Indian Journal of History of Science, 52.1 (2017) 17-27 DOI: 10.16943/ijhs/2017v52i1/41297

Ideas of of Bhāskarācārya in Līlāvatī and Siddhāntaśiromanị

A B Padmanabha Rao∗ (Received 11 June 2016)

Abstract The core concept of is motivated by the dynamic examples of astronomy dealing with instanta- neous velocities of planets. The first attempt at formalisation of these ideas was made during the periods of Bhāskarācārya and Mādhava and, and G F Leibniz developing the entire Calculus, and later Cauchy laying the foundation for modern Calculus based on the rigorous treatment of the concept of . In this paper Bhāskarācārya's algorithm to deal with expressions involving multiples of zero (treated as infinitesimals) and zero-divisors (zero as divisors), is considered with a brief reference to the similarity it bears with the ideas of Newton and Leibniz. Bhāskarācārya says that this is of great use in Astronomy.1 Therefore, the concepts suggested in his Līlāvatī and implied in his geometric treat- ment of instantaneous -difference equivalent of the d sin θ = cos θ dθ, (in Leibniz′s notation) are considered in some detail as given in Ganeśadaivajña'ṣ own commentary (Buddhiviāsinī) and his commentary (Vāsanābhāsya)̣ of Siddhāntaśiromanị .

Keywords: Calculus, Infinitesimal, Bhāskarācārya, Vāsanābhāsya,̣ Khaguna,̣ Khahara, , Limit.

1 INTRODUCTION of inscribed regular polygon of large of sides leading to the idea of π as a limit of a Though the concept of infinitesimals is a develop- . ment of mathematics of relatively recent times (17th century) it has its roots in the ancient world of 3. The area of a circle as the sum of indefinitely mathematics in the form of intuitive ideas prevalent large number of infinitely small sectors lead- in Greek, Arabic, Chinese and Indian civilizations ing to the idea of integration as a limit of the among others. sum. However, these concepts were of primitive na- These are static examples. The core concept of ture based on geometric ideas such as: Calculus is motivated by the dynamic examples of 1. The as a secant in its limiting position astronomy dealing with instantaneous velocities of leading to the idea of as . planets. The earliest development of this was dur- ing the periods of Bhāskarācārya (12th century ce) 2. The circumference as the limit of the perimeter and Mādhava of Kerala School of Mathematics (14th ∗The author was a Professor of Mathematics and Dean of Science Faculty, Walchand College of Arts and Science, Solapur, Maha- rashtra. The article is received from his daughter for considering its publication in IJHS after the demise of the author. 1 The statement found in Vāsanābhāsyạ of Bhāskara while commenting on verse 47 of Līlāvatī goes as follows: अ गणत हगणते महानुपयोगः। 18 INDIAN JOURNAL OF HISTORY OF SCIENCE century ce) and, Sir Isaac Newton and G F Leibniz fluents, the time-rates of changes x˙ and y˙, (17th century ce), the founders of Calculus, and much the , and xo˙ and yo˙ are the moments of yo˙ later, D'Alembert and Cauchy, who laid the theoret- fluxions,3 o being an infinitesimal. is called ical foundation for modern Calculus without involv- xo˙ the ultimate ratio, thereby implying that o ing infinitesimals. The earlier conceptual difficul- 4 ultimately equals zero, and it[ is] denoted by ties of Calculus were due to the controvertial inde- y˙ y˙ 0 , called the derivative. yo˙ = xo˙ is the terminate form 0 , which arises, for example, from x˙ x˙ the intuitive concept of infinitesimals as explained moment of fluxion y˙. below. Leibniz's Differentials: The infinitesimals of the 2 NEWTON'S NOTIONS OF INFINITESIMALS variables x and y are called the differentials dx and dy, where dy = f ′(x)dx. One may treat Newton calls the variables x and y fluents and them as the ultimate (indivisible) things.5 dy considers them as moving points on a curve, x˙ and is the quotient of differentials or differen- y˙ being rates of motion called fluxions (instanta- dx [ ] dy neous velocities). He calls the products xo˙ and tial coefficient as in dy = dx. This is yo˙ the moments of fluxions (which are in- dx a useful tool where dx (≠ 0) is finite though stantaneous displacements or increments) where o sufficiently small. is infinitesimally small.2 The limit of the ratio yo˙ of ultimate is what he calls as Euler's Differentials: The differential dx can di- xo˙ dy minish indefinitely till it equals6 0. ultimate ratio (the so called ). Since the ratio dx [ ] dy of fluxions is same as that of moments of fluxions, dy = f(x + dx) − f(x) = dx, in later years there was confusion between these two dx terms, and the term fluxion itself was indiscrimi- [ ] dy nately used to denote the increments Hayes, A trea- where can be expressed in the form tise on fluxions. For convenience he denotes xo˙ sim- dx ply by o and yo˙ by f(x+o)−f(x), where y = f(x). ′ ′ yo˙ f (x, 0) dx f (x, 0) 0 The ultimate ratio at o = 0 assumes the indeter- = , xo˙ dx 0 minate form 0 . 0 virtually defined as f ′(x), whereas dy = f ′(x) dx ≠ 0, for practical calculations in Different views of infinitesimals science. This section compares algorithms of Newton, Leib- niz, Euler and Cauchy. Cauchy's Limits: A h can diminish indef- initely close enough to 0 called its limit. In Newton's Fluxions: Variables x and y are called symbols: h < ϵ (> 0) or lim h = 0, or h → 0, 2Zero: The Biography of A Dangerous IdeaSeife, Zero: The Biography of A Dangerous Idea. o is the lowercase omicron, the first letter o of the Greek word ouden meaning nothing. Greeks denoted zero by the letter o. Here the word small for all practical purposes means, a small-difference, small enough for the accuracy warranted by the technology of those times. The word `small' in italics, in what follows, is used in this sense. 3Newton's algorithm is given in ``Mathematical Principles of Natural Philosophy'' and Heys's ``Treatise on theory of fuxions''Hayes, A treatise on fluxions. 4Quantities and the ratio of the quantities which in any finite time converge continually to equality and before the end of the time approach nearer the one to the other than by any given difference, become ultimately equal at o = 0 Smith, . 5The Calculus Gallery, Masterpieces from Newton to LebsegueDunham, The Calculus Gallery, Masterpieces from Newton to Leb- segue, p. 24. 6Calculus GalleryDunham, The Calculus Gallery, Masterpieces from Newton to Lebsegue, p. 53. IDEAS OF INFINITESIMALS OF BHĀSKARĀCĀRYA 19

f(x + h) − f(x) h ≠ 0. can be expressed in notation, 0 is treated as a symbol denoting an in- h finitesimal. The commentator of Bhāskarācārya's al- the form gebra, i.e., text Bījaganitạ , Krṣ ṇ adaivajñạ 8 explains: ′ f (x + hθ)h ′ The idea of infinitesimals by considering a numerical = f (x + hθ), h example. The example presented by Krṣ ṇ adaivajñạ where 0 < θ < 1, h ≠ 0. is shown in the form of a table below. Just as Multiplier Multiplicand Product f(x + h) − f(x) ( ( ( lim = lim f ′(x + hθ) यथा गुणकः) गुः ) गुणनफल) h→zero h h→zero 12 4 48 = f ′(x). 12 3 36 12 2 24 In short we can as well define 12 1 12 12 1/2 6 f(x + h) − f(x) = f ′(x + hθ), h ≠ 0, 12 1/4 3 h 12 1/12 1 = f ′(x), h = 0. He further comments:

In the above comparisons the infinitesimal nature of अनयैव युा गु परमापचये small quantities are virtually treated as irrelevant, गुणनफलाप परमापचयेन भा। they being either 0 or not 0, as the case may be. परमापचये तैव पयवतीत े गुे गुणनफलमप मेवेत स। 3 BHĀSKARĀCĀRYA'S NOTIONS OF anayaiva yuktyā gunyasyạ paramā- KHAGUNẠ (MULTIPLE OF ZERO) pacaye gunanaphalasyāpị paramā- pacayena bhāvyam. paramāpacaye It had been the practice to write traditional mathe- śūnyataiva paryavasyatīti śūnye gunyẹ matical works (in Sanskrit) invariably in poetic form. gunanaphalamapị śūnyameveti śid- But a poetic form by its very nature could not ad- dham. mit the mathematical symbolism and therefore the author had to resort to algorithmic style. Though By the same logic, if the multiplicand Bhāskarācārya has himself written a commentary (gunyạ ) becomes smaller and smaller so on the astronomical part of his monumental work does the product (gunanaphalạ ) which Siddhāntaśiromanị , in the case of Līlāvatī and Bī- ultimately becomes the smallest (i.e., jaganita,̣ being basic mathematics necessary for his zero) as the multiplicand becomes the work, the demonstrations of proof and the details of smallest. workings were left to the wisdom of eminent com- Similarly this logic is applicable to the multiplier. mentators (such as Ganeśadaivajñạ and Krṣ ṇ adaiva-̣ Thus x0 is treated as an infinitesimal, ever decreas- jña), teachers and gifted students, lest it be too volu- ing , attaining the value zero. Therefore it is 7 minous. natural to define x0 as zero. In modern notation:

lim x0 = zero, and [x0]0=zero = zero. 3.1 Ideas of infinitesimals 0→zero Bhāskarācārya's multiple of zero khagunạ , x0, is the The idea of limit is conveyed by Buddhivilāsinī 9 as counterpart of moment of fluxion. For lack of proper follows: 7SiddhāntaśiromanịSastry, Siddhāntaśiromanị of Bhāskarācārya with Vāsanābhāsyạ , p. 39, śloka 9, Somayaji, Siddhāntaśiromanị of Bhāskarācārya, p. 99, Bhāskarācārya says that he has made his work neither voluminous nor brief, for both the intelligent and the less gifted are to be enlightened. 8Bhāskarāchārya viracita Bījaganitīyaṃ Apte, Bījaganitīyaṃ , p. 137. 9LīlāvatīApte, Līlāvatī (In Sanskrit) , p. 198. 20 INDIAN JOURNAL OF HISTORY OF SCIENCE

x′0 एवं गुणगुणभुजेकनया यावापा- 0 (of the denominator), giving the result = 0 सो बाः ा तावाधये। x′. evaṃ dvigunadviguṇ abhujakṣ e-̣ trakalpanayā yāvaccāpāsanno bāhuh ̣ The above rules are stated briefly by Bhāskarācāryain syāt tāvatsādhayet. his Līlāvatī: Thus by doubling the number of sides of a regular polygon, [inscribed in a circle], ……खगुणः खं खगुण शेषवधौ॥ ४५ ॥ is to be carried out till its side is close े गुणके जाते खं हारेुनदा राशः। enough to the arc containing it. अवकृत एव ेयः ……॥ ४६ ॥ ……khagunaḥ ̣kham,̣ khagunaścintyaścạ In modern ϵ, δ notation it implies: choose the num- śesavidhaụ || 45 || ber of sides so large as to make the side δ very śūnye gunakẹ jāte khaṃ hāraścetpunas- small so that the chord-arc difference ϵ is small tadā rāśih ̣| enough. Bhāskarācārya has not defined the deriva- avikrtạ eva jñeyah……||̣ 46 || tive in general but incorporates the idea of deriva- tive in the concept of instantaneous velocity, in Normally a multiple of zero xo, is zero. his SiddhāntaśiromanịSastry, Siddhāntaśiromanị of Bhāskarācārya with Vāsanābhāsyạ as we shall see Rule 1 If, however, further (mathematical) calcula- later. Bhāskarācārya says:10 tions are there, multiple of zero, x0, should be regarded as not zero. (0 is therefore not zero). अ यावावहासाध बन च खान Buddhivilāsinī Apte, Līlāvatī (In Sanskrit) ex- ताव ताव ुटा ा ा। plains that it is treated as mere composite sym- atra yāvadayāvanmahadvyāsārdhaṃ bol (See Appendix). bahūni ca khanḍ ānị tāvat tāvat sphutạ̄ Rule 2 If multiple of zero be followed by further jyā syāt. x0 operation of , i.e., , it is The larger the radius and the more the 0 number of parts into which an arc is di- to be understood that the multiplicand x re- x0 vided (and hence the smaller the parts), mains unaltered i.e., = x. (Thus x0 is 0 the better shall be the accuracy of the treated as not zero, and the rsine of the arc. is avoided.) Buddhivilāsinī Apte, Līlāvatī (In The concept of infinitesimals is implied here. The Sanskrit) adds: manipulations of multiples of zero and zero-divisors Rule 3 If, however, division by zero is not there, x0 (zero as divisors) are based on two rules: Normally is zero (See Appendix A for complete transla- x0, a multiple of 0, equals 0. tion of the Sanskrit text for these rules). Rule A If the calculations of an expression contain- ing xo and o, on putting o = 0, ultimately re- The following example given in śloka 47 Līlā- 0 vatīApte, Līlāvatī (In Sanskrit) by Bhāskarācārya il- sults in the indeterminate form 0 , thereby elim- inating the very quantities which are to be de- lustrates the infinitesimal nature of multiple of termined then treat xo and o as mere symbols, zero. and do not evaluate them to 0 i.e., xo and o are Example 1 In mathematical terms it amounts to not equal to 0. solving the equation: Rule B The 0 in the numerator of the indeterminate 0 (x0 + x0 )3 form could be reduced to a multiple x′0 of 2 = 63. 0 0 10Siddhāntaśiromanị Sastry, Siddhāntaśiromanị of Bhāskarācārya with Vāsanābhāsyạ , p. 42. IDEAS OF INFINITESIMALS OF BHĀSKARĀCĀRYA 21

It is obvious that Bhāskarācārya could not (x + o)3 − x3 equation = 12. as follows. have meant 0 = zero in this case, for, if it were o so, x which is to be determined, would itself Since there are further calculations, by Rule 1, be eliminated and the problem becomes pur- (3x2 + 3xo + o2)o o ≠ 0. Hence = 12 and by poseless. Since there are further calculations, o 2 2 by Rule 1, x0 is not zero. On simplification we Rule 2, cancelling o, 3x + 3xo + o = 12. x0 get 0 = 14. Therefore by Rule 2 of the above As no further calculations are there, by Rule 3, algorithm, x = 14. o = 0. Hence 3x2 = 12, giving x = ± 2. It is interesting to see how Newton solves the follow- Bhāskarācārya says that Mathematics of zero ing similar example. manipulations are very useful in Astronomy. Example 2 Newton would, for example, solve the We shall consider this now.

3.2 Comparing the algorithms of Newton and Bhāskarācārya

Newton's method Bhāskarācārya's method dy Newton's calculation of derivative dx involves the If in any expression multiples of zero x0 occur, 0 following algorithm. Let xo˙ be an infinitesimal is treated as a mere symbol kept by the side of x increment in x and yo˙ the increment in y. Let o ≠ as a composite symbol x0 ≠ 0. That is if there are 0. Rule (i) calculations to be done, let x0 ≠ zero Rule (i). Hence 0 ≠ zero.

Newton makes′ all the calculations till he reaches Having done the pending operations, if x0 has f (x, o)o x0 the form and cancels o, to get the de- further operation of division* by zero then = o 0 sired ratio of fluxions x, by cancellation Rule (ii). [Since there is further operation of division by 0, yo˙ f ′(x, o)o by Rule (i), x0 ≠ zero therefore 0 ≠ zero. Hence = = f ′(x, o) Rule (ii). xo˙ o cancellation of the two zeroes is justified]. He, however, finds that there are still some in- finitesimals involved in f ′(x, o), which cannot be ignored,** errors being unacceptable in mathe- matics.

When all the calculations are made he says let o = If all the calulations are over, then x0 = 0 (and 0 0 Rule (iii). ( )  = zero) Rule(iii). y˙ [ ] x˙ o y˙ x0 This is same as defining   as . This is same as defining as x. o x˙ 0 0 = zero [ ] o = 0 y˙ ( x˙ )o y˙ x0 o as x˙ . [ 0 ]0=zero as x. o=0

x0 *The two 0s in are treated as same infinitesimals, whereas x0 and 0 are different infinitesimals. 0 ** It is useful to consider infinitely small quantities such that when their ratio is sought they may not be con- sidered zero, but which are rejected11 as often as they occur with the quantities incomparably greaterCajori, A

11virtually treated as zero. 22 INDIAN JOURNAL OF HISTORY OF SCIENCE

History of Mathematics.

0 Rule (ii) anticipates the reduction of 0 to the form In modern notation fluxion stands for δ so that x.0 0 . fluxion of α = δα, fluxion of x = δx and fluxion of y = δy. As the secant KEW approaches the tan- gent KTS the line TEF approches the point K and ul- EOMETRIC TREATMENT OF 4 G timately coincides with it. Also δα approaches zero, INFINITESIMALS and α becomes equal to θ. 4.1 Newton's method Therefore the two dissimilar triangles KEF and KTF are almost similar to the finite △OKB, for We illustrate Newton′s geometric method of finding small δα. So we have the derivative12 in which he considers the tangent as δy FE FT a rotating secant KEW in its limiting position KTS. = ≈ = m. Newton was of the opinion that the fluxions of geo- δx KF KF metric entities such as lines, areas, , etc., can Thus in the limiting case, the slope of the secant = the be obtained and it is not necessary to introduce them slope of the tangent. That is, into geometry13. But he rightly thought it necessary to demonstrate their role, while considering the tan- dy y˙ m = = . gent as a limit of a secant, as shown below. dx x˙ S This is similar to the following geometric treat- T ment of instantneous quanties by Bhāskara. E W δαδα δy K α F 4.2 Bhāskarācārya's method δx Bhāskara considers tātkālikagati or sūksmagatị y y which means instantaneous motion or displacement θ H BD in a given small interval of time, such as daily x O B δx C motion of slow moving planets, or even smaller in- terval of time (see Fig.2), The term tātkālikabhogyakhanḍ aṃ or tātkā- Fig.1 likadorjyayorantaram which stands for the resulting instantaneous rsine-difference TD (See In Fig. 1, consider a point K on the circle, and a Fig. 4) between two successive rsines is given by secant KEW. Here OKS is the tangent at K, and let rδθ BD the angles. ̸ KOB = ̸ TKF = θ. The secant δ(r sin θ) = (r cos θ) = (r cos θ) , (1) KEW is rotated about K till, the two points K and E, r r ultimately coincide and KEW becomes the tangent where KTS, so that in the process the chord KE coincides with the arc KE. In this ultimate position δα is 0 and δ(r sin θ) = r sin (θ + δθ) − r sin θ. α = θ. Newton states that arc KE is the fluxion (in- crement) of arc HK, BC the fluxion of OB, and FE These are used to find the instantaneous posi- that of CF. These fluxions are in fact the infinitesi- tion of planetsJoseph, The Crest of the Peacock. mals of geometric entities. Bhāskarācārya proves the relation (1) in two parts. 12A History of MathematicsCajori, A History of Mathematics, p. 197. 13Ibid., p. 198. IDEAS OF INFINITESIMALS OF BHĀSKARĀCĀRYA 23

Part (i) First he considers sine differences for adyatanaśvastanakendrayorantaraṃ various arc lengths at constant intervals of kendragatih ̣ ′ 225 as given in the rsine-table he had pre- The arc-distance, rδθ, between the posi- pared, and proves geometrically that tions of (centre of) the planet, say, at Sun ′ rise of today and tomorrow, is the daily ′ 225 δ (r sin θ) = (r cos θ) . (2) motion (displacement) of planet. r In the previous paragraph Bhāskarācārya says: The prime ′ on δ indicates constant intervals of ′ 225 . त काल मेऽनया गा हालयतुं- Part (ii) Next he considers sine differences for a युत इत। इयं कल ूला गतः। given at variable intervals of rδθ. tasya kālasya madhye'nayā gatyā gra- Using the interpolation formula he proves first haścālayitumyujyatạ iti. iyaṃ kila sthūlā for small daily motion of anomaly (kendra- gatih.̣ gati), and later for a small displacement 15′ During this interval this daily motion (the radius (bimbārdha) of the Sun), the for- can be used to find planetary position at rδθ mula δ(r sin θ) = (r cos θ) r . any instant. This indeed is approximate.

ताालकभोखकरणायानुपातः। भुजाकरणे योखं तेन सा गुा tātkālikabhogyakhanḍ akaraṇ āyānu-̣ शरदैभा। pātah.̣ bhujajyākaranẹ yadbhogykhanḍ aṃ ̣ tena Use of Proportions to obtain the in- sā gunyạ̄ śaradvidasrairbhājyā. stantaneous sine difference. This (arc rδθ) be multiplied by the ′ In Figure 2 and 3, r = 3438′, rδ′θ = 225′ = rsine-difference δ (r sin θ) (for the arc BD ≈ BD, and TD = r sin(δ′θ) ≈ arc rθ), obtained from the rsine- ′ ′ differences15 prepared (at intervals of rδ θ = 225 . ′ ′ ̸ BOD = δ′θ δ θ = 225 ) and divided by 225. ′ ̸ TDB = θ + δ θ ≈ θ 2 This prescription essentially gives the rsine- rδθ Initially θ = 0, difference δ′(r sin θ) for the arc rθ and the small 225 interval rδθ. Then it is stated: O ′ O θ δ θ r त ताव ताालकभोखकरणायानु-

′ R T≈ δ θr θ पातः। θ D RT P B tatra tāvat tātkālikabhogyakhanḍ a-̣ D APB A karanāyānupātaḥ .̣ Fig.2 Fig.3 To get this instantaneous rsine- Bhāskara's proof of Part (i) difference, here is the method of direct proportions. The proof of (2) as given in Bhāskarācārya's vāsanābhāsyạ (commentary) proceeds as follows. Now let's consider the similar triangles BDT and He commences the proof by difining kendragati:14 BOP (see Figs 3 and 4). With respect to these similar triangles Bhāskara lays down the rule of proportion अतननकेयोररं केगतः। as follows: 14SiddhāntaśiromanịSastry, Siddhāntaśiromanị of Bhāskarācārya with Vāsanābhāsyạ , pp. 52-53, verses 36-37. 15Bhāskarācārya's Siddhāntaśiromanị Sastry, Siddhāntaśiromanị of Bhāskarācārya with Vāsanābhāsyạ , slokas 2-9, p. 40. 24 INDIAN JOURNAL OF HISTORY OF SCIENCE

यद ातुया कोटयां भोख- The result would be the difference in the ं शरदतुं लते तदेया कम- daily motion. कोटायाः शरदा २२५ गुणा हरः। This is just interpolation which follows from the yadi trijyātulyayā kotijyayādyaṃ ̣ bh- Rule of Three or from the almost similar trian- ogyakhanḍ aṃ ̣ śaradvidasratulyaṃ lab- gles BDT and BCSSharma, Siddhāntaśiromanị of hyate tadesṭayạ̄ kimityatra kotijyāyāḥ ̣ Bhāskarācārya similar to the finite triangle POB, see śaradvidasrā 225 gunastrijyạ̄ harah.̣ Fig. 4. The infinitesimal triangles BDT and BCS are (al- It may noted that triangles BDT and BOP are almost most) ∼ to the finite △ BOP, for small δθ and for ′ similar for all θ and the given small interval of 225 . all θ including the initial value θ = 0, Fig. 2. If initial rsine-difference (225') is obtained from It may be noted that an altitude equal to the radius r, then what is the rsine-difference for any desired altitude (r cos θ)? δ′θ △TDB = θ + ≈ θ. See Fig. 3. 2

फलं ताालकं ुटभोख.... Also, phalaṃ tātkālikaṃ sphutabh-̣ ogyakhanḍ aṃ ̣ ... TD = RD − PB ′ The result is the instantaneous rsine- = r sin (θ + δ θ) − r sin θ. difference. Morevoer, ′ 225′ That is, δ (r sin θ) = (r cos θ) r . δθ D ̸ SCB = θ + ≈ θ Bhāskara's proof of Part (ii) 2 ( ) The proof of the result given by (3) commences with BC SC = r sin (θ + δθ) − r sin θ = r cos θ . the following argument. r

तेन केगतगुणनीया शरदैभा। अ शरदमतयोगुणकभाजकयोुा- ाशे कृते केगतेः कोटागुणा हरः ा। फलं अतननकेदोयोररं ̸ BOD = δ′θ = 225′ भवत। O tenakendragatirgunanīyạ̄ śaradvida- θ r srairbhājyā atra śaradvidasramitayo- δθ R T rgunakabhājakayostulyatvānnāśẹ krtẹ ≈θ Q S kendragateh ̣ kotijyāguṇ astrijyạ̄ harah ̣ P θ C B syāt phalaṃ adyatanaśvastanakendra- Fig.4 A dorjyayorantaraṃ bhavati. By this rsine-difference be multiplied Bhāskarācārya applies the above method to find the daily motion16 and divided by 225′. rsine-difference for the radius of Sun's disk, as ex- When we cancel 225′ from the numera- plained in SiddhāntaśiromanịSastry, Siddhāntaśiro- tor and the denominator what we obtain manị of Bhāskarācārya with Vāsanābhāsyạ given be- is the product r cos θ BC divided by r. low: 16The daily motion denored by BC (= rδθ), will be less than 225′ minutes for all planets except in the case of the moon. IDEAS OF INFINITESIMALS OF BHĀSKARĀCĀRYA 25

यद ातुाय कोटौ थमं ाध श- For arbitrarily small interval rδx, by rule of three रदा भोखं तदाभमतायाम क- rδx मत फलं ुटं भोख। तेन गुणतं δ(r sin(x)) = r cos(x) . r बाध शरदैभं। एवं ते शर- दमतयोगुणहरयोनशे कृते बाध को- टागुणाहरः फलं दोयोरर। 4.4 Infinitesimals and zero-manipulations, yadi trijyātulyāyāṃ kotaụ prathamaṃ involved in the geometric treatment by jyārdhaṃ śaradvidasrā bhogyakhanḍ aṃ ̣ Bhāskarācārya tadābhimatāyāmasyāṃ kimiti phalaṃ In Fig. 5 consider the arc BD( = rδθ). This arc rep- sphutaṃ ̣ bhogyakhanḍ am.̣ tena gunitaṃ ̣ resents the tātkālikagati or infinitesimal increment of bimbārdhaṃ śaradvidasrairbhājyam￿̣ arc AB (= rθ). TD which represents the tātkālikab- evaṃ sthite śaradvidasramitayorguna-̣ hogyakhanḍ aṃ is the infinitesimal increment of PB harayornāśe krtẹ bimbārdhasya koti- (= r sin δθ). As the line OD approaches and reaches jyāgunastrijyāharaḥ ̣ phalam dorjyay- OB, the triangle TDB becomes a point right triangle orantaram. TD 0 and assumes the form . For this to have any BD 0 Here Bhāskarācārya uses the rationale exactly as meaning the infinitesimal TD must be, a multiple of given above. 0, say, x0. Hence The only difference here is that rδθ stands for the arc-difference 15′, corresponding to half the width TD x0 = = x, of Sun's disk, giving the r sine-difference relation: BD 0 δ(r sin θ) = r cos(θ)( rδθ ). r by Bhāskara's Rule (ii). This point right triangle TDB, is similar to finite trianlge POB. Thus we have 4.3 Trigonometric method TD OP = , from which we get Bhāskarācārya gives a hint as to how the above BD OB 17 ( ) results may be obtained using the expansion of BD r sin(x + 1)◦. The relation given by Bhāskara is: δ (r sin θ) = r cos θ . r r sin(x)◦ 10r cos(x◦) r sin(x+1)◦ = r sin(x)◦ − + , 6569 573 The rationale behind the above expression can be un- derstand as follows. The sine addition formule was well known. Hence, O ◦ ◦ ◦ ◦ ◦ θ r sin (x + 1) = r sin x cos 1 + r cos x sin 1 . δθ r Here, R T≈ ( ) θ D 6568 θ r cos 1◦ = r ≈ r and r sin (1◦) P 6569 B ( ) Fig.5 A 10 = r ≈ 60′. 573 Bhāskarācārya was indeed referring to these manip- ◦ ◦ ◦ δ(r sin x ) = r sin(x + 1) − r sin x ulations of infinitesimals Apte, Līlāvatī (In Sanskrit) ◦ 60 when he said that these zero manipulations are very = r cos x . r useful in astronomy. 17English translation of SiddhāntaśiromanịWilkinson, Siddhāntaśiromani,̣ An English Translation, ślokas 16 and 17, p. 267. 26 INDIAN JOURNAL OF HISTORY OF SCIENCE

5 CONCLUSION whereas Newton used the method of term by term dif- ferenciation of infinite unmindful of the con- Though there are strking similarities between the al- vergence problem involved in it. Considering these gorithms of Newton and Bhāskarācārya there are dif- facts Bhāskarācārya's bold attempt to give an algo- ferences in perspective. Newton, Leibnz and Euler rithm to deal with infinitesimals and use these ideas and Bhāskarācārya treat the infinitesimal nature of in astronomy is commendable. Highlighting these small quantities virtually as irrelevant. These quan- facts is appropriate in his 900th birth year which is tities are either 0 or not 0, as the case may be. This being presently celebrated. zero-nonzero dichotomy was there which was im- plicit in the case of Newton, Leibnz and Euler and APPENDIX was rather explicit in Bhāskarācārya's case. Unlike Newton Bhāskarācārya does not explic- Buddhivilāsinī 18 makes this more clear, as given be- itly state the infinitesimal nature of multiples of zero. low. The idea of infinitesimals is implied in his treatment of multiples of zero and zero divisors. This was re- शेष वधौ कते सत खगुणः। strcted to the rsine and rcosine functions, and he तथा ह राशेः े गुणके ाे ताो found it handy to use the Rule of Three and the pro- वधेद, तदा खगुणो राशः खं ादत portional properties of similar right triangles. न काय। कु ं ता गुणकाने He uses geometric method ingeniously thereby ा। 0 avoiding the indeterminate form 0 and replaces Rule śesasyạ vidhau kartavye sati kh- of Proporions by the Rule of Three by identifying agunaścintyaḥ .̣ tathā hi rāśeh ̣ śūnye the almost proportional quantities involved, in a way gunakẹ prāpte tasyānyo vidhiścedasti difficult to imagine. In this geometric process are tadā khagunọ rāśih ̣ khaṃ syāditi involved dissimilar right angled triangles tending to na kāryam .kintu śūnyaṃ tatpārśve similarity to a finite triangle, as they converge to a gunakasthānẹ sthāpyam. point. This avoids direct encounter with the indeter- Further operations pending multiple of minate form 0 and gives us an impression that we are 0 zero be given a second thought. That dealing with finite quantities. Newton also says in is, if a number is multiplied by zero and his geometric treatment of slope of a tangent as the there are further operations remaining derivative, infinitesimals are not necessary as they then the multiple of zero is not to be con- can be denoted by finite lengths. strued as zero, but 0 is to be kept by the Bhāskarācārya's use of the concept of infinitesi- side of the number (as a mere symbol) mals was restricted to applications in planetary cal- culations unlike that of Newton of application was wide and varied. In Bhāskarācārya's times due ततः शेषवधाने कृते पुनः खं हरेदा तयोः to lack of proper notations and motivation there was गुणकहरयोुेन नाशः कायः। नो no development of Calculus, as such, though clearly, चे खं हरः, तदा खगुणो राशः खं ा... there was development of Calculus by Mādhava and tatah ̣ śesavidhānẹ krtẹ punah ̣ khaṃ Nīlakanṭ haDunham,̣ The Calculus Gallery, Master- haraścettadā tayoh ̣ śūnyagunaka-̣ pieces from Newton to Lebsegue of Kerala School of harayostulyatvena nāśah ̣kāryah ̣. no cet Mathematics and others, during 15th and 16th cen- khaṃ harah,̣ tadā khagunọ rāśih ̣ khaṃ turies leading to the expansion of sine and inverse syāt. tangent functions etc. If, having done all the operations, there These developments by Mādhava were again a is further operation of division by zero continuation of geometric treatment of infinitesimals, then the denominator and the numerator 18LīlāvatīApte, Līlāvatī (In Sanskrit) pp. 39, 40. IDEAS OF INFINITESIMALS OF BHĀSKARĀCĀRYA 27

being equal, they be cancelled (0 being Joseph, G. G. The Crest of the Peacock. 1998, treated as a mere symbol). If, however, pp. 217, 252, 298–300. there is no division by zero then the mul- Sastry, Pt. Bapudeva, ed. Siddhāntaśiromanị of tiple of zero is to be treated as zero. Bhāskarācārya with Vāsanābhāsyạ . C.I.E. Chaukhamba Sanskrit Sansthan, Varanasi, 2005, pp. 40, 41, 52–53, 227. BIBLIOGRAPHY Seife, C. Zero: The Biography of A Dangerous Idea. Penguin group, 2000. Apte, V. G., ed. Bījaganitīyaṃ . Anandashrama, Sharma, Maithila Pt. Muralidhara, ed. Siddhān- 1930. taśiromanị of Bhāskarācārya. Vidyavilas press, — ed. Līlāvatī (In Sanskrit). Anandashrama, 1937, Banares: Chaukhamba Samskrita series office, pp. 21, 24, 40, 108, 125, 111, 198. 2007, pp. 107, 151–152. Cajori, F. A History of Mathematics. 3rd. 1919. Smith, D.E. History of Mathematics. Vol. 2. Dover Dunham, William. The Calculus Gallery, Master- publications, 1951. pieces from Newton to Lebsegue. Princeton Uni- Somayaji, Arka. Siddhāntaśiromanị of Bhāskarācārya. versity Press, Princeton And Oxford: Princeton Rashtriya Sanskrit Vidyapeetha, Thirupati, 1980. University Press, 2006. Wilkinson, Lancelot. Siddhāntaśiromani,̣ An English Hayes, Charles. A treatise on fluxions. London, 1704, Translation. Calcutta: Baptist Mission Press, pp. 1, 17, 19. 1860.