The Lumber Room

"Consign them to dust and damp by way of preserving them"

“Your monkey did not jump high enough”

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Yesterday, in Futility Closet there was a post:

In Longfellow’s novel Kavanagh, Mr. Churchill reads a word problem to his wife:

“In a lake the bud of a water-lily was observed, one span above the water, and when moved by the gentle breeze, it sunk in the water at two cubits’ distance. Required the depth of the water.”

“That is charming, but must be very difficult,” she says. “I could not answer it.”

Is it? If a span is 9 inches and a cubit is 18 inches, how deep is the water?

The problem is simple enough: if the depth of the water is x inches so that the lotus from bottom to tip is x+9 inches, then x2+362=(x+9)2, which means x=(362-92)/18=135/2=67.5.

More interestingly, as I accidentally recognised (I don’t know how), it is from the Sanskrit mathematics text Lilavati (and also found in the Bījagaṇita) of Bhaskaracharya (Bhaskara II). That entire chapter of Kavanagh is essentially quoting the Lilavati (Kavanagh is written in a somewhat embarrassing tone that perhaps explains why it’s so obscure :p); it’s included later below the horizontal line in this post.

Bhaskaracharya, believed to have lived in the 12th century, is considered the last great Indian mathematician, outside of the Kerala school. Like most Sanskrit texts, the Līlāvati is written in verse, so as to be easier to memorise. Unlike many Sanskrit technical works (or for that matter technical works in any language), however, Bhāskara’s works are not written in the typical dry style, and can veer quite poetic at times. His description of the seasons in one of his astronomical works is one of the few true instances of poetry in the Sanskrit astronomical/mathematical corpus. This particular problem, it happens, is written in the beautiful mandākrānta metre: (If it helps: mandakranta is the metre of the Meghadūta, of “शान्ताकारं भुजगशयनं…”, of “नास्था धर्मे न वसुनिचये…”, etc., and you can listen to a recitation in the Marathi tradition by Ashwini Deo.)

चक्रक्रौञ्चाकुलितसलिले क्वापि दृष्टं तडागे
तोयादूर्ध्वं कमलकलिकाग्रं वितस्तिप्रमाणम्
मन्दं मन्दं चलितमनिलेनाऽऽहतं हस्तयुग्मे
तस्मिन्मग्नं गणक कथय क्षिप्रमम्बुप्रमाणम्

cakra-krauñcākulita-salile kvāpi dṛṣṭaṃ taḍāge
toyād ūrdhvaṃ kamala-kalikāgraṃ vitasti-pramāṇam
mandaṃ mandaṃ calitam anilenāhataṃ hasta-yugme
tasmin magnaṃ gaṇaka kathaya kṣipram ambu-pramāṇam

In a certain lake swarming with geese and cranes,
the tip of a bud of lotus was seen one span above the water.
Forced by the wind, it gradually moved, and was submerged at a distance of two cubits.
O mathematician, tell quickly the depth of the water.

Well, that’s my translation, close to Longfellow’s quoted translation by Taylor and to Colebrooke’s better translation, but I may be wrong, so details for anyone who cares to improve it:

In a certain [kvāpi] pool [taḍāge] whose water [salile] was swarming [ākulita] with ruddy geese [cakra] and curlews [krauñcā],
above the water [toyād ūrdhvaṃ] a lotus-bud-tip [kamala-kalikāgraṃ] at a distance of one span [vitasti-pramāṇam] was seen [dṛṣṭaṃ].
Slowly slowly [mandaṃ mandaṃ] by the wind [anilena] moved [calitam] and forced [āhataṃ],
at a distance of two cubits [hasta-yugme] it got submerged [magnaṃ] in the water [tasmin].
O mathematician [gaṇaka], say [kathaya] quickly [kṣipram] the depth of the water [ambu-pramāṇam].

The structure of the book may be worth remarking on: the general formula for exactly this problem is given first (in more technical terms), and then this problem is given as an example!

Glancing through Longfellow, one finds he’s also written a tiny poem called King Trisanku:

Viswamitra the Magician,
By his spells and incantations,
Up to Indra’s realms elysian
Raised Trisanku, king of nations.

Indra and the gods offended
Hurled him downward, and descending
In the air he hung suspended,
With these equal powers contending.

Thus by aspirations lifted,
By misgivings downward driven,
Human hearts are tossed and drifted
Midway between earth and heaven.

Ho hum (1845 America).

The chapter of Kavanagh below this line.

“I was thinking to-day,” said Mr. Churchill a few minutes afterwards, as he took some papers from a drawer scented with a quince, and arranged them on the study table, while his wife as usual seated herself opposite to him with her work in her hand,—”I was thinking to-day how dull and prosaic the study of mathematics is made in our school-books; as if the grand science of numbers had been discovered and perfected merely to further the purposes of trade.”

“For my part,” answered his wife, “I do not see how you can make mathematics poetical. There is no poetry in them.”

“Ah, that is a very great mistake! There is something divine in the science of numbers. Like God, it holds the sea in the hollow of its hand. It measures the earth; it weighs the stars;it illumines the universe; it is law, it is order, it is beauty. And yet we imagine—that is, most of us—that its highest end and culminating point is book-keeping by double entry. It is our way of teaching it that makes it so prosaic.”

So saying, he arose, and went to one of his book-cases, from the shelf of which he took down a little old quarto volume, and laid it upon the table.

“Now here,” he continued, “is a book of mathematics of quite a different stamp from ours.”

“It looks very old. What is it?”

“It is the Lilawati of Bhascara Acharya, translated from the Sanscrit.”

“It is a pretty name. Pray what does it mean?”

“Lilawati was the name of Bhascara’s daughter; and the book was written to perpetuate it. Here is an account of the whole matter.”

He then opened the volume, and read as follows:—

“It is said that the composing of Lilawati was occasioned by the following circumstance. Lilawati was the name of the author’s daughter, concerning whom it appeared, from the qualities of the Ascendant at her birth, that she was destinedto pass her life unmarried, and to remain without children. The father ascertained a lucky hour for contracting her in marriage, that she might be firmly connected, and have children. It is said that, when that hour approached, he brought his daughter and his intended son near him. He left the hour-cup on the vessel of water, and kept in attendance a time-knowing astrologer, in order that, when the cup should subside in the water, those two precious jewels should be united. But as the intended arrangement was not according to destiny, it happened that the girl, from a curiosity natural to children, looked into the cup to observe the water coming in at the hole; when by chance a pearl separated from her bridal dress, fell into the cup, and, rolling down to the hole, stopped the influx of the water. So the astrologer waited in expectation of the promised hour. When the operation of the cup had thus been delayed beyond all moderate time, the father was in consternation, and examining, he found that a small pearl had stopped the course of the water, and the long-expected hour was passed. In short, the father, thus disappointed, said to his unfortunate daughter, I will write a book of your name, which shall remain to the latest times,—for a good name isa second life, and the groundwork of eternal existence.”

As the school-master read, the eyes of his wife dilated and grew tender, and she said,—

“What a beautiful story! When did it happen?”

“Seven hundred years ago, among the Hindoos.”

“Why not write a poem about it?”

“Because it is already a poem of itself,—one of those things, of which the simplest statement is the best, and which lose by embellishment. The old Hindoo legend, brown with age, would not please me so well if decked in gay colors, and hung round with the tinkling bells of rhyme. Now hear how the book begins.”

Again he read;—

“Salutation to the elephant-headed Being who infuses joy into the minds of his worshippers, who delivers from every difficulty those that call upon him, and whose feet are reverenced by the gods!—Reverence to Ganesa, who is beautiful as the pure purple lotos, and around whose neck the black curling snake winds itself in playful folds!”

“That sounds rather mystical,” said his wife.

“Yes, the book begins with a salutation to the Hindoo deities, as the old Spanish Chronicles begin in the name of God, and the Holy Virgin. And now see how poetical some of the examples are.”

He then turned over the leaves slowly and read,—

“One-third of a collection of beautiful waterlilies is offered to Mahadev, one-fifth to Huri, one-sixth to the Sun, one-fourth to Devi, and six which remain are presented to the spiritual teacher. Required the whole number of water-lilies.”

“That is very pretty,” said the wife, “and would put it into the boy’s heads to bring you pond-lilies.”

“Here is a prettier one still. One-fifth of a hive 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; and one bee continued flying about, attracted on each side by the fragrant Ketaki and the Malati. What was the number of the bees?”

“I am sure I should never be able to tell.”

“Ten times the square root of a flock of geese—”

Here Mrs. Churchill laughed aloud; but he continued very gravely,—

“Ten times the square root of a flock of geese, seeing the clouds collect, flew to the Manus lake; one-eighth of the whole flew from the edge of the water amongst a multitude of water-lilies; and three couple were observed playing in the water. Tell me, my young girl with beautiful locks, what was the whole number of geese?”

“Well, what was it?”

“What should you think?”

“About twenty.”

“No, one hundred and forty-four. Now try another. The square root of half a number of bees, and also eight-ninths of the whole, alighted on the jasmines, and a female bee buzzed responsive to the hum of the male inclosed at night in a water-lily. O, beautiful damsel, tell me the number of bees.”

“That is not there. You made it.”

“No, indeed I did not. I wish I had made it. Look and see.”

He showed her the book, and she read it herself. He then proposed some of the geometrical questions.

“In a lake the bud of a water-lily was observed, one span above the water, and whenmoved by the gentle breeze, it sunk in the water at two cubits’ distance. Required the depth of the water.”

“That is charming, but must be very difficult. I could not answer it.”

“A tree one hundred cubits high is distant from a well two hundred cubits; from this tree one monkey descends and goes to the well; another monkey takes a leap upwards, and then descends by the hypothenuse; and both pass over an equal space. Required the height of the leap.”

“I do not believe you can answer that question yourself, without looking into the book,” said the laughing wife, laying her hand over the solution. “Try it.”

“With great pleasure, my dear child,” cried the confident school-master, taking a pencil and paper. After making a few figures and calculations, he answered,—

“There, my young girl with beautiful locks, there is the answer,—forty cubits.”

His wife removed her hand from the book, and then, clapping both in triumph, she exclaimed,—

“No, you are wrong, you are wrong, my beautiful youth with a bee in your bonnet. It is fifty cubits!”

“Then I must have made some mistake.”

“Of course you did. Your monkey did not jump high enough.”

She signalized his mortifying defeat as if it had been a victory, by showering kisses, like roses, upon his forehead and cheeks, as he passed beneath the triumphal arch-way of her arms, trying in vain to articulate,—

“My dearest Lilawati, what is the whole number of the geese?”

Written by S

Sat, 2011-01-29 at 04:06:14

Posted in mathematics, sanskrit

5 Responses

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  1. Hey how do you type sanskrit in wordpress. I would also like to know.


    Sun, 2011-01-30 at 12:34:24

  2. “In a lake the bud of a water-lily was observed, one span above the water, and when moved by the gentle breeze, it sunk in the water at two cubits’ distance. Required the depth of the water.”

    Question – but how do you infer from this statement that when the lotus sunk at 2 cubits distance, its base remained at the same original place? The base too could have moved a bit, no?

    BTW, I loved this article. I’m impressed that Bhaskara was not only a great mathematician but also a pretty good poet, considering that in India today, there are strict boundaries between ‘arts’ and science. But if you’re creative, you can exhibit that creativity in any area.


    Mon, 2011-10-03 at 15:24:34

    • Well, the lotus plant grows out of the solid soil at the bottom of the lake, so the base is fixed (rooted) and cannot move. Right?
      [Of course, we can imagine alternative scenarios: perhaps the wind blew so strongly that it uprooted the plant and it’s just floating away, but then we cannot solve the problem. :-)]

      Thanks. Yes, the strict boundaries between “art” and “science”/”theory” that we see in modern India and the West are not really found in pre-modern India… consider also philosophers and logicians like Shankara, Vedantadesika, Dharmakirti, etc., who all wrote abstruse highly technical/philosophical works, but also great poetry.
      Creativity, of course, is essential in mathematics — no one would deny that, for instance, Euler or Poincare or Grothendieck were highly creative. What is interesting here with Bhaskara is creativity in versification too! I agree with your observation.


      Tue, 2011-10-04 at 04:35:54

  3. @svat , me stumbling on this blog post via a Google search result made my day. The chakrakrouncha was in my PUC Sanskrit text book and wanted to search for my daughter.


    Sun, 2014-12-14 at 17:30:03

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