ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

FRACTIONS 173
Reduction to lowest terms.—A non-mathematical work,
Tattvārthādhigama-Sūtra-Bhāṣya by Umāsvāti (c.150A.D.) casua-
lly mentions as follows in the context of a philosophic discourse:
Or, as when the expert mathematician, for the purpose
of simplifying operations, removes common factors
from the numerator and denominator of a fraction,
there is no change in the value of the fraction, so....¹
Reduction to common denominator. Whenever we have to
add or subtract fractions, we follow this reduction operation
to a common denominator. Brahmagupta gives the reduction
along with the similar processes :
By the multiplication of the numerator and denomi-
nator of each of the (fractional) quantities by other
denominators, the quantities are reduced to a common
denominator. In addition, the numerators are united,
In subtraction their difference is taken.²
Fractions in combination :—Since there was no proper
symbolism available to these early Indian mathematicians, they
divided combination of fractions into four classes :
Bhāga, prabhāga, bhāgapavāha and bhāga-bhāga.
(i) Bhāga has been mentioned by Brahmagupta (BrSpSi.
XII, 8) thus : ( a/b ± c/d ± e/f ± ........ ) usually written as
┌───┬───┬───┐ ┌───┬────┬────┐
│ a │ c │ e │ or │ a │ .c │ .e │
├───┼───┼───┤ ├───┼────┼────┤
│ b │ d │ f │ │ b │ d │ f │
└───┴───┴───┘ └───┴────┴────┘
where the dots denote subtraction.
(ii) Prabhāga : The form ( a/b of c/d of e/f ......... )
This is written as
┌───┬───┬───┐
│ a │ c │ e │
├───┼───┼───┤
│ b │ d │ f │
└───┴───┴───┘
(iii) Bhāgānubandha : The form ( a + b/c ) is written as
┌───┐
│ a │
├───┤
│ b │
├───┤
│ c │
└───┘

1. II, 52.
2. विपरीतच्छेदगुणा : राश्योश्छेदांशाः समच्छेदाः ।
   संकलितेंऽशा योज्या व्यवकलितेंऽशान्तरं कार्यम् ॥ —BrSpSi. XII. 2.

174 BRAHMAGUPTA AND ARITHMETIC
and the form
p/q + r/s of p/q + t/u of (p/q + r/s of p/q) + .........
is written as
┌───┐
│ p │
│ q │
├───┤
│ r │
│ s │
├───┤
│ t │
│ u │
└───┘
(iv) Bhāgāpavāha, i.e., the form (a - b/c) is written as
┌───┐
│ a │
│ .b│
│ c │
└───┘
and the form p/q - r/s of p/q - t/u of (p/q - r/s of p/q) - ........
is written as
┌───┐
│ p │
│ q │
├───┤
│ .r│
│ s │
├───┤
│ .t│
│ u │
└───┘
(v) Bhāga-bhāga : The form
(a ÷ b/c) or (p/q ÷ r/s)
There does not appear to have been any notation for divi-
sion, such compounds being written as
┌───┐ ┌───┐
│ a │ │ p │
│ b │ or │ q │
│ c │ ├───┤
└───┘ │ r │
│ s │
└───┘
just as for bhāgānubandha. That division is to be performed was
known from the problem, e.g., 1 ÷ 1/6 was written as ṣaḍ-bhāga-
bhāga, i.e., "one-sixth bhāga-bhāga" or "one divided by one-
sixth". It is only in the Bakhshālī Manuscript that the term bha
is sometimes placed before or after the quantity affected.
(vi) Bhāga-mātṛ, i.e., combinations of forms enumerated
above. Mahāvīra, the author of the Gaṇitasārasaṃgraha (850

FRACTIONS 175
A.D.) gives twenty-six variations of this class. We shall illus-
trate it by the following example from Śrīdhara :
What is the result when half, one-fourth of one-fourth,
one divided by one-third, half-plus half of itself, and
one-third diminished by half of itself, are added
together ? (Triśatikā, p. 12).
A modern writer would have written it as :
½+(¼ of ¼)+(1÷⅓)+(½+½ of ½)(⅓—½ of ⅓)
In the old Indian notation, it is written as ·
| 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 4 | 1 | 2 | 3 |
| | | | 3 | 1 | .1 |
| | | | | 2 | 2 |
The defect of the notation is obvious: | 1 | 1 | can be read
| 4 | 4 |
also as ¼+¼ and | 1 | can also be read as 1 ⅓.
| 1 |
| 3 |
And therefore the original meaning is inferred from the
context or from the enunciation of the problem.
The rules for reduction of the first two classes (bhāga and
prabhāga) are those of addition or subtraction and multiplica-
tion. The rule for the reduction of the third (bhāgānubandha)
and fourth (bhāgāpavāha) classes are given by Brahmagupta in
the Brāhmasphuṭa-siddhānta thus :
The (upper) denominator is multiplied by the deno-
minator and the upper numerator by the same (deno-
minator) increased or diminished by its own numera-
tor.¹
“Numerator” is known as “aṁśa” and the “denominator”
as “cheda.”
We give here from Śrīdhara's Pāṭīgaṇita (about 900 A.D.,
according to K.S. Shukla, 750 A.D. according to Datta and
Singh) a rule for reducing a fraction of the bhāgānubandha class
(i.e., a whole number increased by a fraction or a fraction incre-
ased by a fraction itself) :

1. ऊर्ध्वां शारच्छेदगुखास्तृतीयाजातौ द्वयोः पृथक्परयोः ।
   छेदैश्छेदा गुणिताः स्वांशयुतोनैरुपरीमांशाः ॥ —BrSpSi. XII. 9.

176 BRAHMAGUPTA AND ARITHMETIC
In the bhāgānubandha class, the whole number (rūpa-
gaṇa) is multiplied by the denominator (of the frac-
tion) should be increased by the numerator (of the
fraction) or the upper denominator having been
multiplied by the lower denominator, the initial
numerator (i.e. the upper numerator) should be multi-
plied by the sum of the lower numerator and denomi-
nator.¹
(Pātīgaṇita, 39 cf. BrSpSi. XII. 9 (ii); GSS. (iii) 113
This means that
(i) a + b/c = (ac + b)/c
(ii) a/b + c/d of a/b (which was written by Indians in
the style
┌───┐
│ a │
│ b │
│ c │
│ d │
└───┘
is equal to a(d + c)/bd
Addition and Subtraction of Fractions
In the Brāhmasphuṭa-siddhānta, Brahmagupta gives the
rule for the addition and subtraction of fractions :
If the denominators (cheda) of fractions are different
then reduce these fractions to a common denomina-
tor. Now for the additions, unite the numerators and
take their difference in case of subtraction.²
Brahmagupta and Mahāvīra give the method under Bhāga-
jāti.
Multiplication
Brahmagupta says :
The product of the numerators divided by the pro-

1. भागानुबन्धजातौ रूपगणश्छेद सङ्गुणः सांशः ।
   अवरहरन्धोर्ध्वं हरेऽथोंशाद्युतहरघ्न आद्यंशः ॥ —Patīganita 39.
2. विपरीतच्छेदगुणाः राश्योश्छेदांशकाः समच्छेदाः ।
   संकलितेंऽशा योज्या व्यवकलितेऽशान्तरं कार्यम् ॥ —BrSpSi. XII. 2

DIVISION OF FRACTIONS 177
duct of the denominators is the (result of) multiplica-
tion of two or more fractions.¹
While all other writers give the rule in the same way as
Brahmagupta, Mahāvīra in the Gaṇitasārasaṃgraha refers to
cross reduction in order to shorten the work :
In the multiplication of fractions, the numerators are
to be multiplied by the numerators and the denomi-
nators by denominators, after carrying out the pro-
cess of cross reduction, if that be possible.²
Division of Fractions
The Āryabhaṭīya does not explicitly give the rule of divi-
sion, but under the Rule of Three, we have an indication of
this operation. The Rule of Three states the result as (f × i) / p,
where f stands for phala i.e. "fruit", i for icchā, i.e., demand or
requisition, and p for pramāṇa i.e. argument. When these
quantities are fractional, we get an expression of the form
(a/b × c/d) / (m/n)
for the evaluation of which Āryabhaṭa I states :
The multipliers and the divisor are multiplied by the
denominators of each other.
These quantities are written in the following way
┌───────┐
│ a m │
│ b n │
├───────┤
│ c │
│ d │
└───────┘
Transferring the denominators we have
┌───────┐
│ a m │
│ n b │
│ c d │
└───────┘
Performing multiplication, the result is anc / mbd. The above
interpretation of the obscure line in the Āryabhaṭīya is based

1. रूपाणिच्छेद गुण्यान्यंशयुतानि द्वयोर्बहूनां वा ।
   प्रत्युत्पन्नो भवति च्छेदवधेनोद्धृतोऽशवधः ॥ —BrSpSi. XII. 3
2. GSS. p. 25. (2)

178 BRAHMAGUPTA AND ARITHMETIC
on the commentaries of Sūryadēva and Bhāskara I (the commen-
tary of Parameśvara on this line is vague and misleading).
Sūryadeva in this connection says :
Here by the word guṇakāra is meant the multiplier
and multiplicand, i.e., the phala and icchā quantities
that are multiplied together. By Bhāgahāra is meant
the pramāṇa quantity. The denominators of the phala
and icchā are taken to the pramāṇa. The denominator
of the pramāṇa is taken with the phala and icchā. Then
multiplying these, i.e., (the numerators of) the phala
and icchā and this denominator, and dividing by (the
product of) the numbers standing with the pramāṇa
the result is the quotient of the fractions.
Brahmagupta gives the method of division as follows :
The denominator and numerator of the divisor having
been interchanged, the denominator of the dividend is
multiplied by the (new) numerator. Thus division of
*proper fractions is performed.*¹
Square and Square-Root of Fractions
Brahmagupta says as follows in this connection:—
The square of the numerator of a proper fraction divi-
ded by the square of the denominator gives the
square².
This rule of Brahmagupta has been followed by other
authors also. The rule regarding the square-root as given by
Brahmagupta is as follows :
The square-root of the numerator of a proper fraction divi-
ded by the square-root of the denominator gives the square-
root.³
The Rule of Three :
The Indian term in Sanskrit for the Rule of Three is Trai-
rāśika (literally, "three terms"). The term occurs in the Bakh-
śālī Manuscript also, and also in the Āryabhaṭīya, indicating the

1. परिवर्त्य भागहारच्छेदांशौ छेद संगुणच्छेदः ।
   अंशोंशगुणो भाज्यस्य भागहारः सवर्णितयोः ॥ —BrSpSi: XII. 4
2. संवर्गितांशवर्गश्छेदकृतिविभाजितो भवति वर्गः । —BrSpSi. XII. 5 (1)
3. संवर्गितांशमूलं छेदपदेनोद्धृतं मूलम् । —BrSpSi. XII. 5 (2)

THE RULE OF THREE 179
antiquity of the term. Bhāskara in his commentary of the
Āryabhaṭīya gives a justification of the use of this term for the
Rule of Three thus :
Here three quantities are needed (in the statement and
calculation) so the method is called trairāśika (meaning
thereby the "rule of three terms").
The problem of the Rule of Three has the form :
If p (pramāṇa) yields f (phala), what will i (icchā)
yield ?
Āryabhaṭa II (the author of the Mahāsiddhānta, 950 A.D.)
uses the terms māna, vinimaya and icchā, instead of pramāṇa,
phala and icchā respectively. It has also been pointed out by
several authors that the first and third terms are similar, i.e., of
the same denomination.
We shall give here the Rule of Three as given by Āryabhaṭa
I and Brahmagupta :
In the Rule of Three, the phala ("fruit"), being multi-
plied by the icchā ("requisition") is divided by the
pramāṇa ("argument"). The quotient is the fruit
corresponding to the icchā The denominators of one
being multiplied with the other give the multiplier
(i.e. numerator) and the divisor (i.e. denominator).¹
In the Rule of Three pramāṇa ("argument"), phala
("fruit") and icchā ("requisition") are the (given)
terms; the first and the last terms must be similar. The
icchā multiplied by the phala and divided by the
pramāṇa gives the fruit (of the demand).²
Śrīdhara also gives the Rule of Three almost in the same
words. Bhāskara II, Nārāyaṇa and others follow Brahmagupta
and Śrīdhara in the Trairāśika operation. Śrīdhara in his Pāṭīga-
nita says :

1. त्रैराशिकफलराशिं तमथेच्छाराशिनाहतं कृत्वा ।
   लब्धं प्रमाणभजितं तस्मादिच्छाफलमिदं स्यात् ।।
   छेदाः परस्परं हता भवन्ति गुणकार भागहाराणां ।
   छेदगुणं सच्छेदं परस्परं तत्सवर्णत्वम् ।। —Arya. II 26-27.
2. त्रैराशिके प्रमाणं फलमिच्छाद्यन्तयोः सदृशराशी ।
   इच्छाफलेन गुणिता प्रमाणभक्ता फलं भवति ।। —BrSpSi XII. 10

180 BRAHMAGUPTA AND ARITHMETIC
In (solving problems on) the Rule of Three, the argu-
ment (pramāṇa) and the requisition (icchā), which are
of the same denomination, should be set down in
the first and last places; the fruit (phala), which is of
a different denomination, should be set down in the
middle, (this having been done) that (middle quantity
multiplied by the last quantity should be divided by
the first quantity.¹
We shall illustrate the Rule of Three by an example from
the Pāṭīgaṇita (Example 25) :
Example, If 1 pala and 1 karṣa of sandalwood are obtai-
ned for ten and a half paṇas, then for how much will
nine palas and one karṣa (of sandalwood) be obtai-
ned ?²
Here in this Example.
argument = 1 pala and 1 karṣa = 1¼ or 5/4 palas; fruit =
10½ or 21/2 paṇas;
and requisition = 9 palas and 1 karṣa = 9¼ or 37/4 palas.
According to the Rule we shall write them as :

1. आद्यन्तयोस्त्रिराशावभिन्नजाती प्रमाणमिच्छा च ।
   फलमथ विजातीयं तदन्त्यगुणमादिना विभजेत् ॥ —Pāṭīgaṇita 43.
2. चन्दनपलं सकर्षं सार्धैर्यदि लभ्यते पणैर्दशभिः ।
   तत्किं नु लभ्यन्ते पलानि नव कर्षयुक्तानि ॥ —Pāṭīgaṇita Ex. 25.

RULE OF COMPOUND PROPORTION 181
=4 purāṇa, 13 paṇas, 2 kākiṇīs and 16 varāṭakas. (One purāṇa
is equivalent to 16 paṇas; one paṇa is equivalent to 4 kākiṇīs,
and one kākiṇī is equivalent to 20 varāṭakas or cowries.
Inverse Rule of Three
This is known as vyasta-trairāśika (literally meaning "inverse
rule of three terms)". After having described the rule of three,
Brahmagupta proceeds to give an account of this inverse rule
of three :
Divide the phala with icchā and multiply by pramāṇa;
this gives the vyasta-trairāśika inverse rule of three¹.
Here pramāṇa is the argument also known as the first term
and, and phala is the fruit also known as the middle term and
icchā is known as requisition or the last term. As Bhāskara II
clearly states, this rule is applied where with the increase of the
icchā, the phala decreases or with its decrease the phala increases
(Līlāvatī).
Rule of Compound Proportion
Brahmagupta and other writers call the rule of compound
proportions as pañca-rāśika, sapta-rāśika etc., meaning the
rule of five terms, rule of seven terms etc. depending on the
number of terms involved the problems. These are
sometimes grouped under the general application of the "Rule of
Odd Terms". Āryabhaṭa I (499 A.D.) though actually gives the
rule of three appears to have been quite familiar with the rule
of compound proportion also. In fact the difference between
the rule of three and compound proportion is more or less arti-
ficial. This view was expressed by Bhāskara I (525 A.D.) in his
commentary on the Āryabhaṭīya :
Here Ācārya Āryabhaṭa has described the Rule of
Three only. How the well-known Rules of Five etc. are
to be obtained ? I say thus : The Ācārya has described
only the fundamentals of anupāta (proportion). All
others such as the Rule of Five etc. follow from that
fundamental rule of proportion. How ? The Rule of
Five etc. consist of combinations of the Rule of Three.
......In the Rule of Five, there are two Rules of

1. व्यस्त त्रैराशिक फलमिच्छा भक्तः प्रमाण फलघातः ।
   त्रैराशिकादिषु फलं विषमेष्वेकादशान्तेषु ॥ —BrSpSi XII. 11

182 BRAHMAGUPTA AND ARITHMETIC
Three, in the Rule of Seven three Rules of Three, and
so on. This I shall point out in the examples.
Brahmagupta gives the following rule relating to the solu-
tion of problems in compound proportion :
In the case of odd terms beginning with three terms
up to eleven, the result is obtained by transposing
the fruits of both sides, from one side to the other,
and then dividing the product of the larger set of
terms by the product of the smaller set. In all the
fractions, the transposition of denominators, in like
manner, takes place on both sides.¹
This may be illustrated by taking an example from the
commentary of Pṛthūdaka Svāmī on the Brāhmasphuṭasid-
dhānta :
Example —If there is an increase of 10 in 3 months on
100 (niṣkas), what would be the increase on
60 (niṣkas) in 5 months.
Here the Pramāṇa pakṣa (the first set of terms) is
100 niṣkas, 3 months, 10 niṣkas (phala)
The second set or the icchā pakṣa is
60 niṣkas, 5 months, x niṣkas
The terms are written in compartments as below :
| 100 | 60 |
| 3 | 5 |
| 10 | 0 |
In the above 10 (written lowest) is the fruit of the first side
(pramāṇa pakṣa), and there is no fruit on the second side or the
iccā pakṣa. Interchanging the fruits we get
| 100 | 60 |
| 3 | 5 |
| 0 | 10 |
The larger set of terms is on the second side (icchā pakṣa).
The product of the numbers is 3,000. The product of the

1. व्यस्तं त्रैराशिक फलमिच्छा भक्तः प्रमाणफलघातः ।
   त्रैराशिकादिषु फलं विषमेष्वेकादशान्तेषु ॥
   फलसंक्रममुभयतो बहुराशि वधोऽल्पवधहृतो ज्ञेयम् ।
   सकलेष्वेवं भिन्नेष्वथवैतच्छेदसंक्रमकम् ॥
   —BrSpSi: XII. 11-12.

RULE OF THREE AS A PARTICULAR CASE 183
number on the side of the smaller set of terms is 300. Therefore
the required result is 3000 / 300 = 10.
Rule of Three as a Particular Case
According to Brahmagupta, the above method of "com-
pound proportion" may be applied to the Rule of Three. Taking
the example solved under the Rule of Three :
If one pala and one karṣa of sandal wood are obtained
for ten and a half paṇas, for how much will be obtain-
ed nine palas and one karṣa ?
(4 karṣas = 1 pala).
We shall represent them according to the Rule of Com-
pound Proportion as
Pramāṇa pakṣa : 1 pala, 1 karṣa, 10½ paṇa
or 5/4 pala , 21/2 paṇa,
Icchā pakṣa : 9 pala, 1 karṣa, x paṇa
or 37/4 pala , x paṇa
This we shall represent as
┌────┬────┐
│ 5 │ 37 │
│ 4 │ 4 │
├────┼────┤
│ 21 │ 0 │
│ 2 │ │
└────┴────┘
Transposing the fruits, we have
┌────┬────┐
│ 5 │ 37 │
│ 4 │ 4 │
├────┼────┤
│ 0 │ 21 │
│ │ 2 │
└────┴────┘
Transposing denominators
┌────┬────┐
│ 5 │ 37 │
│ 4 │ 4 │
├────┼────┤
│ 0 │ 21 │
│ 2 │ │
└────┴────┘
The product of numbers on the side of the larger set is
divided by the product of the numbers on the side of the smaller
set, 0 in this case is not a number. It is the symbol for the
unknown or absence. Hence the result is :
(37 . 4 . 21) / (5 . 4 . 2) paṇas

184 BRAHMAGUPTA AND ARITHMETIC
The above method of working Rule of Three
is found among Arabs, although it does not seem to
have been used in India after Brahmagupta.
Problem Containing Quadratic Equation
Perhaps Āryabhaṭa I is the first man in the history of
mathematics to give a solution of a quadratic equation (499
A.D.). In his Āryabhaṭīya, he gives a rule for the solution of the
following problem (I am reproducing it as described by Datta
and Singh) :
The principal sum p (=100) is lent for one month
(interest unknown = x). This unknown interest is then
lent out for t(=six) months. After this period, the
original interest (x) plus the interest on this interest
amounts to A(=16). The rate-interest (x) on the
principal (p) is required.
This problem requires the solution of the quadratic
equation :—
tx² + px - AP = 0
which gives x = (-p/2 ± √((p/2)² + Apt)) / t
The negative value of the radical does not give a solution
of the problem; so that the result is
x = (√(Apt + (p/2)²) - p/2) / t
This solution is stated by Āryabhaṭa I in the following
words :
Multiply the sum of the interest on the principal and
the interest (A) by the time (t) and by the principal
(p). Add to this result the square of half the principal
(p/2)². Take square-root this. Subtract half the
principal (p/2) and divide the remainder by the time
(t). The result will be the (unknown) interest (x) on
the principal.¹
Here the Sanskrit terms are mūla for principal and phala
for interest.

1. मूलफलं सफलं कालमूलगुणमर्धमूलकृतियुक्तम् ।
   मूलं मूलार्धोनं कालहृतं स्यात्स्वमूलफलम् ॥ Ārya. II. 25,

A PROBLEM ON INTEREST 185
Brahmagupta (628 A.D.) gives a more general rule : He
enunciates his problem thus :
The principal (p) is lent out for t₁ months and the
unknown interest on this (=x) is lent out for t₂ months
at the same rate and becomes A. To find x.
This evidently gives the quadratic :
x² + (pt₁ / t₂) x - (Apt₁ / t₂) = 0
whose solution is
x = ± √[ Apt₁/t₂ + (pt₁ / 2t₂)² ] - pt₁ / 2t₂
The negative value of the radical does not give a solution
of the problem, so it is discarded.
Brahmagupta states the formula thus :
Multiply the principal (p) by its time (t₁) and divide
by the other time (t₂) (placing the result) at two places.
Multiply the first of these by the mixture (A). Add to
this the square of half the other. Take the square-root of
this (sum). From the result subtract half the other.
This will be the interest (x) on the principal.¹
A Problem on Interest
Brahmagupta gives a solution of a problem on interest :
In what time will a given sum s, the interest on which
for t months is r, become k times itself ?
The rule for the solution of this problem as given by Brah-
magupta is :
The given sum multiplied by its time and divided by
the interest (phala), being multiplied by the factor
(guṇa) less one, is the time (required).²
Miscellaneous Problems
Brahmagupta in his Gaṇitādhyāya of the Brāhmasphuṭa-
siddhānta gives numerous solutions in relation to miscellaneous
problems. Here I shall be quoting a few of the problems which

1. कालप्रमाणघातः परकालहृतो द्विधाऽऽद्यमिश्रवधात् ।
   अन्यार्धकृतियुतात् पदमन्यार्धोनं प्रमाणफलम् ॥ —BrSpSi. XII. 15.
2. कालगुणितं प्रमाणं फलभक्तं व्येकगुणहतं कालः ।
   स्वफलयुतरूपभक्तं मूलफलैक्यं भवति मूलम् ॥ —BrSpSi. XII. 14.

186 BRAHMAGUPTA AND ARITHMETIC
have been quoted by his commentator Pṛthūdaka Svāmī in
connection with one of his karaṇa-sūtra.¹

1. A horse was purchased by (nine) dealers in partner-
   ship, whose contributions were one, etc. up to nine;
   and was sold by them for five less than five hundred.
   Tell me what was each man's share of the sale proceed²
2. Four colleges (mathas), containing an equal num-
   ber of pupils, were invited to partake of a sacrificial
   feast. A fifth, a half, a third and a quarter (of the
   total number of pupils in the college) came from the
   respective colleges to the feast; and added to one, two,
   three and four, they were found to amount to eighty-
   seven; or, with those deducted, they were sixty seven.
   Find the actual number of the pupils that came from
   each college.³
3. Three (unequal) jars of liquid butter, of water and
   of honey, contained thirty-two, sixty and twenty-four
   Pala respectively; the whole was mixed together and
   the jars filled again. Tell me the quantity of butter,
   of water and of honey in each jar⁴.
4. प्रक्षेपयोगहृत्या लब्ध्वा प्रक्षेपका गुणा लाभाः ।
   ऊनाधिकोत्तरस्तुतोनया स्वफलमूनयुत ॥ BrSpSi. XII. 16.
5. एकाद्यैर्नव पर्यन्तैर्वणिग्भिर्मूलराशिभिः ।
   क्रीतो हयोऽसौ विक्रीतः पञ्चोनैः पञ्चभिः शतैः ।
   किमैकैकस्य तत्रासीद् ब्रूहि त्वं मिश्रकान् मम ॥
6. मठस्थानानि चत्वारि छात्राणां समसंख्यया ।
   भोक्तुं संमन्त्रितान्यासन् दीक्षायां किल यज्वना ॥
   पञ्चार्धत्रिचतुर्थांशास्तेभ्यो भोक्तुं समागताः ।
   एकद्वित्रिचतुर्युक्ता दृष्टाशीतिः ससप्तका ॥
   एवोत्तरैरथवा हीना सप्तषष्टिश्चतैऽशकाः ।
   मठेभ्यश्छात्रसंख्यां मे ब्रूहि ये चागता यतः
7. घृतोदक मधूनां ये त्रयः कलसकाः पलैः ।
   रदषष्टिजिनैः पूर्णा एकीभूतास्ततः पुनः ॥
   मिश्रेण पूरिता यावत् तावत् संख्यां न वेद्म्यहम् ।
   घृतोदकमधूनां तामेकैकत्र गतां वद ॥
   —: ० :—

MISCELLANEOUS PROBLEMS 187
Reference
B. Datta and A.N. Singh : History of Hindu Mathematics,
Part I (1962).
K.S. Shukla : The Pāṭīgaṇita of Śrīdharācārya
(1959).
Kern : The Āryabhaṭīya. Bhaṭadīpikā of
Parameśvara (1875).

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CHAPTER IX
Brahmagupta as an
Algebraist
Ancient Indian name for algebra is Bījagaṇita where
bīja means element or analysis and gaṇita stands for the science
of calculation. As early as 860 A.D., Pṛthūdaka Svāmī used this
epithet for algebra in his commentary. Brahmagupta calls
algebra as Kuṭṭakagaṇita or merely kuṭṭaka, a term which was
later on used for "pulveriser" which deals with that special sec-
tion of algebra which is connected with indeterminate equations
of the first degree. Algebra is often also known as avyakta-
gaṇita or the calculations with unknowns, in contrast to arith-
metic which was known as vyakta-gaṇita or the calculations with
knowns.
Algebra goes to Europe from India
In the history of mathematical sciences, as Colebrooke
rightly remarks, it has long been a question to whom the inven-
tion of algebraic analysis is due. There is no doubt that Europe
got algebra from Arabs mediately or immediately. But the
Arabs themselves scarcely pretend to the discovery of algebra.
Colebrooke says that they were not in general inventors but
scholars during the short period of their successful culture of the
sciences; and the germ at least of the algebraic analysis is to be
found among the Greeks in an age not precisely determined, but
more than probably anterior to the earliest dawn of civilisation
among the Arabs; and this science in a more advanced state
subsisted among the Hindus prior to the earliest disclosure of it
by the Arabians to modern Europe. (Colebrooke: Disserta-
tion on the Algebra of the Hindus)¹.
Colebrooke based his observations on the texts he could
procure for his studies. These were: Bhāskara II's Bījagaṇita or
Vījagaṇita (1150 A.D.) and Līlāvatī (1150 A.D.), the Gaṇitādhyāya
and Kuṭṭakādhyāya of Brahmagupta in his famous treatise the
Brahma Siddhānta or rather the Brāhmasphuṭasiddhānta (628

1. Colebrooke, H. T., Miscellaneous Essays, Vol. II, 1872, p. 418.

190 BRAHMAGUPTA AS AN ALGEBRAIST
A.D.). There can be no doubt regarding the age of these two
authors. Bhāskara II completed his great work on the Siddhānta-
śiromaṇi in 1072 Śaka, and Karaṇa-kutūhala a practical astro-
nomical treatise in 1105 Śaka; these dates are based on the
passages given by Bhāskara himself in his works. The Bīja-
gaṇita and the Līlāvatī form parts of the great treatise, the
Siddhānta-śiromaṇi. The genuineness of the text is established,
as Colebrooke says, with no less certainty by numerous commen-
tators in Sanskrit, besides a Persian version of it. Those com-
mentaries comprise a perpetual gloss, in which every passage of
the original is noticed and interpreted : and every word of it is
repeated and explained. From comparison and collation of
various texts, it appears then that the work of Bhāskara, exhibit-
ing the same uniform text which the modern transcripts of it
do, was in the hands of both Muhammedans and Hindus,
between two or three centuries ago : and numerous copies of it
having been diffused throughout India, at an earlier period, as of
a performance held in high estimation, It was the subject of
study and habitual reference in countries and places so remote
from each other as the north and west of India and the Southern
Peninsula.
This though not marking any extraordinary antiquity, nor
approaching to that of the author himself, was a material point
to be determined : as there will be in the sequel, so says Cole-
brooke, occasion to show, that modes of analysis, and in parti-
cular, general methods for the solution of indeterminate prob-
lems both of the first and second degrees, are taught in the
Bīja-gaṇita, and those for the first degrees repeated
in the Līlāvatī, which were unknown to the mathematicians of
the West, until invented anew in the last two centuries by
algebraists of France and England.¹ Bhāskara who himself
flourished more than six hundred and fifty years ago, was in
this respect a compiler and took those methods from Indian
authors as much more ancient than himself.
Regarding the age of the precursors of Bhāskara II, Cole-
brooke says : The age of his precursors cannot be determined
with equal precision. He then proceeds to examine the evidence
as follows :

1. Colebrooke, H. T., Miscellaneous Essays, p. 421.

ALGEBRA GOES TO EUROPE FROM INDIA 191
Towards the close of his treatise on Algebra, Bhāskara
II informs us, that it is compiled and abridged from
the more diffuse works on the same subject, bearing
the names Brāhme (meaning no doubt Brahmagupta),
Śrīdhara and Padmanābha; and in the body of his
treatise, he has cited a passage of Śrīdhara's algebra
and another of Padmanābha. He repeatedly adverts
to preceding writers and refers to them in general
terms, where his commentators understand him to
allude to Āryabhaṭa, to Brahmagupta to the latter's
scholiast Caturveda Pṛthūdaka Svāmī and to the other
writers above mentioned.
Most, if not all, of the treatises, to which he thus
alludes, must have been extant, and in the hands of
his commentators, when they wrote; as appears from
their quotations of them; more especially those of
Brahmagupta and Āryabhaṭa, who are cited, and
particularly the first mentioned, in several instances.
A long and diligent research in various parts of India,
has, however, failed of recovering any part of the
Padmanābha Bīja (or the algebra of Padmanābha) and
of the algebraic and other works of Āryabhaṭa.
But the translator has been more fortunate in regard
to the works of Śrīdhara and Brahmagupta, having in
his collection Śrīdhara's compendium of arithmetic,
and a copy incomplete however, of the text and
scholia of Brahmagupta's Brahmasiddhānta comprising
among other no less interesting matter, a chapter
treating of arithmetic and mensuration; and another,
the subject of which is algebra : both of them fortu-
nately complete. The commentary is a perpetual one;
successively quoting in length each verse of the text;
proceeding to the interpretation of it, word by word;
and subjoining elucidations and remarks; and its colo-
phon, at the close of each chapter, gives the title
of the work and the name of the author. Now the
name which is there given, Caturveda Pṛthudaka
Svāmī, is that of a celebrated scholiast of Brahmagupta,
frequently cited as such by the commentaries of

192 BRAHMAGUPTA AS AN ALGEBRAIST .
Bhāskara and by other astronomical writers; and the
title of the work, Brāhmasiddhānta or sometimes Brāh-
masphuṭasiddhānta, corresponds, in the shorter form,
to the known title of Brahmagupta's treatise in the
usual references to it by Bhāskara's commentators, and
answers, in the longer form, to the designation of it, as
indicated in an introductory couplet which is quoted
from Brahmagupta by Lakṣmīdāsa, a scholiast of Bhās-
kara II. Remarking this coincidence, the translator
proceeded to collate, with the text and commentary,
numerous quotations from both, which he found in
Bhāskara's writings or in those of his expositors. The
result confirmed the indication and established the
identity of both text and scholia as Brahmagupta's
treatise, and the gloss of Pṛthūdaka. The authenticity
of the Brāhmasiddhānta is further confirmed by numer-
ous quotations in the commentary of Bhaṭṭotpala on
the Saṁhitā of Varāhamihira : as the quotations from
the Brāhmasiddhānta, in that commentary, (which is
the work of an author who flourished eight hundred
and fifty years ago) are verified in the copy under
consideration. A few instances of both will suffice,
and cannot fail to produce conviction.
It is confidently concluded, that the chapters on arith-
metic and algebra, fortunately entire in a copy in many
parts imperfect, of Brahmagupta's celebrated work as
here described, are genuine and authentic. It remains
to investigate the age of the author.
Mr. Davis, who first opined to the public a correct
view of the astronomical computations of the Hindus,
is of opinion, that Brahmagupta lived in the seventh
century of the Christian era. Dr. William Hunter, who
resided for some time with a British Embassy at Ujjay-
inī, and made diligent researches into the remains of
Indian science at that ancient seat of Hindu astrono-
mical knowledge, was there furnished, by the learned
astronomers whom he consulted, with the ages of the
principal ancient authorities. They assigned to Brah-
magupta the date of 550 Saka; which answers to A.D.