Bhaskara II’s Lilavati: Complete Reference Guide (Download PDF)

I first encountered the Lilavati as a student in India, and it completely changed how I thought about mathematics education. Here’s a 12th-century textbook that wraps arithmetic, algebra, and geometry inside riddles about bees, snakes, peacocks, and beautiful maidens. Bhaskara II (also known as Bhaskaracharya, 1114-1185 CE) didn’t write a dry reference manual. He wrote poetry that teaches you math. And it’s been studied for over 850 years. The Lilavati book, written by Bhaskaracharya, remains one of the most influential mathematical texts in history, studied across Asia and eventually Europe.

What makes the Lilavati remarkable isn’t just the math itself, though the math is extraordinary. It’s the pedagogical approach. While other mathematicians of the era wrote dense technical manuals, Bhaskara embedded his mathematics in vivid word problems that made abstract concepts tangible and memorable. A calculation about ratios became a puzzle about monkeys playing in a garden. A geometry problem transformed into a peacock hunting a snake. From zero and negative numbers to iterative algorithms and the Chakravala method, the Lilavati contains ideas that anticipated European mathematics by centuries.

This guide covers everything you need to know about Lilavati: the mathematician behind the text, the legend of how it got its name, detailed modern solutions to its most famous problems, the mathematical methods it presents, its lasting influence on world mathematics, and where to find the Lilavati book PDF in English and Sanskrit. I’ve included complete worked solutions so you can follow along with pen and paper.

Bhaskaracharya: The Mathematician Behind Lilavati

Biographical Overview

Attribute	Details
Full Name	Bhaskaracharya (Bhaskara II)
Born	1114 CE; Vijjalavida, Karnataka, India
Died	c. 1185 CE; Ujjain, Madhya Pradesh, India
Father	Mahesvara (mathematician and astronomer)
Position	Head of Astronomical Observatory, Ujjain
Known for	Siddhanta Shiromani, Lilavati, Bijaganita; Chakravala method, early concepts of calculus

Bhaskaracharya was born in 1114 CE in Vijjalavida (in the modern-day Bijapur district of Karnataka, India). His father, Mahesvara, was also a mathematician and astronomer, which meant Bhaskara grew up surrounded by mathematical manuscripts and astronomical instruments. This intellectual environment shaped his extraordinary career.

By the age of 36, Bhaskara II had completed his magnum opus: the Siddhanta Shiromani (“Crown of Treatises”). This was not a single book but four interconnected volumes covering different branches of mathematics and astronomy. The Lilavati mathematician biography typically focuses on his work in calculus, where he discovered differential calculus concepts centuries before Newton and Leibniz formalized them in Europe.

Bhaskara II served as the head of the astronomical observatory at Ujjain – the same prestigious position once held by the legendary mathematician Brahmagupta – until his death around 1185 CE. His works were translated into Persian, Arabic, and eventually European languages, influencing mathematical development across cultures.

Mathematical Achievements

What set Bhaskara apart from his contemporaries was not only the depth of his mathematical knowledge but his writing style. While other mathematicians wrote dry technical manuals, Bhaskara embedded his mathematics in vivid word problems. A calculation about ratios became a puzzle about monkeys playing in a garden. A geometry problem transformed into a peacock hunting a snake. His contributions span multiple domains:

• Arithmetic and Algebra: Systematic treatment of operations with zero, negative numbers, surds (irrational numbers), and indeterminate equations.
• Calculus: Discovery of differential calculus concepts – including the derivative and differential coefficient – centuries before Newton and Leibniz formalized them in Europe.
• Astronomy: Accurate calculations of planetary positions, eclipses, and the length of the tropical year.
• Number Theory: The Chakravala method for solving Pell’s equation, not rediscovered in Europe until Lagrange’s work in the 18th century.

The Siddhanta Shiromani

The Siddhanta Shiromani consists of four parts:

The Lilavati, as the first and most accessible volume, became by far the most widely read and studied. Its influence on mathematics education persisted for over eight centuries across the Indian subcontinent and beyond.

The Legend of Lilavati

The name “Lilavati” means “beautiful” or “playful” in Sanskrit. According to a celebrated legend, it was the name of Bhaskara’s daughter.

The Legend: Bhaskara, being an accomplished astronomer, cast his daughter Lilavati’s horoscope and discovered she had a narrow window for an auspicious marriage. He designed a water clock to track the precise moment. As Lilavati watched the device, a pearl from her dress fell into the water and blocked the hole, stopping the clock. The auspicious moment passed unnoticed. Lilavati’s marriage never happened.

To console his daughter, Bhaskara wrote a mathematics treatise in her name, addressing many problems directly to her with affectionate phrases such as “Oh Lilavati, intelligent girl…” and “Tell me, beautiful one…”

Whether this story is historically accurate remains debated among scholars; some think it is a later embellishment. But the affectionate tone throughout the Lilavati granth original text is undeniable. Bhaskara wanted his mathematics to be engaging, personal, and memorable. This personal touch made the book feel less like a textbook and more like a conversation between father and daughter.

The Structure of Lilavati

The Lilavati contains thirteen chapters, progressing from basic arithmetic to advanced problem-solving techniques. We summarize each chapter below.

Chapter 1: Definitions and Basic Operations

The opening chapter establishes fundamental terms and operations. Bhaskara defines numbers, place values, and the basic operations of addition, subtraction, multiplication, and division. He also introduces the concept of zero, which Indian mathematicians had developed centuries earlier.

Chapter 2: Fractions

Operations with fractions are covered in detail, including addition, subtraction, multiplication, division, and simplification, all presented through practical examples involving trade, measurement, and distribution.

Chapter 3: Zero and Negative Numbers

Bhaskara on Zero: Bhaskara explores the properties of zero, including division by zero. He correctly identifies that \( \dfrac{a}{0} \) leads to infinity (which he calls khahara), writing:

“A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity.”

While not fully rigorous by modern standards, this showed remarkable insight into the concept of limits.

Bhaskara also addresses operations with negative numbers, treating negative quantities as natural mathematical objects with consistent operational rules – something European mathematicians struggled to accept well into the 18th century.

Chapter 4: Square and Cube Roots

Bhaskara presents efficient algorithms for calculating square roots and cube roots of integers and fractions. His iterative methods remained standard practice in Indian mathematics education for centuries.

Chapter 5: Arithmetic Progressions

This section covers sequences where each term increases by a constant difference. Bhaskara provides the formula for the sum of an arithmetic series:

Sum of an Arithmetic Progression:

$$ S_n = \frac{n}{2}\bigl(2a + (n-1)d\bigr) $$

where \( n \) is the number of terms, \( a \) is the first term, and \( d \) is the common difference.

Chapter 6: Geometric Progressions

Bhaskara extends to sequences where each term is multiplied by a constant ratio:

$$ S_n = a \cdot \frac{r^n – 1}{r – 1} $$

where \( r \) is the common ratio. This formula, known in India well before Bhaskara’s time, was presented by him with particularly clear examples.

Chapter 7: Plane Geometry

This substantial chapter covers properties of triangles, quadrilaterals, and circles. Bhaskara presents formulas for areas, perimeters, and relationships between geometric figures.

Chapter 8: Solid Geometry

Volumes and surface areas of three-dimensional shapes including spheres, cones, and pyramids. Bhaskara uses the approximations \( \pi \approx \frac{22}{7} \) (for rough calculations) and the more accurate \( \pi \approx \frac{355}{113} \).

Chapter 9: The Shadow of the Gnomon

This chapter applies geometry to astronomical measurements using shadow calculations. A gnomon is a vertical stick whose shadow length changes with the sun’s position.

Chapter 10: The Pulverizer (Kuttaka)

This technique, refined over centuries by Indian mathematicians from Aryabhata onwards, produces integer solutions when they exist. Bhaskara’s treatment is among the most complete and systematic.

Chapter 11: Combinations and Permutations

Bhaskara presents methods for counting arrangements and selections:

Bhaskara expresses this through examples involving selections of tastes (sweet, sour, bitter, etc.) and provides a systematic method for enumeration.

Chapter 12: Net of Numbers

This chapter covers magic squares and number patterns with specific properties, demonstrating Bhaskara’s interest in recreational mathematics alongside serious computation.

Chapter 13: Miscellaneous Problems

The final chapter presents diverse word problems combining techniques from earlier chapters. Many of the Lilavati‘s most famous puzzles appear here.

Sample Problems from Lilavati with Detailed Solutions

The best way to appreciate the Lilavati is to work through its problems. We present several classic examples with complete modern solutions, mathematical analysis, and comparisons to modern methods.

Problem 1: The Beautiful Maiden’s Riddle

This is perhaps the Lilavati‘s most famous problem.

Solution. The solution requires working backwards through each operation, undoing them one at a time.

Step 1: Start with the final result: \( 2 \).

Step 2: Undo “division by 10”: \( 2 \times 10 = 20 \).

Step 3: Undo “addition of 8”: \( 20 – 8 = 12 \).

Step 4: Undo “square root found”: \( 12^2 = 144 \).

Step 5: Undo “diminished by 52”: \( 144 + 52 = 196 \).

Step 6: Undo “multiplied by itself”: \( \sqrt{196} = 14 \).

Now we must undo the chain of operations applied to the unknown number \( n \):

Step 7: The operations on \( n \) are:

$$ \begin{aligned} \text{Multiply by 3:} &\quad 3n \\ \text{Increase by } \tfrac{3}{4} \text{ of the product:} &\quad 3n + \tfrac{3}{4}(3n) = 3n + \tfrac{9n}{4} = \tfrac{21n}{4} \\ \text{Divide by 7:} &\quad \frac{21n}{4 \times 7} = \frac{3n}{4} \\ \text{Diminish by } \tfrac{1}{3} \text{ of the quotient:} &\quad \frac{3n}{4} – \frac{1}{3} \cdot \frac{3n}{4} = \frac{3n}{4} – \frac{n}{4} = \frac{n}{2} \end{aligned} $$

Step 8: Setting \( \dfrac{n}{2} = 14 \):

$$ \boxed{n = 28} $$

Verification: Starting with \( n = 28 \):

• \( 28 \times 3 = 84 \)
• \( 84 + \frac{3}{4}(84) = 84 + 63 = 147 \)
• \( 147 \div 7 = 21 \)
• \( 21 – \frac{1}{3}(21) = 21 – 7 = 14 \)
• \( 14 \times 14 = 196 \)
• \( 196 – 52 = 144 \)
• \( \sqrt{144} = 12 \)
• \( 12 + 8 = 20 \)
• \( 20 \div 10 = 2 \) ✓

Problem 2: The Peacock and the Snake

Solution. Let \( x \) be the distance from the base of the pillar where they meet.

• The snake travels from 27 feet away toward the base, covering a distance of \( (27 – x) \) feet.
• The peacock flies diagonally from the top of the 9-foot pillar to the meeting point on the ground. By the Pythagorean theorem, the peacock covers \( \sqrt{9^2 + x^2} = \sqrt{81 + x^2} \) feet.

Since both travel at the same speed and meet simultaneously:

$$ 27 – x = \sqrt{81 + x^2} $$

Squaring both sides:

$$ (27 – x)^2 = 81 + x^2 $$

$$ 729 – 54x + x^2 = 81 + x^2 $$

The \( x^2 \) terms cancel:

$$ 729 – 54x = 81 $$

$$ 648 = 54x $$

$$ \boxed{x = 12} $$

Verification:

• Snake’s distance: \( 27 – 12 = 15 \) feet.
• Peacock’s distance: \( \sqrt{81 + 144} = \sqrt{225} = 15 \) feet. ✓

Problem 3: The Bees and the Lotus

Solution. Let \( n \) be the total number of bees.

$$ \begin{aligned} \text{Bees to Kadamba:} &\quad \frac{n}{5} \\ \text{Bees to Silandhara:} &\quad \frac{n}{3} \\ \text{Difference:} &\quad \frac{n}{3} – \frac{n}{5} = \frac{5n – 3n}{15} = \frac{2n}{15} \\ \text{Three times the difference (to arbor):} &\quad 3 \times \frac{2n}{15} = \frac{6n}{15} = \frac{2n}{5} \\ \text{Remaining bee:} &\quad 1 \end{aligned} $$

The sum of all parts equals the total:

$$ \frac{n}{5} + \frac{n}{3} + \frac{2n}{5} + 1 = n $$

Finding a common denominator of 15:

$$ \frac{3n}{15} + \frac{5n}{15} + \frac{6n}{15} + 1 = n $$

$$ \frac{14n}{15} + 1 = n $$

$$ 1 = n – \frac{14n}{15} = \frac{n}{15} $$

$$ \boxed{n = 15} $$

Verification: Kadamba: \( 3 \); Silandhara: \( 5 \); Arbor: \( 6 \); Remaining: \( 1 \). Total: \( 3 + 5 + 6 + 1 = 15 \). ✓

Problem 4: The Broken Bamboo

Solution. Let \( x \) be the height at which the bamboo was broken. Then:

• The standing portion has height \( x \).
• The fallen portion has length \( 32 – x \) (the top part that bent over).
• The fallen portion, the ground, and the standing portion form a right triangle.

By the Pythagorean theorem:

$$ x^2 + 16^2 = (32 – x)^2 $$

$$ x^2 + 256 = 1024 – 64x + x^2 $$

The \( x^2 \) terms cancel:

$$ 256 = 1024 – 64x $$

$$ 64x = 768 $$

$$ \boxed{x = 12} $$

The bamboo was broken at a height of 12 cubits.

Verification: Standing portion: 12. Fallen portion: 20. Check: \( 12^2 + 16^2 = 144 + 256 = 400 = 20^2 \). ✓

Problem 5: The Lotus in the Pool

Solution. Let \( d \) be the depth of the pool. The lotus stem has total length \( d + \frac{1}{2} \) (depth plus the half-cubit above water).

When the lotus is blown to the side:

• The stem now acts as the hypotenuse of a right triangle.
• The vertical leg is \( d \) (the depth).
• The horizontal leg is \( 2 \) cubits.
• The hypotenuse is \( d + \frac{1}{2} \).

By the Pythagorean theorem:

$$ d^2 + 2^2 = \left(d + \frac{1}{2}\right)^2 $$

$$ d^2 + 4 = d^2 + d + \frac{1}{4} $$

$$ 4 = d + \frac{1}{4} $$

$$ d = 4 – \frac{1}{4} = \frac{15}{4} $$

$$ \boxed{d = 3\tfrac{3}{4} \text{ cubits}} $$

Verification: Stem length: \( \frac{15}{4} + \frac{1}{2} = \frac{17}{4} \). Check: \( \left(\frac{15}{4}\right)^2 + 4 = \frac{225}{16} + \frac{64}{16} = \frac{289}{16} = \left(\frac{17}{4}\right)^2 \). ✓

Problem 6: The Monkey and the Tree

Solution. In each leap, the net progress is \( 5 – 2 = 3 \) cubits. However, the final leap requires special attention: once the monkey reaches or exceeds 100 cubits in its upward phase, it does not slip back.

After \( k \) complete “leap-and-slip” cycles, the monkey is at height \( 3k \) cubits. The monkey reaches the top during the upward phase of a leap when:

$$ 3k + 5 \geq 100 $$

$$ 3k \geq 95 $$

$$ k \geq \frac{95}{3} = 31\tfrac{2}{3} $$

So after \( k = 32 \) complete cycles, the monkey would be at \( 3 \times 32 = 96 \) cubits (but this overcounts). Let us reconsider: after 31 complete cycles, the monkey is at \( 3 \times 31 = 93 \) cubits. On the 32nd leap upward, the monkey reaches \( 93 + 5 = 98 \) cubits, then slips to 96. On the 33rd leap upward, the monkey reaches \( 96 + 5 = 101 > 100 \) cubits and has reached the top.

$$ \boxed{\text{The monkey reaches the top in 33 leaps.}} $$

Problem 7: Division of Gems

Solution. This is an arithmetic progression with:

• Common difference \( d = 3 \)
• Last term (smallest) \( a = 3 \)
• Sum \( S_n = 300 \)

The terms in ascending order are: \( 3, 6, 9, \ldots, 3n \) (where \( n \) is the number of friends). The sum is:

$$ S_n = \frac{n}{2}(a + l) = \frac{n}{2}(3 + 3n) = \frac{3n(n+1)}{2} = 300 $$

$$ n(n+1) = 200 $$

We need \( n^2 + n – 200 = 0 \). By the quadratic formula:

$$ n = \frac{-1 + \sqrt{1 + 800}}{2} = \frac{-1 + \sqrt{801}}{2} $$

Since \( \sqrt{801} \approx 28.3 \), we get \( n \approx 13.65 \). Since \( n \) must be a positive integer, we check: \( n = 13 \) gives \( S = \frac{3 \cdot 13 \cdot 14}{2} = 273 \) and \( n = 14 \) gives \( S = \frac{3 \cdot 14 \cdot 15}{2} = 315 \).

Neither gives exactly 300. This suggests we should revisit the problem formulation. If the first friend receives the most and the \( n \)-th receives 3, with the first receiving \( 3 + 3(n-1) = 3n \):

$$ S_n = \frac{n}{2}(3 + 3n) = 300 \implies n(n+1) = 200 $$

Since 200 has no factorization as \( n(n+1) \) for integer \( n \), Bhaskara likely intended a slightly different version. Taking \( a = 3, d = 3 \), and seeking the closest solution: with \( n = 13 \) friends, the total is 273 gems with the largest share being 39.

Problem 8: The Merchant’s Profit

Solution.

$$ \begin{aligned} \text{After first sale:} &\quad 200 \times 1.25 = 250 \\ \text{After second sale:} &\quad 250 \times 1.25 = 312.5 \end{aligned} $$

$$ \boxed{\text{Total wealth} = 312.5 \text{ coins}} $$

In general, after \( n \) rounds of \( p\% \) profit on initial capital \( C \):

$$ \text{Total} = C \left(1 + \frac{p}{100}\right)^n $$

This is the compound interest formula – a concept well understood by Indian mathematicians.

Problem 9: The Square and the Diagonal

Solution. The side length is \( s = 40/4 = 10 \) cubits. By the Pythagorean theorem:

$$ d = s\sqrt{2} = 10\sqrt{2} \approx 14.142 \text{ cubits} $$

Bhaskara would have used the approximation \( \sqrt{2} \approx \frac{577}{408} \) (accurate to six decimal places), giving:

$$ d \approx 10 \times \frac{577}{408} = \frac{5770}{408} = \frac{2885}{204} \approx 14.1422 \text{ cubits} $$

Problem 10: Combinations of Tastes

Solution. The number of combinations is:

$$ \begin{aligned} \binom{6}{1} &= 6 \\ \binom{6}{2} &= 15 \\ \binom{6}{3} &= 20 \\ \binom{6}{4} &= 15 \\ \binom{6}{5} &= 6 \\ \binom{6}{6} &= 1 \end{aligned} $$

The total number of non-empty combinations is:

$$ \sum_{k=1}^{6}\binom{6}{k} = 2^6 – 1 = 63 $$

Bhaskara’s Rule: The total number of non-empty subsets of \( n \) objects is \( 2^n – 1 \). This is equivalent to the binomial theorem identity:

$$ \sum_{k=0}^{n}\binom{n}{k} = 2^n $$

Problem 11: The Cistern Problem

Solution. The rates of filling and draining per hour are:

$$ \begin{aligned} \text{Pipe 1:} &\quad \frac{1}{3} \text{ of the cistern per hour} \\ \text{Pipe 2:} &\quad \frac{1}{5} \text{ of the cistern per hour} \\ \text{Drain:} &\quad -\frac{1}{15} \text{ of the cistern per hour} \end{aligned} $$

Combined rate:

$$ \frac{1}{3} + \frac{1}{5} – \frac{1}{15} = \frac{5 + 3 – 1}{15} = \frac{7}{15} $$

Time to fill:

$$ t = \frac{1}{7/15} = \frac{15}{7} = 2\tfrac{1}{7} \text{ hours} $$

$$ \boxed{t = \frac{15}{7} \approx 2.14 \text{ hours}} $$

Mathematical Methods in the Lilavati

Beyond individual problems, the Lilavati presents systematic mathematical methods that deserve analysis on their own merits.

The Rule of Three (Trairasika)

Bhaskara extends this to the Rule of Five, Rule of Seven, and so on, for problems involving multiple proportional relationships. These are essentially systems of proportions.

The Kuttaka Method for Indeterminate Equations

The Kuttaka (“pulverizer”) is one of the most important algorithms in Indian mathematics. It solves equations of the form \( ax \equiv c \pmod{b} \).

Bhaskara’s Treatment of Surds

Bhaskara systematically handles irrational numbers (surds), providing rules for operations:

Bhaskara’s Rules for Surds:

$$ \begin{aligned} \sqrt{a} \pm \sqrt{b} &= \sqrt{a + b \pm 2\sqrt{ab}} \\ \sqrt{a} \times \sqrt{b} &= \sqrt{ab} \\ \frac{\sqrt{a}}{\sqrt{b}} &= \sqrt{\frac{a}{b}} \end{aligned} $$

These rules enable computation with irrational quantities without requiring decimal approximations.

The Chakravala Method

Though appearing more fully in Bhaskara’s algebraic work (Bijaganita), the Chakravala method represents one of the greatest algorithmic achievements of Indian mathematics.

Concepts Ahead of Their Time

Several ideas in the Lilavati and its companion volumes anticipated later mathematical developments in Europe by centuries.

The Treatment of Zero

Indian mathematicians, including Bhaskara, treated zero as a number with defined properties centuries before this became standard in European mathematics. Bhaskara’s exploration of division by zero, while not fully resolved, showed sophisticated thinking about mathematical limits.

Negative Numbers as Natural Objects

While European mathematicians struggled to accept negative numbers well into the 18th century – with prominent figures like d’Alembert calling them “absurd” – Bhaskara handled them routinely with consistent operational rules.

Bhaskara recognized that “the square of a positive number and the square of a negative number are both positive; and the square root of a positive number is double: positive and negative. There is no square root of a negative number, for it is not a square.”

Iterative Algorithms

Bhaskara’s methods for calculating square roots and solving equations use iterative refinement, anticipating numerical methods used in modern computer algorithms.

Early Differential Calculus

In the astronomical sections of the Siddhanta Shiromani, Bhaskara computes what we would now call the derivative of \( \sin\theta \):

Bhaskara’s Proto-Derivative: Bhaskara observed that the “instantaneous velocity” of a sine function is proportional to the cosine:

$$ \frac{\delta(\sin\theta)}{\delta\theta} \approx \cos\theta $$

This result, expressed in the context of planetary motion, predates Newton and Leibniz by over 500 years.

Lilavati’s Influence on World Mathematics

Transmission to the Islamic World

Persian translations of the Lilavati appeared in the 13th century, making the work accessible throughout the Islamic world. The translator Abu’l Faiz Faizi produced a notable Persian version in 1587 at the court of the Mughal Emperor Akbar.

Influence on European Mathematics

From the Islamic world, Indian mathematical concepts filtered into European mathematics. The Italian mathematician Fibonacci, whose work popularized Hindu-Arabic numerals in Europe, likely encountered Indian mathematical ideas through Arabic intermediaries.

Educational Legacy

The Lilavati book became the standard arithmetic textbook across the Indian subcontinent for centuries. Its influence extended far beyond India. The pedagogical approach in the Lilavati influenced how mathematics was taught:

• The use of engaging word problems to motivate abstract concepts
• Progression from simple to complex topics
• Integration of practical applications with theoretical foundations
• The practice of providing worked examples before exercises

These are now considered educational best practices – and Bhaskara was using them in the 12th century.

In India, the Lilavati granth original remained in classroom use well into the 19th century. British colonial administrators noted with surprise that Indian students were solving problems from an 800-year-old textbook and getting them right.

The Siddhanta Shiromani: Lilavati in Context

Lilavati forms the first part of Bhaskara’s larger work, the Siddhanta Shiromani. Understanding the complete work provides context for Lilavati’s place in Indian mathematics.

Lilavati (Arithmetic)

The section covered in this guide, focusing on arithmetic operations, fractions, series, geometry, and problem-solving techniques.

Bijaganita (Algebra)

The second volume covers algebraic equations, including quadratic equations, equations with multiple unknowns, and the Chakravala method. Bhaskara’s treatment of algebraic problems is remarkably sophisticated.

Grahaganita (Planetary Mathematics)

This astronomical section calculates planetary positions, eclipses, and celestial phenomena. Bhaskara’s methods achieved remarkable accuracy for predicting astronomical events.

Goladhyaya (Spherical Astronomy)

The final section covers the geometry of the celestial sphere, including coordinate systems and trigonometric calculations needed for astronomical observation.

Together, these four sections represent the most complete mathematical and astronomical treatment produced in medieval India.

Comparison with Other Ancient Mathematical Traditions

Indian vs. Greek Mathematics

Aspect	Indian Tradition	Greek Tradition
Emphasis	Computation and algorithms	Proof and geometry
Number concept	Includes zero, negatives	Positive magnitudes only
Algebra	Symbolic, operational	Geometric (areas, lengths)
Proof style	Verification by example	Deductive chains
Notation	Positional decimal system	Alphabetic numerals
\( \pi \) approximation	\( \frac{355}{113} \) (Bhaskara)	\( \frac{22}{7} \) (Archimedes)

Indian vs. Chinese Mathematics

Both traditions independently developed sophisticated mathematics. The Chinese Jiuzhang Suanshu (“Nine Chapters on the Mathematical Art”, c. 200 BCE) shares several problem types with Indian texts, including the broken bamboo problem and linear system solving. The Indian tradition excelled particularly in:

• The development of the decimal place-value system with zero
• Systematic treatment of indeterminate equations (Kuttaka, Chakravala)
• Trigonometric functions and their applications

The Lilavati in Modern Perspective

Relevance to Modern Mathematics Education

The Lilavati‘s approach to mathematical pedagogy remains remarkably relevant:

1. Contextual learning: Every problem is embedded in a real-world or imaginative context, aiding comprehension and retention.
2. Spiral curriculum: Topics are revisited with increasing complexity across chapters.
3. Multiple representations: Problems are expressed verbally (poetically), solved algorithmically, and verified numerically.
4. Engagement: The use of animals, nature, and human characters creates emotional engagement with mathematical content.

Problems That Remain Relevant

Many problem types from the Lilavati appear in modern textbooks and competitions:

• Work-rate problems (pipes and cisterns)
• Mixture problems
• Pursuit problems (the peacock and the snake)
• Number puzzles (the maiden’s riddle)
• Combinatorial counting

The mathematical structures underlying these problems – linear equations, quadratic equations, the Pythagorean theorem, arithmetic and geometric series – form the core of secondary mathematics education worldwide.

Where to Find Lilavati: Resources and Translations

For those wanting to study Lilavati directly, several resources exist.

Lilavati Book PDF in English

H.T. Colebrooke’s 1817 translation, “Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhascara,” remains a scholarly reference. It’s available through various digital archives, including Archive.org. More recent translations include K.S. Patwardhan’s “Lilavati of Bhaskaracarya” (2001), which provides Sanskrit text alongside English translation and mathematical commentary.

Lilavati Book PDF in Sanskrit

The original Sanskrit text is preserved in various editions. The Anandashrama Sanskrit Series published a critical edition. Digital Sanskrit archives also host scanned manuscripts for researchers who can read the original.

Modern Presentations

Several publishers have created modern presentations of Lilavati problems for students. These adapt the word problems for contemporary audiences while preserving the mathematical content. Search for “Arithmetic in Bhaskaracharya Lilavati ppt” and you’ll find educational presentations summarizing key concepts.

Physical Books

For serious study, consider acquiring Patwardhan’s translation from academic publishers. Used copies of older translations occasionally appear on specialized book sites. The Amazon links for both Indian readers and international readers offer the standard scholarly edition.

Download Lilavati

Here are the copyright-free Lilavati PDFs available for free:

Sanskrit Editions

Lilavati of Bhaskaracharya (1908 Khemraj Edition) https://archive.org/details/EtgU_lilavati-of-bhaskaracharya-1908-khemraj Clean Sanskrit text, well-preserved scan from Khemraj publishers.

Lilavati with Kriya Karmakari Commentary (K.V. Sharma Edition) https://archive.org/details/zNEE_lilavati-of-bhaskaracharya-with-kriya-karmakari-of-shankar-and-narayan-by-k.-v.- : Sanskrit text with the famous commentary by Shankar and Narayan. Published by Vishveshvaranand Vaidik Research Institute.

Lilavati Part 1 (Pandit Champaram Mishra) https://archive.org/details/qilP_lilavati-of-bhaskaracharya-part-1-pandit-champaram-mishra

1473 Manuscript with Vrutti Commentary (Asiatic Society of Mumbai) https://archive.org/details/wg1202 : A historical manuscript with commentary by Mosadeva. This one’s special because it dates to 1473 CE, making it one of the oldest surviving copies.

Sanskrit with Hindi Translation

Lilavati with Hindi Tika (Ram Swarup Sharma, 1894) https://archive.org/details/lilavatiofbhaskaracharyahinditikaoframswarupsharmavenkateswarasteampress1894_202003_813_u : High-resolution 600 PPI scan. Sanskrit verses with Hindi explanations.

Lilavati with Hindi Tika (Pandit Ram Sharma, 1931) https://archive.org/details/lilavatiofbhaskaracharyahinditikaofpanditramsharmavenkateswarasteampress1931_285_r

English Translation

Lilavati (Patwardhan, Naimpally, Singh – Motilal Banarsidass) https://archive.org/details/UQXV_lilavati-of-bhaskaracharya-by-krishnaji-shankara-patwardhan-somashekhara-amrita-

This is the scholarly edition with Sanskrit text, English translation, and mathematical commentary. Best for serious study.

The 1908 Khemraj and the Patwardhan editions are probably the most useful for readers. The Khemraj gives clean Sanskrit text, and Patwardhan provides the full scholarly treatment.

My Free Book on Bhaskara II

Why Lilavati Still Matters

Nine centuries after Bhaskara wrote it, Lilavati remains relevant for several reasons.

The problems are genuinely fun. Working through a puzzle about bees and flowers or peacocks and snakes is more engaging than abstract exercises. Mathematics educators continue to study Bhaskara’s approach for insights into making math accessible.

The historical perspective is valuable. Understanding how mathematical ideas developed across cultures challenges the Eurocentric narrative that dominated 20th-century history of mathematics. Indian mathematicians made foundational contributions that influenced global mathematical development.

The mathematical content is solid. Despite the playful presentation, Lilavati covers real mathematics. Students working through these problems develop genuine problem-solving skills. The techniques transfer to modern contexts.

And there’s something beautiful about solving the same problems that students solved in 12th-century India. Mathematics connects across time. The peacock still flies down to catch the snake. The beautiful maiden still poses her riddle. And we still work backwards through the operations to find that the answer is 28.

Try It Yourself: Practice Problems from Lilavati

Test your understanding with these problems in the style of the Lilavati. I’ve included hints and full solutions so you can check your work.

Problem A: The Gardener’s Flowers

Problem B: The Bird, Cat, and Mouse

Problem C: The Number Puzzle

Problem D: Spice Mixtures

Problem E: The Mangoes

“Of a collection of mangoes, the king took one-sixth, the queen one-fifth of the remainder, and the three princes one-fourth, one-third, and one-half of the successive remainders, and the youngest child took the remaining three mangoes. How many mangoes were there?”

Hint: Work backwards from the 3 remaining mangoes. If 3 mangoes represent half of what was left after the fourth prince took his share, the amount before that was 6.

Solution:

Let \( n \) be the total mangoes.

After king takes \( \frac{1}{6} \): remainder is \( \frac{5n}{6} \)

After queen takes \( \frac{1}{5} \) of remainder: \( \frac{5n}{6} – \frac{1}{5} \cdot \frac{5n}{6} = \frac{5n}{6} \cdot \frac{4}{5} = \frac{4n}{6} = \frac{2n}{3} \)

After first prince takes \( \frac{1}{4} \): \( \frac{2n}{3} \cdot \frac{3}{4} = \frac{n}{2} \)

After second prince takes \( \frac{1}{3} \): \( \frac{n}{2} \cdot \frac{2}{3} = \frac{n}{3} \)

After third prince takes \( \frac{1}{2} \): \( \frac{n}{3} \cdot \frac{1}{2} = \frac{n}{6} \)

This equals 3:

$$ \frac{n}{6} = 3 $$

$$ \boxed{n = 18} $$

The collection contained 18 mangoes.

Problem F: The Taller Pillar

“A snake’s hole is at the foot of a pillar 15 feet high, and a peacock is perched on its summit. Seeing a snake at a distance of 45 feet from the base, the peacock swoops down upon it. If they both travel at equal speed, at what distance from the hole is the snake caught?”

Solution:

Let \( x \) be the distance from the pillar where they meet.

Snake travels: \( 45 – x \)

Peacock travels: \( \sqrt{15^2 + x^2} = \sqrt{225 + x^2} \)

Setting equal:

$$ 45 – x = \sqrt{225 + x^2} $$

$$ (45 – x)^2 = 225 + x^2 $$

$$ 2025 – 90x + x^2 = 225 + x^2 $$

$$ 2025 – 90x = 225 $$

$$ 1800 = 90x $$

$$ \boxed{x = 20} $$

They meet 20 feet from the pillar.

Problem G: The Cubic Equation

“What number is that which multiplied by 12 and added to the cube of the number equals 6 times the square of the number?”

Hint: Set up the equation and factor.

Solution:

Let the number be \( n \).

$$ 12n + n^3 = 6n^2 $$

$$ n^3 – 6n^2 + 12n = 0 $$

$$ n(n^2 – 6n + 12) = 0 $$

So \( n = 0 \) or \( n^2 – 6n + 12 = 0 \)

For the quadratic:

$$ n = \frac{6 \pm \sqrt{36 – 48}}{2} = \frac{6 \pm \sqrt{-12}}{2} $$

This gives complex solutions. In Bhaskara’s context, the answer is \( n = 0 \), though he likely expected students to find this through trial with small integers.

Problem H: The Lotus (Variant)

“In a lake, the tip of a lotus bud was seen to extend one cubit above the water’s surface. Forced by the wind, it gradually moved and was submerged at a distance of two cubits from where it originally stood. Find the depth of the water.”

Solution:

Let the depth be \( d \). The lotus stem length is \( d + 1 \) (depth plus the one cubit above water).

When the lotus is pushed sideways 2 cubits and submerged, the stem forms a right triangle with the water depth:

$$ (\text{stem length})^2 = d^2 + 2^2 $$

$$ (d + 1)^2 = d^2 + 4 $$

$$ d^2 + 2d + 1 = d^2 + 4 $$

$$ 2d + 1 = 4 $$

$$ \boxed{d = \frac{3}{2}} $$

The water depth is 1.5 cubits (about 27 inches, assuming an 18-inch cubit).

Appendix A: Timeline of Indian Mathematics

Date	Mathematician	Key Contribution
c. 800 BCE	Shulba Sutras	Pythagorean theorem, \( \sqrt{2} \) approximation
c. 300 BCE	Pingala	Binary numbers, combinatorics
c. 500 CE	Aryabhata	\( \pi \approx 3.1416 \), trigonometry, algebra
c. 600 CE	Bhaskara I	Commentary on Aryabhata
628 CE	Brahmagupta	Zero, negative numbers, Pell’s equation
c. 850 CE	Mahavira	Combinatorics, arithmetic
1114-1185	Bhaskara II	Lilavati, Chakravala, proto-calculus
c. 1350	Narayana Pandit	Magic squares, combinatorics
c. 1400	Madhava	Infinite series for \( \pi \), \( \sin \), \( \cos \)
c. 1500	Nilakantha	Improved series, heliocentrism

Appendix B: Bhaskara’s Approximation for Sine

In the astronomical part of the Siddhanta Shiromani, Bhaskara provides a remarkably accurate rational approximation for the sine function:

Bhaskara I’s Sine Approximation (later refined by Bhaskara II):

$$ \sin\theta \approx \frac{16\theta(\pi – \theta)}{5\pi^2 – 4\theta(\pi – \theta)}, \quad 0 \leq \theta \leq \pi $$

This formula, expressed in terms of degrees in the original, achieves a maximum error of less than 1.9% over the entire range \( [0, \pi] \).

At the key values:

$$ \begin{aligned} \theta = \frac{\pi}{6}: &\quad \text{Formula gives } \frac{16 \cdot \frac{\pi}{6} \cdot \frac{5\pi}{6}}{5\pi^2 – 4 \cdot \frac{\pi}{6} \cdot \frac{5\pi}{6}} = \frac{\frac{80\pi^2}{36}}{5\pi^2 – \frac{20\pi^2}{36}} = \frac{\frac{80}{36}}{5 – \frac{20}{36}} = \frac{\frac{20}{9}}{\frac{160}{36}} = \frac{20}{9} \cdot \frac{36}{160} = \frac{1}{2} \\ \theta = \frac{\pi}{2}: &\quad \text{Formula gives } \frac{16 \cdot \frac{\pi}{2} \cdot \frac{\pi}{2}}{5\pi^2 – 4 \cdot \frac{\pi}{2} \cdot \frac{\pi}{2}} = \frac{4\pi^2}{5\pi^2 – \pi^2} = \frac{4\pi^2}{4\pi^2} = 1 \end{aligned} $$

Both are exact. The formula is a rational function that closely approximates the transcendental sine function – a remarkable achievement for the 12th century.

Conclusion

The Lilavati of Bhaskaracharya stands as a monument to the mathematical genius of ancient India. Written over 850 years ago, it combines rigorous mathematical content with a poetic, engaging style that modern educators can still learn from.

The key contributions of the Lilavati and its companion volumes include:

1. Systematic Arithmetic: Complete treatment of the four operations, fractions, zero, and negative numbers.
2. Elegant Problem Design: Mathematical problems embedded in memorable stories about bees, peacocks, lotuses, and maidens.
3. Algorithmic Thinking: The Kuttaka and Chakravala methods represent algorithmic sophistication not matched in Europe for centuries.
4. Proto-Calculus: Bhaskara’s computation of instantaneous rates of change anticipated differential calculus by 500 years.
5. Educational Innovation: A pedagogical approach that made mathematics accessible, engaging, and memorable.

Bhaskara II’s Lilavati represents mathematics at its most human. It’s technically rigorous yet personally engaging. It addresses the reader directly, tells stories, and makes abstract concepts tangible through vivid imagery.

For students: work through these problems. They’ll sharpen your algebra and geometry skills while connecting you to a mathematical tradition spanning nearly a millennium.

For educators: study Bhaskara’s pedagogical approach. The integration of narrative, personal address, and practical application offers timeless lessons in mathematical communication.

For anyone curious about the history of ideas: Lilavati demonstrates that mathematical creativity flourished across cultures. The Indian mathematical tradition, of which Bhaskara was a leading figure, contributed foundational concepts that shaped global mathematical development.

The Lilavati book written by Bhaskaracharya isn’t just a historical artifact. It’s a living text that still teaches, still challenges, and still delights.

“As the sun eclipses the stars by its brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more if he solves them.”

Brahmagupta, Brahmasphutasiddhanta (628 CE)

References

1. Bhaskara II. Lilavati. Translated by H.T. Colebrooke. London: John Murray, 1817.
2. Bhaskara II. Lilavati of Bhaskaracharya. Translated by K.S. Patwardhan, S.A. Naimpally, and S.L. Singh. Delhi: Motilal Banarsidass, 2001.
3. Datta, B., and Singh, A.N. History of Hindu Mathematics: A Source Book. Parts I and II. Bombay: Asia Publishing House, 1962.
4. Joseph, G.G. The Crest of the Peacock: Non-European Roots of Mathematics. 3rd ed. Princeton University Press, 2011.
5. Katz, V.J. A History of Mathematics: An Introduction. 3rd ed. Addison-Wesley, 2009.
6. Kim, P., and Imhausen, A. “Mathematics in India.” In The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, edited by V.J. Katz. Princeton University Press, 2007.
7. Plofker, K. Mathematics in India. Princeton University Press, 2009.
8. Selenius, C.-O. “Rationale of the Chakravala Process of Jayadeva and Bhaskara II.” Historia Mathematica 2 (1975): 167-184.
9. Srinivasiengar, C.N. The History of Ancient Indian Mathematics. Calcutta: World Press, 1967.
10. Tiwari, G. “A Problem (and Solution) from Bhaskaracharya’s Lilavati.” https://gauravtiwari.org/a-problem-solution-from-bhaskaracharyas-lilavati/

FAQs