Page:Cyclopaedia, Chambers - Supplement, Volume 2.djvu/849

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Vfiicfl meeting each other in contrary directions, do not pre- vail over each other, it cannot be difputed, that bodies which have equal quantities of motion, have alio equal forces ; and confeq.uently that the mowmg forces of bodies are in a com- pound ratio of their mattes and velocities. On the other hand, if it be admitted that a given fpring bent to a given degree, always communicates the fame force to whatever body it be applied to, it is no lefs indiiputable that the forces of moving bodies will be in a compound ratio of their manes and the fquares of their velocities. For the bending of a fpring to the fame degree^ cannot be done by different manes with equal quantities of motion* or a fpring by unbending itfelf cannot communicate to different manes equal quantities of motion ; but the bending or unbending of the fpring always correfponds to and with what the Leibni- tians- call the vis viva ; that is, the product of the mafs Of a body by the fquare of its velocity. This is admitted by the Newtonians, and follows from the. avowed principles of both parties.

Thus, let M and m denote the maffes of two bodies, V and v their respective velocities ; then if any fpring, a crofs-bow for inftance, bent to a certain degree, give the body M a cer- tain velocity V, the fame fpring or crofs-bow, bent to the fame degree, will never give another body m a velocity v, fo that MV fhall be equal to mv, but will always communi- cate fuch a velocity to »/, that MV V fhall be equal to mvv. See the article Spring, Suppl.

This is admitted by the Newtonians, but the conclufion, that the forces of the bodies M and m are equal, is denied. To put an end, therefore, to this controverfy, other prin- ciples muff be found. This has been attempted by feveral au- thors, and we have had no fmall profufion of obfeure meta- phyfics on this occafion. Many fubtile reafonings have been formed froni the nature of action, caufe, effeft, time, fpacc, &c. by which we believe mere readers have been confounded than enlightened ; and after all the controverfy is flill undecided, a rid nuni remain fo while the Newtonians, on one hand, affume, that equal preffures in equal times produce equal moving forces ; and that the Leibnitians, on the contrary maintain, that equal preffures urging a body through equal fpttces produce equal forces. Hence, iUpr.oling equal preffures to act on equal bodies, ci- ther to produce motion in them, or to flop what motion they have, the queftion will be, whether the force generated or ■dejhoyed be proportional to the time the preffure acls, or to the fpace through which it dels. For example, let two equal bodies, with the velocities, as i and 2, afcend againft the action of uni- form gravity, according to Galileo's hypothecs ; it is certain that the body whofe velocity is 2, will refill: the force of gra- vity twice the time that the body whofe velocity is only 1 can : and it is no lefs certain, that the body whofe velocity Is 2, will afcend to four times the height that the other can. So that if we meafure the forces of thefe bodies by the pref- fure and time requifite to deftroy their motion thek forces will be as the velocities of the moving bodies; but if we meaunc the forces by the preffure, and fpace through which it extends, requifite to deftroy thefe forces, we fhall find them proportional to the fquares of the velocities of the moving bodies. This holds in uniform preffures, fuch as gravity is fuppofed to be near the earth ; but if the preffure be not uniform, as it is not in the action of fprings, which prefs more or lefs as they are more or lefs bent, we muff then have recourfe to the fluxions of the fpace and time. Thus if p ftand for the pref- fure, t for the time, s for the fpace ; the fluxion, or infinite- fimal element as fome call it, of the velocity will, according to both parties, be cxprefed by ft. According to the Newto- nians this is alio the fluxion or element of the force ; but ac- cording to the Leibnitians the element of the force is proportion- al to ps'. As to any demon fixation, either that in unform preflurc, en the fame body, the force produced is in propor- tion to the preffure and the time it acts, and in preffures not uniform that the element of the force is proportional to p t ; or that, on the contrary, the force thus produced is propor- tional to the preffure and fpace in the firfr. cafe, or that its element is proportional to ps in the fecond cafe, we have never been fortunate enough to meet with any conclufive ar- gument on either fide : nor do we believe any fuch demon- ilration pofffble, till fomebody (hall be metaphyfician enough to analyfe the notions of force, action, time and fpace farther than has hitherto been done. — [ - See Dan. Bernoulli in Act. Pctrcp. vol. viii. p. 100.]

It has been already mentioned, that fome Leibnitians do not affume it as a firff principle, that action or force is proportional to the preffure and fpacc; but they fay, that a preffure being given, its action will be proportional to the velocity of the point moved by that preffure. Hence they infer, that the whole action of a preffure, is at its intenfity, as the velocity of the point to which it is applied, and as the time the pref- fure acts. And fpacc being as the time and velocity, they conclude, the action of a preffure to be as that preffure and the fpace through which it acts. And hence they infer, that the force communicated by the preffure is alio as the preffure and fpace. Thus, fay they, if a point runs through a determinate fpace AE, and preffes with acertain given /srec Append.

or intenfity of preffure, it will perform the fame action whe-

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tiler it move faft or (low, and therefore the time of the ac. tion in this cafe ought not to be regarded. 'S Gravefande, lib. cit. § 723 — 728.

But the Newtoniam do not fubmit to this reafoning, and in- fill, notwithstanding, that the aflion of the preffure is as the intenfity o£ the preffure, and the time during which it afts, without any regard to the fpace through Which it afts ; and they make it an axiom, that equal preffures in equal times produce equal moving forces.

Leibnitz himfelf affumed it as certainj that the afiion is as the effect by the velocity with which it is produced : and hence he deduces, that the power is as the mafs by the fquare of the velocity : his words, as quoted by Wolrius, are, Calculum virium purarum feu aP.ionnm talent inflituo. Sit fpatimn s, tempus t, velocitas v, carpus c, effeclus e, potattia p, acliii a, in motu ecquabili erit tv ut s, e lit cs, tp lit a. Atque hate quidemfme iemmftratione affumi poftipit. Accedit quod de- monftrandum, ev at a. Huic porropltaima theoremata demon- Jirari pojjimt, e. gr. p ut cv*. Nam tp ut ev: fed e ut cs if s ut tv. Ergo fit tp ut ctv',feup ut cv~. Vide Afl. Petrop. torn. i. p. 232.

But as we cannot pretend to give a full account of all the ar- guments that have been made ufe of in this controverfy, we muft refer the curious to fome of the principal authors on this fubjefl, fuch as Sir Ifaac Newton •, Mr. Mac Laurin », Dr. Jurin % Dr. Pemberton J, Mr. Robins '-, Monf. de Mairan ', and others, on one fide : Melt Leibnitz s, John * and Da- niel Bernoulli ', Herman '■, Poleni ', Wolfius », 'S Gravef- ande n, Camus °, and many more, on the other. But not- withftanding all that has been laid, the difficulty of deter- mining whether the element of the moving force be propor- tional to pt, or to ps ftiil remains, and till that be demon- ftratively decided, we do not fee but the queftion about the meai'ure of theforce of bodies in motion, muft remain undeter- mined. See Dot. Bernoulli, in A&. Petrop. torn. i. p. 131, feq. [ Philof. Tranf. N°. 371. one of the arguments there pro- posed being Sir Ifaacs, according to Dr. Pemberton. h Ace', of Sir. li\ Newton's difcoveries. Fluxions, art. 511, in the notes. Recueil des pieces qui ont emporte le prix, CirV. torn, i. < Philof. Tranfaft. N 476, and in fome other pieces ' Phil. Tranf. N°. 371. ■ Pref. State of the Repub. of Let. May, 1728. ' Mem. de TAcad. Scienc. 1728. i r\i\. Enid. 1686. and Nouv. de ia Rep. Let. S„-pt. 16P6 art. 2. h Difcours fur les loix de la commiTnication du mouvement. oper. torn. iii. & Differt. de vera notione virium vivarum. ib. 1 A3a Petropol. torn. i. p. igj, feq. Hydrodynamica, § i torn. ii. & in Cofmolog. general. n Journ. Lit. and in Phyf. Elem. Mathem. " Mem. de lAcad. des Scienc. 17 28.] Though Leibnitz was the firft that expreily afferted the force of a body in motion to be as the fquare of its velocity, yet Huygens has been thought to have led him into this notion. This eminent mathematician had demonftrated, that in the collifions of two bodies, perfectly elaftic, the fum of the products of the bodies by the fquares of their respective ve-. locities, was the fame after the ffiock as before. And this propofition is fo far general as to obtain in all cdilifions of bodies that are perfectly elaftic. It is aho true wheii bodies of a peifect clafticity ftrike any immoveable ooitacle, as well as when they ftrike one another j or when they are conftrained by any power or reiiftance to move in directions different from there in which they impel one another *. Thefe considerations might have induced Huygens to lay it down as a general rule, that bodies conlrantly preserve their afcenfional force b, that is, the product of their mafs, by the height to which their center of gravity can afcend ; and, therefore, in a given fyftem of bodies the fum of the fquares of their ve- locities will remain the fame and not be altered by the action of the bodies among themielves, nor againft immoveable ob- ftacles. Leibnitz's metaphyseal fyftem led him to think that the fame quantity of action or force fubfifted in the univerfe ; and finding this irnpoiiible, if force were eftimatcd by the quantities of motion L, he adopted Huygens's principle of the preservation of the afcenlionalycnY, and made it the meafure of moving forces. But it is to be obferved, that Huygens's principle is general only when bodies are perfectly elaftic i and in fome other cafes, which Mr. Mac Laurin has endea- voured to diftinguim' 1. — [ See Mac Laurin's Fluxions, art. 571. * Hxc conftanskx eft, corpora fervare vim fuam afcen- dentem, c^ idcirco fummam quadratorum velocitattan ilhrmfem- per manere candem. Hoc aittcm non folum obtinet in ponderibus pendulorurn & percuffionc csrporum durorum, fed in mitltis criti- que mechanicis experimentis. Huygen. Oper. torn. i. p. 24.8. ' Huygens, p. 247. of the fame book, obferves, gheod ftepe pereat pars w.otus, licet banc in allquo cffctlu ede?:do ccif/imi, q/firmare non p/tffvmits, ut in mihis cajibus iercujfunis durorum corpcrum — ita ut minims pro lege natures habendum fit, eandem motus qiiantitatem femper confervari, nifi alicui impendatur & confumatur ; fed base conflans lex eft corpora fervare vim fuam af- cendentem, b'c'. Where it is to be oblerved, that by hard I by
 * Acta Petropol. torn. i. > De Caftellis. " A3a Petropol.