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Lilavati by Bhaskara Acharya
Lilavati is a 12th-century mathematical treatise written by Bhaskara II (Bhaskaracharya), one of India’s greatest mathematicians and astronomers. The book, part of his larger work Siddhanta Shiromani, is a collection of problems in arithmetic, algebra, geometry, and mensuration, presented in poetic form and often framed as riddles and stories addressed to his daughter Lilavati.
Conclusion
Lilavati is both a mathematical manual and a literary masterpiece, blending rigorous calculations with poetic charm. It teaches core arithmetic—fractions, zero, square roots, cube roots—alongside geometry, algebra, permutations, and combinations. The book is especially famous for its word problems involving merchants, travelers, animals, and daily life, which made abstract mathematics relatable. Bhaskara’s explanations of zero and infinity were groundbreaking for the 12th century, influencing later Indian and Islamic scholars. His style encouraged problem-solving, creativity, and logical reasoning, rather than rote memorization. The legacy of Lilavati is its human touch: mathematics is taught not only as rules but also as puzzles for joy and intellectual delight.
Key points
📜 Poetic teaching: The problems are written in verse, making them easy to memorize and aesthetically beautiful.
👧 Dedicated to Lilavati: Tradition says Bhaskara wrote it for his daughter after her marriage was disrupted, though historians debate this as legend.
➗ Arithmetic foundations: Covers addition, subtraction, multiplication, division, fractions, ratios, and proportions.
🔢 Innovations with zero: Explains operations involving zero and discusses the concept of infinity—rare in world mathematics at that time.
📐 Geometry and mensuration: Includes problems on areas, volumes, and practical measurements such as wells, walls, and pillars.
🧩 Algebraic problems: Introduces quadratic equations, simple algebraic identities, and methods of solving unknowns.
🚂 Word problems: Famous riddles involve boats, bees, birds, and travelers—turning abstract numbers into lively stories.
🌍 Astronomical context: Although mainly arithmetic, it links to Bhaskara’s larger astronomical work, showing application to calendars and time.
📚 Pedagogical legacy: It served as the standard arithmetic textbook in India for centuries, influencing Persian, Arabic, and later European translations.
🎭 Blend of art and science: Shows how medieval India united literature, education, and mathematics into one harmonious discipline.
Summary
- Lilavati begins with an invocation to Ganesha and Saraswati, stressing the harmony of divine knowledge and mathematics. The problems are expressed as verses, which could be sung or recited.
- The book presents arithmetic as the foundation of all higher sciences, giving systematic rules for operations with whole numbers, fractions, and ratios.
- Bhaskara includes clear rules for multiplication, division, and handling large numbers—crucial for astronomy, trade, and land measurement.
- Unique emphasis is placed on zero, distinguishing between meaningful operations (like adding zero) and undefined ones (like division by zero).
- Geometry is taught through practical examples: calculating the volume of granaries, the depth of wells, or the area of fields.
- Word problems are a hallmark: examples include bees in flowers, swans in ponds, or merchants dividing wealth. These transformed abstract theory into relatable contexts.
- Algebra is introduced with quadratic equations, often in the form of riddles about ages, distances, or shares of wealth.
- The book touches on permutations and combinations, illustrating the beginnings of combinatorial thought in Indian mathematics.
- Bhaskara emphasizes problem-solving over rote rules, encouraging learners to think critically and derive solutions themselves.
- The legacy of Lilavati extended beyond India—it was translated into Persian, Arabic, and English, admired by mathematicians worldwide for its beauty and depth.
Quotes from
Lilavati
by Bhaskara Acharya 📜
Here’s a curated collection of 18 quotes/problems from Lilavati. These are translations or faithful renderings of the Sanskrit verses, written as poetic riddles blending mathematics with everyday imagery.
🌑 Zero and Infinity
- “A quantity divided by zero becomes a fraction with zero as denominator—
a quantity of infinite magnitude.” - “Divide the stars of the night by zero—
their number becomes countless, infinite.”
🐝 Bees and Flowers
- “Beautiful maiden, tell me: a number of bees equal to the square root of half the swarm
flies to a lotus in bloom.
One-eighth part goes to another flower,
and one bee, being left, hovers in the air.
Tell me, charming girl, the number of bees.”
🦢 Swans on the Lake
- “A number of swans, seeing twice as many more,
went to sport on a lake.
One more swan remained on the shore.
Tell me, Lilavati, how many swans were there?”
💎 Pearls and Division
- “Lovely maiden, a merchant bought a pearl necklace.
He gave half to his wife,
a sixth to his son,
and a twelfth to his daughter.
Six pearls still remained in his hand.
Tell me, how many pearls were there at first?”
🌸 The Lotus and the Lake
- “A lotus, whose stalk is 3 cubits above the water,
bends over when moved by the wind.
Its tip touches the water 9 cubits distant from the base.
Tell me, Lilavati, what is the depth of the water?”
💰 Merchants and Wealth
- “O fair one with enchanting eyes,
a merchant gains 9 coins daily,
but loses 4 coins in the same time.
In how many days will his gain be 105 coins?”
🕊️ Pigeons in a Tree
- “A flock of pigeons, one-third alighted on a branch,
one-fifth flew to a hollow,
one-sixth settled on the ground.
The remaining 30 birds remained in the sky.
Tell me, Lilavati, how many pigeons were in the flock?”
🐪 Caravans and Travel
- “A caravan travels 20 yojanas in a day.
Half the journey was completed in 15 days.
How long will the remainder take?”
➕ Simple Series
- “O Lilavati, intelligent girl, if you know addition and subtraction,
tell me the sum of the amounts 2, 4, 6, 8, up to 20.”
🌾 Granaries and Sharing
- “A man had a heap of grain.
He gave one-third to his friend,
one-sixth to a beggar,
and one-twelfth to the temple.
He still had 15 measures left.
Tell me, Lilavati, the measure of the heap.”
🐦 Birds in Trees
- “Of a flock of birds, half flew to a tree,
a third flew to another,
and the rest, being ten, stayed on the ground.
Tell me, dear maiden, how many were there in the flock?”
🎲 Dice and Gambling
- “A gambler, having lost half his stake and then a fourth more,
still has 6 coins left.
Tell me, beautiful maiden, what was his stake at first?”
🌺 Garland of Flowers
- “If a number of flowers be multiplied by zero,
how many garlands can be made?”
🕊️ Doves and Sharing
- “A flock of doves was seen.
Half as many more joined them,
and then 8 more came.
Their number was then 100.
Tell me, fair maiden, how many were there at first?”
📐 Geometry Riddle
- “A bamboo 32 cubits high was broken by the wind.
Its top touched the ground at a distance of 16 cubits from the root.
Tell me, Lilavati, where was it broken?”
🌙 Phases of the Moon
- “The moon waxes by 1 digit daily for 15 days,
then wanes in the same order.
How many digits does it have at full moon?”
🎭 Playful Identity with Zero
- “O beautiful maiden, if zero be added to a number,
what remains? The number itself remains, unchanged.”
✨ Together, these quotes show how Bhaskara made arithmetic, algebra, geometry, and fractions engaging through poetry and imagination.
