Archive for October, 2014

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

October 21, 2014

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

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

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 3rd 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 2nd 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 3rd 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)3 and so on.

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):

  1. First write down all (‘n’ number of)  b’s as the first combination
  2. In the next line, replace the first ‘b’ with an ‘a
  3. At the same line, replace all letters to the left of this new ‘a’ with ‘b
  4. For the next and the subsequent lines repeat the steps 2 & 3.
  5. 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 bn and ending in an. 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 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 17th century CE, 600 years after Halayudha. We are blaming all the time ‘the lack of scientific temper’ among Indians.


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

Connected Topics:

Meru Prastarah

Baudhayana’s Circles

Square Root of Two

Sine of an Angle

LVN/ Oct, 2014

Idea for a new Eco-Toilet

October 16, 2014

Recently I saw a news item in Times of India (12th Oct 2014) about saving water by peeing in the shower. When I googled it there were many who were supporting this idea for conservation of water. Hence I am encouraged to publish as a blog, my earlier idea of 2010, for an Eco-Toilet.

New Design for an Eco-toilet

 The goal is to conserve the use of water in a flush-out toilet commode.

The idea is to revise the design of flush out toilets. Millions of people use flush-out toilets (Indian or Western Type). While the amount of water required to flush the solid waste is very high, we do not need the same amount of water for urinating purpose. Though a few designs were made earlier to provide two flush systems in the toilet for major and minor uses, these designs were neither popular nor very successful. The amount of water wasted is enormous as minor uses are about ten times more than the major uses in a day. Hence this suggestion is for the following rough design of a new eco-toilet.


The pot can be divided into two compartments 1 and 2. The outlets of pot-1 can directly drain into the main drain through a separate smaller s-bend and then to the septic tank. The pot-2 can flush out the solid waste through the normal siphon system. This way pot-1 needs much less water as in any normal urinal. Even a mug of water will do the job. A dual flush system could be an added facility. Both the smaller and bigger s-bends are integrated in the same ceramic mold and will hold water in the bend to seal off septic tank from the toilet

Implementation: This idea can be implemented by all the people who have access to toilets with septic tanks. The sanitary engineers should study this suggestion seriously and come out with more practical and feasible designs. All the commode manufacturers should readily come forward with their own implementation of the design to suit various customer segments. The manufacturers, suppliers and distributors should give hefty volume discounts for mass implementation of this design in all apartment blocks in urban areas. The government may also subsidize the cost for poorer sections of the society, who have common toilets. The removed older commodes may be re-used in public toilets where separate urinals are provided in addition to commodes.

Partners: The commode manufacturers, builders, civil contactors, housing societies, municipality health inspectors, plumbers and masons should all be involved in evolving a suitable design, implementation and practice.

All the new apartment blocks, including those under construction can be asked to implement this design as a precondition to issue of occupation clearance certificate. Apartment blocks in the urban area may be asked to implement the design in a phased manner, say in about three years.

The cost of manufacture of this kind of toilets will only be marginally higher than the normal ones. Depending on the aesthetics of the design to suit different markets, the actual costs may vary. Because of a very large initial demand the cost per unit will come down drastically. With exchange offers and deserving subsidies, the cost may not pose a big problem.

Outcome of this modification: This design of toilet will save a lot of water for our future generations at the same time keeping our toilets adequately clean and hygienic.

Name: L V Nagarajan

City:    Mumbai


Phone: 022- 25259073

LVN/5th May 2010