Math in metre
Jun 01 2015
Ancient Indians were masters in the art of blending mathematics with metrical verses
The art of blending mathematics with poetry, had been in place in India at least from the time of Vedanga Jyotisha (c 1400 BCE) of Lagadha. One of the primary reasons for taking recourse to poetry is to make the medium of communication as beautiful as the message. The ancient Indians also chose to be brief in their style of writing, and certainly avoided excessive verbiage. It is said of Indian grammarians, that even if they could manage to save half a mora or syllable (ardhamatralaghava) from one of their rules, they celebrated it like the birth of a son (putrotsava). Of course, enough care was taken to see that brevity did not mar the clarity or accuracy. The brevity of the magnum opus of Aryabhata (b 476 CE), which presents all of the then known mathematics and astronomy in just 108 verses, remains unparalleled till date.
This trend of composing mathematical literature in the form of metrical composition, that got solidly established around the time of Aryabhata was successfully taken forward by the later astronomers and mathematicians such as Varahamihira, Bhaskara-I, Lalla, Brahmagupta, Sridhara, Mahavira and a host of others. It indeed seems to have reached its pinnacle with the compositions of Bhaskaracarya (b 1114), namely the Lilavati (a treatise on arithmetic and geometry), the Bijaganita (a treatise on algebra) and the Siddhantasiromani (a treatise on astronomy). Whether it is representation of numbers, or prescription of algorithms to solve indeterminate equations, or dealing with the infinite and the infinitesimals — all of them were couched in the form of lucid verses by the ancient Indian savants. We shall illustrate how it was done with a few select examples below.
Ancient Indian way of representing numbers
Besides the word numerals whose earliest occurrences are traceable in the Vedas, many other techniques had been developed by Indian mathematicians and astronomers for representing numbers. The three prominent ones that have been in vogue are: Aryabhatan, Katapayadi and Bhutasankhya. While the latter employs bhutas (literally beings) that have the potential to connote numbers (prthvi = 1, netra = 2, guna = 3, veda = 4 … bha (stars) = 27…), the former two employ vowels and consonants to represent numbers, but in ingeniously different ways. Before proceeding further, we present an interesting example to show how the Bhutasankhya system has been employed to represent quite large numbers. The following verse presenting the value of π correct to 11 decimal places using this system, is ascribed to the 14th century Kerala mathematician, Madhava.
Wise men say that if the diameter happens to be 9 x 10¹ ¹ (nava-nikkarva) units, then the measure of the circumference is vibudhanetragajahihutashanatrigunavedabhavaranabahavah units.
The ‘number-connoting’ words employed in the first half of the verse are: vibudha=33, netra=2, gaja=8, ahi=8, hutashana=3, tri = 3, guna=3, veda=4, bha=27, varana=8 and bahu=2 (here it may be noted that, while decoding the number, the digits are written from right to left). The word nikharva represents 10¹ ¹. Thus the value of π given by the above verse is:
(correct to 11 places).
Though the use of alphabets to denote numbers may be found in other civilisations (Greek and Roman), the technique adopted by Indians is quite distinct from them and stands out quite conspicuously, at least in two respects: (i) the Indian systems implicitly recognised the decimal place-value principle, and (ii) more importantly, the alphabets are combined in such a way that they could generate pronounceable (and many a times even meaningful) words and phrases, as indicated in the example below. This verse, presenting the value π — the most mysterious number — correct to 30 decimal places, is supposedly the composition of Sri Bharati Krishna Tirtha, a 20th century saint-mathematician, who was the Shankaracharya of Govardhana Peetham at Puri during the 1950s. Being composed in anushtubh metre, it has 8 syllables per quarter. When interpreted literally the verse speaks of the glory of Lord Vishnu and Shiva in one stroke.
O, the one who is the fortune honeybunch of the gopis, the one who is the lord of the bull, the one who resides in the interstice of ocean (and the land, at Rameswaram), the one who serves as pothole for the wicked ones (to trap and punish them), and the one who has the dreadful poison in the neck, may you protect me.
Setting aside the literary meaning, if the verse is decoded using the Katapayadi system, then it presents the following digits: 31415926 53589693 23846264 33832792, which actually forms the digits of π/10 (when expressed in decimal notation). At this stage, it would be highly instructive to imagine the problems that one may face, if one were to represent the above number using the cumbersome Roman numeral system!
The name Katapayadi stems from the fact that here groups of Sanskrit alphabets commencing from ka, ta, pa, ya are used to denote the numbers. According to this system, the vowels standing alone, represent the number zero. However, the same vowels in conjunction with the consonants have no numerical significance. It is only the 33 consonants ka, kha, ga…, sa, sa, ha that are associated with the numbers. The mapping of these consonants with different numbers is listed in the table below.
Unlike Katapayadi, in the Aryabhatan system, the rank of the vowel determines the power of 10 (a = 10⁰ , i = 10² , u = 10⁴ and so on), to be tagged to the consonant (representing the digits), and consequently very large numbers of the order of 10¹ ⁶ can be easily represented by a single syllable. For instance, the two-syllabled (synthetic) word khyughr represents
khyu+ghr = (30 + 2)X10⁴ + 4X10⁶ = 4320000.
This, incidentally, corresponds to the number of years in a Mahayuga (the four yugas — krta, treta, dvapara, kali put together). The three- syllabled word kaumudi (literally moonlight), when decoded in Aryabhatan system represents: kau+mu+di=1X10¹⁶+25X10⁴+18X10²= 10000000000251800.
Such a system of representing numbers (that is cute in its own way) was invented by Aryabhata in order to specify large numbers such as revolutions made by planets, their aphelions and nodes, made in a Mahayuga in a very efficient way (with minimal number of syllables in the string).
Solving linear equations
Linear equations are those in which the variable appears in its simplest form as x (not as x² or √x or such). As these equations — in one or many variables — occur in several contexts in our day to day life, it is extremely important that children acquire skills in handling them. Realising their potential application in a variety of fields, Indian mathematical texts devote a few chapters to discuss at length the different techniques by which linear equations can be effectively solved.
Unfortunately, in the standard school curriculum (as the author recalls from personal experience), this topic is dealt with in a drab manner, wherein the students are expected to solve problems monotonously, without having an inkling of their practical application. However, in the Indian math-ematical texts while dealing with this topic, the authors illustrate its application with plenty of very attractive examples drawn from day to day life. Here, we provide a couple of examples from Bhaskara's Lilavati to illustrate the point.
Example 1: Dealing with flora
From a bunch of lotuses (plucked from a lotus pond), one-third were offered to Shiva, one-fifth to Vishnu, one-sixth to the sun, and one-fourth to the doddess (Arya). With the remaining six, the feet to the guru was worshipped. Tell me quickly the total number of lotuses (that were plucked from the pond).
A representation of the huge lotus pond that is conceived by Bhaskaracharya in the above example is depicted in Figure 1. Let x be the total number of lotuses that were plucked from this pond. Then, as per the description given in the above verse, the number of lotuses that were offered to the four deities is
It is further stated that the remaining flowers were offered at the feet of the guru and that the number was 6. Hence we have the following equation,
which on solving gives x = 120.
Example 2: Dealing with fauna
Of a group of elephants, it was noted that half, and one-third of the half went into a cave. One-sixth of them, along with one-seventh of one-sixth were found drinking water in a river. One-eighth, and one-ninth of one-eighth were noted to be sporting in a pond full of lotuses. The king of elephants was found majestically marching with three female elephants following him. (If this was the scenario, let me know), what is the strength of the elephant herd?
If x is the total number of elephants in the herd, the number of elephants belonging to the four different groups engaging themselves in four different activities are denoted by the four terms in equation (1) given below. Since this is equal to the total sum x, we have the following equation
When simplifed and rearranged, the above equation reduces to
which yields x=1008
Solving quadratic equations
Consider a quadratic equation of the form Ax² + Bx = C, where we have put the constant on one side of the equation, and the unknown on the other. The solution to this is presented in the following verse:
But for certain finer details, the solution given here may be represented as:
In order to illustrate the application of the above formula, Bhaskara presents several interesting examples chosen from nature as well as itihasas and puranas. We present below one such example:
Arjuna turning furious in the war, shot a bunch of arrows in order to slay Karna. With half of the arrows in the quiver, he countered the arrows shot by Karna. With four times the square root of the (total) arrows he killed Karna's horses. He finished Shalya (Karna's charioteer) with six arrows. He used one arrow each to destroy the top of the chariot, the ag, and the bow of Karna. Finally, he severed the head of Karna with one arrow. How many arrows did Arjuna discharge?
Let x denote the total number of arrows discharged by Arjuna. The details of the purpose for which the arrows in the quiver were discharged by Arjuna, as narrated in the verse, when expressed in the form of an equation translate to:
With the substitution x=y², this reduces to the following quadratic equation:
whose solutions are y=(10,-2). Of the two, we can only consider the positive value, since the number of arrows shot cannot be negative. Hence, x= 100 is the answer.
Representing infinite series in the form of verses
The tradition of teaching and learning mathematics in metrical form was not confined to arithmetic, algebra and geometry. When the Kerala school of mathematicians arrived around the 14th century CE, whose contributions are marked with the transition from the finite to infinite (a benchmark of calculus), they could easily adapt themselves to comfortably express the infinite series for π and other trigonometric functions in the form of elegant verses. The following verse in arya metre, ascribed to Madhava, presents the infinite series for π:
The diameter multiplied by four and divided by unity (is found and stored). Again the products of the diameter (vyasa) and four are divided by odd numbers like three and five, and the results are subtracted and added in order (to the earlier stored result).
The above series generally goes by the name of Gregory-Leibniz series in mathematical literature, though Gregory and Leibniz (the famous European mathematicians of the 17th century) were to come almost three centuries after Madhava!
While introducing his work, Ganitasarasangraha (a compendium of the essence of mathematics), the 9th century Jaina mathematician, Mahavira, in order to convey the importance of learning mathematics, presents a long list of different branches of learning where mathematical principles are involved. Having spelled out a dozen topics that include art, dance, drama, economics, music, prosody, medicine and grammar to convey the indispensability of mathematics he finally observes:
Why keep talking much? In all the three worlds consisting of living and non-living entities, whatever be the transaction, it cannot be executed without mathematics!
This being the importance of mathematics, the surveys conducted by educationists today indicate that only a small fraction of primary and secondary level students really enjoy learning mathematics. The rest of them either find it boring, or consider it to be a burden imposed upon them —let alone appreciating the beauty (rhythm and rigour) of mathematics. Why is it the case that only a small section of students enjoys mathematics, whereas a large section shies away from it? While there could be many reasons, one of the primary ones has to do with the irksome way in which mathematics is being taught in the schools.
Many instructors of mathematics mechanically teach a large number of rules and formulae without providing any kind of historical background or motivation to learn the topic. Even in the tutorial sessions (classwork/homework), students are expected to work out exercises that merely require application of such formulae in its various avatars.
Consequently, they hardly get an opportunity to appreciate, the relevance of the principles and formulae they learn. Unless the tutor presents examples from practical life, the children wouldn’t find the subject interesting, and it is quite natural for them to feel exasperated while learning ‘abstract’ mathematics. On the other hand, if one could present illustrative examples —drawn from various aspects of day to day life as cited above —the level of abstraction in learning mathematics can be significantly brought down.
Taking their cue in Sanskrit, attempts have been made to blend mathematics with poetry in other local languages such as Telugu, Bengali, Tamil and Kannada, and they have been quite successful too. Mathematics presented in metrical form, when included in the school curricula (at least excerpts from them) is likely to reduce the burden of children in trying to remember and recollect various formulae, which is an indispensible part of their learning.
Moreover, the very act of memorising verses (poetry), and recalling them to retrieve the formulae coded in them, is by itself a very productive activity of the brain. Besides contributing to language skills, this process would indeed generate a lot of fun, and is likely make more students experience an Aha! moment rather than an Oho! one while learning mathematics, and it is definitely worth trying.
(K Ramasubramanian is a professor at Cell for Indian Science and Technology in Sanskrit (CISTS), dept of Humanities and Social Sciences, IIT Bombay)
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