The early history of convergence acceleration methods
Naoki Osada
Department of Information and Sciences, Tokyo Woman’s Christian University
2-6-1 Zempukuji, Suginami-ku, Tokyo, 167-8585 JAPANosada@cis.twcu.ac.jp
Dedicated to the 70th birthday of Claude Brezinski
Abstract
The history of convergence acceleration methods in the 17th century are surveyed. Acceleration
methods are classiﬁed into three categories.

1

Introduction

The convergence acceleration methods, or the extrapolation methods, were discovered in the 17th century
in Europe [4][9] and in Japan [3], independently. The purpose of this paper is to solve two questions.
Who are the founders of convergence acceleration methods? What kind of contribution did 17th century
Japanese mathematicians Takakazu Seki and Katahiro Takebe make to convergence acceleration?
We classify the acceleration methods into three categories. I. Acceleration methods of sequences ofintervals; II. Acceleration methods of real sequences with result veriﬁcation; III. Usual (or non-veriﬁed)
acceleration methods of sequences.
In the 17th century the third category contained the following three acceleration methods: 1. The
theorem of Huygens, i.e., the ﬁrst step of the Richardson extrapolation process; 2. The Aitken ∆ 2 process;3. The Richardson extrapolation process.
Almost all convergence acceleration methods in the 17th century were obtained by computing thecircumference of a circle, the length of the arc of a sector, the volume of a sphere, and so on. Therefore
the following notation [9] will be used throughout in this paper. Let Tn be the perimeter of the n-sided
inscribed regular polygon in a circle of diameter 1 and Un be that of circumscribed regular polygon.
(π)(π)
Tn = n sin, Un = n tan.
nn
By the Taylor expansion, we have
(
( ))
(π)
π2
π4
π6
1
=π 1− 2 +−+O
Tn = n sin
,
(1)
n
6n
120n4
5040n6
n8
(
(
))
(π)
π2
2π 4
17π 6
1
Un = n tan=π 1+ 2 +
+
+O
.
(2)
n
3n
15n4
315n6
n8

2

Acceleration of a sequence of intervals

2.1

Deﬁnitions

Let {In } be a sequence of closed intervals such that (I0 ⊃)I1 ⊃ I2 ⊃ · · ∩· and the width or the diameter of∞In converges to 0. Then by the completeness there exists unique s ∈ n=1 In . In this case we say thatthe sequence of intervals {In } monotonically converges to s.
Let I be the set of monotonically converging sequences of closed intervals. Let T : I → I be aninterval transformation with T ({[sn , tn ]}) = {[un , vn ]} such that sn ≦ un < s < vn ≦ tn . Suppose that
un , vn are determined only by sn , tn , and also suppose thatv n − un= 0.
lim
n→∞ tn − sn
Then we call T accelerates the sequence of intervals {[sn , tn ]}.
1

2.2

Weighted mean acceleration methods

Let g : N → R+ be a function satisfying lim g(n) = 0. Let {sn }, {tn } be positive sequences convergingn→∞

to the same limit s. Suppose {sn }, {tn } have the asymptotic formulas
sn = s − cg(n) + o(g(n)),

as n → ∞,(3)

tn = s + dg(n) + o(g(n)),

as n → ∞,(4)

where c, d are known positive constants.
We denote weighted mean acceleration methods by
dsn + ctn
,
d+c
d+c
hn =
.
d
c
+
sn
tn
an =

(5)
(6)

By the assumptions (3)(4), we havean = s + o(g(n)),

hn = s + o(g(n)),

as n → ∞. Acceleration methods (5), (6) are called weighted arithmetic mean and weight harmonicmean, respectively.

2.3

Willebrord Snell

From Archimedes (BC287-BC212) to Ludolph van Ceulen (1540-1610), European mathematicians computed π by using inscribed and circumscribed regular polygons n sides each. Van Ceulen computed 36
digits by using 262 -sided polygons. After Van Ceulen’s death, his result was published in 1621 by his
student Willebrord Snell (1580-1626) in Cyclometricus [19]. Van Ceulen’s result was
314159265358979323846264338327950288
14159265358979323846264338327950289<π<3.
10000000000000000000000000000000000
10000000000000000000000000000000000

It was Snell who surpassed Archimedes and Ceulen. In Cyclometricus Snell derived
3 sin θθθ< θ < 2 sin + tan
2 + cos θ3
3(7)

without adequate proof, where θ is the central angle of a sector.
Putting θ = π/n in formula (7), we have32
1
< π < T3n + U3n .
2
1
3
3
+
Tn
Un

(8)

Snell took the same regular polygons in upper and lower bounds of (8), and he computed π by using
3
2
1< π < Tn + U n .
2
1
3
3
+
Tn
Un

(9)

The upper bound of (9) is the weighted arithmetic mean and the lower bound is the weighted harmonicmean of Tn and Un .
By (9) with n = 1073741824(= 230 ), Snell gave
31415926535897932384626433832795028293
1415926535897932384626433832795028958<π<3.
1000000000000000000000000000000000000
1000000000000000000000000000000000000

Snell’s result is 1 digit less than his teacher Van Ceulen. This is, in a sense, his declaration to put an end
to the endless race for geometric computation of π.
2

1.5

G
F

1

D

0.5

0

E

A K

−0.5

L

M

C

B

H

−1
−2

−1.5

−1

−0.5

0

0.5

1

Figure 1: Huygens Theorem XII [7]

By (1)(2), the upper bound of (9) has the asymptotic formula
(
)
2
1
1 π4
1
Tn + U n = π 1 +
+
O(
)
,
3
3
20 n4
n6
and the lower bound of (8) has
)
(
1 π4
−6
+
O(n
)
.
=π 1−
2
1
180 n4
+
Tn
Un
3






[Tn , Un ] 7→ 

3
1 
2
, T + U2
1 3 n 3 n+TnUnSnell’s transformation(10)

is the ﬁrst acceleration method of a sequence of intervals.

2.4

Christiaan Huygens

Christiaan Huygens (1629 - 1695) published De circuli magnitudine inventa (1654) [7][8] in which heproved 16 theorems on a circle, an arc, or inscribed or circumscribed polygons in a circle. Germantranslation of this book can be read in [13].
2.4.1

The upper bound of the length of an arc

Huygens proved the upper bound of the length of an arc in Theorem XII in [7].
Theorem XII Choose a point D on the circumference of a circle and ﬁt the point E on the extensionof the diameter AB such that ED equals to the radius. The extension of ED intersects F and G with thecircumference and the tangent at B, respectively. Then the length of BG is larger than the length of thearc BF. (See Figure 1).
Let C be the center of the circle and θ = ∠BCF. It follows from Theorem XII and Figure 1 that
θ < 2 sinθθ+ tan ,
3
3

which is the upper bound of Snell’s inequality (7).
Huygens wrote at the end of the proof:
This is one of two theorems on which all of Willebrord Snell’s Cyclometricus depend.

3

[7]

1.5

L

1

E

0.5

0

C

B

Q

A

−0.5

−1
−2

−1.5

−1

−0.5

0

0.5

1

Figure 2: Huygens Theorem XIII [7]2.4.2

The lower bound of the length of an arc

Huygens proved the lower bound of the length of an arc in Theorem XIII.
Theorem XIII Choose the point C on the the extension of the diameter AB such that AC equals to
the radius. Let E be any point on the circumference. The extension of CE intersects L with the tangent
at B. Then the length of BL is smaller than the length of the arc BE. (See Figure 2).
Let O be the center of the circle and ∠BOE = θ. Since EQ/CQ = BL/CB,BL =EQ · CB3r sin θ=,
CQ
2 + cos θ

where r is the radius. It follows from Theorem XIII and Figure 2 that
3 sin θ< θ,
2 + cos θwhich is the lower bound of Snell’s inequality (7).

3

Veriﬁed acceleration of a real sequence

3.1

Deﬁnitions

Let C be the set of real convergent sequences. Let T : C → I be a transformation with T ({sn }) ={[un , vn ]}, where {sn } and {[un , vn ]} converge to the same limit s. Suppose thatlimv n − unn→∞ sσ(n) − s= 0,where un , vn are determined by s0 , s1 , . . . , sσ(n) , for some integer σ(n). Then we call T accelerates thesequence {sn } ∈ C with result veriﬁcation, or T is a veriﬁed acceleration method.

3.2

Christian Huygens

The ﬁrst examples of the veriﬁed acceleration of a real sequence are appeared in Theorem XVI and
Problem IV in De circuli magnitudine inventa.
Theorem XVI Let A be any point of upper semicircle, a be the length of the arc AB, s be the sineAM, s′ be the length of the chord AB. (See Figure 3). Then114s′ + s.
s′ + (s′ − s) < a < s′ + (s′ − s) ′33
2s + 3s
4

(11)

A

D

M

B

Figure 3: Huygens Theorem XVI

It follows from (11) that1
4T2n + Tn
1.
T2n + (T2n − Tn ) < π < T2n + (T2n − Tn )3
3
2T2n + 3Tn(12)

By (1), the lower and upper bounds of (12) have the asymptotic formulas
(
( ))
1
π4
1
T2n + (T2n − Tn ) = π 1 −
,
+
O
3
480n4
n6
(
( ))
4T2n + Tn
π6
1
1
=π 1+
+
O
T2n + (T2n − Tn ).
3
2T2n + 3Tn
22400n6
n8
The lower bound of (11) is the Huygens Theorem, i.e. the ﬁrst step of the Richardson extrapolation
process, which will be treated in Section 4.
Huygens’ transformation
{[
]}
1
1
4T2n + Tn
T : {Tn } 7→
T2n + (T2n − Tn ), T2n + (T2n − Tn )3
3
2T2n + 3Tn
is the ﬁrst veriﬁed acceleration method of a real sequence.
In Problem IV Huygens gave better veriﬁed acceleration
T : {Tn } 7→ {[wn , vn ]}

(13)

where
1
un = T2n + (T2n − Tn ),3
1
4T2n + Tn
vn = T2n + (T2n − Tn ),3
2T2n + 3Tn
1
10(T2n + Tn )wn = Tn + (T2n − Tn ).
3
2T2n + 3Tn + 43 (vn − un )See [11]. The width of [wn , vn ] in (13) satisﬁesvn − wn = O(n−6 ).
Huygens gave a numerical result [3.1415926533, 3.1415926538] by (13) with T30 (= 3.1358538980) andT60 (= 3.1401573746).

4

The theorem of Huygens

4.1

Deﬁnitions

If {sn } can be written in the form
sn = s +cd+ 4 + O(n−6 ),2nn5where c, d are constants, thent(1)n = s2n +s2n − sn,3

satisﬁesd+ O(n−6 ).
4n4The elimination of the term involving n−2 is called the theorem of Huygens. Huygens proved it using(1)
Euclidean geometry. The sequence transformation T : {sn } 7→ {tn } is the ﬁrst step of the Richardson
extrapolation process which we will treat in Section 6.
t(1)
n =s−

4.2

Christian Huygens

In De circuli magnitudine inventa C. Huygens gave three types of the theorem of Huygens.
Theorem V (On the area of a circle). Let Sn be the area of n-sided inscribed regular polygon in acircle with area S. Then1
S2n + (S2n − Sn ) < S.
3
Theorem VII (On the circumference of a circle). Let Tn be the circumference of n-sided inscribed
regular polygon in a circle with circumference C. Then1
T2n + (T2n − Tn ) < C.
3
The last one is the lower bound of Theorem XVI, which has been referred in Section 3.2.
Theorem XVI (On the length of an arc). Let A be any point of upper semicircle, a be the lengthof the arc AB, s be the sine AM, s′ be the length of the chord AB. Then114s′ + ss′ + (s′ − s) < a < s′ + (s′ − s) ′.
33
2s + 3s

4.3

Isaac Newton

Isaac Newton (1642-1727) mentioned the theorem of Huygens at least twice. Newton referred the theoremof Huygens in the letter to Michael Dary on 22 January 1675 [21, p.333][23, p.662]. In this letter Newton
applied the theorem of Huygens on a construction of the length of the arc of an ellipse.
Newton also mentioned the theorem of Huygens in the letter epistola prior to Oldenburg on 13 June
1676 [22, pp.39-40][23, p.669]. Newton considered “ For, being given the chord A of an arc and the chord
B of half the arc, to ﬁnd that arc most nearly, take that arc to be z, and the radius of the circle r:” [22]In Figure 4, let C be the center of a sector CbAB. Let A = Bb, B = AB, z = arc bAB and ∠ACB = θ.
A
Since z = 2rθ = 2r sin−1 2r
,
A
z5
z
z
z3
+
− &c.
= sin
=
−
2r
2r
2r 6 · (2r)3120 · (2r)5
Thus
z3
z5
+
− &c,
2
4 × 6r4 × 4 × 120r4
z
z5
z3
B= −
+
− &c.
2 2 × 16 × 6r2
2 × 16 × 16 × 120r4
A=z−

Eliminating the term z 3 /r2 , Newton obtained
8B − A
z5
+ &c,
=z−
3
64 × 120r4
and wrote
6

0.5

0

C

G

B

E

D

A

−0.5

e

b

−1

−2

−1.5

−1

−0.5

0

0.5

1

Figure 4: Epistola Prior [22]that is, 31 (8B − A) = z, with an error of only z 5 /7680r4 in excess; which is the theorem of
Huygens.
[22]
Newton continued, “let C be the center of the circle, d the diameter AK, and x the sagitta AD.” “takeAH, the ﬁfth part of DH” [22]. Take G on the extension of AK such that KG = HC. The extensions of
GB and Gb intersect E and e with the tangent at A, respectively. Then Newton derived
72

 − AE = 16x 5 ± &c
AB525d 2and wrote√the error being only (16x3 /525d3 ) (dx) ± etc, much less indeed than in the theorem of Huygens.
[22]
In modern notation [4]d sin θ(14 + cos θ)dAE ==2(9 + 6 cos θ)
2

(

)
1 7
9
θ−
θ + O(θ ) .
2100

Furthermore Newton wrote “if we make 7AK : 3AH :: DH : n and take KG = CH − n”, that is,
AG =

3
1
12x2
d− x−
,
2
5
175d

“the error will be much smaller still.” [22] In modern notation)(d sin θ(481 + 47 cos θ − 3 cos 2θ)
d
931
9
11
AE =
=
θ−
θ + O(θ ) .
2(306 + 222 cos θ − 3 cos 2θ)214112000
Newton’s method is expanding to inﬁnite series of z/r, (z/r)3 , (z/r)5 , . . . then eliminating terms oflower degree and was not as systematically as the Richardson extrapolation process.

5

Aitken ∆2 process

5.1

Deﬁnitions

The most famous convergence acceleration method is the Aitken ∆2 process. This method is deﬁned by()(sn+1 − sn )2(∆sn )2
tn = s n −= sn −
sn+2 − 2sn+1 + sn
∆2 s n()(sn+1 − sn )(sn+2 − sn+1 )∆sn ∆sn+1= sn+1 += sn+1 +(sn+1 − sn ) − (sn+2 − sn+1 )∆sn − ∆sn+1()
2
2(sn+2 − sn+1 )(∇sn+2 )= sn+2 −= sn+2 −sn+2 − 2sn+1 + sn
∇2 sn+2
sn sn+2 − s2n+1=,
sn+2 − 2sn+1 + sn
7where ∆ and ∇ are the forward and backward diﬀerence operators, respectively. All four formulas aremathematical equivalent.
A.C. Aitken (1895 - 1967) used this method in [1] (1926), so it named after Aitken. The Aitken ∆ 2process was discovered by Japanese Mathematician Takakazu Seki (? - 1708) before 1680.

5.2

The circumference ratio in Japan before Seki√
Until the mid 17th century, Japanese traditional π was 3.16 or 3.162(≒ 10).
A reprint of Chinese mathematical textbook Suanxue Qimeng (Mathematical Enlightenment, 1299)
by Zhu Shijie , was published (1658) in Japan. In this book π = 3/1, π = 157/50(= 3.14) and π =22/7(= 3.142857) are written.
The ﬁrst Japanese mathematician who determined the circumference ratio was Shigekiyo Muramatsu.
In 1663 he computed inscribed 2n -sided regular polygons (n = 3, . . . , 15) in a circle of diameter 1.
Muramatsu obtained T215 = 3.141592648777698869248, and compared with various Chinese values of π,
he determined 3.14. See [6, pp.52-56][18, p.78].
In 1673 Yoshimasu Murase determined π as 3.1416 by using
T217 = T131072 = 131072 × 0.00002396844980842366(= 3.14159265328970596352).
These methods used only inscribed polygons, both Muramatsu and Murase could not determine thenumber of exact digits.

5.3

Takakazu Seki

Takakazu Seki used the Aitken ∆2 process in three purposes.
1 Determining the circumference of a circle, in Katsuyō Sampō Vol.4 (1712).
2 Computing the length of an arc, in Katsuyō Sampō.
3 Determining the volume of a sphere, in Ritsu-Enritsu-Kai (1680), Taisei Sankei Vol.12 (c.1711),
and Katsuyō Sampō.
Ritsu-Enritsu-Kai (Solving the volume of the sphere) is a manuscript of Seki’s draft dated 1680. Taisei
Sankei (Accomplished Mathematical Sutra) is manuscripts of books of twenty volumes and is considered
to be in collaboration with Takakazu Seki, and his disciples Katahiro Takebe (1664-1739), and his elder
brother Kata’akira (or Kataaki) Takebe (1661-1716).
Kata’akira Takebe wrote the story of Taisei Sankei in his Takebe-shi Denki (Biography of the Takebe
family) in 1715. According to Takebe-shi Denki, compiling of Taisei Sankei began in 1683 under thesupervision of Katahiro, and once was assembled and named Sampō Taisei (Accomplished Mathematical
Methods) in about 1695. Kata’akira by oneself resumed to compile Taisei Sankei in 1701 and completed
in about 1711. See [6, pp.108-109].
Katsuyō Sampō (Tying the pivots of Mathmatical Methods) was published by Seki’s disciple Murahide
Araki and was consist of Seki’s posthumous drafts.
5.3.1

Formula of the volume of a sphere

In Ritsu-Enritsu-Kai Seki gave the formula of the volume of a sphere of diameter D. Seki divided thesphere into m, m is even, segments by parallel equidistant planes. Seki computed)())∑ D ( (i − 1)D ((i − 1)DiD
iD
vm = 2
4
D−
+4
D−
,
2m
m
m
m
m
i=1
m/2

with m = 50, 100, 200 and D = 10, and gavea = v50 = 666.4,

b = v100 = 666.6,

Seki wrote in Ritsu-Enritsu-Kai that8c = v200 = 666.65.

(14)

Put the diﬀerence a from b, multiply the diﬀerence b from c, make the numerator by the
product. Put the diﬀerence a from b, subtract the diﬀerence b from c, make the denominatorby the diﬀerence. Multiply b with the denominator, and add the numerator, make numerator
by the summation, divide the numerator by the denominator, chop under 0.005, and obtain
666 23 which is the divided volume.
[14]
This means((b − a) − (c − b))b + (b − a)(c − b)(b − a) − (c − b)(=b+(b − a)(c − b)(b − a) − (c − b))22
= 666 (= D3 ),
3
3

(15)

which is the Aitken ∆2 process. Since the ratio of the areas of the square and the circle at cut plane is
1 : π/4, Seki determined that the volume of the sphere is 16 πD3 .
The volumes a, b, c are represented as
a=

2D3
2D3
−
,
3
3m2

b=

D3
2D3
−
,
3
6m2

c=

D3
2D3
−
.
3
24m2

where m = 50, D = 10. Therefore
D3 D3
2 8m2
2D3
2D3
D3
D3
D3
2D3
=
−
−
.
+ 2m
+
=
3
3
2
2
2
3
6m
3
6m
6m
3
D
D
−
2m2
8m2
This fact was proved [5] by Matsusaburo Fujiwara (1881-1946). He also pointed out that if we divide
2m, 2km, 2k 2 m instead of 50, 100, 200, the formula (15) gives the constant independent of m and k.
From the numerical analysis view point, the formula (14) is the m-panels trapezoidal rule for thenumerical integration
∫ D
2D3
.
4y(D − y)dy =30
Therefore we can say that Seki accelerated the trapezoidal rule by the Aitken ∆ 2 process.
Seki wrote only the formula of (15) in Ritsu-Enritsu-Kai and Katsuyō Sampō, but Katahiro Takebe
wrote in Taisei Sankei Vol. 12 as follows:
Let b − a be the front diﬀerence, c − b the behind diﬀerence. Add the number by applyingzōyaku-jutsu (the art of summation of geometric series) to both diﬀerences to b, we get thedivided volume.
[16][17]
Takebe called b − a, c − b the front diﬀerence and the behind diﬀerence, respectively, and called to obtain(b − a)(c − b)(b − a) − (c − b)zōyaku-jutsu. Therefore Takebe considered that a, b, c approximately equal to the consecutive partial
sums of a geometric series, say
a = k(1 + r + · · · + rn−1 ), b = k(1 + r + · · · + rn ), c = k(1 + r + · · · + rn+1 ).
Since b − a = krn and c − b = krn+1 , we haveb+(b − a)(c − b)krn+1k
= k(1 + r + · · · + rn ) +=.
(b − a) − (c − b)
1−r
1−r

(16)

(See Horiuchi [6, pp.248-249]). When n = 1 Yoshisuke Matsunaga (? - 1747) who was a second-generation
pupil of Seki proved (16) in Kigenkai or Sampō Shūsei [10].

95.3.2Computing the circumference of a circle

In Katsuyō Sampō [15] Seki determined the circumference of a circle by using (see [6, pp.252-253][18,p.111])(T216 − T215 )(T217 − T216 )T216 += (very little less than)3.14159265359,(17)(T216 − T215 ) − (T217 − T216 )which is the Aitken ∆2 process.
By (1),()(T2n − Tn )(T4n − T2n )
π4
11π 6
−8
Vn = T2n +=π 1+++ O(n ) .
(T2n − Tn ) − (T4n − T2n )1920n4516096n6(18)

Therefore the ∆2 process accelerates {T2n }. Moreover the inequality
T2n < π < V2n−2(19)holds. Furthermore {V2n } monotonically decreases and {T2n } monotonically increases, and both converge
to π. Seki did not mention these facts.
By using the deﬁnite circumference of a circle, Ski derived the rational approximate 355/113(=
3.14159292) of π. See [6, p.248]. We consider that Seki gave only 12 digits of π for this purpose.
Seki computed the length of an arc similarly to the circumference of a circle.

6

Richardson extrapolation process

6.1

Deﬁnitions

Suppose a sequence {sn } ∈ S has the asymptotic representationsn = s + c1 λn1 + c2 λn2 + c3 λn3 + . . . + cm λnm + o(λnm ),(20)

where λ1 , λ2 , . . . , λm , (1 > |λ1 | > |λ2 | > . . . > |λm |) are known constants and c1 , c2 , . . . , cm are constants.
(k)
The double sequence {Tn } is deﬁned byTn(0) = sn ,n = 1, 2, . . . ,
λk
(k−1)
(k−1)
Tn(k) = Tn+1 +(T− Tn(k−1) ),1 − λk n+1
Then
Tn(k) = s + ck+1

k∏
λk+1 − λii=1

1 − λik = 1, 2, . . . , m; n = 1, 2, . . . .

(21)

k = 1, 2, . . . , m − 1.

(22)

λnk+1 + o(λnk+1 ),(k)

The procedure to get {Tn } by the sequential elimination of the terms involving λn1 , λn2 , . . . is called
the Richardson extrapolation process.
When a sequence {sn } ∈ S has the asymptotic representation
c1
c2
c3
cm
sn = s + 2 + 4 + 6 + . . . + 2m + o(n−2m ),
(23)
n
n
n
n
where c1 , c2 , . . . , cm are constants, the subsequence {s2n } satisﬁes (20) with λk = 2−2k (k = 1, . . . , m). In
this case the Richardson extrapolation process are written byTn(0) = s2n , n = 1, 2, . . . ,
1
(k−1)
(k−1)
Tn(k) = Tn+1 + 2k
(T
− Tn(k−1) ),2 − 1 n+1

k = 1, 2, . . . ; n = 1, 2, . . . .

(24)

k = 1, 2, . . . , m − 1.

(25)

Since λk = 2−2k , it follows from (22) thatTn(k) = s + (−1)k ck+1 2(k+1)(−k−2n) + o(2(k+1)(−k−2n) ),In 1927 Lewis Fry Richardson [12] (1881-1953) treated (21) and (24). By this reason the sequence(k)transformation {sn } 7→ {Tn } is called the Richardson extrapolation process.
The Richardson extrapolation process was discovered by Seki’s disciple Katahiro Takebe before 1710,
probably before 1695.
10

6.2

Katahiro Takebe(0)

In Taisei Sankei Vol. 12 [16][17], Takebe computed Tn = (T2n )2 = 22n 100 sin2 (π/2n ) (n = 1, . . . , 9), i.e.
(0)

= (T2 )2 = 400,(0)

= (T4 )2 = 800,

T1
T2

···
(0)

T9

= (T512 )2 = (little over than)986.94805369467323179511059076580.

Takebe watched

(0)

(0)

(0)

(0)

Tn+1 − Tn

Tn+2 − Tn+1
(0)

(0)

(0)

(0)

(0)

(0)

tends to 4. Replacing (Tn+1 − Tn )/(Tn+2 − Tn+1 ) by 4 in Master Seki’s method, i.e. the ∆2 process(0)

Tn+1 +(0)(0)(Tn+1 − Tn )(Tn+2 − Tn+1 )
(0)

(0)

(0)

(0)

(Tn+1 − Tn ) − (Tn+2 − Tn+1 )

(0)

(0)

= Tn+1 +(0)

Tn+1 − Tn(0)(0)
Tn+1 −Tn(0)(0)
Tn+2 −Tn+1,−1

Takebe computed(0)(0)
Tn+1 − Tn(0)(1),
Tn = Tn+1 +4−1

n = 1, . . . , 8,

such as(1)

T1= (little over than )933.33333333333333333333333333333,(1)T2 = (very little over than)983.01106693739722922306405501931,

···
(1)

T8

= (little less than)986.96043986022713417377439639209.

Moreover Takebe watched

(1)

(1)

(1)

(1)

(1)

(1)

Tn+1 − Tn

Tn+2 − Tn+1
tends to 16, he computed(1)Tn(2) = Tn+1 +Tn+1 − Tn,
16 − 1n = 1, . . . , 7.

Similarly Takebe watched
(k−1)

(k−1)

(k−1)

(k−1)

Tn+1 − Tn

Tn+2 − Tn+1
tends to 4k , he computed
(k−1)

(k−1)

Tn(k) = Tn+1 +(k−1)

Tn+1 − Tn
4k − 1,

k = 1, . . . , 8; n = 1, . . . , 9 − k,(26)

and got(8)

T1= (little less than)986.96044010893586188344909998747,

and he determined the squared deﬁnite circumference(very little less than)986.9604401089358618834491,
and then he extracted the square root
(little over than)31.41592653589793238462643,
which was his deﬁnite circumference. Takebe’s method is just the Richardson extrapolation process.
11

The squared length of inscribing polygon (T2n )2 = 100 × 22n sin2 (π/2n ) has the asymptotic expansion


∞
2j
∑
π
(T2n )2 = 100π 2 1 +(−1)j2−2jn+2j+1  .
(2j
+
2)!
j=1
By (25), we have
π 20
(8)× 100 × 2−71 ≒ −1.53 × 10−28 ,T1 − 100π 2 ∼ −
20!
√
π 19
(8)
T1 − 10π ∼ −× 10 × 2−72 ≒ −2.43 × 10−30 .
20!
Takebe’s method can give exact 30 decimal digits.
More than two decades after, Takebe gave exact 41 decimal digits in Tetsujutsu Sankei (1722) [20] of
which English translation [?] will be published.
Takebe noted in Tetsujutsu Sankei
Since all kinds of numbers by the art of summation of geometric series were published in
Enritsu (i.e. Taisei Sankei Vol.12 Sec.1), we omitted them here.
[20]
According to Takebe-shi Denki and the above note, Katahiro Takebe discovered the Richardson extrapolation process probably before 1695.

7

Conclusion

In Europe the circumference ratio was computed using inscribed and circumscribed polygons in a circle
from Archimedes to Van Ceulen. In Japan the ratio was computed using inscribed polygons only. W. Snell(1621) and T. Seki (before 1680) made breakthroughs to these methods. Snell used weighted mean
methods and Seki used the ∆2 process. Snell’s methods were proved by C. Huygens (1654) using Euclidean
geometry. Seki’s method was explained and partially proved by Katahiro Takebe and Y. Matsunaga.
The veriﬁed acceleration method was given and proved by C. Huygens.
The theorem of Huygens, the ﬁrst step of the Richardson extrapolation process, was also given andproved by Huygens. I. Newton (1675) treated the theorem of Huygens by inﬁnite series. Newton’s method
was expanding to inﬁnite series then eliminating terms of lower degree and was not as systematically as
the Richardson extrapolation process. Takebe (before 1695) discovered the Richardson extrapolation
process by improving Seki’s ∆2 process.
The ∆2 process was discovered by Seki. This fact was found by Claude Brezinski [2].

References[1] A.C.Aitken, On Bernoulli’s numerical solution of algebraic equations, Proc. Roy. Soc. Edinburgh Ser
A 46 (1926), 289–305.
[2] C. Brezinski, Introduction and Historical Survey, in J.P. Delayer, Sequence Transformations,
Springer, 1988.
[3] C. Brezinski, Some pioneers of extrapolation methods, in The Birth of Numerical Analysis, World
Scientiﬁc 2010.
[4] J. Dutka, Richardson extrapolation and Romberg integration, Historia Mathematica, 11 (1984), 3-21.
[5] M. Fujiwara, Study on the history of Japanese Mathematics (12) (in Japanese), Japan Imperial
Academy, Vol.3 No.2 (1944), 371-433.
[6] A. Horiuchi, Translated by S. Wimmer-Zagier, Japanese Mathematics in the Edo Period (1600-1868),
Birkhäuser, 2010, Les Mathématiques japonaises à l’éoque d’Edo 1600-1868, J. Vrin, Paris,1994.
[7] C. Huygens, De circuli magnitudine inventa, 1654. Kessinger Legacy Reprints.

12[8] C. Huygens, Christiani Hugenii opera varia; Bd. 2, 357-404., 1724.
European Cultural Heritage Online. http://echo.mpiwg-berlin.mpg.de/[9] D.C. Joyce, Survey of extrapolation processes in numerical analysis, SIAM Rev. 13 (1971), 435-490.
[10] Y. Matsunaga, Sampō Shūsei, vol. 8, (manuscript), Tohoku University, Okamoto Collection.
[11] O. Østerby, Archimedes, Huygens, Richardson
http://www.daimi.au.dk/~oleby/notes/arch.psandRomberg,

December

20,

2000.

[12] L.F. Richardson, The diﬀered approach to the limit, Part I – single Lattice, Philos. Trans. Roy. Soc.
London, Ser A 226 (1927), 299-349.
[13] F. Rudio, Archimedes, Huygens, Lambert, Legendre. Vier Abhandlungen Über die Kreismessung,Teubner, 1892.
[14] T. Seki, Ritsu-Enritsu-Kai, (manuscript), 1680. Tohoku University, Kanō Collection.
[15] T. Seki, Katsuyō Sampō, vol. 4, 1712. Tohoku University, Okamoto Collection.
[16] T. Seki, K. Takebe and K. Takebe, Taisei Sankei, vol.12, (manuscript), c.1711. Tohoku University,
Okamoto Collection.
[17] T. Seki, K. Takebe and K. Takebe, Enhō (Taisei Sankei, vol.12), (manuscript), c.1711. Tokyo University, Nanki Library.
[18] D.E. Smith and Y. Mikami, A history of Japanese mathematics, Chicago, Open Court Publishing,1914. http://www.archive.org/details/ahistoryjapanes00mikagoog[19] W. Snellius, Cyclometricus, 1621. Kessinger Legacy Reprints.
[20] K. Takebe, Tetsujutsu Sankei, (manuscript), 1722. National Archives of Japan, the Cabinet Library.
[21] H.W. Turnbull, The Correspondences of Isaac Newton, vol.1, Cambridge University Press, 1959.
[22] H.W. Turnbull, The Correspondences of Isaac Newton, vol.2, Cambridge University Press, 1960.
[23] D.T. Whiteside, The Mathematical Papers of Isaac Newton, Vol. IV, Cambridge University Press,
1971.

13