Indian Journal of History of Science, 52.1 (2017) 17-27 DOI: 10.16943/ijhs/2017v52i1/41297 Ideas of Infinitesimals of Bhāskarācārya in Līlāvatī and Siddhāntaśiromanị A B Padmanabha Rao∗ (Received 11 June 2016) Abstract The core concept of Calculus 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, Isaac Newton 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 limit. 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 mathematics is of great use in Astronomy.1 Therefore, the infinitesimal concepts suggested in his Līlāvatī and implied in his geometric treat- ment of instantaneous sine-difference equivalent of the differential equation 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, Fluxion, Limit. 1 INTRODUCTION of inscribed regular polygon of large number of sides leading to the idea of π as a limit of a Though the concept of infinitesimals is a develop- sequence. 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 tangent as a secant in its limiting position astronomy dealing with instantaneous velocities of leading to the idea of slope as derivative. 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 fluxions, 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 quantities 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 variable 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, History of Mathematics. 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 quantity, 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.