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

CLASSIFICATION OF EQUATIONS 207
cubic : ghana
biquadratic : varga-varga
Brahmagupta classified them as
(i) equations in one unknown quantity : eka-varṇa
samīkaraṇa.
(ii) equations in several unknowns : aneka-varṇa samī-
karaṇa.
(iii) equations involving products of unknowns :
bhāvita.
Eka-varṇa samīkaraṇas (equations with one unknown) are
further divided into (i) linear equations, and (ii) quadratic equa-
tions (avyakta-varga samīkaraṇa).
Pṛthūdaka Svāmī has classified equations in a different
manner as follows :
(i) linear equations with one unknown : eka-varṇa
samīkaraṇa.
(ii) linear equations with more unknowns : aneka-
varṇa samīkaraṇa.
(iii) equations with one, two or more unknowns in their
second or higher powers : madhyamāharaṇa.
(iv) equations involving products of unknowns : bhā-
vita.
As the method of solution of an equation of the third class
(i.e. equations with one or several unknowns in their second or
higher powers) is based upon the principle of the elimination
of the middle term, that class is called by the term madhyama
(middle) āharaṇa (elimination). The classification of Brahma-
gupta and Pṛthūdaka Svāmī more or less received recognition by
later writers on algebra as Bhāskara II and others.
Linear Equations with
One Unknown and Their Solutions
The first solution of a linear equation with one unknown is
obtainable in the Śulba Sūtras but not through an algebraic
process,—the Śulba process is geometrical. It is said that there
is a reference in the Sthānāṅga Sūtra (c. 300 B.C.) to a linear
equation by its name yāvat-tāvat. There has been a good deal of

208 BRAHMAGUPTA AS AN ALGEBRAIST
controversy regarding the date of the Bakhaśālī Manuscript
where we have definitely a method of solving linear equations
by the Rule of False Position. It would be interesting to give
an account of this rule here by taking an illustration from the
Bakhaśālī Manuscript.
Problem :
The amount given to the first is not known.
The second is given twice as much as the first; the third
thrice as much as the second; and the fourth four times
as much as the third. The total amount distributed is
132. What is the amount of the first ?
(BMS. Folio 23, recto)
In modern algebraic language, the solution of the problem
would be given by the equation
x + 2x + 6x + 24x = 132
where x is the amount given to the first.
The solution of this equation is given as follows in the
Bakhaśālī Manuscript :
‘Putting any desired quantity in the vacant place’; any
desired quantity is || 1 ||, ‘then construct the series’
| 1 | 2 | 2 | 3 | 6 | 4 |
| 1 | 1 | 1 | 1 | 1 | 1 |
‘multiplied’ || 1 | 2 | 6 | 24 | ; ‘added’ 33. ‘Divide the
visible quantity’ | 132 / 33 ; which) on reduction be-
comes | 4 / 1 |. (This is) the amount given (to the first)
(BMS. Folio 23, recto)
The Rule of False Position may be regarded as an early
stage of the development of the science of algebra, since no
symbol could have been evolved for an unknown quantity. As
soon as the system of notations was introduced, the application
of this Rule was no longer considered as necessary. Thus we
find that Āryabhaṭa I does not mention of this Rule.
Āryabhaṭa I states as follows regarding the solution of
linear equations :
The difference of the known ‘amounts’ (rūpaka) relat-
ing to two persons should be divided by the difference

LINEAR EQUATIONS 209
of the coefficients on the unknown (gulikā). The
quotient will be the value of the unknown (gulikā), if
their possessions be equal.¹
The original verse contains the term "gulikāntara" which
has been here translated as the difference of the coefficients
of the unknowns. We have already stated earlier that Āryabhaṭa
uses the term gulikā or shot for an unknown quantity, (gulikāntara
literally means only the difference of unknowns). This practice
is also followed by other Indian algebraists. Pṛthudaka Svāmī
rightly observed that according to the usual practice in this
country, "the coefficient of the square of the unknown is called
the square (of the unknown) and the coefficient of the (simple)
unknown is called the unknown.²
The rule given by Āryabhaṭa, then, contemplates a problem
of this kind :
Two persons, who are equally rich, possess respectively
a, b times a certain unknown amount together with
c, d units of money in cash. What is that amount ?
Now if x be the amount unknown, then according to the
problem
ax + c = bx + d
Thence
x = (d - c) / (a - b)
Āryabhaṭa has merely expressed this solution in his langu-
age.
Regarding the solution of linear equations, Brahmagupta
says :
In a (linear) equation in one unknown, the difference
of the known terms taken in the reverse order, divided
by the difference of the coefficients of the unknown
(is the value of the unknown).³

1. गुलिकान्तरेण विभजेद् द्वयोः पुरुषयोस्तु रूपकविशेषम् ।
   लब्धं गुलिकामूल्यं यद्यर्थं कृतं भवति तुल्यम् ॥ —Ārya. II. 30
2. BrSpSi, XVIII. 44 (com.)
3. अव्यक्तान्तरभक्तं व्यस्तं रूपान्तरं समेऽन्यवतः ।
   वर्गाव्यक्ताः शोध्या यस्माद् रूपाणि तदधस्तात् ॥ —BrSpSi. XVIII. 43

210 BRAHMAGUPTA AS AN ALGEBRAIST
Similar solutions have been offered by the other Indian
algebraists who followed Brahmagupta like Śrīpati, Bhāskara II
and Nārāyaṇa. Here again, we take a problem proposed by
Brahmagupta in this connection :
Problem :
Tell the number of elapsed days for the
time when four times the twelfth part of
the residual degrees increased by one, plus
eight will be equal to the residual degrees
plus one.¹
Pṛthūdaka Svāmī has solved this problem as follows :
Here the residual degrees are (put as) yāvat-tāvat, yā;
increased by one, yā 1 rū 1; twelfth part of it, (yā 1 rū 1) / 12 ;
four times this, (yā 1 rū 1) / 3 ; plus the absolute quantity
eight, (yā 1 rū 25) / 3 . This is equal to the residual degrees
plus unity. The statement of both sides tripled is
yā 1 rū 25
yā 3 rū 3
This difference between the coefficients of the unknown
is 2. By this the difference of the absolute terms,
namely 22, being divided, is produced the residual of
the degrees of the Sun, 11. These residual degrees should
be known to be irreducible. The elapsed days can be
deduced then, (proceeding) as before.
If put in the modern notations, it means the solution of the
equation :
4/12 (x+1)+8=x+1,
from which we have
x+25=3x+3
or 2x=22
or x=11.
Rule of Concurrence or Saṁkramaṇa
Brahmagupta has included this rule in algebra, whereas
other Indian mathematicians included it in arithmetic. Saṁ-

1. सैकादंशकशेषाद् द्वादशभागश्चतुर्गुणोऽष्टयुतः ।
   सैकांशशेषतुल्यो यदा तदाऽहर्गणं कथय ॥ —BrSpSi. XVIII. 46

RULE OF CONCURRENCE 211
kramaṇa is the solution of the simultaneous equations of the
type :
x + y = a
x - y = b
Brahmagupta's rule for solution is:
The sum is increased and diminished by the difference
and divided by two; (the result will be the two un-
known quantities) : (this is) concurrence (Saṁkra-
maṇa).¹
Brahmagupta restates this rule in the form of a problem
and its solution :
The sum and difference of the residues of two (heavenly
bodies) are known in degrees and minutes. What are
the residues? The difference is both added to and
subtracted from the sum, and halved; (the results are)
the residues.²
Linear Equations with Several Unknowns
The first mention of a solution of the problem with more
than one unknown is found in the Bakhshālī Manuscript, and a
system of linear equations of this type is solved in the Bakhshālī
treatise substantially by the False Position Rule.
A generalised system of linear equations will be
b₁Σx - c₁x₁ = a₁, b₂Σx - c₂x₂ = a₂,......,
bₙ Σx - cₙxₙ = aₙ
Therefore
Σx = Σ(a/c) / (Σ(b/c) - 1)
Hence
xᵣ = (bᵣ / cᵣ) × [Σ(a/c) / (Σ(b/c) - 1)] - aᵣ / cᵣ
r = 1, 2, 3......, n
One particular case, where b₁ = b₂ = b₃ = ...... = bₙ = 1 and c₁ =
c₂ = c₃ = ...... = cₙ = c has been treated by Brahmagupta at one
place. He gives the rule as follows :

1. योगोऽन्तरयुतहीनो द्विहतः संक्रमणमन्तरविभक्तं वा ।
   वर्गान्तरमन्तरयुतहीनं द्विहृतं विषमकर्म ॥ — BrSpSi. XVIII. 36
2. भागकला विकलैक्यं दृष्ट्वा विकलान्तरं च के शेषे ।
   ऐक्यं द्विधाऽन्तराधिक हीनं च विभाजितं शेषे ॥ — BrSpSi. XVIII. 96

212 BRAHMAGUPTA AS AN ALGEBRAIST
The total value (of the unknown quantities) plus or
minus the individual values (of the unknowns) multi-
plied by an optional number deing severally (given),
the sum (of the given quantities) divided by the num-
ber of unknowns increased or decreased by the multi-
plier will be the total value; thence the rest (can be
determined).¹
Σx±cx₁=a₁, Σx±cx₂=a₂, Σx±cx₃=a₃,......
Σx±cx =an
Therefore
Σx = (a₁ + a₂ + a₃ + ...... + an) / (n ± c)
Hence
x₁ = 1/c ( ±a₁ ∓ (a₁ + a₂ + a₃ + ...... + an) / (n ± c) );
and so on for x₂, x₃ etc.
Now we shall give the rule enunciated by Brahmagupta
for solving linear equations involving several unknowns :
Removing the other unknowns from (the side of) the

QUADRATIC EQUATIONS 213
be performed all operations conformably to the state-
ment of the example and thus should be carefully
framed two or more sides and also equations. Equi-clear-
ance should be made first between two and two of
them and so on to the last : from one side one un-
known should be cleared, other unknowns reduced to
a common denominator and also the absolute numbers
should be cleared from the side opposite. The
residue of other unknowns being divided by the
residual coefficient of the first unknown will give the
value of the first unknown. If there be
obtained several such values, then with two and two
of them, equations should be formed after reduction to
common denominators. Proceeding in this way to the
end find out the value of one unknown. If that value
be (in terms of) another unknown then the coefficients
of those two will be reciprocally the values of the two
unknowns. If, however, there be present more un-
knows in that value, the method of the pulveriser
should be employed. Arbitrary values may then be
assumed for some of the unknowns.
Datta and Singh have said that the above rule of Brahma-
gupta, and also the one indicated in the commentary of Pṛthudaka
Svāmī, embraces the solution of indeterminate as well as the
determinate equations. In fact, all the examples given by
Brahmagupta in illustration of the rule are of indeterminate
character. So far as the determinate simultaneous equations are
concerned, Brahmagupta's method for solving them will be easily
recognised to be the same as our present one.
Quadratic Equations
The geometrical solution of a quadratic equation in this
country would take us to the Vedic Śulba period. The Bakha-
śālī Manuscript also contains certain problems which need the
solving of quadratic equations. I shall quote one out of the
numerous available :
A certain person travels s yojana on the first day and b
yojana more on each successive day. Another who
travels at the uniform rate of S yojana per day, has a
start of t days. When will the first man overtake the
second ?

214 BRAHMAGUPTA AS AN ALGEBRAIST
This problem would today be expressed in terms of the
following equation :
S(t+x)=x {s+((x-1)/2)b},
where x is the number of days after which the first
overtakes the second. We may write this equation
as
bx²-{2(S-s)+b}x=2tS
whence the value x would be after solving the quadra-
tic :
x= (√{2(S-s)+b}²+8bts+{2(S-s)+b}) / 2b
The Bakhasālī Manuscript gives this solution as follows :
The daily travel (S) diminished by the march of the
first day (s) is doubled; this is increased by the common
increment (b). That (sum) multiplied by itself is
designated (as the kṣepa quantity). The product of the
daily travel and the start (t) being multiplied by
eight times the common increment, the kṣepa quantity
is added. The square-root of this (is increased by the
kṣepā quantity; the sum divided by twice the common
increment will give the required number of days).
(BMS. Folio 5, recto)
Āryabhaṭa I (499 A.D.) is regarded as the founder of
algebra, since he gives the solutions of a few quandratic problems.
For example, to find the number of terms of an arithmetical pro-
gression (A.P.), he gives the following rule :
The sum of the series multiplied by eight times the
common difference is added by the square of the dif-
ference between twice the first term and the common
difference: the square-root (of the result) is diminished
by twice the first term and (then) divided by the com-
mon difference : half of this quotient plus unity is the
number of terms.¹
In the modern notations of algebra, the solution would be
expressed as follows :

1. गच्छोऽष्टोत्तर गुणिताद् द्विगुणाद्युत्तर विशेषवर्गयुतात् ।
   मूलं द्विगुणाद्यूनं स्वोत्तर भजितं सरूपार्थं ॥ —Ārya. II, 20

QUADRATIC EQUATIONS 215
n = 1/2 { [√(8bs + (2a - b)²) - 2a] / b + 1 }
There is another certain interest problem¹, the solution of
which has been provided in the Āryabhaṭīya as
x = [√(Apt + (p/2)²) - p/2] / t
which is the solution of the quadratic equation :
tx² + px - Ap = 0
Āyabhaṭa I has thus given the solutions of a few quadratic
equations, but he nowhere gives the procedure of solving these
equations.
We give here the Rules of Brahmagupta for the solution of
quadratic equations. He undoubtedly is not the discoverer of
these rules; but perhaps for the first time in the history of algebra
we find the process of solving a quadratic equation so clearly
indicated.
First Rule :
The quadratic : the absolute quantities multiplied by
four times the coefficient of the square of the unknown
are increased by the square of the coefficient of the
middle (i.e. unknown); the square-root of the
result being diminished by the coefficient of the middle
and divided by twice the coefficient of the square of
the unknown, is (the value of) the middle."²
This expressed in the modern notations would mean
x = [√(4ac + b²) - b] / 2a
It would be noted that in this rule, Brahmagupta has emp-
loyed the term madhya (middle) to imply the simple unknown
as well as its coefficient. The origin of the term is doubtless
connected with the mode of writing the quadratic equation in
the form
ax² + bx + 0 = 0x² + 0x + c
so that there are three terms on each side of the equation.

1. मूलफलं सफलं कालमूलं गुणमर्धमूल कृति युक्तम् ।
   मूलं मूलाधोनं कालहृतं स्यात्स्वमूलफलम् ॥ —Ārya. II. 25.
2. वर्गं चतुर्गुणितानां रूपाणां मध्यवर्गसहितानाम् ।
   मूलं मध्येनोनं वर्ग द्विगुणोद्धृतं मध्यः ॥ —BrSpSi. XVIII. 44.

216 BRAHMAGUPTA AS AN ALGEBRAIST
Second Rule :
The absolute term multiplied by the coefficient of the
square of the unknown is increased by the square of
half the coefficient of the unknown; the square-root of
the result diminished by half the coefficient of the
unknown and divided by the coefficient of the square
of the unknown is the unknown.²
This when expressed in the modern algebraic notations
would be
x = [√(ac+(b/2)²) - (b/2)] / a
Here if the quadratic equation is
ax² + bx + c = 0
the 'absolute term' is c (the one without the unknown x), 'the
coefficient of the square of unknown' means the coefficient of x²,
i.e. a, and the 'coefficient of the unknown' means the coefficient
of x, i.e. b.
The above two methods of Brahmagupta are exactly the
same as were suggested by Āryabhaṭa I.
The root of the quadratic equation for the number of terms
of an arithmetic progression (A.P.) is given by Brahmagupta
according to the first rule ² :
n = [√(8bs+(2a-b)²) - (2a-b)] / 2b
Third Rule :
Brahmagupta also suggests a Third Rule which is very
much the same as is used commonly now. Though it
has not been expressedly suggested as a new rule, we
find its application in a few instances. For example this
rule has been suggested in connection with the following
problem on interest :
A certain sum (p) is lent out for a period (t₁);
the interest accrued (x) is lent out again at this

1. वर्गाहत रूपाणामव्यक्ताधैंकृति संयुतानां यत् ।
   पदमव्यक्ताधोनं तद्वर्ग विभक्तमव्यक्तः ॥ —BrSpSi. XVIII. 45
2. उत्तरहीनद्विगुणादि शेषवर्गं धनोत्तराष्टवधे ।
   प्रक्षिप्य पदं शेषोनं द्विगुणोत्तरहृतं गच्छः ॥ —BrSpSi. XII. 18

QUADRATIC EQUATIONS 217
rate of interest for another period (t₂) and the
total amount is A. Find x.
The equation for determining x is
(t₂ / pt₁) x² + x = A.
The solution of this equation would be :
x = √((pt₁ / 2t₂)² + (A / t₂) · pt₁) - pt₁ / 2t₂
Brahmagupta has stated the result in exactly the same form.
Pṛthūdaka Svāmī has illustrated it in solving the following pro-
blem of interest :
Problem :
A sum of five hundred paṇas (p) is lent out
for a period of 4 months (t₁); the interest
accrued (x) is lent out again at this rate of
interest for another period of 10 months (t₂)
and the total amount is 78 (A). Give the
pramāṇa-phala, i.e., the interest accrued x.
Here pramāṇa-kāla (t₁) = 4 months
pramāṇa-dhana (p) = 500 paṇas
para-kāla (t₂), the subsequent period = 10 months
miśra dhana or the total interest accrued (A) = 78 paṇas.
Brahmagupta states his solution of such quadratics like
this :
Take the product of the pramāṇa-dhana (p) or the sum
originally lent out and pramāṇa-kāla, i.e. the period for
which originally lent out (t₁); and divide by the para-
kāla or the subsequent time (t₂); place this result at
two places. Multiply the one placed at the first place
with the miśra-dhana (A), that is with the total inter-
est accrued; in this product add the square of half the
one placed in the second place; now take the square-
root of it, and from it subtract half of the one placed
at the second place.¹

1. कालप्रमाणघातः परकालहृतो द्विधाऽऽप्तमिश्रवधात् ।
   अन्यार्धकृति युतात् पदमन्यार्धोनं प्रमाण फलम् ॥ —BrSpSi. XII. 15.

218 BRAHMAGUPTA AS AN ALGEBRAIST
Thus in the above example the product of pramāṇa-dhana
and pramāṇa kāla divided by parakāla is (pt₁/t₂)—is (500 × 4) / 10 = 200.
This is first multiplied by the total interest accrued (A); it
becomes 200 × 78 = 15600. To this is now added square of half
of 200 (which is 10000) ; it becomes 15600 plus 10000 = 25600. Its
square-root is taken which is 160. From this is subtracted half of
the quantity (i.e. half of 200 which is 100). Thus 160–100 = 60,
which is the answer. It was the interest which first accrued (x).
Another Quadratic Problem :
Brahmagupta refers to an astronomical problem which
involves the quadratic equation
(72 + a²)x² ∓ 24 apx = 144 ( R² / 2 - p² ),
where a = agra (the sine of the amplitude of the Sun), b = palabha
(the equinoctial shadow of a gnomon 12 aṅguli long), R = radius,
and x = koṇaśaṅku (sine of the altitude of the Sun when his
altitude is 45°). Dividing out by (72 + a²) we have
x² ∓ 2mx = n,
where
m = 12 ap / (72 + a²), n = 144(R²/2 - p²) / (72 + a²)
Therefore we have
x = √(m² + n) ± m,
as stated by Brahmagupta. We find the same result in the
Sūrya-siddhānta and in the text of Śrīpati. Āryabhaṭa II (1150)
also followed the method of Āryabhaṭa I and Brahmagupta in
solving a quadratic equation in connection with finding out the
number of terms in an arithmetical progression (A.P.) whose first
term is (a), common difference is b and the sum is s. The number
of terms n is given by¹
n = [√(2bs + (a - b/2)²) - a + b/2] / b
Two Roots of a Quadratic Equation and Brahmagupta
A quadratic equation has two roots. This must have been
known to Indian algebraists even at a very early stage. Bhāskara
II in his Bījagaṇita has quoted a rule ascribed to an ancient
writer Padmanābha whose works are not available now :

1. Mahāsiddhānta. Bhāskara II, XV. 50

TWO ROOTS OF A QUADRATIC EQUATION 219
If (after extracting roots) the square-root of the absolute
side (of the quadratic) be less than the negative abso-
lute term on the other side, then taking it negative as
well as positive two values (of the unknown) are
found¹.
The term used here is dvividhotpadyate mitiḥ which means
that two values are obtained.
The existence of two roots of a quadratic equation appears
to have been known also to Brahmagupta (628 A.D.). In illustra-
tion of his rules for the solution of a quadratic he has stated two
problems involving practically the same equation :
Problem I : The square-root of the residue of the
revolution of the Sun less 2 is diminished by 1, multi-
plied by 10 and added by 2; when will this be equal
to the residue of the revolution of the Sun less 1, on
Wednesday ?²
Problem II : When will the square of one-fourth the
residue of the exceeding months less three, be equal to
the residue of the exceeding months ?
We shall follow Pṛthūdaka Svāmī in solving the Problem I.
In this problem the residue of the revolutions of the Sun
may be supposed to be x² + 2; then by the question, we have
10 (x - 1) + 2 = x² + 1,
or x² - 10x = -9
Again in Problem II, if we put 4x for the residue of the
exceeding month, then we have
(x - 3)² = 4x
or x² - 10x = -9.
Now by the second rule of Brahmagupta, retaining both
the signs of the radical, we get :
x = 5 ± √(25 - 9) = 9 or 1.

1. व्यक्त पक्षस्य चेन्मूलमन्यपक्षर्णरूपतः ।
   अल्पं धनर्णगं कृत्वा द्विविधोत्पद्यते मितिः ॥ —Bhāskara, Bījagaṇita
2. मण्डलशेषाद् द्व्यूनान्मूलं व्येकं दशाहतं द्वियुतम् ।
   मण्डलशेषं व्येकं भानोर्ज्ञादिने कदा भवति ॥ —BrSpSi. XVIII. 49
3. अधिमासशेषपादात् त्र्यूनाद्वर्गोऽधिमासशेषसमः ।
   अवमावशेषतो वावमशेषसमः कदा भवति ॥ —BrSpSi. XVIII: 50.

220 BRAHMAGUPTA AS AN ALGEBRAIST
As shown by Pṛthudaka Svāmī, the first value is taken by
Brahmagupta for the Problem I and second value for the problem
II. Thus it is quite clear that Brahmagupta uses sometimes the
positive and at other times the negative sign with the radical.
Hence we shall say that Brahmagupta knew that a quadratic
equation would have two roots, and according to the requisite-
ness of the problem, one value out of the two would be utilised.
Simultaneous Quadratic Equations
Indian authors usually treated problems involving various
forms of simultaneous quadratic equations.
(i) x — y = d } (ii) x + y = a }
xy = b } xy = b }
(iii) x² + y² = c } (iv) x² + y² = c }
xy = b } x + y = a }
For the solution of the combination (i), Āryabhaṭa I gives
the following rule in his Āryabhaṭīya .
The square-root of four times the product (of two quan-
tities) added with the square of their difference, being
added and diminished by their difference and halved
gives the two multiplicands.¹
This means that
x = ½ √(d² + 4b + d) , y = ½ (√(d² + 4b) - d)
For the solution of the same combination, Brahmagupta
states as follows :
The square-root of the sum of the square of the diffe-
rence of the residues and two squared times the product
of the residues, being added and subtracted by the
difference of the residues, and halved (gives) the desi-
red residues severally.²
(Here by difference of the residues is mesnt x — y; and by
product of the residues is meant xy.)
Brahmagupta does not seem to give the solution for simulta-
neous equations of the combination (ii). Mahāvīra (850 A.D.)

1. द्विकृति गुणात्संवर्गाद् द्वयन्तरवर्गेण संयुतान्मूलम् ।
   अन्तरयुक्तं हीनं तद्गुणकारद्वयं दलितम् ॥ —Ārya. II. 24
2. शेषवधाद् द्वि कृति गुणात् शेषान्तर वर्ग संयुतान्मूलम् ।
   शेषान्तरेण युक्तं दलितं शेषे पृथगभीष्टे ॥ —Br SpSi. XVIII. 99

SIMULTANEOUS QUADRATIC EQUATION 221
has given the solution :
Subtract four times the area (of a rectangle) from the
square of the semi-perimeter then by saṅkramaṇa bet-
ween the square-root of that (remainder) and the semi-
perimeter, the base and the upright are obtained.¹
(GSS. VII. 129½)
This expressed in the modern notations would be :
x = ½(a + √(a² — 4b)), y = ½(a — √(a² — 4b))
For the combination (iii), Mahāvīra in his Gaṇita-Sāra-
Saṃgraha gives the following rule :
Add to and subtract twice the area (of a rectangle) from
the square of the diagonal and extract the square-roots.
By saṅkramaṇa between the greater and lesser of these
(roots), the side and upright (are found).²
This put in modern notations would be :
x = ½ √(c + 2b) + √(c — 2b)).
y = ½ (√(c + 2b) — √(c — 2b)).
For the combination (iv), Āryabhaṭa I gives the following
rule :
From the square of the sum (of two quantities) subtract
the sum of their squares. Half of the remainder is
their product.³
The remaining operations will be similar to those for the
equations (ii); so that
x = ½ (a + √(2c — a²)), y = ½ (a — √(2c — a²)).
Brahmagupta in this connection says :
Subtract the square of the sum from twice the sum of
squares; the square-root of the remainder being added
to and subtracted from the sum and halved, (gives)
the desired residues.⁴

1. GSS. VII. 129½
2. GSS. VII. 127½
3. संपर्कस्य हि वर्गाद्विशोधयेदेव वर्गसंपर्कम् ।
   यत्तस्य भवत्यर्धं विद्याद् गुणकारसंवर्गम् ॥ —Ārya. II. 23
4. कृति संयोगाद् द्विगुणाद्युति वर्गं प्रोह्य शेष मूलं यत् ।
   तेन युतोनो योगो दलितः शेषे पृथगभीष्टे ॥ —BrSpSi. XVIII. 98

222 BRAHMAGUPTA AS AN ALGEBRAIST
These equations have also been treated by Mahāvīra,
Bhāskara II and Nārāyaṇa. Nārāyaṇa has attempted two other
forms of quadratic equations :
(v) x² + y² = c } (vi) x² - y² = m }
x - y = d } xy = b }
For their solutions, see Datta and Singh, Algebra, P. 84.
Rule of Dissimilar Operations :
Datta and Singh say that the process of solving the follow-
ing two particular cases of simultaneous quadratic equations was
distinguished by most Indian mathematicians by the special
designation viṣama-karma or dissimilar operation :
(i) x² - y² = m } (ii) x² - y² = m }
x - y = n } x + y = p }
These equations have been regarded by these mathematicians
as if of fundamental importance. They have given the following
solutions (expressed in modern algebraic symbols) :
For the combination (i) :
x = ½ (m/n + n), y = ½ (m/n - n),
For the combination (ii) :
x = ½ (p + m/p), y = ½ (p - m/p).
We shall express these solutions as follows in the words of
Brahmagupta :
The difference of the squares (of the unknowns) is divi-
ded by the difference of the unknowns and the quotient
is increased and diminished by the difference and
divided by two ; (the results will be the two unknown
quantities) ; (this is) dissimilar operation.
The same rule is restated by him on a different occasion in
the course of solving a problem.
If then the difference of their squares, also the differe-
'nce of them (are given) ; the difference of the squares

1. योगोऽन्तरयुग्हीनो द्विहतः संक्रमणमन्तर विभक्तं वा ।
   वर्गान्तरमन्तर-युतहीनं द्विहतं विषमकर्म । BrSpSi. XVIII. 36

RULE OF DISSIMILAR OPERATIONS 223
is divided by the difference of them, and this (latter)
is added to and subtracted from the quotient and then
divided by two ; (the results are) the residues whence
the number of elapsed days (can be found).¹
This viṣama-karma or dissimilar operation has been descri-
bed by other Indian algebraists also, as Āryabhaṭa II (Mahāsi-
ddhānta, XVII, 22); Śrīpati (Siddhānta-śekhara, XIV. 13) ; Bhā-
skara II (Līlāvatī) and Nārāyaṇa (Gaṇita-kaumudī, I, 32).
Indeterminate Equations of the First Degree
Āryabhaṭa I should be given the credit of giving for the
first time a treatment of the indeterminate equation of the first
degree. In his Āryabhaṭīya, we find a method for obtaining the
general solution in positive integers of the simple indeterminate
equation :
by – ax = c
for integral values of a,b,c, and further indicated how to
extend it to get positive integral solutions of simultaneous indeter-
minate equations of the first degree. His disciple Bhāskara I
(522) showed that the same method might be applied to solve :
by – ax = – c
and further that the solution of this equation would follow
that of by – ax = – 1. These methods of Āryabhaṭa I and Bhā-
skara I have also been adopted by Brahmagupta, and in certain
cases, the improvement were suggested by Āryabhaṭa II in the
middle of the tenth century A.D.
The problems which were treated by ancient Indian algeb-
raists and which led them to the investigation of the simple inde-
terminate equation of the first degree may be classified under
three heads :
Class I : To find a number N which being divided by
two given numbers (a,b) will leave two given remain-
ders (R₁, R₂).
Thus we have :
N = ax + R₁ = by + R₂

1. तद्वर्गान्तरमाधे तदन्तरं चान्तरोद्धृतयुतोनम् ।
   वर्गान्तरं विभक्तं द्वाभ्यां शेषे ततोद्युगणः । BrSpSi, XVIII, 97

224 BRAHMAGUPTA AS AN ALGEBRAIST
Hence ' by—ax = R₁—R₂
Putting c = R₁ ∽ R₂
we get by—ax = ± c
the upper or lower sign being taken according as R₁ is grea-
ter than or less than R₂.
Class II : To find a number (x) such that its product
with a given number (a) being increased or
decreased by another given number (δ) and
then divided by a third given number (β)
will leave no remainder.
This means that in other words, we shall have to get the
solution of :
ax ± r
—————— = y
β
in positive integers.
Class III : Here we have to deal with an equation of the
form :
by ÷ ax = ± c
Kuṭṭaka, Kuṭṭākāra and Kuṭṭa : These are the three terms
which Brahmagupta has used in regards to the subject of indeter-
minate analysis of the first degree. Āryabhaṭa I has also descri-
bed this method in brief, but he does not use the word kuṭṭaka.
In the Mahābhāskarīya of Bhāskara I we have the terms kuṭṭā-
kāra and kuṭṭa (522 A.D.) MBh. I. 41,49). These words have
been translated into English as pulveriser or grinder. According
to Datta and Singh, the Hindu method of solving the equation
by-ax= ± c is essentially based on a process of deriving from it
successively other similar equations in which the values of the
coefficients (a,b) become smaller and smaller. Thus the process
is indeed the same as that of breaking a whole thing into smaller
pieces, and this accounts for its name kuṭṭaka or ‘pulveriser’.
In the problems of the Class I, the quantities (a and b) are
called’ divisors’ bhāgahāra, bhajaka, cheda etc.) and R₁ and R₂
as ‘remainders’ (agra or śeṣa etc.). while in a problem of the
Class II ; β is ordinarily called the ‘divisor’ (bhāgahāra or bhā-
jaka) and γ the ‘interpolator’ kṣepa, kṣepaka etc.) ; here a is called
the ‘dividend’ (bhājya), the unkown quantity to be found (x) is
called the ‘multiplier’ or (guṇaka or guṇakāra etc) and y the

INDETERMINATE EQUATIONS 225
quotient or phala. In later years, Mahāvīra has called the unknown
number (x) as rāśi.
Preliminary Operations
in Kuṭṭaka-Karma
Usually it has been suggested that in order that an equation
of the form
by − ax = ± c or by + ax = ± c
may be amenable to solution, the two numbers a and b must not
have a common divisor; for otherwise, the equation would be
absurd, unless the number c had the same common divisor. So
before the rules which we shall give hereafter, could be applied,
the numbers a, b, c must be made prime (dṛḍha or firm; niccheda
or having no divisor, or nirapavarta, meaning irreducible to
each other.
In this connection Bhāskara I writes :
The dividend and divisor will become prime to each
other on being divided by the residue of their mutual
division. The operation of the pulveriser should be
considered in relation to them.¹
Similarly we find in the writings of Brahmagupta :
Divide the multiplier and the divisor mutually and
find the last residue; those quantities being divided by
the residue will be prime to each other.²
Āryabhaṭa's Rule : Āryabhaṭa I is probably the first Indian
writer on this subject, but the operation given by him is rather
obscure. His disciple Bhāskara I has given the solution of inde-
terminate equations of the first degree in more satisfactory langu-
age. We shall give here the translation of Āryabhaṭa's verse from
the Āryabhaṭīya, as rendered by Bibhutibhusan Datta, because
other translations of this verse do very often confuse the sense :
Divide the divisor corresponding to the greater remain-
der by the divisor corresponding to the smaller remain-

1. भूदिनेष्टगणा-न्योन्य भक्तशेषेण भाजितौ ।
   हारभाज्यौ दृढौ स्यातां कुट्टाकारं तयोर्विदुः —MBh, I. 41
2. हतयोः परस्परं यच्छेषं गुणकारभागहारकयोः ।
   तेन हृतौ निश्छेदौ तावेव परस्परं हतयोः । —BrSpSi. XVIII. 9.

224 BRAHMAGUPTA AS AN ALGEBRAIST
Hence ' by—ax=R₁—R₂
Putting c=R₁∽R₂
we get by-ax=±c
the upper or lower sign being taken according as R₁ is grea-
ter than or less than R₂.
Class II : To find a number (x) such that its product
with a given number (a) being increased or
decreased by another given number (8) and
then divided by a third given number (β)
will leave no remainder.
This means that in other words, we shall have to get the
solution of :
ax±r
———— = y
β
in positive integers.
Class III : Here we have to deal with an equation of the
form :
by ÷ ax= ± c
Kuṭṭaka, Kuṭṭākāra and Kuṭṭa : These are the three terms
which Brahmagupta has used in regards to the subject of indeter-
minate analysis of the first degree. Āryabhaṭa I has also descri-
bed this method in brief, but he does not use the word kuṭṭaka.
In the Mahābhāskarīya of Bhāskara I we have the terms kuṭṭā-
kāra and kuṭṭa (522 A.D.) MBh. I. 41,49). These words have
been translated into English as pulveriser or grinder. According
to Datta and Singh, the Hindu method of solving the equation
by-ax= ± c is essentially based on a process of deriving from it
successively other similar equations in which the values of the
coefficients (a,b) become smaller and smaller. Thus the process
is indeed the same as that of breaking a whole thing into smaller
pieces, and this accounts for its name kuṭṭaka or 'pulveriser'.
In the problems of the Class I, the quantities (a and b) are
called' divisors' bhāgahāra, bhajaka, cheda etc.) and R₁ and R₂
as 'remainders' (agra or seṣa etc.). while in a problem of the
Class II ; β is ordinarily called the 'divisor' (bhāgahāra or bhā-
jaka) and Y the 'interpolator' kṣepa, kṣepaka etc.) ; here a is called
the 'dividend' (bhāj