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Mathematics in Ancient and Medieval India c7.s2: Progressive Series

Mathematics in Ancient and Medieval India c7.s2

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Progressive Series

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The usual term for the series is *śreḍhī*. This term is used in almost all mathematical literatures from 5th century A. D. onwards. The other terms used in the *Bakhshālī Ms.*² are *saṃkalita*, *varga*³, *rupona*⁴. Another peculiar term found in the *Bakhshālī Ms.* is *pārtha*. In the *Bṛhaddhārā*⁵, the term *dhārā* ( ladder ) is used. The initial term in an A. P. is *ādi*, *mukha*, *badan* or *baktra* all meaning face. *Antya* or *anta* used to denote the last term; *caya*, *pracaya* or *uttara* stands for the common difference, *gaccha* and *pada* denotes the number of terms, the middle term is *madhya*; the sum of progression is *sarvadhana*, *śreḍhīphala*, *guṇita* or *citi*.

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² Folio 23 Recto; folio 4 verso etc.

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³ Here it means ‘group’.

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⁴ Hoernle ( *Indian Antiquary*, 17, p. 47 ) and Kaye ( vide his edition of *Bakhshālī Ms.*, p. 73 ) think that the origin of the term lies in the fact that ‘the rule in question began with the term *rupona* which corresponds to the ( t-1 ) of the formula’.

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⁵ The *Bṛhaddhārā-Parikarma*, a treatise on series is extant now. Certain portion of the work appears in the *Trilokasāra* of Nemicandra.

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The term used for the sum of G. P. in the *Trilokasāra*⁶ ( 978 A. D. ) is *guṇa-gaṇita* or *guṇa-saṃkalita*. The terms *śreḍhī*, *ādi*, *gaccha*, *uttara*, *guṇita* were standardised long ago in the Jain canons (2nd century A. D. ).

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⁶ Rule 231,

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In Indian treatises, three types of series occur viz. arithmetic, complex and geometric series. The classification appeared late in the 9th century A. D. in the *Gaṇitasārasaṃgraha* of Mahāvīra (850 A. D. ).

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*Arithmetic Progression*: Indian interest in the series began quite early in the Vedic age. A large number of Vedic series are mentioned in the *Taittirīya Saṃhitā*¹, *Vājasaneya Saṃhitā*² and *Pañcaviṃsa Brāhmaṇa*³. Some of the few such series are found in the *Bṛhaddevatā*⁴, *Śatapatha Brāhmaṇa*⁵, *Baudhāyana Śulba*⁶, Buddhist work *Dīgha Nikāya*⁷ and a Jain work *Antagoḍa Dasāo*⁸. The general rule for the summation of an A. P. : S = a + (a + b) + (a + 2b) + + [a + (n - 1)b] where S denotes the sum, a = first term, b = common difference and n = number of terms, is found in the *Bakhshālī Ms* ⁹ as follows: S = n [a + (n - 1) b/2 ]. Āryabhaṭa I¹⁰ (476 A. D. ) gave the above form as well as the form : S = n/2 (a + b). The same rule appears in the works of Brahmagupta¹ (628 A. D.), Mahāvīra² (850 A. D.), Śrīdhara³, Nemicandra⁴ and Bhāskara II⁵ (1150 A. D.). Āryabhaṭa I⁶ and all the above writers with the exception of Nemicandra gave also the sum of the following particular case :

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¹ 7.2. 11–17.

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² 18. 24.25.

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³ 18.3.

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⁴ Vide Macdonall's English translation, 1904.

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⁵ 10.5.4.7.

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⁶ Datta, B. *The Science of the Śulba*, Calcutta University, p. 217-18, 1932.

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⁷ David, T. W. R. *Dialogue of the Buddha*, 3, p. 70–72, 1928.

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⁸ Ed. by L. D. Barnett, p. 102-6, 1907.

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⁹ *Bulletin of the Calcutta Mathematical Society*, 21, p. 1–60.

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¹⁰ *Āryabhaṭīya-gaṇita*, v. 19.

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¹ Colebrooke, H. T, *Hindu Algebra*, p. 290.

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³ *Triśatikā*, Sudhākara's edition, rule 39.

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⁴ *Trilokasāra*, rule 164.

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⁵ Colebrooke's *Hindu Algebra*, p. 53.

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⁶ *Āryabhaṭīya-gaṇita*, v. 22; *Brāhmasphuṭasiddhānta* of Brahmagupta, ch. 12, v. 20.

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$1+2+3+......+n = n/2(n+1).$

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Āryabhaṭa I calculated the value of unknown number of terms n for an A. P. series when its sum S, first term a, and commn difference b, are known with the help of quadratic equation as follows :

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$n = ½[(√(8bS+(2a-b)²)-2a)/b+1]$

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² *Gaṇitasārasaṃgraha*, ch. 2, sl. 61, 62.

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The same was also given by Brahmagupta⁷ (628 A. D.), Mahāvīra⁸ (850 A. D.) and Bhāskara II⁹ (1150 A. D.).

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⁷ Vide Colebrooke's *Hindu Algebra*, p. 291.

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⁸ *Gaṇitasārasaṃgraha*, ch. 2, v. 69.

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⁹ *Līlāvatī*, rule 12 vide H. C. Banerjee's edition .

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*Complex series* : (a) Āryabhaṭa I¹⁰ (499 A. D.), Brahmagupta¹¹, Mahāvīra¹² and also Bhāskara II¹³ gave the summation of the series of squares and cubes of n natural numbers as follows :

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¹⁰ *Āryabhaṭīya-gaṇita*, v. 22, vide Clark's ed.

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¹¹ Vide Colebrooke's *Hindu Algebra*, p. 291.

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¹² *Gaṇitasārasaṃgraha*, ch. 6, v. 296, 301.

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¹³ Vide Colebrooke's *Hindu Algebra*, p. 53.

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$1² + 2² + 3² + ...... + n² = (n (n+1) (2n+1))/6$

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$1³ + 2³ + 3³ + ..... + n³ = [(n (n+1))/2]²$

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The formula for the sum of the cubes of the n natural numbers appears in *Codex Arcerianus*¹ (6th century A.D.). The Arabic Scholar āl-Karkhī² (c. 1020 A.D.) gave and supplied proofs to the summation to the above two series.

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¹ Smith, D. E. *History of Hindu Mathematics*, 2, p. 504, Dover Publication.

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Reference may be made here that the Jain mathematician Mahāvīra (850 A.D.) also gave the results of the generalised (algebraic) summation of the square and cube of the series a, a+b, a+2b,......corresponding to Āryabhaṭa I's work in natural number as follows : $a² + (a+b)² + (a+2b)² + .........to n terms$ $= n [((2n-1) b²)/b + ab (n-1) + a²]$ and a³ + (a+b)³ + (a+2b)³ + ......to n terms $= S²b + Sa (a-b) if a > b$ $= S²b - Sa (a-b) if a < b$ where S = a + (a+b) + (a+2b) + ......to n terms.

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² Cajori, F. *A History of Mathematics*, 2nd revised enlarged edition, p. 100.

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(b) Āryabhaṭa I³ (476 A.D.) gave the summation of the sums of n times the 1st term of n natural numbers, (n-1) times the 2nd term of n natural numbers, (n-2) times the 3rd term of natural numbers……etc. Thus if \begin{align*} S_1 &= n.1 \\ S_2 &= (n-1).2 \\ S_3 &= (n-2).3 \\ & \dots\dots\dots\dots\dots\dots \\ & \dots\dots\dots\dots\dots\dots \\ S_{n} &= [ n-(n-1) ].n \end{align*} then according to Āryabhaṭa I, \begin{align*} & S_1 + S*2 + \dots \dots + S*{n} \\ = \sum_{n=1}n S_{n} & \\ = n.1 + (n-1).2 & + (n-2).3 + \dots\dots + \\ & [n-(n-1)].n \\ = 1 + (1+2) & + (1+2+3) + \dots + (1+2+\dots + n) \\ = (n(n+1)(n+2))/6 & \end{align*}

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³ *Āryabhaṭīya-gaṇita*, v. 21.

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Nārāyaṇa ( 1356 A. D. ) calls this as repeated sum or *Vārasaṁkalita*. According to Nārāyaṇa,

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*Vārasaṁkalita* of 1st order of n natural numbers \begin{align*} &= 1 + 2 + 3 + \dots\dots + n \\ &= (n(n+1))/2 \end{align*}

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*Vārasaṁkalita* of 2nd order of n natural numbers \begin{align*} &= 1 + (1+2) + (1+2+3) + \dots n terms \\ &= \Sigma \ (n(n+1))/2 = (n(n+1)\ (n+2))/6 \end{align*}

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Āryabhaṭa I gave the result on'y upto *Vārasaṁkalita* of 2nd order, but Nārāyaṇa proceeded further.

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His result for the *Vārasaṃkalita* of rth order of a natural numbers¹ is :

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¹ *ekādhikavāramitāḥ padādirupottarāḥ pṛthaktem'śāḥ ।* *ekādyekacayahaśāstaddhāto vārasaṃkalitam ॥* ( *Gaṇitakaumudī*, Part II, p. 411 ). "The number beginning with the number of terms in the series increasing by one and equal in number to one more than the number representing the order of summation separately form the numerators. The corresponding denominators are the natural number beginning with the product of these ( fractions ) is the *Vārasaṃkalita*".

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$(n(n + 1) (n + 2) (n + r))/(1.².³.4 (r + 1 ))$ $= (n(n + 1) (n + 2) (n + r))/((r + 1) !)$

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In a similar manner Mahāvīra² ( 850 A. D. ) also has given an incorrect result of *Vārasaṃkalita* of 2nd order using the series a, a + b, a + 2b, in place of the series in n natural numbers. The result is

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$n/2 [ ((2n-1)b²)/6 + b/2 + ab ](n + 1) + a(a + 1)$

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² *Gaṇitasārasaṃgraha*, ch. 6, rule 305½.

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The correct result is (n(n + 1))/2 [ a + (b(n - 1))/3 ]

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Nārāyaṇa³ calculated the *Vārasaṃkalita* of rth order by employing the general algebraic progressive series like a, a + b, a + 2b. The given result is :

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³ *Gaṇitakaumudī*, Part I', p. 124. Vide editor's note. For edition, see *General Bibliography*.

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$a. n-1S_r. (r+1)/(n-1) + b. n-1S_r$ when a, b and n are the 1st term, common difference and number of terms of an A. P. Nārāyaṇa represented the parallel sides of a trapezia by a-b/2 and nb+a-b/2 and the altitude by n. Arithmetic Progression with sum=0 are diagrammatically represented by joining two equal isosceles inverted triangles with their vertices.

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( c ) In the *Bakhshālī Ms.*¹ certain elementary cases of complex series without sum are mentioned. These series are classified into two types viz. ( i ) *yutivarga-krama* and ( ii ) *yutagaṇita-yutakrama*. Mahāvīra² ( 850 A. D. ) gave the sum of the following complex series which may be considered as one of his highest contribution in this branch : $a+(ar)+[ (ar)r ]+$ $[ (ar)r r ]+ n terms$ $=S ±(((S′)/a-n )m)/(r-1)--$ $where S=a+ar+ar²+ n terms.$

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*Geometrical Progression* : Earliest expression of an idea representing G. P. occurs in the *Chandaḥ-sūtra*³ ( rule of metres ) of Piṅgala ( c. 200 B. C. ) where the series : 1, 2¹, 2², 2³...is found. Mahāvīra⁴ ( 850

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¹ Datta, B. 'The Bakhshālī Mathematics', *Bulletin of the Calcutta Mathematical Society*, 21, p. 30-31, 1929.

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² *Gaṇitasārasaṃgraha*, ch. 2, rule 314.

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³ *Chandaḥ-sūtra* of Piṅgala, rules 8, 28, 29, 30, 31, 32.

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⁴ *Gaṇitasārasaṃgraha*, ch. 2, rules 94 and 93.

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A. D. ) gives the generalised result of the sum of the n terms of the G. P. as follows : $a + ar + ar² + ...... to n terms = (a(rn - 1))/(r - 1)$

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This formula reappears also in the works of Pṛthudakasvāmī¹ ( 864 A.D. ) and Nemicandra² (978 A. D.).

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¹ Colebrooke, *Hindu Algebra*. p. 291.

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² *Trilokasāra*, rule 231.

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Mahāvīra gave also the form S = (rarn-1 - a)/(r-1)

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The rule appears amongst the Arabians in the work of āl-Birūnī's *India* in connection with a chess-board problem which he might have learnt during his tour in India³.

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³ Smith, D. E. *Ibid.*, 2, p. 502, Dover Publication.

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From the above, Mahāvīra deduced three expressions⁴ also for finding the common ratio (r), first term ( a ) and number of terms (n).

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⁴ *Gaṇitasārasaṃgraha*, ch. 2, rules 97, 98 and 101.

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