Sanskrit is not only the language of Vedas, Itihāsas , Purānas Āyurveda ,ślokas and so on but also the language of learning Sciences including mathematics, astronomy, chemistry, biology etc.. Most of the adults and children of our nation do not even know that a vast ancient scientific literature including books on mathematics, science etc. exist in India. Some of the Sanskrit printed texts in Ancient Indian Mathematics (AIM) and astronomy are:

*Ā**ryabha**ṭī**ya* (499 AD) of Āryabha*ṭ*a, *Brāhmasphu**ṭ**asiddhānta (*628 AD) of Brahmagupta, *Ā**ryabha**ṭī**ya* *bhāṣya, Mahābhāskarīya and Laghubhāskarīya* (629 AD) of Bhāskara I, *Pāṭīganitam *of Śrīdhara (750 AD), *The Ga**ë**ita sāra saṅgraha *of Mahāvīrācārya, Siddhāntaśiromaṇi, *L**ī**lāvat**ī* and *B**ī**ja- Ganitam* (1150 AD) of Bhāskara II, *Ga**ë**ita Kaumud**ī* (1346 AD) of Nārāyana etc. Apart from these there are quite a few works of Jain mathematicians and Kerala astronomers and many more commentaries in Sanskrit. Some of the valuable contributions of Sanskrit texts to mathematics are discussed here.

PLACE VALUE SYSTEM AND ZERO

Place Value System and Zero are great contributions of India , to the world of mathematics and hence the whole world must be grateful to India and Sanskrit.

P.S.Laplace (1749-1827 AD), one of the greatest French mathematicians, says,” The idea of expressing all quantities by nine figures (or digits) whereby is imparted to them both an absolute value and one by position is so simple that this very simplicity is the reason for our not being sufficiently aware how much admiration it deserves.”

Prof. G.B.Halstead ( 1853-1922 AD), an American mathematician, proves that zero existed in India even before 200 BC*. i.e*. around 2200 years ago.

The use of Place Value System in India is found in the writings of Vasumitra ( 1^{st} century AD) i.e. around 2000 years ago.

In Indian Place Value System, numbers are written in a simpler way compared to other systems. For example, 3867 in Place Value System is same as MMMDCCCLXVII in Roman System. It is still more difficult to express large numbers in this system and perform basic operations like multiplication and division with these numbers. Due to its simplicity Indian Place Value System is accepted throughout the world.

In the texts of Vedas, terms denoting powers of 10 upto 10^{12}, were given by Vedic seer Medhatithi. They are *Eka* =1, *Daśa* =10, *Śata* = 100, *Sahasra* = 1000, *Ayuta* = 10^{4}, *Niyuta* = 10^{5}, *Prayuta *= 10^{6}, *Arbuda* = 10^{7}, *Nyarbuda* = 10^{8}, *Samudra* = 10^{9} , *Madhya* = 10^{10}, *Anta* = 10^{11}, *Parārdha *= 10^{12} and so on.

The great Indian astronomer and mathematician named Āryabhaṭa, who was born in 476 AD, wrote a commendable work called *Āryabhaṭīya *at the age of 23 years. He gives 10 places (number names) of the Place Value System. ( Ā.II.2 )

@k< dz c zt< c shö< Tvyutinyute twa àyutm!,

kaeq(buRd< c v&Nd< Swanat! Swan< dzgu[< Syat!.

*Eka* (unit),*daśa* ( ten), *śata* ( hundred), *sahasra* ( thousand), *ayuta* ( ten thousand), *niyuta* (lakh), *prayuta* (ten lakh), *koṭi* (crore ), *arbuda* ( ten crore ), *vṛnda* (hundred crore ) are, respectively, from place to place, each ten times the preceding.

Dr.Reno of math-forum of Drexel University writes, ‘ It is from the Indians that we get our present-day symbol for zero (0)…and by 500 AD, Indians have invented as base 10 system that had unique symbols for the numbers 1 through 9, employed a place value notation and used a zero….If we do not have this notion of place value , addition would be a big pain”

George Ifrah in his *Universal History Of Numbers* (published by Harvill) says “… .the discovery of zero and the Place Value System were inventions unique to Indian civilization”.

The above statements reveal the major important contribution of Indian mathematics to the world.

FRACTIONS AND RATIONAL NUMBERS

Liz Rumfrey in his article on *History of Fractions *says, “ Fractions as we use them today didn’t exist in Europe until the 17^{th} century… The format we know today comes directly from the work of the Indian civilization. It was only through the trading of the Arabs that these Indian numerals were spread to Arabia.”

While up to 17^{th} century fractions were not known in the west, there are evidences right from the Vedic age (earlier to 2000 BC) that they were used in India. Simple fractions ½ ( *ardha *), 1/3 (*tṛtīya*), ¼ ( *pāda*), 1/8 (*śapha*), 1/12( *kuṣṭha*), 1/16 ( *kalā*) are found in various *saṁhitas * and ¾ ( tripāda) occurs in the Ṛgveda.

Śrīdhara (750 AD) in his *Pāṭīga**n**itam* (*P* ), gives the following advanced expressions in fraction in the following example. ( *P*.ex. 32)

सार्द्धरूपत्रितयं स्वपादसहितं स्वषष्ठसंयुक्तम् ।

अर्द्धस्वत्र्यंशयुतं स्वपादसहितं च किं योगे ॥

The sum is “What is the sum of

and

The sum is “What is the sum of

What is the sum of

(3+1/2)+1/4 of (3+1/2)+ 1/6 {(3+1/2)+1/4 of (3+1/2)}

and

1/2+1/3 of 1/2+ 1/4 of (1/2+1/3 of 1/2)

?”

It is said that Carl Friedrich Gauss (1777-1855 AD) has discovered Fractions and Rational Numbers. In India *Pāṭīganitam *(*P*) (750AD) of Śrīdhara describes six classes of operations on rational numbers (fractions) namely *bhāga*, *prabhāga*, *bhāga* *bhāga*, *bhāgānubandha* , *bhāgāpavāha *and* bhāgamātā* ( a blending of two or more fractions of previous forms.)

Bhāga: a/b + (c )/d = ( ad +bc)/bd prabhāga: a/b of c/d of e/f = (ace )/bdf bhāga bhāga: a ÷ b/c = ac/b

bhāgānubandha: i) a + (b )/c = (ac +b)/c ii) a/b + c/d of a/b = a(d+c)/bd

bhāgāpavāha: i) a-(b )/c = (ac-b)/c ii) a/b – c/d of a/b = a(d-c)/bd

*Ganita sāra saṅgraha* (*GSS*) (850 AD ) of Mahāvīrācārya gives variety of examples on operations on Rational numbers. An example with fractional terms in a series (II.ex.23)

द्वित्र्यंशष्षड्भागस्त्रिचरणभागो मुखं चयो गच्छः।

द्वौ पञ्चमौ त्रिपादो द्वित्र्यंशोऽन्यस्य कथय किं वित्तम्॥

Tell me what the sum is (in relation to a series) of which and are the first term, the common difference and the number of terms in order; as also in relation to another of which and constitute these elements.

**IRRATIONAL NUMBERS.**

If a square of side 1 unit is constructed then its diagonal represents √2, which can be measured. To get the value by calculation, the following rule is prescribed in *Baudāyana Śulba Sūtra* (*BSS .*1.61) which is dated earlier than 800 BC.

smSy iÖkr[I, àma[< t&tIyen vxRyet! t½ ctuweRn AaTmctuiô<zaenen,

This rule gives the approximation:

which is nearly = 1.4142157

Nārāyaëa (1346 AD) in his *Ga**ë**ita Kaumudī*, gives rules for finding the factors of square numbers and non-square numbers. These are rediscovered by French mathematician Fermat only in 1643 AD, as written by L.E.Dickson in *History Of The Theory Of Numbers*.

INTEGERS

Uncyclopedia . wikia. Com. gives a story about the invention of the integers by a person named Arbermouth Holst when he was doing his famous experiments on fluffy bunnies in 1563. He discovered that the bunnies had multiplied and the number of bunnies obeyed the new laws of integer numbers. But in India in the 7^{th} century itself, there was a well-established integer system. Brahmagupta in his *Brāhma Sphuṭa Siddhānta (Br.Sp.Si ) *(628 AD) has given clear rule ( XIII.33) about the operations on integers.

\[m&[xnyae”aRtae xnm&[yaexRnvxae xn< Évit,

zUNy[Ryae> oxnyae> ozUNyyaevaRvx> zUNym!.

The following rules are contained in this one verse:

(+a) x(-b) = (-a) x (b) = -ab

(+a) x(+b) = (-a) x (-b) = +ab

-a x 0 = 0 x (+b) = 0 x 0 = 0

Like this he gives rules for all operations on integers. In 9^{th} century in Bhagdad Al-Khwarizimi (780-850 AD) acknowledged that he derived the ideas from the work of Brahmagupta.

Dr. Peterson of Mathforum of Drexel University writes, “A good case could be made that positive and negative numbers did not actually become part of a single ‘number line’ until the 1700’s or 1800’s.”

It is considered that Max Beberman seems to have originated the use of the number line in 1950’s. Leo Rogers says that the English mathematician John Wallis (1616-1703AD) is credited with giving some meaning to the negative numbers by inventing the number line.

But in India, we get the written evidence ( the use of number line must have existed before this) from the *Bīja pallava* of Kṛṣëa Daivajïa (16^{th} century AD), which clearly describes a number line for operations on Integers. He explains the operation of integers with a number line as follows:

tÇEkreoa iSwta iÖtIya idkœ ivprIta idigTyuCyte, ywa pUvRivprIta piíma idkœ, ywa %ÄridiGvprIta di][a idigTyaid, twa c pUvaRprdezyaemRXye @ktrSy xnTve kiLpt< t< àit tidtrSy \[Tvm!,

There, a line is placed with second direction opposite to the first direction. West is in the contrary direction to the east and south to north. Thus of two places situated in the east and west, if one is taken to be positive, the other is relatively negative.

Some students try to avoid mathematics as they feel it is dry, abstract and nothing ornamental in it. Our ancient mathematicians are poets as well and they give lively and lovely illustrations from life situations, which will create interest and enthusiasm among students, as they can enjoy mathematics and poetic beauty simultaneously. An example from *GSS* (IV.6) is given here.

Out of a collection of excellent bees, (1/6) took delight in *pāṭali* trees, (1/3) in *kadamba* tree, (1/4) in mango tree, (1/5) in a *campaka* tree with blossoms fully opened; (1/36) in a collection of full-blown lotuses, opened by the rays of the sun; and a single intoxicated bee has been circling in the sky. What is the number of bees in the collection?

We have very interesting problems in *L**ī**lāvat**ī*.

Direct variation, Inverse variation and Chain rule which are dealt with in school are termed as *Trairāśika* (Rule of Three), *Vyasta Trairāśika* (Inverse Rule of Three) and *Pa**ï**carāśika *etc. respectively.

ALGEBRA

The origin of Indian Algebra can be definitely traced back to the period of *Śulba *(800BC) and the *Brāhma**ë**a* (2000BC). The geometrical method of transformation of a square in to a rectangle having a given side, which is described in the *Śulba *is obviously equivalent to the solution of a linear equation in one unknown *viz*., *ax= c ^{2}* . Each geometrical transformation corresponds to an algebraic equation. The equation is called by Brahmagupta (628 AD) as

*sama*–

*kara*

*ë*

*a*or

*samīkara*

*ë*

*a.*Addition is indicated by symbol ‘

*yu*’(

*yuta)*, Subtraction by ‘

*ṛ*’ (

*ṛ*

*ë*

*a*), multiplication by ‘

*gu*’(

*gu*

*ë*

*a*) and division by ‘

*bhā*’(

*bhāga,bhājita*). Ancient mathematicians used the names of colours such as

*kālaka(ka), nīlaka(nī) ,pīta(pī)*etc and

*yāvat tāvat(ya)*for denoting the unknowns. Brahmagupta and other mathematicians have given rules and examples of all operations including squaring, cubing, finding square roots

*etc*. with the unknown terms.

One example for multiplication from *Bījaga**ë**ita* of Bhāskara II(1150AD) is given here:

Tell at once, o learned, the result of multiplying five *yāvat tāvat *minus one known quantity by three *yāvat tāvat *plus two known.

Here multiplicand is *y*ā 5 *rū *-1; multiplier *yā* 3 *rū* 2 and the product is *yā va* 15 *yā* 7 rū -2 This means (5*x* -1)(3*x* + 2) = 15*x*^{2} + 7*x* -2.

EQUATIONS.

In AIM, the solving of equations is referred to as *samaśodhana* .The earliest Indian classification of equations is given in a canonical work of *Circa *(300 BC) as *yāvat tāvat *(simple), *varga* (quadratic), *Ghana *(cubic) and *varga varga* (biquadratic). Brahmagupta (628AD) has classified as i) *Eka var**ë**a samīkara**ë**a* ( equations in one unknown)- simple and quadratic ii) *Aneka var**ë**a samīkara**ë**a* (equations in several unknowns) iii) *Bhāvita* (equations involving product of unknowns). One more *madhyamāhara**ë**a* ( elimination of the middle term) is included by others.

SIMULTANEOUS EQUATIONS.

In 499 AD itself Āryabhaṭa in his *Āryabhaṭīya,* gives a rule(Ā.II.24) for finding quantities from their difference and product.

iÖk«itgu[at! s<vgaRdœ ÖyNtrvgeR[ s<yutaNmUlm!,

ANtryu´< hIn< tdœ gu[karÖy< diltm!.

This means that if, *x- y = a* and *xy = b*; then

;

Mahāvīra in *Ga**ë**ita Sāra Saṅgraha* (*GSS*) ( 850 AD) gives a rule for the solution of simultaneous linear equations. ( GSS.VI.139½).

Jyeó¹mharazejR”Ny)ltaiftaenmpnIy,

)lvgRze;Éagae Jyeóa”aeR=Nyae gu[Sy ivprItm!.

The rule is represented algebraically as follows: If *ax + by = m* and *bx + ay = n*, are the given equations, then and

The method used today also gives the same result. Without knowing the method or proof Mahāvīra could not have given this result.

QUADRATIC EQUATIONS.

The altar-construction of Vedic Hindus (800 BC) involved the solution of the complete quadratic equation *ax ^{2}+ bx + c = 0,* as well as the pure quadratic

*ax*. The general solution of the simple quadratic equation

^{2}= c*4b*, is found in the early canonical works of the jains ( 500-300 BC) and also in the

^{2}– 4db = – c^{2}*Tattvārthādhigama-sūtra*of Umāsvatī (150 BC) as

.

*Bakṣāli Manuscript *(*BM*) (7^{th} Century AD) gives problems and solutions involving quadratic equations. Āryabhaṭa I (499 AD), Brahmagupta (628 AD), Mahāvīrācārya (850 AD), Āryabhaṭa II (950 AD), Bhāskara II (1150 AD) and many other Indian mathematicians have given innovative rules to solve quadratic equations. The methods include completing the squares, factorizing, formulae method etc., which are followed by us even today. But in Europe, it is said that only in 12^{th}century the complete solution of quadratic equation was given. Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book *Liberembadorum * published in 1145, which is the first book published in Europe to give the complete solution of the quadratic equation.

The solution of indeterminate equation of the form N*x*^{2} + 1 = y^{2} is given by Brahmagupta and others. This is solved completely by the Indian Cyclic method (*Cakravāla*) in his *Bījaga**ë**itam *by Bhāskara II. He has solved the example 61x^{2} + 1 = y^{2}, which was proposed by Fermat in 1657 AD, as a challenge problem to Frenicle. Euler solved it in 1732 AD , but this was solved by great Indian mathematicians at least 600 years earlier.

GEOMETRY

Every text in AIM deals with a topic, *Kṣetra vyavahāra*, which includes area, volume and geometrical results. Areas of triangle, trapezium and circle are given by Āryabhaṭa(499 AD). The rule for circumference- diameter ratio is given (Ā.II.10)

cturixk< ztmògu[< Öa;iòStwa shöa[am!,

AyutÖyiv:kMÉSyasÚae v&Äpir[ah>.

100 plus 4, multiplied by 8 and added to 60000; this is the approximate measure of the circumference of a circle whose diameter is 20000.

This gives,

This value first occurs in *Āryabhaṭīya*. It is noteworthy that Āryabhaṭa has specified the above value as approximate (āsanna).

GEOMETRICAL CONSTRUCTIONS.

*Śulba sūtras* (earlier to 800BC) are the oldest geometrical treatises available which represent the traditional Indian mathematics.This has been developed for construction and transformation of vedic altars of various shapes. Baudhāyana, Āpastamba, Kātyāyana, Mānava and many others are the authors of the *Śulba sūtras.*

Most of the constructions that we do today are given in the *Śulba sūtras.*Some of them are i) dividing a line segment, circle and triangle in to equal parts; ii) constructing a line at right angles to a given line at a point on it and from a point outside; iii) constructing a square of given side, a square of area equal to the sum of two different squares, a square equal to a given rectangle or a triangle or two pentagons or a rhombus; iv) constructing a parallelogram, a trapezium, a rectangle a rhombus v) constructing a circle equal in area to a square and *vice versa* and so on.

THEOREMS.

Most of the theorems that the school students study about the quadrilaterals, namely the parallelogram, rectangle, rhombus and the triangles are expressly stated or implied in the methods of construction of the altars in the *Śulba sūtras.*(earlier than 800BC). At least 500 years later than this Euclid, a Greek mathematician in his book called *Elements* ( 300 BC) has written these results.

THEOREM OF THE SQUARE OF THE DIAGONAL.

BSS (I.48)(800 BC) gives the result, which is known now as Pythagoras Theorem of the Greek philisopher Pythagoras (6^{th} century BC), as

dI”RcturïSyaú[yar¾u> pañRmanI ityR’œmanI c yTp&wGÉUte k…étStÊÉy< kraeit,

‘ The diagonal of a rectangle produces both the areas which are produced separately by its length and breadth.’This theorem has to be called as *bhuja-koṭi kar**ë**a nyāya* as our ancient mathematicians called it or as *Śulba* theorem instead of Pythagoras theorem, because the first enunciation of the theorem in its general form is found in the *Śulba*.

The short proof of this theorem is given by Bhāskara II (1150 AD)in his Bījagaëitam ( verse 129) This is geometrico- algebrical proof. Such a proof was rediscovered in Europe by Wallis, five centuries later, in 1693 AD.

There are interesting rules and recreational examples given in *Līlāvatī* from verses 133 to 165, which will catch the attention of the students.

TRIANGLES AND QUADRILATERALS.

Brahmagupta in his *Brāhma Sphuṭa Siddhānta*(628 AD) ,was the first to give the exact formula for the area of a( cyclic) quadrilateral.

Éujyaegax¡ ctuòy Éujaen”ataÝd< sUúmm!,

‘The square root of the product of four factors formed by the semi- perimeter which is diminished by each side is the exact area (of a cyclic quadrilateral)’ This is same as,

Area of (cyclic) quadrilateral , where

a,b,c,d are the sides of the quadrilateral and s = ½ ( a+b+c+d)

C.B.Boyer considers this formula to be ‘perhaps most beautiful result.’

Area of triangle , where ‘d’ is absent. This is called as ‘Heron’s formula’ now.

Brahmagupta was the first to give the expression for the diagonals of a quadrilateral. (Br.Sp.Si.XII.28)

M.Eves in ‘*An Introduction To History Of Mathematics*’(New York,1969) says about the above result as ‘most remarkable in Hindu geometry and solitary in its excellence.’

‘The above formula was rediscovered in Europe only a thousand years later by W.Snell about 1619 AD.’ ( by D.E.Smith in *History Of Mathematics*, New York, 1958)

ARITHMETIC PROGRESSIONS.

In the vedic texts (2000-800BC) several sequences occur such as 1,3,5,…19; 19,29,…99; 2,4,6…18; 20,30,… 100 and so on.

Sum of a series in A.P. is given by Āryabhaṭa (499 AD)(Ā.II.19)

#ò< Vyek< dilt< spUvRmuÄrgu[< smuomXym!,

#ògui[timòxn< Tvwava*Nt< pdaxRhtm!.

Commentator Bhāskara I (629 AD) says that five formulae are severally set out here. They are obtained by suitable combination of the text.

Let an arithmetic series be a, a+d, a+2d,…

- i) The arithmetic mean of ‘n’ terms a, a+d, a+2d,… is = a + ½ (n – 1)d
- ii) The sum of the series with ‘n’ terms a, a+d, a+2d,… is = n{ a + ½ (n – 1)d}

iii) The formula for the ‘n’ th term = a + { (1 -1) + (n-1)}d

- iv) The arithmetic mean of the ‘n’ terms beginning with (p+1)th term

(a+pd), a+ (p+1)d,…a+(p+n-1)d is = a + { ½(n – 1 ) +p}d

- v) The sum of the ‘n’ terms beginning with (p+1)th term

Thus Āryabhaṭa gives the above five formulae in a single verse. This is an example to show how Sanskrit can convey the rules in short and crisp form. The same formulae are used even today.

TRIGONOMETRY.

A great contribution of India to the world of Mathematics is the invention of the basic trigonometric function called Sine (*jyā*). This came from discussions on astronomy regarding the position of planets on the celestial circle. The superior Indian Sine excelled the Greek chord and gave rise to a fully developed science of Trigonometry. The Indian sine of an arc in a circle was defined as the length of half the chord of twice the arc.

*Jīvā* = *jyā* (a) = (chord of 2a)½

= PN= R sin ө

*koṭi jyā* = PM = R cos ө

* *

JOURNEY OF INDIAN SINE(*JῙVĀ*).

When the Indian astronomical works were translated into Arabic at Baghdad in the eighth century A.D., the Sanskrit word *Jīvā* itself was adopted as a technical term for the Indian sine function which was not known to the Arabs earlier. They wrote it as *Jib* or *Jyab* and read them as *Jaib*, which means bosom or pocket in Arabic. When the Arabic works were further translated in to Latin, in Spain, *Jaib* was substituted by *Sinus* (literally, fold or bosom etc.), whence came the anglicized form ‘sine’. Thus an original Indian mathematical term *jyā* or *jīvā*, after its long journey, to west Asia, Spain and England came back to India, after more than a thousand years, during the British Rule as ‘sine.’

CONCLUSION

AIM which is known only to a closed circle must be made available to all. Ancient Indian Mathematics, this name may make the people think that there is nothing modern in it. But all modern mathematics is contained in our ancient Sanskrit texts. Introducing AIM in the mathematics text books will help the children

- i) to know about the greatness of ancient Indian mathematicians and advanced mathematics that existed in India centuries ago, while many other country men did not know the basics of these.
- ii) to feel proud of their country which leads to love towards their country and patriotism and work for their country,

iii) to boost up their self-confidence when they think that if their ancestors could excel the rest of the world, they also can and

- iv) to initiate them into new ideas and research.

By including AIM in school curricula, it will be read by the students, their teachers and parents. Thus many people will come to know about the glory of our Indian mathematicians. Hence including AIM in school curricula is necessary to propagate this treasure of knowledge to the world. There are a lot of works exist in manuscript form which have to be critically edited. There are a lot of edited books which are to be studied so that the existing wealth of knowledge is to be dug out and presented to the people of the world.

**REFERENCES**

- Āpte, V.G.
*; Līl**ä**vatī*with commentaries*Buddhivilāsinī*and*Līlāvativivaraṇam*, Ānandāśramasamskṛtagranthāvali, 107, - Bag, A.K.,
*Mathematics in Ancient and Medieval India,*Chaukhambha Orientalia, Varanasi, Delhi, 1979. *Brāhmasphuṭasiddhānta*of Brahmagupta, Vol. IV, ed. Acharyavara Ram Swarup Sharma, Institute of Astronomical and Sanskrit Research, New Delhi, 1966.*Colebrooke’s Translation of The Līlāvatī,*Asian Educational Services, New Delhi, Madras, 1993.- Datta Bibhuti bhushan and Avadesh Narayan Singh,
*History of Hindu Mathematics,*I and II, Bharatiya Kala Prakashan, Delhi, 2001. - Heroor, D.Venugopal,
*The History Of Mathematics And Mathematicians Of India,*Bengaluru, 2006. - Mangal Deva Shastri (ed.),
*The GaṇitaKaumudī*, The Princess of Wales Sarasvati Bhavana Texts, No,57, Part ii) - Panicker,V.B.
*Bhāskarācārya’s Bīja Ga**ë**itam,*Bharatiya Vidya Bhavan, Mumbai, 2006. *Pāṭīga**ë**itam*of Śrīdhara, The Fine Press, Lucknow, 1959.- Patwardhan, Krishnaji Shankara, Somashekhara Amrita Naimpally and Shyam Lal Singh,
*Līl**ä**vatī of Bhāskarācārya*, , Motilal Banarsidass, Delhi, 2001. - V.,
*A Critical Study of the Commentary Buddhivilāsinī of Gaṇeśa Daivajña on Līlāvatī of Bhāskara II*, Unpublished Ph.D. thesis, University of Madras, 2016. - K.V.,
*Līl**ä**vatī*of Bhāskarācārya with*Kriyākramakarī*of Śaṅkara and Nārāyaṇa, ed. Vishveshvaranand and Vedic Research Institute, Hoshiarpur, 1975. - S.N and A.K.Bag,
*The Śulbasūtra of Baudhāyana, Āpastamba, Kātyāyana and Mānava*with English translation and commentary, Indian National Science Academy, New Delhi, 1983. - Shukla, K.S. ,
*Āryabhaṭīya*of Āryabhaṭa, ed. By Indian National Science Academy, New Delhi, 1976. *The Ga**ë**ita sāra saṅgraha*of Mahāvīrācārya, ed.by M.Rangācācārya, Cosmo Publication, New Delhi,2011.

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