Definition. 1. MECHANICS is the science which enquires 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.

2. 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 catapult, 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 catapultte and ballista, 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 their stupendous fabrics, must have required an accumulation of mechanical power which is not in the possession of modern architects.

3. 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 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 has given some vague observations on the force of impulse, tending to point out the difference between impulse and prelure. 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.

4. The notions 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 hydraulics. In his two books, De Equiponderantibus, he has demonstrated that when a balance with unequal arms, is in equilibrium, by means of two weights in its 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 referable to the lever, might have been deduced as corollaries; but Archimedes did not follow 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 prelure 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 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 which may be conceived as made up of a number of lesser bodies, a centre of pressure or gravity. This discovery 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. The theory of the inclined plane, the pulley, the axis in peritrochio, the screw, and the wedge, which was first published in the eighth book of Pappus's mathematical collections, is generally attributed to Archimedes. It appears also from Plutarch and other ancient authors, that a greater number of machines which have not reached our times was 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.

5. 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 with which they display, or the useful purposes to which they are subservient. In the infancy of the Roman republic the corn was ground by hand-mills consisting of two millstones, one of which was moveable, and the other at rest. 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 there is reason to believe that water-mills were invented during the reign of Augustus; but it is strange that in the description given of them by Vitruvius, who lived under that emperor, they are not mentioned as of recent origin. The invention of wind-mills is of a later date. According to some authors, they were first used in France in the fifth 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.

6. The science of mechanics seems to have been stationary till the end of the 16th century. In 1577 a treatise on mechanics was published by Guidus Ubaldis, but it contained merely the discoveries of Archimedes. Simon Stevinus, however, a Dutch mathematician, contributed greatly to the progress of the science. He discovered the parallelogram of forces; and has demonstrated in his 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 laws of equilibrium in bodies placed on an inclined plane. His works were reprinted in the Dutch language in 1605. They were translated into Latin in 1658, 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.

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

8. 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, Dialogus II. The subject was afterwards fully discussed in another, entitled Dijcursus et Demonstrationes 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 fix theorems, containing 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 incessantly destroyed by the resistance of the table which prevents it from yielding to them. But where 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 air. It therefore follows, that a body falling freely is uniformly accelerated, or receives equal increments of velocity in equal times. Having established this as a definition, 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 on inclined planes.

9. The theory of Galileo was embraced by his pupil Torricelli, who illustrated and extended it in his excellent work entitled De motu gravium naturalium 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.

10. It was about this time that steam began to be employed

History. 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; but it is almost certain, that about 34 years before the date of the marquis's invention, and about 61 years before the construction of Papin's digester, steam was employed as the impelling power of a flampling engine by one Brances an Italian, who published an account of his invention in 1629. It is extremely probable, however, that the marquis of Worcester had never seen the work of Brances, and that the fire-engine which he mentions in his Century of Inventions was the result of his own ingenuity. 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 35 feet. The steam-engine received great improvements from our countrymen Newcomen, Brighton, and Blakey; but it was brought to its present state of perfection by Mr Watt of Birmingham, one of the most accomplished engineers of the present 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 the steam-engine was not only invented, but has received all its improvements, in our own country.

Discoveries of Huygens. 1673. 11. The success of Galileo in investigating the doctrine of rectilineal motion, induced the illustrious Huygens to turn his attention to curvilinear motion. In his celebrated work 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 absciss; 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 disquisitions, 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.

The laws of collision discovered by Wallis, Huygens, and Wren, 1661. 12. 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 1688, and appeared in the 43d and 46th numbers of their Transactions. The rules given by Wallis and Wren are published in No 43, pp. 864 and 867, and those of Huygens in No 47, 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 respective 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.

13. The attention of philosophers was at this time directed to the two mechanical problems proposed by Merennus 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 its motion is destroyed. These two problems were at first discussed by Descartes, Huygens, and Roberval, but the methods which they employed were far from being correct. The first solution of the problem on the centre of oscillation was given by Huygens. He affirmed 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 the conservation 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 Bernouilli by the principle of the lever.

Works of Robervli. 14. In 1666, a treatise De Vi Percussione, was publ. 1666. lished by J. Alphons Borelli, and in 1686, another work, De Motions Naturalibus à Gravitate Pendentibus; but he added nothing to the science of mechanics. His ingenious work, De Motu Animalium, however, 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.

15. 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 Mecanique, 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 Bernouilli'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.

16. 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 underfoot 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, § 279, 280, 281.) The principle of Parent was also applied by him to the construction of windmills. 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 54½ degrees to the axis of rotation, or, when the angle of weather is 35½ degrees. This conclusion, however, is subject to modifications which will be pointed out in a subsequent part of this article.

17. The Traite de Mecanique of De la Hire, published separately in 1695, and in the 9th volume of the Memoirs of the French Academy from 1666 to 1699, contains the general properties of the mechanical powers, and the description of several ingenious and useful machines. But it is chiefly remarkable for the Traite des Epicycloides, 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 lesser 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 flammers, and the lifting eogs of forge hammers; and as the epicycloidal teeth when properly formed roll upon one 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 very certainly due to Olaus Roemer, the celebrated Danish astronomer, who discovered the successive propagation of light. It is expressly stated by Leibnitz * in his letters to John Bernouilli, that Roemer communicated to him the discovery 20 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. Roemer'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 40 years afterwards gave a complete and accurate theory of the teeth of wheels, was unacquainted with the pretensions of Roemer, and ascribes the discovery to De la Hire.

18. The publication of Newton's Principia contributed greatly to the progress of mechanics. His discoveries concerning the curvilinear motion 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 Mecanique Celeste of La Place will be a standing monument of the extension which mechanics has received from the theory of gravity. The important mechanical principle of the conservation of the motion of the centre of gravity is also due to Newton. He has demonstrated in his Principia, that the state 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.

19. 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 fame. This 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 Bernouilli, 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 vortices, 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.

20. 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 given by Daniel Bernouilli in the Commentaries of Peterburgh 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. Fofeneix 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 Bernouilli 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 Mecanique Celeste.

21. About the beginning of the 18th century, the celebrated dispute about the measure of active forces was keenly agitated among philosophers. The first spark of this war, which for 40 years England maintained single-handed against all the genius of the continent, was excited by Leibnitz. In the Leipzig 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 Cottillon, 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 Bernouilli, Herman, Gravelende, Muffchenbroeck, Polemi, Wolff, and Bulfinger; and the opinion of Descartes by MacLaurin, Stirling, Clarke, Deaguliers, 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, 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.

22. In 1743, D'Alembert published his Traité de D'Alembert's principles of dynamics. This principle was first employed by James Bernouilli 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 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 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.

23. Another general principle in dynamics was Euler, about this time discovered separately by Euler, Daniel Bernouilli, and the chevalier D'Arcy, and received the name of the conservation of the momentum of rotary motion. According to the two first philosophers, the conservation 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 that centre, is always independent 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 Bernouilli 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, while 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 shewed, that the sum of the products of the masses 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 generalisation of Newton's theorem about the areas described by the planetary bodies, with that of Euler and Bernouilli, 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'Arsay, 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.

24. 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 Leipzig acts for 1751, who not only attempted to shew its falsity, but affirmed that Leibnitz had first described it in 1707 in a letter to Herman. 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 gave importance to a controversy which was too personal to merit the attention which it received.

25. In his Traité des Isoperimetries, printed at Lausanne in 1744, Euler extended the principle of least action, and shewed, "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 inflected bodies, was generalised 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 shewn†, 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.

26. An important discovery in rotatory motion, was at this time made by Professor Segner. In a paper, entitled Specimen Theoriei Turbinum, 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 at rest or moving uniformly in a straight line.

27. 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 law of magnetic action, and in finding the laws of the repulsion 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.—p. 246. 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 into conducting his experiments on a large scale, he has the subject 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 fix days, it could scarcely be moved with a power of 622 pounds†.

28. 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 inflected mechanics 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 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 Mecanique Analytique of 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 Mecanique Philosophique, of the same author, published in 1799, contains 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 Mecanique Philosophique is a profound work, in which, without the aid of a single diagram, he gives all the formulae, 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.—The Traité de Mecanique Elementaire, by M. Franceur, published in 1802 in one volume octavo, is an excellent abridgement of the works of Prony, and is intended as an introduction to the Mecanique Philosophique of that author, to the Mecanique Analytique of Lagrange, and to the Mecanique Celeste of Laplace.—None of these works have been translated into English; but their place is well supplied by a Treatise on Mechanics Theoretical, Practical, and Descriptive, by Olinthus Gregory, A. M. published in 1860, and containing a complete view of the latest improvements, both in the theory and practice of mechanics.

PART I. THEORY OF MECHANICS.

29. THE theory of mechanics properly comprehends, 1. Dynamics. 2. The motion of projectiles. 3. The theory of 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 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, 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 Simple Machines, or the Mechanical Powers.

30. THE simple machines have been generally reckoned fix in number. 1. The lever; 2. The wheel and axle, or axis in peritrochio; 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 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 simple machines, therefore, we cannot, in strict propriety, include any of the mechanical powers, excepting the lever, the inclined plane, and the rope-machine.

DEFINITIONS.

31. 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.

32. 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.

33. DEFINITIONS. A lever is an inflexible bar or Levers divided moving freely round a point, called its fulcrum, or divided into centre of motion. three kinds.

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 steelyards, scifars, pincers, &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, 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, sheers for sheep, &c. The bones 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. 7. & 8. and Borelli de Motu Animalium.

AXIOMS.

34. 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 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 exert their influence upon the lever in precisely the same circumstances.

35. Axiom 2. If two equal weights are placed at the extremities of a lever supported by two fulcra; and if these fulcra are at equal distances from the weights, or the extremities of the lever; the pressure upon the fulcra 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 fulcra support both the equal weights, the pressure upon each must be equal to one of the weights.

Proposition I.

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

37. Case 1. When 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 fulcra, fF, 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 directions AC, BD, making the angles CAB, DBA equal, these weights will be in equilibrium, 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 equilibrium. Then it is obvious that since FB = fF, the weight E of one pound acting upwards at the point f, so that the angle DFf = 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 equilibrium with a weight of one pound placed at twice the distance of the former from the fulcrum. But 2 : 1 = 2 FB 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.

38. Case 2. When weights act on the same side of the fulcrum.

Let AB be a lever in equilibrium 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 equilibrium, 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.

39. Again, let AB be the same lever supported by the fulcra 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 equilibrium 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 P fF may be equal to DBA. Now, (Axiom 1.) a weight E of one pound acting upward at f will be in equilibrium with a weight E' of one pound acting downwards at f'; f'f being equal to fF', and therefore by removing E from the point f and substituting E at the point f', an equilibrium will still obtain. But since fF' = 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.

40. By making FA in fig. 2. equal to 2FB it may be shewn, 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.

41. 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 f'A 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 equilibrium, they are reciprocally proportional to their distances from the centre of motion. Q. E. D.

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

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.

43. Cor. 2. If a weight W be supported by a horizontal lever resting on the fulcra A, B, the pressure up on 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 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.

44. 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 (Geometry, Sect. iv. Theor. 8.) \( P \times FA = W \times FB \), when an equilibrium takes place,

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

\[ \begin{align*} FA &= \frac{W \times FB}{P} \\ FB &= \frac{P \times FA}{W} \end{align*} \]

45. Cor. 4. We have already seen (Axiom 2.) that when the power and the weight are on contrary sides of the fulcrum, the prelude 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 prelude which it supports will be \( P-W \), or the difference of their weights.

46. 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 equilibrium (Cor. 3.) \( W : P = FA : FB \), or by alternation \( W : FA = P : FB \), and if w be another value of W and f a another value of FA, we shall also have \( w : p = f : a \) or \( w : f = P : FB \), consequently (Euclid, Book v. Prop. xi. and xvi.) \( W : w = FA : f \), 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 figure 6. only the straight lines AF, FB are in that case the length of the arm.

47. Cor. 7. Since by the last corollary \( FA : f = w : W \), it follows that in the Roman flatera or fleelyard, 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 fleelyard is represented in fig. 7, where AB is the lever with unequal arms AF, FB, and F the centre of motion. The body W, whose weight is to be found, is suspended at the extremity B 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 EB will be a scale of equal parts of which EB 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 12 division, we shall have \( W = 12 \) P, and if \( P = 1 \) pound, \( W = 12 \) pounds. But when the gravity of the lever is considered, which must be done in the real fleelyard, 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 fleelyard and are in equilibrium, 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 (Geometry, Sect. iv. Theor. 8.) \( W : DC = P : BF \). By taking different values of the variable quantities W and DC as w and d c, we shall have \( w : d c = P : BF \), consequently (Euclid, B. V. Prop. xi. and xvi.) \( W : w = DC : d s \), 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 fleelyards, 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 fleelyard the body to be weighed and the constant weight are fixed at the extremities of the fleelyard, but in 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 fleelyards 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 fleelyard may be seen in De la Hire's Traite de Mecanique, Prop. 36, 37, 38.

PROP. II.

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

49. 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 \( a = c f = f d = d b \). Then if f be the centre of motion or point of suspension, the cylinder A B will be in equilibrium; 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, and their distances c f, d f 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 e, 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. I. 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 the whole distance AF, we may substitute \( \frac{1}{2} m \times AF \) instead of \( m \times AF \), and the equation becomes

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

Hence

\[ \begin{align*} P &= \frac{W + \frac{1}{2} n \times FB}{AF - \frac{1}{2} m} \\ W &= \frac{P + \frac{1}{2} m \times AF}{FB} - \frac{1}{2} n \\ m &= \frac{W + \frac{1}{2} n \times 2FB}{AF} - 2P \\ n &= \frac{P + \frac{1}{2} m \times 2AF}{FB} - 2W \\ AF &= \frac{W + \frac{1}{2} n \times FB}{P + \frac{1}{2} m} \\ FB &= \frac{P + \frac{1}{2} m \times AF}{W + \frac{1}{2} m}. \end{align*} \]

50. 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.

Prop. III.

51. 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.

52. Let AFB be a lever of any form, F its 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 PK 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, § 140.) 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 shewn that BG is the part of the force BK which tends to move the lever round F. Now suppose AF produced to B, FB being made equal to FB and B'G' = BG. Then by Prop. I. AC : B'G' = FB' : FA; but by Axiom 1. the effort of BG to turn the lever round F is equal to the effort of the equal force B'G' 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 F and M are both right, and on account of the parallels DF, AC, MAC = ADF; therefore AC : AE = FM : FA, and AC × FA = AE × FM. For the same reason in the similar triangles BGK, BFN we have BG : BK = FN : FB, and BK × FN = BG × FB.

Hence AE × FM = BK × FN, and AE : BK or P : W = FM : FM. Q. E. D.

53. Cor. 1. The forces P and W are reciprocally corollaries, 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 FA m, and FN the sine of the angle FB n, FA, FB being made the radii;—therefore P : W = Sin. FB n : Sin. FA m, or P : W = \(\frac{1}{\text{Sin. FA m}} : \frac{1}{\text{Sin. FB n}}\). Since FA : FM = Rad. : Sin. FA m, we have FM = \(\frac{FA \times \text{Sin. FA m}}{\text{Rad.}}\); and since FB : FN = Rad. : Sin. FB n, we have FM = \(\frac{FB \times \text{Sin. FB n}}{\text{Rad.}}\), but in the case of an equilibrium P : W = FN : FM, consequently P : W = \(\frac{FB \times \text{Sin. FB n}}{\text{Rad.}} : \frac{FA \times \text{Sin. FA m}}{\text{Rad.}}\); and since magnitudes have the same ratio as their equimultiples, P : W = EB × Sin. EB n : FA × Sin. FA m.

54. 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 m E, n K 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 from it 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 pulled 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.

55. Cor. 3. If AE and BK the directions in which the forces P, W are exerted be produced till they meet at L; and if from the fulcrum F 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 = fin. LFS : fin. LFS; but on account of the parallels FS, AL the angle LFS = FLA, and FL 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.

56. 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.

57. Cor. 5. If two forces act in a parallel direction upon an angular lever whose fulcrum is its angular point, 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 P m, W n; then if the line FD, parallel to P m or W n, cut AB in such a manner that DB : DA = P : W, the forces will be in equilibrium. Draw F m perpendicular to P m, and produce it to n; then since A m, B n are parallel, m n will also be perpendicular to B n, and by the proposition (Art. 51.) F n : F m = P : W. Now, if through F, there be drawn m' n' parallel to AB, the triangles F m m', F n n' will be similar, and we shall have F n : F m = F n' : F m', but on account of the parallels AB, m' n'; F n' : F m' = DB : DA, therefore DB : DA = P : W.

58. Cor. 6. Let CB 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 this body may be regarded as a lever whose fulcrum is F, the forces will be in equilibrium when P : W = F n : F m the perpendiculars on the directions in which the forces act.

59. 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 A m, A n, these forces will be in equilibrium when they are to one another as the perpendiculars F n, F m.—For by cor. 2. the forces may be considered as applied at m and n, and m F n may be regarded as the lever ; but by the proposition (Art. 51.) P : W = F n : F m ; F n being perpendiculars upon A m, A n.

60. 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 F m, F n 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 = F n : F m. Since F m 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, F m = 0, and F m × P = 0.

61. 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 AD were a solid mass acting upon the lever in the direction of gravity. Consequently if P' p be drawn perpendicular from the point P' to FC, F p will be the lever with which the man in the scale tends to turn the lever round the fulcrum ; and as F p 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 F p' becomes the lever by which it acts.

62. Cor. 10. If a weight W be supported by an inclined lever resting on the fulcra A, B, the prelure upon A is to that upon B inversely, as A f is to f b, 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 prelure upon A ; then B being the centre of motion, and m n being drawn through F perpendicular to the direction of the forces A m, E f, and consequently parallel to A b, we have (Art. 51.) P : W = F n : F m = f b : f A, that is, the prelure upon A is to the prelure upon B inversely as A f is to f b.

Scholium.

63. 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 † 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 supposes 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 AE be a lever with equal weights C, D, supported on the fulcra f, F, so that Theory. A f = 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 weight to turn the lever will be the same to whatever of these radii it is applied. It appears, however, from Art. 54. 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 \(2\,P = P + W\). 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 \(2\,P\) or \(P + W\), a weight E equal to \(2\,P\), 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 \(E = 2\,P = 2\,W\), and \(AB = 2\,AF\), 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 \(= 2\,P = P + W\); 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 \(2\,W\), the force W obviously produces a pressure \(= 2\,W\) 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 Foncenneix in the Milcellan. Jour. tom. ii. p. 321. which was afterwards improved by D'Alembert in the Mem. de l'Acad. 1769. p. 283.

Prop. IV.

64. When several levers AB, a b, a B, 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 is an equilibrium when \(P : W = BF \times bf \times \beta \varphi : AF \times af \times \alpha \varphi\).

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

\[ \begin{align*} P : M &= BF : AF \\ M : N &= bf : af \\ N : W &= \beta \varphi : \alpha \varphi. \end{align*} \]

Consequently by composition

\[ P : W = BF \times bf \times \beta \varphi : AF \times af \times \alpha \varphi. \]

Prop. V.

65. To explain the new property of the lever discovered by M. Zépinus, 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 YA, VB. 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 HF, dividing AB into two parts AF, BF. Let UM 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 VA be resolved into AE and YF; and the force VB into BI and VI.—Then the lever cannot be in equilibrium till

I. \(EA \times f \times A + IB \times f \times B\) 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 till

\[ Tang.\varphi \times P \times Af \times Cot.\alpha + Tang.\varphi \times W \times Bf \times Cot.\beta = W \times Bf \times Sin.\beta - P \times Af \times Sin.\alpha. \] III. And putting \( a, b \) for the arms AF, BF, and \( m, n \) for the angles EAB, EBA, there cannot be an equilibrium unless

\[ \tan \varphi = \frac{\overline{W} \cdot b (\sin \beta \times \cos \alpha - \sin \alpha \times \cos \beta)}{P \cdot a (\cos \alpha \times \cos m + \sin \alpha \times \sin m) + \overline{W} \cdot b (\cos \beta \times \cos n + \sin \beta \times \sin n)} \]

As the demonstrations of these different cases are far from being elementary, we shall only refer the reader to the memoir upon this subject given by Æpinus in the Nov. Comment. Petropol. tom. viii. p. 271.

SCHOLIUM.

66. This property of the lever was only considered by Æpinus in the case of a rectilineal lever with equal arms; but was extended by J. H. Van Swinden. When the lever is rectilineal and with equal arms, we have \( AF = FB = AF = BF \), and also \( m = n = 0 \), so that, if the last formula is suited to these conditions, we shall have the formula of Æpinus.

PROP. VI.

67. 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 AFA', 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 AEF is exceedingly small. Join AA', BB', and from A' and B' draw A'x, 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 A x is the space described by A in the direction AD, and B z 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 A x measures the approach of the point A to the plane passing through P; and for the same reason B z measures the approach of the point B to a plane passing through W at right angles to WB. Therefore A x, B z represent the spaces uniformly and simultaneously described by the points A, B, and may therefore be taken to denote the velocities of these points (Dynamics, § 14); consequently the velocity of A : the velocity of B = A x : B z. Now, in the triangles A x A', Fm A, the exterior angle x AF = A m F + m F, A (Euclid, B. I. Prop. 32.) and A'AF = A m F, because AFA' is so exceedingly small that A'AF is sensibly perpendicular to AF; consequently x A A' = A E m: and as the angles at x and m are right, the triangles A x A', A m F are similar (Geometry, Theor. XX, Sect. IV.).

Therefore, A x : A A' = Fm : FA, and in the similar triangles AFA', BFB', AA' : BB' = FA : FB; and in the similar triangles BB'z, BFn, BB' : Bz = FB : Fn, therefore by composition we have A x : B z = Fm : Fn.

But by Proposition II. P : W = F n : F m, consequently A x : B z = 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.

68. Cor. Since A x : B z = W : P we have A x × P = B z × W, that is, the momenta of the power and weight are equal.

SECT. II. On the Inclined Plane.

69. 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 the length of the plane, AC its height, and BC the length of its base.

70. 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.

71. 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 ND, 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 (Art. 59.) P : W = F n : F m, that is, as the perpendiculars drawn from the fulcrum to the direction in which the forces act. Now FA being radius, F n is the sine of the angle FAB, and F m 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.

72. Cor. 1. When the power acts parallel to the plane in the direction \( AE' \), \( P \) is to \( W \) as \( EA \) to \( En \), that is, as 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.

73. Cor. 2. When the power acts in a vertical line \( Aa \), \( Fm \) 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.

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

75. 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.

76. 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 (\( \text{Dynamics}, \S\ 149. \)), 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 \( P \) in the direction \( AF \). Therefore, since these forces are in equilibrium, and since \( Af \) is parallel to \( nF \), and \( Ff \) to \( An \), the three sides \( AF, Af, Ff \), will represent the three forces (\( \text{Dynamics}, \S\ 144. \)). But the triangle \( AFF \) is similar to \( AnF \), that is, to \( MNO \), for it was already shewn 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 \( Ff \) the energy of the power, these magnitudes will also be represented in the similar triangle \( MNO \) by the sides \( MN, MO, NO \).

77. 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.

78. 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 \( F \), 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, (\( \text{Dynamics}, \S\ 144. \)), 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 \( BC \),

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

But on account of the parallels \( AB, DF \), 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.

79. Cor. 1. 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.

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

81. 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.

82. 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 (\( \text{Geometry}, \text{Sect. II. Theor. 17.} \)), 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 triangle \( DBE \) is equal to \( DFE \), (\( \text{Geometry}, \text{Sect. II. Theor. 15.} \)) and \( DBE = DCH \), (\( \text{Geometry}, \text{Sect. IV. Theor. 23.} \)) and \( FDE = DHC \), (\( \text{Geometry}, \text{Sect. I. Theor. 21.} \)) 21.) the triangles DFE, DHC are similar, and (Geometry, Sect. IV. Theor. 20.) DF : DE = DH : HC. But DH : HC = fin. DCH : fin. HDC, that is, (art. 71.) DF : DE, or the velocity of the power to the velocity of the weight, as W : P. Q. E. D.

Scholium.

83. 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 42 miles on the same level, it is joined to a subterraneous navigation about 12 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 flairs, and a road that is not level, 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.

Sect. III. On the Rope Machine.

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

Prop. I.

85. 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 = fin. CBD : fin. 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 (Dynamics, § 144.), for AB, CB are the directions of the forces P and p. We have therefore P : p = DE : BE; but DE : BE = fin. DBE : fin. BDE, and on account of the parallels DE, AB, the angle BDE = ABD, consequently P : p = fin. DBE : fin. BDE.

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

87. Cor. 2. Any of the powers is to the weight, as the fine of the angle which the other makes with the direction of the weight, is to the fine of the angles which the power makes with one another. For since DB represents the weight, and BE the power P, we have BE : BD = fin. BDE : fin. 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 = fin. ABF : fin. ABC. In the same way it may be shewn that P : W = fin. CBF : fin. ABC. Hence we have P + p : W = fin. CBF + fin. ABF : fin. 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 fine of the angle which the powers make with one another.

88. Cor. 3. The two powers P, p, are also directly proportional to the cofecants of the angles formed by the direction of the powers with the direction of the weight. For since P : p = fin. DBE : fin. BDE, and by the principles of trigonometry, fin. DBE : fin. DBE = cofec. BDE : cofec. DBE, we have P : p = cofec. ABF : cofec. CBF. 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 cofecant of any angle is just the secant of its complement, therefore P : p = sec. BAF : sec. BCF.

Chap. II. On Compound Machines.

89. 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.

90. The wheel and axle, or the axis in peritrochio, Fig. 10, is represented in fig. 9, and consists of a wheel A.B, and cylinder CD 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 CD in a plane perpendicular to its axis. In this machine a winch or handle EH 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 diate from the centre C, and if a rope carrying the power P were to pass over their extremities, and extricate itself from the defending levers when they come into a horizontal position.

91. Axiom. 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.

92. 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 wheel, 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 (Art. 36.) \( P : W = FB : FA \), that is, the power is to the weight as the radius of the axle is to the radius of the wheel.

Corollaries.

93. Cor. 1. If the power and weight act obliquely to the arms of the lever in the directions Ap, Bw, draw Fm Fn perpendicular to Ap and Bw, and in the case of the lever (Art. 51.) there will be an equilibrium when \( P : W = Fn : Fm \). 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.

94. Cor. 2. If the plane of the wheel is inclined to the axle at any angle x, there will be an equilibrium when \( P : W = \) femidiameter of the axle : fin. x.

95. 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 \( P : W = FB + \frac{t}{2} : FA + \frac{T}{2} \).

96. 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 Art. 63. for the combination of Levers.

97. 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.

98. 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.

99. 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 watch 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 smallest part of the fusee when its force is greatest, and upon the largest part of the fusee when its force is least, for the equal 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 Mathemat. par M. Parent, tom. ii. p. 678.; Traité d'Horlogerie, par M. Berthoud, tom. i. chap. 26.; and Traité de Mecanique, par M. de la Hire, prop. 72.

Sect. II. On the Pulley.

100. Definition.—The pulley is a machine composed of a wheel with a groove in its circumference, pulley, and a rope which passes round this groove. The wheel moves on an axis 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. 101. 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.

102. 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 PBAW, passing over the pulley AA; then it is obvious that whatever force is exerted by P in the direction PBA, the same force must be exerted in the opposite direction WBA, 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 \( P : W = 1 : 1 \).

103. CASE 2. In the single moveable pulley, 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 CF, p, HGD 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 CF p 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 \( P : W = 1 : 2 \).

104. 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. Consequently if there be ten ropes supporting the weight, each sustains \( \frac{1}{10} \)th part of the weight, and therefore \( P = \frac{1}{10} W \), or \( W = 10P \), which gives us \( P : W = 1 : 10 \).—The pulley in fig. 15, is the patent pulley invented by Mr White, in which the lateral friction and shaking motion is considerably removed.

PROF. II.

105. 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. 17, where the rope which carries the power P passes over the fixed pulley M, and beneath the moveable pulley A, to the hook E where it is fixed. Another rope fixed at A passes over B and is fixed at F, and so on with the rest. Then by Art. 103.

\[ P : \text{the weight at } A = 1 : 2 \\ \text{The weight at } A : \text{the weight at } B = 1 : 2 \\ \text{The weight at } B : \text{the weight at } C = 1 : 2 \\ \text{The weight at } C : \text{the weight at } D \text{ or } W = 1 : 2 ; \text{ and therefore by composition} \\ P : W = 1 : 2 \times 2 \times 2 \times 2 \text{ or } P : W = 1 : 16. Q.E.D. \]

PROF. III.

106. In a system of moveable pulleys 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 \( P : W = 1 : 2^n - 1 \).

PROF. IV.

107. In the system of pulleys represented in fig. 19, 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 \( P : W = 1 : 4 \).

PROF. V.

108. In the system of pulleys represented in fig. Fig 20, 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 B and 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. pounds, and since it is drawn upwards by the ropes CG, DF, with a force of two 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.

109. 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 sustained 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, WI will respectively represent the forces P, p, W or P : p : W=WB : BI : WI, DYNAMICS, Art. 144. 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 is, P : p : W = CD : DE : EC; but CD, DE are equal to radius, and EC is obviously the chord of the arch CFE, therefore P : W or p : W as radius is to the chord of the arch with which the rope is in contact.

110. 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.

111. By means of this proposition and corollary, the proportion between the powers and the weight in the various systems of pulleys, represented in fig. 12, 13, 14, 15, 16, 17, 18, 19, 20, when the ropes are not parallel, may be easily found.

PROP. VII.

112. 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 2 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,

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

PROP. VIII.

113. 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.

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

115. CASE 2. In the single moveable pulley, when Fig. 13, the weight W is raised one inch, the ropes become one inch shorter; and since the rope has always the same weight, the power must describe two inches, therefore velocity P : velocity W = W : P.

116. CASE 3. In the combination of pulleys, in Figs 14, 15, figs. 14, 15, 16, when the weight rises one inch, each 16, of the four strings becomes an inch shorter, so that P must describe four inches, as the length of the rope is invariably; consequently velocity P : velocity W = W : P.

117. CASE 4. In the system exhibited in fig. 17, it is evident, that when the weight W rises one inch, the rope DC is lengthened two inches, the rope CB four inches, the rope BA eight inches, and the rope AFP, to which the power is sustained, 16 inches; so that since the power of this pulley is as 16 to 1, we have velocity P : velocity W = W : P.

118. CASE 5. In the combination of pulleys, represented in fig. 18, 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, CB becomes one inch longer; while BE shortens one inch, BC becomes one inch longer, and CP two inches longer (art. 110.); and while AF 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.

119. CASE 6. In the system of pulleys, called the Spanish barton, fig. 19, 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 (art. 115.); 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.

120. CASE 7. In the other Spanish barton, in fig. 20, 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, 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.

121. 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 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, fwords, chisels, teeth, &c. properly belong to this mechanical power, when they act in a direction at right angles to the cutting surface; for when they act obliquely, in which case their power is increased, their operation resembles more the action of a saw.

PROP. I.

122. 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 CB, 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, RDN, 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 FC = GE, and FGC, ABC are both right angles, therefore FE is parallel to AB.—Now the force MD is reducible 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 shewn that EG, GD are the only effective forces which result from the force ND. But the forces FG, FG 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 = fin. DFG : radius, or as (Euclid, vi. 8.) fin. DCF : radius, and DF : MD = fin. DMF : radius; therefore by composition, LG : MD = fin. DCF × fin. DMF : rad. × rad. or rad.[P. But, LG : 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 fin. DCF × fin. DMF is to the square of the radius.

123. Cor. 1. 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 fin. DCF × rad. is to rad.[P, that is, as fin. DCF is to radius, or as AD half the back of the wedge is to AC the length of the wedge.

124. 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.

125. 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 fin. ACD² is to radius[P, that is, as the square of AD half the back of the wedge is to the square of AC the length of the wedge.

126. 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 fin. ACD × cos. ACD is to the square of the radius, that is, as DA × DC is to AC². But in the similar triangles DAF, DAC, we have DF : DA = DC : AC, and DF × AC = DA × DC, consequently the force upon the back of the wedge is to the sum of the resistances as DF × AC is to AC², that is, as DF : AC.

PROP. II.

127. If, on account of the friction of the wedge, Fig or any other cause, the resistances are wholly ineffective, 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, which will cut DC perpendicularly at the point 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 \(2\ \mathrm{HD}\). Consequently, the force upon the back is to the sum of the resistances as \(2\ \mathrm{HD}\) is to \(2\ \mathrm{MD}\), or as \(\mathrm{HD}\) is to \(\mathrm{MD}\). But the angle \(\mathrm{ADM}\), which the direction of the forces makes with the back of the wedge, is equal to \(\mathrm{DMN}\), and \(\mathrm{HD}\) is the sine of that angle, \(\mathrm{MD}\) being radius, therefore the force upon the back is to the sum of the resistances as \(\sin\ \mathrm{ADM} : \mathrm{radius}\).

Q. E. D.

Corollaries.

128. Cor. 1. Since the angle \(\mathrm{AMD} = \mathrm{MDC} + \mathrm{MCD}\), the angle \(\mathrm{MDC}\) is the difference between \(\mathrm{MCD}\) the semiangle of the wedge, and \(\mathrm{AMD}\) the angle which the direction of the resisting forces makes with the face of the wedge, and since \(\mathrm{HD}\) is the cosine of that angle, \(\mathrm{MD}\) being radius, we have the force upon the back to the sum of the resistances, as the cosine of the difference between the semiangle 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.

129. 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, § 144. Cor. 2, where it is shewn 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.

130. 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 \(\mathrm{DM}\) to K, and draw \(\mathrm{CK}\) perpendicular to \(\mathrm{DK}\). Then, by Art. 122, the power is to the resistance as \(\mathrm{MD} : \mathrm{DH}\). 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 \(\mathrm{DC} : \mathrm{DK}\); that is, on account of the equiangular triangles \(\mathrm{DHM}, \mathrm{DKC}\), as \(\mathrm{MD} : \mathrm{DH}\); that is, as the resistance is to the power.

Sect. IV. On the Screw.

131. 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 a b c (fig. 4.) be the inclined plane, whose base is a c and altitude b c, and let it be wrapped round the cylinder MN (fig. 5.) of such a size that the points a, c may coincide. The surface a b of the plane (fig. 4.) will evidently form the spiral thread a d e b (fig. 5.), and a b the distance between the threads will be equal to b c (fig. 4.) the altitude of the plane, and the circumference of the screw MN will be equal to a c the base of the plane. If any body, therefore, is made to rise along the plane adeb in fig. 5., 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. 4.).

132. A male screw with triangular threads is represented by AB (fig. 6.), and its corresponding female screw by AB (fig. 7.). A male screw with quadrangular threads is exhibited in fig. 8., and the female screw in which it works in fig. 9. The friction is considerably less in quadrangular than in triangular threads, though, when the screw is made of wood, the triangular threads should 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 lever fixed in its head. This is shewn in fig. 10. 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.

133. 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. Fig. 8. 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 (Art. 72.) P : W, as the altitude of the plane is to the base, or (Art. 131.) 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 (Art. 36.), we have P' : P = CA : BA, that is, as FKCF is to the circumference HBG; but it was shewn 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.

134. Cor. 1. It is evident from the proposition that Theory. 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, \( P : p = \frac{D}{d} : \frac{T}{t} \), or P varies as \( \frac{D}{T} \).

Prop. II.

135. 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. 12, 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 handles or winches P, 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 (Art. 92.) 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.

136. 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. 11. 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 Art. 133.), as the weight is to the power.

Prop. IV.

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

Let the screw CD work in the plate of metal BA, 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 \( \frac{n+1}{n} \) threads in an inch, and into this female screw is introduced a male screw DE, having, of course, \( \frac{n+1}{n} \) threads in an inch. The screw DE is prevented from moving round with CD by the frame ABGF and the cross bar ab, but is permitted to ascend and descend without a 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 \( P : W = \frac{1}{n+1 \times n} : m \); or \( P : W = 1 : m \times n + 1 \).

138. 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 \( \frac{12}{13} \) of an inch, and the consequence of both these motions will be that the point E is advanced \( \frac{1}{13} \) 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 \( \frac{1}{13} \) of an inch. Since, therefore, it has advanced \( \frac{1}{13} \) of an inch in 12 turns, it will describe only \( \frac{1}{13} \) of \( \frac{1}{13} \), or \( \frac{1}{169} \) of an inch uniformly at one turn; but if the length of the lever CK is 8 inches, its extremity K will describe, in the same time, a space equal to \( 16 \times 3.1416 = 50.2656 \) inches, the circumference of the circle described by K; therefore the velocity of the weight is to the velocity of the power, as \( \frac{1}{169} \) of an inch is to 50.2656 inches, or as 1 is to 7841.4336, that is, (Art. 136.) \( P : W = 1 : 7841.4336 \). Hence the force of this double screw is much greater than that of the common screw, for a common one with a lever 8 inches long must have 156 threads in an inch to give the same power, which would render it too weak to overcome any considerable resistance.

139. Mr Hunter proposes * to connect with his * Phil. double screws, a wheel and a lantern, which are put in motion by a winch or handle. The power of this compound machine is so great, that a man, by exerting a force of 32 pounds at the winch, will produce an effect of 172100 pounds; and if we suppose \( \frac{2}{3} \) of this effect to be destroyed by friction, there will remain an effect of 57600 pounds.—In some screws it would be advantageous, instead of perforating the male screw CD, to have two cylindrical screws of different kinds at different parts of the same axis.

Scholium.

140. The screw is of extensive use as a mechanical power, when a very great pressure is required, and is very successfully employed in the printing press. In the press which is used for coining money, the power of the screw is advantageously combined with an impulsive force, which is conveyed to the forew by the intervention of a lever. The screw is also employed for raising water, in which form it is called the screw of Archimedes (Hydrodynamics, § 328); and it has been lately employed in the flour mills in America for pushing the flour which comes from the millstones, 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.

141. 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. See MICROMETER.

Sect. V. On the Balance.

142. 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. 1. 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, 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.

143. To determine the conditions of equilibrium Fig. 2. in a physical balance.

Let AOB 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 of the whole beam, when unloaded; we shall have in the case of an equilibrium,

I. \( p + l \times AC = p + l \times BC + S \times Cc \); 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 \( = S \times Cc \), the distance of its centre of gravity from the vertical line passing through the centre of motion O.

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

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

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 \( BAA = \varphi \) 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 O z, the parts \( \frac{OG}{Og} \) and \( \frac{Gg}{Og} \) will be in equilibrio, and we shall have,

IV. \( p + L \times AO \times \sin. \lambda A O + S \times OG \times \sin. \varphi = p + l + w \times BO \times \sin. ABO + S \times Cc \times \cos. \varphi. \)

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

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

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 v, 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

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

144. 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 in another memoir upon the same subject by Kuhne in the Verfuche der naturforschende gesellschaft in Dantzig, tom. i. p. 1.—See also Henner's Curfus Matheseos applicatae, tom. i. § 123. From the preceding formulae, the following practical corollaries may be deduced.

145. Cor. 1. The arms of the balance must be ex- 149. Cor. 5. The sensibility of the balance will increase, the nearer that the centre of gravity approaches to the centre of motion.

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

Scholium.

151. A balance with all the properties mentioned in the preceding corollaries, has been invented by M. Kuhne, and described in the work already quoted (Art. 144.). 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.

152. In order to get rid of the difficulties which attend the construction of the tongue, the handle, and the arms of 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.

153. The balance invented by Ludlam, and described in the Philosophical Transactions for 1765, No 55, depends upon AEpinus's property of the lever, which we have explained in Art. 65. 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 weighed, is suspended by a thread from B. Hence it is obvious, that with different bodies the lever AFB will have different degrees of inclination, and the index or tongue LFf, which is perpendicular to AB, will form different angles ZFL, b Ff with the line of direction ZF b. Now, by Art. 57. and by substituting for b B, b A the fines of the angles F b B, F b A, to which they are proportional, and also by taking instead of F b B the difference of the angles f FB, f Fb, and instead of AF b, the sum of these angles, we shall have

\[ \text{Tang. } f F b = \frac{P-W}{P+W} \times \text{Tang. } \frac{AFB}{2}, \]

whence, by transposition, and by Geometry, Theor. VIII. Sect. IV.

\[ \frac{P+W}{P-W} = \text{Tang. } \frac{AFB}{2} : \text{Tang. } f Fb. \]

Hence, when the angle formed by the arms of the balance, and the angle of aberration f F b or ZF l, are known, the weights may be found, and vice versa.

Chap. IV. On the Centre of Inertia, or Gravity.

154. 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 plane surfaces bounded by right lines, and also of some solids may be easily determined by the 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.

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

Let ABCD 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 (Art. 36.), and the preasure upon F will be equal to A+B. Join FC, and find the point f, so that A+B:C=C:f; fF; the bodies A, B, C will consequently be in equilibrium upon the point f, which will sustain a preasure equal to A+B+C. Join Df, and take the point φ, so that A+B+C:D=φ D:φf; the bodies A, B, C, D will therefore be in equilibrium about the point φ, which will be their common centre of inertia, and which supports a weight equal to A+B+C+D. In the same manner we may find the centre of inertia of any system of bodies, by merely connecting the last fulcrum with the next body by an inflexible right line, and finding a new fulcrum from the magnitude of the opposite weights which it is to sustain.

156. Cor. 1. If the weights of the bodies A, B, C, D be increased or diminished in a given ratio, the centre of inertia of the system will not be changed, for the positions of the points F, f, φ are determined by the relative and not by the absolute weights of the bodies.

157. Cor. 2. A motion of rotation cannot be communicated to a body by means of a force acting upon its centre of inertia; for the resistances which the inertia of each particle opposes to the communication of motion act in parallel directions, and as they are proportional to the weights of the particles, they will be in equilibrium about the centre of gravity.

Prop. II.

158. 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 (Art. 36.) \(A \times A \varphi + B \times B \varphi = C \times C \varphi + D \times D \varphi + E \times E \varphi\); but since \(X \varphi - XA = A \varphi\), and \(X \varphi - XB = B \varphi\), and so on with the rest, we have by substitution \(A \times X \varphi - XA + B \times X \varphi - XB = C \times X \varphi - XC + D \times X \varphi - XD + E \times X \varphi - XE\). Hence by multiplying and transposing, we obtain \(A \times X \varphi + B \times X \varphi + C \times X \varphi + D \times X \varphi + E \times X \varphi = A \times XA + B \times XB + C \times XC + D \times XD + E \times XE\), then dividing by \(A+B+C+D+E\), we have

\[ X \varphi \]

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

Now \( A \times XA ; B \times XB, &c. \) 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 \( \varphi \) is equal to the sum of the momenta of all the weights divided by the sum of the weights.

159. Cor. 1. If the point \( X \) had been taken between \( A \) and \( E \), at \( x \) for example, then the quantity \( A \times XA \) would have been reckoned negative, as lying on a different side of the point \( X \).

160. 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. III.

161. 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 Art. 155. find \( F \) the 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 \( Aa, Ff, Bb, Gg, Cc \) perpendicular to the plane \( MN \). Through \( F \) draw \( x F y \), meeting \( Aa \) produced in \( x \), and \( Bb \) in \( y \), then in the similar triangles \( AxF, ByF \), we have \( A : x = By : AF \); that is, (Art. 155.) as \( B : A \), hence \( A \times x = B \times By \), that is, \( A \times x - Aa = B \times Bb - yb \), or on account of the equality of the lines \( x a, Ff, Bb \); \( A \times Ff - Aa = B \times Bb - Ff \), therefore, by multiplying and transposing, we have \( A + B \times Ff = A \times Aa + B \times Bb \). In the very same way, by drawing \( w G z \) parallel to the plane, it may be shewn that \( A + B + C \times Gg = A \times Aa + B \times Bb + C \times Cc \). Q. E. D.

162. Cor. By dividing by \( A + B + C \) we have

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

PROP. IV.

163. 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 is equal.

PROP. V.

164. To find the centre of inertia of a parallelogram.

Let \( ABCD \) be a parallelogram of uniform density, Fig. 7. bisect \( AB \) in \( F \), and having drawn \( Ff \) parallel to \( AC \) or \( BD \), bisect it in \( \varphi \); the point \( \varphi \) 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 Art. 155. 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 shewn 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 Art. 165. its centre of inertia will be in \( \varphi \), its middle point.

PROP. VI.

165. To find the centre of inertia of a triangle.

Let \( ABC \) be a triangle of uniform density, and let Fig. 2. \( AB, BC \) be bisection 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 \( BC, bc, \beta x \); but in the similar triangles \( ADC, Acc ; AD : DC = A : ec \), and in the triangles \( ADC, ADB, Aeb ; BD : DA = be : eA \); hence by composition \( BD : DC = be : ec \); but \( BD \) and \( DC \) are equal; therefore, \( be = ec \); and the line \( bc \), supposed to consist of material particles, will be in equilibrium about \( e \). In the same way it may be shewn that every other line \( \beta x \) 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, (GEOMETRY, Sect. IV. Theor. 18.) \( ED \) 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 \( AF = \frac{2}{3} AD \).

166. Cor. 1. By GEOMETRY, Theor. 16. Sect. IV. we have

\[ AB^2 + AC^2 = 2BD^2 + 2AB^2 (\frac{1}{2}BC^2 + \frac{2}{3}AF^2) \\ AB^2 + BC^2 = 2CC^2 + 2BG^2 = \frac{4}{3}AC^2 + \frac{2}{3}CF^2 \\ AC^2 + BC^2 = 2AE^2 + 2EC^2 = \frac{4}{3}AB^2 + \frac{2}{3}BF^2 \] Theory. By adding these three equations, and removing the fractions, we have \( AB^2 + BC^2 + AC^2 = 3 AF^2 + 3 CF^2 + 3 BF^2 \), 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.

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

Prop. VII.

168. 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 m n, so that \( F : n = \text{triangle ABC} : \text{triangle ACE} \), then F will be the centre of gravity of these triangles. Join Fo, and find a point f, so that fo : Ff = 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.

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

Let the pyramid be triangular, as ABCD, fig. 10. Bisect BD in F, and join CF and FA. Make Ff = \( \frac{1}{3} \) of FC, and F \( \varphi = \frac{1}{3} \) of FA, and draw f \( \varphi \). It is evident, from Art. 159, 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 ABC; for, by taking any section b c d, and joining c m, it may be shewn, that \( b m = m d \), and \( m n = \frac{1}{3} m c \), so that n is the centre of gravity of the section b c d. It follows, therefore, that Af will pass through the centre of gravity of the pyramid. In the same way it may be shewn, by considering ABD as the base, and D the vertex, and making F \( \varphi = \frac{1}{3} \)FA, that the centre of gravity lies in the line \( \varphi \)C. But, as the lines Af, \( \varphi \)C 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 Ff = \( \frac{1}{3} \)FC, and F \( \varphi = \frac{1}{3} \)FA, we have F \( \varphi \) : FA = Ff : FC, therefore (Geometry, Theor. 8, Sect. IV.) \( \varphi \)f is parallel to AC. The triangle \( \varphi \)fH will consequently be similar to AHC, and \( H \varphi : HC = HF : HA = \varphi : AC = 1 : 3 \); therefore \( \varphi = \frac{1}{3} \)HC = \( \frac{1}{3} \)C, and \( fH = \frac{1}{3} AH = \frac{1}{3} AF \).

170. 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.

171. Cor. 1. 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.

172. Cor. 2. By proceeding as in Art. 160, 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 fix edges of the pyramids. A demonstration of these theorems may be seen in Gregory's Mechanics, vol. i. p. 59, 60.

173. In order to shew the application of the doctrine of fluxions to the determination of the centre of inertia of curve lines, areas, solids, and the surfaces fig. 11. 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 biflection 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 D dg G, we shall find its centre of inertia by multiplying each weight by its distance from any line mn parallel to the ordinates, and dividing the sum of all these products by the sum of all the particles, Art. 158. Thus, let x denote the distance EB, then its fluxion \( \dot{x} \) will be the breadth of the element or small weight D dg G, and \( \dot{x} \times DG \) will represent the weight, and the fluent of this quantity will be the sum of all the weights. Again, if we multiply the weight \( \dot{x} \times DG \) by \( x = EB \) its distance from the point B, we shall have the momentum of that weight \( = \dot{x} \times \dot{x} \times DG \), 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 Art. 158, 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 \( FB = \frac{\text{fluent of } \dot{x} \times \dot{x} \times DG}{\text{fluent of } \dot{x} \times DG} \), or calling y the ordinate DE, we have \( DG = 2y \), and \( FB = \frac{\text{fluent of } x \times y \times \dot{x}}{\text{fluent of } 2y \times \dot{x}} \), or \( FB = \frac{\text{fluent of } xy \times \dot{x}}{\text{fluent of } y \times \dot{x}} \) in the case of areas.

174. In the case of solids generated by rotation, the element or small weight \( F \times \dot{x} \times DG \) will be a circular section,

174. In finding the centre of inertia of the surfaces of solids, the elements or small weights are the circumferences of circles, whose radii are the ordinates of the curve by whose revolution the solid is generated. Now, the surface of the solid may be conceived to be generated by the circumference of a circle increasing gradually from B towards A and C; making \( x \) therefore equal to BD, its fluxion \( \dot{x} \) multiplied into the periphery of the circle whose diameter is DG, that is, \( 2\pi y \dot{x} \) will express the elementary surface or small weight whose diameter is DG. Then, since \( x \times 2\pi y \dot{x} \), or \( 2\pi xy \dot{x} \), will be the momentum of the elementary weight, we shall have \( \text{FB} = \frac{\text{fluent of } 2\pi xy \dot{x}}{\text{fluent of } 2\pi y \dot{x}} \), and dividing by \( 2\pi \), we obtain \( \text{FB} = \frac{\text{fluent of } xy \dot{x}}{\text{fluent of } y \dot{x}} \).

175. If the body, whose centre of inertia is to be found, be a curve line, as GBD, then it is manifest that the small weights will be expressed by the fluxion of GBD, that is, by \( 2z \), since \( GBD = 2BD = 2z \); consequently their momenta will be \( 2x \dot{z} \), and we shall have \( \text{FB} = \frac{\text{fluent } 2x \dot{z}}{\text{fluent } 2z} = \frac{\text{fluent } x \dot{z}}{\text{fluent } z} = \frac{\text{fluent } x \dot{z}}{\infty} \).

PROP. IX.

177. To find the centre of inertia of a circular segment.

Let AE=x, FC=y, and AD the radius of the circle=R, consequently ME=2R-EA. Then, since by the property of the circle (Geometry, Theor. 28. Sect. IV.) \( ME \times EA = BE^2 \), we have, by substitution, \( BE^2 = 2R \times EA - EA \times EA \), or \( y^2 = 2R x - x^2 \); hence \( y = \sqrt{2R x - x^2} \). Now, by Art. 174, we have the distance of the centre of gravity from A, that is,

\( AG = \frac{\text{fluent } xy \dot{x}}{\text{fluent } y \dot{x}} \); but the fluent of \( y \dot{x} \) or the sum of all the weights, is equal to the area of half the segment ABEC; therefore \( AG = \frac{\text{fluent } xy \dot{x}}{\frac{1}{4}ABEC} \). Then, by substituting instead of \( y \), in this equation, the value of it deduced from the property of the circle, we have

\( AG = \frac{\text{fluent of } x \dot{x} \sqrt{2R x - x^2}}{ABEC} \); or, in order to find GD the distance of the centre of gravity from the centre, we must substitute instead of \( x \) (without the vinculum) its value \( R-x \), and we have \( GD = \frac{\text{fluent } (R-x)(2R x - x^2)}{\frac{1}{4}ABEC} \). Now, in order to find the fluxion of the numerator of the preceding fraction, assume \( x = 2R - x' \), and \( x'^2 = \sqrt{2R x - x^2} \), and by taking the fluxion, we have \( x' = 2R - 2x'x = 2R - 2x'x' \); but this quantity is double of the first term of the numerator, therefore \( \frac{x'}{2} = \frac{R-x}{2} \times x' \). By substituting these values in the fractional formula, we obtain \( GD = \frac{\text{fluent } \frac{x'}{2} \times \frac{x'}{2} = \frac{x'^2}{3}}{\frac{1}{4}ABEC} = \frac{\sqrt{2R x - x^2} \times \frac{1}{3}}{\frac{1}{4}ABEC} \); but since \( y = 2Rx - xx' \), we have, by raising both sides to the third power, \( y^3 = 2R x - xx' \frac{1}{3} \); therefore \( GD = \frac{\frac{1}{4}y^3}{\frac{1}{4}ABEC} = \frac{\frac{1}{4}y^3}{\frac{1}{4}ABEC} = \frac{\frac{1}{4}(2y)^3}{ABEC} \), 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.

178. Cor. When the segment becomes a semicircle we have \( 2y = 2r \); and therefore \( GD = \frac{\frac{1}{8}(2r)^3}{ABEC} = \frac{(2r)^3}{12ABEC} = \frac{8r^3}{12ABEC} = \frac{r^3}{\frac{1}{4}ABEC} \), 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 one and a half times the area of the segment.

PROP. X.

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

Let ABDC be the sector of the circle. By Art. 157, 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 \( ABEC : BCB = m : Gn \), then the point n will be the centre of gravity of the sector.—By proceeding in this way, it will be found that D n, 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.

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

Since \( y = a x^n \), multiply both terms by \( x \dot{x} \), and \( x \) separately, and we have \( y \dot{x} = a x^{n+1} \dot{x} \), and \( y = a x^n \dot{x} \). But, by Art. 174, we have \( \text{FB} = \frac{\text{fluent of } xy \dot{x}}{\text{fluent of } y \dot{x}} \), therefore, by substituting the preceding values of \( y \dot{x} \) and \( yx \) in the formula, we obtain \( \text{FB} = \frac{\text{fluent of } a x^{n+1} \dot{x}}{\text{fluent of } a x^n \dot{x}} \), and Theory. and by taking the fluents it becomes

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

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

When \( n \) is equal to 1, then \( y = a x \), and the parabola degenerates into a triangle, in which case \( FB = \frac{1}{2} x \), as in Art. 165.

Prop. XII.

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

Since \( y = a x^n \), square both sides, and we have \( y^2 = a^2 x^{2n} \); then multiply both sides by \( x \dot{x} \), and \( \dot{x} \) separately, we obtain \( y^2 x \dot{x} = a^2 x^{2n+1} \dot{x} \), and \( y^2 \dot{x} = a^2 x^{2n} \dot{x} \). But, by Art. 174, we have \( FB = \frac{\text{fluent of } y^2 x \dot{x}}{\text{fluent of } y^2 \dot{x}} \); therefore, by substituting the preceding values of \( y^2 x \dot{x} \), and \( y^2 \dot{x} \) in that formula, we obtain \( FB = \frac{\text{fluent of } a^2 x^{2n+1} \dot{x}}{\text{fluent of } a^2 x^{2n} \dot{x}} \), and by taking the fluents we shall have

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

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

When \( n = 1 \), the solid becomes a cone, and \( FB = \frac{1}{4} x \), as in Art. 171.

Prop. XIII.

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

Let BMNC be a section of the spherical surface comprehended between the planes BC, MN, and let EP = x, EC = y, DC = R, and z = the arc CN. Suppose the abscissa EP to increase by the small quantity E o, draw o parallel to EC, C s parallel to E o, and Cr perpendicular to DC; then it is evident, that in the similar triangles CDE, C s r, EC : DC = C r : Cr, that is, \( y : R = C s : C r \); but Cr is the fluxion of the arc NC, and C s the fluxion of the abscissa PE; therefore \( y : R = z : x \), and \( z y = R x \), and \( z = \frac{R x}{y} \). Now, by Art. 175, \( FB = \frac{\text{fluent of } x y z}{\text{fluent of } x y \dot{z}} \), therefore, by substituting the preceding value of \( z \) in this formula, we obtain \( FB = \frac{\text{fluent of } R x \dot{x}}{\text{fluent of } R \dot{x}} \), for

\[ \frac{R x \dot{x} \dot{z}}{y} = \frac{R x \dot{x} \dot{z}}{R y \dot{x} \dot{z}} \quad \text{(and dividing by } y \dot{z}) = \frac{R x \dot{x}}{R \dot{x}}. \]

By taking the fluents we obtain \( FB = \frac{1}{2} \frac{R x}{R} = \frac{1}{2} x \), a fluent which requires no correction, as the other quantities vanish at the same time with \( x \).

183. 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.

184. 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.

185. To find the centre of gravity of a circular arc.

Let BAC be the circular arc, it is required to Fig. 13. 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 semiarcs 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 shewn, as in the last proposition, that \( y : R = x : z \); hence \( z y = R x \). But, by Art. 176, we have \( FB = \frac{\text{fluent of } y z}{z} \), \( y \) being in this case equal to \( x \) in the formula in Art. 176. and substituting the preceding value of \( y \dot{z} \), it becomes \( FB = \frac{\text{fluent of } R x}{z} \), and, taking the fluent, we have \( FB = \frac{R x}{z} \), which requires no correction, as the fluent of \( y \dot{z} \) 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 \( d = \frac{R x}{z} \), and \( d x = R x \); hence \( z : x = R : d \), or \( 2 Z : 2 x = R : d \), 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.

186. When the arc BAC becomes a femicircle, PC or x is equal to DG or radius, so that we have \( 2 x : 2 R = R : d \), or \( 4 Z : 4 R = R : d \); but \( 4 x \) is equal to the whole circumference of the circle, and \( 4 R \) is Theory. is equal to twice the diameter; therefore, \(3.141593 : 2 = R : d\); hence \(d = \frac{2R}{3.141593} = .63662R\).

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

SCHOLIUM I.

188. From the specimens which the preceding propositions contain of the application of the formulae in Articles 173, 174, 175, 176, 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 Simfon*. 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 with to prosecute the subject.

* Mathematical Differen- tions, p. 109.

Position of the centre of inertia in bodies of various forms.

189. 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 propositions.

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 \(d = \frac{2C^3}{A}\), 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.

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 vered fine; 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 vered fine 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 \(\frac{Rx}{x}\), where \(R\) is the radius, \(x\) the semichord, and \(x\) 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, \(a\) the arc, and \(c\) its chord.

20. In a spherical sector, composed of a cone and a 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 vered fine 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 axis 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 \(\frac{4m - 3x}{6m - 4x} \times x\), 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 spheroid is distant from its vertex by a quantity equal to \( \frac{4n-3x}{6m-4x} \times x \), 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 surfold parabola six-elevenths of the axis.

28. In the common semi-parabola, 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 \( \frac{1}{3} \) of the greatest ordinate.

29. In the common paraboloid, the distance of the centre of inertia from its axis, is equal to \( \frac{1}{3} \) of the axis.

30. In the common hyperboloid, the distance of the centre of inertia from the vertex is equal \( \frac{4a+3x}{6a+4x} \times x \), 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 \( \frac{2R^2+r^2}{R^2+Rr+r^2} \times \frac{h}{4} \), 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 \( \frac{3R^2+2Rr+r^2}{R^2+Rr+r^2} \times \frac{h}{4} \) 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. XIV.

190. If a quantity of motion be communicated to a system of bodies, the centre of gravity 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 f the centre of gravity of the whole system, as determined by Art. 155. Then if the body A receives such a momentum as to make it move to a in a second, join F a, and take a point φ so that F φ : φ a = Ff : fa, φ will now be the centre of gravity of the system, fφ the path of that centre will be parallel to A a, and fφ will be to A a 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 B b in a second; and having drawn φ G parallel to B b, take a point G, so that φ G : B b = B : A+B+C, and G will be the centre of gravity of the bodies after B has moved to b. In the same 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 C c. 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 fφ 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 fφ is to A a; therefore A a and fφ are described by A and by the body at f in equal times, and the body at f will be found at φ at the end of a second. In the same way it may be shewn, 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.

191. 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.

192. 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 be two bodies whose common centre of gravity is F ; and let the points b, c, be taken, so that B β : C z = C : B, the spaces B β, C z will represent the mutual action of the bodies B, C, that is, B β will represent the action of C upon B, or the motion which is the result of that action, and C z 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 (Art. 155.) B : C=FC : FB, but B : C=C z : B β, therefore FC : FB=C z : B β; but C z is a magnitude taken from FC, and B β is a magnitude taken from FB, consequently (Playfair's Euclid, Book V. Prop. 19.) the remainder x F : β F =FC : FB, that is, x F : β F=B : C, that is, (Art. 155.) 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. XV.

193. If a body is placed upon a horizontal plane, or suspended by two threads, it cannot be in equilibrium. equilibrio unless a perpendicular drawn from the centre of gravity to the horizontal plane, or to 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.

194. 1. 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 preffing the body upon the horizontal plane ED.

195. 2. Let the body ACBD be suspended at the points f, φ by the threads h f, h φ, and let G be the centre of gravity of the body. Join G φ, G f, draw f φ parallel to the horizon, and through G draw n o parallel to f φ. Continue h f, h φ to o and n, and draw G i perpendicular to f φ, the body AB cannot be in equilibrium unless the point i falls upon the horizontal line f φ 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, φ; if therefore the body is not in equilibrium, its centre of gravity must defend either towards m or n; let it defend towards m till it rests at the point γ, then γ f = f G ; but γ φ is greater than G φ (Euclid, Book I. Prop. 7.) which is absurd, therefore the point G cannot defend, that is, the body is in equilibrium. It may be thrown in the same way, that it will be in equilibrium when G is any where between n and o, that is, when the perpendicular let fall from G cuts the horizontal line f φ that lies between the threads. If the body be supported by the two threads HE, h f, so that the perpendicular G i falls without the line f F, the body is not in equilibrium, 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 × Gm, it will therefore defend, and as its distance from f cannot change, the point f will rise, and the thread h f will be relaxed. When G arrives at m the perpendicular G m vanishes, and G has no power to turn round F. The body AB therefore cannot be in equilibrium till the perpendicular G i falls within f F, which it does as soon as it arrives at m.

196. 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 line of direction must always fall without the base.

197. Cor. 2. The higher the centre of gravity of Fig. 17. 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 C as a fulcrum, the power with which it will resist this force is inversely as FC; then, if the centre of gravity is raised to f, f C will be greater than FC, and the power with which it resists being overturned is diminished, that is, the body is the more easily overturned the higher that its centre of gravity is placed.

198. Cor. 3. If a body be supported 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 h f', h' φ 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 G i, GF must coincide with m F'; but HF and m F' are both perpendicular to the horizontal line f φ, therefore the centre of gravity G is in the direction of the thread HF.

199. Cor. 4. If the bodies A, B, C, fig. 18. be supported 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. 19, the bodies A, B Fig. 19. 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.

200. We have seen in the preceding proposition and different 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. 20, 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 m n, 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, when resting on the extremity of its conjugate axis, has a stable equilibrium, but when resting on its transverse axis as at B, its equilibri- um 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 will have its equilibrium liable 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.

Fig. 22.

Mechanical method of finding the centre of gravity.

Prop. XVI.

1. To find the centre of inertia mechanically.

201. 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, 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.

202. 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 somewherer 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.

203. 3. If the body is so flexible that it can neither be suspended by a thread nor balance 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.

204. 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.

205. 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 gravity.

Prop. XVII.

Centro-baryc method of Guldinus.

206 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 surface or 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 gravity of the generating line or surface.

Let ABCD be the plane surface by whose revolution round the axis MPN is generated the solid a D, contained by the parallelograms ABCD, a b c d, and by the areas a AC c, b BD d, and a AB b, c CD d; let G be the centre of gravity of ABCD, then the solid a D shall be equal to a solid whose base is ABCD, and whose altitude is a line equal to G g, the space described by its centre of gravity G. It is evident from Art. 161. 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 product 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 a D 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, π, &c. be the particles of the surface ABCD; D, d, δ their distance from the centre of rotation P, and A, a, α, the arches which they describe, while GP is the distance of the centre of gravity of the surface ABCD from the centre P, and G g the arch described by it. Then by Art. 161. \( \text{P} \times \text{D} + \text{p} \times \text{d} + \pi + \delta = \text{P} + \text{p} + \pi \times \text{GP} \), but \( \text{D}: \delta : \text{GP} = \text{A}: a : \text{Gg} \), therefore \( \text{P} \times \text{A} + \text{p} \times \alpha + \pi \times \alpha = \text{P} + \text{p} + \pi \times \text{Gg} \). But \( \text{P} \times \text{A} + \text{p} \times \alpha + \pi \times \alpha \) &c. make up the whole solid a D, and \( \text{P} + \text{p} + \pi \), &c. make up the whole surface ABCD; therefore the solid a D is equal to the generating surface ABCD multiplied by the path of its centre of gravity. (C. E. D.)

207. Cor. 1. Let us suppose the circle BACO to be generated by the revolution of the line DA round the Fig. 12; 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 femicircumference, which coincides with the result obtained from the principles of geometry. See Playfair's Geometry, Supp. B. 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.

208. 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 1649, 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. Leipf. 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 Gravitatis, Hutton's Mensuration, Prony's Architecture Hydraulique, vol. i. p. 88. and Gregory's Mechanics, vol. i. p. 64.

PROP. XVIII.

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

On the motion of animals.

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 putting 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.

210. 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.

211. 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 Fergulon's Lectures, vol. ii.).

212. 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*.

213. We have already seen (Art. 197.) 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 prelled to the ground.

214. A knowledge of the laws of the centre of gravity enables us to explain the experiment represented in fig. 24, 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 Art. 199. will be in equilibrium when their common centre of gravity C is in the same vertical line with the point of support E.

215. 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 may be moved from G towards g. Hence

* See Dr T. Young's Natural Philosophy, vol. i. p. 64. 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.

216. Upon the same principle, a double scalene cone may be made to ascend an inclined plane without being loaded with a weight. In fig. 26. 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 ADEF 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 27. 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 will ascend the plane upon the same principles, and under the same conditions, as those 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.

217. When a body moves along an inclined plane, the force which accelerates or retards its motion, is to the whole force of gravity as the height of the plane is to its length, or as the sine of its inclination is to radius.

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 re-action 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 = AB : AC, that is, the accelerating force which makes the body defend 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.

218. Cor. 1. 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 × \( \frac{H}{L} \), or, since g : A = radius : fin. ACB, and A = g × fin. ACB. Now, from the principles of DYNAMICS, s = \( \frac{1}{2} g t^2 \), v = \( g t = \sqrt{2g s} \), and \( t = \frac{v}{g} = \sqrt{\frac{2s}{g}} \), where s is the space described, g the force of gravity, or 32\( \frac{1}{4} \) feet, v the velocity, and t the time. Making \( \phi \), therefore, equal to ACB, and substituting the value of A instead of g in the preceding equation, we shall have s' = fin. \( \phi \times \frac{t^2}{2} \); v' = g fin. \( \phi \) = \( \sqrt{2g s'} \) fin. \( \phi \); and \( t = \frac{v'}{g \text{ fin. } \phi} = \sqrt{\frac{2s'}{g \text{ fin. } \phi}} \).

219. 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, 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 the semicircle Amn; 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 A m, 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.

220. 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 = \( \sqrt{2g s} \) fin. \( \phi \), and the velocity of a body falling in the vertical line AB is v' = \( \sqrt{2g s'} \). But, since v = v', we have \( \sqrt{2g s} \) fin. \( \phi \) = \( \sqrt{2g s'} \) or \( 2g s \) fin. \( \phi = 2g s' \), and dividing by \( 2g \), s fin. \( \phi = s' \), consequently s : s' = fin. \( \phi \) : 1, or AB : AC = fin. DAB : radius. Therefore, when v = v', that is, when the velocities of the two bodies are equal, the spaces described are as fin. DAB : radius, which can only happen when BC is perpendicular to AB. In the same way it may be shewn that the velocity at F is equal to the velocity at C, therefore the velocity at C is equal to the velocity at F.

221. 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 \( t^2 : t'^2 = \frac{s}{\sin \varphi} : \frac{AC}{\sin \varphi} : AB \). But \( \frac{AB}{AC} = \sin \varphi \); therefore,

\( t^2 : t'^2 = AC^2 : AB^2 \), and taking equal multiples of these two last terms, that is, multiplying them by AB, we have \( t^2 : t'^2 = AC^2 : AB^2 \), or \( t : t' = AC : AB \). Hence the times of descending along AF and AC, are as AF and AC.

222. 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.

Prop. II.

223. 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 DN, and let the body be let fall from the point A, so as to move along these planes, without loosing 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 Art. 220, 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 \( GE = Ce \), their respective velocities will be equal when they arrive at the points D, F in the horizontal plane. The velocity, therefore, acquired by the body falling along the planes AB, BC, CD is equal to that which is acquired by the same body falling through the vertical line AF.

224. 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 vered fine, or the velocity lost in passing from one plane to another, is diminished without limit (A), consequently, abstracting from friction, a body will ascend or descend a curve surface without loosing any of its velocity from the curvature of the surface.

225. 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, § 26, 51.; for the body experiences the same decrements of velocity in its ascent, as it received increments in its descent.

226. 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 reaction of the curve surface is in a line perpendicular to the curve.

Scholium.

227. 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 vered 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 by the velocity which the body had when it left the preceding plane.

Prop. III.

228. The times of descending two systems of inclined planes similar and similarly situated, are in the subduplicate ratio of their lengths.

Let AB, BC, CD, and ab, bc, cd be the similar systems of inclined planes, and let T be the time of descending ABCD, and t the time of descending abcd.

By Cor. 4. Prop. 1. we have

\[ \text{Time along } AB : \text{Time along } Ac = AB : Ac, \] \[ \text{Time along } ab : \text{Time along } as = ab : ab, \]

But, on account of the similar triangles AB c, ab β, we have,

\[ AB : Ac = ab : ab. \]

Hence (Euclid, Book v. Prop. 11. 16.)

\[ \text{Time along } AB : \text{Time along } ab = \text{Time along } Ac : \text{Time along } as. \]

(A) See Wood's Principles of Mechanics, p. 58. note; and also Gregory's Mechanics, vol. i. p. 112. where this corollary is demonstrated by the method of fluxions. Theory. In the same way it may be shewn, that

Time along BC : Time along bc = Time along cG : Time along β z, Time along CD : Time along cd = Time along GF : Time along αf.

Then, by Geometry, Sect. III. Theorem VIII.

Time along AB + BC + CD : Time along ab + bc + cd = Time along Ac + cG + GF : Time along αβ + βx + xf, that is,

Time along \( \overline{AB+BC+CD} \) : Time along \( ab+bc+cd \) = Time along AF : Time along af.

But by Dynamics § 37, 2,

Time along AF : Time along af = \( \sqrt{AF} : \sqrt{af} \),

Therefore, Euclid, B. V. Prop. 11.

Time along \( \overline{AB+BC+CD} \) : Time along \( ab+bc+cd = \sqrt{AF} : \sqrt{af} \). Q. E. D.

But by similar triangles, &c.

\( \sqrt{AF} : \sqrt{af} = \sqrt{AB+BC+CD} : \sqrt{ab+bc+cd} \).

Therefore,

Time along \( \overline{AB+BC+CD} \) : Time along \( ab+bc+cd = \sqrt{AB+BC+CD} : \sqrt{ab+bc+cd} \). Q. E. D.

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

230. 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.

231. 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 q FZ be a femicycloid, and A'D', C'F' two parallel and vertical ordinates at an infinitely small distance. Draw the ordinate B'E' an arithmetical mean between the ordinates A'D' and C'F', and from F', E' draw Fv, E'u perpendicular to BT, C'E'. Make C'F' = a, B'E' = b, E'u = c, C'B' = m, B'A' = n. Then since F'E' may be considered as a straight line, and since B'C' = Fv, we have (Euclid, B. I. Prop. 47.) \( F'E' = \sqrt{m^2 + c^2} \), and since Fv = E'u, \( E'D' = \sqrt{n^2 + c^2} \).

Now the velocities at F' and E' vary as \( \sqrt{a} \) and \( \sqrt{b} \), and F'E', 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 \( \frac{\sqrt{m^2 + c^2}}{\sqrt{a}} \), and the time of describing E'D' is \( \frac{\sqrt{n^2 + c^2}}{\sqrt{b}} \), consequently, the time of describing FD must be \( \frac{m^2 + c^2}{a^{\frac{3}{2}}} + \frac{n^2 + c^2}{b^{\frac{3}{2}}} \).

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

\[ \frac{2m \dot{m}}{2 \sqrt{a \times m^2 + c^2}} + \frac{2n \dot{n}}{2 \sqrt{b \times n^2 + c^2}} = 0. \]

But since CA is invariable \( m+n \) is invariable, and therefore its fluxion \( \dot{m} + \dot{n} = 0 \), or \( \dot{m} = -\dot{n} \) and \( \dot{m} = -\dot{n} \), therefore by transposing the second member of the preceding equation, and substituting these values of \( \dot{m} \) and \( \dot{n} \), it becomes \( \frac{m}{\sqrt{a \times m^2 + c^2}} = \frac{n}{\sqrt{b \times n^2 + c^2}} \).

Let us now call the variable absciss qC' = x, the ordinate C'F' = y, and the arc qF' = z, then m and n are fluxions of x, and F'E' is the increment of q F or z, when y is equal to a, and E'D' the increment of q F or z, when y is equal to b, therefore by substituting these values in the preceding equation, we obtain \( \frac{x'}{\sqrt{y^2 + z^2}} = \frac{x'}{\sqrt{y'^2 + z'^2}} \), which shews that this quantity is constant, and gives us the following analogy, \( x' : x' = 1 : \sqrt{y} \). 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. B. VI. Prop. 8.), therefore, we have FE : E'z = AO : AF; but FE = v, E'z = v, AO = 1 and AF = \( \sqrt{y} \), consequently \( x' : x' = 1 : \sqrt{y} \), which coincides with the analogy already obtained, and being the property of the cycloid shews that the curve of quickest descent is an inverted cycloidal arc.

Properties of the Cycloid.

Definition.—If a circle NOP be so placed as Fig. 3. to be in contact with the line AD, and be made to roll along that line from D towards A, till the fame 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. 232. 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. 4, be placed in the inverted position AB, DB (fig. 4.), 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 femicycloid AF, similar and equal to BA. In the same way the femicycloid DF, produced by the evolution of the string BP from the femicycloid 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 BF, 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 Pp 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 BF.

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.

233. 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.

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

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

Prop. V.

236. The velocity of a cycloidal pendulum BP 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, § 37.), therefore the velocity of the pendulum P, when it reaches F, varies as \( \sqrt{EF} \). But (Geometry, Sect. IV. Theor. 23. and 8.) FE varies as \( \frac{FN^2}{FC} \), and since FC is a constant quantity, FE will vary as FN² varies, or, to adopt the notation used in the article Dynamics, \( FE = \frac{FN^2}{r} \), or \( \sqrt{FE} = \frac{FN}{r} \), 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 (Art. 232. No 5.) FN is equal to half FP. Q. E. D.

Prop. VI.

237. If the pendulum begins its oscillation 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 draw p F q parallel to AD, and with a radius equal to the cycloidal arc FP, describe the semicircle p o q. Make r r equal to the arc PR of the cycloid, and through r draw r m perpendicular to p F. Through the points P, R draw PE, RT parallel to AD, and cutting the generating circle CNF in the points N, S.—By Prop. 4. the velocity at R varies as \( \sqrt{ET} \), 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 \( \sqrt{4\, FN^2 - 4\, FS^2} \), that is (Art. 232. No 5.) as \( \sqrt{FP^2 - FR^2} \). But if p r 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 \( \sqrt{Fr^2 - Fr^2} \), but (Euclid, 47. I.) \( rm = \sqrt{Fr^2 - Fr^2} \), 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.

238. 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 fine is also equal to radius or FM.

Prop. VII.

239. 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 p o q, uniformly with the velocity which the pendulum acquires at the point F.

Take any infinitely small arc RV, and making rv equal to it, draw v o parallel to rm, and mn to rv. Now, by the last proposition, and by Dynamics, Art. 28.; the velocity with which RV is described is to the velocity with which mo is described as rm is to FM, that: that is as \( \frac{RV}{rm} : \frac{mo}{Fm'} \) or as \( \frac{mn}{rm} : \frac{mo}{Fm'} \), for \( m n = r v = RV \).

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 fame 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 p o q is uniformly described with the velocity acquired at F.

PROP. VIII.

240. 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, 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 (Dynamics, § 37. No. 6.) in the same time that it would descend NF or CF (Art. 219.). Calling T therefore the time of an oscillation, and t the time of descent along the axis, we have, by the preceding proposition,

T = time along p o q, with the velocity at F, and by the preceding paragraph, t = time along Fp, with the same velocity; therefore

\( T : t = \) time along p o q with velocity at V : time along Fp with the same velocity ; that is, \( T : t = p o q : Fp = 2p o q : 2Fp \) = the circumference of a circle : its diameter.

241. 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.

242. Cor. 2. The oscillations in a small circular arc whose radius is BF, and in an equal arc of the cycloid, being isochronous (Art. 232. No 8.), 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.

243. 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 \( T = \frac{\sqrt{L}}{\sqrt{l}} ; t = \frac{T \times \sqrt{l}}{\sqrt{L}} ; l = \frac{T \times \sqrt{L}}{T} \), and \( L = \sqrt{T \times \sqrt{l}} \), 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.

244. 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 \( \frac{\sqrt{s}}{\sqrt{g}} \), 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.

245. Cor. 5. Since \( T = \frac{\sqrt{L}}{\sqrt{g}} \), \( \sqrt{g} \times T = \sqrt{L} \); and if the time of oscillation is 1 second, we have \( \sqrt{g} = \frac{\sqrt{L}}{T} \), or \( g = \frac{L}{T^2} \), that is, the force of gravity in different latitudes varies as the length of a pendulum that vibrates seconds.

246. 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 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.

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

Since by Proposition 8. 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 3.14159^2 is to 1. Now it appears from the experiments of Mr Whitehurst, that the length of a pendulum which vibrates seconds at London at 113 feet above the level of the sea, in a temperature of Theory. 60° of Fahrenheit, and when the barometer is 30 inches, is 39.1196 inches; hence \( \frac{1^{\circ} : 3.14159^2}{39.1196} = \frac{39.1196}{2} : 19.5508 \times 3.14159^2 = 16.087 \) feet the space required.

The methods of determining the centre of oscillation, gyration, and percussion, properly belong to this chapter, but they have been already given in the article Rotation, to which we must refer the reader who wishes to prosecute the subject.

Chap. VI. On the Collision or Impulsion of Bodies.

248. 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 impell the other. This effect has been distinguished by the names collision, impulsion or impulse, percussion, and impact.

249. Def. 2. The collision or impulsion 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.

250. 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.

251. 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.

252. Two hard bodies B, B' with velocities V, V', striking each other perpendicularly, will be at rest after impulse, 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 impulse, 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 2 B', 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.

253. 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 \( v \times B + B' = BV \) is the quantity of motion after impulse, therefore \( v \times B + B' = BV \), 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 movable 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.

254. 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 Cafe 1.

255. Cor. 1. If \(B = B'\), and the bodies mutually approach each other, the equation in Cafe 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.

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

257. Cor. 3. When the bodies move in the same direction, we have, by Cafe 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 \(B + B': B = V - V'\); 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 \(V - v = \frac{BV + B'V'}{B + B'}\) or \(\frac{B'V' - BV}{B + B'}\); 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 shewn when the bodies move in opposite directions, in which case their relative velocity is \(V + V'\).

Prop. III.

258. To determine the velocities of two elastic bodies after impulse.

If an elastic body strikes a hard and immoveable 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 is 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 impulse, therefore, it either moves in the direction opposite to that of the common momentum, or its motion in the same direction as that of the common momentum is diminished, or it is stopped altogether, according as the force of restitution is greater, less, or equal to the common momentum.

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

260. 1. If \(B\) follows \(B'\), then \(V\) is greater than \(V'\), and when \(B\) 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 onward 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 \(2 V - 2 v\), and the whole velocity gained by \(B'\) is \(2 v - 2 V'\). Now, subtracting this loss from the original velocity of \(B\), we have \(V - 2 V + 2 v\), for the velocity of \(B\) after impact, and adding the velocity gained by \(B\) to its original velocity, we have \(V' + 2 v - 2 V'\) for the velocity of \(B'\) after impact; hence we have \[ v = V - 2 V + 2 v = 2 v - V \] \[ v'' = V' + 2 v - 2 V' = 2 v - V. \] Now, substituting in these equations, the value of \(v\) as found in Cafe 2. Prop. 2, we obtain \[ v' = \frac{BV - B'V + 2 B'V'}{B + B'} \] \[ v'' = \frac{B'V' - B'V' + 2 BV}{B + B'} \]

261. 2. When the bodies move in opposite directions or mutually approach each other, the body \(B\) is in precisely the same circumstances as in the preceding case; but the body B' loses a part of its velocity equal to \( \frac{2v + 2V - V'}{2} \). 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 Cafe 3. Prop. 2. 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.

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

From the preceding equation the following corollaries may be deduced.

262. 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.

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

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

264. 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 formula, 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'.

265. 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.

266. 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.

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

268. 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'a velocity, and B' will move on with B's velocity. For in the formulae in Cafe 1. when B=B, we have v=V' and v''=V.

269. Cor. 8. When the bodies move in opposite directions, or mutually approach 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 Cafe 2. with the inferior signs, when B=B' and V=V', we have v'=-V and v''=V'.

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

271. 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.

272. Cor. 11. When the bodies move in opposite directions, the body which is struck, and the body which strikes it, will flop, continue their motion, or return backwards, according as BV—B'V is equal to, or greater or less than 2B'V'.

273. 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 the different cases V=V', and the relative velocity after impact is v'—v''=V—V'.

274. Cor. 13. By reasoning similar to that which was employed in Prop. 2, Cor. 3. it may be shewn 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.

275. Cor. 14. The vis viva, 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, B v'^2+B' v''^2=B V^2+B' V'^2. From the formulae at the end of Cafe 2. we obtain

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

276. Cor. 14. If several equal elastic bodies B, B'', B''', &c. are in contact, and placed in the same straight line, and if another elastic body s 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 s. By Art. 264, B will transfer Theory. 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 β.

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

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

Prop. IV.

279. To determine the velocities of two imperfectly elastic bodies after impulse, 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 Café 1. Prop. 8. that in consequence of the force of compression alone we have,

\[ V - v = \text{velocity lost by B} \\ v - V' = \text{velocity gained by B'} \]

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

\[ 1 : n = V - v : nV - nv, \text{the velocity lost by B} \\ 1 : n = v - V' : nv - nV', \text{the velocity gained by B} \]

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

\[ 1 + n V - 1 + n v, \text{the whole velocity lost by B}, \\ 1 + n v - 1 + n V', \text{the whole velocity gained by B}. \]

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

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

Now by substituting in the place of v its value as determined in Café 2. Prop. 2. we obtain

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

280. 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.

281. 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 \( v'' - v' = V' - V - \frac{1 + n \times B'V - B'V'}{B + B'} - \frac{1 + n \times BV - B'V}{B + B'} = V - V' - \frac{1 + n \times B + B' \times V - V'}{B + B'} \), dividing by \( B + B' \) we have \( v'' - v' = V - V' - V + V - V' + n \times V - V' = n \times V - V' \), 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.

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

Prop. V.

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

284. Case 1. When the impinging body is perfectly hard. Let AB be the hard and immoveable plane, body is perfectly hard, 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 = radius : cof. CDM, therefore since MD = CF the fine of the angle of incidence CDF, the velocity before impact is to the velocity after impact, as radius is to the fine 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.

285. 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 CDF is equal to the angle EDF, therefore, when an elastic body impinges obliquely against an immovable plane, it will be reflected from the plane, so that the angle of reflexion 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.

286. Case 3. When the impinging body is imperfectly elastic. In DF take a point m, so that DF is to D m as the force of compression is to the force of restitution or elasticity, and having drawn me parallel to DB, and meeting NE in e, join 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 DF 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 D m, perpendicular to the plane, while, by the velocity MD = DN = m e, it would be carried in the same time along m e, consequently, by means of these two velocities, the body will describe De, the diagonal of the parallelogram D m e N. The velocity, therefore, before impact is to the velocity after impact as DC : De, or as DE : De, or as fin. De E, fin. De e, or as fin. D e m : fin. De e, or as fin. FD e : fin. FDE. Now, by producing De so as to meet the line CE, produced in G, we have, on account of the parallels FE, m e, D m : DF = m e : 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 reflexion FDG : Therefore D m : DF = tang. CDF : tang. 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 reflexion 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 reflexion, as the sine of the angle of reflexion is to the sine of the angle of incidence.

Scholium.

287. 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 reflexion, are regarded as perfectly elastic bodies.

Prop. VI.

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

289. Case 1. When there is only one reflexion. 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 reflexion from the plane AB. From C let fall CH perpendicular to AB, continue it towards C till HG = CH, and join G, E 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 reflexion, 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 reflexion; consequently by Prop. 4, a body moving from C and impinging on the plane at D will, after reflexion, move in the line DE, and strike the body at E.

290. Case 2. When there are two reflexions. Let AB, BL, be the two immovable planes, C the place from which the impinging body is to move, and F the body which it is to strike after reflexion 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 D will be the point of the plane, against which the body at C must impinge, in order to strike the body at F. By reasoning as in the preceding case, it may be shewn that the angle CDH = EDB, therefore DE will be the path of the body after the first reflexion. Now, the triangles GEM, EMN are equiangular, because GM = MN, and the angles at M right, therefore DEB = FEL, that is, the body after reflexion at E will strike the body placed at F.

Prop. VII.

291. 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 I. 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. 3.; let these velocities be represented by AN, BP. Take AF = CI and BH = DM, 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 Theory. parallelogram NF; and for the same reason the body B will describe the diagonal BQ of the parallelogram PH.

292. 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 DC may be resolved into the forces EC, CF, of which EC is employed to move the ball C in the direction CB and with the velocity EC; but the force CF 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.

293. In the preceding proposition, we have endeavoured to give a short and perspicuous view of the common theory of impulsion. The limits of this article will not permit us to enter upon those interesting speculations to which this subject has given rise; but those who are anxious to pursue them will find ample assistance in the article IMPULSION, in the Supplement to the last edition of this work, where Dr Robison has treated the subject with his usual ability. 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 impulsive, 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 shew 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.

294. 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 Examen Maritime, a Spanish work which has been translated with additions by M. L'Eveque, under the title of Examen maritime, theorique et pratique, ou Traite de mecanique, applique a la construction, et a la manœuvre des vaisseaux et autres batimens. 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. We shall endeavour, under the article PERCUSSION, to give a short account of this interesting theory, which has been found to accord with the most accurate experiments.

295. 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 Leilie) 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. 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." Enquiry into the Nature of Heat, p. 127, 8.

296. 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, of motion 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.

297. An apparatus for making experiments on the Apparatus collision of bodies is represented in fig. 12. The for experiments 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 MN which is generally 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 Effects of Machines.

298. 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 situation 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.

299. Def. 1.—In a machine performing work, the powers employed to begin and continue the motion of the machine, are called the first movers, the movers of 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. If 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, fix feet from F, be equal to 50 pounds, or \( \pi \), 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, 5 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 A u, C w 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=\( 2\frac{1}{2} \) 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 feet 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 199\( \frac{1}{2} \) pounds acting at the distance of 2.5 feet from F. The mechanical energy, therefore, of the sum of all the resistances will be \( 2.5 \times 199\frac{1}{2} = 398.75 \), while the energy of the moving force, or the sum of all the moving powers, is equal to 500.

300. 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. 1. G is the impelled point of the machine, and D the working point. Had a single force \( \pi \) been applied at the point B to raise a single weight u, acting Theory. 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.

301. 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.

302. 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.

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

Explanations of symbols.

304. 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, (Art. 90.) 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.

305. 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.

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

307. 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

\[ \begin{align*} x &= \text{the velocity of the impelled point or the radius of the wheel, or the length of the lever by which the power acts.} \\ y &= \text{the velocity of the working point, or the radius of the axle, or the length of the lever by which the resistance acts against the power.} \\ P &= \text{the pressure exerted by the power at the impelled point of the machine.} \\ R &= \text{the pressure which the resistance arising from the work to be performed exerts at the working point of the machine.} \\ a &= \text{the inertia of the power } P, \text{ or the quantity of matter to which it must communicate the velocity of the impelled point.} \\ b &= \text{the inertia of the resistance } R, \text{ or the quantity of matter which it must move with the velocity of the working point before any work is performed.} \\ \phi &= \text{a quantity of matter which, if placed at the working point of the machine, would oppose the same resistance to the moving power as that which arises from the friction of the communicating parts.} \\ m &= \text{the quantity of matter which, if placed at the working point of the machine, would oppose the same resistance to the production of an angular motion, that is opposed by the inertia of the various parts of which the machine is composed. Hence, by the principles of rotation, we have} \\ &\quad my^2 = \text{the momentum of inertia of the machine.} \end{align*} \]

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

Prop. I.

308. 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, by Art. 36. 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

Theory. to that part of the power P which balances the resistance R. Hence, \( P - \frac{R\ y}{x} \) will be the effective force exerted by the power P, which, multiplied by x, its distance from the centre of motion, gives \( P x - R y \) for 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 \( \varphi \) 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 \( P x - R y \), and therefore \( P x - R y - \varphi y \) is the motive force exerted by P. Now, the momentum of the inertia of the power P, or the force with which it rests being put in motion, is \( ax^2 \), and the momentum of inertia of the resistance R is \( by^2 \), while the momentum of inertia of the machine is \( my^2 \). Therefore, the sum of these momenta, viz. \( ax^2 + by^2 + my^2 \) is the mass to be put in motion by the power P. But, by DYNAMICS, § 167, 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, is \( \frac{P x - R y - \varphi y}{ax^2 + by^2 + my^2} \). 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,

\[ \frac{P x^2 - R x y - \varphi x y}{ax^2 + by^2 + my^2} = \text{the velocity of the impelled point}, \]

and

\[ \frac{P x y - R y^2 - \varphi y^2}{ax^2 + by^2 + my^2} = \text{the velocity of the working point} \]

of the machine; and multiplying by R, we have from

Def. 4. \( \frac{P x y R - R^2 y^2 - \varphi R y^2}{ax^2 + by^2 + my^2} \) = the work performed.

309. But as forces are proportional to the velocities generated by them in equal times (DYNAMICS, § 153. Cor. 4. § 159.), the preceding quantities will represent the accelerating forces. Now, the velocities are as the forces and times jointly (DYNAMICS, § 153.), that is, \( v = F t \), or \( s = g t F \); but F, the accelerating force, which generates the velocity of the impelled point, is represented by the formula \( \frac{P x^2 - R x y - \varphi x y}{ax^2 + by^2 + my^2} \). Therefore, v, or the absolute velocity of the impelled point, is \( \frac{P x^2 - R x y - \varphi x y}{ax^2 + by^2 + my^2} \times g t \), and the absolute velocity of the working point \( \frac{P x y - R y^2 - \varphi y^2}{ax^2 + by^2 + my^2} \times g t \). 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,

\[ \frac{P x y R - R^2 y^2 - \varphi R y^2}{ax^2 + by^2 + my^2} \times g t. \]

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

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

When \( R = 0 \), we have

\[ y = \frac{\sqrt{a^2 \varphi + P^2 a x m + b^2 l^2 - a \varphi}}{P m + P b} \times x. \]

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

\[ y = \frac{\sqrt{a^2 R^2 + P^2 a x m + b^2 l^2 - a R}}{P m + P b} \times x. \]

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

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

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,

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

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

\[ y = \frac{\sqrt{P^2 R^2 + P^2 R l^2 - P R}}{P R} \times x = \sqrt{\frac{P^2 R^2 + P^2 R}{P^2 R^2} - \frac{P R}{P R}} \times x \]

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

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 \( x = 1 \),

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

These various formulæ, the application of which to particular cases shall 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.

310. 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 enable us to obtain a maximum effect. The method of determining this will be shewn in the following proposition.

PROP. II.

311. 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.

312. CASE 1. When R is the only quantity which is variable, the fluxion of the formula

\[ \frac{P x y R - R^2 y^2 - \varphi R y^2}{a x^2 + b y^2 + m y^2}, \]

which represents the work performed, is equal to the fluxion of the numerator, because the denominator is constant, that is, \(P x y R - 2 R R y^2 - \varphi R y^2 = 0\), and, dividing by \(R\); \(P x y - 2 R y^2 - \varphi y^2 = 0\), hence \(2 R y^2 = P x y - \varphi y^2\), and \(R = \frac{P x y - \varphi y^2}{2 y^2}\), which, divided by y, gives \(R = \frac{P x - \varphi y}{2 y}\).

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, \(\frac{1}{3} R\); and calling \(Z = \frac{1}{3} R\), and omitting \(\varphi y\), we have for the resistance \(R + \frac{1}{3} R\), or \(\frac{4}{3} R = \frac{P x}{2 y}\), or \(R = \left( \frac{P x}{2 y} \right) \div \frac{4}{3}\), and making \(P = 1\), and \(x = 1\), we have \(R = \left( \frac{1}{2 y} \right) \div \frac{4}{3}\), so that, abstracting from the quotient \(\frac{1}{2 y}\), 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.

313. 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 d R, where d may be either an integer or a fraction. In order to simplify the investigation, we may consider the fraction \(\varphi\) 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 \(P - \varphi\). In the same way we may consider the momentum of the machine's inertia applied to the impelled point, that is, instead of \(m y^2\) it may be made \(a x^2\). Now making \(P - \varphi\), 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,

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

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

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

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 \(R = \frac{q^2 + q d y^2}{d y^2} - q\).

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

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

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.

314. Those who wish to prosecute this interesting subject may consult the different papers of Euler in the Comment. Petropol. vol. x. p. 83, 1743, and in the Comment. Nov. Petropol. vol. iii. and viii. In the article. MACHINERY in the Supplement to the last edition of this Work, the 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 formula. The subject has been also ably discussed by Professor Leflie in a paper published in 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, is 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 will produce a maximum effect.

315. 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 \(R \times 1 + \sqrt{2}; R \times \frac{1}{2} + \sqrt{\frac{1}{2}}; R \times \frac{1}{3} + \sqrt{\frac{1}{3}}; R \times \frac{1}{4} + \sqrt{\frac{1}{4}}; R \times \frac{1}{5} + \sqrt{\frac{1}{5}}; R \times \frac{1}{6} + \sqrt{\frac{1}{6}}; \ldots\), 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 Mr Leflie's paper, that in whatever way the maximum be procured, the force which impells the the weight can never amount to one-fourth part of the direct action of the power; and that in machines where the velocity of the power is great, we may disregard the momenta of the connecting parts, and consider the force which ought to be employed as double of what is barely able to maintain the equilibrium.

CHAP. VIII. On the Equilibrium of Arches, Piers, and Domes.

316. DEF. 1. An arch is represented in fig. 3. by the assemblage of stones a b, c d, e f, &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, C its altitude, b its crown, a b the keystone, the curve or lower surface A b B the 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 bridge.

317. DEF. 2. A catenarian curve is the curve formed by any line or cord perfectly flexible, and suspended by its extremities. Thus if the chain ACB be 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.

318. 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 prelures 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 was given by the ingenious Dr Hook, and is contained in the following proposition.

PROP. I.

319. 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 be a number of spheres or beads connected by a string, and suspended by their extremities A, B; they will form a catenarian curve A a b c B, 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 c 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 b c d e 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. 6. 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.

320. Let us now suppose that it is required to form an equilibrated arch, whose span is AB, whose altitude is D k, 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 increase in density from k towards a, the catenarian curve will grow less concave at its vertex e, and more concave towards its extremities A, B. Let us then suppose that the densities of the spheres a, b, c, d, e, &c. are respectively as a m, b n, c o, d p, e q, &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. 6. 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 b n, c o, d p, &c. Then it is obvious that the prelude of each sphere when thus loaded upon that which is contiguous to it, is precisely equal to the prelude 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 prelures, the arch composed of these spheres and supporting TUVB k A 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.

321. Hence we have the following practical method of ascertaining the form of an equilibrated arch, whose span is AB, and altitude D k, and which is to support a roadway of the form T'U'V'. Let a chain A a b c k B, of uniform density, be suspended from the points A, B, so that it forms a catenary whose altitude is D k, the required height of the arch. Divide AB into any number of equal parts, suppose eight, and let the vertical lines 1 m, 2 n, 3 o, 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 equilibrium, the extremities of the chains may lie in the line T'U'V'; then the form which the catenary A k B now assumes, will be the form of an equilibrated arch, which, when inverted like AKB, will support the roadway TUV, similar to T'U'V'. Theory. This is obvious from the last paragraph, for the pieces of chain a m, b n, c o, k U, &c. are forces acting upon the points a, b, c, k of the catenary, and are proportional to a m, b n, c o, &c. the distances of the points a, b, c, k, &c. from the roadway.

322. 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 A B. 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.

323. 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.

324. 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 Fluxions, published in 1742, and afterwards by Dr Hutton in his excellent work on bridges.

Prop. II.

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

Let the lines A D, D E, E B, B F, F G, G H 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 D d, E e, B b, &c. and taking D p of any length, make E r equal to D p, &c. and complete the parallelograms p e, q e. Again, make B s = q e, and complete the parallelogram r s; in like manner make F k = r b, and complete the parallelogram F f; 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 C D, D E, E B, &c. can move round the angular points D, E, B, F, &c. the extremities A, C being immovable; and that forces proportional to D d, E e, B b, &c. are exerted upon the points D, E, B, F, &c. and in the direction D d, E e, &c. Now, by the resolution of forces, the force D d may be resolved into the forces D c, D p, the force E e into the forces E q, E r, and the force B b into the forces B s, B t, and so on with the rest. The force D c 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 D p tends to bring the point D towards E, and to enlarge the angle A D E; this force, however, is destroyed by the equal and opposite force E q, and in the same way the forces E r, B t, F x are destroyed by the equal and opposite forces B s, F k, G v; while the remaining force G w is destroyed by the resistance of the plane which supports the point C. When the lines A D, D E, &c. therefore are acted upon by vertical forces proportional to D d, E e, B b, &c. these forces are all destroyed by equal and opposite ones, and the lines will remain in equilibrium.

326. Now the force D c : D p or E q = fin. c d D or d D p : fin. A D d, that is, by taking the reciprocals

\[ \frac{D c}{E q} = \frac{1}{\text{fin. } A D d} : \frac{1}{\text{fin. } d D p} \]

and for the same reason

\[ \frac{E q}{B s} = \frac{1}{\text{fin. } E q} : \frac{1}{\text{fin. } B s}. \]

Hence

\[ \frac{E q}{\text{fin. } E q} = \frac{1}{\text{fin. } E e q}. \]

Now, since E q : E e = fin. E e q : fin. E q e, we have:

\[ E e = \frac{E q \times \text{fin. } E q e}{\text{fin. } E e q}, \]

that is, since D E m = E q e, and

\[ e E B = E e q ; \quad E e = \frac{E q \times \text{fin. } D E m}{\text{fin. } e E B}. \]

But \( E q = \frac{1}{\text{fin. } E e q} \), therefore, by substitution, we obtain

\[ E e = \frac{\text{fin. } D E m}{\text{fin. } E e q \times \text{fin. } e E B}. \]

Now, as the same reasoning may be employed to find D d, B b, &c. we have obtained expressions of the forces which, when acting at the angular points D, E, B, &c. keep the whole in equilibrium, and these expressions are in terms of the angles which the lines D E, E B, &c. form with the direction of the forces. If the lines A D, D E, &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 m E will be a tangent to the curve at the point E, and D E m will be the angle of contact. The line E q being now infinitely small will coincide with E m, and therefore the angles E q and e E B or E e q will be equal to the angle e E m, 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:

\[ E e = \frac{\text{fin. } D E m}{\text{fin. } e E m \times \text{fin. } e E m} : \frac{\text{fin. } D E m}{\text{fin. } e E m^2}. \]

But: 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,

\[ E c = \frac{1}{\text{radius of curvature} \times \sin e \cdot E m^2} \]

In order to get rid of the confusion in fig. 8, where the arch is a polygon, let us suppose ABC, fig. 9, to be the curve, \( mn \) a tangent to any point E, and E e 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.

327. Corollary. Let us now suppose that the arch ABC supports a mass of homogeneous materials lying between the roadway TUV and the arch AEBC; and the whole being supposed in equilibrium, let us determine the weight which presses on the point E. The weight of the superincumbent column E c b d varies as \( E c \times E d \), but \( g d = E d \times \sin. \overline{E E_g} \), E d being radius, and \( E E_g = E n B \), on account of the parallels E c, UB, therefore the weight of the column E c b d varies as \( E c \times E d \times \sin. E n B \), that is, as \( E c \times \sin. E n B \), because E d is a constant quantity; but the pressure at E was proved to vary as

\[ \frac{1}{\text{radius curvature} \times \sin. e \cdot E m^2}, \]

therefore the weight of the column E c b d or E c \times \sin. E n B varies also as this quantity, that is,

\[ E c \times \sin. E n B = \frac{1}{\text{radius curvature} \times \sin. e \cdot E m^2}. \]

But as the angle E n B is equal to the angle e E m, we shall have, by substitution and division,

\[ E c = \frac{1}{\text{radius curvature} \times \sin. e \cdot E m^2}, \]

that is,

When an arch supports a roadway, the pressure exerted upon 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.

328. Having thus obtained an expression for E c, 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 arch of a circle whose centre is F. Let the radius = R, BD = verified fine, BE=x, DF=col. BE=b, BU=m. Draw the tangent GE, and through E the vertical line ce, which will be parallel to BE. Then since GEF is a right angle, and e EF=EFB, the angle GE e is the complement of EFB, therefore, fin. GE=col. EFB=FD. But, in the present case, the radius of curvature is the radius of the arch, or R, therefore, \( E c = \frac{1}{R \times \sin. GE e} \), or by substitution, \( E c = \frac{1}{R \cdot b^3} \), that is, since R is constant, \( E c = \frac{1}{b^3} \). But when the point E coincides with B, the cosine b becomes equal to radius; therefore, in that case \( E c = \frac{1}{R^3} \), and E c becomes \( BU = m \), hence \( \frac{1}{R^3} : \frac{1}{b^3} = m : E c \), and by Geometry, Theor. 8. Sect. IV. and Division, we have \( E c = \frac{m \cdot b^3}{R^3} \). Now, by the notation R : b = BF : DF; therefore \( R^3 : b^3 = BF^3 : DF^3 \), hence \( \frac{R^3}{b^3} = \frac{BF^3}{DF^3} \), and multiplying each side by m, we have \( \frac{m \cdot R^3}{b^3} = \frac{m \cdot BF^3}{DF^3} \); but \( \frac{m \cdot R^3}{b^3} = E c \), therefore the vertical distance of the surface of the roadway from the point F, or \( E c = \frac{m \cdot BF^3}{DF^3} \).

When the point E coincides with B, BF=DF, and E c=BU. When E coincides with A, the cosine DF vanishes, and therefore E c, or the distance of the point A from the extrados or roadway, is infinite. The curve VU c T, 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 equilibration. 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 roadway horizontal as t U v, and making the density of the superincumbent columns P n, E q, which press upon the points P, E respectively, in the ratio of P p, E c, the distances of these points from the curvilinear roadway.

329. 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 B d, and if points a, b are taken in the lines P p, E c, so that P a: P p = E b: E c = B d: BU, and so on with all the points in the arch; and if a new roadway v d b a t be drawn through these points, the equilibrium of the arch will still continue, for the various pressures which it sustains, though they are diminished, preserve the same proportion.

330. 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 U c T cuts the horizontal line t U v, the point H coincides with U. By substituting BF=BD instead of DF in the value of E c, formerly determined, and by putting BD=y, we have \( E c = \frac{m \cdot R^3}{R - y^3} \). But when H coincides with U, c coincides with o, and therefore \( E c = E c = BD + BU = y + m \), consequently, \( \frac{m \cdot R^3}{R - y^3} = y + m \), and multiplying by \( R - y^3 \), we have \( m \cdot R^3 - m \cdot R \cdot y^3 + m \cdot R - m \cdot y^3 \), or \( m \cdot R^3 + m \cdot R - m \cdot y^3 = y \cdot R - y^3 \), and, dividing by the coefficients of m, we have

\[ m = \frac{y \cdot R - y^3}{R^3 - R \cdot y^3}, \]

that is,

The thickness of the roadway above the keystone, when the extrados is a straight line, is equal to the quotient arising Theory. 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.

331. When the arch is a semicircle \( R - y \) vanishes, and \( m \) becomes equal to 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. \( 60^\circ \), we have \( m = \frac{r}{2} \) of the span, \( 90^\circ \), we have \( m = \frac{r}{4} \) of the span, or \( 110^\circ \), we have \( m = \frac{r}{7} \) of the span nearly.

The two first arches of \( 60^\circ \) and \( 90^\circ \), manifestly give too great a thickness to the part \( BU \) or \( m \). In the third arch of \( 110^\circ \), 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 \( 110^\circ \), the thickness of \( BU \) or \( m \) would be too small. It is obvious, however, that an arch of \( 110^\circ \) is not an arch of perfect equilibration, for this can be the case only when the roadway has the form \( U \times r \). 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 \( r \times U \). 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.

332. 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 their when the transverse axis is horizontal; for as it is tranverse much flatter, the point of contrary flexure in the extra-axis is hori-dos 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.

333. 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.

334. When the form of the roadway is given, the On the me-shape of the intrados for an arch of equilibration may chanical be determined. As the investigation is very difficult, curve of unlefs 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 equilibri-um of the arch is imperfect; but the curve de-scribed by the following formula is an arch of perfect equilibration, and has been called the mechanical curve of equilibration.

\[ ED = AF \times \frac{\text{Hyperbol. log. } \frac{BU + BD + \sqrt{2 \, BU \times BD + BD^2}}{BU}}{\text{Hyperbol. log. } \frac{BU + BF + \sqrt{2 \, BU \times BF + BF^2}}{BU}} \]

From this formula, which corresponds with figure 11. Dr Hutton has computed the following table, contain-ing the values of \( cU \) and \( cE \), for an arch whose span \( AC \) is 100, whose height \( BF \) is 40, and whose thick-ness 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 \( cU \) and \( cE \) must be increased or diminished in the same ratio as these numbers.

<table> <tr> <th>Value of cU.</th> <th>Value of cE.</th> <th>Value of cU.</th> <th>Value of cE.</th> <th>Value of cU.</th> <th>Value of cE.</th> <th>Value of cU.</th> <th>Value of cE.</th> <th>Value of cU.</th> <th>Value of cE.</th> </tr> <tr> <td>0</td><td>6.000</td><td>15</td><td>8.120</td><td>24</td><td>11.911</td><td>33</td><td>18.627</td><td>42</td><td>29.919</td> </tr> <tr> <td>2</td><td>6.035</td><td>16</td><td>8.430</td><td>25</td><td>12.480</td><td>34</td><td>19.617</td><td>43</td><td>31.563</td> </tr> <tr> <td>4</td><td>6.144</td><td>17</td><td>8.766</td><td>26</td><td>13.106</td><td>35</td><td>20.665</td><td>44</td><td>33.299</td> </tr> <tr> <td>6</td><td>6.324</td><td>18</td><td>9.168</td><td>27</td><td>13.761</td><td>36</td><td>21.774</td><td>45</td><td>35.135</td> </tr> <tr> <td>8</td><td>6.580</td><td>19</td><td>9.517</td><td>28</td><td>14.457</td><td>37</td><td>22.948</td><td>46</td><td>37.075</td> </tr> <tr> <td>10</td><td>6.914</td><td>20</td><td>9.934</td><td>29</td><td>15.196</td><td>38</td><td>24.190</td><td>47</td><td>39.126</td> </tr> <tr> <td>12</td><td>7.330</td><td>21</td><td>10.381</td><td>30</td><td>15.980</td><td>39</td><td>25.505</td><td>48</td><td>41.293</td> </tr> <tr> <td>13</td><td>7.571</td><td>22</td><td>10.858</td><td>31</td><td>16.811</td><td>40</td><td>26.894</td><td>49</td><td>43.581</td> </tr> <tr> <td>14</td><td>7.834</td><td>23</td><td>11.368</td><td>32</td><td>17.693</td><td>41</td><td>28.364</td><td>50</td><td>46.000</td> </tr> </table>

TABLE for constructing the Curve of Equilibration, when the span, height, and thickness at the crown, are as the numbers 100, 40, and 6. 335. The construction of arches has also been deduced from considering the arch-stones as frutums* of polished wedges without friction, which endeavour to force their way through the arch. This principle has been adopted by Belidor, Parent, Boisut, 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 between those which are contiguous to them, without friction, they are bound together by the strong cement, and sometimes connected by iron pins or wedges. The theory likewise requires, that the weight of the arch must regularly increase as the portion of the vertical tangent cut off by lines drawn from a given point in a direction parallel to that of the joints, and therefore either the density or the magnitude of the arch-stones must be very great at the spring of the arch, where the portion of the vertical tangent is a maximum. Those who wish to be acquainted with the mode of investigation, by which the equilibrium of arches is established in this theory, may consult Prony's Architecture Hydraulique, tom. i. p. 152.

On the Construction of Piers and Abutments.

336. 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 semiarcs, and which tend either to overthrow 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 the pressure; and as the lateral thrust of the arch which this adhesion 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 semarch, because it is necessary that the pier should have sufficient stability to resist the lateral thrust of one semarch, in case of the failure of the opposite one. Let us therefore consider the strength of the pier which will prevent it from being overthrown.

337. For this purpose, let ABC be an arch, MHTO the pier, and BUHA the loaded semarch, 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. 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 in 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 (Art. 164.). 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 \( \times \frac{1}{2} \)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*, which, by reduction, becomes OM

\( = \sqrt{\frac{2 \ AM \times W \times KA}{MH \times GK}} \). 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 BVHA of a piece of card, and to find its centre of gravity G by the rules given in Articles 201, 202, 203. 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.

338. 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 those 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. On the Construction of Domes.

339. Definition. A dome, cupola, or vault, is an arched roof, either of a spherical, conoidal, or spheroidal form.

The following proposition, taken from Dr Robison's article upon this subject, in the Supplement to the late edition of the Encyclopedia Britannica, contains a very brief view of the theory of domes.

Proposition.

340. "To determine the thickness of a dome vaulting when the curve is given, or the curve when the thickness is given.

"Let B b A, figure 1. be the curve which produces the dome by revolving round the vertical axis AD. We shall suppose this curve to be drawn through the middle of all the arch-stones, and that the courling or horizontal joints are every where perpendicular to the curve. We shall suppose (as is always the case) that the thicknesses KL, HI, &c. of the arch-stones is very small, in comparison with the dimensions of the arch. If we consider any portion HA h of the dome, it is plain that it prelifes on the course, of which HL is an arch-stone, in a direction b C perpendicular to the joint HI, or in the direction of the next superior element b b of the curve. As we proceed downwards, course after course, we see plainly that this direction must change, because the weight of each course is superadded to that of the portion above it, to complete the pressure on the course below. Through B draw the vertical line BCG, meeting b b, produced in C. We may take bc to express the preliffe of all that is above it, propagated in this direction to the joint KL. We may also suppose the weight of the course HL united in b, and acting on the vertical. Let it be represented by b F. If we form the parallelogram b FGC, the diagonal b G will represent the direction and intensity of the whole preliffe on the joint KL. Thus it appears that this preliffe is continually changing its direction, and that the line, which will always coincide with it, must be a curve concave downward. If this be precisely the curve of the dome, it will be an equilibrated vaulting; but so far from being the strongest form, it is the weakest, and it is the limit to an infinity of others, which are all stronger than it. This will appear evident, if we suppose that b G does not coincide with the curve A b B, but passes without it. As we suppose the arch-stones to be exceedingly thin from inside to outside, it is plain that this dome cannot stand, and that the weight of the upper part will press it down, and spring the vaulting outwards at the joint KL. But let us suppose, on the other hand, that b G falls within the curvilineal element b B. This evidently tends to pull the arch-stone inward, toward the axis, and would cause it to slide in, since the joints are supposed perfectly smooth and slipping. But since this takes place equally in every stone of this course, they must all abut on each other in the vertical joints, squeezing them firmly together. Therefore, revolving the thrust b G into two, one of which is perpendicular to the joint KL, and the other parallel to it, we see that this last thrust is withstood by the vertical joints all around, and there remains only the thrust in the direction of the curve. Such a dome must therefore be firmer than an equilibrated dome, and cannot be so easily broken by overloading the upper part. When the curve is concave upwards, as in the lower part of the figure, the line b C always falls below B b, and the point C below B. When the curve is concave downwards, as in the upper part of the figure, 'b C' passes above, or without b B. The curvature may be so abrupt, that even 'b G' shall pass without 'b B', and the point G' is above B'. It is also evident that the force which thus binds the stones of a horizontal course together, by pushing them towards the axis, will be greater in flat domes than in those that are more convex; that it will be still greater in a cone; and greater still in a curve whose convexity is turned inwards: for in this last case the line b G will deviate most remarkably from the curve. Such a dome will stand (having polished joints) if the curve springs from the base with any elevation, however small; nay, since the friction of two pieces of stone is not less than half of their mutual preliffe, such a dome will stand, although the tangent to the curve at the bottom should be horizontal, provided that the horizontal thrust be double the weight of the dome, which may easily be the case if it do not rise high.

"Thus we see that the liability of a dome depends on very different principles from that of a common arch, and is in general much greater. It differs also in another very important circumstance, viz. that it may be open in the middle: for the uppermost course, by tending equally in every part to slide in toward the axis, prelifes all together in the vertical joints, and acts on the next course like the key-stone of a common arch. Therefore an arch of equilibrium, which is the weakest of all, may be open in the middle, and carry at top another building, such as a lantern, if its weight do not exceed that of the circular segment of the dome that is omitted. A greater load than this would indeed break the dome, by causing it to spring up in some of the lower courses; but this load may be increased if the curve is flatter than the curve of equilibrium: and any load whatever, which will not crush the stones to powder, may be set on a truncate cone, or on a dome formed by a curve that is convex toward the axis; provided always that the foundation be effectually prevented from flying out, either by a hoop or by a sufficient mass of solid pier on which it is set.

"We have seen that if b G, the thrust compounded of the thrust b C, exerted by all the courses above HILK, and if the force b F, or the weight of that course, be everywhere coincident with b B, the element of the curve, we shall have an equilibrated dome; if it falls within it, we have a dome which will bear a greater load; and if it falls without it, the dome will break at the joint. We must endeavour to get analytical expressions of these conditions. Therefore draw the ordinates b b', BD'B', C d C". Let the tangents at b and b' meet the axis in M, and make MO, MP, each equal to b c, and complete the parallelogram MONP, and draw OQ perpendicular to the axis, and produce b F, cutting the ordinates in E and e. It is plain that MN Theory. is to MO as the weight of the arch HA h to the thrust bc which it exerts on the joint KL (this thrust being propagated through the course of HILK); and that MQ, or its equal b e, or d d, may represent the weight of the half AH.

"Let AD be called x, and DB be called y. Then b e = x, and e C = y (because b c is in the direction of the element b b). It is also plain, that if we make y constant, BC is the second fluxion of x, or BC = a; and b e and BE may be considered as equal, and taken indiscriminately for x. We have also b C = \( \sqrt{x^2 + y^2} \). Let d be the depth or thickness HI of the arch-stones. Then \( d \sqrt{x^2 + y^2} \) will represent the trapezium HL; and since the circumference of each course increases in the proportion of the radius y, \( d y \sqrt{x^2 + y^2} \) will express the whole course. If \( f \) be taken to represent the sum or aggregate of the quantities annexed to it, the formula will be analogous to the fluent of a fluxion, and \( \int d y \sqrt{x^2 + y^2} \) will represent the whole mass, and also the weight of the vaulting, down to the joint HI. Therefore we have this proportion, \( \int d y \sqrt{x^2 + y^2} : d y \sqrt{x^2 + y^2} = b F : b F = b e : C G = d d : C G = x : C G \). Therefore \( C G = \frac{d y \dot{x} \sqrt{x^2 + y^2}}{\int d y \sqrt{x^2 + y^2}} \).

"If the curvature of the dome be precisely such as puts it in equilibrium, but without any mutual pressure in the vertical joints, this value of OG must be equal to CB, or to x, the point G coinciding with B. This condition will be expressed by the equation \( \frac{d y \dot{x} \sqrt{x^2 + y^2}}{\int d y \sqrt{x^2 + y^2}} = x \), or, more conveniently, by \( \frac{d y \sqrt{x^2 + y^2}}{\int d y \sqrt{x^2 + y^2}} = \frac{\dot{x}}{x} \). But this form gives only a tottering equilibrium, independent of the friction of the joints and the cohesion of the cement. An equilibrium, accompanied by some firm stability, produced by the mutual pressure of the vertical joints, may be expressed by the formula

\[ \frac{d y \sqrt{x^2 + y^2}}{\int d y \sqrt{x^2 + y^2}} = \frac{\dot{x}}{x}, \quad \text{or by} \quad \frac{d y \sqrt{x^2 + y^2}}{\int d y \sqrt{x^2 + y^2}} = \frac{\dot{x}}{x} + \frac{i}{t}, \]

where t is some variable positive quantity, which increases when x increases. This last equation will also express the equilibrated dome, if t be a constant quantity, because in this case \( \frac{i}{t} \) is = 0.

"Since a firm stability requires that \( \frac{d y \dot{x} \sqrt{x^2 + y^2}}{\int d y \sqrt{x^2 + y^2}} \) shall be greater than \( \frac{\dot{x}}{x} \), and CG must be greater than CB: Hence we learn, that figures of too great curvature, whose fides descend too rapidly, are improper. Also, since stability requires that we have

\[ \frac{d y \dot{x} \sqrt{x^2 + y^2}}{\int d y \sqrt{x^2 + y^2}} \]

greater than \( \frac{d y \sqrt{x^2 + y^2}}{\int d y \sqrt{x^2 + y^2}} \), we learn that the upper part of the dome must not be made very heavy. This, by diminishing the proportion of b F to b C, diminishes the angle c b G, and may set the point G above B, which will infallibly spring the dome in that place. We see here also, that the algebraic analysis expresses that peculiarity of dome-vaulting, that the weight of the upper part may even be suppressed.

"The fluent of the equation

\[ \frac{d y \sqrt{x^2 + y^2}}{\int d y \sqrt{x^2 + y^2}} = \frac{\dot{x}}{x} + \frac{i}{t} \]

is most easily found. It is \( L \int d y \sqrt{x^2 + y^2} = L \dot{x} + L t \), where L is the hyperbolic logarithm of the quantity annexed to it. If we consider y as constant, and correct the fluent so as to make it nothing at the vertex, it may be expressed thus, \( L \int d y \sqrt{x^2 + y^2} - L a = L \dot{x} - L y + L t \). This gives us \( L \int \frac{d y \sqrt{x^2 + y^2}}{a} = L \frac{\dot{x}}{y} t \), and therefore \( \int \frac{d y \sqrt{x^2 + y^2}}{a} = t \frac{\dot{x}}{y} \).

"This last equation will easily give us the depth of vaulting, or thickness d of the arch, when the curve is given. For its fluxion is \( \frac{d y \sqrt{x^2 + y^2}}{a} = \frac{i \dot{x} + t \ddot{x}}{y} \), and

\[ d = \frac{a \dot{x} + a t \ddot{x}}{y \dot{y} \sqrt{x^2 + y^2}}, \]

which is all expressed in known quantities; for we may put in place of t any power or function of x or of y, and thus convert the expression into another, which will still be applicable to all sorts of curves.

"Instead of the second member \( \frac{\dot{x}}{x} + \frac{i}{t} \), we might employ \( \frac{p \dot{x}}{x} \), where p is some number greater than unity. This will evidently give a dome having stability; because the original formula \( \frac{d y \dot{x} \sqrt{x^2 + y^2}}{\int d y \sqrt{x^2 + y^2}} \) will then be greater than \( \frac{\dot{x}}{x} \). This will give \( d = \frac{p a \dot{x}^{p-1} \ddot{x}}{y \dot{y} \sqrt{x^2 + y^2}} \).

Each of these forms has its advantages when applied to particular cases. Each of them also gives \( d = \frac{a \dot{x}}{y \dot{y} \sqrt{x^2 + y^2}} \)

when the curvature is such as is in precise equilibrium. And, lastly, if d be constant, that is, if the vaulting be of uniform thickness, we obtain the form of the curve, because then the relation of \( \dot{x} \) to \( x \) and to \( \dot{y} \) is given.

"The chief use of this analysis is to discover what curves are improper for domes, or what portions of given curves may be employed with safety. Domes are are generally built for ornament; and we see that there is great room for indulging our fancy in the choice. All curves which are concave outwards will give domes of great firmness: they are also beautiful. The Gothic dome, whose outline is an undulated curve, may be made abundantly firm, especially if the upper part be convex and the lower concave outwards.

"The chief difficulty in the case of this analysis arises from the necessity of expressing the weight of the incumbent part, or \( \int dy \sqrt{x^2 + y^2} \). This requires the measurement of the conoidal surface, which, in most cases, can be had only by approximation by means of infinite series.

"The surface of any circular portion of a sphere is very easily had, being equal to the circle described with a radius equal to the chord of half the arch. This radius is evidently \( = \sqrt{x^2 + y^2} \).

"In order to discover what portion of a hemisphere may be employed (for it is evident we cannot employ the whole) when the thickness of the vaulting is uniform, we may recur to the equation or formula

\[ \frac{dy \cdot x}{x} \sqrt{x^2 - y^2} = \int dy \sqrt{x^2 + y^2}. \]

Let \( a \) be the radius of the hemisphere. We have \( x = \frac{a \cdot y}{\sqrt{a^2 - y^2}} \), and \( x' = \frac{a^2 y'}{\sqrt{a^2 - y^2}} \). Substituting these values in the formula, we obtain the fluent of the second member

\[ = a^3 - a^2 \sqrt{a^2 - y^2}, \quad \text{and} \quad y = a \sqrt{-\frac{x^2}{a^2} + \frac{1}{4}}. \]

Therefore if the radius of the sphere be 1, the half breadth of the dome must not exceed \( \sqrt{\frac{1}{8} \times \sqrt{\frac{1}{4}}} \), or 0.786, and the height will be .618. The arch from the vertex is about 51° 49'. Much more of the hemisphere cannot stand, even though aided by the cement, and by the friction of the couring joints. This last circumstance, by giving connection to the upper parts, causes the whole to press more vertically on the course below, and thus diminishes the outward thrust; but it at the same time diminishes the mutual abutment of the vertical joints, which is a great cause of firmness in the vaulting. A Gothic dome, of which the upper part is a portion of a sphere not exceeding 45° from the vertex, and the lower part is concave outwards, will be very strong, and not ungraceful.

"341. Persuaded that what has been said on the subject convinces the reader that a vaulting perfectly equilibrated throughout is by no means the best form, provided that the base is secured from separating, we think it unnecessary to give the investigation of that form, which has a considerable intricacy, and shall merely give its dimensions. The thickness is supposed uniform. The numbers in the first column of the table express the portion of the axis counted from the vertex, and those of the second column are the length of the ordinates.

<table> <tr> <th>AD</th><th>DB</th><th>AD</th><th>DB</th><th>AD</th><th>DB</th> </tr> <tr> <td>0.4</td><td>100</td><td>610.4</td><td>1080</td><td>2990</td><td>1560</td> </tr> <tr> <td>3.4</td><td>200</td><td>744</td><td>1140</td><td>3442</td><td>1600</td> </tr> <tr> <td>11.4</td><td>300</td><td>904</td><td>1200</td><td>3972</td><td>1640</td> </tr> <tr> <td>26.6</td><td>400</td><td>1100</td><td>1260</td><td>4432</td><td>1670</td> </tr> <tr> <td>52.4</td><td>500</td><td>1336</td><td>1320</td><td>4952</td><td>1700</td> </tr> <tr> <td>91.4</td><td>600</td><td>1522</td><td>1360</td><td>5336</td><td>1720</td> </tr> <tr> <td>146.8</td><td>700</td><td>1738</td><td>1400</td><td>5756</td><td>1740</td> </tr> <tr> <td>223.4</td><td>800</td><td>1984</td><td>1440</td><td>6214</td><td>1760</td> </tr> <tr> <td>326.6</td><td>900</td><td>2270</td><td>1480</td><td>6714</td><td>1780</td> </tr> <tr> <td>475.4</td><td>1000</td><td>2602</td><td>1520</td><td>7260</td><td>1800</td> </tr> </table>

"The curve formed according to these dimensions will not appear very graceful, because there is an abrupt change in its curvature at a small distance from its vertex; it, however, the middle be occupied by a lantern of equal or of smaller weight than the part whose place it supplies, the whole will be elegant, and free from this defect.

"The connexion of the parts arising from cement and from friction has a great effect on dome-vaulting. In the same way as in common arches and cylindrical vaulting, it enables an overload on one place to break the dome in a distant place. But the resistance to this effect is much greater in dome-vaulting, because it operates all round the overloaded part. Hence it happens that domes are much less shattered by partial violence, such as the falling of a bomb, or the like. Large holes may be broken in them without much affecting the rest; but, on the other hand, it greatly diminishes the strength which should be derived from the mutual preasure in the vertical joints. Friction prevents the sliding in of the arch-stones which produces this mutual preasure in the vertical joints, except in the very highest courses, and even there it greatly diminishes it. These causes make a great change in the form which gives the greatest strength; and as their laws of action are but very imperfectly understood as yet, it is perhaps impossible, in the present state of our knowledge, to determine this form with tolerable precision. We see plainly, however, that it allows a greater deviation from the best form than the other kind of vaulting; and domes may be made to rise perpendicular to the horizon at the base, although of no great thickness; a thing which must not be attempted in a plane arch. The immense addition of strength which may be derived from hooping largely compensates for all defects; and there is hardly any bounds to the extent to which a very thin dome-vaulting may be carried, when it is hooped or framed in the direction of the horizontal courses. The roof of the Halle du Bled at Paris is but a foot thick, and its diameter is more than 200, yet it appears to have abundant strength."

SCHOLIUM.

342. The section of the dome of St Paul's cathedral is part of an ellipse whose conjugate diameter is parallel to the horizon. It is built of wood, and confined by strong iron chains; and is supported by carpentry resting on a cone of brick work.

CHAP. IX. On the Force of Torsion.

343. Definition. Let g 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 e o, be suspended at the lower extremity of the wire; and let the axis of the cylinder, or the wire g a 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 e o may describe the arch ON, the wire g a 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 e o will therefore move backwards from N, and oscillate round the axis 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.

344. To deduce formulae for the oscillatory motion of the cylinder, on the supposition that the reaction 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. 2, 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 BC b=M; then AC b=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 \( n \times \overline{A-M} \), n being a constant coefficient, 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 AC b, and CD the distance of the point d from the axis of rotation C, we shall have, by the principles of Dynamics,

\[ n \times \overline{A-M} \times i = \int dr \dot{v}. \]

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 \( v : u = r : a \), and \( v = \frac{r u}{a} \). Now by substituting the fluxion of this value of v in the place of \( \dot{v} \) in the preceding formula, we have

\[ n \times \overline{A-M} \times i = \int \frac{dr^2}{a}; \]

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

\[ n \times \overline{A-M} \times \frac{a \dot{M}}{u} = \int \frac{dr^2}{a}, \]

whose fluent is

\[ n \times \overline{2AM-M^2} = u \int \frac{dr^2}{a^3}. \]

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

\[ \sqrt{n} \times \sqrt{2AM-M^2} = u \int \frac{dr^2}{a^3} \right)^{\frac{1}{2}}. \]

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

\[ \frac{a \dot{M}}{u} \times u \times \int \frac{dr^2}{a^3} \right)^{\frac{1}{2}} = \frac{a \dot{M} \times \frac{1}{a} \times \int \frac{dr^2}{a^3} \right)^{\frac{1}{2}}}{\sqrt{n} \times \sqrt{2AM-M^2}} \]

\[ = \frac{\dot{M} \times \int \frac{dr^2}{a^3} \right)^{\frac{1}{2}}}{\sqrt{n} \times \sqrt{2AM-M^2}}. \]

Therefore, since \( i = \frac{a \dot{M}}{u} \), we shall have

\[ i = \frac{\dot{M} \times \int \frac{dr^2}{a^3} \right)^{\frac{1}{2}}}{\sqrt{n} \times \sqrt{2AM-M^2}}, \text{ or } \\ i = \frac{\dot{M}}{\sqrt{2AM-M^2}} \times \int \frac{dr^2}{n} \right)^{\frac{1}{2}}. \]

But \( \frac{\dot{M}}{\sqrt{2AM-M^2}} \) represents an arch or angle whose radius is A and whose vered 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

\[ T = \int \frac{dr^2}{n} \right)^{\frac{1}{2}} \times 180°. \]

345. In order to compare the force of torsion with the force of gravity in a pendulum, we have for the time 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

\[ \int \frac{dr^2}{n} \right)^{\frac{1}{2}} 180° = \frac{l}{g} \times 180°. \]

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

\[ \int \frac{dr^2}{n} \right)^{\frac{1}{2}} = \frac{l}{g}. \]

We must therefore find for a cylinder the value of \( \int r^2 \), 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, l=its length, d=its density; then we shall have for the area of its base \( \frac{a^2 \pi}{2} \), which multiplied by \( \lambda \) gives the solid content of the cylinder \( = \frac{a^2 \pi \lambda}{2} \), and this multiplied by Theory. d gives \( \frac{\alpha^2 \pi \lambda d}{2} \) 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 \( \int p r^2 = \frac{\alpha^4 \pi^2 \lambda d}{4} \). But the number of particles in the cylinder, or the mass \( \mu \) of the cylinder, is \( \frac{\alpha^2 \pi \lambda d}{2} \), therefore substituting \( \mu \), instead of this value of it in the preceding equation, we have \( \int p r^2 = \frac{\mu \alpha^2}{2} \), and, dividing both sides by \( n \), we have

\[ \frac{\int p r^2}{n} = \frac{\mu \alpha^2}{2 n}, \]

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

\[ \left( \frac{\int p r^2}{n} \right)^{1/2} \times 180^\circ = \frac{\alpha^2}{2 n}^{1/2} \times 180^\circ. \]

Therefore

\[ T = \frac{\mu \alpha^2}{2 n}^{1/2} \times 180, \]

and since \( \int \frac{p r^2}{r} = \frac{l}{g} \),

\[ \frac{\mu \alpha^2}{2 n} = \frac{l}{g}, \]

and by reduction \( n = \frac{g \mu \alpha^2}{2 l} \). But \( g \mu \) is the weight W of the cylinder, therefore, by substituting W instead of \( g \mu \), we obtain \( n = \frac{P \alpha^2}{2 l} \), 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 × L = n × A—M.

346. 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

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

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

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

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.

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

\[ n \times 2 AM - M^2 + \frac{2m \times A - M'}{r+1} + \frac{2m A^{r+1}}{r+1} = u^2 \int \frac{p r^2}{a^2} \]

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

\[ n A^2 - \frac{2m A^{r+1}}{r+1} = UU \int \frac{p r^2}{a^2}, \]

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

\[ U^2 = \frac{n A^2 - \frac{2m A^{r+1}}{r+1}}{\int \frac{p r^2}{a^2}}. \]

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

\[ \frac{n M'^2}{2} + \frac{m M'^{r+1}}{r+1} = \frac{U^2 - u^2}{2} \times \int \frac{p r^2}{a^2}. \]

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 unified,

\[ A - M' = \frac{2m}{n \times r+2} \times \frac{A^{r+1} + M'^{r+1}}{A + M'}, \]

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 \( A - M' \)

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

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.

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

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

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

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

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

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

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}{r+1} + \frac{2m'A'}{r'+1}.

349. 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'}{r+1} + \frac{2m'A'}{r'+1} + \frac{2m''A''}{r''+1} + \frac{2m'''A'''}{r'''+1}, &c.

350. Having thus given after Coulomb, the mode of deducing formulae 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. 2. 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 formulae, 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 formulae, according to what laws this elastic force of torsion was changed.

351. From a great number of experiments it appeared, that when the angle of torsion was not very great, the oscillations were sensibly isochronous; and therefore it may be regarded as a fundamental law, That for all metallic wires, when the angles of torsion are not very great, the force of torsion is sensibly proportional to the angle of torsion. Hence, as the preceding formulae are founded on this supposition, they may be safely applied to the experiments.

352. In all the experiments, a cylinder of two pounds weight oscillated in twice the time employed by a cylinder which weighed only half a pound; and therefore the duration of the oscillations is as the square root of the weights of the oscillating cylinders. Consequently the tension of the wires has no sensible influence upon the force of torsion. If the tensions however be very great relative to the strength of the metal, the force of torsion does suffer a change; for when the weight of the cylinder, and consequently the tension of the wire, is increased, the wire is lengthened, and as this diminishes the diameter of the wire, the duration of the oscillation must evidently be affected.

353. When the lengths of the wires are varied without changing their diameters or the weights of the cylinders, the times of the same number of oscillations are as the square roots of the lengths of the wires, a result also deducible from theory.

354. 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 varied 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 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.

355. 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

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

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

356. 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.—When the angle of torsion was below 45°, the decrements of the amplitudes of the oscillations were nearly proportional to the amplitudes of the angle of torsion; but when the angle exceeded 45°, the decrements increased in a much greater ratio.—The centre of torsion did not begin to advance or be displaced till the angle of torsion was nearly a femicircle: its displacement was very irregular till the angle was one circle and 10 degrees, but beyond this angle the torsion remained nearly the same for all angles.

357. 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. PART II. ON THE CONSTRUCTION OF MACHINERY.

358. WE have already seen, 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 this 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.

359. 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 rests 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 92, 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 φ 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.

360. 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 = 10, and m = 1.

<table> <tr> <th>Proportional value of the impelling power, or P</th> <th>Value of the velocities of the working point or y; or of the lever by which the resistance acts, that of x being 1.</th> <th>Proportional value of the impelling power, or P</th> <th>Value of the velocities of the working point or y; or of the lever by which the resistance acts, that of x being 1.</th> </tr> <tr><td>1</td><td>0.48829</td><td>20</td><td>0.732051</td></tr> <tr><td>2</td><td>0.95445</td><td>21</td><td>0.760682</td></tr> <tr><td>3</td><td>1.40175</td><td>22</td><td>0.788854</td></tr> <tr><td>4</td><td>1.83216</td><td>23</td><td>0.816590</td></tr> <tr><td>5</td><td>2.24745</td><td>24</td><td>0.843900</td></tr> <tr><td>6</td><td>2.64911</td><td>25</td><td>0.870800</td></tr> <tr><td>7</td><td>3.03841</td><td>26</td><td>0.897300</td></tr> <tr><td>8</td><td>3.41641</td><td>27</td><td>0.923500</td></tr> <tr><td>9</td><td>3.78405</td><td>28</td><td>0.949400</td></tr> <tr><td>10</td><td>4.14214</td><td>29</td><td>0.974800</td></tr> <tr><td>11</td><td>4.49138</td><td>30</td><td>1.000000</td></tr> <tr><td>12</td><td>4.83240</td><td>40</td><td>1.236200</td></tr> <tr><td>13</td><td>5.16575</td><td>50</td><td>1.449500</td></tr> <tr><td>14</td><td>5.49193</td><td>60</td><td>1.645700</td></tr> <tr><td>15</td><td>5.81139</td><td>70</td><td>1.828400</td></tr> <tr><td>16</td><td>6.12451</td><td>80</td><td>2.000000</td></tr> <tr><td>17</td><td>6.43168</td><td>90</td><td>2.162300</td></tr> <tr><td>18</td><td>6.73320</td><td>100</td><td>2.310000</td></tr> <tr><td>19</td><td>7.02938</td><td></td><td></td></tr> </table>

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 this work; that is, the diameter of the axle, that of the wheel being 6. Here the power is evidently 3 cubic feet, while the resistance is only one cubic foot, therefore P = 3R; but in the preceding table Practical R=10, consequently P=3×10=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.

361. Instead of using the preceding table, we might find the best proportion between x and y by a kind of tentative process, from the formula \( \frac{P \times R y - R^2 y^2}{P x^2 + R y^2} \), which expresses the work performed. This method is indeed tedious; and we mention it only for the sake of showing the conformity of the results, and of proving that there is a certain proportion between x and y which gives a maximum effect. Let x=6, as in the preceding paragraph, and let us suppose y to be successively 5, 6, and 7, in order to see which of these values is the best. Since P=3, R=1, and x=6, we have

When \( y=5 \) \( \frac{P \times R y - R^2 y^2}{P x^2 + R y^2} = \frac{3 \times 6 \times 1 \times 5 - 1 \times 5 \times 5}{3 \times 6 \times 6 + 1 \times 5 \times 5} = \frac{65}{133} = 0.488 \)

When \( y=6 \) \( \frac{P \times R y - R^2 y^2}{P x^2 + R y^2} = \frac{3 \times 6 \times 1 \times 6 - 1 \times 6 \times 6}{3 \times 6 \times 6 + 1 \times 6 \times 6} = \frac{72}{144} = 0.500 \)

When \( y=7 \) \( \frac{P \times R y - R^2 y^2}{P x^2 + R y^2} = \frac{3 \times 6 \times 1 \times 7 - 1 \times 7 \times 7}{3 \times 6 \times 6 + 1 \times 7 \times 7} = \frac{77}{157} = 0.49045 \)

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.

To find the best proportion between the power and the resistance.

362. 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. 2. Chap. 7. 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.

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.

<table> <tr> <th>Values of y, or the velocity of the working point; x being equal to 1.</th> <th>Values of R, or the resistance to be overcome, P being = 1.</th> <th>Ratio of R to the resistance which would balance P.</th> <th>Values of y, or the velocity of the working point; x being equal to 1.</th> <th>Values of R, or the resistance to be overcome, P being = 1.</th> <th>Ratio of R to the resistance which would balance P.</th> </tr> <tr> <td>\( \frac{1}{2} \)</td> <td>1.8885</td> <td>0.4724 to 1</td> <td>7</td> <td>0.03731</td> <td>0.26117 to 1</td> </tr> <tr> <td>\( \frac{1}{3} \)</td> <td>1.3928</td> <td>0.4639</td> <td>8</td> <td>0.03125</td> <td>0.25000</td> </tr> <tr> <td>\( \frac{1}{4} \)</td> <td>0.8086</td> <td>0.4493</td> <td>9</td> <td>0.02669</td> <td>0.24021</td> </tr> <tr> <td>1</td> <td>0.4142</td> <td>0.4142</td> <td>10</td> <td>0.02317</td> <td>0.23170</td> </tr> <tr> <td>2</td> <td>0.1830</td> <td>0.3660</td> <td>11</td> <td>0.02037</td> <td>0.22407</td> </tr> <tr> <td>3</td> <td>0.1111</td> <td>0.3333</td> <td>12</td> <td>0.01809</td> <td>0.21708</td> </tr> <tr> <td>4</td> <td>0.0772</td> <td>0.3088</td> <td>13</td> <td>0.01622</td> <td>0.21086</td> </tr> <tr> <td>5</td> <td>0.0580</td> <td>0.2900</td> <td>14</td> <td>0.01466</td> <td>0.20524</td> </tr> <tr> <td>6</td> <td>0.0457</td> <td>0.2742</td> <td>15</td> <td>0.01333</td> <td>0.19995</td> </tr> </table>

364. 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.

365. The same result might be obtained in a more circuitous method by means of the formula \( \frac{P \times R y - R^2 y^2}{P x^2 + R y^2} \), 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 \( \frac{10 \times 4.142 - 4.142^2}{10 + 4.142} = \frac{24.26384}{14.142} = 1.7157 \).

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.

366. 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.

367. 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 judgement 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 fix 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.

368. The advantages of simplifying machinery are well exemplified in the following capstan, which unites great strength and simplicity. It is represented in fig. 4. 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 other 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, 62 inches of rope, equal to the circumference of the cylinder, will be gathered upon the cylinder D, and 60 inches will be unwound 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 of this quantity, or 1.5 inches, 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 Practical inches; and the space described by the pulley, or load to be overcome, would have been 35 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 capstanes, the weights which they will raise will be as 1 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 proportionably 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 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. (Appendix to Ferguson's Lectures, vol. ii.).

369. The principle on which the preceding capstan is constructed, might be applied with great advantage when two separate axles AC, BD are driven by means of the winch H 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 W will be stationary, as the rope wended about one axle will be always equal to what is unwound from the other. If the wheels have different diameters, or different numbers of teeth, the quantity of rope wound round the one axle will exceed what is unwound from the other, and the weight will be raised.

CHAP. III. On the Nature of Friction and the Method of diminishing its effects in Machinery; and on the Rigidity of Ropes.

370. 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.

371. The subject of friction has been examined at great length by Amontons, Bullinger, Parent, Euler, Mechanics, and Bouffet, and has lately occupied the attention of our ingenious countryman Mr Vince of Cambridge.

He found that the friction of hard bodies in motion is an uniformly retarding force, and that the Vince's experiment of friction considered as equivalent to a weight drawing the body backwards is equal to \( \frac{M + \overline{W} \times S}{g t^2} \),

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 incalculates 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 has different surfaces applied to the plane on which it moves, but that the smallest surfaces will have the least friction.

372. Notwithstanding the attempts of preceding philosophers to unfold the nature of friction, it was referred 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 large 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 the interesting results to which they led.

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 shewn that there are exceptions to this rule. He found, for example, that the friction of oak upon oak was equal to \( \frac{1}{2.34} \) of the force of friction; the friction of pine against pine \( \frac{1}{1.78} \), and that of oak against pine \( \frac{1}{1.5} \). The friction of oak against copper was \( \frac{1}{5.5} \), 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 shewn that the friction is in this case sometimes the smallest. When the bodies moved in the direction of their fibres, the friction was \( \frac{1}{2.34} \) of the force with which they were pressed together; but when the motion was contrary to the course of the fibres, the friction was only \( \frac{1}{3.76} \).

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 fix days; and it is very remarkable, that when wooden surfaces were anointed with tallow, the time requisite for producing the greatest quantity of friction is increased. The increase of friction which is generated by prolonging the time of contact is so great, that a body weighing 1630 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 fix 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 greave as an unguent, and was prolonged to five or fix days by beaferneing 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

\[ \frac{1}{2.46}, \frac{1}{2.16}, \]

of the force of pression; and when a less surface and other weights were used, the friction was

\[ \frac{1}{2.21}, \frac{1}{2.36}, \frac{1}{2.42}, \]

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 \( \frac{1}{1.78} \) of the force of pression; on the contrary, when iron moves upon brafs, the least friction is produced, and it amounts to \( \frac{1}{4} \) of the force of pression.

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 \( \frac{1}{2.28} \) at a medium; but when a small surface was used, the friction instead of being greater, as might have been expected, was only \( \frac{1}{2.39} \).

6. Friction for the most part is not augmented by an increase of velocity. In some cases, it is diminished by an augmentation of celerity.—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 preliefed, 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 pression. 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 beaferneed 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 \( \frac{1}{56} \) of the force of pression. In cylinders of elm, the friction was greater by \( \frac{2}{3} \), and was scarcely diminished by the interposition of tallow.

373. From a variety of experiments on the friction of the axes of pulleys, Coulomb obtained the following results.—When an iron axle moved in a brafs bush the friction was \( \frac{1}{5} \) of the pression; but when the bush was beaferneed with very clean tallow, the friction was only \( \frac{1}{17} \); when swine's greave was interposed, the friction amounted to \( \frac{1}{8.5} \); and when olive oil was employed as an unguent, the friction was never less than \( \frac{1}{7.5} \). When the axis was of green oak, and the bush of guaiacum wood, the friction was \( \frac{1}{56} \) when tallow was interposed; but when the tallow was removed, so that a small quantity only covered the surface, the friction was increased to \( \frac{1}{17} \). When the bush was made of elm, the friction was in similar circumstances \( \frac{1}{17} \) and \( \frac{1}{56} \), which is the least of all. If the axle be made of box, and the bush of guaiacum wood, the friction will be \( \frac{1}{7} \) and \( \frac{1}{17} \), circumstances being the same as before. If the axle be of boxwood, and the bush of elm, the friction will be \( \frac{1}{5} \) and \( \frac{1}{15} \); and if the axle be of iron and the bush of elm, the friction will be \( \frac{1}{10} \) of the force of pression.

374. 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 lockets or bushes which sustain the gudgeons of large wheels, and the axles of wheel carriages. Cafatus seems to have been the first who recommended this apparatus. It was afterwards mentioned by Sturmius and Wolfus; 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 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 friction which arises from the friction of the gudgeon, &c. will be extremely trifling.

375. 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 brafs, 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 overholt 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.

376. When the moving force is not exerted in a perpendicular direction, but obliquely as in underholt 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 axes.

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.

377. Mr Gottlieb, 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 five 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-carriges are at rest, Mr Gottlieb 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.

378. Mr Gamott 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 Fig. 7. apparatus is represented in fig. 7, where ABCD is the metallic ring inserted in the nave of the wheel. 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 brafs 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.

379. As it appears from the experiments of Coulomb, that the least friction is generated when polished iron moves upon brafs, 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 brafs.

380. When every mechanical contrivance has been adopted for diminishing 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 wine's greese 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 wine's greese. 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 greased 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.

381. 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." Appendix to Ferguon's Lectures, vol. ii.

382. 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 stiffness 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 subsequent philosophers, but particularly by M. Coulomb, who has investigated the sub-

* Mem. Acad. 1699, p. 217. ject 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 twitted.

2. The rigidity of ropes increases in the duplicate ratio of their diameters. According to Amontons and Delaguliers, 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 Delaguliers 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. 9.)

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 \( \frac{a D^n}{r} \) is a constant quantity independent of the tension, \( a \) being a constant quantity determined by experiment, \( D^n \) 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 \( T \times \frac{b D^n}{r} \), where \( T \) is the tension of the rope, \( b \) a constant quantity, and \( D^n \) and \( r \) the same as before. Hence the complete formula is \( \frac{a D^n}{r} + T \times \frac{b D^n}{r} = \frac{D^n}{r} \times a + T b \).

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.

CHAP. IV. On the Nature and Advantages of Fly Wheels.

383. A FLY, in mechanics, is a heavy wheel or cylinder which moves rapidly upon its axis, and is applied to machines for the purpose of rendering uniform a defultory 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 in the heat of 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 draw it. But if a fly, consisting either of cross bars, or a maffy 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 renew 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.

384. We have already observed that a defultory 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 defultory 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.

385. 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 is 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.

386. 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 shewing the propriety of their application in particular cases.

387. In the description of Vaulone's pile engine *, the reader will observe a striking instance of the utility of fly wheels. The ram Q is raised between the guides bb by means of horses acting against the levers S, S; but as soon as the ram is elevated to the top of the guides, and discharged from the follower G, the resistance against which the horses have been exerting their force is suddenly removed, and they would instantaneously tumble down, were it not for the fly O. This fly is connected with the drum B by means of the trundle X, and as it is moving with a very great force, it opposes a sufficient resistance to the action of the horses, till the ram is again taken up by the follower.

388. 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 maffy fly wheel of a great diameter, revolving with rapidity, till the moving power again resumes its energy.

389. If the moving power is 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 pulled 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.

390. 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 20 pounds applied for the space of 37 seconds to the circumference of a cylinder 20 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 10 seconds, the same effect would be produced if the cylinder was driven by a man who constantly exerted a force of 20 pounds at a winch one foot long (n).

391. 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

* See Part III. Plate CCXXXIX fig. r. See his Treatise on Rectilineal and Rotatory Motion.

Practical extremity of a piece of wood two feet long, and different mechanics charge 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.

392. 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 in the end 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.

393. 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 positions depend 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.

394. 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. 8. 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 sheave 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 frames CD and QR prevent 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 recess, the points C, 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.

CHAP. V. On the Teeth of Wheels, and the Wipers of Stampers.

395. 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 shew, 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 shew, 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 infinite Fig. 9. ly 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 nb 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 a = mn and a = 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.

396. In illustrating the application of this property of the epicycloid, which was discovered by Olaus 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.

397. 1. The first of these modes of action is represented first made in fig. 1, where B is the centre of the wheel (p), 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

(p) In figs. 1, 2, 3, 4, the letter B is supposed to be placed at the centre of the wheels. 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 \( a d h \); and the faces \( a b \) of the teeth of the wheel should be portions of an exterior epicycloid formed by the same generating circle rolling upon the convex superficies \( o d p \) of the wheel.

398. 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 \( a d h \) 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 \( o d p \) of the wheel. This rectilineal form of the teeth is exhibited in fig. 2, and is perhaps the most advantageous, as it requires less trouble, and may be executed with greater accuracy, than if the epicycloidal form had been employed, though the teeth are evidently weaker than those in fig. 1; it is recommended both by De la Hire and Camus as particularly advantageous in clock and watch work.

Fig. 2.

Fig. 1.

399. The attentive reader will perceive from fig. 1, 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 \( A B \); 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 \( A B \) 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 a matter therefore of great importance to determine with accuracy the relative radii of the wheel and pinion.

400. For this purpose, let \( A \), fig. 2, be the pinion having the acting faces of its leaves straight lines tending to the centre, and \( B \) the centre of the wheel. \( A B \) will be the distance of their centres. Then as the tooth \( C \) supposed to act upon the leaf \( A m \) till it arrives at the line \( A B \), it ought not to quit \( A m \) till the following tooth \( F \) has reached the line \( A B \). But since the tooth always acts in the direction of a line drawn perpendicular to the face of the leaf \( A m \) from the point of contact, the line \( C H \), drawn at right angles to the face of the leaf \( A m \), will determine the extremity of the tooth \( C D \), or the last part of it which should act upon the leaf \( A m \), and will also mark out \( C D \) for the depth of the tooth. Now, in order to find \( A H, H B \), and \( C D \), 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. r8.) \( a + b : b - c : y \) (c being equal to \( x + y \)), and consequently the radius of the pinion, viz. \( y = \frac{c b}{a + b} \); then by inverting the first analogy, we have \( b : a = y : x \), and consequently the radius of the wheel, viz. \( x = \frac{a y}{b} \); \( y \) being now a known number.

Now, in the triangle \( A H C \), right-angled at \( C \), the side \( A H \) is known, and likewise all the angles (HAC being equal to \( \frac{360}{b} \)); the side \( A C \), therefore, may be found by plain trigonometry. Then, in the triangle \( A C B \), the \( \angle CAB \), equal to HAC, is known, and also the sides \( A B, AC \), 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.

401. 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.

402. 2. The second mode of action is not so advantageous as that which we have been considering, and should, mode of if possible, always be avoided. It is represented in action. fig. 3, where \( A \) is the centre of the pinion, \( B \) that of the wheel, and \( A B \) the line of 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 \( RA \); 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 \( C, F \), 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 \( B o \) of the circle \( o p \), rolling upon the circumference \( m n \) 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 rating circle rolling within the concave superficies of the circle o p, 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 m n of the pinion.

493. 3. The third mode of action, which is represented in fig. 4, is a combination of the two first modes, and consequently partakes of the advantages and disadvantages of each. It is evident from the figure that the portion e b of the tooth acts upon the part b c of the leaf till they reach the line of centres A B, and that the part e d of the tooth acts upon the portion b a of the leaf after they have passed this line. Hence the acting parts e h and b c must be formed according to the directions given for the first mode of action, and the remaining parts e d, b a, must have that curvature which the second mode of action requires; consequently e h should be part of an interior epicycloid formed by any generating circle rolling on the concave circumference m n of the wheel, and the corresponding part b c of the leaf should be part of an exterior epicycloid formed by the same generating circle rolling upon b E O, the convex circumference of the pinion: the remaining part e d of the tooth should be a portion of an exterior epicycloid, engendered by any generating circle rolling upon e L, the concave superficies of the wheel: and the corresponding part b a of the leaf should be part of an interior epicycloid described by the same generating circle, rolling along the concave side b E O 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 B C, for describing the interior epicycloid e h and the exterior one b c, and a generating circle, whose diameter is equal to AC, the radius of the pinion, for describing the interior epicycloid b a, and the exterior one e d. In this case the two interior epicycloids e h, b a, will be straight lines tending to the centres B and A, and the labour of the mechanic will by this means be greatly abridged.

494. 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 \( \frac{360}{b} \), and LD, or \( \angle LBD \), will be equal to \( \frac{360}{a} \). 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 \( \frac{360}{a} \), therefore by substitution \( LC = \frac{360 \times n}{a \times m} \).

Now, in the triangle APB, AB is known, and also Practical PB, which is the cofine of the angle ABD, PC being perpendicular to DB; AP or the radius of the pinion therefore may be found by plane 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 (r); 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 shewn to be \( \frac{360}{b} \), therefore, by substitution, \( CS = \frac{360 \times n}{b \times m} \). Now the arch ES, or \( \angle EAS \), being equal to \( \frac{360}{b} \), and CS, or \( \angle CAS \), being equal to \( \frac{360 \times n}{b \times m} \), their difference EC, or the angle EAC, will be equal to \( \frac{360}{b} - \frac{360 \times n}{b \times m} \), or \( \frac{360^{\circ} \times m - n}{b \times m} \). 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 cofine 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 shewn that AC, the radius of what is called the primitive pinion, was equal to \( \frac{c \times b}{a + b} \), 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.

495. Another form for the teeth of wheels, different from any which we have mentioned, has been recommended by Dr Robison. He shews 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. 1. let AB be a portion of the wheel on

(E) The letter L marks the intersection of the line BL with the arch e m, and the letter E the intersection of the arch b O with the upper surface of the leaf m. The letters D and S correspond with L and E respectively, and P with I. which the tooth is to be fixed, and let A p a 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 a b c d e h, by unlapping the thread gradually from the circumference A p m. 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 \( \frac{1}{60} \)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.

406. 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. 6, 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 brads, or copper, or tinplate, a b, 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 C, so that the nail a may rise gradually, and the point of contact between the two circular segments may advance towards m; the curve a b described upon the brads 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 brads on the left hand of the epicycloid, and the remaining concave arch or gage a b 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. 7, where CD is the generating circle, F the concave circular base, MN the piece of wood on which this base is fixed, and c d the interior epicycloid formed upon the plate of brads, 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 of the preceding leaf, the height, breadth, and distance of teeth, the teeth must be properly proportioned. For this purpose the pitch-line or circumference of the wheel, which is represented in fig. 2. and 3. 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 \( \frac{3}{7} \) for the diameter of the cylindrical flave 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 in Art. 400, 404. 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.

407. The principle of bevelled wheels was pointed out bevelled by De la Hire, so long ago as the end of the 17th century. It consists in one fluted or toothed cone acting upon another, as is represented in fig. 8. where the cone OD Fig. 8. drives the cone OC, conveying its motion in the direction OC. If these cones be cut parallel to their bases 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. 9. where CD and DE Fig. 9. are but very small portions of the imaginary cones ACD and ADE.

408. 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. 10. be the axis Fig. 10. 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 c d parallel to CD at any convenient distance, and draw a b parallel to AB at four times that distance, then the lines i m and i n drawn perpendicular to AB and CD respectively, will mark the situation and size of the wheels required. In this case the cones are O n i and O m i, and s r n i, r p m i, 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 wheels, indeed, must be formed by the same rules which 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. 10, the part of the tooth at r must be more incurvated than the part at i, as is evident from the inspection of fig. 9.; and the epicycloid for the part i must be formed by means of circles whose diameters are i m and r f, while the epicycloid for the part r must be generated by circles whose diameters are C n and D d.

409. Let us suppose a plane to pass through the points O, A, D; the lines A B, A O, will evidently be in this plane, which may be called the plane of centres. Now, when the teeth of the wheel D E, which is supposed to drive C D the smallest of the two, commence their action on the teeth of C D, when they arrive at the plane of centres, and continue their action after they have passed this plane, the curve given to the teeth of C D at C, should be a portion of an interior epicycloid formed by any generating circle rolling on the concave superficies of a circle whose diameter is twice C n perpendicular to C A, and the curvature of the teeth at i should be part of a similar epicycloid, formed upon a circle, whose diameter is twice i m. The curvature of the teeth of the wheel D E 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 D d perpendicular to D A; and the epicycloid for the teeth at F is formed in the same way, only instead of twice D d, the diameter of the circle must be twice E f. 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. 8. When the small wheel C D impels the large one D E, the epicycloids which were formerly given to C D must be given to D E, and those which were given to D E must be transferred to C D.

410. The wheel represented in fig. 11. is sometimes called a crown wheel, though it is evident from the figure that it belongs to that species of wheels which we just been considering; for the acting surfaces of the teeth both of the wheel M B and of the pinion E D G are directed to C the common vertex of the two cones C M B, C E G. In this case the rules for bevelled wheels must be adopted, in which A S is to be considered as the radius of the wheel for the profile of the tooth at A, and M N 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 M P, at right angles to M C the surfaces of the cone. When the vertex C of the cone M C G approaches to N till it be in the fame plane 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 C E G 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.

411. In fig. 12. let A B 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 A C. 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 p q; but as p q 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 o p, therefore, will be part of a cycloid formed by any generating circle, and m n, the acting face of the teeth of the wheel, must be an exterior epicycloid produced by the same generating circle rolling on m r the convex surface of the wheel. If it is required to make o p a straight line, as in the figure, then m n must be an involute of the circle m r formed in the manner represented in fig. 5.

412. Fig. 12. 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 D E the wheel; e h therefore must be part of an interior epicycloid formed by any generating circle rolling on the concave side e x of the wheel, and b c must be an exterior epicycloid produced by the same generating circle rolling upon the circumference of the rack. The remaining part c d of the teeth of the wheel must be an exterior epicycloid described by any generating circle moving upon the convex side e x, and b a 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 b c and the interior one b a 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 D x the radius of the wheel, for describing the interior epicycloid e h, and the exterior one b c; and a generating circle whose diameter is equal to the radius of the rack, for describing the interior epicycloid a b, and the exterior one d e; a b and e h, therefore, will be straight lines, and b c will be a cycloid, and d e an involute of the circle e x, the radius of the rack being infinitely great.

413. In the same manner may the form of the teeth of rack-work 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 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 pledge hammer in a forge, &c.

414. 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, bc, 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.

415. 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. 14, 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, s, u. Take Bm, and set it from q to r: Take Bn and set it from s to t: Take Bo and set it from u to v, and lastly set Bp from M to E. Through the points r, 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 B, N, O, P, Q, 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 Dc; the extremity of the lever, therefore, will be found in the point c, having moved through Bc. 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, cb, ba, ac, (F) in the same time that the power has moved through the equal spaces qB, s q, u s, M u. The lever, therefore, has been raised uniformly, the ratio between the velocity of the power, and that of the weight, remaining always the fame.

416. If the wheel D turn in a contrary direction, according to the letters MHIB, we must divide the semicircle BH EM, into as many equal parts as the chord cB, viz. in the points e, g, h. Then, having set the arch Bm from e to d, the arch Bn from g to f, and the rest in a similar manner, draw through the points d, f, h, E, the indefinite lines DR, DS, DT, DQ: make DR equal to Dc; 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 c, 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 shewn, that the lever will rise through the equal heights cb, 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 Bc is vertical, and passes, when prolonged, through D, the centre

(f) The arches Bc, cb, &c. are not equal; but the perpendiculars let fall from the points c, a, b, &c. upon the horizontal lines, passing through ab, &c. are equal, being proportional to the equal lines c1, 1, 2. Eucl. VI. 2. centre of the circle (c). 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 4, 6, or 8 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, \( \frac{1}{3} \), \( \frac{2}{3} \), or \( \frac{5}{7} \) 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.

417. 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 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 Ee F. 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 r, 2; and through the points r, 2, draw r a 2 b, parallel to the horizontal line AB, cutting the arch CB in the points a, b, through which draw a A, b A. Upon D as a centre, with the distance DE, describe the arch E i e F, and upon A as a centre, with the distance Practical Mechanics, A D, describe the arch x O D, cutting the arch E i e F in the point e. Divide the arches E i e, and F r e, each into the same number of equal parts as the perpendicular c H, in the points k, i, s, m, and through these points about the centre A, describe the arches k g s, g r, m u. Take x x and set it from k to l, and take x f, and set it from i to h. Take r q also, and let it from r to t, and set n m from o to p, and d c from e to O. Then through the points E, I, h, O, and O, t, p, F, draw the two curves E / h O, and O t p F, 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 E k, or rather E l, the point l will be in k, and the point x in x, because x x is equal to k l, and the lever will have the position A b. 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 A a. 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.

CHAP. VI. On the First Movers of Machinery.

418. The powers which are generally employed as the first movers of machines are water, wind, steam, and animal exertion. The mode of employing water as an impelling power has already been given at great length in the article HYDRODYNAMICS. The application of wind to turn machinery will be discussed in the chapter on Windmills; and what regards steam will be more properly introduced into the article STEAM-Engine. At present, therefore, we shall only make a few general remarks on the strength of men and horses; and conclude with a general view of the relative powers of the first movers of machinery. The following table contains the weight which a man is able to raise through a certain height in a certain time, according to different authors.

<table> <tr> <th>Number of pounds raised.</th> <th>Height to which the weight is raised.</th> <th>Time in which it is raised.</th> <th>Duration of the Work.</th> <th>Names of the authors.</th> </tr> <tr> <td>1000</td> <td>180</td> <td>60 minutes</td> <td></td> <td>Euler</td> </tr> <tr> <td>60</td> <td>1</td> <td>1 second</td> <td>8 hours</td> <td>Bernouilli</td> </tr> <tr> <td>25</td> <td>220</td> <td>145 seconds</td> <td></td> <td>Amontons</td> </tr> <tr> <td>170</td> <td>1</td> <td>1 second</td> <td>half an hour</td> <td>Coulomb</td> </tr> <tr> <td>1000</td> <td>330</td> <td>60 minutes</td> <td></td> <td>Defaguliers</td> </tr> <tr> <td>1000</td> <td>225</td> <td>60 minutes</td> <td></td> <td>Smcaton</td> </tr> <tr> <td>30</td> <td>3½</td> <td>1 second</td> <td>10 hours</td> <td>Emerson</td> </tr> <tr> <td>29 or 30</td> <td>2.45 feet</td> <td>1 second</td> <td></td> <td>Schulze</td> </tr> </table>

(6) In the figure we have taken the point B in a disadvantageous position, because the intersections are in this case more distinct. 419. According to Amontons, a man weighing 133 pounds French, ascended 62 feet French by steps in 34 seconds, but was completely exhausted. The same author informs us that a fawer made 200 strokes of 18 inches French each, with a force of 25 pounds, in 145 seconds; but that he could not have continued the exertion above three minutes.

420. It appears from the observations of Defaguliers, that an ordinary man can, for the space of ten hours, turn a winch with a force of 30 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 70 pounds more easily than one man can raise 30. 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. Defaguliers also found, that a man may exert a force of 80 pounds with a fly when the motion is pretty quick, and that by means of a good common pump, he may raise a hoghead of water 10 feet high in a minute, and continue the exertion during a whole day.

421. A variety of interesting experiments upon the force of men were made by the learned M. Coulomb. He found that the quantity of action of a man who ascended stairs with nothing but his own weight, was double that of a man loaded with 223 pounds avoirdupois, both of them continuing the exertion for a day. In this case the total or absolute effect of the unloaded man is the greatest possible; but the useful effect which he produces is nothing. In the same way, if he were loaded to such a degree that he was almost incapable of moving, the useful effect would be nothing. Hence there is a certain load with which the man will produce the greatest useful effect. This load M. Coulomb found to be 173.8 pounds avoirdupois, upon the supposition that the man is to ascend stairs, and continue the exertion during a whole day. When thus loaded, the quantity of action exerted by the labourer is equivalent to 183.66 pounds avoirdupois raised through 3282 feet. This method of working is however attended with a loss of three-fourths of the total action of the workman.—It appears also from Coulomb's experiments, that a man going up stairs for a day raises 205 chillogrammes (a chillogramme is equal to three ounces five drams avoirdupois) to the height of a chilometre (a chilometre is equal to 30571 English inches); that a man carrying wood up stairs raises, together with his own weight, 109 chillogrammes to one chilometre,—that a man weighing 150 pounds French, can ascend by flairs three feet French in a second, for the space of 15 or 20 seconds;—that a man cultivating the ground performs \( \frac{1}{10} \) as much labour as a man ascending stairs, and that his quantity of action is equal to 328 pounds avoirdupois raised through the space of 3282 feet;—that a man with a winch does \( \frac{6}{7} \) as much as by ascending stairs;—and that in a pile-engine, a man by means of a rope drawn horizontally, raised for the space of five hours 55\( \frac{1}{2} \) pounds French through one foot French in a second.—When men walk on a horizontal road, Coulomb found that the quantity of action was a maximum when they were loaded, and that this maximum quantity of action is to that which is exerted by a man loaded with 192.5 pounds avoirdupois as 7 to 4.—The weight which a man ought to carry in order that the useful effect may be a maximum, is 165.3 pounds avoirdupois. When the workman, however, returns unloaded for a new burden, he must carry 200.7 pounds avoirdupois.

422. According to Dr Robison a feeble old man raised seven cubic feet of water=437.5 pounds avoirdupois, 11\( \frac{1}{2} \) 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 30 pounds, raised 9\( \frac{1}{2} \) cubic feet of water =578.1 pounds avoirdupois, 11\( \frac{1}{2} \) feet high, for 10 hours a day, without being fatigued.

423. 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, 4995.

424. According to Defaguliers and Smeaton, the power of one horse is equal to the power of five men. Several French authors suppose a horse equal to seven horses, while M. Schulze considers one horse as equivalent to 14 men.—Two horses, according to the experiment of Amontons, exerted a force of 150 pounds French, when yoked in a plough. According to Defaguliers, a horse is capable of drawing, with a force of 200 pounds, two miles and a half an hour, and of continuing this action eight hours in the day. When the force is 240 pounds he can work only fix hours. It appears from Smeaton's reports, that by means of pumps a horse can raise 250 hogheads of water, 10 feet high, in an hour.—The most disadvantageous way of employing the power of a horse is to make him carry a load up an inclined plane, for it was observed by De la Hire, that three men, with 100 pounds each, will go faster up the inclined plane than a horse with 300 pounds. When the horse walks on a good road, and is loaded with about two hundred weight, he may easily travel 25 miles in the space of seven or eight hours.

425. 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 eight inches in diameter, and an improved engine with a cylinder 6.12 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 falls in their common position with a radius of nine feet and a half,—that Dutch falls in their best position with a radius of eight feet, and that his enlarged falls 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 four first columns of which are taken from Mr Fenwick's Essays on Practical Mechanics.

TABLE shewing the relative strength of Overshot Wheels, Steam Engines, Horses, Men, and Wind-mills of different kinds.

<table> <tr> <th>Number of ale gallons delivered on an overshot wheel, 10 feet in diameter, every minute.</th> <th>Diameter of the cylinder in the common steam-engine, in inches.</th> <th>Diameter of the cylinder of the improved steam-engine, in inches.</th> <th>Number of horses working 12 hours per day, and moving at the rate of two miles per hour.</th> <th>Number of men working 12 hours a-day.</th> <th>Radius of Dutch falls in their common position in feet.</th> <th>Radius of Dutch falls in their best position, in feet.</th> <th>Radius of Mr Smeaton's enlarged falls, in feet.</th> <th>Height to which these different powers will raise 1000 pounds avar-dupois in a minute.</th> </tr> <tr><td>230</td><td>8.</td><td>6.12</td><td>1</td><td>5</td><td>21.24</td><td>17.89</td><td>15.65</td><td>13</td></tr> <tr><td>399</td><td>9.5</td><td>7.8</td><td>2</td><td>10</td><td>30.04</td><td>25.30</td><td>22.13</td><td>26</td></tr> <tr><td>528</td><td>10.5</td><td>8.2</td><td>3</td><td>15</td><td>36.80</td><td>30.98</td><td>27.11</td><td>39</td></tr> <tr><td>660</td><td>11.5</td><td>8.8</td><td>4</td><td>20</td><td>42.48</td><td>35.78</td><td>31.30</td><td>52</td></tr> <tr><td>790</td><td>12.5</td><td>9.35</td><td>5</td><td>25</td><td>47.50</td><td>40.00</td><td>35.00</td><td>65</td></tr> <tr><td>970</td><td>14.</td><td>10.55</td><td>6</td><td>30</td><td>52.03</td><td>43.82</td><td>38.34</td><td>78</td></tr> <tr><td>1170</td><td>15.4</td><td>11.75</td><td>7</td><td>35</td><td>56.90</td><td>47.33</td><td>41.41</td><td>90</td></tr> <tr><td>1350</td><td>16.8</td><td>12.8</td><td>8</td><td>40</td><td>60.09</td><td>50.60</td><td>44.27</td><td>104</td></tr> <tr><td>1445</td><td>17.3</td><td>13.6</td><td>9</td><td>45</td><td>63.73</td><td>53.66</td><td>46.96</td><td>117</td></tr> <tr><td>1584</td><td>18.5</td><td>14.2</td><td>10</td><td>50</td><td>67.17</td><td>56.57</td><td>49.50</td><td>130</td></tr> <tr><td>1740</td><td>19.4</td><td>14.8</td><td>11</td><td>55</td><td>70.46</td><td>59.33</td><td>51.91</td><td>143</td></tr> <tr><td>1900</td><td>20.2</td><td>15.2</td><td>12</td><td>60</td><td>73.59</td><td>61.97</td><td>54.22</td><td>156</td></tr> <tr><td>2100</td><td>21.</td><td>16.2</td><td>13</td><td>65</td><td>76.59</td><td>64.5</td><td>56.43</td><td>169</td></tr> <tr><td>2300</td><td>22.</td><td>17.</td><td>14</td><td>70</td><td>79.49</td><td>66.94</td><td>58.57</td><td>182</td></tr> <tr><td>2500</td><td>23.1</td><td>17.8</td><td>15</td><td>75</td><td>82.27</td><td>69.28</td><td>60.62</td><td>195</td></tr> <tr><td>2686</td><td>23.9</td><td>18.3</td><td>16</td><td>80</td><td>84.97</td><td>71.55</td><td>62.61</td><td>208</td></tr> <tr><td>2870</td><td>24.7</td><td>19.</td><td>17</td><td>85</td><td>87.07</td><td>73.32</td><td>64.16</td><td>221</td></tr> <tr><td>3055</td><td>25.5</td><td>19.6</td><td>18</td><td>90</td><td>90.13</td><td>75.90</td><td>67.41</td><td>234</td></tr> <tr><td>3240</td><td>26.25</td><td>20.1</td><td>19</td><td>95</td><td>92.60</td><td>77.98</td><td>68.23</td><td>247</td></tr> <tr><td>3430</td><td>27.</td><td>20.7</td><td>20</td><td>100</td><td>95.00</td><td>80.00</td><td>70.00</td><td>260</td></tr> <tr><td>3750</td><td>28.5</td><td>21.2</td><td>22</td><td>110</td><td>99.64</td><td>83.95</td><td>73.42</td><td>286</td></tr> <tr><td>4000</td><td>29.8</td><td>23.</td><td>24</td><td>120</td><td>104.06</td><td>87.63</td><td>76.68</td><td>312</td></tr> <tr><td>4460</td><td>31.1</td><td>23.9</td><td>26</td><td>130</td><td>108.32</td><td>91.22</td><td>79.81</td><td>338</td></tr> <tr><td>4850</td><td>32.4</td><td>24.7</td><td>28</td><td>140</td><td>112.20</td><td>94.66</td><td>82.82</td><td>364</td></tr> <tr><td>5250</td><td>33.6</td><td>25.5</td><td>30</td><td>150</td><td>116.35</td><td>97.98</td><td>85.73</td><td>390</td></tr> </table>

426. Dutch falls are always constructed so that the angle of weather may diminish from the centre to the extremity of the fall. 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 falls are in their best position when their extremities make an angle of seven degrees with the plane of their motion. Mr Smeaton's enlarged falls are Dutch falls in their best position, but enlarged at their extremities.

427. It appears from M. Coulomb's experiments on Dutch wind-mills, with rectangular sails, that when the distance between the extremities of two opposite falls is 66 feet French, and the breadth of each fall 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.

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.

CHAP VII. On the Construction of Wind-mills.

428. A WIND-MILL is represented in fig. 1, where MN is the circular building that contains the machinery, E the extremity of the windshaft, or principal axis, which is generally inclined from 8 to 15 degrees to the horizon; and EA, EB, EC, ED four rectangular frames upon which falls of cloth of the same form are stretched. At the lower extremity G of the falls their surface is inclined to the axis 72°; and at their farthest extremities A, D, &c. the inclination of the fall is about 83°. Now, when the falls are adjusted to the wind, which happens when the wind blows in the direction of the windshaft E, the impulse of the wind Mechanics.

Practical upon the oblique sails may be resolved into two forces, one of which acts at right angles to the windshaft, 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.

429. That the wind may act with the greatest efficacy upon the sails, the windshaft must have the same direction as the wind. But as this direction is perpetually changing, some apparatus is necessary for bringing the windshaft and 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 windshaft. 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 windshaft 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.

Methods of turning the sails to the wind.

On the Form and Position of Wind-mill Sails.

430. It appears from the investigations of Parent, that a maximum effect will be produced when the sails are inclined 54° 17' degrees to the axis of rotation, or when the angle of weather is 35° 46' (G) degrees. 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 sails move with greater rapidity than the parts nearer the centre, 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, that the most advantageous angle of weather for the sails of wind-mills is 35° 46' degrees for that part of the sail which is nearest the centre of rotation, and that the sail has every where 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 to 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 Euler's 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 \( \sqrt{2 + \frac{9cc}{4aa} + \frac{3c}{2a}} \) to 1.

431. In order to apply this theorem, let us suppose that the radius or whip ED of the sail \( \alpha \beta \gamma \), is divided in to fix 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 \( \sqrt{2 + \frac{9}{4} + \frac{3}{2}} = 3.561 \) = 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 \( \frac{1}{8} \) of the radius sm, \( c = \frac{a}{3} \); at \( \frac{2}{5} \) of the radius, \( c = \frac{2a}{3} \); at \( \frac{4}{5} \), \( c = \frac{4a}{3} \); at \( \frac{5}{8} \), \( c = \frac{5a}{3} \); 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.

(c) The weather of the sails is the angle which the surface forms with the plane in which they move, and is equal to the complement of the angle which that surface forms with the axis. <table> <tr> <th>Parts of the radius from the centre of motion at E.</th> <th>Velocity of the sail at the distances—or values of c.</th> <th>Angle made with the axis.</th> <th>Angle of weather.</th> </tr> <tr> <td>\( \frac{1}{2} \)</td> <td>\( \frac{a}{3} \)</td> <td>63</td> <td>26</td> <td>26</td> <td>34</td> </tr> <tr> <td>\( \frac{2}{3} \)</td> <td>\( 2a \)</td> <td>69</td> <td>54</td> <td>20</td> <td>6</td> </tr> <tr> <td>\( \frac{3}{8} \) or \( \frac{1}{4} \)</td> <td>\( a \)</td> <td>74</td> <td>19</td> <td>15</td> <td>4</td> </tr> <tr> <td>\( \frac{4}{8} \) or \( \frac{1}{2} \)</td> <td>\( 4a \)</td> <td>77</td> <td>20</td> <td>12</td> <td>40</td> </tr> <tr> <td>\( \frac{5}{8} \)</td> <td>\( 5a \)</td> <td>79</td> <td>27</td> <td>10</td> <td>33</td> </tr> <tr> <td>1</td> <td>\( 2a \)</td> <td>81</td> <td>0</td> <td>9</td> <td>0</td> </tr> </table>

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, falls, according to Smeaton.

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: 1st, 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 1/3: 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 fails 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, are considerably quicker than the velocity of the wind.

432. Mr Smeaton found, from a variety of experiments, that the common practice of inclining plane sails from 72° to 75° to the axis, was much more efficacious than the angle 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 α β, in fig. 2, so that α β was one-third of the radius ED, and α D to D β 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.

<table> <tr> <th>Parts of the radius EA, which is divided into 6 parts.</th> <th>Angle with the axis.</th> <th>Angle of weather.</th> </tr> <tr> <td>1</td> <td>72</td> <td>18</td> </tr> <tr> <td>2</td> <td>71</td> <td>19</td> </tr> <tr> <td>3</td> <td>72</td> <td>18 middle</td> </tr> <tr> <td>4</td> <td>74</td> <td>16</td> </tr> <tr> <td>5</td> <td>77 1/2</td> <td>12 1/2</td> </tr> <tr> <td>6</td> <td>83</td> <td>7</td> </tr> </table>

If the radius ED of the sail be 30 feet, then the sail will commence at \( \frac{1}{3} \) 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.

On the Effect of Wind-mill Sails.

433. The following maxims deduced by Mr 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, falls, according to Smeaton.

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: 1st, 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 1/3: 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 fails 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, are considerably quicker than the velocity of the wind.

434. A new mode of constructing the sails of wind-mills has been recently given by Mr Sutton, and fully described by Mr Heselden 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; but the subject shall be resumed under the article Windmill. It may be proper however to remark that Mr Sutton gives his sails the form represented in fig. 4, 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. 3. shews 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 to a state of rest is particularly ingenious. On Horizontal Wind-mills.

435. 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 affirms that horizontal wind-mills have only \( \frac{1}{8} \) or \( \frac{1}{16} \) of the power of vertical ones, he certainly forms too low an estimate of their power. Mr Beaton, on the contrary, who has received a patent for the construction of a new horizontal wind-mill, 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.

436. In fig. 4. CK is the windshaft, which moves upon pivots. Four cross bars, CA, CD, IB, FG, are fixed to this arbor, which carry the frames APIB, DEFG. The fails 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 fails are represented; but there are always other two placed at right angles to these. Now, let the fails be exposed to the wind, and it will be evident that no motion will ensue; for the force of the wind upon the fail AI, is counteracted by an equal and opposite force upon the fail EG. In order then, that the wind may communicate motion to the machine, the force upon the returning fail 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 Beaton, from the inconvenience and expense of the machinery and attendance requisite for turning the screens into their proper positions. Notwithstanding this objection, however, I am disposed to think that this is the best method of diminishing the action of the wind upon the returning fails, for the moveable screen may easily be 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 fails 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.

437. The mode of bringing the fails back against the wind, which Mr Beaton invented, is, perhaps, the simplest and best of the kind. He makes each fail AI to consist of six or eight flaps or vanes, AP b 1, b 1 c 2, &c. moving upon hinges represented by the dark lines, AP, b 1, c 2, &c. so that the lower side b 1, of the first flap overlaps the hinge or higher side of the second flap, and so on. When the wind, therefore, acts upon the fail AI, each flap will press upon the hinge of the one immediately below it, and the whole surface of the fail will be exposed to its action. But when the fail AI returns against the wind, the flaps will revolve round upon their hinges, and present only their edges to the Practical wind, as is represented at EG, so that the resistance occasioned by the return of the fail must be greatly diminished, and the motion will be continued by the great superiority of force exerted upon the fails in the position AI. In computing the force of the wind upon the fail AI, and the resistance opposed to it by the edges of the flaps in EG, Mr Beaton finds, that when the pressure upon the former is 1872 pounds, the resistance opposed by the latter is only about 36 pounds, or \( \frac{1}{52} \) part of the whole force; but he neglects the action of the wind upon the arms CA, &c. and the frames which carry the fails, 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 fail, with the whole resistance which these arms and the edges of the flaps oppose to the motion of the windmill. By inspecting fig. 4. it will appear, that if the force upon the edges of the flaps, which Mr Beaton supposed to be 12 in number, amounts to 36 pounds, the force spent upon the bars CD, DG, GF, FE, &c. cannot be less than 60 pounds. Now, since these bars are acted upon with an equal force, when the fails have the position AI, 1872 + 60 = 1932 will be the force exerted upon the fail 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 \( \frac{1}{20} \) of 1932, instead of \( \frac{1}{52} \) as computed by Mr Beaton. Hence we may see the probable advantages of a screen over moveable flaps, as it will preserve not only the fails, but the arms and the frame which support it, from the action of the wind.

438. We shall now conclude this chapter with a comparison of the power of horizontal and vertical forms between vertical and horizontal wind-mills. It was already stated, that Mr Smeaton rather underrated the former, while he maintained that they have only \( \frac{1}{8} \) or \( \frac{1}{16} \) the power of the latter. He observes, that when the vanes of a horizontal and a vertical mill are of the same dimensions, the power of the latter is four times that of the former, because, in the first case, only one fail is acted upon at once, while, in the second case, all the four receive the impulse of the wind. This, however, is not strictly true, since the vertical fails are all oblique to the direction of the wind. Let us suppose that the area of each fail is 100 square feet; then the power of the horizontal fail will be 100, and the power of a vertical fail may be called 100 × sine 70° (70° being the common angle of inclination) = 88 nearly; but since there are four vertical fails, the power of them all will be 4 × 88 = 352; so that the power of the horizontal fail 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 also observes, that if we consider the farther disadvantage which arises from the difficulty of getting the fails back against the wind, we need not wonder if horizontal wind-mills have only about \( \frac{1}{8} \) or \( \frac{1}{16} \) the power of the common sort. We have already seen, that the resistance occasioned by the return of the fails, amounts to \( \frac{1}{52} \) 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 \( \frac{1}{4} \) 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 windshaft 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 \( \frac{1}{2} \) or \( \frac{1}{4} \) 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. 4. instead of the position CA dm, CD o n, their power will be greatly increased, though the quantity of surface is the same, because the part CP 3 m being transferred to BI 3 d, has much more power to turn the sails.

CHAP. VIII. On the Construction of Wheel Carriages.

439. It is evident from Art. 60. that when a wheel furmounts 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 retit the motion of wheel carriages. 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 shewn, when we treat of the line of traffic, so that we may safely affirm, that the diameter of wheels should never be greater than fix feet. The fore wheels of our carriages are still unaccountably small, and it is not uncommon to see carts moving upon wheels scarcely 14 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 14 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.

440. The next thing to be determined is the shape of the wheels. Now it is certainly a matter of surmise how the unnatural shape which is at present given to them could ever have been brought into use. A cylindrical wheel, with the spokes perpendicular to the naves, is undoubtedly the form which every mechanic would give to his wheels, before he had heard of the pretended advantages of concave or dishing wheels, or those which have inclined spokes and conical rims. It has been alleged, indeed, that the form represented in fig. 5, when A r, B s is the conical rim, and o A, p B 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 felles from rubbing against the load or the fides 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. Now it is evident that the second of these advantages is very trifling, and may be obtained, when required, by interposing a piece of board between the wheel and the load.

441. The other two advantages exist only in very bad roads; and if they are necessary, which we much question, 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.

Upon these observations we might rest the opinion which we have been maintaining, and appeal for its truth to the judgment of every intelligent and unbiased mind; but we shall go a step farther, and endeavour to shew that concave dishing wheels are more expensive, more injurious to the roads, more liable to be broken by accidents, and less durable in general, than those wheels in which the spokes are perpendicular to the naves. By inspecting fig. 5, it will appear that the whole of the pressure which the wheel AB sustains is exerted along the inclined spoke p s, and therefore acts obliquely upon the level ground n D, whether the rims are conical or cylindrical. This oblique action mult 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 taking place; but in consequence of the oblique pressure, the stones between s and D will at least be loosed, and by admitting the rain the whole of the road will be materially damaged. But when the spokes are perpendicular to the nave as pn, and when the rims mA, nB 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.

442. 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 hypothenuse oA of the triangle oAn exceeds the side om; and therefore the weight and the resistance of such wheels must be proportionably 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.

443. We shall now discuss the subject of concave wheels with one observation more, and we beg the reader's attention to it, because it appears to be decisive of the question. The obstacles which carriages have to encounter, are almost never spherical protuberances that permit the elevated wheel to return 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 velocity and 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.

444. 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 A r, B s, fig. 5, which not only increases the disadvantages which we have affered to them, but adds many more to the number. Mr. Cumming, in a late Treatise on Wheel Carriages, solely devoted to the consideration of this single point, has shewn with great ability the disadvantages of conical rims, and the propriety of making them cylindrical; but we are of opinion that he has affered 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 so numerous and palpable, that it is wonderful how they should have been so long overlooked. 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 fides of deep ruts.

445. The shape of the wheels being thus determined, we must now attend to some particular parts of their construction. The 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. Mr. Walker, indeed, attempts to throw ridicule upon the act of parliament which enjoined the use of broad wheels; but he does not assign any sufficient reason for his opinion, and ought to have known that several excellent and well devised experiments were lately instituted by Boulard and Margueron, which evince in the most satisfactory manner the great utility of broad wheels. Upon this subject an observation occurs to us, which has not been generally attended to, and which appears to remove all the objections which can be urged against broad rims. 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 fifteen hundred weight 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 Practical over the stones has four points of support, each stone will bear a weight of two hundred weight, 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 now only two points of support, each stone will be pressed with a weight of four hundred weight, 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.

446. 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 felles with nails of such a kind that their heads may not rise above the surface of the rims. In some military waggons 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 felles on which the rims are fixed should in carriages be three inches and a fourth deep, and in waggons 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 infuse itself between the spoke and the naves, and prevent that close adhesion which is necessary to the strength of the wheel.

On the Position of the Wheels.

447. 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 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 fix inches in diameter, the distance of the wheels at top was fully fix 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.

448. 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 linch pin, and the innermost would rest against the shoulder of the axle-tree. In rectilineal 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.

449. M. Camus attempted to shew 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 the figure) Fig. 6, 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. Deparcieux, 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 shewn 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. 6, 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 chest or centre of gravity, and CE the level road. Then AFC will represent the crooked lever by which the horse acts, which is equivalent to the straight one AC. But when the horse's weight acts downwards at A, so as to drag forward the rope AD and raise the load P, CE will represent the power of the lever in this position, or the lever of the horse's weight, and CF the lever by which it is resisted by the load, or the lever of resistance. Now if the horse lowers its centre of gravity A, which it always does when it pulls hard, it is evident that CE, the lever of its weight, will be increased, while CF the lever of its resistance will be diminished, for the line of traction AD will approach nearer to CE. Hence we see the great benefit which may be derived from large horses; for the lever AC necessarily increases with their size, and their power is always proportioned to the length of this lever, their weight remaining the same. Large horses, therefore, and other animals, will draw more than small ones, even though they have less muscular force, force, and are unable to carry such a heavy burden. The force of the muscles tends only to make the horse carry continually forward his centre of gravity, or, in other words, the weight of the animal produces the draught, and the play and force of its muscles serve to continue it.

450. From these remarks, then, we may deduce the proper position of the line of traction. When the line of traction is horizontal, as A D, the lever of resistance is CF; but if this line is oblique to the horizon, as A d, the lever of resistance is diminished to C f, while the lever of the horse's weight always remains the same. Hence it appears, that inclined traces are much more advantageous than horizontal ones, as they uniformly diminish the resistance to be overcome. Deparcieux, however, has investigated experimentally the most favourable angle of inclination, and found, that when the angle DAF made by the trace A d and a horizontal line is fourteen or fifteen degrees, the horses pulled with the greatest facility and force. This value of the angle of draught will require the weight of the spring-tree bar, to which the traces are attached in four-wheeled carriages, to be one-half of the height of that part of the horse's breast to which the fore end of the traces is connected.

451. When several horses are yoked in the same carriage as represented in fig. 7, and when the declivity changes, 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 RF'VE'; and from E draw E m n 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, F n, F' m parallel to EA, and F φ, F' φ' parallel to E n. 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 E n, E φ, one of which E n is solely employed in dragging the cart up the inclined plane DA, while the other E φ 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 RF'VE', and F'E'=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 E m, E f, 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 E m to E n, and the forces which press the horse upon the ground as E f to E φ, or as F' m to F n. Now E φ=F n=FE×fin. n E φ; hence F φ=FE×fin. (n E g'—FE g') (g' E being parallel to AB') and E n=EF×cof. (n E g'—FE g'). In like manner we have E f=FE×fin. (n E g'—F'E g'), and E m=EF×cof. (n E g'—F'E g'). Now fin. FE g'=fin. FE g= \frac{R g}{ER}, and fin. FE g'=\frac{R' g'}{ER'}=\frac{R g}{ER'}; but R g =R' g'—BR—EQ=BR—BR×cof. n E g'=BR×(1—cof. n E g'). By substituting this value in the equations which contain the values of E φ, E n, E f, E m, and considering that the angles FE g', F'E g' are always so small that their arcs differ very little from their sines, we have FE g=\frac{BR×1—cof. n E g}{ER}, and

\[ F'E g'=\frac{BR×1—cof. n E g}{ER'} \]

By substituting these values in the preceding equations, we have

\[ E φ=EF×fin.\ (n E g—\frac{BR×1—cof. n E g}{ER}), \] \[ E f n=EF×fin.\ (n E g—\frac{BR×1—cof. n E g}{ER'}), \] \[ E n=EF×cof.\ (n E g—\frac{BR×1—cof. n E g}{ER}), \] \[ E m=EF×cof.\ (n E g—\frac{BR×1—cof. n E g}{ER'}). \]

If AB is horizontal, and the declivity AD=\frac{2}{3}, we shall have n E g=9° 28', or in parts of the radius=0.16522, and cof. n E g=0.98638. Then, if EF=200 pounds, BR=3\frac{1}{3} feet, ER=8 feet, ER'=12 feet, then we shall have from the preceding formulae, E φ=31.716 pounds, E f=32.350 pounds, E n=197.470 pounds, and E m=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 0.666 pounds.

On the Position of the Centre of Gravity, and the manner of disposing the load.

452. If the axle-tree of a two-wheeled carriage passes 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 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 prelled to it, and he will sometimes bear a great proportion of the load when he should rather be relieved of it.

453. 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 prelled to the ground.—Now this may be in some measure effected in the following manner. Let BCN 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 prelled 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 prelure 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, b A will represent its prelure 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 a A will be the prelure 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 c A will represent the prelure 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 b a and b m, 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 roads, 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 b A should be one foot, which, being one-eighth of DA, will make the prelure upon D exactly 50 pounds. If the road slopes four inches in a foot, b m must be four inches, or the angle b A m 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.

454. When carts are not made in this manner, we may, in some degree, obtain the same end by judiciously disposing the load. 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 on-

ly lower the point o, but will bring it forward, and nearer the proper position m. Part of the load, too, might be fulpended below the fore part of the carriage in dry weather, and the centre of gravity would approach till 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 equalised.

Description of different Carriages.

455. In figure 8. 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 shewn in figure 9., where AA is a small axis fixed into the box, and B 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 centre. 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 tredle 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 tredle rises, and the crank F rising along with it, takes into the teeth of the wheel H, so that when the elevated tredle 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 tredles, the machine advances with a regular pace.

456. 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 Nafmyth of Edinburgh, a gentleman whose mechanical genius is scarcely inferior to his talents as a painter. The pulley B 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 five inches in diameter, and one and a half broad.

475. A carriage driven by the action of the wind is exhibited in fig. 10. 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 fledge 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 21 miles an hour with a very strong wind.

458. The carriage represented in fig. 11. is made so as to fail against the wind by means of the spiral sails Description 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.

Fig. 12. 459. A carriage which cannot be overturned is represented in figure 12, 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. Description of Machines. 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.

CHAP. I. Machines which illustrate the doctrines of Mechanics, or are connected with them.

1. Atwood's Machine.

Atwood's machine, Plate CCCXXV, Fig. 1, 2, 3, &c.

460. THE ingenious machine invented by Mr Atwood for illustrating the doctrines of accelerated and retarded motion, is represented in figs. 1, 2, 3, 4, 5, 6, and enables us to discover, 1. The quantity of matter moved. 2. The moving force. 3. The space described. 4. The time of description; and, 5. The velocity acquired at the end of that time.

461. 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 to the extremities of a very fine silk thread: this thread is stretched over a wheel or fixed pulley a b c d, moveable round a horizontal axis: the two weights A, B being equal, and acting against each other, remain in equilibrium; and when the least weight is superadded to either (letting 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, 1. The wheel a b c d, over which the thread sustaining A and B passes. 2. The four friction wheels on which the axle of the wheel a b c d rests. 3. The thread by which the bodies A and B are connected, so as when set in motion to move with equal velocities. The weight and inertia of the thread are too small to have any sensible effect on the experiments; but the inertia of the other materials constitute a considerable proportion of the mass moved, and must therefore be taken into account. Since when A and B are put in motion, they must move with a velocity equal to that of the circumference of the wheel a b c d to which the thread is applied; it follows, that if the whole mass of the wheels were accumulated in this circumference, its inertia would be truly estimated by the quantity of matter moved; but since the parts of the wheels move with different velocities, their effects in resisting the communication of motion to A and B by their inertia will be different; those parts which are furthest from the axis resisting more than those which revolve nearer in a duplicate proportion of those distances, (see Rotation). If the figures of the wheels were regular, the distances of their centres of gyration from their axes of motion would be given, and consequently an equivalent weight, which being accumulated uniformly in the circumference a b c d, would exert an inertia equal to that of the wheels in their constructed form, would also be given. But as the figures are irregular, recourse must be had to experiment, to assign that quantity of matter, which being accumulated uniformly in the circumference of the wheel a b c d, would resist the communication of motion to A in the same manner as the wheels.

In order to ascertain the inertia of the wheel a b c d, with that of the friction wheels, the weights AB being removed, the following experiment was made.

A weight of 39 grains was affixed to a silk thread of inconsiderable weight; this thread being wound round the wheel a b c d, the weight 30 grains by descending from rest communicated motion to the wheel, and by many trials was observed to describe a space of about 38 1/2 inches in 3 seconds. From these data the equivalent mass or inertia of the wheels will be known from this rule.

Let a weight P, fig. 2, be applied to communicate motion to a system of bodies by means of a very slender and flexible thread going round the wheel SLDIM, through the centre of which the axis passes (G being the common centre of gravity, R the centre of gravity of the matter contained in this line, and O the centre of oscillation). Let this weight descend from rest through any convenient space s inches, and let the observed time of its descent be t seconds; then if l be the space through which bodies descend freely by gravity in one second, the equivalent weight sought \( \frac{W \times SR \times SO}{SL^2} = \frac{P \times t^2}{s} - P. \)

Here we have \( p = 39 \) grains, \( t = 3 \) seconds, \( l = 193 \) inches, \( s = 38.5 \) inches; and \( \frac{P \times t^2}{s} - P = \frac{30 \times 9 \times 193}{385} = 30 \div 1323 \) grains, or 2 1/3 ounces.

This is the inertia equivalent to that of the wheel a b c d, and the friction wheels together: for the rule extends to the estimation of the inertia of the mass contained in all the wheels.

The resistance to motion therefore arising from the wheel's inertia, will be the same as if they were absolutely Description absolutely removed, and a mass of 2 1/2 ounces uniformly accumulated in the circumference of the wheel a b c d. This being premised, let the boxes A and B be replaced, being suspended by the silk thread over the wheel or pulley a b c d, and balancing each other; suppose that any weight m be added to A so that it shall descend, the exact quantity of matter moved, during the descent of the weight A, will be ascertained, for the whole mass will be A + B + m + 2 1/2 oz.

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 1/4 of an ounce, and is represented by the letter m. The inertia of the wheels being therefore = 2 1/2 ounces, will be denoted by 11 m. 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 1/2 oz. or, according to the preceding estimation, the weight of each box will be denoted by 6 m; these boxes contain such weights as are represented by fig. 3, each of which weighs an ounce, so as to be equivalent to 4 m; other weights of 1/2 oz., 2 m, 1/2 m, and aliquot parts of m, such as 3/4 m, 1/4 m, may be also included in the boxes, according to the conditions of the different experiments hereafter described.

If 4 1/2 oz. or 19 m, be included in either box, this with the weight of the box itself will be 25 m; so that when the weights A and B, each being 25 m, &c. balanced in the manner above represented, their whole mass will be 50 m, which being added to the inertia of the wheels 11 m, the sum will be 61 m. Moreover, three circular weights, such as that which is represented at fig. 4, are constructed; each of which = 1/4 oz. or m: if one of these be added to A and one to B, the whole mass will now become 63 m, perfectly in equilibrium, and moveable by the least weight added to either (letting 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 63 m, existing in free space and without gravity.

462. 2. The moving force. Since the weight of any substance is constant, and the exact quantity of it easily estimated, it will be convenient here to apply a weight to the mass A as a moving force: thus, when the system consists of a mass = 63 m, according to the preceding description, the whole being perfectly balanced, let a weight 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 63 m, the whole quantity of matter moved will now become 64 m; and the moving force being = m, this will give the force which accelerates the descent of \( \Delta = \frac{m}{64\ m} \), 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 3 m be all placed on A, the moving force will become 3 m, and the mass moved 64 m as before, and the force which accelerates the descent of A = \( \frac{3\ m}{64\ m} = \frac{3}{64} \) parts of the force by which gravity accelerates falling bodies.

Suppose it were required to make the moving force 2 m, 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 61 m: let \( \frac{1}{4} \) m, fig. 5, be added to A, and \( \frac{1}{4} \) m to B, the equilibrium will be preserved, and the mass moved will be 62 m; now let 2 m be added to A, the moving force will be 2 m, and the mass moved 64 m as before; wherefore the force of acceleration = \( \frac{1}{17} \) part of the acceleration of gravity. These alterations in the moving force may be easily made in the more elementary experiments, there being no necessity for altering the contents of the boxes A and B: but the proportion and absolute quantities of the moving force and mass moved, may be of any assigned magnitude, according to the conditions of the proposition to be illustrated.

463. 3. Of the space described. The body A, fig. 1. Fig. 1. descends in a vertical line; and a scale about 64 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 estimated from o on the scale, when the bottom of the box A is on a level with o. 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 o; so that by altering the position of the stage, the space described from rest may be of any given magnitude less than 64 inches.

464. 4. The time of description is observed by a pendulum, vibrating seconds; and the experiments intended to illustrate the elementary propositions, may easily 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 o on the scale, rest on a flat rod, held in the hand horizontally; its extremity being coincident with o, 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.

465. 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 Description 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 defending 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 = 25 m each, so as together to be = 50 m; this with the wheel's inertia 11 m will make 61 m; 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 63 m. If a weight m be added to A, motion will be communicated, the moving force being m, and the mass moved 64 m. 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 o on the scale, the whole mass being as described above = 63 m, 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. 6, to have defended 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 defends, 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.

466. Mr Atwood's machine is also useful for estimating experimentally the velocities communicated by the impact of bodies elastic and nonelastic; the quantity of resistance opposed by fluids, as well as for various other purposes. These uses we shall not insist on; but the properties of retarded motion being a part of the present subject, it may be necessary to show in what manner the motion of bodies resisted by constant forces are reduced to experiment by means of the instrument above described, with as great ease and precision as the properties of bodies uniformly accelerated. A single instance will be sufficient: Thus, suppose the mass contained in the weights A and B, fig. 6, and the wheels to be 61 m, when perfectly in equilibrium; let a circular 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 64 m: 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 62 m, the force of retardation will be \( \frac{7}{9} \) part of that force where-by 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 already described.

In the preceding descriptions, two suppositions have been assumed, neither of which is mathematically true: but it might be easily shewn that they are so in a physical sense; the errors occasioned by them being infensible in practice.

2. Machine for illustrating the Theory of the Wedge.

467. 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 6c°, 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 M, 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 uv, xw, 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 prelude of the weight p, whereas, in the real wedge, the impelling power is always an impulsive force which is infinitely more powerful.

3. Machine for illustrating the effects of the centrifugal force in flattening the poles of the Earth.

468. 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, wheninking, 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 larger than the polar diameter.

4. Machine for trying the Strength of Materials.

469. The piece of wood, whose strength is to be of materials, Description tried, is represented by LF, 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 Nat. Philos. vol. i. p. 768. from which this drawing and description are taken.

5. Machine in which all the Mechanical powers are combined.

473. The lever AB, whose centre of motion is C, is fixed to the endless screw DE, which drives the wheel and axle FHG. Round the axle G is coiled a rope GHI, which passes round the four pulleys K, L, m, n, and is fixed to a hook at m on the lower block, which carries the weight W. When equal weights are suspended on the lever at equal distances from the fulcrum C, the lever becomes a balance, and the wedge and inclined plane are evidently included in the endless screw DE. If the wheel F has 30 teeth, if the lever AB is equal to twice the diameter of the wheel FH, and if the diameter of the axle G is one-tenth of the diameter of the wheel, a power of 1 exerted at P will raise a weight of 2400 suspended at the lower block of the four pulleys.

6. Fidler's Balance.

471. The balance represented in fig. 3. was made by Fidler for the Royal Institution, and does not differ much from those which have been constructed by Ramden and Troughton. The middle column A can be raised at pleasure by the nut B, and supports the round ends of the axis in the forks at its upper extremity, in order to remove the pressure on the sharp edges of the axis within the forks. C and D are pillars which occasionally support the scales, and may be elevated or depressed by turning the nut E. The screw F raises or depresses a weight within the conical beam, for the purpose of regulating the position of the centre of gravity. The graduated arc G measures the extent of the vibrations. See Young's Nat. Philos. vol. i. p. 765.

7. Improvement on the Balance.

472. An improvement on the balance is represented in fig. 4. where DC is a micrometer screw fixed to the arm FA, so that when it is turned round by the nut D, it neither approaches to, nor recedes from, the centre of motion F. The screw DC works in a female screw in the small weight n, and by revolving in one direction, carries this weight from S to R, and thus gives the preponderance to the scale G. The reception of the weight n from the centre F is measured as in the common micrometer, and a weight x placed in the scale suspended at A, will be in equilibrium with n placed at any distance S n, when \( x = \frac{S n \times n}{FA} \).—Appendix to Ferguson's Lectures.

8. Machine for shewing the Composition of Forces.

473. The part BEFC is made to draw other parts into the wooden square ABCD. The pulley H is joined to the BEFC so as to turn on an axis which will be at H when composition of the square BEFC is pushed in, and at p when it is drawn out. A ball G is made to slide on the wire k which is fixed to PEFC, and the thread m attached to the ball goes over the pulley I, where it is fixed. Now, when the piece BEFC is pulled out, the pulley, wire, and ball, move along with it, in 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 G g, the diagonal of a parallelogram whose sides are GH, GC.

8. Smeaton's Machine for experiments on Windmill Sails.

474. In the experiments with this machine, the sails Apparatus were carried round in the circumference of a circle, so for wind that the same effect was produced as if the wind had mills struck the sails at rest with the velocity which was then given them. In the pyramid frame ABC is fixed to the axis DE, which carries the arm FG with the sails G1. 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.

9. Smeaton's Machine for experiments on Rotatory Motion.

475. This machine is exhibited in fig. 1. where the vertical axis NB is turned by the rope M passing over for rotating the pulley 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 Fig. 1. 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. CHAP. II. Machines for various purposes.

1. Prony's Condenser of Forces.

476. The object of this machine is to obtain a maximum effect from an impelling power which is subject to variation in its intensity. Let us suppose that wind is the first mover, and that O, O is the vertical axis of a windmill; e, e, e, e, e, are several radii issuing from this axis, and carrying a wiper b d, which acts upon the corresponding wipers a f, and gives a motion of rotation to the axes a, a, a, a to which they are attached. The wipers b d, a f must be so constructed that when b d ceases to press on one wiper a f, it shall at the same moment begin to act upon the next wiper. Each of the axes a, a, a, a, carries a drum t t r r, round which is coiled a cord t p F, passing over the pulley p, and supporting a weight Q which can be placed at different distances from G on the lever FG. The axes a, a, a, a also pass through the pinions q q, to which they are not fixed; but these pinions carry ratchet wheels that bear against the teeth r r, so that when the weight Q rises, the rope merely coils round the drum without moving the pinion q q. But when the wiper b d ceases to act upon a f, the weight Q descends, and then the toothed wheel r r acts against the ratchet, so that Q cannot descend without turning the pinion q q along with the drum. The pinion q q drives the wheel a b, which again drives the wheel C E by means of the bevelled teeth C D, and elevates the load at P. Hence, when the axis O O is put in motion by the wind acting on the sails, it will first raise a number of weights Q sufficient to put the machine in motion, and will continue to raise new weights while those before raised are fallen, so that the motion once impressed will be continued.

2. Portable Stone Crane, for loading and unloading Carts.

477. This crane is mounted on a wooden frame, and is so constructed that it may be taken to pieces. The frame A, A, A is about ten feet high, nine feet long and nine feet wide. The wheels B, B are of iron, and about three feet in diameter. The pinion D that is fixed to the axis of the first wheel B is eight inches diameter, and the other pinion C is about the same diameter. When the stones are suspended to the rope that coils round the barrel, the workman turns a winch on the axis of the wheel C, and raises or lowers the weight according to the direction in which he turns it.

3. Portable Cellar Crane.

478. This crane is represented in fig. 5, where A, A are two wooden supports about six feet high, which are jointed at E, and connected by the iron cylinder C and the wooden bar D. The supports A, A are fastened to the edge of the cellar by the iron prongs E, E, and the two ropes which support the barrel and pass round it are fixed to the iron clamp G, G. These ropes coil round the cylindrical bar F, which is put in motion by the winch K, driving the pinion I about four inches diameter, which gives motion to the wheel H, about three feet in diameter. The barrel, therefore, will rise or fall according to the direction in which the winch is moved.

4. Weighing Crane.

479. This crane represented in fig. 6, was invented by Mr Andrews, and weighs the body at the time that it is raising it. The weight W is elevated by means of the levers M, N, O, P, which coil the rope HR round the barrel H. The jib ED stands on a horizontal beam moveable in a vertical plane round the centre crane FA, and the distance of the upright beam E from the centre of motion A is \( \frac{1}{20} \) of BF. The weight of the body W is then ascertained by the weight at B, which keeps it in equilibrium. The piece of wood C projects from the vertical beam CT, in order to prevent the beam from rising too high.

5. Gilpin's Crane.

480. In fig. 1, where this machine is represented, Gilpin's AB is the perpendicular stand, formed of two oaken crane planks let into cast iron mortises C, D : Between these Plate planks is fixed the barrel E with spiral grooves on its surface, on which the chain RL winds. When the 4 & 5 winch N is put in motion it drives the pinion O, which again drives the wheel P, on whose axis is fixed the barrel F, so that the chain is coiled round the barrel and the weight raised. A section of this part of the machinery is shewn in fig. 2. Figure 3, shews an enlarged view of part of the barrel, and part of the chain lying in its proper position in the spiral grooves or channels. In order to prevent the chain from twisting when it is wound upon the barrel, the lower edge of one link lies in the groove, and the next link upon the surface of the barrel. This will be better understood from fig. 4, which is a section of the barrel F, and shews the manner in which one link lies within it, and the other link on its outside. The old method of working chains is exhibited in fig. 5. For a full account of this useful invention, see Nicholson's Journal, vol. xv. p. 126.

6. Bramah's Jib for Cranes.

481. The nature of this invention, for which we are Bramah's indebted to the ingenious Mr Bramah, may be easily understood from a bare inspection of fig. 6, which represents a jib attached to the wall of a warehouse. The jib turns on a perforated axis or pillar. The rope by which the weight is raised after passing over two pulleys, goes through the perforated axis, and is conducted over another pulley to the barrel of the crane, which is not represented in the figure. In jibs of the common construction which turn in two solid gudgeons, the rope passes over the upper gudgeon, and is confined between two vertical rollers; but the bending of the rope occasions a great deal of friction, and produces a constant effort to bring the arm of the jib into a position parallel to the inner part of the rope.

7. Gottlieb's Carriage Crane.

482. This machine, which is useful for carrying large Plate stones where carts and horses cannot be easily obtained, consists of two sets of crane wheels applied to the two carriage sets of wheels belonging to the carriage, so that two crane men, one acting at each winch A, A give motion to the loaded carriage. The pinion B, fix inches in diameter turns the wheel-C, three feet in diameter. The wheel C gives motion to the pinion D one foot in diameter,

Description meter, which works into two wheels E, E three feet of Machines. six inches diameter, and are fixed on the wheels of the carriage.

8. Common Jack.

483. The common worm jack is represented in fig. 8, and is impelled by the weight W, which is suspended to a rope passing through the pulleys V, R, and rolling round the barrel Q. When the barrel is put in motion by the action of the weight, it drives the wheel KL of 60 teeth, by means of a catch fixed to AB, which lays hold of the cros bars in KL. The wheel KL drives the pinion M of 15 teeth, fixed on the axis of the wheel N of 30 teeth, which gives motion to the endless screw O, and the fly-wheel P. On the axis of the wheel KL is fixed the pulley DG, which, by means of a rope, gives motion to the spit. The axis ET is fixed in the barrel AC; and as this axis is hollow, both it and the barrel turn round upon the axis FD, so that the rope may be coiled round the barrel by the winch H without moving the wheel K.

9. Loading and Unloading Machine.

484. This portable machine, invented by Mr Davis of Windsor, is put in motion by the winch A, which drives the two endless screws C, C. These screws move the wheels E, E, and consequently the barrels connected with them, so that the ropes F, F passing over the pulleys G, G are coiled round the barrels, and the load H which these ropes support is raised into the frame R, R, which shews a part of the cart. The barrels and wheels are contained in an iron box L, the sides of which are removed in the figure.

10. Vauloue's Pile Engine.

485. The horses which work this engine are yoked at S, S, and by moving the wheel B and drum C, which are locked together, raise the follower GH, (carrying the ram Q by the handle R), by means of the rope HH which coils round the drum. When the follower G reaches the top of the frame, the upper legs of the tongs H are closed by pressing against the adjacent beams; and their lower legs are opened, so that they drop the ram Q, which falls and strikes the pile. When G is at the top of the frame, the crooked handle 6, of the follower G, presses against the cords a, a, which raise the end of the lever L (see fig. 2.) round m as a centre, and by depressing the extremity N, and consequently the bar S, S, unlock the drum C and the wheel B, so that the follower G falls by its weight and seizes the ram R. As soon as the follower drops, the Description horses would tumble down, having no resistance to overcome, were not this prevented by the fly O, which is moved by the wheel B and trundle X, and opposes a sufficient resistance to the horses till the follower again seizes the ram. When the follower falls, the weight L (fig. 2.) pushes up the bolt Y into the drum C, and locks the wheel and the drum;—and the same operation is afterwards repeated. See Ferguson's Lecl. vol. i. p. 118.

11. Bunce's Pile Engine.

486. A side view of this engine is shewn in fig. 3, 4. Bunce's It consists of two endless ropes or chains A, connected pile engine, by cros pieces of iron B, B, &c. (fig. 4.) which pass Fig. 3, 4- round the wheel C, the cros pieces falling into corresponding cros 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. 3, 5,) is fixed to one of the cros pieces B by the hook M, the weight of the ram, acting by the rope, moves the lever DK round H, and brings the wheel D to G, so that, by turning the winch, the ram L (fig. 3,) is raised in the vertical line LRG. But when it reaches R, the projecting piece R disengages the ram from the cros 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 fulped 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 cros pieces B, by which the ram is again raised.

487. For the description of a great variety of useful machines, the reader is referred to the second volume of Mr Gregory's Mechanics, and to Dr Young's Natural Philosophy, a work of great merit, which would have been more particularly noticed if it had reached us before the historical part of this article was printed off.—See also HYDRODYNAMICS, MARLY, Machine at, MILL, RAMSDEN, and WATER-WORKS.

INDEX.

A. ABUTMENTS, construction of, No 336 Epinus's property of the lever, 64 P'Almber't principle of dynamics, 22 Arches, equilibrium of, 316 catenarian, 319 of equilibration, 334 Arwood's machine, 460

B. Balance, its properties, 142

Balance, Kuhne's, 252 Magellan's, 153 Ludlam's, 471 Fidler's, 472 improvement on it, 407 Bevelled wheels, 14 Borelli, works of, 368

C. Capstan, description of one, 368 Carriages, wheel, on the construction of, 439

No 151 Cellar crane, 478 Centre of inertia, or gravity, how to find it, 4, 154, 201 Centroburyc method, 206, 208 Collifion, laws of, discovered by Wren, &c. 12 of hard bodies, 248—258 of elastic bodies, 258—279 of imperfectly elastic bodies, 279 Condenser of forces, 476 Conical Conical pendulum, No 394 Conservation of active forces discovered by Huygens, 19 generalized by D. Bernouilli, ib. Coulomb on the force of torsion, 27, 343—357 on friction, 27, 372 on the rigidity of ropes, 382 on the strength of men, 421 Crane, carriage, 482 Gilpin's, 488 cellar, 478 weighing, 479 Crown wheels, 410 Curves, motion of bodies along, 217 of quickest descent, 231 Cycloid, isochronism of, discovered by Huygens, 11 properties of it, 232 the proper curve for the teeth of rackwork, 411 Cycloidal pendulum, properties of, 236 Cylinders, friction of loaded ones, 372 D. Domes, equilibrium of, 339 E. Earth, machine for explaining the flatness at its poles, Epicycloids the proper curves for the teeth of wheels, 395 interior, 397 exterior, 397 method of forming them, 406 F. Fidler's balance, 471 Fly wheels, 383 Forces, active, dispute about them, 377 Friction wheels, 377 Fusee of a watch, its construction, 99 G. Galileo, discoveries of; respecting the science of motion, 8 Gottlieb's antiattrition axletree, 377 Gravity, centre of, 154—201 force of, 247 Gregory, Olinthus, his Treatise on Mechanics recommended, 28 H. de la Hire, on the teeth of wheels, 17 Horset, strength of, 423 method in which they exert it, 449 Huygens, discoveries of, 11, 19 I. Jack, common, 483 Jit, Bramah's, 481 Involution. See Collision. Inclined plane, properties of it, 69 its use in practical mechanics, 83 planes, motion of bodies along, 217 Inertia, centre of, 154—201 Involutes proper for the teeth of wheels, 405 L. Lever, properties of 33—69

Lever, various modes of deducing its fundamental property, No 63 Æpinus's property of compound, 65 M. Machine, Atwood's, 460 loading and unloading, 477 for shewing the flatness at the poles of the earth, 468 for trying the strength of materials, 469 in which all the mechanical powers are combined, 470 for rotatory motion, 475 for weighing, 479 Machinery, construction of, 358 simplification of, 366 first movers of, 418 Machines, simple, 30 compound, 89 maximum effects of, 298, 359 Mechanics defined, 1 history of, 2 treatise on, 2 Mechanical powers, machine in which they are combined, 470 Men, strength of, according to various authors, 418, &c. Merfennus, problems proposed by him, 13 Mills driven by water and wind invented, 5 Motion not propagated instantaneously, 295, 296 N. Newton, Sir Isaac, his discoveries, 18 O. Oscillation of pendulums, 236 P. Parallelogram of forces discovered by Stevinus, 6 demonstrated by D. Bernouilli, 20 of forces, machine for explaining it, 473 Parent on the maximum effects of machines, 16 on wind-mills, 430 Pendulum, conical, 394 Percussion. See Collision. Piers, construction of, 336 Pile engine, Vauloue's, 485 Bunce's, 486 Prony's condenser of forces, 476 Pulley, properties of it, 100 R. Rackwork, teeth of, 411 Reflexion of bodies after collision, 283—293 Rope machine, its properties, 84 Ropes, rigidity of, 382 Rotatory motion, three axes of, discovered by Segner, 26 Smeaton's apparatus for, 475

S. Screw, properties of it, No 131 Mr. Hunter's double one, 137 its use in mechanics, 140 Semicycloid, the curve of swiftest descent, 231 Smeaton on wind-mill fails, 430 Stumpert, on the wipers of, 411 Statera, Roman, Danish and Swedish, 217 Steam, the power of, as a first mover, 423, 427 engine invented, 13 improved by Savary, Newcomen, Watt, &c. ib. Steelyard, 47 Strength of materials, machine for trying it, 469 of men, 418 of horses, 424 Sutton on wind-mill fails, 434 T. Table of the strength of first movers, 425 of the strength of men, 418 Teeth of wheels, how to form them, 395 of rackwork, how to form them, 411 Torricelli, labours of, 9 Torsion, force of, 27, 343—357 Traction, line of, 449 W. Water-mills, invention of, 5 Wedge, properties of it, 121 machine for shewing its properties, 467 Wheel and axle, its properties, 90 carriages, on the construction of, 439 on the position of their wheels, 447 on the size and form of their wheels, 439 on the position of their centre of gravity, 452 Wheels, on the formation of the teeth of, 395 bevelled method of forming their teeth, 407 crown, 410 friction, 374 fly, 383 Wind, the power of, as a first mover, 423, 426 Wind-mills, invention of, 5 description of one, 428 fails, on the position of, 430 on the effect of, 433 form given them by Mr Sutton, 434 horizontal, 435 comparison between horizontal and vertical ones, 438 fails, Smeaton's apparatus for determining their power, 474 Wipers of stampers, how to form them, 419

PLATE CCCXVI.

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

W. Archibald sculp.

PLATE CCCXVII.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9.

W. Archibald sculp*

PLATE CCCXVIII.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11.

W. Archibald sculp.*

PLATE CCCXIX.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11. Fig. 12. Fig. 13.

W. Archibald Sculp.

PLATE CCCXX.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11. Fig. 12. Fig. 13. Fig. 14. Fig. 15. Fig. 16. Fig. 17. Fig. 18. Fig. 19. Fig. 20. Fig. 21. Fig. 22. Fig. 23. Fig. 24. Fig. 25. Fig. 26. Fig. 27.

PLATE CCCXXI.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11. Fig. 12.

W. L. Schubald Sculp.

PLATE CCCXXII.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11.

W. Archibald Sculp. MECHANICS

PLATE CCCXXIII.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9.

Engraved by W.K.D. Linars Edin.

PLATE CCCXXIV.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11. Fig. 12. Fig. 13. Fig. 14. Fig. 15.

W. Archibald sculp. T.

PLATE CCCXXV.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11. Fig. 12.

Inches 134.34 Inches 157.6 Inches 77.17 Inches 185.26 Inches 222.30 38.58 inches Apparent Breadth of the Sail 59.1 inches Absolute breadth of the Sail

Engraved by W. & P. Lines Edinburgh. Fig. 2. Fig. 6. MECHANICS. PLATE CCCXXVI. Fig. 3. Fig. 4. Fig. 7. Fig. 1. Fig. 8. Fig. 5.

W. Arbuthnot Sculp. MECHANICS

PLATE CCCXXXVII.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6.

PLATE CCCXXVIII.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5.

Engd by W. & D. Aikins Edts.

PLATE CCCXXIX.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9.

E. Mitchell sculp.

PLATE CCCXXX.

Vauloue's Pile Engine.

Fig. 1. Fig. 2.

Bunce's Pile Engine.

Fig. 3. Fig. 4. Fig. 5.

Andrew's Weighing Crane.

Fig. 6.

E. Mitchell sculpsit