1. The Mathematics Education SECTION B
!'ol. VI, No. 4, Dec. 1973
0[ l M P $ n S 0F A NCInNT INnIAN MAIH. No,4
Bralrrna$upta's Bule For TIre Volrrme Of
Frusturn Llke Elollds
Dr, 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 versesbut 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 titles Sindhind and Al-srkand respectively.
He further adds : ,,Both these works have been largely used and have exerciseda great
influence. ft rvas on this occasion ( of translation ) that the Arabs first became acquainted
with a scientific s)'stemof astronorny......( Thus ) Brahrnagupta holds a remarkable place in
the history of Eastern civilisation. It rvas he who taught the Arabs astronomy before they
became acquainted with Ptolemy (the great Greek astronomer )".
The twelfth chapter of BSS entitled 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 r
gqdo rrFl+{qrf tugui wq qfuaq}q ttv{rr
qtr .rfqrdr(
fftrleq aq-dqRs?i hfu: tu1 r
rriq
o6E -"qq€Rqi cf{cq |r{fd sit {elrq ilvqtt
Mukhatalyutidalaganitarh vedhaguqarh
vydvahirikarh ga4itam /

2. ll8 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 aritihmetic 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
,-(+)(:+)-^
and c:(Lb+;:!'). h
Thus by using Brahmagupta's fornrula (l) the vo'ume of the truncated wedge will be given by
V : { ab* a ' b * (a } a ' ) (b + b ' )} . (h l 6 ) ......(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 1040A. 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
l/ * a' b (2a'} a)b'l, : (h| 6) ......(5)
- {(.2a ) |

3. R. C. Gupta ll9
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') - u) t,h
(a (b ......(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 a
truncated rvedge ), the formula (6) rvill reduce to
,- 1(t{)'+(r/g) .n
,.';'->' ......(7)
knorvn to the Babl'lonians of very romote times.tr
This r,r'as The Babylonians .(about 2000
B. C. ) also used the approxinrate formulas
. ,L
P_(,.Lt').
h . . . . . . (B )
and
c-(xE, .n ......(e)
for 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
[ / - (o 2 ]a a ' ]a ' r). (h l l ) ......(10)
for the volume of the frustum of a pyramid. This formula is called a masterpiece of
Egyptain Geometryrs.
In the case of the frustum of a circular cone, we shall have according to the defini-
tions of Brahmagupta,
P- -/R+rr '
u
2-l'o
and
Q- (TR'+T1'' n
--
- z )"'
where R and r are the radii of the two ends. So that Brahmagupta's rule (t) will give its
volume as
V = (& 2 !R r* rz ). F rh l S)
which is also mathematically correct.

4. 120 The Mathematics Education
An elegant generalization of Brahmagupta's rule is given by Mahnvira ( about 850
A. D.) in his Ganitasira Saigraha, VI[, 9-12 where he asks us to take as many sectionsof
the solid as we like instead of just nvo extreme sectionsra. Ir{ahivira hirnself given numeri-
cal examples u'hen the sectionsare squares, rectangles, circles and triangles taking upto
three sectionsin sorne cases.
Before concluding, it nia;, be pointec.l
out that the interpretation of Brahmagupta's
rule as given by L. V. Gur.jart 5 is wrong and unnecessarl'. The same 4'rong interpretation
is later on given by Dr. B. Moharr. I o
Referencer
1. SeeA concise trIittory of Sciencein India eclited D. M. Bose and others, Indian
by
National Acaderny,Nerv Delhi, lg7l, p. 166.
2. Siddhantr Sircmani edited by Bapudeva Sastri, Chowkhamba Sanskrit Series Office,
Benares, p.
1929, 2.
3. Brihmaspbula Siddhant editedby R. S. Sharmaand his team in 4 volumes, fnstitute of
Astronomical and Sanskrit Reseatch,New Delhi, 1966; Volume I, p 320of the text and
volume IV, p. 1550. All references BSSare according this edition.
to to
4. Alberuni'e India translated by E. C. Sachau, Indian edition ( 2 volumesin one ), S.
Chandaand Co., Delhi, 1964, Volume I, p. 155.
5. Alberuni'e India Op. Cit., Preface, XXXV and volume II, Annotations, 304.
p. P.
6. BSS,Neu' Delhi edition, 1966, Volume III, pp. 874-875,
7. B. B. I)atta: Scicneeof the lulba, CrlcuttaUniversitl,, Calcurta,1932, 103.
p.
B. Y. Nlikarni: llevelcpment of htathematicsin China aud Japanr Chelsea Publishing
Co.,Neiv York, 1961' 16.
P.
('On the supposed
9. B. B. Datla : Indebtness Brahrnagupta Chiu-ciraog Suan-ohu",
of to
Bull. Cal NIath.Soc..Vc,l.XII (1930), 39-51.
pp.
10. T. L Heath : A Manrral of Greek 5{athemetics,Reprintei!,Dover Publications, New
York, 1963, p.427.
ll. c. B. Bo-ver:a History of ll{arhematics.John 1Viley, New york, 1968, p.42.
12. H. Mirlonick : l'he freasury of Mathematics, 2 Volumes,PenguinBooks1968;Volume
l, P. 77.
13. G.Sarton: {.ncientSciencethrotrghtheGctdenAgeof Greece, HarvardUniversity
Press, Cambridge, Mass., 1959, 40.
p.
14. R. C. Gupta : "Soine fmportant Indian Mathematical Methods as conceivedin the
Sanskrit Language." Paperpresentedat the International Sanskrit Conference, New
Delhi, March 1972, pp. l0-ll.
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.