By Narinder Jarial
Filed under: Awesome Numbers, Chronicles of Bopa Rai, Feminine Power, Mythology
Prelude — The Woman Who Counted the Stars
In the shadow of Ujjain’s ancient observatory, where the stars wheeled like unanswered questions across the monsoon sky, Acharya Lila Devi traced planetary arcs with a reed pen dipped in lampblack.
Born to the legendary Bhāskara II, her legacy was not merely the Lilavati—that luminous treatise on arithmetic and geometry named in her honor—but a defiant intellect. She wove Ptolemy’s epicycles with Brahmagupta’s cyclic quadrilaterals, birthing tables of celestial motion precise enough to predict eclipses from Kashmir to the Coromandel.
In a realm where scrolls were guarded by monasteries and men, she donned the saffron of an Acharya. Her voice cut through assemblies like a comet’s tail, proving that measure belonged not to muscle, but to the mind that held the heavens in balance.
The Poetic Mathematics
To Lila, mathematics was not dry logic; it was the language of nature. She presented problems as poetic challenges, often teasing her stoic guardian and consort, Bopa Rai, to see the world through numbers rather than the sword.
1. The Swarm of Bees (Quadratic Beauty)
One evening, pointing to the jasmine vines, she posed this riddle:
“A swarm of bees… one-fifth of them settled on a Kadamba blossom; one-third on a Silindhri flower; three times the difference between those numbers flew to the Kutaja bloom. One bee remained, hovering in the air, lured by the fragrance of jasmine. Tell me, Bopa, the number of bees.”
The Solution:
Lila sketched the algebra in the dust. Let the total number of bees be $$x$$.
The equation forms as:
$$x – \left(\frac{x}{5} + \frac{x}{3} + 3\left(\frac{x}{3} – \frac{x}{5}\right)\right) = 1$$
Simplifying the term inside the bracket:
$$\text{Difference} = \frac{x}{3} – \frac{x}{5} = \frac{2x}{15}$$
$$3 \times \text{Difference} = \frac{6x}{15} = \frac{2x}{5}$$
Substituting back:
$$x – \left(\frac{x}{5} + \frac{x}{3} + \frac{2x}{5}\right) = 1$$
$$x – \left(\frac{3x + 5x + 6x}{15}\right) = 1$$
$$x – \frac{14x}{15} = 1$$
$$\frac{x}{15} = 1 \implies x = 15$$
The Answer: There were 15 bees. A simpler solution would be to add all the fractions and subtract them from One. This will give a reciprocal of the total number of bees or 1/15.
2. The Beautiful Maiden ( The Method of Inversion)
Lila often taught the art of working backward—a skill Bopa used in tracking enemies.
“A beautiful maiden with beaming eyes asks: which is the number that when multiplied by 3, then increased by three-fourths of the product, divided by 7, diminished by one-third of the quotient, multiplied by itself, diminished by 52, the square root found, the addition of 8, and division by 10 gives the number 2?”
The Solution:
We start from the result (2) and reverse every operation:
- Result: $$2$$
- Multiply by 10: $$2 \times 10 = 20$$
- Subtract 8: $$20 – 8 = 12$$
- Square it: $$12^2 = 144$$
- Add 52: $$144 + 52 = 196$$
- Take Square Root: $$\sqrt{196} = 14$$
- Revert “diminished by 1/3”: This means the number is $$2/3$$ of the previous step. So, $$14 \times \frac{3}{2} = 21$.
- Multiply by 7: $$21 \times 7 = 147$$
- Revert “increased by 3/4”: This is $$1 + 0.75 = 1.75$$ or $$7/4$$. So, $$147 \times \frac{4}{7} = 84$$.
- Divide by 3: $$84 / 3 = 28$$.
The Original Number: 28.
3. The Geometry of the Hunt (The Peacock and the Snake)
The fire at the Tapti ashram had burned down to embers. Bopa Rai sat by the riverbank, sharpening his short sword. Lilavati approached, drawing a diagram in the sand to explain the Pythagorean theorem.
“Imagine, Bopa, a peacock sits atop a pillar of height $$h$$. It spots a snake at a distance $$x$$ from the base, moving toward its hole at the pillar’s foot. The peacock swoops diagonally; the snake moves straight. They move at equal speeds and meet at the same instant. How far from the hole do they meet?”
The Solution:
If they meet at distance $$d$$ from the hole:
- The snake travels: $$x – d$$
- The peacock travels the hypotenuse: $$\sqrt{h^2 + d^2}$$
Since speeds (and thus distances) are equal:
$$\sqrt{h^2 + d^2} = x – d$$
Squaring both sides:
$$h^2 + d^2 = (x – d)^2$$
$$h^2 + d^2 = x^2 – 2xd + d^2$$
The $d^2$ cancels out. Solving for $d$:
$$h^2 = x^2 – 2xd$$
$$2xd = x^2 – h^2$$
$$d = \frac{x^2 – h^2}{2x}$$
“You see, Bopa?” she smiled. “The arrow and the algebra obey the same laws.”
The Pilgrimage to Nalanda
Her pilgrimage to Nalanda came as whispers of smoke gathered on distant horizons. The great vihara was still pulsing with 10,000 scholars debating Greek logic and Gupta sutras amid nine-story libraries heavy with the scent of palm-leaf wisdom.
There, amid the Ratnodadhi’s vaulted halls, Lila borrowed a cart groaning under manuscripts—Aryabhata’s revolutions, Varahamihira’s omens—adding her own signed reckonings on lunar nodes. This was mere months before the Turkish raiders’ flames would claim it all in 1193. As embers rained like fallen gods, she urged her students:
“Knowledge is no fragile vase, but seed scattered on wind — let it root where fire cannot reach.”
The Cartload to Tapti Ashram
Lila had married despite her father’s dire predictions. The horoscopes had screamed of loss, but they had not accounted for Bopa Rai.
The “groom of doom” turned out to be the “anointed protector.” Bopa, a warrior rising rapidly in the hierarchy, had no inkling of his celestial destiny, but he possessed the hallmarks of a man in love. He guarded the ashram’s perimeter, his warrior ascent funding ink and reeds, while Lila laughed at the cosmos’s jest: a groom deemed a curse had become heaven’s unwitting vault-keeper.
From their new sanctuary on the banks of the Tapti, Lila’s students—the proto-cabal—fanned out as wandering bards, embedding sine quadrants in doha verses and lunar nodes in temple carvings.
Lila’s Final Lesson: Stars Over Tapti
Beneath a canopy of unblemished night, where the Tapti flows 724km westward from the Satpura Range to the Arabian Sea, Acharya Lila Devi gathered her pupils atop a mud-ramped gnomon.
“Compute the Moon’s node, Bopa,” she commanded, pointing to the shadow devouring the sister orb. “Use the wisdom of Brahmagupta.”
She wrote the final lesson on the slate of the night:
$$\tan(\phi – \delta) = \frac{h}{s}$$
Where $\phi$ is the latitude (~21°N at Tapti), $\delta$ is the solar declination swinging $\pm 23.5^\circ$, $h$ is the gnomon height, and $s$ is the shadow length.
The cabal recorded the verse by lamplight. The seed was planted. The Renaissance had begun, hidden in the rhyme of a woman who counted the stars.