Lilavati

01Questions to a girl

Many problems begin with an address. Aye bale Lilavati, O girl Lilavati, clever one, add these numbers. O child with eyes like a young deer’s, tell me quickly. A fifth of a swarm of bees flies to a kadamba flower; seven times half the root of a flock of swans walk on the shore of a lake; a peacock on a pillar sees a snake making for its hole; a pearl necklace breaks in a lovers’ quarrel and the pearls scatter over the floor. Each scene hides one rule of arithmetic, and the student who finds it gets the answer.

02The water clock

Who was Lilavati? The tradition is told by Faizi in the preface to his Persian translation of 1587, made for Akbar. Bhaskara’s daughter Lilavati, the katha says, was fated by her horoscope to lose her husband soon after marriage unless she was wed at one exact hour. Bhaskara set a cup with a small hole floating in a vessel of water to mark the moment. Lilavati, leaning over to watch, let a pearl fall from her ornaments; it stopped the hole, the hour passed unseen, and the marriage was made at the wrong time. To console her, her father named his book after her, so that her name would last. The first verse of the book gives a second meaning too: it calls the work lalitya-lilavati, full of grace and play.

03Bhaskara and his book

Bhaskaracharya tells us in the Goladhyaya that he was born in Shaka 1036, which is 1114 CE, and finished the Siddhanta Shiromani at thirty-six, in 1150. He lived at Vijjadavida, near the Sahyadri hills, and his father Maheshvara was his teacher. The Shiromani has four parts: Lilavati on arithmetic and measurement, Bijaganita on algebra, Grahaganita on the planets and Goladhyaya on the sphere. He wrote his own commentary, the Vasana, and later a practical handbook, the Karanakutuhala of 1183. Because he states his dates himself, there is no real debate about them.

04From units to permutations

The book opens with a salutation to Shri Ganesh and the tables: cowries and panas for money, gunjas and mashas for gold, angulas and hastas for length, and the names of place values up to ten to the seventeenth. Then come the eight operations on whole numbers and on fractions, the rules for zero, and the clever methods of the prakirnaka: working backward, assuming a number and correcting it. Proportion, interest and mixtures follow, then series, then the long chapter on plane figures, then excavations, brick piles, sawing, grain heaps and shadows. Kuttaka and the permutation of digits close it, about 270 stanzas in all.

05Bees, swans and a lotus

The bee problem is the one every Indian student once knew. A fifth of the swarm goes to the kadamba, a third to the shilindhra, three times their difference to the kutaja, and one bee is left hovering, pulled between the scent of ketaki and malati. The answer is fifteen. In the swan problem, seven times half the square root of the flock walk on the bank and two pairs play in the water; the flock is sixteen. A lotus stands half a hasta above still water, the wind bends it over until it sinks two hastas away, and the student must find the depth of the lake from the right triangle.

06Circle, sphere and the proud

For the circle Bhaskara gives two values. Multiply the diameter by 3,927 and divide by 1,250 for the fine circumference; multiply by 22 and divide by 7 for the rough one used in daily work. He gives the area of a circle as a quarter of circumference times diameter, the surface of a sphere as four times that area, and rules for chords and arcs. In the last chapter, on permutations, he counts the numbers that can be made from given digits, and then adds a teasing verse: there is no multiplying here, no dividing, no squaring or cubing, yet proud and foolish calculators will surely fall into this net of numbers.

07The chakravala in the Bijaganita

The Lilavati keeps to arithmetic; the harder work is in its sister, the Bijaganita. There Bhaskara gives the chakravala, the cyclic method, which takes any equation of the form Nx² + 1 = y² and walks round and round, improving a guess each time, until a whole-number answer appears. His example is 61x² + 1 = y², whose smallest answer is x = 22,61,53,980 and y = 1,76,63,19,049. In 1657 Pierre de Fermat set the very same equation as a challenge to European mathematicians. Bhaskara had solved it five centuries earlier in a few lines of verse.

08A school at Patne

A stone inscription of 1207 at Patne, near Chalisgaon in Khandesh, records that Changadeva, Bhaskara’s grandson and astrologer to the Yadava king Singhana, founded a matha there to teach the Siddhanta Shiromani and other works of his family. It is a rare record of a school endowed to teach one family’s mathematics. Lilavati went on being taught in pathshalas from Kashi to Kerala into the nineteenth century, and students learnt its verses by heart, as they learnt the Amarakosha.

09Commentaries and translations

More than twenty commentaries survive. In Kerala, Shankara Variyar and Narayana wrote the Kriyakramakari around 1534, carrying the Kerala school’s new series into their notes; Ganesha Daivajna wrote the Buddhivilasini in 1545, and Mahidhara the Lilavativivarana. Faizi’s Persian version of 1587 spread it at the Mughal court. John Taylor published an English translation at Bombay in 1816, and H. T. Colebrooke the Lilavati and Bijaganita together at London in 1817. Translations into Marathi, Hindi, Telugu, Kannada and other languages followed, and the Anandashrama edition of 1937 is still the standard Sanskrit text.

What it contains

1. 1
   Paribhasha

   Tables of units: money in cowries, panas and nishkas, weights of gold and grain, lengths from the angula to the yojana, and volumes. It also names the place values from one up to ten to the seventeenth.

2. 2
   Parikarmashtaka

   The eight basic operations on whole numbers: addition, subtraction, multiplication, division, square, square root, cube and cube root. Several methods of multiplication are shown side by side on one example.

3. 3
   Bhinna and shunya

   The same eight operations on fractions, with the reduction of fractions to a common denominator. Then the rules of zero: anything plus zero is unchanged, anything times zero is zero, and a number divided by zero is called khahara.

4. 4
   Prakirnaka

   Miscellaneous methods: inversion (working backward from the answer), the assumed number (ishtakarma), and problems on sums and differences of squares. The bee and swan problems belong here.

5. 5
   Trairashika

   Proportion: the rule of three and its inverse, then the rules of five, seven, nine and eleven terms. It ends with barter, the exchange of one good for another at given rates.

6. 6
   Mishraka

   Mixtures: simple interest, the division of profit among partners, the filling of a cistern by several pipes, and the purity of mixed gold. It closes with the combinations of flavours and of a palace’s windows.

7. 7
   Shreḍhi

   Arithmetic and geometric series: the sum of the first n numbers, the sum of those sums, sums of squares and cubes. One problem asks what a giver pays in a month if he promises a supplicant two cowries on the first day and double each day after.

8. 8
   Kshetra

   The longest chapter: the right triangle and its sides, triangles and quadrilaterals, the circle with 3,927/1,250, the sphere, chords and arcs. The lotus, the peacock and the broken bamboo are its problems.

9. 9
   Khata, chiti, krakacha, rashi

   Practical volumes: the earth dug from a pit, the bricks in a wall or altar, the area cut by a saw through timber, and the grain in a heap against a wall or in a corner.

10. 10
    Chhaya

    Shadows of a gnomon cast by a lamp: finding the height of the lamp or its distance from two shadows. It is the rule of three applied to similar triangles.

11. 11
    Kuttaka

    Indeterminate equations of the first degree, ax + c = by in whole numbers, solved by repeated division. Bhaskara adds shortcuts and the case of a constant term of zero.

12. 12
    Ankapasha

    Permutations and combinations: how many numbers can be formed from given digits, their sum, and how many have a given digit sum. It ends with the teasing verse on proud calculators and the closing blessing on those who keep Lilavati at their throat.

Famous verses

प्रीतिं भक्तजनस्य यो जनयते विघ्नं विनिघ्नन् स्मृतस्तं वृन्दारकवृन्दवन्दितपदं नत्वा मतङ्गाननम्। पाटीं सद्गणितस्य वच्मि चतुरप्रीतिप्रदां प्रस्फुटां संक्षिप्ताक्षरकोमलामलपदैर्लालित्यलीलावतीम्॥

Bowing to the elephant-faced one, whose feet the hosts of devas worship, who when remembered gives joy to his devotees and destroys obstacles, I set out this method of good calculation, clear, a delight to the clever, in short, soft and spotless words, full of grace and play.Lilavati 1

अये बाले लीलावति मतिमति ब्रूहि सहितान् द्विपञ्चद्वात्रिंशत्त्रिनवतिशताष्टादश दश। शतोपेतानेतानयुतवियुतांश्चापि वद मे यदि व्यक्ते युक्तिव्यवकलनमार्गेऽसि कुशला॥

O girl Lilavati, clever one, tell me the sum of two, five, thirty-two, a hundred and ninety-three, eighteen and ten with a hundred added; and tell me also what is left when they are taken from ten thousand, if you are skilled in adding and subtracting.Lilavati 13

पञ्चांशोऽलिकुलात्कदम्बमगमत्त्र्यंशः शिलीन्ध्रं तयोर्विश्लेषस्त्रिगुणो मृगाक्षि कुटजं दोलायमानोऽपरः। कान्ते केतकमालतीपरिमलप्राप्तैककालप्रियादूताहूत इतस्ततो भ्रमति खे भृङ्गोऽलिसंख्यां वद॥

A fifth of a swarm of bees went to the kadamba, a third to the shilindhra; three times their difference, O deer-eyed one, to the kutaja. One bee was left, swinging this way and that in the air, called at once by the fragrance of ketaki and malati. Tell me, dear one, how many bees there were.Lilavati 55

व्यासे भनन्दाग्निहते विभक्ते खबाणसूर्यैः परिधिः स सूक्ष्मः। द्वाविंशतिघ्ने विहृतेऽथ शैलैः स्थूलोऽथवा स्याद्व्यवहारयोग्यः॥

When the diameter is multiplied by 3,927 and divided by 1,250, the result is the fine circumference. Or, multiplied by 22 and divided by 7, it is the rough one, fit for everyday use.Lilavati 199

येषां सुजातिगुणवर्गविभूषिताङ्गी शुद्धाखिलव्यवहृतिः खलु कण्ठसक्ता। लीलावतीह सरसोक्तिमुदाहरन्ती तेषां सदैव सुखसम्पदुपैति वृद्धिम्॥

Those who hold Lilavati close at the throat, her limbs adorned with good reductions, multiplications and squares, all her procedures pure, speaking in sweet words: their joy and wealth grow always. The verse reads equally as praise of a well-born and virtuous bride.Lilavati 272