A Letter from the Rev. Dr. Samuel Clarke to Mr. Benjamin Hoadly, F. R. S. Occasion'd by thePresent Controversy among Mathematicians, concerning the Proportion of Velocity and Forcein Bodies in MotionAuthor(s): Samuel ClarkeSource: Philosophical Transactions (1683-1775), Vol. 35 (1727 - 1728), pp. 381-388Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/103685 .

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r 38 )

the fineR InleSion O opening ttAw StOlnZ and InteRines tlaey conitairwed a lu>CouslN1dtter) }^e gleateLl Part o:f whicll was Bsood, arld the fine villoas Coat urhi^n i'J

fo vitobie in the-sb nitnals WnS entirtly deAroyod Aw bt)ut an Hour before qze was bit he ll.ad a pierwiciful Nleal of coarfe 13eef, of Wllich there was not tI1e le.aS Anpearance OpeninC tlle lJhorancX tne SPIeara and otller WIelul)ranes loolicd as if in jttied; tlle HeArt W3S turgid wvitil bloodX as >Tere alfo its VeiCels. Tlae \eSels of tlze P2Ierllbranes of ttle Brain made a moR beautiful Figure from the QR<ntity of 131ood coratained in tlletn as did likexrrife elle Blooid-Velrgels of the Nerves i tllere

xnras fa ftnall tantity of AIater between tlle two HeLni- f?heres. Ithe Ble)od contained in t?ne Heart and its Veflels vvas an esren I9laSs about rhe C:Onfi1Rence of Crealn The Cat had upon openinO nearly the falne ^Apw

pearance¢) and lived about five Hours °

-

lI. A Letterfro2sltA2ete1wz f)R*. Sa8,Auel C1ar<e zo AI7^* Ben jamin Hoadiy) F t S occafoszi ffiy the PreJeezt Controlessl$ dYnoRZg XlattpmaticiJXs)

LOJZC&;}rSIi4X t}X S} oA170StV08 of Y-elocity .7zd iorce ;yz Bodc5 in MorionO

S 1@v

1[ T I}as often been obirved xn gener44 that Levr* I iGg -dOES IlOt giNre NIen TnderfYanding, and tIlat the <bfurde{t Thitais in L \Vorld taave been aflerte and lnaintaitzed, by Perfons whofe Ed atat,on and Stu

dies

r 38 )

the fineR InleSion O opening ttAw StOlnZ and InteRines tlaey conitairwed a lu>CouslN1dtter) }^e gleateLl Part o:f whicll was Bsood, arld the fine villoas Coat urhi^n i'J

fo vitobie in the-sb nitnals WnS entirtly deAroyod Aw bt)ut an Hour before qze was bit he ll.ad a pierwiciful Nleal of coarfe 13eef, of Wllich there was not tI1e le.aS Anpearance OpeninC tlle lJhorancX tne SPIeara and otller WIelul)ranes loolicd as if in jttied; tlle HeArt W3S turgid wvitil bloodX as >Tere alfo its VeiCels. Tlae \eSels of tlze P2Ierllbranes of ttle Brain made a moR beautiful Figure from the QR<ntity of 131ood coratained in tlletn as did likexrrife elle Blooid-Velrgels of the Nerves i tllere

xnras fa ftnall tantity of AIater between tlle two HeLni- f?heres. Ithe Ble)od contained in t?ne Heart and its Veflels vvas an esren I9laSs about rhe C:Onfi1Rence of Crealn The Cat had upon openinO nearly the falne ^Apw

pearance¢) and lived about five Hours °

-

lI. A Letterfro2sltA2ete1wz f)R*. Sa8,Auel C1ar<e zo AI7^* Ben jamin Hoadiy) F t S occafoszi ffiy the PreJeezt Controlessl$ dYnoRZg XlattpmaticiJXs)

LOJZC&;}rSIi4X t}X S} oA170StV08 of Y-elocity .7zd iorce ;yz Bodc5 in MorionO

S 1@v

1[ T I}as often been obirved xn gener44 that Levr* I iGg -dOES IlOt giNre NIen TnderfYanding, and tIlat the <bfurde{t Thitais in L \Vorld taave been aflerte and lnaintaitzed, by Perfons whofe Ed atat,on and Stu

dies

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( 38: )

vjies hould: feen to have furniPrhgd them with thegreatw eR ExtenF of Scianceo

Tl<at Knowledge irl mantr E>4awgMhages and 7:ferms of Art, vnd In the HiiRory of- Opinions and Romantk potheJ"es of Pllilofophersg fhJllld fotnetimes be of no Effecz in correEing WIens 7?tigmentv is not fo Inucll to be wonderSd atO But that in MathematicAs vllem- felvesX xvllicll are a redl Scienref and founded in the werefary Xltgre of Things b lnen of very great Abi lities in a6/iraS Goruputation5, when they come tc) apv Iy thofeC8olnputations to the Dlatgre of TJvings) l,6lould perfiti in -maintaining tlle InoR palpa61e ilbJrditiess, and in refufing to fee fotme of the lnoR evidZext and ob- vio Eraths; is very Rrange.

An extraordinary Intance of thisX we hanre Ilad of late Years in srery eminent Az1:atlletuaticransX Mr. Lei§ sitz, WIr. Hermans MrO 'sGraM7efande, and kIr. BerO soli ; urllo (in order to oraife a i)Ji of O?poftion aw gainR Sir If4r Slewton's PhiloSo?Ily, tlle GIory of whicil is tlle Spplicatiofis of abJ?rvS Matbematicks to tlze rneaX sPhaenomena of Nats4re) have for fotne Years lnfiIted with great liagerneE, upon a Principle w}ich fubverts all ScienceX and wlaich tnay eafily be lnadc: appear (even to an ordinary Capacity3 to be contrary to the nevefarv and efentialSat8re of IJhingw

Wllat they contend for, is- That the Boree of any Body in Motio is proportional, not to its Yelvaityn but to the Sqvreof its -Zalvaityg

The XbJxrdty of whicll Notion, I ha1I firR make appear, arld tllen Ihow what it is tllat has led theSe Gentlemen into Errour.

In tllo Nature of Tllings} stis evident every Eged m-uR neceiiarity be proportoxate to tlle CauSe of that

Effc& S

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(383 9 EfeA e tIzat i¢ to the JGiau of tile Caufe5 or tlle

f over exerted at the Siane when the Effedc is pro duced. To fuppofe any EffieA proportional to the Sare or Cgw6e of its C4txJ2 is to fuptpofe tllat an sk7gecS arife$ partly frotn itS Cagfe, and partly frotn * Nothing.

In a Bodv in Motzon, there may be confidPred dio RinAly, the 2uantxty of the Matter, and the h;> sity of the lMoto. Tlle Evrre aridlng froln the 2248--

tity of tbe Matter as its C42Nes tnUR neceXarilv be proportisnaltothe28axtityof theM4tter: Andtlle Forae arifing from the Melority qf the Morion as itS Cvtg0e,mutt neceCarily be proportional to the Melorty Qf the Morion. The mvhole Forae tllerefore arfing firom theSe two Cafet) muflc neceXarily be proportios xal to tlleSe two Catgfes taken togethezt. And there- fiore ln Bodies of eq$al Bigne0s and i)effityj or in one and th-eJAme Body, tlle 2Xantity of Matter continu- ing always the fame, the Forre 1nuR neceSarily be always proportional- tO the Velarity of the ]Wotion 1E the Force svere as the Sqtrare of the Velocity, all that 5Pdrt of tXlle ForreX which was above- vlle EProportiobu of the Meloazy) would arifie citller out of Mlotzing, or (according to Mr. Ieibnitzs PlwiloSophy) out of foane

s Whlctl is ju lite the SSppoftionmadebythofeMathematicians, who have taken it fov ,ratlted, thxlt S ts equal to I>a,inxte, that lS3 that as o m I, fo I iS to Infiaite; tlzat is} that INJNite multiplied lDy 0, isequalo x., or an infinite Ntlmlzet of Nothi7ags equal tosomething; X-hich is palw pably falJ^e. The trxe Proportion- 1SJ not as o to }) fo I tO Ia.finite; but as an InfiXite0smal is to 1s fo is I 20 17X481te. Alld as the Infiniteionaj of an Injiniteiia (thatis. afecondElusion. ortheJ<econdpowerofininitermal) ts to X, fo is I t0 Ingi7XityofInfaites, orthe fecoSdPowerofInli7aite; tllat: is, (forInlance) 'tis asai:e [phyrcal] Liaeto an Iea,6niteSmrfdre, or asa fiite [phyJ>cal] Surface to an i7xfinite sosida And as o (which is lXefvoadall Proportiog lower than the in4nit'th Power of aa inJinoteNmal) is to 1, 4) is I t(>

that wbich is beyond all ProportioD higher thars the Infait?th Power of Iaz,iiZes Whicll cleaxly removes vlle Foundation of a11 the riGculos ConXquen6esJ, which l?ve been drawn fxom the SUppOfiltl9 Qf the foxe-mendsnsdfalre Bxopertsono

X.;virS

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( 384) Ziving Sogl efietltialiy ltelonging to every rParticAe f Matter.

XVhenever vsy EffieA xrhatfoever, ls irl a dspierate Proportio) or as the S2are of any Cafie i >tis ai ways eitller l)ecaue tllere are two Cv;Bfies a&ing ar rtze

fazne Em-e) or tlaat one vszd the J^awe Ga/^e cortirlWes to a&-for a dozz61e 2avfittita of Sige.

The RefJ?ence ma(le to a Body rnoving in ang 4sid SIedigm) is in a dplivate Proportion to ti Veicisy of its WIotion i secauCe) in Proportlo1< tO fS VeSocit> it is rer1Lied by a gre4ter lMTnber sf Parf irZe3 in tl<s fame Titne; and again, in Proportion to its ZeA¢cityX 9cis reGlled by the J4we Partic/e3 Jfngl> sv>11 a greats er Borce, as being to be tnloved out of tLleir Places w1tlo greater Velorzfy.

Light decreafes in a dgplicate P1opoistton Ot ltS Di_ Ilance firotn the Sun ; becfluSe tlle Rays div4tticvte acv cording to tx10 DimenGons ;- accordin^, tO the Ditnono firn Spwards or dsrwnwarats, and accoiding to rlle D ;nenflonSde-ways. But according to tlle third Dirnent tson J2orwards frotn tlae SunX a Ray of Light undergoes no AIteration ; becauSe the Particles) of wI1ich it con S1fls) bei-ng etnitted all of tlaeln with an equal velocity9

continue every wllere at an equal Diflarlce zrotrl eaca otner.

Oge and the J^aene CauSe, aAing in a doBble Quan tityof-Tilne, prodalces the f4we Eieci, as ttf0 equal CauSes aEing in a fngle Quantity of Witnev One atld tllC fdtnC BorceX Sin two Parts of Timev xvill cauSe a Body in NIotion to defcribe theAame SpvreX as doBbAe the Force would do in oxe Part of Witr}e. rlle Space deAcribed therefore by a Body ig Morioz) is not as the Force v but as tlle For.re and tlle Eitne taken togetDer

aA;

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(385 ) A Body3 \Vith ;any the ledgi alrgn4«le gorce, Mrill tnove t1lrO 8?1*l iz fnite 3}4ceg if it taects vzitil Zlo lleElRenceX

inn infiite Timeo And i;l Spces mv?lsere there is an gniw Sorg Re5X0ence to WIotion, the Space dey4ati6ed before theA1oLion ceaCe¢tnuft needs be as tlle Forre and as tho Simetogetlwer. BecauSe a dosbAe Forc-e will carry a l30dy t^ire-s at in tlze */ame Time and utill a1fo CaUre tllu lNtOtiOn tO 1)0 t<sice as long Wge in def roya Awg by aa uniform RefiItence. Tlle Space defcribed therefore before tlle I\4otion ceaCes, is in tllis (,.afe de- snonIErably as tlle Sq¢ar-e of the Force. A Bodz gthrown sgpwardw Witll dogble Force, zxrill be carried yfozir Eiges a.s lligh) before its Motion be ftopp'd by tlle unlfortn Relfiftence of Gravity ; loecauSe tlle doxble Force will carry it twire as high in the lnarwe 7Jime, and tnor>Over require twice the cEime for tlae unifotla ReE1ftence to deAroy tlle I\fotione rtlle Cafe is the falue in vcceler;red NIotic)tl-j in Bodies accelervrea{by a Succefl1on of clattick Itr3prefl1l0nsX or falifityg wYith a WIotion caccelerated iJy tlle uniform Power of Graxrity, or by any other utaifott-n Po^rer zvlaatfoPver. Tlle Space deJoctibed lnu rs needs be as tlle Fsrre, and as tlle 7:itnetwherei¢ tlle Force operates.

Wllat i hae tal s detllontiratedconcerning aty Forve, ronfldered as tlle C4axJe ptoducitl , an EffC& b and conW cern1ng tl1e Tise, during whicvh tlle Force operates; is on ali tH(^.nds cLcknogz1Xd3,ed to be trup colacerning VelocAfy. -Xtad tllerefore Velority and ForreX irl tllis Cl2) aie 0tnv and t1lO fidSlC T11i7. 80 tll2t t3 35.r1 Fvrse to bw as rile Sare of tile MelocityX s to affirtn tlzat tlze Force is eS611 tQ vlie Sare of it fielM:

Nox^7 frotn ilence aOerirs srery clearty t1zo GXroX4 of the Erro-r therc Gprtlemen haNrC fallen into, and

1 f f of

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( 386:) of their MiJ^oappliation Qf the Experttnents tlley build upon.

The EffeA of a lMorve italpreLid on a monreable BodyX is tile Motion of tllat Body firom one Place to anOther. NOR-NJ for(aEmlacla as tlle lWeS cannot but be proportional to its Coa8fie, inellce NIr. LeiZ?vatz (WllOIn

thP otller GentIetnen 11ave fiOilo\073d) cc)ntends tllaE the Spare defcrid b-y a 13ody itl zalling, is prc)portional to the Farre by wllich it i5 itupell'd during its Fall: and tTlat the Forae acquir'd loy a Body in fflllingX is proportional to the Sp4re Pt h4 defcri6ed in itS iail tWIzicll Space being agreed- to be as rlae Sqgare of ttwe Melority (as being proportional to tlle Atelocity and to tlle Sizne taken together) laence they- infer tllat the Forae likexvife is as the Sqvre of the Velocgry.

But frotn xvllat 11as been faid) tiS plain, that tTie Spare deNcribed ln theSe and all other tglle like Cafes} is not as tlle Eorce only) but as the Force and as tlle Fieme 2sbereiS the Force aus v thaL is 3 fflyo as the Sqxare of 7the Forre. For Lhe CaBfe of tile Quantlty of the Space de/>rrgbed) is not bar>ly the 2t,aztity of the Forre, but alra tlle C0XtX247Wre fif tberEa3e wherein the Force aa->. Tlle Forre rllerefore and the Tige taliell together, beillg neceliarily as the sp: de/cribed; as tlle Melocity atld tlle time talen to>;e- therX are OI1 all LIands ncknoxa71eqgbd to bej it XPO110\\JS fllat tllb Yelocita and tlle Force are equal9 and not vlle iCarce as the SSare of the VeSocit.

Wllen two iS;eq¢4l Bodies fa{teried 2t t2 Ends of trlle Jrzns of a Balance of 2zeqq4l Le7gth) coi nter poife etlch other) ind vibra+e 'n eqaaX Sirses i as twey muft neceIfarily do) beinD AJ?ezed to ttile wirlris o-t tile

fiage Balance e wllich is an O7Sferxatiotl lXrir. Lei6aitz lays great Strefs upon * In 6rJJat (:afe indeWd tlle Forrew

\Mill

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a 387) wlll be as tlle Spdres deficribed. But not tl1erefore as the Sqaares of the Veloaitiex. For in That CafeX tile Zelocities themfelves are as tlle Spaces deeNcribed lbecauIe tloe Times are equal.

Wllen a Body projeflced with a dRoabAe Veloctty, en- ters deeper into Snow or fioft Clay) or into a HeapS 0

Jp ritXy or^- eSaJ?ick Parts, tllan ir] Proportion to lts Yeo Socaty b 'tiS DOt becaufe tlle Ftorce is lpore thdn pro

portional to the Velority; but becauSe ihe Depth it penetrates into a foft MediutnX arifes partly frcxm the iDegree of rlle Force or Velocitys and partly from the Titne whereig the Force operates before it be EpWntO

In the Collifon of hard Bodies) 'tis (I tllitak) agreea on all Hands, that 'tis derno>J?rvted by Revgox, and confirmed byEncperie¢ce; tlllat uWrhen.l perfedly hard BallX tnoved vvith whatever Degree of Zelocityt Ilrikes full upon another hard Ball,, equal ill lSneSs and Weit,trlt9 and without any Motion in it ) if tI1e Ballsbe cnelaJflirkX tlley will both go on togetlaer the faIne WaTS d;riding tlle Motion equalLar lDetJeen tlzemw wiill half the Melocit.y tlle firft 1sa11 had (3riginallv : I3ut i: they be perfedily elaJ?ick, the lrlc)Nring 13,i11 xrill cotn lnunicate its wilole Motzoa and Velocity to tlle ru.e icent Ball) and it ftlf lie J?ill in tll<t otllers Pac.->* Were it tsue notwn thc^L tlle } 0rre 3f tllO gosizg Bl

was ( s Lslle Sqaare sf its Yelority; thefe Exp>ri IlOt]tS would tllen ihew twllicll is infinitely abfiurd) tlzab tI1e Force OL tvis inerti in the q?iefrent Ball) the Geac! Force) \'<ft35 aIways p"onrtionas to the syare af the Melocitoy VvIzicll silfie Gentletnen -alS^Cct falzaAiully tO CA11 tsle ;tZSitg Fwrc.n) of tle mcvtAyg Bali? wr latesrc; ltS aTelOci.y Nsleree Or 611C orve in Both l-nigllt J$' ZS reaSonablJ> be i-uppDfed to be as t?ne (tsteS cr t^e fsxa- drato-qaadsfe, or a.ny or/Je7 EPower of tlle Afei jX\t oftlac zvo^bfi¢g B;lll. VIIlich ls turzing t^e -7utwre

F f fz

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( 382 ) af Thiggw- into Ridicule. WIr. Leibnitz) in foLne Letv ters which 11e wrote into EBgland3 intimated that Il-e had a Profpeds of aperpet84l Motion, foundwd on tlae Notion of a Fital Prixciplet or aGive SPower in Matter. But fro;n thelisperilnents nov mentI3n- ed, 'tis evident tllat if the Forre of Bodies in Alovio could be exalted exen to tlze izfnitth f ower of tneir. YelGcity; yeS Ilt1ces to anrrer ttze Phnotnbna of Na- ture Wittl Re5ard to SSioz and Re-aSioz) t1ze fas:ne JForce tnuit neceAarily be aliowcd to all qgiaJve> [X24

dies likewiCe ) it could be of 1\1 Eff8eS. . .

1lI. AJEronomical OS%er>tioN3 JZ42e at vera CruZ)

by Mr. 30fieph FIarris. (S,eli/ed and comnaxnic- ted byEdm. Hallcy, L.L.D. ARron Re. & R. S. S

T H E Latitude of tI1ss Place I t4ound ('xy fe>terai ] diRant ObSerstations lmade by a Quadrant of fi)ur

Seet Radius) ro be x9° ttS N- QI1 MaraD- tr) I7z7* 0e S. tllElee lltdppetl'd 1l0tO .R

c£)nfiderable EiclipSe of t1an SurinX tile greateLR ObScur< tion belN7 S50ut I0-1 bD$gtts; anK1 I2r4^;ting tS<at A/ior>tinbS

arefullr 3dji2,Wtd ttze Penduuzl CloPk, and Ssed a Telefcope tD t}zeirodwx of the fceseoltd Quadrant) I ob rerved it to kegin in Gr ,i3out tho S h;. 1)v S. [erc o1> L-he Sun'3Disk at Sr9 X Wsinutes after ̂ XNroon apparent rrne ; tiwe ftltittAdC o-f the Sun's Center tl<WWla XYwsis 67t 53J.

MSe could rwot detertnotle cxadly tT-1^ lAIicldIe & tllis tEiclipSe: but as nar ?.S sve could jlu.dge) t; tH29, 5st5tiR l >5Out

zh 3qIB ps /t,

( 382 ) af Thiggw- into Ridicule. WIr. Leibnitz) in foLne Letv ters which 11e wrote into EBgland3 intimated that Il-e had a Profpeds of aperpet84l Motion, foundwd on tlae Notion of a Fital Prixciplet or aGive SPower in Matter. But fro;n thelisperilnents nov mentI3n- ed, 'tis evident tllat if the Forre of Bodies in Alovio could be exalted exen to tlze izfnitth f ower of tneir. YelGcity; yeS Ilt1ces to anrrer ttze Phnotnbna of Na- ture Wittl Re5ard to SSioz and Re-aSioz) t1ze fas:ne JForce tnuit neceAarily be aliowcd to all qgiaJve> [X24

dies likewiCe ) it could be of 1\1 Eff8eS. . .

1lI. AJEronomical OS%er>tioN3 JZ42e at vera CruZ)

by Mr. 30fieph FIarris. (S,eli/ed and comnaxnic- ted byEdm. Hallcy, L.L.D. ARron Re. & R. S. S

T H E Latitude of tI1ss Place I t4ound ('xy fe>terai ] diRant ObSerstations lmade by a Quadrant of fi)ur

Seet Radius) ro be x9° ttS N- QI1 MaraD- tr) I7z7* 0e S. tllElee lltdppetl'd 1l0tO .R

c£)nfiderable EiclipSe of t1an SurinX tile greateLR ObScur< tion belN7 S50ut I0-1 bD$gtts; anK1 I2r4^;ting tS<at A/ior>tinbS

arefullr 3dji2,Wtd ttze Penduuzl CloPk, and Ssed a Telefcope tD t}zeirodwx of the fceseoltd Quadrant) I ob rerved it to kegin in Gr ,i3out tho S h;. 1)v S. [erc o1> L-he Sun'3Disk at Sr9 X Wsinutes after ̂ XNroon apparent rrne ; tiwe ftltittAdC o-f the Sun's Center tl<WWla XYwsis 67t 53J.

MSe could rwot detertnotle cxadly tT-1^ lAIicldIe & tllis tEiclipSe: but as nar ?.S sve could jlu.dge) t; tH29, 5st5tiR l >5Out

zh 3qIB ps /t,

1 12¢) 1 12¢)

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