Mathematics in Ancient and Medieval India c9: Chapter VI Infinitesimal Calculus

Mathematics in Ancient and Medieval India c9

Language: English

Chapter VI Infinitesimal Calculus

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The infinitesimal calculus may be described as a mathematical method of calculating results produced by non-uniform continuous changes as occur in nature. The idea of such method first appeard in the works of Greek scholars like Antiphon ( 430 B. C. ), Eudoxus ( 370 B. C. ), Eudemus (335 B. C) and others. Nearest approach to infinitesimal calculus is first found in the work of Archimedes ( 225 B. C. ). A similar approach to calculus in the middle ages relating to idea of infinitesimal appears in the Indian works, *Siddhānta-śiromaṇi* of Bhāskara II ( 1150 A. D. ) and *Yuktibhāṣā* ( *c.* 1500 A. D. ). In Europe in the 17th century A.D. Kepler ( 1604 A. D. ) used the method in a rather primitive form and later in 1629 A.D. Cavaliery made a more clear statement of the idea while describing a method for the calculation of the area of a triangle. In the 17th century A. D., the Japanese scholar Seki Kowa ( 1642-1708 A. D.) deve'oped a form of calculus known as *Yenri* meaning ‘circle principle’ to calculate the area of a circle. The subject was finally developed in the modern form by Newton ( 1664 A. D. ) and Leibniz ( 1676 A. D. ).

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In India, the idea of ‘infinitesimal calculus’ arose as Sastri¹, Seal² and Sengupta³ have shown in the attempts of early astronomers to find :

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¹ Śāstrī, Bapudeva, ‘Bhāskara’s Knowledge of the Differential Calculus’, *Journal of the Asiatic Society of Bengal*, 27, p. 213-16, 1858,

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² Seal, B. N., *Positive Sciences of the Ancient Hindus*, p. 77-80, Motilal Banarasi Dass.

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³ Sengupta, P. C., ‘Infinitesimal Calculus in Indian Mathematics’, *Journal of the Department of Letters*, Calcutta University, 22, p. 1-17, 1932.

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( i ) the instantaneous motion of a planet,
( ii ) the ‘position-angle’ of the ecliptic with any secondary to the equator,
( iii ) the surface and volume of a sphere, and
( iv ) the value of π.

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The concept of differentiation appears in the *Siddhāntaśiromaṇi* of Bhāskara II and it was studied in connection with the problem set in items ( i ) and ( ii ). The ideas of integration and its development are found in the *Siddhāntaśiromaṇi* of Bhāskara II, *Gaṇitakaumudī* of Nārāyaṇa and the *Yuktibhāṣā* in connection with the problems set in ( iii ) and ( iv ).

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*Differential Calculus* :

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( a ) *Modern concept*—The differential calculus is directly concerned with the momentary state of a phenomenon. This momentary state is symbolised by the differential coefficient. In order to render different processes such as change of position, change of motion, of temperature, volume etc. susceptible to mathematical treatment, the magnitude are supposed to change during a series of very short intervals of time. The shorter the interval, the more uniform is the process. The conception is of fundamental importance. The velocity of a train, sixty miles per hour, does not mean that the train can run one mile per minute. The speed of the train might vary in successive minutes, the sum total of which for sixty successive minutes or one hour amount to sixty miles. Hence, the average velocity of the train is one mile per minute. If we continue to reduce the unit of time, the average velocity approaches more and more nearly to the actual velocity of the train during the whole time under consideration. If x₁ be the distance traversed at the time t₁ when the observation commences and x₂ be the distance traversed at the time t₂ then the velocity of the train, v = (x₂ - x₁)/(t₂ - t₁). If t₂ - t₁ be extremely small then (x₂ - x₁)/(t₂ - t₁) may be represented as ()/() where denotes the space traversed in the small unit of time δ t with uniform motion.

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Archimedes was the first to give the elementary concept of differential calculus in connection to finding the tangent to a spiral. But no idea of infinitesimal could be found in his method.

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( b ) *Bhāskara II and his concept of differential*—In the *Spaṣṭādhikāra*¹ chapter of his *Siddhāntaśiromaṇi* Bhāskara II deals with *tātkāliki-gati* ( motion for a moment ) of a planet which differs very little from the instataneous motion of modern concept of differential calculus.

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¹ v. 37. see also his own *Vāsanābhāṣya*.

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He computed the instantaneous motion of a planet, compared its successive positions and regarded its motion as constant during the time interval between two successive positions. According to Seal¹, this interval cannot be greater than a *truṭi* = 1/(33,750) the part of a second, a unit of time introduced by Bhāskara II himself.²

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¹ *Ibid.*, p. 77-78.

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² 1 day = 30 *kṣaṇas*, 1 *kṣaṇa* = 2 *ghaṭikās*, 1 *ghaṭikā* = 30 *kalās*, 1 *kalā* = 30 *kāṣṭhās*, 1 *kāṣṭhā* = 18 *nimeṣas*, 1 *nimeṣa* = 30 *tatparas* and 1 *tatpara* = 100 *truṭis*–(SiddāntasiromaṅKi-āla-mānādhyāya).

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Bhāskara’s concept about the motion of the planets will be clear from his own commentary³ on the verses to calculate the *tātkālikī* motion of a planet. He clearly distinguishes between *sthulagati* (velocity roughly measured) and *sukṣmagati* (velocity measured accurately with reference to indefinitely small quantities of space and time). Here Bhāskara II’s *tātkālikī-gati* may be defined as the ‘infinitesimal’ increment in the planet’s longitude due to ‘infinitesimal’ increment in time.

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³ Vide *Siddhāntaśiromaṇi*, Spaṣṭādhikāra chapter, v. 36–37.

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The following three examples may be quoted from Bhāskara II’s work :

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( i ) Difference between two successive sines given in the table of sines was termed as *bhogya-khaṇḍa* by Bhāskara II. *Bhogya-khaṇḍas* are not equal to each other and the increment of the sine varies with the increment of the arc. Bhāskara II calculated the values of the *bhogya-khaṇḍas* from the *tātkālika-bhogya-khaṇḍa*¹. Śāstrī² pointed out that *tātkālika-bhogya-khaṇḍa* is the increment in the sine deflected in the direction of the tangent i.e. the instaneous motion of the sine.

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¹ *Ibid.*, Spaṣṭādhikāra-Vāsanābhāṣya, v. 36–37.

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² Śāstrī, Bapudeva. *Ibid.*, p. 214.

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To explain this in the quarter of a circle (Fig. 27) let PP₀ = A (or PP₁ = PP₀) and arc XP = y and [Figure: Fig. 27] arc XP₁ = y' say, then P₁S = ' - = p₁Q₁ - PQ = *bhogya-khaṇḍa* of the sine XP and P₀T = P₀Q₀ - PQ = tātkālika-bhogya-khaṇḍa (the arc PP₁ is deflected in the direction of the tangent).

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Now in the two similar triangles POQ and P₀TP $= (PP₀ × OQ)/OP = (A × R )/R = A$ From similar traingles PP₀T and PP₁S (chord PP₁ = PP₀, when the arc is very small) $P₁S = (P₀T × PP₁)/PP₀ = (A · (y'-y))/A = · (y'-y)$ or, ' - = · (y'-y) which is equivalent to or, d() = dy

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Changes in the sine chord (*bhujajyā*) and cosine chord (*koṭijyā*) were also calculated by Bhāskara II for small differences in the arc.

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Let PP₁ be a small increase in the arc PX (Fig. 28) and P₀ be the middle point of the small arc. The perpendicualrs PQ, P₀Q₀, P₁ Q₁ and PS are drawn. From the similar triangles P₁SP and OQ₀P₀, we have P₁S = (OQ₀ · P₁P)/OP and PS = (P₀Q₀ · P₁P)/OP₀.

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Fig. 28

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When the arc PP₁ is very small, the cosine chords at P and P₀ become almost equal and the arc PP₁ coincides with the chord PP₁.

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Hence P₁S ( *bhūjakhaṇḍa* or sine difference ) = (Cosine chord at P × increase in the arc)/radius and PS ( *koṭijyākhaṇḍa* or cosine difference ) = (Sine chord at P × increase in the arc)/radius

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In terms of modern symbols, d(R sin x) = (R cos x dx)/R and d(R cos x) = (R sin x dx)/R

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This theorem with proof is also given in both *Āryabhaṭīyabhāṣya*¹ of Nīlakaṇṭha ( c. 1500 A. D. ) and *Yuktibhāṣā*².

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¹ Vide *Āryabhaṭīya* with Nīlakaṇṭha's commentary under v. 12 of *Gaṇitapāda*.

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² *Ibid.*, p. 165 and 173.

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P. C. Sengupta¹ has show that Mūñjāla and his commentator Prasastidhara were aware of the fact that the tabular differences of ‘sines’ for any given value of the arc are proportional to the cosines. This seems to suggest that Bhāskara II’s result in this connection was anticipated by some few centuries ago by Muñjāla, though no clear statement was given by the latter, as to how it was arrived. It might however be pointed out here that even earlier than that, Brahmagupta (620 A. D.) as acknowledged by Bhāskara II himself calculated the average velocity of the planet Mars based on instantaneous motion.

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¹ *koṭiphalaghni mṛdukendrabhuktistrijyoddhṛtā karki mṛgādikendre ।* *tayā yatonā grahamadhyabhuktistātkālikī mandapariṣphuṭa syāt ॥* (*Siddhāntaśiromaṇi*-Spaṣṭādhikāra, v. 37)

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(ii) Śāstrī has given a clear exposition of the *tātkālikī* motion of a planet as conceived by Bhāskara II as follows :

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Suppose x, x' = the mean longitudes of a planet on two successive days.

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y, y' = the mean anomalies.

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u, u' = true longitudes.

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a = the sine of the greatest equation of the orbit or ecentricity.

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Then x' − x = the mean motion (*madhyagati*) of the planet, y' − y = the motion of the mean anomaly (*manda-kendra*), and u' − u = true motion (*sphuṭagati*) of the planet. Now according to Bhāskara II, the equation of the orbit on the first day = (a sin y)/R when R = radius.

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and that on the next day = (a sin y')/R

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∴ u = x ± (a sin y)/R ... (1)

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and u' = x' ± (a sin y')/R ... (2)

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u'-u = x'-x ± (a (sin y'-sin y))/R

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By (i), u'-u = x'-x ± (a cos y)/R (y'-y) ... (3)

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This is the instantaneous motion of the planet. Equation (3) is just the differential of equation (1),

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for d(u) = d(x ± (a sin y)/R)

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or, du = du ± a/R · (cos y)/R · dy,

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which is equivalent to (3).

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(iii) The third case originated in Bhāskara II's

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Fig. 29

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attempt to find the complement of the angle (Sengupta).

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In the Fig. 29 let EQ = Equator, P its pole CT = Ecliptic, K its pole and N = any point on EQ. ν = the first point of aries. It is required to find the complement of the angle vNS In Bhāskara II's terminology, the 'sine' of the required angle is *ayana-valana*. If any point (say S of the ecliptic has a celestial longitude l and δ be its corresponding declination, we have, (Sin δ)/(Sin ω) = (Sin l)/(Sin 90⁰)i.e. Sin δ = Sin l. Sin ω ... (1)

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Now Bhāskara II's method of finding the *ayanavalana* leads to the differentiation of this equation (1) giving *ayana-valana* to be R (dδ)/dl = (R cos l . R sin ω)/(R cos δ) ( ω = obliquity of the ecliptic )

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Leibniz and Newton perfected this concept of differential calculus as we find in modern form.

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From what has been discussed above, it follows that Spottiswood has little justification to make the observation that Bhāskara II had no knowledge of calculus.

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From consideration of all these facts it may be stated that Bhāskar II's claim as the precursor of Newton and Leibniz in the discovery of the principle of differential calculus as well as in its application to astronomical problems and computations cannot be ignored.

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*Integral Calculus*

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Integral calculus is the method of finding the summation of a series when the number of terms in the series tends to infinity, each term being infinitely small.

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Although the integral calculus, as we know it to-day arose with Newton and Leibniz in the 17th century A. D., the early stages of its beginning have been traced to the works of Greek scholars namely Antiphon (c. 430 B. C.), Euclid (300 B. C.) and Archimedes (3rd century B. C.)¹. It is in the works of Archimedes that we find the nearest approach to the concept of actual integration. He calculated the area of a parabolic segment by dividing it into a very large number of small plane geometrical figures and summing up their areas when they are infinitely small². The invention of the concept of calculus was therefore attributed to Archimedes. In India, several centuries after Archimedes, Bhāskara II (1150 A. D) in his *Golādhyāya* has given a similar method of finding the area of a circle, surface area of a sphere and volume of the sphere which involve the ideas of integration³. In this method he assumed that the more or less correct result is obtained in these cases by summation of a very large number of small units in the form of triangles for the circle with common vertex at the centre and base on the circumference of the circle and in the form of pyramids with their vertices meeting at the centre and bases lying on the surface for the volume of a sphere. For calculating the surface-area of a sphere, the method suggested was the summation of the measure of the annular spaces formed by drawing large number of circles of increasing radii around any point on the surface of the sphere covering upto the section passing through the centre of the sphere and multiplying the result by 2. The area of each annular space was calculated by the rule *lambaguṇam kumukhayogārdhamiti* (i.e., the rule for finding the area of a trapezium). All these obviously involved the idea of summation of the infinitesimal quantities. Hence the results given are in good agreement with the modern values namely,

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¹ Heath, T. L. *The works of Archimedes*, pages cxlii–cliv, 235-36, New Dover Edition.

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² Smith, D. E. *History of Mathematics*, 2, p. 679, Dover Publication.

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³ Sengupta, P. C. ‘Infinitesimal Calculus in *Indian Mathematics*’, *Journal of the Department of Letter*, Calcutta University, 22, p. 1-17, 1932.

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| Area of a circle | = | ² |
| Volume of a sphere | = | 4/3π³ |
| Surface of a sphere | = | 4π² |

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But it must be pointed out here that both Archimedes and Bhāskara II though conscious, that the units must be as small as possible for their summation to approach the correct value, they however were not aware of the method of summation of infinite series i.e., the limit of a sum.

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The idea of such summation of an infinite series or the limit of the sum of its n terms is, in fact, first found in the *Yuktibhāṣā*¹ (16th century A. D.), nearly a century earlier than Newton. For calculating the value of π, the author of *Yuktibhāṣā* considered a square circumscribing a quarter of a circle of radius x. The square was divided by its diagonal. The circumference of the circle included between the diagonal and a side of the square passing through the centre of the circle is ⅛th of the whole circumference. The side of the square covering this portion of circumference was again divided into large number of very small equal parts by lines drawn from the centre of the circle. Let Δ x be the length of each small division on the side of the square which is equal to radius and let n. Δ x = x. In the attempt to calculate the value of : π (since π = circumference/diameter), the author of *Yuktibhāṣā* calculated the value of ⅛th of the circumference by summing up the large number of small chords corresponding to each small equal division of the circle by applying the properties of similar triangles. The expression for the sum of the chords as deduced in *Yuktibhāṣā* is given by :

Linked texts (4)

Āryabhaṭīya — Āryabhaṭa, Sanskrit
Yuktibhāṣā (Gaṇitayuktibhāṣā) — Jyeṣṭhadeva 1520, Malayalam
Siddhāntaśiromaṇi — Bhāskara II, Sanskrit
Gaṇitakaumudī — Nārāyaṇa Paṇḍita (mathematician), Sanskrit