole of the motion of the body is destroyed.

Huygens solves these two problems were at first discussed by Descartes and Roberval, but the methods which they employed were problem of far from being correct. The first solution of the problem on the centre of oscillation was given by Huygens. He assumed as a principle, that if several weights attached to a pendulum descended by the force of gravity, and if at any instant the bodies were detached from one another, and each ascended with the velocity it had acquired by its fall, they would rise to such a height that the centre of gravity of the system in that state would descend to the same height as that from which the centre of gravity of the pendulum had descended. The solution founded on this principle, which was not derived from the fundamental laws of mechanics, did not at first meet with the approbation of philosophers; but it was afterwards demonstrated in the clearest manner, and now forms the principle of Bernoulli's theorem of active forces. The problem of the centre of percussion was not attended with such difficulties. Several incomplete solutions of it were given by different geometers; but it was at last resolved in an accurate and general manner by James Bernoulli by the principle of the lever.

Works of Borelli; In 1666, a treatise, *De Vi Percussionis*, was published by J. Alphonso Borelli, and in 1686 another work, *De Motibus Naturalibus a Gravitate Pendentibus*; but he added nothing to the science of mechanics. His ingenious work, *De Motu Animalium*, however, published in 1681 and 1682, in two parts, is entitled to great praise, for the beautiful application which it contains of the laws of statics to explain the various motions of living agents.

Labour of Varignon; The application of statics to the equilibrium of machines was first made by Varignon in his Project of a new System of Mechanics, published in 1687. The subject was afterwards completely discussed in his *Nouvelle Mécanique*, a posthumous work published in 1725. In this work are given the first notions of the celebrated principle of virtual velocities, from a letter of John Bernoulli's to Varignon in 1717. The virtual velocity of a body is the infinitely small space through which the body excited to move has a tendency to describe in one instant of time. This principle has been successfully applied by Varignon to the equilibrium of all the simple machines. The resistance of solids, which was first treated by Galileo, was discussed more correctly by Leibnitz in the *Acta Eruditorum* for 1687. In the Memoirs of the Academy for 1702, Varignon has taken up the subject, and rendered the theory much more universal.

Parent on the maximum effect of machines. An important step in the construction of machinery was about this time made by Parent. He remarked in general, that if the parts of a machine are so arranged that the velocity of the impelling power becomes greater or less according as the weight put in motion becomes greater or less, there is a certain proportion between the velocity of the impelling power, and that of the weight to be moved, which renders the effect of the machine a maximum or a minimum. He then applies this principle to undershot wheels, and shows that a maximum effect will be produced when the velocity of the stream is equal to thrice the velocity of the wheel. In obtaining this conclusion, Parent supposed that the force of the current upon the wheel is in the duplicate ratio of the relative velocity, which is true only when a single floatboard is impelled by the water. But when more floatboards than one are acted upon at the same time, it is obvious that the momentum of the water is directly as the relative velocity; and by making this substitution in Parent's demonstration, it will be found that a maximum effect is produced when the velocity of the current is double that of the wheel. This result was first obtained by the Chevalier Borda, and has been amply confirmed by the experiments of Smeaton. (See Hydrodynamics.) The principle of Parent was also applied by him to the construction of wind-mills. It had been generally supposed that the most efficacious angle of weather was 45°; but it was demonstrated by the French philosopher that a maximum effect is produced when the sails are inclined 54° degrees to the axis of rotation, or when the angle of weather is 33° degrees. This conclusion, however, is subject to modifications, which will be pointed out in a subsequent part of this article.

The *Traité de Mécanique* of De la Hire, published separately in 1695, and in the 9th volume of the *Mémoires* written of the French Academy from 1666 to 1699, contains the description of several ingenious and useful machines. But it is chiefly remarkable for the *Traité des Épicycloïdes*, which is added to the edition published in the Memoirs of the academy. In his interesting treatise, De la Hire considers the genesis and properties of exterior and interior epicycloids, and demonstrates, that when one wheel is employed to drive another, the one will move sometimes with greater and sometimes with less force, and the other will move sometimes with greater and sometimes with less velocity, unless the teeth of one or both of the wheels be parts of a curve generated like an epicycloid. The same truth is applicable to the formation of the teeth of rackwork, the arms of levers, the wipers of stampers, and the lifting cogs of forge-hammers; and as the epicycloidal teeth, when properly formed, roll one upon another without much friction, the motion of the machine will be uniform and pleasant, its communicating parts will be prevented from wearing, and there will be no unnecessary waste of the impelling power. Although De la Hire was the first who published this important discovery, yet the honour of it is certainly due to Olaus Roëmer, the celebrated Danish astronomer, who discovered the successive propagation of light. It is expressly stated by Leibnitz, in his letters to John Bernoulli, that Roëmer communicated to him the discovery twenty years before the publication of De la Hire's work; but still we have no ground for believing that De la Hire was guilty of plagiarism. Roëmer's researches were not published; and from the complete discussion which the subject has received from the French philosopher, it is not unlikely that he had the merit of being the second inventor. Even Camus, who about forty years afterwards gave a complete and accurate theory of the teeth of wheels, was unacquainted with the pretensions of Roëmer, and ascribes the discovery to De la Hire.

The publication of Newton's *Principia* contributed greatly to the progress of mechanics. His discoveries concerning the curvilinear motions of bodies, combined with the theory of universal gravitation, enabled philosophers to apply the science of mechanics to the phenomena of the heavens, to ascertain the law of the force by which the planets are held in their orbits, and to compute the various irregularities in the solar system which arise from the mutual action of the bodies which compose it. The *Mécanique Céleste* of La Place will be a standing monument of the extension which mechanics has received from the theory of gravity. The important mechanical princi- of the conservation of the motion of the centre of gravity is also due to Newton. He has demonstrated in his Principia, that the condition of the centre of gravity of several bodies, whether in a state of rest or motion, is not affected by the reciprocal action of these bodies, whatever it may be, so that the centre of gravity of the bodies which act upon one another, either by the intervention of levers, or by the laws of attraction, will either remain at rest, or move uniformly in a right line.

We have already seen that the principle of the conservation of active forces was discovered by Huygens when he solved the problem of the centre of oscillation. The principle alluded to consists in this, that in all the actions of bodies upon each other, whether that action consists in the percussion of elastic bodies, or is communicated from one body to another by threads or inflexible rods, the sums of the masses multiplied by the squares of the absolute velocities remain always the same. This important law is easily deducible from two simpler laws admitted in mechanics: 1. That in the collision of elastic bodies, their respective velocities remain the same after impact as they were before it; and, 2. that the quantity of action, or the product of the masses of the impinging bodies, multiplied by the velocity of their centre of gravity, is the same after as before impact. The principle of the conservation of active forces was regarded by its inventor only as a simple mechanical theorem. John Bernoulli, however, considered it as a general law of nature, and applied it to the solution of several problems which could not be resolved by direct methods; but his son Daniel deduced from it the laws of the motion of fluids from vessels, a subject which had been formerly treated in a very vague manner. He afterwards rendered the principle more general, and showed how it could be applied to the motion of bodies influenced by their mutual attractions, or solicited towards fixed centres by forces proportional to any function of the distance.

After the parallelogram of forces had been introduced into statics by Stevinus, it was generally admitted upon the same demonstration which was given for the composition of motion. The first complete demonstration was published by Daniel Bernoulli in the Commentaries of Petersburg for 1726, independent of the consideration of compound motion. This demonstration, which was both long and abstruse, was greatly simplified by D'Alembert in the Memoirs of the Academy for 1769. Fonseca and Riccati have given a very ingenious one in the Memoirs of the Academy of Turin for 1761. This was also improved by D'Alembert, who gave another in the same Memoirs, and a third in his Traité de Dynamique, published in 1743. Dr Robison has combined the demonstrations of Bernoulli and D'Alembert with one by Frisi, and produced one that is more expeditious and simple. La Place has likewise given a demonstration of the parallelogram of forces in his Mécanique Céleste.

About the beginning of the eighteenth century, the celebrated dispute about the measure of active forces was keenly agitated among philosophers. The first spark of this war, which for forty years England maintained single-handed against all the genius of the Continent, was excited by Leibnitz. In the Leipsic Acts for 1686, he asserted that Descartes was mistaken in making the force of bodies proportional to their simple velocity, and maintained that it followed the ratio of the square of the velocity. He showed that a body with a velocity of two feet acquires the power of raising itself to a height four times as great as that to which a body could rise with a velocity of only one foot; and hence he concludes that the force of that body is as the square of its velocity. The Abbé de Cotillon, a zealous Cartesian, allowed the premises of Leibnitz, but denied his conclusion. The body, said he, which moves with a velocity of two feet, will certainly rise to quadruple the height of another body that has only the velocity of one foot; but it will take twice the time to rise to that height, and a quadruple effect, in a double time, is not a quadruple force, but only a double one. The theory of Leibnitz was supported by John Bernoulli, Hermann, Gravessande, Muschenbroeck, Poleni, Wolff, and Bulfinger; and the opinion of Descartes by Maclaurin, Stirling, Clarke, Desaguliers, and other English philosophers. The question was at last involved in metaphysical reasoning; and if the dispute did terminate in favour of either party, the English philosophers were certainly victorious. It appears, in the clearest manner, that the force of a moving body, indicated by the space which it describes, is as the simple velocity, if we consider the space as described in a determinate time; but it is as the square of the velocity if we do not consider the time in which the space is described. The question, therefore, which is chiefly one of words, comes to be this: In estimating the forces of bodies in motion, ought we to take time into consideration? If, with the followers of Leibnitz, we reject this element, then we may maintain that the force of a child is equal to that of a man carrying a load, because the child is also capable of carrying the same load, though in small parts and in a greater length of time.

In 1743, D'Alembert published his Traité de Dynamique, founded upon a new principle in mechanics. This principle, which M. Fontaine seems to have recognized so early as 1739, was first employed by James Bernoulli in his solution of the problem of the centre of oscillation; but D'Alembert had the honour of generalizing it, and giving it all that simplicity and fertility of which it was susceptible. He showed, that in whatever manner the bodies of one system act upon one another, their motions may always be decomposed into two others at every instant, those of the one being destroyed the instant following, and those of the other retained, and that the motions retained are necessarily known from the conditions of equilibrium between those which are destroyed. This principle is evidently a consequence of the laws of motion and equilibrium, and has the advantage of reducing all the problems of dynamics to pure geometry and the principles of statics. By means of it D'Alembert has resolved a number of beautiful problems which had escaped the notice of his predecessors, and particularly that of the precession of the equinoxes, which had occupied the attention of Newton. In his Traité de Dynamique, D'Alembert has likewise reduced the whole of mechanics to three principles, the force of inertia, compound motion, and equilibrium; and has illustrated his views on this subject by that profound and luminous reasoning which characterizes all his writings.

Another general principle in dynamics was about this discovery time discovered separately by Euler, Daniel Bernoulli, and the Chevalier D'Arcey, and received the name of the conservation of the momentum of rotatory motion. According to the first two philosophers, the principle may be thus defined: In the motion of several bodies round a fixed centre, the sum of the products of the mass of each body multiplied by the velocity of its motion round the centre, and by its distance from the centre, is always in-

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1 Mem. de l'Acad. Berlin, 1748. 2 Our countryman Smeaton, and more recently Dr Wollaston and Mr Ewart of Manchester, have adopted the Leibnitzian doctrine. See Phil. Trans. 1776, p. 450; Ibid. 1806; and Manchester Memoirs, vol. ii. p. 105. dependent of the mutual action which the bodies may exert upon each other, and always preserves itself the same, provided the bodies are not influenced by any external cause. This principle was given by Daniel Bernoulli in the Memoirs of the Academy of Berlin for 1746; and in the same year by Euler in the first volume of his works. They were both led to the discovery whilst investigating the motion of several bodies in a tube of a given form, and which can only turn round a fixed point. The principle discovered by the Chevalier D'Arcy was given in a memoir dated 1746, and published in the Memoirs of the Academy for 1747. He showed that the sum of the products of the mass of each body by the area which its radius vector describes round a fixed point, is always proportional to the times. The identity of this principle, which is a generalization of Newton's theorem about the areas described by the planetary bodies, with that of Euler and Bernoulli, will be easily perceived, if we consider that the element of the circular arc, divided by the element of the time, expresses the velocity of circulation, and that the element of the circular arc, multiplied by the distance from the centre, gives the element of the area described round that centre; so that the principle of Euler is only a differential expression of the principle of D'Arcy, which he afterwards expressed in this form, that the sum of the products of the masses of each body by their velocities, and by the perpendiculars drawn from the centre to their lines of direction, is a constant quantity.

The principle of least action, which was first proposed by Maupertuis in 1744, consists in this, that when several bodies, acting upon one another, experience any change in their motion, this change is always such that the quantity of action (or the product of the mass by the space and the velocity) employed by nature to produce it, is the least possible. From this principle Maupertuis deduced the laws of the reflection and refraction of light, and those of the collision of bodies. He afterwards extended its application to the laws of motion, and made the principle so general as to comprehend the laws of equilibrium, the uniform motion of the centre of gravity in the percussion of bodies, and the conservation of active forces. This celebrated principle was attacked by Koenig, professor of mathematics at the Hague, in the Leipsic Acts for 1751, who not only attempted to show its falsity, but asserted that Leibnitz had first described it in 1707 in a letter to Hermann. The paper of Koenig gave rise to a long and violent dispute about the accuracy of the principle, and the authenticity of the letter of Leibnitz. The Academy of Berlin interfered in behalf of their president; and Voltaire, under the influence of personal feelings against Maupertuis, who had introduced the Abbé Raynal into the academy against the wishes of the poet, took the part of Koenig, and assailed his opponent with the warmest invective. Maupertuis, however, kept his ground, and Voltaire was under the necessity of quitting Berlin in 1752.

In his Traité des Isoperimétres, printed at Lausanne in 1744, Euler extended the principle of least action, and showed, "that in the trajectories described by means of central forces, the integral of the velocity, multiplied by the element of the curve, is either a maximum or a minimum." This remarkable property, which Euler recognised only in the case of insulated bodies, was generalized by Lagrange into this new principle, "that the sum of the products of the masses by the integrals of the velocities, multiplied by the elements of the spaces described, is always a maximum or a minimum." In the memoirs of Turin, Lagrange has employed this principle to resolve several difficult problems in dynamics; and he has shown, that when it is combined with the conservation of active forces, and developed according to the rules of his method of variations, it furnishes directly all the equations necessary for the solution of each problem, and gives rise to a simple and general method of treating the various problems concerning the motion of bodies.

An important discovery in rotatory motion was at this time made by Professor Segner of Göttingen. In a paper entitled Specimen Theoriae Turbinum, published at Halle in 1755, he demonstrated, that if a body of any form or magnitude, after it has received rotatory motions in all directions, be left entirely to itself, it will always have three principal axes of rotation; or, in other words, all the rotatory motions with which it is affected may be reduced to three, which are performed round three axes, perpendicular to each, passing through the centre of gravity of the revolving body, and preserving the same position in absolute space, while the centre of gravity is either stationary or moving uniformly in a straight line.

The force of torsion began at this time to be investigated by Coulomb, who published two ingenious papers on the subject, in the Memoirs of the French Academy. He has successfully employed this principle in several physical researches, but particularly in determining the laws of electric and magnetic action, and in finding the laws of the resistance of fluids when the motions are extremely slow. It was by means of an elegant experiment on the principle of torsion that Mr Cavendish determined the mutual attraction of two masses of lead, and thence deduced the mean density of the earth. We are also indebted to Coulomb for a complete set of experiments on the nature and effects of friction. By employing large bodies and ponderous weights, and conducting his experiments on a large scale, he has corrected errors which necessarily arose from the limited experiments of preceding writers; he has brought to light many new and interesting facts, and confirmed others which had hitherto been partially established. The most curious result of these experiments is the effect of time in increasing the friction between two surfaces. In some cases the friction reaches its maximum after the rubbing surfaces have remained in contact for one minute; and in other cases five or six days were necessary before this effect was produced. The increase of friction, which is generated by prolonging the time of contact, is so great, that a body weighing 1650 pounds was moved with a force of 64 pounds when first laid upon the corresponding surface. After remaining in contact for the space of three seconds, 100 pounds were necessary to put it in motion; and when the time was prolonged to six days, it could scarcely be moved with a power of 622 pounds.

Before we conclude this brief sketch of the history of Babbage mechanics, we must give our readers some account of the calculating machine of Mr Babbage, one of the most remarkable inventions of the present age. The earliest calculating machine was invented by the celebrated Pascal, with the view of facilitating certain calculations which his father's official duties required him to make. A series of wheels and cylindrical barrels containing the ten digits were employed, but several of the operations were performed by hand, so that the mechanism gave no security against error, while it furnished its results with only a little more celerity than that with which an expert calculator could have supplied them. Poleni, Moreland, Grillet, Delepresse, and Boittissendeau, made similar but un- Mechanics.

Invention consists of two parts. In the first, a system of signs marks the origin and nature of every motion in the machine, and also the nature of the mechanical connection between the part moving and the part moved. In the second part, a complete period of the machinery is divided into any number of portions of time, so as to exhibit to the eye in a table what every part of the machine is about at each instant of the period. In this way the mind is disencumbered as it were of the mechanism, and occupies itself only with the symbols of its parts. Since Mr Babbage first published an account of his "Method of expressing by Signs the Action of Machinery," he has greatly extended and improved it, and introduced an additional section for representing the process of circulation in machines through which liquids or gases are moved; and it has been applied by a foreigner to express the structure, the operation, and the circulation of the vital system of animals.

One of the most important treatises on the science of motion is the Mechanics of the celebrated Euler, published in 1736. It contains the whole theory of rectilineal and curvilinear motion in an insulated body, affected by any accelerating forces, either in vacuo or in a resisting medium. He uniformly uses the analytical method, and has employed the principle of the vis inertiae, and that of compound motion, for putting his problems into equations. By Euler's the vis inertiae, motion is at every moment of time rectilineal and uniform; and by the principle of compound motion, a body, exposed to the action of any number of forces, tending to alter the quantity and the direction of its motion, will move in such a direction as to reach the very point at which it would have arrived had it obeyed successively each of the forces which act upon it. In the Mécanique Analytique de Lagrange, published in 1788, all the mechanical problems are reduced to general formulae, which, being developed, furnish us with the equations that are necessary for the solution of each problem; and the different principles which have been discovered for facilitating the solutions of mechanical questions are brought under one point of view, and their connection and dependence clearly pointed out. The Architecture Hydraulique, by M. Prony, published in 1790, and the Mécanique Philosophique of the same author, published in 1799, contain all the late improvements in mechanics, and a complete view both of the theory and application of that science. The first of these works is intended chiefly for the use of the engineer, though an extensive acquaintance with the higher geometry is necessary for perusing it with advantage. His Mécanique Philosophique is a profound work, in which, without the aid of a single diagram, he gives all the formulæ, and the various theorems and problems which belong to the sciences of mechanics and hydrodynamics. Every alternate page contains a methodical table of the results obtained in the preceding page, the description of the symbols, and the theorems, problems, and formulæ which may have been obtained. Le Traité de Mécanique Élémentaire, by M. Françoise, published in 1802, in one volume octavo, is an excellent abridgment of the works of Prony, and is intended as an introduction to the Mécanique Philosophique of that author, to the Mécanique Analytique of Lagrange, and to the Mécanique Céleste of Laplace. None of these works have been translated into English; but their place is to a certain extent supplied by a Treatise on Mechanics, Theoretical, Practical, and Descriptive, by Olinthus Gregory, A.M., published in 1806, containing a complete view of the latest improvements, both in the theory and practice of mechanics.

The purpose of all these machines, even if they had answered the purposes for which they were invented, was to perform particular mathematical operations; but Mr Babbage had a much higher object in view. He conceived the idea of calculating numerical tables of every kind by machinery, of tabulating the results, and of printing them without the possibility of an error during the press-work. In order to effect these objects, it became requisite to discover some common principle in numerical tables as the basis of the calculating machinery; and among the properties of numerical functions suited to this purpose, Mr Babbage selected the method of differences. The principle of this method he embodied in wheel-work by a system of mechanical contrivance in itself perfect; but, from the high importance of the objects to be attained, he has added to the machinery a supplementary and independent mechanism, by which all small errors and inequalities arising from any cause whatever shall be expunged.

When the calculations are made free from all error, they are submitted to the printing part of the machinery, by which they are multiplied, without the possibility of an error in any individual copy.

The powers of this extraordinary machinery have greatly exceeded the expectations of its inventor, and their extent seems to be incapable of appreciation. The part of it which has been put up has already tabulated equations so far beyond the grasp of mathematicians, that no remote term of the table can be predicated, nor any function discovered, by which its general law can be expressed.

The calculating part of the machinery occupies a space about ten feet broad, ten feet high, and five feet deep. It consists of seven steel axes erected over one another, each of them carrying eighteen wheels five inches in diameter, having on them barrels one and a half inch in height, and inscribed with the ten digits. The machine calculates to eighteen decimal places, true to the last figure; but, by subsidiary contrivances, it will be possible to calculate even to thirty decimal places. The working drawings which have been necessary in the production of this machine are not only elaborate and complicated, but are so numerous that they cover upwards of a thousand superficial feet.

For the purpose of completing the calculating machine, government has already advanced L.15,000; but not a farthing has been received by Mr Babbage for his own labour. We regret to add, that so great a national undertaking is at a stand; but this regret is diminished by the fact, that Mr Babbage has contrived a machine, much more simple in its construction, and more extensive in its application, and which may be regarded as entirely superseding his first invention. It will therefore be a matter of deep consideration whether Mr Babbage should complete his first machine in conformity with his arrangement with government, or throw it entirely aside as comparatively imperfect, and devote his whole time, with the aid of government, to the completion of his new invention.

In the construction of the calculating machinery, Mr Babbage early felt the embarrassment arising from the impossibility of remembering all the various movements and functions of the different parts of the machinery; and he was therefore led to contrive a system of mechanical notation, capable of being applied to all kinds of machinery. This

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1 Berlin Miscellany, tom. i. p. 317. 2 See Phil. Trans. 1826, part iii. p. 250. The principal works on mechanics of a more modern date are, Wood's Mechanics, Camb., 1809; MM. Lanz et Bettancourt, Essai sur la Composition des Machines, Paris, 1808; Poisson's Traité de Mécanique, 2 vols. 8vo, Paris, 1811; Bridge's Treatise on Mechanics, London, 1814; Hachette's Traité Elementaire des Machines, Paris, 1819; Borgnes, Traité complète de Mecanique appliquée aux Arts, Paris, 1818; Whewell's Elementary Treatise on Mechanics, Camb., 1828; Whewell's First Principles of Mechanics, with historical and practical Illustrations, Cambridge, 1832; Dr. Lardner and Captain Kater's Treatise on Mechanics, London, 1833; Moseley's Treatise on Mechanics applied to the Arts, London, 1834.

PART I.—THEORY OF MECHANICS.

Objects of The theory of mechanics properly comprehends, 1. dynamics; 2. the motion of projectiles; 3. the theory of mechanics. simple machines, or the mechanical powers; 4. the theory of compound machines, and their maximum effects; 5. the doctrine of the centre of gravity; 6. the doctrine of the centre of oscillation, gyration, &c.; 7. the collision of bodies; 8. the theory of rotation; 9. the theory of torsion; 10. the strength of materials; and, 11. the equilibrium of arches and domes. The subjects of Dynamics, Projectiles, Rotation, and Strength of Materials having been already ably treated by Dr Robison, under their respective heads, we shall now direct the attention of the reader to the other branches of theoretical mechanics.

CHAP. I.—ON ELEMENTARY MACHINES, OR THE MECHANICAL POWERS.

Division of machines into elementary and compound. The elementary machines have been generally reckoned six in number, 1. the lever; 2. the wheel and axle, or axis in peritrochion; 3. the pulley; 4. the inclined plane; 5. the wedge; and, 6. the screw: to which some writers on mechanics have added the balance, and others the rope-machine. It is evident, however, that all these machines may be reduced to three, the lever, the inclined plane, and the funicular or rope-machine. The pulley, and the wheel and axle, are obviously composed of an assemblage of levers; the balance is a lever with equal arms; the wedge is composed of two inclined planes, with their bases in contact; and the screw is either a wedge or an inclined plane, wrapped round a cylinder. Under the head of elementary machines, therefore, we cannot, in strict propriety, include any of the mechanical powers, excepting the lever, the inclined plane, and the rope-machine.

DEFINITIONS.

Def. 1. When two forces act against each other by the intervention of a machine, the one force is called the power, and the other the weight. The weight is the resistance to be overcome, or the effect to be produced. The power is the force, whether animate or inanimate, which is employed to overcome that resistance, or to produce the required effect.

Def. 2. The power and weight are said to balance each other, or to be in equilibrio, when the effort of the one to produce motion in one direction is equal to the effort of the other to produce motion in the opposite direction; or when the weight opposes that degree of resistance which is precisely required to destroy the action of the power.

SECT. I.—On the Lever.

Definitions.—1. A lever is an inflexible bar or rod, either straight or curved, moving freely round an immovable point, called its fulcrum, or centre of motion. 2. The parts of a lever which lie between the fulcrum and the points where the power and weight are applied, are called its arms.

Levers have been generally divided into three kinds. In levers of the first kind, the fulcrum is situated between the power and the weight, as in balances, steelyards, scissors, pincers, a poker, a crow-bar, &c. Levers of the second kind have the weight between the power and the fulcrum, as in cutting-knives fastened at the point of the blade, nut-crackers, a door, and a ship's rudder, and in the oars of a boat where the water is regarded as the fulcrum. In levers of the third kind, the power is between the weight and the fulcrum, as in tongs, shears for sheep, the treadle of a turning lathe, &c. In using such levers, rapidity and dispatch are of greater consequence than great power. The limbs of animals are generally considered as levers of the third kind; for the muscles, by the contraction of which the power or moving force is generated, are fixed much nearer to the joints or centres of motion than the centre of gravity of the weight to be raised. On this subject see Paley's Natural Theology (chap. vii. and viii.); Borelli de Motu Animalium; and Bell's Animal Mechanics, in Library of Useful Knowledge.

AXIOMS.

Axiom 1. Equal weights acting at the extremities of equal arms of a straight lever, and having the lines of the direction in which they act at equal angles to these arms, will exert the same effort to turn the lever round its fulcrum. This axiom has been generally restricted to the particular case when the weights act perpendicularly to the arms of the lever; but no reason can be assigned for such a limitation. The truth in the axiom is as self-evident when the angles formed by the arms of the lever and the direction of the forces are 80° as when they are 90°, for in each case the two weights exert their influence upon the lever in precisely the same circumstances.

Axiom 2. If two equal weights are placed at the extremities of a lever supported by two fulcrums; and if these fulcrums are at equal distances from the weights, or the extremities of the lever; the pressure upon the fulcrums will be equal to the sum of the weights, and the pressure upon each fulcrum will be equal to one of the weights. The lever being supposed devoid of weight, it is obvious, that as each fulcrum is similarly situated with respect to both the weights, the pressure upon each must be equal; and as the fulcrums support both the equal weights, the pressure upon each must be equal to one of the weights.

PROPOSITION I.

If two weights or forces acting at equal angles upon a straight lever, devoid of weight, are in equilibrio, they are reciprocally proportional to their distances from the fulcrum.

Case 1. Where the weights act on contrary sides of the fulcrum.

Let AB be a lever devoid of weight, and let it be supported upon the two fulcrums, f, F, situated in such a manner that Af = fF = FB. Then, if two equal weights C, D of one pound each are suspended at the extremities A, B, so as to act in the direction AC, BD, making the angles CAB, DBA equal, these weights will be in equilibrio; for since

Af = FB (Axiom 1), the effort of the weight D to turn the lever round the fulcrum F, will be equal to the effort of the weight C to turn it round the fulcrum f. Now (Axiom 2) the pressure upon the fulcrum f is equal to one pound, therefore if that fulcrum be removed, and a weight E of one pound be made to act upward at the point F, the weights C and D will continue in equilibrio. Then it is obvious, that since FB = Ef, the weight E of one pound acting upwards at the point f, so that the angle DfP = DBA, will have the same effect as an equal weight acting downwards at B. By removing the weight E, therefore, and suspending its equal C at the extremity B, the equilibrium will still be preserved. But the weights D, C, suspended at B, are equal to two pounds, and the weight C is only one pound; and as FA is double of FB, it follows that a weight of two pounds, placed at the end of one arm of a lever, will be in equilibrio with a weight of one pound placed at twice the distance of the former from the fulcrum. But 2 : 1 = 2FB or AF : FB, that is, when the distances are as 2 to 1, an equilibrium takes place if the weights are reciprocally proportional to these distances.

CASE 2. When weights act on the same side of the fulcrum.

Let AB be a lever in equilibrio upon the fulcrum F, and let FA be equal to FB, consequently (Case 1) we must have C = D = 1 pound. Now, as the fulcrum F supports a weight equal to C + D = 2 pounds, the equilibrium will continue if a weight E of two pounds is made to act upwards at the point F; for in this case it supplies the place of the fulcrum. It is obvious also that a fulcrum placed at A or B will supply the place of the weights at these parts without affecting the equilibrium. Let, therefore, the weight D be removed, and let the extremity B rest upon a fulcrum; then, since the lever is in equilibrio, we have a weight E = C + D = 2 pounds acting at F, and balancing a weight C of one pound acting at A. But 2 : 1 = AB : FB, consequently, when there is an equilibrium between two weights C, D acting at the distances 2 and 1 from the fulcrum, and on the same side of the fulcrum, the weights are reciprocally proportional to these distances.

Again, let AB be the same lever, supported by the fulcrum f, F, and let Af = FB and fF = 2FB. Then if two weights C, D of one pound each be suspended at the extremities A, B, they will be in equilibrio as before. But since the fulcrum f supports a pressure of one pound (Axiom 2), the equilibrium will still continue when that fulcrum is removed and a weight of one pound made to act in a contrary direction fP at the point f, so that the angle PfF may be equal to DBA. Now (Axiom 1), a weight E of one pound acting upwards at f will be in equilibrio with a weight E' of one pound acting downwards at f', Ef being equal to Ef', and therefore, by removing E from the point f, and substituting E' at the point f', an equilibrium will still obtain. But since Ef = 2FB, a weight of one pound suspended from f will have the same influence in turning the lever round F as a weight of two pounds suspended at B (Case 2). Let us remove, therefore, the weight E' from f', and substitute a weight G = 2E' so as to act at B. Then, since the equilibrium is not destroyed, we have a weight C of one pound acting at the distance FA, and the weights D + G = 3 pounds acting at the distance FB. But FA = 3FB and D + G = 3C, consequently C : D + G = FB : FA; that is, when the distances from the fulcrum are as 3 to 1, and when an equilibrium exists, the weights are reciprocally proportional to these distances.

By making FA, in fig. 2, equal to 2FB, it may be shown, as in Case 2, that the weights are reciprocally proportional to their distances from the fulcrum, when they act on the same side of the fulcrum, and when the distances are as 3 to 1.

In the same way the demonstration may be extended to any commensurable proportion of the arms, by making EA to FB in that proportion, and keeping FA always equal to FB. Hence we may conclude in general, that when two weights acting at equal angles upon a straight lever devoid of weight are in equilibrio, they are reciprocally proportional to their distances from the centre of motion.

Q. E. D.

Cor. 1. If two weights acting at equal angles upon the Corollaries. arms of a straight lever devoid of weight are reciprocally proportional to their distances from the fulcrum, they will be in equilibrio.

For if an equilibrium does not take place, the proportion of the weights must be altered to procure an equilibrium, and then, contrary to the proposition, the weights would balance each other when they were not reciprocally proportional to their distances from the fulcrum.

Cor. 2. If a weight W be supported by a horizontal lever resting on the fulcrum A, B, the pressure upon A is to the pressure upon B in the inverse ratio of their distances from the point where the weight is suspended, that is, as BF to FA.

For if we suppose B to be the fulcrum, and if, removing the fulcrum A, we support the extremity A of the lever by a weight E equivalent to the weight sustained by the fulcrum A, and acting upwards over the pulley P, then the weight E or that sustained by A : W = BF : BA (Prop. I); and if we conceive A to be the fulcrum, and support the extremity B by a weight F equal to that which was supported by the fulcrum B, we shall have the weight F or the weight sustained by B : W = AF : AB. Hence, ex aequo, the weight sustained by A is to the weight sustained by B as BF is to FA.

Cor. 3. We may now call the two weights P and W the power and the weight, as in fig. 5; and since P : W = FB : FA, we have P × FA = W × FB, when an equilibrium takes place;

consequently \( P = \frac{W \times FB}{FA} \); \( W = \frac{P \times FA}{FB} \);

\( FA = \frac{W \times FB}{P} \);

\( FB = \frac{P \times FA}{W} \).

Cor. 4. We have already seen (Axiom 2), that when the power and the weight are on contrary sides of the fulcrum, the pressure upon the fulcrum is equal to \( P + W \), or the sum of the weights; but it is obvious that when they act on the same side of the fulcrum, the pressure which it supports will be \( P - W \), or the difference of their weights.

Cor. 5. If a weight \( P \) be shifted along the arm of a lever \( AD \), the weight \( W \), which it is capable of balancing at \( A \), will be proportional to \( FA \).

When the weights are in equilibrio (Cor. 3) \( W : P = FA : FB \); or, by alternation, \( W : FA = P : FB \); and if \( w \) be another value of \( W \), and \( fa \) another value of \( FA \), we shall also have \( w : P = fa : FB \); consequently (Euclid, book v. prop. xi. and xvi.) \( W : w = FA : fa \), that is, \( W \) varies as \( FA \).

Cor. 6. It is obvious that the truths in the preceding proposition and corollaries also hold when the lever has the form represented in fig. 6, only the straight lines \( AF, FB \) are in that case the length of the arm.

Cor. 7. Since by the last corollary \( FA : fa = W : w \), it follows, that in the steelyard, which is merely a lever with a long and short arm, having a weight moveable upon the long one, the distances at which the constant weight must be hung are as the weights suspended from the shorter arm. The steelyard is represented in fig. 7, where \( AB \) is the lever, with unequal arms \( AR, FB \), and \( F \) the centre of motion. The body \( W \), whose weight is to be found, is suspended at the extremity \( A \) of the lever, and the constant weight \( P \) is moved along the divided arm \( FB \) till an equilibrium takes place. As soon as this happens, the number placed at the point of suspension \( D \) indicates the weight of the body. If the lever is devoid of weight, it is obvious that the scale \( FB \) will be a scale of equal parts of which \( FA \) is the unit, and that the weight of the body \( W \) will be always equal to the constant weight \( P \) multiplied by the number of divisions between \( P \) and \( F \). Thus, if the equilibrium takes place when \( P \) is pulled out to the 12th division, we shall have \( W = 12P \); and if \( P = 1 \) pound, \( W = 12 \) pounds. But when the gravity of the lever is considered, which must be done in the real steelyard, its arms are generally of unequal weight, and therefore the divisions of the scale must be ascertained by experiment. In order to do this, remove the weight \( P \), and find the point \( C \), at which a weight \( P' \) equal to \( P \) being suspended, will keep the unequal arms in equilibrium; \( C \) will then be the point at which the equal divisions must commence. For when \( W \) and \( P \) are placed upon the steelyard and are in equilibrio, \( W \) balances \( P \) along with a weight which, placed at \( D \), would support \( P \) placed at \( C \): Therefore \( W \times BF = P \times DF + P \times CF \); but \( P \times DF + P \times CF = P \times DC \); consequently \( W \times BF = P \times DC \), and \( W : DC = P : BF \). By taking different values of the variable quantities \( W \) and \( DC \), as \( w \) and \( dc \), we shall have \( w : dc = P : BF \), consequently (Euclid, book v. prop. xi. and xvi.) \( W : w = DC : dc \), that is, the weight of \( W \) varies as \( DC \), and therefore the divisions must commence at \( C \). If the arm \( BF \) had been heavier than \( FA \), which, however, can scarcely happen in practice, the point \( C \) would have been on the other side of \( F \). In constructing steelyards, it might be advisable to make the unequal arms balance each other by placing a weight \( M \) at the extremity of the lighter arm, in which case the scale will begin at \( F \). In the Danish and Swedish steelyard the body to be weighed and the constant weight are fixed at the extremities of the steelyard, but the point of suspension or centre of motion \( F \) moves along the lever till the equilibrium takes place. The point \( F \) then indicates the weight of the body required. There are some steelyards in which the constant weight is fixed to the shorter arm, while the body to be weighed moves upon the longer arm. The method of dividing this and the preceding steelyard may be seen in De la Hire's Traité de Mécanique, prop. 35, 36, 37, 38.

Prop. II.

To find the condition of equilibrium on a straight lever when its gravity is taken into the account.

Let us suppose the lever to be of uniform thickness and density, as \( AB \), fig. 7, and let it be suspended by the points \( c, d \) to another lever \( ab \), considered as without weight, so that \( ac = cf = fd = db \). Then if \( f \) be the centre of motion or point of suspension, the cylinder \( AB \) will be in equilibrio; for the weight \( AB \) may be regarded as composed of a number of pairs of equal weights, equally distant from the centre of motion. For the same reason, if we conceive the cylinder to be cut through at \( F \), the equilibrium will continue, \( c, d \) being now the points at which the weights \( AF, FB \) act, their distances \( cf, df \) from the centre of motion being equal. Consequently, the arms \( AF, FB \) have the same energy in turning the lever round \( f \) as if weights equal to \( AF, FB \) were suspended at the distance of their middle points \( c, d \) from the fulcrum.

Let \( P \), therefore, in fig. 5, be the power, \( W \) the weight, \( m \) the weight of the arm \( AF \), and \( n \) the weight of \( FB \). Then, when there is an equilibrium, we shall have (Prop. Cor. 3) \( P \times AF + m \times \frac{1}{2} AF = W \times FB + n \times \frac{1}{2} FB \); and since the weight \( m \) acting at half the distance \( AF \) is the same as half the weight \( m \) acting at the whole distance \( AF \), we may substitute \( \frac{1}{2} m \times AF \) instead of \( m \times \frac{1}{2} AF \), and the equation becomes \( P + \frac{1}{2} m \times AF = W + \frac{1}{2} n \times FB \). Hence,

\[ P = \frac{W + \frac{1}{2} n \times FB}{AF} - \frac{1}{2} m \]

\[ W = \frac{P + \frac{1}{2} m \times AF}{FB} - \frac{1}{2} n \]

\[ m = \frac{W + \frac{1}{2} n \times 2FB}{AF} - 2P \]

\[ n = \frac{P + \frac{1}{2} m \times 2AF}{FB} - 2W \]

\[ AF = \frac{W + \frac{1}{2} n \times FB}{P + \frac{1}{2} m} \]

\[ FB = \frac{P + \frac{1}{2} m \times AF}{W + \frac{1}{2} n} \]

Cor. If the arms of the lever are not of uniform density and thickness, instead of the distance of their middle points, we must take the distance of their centre of gravity from the fulcrum.

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1 See the article Balance (vol. iv. p. 366, and Plate CV. fig. 6) for a description and drawing of the Danish and Swedish steelyard. For an account of the universal steelyard invented by Mr C. Paul of Geneva, see Lardner's Mechanics, p. 296, or the Philosophical Magazine, vol. iii. If two forces acting in any direction, and in the same plane, upon a lever of any form, are in equilibrium, they will be reciprocally proportional to the perpendiculars let fall from the fulcrum upon the directions in which they act.

Let AFB be a lever of any form, Fits fulcrum, A, B the points to which the forces, or the power P and weight W, are applied, and AE, BK the directions in which these forces act. Make AE to BK as P is to W, and they will therefore represent the forces applied at A and B. Draw AC perpendicular to AF and EC parallel to it, and complete the parallelogram ADEC. In the same way form the parallelogram BGKH. Produce EA and KB towards m and n if necessary, and let fall Fm, Fn perpendicular to AE, BK produced. Then P shall be to W as Fn is to Fm. By the resolution of forces (Dynamics, sec. 46), the force AE is equivalent to forces represented by AD and AC, and acting in these directions. But as AD acts in the direction of the arm AF, it can have no influence in turning the lever round F; and therefore AC represents the portion of the force AD which contributes to produce an angular motion round F. In the same way it may be shown that BG is the part of the force BK which tends to move the lever round F. Now, suppose AP produced to B, FB being made equal to EB and BG' = BG. Then, by Prop. I. AC : BG' = FB' : FA; but, by Axiom I., the effort of BG to turn the lever round F is equal to the effort of the equal force BG' to turn the lever round F; therefore AC : BG = FB : FA, and AC × FA = BG × FB. Now the triangles ACE, AEm are similar, because the angles at C and M are both right, and on account of the parallels FA, EC, mAF = AEC; therefore AC : AE = Fm : FA, and AC × FA = AE × Fm. For the same reason, in the similar triangles BGK, BEm we have BG : BK = Fm : FB, and BK × Fm = BG × FB. Hence AE × Fm = BK × Fm, and AE : BK or P : W = Fm : Fm. Q. E. D.

Cor. 1. The forces P and W (fig. 9 and 10) are reciprocally proportional to the sines of the angles which their directions make with the arms of the lever; for Fm is evidently the sine of the angle FAm, and Fn the sine of the angle FBr, FA, FB being made the radii. Therefore P : W = sin. FBr : sin. FAm, or P : W = \frac{1}{\sin. FAm} : \frac{1}{\sin. FBr}. Since FA : Fm = rad. : sin. FAm, we have Fm = \frac{FA \times \sin. FAm}{rad.}; and since FB : Fn = rad. : sin. FBr, we have Fn = \frac{FB \times \sin. FBr}{rad.}; but in the case of an equilibrium P : W = Fn : Fm, consequently P : W = \frac{FB \times \sin. FBr}{rad.}.

Cor. 2. The energies of the forces P, W to turn the lever round the fulcrum F is the same, at whatever point in the directions mE, nK they are applied; for the perpendiculars to which these energies are proportional remain the same. The truth of this corollary has been assumed as an axiom by some writers on mechanics, who have very readily deduced it from the preceding proposition. But it is very obvious, that the truth assumed as self-evident is nearly equivalent to the truth which it is employed to prove. Those who have adopted this mode of demonstration illustrate their axiom by the case of a solid body that is either pushed in one direction with a straight rod, or drawn by a cord; in both of which cases it is manifest that the effect of the force employed is the same, at whatever part of the rod or string it is applied. But these cases are completely different from that of a body moving round a fixed centre.

Cor. 3. If AE and BK, fig. 10, the directions in which the forces P, W are exerted, be produced till they meet at L; and if from the fulcrum E the line FS be drawn parallel to the direction AL of one force till it meets BL, the direction of the other; then LS, SF will represent the two forces. For as the sides of any triangle are as the sines of the opposite angles LS : SF = sin. LFS : sin. FLS; but on account of the parallels FS, AL the angle LFS = FLA, and EL being radius, Fm is the sine of FLA or LFS, and Fn the sine of FLS; therefore, by substitution, LS : SF = Fm : Fn, that is, as the force W : P.

Cor. 4. If several forces act upon a lever, and keep it in equilibrium, the sum of the products of the forces and the perpendiculars from the fulcrum to the direction of the different forces on one side is equal to the sum of the products on the other. For since the energy of each force to turn the lever is equal to the product of the force and the perpendicular from the fulcrum on the line of its direction; and since, in the case of an equilibrium, the energy of all the forces on one side of the fulcrum must be equal to the energy of all the forces on the other side; the products proportional to their energies must also be equal.

Cor. 5. If two forces act in a parallel direction upon an angular lever whose fulcrum is its angular point, these forces will be in equilibrium when a line drawn from the fulcrum upon the line which joins the two points where the forces are applied, and parallel to the direction of the forces, cuts it in such a manner that the two parts are reciprocally proportional to the forces applied.

Let AFB be the angular lever, whose fulcrum is F, and let the forces P, W be applied at A and B in the parallel directions Pm, Wn; then if the line FD, parallel to Pm or Wn, cut AB in such a manner that DB : DA = P : W, the forces will be in equilibrium. Draw Fm perpendicular to Pm, and produce it to n; then, since Am, Bn are parallel, mn will also be perpendicular to Bn, and, by Prop. III. Fn : Fm = P : W. Now, if through F there be drawn m'n' parallel to AB, the triangles Fm'm', Fm'w will be similar, and we shall have Fn : Fm = Fn' : Fm'; but, on account of the parallels AB, m'n' : Fn' : Fm' = DB : DA; therefore DB : DA = P : W.

Cor. 6. Let CD be a body moveable round its centre of gravity F, and let two forces P, W act upon it at the points A, B in the plane AFB, in the directions AP, BW; then, since

On the Lever.

this body may be regarded as a lever whose fulcrum is F, the forces will be in equilibrium when \( P : W = Fn : Fm \) the perpendiculars on the directions in which the forces act.

Cor. 7. If AB be an inflexible rod moveable round F as a fulcrum, and acted upon by two forces P, W in the directions Am, An, these forces will be in equilibrium when they are to one another as the perpendiculars Fn, Fm. For, by Cor. 2, the forces may be considered as applied at m and n, and mFn may be regarded as the lever; but, by Prop. III. \( P : W = Fn : Fm : Fm, Fa being perpendiculars upon Am, An. \)

Cor. 8. Let DE be a heavy wheel, and FG an obstacle over which it is to be moved by a force P, acting in the direction AH. Join AF, and draw Fn, Fa perpendicular to CA and AH. The weight of the wheel is evidently the weight to be raised, and may be represented by W acting at the point A in the vertical direction AC. We may now consider AF as a lever whose fulcrum is F, and, by Cor. 7, there will be an equilibrium when \( P : W = Fn : Fm \). Since Fn represents the mechanical energy of the power P to turn the wheel round F, it is obvious that when FG is equal to the radius of the wheel, the weight P, however great, has no power to move it over the obstacle; for when \( FG = AC, Fn = 0, \) and \( Fm \times P = 0 \).

Cor. 9. If a man be placed in a pair of scales hung at the extremities of a lever, and is in equilibrium with a weight in the opposite scale, then if he presses against any point in the lever, except that point from which the scale is suspended, the equilibrium will be destroyed. Let CB be the lever in equilibrium, F its fulcrum, and let the scales be suspended from A and B, AP being the scale in which the man is placed. Then, if he presses with his hand or with a rod against D, a point nearer the centre than A, the scale will take the position AP', and the same effect will be produced as if ADP were a solid mass acting upon the lever in the direction of gravity. Consequently, if FP' be drawn perpendicular from the point P' to FC, FP will be the lever with which the man in the scale tends to turn the lever round the fulcrum; and as FP is greater than FA, the man will preponderate. In the same way it may be shown, that if the man in the scale AP presses upwards against a point C, more remote from the fulcrum than A, he will diminish his relative weight, and the scale W will preponderate; for in this case the scale assumes the position AP', and FP' becomes the lever by which it acts.

Con. 10. If a weight W be supported by an inclined lever resting on the fulcrum A, B, the pressure upon A is to that upon B inversely as Af is to fb, the sections of a horizontal line by the vertical direction of the weight W.

Remove the fulcrum A, and support the extremity A by a weight P, equal to the pressure upon A; then B being the centre of motion, and mn being drawn through F perpendicular to the direction of the forces Am, Ef, and consequently parallel to Ab, we have (Prop. III.) \( P : W = Fn : Fm = fb : fa \); that is, the pressure upon A is to the pressure upon B inversely as Af is to fb.

SCHOLIUM.

Various attempts have been made by different writers on mechanics to give a complete and satisfactory demonstration of the fundamental property of the lever. The first of these attempts was made by Archimedes, who assumes as an axiom, that if two equal bodies be placed upon a lever, they will have the same influence in giving it a rotary motion as if they were both placed in the middle part between them. This truth, however, is far from being self-evident, and on this account Mr. Vince has completed the demonstration by making this axiom a preliminary proposition. The demonstration of Galileo's is both simple and elegant, and does not seem to have attracted much notice, though in principle it is exactly the same as that of Archimedes completed by Mr. Vince. Galileo suspends a solid cylinder or prism from a lever by several threads. When the lever is hung by its centre, the whole is in equilibrium. He then supposes the cylinder to be cut into two unequal parts, which from their mode of suspension still retain their position, and then imagines each part of the cylinder to be suspended by its centre from the lever. Here then we have two unequal weights hanging at unequal distances from the centre of suspension, and it follows from the construction that these weights are in the reciprocal ratio of their distances from that centre. Mr. Vince, on the other hand, employs a cylinder balanced on a fulcrum.

He supposes this cylinder divided into unequal parts, and thus concludes, from his preliminary proposition, that these unequal parts have the same effect in turning the lever as if the weight of these parts was placed in their centres, which is done by Galileo by suspending them from their centres. From this the fundamental property of the lever is easily deduced. The next demonstration was given by Huygens, who assumes as an axiom, that if any weight placed upon a lever is removed to a greater distance from the fulcrum, its effort to turn the lever will be increased. This axiom he might have demonstrated thus, and his demonstration would have been completely satisfactory, though it applies only to cases where the arms of the lever are commensurable. Let AB, fig. 1, be a lever with equal weights C, D, supported on the fulcrum f, so that \( Af = FB \); then, as was shown in Prop. I. the weights will be in equilibrium, and each fulcrum will support a weight equal to C or D. By removing the fulcrum f, the weight C must descend, as the equilibrium is destroyed by a weight equal to C acting at f; therefore the weight C, at the distance AF, has a greater effect in turning the lever than an equal weight D placed at a less distance FB. In Sir Isaac Newton's demonstration, it is supposed that if a given weight act in any direction, and if several radii be drawn from the fulcrum to the line of direction, the effort of that

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1 Phil. Trans. 1794, p. 33. 2 Discursus et Demonstrations Mathematic, dial. ii. p. 96. weight to turn the lever will be the same, to whatever of these radii it is applied. It appears, however, from Prop. III. Cor. 2, that this principle is far from being self-evident, and therefore the demonstration which is founded upon it cannot be admitted as satisfactory. The demonstration given by MacLaurin is simple and convincing, and has been highly approved of by Dr T. Young, and other writers on mechanics, though it extends only to any commensurable proportion of the arms. He supposes the lever AB with equal arms to be in equilibrium upon the fulcrum F, by means of the equal forces P, W, in which case the fulcrum F will evidently be pressed down with a weight equal to $2P = P + W$. He then substitutes, instead of the weight P, a fixed obstacle O, which will not destroy the equilibrium, and considers the fulcrum as still loaded with a weight equal to $P + W$. The pressure on F being therefore equal to $2P$ or $P + W$, a weight E equal to $2P$, and acting upwards, is substituted in the room of that pressure, so that the equilibrium will still continue. Here then we have a lever AB of the second kind, influenced by two forces E and W acting at different distances from the fulcrum A; and since $E = 2P = 2W$, and $BB = 2AF$, we have $E : W = AB : AF$, which expresses the fundamental property of the lever. Without objecting to the circumstance that this demonstration applies only to the lever of the second kind, we may be allowed to observe, that it involves an axiom which cannot be called self-evident. It is certainly manifest that when P and W are in equilibrium, the pressure upon the fulcrum is $= 2P = P + W$; but it by no means follows that this pressure remains the same when the fixed obstacle O is substituted in the room of P. On the contrary, the axiom assumed is a result of the proposition which it is employed to prove, or rather it is the proposition itself. For if, when the extremity A bears against the obstacle O, the pressure upon F is equal to $2W$, the force W obviously produces a pressure $= 2W$ at half the distance AB, which is the property to be demonstrated.

The demonstrations given by Mr Landen and Dr Hamilton, the former in his Memoirs, and the latter in his Essays, though in a great measure satisfactory, are long and tedious. In the demonstration of Dr Hamilton, he employs the following proposition: That when a body is at rest, and acted upon by three forces, they will be to one another as the three sides of a triangle parallel to the direction in which the forces act. When the three forces act on one point of a body, the proposition is true; but it is not applicable to the case of a lever where the forces are applied to three different points, and at all events the demonstration does not hold when any two of the forces act in parallel directions. The demonstration which we have given in Prop. I. is new, and different from any that have been noticed. The truths on which it is founded are perfectly axiomatic; and the only objection to which it seems liable is, that the demonstration extends only to a commensurate proportion of the arms of the lever. An analytical demonstration of the fundamental property of the lever was given by Foncineix, which was afterwards improved by D'Alembert in the Mem. de l'Acad. 1769, p. 283.

**Prop. IV.**

If several levers AB, ab, aβ, whose fulcras are F, f, φ, are so combined as to act perpendicularly upon each other, or at equal angles; and if the directions in which the power and weight are applied be also perpendicular to the arms, or at the same angles with them as those at which the levers act upon each other, there will be an equilibrium when $P : W = BF \times bf \times βφ : AF \times af \times αφ$.

Let M be the force which is exerted by the first lever AB upon the second ab, and N the force which is exerted by the second lever ab upon the third aβ, then, by Prop. I.

$$P : M = BF : AF,$$

$$M : N = bf : af,$$

$$N : W = βφ : αφ.$$

Consequently, by composition,

$$P : W = BF \times bf \times βφ : AF \times af \times αφ.$$

**Prop. V.**

To explain the new property of the lever discovered by M. Épinus, and extended by Van Swinden.

Let AFB be any lever whose fulcrum is F, and to whose extremities A, B are applied the forces P, W in the directions AY, BV. Join AB, and produce it on both sides towards E and I. Produce also the lines YA, VB till they meet in H, and from H, through the fulcrum F, draw HEf, dividing AB into two parts Af, Bf. Let UT be a line given in position, and let α, β represent the angles which the direction of the forces YA, VB make with that line. Let YA and VB likewise represent the intensity of the forces P, W, and let YA be resolved into AE and YE, and the force VB into BI and VI. Then the lever cannot be in equilibrium till

I. $EA \times fA + IB \times fB$ is a maximum.

II. Or, putting φ for the angles formed by the lines AB, UT, which the lever, when in equilibrium, makes with the line UM given in position, there cannot be an equilibrium unless

$$\tan \phi \times P \times Af \times \cos \alpha + (\tan \phi \times W \times Bf \times \cos \beta) = (W \times Bf \times \sin \beta) - P \times Af \times \sin \alpha.$$

III. And, putting a, b for the arms AF, BF, and m, n for the angles EAB, EBA, there cannot be an equilibrium unless

$$\tan \phi = \frac{W \cdot b \cdot (\sin \beta \cdot \cos \alpha \cdot \cos m - \sin \alpha \cdot \cos \beta)}{P \cdot a \cdot (\sin \alpha \cdot \cos m - \sin \alpha \cdot \cos \beta)} + \frac{W \cdot b \cdot (\cos \beta \cdot \cos \alpha \cdot \cos n + \sin \beta \cdot \sin \alpha)}{P \cdot a \cdot (\sin \alpha \cdot \cos m - \sin \alpha \cdot \cos \beta)}.$$

**Scholium.**

This property of the lever was only considered by Épinus in the case of a rectilineal lever with equal arms; but was extended by J. H. Van Swinden. When the lever is...

On the inclined plane, we have \( AF = FB = Af \) and also \( m = n = 0 \); so that, if the last formula is suited to these conditions, we shall have the formula of Apollonius.

**Prop. VI.**

If a power and weight acting upon the arms of any lever be in equilibrium, and if the whole be put in motion, the velocity of the power is to the velocity of the weight as the weight is to the power.

Let AFB be any lever whose fulcrum is F, and let the power P and weight W be applied to its extremities A, B, so as to be in equilibrium. Draw \( Fm, Fn \) perpendicular to AD, BE the direction of the forces P, W. Then suppose an uniform angular motion to be given to the lever, so as to make it describe the small angle \( \alpha \); the position of the lever will now be \( A'FB' \), and the directions of the forces P, W will be \( A'D', B'E \), parallel to AD, BE respectively, since the angle \( \alpha \) is exceedingly small. Join \( AA', BB' \), and from \( A' \) and \( B' \) draw \( A'z, B'z \) perpendicular to AD and BE. Now it is obvious, that though the point A has moved through the space \( AA' \) in the same time that the point B has described the space \( BB' \), yet \( Az \) is the space described by A in the direction AD, and \( Bz \) the space described by B in the direction BE. For if we suppose a plane passing through A at right angles to AD, and another through P parallel to the former plane, it is manifest that \( Az \) measures the approach of the point A to the plane passing through P; and for the same reason \( Bz \) measures the approach of the point B to a plane passing through W at right angles to WB. Therefore \( Az, Bz \) represent the spaces uniformly and simultaneously described by the points A, B, and may therefore be taken to denote the velocities of these points; consequently the velocity of A : the velocity of B = \( Az : Bz \). Now, in the triangles \( AxA', FmA \), the exterior angle \( xAF = AmF + mFA \) (Euclid, book i., prop. 32), and \( A'AF = AmF \), because \( \alpha \) is so exceedingly small that \( A'A \) is sensibly perpendicular to \( AF \); consequently \( xAA' = AFm \); and as the angles at \( x \) and \( m \) are right, the triangles \( AxA', AmF \) are similar.

Therefore \( Az : A'A = Fm : FA \); and in the similar triangles \( \alphaFA', BFB', AA': BB' = FA : FB \), and in the similar triangles \( BB'z, BFm, BB': Bz = FB : Fm \); therefore, by composition, we have \( Az : Bz = Fm : Fm \). But, by Proposition II., \( P : W = Fm : Fm \), consequently \( Az : Bz = W : P \); that is, the velocity of the power is to the velocity of the weight as the weight is to the power. Q. E. D.

Cor. Since \( Az : Bz = W : P \), we have \( Az \times P = Bz \times W \); that is, the momenta of the power and weight are equal.

**Sect. II.—On the Inclined Plane.**

Definition. An inclined plane is a plane surface AB, supported at any angle ABC formed with the horizontal plane BC. The inclination of the plane is the angle which one line in the plane AB forms with another in the horizontal plane BC, both these lines being at right angles to the common intersection of the two planes. The line BA is called

In order to understand how the inclined plane acts as a mechanical power, let us suppose it necessary to elevate the weight D from C to A. If this weight is lifted by the arms of a man to the point A, he must support the whole of the load; but when it is rolled up the inclined plane, a considerable part of its weight is supported upon the plane, and therefore a much smaller force is capable of raising it to A.

**Prop. I.**

When any weight W is kept in equilibrium upon an inclined plane by a power P, the power is to the weight as the sine of the plane's inclination is to the sine of the angle which the direction of the power makes with a line at right angles to the plane.

Let MN be the inclined plane, NO a horizontal line, and MNO the inclination of the plane, and let the weight W be sustained upon MN by means of the power P acting in the direction AE. From the point A, the centre of gravity of the weight, draw \( AB \) perpendicular to the horizontal plane NO, and \( AF \) perpendicular to MN; produce EA till it meets the plane in C, and from the point F where the body touches the plane draw \( Fm \) at right angles to AC, and \( Fn \) at right angles to AB. Then, since the whole body may be considered as collected in the centre of gravity A, \( AB \) will be the direction in which it tends to fall, or the direction of the weight, and \( EA \) is the direction of the power; but \( AF \) is a lever whose fulcrum is F, and since it is acted upon by two forces which are in equilibrium, we shall have (Prop. III. Cor. 8) \( P : W = Fn : Fm \), that is, as the perpendiculars drawn from the fulcrum to the direction in which the forces act. Now \( FA \) being radius, \( Fm \) is the sine of the angle \( FAB \), and \( Fn \) is the sine of the angle \( FAC \); but \( FAB \) is equal to MNO the angle of the plane's inclination, on account of the right angles at F and B and the vertical angles at D; and \( FAC \) is the angle which the direction of the power makes with a line perpendicular to the plane; therefore \( P : W \) as the sine of the plane's inclination is to the sine of the angle formed by the direction of the power with a line at right angles to the plane.

Cor. 1. When the power acts parallel to the plane in the direction AE, \( P \) is to \( W \) as \( EA \) to \( Ea \), that is, radius is to the sine of the plane's inclination, or, on account of the similar triangles \( FAn, MNO \), as the length of the plane is to its height. In this case the power acts to the greatest advantage.

Cor. 2. When the power acts in a vertical line \( As \), \( Fa \) becomes equal to or coincides with \( Fn \), and we have \( P : W = Fn : Fn \), that is, the power in this case sustains the whole weight.

Cor. 3. When the power acts parallel to the base of the plane in the direction \( Ae \), \( P : W = Fn : Fm = Fa : Am \).

Cor. 4. When the power acts in the direction \( AF \), perpendicular to the plane, it has no power to resist the gravity of the weight; for the perpendicular from the fulcrum \( F \), to which its energy is proportional, vanishes.

Cor. 5. Since the body W acts upon the plane in a direction \( AF \) perpendicular to the plane's surface (for its force downwards may be resolved into two, one parallel to the plane, and the other perpendicular to it), and since the reaction of the plane must also be perpendicular to its

surface, that is, in the direction FA, then, when the direction of the power is AE parallel to the horizon, the power, the weight, and the pressure upon the plane, will be respectively as the height, the base, and the length of the plane. The weight W is acted upon by three forces; by its own gravity in the direction An, by the reaction of the plane in the direction AF, and by the power of P in the direction AF. Therefore, since these forces are in equilibrium, and since AF is parallel to nF, and Pf to An, the three sides AF, Af, Pf will represent the three forces. But the triangle AFf is similar to AnF, that is, to MNO, for it was already shown that the angle nAF is equal to MNO; therefore, since in the triangle AFf, AF represents the pressure on the plane, Af the weight of the body, and Pf the energy of the power, these magnitudes will also be represented in the similar triangle MNO by the sides MN, MO, NO.

Cor. 6. If a power P and weight W are in equilibrium upon two inclined planes AB, AC; P:W = AB:AC. Let p be the power, which, acting on the weight W in a direction parallel to the plane, would keep it in equilibrium, then we have p:W = AD:AC; but since the string is equally stretched at every point, the same power p will also sustain the power P, consequently P:p = AB:AD, and by composition P:W = AB:AC.

Prop. II.

If a spherical body is supported upon two inclined planes, the pressures upon these planes will be inversely as the sines of their inclination, while the absolute weight of the body is represented by the sine of the angle formed by the two planes.

Let AC, BC be the two inclined planes, and F the spherical body which they support. The whole of its matter being supposed to be collected in its centre of gravity P, its tendency downwards will be in the vertical line FO. The reaction of the planes upon F is evidently in the direction MF, NF perpendicular to the surface of these planes, and therefore we may consider the body F as influenced by three forces acting in the directions FC, FM, FN; but these forces are represented by the sides of the triangle ABC perpendicular to their directions, consequently the absolute weight of the body F, the pressure upon the plane AC, and the pressure upon the plane BC, are respectively as AB, AC, and BC, that is, as the sines of the angles ACD, ABC, BAC; for in every triangle the sides are as the sines of the opposite angles, or, to express it in symbols, W being the absolute weight of the body, w the pressure on AC, and w' the pressure on BC,

\[ \frac{W}{w} : \frac{w'}{w} = \frac{AB}{AC} : \frac{BC}{AC}, \text{ or } \frac{W}{w} : \frac{w'}{w} = \sin ACB : \sin ABC : \sin BAC. \]

But, on account of the parallels AB, DE, the angle ABC = BCF, and BAC = ACD, therefore the pressures upon the planes are inversely as the sines of their inclination, the absolute weight of the body being represented by the sine of the angle formed by the surfaces of the two planes.

Cor. I. Since the two sides of a triangle are greater than the third, the sum of the relative weights supported by the two planes is greater than the absolute weight of the body.

Cor. 2. If the inclination of each plane is 60°, then ACB must also be 60°, and the triangle ABC equilateral, consequently the pressure upon each plane is equal to the absolute weight of the body.

Cor. 3. When the inclination of each plane increases, the pressure which each sustains is also increased; and when their inclination diminishes till it almost vanishes, the pressure upon each plane is one half of the absolute weight of the body F.

Prop. III.

If a body is raised with an uniform motion along an inclined plane, the velocity of the power is to the velocity of the weight as the weight is to the power.

Let the weight W be drawn uniformly up the inclined plane AB, from B to D, by a power whose direction is parallel to DH. Upon DB describe the circle BFEDN, cutting BC in E, and having produced HD to F, join FP, FB, FE, and draw DC perpendicular to BD. Now the angles BFD, BED are right; and therefore, though the power moves through a space equal to BD, yet its velocity in the direction DH is measured by the space FD uniformly described; and for the same reason, though the weight W describes the space BD, yet its velocity in the direction in which it acts, that is, in a vertical direction, is evidently measured by the space DE uniformly described. Then, because the angle DBE is equal to DEF, and DBE = DCH, and FDE = DHC, the triangles DEF, DHC are similar, and DF:DE = DH:HC. But DH:HC = sin DCH : sin HDC, that is (Prop. I.p. 362), DF:DE, or the velocity of the power to the velocity of the weight, as W:P. Q.E.D.

Scholium.

The inclined plane, when combined with other machinery, is often of great use in the elevation of weights. It has been the opinion of some writers, that the huge masses of stone which are found at great altitudes in the splendid remains of Egyptian architecture, were raised upon inclined planes of earth, with the aid of other mechanical powers. This supposition, however, is not probable, as the immense blocks of granite which compose the pyramids of Egypt could not possibly have been raised into their present situation by any combination of the mechanical powers with which we are acquainted. The inclined plane has been very advantageously employed in the Duke of Bridgewater's canal. After this canal has extended forty miles on the same level, it is joined to a subterraneous navigation about twelve miles long, by means of an inclined plane, and this subterraneous portion is again connected by an inclined plane with another subterraneous portion about 106 feet above it. This inclined plane is a stratum of stone which slopes one foot in four, and is about 453 feet long. The boats are conveyed from one portion of the canal to another by means of a windlass, so that a loaded boat descending along the plane turns the axis of the windlass, and raises an empty boat. A pair of stairs, and all roads and railways that are not horizontal, may be regarded as inclined planes; and hence it is a matter of great importance, in carrying a road to the top of a hill, to choose such a line that the declivity may be the least possible. The additional length which, in order to effect this purpose, must sometimes be given to the line of road, is a trifling inconvenience, when compared with the advantages of a gentle declivity. In ascending steep hills, the horses have the sagacity to judge of the inclination of the plane, and facilitate the ascent by winding from one side of the road to another.

**Sect. III.—On the Rope Machine.**

**Definition.** When a body suspended by two or more ropes is sustained by powers which act by the assistance of these ropes, this assemblage of ropes is called a *funicular* or rope machine.

**Prop. I.**

If a weight is in equilibrium with two powers acting on a rope machine, these powers are inversely as the sines of the angles which the ropes form with the direction of the weight.

Let the weight $W$ be suspended from the point $B$, where the ropes $AB$, $BC$ are joined, and let the powers $P$, $p$ acting at the other extremities of the ropes which pass over the pulleys $A$, $C$, keep this weight in equilibrium; we shall have $P : p = \sin CBD : \sin ABD$. Produce $WB$ to $F$, and let $BD$ represent the force exerted by $W$; then, by drawing $DE$ parallel to $AB$, the sides of the triangle $BDE$ will represent the three forces by which the point $B$ is solicited; for $AB$, $CD$ are the directions of the forces $P$ and $p$. We have therefore $P : p = DE : BE$; but $DE : BE = \sin DBE : \sin BDE$; and, on account of the parallels $DE$, $AB$, the angle $BDE = ABD$, consequently $P : p = \sin DBE : \sin ABD$.

**Cor. 1.** When the line joining the pulleys is horizontal, as $AC$, then $P : p = FC : FA$; for $FC$ and $FA$ are evidently the sines of the angles $DBE$, $BDE$.

**Cor. 2.** Any of the powers is to the weight as the sine of the angle which the other makes with the direction of the weight is to the sine of the angles which the powers make with one another. For since $DB$ represents the weight and $BE$ the power $P$, we have $BE : BD = \sin BDE : \sin BED$; but, on account of the parallels $DE$, $AB$, the angle $DEB = ABC$, the angle made by the direction of the powers, consequently $BE : BD$, that is, $p : W = \sin ABF : \sin ABC$. In the same way it may be shown that $P : W = \sin CBF : \sin ABC$. Hence we have $P + p : W = \sin CBF + \sin ABF : \sin ABC$; that is, the sum of the powers is to the weight as the sum of the sines of the angles which the powers make with the direction of the weight is to the sine of the angle which the powers make with one another.

**Cor. 3.** The two powers $P$, $p$ are also directly proportional to the cosecants of the angles formed by the direction of the powers with the direction of the weight. For since $P : p = \sin DBE : \sin BDE$, and, by the principles of trigonometry, $\sin DBE : \sin BDE = \cot \sec BDE : \cot \sec DBE$, we have $P : p = \cot \sec ABF : \cot \sec BCF$. It is also obvious that $P : p$ as the secants of the angles which these powers form with the horizon, since the angles which they make with the horizon are the complements of the angles which they form with the direction of the weight, and the cosecant of any angle is just the secant of its complement, therefore $P : p = \sec BAF : \sec BCF$.

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**CHAP. II.—ON COMPOUND MACHINES.**

**Definition.** Compound machines are those which are composed of two or more simple machines, either of the same or of different kinds. The number of compound machines is unlimited, but those which properly belong to this chapter are, 1. the wheel and axle; 2. the pulley; 3. the wedge; 4. the screw; and 5. the balance.

**Sect. I.—On the Wheel and Axle.**

The *wheel and axle*, or the axis in peritrochion, is represented in fig. 27, and consists of a wheel $AB$ and cylinder $EF$, having the same axis, and moving upon pivots $E$, $F$, placed at the extremity of the cylinder. The power $P$ is most commonly applied to the circumference of the wheel, and acts in the direction of the tangent, while the weight $W$ is elevated by a rope which coils round the cylinder $EF$ in a plane perpendicular to its axis. In this machine a winch or handle at $E$ is sometimes substituted instead of the wheel, and sometimes the power is applied to the levers $S$, $S'$ fixed in the periphery of the wheel; but in all these forms the principle of the machine remains unaltered. That the wheel and axle is an assemblage of levers will be obvious, by considering that the very same effect would be produced if a number of levers were to radiate from the centre of the wheel, and if a rope carrying the power $P$ were to pass over their extremities, and extricate itself from the descending levers when they come into a horizontal position.

**Axion.** The effect of the power to turn the cylinder round its axis is the same at whatever point in the axle it is fixed.

**Prop. I.**

In the wheel and axle the power and weight will be in equilibrium when they are to one another reciprocally as the radii of the circles to which they are applied, or when the power is to the weight as the radius of the axle is to the radius of the wheel.

Let $AD$ be a section of the axle, and $BE$ a section of the axle or cylinder, and let the power $P$ and weight $W$ act in the directions $AP$, $WP$, tangents to the circumferences of the axle and wheel in the points $A$, $B$, by means of ropes winding round these circumferences. As the effect is the same according to the axiom, let the power and weight act in the same plane as they appear to do in the figure, then it is obvious that the effort of the power $P$ and weight $W$ will be the same as if they were suspended at the points $A$, $B$; consequently the machine may be regarded as a lever $AFB$, whose centre of motion is $F$. But since the directions of the power and weight make equal angles with the arms of the lever, we have (Prop. I., p. 356) $P : W = FB : FA$, that is, the power is to the weight as the radius of the axle is to the radius of the wheel.

**Cor. 1.** If the power and weight act obliquely to the arms of the lever in the directions $Ap$, $Bw$, draw $Fa$ perpendicular to $Ap$ and $Bw$, and, as in the case of the lever (Prop. III.), there will be an equilibrium when $P : W = Fa : Fm$. Hence the tangential direction is the most advantageous one in which the power can be applied. for FA is always greater than FM, and the least advantageous direction in which the weight can be applied, for it then opposes the greatest resistance to the power.

Cor. 2. If the plane of the wheel is inclined to the axle at any angle \( x \), there will be an equilibrium when \( P : W = \text{semidiameter of the axle} : \sin x \).

Cor. 3. When the thickness of the rope is of a sensible magnitude, there will be an equilibrium when the power is to the weight as the sum of the radius of the axle and half the thickness of its rope is to the sum of the radius of the wheel and half the thickness of its rope; that is, if \( T \) be the thickness of the rope of the wheel, and \( t \) the thickness of the rope of the axle, there will be an equilibrium when \( P : W = FB + \frac{1}{2}t : FA + \frac{1}{2}T \).

Cor. 4. If a number of wheels and axles are so combined that the periphery of the first axle may act on the periphery of the second wheel, either by means of a string or by teeth fixed in the peripheries of each, and the periphery of the second axle on the periphery of the third wheel, there will be an equilibrium when the power is to the weight as the product of the radii of all the axles is to the product of the radii of all the wheels. This corollary may be demonstrated by the same reasoning which is used in p. 360, for the combination of levers.

Cor. 5. In a combination of wheels, where the motion is communicated by means of teeth, the axle is called the pinion. Since the teeth therefore must be nearly of the same size, both in the wheel and pinion, the number of teeth in each will be as their circumferences, or as their radii; and consequently, in the combination mentioned in the preceding corollary, the power will be to the weight, in the case of an equilibrium, as the product of the number of teeth in all the pinions is to the product of the number of teeth in all the wheels.

Prop. II.

In the wheel and axle the velocity of the weight is to the velocity of the power as the power is to the weight.

If the power is made to rise through a space equal to the circumference of the wheel, the weight will evidently describe a space equal to the circumference of the axle. Hence, calling \( V \) the velocity of the power, \( v \) that of the weight, \( C \) the circumference of the wheel, and \( c \) that of the axle, we have \( V : v = C : c \). But, by the proposition, \( P : W = c : C \); therefore \( P : W = v : V \).

Scholium.

The construction of the main-spring box of the fusee of a watch, round which the chain is coiled, is a beautiful illustration of the principle of the wheel and axle. The spring-box may be considered as the wheel, and the fusee the axle or pinion, to which the chain communicates the motion of the box. The power resides in the spring wound round an axis in the centre of the box, and the weight is applied to the lower circumference of the fusee. As the force of the spring is greatest when it is newly wound up, and gradually decreases as it unwinds itself, it is necessary that the fusee should have different radii, so that the chain may act upon the narrowest part of the fusee when its force is greatest, and upon the widest part of the fusee when its force is least; for the equable motion of the watch requires that the inequality in the action of the spring should be counteracted so as to produce an uniform effect. In order to accomplish this, the general outline of the surface of the fusee must be an Apollonian hyperbola, in which the ordinates are inversely as their respective abscissae.

For further information on this subject, see Recherches des Mathématiques, par M. Parent (tom. ii. p. 678); Traité d'Horlogerie, par M. Berthoud (tom. i. chap. 26); and Traité de Mécanique, par M. de la Hire (prop. 72.)

Sect. II.—On the Pulley.

Definition. The pulley is a machine composed of a wheel with a groove in its circumference, and a rope which passes round this groove. The wheel moves on an axis pulley whose extremities are supported on a kind of frame called the block, to which is generally suspended the weight to be raised. A system of pulleys is called a muffle, which is either fixed or moveable, according as the block which contains the pulleys is fixed or moveable.

Prop. I.

In a single pulley, or system of pulleys, where the different portions of the rope are parallel to each other, and where one extremity of it is fixed, there is an equilibrium when the power is to the weight as unity is to the number of the portions of the rope which support the weight.

Case 1. In the single fixed pulley AA let the power \( P \) and weight \( W \) be equal, and act against each other by means of the rope PA AW passing over the pulley AA; then it is obvious, that whatever force is exerted by \( P \) in the direction PA, the same force must be exerted in the opposite direction WA, consequently these equal and opposite forces must be in equilibrium; and as the weight is supported only by one rope, the proposition is demonstrated, for \( P : W = 1 : 1 \).

Case 2. In the single moveable pulley (fig. 30), where the rope, fastened at H, goes beneath the moveable pulley D and over the fixed pulley C, the weight to be raised is suspended from the centre of the pulley D by the block \( p \), and the power is applied at P in the direction PE. Now it is evident that the portions CP, HD of the rope sustain the weight \( W \), and as they are equally stretched in every point, each must sustain one half of \( W \); but (Case 1) in the single pulley C the rope CEP sustains a weight equal to what the rope CP sustains; that is, it sustains one half of \( W \). Consequently \( P = \frac{1}{2}W \), or \( W = 2P \), when there is an equilibrium; and since the weight \( W \) is supported by two strings, we have \( P : W = 1 : 2 \).

Case 3. When the same rope passes round a number of pulleys, the ropes which support the weight \( W \) are evidently equally stretched in every part, and therefore each of them sustains the same weight. Conse- Mechanics.

On the Pulley.

White's pulley.

If there be ten ropes supporting the weight, each sustains \( \frac{1}{10} \) part of the weight, and therefore \( P : W = 1 : 10 \). The pulley in fig. 32 is the patent pulley invented by Mr White, in which the lateral friction and shaking motion is considerably removed. This ingenious pulley has great theoretical advantages; but it has been found difficult to give the grooves the exact proportions, which depend on the thickness of the rope. If this is not effected, the rope becomes unequally stretched; some parts being in a state of great tension, and others comparatively slack. The rope is also subject to be discharged from the grooves.

Prop. II.

In a system of \( n \) moveable pulleys suspended by separate and parallel ropes, there is an equilibrium when \( P : W = 1 : 2^n \); that is, if there are 4 pulleys \( n = 4 \), and \( P : W = 1 : 2 \times 2 \times 2 \times 2 \), or \( P : W = 1 : 16 \).

This system is represented in fig. 34, where the rope which carries the power \( P \) passes over the fixed pulley \( M \), and beneath the moveable pulley \( A_1 \) to the hook \( E \), where it is fixed. Another rope fixed to a hook below \( A \), passes over \( B \), and is fixed at \( F \), and so on with the rest. Then, by Case 2, Prop. I.

\( P : \) the weight at \( A = 1 : 2 \) The weight at \( A : \) the weight at \( B = 1 : 2 \) The weight at \( B : \) the weight at \( C = 1 : 2 \) The weight at \( C : \) the weight at \( D \) or \( W = 1 : 2 \); and therefore, by composition, \( P : W = 1 : 2 \times 2 \times 2 \times 2 \), or \( P : W = 1 : 16 \). Q. E. D.

Prop. III.

In a system of pulleys, one of which is fixed and the rest moveable, whose number is \( n \), suspended by separate and parallel ropes, whose extremities are fixed to the weight \( W \), there is an equilibrium when \( P : W : 1 : 2^n - 1 \).

In this system of pulleys, the rope which sustains the power \( P \) passes over the pulley \( C \), and is fixed to the weight at \( D \). Another rope attached to the pulley \( C \) passes over the pulley \( B \) and is fixed to the weight at \( E \), and a third rope fastened to \( B \) passes over \( A \) and is fixed at \( F \). Then it is manifest that the rope \( CD \) sustains a weight equal to \( P \); and since the pulley \( C \) is pulled downward with a weight equal to \( 2P \), the rope \( BC \) must support a weight equal to \( 2P \), and the rope \( B \) the same weight; consequently the rope \( AB \) sustains \( 4P \). The whole weight therefore is \( P + 2P + 4P \), and hence \( P : W = P : P + 2P + 4P \), or \( P : W = 1 : 1 + 2 + 4 \), &c. to \( n \) terms, so that \( P : W = 1 : 2^n - 1 \).

Prop. IV.

In the system of pulleys represented in fig. 36, and called a Spanish barton, in which two pulleys are supported by one rope, there is an equilibrium when \( P : W = 1 : 4 \).

In this combination of pulleys, the rope \( AB \) which supports the power \( P \) passes over the moveable pulley \( A \), and beneath \( C \) towards \( H \), where it is fixed. Another rope, attached to the pulley \( A \), passes over the fixed pulley \( B \), and is fastened at \( E \) to the pulley \( C \), which supports the weight \( W \). Then, since the rope \( AP \) supports 1 pound, the rope \( AC \) also supports 1 pound, and therefore the pulley \( A \), or the rope \( BA \), is pulled down with a force of 2 pounds. But the rope \( BDE \) is equally stretched with \( BA \), consequently the pulley \( C \), to which \( DE \) is attached, is pulled upwards with a force of 2 pounds. Now the rope \( AC \) supporting 1 pound, the rope \( GH \) must likewise support 1 pound, consequently, since \( DE \) sustains 2 pounds, \( AC \) 1 pound, and \( HG \) 1 pound, they will together sustain \( W = 4 \) pounds, and therefore \( P : W = 1 : 4 \).

Prop. V.

In the system of pulleys represented in fig. 37, also called a Spanish barton, where two pulleys are supported by one rope, there is an equilibrium when \( P : W = 1 : 5 \).

In this system the rope \( PB \) passes over \( B \) round \( C \), and is fixed at \( E \). Another rope attached to \( B \) passes round \( AF \), and is fixed at \( I \) to the pulley \( CD \), which carries the weight \( W \). Now the rope \( BP \) being stretched with a force of 1 pound, the ropes \( BGC, CDE \), are also stretched with a force of 1 pound each, and the pulley \( CD \) is pulled upwards with a force of 2 pounds. But since the three ropes \( BP, ED, \) and \( GC \), are each stretched with a force of 1 pound, the pulley \( Band \) the rope \( BA \), upon which they all act in one direction, must be pulled down with a force of 3 pounds. Now the rope \( FI \) is equally stretched with \( BA \), consequently it will draw the pulley \( CD \) upwards with a force of 3 pounds; and since it is drawn upwards by the ropes \( CG, DF \), with a force of 2 pounds, the whole force will sustain \( W = 5 \) pounds; but this force of 5 pounds is, by the hypothesis, in equilibrium with \( P \) or 1 pound, consequently \( P : W = 1 : 5 \).

Prop. VI.

When the ropes are not parallel, and when two powers are in equilibrium with a weight by means of a pulley, and have their directions at equal angles to the direction of the weight, each of these powers is to the weight as the radius of the pulley is to the chord of that portion of the pulley's circumference with which the rope is in contact.

Let the weight \( W \) suspended from \( C \) be sustained in equilibrium by two powers \( P, p \), which act by a rope \( PCFE \), passing over the pulley \( CHEF \), and touching the arch \( CFE \) of its circumference. Then, since the angles \( PWD, pWD \) are equal, and the powers \( P, p \) in equilibrium, \( P \) must be equal to \( p \); and making \( WA = WB \), and drawing \( AI \) parallel to \( PW \), and \( BI \) parallel to \( pW \); \( WB, BI \) will respectively represent the forces \( P, p, W \), or \( P : p : W = WB : BI : WL \). Now the triangles \( WBI, CDE \) having their respective sides at right angles to each other, are similar; consequently \( WB : BI : WI = CD : DE : EC \), that

Cor. 1. Any of the powers is also to the weight as radius is to twice the cosine of the angle which either rope makes with the direction of the weight. For since CG is the cosine of DCG, and since CE is double of CG, CE is equal to 2 cosine DCG = 2 cos. PWD; but P : W = CD : CE, hence we have, by substituting the preceding value of CE, P : W = CD or radius : 2 cos. PWD.

Scholium.

By means of this proposition and corollary, the proportion between the powers and the weight in the various systems of pulleys represented in figs. 29, 30, 31, 32, 33, 34, 35, 36, and 37, when the ropes are not parallel, may be easily found.

Prop. VII.

In a system of moveable pulleys, where each has a separate rope, and where the ropes are not parallel, there is an equilibrium when the power is to the weight as radius is to the cosines of half the angles made by the rope of each pulley, multiplied into that power of whose exponent is the number of pulleys.

Let the power P sustain the weight W by means of the pulleys A, B, C; let P, p, π be the different powers which support the pulleys A, B, C, and let MAP, NBA, RCB be the angles formed by the ropes. Then, by the last proposition,

\[ P : p = \text{rad.} : 2 \cos. MAP, \] \[ p : \pi = \text{rad.} : 2 \cos. NBA, \] \[ \pi : W = \text{rad.} : 2 \cos. RCB, \] consequently \( P : W = \text{rad.} : 2 \cos. MAP \times 2 \cos. NBA \times 2 \cos. RCB; \) or, which is the same thing, \( P : W = \text{rad.} : 2 \times 2 \times 2 \times \cos. MAP \times \cos. NBA \times \cos. RCB. \)

Prop. VIII.

In a single pulley, or in a combination of pulleys, the velocity of the power is to the velocity of the weight as the weight is to the power.

Case 1. In the single fixed pulley (fig. 29), it is obvious, that if the weight W is raised uniformly one inch, the power D will also describe one inch, consequently velocity P : velocity of W = W : P.

Case 2. In the single moveable pulley (fig. 30), when the weight W is raised one inch, the ropes become one inch shorter; and since the rope has always the same eight, the power must describe two inches, therefore velocity P : velocity W = W : P.

Case 3. In the combination of pulleys (figs. 31, 32, and 33), when the weight rises one inch, each of the four strings becomes an inch shorter, so that P must describe four inches, the length of the rope is invariable; consequently velocity P : velocity W = W : P.

Case 4. In the system exhibited in fig. 34, it is evident, that when the weight W rises one inch, the rope OC is lengthened two inches, the rope CB four inches, the rope BA eight inches, and the rope AFP, to which the power is suspended, sixteen inches; so that since the power of this pulley is as 16 to 1, we have velocity P : velocity W = W : P.

Case 5. In the combination of pulleys represented in fig. 35, when the weight W rises one inch, all the three ropes CD, BE, AF are each shortened one inch. But while CD shortens one inch, CP becomes one inch longer; while BE shortens one inch, BC becomes one inch longer, and CP two inches longer (Prop. I. Cor. 1); and while AE shortens one inch, AB becomes one inch longer, BC two inches longer, and CP four inches longer; therefore CP is lengthened altogether seven inches, and as the power of the pulley is as 7 to 1, we have, as before, velocity P : velocity W = W : P.

Case 6. In the system of pulleys called the Spanish barton (fig. 36), when the weight W rises one inch, the three ropes AC, DE, HG are each shortened one inch. By the shortening of HG, CA one inch each, the rope AP is lengthened two inches; and by the shortening of DE one inch, BA is lengthened one inch, and AP two inches; consequently, since AP is lengthened in all four inches, and since the power of the pulleys is four, we have velocity P : velocity W = W : P.

Case 7. In the other Spanish barton, in fig. 37, when the weight is elevated one inch, the three ropes DE, IF, CG are each one inch shorter. While ED and CG shorten one inch each, BP is lengthened two inches, and while IF becomes one inch shorter, AB becomes one inch longer; but when AB is lengthened one inch, BP becomes one inch longer, and ED, CG one inch shorter each; and by this shortening of ED, CG, the rope B is lengthened two inches; therefore, since the rope BP is lengthened altogether five inches, and since the pulleys have a power of five, we have, as formerly, velocity P : velocity W = W : P.

Sect. III.—On the Wedge.

Definition. A wedge is a machine composed of two inclined planes with their bases in contact; or, more properly, it is a triangular prism, generated by the motion of a triangle, parallel to itself, along a straight line passing through the vertex of one of its angles. The wedge is called isosceles, rectangular, or scalene, according as the triangle ABC (fig. 40) by which the wedge is generated, is an isosceles, a rectangular, or a scalene triangle. The part AB is called the head or back of the wedge, DC its altitude, and AC, BC its faces. The wedge is generally employed for cleaving wood or for quarrying stones; but all cutting instruments, such as knives, swords, chisels, teeth, nails, pins, needles, awls, &c., properly belong to this mechanical power, whether they act in a direction at right angles, or oblique to the cutting surface.

The wedge is frequently employed in machinery where a great force is to be exerted through a short space. Falling edifices have been thrown back into their perpendicular position by means of wedges; ships are raised in docks by driving wedges beneath their keels; and in America houses are raised by them, and a new story built below.

In the Dutch wind-mills described in a future part of this article, and used for manufacturing oil of colza, the seeds from which the oil is expressed are placed in bags of hair lying between two flat pieces of hard wood. Between each two of these pieces of wood a wedge is inserted, and a stamper, lifted by each revolution of the windshaft, strikes a blow on its head, which compresses the planes, and consequently the bags, to such a degree, that the seeds are squeezed into a mass almost as solid as wood.

A beautiful application of the wedge was made by Mr. Brunel in his earliest attempts to cut veneers. He cut the wood by the application of pressure to the back of the wedge, and the thin veneer coiled itself up like a scroll of paper. The valuable wood was thus cut without any loss. As the strata of the wood, however, were slightly separated by the force which compelled it to roll...

On the Wedge.

Itself up, the London cabinet-makers would not purchase it. This apparent defect was, however, a real advantage, for the glue insinuated itself into the fissures, and united the veneer so firmly to its bed that they were inseparable. We have seen, in Mr Brunel's house at Battersea Bridge, beautiful furniture made of such veneers. It was necessary, however, to yield to the prejudices of others, and Mr Brunel abandoned his method of cutting veneers by the wedge, and resorted to the system of circular saws, which he brought to the highest degree of perfection.

In all wedges which cut by pressure, the cutting edge may be much sharper than when they are urged by impulse. Tools for cutting wood should have an angle of about $30^\circ$, for iron from $50^\circ$ to $60^\circ$, and for brass from $80^\circ$ to $90^\circ$.

Prop. I.

If each of the faces of an isosceles wedge, which are perfectly smooth, meet with an equal resistance from forces acting at equal angles of inclination to their faces, and if a power act perpendicularly upon the back, these forces will be in equilibrium when the power upon the back is to the sum of the resistances upon the sides, as the sine of half the angle of the wedge, multiplied by the sine of the angle at which the resisting forces act upon its faces, is to the square of radius.

Let $ABC$ be the wedge, $AC$, $BC$ its acting faces, and $MD$, $ND$ the directions in which the resisting forces act upon these faces, forming with them the equal angles $DMA$, $DNB$. Draw $CD$, $DF$, $DE$ at right angles to three sides of the wedge, and join $F$, $E$ meeting $CD$ in $G$. On account of the equal triangles $CAD$, $CDB$ (Euclid, book i. prop. 26), $AD = DB$; and in the equal triangles $ADM$, $BDN$, $MD = ND$. In the same way $DF = DE$ and $AF = BE$, therefore $CF = CE$. But in the triangles $CFG$, $CEG$ there are two sides $FC$, $CG$ equal to $EC$, $CG$, and the angle $FCG = ECG$, consequently $FG = GE$, and $FGC$, $ADC$ are both right angles, therefore $FE$ is parallel to $AB$. Now the force $MD$ is resolvable into $DF$, $FM$, of which $FM$ has no effect upon the wedge. But, as the effective force $FD$ is not in direct opposition to the perpendicular force exerted on the back of the wedge, we may resolve it into the two forces $FG$, $GD$, of which $GD$ acts in direct opposition to the power, while $FG$ acts in a direction parallel to the back of the wedge. In the same way it may be shown that $EG$, $GD$ are the only effective forces which result from the force $ND$. But the forces $FG$, $EG$ being equal and opposite, destroy each other; consequently $2GD$ is the force which opposes that which is exerted upon the back of the wedge, and the wedge will be kept at rest if the force upon the back is equal to $2GD$, that is, when the force upon the back is to the sum of the resistances upon the faces as $2GD$ is to $MD + ND$, or as $2GD : 2DM$, or as $GD$ is to $DM$. Now $DG : DF = \sin DFG : \text{radius}$, or as (Euclid, vi. 8) $\sin DCF : \text{radius}$, and $DF : MD = \sin DMF : \text{radius}$; therefore, by composition,

$$DG : MD = \sin DCF \times \sin DMF : \text{rad.} \times \text{rad.} \text{or rad.}^2.$$

But $DG : MD$ as the force upon the back is to the sum of the resistances; therefore the force upon the back is to the sum of the resistances as $\sin DCF \times \sin DMF$ is to the square of the radius.

Corollary. If the direction of the resisting forces is perpendicular to the faces of the wedge, $DMF$ becomes a right angle, and therefore its sine is equal to radius. Consequently we have, in this case, the force upon the back to the sum of the resistances, as $\sin DCF \times \text{rad.}$ is to radius$^2$, that is, as $\sin DCF$ is to radius, or as $AD$ half the back of the wedge is to $AC$ the length of the wedge.

Cor. 2. In the particular case in the proposition, it is obvious that the forces $MF$, $NE$ are not opposed by any other forces, and therefore the force upon the back will not sustain the resisting forces; but in the case in Cor. 2 the forces $MF$, $NE$ vanish, and therefore the other forces will sustain each other.

Cor. 3. If the resisting forces act in a direction perpendicular to $AB$, the angle $DMF$ becomes equal to $ACD$, and therefore the force upon the back is to the sum of the resistances as $\sin ACD$ is to radius$^2$, that is, as the square of $AD$, half the back of the wedge, is to the square of $AC$, the length of the wedge.

Cor. 4. When the direction of the resistances is parallel to the back of the wedge, the angle of inclination $DMC$ becomes the complement of the semi-angle of the wedge, and therefore the force upon the back is to the sum of the resistances as the $\sin ACD \times \cos ACD$ is to AC. But in the similar triangles $DAF$, $DAC$, we have $DF : DA = DC : AC$, and $DF \times AC = DA \times DC$, consequently the force upon the back of the wedge is to the sum of the resistances as $DF \times AC$ is to $AC^2$, that is, as $DF : AC$.

Prop. II.

If, on account of the friction of the wedge, or any other cause, the resistances are wholly effective, that is, if the resisting surfaces adhere to the places to which they are applied without sliding, there will be an equilibrium, when the force upon the back is to the sum of the resistances, as the sine of the acute angle which the direction of the resisting forces makes with the back of the wedge is to radius.

Join $MN$, fig. 40, which will cut $DC$ perpendicularly at the point $H$. Then, since the forces $MD$, $ND$ are resolvable into $MH$, $HD$ and into $NH$, $HD$, and since $MH$, $HN$ destroy each other, the force upon the back is sustained by $2HD$. Consequently, the force upon the back is to the sum of the resistances as $2HD$ is to $2MD$, or as $HD$ is to $MD$. But the angle $ADM$, which the direction of the forces makes with the back of the wedge, is equal to $DMN$, and $HD$ is the sine of that angle, $MD$ being radius; therefore the force upon the back is to the sum of the resistances as $\sin ADM : \text{radius}$. Q.E.D.

Con. Since the angle $AMD = MDC + MCD$, the angle $MDC$ is the difference between $MCD$ the semi-angle of the wedge, and $AMD$ the angle which the direction of the resisting forces makes with the face of the wedge, and since $HD$ is the cosine of that angle, $MD$ being radius, we have the force upon the back to the sum of the resistances, as the cosine of the difference between the semi-angle of the wedge and the angle which the direction of the resisting forces makes with the face of the wedge, is to radius.

Prop. III.

When there is an equilibrium between three forces acting perpendicularly upon the sides of a wedge of any form, the forces are to one another as the sides of the wedge.

This is obvious from Dynamics, sect. 50, et seq., from which it follows, that when three forces are in equilibrium, they are proportional to the sides of a triangle, which are respectively perpendicular to their directions.

Prop. IV. When the power acting upon the back of a wedge is in equilibrium with the resistances opposed to it, the velocity of the power is to the velocity of the resistance as the resistance is to the power.

Produce DM to K (fig. 40), and draw CK perpendicular to DK. Then, by Prop. I., the power is to the resistance as MD : DH. Let the wedge be moved uniformly from D to C, and DK is the space uniformly described by the resisting force in the direction in which it acts; therefore, the velocity of the power is to the velocity of the resistance as DC : DK; that is, on account of the equiangular triangles DHM, DKC, as MD : DH; that is, as the resistance is to the power.

Sect. IV.—On the Screw.

Definition. A screw is a cylinder with an inclined plane wrapped round it, in such a manner, that the surface of the plane is oblique to the axis of the cylinder, and forms the same angle with it in every part of the cylindrical surface. When the inclined plane winds round the exterior surface of a solid cylinder, it is called a male screw; but when it is fixed on the interior circumference of a cylindrical tube, it is called a female screw. In the female screw, the spiral grooves formed by the inclined plane on the surface of the cylindrical tube must be equal in breadth to the inclined plane in the male screw, in order that the one may move freely in the other. By attending to the mode in which the spiral threads are formed by the circumvolution of the inclined plane, it will appear, that if one complete revolution of the inclined plane is developed, its altitude will be to its base as the distance between the threads is to the circumference of the screw. Thus, let abc (fig. 41) be the inclined plane, whose base is ac and altitude bc, and let it be wrapped round the cylinder MN (fig. 42), of such a size that the points a, c may coincide. The surface ab of the plane (fig. 41) will evidently form the spiral thread adeb (fig. 43), and ob, the distance between the threads, will be equal to bc (fig. 41), the altitude of the plane, and the circumference of the screw MN will be equal to ac, the base of the plane. If any body, therefore, is made to rise along the plane adeb in fig. 42, or along the spiral thread of the screw, by a force acting in a direction parallel to adeb, there will be the same proportion between the power and the resistance as if the body ascended the plane abc (fig. 41).

A male screw with triangular threads is represented in fig. 48, and its corresponding female screw in fig. 44. A male screw with quadrangular threads is exhibited in fig. 45, and the female screw in which it works in fig. 46. The friction is considerably less in quadrangular than in triangular threads, though, when the screw is made of wood, the triangular threads would be preferred. When the screws are metallic and large, the threads should be quadrangular; but the triangular form is preferable in small screws. When the screw is employed in practice, the power is always applied to the extremity of a screw-driver or lever placed temporarily or fixed permanently in its head. This is shown in fig. 47, where AB is the lever acting upon the screw BC, which works in a female screw in the block F, and exerts its force in bending the spring CD.

Prop. I. If the screw is employed to overcome any resistance, there will be an equilibrium when the power is to the resistance as the distance between two adjacent threads is to the circumference described by the power.

Let FAK be a section of the screw represented in fig. 45, perpendicular to its axis; CD a portion of the inclined plane which forms the spiral thread, and P the power, which, when applied at C in the plane ACF, will be in equilibrium with a weight upon the inclined plane CD. Then, in the inclined plane, when the direction of the power is parallel to the base, we have (Prop. I. Cor. 1, Sect. II.) P : W, as the altitude of the plane is to the base, or (Sect. IV. Def.) as the distance between two threads is to the whole circumference FKCF. If we suppose another power P' to act at the end of the lever AB, and describe the arch HBG, and that this power produces the same effect at B as the power P did at C, then (Sect. I. Prop. I.) we have P':P = CA : BA, that is, as FKCF is to the whole circumference of the circle HBG; but it was shown before, that P : W as the distance between two contiguous threads is to FKCF; therefore, by composition, P : W as the distance between two threads is to HBG, or the circumference of a circle whose radius is AB. Q. E. D.

Cor. I. It is evident from the proposition, that the power does not in the least depend upon the size of the cylinder FCK, but that it increases with the distance of that point from the centre A, to which the power is applied, and also with the shortness of the distance between the threads. Therefore, if P, p be the powers applied to two different screws, D, d the distances of these powers from the axis, and T, t the distances between the threads; their energy in overcoming a given resistance will be directly as their distances from the axis, and inversely as the distances of their threads, that is, P : p = D : d, or P varies as \( \frac{D}{T} \).

Prop. II. In the endless screw there will be an equilibrium when the power is to the weight as the distance of the threads multiplied by the radius of the axle is to the distance of the power from the axis of the screw multiplied by the radius of the wheel.

The endless screw, which is represented in fig. 49, consists of a screw EF, so combined with the wheel and axle ABC, that the threads of the screw may work in teeth fixed in the periphery of the wheel, and thus communicate the power exerted at the handle or winch P. Let \( W' \) represent the power produced by the screw at the circumference of the wheel; then, by the last proposition, \( P : W' \) as the distance between the threads is to the distance of \( P \) from the axis of the screw; but, in the wheel and axle, \( W' : W \) as the radius of the axle is to the radius of the wheel; therefore, by composition, \( P : W \) as the distances of the threads multiplied by the radius of the axle \( C \) is to the distance of the power \( P \) from the axis multiplied by the radius of the wheel \( AB \).

**Prop. III.**

When there is an equilibrium in the screw, the velocity of the weight is to the velocity of the power as the power is to the weight.

It is obvious from fig. 48, that while the power describes the circumference of the circle \( HBG \) uniformly, the weight uniformly rises through a space equal to the distance between two adjacent threads; therefore the velocity of the power is to the velocity of the weight as the distance between the threads is to the arch described by the power, that is (by Prop. I.), as the weight is to the power.

**Prop. IV.**

To explain the construction and advantages of Mr Hunter's double screw.

Let the screw \( CD \) work in the top \( AB \) of the frame \( ABCD \), and have \( n \) threads in an inch; the cylinder \( CD \), of which this screw is formed, is a hollow tube, which is also formed into a screw having \( n + 1 \) threads in an inch, and into this female screw is introduced a male screw \( DE \), having, of course, \( n + 1 \) threads in an inch. The screw \( DE \) is prevented from moving round with \( CD \) by the frame \( ABCD \) and the cross bar \( ab \), but is permitted to ascend and descend without motion of rotation. Then, by a revolution of the screw \( CD \), the other screw \( DE \) will rise through a space equal to \( \frac{1}{n+1} \times n \), and if the circumference described by the lever \( CK \) be \( m \) inches, we shall have

\[ P : W = \frac{1}{n+1} \times n : m ; \text{ or } P : W = 1 : mn \times n + 1. \]

This reasoning will be more perspicuous by supposing \( n \), or the number of threads in \( CD \), to be 12, and \( n + 1 \), or the number of threads in \( DE \), will consequently be 13. Let us suppose that the handle \( CK \) is turned round 12 times; the screw \( CD \) will evidently ascend through the space of an inch; and if the screw \( DE \) is permitted to have a motion of rotation along with \( CD \), it will also advance an inch. Let the screw \( DE \) be now moved backwards by 12 revolutions, it will evidently describe a space of \( \frac{1}{13} \)ths of an inch, and the consequence of both these motions will be that the point \( E \) is advanced \( \frac{1}{13} \)th of an inch.

But since \( DE \) is prevented from moving round with \( CD \), the same effect will be produced as if it had moved 12 times round with \( CD \), and had been turned 12 times backwards; that is, it will in both cases have advanced \( \frac{1}{13} \)th of an inch. Since, therefore, it has advanced \( \frac{1}{13} \)th of an inch in 12 turns, it will describe only \( \frac{1}{12} \)th of \( \frac{1}{13} \)th.

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**Scholium.**

In employing the screw for producing mechanical effects, it is most useful when a great pressure is to be exerted within a short space. Hence it is used in all presses where liquids or juices are expressed from solid bodies. It is peculiarly useful in coining and in printing, where the pressure of the die upon the metal, or of the types upon the paper, is required only through a very short space. The screw is also employed in compressing cotton into hard dense masses, to reduce it into the smallest bulk for land or sea carriage. The screw is likewise used for raising water, in which form it is called the screw of Archimedes (see Hydrodynamics); and it has been lately employed in the flour-mills in America for pushing the flour which comes from the mill-stones to the end of a long trough, from which it is conveyed to other parts of the machinery, in order to undergo the remaining processes. In this case, the spiral threads are very large in proportion to the cylinder on which they are fixed.

As the lever attached to the extremity of the screw moves through a very great space when compared with the velocity of its other extremity, or of any body which it puts in motion, the screw is of immense use in subdividing any space into a great number of minute parts. Hence it is employed in the engines for dividing mathematical instruments, and in those which have been recently used in the art of engraving. It is likewise of great use in the common wire micrometer, and in the divided object-glass micrometer, instruments to which the science of astronomy has been under great obligations.

**Sect. V.—On the Balance.**

**Definition.** The balance, in a mathematical sense, is a lever of equal arms, for determining the weights of bodies. The physical balance is represented in fig. 51, where \( FA, FB \) are the equal arms of the balance, \( F \) its centre of motion situated a little above the centre of gravity of the arms, \( FD \) the handle, which always retains a vertical position, \( P, W \) the scales suspended from the points \( A, B \), and \( CF \) the tongue or index of the balance.

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1 See Phil. Trans. vol. lxxi. p. 35. 2 Ibid. p. 65. which is exactly perpendicular to the beam \( AB \), and is continued below the centre of motion, so that the momentum of the part below \( F \) is equal and opposite to the momentum of that part which is above it. Since the handle \( FD \), suspended by the hook \( H \), must hang in a vertical line, the tongue \( CF \) will also be vertical when its position coincides with that of \( FD \), and consequently the beam \( AB \), which is perpendicular to \( CF \), must be horizontal. When this happens, the weights in the scale are evidently equal.

**Prop. I.**

To determine the conditions of equilibrium in a physical balance.

Let \( AOB \), fig. 52, be the beam, whose weight is \( S \), and let \( P, Q \) be equal weights expressed by the letter \( p \), and placed in the scales, whose weights are \( L \) and \( l \). Let \( O \) be the centre of motion, and \( g \) the centre of gravity.

IV. \( p + L \times AO \times \sin \lambda AO + S \times OG \times \sin \varphi = p + l + w \times BO \times \sin \varphi \).

But since the sines and cosines of any angles are the same as the sines and cosines of their supplement, we have,

V. \( p + L \times AC \times \cos \varphi - OC \times \sin \varphi + S \times OG \times \sin \varphi = p + l + w \times AC \times \cos \varphi + OC \times \sin \varphi + S \times Ce \times \cos \varphi \).

Hence, by No. III., we have,

VI. \( \tan \varphi = \frac{w \times AC}{2p + L + l + w \times OC + S \times OG} \).

But the force \( w \) with which the balance attempts to recover its horizontal situation, is the excess of momenta with which one arm is moved, above the momenta with which the other arm is moved; therefore

\( w = 2p + L + l + w \times OC + S \times OG \times \sin \varphi \).

A more extended illustration of these conditions of equilibrium will be found in an excellent paper by Euler, published in the Comment. Petropol. tom. x. p. 1, and another memoir upon the same subject by Kuhne in the Versuche der naturforschende Gesellschaft in Danzig, tom. i. § 123. From the preceding formulae, the following practical corollaries may be deduced.

Cor. 1. The arms of the balance must be exactly equal length, which is known by changing the weights in the scales; for if the equilibrium continues, the arms must be equal.

Cor. 2. The sensibility of the balance increases with the length of the arms.

Cor. 3. If the centre of motion coincides with the point and the centre of gravity, the balance will be in equilibrium in any position, and the smallest weight added to one of the scales will bring the beam into a horizontal position. The centre of motion, therefore, should not coincide with the centre of gravity.

Cor. 4. If the centre of motion is in the line which joins the points of suspension, the accuracy of the balance will be increased. The excess of the weights may be easily determined by the inclination of the beam, pointed out by the tongue or index upon a circular arch fixed to the handle, or more accurately by means of two divided arches fixed, near the points of suspension, on a scale independent of the balance. When the value of one of the whole beam, when unloaded; we shall have in the case of an equilibrium,

I. \( p + L \times AC = p + l \times BC + S \times Ce \); for since \( S \) is the weight of the beam, and \( g \) its centre of gravity, its mechanical energy in acting against the weights \( p + L \) is \( = 5 \times Ce \), the distance of its centre of gravity from the vertical line passing through the centre of motion \( O \).

II. But since \( AC = BC \); \( p \times AC - p \times BC = 0 \). Then, after transposition, take this from the equation in No. I., and we shall have,

III. \( L \times BC - L \times AC + S \times Ce \); or \( L - l = \frac{S \times Ce}{AC} \).

Let us now suppose that a small weight \( w \) is placed in the scale \( L \), the line \( AB \) which joins the points of suspension will be no longer horizontal, but will assume an inclined position. Let \( BA \varphi = \varphi \) be the angle which the beam makes with the direction of gravity. Then, by resolving the weight of the beam which acts in the direction \( OZ \), the parts \( \frac{OG}{Og} \) and \( \frac{Gq}{Og} \) will be in equilibrium, and we shall have,

of these divisions is determined experimentally, the rest are easily found, being proportional to the tangents of the inclination of the beam.

Con. 5. The sensibility of the balance will increase, the nearer that the centre of gravity approaches to the centre of motion.

Con. 6. If the centre of gravity is above the centre of motion, the balance is useless.

**Scholium.**

A balance with all the properties mentioned in the preceding corollaries, has been invented by M. Kuhne, and described in the work already quoted in the Proposition. It is so contrived that the points of suspension may be placed either above the centre of motion or below it, or in the line of its axis; the beam is furnished with an index, which points out the proportion of the weights upon a divided scale, and the friction of the axis is diminished by the application of friction wheels.

In order to get rid of the difficulties which attend the Magellan's construction of the tongue, the handle, and the arms of balance, the balance, M. Magellan invented a very accurate and moveable one, in which there is no handle, and where one of the arms acts as a tongue. The body to be weighed and the counterpoise are placed in the same scale, so that it is of little consequence whether the arms of the balance are equal or not. In this balance the centre of motion can be moved to the smallest distance from the centre of gravity. See Journal de Physique, Jan. 1781, tom. xvii. p. 43.

The balance invented by Ludlam, and described in the Ludlam's Philosophical Transactions for 1765, No. 55, depends balance, upon Epinus's property of the lever, which we have explained in Prop. V. Sect. I. The angular lever \( AFB \), in which \( AF = FB \), is moveable round \( f \), which is equidistant from \( A \) and \( B \). The weight \( P \) is suspended by a thread from \( A \); and the body \( W \), which is to be weigh- Mechanics.

On the Centre of Inertia.

Inertia, that with different bodies the lever AFB will have different degrees of inclination, and the index or tongue LEF, which is perpendicular to AB, will form different angles ZFL, bEf with the line of direction ZEb. Now, by Cor. 5, Prop. III. Sect. I., and by substituting for bB, bA the sines of the angles FbB, FbA, to which they are proportional, and also by taking, instead of FbB, the difference of the angles fFB, fFb, and instead of AFb, the sum of these angles, we shall have

\[ \tan fFB = \frac{P - W}{P + W} \times \tan \frac{AFB}{2}; \]

whence, by transposition, and by Geometry,

\[ P + W : P - W = \tan \frac{AFB}{2} : \tan fFb. \]

Hence, when the angle formed by the arms of the balance, and the angle of aberration fFb or ZFL, are known, the weights may be found, and vice versa.

See the article Balance, vol. iv. page 303, for more detailed information on this subject, and for a description of various balances.

CHAP. IV.—ON THE CENTRE OF INERTIA OR GRAVITY.

Definition. The centre of inertia, or the centre of gravity, of any body or system of bodies, is that point upon which the body or system of bodies, when influenced only by the force of gravity, will be in equilibrium in every position. The centre of inertia of plain surfaces bounded by right lines, and also of some solids, may be easily determined by common geometry. The application of the method of fluxions, however, to this branch of mechanics is so simple and beautiful, that we shall also avail ourselves of its assistance. The centre of gravity has been called by some writers the centre of position, and by others the centre of mean distances.

Prop. I.

To find the centre of inertia of any number of bodies, whatever be their position.

Let A, B, C, D be any number of bodies influenced by the force of gravity. Suppose the bodies A, B connected by the inflexible line AB considered as devoid of weight, then find a point F, so that the weight of A : the weight of B = BF : FA. The bodies A, B will therefore be in equilibrium about the point F in every position (Prop. I. Sect. I.), and the pressure upon F will be equal to A + B. Join

\[ X \varphi = \frac{A \times XA + B \times XB + C \times XC + D \times XD + E \times XE}{A + B + C + D + E}. \]

Now, A × XA, B × XB, &c. are evidently the momenta of the bodies A, B, &c. and the divisor A + B + C + D + E is the sum of the weights of all the bodies; therefore the distance of the point X from the centre of gravity ϕ is equal to the sum of the momenta of all the weights divided by the sum of the weights.

Cor. 1. If the point X had been taken between A and E, then the quantity A × XA would have been reckoned negative, as lying on a different side of the point X.

Cor. 2. From this proposition we may deduce a general rule for finding the centre of gravity in any body or system of bodies. Let any point be assumed at the extremity of the system, then the product of the momenta of all the bodies (or the product arising from the continual multiplication of each body by its distance from the point), divided by the sum of the weights of all the bodies, will be a quotient which expresses the distance of the centre of gravity from the point assumed.

Prop. II.

To find the centre of inertia of any number of bodies placed in a straight line.

Let A, B, C, D, E be any number of bodies whose common centre of gravity is ϕ. In the straight line AE take any point X. Then, since all the bodies are in equilibrium about their common centre of gravity ϕ, we have by the property of the lever (Prop. I. Sect. I.), A × XA + B × XB = C × XC + D × XD + E × XE; but since Xϕ = XA = Aϕ, and Xϕ = XB = Bϕ, and so on with the rest, we have by substitution A × XA = XA + B × XB = C × XC + D × XD + E × XE. Hence, by multiplying and transposing, we obtain A × XA + B × XB + C × XC + D × XD + E × XE, then dividing by A + B + C + D + E, we have

\[ X \varphi = \frac{A \times XA + B \times XB + C \times XC + D \times XD + E \times XE}{A + B + C + D + E}. \]

Prop. III.

If, in a system of bodies, a perpendicular be let fall from each upon a given plane, the sum of the products of each body multiplied by its perpendicular distance from the plane, is equal to the sum of all the bodies multiplied by the perpendicular distance of their common centre of inertia from the given plane.

Let A, B, C be the bodies which compose the system, and MN the given plane; by Prop. I. Sect. V. find F the

Fig. 56.

centre of inertia of A and B, and G the centre of gravity of the three bodies; and from A, F, B, G, C draw Aa, Ef, Bb, Gg, Ce perpendicular to the plane MN. Through F draw xfy, meeting Aa produced in x, and Bb in y, then in the similar triangles Axf, Byf, we have Ax : By = AF : BF, that is (Prop. I. Sect. V.), as B : A, hence A × Aa = B × Bb — yb, or, on account of the equality of the lines xa, yf, Bb, A × Ff — Aa = B × Bb — Ff; therefore, by multiplying and transposing, we have A + B × Ff = A × Aa + B × Bb.

In the very same way, by drawing wGz parallel to the plane, it may be shown that A + B + C × Gg = A × Aa + B × Bb + C × Cc. Q.E.D.

Corollary. By dividing by A + B + C we have

\[ G = \frac{A \times Aa + B \times Bb + C \times Cc}{A + B + C}. \]

Prop. IV.

To find the centre of inertia of a straight line composed of material particles.

If we consider the straight line as composed of a number of material particles of the same size and density, it is evident that its centre of inertia will be a point in the line equidistant from its extremities. For if we regard the line as a lever supported upon its middle point as a fulcrum, it will evidently be in equilibrium in every position, as the number of particles or weights on each side of the fulcrum are equal.

Prop. V.

To find the centre of inertia of a parallelogram.

Let ABCD be a parallelogram of uniform density, bisect AB in F, and having drawn Ef parallel to AC or BD, bisect it in ϕ; the point ϕ will be the centre of inertia of the parallelogram.

The parallelogram may be regarded as composed of lines AB, ab parallel to one another, and consisting of material particles of the same size and density. Now, by Prop. I. Sect. V., the centre of inertia of AB is F; and the centre of inertia of ab is c; and in the same way it may be shown that the centre of inertia, of every line of which the surface is composed, lies in the line Ff. But Ff may be considered as composed of a number of material particles of uniform density, each being equal in weight to the particles in the line AB; therefore, by Prop. IV., its centre of inertia will be in ϕ, its middle point.

Prop. VI.

To find the centre of inertia of a triangle.

Let ABC be a triangle of uniform density, and let AB, BC be bisected in the points E, D. Join CE, AD, and the point of intersection F shall be the centre of inertia of the triangle ABC. The triangle may be considered as composed of a number of parallel lines of material particles BC, be, βz; but in the similar triangles ADC, Ace, AD : DC = Ae : ec; and in the triangles ADC, ADB, Ace, BD : DA = be : ea;

hence, by composition, BD : DC = be : ec; but BD and DC are equal, therefore be = ec; and the line be, supposed to consist of material particles, will be in equilibrium about e.

In the same way it may be shown that every other line βz will be in equilibrium about a point situated in the line AD, consequently, the centre of gravity is in that line. For the same reason it follows, that the centre of gravity is in the line CE, that is, it will be in F, the point of intersection of these two lines. In order to determine the relation between FA and FD, join ED; then, since BE = EA, and BD = DC, BE : EA = BD : DC, and consequently ED is parallel to AC, and the triangles BED, BAC similar.

We have, therefore, CA : CB = DE : DB, and by alternation CA : DE = CB : DB, that is, CA : DE = 2 : 1. In the similar triangles CFA, DFE, AF : AC = DF : DE, and by alternation AF : DF = AC : DE, that is, AF : DF = 2 : 1, or AF = \( \frac{2}{3} \) AD.

1. Since

\[ AB^2 + AC^2 = 2BD^2 + 2AB^2 = \frac{1}{2} BC^2 + \frac{3}{2} AF^2, \] \[ AB^2 + BC^2 = 2CC^2 + 2BG^2 = \frac{1}{2} AC^2 + \frac{3}{2} CF^2, \] \[ AC^2 + BC^2 = 2AE + 2EC^2 = \frac{1}{2} AB^2 + \frac{3}{2} BF^2, \]

we have, by adding these three equations, and removing the fractions, \( AB^2 + BC^2 + AC^2 = 3AF^2 + 3CF^2 + 3BF^2 \), or in any plane triangle, the sum of the squares of the three sides is equal to thrice the sum of the squares of the distances of the centre of gravity from each of the angular points.

Corollary 2. By resolving the three quadratic equations in the preceding corollary, we obtain \( AF = \frac{1}{\sqrt{2}} \left( \frac{2}{3} BC^2 - AC^2 \right) \); \( CF = \frac{1}{\sqrt{2}} \left( \frac{2}{3} BA^2 + 2BC^2 - AC^2 \right) \); and \( BF = \frac{1}{\sqrt{2}} \left( \frac{2}{3} BC^2 + 2AC^2 - AB^2 \right) \), formulae which express the distances of the centre of gravity from each of the angular points.

Prop. VII.

To find the centre of inertia of a trapezium or any rectilineal figure.

Let ABCDE be the trapezium, and let it be divided into the triangles ABC, ACE, ECD by the lines AC, EC. By the last proposition find m, n, o, the centres of gravity of the triangles, and take the point F in the line mn, so that Fn : Fm = triangle ABC; triangle ACE, then F will be the centre of gravity of these triangles. Join Fo, and find a point f, so that fo : Ef = triangle ABC + triangle ACE : triangle CED, then all the triangles will be in equilibrium about f; that is, f is the centre of gravity of the rectilineal figure ABCDE. The same method may be employed in finding the centre of gravity of a trapezium, whatever be the number of its sides.

**Prop. VIII.**

To find the centre of inertia of a pyramid with a polygonal base.

Let the pyramid be triangular, as ABCD, fig. 60. Bisect BD in F, and join CF and FA. Make Ff = 1/3 of FC, and Ff = 1/3 of FA, and draw ff. It is evident, from Prop. II. of this chapter, that f is the centre of gravity of the triangular base BCD, and that the line Af, which joins the vertex and the point f, will pass through the centre of gravity of all the triangular laminae or sections of the pyramid parallel to its base BCD; for, by taking any section bed, and joining cm, it may be easily shown that bm = md, and mn = 1/2 mc, so that n is the centre of gravity of the section bed. It follows, therefore, that Af will pass through the centre of gravity of the pyramid. In the same way it may be shown, by considering ABD as the base, and C the vertex, and making Ff = 1/3 FA, that the centre of gravity lies in the line gc. But as the lines Af, fc lie in the plane of the triangle AFC, they must intersect each other; and therefore the point of intersection H will be the centre of inertia of the triangular pyramid. Now, since Ef = 1/3 FC, and Ff = 1/3 FA, we have Ff : FA = Ef : FC, therefore ff is parallel to AC. The triangle gh will consequently be similar to AHC, and Hp : HC = Hf : HA = fc : AC = 1 : 3; therefore Hf = 1/3 HC = 1/3 fc, and fh = 1/3 AH = 1/3 Af.

When the pyramid has a polygonal base, it may be conceived to be formed of a number of triangular pyramids, whose centres of inertia will be in one plane parallel to the base. Their common centre of gravity will therefore be in the same plane, and in the line drawn from the vertex to the centre of gravity of all the triangles which compose the base; the distance of the centre of gravity, therefore, from the vertex, will be equal to three fourths of the altitude of the pyramid.

Cor. I. Hence it is obvious that the centre of gravity of a right cone is a point in its axis whose distance from the vertex is equal to three fourths of the length of the axis; for as this may be demonstrated of a pyramid whose base is a polygon with an infinite number of sides, it must hold also of a right cone, which may be considered as a pyramid of this description.

Cor. 2. By proceeding as in Prop. II. Cor. 1, of this chapter, it will be found, that in a triangular pyramid, the distance of any of the vertices from its centre of inertia is equal to one fourth of the square root of the difference of thrice the sum of the squares of the three edges which meet at that vertex, and the sum of the squares of the other three edges; and likewise, that the sum of the squares of the distances of the centre of inertia from the vertices of any triangular pyramid, is equal to one fourth of the sum of the squares of the six edges of the pyramids. A demonstration of these theorems may be seen in Gregory's Mechanics, vol. i. p. 59, 60.

In order to show the application of the doctrine of fluxions to the determination of the centre of inertia of curve lines, arcs, solids, and the surfaces of solids, let ABC be any curve line whose axis is BR. Then, since the axis bisects all the ordinates DG, AC, each of the ordinates, considered as composed of material particles, will be in equilibrium about their points of bisection E, R; and therefore the centre of inertia of the body will lie in the axis. But if we consider the body as composed of a number of small weights DdgG, we shall find its centre of inertia by multiplying each weight by its distance from any line mm parallel to the ordinates, and dividing the sum of all these products by the sum of all the particles (Prop. II. of this chapter). Thus, let x denote the distance EB, then its fluxion x will be the breadth of the element or small weight DdgG, and x × DG will represent the weight, and the fluent of this quantity will be the sum of all the weights. Again, if we multiply the weight x × DG by x = EB, its distance from the point B, we shall have the momentum of that weight = x × x × DG, and the fluent of this quantity will express the sum of the momenta of all the weights into which the body is divided. But, by Prop. II., the distance of the centre of gravity from a given point B is equal to the sum of all the momenta divided by the sum of all the weights or bodies, that is, if F be the centre of gravity of the body ABC, we have FB = fluent of x × DG fluent of x × DG

or, calling y the ordinate DE, we have DG = 2y, and FB = fluent of x × 2 yx fluent of 2 yx

FB = fluent of x × 2 yx fluent of 2 yx

In the case of solids generated by rotation, the element or small weight x × DG will be a circular section, whose diameter is 2DE = 2y, and since the area of a circle is equal to its circumference multiplied by its diameter, we have (making π = 3.1416) 2 πy²x = the circular section whose diameter is DG; and since x × 2 πy²x, or 2 πy²x, will represent the momentum of the weight, we shall have

FB = fluent of 2 πy²x fluent of 2 πy²x

and dividing by 2 π, we obtain FB = fluent of xy² fluent of xy²

If the body whose centre of inertia is to be found be a curve line, as GBD (fig. 61), then it is manifest that the small weights will be expressed by the fluxion of GBD, that is, by 2 z, since GBD = 2 BD = 2 z; con- To find the centre of inertia of a circular segment.

Let \( AE = x \), \( EC = y \), and \( AD \) the radius of the circle \( = R \); consequently \( OE = 2R - EA \). Then, since by the property of the circle \( OE \times EA = BE^2 \), we have, by substitution,

\[ BE^2 = 2R \times EA - EA \times EA, \]

or \( y^2 = 2Rx - x^2 \); hence \( y = \sqrt{2Rx - x^2} \). Now, by p. 374, col. 2, we have the distance of the centre of gravity from \( A \), that is, \( AG = \frac{\text{fluent } yx}{\text{fluent } yz} \); but the fluent of \( yz \) or the sum of all the weights, is equal to the area of half the segment \( ABEC \); therefore \( AG = \frac{\text{fluent } yx}{\text{fluent } yz} \). Then, by substituting instead of \( y \), in this equation, the value of it deduced from the property of the circle, we have

\[ AG = \frac{\text{fluent } yx}{\text{fluent } yz} = \frac{\text{fluent } yx}{\text{fluent } yz} \cdot \frac{1}{2}ABEC. \]

Then, by substituting instead of \( y \), in this equation, the value of it deduced from the property of the circle, we have

\[ AG = \frac{\text{fluent } yx}{\text{fluent } yz} = \frac{\text{fluent } yx}{\text{fluent } yz} \cdot \frac{1}{2}ABEC. \]

Now, in order to find the fluxion of the numerator of the preceding fraction, assume \( z = 2Rx - x^2 \), and \( z' = \sqrt{2Rx - x^2} \), and by taking the fluxion, we have \( z = 2Rx - 2xz = 2R - 2x \times x \); but this quantity is double of the first term of the numerator, therefore \( \frac{z}{2} = R - x \times x \). By substituting these values in the fractional formula, we obtain

\[ GD = \frac{\text{fluent } yx}{\text{fluent } yz} = \frac{\text{fluent } yx}{\text{fluent } yz} \cdot \frac{1}{2}ABEC. \]

That is, the distance of the centre of gravity of a circular segment from the centre of the circle, is equal to the twelfth part of the cube of twice the ordinate (or the chord of the segment) divided by the area of the segment.

Cor. When the segment becomes a semicircle we have \( 2y = 2r \); and therefore \( GD = \frac{1}{12}ABEC = \frac{1}{12}ABEC \), that is, the distance of the centre of gravity of a semicircle from the centre of the semicircle is equal to the cube of the radius divided by nine and a half times the area of the segment.

Prop. X.

To find the centre of inertia of the sector of a circle.

Let \( ABDC \), fig. 62, be the sector of the circle. By Prop. I. Cor. 1, of this chapter, find \( m \) the centre of inertia of the triangle \( BCD \), and by the last proposition find \( G \) the centre of inertia of the segment; then take a point \( n \) so situated between \( G \) and \( m \), that \( ABEC : BCB = mn : GM \), then the point \( n \) will be the centre of gravity of the sector. By proceeding in this way, it will be found that \( Dn \), or the distance of the centre of gravity of the sector from the centre of the circle, is a fourth proportional to the semicircle, to the semichord, and to two thirds of the radius.

Prop. XI.

To find the centre of inertia of a plane surface bounded by a parabola whose equation is \( y = ax^n \).

Since \( y = ax^n \), multiply both terms by \( ax^n \) and \( x \) separately, and we have \( yx = ax^{n+1} \), and \( yx = ax^{n+1} \). But, by p. 374, col. 2, we have \( FB = \frac{\text{fluent of } yx}{\text{fluent } yx} \); therefore, by substituting the preceding values of \( yx \) and \( yx \) in the formula, we obtain \( FB = \frac{\text{fluent of } ax^{n+1}}{\text{fluent of } ax^{n+1}} \), and by taking the fluents it becomes

\[ FB = \frac{ax^{n+2}}{ax^{n+1}} = \frac{n + 1}{n + 2} \times x. \]

If \( n \), therefore, be equal to \( \frac{1}{2} \), then \( y = ax^{\frac{1}{2}} \); and, squaring both sides, \( y^2 = ax \), which is the equation of the common or Apollonian parabola. Hence \( FB = \frac{3}{2}x \), that is, the distance of the centre of gravity from the vertex is \( \frac{3}{2} \)ths of the axis.

When \( n \) is equal to 1, then \( y = ax \), and the parabola degenerates into a triangle, in which case \( FB = \frac{3}{2}x \), as in Prop. VI. of this chapter.

Prop. XII.

To find the centre of inertia of a solid, generated by the revolution of the preceding curve round its axis.

Since \( y = ax^n \), square both sides, and we have \( y^2 = ax^{2n} \); then multiply both sides by \( ax^n \) and \( x \) separately, we obtain \( y^2x = ax^{2n+1} \), and \( y^2x = ax^{2n+1} \). But, by p. 374, col. 2, we have \( FB = \frac{\text{fluent of } y^2x}{\text{fluent } y^2x} \); therefore, by substituting the preceding values of \( y^2x \) and \( y^2x \) in that formula, we obtain \( FB = \frac{\text{fluent of } ax^{2n+1}}{\text{fluent of } ax^{2n+1}} \), and by taking the fluents we shall have

\[ FB = \frac{ax^{2n+2}}{ax^{2n+1}} = \frac{2n + 1}{2n + 2} \times x. \]

When \( n = \frac{1}{2} \), the solid becomes a common paraboloid, and we obtain \( FB = \frac{3}{2}x \).

When \( n = 1 \), the solid becomes a cone, and \( FB = \frac{3}{2}x \), as in Prop. VIII. Cor. 1, of this chapter.

Prop. XIII.

To find the centre of inertia of a spherical surface or zone, comprehended between two parallel planes, or of the spherical surface of any spherical segment.

Let \( BMNC \), fig. 62, be a section of the spherical surface comprehended between the planes \( BC \), \( MN \), and let \( EP \) On the \( x \), \( EC = y \), \( DC = R \), and \( z = \) the arc \( CN \). Suppose Centre of the abscissa \( EP \) to increase by the small quantity \( Eo \), draw Inertia, or parallel to \( EC \), \( Cs \) parallel to \( Eo \), and \( Cr \) perpendicular to \( DC \); then it is evident, that in the similar triangles \( CDE \), \( Csr \), \( EC : DC = Cs : Cr \), that is, \( y : R = Cs : Cr \); but \( Cr \) is the fluxion of the arc \( NC \), and \( Cs \) the fluxion of the abscissa \( PE \); therefore \( y : R = x : z \), and \( zy = Rx \), and \( z = \frac{Rx}{y} \). Now, by p. 374, col. 2, \( FB = \frac{\text{fluent of } xyz}{\text{fluent of } xyz} \), therefore, by substituting the preceding value of \( z \) in this formula, we obtain \( FB = \frac{\text{fluent of } Rxz}{\text{fluent of } Rx} \); for \( \frac{Rxxz}{Rxz} = \frac{Ryxxz}{Ryxz} \) and (dividing by \( yz \)) \( = \frac{Rxx}{Rx} \). By taking the fluents we obtain \( FB = \frac{1}{2} \frac{Rxx}{Rx} = \frac{1}{2} x \), a fluent which requires no correction, as the other quantities vanish at the same time with \( x \).

When \( DP \) is equal to \( DC \), the solid becomes a spherical segment, and \( EA \) becomes the altitude of the segment, so that universally the centre of gravity of the spherical surface of a spherical segment is in the middle of the line which is the altitude of the segment, or in the middle of the line which joins the centres of the two circles that bound the spherical segment.

When the spherical segment is a hemispheroid, the centre of gravity of its hemispherical surface is obviously at the distance of one half the radius from its centre.

**Prop. XIV.**

To find the centre of inertia of a circular arc.

Let \( ACB \) be the circular arc, it is required to find its centre of inertia, or the distance of the centre of inertia of the half arc \( AC \) from the diameter \( HG \); for it is evident that the line which joins the centres of gravity of each of the semicircles \( AB \), \( AC \) must be parallel to \( HG \), and therefore the distance of their common centre of gravity, which must be in that line, from the line \( HG \), will be equal to the distance of the centre of gravity of the semicircle from the same line. Make \( PC = DE = x \), \( EC = y \), \( DC = DA = R \), and \( AC = z \), then it may be shown, as in the last proposition, that \( y : R = x : z \); hence \( zy = Rx \). But, by p. 374, col. 2, we have \( FB = \frac{\text{fluent of } yz}{z} \), \( y \) being in this case equal to \( x \) in the formula referred to, and, substituting the preceding value of \( yz \), it becomes \( FB = \frac{\text{fluent of } Rx}{z} \); and, taking the fluent, we have \( FB = \frac{Rx}{z} \), which requires no correction, as the fluent of \( yz \) vanishes at the same time with \( x \). Calling \( d \), therefore, the distance of the centre of inertia of the arc \( BAC \) from the centre \( D \), we have \( d = \frac{Rx}{z} \), and \( dz = Rx \); hence \( z : x = R : d \), or \( 2Z : 2x = R : d \), that is, the distance of the centre of inertia of a circular arc from the centre of the circle is a fourth proportional to the arc, the chord of the arc, and radius.

When the arc \( BAC \) becomes a semicircle, \( PC \) or \( z \) is equal to \( DG \) or radius, so that we have \( 2z : 2R = R : d \), or \( 4Z : 4R = R : d \); but \( 4z \) is equal to the whole circumference of the circle, and \( 4R \) is equal to twice the diameter; therefore, \( 3:141593 : 2 = R : d \); hence \( d = \frac{2R}{3:141593} = 63662R \).

When \( y \) is equal to \( 2R \), or when the arc \( ABC \) becomes equal to the whole circumference of the circle, \( x \) vanishes, and \( z = 0 \), and therefore \( \frac{Rx}{z} = 0 \), which shows that the centre of inertia coincides with the centre of the circle.

**Scholium I.**

From the specimens which the preceding propositions contain of the application of the formulae in page 374, col. 2, the reader will find no difficulty in determining the centre of inertia of other surfaces and solids, when he is acquainted with the equation of the curves by which the surfaces are bounded, and by whose revolution the solids are generated.

A knowledge of the nature of these curves, however, is not absolutely necessary for the determination of the centres of inertia of surfaces and solids. A method of finding the centre of gravity, without employing the equation of the bounding curves, was discovered by our countryman Mr Thomas Simon. It was afterwards more fully illustrated by Mr Chapman, in his work *On the Construction of Ships*; by M. Leveque, in his translation of Don George Juan's *Treatise on the Construction and Management of Vessels*; and by M. Prony, in his *Architecture Hydraulique*, tom. i. p. 93; to which we must refer such readers as wish to prosecute the subject.

**Scholium II.**

As it is frequently of great use to know the position of the centre of inertia in bodies of all forms, we shall collect all the leading results which might have been obtained by the method given in the preceding proposition.

1. The centre of inertia of a straight line is in its middle point. 2. The centre of inertia of a parallelogram is in the intersection of its diagonals. 3. The centre of inertia of a triangle is distant from its vertex two thirds of a line drawn from the vertex to the middle of the opposite side. 4. The centre of inertia of a circle and of a regular polygon coincides with the centres of these figures. 5. The centre of inertia of a parallelopiped is in the intersection of the diagonals joining its opposite angles. 6. The centre of inertia of a pyramid is distant from its vertex three fourths of the axis. 7. The centre of inertia of a right cone is in a point in its axis whose distance from the vertex is three fourths of the axis. 8. In the segment of a circle, the centre of inertia is distant from the centre of the circle a twelfth part of the cube of the chord of the segment divided by the area of the segment, or \( d = \frac{C^3}{12A} \), where \( d \) is the distance of the centre of inertia from the centre of the circle, \( C \) is the chord of the segment, and \( A \) its axis.

Mathematical Dissertations, p. 109. 9. In the sector of a circle, the centre of inertia is distant from the centre of the circle by a quantity which is a fourth proportional to the semicircle, the semichord, and two thirds of the radius.

10. In a spherical surface or zone, comprehended between two planes, the centre of inertia is in the middle of the line which joins the centres of the two circular planes by which it is bounded. When one of the circular planes vanishes, the spherical zone becomes the spherical surface of a spherical segment; therefore,

11. In a spherical surface of a spherical segment, the centre of inertia is in the middle of its altitude or versed sine; consequently,

12. The centre of inertia of the surface of a complete sphere coincides with the centre of the sphere.

13. In a spherical segment, the centre of inertia is distant from the vertex by a quantity equal to \( \frac{4a - 3x}{6a - 4x} \times x \), where \( a \) is the diameter of the sphere, and \( x \) the altitude or versed sine of the segment; hence,

14. The centre of inertia of a hemisphere is distant from its vertex by a quantity equal to five eighths of the radius, or it is three eighths of the radius distant from the hemisphere; and,

15. The centre of inertia of a complete sphere coincides with the centre of the sphere.

16. In a circular arc, the centre of inertia is distant from its centre by a quantity equal to \( \frac{Rz}{z} \), where \( R \) is the radius, \( z \) the semichord, and \( z \) the semicircle; hence,

17. In a semicircular arc, the centre of inertia is distant from its centre \( 63662 R \); and,

18. The centre of inertia of the circumference of a circle coincides with the centre of the circle.

19. In a circular sector, the centre of inertia is distant from the centre of the circle \( \frac{2cR}{3a} \), where \( R \) is the radius, \( c \) the arc, and \( c \) its chord.

20. In a spherical sector, composed of a cone and spherical segment, the centre of inertia is distant from the vertex of the segment by a quantity equal to \( \frac{2R + 3x}{8} \), where \( R \) is radius, and \( x \) the altitude or versed sine of the segment.

21. In an ellipsis, the centre of inertia coincides with the centre of the figure.

22. The centre of inertia of an oblate and prolate spheroid, solids generated by the revolution of an ellipse round its lesser and its greater axes respectively, coincides with the centres of the figures.

23. In the segment of an oblate spheroid, the centre of inertia is distant from its vertex by a quantity equal to \( \frac{m - 3x}{6m - 4x} \times x \), where \( m \) is the lesser axis, or axis of rotation, and \( x \) the altitude of the segment; hence,

24. In a hemispheroid, the centre of inertia is distant from its vertex five eighths of the radius.

25. The centre of inertia of the segment of a prolate spheroid is distant from its vertex by a quantity equal to \( \frac{n - 3x}{6n - 4x} \times x \), where \( n \) is the greater axis, or axis of rotation.

26. In the common or Apollonian parabola, the distance of the centre of inertia from its vertex is three fifths of the axis.

27. In the cubical parabola, the distance of the centre of inertia from its vertex is four sevenths of the axis, in the biquadratic parabola five ninths of the axis, and in the solid parabola six elevenths of the axis.

28. In the common semiparabola, the distance of its centre of gravity from the centre of gravity of the whole parabola, in the direction of the ordinate passing through that centre, is three eighths of the greatest ordinate.

29. In the common paraboloid, the distance of the centre of inertia from its axis is equal to two thirds of the axis.

30. In the common hyperboloid, the distance of the centre of inertia from the vertex is equal to \( \frac{4a + 3x}{6a + 4x} \times x \), where \( a \) is the transverse axis of the generating hyperbola, and \( x \) the altitude of the solid.

31. In the frustum of a paraboloid, the distance of the centre of inertia from the centre of the smallest circular end is \( \frac{2R^2 + r^2}{R^2 + r^2} \times h \), where \( h \) is the distance between the centres of the circles which contain the paraboloidal frustum, \( R \) the radius of the greater circle, and \( r \) the radius of the lesser circle.

32. In a conic frustum or truncated cone, the distance of the centre of inertia from the centre of the smallest circular end is \( \frac{3R^2 + 2Rr + r^2}{R^2 + Rr + r^2} \times h \), which represents the distance between the centres of the circles which contain the frustum, and \( R, r \) the radii of the circles.

33. The same formula is applicable to any regular pyramid, \( R \) and \( r \) representing the sides of the two polygons by which it is contained.

**Prop. XV.**

If a quantity of motion be communicated to a system of bodies, the centre of inertia of the system will move in the same direction, and with the same velocity, as if all the bodies were collected in that centre, and received the same quantity of motion in the same direction.

Let A, B, C be the bodies which compose the system, and let F be the centre of gravity of the bodies B, C, and


the centre of gravity of the whole system, as determined by Prop. I. Chap. IV. Then, if the body A receive such a momentum as to make it move to \( a \) in a second, join Fa, and take a point \( \varphi \) so that \( F\varphi : \varphi a = Fa : fa \); \( \varphi \) will now be the centre of gravity of the system, \( f\varphi \) the path of that centre will be parallel to \( Aa \), and \( f\varphi \) will be to \( Aa \) as B is to \( A + B + C \). Let the same quantity of motion be now communicated to B, so as to make it describe the space \( Bb \) in a second; and having drawn \( \varphi G \) parallel to \( Bb \), take a point \( G \), so that \( F\varphi : Bb = B : A + B + C \), and \( G \) will be the centre of gravity of the bodies after B has moved to \( b \). In the same manner it may be found that H will be the common centre of gravity of the bodies after the same quantity of motion has been communicated to C in the direction \( Cc \). Now if the quantity of motion which was communicated to A, B, C separately had been communicated to them at the same instant, they would have been found at the end of a second in the points \( a, b, c \), and their centre of gravity would have been the point H. Let us now suppose the three bodies collected in their common centre of gravity \( f \); the body at \( F \) will be equal to \( A + B + C \), and if the same quantity of motion which made \( A \) move to \( a \) in a second be communicated to the body at \( f \) and in the same direction, it will be found somewhere in the line \( f \phi \) at the end of a second. But as the quantity of motion is equal to the product of the velocity of the body multiplied by its quantity of matter, the velocities are inversely as the quantities of matter, and consequently the velocity of the body at \( f \) is to \( A \)'s velocity as \( A \) is to \( A + B + C \), that is, as \( f \phi \) is to \( Aa \); therefore \( Aa \) and \( f \phi \) are described by \( A \) and by the body at \( f \) in equal times, and the body at \( f \) will be found at \( \phi \) at the end of a second. In the same way it may be shown that the body at \( f \) will be found at \( G \) if it receives the same momentum that was given to \( B \), and in the same direction, and that it will be found at \( H \) after it has received the momentum that was communicated to \( C \); consequently, if it received all these momenta at the same instant, it would have described \( fH \) in a second. Q. E. D.

Cor. 1. If the bodies of a system move uniformly in right lines, their common centre of gravity will either be at rest, or move uniformly in a right line. For if the momenta communicated to the bodies \( A, B, C \) were communicated to a body at \( f = A + B + C \), it will either remain at rest or move uniformly in a straight line. (See Newton's Principia, i. sect. iii. cor. 1.)

Cor. 2. The centre of gravity of any system is not affected by the mutual action of the bodies which compose it. For let \( B \) and \( C \) (fig. 64) be two bodies whose common centre of gravity is \( F \), and let the points \( \beta, z \) be taken, so that \( B\beta : Cz = C : B \); the spaces \( B\beta, Cz \) will represent the mutual action of the bodies \( B, C \), that is, \( B\beta \) will represent the action of \( C \) upon \( B \), or the motion which is the result of that action, and \( Cz \) the action of \( B \) upon \( C \), or the motion which results from it. Then, since \( F \) is the common centre of gravity of \( B \) and \( C \), we have (Prop. I. Chap. IV.) \( B : C = FC : FB \); but \( B : C = Cz : B\beta \), therefore \( FC : FB = Cz : B\beta \). But \( Cz \) is a magnitude taken from \( FC \), and \( B\beta \) is a magnitude taken from \( FB \), consequently (Playfair's Euclid, book v. prop. 19) the remainder \( zF : \beta F = FC : FB \), that is, \( zF : \beta F = B : C \), that is, the point \( F \) continues to be the centre of gravity notwithstanding the action of the bodies \( B, C \). If the system is composed of several bodies, the same thing may be proved of every two of the bodies, and consequently of the whole system. (See D'Alembert's Dynamique, art. 76; and Newton's Principia, i. sect. iii. cor. 4.)

**Prop. XVI.**

If a body is placed upon a horizontal plane, or suspended by two threads, it cannot be in equilibrio unless a perpendicular drawn from the centre of inertia to the horizontal plane, or a horizontal line passing through the two threads, fall within the base of the body, or upon that part of the horizontal line which lies between the threads.

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**Fig. 65.**

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**Fig. 66.**

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**Fig. 67.**

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Let \( ABCD \) be a body placed in the horizontal plane \( CD \), \( G \) its centre of gravity, and \( GE \) a perpendicular drawn to the horizontal line \( DE \). Then the whole matter of the body \( ABCD \) may be conceived as united in its centre of gravity \( G \); and as its tendency downwards is in the vertical line \( GE \), it can descend only by turning round the point \( C \) as a centre. Here then we have a body \( G \) placed at the end of a lever \( GC \), whose fulcrum is \( C \), and its power to turn round \( C \) is represented by the quantity of matter in \( G \) multiplied by the perpendicular \( CE \); let fall from the fulcrum upon its line of direction; and as there is no force to counterbalance this, the body \( G \), and consequently the body \( ABCD \), will fall by turning round \( C \). When the vertical line \( GE \) coincides with \( GC, EC \) vanishes, and the weight of the body concentrated at \( G \) has no power to turn the lever round \( C \), but is supported upon the fulcrum \( C \). When the vertical line \( GE \) (by some writers called the line of direction) falls within the base \( CD \), it is obvious that the weight at \( G \) has no influence in producing a motion round \( C \) or \( D \), but is employed in pressing the body upon the horizontal plane \( ED \).

2. Let the body \( ACBD \) be suspended at the points \( f, \phi \) by the threads \( hf, h\phi \), and let \( G \) be the centre of gravity of the body. Join \( Gf, G\phi \); draw \( f\phi \) parallel to the horizon, and through \( G \) draw no parallel to \( f\phi \). Continue \( hf, h\phi \) to \( o \) and \( n \), and draw \( Gi \) perpendicular to \( f\phi \); the body \( AB \) cannot be in equilibrio unless the point \( i \) falls upon the horizontal line \( f\phi \) which passes through the threads. It is obvious that the centre of gravity can never change its distance from the fixed points of suspension \( f, \phi \); if therefore the body is not in equilibrio, its centre of gravity must descend either towards \( m \) or \( n \); let it descend towards \( m \) till it rests at the point \( \gamma \), then \( \gamma f = \gamma G \); but \( \gamma f \) is greater than \( G\phi \) (Euclid, book i. prop. 7), which is absurd; therefore the point \( G \) cannot descend, that is, the body is in equilibrio. It may be shown in the same way, that it will be in equilibrio when \( G \) is anywhere between \( n \) and \( o \), that is, when the perpendicular let fall from \( G \) cuts the horizontal line \( f\phi \) that lies between the threads. If the body be suspended by the two threads \( HF, hF \), so that the perpendicular \( Gi \) falls without the line \( f\phi \), the body is not in equilibrio, for the centre of gravity \( G \) acting at the end of the lever \( GF \), tends to turn round \( F \) with a power equal to \( G \times Gm \); it will therefore descend, and as its distance from \( f \) cannot change, the point \( f \) will rise, and the thread \( fh \) will be relaxed. When \( G \) arrives at \( m \), the perpendicular \( Gm \) vanishes, and \( G \) has no power to turn round \( F \). The body \( AB \), therefore, cannot be in equilibrio till the perpendicular \( Gi \) falls within \( f\phi \), which it does as soon as it arrives at \( m \).

Cor. 1. If a body is placed upon an inclined plane, supposed without friction, it will slide down the plane when the line of direction falls within its base, and will roll down when this line falls without the base. This is the reason why a sphere or cylinder rolls down an inclined plane; for as they touch the plane only in one point or line, the point of direction must always fall without the base.

Cor. 2. The higher the centre of gravity of a body is, the more easily will it be overturned. For if \( ABCD \) be the body whose centre of gravity is \( F \), and if any force be employed to move it round...

Con. 3. If a body be suspended by one thread, it will not be at rest unless its centre of gravity is in the direction of the thread produced; for when the two threads \( hf \) approach so near each other as to coincide with the single thread \( HE \), the point \( i \) must in the case of an equilibrium fall upon \( F \), and the lines \( Gi \), \( GF \) must coincide with \( mF \); but \( HF \) and \( mF \) are both perpendicular to the horizontal line \( f \); therefore the centre of gravity \( G \) is in the direction of the thread \( HF \).

Cor. 4. If the bodies \( A \), \( B \), \( C \) (fig. 68), be suspended by any point \( F \) from the hook \( H \), they will not be in equilibrium unless their common centre of gravity \( G \) is in the vertical line \( FG \) passing through the point of suspension; and, in fig. 69, the bodies \( A \), \( B \) connected by the bent rod \( AFB \), will not be in equilibrium unless their common centre of gravity \( G \) is in a vertical line passing through \( F \), the point in which the system rests upon the plane \( CD \).

Scholium.

We have seen in the preceding proposition and corollaries, the position which must be given to the centre of gravity in order to procure an equilibrium. It is evident, however, that though the bodies are necessarily at rest, yet they have different degrees of stability, depending on the position of the centre of gravity with regard to the centre of motion. Hence bodies are said to have a stable equilibrium when their centre of gravity cannot move without ascending, or when the path described by their centre of gravity has its concavity upwards,—a tottering equilibrium when the centre of gravity cannot move without descending, or when the path which it describes has its concavity downwards,—and a neutral equilibrium when the body will rest in any position. Thus, in fig. 70, if the vessels \( A \), \( B \) have their handles so placed that in the one the handle \( A \) is fixed above the centre of gravity \( g \), and in the other the handle \( B \) is fixed below the centre of gravity \( g \), then the equilibrium of \( A \) will be stable, and that of \( B \) tottering; for if \( A \) is held by the handle, it will require a considerable force to make its centre of gravity describe the path \( mn \), whereas the smallest force will destroy the equilibrium of \( B \). The vessel \( A \), too, has a constant tendency to recover its equilibrium, and always recovers it as soon as the disturbing force is removed; but the vessel \( B \) has no tendency to do this even when its equilibrium is affected in the smallest degree. For the same reason, the elliptical body \( A \) (fig. 71), when resting on the extremity of its conjugate axis, has a stable equilibrium, but when resting on its transverse axis, as at \( B \), its equilibrium is tottering. The equilibrium of a circle or sphere is always neutral; for when it is disturbed, the body has neither a tendency to fall nor to resume its former situation. A flat body \( A \), supported by a sphere \( B \) (fig. 72), will have its equilibrium stable when its centre of gravity is nearer the point of contact than the centre of the sphere is, and the equilibrium of \( C \) will be tottering when its centre of gravity is farther distant from the surface of the sphere \( D \) than the centre of the sphere is.

Prop. XVII.

1. To find the centre of inertia mechanically.

If the body whose centre of inertia is to be found can be suspended by a thread, then, when the body is in equilibrium, the centre of gravity will be somewhere in the line of finding prolonged if necessary, that is, formed by the thread upon the surface of the body. Let a body be again suspended from another part of its surface, so that the direction of the thread may be nearly at right angles to its former direction; then, as the centre of gravity must also be in the new direction of the thread prolonged, it will be in the point where these two lines intersect each other.

2. If the body is of such a kind that it cannot be conveniently suspended, balance it upon two sharp points, and its centre of motion will be somewhere in the line which joins these points. Balance it a second time upon the sharp points, so that the line which joins the points may be nearly at right angles to the former line. The intersection of these two lines will be the centre of inertia of the body.

3. If the body is so flexible that it can neither be suspended by a thread nor balanced upon points, then let a thin board be balanced upon the points as before, and let the body be so placed upon this board when balanced, that the equilibrium may still continue; then, having found the centre of gravity of the board when loaded with the body, the centre of gravity of the body will be a point on its surface exactly opposite to that centre.

The preceding method, however, only gives us the centre of gravity when the body has no sensible thickness; for when it is of three dimensions, the centre of gravity must be somewhere between the two opposite surfaces.

Definition. The centro-baryc method is the method of determining the areas of surfaces, and the contents of solids, by considering them as generated by motion, and by employing the laws of the centre of inertia.

Prop. XVIII.

If any straight or curve line, or any plane surface bounded by straight or curve lines, revolve round an axis situated in the same plane with the lines or surfaces, the solid thus generated will be respectively equal to a surface or solid whose base is equal to the given line or surface, and whose height is equal to the arc described by the centre of inertia of the generating line or surface. Let \(ABCD\) be the plane surface by whose revolution round the axis MPN is generated the solid \(aD\), contained by the parallelograms \(ABCD\), \(abcd\), and by the areas \(aACe\), \(bBDd\), and \(aABb\), \(cCDD\); let \(G\) be the centre of gravity of \(ABCD\), then the solid \(aD\) shall be equal to a solid whose base is \(ABCD\), and whose altitude is a line equal to \(Gg\), the space described by its centre of gravity \(G\). It is evident from Prop. III. Chap. IV. that the sum of the products of all the particles of the surface \(ABCD\), multiplied by their respective distances from any given point \(P\), is equal to the sum of all the particles multiplied by the distance of their common centre of gravity \(G\) from the same point \(P\). Now every particle of the surface \(ABCD\), during its revolution round the point \(P\), will obviously describe the arch of a circle proportional to the distance of that particle from the point \(P\), which is the centre of all the arches; therefore the sum of the products of all the particles multiplied by the arch described by each of them, will be equal to the sum of the particles multiplied by the arch which their common centre of gravity describes; that is, the solid \(aD\) will be equal to the area of the surface multiplied by the path of its centre of gravity. In order to have a clearer illustration of this reasoning, let \(P, p, \sigma, \&c.\) be the particles of the surface \(ABCD\); \(d, d, \delta\) their distance from the centre of rotation \(P\), and \(A, a, a\) the arches which they describe, while \(GP\) is the distance of the centre of gravity of the surface \(ABCD\) from the centre \(P\), and \(Gg\) the arch described by it. Then, by Prop. III. \(P \times D + p \times d + \sigma + \delta = P + p + \sigma \times GP\); but \(D : d : \delta : GP = A : a : a : Gg\), therefore \(P \times A + p \times a + \sigma \times a = P + p + \sigma \times Gg\). But \(P + A + p \times a + \sigma \times a, \&c.\) make up the whole solid \(aD\), and \(P + p + \sigma, \&c.\) make up the whole surface \(ABCD\); therefore the solid \(aD\) is equal to the generating surface \(ABCD\) multiplied by the path of its centre of gravity. Q.E.D.

Cor. I. Let us suppose the circle BACO (see fig. 62) to be generated by the revolution of the line DA round the point \(D\); then, since the centre of gravity of the line DA is in its middle point \(G\), the path of this centre will be a circumference whose radius is \(DG\), or a line equal to half the circumference BONAB, therefore, by the theorem, the area of the circle BONB will be equal to the radius DA multiplied by the semicircumference, which coincides with the result obtained from the principles of geometry. (See Playfair's Geometry, supp. book i. prop. 5.) In the same way, by means of the preceding theorem, we may readily determine the area of any surface, or the content of any solid that is generated by motion.

Scholium.

The centro-baryc method, which is one of the finest inventions of geometry, was first noticed by Pappus in the preface to the seventh book of his mathematical collections; but it is to Father Guldinus that we are indebted for a more complete discussion of the subject. He published an account of his discovery partly in 1635 and partly in 1640, in his work entitled De Centro Gravitatis, lib. ii. cap. 8, prop. 3, and gave an indirect demonstration of the theorem, by showing the conformity of its results with those which were obtained by other means. Leibnitz demonstrated the theorem in the case of superficies generated by the revolution of curves, but concealed his demonstration (Act. Leips. 1695, p. 493). The theorem of Leibnitz, however, as well as that of Guldinus, was demonstrated by Varignon in the Memoirs of the Academy for 1714, p. 78. Leibnitz observes that the method will still hold, even if the centre round which the revolution is performed be continually changed during the generating motion. For further information on this subject, the reader is referred to Dr Wallis's work De Calculo Centri Gravitationis; Hutton's Mensuration; Prony's Architecture Hydraulique, vol. i. p. 88; and Gregory's Mechanics, vol. i. p. 64.

Prop. XIX.

To show the use of the doctrine of the centre of inertia in the explanation of some mechanical phenomena.

In the equilibrium and motion of animals, we perceive many phenomena deducible from the properties of the centre of gravity. When we endeavour to rise from a chair, we naturally draw our feet inwards, and rest upon their extremities, in order to bring the centre of gravity directly below our feet, and we put the body into that position in which its equilibrium is tottering, a position which renders the smallest force capable of producing motion, or of overturning the body. In this situation, in order to prevent ourselves from falling backwards, we thrust forward the upper part of the body for the purpose of throwing the centre of gravity beyond our feet; and when the equilibrium is thus destroyed, we throw out one of our feet, and gradually raise the centre of gravity till the position of the body is erect. When we walk, the body is thrown into the position of tottering equilibrium by resting it on one foot; this equilibrium is destroyed by pushing forward the centre of gravity, and the body again assumes the position of tottering equilibrium by resting it on the other foot. During this alternate process of creating and destroying a tottering equilibrium, the one foot is placed upon the ground, and the other is raised from it; but in running, which is performed in exactly the same way, both the feet are never on the ground at the same time. At every step there is a short interval during which the runner does not touch the ground at all.

When we ascend an inclined plane the body is thrown farther forward than when we walk on a horizontal one, in order that the line of direction may fall without our feet; and in descending an inclined plane, the body is thrown backward, in order to prevent the line of direction from falling too suddenly without the base. In carrying a burden, the centre of gravity is brought nearer to the burden, so that the line of direction would fall without our feet if we did not naturally lean towards the side opposite to the burden, in order to keep the line of direction within our feet. When the burden is therefore carried on the back, we lean forward; when it is carried in the right arm, we lean towards the left; when it is carried in the left arm, we lean towards the right; and when it is carried before the body, we throw the head backwards.

When a horse walks, he first sets out one of his fore feet and one of his hind feet, suppose the right foot; then at the same instant he throws out his left fore foot and his left hind foot, so as to be supported only by the two right feet. His two right feet are then brought up at the same instant, and he is supported only by his two left feet. When a horse pulls at a load which he can scarcely overcome, he raises both his fore feet, his hind feet become the fulcrum of a lever, and the weight of the horse collected in his centre of gravity acts as a weight upon this lever, and enables him to surmount the obstacle. (See Appendix to Ferguson's Lectures, vol. ii.)

When a rope-dancer balances himself upon the fore part of one foot, he preserves his equilibrium in two ways, either by throwing one of his arms or his elevated foot, or his balancing pole, to the side opposite to that towards which he is beginning to fall, or by shifting the point of his foot on which he rests, to the same side towards which he is apt to fall; for it amounts to the same thing whether he brings the centre of gravity directly above the point of support, or brings the point of support directly below the centre of gravity. For this purpose the convex form of the foot is of great use; for if it had been perfectly flat, the point of support could not have admitted of small variations in its position.

We have already seen (Prop. XVI. Cor. 2) that any body is more easily overturned in proportion to the height of its centre of gravity. Hence it is a matter of great importance that the centre of gravity of all carriages should be placed as low as possible. This may often be effected by a judicious disposition of the load, of which the heaviest materials should always have the lowest place. The present construction of our mail and post coaches is therefore adverse to every principle of science, and the cause of many of those accidents in which the lives of individuals have been lost. The elevated position of the guard, the driver, and the outside passengers, and the two boots which contain the baggage, raises the centre of gravity of the loaded vehicle to a very great height, and renders it much more easily overturned than it would otherwise have been. When any accident of this kind is likely to happen, the passengers should bend as low as possible, and endeavour to throw themselves to the elevated side of the carriage. In two-wheeled carriages, where the horse bears part of the load upon its back, the elevation of the centre of gravity renders the draught more difficult, by throwing a greater proportion of the load upon the horse's back when he is going down hill, and when he has the least occasion for it; and taking the load from the back of the horse when he is going up hill, and requires to be pressed to the ground.

A knowledge of the laws of the centre of gravity enables us to explain the experiment represented in fig. 74, where the vessel of water CG is suspended on a rod AB, passing below its handle, and resting on the end E of the beam DE. The extremity B of the rod AB is supported by another rod BF, which bears against the bottom of the vessel; so that the vessel and the two rods become, as it were, one body, which, by Prop. XVI. Cor. 4, will be in equilibrium when their common centre of gravity C is in the same vertical line with the point of support E.

The cylinder G may be made to ascend the inclined plane ABC by putting a piece of lead or any heavy substance on one side of its axis, so that the centre of gravity is not at G, but at g. Hence it is obvious that the centre of gravity g will descend, and by its descent the body will rise towards A. The inclination of the plane, however, must be such, that before the motion commences, the angles formed by a vertical line drawn from g with a line drawn from G perpendicularly to AB, must be less than the angle of inclination ABC, or, which is the same thing, when the vertical line drawn from g does not cut the line which lies between the point of contact and the centre of the cylinder. When the vertical line, let fall from g, meets the perpendicular line drawn from G to the plane in the point of contact, the cylinder will be in equilibrium on the inclined plane.

Upon the same principle, a double scalene cone may be made to ascend an inclined plane without being loaded with a weight. In fig. 76, let ABC be the section of a double inclined plane, AB, BC being sections of its surfaces perpendicular to the line in which the double scalene cone ADEFC moves. Then, since the centre of gravity of a cone is in the line joining the vertex and the centre of its base, and since the axis of a scalene cone is not perpendicular to its base, the line which joins the centres of both the cones, when in the position represented in the figure, will be above the line which joins the centres of their bases. If the circle, therefore, in fig. 77, represents the base of one of the cones, and C its centre, the line which joins the centres of gravity of the two cones will terminate in some point G at a distance from the centre, and therefore the double cone A will ascend the plane upon the same principles, and under the same conditions, as the loaded cylinder mentioned in the last paragraph.

CHAP. V.—ON THE MOTION OF BODIES ALONG INCLINED PLANES AND CURVES, ON THE CURVE OF SWIFTEST DESCENT, AND ON THE OSCILLATIONS OF PENDULUMS.

PROP. I.

When a body moves along an inclined plane, the force of which accelerates or retards its motion is to the whole bodies on force of gravity as the height of the plane is to its length, inclined planes.

Let ABC be the inclined plane, A the place of the body, and let AB represent the whole force of gravity. The force AB is equivalent to the two forces AD, DB, or AE, AD, of which AD is the force that accelerates the motion of the body down the plane, while AE is destroyed by the resistance or reaction of the plane. The part of the force of gravity, therefore, which makes the body arrive at C, is represented by AD, while the whole force of gravity is represented by AB; but the triangle ABD is equiangular to ABC, and AD : AB = AC : AC, that is, the accelerating force which makes the body descend the inclined plane, is to the whole force of gravity as the height of the plane is to its length, or as the sine of the plane's inclination is to radius; for when AC is radius, AB becomes the sine of the angle ACB.

Cor. I. Since the force of gravity, which is uniform, has a given ratio to the accelerating force, the accelerating force is also uniform; consequently the laws of accelerated and retarded motions, as exhibited in the article DYNAMICS, are also true when the bodies move along inclined planes. If H, therefore, represent the height AB of the plane, L its length AC, g the force of gravity, and A the accelerating force, we shall have, by the proposition, L : H = g : A, hence A = g × H / L; or, since g : A = radius : sin. ACB, A = g × sin. ACB. Now, from the principles of Dynamics, s = 1/2 gt^2, v = gt = √2gs, and t = v/g = √2s/g, where s is the space described, g the force of gravity, or

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1 See Dr T. Young's Natural Philosophy, vol. i. p. 64. On the Motion of Bodies on Inclined Planes.

\[ \phi = \frac{1}{2} gt^2; v' = g \sin \phi t = \sqrt{2gs'} \sin \phi; \text{ and } t = \frac{v'}{g \sin \phi}. \]

Cor. 2. If one body begins to descend through the vertical \( AB \) at the same time that another body descends along the plane \( AC \), when the one is at any point \( m \), the position of the other will be \( n \), which is determined by drawing \( mn \) perpendicular to \( AC \). The forces by which the two bodies are actuated are as \( AB \) to \( AD \), that is, as \( Am \) to \( An \); but forces are measured by the spaces described in the same time; therefore the spaces described in the same time are as \( Am \) to \( An \), that is, as the length of the plane is to its height; for \( Am : An = AC : AB \); consequently, when the body that descends along the vertical line \( AB \) is at \( m \), the other body will be at \( n \).

Through the three points \( A, m, n \) describe a semicircle; then, since \( Amn \) is a right angle, the centre of the semicircle will be in the line \( Am \) (Playfair's Euclid, book iv, prop. 5); consequently, if two bodies descend from the point \( A \) at the same time, the one through the diameter of a circle \( Am \), and the other through any chord \( An \), they will arrive at the points \( m, n \), the extremities of the diameter and of the chord, at the same instant.

It also follows from this corollary, that if from the point \( A \) there be drawn any number of lines making different angles with the diameter \( Am \), and if bodies be let fall from \( A \), so as to move along these lines, they will, at the end of any given time, be found in the circumferences of circles which touch one another in the point \( A \). If the lines are not in the same plane, the bodies will be in the circumferences of spheres which touch each other in the point \( A \).

Cor. 3. If any number of bodies descend from the same point \( A \) along any number of inclined planes \( AC, AF \), their velocities at the points \( C, F \) will be equal. By Cor. 1, the velocity of a body descending the plane \( AC \), is \( v = \sqrt{2gs} \sin \phi \), and the velocity of a body falling in the vertical line \( AB \), is \( v' = \sqrt{2gs} \). But, since \( v = v' \), we have \( \sqrt{2gs} \sin \phi = \sqrt{2gs} \) or \( 2gs \sin \phi = 2gs' \), and dividing by \( 2g \), \( s \sin \phi = 0 \), consequently \( s = s' = \sin \phi : 1 \), or \( AB : AC = \sin DAB : \text{radius} \). Therefore, when \( v = v' \), that is, when the velocities of the two bodies are equal, the spaces described are as \( \sin DAB : \text{radius} \), which can only happen when \( BC \) is perpendicular to \( AB \). In the same way it may be shown that the velocity at \( F \) is equal to the velocity at \( C \), therefore the velocity at \( C \) is equal to the velocity at \( F \).

Cor. 4. The time of descending along \( AC \) is to the time of descending along \( AB \), as \( AC \) is to \( AB \). From the values of \( s, s' \) in Cor. 1, we obtain \( t : t' = \frac{s}{\sin \phi} : s' = \frac{AC}{\sin \phi} : AB \). But \( \frac{AB}{AC} = \sin \phi \), therefore \( t : t' = \frac{AC^2}{AB} : AB \), and taking equal multiples of these two last terms, that is, multiplying them by \( AB \), we have \( t : t' = AC^2 : AB^2 \), or \( t : t' = AC : AB \). Hence the times of descending along \( AF \) and \( AC \) are as \( AF \) and \( AC \).

Cor. 5. The velocities acquired by descending any planes \( AC, AF \) are as the square roots of their altitudes \( AB \). The velocity acquired by falling through \( AB \) is, by the principles of Dynamics, as the square root of \( AB \); and as the velocities at \( F, C \) are equal to that at \( B \), they will also be as the square root of \( AB \).

If a body descend from any point along a number of inclined planes to a horizontal line, its velocity, when it reaches the horizontal line, will be equal to that which it would have acquired by falling in a vertical direction from the given point to the horizontal line.

Let \( AB, BC, CD \), be a number of planes differently inclined to a horizontal line \( DF \), and let the body be let fall from the point \( A \), so as to move along these planes, without losing any of its velocity at the angular points; it will have the same velocity when it reaches the horizontal plane at \( D \), which it would have acquired by falling freely from \( A \) to \( F \). It is manifest from Prop. I, Cor. 3, that the velocity of the body when at \( B \) will be the same as that of another body which had fallen freely from \( A \) to \( c \) in a vertical line. The two bodies set out from \( B \) and \( c \) with the same velocity, and will therefore continue to have the same velocity when they reach the points \( C, G \), because \( cG = BD \). The two bodies again set off from the points \( C, G \) with the same celerity, and since \( GF = Ce \), their respective velocities will be equal when they arrive at the points \( D, F \) in the horizontal plane. The velocity, therefore, acquired by the falling body along the planes \( AB, BC, CD \) is equal to that which is acquired by the same body falling through the vertical line \( AF \).

Cor. 1. As the preceding proposition holds true, whatever be the number of inclined planes which lie between the point \( A \) and the horizontal line, it will hold true also of any curve line which may be considered as made up of an infinite number of straight lines. And since the small planes are diminished without limit, the radius is diminished without limit, and therefore the versed sine, or the velocity lost in passing from one plane to another, is diminished without limit; consequently, abstracting from friction, a body will ascend or descend a curve surface without losing any of its velocity from the curvature of the surface.

Cor. 2. If a body be made to ascend a curve surface, or a system of inclined planes, the vertical height to which it will rise is equal to that through which it must fall in order to acquire the velocity with which it ascended, abstracting from the effects of friction, and the velocity which is lost in passing from one plane to another. This is obvious from Dynamics; for the body experiences the same decrements of velocity in its ascent, as it received increments in its descent.

Cor. 3. The same thing will hold if the body is kept in the curve by a string perpendicular to the curve; for the string sustains that part of the weight which was sustained by the curve, since the re-action of the curve surface is in a line perpendicular to the curve.

Scholium.

It is obvious that the body which moves along the system of inclined planes must lose a part of its velocity in passing from one plane to another. By the resolution of motion it will be found that the velocity acquired by falling through any of the planes, is to the velocity lost in passing to the succeeding one, as radius is to the versed sine of the angle formed by the two planes. Or the velocity with which the body enters upon one plane, is as the cosine of the angle made by the contiguous planes, divided

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1 See Wood's Principles of Mechanics, p. 56, note; and also Gregory's Mechanics, vol. i. p. 112; where this corollary is demonstrated by the method of fluxions.

Prop. III.

The times of descending two systems of inclined planes

By Cor. 4, Prop. I., we have

Time along AB : Time along Ac = AB : Ac, Time along ab : Time along aβ = ab : aβ.

But, on account of the similar triangles ABe, abβ, we have,

AB : Ac = ab : aβ.

Hence (Euclid, book v. prop. 11, 16),

Time along AB : Time alone ab = Time along Ac : Time along aβ.

In the same way it may be shown that

Time along BC : Time along bc = Time along cG : Time along βz, Time along CD : Time along cd = Time along GF : Time along xf.

Then,

Time along AB + BC + CD : Time along ab + bc + cd = Time along Ac + cG + GF : Time along aβ + βz + xf; that is,

Time along AB + BC + CD : Time along ab + bc + cd = Time along AF : Time along af.

But, by Dynamics,

Time along AF : Time along af = √AF : √af,

Therefore, Euclid, book v. prop. 11,

Time along AB + BC + CD : Time along ab + bc + cd = √AF : √af. Q. E. D.

But, by similar triangles, &c.

√AF : √af = √AB + BC + CD : √ab + bc + cd.

Therefore,

Time along AB + BC + CD : Time along ab + bc + cd = √AB + BC + CD : √ab + bc + cd. Q. E. D.

Cor. 1. This proposition holds true of curves, for the reasons mentioned in Prop. 2, Cor. 1.

Cor. 2. The times of descent along similar arcs of a circle are as their radii; for, by the preceding corollary, the times are as the arcs and the arcs are as the radii, therefore the times are as the radii.

Prop. IV.

An inverted semicycloid is the curve of quickest descent, or the curve along which a body must descend in order to move between two points not in a vertical line in the least time possible.

Let qFZ be a semicycloid, and AD', CF' two parallel ad vertical ordinates at an infinitely small distance. Draw the ordinate BE' an arithmetical mean between the ordinates D' and CF', and from F, E' draw F'e, a perpendicular to TF, CE'. Make CP = a, BE' = b, E'e = c, B' = m, BA' = n.

Then, since FE' may be considered as a straight line, and since C = Fe, we have Euclid, book i. prop.

PE = √m² + c², and since F'e = E'u, D' = √n² + c². Now

the velocities at F' and E' vary as √a and √b, and F'E', D' are the elementary spaces described with these velocities; but the times are directly as the square root of the spaces, and inversely as the velocities; therefore the time of describing F'E' is √m² + c² / √a, and the time of describing ED' is √n² + c² / √b, consequently the time of describing F'D' must be m² + c² / a² + n² + c² / b². But the proposition requires that this time should be the least possible, or a minimum; therefore, taking its fluxion and making it equal to 0, we have

\[ \frac{2mm}{\sqrt{a} \times mm + c^2} + \frac{2nn}{\sqrt{b} \times nn + c^2} = 0. \]

But since CA is invariable, m + n is invariable, and therefore its fluxion m + n = 0, or m = -n, and n = -m; therefore, by transposing the second member of the preceding equation, and substituting these values of m and n, it becomes \( \frac{m}{\sqrt{a} \times mm + c^2} = \frac{n}{\sqrt{b} \times nn + c^2} \). Let us now call the variable absciss qC=x, the ordinate CF'=y, and the arc qF'=z, then m and n are fluxions of x, and PE' is the increment of qF' or z when y is equal a, and ED' the increment of qF' or z when y is equal to b, therefore, by substituting these values in the preceding equation, we obtain \( \frac{x'}{\sqrt{yz}} = \frac{x'}{\sqrt{yz}} \), which shows that this quantity is constant, and gives us the following analogy, \( \frac{x'}{\sqrt{yz}} = 1 : \sqrt{y} \). Now in the cycloid \( \sqrt{y} \) is always the chord of the generating circle when the diameter is \( y \), for (by Euclid, book i. prop. 47, book ii. prop. 8, and book iii. prop. 35),

\[ AF = \sqrt{AD} \times AO; \]

and since \( AO = 1 \) and \( AD = y \), we have \( AF = \sqrt{y} \).

But since the arc of the cycloid at \( F \) is perpendicular to the chord \( AF \), the elementary triangle \( FEv \) is similar to \( FDO \) (for \( BE \) is parallel to \( AO \)), and consequently to \( AFO \) (Euclid, book iv. prop. 8), therefore we have \( FE : E'z' = AO : AF \); but \( FE = x', Ev = v, AO = 1, \) and \( AF = \sqrt{y}, \) consequently \( z' : z' = 1 : \sqrt{y}, \) which coincides with the analogy already obtained, and, being the property of the cycloid, shows that the curve of quickest descent is an inverted cycloidal arc.

**Properties of the Cycloid.**

**Definition.** If a circle NOP be so placed as to be in contact with the line AD, and be made to roll along that line from D towards A, till the same point D of the circle touches the other extremity A, the point D will describe a curve DBA, called a cycloid.

The line AD is called the base of the cycloid; the line CB, which bisects AD at right angles, and meets the curve in B, is called the axis, and B the vertex.

The circle NOP is called the generating circle.

1. The base AD is equal to the circumference of the generating circle, and AC is equal to half that circumference.

2. The axis CB is equal to the diameter of the generating circle.

3. If from any point G of the cycloid, there be drawn a straight line GM parallel to AD, and meeting the circle BLC in L, the circular arc BL is equal to the line GL.

4. If the points L, B be joined, and a tangent drawn to the cycloid at the point G, the tangent will be parallel to the chord LB; and the tangent is found by joining G, E, for GE is parallel to LB.

5. The arc BG of the cycloid is double of the chord BL, and the arc BA or BD is equal to twice the axis BC.

6. If the two portions AB, DB of the cycloid in fig. 82 be placed in the inverted position AB, DB (fig. 80), and if a string BP, equal in length to BA, be made to coincide with BA, and then be evolved from it, its extremity P will describe a semicycloid AF, similar and equal to BA.

In the same way the semicycloid DF (fig. 80), produced by the evolution of the string BP from the semicycloid BD, is equal and similar to BD and to AF. Therefore, if BP be a pendulum or weight attached to the extremity of a flexible line BP, which vibrates between the cycloidal cheeks BA, BD, its extremity D will describe a cycloid AFD, equal to that which is composed of the two halves BA, BD.

7. The chord CN is parallel to MP, and MP is perpendicular to the cycloid AFD, at the point P.

8. If \( Pp \) be an infinitely small arc, the perpendicular to the curve drawn from the points \( P, p \) will meet at M, and \( Pp \) may be regarded as a circular arc whose radius is MP. An infinitely small cycloidal arc at F may likewise be considered as a circular arc whose radius is BP.

As these properties of the cycloid are demonstrated in almost every treatise on mechanics, and as their demonstrations more properly belong to geometry than to mechanics, they are purposely omitted to make room for more important matter.

**Definition.** If a body descend from any point of a curve, and ascend in the same curve, till its velocity is destroyed, the body is said to oscillate in that curve; and the time in which this descent and ascent are performed is called the time of an oscillation or vibration.

**Definition.** A cycloidal pendulum is a pendulum which oscillates or vibrates in the arch of a cycloid.

**Definition.** Oscillations which are performed in equal times are said to be isochronous.

**Prop. V.**

The velocity of a cycloidal pendulum BP (see fig. 80) at the point F, varies as the arch which it describes.

The velocity of the pendulum at F is that which it would have acquired by falling through EF (Prop. 2, and Cor 3, Prop. 2), and the velocity of a falling body is as the square root of the space which it describes (Dynamics); therefore the velocity of the pendulum \( P_1 \) when it reaches F, varies as \( \sqrt{EF} \). But (Geometry) \( FE \) varies as \( \frac{FN^2}{FC} \), and since FC is a constant quantity, \( FE \) will vary as \( FN^2 \) varies; or, to adopt the notation used in the article Dynamics, \( FE = \frac{FN^2}{FC} \), or \( \sqrt{FE} = FN \); but the velocity acquired by falling through EF varies as \( \sqrt{FE} \), therefore the velocity of the pendulum at F varies as FN, that is, as FP, for (\$ 5, p. 384), FN is equal to half FP. Q. E. D.

**Prop. VI.**

If the pendulum begins its oscillations from the point P, the velocity of the pendulum at any point R varies as the sine of a circular arc whose radius is FP, and whose versed sine is PR.

Through F, fig. 80, draw \( pFq \) parallel to AD, and with a radius equal to the cycloidal arc FP, describe the semicircle \( pog \). Make \( pr \) equal to the arc PR of the cycloid, and through r draw \( rm \) perpendicular to \( pF \). Through the points P, R draw PE, RT parallel to AD, and cutting the generating circle CNF in the points N, S. By Prop. IV, the velocity at R varies as \( \sqrt{EF} \), that is, as \( \sqrt{EF} - TF \); or, since CF is constant, as \( \sqrt{CF \times EF} - CF \times TF \), that is, as \( \sqrt{FN^2 - FS^2} \). For (Playfair's Euclid, book i. prop. 47, book ii. prop. 7, and book iii. prop. 35) \( FN^2 = CF \times EF, \) and \( FS = CF \times TF, \) that is, as \( \sqrt{4FN^2 - 4FS^2} \), that is (\$ 5, p. 384), as \( \sqrt{FP^2 - FR^2} \). But \( FP \) or \( FM \) was made equal to FP, and \( pr \) being made equal to PR, the remainder \( Fr \) must be equal to FR; therefore the velocity at R varies as \( \sqrt{FM^2 - FR^2} \); but (Euclid, 47, 1) \( rm = \sqrt{FM^2 - FR^2} \), and \( rm \) is by construction equal to the sine of a circular arc whose radius is FP, and versed sine PR; consequently, the velocity at R varies as the sine of that arc. Q. E. D.

**Corollary.** The velocity of the pendulum at F is to the velocity of the pendulum at R, as \( FM : rm \); for the versed sine is in this case equal to radius, and therefore the corresponding arc must be a quadrant whose sine is also equal to radius or \( FM \).

Prop. VII. The time in which the pendulum performs one complete oscillation from P to O, is equal to the time in which a body would describe the semicircle poq, uniformly with the velocity which the pendulum acquires at the point F.

Take any infinitely small arc RV, fig. 80, and making rv equal to it, draw eo parallel to rm, and mn to rv. Now, by the last proposition, and by Dynamics, the velocity with which RV is described is to the velocity with which mo is described as rm is to Fm, that is, as \( \frac{RV}{rm} : \frac{mo}{Fm} \), or as \( \frac{mn}{rm} : \frac{mo}{Fm} \), for \( mn = rv = RV \). But in the similar triangles Fmr, mno, Fm : rm = mo : mn, consequently \( \frac{mn}{rm} = \frac{mo}{Fm} \), therefore the velocity with which RV is described is equal to the velocity with which mo is described, and the times in which these equal spaces are described must likewise be equal. The same thing may be demonstrated of all the other corresponding arcs of the cycloid and circle, and therefore it follows that the time in which the pendulum performs one complete oscillation is equal to the time in which the semicircle poq is uniformly described with the velocity acquired at F.

Prop. VIII. The time in which a cycloidal pendulum performs a complete oscillation is to the time in which a body would fall freely through the axis of the cycloid, as the circumference of a circle is to its diameter.

Since FP = 2FN (see fig. 80), and since the velocity acquired by falling down NF is equal to the velocity acquired by falling down PF, the body, if it continued to move uniformly with this velocity, would describe a space equal to 2PF in the same time that it would descend NF or CF (Chap. V. Prop. I. Cor. 2). Calling T, therefore, the time of an oscillation, and t the time of descent along the axis, we have, by the preceding proposition,

\[ T = \text{time along poq, with the velocity at F}, \]

and by the preceding paragraph,

\[ t = \text{time along FP, with the same velocity}; \text{ therefore } T : t = \text{time along poq with velocity at V : time along FP with the same velocity}; \text{ that is, } T : t = \text{poq : FP} = 2\text{poq : 2FP} = \text{circumference of a circle : its diameter}. \]

Cor. 1. The oscillations in a cycloid are isochronous, that is, they are performed in equal times, whatever be the size of the arc which the pendulum describes. For the time of an oscillation has a constant ratio to the time of descent along the axis, and is therefore an invariable quantity.

Cor. 2. The oscillations in a small circular arc whose radius is BF, and in an equal arc of the cycloid, being isochronous (\$ 8, p. 384), the time of an oscillation in a small circular arc will also be to the time of descent along the axis, as the circumference of a circle is to its diameter.

Cor. 3. Since the length BF of the pendulum is double of the axis CF, the time of an oscillation in a cycloid or small circular arc varies as the time of descending along CF, half the length of the pendulum, the force of gravity being constant. But the time of descent along CF varies as \( \sqrt{CF} \), therefore the time of an oscillation in a small circular or cycloidal arc varies as the square root of half the length of the pendulum, or as the square root of its whole length. If T, t, therefore, be the times of oscillations of two pendulums, and L, l their respective lengths, we have, by this corollary, \( T : t = \sqrt{L} : \sqrt{l} \), and \( T \times \sqrt{l} = t \times \sqrt{L} \); hence \( T = \frac{\sqrt{L}}{\sqrt{t}} \times \frac{T \times \sqrt{l}}{\sqrt{L}} \), \( l = \frac{\sqrt{L}}{\sqrt{T}} \times \frac{\sqrt{L}}{\sqrt{t}} \), and \( L = \frac{\sqrt{T}}{\sqrt{t}} \times \frac{\sqrt{L}}{\sqrt{t}} \), from which we may find the time in which a pendulum of any length will vibrate; a pendulum of 39-2 inches vibrating in one second.

Cor. 4. When the force of gravity varies, which it does in going from the poles to the equator, the time of an oscillation is directly as the square root of the length of the pendulum, and inversely as the square root of the force of gravity. The time of an oscillation varies as the time of descent along half the length of the pendulum, and the time of descent through any space varies as \( \frac{s}{g} \), where s is the space described, and g the force of gravity; but in the present case \( s = \frac{L}{2} \); therefore, by substitution, the time of descent along half the length of the pendulum, or the time of an oscillation, varies as \( \frac{\sqrt{L}}{\sqrt{g}} \), or as \( \frac{\sqrt{L}}{\sqrt{g}} \). Hence

\[ T : t = \frac{\sqrt{L}}{\sqrt{g}} : \frac{\sqrt{l}}{\sqrt{g}}, \]

from which it is easy to deduce equations similar to those given in the preceding corollary.

Cor. 5. Since \( T = \frac{\sqrt{L}}{\sqrt{g}} \times \frac{\sqrt{g}}{\sqrt{L}} \times T = \sqrt{L} \); and if the time of oscillation is 1 second, we have \( \sqrt{g} = \sqrt{L} \), or \( g = L \), that is, the force of gravity in different latitudes varies as the length of a pendulum that vibrates seconds.

Cor. 6. The number of oscillations which a pendulum makes in a given time, and in a given latitude, are in the inverse subduplicate ratio of its length. The number of oscillations n made in a given time are evidently in the inverse ratio of \( \frac{1}{t} \), the time of each oscillation, that is, \( n = \frac{1}{t} \); but, by corollary 3, \( t = \sqrt{l} \), therefore \( n = \frac{1}{\sqrt{l}} \), and \( l = \frac{1}{n^2} \), from which it is easy to find the length of a pendulum which will vibrate any number of times in a given time, or the number of vibrations which a pendulum of a given length will perform in a given time.

Prop. IX. To find the space through which a heavy body will fall in one second by the force of gravity.

Since, by Proposition VIII., the time of an oscillation is to the time along half the length of the pendulum as 3-14159 is to 1, and since the spaces are as the squares of the times, the spaces described by a heavy body in the time of an oscillation will be to half the length of the pendulum as 3-14159 is to 1. Now it appears from the experiments of Mr Whitehurst, that the length of a pendulum which vibrates seconds at 113 feet above the level of the sea, in a temperature of 60° of Fahrenheit, and when the barometer is 30 inches, is 39-1196 inches; hence \( 1 : (3-14159)^2 = \frac{39-1196}{2} : 19-5598 \times (3-14159)^2 = 16-087 \) feet, the space required.

The methods of determining the centre of oscillation, gyration, and percussion, properly belong to this chapter.

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1 The length in vacuo at the level of the sea at London, in latitude 51° 31' 8" 40', according to Captain Kater, is 39-13929. but they have already been given in the article Rotation, to which we must refer the reader who wishes to prosecute the subject.

**CHAP. VI.—ON THE COLLISION OR IMPACT OF BODIES.**

**Def. 1.** When a body moving with a certain velocity strikes another body, either at rest or in motion, the one is said to impinge against, or to impel, the other. This effect has been distinguished by the names collision or impulse, percussion, and impact.

**Def. 2.** The collision or impact of two bodies is said to be direct when the bodies move in the same straight line, or when the point in which they strike each other is in the straight line which joins their centres of gravity. When this is not the case, the impulse is said to be oblique.

**Def. 3.** A hard body is one which is not susceptible of compression by any finite force. An elastic body is one susceptible of compression, which recovers its figure with a force equal to that which compresses it. A soft body is one which does not recover its form after compression. There does not exist in nature any body which is either perfectly hard, perfectly elastic, or perfectly soft. Every body with which we are acquainted possesses elasticity in some degree or other. Diamond, crystal, agate, &c., though among the hardest bodies, are highly elastic; and even clay itself will in some degree recover its figure after compression. It is necessary, however, to consider bodies as hard, soft, or elastic, in order to obtain the limits between which the required results must be contained.

**Def. 4.** The mass of a body is the sum of the material particles of which it is composed; and the momentum, or moving force, or quantity of motion, of any body, is the product arising from multiplying its mass by its velocity.

**Prop. I.**

Two hard bodies \( B, B' \) with velocities \( V, V' \) striking each other perpendicularly, will be at rest after impact, if their velocities are inversely as their masses.

1. When the two bodies are equal, their velocities must be equal in the case of an equilibrium after impulse, and therefore \( B : B' = V : V' \), or \( BV = B'V' \); for if they are not at rest after impact, the one must carry the other along with it. But as their masses and velocities are equal, there can be no reason why the one should carry the other along with it.

2. If the one body is double of the other, or \( B = 2B' \), we should have \( V = 2V' \). Now, instead of \( B \) we may substitute two bodies equal to \( B' \); and instead of \( V \) we may substitute two velocities equal to \( V' \), with which the bodies \( B' \) may be conceived to move; consequently we have \( 2B' \times V = B' \times 2V' \), or \( B' : 2B' = V : 2V' \); but \( 2V \) is the velocity of \( B' \), and \( V \) is the velocity of \( 2B' \); therefore, when one body is double of the other, they will remain at rest when the masses of the bodies are inversely as their velocities.

In the same way the proposition may be demonstrated when the bodies are to one another in any commensurable proportion.

**Prop. II.**

To find the common velocity \( v \) of two hard bodies \( B, B' \) whose velocities are \( V, V' \); after striking each other perpendicularly.

If the bodies have not equal quantities of motion they cannot be in equilibrium after impulse. The one will carry the other along with it, and, in consequence of their hardness, they will remain in contact, and move with a common velocity \( v \).

1. In order to find this, let us first suppose \( B' \) to be at rest and to be struck by \( B \) in motion. The quantity of motion which exists in \( B \) before impulse is \( BV \), and as this is divided between the two bodies after impulse, it must be equal to the quantity of motion after impulse. But \( v \times B + B'v = BV \), and \( v = \frac{BV}{B + B'} \).

2. Let us now suppose that both the bodies are in motion in the same direction that \( B \) follows \( B' \). In order that \( B \) may impel \( B' \), we must have \( V \) greater than \( V' \). Now, we may conceive both the bodies placed upon a plane moving with the velocity \( V' \). The body \( B' \), therefore, whose velocity is \( V' \) equal to that of the plane, will be at rest upon the plane, while the velocity of \( B \) with regard to \( B' \) or the plane, will be \( V - V' \); consequently, the bodies are in the same circumstances as if \( B' \) were at rest, and \( B \) moving with the velocity \( V - V' \). Therefore, by the last case, we have the common velocity of the bodies in the moveable plane \( \frac{BV - B'V'}{B + B'} \); and by adding to this \( V \), the velocity of the plane, we shall have \( v \), or the absolute velocity of the bodies after impulse, \( v = \frac{BV + B'V'}{B + B'} \). Hence the quantity of motion after impact is equal to the sum of the quantities of motion before impact.

3. If the impinging bodies mutually approach each other, we may conceive, as before, that the body \( B' \) is at rest upon a plane which moves with a velocity \( V' \) in an opposite direction to \( V \), and that \( B \) moves on this plane with the velocity \( V + V' \). Then, by Case 1, \( \frac{BV + B'V'}{B + B'} \) will be the common velocity upon the plane after impulse; and adding to this \( V' \), or the velocity of the plane, we shall have \( v \), or the absolute velocity of the bodies after impact, \( v = \frac{BV + B'V'}{B + B'} \). Hence the quantity of motion after impact is equal to the difference of the quantities of motion before impact. It is obvious that \( v \) is positive or negative, according as \( BV \) is greater or less than \( B'V' \); so that when \( BV \) is greater than \( B'V' \), the bodies will move in the direction of \( B' \)'s motion; and when \( BV \) is less than \( B'V' \), the bodies will move in the direction of \( A \)'s motion.

All the three formulae which we have given may be comprehended in the following general formula, \( v = \frac{BV \pm B'V'}{B + B'} \); for when \( B' \) is at rest, \( V' = 0 \), and the formula assumes the form which it has in Case 1.

**Cor. 1.** If \( B = B' \), and the bodies mutually approach each other, the equation in Case 3 becomes \( v = \frac{V - V'}{2} \), or the bodies will move in the direction of the quickest body, with a velocity equal to one half of the difference of their velocities.

**Cor. 2.** If \( V = V' \), and the bodies move in the same direction, the last formula will become \( v = V \times \frac{B - B'}{B + B'} \) or \( v = V \); for in this case there can be no impact, the one body merely following the other in contact with it. When the bodies mutually approach each other, and when \( V = V' \), we have \( v = V \times \frac{B - B'}{B + B'} \).

**Cor. 3.** When the bodies move in the same direction, we have, by Case 2, \( v = \frac{BV + B'V'}{B + B'} \). Now the velocity gained by \( B' \) is evidently \( v - V' \) or \( \frac{BV + B'V'}{B + B'} - V = \frac{BV - B'V'}{B + B'} \); hence \( B + B' : B = V - V' : \frac{BV - BV'}{B + B'} \); but this last term is the velocity gained by \( B \), and \( V - V' \) is the relative velocity of the two bodies. Therefore, in the impact of two hard bodies moving in the same direction, \( B + B' : B \) as the relative velocity of the two bodies is to the velocity gained by \( B' \). It is obvious also that the velocity lost by \( B \) is \( V - v = V - \frac{BV + B'V'}{B + B'} \) or \( \frac{BV - B'V'}{B + B'} \); hence \( B + B' : B' = V - V' : \frac{BV - B'V'}{B + B'} \); but this last term is the velocity lost by \( B \), and \( V - V' \) is the relative velocity of the bodies, therefore, in the impact of two hard bodies, \( B + B' : B' \) as their relative velocity is to the velocity lost by \( B \). The same thing may be shown when the bodies move in opposite directions, in which case their relative velocity is \( V + V' \).

**Prop. III.**

To determine the velocities of two elastic bodies after impact.

If an elastic body strikes a hard and immovable plane, it will, at the instant of collision, be compressed at the place of contact. But as the elastic body instantaneously endeavours to recover its figure, and as this force of restitution is equal and opposite to the force of compression, it will move backwards from the plane in the same direction in which it advanced. If two elastic bodies, with equal momenta, impinge against each other, the effect of their mutual compression is to destroy their relative velocity, and make them move with a common velocity, as in the case of hard bodies. But by the force of restitution, equal to that of compression, the bodies begin to recover their figure; the parts in contact serve mutually as points of support, and the bodies recede from each other. Now, before the force of restitution began to exert itself, the bodies had a tendency to move in one direction with a common momentum; therefore the body, whose effort to recover its figure was in the same direction with that of the common momentum, will move on in that direction, with a momentum or moving force equal to the sum of the force of restitution and the common momentum; while the other body, whose effort to recover from compression in a direction opposite to that of the common momentum, will move with a momentum equal to the difference between its force of restitution and the common momentum, and in the direction of the greatest of these momenta. After impact, therefore, it either moves in the direction opposite to that of the common momentum, or its motion the same direction as that of the common momentum diminished, or it is stopped altogether, according as the force of restitution is greater, less, or equal to the common momentum.

In order to apply these preliminary observations, let us opt the notation in the two preceding propositions, and \( v \) be the common velocity which the bodies would have received after impact if they had been hard, and \( v' \) the velocities which the elastic bodies \( B, B' \) receive after impact.

1. If \( B \) follows \( B' \), then \( V \) is greater than \( V' \), and when has reached \( B' \), they are both compressed at the point of impact. Hence, since \( v \) is the common velocity with which they would advance if the force of restitution were not exerted, we have \( V - v = \) the velocity lost by \( B \), and \( V' - v' = \) the velocity gained by \( B' \), in consequence of compression. But when the bodies strive to recover their form by the force of restitution, the body \( B \) will move backwards in consequence of this force, while \( B' \) will move forward in its former direction with an accelerated velocity.

Hence, from the force of restitution, \( B \) will again lose the velocity \( V - v \), and \( B' \) will, a second time, gain the velocity \( v' - V' \); consequently the whole velocity lost by \( B' \) is \( 2V - 2v \), and the whole velocity gained by \( B' \) is \( 2v - 2V' \). Now, subtracting this loss from the original velocity of \( B \), we have \( V - 2V - 2v \) for the velocity of \( B \) after impact, and adding the velocity gained by \( B' \) to its original velocity, we have \( V' + 2v - 2V' \) for the velocity of \( B' \) after impact; hence we have

\[ v' = V - 2V - 2v = 2v - V, \]

\[ v'' = V' + 2v - 2V' = 2v - V. \]

Now, substituting in these equations the value of \( v \) as found in Case 2, Prop. II., we obtain

\[ v' = \frac{BV - B'V + 2BV'}{B + B'}, \]

\[ v'' = \frac{BV' - B'V + 2BV}{B + B'}. \]

2. When the bodies move in opposite directions, or mutually approach each other, the body \( B \) is precisely the same circumstances as in the preceding case; but the body \( B' \) loses a part of its velocity equal to \( 2v + 2V' - V' \). Hence we have, by the same reasoning that was employed in the preceding case,

\[ v' = 2v - V', \]

\[ v'' = 2v + V'; \]

and by substituting instead of \( v \) its value, as determined in Case 3, Prop. II., or by merely changing the sign of \( V' \) in the two last equations in the preceding corollary, we obtain the two following equations, which will answer for both cases, by using the upper sign when the bodies move in the same direction, and the under sign when they move in opposite directions:

\[ v' = \frac{BV - B'V + 2BV'}{B + B'}, \]

\[ v'' = \frac{BV' - B'V + 2BV}{B + B'}. \]

From the preceding equation the following corollaries may be deduced.

Cor. 1. The velocity gained by the body that is struck, and the velocity lost by the impinging body, are twice as great in elastic as they are in hard bodies; for in hard bodies the velocities gained and lost were \( v - V \), and \( V - v \); whereas in elastic bodies the velocities gained and lost were \( 2v - 2V' \), and \( 2V - 2v \).

Cor. 2. If one of the bodies, suppose \( B' \), is at rest, its velocity \( V' = 0 \), and the preceding equation becomes

\[ v' = \frac{VB - VB'}{B + B'}, \quad v'' = \frac{2VB}{B + B'}. \]

Cor. 3. If one of the bodies \( B' \) is at rest, and their masses equal, we have \( B = B' \) and \( V' = 0 \), by substituting which in the preceding formulae, we obtain \( v' = 0 \), and \( v'' = V \); that is, the impinging body \( B \) remains at rest after impact, and the body \( B' \) that is struck when at rest moves on with the velocity of the body \( B \) that struck it, so that there is a complete transfer of \( B \)'s velocity to \( B' \).

Cor. 4. If \( B' \) is at rest and \( B \) greater than \( B' \), both the bodies will move forward in the direction of \( B \)'s motion; for it is obvious from the equations in Cor. 2, that when \( B \) is greater than \( B' \), \( v' \) and \( v'' \) are both positive.

Cor. 5. If \( B' \) is at rest, and \( B \) less than \( B' \), the impinging body \( B \) will return backwards, and the body \( B' \), which is struck, will move forward in the direction in which \( B \) moved before the stroke. For it is evident that when \( B \) is less than \( B' \), \( v' \) is negative and \( v'' \) positive.

Cor. 6. If both the bodies move in the same direction, the body \( B' \), that is struck, will after impact move with greater On the velocity than it had before it. This is obvious from the collision of formula in Case 1 of this proposition.

Cor. 7. If the bodies move in the same direction, and if \( B = B' \), there will at the moment of impact be a mutual transfer of velocities, that is, \( B \) will move on with \( B' \)'s velocity, and \( B' \) will move on with \( B \)'s velocity. For in the formulae in Case 1, when \( B = B' \), we have \( v' = V' \) and \( v'' = V \).

Cor. 8. When the bodies move in opposite directions, or mutually approach each other, and when \( B = B' \) and \( V = V' \), both the bodies will recoil or move backwards after impact with the same velocities which they had before impact. For in the formulae in Case 2, with the inferior signs, when \( B = B' \) and \( V = V' \), we have \( v' = -V \) and \( v'' = V' \).

Cor. 9. If the bodies move in opposite directions, and \( V = V' \), we have \( v' = V \times \frac{B-3B'}{B+B'} \), and \( v'' = V \times \frac{3B-B'}{B+B'} \).

Hence it is obvious, that if \( B = 3B' \), or if one of the impinging bodies is thrice as great as the other, the greatest will be stopped, and the smallest will recoil with a velocity double of that which it had before impact. For since \( B = 3B' \), by substituting this value of \( B \) in the preceding equations, we obtain \( v' = 0 \), and \( v'' = 2V \).

Cor. 10. If the impinging bodies move in opposite directions, and if \( B = B' \), they will both recoil after a mutual exchange of velocities. For when \( B = B' \), we have \( v' = -V \), and \( v'' = V \).

Cor. 11. When the bodies move in opposite directions, the body which is struck, and the body which strikes it, will stop, continue their motion, or return backwards, according as \( BV - BV' \) is equal to or greater or less than \( 2BV' \).

Cor. 12. The relative velocity of the bodies after impact is equal to their relative velocity before impact, or, which is the same thing, at equal instants before and after impact, the distance of the bodies from each other is the same. For in the different cases we have \( v' = 2v - V \); \( v'' = 2v' - V' \). But the relative velocity before impact is in different cases \( V = V' \), and the relative velocity after impact is \( v' = v'' = V - V' \).

Cor. 13. By reasoning similar to that which was employed in Prop. 2, Cor. 3, it may be shown that \( B + B' : 2B \) as their relative velocity before impact is to the velocity gained by \( B' \) in the direction of \( B \)'s motion; and \( B + B' : 2B' \) as their relative velocity before impact is to the velocity lost by \( B \) in the direction of \( A \)'s motion.

Cor. 14. The ris vis, or the sum of the products of each body multiplied by the square of its velocity, is the same before and after impact, that is, \( Bv^2 + Bv'^2 = BV^2 + BV'^2 \). From the formulae at the end of Case 2, we obtain

\[ Bv^2 = \frac{B-B'^2}{B+B'^2} \times \frac{BV^2 + BV'^2}{B+B'^2} \]

\[ Bv'^2 = \frac{4BB' \times BV^2 + BV'^2}{B+B'^2}, \text{ hence their sum } Bv^2 + Bv'^2 = \frac{B-B'^2}{B+B'^2} \times \frac{BV^2 + BV'^2}{B+B'^2} + \frac{4BB' \times BV^2 + BV'^2}{B+B'^2} = \frac{BV^2 + BV'^2}{B+B'^2}. \]

Cor. 14. If several equal elastic bodies \( B, B', B'', B''' \), &c., are in contact, and placed in the same straight line, and if another elastic body \( B \) of the same magnitude impinges against \( B \), they will remain at rest, except the last body \( B''' \), which will move on with the velocity of \( B \). By Cor. 3, Prop. III., \( B \) will transfer to \( B' \) all its velocity, and therefore \( B \) will be at rest; in the same way \( B' \) will transfer to \( B'' \) all its velocity, and \( B'' \) will remain at rest; and so on with the rest; but when the last body \( B''' \) is set in motion, there is no other body to which its velocity can be transferred, and therefore it will move on with the velocity which it received from \( B'' \), that is, with the velocity of \( B \).

Cor. 15. If the bodies decrease in size from \( B \) to \( B'' \), they will all move in the direction of the impinging body \( B \), and the velocity communicated to each body will be greater than that which is communicated to the preceding body.

Cor. 16. If the bodies increase in magnitude, they will all recoil, or move in a direction opposite to that of \( B \), excepting the last, and the velocity communicated to each body will be less than that which is communicated to the preceding body.

Prop. IV.

To determine the velocities of two imperfectly elastic bodies after impact, the force of compression being in a given ratio to the force of restitution or elasticity.

Let \( B, B' \) be the two bodies, \( V, V' \) their velocities before impact, \( v', v'' \) their velocities after impact, and \( 1:n \) as the force of compression is to that of restitution. It is evident from Case I.; Prop. VIII. that in consequence of the force of compression alone we have

\[ V-v = \text{velocity lost by } B \quad \text{from compression.} \]

\[ v-V' = \text{velocity gained by } B' \quad \text{from compression.} \]

But the velocity which \( B \) loses and \( B' \) gains by the force of compression will be to the velocity which \( B \) loses and \( B' \) gains by the force of restitution or elasticity as \( 1:n \); hence

\[ 1:n = V-v : nv - nv, \text{ the velocity lost by } B \quad \text{from elasticity; } \]

\[ 1:n = v-V' : nv - nv, \text{ the velocity gained by } B' \quad \text{from elasticity; } \]

therefore, by adding together the two portions of velocity lost by \( B \), and also those gained by \( B' \), we obtain

\[ 1 + nv - 1 + nv, \text{ the whole velocity lost by } B, \]

\[ 1 + nv - 1 + nv, \text{ the whole velocity gained by } B. \]

Hence, by subtracting the velocity lost by \( B \) in consequence of collision from its velocity before impact, we shall have \( v' \) or the velocity of \( B \) after impact; and by adding the velocity gained by \( B' \) after collision to its velocity before impact, we shall find \( v'' \) or the velocity of \( B' \) after impact; thus

\[ v' = V - 1 + nv - 1 + nv, \text{ the velocity of } B \text{ after impact.} \]

\[ v'' = V' + 1 + nv - 1 + nv, \text{ the velocity of } B' \text{ after impact.} \]

Now, by substituting in the place of \( v \) its value as determined in Case 2, Prop. II., we obtain

\[ v' = V - 1 + n \times \frac{BV - BV'}{B+B'}, \quad v'' = V' + 1 + n \times \frac{BV - BV'}{B+B'}. \]

Cor. 1. Hence, by converting the preceding equation into analogies, \( B + B : 1 + n \times B \) as the relative velocity of the bodies before impact is to the velocity gained by \( B' \) in the direction of \( B \)'s motion; and \( B + B' : 1 + n \times B' \) as the relative velocity of the bodies before impact is to the velocity lost by \( B \).

Cor. 2. The relative velocity before impact is to the relative velocity after impact as the force of compression is to the force of restitution, or as \( 1:n \).

The relative velocity after impact is \( v'' - v' \), or, taking the preceding values of these quantities, \( v'' - v' = V - \frac{1 + n \times BV - BV'}{B+B'} - V + \frac{1 + n \times B+B' \times V - V'}{B+B'} \); dividing by \( B+B' \), we have \( v' = V - V + V - V' + n \times V - V' = n \times V - V' \) = the relative velocity after impact. But the relative velocity before impact is \( V - V' \), and \( V - V' : n \times V - V' = 1 : n \). Q.E.D. The quantity \( V' \) has evidently the negative sign when the bodies move in opposite directions.

Cor. 3. Hence, from the velocities before and after impact, we may determine the force of restitution or elasticity.

**Prop. V.**

To find the velocity of a body, and the direction in which it moves after impinging upon a hard and immoveable plane.

**Case 1. When the impinging body is perfectly hard.** Let \( AB \) be the hard and immoveable plane, and let the impinging body move towards \( AB \) in the direction \( CD \), and with a velocity represented by \( CD \). Then the velocity \( CD \) may be resolved into the two velocities \( CM, MD, \) or \( MD, FD; CM, DF \) being a parallelogram. But the part of the velocity \( FD \), which carries the body in a line perpendicular to the plane, is completely destroyed by impact, while the other part of the velocity \( MD \), which carries the body in a line parallel to the plane, will not be affected by the collision, therefore the body will, after impact, move along the plane with the velocity \( MD \). Now, \( CD : MD = \text{radius} : \cos \angle CDM \); therefore, since \( MD = CF \), the sine of the angle of incidence \( CDF \), the velocity before impact is to the velocity after impact, as radius is to the sine of the angle of incidence; and since \( AM = CD - MD \), the velocity before impact is to the velocity lost by impact, as radius is to the versed sine of the complement of the angle of incidence.

**Case 2. When the impinging body is perfectly elastic.** Let the body move in the direction \( CD \) with a velocity represented by \( CD \), which, as formerly, may be resolved to \( MD, FD \). The part of the velocity \( MD \) remains after impact, and tends to carry the body parallel to the plane. The other part of the velocity \( FD \) is destroyed by compression; but the force of restitution or elasticity will generate a velocity equal to \( FD \), but in the opposite direction \( DF \). Consequently the impinging body after impact is solicited by two velocities, one of which would carry it uniformly from \( D \) to \( F \) in the same time that the other would carry it uniformly from \( M \) to \( D \), or from \( D \) to \( N \); the body will, therefore, move along \( DE \), the diagonal of the parallelogram \( DFEN \), which is equal to the parallelogram \( DFCM \). Hence the angle \( GDF \) is equal to the angle \( EDF \), therefore, when an elastic body impinges obliquely against an immoveable plane, it will be reflected from the plane, so that the angle of reflection is equal to the angle of incidence. Since \( CD, DE \) are equal spaces described in equal times, the velocity of the body after impact will be equal to its velocity before impact.

**Case 3. When the impinging body is imperfectly elastic.** In \( DF \) take a point \( m \), so that \( DF \) is to \( DM \) as the force of compression is to the force of restitution or elasticity, and having drawn \( me \) parallel to \( DB \), and meeting \( NE \) in \( De \); then, if the impinging body approach the plane in the direction \( CD \), with a velocity represented by \( CD \), \( De \) will be the direction in which it will move after impact. Immediately after compression, the velocity \( MD \) is destroyed as in the last case, while the velocity \( MD \) tends to carry the body parallel to the plane. But, by the force of restitution, the body would be carried uniformly along \( DM \), perpendicular to the plane, while, by the velocity \( MD = DN = me \), it would be carried in the same time along \( me \); consequently, by means of these two velocities, the body will describe \( De \), the diagonal of the parallelogram \( DmEN \). The velocity, therefore, before impact is to the velocity after impact as \( DC : De \), or as \( DE : De \), or as \( sin. DEe, sin. DEe \), or as \( sin. Dem : sin. DEe \), or as \( sin. FDe : sin. FDE \). Now, by producing \( De \) so as to meet the line \( CE \) produced in \( G \), we have, on account of the parallels \( FE, me, DM : DF = me : FG \); but \( FD \) being radius, \( FE \) is the tangent of \( FDE \), or \( FDG \) the angle of incidence, and \( FDG \) is the tangent of the angle of reflection \( FDG \); therefore \( DM : DF = tan. \angle CDF : tan. \angle FDG \). Consequently, when an imperfectly elastic body impinges against a plane, it will be reflected in such a manner that the tangent of the angle of reflection is to the tangent of the angle of incidence, as the force of compression is to the force of restitution or elasticity; and the velocity before incidence will be to the velocity after reflection, as the sine of the angle of reflection is to the sine of the angle of incidence.

**Scholium.**

When the surface against which the body impinges is curved, we must conceive a plane touching the surface at the place of incidence, and then apply the rules in the preceding proposition. The doctrine of the oblique collision of bodies is of great use both in acoustics and optics, where the material particles which suffer reflection are regarded as perfectly elastic bodies.

**Prop. VI.**

To find the point of an immoveable plane which an elastic body moving from a given place must strike, in order that it may, after reflection, either from one or two planes, impinge against another body whose position is given.

**Case 1. When there is only one reflection.** Let \( C \) be the place from which the impinging body is to move, and let \( E \) be the body which is to be struck after reflection from the plane \( AB \). From \( C \) let fall \( CH \) perpendicular to \( AB \); continue it towards \( C \) till \( HG = CH \), and join \( GE \) by the line \( GDE \); the point \( D \) where this line cuts the plane is the place against which the body at \( C \) must impinge, in order that, after reflection, it may strike the body at \( E \). The triangles \( CDH, HDG \) are equiangular, because two sides and one angle of each are respectively equal, therefore the angles \( DCH, DHG \) are equal. But on account of the parallels \( FD, CG \), the angle \( EDF = DGC = DCH \), and \( DCH = FDC \), therefore the angle of incidence \( FDC = FDE \) the angle of reflection; consequently, by Prop. IV, a body moving from \( C \) and impinging on the plane at \( D \) will, after reflection, move in the line \( DE \), and strike the body at \( E \).

**Case 2. When there are two reflections.** Let \( AB, BL \) be the two immoveable planes, \( C \) the place from which the impinging body is to move, and \( F \) the body which it is to strike after reflection from the two planes; it is required to find the point of impact \( D \). Draw \( CHG \) perpendicular to \( AB \), so that \( HG = CH \). Through \( G \) draw \( GMN \) parallel to \( AB \), cutting \( LB \) produced in \( M \), and make \( GM = MN \). Join \( N, F \), and from the point \( E \), where \( NF \) cuts the plane \( BL \), draw \( EG \), joining the points \( EG \); the point will be the point of the plane against which the body at \( C \) must impinge, in order to strike the body... On the Collision of Bodies.

By reasoning as in the preceding case, it may be shown that the angle CDA = EDB; therefore DE will be the path of the body after the first reflection. Now, the triangles GEM, EMN are equiangular, because GM = MN, and the angles at M right, therefore DEB = FEL, that is, the body after reflection at E will strike the body placed at F.

Prop. VII.

To determine the motions of two spherical bodies which impinge obliquely upon each other, when their motion, quantities of matter, and radii, are given.

Let A, B be the two bodies, and let CA, DB be the directions in which they move before impact, and let these lines represent their respective velocities. Join A, B the centres of the bodies, and produce it both ways to K and L. Draw LM perpendicular to IK, and it will touch the bodies at the point of impact. Now, the velocity CA may be resolved into the two velocities CI, IA, and the velocity DB into the velocities DK, KB; but CA and DB are given, and also the angles CAI, DBK, consequently CI and IA, and DK and KB, may be found. The velocities CI, DK, which are parallel to the plane, will not be altered by collision, therefore IA, KB are the velocities with which the bodies directly impinge upon each other, consequently their effects, or the velocities after impact, may be found from Prop. III. Let these velocities be represented by AN, BP. Take AF = CI and BH = DK, and having completed the parallelograms AFON, BPQH, draw the diagonals AO, BQ. Then, since the body A is carried parallel to the line LM with a velocity CI = AF, and from the line LM, by the velocity AN, it will describe AO, the diagonal of the parallelogram NF; and for the same reason the body B will describe the diagonal BQ of the parallelogram PH.

Corollary. If A = B, and if the body which is struck moves in a given direction and with a given velocity after impact, the direction of the impinging body, and the velocity of its motion, may be easily found. Let the body D impinge against the equal body C; and let CB be the direction in which C moves after impact; it is required to find the direction in which D will move. Draw DC, touching the ball C at c, the place where the ball D impinges; produce BC to E, and through c draw AcF perpendicular to EB, and complete the rectangle FE. The force De may be resolved into the forces Ec, eF, of which Ec is employed to move the ball C in the direction CB and with the velocity Ec; but the force eF has no share in the impulse, and is wholly employed in making the body D move in the direction CA, and with the velocity CF.

Scholium.

In the preceding proposition, we have endeavoured to give a short and perspicuous view of the common theory of impulse. The limits of this article will not permit us to enter upon those interesting speculations to which this subject has given rise. It may be proper however to remark that all the phenomena of impulse, as well as pressure, are owing to the existence of forces which prevent the particles of matter from coming into mathematical contact. The body which is struck, in the case of collision, is put in motion by the mutual repulsion of the material particles at the point of impact, while the velocity of the impinging body is diminished by the same cause. Hence we see the absurdity of referring all motion to impulse, or of attempting to account for the phenomena of gravitation, electricity, and magnetism, by the intervention of any invisible fluid. Even if the supposition that such a medium exists were not gratuitous, it would be impossible to show that its particles, by means of which the impulse is conveyed, are in contact with the particles of the body to which that impulse is communicated.

A physico-mathematical theory of percussion, in which the impinging bodies are considered as imperfectly elastic, has been lately given by Don Georges Juan, in his Elementos Marítimos, a Spanish work which has been translated with additions by M. l'Evêque, under the title of Éléménts de navigation maritime, théorique et pratique, ou Traité de Mécanique, appliqué à la construction et à la manœuvre des vaisseaux, et autres bâtiments. This theory has been embraced by many eminent French philosophers, and may be seen in Prony's Architecture Hydraulique, vol. i. p. 208; and in Gregory's Mechanics, vol. i. p. 291.

In some cases of collision the results of experiments are rather at variance with those of theory, in consequence of the communication of motion not being exactly instantaneous. "If an ivory ball (says Mr Leslie) strikes against another of equal weight, there should, according to the common theory, be an exact transfer of motion. But if the velocity of the impinging ball be very considerable, so far from stopping suddenly, it will recoil back again with the same force, while the ball which is struck will remain at rest; the reason is, that the shock is so momentary as not to permit the communication of impulse to the whole mass of the second ball; a small spot only is affected, and the consequence is therefore the same as if the ball had impinged against an immoveable wall. On a perfect acquaintance with such facts depends, in a great measure, the skill of the billiard player. It is on a similar principle that a bullet fired against a door which hangs freely on its hinges will perforate without agitating it in the least."

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When a bullet is fired through a sheet of paper hung loosely, the paper scarcely receives a perceptible motion from the momentum of the bullet. Professor Whewell has given the following excellent illustration of this experiment: "The reason," says Professor Whewell, "is, that the bullet acts upon the paper only during the very short time which it employs in passing through it. If the bullet have a velocity of 1000 feet a second, and the paper be one thousandth of an inch in thickness, the time of the action is only \( \frac{1}{1000} \times \frac{1}{1000} \) of a second, or \( \frac{1}{1000} \times \frac{1}{1000} \) of a second. If we suppose that the bullet shot into a solid mass of paper would have lost all its velocity by penetrating one foot, this penetration would have occupied the \( \frac{1}{1000} \) of a second; and hence, in the \( \frac{1}{1000} \) of a second, it would, in the same substance, lose only the \( \frac{1}{1000} \) of its momentum, or the momentum corresponding to a velocity of half an inch a second; and the paper would gain the amount of momentum which the bullet thus loses. If the paper be one twenty-fourth of the weight of the bullet, the paper will acquire a velocity of a foot a second; and therefore in \( \frac{1}{1000} \) of a second, which is a portion of time quite perceptible, it would only move through one inch. And as the resistance of the air upon a sheet of paper is very considerable, the whole velocity would be destroyed almost before the motion could be observed." Nay, a pellet of clay, a bit of tallow, or even a small bag of water, discharged from a pistol, will produce the same effect. In all these instances the impression of the stroke is confined to a single spot, and no sufficient time is allowed for diffusing its action over the extent of the door. If a large stone be thrown with equal momentum, and consequently with smaller velocity, the effect will be totally reversed, the door will turn on its hinges, and yet scarcely a dent will be made on its surface. Hence likewise the theory of most of the tools, and their mode of application in the mechanical arts: the chisel, the saw, the file, the scythe, the hedge-bill, &c. In the process of cutting, the object is to concentrate the force in a very narrow space, and this is effected by giving the instrument a rapid motion. Hence, too, the reason why only a small hammer is used in riveting, and why a mallet is preferred for driving wedges. (Inquiry into the Nature of Heat, p. 127-8.)

The successive propagation of motion may be illustrated by a very simple experiment. Take two balls A, B, of which B is very large when compared with A, and connect them by a string S passing over the pulley P. If the ball B is lifted up towards S and allowed to fall by its own weight, instead of bringing the little ball A along with it, as might have been expected, the string will break at P. Here it is evident that the motion is not propagated instantaneously, for the string is broken before the motion is communicated to the portion of the string between P and A.

An apparatus for making experiments on the collision of bodies is represented in fig. 89. The impinging bodies are suspended by threads like pendulums, and as the velocities acquired by descending through the arches of circles are in the ratio of their chords, the velocities of the impinging bodies may be easily ascertained. The apparatus is therefore furnished with a graduated arch IN, which is gradually divided into equal parts, though it would be more convenient to place the divisions at the extremities of arcs whose chords are expressed by the corresponding numbers. The balls that are not used may be placed behind the arc, as at m and n; and in order to give variety to the experiments, the balls may be of different sizes. Sometimes a dish like G is attached to the extremities of the strings, for the purpose of holding argillaceous balls, and balls of wax softened with a quantity of oil equal to one fourth part of their weight. (See Smeaton's experiments on the Collision of Bodies.)

CHAP. VII.—ON THE MAXIMUM EFFECT OF MACHINES.

We have already seen in some of the preceding chapters, that when two bodies act upon each other by the intervention either of a simple or compound machine, there is an equilibrium when the velocity of the power is to the velocity of the weight as the weight is to the power. In this position of equilibrium, therefore, the velocity of the weight is nothing, and the power has no effect in raising the weight,—or, in other words, the machine performs no work. When the weight to be raised is infinitely small, the velocity is the greatest possible; but in this case likewise the machine performs no work. In every other case, however, between these two extremes, some work will be performed. In order to illustrate this more clearly, let us suppose a man employed in raising a weight by means of a lever with equal arms; and that he exerts a force upon the extremity of the lever equivalent to 50 pounds. If the weight to be raised is also 50 pounds, there will be an equilibrium between the force of the man and the weight to be raised, the machine will remain at rest, and no work will be performed. If the man exert an additional force of one pound, or if his whole force is 51 pounds, the equilibrium will be destroyed, the weight will rise with a very slow motion, and the machine will therefore perform some work. When the motion of the machine therefore is = 0, the work performed is also nothing; and when the machine is in such a state that the power preponderates, the work performed increases. Let us now suppose that the weight suspended from the lever is infinitely small, the motion of the machine will then be the greatest possible; but no work will be performed. If the weight, however, is increased, the motion of the machine will be diminished, and work will be performed. Here then it is evident that the work performed increases from nothing when the velocity is a maximum, and decreases to nothing when the velocity is a minimum. There must therefore be a particular velocity when the work performed is a maximum, and this particular velocity it is our present object to determine. Sometimes, indeed, the velocities of the machine are determined by its structure, and therefore it is out of the power of the mechanic to obtain a maximum effect by properly proportioning them. The same object, however, may be obtained by making the work to be performed, or the resistance to be overcome, in a certain proportion to the power which is employed to perform the work or overcome the resistance.

Def. 1. In a machine performing work, the powers employed to begin and continue the motion of the machine are called the first movers, or the moving powers, and those powers which oppose the production and continuance of motion are called resistances. The friction of the machine, the inertia of its parts, and the work to be performed, all oppose the production and continuance of motion, and are therefore the resistances to be overcome. When various powers act at the same time, and in different directions, the equivalent force which results from their combined action is called the moving force, and the force resulting from all the resisting forces, the resistance.

On the Maximum Effect of Machines.

The machine, for example, is a lever AB moving round the centre F, by means of which, two men raise water out of two pump barrels by the chains Au, Cw attached to the pistons, and passing over the arched heads or circular sectors M, N, for the purpose of giving the pistons and chains a vertical motion. Let the force of the man at B, six feet from F, be equal to 50 pounds, or \( \sigma \), his mechanical energy to turn the lever is \( 6 \times 50 = 300 \). Let the force of the other man applied at E, four feet from F, be also equal to 50 pounds, or \( p \). His mechanical energy will be \( 4 \times 50 = 200 \), so that the whole moving power is equal to \( 300 + 200 = 500 \). But if the two forces of 50 pounds, instead of being applied at two different distances from F, had been applied at the same point G, five feet from F, their energy to turn the lever would have been the same; for \( 5 \times 50 + 50 = 500 \). In the present case, therefore, the moving force is equivalent to \( P \times GF \), or a force of 100 pounds acting at a distance of five feet from the centre of motion. Now let us suppose that each piston Au, Cw raises 60 pounds of water, equivalent to the weights \( u, w \), and that \( CF = 2 \) feet, and \( AF = 3 \) feet, then the mechanical energy of these weights will be respectively \( 2 \times 60 = 120 \), and \( 3 \times 60 = 180 \), and the sum of their energies \( = 300 \). But two forces of 60 pounds each, acting at the distances two feet and three feet from F, are equivalent to their sum \( = 120 \) pounds acting at a distance of two feet and a half from F, for \( 2 \frac{1}{2} \times 120 = 300 \); therefore the resistance arising from the work to be performed, or from the water raised in the pump barrels, is equal to a weight \( P \) of 120 pounds acting at the distance \( DF = 2 \frac{1}{2} \) feet. But in addition to the resistance arising from the work to be performed, the two men have to overcome the resistance arising from the friction of the piston in the barrels, which we may suppose equivalent to \( f, \varphi \), each equal to 10 pounds, acting at the points A, C; but these forces are equivalent to 20 pounds, or \( f + \varphi \) acting at D, therefore the resistance arising from the work and from friction is equal to 140 pounds acting at the distance \( DF = 2 \) feet and a half.

While the two men are employed in overcoming these resistances, they have also to contend against the inertia of the beam AF, and that of the chains and pistons, which we may suppose equal to 20 pounds when collected in their centre of gravity \( g \), whose distance from F is 2-2 feet; but a weight of 20 pounds acting at the distance of 2-2 is equivalent to a weight of 19\(\frac{1}{2}\) pounds acting at the distance of 2-5 feet, or DF; consequently the sum of all the resistances, when reduced to the same point D of the lever, is equal to 159\(\frac{1}{2}\) pounds acting at the distance of 2-5 feet from F. The mechanical energy, therefore, of the sum of all the resistances will be \( 2 \times 2 \times 159\frac{1}{2} = 398\frac{1}{2} \), while the energy of the moving force, or the sum of all the moving powers, is equal to 500.

DEF. 2. The impelled point of a machine is that point to which the moving power is applied if there is only one power, or that point to which all the moving powers are reduced, or at which the moving force is supposed to act. The working point of a machine is that point at which the resistance acts if it is single, or that point to which all the resistances are reduced, and at which they are supposed to act when combined. Thus, in fig. 90, G is the impelled point of the machine, and D the working point. Had a single force \( \sigma \) been applied at the point B to raise a single weight \( u \), acting at the point A, then B would have been the impelled point and A the working point of the machine. In the wheel and axle, the point of the wheel at which the rope touches its circumference is the impelled point, while the working point is that point in the circumference of the axle where the rope which carries the weight is in contact with it.

DEF. 3. The velocity of the moving power, and the velocity of the resistance, are respectively the same as the velocity of the impelled point and the velocity of the working point.

DEF. 4. The effect of a machine, or the work performed, is equal to the resistance multiplied by the velocity of the working point; for when any machine raises a mass of matter to a given height in a certain time, the effect produced is measured by the product of the mass and the height through which it rises, that is, by the product of the mass by the velocity with which it moves.

DEF. 5. The momentum of impulse is equal to the moving force multiplied by the velocity of the impelled point.

In any machine that has a motion of rotation, let \( x \) be the velocity of the impelled point, and \( y \) the velocity of the working point. When the machine is a lever, \( x, y \) will express the perpendiculars let fall from the centre of motion upon the line of direction in which the forces act; and if the machine is a wheel and axle, \( x, y \) will represent the diameters of the wheel and the axle respectively. In compound machines, which may be regarded as composed of levers (Chap. II. Sect. I.), \( x \) will represent the sum of all the levers by which the power acts, and \( y \) the sum of all the levers by which the resistance acts.

Let \( P \) be the real pressure which the moving power exerts at the impelled point of the machine, and \( R \) the actual pressure which the mere resistance of the work to be performed exerts at the working point, or which it directly opposes to the exertion of the power. Let \( a \) be the inertia of the power \( P \), or the mass of matter which the power \( P \) must move with the velocity of the impelled point in order that \( P \) may exert its pressure at the impelled point; and let \( b \) be the inertia of the resistance \( R \), or the mass of matter which must be moved with the velocity of the working point in the performance of the work.

Since the resistance arising from the friction of the communicating parts is an uniformly retarding force, it may be measured by a weight \( \varphi \) acting at the working point of the machine, which will oppose the same resistance to the moving power as the friction of the parts.

Let \( m \) be the inertia of the machine, or rather that quantity of matter which, acting at the working point of the machine, will require the same part of the moving force to give it an angular motion; then, since \( y \) represents the arm of the lever by which the resistance acts, or the distance of the working point from the centre of motion; and since the momentum of inertia, or the momentum with which any mass revolving round a centre resists being put in motion, is equal to its quantity of matter multiplied by the square of its distance from its centre of motion (see article Rotation), we have \( my^2 \) for the momentum of inertia of the machine. It is obvious that every machine opposes a certain resistance to any force that endeavours to give it an angular motion, and that this resistance will increase with the inertia of its parts. It is easy, therefore, to find a quantity of matter which, when placed at any part of the machine, will oppose the same resistance to an angular motion as the combined inertia of the various parts of the machine. This is the quantity of matter which we have called \( m \), and which we have supposed to act at the working point, because to that point all the other resistances have been reduced. Collecting the symbols, therefore, we have

\[ x = \text{the velocity of the impelled point, or the radius of the wheel, or the length of the lever by which the power acts.} \]

\[ y = \text{the velocity of the working point, or the radius of the axle, or the length of the lever by which the resistance acts against the power.} \] P = the pressure exerted by the power at the impelled point of the machine.

R = the pressure which the resistance arising from the work to be performed exerts at the working point of the machine.

a = the inertia of the power P, or the quantity of matter to which it must communicate the velocity of the impelled point.

b = the inertia of the resistance R, or the quantity of matter which it must move with the velocity of the working point before any work is performed.

φ = a quantity of matter which, if placed at the working point of the machine, would oppose the same resistance to the moving power as that which arises from the friction of the communicating parts.

m = the quantity of matter which, if placed at the working point of the machine, would oppose the same resistance to the production of an angular motion, that is opposed by the inertia of the various parts of which the machine is composed. Hence, by the principles of rotation, we have

\[ m y^2 \] = the momentum of inertia of the machine.

We are now prepared for determining the conditions of construction which will enable any machine to produce a maximum effect.

**Prop. I.**

To determine the velocities which must be given to the impelled and working points of a machine, or the ratio of the levers by which the power and resistance ought to act, in order to obtain a maximum effect.

Let AB be a lever, whose fulcrum is F, and to whose extremity B is applied the power P to overcome the resistance R, and let FB = x, and FA = y. Then we shall have, from the following analogy, the weight which, placed at B, would be in equilibrium with R; \( x : y = R : \)

\[ \frac{Ry}{x} \] = the weight which will keep R in equilibrium, or the weight which is equal to that part of the power P which balances the resistance R. Hence, \( P - \frac{Ry}{x} \) will be the effective force exerted by the power P, which, multiplied by x, its distance from the centre of motion, gives \( Px - Ry \) or the force which is exerted in giving an angular motion to the power and resistance. But the resistance of friction was supposed equal to the weight φ acting at the working point, or at the distance FA or y; consequently, \( \varphi y \) will be the resistance which friction opposes to the force \( Px - Ry \), and therefore \( Px - Ry - \varphi y \) is the motive force exerted by P. Now, the momentum of the inertia of the power \( ax^2 + by^2 + my^2 \) is the mass to be put in motion by the power P. But the velocity generated in a given time is directly as the motive force, and inversely as the quantity of matter to which that force is applied. Hence the angular velocity, or the number of turns which the machine will make in a given time,

\[ \text{On the Maximum Effect of Machines.} \]

But in every rotatory machine the velocities of its different parts are as their distance from the axis; hence we shall have the velocities of the impelled and working points of the machine, by multiplying the angular velocity by \( x, y \), the distances of the impelled and working points of the machine from the centre of motion. Therefore,

\[ \frac{Px^2 - Rxy - \varphi xy}{ax^2 + by^2 + my^2} \] = the velocity of the impelled point, and

\[ \frac{Pxy - Ry^2 - \varphi y^2}{ax^2 + by^2 + my^2} \] = the velocity of the working point of the machine; and multiplying by R, we have, from Def. 4,

\[ \frac{PxyR - Ry^2 - \varphi Ry^2}{ax^2 + by^2 + my^2} \] = the work performed.

But as forces are proportional to the velocities generated by them in equal times, the preceding quantities will represent the accelerating forces. Now, the velocities are as the forces and times jointly; that is, \( v = Ft \), or \( v = gtF \); but F, the accelerating force, which generates the velocity of the impelled point, is represented by the formula

\[ \frac{Px^2 - Rxy - \varphi xy}{ax^2 + by^2 + my^2} \times gt \] = the absolute velocity of the impelled point, is

\[ \frac{Pxy - Ry^2 - \varphi y^2}{ax^2 + by^2 + my^2} \times gt \] = the absolute velocity of the working point

\[ \times gt \] Again, by Def. 4, the effect of a machine, or the work performed, is equal to the resistance of the work multiplied by the velocity; consequently, since R is the work, we have for the performance of the machine,

\[ \frac{PxyR - Ry^2 - \varphi Ry^2}{ax^2 + by^2 + my^2} \times gt \] = the work performed.

Now, considering y as the variable quantity, and making the fluxion of the preceding formula = 0, we shall find that the performance of the machine is a maximum, when

\[ y = \frac{a^2 \times R + \varphi^2 + P^2a \times m + b^2}{Pm + Pb} \times x. \]

When R = 0, we have

\[ y = \frac{a^2 \times R + \varphi^2 + P^2a \times m + b^2}{Pm + Pb} \times x. \]

When \( \varphi = 0 \), the first formula becomes

\[ y = \frac{a^2 \times R + \varphi^2 + P^2a \times m + b^2}{Pm + Pb} \times x. \]

When both R and \( \varphi = 0 \), we have, after reduction,

\[ y = \frac{\sqrt{a}}{\sqrt{m + b}} \times x. \]

When \( b = 0 \), the first formula becomes

\[ y = \frac{a^2 \times R + \varphi^2 + P^2a \times m + b^2}{Pm + Pb} \times x. \]

When R, \( \varphi \), and \( b = 0 \), we have

\[ y = \frac{\sqrt{a}}{\sqrt{m}} \times x. \]

When \( a : b = P : R \), we have, by substituting P and R instead of \( a \) and \( b \),

\[ y = \frac{P^2 \times R + \varphi^2 + P^2 \times m + R^2}{Pm + PR} \times x. \]

When \( Pm \) and \( \varphi = 0 \), the last formula becomes

\[ y = \frac{P^2 \times R + \varphi^2 + P^2 \times m + R^2}{PR} \times x = \frac{\sqrt{P^2 \times R + \varphi^2 + P^2 \times m + R^2}}{PR} \times x. \]

\[ x = \sqrt{\frac{P}{R + 1}} - 1. \]

On the Maximum Effect of Machines, and when \( x = 1 \) and \( R = 1 \), we have

\[ y = \sqrt{P + 1} - 1; \]

and when \( P = 1 \) and \( x = 1 \), we obtain

\[ y = \sqrt{\frac{1}{R} + 1} - 1. \]

When \( x = 1 \),

\[ y = \sqrt{\frac{1}{R} + 1} - 1. \]

These various formulae, the application of which to particular cases will be shown in the practical part of this article, give us values of \( y \) for almost every species of machinery; so that the mechanic may easily determine the velocities which must be given to the impelled and working points of the machine in order to produce a maximum effect.

When the machine, however, is already constructed, the velocities of the impelled and working points cannot be changed without altering the structure of the machine; and therefore we must find the ratio between the power and resistance, which will enable us to obtain a maximum effect. The method of determining this will be shown in the following proposition.

**Prop. II.**

To determine the ratio between the power and the resistance of a machine when its performance is a maximum.

Since the structure of the machine is given, the values of \( x, y \) are known, and therefore we have to determine the relative values of \( P \) and \( R \) when the effect of the machine is a maximum. This would be easily done by making \( R \) variable in the formula which expresses the performance of the machine, and making its fluxion equal to 0, if none of the other quantities varied along with \( R \). It often happens, however, that while \( R \) varies, the mass \( b \) suffers a considerable change, though in other cases the change induced upon \( b \) is too unimportant to merit notice. This proposition, therefore, admits of two cases; 1. when the change upon \( b \) is so small that it may be safely omitted in the investigation; and, 2. when the change upon \( b \) is sufficiently great to require attention.

**Case 1.** When \( R \) is the only quantity which is variable, the fluxion of the formula

\[ \frac{PxyR - R^2y^2 - \phi Ry^2}{ax^2 + by^2 + my^2}, \]

which represents the work performed, is equal to the fluxion of the numerator, because the denominator is constant, that is,

\[ PxyR - 2RRy^2 - \phi Ry^2 = 0; \]

and, dividing by \( R \),

\[ Pxy - 2Ry^2 - \phi y^2 = 0; \]

hence \( 2Ry^2 = Pxy - \phi y^2 \), and

\[ R = \frac{Pxy - \phi y^2}{2y^2}, \]

which, divided by \( y \), gives \( R = \frac{Px - \phi y}{2y}. \)

Now, according to the experiments of Coulomb, the friction is, in general, proportional to the resisting pressure, or a certain part of that pressure, for example, \( \frac{1}{2} R \); and calling \( Z = \frac{1}{2} R \), and omitting \( \phi y \), we have for the resistance \( R + \frac{1}{2} R = \frac{Px - \phi y}{2y} \), or \( R = \frac{Px}{2y} + \frac{\phi y}{2y} \); and

making \( P = 1 \), and \( x = 1 \), we have \( R = \left( \frac{1}{2y} \right) + \frac{\phi y}{2y} \), so that, abstracting from the quotient \( \frac{\phi y}{2y} \), which, being little greater than 1, will not alter the result, the resistance should be one half of the force which would keep the impelling power in equilibrium.

**Case 2.** When \( b \) varies at the same time with \( R \), it will in most cases vary in the same proportions, and therefore may be represented by any multiple of \( R \), as \( dR \).

where \( d \) may be either an integer or a fraction. In order to simplify the investigation, we may consider the fraction \( \frac{d}{R} \) as a resistance diminishing the impelling power, instead of regarding it as a resistance to be added to the other resisting forces. Thus the impelling power \( P \) will become \( P - \phi \). In the same way we may consider the momentum of the machine's inertia applied to the impelled point, that is, instead of \( my^2 \), it may be made \( mx^2 \). Now, making \( P - \phi \), or the impelling power \( = 1 \), and making \( x = 1 \), we shall have by these substitutions in the formula which expresses the effect of the machine,

\[ \frac{Ry - R^2y^2}{a + m + dRy^2}, \]

or, for the sake of simplicity, making \( a + m = q \), we have for the performance of the machine

\[ \frac{Ry - R^2y^2}{q + dRy^2}; \]

then, since \( R \) is the variable quantity, we shall find, after making the fluxion of this formula \( = 0 \), that the performance is a maximum when \( R = \frac{q^2 + qdy^2}{dy^2} - q \).

When \( b = R \), then \( d = 1 \), and we shall have

\[ R = \frac{q^2 + qdy^2}{y^2} - q. \]

When \( a = P \) and \( P = 1 \), and when \( m \), the inertia of the machine, \( = 0 \), we shall have \( a + m = 1 = q \), and then the formula becomes

\[ R = \frac{y + 1}{y^2} - 1. \]

When \( y = x \), then \( y = 1 \), and

\[ R = \frac{1 + 1}{1} - 1 = 0.4142. \]

**Scholium.**

Those who wish to prosecute this interesting subject may consult the different papers of Euler in the Comment. Petropol. vol. x. p. 80, 1743, and in the Comment. Nov. Petropol. vol. iii. and viii. This subject has been treated with great ability by Dr. Robison, though he has omitted the various steps in the investigation which conduct to the leading formulæ. The subject has been also ably discussed by Sir John Leslie, in a paper published in the first edition of the appendix to Ferguson's Lectures, vol. ii. p. 353; and as the results of his investigations may be of great use in practice, we shall here present the reader with a short abstract of them.

If the resistance is equal to the power, or double, triple, or quadruple, &c. a maximum effect will be produced when the velocity of the power, or its distance from the centre of motion, is \( 1 + \sqrt{2}, 2 + \sqrt{6}, 3 + \sqrt{12}, 4 + \sqrt{20}, 5 + \sqrt{30}, 6 + \sqrt{42} \), that of the weight being 1, &c. If the resistance is very great compared with the power, the velocity should at least be double of that which would procure an equilibrium, in order that the machine may produce a maximum effect.

If the velocity of the power, or its distance from the centre of motion, be equal to, double, triple, quadruple, &c. &c. of the velocity of the weight or resistance, a maximum effect will be produced when the power \( P \) is equal to \( R \times 1 + \sqrt{2}, R \times \sqrt{2} + \sqrt{3}, R \times \sqrt{3} + \sqrt{4}, R \times \sqrt{4} + \sqrt{5}, R \times \sqrt{5} + \sqrt{6}, \) &c. where \( R \) is the resistance or weight to be raised. If the velocity of the power be very large, a maximum effect will be produced when the power \( P \) is at least double of that which would procure an equilibrium. It appears also, from Sir John Leslie's paper, that in whatever way the maximum be procured, the force which impels the weight can never amount to one fourth part of the direct action of the power; and that, in ma- CHAP. VIII.—ON THE EQUILIBRIUM OF ARCHES, PIERS, AND ABUTMENTS.

DEF. 1. An arch is represented in fig. 92 by the assemblage of stones ab, cd, ef, &c., forming the mass ABMN, whose inferior surface is the portion of a curve. The parts A, B are called the spring of the arch, the line AB the span of the arch, Cb its altitude, b its crown, ob the keystone, the curve or lower surface ABB be intrados, and the roadway TUV the extrados; PQ, RS, the piers when they stand between two arches, and the abutments when they are at the extremities of the ridge.

DEF. 2. A catenarian curve is the curve formed by any line or cord perfectly flexible, and suspended by its extremities. Thus the chain ACB is suspended by its extremities A, B, it will, by the action of gravity upon all its parts, assume the form ACB, which is called the catenary, or catenarian curve.

There are three modes of determining the construction of arches; the first of which is to consider the arch as an inverted catenary; the second is to establish an equilibrium between the vertical pressures of all the materials between the intrados and extrados; and the third is to regard the different arch-stones as portions of wedges without friction, which endeavour by their own weight to force their way through the arch. The first of these methods is given by the ingenious Dr Hook, and is contained in the following proposition.

PROP. I.

To determine the form of an arch by considering it as an inverted catenary, when its span, its altitude, and the form of the roadway or extrados, are given.

Let a, b, c, d, &c., be a number of spheres or beads connected by a string, and suspended by their extremities A, B; they will form a catenarian curve AobeB, and be in equilibrium by the action of gravity. Each sphere is acted upon by two forces; at its lower point by the weight of the spheres immediately below it, and at its upper point by the weight of the same spheres added to that of the sphere itself; that is, any sphere e is in equilibrium from the result of two forces, one of which is produced by the weights of c, d, e acting at the lower point of b, while the other force arises from the weight of bede acting at its upper point. The equilibrium of this chain of spheres is evidently of the stable kind, as it will immediately recover its position when the equilibrium is disturbed. Let us now suppose this arch inverted, so as to stand in a vertical plane, as in fig. 95. It will still preserve its equilibrium. For the relative positions of the lines which mark the directions remain unchanged by inverting the curve, the force of gravity continues the same, and therefore the result of these forces will be the same, and the arch will be in equilibrium. The equilibrium, however, which the arch now possesses is of the tottering kind, so that the least disturbing force will destroy it, and it will consequently be unable to support any other weight but its own.

Let us now suppose that it is required to form an equilibrated arch, whose span is AB, whose altitude is Dk, and which will support the materials of a roadway, whose form TUV is given. It is obvious, that if the spheres a, b, c, d, &c., increase in density from k towards a, the catenarian curve will grow less concave at its vertex e, and more concave toward its extremities A, B. Let us then suppose that the densities of the spheres a, b, c, d, e, &c., are respectively as am, bm, cm, dm, em, &c., the vertical distances of their respective centres from the roadway TUV, the arch will have a form different from that which it would have assumed if the spheres were of equal density, and will be in equilibrium when inverted as in fig. 95. Now, in place of the spheres a, b, c, d, e, &c., of different densities, let us substitute spheres of the same density, and having the same position as those of different densities; let us then load the sphere a with a weight which, when combined with the weight of a, will be equal to the weight of the corresponding sphere a, that had a greater density; and let us load the other spheres b, c, d, &c., with weights proportional to bm, cm, dm, &c. Then it is obvious that the pressure of each sphere when thus loaded upon that which is contiguous to it, is precisely equal to the pressure of the spheres of different densities upon each other, because the density of these spheres varied as their distances from the roadway. But the arch composed of spheres of different densities was in equilibrium when inverted, therefore, since the loaded spheres of the same density have the same position and exert the same pressures, the arch composed of these spheres, and supporting TUVBA, composed of homogeneous materials, will be in equilibrium. Hence a roadway of a given form, and composed of homogeneous materials, will be supported by an arch whose form is that of a catenary, each of whose points varies in density as their distance from the surface of the roadway; or, which is the same thing, a roadway of a given form, and composed of homogeneous materials, will be supported by an arch whose form is that of a catenary, each of whose points is acted upon by forces proportional to the distances of these points from the surface of the roadway.

Hence we have the following practical method of ascertaining the form of an equilibrated arch, whose span is AB, and altitude Dk, and which is to support a roadway of the form TUV. Let a chain AobeB, of uniform density, be suspended from the points A, B, so that it forms a catenary whose altitude is Dk, the required height of the arch. Divide AB into any number of equal parts, suppose eight, and let the vertical lines 1m, 2n, 3o, drawn from these points, intersect the catenary in the points a, b, c.

From the points a, b, c, k, r, s, t, suspend pieces of chain of uniform density, and form them of such a length, that when the whole is in equilibrio, the extremities of the chains may lie in the line TUV; fig. 95; then the form which the catenary AkB now assumes will be the form of an equilibrated arch, which, when inverted like AKB, will support the roadway TUV, similar to TUV, fig. 95. This is obvious from the last paragraph, for the pieces of chain am, bm, cm, ku, &c. are forces acting upon the points a, b, c, k of the catenary, and are proportional to am, bm, cm, &c. the distances of the points a, b, c, k, &c. from the roadway.

An arch of this construction will evidently answer for a bridge, in which the weight of the materials between the roadway and the arch-stones is to the weight of the arch-stones as the weight of all the pieces of chain suspended from a, b, c, &c. is to the weight of the chain AAB. As the ratio, however, of the weight of the arch-stones to the weight of the superincumbent materials is not known, we may assume a convenient thickness for the arch-stones; and if from this assumed thickness their weight be computed, and be found to have the required ratio to the weight of the incumbent mass, the curve already found will be a proper form for the arch. But if the ratio is different from that of the weight of the whole chain to the weight of the suspended chains, it may be easily computed how much must be added to or subtracted from the pieces of chain, in order to make the ratios equal. The new curve which the catenary then assumes, in consequence of the change upon the length of the suspended chains, will be the form of an equilibrated arch, the weight of whose arch-stones is equal to that which we assumed.

**Scholium.**

In most cases the catenarian curve thus determined will approach very near to a circular arc equal to 120 degrees, which springs from the piers so as to form an angle of 60 degrees with the horizon. The form of the arch, however, as determined in the preceding proposition, is suited only to those cases in which the superincumbent materials exert a vertical pressure. A quantity of loose earth and gravel exerts a pressure in almost every direction, and therefore tends to destroy the equilibrium of a catenarian arch. This tendency, however, may be removed by giving the arch a greater curvature towards the piers. This will make it approach to the form of an ellipsis, and make it spring more vertically from the piers or abutments.

We shall now proceed to deduce the form of an arch and its roadway, by establishing an equilibrium among the weights of all the materials between the arch and the roadway. This method was given by Emerson in his *Flexions*, published in 1742; and afterwards by Dr Hutton in his excellent work *On Bridges*.

**Prop. II.**

To determine the form of the roadway or extrados, when the form of the arch or intrados is given.

Let the lines AD, DE, EB, BF, FG, GH lie in the same plane, and let them be placed perpendicular to the horizon. From the points D, E, B, &c. draw the vertical lines Dd, Ee, Bb, &c. and taking Dp of any length, make Er equal to Dp, &c. and complete the parallelograms pe, rq. Again, make Bs = ge, and complete the parallelogram ts; in like manner, make Fk = sb, and complete the parallelogram Ef; and so on with all the other lines, making the side of each parallelogram equal to that side of the preceding parallelogram which is parallel to it. Let us now suppose that the lines CD, DE, EB, &c. can move round the angular points D, E, B, F, &c. the extremities A, C being immovable; and that forces proportional to Dd, Ee, Bb, &c. are exerted upon the points D, E, B, F, &c. and in the direction Dd, Ee, &c. Now, by the resolution of forces, the force Dd may be resolved into the forces De, Dp, the force Ee into the forces Eg, Er, and the force Bb into the forces Bs, Br, and so on with the rest. The force De produces no other effect than to press the point A on the plane on which it rests, and is therefore destroyed by the resistance of that plane; but the remaining force Dp tends to bring the point D towards E, and to enlarge the angle ADE; this force, however, is destroyed by the equal and opposite force Eg, and in the same way the forces Er, Br, Fr are destroyed by the equal and opposite forces Bs, Fk, Gr, while the remaining force Gw is destroyed by the resistance of the plane which supports the point C. When the lines AD, DE, &c. therefore, are acted upon by vertical forces proportional to Dd, Ee, Bb, &c. these forces are all destroyed by equal and opposite ones, and the lines will remain in equilibrio.

Now the force De : Dp or Eg = sin. cdD or dDp : sin. ADd, that is, by taking the reciprocals,

\[ De : Eg = \frac{1}{\sin. ADd} : \frac{1}{\sin. dDp} \]

and, for the same reason,

\[ Eq : Bs = \frac{1}{\sin. Eq} : \frac{1}{\sin. bBs} \]

Hence

\[ Eq = \frac{1}{\sin. Eq} \]

Now, since Eq : Ee = sin. Eq : sin. Ege, we have Ee = Eq × sin. Ege

\[ Ee = \frac{\sin. Eq}{\sin. eEB} \times \frac{\sin. DEM}{\sin. eEB} \]

But Eq = 1 / sin. Eq, therefore, by substitution, we obtain

\[ Ee = \frac{\sin. DEM}{\sin. Eq \times \sin. eEB} \]

Now, as the same reasoning may be employed to find Dd, Bb, &c. we have obtained expressions of the forces which, when acting at the angular points D, E, B, &c. keep the whole in equilibrio, and these expressions are in terms of the angles which the lines DE, EB, &c. form with the direction of the forces. If the lines AD, DE, &c. be increased in number so that they may form a polygon with an infinite number of sides, which will not differ from a curve line, then the forces will act at every point of the curve, and the line mE will be a tangent to the curve at the point E.

and \( DE_m \) will be the angle of contact. The line \( E_q \) being now infinitely small, will coincide with \( E_m \), and therefore the angles \( eE_q \) and \( eEB \) or \( Eeq \) will be equal to the angle \( eEm \), and consequently their sines will be equal. Therefore, by making these substitutions in the last formula, we have an expression of the force at every point of the curve, thus,

\[ E_e = \frac{\sin DE_m}{\sin eEm} \times \frac{\sin eEm}{\sin eEm^2} \]

But the angle of contact \( DE_m \) varies with the curvature at the point \( E \), and the curvature varies as the reciprocal of the radius of curvature, therefore the angle of contact varies as the reciprocal of the radius of curvature; hence, by substitution,

\[ E_e = \frac{1}{\text{radius of curvature}} \times \sin eEm^2 \]

In order to get rid of the confusion in fig. 97, where the arch is a polygon, let us suppose ABC, fig. 98, to be the curve, \( mn \) a tangent to any point \( E \), and \( Ea \) a vertical line; then the pressure at any point of the arch is reciprocally as the radius of curvature at that point, and the square of the sine of the angle which the tangent to that point of the curve forms with a vertical line.

COROLLARY. Let us now suppose that the arch ABC supports a mass of homogeneous materials lying between the roadway TUV and the arch ABB'C'; and the whole being supposed in equilibrium, let us determine the weight which presses on the point \( E \). The weight of the superincumbent column \( EcEb \) varies as \( Ec \times gd \); but \( gd = Ed \times \sin E_q \), \( Ed \) being radius, and \( dEg = EnB \), on account of the parallels \( Ec \) and \( UB \), therefore the weight of the column \( EcEb \) varies as \( Ec \times Ed \times \sin EnB \), that is, as \( Ec \times \sin EnB \), because \( Ed \) is a constant quantity; but the pressure at \( E \) was proved to vary as

\[ \frac{1}{\text{radius curvature}} \times \sin eEm^2 \]

therefore the weight of the column \( EcEb \) or \( Ec \times \sin EnB \) varies also as this quantity, that is,

\[ Ec \times \sin EnB = \frac{1}{\text{radius curvature}} \times \sin eEm^2 \]

but as the angle \( EnB \) is equal to the angle \( eEm \), we shall have, by substitution and division,

\[ \frac{1}{\text{radius curvature}} \times \sin eEm^2 \]

When an arch supports a roadway, the pressure exerted on any point of it is reciprocally as the radius of curvature, and the cube of the sine of the angle which the tangent to that point forms with a vertical line.

Having thus obtained an expression for \( Ec \), we shall proceed to show the application of the formula to the case when the arch is a portion of a circle.

Let \( EB \) be the chord of a circle whose centre is \( F \), let the radius \( R \), \( D = \text{versed sine}, E = x, DF = \cos E = b, BU = m \), draw the tangent \( E_a \) and through \( E_a \) the vertical line \( ea \), which will be parallel to \( BE \). Then, since \( GEF \) is a right angle, and \( eEF = EFB \), the angle \( GEe \) is the complement of \( EFB \), therefore \( \sin GEe = \cos EFB = FD \). But, in the present case, the radius of curvature is the radius of the arch, or \( R \), therefore \( Ec = \frac{1}{R \times \sin GEe} \) or, by substitution, \( Ec = \frac{1}{Rb^3} \), that is, since \( R \) is constant, \( Ec = \frac{1}{b^3} \). But when the point \( E \) coincides with \( B \), the cosine \( b \) becomes equal to radius; therefore, in that case, \( Ec = \frac{1}{R^3} \), and \( Ec \) becomes \( BU = m \),

\[ \frac{1}{R^3} : \frac{1}{b^3} = m : Ec \quad \text{and} \quad Ec = \frac{mb^3}{b^3} \]

Now, by the notation \( R : b = BF : DF \); therefore \( R^3 : b^3 = BF^3 : DF^3 \), hence

\[ R^3 = \frac{mBF^3}{DF^3} \]

and, multiplying each side by \( m \), we have

\[ \frac{mR^3}{b^3} = \frac{mBF^3}{DF^3} ; \quad \text{but} \quad \frac{mR^3}{b^3} = Ec, \quad \text{therefore the vertical distance of the surface of the roadway from the point \( F \), or} \]

\[ Ec = \frac{mBF^3}{DF^3} = \frac{BU \times BF^3}{DF^3} \]

When the point \( E \) coincides with \( B \), \( BF = DF \), and \( Ec = BU \). When \( E \) coincides with \( A \), the cosine \( DF \) vanishes, and therefore \( Ec \), or the distance of the point \( A \) from the extrados or roadway, is infinite. The curve \( VUcT \), therefore, will run up to an infinite height, approaching continually to a vertical line drawn from \( A \), which will be its asymptote. Such a form of the extrados, however, is inadmissible in practice; and therefore a semicircular arch is not an arch of equilibrium. When the arch is less than a semicircle, as PBR, the curve terminates in the point \( p \); and as it does not rise very much above a horizontal line passing through \( U \) when the arch is small, we might produce a perfect equilibrium by making the horizontal, as \( Uv \), and making the density of the superincumbent columns \( Pr, Eo \), which press upon the points \( P, E \) respectively, in the ratio of \( Pp, Ec \), the distances of these points from the curvilinear roadway.

The inconvenience, however, arising from the inflexion of the extrados, may be considerably removed by throwing the point of contrary flexure to a greater distance, which may be done by diminishing \( BU \), the thickness of the incumbent mass above the keystone. Thus, if \( BU \) is diminished to \( Bd \), and if points \( a, b \) are taken in the lines \( Pp, Ec \), so that \( Pa : Pp = Eb : Ec = Bd : BU \), and so on with all the points in the arch, and if a new roadway \( edba \) be drawn through these points, the equilibrium of the arch will still continue, for the various pressures which it sustained, though they are diminished, preserve the same proportion.

Let us suppose it necessary to have the extrados a horizontal line, and let it be required to find \( BU = m \) when there is an equilibrium. In this case the point \( H \) coincides with \( U \); or rather, when the curve \( UcT \) cuts the horizontal line \( Uv \), the point \( H \) coincides with \( U \). By substituting \( BF - BD \) instead of \( DF \) in the value \( Ec \), formerly determined, and by putting \( BD = y \), we have

\[ Ec = \frac{mR^3}{R - y^3} \]

But when \( H \) coincides with \( U \), \( c \) coincides with \( o \), and therefore \( Eo = Ec = BD + BU = y + m \),

\[ \frac{mR^3}{R - y^3} = y + m ; \quad \text{and, multiplying by} \quad R - y^3, \]

we have \( mR^3 = y \times R - y^3 + m \times R - y^3 \), or \( mR^3 + m \)

On the Equilibrium of Arches.

\[ \times R - y^3 = y \times R - y^3; \text{ and, dividing by the co-efficients of } m, \text{ we have} \]

\[ m = \frac{y \times R - y^3}{R^3 - R - y^3}; \text{ that is,} \]

The thickness of the roadway above the keystone, when the extrados is a straight line, is equal to the quotient arising from multiplying the versed sine of half the arch by the cube of its cosine, and dividing this product by the difference between the cube of the radius and the cube of the cosine; or, to change the expressions, the thickness of the roadway above the keystone, when the roadway is a straight line, is equal to the quotient arising from multiplying the height of the arch by the cube of the difference between the radius of the arch and its height, and dividing this product by the difference between the cube of the radius and the cube of the difference between the radius and the height of the arch.

When the arch is a semicircle, \( R - y \) vanishes, and \( m \) becomes equal 0, so that the semicircular arch is evidently inadmissible. But when the arch is less than a semicircle, the value of \( m \) will be finite. Thus, if the arches are respectively

- Arch. 60°, we have \( m = \frac{1}{2} \) of the span, - 90°, we have \( m = \frac{1}{3} \) of the span, or - 110°, we have \( m = \frac{1}{4} \) of the span nearly.

The two first arches of 60° and 90° manifestly give too great a thickness to the part BU or \( m \). In the third arch of 110°, the thickness of BD is nearly what is given to it by good architects, and is therefore the best in practice; for if the arch were made greater than 110°, the thickness of BU or \( m \) would be too small. It is obvious, however, that an arch of 110° is not an arch of perfect equilibration, for this can be the case only when the roadway has the form Uzr, fig. 95. When the roadway, therefore, is horizontal, as Ur, there is an unbalanced pressure on both sides of the keystone, produced by the weight of the materials in the mixtilinear space \( r = U \). It is indeed very small, and might be counteracted by making the materials below Z lighter than those below U; but the unbalanced pressure is so trifling that it may be safely neglected. We may therefore conclude, that when the arch is to be circular with a horizontal roadway, an arch of 110 degrees approaches nearest to an arch of equilibration.

When the arch is elliptical, it will be found, as in the circle, that \( m = \frac{y \times R - y^3}{R^3 - R - y^3} \). An elliptical arch, however, has the advantage of a circular one, when the transverse axis is horizontal; for, as it is much flatter, the point of contrary flexure in the extrados is thrown at a greater distance, and therefore it will, with less inconvenience, admit of a horizontal roadway. Elliptical arches have also the advantage of being more elegant, and likewise require less labour and materials.

The cycloidal arch is likewise superior to a circular one, but inferior to those which are elliptical. Parabolic, hyperbolic, and catenarian arches may be employed when the bridge has only one arch, and is to rise high; but in other cases they are inadmissible. The method of determining the roadway for all these forms of arches will be found in Dr Hutton's excellent work On the Principles of Bridges, p. 3. See also Emerson's Miscellanies, p. 156; and his work On Fluxions, published in 1742.

When the form of the roadway is given, the shape of the intrados for an arch of equilibration may be determined. As the investigation is very difficult, unless when the roadway is a horizontal line, we shall merely give the formula, which will enable any person to construct the curve. In all other curves the equilibrium of the arch is imperfect; but the curve described by the following formula is an arch of perfect equilibration, and has been called the mechanical curve of equilibration:

\[ ED = AF \times \frac{BU + BD + \sqrt{2}BU \times BD + BD^2}{BU} \]

From this formula, which corresponds with fig. 100, Dr Hutton has computed the following table, containing the values of \( eU \) and \( eE \), for an arch whose span AC is 100, whose height BF is 40, and whose thickness at the crown or BU is 6. The table will answer for any other arch whose span and thickness are as the numbers 100, 40, 6; only the values of \( eU \) and \( eE \) must be increased or diminished in the same ratio as these numbers.

Table for constructing the Curve of Equilibration, when the Span, Height, and Thickness at the Crown, are as the Numbers 100, 40, and 6.

| Value of \( eU \) | Value of \( eE \) | Value of \( eU \) | Value of \( eE \) | Value of \( eU \) | Value of \( eE \) | Value of \( eU \) | Value of \( eE \) | Value of \( eU \) | Value of \( eE \) | |------------------|------------------|------------------|------------------|------------------|------------------|------------------|------------------|------------------|------------------| | 0 | 6-000 | 15 | 8-120 | 24 | 11-911 | 33 | 18-627 | 42 | 29-919 | | 2 | 6-035 | 16 | 8-490 | 25 | 12-489 | 34 | 19-617 | 43 | 31-563 | | 4 | 6-144 | 17 | 8-766 | 26 | 13-106 | 35 | 20-665 | 44 | 33-299 | | 6 | 6-324 | 18 | 9-168 | 27 | 13-761 | 36 | 21-774 | 45 | 35-135 | | 8 | 6-580 | 19 | 9-517 | 28 | 14-457 | 37 | 22-948 | 46 | 37-075 | | 10 | 6-914 | 20 | 9-934 | 29 | 15-196 | 38 | 24-150 | 47 | 39-126 | | 12 | 7-330 | 21 | 10-881 | 30 | 15-980 | 39 | 25-505 | 48 | 41-293 | | 13 | 7-571 | 22 | 10-858 | 31 | 16-811 | 40 | 26-694 | 49 | 43-581 | | 14 | 7-834 | 23 | 11-368 | 32 | 17-693 | 41 | 28-364 | 50 | 46-000 |

The construction of arches has also been deduced from considering the arch-stones as frustums of polished wedges without friction, which endeavour to force their way through the arch. This principle has been adopted by Belidor, Parent, Bossut, Prony, and other French philosophers, and likewise by our ingenious countryman the late Mr Atwood. This theory, however, is more plausible than useful. So far from the arch-stones having liberty to slide On the Construction of Piers and Abutments.

In the construction of piers and abutments, there are two circumstances which claim our attention. 1. The strength that must be given to them in order to resist the lateral thrust which they sustain from the adjacent semicircles, and which tend either to overturn them or make them slide upon their base. 2. The form which must be given to their extremities, so that the force of the current may be a minimum. The adhesion of the pier to the place on which it rests is always much greater than one third of its pressure; and as the lateral thrust of the arch which it supports resists is oblique to the horizon, and may be resolved into two forces, one of which is horizontal, and the other vertical, we have the vertical portion of the lateral thrust, the weight of the pier, and the friction on its base, combined in resisting the horizontal portion of the lateral thrust which tends to make the pier slide upon its base, so that there is no danger of the pier yielding to such a pressure. We do not here consider, that the lateral thrust which tends to give a horizontal motion to the pier is completely counteracted by the lateral thrust of the opposite semicircle, because it is necessary that the pier should have sufficient stability to resist the lateral thrust of one semicircle, in case of the failure of the opposite one. Let us therefore consider the strength of the pier which will prevent it from being overturned.

For this purpose, let ABC, fig. 100, be an arch, MHTO the pier, and BUHA the loaded semicircle, whose pressure tends to overturn the pier. Let G be the centre of gravity of the mass BUHA. Join GA, and from G draw GK perpendicular to AC, fig. 100. Then, since the whole pressure of the arch is exerted at its spring A, and since this pressure is the same as if the whole weight of the arch were collected into the point G, GA will be the direction in which the weight of the arch and the superincumbent mass acts upon the point A. Now, by Dynamics, the force GA may be resolved into the two forces GK, KA, one of which, KA, endeavours to give the pier a motion of rotation about the point O, while the other, GK, denotes the weight of the loaded arch in the direction GK. Putting W, therefore, for the weight or area of the superincumbent mass, we have GK : KA = W : \(\frac{W \times KA}{GK}\), the pressure upon A.

Now, as this force tends to turn the pier round O by means of the lever OA, and as ON = AM is the perpendicular from the centre of motion upon the line of direction, we have AM \(\times\) \(\frac{W \times KA}{GK}\) for the force which tends to overturn the pier. Now, the force which is opposed to this is the weight of the pier MHTO collected at its centre of gravity g, which acts by the vertical lever OM = \(\frac{1}{2}\) OM, because g is in the centre of the rectangle TM. But the weight or area of the pier may be represented by OM \(\times\) MH; therefore the force which resists the lateral thrust of the loaded arch is OM \(\times\) MH > \(\frac{1}{2}\) OM, or \(\frac{1}{2}\) MH \(\times\) OM. Now, in the case of an equilibrium between these opposing forces, we have AM \(\times\) \(\frac{W \times KA}{GK}\) = \(\frac{1}{2}\) MH \(\times\) OM\(^2\), which, by reduction, becomes OM = \(\sqrt{\frac{2AM \times W \times KA}{MH \times GK}}\). This formula gives us the breadth of the pier which is capable of balancing the lateral thrust, and therefore OM must be taken a little greater than the preceding value. In practice, OM is generally between one fifth and one seventh of AC, the span of the arch. The method of finding the centre of gravity G of the loaded arch, whether the arch is in perfect equilibrium or not, may be seen in Dr Hutton's work already quoted, p. 49. A very simple method of doing this is to form the part BUHA of a piece of card, and to find its centre of gravity G by the rules already given. This indeed supposes all the materials to be homogeneous; but if they are of various kinds, we can load the arch made of card in a similar manner, and determine its centre of gravity as before.

The limits of this article will not permit us to apply the method of fluxions to the determination of the form which should be given to the ends of the pier, in order that the impulse of the current may be the least possible. The theory of the resistance of fluids, indeed, differs so widely from experiment, that such an investigation would, in this place, be of little practical utility. It may be sufficient merely to remark, that the pier should have an angular form, and that the impulse of the current will be diminished as the angle is more acute. When the ends are semicircular, the impulse of the stream is reduced to one half; and though a triangular termination of the piers reduces the impulse still more, yet semicircular ends are more pleasing to the eye, and are particularly advantageous when small vessels have occasion to pass the arch. When such vessels happen to impinge against the piers, the semicircular ends are more able to bear the shock, and do less injury to the vessel, while the additional quantity of masonry will give greater stability to the pier.

For fuller details on the subject of this chapter, and on that of domes, cupolas, and octagonal pyramids, see the article Arch in this work, vol. iii. p. 378.

CHAP. IX.—ON THE FORCE OF TORSION.

Definition. Let ga be a metallic wire firmly fixed in the pincers g, by means of the screw s; let the cylindrical weight P, furnished with an index o, be suspended at the lower extremity of the wire; and let the axis of the cylinder, or the wire ga produced, terminate in the centre of the divided circle MNO. Then, if the cylinder P is made to move round its axis so that the index o may describe the arch ON, the wire ga will be twisted. If the cylinder be now left to itself, the wire will, in consequence of its elasticity, endeavour to recover its form; the index o will therefore... move backwards from N, and oscillate round the axis of Force of the cylinder. The force which produces these oscillations is called the force of torsion, and the angle measured by the arch ON is called the angle of torsion.

Prop. I.

To deduce formulae for the oscillatory motion of the cylinder, on the supposition that the re-action of the force of torsion is proportional to the angle of torsion, or nearly proportional to it.

Let PQ be a section of the cylinder P in fig. 101, and let all the elements of the cylinder be projected upon this circular section in d, d', d''. Let ACB, the primitive angle of torsion, be called A, and let this angle, after the time t, become ACb, so that it has been diminished by the angle BCb = M; then ACb = A - M = the angle of torsion after the time t.

Since the force of torsion is supposed to be proportional to the angle of torsion, the momentum of the force of torsion must be some multiple of that angle, or n × A - M, n being a constant co-efficient, whose value depends on the nature, length, and thickness of the metallic wire. If, therefore, we call v the velocity of any point d at the end of the time t, when the angle of torsion becomes ACb, and r = Cd the distance of the point d from the axis of rotation C, we shall have, by the principles of dynamics,

\[ n \times A - M \times i = \int dr \cdot \]

But if CD, the radius of the cylinder, be equal a, and if u be the velocity of the point D after the time t, we have evidently \( v = \frac{ru}{a} \). Now, by substituting the fluxion of this value of v in the place of v in the preceding formula, we have

\[ n \times A - M \times i = \int \frac{dr}{a} \]

and since \( i = \frac{aM}{u} \), we have, by substitution,

\[ n \times A - M \times \frac{aM}{u} = \int \frac{dr}{a} \]

whose fluent is

\[ n \times \sqrt{2AM - M^2} = u \int \frac{dr}{a} \]

Taking the square root of both sides of the equation, we have

\[ \sqrt{n} \times \sqrt{2AM - M^2} = u \int \frac{dr}{a} \]

Multiplying both sides by \( \frac{aM}{u} \), and dividing by \( \sqrt{n} \times \sqrt{2AM - M^2} \), the equation becomes

\[ \frac{aM}{u} \times u \int \frac{dr}{a^2} = \frac{aM}{u} \times \int \frac{dr}{a^2} \]

\[ = \frac{M}{\sqrt{n}} \times \sqrt{2AM - M^2} \]

Therefore, since \( i = \frac{aM}{u} \), we shall have

\[ i = \frac{M}{\sqrt{n}} \times \sqrt{2AM - M^2} \]

or

\[ i = \frac{M}{\sqrt{n}} \times \sqrt{2AM - M^2} \]

But \( \frac{M}{\sqrt{2AM - M^2}} \) represents an arch or angle whose radius is A, and whose versed sine is M, which arch vanishes when M = 0, and which becomes equal to 90° when M = A. Therefore the time of a complete oscillation will be

\[ T = \int \frac{pr^2}{n} \times 180° \]

In order to compare the force of torsion with the force of gravity in a pendulum, we have for the time T of a complete oscillation of a pendulum whose length is l, g being the force of gravity,

\[ T = \frac{l}{g} \times 180° \]

Therefore, since the time in which the cylinder oscillates must be equal to the time in which the pendulum oscillates, we have

\[ \int \frac{pr^2}{n} \times 180° = \frac{l}{g} \times 180° \]

Hence, dividing by 180°, and squaring both sides, we obtain

\[ \int \frac{pr^2}{n} = \frac{l}{g} \]

We must therefore find for a cylinder the value of \( \int pr^2 \), or the sum of all the particles multiplied by the squares of their distances from the axis. Now, if we make \( \pi = 6.28318 \) the ratio of the circumference of a circle to its radius, \( a = \) radius of the cylinder, \( \lambda = \) its length, \( d = \) its density; then we shall have for the area of its base \( \frac{\pi a^2}{4} \), which, multiplied by \( \lambda \), gives the solid content of the cylinder \( = \frac{\pi a^2 \lambda}{2} \), and this multiplied by \( d \) gives \( \frac{\pi a^2 \lambda d}{2} \) for the sum of all its particles. But as this is to be multiplied by the sum of the squares of all the distances of the particles from the centre C, we shall have \( \int pr^2 = \frac{\pi a^2 \lambda d}{4} \).

But the number of particles in the cylinder, or the mass \( \mu \) of the cylinder, is \( \frac{\pi a^2 \lambda d}{2} \), therefore, substituting \( \mu \) instead of this value of it in the preceding equation, we have

\[ \int pr^2 = \frac{\mu a^2}{2n} \]

and, dividing both sides by \( n \), we have

\[ \int pr^2 = \frac{\mu a^2}{2n} \]

and, extracting the square root, and multiplying by 180, it becomes

\[ \int \frac{pr^2}{n} \times 180° = \frac{\mu a^2}{2n} \times 180° \]

Therefore

\[ T = \frac{\mu a^2}{2n} \times 180° \]

and since \( \int pr^2 = \frac{l}{g} \), \( \frac{\mu a^2}{2n} = \frac{l}{g} \), and, by reduction, \( n = \frac{g \mu a^2}{2l} \). But \( g \mu \) is the weight W of the cylinder, there- fore, by substituting \( W \) instead of \( gu \), we obtain \( n = \frac{Pa}{2t} \), a very simple formula for determining the value of \( n \) from experiments.

If it were required to find a weight \( Q \), which, acting at the extremity of a lever \( L \), would have a momentum equal to the momentum of the force of torsion when the angle of torsion is \( A - M \), we must make \( Q \times L = n \times A - M \).

In the preceding investigation we have supposed, what is conformable to experiment, that the force of torsion is proportional to the angle of torsion, which gives us \( n \times A - M \) for the momentum of that force. Let us now suppose that this momentum is altered by any quantity \( S \), then the momentum of the force of torsion will become \( n \times A - M - S \) and the general equation will assume this form,

\[ n \times A - M - S \times i = u \int \frac{pr^2}{a} \]

and, by substituting in place of \( i \) its value \( \frac{aM}{u} \), and taking the fluent, we have

\[ n \times 2AM - M^2 - 2S \int SM = u^2 \int \frac{dr^2}{a^2}. \]

Now, in order to find the value of \( T \) or a complete oscillation, we must divide the oscillation into two parts, the first from \( B \) to \( A \), where the force of torsion accelerates the velocity \( u \), while the retarding force, arising from the resistance of the air and the imperfection of elasticity, diminishes the velocity \( u \); and the second from \( A \) to \( B' \), where the force of torsion, as well as the other forces, concur in diminishing \( u \) or retarding the motion.

Ex. 1. If \( S = m \times A - M' \), we shall have for the state of motion in the first portion \( BA \)

\[ n \times 2AM - M^2 + \frac{2M \times A - M'}{\nu + 1} = \frac{2mA + 1}{\nu + 1} = u \int \frac{pr^2}{a^2}, \]

Hence, when the angle of torsion becomes equal to nothing, or \( A - M = 0 \), we have

\[ nA^2 = \frac{2mA + 1}{\nu + 1} = UU \int \frac{pr^2}{a^2}, \]

which, dividing by \( \int \frac{pr^2}{a^2} \), becomes

\[ U^2 = \frac{2mA + 1}{\nu + 1}. \]

Let us now consider the other part of the motion from \( A \) to \( B' \), and suppose the angle \( AC' = M' \), we shall find, by calling \( U \) the velocity of the point \( A \),

\[ \frac{nM^2}{2} + \frac{mM + 1}{\nu + 1} = \frac{U^2 - u^2}{2} \times \int \frac{pr^2}{a^2}. \]

Then, by substituting instead of \( U \) its value as lately found, and taking the fluents, we shall have, when the velocity vanishes, or when the oscillation is finished,

\[ A - M' = \frac{2m}{n \times \nu + 2} \times \frac{A + 1 + M + 1}{A + M}; \]

and if the retarding forces are such that at each oscillation the amplitude is a little diminished, we shall have for the approximate value of \( A - M' \)

\[ A - M' = \frac{2mA}{n \times \nu + 1}, \]

and if the angle \( A - M' \) is so small that it may be treated as a common fluxional quantity, we shall then have for any number of oscillations

\[ N \times \frac{2m}{n \times \nu + 1} = \frac{1}{\nu - 1} \times \frac{1}{M - 1} - \frac{1}{A - 1}, \]

where \( M \) represents the angle to which \( A \) becomes equal after any number of oscillations \( N \). Hence we obtain

\[ M = \frac{1}{\left( N \times \frac{2m}{n \times \nu + 1} + \frac{1}{A - 1} \right) \times \frac{1}{\nu - 1}}, \]

which determines the value of \( M \) after any number of oscillations \( N \).

Ex. 2. If \( S = m \times A - M' + m' \times A - M' \), \( m' \) and \( \nu \) being different values of \( m \) and \( \nu \), we shall obtain, by following the mode of investigation in the last example,

\[ n \times A - M = \frac{2m}{\nu + 1} \times \frac{A + 1 + M + 1}{A + M} \times \frac{2m'}{\nu + 1} \times \frac{A' + 1 + M' + 1}{A' + M}; \]

and if the retarding force is much less than the force of torsion, we shall have for an approximate value of

\[ n \times A - M = \frac{2mA}{\nu + 1} + \frac{2mA'}{\nu + 1}. \]

Ex. 3. In general, if \( S = m \times A - M' + m' \times A - M' + m'' \times A - M'' + m''' \times A - M''' \), &c., we shall always have for an oscillation, when \( S \) is smaller than the force of torsion,

\[ n \times A - M = \frac{2mA}{\nu + 1} + \frac{2mA'}{\nu + 1} + \frac{2mA''}{\nu + 1} + \frac{2mA'''}{\nu + 1}, &c. \]

Having thus given, after Coulomb, the mode of deducing formulæ for the oscillatory motion of the cylinder, we shall proceed to give an account of the results of his experiments.

In these experiments M. Coulomb employed the torsion balance represented in fig. 101, in which he suspended cylinders of different weights from iron and brass wires of different lengths and thicknesses; and, by observing carefully the duration of a certain number of oscillations, he was enabled to determine, by means of the preceding formulæ, the laws of the force of torsion relative to the length, the thickness, and the nature of the wires employed. If the elasticity of the metallic wires had been perfect, and if the air opposed no resistance to the oscillating cylinder, it would continue to oscillate till its motion was stopped. The diminution of the amplitudes of the oscillations, therefore, being produced solely by the imperfection of elasticity, and by the resistance of the air, M. Coulomb was enabled, by observing the successive diminution of the amplitude of the oscillation, and by subtracting the part of the change which was due to the resistance of the air, to ascertain, with the assistance of the preceding formulæ, according to what laws this elastic force of torsion was changed.

These experiments were made with iron and brass wires of the kind described in the following table:

| Nature of the Wires | No. of the Wires in Commerce | Weight under Six Feet of each Wire | Weight under which the Wires broke | |---------------------|-------------------------------|----------------------------------|----------------------------------| | Iron harp-sichord wires | 12 | 5 | 3 0 | | | 7 | 14 | 10 0 | | | 1 | 56 | 33 0 | | | 12 | 5 | 2 3 | | Brass wires | 7 | 18-5 | 14 0 | | | 1 | 66 | 22 0 |

These wires were fixed one after another on the torsion balance, shown in fig. 101, and were made to suspend cylinders nineteen lines in diameter, and of different weights. The circumstances under which the experiments were made, and the results which Coulomb obtained, are given in the On the Force of Torsion.

The angles of torsion were always so small that the particles of the twisted wires returned to their original state. This was not the case, however, at angles greater than those in the table.

| Nature of the Wires | No. of Wires in Commerce | Length of Wires, in Inches | Weight of Suspended Centre of Torsion | Limit of the Cylinder for Inconceivable Oscillations | Time of Twenty Successive Oscillations | |---------------------|--------------------------|---------------------------|--------------------------------------|---------------------------------------------------|----------------------------------| | Harpsichord iron wire | 12 | 9 | 0.5 | 180° | 120° | | | 7 | 9 | 2.0 | 109° | 42 | | | 7 | 9 | 2.0 | 109° | 45 | | | 12 | 9 | 2.0 | 360° | 220 | | | 12 | 9 | 2.0 | 360° | 442 | | Brass wires | 7 | 9 | 0.5 | 360° | 57 | | | 7 | 9 | 2.0 | 360° | 110 | | | 7 | 36 | 2.0 | 1080° | 222 | | | 1 | 9 | 2.0 | 50 | 32 |

As the twenty oscillations of the cylinder were in these experiments sensibly isochronous, Coulomb considered the supposition on which the preceding formulae are founded, as a fundamental law, that in all metallic wires, when the angles of torsion are not very great, the force of torsion is sensibly proportional to the angle of torsion; and hence the formulae may be safely used in the discussion of the experiments.

It appears from the first and second, third and fourth, seventh and eighth, and ninth and tenth experiments, that when the suspended weight was two pounds, the oscillations were performed in the times

\[242^\circ, 85^\circ, 442^\circ,\] and \(110^\circ\),

while with weights of half a pound the times were

\[120^\circ, 43^\circ, 220^\circ,\] and \(57^\circ\),

which are nearly one half of the former. But \(1 : 2 = \sqrt{\frac{1}{2}} : \sqrt{\frac{1}{4}}\). Consequently the durations of the oscillations are proportional to the square roots of the suspended cylinders.

Hence it appears that the different degrees of tension produced by the different weights used in these experiments has no sensible influence on the re-action of the force of torsion.

If we vary the lengths of the wires without changing their diameters, or the suspending weights, as in experiments ninth and tenth, where the lengths are as 9 to 36, or 1 : 4, and the times 110° and 222°, or as 1 : 2, it follows that the times of the same number of oscillations are as the square roots of the lengths of the wires, as the theory indicates.

When the diameters of the wires are varied without changing their lengths, or the weight of the cylinders, the momentum of the force of torsion varies as the fourth power of the diameters of the wires. Now this result is perfectly conformable to theory; for if we suppose two wires of the same substance and of the same length, but having their diameters as one to two, it is obvious, that in the wire whose diameter is double that of the other, there are four times as many parts extended by torsion as in the smaller wire, and that the mean extension of all these parts will be proportional to the diameter of a wire, the same as the mean arm of a lever is relative to the axis of rotation. Hence it appears that, according to theory, the force of torsion of two wires of the same nature and of the same length, but of different diameters, is proportional to the fourth power of their diameter.

From this it follows in general, that in metallic wires the momentum of torsion is directly in the compound ratio of the angle of torsion and the fourth power of their diameter, and inversely as the length of the wires. If \(a\), therefore, be the angle of torsion, \(l\) the length of the thread, \(d\) its diameter, and \(F\) the force of torsion, we shall have

\[F = \frac{m ad^4}{l},\]

where \(m\) is a constant co-efficient for wires of the same metal, depending on the tenacity of the metal, and deducible from experiment.

When the angle of torsion is not great relative to the length of the wire, the index of the cylinder returns to the position which it had before the torsion took place, or, in other words, the wire untwists itself by the same quantity by which it had been twisted. But when the angle of torsion is very great, the wire does not completely untwist itself, and therefore the centre of torsion will have advanced by a quantity equal to that which it has not untwisted. If the angle of torsion, for example, is \(180^\circ\), and if after the oscillations have ceased the index returns only \(170^\circ\) in place of \(180^\circ\), the centre of torsion is said to have advanced \(10^\circ\), or to be displaced by that quantity.

In his experiments on this subject, M. Coulomb used the iron wire No. 1, \(6\frac{1}{2}\) inches long, and supporting a weight of two pounds. The following were his results:

| Angles of Torsion | No. of Oscillations | Degrees Lost | |-------------------|--------------------|-------------| | 90° | 3 | 10° | | 45° | 10 | 10 | | 22° | 23 | 10 | | 14° | 46 | 10 |

When the wire was twisted through a greater angle than \(90^\circ\), the centre of torsion suffered a displacement which increased with the angle of torsion, as shown in the following table:

| Angles of Torsion | Angle through which the Wire untwists itself | Successive Displacement of the Centre of Torsion | Total Displacement | |-------------------|---------------------------------------------|-------------------------------------------------|-------------------| | 1 circle | 172° | 8° | 8° | | 1 circle | 310° | 50° | 58° | | 2 | 410° | 310° | 1 circle + 8° | | 3 | 420° | 1 circle + 309° | 2 ... + 309° | | 4 | 430° | 2 ... + 290° | 5 ... + 238° | | 5 | 440° | 3 ... + 230° | 9 ... + 138° | | 6 | 460° | 4 ... + 260° | 14 ... + 58° | | 10 | 490° | 8 ... + 240° | 22 ... + 293° |

14 wire split longitudinally.

In this table the first column contains the angle through which the index has been turned, the second shows the angle through which the wire has untwisted itself. The differences between the first column and the second constitute the third column, which shows the angle which the wire wanted of returning to its primitive state, or the displacement of the centre of torsion at each successive experiment. The fourth column shows the total displacement of the centre of torsion, and is obtained by adding any one displacement in column 3 to all that precede it.

From these experiments it appears that for angles of torsion below \(45^\circ\), the decrements of the amplitudes of the oscillations were nearly proportional to the amplitudes of the angles of torsion, and that when the angles exceeded \(45^\circ\), the decrements increased in a much greater ratio. From the results in the second table, it appears that the centre of torsion did not begin to advance or be displaced till the angle of torsion was nearly \(180^\circ\); that this displacement increases in proportion as the thread is twisted; that it is irregular till the angle of torsion is \(310^\circ\); and that for greater angles the re-action of torsion remained nearly the same for all angles of torsion.

From these and other experiments, which our limits will not permit us to give, Coulomb has deduced the following theory of elasticity and cohesion in metals.

In all metallic wires the forces necessary for compressing or dilating the integrant particles are proportional to the compressions or the dilatations they experience, or their elasticity is perfect. Cohesion, however, which is different from elasticity, unites these particles together. In small torsions, when the twisting force is less than that of cohesion, the integrant particles are elongated or compressed without any change of place in the points by which they adhere. But when the torsions are so great that the force of compression or dilatation is equal to the cohesion of the particles, these particles ought to separate, or slide upon one another. This sliding of the particles takes place in all ductile bodies; but if by this cause the body is compressed, the extent of the points of contact and the extent of the field of elasticity become greater. As the particles have a given figure, the extent of the points of contact cannot augment but to a certain degree, at which the body breaks.

In confirmation of this theory, Coulomb made the following experiment, in which he varied the cohesion without changing the elasticity. A copper wire which broke with 22 lbs. was brought to the temper of a white heat, which reduced its cohesion so much that it could scarcely support 12 or 14 lbs. The two wires, however, when twisted through the same angle by the same weight, performed the same number of oscillations in the same time, though the cohesion was only half as great in the one wire as in the other, and the amplitude of elasticity diminished in the same proportion.

Coulomb confirmed these views by some ingenious experiments on the flexion of steel plates in different states with respect to cohesion, and he extends his views of the constitution of metals to all bodies whatever. Their ingranant particles have always a perfect elasticity; but bodies are hard, soft, or fluid, according to the cohesion of these particles. If they can slide upon one another, as in hard bodies, without any sensible change of distance, the body will be ductile or malleable; but if they cannot thus slide without their distance being sensibly altered, the body will break when the compressing or dilating force is equal to the cohesion.

The resistance of solid bodies had never undergone that careful investigation which its importance demanded. Mr. Bevan, however, has supplied in a great measure this defect, by a very complete series of frequently-repeated experiments on woods and metals. The following table contains the results of his experiments.

Table showing the Modulus of Torsion in Woods.

| Species of Wood | Specific Gravity of Wood | Modulus of Torsion | Remarks | |-----------------|--------------------------|--------------------|---------| | Pear | .72 | 18115 | | | Pine, St Petersberg | | 10500 Fresh | | | Memel | | 13900 Four or five years old | | | American | | 15600 | | | Plane | .59 | 17617 | | | Plum | .79 | 23700 | | | Poplar | .333 | 9473 | | | Satin wood | 1.02 | 39000 | | | Sallow | | 16600 | | | Sycamore | | 22200 | | | Teak | African | 16390 Old, and partially decayed | | | Walnut | | 27300 | | | | | 19754 | |

Mr. Bevan has given the following formula for finding the deflection D, or the quantity of twisting, in inches and decimals, viz. \( D = \frac{r^2lw}{d^2T} \), \( l \) being the length of a prismatic shaft strained by a given weight or force \( w \) in pounds avoirdupois, acting at right angles to the axis of the prism, and by a given leverage \( r \), the side of the square shaft being \( d \), and \( T \) being the modulus of torsion as given in the preceding table, \( h, r, D, \) and \( d \) being all in inches and decimals.

If the section of the prism is a rectangle whose breadth is \( b \) and depth \( d \), then the formula becomes \( D = \frac{(d+b)r^2lw}{2bd^2T} \).

If an angular measure of the torsion is required, let \( \Delta \) be the number of degrees, \( \varphi = 57.29578 \), then \( \Delta = \frac{r^2lw}{d^2T} \); or if \( T = t \), then \( \Delta = \frac{r^2lw}{d^2t} \). Thus, for wrought iron and steel,

\[ \Delta = \frac{r^2lw}{31000d^2}, \text{ and for cast iron, } \Delta = \frac{r^2lw}{16600d^2}. \]

In exemplifying the application of the first formula, Mr. Bevan takes a square shaft of English oak fifty inches long and six inches by six inches, subject to a strain of 3000 lbs. at the circumference of a wheel two feet in diameter, or having a leverage of twelve inches; then \( r = 12, l = 50, w = 3000, d = 6, \) and \( T = 20000 \); then we shall have \( D = 0.83 \), or about \( \frac{1}{12} \)th of an inch. If the weight or force \( w \) were 300 lbs., then, as the deflection is directly as the force, the deflection will be \( \frac{1}{12} \)th of an inch.

Mr Bevan observed, in a great many of his experiments, that the modulus of torsion bears a close relation to the weight of the wood when dry, whatever be the kind of wood employed; and he has given the following formula for determining the deflection \( D \) from the specific gravity

\[ D = \frac{r^2lw}{30000d^2}. \]

The amount of deflection thus obtained he considers sufficiently near the truth for practical purposes.

The following are Mr Bevan's results for metals:

Table of the Modulus of Torsion for Metals.

| Specific Gravity | Modulus of Torsion | |------------------|--------------------| | Iron, wrought, English | 1810000 | | Iron, wrought, English | 1740000 | | Iron, thin hooping | 1916000 | | Steel | 1984000 | | Steel | 1648000 | | Steel | 1618000 | | Iron, cylindrical | 1910000 | | Iron, cylindrical | 1700000 |

---

1 Phil. Trans. 1829, p. 127-132; or Edinburgh Journal of Science, vol. i. p. 340, New Series.

Specific Gravity. Modulus of Torsion.

Iron, square ........................................... 1617000 Iron, square ........................................... 1667000 Iron, square ........................................... 1951000

Mean of iron and steel .................................. 1779000 Cast iron .................................................. 940000 Cast iron .................................................. 963000 Cast iron .................................................. 952000

Mean of cast iron ....................................... 7163 951600 Bell metal .................................................. 8531 818000

By comparing the numbers in the preceding table with the modulus of elasticity of the same metals, Mr Bevan has found that the modulus of torsion is one sixteenth of the modulus of elasticity in metals.

The theory of torsion is particularly useful in delicate researches, where small forces are to be ascertained with a precision which cannot be obtained by ordinary means. It has been successfully employed by Coulomb in discovering the laws of the forces of electricity and magnetism, and in determining the resistance of fluids when the velocities are very small. Its application to these purposes will be found in our articles on ELECTRICITY, HYDRODYNAMICS, and MAGNETISM.

PART II.—ON THE CONSTRUCTION OF MACHINERY.

We have already stated, when considering the maximum effects of machines, the various causes which affect their performance. It appeared from that investigation, that there must be a certain relation between the velocities of the impelled and working points of a machine, or between the power and the resistance to be overcome, before it can produce a maximum effect, and therefore it must be the first object of the engineer to ascertain that velocity, and to employ it in the construction of his machine. The performance of the machine is also influenced by the friction and inertia of its various parts; and as both these act as resistances, and therefore destroy a considerable portion of the impelling power, it becomes an object of great importance to attend to the simplification of the machinery, and to ascertain the nature of friction, so as to diminish its effect, either by the application of unguents or by mechanical contrivances. Since the impelled and working points of a machine are generally connected by means of toothed wheels, the teeth must be formed in such a manner that the wheels may always act upon each other with the same force, otherwise the velocity of the machine will be variable, and its structure soon injured by the irregularity of its motion. The irregular motion of machines sometimes arises from the nature of the machinery, from an inequality in the resistance to be overcome, and from the nature of the impelling power. In large machines, the momenta of their parts are generally sufficient to equalize these irregularities; but in machines of a small size, and in those where the irregularities are considerable, we must employ fly-wheels for regulating and rendering uniform their variable movements. These various subjects, and others intimately connected with them, we shall now proceed to discuss in their order.

CHAP. I.—ON THE PROPORTION BETWEEN THE VELOCITY OF THE IMPELLED AND WORKING POINTS OF MACHINES, AND BETWEEN THE POWER AND RESISTANCE, IN ORDER THAT THEY MAY PERFORM THE GREATEST WORK.

In the chapter on the maximum effect of machines we have deduced formulae containing \( x \) and \( y \), the velocities of the impelled and working points of the machines, and including every circumstance which can affect their motion. The formula which exhibits the value of \( y \), or the velocity of the working point, assumes various forms, according as we neglect one or more of the elements of which it is composed. When the work to be performed resists only by its inertia, which is the case in urging round a millstone or heavy fly, the quantity \( R \) may be neglected, and the second formula (page 393, col. 2) should be employed. In small machines, and particularly in those where the motion is conveyed by wheels with epicycloidal teeth, the friction is very trifling, and the element \( p \) may be safely omitted. In corn and saw mills, the quantity \( b \), or the inertia of the resistance, may be left out of the formula, as the motion communicated to the flour or to the saw-dust is too small to be subjected to computation. In machines where one heavy body is employed to raise another merely by its weight, the inertia of the power and the resistance, viz. \( a, b \), are proportional to \( P, R \), the powers and resistances themselves, and consequently \( P, R \) may be substituted in the formula in the place of \( a, b \). The engineer therefore must consider, before he construct his machine, what elements should enter into the formula, and what should be omitted, in order that he may adapt it to the circumstances of the case, and obtain from his machine the greatest possible effect.

When the inertia of the power and that of the resistance are proportional to the power and resistance themselves, and when the inertia and friction of the machine may be omitted, the formula becomes \( y = \sqrt{\frac{P}{R}} + 1 - 1 \); from which the following table is computed, which contains the values of \( y \) for different values of \( P; R \) being supposed \( m = 10 \), and \( m = 1 \).

| Proportional Value of the Impelling Power, or \( P \) | Value of the Velocities of the Working Point, or \( y \); or of the Lever by which the resistance acts, that of \( x \) being 1. | Proportional Value of the Impelling Power, or \( P \) | Value of the Velocities of the Working Point, or \( y \); or of the Lever by which the resistance acts, that of \( x \) being 1. | |---|---|---|---| | 1 | 0-048809 | 20 | 0-732051 | | 2 | 0-095445 | 21 | 0-760682 | | 3 | 0-140175 | 22 | 0-788854 | | 4 | 0-183216 | 23 | 0-816590 | | 5 | 0-224745 | 24 | 0-843900 | | 6 | 0-264911 | 25 | 0-870800 | | 7 | 0-303841 | 26 | 0-897300 | | 8 | 0-341641 | 27 | 0-923500 | | 9 | 0-378405 | 28 | 0-949400 | | 10 | 0-414214 | 29 | 0-974800 | | 11 | 0-449138 | 30 | 1-000000 | | 12 | 0-483240 | 40 | 1-236200 | | 13 | 0-516575 | 50 | 1-449500 | | 14 | 0-549193 | 60 | 1-645700 | | 15 | 0-581139 | 70 | 1-828400 | | 16 | 0-612451 | 80 | 2-000000 | | 17 | 0-643168 | 90 | 2-162300 | | 18 | 0-673320 | 100 | 2-316600 | | 19 | 0-702938 | | | In order to explain the use of this table, let us suppose that it is required to raise one cubic foot of water in a second, by means of a stream which discharges three cubic feet of water in a second; and let it be required to find the construction of a wheel and axle for performing his work; that is, the diameter of the axle, that of the wheel being 6. Here the power is evidently three cubic feet, while the resistance is only one cubic foot, therefore \( P = 3 R \); but in the preceding table \( R = 10 \), consequently \( P = 3 \times 10 = 30 \). But it appears from the table that when \( P = 30 \), \( y \) or the diameter of the axle is 1, upon the supposition that the diameter \( x \) of the wheel is 1; but as \( x \) must be 6, we shall have \( y = 6 \).

\[ \frac{P_x R_y - R^2 y^2}{P_x^2 + R_y^2} = \frac{3 \times 6 \times 1 \times 5 - 1 \times 5 \times 5}{3 \times 6 \times 6 + 1 \times 5 \times 5} = \frac{65}{133} = 0.488; \]

When \( y = 5 \),

\[ \frac{P_x R_y - R^2 y^2}{P_x^2 + R_y^2} = \frac{3 \times 6 \times 1 \times 6 - 1 \times 6 \times 6}{3 \times 6 \times 6 + 1 \times 6 \times 6} = \frac{72}{144} = 0.500; \]

When \( y = 6 \),

\[ \frac{P_x R_y - R^2 y^2}{P_x^2 + R_y^2} = \frac{3 \times 6 \times 1 \times 7 - 1 \times 7 \times 7}{3 \times 6 \times 6 + 1 \times 7 \times 7} = \frac{77}{157} = 0.49045. \]

It appears, therefore, that when \( y = 5, 6, 7 \), the work performed is 0.488, 0.500, 0.49045; so that the effect is a maximum when \( y = 6 \), a result similar to what was obtained from the table.

When the machine is already constructed, \( x \) and \( y \) cannot be varied so as to obtain a maximum effect. The same object, however, will be gained by properly adjusting the power to the work when the work cannot be altered, or the work to the power when the power is determinate. The formulae in Prop. II. Chap. VII. exhibit the values of \( R \) under many circumstances, and it depends on the judgment of the engineer to select such of them as are adapted to all the conditions of the case.

The following table is founded on the formula

\[ R = \sqrt{\frac{y + 1}{y^2}} - 1, \]

which answers to the case where the inertia of the impelling power is the same with its pressure, and where the inertia and the friction of the machine may be safely neglected. The second column contains the different values of \( R \) corresponding to the values of \( y \) in the first column. The numbers in the third column show the ratio of \( y \) to \( R \), or they have the same proportion to 1 which \( R \) has to the resistance which will balance \( P \). In the table it is supposed that \( P = 1 \) and \( x = 1 \).

### Table containing the best Proportions between the Power and the Resistance, the Inertia of the Impelling Power being the same with its Pressure, and the Friction and Inertia of the Machine being omitted.

| Values of \( y \), or the Velocity of the Working Point, \( x \) being equal to 1. | Values of \( R \), or the resistance to be overcome, \( P \) being = 1. | Ratio of \( R \) to the resistance which would balance \( P \). | Values of \( y \), or the Velocity of the Working Point, \( x \) being equal to 1. | Values of \( R \), or the resistance to be overcome, \( P \) being = 1. | Ratio of \( R \) to the resistance which would balance \( P \). | |---|---|---|---|---|---| | 1 | 1-8885 | 0-4724 to 1 | 7 | 0-03731 | 0-26117 to 1 | | 2 | 1-3928 | 0-4639 ... | 8 | 0-03125 | 0-25000 ... | | 3 | 0-8986 | 0-4493 ... | 9 | 0-02669 | 0-24021 ... | | 4 | 0-4142 | 0-4142 ... | 10 | 0-02317 | 0-23170 ... | | 5 | 0-1830 | 0-3660 ... | 11 | 0-02037 | 0-22407 ... | | 6 | 0-1111 | 0-3333 ... | 12 | 0-01809 | 0-21708 ... | | 7 | 0-0772 | 0-3088 ... | 13 | 0-01622 | 0-21086 ... | | 8 | 0-0580 | 0-2900 ... | 14 | 0-01466 | 0-20524 ... | | 9 | 0-0457 | 0-2742 ... | 15 | 0-01333 | 0-19995 ... |

To exemplify the use of the preceding table, let us suppose that we are to raise water by means of a simple pulley and bucket, with a power = 10, and that it is required to find the resistance \( R \), or the quantity of water which must be put into the bucket, in order that the work performed may be a maximum. In the simple pulley \( x, y \), the arms of the vertical levers, or the velocities of the impelled and working points, are equal; and since \( x \) is supposed in the table to be = 1, we have \( y = 1 \), which corresponds in the table with 0-4142, the value of \( R \), \( P \) being = 1 in the table: but in the present case \( P = 10 \), therefore \( 10 : 1 = 0-4142 : 4-142 \), the value of \( R \) when \( P = 10 \).

The same result might be obtained in a more circuituous method by means of the formula

\[ \frac{P_x R_y - R^2 y^2}{P_x^2 + R_y^2}, \]

which expresses the performance of the machine. Thus, let \( x = 1 \); \( y = 1 \); \( P = 10 \), and let us suppose \( R \) successively equal to 3, 4, 4-142, 5; so that we may determine which of these values gives the greatest performance.

When \( R = 3 \), the preceding formula becomes

\[ \frac{10 \times 3 - 3 \times 3}{10 + 3} = \frac{21}{13} = 1.6154. \]

When \( R = 4 \), the formula becomes

\[ \frac{10 \times 4 - 4 \times 4}{10 + 4} = \frac{24}{14} = 1.7143. \] When $R = 4.142$, the formula becomes

$$\frac{10 \times 4.142 - 4.142^2}{10 \times 4.142} = \frac{24.26384}{14.142} = 1.7157.$$

When $R = 5$, the formula becomes

$$\frac{10 \times 5 - 5 \times 5}{10 + 5} = \frac{25}{15} = 1.6666.$$

Hence it appears, that when $R = 3, 4, 4.142, 5$, the work performed is respectively $= 1.6154, 1.7143, 1.7157, 1.6666$; so that the work performed is a maximum when $R$ is $= 4.142$, the same result which was obtained from the table.

**CHAP. II.—ON THE SIMPLIFICATION OF MACHINERY.**

As the inertia of every machine adds greatly to the resistance to be overcome, and as the friction of the communicating parts is proportional to the pressure, it becomes a matter of great practical importance, that the different parts of a machine should be proportioned to the strains to which they are exposed. If the beam of a steam-engine, for example, is larger than what is necessary, an immense portion of the impelling power must be destroyed at every stroke of the piston, by dragging the superfluous mass from a state of rest into motion; the pressure upon the gudgeons will also be increased, and their friction in their sockets proportionally enlarged. The engineer, therefore, should be well acquainted with the strength of the materials of which the machine is to be constructed, and should frame its different parts in such a manner that they may not be heavier than what is necessary for resisting the forces with which they are urged. When the motions of the machine are necessarily irregular, and when the machine may be exposed to accidental strains, the parts must be made considerably stronger than what is necessary for resisting its ordinary strains; but it is not often that such a precaution should be observed. The gudgeons of water-wheels, and of the beams of steam-engines, ought to be made as short and small as possible, as the friction increases with the rubbing surfaces. This is very seldom attended to in the construction of water-wheels. The diameter of the gudgeons is frequently thrice as large as what is necessary for supporting the weight of the wheel.

In the construction of machinery we must not only attend to the simplification of the parts, but also to the number of these parts, and the mode of connecting them. From the nature and quantity of the work to be performed, it is easy to ascertain the velocity of the working point which is most proper for performing it. Now this velocity may be procured in a variety of ways, either by a perplexing multiplicity of wheels, or by more simple combinations. The choice of these combinations must be left solely to the judgment of the engineer, as no general rules can be laid down to direct him. It may be useful, however, to remark, that the power should always be applied as near as possible to the working point of the machine, and that when one wheel drives another, the diameter of the one should never be great when the diameter of the other is very small. The size of wheels is often determined from the strains to which they are exposed. If, for example, we are obliged to give a certain velocity to an axle by means of a wheel with 120 teeth, and if the force with which this wheel is urged requires the teeth to be at least one inch thick in order to prevent them from breaking, we shall be obliged to make its diameter at least seven feet; for supposing the spaces between the teeth to be equal to the thickness of the teeth, the circumference of the wheel must at least be equal to $120 + 120 = 240$ inches, the sum of the teeth and their intervals, which gives a diameter of six feet eight inches. There are some cases where our choice of combination must be directed by the nature of the machinery. If the work to be performed is a load raised with a certain velocity by means of a rope winding round a hollow drum, and if the simplest combination of mechanical powers for producing this velocity should give a small diameter to the drum, then this combination must give way to another which corresponds with a larger size of the drum; for, on account of the inflexibility of the ropes, a great portion of the impelling power would be wasted in winding them about the circumference of a small drum.

The advantages of simplifying machinery are well exemplified in the following capstan, which unites great strength and simplicity. It is represented in fig. 103, where AD is a compound barrel composed of two cylinders of different radii. The rope DEC is fixed at the extremity of the cylinder D; and after passing over the pulley E, which is attached to the load by means of the hook F, it is coiled round the cylinder D, and fixed at its upper end. The capstan bar AB urges the compound barrel CD about its axis, so that while the rope coils round the cylinder D it unwinds itself from the cylinder C. Let us suppose that the diameter of the part D of the barrel is 21 inches, while the diameter of the part C is only 20 inches, and let the pulley E be 20 inches in diameter. When the barrel AD, therefore, has performed one complete revolution by the pressure exerted at B, 63 inches of rope, equal to the circumference of the cylinder, will be gathered upon the cylinder D, and 60 inches will be unwinded from the cylinder C. The quantity of wound rope, therefore, exceeds the quantity that is unwound by $63 - 60 = 3$ inches, the difference of their respective perimeters; and the half this quantity, or $\frac{1}{2}$ inch, will be the space through which the load or pulley E moves by one turn of the bar. If a simple capstan of the same dimensions had been employed, the length of rope coiled round the barrel would have been 60 inches; and the space described by the pulley, or load to be overcome, would have been 30 inches. Now, as the power is to the weight as the velocity of the weight is to the velocity of the power, and as the velocity of the power is the same in both capstans, the weights which they will raise will be as $1 \frac{1}{2}$ to 30. If it is wished to double the power of the machine, we have only to cover the cylinder C with lathes a quarter of an inch thick, so that the difference between the radii of each cylinder may be half as little as before; for it is obvious that the power of the capstan increases as the difference between the radii of the cylinders is diminished. As we increase the power, therefore, we increase the strength of our machine, while all other engines are proportionally enfeebled by an augmentation of power. Were we, for example, to increase the power of the common capstan, we must diminish the barrel in the same proportion, supposing the bar AB not to admit of being lengthened, which will not only diminish its strength, but destroy much of its power by the additional flexure of the rope. This capstan may be easily converted into a crane by giving the compound barrel a horizontal position, and substituting a winch instead of the bar AB. The superiority of such a crane... The above the common ones does not require to be pointed out; but it has this additional advantage, that it allows the weight to stop at any part of its progress, without the aid of a ratchet-wheel and catch, because the two parts of the rope pull on the contrary sides of the barrel. The rope indeed which coils round the larger part of the barrel acts with a larger lever, and consequently with greater force, than the other; but as this excess of force is not sufficient to overcome the friction of the machine, the weight will remain stationary in any part of its path.

The principle on which the preceding capstan is constructed might be applied with great advantage when two separate axles AB, CD are driven by means of the winch and the wheels B and A. It is evident that when the winch is turned round in one direction, the rope R is unwound from the axle BD; the wheel B drives the wheel A, so that the axle AC moves in a direction opposite to that of BD, and the rope is coiled round the axle AC. If the wheels A, B are of the same diameter and the same number of teeth, the weight will be stationary, as the rope wound about one axle will be always equal to what is unwound on the other. If the wheels have different diameters, or different numbers of teeth, the quantity of rope wound on the one axle will exceed what is unwound from the other, and the weight will be raised.

**AP. III.—ON THE LAWS OF FRICTION, THE FRICTION OF AXLES, THE METHOD OF DIMINISHING THE EFFECTS OF FRICTION IN MACHINERY; AND ON THE RIGIDITY OF ROPE.**

The friction generated in the communicating parts of machinery opposes such a resistance to the impelling power, and is so injurious to the machine itself, that an acquaintance with the nature and effects of this retarding force, and with the method of diminishing its effects on machinery, is of infinite importance to the practical mechanic.

The subject of friction has been examined at great length by Amontons, Bulfinger, Parent, Euler, and Bossut, and more recently occupied the attention of the late Mr. Prince of Cambridge.

He found that the friction of hard bodies in motion is an uniformly retarding force, and that the quantity of friction considered as equivalent to a weight drawing the body backwards is equal to \( M + \frac{W \times S}{gt} \), where \( M \) is the moving force expressed by its weight, \( W \) the weight of the body upon the horizontal plane, \( S \) the space through which the moving force or weight descended in the time \( t \), and \( g = 16.087 \) feet the force of gravity. Mr. Vince also found that the quantity of friction increases in a less ratio than the quantity of matter or weight of the body, and that the friction of a body does not continue the same when it is different surfaces applied to the plane on which it lies, but that the smallest surfaces will have the least friction.

Notwithstanding the attempts of preceding philosophers to unfold the nature of friction, it was reserved for the celebrated Coulomb to surmount the difficulties which are inseparable from such an investigation, and to give an accurate and satisfactory view of this difficult branch of mechanical philosophy. By employing large bodies, and conducting his experiments on a great scale, he has corrected several errors which arose from the limited experiments of others; he has brought to light many new and striking phenomena, and confirmed others which were hitherto but partially established. As it would be foreign to the nature of this work to follow this ingenious philosopher through his numerous and varied experiments, we shall only present the reader with a description of his apparatus, and an account of the interesting results which he obtained.

The apparatus used by Coulomb in his experiments on friction, consists of a solid table resting upon very strong legs. The plank cd, fig. 105, and d'd', fig. 106, which constitutes the table, is eight feet long, two wide, and three inches thick. Two pieces of oak AB, A'B', twelve feet long and eight inches thick, are placed on the table in the direction of its length, and at the distance of three inches from each other. A pulley H of hard wood, a foot in diameter, and of fourteen pounds weight, is fixed upon an axle of green oak ten lines in diameter, and moving in sockets at the ends B, B' of the two pieces of oak. A cord passing over this pulley carries at one end a scale P, and is fixed at the other to a sledge S, fig. 106, which runs upon the table. The scale P, for carrying weights, is capable of descending four feet into a pit cut beneath it. A small horizontal wheel and axle is placed at the other ends A, A' of the pieces of oak AB, A'B'. Above these pieces is fixed another plank ad'bd', eight feet long, sixteen inches wide, and three inches thick, having its upper surface smoothly planed. Sledges of the form shown in the annexed figures are made to slide upon this plank. The sledge ABDC, fig. 107, is a plank eighteen inches wide, but having its length variable; and there is nailed beneath it two pieces ACmm', BDnn', so that when this sledge is placed upon the fixed plank ad'bd', it may be retained by these pieces on... both sides with a play of two or three lines, so that it may move undisturbed in the direction of the plank. When the touching surfaces are required to be smaller, other pieces of different widths are nailed upon the plank ABDC, and their ends rounded for receiving the nuts, to prevent them from rubbing on the plank. To one of the hooks h is fixed the pulley cord, and to the other h the cord which goes round the wheel and axle already mentioned, and the use of which is to bring back the sledge to the side AA' of the apparatus. In some of the experiments a steel-yard was used, as shown in fig. 106.

1. The friction of homogeneous bodies, or bodies of the same kind, moving upon one another, is generally supposed to be greater than that of heterogeneous bodies; but Coulomb has shown that there are exceptions to this rule. He found, for example, that the co-efficient of friction of oak upon oak was equal to $\frac{1}{234}$ of the force of pressure, or $\frac{1}{239}$ when the surfaces were very small; the friction of pine against pine $\frac{1}{178}$, that of oak against pine $\frac{1}{15}$, and that of elm against elm $\frac{1}{218}$. With elm alone in small pressures the friction increases with the velocity. The friction of oak against copper was $\frac{1}{55}$, and that of oak against iron nearly the same.

2. It was generally supposed, that in the case of wood, the friction is greatest when the bodies are dragged contrary to the course of their fibres; but Coulomb has shown that the friction is in this case sometimes the smallest. When the bodies moved in the direction of their fibres, the friction in the case of oak upon oak was $\frac{1}{234}$ of the force with which they were pressed together; but when the motion was contrary to the courses of the fibres, the friction was only $\frac{1}{376}$.

3. The longer the rubbing surfaces remain in contact, the greater is their friction.—When wood was moved upon wood, according to the direction of the fibres, the friction was increased by keeping the surfaces in contact for a few seconds; and when the time was prolonged to a minute, the friction seemed to have reached its farthest limit. But when the motion was contrary to the course of the fibres, a greater time was necessary before the friction arrived at its maximum. When wood was moved upon metal, the friction did not attain its maximum till the surfaces continued in contact for five or six days; and it is very remarkable, that when wooden surfaces were anointed with tallow, the time requisite for producing the greatest quantity of friction was increased. The increase of friction which is generated by prolonging the time of contact is so great, that a body weighing 1650 pounds was moved with a force of 64 pounds when first laid upon its corresponding surface. After having remained in contact for the space of three seconds, it required 160 pounds to put it in motion; and, when the time was prolonged to six days, it could scarcely be moved with a force of 622 pounds. When the surfaces of metallic bodies were moved upon one another, the time of producing a maximum of friction was not changed by the interposition of olive oil; it was increased, however, by employing swine's grease as an unguent, and was prolonged to five or six days by besmearing the surfaces with tallow.

4. Friction is in general proportional to the force with which the rubbing surfaces are pressed together; and is, for the most part, equal to between $\frac{1}{2}$ and $\frac{1}{4}$ of that force.—In order to prove the first part of this proposition, Coulomb employed a large piece of wood, whose surface contained three square feet, and loaded it successively with 74 pounds, 874 pounds, and 2474 pounds. In these cases the friction was successively $\frac{1}{246}$, $\frac{1}{216}$, $\frac{1}{221}$, of the force of pressure; and when a less surface and other weights were used, the friction was $\frac{1}{236}$, $\frac{1}{242}$, $\frac{1}{240}$. Similar results were obtained in all Coulomb's experiments, even when metallic surfaces were employed. The second part of the proposition has also been established by Coulomb. He found that the greatest friction is engendered when oak moves upon pine, and that it amounts to $\frac{1}{175}$ of the force of pressure; on the contrary, when iron moves upon brass, the least friction is produced, and it amounts to $\frac{1}{4}$ of the force of pressure.

5. Friction is in general not increased by augmenting the rubbing surfaces.—When a superficies of three feet square was employed, the friction, with different weights, was $\frac{1}{228}$ at a medium; but when a small surface was used, the friction, instead of being greater, as might have been expected, was only $\frac{1}{239}$.

6. Friction for the most part is not augmented by an increase of velocity. In some cases, it is diminished by an augmentation of velocity.—M. Coulomb found, that when wood moved upon wood in the direction of the fibres, the friction was a constant quantity, however much the velocity was varied; but that when the surfaces were very small in respect to the force with which they were pressed, the friction was diminished by augmenting the rapidity; the friction, on the contrary, was increased when the surfaces were very large when compared with the force of pressure. When the wood was moved contrary to the direction of its fibres, the friction in every case remained the same. If wood be moved upon metals, the friction is greatly increased by an increase of velocity; and when metals move upon wood besmeared with tallow, the friction is still augmented by adding to the velocity. When metals move upon metals, the friction is always a constant quantity; but when heterogeneous substances are employed which are not bedaubed with tallow, the friction is so increased with the velocity, as to form an arithmetical progression when the velocities form a geometrical one.

7. The friction of loaded cylinders rolling upon a horizontal plane, is in the direct ratio of their weights, and the inverse ratio of their diameters.—In Coulomb's experiments, the friction of cylinders of guaiacum wood, which were two inches in diameter, and were loaded with 1000 pounds, was 18 pounds, or $\frac{1}{55}$ of the force of pressure. In cylinders of elm, the friction was greater by $\frac{3}{4}$, and was scarcely diminished by the interposition of tallow.

The following tables contain the most important results of Coulomb's researches, which we have arranged under three heads: 1. Woods upon Woods; 2. Metals upon Metals; and, 3. Metals upon Woods.

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1 The Abbé Bossut had anticipated this curious result, from theoretical considerations. I.—Friction of Woods upon Woods.

1. Friction of oak upon oak, when freshly greased with tallow at every experiment.

| Pressure | Friction in Parts of the Pressure | Pressure | Co-efficient of Friction | |----------|----------------------------------|----------|-------------------------| | 50 pounds | 7-7 | 850 pounds | 23-6 | | 250 pounds | 18-5 | 1650 pounds | 25-8 | | 50 pounds | 21-5 | 3250 pounds | 27-6 |

The increase of friction produced by a diminution of pressure is here very remarkable. Coulomb ascribes it to the cohesion of the parts of the tallow, the influence of which increases as the pressure is diminished. He considers five pounds as a measure of this cohesion, and deduces $\frac{1}{28-3}$ as the relation of the friction to the pressure when corrected for the cohesion of the tallow.

Friction of oak upon oak, greased with tallow, and depending on the time of contact. The area of the rubbing surface was 180 square inches.

| Pressure | Time of Contact | Co-efficient of Friction | |----------|-----------------|--------------------------| | 47 pounds | 0 minutes | 7-7 | | | 4 minutes | 5-87 | | | 2 hours | 5-25 | | | 0 seconds | 25-8 | | | 3 seconds | 10-3 | | | 15 seconds | 7-9 | | 1650 pounds | 1 minute | 5-82 | | | 4 minutes | 5-25 | | | 2 hours | 3-65 | | | 6 days | 2-65 | | | 0 seconds | 27-1 | | | 3 seconds | 10-16 | | | 15 seconds | 9-16 | | | 1 minute | 7-87 | | | 4 minutes | 5-48 | | | 1 hour | 3-7 | | | 2 hours | 3-53 | | | Five days | 2-66 |

The following results were obtained with still higher pressures, and with tallow that had been laid on and used for some time, without being renewed in the course of the experiments:

| Pressure | Time of Contact | Co-efficient of Friction | |----------|-----------------|--------------------------| | 2310 pounds | 0 minutes | 1 | | | 2 minutes | 12-3 | | | 1 hour | 5-9 | | | 16 hours | 5-12 | | | 0 minutes | 4-5 | | | 2 minutes | 7-35 | | | 4 minutes | 6-71 | | 5810 pounds | 9 minutes | 6-12 | | | 26 minutes | 5-6 | | | 1 hour | 4-9 | | | 16 hours | 3-8 |

3. Friction of oak upon oak, in the direction of the fibres, and when the surfaces are in motion, and without grease.

| Pressure | Co-efficient of Friction | |----------|--------------------------| | 25 pounds | 5-7 | | 188 pounds | 9-4 | | 291 pounds | 9-5 | | 1788 pounds | 9-2 | | 6588 pounds | 10-4 |

Surfaces reduced almost to lines:

| Pressure | Co-efficient of Friction | |----------|--------------------------| | 47 pounds | 10-4 | | 447 pounds | 12-4 | | 847 pounds | 14-6 |

When the velocities varied from nothing to four feet in four or five seconds, the results were not sensibly changed.

II.—Friction of Metals upon Metals.

1. Friction of iron upon iron, and upon brass, after a certain time of rest.

The rulers of iron used were four feet long and two inches wide, attached to the fixed plank of the apparatus. Other four rulers were employed, two of iron and two of brass, fifteen inches long and eighteen lines wide. angles of the rulers were rounded, and the rubbing surfaces were forty-five square inches.

| Pressure | Co-efficient of Friction | |----------|--------------------------| | Iron upon iron | 53 pounds | 1/3.5 | | Iron upon brass | 52 pounds | 1/4.2 |

The friction is here independent of the extent of the rubbing surfaces, and Coulomb found that it did not vary with the velocities. In the case of iron upon brass, when the surfaces are extremely small, the friction varies from \( \frac{1}{4.1} \) to \( \frac{1}{6.1} \); but it does not reach this last ratio till the friction has been continued above an hour, when the iron and brass have taken their highest polish, and become free of scratches.

2. Friction of iron upon brass when greased with tallow.

| Pressure | Time of Contact | Co-efficient of Friction | |----------|-----------------|--------------------------| | 50 pounds | 0 minutes | 1 | | | 4 minutes | 7.14 | | | 30 minutes | 7.14 | | Brass upon iron | 450 pounds | 4 minutes | 9.37 | | | 2 hours | 9.37 | | | 0 minutes | 11 | | | 3 minutes | 10.4 | | | 4 hours | 9.8 | | | 4 days | 9.8 |

The great degree of friction produced with the small pressure of fifty pounds, is owing to the cohesion of the tallow, which amounted to \( \frac{1}{11} \) pounds, which makes the friction \( \frac{1}{11} \) at the shortest time of contact, and \( \frac{1}{9.5} \) when it had reached its maximum.

2. Friction of iron upon iron, and upon brass, when highly polished.

After the surfaces were polished as highly as possible, and greased with oil or tallow, they were attached to the sledge, and made to rest upon one another under a great pressure for half an hour, the grease being from time to time renewed, till it had penetrated the pores of the metal, and in this manner communicated to the rulers a degree of polish which they could not have received from any other method. The size of the rulers was the same as in section 1, and the areas in contact the same. The velocity of the sledge was below an inch in a second. The friction was at first uncertain, but it became more regular as the polish increased.

In these experiments the friction does not obtain its maximum till after a long time.

2. Friction of brass upon oak.

The results were nearly the same as with iron upon oak. The friction, however, increased more slowly with the time, and at its maximum it was \( \frac{1}{5.5} \).

3. Friction of iron upon oak, the surfaces not being greased.

When the velocity was insensible, the results were,

| Pressure | Co-efficient of Friction | |----------|--------------------------| | 53 pounds | 1/118 | | 453 pounds | 1/129 | | 853 pounds | 1/127 | | 1653 pounds | 1/132 |

Iron upon oak

When the velocity was one foot per second, the results were,

| Pressure | Co-efficient of Friction | |----------|--------------------------| | 53 pounds | 1/59 | | 453 pounds | 1/58 | | 853 pounds | 1/55 | | 1653 pounds | 1/63 |

Hence it follows, 1st, that the friction is nearly constant whatever be the pressure, the velocity being the same; 2nd, that the friction increases greatly with the velocity; 3rd, that the frictions increase in an arithmetical progression when the velocities increase in a geometrical one. Coulomb likewise found that when the pressure and velocity were the same, the friction was the same for large and small surfaces.

4. Friction of iron and brass on oak, the surfaces being greased.

Friction is greatly diminished when the metals rub upon wood greased with fatty bodies, and small velocities may be in this case produced with less force than in any other kind of friction.

When the velocities are greater, the friction increases rapidly with the velocity; as in the case already stated when dry metals rub upon wood. With small velocities the following results were obtained:

| Pressure | Co-efficient of Friction | |----------|--------------------------| | Iron upon oak | 1/1550 pounds | 1/351 | | Brass upon oak | 1/1650 pounds | 1/471 |

These results are highly important in practice; but the grease must not be allowed to acquire any consistency, otherwise the friction increases very perceptibly. It must therefore be removed, and renewed when necessary. In order to obtain an estimate of the influence of this thickening of the grease, Coulomb caused the sledge, when carrying the plates of brass, to move fifteen times over the fixed oaken plank, without removing or renewing the tallow. The consequence of this was, that the friction was triple of that which was necessary to give the sledge an insensible velocity when the tallow coating was new. The velocity of the sledge diminished, or the friction increased, at every trial; and at the fifteenth trial the sledge ceased to move. Hence it follows that the tallow actually increases the friction, or is injurious, when the surfaces move a long time without a renewal of the grease.

5. Friction of iron upon oak greased, the surfaces being very small, and the friction across the grain of the wood.

The surfaces were covered with tallow, and afterwards wiped, in order merely to leave them greasy.

| Pressure | Co-efficient of Friction | |----------|--------------------------| | 47 pounds | 1/134 | | 447 pounds | 1/149 | | 1647 pounds | 1/143 |

The results were nearly the same when the surfaces were well tallowed. The ratio of the friction to the pressure is constant, and not much influenced by the velocity. This case is analogous to the case of axles of iron turning upon wood.

From the preceding experiments, Coulomb has drawn the following curious inferences respecting the structure of the surfaces of woods and metals.

He supposes that the fibres of wood are as shown in fig. 109, which represents one surface lying upon another.

When the fibres are bent by the application of a force to the upper surface, but before the fibres are detached from one another, he supposes the two surfaces to be like fig. 110.

When the surfaces are detached, and the upper surface is in motion over the under one, he supposes the surface to be as in fig. 111.

He considers the surfaces of metals to be more smooth than those of wood, and to have the undulated structure shown in fig. 112.

A series of very valuable experiments have recently been made upon friction by Mr George Rennie, under Rennie's pressures of thirty-six pounds to the square inch, and within the limits of abrasion of the softest substance. The following table contains a general abstract of results collected from the different tables. The co-efficient of friction is contained in the first column, and the limiting angle of resistance in the second. This angle has for its tangent the numbers in the first column, and is that angle at and beneath which the moving surface will not begin to slide.

Tabular View of Mr Rennie's Experiments on Friction.

| Names of the Substances | Co-efficient of Friction | Limiting Angle of Resistance | |-------------------------|--------------------------|-----------------------------| | Steel upon ice | 1/69-81 | 0° 49' | | Ice upon ice | 1/58-00 | 1° 35' | | Hard wood upon hard wood| 1/7-73 | 7° 22' |

1 Phil. Trans. 1829, p. 169. 2 Professor Moseley's Mechanics applied to the Arts, p. 47. In Professor Moseley's table, there are some oversights near the beginning, which we have corrected here. ### Names of the Substances

| Substance | Co-efficient of Friction | Limiting Angle of Resistance | |----------------------------|---------------------------|-----------------------------| | Brass upon wrought iron | 1 | 7° 43' | | Brass upon cast iron | 1 | 8° | | Brass upon steel | 1 | 7° 54' | | Soft steel upon soft steel | 1 | 8° 18' | | Cast iron upon steel | 1 | 8° 36' | | Wrought iron upon wrought iron | 1 | 9° 5 | | Cast iron upon cast iron | 1 | 9° 17' | | Hard brass upon cast iron | 1 | 9° 27' | | Cast iron upon wrought iron | 1 | 9° 40' | | Brass upon brass | 1 | 9° 57' | | Tin upon cast iron | 1 | 10° 8 | | Tin upon wrought iron | 1 | 10° 15' | | Soft steel upon wrought iron | 1 | 10° 53' | | Leather upon iron | 1 | 14° 21' | | Tin upon tin | 1 | 14° 49' | | Granite upon granite | 1 | 16° 52' | | Yellow deal upon yellow deal | 1 | 19° 9 | | Sandstone upon sandstone | 1 | 19° 59' | | Woollen cloth upon woollen cloth | 1 | 23° 30' |

Mr. Rennie has drawn the following conclusions from his experiments:

1. The laws which govern the retardation of bodies gliding over each other are as the nature of the bodies. 2. In fibrous substances, such as cloth, friction is increased by surface and time, and diminished by pressure and velocity. 3. In harder substances, such as woods, metals, and stones, and within the limits of abrasion, the friction is directly as the pressure, without regard to surface, time, or velocity. 4. With dissimilar substances, the friction will be determined by the limit of abrasion of the softer substance. 5. Friction is greatest with soft, and least with hard, substances. 6. The effect of unguents is as the nature of the unguents, without reference to the substances to which they are applied. 7. The very soft woods, stones, and metals, approximate to the laws which regulate the friction of fibrous substances.

In comparing his results with those of Coulomb, Mr. Rennie found that the differences relate chiefly to time, and he conceives that the small pressures used by the French philosopher (varying from one pound to forty-five pounds per square inch) account for these differences.

The selection of substances used by Mr. Rennie was made on the following grounds:

1. Ice, in reference to its resistance to sledges, skates, &c. 2. Leather, from its utility in the pistons of pumps. 3. Stones, in reference to their use in arches and buildings. 4. Cloth, from its remarkable properties of resistance in opposition to the law observed by solids. 5. Woods and metals, from their universal application to machinery, carriages, and railways.

Mr. Rennie's experiments on leather are very interesting. When nine square inches of leather, soaked in water, moved over a plate of iron, seven pounds barely kept it in motion fifty-six pounds after starting with the hand. After a contact of five minutes twenty-nine pounds were required to start it; twenty-eight pounds barely kept it in motion sixty-four pounds after starting it; and after being one minute in contact it took forty-two pounds to start it.

Where a surface of four and half inches area (one and a half by three) was used, six and a half pounds barely kept it in motion thirty-six pounds after starting it. After remaining five minutes it required twenty-one pounds to start it. Hence the friction of hide leather soaked in water is greatly increased by time and weight, a fact which explains the enormous friction evinced in the pistons of pumps when first put in motion.

When the hide leather moved dry over a surface of cast iron, the friction evinced from one fourth to nearly one sixth of the pressure, and was, ceteris paribus, diminished by a diminution of surface. The surfaces employed were two and a fourth, four and a half, six and three fourths, and nine square inches.

The following are the angles at which different stones commence gliding.

- Well-dressed stones: 28° to 36° Rondelet. - Granite voussoirs of the New London Bridge, well dressed, and without mortar: 33° to 34° Rennie. - With fresh and finely ground mortar: 25° to 26° ditto. - Sandstone voussoirs from Bramley, Fall, and Whitby: 35° to 36° - With mortar: 33° to 34°

According to Morisot, granite resists abrasion twelve times more than lias, while it possesses a repulsive power only three times greater. According to Boistard, the friction of hard calcareous stones is 0°78.

### On the Friction of Axes

In order to ascertain the friction experienced by the axes of machinery in motion, Coulomb used axes of iron moving in boxes of brass. The iron axis had a diameter of nineteen lines, and a play of one and three-fourths lines in the brass box. In order to give the rubbing surfaces the highest polish, the iron axis was made to work a considerable time in the brass box. The following is a brief abstract of the results which were obtained.

| Pressure on Axis | Kind of Greasing | Co-efficient of Friction | |------------------|------------------|--------------------------| | 226 lbs. | No greasing | 1 | | 424 lbs. | Ditto | 1 | | 825 lbs. | Ditto | 1 | | Pressure on Axis | Kind of Greasing | Co-efficient of Friction | |-----------------|------------------|--------------------------| | 216 lbs. | Greased with tallow | 1/123 | | 420 lbs. | Ditto | 1/116 | | 827 lbs. | Ditto | 1/115 | | 117 lbs. | With cart-grease | 1/67 | | 218 lbs. | Ditto | 1/84 | | 320 lbs. | Ditto | 1/8 | | 429 lbs. | Ditto | 1/85 | | 831 lbs. | Ditto | 1/83 |

The cart grease of the preceding experiments was wiped off, the pores of the metal remaining unctuous.

200 to 1200 lbs. ........................................... 1/79

The surface of the metals was now fresh greased with olive oil,

200 to 1200 lbs. ........................................... 1/75 to 1/79

When the greasing had not been renewed for a long time, though the machine had been much used,

200 to 1200 lbs. ........................................... 1/75

From these experiments it appears that tallow was much more efficacious than cart-grease in diminishing the friction.

When the axes and the boxes or cheeks were made of different kinds of wood, the following results were obtained. The axes were three inches in diameter, and sometimes moveable and sometimes fixed (in both which cases the friction was the same, and the touching surfaces were carefully smoothed).

| Nature of the Axis | Nature of the Box | Co-efficient of Friction | |--------------------|-------------------|---------------------------| | Iota oak running in a box | Lignum vitae | 1/263 | | Ditto, the coating of tallow wiped off, and the surfaces being greasy | ditto | 1/167 | | Ditto, after being used several times without renewing the tallow | ditto | 1/125 | | Ditto, coated with tallow, and running in a box of | Elm | 1/333 | | Ditto, both axis and box wiped, the surfaces being greasy | ditto | 1/20 | | Oak-wood coated with tallow, and running in a box of | Lignum vitae | 1/23 | | Ditto, both axis and box wiped, the surfaces being greasy | ditto | 1/143 | | Ditto, coated with tallow, and running in a box of | Elm | 1/286 | | Ditto, both axis and box wiped, and surfaces remaining greasy | ditto | 1/20 |

Axis of iron, coating of tallow wiped off, and the axle turned for some time.

Lignum vitae, 1/20

In all these experiments, the friction was least when the surfaces were merely greasy, and not coated with tallow. Unless in the first instants of rest, the velocity did not seem to affect the friction.

A series of valuable experiments on the friction of Mr G. axles, both with and without unguents, was made by Mr Rennie's George Rennie. The weights with which he loaded the experiments on axle were generally about ten or eleven hundredweight; the friction and in some cases he varied it down to one hundredweight of axles.

Without unguents the friction of gun-metal on cast iron, when loaded with weights varying from one to ten hundredweight, varied from 1/765 to 1/470, and it was scarcely affected by length of time.

With yellow brass on cast iron the friction was greater, viz. 1/411.

With cast iron on cast iron the friction was 1/6.

With black lead and cast-iron on cast iron, the friction was diminished to 1/765.

With black lead and gun-metal on cast iron the friction was 1/724.

With black lead and yellow brass on cast iron the friction varied from 1/759 to 1/680, as the pressure increased from one to eleven hundredweight.

With oil, gun-metal on cast iron gave the friction 1/503.

With oil, yellow brass on cast iron gave the friction as follows:

| Weight on Axle | Co-efficient of Friction | Weight on Axle | Co-efficient of Friction | |---------------|--------------------------|---------------|--------------------------| | ½ cwt | 1/3733 | 4 cwt | 1/1828 | | 1 cwt | 1/3200 | 5 cwt | 1/1914 | | 2 cwt | 1/3200 | 10 cwt | 1/578 | | 3 cwt | 1/2036 | 11 cwt | 1/613 |

This great increase of friction with the weight Mr Rennie ascribes to the oil "being less fluid and sensible in the one case, and more capable of preventing the contact of metals in the other."

With oil, cast iron on cast iron gave the friction 1/867.

With hog's lard, cast iron on cast iron gave the friction 1/955.

With hog's lard, yellow brass on cast iron, the results were as follows:

| Weight on Axle | Co-efficient of Friction | Weight on Axle | Co-efficient of Friction | |---------------|--------------------------|---------------|--------------------------| | ½ cwt | 1/3446 | 4 cwt | 1/1041 | | 1 cwt | 1/3657 | 5 cwt | 1/1178 | | 2 cwt | 1/2986 | 10 cwt | 1/929 | | 3 cwt | 1/1460 | | | With hog's lard, gun-metal on cast iron, the friction was 1/839 with a pressure of ten hundredweight.

With anti-attrition composition, yellow brass on cast iron, the results were as follows:

| Weight | Co-efficient of Friction | Weight | Co-efficient of Friction | |--------|--------------------------|--------|--------------------------| | 1 cwt. | 14·93 | 5 cwt. | 38·62 | | 2 cwt. | 24·88 | 10 cwt.| 58·9 | | 3 cwt. | 32·00 | 10 cwt.| 47·65 | | 4 cwt. | 35·84 | 10 cwt.| 56·00 |

With tallow, yellow brass on cast iron, the friction varied from 1/36·57 to 1/40·72, as the pressure varied from 1 to 5 hundredweight.

With soft soap, yellow brass on cast iron, the friction varied from 1/26·35 to 1/37·96, as the pressure varied from ½ to 5 hundredweight.

With soft soap and black lead, yellow brass on cast iron, the friction varied from 1/10·18 to 1/23·82, as the pressure varied from ½ to 5 hundredweight.

From these various results, which we have abridged as much as possible, Mr Rennie concludes "that the diminution of friction by unguents varies as the insistent weights and nature of the unguents: the lighter the weight, the finer and more fluid should be the unguent, and vice versa."

On the Friction of Pivots.

In the construction of clocks and watches, and all delicate pieces of machinery, but particularly in the formation of compasses, the proper form of pivots, as well as the material of which they are made, and in which they move, are subjects of the greatest importance.

All that we know on this subject is due to the celebrated Coulomb, who made two different sets of experiments on the friction of pivots. The last was made with most accuracy, and in vacuo, to get rid of the effects of the air's resistance.

The apparatus used by Coulomb is shown in the annexed figure, where akhb is a brass wire fork, fixed at d to a concave lens of glass, whose radius is about two or three lines. Two metallic plates a, b are fixed to the lower ends of the bent wire, and the whole moves upon a pivot beneath d, at the upper end of the needle gd, which was made of tempered steel, and fixed into the cylindrical stand. A rotatory motion is given to the fork by a hooked rod ef, and the whole is enclosed in an air-tight glass receiver AB, in which a vacuum is made in the usual manner by an air-pump. Coulomb found that when ab was two inches, the time of the fork's revolution was seven or eight seconds, and the weight of a and b about five or six gros (a gros is about the eighth part of an ounce). It is unnecessary to perform the experiments and in vacuo, though it is proper to protect the apparatus from currents of air, by placing it under a receiver.

In order to compare the experimental results, Coulomb investigated the following formula:

\[ A = \frac{b^2}{X} \times \int \frac{\mu r^2}{a} \]

in which A is the momentum of friction, b the primitive velocity, X the space described from the beginning to the end of the motion, \( \int \frac{\mu r^2}{a} \) the sum of each molecule multiplied into the square of its distance r from the axis of motion, and divided by the distance a from the axis of motion of a point whose velocity is b. When the same receiver is used, \( \int \frac{\mu r^2}{a} \) is a constant quantity.

With a different apparatus Coulomb made the following preliminary experiments. He took a glass receiver, four inches wide, five inches high, and five ounces in weight, and having caused this to revolve upon a pivot, he observed the time employed in making the four or five first turns, from a mean of which he obtained the primitive velocity b, and he then counted the number of revolutions performed by the revolving receiver till it stopped. Having found that the resistance of the air bore no sensible proportion to that of friction, the preceding results enabled him to calculate the friction by which the receiver was brought to rest. The following were the results:

| Time of performing One Revolution | Number of Revolutions performed before the Apparatus stopped | Values of \( \frac{b^2}{X} \) | |-----------------------------------|---------------------------------------------------------------|---------------------------| | 4 seconds | 34·1 | 547 | | 6·25 seconds | 14·1 | 550 | | 11 seconds | 4·6 | 557 |

Now, as the second term of the above expression for the momentum of friction is constant, and as the value of \( \frac{b^2}{X} \), the first term, appears to be constant in the preceding experiments, the average of which is \( \frac{1}{551} \), A or the momentum of friction is also constant; and hence Coulomb drew the conclusion, that the friction of pivots is independent of the velocity, and must therefore depend on the pressure.

In order to determine the friction of pivots when they supported planes of different materials, Coulomb used the apparatus represented above. The fork akhb was nine inches long, the distance of the branches akhb twenty-four lines, the curve cdh was a semicircle of about three inches long, and the branches ah, hb three inches long. The pieces of metal a, b, and the plane d, were attached to the wire by wax. At g, the top of the cylindrical support, was fixed a small needle of tempered steel, whose point could be made more or less sharp or obtuse, as the nature of the planes and the magnitude of the pressure required. The angle of the point at g was, in the following experiments, eighteen or twenty degrees, the weight of a and b was a quarter of an ounce, and that of the fork about one fifth of an ounce.

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1 After remaining 41 hours in a state of rest with ten hundredweight. 2 Fresh composition having been applied. 3 See Mem. Acad. Par. 1790, p. 451, 452. | Nature of the Planes | Time of performing one Revolution | Number of Revolutions performed before the Apparatus stopped | Values of $\frac{b^2}{X}$ | Momentum of Friction | |----------------------|----------------------------------|-------------------------------------------------------------|--------------------------|---------------------| | Steel, plane at d... | 17 seconds | 15 seconds | 1 | 510 | | Tempered and polished| 8 seconds | 7-25 seconds | 1 | 464 | | Glass, plane | 8-75 seconds | 7-5 seconds | 1 | 570 | | Highly polished | 4-25 seconds | 2-9 seconds | 1 | 589 | | Rock crystal, plane | 13 seconds | 4-62 seconds | 1 | 781 | | Highly polished | 14-5 seconds | 3-75 seconds | 1 | 787 | | Agate, plane | 9 seconds | 10-5 seconds | 1 | 851 | | Highly polished | 15 seconds | 3-5 seconds | 1 | 844 | | Garnet, plane | 12 seconds | 7 seconds | 1 | 1008 | | Highly polished | 23 seconds | 2 seconds | 1 | 1050 |

From these experiments, it is evident that garnet is the best material for the cups of pivots, agate the next, rock crystal the next, glass the next, and steel the worst.

Coulomb next proceeded to determine the effects produced by giving different angles to the pivots of the needle, the other circumstances remaining as before.

### With a Plane of Polished Glass

| Pressure on the Pivot, in Ounces | Time of performing One Revolution | Number of Revolutions performed before the Apparatus stopped | Values of $\frac{b^2}{X}$ | |---------------------------------|----------------------------------|-------------------------------------------------------------|--------------------------| | 0-41 | 24 seconds | 2 | 1 | | | 14 seconds | 5-75 | 1 | | | 10 seconds | 11-75 | 1 | | | 9 seconds | 10-75 | 1 | | | 13 seconds | 4-25 | 1 |

### With a Plane of Garnet

| Pressure on the Pivot, in Ounces | Time of performing One Revolution | Number of Revolutions performed before the Apparatus stopped | Values of $\frac{b^2}{X}$ | |---------------------------------|----------------------------------|-------------------------------------------------------------|--------------------------| | 0-67 | | | 1 | | | 2-08 | | 1 |

Hence it may be shown, that the momentum of friction is proportional to the $\frac{3}{2}$ power of the pressure. When the pressure was very considerable, and the pivot shaped to any angle, the friction varied nearly as the pressure. These results were deduced from the experiments with the glass plane, as the friction was greater and the results more regular with it than with either garnet or agate.

Coulomb found the cups made by the best workmen very irregular in their curvature, and producing a friction three or even four times greater than that of well-polished planes of the same substance. With such cups the angles of the pivots should be diminished.

### On the Friction of Machines

In the year 1786, and at subsequent periods, the late Mr John Rennie made several experiments on the friction and resistance of heavy machinery. He found that an augmentation of resistance took place with the quantity of machinery put in motion; in one instance in the ratio of one to five, when from one fifth to one tenth of the power expended was absorbed; an anomaly which Mr George Rennie ascribes to the irregularity of the movement, and to the difficulty of producing simultaneous actions in com- The following experiments were made by Mr George Rennie:

1. Twenty-one cwt. was suspended at each extremity of a chain passing over two cast-iron sheaves two feet in diameter, with wrought-iron axles working in brass bearings oiled, and twelve feet ten inches apart. It was disturbed by three cwt. or 1/4th of the total weight. Another double-purchased crane gave one ninth.

2. A weight of 7057 lbs. suspended to a double-purchased crane indicated \( \frac{1}{7.62} \) for the friction. Another similar one indicated one ninth. In one of the corn-mills at Deptford, it required one tenth of the weight of the mass to overcome the inertia and friction of the bearings and tangential surfaces. The pressures in this case varied from twenty-eight lbs. to eight cwt. per inch, and the velocities of the surfaces from 50 to 120 feet per minute.

Mr Rennie remarks, that it has been usual to deduct one fourth of the power expended for friction; but though this may hold in machines newly set in motion, he conceives that the proportion is much less when the bearings are properly proportioned to the weights of the parts, and their surfaces kept from contact by unguents.

Having thus considered the nature and effects of friction, we shall now attend to the method of lessening the resistance which it opposes to the motion of machines. The most efficacious mode of accomplishing this, is to convert that species of friction which arises from one body being dragged over another, into that which is occasioned by one body rolling upon another. As this will always diminish the resistance, it may be easily effected by applying wheels or rollers to the sockets or bushes which sustain the gudgeons of large wheels and the axles of wheel-carriages. Casatus seems to have been the first who recommended this apparatus. It was afterwards mentioned by Sturmius and Wolfius, but was not used in practice till Sully applied it to clocks in the year 1716, and Mondran to cranes in 1725. Notwithstanding these solitary attempts to introduce friction-wheels, they seem to have attracted little notice, till the celebrated Euler examined and explained, with his usual accuracy, their nature and advantages. The diameter of the gudgeons and pivots should be made as small as the weight of the wheel and the impelling force will permit. The gudgeons should rest upon wheels as large as circumstances will allow, having their axes as near each other as possible, but no thicker than what is absolutely necessary to sustain the superincumbent weight. When these precautions are properly attended to, the resistance which arises from the friction of the gudgeon, &c. will be extremely trifling.

The effects of friction may likewise in some measure be removed by a judicious application of the impelling power, and by proportioning the size of the friction-wheels to the pressure which they severally sustain. If we suppose, for example, that the weight of a wheel, whose iron gudgeons move in bushes of brass, is 100 pounds, then the friction arising from both its gudgeons will be equivalent to 25 pounds. If we suppose also that a force equal to 40 pounds is employed to impel the wheel, and acts in the direction of gravity, as in the cases of overshot-wheels, the pressure of the gudgeons upon their supports will then be 140 pounds and the friction 35 pounds. But if the force of 40 pounds could be applied in such a manner as to act in direct opposition to the wheel's weight, the pressure of the gudgeons upon their supports would be 100 — 40, or 60 pounds, and the friction only 15 pounds. It is impossible, indeed, to make the moving force act in direct opposition to the gravity of the wheel in the case of water-mills, and it is often impracticable for the engineer to apply the impelling power but in a given way; but there are many cases in which the moving force may be so exerted as at least not to increase the friction which arises from the wheel's weight.

When the moving force is not exerted in a perpendicular direction, but obliquely, as in undershot-wheels, the gudgeon will press with greater force on one part of the socket than on any other part. This point will evidently be on the side of the bush opposite to that where the power is applied; and its distance from the lowest point of the socket, which is supposed circular and concentric with the gudgeon, being called \( x \), we shall have \( \tan x = \frac{H}{V} \); that is, the tangent of the arch contained between the point of greatest pressure and the lowest point of the bush, is equal to the sum of all the horizontal forces divided by the sum of all the vertical forces and the weight of the wheel, \( H \) representing the former, and \( V \) the latter quantities. The point of greatest pressure being thus determined, the gudgeon must be supported at that part by the largest friction-wheel, in order to equalize the friction upon their axles.

The application of these general principles to particular cases is so simple as not to require any illustration. To aid the conceptions, however, of the practical mechanic, we may mention two cases in which friction-wheels have been successfully employed.

Mr Gotthlieb, the constructor of a new crane, has received a patent for what he calls an anti-attrition axle-tree, the beneficial effects of which he has ascertained by a variety of trials. It consists of a steel roller \( R \), about four or six inches long, which turns within a groove cut in the inferior part of the axle-tree \( C \), which runs in the nave \( AB \) of the wheel. When the wheel-carriages are at rest, Mr Gotthlieb has given the friction-wheel its proper position; but it is evident that the point of greatest pressure will change when they are put in motion, and will be nearer the front of the carriage. This point, however, will vary with the weight of the load; but it is sufficiently obvious that the friction-roller should be at a little distance from the lowest point of the axle-tree.

Mr Gamett of Bristol has applied friction-rollers in a different manner, which does not, like the preceding method, weaken the axle-tree. Instead of fixing them in the iron part of the axle, he leaves a space between the nave and the axis, to be filled with equal rollers almost touching each other. A section of this apparatus is represented in fig. 115, where \( ABCD \) is the metallic ring inserted in the nave of the wheel. The

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1 An example of the application of friction-wheels is to be found in Atwood's machine, described in a subsequent part of this Article.

The axle-tree is represented at E, placed between the friction rollers I, I, I, made of metal, and having their axes inserted into a circle of brass, which passes through their centres. The circles are riveted together by means of bolts passing between the rollers, in order to keep them separate and parallel.

As it appears from the experiments of Coulomb that the least friction is generated when polished iron moves upon brass, the gudgeons and pivots of wheels, and the axles of friction-rollers, should all be made of polished iron; and the bushes in which these gudgeons move, and the friction-wheels, should be formed of polished brass.

When every mechanical contrivance has been adopted or diminished the obstruction which arises from the attrition of the communicating parts, it may be still farther removed by the judicious application of unguents. The most proper for this purpose are swine's grease and tallow when the surfaces are made of wood, and oil when they are of metal. When the force with which the surfaces are pressed together is very great, tallow will diminish the friction more than swine's grease. When the wooden surfaces are very small, unguents will lessen their friction a little; but it will be greatly diminished if wood moves upon metal creased with tallow. If the velocities, however, are increased, or the unguent not often enough renewed, in both these cases, but particularly in the last, the unguent will be more injurious than useful. The best mode of applying it is to cover the rubbing surfaces with as thin a stratum as possible, for the friction will then be a constant quantity, and will not be increased by an augmentation of velocity.

In small works of wood, the interposition of the powder of black lead has been found very useful in relieving the motion. The ropes of pulleys should be rubbed with tallow, and whenever the screw is used, the square threads should be preferred.

A very important unguent for diminishing the friction of machinery was accidentally discovered at Lowell in North America. The substance is steatite or soapstone, pulverised and mixed with oil, tallow, or tar, according to use to which it is to be applied. It has been used most successfully in all kinds of machinery, and it is said to be equally applicable to carriage-wheels. The following fact, stated by Mr Moody, the superintendent of the tars near Boston, will show the value of steatite as an unguent. A horizontal balance-wheel of 14 tons runs on a cap 5 inches in diameter, and revolves from 75 to 125 times in a minute. This wheel is connected with the rolling machine, and though 100 tons of iron are rolled in this machine in a minute, yet the balance-wheel has sometimes been used from three to five weeks without inconvenience before the soapstone has been renewed. When machinery begins to be heated, its application is said to succeed when all other unguents fail. In the case of carriage-wheels being set on fire by friction, the application of cow's dung is particularly effectual. A more recently-discovered unguent is mentioned in our chapter on wheel-carriages.

When ropes pass over cylinders or pulleys, a considerable force is necessary to bend them into the form of the circumference round which they are coiled. The force which is necessary to overcome this resistance is called the fitness or rigidity of the ropes. This important subject was first examined by Amontons, who contrived an ingenious apparatus for ascertaining the rigidity of ropes. His experiments were repeated, and confirmed in part, by Desaguliers and others, but particularly by M. Coulomb, who investigated the subject with more care and success than any of his predecessors. His experiments were made both with the apparatus of Amontons and with one of his own invention; and as there was no great discrepancy in the results, he was authorized to place more confidence in his experiments. The limits of this article will not permit us to give an account of the manner in which the experiments were conducted, or even to give a detailed view of the various conclusions which were obtained. We can only present the reader with some of those leading results which may be useful in the construction of machinery.

1. The rigidity of ropes increases, the more that the fibres of which they are composed are twisted.

2. The rigidity of ropes increases in the duplicate ratio of their diameters. According to Amontons and Desaguliers, the rigidity increases in the simple ratio of the diameters of the ropes; but this probably arose from the flexibility of the ropes which they employed; for Desaguliers remarks, that when he used a rope whose diameter was half an inch, its rigidity was increased in a greater proportion; so that it is probable that if they had employed ropes from two to four inches in diameter, like those used by Coulomb, they would have obtained similar results. (See No. 5.)

3. The rigidity of ropes is in the simple and direct ratio of their tension.

4. The rigidity of ropes is in the inverse ratio of the diameters of the cylinders round which they are coiled.

5. In general, the rigidity of ropes is directly as their tensions and the squares of their diameters, and inversely as the diameters of the cylinders round which they are wound.

6. The rigidity of ropes increases so little with the velocity of the machine, that it need not be taken into the account when computing the effects of machines.

7. The rigidity of small ropes is diminished when penetrated with moisture; but when the ropes are thick, their rigidity is increased.

8. The rigidity of ropes is increased and their strength diminished when they are covered with pitch; but when ropes of this kind are alternately immersed in the sea and exposed to the air, they last longer than when they are not pitched. This increase of rigidity, however, is not so perceptible in small ropes as in those which are pretty thick.

9. The rigidity of ropes covered with pitch is a sixth part greater during frost than in the middle of summer, but this increase of rigidity does not follow the ratio of their tensions.

10. The resistance to be overcome in bending a rope over a pulley or cylinder may be represented by a formula composed of two terms. The first term \( \frac{aDn}{r} \) is a constant quantity independent of the tension, \( a \) being a constant quantity determined by experiment, \( Dn \) a power of the diameter \( D \) of the rope, and \( r \) the radius of the pulley or cylinder round which the rope is coiled. The second term of the formula is \( T \times \frac{bDn}{r} \), where \( T \) is the tension of the rope, \( b \) a constant quantity, and \( Dn \) and \( r \) the same as before. Hence the complete formula is

\[ \frac{aDn}{r} + T \times \frac{bDn}{r}, \text{ or } \frac{Dn}{r} \times a + Tb. \]

The exponent \( n \) of the quantity \( D \) diminishes with the flexibility of the rope, but is generally equal to 1.7 or 1.8; or, as in No. 2, the rigidity is nearly in the duplicate ratio of the diameter of the rope. When the cord is much used, its flexibility is increased, and \( n \) becomes equal to 1.5 or 1.4.

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1 Appendix to Ferguson's Lectures, vol. ii. 2 Memoir Acad. 1699, p. 217. 3 Professor Silliman's Journal, No. 27, p. 192. 4 Course of Nat. Phil. vol. i. p. 243. A fly, in mechanics, is a heavy wheel or cylinder, which moves rapidly on its axis, and is applied to machines for the purpose of rendering uniform a desultory or reciprocating motion, arising either from the nature of the machinery, from an inequality in the resistance to be overcome, or from an irregular application of the impelling power. When the first mover is inanimate, as wind, water, and steam, an inequality of force obviously arises from a variation in the velocity of the wind, from an increase or decrease of water occasioned by sudden rains, or from an augmentation or diminution of the steam in the boiler, produced by a variation of heat in the furnace; and accordingly various methods have been adopted for regulating the action of these variable powers. The same inequality of force obtains when machines are moved by horses or men. Every animal exerts its greatest strength when first set to work. After pulling for some time, its strength will be impaired; and when the resistance is great, it will take frequent though short relaxations, and then commence its labour with renovated vigour. These intervals of rest and vigorous exertion must always produce a variation in the velocity of the machine, which ought particularly to be avoided, as being detrimental to the communicating parts as well as the performance of the machine, and injurious to the animal which is employed to drive it. But if a fly, consisting either of cross bars or a massy circular rim, be connected with the machinery, all these inconveniences will be removed. As every fly-wheel must revolve with great rapidity, the momentum of its circumference must be very considerable, and will consequently resist every attempt either to accelerate or retard its motion. When the machine therefore has been put in motion, the fly-wheel will be whirling with an uniform celerity, and with a force capable of continuing that celerity when there is any relaxation in the impelling power. After a short rest the animal renews his efforts; but the machine is now moving with its former velocity, and these fresh efforts will have a tendency to increase that velocity. The fly, however, now acts as a resisting power, receives the greatest part of the superfluous motion, and causes the machinery to preserve its original celerity. In this way the fly secures to the engine an uniform motion, whether the animal takes occasional relaxations or exerts his force with redoubled ardour.

We have already observed that a desultory or variable motion frequently arises from the inequality of the resistance, or work to be performed. This is particularly manifest in thrashing mills on a small scale, which are driven by water. When the corn is laid unequally on the feeding board, so that too much is taken in by the fluted rollers, this increase of resistance instantly affects the machinery, and communicates a desultory or irregular motion even to the water-wheel or first mover. This variation in the velocity of the impelling power may be distinctly perceived by the ear in a calm evening when the machine is at work. The best method of correcting these irregularities is to employ a fly-wheel, which will regulate the motion of the machine when the resistance is either augmented or diminished. In machines built upon a large scale there is no necessity for the interposition of a fly, as the inertia of the machinery supplies its place, and resists every change of motion that may be generated by an unequal admission of the corn.

A variation in the velocity of engines arises also from the nature of the machinery. Let us suppose that a weight of 1000 pounds is to be raised from the bottom of a well 50 feet, by means of a bucket attached to an iron chain which winds round a barrel or cylinder, and that every foot length of this chain weighs two pounds. It is evident that the resistance to be overcome in the first moment is 1000 pounds added to 50 pounds the weight of this chain, and that this resistance diminishes gradually as the chain coils round the cylinder, till it is only 1000 pounds when the chain is completely wound up. The resistance therefore decreases from 1050 to 1000 pounds; and if the impelling power is inanimate, the velocity of the bucket will gradually increase; but if an animal be employed, it will generally proportion its action to the resisting load, and must therefore pull with a greater or less force according as the bucket is near the bottom or top of the well. In this case, however, the assistance of a fly may be dispensed with, because the resistance diminishes uniformly, and may be rendered constant by making the barrel conical, so that the chain may wind upon the part nearest the vertex at the commencement of the motion, the diameter of the barrel gradually increasing as the weight diminishes. In this way the variable resistance will be equalized much better than by the application of a fly-wheel, for the fly, having no motion of its own, must necessarily waste the impelling power.

Having thus pointed out the chief causes of variation in the velocity of machines, and the method of rendering it uniform by the intervention of fly-wheels, the utility, and in some instances the necessity, of this piece of mechanism, may be more obviously illustrated by showing the propriety of their application in particular cases.

When machinery is driven by a single-stroke steam-engine, there is such an inequality in the impelling power, that for two or three seconds it does not act at all. During this interval of inactivity the machinery would necessarily stop, were it not impelled by a massy fly-wheel of a great diameter, revolving with rapidity, till the moving power again resumes its energy.

If the moving power be a man acting with a handle or winch, it is subject to great inequalities. The greatest force is exerted when the man pulls the handle upwards from the height of his knee, and he acts with the least force when the handle, being in a vertical position, is thrust from him in a horizontal direction. The force is again increased when the handle is pushed downwards by the man's weight, and it is diminished when the handle, being at its lowest point, is pulled towards him horizontally. But when a fly is properly connected with the machinery, these irregular exertions are equalized, the velocity becomes uniform, and the load is raised with an equable and steady motion.

In many cases, where the impelling force is alternately augmented and diminished, the performance of the machine may be increased by rendering the resistance unequal, and accommodating it to the inequalities of the moving power. Dr Robison observes, that "there are some beautiful specimens of this kind of adjustment in the mechanism of animal bodies."

Besides the utility of fly-wheels as regulators of machinery, they have been employed for accumulating or collecting power. If motion is communicated to a fly-wheel by means of a small force, and if this force is continued till the wheel has acquired a great velocity, such a quantity of motion will be accumulated in its circumference, as to overcome resistances, and produce effects which could never have been accomplished by the original force. So great is this accumulation of power, that a force equivalent to twenty pounds applied for the space of thirty-seven seconds to the circumference of a cylinder twenty feet diameter, which weighs 4713 pounds, would, at the distance of one foot from the centre, give an impulse to a musket-ball equal to what it receives from a full charge of gunpowder. In the space of six minutes and ten seconds, the same effect would be produced if the cylinder were driven by a man who constantly exerted a force of twenty pounds at a winch one foot long.

This accumulation of power is finely exemplified in the sling. When the thong which contains the stone is swung round the head of the slinger, the force of the hand is continually accumulating in the revolving stone, till it is discharged with a degree of rapidity which it could never have received from the force of the hand alone. When a stone is projected from the hand itself, there is even then a certain degree of force accumulated, though the stone only moves through the arch of a circle. If we fix the stone in an opening at the extremity of a piece of wood two feet long, and discharge it in the usual way, there will be more force accumulated than with the hand alone, for the stone describes a larger arch in the same time, and must therefore be projected with greater force.

When coins or medals are struck, a very considerable accumulation of power is necessary, and this is effected by means of a fly. The force is first accumulated in weights fixed on the rim of the fly. This force is communicated to two levers, by which it is farther condensed; and from these levers it is transmitted to a screw, by which it suffers a second condensation. The stamp is then impressed on the coin or medal by means of this force, which was first accumulated by the fly, and afterwards augmented by the intervention of two mechanical powers. Hence it follows, that if the fly-wheel, thus loaded with accumulated power, should be detached from the steam-engine or the first mover, whose power it regulates and accumulates, it will go on for some time to do the work of the machine after the prime mover has ceased to act. This effect is finely illustrated by a fact stated by Mr Babbage: "The powerful effect," says he, "of a large fly-wheel, when its force can be concentrated in a point, was curiously illustrated at one of the largest of our manufactories. The proprietor was showing to a friend the method of punching holes in iron plates for the boilers of steam-engines. He held in his hand a piece of sheet iron three eighths of an inch thick, which he placed under the punch. Observing, after several holes had been made, that the punch made its perforations more and more slowly, he called to the engine-man to know what made the engine work so sluggishly, when it was found that the fly-wheel and punching apparatus had been detached from the steam engine just at the commencement of the experiment."

Notwithstanding the great advantage of fly-wheels, both as regulators of machines and collectors of power, their utility wholly depends upon the position which is assigned them relative to the impelled and working points of the engine. For this purpose no particular rules can be laid down, as their position depends altogether on the nature of the machinery. We may observe, however, in general, that when fly-wheels are employed to regulate machinery, they should be near the impelling power; and when used to accumulate force in the working point, they should not be far distant from it. In hand-mills for grinding corn the fly is for the most part very injudiciously fixed on the axis to which the winch is attached, whereas it should always be fastened to the upper millstone, so as to revolve with the same rapidity. In the first position, indeed, it must equalize the varying efforts of the power which moves the winch; but when it is attached to the turning millstone, it not only does this, but contributes very effectually to the grinding of the corn.

A new kind of fly, called a conical pendulum, has been ingeniously employed by Mr Watt for procuring a determinate velocity at the working point of his steam-engine.

It is represented in fig. 116, where AB is a vertical axis moving upon pivots, and driven by means of a rope passing from the axis of the large fly over the sheaf EF. The large balls M, N are fixed to the rods NG, MH, which have an angular motion round P, and are connected by joints at G and H, with the rods GK, HK attached to the extremity of the lever KL, whose centre of motion is L, and whose other extremity is connected with the cock which admits the steam into the cylinder. The frame CD prevents the balls from receding too far from the axis, or from approaching too near it. Now when this conical pendulum is put in motion, the centrifugal force of the balls M, N makes them recede from the axis AB. In consequence of this recession, the points G, H, K are depressed, and the other extremity of the lever is raised, and the cock admits a certain quantity of steam into the cylinder. When the velocity of the fly is by any means increased, the balls recede still farther from the axis, the extremity of the lever is raised higher, and the cock closes a little and diminishes the supply of steam. From this diminution in the impelling power, the velocity of the fly and the conical pendulum decreases, and the balls resume their former position. In this way, when there is any increase or diminution in the velocity of the fly, the corresponding increase or diminution in the centrifugal force of the balls raises or depresses the arm of the lever, admits a greater or a less quantity of steam into the cylinder, and restores to the engine its former velocity.

In like manner, four iron balls have been applied to the axis of a millstone to regulate its motion.

Among the contrivances analogous to fly-wheels, the Prony's most useful and ingenious is that of Baron Prony, to which condenser he has given the name of the Condenser of Forces. The problem which Prony proposes to solve is, to transmit to any machine whose construction is determinate the action of the first mover, and to fulfil the following conditions:

1. To vary speedily and easily the resistance which the first mover is intended to overcome. 2. To preserve the resistance constant till it is proper to increase or diminish it. 3. To preserve the velocity of the machine constant under the most sudden variations of the first mover.

The section and plan of a machine for applying his invention to a wind-mill is shown in figs. 6 and 7 of Plate CCCLL. The wind-mill sails, which are not shown in the figures, are supposed to give motion to the vertical arbor O, which carries a quadrangular frame, at each angle of which is fixed a curved wire of iron or steel bd. Round the main arbor O are situated several vertical arbors a, a, a, a (eight being shown in fig. 2), each carrying a curved piece of iron or steel af, so constructed that when the arbor O revolves, all the other arbors a will revolve also by the action of the arms bd, &c. upon af, &c.; the succeeding curve bd beginning to act upon the corresponding one af, when the preceding bd has just quitted its corresponding one af. The same effect may be produced by toothed wheels upon all the arbors, as shown in fig. 2.

Each arbor a, a, a, a, the number of which depends on the circumstances of the case, carries a drum ttrr, round which is coiled a cord passing over a pulley p, and sustaining a weight Q, which slides upon the lever FG, upon any part.

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1 This has been demonstrated by Mr Atwood. See his Treatise on Rectilineal and Rotatory Motion. 2 Economy of Manufactures, § 20. 3 See Lanz and Betancourt's Essai sur Machines, p. 84. On the of which it may be fixed, G being the fulcrum of the lever. Teeth of The axis a passes loosely through the pinion gg, which carries a click or ratchet bearing against teeth upon the side rr of the drum, so that when the weight Q tends to rise, the ratchet gives way, gg is thrown loose, and the weight Q continues to rise. When the arm bd has ceased to act upon gf, the weight Q rises no further, but tends to descend. The ratchet is then thrown into the teeth of rr, so that, in descending, Q turns the pinion gg along with the drum ttrr. The very same thing is going on with all the other arbors, which are raising other weights Q, which re-descend in their turn, and give motion to their corresponding pinions gg.

Now gg drives the wheel AB, which, by means of the bevelled wheel on its lower side GD acting upon the bevelled wheel CF, raises the bucket or weight P, which is the work to be performed. Hence the descent of all the weights Q, Q, &c. will concur in raising P. By regulating the distance of the weights Q from the centre of motion G, the proper ratio between the power of the work to produce a maximum effect may be obtained.

Baron Prony justly remarks, that advantage may be taken of the weakest breezes to obtain a certain effect, when all other wind-mills are inactive.¹

A very ingenious regulator for equalizing the velocity of machinery has been described by the Rev. W. Cecil, of Magdalen College, Cambridge. If we suppose two wheels to be so connected that an increase in the velocity of the first is accompanied with an increase in the velocity of the second in the same ratio, and if these wheels have another connection, by means of which an increase in the velocity of the first is accompanied by an increase in the velocity of the second in a higher ratio, then it will be impossible that any increase should take place, as it would require the second to move with two different velocities at once. These conditions may be fulfilled by connecting the wheels in the first instance by common tooth-work, and, in the next place, by another toothed wheel, which slides into different positions as the centrifugal force varies. By these means Mr Cecil gets a regulator which opposes no resistance up to a certain velocity, but which, at all greater velocities, presents an insurmountable resistance to any augmentation of velocity, and which may be readily united to any revolving machinery, whatever be its construction and power, the velocity of which it is required to equalize.

CHAP. V.—ON THE TEETH OF WHEELS AND THE WIPERS OF STAMPERS.

In the construction of machines, we must not only attend to the form and number of their parts, but also to the mode by which they are to be connected. It would be easy to show, did the limits of this article permit it, that, when one wheel impels another, the impelling power will sometimes act with greater and sometimes with less force, unless the teeth of one or both of the wheels be parts of a curve generated after the manner of an epicycloid, by the revolution of one circle along the convex or concave side of another. It may be sufficient to show that, when one wheel impels another by the action of epicycloidal teeth, their motion will be uniform. Let the wheel CD drive the wheel AB by means of the epicycloidal teeth mp, nq, or, acting upon the infinitely small pins or spindles a, b, c; and let the epicycloids mp, nq, &c. be generated by the circumference of the wheel AB, rolling upon the convex circumference of the wheel CD. From the formation of the epicycloid, it is obvious that the arch ab is equal to mn, and the arch ac to mo; for during the formation of the part ab of the epicycloid nq, every point of the arch ab is applied to every point of the arch mn; and the same happens during the formation of the part co of the epicycloid or. Let us now suppose that the tooth mp begins to act on the pin a, and that b, c are successive positions of the pin a after a certain time; then nq, or will be the positions of the tooth mp after the same time; but ab = mn and ac = mo, therefore the wheels AB, CD, when the arch is driven by epicycloidal teeth, move through equal spaces in equal times, that is, the force of the wheel CD, and the velocity of the wheel AB, are always uniform.

In illustrating the application of this property of the epicycloid, which was discovered by Olaf Roemer, the celebrated Danish astronomer, we shall call the small wheel the pinion, and its teeth the leaves of the pinion. The line which joins the centre of the wheel and pinion is called the line of centres. There are three different ways in which the teeth of one wheel may drive another, and each of these modes of action requires a different form for the teeth.

1. When the action is begun and completed after the teeth have passed the line of centres. 2. When the action is begun and completed before they reach the line of centres. 3. When the action is carried on, on both sides of the line of centres.

The first of these modes of action is represented in Fig. 118, where B is the centre of the wheel, A that of the pinion, and AB the line of centres. It is evident from the figure, that the part b of the tooth ab of the wheel does not act on the leaf m of the pinion till they arrive at the line of centres AB, and that all the action is carried on after they have passed this line, and is completed when the leaf m comes into the situation n. 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 superficies of the pinion, or within the circle adh; and the faces ab of the teeth of the wheel should be portions of an exterior epicycloid, formed by the same generating circle rolling upon the convex superficies odp of the wheel.

But when one circle rolls within another whose diameter is double that of the rolling circle, the line generated by any point of the latter is a straight line tending to the centre of the larger circle. Therefore, if the generating circle above mentioned should be taken with its diameter equal to the radius of the pinion, and be made to roll upon the concave superficies adh of the pinion, it will generate a straight line tending to the pinion's centre, which will be the form of the faces of its leaves; and the teeth of the wheel will be exterior epicycloids, formed by a generating circle whose diameter is equal to the radius of the pinion, rolling upon the convex superficies odp of the wheel. This rectilineal form of the teeth is exhibited in

¹ See Annales des Arts et Manufactures, tom. xix. ² In figs. 109, 110, 111, 112, the letter B is supposed to be placed at the centre of the wheels.

The attentive reader will perceive from fig. 118, that in order to prevent the teeth of the wheel from acting upon the leaves of the pinion before they reach the line of centres AB, 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 AB is given. But in machinery the number of leaves and teeth is always known from the velocity which is required at the working point of the machine; it becomes therefore a matter of great importance to determine with accuracy the relative radii of the wheel and pinion.

For this purpose, let A, fig. 119, be the pinion having the acting faces of its leaves straight lines tending to the centre, and B the centre of the wheel. AB will be the distance of their centres.

Then, as the tooth C is supposed not to act upon the leaf Am till it arrives at the line AB, it ought not to quit Am till the following tooth F has reached the line AB. But since the tooth always acts in the direction of a line drawn perpendicular to the face of the leaf Am from the point of contact, the line CH, drawn at right angles to the face of the leaf Am, will determine the extremity of the tooth CD, or the last part of it which should act upon the leaf Am, and will also mark out CD for the depth of the tooth. Now, in order to find AH, HB, and CD, put a for the number of teeth in the wheel, b for the number of leaves in the pinion, c for the distance of the pivots A and B, and let x be the radius of the wheel, and y that of the pinion. Then, since the circumference of the wheel is to the circumference of the pinion as the number of teeth in the one to the number of leaves in the other, and as the circumferences of circles are proportional to their radii, we shall have \(a : b = x : y\); then, by composition (Eucl. v. 18), \(a + b : b = c : y\) (c being equal to \(x + y\)), and consequently the radius of the pinion, viz. \(y = \frac{cb}{a + b}\); then, by inverting the first analogy, we have \(b : a = y : x\); and consequently the radius of the wheel, viz. \(x = \frac{ay}{b}\) \(y\) being now a known number.

Now, in the triangle AHC, right-angled at C, the side AH is known, and likewise all the angles (HAC being equal to \(\frac{360}{b}\)); the side AC, therefore, may be found by plain trigonometry. Then, in the triangle ACB, the \(\angle CAB\), equal to HAC, is known, and also the sides AB, AC, which contain it; the third side, therefore, viz. CB, may be determined; from which DB, equal to HB, already found, being subtracted, there will remain CD for the depth of the teeth. When the action is carried on after the line of centres, it often happens that the teeth will not work in the hollows of the leaves. In order to prevent this, the \(\angle CBH\) must always be greater than half the \(\angle HBP\). The \(\angle HBP\) is equal to 360 degrees, divided by the number of teeth in the wheel, and CBH is easily found by plane trigonometry.

If the teeth of wheels 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 nearly 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 faces of the teeth which theory requires, a quantity of friction will remain after every precaution has been taken in the formation of the communicating parts.

2. The second mode of action is not so advantageous as that which we have been considering, and should, if possible, always be avoided. It is represented in fig. 120, where A is the centre of the pinion, B that of the wheel, and AB the line of the centres. It is evident from the figure, that the tooth C of the wheel acts upon the leaf D of the pinion before they arrive at the line BA; that it quits the leaf when they reach this line, and have assumed the position of E and F; and that the tooth C works deeper and deeper between the leaves of the pinion, the nearer it comes to the line of centres. From this last circumstance a considerable quantity of friction arises, because the tooth C does not, as before, roll upon the leaf D, but slides upon it; and from the same cause the pinion soon becomes foul, as the dust which lies upon the acting faces of the leaves is pushed into the interjacent hollows. One advantage, however, attends this mode of action: it allows us to make the teeth of the large wheel rectilineal, and thus renders the labour of the mechanic less, and the accuracy of his work greater, than if they had been of a curvilinear form. If the teeth C, E therefore of the wheel BC are made rectilineal, having their surfaces directed to the wheel's centre, the acting faces of the leaves D, F, &c. must be epicycloids formed by a generating circle whose diameter is equal to the radius bo of the circle op, rolling upon the circumference mn of the pinion A. But if the teeth of the wheel and the leaves of the pinion are made curvilinear as in the figure, the faces of the teeth of the wheel must be portions of an interior epicycloid formed by any generating circle rolling within the concave superificies of the circle op, and the faces of the pinion's leaves must be portions of an exterior epicycloid produced by rolling the same generating circle upon the convex circumference mn of the pinion.

3. The third mode of action, which is represented in fig. 121, is a combination of the first two modes, and consequently partakes of the advantages and disadvantages of each. It is evident from the figure that the portion eh of the tooth acts upon the part be of the leaf till they reach the line of centres AB, and that the part ed of the tooth acts upon the portion ba of the leaf after they have passed this line. Hence the acting parts eh and be must be formed according to the directions given for the first mode of action, and the remaining parts ed, ba, must have that curvature which the second mode of action requires; consequently eh should be part of an interior epicycloid formed by any generating circle rolling on the concave circumference mn of the wheel, and the corresponding part be of the leaf should be part of an exterior epicycloid formed by the same generating circle rolling upon bEo, the convex circumference of the pinion; the remaining part ed of the tooth should be a portion of an exterior epicycloid, engendered by any generating circle rolling upon eL, the concave superificies of the wheel; and the corresponding part ba of the leaf should be part of an interior epicycloid described by the same generating circle, rolling along the On the concave side bEo of the pinion. As it would be extremely troublesome, however, to give this double curvature to the acting faces of the teeth, it will be proper to use a generating circle, whose diameter is equal to the radius of the wheel BC, for describing the interior epicycloid eb, and the exterior one be; and a generating circle, whose diameter is equal to AC, the radius of the pinion, for describing the interior epicycloid ba, and the exterior one cd. In this case the two interior epicycloids eb, ba will be straight lines leading to the centres B and A, and the labour of the mechanic will by this means be greatly abridged.

In order to find the relative diameters of the wheel and pinion, when the number of teeth in the one and the number of leaves in the other are given, and when the distance of their centres is also given, and the ratio of ES to CS, let \(a\) be the number of teeth in the wheel, \(b\) the number of leaves in the pinion, \(c\) the distance of the pivots A, B, and let \(m\) be to \(n\) as ES to CS; then the arch ES, or \(\angle SAE\), will be equal to \(\frac{360^\circ}{b}\), and LD, or \(\angle LBD\), will be equal to \(\frac{360^\circ}{a}\). But ES : CS = \(m : n\); consequently LD : LC = \(m : n\), therefore (Eucl. vi. 16) LC \(\times m = LD \times n\), and LC = \(\frac{LD \times n}{m}\); but LD is equal to \(\frac{360^\circ}{a}\), therefore, by substitution, LC = \(\frac{360 \times n}{am}\).

Now, in the triangle APB, AB is known, and also PB, which is the cosine of the angle ABD, PC being perpendicular to DB; AP or the radius of the pinion, therefore, may be found by plain trigonometry. The reader will observe that the point P marks out the parts of the tooth D and the leaf SP where they commence their action; and the point I marks out the parts where their mutual action ceases; AP therefore is the proper radius of the pinion, and BI the proper radius of the wheel, the parts of the tooth L without the point I, and of the leaf SP without the point P, being superfluous. Now, to find BI, we have

\[ES : CS = m : n,\] and \(CS = \frac{ES \times n}{m}\); but ES was shown to be \(\frac{360}{b}\), therefore, by substitution, \(CS = \frac{360 \times n}{bm}\).

Now the arch ES, or \(\angle EAS\), being equal to \(\frac{360}{b}\), and CS, or \(\angle CAS\) being equal to \(\frac{360 \times n}{bm}\), their difference EC, or the angle EAC, will be equal to \(\frac{360}{b} - \frac{360 \times n}{bm}\), or \(\frac{360 \times m - n}{bm}\). The \(\angle EAC\) being thus found, the triangle EAB, or IAB, which is almost equal to it, is known, because AB is given, and likewise AI, which is equal to the cosine of the angle IAB, AC being radius, and AIC being a right angle, consequently IB the radius of the wheel may be found by trigonometry. It was formerly shown that AC, the radius of what is called the primitive pinion, was equal to \(\frac{cb}{a+b}\), and that BC the radius of the primitive wheel was equal to \(\frac{AC \times a}{b}\). If then we subtract AC or AS from AP, we shall have the quantity SP, which must be added to the radius of the primitive pinion; and if we take the difference of BC (or BL) and DE, the quantity LE will be found, which must be added to the radius of the primitive wheel. We have all along supposed that the wheel drives the pinion, and have given the proper form of the teeth upon this supposition. But when the pinion drives the wheel, the form which was given to the teeth of the wheel in the first case must in this be given to the leaves of the pinion; and the shape which was formerly given to the leaves of the pinion must now be transferred to the teeth of the wheel.

Another form for the teeth of wheels, different from any which we have mentioned, has been recommended by Dr Robison. He shows that a perfect uniformity of action may be secured, by making the acting faces of the teeth involutes of the wheel's circumference, which are nothing more than epicycloids, the centres of whose generating circles are infinitely distant. Thus, in fig. 122, let AB be a portion of the wheel on which the tooth is to be fixed, and let Apa be a thread lapped round its circumference, having a loop-hole at its extremity a. In this loop-hole fix the pin a, and with it describe the curve or involute abcdeh, by unlapping the thread gradually from the circumference Apm. This curve will be the proper shape for the teeth of a wheel whose diameter is AB. Dr Robison observes, that as this form admits of several teeth to be acting at the same time (twice the number that can be admitted in M. de la Hire's method), the pressure is divided among several teeth, and the quantity upon any one of them is so diminished, that those dents and impressions which they unavoidably make upon each other are partly prevented. He candidly allows, however, that the teeth thus formed are not completely free from sliding and friction, though this slide is only \(\frac{1}{40}\)th of an inch when a tooth three inches long fixed on a wheel ten feet in diameter drives another wheel whose diameter is two feet. (Append. to Ferguson's Lectures.)

On the Formation of Exterior and Interior Epicycloids, and on the Disposition of the Teeth on the Wheel's Circumference.

Nothing can be of greater importance to the practical mechanic, than to have a method of drawing epicycloids with facility and accuracy: the following, we trust, is the most simple mechanical method that can be employed.—Take a piece of plain wood GH, fig. 123, and fix upon it another piece of wood E, having its circumference mb of the same curvature as the circular base upon which the generating circle AB is to roll. When the generating circle is large, the segment B will be sufficient: in any part of the circumference of this segment fix a sharp-pointed nail a, sloping in such a manner that the distance of its point from the centre of the circle may be exactly equal to its radius; and fasten to the board GH a piece of thin brass, or copper, or tin plate, ab, distinguished by the dotted lines. Place the segment B in such a position that the point of the nail a may be upon the point b, and roll the segment towards G, so that the nail a may rise gradually, and the... point of contact between the two circular segments may advance towards m; the curve ab described upon the brass plate will be an accurate exterior epicycloid. In order to prevent the segments from sliding, their peripheries should be rubbed with rosin or chalk, or a number of small iron points may be fixed on the circumference of the generating segment. Remove, with a file, the part of the brass on the left hand of the epicycloid, and the remaining concave arch or gage ab will be a pattern tooth, by means of which all the rest may be easily formed. When an interior epicycloid is wanted, the concave side of its circular base must be used. The method of describing it is represented in fig. 124, where CD is the generating circle, F the concave circular base, MN the piece of wood on which this base is fixed, and cd the interior epicycloid formed upon the plate of brass by rolling the generating circle C, or the generating segment D, towards the right hand. The cycloid, which is useful in forming the teeth of rack-work, is generated precisely in the same manner, with this difference only, that the base on which the generating circle rolls must be a straight line.

In order that the teeth may not embarrass one another before their action commences, and that one tooth may begin to act upon its corresponding leaf of the pinion before the preceding tooth has ceased to act upon the preceding leaf; the height, breadth, and distance of the teeth must be properly proportioned. For this purpose the pitch-line or circumference of the wheel, which is represented in figs. 119 and 120, by the dotted arches, must be divided into as many equal spaces as the number of teeth which the wheel is to carry. Divide each of these spaces into 16 equal parts; allow 7 of these for the greatest breadth of the teeth, and 9 for the distance between each; or the distance of the teeth may be made equal to their breadth. If the wheel drive a trundle, each space should be divided into 7 equal parts, and 3 of these allotted for the thickness of the tooth, and 3½ for the diameter of the cylindrical stave of the trundle. If each of the spaces already mentioned, or if the distance between the centres of each tooth, be divided into three equal parts, the height of the teeth must be equal to two of these. These distances and heights, however, vary according to the mode of action which is employed. The teeth should be rounded off at the extremities, and the radius of the wheel made a little larger than that which is deduced from the rules p. 421-22. But when the pinion drives the wheel, a small addition should be made to the radius of the pinion.

On the Nature of Bevelled Wheels, and the Method of giving an Epicycloidal Form to their Teeth.

The principle of bevelled wheels was pointed out by De la Hire, so long ago as the end of the seventeenth century, t consists in one fluted or toothed cone acting upon another, as is represented in fig. 125, where the cone OD drives the cone OC, conveying its motion in the direction OC. If these cones be cut parallel to their bases, as at A and B, and if the two small cones between AB and O be removed, the remaining parts AC and BD may be considered as two bevelled wheels, and BD will act upon AC in the very same manner, and with the same effect, that the whole cone OD acted upon the whole cone OC. If the section be made nearer the bases of the cones, the same effect will be produced. This is the case in fig. 126, where CD and DE are but very small portions of the imaginary cones ACD and ADE.

In order to convey motion in any given direction, and determine the relative size and situation of the wheels for this purpose, let AB, fig. 127, be the axis of a wheel, and CD the given direction in which it is required to convey the motion by means of a wheel fixed upon the axis AB, and acting upon another wheel fixed on the axis CD; and let us suppose that the axis CD must have four times the velocity of AB, or must perform four revolutions while AB performs one. Then the number of teeth in the wheel fixed upon AB must be four times greater than the number of teeth in the wheel fixed upon CD, and their radii must have the same proportion. Draw ed parallel to CD at any convenient distance, and draw ef parallel to AB at four times that distance, then the lines im and in drawn perpendicular to AB and CD respectively will mark the situation and size of the wheels required. In this case the cones are Oni and Oni, and smi, rpmi are the portions of them that are employed.

The formation of the teeth of bevelled wheels is more difficult than one would at first imagine. The teeth of such motion of wheels, indeed, must be formed by the same rules which their teeth have been given for other wheels; but since different parts of the same tooth are at different distances from the axis, these parts must have the curvature of their acting surfaces proportioned to that distance. Thus, in fig. 127, the part of the tooth at r must be more incurvated than the part at i, as is evident from the inspection of fig. 126; and the epicycloid for the part i must be formed by means of circles whose diameters are im and If, while the epicycloid for the part r must be generated by circles whose diameters are Ca and Dd.

Let us suppose a plane to pass through the points O, A, D; the lines AB, AO will evidently be in this plane, which may be called the plane of centres. Now, when the teeth of the wheel DE, which is supposed to drive CD the smallest of the two, commence their action on the teeth of CD when they arrive at the plane of centres, and continue their action after they have passed this plane, the curve given to the teeth CD at C, should be a portion of an interior epicycloid formed by any generating circle rolling on the concave supercies of a circle whose diameter is twice Ca perpendicular to CA, and the curvature of the teeth at i should be part of a similar epicycloid formed upon a circle whose diameter is twice im. The curvature of the teeth of the wheel DE, at D, should be part of an exterior epicycloid formed by the same generating circle rolling upon the concave circumference of a circle whose diameter is twice Dd perpendicular to DA; and the epicycloid for the teeth at F is formed in the same way, only, instead of twice Dd, the diameter of the circle must be twice Ef. When any other mode of action is adopted, the teeth are to be formed in the same manner that we have pointed out for common wheels, with this difference only, that different epicycloids are necessary for the parts F and D. It may be sufficient, however, to find the form of the teeth at F, as the remaining part of the tooth may be shaped by directing a straight ruler from different points of the epicycloid at F to the centre A, and filing the tooth till every part of its acting surface coincide with the side of the ruler. The reason of this operation will be obvious by attending to the shape of the tooth in fig. 125. When the small wheel CD impels the large one DE, the epicycloids which were formerly given to CD must be given to DE, and those which were given to DE must be transferred to CD.

The wheel represented in fig. 128 is sometimes called a crown wheel, though it is evident from the figure that it belongs to that species of wheels which we have just been considering; for the acting surfaces of the teeth both of the wheel MB and of the pinion EDG are directed to C the common vertex of the two cones CMB, CEG. In this case the rules for bevelled wheels must be adopted, in which AS is to be considered as the radius of the wheel for the profile of the tooth at A, and MN as its radius for the profile of the tooth at M; and the epicycloids thus formed will be the sections or profiles of the teeth in the direction MP, at right angles to MC the surfaces of the cone. When the vertex C of the cone MCG approaches to N till it be in the same place with the points M, G, some of the curves will be cycloids and others involutes, as in the case of rack-work, for then the cone CEG will revolve upon a plane surface. (Appendix to Ferguson's Lectures.)

Sect. II.—On the Wipers of Stampers, &c. the Teeth of Rack-work, &c. &c.

In fig. 129 let AB be the wheel which is employed to elevate the rack C, and let their mutual action not commence till the acting teeth have reached the line of centres AC. In this case C becomes as it were the pinion or wheel driven, and the acting faces of its teeth must be interior epicycloids formed by any generating circle rolling within the circumference pg; but as pg is a straight line, these interior epicycloids will be cycloids, or curves generated by a point in the circumference of a circle rolling upon a straight line or plane surface. The acting face op, therefore, will be part of a cycloid formed by any generating circle, and mn, the acting face of the teeth of the wheel, must be an exterior epicycloid produced by the same generating circle rolling on mr the convex surface of the wheel. If it be required to make op a straight line, as in the figure, then mn must be an involute of the circle mr formed in the manner represented in fig. 122.

Fig. 129 likewise represents a wheel depressing the rack C when the third mode of action is used. In this case also C becomes the pinion, and DE the wheel; cd therefore must be part of an interior epicycloid formed by any generating circle rolling on the concave side ex of the wheel, and be must be an exterior epicycloid produced by the same generating circle rolling upon the circumference of the rack. The remaining part cd of the teeth of the wheel must be an exterior epicycloid described by any generating circle moving upon the convex side ex, and ba must be an interior epicycloid engendered by the same generating circle rolling within the circumference of the rack. But as the circumference of the rack is in this case a straight line, the exterior epicycloid be and the interior one ba will be cycloids formed by the same generating circles which are employed in describing the other epicycloids. Since it would be difficult, however, as has already been remarked, to give this compound curvature to the teeth of the wheel and rack, we may use a generating circle whose diameter is equal to Dz the radius of the wheel, for describing the interior epicycloid eb, and the exterior one be; and a generating circle whose diameter is equal to the radius of the rack, for describing the interior epicycloid ab, and the exterior one de; ab and eb, therefore, will be straight lines, and be will be a cycloid and de an involute of the circle ez, the radius of the wheel being infinitely great.

In the same manner may the form of the teeth of rackwork be determined when the second mode of action is employed, and when the teeth of the wheel or rack are circular or rectilineal. But if the rack be part of a circle, it must have the same form for its teeth as that of a wheel of the same diameter with the circle of which it is a part.

In machinery, where large weights are to be raised, such as fulling-mills, mills for pounding ore, &c., or where large form pistons are to be elevated by the arms of levers, it is of the greatest consequence that the power should raise the weight with an uniform force and velocity; and this can be effected only by giving a proper form to the wiper.

Now there are two cases in which this uniformity of motion may be required, and each of these demands a different form for the communicating parts: 1. When the weight is to be raised vertically, as the piston of a pump, &c.; 2. when the weight to be raised or depressed moves upon a centre, and rises or falls in the arch of a circle, such as the sledge-hammer in a forge, &c.

1. Let AH be a wheel moved by any power which is sufficient to raise the weight MN by its extremity O, from O to e, in the same time that the wheel moves round one fourth of its circumference; it is required to fix upon its rim a wing OBCDEH, which shall produce this effect with an uniform effort. Divide the quadrant OH into any number of equal parts Om, mn, &c.; the more the better, and Oe into the same number Ob, be, cd, &c.; and through the points m, n, p, H draw the indefinite lines AB, AC, AD, AE, and make AB equal to Ab, AC to Ac, AD to Ad, and AE to Ae; then through the points O, B, C, D, E, draw the curve OBCDE, which is a portion of the spiral of Archimedes, and will be the proper form for the wiper or wing OHE. It is evident, that when the point m has arrived at O, the extremity of the weight will have arrived at b, because AB is equal to Ab; and, for the same reason, when the points n, p, H have successively arrived at O, the extremity of the weight will have arrived at the corresponding points c, d, e. The motion, therefore, will be uniform, because the space described by the weight is proportional to the space described by the moving power, Ob being to Oc as Om to On. If it be required to raise the weight MN with an accelerated or retarded motion, we have only to divide the line Oe according to the law of acceleration or retardation, and divide the curve OBCDE as before.

2. When the lever moves upon a centre, the weight will rise in the arch of a circle, and consequently a new form must be given to the wipers or wings. Let AB, fig. 131, be a lever lying horizontally, which it is required to raise uniformly through the arch BC into the position AC, by means of the wheel BFH furnished with the wing BNOP, which acts upon the extremity C of the lever; and let it be required to raise it through BC in the same time that the wheel BFH moves through one half of its circumference; that is, while the point M moves to B in the direction MFB. Divide the chord CB into any number of equal parts, the more the better, in the points 1, 2, 3, and draw the lines 1a 2b, 3c parallel to AB, or a horizontal line passing through the point B, and meeting the arch CB in the points a, b, c. Draw the lines CD, aD, bD, cD, and BD, cutting the circle BFH in the points m, n, o, p.

Having drawn the diameter BM, divide the semicircle BFM into as many equal parts as the chord CB, in the points q, r, s. Take Bm, and set it from q to r; take Bu and set it from s to t; take Bo and set it from u to v; and lastly, set Rp from M to E. Through the points r, s, t, v, E, draw the indefinite lines DN, DO, DP, DQ, and make DN equal to De, DO equal to Db, DP equal to Da, and DQ equal to DC. Then through the points Q, P, O, N, B, draw the spiral BNOPQ, which will be the proper form for the wing of the wheel when it moves in the direction EMB.

That the spiral BNO will raise the lever AC, with an uniform motion, by acting upon its extremity c, will appear from the slightest attention to the construction of the figure. It is evident, that when the point q arrives at B, the point r will be in m, because Bm is equal to qr, and the point N will be at c, because DN is equal to De; the extremity of the lever therefore will be found in the point c, having moved through Be. In like manner, when the point s has arrived at B, the point t will be at n, and the point O in b, where the extremity of the lever will now be found; and so on with the rest, till the point M has arrived at B. The point E will then be in p, and the point Q in C; so that the lever will now have the position AC, having moved through the equal heights Be, eb, ba, ae; in the same time that the power has moved through the equal spaces qB, rq, us, Mu. The lever therefore has been raised uniformly, the ratio between the velocity of the power and that of the weight remaining always the same.

If the wheel D turn in a contrary direction, according to the letters MHB, we must divide the semicircle BHEM, into as many equal parts as the chord eB, viz. in the points e, g, h. Then, having set the arch Bm from e to d, the arch Ba from g to f, and the rest in a similar manner, draw, through the points d, , h, E, the indefinite lines DR, DS, DT, DQ: make DR equal to De, DS equal to Db, DT equal to Da, and DQ equal to DC; and through the points B, R, S, T, Q, describe the spiral BRSTQ, which will be the proper form for the wing when the wheel turns in the direction MEB. For, when the point e arrives at B, the point d will be in m, and R in e, where the extremity of the lever will now be found, having moved through Be in the same time that the power or wheel has moved through the division eB. In the same manner it may be shown, that the lever will rise through the equal heights eb, ba, ac, in the same time that the power moves through the corresponding spaces eg, gi, iM. The motion of the lever, therefore, and also that of the power, are always uniform. Of all the positions that can be given to the point B, the most disadvantageous are those which are nearest the points F, H; and the most advantageous position is when the chord Be is vertical, and passes, when prolonged, through D, the centre of the circle. In this particular case, the two curves have equal bases, though they differ a little in point of curvature. The farther that the centre A is distant, the nearer do these curves resemble each other; and if it were infinitely distant, they would be exactly similar, and would be the spirals of Archimedes, as the extremity c would in this case rise perpendicularly.

It will be easily perceived that four, six, or eight wings may be placed upon the circumference of the circle, and may be formed by dividing into the same number of equal parts as the chord BC, th, th, or th of the circumference, instead of the semicircle BFM.

That the wing BNO may not act upon any part of the lever between A and C, the arm AC should be bent; and that the friction may be diminished as much as possible, a roller should be fixed upon its extremity C. When a roller is used, however, a curve must always be drawn parallel to the spiral described according to the preceding method, the distance between it and the spiral being everywhere equal to the radius of the roller.

If it should be required to raise the lever with an accelerated or retarded motion, we have only to divide the chord BC, according to the degree of retardation or acceleration required, and the circle into the same number of equal parts as before.

As it is frequently more convenient to raise or depress weights by the extremity of a constant radius, furnished with a roller, instead of wings fixed upon the periphery of a wheel, we shall now proceed to determine the curve which must be given to the arm of the lever which is to be raised and depressed, in order that this elevation or depression may be effected with an uniform motion.

Let AB, fig. 132, be a lever, which it is required to raise uniformly through the arch BC, into the position AC, by means of the arm or constant radius DE, moving upon D as a centre, in the same time that the extremity E describes the arch EeF. From the point C draw CH at right angles to AB, and divide it into any number of equal parts, suppose three, in the points 1, 2; and through the points 1, 2, draw le2b, parallel to the horizontal line AB, cutting the arch CB in the points a, b, through which draw aa, bA. Upon D as a centre, with the distance DE, describe the arch EieF, and upon A as a centre, with the distance AD, describe the arch eOD, cutting the arch EieF in the point e. Divide the arches Eie and eoe, each into the same number of equal parts as the perpendicular eH, in the points h, i, s, m, and through these points about the centre A describe the arches ks, ig, qr, mn. Take xx

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1 The arches Be, eb, &c. are not equal; but the perpendiculars let fall from the points e, a, b, &c. upon the horizontal lines passing through a, b, &c. are equal, being proportional to equal lines c1, 1, 2. Eucl. vi. 2.

2 In the figure, we have taken the point B in a disadvantageous position, because the intersections are in this case more distinct. On the and set it from k to l, and take gf; and set it from i to h. Wipers of Take rq, also, and set it from s to t, and set nm from o to Stampers, p, and de from e to O. Then through the points E, l, h, O, and O, t, p, F, draw the two curves ElhO, and OtypF, which will be the proper form that must be given to the arm of the lever. If the handle DE moves from E towards F, the curve EO must be used; but if in the contrary direction, we must employ the curve OF.

It is evident, that when the extremity E of the handle DE has run through the arch Eh, or rather El, the point l will be in k, and the point z in x, because zx is equal to hl, and the lever will have the position Ab. For the same reason, when the extremity E of the handle has arrived at i, the point h will be in i, and the point g in f, and the lever will be raised to the position Aa. Thus it appears, that the motion of the power and the weight are always proportional. When a roller is fixed at E, a curve parallel to EO, or OF, must be drawn as formerly. (See Appendix to Ferguson's Lectures.)

| Number of Pounds raised. | Height to which the Weight is raised. | Time in which it is raised. | Duration of the Work. | Names of the Authors. | |--------------------------|--------------------------------------|---------------------------|-----------------------|---------------------| | 1000 | 180 | 60 minutes | Euler | | 60 | 1 | 1 second | Bernoulli | | 25 | 220 | 145 seconds | Amontons | | 170 | 330 | 60 minutes | Coulomb | | 1000 | 225 | 60 minutes | Desaguliers | | 30 | 3½ | 1 second | Smeaton | | 26 or 30 | 2½ feet | 1 second | Emerson | | | | | Schulze |

According to Amontons, a man weighing 133 pounds French ascended sixty-two feet French, by steps, in thirty-four seconds, but was completely exhausted. The same author informs us that a sawer made 200 strokes of eighteen inches French each, with a force of twenty-five pounds, in 145 seconds; but that he could not have continued the exertion above three minutes.

It appears from the observations of Desaguliers, that an ordinary man can, for the space of ten hours, turn a winch with a force of thirty pounds, and with a velocity of two feet and a half per second; and that two men working at a windlass with handles at right angles to each other can raise seventy pounds more easily than one man can raise thirty. The reason of this is, that when there is only one man, he exerts variable efforts at different positions of the handle, and therefore the motion of the windlass is irregular; whereas, in the case of two men, with handles at right angles, the effect of the one man is greatest when the effect of the other is least, and therefore the motion of the machine is more uniform, and will perform more work. Desaguliers also found, that a man may exert a force of eighty pounds with a fly when the motion is pretty quick; and that, by means of a good common pump, he may raise a hogshead of water ten feet high in a minute, and continue the exertion during a whole day.

According to Dr Robison, a feeble old man raised seven cubic feet of water, = 437·5 pounds avoirdupois, 11½ feet high, in one minute, for eight or ten hours a day, by walking backwards and forwards on a lever; and a young man weighing 135 pounds, and carrying thirty pounds, raised 9½ cubic feet of water, = 578·5 pounds avoirdupois, 11½ feet high, for ten hours a day, without being fatigued.

From the experiments of Mr Buchanan, it appears that the forces exerted by a man pumping, acting at a winch, ringing, and rowing, are as the numbers 1742, 2856, 3883, and 4095.

The most interesting experiments on the strength of men were made by M. Coulomb; and as they are highly valuable, we shall endeavour to give a condensed account of them.

1. His first object was to ascertain the quantity of action which a man furnished while ascending a stair unloaded.

A man whose weight may be taken at seventy kilogrammes ascended the stairs of a dwelling-house at the rate of fourteen metres, provided he did not ascend more than twenty or thirty metres. His quantity of action is therefore 980 kilogrammes raised one metre in one minute; and if he continue at the work four hours, it will be 235,200 kilogrammes raised one metre in four hours.

The party of sailors who accompanied the Chevalier Borda to the top of the Peak of Tenerife ascended 2923 metres in 7½ hours. Hence their quantity of action in a day was 204,610 kilogrammes raised one metre in 7½ hours.

Coulomb therefore considers 205 kilogrammes raised one kilometre as the quantity of action furnished by a man ascending a stair.

2. When a man, loaded with firewood, ascended a stair, his quantity of action was 109 kilogrammes raised one kilometre. Another man of unusual strength furnished 129 kilogrammes through one kilometre; but he had over-exerted himself.

But even taking this extreme result, it follows that the quantity of action of a man unloaded is to that when he is loaded as 205 to 129, or as 16 to 10.

3. In order to determine the load which a man ought to

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1 The elastic power of heated air has been ingeniously employed as the first mover of machines, by the Reverend Mr Stirling; and the still more intense power of liquefied carbonic acid has been tried, though unsuccessfully, by Mr Brunel; but our limits will not allow us to give any description of the mechanism by which these powers are applied. (See page 436, column second.) carry in order to produce a maximum useful effect, Coulomb remarks, that the only useful effect furnished by a man when loaded is the conveyance of the load. If his load was nothing, his actual quantity of action would be a maximum, but the useful effect would be nothing; and in like manner, if his load was 150 kilogrammes, he could scarcely move it, and the useful effect would also be nothing. Between 0 and 150 kilogrammes there will be some load by which the useful effect will be a maximum. Now, since a man unloaded raised 205 kilogrammes through one kilometre in a day, while a man loaded with 68 kilogrammes raised only 109 kilogrammes through one kilometre, it follows that 68 kilogrammes diminished his quantity of daily work \(205 - 109 = 96\) kilogrammes through one kilometre.

Let us now, as Coulomb has done, suppose the quantities of action lost proportional to the loads; then, calling \(P\) the load, and \(q\) the quantity of action lost, we shall have \(P : q = 68 : 96\), and \(q = P \frac{96}{68} = 1.41P\), or 1.41 kilometres multiplied by \(P\). Hence the quantity of action furnished under the load \(P\) will be \(205 - 1.41P\). If \(h\) is the height through which the man is able to ascend in a day with the load \(P\), then \(Ph\) will be the useful effect of his work, and \((70 + P)h\) will be his total quantity of daily action, 70 kilogrammes being taken as his weight. Hence we obtain

\[ (70 + P)h = 205 - 1.41P, \]

and, by reduction,

\[ Ph = P \frac{(205 - 1.41P)}{70 + P}; \]

or,

\[ Ph = P \frac{(a - bP)}{W + P}, \]

making \(a = 205\), \(b = 1.41\), and \(W = 70\), we have

\[ Ph = P \frac{(a - bP)}{W + P}, \]

which will be a maximum when

\[ P = W \left(\sqrt{1 + \frac{a}{bW}} - 1\right) = 0.754W = 53\text{ kilogrammes}. \]

If this value of \(P\) is now substituted in the formula,

\[ Ph = P \frac{(205 - 1.41P)}{70 + P}, \]

we shall have \(Ph = 56\) kilogrammes raised through 1 kilometre in a day as the measure of the useful effect of a man when carrying a load up a stair.

Hence this mode of employing the strength of a man consumes nearly \(3/4\)ths of his real action, his useful effect being only 56, while his total effect is 205; and, therefore, the expense of employing him will be four times more than that of a man who, having ascended the stair unloaded, raises a weight (by his own descent by gravity) through the height to which he ascended.

If we suppose the man to be so loaded as to perform no work, a case which happens when he bears the greatest weight he can carry, then \(205 - 1.41P = 0\), and \(P = 145\) kilogrammes, which is the weight which an ordinary man can barely carry.

Hence, the formulæ of Coulomb above given correspond to the maximum of the total action of a man ascending a stair unloaded, to the minimum of action when he is loaded with a weight which he can barely carry, and to an intermediate quantity of action of 68 kilogrammes, the ordinary load with which a man can ascend stairs.

Sir David Brewster has greatly simplified the formulæ given by Coulomb, by considering what Coulomb found to be nearly the case, that the quantity of useful effect is only \(1/4\) of the total quantity of action, and making \(1.41 = 1\).

By these substitutions the formula \(Ph \frac{(a - bP)}{W + P}\) becomes

\[ Ph = \frac{(3W - 1.41P)P}{W + P}, \quad \text{and} \quad h = \frac{3W - 1.41P}{W + P}, \]

which becomes a maximum when \(P = W(\sqrt{3} - 1) = 0.732W\).

When \(P = 2W\), we obtain \(3W - 1.41P = 0\), and \(h = 0\); from which it follows, that when \(P\) or the load is equal to \(2W\), or twice the weight of the man, or \(= 2 \times 70 = 140\) kilogrammes, he loses the power of ascending.

When \(P = W\), or when the load is equal to the weight of the man, the mean quantity of action will be reduced one half. This agrees with the observation made by Coulomb, that the reduction of action was in this case from 205 to 96 when the load was 68, the value of \(w\) being 70 kilogrammes.

5. The next case discussed by Coulomb is that of a man walking along a horizontal road either unloaded or loaded.

A man walking unloaded for several days in succession can accomplish easily 50 kilometres in a day, which gives a quantity of action equal to 3500 kilogrammes carried 1 kilometre.

Porters loaded with 58 kilogrammes of furniture had a quantity of action of 2048 kilogrammes carried 1 kilometre. Hence the quantity of action in these two cases, without and with a load of 58 kilogrammes, is as 7 to 4.

When the load was 44 kilogrammes, the quantity of action was 2166 kilogrammes carried 1 kilometre.

When the load is 58 kilogrammes, Coulomb assumes as the best measure of the quantity of action 2000 kilogrammes carried 1 kilometre.

But the quantity of action lost in carrying a load is 3500 - 2000 = 1500 kilogrammes carried 1 kilometre; and assuming as before that the losses \(q\) are proportional to the loads \(P\), we shall have \(58 : P = 1500 : q\), and \(q = P \frac{1500}{58} = 25.86P\), and the real quantity of action will be

\[ 3500 - 25.86P; \]

and making this equal to 0, we have \(P = 135.4\) kilogrammes, which coincides nearly with 140, the result of our simplified formula, as the greatest load which a man can carry.

Hence Coulomb obtains his general formula in the following manner:

Calling \(d\) the distance through which a man can carry the load \(P\), and \(W\) the weight of the man, we have \((P + W)d\) for his quantity of action. Hence we obtain \((P + W)d = 3500 - 25.86P\);

\[ Pd = \frac{(3500 - 25.86P)}{P + W}; \]

or,

\[ Pd = \frac{(a - bP)}{P + W}, \]

making \(3500 = a\), \(25.86 = b\), we have \(Pd = \frac{(a - bP)}{P + W}\),

which becomes a maximum when \(P = W(\sqrt{1 + \frac{a}{bW}} - 1)\),

or \(P = 0.72W\).

Sir David Brewster has simplified these formulæ, and reduced them to the very same as those formerly given, by considering that \(3500 = 50 \times 70\), that is, \(a = 50W\), and by making the co-efficient of \(P\) 25 instead of 25.86. By such substitutions we have \(Pd = \frac{(50W - 25P)}{P + W}\), and

\[ d = \frac{50W - 25P}{P + W}, \]

which becomes a maximum when \(P = W(\sqrt{3} - 1) = 0.732W\), the very same result as when the man ascended stairs.

When \(P = 2W\), then \(50W - 25P = 0\), and \(d = 0\); which shows that when the load is equal to twice the weight of the man, he can no longer carry it through any distance.

The following is a comparison of these results with one another and with those of Coulomb: Nature of the Work. Values of P.

Coulomb. Brewster.

In ascending a height by stairs...0-754 W...0-732

In travelling along a horizontal road...0-72 W...0-732

Mean...0-737...0-732

6. Coulomb's next case is, when the porters return unloaded to carry away a new load.

If we call the distance which a man can travel unloaded D, and x the distance through which he travels unloaded during the day's work, then WD will be the quantity of action when he travels the whole day unloaded, and Wx a portion of his day's journey. Then \( \frac{Wx}{WD} = \frac{x}{D} \) will be the portion of his day's work when he is unloaded, the whole day's work being unity; for \( \frac{x}{D} = 1 \) if \( x = D \). But as the labourer walks over the same space \( x \) loaded, and as his daily action when carrying a load P is 3500 — 25-86 P, the portion of his action under the load P will be \((P + W)x\), and the ratio of this quantity to the daily quantity of action, or \( \frac{(P + W)x}{3500 - 25-86 P} \), will indicate the part of his daily labour which he furnishes under the load P. The sum of these two quantities, namely, when loaded and when unloaded, will be equal to the labour of the entire day, or \( \frac{x}{D} + \frac{(P + W)x}{3500 - 25-86 P} = 1 \).

But if we suppose D to be fifty kilometres, and W = 70 kilogrammes, as before, we have WD = 3500; then, calling 25-86 = b, we shall have, after reduction,

\[ Px = \frac{(WD^2 - bDP)P}{2WD + (D - b)P} \]

which is the useful portion furnished by the man in one day.

Calling \( WD^2 = a \), \( bD = c \), \( 2WD = e \), and \( D - b = f \), we obtain \( Px = \frac{aP - cP}{e + fP} \), which becomes a maximum when \( P = \frac{c}{f} + \left( \sqrt{1 + \frac{fa}{ce}} - 1 \right) \) or \( P = 61-25 \), and \( P = 0-9 \) W.

The preceding formulae have likewise been simplified by Sir David Brewster, by considering that WD = 50 W, that D = 50, and by taking \( b = 25 \) in place of 25-86, so that \( b = \frac{1}{2}D \). By substituting these values, we obtain

\[ Px = \frac{(2500 W - 25D^2P)P}{\frac{1}{2}D(100 W + P)} \]

which becomes a maximum when \( P = 280 (\sqrt{1} - 1) \).

Hence

\[ P = 62-928 \text{ kilogrammes}, \text{ or } 0-8989 \text{ W}; \]

a load very nearly equal to that which porters are in the practice of carrying under similar circumstances.

If we substitute different values of P in Coulomb's formula from 58 to 65, we shall find that Px, the maximum of useful effect, varies from 690 to 691, which shows that a variation of several kilogrammes in the load occasions a very trifling change in the useful effect.

From a comparison of the preceding results, it follows that the quantity of action of a man travelling unloaded is to that when he travels loaded, under the circumstances of the present case, as 505 to 100, or as 5 to 1 nearly.

Coulomb has made a very interesting comparison of the quantity of fatigue undergone by a man ascending a stair and travelling along a horizontal road. The quantity of action in these two cases has been shown to be as 205 to 3500, or 1 to 17. Now, the ordinary height of a step is about 135 millimetres, or 5\(\frac{1}{2}\) English inches, and its width thrice as great, or about 16\(\frac{1}{2}\) English inches. Hence \( 17 \times 135 = 2295 \) millimetres, is the length of a horizontal road on which the man can walk with the same fatigue as in ascending 135 millimetres. But the force of a man is 650 millimetres, or twenty-six inches, consequently the man experiences the same fatigue in ascending 135 millimetres (5\(\frac{1}{2}\) inches) as in advancing horizontally three paces and a half, or seven feet seven inches English measure.

The following table exhibits the preceding results in a condensed form.

| Nature of the Motion | Nature of the Work | Nature of the Mechanical Effect | |----------------------|--------------------|--------------------------------| | Walking | Unloaded | Ground. Kilogrammes. Level...2500 raised 1 kilometre. | | | Loaded with 58 kilog., and returning | | | | unloaded for another burden. The useful effect is... | Level...692-4...1 do. | | Walking | Always loaded | Level...912...1 do. | | Walking | With a loaded wheel-barrow. The useful effect is... | Level...1022-7...1 do. | | Walking up a stair | Unloaded. The mechanical effect is... | 205...1 do. | | Standing | Raising a weight to drive piles. The useful effect is... | 752...1 do. | | Standing | Raising a weight to coin money. The useful effect is... | 395...1 do. | | Walking up a stair | Loaded with 58 kilogrammes. The useful effect is... | 56...1 do. | | Standing | Raising water from a well by a double bucket. The useful effect is... | 71...1 do. | | Standing | Pumping | 190...1 do. | | Standing | Turning a winch | 116...1 do. | | Standing | Reaping | 259...1 do. | | Standing | Rowing | 273...1 do. | | Digging with a spade | | 623...1 do. | The following important results have been given by M. Hachette:

1. A man walking up an inclined plane of 14 centimetres to 1 metre with a load of 7½ kilog., and himself weighing 70 kilog., and walking 7½ hours daily.................................................. $25 raised 1 kilom.

Mechanical Effect.

Kilogrammes.

On the First Movers of Machinery.

2. A man walking loaded in a mountainous country................................................................. 140 1 do.

3. A porter carrying wood up a stair, including his weight....................................................... 112 to 120 1 do.

4. A porter carrying coals up a stair, his weight included......................................................... 40 1 do.

5. A man raising a weight to drive piles...................................................................................... 48 1 do.

6. A man drawing a boat *a la bricole*....................................................................................... 110 1 do.

In the following results, given also by M. Hachette, the numbers are dynamical units or cubic metres of water carried a metre on a horizontal road.

1. A soldier loaded with from 20 to 25 kilog. travelling 20 kilometres a day............................. 1800 to 1900 raised 1 kilom.

2. A Roman soldier travelling 40 kilometres a day by forced marches........................................ 4400 to 4800 1 do.

3. Hawker with crotchets, his weight not included........................................................................ 792 to 880 1 do.

4. Porters drawing a 4-wheel waggon over unequal ground......................................................... 626 1 do.

5. Do. on horizontal planks........................................................................................................... 900 to 1000 1 do.

6. Ditto on the ground, with superficial inequalities.................................................................... 600 1 do.

7. A man drawing a boat in a canal, or 50,000 kilog. transported 11 kilometres.......................... 550,000 1 do.

Among the various ways of applying the strength of men to produce great mechanical effects, the following method of weighing anchors, used by the natives of the coast of Coromandel, is extremely interesting and instructive. It was communicated to the writer of this article many years ago, by a celebrated naval officer, who himself witnessed the process of raising the anchor of his majesty's ship Minden, in September 1814.

The natives, who had been permitted by Sir Samuel Hood to show their method of weighing anchors, formed a raft about three or four feet in diameter, by lashing together, in the form of a rude cylinder, a number of spars, such as top-masts, jib-booms, &c. The buoy-rope of the anchor was wound round the middle of this, and made fast. Thirty small ropes having been fixed to the raft, were coiled round it several times in an opposite way to the buoy rope, so as to form "slew-ropes," or turning ropes. Sixty natives then got upon the spar, and pulled by the ropes so as to turn the raft of spars round. The buoy-rope soon became tight, and any farther rotation of the spar was prevented. The natives now held the slew-ropes tight and firm in both hands, and, standing erect and in a line on the top of the spar, the whole sixty, at the word of command, threw themselves suddenly backwards, so that they all fell flat on the surface of the water at the same instant. By this act the raft of spars turned round one quarter of a circle from the vertical to the horizontal position of the men. This operation did not, of course, start the anchor, but it tightened the buoy-rope to such a degree as to require considerable force to prevent the raft from turning back again. The next turn was made by alternate pairs of the men keeping their horizontal position on the water, while the intermediate pairs climbed up by means of their slew-ropes to the top of the raft. When there, they again tightened their ropes, threw themselves on their backs as before, and then turned the raft another quarter of a circle, the other men "taking in the slack," or tightening their slew-ropes, during the operation. The same operation was repeated by the other half of the party, till the anchor was fairly lifted off the ground. All the natives then continued stretched on the water till the boats towed the raft and the anchor into deep water, when it was hove in the usual way. The anchor was three tons in weight.

When the anchor is lifted off the ground the raft has a tendency to turn back again. This actually took place, in consequence of some of the natives getting tired and letting go their hold; and the rest not being able to overcome the additional load thus suddenly thrown upon them, the anchor sunk to the bottom, and gave such a rapid whirl to the raft, that some of the natives were actually carried round along with it. This might be prevented, as our correspondent suggests, by having two buoy-ropes in place of one, and coiling them round the raft in opposite directions, the rope that slackens or uncoils itself by each quarter turn of the raft being again tightened. This too would allow the whole strength of the party, in place of half of it, to be employed in the operation. A water windlass, to be worked by men in boats, might, as the narrator suggests, be contrived so as to be used when amphibious workmen, such as those of the coast of Coromandel, cannot be had.

In the above process the cylindrical raft is the axle, the men the handspikes, and their weight the moving power. The less the diameter of the spar round which the rope coils, the greater is the power; and the greater the diameter of the part upon which the men stand, the larger is the lever by which they act. The raft should therefore be constructed to give this advantage; and there might be two cylinders of different diameters for the two buoy-ropes, upon the principle of the wheel and axle, shown in fig. 27.

The strength of men varies with the climate and the food. Coulomb found that the grenadier performed one third more work than the other companies. "I have executed," he says, "great works at Martinique by the troops when the thermometer rarely stood below 68° of Fahrenheit, and I have executed in France the same kind of work by troops; and I am assured that under the 14th degree of latitude, where the men are almost always inundated by perspiration, they are not capable of one half the quantity of daily work which they can furnish in our climate." The following measures of the strength of different races of men were made by M. Peron, with Regnier's dynamometer.

| Country | Strength | |------------------|----------| | England | 71·4 | | France | 69·2 | | Timor | 58·7 | | Van Diemen's Land| 51·8 | | New Holland | 50·6 |

Average strength: 60·34

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1 This case is No. 6 of Coulomb's table, where the porter's weight was not included. 2 This result was obtained by M. Lamande, from the work of thirty-eight men, who raised by a pulley a weight of 567 kilog. to drive piles. 3 Guineau, Essai sur les Machines, p. 271. The ground was often clayey. On the Strength of Horses, &c.

Very few accurate observations have been made on the quantity of work done by different quadrupeds, and it would therefore be desirable if the subject were taken up in reference to the different breeds of horses, and the various purposes for which they are employed.

The following table contains several accurate results collected by M. Hachette. The daily action is measured by the draught and the distance travelled over, and the useful effect is measured in dynamical units, each of which is the weight of a cubic metre of water carried a metre upon a horizontal road.

| Kilog. | Kilom. | Daily action | Useful Effect Dyn. Units | |--------|--------|--------------|-------------------------| | 1. A cart-horse, 140 × 40 = 5000 | 2. A post-horse, 90 × 38 = 3420 | 3. A horse moving in a circle, and working a pump | 585 | | 4. A horse raising plaster, 12 hours daily | 1694 | 842 | | 5. A horse working at a pump | Mean of three horses | 1185 | 595 | | 6. A horse raising water by a pump. Mean of eight horses | 2948 | 675 | | 7. A horse raising coals. Mean of two horses. Doubtful | 780 | | 8. A horse drawing a load of 150,000 kilogrammes eight kilometres | 800 | | 9. The force of a horse acting 24 hours is equal to | 5974 dyn. units |

The comparative strength of horses and men has been variously given by different authors, and the discrepancy is so great as to render more necessary the experimental inquiry which we have above recommended. The following table will justify this observation:

One horse is equal to 14 men. Schulze. One horse is equal to 7 men. Bossut. One horse is equal to 5 men. Desaguliers and Smeaton. One ass is equal to 2 men. Bossut.

The following results have been given by Desaguliers: One horse can draw 200 pounds two and a half miles per hour for eight hours a day. If the load is 240 pounds, he can work only six hours. One horse can draw from 450 to 750 pounds up a steep hill, which is more than three men can carry up it. One powerful horse can draw 2000 pounds up a short and steep hill.

One horse has been known to carry 650 or 700 pounds for seven or eight miles without a rest, and to continue this as its ordinary work.

One horse carried 1232 pounds, or eleven hundred-weight, of iron, for eight miles.

In these, as well as in the following results taken from different authors, nothing is said of the nature of the road, which is an essential element in the consideration of this subject.

Two horses, according to Amontons, when ploughing, can exert a force of 150 pounds, or seventy-five pounds each.

One horse, according to Smeaton, can raise 250 hogsheads of water in an hour by means of pumps.

One horse, according to Mr Fenwick, can raise 1000 pounds avoidupois thirteen feet per minute for twelve hours, the horse moving about two miles an hour, by means of a wheel and axle.

One horse, according to Mr Fenwick, can exert a force of seventy-five pounds, moving thirteen feet per minute, and for nine and a half hours.

Four horses, according to Regnier, had a mean draught of thirty-six myriogrammes in 794 hours.

One mule, according to Cazand, can, in the West Indies, exert a force of 150 pounds, walking three feet in a second, and working two hours out of about eighteen.

One horse is supposed to be able to drive 1000 spindles with preparation of cotton water-twist, 1000 spindles with preparation of cotton mule yarn, and seventy-five spindles with preparation of flax yarn.

One horse can draw 140 pounds with the velocity of 200 feet per minute.

The force of a horse has been estimated as follows:

Useful Mechanical Effect.

44,000 pounds raised one foot in one minute. Desaguliers. 29,916 pounds raised one foot in one minute. Smeaton. 33,000 pounds raised one foot in one minute. Watt.

Methods of applying the Power of Men and Horses to drive Machinery.

The methods of applying the power of men and horses to drive ordinary machines is so familiar to every person, that it would be unprofitable to give any particular description of it. Nothing is more familiar to us than a man working at a pump, either by a lever or by a winch turning an axle, or driving round a vertical axis by a lever, and continually walking in a circle. The application of one or more horses to a thrashing-mill, where the horses walk round in a circle, yoked to the extremities of levers which drive a vertical axis, is equally familiar. There are some methods, however, of applying animal power, which, though now little used, are historically interesting, and may in some cases be still resorted to with advantage.

The simplest of these is the tread-mill or tread-wheel, which has within the last twenty years become common in our jails and bridewells, as an instrument of punishment as well as of useful labour. It is shown in the annexed figure, where one or more men are seen climbing as it were up steps on the circumference of a large wheel, whose axis is thus put in motion. The men hold by a fixed rail, and as their weight presses down the step upon which they tread, they ascend the next step, and thus drive the wheel. The muscular force used by the men re-acts principally on the horizontal bar on which they rest, and therefore their weight is the principal source of the moving force.

Another method of applying the force of a man is shown in fig. 134, where the man climbing on a rope employs both his weight and his muscular force in turning round the pulleys round which the rope passes. The re-action of his muscular force is certainly in this case borne by the machine, as Professor Moseley remarks; but notwithstanding this, we cannot agree with him in the conclusion, that it "appears to possess great advantages over the tread-wheel, in the economy of force, space, and materials." In the first place, the weight of the man does not act vertically, and a considerable part of it is injuriously employed in bending the rope and increasing the friction, by pressing it against the rims of the pulley. The friction of two axles is also a disadvantage, and even though the mechanical force of the man were greater during a short time than on the tread-wheel, A very excellent way of employing the power of a man, to make him ascend steps on the inside of a large vertical wheel, a method which we saw in action not many years ago in London, where the force of the man was thus employed to work a crane.

An ancient method of employing horses or oxen to work a crane is shown in the annexed figure, where the animal constantly ascends the inclined plane of a wheel placed obliquely to the horizon, and drives it by the action of its feet against low steps of wood crossing its path.

Another method is shown in fig. 136, where a horse tied to a post has its hind feet resting on low steps on the outside of a wheel, while its fore feet rest upon a raised plank beside the post. In this case, both the weight of the horse and the muscular force of its legs are combined in giving a rotary motion to the wheel or cylinder.

It is a remarkable fact, that so recently as November 1830, Mr. Bramley and Lieutenant Parker took out a patent for various contrivances for applying the power of men and horses to impel carriages on railways. Their first plan is to place a horse upon two tread-wheels or drivers, having raised edges or ribs on their circumferences, against which the fore and hind feet of the horse are intended to press when in the act of stepping, in order to make the wheels revolve by the weight of his body, and by trains of toothed wheels to the running wheels of the carriage. The horse being harnessed to the sides of the frame in which the wheels are mounted, it is intended that he shall impel the wheel also by his muscular power.

The next contrivance is more novel and grotesque, and even if it had more merit than it has, we should be ashamed to see our fellow-creatures lying prostrate on the ground performing the functions of the beasts that perish. In order to impel a railway carriage, a man is placed in a horizontal position in the carriage, supported by his body lying upon a rest. His feet are connected to stirrups, and his legs being put in motion, as in the act of swimming, he communicates a reciprocating motion, which, by intervention of rods and cranks, turns the running wheels. But while our unfortunate friend is labouring with his feet, some occupation is found also for his head, and he is entrusted with the duty of directing with his hands the steering apparatus in front.

A third plan of these gentlemen is to unite with the horizontal man a perpendicular one, who is to aid his neighbour by stepping upon treadles connected by rods and cranks with the running wheels. Such contrivances might sometimes be useful as supplementary ones, when the moving power fails; and the passengers might in turn assist in driving the carriage to its destination.

When horses are yoked in thrashing-mills, they generally pull with unequal force, and too much work is done by those which are best disposed for their work. A very ingenious contrivance for regulating the action of horses under these circumstances, was invented by Mr Walter Samuel, blacksmith at Niddry, in Linlithgowshire, and is now in actual use in many parts of Scotland. In the annexed figure, it is represented as applied to four horses.

Fig. 137. one horse, as that at 7, 8; an iron hook upon the rope F is hooked into the eye of a bolt fastened to the arm, as at R.

Another contrivance connected with the draught of horses deserves a place in this chapter. It is an invention of Baron de Prony for unyoking horses, to prevent them from injuring the machine when its motion is accidentally obstructed. It may also be employed for limiting the efforts of horses thus employed. In the annexed figure, ropes

Fig. 138.

a, a, a pass over friction-pulleys in the openings b, b of the frame dd, which is attached to the end of the lever cc fixed to the vertical axis which gives motion to the machine. Each rope terminates in an eye, which lays hold of a pin on an arbor e, which revolves upon pivots. A rope coiled round this arbor passes over pulleys pp, and suspends along the vertical axis the weight w, which forms the resistance which we wish to oppose to the efforts of the horse. Let it be twenty lbs, and let a stick obstruct the machine. The horse acts against it, and might break the machine; but the instant his effect exceeds w or twenty lbs, the arbor e will describe a quarter of a circle, and the pins having quitted their vertical position, the ends of the ropes which were there united escape from one another, the animal is fixed, and the machine stops.

CHAP. VII.—ON THE CONSTRUCTION OF WIND-MILLS.

A wind-mill is represented in fig. 139, where MN is the circular building that contains the machinery, E the extremity of the wind-shaft or principal axis, which is generally inclined from 8 to 15 degrees to the horizon; and EA, EB, EC, ED, EF five rectangular frames, upon which sails of cloth of the same form are stretched. At the lower extremity G of the sails their surface is inclined to the axis 72°; and at their farthest extremities A, D, &c. the inclination of the sail is about 83°. Now, when the sails are adjusted to the wind, which happens when the wind blows in the direction of the wind-shaft E, the impulse of the wind upon the oblique sails may be resolved into two forces, one of which acts at right angles to the wind-shaft, and is therefore employed solely in giving a motion of rotation to the sails, and the axis upon which they are fixed. When the mill is used for grinding corn, a crown-wheel, fixed to the principal axis E, gives motion to a lantern or trundle, whose axis carries the moveable millstone.

That the wind may act with the greatest efficacy upon the sails, the wind-shaft must have the same direction as of the wind. But as this direction is perpetually changing, some apparatus is necessary for bringing the wind-shaft and the sails into their proper position. This is sometimes effected by supporting the machinery on a strong vertical axis, whose pivot moves in a brass socket firmly fixed into the ground, so that the whole machine, by means of a lever, may be made to revolve upon this axis, and be properly adjusted to the direction of the wind. Most wind-mills, however, are furnished with a moveable roof, which revolves upon friction-rollers inserted in the fixed kerb of the mill; and the adjustment is effected by the assistance of a simple lever. As both these methods of adjustment require the assistance of men, it would be very desirable that the same effect should be produced solely by the action of the wind. This may be done by fixing a large wooden vane or weather-cock at the extremity of a long horizontal arm which lies in the same vertical plane with the wind-shaft. By this means, when the surface of the vane, and its distance from the centre of motion, are sufficiently great, a very gentle breeze will exert a sufficient force upon the vane to turn the machinery, and will always bring the sails and wind-shaft to their proper position. This weather-cock, it is evident, may be applied either to machines which have a moveable roof, or which revolve upon a vertical arbor.

On the Form and Position of Wind-mill Sails.

From an examination of more than fifty wind-mills in the vicinity of Lille, Coulomb has furnished us with the dimensions of the sails of the best Dutch mills. As these mills performed nearly the same amount of work when the velocity of the wind was about 19 feet per second, although there were slight differences in the arrangement of their sails, as well as in the inclination of their wind-shafts, Coulomb concluded that they were so constructed as to produce nearly a maximum effect.

In these mills the distance of the extremities of two opposite sails was sixty-six feet. The form of the sails was that of a rectangle, and their width was six feet, five feet of which was covered with cloth stretched on a frame, the remaining foot being a light board. The sail commenced 6 feet from the wind-shaft, so that the length AG was \( \frac{bb}{2} - 6 = 27 \) feet. The junction of the board with the cloth formed, on the side which faces the wind, a curve perceptibly concave at the commencement of the sail, and the curvature gradually diminished towards the extremity, where it vanished. Although every line of the surface of the cloth is curved, yet we may consider the surface of the sail as consisting of straight lines having different inclinations to the plane of the sail's motion. The inclination of these lines to this plane is called the angle of weather; and in Dutch sails this angle is 30° at the commencement G of the sail, diminishing gradually to the extremity D, where in some cases it is 12°, and in others 6°; the inclination of the wind-shaft varying from 8° to 15°.

It appears from the investigations of Parent and Belidor, that a maximum effect will be produced when the sails are inclined 54° 44' 13" to the axis of rotation, or when the angle of weather is 35° 15' 47". In obtaining this conclusion, however, M. Parent has assumed data which are inadmissible, and has neglected several circumstances which must materially affect the result of his investigations. The angle of inclination assigned by Parent is certainly the most efficacious for giving motion to the sails from a state of rest, and for preventing them from stopping when in motion; but he has not considered that the action of the wind upon a sail at rest is different from its action upon a sail in motion; for, since the extremities of the sail move with greater rapidity than the parts nearer the centre (in the Dutch mills as 33 to 6), the angle of weather should be greater towards the centre than at the extremity, and should vary with the velocity of each part of the sail. The reason of this is very obvious. It has been demonstrated by Bossut, and established by experience, that when any fluid acts upon a plain surface, the force of impulsion is always exerted most advantageously when the impelled surface is in a state of rest, and that this force diminishes as the velocity of the surface increases.

Now, let us suppose, with Parent and Belidor, that the most advantageous angle of weather for the sails of windmills is 35° 45' for that part of the sail which is nearest the centre of rotation, and that the sail has everywhere this angle of weather; then, since the extremity of the sail moves with the greatest velocity, it will in a manner withdraw itself from the action of the wind, or, to speak more properly, it will not receive the impulse of the wind so advantageously as those parts of the sail which have a less degree of velocity. In order, therefore, to counteract this diminution of force, we must make the wind act more perpendicularly upon the sail, by diminishing its obliquity or its angle of weather. But since the velocity of every part of the sail is proportional to its distance from the centre of motion, every elementary portion of it must have a different angle of weather diminishing from the centre to the extremity of the sail. The law or rate of diminution, however, is still to be discovered, and we are fortunately in possession of a theorem of Euler's, afterwards given by Maclaurin, which determines this law of variation. Let \( a \) represent the velocity of the wind, and \( c \) the velocity of any given part of the sail; then the effort of the wind upon that part of the sail will be greatest when the tangent of the angle of the wind's incidence, or of the sail's inclination to the axis, is to radius, as \( \sqrt{2 + \frac{9c^2}{4aa} + \frac{3c}{2a}} \) to 1.

In order to apply this theorem, let us suppose that the radius or whip ED of the sail \( a\beta y \), is divided into six equal parts; that the point \( n \) is equidistant from E and D, and is the point of the sail which has the same velocity as the wind; then, in the preceding theorem, we shall have \( c = a \) when the sail is loaded to a maximum; and therefore the tangent of the angle which the surface of the sail at \( n \) makes with the axis when \( a = 1 \), will be \( \sqrt{2 + \frac{9}{4} + \frac{3}{2}} = 3:561 = \text{tangent of } 74° 19' \), which gives 15° 41' for the angle of weather at the point \( n \). Since, at \( \frac{1}{2} \) of the radius \( c = a \), and since \( c \) is proportional to the distance of the corresponding part of the sail from the centre, we will have, at \( \frac{1}{2} \) of the radius, \( c = \frac{a}{3} \); at \( \frac{2}{3} \) of the radius, \( c = \frac{2a}{3} \); at \( \frac{3}{3} \), \( c = \frac{4a}{3} \); at \( \frac{4}{3} \), \( c = \frac{5a}{3} \); and at the extremity of the radius, \( c = 2a \). By substituting these different values of \( c \) instead of \( c \) in the theorem, and by making \( a = 1 \), the following table will be obtained, which exhibits the angles of inclination and weather which must be given to different parts of the sails.

| Parts of the Radius from the Centre of Motion at E. | Velocity of the Sail at these Distances, or Values of \( c \) | Angle made with the Axis | Angle of Weather | |---------------------------------------------------|-------------------------------------------------|------------------------|----------------| | \( \frac{1}{2} \) | \( \frac{a}{3} \) | 63 | 26 | | \( \frac{2}{3} \) | \( \frac{2a}{3} \) | 69 | 54 | | \( \frac{3}{3} \) or \( \frac{1}{2} \) | \( \frac{a}{3} \) | 74 | 19 | | \( \frac{4}{3} \) or \( \frac{2}{3} \) | \( \frac{4a}{3} \) | 77 | 20 | | \( \frac{5}{3} \) | \( \frac{5a}{3} \) | 79 | 27 | | 1 | \( 2a \) | 81 | 0 |

Mr Smeaton found, from a variety of experiments, that Results of the common practice of inclining plane sails from 72° to Smeaton's 75° to the axis, was much more efficacious than the angle experimentally assigned by Parent, the effect being as 45 to 31. When the sails were weathered in the Dutch manner, that is, when their surfaces were concave to the wind, and when the angle of inclination increased towards their extremities, they produced a greater effect than when they were weathered either in the common way, or according to Euler's theorem. But when the sails were enlarged at their extremities, as represented at \( \alpha \beta \) in fig. 140, so that \( \alpha \beta \) was one third of the radius ED, and \( aD \) to \( D\beta \) as 5 to 3, their power was greatest of all, though the surface acted upon by the wind remained the same. If the sails be farther enlarged, the effect is not increased in proportion to the surface; and besides, when the quantity of cloth is great, the machine is much exposed to injury by sudden squalls of wind. In Mr Smeaton's experiments, the angle of weather varied with the distance from the axis; and it appeared from several trials, that the most efficacious angles were those in the following table.

| Parts of the Radius ED, which is divided into six parts. | Angle with the Axis | Angle of Weather | |----------------------------------------------------------|--------------------|-----------------| | 1 | 72 | 18 | | 2 | 71 | 19 | | 3 | 72 | 18 middle | | 4 | 74 | 16 | | 5 | 77½ | 12½ | | 6 | 83 | 7 |

If the radius ED of the sail be 30 feet, then the sail will commence at \( \frac{1}{2} \) ED, or 5 feet from the axis, where the angle of inclination will be 72°. At \( \frac{2}{3} \) ED, or 10 feet from the axis, the angle will be 71°, and so on.

It has been proposed by Euler and other writers to make the sails elliptical sectors, and to intercept the whole cy- On the Relative and Absolute Effects of Wind-Mill Sails.

The following maxims, deduced by Smeaton from his experiments, contain the most accurate information upon this subject.

Maxim 1. The velocity of wind-mill sails, whether unloaded or loaded, so as to produce a maximum effect, is nearly as the velocity of the wind, their shape and position being the same.

Maxim 2. The load at the maximum is nearly, but somewhat less than, as the square of the velocity of the wind, the shape and position of the sails being the same.

Maxim 3. The effects of the same sails at a maximum are nearly, but somewhat less than, as the cubes of the velocity of the wind.

Maxim 4. The load of the same sails at the maximum is nearly as the squares, and their effects as the cubes, of their number of turns in a given time.

Maxim 5. When sails are loaded so as to produce a maximum at a given velocity, and the velocity of the wind increases, the load continuing the same,—i.e., the increase of effect, when the increase of the velocity of the wind is small, will be nearly as the squares of those velocities; 2dly, when the velocity of the wind is double, the effects will be nearly as 10 : 27 ; but, 3dly, when the velocities compared are more than double of that where the given load produces a maximum, the effects increase nearly in the simple ratio of the velocity of the wind.

Maxim 6. In sails where the figure and positions are similar, and the velocity of the wind the same, the number of turns in a given time will be reciprocally as the radius or length of the sail.

Maxim 7. The load at a maximum that sails of a similar figure and position will overcome at a given distance from the centre of motion, will be as the cube of the radius.

Maxim 8. The effects of sails of similar figure and position are as the square of the radius.

Maxim 9. The velocity of the extremities of Dutch sails, as well as of the enlarged sails, in all their usual positions when unloaded, or even loaded to a maximum, is considerably quicker than the velocity of the wind.

The following very important results were obtained by M. Coulomb, in examining the wind-mills at Lille, in the Netherlands. These mills were made for the manufacture of oil of rape-seed, and were constructed in the following manner: Seven stampers for bruising the seeds were raised by means of an arbor carrying 14 wipers. Five of the stampers were of oak, each from 20 to 22 feet long, and from 9 to 11 inches square; their lower ends were shod with iron, weighing from 50 to 60 lbs.; and each stamper weighed about 1020 lbs. The other two stampers were of the same length, but only from 6 to 7 inches square, and weighed about 500 lbs. each: their purpose was to lock and unlock the wedges used in expressing the oil. One of these two only acted at once, but the rest acted together.

| Velocity of Wind in Feet per Second | Revolutions of Sails per Minute | Number of Stampers used | Weight of Stampers in lbs. | Work done in Twenty-four Hours | |-------------------------------------|---------------------------------|------------------------|--------------------------|-------------------------------| | 7 | 5 | No stampers raised | No work done | Striking two blows each | | | | | | 18 inches high |

One ton is equal to 100 kilogrammes.

* All the sails were carried, the wind blew uniformly, and the velocity seemed to be the best suited to the machine.

† An area of two inches was taken in at the extremity of each sail.

‡ In this experiment the mill carried all her sails; but the flour was heated considerably, and in such cases the millers change from time to time the kind of grain to be ground, to refresh the mill, as they term it.

By taking an average of the work performed during several years, Coulomb found that each wind-mill manufactured annually 400 tons of oil. About one sixth of the force communicated to the stampers he conceives to be absorbed by their blows and the action of the wipers, including friction. Hence he calculates that 100 kilogrammes or one ton of oil requires for its manufacture 12,500 dynamical units.

M. Halle at Lille found that a steam-engine of ten-horse power prepared in twenty-four hours 500 kilogrammes of oil, which gives fifty kilogrammes for each horse. But we have already seen that the power of a horse working twenty-four hours is equal to 5974 dynamical units. Hence

\[ \text{Kilogrammes} = \frac{5974}{100} = 119.48 \text{ units}, \]

agreeing very nearly with 12,500, the result obtained by Coulomb.

By supposing three tons to be the average work of one mill during an average state of the wind, we have

\[ \frac{3}{400} = 133 \text{ for the number of days of twenty-four hours to which the whole wind of the year at Lille is equivalent, blowing at the rate of a little less than four metres per second.} \]

Coulomb likewise examined the effect of wind-mills in grinding corn.

The millstone performed five revolutions in the same time that the sails and vertical arbor performed one.

| Velocity of Wind in Feet per Second | Revolutions of Sails in a Minute | Ditto of the Millstone | Work performed in Twenty-four Hours, or Flour ground | |-------------------------------------|---------------------------------|------------------------|----------------------------------------------------| | 4 metres | 11½ | 57½ | 10,000 kilogs. | | 9½ | 22 | 110 | 21,600‡ |

From these and various other experiments, M. Hachette has drawn the conclusion, that from 500 to 550 dynamical units are required to grind 100 kilogrammes of corn.

M. Lulofs of Leyden examined a Dutch wind-mill that Effects in drained marshes, and which had four rectangular sails set raising water at a weather of 17°. It raised 1500 cubic feet of water four feet high in a minute when the velocity of the wind was 30 feet per second.

Coulomb found that in the Dutch mills, whose dimensions are given in page 432, the effect was 1000 lbs. raised 218 feet in a minute when the wind blew 20 feet in a second. He computed also that the force lost by friction was 1000 lbs. raised 18½ feet per minute; and that the force consumed by the action of the wipers on the stampers was equal to 1000 lbs. raised 16½ feet in a minute. Hence it follows that the total force of the wind was equivalent to 1000 lbs. raised 218 + 16½ + 18½ = 253 feet in a minute.

Both Smeaton and Coulomb have attempted to determine the ratio between the velocity of the wind and that On the Construction of Wind-Mills.

Smeaton has given the following results from Dutch sails in their common position, when they had a radius of 50 feet.

| Velocity of Wind per Hour | Revolutions of the Arbor in a Minute | Ratio of these two Velocities | |---------------------------|-------------------------------------|-------------------------------| | 2 miles | 3 | 0·666 to 1 | | 4 | 5 | 0·800 to 1 | | 5 | 6 | 0·833 to 1 |

Coulomb's results, which are as follows, differ widely from those of Smeaton, though they are susceptible of explanation.

| Velocity of Wind per Hour | Velocity of the Wind per Second | Revolutions of the Arbor in a Minute | Ratio of these two Velocities | |---------------------------|---------------------------------|-------------------------------------|-------------------------------| | 5·1 miles | 2·27 metres | 3 | 1·7 to 1 | | 9·14 | 4·06 | 7½ | 1·22 to 1 | | 14·6 | 6·5 | 13 | 1·05 to 1 | | 20·4 | 9·1 | 17½ | 1·17 to 1 | | 13·0 | 5·8 | 11½ | 1·13 to 1 | | 20·4 | 9·1 | 22 | 0·93 to 1 |

In order to account for the discrepancies between these results and those of Smeaton, we may observe, that in the two first experiments, where the difference is the greatest, viz. 1·7 to 1, and 1·22 to 1, the mill was not doing its full work. The average of the ratios in the other four experiments, where the mill was doing full work, is 1·07, which is less different from Smeaton's results.

The principal cause of the discrepancy, however, is that the ratio increases, as in all Smeaton's experiments, with the velocity of the wind; and as Smeaton's highest velocity was only five miles per hour, whereas the average velocity in the four last experiments of Coulomb was so high as 17·6 miles per hour, it is not difficult to understand how the ratios deduced from them should be so high.

A new mode of constructing the sails of wind-mills has been recently given by Mr Sutton, and fully described by Mr Hesleden of Barton, in a work exclusively devoted to the subject.

The limits of this article will not permit us to enter into any discussion respecting the principles upon which Mr Sutton's gravitated sails are constructed. It may be proper, however, to remark, that Mr Sutton gives his sails the form represented in fig. 141, and makes the angle of weather at the point M, equidistant from A and B, equal to 22° 30'. The inclination of the sail at any other point N of the sail, is an angle whose sine is the distance of that point from the centre of motion A, the radius being the breadth of the sail at that point. Fig. 141 shows the angles at the different points of the sail, and the apparent and absolute breadths of the sail at these points. Mr Sutton's mode of regulating the velocity of the sails, and of bringing them into a state of rest, is particularly ingenious.

ON HORIZONTAL WIND-MILLS.

Various opinions have been entertained respecting the relative advantages of horizontal and vertical wind-mills. Mr Smeaton, with great justice, gives a decided preference to the latter; but when he asserts that horizontal wind-mills have only \( \frac{1}{3} \) or \( \frac{1}{10} \) of the power of vertical ones, he certainly forms too low an estimate of their power. Mr Beatson, on the contrary, who has received a patent for the construction of a new horizontal windmill, seems to be prejudiced in their favour, and greatly exaggerates their comparative value. From an impartial investigation, it will probably appear that the truth lies between these two opposite opinions; but before entering on this discussion, we must first consider the nature and form of horizontal wind-mills.

In fig. 142 CK is the wind-shaft, which moves upon pivots. Four cross bars, CA, CD, IB, FG, are fixed to this arbor, which carry the frames AP, IB, DEFG. The sails AI, EG, are stretched upon these frames, and are carried round the axis CK, by the perpendicular impulse of the wind. Upon the axis CK a toothed wheel is fixed, which gives motion to the particular machinery that is employed. In the figure only two sails are represented; but there are always other two placed at right angles to these.

Now, let the sails be exposed to the wind, and it will be evident that no motion will ensue; for the force of the wind upon the sail AI, is counteracted by an equal and opposite force upon the sail EG. In order, then, that the wind may communicate motion to the machine, the force upon the returning sail EG must either be removed by screening it from the wind, or diminished by making it present a less surface when returning against the wind.

The first of these methods is adopted in Tartary, and in some provinces of Spain; but is objected to by Mr Beatson, from the inconvenience and expense of the machinery and attendance requisite for turning the screens into their proper positions. Notwithstanding this objection, however, we are disposed to think that this is the best method of diminishing the action of the wind upon the returning sails, for the moveable screen may be easily made to follow the direction of the wind, and assume its proper position, by means of a large wooden weathercock, without the aid either of men or machinery. It is true, indeed, that the resistance of the air in the returning sails is not completely removed; but it is at least as much diminished as it can be by any method hitherto proposed. Besides, when this plan is resorted to, there is no occasion for any moveable flaps and hinges, which must add greatly to the expense of every other method.

The mode of bringing the sails back against the wind, Beatson's which Mr Beatson invented, is, perhaps, the simplest and best of the kind. He makes each sail AI to consist of six or eight flaps or vanes, APb1, b1e2, &c., moving upon hinges represented by the dark lines AP, b1, e2, &c., so that the lower side b1 of the first flap overlaps the hinge or higher side of the second flap, and so on. When the wind, therefore, acts upon the sail AI, each flap will press upon the hinge of the one immediately below it, and the whole surface of the sail will be exposed to its action. But when the sail AI returns against the wind, the flaps will revolve round upon their hinges, and present only their edges to the wind, as is represented at EG, so that the resistance occasioned by the return of the sail must be greatly diminished, and the motion will be continued by the great superiority of force exerted upon the sails in the position AI. In computing the force of the wind upon the sail AI, and the resistance opposed to it by the edges of the flaps in EG, Mr Beatson finds, that when the pressure upon the former is 1872 pounds, the resistance opposed by the latter is only about thirty-six pounds, or \( \frac{1}{50} \) part of the whole force; but he neglects the action of the wind upon the arms CA, &c., and the frames which carry the sails, because they expose the same surface in the position AI, as in the position EG. This omission, however, has a tendency to mislead us in the present case, as we shall now see; for we ought to compare the whole force exerted upon the arms, as well as the sail, with the whole resistance which these arms and the edges of the flaps oppose to the motion of the wind-mill. By inspecting fig. 142 it will appear, that if the force upon the edges of the flaps, which Mr Beatson supposed to be twelve in number, amounts to thirty-six pounds, the force spent upon the bars CD, DG, GF, FE, &c., cannot be less than sixty pounds. Now, since these bars are acted upon with an equal force when the sails have the position AI, $1872 + 60 = 1932$ will be the force exerted upon the sail AI and its appendages, while the opposite force upon the bars and edges of the flaps, when returning against the wind, will be $36 + 60 = 96$ pounds, which is nearly $\frac{3}{10}$ of 1932, instead of $\frac{1}{2}$, as computed by Mr Beatson. Hence we may see the probable advantages of a screen over moveable flaps, as it will preserve not only the sails, but the arms and the frame which support it, from the action of the wind.

A very ingenious method of bringing the sails back against the wind has been invented by Mr G. Buchanan of Edinburgh, and is represented in the annexed figure. The sails are attached, as usual, to the horizontal arms of the vertical axis CBA, and each of them moves horizontally round its own axis, as shown at H, in an opposite direction to, and with half the angular direction of, the horizontal arms. In order to effect this, a toothed wheel ab is attached to the frame EF, and is therefore stationary while the mill revolves; bc is a toothed pinion working in ab, and moved by its axis cd, while this axis, fixed to the arm HI, revolves with the principal axis CBA, through which it passes at h. Two pinions f, g, are fixed to the ends of the axis cd, and work in the two wheels h, i, which are attached to the axis of the sails. They will, therefore, move round these sails in the same time, and with a velocity depending on the relative number of teeth in the wheels and pinions, and so adjusted as to give them half the angular velocity of the principal axis CBA. The wheel ab is made moveable round its axis, so that it may be shifted into any position, and keep the sails always to the wind.

By these means, when one of the sails M, is fully exposed to the action of the wind, the opposite one N, will only present its edge to the wind. As M advances, it gradually turns itself away from the wind, till it comes round into the position N, when the wind ceases to have any power over it. All the other sails do the same, and it will readily be seen, by constructing a diagram of the relative position of the sails at any point of their revolution, that the area of the returning sails exposed to the wind is always much less than the area of the sails urged forward by the wind.

We shall now conclude this chapter with a comparison of the power of horizontal and vertical wind-mills. It has been already stated, that Mr Smeaton rather underrated the former, when he maintained that they have only one eighth or one tenth the power of the latter. He observes, that when the vanes of a horizontal and a vertical mill are equal in dimensions, the power of the latter is four times that of the former, because, in the first case, only one sail is acted upon at once by the wind, while, in the second, all the four receive its impulse. This, however, is not strictly true, since the vertical sails are oblique to the direction of the wind. Let us suppose that the area of each sail is 100 square feet; then the power of the horizontal one will be 100, and the power of a vertical one may be called $100 \times \sin(70^\circ)$ ($70^\circ$ being the common angle of inclination) = 88 nearly; but since there are four vertical sails, the power of them all will be $4 \times 88 = 352$; so that the power of the horizontal sail is to that of the four vertical ones as 1 to 3.52, and not as 1 to 4, according to Mr Smeaton. But Mr Smeaton likewise observes, that if we consider the disadvantage which arises from the difficulty of getting the sails back against the wind, we need not wonder if horizontal wind-mills have only about one eighth or one tenth the power of the common sort. We have already seen, that the resistance occasioned by the return of the sails amounts to one twentieth of the whole force which they receive; by subtracting $\frac{1}{20}$ therefore, from

$$\frac{1}{3.52}$$ we shall find that the power of horizontal wind-mills is only $\frac{1.03}{4.40}$, or little more than one fourth that of vertical ones. This calculation proceeds upon a supposition that the whole force exerted upon vertical sails is employed in turning them round the axis of motion, whereas a considerable part of this force is lost in pressing the pivot of the axis or wind-shaft against its gudgeon. Mr Smeaton has overlooked this circumstance, otherwise he could never have maintained that the power of four vertical sails was quadruple the power of one horizontal sail, the dimensions of each being the same. Taking this circumstance into the account, we cannot be far wrong in saying, that, in theory at least, if not in practice, the power of a horizontal wind-mill is about one third or one fourth of the power of a vertical one when the quantity of surface and the form of the sails is the same, and when every part of the horizontal sails has the same distance from the axis of motion as the corresponding parts of the vertical sails. But if the horizontal sails have the position AI, EG, in fig. 142, instead of the position CADm, CDom, their power will be greatly increased, though the quantity of surface is the same, because the part CP3m, being transferred to BI3d, has much more power to turn the sails.

Mechanical Agents not generally introduced.

In giving an account of the mechanical agents which are or may be employed to move machinery, we must enumerate elastic springs, heated air, explosive mixtures, and carbonic acid gas.

The spring is so common a first mover in our watches and clocks, and the method of rendering its varying force uniform by the fusee so well known, that it would be a waste of time to describe it here, and the more so as it will be referred to in another part of the work. See Scholium, p. 365. Heated air has been occasionally employed as a source of mechanical power. We have seen the model of a pump, in which the air in the barrel was rarefied by burning wood shavings at the top of the barrel, an air-tight cap being put on when the rarefaction was supposed to be a maximum. A certain quantity of water was thus raised above the valve at the bottom of the barrel, and the operation was repeated till the water rose to the desired height.

About ten or fifteen years ago a patent was taken out for a heated air-engine, by the Rev. Mr Stirling; and we believe that several such engines were actually constructed, and performed useful work, at Perth, Glasgow, and other places. The air was heated in an iron cylinder kept at a red heat, and after acting mechanically by its expansive force, it was cooled and returned to the cylinder by a very ingenious contrivance. The practical difficulty, however, which was encountered arose from the speedy destruction of the iron cylinder.

In the year 1824, Mr Samuel Brown took out a patent for what he called an atmospherical engine, in which a vacuum was created by burning coal or oil gas within the cylinder, and thus consuming the atmospherical air. The statements of the power of this engine were greatly exaggerated, and high expectations were formed of it, but we believe that it turned out a total failure.

The Rev. Mr Cecil of Cambridge had, previously to Mr Brown's patent, invented an engine in which the power was obtained by taking advantage of the vacuum created by the explosion of a mixture of hydrogen gas and common air; and he also suggested that the expansive force of this explosion might be employed as the first mover of machinery. A mixture of one measure of hydrogen with two and a half measures of common air expands to three times its bulk on ignition, and then collapses instantly to about one half of the original volume of the mixture. If, therefore, we take a cylindrical vessel divided at one third of its length into two portions by a throttle valve (one which moves round its axis in the line of its diameter); if there be a solid piston in the shorter portion, and if at the end of the larger portion there is placed a light valve opening easily outward; then, supposing the throttle valve closed, and the piston lying down upon it, if the piston be now raised, and the mixed gases be permitted to flow in to supply the void left by the piston till it reach the end of its stroke; if at this time the valve be opened, and at the same instant a jet of flame be introduced at a touch-hole in the side of the cylinder to fire the explosive mixture, then the explosion will drive out the common air by the end valve from the larger portion, the valve will close, and the pressure of the atmosphere into the rare medium in the cylinder will force down the piston until it reaches nearly to the middle valve, when the mixed gases are again admitted, the middle valve closed, and the end one opened. The ascent of the piston will of course be produced by the action of a fly-wheel put in motion by the downward stroke.

In this machine the expansive force of the explosion does not produce the moving power, as it might be made to do. It merely drives out the common air, and produces a partial vacuum. The pressure of the atmosphere is in reality the moving power.

When Mr Faraday had made his beautiful discovery of the liquefaction of chlorine gas, Sir Humphry Davy communicated to the Royal Society of London a paper "On the application of liquids formed by the condensation of gases as mechanical agents." In this way the attention of Mr Brunel was turned to the application of liquid carbonic acid gas as a first mover of machinery. Its remarkable powers of expansion by small degrees of heat pointed it out as peculiarly adapted for mechanical purposes, and the highest expectations were excited when Mr Brunel directed his great mechanical genius to the subject. He soon contrived a suitable mechanism for the application of this extraordinary agent, and took out a patent for it in England, and another in France in concert with MM. Termaux and Delessert. In the apparatus which he prepared, the liquefied acid was at the temperature of 50°, and under the pressure of 30 atmospheres. The liquid gas was contained in two cylinders placed at the two extremities of the apparatus, and put in communication. The destruction of the equilibrium was effected by varying the temperature of the liquid in one of the condensers. Such, however, was the effect of the heat upon the liquid, that for a rise of temperature of 180° we obtain a pressure of 90 atmospheres, an enormous power, which having no resistance but that of the gas in the other condenser, tends to displace a piston with a force of 90 — 30 = 60 atmospheres. Of this engine Mr Brunel constructed a working model, and he afterwards began a machine with a power of eight horses. In the course of his experiments, however, with this machine, he met with anomalies in the law of expansion of the liquefied gas which seemed to throw insuperable difficulties in his way, and led him to abandon his undertaking.

At the time that Mr Brunel was occupied with this subject, Mr Faraday had obtained it only in small quantities; but M. Thilorier has lately discovered a new process, by which it may be obtained in quantities in a very few moments. From this cause M. Thilorier has had an opportunity of studying its properties; and we hope it may yet be found to realise at least some of the anticipations of earlier chemists. He found that the pressure of the vapour formed by the liquefied gas from 32° to 86° Fahr. amounts to from 36 to 73 atmospheres, and its volume from 20 to 29, the pressure being equal to an increase of one atmosphere to every centigrade degree, and the expansion being four times greater than that of atmospheric air. At 86° its specific gravity is 0-60, at 32° it is 0-83, and at — 68° it is 0-90.

Comparison of the Quantity of Action furnished by different Mechanical Agents.

When a horse is employed in raising coals by means of a wheel and axle, and moves at the rate of about two miles an hour, Mr Fenwick found that he could continue at work 12 hours each day, two and a half of which were spent in short intervals of rest, when he raised a load of 1000 pounds avoirdupois, with a velocity of 13 feet per minute; and that he will exert a force of 75 pounds for nine hours and a half when moving with the same velocity. Mr Fenwick also found that 230 ale gallons of water delivered every minute on an overshot water-wheel, 10 feet in diameter; that a common steam-engine, with a cylinder 8 inches in diameter, and an improved engine with a cylinder 6-1/2 inches in diameter, will do the work of one horse, that is, will raise a weight of 1000 pounds avoirdupois through the height of 13 feet in a minute. It appears from Mr Smeaton's experiments, that Dutch sails in their common position with a radius of nine feet and a half, that Dutch sails in their best position with a radius of eight feet, and that his enlarged sails with a radius of seven feet, perform the same work as one man, or perform one fifth part of the work of a horse. Upon these facts we have constructed the following table, the first four columns of which are taken from Mr Fenwick's Essays on Practical Mechanics.

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1 See Newton's Journal of Arts and Sciences, vol. viii. No. 44, p. 57; and Edinburgh Journal of Science, vol. i. p. 339. 2 On the application of hydrogen gas to produce a moving power, Cambridge Transactions, vol. i. 3 See Edinburgh Journal of Science, No. ix. p. 168; London and Edinburgh Philosophical Magazine, June 1836, No. 56, p. 583; and Le Globe, tom. iii. No. 29. ### Table showing the relative Strength of Overshot-Wheels, Steam-Engines, Horses, Men, and Wind-Mills of different kinds.

| Number of Ale Gallons delivered on an Overshot Wheel, 10 feet in diameter, every Minute | Diameter of the Cylinder in the Common Steam-engine, in inches | Diameter of the Cylinder of the Improved Steam-engine, in inches | Number of Horses working 12 Hours per Day, and moving at the rate of two Miles per Hour | Number of Men working 12 Hours a Day | Radius of Dutch Sails in their common position, in feet | Radius of Dutch Sails in their best position, in feet | Radius of Mr Smeaton's enlarged Sails, in feet | Height to which these different Powers will raise 1000 pounds avordupois in a Minute | |---|---|---|---|---|---|---|---|---| | 230 | 8 | 6-12 | 1 | 5 | 21-24 | 17-89 | 15-65 | 13 | | 390 | 9-5 | 7-8 | 2 | 10 | 30-04 | 25-30 | 22-13 | 26 | | 528 | 10-5 | 8-2 | 3 | 15 | 36-80 | 30-98 | 27-11 | 39 | | 660 | 11-5 | 8-6 | 4 | 20 | 42-48 | 35-78 | 31-30 | 52 | | 790 | 12-5 | 9-35 | 5 | 25 | 47-50 | 40-00 | 35-00 | 65 | | 970 | 14 | 10-55 | 6 | 30 | 52-03 | 43-82 | 38-34 | 78 | | 1170 | 15-4 | 11-75 | 7 | 35 | 56-90 | 47-33 | 41-41 | 90 | | 1350 | 16-8 | 12-8 | 8 | 40 | 60-09 | 50-60 | 44-27 | 104 | | 1445 | 17-3 | 13-6 | 9 | 45 | 63-73 | 53-66 | 46-96 | 117 | | 1584 | 18-5 | 14-2 | 10 | 50 | 67-17 | 56-57 | 49-50 | 130 | | 1740 | 19-4 | 14-8 | 11 | 55 | 70-46 | 59-33 | 51-91 | 143 | | 1900 | 20-2 | 15-2 | 12 | 60 | 73-59 | 61-97 | 54-22 | 156 | | 2100 | 21 | 16-2 | 13 | 65 | 76-59 | 64-50 | 56-43 | 169 | | 2300 | 22 | 17 | 14 | 70 | 79-49 | 66-94 | 58-57 | 182 | | 2500 | 23-1 | 17-8 | 15 | 75 | 82-27 | 69-28 | 60-62 | 195 | | 2686 | 23-9 | 18-3 | 16 | 80 | 84-97 | 71-55 | 62-61 | 208 | | 2870 | 24-7 | 19 | 17 | 85 | 87-07 | 73-32 | 64-16 | 221 | | 3055 | 25-5 | 19-6 | 18 | 90 | 90-13 | 75-90 | 67-41 | 234 | | 3240 | 26-25 | 20-1 | 19 | 95 | 92-60 | 77-98 | 68-23 | 247 | | 3420 | 27 | 20-7 | 20 | 100 | 95-00 | 80-00 | 70-00 | 260 | | 3750 | 28-5 | 22-2 | 22 | 110 | 99-64 | 83-90 | 73-42 | 286 | | 4000 | 29-8 | 23 | 24 | 120 | 104-06 | 87-63 | 76-68 | 312 | | 4460 | 31-1 | 23-9 | 26 | 130 | 108-82 | 91-22 | 79-81 | 338 | | 4850 | 32-4 | 24-7 | 28 | 140 | 112-20 | 94-66 | 82-82 | 364 | | 5250 | 33-6 | 25-5 | 30 | 150 | 116-35 | 97-98 | 85-73 | 390 |

Dutch sails are always constructed so that the angle of weather may diminish from the centre to the extremity of the sail. They are concave to the wind, and are in their *common position* when their extremities are parallel to the plane in which they move, or perpendicular to the direction of the wind. Dutch sails are in their *best position* when their extremities make an angle of seven degrees with the plane of their motion. Mr Smeaton's enlarged sails are Dutch sails in their best position, but enlarged at their extremities.

It appears from M. Coulomb's experiments on Dutch wind-mills with rectangular sails, that when the distance between the extremities of two opposite sails is 66 feet French, and the breadth of each sail six feet, a wind moving at the rate of 20 feet per second will produce an effect equivalent to 1000 pounds raised through the space of 218 feet in a minute.

It will be seen in our article *Steam-Engine*, that the efficiency of a steam-engine is equal to the column of water in pounds raised at each stroke, multiplied by the length of the stroke and the number of strokes in a given time; and that what is called the *duty* of a steam-engine is equal to the same quantity divided by the number of bushels of coal consumed in the same time.

One of the old standard engines, as they were made in 1778, wrought pumps 17 inches in diameter at the depth of 58 fathoms, at the rate of six strokes in a minute, each stroke being 5½ feet long. Its consumption of coal was 14,080 bushels in 61 days, hence its duty was 6,901,753.

In the year 1793, the improvements of Messrs Boulton and Watt were so important, that the average duty of 17 engines of their construction was 19,569,000, being an increase of duty in the ratio of 28 to 10.

Since that time the use of steam of high temperature, the construction of fire-places with superior drafts, or iron tubes passing through the boiler, and other improvements in the execution of the machinery, have given such additional power to the steam-engine, that in 1829 the highest monthly average duty of one of the engines was 79,000,000, surpassing in duty the engine of 1778 in the ratio of 11½ to 1, and those of 1793 as 4 to 1.

Mr Taylor expresses the improvement on steam-engines by the number of bushels of coal that are requisite to perform a given piece of work.

With the early steam-engine..........................16 bushels. During Boulton and Watt's patent......................4 About the year 1825........................................2 The best Cornish engines.................................1

The following interesting comparisons have been given by Sir John Herschel. "It is well known," says he, "to modern engineers, that there is virtue in a bushel of coals, properly consumed, to raise seventy millions of pounds weight a foot high. This is actually the average effect of an engine at this moment working in Cornwall. Let us pause

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1 According to Watt and Boulton, one of their steam-engines, with a cylinder 31 inches in diameter, and which makes 17 double strokes per minute, is equivalent to 40 horses working day and night; that is, to 101 horses working nine hours and a half, the time of constant exertion in the preceding table. When the cylinder is 19 inches in diameter, and the engine makes 25 strokes of four feet each per minute, its power is equivalent to twelve horses working constantly, or thirty horses working nine hours and a half; and when the cylinder is 24 inches in diameter, and the engine makes 22 strokes of five feet each in a minute, its power is equal to that of 20 horses working constantly, or 50 horses working nine hours and a half. a moment, and consider what this is equivalent to in matters of practice.

"The ascent of Mont Blanc from the valley of Chamouni is considered, and with justice, as the most toilsome feat that a strong man can execute in two days. The combustion of two pounds of coal would place him on the summit.

"The Menai Bridge, one of the most stupendous works of art that has been raised by man in modern ages, consists of a mass of iron not less than four millions of pounds in weight, suspended at a medium height of about 120 feet above the sea. The consumption of seven bushels of coal would suffice to raise it to the place where it hangs.

"The great pyramid of Egypt is composed of granite. It is 700 feet in the side of its base, and 500 in perpendicular height, and stands on eleven acres of ground. Its weight is therefore 12,760 millions of pounds, at a medium height of 125 feet; consequently it would be raised by the effort of about 630 chaldrons of coal, a quantity consumed in some founderies in a week.

"The annual consumption of coal in London is estimated at 1,500,000 chaldrons. The effort of this quantity would suffice to raise a cubical block of marble, 2200 feet in the side, through a space equal to its own height, or to pile one such mountain upon another.

"The Monte Nuovo, near Pozzuoli (which was erupted in a single night by volcanic fire), might have been raised by such an effort from a depth of 40,000 feet, or about eight miles. It will be observed, that in the above statement the inherent power of fuel is, of necessity, greatly underrated. It is not pretended by engineers that the economy of fuel is yet pushed to its utmost limit, or that the whole effective power is obtained in any application of fire yet devised; so that were we to say 100 millions instead of 70, we should probably be nearer the truth."

CHAP. VIII.—ON THE CONSTRUCTION OF WHEEL-CARRIAGES.

It is evident, from Cor. S, p. 360, that when a wheel surmounts an obstacle, it acts as a lever of the first kind, and that its power to overcome such resistances increases with its diameter. The power of the force P, for example, to raise the wheel NB over the eminence C, is proportional to the vertical lever FC, which increases with the diameter of the wheel, while the lever of resistance FA, by which the weight of the wheel acts, remains unchanged; hence we see the advantages of large wheels for overcoming such obstacles as generally resist the motion of wheel-carriges. There are some circumstances, however, which, independent of the additional weight and expense of large wheels, prescribe limits to their size. If the radius AC of the wheel exceeds the height of that part of the horse to which the traces are attached, the line of traction DA will be oblique to the horizon, and part of the power P will be employed in pressing the wheel upon the ground. A wheel exceeding four and a half feet radius, which is the general distance from the ground of that part of the horse to which the traces are attached, has still the advantage of a smaller wheel; but when we consider that the traces or poles of the cart will, in this case, rub against the flanks of the horses, so that the power of the wheel is diminished by the increase of its weight, we shall be convinced that no power is gained by making the radius of the wheels greater than four and a half feet. Even this size is too great, as shall be afterwards shown when we treat of the line of traction, so that we may safely assert that the diameters of wheels should never be greater than six feet. The fore-wheels of our carriages are still unaccountably small, and it is not uncommon to see carts moving upon wheels scarcely fourteen inches in diameter. The convenience of turning is urged as the reason for diminishing the fore-wheels of carriages, and the facility of loading the cart is considered as a sufficient reason for using wheels so small as fourteen inches. The first of these advantages, however, may be obtained by going to the end of a street, or to a proper place for turning the carriage; and a few additional turns of a windlass will be sufficient to convey the heaviest loads into carts mounted on high wheels.

The next thing to be determined is the shape of the wheels. A cylindrical wheel, with the spokes perpendicular to the naves, is perhaps the form which every mechanic would give to his wheels, before he had heard of the advantages of concave or dishing wheels, or those which have inclined spokes and conical rims.

It has been alleged, indeed, and with truth, that the form represented in fig. 145, where Ar, Bs is the conical rim, and oA, pB the inclined spokes, renders the wheel stronger than it would otherwise be; that by extending the base of the carriage it prevents it from being overturned; that it hinders the felloes from rubbing against the load or the sides of the cart; and that when one wheel falls into a rut, and therefore supports more than one half of the load, the spokes are brought into a vertical position, which renders them more capable of sustaining the additional weight.

Two of these advantages exist only in very bad roads; and if they are necessary, in a country like this, where the roads are so excellently made and so regularly repaired, they can easily be procured, by making the axle-tree a few inches longer, and increasing the strength of the spokes. But it is allowed on all hands that perpendicular spokes are preferable on level ground. The inclination of the spokes, therefore, which renders concave wheels advantageous in rugged and unequal roads, renders them disadvantageous when the roads are in good order; and where the good roads are more numerous than the bad ones, as they certainly are in this country, the disadvantages of concave wheels must overbalance their advantages. It is true, indeed, that in concave wheels the spokes are in their strongest position when they are exposed to the severest strains, that is, when one wheel is in a deep rut, and sustains more than one half of the load; but it is equally true, that on level ground, where the spokes are in their weakest position, a less severe strain, by continuing for a much longer time, may be equally if not more detrimental to the wheel.

We shall now endeavour to show that concave dishing wheels are more expensive, more injurious to the roads, and more liable to be broken by accidents, than those wheels in which the spokes are perpendicular to the naves. By inspecting fig. 145, it will appear that the whole of the pressure which the wheel AB sustains is exerted along the inclined spoke ps, and therefore acts obliquely on the level ground nd, whether the rims are conical or cylindrical. This oblique action must necessarily injure the roads, by loosening the stones more between B and D than between B and n; and if the load were sufficiently great, the stones would start up between s and D. The texture of the roads, indeed, is sufficiently firm to prevent this from tak- ing place; but, in consequence of the oblique pressure, the stones between s and D will at least be loosened, and, by admitting the rain, the whole of the road will be materially damaged. But when the spokes are perpendicular to the nave, as pa, and when the rims mA, MB are cylindrical, or parallel to the ground, the weight sustained by the wheel will act perpendicularly upon the road; and however much that weight is increased, its action can have no tendency to derange the materials of which it is composed, but is rather calculated to consolidate them, and render the road more firm and durable.

It was observed that concave wheels are more expensive than plane ones. This additional expense arises from the greater quantity of wood and workmanship which the former require; for in order that dishing wheels may be of the same perpendicular height as plane ones, the spokes of the former must exceed in length those of the latter, as much as the hypotenuse oA of the triangle oAm exceeds the side om; and therefore the weight and the resistance of such wheels must be proportionally great. The inclined spokes, too, cannot be formed nor inserted with such facility as perpendicular ones. The extremity of the spoke which is fixed into the nave is inserted at right angles to it, in the direction op; and if the rims are cylindrical, the other spoke should be inserted in a similar manner, while the intermediate portion has an inclined position. There are therefore two flexures or bendings in the spokes of concave wheels, which requires them to be formed out of a larger piece of wood than if they had no such flexures, and renders them liable to be broken by any sudden strain at the points of flexure.

The obstacles which carriages have to encounter are almost never spherical protuberances, that permit the elevated wheel to resume by degrees its horizontal position. They are generally of such a nature that the wheel is instantaneously precipitated from their top to the level ground. Now the momentum with which the wheel strikes the ground is very great, arising from a successive accumulation of force. The velocity of the elevated wheel is considerable when it reaches the top of the eminence; and while it is tumbling into the level ground it is receiving gradually that proportion of the load which was transferred to the other wheel, till, having recovered the whole, it impinges against the ground with great force. But in concave wheels the spoke which then strikes the ground is in its weakest position, and therefore much more liable to be broken by the impetus of the fall than the spokes of the lowest wheel by the mere transference of additional weight; whereas, if the spokes be perpendicular to the nave, they receive this sudden shock in their strongest position, and are in no danger of giving way to the strain.

In the preceding observations we have supposed the rims of the wheels to be cylindrical. In concave wheels, however, the rims are uniformly made of a conical form, as Ar, Bz, fig. 145, which increases the disadvantages which we have ascribed to them. Mr Cumming, in a late Treatise on Wheel-Carriages, points out the disadvantages of conical rims, and the propriety of making them cylindrical; but we are of opinion that he has ascribed to conical rims several disadvantages which arise chiefly from an inclination of the spokes. He insists much upon the injury done to the roads by the use of conical rims; yet though we are convinced that they are more injurious to pavements and highways than cylindrical rims, we are equally convinced that this injury is occasioned chiefly by the oblique pressure of the inclined spokes. The defects of conical rims are numerous and obvious. Every cone that is put in motion upon a plane surface will revolve round its vertex, and if force is employed to confine it to a straight line, the smaller parts of the cone will be dragged along the ground, and the friction greatly increased. Now when a carriage moves upon conical wheels, one part of the cone rolls while the other is dragged along, and though confined to a rectilineal direction by external force, their natural tendency to revolve round their vertex occasions a great and continued friction upon the linch-pin, the shoulder of the axle-tree, and the sides of deep ruts.

Mr Edgeworth made some interesting experiments on this subject. He compared a pair of conical wheels, whose inside diameter was thirty-three, while their outside diameter was twenty-seven, with a pair of cylindrical ones, the diameter of which was as thirty-four. When they moved on smooth deal-boards, the conical wheels required an addition of 50 per cent. to the moving power in order to make their velocity equal to that of the cylindrical ones. On a fine gravel road, the addition required was only eight per cent. while on a newly-made coarse road there was a perceptible difference between the two kinds of wheels, although they were only eight inches high and four broad.

A very able paper on the wheels of carriages, containing views different from those we have given above, has been recently published by Mr James Walker, civil engineer, and containing the substance of his examination before the House of Commons. When the road is perfectly smooth, and the motion of the carriage regular, and without any tendency to strain in a lateral direction, he considers the cylindrical wheel as decidedly the best, "because the power of the spokes of the wheel to oppose such a pressure is greater when they are perfectly upright under their load, and because the joints, being square, can be made with greater care and accuracy than bevelled ones." As roads, however, are not of this description, he considers the cylindrical wheel as deficient in bracing; and he remarks, "that wheels so constructed soon get loose at their joints; that the whole wheel gets crippled in consequence; and that, for durability under heavy work and rough roads, such wheels will not answer the purpose." Besides the strength arising from the bracing of the conical wheel and bent axle, Mr Walker ascribes to it the following advantages. As these wheels project outwards at top, the rims of carriage-wheels coming into collision with each other in passing will meet in an oblique direction; and unless the wheels get locked, the lightest of the two carriages is pushed off laterally, and the abrasion of the fellies may be the only damage. With cylindrical wheels, on the contrary, he conceives that the effect of a collision would be much more serious, one of them being either brought to a dead stand, or some part of it or of the harness broken. Mr Walker remarks also, that in light carriages the dished wheels throw off the mud from the traveller. When the tire of the wheel consists of a single hoop of iron, Mr Walker states that it increases in length by beating against the road; that in cylindrical wheels the result of this will be, either that the iron rim will get loose upon the fellies, or the spokes will be loose in the nave; whereas, in dished wheels, the only effect of an expansion of the tire will be that the spokes will become more upright, or the wheel a little less dishing. This argument, however, we must observe, falls to the ground when the tire consists of separate segments.

The amount of Mr Walker's argument in favour of dished wheels is, that such wheels are a firmer and more durable piece of carpentry. This argument, however, loses all its force if cylindrical wheels could be made as strong as dished ones by any new combination of spokes and fellies, which have the effect of bracing. That such a combination is possible we cannot doubt; and when we consider the smooth and level surface which prevails under an im-

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1 Edinburgh Journal of Science, vol. i. p. 274. proved system of road-making, we should be disposed still to adopt the cylindrical wheel for the sake of the horse that is to draw it, or rather for the purpose of obtaining a greater load from the same horse, even though the wheels might not last so long as dished ones. If the felloes were connected with the nave by means of spokes equally but oppositely inclined to the strongest axle, we are persuaded that a cylindrical wheel might be made as firm as a dish'd wheel, and of even greater strength in resisting any sudden impulse against the road.

The shape of the wheels being thus considered, we must now attend to some particular parts of their construction. The tire or iron plates of which the rims are composed should never be less than three inches in breadth, as narrow rims sink deep into the ground, and therefore injure the roads and fatigue the horses. When any load is supported upon two points, each point supports one half of the weight; if the points are increased to four, each will sustain one fourth of the load, and so on; the pressure upon each point of support diminishing as the number of points increases. If a weight therefore is supported by a broad surface, the points of support are infinite in number, and each of them will bear an infinitely small portion of the load; and, in the same way, every finite portion of this surface will sustain a part of the weight inversely proportional to the number of similar portions which the surface contains. Let us now suppose that a cart carrying a load of sixteen hundredweight is supported upon wheels whose rims are four inches in breadth, and that one of the wheels passes over four stones, each of them an inch broad and equally high, and capable of being pulverized only by a pressure of four hundred pounds weight. Then as each wheel sustains one half of the load, and as the wheel which passes over the stones has four points of support, each stone will bear a weight of two hundredweight, and therefore will not be broken. But if the same cart, with rims only two inches in breadth, should pass the same way, it will cover only two of the stones; and the wheel having only two points of support, each stone will be pressed with a weight of four hundredweight, and will therefore be reduced to powder. Hence we may infer that narrow wheels are in another point of view injurious to the roads, by pulverizing the materials of which they are composed.

As the rims of wheels wear soonest at their edges, they should be made thinner in the middle, and ought to be fastened to the felloes with nails of such a kind that their heads may not rise above the surface of the rims. In some military wagons we have seen the heads of these nails rising an inch above the rims, which not only destroys the pavements of streets, but opposes a continual resistance to the motion of the wheel. If these nails were eight in number, the wheel would experience the same resistance as if it had to surmount eight obstacles, one inch high, during every revolution. The felloes on which the rims are fixed should in carriages be three inches and a fourth deep, and in wagons four inches. The naves should be thickest at the place where the spokes are inserted; and the holes in which the spokes are placed should not be bored quite through, as the grease upon the axle-tree would insinuate itself between the spoke and the nave, and prevent that close adhesion which is necessary to the strength of the wheel.

On the Position of the Wheels.

It must naturally occur to every person reflecting upon this subject, that the axle-trees should be straight and the wheels perfectly parallel, so that they may not be wider at their highest than at their lowest point, whether they are of a conical or of a cylindrical form. In this country, however, the wheels are always made concave, and the ends of the axle-trees are universally bent downwards, in order to make them spread at the top and approach nearer below. In some carriages which we have examined, where the wheels were only four feet six inches in diameter, the distance of the wheels at top was fully six feet, and their distance below only four feet eight inches. By this foolish practice the very advantages which may be derived from the concavity of the wheels are completely taken away, while many of the disadvantages remain; more room is taken up in the coach-house, and the carriage is more liable to be overturned by the contraction of its base.

With some mechanics it is a practice to bend the ends of the axle-trees forwards, and thus make the wheels wider behind than before. This blunder has been strenuously defended by Mr Henry Beighton, who maintains that wheels in this position are more favourable for turning; since, when the wheels are parallel, the outermost when turning would press against the lynch-pin, and the innermost would rest against the shoulder of the axle-tree. In rectilinear motions, however, these converging wheels engender a great deal of friction both on the axle and the ground, and must therefore be more disadvantageous than parallel ones.

On the Line of Traction, and the Method by which Horses exert their Strength.

M. Camus attempted to show that the line of traction should always be parallel to the ground on which the carriage is moving, both because the horse can exert his greatest strength in this direction, and because the line of draught, being perpendicular to the vertical spoke of the wheel, acts with the largest possible lever. M. Couplet, however, considering that the roads are never perfectly level, and that the wheels are constantly surmounting small eminences even in the best of roads, recommends the line of traction to be oblique to the horizon. By this means the line of draught HA (which is by far too much inclined in fig. 144) will in general be perpendicular to the lever AC which mounts the eminence, and will therefore act with the longest lever when there is the greatest necessity for it. We ought to consider, also, that when a horse pulls hard against any load, he always brings his breast nearer the ground; and therefore it follows, that if a horizontal line of traction is preferable to all others, the direction of the traces should be inclined to the horizon when the horse is at rest, in order that it may be horizontal when he lowers his breast and exerts his utmost force. The particular manner, however, in which living agents exert their strength against great loads, seems to have been unknown both to Camus and Couplet, and to many succeeding writers upon this subject. It is to M. Depareieux, an excellent philosopher and ingenious mechanic, that we are indebted for the only accurate information with which we are furnished; and we are sorry to see that philosophers who flourished after him have overlooked his important instructions. In his memoir on the draught of horses he has shown, in the most satisfactory manner, that animals draw by their weight, and not by the force of their muscles. In four-footed animals, the hinder feet is the fulcrum of the lever by which their weight acts against the load, and when the animal pulls hard, it depresses its chest, and thus increases the lever of its weight, and diminishes the lever by which the load resists its efforts. Thus, in fig. 144, let P be the load, AD the line of traction, and let us suppose FC to be the hinder leg of the horse, and AE part of its body, A its On the construction of wheel-carriages.

On the centre of gravity, and the manner of disposing the load.

If the axle-tree of a two-wheeled carriage pass through the centre of gravity of the load, the carriage will be in equilibrium in every position in which it can be placed with respect to the axle-tree; and in going up and down hill the whole load will be sustained by the wheels, and will have no tendency either to press the horse to the ground or to raise him from it. But if the centre of gravity is above the axle-tree, as it must necessarily be, according to the present construction of wheel-carriages, a great part of the load will be thrown on the back of the horses from

the length of the traces has a considerable influence upon the draught. From the point E, where the traces are fastened to the horse next the load, draw ER to the same point in the second horse R, and let R' be another position of the second horse; it is required to find the difference of effect that will be produced by placing the second horse at R or at R', or the comparative advantages of short and long traces. From R', the point where the traces are fixed, draw R'FE'; and from E draw EEm parallel to the declivity DA. Take EF = EF' to represent the power of the horse in the direction of the traces, which will be the same whether he is yoked at R or at R'; draw EA perpendicular to DA, Fn, Fm parallel to EA, and EF, EF' parallel to En. Then, since the second horse when at R pulls with a force represented by FE, in the direction FE, we may resolve this force into the two forces En, Ef, one of which En is solely employed in dragging the cart up the inclined plane DA, while the other Ef is solely employed in pressing the first horse E to the ground. Let the horse be now removed from R to R', the direction of the traces becomes R'F, and FE = FE' is the power exerted by the horse at R' and the direction in which it is exerted. But this force is equivalent to the forces Em, Ef, the first of which acts directly against the load, while the other presses the horse against the ground. Hence we see the disadvantages of long traces, for the force which draws the load when the horse is at R' is to the force when the horse is at R, as En to En, and the forces which press the horse upon the ground as Ef to ef, or as Fm to Fm. Now Ef = En = FE × sin. nEF; hence Ef = FE × sin. (nEf' − FE') (g' being parallel to AB') and En = EF × cos. (nEf' − FE'). In like manner we have Ef = FE × sin. (nEf' − FE'), and Em = EF × cos. (nEf' − FE'). Now sin. FEg' = sin. FEg = Rg/ER', and sin. FEg' = Rg'/ER'; but Rg = Rg'/BR = EQ = BR − BR × cos. nEf' = BR × (1 − cos. nEf). By substituting this value in the equations which contain the values of Ef, En, Ef', Em, and considering that the angles FEg', FEg are always so small that their arcs differ very little from their sines, we have

\[ FEg = \frac{BR \times 1 - \cos. nEf}{ER}, \]

and

\[ FEg' = \frac{BR \times 1 \cos. nEf}{ER}. \]

By substituting these values in the preceding equations, we have

\[ En = EF \times \sin. (nEf - \frac{BR \times 1 - \cos. nEf}{ER}), \]

\[ Ef = EF \times \sin. (nEf - \frac{BR \times 1 - \cos. nEf}{ER}), \]

\[ Em = EF \times \cos. (nEf - \frac{BR \times 1 - \cos. nEf}{ER}). \]

If AB is horizontal, and the declivity AD = 1°, we shall have nEf = 9° 28', or in parts of the radius = 0.16522, and cos. nEf = 0.96538. Then, if EF = 200 pounds, BR = 3½ feet, ER = 8 feet, ER' = 12 feet, then we shall have, from the preceding formula, Fe = 31.716 pounds, Ef = 32.350 pounds, En = 197.470 pounds, and Em = 197.404. Hence an additional length of four feet to traces eight feet long presses the horse E to the ground with an additional force of 32.250 − 31.716 = 0.534 pounds, and diminishes the effect of the other horse by 0.066 pounds.

On the Position of the Centre of Gravity, and the Manner of disposing the Load.

If the axle-tree of a two-wheeled carriage pass through the centre of gravity of the load, the carriage will be in equilibrium in every position in which it can be placed with respect to the axle-tree; and in going up and down hill the whole load will be sustained by the wheels, and will have no tendency either to press the horse to the ground or to raise him from it. But if the centre of gravity is above the axle-tree, as it must necessarily be, according to the present construction of wheel-carriages, a great part of the load will be thrown on the back of the horses from the wheels when going down a steep road, and thus tend to accelerate the motion of the carriage, which the animal is striving to prevent; while, in ascending steep roads, a part of the load will be thrown behind the wheels, and tend to raise the horse from the ground, when there is the greatest necessity for some weight on his back to enable him to fix his feet in the earth, and overcome the great resistance which is occasioned by the steepness of the road. On the contrary, if the centre of gravity is below the axle, the horse will be pressed to the ground in going up hill, and lifted from it when going down. In all these cases, therefore, where the centre of gravity is either on the axle-tree or directly above it or below, the horse will bear no part of the load in level ground. In some situations the animal will be lifted from the ground when there is the greatest necessity for his being pressed to it; and he will sometimes bear a great proportion of the load when he should rather be relieved of it.

The only way of remedying these evils is to assign such a position to the centre of gravity, that the horse may bear some portion of the weight when he must exert great force against the load, that is, in level ground, and when he is ascending steep roads; for no animal can pull with its greatest effort unless it is pressed to the ground. Now this may be in some measure effected in the following manner. Let BCN (fig. 144) be the wheel of a cart, AD one of the shafts, D that part of it where the cart is suspended on the back of the horse, and A the axle-tree; then, if the centre of gravity of the load is placed at m, a point equidistant from the two wheels, but below the line DA, and before the axle-tree, the horse will bear a certain weight on level ground, a greater weight when he is going up hill and has more occasion for it, and less weight when he is going down hill, and does not require to be pressed to the ground: all this will be evident from the figure. When we recollect that the shaft DA is horizontal, the centre of gravity will press more upon the point of suspension D the nearer it comes to it; or the pressure upon D, or the horse's back, will be proportional to the distance of the centre of gravity from A. If m, therefore, be the centre of gravity, bA will represent its pressure upon D, when the shaft DA is horizontal. When the cart is ascending a steep road, AH will be the position of the shaft, the centre of gravity will be raised to a, and aA will be the pressure upon D. But if the cart is going down hill, AC will be the position of the shaft, the centre of gravity will be depressed to n, and cA will represent the pressure upon the horse's back. The weight sustained by the horse, therefore, is properly regulated by placing the centre of gravity at m. We have still, however, to determine the proper length of ba and bm, the distance of the centre of gravity from the axle, and from the horizontal line DA; but as these depend upon the nature and inclination of the road, upon the length of the shaft DA, which depends on the size of the horse, on the magnitude of the load, and on other variable circumstances, it would be impossible to fix their value. If the load, along with the cart, weighs 400 pounds, if the distance DA be eight feet, and if the horse should bear 50 pounds of the weight; then bA should be one foot, which, being one eighth of DA, will make the pressure upon D exactly 50 pounds. If the road slopes four inches in a foot, bm must be four inches, or the angle bAm should be equal to the inclination of the road; for then the point m will rise to a when ascending such a road, and will press with its greatest force on the back of the horse.

When carts are not made in this manner, we may in some degree obtain the same end by judiciously disposing

Let us suppose that the centre of gravity is at O when the cart is loaded with homogeneous materials, such as sand, lime, &c.; then if the load is to consist of heterogeneous substances, or bodies of different weights, we should place the heaviest at the bottom and nearest the front, which will not only lower the point o, but will bring it forward, and nearer the proper position m. Part of the load, too, might be suspended below the fore part of the carriage in dry weather, and the centre of gravity would approach still nearer the point m. When the point m is thus depressed, the weight on the horse is not only judiciously regulated, but the cart would be prevented from overturning; and in rugged roads the weight sustained by each wheel would be in a great degree equalized.

Composition for diminishing the Friction of the Wheels of Carriages, &c.

The following composition has been very recently discovered by Mr Partridge, and secured by patent.

"To about twenty gallons of clear soft water apply three or four pounds of fresh lime, put them into a cask, and stir them well about; then let it stand about twenty-four hours, or until the water is quite clear, when it may be drawn off as wanted. The lime-water should be stirred up every five or six days, that the water may be kept fully impregnated with the lime. To one part or proportion of this solution of lime water add one part or proportion of olive oil. These ingredients are to be placed in an open vessel, and whisked, beat, or stirred well together until they are completely blended; or they may be placed in a bottle or jar, and well shaken, until the same effect takes place. The composition of lime-water and oil will then have the consistency and appearance of thick cream; and if the composition made from the above proportions should be found too thick, it may be thinned or reduced in consistency by adding a little more oil; and I would remark, that the composition should be well shaken or mixed up before using; and the lime-water should be kept free from air, and as cool as possible.

"For axle-trees and other bearings this composition will be found superior to pure oil, and will not be consumed so rapidly.

"To render this composition more applicable to the lubrication of cogs or teeth of wheels, Mr Partridge prefers whale or other common oil, which may be used in one part or proportion to two parts or proportions of lime-water; and when perfectly commixed, as before stated, to it is to be added palm oil or tallow, or both, and well rubbed or ground therewith until the composition assumes the appearance of thick paste, when a small quantity of carbonaceous matter, such as plumbago, black lead, or soot, may be added, and when incorporated with each other, the composition will be fit for use."

Description of different Carriages.

In figure 147 is represented a carriage invented by Mr Richard, a physician in Rochelle, which moves without horses, merely by the exertion of the passengers. The machinery by which this is effected is placed in a box behind the carriage, and is shown in fig. 148, where AA is a small axis fixed into the box, and E a pulley over which a rope passes, whose two extremities are tied to the ends of the levers or tredles C, D: the other ends of the levers are fixed by joints to the cross-beam MN. The cranks FF are fixed to the axle KL, and move upon it as a

Fig. 147.

Each of them has a detent tooth at F, which catches in the teeth of the wheels H, H, so that they can move from F to H without moving the wheel, but the detent tooth catches in the teeth of the wheels when the cranks are brought backward, and therefore bring the wheel along with them. When the foot of the passenger, therefore, is placed upon the treddle D, it brings down the crank F, and along with it the wheel H, so that the large wheels fixed on the same axis perform part of a revolution; but when D is depressed, the rope DA descends, the extremity C of the other treddle rises, and the crank F rising along with it, takes into the teeth of the wheel H, so that when the elevated treddle C is depressed, the wheels H, H, and consequently the wheels I, I, perform another part of a revolution. In this way, by continuing to work at the treddles, the machine advances with a regular pace.

A carriage of this kind, where the mechanism is much more simple and beautiful than that which we have described, has been lately invented and constructed by Mr Nasmyth of Edinburgh, a gentleman whose mechanical genius is scarcely inferior to his talents as a painter.

Fig. 148.

Fig. 149.

On the Construction of Wheel-Carriages.

Pulley E and axle AA are rendered unnecessary; leather straps are substituted in place of the cranks F, F, and the whole mechanism is contained in two small cylindrical boxes about six inches in diameter and one and a half broad.

A carriage driven by the action of the wind is exhibited in fig. 149. It is fixed on four wheels, and moved by the impulse of the wind upon the sails C, D, being guided by the rudder E. Carriages of this kind will answer very well in a level country, where the roads are good and the wind fair; and are said to be much used in China. In Holland they sometimes use similar vehicles for travelling upon the ice; but they have a sledge instead of wheels, so that if the ice should happen to break, there will be no danger of sinking. Stephinus, a Dutchman, is said to have constructed one of these carriages with wheels, which travelled at the rate of twenty-one miles an hour with a very strong wind.

The carriage represented in fig. 150 is made so as to sail against the wind by means of the spiral sails E, F, G, H, one of which, F, is expanded by the wind. The impulse of the wind upon the sails gives a rotatory motion to the axle M, furnished with a cog-wheel K, whose trundles act upon teeth placed on the inside of the fore-wheels.

A carriage which cannot be overturned is represented in figure 151, where AB is the body of the carriage, consisting of a hollow globe, made of leather or wood, at the bottom of which is placed an immoveable weight proportioned to the load which the carriage is to bear. Two horizontal circles of iron D, E, connected with bars HI, and two vertical circles F, G, surround the globe; and the wheels are fastened by a handle K to the perpendicular bars HI. Then, since the body of the carriage moves freely in every direction within the iron circles, the centre of gravity will always be near C, and the carriage will preserve an upright position even if the wheels and frame were overturned. PART III.—DESCRIPTION OF MACHINES.

As machines for various purposes in the arts have already been described in their appropriate places, we shall confine ourselves, in the present division of this article, to an account of Mr Babbage's mechanical notation; a description of the elements which enter into the composition of all machines; of machines which are illustrative of the general doctrines of mechanics; and of a few others which have not been described in any previous part of this work.

CHAP. I.—ON MECHANICAL NOTATION, OR THE METHOD OF EXPRESSING BY SIGNS THE ACTION OF MACHINERY.

We have already given a general notice of this important method, invented by Mr Babbage, and pointed out its great importance in mechanics. We shall now endeavour to explain it more minutely, and show the method of applying it to a machine of a given structure. The first step in this method consists in drawing, on a sheet of paper, as many vertical lines as there are pieces or moving parts in the machine; to each of these pieces its name must be affixed, and the letter which marks it out in the drawing. These vertical lines must then be crossed with horizontal lines, upon which we must write,

1. The nature of the motion of each piece, indicated by a conventional sign. 2. The number of teeth in the wheels, pinions, or sectors, or the number of pins or studs on any revolving barrel. 3. The velocities, whether absolute or relative, of the several parts of the machine. 4. The mode in which the motion is communicated or transmitted. 5. The modes of adjustment, some of which are permanent and made by the mechanist, while others depend on the nature of the work they are intended to perform, such as the distance between the mill-stones in a flour-mill, on which the fineness of the flour depends, or such as the winding up of the weight or spring, which may be called a periodic adjustment. 6. The times or durations of the movements.

Mr Babbage employs two classes of signs, viz., those for indicating the peculiar nature of the means by which the motion is transmitted, and those which denote the state of motion or rest of every particular part of the machine. The first of those classes are as follows:

1. When one piece receives its motion from another, in consequence of being permanently fixed to it, such as a pin upon a wheel, or a wheel and pinion on the same axis. This may be indicated by an arrow with a bar at the end, thus,

2. When one piece is moved by another, so that the motion of the one is the necessary result of the motion of the other, but without any permanent connection, as when a pinion drives a wheel, the motion may be indicated by an arrow without a bar, thus,

3. When the piece moved is attached to the piece that moves it by stiff friction, the sign is an arrow formed by a line interrupted by dots, thus,

4. When the piece moved does not always move with the piece that moves it, as in the case of a stent lifting a ball once in the course of its revolution, the sign may be an arrow half full and half dotted, thus,

5. When one wheel is connected with another by a ratchet, as the great wheel of a clock is attached to the fusee, we may use a dotted arrow with a ratchet-hook at its end, thus,

The second class of signs are as follows:

1. Unbroken lines indicate motion. 2. Lines on the right side indicate motion from right to left. 3. Lines on the left side indicate motion from left to right. 4. Parallel straight lines denote uniform motion. 5. Curved lines denote a variable velocity, and the ordinates of the curve should express the velocity. 6. When the motion begins at a fixed point, and its termination is uncertain, the line must begin with a small cross line. 7. When the commencement of the motion is uncertain, and its termination fixed, the line must end with a small cross line. 8. Dotted lines imply rest. 9. If the thing be indicated by a check, bolt, or valve, its dotted line should be on the right side if it is out of action, unbolted, or open, and on the left side if the reverse. 10. If a bolt can rest in three positions; 1st, bolted on the right side; 2d, bolted on the left side; 3d, unbolted; the figures are,

1. Bolted on the right side. 2. Bolted on the left side. 3. Unbolted, the two lines being nearer the indicating line.

As illustrations of the application of this method, Mr Babbage has selected the common eight-day clock and the hydraulic ram; but these examples are a little too complicated, especially that of the clock, which would require several plates to make it intelligible. We have therefore taken the case of an ideal but simple machine, which consists of a large wheel driving a system of levers and stampers, also a large beam driven by a crank. The motion is communicated by means of ropes. The system of signs is shown in Plate CCCXLVIII.

CHAP. II.—ON THE ELEMENTS OF MACHINERY.

The simple mechanical powers, which we have already fully described, are in reality machines or instruments by which useful mechanical effects are produced; but a machine is, properly speaking, a combination of the simple mechanical powers, arranged in such a manner as to produce the desired effect. The nature of this combination depends not only on the kind of mechanical agent which is to be the first mover, and on the nature of the work to be performed, but often on the locality of the machine itself, and other accidental causes; and the skill of the engineer is pre-eminently conspicuous in the adaptation of his machinery to these secondary conditions.

The principal elements of machinery are cylindrical and conical or bevelled wheels and pinions, and rack-work. When an axis is to receive motion from another axis lying parallel to it, the effect is produced by placing a cylindrical wheel upon each, as in fig. 104, so that half the sum of the diameters of the two wheels may be equal to the distance of the axes, and making the number of teeth in each wheel inversely proportional to the respective velocities. which are required in the two axes. If the two axes are inclined to each other, then conical wheels are used, as shown in figs. 125, 126, and 128.

But it is not always necessary that the wheels move one another by means of teeth. In machines where the resistance of the work is trifling, one wheel may drive another by the mere friction of their circumferences. This may be done in several ways.

1. The circumference of the wheels may be faced with buff leather, the friction of which upon itself is considerable. We have seen this plan adopted in a pair of bellows, where the wind is propelled by a revolving vane which is driven by a winch and two intermediate wheels faced with buff. The circumferences of the wheels are pressed against each other by a spring, in order to maintain sufficient contact for the purpose. Rims made of caoutchouc would answer the purpose much better.

2. The circumference of the wheels may be faced with wood cut across the grain. Dr O. Gregory informs us that this plan was actually adopted in a saw-mill at Southampton, where all the wheels impelled one another by the contact of the end grain of the wood. The machine did its work with very little noise, and wore very well, having been in use for twenty years.

The same effect might be produced by putting a circumference of metal containing a great number of small blunt spikes or points upon the impelling wheel, and a rim of hard wood upon the other, upon which these blunt spikes act in turning it.

When one wheel drives another, they necessarily turn in opposite directions. If it is required that they should move in the same direction, the best way is to unite them by a rope or leather belt which does not cross itself at any point between the wheels, for if the belt crosses, the wheels will move in opposite directions.

The great use of ropes or belts is to communicate motion from one axle to another, at such a distance that two wheels uniting the axles would be ridiculously large. In belts the motion is conveyed by friction. The friction between the impelling wheel and the belt puts the belt in motion, and the friction of the belt upon the impelled wheel moves it in the same manner.

The great disadvantage of ropes and belts however is, that they become slack, and thus lose their power of turning the wheel. This evil may be partially removed by increasing the friction by rubbing them with chalk, or, what is still better, by shortening them. Another method equally effectual, is to have upon one of the wheels grooves of different diameters, so that a slack belt may be tightened by slipping it upon a groove of larger diameter. This however alters the velocity of the axle upon which this change has been made; and the only way to remove this evil, or to tighten a belt without altering the relative velocity of the two axles, is to have grooves of corresponding diameters on both wheels, so that the loose belt may be half tightened by shifting it to a groove of larger diameter on one wheel, and the tightening completed by shifting it also to a groove of larger diameter on the other wheel. The best method, however, of effecting the object in question would be to have one or more caoutchouc springs forming part of the belt.

From these remarks it will be obvious, that in nice machinery, where regular movements are essential to its performance, or in cases where the work occasions a great strain upon the machinery, belts and ropes cannot with propriety or safety be employed. In such cases chains of various forms are substituted in their place. These chains are sometimes jointed like those of a watch, and convey the motion by friction; but in general it is necessary to adopt quite a different kind of chain, and to make them lay hold of pins or hooks projecting from the circumference which they put in motion. Two of these chains are shown in the annexed figures. In fig. 152 the links of the chain AC lay hold of the pins p, p, projecting from the circumference of the wheel DE.

In fig. 153, metallic teeth t, t, t, project from the jointed chain ABC, and enter into grooves or indentations in the circumference of the wheel. Various chains used in machinery are shown in the annexed figure. That marked A was invented and used by the great mechanist Vaucanson.

One of the simplest and most durable expedients for communicating motion is the crank, which is nothing more than a lever fixed upon the axis, which is to be moved by a force applied to the end of the lever. This lever may be fixed at the end of the axis, as in the common grindstone, or in the electrical machine, where wheels are not used; or it may be fixed upon a wheel carried by the axle, when the object of the mechanism is to put a wheel in motion. This is the case in the old cylindrical electrical machines, where a handle is fixed upon the face of the largest wheel, employed to give velocity to the cylinder. The crank lever may be fixed in the middle of an axis; but in this case the axis must necessarily be bent, and the force applied at the part projecting from the axle, as shown in fig. 155, where mn is the axis bent twice, at a and c, so as to form a double crank. Now if one man were to apply his force at a and another at c, they would work with a lever whose length was equal to the distance of a and e from the axle or line mn. A triple crank is shown at fig. 156; and a variable single crank in fig. 157 where mn is the axle, a the point where the power is applied, and c, d, e, f, four parallel bars, which, by pins passing through the small holes, may be fixed in any position, so as to vary the distance of a from mn. In place of applying the power of a man at a, the power of any other mechanical agent may be made to turn the axle mn, by means of rods ba, de, fig. 155, &c. which have a reciprocating or a backward and forward rectilineal motion. This is exemplified in

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1 In a turning-lathe or an electrical machine, where a change of velocity does no harm, this contrivance answers very well. the grind-stone and the turning lathe, where the power is the downward pressure of the foot of a man upon a foot-board, as in fig. 158, where A may be a grindstone, or a wheel driving a pinion.

When the power applied at \(a\) is obtained from the reciprocating motion of a rod, as in the case of a steam-engine, it acts as in the annexed figure, where \(b\) is the crank fixed near the circumference of the wheel \(A\), and \(MN\) a vertical rod moving upwards and downwards between the guides \(cd\), \(ef\), and connected with the crank by the bar \(ab\).

When a man and a power moving in a fixed direction are applied to the same crank, the mode of action is in the two cases essentially distinct. In every mode of action both the man and the power push in one semicircumference of the circle, which the end of the crank describes, and pull in the other semicircumference; and it is very obvious, that in every position of the crank, both powers must act with different degrees of effect. When the crank lever is horizontal, and the man pushes it down with the aid of his weight, his force is a maximum; and when he pulls it upwards in the next horizontal position, his force is very great. At all intermediate positions he works to a disadvantage; but there are no positions of the crank in which the man has no power to turn it, because he is able to apply his power in any direction he chooses. This, however, is impossible when the man works by a vertical pressure, or a pressure given in direction upon a foot-board, or when the crank is moved by a reciprocating rod. In these cases, as will be seen at \(A\), \(B\), \(C\), \(D\) in the annexed figures, there are two opposite positions in which the rod has no power to move the crank. This happens in the positions \(C\) and \(D\), where the centre \(m\) is between \(a\) and \(b\), or when \(a\) is between \(m\) and \(b\). In the other positions shown in the figures, the power of the rod \(ba\) to move the crank is proportional to the perpendicular let fall upon \(ab\) from the centre \(m\); and when that perpendicular coincides with \(ma\), or when \(ma\) is perpendicular to \(ba\), the power is a maximum either in pushing or pulling the lever.

A combination of two cranks has been used for producing a high degree of pressure. This will be understood from the annexed figure, where the same lever \(BC\) unites two crank levers \(AB\), \(CD\), which move round joints or centres at \(A\), \(B\), \(C\), and \(D\). If we suppose a moving force applied to \(DC\), to make it turn round \(D\) from right to left, it is obvious that the lever \(BC\) will be pulled in the direction \(BC\) with the greatest force, when \(BCD\) is a straight line. The force thus exerted acts upon the lever \(AB\) at \(B\), and it will exert the greatest force when \(CB\) is perpendicular to \(BA\). The force therefore originally applied to move \(CD\) will be very great when exerted at \(A\). This combination of cranks is used in the Stanhope press, in which the axle \(C\) drives a screw, which presses the paper with great force upon the types.

Another contrivance similar in principle to the crank is frequently introduced into machinery. It is called the Eccentric, and is shown in fig. 165, where \(AB\) is a rod which is to be raised and depressed between the guides \(mn\), \(op\), by means of a continued circular motion. Let \(C\) be the centre of this circular motion, or a section of the revolving axis. Upon this axis \(C\) is fixed a circle \(DE\) eccentric to it, \(O\) being its axis, and \(CO\) the degree of eccentricity. This circle \(DE\) works in a circular opening of the frame \(BED\) fixed to \(BA\) by a joint at \(B\). As the axis \(C\) and the circle \(DE\) form one piece, and as \(C\) is fixed in position, the effect of turning the axis \(C\) must be to raise and depress \(BA\) between the guides, the power being always proportional, as in the crank, to a perpendicular let fall from \(C\) upon \(BD\). When this perpendicular vanishes, which happens when the line \(CO\) coincides with \(BO\), the rod \(A\) will be at its highest or its lowest point.

An ingenious contrivance for conveying motion is known by the name of Hooke's Universal Joint. The driving axis Universal joint and the axis to be driven are shown at \(A\), \(B\). Each of them terminates in semicircles, joining two rectangular bars, \(CD\), \(EF\), by means of pivots at their extremities \(C\), \(D\), \(E\), \(F\). It is obvious that when \(A\) is made to revolve, \(B\) will revolve with the same velocity; for if \(A\) moves round without changing its direction, the points \(C\), \(D\) must move in a circle round the intersection of the crossed bars, and this motion of the cross will make the points \(E\), \(F\) move round the same intersection, in consequence of which the shaft \(B\) will revolve, the two shafts turning in the very same manner as if each had a pivot at the intersection, and were impelled with equal velocities. The one axis \(A\) will not drive the other \(B\) if their vertical inclination is less than 140°.

When the inclination of the axis is between 50° and 90°, a double universal joint is used, consisting of four semicircles and two crosses, and acting in the same manner as the other. It is shown in the annexed figure. Another form of the apparatus is shown in the annexed figure. Universal joints may be constructed by fastening four pins 90° from one another on the circumference of a hoop or solid ball.

Universal joints are generally used in cotton-mills, where it is necessary to convey the motion through great distances. In this case it is advisable to divide the rods into lengths, and connect them by these joints. Universal joints are also much used in large telescopes upon equatorial axes, in order to give the vertical and horizontal motions to the tube while the observer continues to look through it.

Another instrument for connecting the parts of machines is called the Cam, or Camb, or Wiper, and the general object of it is to convert a circular into a rectilineal motion. It is extensively used in the tambouring machinery, and in machines for making lace. The different changes of motion which are most useful in machinery may be classified as follows:

1. A continued circular motion into a rectilineal, alternating, or vibrating motion. 2. A continued circular motion into a circular alternating motion. 3. A continued rectilineal into a continued circular motion. 4. A rectilineal alternating into a circular alternating motion.

1. A continued circular into a rectilineal alternating motion.

The simplest of all the contrivances for this purpose is the cam, already referred to. The cam may have one, two, or three, or more wipers, according as we wish to have one, two, three or more alternations of the required motion during one revolution of the axis of the cam. If the alternating or vibrating rod stand vertically, it descends by its own weight, and is lifted by each succeeding cam. If a pause is necessary after each descent, then there must be a blank space in the revolving wheel, to allow the necessary time to elapse before the next cam come round to elevate the stamper.

If the alternating rod be placed horizontally, as \(ab\), which moves between the guides \(m, n\), its extremity \(b\) presses against the spring \(s\), so as to push the rod back again upon the succeeding wiper, which is placed on the lateral face of the wheel \(AB\), which is driven by a handle \(H\). This contrivance is used in the manufacture or fishing nets, and in a machine for pricking holes in leather for making cards. In this last case the prickers will be placed at the end \(b\), and the spring \(s\) will bear against some projecting shoulder of the bar. If all the teeth in \(AB\) were reduced to one, they would form a circular inclined plane, and one vibration of \(ab\) would be performed during each revolution of the wheel.

Another contrivance for this purpose, namely, a double rack and pinion, is shown in fig. 170. A pinion \(P\), works in the teeth of the rack \(AB\), with circular ends. This pinion can move freely in a groove \(mn\), cut out of the cross piece \(CD\). When the end \(B\) comes up to the pinion \(P\), by the revolution of the latter, a projecting piece \(a\), presses against the spring \(s\); and the pinion descending in the groove \(mn\), enters the lower side of the rack, and carries it back, so that an alternating rectilineal motion is maintained.

Another ingenious contrivance of the same kind is shown in the annexed figure. In this case the pinion \(P\), is fixed, and the rack is moveable by means of two jointed rods \(ab, cd\). When the pinion is at the circular end of the rack, the rack receives a small lateral motion from \(a b\) and \(c d\), which brings forward the other side of the rack within the action of the pinion, in order to produce the returning motion.

An elegant piece of mechanism, invented by M. Zureda, is shown in the annexed figure. For the purpose of giving an alternating horizontal motion to the rod \(ab\), a handle \(H\), or any other means, turns the cylinder \(AB\) round its axis. Two opposite grooves, like the threads of a female screw, are cut in its surface, so as to unite or run into one another at both ends of the cylinder. The end \(b\) of the rod \(ab\), fits this groove, while the end \(a\) moves up and down in a groove in the frame \(CD\). As the cylinder revolves, the end \(b\) follows the spiral direction of the groove, till it reaches the end \(B\), and it then enters the opposite groove, by which it returns to its original place, as in the figure. If a cone be substituted for the cylinder, the path of \(ab\) or the degree of its inclination to the revolving axis, is determined by the angle of the cone.

Another contrivance for producing a vertical alternating motion is shown in fig. 173, where \(ab\) is a rod moving vertically between guides. A wheel \(AB\), put in motion by a handle or other means, carries a projecting pin or cylinder \(p\), which moves with a little play in the curvilinear groove \(DpED\). As the wheel \(AB\) turns, the pin \(p\), carried along with it, acts on the under side of the curve, and depresses the rod; and when the pin \(p\) has reached its lowest point, it begins to raise the rod \(ab\), by acting on the upper side \(DpEd\) of the groove. If the groove \(DE\) had been a straight line, the ascent or descent of \(ab\) would have been variable, being equal to the versed sine of the arch described by the wheel.

When a circle revolves in the inside of another of twice the diameter, any point of the smaller circle will describe a straight line, which is the diameter of the larger circle. This straight line is an epicycloid; and this beautiful property has been embodied as a piece of mechanism, for converting a continuous circular into a rectilineal alternating motion. The mechanical expression of the geometrical property is shown in fig. 174, Mechanics.

Contrivance where AB is a concave toothed wheel, in which another wheel of half its diameter works. If the wheel C is made to revolve within the other by means of a winch, any point in its circumference will describe a straight line, and consequently any rod fixed to that point, and lying in the direction of the diameter which the point describes, will have a rectilinear alternating motion.

The ingenious method of M. Berthelot of obtaining a continued circular from an alternating motion is shewn in fig. 175. The alternating motion is communicated from the frame OOGG to hooks in which are fixed the ends of two ropes XX, YY, which pass round the two wheels A, B, which move freely on the axis mn, in one direction, but carry the axle along with them in another direction. This is effected by ratchet wheels and clicks, one of which is placed on the inner side of A, and the other on the outer side of B. By moving the frame to the right, the wheel A being unable to move upon the axis mn, carries it round with it; and by moving the frame to the left, the wheel A being no longer fixed by the click and ratchet turns freely on the axle mn, but the other B being now locked to the axle, carries the axle mn along with it, and in the same direction as the wheel A did, so that the axle mn then receives a continued rotatory motion from the reciprocation of the frame OOG.

M. Betancourt's machine for converting continuous circular motions into rectilineal alternating motions of a given length, is shewn in the annexed figure, where ab is the revolving axis, the pivot of which below b is supported by the piece of wood e furnished with two rollers which enter into a groove (See fig. 177) in the box m'n', so that by the motion of e to the right or the left, the axis ab can approach one after another to the toothed wheels F, G, and drive them by the endless screw h. Two clicks a'b' cd, turning round their axes a' and c are fixed to e, and these clicks carry two arms b' and d perpendicular to them. A metallic bar mn (fig. 177), with a rectangular branch u is fixed at the end of an axis gf, which carries a double fork il; at the end g of the axis gf, is fixed a counterweight P. Two square pins xy, whose projection is equal in thickness to the clicks a'b' cd, are fixed in the box m'n'. Let us now suppose that the alternating motion to be obtained from the rotation of the axis ab is that of two buckets (one of which is seen at S, fig. 176, the Contrivance for changing motion).

Fig. 175.

Fig. 176.

Fig. 177.

A continued circular into an alternating circular motion, or the reverse.

A very ingenious piece of mechanism for converting an alternating circular into a continued circular motion, is shewn in the annexed figure. The axis AB is made to move as if driven by a pendulum swinging at A. This axle carries two ratchet wheels a a with their teeth oppositely inclined, and also two bevelled tooth wheels C, D, which are however only slightly fixed to it by friction. These last wheels carry two catches p, q, so that when one or other falls into the teeth of the ratchet wheels a a, this wheel moves round along with the axle, while the other wheel, which is not connected with its catch, moves round by any force greater than its friction. The bevelled wheel E, which is to receive a continued circular motion, is engaged by its teeth both in C and D.

If we suppose the axis AB to swing so that m and C are fixed by the catch p, the wheel C will turn with the axle, and the wheel E will move in the same direction; and as the teeth on the right hand side of E are engaged on those of D, the latter will turn round upon the axle AB. When AB returns, or its motion is reversed, the teeth of a will move against the catch q, the wheel D will move with the axle AB, and E will be driven in the same direction as before, the wheel C now moving loosely upon AB. By changing the place of the ratchet wheel and catches, the motion of E would be reversed.

Another example of the change of motion under our consideration is seen when wipers fixed upon a revolving wheel raise a forge hammer moveable about a centre.

Another contrivance of this kind is shewn in the annexed figure, where AB is a one toothed wheel, the tooth being an inclined plane winding round its axis. This plane presses against the extremity a of a crooked lever abc, and gives it an alternating circular motion.

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1 See Borgnis Essai, &c. p. 370. 2 Lanz and Betancourt's Essai sur le Composition des Machines, p. 82, 83. Contrivances for changing motion.

M. St. Cyr has produced an alternating circular motion, so that the velocity may vary according to any required law by the mechanism in fig. 180; and he has employed the principle used in the construction of equation clocks. An equation curve BCD is fixed on the annual wheel A, and in the curve is formed a groove in which the pivot E can move. This pivot is united with the levers EF, EG, the last being fixed to the cannon H which carries the minute hand HI, so that it follows the vibration of the curve in more than one half of the circumference of the minute dial which is sufficient to mark the inequalities arising from the equation.

Another pretty contrivance is shewn in fig. 181, where A, a wheel partly toothed drives two toothed wheels B, C, on the same axis DE. After the toothed part A has driven B through a certain angle, it quits it and drives C in the opposite direction; thus giving an alternate circular motion to DE.

3. A continued rectilineal into a continued circular motion.

This change of motion is easily effected by means of a rectilineal rack acting upon a toothed wheel; but the continued circular motion of the wheel must terminate when it has made as many revolutions as the number of teeth in the wheel are contained in the number of teeth in the rack.

4. A rectilineal alternating into a circular motion, or the reverse.

This change of motion is exemplified in the drill-bow of watchmakers and smiths, and in the handles of pumps. In the first the rectilineal alternating motion of the bow gives a circular alternating motion to an axle round which the bow-string is coiled. The extent of angular motion of the axle is determined by the number of times that the circumference of the pulley upon the axle is contained in the length of the stroke of the bow. In pumps the handle moves through an arch of a circle when the piston rises or falls in a straight line.

The various contrivances used by the early constructors of steam engines, and the more elegant ones invented by Mr. Watt for giving an alternating motion to the beam by the rectilineal movements of the piston belong to this class of contrivances, and will be fully described under the article Steam Engine.

In the simple piece of mechanism shewn in fig. 182; the bar AB attached to the axis C by two ropes AC, BC rises and falls by giving it a circular motion which twists the ropes, and makes it rise towards C. In untwisting them by the opposite motion of AB the bar descends. A drill D may thus be put in motion.

The mechanism shewn in fig. 183 produces an analogous effect. The lever AB vibrates round C as a centre. The ends of a rope FMLD are fixed at F, and D, and pass over two pulleys M, N. By depressing the end B of the lever, the point L of the rope will move towards N; and by again raising B it will return. A force applied at L to give a motion in the direction MN will again produce an oscillation of the lever AB round C.

A combination of levers called zig-zag, or lazy tongs, or scissors, shewn in fig. 184, is of the same kind. Joints being placed at the intersection of all the pieces which compose it, it is evident that if the ends MN open or shut by a circular motion, the points A, B will approach to, or recede from, one another. These lazy tongs are ingeniously applied by Mr. Aldous of Clapton, for conveying the motion of the beam of his steam engine to the crank which gives the circular motion. See the London Journal, or Repertory of Arts, No. 55, Oct. 1836, p. 57.

CHAP. IV.—DESCRIPTION OF MACHINES WHICH ILLUSTRATE THE DOCTRINES OF MECHANICS, OR ARE CONNECTED WITH THEM.

Atwood's Machine.

The machine invented by Mr. Atwood for illustrating the Atwood doctrines of accelerated and retarded motion, is represented in figs. 1, 2, 3, 4, 5, 6, Plate CCCXLIX., and enables us to discover, 1st, the quantity of matter moved; 2d, the moving force; 3d, the space described; 4th, the time of description; and, 5th, the velocity acquired at the end of that time.

1. Of the quantity of matter moved.—In order to observe the effects of the moving force, which is the object of any experiment, the interference of all other forces should be prevented; the quantity of matter moved, therefore, considering it before any impelling force has been applied, should be without weight; for though it be impossible to abstract weight from any substance whatever, yet it may be so counteracted as to produce no sensible effect. Thus, in the machine fig. 1, A, B represent two equal weights affixed Fig. 1, to the extremities of a very fine silk thread; this thread is stretched over a wheel or fixed pulley abcd, moveable round a horizontal axis: the two weights A, B being equal, and acting against each other, remain in equilibrio; and when the least weight is superadded to either (setting aside the effects of friction), it will preponderate. When A, B are set in motion by the action of any weight m, the sum A+B+m, would constitute the whole mass moved, but for the inertia of the materials which must necessarily be used in the communication of motion. These materials consist of, 1st, the

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1 See Machines Approvées par l'Académie, tom. iii. No. 146; tom. iv. No. 235, 269, 278; tom. viii. No. 488, 495. 2 Ibid. tom. vi. No. 429. Borgnis Essai, &c. p. 163, 169. In order to avoid troublesome computations in adjusting the quantities of matter moved and the moving forces, some determinate weight of convenient magnitude may be assumed as a standard; to which all the others are referred. This standard weight in the subsequent experiments is a quarter of an ounce, and is represented by the letter \( m \). The inertia of the wheels being therefore \( =2\frac{3}{4} \) ounces, will be denoted by \( 11m \). A and B are two boxes constructed so as to contain different quantities of matter, according as the experiment may require them to be varied; the weight of each box, including the hook to which it is suspended \( =1\frac{1}{2} \) oz., or, according to the preceding estimation, the weight of each box will be denoted by \( 6m \); these boxes contain such weights as are represented by fig. 3, each of which weighs an ounce, so as to be equivalent to \( 4m \); other weights of \( \frac{1}{2} \) oz., \( 2m \), \( \frac{1}{4}m \), and aliquot parts of \( m \), such as \( \frac{1}{2}m \), \( \frac{1}{4}m \), may be also included in the boxes, according to the conditions of the different experiments hereafter described.

If \( 4\frac{3}{4} \) oz. or \( 19m \) be included in either box, this, with the weight of the box itself, will be \( 25m \); so that when the weights A and B, each being \( 25m \), are balanced in the manner above represented, their whole mass will be \( 50m \), which being added to the inertia of the wheels \( 11m \), the sum will be \( 61m \). Moreover, three circular weights, such as that which is represented at fig. 4, are constructed, each of which \( =\frac{1}{4} \) oz., or \( m \); if one of these be added to A and one to B, the whole mass will now become \( 63m \), perfectly in equilibrium, and moveable by the least weight added to either, (setting aside the effects of friction), in the same manner precisely as if the same weight or force were applied to communicate motion to the mass \( 63m \), existing in free space and without gravity.

2. The moving force.—It will be convenient here to apply a weight to the mass A as a moving force. When the system consists of a mass \( =63m \), according to the preceding description, the whole being perfectly balanced, let a weight \( \frac{1}{4} \) oz., or \( m \), such as is represented in fig. 5, be applied on the mass A; this will communicate motion to the whole system; by adding a quantity of matter \( m \) to the former mass \( 63m \), the whole quantity of matter moved will now become \( 64m \); and the moving force being \( =m \), this will give the force which accelerates the descent of \( A = \frac{m}{64m} \), or \( \frac{1}{64} \) part of the accelerating force of gravity.

By the preceding construction, the moving force may be altered without altering the mass moved; for suppose the three weights \( m \), two of which are placed on A and one on B, to be removed, then will A balance B. If the weights \( 3m \) be all placed on A, the moving force will become \( 3m \), and the mass moved \( 64m \) as before, and the force which accelerates the descent of \( A = \frac{3m}{64m} = \frac{3}{64} \) parts of the force by which gravity accelerates falling bodies.

Suppose it were required to make the moving force \( 2m \), the mass moved continuing the same. Let the three weights, each of which \( =m \), be removed; A and B will balance each other; and the whole mass will be \( 61m \); let \( \frac{1}{2}m \), fig. 5, be added to A, and \( \frac{1}{2}m \) to B, the equilibrium will be preserved, and the mass moved will be \( 62m \); now let \( 2m \) be added to A, the moving force will be \( 2m \), and the mass moved \( 64m \) as before; wherefore the force of acceleration \( =\frac{1}{64} \) part of the acceleration of gravity.

3. Of the space described. The body A, fig. 1, descends in a vertical line; and a scale about sixty-four inches in length divided into inches and tenths of an inch is adjusted vertical, and so placed that the descending weight A may fall in the middle of a square stage, fixed to receive it at the end of the descent; the beginning of the descent is esti- Description mated from 0 on the scale, when the bottom of the box A is on a level with 0. The descent of A is terminated when the bottom of the box strikes the stage, which may be fixed at different distances from the point 0; so that by altering the position of the stage, the space described from rest may be of any given magnitude less than sixty-four inches.

4. The time of description is observed by a seconds pendulum; and the experiments may be so constructed that the time of motion shall be a whole number of seconds. The estimation of the time, therefore, admits of considerable exactness, provided the observer takes care to let the bottom of the box A begin its descent precisely at any beat of the pendulum; then the coincidence of the stroke of the box against the stage, and the beat of the pendulum at the end of the time of motion, will show how nearly the experiment and the theory agree. There might be various devices for letting the weight A begin its descent at the instant of a beat of the pendulum W; for instance, let the bottom of the box A, when at 0 on the scale, rest on a flat rod, held in the hand horizontally; its extremity being coincident with 0, by attending to the beats of the pendulum; and with a little practice, the rod which supports the box A may be removed at the moment the pendulum beats, so that the descent of A shall commence at the same instant.

5. Of the velocity acquired. It remains only to describe in what manner the velocity acquired by the descending weight A, at any given point of its path, is made evident to the senses. The velocity of A's descent being continually accelerated will be the same in two points of the space described. This is occasioned by the constant action of the moving force; and since the velocity of A at any instant is measured by the space which would be described by it moving uniformly for a given time with the velocity it had acquired at that instant, this measure cannot be experimentally obtained, except by removing the force by which the descending body's acceleration was caused.

In order to show in what manner this is effected particularly, let us again suppose the boxes A and B=25m each, so as together to be =50m; this with the wheel's inertia 11m will make 61m; now let m be added to A, and an equal weight m to B; these bodies will balance each other, and the whole mass will be 63m. If a weight m be added to A, motion will be communicated, the moving force being m, and the mass moved 64m. In estimating the moving force, the circular weight =m was made use of as a moving force; but for the present purpose of showing the velocity acquired it will be convenient to use a flat rod, the weight of which is also =m. Let the bottom of the box A be placed on a level with 0 on the scale, the whole mass being as described above =63m, perfectly balanced. Now let the rod, the weight of which =m, be placed on the upper surface of A; this body will descend along the scale in the same manner as when the moving force was applied in the form of a circular weight. Suppose the mass A, fig. 5, to have descended by constant acceleration of the force of m, for any given time, or through a given space: let a circular frame be so affixed to the scale, contiguous to which the weight descends, that A may pass centrally through it, and that this circular frame may intercept the rod m by which the body A has been accelerated from rest. After the moving force m has been intercepted at the end of the given space or time, there will be no force operating on any part of the system which can accelerate or retard its motion; this being the case, the weight A, the instant after m has been removed, must proceed uniformly with the velocity which it had acquired that instant: in the subsequent part of its descent, the velocity being uniform will be measured by space described in any convenient number of seconds.

This machine is useful for estimating the velocities communicated by the impact of elastic and nonelastic bodies; the resistance opposed by fluids, as well as for various other purposes. It may be necessary to show in what manner of the motion of bodies resisted by constant forces are reduced to experiment by this machine. Suppose the mass contained in the weights A and B; fig. 5, and the wheels to be 61m, when in equilibrium; let a weight m be applied to B, and let two long weights or rods, each =m, be applied to A, then will A descend by the action of the moving force m, the mass moved being 64m; suppose that when it has described any given space by constant acceleration, the two rods m are intercepted by the circular frame above described, while A is descending through it, the velocity acquired by that descent is known; and when the two rods are intercepted, the weight A will begin to move on with the velocity acquired, being now retarded by the constant force m; and since the mass moved is 62m, the force of retardation will be \( \frac{1}{2} \) part of that force whereby gravity retards bodies thrown perpendicularly upwards. The weight A will therefore proceed along the graduated scale in its descent, with an uniformly retarded motion, and the spaces described, times of motion, and velocities destroyed by the resisting force, will be subject to the same measures as in the examples of accelerated motion.

In this description, three suppositions have been assumed, which though physically are not mathematically true.

1. It has been assumed that the force which communicates motion to the system, is constant, which is true only when the line which suspends the weights A and B is without weight. The effect of the string's weight, however, is so trivial that if, in a particular case, we calculate the time of descent we shall find it to be 3.9896 seconds; and when the string's weight is considered, 4.0208 seconds, giving a difference of only three hundredths of a second, which is unappreciable by observation.

2. It has been also supposed that the bodies move in vacuo, but as the greatest velocity is only 26 inches in a second, the time of descent cannot be increased by the air's resistance, the two hundred and fortieth part of the whole.

3. The machinery has been supposed without friction, but the effect of the friction is such that 1\( \frac{1}{2} \) grains will destroy the equilibrium of the weights A and B. In experiments, on retarded motion, where friction does become sensible, it may be corrected by adding 1\( \frac{1}{2} \) or 2 grains to the descending body, or a little more than will destroy the equilibrium of A and B.

In constructing one of these machines for the Faculty of Sciences at Paris, Mr. Fortin has made a considerable improvement upon it by adding a detent for causing the weight A to begin its descent at the instant that the second's hand marks 0" on the dial plate.

This contrivance will be understood from the annexed figures. Two pillars of brass e,f fixed beneath the platform CD support a horizontal axis gh perpendicular to two arms of a lever, one of which is larger than the other, the use of the shortest being to support the weight A in its first position. (See fig. 186.) The longer lever being at first detained by a catch, disengages Description itself and makes the axis gh turn. The short lever turning at the same time leaves its first position, and the weight A, no longer sustained by the short lever, begins to move.

In fig. 186, which is a section at right angles, to gh the axis gh is projected into g. The short and long levers gh, gh are shown by the dotted lines as in their first position the first is sustaining A, and in their second position by gl', gh' the end k of the larger one having descended to k' and being detained at k' by a fork.

The disengagement of the end k of the long lever gh at the instant of the second's hand pointing to zero in the time-plate is thus effected. A lever with two arms is fixed on a horizontal axis attached to the platform which supports the pendulum. The inner arm of this lever im is held by the plate, and the outer one nr consisting of two branches mn, nr united by a joint or pivot about which the branch mn may turn. The other branch nr, which is a flat surface, can move only horizontally in the groove of a pulley t. A vertical iron rod v, v' is capable of taking these two positions corresponding to the two positions of the moveable plate nr. In the first position v it passes through a cut out part of the plate nr, and in the second position v' its extremity bears against the full or uncut part of the branch nr. In the position v' of the rod the end k of lever gh is engaged in the fork v' at the end of the rod. A wiper, seen in fig. 186, fixed to the toothed wheel of the pendulum acts upon the end i of lever mi. The joint takes the position n' fig. 185; the branch nr slides horizontally in the same direction, and o', no longer supported by the uncut part of nr, falls through the part that is cut out and takes the position v when it is stopped by a horizontal rod xw fixed between the upright bars Px, yz, the one rising from the pendulum box P, and the other from a shelf y below the pendulum. For the purpose of bringing the end k' of the lever gh' into the fork v of the vertical rod, a thread attached to k passes first through the fork, and next through an eye z of the bent part of the rod yz. Another thread goes to the same eye from the fork, and by successively drawing these two threads the plate nr is pushed at the end r towards yz, and the vertical rod v' is then supported by the uncut part. In order to make the second's hand point to 0" on the dial plate when the wiper sets off the detent i fig. 187, let us suppose that the second's hand indicates 15" on the dial plate when the detent sets out, or rather when the vertical rod which supports the end of the lever falls, we have only to turn the second's hand 15" upon its axis in a direction opposite to that of the ratched wheel, and the object in view will be obtained.

Dunn's improved Atwood's Machine.

Mr. Dunn of Edinburgh has introduced two very valuable improvements into this machine. The first consists in selecting a line for suspending the weights of as delicate a texture, and possessing as little rigidity as possible. He was led to this improvement in consequence of finding by experiment that the greater proportion of the whole resistance of a well made Atwood's Machine is to be ascribed to the rigidity of the cord employed; a circumstance which Mr. Atwood had entirely overlooked. The second improvement consists in supporting the axis of the large wheel or pully in the manner shewn in fig. 6, plate CCCXLIX, which represents the machine removed from the pillar on which it must be raised when used. The mode of supporting the axis will be more readily perceived in the annexed figure, where AB represents the axis terminating in finely finished conical points of tempered steel resting as represented on the sides of the conical agate caps C, D, care being taken to adapt the form of the points of the axis and the cups to each other, so that the portions of the axis in contact with the agates may be of the smallest possible size.

The agate caps are inserted into the ends of the large milled screws AB in the frame of the instrument, fig. 6, plate CCCXLIX. When the wheel is put into its place the screws A, B are turned forwards until they approach sufficiently near each other to prevent the wheel being thrown out of its place, but not so near as to force the points of the axis against the bottoms of the caps. In a number of experiments made by a committee of the Society of Arts for Scotland in 1828, appointed to investigate the merits of Mr. Dunn's machine, it was found that when the wheel weighed 4458 grains (including the cord) the strings being loaded with 60 grains each, the addition of 1/6 of a grain produced a very sensible motion; and when the load of each string was increased to 2423 grains, 1/6 of a grain were sufficient. Hence it would appear from these data, that under equal pressures the rigidity was three times the friction.

These results, however, when used for computing the momentum of the wheel gave an inertia much greater than what its form and weight would have sanctioned, and the experiments were repeated and a few others added. With these the results were still discordant, and it was at last found that they had been materially affected by the vibrations of the floor on which the machine stood. In order to overcome this difficulty another set of experiments was instituted for detecting the amount of resistance by observing its effects upon motions caused by different accelerating forces. In all these the space described was seven feet, and the time was measured by a metronome adjusted to 2 seconds. When one of the ends was loaded with 40, and the other with 37 grains, the time was 26 1/2 seconds; and each end being loaded with 2423 grains, and an additional weight of 10 grains superadded to one, the time was 20 seconds, while when the accelerating force was 100 grains, it was only 5 1/2 seconds.

From these experiments the whole resistance was found to be 2 1/2 grains, (1/2 grains more than that given by Atwood for the resistance of his machine), and the inertia came out exactly to 2000 grains. It is probable, however, that of the 2 grains of friction allowed by Atwood, 1 1/2 was owing to the greater rigidity of his cord, and only 1/2 grain to real friction. Such a near coincidence in the performance of the two machines, shews that for all practical purposes the simple machine of Mr. Dunn is as good as the more complex one originally invented by Mr. Atwood, and will, from its cheapness, be used in preference, except, perhaps, in the few cases where the machine is desired to be the best possible, and without any regard to the

Description cost. But, even in this case, the greater liability to be affected by dust, may render the advantages of the original machines form doubtful. The object still to be aimed at is the reduction of the stiffness of the silk line, for till that is still further reduced the diminution of the friction, which is the smaller portion of the whole resistance, is less essential, and Mr. Dunn, in partly disregarding it, has effected an important practical improvement.

The theory of this machine is a particular one of D'Alembert's principle of the distribution of motion, viz. that in which the two inclined planes have a vertical position. See Poisson's Traité de Mécanique, p. 46.

Machine for illustrating the Theory of the Wedge.

This machine is represented in fig. 7, where KILM and LMNO are two flat pieces of wood joined together by a hinge at LM; P is a graduated arch on which these pieces of wood can be moved so as to subtend any angle not greater than 60°, and a, b two screws for fixing them at the required angle. The back of the wedge will therefore be represented by IKNO, its sharp edge by LM, and its two sides by KILM, LMNO. The weight p suspended to the wedge by the hook T, and the weight of the wedge itself, may be considered as the force employed to drive the wedge. The wooden cylinders AB, CD, have their extremities made like two flat circular plates to prevent the wedge from slipping off at one side. To the pivots of these cylinders, two of which are represented at e and f, are fastened the cords EW, fU, CV, AX, which passing over the pulleys U, V, X, W are fastened to the two bars v, x, w, on which any equal weights Y, Z may be hung at pleasure. The tendency of these weights is evidently to draw the cylinders towards each other, and they may therefore be regarded as the resistance of the wood acting against the sides of the wedge. The cylinders themselves are suspended by their pivots to the threads E, F, G, H, which may be fixed to the ceiling of the room, or to the horizontal beam of a frame made on purpose. By placing various equal weights at Y and Z, it may be easy to determine the proportion between the power and the resistance when the wedge is in equilibrium. In this machine the impelling power is the pressure of the weight p, whereas, in the real wedge, the impelling power is always an impulsive force which is infinitely more powerful.

Machine for illustrating the effects of the centrifugal force in flattening the poles of the Earth.

Fig. 8. represents this machine, which consists of two flexible circular hoops, AB and CD, crossing one another at right angles, and fixed to the vertical axis EF at its lower extremity, but left loose at the pole or intersection e. If this axis be made to revolve rapidly by means of the winch m, and the wheel and pinion n, o, the middle parts A, B, C, D will, by their centrifugal force, swell out and strike against the frame at F and G; if the pole e, when sinking, is not stopped by means of a pin E fixed in the vertical axis. The hoops, therefore, will have a spheroidal form; the equatorial being longer than the polar diameter.

Machine for trying the Strength of Materials.

The piece of wood, whose strength is to be tried, is represented by EF, and the force is applied to it by means of the winch A, which winds up the rope BC, passing over the pulley n, and below the pulley m, and attached to the point D of the beam EF. The pulleys slide on two parallel bars fixed in a frame, held down by a projecting point, at G, of the lever GR, which is graduated like a steelyard, and measures the force employed. The beam EF is held by a double vice IK with four screws, two of which are invisible. When a wire is to be torn it is fixed to the cross bar LM; and when any body is to be crushed, it must be placed beneath the lever NO, the rope BC being fixed to the hook N, and the end O being held down by the click which acts on the double ratchet OP. The lever is double from O to Q, and acts on the body by a loop fixed to it by a pin. See Young's Natural Philosophy, vol. i. p. 768, from which this drawing and description are taken.

Machine for shewing the Composition of Forces.

The part BEFC is made to draw out or push into the Machine square ABCD. The pulley H is joined to BEFC, for show so as to turn on an axis which will be at H when the square ing the BEFC is pushed in, and at h when it is drawn out. A ball composition G is made to slide on the wire k, which is fixed to BEFC forces, and the thread m attached to the ball goes over the pulley to I, where it is fixed. Now, when the piece BEFC is Plate ccl pulled out, the pulley, wire, and ball move along with it, in Fig. 2. the direction DCF, and it is evident that the ball G will slide gradually up the wire k. It is therefore acted upon by two forces: one in the direction GH, and the other in the direction GC, and will be found at the end of the motion at g, having moved in the direction Gg, the diagonal of a parallelogram whose sides are GH, GC.

Smeaton's Machine for experiments on Windmill Sails.

In the experiments with this machine, the sails were carried round in the circumference of a circle, so that the same force wind effect was produced as if the wind had struck the sails at mills rest with the velocity which was thus given them. In the Plate ccl. pyramidal frame ABC is fixed to the axis DE, which carries the arm FG with the sails GL. By pulling the rope Z, which coils round the barrel H, a motion of rotation is given to the sails, so that they revolve in the circumference of a circle, whose radius is DI. At L is fixed a cord which passes round the pulleys M, N, O, and coils round a small cylinder on the axis of the sails, and raises the scale C, in which different weights are placed for trying the power of the sails, and which, being in the direction of the axis DE, is not affected by the circular motion of the arm DG. The scale C is kept steady by the pillars Q, R, and prevented from swinging by the chains S, T, which hang loosely round the pillars. VX is a pendulum composed of two leaden balls moveable upon a wooden rod, so that they can be adjusted to vibrate in any given time. The pendulum hangs upon a cylindrical wire, on which it vibrates as on a rolling axis.

Smeaton's Machine for experiments on Rotatory Motion.

This machine is exhibited in fig. 1, where the vertical Apparatus axis NB is turned by the rope M passing over the pulley for rotatory R, and carrying the scale S. The axis NB carries two equal leaden weights K, D, moveable at pleasure on the horizontal bar HI. The upper part N of the axis is one-half the diameter of the part M, so that when the rope is made to wind round N, it acts at half the distance from the axis, at which it acts when coiled round M. When the rope is wound round N, the same force will produce in the same time but half the velocity which is produced when the rope coils round M, the situation of the leaden weights being the same; but when the weights K, L are removed to a double distance from the axis, a quadruple force will be required in order to produce an equal angular velocity in a given time.

Machine for Illustrating the Parallelogram of Forces.

This apparatus, described by Professor Moseley, is shown Description in the annexed figure 189, where AB is a circular frame or ring of wood supported vertically upon a stand BCD. Three moveable pulleys, P₁, P₂, P₃, made with as little friction as possible, are so constructed that they can be placed on any part of the circumference of the ring AB, the wheels being parallel to the surface of the ring. Three fine silken strings, united at one point O, are made to pass over the pulleys in the lines OP₁, OP₂, OP₃, and to support at their other extremities the weights W₁, W₂, W₃. After a certain time these weights will come into a position of equilibrium, when the three pieces acting at O will balance each other. Fill up the interior part of the ring AB with a board slightly depressed beneath the nearer surface of the ring, so that the strings may not rub against it, and draw lines upon the board in the direction of the strings OP₁, OP₂, OP₃; and taking the tenth of an inch, or any other unit of magnitude, divide the lines OP₁, OP₂, OP₃, into tenths. Then, if there are six units of weight in W₁ (ounces or any other unit), make OP₁ = 6-10ths, and OQ as many tenths as there are units or ounces in W₂, and complete the parallelogram OPQR, by drawing lines upon the board. When this is done, we shall find that there will be as many tenths of an inch in the diagonal OR, as there are units of weight or ounces in the weight W₂, which balances the forces produced along the lines OP₁, OP₂, or their compounded or united action in the diagonal OR. This diagonal OR will be found to lie in the same straight line with OP₃. Hence it follows that whatever be the weights W₁, W₂, W₃, or the position of the pulleys P₁, P₂, P₃, two forces acting along the sides of a parallelogram OP₁, OP₂, and proportional to these sides, are equal to a force acting along its diagonal, and proportional to it.

In place of using strings which cannot easily be graduated, and therefore renders it necessary to note the graduations on a board behind them, the apparatus shewn in the annexed figure has been substituted in the class of Experimental Philosophy in King's College, London. A parallelogram OPQR is formed of thin slips of boxwood, so as to be very light, and is divided into inches and tenths. The slips are united at the angles by moveable joints, and the joints P and Q are made to slide along either of the sides which they unite. A slip of wood RC moves freely along with OP and OQ, on the joint O. An inch, or any part of it, being taken as the unit of length, and an ounce as that of weight, the joint P is made to slide along OP, until OP contains as many inches as there are ounces in the weight W₁; and Q is moved till OQ contains as many inches as W₂ contains ounces. By means of other sliding joints, PR and PQ are made equal to OQ and OP. Strings are then fastened to the ends A, B, C of the slips OP, OQ, and R'O, and these are passed over the pulleys P₁, P₂, P₃ of the wooden ring in the preceding instrument, and made to suspend the weights W₁, W₂, W₃. In a certain time, as before, the system of forces will come into a state of equilibrium; and when this has taken place, it will be found that the slip OR' has taken the position of the diagonal OR, and that it contains as many inches as there are ounces in W₃. This instrument was made by Messrs. Watkins and Hill, philosophical instrument makers, Charing Cross.

Apparatus for Illustrating the Equilibrium of a number of Forces.

Professor Moseley, in his excellent work on mechanics, has described an instructive apparatus for exhibiting the equilibrium of a number of forces applied to different points of a body, but acting all in one place. This apparatus is shewn in the annexed figure, where ABC is a smooth flat board, resting on three small ivory balls, laid on a smooth horizontal table, and round the edge of the table are fixed a number of pulleys, P₁, P₂, P₃, the plane of their action being perpendicular to that of the table, and each having the highest point of its circumference or groove level with the surface of the board ABC. Assume any points on the board, viz., P₁, P₂, P₃, P₄, and having fixed strings at them, make these strings pass over the pulleys P₁, P₂, P₃, P₄, &c., and suspend weights at the other ends, represented by the letters P₁, P₂, P₃, &c. If this system of forces is left to take up a position of equilibrium, the following relation will be found between the directions of the forces and their measures P₁, P₂, P₃, &c. If from any point M on the surface of the board ABC, we draw perpendiculars Mm₁, Mm₂, Mm₃, &c., upon the directions P₁, P₂, P₃, &c., of the forces P₁, P₂, P₃, &c., we shall find that the sum of the products arising from multiplying the lengths of the perpendiculars Mm₁, Mm₂, &c., by the forces P₁, P₂, &c., whose direction they are drawn, taken in reference to those forces which tend to turn the system about that point, will be equal to the sum of these, taken in reference to the forces tending to turn it in the opposite direction. In the present figure, for example, we shall find

\[ P₁ \times Mm₁ + P₂ \times Mm₂ + P₃ \times Mm₃ = P₄ \times Mm₄ + P₅ \times Mm₅. \]

Another property of such forces is deducible from this apparatus. If the forces P₁, P₂, &c., are all made to act upon a single point, but parallel to their present directions, they will all be in equilibrio, and the point remain at rest.

When a system of forces P₁, P₂, P₃, and P₄, is not in equilibrium, we may determine the magnitude and direction of a fifth force P₅, which shall place them in equilibrium. On the surface of the board take any point N, and draw Na parallel to P₁p₁, so as to represent the force P₁; and in like manner n₁n₂, n₂n₃, n₃n₄, representing the other three forces P₂, P₃, and P₄; then, if we join N and n₄, the line Nn₄ will represent in magnitude, and be parallel in direction to P₅, the force required to balance the other force.

We must now find where to apply this balancing force, in order to satisfy the condition expressed in the preceding formula. This will be effected by applying it parallel to Nn₄, and at such a distance from M, that the product of P₅ × Mm₅ may be equal to the sum of the similar products of all the other forces. The distance Mm₅ may be easily found, by dividing the difference of the sum of the products about M by the force P₅. If we now draw through M a line Mm₅, equal to the distance found above, and perpendicular to Nn₄, then a line P₅p₅, drawn perpendicular to this, is that in which the force must be applied. The force P₂, now found, is the resultant of all the other forces; for

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1 Treatise on Mechanics applied to the Arts, p. 20. Description of Mr. Bate's Balance for verifying the National Standard Bushel.

When Captain Kater and Mr. Bate were employed by government in the adjustment and construction of the new standard weights and measures, they were under the necessity of constructing a balance capable of determining a weight equal to that of the standard bushel measure, which was about 170 lbs., together with the eighty lbs. of water it should contain, which made altogether a weight of 250 lbs. In this difficult task Mr. Bate has exhibited great skill and sagacity, in combining with strength and solidity of structure that delicacy of adjustment which the object he had in view rendered absolutely necessary.

Mr. Bate's balance is represented in the annexed figure.

Fig. 192.

In the construction of the beam he first tried cast iron; but though it was as light as was consistent with the required degree of strength, the inertia of the mass was so considerable, that much time would have been lost before the balance would have answered to the small differences he wished to ascertain. Lightness being essentially necessary to the sensibility of the balance, and bulk being also very desirable, in order to preclude such errors as might arise from partial alterations of temperature in the beam, he determined to employ dry mahogany. The beam was therefore made of a plank of this wood, about seventy inches long, twenty-two inches wide, and 2½ thick, tapering from the middle to the extremities. An opening was cut in the centre of gravity of the beam, as at L, and strong blocks screwed to each side of the plank, to form a bearing for the back of a knife edge which passed through the centre. Similar blocks were likewise screwed to each side of the plank at its extremities, to form a bearing for the knife edges which support the scales or pans.

In order that the whole weight should not be thrown upon short portions of the knife edges, as was usually the case, by which these edges were exposed to injury, as well as to a change of form, Mr. Bate made his knife edges long, the central one being six inches, and the other two five inches in length. They were triangular prisms, with equal sides, each being 3-4ths of an inch long, very carefully finished. The angles were of course 60° each, but they were ultimately brought to an angle of 120°, which prevented all risk of injury, without impeding the rotation of the beam.

The central knife edge, and the mode of attaching it, and the plane which supports it to the beam L, and to the frame N, are shown in the annexed figure. The knife edge F is screwed to a thick plate of brass L, the touching surfaces having been previously ground together.

The support M upon which the knife edge rested, and revolving through its whole length, was formed of a plate of polished hard steel screwed to a block of cast iron; and this block was passed through the opening in the centre of the beam already mentioned, and properly fixed to the frame of cast iron CC, fig. 192. By means of screws s, s, the knife edge is placed as accurately as possible at right angles to the surface of the beam, and by means of other screws, not seen in the figure, it is adjusted so as to be slightly above the centre of its axis. In the cross horizontal bar, which is sustained by the columns CC, and which carries the steel plane M, there is an aperture, through which a fork-shaped piece of brass N passes, forming part of the stand DED. This fork is completely detached from the beam when the balance is in use; and by the motion of the handle H, it can be raised so as to make the fork catch a piece L (see fig. 193), projecting from the piece of brass which carries the knife edge. Two similar hooks are placed at the ends of the frame DED, by which the other two knife edges may be raised above their supporting planes by the motion of the handle H; and by this means the three knife edges are protected from that injury which they would receive from always resting upon the steel planes.

The mechanism of the ends of the beam for carrying the knife edges and their steel planes, is shown in the annexed figure. These pieces project from the ends of the beam, and are placed at exactly equal distances from the fulcrum, or central knife-edge. In these pieces the knife edge F has its edge turned upwards. The scales or pans are each attached by a hook to the lower end of stirrups, shewn at S', which receive between their two uppermost branches the extremity F' of the beam. The steel plane M' is fixed to the upper branches of the stirrup, and rests upon the knife edge when the balance is in use. The knife edges are adjusted, like the central one, by small screws, which give it a horizontal and a vertical motion. The adjustment of the central knife edge to a point a little above the centre of gravity of the beam, is facilitated by means of small weights which screw on the wires seen between S' and N', in fig. 192; and by screwing these nearer to or farther from the fulcrum, the centre of gravity is moved into its required place.

Since this balance was described in the Philosophical Transactions for 1826, Mr. Bate has made a very great improvement upon it. By an ingenious device, the beam and scales are in the first instance suspended on cylindrical axes, and afterwards by continuing the motion of the handle H, they are made to rest upon the knife edges. In this way the weights in the opposite scales may be nearly balanced, while the beam and scales rest upon the cylindrical axis, and the extremely delicate part of the operation will then be completed in a shorter time when the beam and scales are put upon the knife edges.

The performance of this balance was fully equal to the expectations of Captain Kater and Mr. Bate. With 250 lbs. in each scale, the addition of a single grain produced an immediate variation in the index of one twentieth of an inch, to a radius of fifty inches. Hence the balance was sensible to the \( \frac{1}{17} \) part of the weight.

Description of Dr. Black's simple and delicate Balance.

This simple and accurate instrument which any person may readily construct for himself, was invented by the celebrated Dr. Black, and an account of it communicated to James Smithson, Esq. The beam AB was a thin piece of fir wood not thicker than a shilling, a foot long, \( \frac{1}{3}-\frac{1}{10} \)ths of an inch thick at each end, and twice that thickness at

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1 See Phil. Trans. 1826, Part I. p. 11, and Moseley's Treatise on Mechanics, p. 70. Description of the middle. The whole length was divided into twenty parts by transverse lines, and each of these subdivided into halves and quarters. One of the smallest needles \( mn \) was fixed across the middle with wax for the horizontal axis.

A piece of plate brass, CD is bent so as to form three sides of a cube. It rests on its middle side, and the two vertical sides form the fulcrum on which the needle \( mn \) rests.

The height of \( mn \) above the table is only \( 1\frac{1}{2} \) or two-tenths of an inch; so as to allow only a very limited play to the beam AB. The edges upon which the needle rests are ground at the same time on a plain surface. The weights were one globe of gold weighing one grain, and two or three others one-tenth of a grain each, along with a number of small rings of fine brass wire formed by coiling it round a thicker brass wire into a close spiral. The end of the wire being tied hard with a waxed thread, the coiled wire was put into a vice, and a sharp knife being applied and struck with a hammer, a great number of the coils were cut at one stroke, and were found to be as equal to one another as could be wished.

These weights were about the 1-30th of a grain each, and by means of them the weight of any body from one grain to the 1/36th of a grain could be readily weighed. By using a thinner beam and grinding the needle to an edge, a still more delicate balance could be made, and a paper scale added if necessary. Captain Kater has found that a balance of this kind is sensible to the 1/36th part Fig. 185. of a grain when loaded with ten grains. He found it necessary, however, when accuracy was essential, to use a scale of thin card paper of the form shewn in the annexed figure, a thread being passed through the two ends in order to bring them together.

**Description of a Chinese Mangle.**

This simple and ingenious piece of machinery is represented in fig. 197, which the writer of this article found on a large scale on a series of paper-hangings for rooms. The late Andw. Waddel, Esq., of Hermitage Hill, who had seen it at work at Canton in 1786, had previously furnished us with a drawing of it.

On the floor FF, which is paved with tiles, was a concavity C lined with hard wood. A roller R, having the cloth to be mangled wrapped round it, was placed in this concavity. The weight used was a sandstone S of about 11 cwt., and shaped as at A, so as to leave the line of direction from its centre of gravity, falling within either of its two bases. By resting on the timber frame MN, the workman steps on the uppermost end of the stone S, and causes its under surface to descend gently upon the rolled up cloth R. He is then in the position shewn in the figure, and by the alternate pressure of each foot he gives the stone an oscillating motion which moves the roller over the cloth and the whole of the smooth concave surface C, and with any degree of velocity that he chooses, till the mangling of the cloth be completed.

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**Account of the Mechanical Process of Cutting Steel with Soft Iron.**

As this process seems to depend on a new principle of mechanical action, it may probably be modified and extended, so as to produce effects which have not hitherto been contemplated. The process to which we allude seems to have been invented by the Shakers in America. It consists in cutting the hardest steel by means of a buzz, or wheel of soft iron, made to revolve with great rapidity. By this means a file may be cut in two, while the soft iron plate is not in the least degree impressed by the file. Mr. Barnes cut a saw plate and formed the teeth by the same means.

In the process a band of intense fire appeared round the soft iron wheel, which emitted sparks with great and continued violence. Professor Silliman was of opinion that this was nothing more than "a peculiar method of cutting red hot, or possibly white hot steel; for the mechanical force produces these degrees of heat, and the steel loses its temper at the place of action." (Silliman's Journal, vol. viii. p. 342.) But it appears from a very careful examination of the process by MM. Darien, and Colladon of Geneva, that the process has a more recondite origin. The following is a general abstract of the results which they obtained:

1. Having found that the iron wheel was covered with small fragments of the steel, they could see by a microscope no appearance of softening; on the contrary, they found these fragments as hard as the best tempered steel.

2. Having fitted up a lathe by which they could give a determinate velocity to the iron wheel, they found that with a velocity of 34 feet per second, an iron wheel was easily cut by a steel graver without any reaction on the graver. With a velocity of 34 feet 9 inches the iron was less attacked, and the graver began to experience the impression from the iron. At a velocity of 35 feet 1 inch, the action of the iron on the graver was decided, and increased with greater velocities, till at a velocity of 76 feet per second, the iron was no longer touched by the steel, while the steel was cut with the greatest violence.

3. In order to determine the effect of softening or annealing on the steel, our authors examined the fragments of steel detached from the graver at different velocities, from 40 to 100 feet per second; and in every case when the iron was only touched for an instant with the steel graver, the latter exhibited no trace of annealing; but when the graver was long and strongly pressed it sometimes became red hot. In that case, however, the fracture of the steel became quite different, and the action upon it was rather diminished than increased.

4. Having thus ascertained that the effect is not owing to the annealing of the steel, and found that the effect was not increased by the fragments of steel, which after some time collect on the iron wheel, our authors justly suppose that the whole effect is directly mechanical, arising from the brittleness of the steel, which is torn asunder before it has time to introduce itself amongst the molecules of the soft iron; and they consider it as analogous to the penetration of wood by a ball of tallow.

5. Upon using wheels made of a mixture of copper and iron, and wheels of pure copper, no effect was produced by them on the graver, though they cut different alloys harder than themselves.

In these experiments, however, a very remarkable effect occurred. Little or no heat was generated, when files and steel springs were held firmly against the revolving copper wheel; and our authors observed several other curious facts, which they mean to study with greater care, connected with the production of heat by the friction of metals. See Edinburgh Journal of Science, vol. i. p. 341.; Bibliothèque Universelle, April 1824, p. 283—290, and Silliman's Journal, vol. vi. p. 336—354.

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1 Annals of Philosophy, N. S. vol. x. p. 52. 2 Landers Mechanics, p. 193. The simple instrument called a Lewis for lifting loose stones has been long known and employed in architecture. A much more simple and efficacious contrivance was nearly forty years ago invented by Mr. Richardson of Keswick, and called the lifting plug. It excited little notice till it was made known in Scotland by Mr. Spottiswoode of Spottiswoode, who used it to a greater extent than any other person, and found it an invaluable apparatus for pulling large stones out of the soil.

A cylindrical hole about two inches deep is cut vertically out of the stone by the steel boring chisel used by masons. A cylindrical plug of common iron, a little less than that of the hole, is then driven into it, about an inch deep, by two or three smart blows of a hammer. By using a common windlass or pulley attached by a hook carried by the plug, the heaviest stones may be raised, and the largest masses of stone torn out of the ground by no other fastening than the adhesion of the plug to the hole. When the stone is raised to its place, the pulley may be detached from it by giving a sharp stroke or two from a hammer upon the stone. The elasticity of the stone is no doubt the cause of the adhesion of the plug, the stone grasping the plug as wood does a polished nail; but another cause undoubtedly is the enormous friction occasioned by the particles of the stone being jammed into the surface of the soft iron plug. It has been thought by some that the first of these causes is the only one, and that the circumstance of the plug being detached by a slight vibratory motion of the stone is a proof of this; but if the two causes which we have mentioned, concur in producing the adhesion of the plug, then it is obvious that when one of these causes of adhesion is removed by the vibrations of the stone, the other being inadequate to sustain the weight, the stone must quit its hold of the plug. Independently of this argument, however, it appears to us that the same cause which releases the plug from the influence of elasticity, tends also to relieve it from the abraded material which unites the surface of the stone with the surface of the iron. In all vibratory movements there is an alternate contraction and extension of the vibrating body, and in one of these phases no doubt the great weight of the stone overcame the diminished force by which it was previously sustained. A drawing and description of the machine for raising stones, by Professor Low, will be found in the *Edin. Phil. Journal*, first series, vol. iv. p. 281.

The subject of the adhesion of nails in wood is so intimately connected with the principles of the lifting plug, that our readers will, we are sure, thank us for any information on the subject. The only person who has directly investigated the subject is Mr. B. Bevan, who constructed a machine for measuring the force with which nails adhere to wood into which they are driven. He employed it in drawing out nails of different lengths, from a quarter of an inch to 2½ inches long, from dry Christiania deal, at right angles to the grain of the wood. The following table contains the results of his experiments:

| Kind of Nail | No. of Nails | Inches to the Pound | Inches driven into the wood | Pounds required to extract them | |--------------|-------------|---------------------|---------------------------|-------------------------------| | Fine sprigs | 4560 | 0·44 | 0·40 | 22 | | Do. | 3200 | 0·63 | 0·44 | 37 | | Threepenny brads | 618 | 1·28 | 0·50 | 58 | | Cast iron nails | 380 | 1·00 | 0·50 | 72 | | Sixpenny nails | 73 | 2·50 | 1·00 | 187 | | Do. | | | 1·50 | 327 | | Fivepenny | 139 | 2·00 | 1·50 | 320 |

The weights necessary to press a sixpenny nail into dry Christiania deal, to different depths, are as follows:

| Pressure required | Depth in the wood | |------------------|------------------| | 24 lbs | 0·25 inches | | 76 | 0·50 | | 235 | 1·00 | | 400 | 1·50 | | 610 | 2·00 |

Mr. Bevan extended his experiments to screw nails. Those which he used were about two inches long, ⅝ of an inch diameter at the exterior of the threads, ⅛ at the bottom; the depth of the worm or thread was ⅛ of an inch and the number of threads in an inch twelve. They were screwed through pieces of wood exactly half an inch thick, and drawn out by the weights stated in the following table:

- Dry Sycamore: 830 lbs. - Dry sound Ash: 790 lbs. - Dry Beech: 790 lbs. - Dry Mahogany: 770 lbs. - Dry Oak: 760 lbs. - Dry Elm: 635 lbs. - Dry Beech (another kind): 460 lbs.

When the wood was deal, or any of the softer woods, the force required to draw out the nails was only about one-half of the above weights. The mean of the first six experiments is 762 lbs., and of all the experiments, 719. Hence we may infer that the force required in the softer woods is about 370 lbs.

**Bunce's Pile Engine.**

A side view of this engine is shewn in figs. 5 and 6. It consists of two endless ropes or chains A, connected by cross pieces of iron B, B, &c. (fig. 6.) which pass round the wheel C, the cross pieces falling into corresponding cross grooves, cut in the periphery of the wheel. When the man at S, therefore, drives the wheel m by means of the pinion p, he moves also the wheel C fixed on the axis of m, and makes the double ropes revolve upon the wheels C, D. The wheel D is fixed at the end of a lever DHK, whose centre of motion is H, a fixed point in the beam FT. Now, when the ram L (fig. 5, 6,) is fixed to one of the cross pieces B by the hook M, the weight of the ram, acting by the rope, moves the lever D K round H, and brings the wheel D to G, so that, by turning the winch, the ram L (fig. 5,) is raised in the vertical line LRG. But when it reaches R, the projecting piece R disengages the ram from the cross piece B, by striking the bar Q; and as the weight is removed from the extremity D of the lever, the counterpoise I brings it back from G to its old position at F, and the ram falls without interfering with the chain. When the hook is descending, it is prevented from catching the rope by means of the piece of wood N suspended from the hook M at O; for being specifically lighter than the iron weight L, and moving with less velocity, it does not come in contact with L till the ram is stopped at the end of its path. When N, therefore, falls upon L, it depresses the extremity M of the hook, and therefore brings the hook over one of the cross pieces B, by which the ram is again raised.

**Fixed Iron Crane.**

Plate CCCLII. figs. 4 to 7, shew the elevation, plan, section, and outer frame of a fixed iron crane or compound winch, used for raising heavy weights to a considerable height, as it is employed at the present day for the erection of buildings, &c. The triangular framings FFF, &c., are of cast iron, and give support to three horizontal revolving shafts of malleable iron running in cylindrical journals. The princi-

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1 See *Phil. Mag.* vol. iiiii. p. 168, and *Edin. Journal of Science*, vol. i. p. 150. 2 *Phil. Mag.* Oct. 1827, p. 291, and *Edin. Journal of Science*, No. xv. p. 150. Mechain's shaft is in the axis AX of the hollow cast iron barrel BB, to which one end of the rope is attached, so as to be coiled and uncoiled by the revolutions of the barrel, in opposite directions. The barrel carries with it a toothed wheel WW, connected with a pinion V placed on a secondary axis QM in such a manner as to revolve freely upon it, or be carried round along with it at pleasure. On the end of this axis QM are placed the crank handles, to which the power may be applied through the pinion V to the wheel W and the barrel B directly. But if a more powerful and slower motion be required, the means of obtaining it are provided in connection with a third axis PN; this axis is immediately connected with the wheel W by the pinion V, which revolves with it, and turns the axle PN, and with it the secondary wheel VV. But the circumference of VV is acted on by the teeth of the pinion TT, which is also made to revolve independently of its own axle, or in connection with it, by means of the lever LL and the clutches CC. The change of power is effected in the following manner: The drawings represent the pinions Y and T both disengaged from their axle, and revolving freely around it in oiled bushes; but if the smaller power is to be used, the attendant lays hold of the lever handle L, and drawing it towards T, engages the clutch towards Y with the pinion Y, and as the clutches C and C are feathered on the axle, they must revolve with it, and carry round that pinion with which they are engaged. In this way, the pinion Y being made to give the motion of the crank handles directly to the barrel, give the less power; but if the greater power and slower motion are to be used, the clutch at Y is disengaged, the lever is reversed, the pinion T is caught and carried round by the other clutch, and thus the motion of the axis QM is transferred through the wheel VV to the axis PN, Mechan. and to the pinion V, which works on the primary wheel WW, and the power is increased in the ratio of V to T. When the lever L is placed, as in the figures, so that neither pinion is in gear with its axle, the barrel is free to revolve in the direction of any force applied to it, so that the weight may be run down with great rapidity; and the following contrivance gives a regulating power to this motion: An iron strap SSS embraces closely a wooden drum on the axis of BB, and is attached at both ends to bolts on either side of the fulcrum of the bar HI, which has a handle at H, at a considerable distance from the fulcrum, so that by depressing H, the drum may be firmly embraced by the friction strap SSS, and stopped at any point in its descent, or moderated to the requisite degree. This last appendage is of great consequence in saving time, and preventing the danger attendant on lowering the weights with great velocity.

Revolving Jib Crane.

Plate CCCLII.figs.1, 2, and 3, shew the plan, side elevation and back elevation of the revolving jib-crane or wharf-crane. Its wheels give a variable power in the same way as those in figs. 4, 5, 6, and 7, plate 351.

Fig. 4, is a detached portable equilibrium crane; its pyramidal iron base merely rests on the ground, and may even run on a road or railway, and when very great weights are raised, counterpoises may be placed on the platform with the attendants.

For the description of a great variety of useful machines, the reader is referred to the second volume of Gregory's Mechanics, and to Dr. Young's Natural Philosophy. See also HYDRODYNAMICS.

MECHAIN, Peter Francis Andrew, a well-known practical astronomer and geographer, was born at Laon, on the 16th April 1744. His father was an architect, and educated him with the intention of making him his successor in his business. He afterwards took charge of two young men at Sens, as their private tutor, and accidentally became acquainted with Lalande, under whose patronage he was subsequently brought forward as an observer, surveyor, and computor. He made two voyages with M. de la Bretonnière, and assisted him in surveying some parts of the coast of France. He was afterwards employed in various computations by the Marquis de Chabert, and the Duc D'Ayen. Having obtained a prize from the Academy of Sciences in 1782, for a Memoir on Comets, he became a member of the Academy the same year. About the year 1785, he undertook the publication of the Connaissance des Temps, and continued it till he was employed in geodetical operations at a distance from Paris. He was appointed member of a committee, along with Cassini, de Thury, and Legendre, to meet the English astronomers for the determination of the relative situation of the observatories, which had been proposed by Cassini. It was in these operations that he first brought Bord's circle into general use. In 1791 he was appointed, in conjunction with Delambre, to execute the intentions of the Constituent Assembly, with regard to the determination of a basis of linear measures. A variety of delays and difficulties occurred in these operations. In Spain he was wounded in the head and side, by an accident which occurred whilst he was inspecting a water-wheel; and the political circumstances of the times produced many embarrassments, which caused him to linger in Italy perhaps a little longer than was actually necessary; but the establishment of the Bureau des Longitudes, and his nomination as a member of it, determined his immediate return to Paris. He was now director of the Observatory, and he entered with great zeal on a series of observations, which were to rival those of Flamsteed, of Bradley, and of Maskelyne; but he seems to have been a little tired of the confinement, and he readily accepted, or rather solicited, the appointment to assist in the measurements required for the still farther extension of the arc of the meridian to the south of Barcelona. But the secret motive for his seeking this humbler employment appears to have been a desire to remove some doubts which he entertained respecting the latitude of Barcelona, as it appeared after his death from his papers, that there had been a discordance of 3° in some observations which he had not made public. In this unfortunate undertaking, he paid a heavy penalty for any want of candour which may have been attributed to the concealment. Shipwreck and disease awaited him; and he died at last, on the 20th of September 1805, of a fever, which fatigue and a bad climate had brought on. From the time of his accident in Spain he had become habitually melancholy and timid, though regardless of personal danger in the pursuit of his professional objects. His whole time was occupied in observing and calculating; he published little; and never ventured to advance any reflections on the subjects which employed him, being probably more in the habit of acting than of speculating. He married, in 1777, Mademoiselle Thérèse Marion, with whom he had become acquainted at Versailles. This connexion was in every respect happy; he was indebted to it for a competent fortune, and he left a daughter and two sons.

1. Of his publications the most important are to be found in the Mémoires des Sciences étrangères; that is, besides some Observations of eclipses and occultations, a Memoir on the comets of 1532 and 1661, showing that they are not the same; and their non-appearance seven or eight years afterwards fully justified his conclusions, and the adjudication of the prize. 2. In the Memoirs of the Academy, from 1782 to 1784, there are several of his Observations of transits, eclipses, occultations, and comets. 3. There are also some letters of Méchain in Zach's Geographical Ephemerides about 1800, on the instruments of the Parisian Observatory, and on other subjects. 4. He edited the Connaissance des Temps, from 1786 Mechanics' Institutions.

MECHANICS' INSTITUTIONS have for their object the instruction of the working classes in those branches of science and art, which are of practical application in the exercise of their several occupations.

It has been a subject of very considerable controversy to whom the honour of having originated these institutions ought to be attributed. Some have contended that to Professor Anderson, the founder of the institution in Glasgow, which bears his name, it is alone due; because in addition to the clauses in his will, which especially provide for the formation of a class of manufacturers and artificers, he had for a period of thirty years, whilst professor of natural philosophy in the university of that city, opened a class to which manufacturers and others had access, for the purpose of becoming acquainted with the fundamental principles of experimental physics. Others, again, have affirmed that although Professor Anderson had such an intention in view when he framed his will, it was only through the exertions of Dr. Birkbeck, the second professor of the Andersonian Institution, that the project was carried into complete and practical effect. Without going into the merits of this question, however, it appears to us that to Professor Anderson is certainly due the merit of having originated the idea of instructing that previously neglected but valuable portion of the community. But whilst we give all the weight to which that admission is entitled, we are of opinion that in so far as regards the usefulness of the idea when carried into effect, the merit is due to Dr. Birkbeck. Professor Anderson was in the frequent habit of visiting and spending several hours in the workshops of the mechanics of Glasgow, and freely entering into conversation with them; amongst others he was often found at the workshop of our illustrious countryman James Watt, then a watchmaker in Glasgow; and it was in consequence of the avidity with which the workmen availed themselves of the theoretical information which on these visits he was in the constant practice of imparting, that he first conceived the idea of bringing within their reach those principles of science, the attainment of which had hitherto been confined to the higher and wealthier classes.

He then commenced within the walls of the university a course of lectures on natural philosophy, for the especial benefit of that class of society, and divided the course into two branches; one meeting four times a-week, in which the mathematical principles of the science were considered apart from experimental illustration; and the other, which met twice a-week, consisting entirely of those branches which admitted of such illustration, and in which the principles of mathematics were less called into operation. This latter class was regularly and numerously attended by a great many manufacturers and others engaged in operative occupations, and was continued for several years with great success; and he called it his antitoga class, in contradiction to his regular academic attendance, which was denominated the toga class. It was in consequence of the decided improvement in the arts and manufactures of Glasgow, which from that time became very evident, and which Professor Anderson justly attributed to his instructions, that he conceived the truly philanthropic intention of bequeathing his ample fortune for "the good of mankind and the improvement of science." In the provisions of his will there was an especial reference to the formation of classes upon the principle of that which he had himself so successfully taught within the walls of the university.

Professor Anderson died on the 16th of January 1796; and at a meeting of his executors, held on the 23d of March following, it was resolved to carry his intentions into immediate effect. Accordingly on the 21st of September of the same year, Dr. Thomas Garnett was elected first professor of natural philosophy.

The lectures however were at first only popular. Dr. Garnett, in 1800, was appointed to the chair of natural philosophy in the Royal Institution of Great Britain, founded at the suggestion of Count Rumford and Sir Joseph Banks; and Dr. George Birkbeck was elected in his stead. It was then, upon the suggestion of Dr. Birkbeck, that the trustees were enabled to give full effect to Professor Anderson's favourite idea, as expressed in his will, of affording instruction in experimental philosophy to the artizans of Glasgow. Such was the origin of the first mechanics' class, which has now been taught for upwards of thirty-six years with increasing success and advantage to this interesting class of students.

But important as was this era in the history of the class which was thus instituted, it was not until upwards of twenty years afterwards that the example was followed by any other city in the kingdom. At length, however, in 1821, the School of Arts for the instruction of mechanics was formed in Edinburgh; and within a year or two afterwards, the example was followed, and another institution of the same nature was originated in Glasgow, in consequence of some differences which had arisen between Dr. Ure, (Dr. Birkbeck's successor in the Andersonian Institution), and the students who attended his operative class. A great many of these left the Andersonian and founded the Mechanics' Institution, of which Dr. Birkbeck, at their solicitation, consented to become the patron. About the same time the London Mechanics' Institution was instituted, and within a very few years after this period there was hardly a city of any note in the kingdom which had not institutions of a kindred description.

As might have been expected, there exists a variety of opinions both as to the precise kind of instruction which these institutions ought to afford in order to carry their objects into the most complete effect, and also in regard to the manner in which they ought to be governed. Some conceive that the course of instruction ought to be of a popular nature, admitting amongst the lectures upon the graver and more abstruse sciences, prelections of a lighter and more attractive kind; whilst others are of opinion that the course ought to be strictly confined to those branches of physical science alone which are of practical application in the arts or manufactures. The management of several, indeed of the greater number, is entirely in the hands of the mechanics themselves; some have an admixture of these with the upper classes of society, whilst in others of a third description, the government is entirely vested in the latter, or wealthier classes.

Favourable as we are to the universal diffusion of knowledge amongst the great mass of the people, we conceive, that the paramount object which these institutions have in view, namely, the advancement of the arts and manufactures, will be most decidedly and advantageously promoted, by confining the attention of the student to those branches of the physical sciences which bear more especially on the great majority of the useful arts. We have seen and examined carefully the plans of study as followed by most of the leading institutions of this kind throughout the kingdom; but, whilst we generally approve of them all, we shall confine ourselves to a more extended outline of the course followed by the Edinburgh School of Arts, as being that which seems to us most likely to promote the great objects they all have in view. It being self-evident that the elements of mathematics, mechanical philosophy, and chemistry, are of the most general application to the useful arts, it has been the

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1 Professor John Anderson was the eldest son of the Rev. James Anderson, minister of the parish of Roseneath, in Dumbartonshire, and was born there in 1726. Mechanics' principal aim of the directors of that institution to urge the necessity of acquiring a knowledge of these upon the notice of the student by every possible means; and especially that they should thoroughly understand the fundamental principles of mathematics, both from the intimate connexion these have with all the physical sciences, and as supplying a course of mental discipline, highly favourable to the successful prosecution, at any after period of life, of those studies the principles of which are more immediately involved in the ordinary occupations of life, and by the application of which alone any decided improvements in the arts may reasonably be looked for. And in order to encourage the student to adhere steadily to a certain course of study, an honorary certificate or diploma was instituted, conferring on the successful candidate the privilege of availing himself gratuitously, during his life, of all advantages of the institution. That course of study is as follows:

I. A junior class of mathematics meeting twice a-week, in which the following branches are taught: Arithmetic, including vulgar and decimal fractions; Algebra, as far as simple and quadratic equations; Geometry, first and second books of Euclid. In this class, a portion of each hour of teaching is devoted to exercises and examinations.

II. A senior class of mathematics meeting once a-week, in which the following branches are taught: Geometry, the remaining books of Euclid; Logarithms; Mensuration and Trigonometry, with their various practical applications. In this class, also, a portion of each hour of teaching is devoted to exercises and examinations.

III. A class of Natural Philosophy, illustrated by experiments, meeting once a-week, in which the following branches are taught: Mechanics, including statics and dynamics; Hydrostatics, Hydraulics, Pneumatics, and Optics, with such additional matter as time may permit of. A part of every fourth lecture of this course is devoted to examinations on the subjects treated of in the three preceding lectures.

IV. A class of Theoretical Chemistry, in which the principles of the science are taught, together with their application to the principal arts and manufactures in the processes of which chemical principles are involved. This class meets once-a-week; and a part of every fourth lecture of this course is devoted to examinations upon the subjects of the three preceding lectures.

V. It is of course left optional to the students to attend as many of the classes in one session as they please, and also to attend them separately in whatever order they prefer. But the directors, impressed with the benefits which may arise from such a course of study, earnestly recommend those who desire fully to avail themselves of the benefits of the institution, to pursue their studies on the following plan, viz. During the first year, to attend the junior mathematical class alone; during the second year, to attend the senior mathematical and the chemistry classes; and during the third year, to attend the natural philosophy and the chemistry classes.

It was therefore with the view of encouraging students to pursue this regular and systematic plan of education, that the directors instituted the honorary certificate above mentioned.

This course of study has for the last four years been most steadily persevered in, and the acquirements of the students who obtained the certificates, after the most searching examinations, have completely satisfied the directors and other competent judges who were present on these occasions, that the great object of the institution had, in their case at least, been attained, and that the information acquired by them was sound and practical.

Within the last year an addition to the course of study afforded by this institution has been made, and it promises to have the most beneficial effect on the advancement of those branches of art, in which the principles of Design are in any way involved, viz. a class for Ornamental Modelling.

It is by no means intended in this class to open a nursery for Mechanics' artists, but only to teach those engaged in trades where ornament is in the least concerned, such as silver-chasers, jewellers, plasterers, glass-cutters, brassfounders, smiths, die-sinkers, and a variety of others, to form their ideas of design on the most approved models of ancient and modern excellence; to enable them, instead of servilely imitating existing patterns, with a vague uncertainty and want of acquaintance with the laws which regulate art, to classify their labours, and by accustoming their minds to contemplate and imitate acknowledged excellence, gradually to lead the way to that creative skill in the execution of designs, which may eventually lead to the commencement of a new era in such departments of art.

The fee for each of the four classes above mentioned separately, the junior mathematics, the senior mathematics, the natural philosophy, and the chemistry, is five shillings. A ticket which gives admission to all the lectures, twelve shillings. The privileges of the library is extended to all students, whether attending one or more classes.

When these institutions were first originated, a great many excellent persons were led to withhold from them their countenance and support, fearing that the moral effect which they might have on the minds of the working classes would be unfavourable to the reception of those great and fundamental principles of religion, without which all human acquirements sink into insignificance; and that they would only fit their pupils more thoroughly for becoming the dupes and tools of designing men. Our opinion, however, of their effects upon society at large, is very different indeed; for by invigorating the faculties and enlarging the information of the student, this training serves the double purpose of enabling them to understand the evidences and truths of religion more easily and more thoroughly than they could otherwise have done, and of fortifying them more strongly against those sophistries by which its enemies so often and so fatally perplex the ignorant and the simple. It is true that knowledge is not always accompanied with such results; but its natural tendency is to counteract and prevent the very evils which have been apprehended as likely to result from it. Proceeding on the supposition that the young men who attend these institutions have previously, under the parental roof and at the schools where they obtained their elementary education, been taught the principles of religion, the mental discipline which they receive therein, and the course of study they are encouraged to pursue, furnish them with the means of cherishing a more enlightened, vigorous, and steady attachment to those great doctrines, which constitute the perfection and the glory of all science; and as industry is one of the best safeguards of moral conduct, they will operate most beneficially in that respect on those whose chief danger arises from the temptations which beset them at their leisure hours. Being necessarily so much occupied with objects of sense, and so little accustomed to purely mental exercises, when they seek for enjoyment, they are apt to look for it in the indulgence of mere appetite. Now these institutions are calculated to rescue them from the power of such temptations, not only by filling up their spare time with innocent and laudable pursuits, and providing them with occupations which elevate them far above the grossness of sensuality, but by training them up to habits of purity, sobriety, and correct deportment.

It is pleasing, therefore, to be enabled to say that the progress and advancement of the arts since the commencement of these institutions, has been most striking. They have attained an elevation, and partaken of a character which was formerly unknown; and since that period we may date the discovery and improvement of several of the most widely useful inventions, and the consequent dissemination of those comforts and elegancies of life, which have had such an influence on the whole social system. MECHLIN or MALINES, a circle of the province of Antwerp, in the Netherlands, divided into five cantons and thirty-eight communes, containing 87,800 inhabitants. The capital is the city of the same name, situated in a fertile plain, watered by the river Dyle, and connected by a canal with Louvain. It is the seat of an archbishop, for whom there is a magnificent palace, and a cathedral, with a tower 350 feet in length. Some of the other public edifices are on an extended scale. There is an ecclesiastical seminary, and a convent of Beguines, wherein formerly were 800 professed females; and though the number is now reduced, it is both respectable and useful, as it affords visitors to the poor and the sick, and relief to the distressed, in an extensive and gratuitous way. It contains 3000 houses, and about 20,000 inhabitants, who are very industrious. There are numerous hat-makers, and weavers of woollen cloths and blankets. The most distinguished of the fabrics is that of thread lace of the finest description, which bears the name of the city, and is highly esteemed in every part of the world where it is known. Lat. 51. 15. 2. N. Long. 4. 23. 39. E.