Mathematical Ideas in Ancient Indian Poetry
Year: 2013 Authors: Sarah Glaz
Core claim
Ancient Indian poetic and religious texts preserve early mathematical ideas and can also serve as pedagogical material for teaching mathematics.
Topics
ancient Indian poetry, calendrical calculation, sacred geometry, mathematical poetry
Domains
number systems, geometry, calendar arithmetic, fractions, poetry, literary studies, sacred architecture, pedagogy
Methods
textual analysis, historical interpretation, quoted literary examples, cross-disciplinary comparison
Media
Sanskrit hymns, verse translations, fire-altar diagrams, poetic excerpts
Paper text
The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture
Mathematical Ideas in Ancient Indian Poetry
Sarah Glaz Department of Mathematics University of Connecticut Storrs, CT 06269, USA E-mail: Sarah.Glaz@uconn.edu
Abstract
Modern mathematics owes a big debt to India’s contributions to the subject. Of particular importance is the decimal, place value number system that appeared in India during the Vedic period or soon after, circa 1300 BC to 300 AD, and made its way to Europe during the Middle Ages. That period of time in India also produced a heady mixture of poetic works: poems, songs, grand epics, biographies and books of instruction in verse covering millions of pages. Mathematical ideas are interwoven into the metaphysical, religious and aesthetic fabric of many of these works. This article brings a selection of poems from that time period that provides a taste of ancient India’s mathematical preoccupations in their cultural and esthetic context. They also highlight India’s mathematical accomplishments of the period, and uncover instances where seeds of future mathematical concepts made their first appearance. The concluding remarks touch lightly on current Indian-inspired uses of mathematical poetry as a pedagogical tool.
The Vedas and Supplementary Texts: Calendrical Calculations and Geometry
The most ancient Indian literary works are the four Vedas. Controversially dated around 1300 BC, the oldest of the four, the Rig Veda, is a collection of 1028 hymns written in classical Sanskrit. Mathematical ideas involving astronomy and time reckoning appear in a number of the hymns, particularly those describing the creation of the world. Contrary to other cultures, in Indian creation myths the world has no absolute beginning—a new world emerges from an already existing one. The creation is a sacrificial act in which the old gods sacrifice a primordial man, Purusha. The world, the seasons, the celestial bodies which start time reckoning of this universe’s time cycle, and the new gods themselves, are made out of various parts of Purusha’s body. Below is a fragment from a creation hymn from Rig Veda [21]. Note the precise mathematical measurements, including a very early mention of fractions:
From: Rig Veda (Book 10, Hymn 90): Purusha
A thousand heads hath Puruha, a thousand eyes, a thousand feet.
On every side pervading earth he fills a space ten fingers wide.
All creatures are one-fourth of him, three-fourth eternal life in heaven.
With three-fourths Purusha went up: one fourth of him again was here.
Forth from his navel came mid-air; the sky was fashioned from his head;
Earth from his feet, and from his ear the regions. Thus they formed the worlds.
An important task of ancient mathematics was to develop accurate ways of counting time. The ancient Indian calendar was a lunisolar calendar with intercalated months in five year cycles. In addition, each region implemented a number of changes based on local astronomical and religious customs making calendrical calculations very complicated. Reckoning of sacred time, the time that returns unchanged year after year at each religious holiday, was interwoven into the construction of sacrificial altars. In addition
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to the four Vedas, the Vedic literature includes a large number of supplementary texts. One of these texts, the Satapatha Brahmana (c. 600-500 BC) [22], contains verses of instruction for the construction of sacrificial altars to the fire god, Agni. Mathematically, two interesting features stand out: the interweaving of the notion of sacred time reckoning into altar construction and the precise and extensive plane geometry necessary for the constructions of the altars themselves.
A public fire altar was built in the shape of a falcon consisting of bricks of various geometric shapes. The fire altar was identified with a fixed unit of time—the year. Elements in the construction of the altar corresponded to calendrical elements of the year. Mircia Eliade explained the process as follows: “…with the building of each fire altar… the year is built too,…time is regenerated by being created anew” [4]. In the poem fragment below, the altar is the year with 360 enclosing bricks corresponding to nights, 360 Yagushmat
bricks identified with days, and 36 bricks that are “over” corresponding to the intercalary month. “Space-filler” refers to ruler of the world.
From: Satapatha Brahmana (X Kanda, 5 Adhyaya, 4 Brahmana)
But, indeed, that Fire-altar also is the Year, the nights are its enclosing-stones, and there are three hundred and sixty of these, because there are three hundred and sixty nights in the year; and the days are its Yagushmatī bricks, for there are three hundred and sixty of these, and three hundred and sixty days in the year; and those thirty-six bricks which are over are the thirteenth month, the body of the year and the altar, the half-months and months, there being twenty-four half-months,
and twelve months. And what there is between day and night that is the Suddahas; and what food there is in the days and nights is the earth-fillings, the oblations, and the fire-logs; and what is called “days and nights” that constitutes the space-filling brick:—thus this comes to make up the whole Agni, and the whole Agni comes to be the space-filler and, verily, whosoever knows this, thus comes to be that whole Agni who is the space-filler.
Satapatha Brahmana’s instructions for the construction of fire altars referred not only to the number of bricks that went into each construction, but also to the proportions among various parts of the falcon shaped altar. To achieve specified area proportions, it was necessary to convert some geometric shapes into others without changing the areas involved. This was done by a cut-and-paste plane geometry technique that involved precise measurements and sophisticated geometric considerations. For example, particular cases of the result known to us as Pythagoras Theorem were employed. The second stanza of the fragment below explains how to crop two pieces from the top of both sides of a rectangle and “glue” them to the bottom of the sides in order to get a trapeze of the same area. This constructs the falcon’s tail and the first step in the construction of each wing. In the next step, not reproduced here, each wing was given a “bent,” that is, each wing changed shape from a trapeze into two “glued” parallelograms.
From: Satapatha Brahmana (X Kanda, 2 Adhyaya, 1 Brahmana)
Pragapati was desirous of going up to the world of heaven; but Pragapati, indeed, is all the sacrificial animals—man, horse, bull, ram, and he-goat:—by means of these forms he could not do so. He saw this bird-like body, the fire-altar, and constructed it. He attempted to fly up, without contracting and expanding the wings, but could not do so. By contracting and expanding the wings he did fly up: whence even to this day birds can only fly up when they contract their wings and
spread their feathers.
He contracts the right wing inside on both sides by just four finger-breadths, and expands it outside on both sides by four finger-breadths; he thus expands it by just as much as he contracts it; and thus, indeed, he neither exceeds its proper size nor does he make it too small. In the same way in regard to the tail, and in the same way in regard to the left wing.
Mathematical Ideas in Ancient Indian Poetry
The Grand Epics: Mathematical Games, Probability and Statistics
The 5th to 4th century BC saw the beginning of the writing of the two major Sanskrit epics of ancient India, the Mahābhārata and the Rāmāyana. Written in verse, both books include narratives of grand proportions along with philosophical and devotional material, and offer tantalizing glimpses of mathematical interests in lush and exotic settings. The Mahābhārata in its longest version consists of 200,000 lines of verse. Among the principal narratives of the Mahābhārata is the story of King Nala and his wife Damayanti.
The story begins when King Nala looses everything, including his kingdom, in a dice game. He abandons his faithful and loving wife, Damayanti, for her own good. After a series of misadventures, Damayanti is reunited with her parents and her two children. Meanwhile Nala, under the assumed name Vāhuka, obtains a position as cook and charioteer of the King of the Forest, Rituparna. Damayanti devises an ingenious plan for bringing Rituparna, and therefore his driver, Nala, to her father’s castle. A “mathematical incident” occurs on the way, after which Nala arrives free of the passion for gambling and they live happily ever after. Below are three verse fragments, with commentary, describing the “mathematical incident” as rendered in English by Sir Edwin Arnold (1832 - 1904)[20]:
From: Mahābhārata: Nala and Damayanti
A little onward Rituparna saw Within the wood a tall Myrobolan Heavy with fruit; hereat, eager he cried:— “Now, Vāhuka, my skill thou mayest behold In the Arithmetic. All arts no man knows; Each hath his wisdom, but in one man’s wit Is perfect gift of one thing, and not more. From yonder tree how many leaves and fruits,
Think’st thou, lie fall’n there upon the earth? Just one above a thousand of the leaves, And one above a hundred of the fruits; And on those two limbs hang, of dancing leaves, Five crores exact; and shouldst thou pluck yon boughs, Together with their shoots, on those twain boughs Swing twice a thousand nuts and ninety-five!”
A “crore” is equal to 10,000,000. In this stanza Rituparna tells Nala the number of fruits (nuts) and leaves on the ground, the number of leaves on “two limbs,” and the number of nuts on “twain boughs.” All this, without apparently doing any counting. Statistics is the modern mathematical discipline that investigates procedures of inference of the sum total by examining small samples. This is the first known mention of statistical inference in literature. Intrigued, Nala stops the carriage to check the prediction by counting fruits and leaves. The numbers match and the following dialog occurs:
To Rituparna spake: “Lo, as thou saidst So many fruits there be upon this bough! Exceeding marvelous is this thy gift, I burn to know such learning, how it comes.”
Answered the Raja, for his journey fain:— “My mind is quick with numbers, skilled to count; I have the science.”
“Give it me, dear Lord!”
Vāhuka cried: “teach me, I pray, this lore, And take from me my skill in horse-taming.”
Quoth Rituparna—impatient to proceed Yet of such skill desirous: “Be it so! As thou hast prayed, receive my secret art, Exchanging with me here thy mastery Of horses.”
As soon as Nala learned the “secret art” he became cured of his addiction to gambling. Thus is the power of mathematics!
Thereupon did he impart His rule of numbers, taking Nala’s too. But wonderful! So soon as Nala knew
From the afflicted Prince That bitter plague of Kali passed away.
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In addition to statistics, the story of Nala and Damayanti anticipates the development of other modern branches of mathematics. In the first part of the story there is a description of the game of chance in which Nala lost his kingdom:
That hour there sat with Nala, Pushkara His brother; and the evil spirit hissed Into the ear of Pushkara: “Ehi! Arise, and challenge Nala at the dice. Throw with the Prince! it may be thou shalt win (Luck helping thee, and I) Nishadha’s throne, Town, treasures, palace—thou mayst gain them all.”
And Pushkara, hearing Kali’s evil voice, Made near to Nala, with the dice in hand (A great piece for the “Bull,” and little ones For “Cows,” and Kali hiding in the Bull). So Pushkara came to Nala’s side and said:— “Play with me, brother, at the “Cows and Bull;”
It seems that the game was played with a number of dice or pieces, a large one “bull” and a few small ones “cows.” The game is not identifiable, but dice games gave rise to the need of calculating odds of winning and thus set the stage for the development of the modern mathematical area of Probability. This game is of mathematical interest in another way as well, since it appears to be a board game played with a number of pieces, and as such has some similarity, and perhaps is a precursor, to chess. It is generally believed that the game of chess originated in India. The Indian Sanskrit name for chess was “shatranga,” meaning four “anga” (detachments) following the arrangement of the troops in the Battle of Kurukshetra described in the Mahābhārata. The game was originally played with four armies and a dice [6]. Chess is a particularly mathematical game, but all games involving strategy, contributed to the development of the modern branch of mathematics called Game Theory, which is concerned, among other things, with determining winning strategies. It is rare to find references to games in ancient manuscripts. Perhaps the earliest such mention is a reference to “draughts” (checkers) in The Papyrus of Ani, The Egyptian Book of the Dead (c. 1400 BC) [5], where the title of Plate VII reads: “…the forms of existence which pleased the deceased, of playing at draughts and sitting in the Seh Hall…”
Two men playing chess, century Persian (see [13])
In another grand epic, the Shahnama (Epic of the Kings), the Persian poet, Abu’l Qasim Firdausí (932-1025 AD), tells the story of the introduction of chess into Persia around 550 AD, and the legend of the game’s origin: Gav and Talhand, two sons of an Indian queen, quarreled about the succession to her throne. A battle ensued and Talhand perished. To clear his name and to console his mother, Gav asked the sages of the region to invent a game which will show every move of the decisive battle and in this way prove him innocent of his brother’s death. A small fragment from Shahnama is reproduced below [7]. Firdausí claims that the original game of chess was played with two armies on a chess board of
100 squares, which decreased to 64 by the time chess reached Persia.
From: Shahnama: The Story of Gav and Talhand and the Invention of Chess by Abu’l Qasim Firdausi
Those men of wisdom called for ebony, And two of them—ingenious councilors— Constructed of that wood a board foursquare To represent the trench and battlefield, And with both armies drawn up face to face. A hundred squares were traced upon the board, So that the kings and soldiers might manoeuvre. Two hosts were carved of teak and ivory, And two proud kings with crowns and Grace divine. Both horse and foot were represented there, And drawn up in two ranks in war-array,
The steeds, the elephants, the ministers, And warriors charging at the enemy, All combating as is the use in war, One in offence, another in defense.
King Gav, the great and good, affected much The game of chess suggested through Talhand; His mother studied it. Her heart was filled With anguish for that prince. Both night and day She sat possessed by passion and by pain, With both her eyes intent upon the game.
Mathematical Ideas in Ancient Indian Poetry
Buddhism and Jainism: The Number System and the Transfinite
Around 500 BC India went through a political and religious upheaval. Indian states were established, and two new religions, Buddhism and Jainism, rebelling against Vedic values and caste system, came into prominence. Buddhism was founded by Prince Siddhartha Gautama, called Buddha (620-543 BC). Jainism, believed to be of ancient origin, venerated the last enlightened teacher, Prince Vardhamana, called Mahavira (599-527 BC). The two religions have many aspects in common, among them the incorporation of scientific and mathematical ideas into their philosophical and metaphysical systems. Both leaders were believed to be scholars, well versed in sciences and the other learned subjects of the day. In Kalpa Sutra [19], the story of Mahavira’s life, it is said that he was “… versed in the philosophy of the sixty categories, and well-grounded in arithmetic, in phonetics, ceremonial, grammar, metre, etymology, and astronomy.” A charming story appearing in Lalitavistara [17], the story of Buddha’s life written around 100 AD, shows Buddha’s early prowess with mathematics. The verse fragment below, from Edwin Arnold’s Light of Asia [1], is based on verses from Lalitavistara:
From: The Light of Asia (Book the First): The Education of Buddha by Edwin Arnold
And Viswamitra said, “It is enough, Let us to numbers.
After me repeat Your numeration till we reach the Lakh, One, two, three, four, to ten, and then by tens To hundreds, thousands.” After him the child Named digits, decades, centuries; nor paused, The round lakh reached, but softly murmured on “Then comes the kôti, nahut, ninnahut, Khamba, viskhamba, abab, attata, To kumuds, gundhikas, and utpalas, By pundarīkas unto padumas, Which last is how you count the utmost grains Of Hastagiri ground to finest dust; But beyond that a numeration is, The Kātha, used to count the stars of night; The Kôti-Kātha, for the ocean drops; Ingga, the calculus of circulars; Sarvanikchepa, by the which you deal With all the sands of Gunga, till we come To Antah-Kalpas, where the unit is The sands of ten crore Gungas. If one seeks More comprehensive scale, th’ arithmetic mounts By the Asankya, which is the tale Of all the drops that in ten thousand years Would fall on all the worlds by daily rain; Thence unto Maha Kalpas, by the which The Gods compute their future and their past.
“Tis good,” the Sage rejoined, “Most noble Prince, If these thou know’st, needs it that I should teach The mensuration of the lineal?” Humbly the boy replied, “Acharya!” “Be pleased to hear me. Paramānus ten A parasukshma make; ten of those build The trasarene, and seven trasarenes One mote’s-length floating in the beam, seven motes The whisker-point of mouse, and ten of these One likhya; likhyas ten a yuka, ten Yukas a heart of barley, which is held Seven times a wasp-waist; so unto the grain Of mung and mustard and the barley-corn, Whereof ten give the finger-joint, twelve joints The span, wherefrom we reach the cubit, staff, Bow-length, lance-length; while twenty lengths of lance Mete what is named a ‘breath,’ which is to say Such space as man may stride with lungs once filled, Whereof a gow is forty, four times that A yōjana; and, Master! if it please, I shall recite how many sun-motes lie From end to end within a yōjana.” Thereat, with instant skill, the little Prince Pronounced the total of the atoms true.
The above poem starts with young Buddha being asked to name all numbers up to “lakh,” which means 100,000. But he continues beyond the lakh to “kôti,” which equals , and through increasing powers of 10 to . Such large numbers had names, but the names were not standard, and it is difficult
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to say which is which in this poem. In the next stanza, young Buddha enumerates units of lengths up to a “yōjana,” which is about 9 miles, and then apparently he also names the number of atoms in a yōjana. The number does not appear in Arnold’s English translation/transformation of the poem, but Lalitavistara cites it as 384,000 × 10⁷.
The cultural phenomena encountered here are deep respect for mathematical ability and a passion for large numbers. Large numbers came naturally to a culture where people were believed to live through multiple cycles of existence spanning seemingly never-ending periods of time. This may have been a motivating force behind the development of India’s number system. India needed an adequate way to express and work with astronomically large numbers and therefore developed the decimal, place value numeral system and a notation for numbers that is conductive to arithmetic operations. In a place value system 5, for example, in the first place means 5, while 5 in 56 means 50, and 5 in 546 means 500. It was therefore inevitable that the development of such a numeral system introduced and used 0, the way we do today, as a number of 0 value. An earlier version of a place value system, but with base 60, appeared in Babylonian mathematics, and zero was used by the Babylonians to denote an empty space. The Chinese also claim priority for the invention of the decimal, place value system, using 0 as a number [16]. What is not controversial is that the Indian decimal, place value system (including zero) and notation for numbers, made its way to the Arab world, and through it, during the Middle Ages, traveled to Europe, where it interacted with the Greek method of mathematical deduction, to build the basis of present day mathematics [12, 14, 23]. Nobel Prize laureate Wislawa Szymborska’s poem [24] reproduced below gives voice to the historical uncertainty of the origin of the number 0.
A Poem in Honor of
by Wislawa Szymborska
Once, upon a time, invented zero.
In an uncertain country. Under a star
which may be dark by now. Bounded by dates,
but no one would swear to them. Without a name,
not even a contentious one. Leaving behind
no golden words beneath his zero
about life being like. Nor any legends:
that one day he appended zero
to a picked rose and tied it up into a bouquet;
that when he was about to die, he rode off into the desert
on a hundred-humped camel; that he fell asleep
in the shadow of the palm of primacy; that he will awaken
when everything has been counted,
down to the last grain of sand. What a man.
Slipping into the fissure between fact and fiction,
he has escaped our notice. Resistant
to every fate. He sheds
every form I give him.
Silence has closed over him, his voice leaving no scar.
The absence has taken on the look of the horizon.
Zero writes itself.
Jainism’s early texts exhibit the same passion for large numbers, but favor their expression as powers of 2 rather than powers of 10. Mathematically, an interesting addition is the development of an intuitive notion of numeral infinity, with several categories of infinity. We start with a fragment from the Jainist text Tattvārtha Sutra, That Which Is [25] describing what seems to be an infinite chain of concentric circles:
Mathematical Ideas in Ancient Indian Poetry
From: Tattvārtha Sutra: The Lower and Middle Regions
by Acharya Umāsvāti
There are islands and oceans that bear propitious names such as Jambu Island, Lavana Ocean and so on. The islands and oceans are concentric rings, the succeeding ring being double the preceding one in breadth. At the centre of these islands and oceans is the round island Jambu with a diameter of 100,000 yōjanas and Mount Meru at its navel.
There are seven continents on Jambu Island: Bharata, Haimavata, Hari, Videha, Ramyaka, Hairanyavata and Airavata. The six mountains that extend from east to west and divide the seven continents are Himavan, Mahahimavan, Nisadha, Nila, Rumkin and Sidharin. The mountains are, respectively, as golden as Chinese silk, as white as the Arjuna tree, as crimson as the rising sun, as blue as sapphire, as white as silver, as golden as Chinese silk.
Continuing the description of islands and oceans, Tattvārtha Sutra states that the number of islands and oceans is “innumerable.” Some historians argue that “innumerable” means “countable infinite.” But Tattvārtha Sutra also mentions that the concentric circles of islands and oceans stops at an ocean called Svayambhuramana, and thus the number of islands and oceans cannot be infinite. More likely, “innumerable” meant a very large number, perhaps large enough that it was not yet given a name. Of interest is also the following continuation of the treatment of infinity:
From: Tattvārtha Sutra: Substances
by Acharya Umāsvāti
There are innumerable soul units in a soul.
There are an infinite number of space units in space.
The number of units in clusters of matter may be numerable, innumerable or infinite.
The definitions of numerable, innumerable and infinite in Tattvārtha Sutra are obtained through a recursive process, via the distribution of mustard seeds among the concentric rings of islands and oceans described above. This is sophisticated mathematical thinking, but it is not mathematically precise in the mathematical sense of today. If we accept that “innumerable” is indeed countable infinity, then the recursive procedure leading to Tattvārtha Sutra’s “infinite” gives a cardinality of , which is indeed infinite and uncountable. What we have here is an intuitive notion of two kinds of infinity. It took more than 1000 years longer for Cantor to create set theory and with it the precise mathematical tools required to describe various kinds of infinity.
Concluding Remarks
The Vedic period, the most ancient time considered in this article, was not the oldest civilization known to have inhabited the Indus Valley. Traces of the Harappan period, a five thousand years old civilization, were uncovered in several places in India. This civilization seems to have developed a certain amount of mathematics as seen in the mensuration and weighting devises found at excavation sites. They also developed a pictographic form of script, which unfortunately is still undeciphered [12, 14]. Perhaps poetic works older than Rig Veda, containing seeds of mathematical concepts, are still waiting to be decoded. Due to space restrictions, this article stops at the time the two great religions of India, Buddhism and Jainism, came into maturity—around 300 AD. But mathematics in poetic form has not ended its appearance in India at that time. In fact the golden age of Indian mathematics with its cultural tradition of recording mathematical results and problems in verse occurred during the Middle Ages. Some of the most charming mathematical poems come from this tradition. For example, Bhaskara (1114-1185 AD), the best known of medieval Indian mathematicians, wrote an algebra book intended for the education of his daughter, Lilavati. The book’s title is also Lilavati (meaning “the beautiful”), and it was written entirely in verse [3, 12, 11].
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Mathematics written in poetic form appears to this day, often in the service of mathematical pedagogy. Inspired by the mathematical poetry of medieval India, Barbara Jur [15] encouraged her algebra class to compose word-problems in poetry. The results have both mathematical and poetic merit. Jur’s motivation was to enrich teaching by engagement, but in articles [2, 18] we find reports of such poetry writing experiments conducted in Pre-Calculus, Calculus, and Statistics classes that conclude that poetry writing in mathematics classes strengthens students’ understanding and integration of the subject matter. The mathematical poetry of medieval India and the difficulties students have with word-problems in algebra feature in another article describing the use of poetry in a college algebra course [8]. Glaz and Liang [8] used poems from Lilavati and other historical sources to ease the difficulties students have with the transition between word-problems representing natural phenomena and the corresponding mathematical models—the equations representing the phenomena. The process yielded additional pedagogical benefits, such as the strengthening of students’ number sense and mathematical intuition and the enhancement of retention and integration of the material [8]. A more extensive survey of the uses of poetry in mathematical pedagogy, as well as the poem The Enigmatic Number e, may be found in [9], while [10, 11] are sources for additional mathematical poetry. The connections between history, poetry, mathematics, and pedagogy unfold like the Indian myth of creation, with no absolute beginning and elements of surprise in the future.
Acknowledgements
The author gratefully acknowledges permission to reprint from J. Trzeciak for “A Poem in Honor of” by W. Szymborska.
References
[1] E. Arnold, The Light of Asia (The Life of Gautama Buddha), Robert Brothers, Boston, 1891 [2] P. Bahls, Math and metaphor: Using poetry to teach college mathematics, Writing Across the Curriculum J. 20, 75-90, 2009 [3] Bhaskara, Lilavati, Colebrooke (tr.), H. C. Banerji (commentator), Asian Educational Services, New Delhi, 1993 [4] M. Eliade, The Sacred and the Profane: The Nature of Religion, W. R. Trask (tr.), Harvest/HBJ Pub., 1957 [5] The Egyptian Book of the Dead (The Papyrus of Ani), E.A. W. Budge (tr.), Dover Pub. Inc., New York, 1967 [6] G. Ferlito & A. Sanvito, On the Origins of Chess, The Pergamon Chess Monthly 55 (6), 1990 [7] A. Q. Firdausi, Shāhnāma (The Epic of the Kings), A. G. Warner & E. Warner (tr.), Trench, Trübner $ Co., London, 1915 [8] S. Glaz & S. Liang, Modeling with Poetry in an Introductory College Algebra Course and Beyond, J. of Mathematics and the Arts 3, 123-133, 2009 [9] S. Glaz, The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom, MAA Loci: Convergence, DOI 10.4169/loci 003482, 2010 [10] S. Glaz, Poetry Inspired by Mathematics: A Brief Journey through History, J. of Mathematics and the Arts 5, 171-183, 2011 [11] S. Glaz & J. Growney (ed.), Strange Attractors: Poems of Love and Mathematics, CRC Press/A K Peters, 2008 [12] Georges Ifrah, The Universal History of Numbers, D. Bellos, E.F. Harding, S. Wood, & I. Monk (tr.), Wiley & Sons, Inc., New York, 2000 [13] N. A. R. Jami, Haft Awrang (Seven Thrones). Image is a detail from a full page 16th century illustration of this 15th century poetry book. May be found online at: http://www.asia.si.edu/exhibitions/online/loveyearning/base.html and other sites [14] G. G. Joseph, The Crest of the Peacock, Non-European Roots of Mathematics, Princeton University Press, Princeton, 2011 [15] B. A. Jur, The Poetry of Mathematics, Primus 1, 75-80, 1991 [16] R. Kaplan, The Nothing That Is, A Natural History of Zero, Oxford University Press, Oxford, 1999 [17] Lalitavistara Sutra, G. Bays (tr.), Dharma Pub., Berkeley, 1984 [18] C. C. Patterson, and J. W. Patterson, Poetry Writing in Quantitative Courses, Decision Sciences J. of Innovative Education 7, 233-238, 2009 [19] Kalpa Sutra, H. Jacobi (tr.), The Sacred Books of the East, Oxford at the Clarendon Press, London, 1884 [20] Mahābhārata: Nala and Damayanti, (E. Arnold, tr.), Little, Brown, and Co., Boston, 1907 [21] Rig Veda, R.T.H. Griffith (tr.), Chowkhamba Sanskrit Series Office Vol. XXXV, Vidyavilas Press, Varanasi, 1971 [22] Satapatha Brahmana, J. Eggling (tr.), The Sacred Books of the East, Oxford at the Clarendon Press, London, 1897 [23] D.E. Smith, History of Mathematics, Vol. I & II, Dover Pub., New York, 1958 [24] W. Szymborska, Miracle Fair, J. Trzeciak (tr.), W.W. Norton & Co., New York, 2001 [25] A. Umāsvāti, Tattvārtha Sutra, That Which Is, N. Tatia (tr.), Harper Collins Pub. San Francisco, 1994