Appendix: Daniel Bernoulli's Papers on the Hanging …978-1-4613-9461-7/1.pdf · Appendix: Daniel Bernoulli's Papers on the Hanging Chain and the Linked Pendulum * ... In the translations

Appendix: Daniel Bernoulli's Papers on the Hanging Chain and the

Linked Pendulum *

* Daniel Bernoulli [4,5]. The original publications are reproduced on pp. 125-155. Our translations are given on pp. 156-176. In the translations we have corrected some minor errors, indicated by [ J. We have not attempted to correct all such errors. Figures appear at the end of each paper.

Theoremata de Oscillationibus Corporum 125

COMMENTARII ACADEMIAE /.~

GI . SCI E N T I A R V M c) ,t~['I·Si;

IMPERIALI ~kl' PETROPOLIT ANAE.

TOM VS VI. AD ANNOS tbbCCXXXII. & cbbCCXXXIlI.

PETROPOl.,l, TYPIS ACADEMIAE. cbbccxx:n ~:

126

'tabula VII.

Appendix

108 THEOREllfAT A

Donie/is Bernoulli THEOREMA1'A DE OSCILL1\'TIONIBVS CORPORVM FILO FLEXILI CON ...

NEXORVM ET CATENAE VERTICALITER SVSPENSAE~

Introduaio ad Argumentum ..

THeOriae ofcillationum, quas adhuc Audores pro· corporibus dederunt folidis, inuariatum partiuffi

< fitum in illis ponnne, it~l ut fingula commnni motu anguhri ferantllr. Corpora autem, quae ex fil0 llexili fuspenduntur, aliam pof1:ulant theoriam, nee {iIfficere ad id negotium videntur principia communi .. ter in mechanica 'adhiberi folita, incerto nempe fim " quem corpora inter ie habe:ll1t, eodemquc continue variabilL De his cogitandi anGlm mihi aliquando dc<dir catcn:l verticalitcr 1iIspenfl et motibus osciUatoriis agit:lt:l, hancql1e ttmc videns motibus valdc irrcguhri~ bus i,H~bri, primo mcntem fubiit" ad qmmnam curuam catena cifet infledenda, vt omnibus cins partibllS fimni moucri inCIpieutibus hae quoque ,'l1a in fitllm peruenireut lineae verticalis per punCtum filspenfionis transeniltis: hoc modo ofcillationcs aeqmbiles fore intei1exi atque tales qnartlm tunpofa dcfiniri pOlfe:l t: Mox vero fcnfi ditIicilc cifi.: h~lllC dctcrminare Cli,'u::m ~ ni 'j l:i~qllj[;ti()ni:\ in;tium fiat :l cafiou., fin: ')!ljlhl1j~. o filS it:,quc l;un Ius mcdiLltiones :\ corpori."ij.: "lIO:<,;:-: 1:1 fle,,;l", 1 "" • I <b " < ,MoO • .i),.I,< t 1:, U.:X:l CHtlllt':l co 1:lCrClltl U:i; rom .. , C',:l

n):L;~-'

Theoremata de Oscillationibus Corporum

DE OSCILLATIONIBJ"S CORPORl' ftf.

r:onfideraui moxque qmtuor, ct tandem numcrl1m cor~l1n di1bntiasquc qi.uicscunqnc; emr..quc numcrum eorporum infinitum [;lCcrem, "idi Jcmllm natur:lm o<,cillantis Cltt!IUC fiue :leqU:11i~ fine inaequalis crallitici lc'd ,biql:C perfcCtc ftexilis. Suo fingula percurram online; ckmonfkltioncs autem qnas nunc adornarc norr ,',leat in ai,i .. am occafiollum refcruabo. In {{)Jmione Houis ,-ftlS film pril1cipil~, proptereaquc yolm theorem a t:l experimcntis confirmare, ne de eorum veritate dUDium cile po[f-:t, iis prae[ertim, qui hisce rebus ii};l natura aliquanto difficilioribus non omnem dare poterunt animi attentionem, quique fie [;lciJe in [;llEnn incidere poficnt folutionem. Caeterum alias ofcillationes non confiderabimus, quam quae mini mac fint et jfochronae: pro experi mentis tamen fine notabjIi errore paulo majores illas cfficere lice bit.

Theorema To

127

z. Fuerit jiTltln perfeae ji'::rile non graue AHF jiUpcl!- Fig, I, :..'

Jum C.l' pztllc10 A bab::aJqne in H et F dw} all(~-a:a jJOll-d"'ra aequalia ~ tantulll autem diflet ("f)}j)UJ illfcriu.i {/ fu-p:riori qual1tlllil bol,' a ,PZtJl{fo jitjpaifiolliJ, Sit lYJJ"FJ Ii--rca ABC ri)er~icaliJ, et au bat,,' corl)f)ra H et F t;,,'duti hlil1ite jJarttm dijlt:nt; Denique du("alltw' bori'::'/)J!ta/cJ mi-nimae HE et F C: Diro ji amho t,,'arpora jilllul oj:i//ari illcipial1t, fore rvt eod::m temporiJ pUiu70 perllclIi(mt iii fl-· tI/m lineae rcertiralis atque bo,,' 1Jnci? oji:i1/atiollfJ fl!m 'lJJlijo}"llliter pe;jiciallt, cum jimlitul' C F : BH= I - +, 1/ 2 : I"

C oro1hrium. ~. Igitur dUODU:i modis of..:ili.uioncs fium vnihh'--

o 3 m(;s,~

128 Appendix

110 THEOREMATA

mes' nempe cum fumitur, vt figura prima oficndit, , ~

C F =C I + -V 2 ) B H; tum etiam cum ad llormam li-gurae fecl1l1dae fit C F = (I - y' 2 )BH.

Theorema 2. "!-. Fallis oj:illatlonibus cor porum H et F vniftr

mibus, erit longitudo penduli fimp/ids tauto(:hroni - Z -:-.." 2 •

A H tL'cl 41 t'8' A C, vbi jignum l!/fir111atiztU11z tL'ttlct pro ofi:il/ationibuJ (:ontrtlriis jigurac JecU11dae, fignum negatiUUIn pro (,'ol1jpi, 'antihus jigurat pri11lat.

Corollarium. ;. M nIto itaque celerius ofcillationcs contrarilC

abfoluuntur, quam con(pirantcs: illarum cnim 23 I numerabis, dum hac ccnties fherint replicatae. Confpimntes autem parum dlfferunt ab iis quae fierent 1ilb iisdem circumfl:mtiis pofito filo A H F rigido: paullo tamen celcrius ofcill:mtur corpora in fila rigido quam flex iIi , crnnt ncmpe numeri ofcillationum aequali tempore pcr:tchrum praeterpropter. vt 1012 ad 1000.

Scholium. 6. Vt ad experientiam reuocarem hasce pro po

fitiones, vfils fiun globis pillmbeis pcr£eCtc acqualilms, qui dum fimdcrcntur in medio tcnui foramine perfurati m:meb:1I1t; traieCto filo ferkeo globisquc ope 110-

dorum firmatis ita vt inferior duplo magis difhrct a punteo filspcnfionil:i quam filperior. Digitis dcduxi globum infenorem in limm F tenfo filo: mox o{cilb6ones ficb:mt vnifi)rmes, et ope dillifionum in p:lricrc ta,"brum difiiucte cognolli excurfioncs corporum II ct

F in


DE OSCILLATIONIBJ?S CORroW M. III

F in figura I. filiffe vt 100 ad 2o.p, id eft, vt I ad

I + V 2 (§. 2. ): Numerus etiam ofcillationum datQ

tempori conuelliens accurate refpondit longitudirW. pen-

dllli funplicis ifochroni 2 I .f:2 A H in prop. +. dcfinitae.

Deinde fa~a FC=(I-V2o)BH 10 figura recunda, de

tinlli m.lOibus globos in fitu F et H illosque mox e~

dem temporis punao dimifi: ofeillationes ortae Cunt

fie Catis vniformes feeus atque fiebat cum alia Ffopor

tione dif1:antiae FC et HB fumerentur: numerus ofcil ..

lationum accurate rurfilS fuit, qui conueniret longitu-

dini penduli fimplicis 2~"2AH iCochroni prop. +.

Theorema j.

GENERALE PRO DVOBVS CORPORIBVS. 7. Fuerit iam pars fili AH=I; HF==L; pon

ius corporis H = m t alterizLuJUB F == M: dieD fore oji;illationes vniformes fi ./it

C F - mL-ml+1IlL+M!+"(+mMLL+(ml+ML+~il-mL)2) B H - 2M1 )( •

/Ongitudinem autem penduli limp/ids ifoehroni fore __ _ 2mLI __

-ml+ml+Ml-t-ML+V(+m.MLL+(m4-ML+Ml mL)2),

aut, pof1l0 L+I=A ct M+m== tJ., -.!.. 2 ",>.1-2 m II -1i>':t:y'( .... tLA.A.- ... m .... Z),+ ... "' .... Ilt

Theorema 4-

129

8. Si loco drmum corP()rutn aelJualiuffl POtltlfltur tria tantum a Je inuicem diflantia, lJuantu11Z jupremtlt1l . a punllo fuJpenfionis A diJlat, pote~'u1ft tribuJ dit!crjis Flili- 3- 4· S·

'm1dis oj&'il/ationcs fieri vniformes: pr:rmls ejl quem Jtf;1/~a , teJ1Ja

130 Appendix

It! THEORE}J1ATA

tertia indii:at, cum pofita B H == I, JU7IIittir C F == ~ I

~92 It 1J G == 3, 9 22 : j;!(,'undus" qUi jig-ura quarta rc-PJ'aej~tatur, obtill:tur Jat;iend'l C F.:=:' ~.' ~ 5 3 et ])G -,' ~. _ I , 04+ (II,' ti/dJIIS (;lilll jtt, q;t In jigwa qUinta, Clt

=-0,6+5 et DG=o, 122.

E1\ nempe C F aequalis accipienda tribus radicibus hu-ius aequutionis

+.1" -I ".1'.,,'+ 3 x+ S == 0,

t\lmque pro qlt.lui& r,ldice fumcnda eft DG::::. ~X.1·- 2X-.2.

Thcorema I). 9. Films, tV! mfJI/{} rJiilJlm, oJdllationibus 'V11ifor11li ...

bus; erit /ollgitud() penduli fimplil:is ifochroni in cajuj(r.:u-rae tertia:! prorime aeljulllis .2, 406 AH; in ,;ajupglJrae Ijnartae = 0 , +36 A H et in (:a[u figurae quinta! ::: 0, 1,9 A H; Eft fi:i1ic:et fongitltd I penduli ijochroni = s~~AH, poJito rur!u; 4X3 -- 1.2 .tX+ 3.'l.'+ 8=0.

Scholium. 10. Ambo haec theoremata experimento accur:1te

col}firrn:mi in caCu figurae tertiae, deduao tantl1m corpore intimo extra fitum lineae verticalis mox dimittendo; quamllis enim in primis oici11ationibus inaeqmlita~ quaedam fentiri potucrit, t:lmen haec 1il:1 fponte et citillirne abiit, it;\ vt excurfiones fillgulorum C01'

pomm pluribus vicibus filCcefliuis, quantum oculis dis-· cerni poterat, eaedem manerent; fum tis :luteIn carundem menfi.1ris, talis inter cas repcrta fizit proportio

qU:l-


DE OSCILLATIONIBVS CORPORVlJf.

qU!l.lem theorema 4. indicat: numerus quoque ofcillationum perfeC:te refpondit theoremati quinto: duo reliqui caLiIs maiorem induftriam requinmt: potui tamen. vtriusque generis ofcillationes fatis exaae etficere, vt veritas theorematis quinti appareret.

Theorema 6. GENERALE PRO TRillVS CORPORIBVS.

I I. Fuerint nunc rurJus pontlera corporum H, F, G fualiaCU1l4uc flmulque dfftantiae eorUlldem a punllo Jlffpenfionis rationem habuerint qualetncMnque: fit nempe pondus corporis H = m, corporis F = M, corporisl/.ue G = jJ. ; tuml/.uc A H = I, HF = L et F G = A; dieo ofoillationes 'Vnijormes /uturas effe fi poJita BH= I, G F=x filii ( MMIA+MjJ.IA)xx+(mM/A+tlljJ./L-m MLA - M M lA - M M L A + 111 tJ-l~ - M jJ.IA - M fJ. LA) x -mjJ./A - mMIA) x( (MIA+ fJ.1A)x-mLA-M/'A - M L A - jJ.IA - fL LA + m IL ) =m m jJ. IlL L x. fimulque Jumatur pro I/.uauis radice

G MM>" »>.. >.. M>.. M>" 1I1MA MM>" D = (tJlJl,L+mL)XX+(I+1:+JI,L-JI,Z -mJ,LL-mJl,l -r:!?- _ 111>' ) X _ M>' _ ~. mL ml Jl,L L

Corollaria. J 2. I. Ponatl1f ma{f.'l corporis infimi fL =.0, di

nidaturque, aequatio fundamentalis fuperioris paragraphi per filCtorem alterum ceu radicem inutilem; L'lerOr igitur pnor erit = 0, hincql1e habebitur Mix x + (m /mL-MI-ML )x-11I1;::::.0, vel

mL_m!+\IL+].11+"(4mMll+(ml-mL-~t1-ML)t) • . '\.' = ---.. . 2 }l-Z --. ,

Tom V 1. P NOL

131

132 Appendix

THEOREMATA

Notandum autem en, non differre hunc valorem ab ilio quem dedimus in theoremate tertio, quamuis quan ... titates ab vtraqne parte figno I'adicali inuolutae diuerfam habeant formam.

IL Si vero maffa corporis medii indicata plr M ponamr =0', nmc, vt appareat confcnfus inter theorema tertium et fexnlm, erit in hoc pofieriori intelligendum per L + A et lJ., quod defigoatum filit in altero per L et M, ipfaque linea D G in praecedente paragrapho definita comparanda erit cum linea C F ad theorema tertium pertinente. Ad haec qui ani mum aduerterit, vtriusque theorem-atis aequationes easdem cae. reperiet inftituto calculo.

Ill. Denique crun ponitur corpus ulmmum H indic:ltum per m = 0, poten in aequatione fimdamentali t. I I. "fterque faCtor poni = 0, et Ttroque modo obtmetur C F feu x= I + r, prouti natur.a rei pofiui3.r, quia tlmc lineae A H et H F, vt patet, debent_ in direCtum hlcere. Excurfib autem corpori~ infimi ex ae-quatione dignofci non poteft, nifi id particulari methodo fiat. Ita nee o[cillatioaes definiri immediate po[~ funt per theorem a fextum" €Um duo corpora vniuntur euanefeente alterutr.a longitudinum L vel A •.

IV. FIeri pote(\: in figunr qmrrta, vt fit CF=o, quo in cafu, quia durante tota ofcillatioue difbntia.e corporllm a. linea verticali e:l11dem perpetuo inter fe rationem feruant, corpus me.dium F quiefcit ,_ dum ambo t.eliqua hine inde agitantur; atque. tunc pcrfpicuum eft, longitudinem penduli ifochroni fore == A, quia (orpus,infimwn 'Vcluti ex 'punC'to fixo C [UlpC~ifilr.l o1(i1-

latur 1


DE OSCILLATIONIBVS CORPORV M. I I)

]~lttlr; Iftc vero cafilS, de quo loquimur, obtinetur po .. mIL

nendo .t' = 0, feu :r == 7nL-t-M.1-t-1I\L~ l+I-loL'

Theorema 7. Q,VOD GENERALITER PENDVLVM TAVTO ...

CHRONVM PRO TRIBVS GORPORIBVS -OSCILLANTIBVS DEFINIT. .

13. Retenlis denomittationibus et aequationibus theorematis jeJ.1i di{/o oJdllationes jingulorum uorporum ifochronas fore t:um ojdllationibus penduli /impli(";s, cuius longitutlo

mIL

Corol1arium. J +. In ca1h x = 0, quem modo alJegauimus, nt

1 · d li 'r. h . mlL ongltudo pen u llOC rom = tnL+Mi+itL+lIol-t-lIoy=A, quod conuenit cum Coroll. +. Theorem. 6. Si praeterea ponatur L:::: I, fit longitudo penduli ifochroni feu ~ = m+:~: Conuenit hoc cum problemate I. quod Pater mcus in Comment. A(/ad. Petrop. Tom. III p. IS. dedit: idque vnicuique manifeftum erit, qui confidera ... bit pondus P, quod ibi ab vna parte chordae eft appenfilrn, bie eife furnmam pOllderum G cc F audilm dimidio pondere H.

, Scholium Generale. I;. Poffurn fimiles aequationcs d~lfe pro quatuor,

quinque et quot libucrit corporibus: fcroper autcm ae .. quatio ad tot a{furgit dirnenfiones quot 1iU1t corpora,

P 2. et

133

134 Appendix

116 THEOREMATA

et eft plerumque admodum prolixa: :ltt:lmen quia acquatio finalis orimr ex pluribus aequat:Ol:ibus radicalibus linearibus, lex apparet ex methodo q1.l~l vfus film, cuius auxilio ex tempore omnia determinari poffunt, quae ad aeql1ationem determinandam concnrrunt.

1'heorema &. DE FIGVRA CATENAE VNIFORMITER

OSCILJ..ANTIS. FiS· 6. 16. Sit catena A C rvniformiter grauis ct perfec1e

flexilis JuJpenJa de puntlo A, eaque ojcillatlOllt's fa"ere vniformes intel/igatur; peruenerit catena in filum AMF; fueritque /ongitudo ,'atmae = I ~ longitudo fuiu.rmnque par-tis F M=x, Jumaturque 11 eius va/oris, rz:t Jit

I + Il 1~ 14 p. I-n- 4 nn 4.9.n3+4 9.16·.n~-+-9I6.2~.n5-+-etc.=o: Ponatur porro dijlantia (xtI'emi pundi F ab linea verticali = I, di(;o flJre diflantiam puna; vbkunque rlj}imlti 1\1 nb eadem /in.'a rvertjet/li aequa/em

:c + xx xi' ;1:4 x 5

1- il 4 1tn - 'i-.511~l+ 't-.,.16~- 4'9. 16 . .!s ,,$ +etc.

Scholium. Ii. Per methodum, quam dedi in Comm. A"·,,d.

Petrop. Tom. V. de rf!Jolutio~ aeqltfltifJllu11l fille Ji11~ prfJgi'edientium, innenitur breniflimo calculo n = proxime 0 ,

691 I: Igitur fi filerit v. gr. punCtnm M in mcdio ca·· tcnae, illud diftabit a linea vertic:lli practcrpropter limbus quintis,. vel accur:ltius, trccentis nOlugi nta (Cto partibu) millcfimis diibnti:1C pundi infimi F :lL e:lGCm ljncll verticolli. H~lbct ;H1tcm littera 11 illllniw:> V,I!O!'~:S :lEos. Th~;


DE OSCILLATIONIBVS CORPOW M. 117

Theorema 9. 18. Seruatis pojitionibus theorematis o{/aui, dieo

/ongitudinem pcnduli Jimplir:is ijo/:hroni CUilI ofi,'illallte (,'atena e;je == n , feu lubtangenti C P (,'UI'uac A F in injim(J punClo F; aut pro.r:i1ne aelJuaiem fel.'("C1ltiS nonaginta et 'lI'1Zi partibus mi/lejimis totius calenae in caJu jigurile

Jextae. Corollarium.

19. Tardius igitur hoc modo ofcilh1tur catena,. quam bacullls rigidus aequabilis craffitiei, eiusdem cum catena fiexili longitudinis, huius enim ofcilhtiones ifochronae funt cum ofcillationibus penduli fimplicis , quod in longitudinc duos baculi trientes habet.

Scholium I. 20. Poftquam plurimos globulos plumbeos aequa

les ad diftantias minimas aequales filo connexi, 'Vt in paragrapho fexto didum, eo catenae loco Vfi.lS fum ad experimentum inltitucndum: filum haque globis 0-

neratum ex pundo firma filfpendi ~ dedudaque ad la ... ttlS extremitate F, eaque rurfils dimiffit rationcm ob-feru.1ui, ofciUationibus iam vniformibus fadis,- inter diftantias punCl:i cxtrcmi F et medii M a linea verticali A C, eamque rationem eandem deprehendi" quae paragrapho dcdmo feptimo indicatur: numerum quoque ofcillationum· conuenire obferuaui Cllm longitudine penduli fimplkis ifochroni, quae in theoremate UOllO exhlbetur.

P 3- Scho~

135

136

['18

Appendix

l'HEORE}J-l.AT A

Scholium 2. 2 I. Q..uia 'aequatio in theoremate oceauo exhibi ...

ta, nempe Z Jl F + Z+ IS

I -n---1-+rm-+.9. n3 +.9.16 n+ - +. 9.16·.2~ .nS +etc.~o. habet inlinitas radk.es reales, ideoque catena· infinitis modis' infleeti poteft, vt ofcillationes fiant vniformes: fern per autem littera n minorem atque minorem Y1-

lorem affilmit, ita vt tandem pene euanefcat, eftqne longitudo penduli fimplicis ifochroni conHanter == n , feu fubtangenti C P: vnde edam ofcillationes t.'1.ndem fient velnti infinite celeres. Cauls qui fingi poffi1l1t omnes hue redeu"nt, primo, vt catena lineam verticalem in alio puncto non interfecet praeter punctum Ulfpenfionis, qui reprae1entatur figura fexta et pro quo conuemt longitlldo penduli fimplicis ifochroni '2 =0, 691 I, vt vidimus in antecedentibus: vel vt caten:lliIleum verticalem in vno infllper punteo immobili [ecet,. qualem figura feptima indicat, vbi praediCtum interfeetionis punCtum eft B: in hoc caCu eft longimJo penduli itochroni n == 0, 131, et ofcillationes nnmero viginti tres fient, dum in caUl figurae fexme decem abfolnnnmr: linea C B erit pro xi me == 0, 191: CN pl1ndo maximae excurfionis M conucniens == 0, 4-7 I: ipCaqne M N pmeterpropter == f F C. PoO: hune ca-fum'1eqnitl1r ille, qui figura otbua fifl:itnr: vbi line:\ vertie~\Us in duobllS punCtis fixis B ct G ':1 catena of ... cillante interCecatur: deinde cum tres finnt intcrlc{tin-· nes et fie porro. Arcus inter duo imeril-G.i[)llis p~ln·

ttl proximl illcepti co nuiorcs funt, (lila J.blb P()~ li ti :


DE OSCILLDIONIBVS CORPORJl'M. II!)

fiti : In catena autem infinite quafi longa arcus filmmus nOll differt fenfibiliter a figura chordae ron ... ficae tenfile, quia _pondus Hlius arcus veluti nullum eft refpectu ponderis catenae totius. Neque difficile eife theoriam chordarum muficarum ex theoria ifia deducere" quae plane conuenit cum illis ,. quas Taylorus et Pater meus dederunt,. primus in trat/at. Juo de meth1do incrementorum, alter in Commenf. Acad. S,·. Petrop. Tom. III. SimIles quoque inter[ectiones in chordis muficis, quae in catenis vibratis effici poffe experimentum docet, quod charttlla chordae quibusdam in lods impofita non decidat, cum chorda annexa 'Vi~ bratur~

Theorema. 10. '% 2. Si catena ACe fib L A non graui fuJpcnfa

lucrit, ponaturque longitudo paJ1is ad libitum ~jJumtae F M :::::: x: dijlantia Jupremi punlli N a linea vertkali = ~: fkque porro n fumatur eills valoris vt jit

(l+Al+(lI+".I~ _.(13+ 31l~-t-W+4P~_ ...,._ 1- n 't nrL 4-.~.711 +.9.16n+ e~\..._o,

dh·t) in ofi:illatimibus tV1zijormibus fore (vbique dijldntidln pun!..,,!i" M a linea tVertit:a/i aequo/em, ( r X:IC Xl + x+ I + 11 1-n-+4nli-+.9ns ~n+-e[c.) f3: (I-It 41111

!l Z" 4. \I.nl++.9.1611+-etC.)

IIdeofJue dijlantiam plmc1i btjimi F futuram effe aequal:?11t 1 Zl Zl 1+

(3: (I -n+~nll--+.9nl+4.9.16 n4-- ctC.1.

Erj' jJflrrrJ hngimdo penduli ji11lplidJ ijh'broni,) 'i.:t a:;tt'.'z = 11, Itt in l'a.fit .fimplici!1i1m proximc, ==

2.!...22: i~~" }.+ Ii.3 ~L2:>'+ I 4- + O:A 3+ ~ i (j A .;0'

J ("41'-t-!l1 :'11,,+ I I ) ;;I/,)'.+S 1 ¢",3

138

I~O

Appendix

THEOREMA1'A

Corollarium. ~~. Sit v. gr. longitudo fiIi eadem qUAe longitu ...

do catenae, id eft, / = A, erit Iongitudo penduli fim .. plicis ifoehroni feu n proxime = I, ;6 /~ difl::.mtiaeque pundorum extremorum F et N a linea 'Verticali fe fere habebunt 'Vt I I ad I: plures tamen praeter hune alii cafus fatisfacient fimiles illis, quos in paragrapho 21.

enumemuimus: ita poit dictum cafum tequitur js, quo fit n fere = f/; punctumque C excurfiones contrarias fadt cum puncto A atque triplo m~\iores.

Theorema II. Fig. 10. ~+. Pojitis omnibus vt in theoremate dedmo, ji (,'a--

tena in origine A pondere onerata lucrit tanto, quantum ineft catenae parti /ongitudinis L; erunt omnia vt in eodem theoremate decimo, ji modo ,nunc fiat

{Lh+U-+-ll-t-lX'+ (",IL>.+H+ 2llA) (911l)'+LZ3+I++ 111).) J -' 3(1.+1) ",M'(!.+Zl - ---:t.-9-Ii3(L+i) .

+ (1613+LA.+Ll++J5+",z X) -et· - 0 +.9. I 6 "+(l.+l) (;. - ,

F ' L- -I fi 21 II '1ls , uertt v.gr, _A_, et et 1- n-+1ii- 36",+ I tl+ hi- h' h ,1. b" 1 Ti6n+ -~,;C, =0, zncque alle, ttur proxznze n=I, 37 ;

arculi autem a punltis (t'ate1Zae injimo et Juprenzo defi'ripti ~ vt 100 ad 39,

Theorema 12. GENERALE PRO CATENIS VTCVNQVE IN~

AEQ.VALITER CRASSIS ATQ.VE GRAVIBVS.

2;, Fuerit deniquc catena rvtl:lmque il1aeql!a!is jl/ittlurae ita vt pojito longitudille partis t:ah'lUle F 1"1 =:r ,

fit


DE OSCILLATIONIBVS COR PORT" M. l~I

fit pol1dus eius ~, bitdligmdo per ~ quale11U'lmque fimllionem ipjius .1.': tl,JOt:8Iur porro difltll1tia prmfli M ad libitll11l n.Du1l1ti a lincn t[.,'ertit:ali y: dh:o curutlturam F MN hat; dr:fil1iri aequatione, fimlta ax pro conjlante, fy d~ =-~?: huh'que aetjuationi pojlquam in quolihe! ca.fit partit:ulari rdle Jati·ifaClum fiterit, fore longitlldb1em pe11duli jimplids 'ijo"hroni = n.

Corollaria. ?2.6. 1. In catenis aequabilis cnffitiei quurum pondus

integrum = I, eft ~ = y-: pro l1is igitur talis inferuit aequatio Jy d x = - n;~, eK qua omnia deduci pofiimt, <JlL'le a paragrapho decimo fexto ad vigefimum tertium dida funt.

II. Fuerit pondlls catenae integrae rurfus = I:

longitudo eius = I: fitque vbique ~ =f-?-; 'eront difi.'lntiae punctorum F et M a linea verticali vt 1: ad fummam huius feriei

:c ::ex x3 x+ x 5

I -n+3nii-·i~6nr+3. 6. lon+ 3.6. IO.! ~ nS + etc. Eft autemn longitudo talis, vt fit

I Il II l+ IS _ I - n+ 3 nn.- ].6iil-t-g:6 IOn.+ 3.6. I o. I ~ nS + etc. - o .

.cui conditioni proxime iatisfit, cum fumitur 11_ ~c5- I; tantaquc cit longitudo pellduli fllllplIcis ifOchroni: erit antem ex.curfio punch infimi F fere tripla eius quam £'lcit punch(m medium M.

Schol iUlll Generale. ~ i. In omnibus qll:lS confidcrallimm ofcilbtioni

ims dlftlJ1ti'lC fingniorum plIncrorum a linc:l wrrictli; T~m. VI. ~ Q quai!

139

140 Appendix

t~~ THEOUMAT.A. DE OSCILLAT. CORl'OR.

quafi infinite pantae cenfendae flmt ratione longitudinis fili corpora conneCtentis aut catenae, imo ctiam ratione arcuum catcnae, de qui bus in para~mpho 21.

diximus, ita vt v. gr. in figura feptima etiam diftantia F C infinities minor eife debeat linea C B , ad quod animus eil aduertendus in inftituendis experimcntis, quamuis a magnitudine ofcillationum non facile error 2dmodUln notabilis oriatur.

Si haec pendula in turbinem agantur, eandem figuram indueDt, quam ipfis in ofcillationibus affignauimus, et gyros fuos duplo abfoluent tempore, quo ditlatiODli in eod.em periiQunt plano.


A A A L

h: { J~ 7~ I ..& .9 /;- {p

p

.M N .- M A N A N

c

142 Appendix

COMMENT ARII ACADEMIAE

SCIENTIARVM

IMPERIALIS PETROPOLITAN AE.

TOM VS VII. AD ANNOS cbbcCXXXIV. & cbbccxxxv.

#Ai/~.

PEr R () r ') t T ,

De Oscillationibus Corporum Filo Flexili Connexorum 143

16" DE OSCILLATIONIBVS CORPORV M

Danieli1 Bernoulli DEMO NSTRATIONES

THEOREMATVM SVORVM DE

OSCILLATIONIBVS CORPORVM FILO FLEXILI CONNEXORVM. ET CATENAE

VEa TlCALITER SVSfENSAE~

l~

'labula IXDEdi nuper rheoremata de ofcilfationibus- corporum filo flaili connexorum: demonftrationci

autem, q\.las tum noR vllcabat in Drdinem redigere, nunc paoUllD plus otii mallS eD libentius cum publicD commnnicabo, quod multorum aliDrum fimilium problema nun folutiD inde peti pofiit, eDrum praefertim in qaibm motu& pminm nDn runt inter fe paralle1L Inter huiusmodi pl'Dblemata facillimum eil atque a muftis iam diu folutum, qUDd circa centra ofcillationis inuenkmb verfatur. Ad ea quoque pertinet problema de motu mixto determinando, quo corpus ex pluribus diuerfae gralIita~is fpecificde partibus cDmpofitum in fluidD defcen .. dit: pertinent porro theDremata, quae in Comment:lr~ Tom. II. p. 200. a Patte cum publico commnnicata fue· mnt: Praefertim autem methDdus, quam mox exhibebo, cum (ucceifu adhibetur, quando in fyflemate corpormn plurium lege aliqua inter fe connexorum, fitus vnius ax fitu alterius cog~lito 110n pDtcil immediate dctermi-

nan,

144 Appendix

1'ILO FLEXILI CONNE}{OIVM &c. l.'6S

.ari, Teluti cum corpus Cuper hypothenu[a triangnli ill horizonte mobilis delcendit; hie enim fi Tel noueris fitum corporis in hypothenufl, ipfius tamen trianguli 'tus in horizonte incognitus mallel~it nifi hUDe aliunde determinare Cci~s. Problema hoc pollremum aliquando Patd meo propofueram et plane inter fe conUenerlll1t folutiones nollme; Earn, quae a Patre profe&a eft, Academia Commentariis fuis inred curauic, vid. Tom. V. p. I I • Quae ad hane claifem pertinent, nouam mechanicae partem efficiunt: Principiurn autern, quo vli foleo ad huiusmodi problernata foluenda, tale eft ;

Pura in fyfl:emate ad momentum temporis corpora. lingula a fe inuicem folui, nulla faCta attentione ad mo-tum urn acquifitwn, quia hie de acceleratione feu mlltatione motus elementari tantum fermo eft: ita quolibet corpore fitum foum mutante, fyr1:ema aliam accepit fi.gurarn, quam non-folutum habere debebat: Igitllr tinge cauam mechanicam quamcunque fyftema in debitam £guram reftituentem atque, rurCus inquiro in rnutationem fitus ab hac reftitutione ortam in quolibet corpore; et ex 'Vtraque mutatione intelliges mutationem firus in [yftemate non Coluto, indeque accelerationem ret:udatio· nemue veram cuiusuis corporis ad [yaerna pertinent is ob~inebis.

Quomodo haec regula ad praefens noftmm de oCcillationibus corporum filo flexili ligatorum aut cltenae ye~ticaliter hlspenCae determinandis, negotium appliclllda fit, hie docebo, ali:1 occafione idem fortaffe edam

X .a m01l-

De Oscillationibus Corporum Filo F1exili Connexorum 145

16+ DE OSCILLATIONIBVS CORPORV}}[

monftraturus in problcrnaris aliis partim iam a Patre men tr<lchltis partimque nouis.

riSU1'1 I. II. Sit filum A H F grilUitatis expers, dl10hus onc-ratum ponderibus in H et F, e punCto fixo A [tl'ipCn~l: faciant corpora ofcillationes veluti infinite parms, {1·ltque eorundem difiantiae a linea verticali A C, vt 2 M I ad mL-ml+ML+M/+ V (+mMLL+C11l1+ML + M 1- m L Y' ). Dernonflrandum eft, ofdllationc s in vtroque corpore fore ifoehronas. Valores litteranun 111 ~ M , I et L infra dabuntur. .

Em nt ofcillationes ifochronae, fi fuerint vires acce1eratrices in corporibus vt difiantiae eorundem a linea verticali; nee enim diffcrunt diflantiae hae ab arcublls defcribendis: Has igitur vires acceler:urices definiemus: ponatur pars fili H F extendi filCillime, ita vt corpus F nihil amplius retineat: aecelerabitur corpus ifiud vcrticaliter deorfum a grauitatis vi naturali: finge ita accelerari vt perueniat dato tern pusculo ex F in E, dum eodem temporis puncta alterum corplis .filo A H alligatum !1.rclllum H L defcribit: ducbe iam intelligantur horizon~ales L B et E C, quae quamuis cen infinite par .. uae eonfiderentllr, fint ramen arwIo L H . infinities maiores. Apparet ex mech:lnicis et theoria infinite parllorum, fore H L = ~ ~ x FE. Pofitis igitur corporibus in L et E ducrisque redis A L et L E, (fit quidcm filum A L inuariat:lc longitudinis, alrerum autem LE iam maioris eric longitudinis quam fllcrac in {jeu H F: COIlci piatllr igitur «lUI;l) qW1C filum L E contrah:lt ad liu m

lon-

146 Appendix

FILO FLEXILI CONNEXORVM (:y(,'. 165

longirudincm naturalem: dico ab iib contraerione corpus ex E eleuatum iri vsquc in u, altcrnmque retmctllm ex L in 11: fpatiola E u, L 11 detcrminablmus, poftqnam monuero, quod, ducb minima reera F u, "eme aC$.=elemtiones, quae durante affim1to tempusclllo acceffenmt, feu ipl:1e etiam vires acceleratrices rationem habitllrae fint in corporibus H et F vt H 11 et F u. Sed vt ratio intelligatllr inter H Il et F u, £lciemus A H feu A L == I: H F feu n u = L: maffil in corpore iilperiori =nz; in inferiori = M. Prodllcatllr A L et ex E in illam perpendicularis ED demittatur. Denique ducantllr horizontalis H G et venicalis. F G: erit F tt ad n u perpendiclliaris cenfenda atque tri:mgulum minimum FEu triangulo H F G fimile, ipfaque E tt lineolae FE aequalis:

vnde fi ponatllr B L = 1 ; DE ==x 7' erit M C = I +fr EC:=:x+ I+~; HG=CE-BL=x+f; hinc

Fu=ct+f)xFE: Sllpcreft vt definiatur H n; NotemI'" quod filum L E, dum contrahitnr, corpus E direere filffilm trahir; dum corpus alterum L oblique ad dircerionem filam L n retrahit: hoc igitur titulo erit L n ad E u vt DEad LE feu vt x ad L: fed eft practcrea L 11 ad E u rcciproce vt m:lff,l corporis L ad ma1tlll1 corporis E, id eft, dircere vt M ad In: compoDt:! ratione erit L u: F u = M x:

mL; vndc pofit:! FE pro Eu, erit Ln=~~xFE; et (lui:l' H 1Z = H L - L 11, fcqnitllr fi)re

H n = ( i - ~1 ) x FE: fill1t i.::;imr vires aClcler:miccs in corporibus II et F,

Xj vt


165 DE OSCPLLATIONIBVS CORPORPM

yt t + [- ad t- :~: ponantur hae "fires ad ifochronismum obtinendllm proportionales fpatiis defcribendis L B et EC,

:x: M% ( +1. fen fiat (t+:L):(t-m-iJ=I: X+I "7), atqlle re-perietur fach reduCtione

m.L_m.l_ML-MH-v(4m.MU-t-(m.1-t-lIU.+Ml-m r.~·] .1:=:: - 2ML

Huic autem fi add:lttu M C (ell I + f, habebitur· C E - ~1-+-ML+Ml:!: v(.m.lolLL+(ml-t-ML -+-M1 -m.L)2) x B L

--- 2Ml ,

plane vt habet in parte huillS argllmenti prima theorelila twtium Prop. 7.

III. Pofitis iisdem, erit longitl1do ptndLlli fimplicis ifochroni aeql1.11is

2m.Ll mL+ml+MI+ML+i[+m.MLL+(ml-+-ML+M.l_mL)2] ,

mius rei rationem intelliges ex eo, quod fi pendulum fimplex longitudinis A H feu I confid:retur, fit vis ac.celeratrix in hoc pendulo fimplici ad vim acceleratricem corporis H in pendulo noUro compofito vt HI ad HIt, id eft t vt t ad i - :~,: funt alltem longitudines pendulorum· ifoch~onorum in reciproca ratione virium ac-celeratridum; Erit igitur longitudo pendl1li ql1aefiti ad longitudinem A H vt t au t - ~i: vnde inuenitur longi-

d d I· 't' I . m.Ll fi tu 0 pen u 1 1 OC lrOll1 == m i. ":-MZx, et po Ito valore pro X fupra inuento, erit eadem longirudo, vt dictum eft, aequalii .

• m.L? T1IL-+m;+~L+~L+-~[ ... m~~;:-:-'lLL:--+--:(m!-+-)1l-+-Ml-mL)%j

Fjpra s. IV. Si filum A G fit tribus Ollcratum corporiblls in H, F ct G, oiCili:ltiullCi folcicntibus valdc p:truas Ct ifo ..

chrv-

148 Appendix

FILO FLEX ILl CONNEXORVM be. 167

chron:1.s, pOn:1.tnrque m~l{fa corporis filprcmi = m; medii =1\1 ct infirni= p..: diHantia AH =1; HF = L; F G -= A; di{bntia corporis H a linea -vcrticali A P == I; difiantia vero corporis F ab eadem linea vertic ali = J; erit

( ( M MIA + M ,... lA ) 55+ 111 M IA + 111 p..1 L - til M LA - M M IA - M M LA -1- m,... A - M p..IA - M p- LA) J

-111 p.lA - mMIA) x ((MIA + tJ.IA )S-11Z LA - MIA -MLA- p./A-~L~+ miL )==m111p.IILL5.

Difi:mtia autem corporis infimi a linea ycrticali erit pro quanis r.dice ipfius S aeqlUlis 101~A MA ( ).. 101). M). MM). MM'X 101). (m~ + mL ) S S + 1,+ 1. + ~L - -;:iT - m/oioL - m/iJ - mL- -

~1). ) 101).. _~ mL S - p.L. L- •

Haec vt demonltrentnr 1 ponatur mrfus filum inn .. lI1um F G facillime extendi atque fic corpus G vi grauitatis natumli acceleracum, afiumto aliquo tempnsculo infinite paruo verticaliter defcendere ex G in s, dum interea ambo corpora filperior:l accelerentur, vti in figur.t prima, faciendo arcllios {nos H 1J et F u. Patct antern, fi G s in figur;J fecunda aequalis pomtur deICenfui F'E in fignr:t prima, fore pariter arculos H net Fit idem in vtraque fignra; erit igitur per praecedcntcm p.l-

ragr;tphum Hn=(1-~~)xGJ' et Fu=Cl+yxGJ, intelligendo per x lincolam u M perpendicubritcr ad prolong;lt:lm A 1l due'bm, prouti deinceps per y intelligerons lineobm y q), quae pcrpendicubris ell: ad prolon,:lum nit: ilffi dUC;1l1tuI horizontalis F Q. ac vcrticillis QG ,.

liHnt:lllllo..

De OsciIlationibus Corporum Filo Flexili Connexorum 149

168 DE OSCILLATIONIBVS CORPORV M

fumt<lque uy=FG, dlleatur Gy. His ita ad ealculum praep~lfatis, nunc ruruls finge.ndum efi:, rccbm U J, in prifi:inam longitudinem F G eontrahi: ita eleuabitur cor· pus ex s in y vel in r (eft autem yr nulla pme Gy); corpora aucem fuperiora iterum retrahentur ex 11 in (J

ct ex u in m: atque fie tandem patet fore viresacceleratrices in fingulii ~orporibus fecundum dircCtiones fuas naturales ad vim gr;luitatis naturalem, vt fe habent H 0, F nz et Gy ad G s; fuperefi: igitur vt fingula haec elementa exprimantur, probe obferuato arcllios H 11, Fit etc. nuUos eife prae difiantiis corporum a linca ,'crti-cali. Inuenietur autem rcae infi:ituto cakulo F L == ~ -i-~.\'+y: et quia FG:FQ=Gs:Gy, erit

Gy=Ct+f+t)xGs. lam porro ql1aerendum efi:, quantae filtllrae fint retrogmdationes corporum in u et n pofitorum, qllae fiunt, dum corpus infimum ex s in y :tut in r eleuatnr. Notetur potentiam filum U J contrahentem vbique aequaliter difIundi. Erit igitur rurius vt in fuperiori par:lgrapho U 111 ad sy feu ad G s in ratione compofita ex tj,,'Y ad uy et rtaffi1e fL ad maffilm 1\1: vnde inuenitnr um = ~1 x G s, qua ulbtracta ab F u feu ab (t + r ) x G s, oritur

Fm=(f+ I--~i~JxGJ. Dellique quia ab eo, quod corpus medium ex U 111 1JZ

cedit, nihil patimr corpus ltlpremUm, erit, "t :1ntCl,

no ad y s fen G s in ratione compofita ex 1\1 u ad u!z et maifae p. ad m:lfClm 111; vndc 1Z 0 = ~1~ x G.i: luc:.,tlC

ii:b-

150 Appendix

FILO FLEXILI CONNEXORVlrf (;ye. 169

fublata ab 11 H feu ab (1- !~) x G J t fiet.

H 0 - ( '- _ M:c - ~) x G J. -z mL mL

Po!1:quam fie aeeelerationes corporum fingulorum in veris fuis direCl:ionibus inuenimus, crunt hae difhntiis {ilis ab linea verticali J' P, u C et 11 B fen quantitatibus el + L >.. >.. L T';-Z-+.\'+LX+Y), (1+7+'\') et (I) pro-portionales filciendae ifochronismi ergo: Ita duae aeql1ationes obtinebuntur valores x et y determin:ll1tes: atque fi deinde pOl1atur I + t + x = .f , inuenietur aequatio pro s eadem, quam fupra recenfuimus, quamque demon!1:ralldam fuscepimus.

V. Aeeeleratio corporis H expretfa per H 0 feu per ( 1-:~ - ~~ ) x Gsell: ad accelerationem eiusdem corporis, abfentibus duobus inferioribus expreffilm per 1 x G J,

'Vt 1-- :~ - ~ ad 1, feu vt m L - M l x - p..l x ad m L. Sequitur inde longitudinem penduli ifochroni eife.

mLZ mL Ml;t-,.u~

Rmc autem non ditferre ab illa, qt1:1e in parte prirna, propofitione decim.l tertia data filit, videbis fi ibi, prouti tlebe a nobis dcnomil1!1tiones poll:lllant, intclli-gas per x, quod hie per s feu per 1 -t- t + x.

VI. Sint iam plura ct quotcunqnc volueris Corpor~l, Fig.ura 3.

vcluti B, C, D , E, F prodtlc:mtur fingnb fila dcfigncn-tllrquc finus angtl\orum BAN (A N ell vertical is) C B G, DCH, EDL,FEl'vl per p, q,I',.r,t; ma{f:\c autem corporurn per ipfas littcras iisJcm appofitas dCllotcl1tur, 1'011/. V II. Y dico

De Oscillationibus Corporum Filo FIexili Connexorum 151

1'70 DE OSCILLATIONIBVS CORPOR1- M

dieo pouta vi grauitatis naturali = I, vires aeeelemtrices corporllm fecundllm fuas dh·ed:iones fOIe vt fcquitur.

. B -p C+D+E+F IJ In - - B ,

in C=p+q_D+~F.r, in D=p+4+r-~F s., in E=p+q+r+s--it., in F=p+q+r+s+t;

Veram hane eife virium aceeleratricillm legem, pertipies fi fextum corpus filO filo inferills adhue appcndi ponas, ea1culumque deinde infiituas, vt fecimlls ratione trium corporllm paragrapbo quarto, fingendo fcilicet, corpus infimum natllrali grauitatis vi verticaliter deorfum aeeeleruri, reliquis interim fecundun1 filam indolem vibratis, idemque corpus mox a contraetionc fiII iterllm eleuari: im eoim legem hune acccicrationmn mtne expofitam a quinque corporibus ad [ex, et indead feptem atql1e fie quousque libuerit reete continuuri 'ViJcbis.

Ex utfumtis alltem llllgllIorum allglllorom fillibus, dedllcuntnr corporum a linea vcrticali dithntiae, ac fi qU:lmuis difi::mti'.lm vi acccleratrici rc(pondcmi proF 0 rtl 0-

n:tlem £lci:1S, h:lbebis tot aeqnationes quot incflgnitls, fie vt oqmb denique defidcrata inde rcae definiri pollint.

VII. Putl nllnc corpom effe nurnero infinita et Figlift ... aequ:.llia, difi:;U1tii~ minimis et !leql1alibus a fc il1yiccm

pofita: ideam habebis C:ltenae vnifi>rmis ab VlU cxttenIit~\te .filspenfae, qU:llh eft A C vel A F: In luc c1e-

152 Appendix

FILO FI..IEXILI CONNEXORVM (ye. IiI

mentum confideretur infinite partIum 1\1 11Z vel N 11

dudis M N et m n ad A C perpendicubri bus c t lit ~ eidem A C paralIeIa: pon:ttur A m vel An = s ( n(e cnim differunr quia infinite parum difiant); 111M \'el n N = d s, quod elementum conftans affilmntur: longltudo catenae integrae A F == I; mil --y; M 0 =::dy: erit (pofitil vi grauitatis naturah == I ) per praeccdens theorema 'Vis accelemtrix in tlZ aequalis iummae omnium finul1m angulorum contadus, qui ftlllt inter A et m, diminutae tertia proportionali corpusculi in 111 , ftlmmae omnium corpusculorum in M F et finus anguli contac-

tus in M: fie igitur habetur vis acceleratrix in M ==JdJt-(l-s)ddy (), . 'f h' ft I .

- --;[$;:-' x!-lJa vero J oc romsmllS po u at, vt VIS ac-celeratrix fit proportionalis applicatae M N, erit affim1ta n

ft JddY (1-s) ddy Y fi . I pro con ante ds - -asr--= Ii: umatur 1l1tegra e ter-mini primi fine additione conftantis, quia hie nulla fumenda eft: fie fiet ~ _ (l-;:~ddy == - ~. Denique ponatllr I fc F M C N . _dY xddy Y Ii - s eu aut == x, et ent di" - ax: ==n' lUe

ndydx+nxddy=-yd."/ , quae aeqllatio denotat naturam curuae AF: qlloniam vew integralis eius non apparet, porui

y == ct. - t; x - 'Y .1.' X - d .. t'~ - e x~ - etc. , dy=- t;dJ: - 2. 'Y xdx- 36 xxdx- 4E .1:T d.1.'- etc.

- d d.}' = - 2 'Y d x! - .2. , 3 . 6 J: d X Z - 3·4· e .1: .l.' d x' - etc.

Hisgue v;lloribus fubftitutis diuiflql1e deinde aequ;ltione per d x! , oritur


17'). DE OSCILLATIONIBVS CORPORV M

- ~ - 2 yx- 3 0 X,,\'-+e.\·~ -etc. " 3 - 2 'Y x - .2. 3 0 X .t' - 3 . + e..t' - etc. == ~ ,.

+ ~ _ ~ x - Xx x - t x! - etc. n n' 71 11

cui aequationi fatisfit ponendo a == I ; r;' == ~; y = ::~; ~ I -, t d -= ... 9711 ; e=="o5> 16714 e c. vn e

x +- xx x:r + -,-,-xf._..,. t y= I-'n 0 ;vi - -';'5>,,3 4. 5>.IGn" -c C. "hi per 1 intelligenda eft diftantia puncH infimi F a -verticali: et quia pofit~ x=l,. ell y=o erit fimul

I 11 P 1· I -Ii +:;nn- "05>711 +' H.,6n4 - etc.=o;

Hine deriuandus efl: valor litterae n, qui exprimct 10ngitudinem filbt~lllgentis in F. Haec demonftr:ll1t verit:.Item theorematis,. quod in praecedente diifertatione o&mum eft.

VIII. Vt habeatllr longitudo penduli ifochroni, quaerenda en.: vis acceler:ltrix in punCto F, quae per §. VI. erit acqualis fummae fill11um omnium :mgulorum contaCtus ab A vsque in F, id eft=J~!'2. feu== d.:.r; ponendo fimul x=:: 0; et hine fit -A~= ~'. Eft itaque vis acceleratrix in F ad vim acce1crarricem natura1em vt 1 ad, n; fi vera pendulum fimplex longitudinis I h~heatur, erit vis acceleratrLx in illo=+ filb cadem dillantia a. linea vel'ticali; ergo vis a(celeratrix in cxtremitllte catenae ell ad vim accelcratricem in pendu-10 fimpliCi eiusdem longitlldinis, vt I ad 11: Hincque erit longitlldo penduli fimplicis cum catena fimul vibr:mtis = rt, vt habet theorema lIonum in praeffilffa.. dUfertatione • .

IX.

154 Appendix

FlI,o FLEXILI CONNEXORV M (:re. I73

IX. Theoremata autem dccimum et vlldccirnum Tnice pendent a debit.le confbntis additiol1c, caque proindc cell nimis nunc flcilia hie ilon attingam: ted dnodecimum ex §. VI rudus, hUlle in modum deducemr.

Corpuscula nunc confiderentur infinita et -aequali ~ bus dHbntiolis :\ fe inuicem pofita, fed inaequalis ponderis: ita habebitur idea catenae pro lubitu inaequa liter cruftle; fit haec it.t formata, vt longitudini F M (x) pondlls ref pondeat ~, denotante ~ nmetionem qualemcunqlle ipfillS x: Erit (per §. VI.) vis ac-celeratrtx in M J-;xil2. - *!~ = t, vel quia d x con" n.~ • -d:J. l;dd~ - :J . d~;r + ~ dd -I",ns, ent dx - - d;dx -iL , aut n c; fly 11 c; 'J -- .y d; d . ..;, vel deniql1e

-~:J Jydx, vt fert theorema duodecimum, de quo {ermo ernt: demonftratio magis fiet intelligibi1i&, fi fimul confe,~tur paragraphus ieptimus.

DE

J , , "

c


, 'vII/Nt, 'N/'-=.-f.'.r./. /.: ,'.7:'/17 W, ZV:IX 1'- ' /t~2.

N

A

.

\ I \ I \ I \

\

"

C ~ F '-'to' ,'+, ,,'

F ~l

Danielis Bernoulli THEOREMATA DE OSCILLA

TIONIBVS CORPORVM FILO FLEXILI CONNEXORVM ET CATENAE VERTICA

LITER SVSPENSAE

Introduction to the Argument

T heories of oscillation for extended objects which authors have given up to the present time presuppose that the relative positions of the

parts are fixed so that they are all carried with a common angular motion. But bodies suspended from a flexible thread require another theory. The principles usually used in mechanics do not seem to be sufficient for this business, since, of course, the relative positions of the bodies are variable and continually changing. A chain hanging vertically and agitated with an oscillatory motion once gave me an opportunity for thinking about these things. Seeing that the chain is thrown with an exceedingly irregular motion it occur.red to me to consider how the chain should be curved so that if all its parts begin their motion simultaneously they will arrive simultaneously at the \(ertical line going through the point of suspension. I realized that oscillations of this kind would be regular and such that the times [of oscillation] could be determined. Soon, however, I found that this curve is difficult to determine unless one begins the investigation with the simplest cases. And so I began these considerations with two bodies held together a given distance apart by a flexible thread; then I considered three, and soon four, and finally any number of bodies any distance apart. And when I made the number of bodies infinite, I saw at last the nature of the oscillating chain whether equal or unequal in thickness but everywhere perfectly flexible. Let me go through each successive case. But I shall reserve for another occasion demonstrations which are not worth furnishing now. To find the solution I used new principles, and therefore I wanted to confirm the theorems with experiments, so that even people who have not been able to give their full attention to these esoteric matters (and who could thus easily come upon a false solution) would have no doubt about their truth.

Theorems on the Oscillations of Bodies 157

We will not consider oscillations other than those that are minimal and isochronous; for the experiments, however, it will be possible to use somewhat larger ones without noticeable error.

Theorem 1

2. A perfectly flexible weightless string AHF is suspended from the point Fig. 1,2 A. Let two equal weights be attached to it at H and at F: the lower weight is as far from the higher weight as the higher weight is from the point of suspension. Next let ABC be the vertical, and let the displacements of the weights Hand F from this line be as though infinitely small. Finally, let the minimal horizontal lines HB and FC be drawn. I say that if both bodies begin to oscillate simultaneously, they will reach the vertical line at the same time, and in this way carry out their oscillations uniformly, provided one has chosen

CF: BH = 1 ± .J2 : 1

Corollary

3. Therefore uniform oscillations are made in two ways; when CF = (1 + .J2)BH, as the first figure shows, and also when CF = (1- .J2)BH, as in the second figure.

Theorem 2

4. When the oscillations of the bodies Hand F are made uniformly, the length oLthe simple tautochronous pendulum will be = (1/(2±J2))AH or (1/(4±.J8))AC, where the plus sign applies to the contrary oscillations of the second figure, the negative sign for the conspiring oscillations of the first figure. '

Corollary

5. And so contrary oscillations are completed much more quickly than conspiring ones: for you will count [241] contrary oscillations while the conspiring ones complete a hundred. But the conspiring oscillations differ less [than the contrary ones] from those that would be made under the same circumstances if the thread AHF were rigid. Nevertheless, bodies on a rigid thread oscillate a little more quickly than those on a flexible one, for the number of oscillations completed in equal times will be a little bit more than 1012 to 1000.

Fig.3, 4,5

158 Appendix

Scholium

6. So that I might subject these propositions to experiment, I used two perfectly similar lead balls which were perforated [because] while they were being poured I restricted them in the middle to make a hole. The balls were connected by means of knots to a steel thread passing through in such a way that the low one was twice as far from the point of suspension as was the higher one. With my fingers on the tense thread I drew the lower ball to the position F. Soon the oscillations became uniform and with the help of divisions marked on the wall I ascertained that the displacements of the bodies Hand F in fig. 1 were as 100 to 241, that is, as 1 to 1 + J2 (par. 2). Also the number of oscillations occurring in a given time correspond~d accurately to the length of the simple isochron~us pendulum (1/(2-.J2))AH defined in [par.] 4. Next, when FC = (1-.J2)BH as in the second figure, I held the balls by hand at F and H and then released them at the same moment. Reasonably uniform oscillations arose as before when the other prop~rtion of distances FC and HB was used. Again, the number of oscillations agreed accurately with that of the simple isochronous pendulum of length (1/ (2 + .J2))AH in [par.] 4.

Theorem 3 IN GENERAL FOR TWO BODIES

7. Now suppose that the part of the thread AH = I; H F = L; the weight of the body H = m, and of the other F = M. I say that the oscillations will be uniform if

CF _ mL - ml + ML + MI ± .J(4mMLL + (ml + ML + MI- mL)2) -, 2MI xBH.

The length of the simple isochronous pendulum will be

2mLI

,- mL + ml + MI + ML =F .J(4mMLL + (ml + ML + M/- mL)2)

or, supposing L + I = A and M + m = {.t,

2mA/-2mll

Theorem 4

8. If there are three instead of two similar bodies separated from each other by the same distance as that which separates the highest one from the point of suspension A, uniform oscillations are possible in three different


ways. The first is that which the third figure indicates when, assuming BH = 1, one takes CF = 2.292 and DG = 3.922; the second, which is represented by the fourth figure, is obtained by making CF = 1.353 and DG = -1.044; and the third when, as in the fifth figure, CF = -0.645 and DG = 0.122. CF is, of course, found by taking the three roots of the equation

4x 3 -12xx +3x +8 = 0,

and then for any root one takes

DG = 2xx -2x -2.

Theorem 5

9. When uniform oscillations are made as described, the length of the simple isochronous pendulum will be, in the case of the third figure, approximately 2.406AH; in the case of the fourth figure, 0.0436AH; and in the case of the fifth figure, 0.159AH. The length of the isochronous pendulum is, obviously, (1/(5 -2x» xAH, assuming again 4x 3 -12xx +3x +8 = O.

Scholium

10. I accurately confirmed both these theorems by experiment in the case of the third figure, drawing the lowest body away from the vertical line and then letting it go. Although in the first oscillations a certain inequality could' be felt, this spontaneously and very quickly disappeared, so that the displacements of each body remained, as far as could be discerned by eye, the same in many successive vibrations. Moreover, when measurements were made of the same things, a proportion of such a kind as theorem 4 indicates was' found between them. Also the number of oscillations corresponded perfectly to the fifth theorem. The two remaining cases require much work: nevertheless I was able to bring about oscillations of both types well enough that the truth of the fifth theorem was apparent.

Theorem 6 IN GENERAL FOR THREE BODIES

11. Now, the weights of the bodies H, F, G are arbitrary, and also their distances from the point of suspension are in any proportion. Let the weight of the body H = m, of the body F = M, and of the body G = f.L; and then

160 Appendix

AH = I, H F = L, and FG = A ; I say that oscillations will be uniform if assuming BH = 1, [C]F = x, one has

{(MM/A + MILIA )xx + (mMlA + mILIL - mMLA - MMlA - MMLA

+ mILIA - MILIA - MILLA)x - mILIA - mM/A} x {(MIA + ILIA )x

- mLA - MIA - MLA - ILIA - ILLA + mIL} = mmILllLLx

and also one takes for any root

DG=(MMA + MA)xx+(l+~+ MA _ MA _ MMA _ MMA_ MA _ MA)x mILL mL L ILL ILl mILL mILl mL ml

MA A ----ILL L·

Corollaries

12. I. The mass IL of the lowest body is assumed equal to zero, and the fundamental equation of the above paragraph is divided by the second factor which has a non-useful root. The first factor is = 0, and hence one will have Mlxx + (ml- mL - MI- ML)x - ml = 0, or

mL - ml + ML + Ml ± .J(4mMll + (ml- mL - Ml- ML)2) x= 2MI

Moreover, it is to be noted that this value does not differ from that which we gave in the third theorem although the quantities on the inside of the parentheses under the sign of the radical have different signs.

II. If the mass of the middle body indicated by M is assumed = 0, then, in order to have agreement between the third and sixth theorems, L + A and IL in the latter theorem are to be understood as the quantities designated by Land M in the former; and the line DG defined in the preceding paragraph is to be compared with the line CF in the third theorem. Anyone who pays attention to this will discover by doing some calculations that the equations of both theorems are the same.

III. Finally, when the mass m of the highest body H is assumed equal to zero, either one of the factors can be assumed = 0, and in either way one obtains CF or x = 1 + (L/ I), just as the nature of the thing requires, because~ then the lines AH and HF, as is clear, must lie in one direction. But the displacement of the lowest body cannot be determined from the equation unless it is done by some special method. Nor can the oscillations be immediately determined by the sixth theorem, since one or another of the lengths L or A vanishes when two bodies are united.

IV. In the fourth figure one can so arrange things so that CF = 0, in which case, since the distances of the bodies from the vertical line maintain


the same proportion during the whole oscillation, the middle body F remains at rest while both the others are agitated back and forth. And consequently it is clear that the length of the isochronous pendulum will be A, since the lowest body oscillates just as though suspended from a fixed point C; indeed, that case about which we speak is obtained by setting x = ° or rather

[A]= miL mL + MI + ML + ILl + ILL

Theorem 7 WHICH DETERMINES IN GENERAL THE T A VTOCHRONOVS

PENDVLVM FOR THREE OSCILLATING BODIES

13. Retaining the notation and equations of the sixth theorem, I say that the oscillations of each body will be isochronous with the oscillations of a simple pendulum whose length is mILI[mL + (M + IL)(I + L -Ix)]'

Corollary

14. In the case x = 0, which we have selected [par. 12.1V.], the length of the isochronous pendulum is A = mILI(mL + MI + ML + ILl + ILL), which agrees with Cor. 4, Theorem 6. If in addition it is assumed that L = I, the length of the isochronous pendulum is A = mil (m + 2M + 21L): This agrees with Problem 1 which my father gave in Comm. Acad. Petrop. Vol. III, p. 15. And it is oQvious that the weight P which in that problem is attached to one end of the string is here the sum of the weights G and F increased by half the weigh~ H.

General Scholium

15. I can give similar equations for four, five, and however many bodies one would like: but the equation always rises to as high a degree as the number of bodies, and it is generally quite lengthy. But because the final equation arises from a system of linear equations, the law appears from the method I have used, by _means of which all things associated with the equation to be delermined can be found by iteration.

Theorem 8 ON THE SHAPE OF THE VNIFORML Y OSCILLATING CHAIN

16. Let a uniformly heavy and perfectly flexible chain AC be suspended Fig. 6 from the point A, and let it be understood to make uniform oscillations. The

162 Appendix

chain has the extreme position AM F. The length of the chain is I; the length of a part FM = x, and n is assumed to have one of its values, so that

I II 13 14 [s 1--+ 0

n 4nn 4·9·n 3 +4·9·16·n4 4·9·16·25.n s+ etc.=. It is assumed that the distance of the end point F from the vertical line is equal to 1; I say that the distance of an arbitrary point M from the same vertical line equals

x xx x 3 X4 X S

1--+-- + +etc n 4n 2 4'9'n 3 4·9·16n 4 4·9·16·25n 6 •

Scholium

17. By the method that I gave in Comm. Acad. Petrop. Tom. V de resolutione aequationum sine fine progredientium, one finds by a very short calculation that n is approximately 0.691l. Therefore, if, for example, the point M is in the middle of the chain, it will be distant from the vertical line by approximately two fifths or, more accurately, three hundred and ninety eight thousandths of the distance of the lowest point F from the same vertical line. However, n has an infinite number of other values.

Theorem 9

18. Using the notation of the eighth theorem, I say that the length of the simple pendulum isochronous with the oscillating chain is equal to n or rather to the subtangent CP to the curve AF at the lowest point F; or approximately equal to' six hundred and ninety one thousandths at the whole chain in the case of the sixth figure.

Corollary

19. In this mode the chain oscillates more slowly than the rigid rod of uniform thickness of the same length as the flexible chain. For the oscillations of [the rod] are isochronous with the oscillations of a simple pendulum having two thirds the length of the rod.

Scholium 1

20. After I connected a large number of lead balls close together along a thread, as said in the sixth paragraph, I used it as a chain for doing an experiment. I suspended the thread loaded with the balls from a firm point;


and having drawn it aside by the end F and having let it go, I observed the ratio between the distances of the extreme point F and the midpoint M from the vertical line AC, the oscillations now being made uniformly, and I found the same ratio as that which is indicated in the seventeenth paragraph. I also observed that the number of oscillations agreed with the length of the simple isochronous pendulum which is given in the ninth theorem.

Scholium 2

21. Since the equation given in the eighth theorem, that is,

has infinitely many real roots, the chain can be inflected in infinitely many ways to make uniform oscillations. However, n takes on smaller and smaller values, so that in the end it almost vanishes. The length of the simple isochronous pendulum is always equal to n or to the subtangent CP; whence the oscillations will in the end be made as though infinitely fast. All possible cases are given in the following: first, when the chain does not intersect the vertical line at any point other than the point of suspension, which is represented by the sixth figure and for which the length of the simple isochronous pendulum is n = 0.6911, as we saw above. Next, when the chain crosses the vertical line in an additional fixed point, as indicated in the seventh figure, where the already-mentioned point of intersection is B: in this case the length of the isochronous pendulum is n = 0.13/, and twenty three oscillations are completed in the time that ten are completed for the case of the sixth figure; for M the point of maximum displacement, CN corresponds to 0.471 and MN is approximately tFC. After this case comes that that is shown in the eighth figure, where the vertical line is crossed by the oscillating chain at two fixed points Band G. Next [comes the case] when three intersections are made, and so forth. The arcs subtended between adjacent points of intersection are longer corresponding to their being higher. Moreover, in a chain that can be taken as though infinitely long the [shape of the] highest arc does not differ perceptibly from the shape oJ a tense musical string since the weight of that arc is as nothing compared with the weight of the whole chain. Nor is it difficult from this theory to deduce a theory of musical strings which clearly agrees with those that Taylor and my father gave-the first in his work de methodo incrementorum, the other in Comm. Acad. Sc. Petrop. Vol. III. Experiment teaches that intersections similar to those in vibrating chains occur also in musical strings, in fact a small piece of paper placed at certain places on the string does not move when an adjacent string vibrates.

164 Appendix

Theorem 10

Fig_ 9 22_ If the chain AC is suspended from a weightless thread LA, and the distance [to the bottom] of a point chosen arbitrarily is supposed to be FM = x, the distance of the highest point N from the vertical line is (3, and n takes on one of its values so that

1_(l+'\)+(l1+2/,\) (13+ 3/2,\) (14+4/3,\) _ n 4n 2 4-9n 3 + 4-9-16n4 etc_-O,

I say that in uniform oscillations the distance of any point M from the vertical line equals

(XXX x 3 X4 )

1-;;+ 4nn - 4-9n 3+ 4-9-16n4 etc_

I II 13 14 ) X(3: (1--+-----3+ 4 etc_

n 4nn 4-9n 4-9-16n

and thus that the distance of the lowest point F will be equal to

( I II 13 14 ) (3: 1-;;+ 4nn - 4-9-n 3+ 4-9-16n4 etc __

The length of the simple isochronous pendulum will be equal, as before, to n, or rather, in the simplest case, approximately

21114 +844/3,\ + 153611,\,\ + 1440/,\ 3+576'\ 4

304/3+91211'\ + 1152/,\,\ +576,\ 3

Corollary

23_ If, for example, the length of the thread is the same as the length of the chain, that is, I =,\, the length of the simple isochronous pendulum or n will be approximately 1_56/; and the distances of the endpoints F and N from the vertical line will be almost as 11 to 1. In addition to this, however, there will be many other cases similar to those which we listed in paragraph 21: thus, after the case mentioned follows that in which n is approximately t/; and the point C has displacements in the direction contrary. to those of point A and three times greater_

Theorem 11

Fig_ 10 24_ With everything assumed to be as in the tenth theorem, if the chain is loaded at the top A with a weight equal to the weight of a part of length


L of the chain, everything will be again as in the tenth theorem if now

1- (LA +LI +ll +/A) + ([Lf] + 41LA + 13 +211A) (911LA + Ll3 + 14 +3/3A) n(L+l) 4nn(L+/) 4'9n 3 (L+/)

(16/ 3 [. ]LA + Ll4 + IS +4/4A) + 4 etc. = O.

4·9·16n (L+l)

If, for example, L = A = I, and

21 II 7/3 11/4 1--;+ nn -36n3+576n4-etc.=0,

one has n = 1.371 approximately; the small arcs described by the lowest and highest points of the chain will be as 100 to 39.

Theorem 12 IN GENERAL FOR CHAINS OF ARBITRARY

THICKNESS AND WEIGHT

25. Finally consider a chain of non-uniform construction, let ~ be the weight of a length of a part of the chain FM = x, understanding ~ to be any function of that x; let the distance of the arbitrary point M from the vertical line be = y. I say that the curve FMN is defined from the equation J y d~ = -n~ dyl dx, with dx taken as a constant element; and after this equation is satisfied correctly in any particular case, the length of the simple isochronous pendulum will be = n.

Corollaries

26. 1. In chains of uniformly changing thickness the entire weight of which = 1, one ~ has ~ = xl I; for these the equation given by J y dx = -nx dyl dx suffices, from which all the things that have been said in paragraphs sixteen to twenty-three can be deduced.

II. The weight of the entire chain is again = 1; its length = I; and let ~ = xxi II everywhere. The distances of the points F and M from this vertical line will be as 1 to the sum of the series

x xx x 3 X4 XS 1--+----+ + etc

n 3nn 3·6n 3 3·6·10n4 3·6·10·15n s .

Moreover, n is of such length that

I II 13 14 1--+-----+-----,

n 3nn 3·6n 3 3·6·10n 4 3.6 .10.15ns+ etc. = O.

166 Appendix

a condition that is approximately satisfied when n = ~61; and this is the length of the simple isochronous pendulum. Moreover, the displacement of the lowest point F will be approximately three times that of the midpoint M.

General Scholium

27. In all the oscillations that we have considered the distances of all the points from the vertical line are taken to be as though infinitely small in comparison with the length of the thread connecting the bodies or the length of the chain and, indeed, also in comparison with the arc of the chain about which we spoke in paragraph 21, so that, for example, in the seventh figure the distance Fe must be infinitely small in comparison with the line CB; one should pay attention to this in doing experiments, although a very conspicuous error does not arise easily from too great an amplitude in the oscillations.

If these pendula are carried in a revolving motion, they take on the same shape as that which we have assigned to those in oscillations, and they complete their revolutions in twice the time that they would take to perform [semi] oscillations in a plane.


A A A A A

B - H

F

F F G G fig. 1 fig. 2 fig.3 fig. 5

A A A L L

p

A N N - M G

C F fig. 6

F C fig. 7 fig. 8

C fig. 9

F F C fig. 10

Danielis Bernoulli DEMONSTRATIONES

THEOREMATVM SVORVM DE

OSCILLATIONIBVS CORPORVM FILO FLEXILI CONNEXORVM ET CATENAE

VERTICALITER SVSPENSAE

1.

I recently gave some theorems on the oscillations of bodies that are connected by a flexible thread: there was not then time to give the

demonstrations in detail; but having come upon a little more leisure, I will now more cheerfully communicate them to the public, because the solutions of many other similar problems, especially of those in which the motions of the parts are not parallel, can be sought by the same method. Among problems of this kind, the easiest one, already studied by many for a long time, concerns finding the center of oscillation. To this category belongs also the problem of determining the mixed motion with which a body made of many parts having different specific gravities descends in a fluid. Next belong the theorems which were communicated to the public by my father in Comment. Tom. II p. 200. However, the method which I will give presently can be used with success even when, in a system of many bodies connected to each other according to some law, the location of any single [body] cannot be immediately determined from the known location of another, as for example when a body descends along the hypotenuse of a triangle that is free to move horizontally. For in this case even if you know the place of the body on the hypotenuse, the location of the triangle on the horizontal will remain unknown unless you know how to determine this from another source. I once proposed this problem to my father, and our solutions agreed completely. (My father took care to put his solution in the Acad. Comm. See Vol. V, P. 11.) This class of problems takes one into a new domain of mechanics: the principle which I use to solve these problems is the following:

On the Oscillations of Bodies Connected by a Flexible Thread 169

Suppose that in a system, at a moment of time, each body is in turn considered to be free, without paying attention to the motion already acquired, because one is here considering so elementary an acceleration or rather change in motion. But when any body changes its position, the system takes on a shape other than that which it would have had were the body not constrained. Therefore imagine some mechanical cause restoring the system to its original shape. I next inquire into the change of location of any body arising from this restitution. From the sum of the two changes you learn the change of place of a body in a constrained system, and you will thence obtain the true acceleration or retardation of any body that is part of the system.

I will show here how this rule is to be applied to our present problem of determining the oscillations of bodies connected by a flexible thread or of a vertically suspended chain. On another occasion, perhaps, the same thing will be shown in other problems, including some that were already treated by my father and some that are new.

1 II. Let the weightless string AHF, loaded with two weights at Hand F, be suspended from the fixed point A: Let the bodies make oscillations that can be considered to be infinitely small, and let their distances from the vertical line AC be as 2MI to mL - ml - ML + MI ± J(4mMLL + (ml + ML + MI- mL)2). It is to be shown that the oscillations of each body will be isochronous. The values of the letters m, M, I, and L will be given below.

The oscillations will be isochronous if the accelerating forces on the bodies are as their distances from the vertical line since these distances do not differ from those described by the trajectories. Therefore we will determine these accelerating forces: Suppose first that the part of the string H F is easily extended, so that the body F will not be held back. That body will be accelerated,downwards by the natural force of gravity. Imagine that it is accelerated so that it arrives at E from F in a given infinitesimal interval of time, while at the same point of time the other body attached to the string AH describes the arc HL. The horizontals LB and EC are understood to have been drawn; although considered as if infinitely small, they are nevertheless infinitely bigger than the arc LH. It is evident from mechanics and from the theory of the infinitely small that HL = (BL/LA)FE. Assume that the bodies are at Land E, and draw the lines AL and LE; the thread AL will, of course, be of invariant length, but the other, LE, will now be of greater length than it was at HF. Next, imagine that the cause which will contract LE'to its natural length is activated. I say that by that contraction the one body will be raised from E to u, and the other drawn back from L to n: We will determine the elements of distance, Eu and Ln after I point out, having drawn the short line Fu, that the true accelerations that the bodies accwnulate during the time interval, or rather the actual accelerating forces on the bodies Hand F, will be proportional to Hn and Fu. But to find the ratio between Hn and Fu, we shall let AH or AL= I,

170 Appendix

HF or nu = L, the mass of the upper body = m, of the lower = M. Let AL be extended and let the perpendicular ED be dropped to it from E. Finally the horizontal HG and the vertical FG are drawn. Fu will be taken as perpendicular to nu, and the small triangle FEu similar to the triangle HFG, and Eu equal to the line element FE; whence if one assumes that BL = 1 and DE=x; then MC= I+L/I; EC=x+l+L/I; HG=CE-BL=x+L/I; from these equalities

It remains that Hn be determined: It is to be observed that the thread LE, while it is being contracted, draws the body E upwards in the direction [LE]; while it draws the other body L back obliquely in its own direction Ln. Therefore with this said, Ln to Eu will be as DE to LE or, rather, as x to L. But in addition Ln to Eu is, reciprocally, as the mass of the body L to the mass of the body E, that is, directly, as M to m. When the proportions are combined, Ln: Eu = Mx : mL, whence, writing FE for Eu, Ln = (Mx/mL)FE; and since Hn = HL- Ln, it follows that

( 1 MX) Hn = --- xFE. 1 mL

The accelerating forces on the bodies Hand F are therefore as 1/1 + x/ L to 1/1- Mx/ mL: To obtain isochronism, these forces are assumed proportional to the distances described, LB and EC, or rather, suppose that

(.!.+~) . (.!.+ MX) = l' (x + 1 +~) IL'lmL' I'

and when this is solved, one obtains

mL - ml- ML - MI ± .J{4mMLL + (ml + ML + MI- mL)2} x= 2ML

If MC or'l + L/ 1 is added to this, one obtains

CE= mL-ml+ML+MI±.J{4mMLL+(ml+ML+Ml-mL)2} x BL 2ML '

entirely as in the first part of this argument, third theorem, prop. 7. III. I(the same things are assumed, the length of the simple isochronous

pendulum will be equal to

2mLI

mL + ml + Ml + ML +' .J{4mMLL + (ml + ML + MI- mL)2}'

the reason for which you understand from the fact that if a simple pendulum of length AH or 1 were considered, the accelerating force in this simple


pendulum would be to the accelerating force on the body H in our composite pendulum as HI to Hn, that is, as 1/1 to 1/I-Mx/mL: moreover, the lengths of the isochronous pendula are in the reciprocal ratio of the accelerating forces. Therefore the length of the desired pendulum to the length AH will be as 1/1 to 1/1- Mx/ mL, whence is found the length of the isochronous pendulum = mLi/ (mL - Mix), and assuming for x the value found above, the length will be, as was said, equal to

2mLl mL + ml + Ml + ML =F J{4mMLL + (ml + ML + MI- mL)2} ..

IV. If the thread is loaded with three bodies at H, F, and G, making isochronous and extremely small vibrations, and the mass of the highest Fig. 2 body is assumed = m; of the middle body = M and of the lowest = IL: the distances AH = I; HF = L; FG = A: the distance of the body H from the vertical line AP = 1; the distance of the body F from the same vertical line =s; then

{(MM/A + MILIA )ss + (mM/A + mlLIL - mMLA - MM/A - MMLA

+ mILIA - MILIA - MILLA)s - mILIA - mM/A} x {(MIA + ILIA )s

- mLA - MIA - MLA -ILIA -ILLA + miL} = mmlLllLLs.

The distance of the lowest body from the vertical line will be, for any root s, equal to

( MMA + MA) ss +(1 +~+ MA _ MA _ MMA _ MMA _ MA _ MA)s mILL mL L ILL ILl mILL mILl mL ml

MA A ILL L'

In order to demonstrate these things, it is again assumed that the lowest thread FG is easily extended and that thus the body G, accelerated by the natural force of gravity, descends vertically from G to s in a given infinitesimal interval of time while both the higher bodies are accelerated just as in the first figure by making their arcs Hn and Fu. Moreover, it is clear that if Gs in the second figure is assumed equal to the descent FE in the first figure, the arcs Hn and Fu will be the same in both figures; therefore, by the preceding paragraph Hn=(1/I-Mx/mL)xGs and Fu= (1/ 1+ x/ L) x Gs, where x is the infinitesimal interval uM drawn perpendicular to the prolongation of An, just as next we understand by y the line element yv, which is perpendicufar to the prolongation of nu. Now draw the horizontal ~a and the vertical aG, and having taken uy = FG, draw Gy. These things having been prepared· for the calculation, it is now as in the previous case to be imagined that the straight line us is contracted to its original length FG: so the body will be raised from Iii to y or r <however yr is as nothing compared with Gy); the higber bodies are again drawn

172 Appendix

back from n to 0 and from u to m: and thus it is clear that the accelerating force on each body in its natural direction will be to the natural force of gravitation as Ho, Fm and Gy to Gs, respectively. It remains therefore to find an expression for each of these elements, having observed that the arcs Hn, Fu etc. are nothing in comparison with the distances of the bodies from the vertical line. By doing the calculation correctly, one finds that FL = A/I + (A/ L)x + y and, since FG : FQ = Gs: Gy, one obtains

(1 x Y) Gy= -+-+- xGs. 1 L A

Next one has to find the size of the retrogradations of the bodies (which, at this point, are supposed to be at u and n) which are made while the lowest body is raised from s to y or to r. It is to be observed that the tension contracting the thread us is carried equally in the other threads. Therefore again as in the above paragraph um to sy or rather to Gs will be in a ratio composed of vy to uy and of the mass {.L to the mass M: Whence um = ({.Ly/ MA) x Gs, and when this is subtracted from Fu or rather from (1/1 + x/ L) x Gs, there arises

Finally, since the highest body is not affected by the shift of the middle body from u to m, there will be, as before no to ys or Gs in a ratio composed of Mu to un and of the mass {.L to the mass m; whence no = ({.Lx/ ML) x Gs. And subtracting this from nH or from (1/1-Mx/ mL) x Gs, one obtains

( 1 Mx {.LX) Ho= [-mL - mL xGs.

Now that we have thus found the acceleration of all bodies in their true directions, we must make them proportional to their distances from the vertical line, yP, ue, and nB, or rather to the quantities (1 + L/I + A/I + x + (A/L)x+y), (l+L/I+x) and (1), in order to obtain isochronism: Two equations determining the values x and y will be obtained: and if next it is assumed that 1 + L/ 1 + x = s, the same equation for x will be found as that which we examined above and which we undertook to analyze.

v. The acceleration of the body H expressed by Ho or rather by (1/1-Mx/ mL .2- {.Lx/ mL) x Gs is to the acceleration of the same body when the two lower bodies are absent, expressed by (1/ l) x Gs, as 1/1-Mx/mL{.Lx/ mL to 1/1 or rather as mL - MIx - {.Llx to mL. It follows thence that the length of the isochronous pendulum is

mLI mL - MIx - {.Llx


Moreover, that this does not differ from that which was given in the first part in the thirteenth proposition you will see if you replace the x given there by s or by 1 + L/ I + x, just as the definitions made by us require.

VI. Let there now be many bodies and as many as you wish, say B, C, Fig. 3 D, E, F. Let [the line of] each thread be extended and let the sines of the angles BAN (AN is vertical), CBG, DCH, EDL, FEM be designated by p, q, r, s, t; let the masses of the bodies be designated by the same attached letters. I say that, with the natural force of gravity assumed = 1, the accelerating forces of the bodies according to their directions will be as follows.

C+D+E+F on B=p B q,

D+E+F on C = p + q - C r,

E+F on D=p+q+r-Os,

F on E = p + q + r + s - E t,

on F = p + q + r + s + t;

That this is the true law of the accelerating forces you will see if you suppose that a sixth body is attached by its own thread to the lowest body considered here" and if you next do the calculation as we did, in the case of three bodies in the fourth paragraph, by imagining, naturally, that the lowest body is accelerated downwards by the natural force of gravity, leaving the rest' of the system to move according' to its own nature, and then that the same body is elevated again by the contraction of the thread. Thus you will see that this law of acceleration, now accessible, is continued from five bodies to six, and thence to seven and thus to any number one pleases. .

From the sines of the angles the distances of the bodies from the vertical line can be deduced; and if you make the d·istances proportional to the corresponding accelerating forces you will have as many equations as unknowns; thus everything desired can be correctly determined.

VII. Suppose now that the bodies are infinite in number and of equal mass, arranged with infinitesimal and equal distances between them: you Fig. 4 will have the ideal uniform chain suspended from one end, such as AC or AF. In the [figure] the element Mm or Nn is to be considered infinitely small, with MN and mn drawn perpendicular to AC and mo drawn parallel to AC: It is assumed that Am or An = s (these do not differ since they are infinitesimally separated); mM or nN = ds, taken as a constant element; the length of the whole chain AF = I; mn = y; Mo = dy. By the preceding

174 Appendix

theorem (assuming that the natural force of gravity = 1) the accelerating force on m will be equal to the sum of all the sines of the contact angles between A and m, diminished by the third-proportional of the corpuscle at m, the sum of all the corpuscles in MF, and the sine of the contact angle at M.

Thus the accelerating force on M is

Since one requires isochronism, the accelerating force is proportional to the ordinate M N; that is, with n taken as a constant

f ddy - (I-s)2ddy I. ds ds n

The integral of the first term is taken without the addition of a constant, since nothing is to be added here. Thus

dy (1- s) ddy [+] I. ds - ds2 n

Finally suppose that 1- s or FM or eN = x, then -dy/ dx - xddy/ dx 2 = y/ n, or

n dy dX:J- nx ddy = -y dx2,

an equation that indicates the nature of the curve AF: Since its integral is not evident, I assumed

y = a -{3x -yxx -8x3 -ex4 -etc.

dy = -{3 dx -2yx dx -38xx dx -4ex 3 dx -etc.

ddy = -2ydx2-2· 38x dx 2- 3· 4exx dx 2-etc.

When these values are substituted and next the equation divided by dx 2 ,

there arises

-{3 - 2yx - 38xx + 4ex 3 - etc.

-2yx -2· 38xx -3· 4ex 3 -etc.

a{3 y 8 3 +--- X -- xx -- x -etc. = O. n n n n

An equation that one satisfies by assuming a = 1; {3 = l/n; y = -1/4nn; 8 = 1/(4· 9n 3 ); e = -1/(4·9· 16n 4 ) etc. whence


where 1 is to be understood as the distance of the lowest point from the vertical: and since at x = [ it has been supposed that y = 0, there will be also

[ II [3 [4 1--+------3 + 4 etc. = 0.

n 4nn 4· 9n 4· 9 . 16n

From this is to be derived the value of the letter n, which expresses the length of the subtangent at F. These things demonstrate the truth of the theorem which is the eighth in the preceding dissertation.

VIII. In order to obtain the length of the isochronous pendulum, the accelerating force at the point F has to be found; by par. VI. it will be equal to the sum of the sines of all the contact angles from A to F, thus, it is J [ + ]ddYI dx or [ + ] dyl dx, taken at x = 0; and whence [ + ] dYI dx = -11 n. And so the accelerating force at F to the natural accelerating force is as 1 to n. If a simple pendulum has length [, the accelerating force on it is 1 I [ at the same distance from the vertical line ; therefore the accelerating force at the extremity of the chain is to the accelerating force of the simple pendulum of the same length as [ to n; and therefore the length of the simple pendulum isochronous with the chain is n, as in the ninth theorem of the previous dissertation.

IX. The tenth and eleventh theorems depend only on the addition of an appropriate constant, therefore let me not undertake their demonstrations here as this would be excessively easy: but the twelfth will be deduced, again from par. VI, in the following way:

The corpuscles are considered to be infinite in number and assumed to be at equal distances from each other but now of unequal weights. Thus one will have the ideal chain that is unequally thick. Let it be so arranged that g corresponds to the length FM(x), with g designating some function of x. By p~r. VII there will be an accelerating force on M equal to J - ddy I dx - g ddy I dg dx = yin, or since dx is constant, there will be -dyl dx -g ddyldg dx = yin, or n dg dy + ngd dy = -y dg dx, or finally

- n!:y = f ydx,

as the twelfth theorem shows, about which there was discussion. The demonstration will be made more intelligible if the seventh paragraph is considered at the same time.

176 Appendix

A A

A

A

B

F

s fig. 4

E

fig. 3

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Index

Absolute. frequency (experimental determination of) 4, 19,26,45, 101

Archimedes' principle 88 Arch that supports a fluid 17

Beeckman, Isaac (1588-1637) 5 Beggar's Opera, The 21n Bending moment 73 Bernoulli, Daniel (1700 - 1782) 7, 27.,

28n;47, 53-69, 70, 71, 76, 79, 81, 83-92,93-103,104,109,110,122

Bernoulli, Jakob (I) (1654-1705) 16n, 17,32,36

Bernoulli, Johann (I) (1667 - 1748) 2, 7, 8,26,27, 33n, 36, 44, 47-52,53,55, 67,71,75,77,78,81,89,110-122

Bernoulli, Johann II (1710-1790) 14, 28n,77-82

Bessel function 7,53,58,66 Beta function 68 Bouguer, ,Pierre (1698-1758) 88 Boundary conditions (or values) 7,49,

64, 70, 74, 95, 102 Boyle, Robert (1627 -1691) 33n Boyle's law 6, 10, 12,43, 44n, 77 Brachisto<;hrone problem 26 Buoyancy .83

Carillon 93, 101 Carre, Louis (1663-1711) 33n Catenary problem 26 Center of force 94 Center of percussion III n Centrifugal force 120, 121, 122 Cohen, 1. Bernard (1914- ) 21n Conservation of energy 2, 7, 52, 118 Cotes, Roger(1682-1716) 9,13,14,20

Cramer, Gabriel (1704-1752) 27, 3"-36

Curvature 15, 16n, 40, 73, 94

D' Alembert, Jean Le Rond (1717 -1783) 20, 103

Dangling rod 83,89, 107, III De La Croix 88 De Moivre's theorem 61 Diderot, Denis (1713-1784) 20,27 Dilatation 10, 13, 14 Double pendulum 119, 120 Dynamical equations 1, 116, 120

Eigenvalues (or eigenvalue problem) 5, 7 Ellis, Alexander J. (1814-1890) 19n Enlightenment, the 20 Equilibrium configuration 5, 71 Euler, Leonhard (1707-1783) 7,8,20,

27,32,36,37-46,53-69,70-76, 79,81,88,93,95, 100, 103, 104-109,110, 122

Fabri, Honore (1607 -1688) 33n Flexural rigidity 6, 74, 93, 94, 100 Floating body 8, 83 Function theory 2, 104 Functional (calculus) 2,3,6,7, 12

Galilei, Galileo (1564-1642) 5 Galileo's law 55

, Gamma function 68 Gassendi, Pierre (1592-1655) 33n

Index 183

Geometrical (calculus) 2, 3, 12, 67 Glockenspiel lOIn Goldbach, Christian (1690-1764) 62

Hanging chain 7,53,95, 122 Harmonic constant 6, 47, 75 Harmonic force 6, 14, 15, 16, 39, 42n,

72,90,105 Harmonics 21, 35, 92 Hermann, Jakob (1678-1733) 7, 14,

28-32, 35,44n, 50, 78,81,121 Higher modes 2,4,21,45,53,75,76,

92, 93, 95, 97 Hire, Philippe de La (1640-1718) 33n Hooke.'s law 32 Huygens, Christiaan (1629-1695) 4, 5,

30, 33, 52, 73, 112

Intensity (of harmonic force) 6, 117n, 118n

Internal energy 38, 43 Irregular (motions) 6, 92 Isochronism 4,5,6, 18,47, 116

Kinetic energy 50, 121 Kircher, Athanasius (1602-1680) 33n Kline, Morris (1908- ) 68

, Lagrange, Joseph Louis (1736-1813) 27 Laguerre polynomials 7, 53, 58 Lana Terzi, Francesco de 32, 33n Laplace, Pierre Simon, Marquis de

(1749-1827) ,6 Leibniz, Gottfried Wilhelm

(1664-1716) 32 L'Hospital's rule 25,26 Light propagation 9, 77 Linear analysis 5 Linear force 104 Linked compound pendulum 8, 104 Linked pendulum 53, 110, 114, 121

McGuire, J. E., and Rattansi, P. M. 22n Maclaurin, Colin (1698-1746) 20

Mairan, Jean Jacques d 'Ortous de (1678-1771) 14,35,36,81

Manuel, Frank Edward (1910- ) 21n

Many degrees of freedom 1,3,4,6,28, 35

Mersenne's law 4,6,7,9,16, 18, 22, 26, 31,32,33,34,35,64

Metacenter (or metacentric·height) 8, 88 Moment of inertia 88, 100n, 105, 111 Momentum principle 1,2,6,7, )0, 16,

18,29, 31, 32, 42n, 110, 112, 120, 122

Momentum law 2,6, 8, 9, 10, 11, 16, 115, 116, 120

Motte, Andrew (d. 1730) 9 Music 20

Newton, Isaac (1642-1727) 6, 9-14, 21,22,30,32,33,34,36,43,44,77, 78,81

Newton's second law 1 Node (nodal points) 21,35,64,82,97,98

One degree of freedom I, 71, 11 0

Pipes (vibrating air column) 4,26,45,81, 82n

Penduluih condition 5,6,7,8,15,17,18, 39,47,49,53,57,58,59,64,70,71, 75,78,87,93,94,102,105,106,121, 122

Pepusch, John Christopher (1667-1752) 21

Pitch 4, 20, 101, 102 Poisson-Lommel integral

. representation 7, 69 Potential energy 50, 115, 119, 121 Pressure wave 4, 6, 9, 28, 77 Principia (of Newton) 8, 9, 18n, 32 Pythagorean tradition 4, 20, 22

Rameau, Jean-Philippe (1683-1764) 27 Regular motions (see also irregular

motions) 63, 65

184 Index

Rhenish feet 10 1 n Riccati equation 66 Robartes, Francis (1650-1718) 21 Robison, John (1739-1805) 20n, 27 Rocking 8, 71, 73, 87, l04n, 110 Rohault, Jacques (1620-1675) 33n Roots of polynomials 61

Sail filled with water 17 Sauveur, Joseph (1653-1716) 7,23-27,

33n, 35,45, 52, 98n Scruples lOIn Simple harmonic motion 5 Simple isochronous pendulum (length

of) 5, 18, 42n, 57, 60, 72, 95, 119

Simultaneous crossing of the axis 5,6, 18, 47, 116

Small vibrations 5 Sound (see also pressure wave) 33,43 Spontaneous center of rotation III Static equilibrium 8. 39, 70, 71 Strain 9, 10, 12, 14, 40 Struik, Dirk Jan (1894- ) 16n Stukeley, William (1687 - 1765) 21 n Superposition 3,4,8,21,83,89,91,93,

102

Taylor, Brook (1685-1731) 2,6,7, 15-:21,27,31,32,35,43,44,47,48,

Truesdell, Clifford Ambrose ill (1919- ) iv, 14n, 16n, 37, 44, 50n, 70n, 95n, 121n

Trumpet marine 27

Variational methods 8, 39, 76 Velocity of sound (or propagation) 4, 12,

44,79,80 Vibrating bell 36 Vibrating ring 7, 36, 3r Vibrating rod 6, 7, 8, 73, 94 Vibrating string 4, 15,23,28,30,47,75 Vibration theory 3, 5, 6, 7, 36, 70, 83 vis viva 50 Vortices 77

Watson, George Neville (1886- ) 63 Wave equation 10 Wave length 9,30 Weighted chain 107 Westfall, Richard S. (1924- ) 14n Whittaker, Edmund Taylor

(1873-1956) 77n Work 41,119

Young, William 20n Young's modulus 37,40, 42n, 43

50,53,67,71,75,80,92,100 Zeros of Jo 63

Volume 1

Studies in the History of Mathematics and Physical Sciences

A History of Ancient Mathematical Astronomy By O. Neugebauer ISBN 0-387-06995-X

Volume 2 A History of Numerical Analysis from the 16th through the 19th Century By H. H. Goldstine ISBN 0-387-90277-5

Volume 3 I. J. Bienayme: Statistical Theory Anticipated By C. C. Heyde and E. Seneta ISBN 0-387-90261-9

Volume 4 The Tragicomical History of Thermodynamics, 1822-1854 By C. Truesdell ISBN 0-387-90403-3

Volume 5 A History of the Calculus of Variations from the 17th through the 19th Century By H. H. Goldstine ISBN 0-387-90521-9

Volume 6 The Evolution of Dynamics: Vibration Theory from 1687 to 1742 By J. Cannon and S. Dostrovsky ISBN 0-387-90626-6

Appendix: Daniel Bernoulli's Papers on the Hanging …978-1-4613-9461-7/1.pdf · Appendix: Daniel Bernoulli's Papers on the Hanging Chain and the Linked Pendulum * ... In the translations

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