Mathematics in Sanskrit

Mathematics in Sanskrit 4 Minutes

A civilization is known by contributions of learned persons, who contributed to its sustenance, development and progress.

Learnedness has been spoken of as alacrity at 3 R’s – Reading wRiting aRithmetic.

The third R really encompasses a vast field of mathematical knowledge. Indian civilization is very rich in respect of all 3 R’s, certainly the third R – mathematics.

In this category I would like to compile thoughts on mathematics, especially from Sanskrit texts

Mathematics in Sanskrit has millenia-old tradition. 

ज्योतिष् is one of six वेदाङ्गानि specified for comprehensive study of वेदाः. 

When I was a boy, the postman would deliver mail to our house. Some mails addressed to my grandfather would have for him a title वे.शा.सं (वेदशास्त्रसंपन्न). I do recall now that people did consult him for astrological queries. Though more often than not, his responses were extempore or intuitive, I also recall seeing him read the informations and tables in annual edition of पञ्चाङ्गम् the Hindu almanac, Off and on I too would glance through that, without being able to make any sense. But I do recall that there used to be information about the eclipses. How could the almanac give such accurate information of all eclipses due to occur every year ? That is simply by mathematics of astronomy. 

I think that this phrase “Mathematics of astronomy” can be treated as giving an amazing summary view of Mathematics in Sanskrit. 

It is a challenging task to list all Sanskrit texts, which deal with mathematics or have a mathematical reference. I would venture to make a list as follows 

So, to study Mathematics in Sanskrit, there is so much to study ! 

Basically the word गणितम् is from धातुः गण् See in शब्दकल्पद्रुमः – गणितं, त्रि, (गण + कर्म्मणि क्तः ।) कृतगणनम् । तत्पर्य्यायः । संख्यातम् २ । इत्यमरः ।३ । १ । ६४ ॥ (चिन्तितम् । यथा, शिवपुराणे ज्ञानसंहितायाम् । १२ । २७ । “दुःखं च विविधं तत्र गणितं न तया तथा ॥”)

गणितं, क्ली, (गण्यते इति । गण + क्तः ।) अङ्कशास्त्रम् । तत् द्बिविधं व्यक्तं पाटीगणितम् । अव्यक्तं बीजगणितम् । इति लीलावती ॥ गणनम् । यथा, नैषधे । ३ । ४० । “पारे परार्द्धं गणितं यदि स्यात् ॥”

In Apte’s dictionary गणित gaṇita p. p. [गण्-क्त] 1 Counted, numbered, calculated. -2 Regarded, cared for &c; see गण्. -Comp. -दिवसक a. Consisting of a definite number of days; विचाली हि संवत्सरशब्दः सावनोऽपि गणितदिवसकः ŚB. on MS.6.7.39. -तम् 1 Reckoning, calculating. -2 The science of computation, mathematics; (it comprises पाटीगणित or व्यक्तगणित ‘arithmetic’, बीजगणित, ‘algebra’, and रेखागणित geometry’); गणितमथ कलां वैशिकीं हस्तिशिक्षां ज्ञात्वा Mk.1.4. -3 The sum of a progression. -4 A sum (in general). -5 Study, practice; वेत्तुमर्हसि राजेन्द्र स्वाध्यायगणितं महत् Mb.12.62.9.

For counting and calculating, there has to be a number system संख्याशास्त्रम् and there have to be परिमाणानि well-defined units of measurement. 

In the Sanskrit number system, numbers are written in the decimal system, such that when we proceed leftwards from the digit at unit place, the unit value of the place is ten times of the unit value of the place on the right अङ्कानां वामतो गतिः. And unit value of every place has a name such as एक, दश, शत, सहस्र, … right up to 17th place as stated in the two verses (in वाचस्पत्यम् these verses are quoted when detailing the word सङ्ख्या and are said to be taken from  ब्रह्माण्डपुराणम्)

एकं दश शतञ्चैव सहस्रमयुतं तथा । 

लक्षञ्च नियुतञ्चैव कोटिरर्वुदमेव च ॥

वृन्दः खर्व्वो निखर्व्वश्च शङ्खपद्मौ च सागरः ।

अन्त्यं मध्यं परार्द्धञ्च दशवृद्ध्या यथाक्रमम् ॥

Regarding परिमाणानि it is interesting to note for time as a quantity to be measured, the smallest unit is possibly निमिष, which is defined as the time taken by wink of the eye. But since a wink includes both closing and opening of the eyelid, there are also mentions of निमिषार्ध i.e. half of निमिष.  Units larger than निमिष are as specified in the श्लोकः below  

   निमेषा दश चाष्टौ च काष्ठा त्रिंशत्तु ता: कला: । त्रिंशत्कला मुहूर्त:स्यादहोरात्रंस्तु तावत: ॥मनुस्मृतिः १-६४॥ 

18 निमेषा = 1 काष्ठा; 30 काष्ठा = 1 कला; 30 कला = 1 मुहूर्त; 30 मुहूर्त = 1 अहोरात्र i.e. 1 day of 24 hours; has 1440 minutes i.e. 86400 seconds has 30 मुहूर्त-s i.e. 900 कला: i.e. 27000 काष्ठा i.e. 486000 निमेषा, which are 86400 seconds. So 1 second = 5.625 निमेषा or 1 निमेष = 0.177777.. Second. 

Units smaller than निमेष are also mentioned in लीलावती as तत्परा = 1/30 of निमिष and त्रुटि = 1/100 of तत्परा. So 1 त्रुटि = 0.0000592592 of a second. One wonders what made such a small micro-second unit of time to be needed to be defined !  

  • There is also a definition of त्रुटि भागवते । ३ । ११ । ५ । “अणुर्द्वौ परमाणू स्यात् त्रसरेणुस्त्रयः स्मृतः । जालार्करश्म्यवगतः खमेवानुपतन्नगात् । त्रसरेणुस्त्रिकं भुङ्क्ते यः कालः सा त्रुटिःस्मृता ॥” 
  • Two परमाणू-s make one अणु, three अणु-s make one त्रसरेणु, which is what particle one sees floating in a beam of light. Three त्रसरेणु-s make one त्रुटि.

This definition of त्रुटि (18 परमाणू-s) seems to be more related to the size Quantity than Time-quantity. Or maybe, at that micro level the concept of quantities becomes unified.  

Readers must have noted that the above details from मनुस्मृतिः and भागवतम् are outside of the गणितग्रन्थाः listed earlier. 

When studying Sanskrit grammar, I came across a कृत्-प्रत्ययः क्विन् (क्+व्+इं+न्). Interestingly, क्विन् is the grammatical name of a suffix कृत्-प्रत्ययः, which affixes to a धातुः a verbal root. Practically however the process of affixing of क्विन् happens with nothing affixing. So क्विन् is the name of a ‘null set’ a mathematical concept in Set Theory. By Google search on ‘History of Set Theory’, I get “Set theory, however, was founded by a single paper in 1874 by Georg Cantor:” But we have evidence of a null set in 2500+years old Sanskrit grammar !

Looks like mathematics in Sanskrit is in Sanskrit all over. 

It is well-acknowledged that it is mathematics in Sanskrit which gave to the world the concept of शून्यम् zero. Let us proceed much further. 

How I wish that people will pursue study of Mathematics in Sanskrit, such that students of future generations, that too students all over the world will be greatly benefitted. Also researchers will be saved of ‘reinventing the wheel’, not to spend their time and energy in reinventing, what is already there in Mathematics in Sanskrit. 

You may also view and listen to the YouTube video https://youtu.be/zJ7p6pvL_K0

शुभमस्तु !

-o-O-o-

Published by slabhyankar अभ्यंकरकुलोत्पन्नः श्रीपादः

Published

Leave a comment