a new and general method of finding simple and quickly-converging series; by which the proportion of...

A New and General Method of Finding Simple and Quickly-Converging Series; By Which theProportion of the Diameter of a Circle to Its Circumference May Easily be Computed to aGreat Number of Places of Figures. By Charles Hutton, Esq. F. R. S.Author(s): Charles HuttonSource: Philosophical Transactions of the Royal Society of London, Vol. 66 (1776), pp. 476-492Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/106293 .

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XXV-II-t. A n vd geaeral Method of.nding Jimpla vxod f uickly-conXverg1ng Serses; by whicb the .Proporhox of tbg Diameter of a CircSle ta itrs Circgmference Pnay safy be cotuted t a geat lSrtnbewz og PMaaes of Fiy?wr-aws By Chatles Htsttonnt Bq. Fr Rb 5..

t0 THE R-X t. I>kw- I;QRS3LE;Yt. $EC. A. Xe

S x RX B=oyal Mi1. Ac.at. Jan. 25,> I776t

R Maz ) tN a Iate exam-natioxl of the methods of Mrt 1 MACHIN alld otherss for computixlg the pro-

portion of the diameter of za circle to its circumference, I diScovered the- method which is explained in the paper accompanytug this letter This methoul you, SI>RX willv perctive is very genex*$1, and diScover$ many ferses w-hicll are very fit for the abovementioned pllrpofe. If yovs think it has fufficlent merit to entitle it to the honour of being o£ered to the Royal Societys I have takerl the- liberty to inclofe it to yon, requeiting the favour of yow ts prefent k accordlrElys fromy &c.

P S. In the courSe of the lafic year I attendeil fomv veEy mtItiIlbffl experiments on the effefts of th

Xce



jlIr uvtos o Conquergixg Seriew, &c. 477 force of fired gull-powder; asthey Seem to -lead to many important concluFlons} I believe I Ihall, in a ihon tinte} have the honollr of fubmitting to your infpedtion an account of fome of them for the like purpoSe as the following paperW

T H E excellency of this method is primarily oseng to the filmplicity of the feries by which an arc is found fiom its tangent. For if t denote the tangent of an arc v} the radius being I, then it is well known that the arc W will be equal to the infinite feries,

t t3 ts t? t9 t

+ 5 7 + 9 I I '

where the f(srnl is as fimple as can l)e defiredX And it ts evi(lellt, that rlothing farther ts required thatl to contriere matters fo as that the valtle of the quarltity t irl this feries may be both a fmall and a very flmple number. Smallt that the feries may be made to conYerge fufficiently fak and flmple, that the feveral powers of t may be raifed by eafy mllltiplications} (sr eafy diariflolls.

Since the firk diScovery-of the above feries, many have ufed itX and that after different methods, br de terrnining the lengtll of the circurnference to a great number of figures. Among thefe were Drb HALLEYj Mr ABRhRAM SHARP Mr. MACHlN, and others} of our own country; and Mt DE LAGNYx M. EULERt kCx abroad Dr I-IALbEY uSed the arc of 3o°, or I'tth of the circum- forerlce the tangent of svhich being=*/3t, 1)y fubRituting 23

iru the above feries, and mtlltiplying by 6) the femi R r r 2 circum-



473 Mrw BUtTON on

circutferenceis=6 t3t x: I- 313+5 3z-7 3j+9t34' kC. which

felies isn to be fure, very flmple; but its rate of converging

is rlot very greatX on rhich accourlt a great Inariy terlzas

muR be uSed to compute the circtlmierellce to lnally

places of figelres. IJy this very feries, hc>X -ever, the in-

(luItriot s WIr. SETARP computed the circtlmforexlce to 7 X

places of figures; Mr MACEIIN extended it to IOO; and

i. DE LAGN EY, fiill by the fame feries, continued it to

-Xz8 places of figllres. 13ut alt}ough this feries, fron

the I zth part of tlle circunlference, does not converge

very qtlickly, it is, perhaps, the beil aliquot part of tlle

circumference rhich cart be ufed for thIS purpofe; for

ufheIl fmaller arcs, w-hich are exadt aliquot parts, are

uSed, their tangents altlloug}w fmallery are fo mucll more

complexs as to render them, QIl the ssrhole, more aperofe

in the application; this will eafily appear, by infpeEting

fome irlIianc@s that have beer1 giverl by Mr. GARDINERy

iB his editions of S}IERWIN'S Tables. One of thefe me-

thods is from the arc of I8°e the tangent of hich i5

rf I - 221; another is from the arc Of z z°^i the tangent

of which is 42- I ; and a third is from the arc of I 5°>

the tarlgent of whsch }s 2-4/3 -A11 of which are evi-

dently too comples to afford an eafy application to the-

general feries. Irl order to a Itill farther imprc>vement of the method

by the above gerleral feries, MrX MACHIN, by a very fin-

gular and excellent contrivance, has gre-atly reduced the lawbour



Coo;CergU Sries for the CircAe. 479

Iabcur rlaturally attending it. His methotI is explained in Mr. MASERESS Appendix to his Diffiertaton on the uSe of the Negatve Sign in AIgebra; an(l 1 have gisen a11 analyfis of it, or a coiljeAure concernillg tl-e manner ill svhjich it ls probable Etr. M xCI-IIN ckiI'cc)sered it, irl my Treatife on Merlflration xshich, I believeX are the only tsvo books in sthich that methcz(l has been explained, as I Ilever had feen it expla.zed by allY? till I met w-ith WIr. WIRSERESSS hook abovementioneKl on t'ne Ufe of the b;egative Sigrl. For thotlgh tlle feries' diScovered by that nethod svere pllluli2zed by Mt. JONES, in his Synofs Palmariorusg Mathefeos, which sras printed in fhe year I 706, he has givetu- them merely by them{ele7es, with- out the leaIt hillt of the manner in which they ere ob_ taine(l. The refult ffiews,, that tlle proportion of the diameter to the cireumference is equal to that of I to quadruple the fum of the two feries',

f 5-X :-I- 35^ + 55> 75a t* &CX

l 239 X I-3 ,2,39^ t S 239*-7 2396 + 9 239as &Ct

The Ilower of which converges almofc thrice as faR as Dr. HALLEY'S raifed from the tangent of 30° The latter of thefetwo feries converges Rillagreatdeal qllicker; but then the Iarge incompofite number 2 3g,by the reciprocals of the powers of which the Series converges, occafions fuch long,tedious divifionsyas tocounter-balance its quick- neSs of convergency, fo thatthe former feries is fummedS with rather more eafe than the latterX to the fame num-

ber



MF. HTTN-QN 480

ber of places of figures., Mr. JlONES, itl his Synot#w} melltions other feries'befldes thih which hehaxl received from Mr. ItACHIN forthe feurpMe, and iava from the farne principleS But mre may conclude this to be the beR of them all, as he elid aot }?ubliSh any other befldes it.

. 9:IJLER toU}, ill his Itt*QzAvEia iz Aw0fn In>i- torum by a contlrivance 6vxalethin8,- like -Mr. MACHIN'St diScovers tllat 2 arld 3 are the tangents of two arcs the lum ot svhich s jtlk 45°; and that AthereforeX rhe diameter is :to th-e cutafEr¢e as -x to quadmple the fum of the tsro f:eries-', X

f I t I I I J 2 * I - 3 4 + $ 42- 7 43 + 9 44^ tC*w.

l zC I - 3 9 + 5 9z 7z9 ss Both v}wich felsies corlverge much SPcer than l:>r. fIAL- fY'S, and are yet at the fame time made to confirergu by the pos-vel;s of nurubers producin$ ouly ffiort divifions; that is, tdivifilorls performed in one line orwithout wlsiting dosvn arly tlwillg befides the quotients.

I c:^otne now to explain my own method, which, in- deeda bears fome little refemblance to the methods of M.XCHIN and EULER; but then it is more generS, and diScovers as partic.ular cafes of it, both the Seriest of thoie gentlemen alld many others, fome of which are fEtter for tilis isur£wSe than theiz are

Tius tnethod then COlfis in finding out fuch fmall Xll'CS, VdS hav@ for tangents fome fmall alld fimple sulgar katiolls (tlle l (ut1S beillg denoted by a 3>and I;lcoh alfo that

fome



Consergag Series for tbe Circle. 48t- feme multiple of thofe arcs {hall differ from an arc of 450, the tangent of which is equal t-o the radius, by other fmall arcs, whxch aSo ihall have tangents clexloted by other fuch fmall and fimplevulgarfradions. For it isevi- dent, that if fuch a Enall arc can be fovlnd, fome multiple of which has fuch a propoSed diffierence from an arc of 45°, then the lengths of tisefe two fmall arcs m11 beeafily computcal £leomthe geneleal ferie-s, becarlfeof the fmallneEs and fimplicity of their tangellts; after v-hich if tlle pro pel multiple of the fir arc be increaSed or diminiI.led by the other alncJ the rei5xllt will tze the lerqgth of ar] arc of 45°> or gth of the circumference. And. the nzanner in which I diScover fuch arcs is thus: -

Let t tX denoter any two taIlgentsX of which T is the greaterX arul t the lefs; then it is kllown, that the tangent

of the difference ofthe correfponding arcs is equal toIT+ TtSw

Hence, if , the tangent of the fmaller arc, be fllccef iRvely dersoted by each of the fimple fradtions 2 3J 4 5ss

&c. the general expreffilon for the tangent cyf the diXerence betveen the arcs will become refpeEtivelry

I 3T-I 4T-tx ST 1> &c.; fo that if T be p

by any given number, then thefe expreilons will giV¢

the tangent of the difference of the arcs in known numbers, according to tlze values of t, feverally aIIumed re- fpeEtively. And if,¢ in the firk place, T be equi tO I, the tarlgen.t of 45°, the foregoing expresons will give the tangent of an; arc, which is equal to the difference be- tweexs that of 45° and the fixie arc > or that, of which the

twgmt



48 lWr HUTTON tZ

tangent is one of thntlmbers 22 3t, 4^ 52 &G. Then if thc tangent of this difference juR now fourld, be taken for Ts

the fame expreIBlons will give the tangent of an arct hich is equal to the difference between the arc of 45° arldthe double of the firll arc. Abatn, if for T we take the tangent of this laR found clifferetice, then the tore going expreIfions will give tlle tangent of an arc, equal to the differellce betsreen tllat of 4 >° and the triple of the firll arc. An(l a,gain taking this lafl found tmgent for T, the I*ame theorem ssill produce the tangent of an arc equal to the difference between that of 45° and the quadruple of the firlt arc; and fo cenfi -always takillg for t the tangent lalR found, the fame expreIElotls wilt give the tarlgerlt of the difference between the arc of 45° and the next greater multiple of the firIl arc; or that, of which the tangent was a-t firR affumed equa1 to one of the fmall numbers 2 -3, -4, 5, kC. This operation, being continued till fome of the expreIElons give filch a fit, fmall, and flmple fradtion as is required, is thErl at an end, for we have then fourld two fuch fm!all tangents as were required, tiz. the tangent laflc found, and the tangent firIt affumed.

Here follov the feveral operations adapted to the feveral values of t. The letters a, b, c, d, 8cc. slenote the feveral filccetffilve tangents.

I. Take t = , then the theorem 2T+T giVa

r b = 7

Therefore the arc of 45°, or 8th of the circumference, is either



ConverFn, Reriew for tht Cir¢Ze. +83 elther eqtlal to tlut fum of the two arcs of which z and 3

are the tangents or to the differerlce between the arc of -hich the tailgent is 7Ix and the doubleof the arcof wrhich

the tangent is 2I; that is, putting A-the arc of 45°* I I I I I

_ 2 I 3.4 + 5,+4-7 ,43 + 9-44-kC A I I I I I

+ 3 x: I-3 9+5 92-7 93+9-94 &C.

t I-34 + 54Z-743 + 944-&C. r, A-A

t 7 X I-3 4g + 5 49z 7 493 + 9 49+ &c. ftnd the former of theiSe;talues of h iS the fate tth

that before mentioned as gtven by M. EULER; bUt the latter is much better, as the powers of 419 converge much fai:ler than thofe of 91.

cosot Frotn double the former of theSe values of A

fubtra&lillg the latterX the remainder is, r 2 I I T I

4 + 3 X: I - 3 9 + 5 9s 7 91 + 9^94 - &C.

1 +-X * I + --+ t 7 * 3*49 54< 7 49^ 949+ '

svhich is a much better theorem than either of the former.

. If t I)e taken - 3 then the expreIElon 33 + ,- gives, I

.e _ _

_ b I

- 7 Here the value of a-l gives the fame expreSzon fol the valvle of A as the firlzc in the foregoing caKe, ald the N alue

VOL. LXVI. S ff Of



480 Mr. HUTTON on -of b= 7 gives the value of A the yery fame as in the corol- Iary to the cafe above.

3.Taliingl-,theexpleeEson44 T gives

a- 3

b - 2'3

C S

99 d-- 79 - 40r

Here it is evident, that the valtle of C-9%9 iS the fitteR number afforded by thls cafe; and; frona-it it appears, that the arc of 45° is equal to the fum of the arc of which the tangent is 959, aru(l the triple of the arc of which the tangent is 4.

t + 4 x: I - 3.16 + 5.l62 7.l63 9.I6+

l+959X I - @9; + 5 994- 7 g96 + 9 998 -kC.

Which is the beR theorem that 5ve have yet found, be- callfe that the number 99 refolves into the two eafy fadors g and I 1*

4. Let now * be taken _5, and the expreElon 55T+T' wil;l

gve 2

a= -3

b - 7

7

s _ _

_t

- 239 Where



Conevergxng Ser^es for the Crcl. +85

Where it is evident, that the lafc number, or the value of d, is the fitteflc of thofe produced in this cafe, and ftom which it appears, that the arc of 45° is equal to the difference between the arc of which the tangent is 2 3 9 X and quadruple the arc of which the tangent is 5. Or that

t + 5 x : 1- 3 52 t 5'5+ 7,56 9,58 t 239 3.239 5.2394 7.239 9.239

Which is the very theorem that was inverlted by Mr. MACHIN, as we have before mentioned.

5. Take now t= 6 and the exprefl"lon 66T-*- Tl wilI give - 5 a _ 7

3

b 47

9I P - _

3os d 24 I9ZI

^ - .^ 475 I I 767

05 which numbers none, it is evident, are fit for oux purpofe.

S f f 2 6. Againt



486 lZr. HUTTON on

6. Again take t VrX atltl the expreffiion 72 +T will vgirc a _ _

b- 17 3t r

r _ _ tf _ XY

49 d _ aos v _ _

742 -2Ja

f- J s2o3

NeRher we any of thefe fit numbers br our purpoSev

7. In like marmer take t_p fo- lhall 8t-Tt give 7 a = 9

b - 47 79 29t

- _ _

> -679 I697

J _ _b - s7rs

e - 1 78 +} 47529

+* _ 1 4t47

2 -388079

8. And



Conerertsxg Seas for Me CircA 487

8 And if t be aten--9tX the expreffilon9q+T wlll givt _ 4 a_ ffi

b = 9 I I 5

P - _

e -236 799 d-- - -2239 2467

S - _

It475

809 - s - 6 5

_ 4765 g 44I28+

9* Alfosfwe tzlLe -,XO>the exX°T t will gEw

e - t 79

A__

- Ilw

67x ^ _

_ , I 2o9

d = rS44l I336

_ 4I049 * _

1 3905 I

2 7 I439 - J I43IsS9

I 282g3 I

fF I W l'l1:

6 -14587029

Fso. =Xt



488 ^ Bt[T5*0N.OX

< o. Farther if- we take t_, the expreffilon tllt-|--Tt . . .

W1 glve

|,, s a - 6

b - 49 71 234

,* _ _

4I 5 2isg

-A799

e S5

27474 7675}

f _ f 31 Ioug 266286

n _

6 I 75z665 Il7648I

f954s6Q I

T T.taStlys if we talQet-,t;! ,theexpreiNoix'l22t+T gives r

wr _ _

I3

XI3

a-s6?

_4t

t 73 d - 4tg

9X?

4Itt

_

I I423

f _ I41 I87

3I372t

-

^ 1732153 2032499 . _

2 Io9gss7

. afigo43 I

- ' sssax7l83 Here



Converga'ng Series fbr thg CirEte. 4gg

Here it is estidents that none of thefe Iatter cafes affiord: any nutnbers that are fit for this purpofe. And to try any other fraftions lefs than xIa for the vallle of t, does not feenu likely to anfwer anygood purpofe eiJpeciallyX as the divifors, afier 1 2,sbecome too large to te managed in the eafy ray of {llor.t dviflon in one lire.* :

-By the iregsing meanS it appews that ̂ re have dif- covered five diffierent forms of the value of A or 4th of the femi-circumErence, all of which are very proper for readily computing its length, siz. three fims in the firPc cafe and- its coroHary one tn the th¢d cafe and oneS in the fourth cafes Of theSe, the firilc and laR are the fameaas thofe inv«ted by EULER and MACHIN refpec- tively, and the other three are quite new, as far as 1 know.

But another remarkable exell«cy attending the firk three of the before mentioned feries is, that they are capable of being changed into others which not only converge i:Vill faRer but in which the converpng quan -tty {hall be IXOs °t fome tmidple or fub-multiple - of it; and fs3 the powers of it raifed with the utmoR eafe. The Series'> or theorems, hereE meant are theX three ̂

f + 2 x: I-3.4 + 5.4^ t4' 9*4

l + 3 x * I 3.9 + 5.9^ .9' 9794

{ + I 3.4 + 5.4^ 7.4' 9^4+

-7 X. I 3.49. + S*49$ 7X49$ 9*49

3dly



Xr. HUTTON 0X 49

2 ! X I X 3d1y, A-t +% . X _ _ + , _ 7 3 + 4 X &,

l + '7 X : I -3, + 5.49' 2.493 + 9*49+

Now if each of thefe be trans-formeds by means of the differential Seriu frs cor. 3. p. 64. of the late l>Ir. THOMAS .sIMRSON'S MathemadcK DiXenainst they -will become of t}sefe verzr commodious fbrms, sit.

12: I6y 7.1o + gloX kc.

6C + 8r, 8tC

I2: I6y

7*i° + 9.10 + &c

I2: I6y +7.1oo g.loo &'

_+9,o>&c.

6¢o + -9 50 X &

4 8X-4rt

3.IO 5. IO

... 2 P

* T * _ .L 3.I0 5.IO

4 8a

* T + +

* ̂ 3.I0 5.10 4 8a

* T + +

^ 3. IOO 5. IOO . . .

2 i

3.t<2 i a ' #

# I + - | + -t * a*54 > as*52

Where aO C, av 8oct dewtE altays the precedirlg terms in eadh ferxesm

Now it ls evident,, tha£ alEthefe iflter Ees, are nush - eafier than the former ones, ta which they refpeEtively correfpond; > fbr, bemuSe af the powers of I O here con- sernedi we harre little more to do than to divide by the fenes of odd numbus I, 3, 5, 7, 9, &c.

Qf all thefe three forms the iecond is the fittet} for computing the required proponion; becanfe that, of the tsre feriesS Qf tio k fasss -tS*^ralxrmsof theone

J'

o 'Ve

r 4 1 +-o x

I0t A - g

t ,ox

I + 4-X X - 4 S

t--X

f 6 I +-X

3dly, t_4 1 t +5QX



Coevergeng Serees for fhe XrcA. 491 are found from the like tertus of the otherX ly dividing therel^tty XQ anditSseV@rAlccC«lve pOWttS IOO,

soco, &c., that is, the terms of the one confsk of the fame figures as the terms of $he other, oIlly removed a certain number of places farther £owards ithe right, tn the decuple Scale of numbers; and the numlber of places, by Swhich they muR be remoured, is t;he fame as the diflcance of each term from the firk term af the feriesX siz. in the fecond term the figures mufc be moved one place lowery in the third term two, in the fourth term three, &c. fo that the latter feries will conflS of but about half the rlumber of the terms of the former. Thus, then, this method may be fai(I to eieEt the bufinefs by one feries only, in which there is little more to do, than to divide by the feveral numbers , 3 Ss 7 &c.; for as to tlle multiplications by the numbers in the

umerators of the terms, after they become large, they are earlly performed by barely multiplying by the number two, and fubtradring one number from another: for fince every numerator is lefs by two than the double of its denominator, if d denote arly denominator (exclufye always of the powers of I o) then the co-efficient of that

term is 2dd 2> or z_ 2d ' by which the preceding term is to

be multiplied; to do which, therefore, multiply it by two, that is dollble it, and divide that double by the divifor d, and lubtr-adt the quotient from the faid double.

VOL. LXVI. T t t By



492 Mr. HUTtON on Conser^ing Serxes, &c.

13y a pretty exadr eRimate, which I have made, of the proporiion of the trouble or time in computing the circumference by this middle form of the value of A, and by Mr. hIA£lIIN'S theorem, I halre -found, that the com- ptltation by his method requires about 8th or Itoth more time than by mine. And its advantage over arly of the feries' invented by EULER or othersX is Rill aluch more corlfiderable.

XXIX. Xn



a new and general method of finding simple and quickly-converging series; by which the proportion of...

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