Msanies, composition of motion, we shall arrive at a knowledge of every species of curvilinear motion. In fact, whatever may be the forces applied to a body which describes any curve, we may at each instant reduce all these forces into two only, the one tangential, and the other perpendicular to the element of the curve. To the first of these, which is essentially rectilinear for every instant, we may apply the principle of the force of inertia; and to the second, which is expressed by the square of the actual velocity of the body, divided by the radius of the osculatory circle, the theory of central forces in a circle; which revert again to the same principle, i. e. the motion in the direction of the radius of the osculatory

Next

circle.

Newton did not write fully upon the doctrine of Mechanics, considering it as a distinct topic: yet he laid down its first principles, as well as investigated several interesting mechanical problems, which naturally fell in his way, when establishing the nature and investigating the truths of general physics, in his "Principia," pubAD.1687. lished in 1687. Newton also exhibited three celebrated axioms, or laws of motion; which now find their way, either explicitly or implicitely, into every treatise of Mechanics. They are these:

1. Every body continues in its state, whether it be of rest, or of uniform motion in a right line; unless it be compelled to change that state by extraneous forces.

2. The alteration of motion is always proportional to the motive force impressed, and is in the direction of the right line in which that force acts.

3. To every action there is always opposed an equal and opposite re-action.

The mechanical and other momentous verities developed in the Principia, were there given synthetically. Herman also proceeded after the synthetical 1716, manner in his "Phoronomia," a valuable work, though containing but few discoveries, published in 1716.

42.1736.

Euler, in his Mechanics, employs the same principles for putting his problems into equations; but he follows throughout the analytical mode of investigation; which, by reducing all the branches of this doctrine to uniformity, greatly facilitates the connection of the different parts with each other. This work was published in 1736.

But although the method hitherto adopted of laying the foundation of the calculus was convenient, it was not the most perspicuous that might have been used. It is obvious, that instead of the processes employed by most of the foregoing philosophers, the forces and motions at every instant may be resolved into other forces and motions parallel to fixed lines, having determined positions in space. In which case it is only necessary to apply to these motions and forces the equations that involve the principle of the force of inertia, or, as it is commonly termed, the vis inertia, and there will then be no longer any occasion to have recourse to the theorem of Huygens.

Maclaurin, in his "Treatise on Fluxions," first sug.1742. gested this happy idea, which threw a new light on the entire theory of Mechanics. But unfortunately this simple principle has been neglected by later English authors, and much of what our mathematicians at present know and practice of this method, we owe chiefly to the re-importation of it through the medium of modern French works; and many, perhaps,

The

who are admiring the facility which is thus thrown into Mechanics. mechanical investigations of the greatest difficulty, are unconscious that this thought had its origin in our own land. The same fate has attended several other brilliant discoveries in the mathematical sciences. doctrine of increments, due to Dr. Brooke Taylor; the limitation in the irregularities of the orbits of the planets, suggested by Simpson; various important analytical deductions discovered by Dr. Waring, and afterwards re-modelled into the form of the calculus of derivations; and many curious properties of rotatory motion of bodies about fixed axes, resulting from the profound investigations of Landen: are a few of the many instances that might be enumerated of the unpardonable neglect on the part of modern English mathematicians, in yielding to foreign authors the honour of discoveries which are legitimately due to ourselves; and which, for the sake of the scientific character of England, and in pure justice to the memory of these distinguished philosophers, it is our duty to assert and to defend. In stating this, however, we by no means wish to detract from the wellearned fame of the French mathematicians; we only regret that none of our countrymen should have been found to pursue the ideas thus suggested by Englishmen; and that the fruit should not have been matured in the same soil where the seed was first

sown.

Returning from this digression, and referring still to the method of resolution proposed by Maclaurin; it is obvious, that when the body moves constantly in one plane, two fixed axes only will be necessary; and these for the sake of simplicity we may suppose at right angles to each other: but if by the nature of the forces the path of the body be continually changing, so as tó describe a curve of double curvature, three axes must be employed perpendicular to each other, or forming the edges of a right angled parallelopipedon. This method of reducing a problem into equations is now commonly adopted by foreign mathematicians; and the generality which is thus given to mechanical investigations has been the means of obtaining the solution of numerous difficult questions which would in all probability have resisted every other mode of research.

Soon after Huygens first gave his solution of the James problem of the centre of oscillation, to which we have Bernoulli. above referred, his principle was objected to by many A.D.1686. mathematicians; but it was defended by James Bernoulli, in the Leipsic acts for 1686, who there undertook to give a direct demonstration of it by means of the lever. He began by considering only two weights attached to an inflexible rod void of gravity, which was supposed to be put in motion on a horizontal fixed axis. Observing then that the velocity of the weight, nearest to the axis of rotation, must necessarily be less, and that of the other greater, than if each acted on the rod separately, he came to this conclusion; that the force lost and the force gained balance each other, and that consequently the product of the quantity of matter in the one into the velocity lost, and that of the other into the velocity gained, must be inversely proportional to the arms of the lever. This reasoning, though accurate in the main, involved a defect which was first pointed out by de l'Hopital, viz. that James Bernoulli l'Hopital. began by considering the velocities of the two bodies as finite; instead of which he ought only to have considered the elementary velocities, and have compared them with

described in the direction of the forces; and as negative, Mechanic those described in a contrary direction."

When the forces on a point or a body, or a system of bodies, are not so proportioned as to maintain the system in equilibrio, motion necessarily takes place; and the laws of this motion may be deduced by extending the principles which are employed in the investigation of the state of equilibrium. This is the method pursued by Lagrange, in his "Mecanique Analytique," and subsequently by Laplace. Lagrange, with the principle of virtual velocities, combines what is commonly called the principle of D'Alembert, which is extremely simple, and may indeed be considered as an axiom; or we would rather say, a practical modification of Newton's 3d law.

Mechanics, the similar velocities produced every instant by the action of gravitation. The latter author, adopting this method, without departing in any other respect from Bernoulli's principle, arrived at the same conclusion for two bodies; and in order thence to proceed to a third he united the two former at their centre of oscillation, and again proceeded as with two weights, which he afterwards combined in the same manner, and so on. This induced Bernoulli to revive his former solution, in order to extend it generally to any number of bodies; a problem which he finally accomplished, by resolving the motion of each body at every instant into two other motions; the one, that which the body actually takes, and the other that which is destroyed; and thus forming equations which express the condition of equilibrium between the motions lost: by these means the problem is brought within the range of the ordinary laws of statics. The author applied the same principle to several examples, and demonstrated strictly, and in the most evident manner, the proposition which Huygens employed as the basis of his solution, and which, since that time, has been commonly denominated the principle conservatio virium vivarum, i. e. the conservation of living forces; which may be expressed shortly in the following terms: "In the motion of any system of bodies, the sum of the products of the masses by the squares of the velocities at every instant is the same, whether the bodies have descended conjointly in any manner whatever, or have each descended freely through the same vertical heights." And as Bernoulli, after the example of Leibnitz, had assumed the mass into the square of the velocity, as the measure of a force, we see, in the above, the derivation and origin of the denomination "conservatio virium vivarum" given to this principle; though it is not strictly correct with reference to the measure of force now generally adopted. The propriety of the expression however is of but little importance; while the principle which it involves is of great utility, as it affords us the means of immediately forming an equation between the velocities of the bodies and the variable quantities which determine their position in space; so that when, by the nature of the problem, these variable quantities are reduced to one, this equation, is alone sufficient for its solution.

Virtual

Besides the principle of the conservation of the tis velocities. viva, other general properties or principles have been proposed, both prior and subsequently to this, the origin of some of which we have had occasion to notice in our preceding remarks; viz. the principle of virtual velocities; the conservation of the motion of the centre of gravity; the conservation of equal areas; and the principle of least action.

The former, i. e. the principle of virtual velocities, we have already alluded to, in speaking of the discoveries of Galileo; but it was not treated by that author in all the generality which it has since attained. According to the present signification of this expression, it asserts, that "If a system, composed of any number of bodies or points, which are drawn in any direction, by any forces, be in equilibrium; and a small motion be given to the system, by virtue of which every point describes an infinitely small space, which will express its virtual velocity; then the sum of the forces, multiplied each by the space described by the point to which it is applied, in the direction of the force, will be equal to zero; estimating as positive the small spaces

bert.

It is this: "If several bodies have a tendency to D'Alemmotion, with velocities and in directions which they are constrained to change, in consequence of their recipro- A.D. 1743 cal action on each other; then these motions may be considered as compounded of two others: i. e. one which the bodies actually take, and the other such, that had they been acted upon by these alone, they would have remained in equilibrio." Here, since the force by which the second motion is destroyed, arises merely from the connection between the bodies, whatever such force may be, the body effected by it will, by virtue of the 3d law of motion, exert an equal re-action on the other body or bodies of the system, and destroy an equal quantity of motion. D'Alembert's proposition simply equates these equal quantities of motion. This observation we regard as due to the impartiality of historical deduction.

The principle of the conservation of the motion of the Conservacentre of gravity, to which we have already made some tion of the reference, is that which Newton has adopted in his centre of Principia. He there demonstrates that the state of regravity. pose or of motion of the centre of gravity of several bodies is not altered by the reciprocal action of the bodies on each other in any manner whatever; so that the centre of gravity of bodies which act upon each other, either by cords or levers, or by the laws of attraction, independently of any exterior action or obstacle, remains always in repose, or moves uniformly in a right line.

· D'Alembert has also extended this theorem, by showing, that if every body of the system is solicited by a constant accelerating force, acting either in parallel lines or directed towards a fixed point, and varying with the distance; the centre of gravity will describe the same curve as if the bodies were free. It is evident that this principle serves to determine the motion of the centre of gravity independently of the respective motions of the bodies.

As to the third of the above principles, viz. the conConservaservation of equal areas, it is of much more recent date tion of than those to which we have already referred, and ap- equal area pears to have been discovered simultaneously by Euler, Bernoulli, and the chevalier D'Arcy, but under different forms. According to the two former, the principle consists in this: that in the motion of several bodies about a fixed axis, the sum of the products of the mass of each body, by the velocity of rotation round the centre, and by its distance from the same centre, is always independent of any mutual action which the bodies A D.174( may exert on each other, and preserves itself the same as long as there is no exterior action or obstacle. Daniel Bernoulli gave this principle in the first volume of the memoirs of the academy of Berlin in 1746, and D'Alembert the same year in his "Opuscula."

ibanics, principle of D'Arcy, as given in the Memoires de l'Academie des Sciences for 1746, but not published till 1752, is this: that the sum of the products of the mass of each body, by the area traced by its radius vector about a fixed point, is always proportional to the time. This is obviously only a generalization of one of the laws of Kepler, or the beautiful theorem of Newton of the equalities of areas described by centripetal forces: and to perceive its analogy, or rather its identity with that of Euler and Bernoulli, it is sufficient to recollect that the velocity of rotation is expressed by the element of the circular arc, divided by the element of the time, and that the first of these elements, multiplied by the distance from the centre, gives the element of the area described about this centre; so that this last principle is only the differential expression of that of M. D'Arcy. The author afterwards gave this principle another form, which renders it still more analagous to the preceding: i.e." that the sum of the products of the masses into the velocities, and into the perpendiculars drawn from the centre to the direction of the forces, is always a constant quantity." Under this point of view, the author established a species of metaphysical principle, which he calls the conservation of action, as a convenient substitute for the principle of least action. This principle, viz. of the conservation of action, whether we choose to consider it as an ultimate cause, or as a necessary consequence of some other ultimate cause, takes place in every system of bodies which act on each other in any manner whatever, whether by cords, inflexible lines, attractions, &c. or whether solicited by forces directed to a centre; and the same obtains, whether the system be entirely free or constrained to move about the same centre: i. e. "the sum of the product of the masses into the areas described about the centre, and projected on any plane whatever, is always proportional to the time; so that referring these areas to three rectangular planes, three differential equations are obtained of the first order, between the time and the co-ordinates of the curve described by the bodies;" and it is properly in these equations, that the nature of the principle exists. leofThe fourth of the principles above enumerated, viz. the principle of least action, so called by Maupertius, 1744 and which the writings of several eminent authors have since rendered celebrated, considered analytically, is as follows: "that in the motion of bodies which act on one another, the sum of the products of the masses into the velocities, and into the spaces described, is a minimum." From this principle, the author has deduced the laws of reflection and refraction of light; and Laplace has lately shown its application in investigations relative to the properties of doubly refracting crystals.

It ought to be observed, however, that these applications are too partial to establish the truth of a general principle, besides that they are too vague and arbitrary, which renders the consequences uncertain with regard to the principle itself. It ought not therefore to be classed with the other principles above discussed; although there is one point of view in which it may be considered more general than any of them, and which therefore merits the attention of philosophers. Euler gave the first indication of this in a treatise on Isoperimetrical problems printed at Lausanne in 1744; showing that in trajectories described by central forces, the integral of the velocity, multiplied into the element of the curve, is always a maximum or a minimum.

This principle was only known to Euler as appertaining Mechanics. to isolated bodies: but Lagrange afterwards extended it to the motion of bodies which act on each other in any Lagrange. manner whatever; from which there results this new general principle, viz. the sum of the products of the 4.D.1788. masses into the integrals of the velocities, multiplied into the elements of the spaces described, is constantly a maximum or a minimum. By combining this with the principle of ris viva, a solution of many difficult problems in dynamics may be obtained.

Such is a general outline of the chronological developement of the principles of Mechanics, at least so far as the subject can be treated without involving in our sketch much that relates to the differential calculus: for these two subjects are so intimately connected in many respects, or at least all investigations of the higher kind in dynamics, are so essentially dependent upon the principles of the modern analysis, that it is impos sible to enter into a more minute illustration of the former, without embracing more of the latter than properly belongs to an article which professes only to give a concise view of the rise, progress, and present state of the mechanical sciences.

The preceding sketch of the history of Mechanics re- Principal lates rather to the successive improvements that have works. been introduced into that science, than to an account of the several works which have been written on the subject we propose, however, in conclusion, to enumerate some of the most important of these; those to which a reader may with confidence refer for the most accurate information.

Among English works of this kind, are chiefly to be noticed Emerson's Principles of Mechanics, 4to.: Gregory's Treatise of Mechanics, 3 vols. 8vo.: Bridge's Treatise of Mechanics, 8vo.: Parkinson's System of Mechanics, 4to. : and Atwood on the Rectilinear Motion and Rotation of Bodies, 8vo. Emerson's work, besides the theory, contains a variety of information relative to the construction of many useful machines; but Gregory's Mechanics has in a great measure superseded it, by extending both the theory and practice of the science to a length unknown in the days of the former author. For acquiring the first principles of the science, the student might probably consult to advantage Mr. Bridge's book, particularly on account of the numerous practical examples with which it abounds. kinson's Mechanics, and Atwood on the Rectilinear and Rotatory Motion of Bodies, are each highly respectable performances, particularly the latter; although the number of press errors with which it is crowded will be found a great inconvenience to the student.

Par

The above are the principal English works to which we think it requisite to refer: and of foreign authors the following are, we believe, all that it is necessary to enumerate, viz. Euler, "Tractatus de Motu:" Lagrange, "Mecanique Analytique ;"-a work of great merit and originality, but which can only be advantageously consulted by such as are already well read on the subject: Prony "Mecanique Analytique," and "Architecture Hydraulique;" to which we ought also to add the treatises of Bossut, Poisson and Boucharlat: Poisson's work in particular, which is contained in two octavo volumes, we consider to be a very valuable performance, exhibiting in a concise and accurate manner, a complete view of the present improved state of the mechanical sciences,

§ II. STATICS. Definitions, &c.

1. Statics, as we have already stated, is that branch of Mechanics which treats of the equilibrium of solid bodies.

2. Force, or power, is used to denote any cause or agency, which has a tendency to put, or which actually puts a body at rest into motion, or which has a tendency to change or destroy, or which actually changes or destroys that motion which a body may already possess.

Forces may be divided into three distinct classes, viz. impulsive, uniform, and variable.

3. Impulsive forces, are those which act but for an instant, after which they cease to have effect.

4. Uniform forces, are those which act constantly, or incessantly, with the same degree of energy; and these are further denominated accelerating or retarding forces, according as they are employed in the production or in the destruction of motion.

5. Variable forces are those which act incessantly, but instant with different degrees of energy.

every

It is, however, only the first and most general of the above definitions which appertains to the theory of

Statics.

6. A force acting on a material point may either attract or draw towards it, or repel or push from it; the former of these is called an attracting, and the latter a repelling force; but in the following pages, unless the contrary be expressed, an attractive force is to be understood.

7. The effect of a force depends, 1st, upon its intensity; 2d, upon its point of application; 3d, upon the direction in which it is estimated; and 4th, upon its own direction with reference to that in which the estimation is made.

8. The intensity of a force is its greater or less faculty of producing or destroying motion.

9. A material point being urged by any single force, ought necessarily to move in a right line; for no reason can be assigned for its deviation, either to the one side or the other. Or whatever reason we may imagine for a change of direction on the one side, will apply equally to a symmetrical change on the other; whence, as the body cannot move in both these directions at the same time, we conclude, that it will follow neither, and that its entire direction will be a right line.

10. And the right line in which a force acts, is called

14. The unit of force being arbitrary, we may esti- Mechanics mate it by any part of the right line which expresses its direction.

15. When a force is applied to a body, all the points or particles of which are immoveably connected with each other, it is obvious that one point cannot move without a corresponding change in all the others; and consequently a force applied to any point, will have the same effect as if it were applied at any other point taken in the direction of that force, viz. A B C D (fig. 1, plate 1, STATICS), representing any body, and EF the direction of any force acting upon it, the effect of this force will be the same, whether we suppose it attached to the point E, or to any other point, M or N, in the same line of direction.*

It follows from the preceding articles, that if in the line of direction we place an immoveable obstacle, the force can have no effect.

III. On the method of determining the direction of a force in space.

forces.

16. We know, from the principles of geometry, that Direction the position of a point situated in a given plane, or any point in a plane curve, may always be determined by means of two rectangular co-ordinates, which we commonly call the absciss and ordinate; but that to fix the position of a point in space, or to determine any point in a curve of double curvature, three such axes become necessary, and three only are sufficient in all

cases.

In the same manner, the direction of any force acting in a given plane, will be determined by referring it to any two rectangular axes passing through its point of application; and when the plane in which it acts is not given, its direction may still be determined by means of three such rectangular axes.

Let MD (fig. 2) denote the direction of any force, M being its point of application; through which draw the three axes Ma, My, M z, parallel to the coordinates which determine the position of the point D, and which we will suppose to be rectangular; then if the angles a, B, y be known, which the line of direction MD makes with each of those axes, its position will be determined; for there is a necessary relation between these angles, such that whatever may be the value of two of them, the third always depends upon the equation, cos'a + cos' + cos3y = 1,

γ

radius being assumed equal to unity.

is

That is, the angles a and ß being given, the angle y determined from the equation

- cos23.} cos y = ± √ {1 — cos'a The preceding relation may be demonstrated as follows. Let x, y, z (fig. 3) be taken to denote the three co-ordinates MP, PQ, QD, of any point D, taking in the direction MD of the given force: then we shall have in the triangle MDQ, which is right angled at Q,

MD2 + QD2 = MD2:

Again, in the triangle MQP, right angled at P, MQ2 MP + QP2; and substituting this value in the preceding equation,

we obtain

The reader, however, must here divest himself of any idea of resistance or weight in the body; for this being supposed, it will have a motion of rotation, a consideration which belongs to the doctrine of Dynamics.

Conserva

Da of

action.

Mechanics, principle of D'Arcy, as given in the Memoires de l'Academie des Sciences for 1746, but not published till 1752, is this: that the sum of the products of the mass of each body, by the area traced by its radius vector about a fixed point, is always proportional to the time. This is obviously only a generalization of one of the laws of Kepler, or the beautiful theorem of Newton of the equalities of areas described by centripetal forces: and to perceive its analogy, or rather its identity with that of Euler and Bernoulli, it is sufficient to recollect that the velocity of rotation is expressed by the element of the circular arc, divided by the element of the time, and that the first of these elements, multiplied by the distance from the centre, gives the element of the area described about this centre; so that this last principle is only the differential expression of that of M. D'Arcy. The author afterwards gave this principle another form, which renders it still more analagous to the preceding: i.e." that the sum of the products of the masses into the velocities, and into the perpendiculars drawn from the centre to the direction of the forces, is always a constant quantity." Under this point of view, the author established a species of metaphysical principle, which he calls the conservation of action, as a convenient substitute for the principle of least action. This principle, viz. of the conservation of action, whether we choose to consider it as an ultimate cause, or as a necessary consequence of some other ultimate cause, takes place in every system of bodies which act on each other in any manner whatever, whether by cords, inflexible lines, attractions, &c. or whether solicited by forces directed to a centre; and the same obtains, whether the system be entirely free or constrained to move about the same centre: i. e. "the sum of the product of the masses into the areas described about the centre, and projected on any plane whatever, is always proportional to the time; so that referring these areas to three rectangular planes, three differential equations are obtained of the first order, between the time and the co-ordinates of the curve described by the bodies;" and it is properly in these equations, that the nature of the principle exists. Principle of The fourth of the principles above enumerated, viz. est action. the principle of least action, so called by Maupertius, A.D.1744, and which the writings of several eminent authors have since rendered celebrated, considered analytically, is as follows: "that in the motion of bodies which act on one another, the sum of the products of the masses into the velocities, and into the spaces described, is a minimum." From this principle, the author has deduced the laws of reflection and refraction of light; and Laplace has lately shown its application in investigations relative to the properties of doubly refracting crystals.

It ought to be observed, however, that these applications are too partial to establish the truth of a general principle, besides that they are too vague and arbitrary, which renders the consequences uncertain with regard to the principle itself. It ought not therefore to be classed with the other principles above discussed; although there is one point of view in which it may be considered more general than any of them, and which therefore merits the attention of philosophers. Euler gave the first indication of this in a treatise on Isoperimetrical problems printed at Lausanne in 1744; showing that in trajectories described by central forces, the integral of the velocity, multiplied into the element of the is always a maximum or a minimum..

curve,

This principle was only known to Euler as appertaining Mechanics. to isolated bodies: but Lagrange afterwards extended it to the motion of bodies which act on each other in any Lagrange. manner whatever; from which there results this new general principle, viz. the sum of the products of the A.D.1788. masses into the integrals of the velocities, multiplied into the elements of the spaces described, is constantly a maximum or a minimum. By combining this with the principle of ris viva, a solution of many difficult problems in dynamics may be obtained.

Such is a general outline of the chronological developement of the principles of Mechanics, at least so far as the subject can be treated without involving in our sketch much that relates to the differential calculus: for these two subjects are so intimately connected in many respects, or at least all investigations of the higher kind in dynamics, are so essentially dependent upon the principles of the modern analysis, that it is impos sible to enter into a more minute illustration of the former, without embracing more of the latter than properly belongs to an article which professes only to give a concise view of the rise, progress, and present state of the mechanical sciences.

The preceding sketch of the history of Mechanics re- Principal lates rather to the successive improvements that have works. been introduced into that science, than to an account of the several works which have been written on the subject: we propose, however, in conclusion, to enumerate some of the most important of these; those to which a reader may with confidence refer for the most accurate information.

Among English works of this kind, are chiefly to be noticed Emerson's Principles of Mechanics, 4to.: Gregory's Treatise of Mechanics, 3 vols. 8vo.: Bridge's Treatise of Mechanics, 8vo.: Parkinson's System of Mechanics, 4to.: and Atwood on the Rectilinear Motion and Rotation of Bodies, 8vo. Emerson's work, besides the theory, contains a variety of information relative to the construction of many useful machines; but Gregory's Mechanics has in a great measure superseded it, by extending both the theory and practice of the science to a length unknown in the days of the former author. For acquiring the first principles of the science, the student might probably consult to advantage Mr. Bridge's book, particularly on account of the numerous practical examples with which it abounds. Parkinson's Mechanics, and Atwood on the Rectilinear and Rotatory Motion of Bodies, are each highly respectable performances, particularly the latter; although the number of press errors with which it is crowded will be found a great inconvenience to the student.

The above are the principal English works to which we think it requisite to refer: and of foreign authors the following are, we believe, all that it is necessary to enumerate, viz. Euler, "Tractatus de Motu:" Lagrange, "Mecanique Analytique;"-a work of great merit and originality, but which can only be advantageously consulted by such as are already well read on the subject: Prony "Mecanique Analytique," and "Architecture Hydraulique;" to which we ought also to add the treatises of Bossut, Poisson and Boucharlat: Poisson's work in particular, which is contained in two octavo volumes, we consider to be a very valuable performance, exhibiting in a concise and accurate manner, a complete view of the present improved state of the mechanical sciences,