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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$uptas 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 thegreatest scientists of his race and the greatest of his time. The famous Bhdskara II (aboutI150 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 theyear 665 A. D. According to BSS, XXIV, 9, Brahrnagupta also composed a snrall tract calledthe 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 otherwork of BrahmaguPta is known. According to E. C. Sachaus,the BSS and KK were translated into Arabic at Baghdadas 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 greatinfluence. ft rvas on this occasion ( of translation ) that the Arabs first became acquaintedwith a scientific s)stemof astronorny......( Thus ) Brahrnagupta holds a remarkable place inthe history of Eastern civilisation. It rvas he who taught the Arabs astronomy before theybecame 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 ofa frustum like solid whose upper and lorver ends ( or sections) are parallel and of similarshape ( 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 /
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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 bottornmultiplied by the depth ( or height ) ir the Practical Volutne. Half the sum of the areas ofthe 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 bythree. The quotient ( so obtained ) added to the Practical Volume becomes the accuratevolume ( of the pit or solid ). , Let P be the Practical Volurne which is to be calculated by multiplying the height bythe 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 isthe arithmetic mean of the areas of the top and bottom sections of the solid. Then, accord-ing to Brahmaguptas 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 topsection, a and Dbe the corresponding sides of the rectangular bottom section and i be theheight. Here wc shall have ,-(+)(:+)-^and c:(Lb+;:!). hThus by using Brahmaguptas fornrula (l) the voume 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 theauthors of the Sulba Sirtras for getting the approximate volume of the truncated wedgc abouta thousand years before Brahmagupta. Brahmaguptas formula in the reduced form (4) is found in many subsequent Indianworks such as those of Aryabhata II ( about 950 A. D. ) Sripati ( about 1040A. D. ) andBhdskara II. The Chinese mathematical classic Chiu-chan! Suan-shu, which was composedoriginally by Cl ang Tsang ( died 152 B. C. ), contains the formula (4) is the following form8 l/ * a b (2a} a)bl, : (h| 6) ......(5) - {(.2a ) |
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R. C. Gupta ll9But, since Brahmaguptas original rule (l) is more general from lvhich (4) has been derivedas an illustrative example by applying (l) to a particular case, it is difficult to believe thatBrahmagupta got his rule from the Chinese source. On this point readers may refer to adetailed paper of Dr. B. B. Dattae. Heron of Alexandria ( betrveen 150 B. C. and 250 A. D. ) gave the formula for thevolume 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 Herons form is nearsst to Brahmaguptas 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) knorvn to the Babllonians of very romote times.trThis r,ras The Babylonians .(about 2000B. 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 ofEgyptain 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-lo and Q- (TR+T1 n -- - z )" where R and r are the radii of the two ends. So that Brahmaguptas rule (t) will give its volume as V = (& 2 !R r* rz ). F rh l S) which is also mathematically correct.
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120 The Mathematics Education An elegant generalization of Brahmaguptas rule is given by Mahnvira ( about 850A. D.) in his Ganitasira Saigraha, VI[, 9-12 where he asks us to take as many sectionsofthe solid as we like instead of just nvo extreme sectionsra. Ir{ahivira hirnself given numeri-cal examples uhen the sectionsare squares, rectangles, circles and triangles taking uptothree sectionsin sorne cases. Before concluding, it nia;, be pointec.l out that the interpretation of Brahmaguptasrule as given by L. V. Gur.jart 5 is wrong and unnecessarl. The same 4rong interpretationis later on given by Dr. B. Moharr. I o Referencer1. 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 to4. Alberunie India translated by E. C. Sachau, Indian edition ( 2 volumesin one ), S. Chandaand Co., Delhi, 1964, Volume I, p. 155.5. Alberunie 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 supposed9. 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 : lhe 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.
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