Isaac Newton’s ‘Of Quadrature by Ordinates’lab.twcu.ac.jp/osada/of_quadrature_rev2.pdf · Isaac Newton’s ‘Of Quadrature by Ordinates’ 3 D B A M Fig. 1 Theorem XVI in De

Archive for History of Exact Sciences manuscript No.(will be inserted by the editor)

Isaac Newton’s ‘Of Quadrature by Ordinates’

Naoki Osada

Abstract In Of Quadrature by Ordinates (1695), Isaac Newton tried two methodsfor obtaining the Newton-Cotes formulae. The first method is extrapolation andthe second one is the method of undetermined coefficients using the quadrature ofmonomials. The first method provides n ordinates Newton-Cotes formulae only forcases in which n = 3, 4 and 5. However this method provides another importantformulae if the ratios of errors are corrected. It is proved that the second methodis correct and provides the Newton-Cotes formulae. Present significance of each ofthe methods is given.

Keywords Isaac Newton · quadrature · ordinate · Newton-Cotes formula ·interpolation · extrapolation · theorem of Huygens · Christiaan Huygens · JamesGregory

1 Introduction

Isaac Newton proposed the principle of numerical integration by polynomial in-terpolation in Cor of Lemma V of Book III of his Principia (1687).

Hence the areas of all curves may be nearly found; for if some number ofpoints of the curve to be squared are found, and a parabola be supposedto be drawn through those points, the area of this parabola will be nearlythe same with the area of curvilinear figure proposed to be squared: butthe parabola can be always squared geometrically by methods generallyknown. [1, p.500]

Numerical integration formulae based on this idea of Newton’s are called of in-terpolatory type. As an example, Newton gave the formula for four equidistantordinates in Scholium of Prop VI of his Methodus Differentialis (1711) [8].

Naoki OsadaTokyo Woman’s Christian University,2-6-1, Zempukuji, Suginami, Tokyo, 167-8585 JapanE-mail: [email protected]

2 Naoki Osada

If there are four ordinates at equal intervals, let A be the sum of the firstand fourth, B the sum of the second and third, and R the interval betweenthe first and fourth, then the central ordinate will be 9B−A

16 , and the area

between the first and fourth ordinates will be A+3B8 R. [3]

This formula is so-called Simpson 3/8 rule. Roger Cotes published similar rulesfor the areas included between the curves of equidistant ordinates up to elevenordinates in his posthumous Harmonia Mensurarum (1722). Therefore numericalintegration formulae by interpolation with equidistant ordinates are called theNewton-Cotes formulae or the Newton-Cotes rules.

The numerical integration formula of interpolatory type is usually applied notto the whole interval of integration, but to m equal subintervals into which theinterval is divided. The full integral is approximated by the sum of the approxi-mations to the integrals on the subintervals. This method is called the m panelscomposite formula.

In the manuscript Of Quadrature by Ordinates (1695), which is abbreviated hereto Of Quadrature, Newton tried two methods to obtain Newton-Cotes formulae.The first method is extrapolation and the second is the method of undeterminedcoefficients using the quadrature of monomials.

The first method provides n ordinates Newton-Cotes formulae only for casesin which n = 3, 4 and 5. In using the first method it is necessary to know theratio of errors of the single Newton-Cotes formula for the corresponding compositeformula, but Newton’s ratios of errors were incorrect. D.T. Whiteside called thisfirst method a “Newton’s clever (but not wholly exact) approach to achieving theCotesian formulas” [18, p.xliv].

The second method has been considered as mere check of the formula. White-side noted “If Newton had gone on similarly to check the accuracy of the area-approximation in his preceding ‘Cas. 5’, he would (see note (12) above) have hada shock!” [18, p.698].

In this paper we study Newton’s two methods for obtaining the Newton-Cotesformulae and re-evaluate the manuscript Of Quadrature.

Since Newton used the theorem of Huygens, i.e. the first step of the Richardsonextrapolation process, and the ratios of errors of the single to composite Newton-Cotes formulae, we will survey the theorem of Huygens in Section 2 and the errorterms of the composite Newton-Cotes formulae in Section 3.

In Sections 4 and 7 we will annotate Of Quadrature and will quote the full-text of the Of Quadrature from Whiteside’s translation [18]. We will also includeNewton’s original Latin in footnotes. In Section 5 we will deal with Newton’sincorrect ratios of single to composite Newton-Cotes formulae. In Section 6 wewill describe the modern significance of Newton’s numerical integration formulaeby extrapolation. In Section 7 we will prove that the Newton-Cotes formulae aredetermined by the second method.

2 The theorem of Huygens

2.1 De circuli magnitudine inventa by C. Huygens

In De circuli magnitudine inventa [5] (1654), Christiaan Huygens proved TheoremXVI:

Isaac Newton’s ‘Of Quadrature by Ordinates’ 3

D B

A

M

Fig. 1 Theorem XVI in De circuli magnitudine inventa by Christiaan Huygens

Any arc less than a semicircle is greater than its subtense together withone third of the difference, by which the subtense exceeds the sine, andless than the subtense together with a quantity, which is to the said third,as four times the subtense added to the sine is to twice the subtense withthree times the sine. [6]

Theorem XVI means [12, p.40]

s′ +1

3(s′ − s) < a < s′ +

1

3(s′ − s)

4s′ + s

2s′ + 3s, (1)

where a is the length of any arc ÄAB, s is the sine AM, and s′ is the subtense(chord)AB in Figure 1.

Let Tn be the circumference of n-sided inscribed regular polygon in a circlewith diameter 1. The inequality (1) implies

T2n +1

3(T2n − Tn) < π < T2n +

1

3(T2n − Tn)

4T2n + Tn

2T2n + 3Tn. (2)

By the Taylor expansion we have

Tn = n sinπ

n= π

„

1 − π2

6n2+

π4

120n4+ O

„

1

n6

««

. (3)

It follows from (3) that

T2n +1

3(T2n − Tn) = π

„

1 − π4

480n4+ O

„

1

n6

««

,

T2n +1

3(T2n − Tn)

4T2n + Tn

2T2n + 3Tn= π

„

1 +π6

22400n6+ O

„

1

n8

««

.

The lower bounds of (1) and (2) are the first step of the Richardson extrapolationwhich was called the theorem of Huygens by Newton. For details, see [9].

4 Naoki Osada

2.2 Newton’s application of the theorem of Huygens

Newton referred the theorem of Huygens in the letter to Michael Dary on 22January 1675 [14, p.333] [17, p.662]. In this letter Newton applied the theorem ofHuygens to a construction of the length of the arc of an ellipse. Newton wrote

This is derived from Hugenius’ Quadrature of ye Circle, and I believe ap-proaches ye Ellipsis as near as his doth the Circle. [14]

Newton also mentioned the theorem of Huygens in the letter epistola prior toHenry Oldenburg on 13 June 1676 [15, pp.39-40] [17, p.669]. Let A be the chordof an arc, B the chord of half the arc. Let z be the length of the arc, and r be theradius of the circle. Then Newton derived

A = z − z3

24r2+

z5

1920r4− &c,

B =z

2− z3

192r2+

z5

61440r4− &c.

Newton eliminated the terms z3/r2 in A, B, and obtained

8B − A

3= z − z5

7680r4+ &c.

Newton stated:

that is, 13 (8B − A) = z, with an error of only z5/7680r4 in excess; which is

the theorem of Huygens. [15]

3 Error terms in the Newton-Cotes formulae

3.1 The trapezoidal rule of James Gregory

James Gregory discussed numerical integration in his Exercitationes geometricae

(1668) [4]. In Figure 2 he divided the equally line segment AI into minimumparts. If the second differences of ordinates BP, CO, · · · , HI are equal to Z, Gregorystated:

GHQ =HQ × GQ

2− Z × GQ

12. (4)

By adding the square GQIK to both sides of (4) we have

GHIK =(HI + GK) × GQ

2− Z × GQ

12, (5)

which is the special case of the error term of the trapezoidal rule.Whiteside [16, p.249] pointed out that (5) implies the Simpson rule as follows:

FHIL = FGKL + GHIK

=1

2(FL + GK)FR +

1

2(GK + HI)GQ − 1

6(FL − 2GK + HI)GQ

=1

3(FL + 4GK + HI)GQ =

1

6(FL + 4GK + HI)LI.


−1 0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

I K L M N O P

H

G

F

E

D

C

BA

Q

R

S

T

V

X

Fig. 2 Fig. 8 in Exercitationes Geometricae by James Gregory

3.2 Composite Newton-Cotes formulae

In the remainder of this paper, we shall use the following notation and well knownpropositions and their corollaries. See, for example, [2]. Let f(x) be a continuous

function on [a, b]. Put I =R ba f(x)dx, and let n denote the number of interpolation

points or that of ordinates.

Proposition 1 (n = 2) Suppose f(x) is of class C2 on [a, b].

(i) The trapezoidal rule.

T1 =b − a

2(f(a) + f(b)) , I = T1 − (b − a)3

12f ′′(ξ), a < ξ < b.

(ii) The m panels composite trapezoidal rule. Let h = (b − a)/m, xi = a + ih (i =0, . . . , m).

Tm =h

2

f(x0) + 2m−1X

i=1

f(xi) + f(xm)

!

,

I = Tm − (b − a)3

12m2f ′′(ξ), a < ξ < b.

Corollary 1 Under the conditions of Proposition 1, the ratio of errors is as follows:

T1 − I : Tm − I ≈ m2 : 1.


(i) The Simpson rule.

S1 =b − a

6(f(a) + 4f(a + (b − a)/2) + f(b)) ,

I = S1 − (b − a)5

2880f (4)(ξ), a < ξ < b.

6 Naoki Osada

(ii) The m panels composite Simpson rule. Let h = (b − a)/(2m), xi = a + ih (i =0, . . . , 2m).

Sm =h

3

f(x0) + 4mX

i=1

f(x2i−1) + 2m−1X

i=1

f(x2i) + f(x2m)

!

,

I = Sm − (b − a)5

2880m4f (4)(ξ), a < ξ < b.


S1 − I : Sm − I ≈ m4 : 1.


(i) The Simpson 3/8 rule.

N1 =b − a

8(f(a) + 3f(a + (b − a)/3) + 3f(a + 2(b − a)/3) + f(b)) ,

I = N1 − (b − a)5

6480f (4)(ξ), a < ξ < b.

(ii) The m panels composite Simpson 3/8 rule. Let h = (b−a)/(3m), xi = a+ih (i =0, . . . , 3m).

Nm =3h

8

f(x0) + 3mX

i=1

f(x3i−2) + 3mX

i=1

f(x3i−1) + 2m−1X

i=1

f(x3i) + f(x3m)

!

,

I = Nm − (b − a)5

6480m4f (4)(ξ), a < ξ < b.


N1 − I : Nm − I ≈ m4 : 1.


(i) The Boole rule.

B1 =b − a

90(7f(a) + 32f(a + (b − a)/4) + 12f(a + (b − a)/2)

+32f(a + 3(b − a)/4) + 7f(b))

I = B1 − (b − a)7

1935360f (6)(ξ), a < ξ < b.

(ii) The m panels composite Boole rule. Let h = (b − a)/(4m), xi = a + ih (i =0, . . . , 4m).

Bm =2h

45

7f(x0) + 32mX

i=1

f(x4i−3) + 12mX

i=1

f(x4i−2) + 32mX

i=1

f(x4i−1)

+14m−1X

i=1

f(x4i) + 7f(x4m)

!

,

I = Bm − (b − a)7

1935360m6f (6)(ξ), a < ξ < b.


B1 − I : Bm − I ≈ m6 : 1.


−1 0 1 2 3 4 5 6 7 8 9

−1

0

1

2

3

4

5

A B C D E F G H I

KL

MN

OP Q R S

Fig. 3 The figure in Of Quadrature by Ordinates [18, p.690]

4 The first method: Numerical integration by extrapolation

In the manuscript Of Quadrature by Ordinates, Newton gave two methods for ob-taining Newton-Cotes formulae. The first method is extrapolation and the secondone is the method of undetermined coefficients using the quadrature of monomials.We study the former in this section and the latter in Section 7.

If upon the base A at equal distances be erected ordinates to any CurveAK, BL, CM &c the Curve may be ye Ordinates be squared quamproxime

as follows.1

Case 1. If there be given but two ordinates AK and BL, make the area(AKLB) = 1

2 (AK + BL)AB.Case 2. If there be given three AK, BL and CM , say that

1

2(AK + CM)AC = ¤(AM) (6)

and again, by Case 1,

1

2(1

2(AK + BL) +

1

2(BL + CM))AC =

1

4(AK + 2BL + CM)AC = ¤(AM),

(7)and that the error in the former solution is to the error in the latter as AC2

to AB2 or 4 to 1, and hence the difference 14 (AK − 2BL + CM)AC of the

solutions is to the error in the latter as 3 to 1, and the error in the latterwill be

1

12(AK − 2BL + CM)AC. (8)

Take away this error and the latter solution will come to be

1

6(AK + 4BL + CM)AC = ¤(AM), (9)

1 Newton did not translate the first sentence into Latin.

8 Naoki Osada

the solution required.2

Newton gave the single trapezoidal rule in Case 1. In Case 2 Newton appliedthe single trapezoidal rule (6) and 2 panels composite trapezoidal rule (7) onI = area(AKMC). Let T1 = 1

2 (AK + CM)AC and T2 = 14 (AK + 2BL + CM)AC.

Newton used Corollary 1 which follows from Gregory’s (5). Suppose Z = CM−2BL+AK = DN−2CM +BL = EO−2DN +CM, we have 4Z = EO−2CM +AK.

Then

T1 − I =1

2(AK + CM)AC − ¤(AM) =

1

12(EO − 2CM + AK) · AC =

1

3Z · AC.

On the other hand

T2 − I =1

4(AK + 2BL + CM)AC − ¤(AM) =

1

12Z(AB + BC) =

1

12Z · AC.

ThereforeT1 − I : T2 − I = 4 : 1. (10)

For m > 2, we can derive similarly that the error in the single trapezoidal rule isto the error in the m panels composite trapezoidal rule as m2 to 1.

Newton derived T1 − T2 : T2 − I = 3 : 1 from (10). Since T1 − T2 = 14 (AK −

2BL + CM)AC, Newton obtained

T2 − I =1

12(AK − 2BL + CM)AC.

Therefore

T2 − 1

12(AK − 2BL + CM)AC =

1

6(AK + 4BL + CM)AC,

which is the Simpson rule.Newton derived S1 = 1

6 (AK + 4BL + CM)AC from

S1 = T2 − 1

3(T1 − T2), (11)

which is the theorem of Huygens, i.e. the first step of the Richardson extrapolationprocess.

Newton continued:

2 Cas. 1. Si dentur duæ tantu ordinatæ AK, BL fac aream

AKLB =AK + BL

2AB

Cas. 2. Si dentur tres AK, BL, CM , dicAK + CM

2AC = ¤AM, et rursus

AK + BL

4+

BL + CM

4in AC = AK + 2BL + CM in 1

4AC = ¤AM (per Cas. 1) et errorem solutionis

prioris esse ad errorem solutionis posterioris ut ACq ad ABq seu 4 ad 1 adeoque solutionum

differentiamAK − 2BL + CM

4AC esse ad errorem posterioris ut 3 ad 1, et error posterioris

eritAK − 2BL + CM

12AC. Aufer hunc errorem et solutio posterior evadet

AK + 4BL + CM

6AC = ¤AM. Solutio quæsita.


Case 3. If there be given for ordinates AK, BL, CM and DN , say that

1

2(AK + DN)AD = ¤(AN) (12)

likewise, that

1

3(1

2(AK + BL) +

1

2(BL + CM) +

1

2(CM + DN))AD,

that is,1

6(AK + 2BL + 2CM + DN)AD = ¤(AN). (13)

The errors in the solutions will be as AD2 to AB2 or 9 to 1, and hence the

difference in the errors — which is the difference1

6(2AK − 2BL − 2CM +

2DN)AD in the solutions — will be to the error in the latter as 8 to 1.Take away this error and the latter will remain as

1

8(AK + 3BL + 3CM + DN)AD = ¤(AN).3 (14)

Newton applied the 3 panels composite trapezoidal rule on I = area(AKND).Let T1 = 1

2 (AK + DN)AD and T3 = 16 (AK + 2BL + 2CM + DN)AD. By T1 − I :

T3 − I = 32 : 1, Newton derived T1 − T3 : T3 − I = 8 : 1. Using T1 − T3 =13 (AK − BL − CM + DN)AD, Newton obtained

T3 − T1 − T3

8=

1

8(AK + 3BL + 3CM + DN)AD, (15)

and he gave (15) in his Methodus Differentialis. Newton’s derivation

N1 = T3 − T1 − T3

8(16)

is the theorem of Huygens.Newton continued:

Case 4. If there be given five ordinates, say (by Case 2) that

1

6(AK + 4CM + EO)AE = ¤(AO), (17)

likewise that

1

2(1

6(AK + 4BL + CM)AC +

1

6(CM + 4DN + EO))AE = ¤(AO),

3 Cas. 3. Si dentur 4 Ordinatæ AK, BL, CM, DN : dicAK + DN

2AD = ¤AN. Item

AK + BL

6+

BL + CM

6+

CM + DN

6in AD (id est

AK + 2BL + 2CM + DN

6AD)= ¤AN.

Et solutionu errores erunt ut ADq ad ABq seu 9 ad 1 adeoque errorum differentia (quæ est

solutionu differentia2AK − 2BL − 2CM + 2DN

8AD) erit ad errorem posterioris ut 8 ad 1.

Aufer hunc errorem et posterior manebit

AK + 3BL + 3CM + DN

8AD = ¤AN.

10 Naoki Osada

and that the errors are as AE2 to AB2 or 16 to 1; then, since the differencein the errors is 1

12 (AK − 4BL + 6CM − 4DN + EO)AE, the error in thelesser will be 1

180 (AK − 4BL + 6CM − 4DN + EO)AE, and when this istaken away there will remain

1

90(7AK + 32BL + 12CM + 32DN + 7EO)AE = ¤(AO).4 (18)

In Case 4, Newton applied the single and 2 panels Simpson rule on I =area(AKOE). Let S1 = 1

6 (AK +4CM +EO)AE and S2 = 112 (AK +4BL+2CM +

4DN + EO)AE. Newton wrote S1 − I : S2 − I = AE2 : AB2 = 16 : 1, which iscorrect as good luck. The exact ratio is S1 − I : S2 − I = 24 : 14 = 16 : 1 byCorollary 2. Similarly to Case 3, from S1 − S2 : S2 − I = 15 : 1 and

S1 − S2 =1

12(AK − 4BL + 6CM − 4DN + EO)AE,

Newton derived

S2 − S1 − S2

15=

1

90(7AK + 32BL + 12CM + 32DN + 7EO)AE,

which is the Boole rule. Newton’s derivation

B1 = S2 − S1 − S2

15(19)

is the theorem of Huygens.Newton continued:

Case 5. In the same way, if there be seven ordinates there will come to be

1

280(17AK+54BL+51CM +36DN +51EO+54FP +17GQ)AG = ¤(AQ).5

(20)

4 Cas. 4. Si dentur 5 Ordinatæ, dic (per Cas.2)

AK + 4CM + EO

6AE = ¤AO.

ItemAK + 4BL + CM

12+

CM + 4DN + EO

12in AE = ¤AO et errores esse ut AEq ad ABq

seu 16 ad 1[,] et cum errorum differentia sit

AK − 4BL + 6CM − 4DN + EO

12AE

error minoris eritAK − 4BL + 6CM − 4DN + EO

180AE quem aufer et manebit

7AK + 32BL + 12CM + 32DN + 7EO

90AE = ¤AO.

5 Cas. 5. Eodem modo si dentur 7 Ordinatæ, fiet

17AK + 54BL + 51CM + 36DN + 51EO + 54FP + 17GQ

280AG = ¤AQ.


Case 6. While if there be given nine there will come

1

5670(217AK + 1024BL + 352CM + 1024DN + 436EO

+ 1024FP + 352GQ + 1024HR + 217IS)AI = ¤(AS).

(21)

6

The formula (20) is derived by

N2 − 1

62 − 1(N1 − N2), (22)

but N1 − I : N2 − I = 36 : 1 is incorrect. Newton derived the incorrect formula(20) from the mistaken ratio N2 − N1 : N2 − I = 35 : 1.

By Corollary 3, the exact ratio is N1 − I : N2 − I = 24 : 1 = 16 : 1. Correctedextrapolation is

N2−1

15(N1−N2) =

1

120(7AK+24BL+21CM +16DN +21EO+24FP +7GQ)AG,

(23)which was pointed out by Whiteside [18, p.695]. As Whiteside noted the rule (23)is distinct from the 7 ordinates Newton-Cotes formula.

By Corollary 4, B1 − I : B2 − I = 26 : 16 = 64 : 1. In this case Newton’s ratiois perhaps B1 − I : B2 − I = AI2 : AB2 = 64 : 1, which is correct but only by goodluck. From B2 − B1 : B2 − I = 63 : 1, Newton derived

R1 = B2 − 1

63(B1 − B2)

=1

5670(217AK + 1024BL + 352CM + 1024DN + 436EO

+ 1024FP + 352GQ + 1024HR + 217IS)AI, (24)

as will be mentioned in Section 6.

Newton concluded:

These are quadratures of the parabola which passes through the end-pointsof all the ordinates. 7

This sentence makes it apparent that Newton mistakenly thought that thenumerical integration formulae obtained by extrapolation coincide with Newton-Cotes formulae.

6 Cas. 6. Et si dentur 9, fiet

217AK + 1024BL + 352CM + 1024DN + 436EO

+1024FP + 352GQ + 1024HR + 217IS

5670AI = ¤AS.

7 Hæ sunt quadraturæ Parabolæ quæ per terminos Ordinatarum omniu transit.

12 Naoki Osada

5 Incorrect ratio of errors by Newton

Newton used incorrect ratios of errors in the single to composite Newton-Cotesformulae in Of Quadrature as follows:

Case 2 T1 − ¤(AM) : T2 − ¤(AM) = AC2 : AB2 = 4 : 1.

Case 3 T1 − ¤(AN) : T3 − ¤(AN) = AD2 : AB2 = 9 : 1.

Case 4 S1 − ¤(AO) : S2 − ¤(AO) = AE2 : AB2 = 16 : 1.

Case 5 N1 − ¤(AQ) : N2 − ¤(AQ) = AG2 : AB2 = 36 : 1.

Case 6 B1 − ¤(AS) : B2 − ¤(AS) = AI2 : AB2 = 64 : 1.

Let Qm be m panels composite Newton-Cotes formula of n ordinates for I =R ba f(x)dx. Newton seemed to believe the following incorrect ratio of error in the

single to composite Newton-Cotes formula:

Q1 − I : Qm − I ≈ (n − 1)2m2 : 1. (incorrect form)

The exact ratio is

Q1 − I : Qm − I ≈

mn+1 : 1. if n is odd,mn : 1. if n is even.

See [2, p.79] for the proof. For n > 2, the couples (n, m) that Newton’s ratios arecorrect by good luck are (n, m) = (3, 2), (4, 3), and (5, 2). The only incorrect casethat Newton used is (n, m) = (4, 2), i.e. N1 − I : N2 − I = 36 : 1.

Since “With Gregory’s Exercitationes Newton was well familiar” [18, p.693],Newton knew the ratio of errors in the single to composite trapezoidal rule. Newtonmight generalize this case, i.e. n = 2. Whiteside noted “Newton in sequel seemsto adopt the argument per analogiam that, where n + 1 equidistant ordinates aregiven, the corresponding ratio of the errors is n2 : 1, but this will fail him in ‘Cas.5”’[18, p.692].

6 Rediscovery of Newton’s integration formulae by extrapolation

After 1900, Newton’s numerical integration formulae by extrapolation have beenrediscovered.

In 1900, J.F. Sheppard [13] gave

Sm/2 =4Tm − Tm/2

3, (25)

for cases in which m is even. When m = 2, (25) coincides with Newton’s (11). Forcases in which m is a multiple of 3, Sheppard also gave

Nm/3 =9Tm − Tm/3

8. (26)

When m = 3, (26) coincides with Newton’s (16).In 1933 L.M. Milne-Thomson [7, p.199] gave an exercise “By eliminating the

error term in h4 between two Simpson formulae of 2p + 1 and 4p + 1 ordinates,obtain the Newton-Cotes’ formula for n = 5.” This exercise for p = 1 coincideswith Newton’s (19).


In 1955 W. Romberg [11] proposed the Romberg integration which is an ap-plication of Richardson extrapolation process to T1, T2, T4, T8, . . ., i.e.

Sm =4T2m − Tm

3, m = 1, 2, 4, . . . ,

Bm =16S2m − Sm

15, m = 1, 2, 4, . . . ,

Rm =64B2m − Bm

63, m = 1, 2, 4, . . . .

This process is computed from top to bottom and from left to right in the followingtriangular array:

T1

T2 S1

T4 S2 B1

T8 S4 B2 R1

.... . .

Newton’s 9 ordinates rule (24) is appeared in the fourth column. We remark thatthe fourth, fifth, ... columns are not composite Newton-Cotes formulae. See [10,p.124].

7 The second method: Determination of coefficients in Newton-Cotes

formulae

Newton proposed a method of determining the values of coefficients in Newton-Cotes formulae in the last part of Of Quadrature. Newton stated:

These quadratures may also be ascertained in this manner. Let a be thesum of the first and last ordinates, b the sum of the second and next to thelast ones, c that of the third and next but one to last, d the sum of thefourth from the beginning and the fourth from the end, e the sum of the fifthfrom the beginning and the fifth from the end, ..., m the middle ordinate

and A the abscissa, and let there beza + yb + xc + vd + te + sm

2z + 2y + 2x + 2v + 2t + sA = ¤re

required. Then in place of the ordinates write first an equal number of theopening terms of this series of square numbers 0, 1, 4, 9, 16, 25, . . .; next, asmany first terms of this one of cubes 0, 1, 8, 27, 64, 125, . . .; thereafter thesame number of this one of fourth powers 0, 1, 16, 81, 256, . . . , so proceeding(if need be) to as many series, less one, as there are unknown quantitiesz, y, x, v, . . . , and in place of the quadrature required write the quadrature ofthe parabola to which these ordinates accord. There will ensure equationsfrom which, when they are collated,z, y, x, . . . will be determined.8

8 Investigantur etiam hæ quadraturæ in hunc modum. Sit a summa Ordinatæ primæ et ul-timæ, b summa secundæ et penultimæ[,] c tertiæ & antepenultimæ[,] d summa 4tae a principioet quartæ a fine[,] [e] summa quintæ a principio et quintæ a fine &c[,] m Ordinata media etAbscissa A et sit

za + yb + xc + vd + te + sm

2z + 2y + 2x + 2v + 2t + sA = ¤æ quæsitæ,

14 Naoki Osada

For instance, if there be four ordinates, I set a = 0 + 9, b = 1 + 4, A = 3

and the quadrature = 9. In this way there comes to be9z + 5y

2z + 2y× 3 = 9

and thence y = 3z. Therefore in place of y I write 3z and the equationza + yb

2z + 2yA = ¤ becomes 1

8 (a + 3b)A = ¤, as in the third case.9

Again, if there be five ordinates, I set a = 0 + 16, b = 1 + 9, c = 4, A = 4

and ¤ = 643 , and thus there comes to be

16z + 10y + 4s

2z + 2y + s× 4 =

64

3, that is,

8z = y +2s. Again I set a = 0+64, b = 1+27, c = 8, A = 4 and ¤ = 64, and

in this way there comes64z + 28y + 8s

2z + 2y + s× 4 = 64, or 8z = y + 2s as above.

I therefore put [a = 0 + 256, b = 1 + 81, c = 16, A = 4 and the quadrature

=1024

5. And in this way there comes to be

256z + 82y + 16s

2z + 2y + s× 4 =

1024

5or 384z−88s = 51y = 408z−102s, that is, 14s = 24z or 7s = 12z and hence7y = 32z. Accordingly, in place of s and y I write 12

7 z and 327 z respectively,

and the equationza + yb + sm

2z + 2y + sA = ¤ becomes 1

90 (7a + 32b + 12m)[A] = ¤,

as in the fourth case.] 10

Newton proposed the second method under two assumptions.

1. The coefficients of the Newton-Cotes formula are symmetric about the coeffi-cient of the the central ordinate.

et pro ordinatis primo scribantur termini totidem primi hujus seriei numeroru quadratorum0.1.4, 9.16.25 &c[,] dein termini totidem primi hujus cubicorum 0.1.8.27.64.125 &c[,] deintotidem hujus quadrato-quadraticoru 0.1.16.81.256.&c pergendo (si opus est) ad tot seriesuna dempta quot sunt incognitæ quantitates z, y, x, v &c, et pro quadratura quæsita scribequadraturam Parabolæ cui hæ ordinatæ congruunt. Et provenient æquationes ex quibus collatisdeterminabuntur z, y, x &c.

9 Ut si Ordinatæ sint quatuor, pono a = 0 + 9 & b = 1 + 4.A = 3 et quadraturam= 9.

Sic fit9z + 5y

2z + 2y× 3 = 9. et inde y = 3z. Igitur pro y scribo 3z et æquatio

za + yb

2z + 2yA = ¤ fit

a + 3b

8A = ¤. ut in casu tertio.

10 Rursus si ordinatæ sint quinque pono a = 0 + 16, b = 1 + 9, c = 4.A = 4, ¤ = 643

. Et sic fit16z + 10y + 4s

2z + 2y + s× 4 =

64

3seu 8z = y + 2s. Rursus pono

a = 0 + 64, b = 1 + 27, c = 8, A = 4 et ¤ = 64.

et sic fit64z + 28y + 8s

2z + 2y + s× 4 = 64. seu 8z = y + 2s ut supra. Pono igitur [a = 0 + 256, b =

1 + 81, c = 16, A = 4 et ¤ =1024

5. Et sic fit

256z + 82y + 16s

2z + 2y + s× 4 =

1024

5

seu 384z− 88s = 51y = 408z − 102s, hoc est 14s = 24z sive 7s = 12z adeoque 7y = 32z. Igiturpro s et y scribo 12

7z et 32

7z respective et æquatio

za + yb + sm

2z + 2y + sA = ¤ fit

7a + 32b + 12m

90[A] = ¤

ut in casu quarto.]


2. The n ordinates Newton-Cotes formula is exact for 1, x2, . . . , xn−1. When n isodd, this is exact for 1, x2, . . . , xn

The first assumption is correct. See, [2, pp.77-78] for the proof. The second assump-tion follows from Proposition 2(i), 3(i), and 4(i), for n = 3, 4, and 5, respectively.For the cases in which n is greater than 5, see [2, pp.79].

Let n be a fixed odd number greater than 2. Let xi = x0 + ih (i = 0, . . . , n− 1)be equidistant abscissae yi (i = 0, . . . , n − 1) the corresponding ordinates. By theassumptions, Newton put the n ordinates Newton-Cotes formula

R

Pl−1i=0 ci(yi + yn−i−1) + clyl

2Pl−1

i=0 ci + clyl

, (27)

where ci (i = 0, . . . , l) are constants and l = (n − 1)/2. The second assumptionimplies

Pl−1i=0 ci(i

k + (n − i − 1)k) + cllk

2Pl−1

i=0 ci + cl

(n − 1) =

Z n−1

0xkdx

=(n − 1)k+1

k + 1, k = 2, . . . , l + 2. (28)

We remark that l + 2 5 n if n = 3. By solving the simultaneous equations (28),Newton represented ci (i = 1, . . . , (n − 1)/2) in c0. By substituting ci, (i > 0) to(27), Newton reduced the fraction and obtained the quadrature formula.

We now prove that the second method provides the n ordinates Newton-Cotesformula. Let

x0 =

0

B

B

B

@

10...0

1

C

C

C

A

∈ Rn, b =

0

B

B

B

B

B

B

@

113 (n − 1)214 (n − 1)3

...1

n+1 (n − 1)n

1

C

C

C

C

C

C

A

∈ Rn,

and

xi =

0

B

B

B

@

1i2

...in

1

C

C

C

A

∈ Rn, i = 1, . . . , n − 1.

Since the matrix (x0, x1, · · · , xn−1) is nonsingular, there is the unique solution(d0, . . . , dn−1) of the equation

d0x0 + · · · + dn−1xn−1 = b. (29)

The equation (29) is equivalent to

n−1X

i=0

di = 1,

n−1X

i=1

diik =

1

k + 1(n − 1)k, k = 2, . . . , n.

16 Naoki Osada

By Proposition 5 which we will cite in the end of this section,

R

n−1X

i=0

diyi

is the n ordinates Newton-Cotes formula. By the uniqueness and the symmetry ofthe coefficients, we have

d0 6= 0,

di = dn−1−i, i = 0, . . . , l − 1.

Take any nonzero number c0. Put

ci =c0d0

di, i = 1, . . . , n − 1.

Then for k = 2, . . . , n,

Pn−1i=1 cii

k

Pn−1i=0 ci

=

Pn−1i=1 dii

k

Pn−1i=0 di

=1

k + 1(n − 1)k,

and ci = cn−i−1 (i = 0, . . . , l − 1). By Proposition 5,

R

Pni=0 ciyiPn−1

i=0 ci

is the n ordinates Newton-Cotes formula. Therefore the second method providesthe n ordinates Newton-Cotes formula.

When n is even, Newton did not explain in general, but he gave the fourordinates rule, i.e. the Simpson 3/8 rule. In this example we can see the secondmethod. Starting from

R

Pl−1i=0 ci(yi + yn−i−1)

2Pl−1

i=0 ci

with l = n/2 and solving

Pl−1i=0 ci(i

k + (n − i − 1)k)

2Pl−1

i=0 ci

=(n − 1)k

k + 1, k = 2, . . . , l + 1.

For the rest of these the determination of the coefficients is similar to the odd case.

Although Newton wrote “These quadratures may also be ascertained in thismanner”, the Newton-Cotes formulae can be determined by the second method.

In the 20th century Newton’s idea has been formulated as the following Propo-sition 5 which is used for determining the integration formulae of interpolatorytype, such as Gaussian integration.

Proposition 5 Let x0, . . . , xn−1 be fixed and distinct points lying between a andb prescribed in advance. The two ways of determining w0, . . . , wn−1, (i) and (ii)yield the same numbers.


(i) Interpolate to the function f(x) at the points x0, . . . , xn−1 by a polynomial ofdegree at most n−1. Then integrate the interpolation polynomial, and expressthe result in the form

Z b

af(x)dx ≈

n−1X

i=0

wif(xi).

(ii) Select the constants w0, . . . , wn−1 so that

Z b

af(x)dx =

n−1X

i=0

wif(xi),

for f(x) = 1, x, . . . , xn−1.

Proof. See [2, p.74].

8 Conclusion

In Of Quadrature by Ordinates Newton tried two methods for obtaining the Newton-Cotes formulae. The first method is extrapolation and the second one is the methodof undetermined coefficients using the quadrature of monomials.

The first method provides n ordinates Newton-Cotes formulae only for cases inwhich n = 3, 4 and 5. However this method provides another important formulaeif the ratios of errors are corrected. Newton’s numerical quadrature by extrapola-tion,i.e. the first method, is not systematic but can be said to be a germ of theRomberg integration.

The second method is proposed under two assumptions. We proved that twoassumptions are correct and that the second method provides the Newton-Cotesformulae. In the 20th century the second method has been formulated and usedfor determining the integration formulae of interpolatory type, such as Gaussianintegration.

References

1. Florian Cajori, Sir Isaac Newton’s Mathematical Principles, Of Natural Philosophy andhis system of the world, University of California Press, 1947.

2. Philip Davis and Philip Rabinowitz, Methods of Numerical Integration, 2nd. ed. AcademicPress 1984.

3. Duncan C. Fraser, Newton’s Interpolation Formulas, London, 1927, Reprinted in Bibliolife.4. James Gregory, Exercitationes geometricae, 1668. http://books.google.co.uk/5. Christian Huygens, De Circuli Magnitudine Inventa. Lugd. Batavia. (1654), Kessinger

Legacy Reprints (2011)6. R.M. Milne, Extension of Huygens’ approximation to a circle arc, Mathematical Gazette

2(1903), 309-311.7. L.M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, 1933.8. Isaac Newton, Methodus Differentialis, 1711, in Isaaci Newtoni, Equitis Aurati, Opuscula

Mathematica, philosophica et philologica, Collegit partımque Latine vertit ac recensuitJon. Castillioneus Jurisconsultus, Lausannæ & Geneva, Apud Marcum-Michaelem Bous-quet, 1744.

9. Naoki Osada, The early history of convergence acceleration methods, Numerical Algo-rithms, Vol.60(2012), 205-221, DOI 10.1007/s11075-012-9539-0

10. Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 2nd. ed.,McGraw-Hill, 1978.

18 Naoki Osada

11. Werner Romberg, Vereinfachte numerishe Integration, Det Kongelige Norske VidenskabersSelskabs Forhandlinger, Bind 28(1955), 30-36.

12. Ferdinand Rudio, Archimedes,Huygens,Lambert,Legendre. Vier Abhandlungen Uber dieKreismessung, Teubner, 1892.

13. W.F. Sheppard, Some quadrature formulae, Proc. London Math. Soc. 32(1900), 258-277.14. Herbert Westren Turnbull, The Correspondences of Isaac Newton, vol. 1. Cambridge Uni-

versity Press, 1959.15. Herbert Westren Turnbull, The Correspondences of Isaac Newton, vol. 2. Cambridge Uni-

versity Press, 1960.16. Derek T. Whiteside, Patterns of Mathematical Thought in the later seventeenth Century,

Archive Hist. Exact Sci., 1(1961), 179-388.17. Derek T. Whiteside, The Mathematical Papers of Isaac Newton, Vol. IV, Cambridge Uni-

versity Press, 1971.18. Derek T. Whiteside, The Mathematical Papers of Isaac Newton, Vol. VII, Cambridge

University Press, 1976.

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Isaac Newton’s ‘Of Quadrature by Ordinates’lab.twcu.ac.jp/osada/of_quadrature_rev2.pdf · Isaac Newton’s ‘Of Quadrature by Ordinates’ 3 D B A M Fig. 1 Theorem XVI in De