FOR
FOR
to which Mr. Euler obferves, that if we give the name of force to fuch caufes only as can change the ftate of bodies, the inertia^ by which all bodies remain in their ftate of mo- tion, or reft, cannot properly be called force \ although a true force may fome times be the refill t of it. For when the iner- tia preferves a body in its ftate of reft, or an uniform and di- rect progrefiion, the fame inertia may be the caufe of a change in the ftate of other bodies : fo that tho' the name of force does not belong to this inertia, with refpect to the bo- dy wherein it refides, yet it may change into a force with refpect to other bodies. Mr. Euler 1 even thinks it probable that all the changes which happen in the world, arifc, with- out exception, from the inertia of bodies, and that there are no other forces in nature than what are excited by this in- ertia. — [ a In Mem. Acad. Berlin. 1745. p. 22.] To illuftratc this, he confiders a body A (fig. I.) moving with a given velocity In the direction ac : fo long as this bo- dy meets with no obftacle, its motion will continue with the fame velocity, and in the fame direction, and thus it wil! perfevere in the fame ftate, the caufe of this perfeverance be- ing the inertia of the body. But fuppofing another body B at reft, and that the body A has approached fo near to B, that their extremities touch j what will happen ? The bodies
Fig. I.
G> — o
being impenetrable, it is plain that A cannot remain in its ftate, without difturbing that of B ; iince A mult, in order to continue its motion, either drive B before it with a velo- city equal to, or greater than its own, or it mure turn afide. Alfo the body B cannot remain in its ftate of repofe, unlefs A flops, or returns, or deflects to one fide. All this clear- ly ftlews, that thefe two bodies cannot at the fame time pre- ferve their ftate. When A touches B, the ftate of one or both mull be changed, fmce both equally endeavour to preferve their ftate : there can be no reafon why the one fhould fuffer a change rather than the other, and therefore the ftate of both muft be altered. But whatever change happens in this refpect it muft arife from the vis inertia ; for when B's ftate of reft is changed to motion, the caufe is the inertia of A ; fince B would have remained eternally at reft had it not been for the appulfe of A. In like manner, the caufe of the change which happens in the motion of the body A, can be nothing elfe than the inertia of B, fince without this A. would have pre- ferved its motion without any alteration. The inertia being the caufe of the perfeverance of a body in its ftate, cannot be conceived but as a principle of refinance to any change of ftate ; fince we could not hy that a body has power of remain- ing in its ftate, if it gave way without refiftance to any caufe endeavouring to alter that ftate. This confideration autho- rifes the giving the name of farce to the inertia, taking the term force in an extenfive fenfe. When then the body A i deavours by its inertia to preferve its uniform rectilinear mo- tion, it has at the fame time the free to refill: all obftacles ; and the body B, the inertia of which exerts itfelf in the pre- fervation of its ftate cf reft, has a force by which it refills all caufes endeavouring to draw it out of that ftate. Hence in the fhock of thefe two bodies, both being unable to preferve their ftate, becaufe of their impenetrability, and the inertia of each refilling a change, this inertia of the one muft produce a change in the other ; therefore, though the inertia cannot be called a force with refpect to the body in which it refides, becaufe it only produces a prefervation of its ftate, yet with refpect to ether bodies it may become a true force, by which 'their ftate is changed. Now as many bodies mud fhock each
• 'other in a world full of bodies differently moved, and that fome muft hinder others from pcrfevering in their ftate, it follows, that the (late of all thole bodies muft undergo perpetual changes ; and the caufe of all thefe will be the inertia by which all bo- dies have a tendency to preferve their ftate. Nor do the changes which continually happen in the world oblige us to afcribe moving forces to bodies, different from their inertia, fince
this alone "may produce all the. alterations we obferve b .
[ b Ibid. p. 23, 24.]
The inertia, as well as impenetrability and extenfion, is an univcri'al property of all matter ; and this inertia is propor- tional to the ma'fs or quantity of matter. A body, whether ' at reft or in motion, has the fame inertia, or the fame power or force to preferve its ftate. This inertia is the caufe of a body's refilling cither to a change of its velocity, or of its di- rection: and from hence arifes the two kinds of forces before mentioned'. — [ « Ibid. p. 25.]
Central Force. It may be proper here to fubjoin fomething on the head of centripetal and centrifugal, or, in one word, central forces, to what is laid in the Cyclopaedia, under the head Central. The doctrine of central or centripetal and centri- fugal forces has been much cultivated by mathematicians, as being of extenfive ufe in the theory of gravity and other phy- fico-mathematical iciences. In this doctrine it is fuppolcd, that a body at reft never moves
itfelf : and that a body in motion never changes the velocity or direction of its motion of itfelf; but that every motion would continue uniform, and its direction rectilinear, unlefs fome external force or refiftance affected it. Hence, when a body at reft always tends to move, or when the velocity of any rectilinear motion is accelerated continually, or when the di- rection of a motion is continually varied and a curve line de- fcribed, thefe are fuppofed to proceed equally from the influ- ence of fome power that acts inceflantly i which may be mea- fured either by the preffure of the quiefcent body againft the obftacle that hinders it to move, in the firft cale ; or by the acceleration of the motion, in the fecond ; or by the flexure of the curve defcribed, in the third cafe ; due regard beintr had tq the time in which thefe effects are produced, and the other circumftances, according to the principles of mechanics. Effe£ts of the power of gravity of each kind fall under our conflant obfervation near the furface of the earth ; for the fame power which renders bodies heavy while they are at reft, ac- celerates them when they defcend perpendicularly, and bends their motion into a curve line when they are projected in any other direftion than that of their gravity. But we can judge of the powers that act on the celcftial bodies by eHcAs of the laft kind only. And hence it is that the doctrine of central forces is of fo much ufe in the theory of the planetary mo- tions.
Sir Ifaac Newton has treated of central forces in book i. § 2. of his principles. Mr. De Moivre has given the following ele- gant general theorem relating to the fame fubjefl, in the Phil. Tranf. and in his Mifcel. Analyt. p. 231. Let M P Q_ be any given curve in the perimeter of which a body moves : let P be the place of the body in the curve, S the center of forces, PG the radius of concavity or curva- ture, S T the perpendicular drawn from the center of forces to the tangent of the curve in P ; then will the centripetal force
_ SP PGXSTcub.
be every where proportional to the quantity See Mifcel. Analyt. p. 231.
Fig. II.
What is here called the center of force, is the point,to which the central force is always directed.
Monf. Varignon has alfo given two general theorems on this fubject in the Memoirs of the Acad. Scienc. an. 1700, 1701, and has fhewn their application to the motions of the planets. See alfo the fame Memoirs, an. 1706, 1710. Mr. Mac Laurin has alfo treated the fubject of central forces very fully in his Fluxions, from art. 416 to 493. where he gives a great variety of expreflions for thefe forces, and feveral elegant methods of inveftigating them.
Sir Ifaac Newton has demonftrated " this fundamental theorem of central forces, that the areas which revolving bodies defcribe by radii drawn to an immoveable center, lie in the fame im- moveable planes, and are proportional to the times in which they are defcribed. — [ * Princip. lib. i. prop. I.] A late eminent mathematician obferves that this law, which is originally Kepler's, is the only general principle in the do- ctrine of centripetal forces ; but fince this law, as Sir Ifaac Newton himielf has proved, cannot hold whenever a body- has a gravity or force to any other than one and the fame point, there feems to be wanting fome law that may ferve to explain the motions of the moon and fatellites which have a gravity towards two different centers : the law he lays down for this purpofe is,
That where a body is deflected by two forces tending conilant- ly to two fixed points, it will defcribe by lines drawn from the two fixed points, equal folids in equal times, about the line joining thole fixed points. See Macbin, of the Laws of the Moon's Motion, in the poftfeript. This fllort treatife is pubhfhed at the end of the Englilh tranflation of Sir Ifaac Newton's Principles. Force, in praaical mechanics. We have feveral curious as well as uleful obligations in Defagulierss Experimental I hiloiophy, concerning the comparative forces of men and hories, and the bell way of applying them. An horle draws with the greateft advantage when the line of dircflion is level with his breaft : in (uch a fituation, he is able to draw 20olb. eight hours a day, walking about two miles and an half an hour. And if the fame horfe is made to draw 2401b. he can work but fix hours a day, and cannot go quite fo fall. On a carriage, indeed, where friction alone is to be overcome, a middling horfe will draw icoolb. But the bell way to try a 2 horfe's
