preservation of: he cannot for his life do it Addison). 39. FOR all. Notwithstanding (South).

FOR. Conj. 1. The word by which the reason is given of something advanced before (Cowley). 2. Because; on this account that (Spenser). 3. FOR as much. In regard to that; in consideration of (Hooker). 4. FOR why. Because; for this reason that (Knoll.). To FORAGE. v. a. (from foris, abroad, Lat.) 1. To wander far; to rove at a distance: not in use (Shakspeare). 2. To wander in search of provisions (Denham). 3. To ravage; to feed on spoil (Shakspeare).

To FO'RAGE. v. a. To plunder; to strip; to spoil (South).

FO'RAGE. s. (fourage, Germ. and French, from foris, Latin.) 1. Search of provisions; the act of feeding abroad (Milton). 2. Provisions sought abroad (Dryden). 3. Provisions in general (Dryden).

FORA MEN. foramen, from foro, to pierce.) A little opening.

FORA'MEN CECUM. A single opening in the basis of the cranium between the ethnoid and the frontal bone, that gives exit to a small vein.

FORA'MEN OVALE. The opening between the two auricles of the heart of the foetus. See also INNOMINATUM OS.

FORAMINA LACERA IN BASI CRANII, A pair of foramina in the basis of the cranium, through which the internal jugular veins and the eighth pair of accessory nerves pass. FORA'MINOUS. a. (from foramen, Lat.) Full of holes; porous (Bacon).

T. FORBEAR. v. n. pret. I forbore, anciently forbare; part. forborn. (Ƒoɲbæɲan, Saxon.) 1. To cease from any thing; to intermit. 2. To pause; to delay (Shakspeare). 3. To omit voluntarily; to abstain (Denham). 4. To restrain any violence of temper; to be patient (Proverbs).

To FORBEAR. ". a. 1. To decline; to avoid voluntarily (Waller). 2. To abstain from; to omit (Clarendon). 3. To spare; to treat with clemency (Ephesians). 4. To withhold (Chronicles).

FORBEARANCE. s. (from forbear.) 1. The care of avoiding or shunning any thing; negation of practice (South). 2. Intermission of something. 3. Command of temper (Shakspeare). 4. Lenity; delay of punishment; mildness (Rogers).

FORBEARER. s. (from forbear.) An intermitter; interceptor of any thing (Tusser). To FORBID. v. a. pret. I forbade; part. forbidden or forbid (popbeovan, Sax.) 1. To prohibit; to interdict any thing (Shaksp.). 2. To command to forbear any thing (Sidney). 3. To oppose; to hinder (Dryden). 4. To accurse; to blast: obsolete (Shakspeare).

To FORBI'D. v. n. To utter a prohibition (Shakspeare).

FORBIDDANCE. s. (from forbid.) Prohibition; edict against any thing (Milton). FORBIDDENLY. ad. (from forbid.) In an unlawful manner (Shakspeare).

VOL. V.

FORBIDDER. s. (from forbid.) One that prohibits (Brown).

FORBIDDING. particip. a. (from forbid.) Raising abhorrence (Aaron Hill).

FORCE. s. (force, French.) 1. Strength; vigour; might (Donne). 2. Violence (Dryden). 3. Virtue; efficacy (Locke). 4. Va◄ liduess; power of law (Denham). 5. Armament; warlike preparation (Waller). 6. Destiny; necessity; fatal compulsion. To FORCE. v. a. (from the noun.) 1. To compel; to constrain (Swift). 2. To overpower by strength (Milton). 3. To impel to press; to draw or push by main strength (Dryden). 4. To enforce; to urge (Milton). 5. To drive by violence or power (Decay of Piety). 6. To gain by violence or power (Dryden). 7. To storm; to enter by violence (Waller). 8. To ravish; to violate by force (Dryden). 9. To constrain; to distort (Ad dison). 10. To man; to strengthen by soldiers (Raleigh). 11. To FORCE out. extort (Atterbury).

To

To FORCE. v. n. To lay stress upon (Camden).

FORCE, OF POWER, in physics, is that which causes a change in the state of a body, whether that state be rest or motion.

We speak here of proximate causes, for it is not the business of mechanics to search into the essential and hidden causes of motion. The enquiry whether they are material or spiritual may exercise the talents of ingenious speculatists, and may, perhaps, be of some importance in a moral point

of view; but certainly forms no part of the prin

ciples of mechanical science. The muscular power of animals, as likewise pressure, impact, gravity, electricity, &c. are by us looked upon as forces, or sources of motion; for it is an incontrovertible fact that bodies exposed to the free action of either of these are put into motion, or have the state of their motion changed. All forces, however various, are measured by the effects they produce in like circumstances; whether the effects be creating, accelerating, retarding, or deflecting motious: the effect of some general and commonly observed force is taken for unity; and with this any others may be compared, and their proportions represented by numbers or by lines: in this point of view they are considered by the mathematician; all else falls within the province of the universal philosopher or the metaphysician. When we say that a force is represented by a right line AB it is to be understood that it would cause a material point situated at rest in A, to run over the line AB (which we name the direction of the power) so as to arrive at B, at the end of a given time; while another power should cause the same point to have moved a greater or less distance from A in the same time.

Mechanical forces may be reduced to two sorts; one of a body at rest, the other of a body in motion.

The force of a body at rest, is that which we conceive to be in a body lying li on a table, or hanging by a rope, or supported by a spring, &c.; and this is called by the names of pressure, tension, force, or vis mortua, solicitatio, conatus movendi, conamen, &c.; which kind of force may be always measured by a weight, viz. the weight that sustains it. To this class of forces may also be referred

D

centripetal and centrifugal forces, though they reside in a body in motion; because these forces are homogeneous to weights, pressures, or tensions of any kind. The pressure, or force of gravity in any body, is proportional to the quantity of matter

in it.

The force of a body in motion, is a power residing in that body, so long as it continues its motion; by means of which, it is able to remove obstacles lying in its way; to lessen, destroy, or overcome the force of any other moving body, which meets it in an opposite direction; or to surmount any the largest dead pressure or resistance, as tension, gravity, friction, &c. for some time; but which will be lessened or destroyed by such resistance as lessens or destroys the motion of the body. This is called vis motrix, moving force, or motive force, and by some late writers vis viva, to distinguish it from the vis mortua, spoken of before.

Concerning the measure of moving force mathematicians have been divided into two parties. It is allowed on both hands, that the measure of this force depends partly upon the mass of matter in the body, and partly upon the velocity with which it moves: the point in dispute is, whether the force varies as the velocity, or as the square of the velocity.

Descartes, and all the writers of his time, assumed the velocity produced in a body as the measure of the force which produces it; and observing that a body, in consequence of its being in motion, produces changes in the state or motion of other bodies, and that these changes are in the proportion of the velocity of the changing body, they as serted that there is in a moving body a vis insita, an inherent force, and that this is proportional to its velocity; saying that its force is twice or thrice as great, when it moves twice or thrice as fast at one time as at another. But Leibnitz observed, that a body which moves twice as fast, rises four times as high, against the uniform action of gravity; that it penetrates four times as deep into a piece of uniform clay; that it bends four times as many springs, or a spring four times as strong to the same degree; and produces a great many effects which are four times greater than those produced by a body which has half the the initial velocity. If the velocity be triple, quadruple, &c. the effects are nine times, 16 times, &c. greater; and, in short, are proportional, not to the velocity, but to its square. This observation had been made before by Dr. Hooke, who has enumerated a prodigious variety of important cases in which this proportion of effect is observed. Leibnitz, therefore, affirmed that the force inherent in a moving body is proportional to the square of the velocity. It is evident that a body, moving with the same velocity, has the same inherent force, whether this be employed to move another body, to bend springs, to rise in opposition to gravity, or to penetrate a mass of soft matter. Therefore these measures, which are so widely different, while each is agreeable to a numerous class of facts, are not measures of this something inherent in the moving body which we call its force, but are the measures of its exertions when modified according to the circumstances of the case; or, to speak still more cautiously and securely, they are the measures of certain classes of phenomena consequent on the action of a moving body. It is in vain, therefore, to attempt to support either of them by demonstration. The measure itself is nothing but a definition. The Cartesian calls that a double force which produces a

double velocity in the body on which it acts. The Leibs nitzian calls that a quadruple force which makes a quadruple penetration. The reasonings of both in the demonstration of a proposition in dynamics may be the same, as also the result, though expressed in different numbers.

But the two measures are far from being equally proper; for the Leibnitzian measure obliges us to do continual violence to the common use of words. When two bodies moving in opposite directions meet, strike each other, and stop, all men will say that their forces are equal, because they have the best test of equality which we can devise. Or when two bodies in motion strike the parts of a machine, such as the opposite arms of a lever, and are thus brought completely to rest, we and all men will pronounce their mutual energies by the intervention of the machine to be equal. Now, in all these cases, it is well known that a perfect equality is found in the products of the quantities of matter and velocity. Thus a ball of two pounds, moving with the velocity of four feet in a second, will stop a ball of eight pounds moving with the velocity of one foot per second. But the followers of Leibnitz say, that the force of the first ball is four times that of the second.

All parties are agreed in calling gravity an uni form or invariable accelerating force; and the definition which they give of such a force is, that it always produces the same acceleration, that is, equal accelerations in equal times, and therefore produces augmentations of velocity proportionable to the times in which they are produced. The only effect ascribed to this force, and consequently the only thing which indicates, characterises, and measures it, is the augmentation of velocity. What is this velocity, considered not merely as a mathematical term, but as a phenomenon, as an event, a production by the operation of a natural cause? It cannot be conceived any other way than as a determination to move on for ever at a certain rate, if nothing shall change it. We cannot conceive this very clearly. We feel ourselves forced to animate, as it were, the body, and give it not only a will and intention to move in this manner, but a real exertion of some faculty in consequence of this determination of mind. We are conscious of such a train of operations in ourselves; and the last step of this train is the exertion of energy of some natural faculty, which we, in the utmost propriety of language, call force. By such analogical conception we suppose a sontething, an energy inherent in the moving body; and its only office is the production and continuation of this motion, as in our own case. Scientific curiosity was among our latest wants, and language was formed long before its appearance: as we formed analogical conceptions, we contented ourselves with the words already familiar to us, and to this something we gave the name FORCE, which expressed that energy in ourselves which bears some resemblance (in office at least) to the determination of a body to move on at a certain rate. This sort of allegory pervades the whole of our conceptions of natural operations, and we can hardly think or speak of any operation without a language, which supposes the animation of matter. And, in the present case, there are so many points of resemblance between the effects of our exertions and the operations of nature, that the language is most expressive, and has the strongest appearance of propriety. By exerting our force, we not only move and keep in motion, but we move other

bedies. Just so a ball not only moves, but puts other bodies in motion, or penetrates them, &c.This is the origin of that conception which so forcibly obtrudes itself into our thoughts, that there is inherent in a moving body a force by which it produces changes in other bodies. No such thing appears in the same body if it be not in motion. We therefore conclude, that it is the production of the moving force, whatever that has been. If so, it must be conceived as proportional to its producing cause. Now this force, thus produced or exerted in the moving body, is only another way of conceiving that determination which we call velocity, when it is conceived as a natural event. We can form no other notion of it. The vis insita, the determination to move at a certain rate, and the velocity, are one and the same thing, considered in different relations.

Therefore the vis insita corpori moventi, the determination to move at a certain rate, and the velocity, should have one and the same measure, or any one of them may be taken for the measure of the other. The velocity being an object of perception, is therefore a proper measure of the inherent force; and the propriety is more evident by the perfect agreement of this use of the words with cominon language. For we conceive and express the action of gravity as uniform, when we think and say that its effects are proportional to the times of its action. Now all agree, that the velocity produced by gravity is proportional to the time of its action. And thus the measure of force, in reference to its producing cause, perfectly agrees with its measure, independent of this consideration.

But this agreement is totally lost in the Leibnitzian doctrine; for the body which has fallen four times as far, and has sustained the action of gravity twice as long, is said to have four times the force.

The quaintness and continual paradox of expression which this measure of inherent force leads us into, would have quickly exploded it, had it not been that its chief abettors were leagued in a keen and acrimonious warfare with the British mathematicians who supported the claim of Sir Isaac Newton to the invention of fluxions. They rejoiced to find in the elegant writings of Huyghens a physical principle of great extent, such as this is, which could be set in comparison with some of the wonderful discoveries in Newton's Principia. The fact, that in the mutual actions of bodies on each other, the products of the masses and the squares of the velocities remain always the same (which they call the conservatio virium vivarum), is of almost universal extent; and the knowledge of it enabled them to give ready and elegant solutions of the most abstruse and intricate problems, by which they acquired a great and deserved celebrity. Dr. Robert Hooke, whose observation hardly any thing escaped, was the first (long before Huyghens) who remarked that in all the cases of the gradual production and extinction of motion, the sensible phenomenon is proportional to the square of the produced or extinguished velocity.

John Bernoulli brought all these facts together, and systematized them according to the principle advanced by Huyghens in his treatise on the centre of oscillation. He, and Daniel Bernoulli, gave most beautiful specimens of the prodigious use of this principle, for the solution of difficult physical problems, in their dissertations on the motion and impulse of fluids, and on the communication of motion. It was, however, very early objected to them (we think by Marquis Poleni), that in the

collision of bodies, perfectly hard, there was ne such conservatio virium vivarum; and that, in this case, the forces must be acknowledged to be proportional to the velocities. The objections were unanswerable. But John Bernoulli evaded their force, by affirming, that there were and could be no bodies perfectly hard. This was the origin of another celebrated doctrine, on which Leibnitz greatly plumed himself, the Law of Continuity, viz. that nothing is observed to change abruptly, or per saltum. But no one will pretend to say that a perfectly hard body is an inconceivable thing; on the contrary, all will allow, that softness and compressibility are adjunct ideas, and not in the least necessary to the conception of a particle of matter; nay, totally incompatible with our notion of an ultimate atom.

Sir Isaac Newton never could be provoked to engage in this dispute. He always considered it as a wilful abuse of words, and unworthy of his attention. He guarded against all possibility of cavil, by giving the most precise and perspicuous definitions of those measures of forces, and all other quantities which he had occasion to consider, and by carefully adhering to them. And in one proposition of about 20 lines, viz. the 59th of the 1st book of the Principia, he explained every phenomenon adduced in support of the Leibnitzian doctrine, shewing them to be immediate consequences of the action of a force measured by the velocity which it produces or extinguishes. There it appears that the heights to which bodies will rise in opposition to the uniform action of gravity are as the squares of the initial velocities: so are the depths to which they will penetrate uniformly resisting matter: so is the number of equal springs which they will bend to the same degree, &c. We have had occasion to mention this proposition as the most extensively useful of all Newton's discoveries. It is this which gives the immediate application of mechanical principles to the explanation of natural phenomena. It is incessantly employed in every problem by the very persons who hold by the other measures of forces, although such conduct is virtually giving up that measure. They all adopt, in every investigation, the two theorems fi=v, and fs=vv; both of which suppose an accelerating force ƒ proportional to the velocity which it produces by its uniform action during the time 1, and the theorem ƒ ƒs=v2 is the 39th 1. Princip. and is the conservatio virium vivarum.

Let a certain force Q, such, for instance, as would propel a body B with a velocity U, be capable, by its instantaneous action, of raising a mass M, whose weight is W to a certain height H; and let g denote the force of gravity, while t is an evanescent element of time. Then, that which has been employed to raise W, to the height H, will be equivalent to WH, this being the effect produced. But H, being a space run over, may be expressed by the product of a velocity V and a time T; and, on the other hand, we have W=gM giM

=

where gt is manifestly the velocity V', which would be generated by gravity in the element of time i. Consequently WH=V'M

T

× VT

=VVM‡=»M‡ -u being the mean propor

D2

tional between the velocities V and V': and since T and are homogeneous quantities, we shall have WH Mu, the original force being thus fesolved into the product of a mass by the square of a velocity, conformably to the notion attached by most foreigners to the term vis viva. This force is, notwithstanding, measured by the product BU above: so that the warm discussions on the measure of the force of moving bodies, is reduced to a dispute about words.

Mr. Robins, in his remarks on J. Bernoulli's treatise, intitled, Discours sur les Loix de la Communication du Mouvement, informs us, that Leib nitz adopted this opinion through mistake; for though he maintained that the quantity of force is always the same in the universe, he endeavours to expose the error of Des Cartes, who also as serted, that the quantity of motion is always the same; and in his discourse on this subject in the Acta Eruditorum for 1686, he says that it is agreed, on by the Cartesians, and all other philosophers and mathematicians, that there is the same force requisite to raise a body of 1 pound to the height of 4 yards, as to raise a body of 4 pounds to the height of 1 yard; but being shewn how much he was mistaking in taking that for the common opinion, which would, if allowed, prove the force of the body to be as the square of the velocity it moved with, he afterwards, rather than own himself capable of such a mistake, endeavoured to defend it as true; since he found it was the necessary consequence of what he had once asserted; and maintained, that the force of a body in motion was proportional to the height from which it must fall, to acquire that velocity; and the heights be ing as the squares of the velocities, the forces would be as the masses multiplied by them; whereas, when a body descends by its gravity, or is projected perpendicularly upwards, its motion may be considered as the sum of the uniform and continual impulses of the power of gravity, during its falling in the former case, and till they extinguish it in the latter. Thus when a body is projected upwards with a double velocity, these uniform impulses must be continued for a double time, in order to destroy the motion of the body; and hence it follows, that the body, by setting out with a double velocity, and ascending for a double time, must arise to a quadruple height, before its motion is exhausted. But this proves, that a body with a double velocity moves with a double force, since it is produced or destroyed by the same uniform power continued for a double time, and not with a quadruple force, though it rises to a quadruple height; so that the error of Leibnitz consisted in his not considering the time, since the velocities alone are not the causes of the spaces described, but the times and the velocities together; yet this is the fallacious argument on which he first built, his new doctrine; and those which have been since much insisted on, and derived from the indentings or hollows produced in soft bodies by others falling into them, are much of the same kind. Robins's Tracts, vol. 2. p. 178. See also farther on this subject Dr. Reid's valuable disquisition on Quantity, in vol. 45, of the Philosophical Transactions, or New Abridgment, vol. 9, p. 562.

and mortua, or living and dead force, will vanish; since a pressure may always be assigned, which in the same time, however little, shall produce the same effect. If then the vis viva be homogeneous to the vis mortua, and having a perfect measure and knowledge of the latter, we need require no other measure of the former than that which is derived from the vis mortua equivalent to it.

Now that the change in the state of two bodies, by their shock, does not happen in an instant, appears evidently from the experiments made on soft bodies: in these, percussion forms a small cavity, visible after the shock, if the bodies have no elasticity. Such a cavity cannot certainly be made in an instant. And if the shock of soft bodies require a determinate time, we must certainly say as much of the hardest, though this time may be so small as to be beyond all our ideas. Neither can an instantaneous shock agree with that constant law of nature, by virtue of which nothing is performed per saltum. But it is needless to insist farther upon this, since the duration of any shock may be determined from the most certain prin ciples.

There can be no shock or collision of bodies, without their making mutual impressions on each, other: these impressions will be greater or less, according as the bodies are more or less soft, other circumstances being the same. In bodies, called hard, the impressions are small; but a perfect hardness, which admits of no impression, seems inconsistent with the laws of nature; so that while the collision lasts, the action of bodies is the result of their mutually pressing each other. This pressure changes their state; and the forces exerted in percussion are really pressures, and truly vires mortuæ, if we will use this expression, which is no longer proper, since the pretended infinite difference between the vires vivæ and mortuæ ceases.

The force of percussion, resulting from the pressures that bodies exert on each other, while the collision lasts, may be perfectly known, if these pressures be determined for every instant of the shock. The mutual action of the bodies begins the first moment of their contact; and is then. least; after which this action increases, and becomes greatest when the reciprocal impressions are strongest. If the bodies have no clasticity, and, the impressions they have received remain, the forces will then cease. But if the bodies be elastic, and the parts compressed restore themselves to their former state, then will the bodies continue to press each other till they separate. To comprehend, therefore, perfectly, the force of percussion, it is requisite first to define the time the shock lasts, and then to assign the pressure corresponding to each instant of this time; and as the effect of pressures in changing the state of any body may be known, we may, thence come at the true cause of the change of motion arising from collision. The force of percussion, therefore, is no more than the operation of a variable pressure during a given, time; and, to measure this force, we must have, regard to the time, and to the variations according, to which the pressure increases and decreases.