Having bowed to the deity, whose head is like an elephant; whose feet are adorned by gods; who, when called to mind, relieves his votaries from embarrassment; and bestows happiness on his worshipers; I propound this easy process of computation, delightful by its elegance, perspicuous with words concise, soft and correct, and pleasing to the learned.
So begins Bhaskara’s Lilavati in Henry T. Colebrooke’s classic late 19th century translation. There’s an earlier Bhaskara, also famous, so Lilavati’s Bhaskara is sometimes referred to as Bhaskara II or Bhaskaracharya. He was born in 1114 CE in Vijayapura (modern-day Bijapur, Karnataka), India. It was a quiet time. The quiet, that is, of a hurricane’s eye. In 1114 CE, Angkor Wat was still an idea searching for its stone, but its future builder, Suryavarman II, had just ascended the throne.
Genghis Khan would be born some 50 years later. The West hadn’t rediscovered Euclid and Aristotle yet, but Adelard of Bath and his students were in Syria poring over the Arabic texts that would eventually ignite the Western Renaissance. Muslims had gained a foothold in Gujarat, and in the 13th through 15th centuries, they would both ravage and reinvigorate the subcontinent. And way north of Vijayapura, about 400 miles from Delhi, the last of the magnificent Khajuraho temples was being built.
For the duffer
Why did Bhaskara write Lilavati? Simple. To teach duffers. As he concludes in Bijaganita (one of his six works):
“A morsel of tuition conveys knowledge to a comprehensive mind; and having reached it, expands of its own impulse. As oil poured upon water, as a secret entrusted to the vile, as alms bestowed upon the worthy, however little, so does knowledge infused into a wise mind spread by intrinsic force…. What is there unknown to the intelligent? Therefore for the dull alone it is set forth.”
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Okay, that should put students in their proper place. Gurus have traditionally taken a dim view of teaching anyone anything. Almost 650 years after Bhaskara, Edward Gibbons was even more pessimistic in his Decline and Fall: “The power of instruction is seldom of much efficacy except in those happy dispositions where it is almost superfluous.”
He was referring to Commodus, the thuggish son of the revered philosopher-king Marcus Auerlius. But Gibbons’ conclusion has continued to resonate in modern times. The late Richard Feynman—by all accounts, a great teacher—cites Gibbons with glum relish in the preface to his celebrated Lectures.
“The problems are generally addressed to one Lilavati, traditionally taken to be either his wife or his daughter. Tradition is as good a reason as any because there’s no other reason to support the claim.”
Lilavati is a collection of worked-out examples in algebra and geometry. The level of mathematics ranges between high-school algebra and freshman pre-calculus. In its time, it represented the height of 12th century mathematics. The problems are generally addressed to one Lilavati, traditionally taken to be either his wife or his daughter. Tradition is as good a reason as any because there’s no other reason to support the claim. To modern ears, the opening phrases of some of the problems are decidedly intimate. Problem 2.2.16 begins with:
“Beautiful and dear Lilavati, whose eyes are like a fawn’s….” Then there’s Problem 3.1.49, which begins: “Pretty girl, with tremulous eyes, if thou know the correct method of inversion….”
And how can I overlook Problem 3.5.68?
“The square root of half the number of a swarm of bees is gone to a shrub of jasmine; and so are eight-ninths of the whole swarm: a female is buzzing to one remaining male that is humming within a lotus in which he is confined, having [been] allured to it by its fragrance at night. Say, lovely woman, the number of bees.”
Colebrooke, thorough as always, notes that the “jasmine” referred to is “jasminum grandiflorum”. And Ganesa Daivajna, in his Buddhivilasini (1545 CE), supplies some context: “…the lotus being open at night and closed in the day, the bee might be caught in it.” Indeed. The good professor’s concern is not misplaced. The hazards of being a bee are many. For poor Colebrooke, the text must have made many a warm Calcutta night even warmer.
Walk in the park
In Bhaskara’s time, “shyness”—at least in Sanskrit poetics—was as much a performance, as say, cockfighting. Bees, lotuses, plantain stalks, jasmine, etc. are all conventional metaphors in Sanskrit poetry. For example, here’s one of Vittoka’s verses (verse 387 in Ingall’s translation of Vidyakara’s Treasury):
“I saw a golden staff wandering in the park
on which a wondrous lotus ever bloomed;
herein two bees; above a moon half full;
and still above, massed darkness,
which yet remained both day and night.”
In case you’re wondering, the poet is talking about a young woman. The golden staff is her body, the lotus is her face, the two bees are her eyes, the “moon half full” is her fair forehead, and the massed darkness, both day and night, is her bejewelled hair. Sometimes the eyes are waterlilies, the breasts are always like golden pots, plantain stalks refer to thighs, and so on and so forth. When done poorly, Sanskrit poetry is akin to medieval European painting with its relentless in-your-face symbolism.
Artist Hemen Mazumdar’s untitled oil popularly known as “Woman in Moonlight”, painted in the 1930s. The sensuous woman would remind one of the nayikas of Sanskrit poetry.
| Photo Credit:
By Special Arrangement
But I digress. What do we want from a teacher? Jean-Jacques Rousseau, who offloaded his own kids in an orphanage because he was busy writing a book on raising children, put it bluntly: “The art of teaching consists in making the pupil wish to learn.” By that criterion, Bhaskara’s problem exercises excite a real passion for mathematics. Who, but one with an adamantine heart, can resist the call of a verse such as this:
“Whilst making love a necklace broke.
A row of pearls mislaid.
One third fell to the floor.
One fifth upon the bed.
The young woman saved one sixth of them.
One tenth were caught by her lover.
If six pearls remained upon the string
How many pearls were there altogether?”
The above problem is not from Lilavati proper. It’s from Manoranjana, a commentary written by one Ramakrishna Deva. Colebrooke spotted its connection with Bhaskara’s Problem 3.2.54 and decided—god bless him—to footnote it.
Lilavati is not exactly Aguilera-dirrty. I’m sorry to report that problems involving randy geese, bees trapped in lotuses, girls with tremulous eyes, etc. are relatively few in number. One of the problems (3.6.75) is downright chilling:
“If a female slave, sixteen years of age, brings thirty-two (nishkas), what will one aged twenty cost? If an ox which has been worked two years sells for four nishkas, what will one which has been worked six years cost?”
But most of the problems would fit in today’s high-school algebra textbooks. They are the kind of problems that Frigyes Karinthy made such fun of in his delightful short story, “The Refund”:
“If we represent the speed of light by X and the distance of the star Sirius from the Sun by Y, what is the circumference of a one-hundred-and-nine-sided regular polyhedron whose surface area coincides with that of the hip-pocket of a state railway employee, whose wife has been deceiving him for two years and eleven months with a regimental sergeant major of hussars.”
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If you answered “28 apricots”, you are wrong! It’s 27 apricots!
Lilavati is now a mood, a symbol, a place, and an attitude; a woman. For me, it’s hard to believe it was written almost a thousand years ago. Much has changed in the world since it was written. These days, it is kids who despair about educating the adults. Angkor Wat and Khajuraho are now ruin and memory.
But poetry endures and the time-laden ponds remain. So do the lotuses and bees, forever getting entangled in signs, sighs, and mathematics.
Anil Menon is a novelist, whose most recent work is The Coincidence Plot.
source: https://frontline.thehindu.com/society/bhaskara-lilavati-ancient-indian-mathematics-poetry/article69261292.ece
