de inventione centri oscillationis- per brook taylor armig. regal. societat. sodal

De Inventione Centri Oscillationis- Per Brook Taylor Armig. Regal. Societat. SodalAuthor(s): Brook TaylorSource: Philosophical Transactions (1683-1775), Vol. 28 (1713), pp. 11-21Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/103176 .

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- -

II. I)e Inaentione Centri 0fictllationis. 9ter E3rook Taylor Wrmgg. Reg410 30cietata Soddl.

DeEnitio.

EJZ Centrug Oficillationis ptgnXg toddaa za carpare peno deZoX cGWgtws vtbrattanes yXgglx eode7 modo atq; endea tetpore pervgttntxr a; # iQ7nd Jolaw dd eandeg ¢iff tzarz a pard fipenfonxs jilo RJpewdere;r

)ER fe vix fatis manifeRum eft in corpore aliquo dari 1 hajuliodi purlEum: utpote cujus acctleratio de- beatX (per ha¢ defi) itl omnibus inclinationibus corporis penduli ad Horitontem} perinde eSc, ac & a proprta tanw turrJ gravitate urgeatllr; reliquis particulis totius corpoX ris ejus rnotum proprillm haud perturbantibu¢. Itaq; x ordirae ad inventionern hu jus Centri} przmittenda ek una atq; altetA prOpQ6t1Q) unde confiet tale pun&um dari.

Prop. t ProbQ t

liB ¢arporz Ofcillantis datAi qt¢dvzs i7ratXtx0ne dd fior;; zontews ir7oeaxrepgthnz cuymmw<veleratiotristWefi>, ac Ji ab ipy<Xffl pr)prix tant&{n grvextate argeattxr.

Sit A B9corporis propolSti fe&o in plano ad HO:IT zontem perpendiculari in qus movetur centrum gravi tatis X, centro fuEpenfiorlis exiftente C. Di(tingllatalr corpus sa ele-menta priEmarica plano ̂ B D perpendi-

C 2 cularta,"

( tt )



( A z ) culal^Xa, asdcoque Hor;Zont _}t t fempicr paralXela; ut facile =X.St > patesit ex mo*n centri gravi-

nk / / tatis G An pla;zo illo A B D. I P ?\//; if Arq ob hujuEmodi litLxm, G / tale elemen,t17m quodris fipe- O /2 dEari pot v-anquam putiEum X PhsZEcam p iti plano eedem

AB3) ad p£z2utM X 10cRs tum Pveducaur ltAq; COrptlS 3Propofitllm in planum Phyficll) A B D conRans ex hu- s<wliedi particulis p.

In hoc plano ut inveniatur pundalm ( }t CU)S acceleratio pro;ra non mutatUr ab aEtionibus particularum reliqual um atendendunl cft ad \ ires particulz clJ ju6vis fingularis p in pUtl&o z fitr. Natn ex hiSce stirtbus con )un2Is oritur pani totiuS motus abfolutus; cujusqpe dattlr motels punAi cz3ufvis propofiti; uncle viclm invenitur pllnEbum C't7}Us ]OtUS eS datus.

At urgerur pareicula p a vi oropri gratitatis; quz fi partium colafio diffic)lveretllr; jn dato tempore tllinilzlof datam prodeceret accel£2rseionem nzotQs in ptrpendiculari ad Hor^XZon£eni zay Ad- C Z duc normalem y x & refol- vetur acceleratio Z y in partees 7 X 2 X y. 1D COrporiS rigiditatem, tollitur vis z x pPr rcl;Rentam punCti C. At Yi reliquaxy trabitur fpaviurn>ABD ingttrum circa punEtum C i & duEa horizontali C c 8 perpendicular; z s, erit ea ut S : Nempe ob . gravttatis vim d-atarn, & Emilia triangula X y Z & s C- z. Ergo . ss particula p ad movendutn; Epatium ABL) elt ue C z X pe>,

Ad has vires in unum colltgendac fit O punAum ilF sariabiles in line;i ad lbitutn duAa gt ad difEantiamad- hac in¢ognitam C C. Tum erit YiS partsula p ad mo- - sendu



( a)

renJum punAum O>t3cco x c-z X p taoc eR ut ccox p

Accelerarls autmX quam tribuit p eidem pun&{o , erit ut

C C} ̂ cc s. Itaq; applicata vi illa C) x p ad hanc ac-

C O' M C s . . C t (1 : celeratsollem C z q- o trltqtlotlenScoqpXppartesu-

las qutX fil ln ipCo pun9o {) Sngatur moveri ctm eadem

acceleratione X e-undem- omnltlo produceret 7, q:

motumX quem ln eodWm pun&o O producit particula p.

Hinc detnum reducitur Prolzlenza ad nwotuut-n Thtornla

noti{limum: Applicati enim fumma viriUzjC () x i) ad fum- * Czq.: . v

mampartlctllarllsCO- x pX erlt quOtlells acceleratlo

abSoluta pun&i O. Dei^ dtlAa pcjrpendiculari O o, & pofita llac acceleratiotle aquali datt accelerationi

CO ipfius punAi O:dabit-ur di{Eantfa CO. Sit enim CO_ d,

& (iuxta metllodum. Flrxvonaw) C s x P!-1VI & CM- Z q:

x p C. Tum ob C0 invar>abtlern crit futnZlgl c)a- nium Viriunl CO x p Mor & fu[lma oium part-

COq XP=cOq Unde,appltcataSlmmfi,

tuomentorum ad fumnlam-corporum, erit - x C O _ d

adeoq; C C) = M . Inventis igituu C 8t M, per F1gxtae as mvthodum irlverfam dabitur C: O.;QL E. 1*^

Cor. A centro grargatis G ad horizontalem Co dXu£ perpendicularcx G g, & Elt corpus ipfum A B C-AN



( I4 ) Tum ex notiElms indolecentri gravitatis erit M= (t x A. U£de eR C V = * Cg x A

Prop o. Theor. ̂

WrdeM toSis. vuxrdr terh O xn reUd C G tvanDe



( 5) = c G q : + G z q + 2 c Gx Gf C

E(t ergo C _ (aggregato sm- e;< nium C z q : x p =) aggregato J,/XX omniumCGq: x p + Gzq: x p / \ /i -2 CGx GF x p+2 CG x =/ JX

Gfx p. Atobcentrumgravita- /<

tis G, tA aggregatum emnium \/ 2 C G x G F x p- aggregato ti omniUm 2 C G x G f x. Quare ePcs C _ aggregato omnium CGq-. X p +Cz q; "-p -- CG q x A+D. At enim

P CGxA ErgoCO G tCG#A QL E. D;

Cor Hinc datur parallelogrammum CG x G 00 E(t enim G° =G-G A- Ae danturA& D. Q-aare datur

(: G x G O = A .

Prop-* 4. Theor. g.

Iefdetn pofti F in pxn@ O sonJ2itaater partxcta pZJyrcJ

C O Ao qeze propria gravitate agitata Orcillet circa pxn&X:n C; ftat-ij A B C atAttWS perinde ognino eritv

asX agitaretar ab OfciZlatione ip#a corporis S e

Con(lat tam ex Natura centri gravitavis, quam per Prob

** EIlenim - (;0 aggregatum omnium (;0 q P C

C O q O

toP° 5t



( 1d)

Prop. - 5. Prob. 2^ Datt sorporis czxjs*rvis wagnitgdine As ceatro grtitatis-

Gt @ pxnGof0,asafanis C. Itwvenie ejezfJew cen trxaw Of iEatiosiw O.

Fit ser Tl>cr I. inveniendo quantitat¢m t 5 vei per T.tor . qua-endo qllantitatem Do

SrSoliBnsX

Ad inat.llri {x?¢m ca(vlXsum in ca¢u ptrticui>in ;'fgvn;1n t{) q2Jueita3 4& \?et 3 pl70llt lg3s¢it nat:ra figua pror pofi1tz Dv.-in dati t3RUtR1 a}CerutâX arfPra it>;ass dja5t,SLtr

p-er zquatzorietr, (PrGt. 3vD C-C G < z A D. LIt,-

de etarn da-sr pgr. C G x G O- A (Sr. Pra,. g.)

C £ q :. Cu)us ope ex datis centro grav;a- tis S ;:alln&C3 fuf) ntioni-s, .atur centtutu (3tCcislatit3nis p¢t

fiQ}am QiVi{I3r}em. Qua: in quo1ibeL exempJo fenzper cnwmmod fElmum frit 110& parcll-30grammum primtim

erueren sfel per ÔMpUtUNl ip6Us D) rel per quattt taiem Cn ts ;lo.ae a,Ii=pFione tentri furpe. fi ;>e ;s,

g: Qup;s eCtX ut 1zc exalplis aliquot A iilul'F-rertlu*>

// % E.r. x. Stt fii7;ura * z r(rtpe fi ; p vv5 nsis > o D A D C, ctx31.lBb-.lSs e;r p,, A D, JAtcu

vS tus.<s oentr] gr.awttatis in pI;-<r O tr lefLun- A 3 -te pel verticem C & dllmetrult baf.l;

E F lateri A B parallejam. Ad calculum colr3mod;Elmn inft tu;^n-

hi dumX fit infe vertes Ca ( ntrtltn lutp-n- ,*t/ \ E1onis. Tum ad medum Pro6. X . redt

e// 1S\ catur figllta az1 planurn ThlwE, ua3 tRi i = } \ anguli lfnrce lis c E F, in qu-) e r pa,âl-

/ tQ \ lela{<SlEF repr.Crl*at lln+*arrl phySl?m X u J>- cx partculis p cotnpoEltam. Sir C H = as

H F



( 17) H F b, & C h-X3 Tum ex natura Eglzrx erte e h = b x) 8 panicula p {;ta ad punEtum z erit ue x ; svel a

Potills, fa&o h x vX erit v x elementi prifmatici baXs, -& p erit ut v x x. Urlde crit {> = C; z q: x v X x v x xt

+X Y V2 S Ideot; Cumna otnnivm C z q * x p in lirlea

h z erit v x x3 + x x ) & in Sitlea e f (pro v po bxX . 6 b 24 2 b3 . nendo- erlt fiumma llla --x x X4 l1nde a , s a3

itexum capiendo fluetltem, & pro x fcibendo 39 erit C 6 b a2 +" 2 b3 x a2. ER autem pyralIIiz tRfia A

- v & di0antia cetztri gravitatis G a tcrtice G

cftC:G--a. UndeA-CGq: =A. CSSGO g a+ r6bz _

. .

8o

Ex. 2. Sit figura propofita Conus reEss defcriptus row tatone triangllli irOfiCElIS E C F clrca perpendiculum , u

Hic iterum Cllmpto vertice C pro centro fufp!onSonis, & fattisCH-a, HE=b Ch=s, hz-, ut

4 b-b * * * fUpR3;tritp-2xlrX/aaXx-vv;undec-- 25rx

x x x + v v x f a a x x-v Y. Sit B [egmentum cir-

culi daametro e f defcripti, quod ad jacet AbSciSx h z ar,

& or D



( 18 ) ; , v s

& ()rdinatx v sx * vr; bum erit fumma emniut aa C z q : x p in red[a h z-2 : x 4 4 a X2 B-v x v

X a2 X2 v^| * Et quando v-c Ia, erit halc filmma

2 x x t a - b s^ Z > C jus duplv)rn 4 a + b x x2 B el}

p>§ iplluS b iE tct4+/%; t f EA artem area B ut X2 ; 6t

ergo B = c x2; atq; })ais ills ipfius C erit 4 a t 6 s

$ C X X4 Unde capsendo fluentem crie C _ 4 a + b x c <>.

ELt autem conus ipSe A - v c a3t & C Q -3 :1. Unde

C C r;;q -D= 3 a E I2 b +

At; ad hune- modam proceditcalcuXus tn a1;)s figurs9 uti rationes C h ad h e3 & h z ad p funt maDis come EzoGrata

Ex g Ilt pateat ratio caiculi quaNtitati5 Df St figurs propo&ta paralle]epipedorlS,

2 B cujus facies Horxzenti per-

8 r . pendicularis, & paraN!;ia plae

}£ _ / - F no motAs centri gravitat D cN

4strs**'¢{' < w A B D Duc dwamerros E F >' I , _ & [! 19 & fit altitude esc

2 b at Z C mentorump:: &fiterpa

rallela H l ; & G F a, o H bs Gs sb 8e s z = Yw Tum crit D = v x Xx

+ x v v r. Unde iptiU5 D pars in rel t r el it ̂ b x x b + 2 bx y at$ iterllm I;xmende fluentis duplum erXt



( 19)

s _ + - - 4 * Atut ctt s4 ; 4 a b w unds t* 2

s_ t /

U a3--,-Ub t

-Sx _ *- J:) B quad

A S I2

Ex. 4* Slt ultimum esemp!umk sn S?sstta*} cuus circu uv maxi¢mus Bc rs ctiameter A B, & centrums G Turn. duttis lineis u; in + Schcmate fatis patent erit D = G s cli: kf , > i * p + Gm q: X p At fsamma otn r 1 z 8

G s q . du2utn in areatn circuli dia- W J metro t r deScripti. Iveal futulua \ / omnium & M q . x p in redA k i tA § G m q: x aream circuli diametro k i cletcripki.Unde [tatim corlFat tfie D - quater fluenti ipEas G s g: in aream circuli cujus dsatucter cS £ r Sit crgo c area ctrc: li cs jus radii quadratun] eft xX & fit G A _ a,, &Gs-- x, Tumerito = 4xxx X cascx;x

4c aaX X2 4c x 2k4. Uade fumendo flubntemj 8s

ficiendo x at erit D--c a5^ ER autem A= 4 c a XS 3

[3 2 Unde---a a ft 5

Ob qalEnitatem tolutioBis libht his fubJuaDer: Probs lema de inrentiorle CJ>entri PercuElonts

Prop 6. Prob.

COYpOtZ Ct8J#AXS CitCd ddt" panJ tt8Dtttiv XtCSie Cea trxa Perv/<4Fiio7Zxs; p#^g#Ew frAlgret tz1cS at Srpaw i i/lgd i^wltizygerw b cadefw qer rfgZ 4 'g>^

pttYiOtFt e Cftxd 1?XA tef/@ illGC itCli#^4

Primum conlllae hec 1?U?a5um quz*i n6\?5r^5 lZ aplno W10ttiS CCtlttti gtAYitUlti$* Si ¢ninz 9 >rpe^> ;folvatur iz tt

D a lemeata



ement>- pti%atlca ,< p1g&vo K26¢ no^smatxas

_. a r

\ tel cetur ea t; e0tU

/\ G > li5; parailC;:) t an

//-*2l \ de 'tCta eX s

C i7 20 / } t rag) par s c i 0 AuS

t ig V;;< / p]an1 tls8t tqa_

t jX / Xiti / I3;a>Ct;perreg

v Q,< / fientxan] i2tsm itl

\//1 ZB boc planen corpo.

_ / / ris )unEtum nulOum

4 / / de eo pelletur. Sie

ergo planum illud

A>Bs ad quvd re-

ducetllr corpus per contraaionem elemeAtorum pri(ma-*

teor:;m tn pZrtCUlSs p ad pun&a z fitse, t1t in Prab. I

Ifl lloc p ane i1 C centrum r<3talioniS; a.u;. fattem ejus

proje&JC) fn&a pPr li)eam pAs+ediculaem ill hoc planum denlfi<e; & {it Q p-ndutn qutErum Per C} duc ad libittlm C { in quS fajme pund:ta duox&(> ita 1ltF

duEis z Q & t %.fit alrglJlus G x (2 ot3fllrUS & tgulUs

¢{Q ac;tus: atque in p1neis z a{fint articuls2

p & t Tum ad C f duAis norn)alibu) z r 8 e r, qux

fi.;t ad inuicem ur C z1 ad C (, i js pwaSentabuntur

Vel *t'itatP5 abSoiurx particularurn p &f. At hatum

Wejocit.3tuZ partes qut int in diredionibus z Q & t- %

tolluntur p;r refiftentiam punEtiQo Ad Q4x & QE

duc rlornales w D & C dX & ob angllloS tqualeS

z C D = r x Q) & f C d-r e Q veloctratu¢l partes re°

liqu) in diredi ntbus iplSs Qz & Q e perpendiculari

b1s) eunt ut z i) a e dX Urede habita ratif)nn diAantiaw

Xum QX & Qfs erunt vires particularum p& t ad trlo

vcndum faFium A B in partes cons ral-;ias, ut D t x z Q X pX

& d e ^ f a x p At per con -itiones Probletnatis devent

imm;t hu3019edx contrariarum viriutn ei inSter §e aw

wa1c3} .



( 21 )

(:)S atl>ulos ad D & d xt5Os, fU21t purd?ca D &t d a.ct circumLercntianl cicuRi dian-satro C QdetiqCrpri. Sti itliu$ cire-,li centrp m EX Tum d>Etjes E z a E 0 ci^ctelo occur renziot1s in F Es JX f & it crit i) z X ^ Q = F z x z I Et q *-Etsq -EQq:-Ez q:&t d e x {Q-Eeqe _- E Qq : 2]-ore erit f^m.>.*na onJi¢in1 Ed t: x p E: z q : x p - Xurrln emniurn E g q X T-E: Q c1 )t T > & termirlis tranfpotitis)faa73ma Gmniu-n E Qq: x p + T

fummaeomniu-n E zq: )t p + Ef q :x,> liocett) fi

p ponatur tam pio partic..a p s>ta circtllun q¢Xanz prs pRRtiCUl2 x extra titU01ll2] tr]t futlim on1niun] E Qq: X p -=fumul1se em;iurn £ z q : x p. Acl C q duc normalem z s. Tum crt E z q: _ C: zq: + ECq: -- Q C x C sv Qus nal}re ipSus E zq: ei fub fii, uto, & aquationc deSite tradata, tandern itive-*

nies (ummam ornnum C Q x C s x p _ furnmt om njum C z q -K p. Unde v C Q

faJmmx omniu.n C z q : x p . . * At enlm eR fumma

fumm: olunsuna (J s x p emnium C z q: x p ipfa quantitas C itl calculo centri OScillationis : 8 rl centrum y;ravitatis &t G) & ad C Q_ ducatur rjormalis Gg) & corpus ipfum dicatur As erit fumma oznnia;m CsXp=GgxA tbInde cSt- CQ

ro

= C-g A Sit centrllm OCcillationis O; tarn per

Theor- xw erit C °-CG A Unde eR C g CG w

60 : C Q Quare per O du&a ad C ̂ O perpendicularis tr>anfibit per purldum Q. Q. E. I

l-El. FE,pifo./-S.



de inventione centri oscillationis- per brook taylor armig. regal. societat. sodal

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