The Mathematics Education

! 'ol. VI, No. 4, Dec. 1973SECTION B

0[ lMP$nS 0F ANCInNT INnIAN MAIH. No,4Bralrrna$upta's Bule For TIre Volrrme Of

Frusturn Llke ElolldsDr, R. C. Glupte, Atsistant Profcssor of Malhcnatics Birla Inilitutc of Tachnologl

Mctra, Ranchi (Bihar )

George Sartonrl a great historian of science, described Brahmagupta as 'one of the

greatest scientists of his race and the greatest of his time'. The famous Bhdskara II (about

I150 A. D.) called Brahmagupta 'jewel among the mathematicians, (ganakacakrdcfldamani)2.

It is well known that Brahmagupta (born 598 A. D.) wrote his voluminous l:rdhmas.

phuta Sirldh6nta (:BSS) in the year 628 A. D. and the Khanda Khadyaka ( =KK ) in the

year 665 A. D. According to BSS, XXIV, 9, Brahrnagupta also composed a snrall tract called

the Dbyina Graha in 72 verses but the author did not include this in his BSS of 24 chapterss.

Alberuni ( about 1030 A. D. ) '!, however, regarded it as the 25th chapter of BSS. No other

work of BrahmaguPta is known.

According to E. C. Sachaus, the BSS and KK were translated into Arabic at Baghdad

as early as the eighth cerrtury A. D. under the tit les Sindhind and Al-srkand respectively.

He further adds : ,,Both these works have been largely used and have exercised a great

influence. ft rvas on this occasion ( of translation ) that the Arabs first became acquainted

with a scientif ic s)'stem of astronorny......( Thus ) Brahrnagupta holds a remarkable place in

the history of Eastern civil isation. It rvas he who taught the Arabs astronomy before they

became acquainted with Ptolemy (the great Greek astronomer )".

The twelfth chapter of BSS entit led Ganitadhydya is devoted to elementary mathe-

matics. Verses 45 and 46 of this chapter give a general method of calculating the volume of

a frustum like solid whose upper and lorver ends ( or sections ) are parallel and of similar

shape ( and similarly situated ). The commonly accepted Sanskrit text of BSS, XII, 4546,

may be taken aso

gqildrgfu{q{rFrd +{TJi aqr€rFTd' qFieq rgqdo rrFl+{qrf tugui wq qfuaq}q ttv{rrqtr .rfqrdr( fftrleq aq-dqRs?i rriq hfu: tu1 ro6E -"qq€Rqi cf{cq |r{fd sit {elrq ilvqtt

Mukhatalyutidalaganitarh vedhaguqarh

vydvahirikarh ga4itam /

l l8 The Mathematio Education

Mukhatalaganitaikyardham vedhagu4am

syid gar.ritamauffam | | 45 | |

Autraganltld viSodhya vyavahdraphalani

bhajet tribhib 5e9am /

Labdham vyavahiraphale praksiphy

bhavati Phalam snkqmam //46//

.The area computed from half the sum of the (linear dimensions of the) top and bottorn

multiplied by the depth ( or height ) ir the Practical Volutne. Half the sum of the areas of

the top and bottom multiplied by the depth ( or height ) become the Autra ( Gross ? ) Vo-

lume. From the Autra Volume subtract the Practical Volume and divide the remainder by

three. The quotient ( so obtained ) added to the Practical Volume becomes the accurate

volume ( of the pit or solid ). '

, Let P be the Practical Volurne which is to be calculated by multiplying the height by

the area of a similar section whose linear dimensions are the arit ihmetic means of the corres-

ponding linear dimensions of the top and bottom sections of the solid. Let G be the Gross

(or Austra) Volume which is to be calculated by multiplying the height by the area which is

the arithmetic mean of the areas of the top and bottom sections of the solid. Then, accord-

ing to Brahmagupta's rule, we have the accurate volume Z to be given by

V -P* (G-P)13=(2P +G)13 . . . . . . ( l )

In the case of a frustum of a wedge, let a and 6 be the sides of the rectangular top

section, a' and D'be the corresponding sides of the rectangular bottom section and i be the

height. Here wc shall have,-(+)(:+)-^c:(Lb+;:! '). h

Thus by using Brahmagupta's fornrula (l) the vo'ume of the truncated wedge wil l be given by

V:{ab*a'b *(a}a ' ) (b+b')} . (h l6) . . . . . . (4)

which is mathematicallY correct.

According to B. B. DattaT, a rule eguivalent to the formula (2) was used qy the

authors of the Sulba Sirtras for getting the approximate volume of the truncated wedgc about

a thousand years before Brahmagupta.

Brahmagupta's formula in the reduced form (4) is found in many subsequent Indian

works such as those of Aryabhata II ( about 950 A. D. ) Sripati ( about 1040 A. D. ) and

Bhdskara II. The Chinese mathematical classic Chiu-chan! Suan-shu, which was composed

originally by Cl ang T'sang ( died 152 B. C. ), contains the formula (4) is the following form8

and

l/ - {(.2a * a' ) b | (2a' } a) b' l, : (h | 6) . . . . . . (5)

R. C. Gupta l l9

But, since Brahmagupta's original rule (l) is more general from lvhich (4) has been derived

as an illustrative example by applying (l) to a particular case, it is difficult to believe that

Brahmagupta got his rule from the Chinese source. On this point readers may refer to a

detailed paper of Dr. B. B. Dattae.

Heron of Alexandria ( betrveen 150 B. C. and 250 A. D. ) gave the formula for the

volume of the truncated rvedge aslo.

v: (, ( "\n ) t'-l#) * e tt2) (a - a') (b - u) t, h . . . . . . (6)

Out of (4), (5) and (6), the Heron's form is nearsst to Brahmagupta's rule (l).

For the frustum of a p;,ramid with square base ( which is just a particular case of atruncated rvedge ), the formula (6) rvill reduce to

,- 1(t{) '+(r/g) ,. ' ; '->' \.n. . . . . . (7)

This r,r 'as knorvn to the Babl' lonians of very romote times.tr The Babylonians .(about 2000

B. C. ) also used the approxinrate formulas. ,L

P _(,.Lt'). h . . . . . . (B)

. . . . . .(e)and

c-(xE, \ .nfor the volume of a truncated pyramid with sguare base.

The Moscow Papyrus (about lB50 B. C.), a manual of ancient Egyptian mathematics,is also reported to have used the eguivalent of the formular2

[ / - (o2]aa' ]a ' r ) . (h l l ) . . . . . . (10)

for the volume of the frustum of a pyramid. This formula is called a masterpiece ofEgyptain Geometryrs.

In the case of the frustum of a circular cone, we shall have according to the defini-tions of Brahmagupta,

P- - /R+r\r 'u\ 2- l 'oand

Q- (TR'+T1'\ ' n\- z -- )"'where R and r are the radii of the two ends. So that Brahmagupta's rule (t) wil l give itsvolume as

V=(&2!Rr*rz) . FrhlS)

which is also mathematically correct.

120 The Mathematics Education

An elegant generalization of Brahmagupta's rule is given by Mahnvira ( about 850A. D.) in his Ganitasira Saigraha, VI[, 9-12 where he asks us to take as many sections of

the solid as we like instead of just nvo extreme sectionsra. Ir{ahivira hirnself given numeri-cal examples u'hen the sections are squares, rectangles, circles and triangles taking uptothree sections in sorne cases.

Before concluding, it nia;, be pointec.l out that therule as given by L. V. Gur.jart 5 is wrong and unnecessarl '.is later on given by Dr. B. Moharr. I o

interpretation of Brahmagupta'sThe same \4'rong interpretation

Referencer

1. See A concise trIittory of Science in India eclited by D. M. Bose and others, IndianNational Acaderny, Nerv Delhi, lg7l, p. 166.

2. Siddhantr Sircmani edited by Bapudeva Sastri, Chowkhamba Sanskrit Series Office,Benares, 1929, p. 2.

3. Brihmaspbula Siddhant edited by R. S. Sharma and his team in 4 volumes, fnstitute ofAstronomical and Sanskrit Reseatch, New Delhi, 1966; Volume I, p 320 of the text andvolume IV, p. 1550. All references to BSS are according to this edition.

4. Alberuni'e India translated by E. C. Sachau, Indian edition ( 2 volumes in one ), S.Chanda and Co., Delhi, 1964, Volume I, p. 155.

5. Alberuni'e India Op. Cit., Preface, p. XXXV and volume II, Annotations, P. 304.

6. BSS, Neu' Delhi edition, 1966, Volume III, pp. 874-875,

7. B. B. I )at ta: Scicnee of the lulba, Crlcutta Universi t l , , Calcurta, 1932, p. 103.

B. Y. Nl ikarni : l levelcpment of htathematics in China aud Japanr Chelsea Publ ishingCo., Neiv York, 1961' P. 16.

9. B. B. Datla : ('On the supposed Indebtness of Brahrnagupta to Chiu-ciraog Suan-ohu",Bul l . Cal NIath. Soc.. Vc, l . XII (1930), pp. 39-51.

10. T. L Heath : A Manrral of Greek 5{athemetics, Reprintei!, Dover Publications, NewYork, 1963, p.427.

l l . c. B. Bo-ver: a History of l l {arhematics. John 1Vi ley, New york, 1968, p.42.12. H. Mirlonick : l 'he freasury of Mathematics, 2 Volumes, Penguin Books 1968; Volume

l, P. 77.13. G.Sarton: { .ncientSciencethrotrghtheGctdenAgeof Greece, HarvardUniversi ty

Press, Cambridge, Mass., 1959, p. 40.14. R. C. Gupta : "Soine fmportant Indian Mathematical Methods as conceived in the

Sanskrit Language." Paper presented at the International Sanskrit Conference, NewDelhi , March 1972, pp. l0- l l .

15. L. V. Gurjar : Ancient rndian Mathematics and Vedha, poona, 1947, pp. Bg_Bg.16. B. Mohan : Hirtory of Mathematic. ( in llindi ), Lucknorv, 1965, p. 276.