Meru Prastarah (or Pascal’s Triangle ?!?)

Original Writer : Ivanga.wordpress.com (Below is a copy for reading)

Let me start with an ancient (1000 CE) Sanskrit text as below:

*अतोऽनेकद्वित्रिलघुक्रियासिद्यर्थ यावदभिमतं प्रथमप्रस्तारवन्मेरुप्रस्तारं दर्शयति-*

*परे पूर्णमिति ॥ ८। ३५ ॥*

*उपरिष्टादेकं चतुरस्रकोष्ठं लिखित्वा तस्याधस्तादुभयतोऽर्धनिष्क्रान्तं कोष्ठकद्वयं लिखेत्।**तस्याप्यधस्तात्रयं तस्याप्यधस्ताच्चतुष्टयं यावदभिमतं स्थानमिति मेरुप्रस्तारः ॥**तस्य प्रथमे कोष्ठे एकसंख्यां व्यवस्थाप्य लक्षणमिदं प्रवर्तयेत्। तत्र परे कोष्ठे यद्वृत्तसंख्याजातं तत् पूर्वकोष्टयोः पूर्णं निवेशयेत्।**तत्रोभयोः कोष्ठकयोरेकैकमंगं दद्यात्, मध्ये कोष्ठे तु परकोष्टद्वयांकमेकीकृत्य पूर्ण निवेशयेदिति पूर्णशब्दार्थः।**चतुर्थ्यां पंक्तावपि पर्यन्तकोष्ठयोरेकैकमेव स्थापयेत्। मध्यमकोष्ठयोस्तु परकोष्ठद्वयांकमेकीकृत्य पूर्णं त्रिसंख्यारूपं स्थापयेत्।**उत्तरत्राप्ययमेव न्यासः। तत्र द्विकोष्ठायां पंक्तौ एकाक्षरस्य विन्यासः। तत्रैकगुर्वेकलघुवृत्तं भवति। तृतीयायां पंक्तौ द्वयक्षरस्य प्रस्तारः।**तत्रैकं सर्वगुरु, द्वे एकलघुनी, एकं सर्वलघ्विति कोष्ठक्रमेण वृत्तानि भवन्ति ॥ चतुर्थ्यां पंक्तौ त्यक्षरस्य प्रस्तारः।**तत्रैकं सर्वगुरु त्रीण्येकलघूनि त्रीणि द्विलघूनि एकं सर्वलघु ॥ तथा पंचमादिपंक्तावपि सर्वगुर्वादिसर्वलघ्वन्तमेकद्वयादिलघु द्रष्टव्यमिति ॥*^{[2]}

**The above text in Devanagari script is given below in Roman script:**

Anena ekadvyaadilaghukriyasiddhyartham, yaavadabhimatam prathama prastaravat meruprastaram darsayati, uparistadekam chaturasrakoshtam likhitva, tasya adhastat ubhayatordhaniskrantam koshtadwayam likhet, tasyapiadhastatrayam tasyapiadhastatccaturtyamevam yaavadabhimatam sthanamiti meruprastarah tasya prathame koshte ekasamkhyam vyasthapyalakshanamidam pravarttayet, tatra dvikoshtaayaampanktau ubhayo koshtayorekaikamankam dadyaat, tatastritiyaayaam panktau, paryantakoshtayorekaikamankam dadyaat, madhyamakoshtethuparikoshtadvayaankamekikrtya purnam nivesayediti purnasabdarthah, chaturtyampanktavapi, paryantakoshtayorekaikamankam sthapayet, madhyamakoshtayothuparakoshtadvayaankamekikrtya purnam trisamkhya rupam sthapayeth, uttaraataraapyevameva nyaasah, tatra dwikoshtaayaam pankatauekaakshrasya prastaarah,……. tritiyayaam pankatau, dviakshrasya prastaarah, chaturtyaam pankatau, triakshrasya prastaarah, ….

The above is not in praise of any god of Jain, Budha or Hindu religion. It is not a religious text at all. It is a text describing a method for constructing a mathematical table. Ancient Indian Mathematician Pingala (200 BC) in his Chandahsutra had given the rules for formation of different chandahs (≈ musical meters) for Sanskrit prosody. Another ancient Indian mathematician Halayudha (1000 CE) has given the explanation and commentary on this work by Pingala. Given above is a selected portion of his commentary. For some reasons unknown to me, ancient Sanskrit texts always use composite words very frequently. These words need to be broken into individual words properly to obtain the intended meaning of these words. Here is an attempt to translate the above text into English with proper separation of words.

Anena ekadvyaadi laghu kriya siddhyartham, yaavadabhi matam

(To get every combination of one, two, etc. syllables as required)

Prathama prastaravat meru prastaram darsayati.

(from first row onwards , the meru tabulation will be shown below)

Uparihi tad ekam Chaturasrakoshtam likhitva,

(At the top itself one square cell is drawn)

Tasya adhah tat ubhayato ardhani skrantam

(Below this row let us have a pair, half over lapping)

Koshtadwayam likhet.

(Two cells are drawn)

Tasyapi adhah tat trayam

(Again the row below will have three)

Tasyapi adhah tat chaturtyam,

(Again its next line will have four)

Evam yaavadabhi matam sthanam

(same way, up to the required stage, cells are constructed)

iti meru prastarah.

(This is called Meru Prastara or Meru-Tabulation)

Tasya prathame koshte eka samkhyam

(Its first stage-cell will hold the number 1)

Vyvasthapya lakshanamidam pravarttayet

(From here on, the following is the way it grows)

Tatra dvikoshtaayaam panktau

(in its twin-cell row)

ubhayo Koshtayoh eka ekam ankam dadyaat

(the pair of cells holds numbers 1,1)

Tatah tritiyaayaam panktau, paryanta Koshtayoh Eka ekam ankam dadyaat

(then in the 3^{rd} row, the extreme cells will hold numbers 1,1)

Madyama koshteth, upari koshtadvayah ankam eki krtya purnam nivesayeth

(middle cell takes the added value of the two cells above)

Iti purnasabdarthah

(Thus completes the table for 2^{nd} power)

Chaturtyam panktau api, paryanta Koshtayoh eka ekam ankam sthapayet

(then in the 4th row also, the extreme cells will hold numbers 1,1)

Madyama koshtayoth, upara koshtadvayah ankam eki krtya purnam

(middle cells take the added values of the two cells above each)

Trisamkhya rupam sthapayeth

(this completes the 3^{rd} power)

Uttara utaaro api evameva nyaasah,

(next and next stages also follow the same rule)

tatra dwikoshtaayaam pankatau, eka akshrasya prastaarah

(Here the twin-cell row gives one syllable table)

tritiyaam pankatau, dvi akshrasya prastaarah

(the 3rd row gives two syllables table)

chaturtyaam pankatau, tri akshrasya prastaarah

(Thus 4th row gives three syllables table)

And so on.

If we follow the above step by step construction given so clearly by Halayudha (1000 CE), we get the above pyramid or Meru in Sanskrit, (stands for a mountain with a peak). What do we have here? It is the same as **Pascal’s Triangle,** “discovered” by Blaise Pascal (1623-1662 CE).

This table gives in every nth line the coefficients **(a+b)**(n-1)**. i.e. the second line gives coefficients of **(a+b)** as **1,1**; the second line gives **1,2,1**, as coefficients of **(a+b) ^{2}**.; the third line gives

**1,3,3,1**as coefficients of

**(a+b)**and so on.

^{3}However Halayudha gives credit for this table to Pingala (200 BC). He claims to have derived this table from Pingala’s cryptic clue which he translates to a set of rules, as below (with **a** and **b** as the two syllables to be combined, in any n-syllable chandah):

- First write down all (‘n’ number of)
**b**’s as the first combination - In the next line, replace the first ‘
**b**’ with an ‘**a**’ - At the same line, replace all letters to the left of this new ‘
**a**’ with ‘**b**’ - For the next and the subsequent lines repeat the steps 2 & 3.
- Continue as above till we arrive at a line with all a’s,

This can be clearly seen as a binomial expansion **(a+b) ^{n}** staring with

**b**and ending in

^{n}**a**. Halayudha later puts these results on a table known as Meru Prasthara. He later gives a step-by-step method as above, for constructing this table without specifically going through the above rules. This

^{n}**Meru Prastarah**traveled to China and the Chinese mathematician Yang Hui reported it in the thirteenth century, although his work was unknown in Europe until relatively recent times. The

**Meru Prastarah**traveled to Europe a little later through Arabia, Egypt and Greece and gets “discovered” by Pascal in 17

^{th}century CE, 600 years after Halayudha. We are blaming all the time ‘the lack of scientific temper’ among Indians.

**Ref:**

- Binomial Theorem in Ancient India – By Amulya Kumar Bag, History of Science, Ancient Period Unit II, No.1, Park Street, Calcutta-16 (1966)
- Journey Through Genius – The Great Theorems Of Mathematics – by William Dunham – Wiley Science Editions, John Wiley & Sons Inc.(1990)
- Probability in Ancient India, by C K Raju., ckraju.net, 2011.