Bhaskara II’s Lilavati: The Complete Reference Guide

Lilavati is a 12th-century mathematics treatise that reads like poetry. Written by Bhaskara II (also known as Bhaskaracharya), it wraps arithmetic, algebra, and geometry inside riddles about bees, snakes, peacocks, and beautiful maidens. The Lilavati book, written by Bhaskaracharya, remains one of the most influential mathematical texts in history, studied for over 850 years across Asia and eventually Europe.

This guide covers everything you need to know about Lilavati: the mathematician behind it, the legend of its creation, its mathematical content, sample problems with solutions, and where to find the Lilavati book PDF in English and Sanskrit.

Bhaskaracharya: The Mathematician Behind Lilavati

Bhaskaracharya, also known as Bhaskara II, was born in 1114 CE in Vijjalavida (modern-day Bijapur district, Karnataka, India). His father, Mahesvara, was also a mathematician and astronomer, which meant Bhaskara grew up surrounded by mathematical manuscripts and astronomical instruments.

By the time he was 36, Bhaskara II had completed his magnum opus: the Siddhānta Shiromani (“Crown of Treatises”). This wasn’t 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 position once held by the legendary mathematician Brahmagupta. He remained there until his death around 1185 CE. His works were translated into Persian, Arabic, and eventually European languages, influencing mathematical development across cultures.

What set Bhaskara apart from his contemporaries was 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. This approach made his work accessible to students who might otherwise struggle with abstract concepts.

The Legend of Lilavati: A Father’s Gift to His Daughter

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

The story goes like this: Bhaskara, being an accomplished astronomer, cast his daughter’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. The text contains passages like “Oh Lilavati, intelligent girl…” and “Tell me, beautiful one…” This personal touch made the book feel less like a textbook and more like a conversation between father and daughter.

Whether this story is historically accurate remains debated. Some scholars think it’s 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.

The Structure of Lilavati: Thirteen Chapters of Mathematical Knowledge

The Lilavati book contains thirteen chapters, progressing from basic arithmetic to advanced problem-solving techniques. Here’s what each section covers.

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

Bhaskara covers operations with fractions in detail, including addition, subtraction, multiplication, division, and simplification. He presents these through practical examples involving trade, measurement, and distribution.

Chapter 3: Zero and Negative Numbers

This chapter explores the properties of zero, including the famous (and mathematically problematic) division by zero. Bhaskara correctly identifies that \(\frac{a}{0}\) leads to infinity, though his treatment isn’t fully rigorous by modern standards. He also addresses operations with negative numbers.

Chapter 4: Square and Cube Roots

Bhaskara presents algorithms for calculating square roots and cube roots of integers and fractions. His 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 formulas for the sum of arithmetic series:

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

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. The sum formula for a geometric series appears as:

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

Where \(r\) is the common ratio.

Chapter 7: Plane Geometry

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

For a triangle with sides \(a\), \(b\), and \(c\), Bhaskara uses Heron’s formula (known in India before Heron):

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

Where \(s = \frac{a+b+c}{2}\) is the semi-perimeter.

Chapter 8: Solid Geometry

Bhaskara covers volumes and surface areas of three-dimensional shapes including spheres, cones, and pyramids. His approximation of \(\pi\) as \(\frac{22}{7}\) and the more accurate \(\frac{355}{113}\) appears in these calculations.

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. Bhaskara shows how to calculate solar angles, time, and geographical latitude from shadow measurements.

Chapter 10: The Pulverizer (Kuṭṭaka)

The Kuṭṭaka is a method for solving indeterminate equations of the form:

$$ax + by = c$$

This technique, refined over centuries by Indian mathematicians, 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. The formula for combinations appears as:

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

Though expressed in different notation.

Chapter 12: Net of Numbers

This chapter covers magic squares and number patterns with specific properties.

Chapter 13: Miscellaneous Problems

The final chapter presents diverse word problems that combine techniques from earlier chapters. Many of Lilavati’s most famous puzzles appear here.

Sample Problems from Lilavati with Solutions

The best way to understand Lilavati is to work through its problems. Here are three classic examples with complete solutions.

Problem 1: The Beautiful Maiden’s Riddle

This is perhaps Lilavati’s most famous problem:

“A beautiful maiden, with beaming eyes, asks: which is the number that, 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 requires working backwards through each operation.

Start with the final result: 2

Multiply by 10: \(2 \times 10 = 20\)

Subtract 8: \(20 – 8 = 12\)

Square: \(12^2 = 144\)

Add 52: \(144 + 52 = 196\)

Now, “multiplied by itself” means we need \(\sqrt{196} = 14\)

For the remaining operations, let the unknown number be \(n\).

The problem states: multiply by 3, add three-fourths of the product, divide by 7, subtract one-third of the quotient.

Multiply by 3: \(3n\)

Add three-fourths of the product: \(3n + \frac{3}{4}(3n) = 3n + \frac{9n}{4} = \frac{21n}{4}\)

Divide by 7: \(\frac{21n}{4 \times 7} = \frac{3n}{4}\)

Subtract one-third of the quotient: \(\frac{3n}{4} – \frac{1}{3} \cdot \frac{3n}{4} = \frac{3n}{4} – \frac{n}{4} = \frac{2n}{4} = \frac{n}{2}\)

Setting this equal to 14:

$$\frac{n}{2} = 14$$

$$n = 28$$

The number is 28. You can verify by working forward through all the operations.

Problem 2: The Peacock and the Snake

“A peacock sits on a pillar 9 feet high. Seeing a snake coming towards its hole at the base of the pillar from a distance of 27 feet, the peacock flies down to catch it. If both move at the same speed, at what distance from the pillar do they meet?”

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

The snake travels from 27 feet away toward the base, covering \((27 – x)\) feet.

The peacock flies diagonally from the top of the 9-foot pillar to the meeting point on the ground. By the Pythagorean theorem, this distance is \(\sqrt{9^2 + x^2} = \sqrt{81 + x^2}\).

Since both travel at the same speed and meet simultaneously:

$$27 – x = \sqrt{81 + x^2}$$

Square both sides:

$$(27 – x)^2 = 81 + x^2$$

$$729 – 54x + x^2 = 81 + x^2$$

The \(x^2\) terms cancel:

$$729 – 54x = 81$$

$$648 = 54x$$

$$x = 12$$

They meet 12 feet from the pillar. The snake travels 15 feet \((27 – 12)\), and the peacock travels \(\sqrt{81 + 144} = \sqrt{225} = 15\) feet. Same distance, confirming our answer.

Problem 3: The Bees and the Lotus

“One-fifth of a swarm of bees flew to the Kadamba flower, one-third flew to the Silandhara, three times the difference of these two numbers flew to an arbor. One bee remained, hovering and attracted on each side by the fragrant ketaki and the malati. What is the number of bees?”

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

Bees to Kadamba: \(\frac{n}{5}\)

Bees to Silandhara: \(\frac{n}{3}\)

Difference: \(\frac{n}{3} – \frac{n}{5} = \frac{5n – 3n}{15} = \frac{2n}{15}\)

Three times the difference (to arbor): \(3 \times \frac{2n}{15} = \frac{6n}{15} = \frac{2n}{5}\)

Remaining bee: 1

The sum equals the total:

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

Finding a common denominator of 15:

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

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

$$1 = n – \frac{14n}{15}$$

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

$$n = 15$$

The swarm contains 15 bees.

Mathematical Concepts in Lilavati That Were Ahead of Their Time

Several ideas in Lilavati anticipated later mathematical developments in Europe and elsewhere.

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

While European mathematicians struggled to accept negative numbers well into the 18th century, Bhaskara handled them routinely. His work treats negative quantities as natural mathematical objects with consistent operational rules.

Iterative Algorithms

Bhaskara’s methods for calculating square roots and solving equations use iterative refinement, a technique that became central to computational mathematics. His approach anticipates numerical methods used in modern computer algorithms.

The Chakravala Method

Though appearing more fully in Bhaskara’s algebraic work (Bijaganita), the Chakravala method for solving Pell’s equation represents an algorithmic breakthrough. This “cyclic method” solves equations of the form:

$$x^2 – Dy^2 = 1$$

The algorithm wasn’t rediscovered in Europe until Lagrange’s work in the 18th century.

Lilavati’s Influence on Mathematics Education

The Lilavati book became the standard arithmetic textbook across the Indian subcontinent for centuries. Its influence extended far beyond India.

Persian translations appeared in the 13th century, making the work accessible throughout the Islamic world. From there, some 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.

The pedagogical approach in Lilavati influenced how mathematics was taught. The use of engaging word problems, the progression from simple to complex, and the integration of practical applications with theoretical concepts all became educational best practices that persist today.

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 Siddhānta Shiromani: Lilavati in Context

Lilavati forms the first part of Bhaskara’s larger work, the Siddhānta 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.

Golādhyāya (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.

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 Ānandāśrama 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 your readers. The Khemraj gives clean Sanskrit text, and Patwardhan provides the full scholarly treatment.

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

Here are additional problems from Lilavati for you to solve. I’ve included hints and full solutions so you can check your work.

Problem A: 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:

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

$$n = 18$$

The collection contained 18 mangoes.

Problem B: 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?”

Hint: Same setup as the earlier peacock problem, just different numbers.

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:

$$45 – x = \sqrt{225 + x^2}$$

$$(45 – x)^2 = 225 + x^2$$

$$2025 – 90x + x^2 = 225 + x^2$$

$$2025 – 90x = 225$$

$$1800 = 90x$$

$$x = 20$$

They meet 20 feet from the pillar.

Problem C: 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 D: The Lotus

“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:

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

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

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

$$2d + 1 = 4$$

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

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

Conclusion

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.

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